text stringlengths 4 2.78M |
|---|
---
abstract: 'In this work, we investigated the feasibility of applying deep learning techniques to solve Poisson’s equation. A deep convolutional neural network is set up to predict the distribution of electric potential in 2D or 3D cases. With proper training data generated from a finite difference solver, the strong approximation capability of the deep convolutional neural network allows it to make correct prediction given information of the source and distribution of permittivity. With applications of L2 regularization, numerical experiments show that the predication error of 2D cases can reach below 1.5% and the predication of 3D cases can reach below 3%, with a significant reduction in CPU time compared with the traditional solver based on finite difference methods.'
author:
-
title: 'Study on a Poisson’s Equation Solver Based On Deep Learning Technique'
---
Deep Learning; Poisson’s Equation; Finite Difference Method; Convolutional Neural Network; L2 regularization.
Introduction
============
Computational electromagnetic simulation has been widely used in research and engineering, such as antenna and circuit design, target detection, geophysical exploration, nano-optics, and many other related areas [@Chew2001]. Computational electromagnetic algorithms serve as the kernel of simulation. They solve Maxwell’s equations under various materials and different boundary conditions. In these algorithms, the domain of simulation is usually discretized into small subdomains and the partial differential equations is converted from continuous into discrete forms, usually as matrix equations. These matrix equations are either solved using direct solvers like LU decomposition, or iterative solvers such as the conjugate gradient method [@Golub1996]. Typical methods in computational electromagnetics include finite difference method (FDM) [@Taflove2005], finite element method (FEM) [@Jin2014], method of moments (MOM) [@Harrington1993], and etc. Practical models are usually partitioned into thousands or millions of subdomains, and matrix equations with millions of unknowns are solved on computers. This usually requires a large amount of CPU time and memory. Therefore, it is still very challenging to use full-wave computational electromagnetic solvers to applications that require real-time responses, such as radar imaging, biomedical monitoring, fluid detection, non-destructive testing, etc. The speed of electromagnetic simulation still cannot meet the demand of these applications.
One method of acceleration is to divide the entire computation into offline and online processes. In the offline process, a set of models are computed and the results are stored in the memory or computer hard disk. Then in the online process, solutions can be interpolated from the pre-computed results. These methods include the model order reduction [@Wilhelmus2008], the characteristic basis function [@prakash2003characteristic], the reduced basis method [@Noor1980; @dang2017quasi], and etc. The idea of these schemes is to pay more memory in return for faster speed. Moreover, artificial neural network has also been used to optimize circuit[@zaabab1995neural] and accelerate the design of RF and microwave components [@Zhang2000][@zhang2003artificial]. However, the extension capability is still limited for most of these methods, and they are mainly used to describe systems with few parameters.
With rapid development of big data technology and high performance computing, deep learning methods have been applied in many areas and significantly improve the performance of voice and image processing [@Hinton2006; @LuCun2015]. These dramatic improvements rely on the strong approximation capability of deep neural networks. Recently, researchers have applied the deep neural networks to approximate complex physical systems [@Ehrhardt2017][@lerer2016learning], such as fluid dynamics [@Tompson2016; @Guo2016], Schrödinger equations [@Mills2017] and rigid body motion[@Byravan2017SE3]. In these works, the deep neural networks “learn” from data simulated with traditional solvers. Then it can predict field distribution in a domain with thousands or millions unknowns. Furthermore, it has also been applied in capacitance extraction with some promising results [@yao2016machine]. The flexility in modeling different scenarios is also significantly improved compared with traditional techniques using artificial neural networks.
In this study, we investigate the feasibility of using deep learning techniques to accelerate electromagnetic simulation. As a starting point, we aim to compute 2D or 3D electric potential distribution by solving the 2D or 3D Poisson’s equation. We extended the deep neural network structure in [@Tompson2016] and proposed a approximation model based on fully convolutional network [@long2015fully]. We apply L2 regularization[@ng2004feature] in the objective function in order to prevent over-fitting and improve the prediction accuracy. In the offline training stage, a finite-difference solver is used to model inhomogeneous permittivity distribution and point-source excitation at different locations, the permittivity distribution, excitation, and potential field are used as training data set. The input data include the permittivity distribution and the location of excitation, the output data is the electric potential of the computation domain. Then in the online stage, the network can mimic the solving process and correctly predict the electric potential distribution in the domain. Different from traditional algorithms, the method proposed in this paper is an end-to-end simulation driven by data. The computational complexity of the network is fixed and much smaller than that of traditional algorithms, such as the finite-difference method. Preliminary numerical studies also support our observations.
This paper is organized as follows: In Section 2 we introduce the data model and the deep convolutional neural network model used in the computation. In Section 3 we show more details of preliminary numerical examples and compare the accuracy and computing time with the algorithm using finite-difference method. Conclusions are drawn in Section 4.
Formulation
===========
Finite Difference Method Model
------------------------------
The electrostatic potential in the region of computation with Dirichlet boundary condition can be described as
$$\nabla\cdot(\varepsilon({\mbox{$\mathbf{r}$}})\nabla\phi({\mbox{$\mathbf{r}$}}))=-\rho({\mbox{$\mathbf{r}$}}) \,,
\label{eq10}$$
$$\phi|_{\partial D}=0 \,,
\label{eq20}$$
where $\phi({\mbox{$\mathbf{r}$}})$ is the electric potential in Domain $D$, $\rho({\mbox{$\mathbf{r}$}})$ represents distribution of electric charges, and $\varepsilon({\mbox{$\mathbf{r}$}})$ represents dielectric constant. describes the Dirichlet boundary condition, which enforces the value of potential to be zero along the boundary.
The above equations are solved using the finite difference method. The domain of computation is partitioned into subdomains using Cartesian grids. The electric potential and electric charge density in each subdomain is assumed constant. Central difference scheme is used to approximate the derivative in . If the computation domain is 2D, then we can write as $$\begin{split}
&\frac{\varepsilon_{i+\frac{1}{2},j}\frac{\phi_{i+1,j}-\phi_{i,j}}{\Delta x}-\varepsilon_{i-\frac{1}{2},j}\frac{\phi_{i,j}-\phi_{i-1,j}}{\Delta x}}{\Delta x}+\\
&\frac{\varepsilon_{i,j+\frac{1}{2}}\frac{\phi_{i,j+1}-\phi_{i,j}}{\Delta y}-\varepsilon_{i,j-\frac{1}{2}}\frac{\phi_{i,j}-\phi_{i,j-1}}{\Delta y}}{\Delta y}=-\rho_{i,j}
\end{split}
\label{eq30}$$ and $$\varepsilon_{i+a\frac{1}{2},j+a\frac{1}{2}}=\frac{\varepsilon_{i,j}+\varepsilon_{i+a,j+b}}{2} , a \in \{-1, 1\}
\label{eq40}$$ where $(i,j)$ represents the location of the subdomain in the grid. if the comptation domain is 3D, then can be written as $$\begin{split}
&\frac{\varepsilon_{i+\frac{1}{2},j,k}\frac{\phi_{i+1,j,k}-\phi_{i,j,k}}{\Delta x}-\varepsilon_{i-\frac{1}{2},j,k}\frac{\phi_{i,j,k}-\phi_{i-1,j,k}}{\Delta x}}{\Delta x}+\\
&\frac{\varepsilon_{i,j+\frac{1}{2},k}\frac{\phi_{i,j+1,k}-\phi_{i,j,k}}{\Delta y}-\varepsilon_{i,j-\frac{1}{2},k}\frac{\phi_{i,j,k}-\phi_{i,j-1,k}}{\Delta y}}{\Delta y}+\\
&\frac{\varepsilon_{i,j,k+\frac{1}{2}}\frac{\phi_{i,j,k+1}-\phi_{i,j,k}}{\Delta z}-\varepsilon_{i,j,k-\frac{1}{2}}\frac{\phi_{i,j,k}-\phi_{i,j,k-1}}{\Delta z}}{\Delta z}=-\rho_{i,j}
\end{split}
\label{eq30}$$ and $$\varepsilon_{i+a\frac{1}{2},j+b\frac{1}{2},k+c\frac{1}{2}}=\frac{\varepsilon_{i,j,k}+\varepsilon_{i+a,j+b,k+c}}{2} , a,b,c \in \{-1,0,1\}$$ where $(i,j,k)$ represents the location of the subdomain in the grid. The above equation in each subdomain construct a linear system of equations $\bar{\bar{A}}\cdot{\mbox{$\boldsymbol{\bar{\phi}}$}} = - {\mbox{$\boldsymbol{\bar{\rho}}$}}$, where $\bar{\bar{A}}$ is symmetric and positive semi-definite. LU decomposition or conjugate gradient method can be applied to solve this equation.
![2D Modeling Setup: Yellow points are the 11 positions for source , blue area is where we predict[]{data-label="2dmodel"}](2dmodel_t.jpg){height="3cm"}
![3D Modeling Setup: Yellow points are the 11 positions for source , blue area is where we predict[]{data-label="3dmodel"}](3dmodel_t.jpg){height="4cm"}
in [1,...,5]{}[ ]{}
in [1,...,3]{}[ ]{}
{width="12.7cm" height="8cm"}
ConvNet model
-------------
Neural networks have excellent performance for function fitting. One can approximate a complex function using powerful function fitting method based on deep neural networks [@Tompson2016]. Convolutional neural networks (CNN) have excellent performance in learning geometry and predicting per-piexl in images[@lawrence1997face][@krizhevsky2012imagenet] and fully convolutional networks have been proven powerful models of pixelwise prediction. In this paper, the problem that we model has various locations of the excitation and dielectric constant distribution, so the geometric characteristics of this problem are obvious. They all need to be considered in the design of the network layers. Therefore, the input of the network includes distribution of electrical permittivity and the source information. The electrical permittivity distribution is represented as a two-dimensional or three-dimensional array with every element $(i,j)$ or $(i,j,k)$ represents electrical permittivity at grid $(i,j)$ or $(i,j,k)$. The source information is also represented by a two-dimensional or three-dimensional array, in which every element represents the distance between the source and a single grid. The distance function can provide a good representation of the universial source information and if the case is 2D, then the distance function can be written as
$$f(i,j)_n=\sqrt{(i-i_n )^2+(j-j_n )^2}, n \in \{1,2...,11\}
\label{eq50}$$
where $i$,$j$ is the location of grids in the predicted area and $i_n$,$j_n$ is source excitation’s location that has 11 different positions. if the case is 3D, then the distance function can be written as
$$\begin{split}
&f(i,j,k)_n=\sqrt{(i-i_n )^2+(j-j_n )^2+(k-k_n )^2},\\
& \quad\quad\quad\quad \quad\quad\quad\quad \quad\quad\quad\quad \quad\quad\quad\quad n \in \{1,2...,11\}
\end{split}$$
where $i$,$j$,$k$ is the location of grids in the predicted area and $i_n$,$j_n$,$k_n$ is source excitation’s location that has 11 different positions. The setup of deep neural network is based on optimization process that adjusts the network parameters to minimize the difference between “true” values of function and the one predicted by the network on a set of sampling points. In this problem, the loss function in optimization is defined to measure the difference between the logarithm of predicted potential and the one obtained by FDM, it can be written as
$$loss_{obj}=\|log_{10}(\phi)-log_{10}(\widehat{\phi}) \|^2\,,$$
and L2 regularization expression is included in the cost function to prevent over-fitting, the final cost function is written as: $$f_{obj}=loss_{obj}+ \frac{\lambda}{2n}\sum_{w}w^{2}\,,
\label{eq60}$$ where $\phi$ is the predicted potential, $\widehat{\phi}$ is the potential solved by FDM, $\lambda$ is a hyperparameter, *n* is the amount of training samples, *w* is weights of the network. The use of logarithm of potential is to avoid the instability in optimization due the fast attenuation in the distribution of electrical potential. It can also help to improve the accuracy of prediction. L2 regularization implies a tendency to train as small a weight as possible and retain a larger weight that can be guaranteed to significantly reduce the original loss. The internal structure of fully convolution neural network model is shown in the . It consists of seven stages of convolution and Rectifying Linear layers (ReLU)[@glorot2011deep] but there are no pooling layers beacuse features in this problem are not complicated. The input data of ConvNet model includes the permittivity distribution and location of excitation expressed as the distance function, the output data is the predicted electric potential of the computation domain.
in [1,...,3]{}[ ]{}
One result of scenario 1: *Left*: FDM result, *Midlle*: ConvNet result, *Right*: error distribution
in [1,...,3]{}[ ]{}
One result of scenario 2: *Left*: FDM result, *Midlle*: ConvNet result, *Right*: error distribution
in [1,...,3]{}[ ]{}
One result of scenario 3: *Left*: FDM result, *Midlle*: ConvNet result, *Right*: error distribution
in [1,...,3]{}[ ]{}
One result of scenario 4: *Left*: FDM result, *Midlle*: ConvNet result, *Right*: error distribution
in [1,...,3]{}[ ]{}
One result of scenario 5: *Left*: FDM result, *Midlle*: ConvNet result, *Right*: error distribution
Results And Analysis
====================
In this study, we solve the electrostatic problem in a square region (2D case) or a cube region (3D case). In 2D cases, a square region is partitioned into $64\times 64$ grids, as shown in . The yellow points indicate the location of sampled excitation. In 3D cases, a cube region is partioned into $64\times 64\times64$ grids, as shown in .The yellow points indicate the location of sampled excitation and the value of excitation is fixed at -10. We aim to solve the potential field in the region of $32\times 32$ or $32\times32\times32$ colored in blue.
In 2D case, we try 6 possible scenarios as shown in c and the background’s permittivity of all scenarios is 1:
scenario 1
----------
Scenario 1 has a single ellipse located in the center of the square region. This ellipse has different shapes whose semi-axis varies from 1 to 20 and rotation angle is randomly chosen between $\frac{\pi}{20}$ and $\pi$. The permittivity values of the target is randomly selected from \[0.125,0.25,0.5,2,4,6\].
scenario 2
----------
Scenario 2 divides the square region into four identical parts and each part has a ellipse whose semi-axis varies from 1 to 8 and rotation angle is randomly chosen between $\frac{\pi}{20}$ and $\pi$. The four ellipses have different shapes but their permittivity valuse are the same and randomly selected from \[0.125,0.25,0.5,2,4,6\].
scenario 3
----------
Scenario 3 divides the square region into four identical parts and each part has a ellipse whose semi-axis varies from 1 to 8 and rotation angle is randomly chosen between $\frac{\pi}{20}$ and $\pi$. The four ellipses have different shapes and their permittivity valuse are different and randomly selected from \[0.125,0.25,0.5,2,4,6\].
scenario 4
----------
Scenario 4 has a single ellipse whose location moves in a small range. This ellipse has different shapes whose semi-axis varies from 1 to 12 and rotation angle is randomly chosen between $\frac{\pi}{20}$ and $\pi$. The permittivity values of the target is randomly selected from \[0.125,0.25,0.5,2,4,6\].
scenario 5
----------
scenario 5 has no special shapes and the predicted region is the region of $32\times 32$ colored in blue. The permittivity values of every four grids in the target is randomly chosen from 0.125 to 6.
scenario 6
----------
Scenario 6 includes scenario 1 to 5.
in [1,...,6]{}[ ]{}
In the 3D case, ellipsoids with different shapes are located inside the predicted region. As shown in , their three semi-axis varies from 1 to 20.
The convolution neural network takes two $64\times 64$ or $64\times64\times64$ arrays as input, as depicted in , and output is a $32\times32$ or $32\times32\times32$ array representing the field in the region of investigation. The training and testing data for the network are obtained by the finite-difference solver. For 2D cases, We use 8000 samples for training and 2000 samples for testing in sceranio 1 to 5 and 40000 samples for training and 10000 samples for testing in scenario 6; for 3D cases, we use 4000 samples for training and 1000 samples for testing. The ConvNet model was implemented in Tensorflow and an Nvidia K80 GPU card is used for computation. The Adam[@kingma2014adam] Optimizer is used to optimize objective function in .
For more detailed comparison, we use relative error in the ConvNet model to measure the accuracy of the prediction. We first compute the difference between the ConvNet model predicted potential and the FDM generated potential. For a subdomain, the relative error is defined as $$err(i,j)\ or\ err(i,j,k)=\frac{|\phi_{ConvNet}-\phi_{FDM}|}{\phi_{FDM}} \,,$$ where $\phi_{ConvNet}$ and $\phi_{FDM}$ are predicted and “true” potential field, respectively. The average relative error of the $n$-th testing case is the mean value of relative error in all subdomains: $$err_{aver_n}=20\lg10(\frac{\sum_{i}\sum_{j}err(i,j)}{\sum_{i}\sum_{j} 1 }), for\ 2D\ case$$ $$err_{aver_n}=20\lg10(\frac{\sum_{i}\sum_{j}\sum_{k}err(i,j)}{\sum_{i}\sum_{j}\sum_{k} 1 }), for\ 3D\ case$$
shows one result of scenario 1 to 5 in 2D cases, which is randomly chosen from the testing samples. It can be observed that the predicted potential field distribution agrees well with the one computed by finite difference method. The final average relative error of the prediction in scenario 1 to 5 by ConvNet model is -41dB, -41dB, -38dB, -38dB, -40dB respectivly. And in the scenario 6, the final average relative error of the prediction is -38dB. The proposed ConvNet model shows a good prediciton capability and good generalization ability for 2D cases.
![Loss curves of trainning and testing in 2D cases []{data-label="traintestloss"}](traintestloss.jpg){width="6cm"}
The result of 3D cases is visualized in . The difference between ConvNet model’s prediction and FDM results is little and the final average relative error is -31dB. The ConvNet model can do good predictions on 3D cases which are more complicated than 2D cases. The good prediction capability and generalization ability of proposed ConvNet model is verified.
shows that in 2D cases, the curve of testing loss agrees well with training loss’s curve, which means the ConvNet model do not over-fit the training data.
Using this model, the CPU time is reduced significantly for 2D cases and 3D cases. For example, using FDM to obtain 2000 sets of 2D potential distribution takes 16s but using ConvNet model only takes 0.13s, and using FDM to obtain 5 sets of 3D potential distribution takes 292s but using ConvNet model only takes 1.2s. This indicates the possibility to build a realtime electromagnetic simulator.
Conclusion
==========
In this study, we investigate the possibility of using deep learning techniques to reduce the computational complexity in electromagnetic simulation. Here we compute the 2D amd 3D electrostatic problem as an example. By building up a proper convolutional neural network, we manage to correctly predict the potential field with the average relative error below 1.5% in 2D cases and below 3% in 3D cases. Moreover, the computational time is significantly reduced. This study shows that it may be possible to take advantage of the flexibility in deep neural networks and build up a fast electromagnetic solver that may provide realtime responses. In the future work, we will further improve the accurracy of 3D cases’ preidiction and try to build a fast electromagnetic realtime simulator.
[10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{}
W. Chew, E. Michielssen, J. M. Song, and J. M. Jin, Eds., *Fast and Efficient Algorithms in Computational Electromagnetics*.1em plus 0.5em minus 0.4emNorwood, MA, USA: Artech House, Inc., 2001.
G. H. Golub and C. F. V. Loan, *Matrix Computations*, 3rd ed.1em plus 0.5em minus 0.4emJohns Hopkins University Press, 1996.
A. Taflove and S. C. Hagness, *Computational Electrodynamics: The Finite-Difference Time-Domain Method*, 3rd ed.1em plus 0.5em minus 0.4emArtech House, 2005.
J.-M. Jin, *The Finite Element Method in Electromagnetics*, 3rd ed. 1em plus 0.5em minus 0.4emWiley-IEEE Press, 2014.
R. F. Harrington, *Field Computation by Moment Methods*.1em plus 0.5em minus 0.4emWiley-IEEE Press, 1993.
W. H. Schilders, H. A. van der Vorst, and J. Rommes, Eds., *Model Order Reduction: Theory, Research Aspects and Applications*.1em plus 0.5em minus 0.4emSpringer, 2008.
V. Prakash, , and R. Mittra, “Characteristic basis function method: A new technique for efficient solution of method of moments matrix equations,” *Microwave and Optical Technology Letters*, vol. 36, no. 2, pp. 95–100, 2003.
A. K. Noor and J. M. Peters, “Reduced basis technique for nonlinear analysis of structures,” *AIAA Journal*, vol. 18, no. 4, pp. 455–462, 1980.
X. Dang, M. Li, F. Yang, and S. Xu, “Quasi-periodic array modeling using reduced basis method,” *IEEE Antennas and Wireless Propagation Letters*, vol. 16, pp. 825–828, 2017.
A. H. Zaabab, Q.-J. Zhang, and M. Nakhla, “A neural network modeling approach to circuit optimization and statistical design,” *IEEE Transactions on Microwave Theory and Techniques*, vol. 43, no. 6, pp. 1349–1358, 1995.
Q. J. Zhang and K. C. Gupta, *Neural Networks for RF and Microwave Design*.1em plus 0.5em minus 0.4emArtech House, 2000.
Q.-J. Zhang, K. C. Gupta, and V. K. Devabhaktuni, “Artificial neural networks for rf and microwave design-from theory to practice,” *IEEE transactions on microwave theory and techniques*, vol. 51, no. 4, pp. 1339–1350, 2003.
G. E. Hinton and R. R. Salakhutdinov, “Reducing the dimensionality of data with neural networks,” *Science*, vol. 313, no. 5786, p. 504.
Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” *Nature*, vol. 521, no. 7553, p. 436.
S. Ehrhardt, A. Monszpart, N. J. Mitra, and A. Vedaldi, “Learning [A]{} physical long-term predictor,” *CoRR*, vol. abs/1703.00247, 2017. \[Online\]. Available: <http://arxiv.org/abs/1703.00247>
A. Lerer, S. Gross, and R. Fergus, “Learning physical intuition of block towers by example,” *arXiv preprint arXiv:1603.01312*, 2016.
J. Tompson, K. Schlachter, P. Sprechmann, and K. Perlin, “Accelerating eulerian fluid simulation with convolutional networks,” *CoRR*, vol. abs/1607.03597, 2016. \[Online\]. Available: <http://arxiv.org/abs/1607.03597>
X. Guo, W. Li, and F. Iorio, “Convolutional neural networks for steady flow approximation,” in *Proceedings of the 22Nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining*, ser. KDD ’16.1em plus 0.5em minus 0.4emNew York, NY, USA: ACM, 2016, pp. 481–490. \[Online\]. Available: <http://doi.acm.org/10.1145/2939672.2939738>
K. [Mills]{}, M. [Spanner]{}, and I. [Tamblyn]{}, “[Deep learning and the Schrödinger equation]{},” *ArXiv e-prints*, Feb. 2017.
A. Byravan and D. Fox, “Se3-nets: Learning rigid body motion using deep neural networks,” in *Robotics and Automation (ICRA), 2017 IEEE International Conference on*.1em plus 0.5em minus 0.4emIEEE, 2017, pp. 173–180.
H. M. Yao, Y. W. Qin, and L. J. Jiang, “Machine learning based mom (ml-mom) for parasitic capacitance extractions,” in *Electrical Design of Advanced Packaging and Systems (EDAPS), 2016 IEEE*.1em plus 0.5em minus 0.4emIEEE, 2016, pp. 171–173.
J. Long, E. Shelhamer, and T. Darrell, “Fully convolutional networks for semantic segmentation,” in *Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*, 2015, pp. 3431–3440.
A. Y. Ng, “Feature selection, l 1 vs. l 2 regularization, and rotational invariance,” in *Proceedings of the twenty-first international conference on Machine learning*.1em plus 0.5em minus 0.4emACM, 2004, p. 78.
S. Lawrence, C. L. Giles, A. C. Tsoi, and A. D. Back, “Face recognition: A convolutional neural-network approach,” *IEEE transactions on neural networks*, vol. 8, no. 1, pp. 98–113, 1997.
A. Krizhevsky, I. Sutskever, and G. E. Hinton, “Imagenet classification with deep convolutional neural networks,” in *Advances in neural information processing systems*, 2012, pp. 1097–1105.
X. Glorot, A. Bordes, and Y. Bengio, “Deep sparse rectifier neural networks,” in *Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics*, 2011, pp. 315–323.
D. Kingma and J. Ba, “Adam: A method for stochastic optimization,” *arXiv preprint arXiv:1412.6980*, 2014.
|
---
abstract: 'We carried out global three-dimensional resistive magnetohydrodynamic simulations of galactic gaseous disks to investigate how the galactic magnetic fields are amplified and maintained. We adopt a steady axisymmetric gravitational potential given by @Miy75 and [-@Miy80]. As the initial condition, we assume a warm ($T\sim10^{5}\,\, \mathrm{K}$) rotating gas torus centered at $\varpi=10\,\, \mathrm{kpc}$ threaded by weak azimuthal magnetic fields. Numerical results indicate that in differentially rotating galactic gaseous disks, magnetic fields are amplified due to magneto-rotational instability and magnetic turbulence develops. After the amplification of magnetic energy saturates, the disk stays in a quasi-steady state. The mean azimuthal magnetic field increases with time and shows reversals with period of 1Gyr (2Gyr for a full cycle). The amplitude of $B_{\varphi}$ near the equatorial plane is $B_{\varphi}\sim1.5\mu\,\, \mathrm{G}$ at $\varpi=5\,\, \mathrm{kpc}$. The magnetic fields show large fluctuations whose standard deviation is comparable to the mean field. The mean azimuthal magnetic field in the disk corona has direction opposite to the mean magnetic field inside the disk. The mass accretion rate driven by the Maxwell stress is $\sim10^{-3}M_{\odot}/\,\, \mathrm{yr}$ at $\varpi=2.5\,\, \mathrm{kpc}$ when the mass of the initial torus is $\sim5\times10^{8}M_{\odot}$. When we adopt an absorbing boundary condition at $r=0.8\,\, \mathrm{kpc}$, the rotation curve obtained by numerical simulations almost coincides with the rotation curve of the stars and the dark matter. Thus even when magnetic fields are not negligible for gas dynamics of a spiral galaxy, galactic gravitational potential can be derived from observation of rotation curves using gas component of the disk.'
author:
- Hiromitsu Nishikori
- Mami Machida
- Ryoji Matsumoto
title: |
GLOBAL THREE-DIMENSIONAL MHD SIMULATIONS OF GALACTIC GASEOUS DISKS:\
I. AMPLIFICATION OF MEAN MAGNETIC FIELDS IN AXISYMMETRIC GRAVITATIONAL POTENTIAL
---
INTRODUCTION
============
Spiral galaxies have magnetic fields. In our Galaxy, the average magnetic field strength is a few $\mu$G. [e.g., @Sof86; @Bec96]. These magnetic fields have been explained by the dynamo action operating in galactic gas disks [e.g., @Par71]. In conventional theory of galactic dynamos, the following induction equation is solved; $$\frac{\partial\bar{\mbox{\boldmath $B$}}}{\partial t} =
\nabla\times(\mbox{\boldmath $v$}\times\bar{\mbox{\boldmath $B$}})
+\eta \nabla^{2} \bar{\mbox{\boldmath $B$}} +
\nabla\times(\alpha \bar{\mbox{\boldmath$B$}}),
\label{eqn1}$$ where $\eta$ denotes the turbulent magnetic diffusivity, is the mean velocity, and is the mean magnetic field. The last term on the right hand side of equation (\[eqn1\]) is the dynamo term which represents the re-generation of mean magnetic fields ($\alpha$ effect). When velocity fields are given, equation (\[eqn1\]) is linear with . It can be solved with appropriate boundary conditions. This approach is referred to as the [*kinematic dynamo*]{}. The effects of nonlinearities such as the quenching of the $\alpha$-effect for strong magnetic fields and the loss of magnetic flux due to the magnetic buoyancy have also been studied in the framework of the kinematic dynamo theory [e.g., @Sch89; @Bra89; @Bra92].
In 1990s, it became clear that nonlinear dynamical processes are essential for the evolution of magnetic fields in differentially rotating disks. @Bal91 showed that in the presence of weak magnetic fields, differentially rotating gas disks subject to the robust magneto-rotational instability (MRI). This instability is caused by the back-reaction of magnetic fields to the fluid motion. Thus, in order to study the evolution of magnetic fields in differentially rotating disks, we should solve the momentum equation including Lorentz force simultaneously with the induction equation. This approach is called [*dynamical dynamo*]{}. The developments of computational magnetohydrodynamics (MHD) enabled us to study this dynamical dynamo process by direct three-dimensional (3D) global simulations.
In the context of accretion disks, the nonlinear growth of MRI was first studied by local 3D MHD simulations [e.g., @Haw95; @Mat95; @Bra95]. They showed that MRI drives magnetic turbulence in accretion disks and amplifies magnetic fields. Subsequently, global 3D MHD simulations of initially weakly magnetized rotating torus were carried out [e.g., @Mat99; @Haw00; @Mac00]. They showed that the magnetic field amplification saturates when $\beta=P_{gas}/P_{mag} \sim 10$. The magnetic fields maintained in the quasi-steady state are mainly toroidal but have turbulent poloidal components. They also showed that in the innermost region of black hole accretion disks where radial infall of disk gas becomes important, spiral magnetic fields which accompany radial field reversals are created [e.g., @Mac03].
In numerical studies of accretion disks, the gravitational field is assumed to be that of the central point mass. It is straightforward to extend this gravitational potential to more general potentials such as those of spiral galaxies. Since the galactic gaseous disk is also a differentially rotating disk, it subjects to the MRI. The growth of MRI and generation of MHD turbulence in galactic HI disks were discussed by @Sel99. @Kim03 carried out three-dimensional local MHD simulations of galactic gaseous disks and showed that MRI drives interstellar turbulence and helps forming the giant molecular clouds. The coupling of the thermal instability and MRI in the interstellar space has been studied by @Pio04.
@Dzi03 reported the results of global three-dimensional MHD simulations of the galactic gaseous disks including the dynamo $\alpha$ term. More recently, @Dzi04 carried out global three-dimensional MHD simulations of galactic gaseous disks without assuming the dynamo $\alpha$ term starting from vertical magnetic fields and showed that mean magnetic fields and interstellar turbulence are generated. However, their simulation area was limited to 5kpc in the radial direction and 1kpc in the vertical direction. @Kit04 carried out linear but global analysis for MRI in a disk geometry and showed that MRI can amplify very weak seed magnetic fields in proto-galaxies.
In this paper, we report the results of global 3D MHD simulations of galactic gaseous disks by using the axisymmetric gravitational potential created by stars and the dark matter. We solve the nonlinear MHD equations without introducing the phenomenological dynamo $\alpha$ parameter. The initial magnetic field is assumed to be toroidal. This initial condition enables us to start the simulation without much disturbing the interface between the disk and the disk corona. When we start from vertical magnetic fields, efficient angular momentum loss near the surface of the initial torus drives an avalanche-like surface accretion, which deforms magnetic field lines into an hourglass shape and produces magnetocentrifugally driven outflows [e.g., @Mat96]. For simplicity we neglect the multi-phase nature of galactic gaseous disk, non-axisymmetric spiral gravitational potential, self gravity and supernova explosions. These processes will be incorporated in subsequent papers.
In section 2, we present numerical methods. Numerical results are given in section 3. Section 4 is devoted for summary and discussion.
NUMERICAL METHODS
=================
Basic Equations
---------------
The basic equations we solve are resistive MHD equations,
$$\frac{\partial \rho}{\partial t} + \nabla \cdot
(\rho \mbox{\boldmath $v$})=0, \label{eqn:eqn2.1}$$
$$\rho\left[\frac{\partial\mbox{\boldmath $v$}}{\partial t} +
(\mbox{\boldmath $v$}\cdot\nabla)\mbox{\boldmath $v$}\right] =
- \nabla P + \frac{1}{4\pi}(\nabla \times \mbox{\boldmath $B$}) \times
\mbox{\boldmath $B$}
- \rho \nabla \phi,
\label{eqn:eqn2.2}$$
$$\frac{\partial\mbox{\boldmath $B$}}{\partial t} =
\nabla\times(\mbox{\boldmath $v$}\times\mbox{\boldmath $B$}
-\eta \nabla \times \mbox{\boldmath $B$}),
\label{eqn:eqn2.3}$$
$$\begin{aligned}
\frac{\partial}{\partial t}
\left(\frac{1}{2}\rho v^2 + \frac{B^2}{8\pi} + \frac{P}{\gamma -1}\right)
\hspace{35mm} \nonumber\\
+ \nabla\cdot\left[\left(\frac{1}{2}\rho\mbox{\boldmath $v$}^2 +
\frac{\gamma P}{\gamma -1}\right)\mbox{\boldmath $v$} +
\frac{c}{4\pi}\mbox{\boldmath $E$}\times\mbox{\boldmath $B$}\right]
\nonumber\\
= - \rho \mbox{\boldmath $v$} \nabla \phi .
\label{eqn:eqn2.4}\end{aligned}$$
In these equations, $\rho$, $p$, $\mbox{\boldmath $B$}$, $\mbox{\boldmath $v$}$, $\phi$ and $\gamma$ are the density, pressure, magnetic fields, velocity, gravitational potential and the specific heat ratio, respectively. The electric field is related to the magnetic field by = ($-$$\times$ + $\eta$$\nabla$$\times$)/$c$. We assume the anomalous resistivity $\eta$ $$\eta = \left\{
\begin{array}{ll}
\eta_{0} (v_{d}/v_{c}-1)^2, & (\mbox{when}\: v_{d} > v_{c}), \\
0, & (\mbox{when}\: v_{d} < v_{c}),
\end{array}
\right.$$ where $v_{d}=j / \rho$ is the electron-ion drift velocity and $v_{c}$ is the critical velocity above which anomalous resistivity sets in [@Yok94].
We adopted cylindrical coordinates $(\varpi, \varphi, z)$. For normalization, we take the unit radius $\varpi_{0} = 1 \,\, \mathrm{kpc}$, and unit velocity $v_{0} = 255 \,\, \mathrm{km/s}$. The unit time is $t_{0} = \varpi_{0} / v_{0} = 3.8 \times 10^{6}$ years. For specific heat ratio and resistivity, we adopt $\gamma = 5 / 3$ and $\eta_{0}=0.01 v_{0}\varpi_{0}$, respectively.
Initial Condition
-----------------
As the initial density distribution, we assume a rotating equilibrium torus threaded by azimuthal magnetic fields embedded in the isothermal, non-rotating, spherical hot halo [@Oka89]. We assume that the gas torus has constant specific angular momentum $L$ and assume polytropic equation of state, $P = K \rho^{\gamma}$ where $K$ is a constant. This initial condition simulates the situation that after a spiral galaxy is formed, the interstellar gas is supplied either by accretion of intergalactic gas which has some specific angular momentum or by supernova explosions following bursts of star formation inside the galaxy at some radius. As we shall show later, the angular momentum distribution of the gas settles into that determined by the galactic gravitational potential. Thus, the final distribution of density and angular momentum is not sensitive to the initial distribution.
We assume that the Alfvén speed $v_{A}$ is a function of $\varpi B_{\varphi}$, $$v_{A}^{2} \equiv \frac{B_{\varphi}^2}{4\pi\rho} =
\frac{(4 \pi \mathcal{H})^{1/\mu}}{4\pi}
(\varpi B_{\varphi})^{2(\mu-1)/\mu},
\label{eqn:Alfven}$$ where $B_{\varphi}$ is the toroidal magnetic field and $\mathcal{H}$ and $\mu$ are constants. We take $\mu = \gamma$. Using these assumptions, we can integrate the equation of motion into the potential form, $$\begin{aligned}
\Psi (\varpi , z) = \phi + \frac{L^2}{2\varpi^2} +
\frac{1}{\gamma -1}v_{s}^{2} +
\frac{\gamma}{2(\gamma-1)}v_{A}^{2}
= \Psi_{b} \nonumber \\
= constant,
\label{eqn:pot1}\end{aligned}$$ where $v_{s} = (\gamma P / \rho)^{1/2}$ is the sound speed and $\Psi_{b} = \Psi (\varpi_{b},0)$. We take the reference radius $\varpi_{b}$ at the radius of the density maximum of the torus. We adopt $\varpi_{b} = 10 \,\, \mathrm{kpc}$ and set $\rho(\varpi_{b},0) = \rho_{b}$. Using equation (\[eqn:pot1\]), we obtain the density distribution
$$\rho = \rho_{b}\left[
\frac{\,\, \mathrm{max} \{ \Psi_{b} -\phi (\varpi,z) - L^{2}/(2\varpi^{2}),0 \} }
{K\{\gamma/(\gamma -1)\}\left( 1+\beta_{b}^{-1}\varpi^{2(\gamma - 1)}
/\varpi_{b}^{2(\gamma - 1)}\right)}
\right]^{1/(\gamma - 1)},
\label{eqn:density}$$
where $\beta_{b} = (2K/\mathcal{H}) / \varpi_{b}^{2(\gamma - 1)}$ is the ratio of gas pressure to magnetic pressure at $(\varpi,z) = (\varpi_{b},0)$. The thermal energy of the torus is parameterized by $$E_{th} =\frac{c_{sb}^{2}}{\gamma v_{0}^{2}}
=0.05,
\label{eqn:ethermal}$$ where $c_{sb}$ is the sound speed at $(\varpi, z) = (\varpi_{b}, 0)$. This sound speed corresponds to the temperature $T_{b} \sim 2 \times 10^{5} \,\, \mathrm{K}$. This means that we assume a warm torus as the initial condition. For unit density, we take $\rho_{b} = 3 \times 10^{-25} \,\, \mathrm{g/cm}^{3}$. The initial gas pressure at $(\varpi, z) = (\varpi_{b}, 0)$ is $$\begin{aligned}
P_{b} &=& P(\varpi_{b},0) = \frac{1}{\gamma}\rho_{b}c_{sb}^{2}
\nonumber \\
&=& 10^{-11}\left(\frac{\rho_{b}}{3 \times 10^{-25}\,\, \mathrm{g/cm^{3}}}\right)
\left(\frac{T_{b}}{2 \times 10^{5}\,\, \mathrm{K}}\right)
\,\, \mathrm{dyne/cm^{2}}.
\label{eqn:pressure}\end{aligned}$$ The density of the corona $\rho_{c}$ is given by $$\rho_{c}
=10^{-4}\rho_{b}\,\, \mathrm{exp}\left[ -\frac{\gamma}{c_{sc}^{2}}
(\phi - \phi_{b})\right],
\label{eqn:halo}$$ where $c_{sc}$ is the sound speed in the corona and $\phi_{b}$ is the gravitational potential at $(\varpi, z) = (\varpi_{b}, 0)$. We take the characteristic coronal pressure $P_{c} = \rho_{c}c_{sc}^{2} \sim 10^{-4}\rho_{b}c_{sc}^{2} = 3 \times 10^{-3}P_{b}$.
As the gravitational potential $\phi$, we adopted a model of our galaxy given by @Miy75 and @Miy80. They assumed centrifugal balance only in the galactic plane. To reproduce the galactic rotation curve, three different sets of parameters $(a_{1}, b_{1}, M_{1})$, $(a_{2}, b_{2}, M_{2})$, $(a_{3}, b_{3}, M_{3})$ are introduced in the gravitational potential as follows, $$\phi (\varpi , z) =
\sum_{i=1}^{3}
\frac{GM_{i}}{[\varpi^2 +
\{a_{i} + (z^{2} + b_{i}^{2})^{0.5}\}^{2}]^{0.5}}.
\label{eqn:MNpotential}$$ Here $i = 1$ and $2$ take into account the gravitational field by bulge and disk stars, respectively. The third component $i = 3$ is the contribution by dark matter. In this equation, the constants $a_{i}$, $b_{i}$ and $M_{i}$ have dimension of length and mass, respectively, and $G$ is the gravitational constant. This potential is axisymmetric. The values of the constants are given in Table \[tbl:tbl1\]. Model I and II are models without dark matter and model III-VII include the dark matter.
Figure \[fig:f1\] shows the isocontours of the gravitational potential for model I. Figure \[fig:f2\] shows the isocontours of the initial density and toroidal magnetic field for model I.
Simulation Code
---------------
We solved the MHD equations by using a three-dimensional MHD code based on a modified Lax-Wendroff method [@Rub67] with artificial viscosity [@Ric67]. This code was originally developed to carry out two-dimensional axisymmetric global MHD simulations of jet formation from accretion disks [e.g., @Shi85; @Mat96; @Hay96]. The MHD code was extended to three-dimensions and has been used to carry out global three-dimensional MHD simulations of accretion disks (@Mac00; @Mac03 [@Kat04]).
Simulation Region and Boundary Conditions
-----------------------------------------
The number of mesh points is $(N_{\varpi}, N_{\varphi}, N_{z}) = (250, 64, 319)$ for model I-IV. The grid size is $\Delta \varpi = 0.05, \Delta z = 0.01$ for $0 \leq \varpi < 6.0$, $0 \leq z < 2.0$, and otherwise increases with $\varpi$ and $z$ as $\Delta z_{k+1}= 1.05 \Delta z_{k}$. We used this nonuniform mesh to concentrate meshes near the galactic plane and near the central region. The size of the simulation box is $0 \,\, \mathrm{kpc} < \varpi < 56 \,\, \mathrm{kpc}$, and $0 \,\, \mathrm{kpc} < z < 10 \,\, \mathrm{kpc}$. We used this large simulation box in order to simulate the buoyant rise of magnetic flux into the galactic corona and to reduce the effects of the outer boundary condition.
We imposed free boundary conditions at $\varpi = 56 \,\, \mathrm{kpc}$ and $z = 10 \,\, \mathrm{kpc}$ where waves can be transmitted. For azimuthal direction, we include full circle ($0 \leq \varphi < 2 \pi$) for models I-IV. We carried out full circle simulations because we are interested in the formation of large-scale mean magnetic fields. We should note that the amplitude of modes with low azimuthal mode numbers ($m = 1$ or $m = 2$) are often large in global simulations of accretion disks [e.g., @Mac03]. In order to study the effects of the azimuthal resolution and the azimuthal size of the simulation region, we carried out simulations for model V, VI and VII, in which the azimuthal simulation region is limited to 0 $\leq \varphi < \pi/2$. Periodic boundary conditions are imposed in the azimuthal direction. The other parameters are the same as those for model III (Table \[tbl:tbl2\]).
We assume symmetric boundary condition at the equatorial plane $z = 0$ and simulated only the upper half space. The physical quantities at rotation axis ($\varpi = 0$) are computed by averaging those at the mesh point next to the axis. In models I and III-VII, we imposed absorbing boundary condition at $r = (\varpi^{2}+z^{2})^{1/2} = r_{in} = 0.8 \,\, \mathrm{kpc}$. We introduce a damping layer inside $r = r_{in}$. In this layer, the deviation of physical quantity $q$ from initial value $q_{0}$ is reduced after each time step with a damping rate $a$, $$q'=q-a(q-q_{0}).$$ Here we take $$a =
0.1 \left( 1.0- \tanh \frac{r-0.8+5 \Delta \varpi}{2 \Delta \varpi } \right).$$ This damping layer serves as the non-reflecting boundary which absorbs accreting mass and waves propagating into $r = 0.8$. This boundary condition is adopted in our standard model because the gas accreted to the central region of the galaxy will be converted to stars or absorbed by the central massive black hole. In model II, we imposed no specific condition at $r = r_{in}$. Thus the accreting mass piles up in the central region.
Initial Perturbation
--------------------
To initiate the evolution, we imposed random perturbation for the azimuthal velocity with maximum amplitude $0.01 v_{0}$. We checked the stability of the initial state by carrying out numerical simulations without including magnetic fields and without imposing velocity perturbations. Since the boundary between the rotating disk and the static corona is not exactly in hydrostatic equilibrium, small motions with maximum velocity $0.05 v_{0}$ appears in this region. The radial velocity in the dense region of the torus is less than $0.001 v_{0}$. The radial flow in the boundary layer between the disk and the corona slightly modifies the density distribution and smears the difference of the rotation speed between the disk and the corona. However, the flow does not significantly modify the density distribution of the main part of the torus. At $t = 2 \,\, \mathrm{Gyr}$, the maximum radial speed is about $0.0045 v_{0}$ even at the interface between the disk and the corona. The torus stays in the equilibrium state for time scale longer than $2.3 \,\, \mathrm{Gyrs}$.
NUMERICAL RESULTS
=================
Time Evolution of the Model with Absorbing Inner Boundary Condition
-------------------------------------------------------------------
First we present the results of Model I (the standard model). The inner boundary is treated as an absorbing boundary where accreting mass and magnetic fields are absorbed. Figure \[fig:f3\] shows the time evolution of model I. Color contours show density distribution and solid curves show magnetic field lines. The left panels show the isosurface of the density ($\rho / \rho_{b} = 10^{-1.5}$), density distribution at the equatorial plane and magnetic field lines. We plotted the region $-15 \,\, \mathrm{kpc} < x, y, z < 15 \,\, \mathrm{kpc}$. The right panels show the equatorial density and magnetic field lines projected onto the equatorial plane. As the MRI grows, matter accretes to the central region by losing the angular momentum. The initial torus deforms itself into a flat disk. On the other hand, the matter in the outer part of the disk gets angular momentum and expands. The magnetic fields globally show spiral structure and locally show turbulent structure.
Figure \[fig:f4\] shows the close up view of the density distribution and magnetic field lines projected onto the equatorial plane at $t = 1000 t_{0} (= 3.8 \,\, \mathrm{Gyr})$. The box size is $-15 \,\, \mathrm{kpc} < x,y < 15 \,\, \mathrm{kpc}$. The density distribution shows weak non-axisymmetric spiral pattern.
Figure \[fig:f5\][*a*]{} shows the time evolution of magnetic energy for model I averaged in $2 \,\, \mathrm{kpc} < \varpi < 5 \,\, \mathrm{kpc}$, $0 \,\, \mathrm{kpc} < z < 1 \,\, \mathrm{kpc}$ and $0 \leq \varphi < 2\pi$. Figure \[fig:f5\][*b*]{} shows the plasma $\beta$ averaged in the same region. As the MRI grows in the disk, the magnetic energy increases and its strength saturates when $\log [(B^{2} / 8\pi) / P_{b}] \sim -1.5$. For our Galaxy, this saturation level corresponds to $B \sim 3 \mu \,\, \mathrm{G}$. The magnetic energy is maintained for time scale at least $2 \,\, \mathrm{Gyrs}$. As magnetic energy is amplified, the average value of plasma $\beta \equiv \langle P \rangle / \langle B^{2} / 8\pi \rangle$ decreases and stays around $\langle \beta \rangle \sim 20$ where $\langle \beta \rangle$ is the spatial average of local plasma $\beta$ in $2.0 \,\, \mathrm{kpc} < \varpi < 5.0 \,\, \mathrm{kpc}$, $0 \,\, \mathrm{kpc} < z < 1.0 \,\, \mathrm{kpc}$ and $0 \leq \varphi < 2 \pi$. This saturation level of magnetic energy is smaller than that observed in our galaxy. Further amplification of magnetic energy by, for example, supernova explosions may be necessary to explain the strength of magnetic fields in our galaxy.
Magnetic field lines shown in the left panels of figure \[fig:f3\] indicate that magnetic fields initially confined in the disk emerge from the disk and form extended magnetized corona. Figure \[fig:f6\] shows the vertical distribution of the gas pressure and the magnetic pressure at $\varpi = 3 \,\, \mathrm{kpc}$, $5 \,\, \mathrm{kpc}$ and $10 \,\, \mathrm{kpc}$ at $t = 0$ and $t = 1000t_{0} ( = 3.8 \,\, \mathrm{Gyr})$. The magnetic flux initially confined in the torus fills almost the whole simulation region. The plasma $\beta$ in the corona is $1 < \beta < 10$ in $1 \,\, \mathrm{kpc} < z < 10 \,\, \mathrm{kpc}$.
Magnetic fields emerge from the disk to the corona due to the growth of the Parker instability [@Par66]. Nonlinear growth of Parker instability in disks was studied by @Mat88 by two-dimensional MHD simulations taking into account the vertical variation of the gravitational field. The effects of the differential rotation and MRI on the Parker instability were studied by @Fog94 [@Fog95]. @Mil00 carried out local three-dimensional MHD simulations of gravitationally stratified differentially rotating disks and showed that strongly magnetized corona is created due to the buoyant rise of magnetic flux from the disk to the corona. @Mac00 showed by global three-dimensional MHD simulations that buoyantly rising magnetic loops are formed above accretion disks. Our numerical results are consistent with these previous results.
Figure \[fig:f7\] shows the radial distribution of azimuthal velocity $v_{\varphi}$ and the density $\rho$ averaged in $0 \,\, \mathrm{kpc} < z < 0.3 \,\, \mathrm{kpc}$ and $0 \leq \varphi < 2\pi$ at $t = 3.8 \,\, \mathrm{Gyr}$. In the inner region ($\varpi < 10 \,\, \mathrm{kpc}$), the radial profile of azimuthal velocity of model I is similar to that observed in our Galaxy. The density distribution in Figure \[fig:f7\][*b*]{} indicates that the initial torus spreads and becomes flat in the inner region.
Figure \[fig:f8\][*a*]{} shows the time evolution of the ratio of Maxwell stress averaged in $2 \,\, \mathrm{kpc} < \varpi < 5 \,\, \mathrm{kpc}$, $0 \,\, \mathrm{kpc} < z < 1 \,\, \mathrm{kpc}$ and $0 \leq \varphi < 2 \pi$ to the initial gas pressure at $\varpi = 10 \,\, \mathrm{kpc}$, $$\alpha_{B} \equiv \left\langle - \frac{B_{\varpi} B_{\varphi}}{4 \pi P_{b}}\right\rangle
.
\label{eqn:Maxwell_stress}$$ The Maxwell stress increases as the gas accretes and later decreases gradually.
Figure \[fig:f8\][*b*]{} shows the time evolution of the accretion rate at $\varpi = 2.5$kpc defined by $$\dot M_{\varpi=2.5} =
- \int_{0}^{2\pi} \int_{0}^{2.5kpc} \rho v_{\varpi} \varpi d \varphi dz.
\label{eqn:M_dot}$$ The unit of the accretion rate is $1 M_{\odot} / \,\, \mathrm{yr}$ when $\rho_{b} = 3 \times 10^{-25} \,\, \mathrm{g/cm}^{3}$. Numerical result indicates that $\dot M \sim 10^{-3}M_{\odot} / \,\, \mathrm{yr}$. Since the mass of the initial torus is $M_{torus} = 5 \times 10^{8}M_{\odot}$, mass accretion takes place for time scale longer than $10 \,\, \mathrm{Gyr}$.
Effects of the Central Absorber and Dark Matter
-----------------------------------------------
In this subsection, we show the effects of the central absorber and the dark matter. In model I, we assumed that the interstellar matter accreting to the central region are absorbed at $r = 0.8 \,\, \mathrm{kpc}$. This mimics the existence of the central black hole or the efficient conversion of gas to stars in the central region. When such absorber does not exist, accreting matter piles up in the central region. As a reference model, we carried out a simulation of model II, in which we allow the accreting mass to pile up in the central region.
Figure \[fig:f9\] shows the density distribution and magnetic field lines at $t = 1000t_{0} ( = 3.8 \,\, \mathrm{Gyr})$ for model II. Since accreted matter accumulates in the central region, it forms a high density bulge.
Figure \[fig:f10\] shows the distribution of density and magnetic field lines at $t = 3.8 \,\, \mathrm{Gyr}$ for model III. In this model, the effect of dark matter is included. Absorbing boundary condition is imposed at $r = 0.8 \,\, \mathrm{kpc}$. The density distribution and magnetic field lines are similar to those of model I.
Figures \[fig:f11\][*a*]{} and \[fig:f11\][*b*]{} show the time evolution of magnetic energy and plasma $\beta$ averaged in $2 \,\, \mathrm{kpc} < \varpi < 5 \,\, \mathrm{kpc}$, $0 \,\, \mathrm{kpc} < z < 1 \,\, \mathrm{kpc}$ and $0 \leq \varphi < 2 \pi$, for models I, II and III. The magnetic energy saturates at approximately the same level for all models. The magnetic energy is maintained more than 2Gyrs. However, after significant amount of gas accretes, the value of plasma $\beta$ in model II becomes larger than that of other models, because in model II, the gas density and pressure increase with time. Figure \[fig:f11\][*c*]{} shows the ratio of Maxwell stress to gas pressure averaged in $2 \,\, \mathrm{kpc} < \varpi < 5 \,\, \mathrm{kpc}$, $0 \,\, \mathrm{kpc} < z < 1 \,\, \mathrm{kpc}$ and $0 \leq \varphi < 2 \pi$ for model I, II and III. The time evolution of Maxwell stress has almost the same profile in all models. Figure \[fig:f11\][*d*]{} shows the time evolution of the accretion rate at $\varpi = 2.5 \,\, \mathrm{kpc}$ for these models. In model II, the accretion rate is about 10 times larger than that of model I and model III because the density near $\varpi = 2.5 \,\, \mathrm{kpc}$ of model II is larger than the density of model I and model III (see figure \[fig:f12\][*b*]{}).
Figure \[fig:f12\] shows the distribution of azimuthal velocity $v_{\varphi}$ and density $\rho$ averaged in $0 \,\, \mathrm{kpc} < z < 0.3 \,\, \mathrm{kpc}$ and $0 \leq \varphi < 2 \pi$ at $t = 3.8 \,\, \mathrm{Gyr}$. The azimuthal velocity of model I and model III is similar to the observed profile of our Galaxy in the inner region. The velocity profile of model III coincides with the observed profile in all region because the effects of dark matter are included. On the other hand, the velocity distribution of model II is different from the observed rotation curve. In the inner region, the density of model II is ten times larger than that of model I and model III because the accreted gas piles up in the central region. When mass piles up in the central region, since the pressure gradient force in the radial direction reduces the effective gravity in the radial direction, the equilibrium rotation speed becomes much less than the observed rotation speed. Thus, the existence of the central absorber or conversion of infalling gas to stars are essential to reproduce the observed rotation curve. In the outer region ($8 \,\, \mathrm{kpc} < \varpi$) velocity and density of model I and model II almost coincide.
Spatial and Temporal Reversals of Mean Magnetic Fields {#sec:mmf}
------------------------------------------------------
Figure \[fig:f13\] shows the magnetic field lines depicted by mean magnetic fields for model III at $t = 3.8 \,\, \mathrm{Gyr}$. Regions colored in orange or blue show domains where mean azimuthal magnetic field at $z = 0.25\,\, \mathrm{kpc}$ is positive or negative, respectively. The box size is $30 \,\, \mathrm{kpc} \times 30 \,\, \mathrm{kpc}$. We computed the mean magnetic field at each grid point $${\bar{\mbox{\boldmath $B$}}}(\varpi_{i},\varphi_{j},z_{k}) =
\frac{
\sum^{i+10}_{l=i-10} \sum^{j+3}_{m=j-3} \sum^{k+10}_{n=k-10}
\mbox{\boldmath $B$} (\varpi_{l},\varphi_{m},z_{n})
}
{21 \times 7 \times 21},$$ by averaging magnetic fields inside $\pm 10$ meshes in $\varpi$ and $z$ directions, and $\pm 3$ meshes in $\varphi$ direction. The mean magnetic field is mainly toroidal but shows reversals of toroidal components around $\varpi \sim 5 \,\, \mathrm{kpc}$ at this time.
The azimuthal direction of mean magnetic fields in the disk changes with radius in regions where channel-like flow develops. @Haw92 showed by axisymmetric simulations that channel-like flows appear in the nonlinear stage of MRI. Even when the unperturbed magnetic field is purely azimuthal, spiral channel flows appear as a result of the nonlinear growth of the non-axisymmetric MRI [@Mac03]. Since such magnetic channels move inward as the infalling gas loses angular momentum, the radius of the field reversal changes with time.
Figure \[fig:f14\] shows the azimuthally averaged mean toroidal field $\langle \bar{B_{\varphi}} \rangle$ (mean field) and the standard deviation of azimuthal field $\sqrt{\langle (B_{\varphi}-{\bar{B_{\varphi}}})^{2}} \rangle$ (fluctuating field) at $t = 3.1\,\, \mathrm{Gyr}$ where $\langle$ $\rangle$ denotes the spatial average. The dashed, dotted and solid curves show the distribution of mean azimuthal magnetic fields averaged in azimuthal direction ($0 \leq \varphi < 2 \pi$) and in $0.1 \,\, \mathrm{kpc} < z < 1.0 \,\, \mathrm{kpc}$ (equatorial region), $1.0 \,\, \mathrm{kpc} < z < 3.0 \,\, \mathrm{kpc}$ (coronal region) and $0.1 \,\, \mathrm{kpc} < z < 3.0 \,\, \mathrm{kpc}$, respectively. The vertical bars show the standard deviation of the azimuthal field. The strength of fluctuating field is comparable to the mean magnetic field. The dash-dotted curve shows the initial distribution of azimuthal mean magnetic fields averaged in 0.3kpc $< z <$ 2.0kpc and 0 $\leq \varphi < 2 \pi$. The strength of equatorial azimuthal magnetic field ($0.1 \,\, \mathrm{kpc} < z < 1 \,\, \mathrm{kpc}$) is larger than the initial azimuthal field by factor 2.
The mean azimuthal magnetic field in the coronal region ($1.0 \,\, \mathrm{kpc} < z < 3.0 \,\, \mathrm{kpc}$) has direction opposite to that in the disk region. The total magnetic flux threading the [*$\varpi$-z*]{} plane is almost conserved.
Figure \[fig:f15\] shows the spatial distribution of mean azimuthal magnetic fields for model III at $t = 590t_{0} (=2.2
\,\, \mathrm{Gyr})$ and $826t_{0} (= 3.1 \,\, \mathrm{Gyr})$. Blue and orange indicate regions where mean magnetic field $\bar{B_{y}}$ threading the $y = 0$ plane is positive or negative, respectively. Arrows show the direction of magnetic fields. These panels show that the mean azimuthal field $\bar{B_{y}}$ threading the $y=0$ plane reverses its direction in the coronal region. The maximum height of the magnetized region (the wavefront of the azimuthal magnetic field) at $\varpi = 10 \,\, \mathrm{kpc}$ locates at $z =
4 \,\, \mathrm{kpc}$ at $t = 2.2 \,\, \mathrm{Gyr}$ and $z \sim 10 \,\, \mathrm{kpc}$ at $t = 3.1 \,\, \mathrm{Gyr}$. It indicates that the azimuthal magnetic flux is rising.
The seeds for the reversals of the azimuthal magnetic fields are generated during the growth of MRI because when the initial magnetic field is purely azimuthal, the growth rate of the MRI is larger when $q = (k_{y}/k_{z})^{2}$ is small, where $k_{y}$ and $k_{z}$ are the wavenumber in the azimuthal direction and the vertical direction, respectively [e.g., @Mat95]. Thus, the azimuthal direction of the perturbed magnetic field changes with height. In the nonlinear stage when the vertical magnetic fields are produced by the buoyant rise of the azimuthal magnetic flux, the formation of the channel flows [e.g., @Haw92] due to the nonlinear growth of MRI also produces the reversals of azimuthal magnetic fields.
Figure \[fig:f16\][*a*]{} shows the time variation of the vertical distribution of azimuthally averaged magnetic field $\langle B_{\varphi}
\rangle$ for model III at $\varpi = 10 \,\, \mathrm{kpc}$. The regions where $\langle B_{\varphi} \rangle < 0$ are not plotted. The wavefront of the azimuthal flux propagates in the vertical direction with speed $v \sim 8 \,\, \mathrm{kpc}/1.5 \,\, \mathrm{Gyr} \sim 5
\,\, \mathrm{km/s}$. This speed is comparable to the Alfvén speed. The vertical distribution of the azimuthal field $\langle B_{\varphi} \rangle$ approximately follows the self-similar solution of the nonlinear Parker instability $\langle B_{\varphi} \rangle \propto z^{-1}$ [@Shi89]. Figure \[fig:f16\][*b*]{} shows the time evolution of the height of the wavefront of the azimuthal magnetic flux (solid curve) and the height of reversals of azimuthal magnetic fields (dashed curve and dash-dotted curve) at $\varpi = 10 \,\, \mathrm{kpc}$. The exponential increase of the height of the wavefront is consistent with the self-similar solutions of the nonlinear Parker instability [@Shi89]. After an azimuthal magnetic flux with one direction rises, another azimuthal magnetic flux with opposite direction rises. The MRI unstable disk continuously feeds the corona with magnetic fields.
In figure \[fig:f17\] we plotted the time variation of mean azimuthal field of model III. Magnetic fields are averaged in azimuthal direction ($0 \leq \varphi < 2 \pi$), radial direction ($5 \,\, \mathrm{kpc} \leq \varpi < 6 \,\, \mathrm{kpc}$), and in $0.1 \,\, \mathrm{kpc} < z < 1.0 \,\, \mathrm{kpc}$ (equatorial region), $1.0 \,\, \mathrm{kpc} < z < 3.0 \,\, \mathrm{kpc}$ (coronal region) and $0.1 \,\, \mathrm{kpc} < z < 3.0 \,\, \mathrm{kpc}$ (disk + corona), respectively. The equatorial magnetic field in $5\,\, \mathrm{kpc}<\varpi<6\,\, \mathrm{kpc}$ changes direction with time. The azimuthal magnetic fields reverse their direction with timescale $t \sim 300t_{0} (\sim 1 \,\, \mathrm{Gyr})$. This timescale is comparable to that of the buoyant rise of azimuthal magnetic flux, which takes place in $t \sim 10 H/v_{A}$, [e.g., @Par66], where $H$ is the half thickness of the disk ($\sim 0.5 \,\, \mathrm{kpc}$), and $v_{A}$ is the Alfvén speed around the disk-corona interface ($v_{A} \sim 5 \,\, \mathrm{km/s}$).
Here we would like to point out that the amplification of equatorial magnetic field is enabled by the escape of magnetic flux from the disk to the corona. When the total azimuthal magnetic flux is conserved, the azimuthal magnetic flux inside the disk can increase when magnetic flux escapes from the disk to the corona. Although azimuthal magnetic flux is not exactly conserved in resistive MHD simulations, the flux is nearly conserved because $\eta$ is small. Thus the buoyant escape of azimuthal magnetic flux from the disk to the corona is compensated by the amplification of azimuthal magnetic flux inside the disk with polarity opposite to that in the corona.
Dependence on the Strength of Initial Magnetic Fields {#sec:dep}
-----------------------------------------------------
We also carried out a simulation (model IV) starting from the initially weaker magnetic field ($\beta_{b} = 1000$). In this model, the magnetic energy saturates at almost the same level as in model III ($B_{\varphi} \sim 1 \mu \,\, \mathrm{G}$ at $\varpi \sim 5 \,\, \mathrm{kpc}$). Figure \[fig:f18\] shows the spatial distribution of mean azimuthal magnetic fields for model IV at $t = 824t_{0} (= 3.1 \,\, \mathrm{Gyr})$ and at $t = 1115t_{0} (= 4.2 \,\, \mathrm{Gyr})$. Blue and orange indicate regions where mean magnetic field $\bar{B_{\varphi}}$ threading the $y = 0$ plane is positive or negative, respectively. Again, the azimuthal magnetic field changes direction with radius and with height. The striped distribution of azimuthal magnetic fields and the vertical propagation of the reversals of azimuthal magnetic fields are similar to those in model III (figure \[fig:f15\]).
In Figure \[fig:f19\] we plotted the time variation of mean azimuthal field of model IV. Magnetic fields are averaged in azimuthal direction ($0 \leq \varphi < 2 \pi$), radial direction ($5 \,\, \mathrm{kpc} \leq \varpi < 6 \,\, \mathrm{kpc}$), and in $0.1 \,\, \mathrm{kpc} < z < 1.0 \,\, \mathrm{kpc}$ (equatorial region), $1.0 \,\, \mathrm{kpc} < z < 3.0 \,\, \mathrm{kpc}$ (coronal region) and $0.1 \,\, \mathrm{kpc} < z < 3.0 \,\, \mathrm{kpc}$ (disk + corona), respectively. These results indicate that the direction of mean azimuthal magnetic field reverses several times. The amplitude of oscillation of azimuthal magnetic fields increases with time. The interval of reversal of azimuthal magnetic fields is $\sim 1
\,\, \mathrm{Gyr}$ (2 Gyr for a full cycle). This timescale coincides with that in model III, in which the initial plasma $\beta$ is 100. The direction of azimuthal magnetic field in the coronal region is opposite to that of the equatorial region.
Dependence on the Size of the Azimuthal Region and Mesh Size
------------------------------------------------------------
In order to study the dependence of numerical results on the size of the azimuthal simulation region and mesh sizes in the azimuthal direction, we carried out simulations in which we used 64 (model V), 32 (model VI) and 16 (model VII) grid points in $0 \leq \varphi < \pi/2$. Other parameters are the same as those in model III.
Figure \[fig:f20\][*a*]{} shows the early stage ($t < 600t_{0}$) of the magnetic field amplification in $2 \,\, \mathrm{kpc} < \varpi < 5
\,\, \mathrm{kpc}$ and figure \[fig:f20\][*b*]{} shows the time variation of magnetic energy in the nonlinear saturation stage. We found that in the early stage ($t < 600t_{0}$), the growth of the magnetic energy in model VII is similar to that of model III, which has the same azimuthal grid resolution. In the nonlinear saturation stage (figure \[fig:f20\][*b*]{}), however, the magnetic energy in model VII decreases and becomes smaller than that in model III. These results are consistent with the results of local 3D MHD simulations of accretion disks [@Haw95], in which they showed that the nonlinear saturation level of the magnetic energy increases with the size of the simulation region.
When the azimuthal size of the simulation region is fixed (model V, VI and VII), the saturation level of the magnetic energy ($t > 600t_{0}$) is larger in the high resolution model (model V) than that in low resolution models (model VI and VII). This result is also consistent with the results of local 3D MHD simulations of the growth of MRI in accretion disks [e.g., @Haw95; @Haw96].
The time evolution of the magnetic energy before its saturation (figure \[fig:f20\][*a*]{}) depends on how the gas infalls and carries in magnetic fields. The earlier rise of magnetic energy in the highest resolution model (model V) is due to the faster growth of the MRI in the torus. The growth rate of the MRI is sensitive to the azimuthal grid resolution because the most unstable wavenumber $k_{\parallel} \sim \Omega / v_{A}$, where $k_{\parallel}$ is the wavenumber parallel to the unperturbed magnetic fields [e.g., @Bal92] is hardly resolved in low resolution simulations. When $\beta \sim 100$, the most unstable wavelength is $\lambda_{max} \sim 2\pi v_{A} / \Omega \sim 2\pi H(2/\beta)^{1/2} \sim H$ where $H$ is the disk half thickness. Thus, $\lambda_{max} \sim 0.5 \,\, \mathrm{kpc} \sim (\pi/2) \times 10
\,\, \mathrm{kpc} /30$. We need at least 120 mesh points in $0 < \varphi <
\pi/2$ when one wavelength is resolved by 4 mesh points. Thus, in all models we adopted in this paper, the fastest growing mode is not numerically resolved. These estimates indicate that the growth rate of MRI is affected by the grid size and that the growth rate in models III, VI and VII at $\varpi
= 10 \,\, \mathrm{kpc}$ are smaller than that in model V. Therefore, the infall of the torus gas is delayed in low-resolution models.
Let us discuss why magnetic energy in low resolution models (model III and VII) is larger than that in high resolution models (model V and VI) in the nonlinear stage before the saturation ($t < 600t_{0}$). These magnetic fields are carried into $2 \,\, \mathrm{kpc} < \varpi < 5
\,\, \mathrm{kpc}$ with the gas infalling along the channels of magnetic fields. Since non-axisymmetric parasitic instabilities [@Goo94] and magnetic reconnection [e.g., @San01; @Mac03] which break up the channel flow can be captured better in high resolution simulations, the growth rate of the magnetic energy becomes smaller in high resolution models than that in low resolution models.
Figure \[fig:f21\][*a*]{} compares the vertical distribution of azimuthal magnetic fields at $\varpi = 10 \,\, \mathrm{kpc}$ at $t = 800t_{0}$ in model III, V, VI and VII. The region where $B_{\varphi} < 0$ is not plotted. The azimuthal magnetic flux buoyantly rises into the corona and forms extended magnetized region. Numerical results indicate that the magnetic flux rises faster into the corona in the high resolution model (model V) than low resolution models (model VI and VII). High resolution simulations could resolve the most unstable wavelength of the Parker instability, $\lambda \sim 10 H \sim 5 \,\, \mathrm{kpc}$ in our simulation model [e.g., @Par66; @Mat88]. Thus, one wavelength of the Parker unstable loop is well resolved in high resolution model with 64 grid points in $0 \leq \varphi < \pi/2$ at $\varpi = 10 \,\, \mathrm{kpc}$ (model V) but hardly resolved in the low resolution model (model VII). This may be the reason why toroidal magnetic flux rises faster in model V than in other models (figure \[fig:f21\][*b*]{}).
When the azimuthal resolution is fixed (model III and VII), the magnetic flux rises faster in the model with larger azimuthal simulation region (model III). This dependence on the azimuthal simulation region comes from the horizontal expansion of the rising magnetic loops [e.g., @Shi89]. When the horizontal simulation region is restricted, since magnetic tension prevents the escape of the magnetic flux, the speed of the buoyantly rising magnetic loops is reduced.
Difference Between the Rotation Curve of Dark Matter and Gas
------------------------------------------------------------
Figure \[fig:f22\] compares the rotation curve of stars and dark matter computed from the gravitational potential (dashed curve) and the rotation curves for gas (solid curve) obtained from simulations for model III. The rotation curves of our Galaxy interior to the Sun has been obtained by HI gas [e.g., @Gun79], and CO [@Bur78] observations. The rotation speed in $5-10 \,\, \mathrm{kpc}$ from the galactic center is $220-260 \,\, \mathrm{km/s}$ and similar to the rotation speed shown in figure \[fig:f22\]. The rotation speed of our galaxy distant from the galactic center ($\varpi > 10 \,\, \mathrm{kpc}$) is obtained by radio observation of HI gas [e.g., @Mer92]. The rotation speed is nearly flat and $v_{\varphi} \sim 240 \,\, \mathrm{km/s}$. The flat rotation curve is similar to the rotation curve in the outer part of external spiral galaxies [e.g., @Rub85]. Our numerical result reproduces these flat rotation curves when we use the gravitational potential which includes the dark matter.
Since the rotation curve of our galaxy distant from the galactic center is obtained by radio observations of HI gas, the observationally measured rotation curve is that for the gas. The rotation curve of the neutral hydrogen does not necessarily coincide with that of stars and the dark matter. Since they are almost frozen to the magnetic fields, if the magnetic arm rotates with angular speed different from the dark matter, the neutral hydrogen is not a good tracer of mass distribution in our Galaxy. @Bat92 suggested that when magnetic pinch force is large enough, they can explain the observed rotation curve of the Galaxy without dark matter. On the other hand, @San04 reported that the contribution of the magnetic field to the rotation curve is up to 20 km/s and it is not sufficient to banish the effect of dark matter. Our numerical results indicate that even in the presence of dynamically non-negligible magnetic fields, gas component shows a flat rotation curve similar to that observed by neutral hydrogen. Numerical results for model I indicates that if the dark matter does not exist, gas rotation curve decreases with radius. Thus, the flat rotation curve obtained by observations of neutral hydrogen really indicates the existence of dark matter.
Summary and Discussion
======================
We carried out global three-dimensional magnetohydrodynamic (MHD) simulations of galactic gaseous disks to investigate how the galactic magnetic fields are amplified and maintained. We showed the dependence of numerical results on physical conditions in the innermost region, gravitational potential, and strength of initial magnetic fields.
In all models, the magnetorotational instability (MRI) grows in the disk and the magnetic energy is amplified and maintained for more than $5 \,\, \mathrm{Gyrs}$. The saturation level of magnetic fields, however, is smaller than that expected from observations ($B \sim$ several $\mu G$) at $\varpi = 10 \,\, \mathrm{kpc}$. The multiphase nature of interstellar space, supernova explosions, cosmic rays and/or non-axisymmetric spiral potential may be important to further amplify magnetic fields.
Numerical results indicate that mean azimuthal field inside the galactic disk is amplified and that its direction changes quasi-periodically. The amplitude of the oscillation of the azimuthal field increases with time. The interval of reversal of equatorial azimuthal field is $\sim 1 \,\, \mathrm{Gyr}$. This timescale is comparable to the timescale of the buoyant rise of azimuthal magnetic flux from the disk to the corona. The amplitude of the fluctuating field is comparable to the mean magnetic field. The strength of the mean field only weakly depends on the initial strength of magnetic fields. In the coronal region, the mean azimuthal field has direction opposite to the mean field of the disk.
Mean magnetic fields obtained from numerical simulations also show reversals in the radial direction near the equatorial plane around $\varpi = 5 \,\, \mathrm{kpc}$ at $t = 3.8 \,\, \mathrm{Gyr}$. The radius of the field reversal changes with time because the spiral magnetic channels which produce the reversal of azimuthal fields in the radial direction move inward as the disk gas accretes toward the galactic center.
In the Galactic plane, magnetic field reversals are observed by Rotation Measure (RMs) of pulsars [e.g., @Han02]. The reversals of mean magnetic fields are observed in our Galaxy between the local Orion arm and the inner Sagittarius arm [@Sim79]. More recent analysis imply axisymmetric field with two reversals (@Ran89 [@Ran94]; see also @Bec96). Simulation results of model III (Figure \[fig:f13\]) show reversals of mean magnetic fields near the equatorial plane around $\varpi = 5 \,\, \mathrm{kpc}$. Note that RMs measure the mean magnetic field in some specific direction from the Earth. When we draw lines from a point in Figure \[fig:f13\], we can see field reversals in several directions. Since the magnetic fields inside the galactic disk is turbulent, more and more field reversals will be observed as angular resolution of RMs increase.
In conventional theories of $\alpha \omega$-dynamos, various kinds of spatial and temporal oscillations appear depending on the dynamo number $D = C_{\alpha} C_{\Omega}$ where $C_\Omega = H^{2} \Omega/ \eta$ ($H$ is the thickness of the disk, $\Omega$ is the rotation angular speed and $\eta$ is the magnetic diffusivity) and $C_\alpha = H \alpha/\eta$ [e.g., @Ste88]. In our global MHD simulations, we do not need to introduce the dynamo $\alpha$ parameter. The amplification of magnetic fields due to MRI and the reproduction of toroidal fields through gas motions are self-consistently incorporated in our simulations. We introduced anomalous resistivity $\eta$ to handle magnetic reconnection which takes place in local region where current sheets are formed, but the turbulent diffusivity is not explicitly assumed. We showed by direct 3D simulations that mean toroidal magnetic fields show reversals both in space and time.
@Tou92 constructed phenomenological models of disk dynamos by taking into account the growth of magnetic fields due to MRI, escape of magnetic flux by the Parker instability, and the dissipation of magnetic fields by magnetic reconnection. They showed that the magnetic fields in the disk oscillate quasi-periodically with period comparable to the time scale of the buoyant escape of the magnetic flux. This timescale is consistent with the timescale of field reversals obtained from our MHD simulations.
Let us discuss the similarities and dissimilarities of the magnetic activity of galactic gas disks produced by our numerical simulations and those in the Sun. Both objects are differentially rotating plasma with high conductivity in which magnetic fields are almost frozen to the rotating plasma. Thus the magnetic fields can be deformed and amplified by the plasma motion.
The Sun is a slow rotator in which the centrifugal force is much smaller than the gravity, thus its atmosphere is confined by the gravity. The convective motions and/or the differential rotation in the convection region may be the source of magnetic field amplification in the Sun. However, since the magnetic flux tube in the convection region buoyantly rises with a timescale of the order of a month [e. g., @Par75], much shorter than the solar magnetic cycle ($\sim 22 \,\, \mathrm{year}$), solar dynamo activity is considered to be operating in the interface between the convection zone and the radiative zone, where magnetic buoyancy is small and differential rotation exists [e. g., @Spi80; @Sch83]. On the other hand, galactic gas disks are supported by the rotation and subject to differential rotation, which induces the growth of MRI in the whole disk. In galactic gas disks, since the gas disk is convectively stable, the buoyant escape of the magnetic flux takes place with timescale $t_{buoy} \sim 10 H /v_{A} \sim 10 (1/\Omega)(\beta/2)^{1/2}$. This timescale is longer than the growth time of MRI $t_{MRI} \sim 1/\Omega$. Thus the magnetic fields can be amplified in the disk before escaping to the corona.
Numerical results indicate that the total azimuthal magnetic flux is nearly conserved. When turbulent diffusivity exists, the azimuthal magnetic flux is not necessarily conserved. In our simulations, however, since magnetic diffusion only exists in the local current sheet, the magnetic flux is almost conserved. We found that the toroidal magnetic flux in the disk buoyantly rises into the corona. This result indicates that the buoyant escape of the magnetic flux from the disk enables the amplification of mean magnetic fields in the galactic gas disk. When the magnetic diffusion is not negligible, the total azimuthal magnetic flux is not conserved but the magnetic helicity is conserved. @Bla03 pointed out the importance of helicity conservation and the role of coronal mass ejections, which help sustaining the solar dynamo cycle. In our simulations, buoyantly rising magnetic loops carry helicity as well as the magnetic flux threading the disk. This escape of magnetic flux and helicity enable the disk to amplify the azimuthal magnetic flux in the equatorial region.
We showed that when we remove the central absorber at $r = 0.8 \,\, \mathrm{kpc}$, formation of dense gas bulge forces the rotation curve to deviate from that expected from the gravitational potential (model II). This indicates that in the central region of the galaxy, the gas absorption or the conversion of the accumulated gas to stars are essential to explain the observed rotation curve. The mass accretion rate to the central region is $\sim 10^{-3} M_{\odot}/ \,\, \mathrm{yr}$ when we adopt the absorbing boundary condition. This accretion rate depends on the initial density of the torus, which can be formed by the infall of intergalactic matter or by the supernova explosions after bursts of star formation in the certain radius of the disk. When the galaxy was more gas rich in the early stage of its evolution, the accretion rate should be much higher.
We also showed that the gas rotation curve approximately coincides with the rotation curve for stars and dark matter even when magnetic fields are dynamically important. This justifies us to use the gas rotation curve to estimate the distribution of the dark matter.
We thank Drs. T. Kuwabara, S. Miyaji and K. Shibata for discussion. Discussions at IAU XXXth General Assembly, 2003 and the workshop on Magnetic Fields in the Universe, 2004 helped us to complete this work. This work is supported by the Grants-in Aid of Ministry of Education, Science, Sports and Culture (16340052, P.I., R. Matsumoto) the priority research project of the Graduate School of Science and Technology, Chiba University, ACT-JST of the Japan Science and Technology corporation, and UK-Japan collaboration on magnetic activities of the Sun, Stars and accretion disks (P.I., K. Shibata and N. Weiss). Numerical computations were carried out on VPP5000 at the Astronomical Data Analysis Center of the National Astronomical Observatory, Japan (NAOJ).
Balbus, S. A., & Hawley, J. F. 1991, , 376, 214 Balbus, S. A., & Hawley, J. F. 1992, , 400, 610 Battaner, E., Garrido, J. L., Membrado, M., & Florido, E. 1992, , 360, 652 Beck, R., Brandenburg, A., Moss, D., Shukurov, A., & Sokoloff, D. 1996, , 34, 155 Blackman, E. G., & Brandenburg, A. 2003, , 584, 99 Brandenburg, A., Krause, F. Meinel, R., Moss, D., & Tuominen, I. 1989, , 213, 411 Brandenburg, A., Donner, K. J.,Moss, D., Shukurov, A., Sokolov, D. D., & Tuominen, I. 1992, , 259, 453 Brandenburg, A., Nordlund, A., Stein, R. F., & Torkelsson, U. 1995, , 446, 741 Burton, W. B., & Gordon, M. A. 1978, , 63, 7 Dziourkevitch, N., & Elstner, D. 2003, , 284, 757 Dziourkevitch, N., Elstner, D., & Rüdiger, G. 2004, , 423, L29 Foglizzo, T., & Tagger, M. 1994, , 287, 297 Foglizzo, T., & Tagger, M. 1995, , 301, 293 Goodman, J.,& Xu, G. 1994, , 432, 213 Gunn, J. E., Knapp, G. R., & Tremaine, S. D. 1979, , 84, 1181 Han, J. L., Manchester, R. N., Lyne, A. G., & Qiao, G. J. 2002, , 570, L17 Hawley, J. F., & Balbus, S. A. 1992, , 400, 595 Hawley, J. F., Gammie, C. F., & Balbus, S. A. 1995, , 440, 742 Hawley, J. F., Gammie, C. F., & Balbus, S. A. 1996, , 464, 690 Hawley, J. F. 2000, , 528, 462 Hayashi, M. R., Shibata, K., & Matsumoto, R. 1996, , 468L, 37 Kato, Y., Mineshige, S., & Shibata, K. 2004, , 605, 307 Kim, W. T., Ostriker, E. C., & Stone, J. M. 2003, , 599, 1157 Kitchatinov, L. L., & Rüdiger, G. 2004, , 424, 565 Machida, M., Hayashi, M. R., & Matsumoto, R. 2000, , 532, L67 Machida, M., & Matsumoto, R. 2003, , 585, 429 Matsumoto, R., Horiuchi, T., Shibata, K., & Hanawa, T. 1988, , 40, 171 Matsumoto, R., & Tajima, T. 1995, , 445, 767 Matsumoto, R., Uchida, Y., Hirose, S., Shibata, K., Hayashi, M. R., Ferrari, A., Bodo, G., & Norman, C. 1996, , 461, 115 Matsumoto, R. 1999, in Numerical Astrophysics, ed. Miyama, S. M., Tomisaka, K., & Hanawa, T. (Amsterdam: Kluwer Academic Publishers) 195 Merrifield, M. R. 1992, , 103, 1552 Miller, K. A., & Stone, J. M. 2000, , 534, 398 Miyamoto, M., & Nagai, R. 1975, , 27, 533 Miyamoto, M., Satoh, C., & Ohashi, M. 1980, , 90, 215 Okada, R., Fukue, J., & Matsumoto, R. 1989, , 41, 133 Parker, E. N. 1966, , 145, 811 Parker, E. N. 1971, , 163, 255 Parker, E. N. 1975, , 198, 205 Piontek, R. A., & Ostriker, E. C. 2004, , 601, 905 Rand, R. J., & Kulkarni, S. R. 1989, , 21, 1188 Rand, R. J., & Lyne, A. G. 1994, , 268, 497 Richtmyer, R., O., & Morton, K., W. 1967, Differential Methods for Initial Value Probrem (2d ed., New York: Wiley) Rubin, E., & Burstein, S., Z. 1967, J. Comput. Phys., 2, 178 Rubin, V. C., Burstein, D., Ford, W. K., & Thonnard, N. 1985, , 289, 81 Sánchez-Salcedo, F. J., & Reyes-Ruiz, M. 2004, , 607, 247 Sano, T., & Inutsuka, S. 2001, , 561, 179 Schmitt, D., & Schüssler M., 1989, , 223, 343 Schmitt, J. H. M. M., & Rosner, R. 1983, , 265, 901 Sellwood, J. A., & Balbus, S. A., 1999, , 511, 660 Shibata, K., & Uchida, Y. 1985, , 37, 31 Shibata, K., Tajima, T., Steinolfson, R. S. & Matsumoto, R. 1989, , 345, 584 Simard-Normandin, M., & Kronberg, P. P. 1979, , 279, 115 Sofue, Y., Fujimoto, M., & Wielebinski, R. 1986, , 24, 459 Spiegel, E. A., & Weiss, N. O. 1980, , 287, 616 Stepinski, T. F., & Levy, E. H. 1988, , 331, 416 Tout, C. A., & Pringle, J. E. 1992, , 259, 604 Yokoyama, T., & Shibata, K. 1994, , 436, L197
[lccccccccccc]{} Model I &$0.0$&$7.258$&$0.0$&$0.495$&$0.520$&$0.0$&$2.05$&$25.47$&$0.0$&100&Absorbing\
Model II &$0.0$&$7.258$&$0.0$&$0.495$&$0.520$&$0.0$&$2.05$&$25.47$&$0.0$&100&Non-Absorbing\
Model III &$0.0$&$6.2$&$0.0$&$0.47$&$0.15$&$31.2$&$1.95$&$17.4$&$73.5$&100&Absorbing\
Model IV &$0.0$&$6.2$&$0.0$&$0.47$&$0.15$&$31.2$&$1.95$&$17.4$&$73.5$&1000&Absorbing\
Model V-VII &$0.0$&$6.2$&$0.0$&$0.47$&$0.15$&$31.2$&$1.95$&$17.4$&$73.5$&100&Absorbing\
\[tbl:tbl1\]
[llcc]{}
Model I-IV &$\pi/32$ &64 &$0 \leq \varphi < 2\pi$\
Model V &$\pi/128$ &64 &$0 \leq \varphi < \pi/2$\
Model VI &$\pi/64$ &32 &$0 \leq \varphi < \pi/2$\
Model VII &$\pi/32$ &16 &$0 \leq \varphi < \pi/2$\
\[tbl:tbl2\]
|
---
abstract: 'The structure of molecular clouds can be characterized with the probability distribution function (PDF) of the mass surface density. In particular, the properties of the distribution can reveal the nature of the turbulence and star formation present inside the molecular cloud. In this paper, we explore how these structural characteristics evolve with time and also how they relate to various cloud properties as measured from a sample of synthetic column density maps of molecular clouds. We find that, as a cloud evolves, the peak of its column density PDF will shift to surface densities below the observational threshold for detection, resulting in an underlying lognormal distribution which has been effectively lost at late times. Our results explain why certain observations of actively star-forming, dynamically older clouds, such as the Orion molecular cloud, do not appear to have any evidence of a lognormal distribution in their column density PDFs. We also study the evolution of the slope and deviation point of the power-law tails for our sample of simulated clouds and show that both properties trend towards constant values, thus linking the column density structure of the molecular cloud to the surface density threshold for star formation.'
author:
- |
Rachel L. Ward, James Wadsley, and Alison Sills\
Department of Physics and Astronomy, McMaster University, Hamilton, ON, L8S 4M1, Canada
title: Evolving Molecular Cloud Structure and the Column Density Probability Distribution Function
---
\[firstpage\]
ISM: clouds – ISM: evolution – ISM: kinematics and dynamics – ISM: structure – stars: formation
Introduction {#sec:Intro}
============
Giant molecular clouds (GMCs) are large regions of gas and dust, which undergo local gravitational collapse to form dense star-forming cores. The morphology and structure of GMCs are significantly affected by the presence of turbulence which is highly supersonic on large-scales and leads to the formation of density enhancements such as filaments, clumps, and cores within which star formation occurs.
The probability distribution function (PDF) of the mass density is one representation of the role of supersonic turbulence on the structure of molecular clouds. Many numerical studies [e.g. @vazquez1994; @NP99; @PN02; @ostriker2001; @KM05; @HC08] have shown that a lognormal shape is expected for the density PDF of an isothermal, supersonic turbulent gas.
Recent observations by @kain2009 showed that column density PDFs for molecular clouds also exhibit a lognormal distribution. However, they observed this distribution only in quiescent clouds which are not actively undergoing star formation. The column density PDFs for star-forming molecular clouds, such as the Taurus Molecular Cloud, have an underlying lognormal shape with the addition of a power-law tail at high column densities. This distribution is reminiscent of the stellar initial mass function (IMF) whose form is often approximated either by a lognormal [@chabrier] or by a power-law [@kroupa; @salpeter] at the high-mass end of the spectrum. These results suggest that characterising the shape of the density and column density PDF of molecular clouds is crucial for understanding the origin of the stellar IMF.
Several numerical studies [@balle2011; @knw11; @tassis] have subsequently confirmed the presence of a power-law tail in the density and column density PDFs from three-dimensional simulations of molecular clouds collapsing to form stars. These authors argue that the tail develops over time and its strength grows as more stars are formed and as gravity becomes dominant over turbulence.
@kain2011 explored the nature of the transition from a lognormal shape at low column densities to a power-law shape at higher column densities for actively star-forming clouds. The authors showed that the changing shape of the PDF can be interpreted as the transition from a diffuse cloud dominated by turbulence to a clumpy gravitationally-dominated structure. @kain2011 showed that the ‘structural transition’ of the PDF offers a physical explanation for the star formation threshold and for the correlation between the star formation rate and the mass of dense gas above the threshold in molecular clouds [@lada2010]. Throughout this paper, we will refer to the threshold at which the structural transition occurs as $\Sigma_{\text{tail}}$ where all gas above this threshold is considered dense enough to form stars.
There are several properties of the distribution which characterise the shape of the column density PDF for molecular clouds. In this paper, we explore how these structural characteristics evolve with time and also how they relate to various cloud properties as measured from a sample of synthetic column density maps of bound and unbound molecular clouds. Column density PDFs derived from molecular cloud observations can be ordered into an assumed sequence to demonstrate the evolutionary process from lognormal shape to the development of a power-law tail. Simulators have shown this to be the case; however, the evolution of column density PDFs of simulated clouds in an observational context has not yet been considered. Observational limits and thresholds naturally interfere with estimates of cloud properties. In this paper, we show that thresholds can significantly affect the way we interpret molecular cloud structure and evolution, particularly for older, actively star-forming clouds like the Orion molecular cloud.
In Section \[sec:methods\], we review the details of our simulations, the methods used to produce our synthetic column density maps, and the criteria for cloud selection first outlined in @paper1. We describe our method of tracking clouds through time and present our findings on the evolution of their structural properties in Section \[sec:results\]. Lastly, in Section \[sec:discussion\], we summarize our results and discuss the implications of our conclusions.
Methods {#sec:methods}
=======
We use simulations of molecular clouds to study the evolution of their structure and its effect on star formation. Simulations allow us to explore the three-dimensional properties of the cloud in order to better understand the two-dimensional properties which are observed. We created column density projections of our simulations to compare more directly to observations. The dynamic properties of our synthetic clouds were the subject of a previous study by @paper1 and full details of the simulations, column density maps, and the cloud selection process are provided in that paper. What follows is a brief overview highlighting the key details of our method.
---- -------- ---------------------- ---------------------- ---------- --------------------------- -- --
Id Radius $\sigma_{3\text{D}}$ n$_{\text{initial}}$ t$_{ff}$ $\alpha_{\text{initial}}$
(pc) (km s$^{-1}$) (cm$^{-3}$) (Myr)
9 30 2.93 7.69 22.4 2
10 21.2 3.49 21.8 13.3 2
11 15 4.15 61.5 7.9 2
12 10.6 4.93 174 4.7 2
13 30 2.07 7.69 22.4 1
14 21.2 2.47 21.8 13.3 1
15 15 2.93 61.5 7.9 1
16 10.6 3.49 174 4.7 1
---- -------- ---------------------- ---------------------- ---------- --------------------------- -- --
: Initial Conditions for the Simulations \[table:initcondit\]
Our synthetic clouds were selected from initially bound 50 000 M$_{\odot}$ regions of the ISM, simulated using the smoothed particle hydrodynamics code <span style="font-variant:small-caps;">Gasoline</span> [@gasoline]. Throughout this paper we only consider the cases where the initial global virial parameter of the volumes is less than or equal to 2 (or gravitationally bound) to avoid rapid dispersal due to high turbulent motions. The initial conditions for these cases cover a range of initial densities and Mach numbers listed in Table 1 of @paper1 for volumes labelled by IDs 9 - 16. This table with the relevant cases has been reproduced for this work (Table 1). Our particle mass is 8.85 $\times$ 10$^{-3}$ M$_{\odot}$ with a mass resolution limit for fragmentation of 0.6 M$_{\odot}$, resulting in simulations which are highly resolved and include only gravity and decaying turbulence with large-scale modes in non-periodic boxes. We use the @BB05 equation of state, whose opacity limit for fragmentation is greater than our maximum gas density, resulting in simulations which are entirely optically thin and isothermal at 10 K. As a result, our column density projections are equivalent to synthetic observations derived from radiative transfer calculations in the limit of optically thin gas. Detailed descriptions of the methods used to produce our synthetic column density maps can be found in @paper1 [@ward2012]. Approximately 60 clouds at various stages of their evolution were selected from the column density maps of each turbulent region using surface density contours to identify the peak emission. The selection process ensures that the clouds in our sample have column densities which exceed the lower limit for detection in dust extinction measurements [A$_{\text{v}}$ $\sim$ 0.5; @beaumont2012; @kain2009]. This limit also approximates the threshold at which atomic hydrogen begins to self-shield against UV radiation ($\sim$ 10$^{20-21}$ cm$^{-2}$) leading to the formation of molecular hydrogen. These synthetic clouds have sizes and surface densities which are analogous to observed molecular clouds, allowing for a more direct comparison to observations. The spatial resolution of our column density maps is 6000 AU px$^{-1}$, which corresponds to an angular resolution of $\sim$ 45” for a cloud at a distance of 140 pc, such as the Taurus molecular cloud [@nutter].
Although the simulation volumes are bound overall with global virial parameters $\leq$ 2, the clouds within the volume have local virial parameters ranging from 0.5 to 6.5 due to localised regions of collapse and turbulence. Of the 58 clouds in our sample, over 30% are gravitationally bound with local virial parameters $\leq$ 2. This allows us to explore the structural evolution of both bound and unbound clouds, since the local virial parameters of the clouds remain steady and evolve minimally for the duration of the simulations.
We used sink particles to represent the high density regions of the simulations (inserted at densities greater than 10$^6$ cm$^{-3}$) based on the formation criteria of @federrath10. Each sink corresponds to a star-forming core or a cluster-forming core with a radius of 500 AU. We find that all of our clouds eventually form stars regardless of whether or not they are bound overall. Since our simulations do not model radiative feedback, we do not include clouds in our study which have converted more than 35% of their mass to stars.
Results {#sec:results}
=======
In order to compare directly to observations, we only included the pixels from our maps with surface densities greater than $\sim$ 10 M$_{\odot}$ pc$^{-2}$. This minimum surface density corresponds to the threshold for detection in extinction maps – A$_{\text{v}}$ $\sim$ 0.5 [@kain2009] where the conversion factor is $\Sigma$/A$_{\text{v}}$ = 20 M$_{\odot}$ pc$^{-2}$ mag$^{-1}$ [@bohlin; @lombardi2010; @kain2011; @lombardi2014]. We also removed the sink particles from our column density maps, which is common practice for the case of embedded stars in studies of molecular clouds (in CO and extinction-based data) due to their tendency to bias measurements [@kain2009; @goodman09; @KFH13].
To demonstrate that we can sensibly track the evolution of our clouds, a bound 10$^4$ M$_{\odot}$ cloud selected at t = 4.0 Myr is shown in Figures \[fg:evolclouds\_maps\] and \[fg:evolclouds\_pdfs\] to evolve in two possible ways. For the true evolution (shown by the solid arrow), gas particles present in a column density map at t = 4.0 Myr are marked using their particle ids from the simulation, extracted from the simulation volume, and evolved forward in time. This allows us to study the dynamics of the particles, as they are isolated from the surrounding medium. For the perceived evolution (shown by the dotted arrow), we take two independently selected clouds: one in the 4.0 Myr snapshot and one in the 5.0 Myr snapshot. If two clouds selected in this way have x-y positions in the map which are close, we claim that the older cloud is the evolved version of the younger cloud. This method based on appearance is analogous to what observers do to better understand cloud evolution. We note from Figure \[fg:evolclouds\_maps\] that the two outcomes are quite alike in appearance; however, they are also similar in their bulk properties, such as their maximum surface densities and in the amount of mass in dense gas. We also find that both methods of tracking the evolution of the cloud result in column density PDFs that are in agreement within uncertainties as seen in Figure \[fg:evolclouds\_pdfs\]. Therefore, our method of tracking cloud evolution is robust as the effects of the surrounding environment are minor and do not significantly influence the structural evolution of the cloud during the timescales of interest (on the order of a free-fall time, t$_{ff}$). We will explore the effect of environment on longer timescales in future work for clouds formed in a galactic disc.
We follow the evolution of each of our $\sim$ 60 clouds forward in time, producing a column density map after each 1 Myr time interval. For each column density map, we produced a corresponding column density PDF plotted over a range chosen to match the observations of @kain2009. To study the evolution of the molecular cloud structure in detail, we focus on the defining characteristics of the column density PDF. The shape of the PDF is characterised by four key structural properties: its width ($\sigma$), peak location ($\Sigma_{peak}$), the slope of the power-law (m), and the deviation point ($\Sigma_{tail}$). These properties are identified in the diagram shown in Figure \[fg:Npdf\].
![\[fg:Npdf\] Schematic diagram of the mass surface density probability distribution function (PDF). The structural properties of interest are the peak, $\Sigma_{\text{peak}}$, and width of the distribution, $\sigma$, the slope of the power-law tail, m, and the surface density at which the power-law tail first forms, $\Sigma_{\text{tail}}$. ](fig3.pdf)
The peak and width of the distribution are determined by fitting the lognormal function $$p(\zeta) = \frac{1}{\sqrt{2\pi\sigma_{\zeta}^2}} \exp{\left[-\frac{(\zeta-\mu)^2}{2\sigma_{\zeta}^2}\right]}$$ to the low extinction range of the PDF between $\zeta = \ln(A_{V}/\langle A_{V}\rangle)$ = \[-1,1\] [@kain2009; @schneider2014] where $\mu$ and $\sigma_{\zeta}$ are the mean logarithmic column density and dispersion respectively. The deviation point, $\Sigma_{\text{tail}}$, is defined as the value of $\zeta$ where the data deviates from the lognormal fit by a factor of 1.25 (dashed lines in Figure \[fg:evolclouds\_pdfs\]). This definition of the deviation point coincides with the values which would be identified by eye. The power-law slopes are fit between the deviation point and the surface density at which $N/N_{peak}$ = 0.001, below which the signal would be too low to be detected observationally.
At early times (t $\sim$ 0.1 t$_{ff}$), we find that the clouds appear diffuse and turbulent, exhibiting a lognormal distribution. At later times after the sinks have begun to form (t $\sim$ 0.6 t$_{ff}$ ), the distributions have developed a power-law tail along with an underlying lognormal shape, which is significantly wider with a lower peak column density (Figure \[fg:evolpdf\]). To illustrate the characteristics of the structural transition and the times at which these transitions occur, we plotted the evolution of the four column density PDF properties for our clouds in Figure \[fg:evols\].
Clouds which could not be fit by a lognormal due to evolution of the peak to extinction values below the minimum threshold were excluded from Figures \[fg:evols\](a) and \[fg:evols\](b) and are not considered in subsequent sections. Since it was still possible to fit power-law slopes to these clouds, they are included in Figures \[fg:evols\](c) and \[fg:evols\](d) and are considered in later discussions.
In Figure \[fg:evols\](a), we see that the peak of the distribution decreases as a function of time. We see this trend for both bound and unbound clouds. However, the peak of the distribution cannot evolve to surface densities any lower than the imposed threshold. Therefore, there may be a significant amount of mass in clouds which exists below the threshold for detection. This indicates that observational limitations can greatly impact the number of molecular clouds identified at late stages of their evolution.
@lombardi2014 find no evidence of lognormality in their PDFs of the Orion star-forming region using data from Herschel and Planck. The authors suggest that the lognormal could be confined to low column densities ($<$ A$_V$ $\sim$ 1 mag) below what they are able to detect and argue that the power-law regime dominates and characterises the majority of cloud structure. We show that this is indeed the case for ‘old’ clouds – clouds which have been actively star-forming for several Myrs, like Orion. Although stars are consuming gas, the amount of dense gas (n $>$ 10$^4$ cm$^{-3}$) available for star formation tends to remain relatively constant over time due to accretion and dispersal. These clouds have begun dispersing the low column density gas and are rapidly converting high column density gas into stars, the net result of which is a lowering of the overall peak column density of the lognormal distribution. The younger (or less evolved) a cloud is, the more likely a lognormal distribution will be observed in its PDF. Alternatively, older star-forming regions will be dominated solely by a power-law as their distributions have evolved to peak below the current limits of detection by observations. Figure \[fg:thresh\] shows the PDF for a cloud in our sample which would be identified as having a power-law across the entire range of column densities within the observed limits ($A_{\text{V}}$ $>$ 1 mag); however, we can see that the lognormal is still present, but now shifted below the lower limit for detection (dashed vertical line).
![\[fg:thresh\] Column density PDF for a sample cloud at a late stage of its evolution ($t$ $>$ $t_{ff}$). The solid line shows the lognormal fit to the distribution and the dash-dotted line shows the power-law fit to the tail of the distribution. The dashed vertical line at $A_{\text{V}}$ = 1 mag represents the lower limit for detection adopted by @lombardi2014. ](fig6_color.pdf)
We explore the effect of a threshold further by measuring the mean mass surface densities, $\langle\Sigma\rangle$, for our sample of simulated clouds assuming five possible extinction thresholds (A$_{\text{v,th}}$ (mag) = 0.5, 1.0, 1.5, 4, 8) and comparing them to the mean mass surface densities of observed molecular clouds. Only pixels with column densities above a given threshold contribute to estimates of $\langle\Sigma\rangle$. Using near-IR extinction observations of molecular clouds, @lombardi2010 measured $\langle\Sigma\rangle$ for five different K-band extinction thresholds: A$_{\text{K,th}}$ = \[0.1, 0.2, 0.5, 1.0, 1.5\] [$A_{\text{K}}/A_{\text{V}}$ = 0.11; @RL85] and found that different clouds will have a constant column density for a given threshold and the value of the column density is dependent on the threshold. Figure \[fg:sigAth\] shows the agreement between our results (black stars) and the observational estimates of $\langle\Sigma\rangle$ by @lombardi2010 (black dots) along with their corresponding fit, $\Sigma = 265 \text{ M}_{\odot} \text{ pc}^{-2} (A_{th}/\text{mag})^{0.8}$ (black dashed line). The most recent estimates for mean surface densities of nearby clouds from Schneider et al. (2014) are also shown (red circles).
![\[fg:sigAth\] Mean mass surface density as a function of the threshold, given by the corresponding extinction threshold, A$_{\text{v,th}}$. The black stars represent the mean mass surface densities of our sample of simulated molecular clouds for five thresholds corresponding to A$_{\text{v,th}}$ = \[0.5, 1, 1.5, 4, 8\]. The black dots represent the mean mass surface densities for a sample of *observed* clouds obtained from Table 1 of @lombardi2010. The black dashed line is a power-law fit to the observed data from @lombardi2010. The grey lines are analytic calculations of the mean surface density as a function of the threshold assuming a lognormal column density probability distribution function (PDF) with a peak at A$_{\text{v,peak}}$ = 1 mag (solid line) and at A$_{\text{v,peak}}$ = 0.3 mag (dash-dotted line) and standard deviations of $\sigma$ = \[0.6, 0.9, 1.2\] [@lombardi2010; @balle2012]. ](fig7_color.pdf)
@lombardi2010 argue that their results are a consequence of cloud structure described by a lognormal column density PDF. However, @beaumont2012 and @balle2012 both suggest that alternate forms of the column density PDF could also produce these results provided that the chosen thresholds are near or above the peak of the distribution. In particular, @balle2012 explored the consequences of several different functional forms of the column density PDF. While they argue that the results of @lombardi2010 are an effect of thresholding the surface density, their analytic models do show that a lognormal column density PDF with or without a power-law tail can fit the observations. In Figure \[fg:sigAth\], we plotted six possible cases of their model for a lognormal-shaped column density PDF, shown as grey lines, where
$$\begin{split}
\langle\Sigma(\text{A}_{\text{v,th}})\rangle = (17.5 \text{M}_{\odot} \text{pc}^{-2})\text{A}_{\text{peak}}\exp(\frac{\sigma^2}{2}) \\
\times \frac{1-\text{erf}\{[\text{ln}(\text{A}_{\text{v,th}}/\text{A}_{\text{peak}})-\sigma^2]/\sqrt{2}\sigma\}}{1-\text{erf}[\text{ln}(\text{A}_{\text{v,th}}/\text{A}_{\text{peak}})/\sqrt{2}\sigma]}
\label{eq:1}
\end{split}$$
[@balle2012] with A$_{\text{peak}}$ = 1 mag (shown as solid lines) or A$_{\text{peak}}$ = 0.3 mag (shown as dash-dotted lines) and $\sigma$ = 0.6, 0.9, and 1.2 for the widths of each A$_{\text{peak}}$ distribution. We find that a lognormal distribution is consistent with both the observational data of @lombardi2010 and our simulated clouds. The analytic solutions plateau for extinction thresholds less than the peak for the A$_{\text{peak}}$ = 1 mag case. However, once the peak of the distribution evolves to lower surface densities (and correspondingly lower extinctions) such as for the A$_{\text{peak}}$ = 0.3 mag case, the mean surface density is always a function of the threshold across the entire range of threshold values.
We have shown that due to the peaked distribution of the column density PDF, great care must be taken when determining the mean mass surface density as the estimate depends sensitively on the choice of threshold. Clouds disperse as they evolve, resulting in localised high density regions within a cloud dominated by low-density molecular gas. Provided that the threshold is below or near the peak of the distribution, the measure of mean cloud mass surface density can be trusted; however at late times, many of the cloud properties, including the mass and area, will be largely underestimated as they now have evolved to become a function of the threshold.
Cloud dispersal can also lead to a wider lognormal surface density PDF. This widening, seen in Figure \[fg:evols\](b), is expected for clouds with increasing Mach numbers (see section \[sec:discussion\]) and is in agreement with @balle2011 and @tassis who have shown that the distribution will widen prior to the development of a power-law tail.
The power-law tail is an interesting feature of the column density PDF. Many authors [e.g. @knw11; @tassis; @balle2011] have shown that the development of a power-law tail occurs as the cloud evolves from being turbulence-dominated to being gravity-dominated. However, there is still a question as to whether or not the slope of the tail is universal. @froebrich07 attribute a variation in the slope to cloud distance based on 2MASS observations of molecular clouds. @balle2011 also argue against a universal slope as they find variation due to projection in their simulations of clouds formed at the interface of colliding streams. However, neither of these interpretations account for cloud age and evolution.
In Figure \[fg:evols\](c), we show the slopes of the power-law tails obtained from the column density PDFs of our simulated clouds as a function of time. At early times, the slopes are very steep, as they coincide with the lognormal fits at high column densities. As the clouds evolve, the slopes become more shallow, eventually converging to a power-law slope of approximately -2. This result is in agreement with @knw11 who also do not see large variation in their power-law slopes with a range of values between -2.8 and -2. Previous studies which have found power-law tails with varying slopes are consistent with our findings for clouds at early stages of their evolution, but our results show that the slopes trend towards a constant value at late times. A universal slope for the high-mass end of molecular cloud structure is appealing as it links the column density distribution of the gas to the initial mass function (IMF) of stars. However, this slope is largely dependent on the point, $\Sigma_{\text{tail}}$, where the tail begins its divergence from the lognormal distribution. Determining whether or not the deviation point is constant will aid in our understanding of the universality of the slope and its relation to star formation.
Figure \[fg:evols\](d) shows the evolution of the deviation point as a function of time. We find a decreasing trend towards constant values of the deviation point, consistent with observations [@kain2011; @schneider2014]. The reason for this is that the peak of the distribution is also changing with time (Figure \[fg:evols\](a)). As the distribution widens, the peak shifts to lower extinction values, resulting in a relatively constant transition point at $A_{\text{V,tail}}$ $\sim$ 4 mag for the power-law tail at late times. This deviation point is consistent with the value estimated from 2MASS observations by @kain2011 (A$_{\text{v}}^{\text{tail}}$ $\approx$ 2 – 4 mag) and from Herschel observations by @schneider2014 (A$_{\text{v}}^{\text{tail}}$ $\approx$ 4 – 5 mag). Once gravity begins to dominate, the tail deviates from the lognormal and grows to have a shallower slope. The peak of the distribution is simultaneously decreasing so, by the time star formation takes hold, the dense gas component is entirely confined to the tail of the distribution for surface density values greater than a constant deviation point of $A_{\text{V,tail}}$ $>$ 4 mag.
@lada2010 define an extinction threshold for star formation, above which the surface density of the participating gas correlates linearly with the star formation rate. The authors assume that the material at visual extinctions of greater than 7.3 mag corresponds to a gas volume density of 10$^4$ cm$^{-3}$ [@bergin01]. @heiderman2010 also found the presence of a star formation threshold in an independent study of young stellar objects (YSOs) using the Spitzer c2d and Gould Belt surveys; however, this threshold was at a higher visual extinction of 8.6 mag. Both of these estimates of the star formation threshold are greater than the value of the deviation point found in this work and by other authors [@kain2011; @schneider2014]; however, $A_{\text{V}}^{\text{tail}}$ by its construction is dominated by the diffuse component of the underlying lognormal PDF. @kain2011 argue that the surface density where the contribution from the tail dominates more than 90% of the PDF is a more direct comparison to the star formation thresholds. Since we know the volume density of the gas particles in our simulation, we can determine where the dense component of the gas is dominant in our clouds. In Figure \[fg:sfgas\], we compare the mass in high volume density regions to the mass found at high extinction in our own synthetic maps for the two estimates of the star formation threshold.
![\[fg:sfgas\] Mass found in dense gas (n $>$ 10$^4$ cm$^{-3}$) compared to the mass found above an extinction of $A_{\text{V}}$ = 7.3 mag (black diamonds) or A$_{\text{V}}$ = 8.6 mag (red diamonds). The one-to-one line is also shown as a black dashed line. ](fig8_color.pdf)
This figure shows that masses derived from column density maps will be largely overestimated if one assumes that the star formation threshold is an effective tracer of the dense gas regions corresponding to active star formation. We note however, that while in both cases we find that the mass in high extinction gas is much greater than the mass in high density gas, there still appears to be a correlation, particularly for masses greater than 10$^3$ M$_{\odot}$. As the clouds evolve, the mass in dense gas increases and eventually collapses to form stars. This would explain why the correlation is much tighter at higher masses, as it corresponds to later stages of cloud evolution when the cloud begins to actively form stars. This correlation also suggests that we should expect a similar relation as that found by @lada2010, where the star formation rate (SFR) scales as a function of the mass of high extinction material such that, $$SFR \text{ (M}_{\odot} \text{ yr}^{-1}) = 4.6 \pm 2.6 \times 10^{-8} M_{\text{A$_\text{V}$} > 7.3} \text{ (M}_{\odot})$$
We calculated the star formation rate for our simulated clouds by dividing the total mass in stars at each output by the total time elapsed since the beginning of the simulation. Figure \[fg:sfrs\] shows the star formation rate in our clouds as a function of the cloud mass above the extinction threshold $A_{\text{V}}$ = 7.3 mag.
![\[fg:sfrs\] Star formation rate as a function of mass above an extinction threshold of $A_{\text{V}}$ = 7.3 mag. The best fit to the observations found by @lada2010 is also shown as a dashed line. ](fig9R.pdf)
We find that the star formation rate of molecular clouds correlates very well with the mass above a visual extinction of 7.3 mag. We find the best fit to our data is $SFR$ = 10.7 $\times$ 10$^{-8}$ M$_{\text{A$_\text{V}$} > 7.3}$, where the coefficient is a factor of approximately 2 larger than that found by @lada2010. This could be due to a higher star formation efficiency in our simulations than for the observed clouds. Nevertheless, the existence of the correlation confirms the relation between the mass in dense gas and the mass at high extinction as well as the significance of a constant star formation threshold. The evolution towards a constant deviation point is the result of the interplay between all other structural properties of the column density PDF. A constant value for the deviation point at late times leads to a universal slope of the power-law tail: structural properties which are directly linked to the star formation threshold and slope of the high-mass end of the IMF, respectively.
Conclusions {#sec:discussion}
===========
In this paper, we studied the evolution of the column density PDF of molecular clouds using simulations and synthetic column density maps. For bound and unbound clouds which form from a variety of initial conditions, we find that the column density PDF is consistently represented by a lognormal distribution at early times (t $<$ 0.25 t$_{ff}$ ) with a structural transition to a lognormal + power-law tail at late times (t $>$ 0.5 t$_{ff}$ ). The evolution of the four key structural properties ($\sigma$, $\Sigma_{peak}$, m, and $\Sigma_{tail}$) were explored in detail. We found that, as a cloud evolves, the peak of its column density PDF will shift to lower extinction values as the distribution widens regardless of whether or not it is gravitationally bound. This means that the underlying lognormal distribution, which can be present for a cloud at early times, will be lost for that same cloud at late times due to the evolution of the peak below the observational threshold for detection. Our results help to explain why certain observations of actively star-forming, dynamically older clouds, such as the Orion molecular cloud, do not appear to have any evidence of a lognormal distribution in their column density PDFs and are instead represented by a power-law tail over the full range of extinction [@lombardi2014]. We also find an increase in the column density variance, even though we do not have externally driven turbulence in our simulations. Therefore, the widening of the distribution does not depend solely on the supersonic turbulence [see also @tassis], but arises as a result of density perturbations produced by localised gravitational collapse.
By studying the evolution of the slope and deviation point of the power-law tails for our sample of simulated clouds, we also show that the range for both properties narrows as they trend towards constant values, thus linking the column density structure of the molecular cloud to such stellar properties as the initial mass function and the surface density threshold for star formation. Although the mass in high extinction gas is much greater than the mass in dense gas, we still find a correlation between the two, particularly at masses greater than 10$^3$ M$_{\odot}$. We also show that the thresholds used to define the minimum surface density of gas participating in star formation [@lada2010; @heiderman2010] are suitable tracers of the star formation rates in molecular clouds.
A study of the statistics of cloud properties for a large population of molecular clouds formed in a galactic disc will be the subject of future work. This study will allow for a comparison of our results to clouds observed in galactic and extragalactic environments, as well as further explore the implications of a constant star formation threshold on the star formation rates of bound and unbound molecular clouds.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank SHARCNET (Shared Hierarchical Academic Research Computing Network) and Compute/Calcul Canada, which provided dedicated resources to run these simulations. This work was supported by NSERC. J.W. acknowledges support from the Ontario Early Researcher Award (ERA).
, J., [Vázquez-Semadeni]{}, E., [Gazol]{}, A., et al. 2011, MNRAS, 416, 1436
, J., [D’Alessio]{}, P., & [Hartmann]{}, W. L. 2012, MNRAS, 427, 2562
, M. R. & [Bonnell]{}, I. A. 2005, MNRAS, 356, 1201
, C. N., [Goodman]{}, A. A., [Alves]{}, J. F., et al. 2012, MNRAS, 423, 2579
, E. A., [Ciardi]{}, D. R., [Lada]{}, C. J., [Alves]{}, J., & [Lada]{}, E. A. 2001, ApJ, 557, 209
, R. C., [Savage]{}, B. D., & [Drake]{}, J. F. 1978, ApJ, 224, 132
, G. 2003, PASP, 115, 763
, C., [Banerjee]{}, R., [Clark]{}, P. C., & [Klessen]{}, R. S. 2010, ApJ, 713, 269
, D., [Murphy]{}, G C., [Smith]{}, M. D., [Walsh]{}, J. & [del’ Burgo]{}, C. 2007, MNRAS, 378, 1447
, A. A., [Pineda]{}, J. E., & [Schnee]{}, S. L. 2009, ApJ, 692, 91
, A., [Evans]{}, N. J., [Allen]{}, L. E., [Huard]{}, T., & [Heyer]{}, M. 2010, ApJ, 723, 1019
, P. & [Chabrier]{}, G. 2008, ApJ, 684, 395
, J., [Lada]{}, C. J., [Rathborne]{}, J. M., & [Alves]{}, J. F. 2009, A&A, 497, 399
, J., [Beuther]{}, H., [Banerjee]{}, R., [Federrath]{}, C., & [Henning]{}, T. 2011, A&A, 530, A64
, J., [Federrath]{}, C., & [Henning]{}, T. 2013, A&A, 553, 8
, A. G., [Norman]{}, M. L., & [Wagner]{}, R. 2011, ApJ, 727, L20
, P. 2001, MNRAS, 322, 231
, M. R. & [McKee]{}, C. F. 2005, ApJ, 630, 250
, C. J., [Lombardi]{}, M., & [Alves]{}, J. F. 2010, ApJ, 724, 687
, M., [Alves]{}, J., & [Lada]{}, C. J. 2010, A&A, 519, 7
, M., [Bouy]{}, H., [Alves]{}, J., & [Lada]{}, C. J. 2014, A&A, 566, A45
, Å. & [Padoan]{}, P. 1999, Interstellar Turbulence, ed. J. Franco & A. Carraminana, 218
, D., [Kirk]{}, J. M., [Stamatellos]{}, D., & [Ward-Thompson]{}, D. 2008, MNRAS, 384, 755
, E. C., [Stone]{}, J. M., & [Gammie]{}, C. F. 2001, ApJ, 546, 980
, P. & [Nordlund]{}, Å. 2002, ApJ, 576, 870
, G. H. & [Lebofsky]{}, M. J. 1985, ApJ, 288, 618
, E. E. 1955, ApJ, 121, 161
, N., [Ossenkopf]{}, V., [Csengeri]{}, T., et al. 2014, A&A, submitted
, K., [Christie]{}, D. A., [Urban]{}, A., et al. 2010, MNRAS, 408, 1089
, E. 1994, ApJ, 423, 681
, J. W., [Stadel]{}, J., & [Quinn]{}, T. 2004, New Astronomy, 9, 137
, R. L., [Wadsley]{}, J., [Sills]{}, A., & [Petitclerc]{}, N. 2012, ApJ, 756, 119
, R. L., [Wadsley]{}, J., & [Sills]{}, A. 2014, MNRAS, 439, 651
|
---
abstract: 'We study mixing of the Metropolis algorithm for a distribution on the hypercube that corresponds to the Erdős-Rényi random graph with edge probability $p$. This Markov chain has cutoff at $\max\{p,1-p\} n \log n$ with window size $n$, a result proved by Diaconis and Ram (2000) using Fourier analysis. Here we give an alternative proof that relies on coupling and a projection to a two-dimensional Markov chain. This is done in the hope that probabilistic techniques will be easier to generalize to less symmetric distributions. We also describe a close relationship between the Metropolis and Gibbs samplers for this model. Our proof extends to the case where the edge probabilities vary with $n$. In that case, we also show that a natural coordinate wise coupling is sharp if and only if the edge probabilities are of order $1/n$.'
address: University of Chicago
author:
- Winfried Barta
title: 'A probabilistic proof of cutoff in the Metropolis algorithm for the Erdős-Rényi random graph'
---
Markov chain ,hypercube ,Metropolis algorithm ,mixing time ,convergence rate ,cutoff 60J10.
Introduction
============
We are interested in analyzing convergence rates of the random walk Metropolis algorithm for various distributions $\pi$ on the hypercube $\mathcal{X} := \{0,1\}^n$. This Markov chain on $\mathcal{X}$ moves as follows: Given that we are at a state $x \in \mathcal{X}$, we chose one of the $n$ neighbors of $x$ uniformly at random, say $y$, and propose to move from $x$ to $y$. Here $x$ is called a neighbor of $y$, denoted by $x \sim y$, whenever $x$ and $y$ differ in exactly one coordinate. This proposal gets accepted, i.e. we move to $y$, with probability min$\big(1,\pi(y) / \pi(x)\big)$. If it gets rejected, we stay at $x$. This transition rule ensures that we have detailed balance $$\pi(x) \, P(x,y) = \pi(y) \, P(y,x) \qquad\mbox{for all $x,y \in \mathcal{X}$} .$$ If $\pi$ is positive on the entire state space, then the chain is irreducible and its unique stationary distribution is $\pi$. Often it will be convenient to make the chain *lazy*. This corresponds to flipping a fair coin independently at each step. If it comes up heads, we stay where we are; if it comes up tails, we move according to the rule specified above.
We are interested in distributions $\pi$ that are *unimodal* and *radially symmetric* with respect to their mode. By this we mean that there exists a state, $z$ say, that has highest mass under $\pi$; any two states with the same distance to $z$ have the same $\pi$-mass; and this $\pi$-mass decreases with distance to $z$. That is, we have $$\pi(x) = \pi(y) \qquad\mbox{whenever $d(x,z)=d(y,z)$} ,$$ and $$\pi(x) \leq \pi(y) \qquad\mbox{whenever $d(x,z) > d(y,z)$} .$$ Here, $d(x,z) := \sum_{i=1}^n |x_i - z_i|$ is the graph distance (Hamming distance) of $x$ and $z$, i.e the number of coordinates where $x$ and $z$ differ.
By relabeling the states, we may (and will) assume that the mode $z$ of $\pi$ is at $\underline{0}=(0,0,...,0)$. This ensures that $\pi$ is constant on level sets $L(k) := \{x \in \mathcal{X}: S(x)=k\}$ where $k \in \{0,1,...,n\}$ and $S(x):=\sum_{i=1}^n x_i$ is the number of ones in $x$. Hence, in the Metropolis algorithm we will always accept downward moves $x \rightarrow y$ where $S(x)>S(y)$, and we will accept upward moves $x \rightarrow y$ where $S(x)<S(y)$ with probability $\theta_{S(x)}$, where we write $\theta_k := \pi(v)/\pi(w)$, where $v,w$ are some (any) states such that $S(v)=k+1$ and $S(w)=k$.
For concreteness, the transition kernel for the lazy random walk Metropolis Hastings algorithm is $$\label{transition_kernel_lRW-MH}
P(x,y) = \left\{ \begin{array}{l@{\quad:\quad}l}
\frac{1}{2n} & x \sim y, S(x) > S(y) , \\
\frac{1}{2n} \theta_{S(x)} & x \sim y, S(x) < S(y) , \\
\frac{1}{2} + \frac{n-S(x)}{2n} (1-\theta_{S(x)}) & x = y , \\
0 & \mbox{otherwise} .\\
\end{array} \right.$$ Note that a consequence of radial symmetry is that the projection $S(X_t)$ of the Markov chain $(X_t)$ is also Markov, since for all $x,y$ we then have $$P(x,[y]) = P(x',[y]) \qquad\mbox{for all } x' \sim_S x .$$ Here we write $x \sim_S y$ whenever $S(x)=S(y)$ and $[y] := \{z: S(z)=S(y)\}$ for $y \in \mathcal{X}$ are the equivalence classes of the relation $\sim_S$. See [@LevinPeresWilmer2009 Lemma 2.5 on page 25].
We measure distance to stationarity by total variation: $$||P^t(x,\cdot) - \pi||_{TV} := \max_{A \subset \mathcal{X}} \; \left(P^t(x,A)-\pi(A)\right) ,$$ and we are interested in this distance from the worst starting point: $$d(t) := \max_{x \in \mathcal{X}} ||P^t(x,\cdot) - \pi|| .$$ The mixing time for a parameter $\varepsilon \in (0,1)$ is defined as $$t_{\rm mix}(\varepsilon) := \min\{t \geq 0: d(t) \leq \varepsilon\}$$ and we write $t_{\rm mix}$ for $t_{\rm mix}(1/4)$. We are interested in the behavior of $t_{\rm mix}$ as the dimension $n$ of the hypercube goes to infinity.
An interesting phenomenon is that for some chains the mixing time $t_{\rm mix}(\varepsilon)$ doesn’t depend on the parameter $\varepsilon$ (asymptotically, as $n$ goes to infinity). We say a sequence $\left(\mathbf{X}^{(n)}\right)_{n \in \mathbb{N}}$ of Markov chains $\mathbf{X}^{(n)} = \big(X_t^{(n)}\big)_{t=0,1,...}$ on $\{0,1\}^n$ has a *cutoff* (at $t^{(n)}_{\rm mix})$ if, for all $\varepsilon \in (0,1)$, $$\lim_{n \rightarrow \infty} \frac{t^{(n)}_{\rm mix}(\varepsilon)}{t^{(n)}_{\rm mix}(1-\varepsilon)} = 1 .$$ Here, $t_{\rm mix}^{(n)}(\varepsilon)$ denotes the $\varepsilon$-mixing time of the $n^{th}$ chain $(X_t^{(n)})_{t=0,1,...}$. This is equivalent to $$\label{lemma18.1}
\lim_{n \rightarrow \infty} d_n\big(c \, t_{\rm mix}^{(n)}\big) = \left\{ \begin{array}{l@{\quad:\quad}l}
1 & \mbox{if } \; c < 1 , \\
0 & \mbox{if } \; c > 1 .
\end{array}
\right.$$ So the function $d_n(\cdot)$, the total variation distance to stationarity from the worst starting point for the $n^{th}$ chain, approaches a step function as $n$ goes to infinity (if we rescale time by $t_{\rm mix}^{(n)}$). For a proof of this equivalence see [@LevinPeresWilmer2009 Lemma 18.1 on page 247], from where we also borrow the notation. For an overview of the cutoff phenomenon, see [@Diaconis1996].
Sometimes it is possible to analyze more precisely what happens for $c=1$ in (\[lemma18.1\]). We say a sequence of Markov chains has a cutoff with *window size* $(w_n)$, if $w_n \in o(t_{\rm mix}^{(n)})$ and $$\begin{aligned}
\lim_{\alpha \rightarrow \infty} \, \liminf_{n \rightarrow \infty} \; d_n\big(t_{\rm mix}^{(n)} - \alpha w_n\big) & = & 1 , \\
\lim_{\alpha \rightarrow \infty} \, \limsup_{n \rightarrow \infty} \; d_n\big(t_{\rm mix}^{(n)} + \alpha w_n\big) & = & 0 . \end{aligned}$$ For an introduction to Markov chains and mixing times see the book by Levin, Peres and Wilmer [@LevinPeresWilmer2009], from which we borrow heavily.
The Erdős-Rényi random graph model
==================================
The easiest model in the class of unimodal and radially symmetric distributions $\pi$ on the hypercube arises when we have $\theta_k = \theta \in (0,1]$ for all $k$, i.e. the acceptance probabilities for upward moves are constant across level sets. That is, $\pi(x) = \theta^{S(x)} (1+\theta)^{-n}$. The current paper will focus on this model. The case where the neighboring odds ratios $\theta=\theta_k=\pi(v)/\pi(w)$ are bigger than one, where $S(v)=k+1$ and $S(w)=k$, would correspond to the mode of $\pi$ being at $\underline{1} = (1,1,...,1)$ instead of $\underline{0} = (0,...,0)$. By symmetry, this gives rise to nothing new, so we will assume $\theta \in (0,1]$ henceforth. The case where $\theta=1$ corresponds to $\pi$ being the uniform distribution.
If we have $n = {\nu \choose 2}$ and identify the list of coordinates of the hypercube with the list of potential edges of a graph on $\nu$ vertices, then the hypercube represents the space of all possible (simple) graphs on $\nu$ vertices: a one indicates that a certain edge is present in the graph; a zero indicates that it is absent. Since $\pi(x) = \left(\frac{\theta}{1+\theta}\right)^{\!S(x)} \left(1 - \frac{\theta}{1+\theta}\right)^{\!n-S(x)}$ for all $x \in \{0,1\}^n$, this distribution $\pi$ corresponds to the Erdős-Rényi random graph model with parameter $p:=\frac{\theta}{1+\theta}$. This is the probability distribution on (simple) graphs on $\nu$ vertices where each of the $n={\nu \choose 2}$ potential edges is present independently with probability $p$.
The case where $\pi$ is the uniform distribution ($\theta=1$) corresponds to the Erdős-Rényi model with edge probability $\theta / (1+\theta) = 1/2$. For this model it is well known that the (non-lazy) random walk Metropolis algorithm has cutoff at $(1/4) n \log n$ with a window of size $n$. This was proved by Aldous and Diaconis [@Aldous1983; @DiaconisShahshahani1987] using Fourier analysis. The shape of the cutoff was studied in more detail by Diaconis, Graham and Morrison [@DiaconisGrahamMorrison1990], whereas Levin, Peres and Wilmer [@LevinPeresWilmer2009 Theorem 18.3 on page 251] give a probabilistic proof.
The result has been generalized to edge probabilities different from $1/2$: For general $\theta \in (0,1]$, it is known that the non-lazy version of the random walk Metropolis chain has cutoff at $\frac{1}{2(1+\theta)} n \log n$ with window size $n$. This was derived using Fourier analysis by Diaconis and Ram [@DiaconisRam2000 Theorem 5.4 on page 177]. See also the work of Diaconis and Hanlon [@DiaconisHanlon1992 Theorem 2 on page 104] who explicitly calculate the eigenvalues and eigenvectors of the transition kernel of the projection $S(X_t)$. Ross and Xu [@RossXu1994] view the Metropolis chain for this model as a random walk on a hypergroup deformation of the hypercube, and proceed by performing Fourier analysis of this random walk. See also [@DiaconisSaloff-Coste2006 page 2117] for a discussion of these and some related results. For the lazy version of this chain for $\theta \in (0,1]$, we therefore get cutoff at $\frac{1}{1+\theta} n \log n$ with window size $n$. This follows from [@DiaconisRam2000]: For the upper bound, note that running the lazy chain for $2t$ steps corresponds to running the non-lazy chain for $T \sim Bin(2t,1/2)$ steps. So the standard deviation of $T$ is of order ${\mathop{\mathrm{O}}\bigl(\sqrt{t}\bigr)} = {\mathop{\mathrm{O}}\bigl(\sqrt{n \log n}\bigr)}$, which is easily absorbed into the window size term of order $n$. For the lower bound, note that making the chain lazy sends an eigenvalue $\lambda$ of the transition kernel to $(\lambda+1)/2$, while leaving the associated eigenfunction unchanged. Applying the results and methods of [@DiaconisRam2000 page 178] then completes the proof.
In this paper we will give an alternative proof of this result, generalizing the methods used by Levin, Peres and Wilmer [@LevinPeresWilmer2009 Theorem 18.3 on page 251] for the case where $\theta=1$. This is done in the hope that a probabilistic proof will be easier to generalize to less symmetric models, where Fourier analysis might be harder to apply. To be specific, we will proof the following result:
\[theta\_model\] The lazy random walk Metropolis chain for $\pi(x) = \theta^{S(x)} (1+\theta)^{-n}$ on $\{0,1\}^n$ has cutoff at $\frac{1}{1+\theta} n \log n$ with a window of size $n$.
Let $n := {\nu \choose 2}$ and let $\pi(x) = p^{S(x)} (1-p)^{n-S(x)}$ be the Erdős-Rényi random graph model on $\nu$ vertices with parameter $p \in (0,1)$. The lazy random walk Metropolis chain for this model has cutoff at $\max\{p, 1-p\} n \log n$ with a window of size $n$.
[**Proof of the Corollary.**]{} Let $\theta:=p/(1-p)$. As mentioned above, by relabeling states (switching zeros and ones) if necessary, we may assume $\theta \in (0,1]$. Since $$p^{S(x)} (1-p)^{n-S(x)} = \left(\frac{\theta}{1+\theta}\right)^{\!S(x)} \left(1 - \frac{\theta}{1+\theta}\right)^{\!n-S(x)}
= \theta^{S(x)} (1+\theta)^{-n}$$ and $1/(1+\theta) = 1-p = \max\{p, 1-p\}$, the result follows from the Theorem. $\; \Box$
Lower bound {#sec_lower_bound_1}
===========
Our proof for the lower bound part of Theorem \[theta\_model\] generalizes the proof given by Levin, Peres and Wilmer [@LevinPeresWilmer2009 Proposition 7.13 on page 95] for the case where the stationary distribution $\pi$ is uniform ($\theta = 1$). It is based on the method of distinguishing statistics, described as follows in [@LevinPeresWilmer2009 Proposition 7.8 on page 92]:
\[dist\_stat\] Let $\mu$ and $\nu$ be two probability distributions on $\mathcal{X}$, and let $S$ be a real-valued function on $\mathcal{X}$. If $$|E_{\mu}(S) - E_{\nu}(S)| \geq r \sigma ,$$ where $\sigma^2 = [{\rm Var}_{\mu}(S) + {\rm Var}_{\nu}(S)]/2$, then $$||\mu - \nu||_{TV} \geq 1 - \frac{4}{4+r^2} .$$
Here $E_{\mu}(S) := \sum_{x \in \mathcal{X}} S(x) \mu(x)$ denotes the expectation of $S$ under $\mu$, and likewise for $\nu$. So if we can find a real function on the state space $\mathcal{X}$ such that its expectations under $P^t(x,\cdot)$ and $\pi$ are still very different on the scale of the square root of their average variance after $t$ steps of the chain, then we have demonstrated that $||P^t(x,\cdot) - \pi||$ must still be large.
A natural choice for the distinguishing statistic is the number of ones in a state, $S(x) := \sum_{i=1}^n x_i$ for $x \in \{0,1\}^n$. Therefore we have to analyze the one-dimensional projection $S(X_t) =: S_t$ of our Markov chain $(X_t)$. As mentioned in the introduction, this is again a Markov chain whose transition probabilities satisfy $P(k,l) = P(x,S^{-1}(l)) = P(x,[y])$ for any $x,y$ with $S(x)=k$ and $S(y)=l$. As before, $[y] = \{z \in \{0,1\}^n \, : \, S(z)=S(y)\}$ denotes the equivalence class of all states with the same number of ones as $y$. Because of their different domains it should not lead to confusion that we are using the same notation $P(\cdot,\cdot)$ for the transition probabilities of the original chain and the projected chain.
Similarly, the stationary distribution $\pi_S$ of $(S_t)$ is the push-forward $\pi_S := \pi S^{-1}$ of $\pi$ under $S$. This entails $$\begin{aligned}
\pi_S(k) & = & {n \choose k} \theta^k (1+\theta)^{-n} \\
& = & {n \choose k} \left( \frac{\theta}{1+\theta} \right)^{\!k} \left( 1 - \frac{\theta}{1+\theta} \right)^{\!n-k} \\
& = & \mbox{Binomial}\left(n, \frac{\theta}{1+\theta}\right)(k) .\end{aligned}$$ Therefore we get the expectation and variance of $S \sim \pi_S$ as $$E_{\pi_S} S = \frac{n \; \theta}{1 + \theta}, \;\;\;\;\;
{\rm Var}_{\pi_S} S = \frac{n \; \theta}{(1 + \theta)^2} .$$
The chain $(S_t)$ is a birth and death chain on $\{0,1,...,n\}$ with transition probabilities $$\begin{aligned}
\label{transition_kernel_1d}
P(S_{t+1} = k+1 | S_t = k) & = & \left( 1 - \frac{k}{n} \right) \frac{\theta}{2} , \nonumber \\
P(S_{t+1} = k | S_t = k) & = & \frac{1}{2} + \left( 1 - \frac{k}{n} \right) \frac{1-\theta}{2} , \\
P(S_{t+1} = k-1 | S_t = k) & = & \frac{k}{2n} . \nonumber\end{aligned}$$ We begin by calculating its expectation after $t$ steps starting from $k$. Note that for all $t$ we get $$S_{t+1} - S_t = \left\{ \begin{array}{r@{\quad\mbox{with probability }\quad}l}
1 & \left( 1 - \frac{S_t}{n} \right) \frac{\theta}{2} , \\
-1 & \frac{S_t}{2n} , \\
\end{array} \right.$$ and $S_{t+1} - S_t = 0$ otherwise. So $$E[S_{t+1} - S_t | S_t] = \left( 1 - \frac{S_t}{n} \right) \frac{\theta}{2} - \frac{S_t}{2n} =
\frac{\theta}{2} - S_t \frac{1+\theta}{2n} ,$$ and therefore $$E[S_{t+1} | S_t] =
\frac{\theta}{2} + \left( 1 - \frac{1+\theta}{2n} \right) S_t .$$ By taking expectation $E_k$ with respect to the starting state $k$ we get for all $t,k$ that $$\label{exp3}
E_k (S_{t+1}) = \frac{\theta}{2} + \left( 1 - \frac{1+\theta}{2n} \right) E_k (S_t) .$$ By induction on $t$ this leads to the following result:
\[exp\_theta\] The projected chain $S_t := S(X_t)$ of our lazy Metropolis chain $(X_t)$ has for all $k=0,1,...,n$ and all $t \in \mathbb{N}$ $$\label{expect_St}
E_k (S_t) = \frac{n \theta}{1 + \theta} \left( 1 - \gamma^t \right) + k \gamma^t ,$$ where $\gamma := \gamma_{n,\theta} := 1 - \frac{1+\theta}{2n}$.
Note that the expected location $E_k S_t$ is a convex combination of the starting state $S_0=k$ and the stationary mean $E_{\pi_S} S$, with relative weights $\gamma^t$ and $1-\gamma^t$ respectively.
\[gamma\^u\] By expanding $\log(\gamma)$ about one, it is easy to see that for $u := u_{n,\theta} := \frac{1}{1+\theta} n \log n$ we get $$\gamma^u \sim n^{-1/2} ,$$ by which we mean that $\lim_{n \rightarrow \infty} \frac{\gamma^u}{n^{-1/2}} = 1$.
It remains to bound the variance of $S_t$. Since we want a lower bound on $$d(t) = \sup_x || P^t(x, \cdot) - \pi || \geq || P^t(\underline{1}, \cdot) - \pi || ,$$ it’s enough to consider the starting state $\underline{1} = (1,...,1)$ of all ones. For this, first note that we can run the chain $X_t$ in the following way. For $t=0,1,2,...$, given we are at state $X_t$ at time $t$:
- Pick a coordinate $i \in [n]$ uniformly at random, independent of all previous choices.
- Draw $U_t \sim$ Uniform$[0,1]$, independent of all previous choices.
- Set $X_{t+1}^{(j)} := X_t^{(j)}$ for $j \ne i$, and set the $i^{th}$ coordinate of $X_{t+1}$ to
$$X_{t+1}^{(i)} := \left\{ \begin{array}{l@{\quad:\quad}l}
1 & 0 \leq U_t \leq \frac{\theta}{2} , \\
X_t^{(i)} & \frac{\theta}{2} < U_t \leq \frac{1}{2} , \\
0 & \frac{1}{2} < U_t \leq 1 .\\
\end{array} \right.$$ Now say that a coordinate $j$ has been *refreshed* by time $t$, if coordinate $j$ was selected at some time $s<t$ *and* $U_s \notin \left( \frac{\theta}{2}, \frac{1}{2} \right]$. Let $R_t$ be the number of coordinates not refreshed by time t. We can study the expectation and variance of $R_t$ with a natural modification of the classical coupon collector problem, where coordinate $j$ being refreshed corresponds to coupon $j$ being collected. This leads to the following result, with a proof analogous to the one for [@LevinPeresWilmer2009 Lemma 7.12 on page 94]:
\[coupon\_collector\] Consider the coupon collector problem with $n$ distinct coupon types, where at each trial, with probability $\frac{1-\theta}{2}$ we get *no* coupon, and with probability $1 - \frac{1-\theta}{2} = \frac{1+\theta}{2}$ we get a coupon chosen (independently and) uniformly at random. Let $I_j(t)$ be the indicator of the event that the $j^{th}$ coupon has *not* been collected by time $t$. Let $R_t := \sum_{j=1}^n I_j(t)$ be the number of coupon types not collected by time $t$. The random variables $I_j(t)$ are negatively correlated, and letting $\gamma := 1 - \frac{1+\theta}{2n}$, we get for $t \geq 0$ that $$\begin{aligned}
E(R_t) & = & n \gamma^t , \\
{\rm Var}(R_t) & \leq & n \gamma^t \,(1-\gamma^t) \leq n \gamma^t . \end{aligned}$$
If we start the chain at $X_0 = \underline{1}$, then the conditional distribution of $S_t := S(X_t)$ given $R_t = r$ is the same as that of $r + B$, where $B \sim \mbox{Binomial}(n-r, \frac{\theta}{1+\theta})$. Therefore, $$E_{\underline{1}} [S_t \, | \, R_t] = R_t + (n - R_t) \frac{\theta}{1+\theta} = \frac{R_t + n \theta}{1+\theta} ,$$ so by taking expectation we get $$E_{\underline{1}} [S_t] = \frac{E [R_t] + n \theta}{1+\theta} = \frac{n \gamma^t + n \theta}{1+\theta}
\left( = \frac{n \theta}{1+\theta} (1-\gamma^t) + n \gamma^t \right) ,$$ confirming our result (\[expect\_St\]) for general starting states $k$ for the special case $k=n$. Furthermore, since $${\rm Var}_{\underline{1}} [S_t] = {\rm Var}\left[ E_{\underline{1}} (S_t \, | \, R_t) \right]
+ E\left[ {\rm Var}_{\underline{1}} (S_t \, | \, R_t) \right] ,$$ we get $$\begin{aligned}
{\rm Var}_{\underline{1}} [S_t] & = & {\rm Var}\left[ \frac{R_t + n \theta}{1+\theta} \right] + E\left[ {\rm Var} \Big(\mbox{Binomial}(n-R_t, \frac{\theta}{1+\theta})\Big) \right] \\
& = & \frac{1}{(1+\theta)^2} \, {\rm Var} [R_t] + \frac{1}{(1+\theta)^2} \, (n - E [R_t]) \, \theta \\
& \leq & \frac{1}{(1+\theta)^2} \left[ n \gamma^t + (n - n \gamma^t) \theta \right] \\
& \leq & \frac{n}{(1+\theta)^2} . \end{aligned}$$ To apply Proposition \[dist\_stat\], observe that $$\sigma^2 := \frac{{\rm Var}_{P^t(\underline{1}, \cdot)} S + {\rm Var}_{\pi} S}{2} \leq
{\max\{{\rm Var}_{\underline{1}} S_t, {\rm Var}_{\pi_S} S\}} \leq \frac{n}{(1+\theta)^2} .$$ So for $\alpha>0, t := t_{n, \alpha} := \frac{1}{1+\theta} n \log n - \alpha n$ and $\gamma = 1 - \frac{1+\theta}{2n}$, we get for any fixed $\varepsilon>0$ and large $n$ that $$\begin{aligned}
\left| E_{\underline{1}} S_t - E_{\pi} S \right| & = & \left| \frac{n (\theta + \gamma^t)}{1 + \theta} - \frac{n \theta}{1+\theta} \right| \\
& = & \frac{n}{1+\theta} \left( 1 - \frac{1+\theta}{2n} \right)^{\!t} \\
& \geq & \sigma \sqrt{n} \left( 1 - \frac{1+\theta}{2n} \right)^{\!n \left( \frac{1}{1+\theta} \log n - \alpha \right)} \\
& \geq & \sigma \sqrt{n} (1-\varepsilon) \exp\left\{- \frac{1+\theta}{2} \left( \frac{1}{1+\theta} \log n - \alpha \right)\right\} \\
& = & \sigma (1-\varepsilon) \exp\left\{\alpha \frac{1+\theta}{2}\right\} \\
& =: & \sigma \, r_{\alpha} .\end{aligned}$$ By Proposition \[dist\_stat\], this means $d(t) \geq ||P^t(\underline{1},\cdot) - \pi|| \geq 1 - \frac{4}{4 + r_{\alpha}^2}$, and therefore $$\lim_{\alpha \rightarrow \infty} \liminf_{n \rightarrow \infty} \, d(t_{n, \alpha}) \geq
\lim_{\alpha \rightarrow \infty} 1 - \frac{4}{4 + r_\alpha^2} = 1 .$$ This finishes the proof of the lower bound part of Theorem \[theta\_model\]. $\;\; \Box$
Lower bound, alternative proof {#sec_azuma}
==============================
The previous proof for the lower bound part of Theorem \[theta\_model\] might be hard to generalize to distributions $\pi$ where $\theta_k$ is not constant in $k$. (Recall that $\theta_k := \pi(v)/\pi(w)$ for any $v,w$ such that $S(v)=k+1, S(w)=k$.) Therefore, we give here an alternative proof for this result. For this we use a modified version of the method of distinguishing statistics (Proposition \[dist\_stat\]), that avoids the need to approximate the variance of the statistic $S$ under $P^t(x,\cdot)$. Instead, we use the fact that our chain only makes *local moves* to argue that $S_t = S(X_t)$ must be concentrated about its mean under $P^t(x,\cdot)$. This will follow from Azuma’s inequality, applied to the conditional expectation martingale formed by $Y_i := E [S_t \, | \, S_{0:i}]$, for $i=0,1,...,t$, since we can show that $(Y_i)$ has bounded differences. Here we write $S_{k:l} := (S_i)_{k \leq i \leq l}$.
\[conc\_theta\] Let $(X_t)$ be the lazy random walk Metropolis chain on $\{0,1\}^n$ for $\pi(x) = \theta^{S(x)} (1+\theta)^{-n}$. Let $S_t := S(X_t)$ and $v:=S_0=S(X_0)$. Then for any $s > 0$ and $\gamma := 1 - \frac{1+\theta}{2n}$ we get $$\begin{aligned}
P_v\{S_t \geq E_v S_t + s\} & \leq & e^{-\frac{2}{9}s^2(1-\gamma^2)} \qquad\mbox{ and } \\
P_v\{S_t \leq E_v S_t - s\} & \leq & e^{-\frac{2}{9}s^2(1-\gamma^2)} .\end{aligned}$$
**Proof.** Fix any $t \in \mathbb{N}$. Since $(S_i)$ is a Markov chain, we get from Proposition \[exp\_theta\] and the Markov property that for any $i=0,1,...,t$ $$E[S_t \, | \, S_{0:i}] = E_{S_i} \, [S_{t-i}] = \gamma^{t-i} \left[ S_i - \frac{n \theta}{1+\theta} \right] + \frac{n \theta}{1+\theta} .$$ Therefore for any $i=1,2,...,t$, we get $$\begin{aligned}
E[S_t \, | \, S_{0:i}] - E[S_t \, | \, S_{0:(i-1)}] & = & \gamma^{t-i} \left[ S_i - \frac{n \theta}{1+\theta} \right] - \gamma^{t-(i-1)} \left[ S_{i-1} - \frac{n \theta}{1+\theta} \right] \\
& = & \gamma^{t-i} \left[ S_i - S_{i-1} + (1-\gamma) S_{i-1} - (1-\gamma) \frac{n \theta}{1+\theta} \right] \\
& = & \gamma^{t-i} \left[ S_i - S_{i-1} + \frac{1+\theta}{2n} S_{i-1} - \frac{\theta}{2} \right] .\end{aligned}$$ This means the martingale $(Y_i)$ has bounded differences $$| E[S_t \, | \, S_{0:i}] - E[S_t \, | \, S_{0:(i-1)}] | \leq \frac{3}{2} \gamma^{t-i} ,$$ since we have $S_i \in [0,n]$ and $|S_i - S_{i-1}| \leq 1$, and also $\theta \in (0,1]$. Applying Azuma’s inequality now gives the result, since we have $$c := 4 \sum_{i=1}^t \left(\frac{3}{2} \gamma ^{t-i}\right)^{\!2} = 9 \sum_{i=0}^{t-1} \gamma^{2i} \leq \frac{9}{1-\gamma^2} . \Box$$
Now we use this concentration result for $(S_t)$ to establish the lower bound on the mixing time for this Markov chain. Fix $\alpha > 0$ and set $t := t_{n,\alpha} := \frac{1}{1+\theta} n \log n - \alpha n$. We need to show that $$\label{theta_lbb}
\lim_{\alpha \rightarrow \infty} \liminf_{n \rightarrow \infty} d(t_{n,\alpha}) = 1 .$$ Denote with $S_t := S(X_t)$ the projection of our chain under $S$, started at $X_0 = \underline{\mathbf{1}}$, so that $S_0=n$ and note that for any $r \in [0,n]$ we get $$\begin{aligned}
d(t) & \geq & ||P^t(\underline{1},\cdot) - \pi|| \\
& \geq & sup_{L \subset [n]} P^t(\underline{1},S^{-1}(L)) - \pi(S^{-1}(L)) \\
& \geq & 1 - P_n\{S_t < r\} - P\{B_n \geq r\} .\end{aligned}$$ Here the second inequality comes from noting that the supremum on the right is over a subset of events from the supremum defining the total variation distance on the left. Therefore, total variation distance can only decrease after projecting down. For the last inequality we took the event $L := \{\lceil r \rceil ,\lceil r \rceil +1,...,n\}$ and let $B_n$ stand for a random variable with distribution $\pi_S=\mbox{Binomial}(n,\frac{\theta}{1+\theta})$. Our task is to find $r \in [0,n]$ such that $$\begin{aligned}
\label{conc_th}
\lim_{\alpha \rightarrow \infty} \limsup_{n \rightarrow \infty} P_n\{S_t < r\} & = & 0 \qquad\mbox{ and } \\
\label{conc_Bin}
\lim_{\alpha \rightarrow \infty} \limsup_{n \rightarrow \infty} P\{B_n \geq r\} & = & 0 . \end{aligned}$$ Suppressing rounding issues from the notation, we can pick $r := np + \sqrt{\alpha n}$, where we write $p := \frac{\theta}{1+\theta}$. To establish (\[conc\_th\]), we use the concentration result for $S_t$ from Proposition \[conc\_theta\]. Using the results from Proposition \[exp\_theta\] and Remark \[gamma\^u\], this shows that for all $\epsilon'>0$ and large $n$ we get $$\begin{aligned}
P_n\{S_t < r\}& = & P_n\{S_t - E_n S_t < -((1-p)n \gamma^t - \sqrt{\alpha n})\} \\
& \leq & P_n\left\{S_t - E_n S_t < -\left((1-\epsilon')(1-p)\sqrt{n} e^{\alpha (1+\theta)/2} - \sqrt{\alpha n}\right)\right\} \\
& \leq & \exp\left\{- \frac{2}{9} n \left( (1-\epsilon')(1-p) e^{\alpha (1+\theta)/2} - \sqrt{\alpha} \right)^2 (1-\gamma^2)\right\} \\
& \leq & \exp\left\{- \frac{2}{9} \left( (1-\epsilon')(1-p) e^{\alpha (1+\theta)/2} - \sqrt{\alpha} \right)^2 (1+\theta) \gamma\right\} .\end{aligned}$$ Here we use the fact that $1-\gamma^2 \geq \frac{1+\theta}{n} \gamma$ for the last inequality. Taking limits establishes (\[conc\_th\]). The result (\[conc\_Bin\]) follows from Chebychev’s inequality applied to the Binomial($n,p$) random variable $B_n$. $\Box$
Upper bound
===========
To establish the upper bound part of Theorem \[theta\_model\], we use a two-stage coupling procedure, generalizing the one given by Levin, Peres and Wilmer [@LevinPeresWilmer2009 Theorem 18.3 on page 251] for the case where $\pi$ is uniform ($\theta=1$). In that case, it is enough to study the one-dimensional projection $X_t \mapsto S(X_t)$, since due to symmetry, total variation distance to stationarity is the same from any starting state. Therefore, we may start our chain at the state of all zeros (or all ones), in which case total variation distance to stationarity doesn’t change under the projection to the number of ones.
For general $\theta \in (0,1]$, total variation distance to stationarity is not necessarily the same from any starting state, and we weren’t able to prove that the state of all zeros (or all ones) is a worst starting state. So it’s not clear to us whether the problem of upper bounding the mixing time can still be reduced to a one-dimensional problem here. However, we were able to get a sharp upper bound on the mixing time using a *two*-dimensional projection that depends on the starting state $X_0=x$. For this, consider $Z := Z_x \colon \{0,1\}^n \rightarrow \{0,1,...,n\} \times \{0,1,...,n\}$, $$X_t \mapsto Z_x(X_t) := Z_t := \left( S(X_t), d(x,X_t) \right) ,$$ where $d(x,y)$ is the number of coordinates where $x$ and $y$ disagree, and $x=X_0$ is the starting state of the chain. In words, we project down to the number of ones, $S(X_t)$, and the distance to the starting state, $d(X_t,X_0)$. This two-dimensional process $(Z_t)$ is similar to the two-coordinate chain used in [@LevinLuczakPeres2010 section 3] in the study of Glauber dynamics for the mean-field Ising model.
Our proof proceeds by showing that $(Z_t)$ is a (two-dimensional) birth and death chain with the same total variation distance to its stationary distribution as the original chain $(X_t)$. Bounding this distance is then achieved by a two-stage coupling procedure using an independence coupling of two versions of the chain. The first stage (*drift*-regime) brings them close together in expectation after $\frac{1}{1+\theta} n \log n$ steps due to the drift towards the mean in this birth and death chain. In the second stage (*entropy*-regime) the drift of $(Z_t)$ is likely weak, so a comparison to the behavior of simple random walk shows that the chains can be made to coalesce after an additional $\alpha n$ steps with high probability.
Properties of the projection
----------------------------
We begin by establishing some properties of the two-dimensional projection $Z_t := Z_x(X_t)$ of $X_t$, for which we need some more notation. Fix $x,z \in \{0,1\}^n$ and let $S(x)=k, S(z)=l, d(x,z)=l'$. Define $$\begin{aligned}
G & := & G(x,z) := \{i \in [n] \, : \, x^{(i)}=0, z^{(i)}=0\} , \\
N & := & N(x,z) := \{i \in [n] \, : \, x^{(i)}=0, z^{(i)}=1\} , \\
E & := & E(x,z) := \{i \in [n] \, : \, x^{(i)}=1, z^{(i)}=0\} , \\
F & := & F(x,z) := \{i \in [n] \, : \, x^{(i)}=1, z^{(i)}=1\} .\end{aligned}$$ When clear from the context, we might suppress the dependence of $G, N, E, F$ on $x, z$ in the notation. For the number of elements in these four sets we get $$\begin{aligned}
\label{nG}
\#G & = &n - \frac{l + l' + k}{2} , \nonumber \\
\#N & = & \frac{l + l' - k}{2} , \\
\#E & = &\frac{l' + k - l}{2} , \nonumber \\
\#F & = & \frac{l - (l' - k)}{2} \nonumber .\end{aligned}$$ This follows from $\#N+\#E=l', \#N+\#F=l, \#E+\#F=k, \#G+\#N=n-k$, e.g. by starting with the observation $$l' - l + \#F = l' - \#N = \#E = k - \#F ,$$ which gives the last equality $2 \#F = l - (l' - k)$ above. The remaining equalities then follow. This is probably best understood by looking at an example. Suppose $$\begin{aligned}
x & = & 0 0 0 0 0 \, 0 0 0 \, 1 1 \, 1 ,\\
z & = & 0 0 0 0 0 \, 1 1 1 \, 0 0 \, 1 .\end{aligned}$$ Then $n=11, S(x)=k=3, S(z)=l=4, d(x,z)=l'=5$ and $\#G=5, \#N=3, \#E=2, \#F=1$. In general, we have a one-to-one correspondence between $(n,k,l,l')$ and $(\#G,\#N,\#E,\#F)$. Note that the formulas for $\#G,\#N,\#E,\#F$ above all give integers because $l, l'-k, l'+k$ always have the same parity.
\[const\_2d\] For each $x \in \{0,1\}^n$ the $t$-step transition probabilities $P^t(x,\cdot)$ are constant on the level sets $$\mathcal{X}(l,l',x) := \left\{ z \in \{0,1\}^n \, : \, Z_x(z)=(l,l') \right\}$$ for all $l,l' \in \{0,1,...,n\}$.
[**Proof.**]{} Let a permutation $\phi$ act on $\{0,1\}^n$ by mapping $x=(x_1,...,x_n)$ to $\phi(x)=(x_{\phi(1)},...,x_{\phi(n)})$. Fix $x \in \{0,1\}^n$. By symmetry, we get $$\begin{aligned}
P(x,y) & = & P\big( \phi(x),\phi(y)\big) , \\
P^t(x,y) & = & P^t\big(\phi(x),\phi(y)\big) , \\
P^t(x,A) & = & P^t\big(\phi(x),\phi(A)\big) , \end{aligned}$$ for all states $y$, all times $t$, and all subsets $A$ of $\{0,1\}^n$. Now fix $l,l' \in \{0,1,...,n\}$ and pick any $y,z \in \mathcal{X}(l,l',x)$. Then there exists a permutation $\phi$ that maps $y$ to $z$ and leaves $x$ fixed; $\phi$ maps $\{i : x_i = 0\}$ to itself and $\{i : x_i = 1\}$ to itself. Thus we get $$P^t(x,y) = P^t\big(\phi(x),\phi(y)\big) = P^t(x,z)$$ as desired. $\Box$
\[proj\_2d\] The projection $Z_t := Z_x(X_t) := \big(S(X_t), d(x,X_t)\big)$ is Markov and we get $$|| P^t(x,\cdot) - \pi || = || \mathcal{D}(_kZ_t) - \pi_{Z_x} || ,$$ where $x=X_0$ and $k=S(x)$. Here, $\mathcal{D}(_kZ_t)$ denotes the distribution of the chain $(Z_t)$ at time $t$ when started at $Z_0 = (k,0)$ and $\pi_{Z_x} := \pi Z_x^{-1}$ is the stationary distribution of the chain $(Z_t)$.
[**Proof.**]{} Fix $X_0 = x$ with $S(x)=k$ and consider the equivalence relation $\sim_x$ corresponding to the classes $\mathcal{X}(l,l',x)$, for $l,l' \in \{0,1,...,n\}$. That is, for $y,z \in \{0,1\}^n$ we have $$y \sim_x z \; \Leftrightarrow \; Z_x(y) = Z_x(z) .$$ Then the projection $(Z_t)$ of $(X_t)$ is a Markov chain, if $$\label{proj_markov}
P\big(y,\mathcal{X}(l,l',x)\big) = P\big(z,\mathcal{X}(l,l',x)\big)$$ for all $l,l' \in \{0,1,...,n\}$ and all $y,z \in \{0,1\}^n$ such that $y \sim_x z$. So fix any $y \sim_x z$ and any $l,l' \in \{0,1,...,n\}$. Then there exists a permutation $\phi$ of $1 \colon \!n$ such that $\phi(y)=z$ and $\phi(\mathcal{X}(l,l',x)) = \mathcal{X}(l,l',x)$. Consequently, we get $$P\big(y,\mathcal{X}(l,l',x)\big) = P\big(\phi(y),\phi(\mathcal{X}(l,l',x))\big) = P\big(z,\mathcal{X}(l,l',x)\big) ,$$ showing that $(Z_t)$ is Markov.
Total variation distance to stationarity remains unchanged under the projection $Z_x$ because both $P^t(x,\cdot)$ and $\pi$ are constant on sets $\mathcal{X}(l,l',x)$ for $l,l' \in \{0,1,...,n\}$ by Proposition \[const\_2d\]. That allows us to pull out the absolute values from the inner sum in the second equation below, because all the terms in the sum are equal: $$\begin{aligned}
|| P^t(x,\cdot) - \pi || & = & \frac{1}{2} \sum_l \sum_{l'} \sum_{z \in \mathcal{X}(l,l',x)} \left| P^t(x,z) - \pi(z) \right| \\
& = & \frac{1}{2} \sum_l \sum_{l'} \left| \sum_{z \in \mathcal{X}(l,l',x)} P^t(x,z) - \pi(z) \right| \\
& = & \frac{1}{2} \sum_l \sum_{l'} \left| P^t\big(x, Z_x^{-1}(l,l')\big) - \pi\big(Z_x^{-1}(l,l')\big) \right| \\
& = & || P^t(x,\cdot) Z_x^{-1} - \pi Z_x^{-1} || .\end{aligned}$$ Clearly, $P^t(x, \cdot) Z_x^{-1} = \mathcal{D}(_kZ_t)$, and the fact that $\pi_{Z_x} := \pi Z_x^{-1}$ is stationary for $(Z_t)$ is an elementary calculation. $\; \Box$
Reparametrization
-----------------
For any fixed $X_0=x$ with $S(x)=k$ the projected chain $(Z_t) = (Z_x(X_t))$ has the following transition kernel: $$\label{transition_kernel_2d}
P((l,l'),(h,h')) = \left\{ \begin{array}{l@{\quad:\quad}l}
\frac{2n - (l'+l+k)}{4n} \theta & h=l+1, h'=l'+1 , \\
\frac{l'+l-k}{4n} & h=l-1, h'=l'-1 , \\
\frac{k+l'-l}{4n} \theta & h=l+1, h'=l'-1 , \\
\frac{k-(l'-l)}{4n} & h=l-1, h'=l'+1 , \\
\frac{1}{2} + \frac{n-l}{2n} (1-\theta) & h=l, \;\;\;\;\;\; h'=l' , \\
0 & \mbox{otherwise} . \\
\end{array} \right.$$ This follows from (\[nG\]) together with the transition rule of the original chain $(X_t)$. For example, $$P((l,l'),(l+1,l'+1)) = P\{ \mbox{pick } i \in G \mbox{ and flip } 0 \rightarrow 1\} = \#G \frac{1}{2n} \theta .$$ For $k \leq \frac{n}{2}$ the state space of this chain is $$\begin{array}{ll@{\quad:\quad}l}
\{(l,l') \in \{0,1,...,n\}^2 \, : \, &
l' \in k + \{-l, -l+2, ..., l\} & \mbox{ for } l \leq k , \\
& l' \in k + \{l-2k, l-2k+2,...,l\} & \mbox{ for } k < l < n-k , \\
& l' \in n-k + \{l-n, l-n+2,...,n-l\} & \mbox{ for } n-k \leq l
\} .\\
\end{array}$$ A similar result holds for $k \geq \frac{n}{2}$. In both cases, after reparametrizing $$(l,l') \mapsto (l'-l, l'+l) =: (r,r') ,$$ the state space becomes $\{(r,r') \, : \, r \in \{-k,-k+2,...,k\}, r' \in \{k,k+2,...,2n-k\}$. The boundaries $-k \leq r \leq k$ and $k \leq r' \leq 2n-k$ here can also be confirmed like this: By definition we have $$\begin{aligned}
r & = & l'-l = N+E-(N+F) = E-F \;\;\;\;\;\; \mbox{ and} \\
r' & = & l'+l = N+E+(N+F) = 2N+k . \end{aligned}$$ Since $F \geq 0$, we get $r=E-F \leq E \leq E+F = k$. Also, since $E \geq 0$ and $F \leq k$, we get $r=E-F \geq -F \geq -k$. Similarly, since $N \leq n-k$, we get $r'=2N+k \leq 2(n-k)+k=2n-k$. And since $N \geq 0$, we get $r'=2N+k \geq k$.
The transition kernel in this new parametrization becomes $$\label{transition_kernel_2dr}
P((r,r'),(s,s')) = \left\{ \begin{array}{l@{\quad:\quad}l}
\frac{2n - (r'+k)}{4n} \theta & s=r, \;\;\;\;\;\;\, s'=r'+2 , \\
\frac{r'-k}{4n} & s=r, \;\;\;\;\;\;\, s'=r'-2 , \\
\frac{k+r}{4n} \theta & s=r-2, \, s'=r' , \\
\frac{k-r}{4n} & s=r+2, \, s'=r' , \\
\frac{1}{2} + \frac{2n-(r'-r)}{4n} (1-\theta) & s=r, \;\;\;\;\;\;\,\, s'=r' , \\
0 & \mbox{otherwise} .\\
\end{array} \right.$$ So the chain $(Z_t)$ can be viewed as a birth and death chain on a rectangle in $\mathbb{Z}^2$. A useful feature of the parametrization (\[transition\_kernel\_2dr\]) is that here the probability of moving up (down) in the $r$-dimension only depends on the current location in that dimension: it only depends on $r$, not on $r'$. Similarly, the probability of moving up (down) in the $r'$-dimension only depends on $r'$, not on $r$. Therefore, the problem of coupling two versions of this chain can be split up into coupling two one-dimensional processes. For this reason we will use the parametrization (\[transition\_kernel\_2dr\]) for the rest of the paper. Note that the chain $(Z_t)$ and its stationary distribution depend on the initial state $x$ (and its number of ones $k$) used for the projection.
Expected location of $(Z_t)$
----------------------------
We now calculate the expected location of the chain $(Z_t)$ after $t$ steps when started at $Z_0 = (r,r')$ . Similar to the one-dimensional projection $S(X_t)$ that we analyzed for the lower bound, this expectation can be calculated explicitly by induction on $t$, since the transition probabilities (\[transition\_kernel\_2dr\]) are all linear in the current location $(r,r')$ of the chain.
Fix $X_0=x \in \{0,1\}^n$ and let $k=S(x), Z_t=Z_x(X_t)=(S(X_t),d(x,X_t))$, so that $Z_0=(k,0)$. Denote with $Z_t = (Z_t^{(r)},Z_t^{(r')})$ the coordinates of the chain in the new parametrization (\[transition\_kernel\_2dr\]), so that $(Z_0^{(r)},Z_0^{(r')}) = (-k,k)$, and write $E_k$ for the expectation operator given this starting state. Then for this parametrization, $$Z_{t+1} - Z_t = \left\{ \begin{array}{l@{\quad:\quad}l}
(0,2) & \mbox{ with probability } \;\; \frac{2n - (k+Z_t^{(r')})}{4n} \, \theta , \\
(0,-2) & \mbox{ with probability } \;\; \frac{Z_t^{(r')}-k}{4n} , \\
(-2,0) & \mbox{ with probability } \;\; \frac{k+Z_t^{(r)}}{4n} \, \theta , \\
(2,0) & \mbox{ with probability } \;\; \frac{k-Z_t^{(r)}}{4n} , \\
(0,0) & \mbox{ otherwise } . \\
\end{array} \right.$$ Therefore, $$\begin{aligned}
\label{drift_function}
& & \hspace{-3em} E[Z_{t+1}-Z_t \, | \, Z_t] \nonumber\\
& = & \left( -2 \, \frac{k+Z_t^{(r)}}{4n} \, \theta + 2 \, \frac{k-Z_t^{(r)}}{4n} \, , \,
2 \, \frac{2n - (k+Z_t^{(r')})}{4n} \, \theta - 2 \frac{Z_t^{(r')}-k}{4n} \right) \nonumber \\
& = & \left( \frac{k(1-\theta)-Z_t^{(r)}(1+\theta)}{2n}, \frac{2n\theta + k(1-\theta)-Z_t^{(r')}(1+\theta)}{2n} \right) , \end{aligned}$$ so that $E[Z_{t+1} \, | \, Z_t]$ is equal to $$\left( \frac{k(1-\theta)+[2n - (1+\theta)] Z_t^{(r)}}{2n}, \frac{2n\theta + k(1-\theta) + [2n - (1+\theta)] Z_t^{(r')}}{2n} \right) .$$ By taking expectation, we get $$\begin{aligned}
\label{indstep2}
& & \hspace{-2em} E_k [Z_{t+1}] \nonumber \\
& = & \left( \frac{k(1-\theta)+[2n - (1+\theta)] E_k Z_t^{(r)}}{2n}, \frac{2n\theta + k(1-\theta) + [2n - (1+\theta)] E_k Z_t^{(r')}}{2n} \right) \; \nonumber \\
& =: & \left( \beta + \gamma \, E_k Z_t^{(r)}, \theta + \beta + \gamma \, E_k Z_t^{(r')} \right) .\end{aligned}$$ By induction on $t$, this leads to a proof of the following result:
\[exp\_2d\] Let $Z_t = Z_x(X_t) = \big(S(X_t),d(x,X_t)\big)$ be the two-dimensional projection of the lazy random walk Metropolis chain $(X_t)$ for $\pi(x) = \theta^{S(x)} (1+\theta)^{-n}$, started at $X_0 = x \in \{0,1\}^n$ with $S(x)=k$. Then, in the parametrization (\[transition\_kernel\_2dr\]) and for any $t \in \mathbb{N}$, we get $$E_k \left(Z_t^{(r)},Z_t^{(r')}\right) = \left( \frac{2n\beta}{1+\theta} (1-\gamma^t) - k \gamma^t \, , \,
\frac{2n (\theta + \beta)}{1+\theta} (1 - \gamma^t) + k \gamma^t \right),$$ where $\beta := \beta_{n,k,\theta} := \frac{k}{2n} (1-\theta)$ and $\gamma := \gamma_{n,\theta} := 1 - \frac{1+\theta}{2n}$.
[**Proof.**]{} The claim is true for $t=0$. Now suppose it is true for $t$. Then by (\[indstep2\]) we get that $E_k \left(Z_{t+1}^{(r)}, Z_{t+1}^{(r')}\right)$ is equal to $$\begin{aligned}
& & \left( \beta + \gamma \left[ \frac{2n\beta}{1+\theta} (1-\gamma^t) - k \gamma^t \right] \, , \, \theta + \beta + \gamma \left[ \frac{2n(\theta+\beta)}{1+\theta} (1-\gamma^t) + k \gamma^t \right] \right) \\
& = & \left( \beta + \frac{2n\beta}{1+\theta} \gamma - \frac{2n\beta}{1+\theta} \gamma^{t+1} - k \gamma^{t+1} \, , \,
\theta + \beta + \frac{2n(\theta+\beta)}{1+\theta} \gamma - \frac{2n(\theta+\beta)}{1+\theta} \gamma^{t+1} + k \gamma^{t+1} \right) \\
& = & \left( \frac{2n\beta}{1+\theta} \left[ \frac{1+\theta}{2n} + \gamma \right] - \frac{2n\beta}{1+\theta} \gamma^{t+1} - k \gamma^{t+1} \, , \,
\frac{2n(\theta+\beta)}{1+\theta} \left[ \frac{1+\theta}{2n} + \gamma \right] - \frac{2n(\theta+\beta)}{1+\theta} \gamma^{t+1} + k \gamma^{t+1} \right) \\
& = & \left( \frac{2n\beta}{1+\theta} (1-\gamma^{t+1}) - k \gamma^{t+1} \, , \,
\frac{2n(\theta+\beta)}{1+\theta} (1-\gamma^{t+1}) + k \gamma^{t+1} \right) .\end{aligned}$$ So the claim is also true for $t+1$. $\; \Box$
\[stat\_exp\_2dr\] For the expectation under stationarity, in the parametrization (\[transition\_kernel\_2dr\]) of our two-dimensional chain $(Z_t)$, we get $$E_{\pi} \left( Z^{(r)}, Z^{(r')} \right) = \left( \frac{2n\beta}{1+\theta}\, , \, \frac{2n(\theta+\beta)}{1+\theta} \right)
= \left( k \frac{1-\theta}{1+\theta} \, , \, \frac{2n\theta}{1+\theta} + k \frac{1-\theta}{1+\theta} \right) .$$
[**Proof.**]{} Since the Markov chain $(Z_t)$ is irreducible and aperiodic, it converges to its unique stationary distribution as $t$ goes to infinity for fixed $n,k$. Since the state space is finite, this convergence also holds for expectations. So by the Proposition, we get $$\begin{aligned}
E_{\pi} \left( Z^{(r)}, Z^{(r')} \right) & = & \lim_{t \rightarrow \infty} E_k \left( Z_t^{(r)},Z_t^{(r')} \right) \\
& = & \left( \frac{2n\beta}{1+\theta} \, , \, \frac{2n(\theta+\beta)}{1+\theta} \right) ,\end{aligned}$$ since for fixed $n,k$ we have $\gamma^t \rightarrow 0$ as $t$ goes to infinity. $\;\;\; \Box$
By reversing the linear transformation $(l,l') \mapsto (l'-l,l'+l)$, we immediately get $$E_{\pi} Z = \left( \frac{n\theta}{1+\theta} \, , \, \frac{n\theta + k(1-\theta)}{1+\theta} \right)$$ for the expectation under stationarity in the original parametrization (\[transition\_kernel\_2d\]) of our two-dimensional chain $(Z_t)$. For the first coordinate this confirms what we already know from $S(X) \sim \mbox{Binomial}(n, \frac{\theta}{1+\theta})$ under stationarity.
\[exp\_2dr\_general\] By exactly the same proof we get for a general starting state $(v,v')$ in the new parametrization (\[transition\_kernel\_2dr\]) $$E_{(v,v')} \left(Z_t^{(r)},Z_t^{(r')}\right) = \left( \frac{2n\beta}{1+\theta} (1-\gamma^t) + v \, \gamma^t \, , \,
\frac{2n (\theta + \beta)}{1+\theta} (1 - \gamma^t) + v' \gamma^t \right) .$$ Note that the expected location of $Z_t$ after $t$ steps is a convex combination of the starting state $Z_0=(v,v')$ and the stationary mean $E_{\pi_{Z_x}} Z$, with relative weights $\gamma^t$ and $1-\gamma^t$ respectively.
\[end\_stage\_one\] From Remark \[gamma\^u\] on page , we know that for $u := u_{n,\theta} := \frac{1}{1+\theta} n \log n$ we get $\gamma^u \sim n^{-1/2}$. Therefore, Proposition \[exp\_2d\] and Corollary \[stat\_exp\_2dr\] imply that for any starting state the expected location of the chain $(Z_t)$ after $u$ steps is within $O(\sqrt{n}\,)$ of the expected location of the chain under stationarity.
We now want to show that an additional $\alpha n$ number of steps is enough to couple (with high probability) two chains that are at distance $O(\sqrt{n}\,)$ of their stationary mean. This will follow from a corresponding result for simple random walk on $\mathbb{Z}^2$, since close to the stationary mean we are now in the “entropy regime” where the drift of the chain $(Z_t)$ is negligible.
Burn-in
-------
But first we show that by running the chain for an initial $\alpha n$ steps (burn-in period), we may assume that the number of ones in the state that is used in the two-dimensional projection is close to its stationary mean. Fix $\delta>0$ and let $p := \frac{\theta}{1+\theta}$. We will show that $$\label{burnin_1}
\max_{x \in \{0,1\}^n} || P^{\alpha n + t}(x,\cdot) - \pi || \leq \max_{y: S(y) \in n (p \pm \delta)} || P^{t}(y,\cdot) - \pi || + o(1) ,$$ where the $o(1)$ term goes to zero as $n$ goes to infinity (uniformly in $t$). To see this, we condition on where we are after the first $\alpha n$ steps: $$\begin{aligned}
& & || P^{\alpha n + t}(x,\cdot) - \pi || \\
& = & || \sum_{y} P^{\alpha n}(x,y) \left[ P^{t}(y,\cdot) - \pi \right] || \\
& \leq & \sum_{y: S(y) \in n(p \pm \delta)} P^{\alpha n}(x,y) \; || P^{t}(y,\cdot) - \pi || +
\sum_{y: S(y) \notin n(p \pm \delta)} P^{\alpha n}(x,y) \; || P^{t}(y,\cdot) - \pi || \\
& \leq & \max_{y: S(y) \in n(p \pm \delta)} || P^{t}(y,\cdot) - \pi || +
P_x \{S(X_{\alpha n}) \notin n(p \pm \delta)\} .\end{aligned}$$ The last term on the right hand side doesn’t depend on $t$ and is in fact in $o(1)$, uniformly in $x \in \mathcal{X}$, because when writing $S_{\alpha n} := S(X_{\alpha n})$ we get for large $\alpha$ $$\begin{aligned}
P_x \{S(X_{\alpha n}) \notin n(p \pm \delta)\} & = & P_x \left\{ | S_{\alpha n} - E_x S_{\alpha n} + E_x S_{\alpha n} - n p \, | \geq \delta n \right\} \\
& \leq & P_x \left\{ | S_{\alpha n} - E_x S_{\alpha n} | \geq \delta n - | E_x S_{\alpha n} - np \, | \right\} \\
& \leq & P_x \left\{ | S_{\alpha n} - E_x S_{\alpha n} | \geq n \left( \delta - \exp{\left\{-\frac{1+\theta}{2} \alpha\right\}} \right) \right\} \\
& \leq & 2 \exp{\left\{- \frac{2}{9} \frac{n^2 \left( \delta - \exp{\left\{-\frac{1+\theta}{2} \alpha \right\}} \right)^{\!2}}{{\mathop{\mathrm{O}}\bigl(n\bigr)}} \right\}} \\
& = & o(1) .\end{aligned}$$ Here the second inequality holds since we get from Proposition \[exp\_theta\] that $|E_x S_{\alpha n} - np| \leq n \gamma^{\alpha n} \leq n \exp{\{-\frac{1+\theta}{2} \alpha\}}$. The third inequality follows from our concentration result based on Azuma’s inequality (Proposition \[conc\_theta\]). This proves (\[burnin\_1\]). So after an initial $\alpha n$ steps, we may assume that the number of ones is within $\delta n$ of its stationary mean $np$.
Coupling
--------
Now we project down to our two-dimensional chain: Fix any $y \in \{0,1\}^n$ such that $k:=S(y) \in n(p \pm \delta)$. Writing the transition kernel in the parametrization (\[transition\_kernel\_2dr\]), we get from Corollary \[proj\_2d\] that $$\begin{aligned}
\label{burnin_1a}
|| P^{t}(y,\cdot) - \pi || & = & || \mathcal{D}(_{S(y)}Z_{t}) - \pi_{Z_y} || \nonumber \\
& = & || P^{t}((-k,k),\cdot) - \pi_{Z_y} || \nonumber \\
& \leq & \max_{(v,v')} || P^{t}\big((v,v'),\cdot\big) - \pi_{Z_y} || \\
& \leq & \max_{(v,v'),(w,w')} || P^{t}\big((v,v'),\cdot\big) - P^{t}\big((w,w'),\cdot\big) || \nonumber \\
& \leq & \max_{(v,v'),(w,w')} P_{(v,v'),(w,w')} \left\{ \tau > t \right\} \nonumber .\end{aligned}$$ The maxima here are over the entire state space of the two-dimensional chain $(v,v'),(w,w') \in \{-k,...,k\} \times \{k,...,2n-k\}$. As is well known, the second inequality above follows from stationarity of $\pi_{Z_y}$; see for instance [@LevinPeresWilmer2009 Lemma 4.11 on page 53]. The last inequality is the coupling inequality, where $\tau := \min \{j \geq 0 \, : \, Z_j = Y_j\}$ is the coupling time in the coupling $(Z_j,Y_j)$ that we are about to describe.
Fix any $(v,v'),(w,w') \in \{-k,...,k\} \times \{k,...,2n-k\}$ and set $Z_0 := (v,v')$ and $Y_0 := (w,w')$. Let $t=s+u$, where $s := \frac{1}{1+\theta} n \log n$ and $u := \alpha n$. Throughout all steps $j=1,2,...,s,...,t$ we use an (alternating) independence coupling. By this we mean that at each step we flip a fair coin to decide which chain to move according to the non-lazy version of its transition kernel. The other chain stays at its current location. Here, if $P$ is the transition probability matrix (\[transition\_kernel\_2dr\]) of the chain $(Z_j)$, then $P' := 2P-I$ is its non-lazy version, where $I$ is the identity. However, if $Z_j$ and $Y_j$ ever agree in the $r$ (or $r'$) coordinate, we modify the coupling so that they agree in that coordinate forever after. This is possible since the probability of moving up (or down) in the $r$-coordinate does not depend on the current location in the $r'$-coordinate. Similarly, the probability of moving up (or down) in the $r'$-coordinate does not depend on the current location in the $r$-coordinate. This is easily seen from the transition kernel (\[transition\_kernel\_2dr\]).
We could implement this change as follows: Suppose $Y_j^{(r)}=Z_j^{(r)}$. Flip fair coin number one; if it comes up heads, try to move $Y_j$ according to its non-lazy transition rule. If that would result in $Y_j$ moving up (or down) in the $r$-coordinate, flip another fair coin. If it comes up heads, move $Y_j^{(r)}$ accordingly and move $Z_j^{(r)}$ in the same way; if it comes up tails, reject the move. If fair coin number one comes up tails, try to move $Z_j$ according to its non-lazy transition rule. If that would result in $Z_j$ moving up (or down) in the $r$-coordinate, flip another fair coin. If it comes up heads, move $Z_j^{(r)}$ accordingly and move $Y_j^{(r)}$ in the same way; if it comes up tails, reject the move. Similarly for $Y_j^{(r')}=Z_j^{(r')}$.
For steps $j=1,...,s$ we exploit the drift of both chains towards their stationary mean and use our calculation of the expected location of the two chains after $s$ steps. From Remark \[end\_stage\_one\] we know that at time $s$ the two chains will be within ${\mathop{\mathrm{O}}\bigl(\sqrt{n}\,\bigr)}$ of their stationary means in both coordinates in expectation. For steps $j=s+1,...,t$, the drift is likely to be weak, so we can compare our two chains to two simple random walks on $\mathbb{Z}^2$. We want to show that in additional $\alpha n$ steps the two chains will coalesce with high probability. In [@Barta2012] we construct an explicit coupling of $(Z_j,Y_j)$ with two lazy simple random walks $(V_j,W_j)$ to achieve this. Here we use a more abstract result due to Levin, Peres and Wilmer [@LevinPeresWilmer2009 Proposition 17.20 on page 240] that is based on martingale theory.
\[superMG\] Let $(R_j)_{j \geq 0}$ be a non-negative supermartingale, adapted to the sequence $(W_j)$, and let $\tau$ be a stopping time for the sequence $(W_j)$. Suppose that
(i) $R_0 = k$ ,
(ii) $R_{j+1} - R_j \leq B$ ,
(iii) there exists a constant $\sigma^2 > 0$ such that ${\rm Var}(R_{j+1} \, | \, W_0,W_1,...,W_j) \geq \sigma^2$ on the event $\{\tau > j\}$.
If $u > 12B^2/\sigma^2$, then $$P_k \{\tau > u\} \leq \frac{4 k}{\sigma \sqrt{u}} .$$
We will apply this result to the two difference processes formed by $$R_j := Z_j^{(r)} - Y_j^{(r)} \;\;\; \mbox{ and } \;\;\; R_j' := Z_j^{(r')} - Y_j^{(r')} .$$ It will later be convenient to restart time at $s$, so let $$\begin{aligned}
\tau_r := \inf \{j \geq 0 : Z_{s+j}^{(r)} = Y_{s+j}^{(r)} \} , \\
\tau_{r'} := \inf \{j \geq 0 : Z_{s+j}^{(r')} = Y_{s+j}^{(r')} \} , \end{aligned}$$ so that on the event $\{\tau>s\}$, the coupling time is $$\label{tau_tau_r_tau_r_prime}
\tau := \inf \{i \geq 0 : Z_i = Y_i \} = s + (\tau_r \vee \tau_{r'}) \leq s + \tau_r + \tau_{r'} .$$ To show that the assumptions of Proposition \[superMG\] are satisfied by $(R_j)$ and $\tau_r$, starting at time $j=s$, we use the sequence $W_j := (Z_j,Y_j)$ and write $\mathcal{F}_j$ for the sigma algebra generated by $(W_i : i=0,1,...,j)$. First note that, without loss of generality, we may assume that $R_s \geq 0$. If not, simply replace $R_j$ by $-R_j$. By construction of our coupling, it then follows that $R_j \geq 0$ for all $j \geq 0$. Furthermore, the increments of $(R_j)$ are bounded above by $B := 2$, $$\begin{aligned}
R_{j+1} - R_j & = & \left( Z_{j+1}^{(r)} - Z_j^{(r)} \right) - \left( Y_{j+1}^{(r)} - Y_j^{(r)} \right) \\
& \leq & 2 ,\end{aligned}$$ since in our coupling only one of the two chains $(Z_j)$ and $(Y_j)$ can move at each step, and their (absolute) step sizes are both bounded by two.
To see that $(R_j)$ is a supermartingale, let $$D(Z_j^{(r)}) := E [Z_{j+1}^{(r)} - Z_j^{(r)} \; | \; Z_j]$$ be the drift of the coordinate $Z_j^{(r)}$ at time $j$, and similarly for $Y_j^{(r)}$. Then $$E[R_{j+1} - R_j \; | \; \mathcal{F}_j] = D(Z_j^{(r)}) - D(Y_j^{(r)}) \leq 0 .$$ This follows, since by (\[drift\_function\]), the drift function $D(\cdot)$ is decreasing (non-increasing), and we have $Z_j^{(r)} \geq Y_j^{(r)}$ by assumption. It remains to show that the conditional one-step variances of $(R_j)$ are bounded away from zero. Since $A := R_{j+1} - R_j \in \{-2,0,2\}$, we have to lower bound $$\begin{aligned}
\label{var_bound_R}
& & \hspace{-3em} \frac{1}{4} {\rm Var} [R_{j+1} \; | \; \mathcal{F}_j] \nonumber \\
& = & \frac{1}{4} {\rm Var} [R_{j+1} - R_j \; | \; \mathcal{F}_j] \\
& = & \left[ \tilde{P}(A=2) + \tilde{P}(A=-2) \right] - \left[ \tilde{P}(A=2) - \tilde{P}(A=-2) \right]^2 \nonumber \\
& = & \tilde{P}(A=2) \tilde{P}(A \neq 2) + \tilde{P}(A=-2) \tilde{P}(A \neq -2) + 2 \tilde{P}(A=2) \tilde{P}(A=-2) . \nonumber\end{aligned}$$ Here we write $\tilde{P} := P(\cdot \; | \; \mathcal{F}_j)$. From the transition probabilities (\[transition\_kernel\_2dr\]) we get $$\tilde{P}(A=2) = \frac{1}{4n} \left[\left( k-Z_j^{(r)} \right) + \left( k + Y_j^{(r)} \right) \theta \right] ,$$ and $$\tilde{P}(A=-2) = \frac{1}{4n} \left[\left( k+Z_j^{(r)} \right) \theta + \left( k - Y_j^{(r)} \right) \right] .$$ From the burn-in (\[burnin\_1\]), we may assume $k \in n(p \pm \delta)$, where $\delta > 0$ is a small constant. Since $Z_j^{(r)}, Y_j^{(r)} \in [-k,k]$, this implies that $\tilde{P}(A=2)$ and $\tilde{P}(A=-2)$ are both bounded above by $\frac{k}{n} \frac{1+\theta}{2} \leq p+\delta$, and therefore $\tilde{P}(A \neq 2) \geq 1 - p - \delta$ and $\tilde{P}(A \neq -2) \geq 1 - p - \delta$. Consequently, the right hand side in (\[var\_bound\_R\]) is bounded below by $$\left( 1-p-\delta \right) \, \left( \tilde{P}(A=2) + \tilde{P}(A=-2) \right) .$$ Since we have the bound $$\begin{aligned}
\tilde{P}(A=2) + \tilde{P}(A=-2) & = &
\frac{1}{4n} \left[ 2k(1+\theta) - (1-\theta) \left( Z_j^{(r)} + Y_j^{(r)} \right) \right] \\
& \geq & \frac{1}{4n} \left[ 2k(1+\theta) + 2k(1-\theta) \right] \\
& \geq & \frac{k}{n} \theta \\
& \geq & (p - \delta) \theta ,\end{aligned}$$ we get $${\rm Var}(R_{j+1} \; | \; \mathcal{F}_j) \geq \sigma^2 > 0 ,$$ for $\sigma^2 := 4 (1-p-\delta) (p-\delta) \theta$, as required.
Analogously it can be shown that $(R_j')$ and $\tau_{r'}$ also satisfies the assumptions of Proposition \[superMG\] with the same $\sigma^2$. Since a modification of this argument will be needed in section \[sec\_theta\_n\], we briefly sketch it here. Write $A' := R_{j+1}' - R_j'$, so we have to bound $$\tilde{P}(A'=2) = \frac{1}{4n} \left[ \left( 2n - (Z_j^{r'}+k) \right) \theta + Y_j^{r'} - k \right]$$ and $$\tilde{P}(A'=2) = \frac{1}{4n} \left[ \left( 2n - (Y_j^{r'}+k) \right) \theta + Z_j^{r'} - k \right] .$$ Since $Z_j^{r'}, Y_j^{r'}$ always lie in $[k,2n-k]$, we get that $\tilde{P}(A'=2)$ and $\tilde{P}(A'=-2)$ are both bounded above by $\frac{n-k}{n} \frac{1+\theta}{2} \leq 1-(p-\delta)$. For the lower bound, we get $$\begin{aligned}
\tilde{P}(A'=2) + \tilde{P}(A'=-2)
& = & \frac{1}{4n} \left[ (Z_j^{r'} + Y_j^{r'})(1-\theta) - 2k(1+\theta) + 4n \theta \right] \\
& \geq & \frac{1}{4n} \left[ 2k(1-\theta) - 2k(1+\theta) + 4n \theta \right] \\
& = & \frac{n-k}{n} \theta \\
& \geq & (1 - p -\delta) \theta .\end{aligned}$$ In the same way as above for $(R_j)$, this shows that ${\rm Var}(R_{j+1}' | \mathcal{F}_j) \geq \sigma^2 := 4(1-p-\delta)(p-\delta)\theta > 0$ in this case as well.
Applying Proposition \[superMG\], we get for $u > 48/\sigma^2$, $$\begin{aligned}
\label{entropy_result}
P_{Z_s,Y_s} \{ \tau_r > u \} & \leq & \frac{4 |R_s|}{\sigma \sqrt{u}} \;\;\; \mbox{ and} \\
P_{Z_s,Y_s} \{ \tau_{r'} > u \} & \leq & \frac{4 |R_s'|}{\sigma \sqrt{u}} . \nonumber\end{aligned}$$
Now we put things together. By conditioning on where we are after the first $s$ steps (the drift regime of our coupling), we get $$\begin{aligned}
P_{(v,v'),(w,w')} \{ \tau > s + u \, | \, Z_s, Y_s\} & = & \mathbf{1}\{\tau > s\} P_{(v,v'),(w,w')} \{ \tau > u \; | \; Z_s,Y_s\} \\
& \leq & P_{Z_s,Y_s} \{ \tau_r + \tau_{r'} > u \} \\
& \leq & P_{Z_s,Y_s} \{\tau_r > u/2\} + P_{Z_s,Y_s} \{\tau_{r'} > u/2\} \\
& \leq & \frac{4 \sqrt{2}}{\sigma} \; \frac{|R_s| + |R_s'|}{\sqrt{u}} .\end{aligned}$$ Here the first inequality comes from (\[tau\_tau\_r\_tau\_r\_prime\]), where we restart time at $s$. The last inequality is (\[entropy\_result\]). By taking expectation, we get for some positive constants $C,C'$ and large $n$ that $$\begin{aligned}
P_{(v,v'),(w,w')} \{ \tau > s+u \} & \leq &
\frac{4 \sqrt{2}}{\sigma \sqrt{u}} \; \left[ E_{(v,v'),(w,w')} |R_s| + E_{(v,v'),(w,w')} |R_s'| \right] \\
& = & \frac{4 \sqrt{2}}{\sigma \sqrt{u}} \; \left[ |E_{(v,v'),(w,w')} R_s| + |E_{(v,v'),(w,w')} R_s'| \right] \\
& \leq & \frac{4 \sqrt{2}}{\sigma} \; \frac{C \sqrt{n} + C' \sqrt{n}}{\sqrt{\alpha n}} .\end{aligned}$$ For the equality above, note that we can move the absolute values outside the expectations, since in our coupling the sign of $R_j$ (respectively $R_j'$) can never switch between plus and minus. The second inequality above comes from Remark \[end\_stage\_one\]. Combining this with (\[burnin\_1\]) and (\[burnin\_1a\]), we get $$\begin{aligned}
d(t) & := & \max_x ||P^t(x,\cdot) - \pi|| \\
& \leq & \max_{(v,v'),(w,w')} P_{(v,v'),(w,w')} \{\tau > t\} + {\mathop{\mathrm{o}}\bigl(1\bigr)} \\
& \leq & \frac{4 \sqrt{2} (C+C')}{\sigma \sqrt{\alpha}} + {\mathop{\mathrm{o}}\bigl(1\bigr)} ,\end{aligned}$$ which implies $$\lim_{\alpha \rightarrow \infty} \limsup_{n \rightarrow \infty} d(t) = 0 .$$ This finishes the proof of the upper bound part of Theorem \[theta\_model\]. $\Box$
Metropolis vs. Gibbs {#RWM_GS}
=====================
Since the coordinates are independent under the distribution $\pi(x) = \theta^{S(x)} (1+\theta)^{-n}$ on $\mathcal{X}=\{0,1\}^n$, where $\theta \in (0,1]$, there exists a very close connection between the random walk Metropolis algorithm for $\pi$, that we studied above, and the Gibbs sampler for the distribution $\pi$. The (random scan) Gibbs sampler for the distribution $\pi$, also known as Glauber dynamics in statistical physics, is also a Markov chain $(X_t)$ on $\mathcal{X}$ that converges to $\pi$ as the number of steps $t$ goes to infinity. It evolves as follows: Given we are at $X_t=x$, draw a coordinate $i \in [n]$ uniformly at random. Then set $X_{t+1}^{(j)} := X_t^{(j)}$ for all $j \neq i$, and set $X_{t+1}^{(i)} := B$, where $B \in \{0,1\}$ is a draw from the conditional distribution of $\pi$ for coordinate $i$, given the values of $x$ at all other coordinates $j \neq i$. Since the coordinates are all i.i.d. Ber$(p)$ under $\pi$, where $p=\frac{\theta}{1+\theta}$, the random variable $B$ above follows a Ber$(p)$ distribution and is independent of $X_t$.
Let $\mathbb{P}$ be the transition probability matrix of the (non-lazy) random walk Metropolis algorithm for $\pi$. Similarly, let $\mathbb{Q}$ be the transition probability matrix of the (non-lazy) Gibbs sampler for $\pi$. Fix any $q \in [0,1]$. Then the *lazy(q) random walk Metropolis algorithm for $\pi$* refers to the Markov chain on $\mathcal{X}$ with transition probability matrix $(1-q) \mathbb{P} + q I$. Similarly, the *lazy($q$) Gibbs sampler for $\pi$* refers to the Markov chain on $\mathcal{X}$ with transition probability matrix $(1-q) \mathbb{Q} + q I$. Here $I$ is the (appropriately sized) identity matrix. We refer to $q$ as the laziness-factor of the chain in question. When no value $q$ is specified, a lazy chain is understood to be lazy$(1/2)$.
The lazy$(q)$ random walk Metropolis algorithm for $\pi$ simply refers to the Markov chain on $\mathcal{X}$, where at each step with probability $q$ we stay where we are, and with probability $1-q$ we make one step according to the (non-lazy) random walk Metropolis algorithm. Similarly for the Gibbs sampler. For example, the lazy random walk Metropolis algorithm studied before corresponds to the lazy($1/2$) random walk Metropolis algorithm, whereas the lazy($0$) Gibbs sampler refers to the non-lazy Gibbs sampler for $\pi$.
With this definition, we get the following relationship between the random walk Metropolis algorithm and the Gibbs sampler for $\pi$.
\[Metropolis\_Gibbs\] Fix any $\theta \in (0,1]$ and consider the distribution $\pi=\pi_{\theta}$ on $\mathcal{X}=\{0,1\}^n$ defined by $\pi(x) = \theta^{S(x)} (1+\theta)^{-n}$. Then the lazy $\!(1/2)$ random walk Metropolis algorithm for $\pi$ follows the same law as the lazy $\!\!\big((1-\theta)/2\big)$ Gibbs sampler for $\pi$.
For $\theta = 1$ this shows that the lazy random walk Metropolis algorithm for the uniform distribution on the hypercube is equal to the non-lazy Gibbs sampler for the uniform distribution. If we say that a non-lazy chain “runs twice as fast” as its lazy ($q=1/2$) version, then for $\theta=1$ the lazy random walk Metropolis algorithm “runs twice as fast” as the lazy Gibbs sampler. The above Proposition shows that this “speed advantage” of the Metropolis algorithm disappears as $\theta \downarrow 0$. The reason for this is that the random walk proposal in the Metropolis algorithm is less and less well adapted to $\pi$ as $\theta$ goes down to zero.
**Proof of Proposition \[Metropolis\_Gibbs\].** Fix any $\theta \in (0,1]$. The lazy($\frac{1}{2}$) random walk Metropolis algorithm for $\pi$ has the following transition kernel: $$\label{transition_kernel_lRW-MH-theta}
\mathbb{P}(x,y) = \left\{ \begin{array}{l@{\quad:\quad}l}
\frac{1}{2n} & x \sim y, S(x) > S(y) , \\
\frac{1}{2n} \theta & x \sim y, S(x) < S(y) , \\
\frac{1}{2} + \frac{n-S(x)}{2n} (1-\theta) & x = y , \\
0 & \mbox{otherwise} .\\
\end{array} \right.$$ As noted before, it’s easy to see that we can run the corresponding chain $(X_t)$ in the following way. Given we are at state $X_t$ at time $t$:
- Pick a coordinate $i \in [n]$ uniformly at random, independent of all previous choices.
- Draw $U_t \sim$ Uniform$[0,1]$, independent of all previous choices.
- Set $X_{t+1}^{(j)} := X_t^{(j)}$ for $j \ne i$, and set the $i^{th}$ coordinate of $X_{t+1}$ to
$$X_{t+1}^{(i)} := \left\{ \begin{array}{l@{\quad:\quad}l}
1 & 0 \leq U_t \leq \frac{\theta}{2} , \\
X_t^{(i)} & \frac{\theta}{2} < U_t \leq \frac{1}{2} , \\
0 & \frac{1}{2} < U_t \leq 1 .\\
\end{array} \right.$$ However, this corresponds exactly to the transition rule of the lazy$(\frac{1-\theta}{2})$ Gibbs sampler: If $U_t \in (\frac{\theta}{2},\frac{1}{2}]$, that is, with probability $(1-\theta)/2$, we stay where we are; and if $U_t \notin (\frac{\theta}{2},\frac{1}{2}]$, that is, with probability $1 - (1-\theta)/2 = (1+\theta)/2$, we pick one of the $n$ coordinates uniformly at random and replace it with an independent Bernoulli$(\frac{\theta}{1+\theta})$ random variable, since we get $P(U_t \leq \theta/2 \, | \, U_t \notin (\frac{\theta}{2},\frac{1}{2}]) = \frac{\theta}{1+\theta}$. $\Box$
Edge probabilities varying with $n$ {#sec_theta_n}
===================================
So far we considered the model $\pi(x) = \theta^{S(x)} (1+\theta)^{-n}$ where $\theta \in (0,1]$ is fixed as $n$ goes to infinity. In this section we consider the case where $\theta = \theta_n$ is allowed to vary with $n$. As mentioned in [@DiaconisSaloff-Coste2006 page 2118], the methods and results of [@DiaconisRam2000] imply that the non-lazy version of the Metropolis algorithm for this model has cutoff at $(1+\theta_n)^{-1} n \min\{ \log n, \log \sqrt{n / \theta_n}\}$ with window size $n$. By the same argument as for Theorem \[theta\_model\], this implies the following result for the lazy version:
\[theta\_n\_model\] The lazy random walk Metropolis chain for $\pi(x) = \theta_n^{S(x)} (1+\theta_n)^{-n}$ on $\{0,1\}^n$, where $\theta_n \in (0,1]$, has cutoff at $\frac{2}{1+\theta} n \min\{ \log n, \log \sqrt{n / \theta_n}\}$ with a window of size $n$.
Let $n := {\nu \choose 2}$ and let $\pi(x) = p_n^{S(x)} (1-p_n)^{n-S(x)}$ be the Erdős-Rényi random graph model on $\nu$ vertices with parameter $p_n \in (0,1)$. The lazy random walk Metropolis chain for this model has cutoff at $$2 \mspace{0.5mu} \rho \mspace{1.5mu} n \min\left\{\log n, \log \sqrt{\frac{n \rho}{1-\rho}}\right\}$$ with a window of size $n$, where $\rho := \max \{p, 1-p\}$.
By absorbing $- (1+\theta)^{-1} n \log \theta$ into the window size term, Theorem \[theta\_n\_model\] is seen to be consistent with the result from Theorem \[theta\_model\] for constant $\theta$. We note that the expression for the mixing time changes when $\theta_n \leq 1/n$, corresponding to $p_n := \theta_n/(1+\theta_n) \leq 1/(n+1)$, which is right about where the edge probability $p_n$ passes the critical threshold of $1/n$ for the existence of a giant component in the Erdős-Rényi random graph.
**Proof of Theorem \[theta\_n\_model\].** In the remainder of this section we will extend our proof of Theorem \[theta\_model\] to the case where $\theta$ is allowed to vary with $n$, and prove Theorem \[theta\_n\_model\]. For the lower bound part of this cutoff result, we can follow the proof given in section \[sec\_lower\_bound\_1\]. Propositions \[exp\_theta\] and \[coupon\_collector\] including their proofs stay true as stated when $\theta_n$ is allowed to vary with $n$. After that, use the bound $${\rm Var}_{\underline{1}} S_t \leq \frac{n}{(1+\theta_n)^2} \left[ \gamma^t + \theta_n \right] ,$$ which leads to the bound $$\sigma^2 := 1/2 \left( {\rm Var}_{\underline{1}} S_t + {\rm Var}_{\pi_S} S \right) \leq
\frac{n}{(1+\theta_n)^2} \left[ \gamma^t + \theta_n \right] .$$ Then we get $$\begin{aligned}
|E_{\underline{1}} S_t - E_{\pi_S} S| & = & \frac{n}{1+\theta_n} \gamma^t \nonumber \\
& \geq & \sigma \frac{\sqrt{n} \gamma^t}{\sqrt{\gamma^t + \theta_n}} . \label{expectation_difference_theta_n}\end{aligned}$$ For $t := \frac{2}{1+\theta_n} n \min\{ \log n, \log \sqrt{n/\theta_n}\} - \alpha n$ and $\gamma := 1 - \frac{1+\theta_n}{2n}$ as before, we get $$\gamma^t \sim \max\{ 1/n, \sqrt{\theta_n/n} \} \; e^{\frac{1+\theta_n}{2} \alpha} ,$$ which implies $$\lim_{\alpha \rightarrow \infty} \liminf_{n \rightarrow \infty} \left( \frac{\sqrt{n} \gamma^t}{\sqrt{\gamma^t + \theta_n}} \right)^{\!2} = \infty .$$ The lower bound result now follows from (\[expectation\_difference\_theta\_n\]) and Proposition \[dist\_stat\].
For the upper bound part of the cutoff result, we distinguish two cases, according to which term attains the minimum in the expression for the mixing time. For $\theta_n \leq 1/n$, it is enough to consider a coordinate wise coupling: To update two copies $(X_t), (Y_t)$ of the chain, use the construction given before Proposition \[coupon\_collector\] to run the chains, but update *the same* coordinate $i \in [n]$ and use *the same* uniform random variable $U_t$ *for both* $(X_t)$ and $(Y_t)$. If $R_t$ is the number of coordinates not refreshed by time $t$, as defined before Proposition \[coupon\_collector\], then for for $t := \frac{2}{1+\theta_n} n \log n + \alpha n$ we get $$\begin{aligned}
d(t) & = & \max_{x} || P^t(x,\cdot) - \pi || \\
& \leq & \max_{x,y} || P^t(x,\cdot) - P^t(y,\cdot) || \\
& \leq & \max_{x,y} P_{x,y}(X_t \neq Y_t) \\
& \leq & P (R_t \geq 1) \\
& \leq & E R_t \\
& = & n \gamma^t \\
& \sim & e^{-\frac{1+\theta_n}{2} \alpha} , \end{aligned}$$ which proves that $\lim_{\alpha \rightarrow \infty} \limsup_{n \rightarrow \infty} d(t) = 0$, as required.
For $\theta_n > 1/n$, this coordinate wise coupling is generally not sharp enough to establish cutoff. To see this, note that for $t=\frac{2}{1+\theta_n} n \log \sqrt{n/\theta_n} + \alpha n$ we get $n \gamma^t \sim \sqrt{n \theta_n} e^{-\alpha (1+\theta_n)/2}$, which grows without bound unless $\theta_n \in {\mathop{\mathrm{O}}\bigl(1/n\bigr)}$. However, a slight modification of our proof for constant $\theta$ works in this case. With the exception of Remark \[end\_stage\_one\], everything in this proof stays true until the application of Proposition \[superMG\]. Restarting time after a burn-in of $\alpha n$ steps, we may assume that the number of ones of the state $X_0$ that we use for the two dimensional projection satisfies $k := S(X_0) \in n (p_n \pm \delta)$, for some small constant $\delta > 0$. Again, we write $p_n := \theta_n/(1+\theta_n)$. Note that $p_n-\delta$ can be negative in this case. The two difference processes formed by $$R_j := Z_j^{(r)} - Y_j^{(r)} \;\;\; \mbox{ and } \;\;\; R_j' := Z_j^{(r')} - Y_j^{(r')}$$ still satisfy all the assumptions of Proposition \[superMG\], except that the conditional one step variances can no longer be bounded away from zero. Therefore, here we modify the argument given for constant $\theta$, treating the $r$ and $r'\!$-coordinates separately.
The width of the domain $[k,2n-k]$ of the $r'\!$-coordinate of our chain $(Z_t)=\left(Z_t^{(r)}, Z_t^{(r')}\right)$ is of order $n$, as before, and we can still apply Proposition \[superMG\]. To bound the conditional one step variances, using the same notation as above, we get that $\tilde{P}(A'=2)$ and $\tilde{P}(A'=-2)$ are both bounded above by $\frac{n-k}{n} \frac{1+\theta}{2} \leq 1-\delta'$, for some constant $\delta'>0$ that doesn’t depend on $n$. To see this inequality, first note that both terms on the left hand side are in $[0,1]$. Let $\delta'' := 2\delta/(1-\delta)$, where $\delta>0$ is the constant we used in the burn-in to ensure that the number of ones $k$ of the state used in the two dimensional projection satisfies $k \in n(p \pm \delta)$. If $\theta_n < \delta''$, we get $\frac{1+\theta_n}{2} < \frac{1+\delta''}{2}$, so we can pick $\delta' := (1-\delta'')/2$. If $\theta_n \geq \delta''$, we get $p_n - \delta \geq \frac{\delta''}{1+\delta''} - \delta > 0$, so here we can pick $\delta' := \frac{\delta''}{1+\delta''} - \delta > 0$. For the lower bound, we get as before $$\tilde{P}(A'=2) + \tilde{P}(A'=-2) \geq (1 - p - \delta) \theta_n .$$ That means our variance bound ${\rm Var}(R_{j+1}' | \mathcal{F}_j) \geq \sigma^2_n := 4 \delta' (1-p-\delta) \theta_n$ depends on $n$ and can’t be bounded away from zero, as before. However, for some constant $C>0$ and large $\alpha$ we still get $u := \alpha n > C/\theta_n = 12 B^2 / \sigma_n^2$, as required, since we have $\theta_n > 1/n$. Applying Proposition \[superMG\] to $(R_j')$ and $\tau_{r'}$ entails $$\label{r'_mod1}
P_{Z_s,Y_s} \{ \tau_{r'} > u/2 \} \leq \frac{4 |R_s'|}{\sigma_n \sqrt{u/2}} .$$ Furthermore, for $s := 2/(1+\theta_n) n \log \sqrt{n/\theta_n}$ we get $n \gamma^s \sim \sqrt{n \theta_n}$, and therefore, for some constant $c>0$, $$\label{r'_mod2}
\frac{|E R_s'|}{\sigma_n \sqrt{u}} \leq \frac{n \gamma^s}{\sqrt{c \theta_n \alpha n}} \sim \frac{1}{\sqrt{\alpha c}} ,$$ as required in this context.
On the other hand, the width of the domain $[-k,k]$ for the $r$-coordinate of our chain $(Z_t)$ is not necessarily of order $n$, as it was for fixed $\theta$. As before, for $(R_j)$ and $\tau_r$ we get that $\tilde{P}(A=2)$ and $\tilde{P}(A+-2)$ are both upper bounded by $\frac{k}{n} \frac{1+\theta_n}{2} \leq p + \delta$, and further $$\tilde{P}(A=2) + \tilde{P}(A=-2) \geq \frac{k}{n} \theta_n .$$ Therefore, we get the variance bound $${\rm Var} \left[ R_{j+1} | \mathcal{F}_j \right] \geq \sigma_n^2 := 4 (1-p_n-\delta) \frac{k}{n} \theta_n .$$ This violates the assumption $\alpha n =: u > 12 B^2/\sigma_n^2$ of Proposition \[superMG\]. Therefore, we use here a slightly modified version of Proposition \[superMG\], where we replace the last sentence in its statement by the following one: If $h \geq 2B$ then for any $u>0$ we get $$P_k\{\tau > u\} \leq \frac{k}{h} + \frac{3kh}{u \sigma^2} .$$ This version directly follows from the proof or the original version as given by Levin, Peres and Wilmer [@LevinPeresWilmer2009 page 346]. Minimizing the right hand side above by choosing $h:=\sqrt{u \sigma^2/3}$ leads to the original version of Proposition \[superMG\], which we can’t apply in our context. Instead, here we choose $h:=\sqrt{\alpha n \theta_n}$. Since $\theta_n>1/n$, for large $\alpha$ we get $h \geq 2B = 4$, as required. Applying this modified version of Proposition \[superMG\] to $(R_j)$ and $\tau_r$, we get $$\begin{aligned}
P_{Z_s,Y_s} \{\tau_r > u/2\} & \leq & \frac{|R_s|}{h} + \frac{3|R_s|h}{\sigma^2_n u/2} \\
& \leq & \frac{|R_s|}{\sqrt{\alpha n \theta_n}} + \frac{6|R_s|\sqrt{\alpha n \theta_n}}{\alpha n \theta_n c k/n} \\
& = & \frac{|R_s|}{\sqrt{\alpha n \theta_n}} \left[ 1 + \frac{6n}{ck} \right] .\end{aligned}$$ Here, we used $u:=\alpha n$ and $\sigma_n^2 \geq c \theta_n k/n$ for some constant $c>0$. By setting $s:=2/(1+\theta_n) n \log \sqrt{n/\theta_n}$ and taking expectation, this leads to the bound $$\begin{aligned}
\frac{E_{(v,v'),(w,w')} |R_s|}{\sqrt{\alpha n \theta_n}} \left[ 1 + \frac{6n}{ck} \right] & = &
\frac{|E_{(v,v'),(w,w')} R_s|}{\sqrt{\alpha n \theta_n}} \left[ 1 + \frac{6n}{ck} \right] \\
& \leq & \frac{2k\gamma^s}{\sqrt{\alpha n \theta_n}} \left[ 1 + \frac{6n}{ck} \right] \\
& \sim & \frac{2\sqrt{n\theta_n} k/n}{\sqrt{\alpha n \theta_n}} \left[ 1 + \frac{6n}{ck} \right] \\
& \leq & \frac{2 + 12/c}{\sqrt{\alpha}} .\end{aligned}$$ Here, the first equality follows by the same argument as in the case for constant $\theta$. The first inequality follows from Remark \[exp\_2dr\_general\], using the fact that the $r$-coordinates live in the domain $[-k,k]$. The asymptotic equality comes from $n \gamma^s \sim \sqrt{n \theta_n}$. In the same way as for constant $\theta$, we can now combine this result for $(R_j)$ and $\tau_r$ with the corresponding results (\[r’\_mod1\]) and (\[r’\_mod2\]) for $(R_j')$ and $\tau_{r'}$ to establish the upper bound part of the cutoff result. This finishes the proof. $\Box$
Acknowledgments: {#acknowledgments .unnumbered}
================
I would like to thank my PhD advisor Steven Lalley for his invaluable guidance and many helpful discussions and suggestions. I also would like to thank Michael Wichura for a very careful reading of an earlier version of this paper that lead to significant improvements.
[99]{} D. Aldous, Random walks on finite groups and rapidly mixing Markov chains, Springer Lecture Notes Math. 986 (1983) 243-297.
W. Barta, Random walk Metropolis chains on the hypercube, PhD thesis, Department of Statistics, University of Chicago (2012).
P. Diaconis, The cutoff phenomenon in finite Markov chains, Proc. Natl. Acad. Sci. USA 93(Feb) (1996) 1659-1664.
P. Diaconis, R. L. Graham, J. A. Morrison, Asymptotic analysis of a random walk on a hypercube with many dimensions, Rand. Struct. Alg. 1(1) (1990) 51-72.
P. Diaconis, P. Hanlon, Eigen analysis for some examples of the Metropolis algorithm, Contemporary Math. 138 (1992) 99-117.
P. Diaconis, A. Ram, Analysis of systematic scan Metropolis algorithms using Iwahori-Hecke algebra techniques, Michigan Math. J. 48 (2000) 157-190.
P. Diaconis, L. Saloff-Coste, Separation cut-offs for birth and death chains, Ann. Appl. Prob. 16(4) (2006) 2098-2122.
P. Diaconis, M. Shahshahani, Time to reach stationarity in the Bernoulli-Laplace diffusion model, Siam J. Math. Anal. 18(1) (1987).
D. A. Levin, M. J. Luczak, Y. Peres, Glauber dynamics for the mean-field Ising model: cutoff, critical power law, and metastability, Prob. Theory Rel. Fields 146 (2010) 223-265.
D. A. Levin, Y. Peres, E. L. Wilmer, Markov Chains and Mixing Times, AMS, Providence, RI, 2009.
K. A. Ross, D. Xu, Hypergroup deformations and Markov chains, J. Theor. Prob. 7(4) (1994) 813-830.
|
---
abstract: 'An overview is given on those theoretical gravitational lensing results that can be formulated in a spacetime setting, without assuming that the gravitational fields are weak and that the bending angles are small. The first part is devoted to analytical methods considering spacetimes in which the equations for light rays (lightlike geodesics) is completely integrable. This includes spherically symmetric static spacetimes, the Kerr spacetime and plane gravitational waves. The second part is devoted to qualitative methods which give some information on lensing properties without actually solving the equation for lightlike geodesics. This includes Morse theory, methods from differential topology and bifurcation theory.'
address: |
Physics Department,\
Lancaster University,\
Lancaster LA1 4YB, United Kingdom\
author:
- VOLKER PERLICK
title: 'THEORETICAL GRAVITATIONAL LENSING – BEYOND THE WEAK-FIELD SMALL-ANGLE APPROXIMATION'
---
Introduction {#sec:intro}
============
For many years, theoretical work on gravitational lensing was almost exclusively done in an approximation formalism based on the assumptions that the gravitational fields are weak and that the bending angles are small. In its simplest version this formalism gives rise to the traditional “lens equation”, see e.g. Schneider, Ehlers and Falco [@SchneiderEhlersFalco1992], Petters, Levine and Wambsganss [@PettersLevineWambsganss2001] or Wambsganss [@Wambsganss1998]. This lens equation has proven a very powerful tool for investigating gravitational lensing situations.
In recent years, however, some gravitational lensing situations have come into the reach of observability for which the weak-deflection approximation is not applicable. In particular, the discovery that there is a black hole at the center of our galaxy, and probably at the center of most galaxies, is an important motivation for investigating the deflection of light rays that come close to a black hole. For such light rays the bending angle may become, in principle, arbitrarily large because light rays can make arbitrarily many turns around the black hole. There are also some other, more hypothetical, astrophysical objects whose lensing properties may lead to observable effects beyond the traditional lens equation, such as wormholes or monopoles.
If one wants to discuss gravitational lensing without the weak-field small-angle approximation, one has to use the formalism of general relativity. In this formalism, light rays are mathematically described as lightlike geodesics of a Lorentzian manifold. The past-oriented lightlike geodesics issuing from an observation event $p_O$ make up the *past light cone* of $p_O$. All information on what can be seen at the celestial sphere of an observer at $p_O$ is coded in the geometry of $p_O$’s past light cone. In this sense, all lensing features are known if the past light cones are known. Determining the past light cone of $p_O$ requires solving the equation for lightlike geodesics through $p_O$. In spacetimes with sufficiently many symmetries this equation is completely integrable, so the light cones can be determined analytically. In spacetimes where the geodesic equation is not completely integrable one may study some qualitative features of the lightlike geodesics without actually solving the geodesic equation, e.g. with the help of methods from global analysis or differential topology. In this sense the theoretical work on gravitational lensing in a Lorentzian geometry setting can be naturally divided into analytical studies and qualitative studies.
As was already mentioned, the main motivation for investigating gravitational lensing in a spacetime setting without the weak-field small-angle approximation comes from observations: There are observational situations, accessible with present or near-future technology, for which this approximation is simply not adequate. However, a second and almost equally important motivation comes from methodology: One gets a much better understanding of the physics behind gravitational lensing if one develops the relevant formalism as far as possible in the general framework of general relativity and introduces approximations only for those applications for which they are really necessary.
This article gives an overview on methods of how to study gravitational lensing in a Lorentzian geometry setting, without the weak-field small-angle approximation. Chapter 2 reviews analytical work in spacetimes where the geodesic equation is completely integrable. This includes all spherically symmetric static spacetimes such as the Schwarzschild spacetime (Section \[subsec:spherical\]), the Kerr spacetime (Section \[subsec:kerr\]), and plane gravitational waves (Section \[subsec:wave\]). Chapter 3 reviews some mathematical methods that give qualitative results on lensing features, e.g. on the number of images, without actually solving the geodesic equation. This includes Morse theory (Section \[subsec:morse\]), methods from differential topology (Section \[subsec:topology\]) and bifurcation theory (Section \[subsec:bifurcation\]). Related material can be found in Ref. .
Analytical Methods {#sec:analytical}
==================
Spherically Symmetric Static Spacetimes {#subsec:spherical}
---------------------------------------
In a spherically symmetric static spacetime, the geodesic equation is completely integrable, so the lightlike geodesics can be explicitly written in terms of integrals. (In the case of the Schwarzschild solution these are elliptic integrals.) This allows to write an *exact lens equation* for such spacetimes.
Lensing without weak-field or small-angle approximations in spherically symmetric static spacetimes was pioneered by Darwin [@Darwin1959; @Darwin1961] and by Atkinson [@Atkinson1965]. Whereas Darwin’s work is restricted to the Schwarzschild spacetime throughout, Atkinson derives all relevant formulas for an unspecified spherically symmetric static spacetime before specializing to the Schwarzschild spacetime in Schwarzschild and in isotropic coordinates. All important features of Schwarzschild lensing are clearly explained in both papers. However, they do not derive anything like a lens equation.
The first version of a lens equation for spherically symmetric static spacetimes that goes beyond the weak-field small-angle approximation was brought forward by Virbhadra, Narasimha and Chitre [@VirbhadraNarasimhaChitre1998] and then, in a modified form, by Virbhadra and Ellis [@VirbhadraEllis2000]. The Virbhadra-Ellis lens equation was originally applied to the Schwarzschild spacetime [@VirbhadraEllis2000] and later also to other spherically symmetric static spacetimes, e.g. to a boson star by Dabrowski and Schunck [@DabrowskiSchunck2000], to a fermion star by Bili[ć]{}, Nikoli[ć]{} and Viollier [@BilicNikolicViollier2000], and to spacetimes with naked singularities by Virbhadra and Ellis [@VirbhadraEllis2002]. The Virbhadra-Ellis lens equation might be called an “almost exact lens equation”. It is approximative insofar as it restricts to asymptotically flat spacetimes, with observer and light source in the asymptotic region and almost aligned (i.e., almost in opposite directions from the center of symmetry). However, the light rays are not restricted to the asymptotic region where the gravitational field is weak. They are rather allowed to enter the central region of the spacetime, on their way from the source to the observer, and to be arbitrarily strongly bent, i.e., to make arbitrarily many turns around the center.
An exact lens equation for spherically symmetric spacetimes, without any restriction on observer or source positions, was suggested by Perlick [@Perlick2004b]. (This can be viewed as a special case of Fritteli and Newman’s exact lens equation in arbitrary spacetimes [@FrittelliNewman1999], see Section \[subsec:topology\].) To make use of the symmetry, one places observer and light source on spheres, rather than on planes as in the traditional lens equation, see Fig. \[fig:sphlens\]. The observer is at $r = r_O$ and the light sources are distributed at $r = r_S$, where $r$ is the radial coordinate in the spherically symmetric spacetime under consideration. For each past-oriented light ray, leaving the observer at an angle $\Theta$ with respect to the radial outward direction, one can integrate the lightlike geodesic equation to get the azimuthal angle $\Phi$ swept out by the light ray before it reaches the sphere $r = r_S$. In this way one gets a lens equation which gives $\Phi$ as a function of $\Theta$ and, thereby, relates the source position in the spacetime to the image position at the observer’s sky. Of course, one has to take into account that a light ray with $\Phi + 2 \pi$ arrives at the same source position as a light ray with $\Phi$. Also, one has to note that light rays may hit the sphere $r = r_S$ several times; if this is the case, the lens equation gives a *multi-valued* map $\Theta \mapsto \Phi$.
The lensing features are nicely illustrated if one plots $\Phi$ as a function of $\Theta$, as given by the lens equation. This is shown for the Schwarzschild spacetime in the top panel of Fig.\[fig:schwlens\]. (Similar pictures for the spacetimes of an Ellis wormhole and of a Barriola-Vilenkin monopole can be found in Ref. .) The multiples of $\pi$ are indicated on the $\Phi$ axis; they correspond to source positions for which the observer sees an Einstein ring. If $\Theta$ approaches a certain value $\delta$, the angle $\Phi$ becomes infinite, corresponding to a light ray that circles around the center forever. This light ray asymptotically spirals towards the photon sphere at $r = 3m$.
A similar behaviour can be observed in any spherically symmetric static spacetime which is asymptotically flat and has an unstable photon sphere. (Here the sphere $r= r_L$ is called a *photon sphere* if a lightlike geodesic remains on this sphere if it starts tangentially to it. Thus, a photon sphere is filled with circular geodesics. A photon sphere is called stable if all lightlike geodesics with initial values close to those of a circular geodesic stay close to the photon sphere; otherwise it is called unstable. The surface $r = 3 m$ in the Schwarzschild spacetime is the best known example of an unstable photon sphere.) For an observer situated between the photon sphere and infinity, the plot of $\Phi$ versus $\Theta$ always looks qualitatively as in Fig. \[fig:schwlens\]. For a certain value of $\Theta$, the angle $\Phi$ diverges, corresponding to a light ray that spirals towards the photon sphere. The divergence is always logarithmic, as was shown by Bozza [@Bozza2002] using the setting of the Virbhadra-Ellis lens equation. Thus, by logarithmically rescaling the $\Theta$ axis, the graph of $\Phi$ versus $\Theta$ asymptotically becomes a straight line. For the Schwarzschild spacetime this is shown in the middle panel of Fig. \[fig:schwlens\]. Following Bozza, this straight line is called the *strong field limit* or the *strong deflection limit*. (The latter name is actually more appropriate; the bending angle goes to infinity but the gravitational field, measured in terms of the tidal force, need not be particularly strong near a photon sphere.) Calculating the strong deflection limit analytically requires some careful handling of integrals with singularities. The relevant formulas were worked out by Bozza [@Bozza2002].
The fact that $\Phi$ goes to infinity implies that the observer sees infinitely many images of each light source. This can be read from the bottom panel of Fig. \[fig:schwlens\]: the dotted line, indicating a particular value of $\mathrm{cos}\, \Phi$, intersects the graph of $\mathrm{cos} \, \Phi$ over $\Theta$ infinitely many times. For light sources exactly on the axis through the observer position ($\,$i.e. $\mathrm{cos} \, \Phi =
\pm 1 \,$), there are infinitely many Einstein rings. For off-axis light sources ($\,$i.e. all other values of $\Phi\,$), there are two infinite sequences of isolated images. They correspond to light rays that make increasingly many turns around the center, in the positive $\varphi$-direction for the first sequence and in the negative $\varphi$-direction for the second sequence. Images corresponding to light rays that make at least one full turn around the center are called *higher-order images*. (The name *relativistic images* is also in use, but this name is a bit misleading because such images would occur in any theory, relativistic or not, in which photons are acted upon by gravity.) Higher-order images have not been observed until now, but there is some hope that they might be observed sometimes in the future. The most promising candidate for such observations is the black hole at the center of our galaxy, followed by the black hole at the center of M31. The perspectives for observing lensed images (secondary and higher order) of stars near these black holes were discussed at this conference in a talk by Mancini, cf. Refs. . Of course, even with the most advanced future technology it will never be possible to see all the infinitely many images because there will always be some limits on sensitivity and resolution power.
Note that, if $\Phi$ is close to a multiple of $2 \pi$, the light returns approximately in the same direction from which it came, after having made one or more turns around the center. This phenomenon, sometimes called *retrolensing*, was discussed e.g. by Stuckey [@Stuckey1993], by Holz and Wheeler [@HolzWheeler2002] and by Eiroa and Torres [@EiroaTorres2004].
If one calculates the strong-deflection limit for observer and light source in the asymptotic region, one always qualitatively gets the same picture as for Schwarzschild, see middle panel of Fig. \[fig:schwlens\]. However, the pitch of the asymptotic straight line and its intersection point with the vertical axis are different for other spacetimes. Thus, the observation of higher-order images would allow to distinguish a Schwarzschild black hole from other black holes, see Bozza [@Bozza2002] for a detailed discussion. For calculations of the strong deflection limit in various black hole spacetimes see e.g. Refs.
The strong deflection limit has also been investigated for wormhole spacetimes [@TejeiroLarranaga2005; @NandiZhangZakharov2006]. The qualitative lensing features of wormholes are quite similar to black holes. The reason is that the spacetime of a wormhole, like that of a black hole, is asymptotically flat and has an unstable photon sphere. (The photon sphere is at the neck of the wormhole.) As a consequence, the lens map for a wormhole is qualitatively similar to that for a black hole if observer and light source are in the same asymptotically flat region, compare the $\Phi$-versus-$\Theta$ plot for the Schwarzschild spacetime given in Fig. \[fig:schwlens\] with that for an Ellis wormhole given in Ref. .
Iyer and Petters [@IyerPetters2006] have recently shown that the strong deflection limit in the Schwarzschild spacetime can be viewed as the leading order term in an infinite perturbation series. A similar perturbation series can be set up with the standard weak deflection limit as the leading order term. They are able to show that both series together provide an approximation that deviates by not more than 1 $\%$ from the exact bending angle for every light ray between the light sphere and infinity.
A completely different approach to analytically evaluating the bending angle formula in spherically symmetric static spacetimes was brought forward by Amore et al. [@AmoreArceo2006; @AmoreArceoFernandez2006; @AmoreCervantesPaceFernandez2006]. This approach, which makes no assumption on either strong or weak bending, converts the formula for the bending angle into a geometrically convergent series whose terms can be calculated analytically. Example calculations demonstrate that typically only a few terms are needed to get accurate results, even for cases where the bending angle is neither particularly small nor particularly large.
An important motivation for studying gravitational lensing in spherically symmetric static metrics is in the fact that this provides a means for testing alternative theories of gravity. A systematic theoretical framework for investigations of this kind was brought forward by Keeton and Petters [@KeetonPetters2005; @KeetonPetters2006a; @KeetonPetters2006b] on which Keeton reported at this conference, see Ref. . The basic idea is to write a spherically symmetric static metric as a Taylor series with respect to the gravitational radius of the central mass. The coefficients of this series are different for different gravitational theories. The linear term corresponds to the weak deflection limit; so far all lensing observations can be satisfactorily explained if only this term is taken into account. However, as outlined in the quoted papers, near-future technology will lead to an accuracy in observation that requires taking second and higher order terms into account. This will give us a new test of Einstein’s theory and of alternative theories of gravity.
Among alternative theories of gravity, braneworld scenarios are particularly fashionable. These scenarios provide spherically symmetric static black hole solutions for which lensing can be studied with the analytical methods discussed above. At this conference Whisker (see Refs. ) and Majumdar (see Refs. ) reported on their recent investigations comparing lensing by braneworld black holes to Schwarzschild lensing.
Kerr Spacetime {#subsec:kerr}
--------------
There is strong evidence for a supermassive black hole at the center of our galaxy, and some observations indicate that its spin is non-negligible. A non-vanishing spin modifies the lensing features of a black hole in a characteristic way. Some spin-dependent observable effects were discussed by Zakharov in his talk at this conference, see Ref. .
The appropriate mathematical model for describing the spacetime around a spinning black hole is the Kerr metric. Its lightlike (and timelike) geodesics have been studied in great detail, see e.g. Chandrasekhar [@Chandrasekhar1983] or O’Neill [@ONeill1995]. In the Kerr spacetime the geodesic equation is completely integrable – the solutions can be written in terms of elliptic integrals – but the analysis of the resulting light paths turns out to be rather involved.
The Kerr metric depends on two parameters, the mass $m$ and the specific angular momentum $a$. In standard Boyer-Lindquist coordinates, the metric reads $$\label{eq:kerr}
g = - \frac{\Delta}{\rho ^2} \, \big( \, dt \, - \,
a \, \mathrm{sin} ^2 \vartheta \, d \varphi \big) ^2 \, + \,
\frac{\mathrm{sin} ^2 \vartheta}{\rho ^2} \, \big(
(r^2 + a^2) \, d \varphi \, - \, a \, dt \, \big) ^2 \, + \,
\frac{\rho ^2}{\Delta} \, dr^2 \, + \, \rho ^2 \, d \vartheta ^2 \, ,$$ where $\rho$ and $\Delta$ are defined by $$\label{eq:rhodelta}
\rho ^2 = r^2 + a^2 \, {\mathrm{cos}} ^2 \vartheta
\quad \text{and} \quad
\Delta = r^2 - 2mr + a^2 \, .$$ For $a=0$ one recovers the Schwarzschild metric. In the following we restrict to the case $a^2 \le m^2$ which describes a black hole, and we ignore the case $m^2 < a^2$ which describes a naked singularity.
We denote the region in the Kerr spacetime where $$\label{eq:K}
\, - \, 2 \, a \, r \, \sqrt{\Delta} \, {\mathrm{sin}} \, \vartheta \: \le
\: 2 \, r \, \Delta - (r-m) \, \rho ^2 \: \le
\: 2 \, a \, r \, \sqrt{\Delta} \, {\mathrm{sin}} \, \vartheta$$ by $K$ and we call this the *photon region*, see Fig. \[fig:KerrK\]. This region is characterized by the fact that it is filled with *spherical lightlike geodesics*, i.e., lightlike geodesics that stay on spheres $r = \,$constant. Along each such geodesic the $\vartheta$ coordinate oscillates between extremal values $- \vartheta _0$ and $\vartheta _0$, whereas the $\varphi$ and $t$ coordinates increase or decrease linearly with the affine parameter. In the Schwarzschild limit $a \to 0$ the photon region $K$ shrinks to the photon sphere $r=3m$.
In the Schwarzschild spacetime light rays that come close to the photon sphere $r=3m$ can make arbitrarily many turns around the center. Similarly, in the Kerr spacetime light rays that come arbitrarily close to the photon region $K$ can make arbitrarily many turns around the center. Thus, one can define a strong deflection limit in the same sense as for spherically symmetric static spacetimes. Light rays that start tangential to the equatorial plane $\vartheta = \pi /2$ remain in that plane. Their analysis is, therefore, not much more difficult than for the spherically symmetric static case. One gets infinitely many images on either side of the center, but in contrast to the spherically symmetric static case the positions of the images are not mirror symmetric. This asymmetry can be interpreted as saying that the light rays are “dragged” along with the rotation of the black hole. The analytic formulas for the strong deflection limit in the equatorial plane of the Kerr metric have been worked out by Bozza [@Bozza2003].
Away from the equatorial plane, the analysis is much more difficult because then a light ray is not confined to a plane. The strong deflection limit was calculated by Bozza, de Luca, Scarpetta and Sereno [@BozzadeLucaScarpettaSereno2005] for the case that the observer is in the equatorial plane (but the light source need not) and by Bozza, de Luca and Scarpetta [@BozzadeLucaScarpetta2006] for the general case. These results were presented at this conference in a talk by de Luca, cf. Ref. . (For another work of Kerr lensing without weak-field assumption, largely based on numerical studies, see V[á]{}zquez and Esteban [@VazquezEsteban2003].) The investigations of the strong deflection limit complements ongoing work on the weak deflection limit in the Kerr metric which was presented by Sereno at this conference, cf. Refs. .
An important aspect of the above-mentioned work on the strong deflection limit in the Kerr metric is in the fact that it analytically corroborates knowledge about the caustics in this spacetime which, in earlier work, has been investigated numerically. If one considers the past light cone of an event $p_O$ in the Kerr spacetime, one finds that it forms infinitely many caustics. By definition, a caustic is a connected subset of the light cone where neighboring light rays from $p_O$ intersect each other, at least to first order. At caustic points, the light cone fails to be an immersed submanifold of the spacetime, forming e.g. cuspidal edges. In the Kerr spacetime the caustics, if projected to three-dimensional space, are tubes with astroid cross sections. A picture can be found in a paper by Blandford and Rauch [@RauchBlandford1994] who determined the caustics numerically. In the Schwarzschild case, $a =0$, the caustics are just radial lines in three-space. With increasing $a$, they open out into tubes with astroid cross-sections, and they start winding around the center. In the limit of an extreme Kerr black hole, $a=m$, they spiral asymptotically towards the horizon. The analysis of the strong deflection limit gives analytic support to these results which have been found from numerical studies. However, what is still missing is a good picture that shows how the caustics are situated inside the light cone. Such pictures can easily be produced for the Schwarzschild case in two different ways: First, one can depict the past light cone of an event in the spacetime, restricting oneself to the equatorial plane, see Fig. \[fig:schwcon\]. Second, one can depict in three-space a series of intersections of the light cone with hypersurfaces $t=\,$constant (*wavefronts*), see Fig. \[fig:schwfrt\]. It would be desirable to have similar pictures for the Kerr metric. Some attemps in this direction have been made, beginning with a 1977 paper by Hanni [@Hanni1977]. At this conference Grave presented computer visualizations for lensing, including a movie that shows the propagation of wave fronts in the Kerr spacetime. A picture of wave fronts in the Kerr spacetime, analogous to the Schwarzschild wave fronts shown in Fig. \[fig:schwfrt\], can be found in Ref. . This picture nicely shows how the twisting effect, governed by the rotation of the Kerr black hole, produces an asymmetry in the wave fronts. However, the resolution of the picture is too low to show the astroid structure of the caustic. Also, it is hard to read from this picture the structure of the caustic near the horizon. Therefore it seems fair to say that until now the structure of the caustics in the Kerr spacetime has not been clearly brought out in visualizations.
Plane Gravitational Waves {#subsec:wave}
-------------------------
Standard textbooks on general relativity usually treat gravitational waves as small perturbations of Minkowski spacetime. In this weak-field approximation, it is easy to study the effect of gravitational waves on light rays. However, the resulting picture is rather incomplete. Several interesting qualitative features are brought out only if one goes beyond the weak-field approximation. This can be done by studying plane gravitational waves in terms of exact solutions to Einstein’s field equation. Because of their high symmetry, plane gravitational wave solutions are over-idealized in view of applications; however, they are highly instructive in view of understanding the focusing properties of gravitational waves.
A *plane gravitational wave* is a spacetime with metric $$\label{eq:wave}
g \, = \, -2 \, du \, dv
\, - \,
\big( \, f(u)\, (x^2-y^2) \, + \,
2 \, g(u) \, x
\, y \, \big)
\, du^2
\, + \, dx^2
\, + \, dy^2$$ where $f(u)^2+g(u)^2$ is not identically zero. For any choice of $f(u)$ and $g(u)$, the metric (\[eq:wave\]) has vanishing Ricci tensor, i.e., Einstein’s vacuum field equation is satisfied. With the four coordinates $u,v,x,y$ ranging over all of $\mathbb{R}^4$, the spacetime is geodesically complete. The spacelike coordinates $x$ and $y$ are transverse to the propagation direction of the gravitational wave.
For a metric of the form (\[eq:wave\]), the geodesic equation is completely integrable, i.e., the light cone structure can be studied excplicitly without approximation. It reveals an interesting focusing property of gravitational waves that is not captured if one restricts to weak fields. This focusing property was discussed in great detail by Ehrlich and Emch [@EhrlichEmch1992; @EhrlichEmch1993], cf. Beem, Ehrlich and Easley [@BeemEhrlichEasley1996], Chapter 13. What one finds is the following. Of all the past-oriented lightlike geodesics issuing from an event $p_O$, one goes straight to infinity whereas all the other ones are being refocused in a spacelike curve $q$ which is completely contained in a hypersurface $u = \mathrm{constant}$, see Fig. \[fig:wave\]. The curve $q$ is the first caustic of the past light cone of $p_O$. This focusing property is universal, i.e., independent of the profile functions $f(u)$ and $g(u)$. It is closely related to the fact that a plane gravitational wave spacetime cannot be globally hyperbolic. The latter fact was discovered by Penrose [@Penrose1965] who studied the focusing property for the special case that $f(u)$ and $g(u)$ are different from zero only in a sufficiently small interval $u_1 < u < u_2$. For this special choice of profile functions there is no second caustic, whereas for other choices of $f(u)$ and $g(u)$ the past-oriented lightlike geodesics issuing from $p_O$ may be refocused another time after passing through $q$. It is geometrically evident from Fig. \[fig:wave\] that an inextendible timelike curve $\gamma _S$ which is sufficiently close to $q$ intersects the past light cone of $p_O$ exactly twice. Thus, an observer at $p_O$ sees exactly two images of a light source with worldline $\gamma _S$. This example demonstrates that, in contrast to the weak-field theory, a transparent gravitational lens need not produce an odd number of images, even in the case of a geodesically complete spacetime with trivial topology.
Qualitative Methods {#sec:qualitative}
===================
Morse Theory {#subsec:morse}
------------
Morse theory is a body of mathematical theorems that give some information on the number of solutions to a variational problem. In applications to lensing, the variational problem is a version of Fermat’s principle and the solutions are the light rays from the source to the observer. Every such light ray corresponds to an image of the source on the observer’s sky. Morse theory can be used to determine whether the number of images is infinite or finite, even or odd. Results of this kind are useful because, if there is an observation which contradicts them, one knows that one has to search for an additional image which has gone unnoticed so far. This may have happened because an image was too faint or too close to another image for being detected.
Morse theory can be applied to gravitational lensing either in the weak-field small-angle approximation, or in a spacetime setting without such approximations. In the following we will be concerned only with the latter, for the former see Petters [@Petters1992] or the respective chapter in Petters, Levine and Wambsganss [@PettersLevineWambsganss2001]. We will concentrate on a version of Morse theory that was brought forward by Uhlenbeck [@Uhlenbeck1975] and applied to gravitational lensing first by McKenzie [@McKenzie1985]. This version is restricted to globally hyperbolic spacetimes throughout. There have been several attempts to formulate a Morse theory in more general spacetimes, starting out from Kovner’s version [@Kovner1990] of Fermat’s principle, see Fig. \[fig:fermat\]. Although this variational principle can be formulated in an arbitrary spacetime, a fully-fledged Morse theory requires additional assumptions which come close to global hyperbolicity. For the most general version available, which applies to some subsets-with-boundary of a stably causal spacetime, see Giannoni, Masiello and Piccione [@GiannoniMasielloPiccione1998].
Following Uhlenbeck [@Uhlenbeck1975] we consider a 4-dimensional Lorentzian manifold $(M,g)$ that admits a foliation into smooth Cauchy surfaces, i.e., a globally hyperbolic spacetime. (The fact that the original definition of global hyperbolicity is equivalent to the existence of a foliation into *smooth* Cauchy surfaces was completely proven only recently by Bernal and S[á]{}nchez [@BernalSanchez2005].) Then $M$ can be written as a product of a three-dimensional manifold $S$, which serves as the prototype for each Cauchy surface, and a time-axis, $$\label{eq:product}
M= S \times {\mathbb{R}} \, .$$ In this spacetime we fix an event $p_O$ and a timelike curve $\gamma _S$, and we make the following additional assumptions. (We are interested in past-oriented lightlike geodesics from $p_O$, whereas Uhlenbeck considered future-oriented lightlike geodesics. Therefore, our assumptions are the time-reversals of her’s.)
- With respect to the splitting (\[eq:product\]), the spacetime has no particle horizons. (An analytical criterion for this to be true is the time-reversed version of Uhlenbeck’s “metric growth condition”.)
- The timelike curve $\gamma _S$ is inextendible and it does not meet the caustic of the past light cone of $p_O$. (This excludes situations where the observer sees an extended image, such as an Einstein ring.)
- There is a sequence $t_n$ with $t_n \to - \infty$ for $n \to \infty$ such that the projection to $S$ of $\gamma _S (t_n)$ has an accumulation point. (Roughly speaking, this condition prohibits that $\gamma _S$ “goes to infinity” in the past.)
Then the Morse inequalities $$\label{eq:Morseineq}
N_k \ge B_k \; \qquad {\text{for all}} \quad k \in {\mathbb{N}}_0$$ and the Morse relation $$\label{eq:Morserel}
\sum_{k=0}^{\infty} (-1)^k N_k =
\sum_{k=0}^{\infty} (-1)^k B_k$$ hold true, where $N_k$ denotes the number of past-pointing lightlike geodesics with index $k$ from $p_O$ to $\gamma _S$, and $B_k$ denotes the $k$-th Betti number of the loop space of $M$. (The index of a geodesic is the number of its conjugate points, counted with multiplicity. The loop space of a connected topological space is the space of all continuous curves connecting two fixed points. For the definition of Betti numbers see, e.g., Frankel [@Frankel1997].)
The sum on the right-hand side of (\[eq:Morserel\]) is, by definition, the Euler characteristic $\chi$ of the loop space of $M$. Hence, (\[eq:Morserel\]) can also be written in the form $$\label{eq:N+-}
N_+ - N_- = \chi \, ,$$ where $N_+$ (respectively $N_-$) denotes the number of past-pointing lightlike geodesics with even (respectively odd) index from $p_O$ to $\gamma _S$.
The Betti numbers of the loop space of $M=S \times {\mathbb{R}}$ are, of course, determined by the topology of $S$. Three cases are to be distinguished.
[**Case A**]{}: $M$ is not simply connected. Then the loop space of $M$ has infinitely many connected components, so $B_0 = \infty$. In this situation (\[eq:Morseineq\]) says that $N_0 = \infty$, i.e., that there are infinitely many past-pointing lightlike geodesics from $p$ to $\gamma$ that are free of conjugate points.
[**Case B**]{}: $M$ is simply connected but not contractible to a point. Then for all but finitely many $k \in {\mathbb{N}}_0$ we have $B_k > 0$. This was proven in a classical paper by Serre [@Serre1951], cf. McKenzie [@McKenzie1985]. In this situation (\[eq:Morseineq\]) implies $N_k > 0$ for all but finitely many $k$. In other words, for almost every positive integer $k$ we can find a past-pointing lightlike geodesic from $p_O$ to $\gamma _S$ with exactly $k$ conjugate points. Hence, there must be infinitely many past-pointing lightlike geodesics from $p_O$ to $\gamma _S$.
[**Case C**]{}: $M$ is contractible to a point. Then the loop space of $M$ is contractible to a point, i.e., $B_0 =1$ and $B_k = 0$ for $k > 0$. In this case (\[eq:N+-\]) takes the form $N_+ - N_-
= 1$ which implies that the total number $N_+ + N_- = 2 N_- + 1$ of past-pointing lightlike geodesics from $p_O$ to $\gamma _S$ is (infinite or) odd.
Case C has relevance for asymptotically simple and empty spacetimes. These spacetimes describe the gravitational field around a transparent isolated body, see Hawking and Ellis [@HawkingEllis1973] for the formal definition. It is well known that asymptotically simple and empty spacetimes are globally hyperbolic, with contractible Cauchy surface, and it is not difficult to verify that they have no particle horizons. Also, it follows from the definition of an asymptotically simple and empty spacetime that $N_+ + N_-$ must be finite. Hence, if $\gamma _S$ satisfies conditions (b) and (c) above, Case C applies and says that the total number of images must be odd. This version of the “odd number theorem” makes no assumption on the weakness of the gravitational field. (It does not apply to the example of a plane gravitational wave, treated in Section \[subsec:wave\], because the latter is not asymptotically simple.) For further discussions of the odd number theorem and Morse theory see McKenzie [@McKenzie1985].
Case B has relevance for black hole spacetimes. In particular, it can be used to give a general proof that a Kerr-Newman black hole produces infinitely many images, with only very mild restrictions on the allowed motion of the light source. Here the globally hyperbolic spacetime to be considered is the domain of outer communication of the black hole, i.e., the region outside of the outer horizon. Its Cauchy surfaces have topology $S^2 \times \mathbb{R}$ which is, indeed, simply connected but not contractible. The details are worked out in Ref. .
Differential Topology {#subsec:topology}
---------------------
The idea of formulating a lens map in an arbitrary general-relativistic spacetime, without weak-field or small-angle approximations, is due to Frittelli and Newman [@FrittelliNewman1999] and was further developed by Ehlers, Frittelli and Newman [@Ehlers2000; @EhlersFrittelliNewman2003]. In this section we briefly review the construction of their lens map and we discuss how methods from differential topology can be applied to it.
In the standard weak-field small-angle formalism for thin lenses the lens map, resulting from the lens equation, is a map from a deflector plane to a source plane, see e.g. Schneider, Ehlers and Falco [@SchneiderEhlersFalco1992], Petters, Levine and Wambsganss [@PettersLevineWambsganss2001] or Wambsganss [@Wambsganss1998]. To define something analogous in an arbitrary spacetime, without approximations, we fix an observation event $p_O$ and we look for analogues of the deflector plane and the source plane. As to the deflector plane, there is an obvious candidate, namely the *celestial sphere* $\mathcal{S}_O$ at $p_O$. This can be defined as the set of all lightlike directions at $p_O$. As to the source plane, however, there is no natural candidate. Following Frittelli, Newman and Ehlers, we choose a timelike 3-dimensional submanifold $\mathcal{T}$ of the spacetime manifold. We assume that $\mathcal{T}$ is of the form $\mathcal{T} = \mathcal{N} \times \mathbb{R}$ where the $\mathbb{R}-$lines are timelike and can be interpreted as the worldlines of light sources. The two-dimensional manifold $\mathcal{N}$ is the analogue of the source plane. In this situation the lens map $f : \mathcal{S}_O
\longrightarrow \mathcal{N}$ is defined by associating each lightlike direction $Y$ at $p_O$ with the geodesic to which it is tangent, extending this lightlike geodesic into the past until it meets $\mathcal{T}$ and then projecting onto $\mathcal{N}$, see Fig. \[fig:lensmap\]. In general, this prescription does not give a well-defined map $f$ since neither existence nor uniqueness of the target value is guaranteed. As to existence, there might be some past-pointing lightlike geodesics from $p_O$ that never reach $\mathcal{T}$. As to uniqueness, one and the same lightlike geodesic might intersect $\mathcal{T}$ several times.
(10,10) (6,0.5, 5,1, 4,1.2, 3,1.2) (6,3.5, 5,4, 4,4.2, 3,4.2) (6,8.5, 5,9, 4,9.2, 3,9.2) (6,3.5, 6,8.5) (5,4, 5,9) (4,4.2, 4,9.2) (3,4.2, 3,9.2) (1.5,8, 4.5,5.2) (1.5,8) (4.5,5.2) (4.5,1.13) (6.4,7)[${\mathcal{T}}$]{} (6.4,0.5)[${\mathcal{N}}$]{} (1.2,8.3)[$p_O$]{} (1.4,7.2)[$Y$]{} (3.9,0.6)[$f(Y)$]{} (4.5,3.2)[(0,-1)[1]{}]{}
However, there are some situations of physical interest where the lens map is well-defined. An important example is the lens map in asymptotically simple and empty spacetimes. This class of spacetimes was mentioned already in Section \[subsec:morse\], and it was indicated how Morse theory can be used to prove that, in such a spacetime, the number of images is odd under very general conditions. The same result can also be found with the help of the Frittelli-Newman lens map. The proof, which is worked out in Ref. , proceeds in two steps. First one considers “light sources at infinity”; in this case the lens map is a map from a two-sphere to a two-sphere, and it can be shown that its mapping degree (Brouwer degree) is equal to $\pm 1$. In the second step it is proven that this result implies an odd number of images for very general light sources in the asymptotically simple and empty spacetime. Ref. contains a number of other applications of differential topology to the lens map.
Clearly, the “exact lens equation” for spherically symmetric static spacetimes, discussed in Section \[subsec:spherical\], defines a lens map which is a special case of the Frittelli-Newman lens map.
Bifurcation Theory {#subsec:bifurcation}
------------------
Bifurcation theory is concerned with variational problems that depend on a real parameter. The goal is to characterize the situation that, in dependence on the parameter, a solution to the variational problem (i.e., a stationary point of the variation functional) bifurcates into two or more solutions. Recently bifurcation theory has been applied to Kovner’s version of Fermat’s principle, recall Fig. \[fig:fermat\], by Giamb[ó]{}, Giannoni and Piccione [@GiamboGiannoniPiccione2004] and by Javaloyes and Piccione [@JavaloyesPiccione2006]. To discuss their approach, we fix a timelike curve $\gamma _S$ and a past-oriented lightlike geodesic $\lambda$ such that $\lambda (0)$ is on $\gamma _S$, see Fig. \[fig:bifurcat\]. Now for every $s<0$ we can consider Fermat’s principle with the given worldline $\gamma _S$ and the observation event $p_O = \lambda (s)$. This gives us a one-parameter family of variational problems, depending on the parameter $s$. For any value of $s$, a section of $\lambda$ is a solution curve of the variational problem. With the help of bifurcation theory one can characterize those values $s_0$ of $s$ where this solution bifurcates into two or more solutions. It is easy to see that a bifurcation can occur only at points $\lambda (s _0)$ which are conjugate to $\lambda (0)$. The latter condition means that $\lambda (0)$ is in the caustic of the past light cone of the event $p_O = \lambda (s _0)$. Bifurcation theory gives us some information on conjugate points that is difficult to get with other methods. In the following we present one of the results that is proven in the quoted papers.
As a preparation, we recall some facts which can be verified with elementary means. Let $\lambda$ be a past-oriented lightlike geodesic and assume that, for some $s_0 < 0$, the point $\lambda (s _0)$ is conjugate to $\lambda (0)$, see again Fig. \[fig:bifurcat\]. Then it is an elementary exercise to verify that, for every $\varepsilon > 0$, the geodesic from $\lambda (s _0 - \varepsilon)$ to $\lambda (0)$ contains a point conjugate to $\lambda
(s_0 - \varepsilon)$. Hence a known result (Theorem 2 in Ref. ) implies that there is a timelike curve $\gamma _S$ through $\lambda (0)$ that can be connected to $\lambda (s_0 - \varepsilon)$ by a second lightlike geodesic $\lambda _{\varepsilon}$.
With the help of bifurcation theory, this statement can be strengthened in the following way, see Proposition 13 in Ref. . For $\varepsilon$ sufficiently small,
- the statement is true for *all* timelike curves $\gamma _S$ through $\lambda (0)$;
- $\lambda _{\varepsilon}$ depends continuously on $\varepsilon$ (i.e., the geodesics $\lambda _{\varepsilon}$ can be parametrized such that the map $(s, \varepsilon) \mapsto \lambda _{\varepsilon} (s)$ is differentiable with respect to $s$ and continuous with respect to $\varepsilon$).
Thus, an observer at $\lambda (s_0 - \varepsilon )$ sees (at least) two images of the light source $\gamma _S$, one along the lightlike geodesic $\lambda$ and one along the lightlike geodesic $\lambda _{\varepsilon}$. For $\varepsilon
\to 0$ these two images move continuously towards each other until they merge for $\varepsilon = 0$. This property is universal, i.e., it holds for all conjugate points.
Concluding Remarks {#sec:conclude}
==================
With an increase in observational accuracy, theoretical gravitational lensing beyond the weak-field and small-angle approximation becomes more and more relevant in view of applications. At this conference, it was already the majority of contributions to the Theoretical Gravitational Lensing workshop that went beyond these approximations. Of course, studying lensing in weak gravitational fields is of remaining value. At this conference this was illustrated, in addition to the talks already mentioned, by Yoo’s talk [@Yoo] on lensing in a clumpy universe, by Miranda’s talk [@Miranda] on lensing by galactic halo structures and by Frutos-Alfaro’s talk [@Frutos] on computer programs for visualizing and modeling gravitational lenses based on the traditional lens equation. Quite generally, weak-field studies have a wide range of applications and they complement calculations in arbitrarily strong gravitational fields in a useful way. From a methodological point of view, it is desirable to develop the general formalism in a spacetime setting without approximations, as far as possible, and to introduce approximations only afterwards for those applications for which they are necessary.
[10]{}
P. Schneider, J. Ehlers and E. Falco, [*Gravitational lenses*]{} (Springer, Heidelberg, 1992).
A. O. Petters, H. Levine and J. Wambsganss, [*Singularity theory and gravitational lensing*]{}, Progress in Mathematical Physics, Vol. 21 (Birkh[ä]{}user, Boston, USA., 2001).
J. Wambsganss, [*Living Rev. Relativity*]{} [**1(12)**]{} (1998),\
http://www.livingreviews.org/lrr-1998-12.
V. Perlick, [*Living Rev. Relativity*]{} [**7(9)**]{} (2004),\
http://www.livingreviews.org/lrr-2004-9.
C. Darwin, [*Proc. Roy. Soc. London*]{} [**A 249**]{}, 180 (1959).
C. Darwin, [*Proc. Roy. Soc. London*]{} [**A 263**]{}, 39 (1961).
R. Atkinson, [*Astron. J.*]{} [**70**]{}, 517 (1965).
K. S. Virbhadra, D. Narasimha and S. M. Chitre, [*Astron. & Astrophys.*]{} [**337**]{}, 1 (1998).
K. S. Virbhadra and G. F. R. Ellis, [*Phys. Rev.*]{} [**D62**]{}, 084003 (2000).
M. D[a]{}browski and F. Schunck, [*Astrophys. J.*]{} [**535**]{}, 316 (2000).
N. Bili[ć]{}, H. Nikoli[ć]{} and R. Viollier, [*Astrophys. J.*]{} [**537**]{}, 909 (2000).
K. S. Virbhadra and G. F. R. Ellis, [*Phys. Rev.*]{} [**D 65**]{}, 103004 (2002).
V. Perlick, [*Phys. Rev.*]{} [**D 69**]{}, 064017 (2004).
S. Frittelli and E. T. Newman, [*Phys. Rev.*]{} [**D 59**]{}, 124001 (1999).
V. Bozza, [*Phys. Rev.*]{} [**D 66**]{}, 103001 (2002).
V. Bozza and L. Mancini, [*Astrophys. J.*]{} [**627**]{}, 790 (2005).
V. Bozza and L. Mancini, in this proceedings volume.
W. M. Stuckey, [*Amer. J. Phys.*]{} [**61**]{}, 448 (1993).
D. E. Holz and J. A. Wheeler, [*Astrophys. J.*]{} [**578**]{}, 330 (2002).
E. Eiroa and D. Torres, [*Phys. Rev.*]{} [**D 69**]{}, 063004 (2004).
E. Eiroa, G. Romero and D. Torres, [*Phys. Rev.*]{} [**D 66**]{}, 024010 (2002).
A. Bhadra, [*Phys. Rev.*]{} [**D 67**]{}, 103009 (2003).
E. Eiroa, [*Phys. Rev.*]{} [**71**]{}, 083010 (2005).
E. Eiroa, [*Phys. Rev.*]{} [**D 73**]{}, 043002 (2006).
J. M. Tejeiro and E. A. Larranaga, http://arXiv.org/abs/gr-qc/0505054.
K. K. Nandi, Z. Yuang-Zhong and A. V. Zakharov, [*Phys. Rev.*]{} [**D 74**]{}, 024020 (2006).
S. V. Iyer and A. O. Petters, http://arXiv.org/abs/gr-qc/0611086.
P. Amore and S. Arceo, [*Phys. Rev.*]{} [**D 73**]{}, 083004 (2006).
P. Amore, S. Arceo and F. Fernandez, [*Phys. Rev.*]{} [**D 74**]{}, 083004 (2006).
P. Amore, M. Cervantes, A. d. Pace and F. Fernandez,\
http://arXiv.org/abs/gr-qc/0610153.
C. R. Keeton and A. O. Petters, [*Phys. Rev.*]{} [**D 72**]{}, 104006 (2005).
C. R. Keeton and A. O. Petters, [*Phys. Rev.*]{} [**D 73**]{}, 044024 (2006).
C. R. Keeton and A. O. Petters, [*Phys. Rev.*]{} [**D 73**]{}, 104032 (2006).
C. R. Keeton and A. O. Petters, in this proceedings volume.
R. Whisker, [*Phys. Rev.*]{} [**71**]{}, 064004 (2005).
R. Whisker, in this proceedings volume.
N. Majumdar, A. S.; Mukherjee, [*Modern Phys. Lett.*]{} [**A 32**]{}, 2487 (2005).
A. Majumdar and N. Mukherjee, in this proceedings volume.
A. F. Zakharov, F. de Paolis, A. A. Nucita and G. Ingrosso, in this proceedings volume.
S. Chandrasekhar, [*The mathematical theory of black holes*]{} (Oxford University Press, Oxford, 1983).
B. O’Neill, [*The geometry of [K]{}err black holes*]{} (A. K. Peters, Wellesley, 1995).
V. Bozza, [*Phys. Rev.*]{} [**D 67**]{}, 103006 (2003).
V. Bozza, F. de Luca, G. Scarpetta and M. Sereno, [*Phys. Rev.*]{} [**72**]{}, 083003 (2005).
V. Bozza, F. de Luca and G. Scarpetta, [*Phys. Rev.*]{} [**74**]{}, 063001 (2006).
F. de Luca, in this proceedings volume.
S. V[á]{}zquez and E. Esteban, [*Nuovo Cimento*]{} [**119**]{}, 489 (2004).
M. Sereno and F. de Luca, [*Phys. Rev.*]{} [**D 74**]{}, 123009 (2006).
M. Sereno, in this proceedings volume.
K. Rauch and R. Blandford, [*Astrophys. J.*]{} [**421**]{}, 46 (1994).
R. S. Hanni, [*Phys. Rev.*]{} [**D 16**]{}, 933 (1977).
F. Grave, F. Frutos-Alfaro, T. M[ü]{}ller and D. Adis, in this proceedings volume.
P. Ehrlich and G. Emch, [*Rev. Math. Phys.*]{} [**4**]{}, 163 (1992), Errata: 4, 501 (1992).
P. Ehrlich and G. Emch, in [*Differential geometry. Pt. 2: Geometry in mathematical physics and related topics*]{}, eds. R. Greene and S. Yau, Proceedings of Symposia in Pure Mathematics, Vol. 54 (American Mathematical Society, Providence, U.S.A., 1993).
J. K. Beem, P. E. Ehrlich and K. L. Easley, [*Global [L]{}orentzian geometry*]{} (Marcel Dekker, New York, 1996).
R. Penrose, [*Rev. Modern Phys.*]{} [**37**]{}, 215 (1965).
A. O. Petters, [*J. Math. Phys.*]{} [**33**]{}, 1915 (1992).
K. Uhlenbeck, [*Topology*]{} [**14**]{}, 69 (1975).
R. McKenzie, [*J. Math. Phys.*]{} [**26**]{}, 1592 (1985).
I. Kovner, [*Astrophys. J.*]{} [**351**]{}, 114 (1990).
F. Giannoni, A. Masiello and P. Piccione, [*Ann. Inst. H. Poincar[é]{}*]{} [ **69**]{}, 359 (1998).
V. Perlick, [*Class. Quant. Grav.*]{} [**7**]{}, 1319 (1990).
G. Temple, [*Proc. Roy. Soc. London*]{} [**A 168**]{}, 122 (1938).
A. Bernal and M. S[á]{}nchez, [*Comm. Math. Phys.*]{} [**257**]{}, 43 (2005).
T. Frankel, [*The geometry of physics*]{} (Cambridge University Press, Cambridge, 1997).
J. P. Serre, [*Ann. Math.*]{} [**54**]{}, 425 (1951).
S. W. Hawking and G. F. R. Ellis, [*The large scale structure of space-time*]{} (Cambridge University Press, Cambridge, 1973).
W. Hasse and V. Perlick, [*J. Math. Phys.*]{} [**47**]{}, 042503 (2006).
J. Ehlers, [*Ann. Phys. (Leipzig)*]{} [**9**]{}, 307 (2000).
J. Ehlers, S. Frittelli and E. Newman, in [*Revisiting the foundations of relativistic physics*]{}, eds. A. Ashtekar, R. Cohen, D. Howard, J. Renn, S. Sarkar and A. Shimony, Boston Studies in the Philosophy of Science, Vol. 234 (Kluwer, Dordrecht, Netherlands; Boston, U.S.A., 2003).
V. Perlick, [*Comm. Math. Phys.*]{} [**220**]{}, 403 (2001).
R. Giamb[ó]{}, F. Giannoni and P. Piccione, [*Nonlinearity*]{} [**17**]{}, 117 (2004).
M. A. Javaloyes and P. Piccione, in [*Variations on a century of relativity*]{}, ed. E. Barletta, Lecture Notes of Seminario Interdisciplinare di Matematica (Universit[á]{} degli [S]{}tudi della [B]{}asilicata, Potenza, 2006).
V. Perlick, [*Class. Quant. Grav.*]{} [**13**]{}, 529 (1996).
C.-M. Yoo, K. Nakao, H. Kozaki and R. Takahashi, in this proceedings volume.
M. Miranda, P. Jetzer and A. Macci[ò]{} in this proceedings volume.
F. Frutos-Alfaro, H. Sol[í]{}s-S[á]{}nchez, in this proceedings volume.
|
---
abstract: 'Multidimensional potential energy landscapes (PELs) have a Gaussian distribution for the energies of the minima, but at the same time the distribution of the hyperareas for the basins of attraction surrounding the minima follows a power-law. To explore how both these features can simultaneously be true, we introduce an “egg-box” model. In these model landscapes, the Gaussian energy distribution is used as a starting point and we examine whether a power-law basin area distribution can arise as a natural consequence through the swallowing up of higher-energy minima by larger low-energy basins when the variance of this Gaussian is increased sufficiently. Although the basin area distribution is substantially broadened by this process, it is insufficient to generate power-laws, highlighting the role played by the inhomogeneous distribution of basins in configuration space for actual PELs.'
address: 'Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ, United Kingdom'
author:
- 'Claire P. Massen'
- 'Jonathan P. K. Doye'
- 'Rupert W. Nash'
title: 'Exploring the origins of the power-law properties of energy landscapes: An egg-box model'
---
Introduction
============
Potential energy landscapes describe how the potential energy of a system depends on the coordinates of its constituent atoms [@Landscapes]. As the thermodynamics and dynamics of a system are controlled by the PEL, there has been much interest in exploring the insights into the behaviour of relatively complex systems that can be obtained by characterizing the features of the PEL. Particular active research areas for this landscape approach have been supercooled liquids and protein folding. For example, the dynamics of fragile liquids change at lower temperatures, as the system explores low-energy regions of the PEL, where the effective activation energies for structural relaxation are larger [@Debenedetti01]. Furthermore, due to the astronomical number of conformations, the Levinthal paradox predicts that protein folding would be extremely unlikely to occur if these conformations were searched randomly [@Levinthal]. However, a funnel on the PEL directs the protein towards its folded state [@Leopold; @Bryngelson95].
Here, our focus is not so much on how to relate the behaviour of the system to the underlying PEL, but rather it is to understand the fundamental properties of PELs better. When classifying high-dimensional PELs it is common to focus on the stationary points of the PEL, particularly the minima and transition states (first-order saddle points). Fundamental results concerning these stationary points include that the number of minima scales exponentially with the system size [@Stillinger82; @Stillinger99; @Tsai93], and that the distribution of energies of the minima is Gaussian. The latter has been found empirically for a range of systems [@Buchner99; @Sciortino99], and can be justified theoretically from the central limit theorem, assuming the system can be broken down into independent subsystems [@Heuer00].
More recently, a network approach has been applied to gain insights into the connectivity of PELs [@Doye02; @Doye05b]. In these “inherent structure” networks, the minima correspond to nodes, and two nodes are joined by an edge if the corresponding minima are directly connected by the two steepest descent pathways going away from a transition state. Such networks are dynamically motivated since if the temperature is not too high, the dynamics can be separated into vibrational motion in the basin surrounding a minimum with hopping between minima via transition state valleys on longer time scales [@Stillinger84; @Schroder98].
Interestingly, for small clusters, for which the complete inherent structure networks can be obtained, these networks have been found to be scale-free, that is the degree (the number of connections to a node) distribution has a power-law tail [@Barabasi99]. For such a topology, most of the nodes have relatively few connections, but there are a small number of hubs that have a large number of connections, and which play an important role in connecting up the landscape. Given that these inherent structure networks are based on the adjacency of basins in configuration space, it may seem surprising that the networks are so heterogeneous, and not more lattice-like. However, the energy of the minima plays a key role, with the lower energy minima acting as hubs in the network, because they have larger basins of attraction, and so can have many transition states along the boundary. Thus this leads to a picture with larger basins surrounded by smaller basins, which are in turn surrounded by smaller basins, and so on. Apollonian packings (see Fig. \[fig:apollo\] for a two-dimensional example) provide a model of how such a hierarchical arrangement can be achieved. Furthermore, the network of contacts between the disks in such a packing is also scale-free [@Andrade05; @Doye05]. The analogy holds in one further way. The distribution of disk areas in the Apollonian packing follows a power-law. Similarly, the distribution of basin areas on a range of PELs has also been found to follow a power-law with exactly the predicted exponent [@Massen05b].
![ An Apollonian packing. Space is filled with different sized disks, starting from an initial configuration where the three larger disks are placed within the bounding circle. The packing is generated iteratively, with the largest disk possible added to each gap at each iteration. This process is continued ad infinitum, thus filling the space with successively smaller and smaller disks. []{data-label="fig:apollo"}](Apcircle.ps){width="6.6cm"}
Despite the success of the Apollonian analogy, one of its deficiencies is that it provides no explanation for the role of energy in the organization of the landscape. In particular, it is not clear how a Gaussian distribution for the energies of the minima is compatible with the power-law area and degree distributions. The broad area distribution implies that there are some very big basins and lots of small basins in the landscape, whereas a Gaussian energy distribution shows much less variation from the mean, and is symmetrical. In order for both of these distributions to occur, the dependence of the basin area on the energy of a minimum is necessarily very steep, with the increase in the area as the energy decreases having to be faster than exponential. The aim of this work is to explore a model landscape (which we call the “egg-box” model) to determine whether the broad area distribution might actually be a natural consequence of the Gaussian energy distribution, or whether it is due to a particular way that PELs are organized. We will describe the nature and resulting properties of our model landscapes in Section \[sec:model\], and discuss the implications of the results in Section \[sec:concs\].
The egg-box model {#sec:model}
=================
![ Examples of a one-dimensional egg-box model with (a) $\sigma^*=0.01$ (b) $\sigma^*=1$ (c) $\sigma^*=100$, where $\sigma^*$ is the standard deviation of the Gaussian energy distribution for the minima in reduced units. Initially there are ten minima, but as $\sigma^*$ increases, successive minima get swallowed up, until there is only one low-energy minimum remaining. []{data-label="fig:sigma"}](sigma.eps){width="8.6cm"}
The idea of our model is to assign minima (which are surrounded by harmonic basins) to particular positions in configuration space, typically a lattice, e.g. a square lattice in two dimensions. If these minima all have the same energy, then each basin of attraction will have the same area, and the same connectivity. This, if you like, is the “egg-box” limit. However, if instead the energies of the minima have a Gaussian distribution of sufficient variance, there is the possibility that some of the higher-energy minima will get swallowed up by the basins of lower energy minima, as illustrated in Fig. \[fig:sigma\]. This process will inevitably lead to a broader area distribution and, for dimension greater than one, a broader degree distribution. But, the question is whether it will be sufficient to generate the power-law area and degree distributions that have been empirically observed for real PELs.
![(Colour online) Variation of the basin area distribution with the width of the Gaussian energy distribution for the one-dimensional egg-box model. (a) Area distributions, (b) cumulative distributions, with the area measured with respect to its mean. Each is for the one dimensional system with 15625 minima initially (some may be eaten). []{data-label="fig:sigma-pa"}](sigma-pa.eps){width="8.6cm"}
More formally, the potential energy for our model is given by $$V({\mathbf r})=\min\left[\left\{V_i({\mathbf r})\right\}\right],$$ where $$V_i({\mathbf r})=E_i+{1\over 2} k \left|{\mathbf r}-{\mathbf r_i}\right|^2,$$ ${\mathbf r_i}$ is the site of a harmonic potential and $E_i$ is the energy at its bottom. Each site will give rise to a minimum on the PEL, except for those sites where $V({\mathbf r_i})<E_i$. For the sites of the initial minima we use a hypercubic lattice, and apply periodic boundary conditions. The parameters of this egg-box model are the lattice spacing $a$, the force constant $k$ of the harmonic potential surrounding each minimum, and the mean $E_0$ and standard deviation $\sigma$ of the Gaussian. We always centre the Gaussian at zero, leaving three remaining variables, two of which define the units of energy and length, with one effective parameter. We use $a$ and $ka^2$ as the units of length and energy respectively, and choose to vary $\sigma^*=\sigma /ka^2$. As already mentioned, as $\sigma^*$ is increased the landscape becomes more heterogeneous (Fig. \[fig:sigma\]). One obvious question is whether this evolution reaches some kind of steady state. The results in Fig. \[fig:sigma-pa\] for a one-dimensional example show that although the mean basin area will continue to increase with $\sigma^*$, the distribution tends to a limiting form. We focus on the landscapes produced in this limit since this is when the area distributions will be at their broadest. Fig. \[fig:sigma-pa\] suggests that landscapes with $\sigma^* = 1$ will be sufficient for this purpose, and this is the value we generally use.
In order to compare to the egg-box model to PELs, we of course need to examine the model in higher dimension. Some of these effects are illustrated in the two-dimensional example shown in Fig. \[fig:2dpic\]. Firstly, the basins exhibit a wider range of areas. However, the area distribution is clearly much narrower than that for the Apollonian packing in Fig. \[fig:apollo\], and there is no sign of any hierarchical ordering. Secondly, as the dimension is increased, more basins are eaten because each basin has more neighbours and the largest possible degree increases.
![ Example of the egg-box model in two dimensions. The grid shows the positions of the initial minima, the black squares represent those that remain at $\sigma^*=1$, and the thick lines represent the boundaries of their basins of attraction. []{data-label="fig:2dpic"}](2dpic.eps){width="7.6cm"}
![(Colour online) Variation of the cumulative basin area distribution with the dimension of configuration space for $d=1$–6. In (a) the distribution is plotted against the absolute area, whereas in (b) the area is measured relative to its mean value. Also shown is the power-law $P_{cum}(A) \sim A^{-1}$ which is the form seen for high-dimensional Apollonian packings [@Doye05] and various PELs [@Massen05b]. []{data-label="fig:dim-pa"}](dim-pa.eps){width="8.6cm"}
Quantitative results have been obtained for systems with up to six dimensions and around 15625 initial minima. For higher dimensions, analytical approaches to determine the shapes of the basins, and hence their areas, become increasingly involved, so we instead use a Monte Carlo approach. A large number of points were chosen at random and the basins to which the points belong were determined. The area of a basin is then proportional to the number of points assigned to that minimum. Area distributions are shown in Fig. \[fig:dim-pa\] for increasing dimension. There is a wider range of basin sizes in higher dimension, where small changes in radius have a larger effect. The average and maximum areas also increase because more basins are eaten.
In the results shown so far, the minima have initially been placed on a lattice. We also consider landscapes where the initial minima have instead been given random positions. As shown in Fig. \[fig:lat-pa\], this broadens the area distribution because the starting points are less uniformly separated. For example, when $\sigma^*=0$, the area distribution is a delta function if the basins start on a hypercubic lattice. However, there will be some heterogeneity if the basins are given random starting points, leading to a broader distribution. This effect is weaker in higher dimensions, and so the choice of the initial sites becomes less important.
) and solid lines are for when they are distributed randomly. []{data-label="fig:lat-pa"}](lat.eps){width="8.6cm"}
We can compare the area distributions seen for the egg-box model to that seen for PELs [@Massen05b] and for the Apollonian packing [@Doye05]. The latter two both follow a power-law with exponent close to $-2$ (or $-1$ for the cumulative distribution), and such a power-law has also been plotted in Fig. \[fig:dim-pa\] for comparison. This form is tangential to the observed basin area distributions, but only for a small range of areas, and the distributions show significant curvature. By contrast, the area distributions for PELs associated with common model liquids followed an approximate power-law over up to 18 decades. However, the configuration space for the latter had 765 dimensions. Therefore, it is natural to ask whether in the limit of much higher dimensions the egg-box model might show similar behaviour.
![(Colour online) (a) Basin area and (b) average radius as a function of energy for dimensions 1–6. Each point represents an average of those minima in a given energy bin. []{data-label="fig:ae"}](ae.eps){width="8.6cm"}
Extending our results to significantly higher dimensions, however, is not feasible because of the rapidly increasing number of lattice points required. Instead, our approach is to see whether the trends with increasing dimension are in the right direction to make a power-law a possibility. In this regard, the variation of basin area with energy of the minimum is particularly informative (Fig. \[fig:ae\]). As expected, low-energy basins are bigger. Furthermore, the energy-dependence of the basin area becomes steeper with increasing dimension, because the maximum area increases, giving a wider range of areas, and $E_{max}$ decreases as high-energy basins are more likely to be eaten. Furthermore, small changes in radius have a larger effect due to the higher dimension.
Most importantly, the curvature of $\log A(E)$ is negative. This contrasts with the PELs studied, which showed the positive curvature that is necessary to give rise to a power-law area distribution. In the egg-box model, as we go to lower energies, the curve gets flatter. Therefore, we have to go to much lower energies to see basins with big areas, but a Gaussian energy distribution means that very low-energy minima are very rare. Furthermore, at high energy, there is a steeper cut-off for the egg-box model, meaning there are fewer small basins. This is in the region of the tail of the Gaussian distribution, where $A(E)$ would need to be shallow in order to see lots of high-energy, low-area basins, as is seen in PELs.
These trends suggest that, with increasing dimension, the egg-box model will not become more like PELs. On the contrary, the negative curvature becomes even more pronounced. This is confirmed on examining the dependence of the radius on energy in Fig. \[fig:ae\](b), where the radius of a basin was calculated from the area by assuming that the basin was hyperspherical. $r(E)$ is almost linear, and seems to be tending towards a limiting form, thus further implying that the curvature of $\log A(E)$ will be even more negative in higher dimension.
![ Cumulative degree distribution for the two-dimensional egg-box model with $\sigma^*=1$. []{data-label="fig:pk"}](pk.eps){width="8.6cm"}
Some network properties for the two-dimensional egg-box have also been studied. Initially, there were 15625 minima. As some of these are eaten, the final number of minima is $N=4140$, so the size of the network is very similar to that of the ISN for the 14-atom cluster LJ$_{14}$ studied previously [@Doye02]. The number of edges in the network is $M=12\,142$, giving an average number of connections per node (average degree) of $\langle k \rangle = 2M/N = 5.87$. This can be directly compared to the ISN of LJ$_{14}$, which has $\langle k \rangle = 29.12$. Therefore, the egg-box model has much lower overall connectivity than the PEL. The maximum connectivity, $k_{max}=13$, is again much lower than that seen for LJ$_{14}$, where the highest-connected basin is connected to 3201 others.
The degree distribution for the 2D egg-box model is roughly exponential, as shown in Fig. \[fig:pk\]. By contrast, the networks associated with PELs and the Apollonian packing have scale-free character, i.e. their degree distributions follow power-laws. This behaviour is due to the presence of some very large basins, with very long boundaries, which have high degree and act as hubs, connecting up the networks. By contrast, the egg-box model features neither very large basins to act as hubs, nor many small basins around their boundaries, as evident from comparing Figs. \[fig:apollo\] and \[fig:2dpic\].
The network of the egg-box model has very long paths between nodes. This result cannot be compared directly to the PELs due to the different sizes and connectivity patterns, both of which affect path lengths. However, if we compare the path length of the egg-box to that of a random network of the same size and having the same degree distribution, we see that the average path length is roughly four times that of the random network. By contrast, for the PEL of clusters LJ$_{7}$ to LJ$_{14}$, the path length is between 1-1.05 times that for a random network. The egg-box model therefore has very long path lengths. The egg-box model is also very highly clustered, i.e. the neighbours of a node are also likely to be connected. Its clustering coefficient [@Watts98] is approximately 30 times that for a random network, compared to roughly 1.6 times for the ISN of LJ$_{14}$. Therefore, with its long path lengths and high clustering, the egg-box model has very similar properties to a lattice. Despite the broadening of the area distribution resulting from the swallowing up of higher-energy minima as $\sigma^*$ is increased, the contact network between the basins does not change its essential character. Although it is possible that these properties may change for higher dimensions, it has been shown that PELs behave differently to high-dimensional lattices, in particular having shorter path lengths [@Doye02; @Doye05b].
Conclusions {#sec:concs}
===========
The aim of this paper was to determine whether the power-law area distribution observed in PELs is a natural consequence of a Gaussian distribution of energies for the minima, or whether it reflects further ordering of the PEL. We do see some of the expected trends. Increasing the width of the Gaussian leads to a broadening of the area distribution, where lower-energy basins have higher area and higher degree. However, this is not sufficient to reproduce the properties of real PELs. In some ways, this might be regarded as a negative result, however, it allows us to get a better understanding of what the key features of real PELs are that lead to the observed properties.
Some assumptions are used in the egg-box model that constitute possible causes of the differences between the resultant model landscapes and PELs. Firstly, the dimension of the models studied is much lower than that for PELs of all but the smallest systems. However, the trends for the dependence of the area distribution on dimension that we observed are not going in the right direction. Thus higher-dimensional model landscapes are unlikely to correct the deficiencies of the lower-dimensional models examined here.
Secondly, we assumed that the basins have a harmonic form. Generally, it is found that applying the harmonic approximation to the basins of real PELs gives a correct qualitative description of the systems’ behaviour [@Doye95; @Sciortino00], and so we think this assumption is unlikely to have a significant effect on our conclusions. Thirdly, we assumed that the force constant $k$ was the same for all basins, independent of the basin energy. However, given that the vibrational frequencies for clusters [@Wales93; @Noya06] and for liquids at constant volume [@Middleton01b] have an opposite dependence on energy, yet the PELs of examples of both of these types of system have been seen to have broad area distributions [@Massen05b], we think this assumption is unlikely to have too much effect.
Fourthly, there is an absence of correlations between neighbouring minima in the egg-box model, i.e. the assigned energy of an initial minimum is independent of that of its neighbours. However in real landscapes, the energies of neighbouring minima are correlated with minima more likely to be connected to others of similar energy [@Doye05b]. The likely effect of such correlations on the current model would be to increase the number of high-energy minima as they are less likely to be swallowed up if they are surrounded by minima of similar energy, and to decrease the area of lower-energy basins as they are more likely to be surrounded by other lower-energy minima restricting the growth of their basins. Thus the dependence of the basin area on energy would be likely to be less steep, and in itself the introduction of correlations into our model is unlikely to lead to a further broadening of the area distribution.
Fifthly, there is the homogeneous distribution of initial minima in the egg-box model. Both the lattice and the random initial configurations lead to an essentially uniform density of points, in contrast with the Apollonian packing which is very heterogeneous, with small basins clustered in the gaps between larger basins. Such an inhomogeneous distribution of minima combined with energy correlations, where the high-energy minima are more closely spaced and the low-energy minima are more separated, would potentially lead to larger areas for the lower-energy minima and a greater number of small high-energy minima, thus impacting on the curvature of $A(E)$ and broadening the area distribution.
Thus, the differing properties of our egg-box model landscapes and real PELs are most likely due to the inhomogeneous distribution of minima in configuration space, and the hierarchical nature of the packing of the basins of attraction on the PEL. This conclusion therefore highlights the role these features play in giving rise to the power-law basin area distribution observed.
But, this of course, begs the question what then are the sources of the inhomogeneity. This will be the subject of future work, but some partial answers come from a further difference between the egg-box model and real PELs, namely the lack of a corresponding real space atomic structure in the egg-box model. For example, high-energy minima on a real PEL will correspond to disordered configurations, where the structure is frustrated. It has been found that such structures have many non-diffusive modes, which have small barriers leading to different minima, but involve no real reordering of atom positions [@Middleton01b]. Thus there will be a high density of high-energy minima with small basins clustered together. By contrast, for lower-energy minima the local environment around each atom is likely to be much closer to optimal, and to go between basins requires following a diffusive mode where there is a significant reordering of the atomic positions, associated with some atoms escaping from their caging environments. Thus, the organization of the PEL and the atomic structure are intimately related.
The authors would like to thank the Royal Society (JPKD) and the EPSRC (CPM) for financial support.
[25]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
, ** (, , ).
, ****, ().
, in **, edited by , , (, , ).
, , , ****, ().
, , , , ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, , , ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, ****, ().
, , , , ****, ().
, ().
, ****, ().
, ****, ().
, , , ****, ().
, ****, ().
, ****, ().
, ****, ().
|
---
abstract: |
For scattering off a smooth, strictly convex obstacle $\Omega \subset \mathbb{R}^d$ with positive curvature, we show that the eigenvalues of the scattering matrix – the phase shifts – equidistribute on the unit circle as the frequency $k \to \infty$ at a rate proportional to $k^{d - 1}$, under a standard condition on the set of closed orbits of the billiard map in the interior. Indeed, in any sector $S
\subset \mathbb{S}^1$ not containing $1$, there are $c_d |S| {\mathrm{Vol}}({\partial}\Omega)\ k^{d - 1} + o(k^{d-1})$ eigenvalues for $k$ large, where $c_d$ is a constant depending only on the dimension. Using this result, the two term asymptotic expansion for the counting function of Dirichlet eigenvalues, and a spectral-duality result of Eckmann-Pillet, we then give an alternative proof of the two term asymptotic of the total scattering phase due to Majda-Ralston [@MajdaRalston1978].
address:
- 'School of Mathematics and Statistics, University of Melbourne'
- 'Laboratoire de Mathematiques d’Orsay, Universite Paris-Sud'
author:
- 'Jesse Gell-Redman'
- Maxime Ingremeau
bibliography:
- 'obstacle.bib'
title: Equidistribution of phase shifts in obstacle scattering
---
Introduction
============
Let $\Omega \subset \mathbb{R}^{d}$ denote a smooth, strictly convex set with positive sectional curvature. We shall write $\Omega^c := {\mathbb{R}}^d\backslash \Omega$. It is well-known (see for instance [@GST §5] or [@dyatlovmathematical §4.4]) that for any $k>0$ and any $\phi_{in}\in C^\infty ({\mathbb{S}}^{d-1})$, there is a unique solution $u\in C^\infty(\overline{\Omega^c})$ to the Dirichlet problem $$(\Delta + k^2)u = 0 \qquad u \rvert_{{\partial}\Omega} = 0$$ such that $$\label{eq:defscattering}
u(x)=|x|^{-(d-1)/2}\big{(} e^{-i k|x|} \phi_{in}(-{\widehat{x}}) +
e^{i k |x|} \phi_{out}({\widehat{x}}) \big{)} + O_{|x|\rightarrow \infty}(|x|^{-(d+1)/2}),$$ where we write ${\widehat{x}}= \frac{x}{|x|}\in {\mathbb{S}}^{d-1}$ and $\Delta =
\sum_{i = 1}^d {\partial}_{x_i}^2$. In particular $\phi_{out}$ is determined by $\phi_{in}$ and we define the *scattering matrix* $S(k)$, which depends on $k$ and $\Omega$, by $$S(k)(\phi_{in}) := e^{i\pi (d-1)/2} \phi_{out}.$$ In fact $S(k)$ extends to a unitary operator acting on $L^2({\mathbb{S}}^{d-1})$ with the property that $S(k) - {\mathrm{Id}}$ is trace class [@taylor:vol2; @RSIII]. Therefore, for any $k>0$, $S(k)$ has purely discrete spectrum, accumulating only at 1, which we denote by $\sigma(S(k)) := \{e^{i
\beta_{k,n}} \}$. *Our aim in this paper will be to study the asymptotic distribution of* the $e^{i \beta_{k,n}}$ as $k\rightarrow \infty$.
One of our main results is an estimate for the number of phase shifts in a sector $S \subset \mathbb{S}^1 \setminus \{ 1 \}$ as $k \to \infty$. Define the counting function $$N_k(\phi_0, \phi_1, \Omega) = N_k(\phi_0, \phi_1) := \# \{ e^{i
\beta_{k,n}} \in \sigma(S(k)) : \phi_0 <
\beta_{k, n} < \phi_1,\ \mathrm{mod}\ 2\pi \}.$$ Letting $\omega_{d-1} = |B^{d-1}|$ where $B^{d-1}$ is the unit ball in $\mathbb{R}^d$, we will prove $$\label{eq:number}
N_{k}(\phi_{0}, \phi_{1}) = \frac{\omega_{d-1}}{(2\pi)^{d-1}} \Big{(}\frac{\phi_{1} - \phi_{0}}{2\pi} \Big{)}
{\mathrm{Vol}}({\partial}\Omega) k^{d-1} + o(k^{d-1}).$$ In particular, the phase shifts accumulate in each sector $S$ at a rate proportional to $k^{d-1}$ as $k \to \infty$ times ${\mathrm{Vol}}({\partial}\Omega)
|S|$. The estimate in follows from Theorem \[th: main theorem\], see Section \[sec:proofs\].
To study the asymptotic distribution of the phase shifts, consider the measure $\mu_{k}$ on the circle $\mathbb{S}^{1}$, defined for continuous functions $f \colon \mathbb{S}^{1} {\longrightarrow}\mathbb{C}$ by $$\label{eq:measure}
{\langle}\mu_{k}, f {\rangle}= \Big{(} \frac{2\pi}{k}\Big{)}^{d - 1} \sum_{\sigma(S(k))}
f(e^{i \beta_{k,n}}).$$ Note that $ {\langle}\mu_{k}, f {\rangle}$ is finite if $1 \not \in \operatorname{supp}f$. The following theorem describes the behavior $\mu_k$ as $k\rightarrow \infty$, provided (\[eq: hypVol3\]) holds, which is a standard assumption on the volume of the periodic points of the inside billiard map. Note that this assumption holds if our smooth convex obstacle, is generic, or is analytic (see the discussion at the end of Section \[sec:dynamics\]).
\[th: main theorem\] Let $\Omega\subset {\mathbb{R}}^d$ be a smooth strictly convex open set, such that (\[eq: hypVol3\]) holds. Then for any $f
\colon \mathbb{S}^{1} {\longrightarrow}\mathbb{C}$ with $\operatorname{supp}f \cap {\left\{ 1 \right\} } =
\varnothing$, we have $$\label{eq:limit_measure}
\lim \limits_{k\rightarrow \infty} {\langle}\mu_{k}, f {\rangle}=
\frac{{\mathrm{Vol}}({\partial}\Omega) \omega_{d-1}}{2\pi} \int_{0}^{2\pi}f(e^{i\theta}) d\theta$$
The factor in front of the integral in arises as the volume of the ‘interacting region’ in phase space of incoming rays from the sphere at infinity that make contact with the obstacle. See Section \[sec:dynamics\] for further description of the classical dynamics. In [@MR3335243], in which the first author and collaborators studied the same problem for semiclassical potential scattering, they defined a measure $\mu_h$, depending on a semiclassical parameter $h \to 0$, analogously to the measure in except they included the volume of the interacting region. Here we prefer not to, so that the dependence on the interacting region appears explicitly in the limit measure.
As an application of the equidistribution of the measure $\mu_k$, we will give an alternative proof of the following result of Majda-Ralston, generalized by Melrose and then by Robert, regarding the asymptotic development of the total scattering phase $$\label{eq:scattering-phase}
s(k) = i \log \det S(k).$$ The scattering phase $s(k)$ can be defined in a natural way so that $s(k) \in C^{\infty}((0, \infty))$.
\[thm:scattering-phase\] Let $\Omega$ be a smoothly bounded, strictly convex obstacle whose set of periodic billiard trajectories has measure zero. Then $$\label{eq:scattering-phase-asymptotics}
s(k) = \frac{\omega_d}{(2\pi)^{d-1}} {\mathrm{Vol}}(\Omega) k^d +
\frac{\omega_{d - 1}}{4 (2\pi)^{d-2}} {\mathrm{Vol}}({\partial}\Omega) k^{d -1} + o(k^{d-1}).$$
In fact, Theorem [@MelroseWeylExterior; @Robert1996] holds for *all smoothly bounded, compact domains* satisfying the stated assumption on the periodic trajectories.
As we describe in Section \[sec:proofs\], the novelty in our proof comes from its use of the explicit relationship between the counting function for the Dirichlet eigenvalues, $$\label{eq:counting-function}
N_D(\lambda_0) := \# \{ 0 < \lambda < \lambda_0 : \exists \phi \in L^2(\Omega),\ \phi\rvert_{{\partial}\Omega} = 0,\ \Delta \phi
= - \lambda^2 \phi, \phi \neq 0 \}.$$ and the scattering phase which arises from the spectral duality result of Eckmann-Pillet [@EP1995]. Indeed, note that the leading order term in is $2\pi$ times the leading order term in Weyl’s law [@ivrii1980second], which is to be expected since, as explained in Section \[sec:proofs\], ‘inside-outside’ duality says that a phase shift makes a complete rotation of the unit circle for each Dirichlet eigenvalue of $\Omega$.
The main technical ingredient in the proof of Theorem \[th: main theorem\] is a trace formula for powers of the scattering amplitude $$\label{eq:scattering-matrix-and-amplitude}
A(k) := S(k) - {\mathrm{Id}}.$$
\[thm:trace-lemma\] Suppose that (\[eq: hypVol3\]) holds. Let $p \in \mathbb{Z}$. Then for all $\varepsilon>0$, we have $$\label{eq: TracePowers}
\mbox{Tr} A^p(k) = (-1)^p
{\mathrm{Vol}}({\partial}\Omega) \omega_{d-1} \Big{(}\frac{k}{2\pi}\Big{)}^{d-1} + O(k^{d - 1 -1/3 + \varepsilon}).$$ In particular, for any trigonometric polynomial $P$ vanishing at $1$ and for the measure $\mu_k$ in , as $k \to \infty$, $$\langle \mu_k, P\rangle =
\frac{{\mathrm{Vol}}({\partial}\Omega) \omega_{d-1}}{2\pi} \int_{{\mathbb{S}}^1} P(\theta) \mathrm{d}\theta + O(k^{-1/3+\varepsilon})$$
As we show in Section \[sec:proofs\], the trace formula in Proposition \[thm:trace-lemma\], together with $k$-dependent bounds for the number of eigenvalues at least a fixed exponentially small distance from $1$, imply Theorem \[th: main theorem\].
It may appear surprising that the bound on the error is much better in (\[eq: TracePowers\]) than in (\[eq:limit\_measure\]). The reason for this is that in (\[eq: TracePowers\]), the constant in the $O(k^{d - 1 -1/3 + \varepsilon})$ depends on the degree of the polynomial $P$, and it grows exponentially with it. Therefore, when we want to approach a continuous function by polynomials to prove Theorem \[th: main theorem\], our control on the remainder becomes much worse. Actually, we believe that, by paying attention to all the constants appearing in the remainders appearing in the proof of Proposition \[thm:trace-lemma\], it should be possible to obtain that, provided $f$ is regular enough, we have $${\langle}\mu_{k}, f {\rangle}=
\frac{{\mathrm{Vol}}({\partial}\Omega) \omega_{d-1}}{2\pi} \int_{0}^{2\pi}f(e^{i\theta}) d\theta + O\big{(}(\log k)^{-\varepsilon}\big{)},$$ for some $\varepsilon>0$.
### Relation to other works {#relation-to-other-works .unnumbered}
Since the pioneering works of Birman, Sobolev, and Yafaev (see for example [@SY1985; @BY1984]), there has been a wealth of literature on the asymptotic behavior of the scattering matrix at high energy, in particular about the distribution of phase shifts. In semi-classical potential scattering, an analogous result for compactly supported potentials was proven by the first author, Hassell, and Zelditch in [@MR3335243] for non-trapping potentials, and was generalized to trapping potentials by the second author in [@I2016]. See [@MR3335243] for a complete literature review of phase shift asymptotics for potential scattering. The behaviour of the phase shifts in the semi-classical limit has been studied in various settings: for magnetic potentials ([@BP2012]), for scattering by radially symmetric potentials, in [@DGHH2013], near resonant energies in [@nakamura2014spectrum]...
The idea of using trace formulae to analyze the asymptotics of the spectra comes from [@Z1992; @Z1997], and was the starting point of [@MR3335243], [@I2016] and of the present paper. The main tool we use here is the Kirchhoff approximation, which was proven in its optimal form in [@Melrose-Taylor-near-peak]. Finally, our proof is simplified by describing the micro-local properties of the scattering matrix in terms of its action on Gaussian states, an approach which was introduced in [@I2016] for potential scattering.
### Organisation of the paper {#organisation-of-the-paper .unnumbered}
In Section \[sec:dynamics\], we will recall a few facts about the classical scattering dynamics, and its links with the interior billiard dynamics. In Section \[sec: tools\], we will recall the main tools we use in the proof of Proposition \[thm:trace-lemma\]. Proposition \[thm:trace-lemma\] is then proved in Section \[sec : proof proposition\]. Finally, we prove Theorems \[th: main theorem\] and \[thm:scattering-phase\] in Section \[sec:proofs\]. The appendix contain elementary facts of semiclassical analysis, a proof of a resolution of identity formula on the sphere, as well as a cumbersome determinant computation.
Classical scattering dynamics and interior dynamics {#sec:dynamics}
===================================================
Let $\omega\in {\mathbb{S}}^{d-1}$ and $\eta\in \omega^\perp\subset {\mathbb{R}}^d$. We will always identify $(\omega,\eta)$ with a point in $T^*{\mathbb{S}}^{d-1}$. Consider the line $L_{(\omega,\eta)}:=\{t\omega +\eta, t\in {\mathbb{R}}\}$. By strict convexity of $\partial \Omega$, it intersects $\partial \Omega$ in zero, one or two points. We define the interaction region, $$\label{eq: def I}
\mathcal{I}:= \{ (\omega,\eta)\in T^*{\mathbb{S}}^{d-1} ; L_{(\omega,\eta)}\cap \partial \Omega \text{ contains two points} \}.$$ If $(\omega,\eta)\in \mathcal{I}$, then there exists $t_1<t_2$ such that $t_i\omega+\eta \in \partial \Omega$ for $i=1,2$, we set (see Figure \[fig: kappa\])
![The construction of the scattering map $\kappa$.[]{data-label="fig: kappa"}](ScOb)
$$\begin{aligned}
x(\omega,\eta)&:= t_1\omega+\eta\in \partial \Omega\\
\omega'(\omega,\eta)&:= \omega- 2 (\omega\cdot \nu_{x(\omega,\eta)}) \nu\\
\eta'(\omega,\eta) &:= x(\omega,\eta) - (\omega'\cdot x(\omega,\eta)) \omega',
\end{aligned}$$
where $\nu_x$ is the outward pointing normal vector at the point $x \in
{\partial}\Omega$. We then set $$\kappa(\omega,\eta) = (\omega',\eta').\label{eq:the-scattering-map}$$ If $(\omega,\eta)\notin \mathcal{I}$, we shall set $\kappa(\omega,\eta) = (\omega,\eta)$. The map $\kappa$ may then be seen as a $C^0$ map $\kappa :
T^*{\mathbb{S}}^{d-1} \rightarrow T^*{\mathbb{S}}^{d-1}$, which is smooth (and even symplectic) away from the glancing set $\partial \Omega$. The fact that interaction region satisfies $${\mathrm{Vol}}(\mathcal{I}) = {\mathrm{Vol}}({\partial}\Omega) \omega_{d-1}$$ follows in a straightforward way from Cauchy’s surface area formula.
For $p\in
\mathbb{Z}\backslash\{0\}$, we will denote by $\mathcal{P}_p\subset
T^*{\mathbb{S}}^{d-1}$ the set of fixed points of $\kappa^p$. Note that we then have $$\mathcal{I}= T^*{\mathbb{S}}^{d-1}\backslash \mathcal{P}_1,$$ and that $\partial \mathcal{P}_1= \partial \mathcal{I}$ is exactly the ‘glancing set’, i.e. the set of $(
\omega, \eta)$ such that $L_{(\omega,\eta)}\cap \partial \Omega$ consists of a single point. We define $$\label{eq:non-trivial-periodic}
\mathcal{P}_p' := \mathcal{P}_p \setminus \mathcal{P}_1,$$ the set of non-trivial glancing periodic points with period $p$, also an invariant subset.
The sets $\mathcal{P}'_p$ will play a central role in our proof, and can be better understood in terms of the periodic points of the interior billiard map, as follows. (Indeed, the relationship between the interior billiard map and the scattering relation is a reflection of the relationship between the interior eigenvalue problem and the phase shifts exemplified in Eckmann-Pillet’s inside-outside duality, as pointed out in [@Sm1992].) Consider the set $\mathcal{O}:=\{(y,\xi)\in S^*\partial \Omega;~ \xi\cdot \nu_y<0\}$. If $(y,\xi)\in \mathcal{O}$, there will be a unique $t>0$ such that $y+t\xi\in \partial \Omega$. We shall then write $y'(y,\xi)= y+ t\xi$, and $\xi'(y,\xi) = \xi - 2 (\xi\cdot \nu_{y'}) \nu_{y'}$. We have $(x',\xi')\in \mathcal{O}$, and we may define $\kappa_{int} : \mathcal{O}\rightarrow \mathcal{O}$ by $\kappa_{int}(x,\xi) = (x',\xi')$. The map $\kappa_{int}$, and we shall denote by $\mathcal{P}_p^{int}$ the set of periodic points of period $p$ of $\kappa_{int}$.
The following elementary lemma makes explicit the link between $\kappa$ and $\kappa_{int}$, as can be seen on Figure \[fig: billiard\].
Let $(\omega,\eta)\in T^*{\mathbb{S}}^{d-1}\backslash \mathcal{P}_1$. We then have $$\kappa_{int} \big{(}(x(\omega,\eta), - \omega\big{)} = \big{(} x(\kappa(\omega,\eta), \omega'(\kappa(\omega,\eta)\big{)}.$$
![The scattering map and the billiard map.[]{data-label="fig: billiard"}](ScOb2)
As a consequence of this lemma, we see that $\mathcal{P}'_p$ is homeomorphic to $\mathcal{P}_p^{int}$.
The volume of the set of fixed points {#the-volume-of-the-set-of-fixed-points .unnumbered}
-------------------------------------
Let us denote by ${\mathrm{Vol}}$ the (symplectic) volume on $T^*{\mathbb{S}}^{d-1}$. We will always assume that we have $$\label{eq: hypVol}
\forall p\in \mathbb{Z}\backslash \{0\}, ~~ {\mathrm{Vol}}(\mathcal{P}'_p) = 0.$$ Let us denote by $d$ the Riemannian distance on $T^*{\mathbb{S}}^{d-1}$. We will often make the following stronger hypothesis.
$$\label{eq: hypVol3}
\forall p\in \mathbb{Z}\backslash \{0\}, ~~ {\mathrm{Vol}}\big{(\{}(\omega,\eta)\in T^*{\mathbb{S}}^{d-1}\backslash \mathcal{P}_1 \text{ such that } d(\kappa^p(\omega,\eta), (\omega,\eta)<\varepsilon)\big{\}}\big{)} = O_{\varepsilon\rightarrow 0} (\varepsilon).$$
These conditions may of course be rephrased in terms of the dynamics of $\kappa_{int}$ on $\mathcal{O}$. If $\mu_\mathcal{O}$ is any Riemannian volume and $d_\mathcal{O}$ is any Riemannian distance on the manifold $\mathcal{O}$, Equation (\[eq: hypVol\]) is equivalent to $$\label{eq: hypVol2}
\forall p\in \mathbb{Z}\backslash \{0\},~~ \mu (\mathcal{P}^{int}_p) = 0,$$ while Equation (\[eq: hypVol3\]) is equivalent to $$\label{eq: hypVol4}
\forall p\in \mathbb{Z}\backslash \{0\}, ~~ \mu_\mathcal{O}\big{(\{}(y,\xi)\in \mathcal{O} \text{ such that } d(\kappa_{int}^p(y,\xi), (y,\xi)<\varepsilon)\big{\}}\big{)} = O_{\varepsilon\rightarrow 0} (\varepsilon).$$
Condition (\[eq: hypVol2\]) is conjectured to hold for all domains $\Omega\subset {\mathbb{R}}^d$, not necessarily convex. This conjecture, known as Ivrii’s conjecture, has implications in terms of remainders for the Weyl’s law for the eigenvalues of the Laplacian (see [@ivrii1980second]). In the generic case, it was shown in [@petkov1988number] that $\mathcal{P}^{int}_p$ is finite for all $p\in \mathbb{Z}\backslash\{0\}$, so that (\[eq: hypVol2\]) holds. Actually, the proof of Petkov and Stoyanov shows that the stronger property (\[eq: hypVol4\]) also holds in the generic case. If the manifold $\partial \mathcal{O}$ is analytic, then the map $\kappa_{int}$ will be analytic, and we can show that (\[eq: hypVol4\]) will hold (see for instance [@safarov1997asymptotic]).
Tools for the proof of Proposition \[thm:trace-lemma\] {#sec: tools}
======================================================
Before proving Proposition \[thm:trace-lemma\], let us recall a few facts we will need in the proof.
An integral representation for the scattering amplitude
-------------------------------------------------------
The operator $A(k)$ introduced in (\[eq:scattering-matrix-and-amplitude\]) can also be defined as follows. Let $v(\cdot; \xi,k)$ be the unique solutions to $$\label{eq:dirichletexterior}
\begin{split}
(\Delta + k^{2}) v &= 0 \\
v \rvert_{{\partial}\Omega} &= -e^{i x \cdot \xi},
\end{split}$$ satisfying the Sommerfeld radiation condition.
$v$ may then be written as $$v(|x|\omega;\xi,k) = |x|^{-(d-1)/2}e^{i k|x|} a(\omega,\xi,k) + O(|x|^{-(d+1)/2}).$$
One can show (see for instance [@HR1976], page 381) that $A(k)$ is given by an integral kernel $$\label{eq:big-a}
A(k)f(\omega)= \int_{\mathbb{S}^{d-1}} a(\omega, \theta, k) f(\theta) dVol_{\mathbb{S}^{d-1}}(\theta),$$ where $a$ satisfies $$\begin{aligned}
\label{eq:little-a}
a(\omega, \theta, k) &= \frac{1}{2i} k^{d-2} (2\pi)^{1-d} \int_{{\partial}\Omega} e^{ik \omega
\cdot y} \frac{{\partial}}{{\partial}\nu} e^{-ik\theta \cdot y} dVol_{{\partial}\Omega}(y) \\
&\qquad + \frac{1}{2i} k^{d-2} (2\pi)^{1-d} \int_{{\partial}\Omega} e^{-ik \theta
\cdot y} \frac{{\partial}}{{\partial}\nu} v(y, -k\omega) dVol_{{\partial}\Omega}(y).\end{aligned}$$
The Kirchhoff approximation
---------------------------
The function ${\partial}_\nu v$ was studied in [@Melrose-Taylor-near-peak], where the authors write $$\label{eq:K-definition}
\frac{{\partial}v(x, -k\omega)}{{\partial}\nu} = K(\omega, x, k) e^{i k x \cdot \omega}.$$ Their main results, or at least what we shall need from them, can be summed up as follows. The definition of the symbol classes $S_\delta$ is recalled in (\[eq:defsymbol\]).
\[th: meltay\] $$K(\omega,x,k) = - i k | \nu_x \cdot \omega| + k E(\omega,x,k),$$ where $\nu_x$ is the outward pointing normal vector at the point $x \in
{\partial}\Omega$, and where $E$ satisfies $$\label{eq:symbolE}
E\in k^{-1/3} S_{1/3}({\mathbb{S}}^{d-1}\times \partial \Omega),$$ Furthermore, for any $\varepsilon>0$, we have that $$\label{eq:NoShadow}
\nu_x\cdot \omega > k^{-1/3+\varepsilon} \Longrightarrow E(\omega,x,k) = O(k^{-\infty}).$$
In particular, we have $$\label{eq:boundE}
||E(\cdot,\cdot,k) ||_{C^0}\leq C k^{-1/3}.$$ We therefore have $$\label{eq : Kirchhoff amplitude}
\begin{aligned}
a(\omega, \theta, k) &= -\frac{1}{2} \Big{(}\frac{k}{2\pi}\Big{)}^{d-1} \int_{{\partial}\Omega}\Big{(} e^{i k
(\omega - \theta) \cdot y} ( - \nu_y \cdot \theta + |\nu_y \cdot
\omega| + E(\omega,y,k))\Big{)} dVol_{{\partial}\Omega}(y).
\end{aligned}$$
The use of Gaussian states
--------------------------
Let $(\omega_0,\eta_0) \in T^*{\mathbb{S}}^{d-1}$. We shall write[^1] $$\phi_{\omega_0,\eta_0}(\omega;k)= \chi \big{(} k^{1/3}|\omega-\omega_0|\big{)} e^{-ik \eta_0 \cdot\omega} e^{-\frac{k}{2} |\omega-\omega_0|^2}.$$ Note that $\|\phi_{\omega_0,\eta_0}\|_{L^2} = O(k^{(1-d)/4})$. We have that, for $T \in \mathcal{L}(L^2(\mathbb{S}^{d-1}))$ a trace class operator, then $$\operatorname{Tr}(T) = c_k \int_{T^*{\mathbb{S}}^{d-1}}d\omega_0 d\eta_0 \langle \phi_{\omega_0,\eta_0}, A \phi_{\omega_0,\eta_0}\rangle_{L^2({\mathbb{S}}^{d-1})},
\label{eq:trace-gaussian-states}$$ where $$c_k= 2^{-(d-1)/2}\big{(}k/(2\pi) \big{)}^{3(d-1)/2} + O_{k\rightarrow \infty}(k^{(3d)/2-2}).$$ The proof of this formula, first given in [@ingremeau2016semi] will be recalled in appendix \[section: appendixGaussian\].
Let $\mathrm{d}$ be some Riemannian distance on $T^*{\mathbb{S}}^{d-1}$. For any $\varepsilon>0$, the set $$\label{eq: def C alpha}
G_\varepsilon=\{(\omega,\eta) \in
T^*{\mathbb{S}}^{d-1} ; \mathrm{d}\big{(}(\omega,\eta), \partial \mathcal{P}_1\big{)} < k^{-1/3+\varepsilon}\}$$ has volume $O(k^{(2/3-\varepsilon) (d-1)})$. Therefore, since $\| S(k) \|_{L^2 {\longrightarrow}L^2} = 1$, we have $$I_\varepsilon (k) := c_k \int_{G_\varepsilon} d\omega_0 d\eta_0 \langle
\phi_{\omega_0,\eta_0}, (S(k)-Id)^p
\phi_{\omega_0,\eta_0}\rangle_{L^2({\mathbb{S}}^{d-1})} = O (k^{(d-1)(2/3+\varepsilon)})$$ In particular, we have $$\label{eq:alph-bound}
I_\varepsilon(k) = O(k^{d - 1 - 1/3+\varepsilon}).$$
The following lemma tells us that the set $G_\varepsilon$ is not far from being stable by $\kappa$.
\[lem: stable glancing\] Let $\varepsilon>0$. There exists $k_\varepsilon$ such that for all $k>k_\varepsilon$, we have $$\begin{aligned}
\kappa(G_\varepsilon)&\subset G_{2\varepsilon}\\
\kappa(T^*{\mathbb{S}}^{d-1} \backslash G_\varepsilon)&\subset T^*{\mathbb{S}}^{d-1}\backslash G_{\varepsilon/2}.
\end{aligned}$$
Let $d \colon \mathcal{I} {\longrightarrow}\mathbb{R}_+$ denote $d(\omega, \eta) ={\mathrm{dist}}_{T^* \mathbb{S}^{d-1}}(
(\omega, \eta),{\partial}\mathcal{I})$, i.e. distance from $(\omega,
\eta)$ to the glancing set. The lemma follows if we can establish the existence of a $C > 0$ such that $$C^{-1} d(\omega, \eta) \le
d(\kappa(\omega, \eta)) < C d(\omega, \eta).$$ Since $\kappa$ is a homeomorphism of $\mathcal{I}$ to itself, it suffices to find a $C$ for which this holds in a neighborhood of the boundary, and we therefore restrict to a neighborhood on which $d$ is smooth. We will compare $d$ to the function ${\widetilde{d}} \colon \mathcal{I} {\longrightarrow}\mathbb{R}_+$ defined by $${\widetilde{d}}(\omega, \eta) = \inf_{(\omega, \eta') \in {\partial}\mathcal{I}} |
\eta - \eta'|,$$ i.e. the distance of rays $(\omega, \eta)$ parallel to $\omega$ to the part of the glancing set parallel to $\omega$. Note that ${\widetilde{d}}$ is smooth in a neighborhood of ${\partial}\mathcal{I}$, vanishes on ${\partial}\mathcal{I}$ and that moreover the derivative $D
{\widetilde{d}}$ is non-zero on ${\partial}\mathcal{I}$. If follows that in a neighborhood of the boundary there is a $C$ so that $${C}^{-1} d(\omega, \eta) \le
{\widetilde{d}}(\omega, \eta) < {C} d(\omega, \eta),$$ and thus the question reduces to proving $${C}^{-1} \tilde{d}(\omega, \eta) \le
\tilde{d}(\kappa(\omega, \eta))< {C} \tilde{d}(\omega, \eta),\label{eq:actualdesire}$$ the original inequality for ${\widetilde{d}}$ instead of $d$.
Given $(\omega, \eta)$, let $\phi(\omega, \eta)$ denote the impact angle, i.e. the angle that the ray $t \omega + \eta$ makes with the tangent plane $T_q {\partial}\Omega$ at the first point of impact $q$. We claim that there is a constant $C > 0$ such that $$C^{-1} \phi(\omega, \eta)^2 < {\widetilde{d}}(\omega, \eta) < C \phi(\omega,
\eta)^2.\label{eq:angleinequality}$$
This inequality implies for the following reason, which is illustrated in Figure \[fig: stableglancing\]. Let us write $(\omega',\eta'):= \kappa (\omega,\eta)$, and $(\omega'',\eta'') := (-\omega',\eta')$. We then have $\tilde{d}(\omega'',\eta'')= \tilde{d}(\omega',\eta')$. But, by , we have that $\tilde{d}(\omega'',\eta'')$ is equivalent to $\phi(\kappa(\omega'', \eta''))^2$, which is equal to $\phi(\kappa(\omega, \eta))^2$. Therefore, using (\[eq:angleinequality\] once more, we have that $\tilde{d}(\omega'',\eta'')$, and hence $\tilde{d}(\kappa(\omega,\eta))$ is equivalent to $\tilde{d}(\omega,\eta)$.
![How to go from to .[]{data-label="fig: stableglancing"}](stableglancing)
We argue by comparison to parabolas. Indeed, assume without loss of generality, e.g. acting by a matrix in $O(d)$, that $(\omega, \eta)$ is incident at a point $q = (x'_0, x_0) \in \mathbb{R}_d =
\mathbb{R}^{d-1}_{x'} \times \mathbb{R}_{x}$ with $x_0 = \max_{(x', x)
\in {\partial}\Omega} x$, we write ${\partial}\Omega$ locally a graph $g(x') = x$ with maximum $x_0$ at $x'_0$. Then the curvature condition on ${\partial}\Omega$ implies that the Hessian satisfies $ - r_1 \le {\partial}^2 g \le - r_2$ for some $r_1 \ge r_2 > 0$. Then for any $1 > \delta > 0$, $ - (1 + \delta) r_1 |x'- x'_0|^2
\le g \le - r_2 / (1 - \delta) |x' - x'_0|^2$ for $(x', x)$ sufficiently close to $(x_0', x_0)$. For $\phi(\omega, \eta)$ sufficiently small, the point $\overline{q}$ in the glancing set of $\omega$ closest to the line $t \omega + \eta$ is bounded in distance by the corresponding glancing points of these two parabolas, and it is straightforward to check that one can obtain by taking $\delta$ small.
A useful change of variables {#sec: changevariable}
----------------------------
For any fixed $\omega_0\in {\mathbb{S}}^{d-1}$, consider the map $$\pi_{\omega_0} : \partial \Omega\ni y\mapsto y- (y\cdot\omega_0)\omega_0\in \omega_0^\perp,$$ and the glancing, illuminated and shadow sets $$\begin{aligned}
Y^{0}_{\omega_0}&:=\{ y\in \partial \Omega; \nu_y\cdot \omega_0= 0\}\\
Y^{-}_{\omega_0}&:=\{ y\in \partial \Omega; \nu_y\cdot \omega_0< 0\}\\
Y^{+}_{\omega_0}&:=\{ y\in \partial \Omega; \nu_y\cdot \omega_0>0\}.
\end{aligned}$$ Note that $\pi_{\omega_0}(Y^{-}_{\omega_0})= \pi_{\omega_0}(Y^{+}_{\omega_0}) =: Z_{\omega_0}$, and that $\pi_{\omega_0}$ is a diffeomorphism from $Y^{-}_{\omega_0}$ to $Z_{\omega_0}$, and from $Y^{+}_{\omega_0}$ to $Z_{\omega_0}$. Let us denote by $y^{-}= y^-_{\omega_0}$ and $y^{+}= y^+_{\omega_0}$ its respective inverses. For any $z\in Z_{\omega_0}$, we have $$\Big{|}\det \Big{(} \frac{\partial y^{\pm}(z) }{\partial
z}\Big{)}\Big{|} = |\omega_0\cdot \nu_{y^{\pm}(z)}|^{-1}.$$
Proof of the Proposition \[thm:trace-lemma\] {#sec : proof proposition}
============================================
First of all, let us note that it is enough to show the result for $p>0$. Indeed, if we show that we have $$\label{eq : tracepoly}
\operatorname{Tr}P( S(k)) = {\mathrm{Vol}}(\mathcal{I}) \Big{(}\frac{k}{2\pi}\Big{)}^{d-1} + O(k^{d-1-1/3+\varepsilon})$$ whenever $P$ is a polynomial of the form $P(X)= (X-Id)^p$, then by linearity, we have the result for any polynomial $P$ vanishing at 1. Now, since $S(k)$ is unitary, we have $$\begin{aligned}
\operatorname{Tr}\big{(} S(k)^{-p}-Id\big{)} &= \sum_n \langle e_n, ( S(k)^{-p}-Id) e_n \rangle~~\text{ for any orthonormal basis } (e_n)\\
&= \sum_n \langle (S(k)^p - Id) e_n, e_n \rangle\\
&= \operatorname{Tr}(S(k)^p-Id),
\end{aligned}$$ so that (\[eq : tracepoly\]) holds when $P(X)= X^p-Id$ for any $p\in
\mathbb{Z}\backslash \{0\}$, and, by linearity, for any trigonometric polynomial.
Therefore, let us fix from now on $p\geq 1$.
We have $$\big{(}A^p(k) \phi_{\omega_0,\eta_0}\big{)}(\alpha_p) = \Big{(}-\frac{1}{2} \Big{(}\frac{k}{2\pi}\Big{)}^{d-1} \Big{)}^p \int_{({\mathbb{S}}^{d-1})^p} \int_{(\partial \Omega)^p} e^{ik \Phi_p(\boldsymbol{\alpha}, \alpha_p,\boldsymbol{y})} a_p(\boldsymbol{\alpha},\alpha_p,\boldsymbol{y}) \mathrm{d}\boldsymbol{\alpha} \mathrm{d} \boldsymbol{y},$$ where $$\label{eq:defPhip}
\Phi_p (\boldsymbol{\alpha},\alpha_p,\boldsymbol{y}):= \frac{i}{2}
|\alpha_0-\omega_0|^2+ \alpha_0\cdot\eta_0 + \sum_{i=0}^{p-1}
(\alpha_i-\alpha_{i+1})\cdot y_i,$$ $$\label{eq: def a}
a_p(\boldsymbol{\alpha},\alpha_p,\boldsymbol{y}) := \chi
(k^{1/3}|\alpha_0-\omega_0|) \prod_{i=0}^{p-1} \big{(}
|\nu_{y_i}\cdot \alpha_i | - \nu_{y_i}\cdot \alpha_{i+1} +
E(\alpha_i,y_i,k)\big{)}.$$
Let us fix $\chi\in C^{\infty}({\mathbb{R}}^+)$, such that $\chi(t)= 1$ if $t\leq 1$, and $\chi(t)=0$ if $t\geq 2$, and let us write $$\label{eq: def rho}
\rho(\boldsymbol{\alpha},\boldsymbol{y}) := \prod_{j=0}^{p-1} \big{(}1-\chi\big{(} k^{1/3} |\nu_{y_j}\cdot \alpha_j|\big{)}\big{)}.$$
We shall write $$\begin{aligned}
A_{glan}^{p,\omega_0,\eta_0} (\alpha_p) &:= \Big{(}-\frac{1}{2} \Big{(}\frac{k}{2\pi}\Big{)}^{d-1} \Big{)}^p \int_{({\mathbb{S}}^{d-1})^p} \int_{(\partial \Omega)^p} e^{ik \Phi_p(\boldsymbol{\alpha}, \alpha_p,\boldsymbol{y})} \big{(}1- \rho(\boldsymbol{\alpha},\boldsymbol{y})\big{)} a_p(\boldsymbol{\alpha},\alpha_p,\boldsymbol{y}) \mathrm{d}\boldsymbol{\alpha} \mathrm{d} \boldsymbol{y}\\
A_{non glan}^{p,\omega_0,\eta_0} (\alpha_p) &:= \Big{(}-\frac{1}{2} \Big{(}\frac{k}{2\pi}\Big{)}^{d-1} \Big{)}^p \int_{({\mathbb{S}}^{d-1})^p} \int_{(\partial \Omega)^p} e^{ik \Phi_p(\boldsymbol{\alpha}, \alpha_p,\boldsymbol{y})} \rho(\boldsymbol{\alpha},\boldsymbol{y}) a_p(\boldsymbol{\alpha},\alpha_p,\boldsymbol{y}) \mathrm{d}\boldsymbol{\alpha} \mathrm{d} \boldsymbol{y},
\end{aligned}$$ so that $$\big{(}A^p(k) \phi_{\omega_0,\eta_0}\big{)}(\alpha_p) = A_{glan}^{p,\omega_0,\eta_0} (\alpha_p) + A_{non glan}^{p,\omega_0,\eta_0} (\alpha_p) .$$
\[lem : glancingset is burried\] Suppose that $(\omega_0,\eta_0)\in T^*{\mathbb{S}}^{d-1}\backslash G_\varepsilon$, with $\varepsilon>0$ small enough. We then have $$\|A_{glan}^{p,\omega_0,\eta_0}\|_{C^0} = O(k^{-\infty}).$$
We shall prove this lemma in Section \[section : killing the glancing\]. Let us now describe the critical set of the phase $\Phi_p$.
The following lemma is elementary, and its result is depicted on Figure \[fig: criticalObs\].
\[lem: critical3\] Let $(\alpha,\eta)\in \mathcal{I}$. Then there are exactly four points $(\alpha',y')\in {\mathbb{S}}^{d-1}\times \partial \Omega$ which satisfy $$\begin{split}
\eta - y' \in \mathbb{R}\alpha,\\
\alpha - \alpha' \in
\mathbb{R}\nu_{y'}. \\
\end{split}$$ We shall denote them by $(\alpha',y') =: \tau^{\epsilon_1,\epsilon_2}(\alpha,\eta)=: \big{(}\alpha^{\epsilon_1,\epsilon_2}(\alpha,\eta),y^{\epsilon_1,\epsilon_2}(\alpha,\eta)\big{)}$, where $\epsilon_1,\epsilon_2\in \{\pm\}$, and they are as follows: $$\begin{aligned}
&\epsilon_1=+ \Longrightarrow \nu_{y'}\cdot \alpha > 0\\
&\epsilon_1=- \Longrightarrow \nu_{y'}\cdot \alpha < 0\\
&\epsilon_2=+ \Longrightarrow |\alpha' - \alpha| = | \nu_{y'}\cdot \alpha|>0\\
&\epsilon_2=- \Longrightarrow \alpha'= \alpha.
\end{aligned}$$ In any case, we have $$\label{eq:equalangles}
|\alpha'\cdot \nu_{y'}| = |\alpha\cdot \nu_{y'}|.$$
Furthermore, let us write $$\tilde{\kappa}^{\epsilon_1,\epsilon_2}(\alpha,\eta) := \big{(}\alpha',\pi_{\alpha'}(y')\big{)}.$$ Then for all $(\alpha,\eta)$, we have $$\label{eq:DifferentKappas}
\begin{aligned}
\tilde{\kappa}^{\pm,-}(\alpha,\eta) = (\alpha,\eta)\\
\tilde{\kappa}^{-,+}(\alpha,\eta) = \kappa(\alpha,\eta)\\
\tilde{\kappa}^{+,+}(\alpha,\eta) = \kappa^{-1}(\alpha,\eta),
\end{aligned}$$ with $\kappa$ as in (\[eq:the-scattering-map\]).
![The solutions to (\[eq:equalangles\]).[]{data-label="fig: criticalObs"}](criticalObstacle)
For all $(\alpha, y)\in {\mathbb{S}}^{d-1}\times \partial \Omega$ such that $\alpha\cdot \nu_y\neq 0$, we shall write $$\tilde{\tau}^{\epsilon_1,\epsilon_2} (\alpha,y):= \tau^{\epsilon_1,\epsilon_2}(\alpha,\pi_{\alpha}(y)).$$
We shall write $$\Sigma_p := \big{(}\{+,-\} \times \{+,-\}\big{)}^p.$$ If $\sigma=(\sigma_0,...,\sigma_{p-1})\in \Sigma_p$, we will write $$\label{eq:deftildekappa}
\tilde{\kappa}^\sigma := \tilde{\kappa}^{\sigma_{p-1}}\circ ... \circ \tilde{\kappa}^{\sigma_0}.$$
\[lem : critical2\] Let $(\omega_0,\eta_0)\in \mathcal{I}$, and let $\sigma\in \Sigma_p$. There exists a unique $(\boldsymbol{\alpha}^\sigma,\alpha_p^\sigma,\boldsymbol{y}^\sigma)$ such that $$\begin{aligned}
\alpha^\sigma_0&= \omega^\sigma_0\\
(\alpha^\sigma_1,y^\sigma_0) &= \tau^{\sigma_0} (\alpha^\sigma_0, \eta_0)\\
\forall k=1,...,p-1,~~ (\alpha^\sigma_{k+1},y^\sigma_k) &= \tilde{\tau}^{\sigma_k} (\alpha^\sigma_k, y^\sigma_{k}).
\end{aligned}$$
We have $$\{ (\boldsymbol{\alpha},\alpha_p,\boldsymbol{y})\in \big{(}{\mathbb{S}}^{d-1}\big{)}^{p+1} \times \big{(} \partial \Omega \big{)}^p ; \Phi_p (\boldsymbol{\alpha},\alpha_p,\boldsymbol{y})=0 \} = \bigcup_{\sigma \in \Sigma_p} (\boldsymbol{\alpha}^{\sigma},\alpha_p^\sigma,\boldsymbol{y}^{\sigma}).$$
Furthermore, for each $\sigma\in \Sigma_p$, we have $\partial \Phi_p (\boldsymbol{\alpha}^\sigma,\alpha^\sigma_p,\boldsymbol{y}^\sigma)= 0$ if and only if we have $$\begin{aligned}
&\alpha_p=\omega_0\\
&\eta_0 = \pi_{\omega_p} (y_{p-1}).
\end{aligned}$$
Note that this condition is equivalent to $$(\omega_0,\eta_0)= \tilde{\kappa}^{\sigma}(\omega_0,\eta_0).$$
Clearly $\alpha_0 = \alpha_p =
\omega_0$. The criticality condition for $\alpha_0$ and $\alpha_p$ give that $y_0, y_{p-1} \in \mathbb{R}\omega_0 -
\eta_0$, and the other equations in ${\partial}\Phi = 0$ imply $$\begin{split}
\alpha_i - \alpha_{i + 1} \in
\mathbb{R}\nu_{y_i},\ \mbox{ for } i = 0, \dots, p-1, \\
y_{i -1} - y_{i} \in \mathbb{R}\alpha_{i},\ \mbox{ for } i = 0, \dots, p-2
\end{split}$$ The lemma then follows from Lemma \[lem: critical3\].
In the sequel, we shall write $$\sigma_0 := (-, - , - , ..., -).$$
Thanks to (\[eq:equalangles\]) and to (\[eq: def rho\]), we see that if $(\omega_0,\eta_0)\in \mathcal{I}\backslash G_\varepsilon$ and $\sigma \neq \sigma'\in \Sigma_p$, then if $\mathrm{d}_p$ is a Riemanian distance on $\big{(}{\mathbb{S}}^{d-1}\big{)}^{p+1} \times \big{(} {\partial}\Omega\big{)}^p$, there exists $C>0$ such that $$\mathrm{d}\Big{(}(\boldsymbol{\alpha}^{\sigma},\alpha_p^\sigma,\boldsymbol{y}^{\sigma}),(\boldsymbol{\alpha}^{\sigma'},\alpha_p^{\sigma'},\boldsymbol{y}^{\sigma'})\Big{)}\geq C k^{-1/3}.$$
We may therefore build a family of functions $\chi_\sigma \in S_{1/3} \big{(} \big{(}{\mathbb{S}}^{d-1}\big{)}^{p+1} \times \big{(} {\partial}\Omega\big{)}^p \big{)}$ such that there exists $c>0$ satisfying $$\forall \sigma,\sigma'\in \Sigma_p, \forall x \in \big{(}{\mathbb{S}}^{d-1}\big{)}^{p+1} \times \big{(} {\partial}\Omega\big{)}^p, \mathrm{d}_p(x, x_\sigma) < c k^{-1/3} \Longrightarrow \chi_{\sigma'}(x) = \delta_{\sigma,\sigma'},$$ and such that $$\label{eq:conditionsupport}
\forall \sigma = (\pm, \bullet)\in \Sigma_p,
\forall (\boldsymbol{\alpha},\alpha_p,\boldsymbol{y}^{\sigma})\in \mathrm{spt} \chi_{\sigma},~ \forall i=0,...,p-1, y_i\in Y^\pm_{\alpha^\sigma_i},$$ where $Y^\pm_{\alpha_i}$ is as in Section \[sec: changevariable\].
In particular, if $(\boldsymbol{\alpha},\alpha_p,\boldsymbol{y}^{\sigma})\in \mathrm{spt} \chi_{\sigma_0}$, then $y_i\in Y^-_{\omega_0}$ for all $i$.
We then set $$\chi_\infty = 1- \sum_{\sigma\in \Sigma_p} \chi_\sigma.$$
We shall then write, for each $\sigma\in \Sigma_p\cup \{\infty\}$ $$\begin{aligned}
A_{\sigma}^{p,\omega_0,\eta_0} (\alpha_p) &:= \Big{(}-\frac{1}{2} \Big{(}\frac{k}{2\pi}\Big{)}^{d-1} \Big{)}^p \int_{({\mathbb{S}}^{d-1})^p} \int_{(\partial \Omega)^p} e^{ik \Phi_p(\boldsymbol{\alpha}, \alpha_p,\boldsymbol{y})} \big{(}1- \rho(\boldsymbol{\alpha},\boldsymbol{y})\big{)} \chi_{\sigma}(\boldsymbol{\alpha},\boldsymbol{y}) a_p(\boldsymbol{\alpha},\alpha_p,\boldsymbol{y}) \mathrm{d}\boldsymbol{\alpha} \mathrm{d} \boldsymbol{y}.
\end{aligned}$$
Lemma \[lemm: nonstat\] implies that $A_\infty^{p,\omega_0,\eta_0}= O(k^{-\infty})$, so that we have $$\begin{aligned}
\operatorname{Tr}((S(k)-Id)^p) & = c_k \int_{\mathcal{I}\backslash G_\varepsilon} d\omega_0 d\eta_0 \langle A^{p,\omega_0,\eta_0}_{\sigma_0}, \phi_{\omega_0,\eta_0}\rangle +
\sum_{\sigma\in \Sigma_p\backslash \{\sigma_0\} } c_k \int_{\mathcal{I}\backslash G_\varepsilon}d\omega_0 d\eta_0 \langle A^{p,\omega_0,\eta_0}_{\sigma_0}, \phi_{\omega_0,\eta_0}\rangle \\
&+ c_k \int_{\mathcal{I}\backslash G_\varepsilon} d\omega_0 d\eta_0 \langle A^{p,\omega_0,\eta_0}_{glan}, \phi_{\omega_0,\eta_0}\rangle \\
&+ c_k \int_{T^*{\mathbb{S}}^{d-1}\backslash(\mathcal{I}\cup G_\varepsilon)} d\omega_0 d\eta_0 \langle \phi_{\omega_0,\eta_0}, (S(k)-Id)^p \phi_{\omega_0,\eta_0}\rangle\\
&+ c_k \int_{ G_\varepsilon} d\omega_0 d\eta_0 \langle \phi_{\omega_0,\eta_0}, (S(k)-Id)^p \phi_{\omega_0,\eta_0}\rangle + O(k^{-\infty}).
\end{aligned}$$
The proof of Proposition \[thm:trace-lemma\] then follows from the following three lemmas, as well as Lemma \[lem : glancingset is burried\] and equation (\[eq:alph-bound\]).
\[lemm:noninteracting\] We have $$\int_{\mathcal{P}_1\backslash G_\varepsilon} d\omega_0 d\eta_0 \langle \phi_{\omega_0,\eta_0}, (S(k)-Id)^p \phi_{\omega_0,\eta_0}\rangle = O(k^{-\infty}).$$
Let $(\omega_0,\eta_0)\in \mathcal{P}_1\backslash G_\varepsilon$. From (\[eq:defPhip\]), we see that $|\partial_{\alpha_0} \Phi_p| = |\pi_{\alpha_0} (\eta_0-y_0)| + |\alpha_0-\omega_0| \geq C \mathrm{d}((\omega_0,\eta_0),\mathcal{I})$, where $\mathrm{d}$ is the Riemannian distance on $T^*{\mathbb{S}}^{d-1}$. Therefore, Lemma \[lemm: nonstat\] implies that we have $$A^p(k)\phi_{\omega_0,\eta_0} = O\big{(} \big{(}k \mathrm{d}((\omega_0,\eta_0)\big{)}^{-\infty}\big{)}.$$ The lemma follows.
The following two lemmas will be proven in sections \[sec:proof offdiagonal\] and \[sec:Proofmainpart\] respectively.
\[lem: offdiagonal\] For any $\sigma\in \Sigma_p\backslash \{\sigma_0\}$, we have $$c_k\int_{\mathcal{I}\backslash G_\varepsilon} \mathrm{d}\omega_0\mathrm{d}\eta_0 \langle A_{\sigma}^{p,\omega_0,\eta_0}, \phi_{\omega_0,\eta_0} \rangle = O(k^{d-1-1/3+\varepsilon}).$$
\[lem: main part\] $$c_k\int_{\mathcal{I}\backslash G_\varepsilon} \mathrm{d}\omega_0\mathrm{d}\eta_0 \langle A_{\sigma_0}^{p,\omega_0,\eta_0}, \phi_{\omega_0,\eta_0} \rangle =(-1)^p \Big{(}\frac{k}{2\pi}\Big{)}^{d-1} {\mathrm{Vol}}(\mathcal{I}) + O(k^{d-1-1/3}).$$
Proofs of the technical lemmas
==============================
Computation of the major contribution: Proof of Lemma \[lem: main part\] {#sec:Proofmainpart}
------------------------------------------------------------------------
Recall that we have $$\langle A_{\sigma_0}^{p,\omega_0,\eta_0}, \phi_{\omega_0,\eta_0} \rangle = \Big{(}-\frac{1}{2} \Big{(}\frac{k}{2\pi}\Big{)}^{d-1} \Big{)}^p
\int_{({\mathbb{S}}^{d-1})^{p+1}} \int_{(\partial \Omega)^p} e^{ik
\tilde{\Phi}_p(\boldsymbol{\alpha},\alpha_p,\boldsymbol{y})}
\tilde{a}_p(\boldsymbol{\alpha},\alpha_p,\boldsymbol{y})
\mathrm{d}\boldsymbol{\alpha} \mathrm{d} \alpha_p
\mathrm{d}\boldsymbol{y},$$ where $$\begin{aligned}
\tilde{\Phi}_p (\boldsymbol{\alpha},\alpha_p,\boldsymbol{y})&:= \frac{i}{2} |\alpha_0-\omega_0|^2+ \frac{i}{2} |\alpha_p-\omega_0|^2 + (\alpha_0- \alpha_p)\cdot \eta_0 + \sum_{i=0}^{p-1} (\alpha_i-\alpha_{i+1})\cdot y_i,\\
\tilde{a}_p(\boldsymbol{\alpha},\alpha_p,\boldsymbol{y}) &:= \big{(}1- \rho(\boldsymbol{\alpha},\boldsymbol{y})\big{)} \chi_{\sigma}(\boldsymbol{\alpha},\boldsymbol{y}) \chi (k^{1/3}|\alpha_0-\omega_0|) \chi
(k^{1/3}|\alpha_p-\omega_0|) \\
&\qquad \qquad \prod_{i=0}^{p-1} \big{(}
|\nu_{y_i}\cdot \alpha_i | - \nu_{y_i}\cdot \alpha_{i+1} +
E(\alpha_i,y_i,k)\big{)}.
\end{aligned}$$
Thanks to (\[eq:conditionsupport\]), we may write $$\langle A_{\sigma_0}^{p,\omega_0,\eta_0}, \phi_{\omega_0,\eta_0}
\rangle = \Big{(}-\frac{1}{2} \Big{(}\frac{k}{2\pi}\Big{)}^{d-1}
\Big{)}^p \int_{({\mathbb{S}}^{d-1})^{p+1}}
\int_{Y_{\omega_0}^{-}}...\int_{Y_{\omega_0}^{-}}e^{ik
\tilde{\Phi}_p(\boldsymbol{\alpha},\alpha_p,\boldsymbol{y})}
\tilde{a}_p(\boldsymbol{\alpha},\alpha_p,\boldsymbol{y})
\mathrm{d}\boldsymbol{\alpha} \mathrm{d} \alpha_p
\mathrm{d}\boldsymbol{y}.$$
We may then carry out $p$ times the changes of variables $y^-$ to obtain $$\langle A_{\sigma_0}^{p,\omega_0,\eta_0}, \phi_{\omega_0,\eta_0} \rangle = \Big{(}-\frac{1}{2} \Big{(}\frac{k}{2\pi}\Big{)}^{d-1} \Big{)}^p\int_{({\mathbb{S}}^{d-1})^{p+1}}
\int_{(Z_{\omega_0})^p} e^{ik
\tilde{\Phi}^-_p(\boldsymbol{\alpha},\alpha_p,\boldsymbol{z})}
\tilde{a}^-_p(\boldsymbol{\alpha},\alpha_p,z))
\mathrm{d}\boldsymbol{\alpha} \mathrm{d} \alpha_p
\mathrm{d}\boldsymbol{z} + O(k^{-\infty}).$$ Here, we wrote $\boldsymbol{y}^-(\boldsymbol{z}):= (y^{-}(z_0),...,y^{-}(z_{p-1}))$, and $$\begin{aligned}
\tilde{\Phi}^-_p(\boldsymbol{\alpha},\alpha_p,\boldsymbol{z})&:=\tilde{\Phi}_p(\boldsymbol{\alpha},\alpha_p,\boldsymbol{y}^-(\boldsymbol{z}))\\
\tilde{a}^-_p(\boldsymbol{\alpha},\alpha_p,\boldsymbol{z}))&:= \frac{\tilde{a}_p(\boldsymbol{\alpha},\alpha_p,\boldsymbol{y}^-(\boldsymbol{z}))}{|\alpha_0\cdot \nu_{y^{-}(z_0)}|... |\alpha_{p-1}\cdot \nu_{y^{-}(z_{p-1})}|}\\
&= \tilde{\chi} (\boldsymbol{\alpha},\alpha_p,\boldsymbol{y}^-(\boldsymbol{z})) \frac{\prod_{i=0}^{p-1} \big{(}
|\nu_{y^-(z_i)}\cdot \alpha_i | - \nu_{y^\sigma(z_i)}\cdot \alpha_{i+1} +
E(\alpha_i,y^-(z_i),k)\big{)}}{|\alpha_0\cdot \nu_{y^{-}(z_0)}|... |\alpha_{p-1}\cdot \nu_{y^{-}(z_{p-1})}|},
\end{aligned}$$ where $\tilde{\chi} (\boldsymbol{\alpha},\alpha_p,\boldsymbol{y}) = \big{(}1- \rho(\boldsymbol{\alpha},\boldsymbol{y})\big{)} \chi_{\sigma_0}(\boldsymbol{\alpha},\boldsymbol{y}) \chi (k^{1/3}|\alpha_0-\omega_0|) \chi
(k^{1/3}|\alpha_p-\omega_0|).$
Note that we have $$\tilde{a}^-_p\in S_{1/3}.$$
Define the *critical set* as $$\mathcal{C}^{-}_p:= \{(\boldsymbol{\alpha},\alpha_p,\boldsymbol{z})\in
({\mathbb{S}}^{d-1})^{p+1} \times (Z_{\omega_0})^p; \partial \tilde{\Phi}_p^- (
\boldsymbol{\alpha},\alpha_p,\boldsymbol{y}^{-}(\boldsymbol{z}))= 0 \text{ and } \Im \tilde{\Phi}_p^-
(\boldsymbol{\alpha},\alpha_p,\boldsymbol{y}^{-}(\boldsymbol{z}))=0 \}.$$
By Lemma \[lem : critical2\], we have that $\mathcal{C}^{-}_p=\{(\omega_0,...,\omega_0, \eta_0,...,\eta_0)\}$.
Note that we have $$\label{eq: OnCrit}
\begin{aligned}
\tilde{\Phi}^-_p(\omega_0,...,\omega_0,\eta_0,...,\eta_0)&= 0\\
\tilde{a}^-_p(\omega_0,...,\omega_0,\eta_0,...,\eta_0) &= \frac{ \big{(}
2\nu_{y^-}(\eta_0)\cdot \omega_0 +
E(\omega_0,y^-(\eta_0),k)\big{)}^p}{|\nu_{y^-(\eta_0)}\cdot \omega_0|^p} \\
&= 2^p + R_p(\omega_0,\eta_0),
\end{aligned}$$ where $$R_p(\omega_0,\eta_0) = \sum_{l=1}^p C_l^p \frac{E(\omega_0,y^-(\eta_0),k)^l}{|\nu_{y^-(\eta_0)}\cdot \omega_0|^l}.$$
### Computation of the Hessian {#computation-of-the-hessian .unnumbered}
\[lem: Hessian\] $$\det \partial^2 \tilde{\Phi}_p^- (\omega_0,...,\omega_0, \eta_0,...,\eta_0) = (2i)^{d-1} (-1)^{p(d-1)}.$$
We have $$\frac{\partial^2 \tilde{\Phi}_p^-}{\partial z_i \partial z_{i'}} = (\alpha_i-\alpha_{i+1})\cdot \frac{\partial^2 y^{\beta_i}}{\partial z_i \partial z_{i'}},$$ which vanishes at $(\omega_0,...,\omega_0, \eta_0,...\eta_0)$. We also have $$\frac{\partial^2 \tilde{\Phi}_p^-}{\partial \alpha_i \partial \alpha_{i'}}(\omega_0,...,\omega_0, \boldsymbol{z}_\beta)= \delta_{i=0} \big{(} i Id + B\big{)}+ \delta_{i'=p} \big{(}i Id - B\big{)},$$ where $B$ is the matrix $\frac{\partial^2}{\partial \alpha^2}\big{(} \alpha \cdot y^-(\nu_0)\big{)}$.
For $i=0,...,p$, $i'=0,...,p-1$, let us define the matrix $$M^{i,i'} = \Big{(}\frac{\partial^2 \tilde{\Phi}_p^\beta}{\partial \alpha_{i,j} \partial z_{i',j'}}(\omega_0,...,\omega_0, \eta_0,...,\eta_0)\Big{)}_{1\leq j,j'\leq d-1}.$$ For $1\leq i\leq p-1$, we have $M^{i,i'}=0$ unless $i' = i$ or $i'=i-1$. We also have $M^{0,i}=0$ if $i\neq 0$, and $M^{p,i}=0$ unless $i=p-1$. For $0\leq i\leq p-1$, we have $M^{i,i} = \boldsymbol{1}_{d-1}$, while $M^{i,i-1}= - \boldsymbol{1}_{d-1}$.
All in all, $ \partial^2 \tilde{\Phi}_p^\beta (\omega_0,...,\omega_0, \eta_0,...,\eta_0) $ can be written as follows, with $(2p+1)^2$ blocks of size $d'=d-1$. $$\begin{pmatrix}
& & & & & & \boldsymbol{1}_{d'} & -\boldsymbol{1}_{d'} & 0 & ... & 0 & 0 & 0 \\
& & & & & & 0 & \boldsymbol{1}_{d'} & -\boldsymbol{1}_{d'} &... & 0 & 0 & 0\\
& &\bold{0}_{d'p\times d'p} && & & &&&... \\
&&&&&& 0 & 0 &0 &...& \boldsymbol{1}_{d'} & -\boldsymbol{1}_{d'} & 0\\
& & & & & & 0 & 0 & 0 & ... & 0 & \boldsymbol{1}_{d'} & -\boldsymbol{1}_{d'}\\
\boldsymbol{1}_{d'} & 0 & 0 & ...& 0 & 0 & i\bold{1}_{d'} +B & 0 & 0 & ... & 0 & 0 & 0\\
-\boldsymbol{1}_{d'}& \boldsymbol{1}_{d'} & 0& ... & 0 & 0 & \\
0 & -\boldsymbol{1}_{d'} & \boldsymbol{1}_{d'} & ... & 0 & 0 & && & \bold{0}_{d'(p+1)\times d'(p-1)} \\
&&&... & & & & \\
0 & 0 &0&... & -\boldsymbol{1}_{d'} & \boldsymbol{1}_{d'} & \\
0 &0 &0 & ...& 0 & -\boldsymbol{1}_{d'} & 0 & 0 & 0 & ... & 0 &0 & i\bold{1}_{d'}-B
\end{pmatrix}.$$ The proof then follows from Lemma \[lem:determinant\].
Lemma \[lem: Hessian\] tells us that $\det \big{(}i^{-1} \partial^2 \tilde{\Phi}_p^\beta (\omega_0,...,\omega_0, \eta_0,...,\eta_0)\big{)} = 2^{d-1}.$ Therefore, using Lemma \[lem: phistat\] combined with (\[eq: OnCrit\]), we get that $$T^p_{diag}(\omega_0,\eta_0,k) = \Big{(}\frac{2\pi}{k}\Big{)}^{(d-1)(2p+1)/2}2^{(d-1)/2} \big{(}2^p + R_p(\omega_0,\eta_0)\big{)} + O\big{(}k^{-(d-1)(2p+1)/2-1/3}\big{)}.$$
All in all, we have $$\begin{aligned}
&c_k\int_{\mathcal{I}\backslash G_\varepsilon} \mathrm{d}\omega_0\mathrm{d}\eta_0 \langle A_{\sigma_0}^{p,\omega_0,\eta_0}, \phi_{\omega_0,\eta_0} \rangle\\
&= (-1)^p \Big{(}2^{-(d-1)/2}\big{(}k/(2\pi) \big{)}^{3(d-1)/2} (\frac{1}{2} k^{d-1} (2\pi)^{1-d} )^p \Big{(}\frac{2\pi}{k}\Big{)}^{(d-1)(2p+1)/2} 2^p {\mathrm{Vol}}(\mathcal{I})2^{(d-1)/2}\Big{)}\Big{(}1+O(k^{-1/3})\Big{)}
\\&= (-1)^p \Big{(}\frac{k}{2\pi}\Big{)}^{d-1} {\mathrm{Vol}}(\mathcal{I}) + O(k^{d-1-1/3}).
\end{aligned}$$
Ruling out the glancing rays: proof of Lemma \[lem : glancingset is burried\] {#section : killing the glancing}
-----------------------------------------------------------------------------
Let us define the operator $S_{glan}$ by $$\big{(}S_{glan}\big{)} f (\theta)= -\frac{1}{2} \Big{(}\frac{k}{2\pi}\Big{)}^{d-1} \int_{{\mathbb{S}}^{d-1}} \int_{{\partial}\Omega} \chi\big{(} k^{1/6} |\nu_{y}\cdot \alpha|\big{)} f(\alpha) a(\alpha,\theta,k) \mathrm{d}\alpha \mathrm{d}y,$$ where $a(\alpha,\theta,k)$ is the scattering amplitude as in (\[eq : Kirchhoff amplitude\]).
We shall also write $$\big{(}S_{nonglan} f\big{)}(\theta) = -\frac{1}{2} \Big{(}\frac{k}{2\pi}\Big{)}^{d-1} \int_{{\mathbb{S}}^{d-1}} \int_{{\partial}\Omega} \big{(}1-\chi\big{(} k^{1/6} |\nu_{y}\cdot \alpha|\big{)}\big{)} f(\alpha) a(\alpha,\theta,k) \mathrm{d}\alpha \mathrm{d}y,$$ so that $$S(k)-Id = S_{glan} + S_{nonglan}.$$
In particular, we have that $$(S_{nonglan})^p \phi_{\omega_0,\eta_0} = A^{p,\omega_0,\eta_0}_{nonglan}.$$
\[lem: dying glancingset\] Let $\varepsilon>0$ be small enough, and let $(\omega_0,\eta_0)\in T^*{\mathbb{S}}^{d-1}\backslash G_{\varepsilon}$. Then $$\|S_{glan} \phi_{\omega_0,\eta_0} \|=O(k^{-\infty}).$$
We have $$S_{glan} \phi_{\omega_0,\eta_0}(\theta) = f_1 (\theta) + f_2 (\theta),$$ where $$f_1(\theta) = -\frac{1}{2} \Big{(}\frac{k}{2\pi}\Big{)}^{d-1} \int_{{\mathbb{S}}^{d-1}} \int_{{\partial}\Omega} e^{i k
(\alpha - \theta) \cdot y} \chi\big{(} k^{1/6} |\nu_{y}\cdot \alpha|\big{)} \chi \big{(} k^{1/3}|\alpha-\omega_0|\big{)} e^{-ik \eta_0 \cdot\alpha} e^{-\frac{k}{2} |\alpha-\omega_0|^2} \big{(}-\nu_y \cdot \theta + E(\alpha,y,k) \big{)} \mathrm{d}\alpha \mathrm{d}y$$ and $$f_2(\theta) = -\frac{1}{2} \Big{(}\frac{k}{2\pi}\Big{)}^{d-1} \int_{{\mathbb{S}}^{d-1}} \int_{{\partial}\Omega} e^{i k
(\alpha - \theta) \cdot y} \chi\big{(} k^{1/6} |\nu_{y}\cdot \alpha|\big{)} \chi \big{(} k^{1/3}|\alpha-\omega_0|\big{)} e^{-ik \eta_0 \cdot\alpha} e^{-\frac{k}{2} |\alpha-\omega_0|^2} |\nu_y \cdot \alpha | \mathrm{d}\alpha \mathrm{d}y$$
$f_1$ is an oscillatory integral, with an amplitude $$\chi\big{(} k^{1/6} |\nu_{y}\cdot \alpha|\big{)} \chi \big{(} k^{1/6}|\alpha-\omega_0|\big{)} \big{(}-\nu_y \cdot \theta + E(\alpha,y,k) \big{)}\in S_{1/3},$$ and a phase $$\varphi(\alpha,y; \theta, \omega_0,\eta_0) = \frac{i}{2} |\alpha-\omega_0|^2 + \alpha \cdot \eta_0 + (\theta-\alpha)\cdot y.$$
We have $|\partial_\alpha \Re \varphi| = |\pi_\alpha (y)- \eta_0|$, with $\pi_\alpha$ as in Section \[sec: changevariable\]. By the assumption we made on $(\omega_0,\eta_0)$, we have $|\partial_\alpha \Re \varphi| \geq c k^{-1/3}$ for some $c>0$. Therefore, we may use Lemma \[lemm: nonstat\] to obtain that $f_1=O(k^{-\infty})$.
To deal with $f_2$, we may use the changes of variables $y^\pm$ introduced above, to write $$\begin{aligned}
f_2(\theta)= -\frac{1}{2} \Big{(}\frac{k}{2\pi}\Big{)}^{d-1}\sum_{\epsilon= \pm} \int_{{\mathbb{S}}^{d-1}} \int_{Z^\epsilon} e^{i k
(\alpha - \theta) \cdot y^\pm(z)} \chi\big{(} k^{1/6} |\nu_{y^\pm(z)}\cdot \alpha|\big{)} \chi \big{(} k^{1/3}|\alpha-\omega_0|\big{)} e^{-ik \eta_0 \cdot\alpha} e^{-\frac{k}{2} |\alpha-\omega_0|^2} \mathrm{d}\alpha \mathrm{d}z.
\end{aligned}$$
Each of the integrals $\int_{{\mathbb{S}}^{d-1}} e^{i k
(\alpha - \theta) \cdot y^\pm(z)} \chi\big{(} k^{1/6} |\nu_{y^\pm(z)}\cdot \alpha|\big{)} \chi \big{(} k^{1/3}|\alpha-\omega_0|\big{)} e^{-ik \eta_0 \cdot\alpha} e^{-\frac{k}{2} |\alpha-\omega_0|^2} \mathrm{d}\alpha$ can be seen as an oscillatory integral with a parameter $z\in Z^\pm$. The amplitude $\alpha \mapsto \chi\big{(} k^{1/6} |\nu_{y^\pm(z)}\cdot \alpha|\big{)} \chi \big{(} k^{1/3}|\alpha-\omega_0|\big{)}$ is in $S_{1/3}$, with bounds on the semi-norms independent of the point $z\in Z^\pm$, while the phase $\varphi (\alpha; z, \theta) = (\alpha - \theta) \cdot y^\pm(z) + \eta_0 \cdot\alpha -\frac{i}{2} |\alpha-\omega_0|^2$ satisfies $|\partial_\alpha \Re \varphi| \geq c k^{-1/2 +\varepsilon}$, with a constant $c$ independent of the point $z$ in the support of the amplitude. We may therefore apply the non-stationary phase lemma \[lemm: nonstat\] to conclude.
\[cor: dying glancing\] Let $(\omega_0,\eta_0)\in T^*{\mathbb{S}}^{d-1}\backslash G_{4\varepsilon}$, and $(\omega_1,\eta_1)\in G_\varepsilon$. We then have $$\langle \phi_{\omega_1,\eta_1}, (S(k)-Id) \phi_{\omega_0,\eta_0} \rangle = O(k^{-\infty}).$$
By Lemma \[lem: stable glancing\], we have that $\kappa(\omega_0,\eta_0)\in T^*{\mathbb{S}}^{d-1}\backslash G_{2\varepsilon}$, so that $$\label{eq: distant points}
\mathrm{d}(\kappa (\omega_0,\eta_0), (\omega_1,\eta_1)) > k^{-1/2+\varepsilon/2}.$$
By the preceding lemma, we have that $$\begin{aligned}
&\langle \phi_{\omega_1,\eta_1}, (S(k)-Id) \phi_{\omega_0,\eta_0} \rangle = \langle \phi_{\omega_1,\eta_1}, S_{nonglan} \phi_{\omega_0,\eta_0} \rangle + O(k^{-\infty})\\
&= \int_{({\mathbb{S}}^{d-1})^2} \int_{{\partial}\Omega} \big{(}1-\chi\big{(} k^{1/3} |\nu_{y}\cdot \alpha_0|\big{)} a(\alpha_0,\alpha_1,y;k) e^{ik \varphi(\alpha_0,\alpha_1,y)} \mathrm{d}\alpha_0\mathrm{d}\alpha_1 \mathrm{d}y + O(k^{-\infty}),
\end{aligned}$$ where $\varphi(\alpha_0,\alpha_1,y) =\frac{i}{2}|\alpha_0-\omega_0|^2 +\frac{i}{2}|\alpha_1-\omega_1|^2 + (\eta_1-y)\cdot \alpha_1+ (y-\eta_0)\cdot \alpha_0$.
By (\[eq: distant points\]), we have that $|\partial \varphi |\geq k^{-1/2+\varepsilon/2}$, and the corollary follows from Lemma \[lemm: nonstat\].
We may now prove Lemma \[lem : glancingset is burried\], which follows from (\[eq: glancingdead\]) below.
Let $(\omega_0,\eta_0)\in T^*{\mathbb{S}}^{d-1}\backslash G_{\varepsilon}$. For any $p\geq 1$, we have $$\label{eq: glancingdead2}
\forall (\omega_1,\eta_1)\in G_{2^{-p-1}\varepsilon}, \langle \phi_{\omega_1,\eta_1}, (S(k)-Id)^p \phi_{\omega_0,\eta_0} = O(k^{-\infty})$$ $$\label{eq: glancingdead}
(S(k)-Id)^p \phi_{\omega_0,\eta_0} = S_{nonglan}^p \phi_{\omega_0,\eta_0} + O(k^{-\infty}).$$ In particular, we have $$\|A_{glan}^{p,\omega_0,\eta_0}\|_{C^0} = O(k^{-\infty}).$$
Let us prove (\[eq: glancingdead2\]) and (\[eq: glancingdead\]) by induction. Lemma \[lem: dying glancingset\] and Corollary \[cor: dying glancing\] imply that (\[eq: glancingdead2\]) and (\[eq: glancingdead\]) are satisfied for $p=1$. Suppose that they are satisfied for some $p\geq 1$.
By Lemma \[lemma: resolution3\], we know that $(S(k)-Id)^{p-1} \phi_{\omega_0,\eta_0}$ can be written as $\int_{T^*{\mathbb{S}}^{d-1}} f_p(\omega,\eta; k) \phi_{\omega,\eta}\mathrm{d}\omega \mathrm{d}\theta$ for a certain function $f_p$. Now, (\[eq: glancingdead2\]) implies that $f_p(\omega,\eta; k) = O(k^{-\infty})$ for $(\omega,\eta)\in G_{2^{-p}\varepsilon}$.
We therefore have $$\begin{aligned}
(S(k)-Id)^p \phi_{\omega_0,\eta_0} = \int_{T^*{\mathbb{S}}^{d-1}\backslash G_{2^{-p}\varepsilon}} f_p(\omega,\eta) (S(k)-Id) \phi_{\omega,\eta}.
\end{aligned}$$
We then deduce (\[eq: glancingdead2\]) and (\[eq: glancingdead\]) by applying Lemma \[lem: dying glancingset\] and Corollary \[cor: dying glancing\].
Proof of Lemma \[lem: offdiagonal\] {#sec:proof offdiagonal}
-----------------------------------
Recall that the statement of Lemma \[lem: offdiagonal\] is the following.
For any $\sigma\in \Sigma_p\backslash \{\sigma_0\}$, for any $\varepsilon>0$, we have $$c_k\int_{\mathcal{I}\backslash G_\varepsilon} \mathrm{d}\omega_0\mathrm{d}\eta_0 \langle A_{\sigma}^{p,\omega_0,\eta_0}, \phi_{\omega_0,\eta_0} \rangle = O_\varepsilon(k^{d-1-1/3+\varepsilon}).$$
For each $i=0,...,p-1$, if $\sigma_i =(\pm, \bullet)$, we set $\beta_i= \pm$. By (\[eq:conditionsupport\]), we have that $$\begin{aligned}
A_{\sigma}^{p,\omega_0,\eta_0} (\alpha_p) &= \Big{(}-\frac{1}{2} \Big{(}\frac{k}{2\pi}\Big{)}^{d-1} \Big{)}^p \int_{({\mathbb{S}}^{d-1})^p} \int_{Y^{\beta_0}}... \int_{Y^{\beta_{p-1}}}e^{ik \Phi_p(\boldsymbol{\alpha}, \alpha_p,\boldsymbol{y})} \big{(}1- \rho(\boldsymbol{\alpha},\boldsymbol{y})\big{)} \chi_{\sigma}(\boldsymbol{\alpha},\boldsymbol{y}) a_p(\boldsymbol{\alpha},\alpha_p,\boldsymbol{y}) \mathrm{d}\boldsymbol{\alpha} \mathrm{d} \boldsymbol{y}\\
&= \Big{(}-\frac{1}{2} \Big{(}\frac{k}{2\pi}\Big{)}^{d-1} \Big{)}^p\int_{({\mathbb{S}}^{d-1})^{p}}
\int_{Z_{\alpha_0^\sigma}}... \int_{Z_{\alpha_{p-1}^\sigma}} e^{ik
\Phi^\sigma_p(\boldsymbol{\alpha},\alpha_p,\boldsymbol{z})}
\tilde{a}^\sigma_p(\boldsymbol{\alpha},\alpha_p,z))
\mathrm{d}\boldsymbol{\alpha}
\mathrm{d}\boldsymbol{z}
\end{aligned}$$
Here, we wrote $\boldsymbol{y}^\sigma(\boldsymbol{z}):= (y_{\alpha_0}^{\beta_0}(z_0),...,y_{\alpha_{p-1}}^{\beta_{p-1}}(z_{p-1}))$, and $$\begin{aligned}
\Phi^\sigma_p(\boldsymbol{\alpha},\alpha_p,\boldsymbol{z})&:=\Phi_p(\boldsymbol{\alpha},\alpha_p,\boldsymbol{y}^\sigma(\boldsymbol{z}))\\
&= \frac{i}{2} |\alpha_0-\omega_0|^2+ \alpha_0\cdot \eta_0 + \sum_{i=0}^{p-1} (\alpha_i-\alpha_{i+1})\cdot y_i^\sigma(z_i),\\
\tilde{a}^\sigma_p(\boldsymbol{\alpha},\alpha_p,\boldsymbol{z}))&:= \frac{\tilde{a}_p(\boldsymbol{\alpha},\alpha_p,\boldsymbol{y}^\sigma(\boldsymbol{z}))}{|\alpha_0\cdot \nu_{y^{\sigma}(z_0)}|... |\alpha_{p-1}\cdot \nu_{y^{\sigma}(z_{p-1})}|}\\
&= \tilde{\chi}_\sigma (\boldsymbol{\alpha},\alpha_p,\boldsymbol{z}) \frac{\prod_{i=0}^{p-1} \big{(}
|\nu_{y_i^\sigma(z_i)}\cdot \alpha_i | - \nu_{y_i^\sigma(z_i)}\cdot \alpha_{i+1} +
E(\alpha_i,y_i^\sigma(z_i),k)\big{)}}{|\alpha_0\cdot \nu_{y^{\sigma}(z_0)}|... |\alpha_{p-1}\cdot \nu_{y^{\sigma}(z_{p-1})}|},
\end{aligned}$$ where $\tilde{\chi}_\sigma (\boldsymbol{\alpha},\alpha_p,\boldsymbol{y}^\sigma(\boldsymbol{z})) = \big{(}1- \rho(\boldsymbol{\alpha},\boldsymbol{y}^\sigma(\boldsymbol{z}))\big{)} \chi_{\sigma}(\boldsymbol{\alpha},\boldsymbol{y}^\sigma(\boldsymbol{z})) \chi (k^{1/3}|\alpha_0-\omega_0|) \chi
(k^{1/3}|\alpha_p-\omega_0|).$
Note that we have $$\tilde{a}^\sigma_p\in S_{1/3}.$$
Let us write $\boldsymbol{z}^\sigma$ for $(\pi_{\alpha^\sigma_j} (y_j^\sigma))_{j=0,...,p-1}$. Note that, if $\mathrm{d}$ denotes the Riemannian distance on $({\mathbb{S}}^{d-1})^{p+1}\times ({\mathbb{R}}^{d-1})^p$, we have $$\label{eq:localised image}
\begin{aligned}
&\forall \varepsilon >0, \forall (\boldsymbol{\alpha},\alpha_p,\boldsymbol{z}) \in \mathrm{spt} \tilde{\chi}_\sigma, \forall k \text{ large enough }\\
&\mathrm{d}(\boldsymbol{\alpha},\alpha_p,\boldsymbol{z}), (\boldsymbol{\alpha}^\sigma,\alpha^\sigma_p,\boldsymbol{z}^\sigma)) > k^{-1/2+\varepsilon} \Longrightarrow |\partial \Phi_p^\sigma (\boldsymbol{\alpha},\alpha_p,\boldsymbol{z}) | > k^{-\frac{1}{2}+\frac{\varepsilon}{2p}}.
\end{aligned}$$
If we write $A_{\sigma}^{p,\omega_0,\eta_0} (\alpha_p) =\Big{(}-\frac{1}{2} \Big{(}\frac{k}{2\pi}\Big{)}^{d-1} \Big{)}^p\big{(} A_\sigma^1 (\alpha_p)+ A_\sigma^2(\sigma_p)\big{)} $ with $$\begin{aligned}
A_\sigma^1(\alpha_p) := &\int_{({\mathbb{S}}^{d-1})^{p}}
\int_{Z_{\alpha_0^\sigma}}... \int_{Z_{\alpha_{p-1}^\sigma}} \chi\big{(}k^{1/2-\varepsilon} \mathrm{d}(\boldsymbol{\alpha},\alpha_p,\boldsymbol{z}), (\boldsymbol{\alpha}^\sigma,\alpha^\sigma_p,\boldsymbol{z}^\sigma))\big{)} e^{ik
\Phi^\sigma_p(\boldsymbol{\alpha},\alpha_p,\boldsymbol{z})}
\tilde{a}^\sigma_p(\boldsymbol{\alpha},\alpha_p,z))
\mathrm{d}\boldsymbol{\alpha}
\mathrm{d}\boldsymbol{z} \\
A_\sigma^2(\alpha_p) := &\int_{({\mathbb{S}}^{d-1})^{p}}
\int_{Z_{\alpha_0^\sigma}}... \int_{Z_{\alpha_{p-1}^\sigma}} \big{(}1-\chi\big{(}k^{1/2-\varepsilon} \mathrm{d}(\boldsymbol{\alpha},\alpha_p,\boldsymbol{z}), (\boldsymbol{\alpha}^\sigma,\alpha^\sigma_p,\boldsymbol{z}^\sigma))\big{)}\big{)} e^{ik
\Phi^\sigma_p(\boldsymbol{\alpha},\alpha_p,\boldsymbol{z})}
\tilde{a}^\sigma_p(\boldsymbol{\alpha},\alpha_p,z))
\mathrm{d}\boldsymbol{\alpha}
\mathrm{d}\boldsymbol{z} ,
\end{aligned}$$ then (\[eq:localised image\]) and Lemma \[lemm: nonstat\] implies that $A_\sigma^2= O(k^{-\infty})$.
On the other hand, the integrand in $A_\sigma^1$ is supported in a set of volume $k^{(-1/2+\varepsilon)2(d-1)p}$. Therefore, there exists $C,c>0$ such that $$\begin{aligned}
c_k\|A_{\sigma}^{p,\omega_0,\eta_0}\|_{C^0}
&\leq C k^{(-1/2+\varepsilon)2(d-1)p} k^{\frac{3}{2} (d-1)} k^{p(d-1)} \| \chi\big{(}k^{1/2-\varepsilon} \mathrm{d}(\boldsymbol{\alpha},\alpha_p,\boldsymbol{z}), (\boldsymbol{\alpha}^\sigma,\alpha^\sigma_p,\boldsymbol{z}^\sigma))\big{)}
\tilde{a}^\sigma_p(\boldsymbol{\alpha},\alpha_p,z))\|_{C_0}\\
&\leq C k^{\frac{3}{2}(d-1) + c\varepsilon} \| \chi\big{(}k^{1/2-\varepsilon} \mathrm{d}(\boldsymbol{\alpha},\alpha_p,\boldsymbol{z}), (\boldsymbol{\alpha}^\sigma,\alpha^\sigma_p,\boldsymbol{z}^\sigma))\big{)}
\tilde{a}^\sigma_p(\boldsymbol{\alpha},\alpha_p,z))\|_{C_0}.
\end{aligned}$$
Let us first prove the lemma when there exists $0\leq i\leq p-1$ such that $\sigma_i = (+, \bullet)$. By assumption, we then have that $\nu_{y^\sigma_i(z_i)}\cdot \alpha^\sigma_i > 0$.
Recall that the function $
\tilde{a}^\sigma_p$ can be written as $$\frac{
|\nu_{y_i^\sigma(z_i)}\cdot \alpha_i | - \nu_{y_i^\sigma(z_i)}\cdot \alpha_{i+1} +
E(\alpha_i,y_i^\sigma(z_i),k)}{|\nu_{y_i^{\sigma}(z_i)}\cdot \alpha_i|} f,$$ with $f\in S_{1/3}$.
Now, by (\[eq:NoShadow\]), we have that $E(\alpha_i,y_i^\sigma(z_i),k)= O(k^{-\infty})$. On the other hand, by the assumption we made on $\sigma$, we have $|\nu_{y_i^\sigma(z_i^\sigma)}\cdot \alpha^\sigma_i |= \nu_{y_i^\sigma(z_i^\sigma)}$. The fact that $\mathrm{d}((z_i,\alpha_i), (z_i^\sigma,\alpha_i^\sigma)) \leq k^{-1/2+\varepsilon}$, combined with the fact that we work on a set where $|\nu_{y_i^{\sigma}(z_i)}\cdot \alpha_i|\geq C k^{-1/6}$ then implies that $$\frac{
|\nu_{y_i^\sigma(z_i)}\cdot \alpha_i | - \nu_{y_i^\sigma(z_i)}\cdot \alpha_{i+1} +
E(\alpha_i,y_i^\sigma(z_i),k)}{|\nu_{y_i^{\sigma}(z_i)}\cdot \alpha_i|} =O(k^{-1/3+\varepsilon}).$$ Therefore, we obtain that
$$\| \chi\big{(}k^{1/2-\varepsilon} \mathrm{d}(\boldsymbol{\alpha},\alpha_p,\boldsymbol{z}), (\boldsymbol{\alpha}^\sigma,\alpha^\sigma_p,\boldsymbol{z}^\sigma))\big{)}
\tilde{a}^\sigma_p(\boldsymbol{\alpha},\alpha_p,z))\|_{C_0} = O(k^{-1/3 + c' \varepsilon}).$$
All in all, we have that $$c_k \Big{(}-\frac{1}{2} \Big{(}\frac{k}{2\pi}\Big{)}^{d-1} \Big{)}^p \|A_\sigma^{2}\|_{C^0} \leq C k^{\frac{3}{2} (d-1) - 1/3 + c''\varepsilon}.$$ Using the fact that $\|\phi_{\omega_0,\eta_0}\|_{L^1} = O(k^{(d-1)/2})$, we get that $$c_k\int_{\mathcal{I}\backslash G_\varepsilon} \mathrm{d}\omega_0\mathrm{d}\eta_0 \langle A_{\sigma}^{p,\omega_0,\eta_0}, \phi_{\omega_0,\eta_0} \rangle = O(k^{d-1-1/3 + c''\varepsilon}),$$ which proves the lemma in this case.
Let us now suppose that $\forall i=0,...,p-1$, we have $\sigma_i=(-,\bullet)$. From (\[eq:DifferentKappas\]), we see that $\tilde{\kappa}^\sigma(\omega_0,\eta_0) = \kappa^{n_0}(\omega_0,\eta_0)$, for some $n_0\in \mathbb{N}$, where $\tilde{\kappa}^\sigma$ is as in (\[eq:deftildekappa\]). The fact that $\sigma\neq \sigma_0$ implies furthermore that $n_0>0$.
Suppose that $(\omega_0,\eta_0)$ is such that for all $n_0=1,...,p-1$, we have $\mathrm{d}((\omega_0,\eta_0),\kappa^{n_0}(\omega_0,\eta_0))> k^{-1/3}$. Then, using (\[eq:localised image\]) and computing $\partial_{\alpha_p} \Phi_p^\sigma$, we see that we have $$\begin{aligned}
&\forall \varepsilon >0, \forall (\boldsymbol{\alpha},\alpha_p,\boldsymbol{z}) \in \mathrm{spt} \tilde{\chi}_\sigma, \forall k \text{ large enough },~~ |\partial \Phi_p^\sigma (\boldsymbol{\alpha},\alpha_p,\boldsymbol{z}) | > k^{-\frac{1}{2}+\frac{\varepsilon}{2p}}.
\end{aligned}$$
Therefore, by Lemma \[lemm: nonstat\], we have that $$\langle A_\sigma^{p,\omega_0,\eta_0},\phi_{\omega_0,\eta_0}\rangle = O(k^{-\infty}).$$
Therefore, if we write $$\mathcal{I}^p_{per}(k) :=\{(\omega_0,\eta_0)\in \mathcal{I} \text{ such that } \exists 1\leq n \leq p-1, \mathrm{d}((\omega_0,\eta_0), \kappa^n(\omega_0,\eta_0))< k^{-1/3}\},$$ we have that
$$c_k\int_{\mathcal{I}\backslash G_\varepsilon} \mathrm{d}\omega_0\mathrm{d}\eta_0 \langle A_{\sigma}^{p,\omega_0,\eta_0}, \phi_{\omega_0,\eta_0} \rangle =c_k\int_{\mathcal{I}_{per}^p(k)} \mathrm{d}\omega_0\mathrm{d}\eta_0 \langle A_{\sigma}^{p,\omega_0,\eta_0}, \phi_{\omega_0,\eta_0} \rangle + O(k^{-\infty}).$$
Now, thanks to (\[eq: hypVol3\]), we know that ${\mathrm{Vol}}(\mathcal{I}^p_{per}(k)) = O(k^{-1/3})$. We deduce from this that $$c_k\int_{\mathcal{I}_{per}^p(k)} \mathrm{d}\omega_0\mathrm{d}\eta_0 \langle A_{\sigma}^{p,\omega_0,\eta_0}, \phi_{\omega_0,\eta_0} \rangle = O(k^{d-1-1/3}).$$ This concludes the proof of the lemma.
Proofs of theorems \[th: main theorem\] and \[thm:scattering-phase\] {#sec:proofs}
====================================================================
We now prove the theorems from the introduction using the trace formula in Proposition \[thm:trace-lemma\]. The proof of Theorem \[th: main theorem\] from Proposition \[thm:trace-lemma\] follows is the same as in [@I2016 §5], but we recall it for the reader’s convenience.
Phase shift asymptotics
-----------------------
Let us define, for any $\alpha>0$, $$C^0_{\alpha}(\mathbb{S}^1)= \{f\in C^0(\mathbb{S}^1;\mathbb{C}); f(z)\big{|}\log |z-1|\big{|}^{\alpha} \text{ is continuous } \}.$$
$$\|f\|_{\alpha} = \sup_{|z|=1, z\neq 1} \big{|}\log |z-1|\big{|}^{\alpha} |f(z)|~~ \text{ for } f\in C^0_{\alpha}(\mathbb{S}^1).$$
Note that $C^0_{\alpha}\subset C^0_{\alpha'}$ if $\alpha>\alpha'$. We will now prove the following theorem, which is a slightly refined version of Theorem \[th: main theorem\].
\[th: main theorem 2\] Let $\Omega\subset {\mathbb{R}}^d$ be a smooth strictly convex open set, such that (\[eq: hypVol3\]) holds. Let $\alpha>d$ and let $f\in C^0_{\alpha}(\mathbb{S}^1)$. Then we have $${\langle}\mu_{k}, f {\rangle}=
\frac{{\mathrm{Vol}}({\partial}\Omega) \omega_{d-1}}{2\pi} \int_{0}^{2\pi}f(e^{i\theta}) d\theta + o(1).$$
Before writing the proof, let us state two technical lemmas. Recall that we denote the eigenvalues of $S(k)$ by $e^{i\beta_{k,n}}$. We shall from now on take the convention that $|e^{i\beta_{k,n}}-1|\geq |e^{i\beta_{k,n+1}}-1|$.
For any $L\geq1$, we shall denote by $N_{L,k}$ the number of $n\in \mathbb{N}$ such that $|e^{i\beta_{k,n}}-1|\geq e^{-Lk}$.
\[lem : bound big shifts\] There exists $C_0>0$ such that for any $L\geq1$ and $k\geq 1$, we have $N_{L,k}\leq C_0 \big{(}Lk\big{)}^{d-1}$.
Thanks to equation (2.3) in [@christiansen2015sharp] (which relies on the methods developed in [@zworski1989sharp]), we have that there exists $C>0$ independent of $k$ and $n$ such that $$\label{eq:christ}
|e^{i\beta_{k,n}}-1|\leq Ck^d \exp \Big{(} Ck- \frac{n^{1/(d-1)}}{C}\Big{)}.$$
In particular, we have that for any $N\geq 1$, $$\begin{aligned}
\prod_{n=1}^N |e^{i\beta_{k,n}}-1| &\leq \Big{(}C k^d\Big{)}^N \exp \Big{(} \frac N C k- \frac{1}{C} \sum_{n=1}^N n^{1/(d-1)}\Big{)}\\
&\leq \big{(}C k^d\big{)}^N \exp \Big{(} N Ck- C' N^{d/(d-1)}\Big{)},
\end{aligned}$$ for some $C'>0$ independent of $k,N$.
Therefore, we have that $$\begin{aligned}
e^{-k LN_{L,k}}&\leq \prod_{n=1}^{N_{L,k}} |e^{i\beta_{k,n}}-1|\\
&\leq \Big{(}C k^d\Big{)}^{N_{L,k}} \exp \Big{(} CkN_{L,k} - C' {N_{L,h}}^{d/(d-1)}\Big{)}.
\end{aligned}$$
By taking logarithms, we get $$\begin{aligned}
-kLN_{L,k}
&\leq {N_{L,k}}\log\Big{(}Ck^d\Big{)} + kC N_{L,k} - C' {N_{L,k}}^{d/(d-1)}.
\end{aligned}$$ The first term in the right hand side is negligible, so we get, by possibly changing slightly the constant $C'$, $$\begin{aligned}
C' {N_{L,k}}^{d/(d-1)}
&\leq k{N_{L,k}} (C+L).
\end{aligned}$$ Therefore, $N_{L,k} \leq \Big{(}\frac{k(C+L)}{C'}\Big{)}^{d-1}\leq C_0 (Lk)^{d-1}$ for some $C_0>0$ large enough, but independent of $L$ and $k$, which concludes the proof of the lemma.
\[lem : boundmeasure\] For any $\alpha>d$, there exists $C_\alpha> 0$ such that for any $f\in C^0_{\alpha}(\mathbb{S}^1)$, we have $$|\langle\mu_{k},f\rangle| \leq C \|f\|_{\alpha}$$
We have $$\label{eq : mu k}
\begin{aligned}
|\langle\mu_{k},f \rangle| &= \Big{(}\frac{2\pi}{k}\Big{)}^{d-1} \Big{|}\sum_{n\in
\mathbb{N}} f(e^{i\beta_{k,n}})\Big{|}\\
&\leq \Big{(}\frac{2\pi}{k}\Big{)}^{d-1}\sum_{|e^{i\beta_{k,n}}-1|\geq e^{-k}} |f(e^{i\beta_{k,n}})| + \Big{(}\frac{2\pi}{k}\Big{)}^{d-1} \sum_{|e^{i\beta_{k,n}}-1|< e^{-k}} |f(e^{i\beta_{k,n}})|.
\end{aligned}$$
Let us consider the first sum. By Lemma \[lem : bound big shifts\], it has at most $C_0 k^{d-1}$ terms. Hence, it is bounded by $$\Big{(}\frac{2\pi}{k}\Big{)}^{d-1} \sum_{|e^{i\beta_{k,n}}-1|\geq e^{-k}} |f(e^{i\beta_{k,n}})|\leq \Big{(}\frac{2\pi}{k}\Big{)}^{d-1} C_0 k^{d-1}\|f\|_{C^0}\leq C \|f\|_{\alpha},$$ for some $C>0$. Let us now consider the second term in (\[eq : mu k\]). For each $p\geq 1$, we denote by $\sigma_{p,k}$ the set of $n\in \mathbb{N}$ such that $e^{-k(p+1)}\leq |e^{i\beta_{k,n}}-1|< e^{-pk}$. By Lemma \[lem : bound big shifts\], $\sigma_{p,k}$ contains at most $C_0 \big{(} k(p+1)\big{)}^{d-1}$ elements. On the other hand, for each $n\in \sigma_{p,k}$, we have $$|f(e^{i\beta_{k,n}})| \leq \|f\|_\alpha \big{|}\log (e^{-pk})\big{|}^{-\alpha}=\frac{\|f\|_\alpha}{(kp)^{\alpha}} .$$ Therefore, we have $$\begin{aligned}
\Big{(}\frac{2\pi}{k}\Big{)}^{d-1}\sum_{|e^{i\beta_{k,n}}-1|< e^{-k}} |f(e^{i\beta_{k,n}})|
&= \Big{(}\frac{2\pi}{k}\Big{)}^{d-1}\sum_{p=1}^{+\infty} \sum_{n\in \sigma_{p,k}} |f(e^{i\beta_{k,n}})| \\
&\leq \Big{(}\frac{2\pi}{k}\Big{)}^{d-1}\sum_{p=1}^{+\infty} C_0 \big{(} p(k+1)\big{)}^{d-1} \frac{\|f\|_\alpha}{(kp)^{\alpha}}\\
&\leq C k^{-\alpha} \|f\|_\alpha,
\end{aligned}$$ for some $C$ independent of $k$. This concludes the proof of the lemma.
We have proved the result for all trigonometric polynomials vanishing at 1 in Proposition \[thm:trace-lemma\]. Let $\alpha>\alpha'>d$, and let $f\in C^0_\alpha\subset C^0_{\alpha'}$. Let us show that $f$ can be approximated by trigonometric polynomials vanishing at 1 in the $C^0_{\alpha'}$ norm, which will conclude the proof of the theorem thanks to Lemma \[lem : boundmeasure\].
Since $f (z) \big{(}1+\big{|}\log |z-1|\big{|}^{2\alpha}\big{)}^{1/2}$ is continuous, we may find a sequence $P_n$ of polynomials such that $$\big{\|}P_n - f (z) \big{(}1+\big{|}\log |z-1|\big{|}^{2\alpha}\big{)}^{1/2}\big{\|}_{C^0}\leq 1/n.$$ Since $f(0)=0$, we may suppose that $P_n(1)=0$. We may also suppose that $P_n'(1)=0$ (for a proof of this fact, see for example [@duren2012invitation Theorem 8, §6]).
Since the function $\big{|}\log |z-1|\big{|}^{\alpha'}\big{(}1+\big{|}\log |z-1|\big{|}^{2\alpha}\big{)}^{-1/2}$ is continuous, we have that $$\label{approche}
\big{\|}P_n \big{|}\log |z-1|\big{|}^{\alpha'}\big{(}1+\big{|}\log |z-1|\big{|}^{2\alpha}\big{)}^{-1/2} - f (z) \big{|}\log |z-1|\big{|}^{\alpha'}\big{\|}_{C^0}\leq C/n.$$
Now, since $P_n(1)=P_n'(1)=0$, the function $P_n /\big{(}(z-1)\big{(}1+\big{|}\log |z-1|\big{|}^{2\alpha}\big{)}^{1/2}\big{)} $ is continuous, and we may find a polynomial $Q_n$ such that $$\Big{\|} \frac{P_n}{(z-1)\big{(}1+\big{|}\log |z-1|\big{|}^{2\alpha}\big{)}^{1/2}} - Q_n\Big{\|}_{C^0}\leq 1/n$$ Since the function $ (z-1)\big{|}\log |z-1|\big{|}^{\alpha'}$ is continuous, we obtain that $$\label{approche2} \Big{\|} P_n\big{|}\log |z-1|\big{|}^{\alpha'}\big{(}1+\big{|}\log |z-1|\big{|}^{2\alpha}\big{)}^{-1/2} - Q_n(z-1)\big{|}\log |z-1|\big{|}^{\alpha'}\Big{\|}_{C^0}\leq C'/n$$
Combining (\[approche\]) and (\[approche2\]), we obtain that $f$ can be approached by $(z-1)Q_n$ in the $C^0_{\alpha'}$ norm. This concludes the proof of Theorem \[th: main theorem 2\].
Total scattering phase
----------------------
We now give our alternative proof of the scattering phase asymptotics in Theorem \[thm:scattering-phase\]. We begin by recalling that the scattering phase $s(k)$ can be defined continuously in such a way that $\lim_{k \to 0^+} s(k) = 0$ and thus defined is in fact smooth for all $k >
0$. We define the ‘reduced’ scattering phase by the sum $$s_{2\pi}(k) = -\sum_{e^{i\beta_{k,n}} \in \sigma(S(k))} \beta_{k,n}$$ where the logarithms of the eigenvalues, the $\beta_{k,n}$ are chosen to take values in $(-2\pi, 0]$. For fixed $k$ the eigenvalues accumulate at $1$ from the bottom half plane and thus contribute positive values to the sum, which is nonetheless finite. A result of Eckmann-Pillet [@EP1995] shows that eigenvalues approach $1$ with positive imaginary part if and only if $k$ approaches a Dirichlet eigenvalue of $\Omega$. In fact, with $N_D(k)$ as in , we have $$s(k) = 2 \pi N_D(k) + s_{2\pi}(k).$$ Under the hypothesis that the measure of the periodic billiard trajectories in $\Omega$ is zero, it is known [@ivrii1980second] that $$N_D(k) = \frac{\omega_d}{(2\pi)^{d}} {\mathrm{Vol}}(\Omega) k^d -
\frac{\omega_{d - 1}}{4 (2\pi)^{d-1}} {\mathrm{Vol}}({\partial}\Omega) k^{d -1} + o(k^{d-1}).$$ Note that this is the same asymptotic expansion as for the scattering phase, with the sign of the second term reversed. We claim that $$s_{2\pi}(k) = s(k) - N_D(k) = \frac{\omega_{d - 1}}{2 (2\pi)^{d-2}} {\mathrm{Vol}}({\partial}\Omega) k^{d -1} + o(k^{d-1}).$$ We will prove this by breaking up the unit circle into $M \in \mathbb{N}$ sectors of size $2\pi/M$ estimating the sum defining $s_{2\pi}(k)$ in these sectors. Namely, let $A_{M,k}(j) = \{- 2\pi j/M < \beta_{k,n} < - 2
\pi (j + 1)/M \}$ and $\alpha_{M, k}(j) = \sum_{A_{M, k}(j)} -
\beta_{M,k}$, so that $$s_{M, k} = \sum_{j = 0}^{M - 1} \alpha_{M, k}(j).$$ We begin with $j = 0$, which is distinct from $j > 0$ since there are infinitely many phase shifts in $A_{M, k}(0)$. We claim that $$|a_{M, k}(0)| \le \frac{C}{M} k^{d - 1}.$$ Indeed, using and a constant $C> 0$ whose value changes from line to line, we see that $$\label{eq:1}
\begin{split}
a_{M, k}(0) &\le \sum_{j = 0}^\infty \sum_{2^{-j}/M \ge
|\beta_{k,n}|/(2\pi) \ge 2^{-(j + 1)}/M } | e^{i \beta_{k,n}} - 1 | \\
&\le \sum_{j = 0}^\infty {\left(}\frac{1}{M} 2^{-j} {\right)}\sum_{2^{-j}/M \ge
|\beta_{k,n}|/(2\pi) \ge 2^{-(j + 1)}/M } 1 \\
&\le \sum_{j = 0}^\infty {\left(}\frac{1}{M} 2^{-j} {\right)}C (k + (j +
1)/k)^{d-1} \\
&\le \frac{k^{d-1}}{M} \sum_{j = 0}^\infty {\left(}2^{-j} {\right)}C (1 + (j
+ 1)/k^2)^{d-1} \le C\frac{ k^{d-1}}{M}
\end{split}$$
For $j > 0$, we estimate $\alpha_{M, k}(j)$ from above and below, and clearly $$\begin{split}
\frac{2\pi j}{M} | A_{M, k}(j) | \le a_{M, k}(j) \le \frac{ 2 \pi (j +
1)}{M} | A_{M, k}(j) |
\end{split}$$ It follows from the , since our sectors are size $2\pi/M$, that for $0 < j \le M -1$, for any $\delta > 0$ and $k > k_{M, \delta}$, there is a constant $C > 0$ such that $${\left(}\frac{\omega_{d-1}}{(2\pi)^{d-1}} \frac{2\pi}{M}
{\mathrm{Vol}}({\partial}\Omega) - \delta {\right)}k^{d-1} \le |A_{M, k}(j)| \le {\left(}\frac{\omega_{d-1}}{(2\pi)^{d-1}} \frac{2\pi}{M}
{\mathrm{Vol}}({\partial}\Omega) + \delta {\right)}k^{d-1}$$
Since $\sum_{j = 1}^{M-1} (j + 1) = M(M - 1)/2$ we have $$\begin{gathered}
{\left(}\frac{\omega_{d-1}}{(2\pi)^{d-2}} \frac 12 {\mathrm{Vol}}({\partial}\Omega) -
C(1/M + \delta M) {\right)}k^{d-1} \le s_{M, k} \\ s_{M, k} \le {\left(}\frac{\omega_{d-1}}{(2\pi)^{d-2}} \frac 12 {\mathrm{Vol}}({\partial}\Omega) + C(1/M +
\delta M) {\right)}k^{d-1},
\end{gathered}$$ and thus $$\begin{split}
\limsup_{k \to \infty} s(k) k^{d-1} &\le
\frac{\omega_{d-1}}{(2\pi)^{d-2}} \frac 12 {\mathrm{Vol}}({\partial}\Omega) + C(1/M +
\delta M), \\ \liminf_{k \to \infty} s(k) k^{d-1} &\ge
\frac{\omega_{d-1}}{(2\pi)^{d-2}} \frac 12 {\mathrm{Vol}}({\partial}\Omega) - C(1/M +
\delta M)
\end{split}$$ for any $M \in \mathbb{N}, \delta > 0$. Taking $\delta = 1/M^2$ and sending $M \to \infty$ gives the result.
Symbol classes and stationary phase
===================================
Let $X$ be a compact manifold, and let $a=(a_k)_{k>0}$ be a family of functions in $C^{\infty}(X)$, and let $0\leq\delta<1/2$. We shall write that $a\in S_{\delta}(X)$ if $$\label{eq:defsymbol}
\forall \beta \in \mathbb{N}^{d-1}, \exists C_{\beta} \text{ such that } \forall x\in X, |\partial^\beta a(x,k)| \leq C_{\beta}k^{|\beta|}.$$ The following lemma follows from [@hormander:vol1 Lemma 7.7.1].
\[lemm: nonstat\] Let $\delta<1/2$. Suppose that $a\in S_{\delta}(X)$, and that $\Phi_k\in S_0(X)$. Suppose that there exists $C, \varepsilon>0$ such that for all $x$ in the support of $a_k$, we have $|\partial \Phi_k(x)|\geq C k^{-1/2+\varepsilon}$. We then have $$\int_X e^{ik \Phi_k(x)} a_k(x) dx = O(k^{-\infty}).$$
The following stationary phase result is a consequence of [@hormander:vol1 Theorem 7.7.5] when the phase $\phi$ is independent of $k$, and its proof is the same when $\phi$ depends on $k$.
\[lem: phistat\] Let $X$ be a $n$ dimensional manifold, and let $X'\Subset X$. Let $a\in S_{1/3}(X')$, let $x_0\in X'$ and let $\phi\in S_{1/3}(X)$ be a function, such that for all, $\Im \phi\geq 0$ in $X'$, $\Im \phi(x_0)=0$, $\nabla \phi (x_0)=0$. We furthermore assume that there exists $c,C>0$ independent of $k$ such that $c\leq |\det \partial^2 \phi (x_0)| \leq C$, and $|x-x_0|/|\nabla \phi(x)|\leq C$. We then have $$\int_X a(x) e^{ik \phi(x)} \mathrm{d}x = \Big{(}\frac{2\pi}{k}\Big{)}^{n/2} |\det (\partial^2 \phi(x_0)/i)|^{-1/2} a(x_0) e^{ik \phi(x_0)} + O\big{(}k^{-n/2-1/3}\big{)}.$$
Traces and Gaussian states {#section: appendixGaussian}
==========================
The following lemma can be found in [@ingremeau2016semi], but we recall its proof for completeness.
\[lemma: resolution3\] Let $f\in L^2({\mathbb{S}}^{d-1})$. We have $$\label{resolution}f(\omega) = c_k \int_{{\mathbb{S}}^{d-1}}\mathrm{d}\xi \int_{\xi^\perp} \mathrm{d}x \phi_{x,\xi}(\omega) \int_{{\mathbb{S}}^{d-1}} \mathrm{d}\omega' \overline{ \phi_{x,\xi}(\omega')} f(\omega'),$$ where $c_k$ is a parameter depending on $k$, with $c_k= 2^{-(d-1)/2}\big{(}\frac{k}{2\pi} \big{)}^{3(d-1)/2} + O_{k\rightarrow \infty}(k^{3d/2-2})$.
Let us write $g(\omega):= \int_{{\mathbb{S}}^{d-1}}\mathrm{d}\xi \int_{\xi^\perp} \mathrm{d}x \phi_{x,\xi}(\omega) \int_{{\mathbb{S}}^{d-1}} \mathrm{d}\omega' \overline{ \phi_{x,\xi}(\omega')} f(\omega') $.
Let us write for all $t\geq 0$ $\tilde{\chi}_k(t) := e^{-k t^2/2} \chi \big{(}k^{1/3} t \big{)}$, so that $\tilde{\chi}_k$ has support in $[0,\frac{3}{4}k^{-1/3})$.
We have $$g(\omega)= \int_{{\mathbb{S}}^{d-1}}\mathrm{d}\xi \int_{{\mathbb{S}}^{d-1}} \mathrm{d}\omega' \tilde{\chi}_k(|\omega-\xi|) \tilde{\chi}_k(|\omega'-\xi|) f(\omega') \int_{\xi^\perp} \mathrm{d}x e^{\frac{i}{h} (-\omega+\omega')\cdot x}.$$
On $\{\omega \in {\mathbb{S}}^{d-1}; |\omega-\xi|\leq 3/4\}$, we may define a local chart by projecting on $\xi^\perp$. This way, we have $\omega=\omega(y)$ with $y\in \xi^\perp$, so that $|y|= |\sin (\widehat{\omega,\xi})|$, where $(\widehat{\omega,\xi})$ denotes the angle between the vectors $\omega$ and $\xi$. We also have $\Big{|} \det \Big{(}\frac{\mathrm{d}\omega}{dy}\Big{)}\Big{|} = |\cos (\widehat{\omega,\xi})|^{d-1}$.
Noting that if $x\in \xi^\perp$, we have $\omega(y)\cdot x= y\cdot x$, we get $$\begin{aligned}
g(\omega)&=\int_{{\mathbb{S}}^{d-1}}\mathrm{d}\xi \int_{\xi^\perp} |\cos (\widehat{\omega(y'),\xi})|^{d-1} \mathrm{d}y' \tilde{\chi}_k(|\omega-\xi|) \tilde{\chi}_k(|\omega(y')-\xi|) f(\omega(y')) \int_{\xi^\perp} \mathrm{d}x e^{\frac{i}{h} (-y_{\omega}+y')\cdot x}\\
&= \Big{(}\frac{2\pi}{k}\Big{)}^{d-1} f(\omega) \int_{{\mathbb{S}}^{d-1}}\mathrm{d}\xi |\cos (\widehat{\omega,\xi})|^{d-1} \tilde{\chi}_k^2(|\omega-\xi|).
\end{aligned}$$ We write $c_k ^{-1}:= (2\pi/k)^{d-1} \int_{{\mathbb{S}}^{d-1}}\mathrm{d}\xi |\cos (\widehat{\omega,\xi})|^{d-1} \tilde{\chi}_k^2(|\omega-\xi|)$, which is independent of $\omega$, since the integrand depends only on $|\omega-\xi|$.
Let us write $r(y)= |\omega(y)-\xi|$. We have $r(y) = y +O(y^2)$.
We have $$\begin{aligned}
c_k^{-1} &= \Big{(}\frac{2\pi}{k}\Big{)}^{d-1} \int_{\xi^\perp} |\cos (\widehat{\omega(y),\xi})|^{d-1} e^{-k|\omega(y)-\xi|^2} \big{|}\chi\big{(}k^{1/3}|\omega(y)-\xi|\big{)}\big{|}^2 \mathrm{d}y\\
&= \Big{(}\frac{2\pi}{k}\Big{)}^{d-1} \int_{\xi^\perp} \Big{|}\cos \Big{(}\widehat{\omega\Big{(}\frac{z}{\sqrt{k}}\Big{)},\xi}\Big{)}\Big{|}^{d-1} e^{-k r(k^{-1/2}z)^2} \Big{|}\chi\Big{(}k^{1/3}r(z/\sqrt{k})\Big{)}\Big{|}^2 k^{(1-d)/2}\mathrm{d}z\\
&= \Big{(}\frac{2\pi}{k}\Big{)}^{d-1} \Big{(} k^{(1-d)/2} \int_{\xi^\perp} e^{-z^2} \mathrm{d}z + O(k^{-d/2})\Big{)}\\
&=2^{-(d-1)/2}\Big{(}\frac{2\pi}{k}\Big{)}^{3(d-1)/2} + O\big{(}k^{-3d/2+1} \big{)}.
\end{aligned}$$
As a corollary of the resolution of identity formula, let us state the following result. The proof is standard (see for instance [@combescure2012coherent 1.2.3]), but we recall it for the reader’s convenience.
Let $A$ be a trace-class operator acting on $L^2({\mathbb{S}}^{d-1})$. We then have $$\mathrm{Tr} A = c_k \int_{{\mathbb{S}}^{d-1}} \mathrm{d}\omega \int_{\omega^\perp} \mathrm{d}\xi \langle\phi_{\omega,\xi}, A \phi_{\omega,\xi}\rangle_{L^2({\mathbb{S}}^{d-1})}.$$
Let $A$ be a trace class operator, and let $(e_n)_{n\in \mathbb{N}}$ be an orthonormal basis of $L^2({\mathbb{S}}^{d-1})$. We have $$\begin{aligned}
\operatorname{Tr}A &= \sum_{n=0}^{+\infty} \langle e_n, A e_n \rangle\\
&= c_k^2 \sum_{n=0}^{+\infty} \int \mathrm{d}\omega \mathrm{d}\eta \mathrm{d}\omega' \mathrm{d}\eta'\langle e_n,\phi_{\omega,\eta}\rangle \langle e_n,\phi_{\omega',\eta'}\rangle \langle \phi_{\omega,\eta} A\phi_{\omega',\eta'} \rangle\\
&= c_k^2 \int \mathrm{d}\omega \mathrm{d}\eta \mathrm{d}\omega' \mathrm{d}\eta' \langle \phi_{\omega',\eta'},\phi_{\omega,\eta}\rangle \langle \phi_{\omega,\eta} A\phi_{\omega',\eta'} \rangle\\
&= c_k \int \mathrm{d}\omega \mathrm{d}\eta \langle \phi_{\omega,\eta} A\phi_{\omega,\eta} \rangle.
\end{aligned}$$ Here, we used Lemma \[lemma: resolution3\] twice to go from the first line to the second, by decomposing $e_n$ in terms of the functions $\phi_{\omega,\eta}$. We then used the fact that $e_n$ is an orthonormal basis to go from the second line to the third one. Finally, we used Lemma \[lemma: resolution3\] to go from the third line to the last one.
Computation of a determinant
============================
\[lem:determinant\] Let $n,p\in \mathbb{N}$, and let $A,B\in M_n(\mathbb{C})$. Consider the matrix $$M:= \begin{pmatrix}
& & & & & & \boldsymbol{1}_n & -\boldsymbol{1}_n & 0 & ... & 0& 0 & 0 \\
& & & & & & 0 & \boldsymbol{1}_n & -\boldsymbol{1}_n &... & 0 & 0 & 0\\
& & \boldsymbol{0}_{np\times np} & && &0 &0&-\boldsymbol{1}_n& ... & 0&0&0 \\
& & & & & && & & ...\\
&&&&&& 0 & 0 &0 &...& \boldsymbol{1}_n & -\boldsymbol{1}_n & 0\\
& & & & & & 0 & 0 & 0 & ... & 0 & \boldsymbol{1}_n & -\boldsymbol{1}_n\\
\boldsymbol{1}_n & 0 & 0 & ...& 0 & 0 & A+B & 0 & 0 & ... & 0 & 0 & 0\\
-\boldsymbol{1}_n& \boldsymbol{1}_n & 0& ... & 0 & 0 & \\
0 & -\boldsymbol{1}_nr & \boldsymbol{1}_n & ... & 0 & 0 & & & \bold{0}_{n(p+1)\times n(p-1)} \\
&&&... & & & & \\
0 & 0 &0& ... & -\boldsymbol{1}_n & \boldsymbol{1}_n & \\
0 &0 & 0 & ...& 0 & -\boldsymbol{1}_n & 0 & 0 & 0 & ... & 0 &0 & A-B
\end{pmatrix}.$$
We then have $$\det M = (-1)^{pn}\det (2A).$$
In the right part of the matrix, we add each block of $n$ columns to the block of $n$ columns on its right, starting from the left. We get $$\det M= \det \begin{pmatrix}
& & & & & & \boldsymbol{1}_n & 0& 0 & ... & 0& 0 & 0 \\
& & & & & & 0 & \boldsymbol{1}_n & 0 &... & 0 & 0 & 0\\
& & \boldsymbol{0}_{np\times np} & && &0 &0&0& ... & 0&0&0 \\
& & & & & && & & ...\\
&&&&&& 0 & 0 &0 &...& \boldsymbol{1}_n &0& 0\\
& & & & & & 0 & 0 & 0 & ... & 0 & \boldsymbol{1}_n & 0\\
\boldsymbol{1}_n & 0 & 0 & 0 & ...& 0 & 0 && & &\boldsymbol{L}\\
-\boldsymbol{1}_n& \boldsymbol{1}_n & 0& 0 & ... & 0 & 0 & \\
0 & -\boldsymbol{1}_d & \boldsymbol{1}_d & 0 & ... & 0 & 0 & & & &\bold{0}_{n(p+1)\times n(p-1)} \\
&&&... & & & \\
0 & 0 &... & & & -\boldsymbol{1}_n & \boldsymbol{1}_n & \\
0 & & & & ...& 0 & -\boldsymbol{1}_n & 0 & 0 & 0 & ... & 0 &0 & A-B
\end{pmatrix},$$
where $\boldsymbol{L}$ is the $n(p+1)\times d$ matrix $$\boldsymbol{L}:= \begin{pmatrix}
A+B & A+B & ... & A+B
\end{pmatrix}.$$
For each $i=1,...,p$, we may use the Laplace formula[^2] for the block of lines number $i$. We obtain that $$\begin{aligned}
\det M&= (-1)^{pn}\det \begin{pmatrix}
\boldsymbol{1}_n & 0 & A+B\\
-\boldsymbol{1}_n& \boldsymbol{1}_n & 0 \\
0 & -\boldsymbol{1}_n & A-B \\
\end{pmatrix}\\
&= (-1)^{pn}\det \begin{pmatrix}
\boldsymbol{1}_n & 0 & A+B\\
0& \boldsymbol{1}_n & A+B \\
0 & -\boldsymbol{1}_n & A-B \\
\end{pmatrix}\\
&= (-1)^{pn}\det \begin{pmatrix}
\boldsymbol{1}_n & A+B \\
-\boldsymbol{1}_n & A-B \\
\end{pmatrix}\\
&= (-1)^{pn} \det (2A).
\end{aligned}$$
[^1]: The $\chi$ is not very important here, and we could take another power of $k$ in it. It is just here to ensure that the integral in (\[eq:trace-gaussian-states\]) makes sense.
[^2]: We actually obtain a factor $(-1)^{(pn)^2)}$ due to all the sign changes in the Laplace formula, which is always equal to $(-1)^{pn}$.
|
---
abstract: 'A short proof of a conjecture of Kropholler is given. This gives a relative version of Stallings’ Theorem on the structure of groups with more than one end. A generalisation of the Almost Stability Theorem is also obtained, that gives information about the structure of the Sageev cubing.'
author:
- 'M.J. Dunwoody'
title: Almost Invariant Sets
---
[Introduction]{} Let $G$ be a group. A subset $A$ of $G$ is said to be [*almost invariant*]{} if the symmetric difference $A + Ag$ is finite for every $g \in G$. In addition $A$ is said to be [*proper*]{} if both $A$ and $A^* = G -A$ are infinite. The group $G$ is said to have more than one end if it has a proper almost invariant subset.
\[SC\] A group $G$ contains a proper almost invariant subset (i.e. it has more than one end) if and only if it has a non-trivial action on a tree with finite edge stabilizers.
This result was proved by Stallings [@[St]] for finitely generated groups and was generalized to all groups by Dicks and Dunwoody [@[DD]]. The action of a group $G$ on a tree is [*trivial*]{} if there is a vertex that is fixed by all of $G$. Every group has a trivial action on a tree.
Let $T$ be a tree with directed edge set $ET$. If $e$ is a directed edge, then let $\bar e$ denote $e$ with the reverse orientation. If $e, f$ are distinct directed edges then write $e >f$ if the smallest subtree of $T$ containing $e$ and $f$ is as below.
(-.5, 2)– (1,2)–(2,1)–(3,2)–(4,1)–(5,2)–(6,1)–(7,2)–(8,1)–(9,2)–(10.5,2) ; (.5,2) node [$>$]{} ; (9.5,2) node [$>$]{} ; (.5,2.1) node \[above\] [$e$]{} ; (9.5,2.1) node \[above\] [$f$]{} ;
Suppose the group $G$ acts on $T$. We say that $g$ [*shifts* ]{} $e$ if either $e >ge$ or $ge > e$. If for some $e \in ET$ and some $g \in G$, $g$ shifts $e$, then $G$ acts non-trivially on a tree $T_e$ obtained by contracting all edges of $T$ not in the orbit of $e$ or $\bar e$. In this action there is just one orbit of edge pairs. Bass-Serre theory tells us that either $G = G_u*_{G_e} G_v$ where $u, v$ are the vertices of $e$ and they are in different orbits in the contracted tree $T_e$, or $G$ is the HNN-group $G = G_u *_{G_e}$ if $u,v$ are in the same $G$-orbit. If either case occurs we say that $G$ splits over $G_e$.
If there is no edge $e$ that is shifted by any $g \in G$, (and $G$ acts without involutions, i.e. there is no $g \in G$ such that $ge = \bar e$) then $G$ must fix a vertex or an end of $T$. If the action is non-trivial, it fixes an end of $T$, i.e. $G$ is a union of an ascending sequence of vertex stabilizers, $G = \bigcup G_{v_n}$, where $v_1, v_2, \dots $ is a sequence of adjacent vertices and $G_{v_1} \leq G_{v_2}\leq \dots $ and $G \not= G_{v_n} $ for any $n$.
Thus Theorem \[SC\] could be restated as
\[[@[St]], [@[DD]]\] A group $G$ contains a proper almost invariant subset (i.e. it has more than one end) if and only if it splits over a finite subgroup or it is countably infinite and locally finite.
The if part of the theorem is fairly easy to prove. We now prove a stronger version of the if part, following [@[D]].
Let $H$ be a subgroup of $G$. A subset $A$ is $H$-[*finite*]{} if $A$ is contained in finitely many right $H$-cosets, i.e. for some finite set $F$, $A \subseteq HF$. A subgroup $K$ is $H$-finite if and only if $H\cap K$ has finite index in $K$. Let $T$ be a $G$-tree and suppose there is an edge $e$ and vertex $v$.
We say that $e$ [*points at*]{} $v$ if there is a subtree of $T$ as below. We write $e \rightarrow v$.
(-.5, 2)– (1,2)–(2,1)–(3,2)–(4,1)–(5,2)–(6,1)–(7,2)–(8,1)–(9,2)–(10.5,2) ; (.5,2) node [$>$]{} ; (.5,2.1) node \[above\] [$e$]{} ; (10.5,2.1) node \[above\] [$v$]{} ; (10.5,2) node [$\bullet$]{} ;
Let $G[e, v] = \{g \in G | e \rightarrow gv\}$.
If $h \in G$, then $G[e, v]h = G[e,h^{-1}v]$, since if $e\rightarrow gv, e \rightarrow gh (h^{-1}v)$.
It follows from this that If $K = G_v$, then $G[e,v]K =G[e,v]$. Also if $H = G_e$, then $HG[e,v] = G[e,v]$.
If $v = \iota e $, then $G_e = H \leq K = G_v$ and if $A = G[e, \iota e]$, then $A= HAK$.
(-2, 2)– (2,2) ; (.0,2) node [$>$]{} ; (2,2) node [$\bullet$]{} ; (-2,2) node [$\bullet$]{} ;
(.5,2.1) node \[above\] [$e$]{} ; (-2,2.1) node \[above\] [$v$]{} ;
Consider the set $Ax, x\in G$. If $g \in A, gx \notin A$ , then $e \rightarrow gv, \bar e \rightarrow gx v$. This means that $e$ is on the directed path joining $gxv$ and $gv$. This happens if and only if $g^{-1}e$ is on the path joining $xv$ and $v$. There are only finitely many directed edges in the $G$-orbit of $e$ in this path. Hence $g^{-1} \in FH$, where $F$ is finite, and $H = G_e$, and $g \in HF^{-1}$. Thus $A -Ax^{-1} = HF^{-1}$, i.e. $A- Ax^{-1}$ is $H$-finite. It follows that both $Ax - A$ and $A-Ax$ are $H$-finite and so $A + Ax$ is $H$-finite for every $x \in G$, i.e. $A$ is an $H$-[*almost invariant set*]{}.
If the action on $T$ is non-trivial, then neither $A$ nor $A^*$ is $H$-finite. We say that $A$ is [*proper*]{}.
Peter Kropholler has conjectured that the following generalization of Theorem \[SC\] is true for finitely generated groups.
\[KC\] Let $G$ be a group and let $H$ be a subgroup. If there is a proper $H$-almost invariant subset $A$ such that $A = AH$, then $G$ has a non-trivial action on a tree in which $H$ fixes a vertex $v$ and every edge incident with $v$ has an $H$-finite stabilizer.
We have seen that the conjecture is true if $H$ has one element. The conjecture has been proved for $H$ and $G$ satisfying extra conditions by Kropholler [@[K90]], Dunwoody and Roller [@[DR]] , Niblo [@[N]] and Kar and Niblo [@[NK]].
If $G$ is the triangle group $G = \langle a, b | a^2 = b^3 = (ab)^7 =1\rangle$, then $G$ has an infinite cyclic subgroup $H$ for which there is a proper $H$-almost invariant set. Note that in this case $G$ has no non-trivial action on a tree, so the condition $A = AH$ is necessary in Conjecture \[KC\].
{width="10cm"}
A discussion of the Kropholler Conjecture is given in [@NS]. I first learned of this conjecture in a letter Peter wrote to me in January 1988, a page of which is shown here.
We give a proof of the conjecture when $G$ is finitely generated over $H$, i.e. it is generated by $H$ together with a finite subset.
I am very grateful to Peter Kropholler for enjoyable discussions and a very helpful email correpondence.
Infinite Networks
=================
Let $X$ be an arbitrary connected simple graph. It is not even assumed that $X$ is locally finite. Let ${\mathcal{B}}X$ be the set of all edge cuts in $X$. Thus if $A \subset VX$, then $A \in {\mathcal{B}}X$ if $\delta A$ is finite. Here $\delta A$ is the set of edges which have one vertex in $A$ and one in $A^*$.
A ray $R$ in $X$ is an infinite sequence $x_1, x_2, \dots $ of distinct vertices such that $x_i, x_{i+1}$ are adjacent for every $i$. If $A$ is an edge cut, and $R$ is a ray, then there exists an integer $N$ such that for $n > N$ either $x_n \in A$ or $x_n \in A^*$. We say that $A$ separates rays $R = (x_n), R' = (x_n')$ if for $n$ large enough either $x_n \in A, x_n' \in A^*$ or $x_n \in A^*, x_n' \in A$. We define $R\sim R'$ if they are not separated by any edge cut. It is easy to show that $\sim $ is an equivalence relation on the set $\Phi X$ of rays in $X$. The set $\Omega X = \Phi X/ \sim $ is the set of [edge]{} ends of $X$. An edge cut $A$ separates ends $\omega , \omega '$ if it separates rays representing $\omega , \omega '$. A cut $A$ separates an end $\omega $ and a vertex $v \in VX$ if for any ray representing $\omega $, $R$ is eventually in $A$ and $v \in A^*$ or vice versa.
We define a network $N$ to be a simple, connected graph $X$ and a map $c : EX \rightarrow \{1, 2, \dots \}$. If $X$ is a network in which each edge has capacity $1$, then ${\mathcal{B}}X$ is the set of edge cuts, and if $A \in {\mathcal{B}}X$, then $c(A) = |\delta A|$.
The following result is proved in [@[D2]].
\[maintheoremB\] Let $N(X)$ be a network in which $X$ is an arbitrary connected graph. For each $n >0$, there is a network $N(T_n)$ based on a tree $T_n$ and a map $\nu : VX\cup \Omega X \rightarrow VT\cup \Omega T$, such that $\nu (VX ) \subset VT$ and $\nu x = \nu y$ for any $x, y \in VX \cup \Omega X$ if and only if $x, y$ are not separated by a cut $A$ with $c(A) \leq n$.
The network $N(T_n)$ is uniquely determined and is invariant under the automorphism group of $N(X)$.
Theorem \[maintheoremB\] is proved by proving the following lemma. Let ${\mathcal{B}}_n X$ be the subring of ${\mathcal{B}}X$ generated by the cuts $A$ such that $c(A) < n$
\[lem2.2\] There is a uniquely defined nested set ${\mathcal{E}}_n$ of generators of ${\mathcal{B}}_n X$, with the following properties:-
- If $G$ is the automorphism group of $N(X)$, then ${\mathcal{E}}_n $ is invariant under $G$.
- For each $i <j$, ${\mathcal{E}}_i \subseteq {\mathcal{E}}_j$.
We will only really be using Theorem \[maintheoremB\] for networks in which every edge has capacity one.
\[maintheoremC\] Let $X$ be a connected graph. There is a uniquely determined sequence of structure trees $T_n$ and a map $\nu : VX\cup \Omega X \rightarrow VT\cup \Omega T$, such that $\nu (VX ) \subset VT$ and $\nu x = \nu y$ for any $x, y \in VX \cup \Omega X$ if and only if $x, y$ are not separated by a cut $A$ with $|\delta A| \leq n$. Each tree $T_n$ admits an action of the automorphism group of $X$.
In this case $ET_n = {\mathcal{E}}_n$.
In any tree $T$ if $p$ is a vertex and $Q$ is a set of unoriented edges, then there is a unique set of vertices $P$ such that $v \in P$ then the geodesic $[v,p]$ contains an odd number of edges from $Q$. We then have $\delta P = Q$. Note that ${\mathcal{B}}T = {\mathcal{B}}_1T$ and every element of ${\mathcal{B}}T$ is uniquely determined by the set $Q$ together with the information for a fixed $p\in VT$ whether $p \in A$ or $p \in A^*$. The vertex $p$ induces an orientation ${\mathcal{O}}_p$ on the set of pairs $\{ e, \bar e\} $ of oriented edges by requiring that $e \in {\mathcal{O}}$ if $e$ points at $p$. For $A \in {\mathcal{B}}T$, $A$ is uniquely determined by $\delta A $ together with the orientation ${\mathcal{O}}_p \cap \delta A$ of the edges of $\delta A$.
In $X$ it is the case that a cut $A$ is uniquely determined by $\delta A$ together with the information for a fixed $p \in VX$ whether $p \in A$ or $p \in A^*$.
Since ${\mathcal{B}}_nX$ is generated by ${\mathcal{E}}_n = ET_n$, the cut $A$ can be expressed in terms of a finite set of oriented edges of $T_n$. This set is not usually uniquely determined. Thus if $\nu $ is not surjective, and $v$ is not in the image of $\nu $, and the set of edges incident with $v$ is finite, then $VX$ is the union of these elements in ${\mathcal{B}}X$. The empty set is the intersection of the complements of these sets. Orienting the edges incident with $v$ towards $v$ gives the empty set and orienting them away from $v$ gives all of $VX$. However there is a canonical way of expressing an element of ${\mathcal{B}}_nX$ in terms of the generating set ${\mathcal{E}}_n$. To see this let $A \in {\mathcal{B}}_nX - {\mathcal{B}}_{n-1}X$. There are only finitely many $C \in {\mathcal{E}}_n$ with which $C$ is not nested. This number is $\mu (A, {\mathcal{E}}_n) = \mu (A)$. We use induction on $\mu (A)$. Our induction hypothesis is that there is a canonically defined way of expressing $A$ in terms of the ${\mathcal{E}}_n$. Any two ways of expressing $A$ in terms of ${\mathcal{E}}_n$ differ by an expression which gives the empty set in terms of ${\mathcal{E}}_n$. Such an expression will correspond to a finite set of vertices each of which has finite degree in $T_n$ and none of which is in the image of $\nu $. The canonical expression is obtained if there is a unique way of saying whether or not each such vertex is in the expression for $A$. Thus the canonical expression for $A$ is determined by a set of vertices of $VT$ which consists of the vertices of $\nu (A)$ together with a recipe for deciding for each vertex which is not in the image of $\nu $ whether it is in the expression for $A$.
Suppose $\mu (A) =0$, so that $A$ is nested with every $C \in {\mathcal{E}}_n$, and neither $A$ nor $A ^*$ is empty. If $A \in {\mathcal{E}}_n$, then this gives an obvious way of expressing $A$ in terms of the ${\mathcal{E}}_n$. If $A$ is not in ${\mathcal{E}}_n$, then it corresponds to a unique vertex $z \in VT_n$. Thus because $\mu (A) = 0$, $A$ induces an orientation of the edges of ${\mathcal{E}}_n$. To see this, let $C \in {\mathcal{E}}_n$, then just one of $C \subset A, C^* \subset A, C\subset A^*, C^* \subset A^*$ holds. From each pair $C, C^*$ we can choose $C$ if $C \subset A$ or $C \subset A^*$ and we choose $C^*$ if $C^*\subset A$ or $C^* \subset A^*$. Let ${\mathcal{O}}$ be this subset of ${\mathcal{E}}$. Then If $C \in {\mathcal{O}}$ and $D \in {\mathcal{E}}$ and $D \subset C$, then $D \in {\mathcal{O}}$. This means that the orientation ${\mathcal{O}}$ determines a vertex $z$ in $VT _n$. Intuitively the edges of ${\mathcal{O}}$ point at the vertex $z$. It can be seen that $A$ or $A^*$ will be the union of finitely many edges $E$ of ${\mathcal{E}}_n = ET_n$, all of which have $\tau E = z$. If $A$ is such a union, then we use this to express $A = C_1\cup C_2, \dots \cup C_k$. If $A$ is not such a union, but $A^* = C_1\cup C_2, \dots \cup C_k$, then we write $A = (C_1\cup C_2, \dots \cup C_k)^* = C_1^*\cap C_2^*\cap \dots \cap C_k^*$. Note that this gives a unique way of expressing cuts corresponding to a vertex $z$ of finite degree not in the image of $\nu$. The vertex $z$ is included in the expression for $A^*$ if and only if only finitely many cuts in ${\mathcal{E}}_n$ incident with $z$ and pointing at $z$ are subsets of $A$. Suppose then that the hypothesis is true for elements $B \in {\mathcal{B}}_nX$ for which $\mu (B) < \mu (A)$. Let $C \in {\mathcal{E}}_n$ be not nested with $A$. Then $ \mu (A\cap C) +\mu (A\cap C^*) \leq \mu (A)$. Thus each of $A\cap C$ and $A\cap C^*$ can be expressed in a unique way in terms of the ${\mathcal{E}}_n$. If at most one of these expressions involves $C$ then we take the expression for $A$ to be the union of the two expressions for $A\cap C$ and $A\cap C^*$. If both of the expressions involve $C$, then we take the expression for $A$ to be the union of the two expression with $C$ deleted. The expression obtained for $A$ is independent of the choice of $C$. In fact the decomposition will involve precisely those $C$ for which $C$ occurs in just one of the decompositions for $A\cap C$ and $A\cap C^*$. We therefore have a canonical decomposition for $A$. To further clarify this proof observe the following. The edges $C$ which are not nested with $A$ form the edge set of a finite subtree $F$ of $T_n$. If $EF \not= \emptyset$ we can choose $C$ so that it is a twig of $F$, i.e. so that one vertex $z$ of $F$ is only incident with a single edge $C$ of $F$. By relabelling $C$ as $C^*$ if necessary we can assume that $\mu (A\cap C) = 0$. The vertex determined by $A\cap C$ as above is $z$, and we have spelled out the recipe for if this vertex is to be included in the expression for $A$. The induction hypothesis gives us a canonical expression for $A\cap C^*$, which together with the expression for $A\cap C$ gives the expression for $A$.
Relative Structure Trees
========================
We prove Conjecture \[KC\] in the case when $G$ is finitely generated over $H$, i.e. $G$ is generated by $H\cup S$ where $S$ is finite.
First, we explain the strategy of the proof. Suppose that we have a non-trivial $G$-tree $T$ in which every edge orbit contains an edge which has an $H$-finite stabiliser, and suppose there is a vertex $\bar o$ fixed by $H$. Let $T_H$ be an $H$-subtree of $T$ containing $\bar o$ and every edge with $H$-finite stabiliser. The action of $H$ on $T_H$ is a trivial action, since it has a vertex fixed by $H$, and so the orbit space $H\backslash T_H$ is a tree, which might well be finite, but must have at least one edge. Our strategy is to show that if $G$ is finitely generated over $H$ and there is an $H$-almost invariant set $A$ satisfying $AH =A$, then we can find a $G$-tree $T$ with the required properties by first deciding what $H\backslash T_H$ must be and then lifting to get $T_H$ and then $T$.
We show that if $G$ is finitely generated over $H$, then there is a $G$-graph $X$ if which there is a vertex with stabiliser $H$ and in which a proper $H$-almost invariant set $A$ satisfying $AH = A$ corresponds to a proper set of vertices with $H$-finite coboundary. It then follows from the theory of [@[D2]], described in the previous section, that there is a sequence of structure trees for $H\backslash X$. We choose one of these to be $H\backslash T_H$, and show that we can lift this to obtain $T_H$ and then $T$ itself.
For example if $G = {H*}_KL$ then there is a $G$-tree $Y$ with one orbit of edges and a vertex $\bar o$ fixed by $H$, and every edge incident with $\bar o$ has $H$-finite stabiliser. Suppose that $K, L$ are such that these are the only edges with $H$-finite stabilisers. Then $H\backslash T_H$ has two vertices and one edge. When we lift to $T_H$ we obtain an $H$-tree of diameter two in which the middle vertex $\bar o$ has stabiliser $H$. The tree $T$ is covered by the translates of $T_H$.
On the other hand, if $G = L*_KH$ where $K$ is finite, and $T$ is as above, then every edge of $T$ is $H$-finite and so $T_H$ is $T$ regarded as an $H$-tree. The fact that our construction gives a canonical construction for $H\backslash T_H$ means that when we lift to $T_H$ and $T$ we will get the unique tree that admits the action of $G$.
We proceed with our proof.
The group $G$ is finitely generated over $H$ if and only if there is a connected $G$-graph $X$ with one orbit of vertices, and finitely many orbits of edges, and there is a vertex $o$ with stabiliser $H$.
Suppose $G$ is generated by $H\cup S$, where $S$ is finite. Let $X$ be the graph with $VX = \{ gH | g \in G\}$ and in which $EX$ is the set of unordered pairs $\{ \{ gH, gsH \}, g \in G, s \in S\} $. We then have that $X$ is vertex transitive, there is a vertex $o = H$ with stabilizer $H$ and $G\backslash X$ is finite. We have to show that $X$ is connected. Let $C$ be the component of $X$ containing $o$. Let $G'$ be the set of those $g \in G$ for which $gH \in C$. Clearly $G'H = G'$ and $G's = G'$ for every $s \in S$. Hence $G' = G$ and $C = X$. Thus $X$ is connected.
Conversely let $X$ be a connected $G$-graph and $VX = Go$ where $G_o = H$. Suppose $EX$ has finitely many $G$-orbits, $Ge_1, Ge_2,
\dots , Ge_r$ where $e_i$ has vertices $o$ and $g_io$. It is not hard to show that $G$ is generated by $H\cup \{ g_1, g_2, \dots , g_r\} $.
Let $A \subset G$ be a proper $H$-almost invariant set satisfying $AH = A$. Let $G$ be finitely generated over $H$, and let $X$ be a $G$-graph as in the last lemma. There is a subset of $VX$ corresponding to $A$, which is also denoted $A$. For any $x \in G$, $A +Ax$ is $H$-finite. In particular this is true if $s \in S$. This means that $\delta A$ is $H$-finite. Note that neither $A$ nor $A^* = VX - A$ is $H$-finite. Thus a proper $H$-almost invariant set corresponds to a proper subset of $VX$ such that $\delta A$ is $H$-finite.
From the previous section (Lemma \[lem2.2\]) we know that ${\mathcal{B}}(H\backslash X)$ has a uniquely determined nested set of generators ${\mathcal{E}}= {\mathcal{E}}(H\backslash X)$. For $E \in {\mathcal{E}}$, let $\bar E \subset VX$ be the set of all $v \in VX$ such that $Hv \in E$. Let $C$ be a component of $\bar E$.
For $h \in H$, $hC = C$ or $hC\cap C = \emptyset$. Also $HC = \bar E$, $h\delta C \cap \delta C =\delta C$ or $h\delta C \cap \delta C = \emptyset$ and $H\backslash \delta C = \delta E$.
Let $h \in H$. Then $hC$ is also a component of $\bar E$, since $HC \subseteq E$. Thus either $hC =C$ or $hC\cap C = \emptyset $. Let $K$ be the stabilizer of $C$ in $H$. if $v \in C$ then $hv \in C$ if and only if $h \in K$. Thus $K\backslash C$ injects into $H\backslash C = E$ and $K \backslash \delta C$ injects into $\delta E$. But $E$ is connected, and so the image $HC$ is $E$. It follows that there is a single $H$-orbit of components.
It follows from the lemma that it is also the case that $C^*$ is connected, since any component of $C^*$ must have coboundary that includes an edge from each orbit of $\delta C$. Let $\bar {\mathcal{E}}(H, X)$ be the set of all such $C$, and let $\bar {\mathcal{E}}_n(H, X)$ be the subset of $\bar {\mathcal{E}}(H, X)$ corresponding to those $C$ for which $\delta C$ lies in at most $n$ $H$-orbits.
the set $\bar {\mathcal{E}}(H,X)$ is a nested set. The set $\bar {\mathcal{E}}_n(H, X)$ is the edge set of an $H$-tree.
Let $C, D \in \bar {\mathcal{E}}_n(H,X)$. Then $HC, HD$ are in the nested set ${\mathcal{E}}$. Suppose $HC \subset HD$, then $C \subset D$ or $C\cap D = \emptyset $. It follows easily that $\bar {\mathcal{E}}(H, X)$ is nested. It was shown in [@[D1]] that a nested set ${\mathcal{E}}$ is the directed edge set of a tree if and only if it satisfies the finite interval condition, i.e. if $C, D \in {\mathcal{E}}$ and $C \subset D$, then there are only finitely many $E \in {\mathcal{E}}$ such that $C \subset E \subset D$. Thus we have to show that $\bar {\mathcal{E}}_n(H, X)$ satisfies the finite interval condition. If $C\subset D$ and $C\subseteq E \subseteq D$ where $C, E,D \in \bar {\mathcal{E}}_n(H,X)$, then $HC \subseteq HE \subseteq HD$. But ${\mathcal{E}}_n(H, X)$ does satisfy the finite interval condition and $HC = HE$ implies $C = E$. Now let $C\cap D = \emptyset $ and suppose that $o = H \in C^*\cap D^*$. There are only finitely many $E \in \bar {\mathcal{E}}_n$ such that $C \subset E$ and $o \in E^*$ or such that $D \subset E^*$ and $o \in E$. Each $E \in \bar {\mathcal{E}}_n$ such that $C \subset E \subset D^*$ has one of these two properties.
Let $\bar T = \bar T(H)$ be the tree constructed in the last Lemma. Let $T = H\backslash \bar T$. Note that in the above $\bar T(H)$ is the Bass-Serre $H$-tree associated with the quotient graph $T(H) = H\backslash \bar T(H)$ and the graph of groups obtained by associating appropriate labels to the edges and vertices of this quotient graph (which is a tree). Clearly the action of $H$ on $T(H)$ is a trivial action in that $H$ fixes the vertex $\bar o = \nu o$. The stabilisers of edges or vertices on a path or ray beginning at $\bar o$ will form a non-increasing sequence of subgroups of $H$.
We now adapt the argument of the previous section to show that if $A \subset VX$ is such that $\delta A$ lies in at most $n$ $H$-orbits, then there is a canonical way of expressing $A$ in terms of the set $\bar {\mathcal{E}}(H, X)$. In this case we have to allow unions of infinitely many elements of the generating set. Our induction hypothesis is that if $\delta A$ lies in at most $n$ $H$-orbits, then $A$ is canonically expressed in terms of $\bar {\mathcal{E}}_n (H, X)$. First note that there are only finitely many $H$-orbits of elements of $\bar {\mathcal{E}}_n = \bar {\mathcal{E}}_n(H,X)$ with which $A$ is not nested. This is because if $C \in \bar {\mathcal{E}}_n$ is not nested with $A$ and $F$ is a finite connected subgraph of $H\backslash X$ containing all the edges of $H\delta A$, then $H\delta C$ must contain an edge of $F$ and there are only finitely many elements of ${\mathcal{E}}_n$ with this property. We now let $\mu (A)$ be the number of $H$-orbits of elements of $\bar {\mathcal{E}}_n$ with which $A$ is not nested. If $\mu (A) = 0$, then $A$ is nested with every $C \in \bar {\mathcal{E}}_n$. This then means that if neither $A$ nor $A^*$ is empty and it is not already in $\bar{\mathcal{E}}_n$, then $A$ determines a vertex $z$ of $\bar T_n$ and either $A$ or $A^*$ is the union (possibly infinite) of edges of $T_n$ that lie in finitely many $H$-orbits. If $A$ is such a union, then we use this union for our canonical expression for $A$. If $A$ is not such a union, then $A^*$ is; we have $A^* = \bigcup \{ C_{\lambda} | \lambda \in \Lambda \}$, where each $C_{\lambda }$ has $\tau C_{\lambda } = z$ and the edges lie in finitely many $H$-orbits. We write $A = ( \bigcup \{ C_{\lambda} | \lambda \in \Lambda \})^* = \bigcap \{ C_{\lambda}^* | \lambda \in \Lambda \}$. Note that this gives a canonical way of expressing cuts corresponding to a vertex that is not in the image of $\nu $ and whose incident edges lie in finitely many $H$-orbits. Suppose then that the hypothesis is true for elements $B$ for which $\mu (B) < \mu (A)$. Let $C \in \bar {\mathcal{E}}_n$ be not nested with $A$. Then $ \mu (A\cap HC) +\mu (A\cap HC^*) \leq \mu (A)$. Thus each of $A\cap HC$ and $A\cap HC^*$ can be expressed in a unique way in terms of the ${\mathcal{E}}_n$. We take the expression for $A$ to be the union of the two expressions for $A\cap HC$ and $A\cap HC^*$ except that we include $hC$ for $h \in H$, only if just one of the two expressions involve $hC$.
If $g \in G$, then $g\bar T(H)$ is a $(gHg^{-1})$-tree. It is the tree $\bar T(gHg^{-1})$ obtained from the $G$-graph $X$ by using the vertex $go$ instead of $o$. We now show that there is a $G$-tree $T$ which contains all of the trees $g\bar T(H)$.
We know that the action of the group $G$ on $X$ is vertex transitive and that $X$ has a vertex $o$ fixed by $H$. Also $G$ is generated by $H\cup S$ where $S$ is finite.
Clearly there is an isomorphism $\alpha _g : \bar T(H) \rightarrow \bar T(gHg^{-1})$ in which $D \mapsto gD$.
Suppose now that $\nu o \not= \nu (go)$. Let $A, B $ be $H$-almost invariant sets satisfying $AH = A, BH = B$ and let $g \in G$. We regard $A,B$ as subsets of $VX$, so that $\delta A$ and $\delta B$ are $H$-finite.
Suppose that $o \in gB^*$ and $go\in A^*$. The following Lemma is due to Kropholler [@[K90]], [@[K91]]. We put $K = gHg^{-1}$.
\[Krop\] In this situation $\delta (A\cap gB)$is $(H\cap K)$-finite.
Let $x \in G$. We show that the symmetric difference $(A\cap gB)x + (A\cap gB)$ is $(H\cap K)$-finite. Since $A, B$ are $H$-almost invariant, there are finite sets $E, F$ such that $A+ Ax \subseteq HE$ and $B+Bx\subseteq HF$. We then have
$$(A\cap gB)x + (A\cap gB) = Ax\cap (gBx +gB) + (Ax + A)\cap gB
= Ax\cap gHF + g(g^{-1}HE\cap B).$$
Now $Ax\cap gHF$ is $K$-finite, but it is also $H$-finite because $gH$ is contained in $A^*$, since $go \in A^*$. A set which is both $H$-finite and $K$-finite is $H\cap K$-finite. Thus $Ax\cap gHF$ is $(H\cap K)$-finite. Similarly using the fact that $g^{-1}o \in B^*$, it follows that $g^{-1}HE\cap B$ is $H\cap (g^{-1}Hg)$-finite, and so $g(g^{-1}HE\cap B)$ is $(H\cap K)$-finite. Thus $A\cap gB$ is $(H\cap K)$-almost invariant. But this means that $\delta (A\cap gB)$ is $(H\cap K)$-finite.
What this Lemma says is that if $A, gB$ are not nested then there is a special corner - sometimes called the [*Kropholler corner*]{} - which is $(H\cap K)$-almost invariant.
Notice that in the above situation all of $\delta A, \delta (A\cap gB^*)$ and $\delta (A\cap gB)$ are $H$-finite. If we take the canonical decomposition for $A$, then it can be obtained from the canonical decompositions for $A\cap gB$ and $A\cap gB^*$ by taking their union and deleting any edge that lies in both. Also $\delta (gB)$ is $K$-finite and the decomposition for $gB$ can be obtained from those for $gB\cap A$ and $gB\cap A^*$. But the edges in the decomposition for $A\cap gB$ which is $(H\cap K)$-almost invariant are the same in both decompositions.
We will now show that it follows from Lemma \[Krop\] that the set $G\bar {\mathcal{E}}_n$ is a nested $G$-set which satisfies the final interval condition, and so it is the edge set of a $G$-tree. We have seen that $\bar {\mathcal{E}}_n$ is a nested $H$-set where ${\mathcal{E}}_n = H\backslash \bar {\mathcal{E}}_n$ is the uniquely determined nested subset of ${\mathcal{B}}_n(H\backslash X)$ that generates ${\mathcal{B}}_n (H\backslash X)$ as an abelian group. It is the edge set of a tree $T _n(H\backslash X)$.
(0,4)–(8,4); (4,0)–(4,8);
(3.6, 7.8) node [$A$]{}; (4.6,7.8) node [$A^*$]{}; (.4, 4.4) node [$gB$]{} ; (.4,3.6) node [$gB^*$]{} ; (6,6) node [$A^*\cap gB$]{}; (2,6) node [$A\cap gB$]{}; (2,2) node [$A\cap gB^*$]{}; (6,2) node [$A^*\cap gB^*$]{};
(10,5)–(13,5)– (13, 8); (10,3)–(13,3)– (13, 0); (18,5)–(15,5)– (15, 8); (18,3)–(15,3)– (15, 0);
(13,3)–(15,5) ; (15,3)–(13,5) ;
(17,6) node [$A^*\cap gB$]{}; (11,6) node [$A\cap gB$]{}; (11,2) node [$A\cap gB^*$]{}; (17,2) node [$A^*\cap gB^*$]{};
(14,7) node [$a$]{}; (14,1) node [$b$]{}; (11, 4) node [$c$]{}; (17, 4) node [$d$]{}; (13.5, 3.5) node [$e$]{}; (14.5, 3.5) node [$f$]{};
.5cm
If $A, B \in \bar {\mathcal{E}}_n$ and $A, gB$ are not nested for some $g \in G$, then by Lemma \[Krop\] there is a corner -the Kropholler corner -, which we take to be $A\cap gB$, for which $\delta (A\cap gB)$ is $(H\cap K)$-finite. We than have canonical decompositions for $A\cap gB$ and $A\cap gB^*$ as above. This is illustrated in Fig \[Cuts\]. The labels $a, b, c, d, e, f$ are for sets of edges joining the indicated corners. In this case the letters do not represent edges of $X$ but elements of $\bar {\mathcal{E}}_n$. Although each $E \in \bar {\mathcal{E}}_n$ comes with a natural direction, in the diagram we only count the unoriented edges, i.e. we count the number of edge pairs $(E, E^*)$. In the diagram, $A\cap gB$ is always taken to be the Kropholler corner. Thus we have that any pair contributing to $a, f$ or $e$ must be $(H\cap K)$-finite. Any pair contributing to $e$ or $b$ must be $H$-finite and any pair contributing to $e$ or $d$ must be $K$-finite.
We have that $a + e + f +b = 1$ and $c + e + f + d =1$. Suppose that the Kropholler corner $A\cap B$ is not empty. It is the case that each of $o$ and $go$ lies in one of the other three corners. We know that $o \in gB^*, go \in A^*$. If $o \in A\cap gB^*$ and $go \in A^*\cap gB$, then $a = c = 1$ and $e =f = b = d =0$ and $A^*\cap gB^* = \emptyset$. If $o \in A^*\cap gB$ and $go \in A^*\cap gB^*$, then $a = d =1$ and $A\cap gB^* = \emptyset$, while if both $o$ and $go$ are in $A^*\cap gB^*$, then either $a =d =1$ and $A\cap gB^* = \emptyset$ or $a =c =1$ and $A^*\cap gB = \emptyset$ or $f =1$ and both $A\cap gB^*$ and $A^*\cap gB$ are empty, so that $A = gB$. In all cases $A, gB$ are nested.
(-.8, 1) – (-.5,2) – (-.5,3)– (-1, 4) – (-1, 5) – (-1.4, 4) – (-1. 6, 3) ; (-2, 2) –(-1.6,3) – (-1.4, 2) ; (-.5,0) – ( -.3, 1) –(-.5, 2)– (1,2)–(2,1)–(3,2)–(4,1)–(5,2)–(6,1)–(7,2)–(8,1)–(9,2)–(10.5,2) ; (-1,5) node [$\bullet$]{} ; (10.5,5) node [$\bullet$]{} ; (10.5,2) –(10.5, 3) –(11,4)–(10.5, 5) – (11.5, 4) –( 11.5, 3) ; (3,-1) –( 2.5, 0) – (3, 2) – (3.5, 0) ; (4.5,-1) –( 4.5, 0) – (5, 2) – (5.5, 0) ;
(-1,5) node \[above\] [$o$]{} ; (10.5,5) node \[above\] [$go$]{} ;
We need also to show that $G\bar {\mathcal{E}}_n$ satisfies the finite interval condition. Let $g \in G$ and let $K = gHg^{-1}$. Consider the union $\bar {\mathcal{E}}\cup g\bar {\mathcal{E}}$. This will be a nested set. In fact it will be the edge set of a tree that is the union of the trees $T(H)$ and $T(K)$. In the diagram the red edges are the edges that are just in $T(H)$. The blue edges are the ones that are in $T(K)$. The brown edges are in both $T(H)$ and $T(K)$. An edge is in the geodesic joining $o$ and $go$ if and only if it has stabiliser containing $H\cap K$, it will also lie in both $T(H)$ and $T(K)$ (i.e. it is coloured brown) if and only if it its stabiliser contains $H\cap K$ as a subgroup of finite index. It may be the case that $T(H)$ and $T(K)$ have no edges in common, i.e. there are no brown edges. An edge lies in both trees if and only if it has a stabiliser that is $(H\cap K)$-finite. It there are such edges then they will be the edge set of a subtree of both trees. They will correspond to the edge set $\bar {\mathcal{E}}(H\cap K)$.
It follows that $T(H)$ is always a subtree of a tree constructed from a subset of $G\bar {\mathcal{E}}_n$ that contains $\bar {\mathcal{E}}_n$. If $T(H)$ and $T(K)$ do have an edge in common, then $T(H)\cup T(K)$ will be a subtree of the tree we are constructing. If $e \in EX$ has vertices $go$ and $ko$ and there is some $C \in G\bar {\mathcal{E}}_n$ that has $e \in \delta C$, then $C \in gET(g^{-1}Hg)\cap kET(k^{-1}Hk)$. If there is no such $C$, i.e. there is no cut $C \in G\bar {\mathcal{E}}_n$ that separates $o$ and $k^{-1}go$ then $T(H) = k^{-1}gT(H)$. As there is a finite path connecting any two vertices $u,v$ in $X$, it can be seen that there are only finitely many edges in $G\bar {\mathcal{E}}_n$ separating $u$ and $v$ since any such edge must separate the vertices of one of the edges in the path. Thus $G{\mathcal{E}}_n$ is the edge set of a tree.
We say that a $G$-tree $T$ is reduced if for every $e \in ET$, with vertices $\iota e$ and $\tau e$ we have that either $\iota e$ and $\tau e$ are in the same orbit, or $G_e$ is a proper subgroup of both $G _{\iota e} $ and $G _{\tau e} $.
\[tree\] Let $G$ be a group that is finitely generated over a subgroup $H$. The following are equivalent:-
- There is a proper $H$-almost invariant set $A = HAK$ with left stabiliser $H$ and right stabiliser $K$, such that $A$ and $gA$ are nested for every $g \in G$.
- There is a reduced $G$-tree $T$ with vertex $v$ and incident edge $e$ such that $G_v = K$ and $G_e = H$.
It is shown that (ii) implies (i) in the Introduction.
Suppose than that we have (i). We will show that there is a $G$-tree - in which $G$ acts on the right - which contains the set $V = \{ Ax | x \in G\}$ as a subset. Let $x \in G$, then $A + Ax$ is a union of finitely many coset $\{ Hg_1, Hg_2, \dots , Hg _k$. Then $\{ g_1 ^{-1}A, g_2^{-1}A, \dots , g_k^{-1}A \}$ is the edge set of a finite tree $F$. We know that the set $\{ gA | g \in G\}$ is the edge set of a $G$-tree $T$ provided we can show that it satisfies the finite interval condition. But this must be the case as the edges separating vertices $A$ and $Ax$ will be the edges of $F$.
Let $G$ be a group and let $H$ be a subgroup, and suppose $G$ is finitely generated over $H$. There is a proper $H$-almost invariant subset $A$ such that $A = AH$, if and only if there is a non-trivial reduced $G$-tree $T$ in which $H$ fixes a vertex and every edge orbit contains an edge with an $H$-finite edge stabilizer.
The only if part of the theorem is proved in Theorem \[tree\]. In fact it is shown there that if $G$ has an action on a tree with the specified properties, then there is a proper $H$ almost invariant set $A$ for which $HAH= A$.
Suppose then that $G$ has an $H$-almost invariant set $A$ such that $AH = A$. Since $G$ is finitely generated over $H$, we can construct the $G$-graph $X$ as above, in which $A$ can be regarded as a set of vertices for which $\delta A$ lies in finitely many $H$-orbits. Let this number of orbits be $n$. Then we have seen that there is a $G$-tree $\bar T_n$ for which $H$ fixes a vertex $\bar o$ and every edge is in the same $G$-orbit as an edge in $\bar T(H)$. The edges in this tree are $H$-finite. The set $A$ has an expression in terms if the edges of $\bar T(H)$. Finally we need to show that the action on $\bar T_n$ is non-trivial. If $G$ fixes $\bar o$, then $\nu (A)$ consists of the single vertex $o$ and so $A$ is not proper. In fact the fact that $A$ is proper ensures that no vertex of $\bar T_n$ is fixed by $G$.
It can be seen from the above that $ \bar T(H) \cap \bar T(g^{-1}Hg) = \bar T(H\cap gHg^{-1})$ so that if $e \in ET(H)$, and $g \in G_e$, then $e\in \bar T(gHg^{-1})$ and so $G_e$ is $H$-finite.
The Kropholler Conjecture follows immediately from the last Theorem.
$H$-almost stability
====================
Let $G$ be a group with subgroup $H$, and let $T$ be a $G$-tree.
Let $ \bar A \subset VT$ be such that $\delta \bar A \subset ET$ consists of finitely many $H$-orbits of edges $e$ such that $G_e$ is $H$-finite. Also let $H$ fix a vertex of $T$. Note that $\delta \bar A$ consists of whole $H$-orbits, so that $e \in \delta \bar A$ implies $he \in \delta \bar A$ for every $h \in H$. The fact that $G_e$ is $H$-finite for $e \in \delta \bar A$ follows from the fact that $\delta \bar A$ is $H$-finite. If $H_e$ is the stabiliser of $e \in \delta \bar A$, then $[G_e : H_e]$ is finite.
Let $v \in VT$, and let $A = A(v) = \{ g \in G | gv \in \bar A\} $. Note that $A(xv) = A(v)x^{-1}$, so that the left action on $T$ becomes a right action on the sets $A(v)$. if $x \in G$ and $[v , xv]$ is the geodesic from $v$ to $xv$, then $g \in A + Ax$ if and only the geodesic $[gv, gxv]$ contains an odd number of edges in $\delta \bar A$. If $[v, xv]$ consists of the edges $e_1, e_2, \dots , e_r$, then $ge_i \in \delta \bar A $ if and only if $Hge_i \in \delta \bar A$. It follows that $H(A + Ax) = A+Ax$. It is also clear that for each $e_i$ there are only finitely many cosets $Hg$ such that $Hge_i \in \delta \bar A$. Thus $A$ is $H$-almost invariant. We also have $A (v)H = A(v)$ if $H$ fixes $v$.
For each $e \in ET$, let $d(e)$ be the number of cosets $Hg$ such that $Hge \in \delta \bar A$. We see that $d(e) = d(xe)$ for every $x \in G$ and so we have a metric on $VT$, that is invariant under the action of $G$. We will show that if $G$ has an $H$-almost invariant set such that $HAH = A$ then there is a $G$-tree with a metric corresponding to this set.
From now on we are interested in the action of $G$ on the set of $H$-almost invariant sets. But note that we are interested in the action by right multiplication. The Almost Stability Theorem [@[DD]], also used the action by right multiplication. Let $A \subset G$ be $H$-almost invariant and let $HA = A$ For the moment we do not assume that $AH = A$.
Let $M = \{ B | B =_a A\} $ so that for $B, C \in M , B +C = HF$ where $F$ is finite.
Note that for $H = \{ 1 \}$ it follows from the Almost Stability Theorem that $M$ is the vertex set of a $G$-tree.
We define a metric on $M$. For $B, C \in M$ define $d(B, C)$ to be the number of $H$-cosets in $B +C$.
This is a metric on $M$, since $(B+C) + (C+D) = (B+D)$, and so an element which is in $B+D$ is in just one of $B+C$ or $C+D$. Thus $d(B,D) \leq d(B,C) + d(C, D)$.
Also $G$ acts on $M$ by right multiplication and this action is by isometries, since $(B+C)z = Bz + Cz$. Let $ \Gamma $ be the graph with $V \Gamma = M$ and two vertices are joined by an edge if they are distance one apart. Every edge in $ \Gamma $ corresponds to a particular $H$-coset. There are exactly $n!$ geodesics joining $B$ and $C$ if $d(B, C) = n$, since a geodesic will correspond to a permutation of the cosets in $B + C$. The vertices of $ \Gamma$ on such a geodesic form the vertices of an $n$-cube.
The edges corresponding to a particular coset $Hb$ disconnect $\Gamma $, since removing this set of edges gives two sets of vertices, $B$ and $B^*$, where $B$ is the set of those $C \in M$ such that $Hb \subset C$.
It has been pointed out to me by Graham Niblo that $\Gamma $ is the $1$-skeleton of the Sageev cubing introduced in [@[Sa]]. For completeness we describe this alternative characterization of $\Gamma $.
Let $G$ be a group with subgroup $H$ and let $A = HA$ be an $H$-almost invariant subset. Let
$$\Sigma = \{ gA | g \in G\} \cup \{gA^* | g \in G \}.$$
We define a graph $\Gamma '$. A vertex $V$ of $\Gamma '$ is a subset of $\Sigma $ satisfying the following conditions:-
- For all $B \in \Gamma '$, exactly one of $B, B^*$ is in $V$.
- If $B \in V, C \in \Sigma $ and $B \subseteq C$, then $C \in V$.
Two vertices are joined by an edge in $\Gamma '$ if they differ by one element of $\Sigma $. For $g \in G$, there is a vertex $V_g$ consisting of all the elements of $\Sigma $ that contain $g$. Then Sageev shows that there is a component $\Gamma ^1$ of $\Gamma '$ that contains all the $V_g$. In fact this graph $\Gamma ^1$ is isomorphic to our $\Gamma $.
By (1) for each $V \in \Sigma $ either $A \in V$ or $A^* \in V$ but not both. Let $\Sigma _A$ be the subset of $\Sigma $ consisting of those $V \in \Sigma $ for which $A \subset V$. The edges joining $\Sigma _A$ and $\Sigma _A^*$ in $\Gamma ^1$ form a hyperplane. Each edge in the hyperplane joins a pair of vertices that differ only on the set $A$. For each $xA$ there is a hyperplane joining vertices that differ only on the set $xA$. Clearly $G$ acts transitively on the set of hyperplanes.
With $V$ as above, consider the subset $A_V$ of $G$ $$A_V =\{ x \in G | x^{-1}A \in V \} .$$ Then $HA_V = A_V$ and $A_{V_1} =A$. Also $A_V +A$ is the union of those cosets $Hx$ for which $V$ and $V_1$ differ on $x^{-1}A$, which is finite. Thus $A_V \in V\Gamma $.
Thus there is a map $V\Gamma ^1 \rightarrow V\Gamma $ in which $V \mapsto A_V$. This map is a $G$-map and an isomorphism of graphs.
If the set $A$ is such that $A$ and $gA$ are nested for every $g \in G$, then there is a $G$-subgraph of $\Gamma _1$ which is a $G$-tree. This will also be true of $\Gamma $.
In $\Gamma $ a hyperplane consists of edges joining those vertices that differ only by a particular coset $Hx$. Every edge of $\Gamma $ belongs to just one hyperplane. The group $G$ acts transitively on hyperplanes. The hyperplane corresponding to $Hx$ has stabilizer $x^{-1}Hx$.
Suppose now that $A$ is $H$-almost invariant with $HAK = A$. Here $H$ is the left stabiliser and $K$ is the right stabiliser of $A$, and we assume that $H \leq K$, so that in particular $HAH = A$. Note that it follows from the fact that $A$ is $H$-almost invariant that it is also $K$ almost invariant. Suppose that $G$ is finitely generated over $K$. We have seen, in the previous section, that there is a $G$-tree $T$ in which $A$ uniquely determines a set $\bar A$ of vertices with $H$-finite coboundary $\delta \bar A$. Here $T = T_n$ for $n$ sufficiently large that in the graph $X$ -as defined in the previous section - the set $\delta \bar A $ is contained in at most $n$ $H$-orbits of edges. Note that if $e$ is an edge of $\bar T (H) = \bar {\mathcal{E}}(H, X)$, then $\delta e$ is $H_e$-finite, and will consist of finitely many $H_e$-orbits. It is then the case that $[G_e: H_e]$ is finite, since $\delta e$ will consist of finitely many $G_e$-orbits each of which is a union of $[G_e:H_e]$ $H_e$-orbits of edges.
We also know that $K$ fixes a vertex $\bar o$ of $T$, and that $H\delta \bar A = \delta \bar A$. Thus $\delta \bar A$ consists of finitely many $H$-orbits of edges. We can contract any edge whose $G$-orbit does not intersect $\delta \bar A$. We will then have a tree that has the properties indicated at the beginning of this section. Thus $ \bar A \subset VT$ is such that $\delta \bar A \subset ET$ consists of finitely many $H$-orbits of edges $e$ such that $G_e$ is $H$-finite. We see that the metric $d$ on $M$ is the same as the metric defined on $VT$. Explicitly we have proved the following theorem in the case when $G$ is finitely generated over $K$.
Let $G$ be a group with subgroup $H$ and let $A = HAK$ where $H \leq K$ and $A$ is $H$-almost invariant. Let $M$ be the $G$-metric space defined above. Then there is a $G$-tree $T$ such that $VT$ is a $G$-subset of $M$ and the metric on $M$ restricts to a geodesic metric on $VT$. If $e \in ET$ then some edge in the $G$-orbit of $e$ has $H$-finite stabiliser.
This is illustrated in Fig 1 and Fig 2.
It remains to sow that that the theorem for arbitrary $G$ follows from the case when $G$ is finitely generated over $K$. Thus if $F$ is a finite subset of $G$, then there is a finite convex subgraph $C$ of $\Gamma $ containing $AF$. We can use the graph $X$ of the previous section for the subgroup $L$ of $G$ generated by $H \cup F$ to construct an $L$-tree which has a subtree $S(F)$ with vertex set contained in $VC$. These subtrees have the nice property that if $F_1 \subset F_2$ then $S(F_1)$ is a subtree of $S(F_2)$. They therefore fit together nicely to give the required $G$-tree. We give a more detailed argument for why this is the case. We follow the approach of [@[C]].
Let $M'$ be the subspace of $M$ consisting of the single $G$-orbit $AG$. Define an inner product on $M'$ by $(B.C)_A = {1\over 2} (d(A, B) +d(A,C) - d(B, C)).$
This turns $M'$ into a $0$-hyperbolic space, i.e. it satisfies the inequality $$(B.C)_A \geq min \{ (B.D)_A, (C.D)_A\}$$ for every $B, C, D \in M'$. This is because we know that if $L \leq G$ is finitely generated over $H$, then there is an $L$-tree which is a subspace of $M$. But $A, B, C,D$ are vertices of such a subtree which is $0$-hyperbolic. It now follows from [@[C]], Chapter 2, Theorem 4.4 that there is a unique ${\mathbb{Z}}$-tree $VT$ (up to isometry) containing $M'$. The subset of $VT$ consisting of vertices of degree larger than $2$ will be the vertices of a $G$-tree and can be regarded as a $G$-subset of $M$ containing $M'$.
(6,2) coordinate (p1); (2,3) coordinate (p2); (5,4) coordinate (p3); (4, 3) coordinate (q1); (3, 5) coordinate (q3); (5.5, 5) coordinate (q4); (1.5, 2) coordinate (q5); (2.5, 1) coordinate (q6); (p1) circle (3pt) ; (p2) circle (3pt); (p3) circle (3pt); (q1) circle (3pt); (q1) circle (2pt);
(3.5, 6) coordinate (p4); (p4) circle (3pt); (3, 2) coordinate (p5); (p5) circle (3pt); (7.5, 2) coordinate (q2); (q2) circle (3pt); (q3) circle (3pt); (q4) circle (3pt); (q5) circle (3pt); (q6) circle (3pt);
(q3)–(p4) ; (q4)–(p3) ; (p2)–(q5)–(p5); (4,3) –(5.1,3) –(6, 2.7) – (6,2) ; (4,3) –(4.9,2.7) –(6, 2.7) ; (4,3) –(4,2.3) –(4.9, 2) – (6,2) ; (4,2.3) –(5.1, 2.3) – (6,2) ; (5.1, 2.3) – (5.1,3) ; (4.9, 2.7) – (4.9,2) ; (4, 3)–(4.5, 3.2) –(5, 4) – (4.5, 3.8) – cycle ; (4, 3)–(3, 3.3) –(2,3) – (3,2.7) – cycle ;
(3.5,6) –(4.6,6) –(5.5, 5.7) – (5.5,5) ; (3.5,6) –(4.4,5.7) –(5.5,5.7) ; (3.5,6) –(3.5,5.3) –(4.4, 5) – (5.5,5) ; (3.5,5.3) –(4.6,5.3) – (5.5,5) ; (4.6, 5.3) – (4.6,6) ; (4.4, 5.7) – (4.4,5) ; (6,2)–(6.75, 2.5) –(7.5,2) – (6.75, 1.5) – cycle ; (2.5,1)–(2.5, 1.5) –(3,2) – (3,1.5) – cycle ;
(p1) circle (2pt); (p2) circle (2pt); (p3) circle (2pt); (p5) circle (2pt); (p4) circle (2pt); (q1) circle (2pt); (q2) circle (2pt); (q3) circle (2pt); (q4) circle (2pt); (q5) circle (2pt); (q6) circle (2pt);
(6,2) coordinate (p1); (2,3) coordinate (p2); (5,4) coordinate (p3); (4, 3) coordinate (q1); (3, 5) coordinate (q3); (5.5, 5) coordinate (q4); (1.5, 2) coordinate (q5); (2.5, 1) coordinate (q6); (p1) circle (3pt) ; (p2) circle (3pt); (p3) circle (3pt); (q1) circle (3pt); (q1) circle (2pt);
(3.5, 6) coordinate (p4); (p4) circle (3pt); (3, 2) coordinate (p5); (p5) circle (3pt); (p1) – (q1); (p2) – (q1); (p3) – (q1); (7.5, 2) coordinate (q2); (q2) circle (3pt); (q3) circle (3pt); (q4) circle (3pt); (q5) circle (3pt); (q6) circle (3pt);
(q3)–(p4)–(q4)–(p3); (p2)–(q5)–(p5)–(q6); (p1) –(q2); (4,3) –(5.1,3) –(6, 2.7) – (6,2) ; (4,3) –(4.9,2.7) –(6, 2.7) ; (4,3) –(4,2.3) –(4.9, 2) – (6,2) ; (4,2.3) –(5.1, 2.3) – (6,2) ; (5.1, 2.3) – (5.1,3) ; (4.9, 2.7) – (4.9,2) ; (4, 3)–(4.5, 3.2) –(5, 4) – (4.5, 3.8) – cycle ; (4, 3)–(3, 3.3) –(2,3) – (3,2.7) – cycle ;
(3.5,6) –(4.6,6) –(5.5, 5.7) – (5.5,5) ; (3.5,6) –(4.4,5.7) –(5.5,5.7) ; (3.5,6) –(3.5,5.3) –(4.4, 5) – (5.5,5) ; (3.5,5.3) –(4.6,5.3) – (5.5,5) ; (4.6, 5.3) – (4.6,6) ; (4.4, 5.7) – (4.4,5) ; (6,2)–(6.75, 2.5) –(7.5,2) – (6.75, 1.5) – cycle ; (2.5,1)–(2.5, 1.5) –(3,2) – (3,1.5) – cycle ;
(p1) circle (2pt); (p2) circle (2pt); (p3) circle (2pt); (p5) circle (2pt); (p4) circle (2pt); (q1) circle (2pt); (q2) circle (2pt); (q3) circle (2pt); (q4) circle (2pt); (q5) circle (2pt); (q6) circle (2pt);
(5, 0) node [Fig 1]{};
(6,2) coordinate (p1); (2,3) coordinate (p2); (5,4) coordinate (p3); (4, 3) coordinate (q1); (3, 5) coordinate (q3); (5.5, 5) coordinate (q4); (1.5, 2) coordinate (q5); (2.5, 1) coordinate (q6); (p1) circle (3pt) ; (p2) circle (3pt); (p3) circle (3pt); (q1) circle (3pt); (q1) circle (2pt);
(3, 5.5) node [$^1$]{}; (4.6, 5.6) node [$^3$]{}; (5.1, 4.5) node [$^1$]{}; (4.4, 3.6) node [$^2$]{}; (5, 2.6) node [$^3$]{}; (3, 3.1) node [$^2$]{}; (2.25,2.1) node [$^1$]{}; (6.75, 2.1) node [$^2$]{}; (1.5, 2.5) node [$^1$]{}; (2.6, 1.5) node [$^2$]{};
(3.5, 6) coordinate (p4); (p4) circle (3pt); (3, 2) coordinate (p5); (p5) circle (3pt); (p1) – (q1); (p2) – (q1); (p3) – (q1); (7.5, 2) coordinate (q2); (q2) circle (3pt); (q3) circle (3pt); (q4) circle (3pt); (q5) circle (3pt); (q6) circle (3pt);
(q3)–(p4)–(q4)–(p3); (p2)–(q5)–(p5)–(q6); (p1) –(q2);
(p1) circle (2pt); (p2) circle (2pt); (p3) circle (2pt); (p5) circle (2pt); (p4) circle (2pt); (q1) circle (2pt); (q2) circle (2pt); (q3) circle (2pt); (q4) circle (2pt); (q5) circle (2pt); (q6) circle (2pt); (5, 0) node [Fig 2]{};
[99]{} Ian Chiswell. [*Introduction to $\Lambda $-trees*]{}. World Scientific, 2001.
Warren Dicks, [*Group, trees and projective modules*]{}, Springer Lecture Notes [**790**]{} 1980
Warren Dicks and M.J.Dunwoody, [*Groups acting on graphs*]{}, Cambridge University Press, 1989. Errata [http://mat.uab.es/\~dicks/]{}
M.J.Dunwoody, [*Accessibility and groups of cohomological dimenstion one*]{}, Proc. London Math. Soc. [**38**]{} (1979) 193-215.
M.J. Dunwoody, [*Structure trees and networks*]{}, arXiv:1311.3929.
M.J.Dunwoody and M.Roller, [*Splitting groups over polycyclic-by-finite subgroups*]{}, Bull. London Math.Soc. [**23**]{} 29-36 (1989).
A.Kar and G.A.Niblo, [*Relative ends $\ell ^2$-invariants and property T*]{}, arXiv:1003.2370.
P.H.Kropholler, [*An analogue of the torus decomposition theorem for certain Poincar' e groups*]{}, Proc. London Math. Soc. (3) [**60**]{} 503-529 (1990).
P.H.Kropholler, [*A group theoretic proof of the torus theorem*]{}, London Math. Soc. Lecture Note Series [**181**]{} 138-158 (1991). G.A. Niblo, [*A geometric proof of StallingsÕ theorem on groups with more than one end,*]{} Geometriae Dedicata [**105**]{}, 61-76 (2004).
G. Niblo, M. Sageev, [*The [K]{}ropholler conjecture*]{}, In *Guido’s Book of Conjectures*, Monographies de L’Enseignement Mathématique, 40. L’Enseignement MathŽmatique, Geneva, 2008.
M.Sageev, [*Ends of group pairs and non-positively curved cube complexes,*]{} Proc. London Math. Soc. (3) [**71**]{}, 585-617 (1995).
J.R. Stallings, [*Group theory and three-dimensional manifolds.*]{} Yale Mathematical Monographs,[**4.**]{} Yale University Press, New Haven, Conn.-London, 1971.
|
---
abstract: 'We review recent work on resonant Andreev tunneling through a strongly interacting quantum dot connected to a normal and to a superconducting lead. We derive a general expression for the current flowing in the structure and discuss the linear and non-linear transport in the nonperturbative regime. New effects associated to the Kondo resonance combined with the two-particle tunneling arise. The Kondo anomaly in the $I-V$ characteristics depends on the relative size of the gap energy and the Kondo temperature.'
address:
- ' $^{a}$ Istituto Nazionale per la Fisica della Materia e Dipartimento di Fisica E. Amaldi, Università di Roma Tre, Via della Vasca Navale 84, I-00146 Roma, Italy'
- ' $^{b}$ Dipartimento di Fisica, Universitá degli Studi di Roma “La Sapienza”, Piazzale Aldo Moro 2, I-00185 Roma, Italy'
author:
- 'R. Raimondi$^{a}$ and P. Schwab$^{b}$'
title: Andreev Tunneling in Strongly Interacting Quantum Dots
---
epsf
[2]{}
Introduction
============
In recent years electrical transport through confined regions has seen impressive theoretical and experimental activity. Several well known phenomena have received renewed attention due to the present possibility of studying them in novel and more controllable situations. For instance, Andreev scattering (Andreev 1964), according to which a particle-like excitation impinging on a normal metal-superconductor interface is reflected back as a hole-like excitation, has been shown to be the key mechanism controlling transport in a variety of hybrid mesoscopic superconducting devices. (For a review see, for example, Hekking, Schön, and Averin 1994, Beenakker 1995, Lambert and Raimondi 1998). At the same time, electron-electron interaction in small confined regions, or quantum dots (QDs), has been shown to lead to the so-called Coulomb blockade of electrical transport. This occurs at low temperatures, when the Fermi energy of the contacts falls in the gap between the ground state energies of the dot with $N$ and $N+1$ electrons (Kastner 1992). However, a QD attached to metallic leads resembles an impurity level in a metal. As a consequence, even in the Coulomb blockade regime, transport will occur due to the Kondo effect (Glazman and Raikh 1988, Ng 1988). This is due to the formation of a spin singlet between the impurity level and the conduction electrons, which gives rise to a quasiparticle peak at the Fermi energy in the dot spectral function. This suggestion has been explored theoretically by several groups (Meir, Wingreen, and Lee 1991,1993, Ng 1993, Levy Yeyati, Martin-Rodero, and Flores 1993, Schoeller and Schön 1994, Hettler and Schoeller 1995). This has lead to the prediction of a zero-bias anomaly in the current voltage characteristics and an increase of the linear conductance in the Coulomb blockade regime for decreasing temperature. Such phenomena have indeed been observed in different QD systems (Ralph and Buhrman 1994, Goldhaber-Gordon et al. 1998, Cronenwett et al. 1998).
What will happen if the QD is coupled to a normal and a superconducting lead as shown schematically in Fig.\[fig\_bag\_dot\]? Does the zero-bias anomaly observed in the N-QD-N case survive in the N-QD-S case? Such a problem has been investigated recently by various groups (Fazio and Raimondi 1998, K. Kang 1999, Schwab and Raimondi 1999, Sun, Wang, and Lin 1999, Clerk, Ambegaokar, Hershfield 1999). In this paper we review mainly the work done in Refs. (Fazio and Raimondi 1998) and (Schwab and Raimondi 1999).
The problem one is faced with has all the difficulty of the original Kondo problem. It is well known that in the limits of high and low temperatures (here high and low are with respect to the Kondo temperature), qualitatively correct results are obtained by means of different techniques. Such an attitude we adopt here, having in mind to analyze the interplay between Andreev scattering and Kondo physics. We use a simple equation-of-motion approach in the perturbative regime above the Kondo temperature. This corresponds to taking into account only the leading logarithmic corrections in the renormalization group sense. The cross-over behaviour to low temperature may be rather complicated, but as happens in the standard Kondo problem, we expect that the extreme low-temperature phase can be described in simple terms. Such a description is obtained by means of the slave-boson technique in a mean-field approximation. In the high temperature regime, we find a suppressed Andreev current at low bias voltage due to the competition between the Coulomb energy and the superconducting proximity effect in the QD. In the low-temperature case, in contrast, our analysis predicts that the linear conductance of the N-QD-S system is enhanced as compared to the normal case and may reach the maximum universal value of $G_{\rm NS}=4e^2/h$ which is twice the maximum for the N-QD-N system.
Before entering the technical details, it is also worthwhile discussing the relevant energy scales. In investigating Andreev scattering at a normal metal-superconductor interface one is often interested in voltages and temperatures well below the energy scale set by the superconducting gap $\Delta$ and one could be tempted to take $\Delta \to \infty$ from the outset. However, the dot charging energy $U$ (see below) introduces another energy scale into the problem, and the physics is different in the two limits $U\gg \Delta $ and $\Delta \gg U$. In order to appreciate this point, let us consider the Hamiltonian of an isolated QD, plus a term which models Andreev scattering in the limit $\Delta \to \infty$ by $H_{\rm A} = t_{\rm A} d^\dagger_\uparrow d^\dagger_\downarrow + c.c $. For this Hamiltonian the off-diagonal element of the inverse dot Green’s function reads $-t_{\rm A}\lbrace 1+U(\epsilon+\epsilon_{\rm d}
+U)/[\epsilon^2-(\epsilon_{\rm d}+U)^2-t_{\rm A}^2]\rbrace$ and vanishes in the $U\to \infty$ limit, [*i.e.*]{}, in the limiting sequence $\Delta \to \infty$ first, $U\to \infty$ after, the induced superconductivity and hence transport in the dot is completely suppressed. In order to have Andreev scattering in the large $U$ limit two electrons have to enter the superconductor without doubly occupying the QD. This can only happen on a time scale of the order $1/\Delta$. We therefore concentrate on the limit $U \gg \Delta$ in the following analysis.
Our paper is organized as follows. In Section II we derive a formula for the current through the dot in the presence of Andreev scattering. Calculating the current explicitly requires the dot Green’s function. In Sections III and IV we present the equation-of-motion and slave-boson approaches, respectively. A few conclusions are drawn in section V.
The Andreev current formula
===========================
In this section we define the system under consideration and derive an expression for the Andreev current which holds in the presence of electron-electron interaction.
The model Hamiltonian for the N-QD-S system is $$H=H_{\rm N} +H_{\rm S}+H_{\rm D}+H_{\rm T,N}+H_{\rm T,S}
\label{model}$$ where $H_{\rm N}$, $H_{\rm S}$, are the Hamiltonians of the normal and the superconducting leads ($ \Delta $ being the superconducting gap), $$\label{nlead}
H_{\rm N}=\sum_{{\bf k},\sigma} \epsilon_{\bf k}
c^{\dagger}_{{\rm N},{\bf k}\sigma} c_{{\rm N} ,{\bf k}\sigma}$$ $$\label{slead}
H_{ \rm S}=\sum_{{\bf k},\sigma} \epsilon_{\bf k} c^{\dagger}_{{\rm S},{\bf k}\sigma}
c_{{\rm S},{\bf k}\sigma}+
\sum_{\bf k} (\Delta c^{\dagger}_{{\rm S},{\bf k}\uparrow}
c^{\dagger}_{{\rm S},-{\bf k}\downarrow}
+ c.c.) \;\;\;\;.$$ If we restrict ourselves to temperatures and bias voltages much smaller than the average level spacing in the QD, then transport occurs through a single level. In this case the Hamiltonian of the quantum dot $H_{\rm D}$ reads $$\label{dot}
H_{\rm D}=\epsilon_{\rm d} d^{\dagger}_{\sigma}d_{\sigma}+Un_{\rm d\uparrow}
n_{\rm d\downarrow} \;\; .$$ The level $\epsilon_{\rm d}$ is assumed to be spin degenerate and the electron-electron interaction is included through the on-site repulsion $U$ ($ \sim 1-5$K for currently fabricated QDs). Experimentally the position of the dot level can be modulated by an external gate voltage. Tunneling between the leads and the dot is described by $H_{\rm T,N}$ and $H_{\rm T,S}$ $$\label{tunn}
H_{{\rm T},\eta}=
\sum_{{\bf k}\sigma} (V_{\eta} c^{\dagger}_{\eta ,{\bf k}\sigma} d_{\sigma}
+ c.c. )$$ where $\eta ={\rm N,S}$ and $V_{\eta}$ is the tunneling amplitude.
The starting point for deriving the current formula is $I = -e ({{\rm d}}/{{\rm d}}t) \langle N_{\rm N} \rangle = {{\rm i}}e\left[N_{\rm N}, H \right]$ where $N_{\rm N}$ is the electron number operator in the normal lead. In the case of a hybrid structure, like the one we are considering now, it turns out to be more convenient to evaluate the current in the normal lead. The average current can be rewritten as follows $$\label{current}
I =
2e{ {{\rm Im}}}\sum_{{\bf k},\sigma}V_{\rm N} \langle c_{{\rm N},{\bf k}\sigma}^{\dagger}
d_{\sigma}\rangle \;\;\; .$$ Since we deal with a nonequilibrium situation we work in the framework of the Keldysh technique, as employed in the literature (Meir and Wingreen 1992). By introducing the Nambu notation $$\Psi_{{\rm N},{\bf k}} =\left( \begin{array}{c}
c_{{\rm N},{\bf k} \uparrow}\\
c^{\dagger}_{{\rm N},-{\bf k}\downarrow}
\end{array}
\right) \;\;\;\;\;\;\;\;
\phi =\left( \begin{array}{c}
d_{\uparrow}\\
d^{\dagger}_{\downarrow}
\end{array}
\right),$$ the average current in eq.(\[current\]) requires the evaluation of the lesser Green’s function $
G^{<}_{\alpha \beta ,{\bf k} } (t,t')= {{\rm i}}\langle \Psi^{\dagger}_{\beta,{\rm N},{\bf k}}(t')
\phi_{\alpha}(t) \rangle
$ which, by means of the Dyson equation, can be expressed in terms of the exact Green’s function $G$ of the QD and the free Green’s function of the normal lead ${ g}_{{\rm N},{\bf k}}$. The lesser component of the Dyson equation reads $$\begin{aligned}
{\hat G}^{<}_{{\bf k}}(\epsilon) &=&
{\hat G}^R(\epsilon)
\left( \begin{array}{c c}
V_{\rm N} & 0\\
0 & V^*_{\rm N}
\end{array} \right)
{\hat g}^<_{{\rm N},{\bf k}}(\epsilon) \nonumber\\
&& +
{\hat G}^<(\epsilon)
\left( \begin{array}{c c}
V_{\rm N} & 0\\
0 & V^*_{\rm N}
\end{array} \right)
{\hat g}^A_{{\rm N} ,{\bf k}}(\epsilon) \;\;\; .\end{aligned}$$ where ${\hat {G}}^{R(A)}$, ${\hat {G}}^<$ are the retarded (advanced) and the lesser Green’s functions of the dot (for example ${\hat {G}}^{R}(t)=
-{{\rm i}}\theta (t)\langle\lbrace \phi (t),\phi^{\dagger} (0)\rbrace \rangle$). Using the relation ${\hat {G}}^<={\hat {G}}^R{\hat {\Sigma}}^<{\hat {G}}^A$ and the expression for the Green’s functions in the normal lead (diagonal in Nambu space) $g^R_{{\rm N},{\bf k}}(\epsilon) = 1/(\epsilon - \epsilon_k +{{\rm i}}\eta)$, $g^<_{{\rm N} ,{\bf k}}(\epsilon) = 2\pi {{\rm i}}f(\epsilon ) \delta (\epsilon - \epsilon_{\rm k})$ ($f(x)$ is the Fermi function), it is possible to rewrite the current in the following form $$\begin{aligned}
\label{current3}
&I = {{\rm i}}e\int_{-\infty}^{\infty}
{{{{\rm d}}\epsilon }\over {2\pi}} \Gamma_{\rm N} {{\rm Tr}}\lbrace {\hat {\tau}}_z {\hat {G}}^R(\epsilon) [
{\hat {\Sigma}}^R(\epsilon) {\hat {f}}_{\rm N} (\epsilon) -&\cr
& {\hat {f}}_{\rm N }(\epsilon){\hat {\Sigma}}^A(\epsilon)+
{\hat {\Sigma}}^<(\epsilon)]{\hat {G}}^A(\epsilon ) \rbrace &\end{aligned}$$ In the formula above, we introduced the elastic rate $\Gamma_{\rm N} (\epsilon ) =2 \pi \sum_{\bf k} |V_{\rm N}|^2
\delta (\epsilon -\epsilon_{\bf k} )$. The diagonal matrix ${\hat {f}}_{\rm N}$ has elements $f_{{\rm N},11}=f(\epsilon -\mu_{\rm N} )$ and $f_{{\rm N},22}=1-f(-\epsilon -\mu_{\rm N} )$. The normal electrode is kept at a chemical potential $\mu_{\rm N}=-eV$ while that of the superconductor is fixed to zero.
Up to now no approximations were involved. In order to determine the current an expression for the self-energy of the QD should be found. A determination of $I$ requires both the lesser and retarded parts of the self-energy. We formulate an ansatz for the lesser Green’s function which is expressed solely in terms of the retarded one. This ansatz automatically guarantees current conservation. This generalizes an ansatz put forward by (Ng 1996) for two normal leads to the calculation of ${\hat {\Sigma}}^< (\epsilon)$ in the presence of a superconducting lead.
In order to understand how the ansatz is constructed it is useful to start from the noninteracting dot. In this case, ${\hat {\Sigma}}^{<,>}(\epsilon)$ can be computed exactly and it is expressed in terms of the retarded and advanced self-energies as $$\label{nonintselfenergy}
{\hat {\Sigma}}^<_0 (\epsilon) = -\sum_{\eta ={\rm N,S}}\left[
{\hat {\Sigma}}_{0,\eta}^R(\epsilon) {\hat {f}}_{\eta} (\epsilon) -
{\hat {f}}_{\eta} (\epsilon){\hat {\Sigma}}_{0,\eta}^A(\epsilon)\right]$$ $$\begin{aligned}
&{\hat {\Sigma}}^>_0 (\epsilon) = -\sum_{\eta ={\rm N,S}}
\{{\hat {\Sigma}}_{0,\eta}^R(\epsilon) [{\hat 1}-{\hat {f}}_{\eta}
(\epsilon)]-&\cr
& [{\hat 1}-{\hat {f}}_{\eta} (\epsilon)]{\hat
{\Sigma}}_{0,\eta}^A(\epsilon)\}.&\end{aligned}$$ In this case (with no interaction) the nonequilibrium self-energy has the same form as in equilibrium but with the Fermi functions of the two leads kept at different chemical potentials. The idea is to assume that even in the presence of interaction the dependence on the Fermi distribution is the same and that both the lesser and the greater functions depend on a single function ${\hat {A}}$ such that $$\label{intself}
{\hat {\Sigma}}^< ={\hat {\Sigma}}^<_0 {\hat {A}}
\;\;\;\;\;\; , \;\;\;\;\;\;
{\hat {\Sigma}}^> ={\hat {\Sigma}}^>_0 {\hat {A}}
.$$ The matrix ${\hat {A}}$ is determined by the condition $${\hat {\Sigma}}^<-{\hat {\Sigma}}^>={\hat {\Sigma}}^R-{\hat {\Sigma}}^A
,$$ which is a general property of the Keldysh Green’s functions. As already mentioned, this ansatz leads to current conservation. Moreover it is exact both in the non-interacting limit, $U=0$, and in absence of superconductivity, $\Delta =0$. As a result $$\label{intselfapp}
{\hat {\Sigma}}^<={\hat {\Sigma}}^<_0 ({\hat {\Sigma}}^R_0 -{\hat {\Sigma}}^A_0)
^{-1}
({\hat {\Sigma}}^R-{\hat {\Sigma}}^A).$$ Eq.(\[intselfapp\]) allows us to evaluate the expression of the current (\[current3\]), once we know the retarded Green’s function of the dot. The expression for the current can be greatly simplified in the relevant limit $U,\Delta \gg k_BT,e V$. In this case the non-interacting self-energy due to the superconducting lead ${\hat {\Sigma}}^{R(A)}_{0,{\rm S}}$ is real and purely off-diagonal whereas that due to the normal lead, ${\hat {\Sigma}}^{R(A)}_{0,{\rm N}}$, is diagonal. Using these properties of the self-energy and substituting eq.(\[intselfapp\]) in eq.(\[current3\]) we obtain the following form for the Andreev current through a QD. $$\label{current4}
I = {{\rm i}}e\int_{-\infty}^{\infty}
{{{{\rm d}}\epsilon }\over {2\pi}} \Gamma_{\rm N} \nonumber \\
{{{\rm Tr}}} \left\{ {\hat {\tau}}_z {\hat G}^R(\epsilon)\left[
{\hat {\Sigma}}^R(\epsilon), {\hat {f}}_{\rm N} (\epsilon) \right]
{\hat G}^A(\epsilon ) \right\} \;\;\; .$$ This equation generalizes the formula obtained by (Beenakker 1992, Claughton, Leadbeater, and Lambert 1995) for the non-interacting case to the case of a strongly interacting dot .
In the following sections we apply this formula to the study of the $I-V$ curves of a N-QD-S system.
Equation of motion
==================
Our aim is to find an equation for the dot’s Green’s function $\hat{G}(t)=-{{\rm i}}\langle T\left[ \phi (t)\phi (0)\right] \rangle$. The retarded and advanced Green’s functions necessary for the evaluation of the current can easily be obtained from the time-ordered one. In this section we will describe the equation-of-motion approach. It is useful to introduce quasiparticle operators by performing a Bogolubov transformation $$\begin{aligned}
\label{eom2a}
\gamma _{\eta ,{\bf k}\sigma } &=&
u_{\eta ,{\bf k} }c_{\eta ,{\bf k}\sigma }+{{\rm sign}}(\sigma )
{{\rm e}}^{-{{\rm i}}\varphi }v_{\eta ,{\bf k}}c_{\eta ,-{\bf k}\overline{\sigma } }^{\dagger }\\
\label{eom2b}
\gamma _{\eta ,-{\bf k}\overline{\sigma } }^{\dagger }
&=&-{{\rm sign}}(\sigma ){{\rm e}}^{{{\rm i}}\varphi }v_{\eta ,{\bf k} }c_{\eta ,{\bf k}\sigma}
+u_{\eta ,{\bf k} }c_{\eta ,-{\bf k}\overline{\sigma }}^{\dagger }\end{aligned}$$ where $u^{2}_{\eta ,{\bf k} }(v^{2}_{\eta ,{\bf k} })
=\left(1 \pm \epsilon_{\bf k}/E_{\eta, \bf k}\right)/2$ , $ E_{\eta,\bf k}=\sqrt{\epsilon_{\bf k}^{2}
+\mid \Delta _{\eta }\mid ^{2}} $, and $ \Delta _{\eta }={{\rm e}}^{{{\rm i}}\varphi }\mid \Delta _{\eta }\mid $. In the normal lead we have $ \Delta _{\rm N}=0 $ and the Bogolubov transformation reduces to the identity transformation. The hopping term becomes in the quasiparticle basis $$\label{eom3}
\hat{T}_{\eta ,{\bf k} }=\left( \begin{array}{cc}
u_{\eta ,{\bf k} }V_{\eta } &
-{{\rm e}}^{{{\rm i}}\varphi }v_{\eta ,{\bf k} }V^{*}_{\eta }\\
-{{\rm e}}^{{{\rm i}}\varphi }v_{\eta ,{\bf k} }V_{\eta }
& -u_{\eta ,{\bf k} }V^{*}_{\eta }
\end{array}
\right) .$$ The Nambu formalism is introduced for the $\gamma$ operator via $\tilde\Psi_{\eta,{\bf k}}=
(\gamma_{\eta,{\bf k}\uparrow },
\gamma^{\dagger}_{{\eta} ,-{\bf k}\downarrow})$. We start by writing down the equation of motion for the operators $\phi$ and $\tilde\Psi$: $$\begin{aligned}
\label{eom6}
{{\rm i}}\frac{\partial }{\partial t}\phi &=&\epsilon _{\rm d}\hat{\sigma }_{z}
+U\hat{\sigma }_{z}\Phi
+\sum _{\eta ,{\bf k} }\hat{T}^{\dagger }_{\eta ,{\bf k}}
\tilde\Psi _{\eta ,{\bf k} }\\
\label{eom7}
{{\rm i}}\frac{\partial }{\partial t}\tilde\Psi _{\eta ,{\bf k}}
&=& E_{\eta, \bf k}\hat{\sigma }_{z}
\tilde\Psi _{\eta ,{\bf k} }+\hat{T}_{\eta ,{\bf k} }\phi. \end{aligned}$$ In the equation for $\phi$, the new operator $ \Phi =(
d_{\downarrow }^{\dagger }d_{\downarrow }d_{\uparrow },
d^{\dagger }_{\downarrow }d^{\dagger }_{\uparrow }d_{\uparrow } )$, appears because of the interaction. It is straightforward to iterate once the equation of motion and get an equation for $\Phi$ $$\begin{aligned}
\label{eom8}
&{{\rm i}}\frac{\partial }{\partial t}\Phi =\left( \epsilon _{\rm d}
+U\right) \hat{\sigma }_{z}\Phi +\sum _{\eta ,{\bf k}}
\left[ \hat{N}\hat{T}_{\eta ,{\bf k} }^{\dagger }\tilde\Psi _{\eta ,{\bf k} }\right.
&\cr
&\left. +d^{\dagger }_{\downarrow }d_{\uparrow }
\left( \hat{T}_{\eta ,{\bf k} }\right) ^{t}
\left( \tilde\Psi _{\eta ,{\bf k} }^{\dagger }\right) ^{t}\right] &
,\end{aligned}$$ where $$\hat{N}=\left( \begin{array}{cc}
d_{\downarrow }^{\dagger }d_{\downarrow },
& d_{\downarrow }d_{\uparrow }\\
d_{\downarrow }^{\dagger }d^{\dagger }_{\uparrow },
& d^{\dagger }_{\uparrow }d_{\uparrow }
\end{array}
\right) .$$ Here the superscript $ t $ means the transpose. In eq.(\[eom8\]) again several new operators have appeared. In order to get a closed system of equations one has to truncate the hierarchy at some point. Following the decoupling procedure used in the absence of superconducting leads (Meir, Wingreen and Lee 1991), we neglect correlations in the leads. To see how this is achieved, consider the general structure of all the new operators entering eq.(\[eom8\]) for $ \Phi $. They are products of two $ d $ and one $ \gamma $ operator. By iterating once more the equation of motion one obtains new terms which contain two $ \gamma $ and one $ d $ operator. The approximation consists then in replacing a pair of two such $ \gamma $ operators with their statistical average. In such a way all the new operators entering the eq.(\[eom8\]) for $ \Phi $ can be expressed in terms of $ \Phi $ itself and $ \phi $. A further approximation is done by considering the limit of large Coulomb interaction, [*i.e.*]{}, $ U\rightarrow \infty $. In this case one can safely neglect, in the third order of iteration of the equation of motion, all operators having two creation or two annihilation $ d $ operators, because they will give rise to terms of the order $ 1/U $. We write here the resulting equations of motion for $ {\hat G} $ and ${\hat G}^{II} =
-{{\rm i}}\left\langle T\left[ \Phi (t)\phi (0)\right] \right\rangle$. They are $$\begin{aligned}
\label{eom21}
\left( \omega \hat\sigma_0
-\epsilon _{\rm d}\hat{\sigma }_{z}\right) {\hat G}(\omega )
&=&\hat\sigma_0+\hat{\Sigma }_{0}(\omega )\hat G(\omega )
+U\hat{\sigma }_{z}{\hat G}^{II}(\omega ) \\
\label{eom22}
U\hat{\sigma }_{z}{\hat G}^{II}(\omega )
&=&-\langle \hat{N}\rangle
+\hat{\Sigma }_{I}(\omega ){\hat G}(\omega )\end{aligned}$$ and the equation for the single particle Green’s function becomes $$\label{eom23}
\left( \omega \hat\sigma_0 -\epsilon _{\rm d}\hat{\sigma }_{z}
-\hat{\sigma }(\omega )\right)
{\hat G}(\omega )
=\hat\sigma_0-\langle \hat{N}\rangle ,$$ where $\hat{\sigma }=\hat{\Sigma }_{0}+\hat{\Sigma }_{I}$. The noninteracting self-energy $ \hat{\Sigma }_{0} $ is obtained as $$\label{eom24}
\hat{\Sigma }_{0}\left( \omega \right)
=-{{{\rm i}}\over 2} \sum _{\eta }\frac{\Gamma_{\eta }}{\sqrt{\omega ^{2}-\Delta ^{2}_{\eta }}}
\left( \begin{array}{cc}
\omega & \Delta _{\eta }\\
-\Delta _{\eta } & -\omega
\end{array}
\right)$$ with $\Gamma_\eta = 2 \pi \sum_{\bf k} |V_\eta |^2
\delta(\epsilon - \epsilon_{\bf k}).$ For the calculation of the interacting self-energy $ \hat{\Sigma }_{I}\left( \omega \right) $ we assume that in the superconducting reservoir there are no quasiparticles present (we consider energies much smaller than the gap $ \Delta $) $ \langle
\gamma^{\dagger }_{{\rm S}, {\bf k}\sigma }\gamma_{{\rm S},{\bf k}\sigma }\rangle
=0 $, and that in the normal reservoir the quasiparticle population follows the Fermi distribution $ \langle
\gamma^{\dagger}_{{\rm N},{\bf k}\sigma }\gamma_{{\rm N},{\bf k}\sigma }
\rangle
=f\left( \epsilon_{\bf k} -\mu_{\rm N}\right) $. We then have $$\label{eom25}
\hat{\Sigma }_{I,{\rm S}}\left( \omega \right)
=V^{2}_{\rm S}\sum _{\bf k}\left( \begin{array}{cc}
\frac{|v_{{\bf k},{\rm S} }|^2}{\omega -E_{{\bf k},{\rm S}}}
& \frac{2u_{{\bf k},{\rm S}}v_{{\bf k},{\rm S}}}{\omega -E_{{\bf k},{\rm S}}}\\
-\frac{2u_{{\bf k},{\rm S}} v_{{\bf k},{\rm S}}}{\omega +E_{{\bf k},{\rm S}}}
& \frac{|v_{{\bf k},{\rm S}}|^2}{\omega +E_{{\bf k},{\rm S}}}
\end{array}
\right)$$
$$\label{eom26}
\hat{\Sigma }_{I,{\rm N}}\left( \omega \right)
=-V^{2}_{\rm N}\sum _{\bf k}\left( \begin{array}{cc}
\frac{f( \epsilon_{\bf k} ) }{\omega -\epsilon_{\bf k}} & 0\\
0 & \frac{f( \epsilon_{\bf k} ) }{\omega +\epsilon_{\bf k}}
\end{array}
\right)
.$$
In what follows, we consider $W \gg\Delta \gg T,\omega$ with $W$ the bandwidth. The self-energy ${\Sigma}_{I,{\rm S},11}$ is weakly energy dependent since $\omega \ll \Delta < E_{{\bf k}, \rm S}$. The imaginary part of the diagonal elements of the self-energy ${\hat \Sigma}_{\rm S}$ vanishes, so quasiparticles present in the dot cannot decay by tunneling into the superconductor. As a result the contribution to the self-energy due to the superconducting lead simply shifts the dot level to the new value $\tilde{\epsilon }_{\rm d} \approx \epsilon_{\rm d} +(\Gamma _{\rm S}/2\pi )
\ln W/\Delta$. The divergence of the level energy renormalization with $\Delta$ reveals that the process occurs via a virtual state in which a quasiparticle is created in the superconductor. Substituting the expression of the QD’s Green’s function in eq.(\[current4\]) we get the result $$\label{current55}
I(V) =\int^{\infty}_{-\infty}~{{\rm d}}\epsilon
{{f(\epsilon -eV)-f(\epsilon +eV)}\over {2e}} G_{\rm NS}(\epsilon )$$ with
$$\label{conductancebis}
G_{\rm NS}(\epsilon )={{4e^2}\over h}
{{2(\Gamma_{\rm N} \Gamma_{\rm S}(\epsilon ) )^2
\left[(\Gamma_{1,{\rm N}} +\Gamma_{2,{\rm N}})/2\Gamma_{\rm N})\right]}\over
{\left[4 (\epsilon-\epsilon_1)(\epsilon+\epsilon_2)
-\Gamma_{1,{\rm N}}\Gamma_{2,{\rm N}}
-\Gamma_{\rm S}^2(\epsilon )\right]^2 +4\left[\Gamma_{1,{\rm N}}(\epsilon+\epsilon_2)+
\Gamma_{2,{\rm N}}(\epsilon-\epsilon_1)\right]^2} }$$
[2]{} where $\epsilon_{1(2)} =\epsilon_{\rm d} \pm {{\rm Re}}{\hat {\sigma}}_{11(22)}$, $\Gamma_{1(2),{\rm N}}=-2{{\rm Im}}{\hat {\sigma}}_{11(22)}$, and $\Gamma_S (\epsilon ) =2{{\rm Re}}{\hat {\sigma}}_{12}=
\Gamma_S 2\epsilon / (\pi \Delta )$. Notice that the anomalous non-interacting $\Sigma_{0,12}$ and interacting $\Sigma_{{\rm I},12}$ self-energies exactly cancel in the zero energy limit, resulting in a linear behaviour for energies smaller than the gap[@correction]. The spectral function $G_{\rm NS}(\epsilon )$ associated with the resonant Andreev tunneling is plotted in Fig. \[fig2\] for various bias voltages. Several features are worth noticing. First, two peaks at $\pm \tilde{\epsilon}$ are due to particle and hole bare levels. Note that in the interacting case the bare level energy includes the renormalization due to the superconducting electrode self-energy as discussed above. Second, at low temperatures the spectral function at the Fermi energy is completely suppressed, in contrast to what happens in N-QD-N case. Quite remarkably, at finite positive (negative) voltages a Kondo peak develops pinned to the Fermi level of the normal metal while a small kink develops at negative (positive) voltages. At finite voltages hole and particle energies differ by $2eV$, and while the electron (hole) is on resonance for positive (negative) voltage, the Andreev reflected hole (electron) is off resonance with respect to the shifted Fermi level.
The differential conductance, for various temperatures, is shown in Fig. \[fig3\]. Lowering the temperature a zero-bias anomaly starts to develop where the conductance is strongly suppressed at low voltages. From eqs.(\[current55\]-\[conductancebis\]) we conclude that the linear conductance is roughly proportional to $T^2/\Delta^2$, so that it seems to be completly suppressed in the zero temperature limit. However, the equation-of-motion approach is quantitatively reliable only above the Kondo temperature. Hence the analysis carried out here applies to the regime $\Delta > T > T_K$. An interesting question to ask is what happens in the opposite regime when the Kondo temperature dominates, i.e., when $T_K > \Delta > T$. This regime cannot be explored by the equation-of-motion approach. For this reason in the next section we will investigate the low temperature regime by means of the slave-boson technique.
Slave boson mean field approximation
====================================
In this section we extend the analysis of the previous section to the extreme low temperature regime. To this end we use slave-boson mean field theory. This approach has been successfully applied to the low temperature properties of a Kondo impurity in the presence of normal (Barnes 1976, Coleman 1984, Read and Newns 1983, Read 1985) as well as for superconducting conduction electrons (Borkowski and Hirschfeld 1994). Despite its simplicity, this method captures the main physical aspects of the Fermi liquid regime at low temperatures, [*i.e.*]{}, the formation of a many-body resonance at the Fermi energy. For this reason it presents a convenient framework in which to study the interplay between Andreev scattering and Coulomb interactions.
Again we consider an infinite on-site repulsion $U$, so processes where the dot level is doubly occupied are excluded. The dot level is represented as $d^\dagger_\sigma = f^\dagger_\sigma b$, where the fermion $f_\sigma$ and the boson $b$ describe the singly occupied and empty dot states. Since the dot is either empty or singly occupied, the constraint $b^\dagger b + \sum_\sigma f^\dagger_\sigma f_\sigma =1$ has to be fulfilled.
In mean field approximation, the operator $b$ is replaced by a $c$-number $b_0$, and the constraint is fulfilled only on average. This is achieved by introducing a chemical potential $\lambda_0$ for the pseudo particles. Notice that one ends up with a non-interacting-like problem with renormalized parameters, [*i.e.*]{}, an energy shift for the dot level $\epsilon_{\rm d} \rightarrow \epsilon_{\rm d} +\lambda_0=\tilde \epsilon_{\rm d}$ and a multiplicatively renormalized tunneling amplitude $V_\eta \rightarrow b_0 V_\eta$.
We discuss the mean field equations and their solution first in equilibrium and then generalize to non-equilibrium. We start from the impurity part of the free energy, which in the presence of both normal and superconducting leads is given by $$\label{sb2}
F= - T\sum_{\epsilon_n} {{\rm Tr}}\ln [
{{\rm i}}\epsilon_n \hat\sigma_0 - \tilde \epsilon_{\rm d} \hat\sigma_z -b_0^2
\hat \Gamma({{\rm i}}\epsilon_n) ]
+\lambda_0 b_0^2 +\epsilon_{\rm d} - \mu
,$$ where $\epsilon_n$ is a fermionic Matsubara frequency, ${\hat {\sigma}}^i$ are the Pauli matrices, and $$\begin{aligned}
\label{sb3}
\hat \Gamma({{\rm i}}\epsilon_n )&= & \sum_{{\bf k}, \eta }
|V_\eta|^2 \hat \sigma_z \hat g_{\eta ,{\bf k}}({{\rm i}}\epsilon_n) \hat \sigma_z\end{aligned}$$ with $\hat g_{\eta ,{\bf k}}$ being the Green’s function of the lead $\eta$.
By minimizing the free energy with respect to $\lambda_0$ and $b_0$ we find the equations $$\begin{aligned}
\label{sb6}
b_0^2 + T\sum_{\epsilon_n} {{\rm Tr}}\left[ \hat {\cal G}({{\rm i}}\epsilon_n)
\hat \sigma_z \right] &=& 0, \\
\label{sb7}
b_0 \lambda_0 + b_0T \sum_{\epsilon_n} {{\rm Tr}}\left[ \hat{\cal G}({{\rm i}}\epsilon_n)
\hat \Gamma({{\rm i}}\epsilon_n) \right] &= &0
,\end{aligned}$$ which have to be solved self-consistently. $\hat{\cal G}({{\rm i}}\epsilon_n)$ is the pseudo fermion Green’s function given by $\hat{\cal G}({{\rm i}}\epsilon_n) =
[{{\rm i}}\epsilon_n \hat\sigma_0 - \tilde \epsilon_{\rm d} \hat\sigma_z -b_0^2
\hat \Gamma({{\rm i}}\epsilon_n)]^{-1}$. Both in the limit of small and large superconducting gap, we are able to solve the mean field equations analytically as demonstrated here below. The first equation, eq.(\[sb6\]), is the constraint. Since the pseudo fermion level is at maximum singly occupied, the renormalized level is above the Fermi energy. In the Kondo limit, where the occupancy is nearly one, we find that $0 < \tilde \epsilon_{\rm d} < b_0^2(\Gamma_{\rm N} + \Gamma_{\rm S}) $, [*i.e.*]{} $\lambda_0 \approx | \epsilon_{\rm d} |$ and $\tilde \epsilon_{\rm d} \approx 0$. The renormalization of the tunneling amplitude is determined from eq.(\[sb7\]). A trivial solution $b_0=0$ always exists. The solutions which minimize the free energy, however, are those with $b_0 \ne 0$. By introducing a flat density of states in the leads and the tunneling rates $\Gamma_\eta= 2\pi N_{0}|V_\eta|^2$, the elements of the matrix $\hat \Gamma ({{\rm i}}\epsilon_n )$ are $\Gamma_{11}=\Gamma_{22}=-{{\rm i}}\gamma_1$ and $\Gamma_{12}=\Gamma_{21}^*=\gamma_2$, where $$\label{sb8}
\gamma_1 = {{{\rm sign}}}(\epsilon_n) {\Gamma_{\rm N} \over 2}
+{\Gamma_{\rm S}\over 2} {\epsilon_n \over \sqrt{\epsilon_n^2 + |\Delta|^2} },~
\gamma_2 ={ \Gamma_{\rm S} \over 2}
{ \Delta \over \sqrt{\epsilon_n^2 + | \Delta|^2} }
.$$ Restricting ourselves to zero temperature, we replace the Matsubara sum in eq.(\[sb7\]) by an integral and obtain $$\label{sb12}
|\epsilon_{\rm d} | = 4 \int_0^W {{{\rm d}}\epsilon \over 2 \pi }
{ \gamma_1 ( \epsilon + b_0^2 \gamma_1 ) + b_0^2 | \gamma_2|^2 \over
( \epsilon+ b_0^2 \gamma_1 )^2 + b_0^4 | \gamma_2 |^2}
,$$ where $W$ is a cut-off of order the band-width. We simplify the integral by approximating $\gamma_1 $ and $\gamma_2$ as $$\label{sb13}
\gamma_1 = \left\{ \begin{array}{ll}
\Gamma_{\rm N}/2 & {\rm for }\,\,\, \epsilon < \Delta \\
(\Gamma_{\rm N}+ \Gamma_{\rm S})/2 & {\rm for }\,\,\, \epsilon > \Delta
\end{array} \right.
,~ \gamma_2 = \left\{
\begin{array}{ll}
\Gamma_{\rm S}/2 & {\rm for} \,\,\, \epsilon < \Delta \\
0 & {\rm for} \,\,\, \epsilon > \Delta
\end{array} \right.
.$$ The result is $$\begin{aligned}
\label{ab15}
|\epsilon_{\rm d}| &= & {\Gamma_{\rm N} \over 2 \pi }
\ln{ (2 \Delta + b_0^2 \Gamma_{\rm N})^2 + b_0^4\Gamma_{\rm S}^2
\over
b_0^4( \Gamma_{\rm N}^2+\Gamma_{\rm S}^2) } \nonumber\\
&& +{\Gamma_{\rm N}+\Gamma_{\rm S} \over 2\pi }
\ln{ 4 W^2 \over (2 \Delta +b_0^2 \Gamma_{\rm S} +b_0^2 \Gamma_{\rm N} )^2 }
,\end{aligned}$$ where we neglect a term proportional to $\Gamma_{\rm S}$, but without any logarithmic factor. If $\Delta $ is much smaller than the Kondo temperature which is given by $T_{\rm K} = b_0^2 \Gamma_{\rm N}+ b_0^2 \Gamma_{\rm S}$, $\Delta$ is negligible. One can then easily solve eq.(\[ab15\]) for $b_0^2$ and obtain the result for two normal leads with total tunneling rate $\Gamma_{\rm N}+\Gamma_{\rm S}$: $$\label{sb16}
b_0^2(\Gamma_{\rm N}+\Gamma_{\rm S})
= 2 W \exp\left( -{\pi |\epsilon_{\rm d}|\over \Gamma_{\rm N} + \Gamma_{\rm S} }
\right)
.$$ In the opposite limit, where $\Delta$ is much larger than $T_{\rm K}$, we find $$\label{sb18}
b_0^2 \sqrt{ \Gamma_{\rm N}^2 + \Gamma_{\rm S}^2 } = 2 W \exp \left( -{\pi }
{ |\epsilon_{\rm d}| - ( \Gamma_{\rm S} / \pi ) \ln(W / \Delta) \over \Gamma_{\rm N} } \right)
.$$ The results agree qualitatively with what we expect from scaling arguments for the Anderson model. In the perturbative regime, a logarithmic correction to $\epsilon_{\rm d}$ has been found (Haldane 1978). This applies to the case of a large gap, since scaling due to the superconducting electrons stops at energies of the order $\Delta$, giving rise to a logarithmic renormalization of $\epsilon_{\rm d}$, as seen in eq.(\[sb18\]). In the case of a small gap, the superconducting lead contributes to scaling down to low energies, where one enters the strong coupling regime. Presumably, the fixed point is still reached for energies of the order of $T_{\rm K}$, much greater than $\Delta$, so that the Kondo temperature does not depend on $\Delta$, as indeed found in eq.(\[sb16\]). Notice that in the presence of normal electrons, we always find a non-trivial solution of the mean field equations. This is to be contrasted with the case of superconducting electrons only, $\Gamma_{\rm N}=0$, where for large gap only the solution $b_0=0$ exists, and there is no Kondo effect (Borkowski and Hirschfeld 1994).
In a non-equilibrium situation, when a voltage is applied between the two leads, the mean field parameters cannot be obtained by minimizing the free energy. However the mean field equations (\[sb6\],\[sb7\]) can also be derived using a self-consistent diagrammatic method (Millis and Lee 1987). Then it is straightforward to generalize to non-equilibrium. The equations read $$\begin{aligned}
\label{sb21}
b_0^2 -{{\rm i}}\int {{{\rm d}}\epsilon \over 2\pi } {{\rm Tr}}\left[ \hat{\cal G }^<(\epsilon )
\hat \sigma_z \right] &=& 0 \\
\label{sb22}
\lambda_0 b_0 -{{\rm i}}b_0 \int{{{\rm d}}\epsilon \over 2 \pi} {{\rm Tr}}\left[
\hat {\cal G}^R(\epsilon) \hat \Gamma^<(\epsilon)
+ \hat{\cal G}^<(\epsilon) \hat \Gamma^A(\epsilon)
\right] &=&0
,\end{aligned}$$ where the lesser Green’s function $ \hat{\cal G}^<(t, t' ) = {{\rm i}}\langle \phi^\dagger (t')
\phi(t) \rangle$ has been introduced, with $\phi=( f_\uparrow, f^\dagger_\downarrow )$. The lesser and advanced matrix $\hat \Gamma$ is defined in analogy to its equilibrium version in eq.(\[sb3\]). To obtain $\hat {\cal G}^<$, we use the general relation $\hat {\cal G}^<= \hat{\cal G}^R \hat \Sigma^< \hat{\cal G}^A$, where at mean-field level $\hat \Sigma^<= b_0^2 \hat \Gamma^< $ and $$\label{sb23}
{\hat {\Gamma}}^< (\epsilon)=-\sum_{\eta, {\bf k} }
|V_\eta|^2{\hat\sigma}_z
\left[ {\hat g }_{\eta,{\bf k}}^R(\epsilon) {\hat {f}}_{\eta} (\epsilon) -
{\hat {f}}_{\eta} (\epsilon){\hat {g}}_{\eta,{\bf k}}^A(\epsilon)\right]
{\hat\sigma}_z.
\label{nonintself}$$ Note that the superconducting lead does not contribute to $\Sigma^< (\epsilon )$ for $| \epsilon | < \Delta$.
We have solved the mean-field equations in the presence of an external voltage numerically. As long as $|e V| \ll T_{\rm K}$ the solution is almost independent of the voltage. For large voltage, $|eV| \gg T_{\rm K}$, we have found that the Kondo peak is pinned to the chemical potential in the normal lead, [*i.e.*]{} $\tilde\epsilon_{\rm d} \to \tilde\epsilon_{\rm d} -eV$, and the peak width is decreased.
The Andreev current can now be determined using the current formula eq.(\[current4\]), with the dot Green’s function $\hat{ G }= b_0^2 \hat {\cal G}$. The spectral function, which is defined as in eq.(\[current55\]), is determined as $$\label{conductance}
G_{{\rm NS}}(\epsilon )= {4 e^2 \over h}
{4 {(\tilde\Gamma_{\rm N} \tilde\Gamma_{\rm S} )^2 }\over
(4\tilde \epsilon^2 -4{ \tilde\epsilon}_{\rm d}^2-
\tilde \Gamma_{\rm N}^2-\tilde\Gamma_{\rm S}^2)^2
+ 16 \tilde\Gamma_{\rm N}^2 \tilde \epsilon^2 }$$ Here the renormalized tunneling rates $\tilde\Gamma_{{\rm S},{\rm N}}=b_0^2\Gamma_{{\rm S},{\rm N}}$, and ${ \tilde {\epsilon}} = \epsilon (1+b_0^2 \Gamma_{\rm S}/2\Delta)$ are introduced. One recovers the current formula for a non-interacting quantum dot (Beenakker 1992), with renormalized parameters which are voltage dependent. On resonance, when $\tilde\epsilon_{\rm d} \approx 0$ and $\epsilon =0$, the small renormalization factor $b_0$ drops out. The differential conductance becomes maximal when $\tilde{\Gamma}_{\rm N}=\tilde{ \Gamma}_{\rm S}$ with $G_{{\rm NS},{\rm max}}=4e^2/h$, twice the maximum for a N-QD-N system. For large voltage $G_{\rm NS}$ drops quickly, since the resonance moves away from zero energy, $|\tilde\epsilon_{\rm d} | \approx |eV|$. $G_{\rm NS}$ as a function of energy is proportional to $(eV)^{-2}$ near $\tilde \epsilon \approx \pm \tilde \epsilon_{\rm d} $ and proportional to $(eV)^{-4}$ near $\epsilon = 0$. As a consequence the current decreases with increasing voltage, leading to a negative differential conductance. This is also demonstrated in Fig. \[fig4\], where the current as a function of voltage obtained using $G_{\rm NS}$ of eq.(\[conductance\]) with the numerically determined, voltage dependent mean field parameters. The results were obtained with $\epsilon_{\rm d }= W/3$, $\Gamma_{\rm N}=\Gamma_{\rm S}=0.14W$, and $\Delta = 0.2W$. For low voltages the current is, to good approximation, given by $I= 4 e^2 V/ h$, whereas the current drops when the voltage exceeds the Kondo temperature.
Finally, we want to comment on the reliability of our results. The success of slave-boson mean field theory stems from the fact that it captures the Fermi-liquid regime at low temperature. If the N-QD-S system scales to a Fermi liquid at low temperature, $G_{\rm NS}$ as given in eq.(\[conductance\]) is exact in the low temperature, low voltage limit. Since it is known that slave boson mean field theory has problems in describing dynamical properties, the results far away from equilibrium need to be treated with caution.
Within the Fermi-liquid point of view, the present mean field approach allows us to estimate the parameters entering eq.(\[conductance\]). In particular, we found that $\Gamma_{\rm N}$ and $\Gamma_{\rm S}$ renormalize equally, but this may no longer be the case when considering higher order corrections. For illustration, we estimate the effect of residual quasiparticle interaction in the limit $\Delta \ll T_{\rm K}$. By assuming an effective quasiparticle interaction of the form $H_{\rm int} = \tilde U n_\uparrow n_\downarrow$, we find to first order in $\tilde U$ no corrections to $\tilde{\Gamma}_{\rm N}$, while, as one could have expected, repulsive quasiparticle interaction suppresses the renormalized coupling to the superconductor, $\tilde{\Gamma}_{\rm S} =b_0^2{\Gamma}_{\rm S}[1-
4(\tilde U /\pi T_{\rm K})(\Delta / T_{\rm K})\ln{T_{\rm K}/2 \Delta}]$.
Conclusions
===========
We have studied Andreev tunneling in a normal metal-quantum dot-superconductor device, and obtained a general formula for the current through the device. We calculated the current voltage characteristics within the equation-of-motion approach, where we found a suppression of the conductance at low temperature. However it is known that this approach does not provide quantitative results in the Kondo regime. Therefore we extended the analysis to the extreme low temperature regime using slave boson mean field theory. In the regime when the superconducting gap is smaller than the Kondo temperature, we found an enhanced Andreev current at low bias voltage due to the Kondo effect. The zero bias conductance is maximum with the universal value $G_{\rm NS}=4e^2/h$ when the renormalized tunneling rates $\tilde \Gamma_{\rm N}$ and $\tilde
\Gamma_{\rm S}$ are equal. We identified the ratio $\Delta/T_{\rm K}$ as an important parameter. In the case $\Delta \ll T_{\rm K}$, the Kondo resonance forms as for two normal leads. The condition $\tilde \Gamma_{\rm N} = \tilde \Gamma_{\rm S}$ coincides with equal bare tunneling rates ${\Gamma}_{\rm N}={\Gamma}_{\rm S}$. In the case of large gap, quasiparticle interaction suppresses $\tilde{\Gamma}_{\rm S}$, nevertheless the conductance maximum condition may be achieved in an asymmetric QD where ${\Gamma}_{\rm S} > {\Gamma}_{\rm N}$.
We thank R. Fazio for many fruitful discussions. We acknowledge the financial support of INFM under the PRA-project “Quantum Transport in Mesoscopic Devices” and the EU TMR programme. We acknowledge a useful correspondence with A. Clerk which made us aware of a mistake in our original manuscript and in (Fazio and Raimondi 1998).
-12pt
A. F. Andreev, Sov. Phys.-JETP [**19**]{}, 1228 (1964). S.E. Barnes, J. Phys. F [**6**]{}, 1375 (1976). C.W.J. Beenakker, Phys. Rev. B [**46**]{}, 12841 (1992). C.W.J. Beenakker in [*Mesoscopic Quantum Physics*]{}, edited by E. Akkermans, G. Montambaux, and J.-L Pichard (North Holland, Amsterdam) 1995. L.S. Borkowski and P.J. Hirschfeld, J. Low Temp. Phys. [**96**]{}, 185 (1994), and references therein. N.R. Claughton, M. Leadbeater, and C.J. Lambert, J. Phys.: Condensed Matter [**7**]{}, 8757 (1995). A. Clerk, V. Ambegaokar and S. Hershfield, cond-mat/9901205. P. Coleman, Phys. Rev. B [**29**]{}, 3035 (1984). S.M. Cronenwett, T.H. Oosterkamp, and L.P. Kouwenhoven, Science [**281**]{}, 540 (1998). R. Fazio and R. Raimondi, Phys. Rev. Lett. [**80**]{}, 2913 (1998). L.I. Glazman and M.E. Raikh, JETP Lett. [**47**]{}, 452 (1988). D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abush-Magder, U. Meirav and M.A. Kastner, Nature [**391**]{}, 156 (1998). F.D. Haldane, Phys. Rev. Lett. [**40**]{}, 416 (1978). , Proceedings of the NATO ARW, F.W.J. Hekking, G. Schön, and D.V. Averin, eds., Physica B [**203**]{} (1994). M.H. Hettler and H. Schoeller, Phys. Rev. Lett. [**74**]{}, 4907 (1995). M.A. Kastner, Rev. Mod. Phys. [**64**]{}, 849 (1992). K. Kang, Phys. Rev. B [**58**]{}, 9641 (1998). C.J. Lambert and R. Raimondi, J. of Phys.: Condens. Matter [**10**]{}, 901-941 (1998). A. Levy Yeyati, A. Martin-Rodero, and F. Flores, Phys. Rev. Lett. [**71**]{}, 2991 (1993). Y. Meir, N.S. Wingreen, and P.A. Lee, Phys. Rev. Lett. [**66**]{}, 3048 (1991). Y. Meir and N.S. Wingreen, Phys. Rev. Lett. [**68**]{}, 2512 (1992). Y. Meir, N.S. Wingreen, and P.A. Lee, Phys. Rev. Lett. [**70**]{}, 2601 (1993). A. Millis and P.A. Lee, Phys. Rev. B [**35**]{}, 3394 (1987). T.K. Ng and P.A. Lee, Phys. Rev. Lett. [**61**]{}, 1768 (1988). T.K. Ng, Phys. Rev. Lett. [**70**]{}, 3635 (1993). T.K. Ng, Phys. Rev. Lett. [**76**]{}, 487 (1996). D.C. Ralph and R.A. Buhrman, Phys. Rev. Lett. [**72**]{}, 3401 (1994). N. Read and D.M. Newns, J. Phys. C: Solid State Phys. [**16**]{}, 3273 (1983); ibidem [**16**]{}, L1055 (1983). N. Read, J. Phys. C: Solid State Phys. [**18**]{}, 2651 (1985). P. Schwab and R. Raimondi, Phys. Rev. B [**59**]{}, 1637 (1999). H. Schoeller and G. Schön, Phys. Rev. B [**50**]{}, 18436 (1994). Q. Sun, J. Wang, and T. Lin, Phys. Rev. B [**59**]{}, 3831 (1999). Note that in (Fazio and Raimondi 1999), due to the overlooking of a factor of two in the interacting self-energy, the anomalous self-energy remained finite in the zero energy limit, leading to a completly different conclusion.
|
\#1 \#1
|
---
abstract: |
I analyze the hierarchy of large cardinals between a supercompact cardinal and an almost-huge cardinal. Many of these cardinals are defined by modifying the definition of a [high-jump]{} cardinal. A high-jump cardinal is defined as the critical point of an elementary embedding $j: V \to M$ such that $M$ is closed under sequences of length $\sup\set{j(f)(\kappa) \st f:
\kappa \to \kappa}$. Some of the other cardinals analyzed include the super-high-jump cardinals, almost-high-jump cardinals, Shelah-for-supercompactness cardinals, Woodin-for-supercompactness cardinals, cardinals, hypercompact cardinals, and enhanced supercompact cardinals. I organize these cardinals in terms of consistency strength and implicational strength. I also analyze the superstrong cardinals, which are weaker than supercompact cardinals but are related to high-jump cardinals. Two of my most important results are as follows.
- cardinals are the same as Woodin-for-supercompactness cardinals.
- There are no excessively hypercompact cardinals.
Furthermore, I prove some results relating high-jump cardinals to forcing, as well as analyzing Laver functions for super-high-jump cardinals.
author:
- Norman Lewis Perlmutter
bibliography:
- 'NormansBiblio.bib'
- 'WorksByNorman.bib'
date: 'Received: date / Accepted: date'
title: 'The large cardinals between supercompact and almost-huge'
---
Introduction
============
The main purpose of this paper is to examine the consistency and implicational strengths of several large cardinals falling between supercompact and almost-huge cardinals. Many of these cardinals are variants of the [high-jump]{} cardinals, which are described in definition \[definition.HJ\]. I will also investigate superstrong cardinals, which are weaker than supercompact cardinals but are closely related to [high-jump]{} cardinals. Many of the cardinals that I will discuss have been used by Apter, Hamkins, and Sargsyan to prove several results about universal indestructibility in [@ApterHamkins99:UniversalIndestructibility], [@ApterSargsyan2007:ReductionInCons(WoodinSC)], [@Apter2008:Reducing(EnhancedSC)], [@ApterSargsyan2010:UniversalIndestructibilty] and [@Apter2011:SomeApplicationsMethod(Hypercompact)]. This paper is adapted from the second chapter of my doctoral dissertation, [@Perlmutter2013:Dissertation].
Perhaps the most interesting result in this paper is the main result of section \[section.WoodinSC=Vopenka\], that a Woodin-for-supercompactness cardinal is equivalent to a cardinal. Another noteworthy result is that there are no excessively hypercompact cardinals, which is proven in section \[section.ExcessivelyHC\].
Recall that an almost-huge cardinal $\kappa$ is characterized by an elementary embedding $j: V \to
M$ with critical point $\kappa$ such that $M$ is closed under $<\!\!j(\kappa)$-sequences in $V$.[^1] Many of the large cardinals that I will discuss here are natural weakenings of an almost-huge cardinal, formed by reducing the level of closure of the target model. Indeed, in my study of these cardinals, a key methodology is to define new large cardinals by weakening, strengthening, or otherwise modifying existing large cardinal definitions. Often, the weaker large cardinals will still be sufficient for proving many of the same results as the stronger large cardinals. Eventually, by repeatedly weakening definitions, one hopes to obtain an equiconsistency, as is done in [@ApterSargsyan2010:UniversalIndestructibilty]. However, in this paper, I focus on the large cardinals themselves rather than their applications.
The chart at the end of the introduction summarizes the relationships between the large cardinals that I discuss in this paper. Most of the remaining sections will be dedicated to proving these relationships. The arrows on the chart represent relationships between the cardinals, as indicated in the key. A solid arrow from $A$ to $B$ indicates a direct implication: every cardinal with property $A$ has property $B$. A dotted arrow means that a cardinal of type $A$ is *strictly* stronger in consistency than a cardinal of type $B$. That is to say, if there is a cardinal of type A, then it is consistent with that there is a cardinal of type B. A double arrow indicates that both of these relationships hold. The arrows are labeled with theorem numbers referring to the theorems, propositions, and corollaries in which the corresponding results are proven. Dashed arrows are labeled with two numbers: one for a theorem demonstrating the consistency implication and one for a theorem demonstrating the failure of the direct implication.
Throughout the paper, I will use the following seed theory notation, which has been popularized by Hamkins, to refer to factor embeddings and the related measures.
Let $j: V \to M$ be an elementary embedding with critical point $\kappa$, and let $\lambda$ be a cardinal greater than $\kappa$. Let $U$ be a normal fine measure on $P_\kappa \lambda$ given by $A \in U \iff j \image \lambda \in j(A)$. Then $U$ is the normal fine measure on $P_\kappa \lambda$ **induced via $j$ by the seed $j \image \lambda$**, and the ultrapower embedding generated by $U$ is the $\lambda$-supercompactness factor embedding of $j$ induced by the seed $j \image
\lambda$.
The organization of the paper is as follows. The sections after section \[section.HJ\] can mostly be read out of order. I have noted the most important dependencies between the sections below. In section \[section.HJ\], I define the clearance of an elementary embedding and use this property to define the [high-jump]{} cardinals. I also define and analyze the related notions of [almost-high-jump]{} cardinals, Shelah-for-supercompactness cardinals, and [high-jump]{} functions. In section \[section.SuperstrongAndClearance\], which depends on section \[section.HJ\], I analyze properties of the clearance of an embedding and prove theorems tying together the ideas of the clearance of an embedding, the [almost-high-jump]{} cardinals, and the superstrong cardinals. The next few sections are arranged mostly by decreasing strength of the large cardinal notions studied. In section \[section.CardinalStrengths\], which depends on section \[section.HJ\] and on lemma \[lemma.VThetaElemMjkappa\], I define and analyze several large cardinals above a cardinal and below an almost-huge cardinal. In section \[section.WoodinSC=Vopenka\], I define the and Woodin-for-supercompactness cardinals and prove that they are equivalent. In section \[section.ExcessivelyHC\], I define the hypercompact cardinals and the excessively hypercompact cardinals, and I show that the existence of an excessively hypercompact cardinal is inconsistent with . In section \[section.EnhancedSC\], I define the enhanced supercompact cardinals and analyze their place in the large cardinal hierarchy. In section \[section.forcing\], which depends on section \[section.HJ\], I consider the relationship between high-jump cardinals and forcing. In section \[section.Laver\], which depends on section \[section.HJ\] and on lemma \[lemma.VThetaElemMjkappa\], I develop analogues of Laver functions for [high-jump]{} cardinals and related cardinals. In section \[section.FurtherDefinitionIdeas\], I review open problems and directions for further research.
I use the label *theorem* to denote very important results. The results labeled as *propositions* vary in their mathematical depth. Some of them might more appropriately be considered as examples or observations.
High-jump cardinals, [almost-high-jump]{} cardinals,\
and Shelah-for-supercompactness cardinals {#section.HJ}
=====================================================
In this section, I define [high-jump]{} cardinals, [almost-high-jump]{} cardinals, and Shelah-for-supercompactness cardinals.[^2] I also give characterizations for these large cardinals in terms of ultrafilters and prove a lemma about factor embeddings that will be very useful for the rest of the paper.
The clearance of an elementary embedding, defined below in definition \[definition.ClearanceHeightAndClosure\], is a key concept for defining several large cardinals. The motivation for defining the clearance is for use as a weaker substitute for $j(\kappa)$ in large cardinal definitions.
\[definition.ClearanceHeightAndClosure\] Let $j: M \to N$ be an elementary embedding with critical point $\kappa$. The **clearance** of $j$ denotes the ordinal $${ \sup\set{j(f)(\kappa) \st f: \kappa \to
\kappa}}.$$
The notation *[clearance]{}* is borrowed from the sport of pole vaulting, where the [clearance]{} is the height of the bar that the pole vaulter must clear. A high-jump embedding is like a pole vaulter: for a cardinal to be high jump, the closure of the embedding must successfully clear the [clearance]{}, as is described precisely in the following definition. The term *[high-jump]{} cardinal* comes from [@ApterHamkins99:UniversalIndestructibility]. However, these cardinals were previously defined in [@SolovayReinhardtKanamori1978:StrongAxiomsOfInfinity p.111], where they were given the designation $A_4$.
\[definition.HJ\] The cardinal $\kappa$ is a **high jump cardinal** if and only if there exists a cardinal $\theta$ and an elementary embedding $j: V \to M$ with critical point $\kappa$ and clearance $\theta$ such that $M^\theta \of M$.
An embedding witnessing that $\kappa$ is [high jump]{} is called a **[high-jump]{} embedding** for $\kappa$. A normal fine measure on some $P_\kappa \lambda$ generating an ultrapower embedding that is a [high-jump]{} embedding is called a **[high-jump]{} measure**.
The clearance of an embedding has strong properties, as I will show in the next section. In particular, I will show in corollary \[Theorem.M\_thetaElemInM\_j(kappa)\] that if $\theta$ is the clearance of *any* elementary embedding $j: V \to N$ with critical point $\kappa$, then $N_\theta \elem N_{j(\kappa)}$.
The following lemma provides a criterion for showing that a factor embedding of a high-jump embeddings is a high-jump embedding. It will be used many times throughout the paper.
\[lemma.FactorEmbeddingJump\] \[lemma.FactorEmbeddingEquality\] \[lemma.HJFactor\] Let $j: V \to M$ be a [high-jump]{} embedding for $\kappa$ with clearance $\theta$. Let $\lambda \geq \theta$ be a cardinal such that $j \image \lambda \in M$. Let $j_0: V \to M_0$ be the factor embedding induced via $j$ by the seed $j \image \lambda$. Then $j_0$ is a [high-jump]{} embedding, and the clearance of $j_0$ is $\theta$.
V & \^j & M &\
&\
M\_0 & &
Let $f: \kappa \to \kappa$ be any function. Referring to the diagram above, note that the critical point of $k$ is greater than $\lambda$. It follows that $$j(f)(\kappa) = (k\circ h)(f)(\kappa) =
\big(k (h(f))\big)(k(\kappa)) = k(h(f)(\kappa)).$$
The ordinal $j(f)(\kappa)$ is less than $\theta$, since $\theta$ is the clearance of $j$. Therefore, again since $\crit(k) > \lambda$, it must be the case that $h(f)(\kappa)
= j(f)(\kappa)$. Since $f$ was arbitrary, the embedding $j_0$ must have the same clearance as the embedding $j$, namely $\theta$. Since $\lambda \geq \theta$, it follows that $j_0$ is a [high-jump]{} embedding.
Next, I provide a combinatorial characterization of [high-jump]{} measures.
\[theorem.HighJumpMeasure\] Given an ordered pair of cardinals $(\kappa, \theta)$, the following are equivalent.
1. \[item.HJemb\] There exists a [high-jump]{} embedding $j: V \to M$ with critical point $\kappa$ such that $M^\theta \of M$ and the clearance of $j$ is at most $\theta$.
2. \[item.HJMeasureLosChar\] There exists a normal fine measure $U$ on $P_\kappa \theta$ such that for every function $f: \kappa \to \kappa$, the set $\set{A \in
P_\kappa
\theta \st f(\ot(A \cap \kappa)) < \ot(A)}$ is a member of $U$. (The operator $\ot$ denotes order type.)
The proof consists of a straightforward argument using the Theorem and lemma \[lemma.FactorEmbeddingEquality\]. For the details, see [@Perlmutter2013:Dissertation lemma 55].
Next, I define the [almost-high-jump]{} cardinals by a slight weakening of the closure property used for defining [high-jump]{} cardinals. An [almost-high-jump]{} cardinal is to a [high-jump]{} cardinal as an almost-huge cardinal is to a huge cardinal.
A cardinal $\kappa$ is **almost high jump** if and only if there exists an elementary embedding $j: V \to M$ with critical point $\kappa$ and clearance $\theta$ such that $M^{<\theta} \of M$. Such an embedding is called an [almost-high-jump]{} embedding for $\kappa$.
Another way to look at the definition of an [almost-high-jump]{} cardinal is as follows. The cardinal $\kappa$ is almost high jump if and only if there exists an elementary embedding $j: V
\to M$ with critical point $\kappa$ such that for every function $f: \kappa \to
\kappa$, the closure property $M^{j(f)(\kappa)} \of M$ holds. The [almost-high-jump]{} cardinals have a combinatorial characterization in terms of coherent sequences of normal fine measures. See [@Perlmutter2013:Dissertation lemma 57] for details.
Weakening the definition of an [almost-high-jump]{} cardinal to allow for distinct embeddings to witness closure with respect to distinct functions $f: \kappa
\to \kappa$ produces the definition of a Shelah-for-supercompactness cardinal. The analogue of this definition for strongness (in place of supercompactness) was originally formulated by Shelah.
A cardinal $\kappa$ is **Shelah for supercompactness** if and only if for every function $f: \kappa \to
\kappa$, there is an elementary embedding $j: V \to M$ such that $M^{j(f)(\kappa)} \of
M$.
Note that an [almost-high-jump]{} cardinal is a uniform version of a Shelah for supercompactness cardinal — with an [almost-high-jump]{} cardinal, one embedding must be the witness for every $f$ uniformly, whereas with a Shelah-for-supercompactness cardinal, each function $f$ may have a separate witnessing embedding.
One might want to define an almost-Shelah-for-supercompactness cardinal by tweaking the above definition to require that the closure of the target model is only $<\!\!j(f)(\kappa)$. However, this definition is actually equivalent to a Shelah-for-supercompactness cardinal, because of the following argument. Let $g: \kappa \to
\kappa$ be given by $g(\alpha) = f(\alpha)^+$. If $j: V \to M$ is an elementary embedding with critical point $\kappa$ such that $M^{<j(g)(\kappa)} \of M$, then $M^{j(f)(\kappa)} \of M$ as well.
In [@Hamkins98:AsYouLikeIt p.201], Hamkins defines a [high-jump]{} function as follows. A **[high-jump]{} function** for a (partially) supercompact cardinal $\kappa$ is a function $f \from \kappa \to \kappa$ such that $j(f)(\kappa) >
\lambda$ whenever $j$ is a $\lambda$-supercompactness embedding on $\kappa$. Hamkins allows for partial functions, but any partial [high-jump]{} function can be extended to a total [high-jump]{} function, so I will assume without loss of generality that [high-jump]{} functions are total. Furthermore, I will extend the definition of a [high-jump]{} function to the vacuous case where $\kappa$ has no supercompactness by saying that in this case, there exists a [high-jump]{} function for $\kappa.$ The following proposition shows that the existence of a [high-jump]{} function for a cardinal $\kappa$ is actually an anti-large-cardinal property.
\[proposition.HJfunction\] Let $\kappa$ be a cardinal. Then there exists a [high-jump]{} function for $\kappa$ if and only if $\kappa$ is not Shelah for supercompactness.
The proof follows immediately from the definitions. The cardinal $\kappa$ is Shelah for supercompactness if and only if $$(\forall f: \kappa \to \kappa) (\exists j: V
\to M \text{ with critical point $\kappa$) such that } M^{j(f)(\kappa)} \of M$$ The logical negation of this statement is $$(*) \qquad (\exists f: \kappa \to \kappa) (\forall j: V
\to M \text{ with critical point $\kappa$) } M^{j(f)(\kappa)} \nsubseteq M$$
The formula (\*) asserts that $f$ is a [high-jump]{} function for $\kappa$.[^3]
The Shelah-for-supercompactness cardinals have an ultrafilter characterization similar to that for high-jump cardinals, given by the following corollary to lemma \[theorem.HighJumpMeasure\].
A cardinal $\kappa$ is Shelah-for-supercompactness if and only if for every function $f: \kappa \to \kappa$, there is a cardinal $\theta$ and a normal fine measure $U$ on $P_\kappa
\theta$ such that the set $\set{A \in P_\kappa
\theta \st f(\ot(A \cap \kappa)) < \ot(A)}$ is a member of $U$.
The proof is very similar to that of lemma \[theorem.HighJumpMeasure\] and is given in [@Perlmutter2013:Dissertation corollary 60].
The clearance, superstrongness embeddings, and related embeddings {#section.SuperstrongAndClearance}
=================================================================
In the large cardinal literature, a cardinal $\kappa$ is **superstrong** if and only if there exists an elementary embedding $j: V \to M$ such that $V_{j(\kappa)} \of M$. A cardinal $\kappa$ is **almost huge** if and only if there exists an elementary embedding $j: V
\to M$ such that $M^{<j(\kappa)} \of M$. The chart in the introduction shows that an almost-huge cardinal is much stronger in consistency strength than a [high-jump]{} cardinal. Remarkably, the analogous situation does not hold in the case of strongness. In theorem \[theorem.SuperstrongCharacterization\], I will show that a superstrong cardinal is equivalent to a high-jump-for-strongness cardinal. Before proving this result, I will prove some facts about the clearance of an embedding and about [almost-high-jump]{} embeddings. I begin with the following lemma.
\[lemma.clearanceBasicsGeneralEmbedding\] \[lemma.CHnotAchieved\] Let $j: V \to M$ be an elementary embedding with critical point $\kappa$ and clearance $\theta$. Then the following conclusions are true.
- There is no function $f : \kappa \to \kappa$ such that $j(f)(\kappa) = \theta.$
- The ordinal $\theta$ is a $\beth$ fixed point in $M$, that is to say, $\beth_\theta^M = \theta$.
- The inequality $\kappa^+
\leq \cof(\theta)
\leq 2^\kappa$ holds in $V$.
To prove the first conclusion, suppose to the contrary that $f$ is a function such that $j(f)(\kappa) =
\theta.$ Let $g: \kappa \to \kappa$ be defined by $g(\alpha) = f(\alpha) + 1$. Then $j(g)(\kappa) = \theta +1 >
\theta$, contradicting the definition of the clearance.
Next, I will show that $\beth_\beta^M< \theta$ for all ordinals $\beta <
\theta$, so that $\beth_\theta^M = \theta$. Let $\beta < \theta$. Then there exists a function $f:
\kappa \to \kappa$ such that $j(f)(\kappa) \geq \beta.$ Let the function $g: \kappa \to \kappa$ be given by $g(\alpha) = \beth_{f(\alpha)}$. Then $\beth_\beta^M \leq j(g)(\kappa)
<
\theta$. It follows that $\theta$ is a $\beth$ fixed point in $M$.
The cofinality of the clearance $\theta$ must be at most $2^\kappa$, because the clearance is defined as the supremum of a set indexed by functions from $\kappa$ to $\kappa$, of which there are $2^\kappa$ many.
Finally, I show that $\cof(\theta) \geq \kappa^+$ by a diagonalization argument. Suppose to the contrary that $\cof(\theta) \leq
\kappa.$ Then there is a sequence $\<f_\alpha>_{\alpha<\kappa}$ of functions on $\kappa$ such that $\theta =
\sup\set{j(f_\alpha)(\kappa) \st \alpha < \kappa}.$ Define a function $g:
\kappa
\to
\kappa$ diagonalizing over these functions. That is to say, given $\beta<
\kappa$, let $g(\beta) = \sup\set{f_\alpha(\beta) + 1 \st \alpha \leq \beta} .$ Then $j(f_\alpha)(\kappa) < j(g)(\kappa) < \theta$ for every $\alpha < \kappa$, contradicting the assumption that $\theta = \sup\set{j(f_\alpha)(\kappa) \st
\alpha < \kappa}$.
The next lemma applies the result of lemma \[lemma.CHnotAchieved\] in the case that $j$ is an [almost-high-jump]{} embedding.
\[lemma.ThetaStrongLimit\] \[lemma.ClearanceFactsAHJEmbedding\] Suppose $j: V \to M$ is an [almost-high-jump]{} embedding with critical point $\kappa$ and clearance $\theta$. Then the following conclusions are true in both $V$ and $M$.
- The cardinal $\theta$ is a singular $\beth$ fixed point.
- The inequality $\kappa^+
\leq \cof(\theta)
\leq 2^\kappa$ holds.
- The cardinal exponentiation identity $\theta^{\kappa} = \theta$ holds.
The proof follows from lemma \[lemma.CHnotAchieved\], along with the fact that $M$ is sufficiently closed so that it agrees with $V$ on cofinalities less than $\theta$ and on cardinal exponentiation below $\theta$.
To show that $\theta^\kappa = \theta$ in both $V$ and $M$, note that $\theta$ is a strong limit in both $V$ and $M$ and $\cof(\theta) > \kappa$ in both $V$ and $M$. The fact that $\theta^{\kappa} = \theta$ in both $V$ and $M$ then follows from a basic theorem of cardinal arithmetic (see [@Jech:SetTheory3rdEdition theorem 5.20]).
With these preliminaries out of the way, I now state the main theorem of this section.
\[theorem.SuperstrongCharacterization\]
A cardinal $\kappa$ is high jump for strongness if and only if $\kappa$ is superstrong. This fact follows from the following stronger but more technical result.
Let $\kappa$ be a cardinal. Let $j: V \to M$ be a high-jump-for-strongness embedding with critical point $\kappa$ and clearance $\theta$. Then $V_\theta \elem M_{j(\kappa)}$, and $j$ has a factor embedding $h: V \to M'$ such that $h$ is a superstrongness embedding with critical point $\kappa$ and such tha $h(\kappa) = \theta$.
Let $j: V \to M$ be a [high-jump]{}-for-strongness embedding with critical point $\kappa$ and clearance $\theta$. I define the seed hull of $\theta$ in $M$, denoted by $X_{\theta}$, as follows.
$$X_{\theta} = \set{j(f)(\alpha) \st \alpha < \theta \text{ and } f \in V
\text{ is
a
function}}.$$ The seed hull $X_{\theta}$ is an elementary substructure of $M$, and setting $M'$ equal to its Mostowski collapse yields the following commutative diagram of elementary embeddings of models of set theory, where $k$ is the inverse of the collapse map.
V & \^j & M\
\^[h]{} & >k\
M’
Next, I will show that the critical point of $k$ is $\theta$, and $k(\theta) =
j(\kappa)$. Since $k$ is the inverse of the Mostowski collapse of $X_\theta$, it suffices to show that the supremum of the ordinals $\beta$ of $X_{\theta}$ below $j(\kappa)$ is $\theta$. Every such ordinal $\beta$ is of the form $j(f)(\alpha)$ for some ordinal $\alpha < \theta$ and some function $f: \kappa \to \kappa$. Fix such an ordinal $\alpha$ and function $f$. Since $\theta$ is the clearance of the embedding $j$, it follows that $\alpha < j(g)(\kappa)$ for some other function $g:\kappa \to \kappa$. Define yet another function $g':
\kappa \to
\kappa$ by $g'(\beta) = \sup\set{f(\gamma) \st \gamma < g(\beta)}$. By the elementarity of $j$, and since $\theta$ is the clearance of the embedding $j$, it follows that $$\label{doohickie}
j(g')(\kappa) = \sup\set{j(f)(\gamma) \st
\gamma < j(g)(\kappa)} < \theta$$ Considering the case $\gamma = \alpha$ in equation \[doohickie\] above, it follows that $j(f)(\alpha) < \theta$. It follows that $\theta$ is the critical point of $k$ and that $k(\theta) = j(\kappa),$ as claimed. Since the diagram above commutes, it further follows that $h(\kappa) = \theta$.
Next, I claim that $V_\theta \of M'$. Towards the proof of this claim, first recall that since $j$ is a [high-jump]{}-for-strongness embedding, $V_\theta =
M_\theta$. Next, let $f: \kappa
\to V_\kappa$ be an enumeration of $V_\kappa$ in $V$ such that whenever $\alpha < \kappa$, it follows that $V_\alpha \of f \image \beth_\alpha$. By lemma \[lemma.CHnotAchieved\], the ordinal $\theta$ is a $\beth$ fixed point in $M$, and so it follows from the definitions of $f$ and of $X_\theta$ that $M_\theta
\of
X_\theta$. Furthermore $M_\theta = V_\theta$, so $V_\theta \of X_\theta.$ Since $M'$ is the Mostowski collapse of $X_\theta$ in $M$, it follows that $V_\theta
\of M'$, as claimed.
Since $V_\theta \of M'$ and $h(\kappa) = \theta$, it follows that $h$ is a superstrongness embedding. Furthermore, the embedding $k$ witnesses that $V_\theta \elem M_{j(\kappa)}$.
A few easy corollaries to theorem \[theorem.SuperstrongCharacterization\] follow. I will label the first corollary as a lemma, because it is a key fact about [almost-high-jump]{} embeddings and will be used in many places in this paper.
\[lemma.VThetaElemMjkappa\] Let $j: V \to M$ be an [almost-high-jump]{} embedding for $\kappa$ with clearance $\theta$. Then $V_\theta \satisfies \ZFC$ and $V_\theta \elem
M_{j(\kappa)}$.
By lemma \[lemma.ThetaStrongLimit\], the clearance $\theta$ of $j$ is a $\beth$ fixed point. Therefore, since $j$ is a $\theta$-supercompactness embedding, it is also a $\theta$-strongness embedding, and thus a [high-jump]{}-for-strongness embedding. It follows immediately from theorem \[theorem.SuperstrongCharacterization\] that $V_\theta \elem
M_{j(\kappa)}$. Moreover, since $j(\kappa)$ is inaccessible in $M$, it follows that $V_\theta \satisfies \ZFC$.
\[Theorem.M\_thetaElemInM\_j(kappa)\] \[Corollary.N\_thetaElemInN\_j(kappa)\] Let $j: V \to M$ be an elementary embedding with critical point $\kappa$ and clearance $\theta$. Then $M_\theta \satisfies \ZFC$ and $M_\theta \elem
M_{j(\kappa)}$.
The same line of reasoning as in the proof of theorem \[theorem.SuperstrongCharacterization\] shows that $M_\theta \elem
M_{j(\kappa)}$, even without the assumption that the embedding $j$ has additional strength.
\[corollary.SuperstrongFactorofHJ\] Every [almost-high-jump]{} embedding has a superstrongness factor embedding.
This follows immediately from theorem \[theorem.SuperstrongCharacterization\], since every [almost-high-jump]{} embedding is also a [high-jump]{}-for-strongness embedding, as was shown in the proof of lemma \[lemma.VThetaElemMjkappa\].
As a closing observation, note that analogues of many of the results in this section can be proven when $V$ is replaced by a more general model, $N$.
Large cardinals strictly above a cardinal {#section.CardinalStrengths}
===========================================
In the next few sections, I define the remaining large cardinals mentioned in the chart from the introduction, and I prove results about their consistency and implicational strengths. The sections are organized in order of strength in the large cardinal hierarchy. In the present section, I consider cardinals stronger than a cardinal but no stronger than an almost-huge cardinal.
I begin by defining the large cardinal notions that I will be analyzing in this section, starting with the [high-jump]{} order and the [super-high-jump]{} cardinals. These definitions are somewhat analogous to the definitions of the many times huge and superhuge cardinals, which are defined in [@BarbanelDipriscoTan84:nHugeSuperhuge].
\[definition.shj\] Given an ordinal $\eta$, the cardinal $\kappa$ has **[high-jump order $\eta$]{}** if and only if there exists a strictly increasing sequence $\<
\theta_\alpha \st \alpha < \eta >$ of ordinals such that for each ordinal $\alpha < \eta$, there exists a [high-jump]{} embedding for $\kappa$ with [clearance]{} $\theta_\alpha$. The cardinal $\kappa$ is **super high jump** if and only if there exist [high-jump]{} embeddings for $\kappa$ of arbitrarily high clearance. (In other words, a [super-high-jump]{} cardinal $\kappa$ has [high-jump order $\ORD$]{}.)
The [almost-high-jump]{} order and the [super-almost-high-jump]{} cardinals are defined similarly to the [high-jump]{} order and the [super-high-jump]{} cardinals, as follows.
Given an ordinal $\eta$, the cardinal $\kappa$ has **almost-high-jump order** $\eta$ if and only if there exists a strictly increasing sequence $\<
\theta_\alpha \st \alpha < \eta >$ of cardinals such that for each ordinal $\alpha < \eta$, there exists an [almost-high-jump]{} embedding for $\kappa$ with [clearance]{} $\theta_\alpha$. The cardinal $\kappa$ is **super almost high jump** if and only if there exist [almost-high-jump]{} embeddings of arbitrarily high clearance for $\kappa$.
It will also be interesting to consider [high-jump]{} embeddings with **excess closure**, that is, embeddings $j: V \to M$ with clearance $\theta$ such that the target model $M$ is closed under sequences of length greater than $\theta$. For instance, [high-jump]{} embeddings with clearance $\theta$ where the target model is closed under sequences of length $2^\theta$ will be fruitful objects of study. An extreme example of excess closure is as follows.
\[definition.StronglySHJ\] The cardinal $\kappa$ is **[high jump]{} with unbounded excess closure** if and only if for some fixed clearance $\theta$, for all cardinals $\lambda \geq \theta$, there is a [high-jump]{} measure on $P_\kappa \lambda$ generating an embedding with clearance $\theta$.
With all of the above definitions given, the time has come to prove many of the simpler consistency strength relations shown on the chart in the introduction, along with some additional related consistency strength relations that are not shown on the chart.
I begin with the following proposition, which involves a [high-jump]{} embedding with a little bit of excess closure. This proposition is a simple example of the use of lemma \[lemma.VThetaElemMjkappa\], which will be used in more complicated arguments later.
\[proposition.HJplus\] Suppose that there exists a pair of cardinals $(\kappa, \theta)$ such that there is a [high-jump]{} embedding $j: V \to M$ with critical point $\kappa$ and clearance $\theta$ and such that $M^{2^\theta} \of M$. Then the cardinal $\kappa$ is [super high jump]{} in the model $V_\theta$, and the cardinal $\kappa$ has [high-jump]{} order $\theta$ in $V$. Furthermore, there are many [super-high-jump]{} cardinals in the models $V_\kappa$, $V_\theta$, and $M_{j(\kappa)}$.
By lemma \[lemma.HJFactor\], there is a factor embedding, $h$, of $j$ such that $h$ has clearance $\theta$ and is generated by a [high-jump]{} measure $U$ on $P_\kappa \theta$. By lemma \[lemma.ClearanceFactsAHJEmbedding\], the cardinal exponentiation identity $\theta^\kappa = \theta$ holds. It follows that the model $M$ is sufficiently closed so that $U \in M$.
In the model $M_{j(\kappa)}$, consider the set of cardinals $\lambda$ such that there is a [high-jump]{} measure generating an embedding with critical point $\kappa$ and clearance $\lambda$. By lemma \[lemma.VThetaElemMjkappa\], the elementarity relation $V_\theta \elem
M_{j(\kappa)}$ holds. It follows that if this set of cardinals is bounded in the model $M_{j(\kappa)}$, then this bound is below $\theta$. But $\theta$ is an element of this set, since $U \in M$. Therefore, the set is unbounded in both $V_\theta$ and $M_{j(\kappa)}$, and in particular, $\kappa$ is a [super-high-jump]{} cardinal in the model $M_{j(\kappa)}$. By reflection, there are many [super-high-jump]{} cardinals in the model $V_\kappa$. By the elementarity of $j$ and since $V_\theta \elem M_{j(\kappa)}$, it follows that there are also many [super-high-jump]{} cardinals in $M_{j(\kappa)}$ and in $V_\theta$. Finally, since $V_\theta \satisfies \ZFC$ and since every [high-jump]{} measure of $V_\theta$ is also a [high-jump]{} measure in $V$, it follows that the cardinal $\kappa$ has [high-jump]{} order $\theta$ in $V$.
In later similar consistency proofs, I will finish the proof with a conclusion about one of $M_{j(\kappa)}$, $V_\kappa$, or $V_\theta$, and leave it to the reader to work out the additional consequences in the other models. Note that the hypothesis of proposition \[proposition.HJplus\] is equivalent to the hypothesis that there for some pair $(\kappa, \theta)$, such that there is a [high-jump]{} measure on $P_\kappa 2^\theta$. This alternative hypothesis follows immediately from the hypothesis of proposition \[proposition.HJplus\]. For the converse, given a pair $(\kappa, \theta)$ such that there is a [high-jump]{} measure on $P_\kappa 2^\theta$, the clearance of the corresponding embedding must be at most $\theta$. If the clearance of this embedding is some $\theta'<
\theta$, then take a $2^{\theta'}$-supercompactness factor embedding and apply lemma \[lemma.HJFactor\].
Next, I will consider elementary embeddings for which the closure of the target model is extremely large compared with the clearance of the embedding, beginning with the [high-jump]{} cardinals with [unbounded excess closure]{}.
\[proposition.AlmostHuge>SSHJ\] Suppose the cardinal $\kappa$ is almost huge. Then in the model $V_\kappa$, there are many cardinals $\delta$ such that $\delta$ is [[high jump]{} with unbounded excess closure]{}
Suppose $\kappa$ is almost huge, witnessed by an elementary embedding $j: V \to M$ with clearance $\theta$. In particular, the embedding $j$ is also a [high-jump]{} embedding. Let $\lambda$ be a cardinal such that $\theta \leq \lambda < j(\kappa)$. The cardinal $j(\kappa)$ is a strong limit cardinal. Therefore, by lemma \[lemma.HJFactor\], the embedding $j$ has a $\lambda$-supercompactness factor embedding with clearance $\theta$ generated by a [high-jump]{} measure on $P_\kappa
\lambda$. This [high-jump]{} measure is an element of $M_{j(\kappa)}$.
Consider a cardinal $\kappa$ such that for all sufficiently large cardinals $\lambda$, there is a [high-jump]{} measure on $P_\kappa \lambda$. It may be possible that such a cardinal is not [[high jump]{} with unbounded excess closure]{}, because the [high-jump]{} measures may not all generate embeddings with the same closure. However, the following proposition shows that these two types of cardinals are equiconsistent.
\[proposition.SSHJEquiconsistentWithHJplusORD\] The following two large cardinal axioms are equiconsistent over .
1. \[item.HJplusORD\] There exists a cardinal $\kappa$ such that for all sufficiently large cardinals $\lambda$, there is a [high-jump]{} measure on $P_\kappa \lambda$.
2. \[item.SSHJ\] There exists a cardinal that is [[high jump]{} with unbounded excess closure]{}
In particular if there are [high-jump]{} measures on $P_\kappa \lambda$ for all sufficiently large cardinals $\lambda$, then either $\kappa$ is [[high jump]{} with unbounded excess closure]{} or else there is a cardinal $\theta$ such that $\kappa$ is [[high jump]{} with unbounded excess closure]{} in the model $V_\theta$.
It is immediate from the definitions that if $\kappa$ is [[high jump]{} with unbounded excess closure]{}, then for all sufficiently large $\lambda$, there is a [high-jump]{} measure on $P_\kappa \lambda$.
For the converse, suppose that for all sufficiently large $\lambda$, there is a [high-jump]{} measure on $P_\kappa \lambda$, but the cardinal $\kappa$ is not [[high jump]{} with unbounded excess closure]{}. Let $\theta_0$ be the minimal cardinal such that for all cardinals $\lambda
\geq
\theta_0$, there is a [high-jump]{} measure on $P_\kappa \lambda$. Since the cardinal $\kappa$ is not [[high jump]{} with unbounded excess closure]{}, these [high-jump]{} measures do not all generate embeddings with clearance $\theta_0$.
None of these measures generates a [high-jump]{} embedding with clearance less than $\theta_0$. If it did, then the minimality of $\theta_0$ would be contradicted by taking factor embeddings and applying lemma \[lemma.HJFactor\].
Accordingly, let $\theta_1$ be the least cardinal above $\theta_0$ such that there is a [high-jump]{} embedding for $\kappa$ with clearance $\theta_1$. Let $j: V \to M$ be a [high-jump]{} embedding for $\kappa$ with clearance $\theta_1$. Then the model $V_{\theta_1}$ satisfies by lemma \[lemma.VThetaElemMjkappa\], and in this model, the cardinal $\kappa$ is [[high jump]{} with unbounded excess closure]{} with respect to the clearance $\theta_0$.
Next, proposition \[proposition.ClosureHierarchy\] shows that the degrees of excess closure of [high-jump]{} embeddings form a hierarchy of consistency strength. In this hierarchy, there are many more cardinals above the ones described in proposition \[proposition.ClosureHierarchy\] and below the [high-jump]{} cardinals with unbounded excess closure. For further details, see [@Perlmutter2013:Dissertation pp. 117-118].
\[proposition.ClosureHierarchy\] Suppose that for some cardinals $\kappa$ and $\theta$ and for some ordinal $\alpha< \theta$, there exists a [high-jump]{} embedding $j: V \to M$ with critical point $\kappa$ and clearance $\theta$ such that the model $M$ is closed under sequences of length $2^{\aleph_{\theta+\alpha}^{<\kappa}}.$ Then in the model $M_{j(\kappa)}$, there are unboundedly many cardinals $\lambda$ such that there is a [high-jump]{} measure on $P_\kappa (\aleph_{\lambda + \alpha})$ generating a [high-jump]{} embedding with critical point $\kappa$ and clearance $\lambda$.
In the model $M_{j(\kappa)}$, consider the set of cardinals $\lambda$ such that there is a [high-jump]{} measure on $P_\kappa (\aleph_{\lambda + \alpha})$ generating a [high-jump]{} embedding with critical point $\kappa$ and clearance $\lambda$. The model $M_{j(\kappa)}$ is sufficiently closed to see that $\theta$ is an element of this set. By lemma \[lemma.VThetaElemMjkappa\], the elementarity relation $V_\theta \elem M_{j(\kappa)}$ holds, so it follows that this set is unbounded in $M_{j(\kappa)}$.
Next, I move on to prove some results lower down in the hierarchy of [high-jump]{} cardinals and related cardinals.
\[proposition.HJorder\] \[proposition.HJplusOrder\] Let $\eta$ and $\eta'$ be ordinals such that $\eta < \eta'$. Suppose the cardinal $\kappa$ has [high-jump]{} order $\eta'$. Then there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that the cardinal $\kappa$ has [high-jump]{} order $\eta$ in $M_{j(\kappa)}$.
The cardinal $\kappa$ has [high-jump]{} order $\eta'$, and this is witnessed by a sequence of clearances $\langle \theta_\alpha \st {\alpha<\eta'\rangle}$. Let $j: V \to M$ be a [high-jump]{} embedding for $\kappa$ with clearance $\theta$ for some $\theta_\alpha$ such that $\alpha \geq \eta$. Then the model $M$ is sufficiently closed so that in $M_{j(\kappa)}$, the cardinal $\kappa$ has [high-jump]{} order $\eta$.
I now move further down the large cardinal hierarchy, to the [almost-high-jump]{} cardinals. Recall from the introduction that [almost-high-jump]{} cardinals are characterized by combinatorially by coherent sequences of normal measures, which are described in detail in [@Perlmutter2013:Dissertation lemma 57].
\[proposition.HJ>SAHJ\] Suppose there is a [high-jump]{} embedding with critical point $\kappa$ and clearance $\theta$. Then $\kappa$ has [almost-high-jump]{} order $\theta$, and in the models $V_\theta$, $M_{j(\kappa)}$, and $V_\kappa$, there are many [super-almost-high-jump]{} cardinals.
Suppose $j: V \to M$ is a [high-jump]{} embedding with critical point $\kappa$ and clearance $\theta$. It follows immediately from definitions that the embedding $j$ also witnesses that $\kappa$ is almost high jump. By corollary \[lemma.ThetaStrongLimit\], the cardinal $\theta$ is a strong limit, and so it follows that the coherent sequence of measures witnessing that there is an [almost-high-jump]{} embedding for $\kappa$ with clearance $\theta$ is an element of $H_{\theta^+}$. This coherent sequence of measures is also an element of $M$, by the closure of $M$. Therefore, the cardinal $\kappa$ is almost high jump in $M$ with clearance $\theta$. Consider the set $\set{\delta \st M_{j(\kappa)} \satisfies
\kappa \text{
almost high jump with clearance }\delta}.$ By theorem \[Theorem.M\_thetaElemInM\_j(kappa)\], if this set has a bound in $M_{j(\kappa)}$, then the bound must be less than $\theta$. It follows that the set is unbounded in $M_{j(\kappa)}$, and so $\kappa$ is super almost high jump in the model $M_{j(\kappa)}$. The other conclusions are immediate.
\[corollary.AHJorder\] Let $\eta < \eta'$ be ordinals. Suppose the cardinal $\kappa$ has [almost-high-jump]{} order $\eta'$. Then there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that the cardinal $\kappa$ is has [almost-high-jump]{} order $\eta$ in $M_{j(\kappa)}$.
The proof follows the same reasoning as the proof of proposition \[proposition.HJorder\], replacing [high-jump]{} embeddings with [almost-high-jump]{} embeddings and [high-jump]{} measures with coherent sequences of measures.
Finally, I reach the Shelah-for-supercompactness cardinals.
\[proposition.AHJ>SSC\] Suppose the cardinal $\kappa$ is [almost-high-jump]{}. Then there are many cardinals below $\kappa$ that are Shelah for supercompactness.
Suppose $j: V \to M$ is an [almost-high-jump]{} embedding for $\kappa$ with clearance $\theta$. I will show that $\kappa$ is Shelah for supercompactness in $M$. Let $f: \kappa \to \kappa$ be a function in $M$. Let $j_0: V \to M_0$ be the $\lambda$-supercompactness factor embedding induced by $j$ via the seed $j \image \lambda$, where $\lambda$ is the maximum of $j(f)(\kappa)$ and $\kappa$. From corollary \[lemma.ThetaStrongLimit\], it follows that $2^{\lambda^{<\kappa}} < \theta$. Therefore, the $\lambda$-supercompactness measure $U$ that generates $j_0$ is an element of $M$. By reasoning similar to the proof of lemma \[lemma.FactorEmbeddingEquality\], it follows that $j(f)(\kappa) =
j_0(f)(\kappa)$. Let $j^M_0$ be the elementary embedding generated by $U$ in $M$. The measure $U$ is an element of $V_\theta = M_\theta$, which satisfies $\ZFC$ by theorem \[Theorem.M\_thetaElemInM\_j(kappa)\]. It follows that $j_0 \restrict V_\theta = j_0^M \restrict
M_\theta$. Therefore, in $M$, the elementary embedding $j_0^M$ witnesses that $\kappa$ is Shelah for supercompactness with respect to the function $f$. Since $f$ was arbitrary, it follows that $\kappa$ is Shelah for supercompactness in $M$.
I wind up the section with a few miscellaneous propositions. Proposition \[proposition.LeastHJnotStr\] shows that several direct implications are lacking from the large cardinal hierarchy.
\[proposition.LeastHJnotStr\] The least [high-jump]{} cardinal is not $\Sigma_2$-reflecting. In particular, it is not supercompact and not even strong. The same is true for the least almost-huge cardinal, the least [almost-high-jump]{} cardinal, and the least Shelah-for-supercompactness cardinal.
All of these cardinals can be characterized by $\Sigma_2$ definitions — they are characterized by a measure or a set of measures with certain combinatorial properties, all of which can be seen from within a particular $V_\alpha$. Since supercompact and strong cardinals are $\Sigma_2$-reflecting, the theorem follows.
The proofs of propositions \[lemma.AHJStrongerThanSC\] and \[proposition.HJO2\] are simple and are left as exercises for the reader. The complete proofs can be found in [@Perlmutter2013:Dissertation propositions 81 and 83]
\[lemma.AHJStrongerThanSC\] \[proposition.AHJStrongerThanSC\] Suppose $j: V \to M$ is an elementary embedding with clearance $\theta$ witnessing that $\kappa$ is [almost-high-jump]{}. Then in the model $V_\kappa$ there are many supercompact cardinals.
The definition of super high jump makes it tempting to think that every cardinal that is both supercompact and high jump is super high jump. However, the following simple proposition shows that this is not the case.
\[proposition.HJO2\] If $\kappa$ is the least cardinal that has [high-jump order $2$]{}, then in $V_\kappa$, there are many cardinals that are both supercompact and high jump but not super high jump.
Proposition \[proposition.HJO2\] shows that below a cardinal of [high-jump]{} order $2$, there are many cardinals that are both [high jump]{} and tall. If a cardinal $\kappa$ is both [high jump]{} and tall, then there are [high-jump]{} embeddings for $\kappa$ such that $j(\kappa)$ is arbitrarily large — this can be seen by taking a [high-jump]{} embedding for $\kappa$ followed by a tallness embedding for $j(\kappa)$. It is easy to see that below a cardinal that is both [high-jump]{} and supercompact, there are many [high-jump]{} cardinals. But it is an open question whether the existence of a cardinal that is both [high-jump]{} and tall is equiconsistent with the existence of a [high-jump]{} cardinal.
Many more definitions could be made along the lines of the ones given in this section, and these definitions would lead to many more questions of consistency strength. In light of proposition \[proposition.HJplus\], the following question comes to mind.
\[question.ClosureTheta+\] What is the consistency strength of the existence of a [high-jump]{} embedding with critical point $\kappa$ and clearance $\theta$ generated by a [high-jump]{} measure on $P_\kappa \theta^+$? Is the existence of such an embedding equiconsistent with the existence of a [high-jump]{} embedding with critical point $\kappa$ and clearance $\theta$ generated by a [high-jump]{} measure on $P_\kappa 2^{\theta}$?
Of course, under , these two types of embeddings are equivalent.
The equivalence of cardinals and Woodin-for-supercompactness cardinals {#section.WoodinSC=Vopenka}
=======================================================================
In this section, I define a Woodin-for-supercompactness cardinal and show that it is strictly weaker than a Shelah-for-supercompactness cardinal. I then show that a cardinal is Woodin for supercompactness if and only if it is a cardinal.
The definition of a Woodin-for-supercompactness cardinal is as follows. This definition is taken from . These cardinals have also been studied by Foreman [@Foreman07:GenericElemEmb3 p.31] and by Fuchs [@Fuchs09:CombinedMaximality p.1043] under the name of Woodinized supercompact cardinals. A Woodin-for-supercompactness cardinal is the analogue of a Woodin cardinal, with supercompactness in place of strongness.
\[definition.WSC\] A cardinal $\delta$ is **Woodin for supercompactness** if and only if for every function $f: \delta
\to \delta$, there exists a cardinal $\kappa < \delta$ such that $\kappa$ is a closure point of $f$ (i.e. $f \image
\kappa \of \kappa$), and there exists an elementary embedding $j:
V \to M$ with critical point $\kappa$ such that $M^{j(f)(\kappa)} \of M$.
Note in particular that every Woodin-for-supercompactness cardinal is also a Woodin cardinal.
Apter and Sargsyan require that $f$ be defined so that $f(\alpha)$ is always a cardinal and so that the elementary embedding $j$ is generated by a supercompactness measure on $P_\kappa \lambda$ for some $\lambda <
\delta$, but this definition is equivalent to the definition given here. The first requirement does not make their definition any weaker, because any function $f$ can be replaced with a function $f'$ given by $f'(\alpha) = |f(\alpha)|^+$. Theorem \[theorem.WSCcharacterization\] below shows that the second requirement does not make their definition any stronger.
The next theorem shows that a Shelah-for-supercompactness cardinal is strictly stronger than a Woodin-for-supercompactness cardinal. The proof is an improvement on lemma 1.1 of [@ApterSargsyan2007:ReductionInCons(WoodinSC)] and uses a similar line of reasoning.
\[proposition.ShelahSCStrongerThanWoodinSC\] \[theorem.ShelahSCStrongerThanWoodinSC\] Suppose the cardinal $\kappa$ is Shelah for supercompactness. Then $\kappa$ is Woodin for supercompactness, and there are many cardinals below $\kappa$ that are Woodin for supercompactness in both $V_\kappa$ and $V$.
Let $\kappa$ be Shelah for supercompactness. The main difficulty is to show that $\kappa$ is Woodin for supercompactness. Once this has been shown, it is immediate that there are many cardinals below $\kappa$ that are Woodin for supercompactness in the model $V_\kappa$, because “The cardinal $\kappa$ is Woodin for supercompactness” is $\Pi_1^1$-definable over $V_\kappa$, and Shelah-for-supercompactness cardinals are $\Pi_1^1$-indescribable, since they are weakly compact. Furthermore, it is easily seen that any cardinal that is Woodin for supercompactness in the model $V_\kappa$ must also be Woodin for supercompactness in $V$.
Towards showing that $\kappa$ is Woodin for supercompactness, let $f: \kappa
\to
\kappa$ be an arbitrary function. I will show that $\kappa$ satisfies the definition of Woodin for supercompactness with respect to $f$. Without loss of generality, I assume that $f$ is nowhere regressive, that is to say, for all $\alpha <
\kappa$ the inequality $\alpha \leq f(\alpha)$ holds. It follows that $\kappa
\leq j(f)(\kappa).$ Let $g: \kappa \to \kappa$ be given by $g(\alpha) =
\scexp{f(\alpha)}{\alpha}$. Let $j:V \to M$ witness that $\kappa$ is Shelah for supercompactness with respect to the function $g$, that is, the embedding $j$ has critical point $\kappa$ and $M^{j(g)(\kappa)} \of M$. Note that $j(g)(\kappa) = \scexp
{(j(f)(\kappa))}{\kappa}$, as calculated in both $M$ and $V$.
Let $U$ be the normal fine measure on $P_\kappa \big(j(f)(\kappa)\big)$ induced via $j$ by the seed $j(f)(\kappa)$. Let $j_U: V \to N$ be the $j(f)(\kappa)$-supercompactness factor embedding induced by $U$. Let $k: N
\to M$ be the elementary embedding such that $k \circ h = j$. The model $M$ is sufficiently closed so that $U \in M$, and the closure of $M$ further guarantees that every function from $P_\kappa\big(j(f)(\kappa)\big)$ to $M$ is an element of $M$. It follows that the elementary embedding induced by $U$ in $M$, as calculated in $M$, is equal to $j_U \restrict M$, as calculated in $V$.
V & \^j & M & \^[j\_U M]{} & N’\
&\
N & &\
& &
Next, I will show that $$\begin{aligned}
\label{thingie}
j(f)(\kappa) = j_U(j(f))(\kappa)\end{aligned}$$
First of all, $f = j(f) \restrict \kappa$. Applying $j_U$ to both sides, it follows that $j_U(f) = j_U(j(f)) \restrict j_U(\kappa),$ and in particular, $j_U(f)(\kappa) = j_U(j(f))(\kappa).$ Furthermore, by reasoning similar to the proof of lemma \[lemma.FactorEmbeddingEquality\], it follows that $j(f)(\kappa) = j_U(f)(\kappa)$, and so statement \[thingie\] is proven.
The embedding $j_U \restrict M$ is generated in $M$ by the measure $U \in M$. Therefore, in $M$, the cardinal $j(\kappa)$ satisfies the definition of Woodin for supercompactness with respect to the function $j(f)$ — this fact is witnessed by the embedding $j_U \restrict M$ along with statement \[thingie\], since $\kappa$ is a closure point of $j(f)$. It follows from the elementarity of $j$ that in $V$, the cardinal $\kappa$ satisfies the definition of Woodin for supercompactness with respect to the function $f$. But $f$ was chosen arbitrarily, so $\kappa$ is Woodin for supercompactness in $V$.
Like Woodin cardinals, Woodin-for-supercompactness cardinals have several alternative characterizations, as described below in theorem \[theorem.WSCcharacterization\]. The proof that these characterizations hold is essentially the same as for the case of Woodin cardinals, (see [@Kanamori:TheHigherInfinite2ed Theorem 26.14]). In order to state them, I need the following definition of $(\gamma, A)$-supercompactness.
Given a set $A$ and a cardinal $\gamma$, the cardinal $\kappa$ is $\mathbf{( \boldsymbol{\gamma}, A)}$**-supercompact** if and only if there is an elementary embedding $j: V \to M$ with critical point $\kappa$ such that
1. $\gamma < j(\kappa)$,
2. $M^\gamma \of M$, and
3. $A \cap V_{\gamma} = j(A) \cap
V_{\gamma}.$
Given cardinals $\kappa$ and $\delta$ such that $\kappa < \delta$, the notation *$\kappa$ is $(<\!\!\delta,
A)$-supercompact* denotes that $\kappa$ is $(\gamma, A)$-supercompact for all cardinals $\gamma < \delta$. The alternative characterizations of Woodin-for-supercompactness cardinals are as follows.
\[theorem.WSCcharacterization\] Given a cardinal $\delta$, the following are equivalent.
1. \[hypothesis.WSCstrict\]
For every function $f: \delta
\to \delta$, there exists a cardinal $\kappa < \delta$ such that $\kappa$ is a closure point of $f$ (i.e. $f \image
\kappa \of \kappa$), and there exists an elementary embedding $h:
V \to M$ with critical point $\kappa$ such that $M^{h(f)(\kappa)} \of M$. Furthermore, the embedding $h$ is generated by a normal fine measure on $P_\kappa \lambda$ for some cardinal $\lambda < \delta$.
2. \[hypothesis.WSC\] The cardinal $\delta$ is Woodin for supercompactness. That is to say, for every function $f: \delta
\to \delta$, there exists a cardinal $\kappa < \delta$ such that $\kappa$ is a closure point of $f$ (i.e. $f \image
\kappa \of \kappa$), and there exists an elementary embedding $j:
V \to M$ with critical point $\kappa$ such that $M^{j(f)(\kappa)} \of M$.
3. \[hypothesis.ManyASC\] For every set $A \of
V_\delta$, the set $$\set{\kappa< \delta \st \kappa \text{ is }
(<\!\!\delta, A) \text{-supercompact}}$$ is stationary in $\delta$.
4. \[hypothesis.OneASC\] For every set of ordinals $A \of
\delta$, the set $$\set{\kappa< \delta \st \kappa \text{ is }
(<\!\!\delta, A) \text{-supercompact}}$$ is nonempty.
This is the analogue of a standard theorem for Woodin cardinals, and the proof is essentially the same as for that theorem. See [@Perlmutter2013:Dissertation theorem 88] for details.
I now shift my attention to the cardinals, which I will eventually show are equivalent to the Woodin-for-supercompactness cardinals.
\[definition.Vopenka\] The cardinal $\delta$ is **** if and only if for every $\delta$-sequence of model-theoretic structures $\langle M_\alpha \st \alpha< \delta
\rangle$ over the same language, with each structure $M_\alpha$ an element of $V_\delta$, there exists an elementary embedding $j: M_\alpha \to M_\beta$ for some ordinals $\alpha < \beta < \delta$.
The following result is well-established in the large cardinal literature.
\[proposition.LeastVopenkaNotWC\] \[proposition.LeastWoodinForSCNotMeas\] The least cardinal that is is not weakly compact.
“The cardinal $\kappa$ is ” is definable by a $\Pi_1^1$ formula over $V_\kappa$, but weakly compact cardinals are $\Pi_1^1$-indescribable.
It is convenient to have a characterization of cardinals in terms of a more limited class of model-theoretic structures. Towards this end, following [@Kanamori:TheHigherInfinite2ed], I make the following definition.
Let $\delta$ be a cardinal. A sequence of model-theoretic structures, $\<
M_\alpha \st \alpha < \delta>$ is a **natural $\delta$-sequence** if and only if the following properties are satisfied. There is a function $f: \delta \to \delta$ such that the domain of $M_\alpha$ is $V_{f(\alpha)}$ and such that whenever $\alpha < \beta < \delta$, are ordinals, it follows that $\alpha < f(\alpha)
\leq f(\beta) < \delta$. Furthermore, each structure $M_\alpha$ is of the form $(V_{f(\alpha)}, \in, \singleton{\alpha}, R_\alpha)$, for some unary relation $R_\alpha \of V_{f(\alpha)}$.
Without loss of generality, $R_\alpha$ may be taken to encode finitely many constants, functions, and relations of any finite arity on $V_{f(\alpha)}.$ I will show in proposition \[proposition.NaturalSequence\] below that it suffices to consider only natural sequences when determining whether the cardinal $\kappa$ is .
It is immediately clear that any sequence of the type specified in definition \[definition.Vopenka\] can be encoded as a natural sequence if the language involved is countable. For larger languages, one might be concerned that the critical point of the embedding would be small enough to mess up the encoding.
To clear up this concern, I will discuss the critical point of an elementary embedding between two elements of a natural sequence. This analysis will also play an important role in the proof of the equivalence between cardinals and Woodin-for-supercompactness cardinals. The inclusion of the constant $\singleton{\alpha}$ in the definition of a natural sequence ensures that any elementary embedding between members of a natural sequence has a critical point.
I begin the analysis of these critical points with the definition of the filter, given below in definition \[definition.VopenkaFilter\]. The filter is actually a filter on $\delta$ if and only if $\delta$ is a cardinal. In this case, the filter is a normal filter and contains every club.[^4] These facts are proven in [@Kanamori:TheHigherInfinite2ed pp.336-337] and [@SolovayReinhardtKanamori1978:StrongAxiomsOfInfinity proposition 6.3].
\[definition.VopenkaFilter\] Let $\delta$ be an inaccessible cardinal. Then the set $X \of \delta$ is a member of the ** filter** on $\delta$ if and only if there exists a natural $\delta$-sequence $\<M_\alpha \st \alpha < \delta>$ such that whenever $j: M_\alpha \to M_\beta$ is an elementary embedding, then the critical point of $j$ is an element of $X$.
Given an elementary embedding $j: V_\alpha \to V_\beta$, the critical point of $j$ must be inaccessible, and so if $\delta$ is a cardinal, then the set of inaccessible cardinals below $\delta$ is a member of the filter on $\delta$. In particular, this implies that $\delta$ is Mahlo, since the filter contains every club.
The next proposition states that it suffices to consider only natural sequences in defining cardinals.
\[proposition.NaturalSequence\] The cardinal $\delta$ is if and only if for every normal $\delta$-sequence $\< M_\alpha \st \alpha < \delta>$, there is an elementary embedding $j: M_\alpha \to M_\beta$ for some ordinals $\alpha, \beta < \delta$.
The forwards direction is immediate. For the converse, the essential observation is that the filter on $\delta$ contains every tail, so that after encoding a sequence of model-theoretic structures as a natural sequence, it is possible to choose an embedding with critical point much larger than the size of the language of the original structures, so that the embedding does not interfere with the encoding.
Kanamori suggests the equivalence of a Woodin-for-supercompactness cardinal to a cardinal in [@Kanamori:TheHigherInfinite2ed p.364]. However, he does not formally define a Woodin-for-supercompactness cardinal, nor does he work out the details of the equivalence.
\[theorem.WSC=Vopenka\] The cardinal $\delta$ is Woodin for supercompactness if and only if $\delta$ is a cardinal. Furthermore, if $\delta$ is a cardinal, then for every set $A
\of
V_\delta$, the set $\set{\kappa < \delta \st \kappa \text{ is $(<\!\!\delta,
A)$-supercompact}}$ is a member of the filter on $\delta$.
For the forward direction, let $\delta$ be Woodin for supercompactness, and let $A = \langle A_\alpha \st \alpha < \delta \rangle$ be a natural $\delta$-sequence. I will show that for some ordinals $\alpha <
\beta < \delta$, there exists an elementary embedding $j:
A_\alpha \to A_\beta.$
Since $\delta$ is Woodin for supercompactness, there is a cardinal $\kappa < \delta$ such that $\kappa$ is $(<\!\!\delta,
A)$-supercompact. Therefore, by choosing a large enough degree of $A$-supercompactness, there is an elementary embedding $j: V \to N$ such that $j(A)_\kappa = A_\kappa$ and $j \restrict A_\kappa \in N$. In $N$, the map $j
\restrict A_\kappa$ is an elementary embedding from $j(A)_\kappa$ to $j(A)_{j(\kappa)}$. So in $N$, there exists an elementary embedding between two elements of $j(A)$. By the elementarity of $j$, there exists an elementary embedding between two elements of $A$ in $V$. It follows that $\delta$ is .
The proof of the converse direction uses some of the same ideas as the proof of proposition 24.14 of [@Kanamori:TheHigherInfinite2ed], which shows that $V_\delta$ contains many extendible cardinals if $\delta$ is . Suppose that the cardinal $\delta$ is , and let $A \of V_\delta$. I will show that there exists a cardinal $\kappa < \delta$ such that $\kappa$ is $(<\!\!\delta,
A)$-supercompact, thereby showing that the cardinal $\delta$ is Woodin for supercompactness. Indeed, I will show that the set of such $\kappa$ is an element of the filter on $\delta$.
Let $g:\delta \to \delta$ be the variation of the failure-of-$A$-supercompactness function described as follows. Given $\xi<
\delta$, let $g(\xi)$ be the least cardinal $\eta > \xi$ such that $\xi$ is not $(\eta, A)$-supercompact. In case no such $\eta$ exists, then set $g(\xi) = \xi$.
Let $C \of \delta$ be the club of closure points of $g$, i.e. $C =
\set{\rho < \delta \st g \image \rho \of \rho}$. Since the filter on $\delta$ contains every club, it follows that the club $C$ is a member of this filer. Therefore, there exists a natural $\delta$-sequence $\<M_\alpha \st \alpha < \delta>$ such that whenever $j: M_\alpha
\to M_\beta$ is an elementary embedding, the critical point of $j$ is an element of $C$.
For each ordinal $\alpha < \delta$, let $\gamma_\alpha$ be the least inaccessible element of $C$ above all the ordinals of $M_\alpha$, and for each ordinal $\alpha < \delta$, let $$N_\alpha = ( V_{\gamma_\alpha}, \in, \singleton{\alpha}, M_\alpha, C \cap
\gamma_\alpha, A \cap
V_{\gamma_\alpha}).$$ Let $j:
N_\alpha \to N_\beta$ be an elementary embedding. It suffices to show that the critical point, $\kappa$, of $j$ is $(<\!\!\delta, A)$-supercompact.
Assume to the contrary that $\kappa$ is not $(<\!\!\delta, A)$-supercompact. Then $\kappa< g(\kappa)$. Furthermore, $g(\kappa) < \gamma_\alpha$, because $\gamma_\alpha \in C$. Since $M_\alpha$ is encoded in $N_\alpha$, it follows from the definition of the sequence $\<M_\alpha>$ that $\kappa \in C$. By the elementarity of $j$, it follows that $j(\kappa) \in C$ as well.
Let $U$ be the normal fine measure on $P_\kappa \big(g(\kappa)\big)$ induced via $j$ by the seed $j \image g(\kappa)$, and let $j_U: V \to N$ be the ultrapower generated by $U$. Using the fact that $\gamma_\alpha$ is inaccessible, the theory of factor embeddings shows that there exists an elementary embedding $k$ so that the following diagram commutes.
V\_[\_]{} & \^j & V\_[j(\_)]{}\
\^[j\_U V\_[\_]{}]{} & \_k &\
N V\_[\_]{} & &\
I claim that the map $j_U : V \to N$ witnesses that $\kappa$ is $(
g(\kappa), A)$-supercompact. This will contradict the definition of $g$, thereby completing the proof. Clearly, this map is a $g(\kappa)$-supercompactness embedding with critical point $\kappa$, and so it suffices to show that $j_U(A)
\cap V_{g(\kappa)} = A \cap V_{g(\kappa)}.$
First of all, since $A \cap V_{\gamma_\alpha}$ is encoded in $N_\alpha$, it follows that $j(A) \cap V_{g(\kappa)} = A \cap V_{g(\kappa)}$. Since the critical point of $k$ is an inaccessible cardinal above $g(\kappa)$, it follows that $j(A)
\cap
V_{g(\kappa)} = j_U(A) \cap
V_{g(\kappa)}$.
There are no excessively hypercompact cardinals. {#section.ExcessivelyHC}
================================================
In definition 1.2 of [@Apter2011:SomeApplicationsMethod(Hypercompact)], Apter defined an excessively hypercompact cardinal as follows.[^5]
\[Apter, [@Apter2011:SomeApplicationsMethod(Hypercompact)]\] \[definition.StronglyHC\] A cardinal $\kappa$ is excessively $0$-hypercompact iff $\kappa$ is supercompact. For $\alpha>0$, a cardinal $\kappa$ is excessively $\alpha$-hypercompact iff for any cardinal $\delta \geq
\kappa$, there is an elementary embedding $j: V \to M$ witnessing the $\delta$-supercompactness of $\kappa$ (i.e. cp$(j)
=
\kappa, j(\kappa) > \delta,$ and $M^\delta \of M$) generated by a supercompact ultrafilter over $P_\kappa(\delta)$ such that $M \satisfies$ “$\kappa$ is excessively $\beta$-hypercompact for every $\beta<\alpha$”. A cardinal $\kappa$ is **excessively hypercompact** iff $\kappa$ is excessively $\alpha$-hypercompact for every ordinal $\alpha$.
Postulating the existence of an excessively hypercompact cardinal leads to a contradiction.
\[theorem.StrongHypercompactDNE\] There are no excessively hypercompact cardinals. In particular, there is no cardinal $\kappa$ such that $\kappa$ is excessively $(2^\kappa)^+$-hypercompact.
Suppose towards a contradiction that $\kappa$ is least such that $\kappa$ is excessively $(2^\kappa)^+$-hypercompact. Apply the definition of excessive hypercompactness in the case $\delta=\kappa$ to obtain an elementary embedding $j:V \to M$ which is witnessed by a normal fine measure on $P_\kappa \kappa$ (which is isomorphic to a normal measure on $\kappa$) such that $\kappa$ is excessively $\alpha$-hypercompact in $M$ for all $\alpha < \big((2^\kappa)^+\big)^V$. This includes all $\alpha<j(\kappa),$ since $j(\kappa)$ has cardinality $2^\kappa$ in $V$. In particular, it includes the case of $\alpha = \big((2^\kappa)^+\big)^M$ , since this is less than $j(\kappa).$ By reflection, there are many $\gamma < \kappa$ such that $\gamma$ is excessively $(2^\gamma)^+$-hypercompact, and this contradicts the minimality of $\kappa.$
In definition \[definition.hypercompact\], I describe a hypercompact cardinal. Apter had erroneously believed that this definition was equivalent to the definition of an excessively hypercompact cardinal.[^6] However, the existence of a hypercompact cardinal is strictly weaker in consistency strength than the existence of a Woodin-for-supercompactness cardinal; I prove this fact in theorem \[theorem.WSC>WHC\]. The proofs in [@Apter2011:SomeApplicationsMethod(Hypercompact)] all work using hypercompact cardinals in place of excessively hypercompact cardinals, so the error in the definition given in that paper did not have severe consequences.[^7]
\[definition.hypercompact\] The hypercompact cardinals are defined recursively as follows. Given any ordinal $\alpha$, the cardinal $\kappa$ is $\alpha$-hypercompact if and only if for every ordinal $\beta < \alpha$ and for every cardinal $\lambda \geq \kappa$, there exists a cardinal $\lambda' \geq \lambda$ and there exists an elementary embedding $j: V \to M$ generated by a normal fine measure on $P_\kappa \lambda'$ such that the cardinal $\kappa$ is $\beta$-hypercompact in $M$. (In particular, every cardinal is $0$-hypercompact, and $1$-hypercompact is equivalent to supercompact.) The cardinal $\kappa$ is **hypercompact** if and only if it is $\beta$-hypercompact for every ordinal $\beta$.
The key difference between the definitions of hypercompact and excessively hypercompact is that in the definition of hypercompact, the embedding $j$ need not be witnessed by a normal fine measure on $P_\kappa \lambda$, but can be witnessed instead by a larger supercompactness measure.[^8]
Note that both the hypercompact cardinals and the excessively hypercompact cardinals are first-order definable in . Formally, the definition of a hypercompact cardinal is by recursion on $\kappa$ as follows. Assuming recursively that the set $$\text{HC}_{<\kappa} := \set{(\alpha, \eta) \st \eta \text{ is }
\alpha\text{-hypercompact and } \eta < \kappa}$$ is already defined, define that $\kappa$ is $\alpha$-hypercompact if and only if for every $\beta < \alpha$ and for every $\lambda \geq \kappa$ there exists $\lambda' \geq \lambda$ and there exists an elementary embedding $j: V \to M$ generated by a normal fine measure on $P_\kappa \lambda'$ such that $(\beta, \kappa) \in j(H_{<\kappa})$. This in turn can be stated formally as a first-order proposition using the Loś theorem, without referring explicitly to the embedding $j$.
I now establish the consistency of a hypercompact cardinal relative to a Woodin-for-supercompactness cardinal. The bold part of the proof emphasizes why the proof would not work to establish the consistency of an excessively hypercompact cardinal.
\[theorem.WSC>WHC\] If the cardinal $\delta$ is Woodin for supercompactness, then in the model $V_\delta$, there is a proper class of hypercompact cardinals.
Suppose $\delta$ is Woodin for supercompactness. Suppose towards a contradiction that the hypercompact cardinals of $V_\delta$ are bounded above by some cardinal $\eta$. Let the function $f: \delta \to
\delta$ be the failure-of-hypercompactness function as defined in the model $V_\delta$. That is to say, for an ordinal $\xi< \delta$, let $f(\xi)$ be the least ordinal $\beta$ such that $\xi$ is not $\beta$-hypercompact in $V_\delta$ if such a $\beta$ exists, and let $f(\xi) = 0$ otherwise.
By theorem \[theorem.WSCcharacterization\], there is a $(<\delta, f)$-supercompact cardinal $\kappa$ above $\eta$, and this fact is witnessed by a collection of elementary embeddings $j_\gamma : V \to M_\gamma$ for $\gamma < \delta$. (The subscripted $\gamma$ serves to index the target model, not to refer to a rank-initial cut thereof. ) **If $\gamma$ is taken to be sufficiently large,** then $(\kappa, f(\kappa)) \in j_\gamma(f)$, and so $j_\gamma(f)(\kappa) = f(\kappa)$. That is to say, in $M_\gamma$, the cardinal $\kappa$ is $\beta$-hypercompact for every $\beta < f(\kappa)$. By taking a factor embedding if necessary, assume that $j_\gamma$ is generated by a normal fine measure $U$ on $P_\kappa \gamma$ such that $U \in V_\delta$. Thus in $V_\delta$, the collection of embeddings $\<j_\gamma>$ witness that $\kappa$ is $f(\kappa)$-hypercompact, contradicting the definition of $f$. This contradiction completes the proof.
Finally, I consider the extent to which the hierarchy of $\beta$-hypercompactness and the hierarchy of excessive $\beta$-hypercompactness coincide for particular small values of $\beta$.
\[theorem.StrongVsWeakHypercompact\] Let $\kappa$ be a cardinal, and let $\beta \leq \kappa^+$ be an ordinal. If $\kappa$ is $\beta$-hypercompact, then for every ordinal $\alpha < \beta$ and for every cardinal $\lambda
\geq \kappa$, there is an elementary embedding $j: V \to M$ generated by a normal fine measure on $P_\kappa \lambda$ such that $\kappa$ is $\alpha$-hypercompact in $M$. Thus, the $\beta$-hypercompactness and excessive $\beta$-hypercompactness hierarchies align below $\kappa^+$.[^9]
The proof is by induction on ordinals $\beta$. Suppose that the cardinal $\kappa$ is $\beta$-hypercompact and that the theorem is true for all $\beta' < \beta$. Let $\lambda \geq \kappa$ be a cardinal, and let $\alpha < \beta$. It suffices to show that there is an elementary embedding $j: V \to M$ generated by a normal fine measure in $V$ on $P_\kappa \lambda$ such that in $M$, the cardinal $\kappa$ is $\alpha$-hypercompact.
By hypothesis, the cardinal $\kappa$ is $\beta$-hypercompact. So for some cardinal $\theta \geq \lambda$, there exists an elementary embedding $j: V \to M$ such that in $M$, the cardinal $\kappa$ is $\alpha$-hypercompact.
Let $j_\lambda : V \to M_\lambda$ be the $\lambda$-supercompactness factor embedding induced via $j$ by the seed $j
\image \lambda$, and let $k: M_\lambda \to M$ be the elementary embedding such that $k \circ j_\lambda = j$, as in the following commutative diagram. To be precise, the embedding $j_\lambda$ is the ultrapower generated by $U_\lambda$, where $U_\lambda$ is the normal fine measure on $P_\kappa
\lambda$ given by $A \in U \iff j \image \lambda \in j(A)$. (The subscript $\lambda$ in $M_\lambda$ serves to index the model $M_\lambda$, not to denote a level of its cumulative hierarchy.)
V & \^j & M\
&\
M\_& &
If $M_{\lambda} = M$, then the existence of the embedding $j_\lambda$ suffices to complete the proof. If $M_\lambda \neq M$, then the elementary embedding $k$ must be nontrivial, and its critical point must be an greater than $\kappa$ and inaccessible in $M_\lambda$. The model $M_\lambda$ agrees with $V$ on $\kappa^+$, so this critical point must be greater than $\kappa^+$. Therefore, it follows from the elementarity of $k$ that $\kappa$ is $\alpha$-hypercompact in $M_\lambda$.
Enhanced supercompact cardinals {#section.EnhancedSC}
===============================
In this brief section, I analyze the consistency strength of an enhanced supercompact cardinal. The definition of an enhanced supercompact cardinal comes from Apter’s paper, [@Apter2008:Reducing(EnhancedSC)].
A cardinal $\kappa$ is **enhanced supercompact** if and only if there exists a strong cardinal $\theta > \kappa$ such that for every cardinal $\lambda > \theta$, there exists a $\lambda$-supercompactness embedding $j: V \to M$ such that $\theta$ is strong in $M$.
Apter required that the embedding $j$ be generated by a normal fine measure on $P_\kappa
\lambda,$. This requirement provides a first-order characterization, but it adds no strength, because one can take a factor embedding.
This next theorem shows that a Woodin-for-supercompactness cardinal is strictly stronger in consistency than an enhanced supercompact cardinal.
\[proposition.WoodinSC>ESC\] \[theorem.WoodinSC>ESC\] Suppose the cardinal $\delta$ is Woodin for supercompactness. Then there are unboundedly many cardinals $\kappa < \delta$ such that $\kappa$ is a limit of cardinals $\eta$ such that there exists an inaccessible cardinal $\beta$ such that $\eta <
\beta < \kappa$, and $$V_\beta \satisfies \eta \text{ is enhanced
supercompact. }$$
The proof follows the same general line of reasoning as theorem 5 of [@Apter2008:Reducing(EnhancedSC)]. Suppose $\delta$ is Woodin for supercompactness. Let $f: \delta \to
\delta$ be given by taking $f(\alpha)$ to be the second strong cardinal of $V_\delta$ greater than $\alpha$. This function is well-defined, since the strong cardinals of $V_\delta$ are unbounded, since $\delta$ is Woodin.
Let $\kappa$ be a closure point of $f$, and let $j:
V \to M$ be an elementary embedding such that $M^{j(f)(\kappa)} \of M$ and $j(f)(\kappa)<
\delta$, i.e. the embedding $j$ witnesses that $\delta$ is Woodin for supercompactness with respect to the function $f$. By theorem \[theorem.WSCcharacterization\], assume without loss of generality that the embedding $j$ is generated by a normal fine measure on $P_\kappa \lambda$ for some cardinal $\lambda < \delta$. It follows that $j(\delta) = \delta$. By the definition of $f$ and the elementarity of $j$, there is a cardinal $\kappa_0$ such that $\kappa < \kappa_0 <
j(f)(\kappa)$, and the cardinal $\kappa_0$ is strong in the model $M_{j(\delta)} = M_\delta$, and furthermore, the cardinal $j(f)(\kappa)$ is strong in the model $M_{\delta}$.[^10]
For each cardinal $\lambda$ such that $\kappa_0 < \lambda <
j(f)(\kappa)$, let $U_\lambda$ be the normal fine measure on $P_\kappa \lambda$ given by $A \in U \iff j \image \lambda \in j(A)$. Let $j_\lambda: V \to
M_\lambda$ be the $\lambda$-supercompactness embedding generated by $U_\lambda$, and let $i: M_\lambda \to M$ be the elementary embedding such that $i \circ j_\lambda = j$. (The subscripted $\lambda$ serves to index the model $M_\lambda$, not to denote a level of its cumulative hierarchy.)
V & \^j & M\
&\
M\_& &
Suppose towards a contradiction that for some cardinal $\gamma$ with $\kappa_0 <
\gamma <
j(f)(\kappa),$ and for some cardinal $\lambda$ such that $\kappa_0 < \lambda <
j(f)(\kappa)$, $$M_\lambda \satisfies \kappa_0 \text{ is not
$\gamma$-strong}.$$
Then by elementarity, $$M \satisfies i(\kappa_0) \text{ is not
$i(\gamma)$-strong}.$$ But $i$ fixes $\kappa_0$, and so this contradicts the fact that the cardinal $\kappa_0$ is strong in $M_\delta$, since $i(\gamma) <
i(j(f)(\kappa)) \leq
j(j(f)(\kappa)) < \delta$. From this contradiction, I conclude that for all cardinals $\gamma$ and $\lambda$, if $\kappa_0 < \gamma <
j(f)(\kappa)$ and $\kappa_0 < \lambda <
j(f)(\kappa)$ then $$M_\lambda \satisfies \kappa_0 \text{ is
$\gamma$-strong}.$$
Finally, from the closure of $M$, it follows that $U_\lambda \in
M_{j(f)(\kappa)}$ for each cardinal $\lambda$ such that $\kappa_0 < \lambda < j(f)(\kappa)$. Furthermore, for each such cardinal $\lambda$, the elementary embedding generated by $U_\lambda$ in the model $M$ is equal to $j_\lambda \restrict M$. Since $\lambda$ was taken to be an arbitrary cardinal between $\kappa_0$ and $j(f)(\kappa)$, it follows that in the model $M_{j(f)(\kappa)}$, the cardinal $\kappa$ is enhanced supercompact.
By reflection, in $V_\kappa$, there are unboundedly many cardinals $\eta$ such that for some inaccessible cardinal $\beta$ with $\eta <
\beta <
\kappa$, $$V_\beta \satisfies \eta \text{ is enhanced supercompact. }$$
By a simple modification to the function $f$, the cardinal $\kappa$ can be made arbitrarily large below $\delta$. The conclusion of the theorem follows.
High-jump cardinals and forcing {#section.forcing}
===============================
In this section, I prove some results about the preservation and destruction of [high-jump]{} cardinals by forcing.
Suppose $j: V \to M$ is a [high-jump]{} embedding, and $V[G]$ is a forcing extension of $V$. Under what conditions does $j$ lift to a [high-jump]{} embedding $j^* : V[G] \to
M[H]$? The conditions under which a supercompactness embedding lifts to a supercompactness embedding have been well-studied in the literature. The following lemma extends these conditions to provide conditions for which a [high-jump]{} embedding lifts to a [high-jump]{} embedding.
\[lemma.HJPreservation\] Suppose $j: V \to M$ is a [high-jump]{} embedding for $\kappa$ with clearance $\theta$. Let $V[G]$ be a forcing extension of $V$, and suppose that $j$ lifts to a $\theta$-supercompactness embedding $j^*: V[G] \to M[H]$.[^11] Let $U$ be the normal measure on $\kappa$ given by $A \in U \iff
\kappa \in j(A)$. If the family of functions $(\kappa^\kappa)^V$ is $\leq_U$-unbounded in $(\kappa^\kappa)^{V[G]},$ then the lifted embedding $j^*$ is a [high-jump]{} embedding. Furthermore, if $M[H]^{\theta^+} \nsubseteq M[H]$ in $V[G]$, then the conclusion can be strengthened to a biconditional: the lifted embedding $j^*$ is a [high-jump]{} embedding if and only if the family of functions $(\kappa^\kappa)^V$ is $\leq_U$-unbounded in $(\kappa^\kappa)^{V[G]}.$
Note that since $U$ is an ultrafilter, the family of functions $(\kappa^\kappa)^V$ is $\leq_U$-unbounded in $(\kappa^\kappa)^{V[G]}$ if and only this family is a dominating family, which is true if and only if the forcing does not add a $U$-dominating function.
To prove the first part of the theorem, assume that the family of functions $(\kappa^\kappa)^V$ is $\leq_U$-unbounded in $(\kappa^\kappa)^{V[G]}.$ In $V[G]$, let $f: \kappa \to
\kappa$. It suffices to show that $j^*(f)(\kappa) < \theta$. Since $(\kappa^\kappa)^V$ is a dominating family, there is a function $g \in (\kappa^\kappa)^V$ such that $f \leq_U g$. It follows that $$j(f)(\kappa) \leq j(g)(\kappa) < \theta,$$ and so the lifted embedding is a [high-jump]{} embedding.
To prove the second part of the theorem, suppose that $M[H]^{\theta+}
\nsubseteq
M[H]$ in $V[G]$, and that $(\kappa^\kappa)^V$ is $\leq_U$-bounded by some function $g \in (\kappa^\kappa)^{V[G]}.$ Then $j^*(g)(\kappa) \geq
j^*(f)(\kappa)$ for every $f \in (\kappa^\kappa)^V$, and so in particular $j^*(g)(\kappa) \geq
\theta$, and so the function $g$ witnesses that $j^*$ is not a [high-jump]{} embedding.
One particular important instance where the class of functions $(\kappa^\kappa)^V$ is unbounded in $(\kappa^\kappa)^{V[G]}$ is if the forcing satisfies the $\kappa$-chain condition.
The biconditional version of the lemma actually holds even holds in many cases where $M[H]$ is closed under sequences of length greater than $\theta$ — given a $g$ such that $j^*(g)(\kappa) \geq \theta$, one can easily modify the function $g$ to produce another function $h$ such that $j^*(h)(\kappa)$ is much larger than $\theta$. For instance, let $h(\alpha)$ be the least measurable cardinal above $g(\alpha)$, so that $j^*(h)(\kappa)$ is the least measurable cardinal of $M[H]$ above $\theta$.
The next theorem addresses the preservation of [high-jump]{} cardinals in the downwards direction.
\[theorem.HJapproxcover\] Suppose $V \of \overline{V}$ satisfies the $\delta$ approximation and cover properties, and for some cardinals $\kappa, \theta > \delta$ there is a [high-jump]{} measure $U$ on $P_\kappa {\theta}$ in $\overline{V}$. Then there is a [high-jump]{} measure on $P_\kappa {\theta}$ in $V$ as well.
Let $j: \overline{V} \to \overline{N}$ be the elementary embedding generated by $U$ in $\overline{V}$. By the proof of corollary 26 of [@Hamkins2003:ExtensionsWithApproximationAndCoverProperties], the restricted embedding $j \restrict V: V \to N$ is amenable with $V$, and $N^\theta \of N$ in $V$. In particular, $j \restrict V$ is a [high-jump]{} embedding. Let $j_0: V \to M$ be the $\theta$-supercompactness factor embedding induced via $j\restrict V$ by the seed $j
\image \theta$. Let $f: \kappa \to \kappa$ be a function. It follows from lemma \[lemma.FactorEmbeddingJump\] applied in $V$ to the embedding $j
\restrict V$ that $j_0$ is a [high-jump]{} embedding. Furthermore, the factor embedding construction ensures that $j_0$ is generated by a measure that is an element of $V$, so the proof is complete.
Next, I show that the previous two results together prove the analogue of the Levy-Solovay theorem for [high-jump]{} cardinals.
\[theorem.HJPreservation\] Let $\P$ be a forcing notion such that $|\P| < \kappa$. Let $G \of \P$ be $V$-generic. Then in $V[G]$, the cardinal $\kappa$ is [high jump]{} if and only if $\kappa$ is [high jump]{} in $V$.
Since the forcing is small, in particular it satisfies the $\kappa$-chain condition, and so every function $f: \kappa \to \kappa$ in $V[G]$ is bounded by such a function in $V$. Thus, the upwards direction of the proof follows from lemma \[lemma.HJPreservation\]. By lemma 13 of [@Hamkins2003:ExtensionsWithApproximationAndCoverProperties], the forcing $\P$ satisfies the $\delta$ approximation and cover properties for some cardinal $\delta < \kappa$. Thus, the downwards direction of the proof follows immediately from theorem \[theorem.HJapproxcover\].
Next, I apply lemma \[lemma.HJPreservation\] to show that the canonical forcing of the preserves [high-jump]{} cardinals.
\[theorem.hjGCH\] Every [high-jump]{} cardinal is preserved by the canonical forcing $\P$ of the . To be precise, the forcing $\P$ is defined as the Easton support product over all infinite cardinals $\delta$ of $\Add(\delta^+, 1)$.
Let $G \of \P$ be $V$-generic. In $V$, let $U$ be a [high-jump]{} measure on $P_\kappa \theta$ for some cardinals $\kappa$ and $\theta$. Let $j_U$ be the [high-jump]{} embedding generated by $U$. It follows from a standard argument that the embedding $j_U$ lifts to a $\theta$-supercompactness embedding $j_U^*: V[G] \to M[H]$.[^12] To complete the proof that [high-jump]{} cardinals are preserved by $\P$, it suffices to show that every function on $f: \kappa \to \kappa$ in $V[G]$ is dominated by such a function in $V$ and then apply lemma \[lemma.HJPreservation\]. Towards this end, note that the forcing $\P$ factors as $\P_{<\!\kappa}\ast \P_{\geq \kappa}$. The first factor satisfies the $\kappa$-chain condition, and so every function $f: \kappa \to \kappa$ added by it is dominated by a ground model function. The second factor is $\leq\!\kappa$-closed, and so it adds no new function $f:
\kappa \to \kappa$.
High-jump cardinals are in general much more fragile than supercompact cardinals, as is shown by the following theorem.
\[theorem.HJFragility\] Let $\kappa$ be a [high-jump]{} cardinal. After forcing with $\Add(\kappa, 1)$ or with $Add(\kappa^+, 1)$, the cardinal $\kappa$ is no longer a [high-jump]{} cardinal.
A recent theorem of Bagaria, Hamkins, and Tsaprounis shows that superstrong cardinals are destroyed by these forcings, among others [@BagariaHamkinsTsaprounis:SuperstrongFragility]. By corollary \[corollary.SuperstrongFactorofHJ\], every [high-jump]{} cardinal is also superstrong, so it follows that these forcings also destroy [high-jump]{} cardinals.
Finally, I show that if the cardinal $\kappa$ is [high jump]{}, then there is a forcing extension where $\kappa$ is still [high jump]{} but is not supercompact.
\[theorem.HJButNotSC\] Suppose there exists a [high-jump]{} measure on $P_\kappa \theta$, and furthermore, the cardinal $\kappa$ is supercompact. Let $\P$ be any forcing smaller than $\kappa$. Let $g \of \P$ be $V$-generic. Let $\Q$ be any nontrivial forcing that is $\leq
\!\!\scexp{\theta}{\kappa}$-closed in $V[g]$, and let $G \of \Q$ be $V[g]$-generic. Then in $V[g][G]$, there is still a [high-jump]{} measure on $P_\kappa
\theta$, but the cardinal $\kappa$ is not supercompact.
Since the forcing $\P$ is small relative to $\kappa$, the cardinal $\kappa$ is still both supercompact and [high jump]{} in $V[g]$. Because of the closure condition on the forcing $\Q$, this forcing does not add any subsets or elements to $P_\kappa
\theta$, nor does it add any new functions $f: \kappa \to \kappa$. Therefore, in $V[g][G]$, the cardinal $\kappa$ is still [high jump]{}. However, since the forcing $\Q$ is nontrivial, there is a cardinal $\lambda > \scexp{\theta}{\kappa}$ such that $\Q$ adds a subset to $\lambda$. By a theorem of Hamkins and Shelah ([@HamkinsShelah98:Dual p.551]), the cardinal $\kappa$ is no longer $\lambda$-supercompact in $V[g][G]$.
Some open questions on the topics of this section are as follows.
\[question.HJcardinalFragility\] Suppose $j: V \to M$ is a [high-jump]{} embedding for $\kappa$ with clearance $\theta$. What types of forcing, if any, preserve the $\theta$-supercompactness of $\kappa$ while destroying the [high-jump]{} cardinal property of $\kappa$?
Question \[question.HJcardinalFragility\] can be further refined to a question about individual embeddings rather than about cardinals.
\[question.HJEmbeddingfragility\] Let $j: V \to M$ be a [high-jump]{} embedding for $\kappa$ generated by a [high-jump]{} measure. Let $\P$ be a forcing notion, and suppose that the embedding $j$ lifts over $\P$ such that the lift is a supercompactness embedding. Under what conditions does the lift fail to be a [high-jump]{} embedding?
Laver functions for [high-jump]{} cardinals {#section.Laver}
===========================================
In this section, I define Laver functions for [super-high-jump]{}cardinals and establish their existence under suitably strong hypotheses. Laver functions were originally defined for supercompact cardinals in [@Laver78].
Given a supercompact cardinal $\kappa$, a supercompactness Laver function for $\kappa$ is a partial function $\ell\from \kappa \to V_\kappa$ such that for every cardinal $\lambda$ and for every set $x \in H_{\lambda^+}$ there is a $\lambda$-supercompactness embedding generated by a normal fine measure on $P_\kappa
\lambda$ such that $j(\ell)(\kappa) = x$. One can also put additional requirements on the domain of a supercompactness Laver function. For instance, one can require that each $\gamma \in \dom(\ell)$ is an inaccessible cardinal such that the closure property $\ell \image \gamma \of V_\gamma$ is satisfied. A [super-high-jump]{} Laver function is defined similarly to a supercompactness Laver function, as follows.
Given a [super-high-jump]{} cardinal $\kappa$, a **[super-high-jump]{} Laver function** for $\kappa$ is a partial function $\ell \from \kappa \to V_\kappa$ satisfying the following properties. For every set $x$, for unboundedly many cardinals $\delta$, there is [high-jump]{} embedding with critical point $\kappa$ and clearance $\delta$, generated by a [high-jump]{} measure, such that $j(\ell)(\kappa) = x$. Furthermore, for every ordinal $\gamma \in \dom(\ell)$, the closure property $\ell \image \gamma \of V_\gamma$ holds.
I will prove that every supercompact cardinal has a supercompactness Laver function anticipating every set, and whenever the cardinal $\kappa$ is $2^{\theta^{<\kappa}}$-supercompact, there is a supercompactness Laver function for $\kappa$ anticipating every set in $H_\theta^+$.
The analysis for [high-jump]{} cardinals is more complicated than in the case of supercompact cardinals, because a supercompactness factor embedding of a [high-jump]{} embedding is not in general a [high-jump]{} embedding. For this reason, the [high-jump]{} cardinals with excess closure are a useful tool. As a warm-up exercise before reading the proof of the existence of [super-high-jump]{} Laver functions, the reader may wish to review proposition \[proposition.HJplus\], which uses a related technique.
\[theorem.SHJLaverFunction\] Let $\kappa$ be a cardinal. Then there exists a partial function $\ell \from
\kappa \to V_\kappa$ such that for all cardinals $\theta$, if there is a [high-jump]{} measure on $P_\kappa 2^\theta$ generating an ultrapower embedding with clearance $\theta$, then in the model $V_\theta$, the function $\ell$ is a [super-high-jump]{} Laver function for $\kappa$.
Define the function $\ell$ recursively as follows. Suppose that $\ell
\restrict \gamma$ has been defined. Define $\ell(\gamma)$ as described in the next paragraph if the relevant hypotheses hold. Otherwise, leave $\gamma$ out of the domain of $\ell$.
Suppose that $\ell \image \gamma \of V_\gamma$ and that furthermore, in the model $V_\kappa$, some set $x$ witnesses that the function $\ell \restrict
\gamma$ is not a [super-high-jump]{} Laver function for $\gamma$. *That is to say, in the model $V_\kappa$, there is a cardinal $\delta_0$ such that for all cardinals $\delta
>
\delta_0$, there is no elementary embedding $j: V \to M$ with critical point $\gamma$ and clearance $\delta$, generated by a [high-jump]{} measure, such that $j(\ell\restrict \gamma)(\gamma) = x$.* Then pick a set $x
\in
V_\kappa$ of minimal $\in$-rank among all sets with this property, and let $\ell(\gamma) = x$.
I now verify that the function $\ell$ has the desired feature. Suppose not. Then there is some cardinal $\theta$ such that there is a [high-jump]{} measure $\mu$ on $P_\kappa
2^\theta$ generating an ultrapower embedding with clearance $\theta$, but in the model $V_\theta$, some set $x$ witnesses that the function $\ell$ fails to be a [super-high-jump]{} Laver function for $\kappa$. Let $j: V \to M$ be the ultrapower generated by $\mu$. By lemma \[lemma.VThetaElemMjkappa\], the elementarity relation $V_\theta \elem M_{j(\kappa)}$ holds. Therefore, in the model $M_{j(\kappa)}$, the set $x$ witnesses that the function $\ell$ is not a [super-high-jump]{} Laver function for $\kappa$. That is to say, in the model $M_{j(\kappa)}$, there is some cardinal $\delta_0$ such that for all cardinals $\delta
> \delta_0$, there does not exist a [high-jump]{}embedding $h$, generated by a [high-jump]{} measure, with critical point $\kappa$ and clearance $\delta$, such that $h(\ell)(\kappa) = x$.
Accordingly, since $j(\ell)
\restrict \kappa = \ell$, it follows from the definition of the function $\ell$ and from the elementarity of the embedding $j$ that $j(\ell)(\kappa)$ is defined and equal to some set $y \in M_{j(\kappa)}$ such that in the model $M_{j(\kappa)}$, the set $y$ witnesses that the function $\ell$ is not a [super-high-jump]{} Laver function for $\kappa$. Furthermore, this set $y$ is of minimal $\in$-rank, and so $y \in V_\theta$, since $V_\theta \elem M_{j(\kappa)}$.
Let $U$ be the $\theta$-supercompactness measure on $P_\kappa \theta$ induced by $j$ via the seed $j \image
\theta$. Let $j_U: V \to N$ be the supercompactness embedding generated by $U$, and let the elementary embedding $k$ be such that the diagram below commutes.
V & \^j & M\
&\
N & &
By reasoning similar to the proof of lemma \[lemma.HJFactor\], the measure $U$ is a [high-jump]{} measure, the clearance of $j_U$ is $\theta$, and $j_U(\ell)(\kappa) = y$. Furthermore, since the model $M_{j(\kappa)}$ is closed in $V$ under sequences of length $2^{\theta}$, and since $\theta^\kappa = \theta$ by lemma \[lemma.ClearanceFactsAHJEmbedding\], it follows that $U \in
M_{j(\kappa)}$, and the model $M_{j(\kappa)}$ correctly computes that $j_U(\ell)(\kappa) = y$. (In the previous sentence, $j_U$ denotes the embedding generated by the measure $U$ in the model $M_{j(\kappa)}$.) However, $\theta > \delta_0$, so this computation contradicts the fact that $y$ is not anticipated in $M_{j(\kappa)}$ by $\ell$ with respect to any [high-jump]{} embedding with clearance greater than $\delta_0$ that is generated by a [high-jump]{} measure.
\[corollary.SHJLaverFunction\] Suppose that for some cardinal $\kappa$, there is an unbounded set of cardinals $\theta$ such that there is a [high-jump]{} measure on $P_\kappa 2^\theta$ generating an ultrapower embedding with clearance $\theta$. Then there exists a [super-high-jump]{} Laver function for $\kappa$ in $V$.
Define the function $\ell$ as in theorem \[theorem.SHJLaverFunction\]. It follows from theorem \[theorem.SHJLaverFunction\] that for arbitrarily large cardinals $\theta$, the function $\ell$ is a [super-high-jump]{} Laver function for $V_\theta$. Furthermore, for such a cardinal $\theta$, if $U
\in V_\theta$ is a [high-jump]{} measure, generating an embedding $j_U: V \to M$, then the embedding generated by the measure $U$ in the model $V_\theta$ is $j_U
\restrict V_\theta$. It follows that $\ell$ is a [super-high-jump]{} Laver function for $\kappa$.
It is also possible to define Laver functions for other large cardinal notions related to [high-jump]{} cardinals, for instance for particular types of [high-jump]{} cardinals with excess closure. For an example, see [@Perlmutter2013:Dissertation theorem 116].
I close the section with a question.
Is it possible to prove the existence or the consistency of a [super-high-jump]{} Laver function from a hypothesis substantially weaker than that of theorem \[theorem.SHJLaverFunction\]?
For instance, it may be possible to force the existence of a [super-high-jump]{} Laver function for $\kappa$, beginning in a model where $\kappa$ is only [super-high-jump]{}.
Ideas for further research {#section.FurtherDefinitionIdeas}
==========================
\[Section.FurtherStudy\] In this section, I review some of the areas for further research discussed in previous sections, and I also suggest a few additional areas for further research.
One relationship between cardinals in the chart is unresolved. I do not know the relationship between enhanced supercompact cardinals and hypercompact cardinals. One established large cardinal is conspicuously missing from my analysis. An extendible cardinal is known to be intermediate in consistency strength between a supercompact cardinal and a cardinal. But I don’t know the relationship between an extendible cardinal and a hypercompact cardinal or an enhanced supercompact cardinal. Furthermore, the $C^{(n)}$-extendible cardinals, introduced by Bagaria in [@Bagaria2012:CnCardinals], fall into the large cardinal hierarchy between an extendible cardinal and a cardinal.
Another possible direction for further research would be to define more large cardinal notions by modifying the definitions that I have already given. One possibility would be to modify the definition of a [high-jump]{} cardinal so that $M$ is closed under $j(f)(\kappa)$-sequences for all $f: \kappa \to ORD$ rather than just for $f: \kappa \to \kappa$. Such a cardinal would be huge, to say the least. This hugeness is witnessed by the case where $f$ is the function with constant value $\kappa$.
As discussed in the conclusion of section \[section.forcing\], more work can also be done on the relationships between [high-jump]{} cardinals and forcing. More generally, there is more work to be done proving forcing results for all of the cardinals discussed in this paper, and in particular, many of the results from section \[section.forcing\] could be extended to apply to other large cardinals in this paper.
Keeping in mind that many of the large cardinals in this paper were first applied towards universal indestructibility results, it is an interesting goal to use the new large cardinals to weaken the hypotheses for universal indestructibility results, with the goal of eventually proving an equiconsistency between a universal indestructibility result and a large cardinal notion. One could also prove new universal indestructibility results. More generally, the large cardinals that I have studied here could be used to weaken the hypotheses or find equiconsistencies for other set-theoretic results as well.
Acknowledgments
===============
I would like to thank my dissertation advisor, Joel David Hamkins, for his advice relating to this research. This paper uses Paul Taylor’s Commutative Diagrams in TeX package.
[^1]: When I speak of an elementary embedding, I always intend to denote an elementary embedding with a critical point between transitive proper class models of $\ZFC$, unless otherwise stated.
[^2]: In many cases, the English usage rule for the punctuation of compound adjectives is to hyphenate compound adjectives coming before a noun, but not compound adjectives coming after a noun. Hence, I will write that $\kappa$ is a high-jump cardinal, but also that the cardinal $\kappa$ is high jump.
[^3]: If one requires that a $\lambda$-supercompactness embedding be generated by a normal fine measure on $P_\kappa\lambda$ rather than simply defining such embeddings by the closure of the target model, then a factor embedding argument is required. See [@Perlmutter2013:Dissertation proposition 59] for details.
[^4]: Every normal *ultrafilter* on a cardinal $\delta$ contains every club. But a normal *filter* on a cardinal $\delta$ contains every club if and only if it contains every tail. The filter on a cardinal is not in general an ultrafilter, by proposition \[proposition.LeastVopenkaNotWC\].
[^5]: At that time, Apter called these cardinals hypercompact rather than excessively hypercompact. But in light of theorem \[theorem.StrongHypercompactDNE\], we now call them excessively hypercompact.
[^6]: personal communication with Apter, 2012
[^7]: personal communication with Apter, 2012.
[^8]: An additional minor difference is that the definition of hypercompact handles limit stages differently from the definition of excessively hypercompact. I made this change in order to unify the definition for the successor and limit stages, and also to define hypercompact cardinals analogously to the Mitchell order.
[^9]: Actually, this alignment is off by one, because the definitions of these hierarchies handle limit stages differently. But this fact is a technical detail not germane to the main idea.
[^10]: Actually, it suffices for the proof that $j(f)(\kappa)$ is inaccessible.
[^11]: By a $\theta$-supercompactness embedding, I simply mean that $M[H]$ is sufficiently closed, not that the embedding is generated by a normal fine measure.
[^12]: For details, see theorem 105 of Hamkins’s unpublished book, *Forcing and Large Cardinals*.
|
---
abstract: 'I performed photometric observations of the poorly studied cataclysmic variable and discovered very deep eclipses. I obtained observations over 14 nights for a total time of 50 hours during a time span of 6 months. Thanks to the long observation interval, I determined the orbital period with high precision, $P_{\rm orb}=0.149\,350\,14\pm0.000\,000\,20$ d. I derived the eclipse ephemeris, which, thanks to the precision of the orbital period, has a formal validity of 300 years. The average eclipse depth was $1.88\pm0.07$ mag. The prominent parts of the eclipses were smooth and symmetrical. The average eclipse width, including extended asymmetric eclipse wings, was $0.18\pm0.01$ phases or $40\pm2$ min. The average orbital light curve did not show a prominent orbital hump. Because no dwarf nova outburst occurred during the 6 months of monitoring, this cataclysmic variable is likely to be a nova-like variable.'
address: 'Astronomical Observatory, Ural Federal University, Lenin Av. 51, Ekaterinburg, 620083, Russia'
author:
- 'V. P. Kozhevnikov'
title: Discovery of deep eclipses in the cataclysmic variable
---
,
Novae, cataclysmic variables ,Binaries: eclipsing ,Stars: individual:
97.10.Sj ,97.30.Qt ,97.80.Gm
Introduction
============
Cataclysmic variables (CVs) are interacting binary stars consisting of a white dwarf and a late-type companion. The late-type companion fills its Roche lobe and transfers material to the white dwarf. Depending on the presence of large outbursts, CVs are subdivided into dwarf novae and nova-like variables. Classical and recurrent novae except for symbiotic stars are also reckoned among CVs. Such novae in quiescence are similar to nova-like variables. In non-magnetic systems, accretion occurs through accretion discs. In nova-like variables, accretion discs are permanently very bright. In dwarf novae, accretion discs are moderately bright during quiescence and very bright during outburst. Accretion discs often reveal bright spots, which form at the place where the accretion stream impacts the disc. These bright spots cause orbital humps in orbital light curves. These humps are more prominent in dwarf novae during quiescence and are less noticeable in nova-like variables. Comprehensive reviews of CVs are given in @ladous94, @warner95 and @hellier01.
If the orbital inclination is high ($i > 60^\circ$), CVs show eclipses (e.g., @ladous94). Eclipsing CVs are important for several reasons. First, eclipses make it possible to accurately determine the orbital period and then study its change. Second, eclipses make it possible to reliably determine the orbital inclination, which is necessary to determine the masses of stellar components using radial velocity measurements (e.g., @hellier01). Third, eclipses of a white dwarf and eclipses of a bright spot distinguishable in light curves make it possible to determine the masses of stellar components using only photometric data. (e.g., @zorotovic11). Finally, eclipses allow one to study the structure and time evolution of accretion discs using eclipse mapping methods (e.g., @baptista04).
Using the Isaac Newton Telescope Photometric H$\alpha$ Survey (IPHAS), @witham07 discovered 11 new CV candidates. @witham07 performed follow-up observations of only three of them. To find the orbital periods, @witham07 performed spectroscopic observations of two CV candidates, and . For both stars, these observations gave ambiguous results due to the aliasing problem (see Fig. 6 in @witham07). Photometric observations of these two CV candidates were not performed. In contrast, the third CV candidate, , was observed by @witham07 only photometrically. They discovered this star to be an eclipsing CV. Later, @aungwerojwit12 refined this result by discovering that is an eclipsing intermediate polar with an eclipse depth of 1.3 mag. Moreover, @witham07 highlighted another system, , which might be a high-luminosity object reminiscent of . Detailed photometric observations of this bright (13.5 mag) CV candidate were not performed. Following the suggestion by @witham07 that may be a member of the V Sge class, I observed it photometrically and ruled out this possibility. Instead, I found that is an ordinary eclipsing CV with an eclipse depth of 0.3 mag [@kozhevnikov14]. Lately, I improved my photometric technique. This allowed me to observe faint stars like and . In @kozhevnikov18, I continued the study of these CV candidates and discovered deep eclipses (2.4 mag) in . Then, I monitored (hereafter ) and also discovered deep eclipses. In this paper I present the results obtained from these observations.
Observations
============
[lcc]{} Date & BJD$_{\rm TDB}$ start & Length\
(UT) & (-2,458,000) & (h)\
February 16 & 166.126823 & 10.2\
February 18 & 168.152645 & 2.8\
August 12 & 343.278899 & 3.2\
August 15 & 346.261779 & 3.8\
August 17 & 348.260525 & 4.0\
August 19 & 350.284742 & 3.2\
August 20 & 351.321358 & 0.8\
September 3 & 365.208981 & 2.1\
September 4 & 366.223281 & 4.3\
September 5 & 367.207106 & 3.1\
September 11 & 373.223956 & 4.3\
September 12 & 374.238767 & 2.1\
September 13 & 375.316815 & 3.9\
September 16 & 378.398499 & 1.7\
I performed the photometric observations of J0130 at Kourovka observatory, Ural Federal University, using the 70-cm Cassegrain telescope and the multi-channel pulse-counting photometer. The observatory is located at a distance of 80 km from Ekaterinburg. The design of the photometer is described in @kozhevnikoviz. Its three channels enable to measure the brightness of two stars and the sky background simultaneously. The mounting of the telescope is equipped with computer-controlled step motors. To maintain precise centring of the two stars in the photometer diaphragms, I use the CCD guiding system. This guiding system and computer-controlled step motors make it possible to perform brightness measurements automatically and nearly continuously. Short interruptions occur according to a computer program when I measure the sky background in all three channels simultaneously. This is necessary to define the differences in the sky background, which are caused by differences in the size of the diaphragms, and to eliminate the effect of faint stars in the sky background channel.
Although the multi-channel photometer provides high-quality photometric data even under unfavourable atmospheric conditions [e.g., @kozhevnikov02], for a long time I could not observe stars fainter than 15 mag that are invisible to the eye. A few years ago, I realized that, with the aid of step motors of the telescope, I can centre two stars in the diaphragms, one of which is invisible, using the coordinates of the invisible star and the coordinates of a nearby reference star. Using this method, I can observe very faint stars up to 20 mag [e.g., @kozhevnikov18].
Photometric observations of were performed in February–September 2018 over 14 nights with a total duration of 50 h. The data were obtained in white light (approximately 3000–8000 Å). The time resolution was 16 s. For and the comparison star, I used diaphragms of $16''$. For the sky background, I used a diaphragm of $30''$. The comparison star is . It has $B=14.1$ mag and $B-R=0.9$ mag. The colour index of is similar to the colour index of this star. According to the USNO-A2.0 catalogue, has $B=17.5$ mag and $B-R=0.6$ mag. Similar colour indexes of these two stars reduce the effect of differential extinction.
I obtained differences of magnitudes of and the comparison star by taking into account differences in light sensitivity between different channels. Using the comparison of the sky background counts in the sky background channel, the sky background counts were corrected for the effect of faint stars, which may accidentally be in the sky background diaphragm. To measure the sky background in stellar diaphragms, I carefully found a place without faint stars using star maps provided by Astronet (http://www.astronet.ru/db/map). These star maps are based on USNO-A2.0 $B$ magnitudes. However, this did not allow me to avoid a faint star in the diaphragm of when I measured the sky background. The counts in the diaphragm of , from which the sky background counts were subtracted, were mainly negative near the mid-eclipses. Unfortunately, I noticed this during the final data processing.
Using Aladin Lite (https://aladin.u-strasbg.fr/AladinLite) and VizieR (http://vizier.u-strasbg.fr/viz-bin/VizieR), I identified this faint star. It is with $G=19.05$ mag [@gaia16; @gaia18]. During discrete measurements of the sky background, this faint star was located in the diaphragm of near its edge. The spectral response of the photomultipliers (S20 photocathodes) suggests that my white light magnitudes are close to Gaia $G$ magnitudes (e.g., @maiz18). Using the G magnitude of this faint star, I estimated that the sky background counts in the diaphragm of should be reduced by 1.5–2.8%. After this correction, most of the counts in the diaphragm of near the mid-eclipses were positive. To evaluate the precision of this correction, I assumed that the faint star was 0.2 mag fainter. This can be caused by poor centring of this star or by a discrepancy between my white light magnitudes and the $G$ magnitudes. Then the average eclipse was 0.15 mag deeper.
A journal of the observations is shown in Table \[journal\]. This table gives BJD$_{\rm TDB}$, which is the Barycentric Julian Date in the Barycentric Dynamical Time (TDB) standard. BJD$_{\rm TDB}$ is uniform and therefore preferred. I calculated BJD$_{\rm TDB}$ using the online calculator (http://astroutils.astronomy.ohio-state.edu/time/) [@eastman10]. In addition, I calculated BJD$_{\rm UTC}$. During my observations, the difference between BJD$_{\rm TDB}$ and BJD$_{\rm UTC}$ was constant. BJD$_{\rm TDB}$ exceeded BJD$_{\rm UTC}$ by 69 s.
Analysis and results
====================
\[eclipses\]
[lccc]{} Date & BJD$_{\rm TDB}$ mid-ecl. & Eclipse depth & Out-of-ecl.\
(UT) & (-2,458,000) & (mag) & magnitude\
February 16 & 166.22433(27) & $1.95\pm0.10$ & 17.5\
February 16 & 166.37327(22) & $1.95\pm0.08$ & 17.5\
February 16 & 166.52195(21) & $1.84\pm0.09$ & 17.5\
February 18 & 168.16491(19) & $2.23\pm0.08$ & 17.4\
August 12 & 343.35331(22) & $1.97\pm0.08$ & 17.1\
August 15 & 346.33969(21) & $1.66\pm0.07$ & 17.3\
August 17 & 348.28208(19) & $1.44\pm0.05$ & 17.3\
August 19 & 350.37213(16) & $2.18\pm0.07$ & 17.6\
September 4 & 366.35288(24) & $1.63\pm0.08$ & 17.6\
September 5 & 367.24909(25) & $2.16\pm0.10$ & 17.4\
September 11 & 373.37170(30) & $1.63\pm0.09$ & 17.7\
September 12 & 374.26806(15) & $2.24\pm0.07$ & 17.6\
September 13 & 375.46400(24) & $1.60\pm0.06$ & 17.7\
September 16 & 378.45038(36) & $1.81\pm0.12$ & 17.6\
Fig. \[figure1\] shows the longest light curve of , in which three consecutive eclipses are obvious. For usability, the differential magnitudes were converted into the magnitudes using the average of the $B$ and $R$ magnitudes of the comparison star. To reduce the photon noise, counts were previously averaged over 128-s time intervals. The photon noise of the out-of-eclipse light curve (rms) is 0.06 mag. The photon noise near the mid-eclipses is undefined. It is obvious that the accuracy of the points near the mid-eclipses depend mainly on the accuracy of subtracting the sky background because the sky background near the mid-eclipses considerably exceeds the star counts. None the less, as seen in Fig. \[figure1\], the eclipses have roughly equal depths.
During observations of , I obtained 14 whole eclipses. Table \[eclipses\] shows their parameters. The out-of-eclipse magnitudes were obtained by averaging of out-of-eclipse light curves. The mid-eclipse times and the eclipse depths were measured using a Gaussian function fitted to the eclipse and two adjacent parts of the out-of-eclipse light curve. The average eclipse depth is $1.88\pm0.07$ mag. As shown above, if I underestimated the sky background excess caused by the faint star, which accidentally was in the diaphragm of when I measured the sky background, the eclipse depth may be several tenths of a magnitude larger. In contrast, if I overestimated the sky background excess, the eclipse depth may be several tenths of a magnitude less. However, in any case, the eclipses in seem very deep.
It is well known that a direct fit of an ephemeris to a series of mid-eclipse times gives the best precision of the eclipse period. Obviously, this method is good when eclipses are evenly distributed in time. This is not consistent with my observations of . Therefore, to find the eclipse period, I use various methods and then compare the results.
Three consecutive eclipses (Fig. \[figure1\]) allow me to find the approximate eclipse period. These measurements are useful because they eliminate the aliasing problem, which might arise as a result of complicated analysis. Using the three mid-eclipse times, which are presented in Table \[eclipses\], I obtained $P_{1}=0.14894\pm0.00035$ d and $P_{2}=0.14869\pm0.00030$ d. The average value is $0.14882\pm0.00024$ d.
To improve the precision of the eclipse period, I used the Analysis of Variance (AoV) method [@schwarzenberg89], which is advantageous in comparison with the Fourier transform for non-sinusoidal signals [@schwarzenberg98]. At first, I used only the data obtained in August–September because they are densely distributed in time. These data contain 10 whole eclipses. Fig \[figure2\]a shows the AoV spectrum of these data. This AoV spectrum reveals the principal peak with no aliases. The precise peak maximum was found using a Gaussian function fitted to the upper part of the peak. The error was defined according to the method of @schwarzenberg91. This method considers both frequency resolution and noise and is appropriate for Fourier power spectra. For AoV spectra, I use it as a tentative one. From this AoV spectrum, the eclipse period is $0.149\,3504\pm0.000\,0029$ d.
Having ascertained that the AoV spectrum of the data of August–September showed no aliases, I calculated the AoV spectrum of all data, which additionally contain 4 eclipses observed in February. As seen in Fig. \[figure2\]b, the AoV spectrum of all data reveals a much narrower principal peak. This AoV spectrum also shows no aliases. From the AoV spectrum presented in Fig. \[figure2\]b, the eclipse period is $0.149\,350\,29\pm0.000\,000\,58$ d. The period precision obtained from all data is 5 times better than the period precision obtained from the data of August–September.
The precision of the eclipse period found from the AoV spectra is sufficient to avoid ambiguity in determining the number of cycles. Therefore I determined the eclipse period using independent pairs of distant eclipses. These four eclipse pairs and four corresponding eclipse periods are presented in Table \[pairs\]. The average period is $0.149\,350\,23\pm0.000\,000\,25$ d. The period error is approximately two times less than the error obtained from the AoV spectrum of all data according to @schwarzenberg91. Hence, the error found from the AoV spectrum is at least two times overestimated.
\[pairs\]
[lccc]{} BJD$_{\rm TDB}$ mid-ecl. 1 & BJD$_{\rm TDB}$ mid-ecl. 2 & Number & Period\
(-2,458,000) & (-2,458,000) & of cycles & (days)\
166.22433(27) & 348.28208(19) & 1219 & 0.14935008(27)\
166.37327(22) & 350.37213(16) & 1232 & 0.14934973(22)\
166.52195(21) & 374.26806(15) & 1391 & 0.14935018(18)\
168.16491(19) & 375.46400(24) & 1388 & 0.14935093(22)\
& & Average period: & 0.14935023(25)\
\[ephemeris\]
[lccc]{} Date & BJD$_{\rm TDB}$ mid-ecl. & Number & O–C $\times 10^{3}$\
(UT) & (-2,458,000) & of cycles & (days)\
February 16 & 166.22433(27) & -1393 & $0.60\pm0.31$\
February 16 & 166.37327(22) & -1392 & $0.19\pm0.27$\
February 16 & 166.52195(21) & -1391 & $-0.47\pm0.26$\
February 18 & 168.16491(19) & -1380 & $-0.37\pm0.25$\
August 12 & 343.35331(22) & -207 & $0.32\pm0.27$\
August 15 & 346.33969(21) & -187 & $-0.31\pm0.26$\
August 17 & 348.28208(19) & -174 & $0.53\pm0.25$\
August 19 & 350.37213(16) & -160 & $-0.32\pm0.22$\
September 4 & 366.35288(24) & -53 & $-0.03\pm0.28$\
September 5 & 367.24909(25) & -47 & $0.08\pm0.30$\
September 11 & 373.37170(30) & -6 & $-0.67\pm0.33$\
September 12 & 374.26806(15) & 0 & $-0.42\pm0.21$\
September 13 & 375.46400(24) & 8 & $0.73\pm0.28$\
September 16 & 378.45038(36) & 28 & $0.10\pm0.39$\
Using the most precise mid-eclipse time (see Table \[eclipses\]) and the most precise eclipse period obtained from pairs of distant eclipses, I obtained the following tentative ephemeris:
[$$\small BJD_{\rm TDB}(\rm mid-ecl.)= 245\,8374.268\,06(15)+0.149\,350\,23(25) {\it E}.
\label{ephemeris1}$$ ]{}
Using this ephemeris, I calculated O – C for 14 eclipses. These O – C obey the linear relation: O – C = 0.00041(15) – 0.0000009(20) [*E*]{}. It reveals a significant displacement along the vertical axis. I corrected the tentative ephemeris using the coefficients of this relation and obtained the final ephemeris:
[$$\small BJD_{\rm TDB}(\rm mid-ecl.)= 245\,8374.268\,47(15)+0.149\,350\,14(20) {\it E}.
\label{ephemeris2}$$ ]{}
Using ephemeris \[ephemeris2\], I calculated O – C and presented them in Table \[ephemeris\] and in Fig. \[figure3\]. As seen, the O – C diagram reveals no slope and displacement. The eclipse period and its rms error in ephemeris \[ephemeris2\] obtained from the linear fit of ephemeris \[ephemeris1\] to the series of mid-eclipse times, are close to the eclipse period and to its rms error obtained from pairs of distant eclipses. The time during which the accumulated error from the period runs up to one oscillation cycle is considered as a formal validity of an ephemeris. Based on the error of the eclipse period, the formal validity of ephemeris \[ephemeris2\] is 300 years (a confidence level of $1\sigma$). This means that ephemeris \[ephemeris2\] can be used without ambiguity in determining the number of cycles during 300 years. Table \[periods\] presents the eclipse periods obtained by different methods. All periods are compatible with each other.
I folded 12 light curves containing 14 whole eclipses. As seen in Fig. \[figure4\], the prominent parts of the eclipses are smooth and symmetrical, and the eclipses have asymmetric extended wings. The eclipse width including the extended eclipse wings is $0.18\pm0.01$ phases or $40\pm2$ min. From the folded light curve, I determined the depth of the average eclipse using a Gaussian function fitted to the average eclipse and two adjacent parts of the out-of-eclipse light curve. The depth of the average eclipse is $1.87\pm0.03$ mag. This depth is nearly the same as I obtained from direct averaging of the depths of individual eclipses. As seen in Fig. \[figure4\], the out-of-eclipse light curve reveals no prominent orbital hump.
\[periods\]
[lll]{} Method & Period & Deviation from\
& (days) & linear fit\
Adjacent eclipses & 0.14882(24) & $2.2\sigma$\
AoV of August–September & 0.1493504(29) & $0.1\sigma$\
AoV of all data & 0.14935029(58) & $0.3\sigma$\
Pairs of distant eclipses & 0.14935023(25) & $0.2\sigma$\
Linear fit of the ephemeris & 0.14935014(20) & –\
All CVs reveal random brightness changes called flickering, which are visible directly in the light curves. Because is a faint star, such brightness changes are difficult to notice due to the large photon noise. None the less, I found a segment of the light curve, in which random brightness changes are clearly visible and can be attributed to flickering (Fig. \[figure5\]). From this segment, I estimated a flickering peak-to-peak amplitude of about 0.2 mag.
To search for rapid coherent oscillations similar to oscillations in intermediate polars, I analysed the out-of-eclipse light curves obtained with a time resolution of 16 s. The careful Fourier analysis did not reveal rapid coherent oscillations exceeding the noise level. The maximum semi-amplitude of the noise peaks was 15 mmag at frequencies higher than 1.3 mHz. At lower frequencies, the noise peaks had noticeably higher semi-amplitudes due to flickering.
At low frequencies, I attempted to find peaks with periods close to the orbital period. Such peaks might be evidence of superhumps. The result was ambiguous. The Fourier power spectrum of the out-of-eclipse light curves obtained in August–September revealed a coherent oscillation with a semi-amplitude of 70 mmag and a period of $0.14440\pm0.00021$ d. This oscillation can be attributed to negative superhumps because its period is 3.3% less than the eclipse period. However, the August data and the September data analysed separately did not show this oscillation.
Discussion
==========
I performed photometric observations of and detected deep eclipses for the first time. By analyzing the light curves contained 14 whole eclipses, I accurately measured the orbital period, $P_{\rm orb}=0.149\,350\,14\pm0.000\,000\,20$ d. The high precision in the period determination was obtained thanks to the large coverage of the observations, which is six months. As seen in Fig. \[figure2\], my data are densely distributed in time to avoid the aliasing problem. The sharp eclipses were also favourable to avoid the aliasing problem. From spectroscopic observations, however, @witham07 suggested a different orbital period, but their periodogram was heavily aliased. This led to a large error.
The reliably determined orbital period and the precise average orbital light curve (Fig. \[figure4\]) allow me to make the assumption about the subtype of . The long orbital period well above the period gap (measured only in 18% of dwarf novae, see @ritter03) and the absence of luminosity variations outside the eclipses are more typical of nova-like variables than of other types of CVs. In addition, during six months of observations, did not show large brightness changes resembling dwarf nova outbursts. Thus, is most probably a nova-like variable.
The average eclipse depth obtained from direct averaging of 14 individual eclipses (Table \[eclipses\]) is $1.88\pm0.07$ mag. This eclipse depth is nearly equal to the eclipse depth in the folded light curve (Fig. \[figure4\]), which is $1.87\pm0.03$ mag. The eclipses discovered in are very deep. Such deep eclipses are more typical of polars because polars are disc-less systems. But I have ruled out that this CV is a polar because it shows no significant brightness variations outside of eclipses (see, e.g., @warner95). In contrast with polars, only 30% of eclipsing nova-like variables and nova remnants show eclipses deeper than 1.8 mag (see @ritter03 and references therein). Their total number is 20. So, is one of seldom disc systems with deep eclipses. This characteristic makes it suitable for future work with eclipse mapping (see, e.g., @baptista04).
Using the precise eclipse period, I derived the eclipse ephemeris with a formal validity of 300 years. This means that ephemeris \[ephemeris2\] allows one to calculate O – C without ambiguity in determining the number of cycles during 300 years. Therefore, ephemeris \[ephemeris2\] is suitable for future investigations of long-term changes of the orbital period using O – C. Such changes might be caused by variations of the rotational oblateness of the late-type companion during its activity cycle (e.g., @rubenstein91 [@applegate92]). Such changes might also be caused by a possible giant planet orbiting around the centre of gravity in (e.g., @bruch14 [@beuermann11]).
Eclipses observed in binary stars make it possible to determine the orbital inclination and then determine the masses of stellar components from radial velocity measurements. In CVs, this task is complicated because the white dwarf spectrum is directly invisible and, in radial velocity measurements, is replaced by the accretion disc spectrum. Because this spectrum can be contaminated with asymmetric disc structures [@robinson92], measurements of radial velocities are difficult. In future spectroscopic observations of , the knowledge of orbital phases defined using ephemeris \[ephemeris2\] can help to identify these structures and thus solve this task (e.g., @hellier01). To determine the masses of stellar components in , radial velocity measurements can be used together with photometric data. Such measurements make it possible to simultaneously determine the masses of stellar components and the orbital inclination (e.g., @szkody93 [@downes86]).
Conclusions
===========
I performed photometric observations of the cataclysmic variable and detected very deep eclipses for the first time. The analysis of these data gives the following results:
1. My prolonged monitoring allowed me to determine the orbital period in with high precision, $P_{\rm orb}=0.149\,350\,14\pm0.000\,000\,20$ d.
2. The average eclipse depth was very large and equal to $1.88\pm0.07$ mag.
3. The prominent parts of the eclipses were smooth and symmetrical.
4. The eclipses revealed extended asymmetric wings.
5. The average eclipse width including extended eclipse wings was $0.18\pm0.01$ phases or $40\pm2$ min.
6. I derived the eclipse ephemeris, which, thanks to the precision of the orbital period, has a formal validity of 300 years. This ephemeris is suitable for future investigations of the orbital period changes.
7. The constant luminosity without outbursts and the absence of orbital humps suggest a nova-like classification.
8. Due to deep eclipses, is suitable to study the accretion disc structure using eclipse mapping methods.
9. Constraints on the inclination obtained from the eclipse will allow phasing the spectroscopic data and determining of the stellar masses with radial velocity measurements.
[**[Acknowledgements]{}**]{}
This work was supported in part by the Ministry of Education and Science (the basic part of the State assignment, RK no. AAAA-A17-117030310283-7) and by the Act 211 of the Government of the Russian Federation, agreement no. 02.A03.21.0006. This research has made use of the SIMBAD database, the NASA Astrophysics Data System (ADS), the International Variable Star Index (VSX) database and the VizeR catalogue access tool. The SIMBAD database is operated at CDS, Strasbourg, France. The VSX database is operated at AAVSO, Cambridge, Massachusetts, USA. The original description of the VizeR service was published in @ochsenbein00. This research has made use of Aladin sky atlas developed at CDS, Strasbourg Observatory, France. This work has made use of data from the European Space Agency (ESA) mission [*Gaia*]{} ([https://www.cosmos.esa.int/gaia]{}), processed by the [*Gaia*]{} Data Processing and Analysis Consortium (DPAC, [https://www.cosmos.esa.int/web/gaia/dpac/consortium]{}). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the [*Gaia*]{} Multilateral Agreement.
[**[References]{}**]{}
Applegate, J. H., 1992. A Mechanism for Orbital Period Modulation in Close Binaries. ApJ 385, 621. https://doi.org/10.1086/170967.
Aungwerojwit, A., Gänsicke, B. T., Wheatley, P. J., Pyrzas, S., Staels, B., Krajci, T., Rodríguez-Gil, P., 2012. IPHAS J062746.41+014811.3: A Deeply Eclipsing Intermediate Polar. ApJ 758, id. 79. https://doi.org/10.1088/0004-637X/758/2/79.
Baptista, R., 2004. What can we learn from accretion disc eclipse mapping experiments? Astron. Nachr. 325, 181. https://doi.org/10.1002/asna.200310212.
Beuermann, K., Buhlmann, J., Diese J., et al., 2011. The giant planet orbiting the cataclysmic binary DP Leonis. A&A 526, id. A53. https://doi.org/10.1051/0004-6361/201015942.
Bruch, A., 2014. Long-term photometry of the eclipsing dwarf nova V893 Scorpii. Orbital period, oscillations, and a possible giant planet. A&A 566, id. A101. https://doi.org/10.1051/0004-6361/201423576.
Downes, R. A., Mateo, M., Szkody, P., Jenner, D. C., Margon, B., 1986. Discovery of a new short-period, eclipsing cataclysmic variable. ApJ 301, 240. https://doi.org/10.1086/163893.
Eastman, J., Siverd, R., Gaudi, B. S., 2010. Achieving better than 1 minute accuracy in the heliocentric and barycentric julian dates. PASP 122, 935. https://doi.org/10.1086/655938.
Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., Brown, A. G. A., et al., 2016. The Gaia mission. A&A 595, id. A1. https://doi.org/10.1051/0004-6361/201629272.
Gaia Collaboration, Brown, A. G. A., Vallenari, A., Prusti, T., et al., 2018. Gaia Data Release 2. Summary of the contents and survey properties. A&A 616, id. A1. https://doi.org/10.1051/0004-6361/201833051.
Hellier, C., 2001. Cataclysmic Variable Stars, Springer.
Kozhevnikov, V. P., 2002. Advantages of multichannel photometers in observations of rapid stellar oscillations and planetary transits. In: Battrick, B., Favata, F., Roxburgh, I. W., Galadi, D. (Eds.), Proceedings of the First Eddington Workshop on Stellar Structure and Habitable Planet Finding, 11 – 15 June 2001, Córdoba, Spain. ESA SP-485, Noordwijk: ESA Publications Division, p. 299.
Kozhevnikov, V. P., 2014. Detection of eclipses in the suspected V Sge star IPHAS J025827.88+635234.9. ApSS 349, 361. https://doi.org/10.1007/s10509-013-1648-2.
Kozhevnikov, V. P., 2018. Discovery of deep eclipses in the cataclysmic variable IPHAS J051814.33+294113.0. ApSS 363, id. 130. https://doi.org/10.1007/s10509-018-3351-9.
Kozhevnikov, V. P., Zakharova, P. E., 2000. The two-star photometer at Kourovka observatory: design and noise analysis. In: Garzon, F., Eiroa, C., de Winter, D., Mahoney, T. J. (Eds.), ASP Conf. Ser. Vol. 219, Disks, Planetesimals and Planets. Astron. Soc. Pac., San Francisco, p. 381.
la Dous, C., 1994. Observations and theory of cataclysmic variables: on progress and problems in understanding dwarf novae and nova-like stars. Space Sci. Rev. 67, 1. https://doi.org/10.1007/BF00750527.
Maíz Apellániz, J., Weiler, M., 2018. Reanalysis of the Gaia Data Release 2 photometric sensitivity curves using HST/STIS spectrophotometry. A&A 619, id. A180. https://doi.org/10.1051/0004-6361/201834051.
Ochsenbein, F., Bauer, P., Marcout, J., 2000. The VizieR database of astronomical catalogues. A&AS 143, 23. https://doi.org/10.1051/aas:2000169.
Ritter, H., Kolb, U., 2003. Catalogue of cataclysmic binaries, low-mass X-ray binaries and related objects (Seventh edition). A&A 404, 301 (update RKcat 7.24, 2017). https://doi.org/10.1051/0004-6361:20030330.
Robinson, E. L., 1992. On the Reliability of White Dwarf Radial Velocity Curves determined from Emission-Line Velocities. In: Vogt, N. (Ed.), ASP Conf. Ser. Vol. 29, Viña Del Mar Workshop on Cataclysmic Variable Stars. Astron. Soc. Pac., San Francisco, p. 3.
Rubenstein, E. P., Patterson, J., Africano, J. l., 1991. The orbital period changes of UX Ursae Majoris. PASP, 103, 258. https://doi.org/10.1086/132945.
Schwarzenberg-Czerny, A., 1989. On the advantage of using analysis of variance for period search. MNRAS 241, 153. https://doi.org/10.1093/mnras/241.2.153.
Schwarzenberg-Czerny, A., 1991. Accuracy of period determination. MNRAS 253, 198. https://doi.org/10.1093/mnras/253.2.198.
Schwarzenberg-Czerny, A., 1998. Period Search in Large Datasets. Baltic Astron. 7, 43. https://doi.org/10.1515/astro-1998-0109.
Szkody, P., Howell, S. B., 1993. A spectroscopic study of DV Ursae Majoris (US 943), AY Piscium (PG 0134+070), and V503 Cygni. ApJ 403, 743. https://doi.org/10.1086/172245.
Warner, B., 1995. Cataclysmic Variable Stars. Cambridge Astrophys. Ser., vol. 28. Cambridge University Press, Cambridge.
Witham, A. R., Knigge, C., Aungwerojwit, A., Drew, J. E., Gänsicke, B. T., Greimel, R., Groot, P. J., Roelofs, G. H. A., Steeghs, D., Woudt, P. A., 2007. Newly discovered cataclysmic variables from the INT/WFC photometric H$\alpha$ survey of the northern Galactic plane. MNRAS 382, 1158. https://doi.org/10.1111/j.1365-2966.2007.12426.x.
Zorotovic, M., Schreiber, M. R., Gänsicke, B. T., 2011. Post common envelope binaries from SDSS. XI. The white dwarf mass distributions of CVs and pre-CVs. A&A 536, id. A42. https://doi.org/10.1051/0004-6361/201116626.
|
---
abstract: 'An [*ab initio*]{} approach to the calculation of excitonic effects in the optical absorption spectra of semiconductors and insulators is formulated. It starts from a quasiparticle bandstructure calculation and is based on the relevant Bethe–Salpeter equation. An application to bulk silicon shows a substantial improvement with respect to previous calculations in the description of the experimental spectrum, for both peak positions and lineshape.'
address:
- ' Laboratoire des Solides Irradiés, URA 1380 CNRS – CEA/CEREM, École Polytechnique, F-91128 Palaiseau, France '
- |
Istituto Nazionale per la Fisica della Materia, Dipartimento di Fisica dell’Università\
di Roma “Tor Vergata”, Via della Ricerca Scientifica, I–00133 Roma, Italy
author:
- Stefan Albrecht and Lucia Reining
- Rodolfo Del Sole and Giovanni Onida
title: ' [*Ab initio*]{} calculation of excitonic effects in the optical spectra of semiconductors '
---
=10000
{#section .unnumbered}
Recent advances in [*ab initio*]{} calculations, mostly Density Functional Theory – Local Density Approximation (DFT-LDA) applications, allow to determine the ground state properties and the Kohn-Sham (KS) electronic structure[@hohk] for even complicated systems. In order to treat excited states, realistic quasiparticle (QP) energies are then in general obtained by applying self-energy corrections to the KS energies, usually evaluated in the $GW$ approximation[@hedin]. Excellent agreement of the resulting bandstructure with experimental data has been found for a wide range of materials[@hyblou; @godshsh]. However, spectroscopic properties involving two-particle excitations are often only poorly described at this one-particle level. The main example is absorption spectroscopy, where a simultaneously created electron–hole pair interacts more or less strongly. As a consequence, in addition to bound exciton states which occur within the gap, the spectral lineshape above the continuous–absorption edge is distorted.
The reported qualitative agreement with experiment of many computed KS-LDA absorption spectra, obtained from one-electron transitions between KS states [@engfa], is indeed due to a partial cancellation between two principal errors: namely, the compensation of the large KS-LDA underestimation of the valence–conduction bandgap, with an overestimation of the absorption onset induced by calculating the dielectric function entirely within the one-particle picture. The situation often worsens when only the first error is corrected by replacing the KS eigenvalues with the realistic QP energies [@na4; @delsgi]. On the other hand, going beyond the one-particle picture through inclusion of local field and/or exchange–correlation effects within DFT-LDA in the calculation of the absorption spectrum does generally not remove the observed discrepancy[@bech]. In fact, most of the residual error stems from the neglect of the electron–hole interaction.
Up to now, excitonic effects have been rarely calculated from first principles. Some information about energetic changes can be extracted from an LDA-based $\Delta$-SCF approach[@mauri]. Large excitonic effects on the spectral properties have been calculated [*ab initio*]{} in the case of a small sodium cluster[@na4]. This approach has consequently been generalized with the calculation of the absorption onset for an infinite system, the Li$_2$O crystal [@li2o]. The calculation of the entire optical spectrum of a solid, finally, still remains a major challenge [@stras]. Quantitatively correct theoretical absorption spectra are indeed needed as a reference for the interpretation and prediction of experimental results.
A paradigmatic case is bulk silicon, which is representative for the group IV, III-V, and II-VI semiconductors. These materials show qualitatively similar optical spectra, with two major structures at 3–5 eV. The first peak (E$_1$) has been interpreted as a M$_1$ type critical point transition, and the second peak (E$_2$) as a M$_2$ type one [@wang]. Theoretical work based on the one-electron approximation, ranging from early empirical pseudopotential approaches [@ppp] to [*ab initio*]{} DFT-LDA work [@bech], all yielded the same qualitative result, i.e. an underestimation of the E$_1$ peak by as much as 50$\%$, reducing it to a weak shoulder of the generally overestimated E$_2$ peak. In order to go beyond, Louie [*et al.*]{} [@louie] included local field effects in the calculation of the dielectric matrix. The resulting spectrum is significantly improved at higher energies (above 15 eV), but not in the region of interest around 4 eV.
Several authors suggested that strong contributions to the E$_1$ peak could arise from saddle point excitons [@phivel; @expe; @HS]. Excitonic effects allowed to explain the measured temperature and pressure dependence of the lineshape and the symmetry in wavelength modulation reflectance spectra [@expe]. Until now, the most sophisticated calculation of excitonic effects on the spectral lineshape of silicon was done by Hanke and Sham [@HS]. They performed a semi-empirical LCAO calculation, including local field effects and the screened electron–hole attraction. As in Ref. [@louie], local field effects alone were shown to transfer oscillator strength to higher energies and hence to increase the discrepancy with experiment at lower energies. On the contrary, the electron–hole interaction shifted the position of the E$_1$ peak to lower energies, and almost doubled its intensity, while the oscillator strength of the higher energy peaks was decreased. The overall agreement with experiment was hence improved, and clear evidence was given for the importance of excitonic effects. However, the final intensity ratio between the E$_1$ and E$_2$ peaks was reversed, in disagreement with the experimental spectrum. As pointed out by Wang [*et al.*]{} [@wang], the reliability of semi-empirical approaches is limited. For instance, there are important differences, already at the one-electron level, between the spectra of Refs. [@louie] and [@HS].
In the [*ab initio*]{} framework, on the other hand, the precision achievable for the computation of electronic spectra is in general still poor when compared with the quality of calculated ground state properties. This work is aimed to shrink this gap, showing how a significant improvement of the [*ab initio*]{} calculation of absorption spectra can be obtained.
The absorption spectrum is given by the imaginary part of the macroscopic dielectric function $\epsilon_M$ $$\epsilon_M(\omega ) = 1 - \lim_{{\bf q} \to 0} v({\bf q}) \hat \chi_{{\bf G}%
=0,{\bf G}^{\prime}=0} ({\bf q};\omega ),$$ where $\hat \chi ({\bf r},{\bf r}^{\prime};\omega ) = -iS({\bf r},{\bf r},%
{\bf r}^{\prime},{\bf r}^{\prime};\omega )$. $S(1,1^{\prime};2,2^{\prime})$ is the part of the two-particle Green’s function which excludes the disconnected term $-G(1,1^{\prime})G(2,2^{\prime})$, and $G(1,1^{\prime})$ is the one-particle Green’s function [@HS]. The notation (1,2) stands for two pairs of space and time coordinates, $({\bf r}_1, t_1;{\bf r}_2,
t_2) $.
Following Ref. [@HS], we start from the Bethe–Salpeter equation for $S,$$$S(1,1^{\prime };2,2^{\prime })=S_{0}(1,1^{\prime };2,2^{\prime
})+S_{0}(1,1^{\prime };3,3^{\prime })\Xi (3,3^{\prime };4,4^{\prime
})S(4,4^{\prime };2,2^{\prime }).$$ Repeated arguments are integrated over. The term $S_{0}(1,1^{\prime
};2,2^{\prime })=G(1^{\prime },2^{\prime })G(2,1)$ yields the polarization function of independent quasiparticles $\chi _{0}$, from which the standard RPA $\epsilon _{M}$ is obtained. The kernel $\Xi $ contains two contributions: $$\Xi (1,1^{\prime },2,2^{\prime })=-i\delta (1,1^{\prime })\delta
(2,2^{\prime })v(1,2)+i\delta (1,2)\delta (1^{\prime },2^{\prime
})W(1,1^{\prime }).$$
Considering the first term in the calculation of $S$ is equivalent to the inclusion of local field effects in the matrix inversion of a standard RPA calculation. In order to obtain the macroscopic dielectric constant, the bare Coulomb interaction $v$ contained in this term must, however, be used without the long range term of vanishing wave vector [@hankedelsolefiorino84]. When spin is not explicitly treated, $v$ gets a factor of two for singlet excitons. In the second term, $W$ is the screened Coulomb attraction between electron and hole. It is obtained as a functional derivative of the self-energy in the $GW$ approximation, neglecting a term $G%
\frac{\delta W}{\delta G}$. This latter term contains information about the change in screening due to the excitation, and is expected to be small [@strinati]. We limit ourselves to static screening, since dynamical effects in the electron–hole screening and in the one particle Green’s function tend to cancel each other [@bech2], which suggests to neglect both of them.
In order to solve Eq. (2), we have to invert a 4-point function. In Ref. [@HS] this has been possible due to the use of a very limited basis set. In an [*ab initio*]{} plane wave calculation, such a procedure is clearly prohibitive, when plane waves are chosen as straightforward basis functions. Instead, the physical picture of interacting electron–hole pairs suggests to use a basis of LDA Bloch functions, $\psi _{n}({\bf r}),$ expecting that only a limited number of electron–hole pairs will contribute to each excitation.
In this basis, $\chi _{0}^{(n_{1},n_{2}),(n_{3},n_{4})}=\delta
_{n_{1},n_{3}}\delta
_{n_{2},n_{4}}(f_{n_{2}}-f_{n_{1}})/(E_{n_{2}}-E_{n_{1}}-\omega )$ and, after solving for $S$, in the case of static screening, equation (2) can be written as $$S_{(n_{1},n_{2}),(n_{3},n_{4})}=(H_{exc}-I\,\omega
)_{(n_{1},n_{2}),(n_{3},n_{4})}^{-1}(f_{n_{4}}-f_{n_{3}}),$$ with $$\begin{aligned}
H_{exc}^{(n_{1},n_{2}),(n_{3},n_{4})} &=&(E_{n_{2}}-E_{n_{1}})\delta
_{n_{1},n_{3}}\delta _{n_{2},n_{4}}-i(f_{n_{2}}-f_{n_{1}})\times \nonumber
\\
&&\int d{\bf r}_{1}\int d{\bf r}_{1}^{\prime }\int d{\bf r}_{2}\int d{\bf r}%
_{2}^{\prime }\ \psi _{n_{1}}({\bf r}_{1})\,\psi _{n_{2}}^{*}({\bf r}%
_{1}^{\prime })\,\Xi ({\bf r}_{1},{\bf r}_{1}^{\prime },{\bf r}_{2},{\bf r}%
_{2}^{\prime })\,\psi _{n_{3}}^{*}({\bf r}_{2})\,\psi _{n_{4}}({\bf r}%
_{2}^{\prime }).\end{aligned}$$ $I$ is the identity operator. The energies $E_{n}$ are the QP levels. Together with the above form of $\chi _{0}$ this is consistent with the use of LDA wavefunctions and updated energy denominators in the Green’s function used to construct the self-energy in the $GW$ calculation. The $f_{n}$ are Fermi-Dirac occupation numbers. We avoid to invert the matrix $%
(H_{exc}-I\,\omega )$ for each absorption frequency $\omega $ by applying the identity $$(H_{exc}-I\,\omega )^{-1}=\sum_{\lambda ,\lambda ^{\prime }}\frac{|\lambda
>M_{\lambda ,\lambda ^{\prime }}^{-1}<\lambda ^{\prime }|}{(E_{\lambda
}-\omega )},$$ which holds for a system of eigenvectors and eigenvalues of a general, non-hermitian matrix defined by $$H_{exc}|\lambda >=E_{\lambda }|\lambda >.$$ $M_{\lambda ,\lambda ^{\prime }}$ is the overlap matrix of the (in general non-orthogonal) eigenstates of $H_{exc}$.
Equation (7) is the effective two-particle Schrödinger equation which we solve by diagonalization. The explicit knowledge of the coupling of the various two-particle channels, given by the coefficients $%
A_{\lambda}^{(n_1,n_2)}$ of the state $|\lambda>$ in our LDA basis, allows to identify the character of each transition. (This analysis would be much more cumbersome if a matrix inversion instead of the spectral representation was chosen, as in Ref. [@HS].)
The macroscopic dielectric function in Eq. (1) is obtained as $$\begin{aligned}
\epsilon _{M}(\omega )=1-\lim_{{\bf q}\to 0}v({\bf q})\sum_{\lambda ,\lambda
^{\prime }}M_{\lambda ,\lambda ^{\prime }}^{-1}
&&\sum_{n_{1},n_{2}}<n_{1}|e^{-i{\bf q\cdot r}}|n_{2}>A_{\lambda
}^{(n_{1},n_{2})}\times \nonumber \\
&&\sum_{n_{3},n_{4}}<n_{4}|e^{+i{\bf q\cdot r}}|n_{3}>A_{\lambda ^{\prime
}}^{*(n_{3},n_{4})}\frac{(f_{n_{4}}-f_{n_{3}})}{(E_{\lambda }-\omega )}.\end{aligned}$$ In practice, the KS eigenvalues and eigenfunctions from a DFT-LDA calculation serve as input to the evaluation of the RPA screened Coulomb interaction $W$ and the $GW$ self-energy $\Sigma $. The KS eigenfunctions, together with the QP energies and $W$, are then used in the exciton calculation. Here each pair of indices ${(n_{1},n_{2})}$ stands for a pair of bands and one Bloch vector ${\bf k}$ in the Brillouin zone (BZ), since we are interested in direct transitions only.
In principle, all combinations of bands should be considered. It can, however, be shown exactly that only pairs containing one filled and one empty LDA state contribute to (8). Still, the portion of the matrix $H_{exc}$ to be considered is in general non-hermitian, being of the form[@ekpa] $${\bf H} = \left(
\begin{array}{ccc}
H^{(v_1,c_1),(v_2,c_2)} & H^{(v_1,c_1),(c_2,v_2)} & \\
- H^{*(v_1,c_1),(c_2,v_2)} & -H^{*(v_1,c_1),(v_2,c_2)} &
\end{array}
\right).$$ The off-diagonal coupling matrices do not contain the QP transition energies, but only the interaction elements, which are much smaller in the case of silicon. Hence, we neglect the latter and separate the Hamiltonian into two block-diagonal parts: the resonant contributions, which are active for positive frequencies, and the antiresonant ones, only contributing to negative frequencies. The matrix of the resonant part by its own is hermitian, and we therefore obtain the simpler formula $$\epsilon _{M}(\omega )=1+\lim_{{\bf q}\to 0}v({\bf q})\sum_{\lambda }\frac{%
\mid \sum_{v,c;{\bf k}}<v,{\bf k}|e^{-i{\bf qr}}|c,{\bf k}>A_{\lambda
}^{(v,c;{\bf k})}\mid ^{2}}{(E_{\lambda }-\omega )}.$$
\(7) and (9) constitute a set of equations which has been frequently used in the non-[*ab initio*]{} framework [@strinati; @bassani]. Here, it appears as a particular approximation to the more general formula (8), with well-defined [*ab initio*]{} ingredients which are consistent with the $GW$ approach.
We evaluate expression (9) for bulk silicon. The DFT-LDA calculation is performed using norm-conserving pseudopotentials[@BACH], an energy cutoff of 15 Ry, and 256 special [**k**]{} points in the BZ [@monk]. Next, $GW$ corrections to the KS band structure are obtained following the approach of [@godneeds]. The quite smooth $GW$ corrections are interpolated for the denser [**k**]{} point mesh needed for the absorption spectrum. We evaluate equations (7) and (9) using different sets containing up to 2048 [**k**]{} points in the BZ. In order to handle such large matrices, the symmetry properties of the crystal are exploited. One has to be very careful in doing so, since the spectrum turns out to be extremely sensitive to any inconsistency in the phases which may appear when wavefunctions are rotated, notably for degenerate bands. A safe way to proceed is to make only partial use of symmetry, considering only those operations which form an abelian subgroup of the point group, and which altogether allow to reconstruct the whole zone from a corresponding reduced part. In the case of silicon, we found it convenient to use the 180$^{0}$ rotations around the $x$ and the $y$ axis, respectively. These two operations $T$ allow us to break the equation $H^{{\bf kk^{\prime }}}A^{{\bf k^{\prime }}%
}=EA^{{\bf k}}$ (band indices have been suppressed, and repeated indices are summed over) into four equations to be used for points ${\bf k}_{i}$ in the reduced zone only. These equations are of the form $h_{{\bf k}_{i}{\bf %
k^{\prime }}_{i}}a^{{\bf k^{\prime }}_{i}}=Ea^{{\bf k}_{i}}$, where $h_{{\bf %
k}_{i}{\bf k^{\prime }}_{i}}=H^{{\bf k}_{i}{\bf k^{\prime }}_{i}}\pm H^{{\bf %
k}_{i}T{\bf k^{\prime }}_{i}}$. The $A^{{\bf k}}$ are then reconstructed from the reduced eigenvectors $a^{{\bf k}_{i}}$. Moreover, we apply time reversal and hermiticity in order to accelerate the calculation of the matrix elements.
A set with 864 [**k**]{} points in the full BZ is used to check the various ingredients of our calculation, in particular the number of bands and the importance of the off-diagonal elements of the inverse dielectric matrix in the evaluation of $W$. In the inset of Fig. 1, the continuous line shows the results of a calculation with 4 valence and 4 conduction bands, and the full $\epsilon ^{-1}$. In the region of interest (below 4.5 eV), a 6 bands calculation (4 valence + 2 conduction, dotted line) appears to be sufficient. Neglecting the off-diagonal elements of $\epsilon _{{\bf GG}
^{\prime }}^{-1}({\bf q})$ yields an indistinguishable curve. We then use 6 bands and the diagonal $\epsilon ^{-1}$ to compute the spectrum with 2048 [**k**]{} points in the full BZ. In the main part of Fig. 1 the experimental spectrum (dotted line) [@aspnes] is compared to: i) an RPA calculation [@NOTA] taking only into account the QP shifts, but not the excitonic or local field effects (short–dashed curve): the result is, as generally observed, in great discrepancy with experiment; ii) a calculation including local field effects (i.e. using equation (3) with $W$ set to 0, long–dashed curve): the agreement is worsened, since the oscillator strength is slightly shifted to higher energies and both the E$_{1}$ and E$_{2}$ peaks are lowered, thus confirming previous findings in the literature [@louie; @HS]; iii) finally, the full calculation including the electron–hole attraction (continuous curve): absolute intensities now agree well with experiment. The remaining slight overestimate is of the order of what has been predicted by Ref. [@bech2] to be the contribution of dynamical effects. More important, the peak positions and the relative intensity of the main structures are both in good agreement with experiment. Also the structure at 3.8 eV, even though slightly overestimated due to a finite ${\bf k}$ point sampling, has been repeatedly confirmed in theoretical and experimental work [@Cheli76].
In conclusion, we have shown how excitonic effects can be included in an [*ab initio*]{} calculation of optical absorption spectra of semiconductors. At the example of bulk silicon, we have demonstrated that good agreement with experiment can be obtained for a case where the inclusion of self-energy and local field effects alone still gives rise to a rather poor theoretical spectrum. In this context, bulk silicon is not particularly easy to handle, since the bottleneck of the calculation is the number of [**k**]{} points (high in silicon, due to large dispersion) and not the energy cutoff. Even though, the computational effort, mostly steming from diagonalization, is reasonable and demands only a few hours on a Cray C98. The present work opens hence the way to first-principles calculations of optical absorption spectra with a precision comparable to that typically achieved in ground state calculations.
We thank F. Bechstedt and R.W. Godby for helpful discussions. This work was supported in part by the European Community programme “Human Capital and Mobility” through Contract No. ERB CHRX CT930337. Computer time on the Cray C98 was granted by IDRIS (Project No. CP9/970544).
P. Hohenberg and W. Kohn, Phys. Rev. [**136**]{}, B864 (1964); W. Kohn and L. Sham, Phys. Rev. [**140**]{}, A1133 (1965).
L. Hedin, Phys. Rev. A [**139**]{}, 796 (1965).
M.S. Hybertsen and S.G. Louie, Phys. Rev. Lett. [**55**]{}, 1418 (1985); Phys. Rev. B [**34**]{}, 5390 (1986).
R.W. Godby, M. Schlüter, and L.J. Sham, Phys. Rev. Lett. [**56**]{}, 2415 (1986); Phys Rev. B [**37**]{}, 10 159 (1988).
See, for example, G.E. Engel and B. Farid, Phys. Rev. B [** 46**]{}, 1812 (1992).
G. Onida, L. Reining, R.W. Godby, R. Del Sole, and W. Andreoni, Phys. Rev. Lett. [**75**]{}, 818 (1995).
R. Del Sole and R. Girlanda, Phys. Rev. B [**48**]{}, 11 789 (1993).
V.I. Gavrilenko and F. Bechstedt, Phys. Rev. B [**54**]{}, 13 416 (1996).
F. Mauri and R. Car, Phys. Rev. Lett. [**75**]{}, 3166 (1995).
S. Albrecht, G. Onida, and L. Reining, Phys. Rev. B [**55**]{}, 10 278 (1997).
The difficulties and some preliminary results were illustrated in Ref.[@stras2].
S. Albrecht, G. Onida, L. Reining, and R. Del Sole, Comp. Mat. Sci. [**572**]{} (1997) (in press).
C.S. Wang and B.M. Klein, Phys. Rev. B. [**24**]{}, 3417 (1981).
D.J. Stukel, R.N. Euwema, T.C. Collins, F. Herman, and R.L. Kortum, Phys. Rev. [**179**]{}, 740 (1969); J.P. Walter, M.L. Cohen, Y. Petroff, and M. Balkanski, Phys. Rev. B [**1**]{}, 2661 (1970).
S.G. Louie, J.R. Chelikowsky, and M.L. Cohen, Phys. Rev. Lett. [**34**]{}, 155 (1975).
J.C. Phillips, Phys. Rev. A [**136**]{}, 1705 (1964); B. Velicky and J. Sak, Phys. Status Solidi [**16**]{}, 147 (1966).
R.R.I. Zucca and Y.R. Shen, Phys. Rev. B [**1**]{}, 2668 (1970); J.E. Rowe, F.H. Pollack, and M. Cardona, Phys. Rev. Lett. [**22**]{}, 933 (1969).
W. Hanke and L.J. Sham, Phys. Rev. Lett. [**43**]{}, 387 (1979); Phys. Rev. B [**21**]{}, 4656 (1980).
W. Hanke and L. J. Sham, Phys. Rev. B [**12**]{}, 4501 (1975); R. Del Sole and E. Fiorino, Phys. Rev. B [**29**]{}, 4631 (1984).
G. Strinati, Phys. Rev. Lett. [**49**]{}, 1519 (1982); Phys. Rev. B [**29**]{}, 5718 (1984).
F. Bechstedt, K. Tenelsen, B. Adolph, and R. Del Sole, Phys. Rev. Lett. [**78**]{}, 1528 (1997).
This form for $H$ was already used by Ekhardt [*et al.*]{} [@ekpa2], who, however, did not give further details.
W. Ekardt and J.M. Pacheco, Phys. Rev. B [**52**]{}, 16 864 (1995).
F. Bassani and G. Pastori Parravicini, [*Electronic States and Optical Transitions in Solids*]{} (Pergamon, Oxford, 1975), Chap. 6.
G.B. Bachelet, D.R. Hamann, and M. Schlüter, Phys. Rev. B [**26**]{}, 4199 (1982).
H.J. Monkhorst and J.D. Pack, Phys. Rev. B [**13**]{}, 5188 (1976).
R.W. Godby and R.J. Needs, Phys. Rev. Lett. [**62**]{}, 1169 (1989).
D.E. Aspnes and A.A. Studna, Phys. Rev. B [**27**]{}, 985 (1983).
For comparison, all curves in the main part of Fig. 1 correspond to calculations with 2048 [**k**]{} points in the full BZ. The calculation without excitonic effects can also be (and usually is) performed with larger [**k**]{} points sets. Doing this, only the high-energy part of the spectra is modified, namely the anomalous peak at about 5 eV, which reduces to a small shoulder at full convergence.
J.R. Chelikowsky and M.L. Cohen, Phys. Rev. B [**14**]{}, 556 (1976), and references therein.
|
---
abstract: 'We prove that Waldhausen $K$-theory, when extended to a very general class of quasicategories, can be described as a Goodwillie differential. In particular, $K$-theory spaces admit canonical (connective) deloopings, and the $K$-theory functor enjoys a simple universal property. Using this, we give new, higher categorical proofs of the Approximation, Additivity, and Fibration Theorems of Waldhausen in this context. As applications of this technology, we study the algebraic $K$-theory of associative rings in a wide range of homotopical contexts and of spectral Deligne–Mumford stacks.'
address: 'Massachusetts Institute of Technology, Department of Mathematics, Building 2, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA'
author:
- Clark Barwick
bibliography:
- 'kthy.bib'
title: 'On the algebraic $K$-theory of higher categories'
---
Introduction
============
We characterize algebraic $K$-theory as a *universal homology theory*, which takes suitable higher categories as input and produces either spaces or spectra as output. What makes $K$-theory a *homology theory* is that it satisfies an excision axiom. This excision axiom is tantamount to what Waldhausen called *additivity*, so that an excisive functor is precisely one that splits short exact sequences. What makes this homology theory *universal* is this: if we write $\iota$ for the functor that carries a higher category to its moduli space of objects, then algebraic $K$-theory is initial among homology theories $F$ that receive a natural transformation ${{\iota}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {F}}$. In the lingo of Tom Goodwillie’s calculus of functors [@MR1076523; @MR1162445; @MR2026544], $K$ is the *linearization* of $\iota$. Algebraic $K$-theory is thus the analog of stable homotopy theory in this new class of categorified homology theories. From this we obtain an explicit universal property that completely characterizes algebraic $K$-theory and permits us to give new, conceptual proofs of the fundamental theorems of Waldhausen $K$-theory.
To get a feeling for this universal property, let’s first contemplate $K_0$. For any ordinary category $C$ with a zero object and a reasonable notion of “short exact sequence” (e.g., an exact category in the sense of Quillen, or a category with cofibrations in the sense of Waldhausen, or a triangulated category in the sense of Verdier), the abelian group $K_0(C)$ is the universal target for Euler characteristics. That is, for any abelian group $A$, the set $\mathrm{Hom}(K_0(C),A)$ is in natural bijection with the set of maps $\phi\colon{{\operatorname{Obj}C}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {A}}$ such that $\phi(X)=\phi(X')+\phi(X'')$ whenever there is a short exact sequence $$X'\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X\ \tikz[baseline]\draw[>=stealth,->>](0,0.5ex)--(0.5,0.5ex);\ X''.$$
We can reinterpret this as a universal property on the entire functor $K_0$, which we’ll regard as valued in the category of *sets*. Just to fix ideas, let’s assume that we are working with the algebraic $K$-theory of categories with cofibrations in the sense of Waldhausen. If $E(C)$ is the category of short exact sequences in a category with cofibrations $C$, then $E(C)$ is also a category with cofibrations. Moreover, for any $C$,
(1) the functors $${{[X'\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X\ \tikz[baseline]\draw[>=stealth,->>](0,0.5ex)--(0.5,0.5ex);\ X'']}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {X'}}\textrm{\quad and\quad}{{[X'\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X\ \tikz[baseline]\draw[>=stealth,->>](0,0.5ex)--(0.5,0.5ex);\ X'']}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {X''}}$$ together induce a bijection ${{K_0(E(C))}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {K_0(C)\times K_0(C)}}$. The functor ${{[X'\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X\ \tikz[baseline]\draw[>=stealth,->>](0,0.5ex)--(0.5,0.5ex);\ X'']}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {X}}$ now gives a commutative monoid structure $K_0(C)\times K_0(C)\cong K_0(E(C))\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ K_0(C)$. With this structure, $K_0$ is an abelian group. We can express this sentiment diagrammatically by saying that \[[\[(1)\]]{}\]
(2) the functors $${{[X'\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X\ \tikz[baseline]\draw[>=stealth,->>](0,0.5ex)--(0.5,0.5ex);\ X'']}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {X'}}\textrm{\quad and\quad}{{[X'\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X\ \tikz[baseline]\draw[>=stealth,->>](0,0.5ex)--(0.5,0.5ex);\ X'']}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {X}}$$ also induce a bijection ${{K_0(E(C))}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {K_0(C)\times K_0(C)}}$.
Now our universal characterization of $K_0$ simply says that we have a natural transformation ${{\operatorname{Obj}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {K_0}}$ that is initial with properties (1) and (2).
For the $K$-theory *spaces* (whose homotopy groups will be the higher $K$-theory groups), we can aim for a homotopical variant of this universal property. We replace the word “bijection” in (1) and (2) with the words “weak equivalence;” a functor satisfying these properties is called an *additive functor*. Instead of a map from the set of objects of the category with cofibrations $C$, we have a map from the *moduli space* of objects — this is the classifying space $N\iota C$ of the groupoid $\iota C\subset C$ consisting of all isomorphisms in $C$. An easy case of our main theorem states that algebraic $K$-theory is initial in the homotopy category of (suitably finitary) additive functors $F$ equipped with a natural transformation ${{N\iota}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {F}}$.
Now let’s enlarge the scope of this story enough to bring in examples such as Waldhausen’s algebraic $K$-theory of spaces by introducing homotopy theory in the source of our $K$-theory functor. We use $\infty$-categories that contain a zero object and suitable cofiber sequences, and we call these *Waldhausen $\infty$-categories*. Our homotopical variants of (1) and (2) still make sense, so we still may speak of *additive functors* from Waldhausen $\infty$-categories to spaces. Moreover, any $\infty$-category has a moduli space of objects, which is given by the maximal $\infty$-groupoid contained therein; this defines a functor $\iota$ from Waldhausen $\infty$-categories to spaces. Our main theorem (§\[sect:univpropKthy\]) is thus the natural extension of the characterization of $K_0$ as the universal target for Euler characteristics:
Algebraic $K$-theory is homotopy-initial among suitably finitary additive functors $F$ equipped with a natural transformation ${{\iota}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {F}}$.
It is well-known that algebraic $K$-theory is hair-raisingly difficult to compute, and that various theories that are easier to compute, such as forms of $\mathrm{THH}$ and $\mathrm{TC}$, are prime targets for “trace maps” [@MR1474979]. The Universal Additivity Theorem actually classifies all such trace maps: for any additive functor $H$, the space of natural transformations ${{K}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {H}}$ is equivalent to the space of natural transformations ${{\iota}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {H}}$. But since $\iota$ is actually represented by the ordinary category $\Gamma^{{\mathrm{op}}}$ of pointed finite sets, it follows from the Yoneda lemma that the space of natural transformations ${{K}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {H}}$ is equivalent to the space $H(\Gamma^{{\mathrm{op}}})$. In particular, by Barratt–Priddy–Quillen, we compute the space of “global operations” on algebraic $K$-theory: $$\operatorname{End}(K)\simeq QS^0.$$
The proof of the Universal Additivity Theorem uses a new way of conceptualizing functors such as algebraic $K$-theory. Namely, we regard algebraic $K$-theory as a homology theory on Waldhausen $\infty$-categories, and we regard additivity as an excision axiom. But this isn’t just some slack-jawed analogy: we’ll actually pass to a homotopy theory on which functors that are $1$-excisive in the sense of Goodwillie (i.e., functors that carry homotopy pushouts to homotopy pullbacks) correspond to additive functors as described above. (And making sense of this homotopy theory forces us to pass to the $\infty$-categorical context.)
The idea here is to regard the homotopy theory ${\mathbf{Wald}_{\infty}}$ of Waldhausen $\infty$-categories as formally analogous to the ordinary category $V(k)$ of vector spaces over a field $k$. The left derived functor of a right exact functor out of $V(k)$ is defined on the *derived category* $\mathrm{D}_{\geq 0}(k)$ of chain complexes whose homology vanishes in negative degrees. Objects of $\mathrm{D}_{\geq 0}(k)$ can be regarded as formal geometric realizations of simplicial vector spaces. Correspondingly, we define a *derived $\infty$-category* ${\mathrm{D}(\mathbf{Wald}_{\infty})}$ of ${\mathbf{Wald}_{\infty}}$, whose objects can be regarded as formal geometric realizations of simplicial Waldhausen $\infty$-categories. This entitles us to speak of the *left derived functor* of a functor defined on ${\mathbf{Wald}_{\infty}}$. Then we suitably localize ${\mathrm{D}(\mathbf{Wald}_{\infty})}$ in order to form a universal homotopy theory ${\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}$ in which exact sequences split; we call this the *fissile derived $\infty$-category*. Our Structure Theorem (Th. \[thm:additiveequiv\]) uncovers the following relationshp between excision on ${\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}$ and additivity:
A suitably finitary functor from Waldhausen $\infty$-categories to spaces is additive in the sense above if and only if its left derived functor factors through an excisive functor on the fissile derived $\infty$-category ${\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}$.
This Structure Theorem is not some dreary abstract formalism: the technology of Goodwillie’s calculus of functors tells us that the way to compute the universal excisive approximation to a functor $F$ is to form the colimit of $\Omega^nF\Sigma^n$ as ${{n}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\infty}}$. This means that as soon as we’ve worked out how to compute the suspension $\Sigma$ in ${\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}$, we’ll have an explicit description of the *additivization* of any functor $\phi$ from ${\mathbf{Wald}_{\infty}}$ to spaces, which is the universal approximation to $\phi$ with an additive functor. And when we apply this additivization to the functor $\iota$, we’ll obtain a formula for the very thing we’re claiming is algebraic $K$-theory: the initial object in the homotopy category of additive functors $F$ equipped with a natural transformation ${{\iota}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {F}}$.
So what *is* $\Sigma$? Here’s the answer: it’s given by the formal geometric realization of Waldhausen’s $S_{\bullet}$ construction (suitably adapted for $\infty$-categories). So the universal homology theory with a map from $\iota$ is given by the formula $${{\mathscr{C}}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{colim}_{n}\Omega^n|\iota S_{\ast}^n(\mathscr{C})|}}.$$ This is exactly Waldhausen’s formula for algebraic $K$-theory, so our Main Theorem is an easy consequence of our Structure Theorem and our computation of $\Sigma$.
Bringing algebraic $K$-theory under the umbrella of Goodwillie’s calculus of functors has a range of exciting consequences, which we are only able to touch upon in this first paper. In particular, three key foundational results of Waldhausen’s algebraic $K$-theory — the Additivity Theorem [@MR86m:18011 Th. 1.4.2] (our version: Cor. \[for:preaddsareeasy\]), the Approximation Theorem [@MR86m:18011 Th. 1.6.7] (our version: Pr. \[prp:approx\]), the Fibration Theorem [@MR86m:18011 Th. 1.6.4] (our version: Pr. \[thm:fibration\]), and the Cofinality Theorem [@MR990574 Th. 2.1] (our version: Th. \[thm:cofinality\]) — are straightforward consequences of general facts about the calculus of functors combined with some observations about the homotopy theory of ${\mathbf{Wald}_{\infty}}$.
To get a glimpse of various bits of our framework at work, we offer two examples that exploit certain features of the algebraic $K$-theory functor of which we are fond. First (§\[sect:example1\]), we apply our foundational work to the study of the connective $K$-theory of $E_1$-algebras in suitable ground $\infty$-categories. We define a notion of a *perfect* left module over an $E_1$-algebra (Df. \[ntn:perfectmodule\]). In the special case of an $E_1$ ring spectrum $\Lambda$, for any set $S$ of homogenous elements of $\pi_{\ast}\Lambda$ that satisfies a left Ore condition, we obtain a fiber sequence of connective spectra $${\mathbf{K}}(\mathbf{Nil}_{(\Lambda,S)}^{\omega})\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{K}}(\Lambda)\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{K}}(\Lambda[S^{-1}]),$$ in which the first term is the $K$-theory of the $\infty$-category of $S$-nilpotent perfect $\Lambda$-modules (Pr. \[prp:protolocforE1rings\]). Such a result — at least in special cases — is surely well-known among experts; see for example [@BM Pr. 1.4 and Pr. 1.5].
Finally (§\[sect:example2\]), we introduce $K$-theory in derived algebraic geometry. In particular, we define the $K$-theory of quasicompact nonconnective spectral Deligne–Mumford stacks (Df. \[dfn:algKthyDMstacks\]). We prove a result analogous to what Thomason called the “proto-localization” theorem [@MR92f:19001 Th. 5.1]; this is a fiber sequence of connective spectra $${\mathbf{K}}(\mathscr{X}\setminus\mathscr{U})\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--(0.75,0.5ex);\ {\mathbf{K}}(\mathscr{X})\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--(0.75,0.5ex);\ {\mathbf{K}}(\mathscr{U})$$ corresponding to a quasicompact open immersion $j\colon{{\mathscr{U}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{X}}}$ of quasicompact, quasiseparated spectral algebraic spaces. Here ${\mathbf{K}}(\mathscr{X}\setminus\mathscr{U})$ is the $K$-theory of the $\infty$-category perfect modules $\mathscr{M}$ on $\mathscr{X}$ such that $j^{\star}\mathscr{M}\simeq0$ (Pr. \[prp:protolocforDMstacks\]). Our proof is new in the details even in the setting originally contemplated by Thomason (though of course the general thrust is the same).
Relation to other work {#relation-to-other-work .unnumbered}
----------------------
Our universal characterization of algebraic $K$-theory has probably been known — perhaps in a more restrictive setting and certainly in a different language — to a variety of experts for many years. In fact, the universal property stated here has endured a lengthy gestation: the first version of this characterization emerged during a question-and-answer session between the author and John Rognes after a talk given by the author at the University of Oslo in 2006.
The idea that algebraic $K$-theory could be characterized via a universal property goes all the way back to the beginnings of the subject, when Grothendieck defined what we today call $K_0$ of an abelian or triangulated category just as we describe above [@MR0116023; @MR50:7133]. The idea that algebraic $K$-theory might be expressible as a linearization was directly inspired by the ICM talk of Tom Goodwillie [@MR1159249] and the remarkable flurry of research into the relationship between algebraic $K$-theory and the calculus of functors — though of course the setting for our Goodwillie derivative is more primitive than the one studied by Goodwillie et al.
But long before that, of course, came the foundational work of Waldhausen [@MR86m:18011]. Since it is known today that relative categories comprise a model for the homotopy theory of $\infty$-categories [@BarKan1011], the work of Waldhausen can be said to represent the first study of the algebraic $K$-theory of higher categories. Furthermore, the idea that the defining property of this algebraic $K$-theory is additivity is strongly suggested by Waldhausen, and this point is driven home in the work of Randy McCarthy [@MR1217072] and Ross Staffeldt [@MR990574], both of whom recognized long ago that the Additivity Theorem is the ur-theorem of algebraic $K$-theory.
In a parallel development, Amnon Neeman has advanced the *algebraic $K$-theory of triangulated categories* [@MR1491990; @MR1604910; @MR1656552; @MR1724625; @MR1793672; @MR1798824; @MR1798828; @MR1828612; @MR2181838] as a way of extracting $K$-theoretic data directly from the triangulated homotopy category of a stable homotopy theory. The idea is that the algebraic $K$-theory of a ring or scheme should by approximation depend (in some sense) only on a derived category of perfect modules; however, this form of $K$-theory has known limitations: an example of Marco Schlichting [@MR1930883] shows that Waldhausen $K$-theory can distinguish stable $\infty$-categories with equivalent triangulated homotopy categories. These limitations are overcome by passing to the *derived $\infty$-category*.
More recently, Bertrand Toën and Gabriele Vezzosi showed [@MR2061207] that the Waldhausen $K$-theory of many of the best-known examples of Waldhausen categories is in fact an invariant of the simplicial localization; thus Toën and Vezzosi are more explicit in identifying higher categories as a natural domain for $K$-theory. In fact, in the final section of [@MR2061207], the authors suggest a strategy for constructing the $K$-theory of a Segal category by means of an “$S_{\scriptstyle{\bullet}}$ construction up to coherent homotopy.” The desired properties of their construction are reflected precisely in our construction $\mathscr{S}$. These insights were explored more deeply in the work of Blumberg and Mandell [@MR2764905]; they give an explicit description of Waldhausen’s $S_{\scriptstyle{\bullet}}$ construction in terms of the mapping spaces of the simplicial localization, and they extend Waldhausen’s approximation theorem to show that in many cases, equivalences of homotopy categories alone are enough to deduce equivalences of $K$-theory spectra.
Even more recent work of Andrew Blumberg, David Gepner, and Gon[ç]{}alo Tabuada [@BGT] has built upon brilliant work of the last of these authors in the context of DG categories [@MR2451292] to produce another universal characterization of the algebraic $K$-theory of stable $\infty$-categories. One of their main results may be summarized by saying that the algebraic $K$-theory of stable $\infty$-categories is a universal additive invariant. They do not deal with general Waldhausen $\infty$-categories, but they also study nonconnective deloopings of $K$-theory, with which we do not contend here.
Finally, we recall that Waldhausen’s formalism for algebraic $K$-theory has of course been applied in the context of associative $S$-algebras by Elmendorf, Kriz, Mandell, and May [@MR97h:55006], and in the context of schemes and algebraic stacks by Thomason and Trobaugh [@MR92f:19001], Toën [@MR2000h:14010], Joshua [@MR1942183; @MR1949305; @MR1967388], and others. The applications of the last two sections of this paper are extensions of their work.
A word on higher categories {#a-word-on-higher-categories .unnumbered}
---------------------------
When we speak of $\infty$-categories in this paper, we mean $\infty$-categories whose $k$-morphisms for $k\geq 2$ are weakly invertible. We will use the *quasicategory* model of this sort of $\infty$-categories. Quasicategories were invented in the 1970s by Boardman and Vogt [@MR0420609], who called them *weak Kan complexes*, and they were studied extensively by Joyal [@Joyal; @Joyal08] and Lurie [@HTT]. We emphasize that quasicategories are but one of an array of equivalent models of $\infty$-categories (including simplicial categories [@MR81h:55018; @MR81h:55019; @MR81m:55018; @MR2276611], Segal categories [@math.AG/9807049; @MR2341955; @simpson:book], and complete Segal spaces [@MR1804411; @MR2439415]), and there is no doubt that the results here could be satisfactorily proved in any one of these models. Indeed, there is a canonical equivalence between any two of these homotopy theories [@JT; @MR2321038; @Bergner] (or any other homotopy theory that satisfies the axioms of [@Toen] or of [@BSP]), through which one can surely translate the main theorems here into theorems in the language of any other model. To underscore this fact, we will frequently use the generic term *$\infty$-category* in lieu of the more specialized term *quasicategory*.
That said, we wish to emphasize that we employ many of the technical details of the particular theory of quasicategories as presented in [@HTT] in a critical way in this paper. In particular, beginning in §\[sect:Waldfib\], the theory of fibrations, developed by Joyal and presented in Chapter 2 of [@HTT], is instrumental to our work here, as it provides a convenient way to finesse the homotopy-coherence issues that would otherwise plague this paper and its author. Indeed, it is the convenience and relative simplicity of this theory that compelled us to work with this model.
Acknowledgments {#acknowledgments .unnumbered}
---------------
There are a lot of people to thank. Without the foundational work of André Joyal and Jacob Lurie on quasicategories, the results here would not admit such simple statements or such straightforward proofs. I thank Jacob also for generously answering a number of questions during the course of the work represented here.
My conversations with Andrew Blumberg over the past few years have been consistently enlightening, and I suspect that a number of the results here amount to elaborations of insights he had long ago. I have also benefitted from conversations with Dan Kan.
In the spring of 2012, I gave a course at MIT on the subject of this paper. During that time, several sharp-eyed students spotted errors, including especially Rune Haugseng, Luis Alexandre Pereira, and Guozhen Wang. I owe them my thanks for their scrupulousness.
John Rognes has declined to be listed as a coauthor, but his influence on this work has been tremendous. He was present at the conception of the main result, and this paper is teeming with insights I inherited from him.
I thank Peter Scholze for noticing an error that led to the inclusion of the Cofinality Theorem \[thm:cofinality\].
Advice from both the incognito referee and Mike Hopkins has led to great improvements in the exposition of this paper.
On a more personal note, I thank Alexandra Sear for her unfailing patience and support during this paper’s ridiculously protracted writing process.
In this part, we introduce the basic input for additive functors, including the form of $K$-theory we study. We begin with the notion of a *pair* of $\infty$-categories, which is nothing more than an $\infty$-category with a subcategory of *cofibrations* that contains the equivalences. Among the pairs of $\infty$-categories, we will then isolate the *Waldhausen $\infty$-categories* as the input for algebraic $K$-theory; these are pairs that contain a zero object in which the cofibrations are stable under pushout. This is the $\infty$-categorical analogue of Waldhausen’s notion of categories with cofibrations.
We will also need to speak of families of Waldhausen $\infty$-categories, which are called *Waldhausen (co)cartesian fibrations*, and which classify functors valued in the $\infty$-category ${\mathbf{Wald}_{\infty}}$ of Waldhausen $\infty$-categories. We study limits and colimits in ${\mathbf{Wald}_{\infty}}$, and we construct the $\infty$-category of *virtual Waldhausen $\infty$-categories*, whose homotopy theory serves as the basis for all the work we do in this paper.
Pairs of $\infty$-categories
============================
The basic input for Waldhausen’s algebraic $K$-theory [@MR86m:18011] is a category equipped with a subcategory of weak equivalences and a subcategory of cofibrations. These data are then required to satisfy sundry axioms, which give what today is often called a *Waldhausen category*.
A category with a subcategory of weak equivalences (or, in the parlance of [@BarKan1011], a *relative category*) is one way of exhibiting a homotopy theory. A *quasicategory* is another. It is known [@BarKan1011 Cor. 6.11] that these two models of a homotopy theory contain essentially the same information. Consequently, if one wishes to employ the flexible techniques of quasicategory theory, one may attempt to replace the category with weak equivalences in Waldhausen’s definition with a single quasicategory.
But what then is to be done with the cofibrations? In Waldhausen’s framework, the specification of a subcategory of cofibrations actually serves two distinct functions.
(1) First, Waldhausen’s Gluing Axiom [@MR86m:18011 §1.2, Weq. 2] ensures that pushouts along these cofibrations are compatible with weak equivalences, so in fact these pushouts are all *homotopy pushouts*. For example, pushouts in the category of simplicial sets along inclusions are homotopy pushouts; consequently, in the Waldhausen category of finite spaces, the cofibrations are monomorphisms.
(2) Second, the cofibrations permit one to restrict attention to the particular class of cofiber sequences one wishes to split in $K$-theory. For example, an exact category is regarded as a Waldhausen category by declaring the cofibrations to be the admissible monomorphisms; consequently, the admissible exact sequences are the only exact sequences that algebraic $K$-theory splits.
In a quasicategory, the first function becomes vacuous, as the only sensible notion of *pushout* in a quasicategory is necessarily homotopy invariant. Thus only the second function for a class of cofibrations in a quasicategory will be relevant. This means, in particular, that we needn’t make any distinction between a cofibration in a quasicategory and a morphism that is *equivalent* to a cofibration. In other words, a suitable class of cofibrations in a quasicategory $C$ will be uniquely specified by a subcategory of the homotopy category $hC$. We will thus define a *pair* of $\infty$-categories as an $\infty$-category along with a subcategory of the homotopy category. Among these pairs, we will isolate the Waldhausen $\infty$-categories in the next section.
In this section, we introduce the homotopy theory of pairs as a stepping stone on the way to defining the critically important homotopy theory of Waldhausen $\infty$-categories. As many constructions in the theory of Waldhausen $\infty$-categories begin with a construction at the level of pairs of $\infty$-categories, it is convenient to establish robust language and notation for these objects. To this end, we begin with a brief discussion of some set-theoretic considerations and a reminder on constructions of $\infty$-categories from simplicial categories and relative categories. We apply these to the construction of an $\infty$-category of $\infty$-categories and — following a short reminder on the notion of a *subcategory* of an $\infty$-category — an $\infty$-category ${\mathbf{Pair}}_{\infty}$ of pairs of $\infty$-categories. Finally, we relate this $\infty$-category of pairs to an $\infty$-category of functors between $\infty$-categories; this permits us to exhibit ${\mathbf{Pair}}_{\infty}$ as a *relative nerve*.
Set theoretic considerations {#set-theoretic-considerations .unnumbered}
----------------------------
In order to circumvent the set-theoretic difficulties arising from the consideration of these $\infty$-categories of $\infty$-categories and the like, we must employ some artifice. Hence to the usual Zermelo–Frankel axioms <span style="font-variant:small-caps;">zfc</span> of set theory (including the Axiom of Choice) we add the following *Universe Axiom* of Grothendieck and Verdier [@MR50:7130 Exp I, §0]. The resulting set theory, called <span style="font-variant:small-caps;">zfcu</span>, will be employed in this paper.
Any set is an element of a universe.
This axiom is independent of the others of <span style="font-variant:small-caps;">zfc</span>, since any universe ${\mathbf{U}}$ is itself a model of Zermelo–Frankel set theory. Equivalently, we assume that for any cardinal $\tau$, there exists a strongly inaccessible cardinal $\kappa$ with $\tau<\kappa$; for any strongly inaccessible cardinal $\kappa$, the set ${\mathbf{V}}_{\kappa}$ of sets whose rank is strictly smaller than $\kappa$ is a universe [@MR0244035].
In addition, we fix, once and for all, three uncountable, strongly inaccessible cardinals $\kappa_0<\kappa_1<\kappa_2$ and the corresponding universes ${\mathbf{V}}_{\kappa_0}\in{\mathbf{V}}_{\kappa_1}\in{\mathbf{V}}_{\kappa_2}$. Now a set, simplicial set, category, etc., will be said to be ***small*** if it is contained in the universe ${\mathbf{V}}_{\kappa_0}$; it will be said to be ***large*** if it is contained in the universe ${\mathbf{V}}_{\kappa_1}$; and it will be said to be ***huge*** if it is contained in the universe ${\mathbf{V}}_{\kappa_2}$. We will say that a set, simplicial set, category, etc., is ***essentially small*** if it is equivalent (in the appropriate sense) to a small one.
Simplicial nerves and relative nerves {#simplicial-nerves-and-relative-nerves .unnumbered}
-------------------------------------
There are essentially two ways in which $\infty$-categories will arise in the sequel. The first of these is as *simplicial categories*. We follow the model of [@HTT Df. 3.0.0.1] for the notation of simplicial nerves.
\[ntn:superscriptsscats\] A ***simplicial category*** — that is, a category enriched in the category of simplicial sets — will frequently be denoted with a superscript $(-)^{\Delta}$.
Suppose ${\mathbf{C}}^{\Delta}$ a simplicial category. Then we write $({\mathbf{C}}^{\Delta})_{0}$ for the ordinary category given by taking the $0$-simplices of the $\operatorname{Mor}$ spaces. That is, $({\mathbf{C}}^{\Delta})_{0}$ is the category whose objects are the objects of ${\mathbf{C}}$, and whose morphisms are given by $$({\mathbf{C}}^{\Delta})_{0}(x,y){\mathrel{\mathop:}=}{\mathbf{C}}^{\Delta}(x,y)_0.$$ If the $\operatorname{Mor}$ spaces of ${\mathbf{C}}^{\Delta}$ are all fibrant, then we will often write $${\mathbf{C}}\textrm{\quad for the simplicial nerve\quad}N({\mathbf{C}}^{\Delta})$$ [@HTT Df. 1.1.5.5], which is an $\infty$-category [@HTT Pr. 1.1.5.10].
It will also be convenient to have a model of various $\infty$-categories as *relative categories* [@BarKan1011]. To make this precise, we recall the following.
\[dfn:relnerve\] A ***relative category*** is an ordinary category $C$ along with a subcategory $wC$ that contains the identity maps of $C$. The maps of $wC$ will be called ***weak equivalences***. A ***relative functor*** ${{(C,wC)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(D,wD)}}$ is a functor ${{C}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {D}}$ that carries $wC$ to $wD$.
Suppose $(C,wC)$ a relative category. A ***relative nerve*** of $(C, wC)$ consists of an $\infty$-category $A$ equipped and a functor $p\colon{{NC}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {A}}$ that satisfies the following universal property. For any $\infty$-category $B$, the induced functor $${{\operatorname{Fun}(A,B)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}(NC,B)}}$$ is fully faithful, and its essential image is the full subcategory spanned by those functors ${{NC}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {B}}$ that carry the edges of $wC$ to equivalences in $B$. We will say that the functor $p$ ***exhibits $A$ as a relative nerve of $(C,wC)$***.
Since relative nerves are defined via a universal property, they are unique up to a contractible choice. Conversely, note that the property of being a relative nerve is invariant under equivalences of $\infty$-categories. That is, if $(C,wC)$ is a relative category, then for any commutative diagram $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{&NC&\\
A'&&A\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-2) edge node[above]{$p'$} (m-2-1)
edge node[above]{$p$} (m-2-3)
(m-2-1) edge node[below]{$\sim$} (m-2-3);
\end{tikzpicture}$$ in which ${{A'}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {A}}$ is an equivalence of $\infty$-categories, the functor $p'$ exhibits $A'$ as a relative nerve of $(C,wC)$ if and only if $p$ exhibits $A$ as a relative nerve of $(C,wC)$.
\[nul:constructrelnerve\] There are several functorial constructions of a relative nerve of a relative category $(C,wC)$, all of which are (necessarily) equivalent.
1. One may form the hammock localization $\mathrm{L}^{\!\mathrm{H}}(C,wC)$ [@MR81h:55019]; then a relative nerve can be constructed as the simplicial nerve of the natural functor ${{C}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {R(\mathrm{L}^{\!\mathrm{H}}(C,wC))}}$, where $R$ denotes any fibrant replacement for the Bergner model structure [@MR2276611].
2. One may mark the edges of $NC$ that correspond to weak equivalences in $C$ to obtain a marked simplicial set [@HTT §3.1]; then one may use the cartesian model structure on marked simplicial sets (over $\Delta^0$) to find a marked anodyne morphism $${{(NC,NwC)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(N(C,wC),\iota N(C,wC))}},$$ wherein $N(C,wC)$ is an $\infty$-category. This map then exhibits the $\infty$-category $N(C,wC)$ as a relative nerve of $(C,wC)$.
3. A relative nerve can be constructed as a fibrant model of the homotopy pushout in the Joyal model structure [@HTT §2.2.5] on simplicial sets of the map $${{\coprod_{\phi\in wC}\Delta^1}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\coprod_{\phi\in wC}\Delta^0}}$$ along the map ${{\coprod_{\phi\in wC}\Delta^1}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {NC}}$.
The $\infty$-category of $\infty$-categories {#the-infty-category-of-infty-categories .unnumbered}
--------------------------------------------
The homotopy theory of $\infty$-categories is encoded first as a simplicial category, and then, by application of the simplicial nerve [@HTT Df. 1.1.5.5], as an $\infty$-category. This is a pattern that we will follow to describe the homotopy theory of pairs of $\infty$-categories below in Nt. \[ntn:pair\].
To begin, recall that an ordinary category $C$ contains a largest subgroupoid, which consists of all objects of $C$ and all isomorphisms between them. The higher categorical analogue of this follows.
\[ntn:interior\] For any $\infty$-category $A$, there exists a simplicial subset $\iota A\subset A$, which is the largest Kan simplicial subset of $A$ [@HTT 1.2.5.3]. We shall call this space the ***interior $\infty$-groupoid of $A$***. The assignment ${{A}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {\iota A}}$ defines a right adjoint $\iota$ to the inclusion functor $u$ from Kan simplicial sets to $\infty$-categories.
We may think of $\iota A$ as the *moduli space of objects* of $A$, to which we alluded in the introduction.
\[ntn:catofcat\] The large simplicial category ${\mathbf{Kan}}^{\Delta}$ is the category of small Kan simplicial sets, with the usual notion of mapping space. The large simplicial category ${\mathbf{Cat}}^{\Delta}_{\infty}$ is defined in the following manner [@HTT Df. 3.0.0.1]. The objects of ${\mathbf{Cat}}^{\Delta}_{\infty}$ are small $\infty$-categories, and for any two $\infty$-categories $A$ and $B$, the morphism space $${\mathbf{Cat}}^{\Delta}_{\infty}(A,B){\mathrel{\mathop:}=}\iota\operatorname{Fun}(A,B)$$ is the interior $\infty$-groupoid of the $\infty$-category $\operatorname{Fun}(A,B)$.
Similarly, we may define the huge simplicial category ${\mathbf{Kan}}(\kappa_1)^{\Delta}$ of large simplicial sets and the huge simplicial category ${\mathbf{Cat}}_{\infty}(\kappa_1)^{\Delta}$ of large $\infty$-categories.
\[nul:Catrelnerve\] Denote by $$w({\mathbf{Kan}}^{\Delta})_0\subset({\mathbf{Kan}}^{\Delta})_0$$ the subcategory of the ordinary category of Kan simplicial sets (Nt. \[ntn:superscriptsscats\]) consisting of weak equivalences of simplicial sets. Then, since $({\mathbf{Kan}}^{\Delta},w({\mathbf{Kan}}^{\Delta})_0)$ is part of a simplicial model structure, it follows that ${\mathbf{Kan}}$ is a relative nerve of $(({\mathbf{Kan}}^{\Delta})_0,w({\mathbf{Kan}}^{\Delta})_0)$. Similarly, if one denotes by $$w({\mathbf{Cat}}_{\infty}^{\Delta})_{0}\subset({\mathbf{Cat}}_{\infty}^{\Delta})_{0}$$ the subcategory of categorical equivalences of $\infty$-categories, then ${\mathbf{Cat}}_{\infty}$ is a relative nerve (Df. \[dfn:relnerve\]) of $({\mathbf{Cat}}^{\Delta}_{\infty})_{0},w({\mathbf{Cat}}^{\Delta}_{\infty})_{0})$. This follows from [@HTT Pr. 3.1.3.5, Pr. 3.1.3.7, Cor. 3.1.4.4].
Since the functors $u$ and $\iota$ (Nt. \[ntn:interior\]) each preserve weak equivalences, they give rise to an adjunction of $\infty$-categories [@HTT Df. 5.2.2.1, Cor. 5.2.4.5] $${{u}\colon{{\mathbf{Kan}}}\ \begin{tikzpicture}[baseline] \draw[>=stealth,->] (0,1ex) -- (0.75,1ex); \draw[>=stealth,->] (0.75,0.25ex) -- (0,0.25ex); \end{tikzpicture}\ {{\mathbf{Cat}}_{\infty}}\colon{\iota}}.$$
Subcategories of $\infty$-categories {#subcategories-of-infty-categories .unnumbered}
------------------------------------
The notion of a *subcategory* of an $\infty$-category is designed to be completely homotopy-invariant. Consequently, given an $\infty$-category $A$ and a simplicial subset $A'\subset A$, we can only call $A'$ a subcategory of $A$ if the following condition holds: any two equivalent morphisms of $A$ both lie in $A'$ just in case either of them does. That is, $A'\subset A$ is completely specified by a subcategory $(hA)'\subset hA$ of the homotopy category $hA$ of $A$.
\[rec:subcats\] Recall [@HTT §1.2.11] that a ***subcategory*** of an $\infty$-category $A$ is a simplicial subset $A'\subset A$ such that for some subcategory $(hA)'$ of the homotopy category $hA$, the square $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{A'&A\\
N(hA)'&N(hA)\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge[right hook->] (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge[right hook->] (m-2-2);
\end{tikzpicture}$$ is a pullback diagram of simplicial sets. In particular, note that a subcategory of an $\infty$-category is uniquely specified by specifying a subcategory of its homotopy category. Note also that any inclusion ${{A'}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {A}}$ of a subcategory is an inner fibration [@HTT Df. 2.0.0.3, Pr. 2.3.1.5].
We will say that $A'\subset A$ is a ***full subcategory*** if $(hA)'\subset hA$ is a full subcategory. In this case, $A'$ is uniquely determined by the set $A'_0$ of vertices of $A'$, and we say that $A'$ is ***spanned by*** the set $A'_0$.
We will say that $A'$ is ***stable under equivalences*** if the subcategory $(hA)'\subset hA$ above can be chosen to be stable under isomorphisms. Note that any inclusion ${{A'}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {A}}$ of a subcategory that is stable under equivalences is a categorical fibration, i.e., a fibration for the Joyal model structure [@HTT Cor. 2.4.6.5].
Pairs of $\infty$-categories {#pairs-of-infty-categories-1 .unnumbered}
----------------------------
Now we are prepared to introduce the notion of a *pair* of $\infty$-categories.
\[dfn:pair\]
1. By a ***pair $(\mathscr{C},\mathscr{C}_{\dag})$ of $\infty$-categories*** (or simply a ***pair***), we shall mean an $\infty$-category $\mathscr{C}$ along with a subcategory $\mathscr{C}_{\dag}\subset\mathscr{C}$ containing the maximal Kan complex $\iota\mathscr{C}\subset\mathscr{C}$. We shall call $\mathscr{C}$ the ***underlying $\infty$-category*** of the pair $(\mathscr{C},\mathscr{C}_{\dag})$. A morphism of $\mathscr{C}_{\dag}$ will be said to be an ***ingressive*** morphism or a ***cofibration***.
2. A ***functor of pairs*** $\psi\colon{{(\mathscr{C},\mathscr{C}_{\dag})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(\mathscr{D},\mathscr{D}_{\dag})}}$ is functor ${{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{D}}}$ that carries cofibrations to cofibations; that is, it is a commutative diagram
$$\label{eqn:functorofpairs}
\begin{tikzpicture}[baseline]
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\mathscr{C}_{\dag}&\mathscr{D}_{\dag}\\
\mathscr{C}&\mathscr{D}\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge node[above]{$\psi_{\dag}$} (m-1-2)
edge[right hook->] (m-2-1)
(m-1-2) edge[left hook->] (m-2-2)
(m-2-1) edge node[below]{$\psi$} (m-2-2);
\end{tikzpicture}$$
of $\infty$-categories.
3. A functor of pairs ${{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{D}}}$ is said to be ***strict*** if a morphism of $\mathscr{C}$ is ingressive just in case its image in $\mathscr{D}$ is so — that is, if the diagram is a pullback diagram in ${\mathbf{Cat}}_{\infty}$.
4. \[item:subpair\] A ***subpair*** of a pair $(\mathscr{C},\mathscr{C}_{\dag})$ is a subcategory $\mathscr{D}\subset\mathscr{C}$ equipped with a pair structure $(\mathscr{D},\mathscr{D}_{\dag})$ such that the inclusion ${{\mathscr{D}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$ is a strict functor of pairs. If the subcategory $\mathscr{D}\subset\mathscr{C}$ is full, then we’ll say that $(\mathscr{D},\mathscr{D}_{\dag})$ is a ***full subpair*** of $(\mathscr{C},\mathscr{C}_{\dag})$.
Since a subcategory of an $\infty$-category is uniquely specified by a subcategory of its homotopy category, and since a morphism of an $\infty$-category is an equivalence if and only if the corresponding morphism of the homotopy category is an isomorphism [@HTT Pr. 1.2.4.1], we deduce that a pair $(\mathscr{C},\mathscr{C}_{\dag})$ of $\infty$-categories may simply be described as an $\infty$-category $\mathscr{C}$ and a subcategory $(h\mathscr{C})_{\dag}\subset h\mathscr{C}$ of the homotopy category that contains all the isomorphisms. In particular, note that $\mathscr{C}_{\dag}$ contains all the objects of $\mathscr{C}$.
Note that pairs are a bit rigid: if $(\mathscr{C},\mathscr{C}_{\dag})$ and $(\mathscr{C},\mathscr{C}_{\dag\dag})$ are two pairs, then any equivalence of $\infty$-categories ${{\mathscr{C}_{\dag}}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\mathscr{C}_{\dag\dag}}}$ that is compatible with the inclusions into $\mathscr{C}$ must be the identity. It follows that for any equivalence of $\infty$-categories ${{C}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {D}}$, the set of pairs with underlying $\infty$-category $C$ is in bijection with the set of pairs with underlying $\infty$-category $D$.
Consequently, we will often identify a pair $(\mathscr{C},\mathscr{C}_{\dag})$ of $\infty$-categories by defining the underlying $\infty$-category $\mathscr{C}$ and then declaring which morphisms of $\mathscr{C}$ are ingressive. As long as the condition given holds for all equivalences and is stable under homotopies between morphisms and under composition, this will specify a well-defined pair of $\infty$-categories.
Suppose $(\mathscr{C},\mathscr{C}_{\dag})$ a pair. Then a cofibration will frequently be denoted by an arrow with a tail: $\!{{}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ {}}\!$. We will often abuse notation by simply writing $\mathscr{C}$ for the pair $(\mathscr{C},\mathscr{C}_{\dag})$.
\[exm:minpairmaxpair\] Any $\infty$-category $C$ can be given the structure of a pair in two ways: the ***minimal pair*** $C^{\flat}{\mathrel{\mathop:}=}(C,\iota C)$ and the ***maximal pair*** $C^{\sharp}{\mathrel{\mathop:}=}(C,C)$.
The $\infty$-category of pairs {#the-infty-category-of-pairs .unnumbered}
------------------------------
We describe an $\infty$-category ${\mathbf{Pair}}_{\infty}$ of pairs of $\infty$-categories in much the same manner as we described the $\infty$-category ${\mathbf{Cat}}_{\infty}$ of $\infty$-categories (Nt. \[ntn:catofcat\]).
\[ntn:pair\] Suppose $\mathscr{C}=(\mathscr{C},\mathscr{C}_{\dag})$ and $\mathscr{D}=(\mathscr{D},\mathscr{D}_{\dag})$ two pairs of $\infty$-categories. Let us denote by $\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{C},\mathscr{D})$ the the full subcategory of the $\infty$-category $\operatorname{Fun}(\mathscr{C},\mathscr{D})$ spanned by the functors ${{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{D}}}$ that carry cofibrations to cofibrations.
The large simplicial category ${\mathbf{Pair}}^{\Delta}_{\infty}$ is defined in the following manner. The objects of ${\mathbf{Pair}}^{\Delta}_{\infty}$ are small pairs of $\infty$-categories, and for any two pairs of $\infty$-categories $\mathscr{C}$ and $\mathscr{D}$, the morphism space ${\mathbf{Pair}}^{\Delta}_{\infty}(\mathscr{C},\mathscr{D})$ is interior $\infty$-groupoid (Nt. \[ntn:interior\]) $${\mathbf{Pair}}^{\Delta}_{\infty}(\mathscr{C},\mathscr{D}){\mathrel{\mathop:}=}\iota\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{C},\mathscr{D}).$$ Note that ${\mathbf{Pair}}^{\Delta}_{\infty}(\mathscr{C},\mathscr{D})$ is the union of connected components of ${\mathbf{Cat}}_{\infty}^{\Delta}(\mathscr{C},\mathscr{D})$ that correspond to functors of pairs.
Now the $\infty$-category ${\mathbf{Pair}}_{\infty}$ is the simplicial nerve of this simplicial category (Nt. \[ntn:superscriptsscats\]).
Pair structures {#pair-structures .unnumbered}
---------------
It will be convenient to describe pairs of $\infty$-categories as certain functors between $\infty$-categories. This will permit us to exhibit ${\mathbf{Pair}}_{\infty}$ as a full subcategory of the arrow $\infty$-category $\operatorname{Fun}(\Delta^{1},{\mathbf{Cat}}_{\infty})$. This description will in fact imply (Pr. \[thm:pairasloc\]) that the $\infty$-category ${\mathbf{Pair}}_{\infty}$ is presentable.
For any simplicial set $X$, write $$\mathscr{O}(X){\mathrel{\mathop:}=}\operatorname{Map}(\Delta^1,X).$$ If $C$ is an $\infty$-category, then $\mathscr{O}(C)=\operatorname{Fun}(\Delta^1,C)$ is an $\infty$-category as well [@HTT Pr. 1.2.7.3]; this is the ***arrow $\infty$-category*** of $C$. (In fact, $\mathscr{O}$ is a right Quillen functor for the Joyal model structure, since this model structure is cartesian.)
Suppose $C$ and $D$ $\infty$-categories. We say that a functor ${{D}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}$ ***exhibits a pair structure on $C$*** if it factors as an equivalence ${{D}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {E}}$ followed by an inclusion ${{E}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {C}}$ of a subcategory such that $(C,E)$ is a pair.
\[lem:pairstruct\] Suppose $C$ and $D$ $\infty$-categories. Then a functor $\psi\colon{{D}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}$ exhibits a pair structure on $C$ if and only if the following conditions are satisfied.
1. The functor $\psi$ induces an equivalence ${{\iota D}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\iota C}}$.
2. The functor $\psi$ is a (homotopy) monomorphism in the $\infty$-category ${\mathbf{Cat}}_{\infty}$; i.e., the diagonal morphism $${{D}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {D\times^h_CD}}$$ in $h{\mathbf{Cat}}_{\infty}$ is an isomorphism.
Clearly any equivalence of $\infty$-categories satisfies these criteria. If $\psi$ is an inclusion of a subcategory such that $(C,D)$ is a pair, then $\psi$, restricted to $\iota D$, is the identity map, and it is an inner fibration such that the diagonal map ${{D}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {D\times_CD}}$ is an isomorphism. This shows that if $\psi$ exhibits a pair structure on $C$, then $\psi$ satisfies the conditions listed.
Conversely, suppose $\psi$ satisfies the conditions listed. Then it is hard not to show that for any objects $x,y\in D$, the functor $\psi$ induces a homotopy monomorphism $${{\operatorname{Map}_D(x,y)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Map}_C(\psi(x),\psi(y))}},$$ whence the natural map $${{\operatorname{Map}_D(x,y)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Map}_{NhD}(x,y)\times^h_{\operatorname{Map}_{NhC}(\psi(x),\psi(y))}\operatorname{Map}_C(\psi(x),\psi(y))}}$$ of the homotopy category $h{\mathbf{Kan}}$ is an isomorphism. This, combined with the fact that the map ${{\iota D}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\iota C}}$ is an equivalence, now implies that the natural map ${{D}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {NhD\times^h_{NhC}C}}$ of $h{\mathbf{Cat}}_{\infty}$ is an isomorphism.
Since isomorphisms in $hC$ are precisely equivalences in $C$, the induced functor of homotopy categories ${{hD}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {hC}}$ identifies $hD$ with a subcategory of $hC$ that contains all the isomorphisms. Denote by $hE\subset hC$ this subcategory. Now let $E$ be the corresponding subcategory of $C$; we thus have a diagram of $\infty$-categories $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{D&E&C\\
NhD&NhE&NhC\\};
\path[>=stealth,->,font=\scriptsize,inner sep=0.75pt]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge[right hook->] (m-1-3)
edge (m-2-2)
(m-1-3) edge (m-2-3)
(m-1-2) edge (m-2-2)
(m-2-1) edge node[below]{$\sim$} (m-2-2)
(m-2-2) edge[right hook->] (m-2-3);
\end{tikzpicture}$$ in which the square on the right and the big rectangle are homotopy pullbacks (for the Joyal model structure). Thus the square on the left is a homotopy pullback as well, and so the functor ${{D}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {E}}$ is an equivalence, giving our desired factorization.
\[prp:PairsubcatOcat\] The functor ${{{\mathbf{Pair}}_{\infty}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{O}({\mathbf{Cat}}_{\infty})}}$ that sends a pair $(\mathscr{C},\mathscr{C}_{\dag})$ to the inclusion functor ${{\mathscr{C}_{\dag}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$ induces an equivalence between ${\mathbf{Pair}}_{\infty}$ and the full subcategory of $\mathscr{O}({\mathbf{Cat}}_{\infty})$ spanned by those functors ${{D}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}$ that exhibit a pair structure on $C$.
The essential surjectivity follows from Lm. \[lem:pairstruct\], and the full faithfulness follows directly from the definitions of the mapping spaces in ${\mathbf{Pair}}_{\infty}^{\Delta}$.
The $\infty$-categories of pairs as a relative nerve {#the-infty-categories-of-pairs-as-a-relative-nerve .unnumbered}
----------------------------------------------------
It will be convenient for us to have a description of ${\mathbf{Pair}}_{\infty}$ as a relative nerve (Df. \[dfn:relnerve\]). First, we record the following trivial result.
\[lem:functorofpairsequiv\] The following are equivalent for a functor of pairs $\psi\colon{{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{D}}}$.
1. The functor of pairs $\psi$ is an equivalence in the $\infty$-category ${\mathbf{Pair}}_{\infty}$.
2. The underlying functor of $\infty$-categories is a categorical equivalence, and $\psi$ is strict.
3. The underlying functor of $\infty$-categories is a categorical equivalence that induces an equivalence $h\mathscr{C}_{\dag}\simeq h\mathscr{D}_{\dag}$.
The equivalence of the first two conditions follows from the equivalence between ${\mathbf{Pair}}_{\infty}$ and a full subcategory of $\mathscr{O}({\mathbf{Cat}}_{\infty})$. The second condition clearly implies the third. To prove that the third condition implies the second, consider the commutative diagram $$\begin{tikzpicture}[cross line/.style={preaction={draw=white, -,
line width=6pt}}]
\matrix(m)[matrix of math nodes,
row sep=2ex, column sep=0.75ex,
text height=1.5ex, text depth=0.25ex]
{&\mathscr{D}_{\dag}&&\mathscr{D}\\
\mathscr{C}_{\dag}&&\mathscr{C}&\\
&Nh\mathscr{D}_{\dag}&&Nh\mathscr{D}.\\
Nh\mathscr{C}_{\dag}&&Nh\mathscr{C}&\\
};
\path[>=stealth,->,font=\scriptsize]
(m-1-2) edge[<-] (m-2-1)
edge (m-3-2)
edge[right hook->] (m-1-4)
(m-3-2) edge[<-] (m-4-1)
edge[right hook->] (m-3-4)
(m-2-1) edge[cross line,right hook->] (m-2-3)
edge (m-4-1)
(m-1-4) edge[<-] (m-2-3)
edge (m-3-4)
(m-4-1) edge[right hook->] (m-4-3)
(m-3-4) edge[<-] (m-4-3)
(m-2-3) edge[cross line] (m-4-3);
\end{tikzpicture}$$ The front and back faces are pullback squares and therefore homotopy pullback squares. Since both ${{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\mathscr{D}}}$ and ${{Nh\mathscr{C}_{\dag}}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {Nh\mathscr{D}_{\dag}}}$ are equivalences, the bottom face is a homotopy pullback as well. Hence the top square is a homotopy pullback. But since $(\mathscr{C},\mathscr{C}_{\dag})$ is a pair, it must be an actual pullback; that is, $\psi$ is strict.
This proposition implies that the $\infty$-category of functors of pairs is compatible with equivalences of pairs.
Suppose $\mathscr{A}$ a pair, and suppose ${{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\mathscr{D}}}$ an equivalence of pairs of $\infty$-categories. Then the induced functor ${{\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{A},\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{A},\mathscr{D})}}$ is an equivalence of $\infty$-categories.
The proposition implies that any homotopy inverse ${{\mathscr{D}}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\mathscr{C}}}$ of the equivalence ${{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\mathscr{D}}}$ of underlying $\infty$-categories must carry cofibrations to cofibrations. This induces a homotopy inverse ${{\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{A},\mathscr{D})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{A},\mathscr{C})}}$, completing the proof.
Furthermore, Pr. \[lem:functorofpairsequiv\] may be combined with Pr. \[prp:PairsubcatOcat\] and \[nul:Catrelnerve\] to yield the following.
\[prp:Pairisrelnerve\] Denote by $w({\mathbf{Pair}}_{\infty}^{\Delta})_0\subset({\mathbf{Pair}}_{\infty}^{\Delta})_0$ the subcategory of the ordinary category of pairs of $\infty$-categories consisting of those functors of pairs ${{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{D}}}$ whose underlying functor of $\infty$-categories is a categorical equivalence that induces an equivalence $h\mathscr{C}_{\dag}\simeq h\mathscr{D}_{\dag}$. Then the $\infty$-category ${\mathbf{Pair}}_{\infty}$ is a relative nerve of the relative category $(({\mathbf{Pair}}_{\infty}^{\Delta})_0,w({\mathbf{Pair}}_{\infty}^{\Delta})_0)$.
The dual picture {#the-dual-picture .unnumbered}
----------------
Let us conclude this section by briefly outlining the dual picture of $\infty$-categories with *fibrations*.
\[dfn:egressives\] Suppose $(\mathscr{C}^{{\mathrm{op}}},(\mathscr{C}^{{\mathrm{op}}})_{\dag})$ a pair. Then write $\mathscr{C}^{\dag}$ for the subcategory $$((\mathscr{C}^{{\mathrm{op}}})_{\dag})^{{\mathrm{op}}}\subset\mathscr{C}.$$ We call the morphisms of $\mathscr{C}^{\dag}$ ***egressive*** morphisms or ***fibrations***. The pair $(\mathscr{C},\mathscr{C}^{\dag})$ will be called the ***opposite pair*** to $(\mathscr{C}^{{\mathrm{op}}},(\mathscr{C}^{{\mathrm{op}}})_{\dag})$. One may abuse terminology slightly by referring to $(\mathscr{C},\mathscr{C}^{\dag})$ as a ***pair structure on $\mathscr{C}^{{\mathrm{op}}}$***.
Suppose $(\mathscr{C}^{{\mathrm{op}}},(\mathscr{C}^{{\mathrm{op}}})_{\dag})$ a pair. Then a fibration of $\mathscr{C}$ will frequently be denoted by a double headed arrow: $\!{{}\ \tikz[baseline]\draw[>=stealth,->>](0,0.5ex)--(0.5,0.5ex);\ {}}\!$. We will often abuse notation by simply writing $\mathscr{C}$ for the opposite pair $(\mathscr{C},\mathscr{C}^{\dag})$.
We summarize this discussion with the following.
\[prp:opinvolonPair\] The formation ${{(\mathscr{C},\mathscr{C}_{\dag})}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {(\mathscr{C}^{{\mathrm{op}}},(\mathscr{C}^{{\mathrm{op}}})^{\dag})}}$ of the opposite pair defines an involution $(-)^{{\mathrm{op}}}$ of the $\infty$-category ${\mathbf{Pair}}_{\infty}$.
Waldhausen $\infty$-categories
==============================
In developing his abstract framework for $K$-theory, Waldhausen introduced first [@MR86m:18011 §1.1] the notion of a *category with cofibrations*, and then [@MR86m:18011 §1.2] layered the added structure of a subcategory of weak equivalences satisfying some additional compatibilities to obtain what today is often called a *Waldhausen category*. This added structure introduces homotopy theory, and Waldhausen required that the structure of a category with cofibrations interacts well with this homotopy theory.
The theory of *Waldhausen $\infty$-categories*, which we introduce in this section, reverses these two priorities. The layer of homotopy theory is already embedded in the implementation of quasicategories. Then, because it is effectively impossible to formulate $\infty$-categorical notions that do *not* interact well with the homotopy theory, we arrive at a suitable definition of Waldhausen $\infty$-categories by writing the quasicategorical analogues of the axioms for Waldhausen’s categories with cofibrations. Consequently, a Waldhausen $\infty$-category will be a pair of $\infty$-categories that enjoys the following properties.
(A) The underlying $\infty$-category admits a *zero object* $0$ such that the morphisms ${{0}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X}}$ are all ingressive.
(B) *Pushouts* of cofibrations exist and are cofibrations.
Limits and colimits in $\infty$-categories {#limits-and-colimits-in-infty-categories .unnumbered}
------------------------------------------
To work with these conditions effectively, it is convenient to fix some notations and terminology for the study of *limits* and *colimits* in $\infty$-categories, as defined in [@HTT §1.2.13].
Recall [@HTT Df. 1.2.12.1] that an object $X$ of an $\infty$-category $C$ is said to be ***initial*** if for any object $Y$ of $C$, the mapping space $\operatorname{Map}(X,Y)$ is weakly contractible. Dually, $X$ is said to be ***terminal*** if for any object $Y$ of $C$, the mapping space $\operatorname{Map}(Y,X)$ is weakly contractible.
A ***zero object*** of an $\infty$-category is an object that is both initial and terminal.
For any simplicial set $K$, one has [@HTT Nt. 1.2.8.4] the ***right cone*** $K^{\rhd}{\mathrel{\mathop:}=}K\star\Delta^0$ and the ***left cone*** $K^{\lhd}{\mathrel{\mathop:}=}\Delta^0\star K$; we write $+\infty$ for the cone point of $K^{\rhd}$, and we write $-\infty$ for the cone point of $K^{\lhd}$.
Just as in ordinary category theory, a colimit and limit in an $\infty$-category can be described as an initial and terminal object of a suitable associated $\infty$-category. For any simplicial set $K$, a ***limit diagram*** in an $\infty$-category $C$ is a diagram $$\overline{p}\colon{{K^{\lhd}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}$$ that is a terminal object in the overcategory $C_{/p}$ [@HTT §1.2.9], where $p=\overline{p}|K$. Dually, a ***colimit diagram*** in an $\infty$-category $C$ is a diagram $$\overline{p}\colon{{K^{\rhd}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}$$ that is a terminal object in the undercategory $C_{p/}$, where $p=\overline{p}|K$.
For any $\infty$-category $A$ and any $\infty$-category $C$, we denote by $$\operatorname{Colim}(A^{\rhd},C)\subset\operatorname{Fun}(A^{\rhd},C)$$ the full subcategory spanned by colimit diagrams ${{A^{\rhd}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}$.
A ***pushout square*** in an $\infty$-category $C$ is a colimit diagram $$X\colon{{(\Lambda^2_0)^{\rhd}\cong\Delta^1\times\Delta^1}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}.$$ Such a diagram may be drawn $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{X_{00}&X_{01}\\
X_{10}&X_{11};\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge (m-2-2);
\end{tikzpicture}$$ the edge ${{X_{10}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X_{11}}}$ is called the ***pushout*** of the edge ${{X_{00}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X_{01}}}$.
\[rec:Ashapedcolim\] A key result of Joyal [@HTT Pr. 1.2.12.9] states that for any functor $\psi\colon{{A}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}$, the fiber of the canonical restriction functor $$\rho\colon{{\operatorname{Colim}(A^{\rhd},C)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}(A,C)}}$$ over $\psi$ is either empty or a contractible Kan space. One says that $C$ ***admits all $A$-shaped colimits*** if the fibers of the functor $\rho$ are all nonempty. In this case, $\rho$ is an equivalence of $\infty$-categories.
More generally, if $\mathscr{A}$ is a family of $\infty$-categories, then one says that $C$ ***admits all $\mathscr{A}$-shaped colimits*** if the fibers of the functor ${{\operatorname{Colim}(A^{\rhd},C)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}(A,C)}}$ are all nonempty for every $A\in\mathscr{A}$.
Finally, if $\mathscr{A}$ is a family of $\infty$-categories, then a functor $f\colon{{C'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}$ will be said to ***preserve all $\mathscr{A}$-shaped colimits*** if for any $A\in\mathscr{A}$, the composite $$\operatorname{Colim}(A^{\rhd},C')\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ \operatorname{Fun}(A^{\rhd},C')\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ \operatorname{Fun}(A^{\rhd},C)$$ factors through $\operatorname{Colim}(A^{\rhd},C)\subset\operatorname{Fun}(A^{\rhd},C)$. We write $\operatorname{Fun}_{\mathscr{A}}(C',C)\subset\operatorname{Fun}(C',C)$ for the full subcategory spanned by those functors that preserve all $\mathscr{A}$-shaped colimits.
Waldhausen $\infty$-categories {#waldhausen-infty-categories-1 .unnumbered}
------------------------------
We are now introduce the notion of *Waldhausen $\infty$-categories*, which are the primary objects of study in this work.
\[dfn:preWald\] A ***Waldhausen $\infty$-category*** $(\mathscr{C},\mathscr{C}_{\dag})$ is a pair of essentially small $\infty$-categories such that the following axioms hold.
1. The $\infty$-category $\mathscr{C}$ contains a zero object.
2. \[item:0toxingressive\] For any zero object $0$, any morphism ${{0}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X}}$ is ingressive.
3. \[item:pushcof\] Pushouts of cofibrations exist. That is, for any diagram $G\colon{{\Lambda^2_0}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$ represented as $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{X&Y\\
X'&\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge[>->] (m-1-2)
edge (m-2-1);
\end{tikzpicture}$$ in which the morphism ${{X}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ {Y}}$ is ingressive, there exists a pushout square $\overline{G}\colon{{(\Lambda^2_0)^{\rhd}\cong\Delta^1\times\Delta^1}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$ extending $G$: $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{X&Y\\
X'&Y'.\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge[>->] (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge (m-2-2);
\end{tikzpicture}$$
4. \[item:pushcofcof\] Pushouts of cofibrations are cofibrations. That is, for any pushout square ${{(\Lambda^2_0)^{\rhd}\cong\Delta^1\times\Delta^1}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$ represented as $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{X&Y\\
X'&Y',\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge[>->] (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge (m-2-2);
\end{tikzpicture}$$ if the morphism ${{X}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ {Y}}$ is ingressive, then so is the morphism ${{X'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y'}}$.
Call a functor of pairs $\psi\colon{{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{D}}}$ between two Waldhausen $\infty$-categories ***exact*** if it satisfies the following conditions. \[[\[(\[dfn:preWald\].1)\]]{}\]
5. The underlying functor of $\psi$ carries zero objects of $\mathscr{C}$ to zero objects of $\mathscr{D}$.
6. \[item:psipreservepushcof\] For any pushout square $F\colon{{(\Lambda^2_0)^{\rhd}\cong\Delta^1\times\Delta^1}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$ represented as $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{X&Y\\
X'&Y'\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge[>->] (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge[>->] (m-2-2);
\end{tikzpicture}$$ in which ${{X}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ {Y}}$ and hence ${{X'}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ {Y'}}$ are ingressive, the induced square $\psi\circ F\colon{{(\Lambda^2_0)^{\rhd}\cong\Delta^1\times\Delta^1}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{D}}}$ represented as $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\psi(X)&\psi(Y)\\
\psi(X')&\psi(Y')\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge[>->] (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge[>->] (m-2-2);
\end{tikzpicture}$$ is a pushout as well.
A ***Waldhausen subcategory*** of a Waldhausen $\infty$-category $\mathscr{C}$ is a subpair $\mathscr{D}\subset\mathscr{C}$ such that $\mathscr{D}$ is a Waldhausen $\infty$-category, and the inclusion ${{\mathscr{D}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$ is exact.
Let us repackage some of these conditions.
\[exm:LambdaDelta\] Denote by $\Lambda_0\mathscr{Q}^2$ the pair $(\Lambda^2_0,\Delta^{\{0,1\}}\sqcup\Delta^{\{2\}})$, which may be represented as $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{0&1\\
2.&\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge[>->] (m-1-2)
edge (m-2-1);
\end{tikzpicture}$$ Denote by $\mathscr{Q}^2$ the pair $$((\Lambda^2_0)^{\rhd},\Delta^{\{0,1\}}\sqcup\Delta^{\{2,\infty\}})\cong(\Delta^1)^{\flat}\times(\Delta^1)^{\sharp}\cong(\Delta^1\times\Delta^1,(\Delta^{\{0\}}\sqcup\Delta^{\{1\}})\times\Delta^1)$$ (Ex. \[exm:minpairmaxpair\]), which may be represented as $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{0&1\\
2&\infty.\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge[>->] (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge[>->] (m-2-2);
\end{tikzpicture}$$ There is an obvious strict inclusion of pairs ${{\Lambda_0\mathscr{Q}^2}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{Q}^2}}$.
Conditions (\[dfn:preWald\].\[item:pushcof\]) and (\[dfn:preWald\].\[item:pushcofcof\]) can be rephrased as the single condition that the functor $${{\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{Q}^2,\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\Lambda_0\mathscr{Q}^2,\mathscr{C})}}$$ induces an equivalence of $\infty$-categories $${{\operatorname{Colim}_{{\mathbf{Pair}}_{\infty}}(\mathscr{Q}^2,\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\Lambda_0\mathscr{Q}^2,\mathscr{C})}}$$ where $\operatorname{Colim}_{{\mathbf{Pair}}_{\infty}}(\mathscr{Q}^2,\mathscr{C})$ denotes the full subcategory of $\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{Q}^2,\mathscr{C})$ spanned by those functors of pairs ${{\mathscr{Q}^2}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$ whose underlying functor ${{(\Lambda^2_0)^{\rhd}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$ is a pushout square.
Condition (\[dfn:preWald\].\[item:psipreservepushcof\]) on a functor of pairs $\psi\colon{{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{D}}}$ between Waldhausen $\infty$-categories is equivalent to the condition that the composite functor $$\operatorname{Colim}_{{\mathbf{Pair}}_{\infty}}(\mathscr{Q}^2,\mathscr{C})\subset\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{Q}^2,\mathscr{C})\ {\tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);}\ \operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{Q}^2,\mathscr{D})$$ factors through the full subcategory $$\operatorname{Colim}_{{\mathbf{Pair}}_{\infty}}(\mathscr{Q}^2,\mathscr{D})\subset\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{Q}^2,\mathscr{D}).$$
Some examples {#some-examples .unnumbered}
-------------
To get a sense for how these axioms apply, let’s give some examples of Waldhausen $\infty$-categories.
\[exm:whenisminandmaxWald\] When equipped with the *minimal* pair structure (Ex. \[exm:minpairmaxpair\]), an $\infty$-category $C$ is a Waldhausen $\infty$-category $C^{\flat}$ if and only if $C$ is a contractible Kan complex.
Equipped with the *maximal* pair structure (Ex. \[exm:minpairmaxpair\]), any $\infty$-category $C$ that admits a zero object and all finite colimits can be regarded as a Waldhausen $\infty$-category $C^{\sharp}$.
\[exm:inftytopoigiveWaldcats\] As a special case of the above, suppose that $\mathscr{E}$ is an $\infty$-topos [@HTT Df. 6.1.0.2]. For example, one may consider the example $\mathscr{E}=\operatorname{Fun}(S,{\mathbf{Kan}})$ for some simplicial set $S$. Then the $\infty$-category $\mathscr{E}^{\omega}_{\ast}$ of compact, pointed objects of $\mathscr{E}$, when equipped with its maximal pair structure, is a Waldhausen $\infty$-category. Its algebraic $K$-theory will be called the *$A$-theory of $\mathscr{E}$*. For any Kan simplicial set $X$, the $A$-theory of the $\infty$-topos $\operatorname{Fun}(X,{\mathbf{Kan}})$ agrees with Waldhausen’s $A$-theory of $X$ (where one defines the latter via the category $\mathscr{R}_{\mathrm{df}}(X)$ of finitely dominated retractive spaces over $X$ [@MR86m:18011 p. 389]). See Ex. \[exm:Atheoryofinfintopoi\] for more.
Any stable $\infty$-category $\mathscr{A}$ [@HA Df. 1.1.1.9], when equipped with its maximal pair structure, is a Waldhausen $\infty$-category. If $\mathscr{A}$ admits a t-structure [@HA Df. 1.2.1.4], then one may define a pair structure on any of the $\infty$-categories $\mathscr{A}_{\leq n}$ by declaring that a morphism ${{X}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y}}$ be ingressive just in case the induced morphism ${{\pi_nX}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\pi_nY}}$ is a monomorphism of the heart $\mathscr{A}^{\heartsuit}$. We study the relationship between the algebraic $K$-theory of these $\infty$-categories to the algebraic $K$-theory of $\mathscr{A}$ itself in a follow-up to this paper [@K21].
\[exm:catwithcofibsisWaldinftycat\] If $(C,\operatorname{cof}C)$ is an ordinary ***category with cofibrations*** in the sense of Waldhausen [@MR86m:18011 §1.1], then the pair $(NC,N(\operatorname{cof}C))$ is easily seen to be a Waldhausen $\infty$-category. If $(C,\operatorname{cof}C,wC)$ is a category with cofibrations and weak equivalences in the sense of Waldhausen [@MR86m:18011 §1.2], then one may endow a relative nerve (Df. \[dfn:relnerve\]) $N(C,wC)$ of the relative category $(C,wC)$ with a pair structure by defining the subcategory $N(C,wC)_{\dag}\subset N(C,wC)$ as the smallest subcategory containing the equivalences and the images of the edges in $NC$ corresponding to cofibrations. In Pr. \[thm:classWaldareWald\], we will show that if $(C,wC)$ is a *partial model category* in which the weak equivalences and trivial cofibrations are part of a three-arrow calculus of fractions, then any relative nerve of $(C,wC)$ is in fact a Waldhausen $\infty$-category with this pair structure.
The $\infty$-category of Waldhausen $\infty$-categories {#the-infty-category-of-waldhausen-infty-categories .unnumbered}
-------------------------------------------------------
We now define the $\infty$-category of Waldhausen $\infty$-categories as a subcategory of the $\infty$-category of pairs.
\[ntn:preWald\]
1. Suppose $\mathscr{C}$ and $\mathscr{D}$ two Waldhausen $\infty$-categories. We denote by $\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{C},\mathscr{D})$ the full subcategory of $\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{C},\mathscr{D})$ spanned by the exact functors ${{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{D}}}$ of Waldhausen $\infty$-categories.
2. Define ${\mathbf{Wald}_{\infty}}^{\Delta}$ as the following simplicial subcategory of ${\mathbf{Pair}}^{\Delta}_{\infty}$. The objects of ${\mathbf{Wald}_{\infty}}^{\Delta}$ are small Waldhausen $\infty$-categories, and for any Waldhausen $\infty$-categories $\mathscr{C}$ and $\mathscr{D}$, the morphism space ${\mathbf{Wald}_{\infty}}^{\Delta}(\mathscr{C},\mathscr{D})$ is defined by the formula $${\mathbf{Wald}_{\infty}}^{\Delta}(\mathscr{C},\mathscr{D}){\mathrel{\mathop:}=}\iota\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{C},\mathscr{D}),$$ or, equivalently, ${\mathbf{Wald}_{\infty}}^{\Delta}(\mathscr{C},\mathscr{D})$ is the union of the connected components of ${\mathbf{Pair}}^{\Delta}_{\infty}(\mathscr{C},\mathscr{D})$ corresponding to the exact morphisms.
3. We now define the $\infty$-category ${\mathbf{Wald}_{\infty}}$ as the simplicial nerve of ${\mathbf{Wald}_{\infty}}^{\Delta}$ (Nt. \[ntn:superscriptsscats\]), or, equivalently, as the subcategory of ${\mathbf{Pair}}_{\infty}$ whose objects are Waldhausen $\infty$-categories and whose morphisms are exact functors.
The subcategory ${\mathbf{Wald}_{\infty}}\subset{\mathbf{Pair}}_{\infty}$ is stable under equivalences.
Suppose $\mathscr{C}$ a Waldhausen $\infty$-category, and suppose $\psi\colon{{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\mathscr{D}}}$ an equivalence of pairs. The functor of pairs $\psi$ induces an equivalence of underlying $\infty$-categories, whence $\mathscr{D}$ admits a zero object as well. We also have, in the notation of Nt. \[exm:LambdaDelta\], a commutative square $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\operatorname{Colim}_{{\mathbf{Pair}}_{\infty}}(\mathscr{Q}^2,\mathscr{C})&\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\Lambda_0\mathscr{Q}^2,\mathscr{C})\\
\operatorname{Colim}_{{\mathbf{Pair}}_{\infty}}(\mathscr{Q}^2,\mathscr{D})&\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\Lambda_0\mathscr{Q}^2,\mathscr{D})\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge (m-2-2);
\end{tikzpicture}$$ in which the top functor is an equivalence since $\mathscr{C}$ is a Waldhausen $\infty$-category, and the vertical functors are equivalences since ${{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\mathscr{D}}}$ is an equivalence of pairs. Hence the bottom functor is an equivalence of $\infty$-categories, whence $\mathscr{D}$ is a Waldhausen $\infty$-category.
Equivalences between maximal Waldhausen $\infty$-categories {#equivalences-between-maximal-waldhausen-infty-categories .unnumbered}
-----------------------------------------------------------
Equivalences between Waldhausen $\infty$-categories with a *maximal* pair structure (Ex \[exm:whenisminandmaxWald\]) are often easy to detect, thanks to the following result.
\[prp:preapprox\] Suppose $\mathscr{C}$ and $\mathscr{D}$ two $\infty$-categories that each contain zero objects and all finite colimits. Regard them as Waldhausen $\infty$-categories equipped with the maximal pair structure . Assume that the suspension functors $\Sigma\colon{{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$ is essentially surjective. Then an exact functor $\psi\colon{{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{D}}}$ is an equivalence if and only if it induces an equivalence of homotopy categories ${{h\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {h\mathscr{D}}}$.
We need only show that $\psi$ is fully faithful. Since $\psi$ preserves all finite colimits [@HTT Cor. 4.4.2.5], it follows that $\psi$ preserves the tensor product with any finite Kan complex [@HTT Cor. 4.4.4.9]. Thus for any finite simplicial set $K$ and any objects $X$ and $Y$ of $\mathscr{C}$, the map $${{[K,\operatorname{Map}_{\mathscr{C}}(X,Y)]}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {[K,\operatorname{Map}_{\mathscr{D}}(\psi(X),\psi(Y))]}}$$ can be identified with the map $${{\pi_0\operatorname{Map}(X\otimes K,Y)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\pi_0\operatorname{Map}(\psi(X\otimes K),\psi(Y))\cong\pi_0\operatorname{Map}(\psi(X)\otimes K,\psi(Y))}}.$$ This map is a bijection for any finite simplicial set $K$. In particular, the map ${{\operatorname{Map}(X,Y)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Map}(\psi(X),\psi(Y))}}$ is a weak homotopy equivalence on the connected components at $0$, whence ${{\operatorname{Map}(\Sigma X,Y)}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\operatorname{Map}(\psi(\Sigma X),\psi(Y))}}$ is an equivalence. Now since every object in $\mathscr{C}$ is a suspension, the functor $\psi$ is fully faithful.
The dual picture {#the-dual-picture-1 .unnumbered}
----------------
Entirely dual to the theory of Waldhausen $\infty$-categories is the theory of *coWaldhausen $\infty$-categories*. We record the definition here; clearly any result or construction in the theory of Waldhausen $\infty$-categories can be immediately dualized.
\[dfn:coWald\]
1. A ***coWaldhausen $\infty$-category*** $(\mathscr{C},\mathscr{C}^{\dag})$ is an opposite pair $(\mathscr{C},\mathscr{C}^{\dag})$ such that the opposite $(\mathscr{C}^{{\mathrm{op}}},(\mathscr{C}^{{\mathrm{op}}})_{\dag})$ is a Waldhausen $\infty$-category.
2. A functor of pairs $\psi\colon{{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{D}}}$ between two coWaldhausen $\infty$-categories is said to be ***exact*** if its opposite $\psi^{{\mathrm{op}}}\colon{{\mathscr{C}^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{D}^{{\mathrm{op}}}}}$ is exact.
\[ntn:precoWald\]
1. Suppose $\mathscr{C}$ and $\mathscr{D}$ two coWaldhausen $\infty$-categories. Denote by $\operatorname{Fun}_{{\mathbf{coWald}}}(\mathscr{C},\mathscr{D})$ the full subcategory of $\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{C},\mathscr{D})$ spanned by the exact morphisms of coWaldhausen $\infty$-categories.
2. Define ${\mathbf{coWald}}^{\Delta}_{\infty}$ as the following large simplicial subcategory of ${\mathbf{Pair}}^{\Delta}_{\infty}$. The objects of ${\mathbf{coWald}}^{\Delta}_{\infty}$ are small coWaldhausen $\infty$-categories, and for any coWaldhausen $\infty$-categories $\mathscr{C}$ and $\mathscr{D}$, the morphism space is defined by the formula $${\mathbf{coWald}}^{\Delta}_{\infty}(\mathscr{C},\mathscr{D}){\mathrel{\mathop:}=}\iota\operatorname{Fun}_{{\mathbf{coWald}}}(\mathscr{C},\mathscr{D}),$$ or equivalently, ${\mathbf{coWald}}^{\Delta}_{\infty}(\mathscr{C},\mathscr{D})$ is the union of the connected components of ${\mathbf{Pair}}^{\Delta}_{\infty}(\mathscr{C},\mathscr{D})$ corresponding to the exact morphisms.
3. We then define an $\infty$-category ${\mathbf{coWald}}$ as the simplicial nerve (Df. \[dfn:relnerve\]) of the simplicial category ${\mathbf{coWald}}^{\Delta}_{\infty}$.
We summarize these constructions with the following.
The opposite involution on ${\mathbf{Pair}}_{\infty}$ restricts to an equivalence between ${\mathbf{Wald}_{\infty}}$ and ${\mathbf{coWald}}$.
Waldhausen fibrations {#sect:Waldfib}
=====================
A key component of Waldhausen’s algebraic $K$-theory of spaces is his $S_{\bullet}$ construction [@MR86m:18011 §1.3]. In effect, this is a diagram of categories $$S\colon{{\Delta^{op}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}}}$$ such that for any object $\mathbf{m}\in\Delta$, the category $S_m$ is the category of filtered spaces $$\ast=X_0\subset X_1\subset\cdots\subset X_m$$ of length $m$, and, for any simplicial operator $[\phi\colon{{\mathbf{n}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{m}}}]\in\Delta$, the induced functor $\phi_!\colon{{S_m}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S_n}}$ carries a filtered space $\ast=X_0\subset X_1\subset\cdots\subset X_m$ to a filtered space $$\ast=X_{\phi(0)}/X_{\phi(0)}\subset X_{\phi(1)}/X_{\phi(0)}\subset\cdots\subset X_{\phi(n)}/X_{\phi(0)}.$$ We will want to construct an $\infty$-categorical variant of $S_{\bullet}$, but there is a little wrinkle here: as written, this is not a functor on the nose. Rather, it is a *pseudofunctor*, because quotients are defined only up to (canonical) isomorphism. To rectify this, Waldhausen constructs [@MR86m:18011 §1.3] an honest functor by replacing each category $S_m$ with a fattening thereof, in which an object is a filtered space $$\ast=X_0\subset X_1\subset\cdots\subset X_m$$ along with compatible choices of all the quotient spaces $X_s/X_t$.
If one wishes to pass to a more homotopical variant of the $S_{\bullet}$ construction, matters become even more complicated. After all, any sequence of simplicial sets $$\ast\simeq X_0\ {\tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);}\ X_1\ {\tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);}\ \cdots\ {\tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);}\ X_m$$ can, up to homotopy, be regarded as a filtered space. To extend the $S_{\bullet}$ construction to accept these objects, a simplicial operator should then induce functor that carries such a sequence to a corresponding sequence of *homotopy* quotients, in which each map is replaced by a cofibration, and the suitable quotients are formed. This now presents not only a functoriality problem but also a homotopy coherence problem, which is precisely solved for Waldhausen categories satisfying a technical hypothesis (functorial factorizations of weak $w$-cofibrations) by means of Blumberg–Mandell’s $S'_{\bullet}$-construction [@BM Df. 2.7].
Unfortunately, these homotopy coherence problems grow less tractable as $K$-theoretic constructions become more involved. For example, if one seeks multiplicative structures on algebraic $K$-theory spectra, it becomes a challenge to perform all the necessary rectifications to turn a suitable pairing of Waldhausen categories into an $E_k$ multipciation on the $K$-theory. The work of Elmendorf and Mandell [@MR2254311] manages the case $k=\infty$ by using more rigid inputs for the $K$-theory functor. More generally, Blumberg and Mandell [@MR2805994 Th. 2.6] generalize this by providing, for any (colored) operad $O$ in categories, an $O$-algebra structure on the $K$-theory of any $O$-algebra in Waldhausen categories.
However, the theory $\infty$-categories provides a powerful alternative to such explicit solutions to homotopy coherence problems. Namely, the theory of *cartesian* and *cocartesian* fibrations allows one, in effect, to leave the homotopy coherence problems *unsolved* yet, at the same time, to work effectively with the resulting objects. For this reason, these concepts play a central role in our work here. (For fully general solutions to the problem of finding $O$ structures on $K$-theory spectra using machinery of the kind developed here, see either Blumberg–Gepner–Tabuada [@BGT2] or [@K3].)
Cocartesian fibrations {#cocartesian-fibrations .unnumbered}
----------------------
The idea goes back at least to Grothendieck (and probably further). If $X\colon{{C}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}}}$ is an (honest) diagram of ordinary categories, then one can define the *Grothendieck construction* of $X$. This is a category $G(X)$ whose objects are pairs $(c,x)$ consisting of an object $c\in C$ and an object $x\in X(c)$, in which a morphism $(f,\phi)\colon{{(d,y)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(c,x)}}$ is a morphism $f\colon{{d}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {c}}$ of $C$ and a morphism $$\phi\colon{{X(f)(y)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {x}}$$ of $X(c)$. There is an obvious forgetful functor $p\colon{{G(X)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}$.
One may now attempt to reverse-engineer the Grothendieck construction by trying to extract the salient features of the forgetful functor $p$ that ensures that it “came from” a diagram of categories. What we may notice is that for any morphism $f\colon{{d}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {c}}$ of $C$ and any object $y\in X(d)$ there is a special morphism $$F=(f,\phi)\colon{{(d,y)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(c,X(f)(y))}}$$ of $G(X)$ in which $$\phi\colon{{X(f)(y)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X(f)(y)}}$$ is simply the identity morphism. This morphism is *initial* among all the morphisms $F'$ of $G(X)$ such that $p(F')=f$; that is, for any morphism $F'$ of $G(X)$ such that $p(F')=f$, there exists a morphism $H$ of $G(X)$ such that $p(H)=\operatorname{id}_c$ such that $F'=H\circ F$.
We call morphisms of $G(X)$ that are initial in this sense *$p$-cocartesian*. Since a $p$-cocartesian edge lying over a morphism ${{d}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {c}}$ is defined by a universal property, it is uniquely specified up to a unique isomorphism lying over $\operatorname{id}_{c}$. The key condition that we are looking for is then that *for any morphism of $C$, there is a $p$-cocartesian morphism lying over it*. A functor $p$ satisfying this condition is called a *Grothendieck opfibration*.
Now for *any* Grothendieck opfibration $q\colon{{D}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}$, let us attempt to extract a functor $Y\colon{{C}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}}}$ whose Grothendieck construction $G(Y)$ is equivalent (as a category over $C$) to $D$. We proceed in the following manner. To any object $c\in C$ assign the fiber $D_c$ of $q$ over $c$. To any morphism $f\colon{{d}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {c}}$ assign a functor $Y(f)\colon{{D_d}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {D_c}}$ that carries any object $y\in D_d$ to the target $Y(f)(y)\in D_c$ of “the” $q$-cocartesian edge lying over $f$. However, the problem is already apparent in the scare quotes around the word “the.” These functors will not be strictly compatible with composition; rather, one will obtain natural isomorphisms $$Y(g\circ f)\simeq Y(g)\circ Y(f)$$ that will satisfy a secondary layer of coherences that make $Y$ into a *pseudofunctor*.
It is in fact possible to rectify any pseudofunctor to an equivalent honest functor, and this gives an honest functor whose Grothendieck construction is equivalent to our original $D$.
In light of all this, three options present themselves for contending with weak diagrams of ordinary categories:
(1) Rectify all pseudofunctors, and keep track of the rectifications as constructions become more involved.
(2) Work systematically with pseudofunctors, verifying all the coherence laws as needed.
(3) Work directly with Grothendieck opfibrations.
Which of these one selects is largely a matter of taste. When we pass to diagrams of higher categories, however, the first two options veer sharply into the realm of impracticality. A pseudofunctor ${{S}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}}}$ has not only a secondary level of coherences, but also an infinite progression of coherences between witnesses of lower-order coherences. Though rectifications of these pseudofunctors do exist (see \[rec:straighten\] below), they are usually not terribly explicit, and it would be an onerous task to keep them all straight.
Fortunately, the last option generalizes quite comfortably to the context of quasicategories, yielding the theory of *cocartesian fibrations*.
\[rec:cocart\] Suppose $p\colon{{X}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ an inner fibration of simplicial sets. Recall [@HTT Rk. 2.4.1.4] that an edge $f\colon{{\Delta^{1}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X}}$ is ***$p$-cocartesian*** just in case, for each integer $n\geq 2$, any extension $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\Delta^{\{0,1\}}&X,\\
\Lambda^n_0&\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge node[above]{$f$} (m-1-2)
edge[right hook->] (m-2-1)
(m-2-1) edge node[below]{$F$} (m-1-2);
\end{tikzpicture}$$ and any solid arrow commutative diagram $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\Lambda^n_0&X\\
\Delta^n&S,\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge node[above]{$F$} (m-1-2)
edge[right hook->] (m-2-1)
(m-1-2) edge node[right]{$p$} (m-2-2)
(m-2-1) edge (m-2-2)
(m-2-1) edge[dotted] node[below]{$\overline{F}$} (m-1-2);
\end{tikzpicture}$$ the dotted arrow $\overline{F}$ exists, rendering the diagram commutative.
We say that $p$ is a ***cocartesian fibration*** [@HTT Df. 2.4.2.1] if, for any edge $\eta\colon{{s}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {t}}$ of $S$ and for every vertex $x\in X_0$ such that $p(x)=s$, there exists a $p$-cocartesian edge $f\colon{{x}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {y}}$ such that $\eta=p(f)$.
***Cartesian edges*** and ***cartesian fibrations*** are defined dually, so that an edge of $X$ is $p$-cartesian just in case the corresponding edge of $X^{{\mathrm{op}}}$ is cocartesian for the inner fibration $p^{{\mathrm{op}}}\colon{{X^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S^{{\mathrm{op}}}}}$, and $p$ is a cartesian fibration just in case $p^{{\mathrm{op}}}$ is a cocartesian fibration.
\[exm:nerveofGrothfib\] A functor $p\colon{{D}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}$ between ordinary categories is a Grothendieck opfibration if and only if the induced functor $N(p)\colon{{ND}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {NC}}$ on nerves is a cocartesian fibration [@HTT Rk 2.4.2.2].
\[exm:scocartfib\] Recall that for any $\infty$-category $C$, we write $\mathscr{O}(C){\mathrel{\mathop:}=}\operatorname{Fun}(\Delta^1,C)$. By [@HTT Cor. 2.4.7.12], evaluation at $0$ defines a cartesian fibration $s\colon{{\mathscr{O}(C)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}$, and evaluation at $1$ defines a cocartesian fibration $t\colon{{\mathscr{O}(C)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}$.
One can ask whether the functor $s\colon{{\mathscr{O}(C)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}$ is also a *cocartesian* fibration. One may observe [@HTT Lm. 6.1.1.1] that an edge ${{\Delta^1}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{O}(C)}}$ is $s$-cocartesian just in case the correponding diagram ${{(\Lambda^2_0)^{\rhd}\cong\Delta^1\times\Delta^1}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}$ is a pushout square.
\[rec:straighten\] Suppose $S$ a simplicial set. Then the collection of cocartesian fibrations to $S$ with small fibers is naturally organized into an $\infty$-category ${\mathbf{Cat}}_{\infty/S}^{{\mathrm{cocart}}}$. To construct it, let ${\mathbf{Cat}}_{\infty}^{{\mathrm{cocart}}}$ be the following subcategory of $\mathscr{O}({\mathbf{Cat}}_{\infty})$: an object ${{X}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {U}}$ of $\mathscr{O}({\mathbf{Cat}}_{\infty})$ lies in ${\mathbf{Cat}}_{\infty}^{{\mathrm{cocart}}}$ if and only if it is a cocartesian fibration, and a morphism ${{p}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {q}}$ in $\mathscr{O}({\mathbf{Cat}}_{\infty})$ between cocartesian fibrations represented as a square $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{X&Y\\
U&V\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge node[above]{$f$} (m-1-2)
edge node[left]{$p$} (m-2-1)
(m-1-2) edge node[right]{$q$} (m-2-2)
(m-2-1) edge (m-2-2);
\end{tikzpicture}$$ lies in in ${\mathbf{Cat}}_{\infty}^{{\mathrm{cocart}}}$ if and only if $f$ carries $p$-cocartesian edges to $q$-cocartesian edges. We now define ${\mathbf{Cat}}_{\infty/S}^{{\mathrm{cocart}}}$ as the fiber over $S$ of the target functor $$t\colon{\mathbf{Cat}}_{\infty}^{{\mathrm{cocart}}}\subset\mathscr{O}({\mathbf{Cat}}_{\infty})\ {\tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);}\ {\mathbf{Cat}}_{\infty}.$$ Equivalently [@HTT Pr. 3.1.3.7], one may describe ${\mathbf{Cat}}_{\infty/S}^{{\mathrm{cocart}}}$ as the simplicial nerve (Nt. \[ntn:superscriptsscats\]) of the (fibrant) simplicial category of marked simplicial sets [@HTT Df. 3.1.0.1] over $S$ that are fibrant for the *cocartesian model structure* — i.e., of the form ${{X^{\natural}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ for ${{X}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ a cocartesian fibration [@HTT Df. 3.1.1.8].
The straightening/unstraightening Quillen equivalence of [@HTT Th. 3.2.0.1] now yields an equivalence of $\infty$-categories $${\mathbf{Cat}}_{\infty/S}^{{\mathrm{cocart}}}\simeq\operatorname{Fun}(S,{\mathbf{Cat}}_{\infty}).$$ So the dictionary between Grothendieck opfibrations and diagrams of categories generalizes gracefully to a dictionary between cocartesian fibrations $p\colon{{X}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ with small fibers and functors ${\mathbf{X}}\colon{{S}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}}}$. As for ordinary categories, for any vertex $s\in S_0$, the value ${\mathbf{X}}(s)$ is equivalent to the fiber $X_s$, and for any edge $\eta\colon{{s}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {t}}$, the functor ${{h{\mathbf{X}}(s)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {h{\mathbf{X}}(t)}}$ assigns to any object $x\in X_s$ an object $y\in X_t$ with the property that there is a cocartesian edge ${{x}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {y}}$ that covers $\eta$. We say that ${\mathbf{X}}$ ***classifies*** $p$ [@HTT Df. 3.3.2.2], and we will abuse terminology slightly by speaking of ***the functor $\eta_{!}\colon{{X_s}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X_t}}$ induced by*** an edge $\eta\colon{{s}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {t}}$ of $S$, even though $\eta_!$ is defined only up to canonical equivalence.
Dually, the collection of cartesian fibrations to $S$ with small fibers is naturally organized into an $\infty$-category ${\mathbf{Cat}}_{\infty/S}^{{\mathrm{cart}}}$, and the straightening/unstraightening Quillen equivalence yields an equivalence of $\infty$-categories $${\mathbf{Cat}}_{\infty/S}^{{\mathrm{cart}}}\simeq\operatorname{Fun}(S^{{\mathrm{op}}},{\mathbf{Cat}}_{\infty}).$$
For any $\infty$-category $C$, the functor ${{C^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}}}$ that classifies the cartesian fibration $s\colon{{\mathscr{O}(C)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}$ is the functor that carries any object $X$ of $C$ to the undercategory $C_{X/}$ and any morphism $f\colon{{Y}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X}}$ to the forgetful functor $f^{\star}\colon{{C_{X/}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C_{Y/}}}$.
If $C$ admits all pushouts, then the cocartesian fibration $s\colon{{\mathscr{O}(C)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}$ is classified by a functor ${{C}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}}}$ that carries any object $X$ of $C$ to the undercategory $C_{X/}$ and any morphism $f\colon{{Y}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X}}$ to the functor $f_{!}\colon{{C_{Y/}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C_{X/}}}$ that is given by pushout along $f$.
\[rec:leftfib\] A cocartesian fibration with the special property that each fiber is a Kan complex — or equivalently, with the special property that the functor that classifies it factors through the full subcategory ${\mathbf{Kan}}\subset{\mathbf{Cat}}_{\infty}$ — is called a ***left fibration***. These are more efficiently described as maps that satisfy the right lifting property with respect to horn inclusions ${{\Lambda^n_k}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\Delta^n}}$ such that $n\geq1$ and $0\leq k\leq n-1$ [@HTT Pr. 2.4.2.4].
For any cocartesian fibration $p\colon{{X}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$, one may consider the smallest simplicial subset $\iota_SX\subset X$ that contains the $p$-cocartesian edges. The restriction $\iota_S(p)\colon{{\iota_SX}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ of $p$ to $\iota_SX$ is a left fibration. The functor ${{S}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$ that classifies $\iota_Sp$ is then the functor given by the composition $$S\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$F$}(0.5,0.5ex);\ {\mathbf{Cat}}_{\infty}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$\iota$}(0.5,0.5ex);\ {\mathbf{Kan}},$$ where $F$ is the functor that classifies $p$.
Let us recall a particularly powerful construction with cartesian and cocartesian fibrations, which will form the cornerstone for our study of filtered objects of Waldhausen $\infty$-categories.
\[rec:htt32213\] Suppose $S$ a simplicial set, and suppose ${\mathbf{X}}\colon{{S^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}}}$ and ${\mathbf{Y}}\colon{{S}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}}}$ two diagrams of $\infty$-categories. Then one may define a functor $$\operatorname{Fun}({\mathbf{X}},{\mathbf{Y}})\colon{{S}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}}}$$ that carries a vertex $s$ of $S$ to the $\infty$-category $\operatorname{Fun}({\mathbf{X}}(s),{\mathbf{Y}}(s))$ and an edge $\eta\colon{{s}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {t}}$ of $S$ to the functor $${{\operatorname{Fun}({\mathbf{X}}(s),{\mathbf{Y}}(s))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}({\mathbf{X}}(t),{\mathbf{Y}}(t))}}$$ given by the assignment ${{F}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Y}}(\eta)\circ F\circ{\mathbf{X}}(\eta)}}$.
If one wishes to work instead with the cartesian and cocartesian fibrations classified by ${\mathbf{X}}$ and ${\mathbf{Y}}$, the following construction provides an elegant way of writing explicitly the cocartesian fibration classified by the functor $\operatorname{Fun}({\mathbf{X}},{\mathbf{Y}})$. If $p\colon{{X}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ is the cartesian fibration classified by ${\mathbf{X}}$ and if $q\colon{{Y}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ is the cocartesian fibration classified by ${\mathbf{Y}}$, one may define a map $r\colon{{T}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ defined by the following universal property: for any map $\sigma\colon{{K}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$, one has a bijection $$\operatorname{Mor}_{S}(K,T)\cong\operatorname{Mor}_{S}(X\times_SK,Y),$$ functorial in $\sigma$. It is then shown in [@HTT Cor. 3.2.2.13] that $p$ is a cocartesian fibration, and an edge $g\colon{{\Delta^1}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {T}}$ is $r$-cocartesian just in case the induced map ${{X\times_S\Delta^1}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y}}$ carries $p$-cartesian edges to $q$-cocartesian edges. The fiber of the map ${{T}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ over a vertex $s$ is the $\infty$-category $\operatorname{Fun}(X_s,Y_s)$, and for any edge $\eta\colon{{s}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {t}}$ of $S$, the functor $\eta_!\colon{{T_s}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {T_t}}$ induced by $\eta$ is equivalent to the functor ${{F}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Y}}(\eta)\circ F\circ{\mathbf{X}}(\eta)}}$ described above.
Pair cartesian and cocartesian fibrations {#pair-cartesian-and-cocartesian-fibrations .unnumbered}
-----------------------------------------
Just as cartesian and cocartesian fibrations are well adapted to the study of weak diagrams of $\infty$-categories, so we will introduce the theory of *Waldhausen cartesian* and *cocartesian fibrations*, which make available a robust notion of weak diagrams of Waldhausen $\infty$-categories. In order to introduce this notion, we first discuss *pair cartesian* and *cocartesian fibations* in some detail. These will provide a notion of weak diagrams of pairs of $\infty$-categories.
\[dfn:paircartfib\] Suppose $S$ an $\infty$-category. Then a ***pair cartesian fibration*** ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ is a pair $\mathscr{X}$ and a morphism of pairs $p\colon{{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S^{\flat}}}$ (where the target is the minimal pair $(S,\iota S)$ — see Ex. \[exm:minpairmaxpair\]) such that the following conditions are satisfied.
1. The underlying functor of $p$ is a cartesian fibration.
2. \[item:etastarisfunctorofpairs\] For any edge $\eta\colon{{s}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {t}}$ of $S$, the induced functor $\eta^{\star}\colon{{\mathscr{X}_t}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{X}_s}}$ carries cofibrations to cofibrations.
Dually, a ***pair cocartesian fibration*** ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ is a pair $\mathscr{X}$ and a morphism of pairs $p\colon{{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S^{\flat}}}$ such that $p^{{\mathrm{op}}}\colon{{\mathscr{X}^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S^{{\mathrm{op}}}}}$ is a pair cartesian fibration.
\[cor:pairfibsarefunsintopair\] If $S$ is an $\infty$-category and $p\colon{{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ is a pair cartesian fibration respectively, a pair cocartesian fibration with small fibers, then the functor ${{S^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}}}$ resp., the functor ${{S}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}}}$ that classifies $p$ lifts to a functor ${{S^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}_{\infty}}}$ resp., ${{S}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}_{\infty}}}$.
We employ the adjunction $(\mathfrak{C},N)$ of [@HTT §1.1.5]. Since ${\mathbf{Pair}}_{\infty}$ and ${\mathbf{Cat}}_{\infty}$ are both defined as simplicial nerves, the data of a lift ${{S^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}_{\infty}}}$ of ${{S^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}}}$ is tantamount to the data of a lift $\overline{{\mathbf{X}}}\colon{{\mathfrak{C}[S]^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}^{\Delta}_{\infty}}}$ of the corresponding simplicial functor ${\mathbf{X}}\colon{{\mathfrak{C}[S]^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}^{\Delta}_{\infty}}}$. Now for any object $s$ of $\mathfrak{C}[S]$, the categories ${\mathbf{X}}(s)$ inherits a pair structure via the canonical equivalence ${\mathbf{X}}(s)\simeq\mathscr{X}_s$. For any two objects $s$ and $t$ of $\mathfrak{C}[S]$, condition (\[dfn:paircartfib\].\[item:etastarisfunctorofpairs\]) ensures that the map $${{\mathfrak{C}[S](t,s)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}^{\Delta}({\mathbf{X}}(s),{\mathbf{X}}(t))}}$$ factors through the simplicial subset (Nt. \[ntn:pair\]) $${\mathbf{Pair}}_{\infty}^{\Delta}({\mathbf{X}}(s),{\mathbf{X}}(t))\subset{\mathbf{Cat}}_{\infty}^{\Delta}({\mathbf{X}}(s),{\mathbf{X}}(t)).$$ This now defines the desired simplicial functor $\overline{{\mathbf{X}}}$.
In the situation of Pr. \[cor:pairfibsarefunsintopair\], we will say that the lifted functor ${{S^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}_{\infty}}}$ respectively, the lifted functor ${{S}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}_{\infty}}}$ ***classifies*** the cartesian resp., cocartesian fibration $p$.
\[cor:paircartfibsbasechange\] The classes of pair cartesian fibrations and pair cocartesian fibrations are each stable under base change. That is, for any pair cartesian \[respectively, cocartesian\] fibration ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ and for any functor $f\colon{{S'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$, if the pullback $\mathscr{X}'{\mathrel{\mathop:}=}\mathscr{X}\times_SS'$ is endowed with the pair structure in which a morphism is ingressive just in case it is carried to an equivalence in $S'$ and to an ingressive morphism of $\mathscr{X}$, then ${{\mathscr{X}'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S'}}$ is a pair cartesian \[resp., cocartesian\] fibration.
We treat the case of pair cartesian fibrations. Cartesian fibrations are stable under pullbacks [@HTT Pr. 2.4.2.3(2)], so it remains to note that for any morphism $\eta\colon{{s}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {t}}$ of $S'$, the induced functor $$\eta^{\star}\simeq f(\eta)^{\star}\colon{{\mathscr{X}'_t\cong\mathscr{X}_{f(t)}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{X}_{f(s)}\cong\mathscr{X}'_s}}$$ carries cofibrations to cofibrations.
The $\infty$-categories of pair (co)cartesian fibrations {#the-infty-categories-of-pair-cocartesian-fibrations .unnumbered}
--------------------------------------------------------
The collection of all pair cocartesian fibrations are organized into an $\infty$-category ${\mathbf{Pair}}_{\infty}^{{\mathrm{cocart}}}$, which is analogous to the $\infty$-category ${\mathbf{Cat}}_{\infty}^{{\mathrm{cocart}}}$ of \[rec:straighten\]. Furthermore, pair cocartesian fibrations with a fixed base $\infty$-category $S$ organize themselves into an $\infty$-category ${\mathbf{Pair}}_{\infty/S}^{{\mathrm{cocart}}}$.
Denote by $${\mathbf{Pair}}_{\infty}^{{\mathrm{cart}}}\textrm{\qquad[respectively, by\quad}{\mathbf{Pair}}_{\infty}^{{\mathrm{cocart}}}\textrm{\quad]}$$ the following subcategory of $\mathscr{O}({\mathbf{Pair}}_{\infty})$. The objects of ${\mathbf{Pair}}_{\infty}^{{\mathrm{cart}}}$ \[resp., ${\mathbf{Pair}}_{\infty}^{{\mathrm{cocart}}}$\] are pair cartesian fibrations (resp., pair cocartesian fibrations) ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$. For any pair cartesian (resp., cocartesian) fibrations $p\colon{{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ and $q\colon{{\mathscr{Y}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {T}}$, a commutative square $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\mathscr{X}&\mathscr{Y}\\
S^{\flat}&T^{\flat}\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge node[above]{$\psi$} (m-1-2)
edge node[left]{$p$} (m-2-1)
(m-1-2) edge node[right]{$q$} (m-2-2)
(m-2-1) edge (m-2-2);
\end{tikzpicture}$$ of pairs of $\infty$-categories is a morphism ${{p}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {q}}$ of ${\mathbf{Pair}}_{\infty}^{{\mathrm{cart}}}$ \[resp., of ${\mathbf{Pair}}_{\infty}^{{\mathrm{cocart}}}$\] if and only if $\psi$ carries $p$-cartesian (resp. $p$-cocartesian) edges to $q$-cartesian (resp. $q$-cocartesian) edges.
By an abuse of notation, we will denote by $(\mathscr{X}/S)$ an object ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ of ${\mathbf{Pair}}_{\infty}^{{\mathrm{cart}}}$ \[resp., of ${\mathbf{Pair}}_{\infty}^{{\mathrm{cocart}}}$\].
The following is immediate from Pr. \[cor:paircartfibsbasechange\] and [@HTT Lm. 6.1.1.1].
The target functors $${{{\mathbf{Pair}}_{\infty}^{{\mathrm{cart}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}}}\textrm{\quad and\quad}{{{\mathbf{Pair}}_{\infty}^{{\mathrm{cocart}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}}}$$ induced by the inclusion $\{1\}\subset\Delta^1$ are both cartesian fibrations.
The fibers of the cartesian fibrations $${{{\mathbf{Pair}}_{\infty}^{{\mathrm{cart}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}}}\textrm{\quad and\quad}{{{\mathbf{Pair}}_{\infty}^{{\mathrm{cocart}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}}}$$ over an object $\{S\}\subset{\mathbf{Cat}}_{\infty}$ will be denoted ${\mathbf{Pair}}_{\infty,/S}^{{\mathrm{cart}}}$ and ${\mathbf{Pair}}_{\infty,/S}^{{\mathrm{cocart}}}$, respectively.
By an abuse of notation, denote by $$({\mathbf{Pair}}_{\infty/S}^{{\mathrm{cart}}})_0\textrm{\qquad[respectively, by\quad}({\mathbf{Pair}}_{\infty/S}^{{\mathrm{cocart}}})_0\textrm{\quad]}$$ the subcategory of the ordinary category $(({\mathbf{Pair}}_{\infty}^{\Delta})_{0}\downarrow S^{\flat})$ whose objects are pair cartesian fibrations \[resp., pair cocartesian fibrations\] ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ and whose morphisms are functors of pairs ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{Y}}}$ over $S$ that carry cartesian morphisms to cartesian morphisms \[resp., that carry cocartesian morphisms to cocartesian morphisms\]. Denote by $$w({\mathbf{Pair}}_{\infty/S}^{{\mathrm{cart}}})_0\subset({\mathbf{Pair}}_{\infty/S}^{{\mathrm{cart}}})_0\textrm{\qquad [resp., by\quad}w({\mathbf{Pair}}_{\infty/S}^{{\mathrm{cocart}}})_0\subset({\mathbf{Pair}}_{\infty/S}^{{\mathrm{cocart}}})_0\textrm{\quad]}$$ the subcategory consisting of those morphisms ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{Y}}}$ such that for any vertex $s\in S_0$, the induced functor ${{\mathscr{X}_s}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{Y}_s}}$ is a weak equivalence of pairs.
\[lem:Paircocartisarelnerve\] For any $\infty$-category $S$, the $\infty$-category ${\mathbf{Pair}}_{\infty/S}^{{\mathrm{cart}}}$ respectively, the $\infty$-category ${\mathbf{Pair}}_{\infty/S}^{{\mathrm{cocart}}}$ is a relative nerve of $$(({\mathbf{Pair}}_{\infty/S}^{{\mathrm{cart}}})_0,\ w({\mathbf{Pair}}_{\infty/S}^{{\mathrm{cart}}})_0)\textrm{\qquad\textup{[}resp., of\quad}(({\mathbf{Pair}}_{\infty/S}^{{\mathrm{cocart}}})_0,\ w({\mathbf{Pair}}_{\infty/S}^{{\mathrm{cocart}}})_0)\textrm{\quad\textup{]}.}$$
To show that ${\mathbf{Pair}}_{\infty/S}^{{\mathrm{cart}}}$ is a relative nerve of $(({\mathbf{Pair}}_{\infty/S}^{{\mathrm{cart}}})_0,w({\mathbf{Pair}}_{\infty/S}^{{\mathrm{cart}}})_0)$, we first note that the analogous result for $\infty$-categories of cartesian fibrations ${{X}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ holds. More precisely, recall (\[rec:straighten\]) that ${\mathbf{Cat}}_{\infty/S}^{{\mathrm{cart}}}$ may be identified with the nerve of the cartesian simplicial model category of marked simplicial sets over $S$, whence it is a relative nerve of the category of cartesian fibrations over $S$, equipped with the cartesian equivalences.
To extend this result to a characterization of ${\mathbf{Pair}}_{\infty/S}^{{\mathrm{cart}}}$ as a relative nerve, it suffices to note that $${\mathbf{Pair}}_{\infty/S}^{{\mathrm{cart}}}\subset{\mathbf{Cat}}_{\infty/S}^{{\mathrm{cart}}}\times_{{\mathbf{Cat}}_{\infty}}{\mathbf{Pair}}_{\infty}$$ is the full subcategory spanned by the pair cartesian fibrations.
We may now employ this lemma to lift the equivalence of $\infty$-categories $${\mathbf{Cat}}_{\infty/S}^{{\mathrm{cart}}}\simeq\operatorname{Fun}(S^{{\mathrm{op}}},{\mathbf{Cat}}_{\infty})$$ of [@HTT §3.2] to an equivalence of $\infty$-categories $${\mathbf{Pair}}_{\infty/S}^{{\mathrm{cart}}}\simeq\operatorname{Fun}(S^{{\mathrm{op}}},{\mathbf{Pair}}_{\infty}).$$
\[prp:WaldcocartisFunSWald\] For any $\infty$-category $S$, the $\infty$-category $\operatorname{Fun}(S^{{\mathrm{op}}},{\mathbf{Pair}}_{\infty})$ respectively, the $\infty$-category $\operatorname{Fun}(S,{\mathbf{Pair}}_{\infty})$ is a relative nerve of $$(({\mathbf{Pair}}_{\infty/S}^{{\mathrm{cart}}})_0,\ w({\mathbf{Pair}}_{\infty/S}^{{\mathrm{cart}}})_0)\textrm{\qquad\textup{[}resp., of\quad}(({\mathbf{Pair}}_{\infty/S}^{{\mathrm{cocart}}})_0,\ w({\mathbf{Pair}}_{\infty/S}^{{\mathrm{cocart}}})_0)\textrm{\quad\textup{]}.}$$
The unstraightening functor of [@HTT §3.2] is a weak equivalence-preserving functor $$\mathrm{Un}^+\colon{{({\mathbf{Cat}}_{\infty}^{\Delta})^{\mathfrak{C}[S]^{{\mathrm{op}}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty/S}^{{\mathrm{cart}}}}}$$ that induces an equivalence of relative nerves. For any simplicial functor $${\mathbf{X}}\colon{{\mathfrak{C}[S]^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}_{\infty}^{\Delta}}},$$ endow the unstraightening $\mathrm{Un}^+({\mathbf{X}})$ with a pair structure by letting $\mathrm{Un}^+({\mathbf{X}})_{\dag}\subset\mathrm{Un}^+({\mathbf{X}})$ be the smallest subcategory containing all the equivalences as well as any cofibration of the of any fiber $\mathrm{Un}^+({\mathbf{X}})_s\cong{\mathbf{X}}(s)$. With this definition, we obtain a weak equivalence-preserving functor $$\mathrm{Un}^+\colon{{({\mathbf{Pair}}_{\infty}^{\Delta})^{\mathfrak{C}[S]^{{\mathrm{op}}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {({\mathbf{Pair}}_{\infty/S}^{{\mathrm{cart}}})_0}}.$$
This functor induces a functor on relative nerves, which is essentially surjective by Pr. \[cor:pairfibsarefunsintopair\]. Moreover, for any simplicial functors $${\mathbf{X}},{\mathbf{Y}}\colon{{\mathfrak{C}[S]^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}_{\infty}^{\Delta}}},$$ the simplicial set $$\operatorname{Map}_{N(({\mathbf{Pair}}_{\infty}^{\Delta})^{\mathfrak{C}[S]^{{\mathrm{op}}}})}({\mathbf{X}},{\mathbf{Y}})$$ may be identified with the simplicial subset of $$\operatorname{Map}_{N(({\mathbf{Cat}}_{\infty}^{\Delta})^{\mathfrak{C}[S]^{{\mathrm{op}}}})}({\mathbf{X}},{\mathbf{Y}})$$ given by the union of the connected components corresponding to natural transformations ${{{\mathbf{X}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Y}}}}$ such that for any $s\in S_0$, the functor ${{{\mathbf{X}}(s)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Y}}(s)}}$ is a functor of pairs. Similarly, the simplicial set $$\operatorname{Map}_{{\mathbf{Pair}}_{\infty/S}^{{\mathrm{cart}}}}(\mathrm{Un}^+({\mathbf{X}}),\mathrm{Un}^+({\mathbf{Y}}))$$ may be identified with the subspace of $$\operatorname{Map}_{{\mathbf{Cat}}_{\infty/S}^{{\mathrm{cart}}}}(\mathrm{Un}^+({\mathbf{X}}),\mathrm{Un}^+({\mathbf{Y}}))$$ given by the union of the connected components corresponding to functors $${{\mathrm{Un}^+({\mathbf{X}})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathrm{Un}^+({\mathbf{Y}})}}$$ over $S$ that send cartesian edges to cartesian edges with the additional property that for any $s\in S_0$, the functor $${{\mathrm{Un}^+({\mathbf{X}})_s\cong{\mathbf{X}}(s)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Y}}(s)\cong\mathrm{Un}^+({\mathbf{Y}})_s}}$$ is a functor of pairs. We thus conclude that $\mathrm{Un}^+$ is fully faithful.
Armed with this, we may characterize colimits of pair cartesian fibrations fiberwise.
\[cor:colimitsofpairfibs\] Suppose $S$ a small $\infty$-category, $K$ a small simplicial set. A functor $\mathscr{X}\colon{{K^{\rhd}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}_{\infty/S}^{{\mathrm{cart}}}}}$ respectively, a functor $\mathscr{X}\colon{{K^{\rhd}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}_{\infty/S}^{{\mathrm{cocart}}}}}$ is a colimit diagram if and only if, for every vertex $s\in S_0$, the induced functor $$\mathscr{X}_s\colon{{K^{\rhd}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}_{\infty}}}$$ is a colimit diagram.
Of course the same characterization of limits holds, but it will not be needed. We will take up the question of the *existence* of colimits in the $\infty$-category ${\mathbf{Pair}}_{\infty}$ in Cor. \[cor:colimsofpairs\] below.
A pair version of {#a-pair-version-of .unnumbered}
------------------
The theory of pair cartesian and cocartesian fibrations is a relatively mild generalization of the theory of cartesian and cocartesian fibrations, and many of the results extend to this setting. In particular, we now set about proving a pair version of \[rec:htt32213\] (i.e., of [@HTT Cor. 3.2.2.13]).
In effect, the objective is to give a fibration-theoretic version of the following observation. For any $\infty$-category $S$, any diagram ${\mathbf{X}}\colon{{S^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}_{\infty}}}$, and any diagram ${\mathbf{Y}}\colon{{S}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}_{\infty}}}$, there is a functor $$\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}({\mathbf{X}},{\mathbf{Y}})\colon{{S}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}}}$$ that carries any object $s$ of $S$ to the $\infty$-category $\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}({\mathbf{X}}(s),{\mathbf{Y}}(s))$.
\[ntn:ordcatofpairs\] Consider the ordinary category $s{\mathbf{Set}}(2)$ of pairs $(V,U)$ consisting of a small simplicial set $U$ and a simplicial subset $U\subset V$.
\[prp:htt32213\] Suppose $p\colon{{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ a pair cartesian fibration, and suppose $q\colon{{\mathscr{Y}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ a pair cocartesian fibration. Let $r\colon{{T_p\mathscr{Y}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ be the map defined by the following universal property. We require, for any simplicial set $K$ and any map $\sigma\colon{{K}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$, a bijection $$\operatorname{Mor}_{S}(K,T_p\mathscr{Y})\cong\operatorname{Mor}_{s{\mathbf{Set}}(2)/(S,\iota S)}((K\times_S\mathscr{X},K\times_S\mathscr{X}_{\dag}),(\mathscr{Y},\mathscr{Y}_{\dag}))$$ , functorial in $\sigma$. Then $r$ is a cocartesian fibration.
We may use [@HTT Cor. 3.2.2.13] to define a cocartesian fibration $r'\colon{{T'_p\mathscr{Y}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ with the universal property $$\operatorname{Mor}_{S}(K,T'_p\mathscr{Y})\cong\operatorname{Mor}_{S}(K\times_S\mathscr{X},\mathscr{Y}).$$ Thus $T'_p\mathscr{Y}$ is an $\infty$-category whose objects are pairs $(s,\phi)$ consisting of an object $s\in S_0$ and a functors $\phi\colon{{\mathscr{X}_s}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{Y}_s}}$, and $T_p\mathscr{Y}\subset T'_p\mathscr{Y}$ is the full subcategory spanned by those pairs $(s,\phi)$ such that $\phi$ is a functor of pairs. An edge ${{(s,\phi)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(t,\psi)}}$ in $T'_p\mathscr{Y}$ over an edge $\eta\colon{{s}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {t}}$ of $S$ is $r'$-cocartesian if and only if the corresponding natural transformation ${{\eta_{\mathscr{Y},!}\circ\phi\circ\eta_{\mathscr{X}}^{\star}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\psi}}$ is an equivalence. Since composites of functors of pairs are again functors of pairs, it follows that if $(s,\phi)$ is an object of $T_p\mathscr{Y}$, then so is $(t,\psi)$, whence it follows that $r$ is a cocartesian fibration.
Suppose that ${\mathbf{X}}$ classifies $p$ and that ${\mathbf{Y}}$ classifies $q$. Since $\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}({\mathbf{X}}(s),{\mathbf{Y}}(s))$ is a full subcategory of $\operatorname{Fun}({\mathbf{X}}(s),{\mathbf{Y}}(s))$, it follows from \[rec:htt32213\] that $T_p\mathscr{Y}$ is in fact classified by $\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}({\mathbf{X}},{\mathbf{Y}})$.
Suppose $S$ an $\infty$-category, and suppose $p\colon{{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ a cartesian fibration. The construction $T_p$ is visibly a functor $${{({\mathbf{Pair}}_{\infty/S}^{{\mathrm{cocart}}})_0}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {({\mathbf{Pair}}_{\infty/S}^{{\mathrm{cocart}}})_0}}.$$ To show that $T_p$ defines a functor of $\infty$-categories ${{{\mathbf{Pair}}_{\infty/S}^{{\mathrm{cocart}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}_{\infty/S}^{{\mathrm{cocart}}}}}$, it suffices by Lm. \[lem:Paircocartisarelnerve\] just to observe that the functor $T_p$ preserves the weak equivalences of ${\mathbf{Pair}}_{\infty/S}^{{\mathrm{cocart}},0}$. Hence we have the following.
\[nul:Tpafunctor\] Suppose $p\colon{{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ a cartesian fibration; then the assignment ${{\mathscr{Y}}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {T_p\mathscr{Y}}}$ defines a functor $${{{\mathbf{Pair}}_{\infty/S}^{{\mathrm{cocart}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}_{\infty/S}^{{\mathrm{cocart}}}}}.$$
Waldhausen cartesian and cocartesian fibrations {#waldhausen-cartesian-and-cocartesian-fibrations .unnumbered}
-----------------------------------------------
Now we have laid the groundwork for our theory of *Waldhausen cartesian* and *cocartesian* fibrations.
\[dfn:Waldcartfib\] Suppose $S$ an $\infty$-category. A ***Waldhausen cartesian fibration*** $p\colon{{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ is a pair cartesian fibration satisfying the following conditions.
1. For any object $s$ of $S$, the pair $$\mathscr{X}_s{\mathrel{\mathop:}=}(\mathscr{X}\times_S\{s\},\mathscr{X}_{\dag}\times_S\{s\})$$ is a Waldhausen $\infty$-category.
2. For any morphism $\eta\colon{{s}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {t}}$, the corresponding functor of pairs $$\eta^{\star}\colon{{\mathscr{X}_t}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{X}_s}}$$ is an exact functor of Waldhausen $\infty$-categories.
Dually, a ***Waldhausen cocartesian fibration*** $p\colon{{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ is a pair cocartesian fibration satisfying the following conditions.
1. For any object $s$ of $S$, the pair $$\mathscr{X}_s{\mathrel{\mathop:}=}(\mathscr{X}\times_S\{s\},\mathscr{X}_{\dag}\times_S\{s\})$$ is a Waldhausen $\infty$-category.
2. For any morphism $\eta\colon{{s}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {t}}$, the corresponding functor of pairs $$\eta_!\colon{{\mathscr{X}_s}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{X}_t}}$$ is an exact functor of Waldhausen $\infty$-categories.
As with pair cartesian fibrations, Waldhausen cartesian fibrations classify functors to ${\mathbf{Wald}_{\infty}}$. The following is an immediate consequence of the definition.
\[prp:Waldcartsclassify\] Suppose $S$ an $\infty$-category. Then a pair cartesian respectively, cocartesian fibration $p\colon{{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ is a Waldhausen cartesian fibration resp., a Waldhausen cocartesian fibration if and only if the functor ${{S^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}_{\infty}}}$ resp., the functor ${{S}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}_{\infty}}}$ that classifies $p$ factors through ${\mathbf{Wald}_{\infty}}\subset{\mathbf{Pair}}_{\infty}$.
\[prp:Waldcartfibsbasechange\] The classes of Waldhausen cartesian fibrations and Waldhausen cocartesian fibrations are each stable under base change. That is, for any Waldhausen cartesian respectively, cocartesian fibration ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ and for any functor $f\colon{{S'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$, if the pullback $\mathscr{X}'{\mathrel{\mathop:}=}\mathscr{X}\times_SS'$ is endowed with the pair structure in which a morphism is ingressive just in case it is carried to an equivalence in $S'$ and to an ingressive morphism of $\mathscr{X}$, then ${{\mathscr{X}'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S'}}$ is a Waldhausen cartesian resp., cocartesian fibration.
We treat the case of Waldhausen cartesian fibrations. By Pr. \[cor:paircartfibsbasechange\], ${{\mathscr{X}'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S'}}$ is a pair cartesian fibration, so it remains to note that for any morphism $\eta\colon{{s}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {t}}$ of $S'$, the induced functor of pairs $$\eta^{\star}\simeq f(\eta)^{\star}\colon{{\mathscr{X}'_t\cong\mathscr{X}_{f(t)}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{X}_{f(s)}\cong\mathscr{X}'_s}}$$ is an exact functor.
Denote by $${\mathbf{Wald}_{\infty}}^{{\mathrm{cart}}}\textrm{\qquad [respectively, by\quad}{\mathbf{Wald}_{\infty}}^{{\mathrm{cocart}}}\textrm{\quad]}$$ the following subcategory of $${\mathbf{Pair}}_{\infty}^{{\mathrm{cart}}}\textrm{\qquad [resp., of\quad}{\mathbf{Pair}}_{\infty}^{{\mathrm{cocart}}}\textrm{\quad].}$$ The objects of ${\mathbf{Wald}_{\infty}}^{{\mathrm{cart}}}$ \[resp., of ${\mathbf{Wald}_{\infty}}^{{\mathrm{cocart}}}$\] are Waldhausen cartesian fibrations \[resp., Waldhausen cocartesian fibrations\] ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$. A morphism $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\mathscr{X}&\mathscr{Y}\\
S^{\flat}&T^{\flat}\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge node[above]{$\psi$} (m-1-2)
edge node[left]{$p$} (m-2-1)
(m-1-2) edge node[right]{$q$} (m-2-2)
(m-2-1) edge node[below]{$\phi$} (m-2-2);
\end{tikzpicture}$$ of ${\mathbf{Pair}}_{\infty}^{{\mathrm{cart}}}$ (resp., ${\mathbf{Pair}}_{\infty}^{{\mathrm{cocart}}}$) is a morphism ${{p}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {q}}$ of the subcategory ${\mathbf{Wald}_{\infty}}^{{\mathrm{cart}}}$ \[resp., of ${\mathbf{Wald}_{\infty}}^{{\mathrm{cocart}}}$\] if and only if $\psi$ induces exact functors ${{\mathscr{X}_s}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{Y}_{\phi(s)}}}$ for every vertex $s\in S_0$.
The following is again a consequence of Pr. \[prp:Waldcartfibsbasechange\] and [@HTT Lm. 6.1.1.1].
\[lem:waldcartdiawald\] The target functors $${{{\mathbf{Wald}_{\infty}}^{{\mathrm{cart}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}}}\textrm{\quad and\quad}{{{\mathbf{Wald}_{\infty}}^{{\mathrm{cocart}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}}}$$ induced by the inclusion $\{1\}\subset\Delta^1$ are both cartesian fibrations.
The fibers of the cartesian fibrations $${{{\mathbf{Wald}_{\infty}}^{{\mathrm{cart}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}}}\textrm{\quad and\quad}{{{\mathbf{Wald}_{\infty}}^{{\mathrm{cocart}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}}}$$ over an object $\{S\}\subset{\mathbf{Cat}}_{\infty}$ will be denoted $\mathbf{Wald}_{\infty/S}^{{\mathrm{cart}}}$ and $\mathbf{Wald}_{\infty/S}^{{\mathrm{cocart}}}$, respectively.
\[prp:Waldstraightening\] The equivalence of $\infty$-categories ${\mathbf{Pair}}_{\infty/S}^{{\mathrm{cart}}}\simeq\operatorname{Fun}(S^{{\mathrm{op}}},{\mathbf{Pair}}_{\infty})$ respectively, the equivalence of $\infty$-categories ${\mathbf{Pair}}_{\infty/S}^{{\mathrm{cocart}}}\simeq\operatorname{Fun}(S,{\mathbf{Pair}}_{\infty})$ of restricts to an equivalence of $\infty$-categories $$\mathbf{Wald}_{\infty/S}^{{\mathrm{cart}}}\simeq\operatorname{Fun}(S^{{\mathrm{op}}},{\mathbf{Wald}_{\infty}})\textrm{\qquad\textup{[}resp.,\quad}\mathbf{Wald}_{\infty/S}^{{\mathrm{cocart}}}\simeq\operatorname{Fun}(S,{\mathbf{Wald}_{\infty}})\textrm{\quad\textup{]}.}$$
We treat the cartesian case. Note that $\mathbf{Wald}_{\infty/S}^{{\mathrm{cart}}}$ is the subcategory of the $\infty$-category ${\mathbf{Pair}}_{\infty/S}^{{\mathrm{cart}}}$ consisting of those objects and morphisms whose image under the equivalence ${\mathbf{Pair}}_{\infty/S}^{{\mathrm{cart}}}\simeq\operatorname{Fun}(S^{{\mathrm{op}}},{\mathbf{Pair}}_{\infty})$, lies in the subcategory $\operatorname{Fun}(S^{{\mathrm{op}}},{\mathbf{Wald}_{\infty}})\subset\operatorname{Fun}(S^{{\mathrm{op}}},{\mathbf{Pair}}_{\infty})$. So one may identify $\mathbf{Wald}_{\infty/S}^{{\mathrm{cart}}}$ as the pullback $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\mathbf{Wald}_{\infty/S}^{{\mathrm{cart}}}&\operatorname{Fun}(S^{{\mathrm{op}}},{\mathbf{Wald}_{\infty}})\\
{\mathbf{Pair}}_{\infty/S}^{{\mathrm{cart}}}&\operatorname{Fun}(S^{{\mathrm{op}}},{\mathbf{Pair}}_{\infty}).\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge[right hook->] (m-2-1)
(m-1-2) edge[right hook->] (m-2-2)
(m-2-1) edge node[below,inner sep=0.6pt]{$\sim$} (m-2-2);
\end{tikzpicture}$$ The result now follows from the fact that because the right-hand vertical map is a categorical fibration (\[rec:subcats\]), this square is a homotopy pullback for the Joyal model structure.
As with pair fibrations (Cor. \[cor:colimitsofpairfibs\]), we employ this result to observe that colimits of Waldhausen cartesian fibrations may be characterized fiberwise.
\[cor:colimsinWaldScocart\] Suppose $S$ a small $\infty$-category, $K$ a small simplicial set. A functor $\mathscr{X}\colon{{K^{\rhd}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{Wald}_{\infty/S}^{{\mathrm{cart}}}}}$ respectively, a functor $\mathscr{X}\colon{{K^{\rhd}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{Wald}_{\infty/S}^{{\mathrm{cocart}}}}}$ is a colimit diagram if and only if, for every vertex $s\in S_0$, the induced functor $$\mathscr{X}_s\colon{{K^{\rhd}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}}}$$ is a colimit diagram.
The derived $\infty$-category of Waldhausen $\infty$-categories
===============================================================
So far, we have built up a language for talking about the $\infty$-categories of interest to $K$-theorists. Now we want to study the $\infty$-category ${\mathbf{Wald}_{\infty}}$ of all these objects in some detail. More importantly, in later sections we’ll need an enlargement of ${\mathbf{Wald}_{\infty}}$ on which we can define suitable *derived functors*.
We take our inspiration from the following construction. Let $V(k)$ denote the ordinary category of vector spaces over a field $k$, and let $\mathrm{D}_{\geq 0}(k)$ be the *connective derived $\infty$-category* of $V(k)$. That is, $\mathrm{D}_{\geq 0}(k)$ is a relative nerve of the relative category of (homologically graded) chain complexes whose homology vanishes in negative degrees, where a weak equivalence is declared to be a quasi-isomorphism.
The connective derived $\infty$-category is the vehicle with which one may define *left derived functors* of right exact functors: one very general way of formulating this is to characterize $\mathrm{D}_{\geq 0}(k)$ as the $\infty$-category obtained from $V(k)$ by adding formal *geometric realizations* — that is, homotopy colimits of simplicial diagrams. More precisely, for any $\infty$-category $C$ that admits all geometric realizations, the functor $${{\operatorname{Fun}(\mathrm{D}_{\geq 0}(k),C)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}(NV(k),C)}}$$ induced by the inclusion ${{NV(k)}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathrm{D}_{\geq0}(k)}}$ restricts to an equivalence from the full subcategory of $\operatorname{Fun}(\mathrm{D}_{\geq 0}(k),C)$ spanned by those functors ${{\mathrm{D}_{\geq0}(k)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}$ that preserve geometric realizations to $\operatorname{Fun}(NV(k),C)$. (This characterization follows from the Dold–Kan correspondence; see [@HA Pr. 1.3.3.8] for a proof.) The objects of $\mathrm{D}_{\geq0}(k)$ can be represented as presheaves (of spaces) on the nerve of the category of *finite-dimensional* vector spaces that carry direct sums to products.
In this section, we wish to mimic this construction, treating the $\infty$-category ${\mathbf{Wald}_{\infty}}$ of Waldhausen $\infty$-categories as formally analogous to the category $V(k)$. We thus define ${\mathrm{D}(\mathbf{Wald}_{\infty})}$ as the $\infty$-category presheaves (of spaces) on the nerve of the category of suitably finite Waldhausen $\infty$-categories that carry direct sums to products. We call these presheaves *virtual Waldhausen $\infty$-categories*. As with $\mathrm{D}_{\geq0}(k)$, virtual Waldhausen $\infty$-categories can be viewed as formal geometric realizations of simplicial Waldhausen $\infty$-categories, and the $\infty$-category ${\mathrm{D}(\mathbf{Wald}_{\infty})}$ enjoys the following universal property: for any $\infty$-category $C$ that admits all geometric realizations, the functor $${{\operatorname{Fun}({\mathrm{D}(\mathbf{Wald}_{\infty})},C)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}({\mathbf{Wald}_{\infty}},C)}}$$ induced by the Yoneda embedding ${{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {{\mathrm{D}(\mathbf{Wald}_{\infty})}}}$ restricts to an equivalence from the full subcategory of $\operatorname{Fun}({\mathrm{D}(\mathbf{Wald}_{\infty})},C)$ spanned by those functors ${{{\mathrm{D}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}$ that preserve geometric realizations to $\operatorname{Fun}({\mathbf{Wald}_{\infty}},C)$.
To get this idea off the ground, it is clear that we must analyze limits and colimits in ${\mathbf{Wald}_{\infty}}$. Along the way, we’ll find that, indeed, ${\mathbf{Wald}_{\infty}}$ is rather a lot like $V(k)$.
Limits and colimits of pairs of $\infty$-categories {#limits-and-colimits-of-pairs-of-infty-categories .unnumbered}
---------------------------------------------------
We first analyze limits and colimits in the $\infty$-category ${\mathbf{Pair}}_{\infty}$.
Suppose $C$ a locally small $\infty$-category [@HTT Df. 5.4.1.3]. For a regular cardinal $\kappa<\kappa_0$, recall [@HTT Df. 5.5.7.1] that $C$ is said to be ***$\kappa$-compactly generated*** (or simply ***compactly generated*** if $\kappa=\omega$) if it is $\kappa$-accessible and admits all small colimits. From this it will follow that $C$ admits all small limits as well. It follows from Simpson’s theorem [@HTT Th. 5.5.1.1] that $C$ is $\kappa$-compactly generated if and only if it is a $\kappa$-accessible localization of the $\infty$-category of presheaves $\mathscr{P}(C_0)=\operatorname{Fun}(C_0^{{\mathrm{op}}},{\mathbf{Kan}})$ of small spaces on some small $\infty$-category $C_0$.
\[thm:pairasloc\] The $\infty$-category ${\mathbf{Pair}}_{\infty}$ is an $\omega$-accessible localization of the arrow $\infty$-category $\mathscr{O}({\mathbf{Cat}}_{\infty})$.
We use \[prp:PairsubcatOcat\] to identify ${\mathbf{Pair}}_{\infty}$ with a full subcategory of $\mathscr{O}({\mathbf{Cat}}_{\infty})$. Now the condition that an object ${{C'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}$ of $\mathscr{O}({\mathbf{Cat}}_{\infty})$ be a monomorphism is equivalent to the demand that the functors $${{\iota C'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\iota C'\times^h_{\iota C}\iota C'}}\textrm{\quad and\quad}{{\iota\mathscr{O}(C')}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\iota\mathscr{O}(C')\times^h_{\iota\mathscr{O}(C)}\iota\mathscr{O}(C')}}$$ be isomorphisms of $h{\mathbf{Cat}}_{\infty}$. This, in turn, is the requirement that the object ${{C'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}$ be $S$-local, where $S$ is the set $$S{\mathrel{\mathop:}=}\left\{
\begin{tikzpicture}[baseline]
\matrix(m)[matrix of math nodes,
row sep=6ex, column sep=6ex,
text height=1.5ex, text depth=0.25ex]
{\Delta^p\sqcup\Delta^p&\Delta^p\\
\Delta^p&\Delta^p\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge node[above]{$\nabla$} (m-1-2)
edge node[left]{$\nabla$} (m-2-1)
(m-1-2) edge[-,double distance=1.5pt] (m-2-2)
(m-2-1) edge[-,double distance=1.5pt] (m-2-2);
\end{tikzpicture}
\ \Big|\quad\mathbf{p}\in\Delta\right\}$$ of morphisms of $\mathscr{O}({\mathbf{Cat}}_{\infty})$. The condition that an object ${{C'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}$ of $\mathscr{O}({\mathbf{Cat}}_{\infty})$ induce an equivalence ${{\iota C'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\iota C}}$ is equivalent to the requirement that it be local with respect to the singleton $$\{\phi\colon{{(\Delta^0)^{\flat}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(\Delta^0)^{\sharp}}}\}$$ (Ex. \[exm:minpairmaxpair\]). Hence ${\mathbf{Pair}}_{\infty}$ is equivalent to the full subcategory of the $S\cup\{\phi\}$-local objects of $\mathscr{O}({\mathbf{Cat}}_{\infty})$. Now it is easy to see that the $S\cup\{\phi\}$-local objects of $\mathscr{O}({\mathbf{Cat}}_{\infty})$ are closed under filtered colimits; hence by [@HTT Pr. 5.5.3.6 and Cor. 5.5.7.3], the $\infty$-category ${\mathbf{Pair}}_{\infty}$ is an $\omega$-accessible localization.
\[cor:paircpctlygen\] The $\infty$-category ${\mathbf{Pair}}_{\infty}$ is compactly generated.
\[cor:limsofpairs\] The $\infty$-category ${\mathbf{Pair}}_{\infty}$ admits all small limits, and the inclusion $${{{\mathbf{Pair}}_{\infty}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{O}({\mathbf{Cat}}_{\infty})}}$$ preserves them.
\[cor:colimsofpairs\] The $\infty$-category ${\mathbf{Pair}}_{\infty}$ admits all small colimits, and the inclusion $${{{\mathbf{Pair}}_{\infty}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{O}({\mathbf{Cat}}_{\infty})}}$$ preserves small filtered colimits.
Any pair $\mathscr{C}$ is the colimit of its compact subpairs.
Limits and filtered colimits of Waldhausen $\infty$-categories {#limits-and-filtered-colimits-of-waldhausen-infty-categories .unnumbered}
--------------------------------------------------------------
Now we construct limits and colimits in ${\mathbf{Wald}_{\infty}}$.
\[thm:Waldlims\] The $\infty$-category ${\mathbf{Wald}_{\infty}}$ admits all small limits, and the inclusion functor ${{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}_{\infty}}}$ preserves them.
We employ [@HTT Pr. 4.4.2.6] to reduce the problem to proving the existence of products and pullbacks in ${\mathbf{Wald}_{\infty}}$. To complete the proof, we make the following observations.
1. Suppose $I$ a set, suppose $(\mathscr{C}_i)_{i\in I}$ an $I$-tuple of pairs of $\infty$-categories, and suppose $\mathscr{C}$ the product of these pairs. If for each $i\in I$, the pair $\mathscr{C}_i$ is a Waldhausen $\infty$-category, then so is $\mathscr{C}$. Moreover, if $\mathscr{D}$ is a Waldhausen $\infty$-category, then a functor of pairs ${{\mathscr{D}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$ is exact if and only if the composite $$\mathscr{D}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ \mathscr{C}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ \mathscr{C}_i$$ is exact for any $i\in I$. This follows directly from the fact that limits and colimits of a product are computed objectwise [@HTT Cor. 5.1.2.3].
2. Suppose $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\mathscr{E}'&\mathscr{F}'\\
\mathscr{E}&\mathscr{F}\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge node[above]{$q'$} (m-1-2)
edge node[left]{$p'$} (m-2-1)
(m-1-2) edge node[right]{$p$} (m-2-2)
(m-2-1) edge node[below]{$q$} (m-2-2);
\end{tikzpicture}$$ a pullback diagram of pairs of $\infty$-categories. Suppose moreover that $\mathscr{E}$, $\mathscr{F}$, and $\mathscr{F}'$ are all Waldhausen $\infty$-categories, and $p$ and $q$ are exact functors. Then $\mathscr{E}'$ is a Waldhausen $\infty$-category, and for any Waldhausen $\infty$-category $\mathscr{D}$, a functor of pairs $\psi\colon{{\mathscr{D}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}}}$ is exact if and only if the composites $p'\circ\psi$ and $q'\circ\psi$ are exact. This follows from [@HTT Lm. 5.4.5.5].
We obtain a similar characterization of filtered colimits in ${\mathbf{Wald}_{\infty}}$.
\[thm:Waldfiltcolims\] The $\infty$-category ${\mathbf{Wald}_{\infty}}$ admits all small filtered colimits, and the inclusion functor ${{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}_{\infty}}}$ preserves them.
Suppose $A$ a filtered $\infty$-category, and suppose ${{A}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}}}$ a functor given by the assignment ${{a}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}_a}}$, and suppose $\mathscr{C}$ the colimit of the composite functor $$A\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{Wald}_{\infty}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{Pair}}_{\infty}.$$ The proof is completed by the following observations.
1. Pushouts of cofibrations in $\mathscr{C}$ exist and are cofibrations. This follows by precisely the same argument as [@HTT Pr. 5.5.7.11].
2. The underlying $\infty$-category $\mathscr{C}$ admits a zero object, which is the initial object of $\mathscr{C}_{\dag}$. This follows from the fact that for any object $a\in A$, the functor ${{\mathscr{C}_a}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$ preserves zero objects, and the functor ${{\mathscr{C}_{a,\dag}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}_{\dag}}}$ preserves initial objects.
Direct sums of Waldhausen $\infty$-categories {#direct-sums-of-waldhausen-infty-categories .unnumbered}
---------------------------------------------
The $\infty$-category ${\mathbf{Wald}_{\infty}}$ also admits finite *direct sums*, i.e., that finite products in ${\mathbf{Wald}_{\infty}}$ are also finite coproducts.
\[item:directsums\] Suppose $C$ is an $\infty$-category. Then $C$ is said to *admit finite direct sums* if the following conditions hold.
1. The $\infty$-category $C$ is pointed.
2. The $\infty$-category $C$ has all finite products and coproducts.
3. For any finite set $I$ and any $I$-tuple $(X_i)_{i\in I}$ of objects of $C$, the map $${{\coprod X_I}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\prod X_I}}$$ in $hC$ — given by the maps $\phi_{ij}\colon{{X_i}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X_j}}$, where $\phi_{ij}$ is zero unless $i=j$, in which case it is the identity — is an isomorphism.
If $C$ admits finite direct sums, then for any finite set $I$ and any $I$-tuple $(X_i)_{i\in I}$ of objects of $C$, we denote by $\bigoplus X_I$ the product (or, equivalently, the coproduct) of the $X_i$.
\[prp:Waldhassums\] The $\infty$-category ${\mathbf{Wald}_{\infty}}$ admits finite direct sums.
The Waldhausen $\infty$-category $\Delta^0$ is a zero object. To complete the proof, it suffices to show that for any finite set $I$ and any $I$-tuple of Waldhausen $\infty$-categories $(\mathscr{C}_i)_{i\in I}$ with product $\mathscr{C}$, the functors $\phi_i\colon{{\mathscr{C}_i}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$ — given by the functors $\phi_{ij}\colon{{\mathscr{C}_i}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}_j}}$, where $\phi_{ij}$ is zero unless $j=i$, in which case it is the identity — are exact and exhibit $\mathscr{C}$ as the *coproduct* of $(\mathscr{C}_i)_{i\in I}$. To prove this, we employ [@HTT Th. 4.2.4.1] to reduce the problem to showing that for any Waldhausen $\infty$-category $\mathscr{D}$, the map $${{{\mathbf{Wald}_{\infty}}^{\Delta}(\mathscr{C},\mathscr{D})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\prod_{i\in I}{\mathbf{Wald}_{\infty}}^{\Delta}(\mathscr{C}_i,\mathscr{D})}}$$ induced by the functor $\phi_i$ is a weak homotopy equivalence. We prove the stronger claim that the functor $$w\colon{{\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{C},\mathscr{D})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\prod_{i\in I}\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{C}_i,\mathscr{D})}}$$ is an equivalence of $\infty$-categories.
For this, consider the following composite $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=-6ex,
text height=1.5ex, text depth=0.25ex]
{&\operatorname{Fun}(\mathscr{C},\operatorname{Fun}(NI,\mathscr{D}))&[14ex]\operatorname{Fun}(\mathscr{C},\operatorname{Colim}((NI)^{\rhd},\mathscr{D}))&\\
\prod_{i\in I}\operatorname{Fun}(\mathscr{C}_i,\mathscr{D})&&&\operatorname{Fun}(\mathscr{C},\mathscr{D})\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-2) edge node[above]{$r$} (m-1-3)
edge[<-,inner sep=4pt] node[left]{$u$} (m-2-1)
(m-1-3) edge[inner sep=4pt] node[right]{$e$} (m-2-4);
\end{tikzpicture}$$ where $u$ is the functor corresponding to the functor $${{\mathscr{C}\times\prod_{i\in I}\operatorname{Fun}(\mathscr{C}_i,\mathscr{D})\cong\prod_{i\in I}(\mathscr{C}_i\times\operatorname{Fun}(\mathscr{C}_i,\mathscr{D}))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\prod_{i\in I}\mathscr{D}}},$$ where $r$ is a section of the trivial fibration $${{\operatorname{Fun}(\mathscr{C},\operatorname{Colim}((NI)^{\rhd},\mathscr{D}))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}(\mathscr{C},\operatorname{Fun}(NI,\mathscr{D}))}},$$ and $e$ is the functor induced by the functor ${{\operatorname{Colim}((NI)^{\rhd},\mathscr{D})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{D}}}$ given by evaluation at the cone point $\infty$. This composite restricts to a functor $$v\colon{{\prod_{i\in I}\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{C}_i,\mathscr{D})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{C},\mathscr{D})}};$$ indeed, one checks directly that if $(\psi_i\colon{{\mathscr{C}_i}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{D}}})_{i\in I}$ is an $I$-tuple of exact functors, then a functor $\psi\colon{{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{D}}}$ that sends a simplex $\sigma=(\sigma_i)_{i\in I}$ to a coproduct $\coprod_{i\in I}\psi_i(\sigma_i)$ in $\mathscr{D}$ is exact, and the situation is similar for natural transformations of exact functors.
We claim that the functor $v$ is a homotopy inverse to $w$. A homotopy $w\circ v\simeq\operatorname{id}$ can be constructed directly from the canonical equivalences $$Y\simeq Y\sqcup\coprod_{i\in I-\{j\}}0_i$$ for any zero objects $0_i$ in $\mathscr{D}$. In the other direction, the existence of a homotopy $v\circ w\simeq\operatorname{id}$ follows from the observation that the natural transformations ${{\phi_i\circ\operatorname{pr}_i}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{id}}}$ exhibit the identity functor on $\mathscr{C}$ as the coproduct $\coprod_{i\in I}\phi_i\circ\operatorname{pr}_i$.
Since any small coproduct can be written as a filtered colimit of finite coproducts, we deduce the following.
The $\infty$-category ${\mathbf{Wald}_{\infty}}$ admits all small coproducts.
Coproducts in ${\mathbf{Wald}_{\infty}}$ enjoy a description reminiscent of the description of coproducts in the category of vector spaces over a field: for any set $I$ and an $I$-tuple $(\mathscr{C}_i)_{i\in I}$ of Waldhausen $\infty$-categories, $\coprod_{i\in I}\mathscr{C}_i$ is equivalent to the full subcategory of $\prod_{i\in I}\mathscr{C}_i$ spanned by those objects $(X_i)_{i\in I}$ such that all but a finite number of the objects $X_i$ are zero objects.
Accessibility of {#accessibility-of .unnumbered}
-----------------
Finally, we set about showing that ${\mathbf{Wald}_{\infty}}$ is an accessible $\infty$-category. In fact, we prove the following stronger result.
\[prp:Waldcpctlygen\] The $\infty$-category ${\mathbf{Wald}_{\infty}}$ is compactly generated.
We wish to show that ${\mathbf{Wald}_{\infty}}$ is an object of the (huge) $\infty$-category ${\mathbf{Pr}}^{\mathrm{R}}_{\omega}$ of large compactly generated $\infty$-categories and limit-preserving, $\omega$-continuous functors [@HTT Df. 5.5.7.5]. Thanks to [@HTT Pr. 5.5.7.6], it suffices for us to exhibit ${\mathbf{Wald}_{\infty}}$ as a suitable limit.
Let $S$ be the $\infty$-category $$(\Delta^1\cup^{\partial\Delta^1}\Delta^1)\cup^{\Delta^{\{0\}}}(\Delta^1\cup^{\partial\Delta^1}\Delta^1).$$ This $\infty$-category is in fact the nerve of an ordinary category with three objects, $o$, $z$, and $w$, two maps ${{o}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {z}}$, two maps ${{o}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {w}}$, and no other nonidentity maps. We may now construct a functor $W\colon{{S}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pr}}^{\mathrm{R}}_{\omega}}}$ in the following manner. Let $W$ assign to $o$ the $\infty$-category ${\mathbf{Pair}}_{\infty}$, to $z$ the $\infty$-category ${\mathbf{Cat}}_{\infty}$, and to $w$ the $\infty$-category ${\mathbf{Cat}}_{\infty}$. The two maps ${{o}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {z}}$ are sent to: (1) the constant functor at the object $\Delta^0\in{\mathbf{Cat}}_{\infty}$, and (2) the functor $${{\mathscr{C}}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Colim}(\Delta^0,\mathscr{C}_{\dag})\times_{\operatorname{Fun}(\Delta^0,\mathscr{C})}\operatorname{Colim}(\Delta^0,\mathscr{C})\times_{\operatorname{Fun}(\Delta^0,\mathscr{C})}\operatorname{Lim}(\Delta^0,\mathscr{C})}}.$$ The two maps ${{o}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {w}}$ are sent to: (1) the functor $${{\mathscr{C}}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\Lambda_0\mathscr{Q}^2,\mathscr{C})}}$$ and (2) the functor $${{\mathscr{C}}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Colim}(\mathscr{Q}^2,\mathscr{C})\times_{\operatorname{Fun}(\mathscr{Q}^2,\mathscr{C})}\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{Q}^2,\mathscr{C})}}.$$ It is not difficult to see that these functors are all limit-preserving and $\omega$-continuous. Now we observe that ${\mathbf{Wald}_{\infty}}$ may be exhibited as a limit of the functor $$S\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$W$}(0.5,0.5ex);\ {\mathbf{Pr}}^{\mathrm{R}}_{\omega}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{Cat}}_{\infty}(\kappa_1),$$ whence we deduce that ${\mathbf{Wald}_{\infty}}$ is compactly generated from [@HTT Pr. 5.5.7.6].
This result shows that in fact the $\infty$-category ${\mathbf{Wald}_{\infty}}$ admits *all* small colimits, not only the filtered ones. However, these other colimits are not preserved by the sorts of invariants in which we are interested, and so we will regard them as pathological. Nevertheless, we will have use for the following.
The $\infty$-category ${\mathbf{Wald}_{\infty}}$ is $\omega$-accessible.
\[cor:WaldisInd\] The $\infty$-category ${\mathbf{Wald}_{\infty}}$ may be identified with the $\mathrm{Ind}$-objects of the full subcategory ${\mathbf{Wald}_{\infty}}^{\omega}\subset{\mathbf{Wald}_{\infty}}$ spanned by the compact Waldhausen $\infty$-categories: $${\mathbf{Wald}_{\infty}}\simeq\operatorname{Ind}({\mathbf{Wald}_{\infty}}^{\omega}).$$
We obtain a further corollary by combining Prs. \[prp:Waldcpctlygen\], \[thm:Waldlims\], and \[thm:Waldfiltcolims\] together with the adjoint functor theorem [@HTT Cor. 5.5.2.9].
\[cor:Wconstruction\] The forgetful functor ${{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}_{\infty}}}$ admits a left adjoint $W\colon{{{\mathbf{Pair}}_{\infty}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}}}$.
Since the opposite functor ${{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{coWald}}}}$ is an equivalence of $\infty$-categories, it follows that this whole crop of structural results also hold for ${\mathbf{coWald}}$. That is, ${\mathbf{coWald}}$ admits all small limits and all small filtered colimits, and the inclusion functor ${{{\mathbf{coWald}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}_{\infty}}}$ preserves each of them. Similarly, ${\mathbf{coWald}}$ admits finite direct sums and all small coproducts, and it is compactly generated.
Virtual Waldhausen $\infty$-categories {#virtual-waldhausen-infty-categories .unnumbered}
--------------------------------------
Now we are prepared to introduce a convenient enlargement of the $\infty$-category ${\mathbf{Wald}_{\infty}}$. In effect, we aim to “correct” the colimits of ${\mathbf{Wald}_{\infty}}$ that we regard as pathological. As with the formation of $\mathrm{D}_{\geq0}(k)$ from $NV(k)$ (see the introduction of this section) — or indeed with the formation of the $\infty$-category of spaces from the nerve of the category of sets —, we will add to ${\mathbf{Wald}_{\infty}}$ formal geometric realizations and nothing more. The result is the *derived $\infty$-category of Waldhausen $\infty$-categories*, whose homotopy theory forms the basis of our work here.
The definition is exactly as for $\mathrm{D}_{\geq0}(k)$:
\[dfn:virtWald\] A ***virtual Waldhausen $\infty$-category*** is a presheaf $$\mathscr{X}\colon{{({\mathbf{Wald}_{\infty}}^{\omega})^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$$ that preserves products.
Denote by $${\mathrm{D}(\mathbf{Wald}_{\infty})}\subset\operatorname{Fun}({\mathbf{Wald}_{\infty}}^{\omega,{\mathrm{op}}},{\mathbf{Kan}})$$ the full subcategory spanned by the virtual Waldhausen $\infty$-categories. In other words, ${\mathrm{D}(\mathbf{Wald}_{\infty})}$ is *nonabelian derived $\infty$-category* of ${\mathbf{Wald}_{\infty}}^{\omega}$ [@HTT §5.5.8]. We simply call ${\mathrm{D}(\mathbf{Wald}_{\infty})}$ the ***derived $\infty$-category of Waldhausen $\infty$-categories***.
For any $\infty$-category $C$, we shall write $\mathscr{P}(C)$ for the $\infty$-category $\operatorname{Fun}(C^{{\mathrm{op}}},{\mathbf{Kan}})$ of presheaves of small spaces on $C$. If $C$ is locally small, then there exists a Yoneda embedding [@HTT Pr. 5.1.3.1] $$j\colon{{C}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{P}(C)}}.$$
\[rec:PABofC\] Suppose $\mathscr{A}\subset\mathscr{B}$ two classes of small simplicial sets, and suppose $C$ an $\infty$-category that admits all $\mathscr{A}$-shaped colimits (\[rec:Ashapedcolim\]). Recall [@HTT §5.3.6] that there exist an $\infty$-category $\mathscr{P}_{\mathscr{A}}^{\mathscr{B}}(C)$ and a fully faithful functor $j\colon{{C}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{P}_{\mathscr{A}}^{\mathscr{B}}(C)}}$ such that for any $\infty$-category $D$ with all $\mathscr{B}$-shaped colimits, $j$ induces an equivalance of $\infty$-categories (\[rec:Ashapedcolim\]) $${{\operatorname{Fun}_{\mathscr{B}}(\mathscr{P}_{\mathscr{A}}^{\mathscr{B}}(C),D)}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\operatorname{Fun}_{\mathscr{A}}(C,D)}}.$$
Recall also [@HTT Nt. 6.1.2.12] that for any $\infty$-category $C$, the colimit of a simplicial diagram $X\colon{{N\Delta^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}$ will be called the ***geometric realization*** of $X$.
\[nul:VWaldasPDKWald\] In the notation of \[rec:PABofC\], the $\infty$-category ${\mathrm{D}(\mathbf{Wald}_{\infty})}$ can be identified with any of the following $\infty$-categories:
1. the $\infty$-category $\mathscr{P}_{\varnothing}^{\{N\Delta^{{\mathrm{op}}}\}}{\mathbf{Wald}_{\infty}}$,
2. the $\infty$-category $\mathscr{P}_{\mathscr{I}}^{\mathscr{J}}{\mathbf{Wald}_{\infty}}$, where $\mathscr{I}$ is the collection of small, filtered simplicial sets and $\mathscr{J}$ is the collection of small, sifted simplicial sets,
3. the $\infty$-category $\mathscr{P}_{\varnothing}^{\mathscr{J}}{\mathbf{Wald}_{\infty}}^{\omega}$, and
4. the $\infty$-category $\mathscr{P}_{\mathscr{D}}^{\mathscr{K}}{\mathbf{Wald}_{\infty}}^{\omega}$, where $\mathscr{D}$ is the collection of finite discrete simplicial sets, and $\mathscr{K}$ is the collection of small simplicial sets.
The equivalence of these characterizations follow directly from Cor. \[cor:WaldisInd\], the description of the nonabelian derived $\infty$-category of [@HTT Pr. 5.5.8.16], the fact that sifted colimits can be decomposed as geometric realizations of filtered colimits [@HTT Pr. 5.5.8.15], and the transitivity assertion of [@HTT Pr. 5.3.6.11].
We may summarize these characterizations by saying that the Yoneda embedding is a fully faithful functor $$j\colon{{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {{\mathrm{D}(\mathbf{Wald}_{\infty})}}}$$ that induces, for any $\infty$-category $E$ that admits geometric realizations, any $\infty$-category $E'$ that admits all sifted colimits, and any $\infty$-category that admits all small colimits, equivalences (\[rec:Ashapedcolim\]) $$\begin{aligned}
&{{\operatorname{Fun}_{\{N\Delta^{{\mathrm{op}}}\}}({\mathrm{D}(\mathbf{Wald}_{\infty})},E)}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\operatorname{Fun}({\mathbf{Wald}_{\infty}},E)}};&\nonumber\\
&{{\operatorname{Fun}_{\mathscr{J}}({\mathrm{D}(\mathbf{Wald}_{\infty})},E')}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\operatorname{Fun}_{\mathscr{I}}({\mathbf{Wald}_{\infty}},E')}};&\nonumber\\
&{{\operatorname{Fun}_{\mathscr{J}}({\mathrm{D}(\mathbf{Wald}_{\infty})},E')}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\operatorname{Fun}_{\mathscr{I}}({\mathbf{Wald}_{\infty}}^{\omega},E')}};&\nonumber\\
&{{\operatorname{Fun}_{\mathscr{K}}({\mathrm{D}(\mathbf{Wald}_{\infty})},E'')}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\operatorname{Fun}_{\mathscr{D}}({\mathbf{Wald}_{\infty}}^{\omega},E'')}}.&\nonumber\end{aligned}$$
\[dfn:lderivWald\] Suppose $E$ an $\infty$-category that admits all sifted colimits. Then a functor $$\Phi\colon{{{\mathrm{D}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {E}}$$ that preserves all sifted colimits will be said to be the ***left derived functor*** of the corresponding $\omega$-continuous functor $\phi=\Phi\circ j\colon{{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {E}}$ (which preserves filtered colimits) or of the further restriction ${{{\mathbf{Wald}_{\infty}}^{\omega}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {E}}$ of $\phi$ to ${\mathbf{Wald}_{\infty}}^{\omega}$.
The $\infty$-category ${\mathrm{D}(\mathbf{Wald}_{\infty})}$ is compactly generated. Moreover, it admits all direct sums, and the inclusion $j$ preserves them.
The first statement is [@HTT Pr. 5.5.8.10(8)]. To see that ${\mathrm{D}(\mathbf{Wald}_{\infty})}$ admits direct sums, we use the fact that we may exhibit any object of ${\mathrm{D}(\mathbf{Wald}_{\infty})}$ as a sifted colimit of compact Waldhausen $\infty$-categories in $\mathscr{P}({\mathbf{Wald}_{\infty}}^{\omega})$ [@HTT Lm. 5.5.8.14]; now since sifted colimits commute with both finite products [@HTT Lm. 5.5.8.11] and coproducts, and since $j$ preserves products and finite coproducts [@HTT Lm. 5.5.8.10(2)], the proof is complete.
Realizations of Waldhausen cocartesian fibrations {#realizations-of-waldhausen-cocartesian-fibrations .unnumbered}
-------------------------------------------------
We now give an explicit construction of colimits in ${\mathrm{D}(\mathbf{Wald}_{\infty})}$ of sifted diagrams of Waldhausen $\infty$-categories when they are exhibited as Waldhausen cocartesian fibrations.
The idea behind our construction comes from the following observation.
\[rec:colimitastotalspace\] For any left fibration $p\colon{{X}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ (\[rec:leftfib\]), the total space $X$ is a model for the colimit of the functor ${{S}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$ that classifies $p$ [@HTT Cor. 3.3.4.6].
If $S$ is an $\infty$-category and ${\mathbf{X}}\colon{{S}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}}}$ is a diagram of Waldhausen $\infty$-categories, then the colimit of the composite $$S\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{${\mathbf{X}}$}(0.5,0.5ex);\ {\mathbf{Wald}_{\infty}}\ \tikz[baseline]\draw[>=stealth,right hook->,font=\scriptsize](0,0.5ex)--node[above]{$j$}(0.5,0.5ex);\ \mathscr{P}({\mathbf{Wald}_{\infty}})$$ is computed objectwise [@HTT Cor. 5.1.2.3]. If $S$ is sifted, then since ${\mathrm{D}(\mathbf{Wald}_{\infty})}\subset\mathscr{P}({\mathbf{Wald}_{\infty}})$ is stable under sifted colimits, the colimit of the composite $$S\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{${\mathbf{X}}$}(0.5,0.5ex);\ {\mathbf{Wald}_{\infty}}\ \tikz[baseline]\draw[>=stealth,right hook->,font=\scriptsize](0,0.5ex)--node[above]{$j$}(0.5,0.5ex);\ {\mathrm{D}(\mathbf{Wald}_{\infty})}$$ is also computed objectwise. That is, for any compact Waldhausen $\infty$-category $\mathscr{C}$, one has $$(\operatorname{colim}_{s\in S}{\mathbf{X}}(s))(\mathscr{C})\simeq\operatorname{colim}_{s\in S}\iota\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{C},{\mathbf{X}}(s)).$$ Suppose that ${\mathbf{X}}$ classifies a Waldhausen cartesian fibration ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$; then we can use the construction described in \[rec:htt32213\] first to produce a cocartesian fibration ${{\mathscr{H}(\mathscr{C},(\mathscr{X}/S))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ classified by the functor ${{s}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{C},{\mathbf{X}}(s))}}$ and then to produce a left fibration (\[rec:leftfib\]) $${{\mathrm{H}(\mathscr{C},(\mathscr{X}/S)){\mathrel{\mathop:}=}\iota_S\mathscr{H}(\mathscr{C},(\mathscr{X}/S))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$$ that classifies the colimit of the composite $$S\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{${\mathbf{X}}$}(0.5,0.5ex);\ {\mathbf{Wald}_{\infty}}\ \tikz[baseline]\draw[>=stealth,right hook->,font=\scriptsize](0,0.5ex)--node[above]{$j$}(0.5,0.5ex);\ \mathscr{P}({\mathbf{Wald}_{\infty}})\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$\operatorname{ev}_{\mathscr{C}}$}(0.75,0.5ex);\ {\mathbf{Kan}}.$$ We can avoid choosing a straightening of the Waldhausen cocartesian fibration by means of the following.
\[ntn:geomrealinWald\] Suppose $S$ a sifted $\infty$-category, and suppose ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ a Waldhausen cocartesian fibration. Then for any compact Waldhausen $\infty$-category $\mathscr{C}$, define a simplicial set $\mathscr{H}'(\mathscr{C},(\mathscr{X}/S))$ over $S$ via the universal property $$\operatorname{Mor}_{S}(K,\mathscr{H}'(\mathscr{C},(\mathscr{X}/S)))\cong\operatorname{Mor}_{S}(\mathscr{C}\times K,\mathscr{X}),$$ functorial in simplicial sets $K$ over $S$. The resulting map $${{\mathscr{H}'(\mathscr{C},(\mathscr{X}/S))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$$ is a cocartesian fibration by \[rec:htt32213\] and [@HTT Cor. 3.2.2.13]. Denote by $\mathscr{H}(\mathscr{C},(\mathscr{X}/S))$ the full subcategory of $\mathscr{H}'(\mathscr{C},(\mathscr{X}/S))$ spanned by those functors ${{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{X}_s}}$ that are exact functors of Waldhausen $\infty$-categories; here too the canonical functor $$p\colon{{\mathscr{H}(\mathscr{C},(\mathscr{X}/S))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$$ is a cocartesian fibration. Now denote by $\mathrm{H}(\mathscr{C},(\mathscr{X}/S))$ the subcategory $$\iota_S\mathscr{H}(\mathscr{C},(\mathscr{X}/S))\subset\mathscr{H}(\mathscr{C},(\mathscr{X}/S))$$ consisting of the $p$-cocartesian morphisms (\[rec:leftfib\]). The functor $$\iota_S(p)\colon{{\mathrm{H}(\mathscr{C},(\mathscr{X}/S))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$$ is now a left fibration.
Of course, we may simply realize the assignment ${{(\mathscr{C},(\mathscr{X}/S))}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {\mathrm{H}(\mathscr{C},(\mathscr{X}/S))}}$ as a functor $$\mathrm{H}\colon{{{\mathbf{Wald}_{\infty}}^{\omega,{\mathrm{op}}}\times\mathbf{Wald}_{\infty/S}^{{\mathrm{cocart}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$$ by choosing both an equivalence ${{\mathbf{Wald}_{\infty/S}^{{\mathrm{cocart}}}}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\operatorname{Fun}(S,{\mathbf{Wald}_{\infty}})}}$ and a colimit functor ${{\operatorname{Fun}(S,{\mathbf{Kan}})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$. We have given this explicit construction of the values of this functor in terms of Waldhausen cocartesian fibrations for later use.
In the meantime, since virtual Waldhausen $\infty$-categories are closed under sifted colimits in $\mathscr{P}({\mathbf{Wald}_{\infty}}^{\omega})$, we have the following.
If $S$ is a small sifted $\infty$-category and if ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ is a Waldhausen cocartesian fibration in which $\mathscr{X}$ is small, then the corresponding functor $\mathrm{H}(-,(\mathscr{X}/S))\colon{{{\mathbf{Wald}_{\infty}}^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$ is a virtual Waldhausen $\infty$-category.
If $S$ is a small sifted $\infty$-category, then functor $$\mathrm{H}\colon{{\mathbf{Wald}_{\infty,/S}^{{\mathrm{cocart}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{P}({\mathbf{Wald}_{\infty}}^{\omega})}}$$ factors through the $\infty$-category of virtual Waldhausen $\infty$-categories: $$|\cdot|_S\colon{{\mathbf{Wald}_{\infty/S}^{{\mathrm{cocart}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathrm{D}(\mathbf{Wald}_{\infty})}}}.$$
A presheaf on ${\mathbf{Wald}_{\infty}}^{\omega}$ lies in the nonabelian derived $\infty$-category just in case it can be written as the geometric realization of a diagram of Ind-objects of ${\mathbf{Wald}_{\infty}}^{\omega}$ [@HTT Lm. 5.5.8.14]. In other words, we have the following.
\[cor:everythingsarealization\] Suppose $\mathscr{X}$ a virtual Waldhausen $\infty$-category. Then there exists a Waldhausen cocartesian fibration ${{\mathscr{Y}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}}}$ and an equivalence $\mathscr{X}\simeq|\mathscr{Y}|_{N\Delta^{{\mathrm{op}}}}$.
\[dfn:realization\] For any small sifted simplicial set and any Waldhausen cocartesian fibration $\mathscr{X}/S$, the virtual Waldhausen $\infty$-category $|\mathscr{X}|_S$ will be called the ***realization*** of $\mathscr{X}/S$.
In this part, we study reduced and finitary functors from ${\mathbf{Wald}_{\infty}}$ to the $\infty$-category of pointed objects of an $\infty$-topos, which we simply call *theories*. We begin by studying the virtual Waldhausen $\infty$-categories of filtered and totally filtered objects of a Waldhausen $\infty$-category. Using these, we study the class of *fissile* virtual Waldhausen $\infty$-categories; these form a localization of ${\mathrm{D}(\mathbf{Wald}_{\infty})}$, and we show that suspension in this $\infty$-category is given by the formation of the virtual Waldhausen $\infty$-category of totally filtered objects, which is in turn an $\infty$-categorical analogue of Waldhausen’s $S_{\bullet}$ construction. We then show that suitable excisive functors on the $\infty$-category of fissile virtual Waldhausen $\infty$-categories correspond to additive theories that satisfy the consequences of an $\infty$-categorical analogue of Waldhausen’s additivity theorem, and we construct an *additivization* as a Goodwillie derivative, employing our newly minted suspension functor.
Filtered objects of Waldhausen $\infty$-categories {#sect:filt}
==================================================
The phenomenon behind additivity is the interaction between a filtered object and its various quotients. For example, for a category $\mathscr{C}$ with cofibrations in the sense of Waldhausen, the universal property of $K_0(\mathscr{C})$ ensures that it regards an object with a filtration of finite length $$X_0\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X_1\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X_n$$ as the sum of the first term $X_0$ and the filtered object obtained by quotienting by $X_0$: $$0\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X_1/X_0\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X_n/X_0,$$ or, by induction, as the sum of $X_0$, $X_1/X_0$, …, $X_n/X_{n-1}$. In order to formulate this condition properly for the entire $K$-theory *space*, it is necessary to study $\infty$-categories of filtered objects in a Waldhausen $\infty$-category and the various quotient functors all as suitable inputs for algebraic $K$-theory. This is the subject of this section.
In particular, for any integer $m\geq 0$ and any Waldhausen $\infty$-category $\mathscr{C}$, we construct a Waldhausen $\infty$-category $\mathscr{F}_m(\mathscr{C})$ of filtered objects of length $m$, and we define not only the exact functors between these Waldhausen $\infty$-categories corresponding to changing the length of the filtration (given by morphisms of $\Delta$), but also sundry quotient functors. Since quotient functors are only defined up to coherent equivalences, we employ the language of Waldhausen (co)cartesian fibrations (§\[sect:Waldfib\]).
After we pass to suitable colimits in ${\mathrm{D}(\mathbf{Wald}_{\infty})}$, we end up with two functors ${{{\mathrm{D}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathrm{D}(\mathbf{Wald}_{\infty})}}}$. The first of these, which we denote $\mathscr{F}$, is a model for the *cone* in ${\mathrm{D}(\mathbf{Wald}_{\infty})}$ (Pr. \[prp:FCisacone\]). The second, which we will denote $\mathscr{S}$, will be a *suspension*, not quite in ${\mathrm{D}(\mathbf{Wald}_{\infty})}$, but in a suitable localization of ${\mathrm{D}(\mathbf{Wald}_{\infty})}$ (Cor. \[cor:Sdotisreallysuspension\]). The study of these functors is thus central to our interpretation of additive functors as excisive functors (Th. \[thm:additiveequiv\]).
The cocartesian fibration of filtered objects {#the-cocartesian-fibration-of-filtered-objects .unnumbered}
---------------------------------------------
Filtered objects are defined in the familiar manner.
A ***filtered object of length $m$*** of a pair of $\infty$-categories $\mathscr{C}$ is a sequence of ingressive morphisms $$X_0\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X_1\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X_m;$$ that is, it is a functor of pairs $X\colon{{(\Delta^m)^{\sharp}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$ (Ex. \[exm:minpairmaxpair\]).
For any morphism $\eta\colon{{\mathbf{m}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{n}}}$ of $\Delta$ and any filtered object $X$ of length $n$, one may precompose $X$ with the induced functor of pairs ${{(\Delta^m)^{\sharp}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(\Delta^n)^{\sharp}}}$ to obtain a filtered object $\psi^{\star}X$ of length $m$: $$X_{\eta(0)}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X_{\eta(1)}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X_{\eta(m)}.$$ One thus obtains a functor ${{N\Delta^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}}}$ that assigns to any object $\mathbf{m}\in\Delta$ the $\infty$-category $\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}((\Delta^m)^{\sharp},\mathscr{C})$. This is all simple enough.
But we will soon be forced to make things more complicated: if $\mathscr{C}$ is a Waldhausen $\infty$-category, we will below have to contemplate not only filtered objects but also *totally filtered objects* in $\mathscr{C}$; these are filtered objects $X$ such that the object $X_0$ is a zero object. The $\infty$-category of totally filtered objects of length $m$ is also functorial in $\mathbf{m}$: for any morphism $\eta\colon{{\mathbf{m}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{n}}}$ of $\Delta$ and any totally filtered object $X$ of length $n$, one may still precompose $X$ with the induced functor of pairs ${{(\Delta^m)^{\sharp}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(\Delta^n)^{\sharp}}}$ to obtain the filtered object $\eta^{\star}X$ of length $m$, and then one may get a *totally* filtered object by forming a quotient by the object $X_{\eta(0)}$: $$0\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X_{\eta(1)}/X_{\eta(0)}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X_{\eta(m)}/X_{\eta(0)}.$$ As we noted just before \[rec:cocart\], this does not specify a functor ${{N\Delta^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}}}$ on the nose, because the formation of quotients is only unique up to canonical equivalences.
This can be repaired in a variety of ways; for example, one may follow in Waldhausen’s footsteps [@MR86m:18011 §1.3] and rectify this construction by choosing all the compatible homotopy quotients at once. (For example, Lurie makes use of Waldhausen’s idea in [@HA §1.2.2].) But this is overkill: the theory of $\infty$-categories is precisely designed to finesse these homotopy coherence problems, and there is a genuine technical advantage in doing so. (For example, the total space of a left fibration is a ready-to-wear model for the homotopy colimit of the functor that classifies it; see \[rec:colimitastotalspace\] or [@HTT Cor. 3.3.4.6].) More specifically, the theory of cocartesian fibrations allows us to work effectively with this construction without solving homotopy coherence problems like this.
To that end, let’s first use Pr. \[prp:htt32213\] to access the cocartesian fibration $${{\mathscr{F}(\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}}}$$ classified by the functor $${{\mathbf{m}}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}((\Delta^m)^{\sharp},\mathscr{C})}}.$$ At no extra cost, for any Waldhausen cocartesian fibration ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ classified by a functor ${\mathbf{X}}\colon{{S}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}}}$, we can actually write down the cocartesian fibration $${{\mathscr{F}(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}\times S}}$$ classified by the functor $${{(\mathbf{m},s)}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}((\Delta^m)^{\sharp},{\mathbf{X}}(s))}}.$$ Once this has been done, we’ll be in a better position to define a Waldhausen cocartesian fibration of totally filtered objects.
The first step to using Pr. \[prp:htt32213\] is to identify the pair cartesian fibration (Df. \[dfn:paircartfib\]) that is classified by the functor ${{\mathbf{m}}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {(\Delta^m)^{\sharp}}}$.
\[ntn:M\] Denote by $\mathrm{M}$ the ordinary category whose objects of $\mathrm{M}$ are pairs $(\mathbf{m},i)$ consisting of an object $\mathbf{m}\in\Delta$ and an element $i\in\mathbf{m}$ and whose morphisms ${{(\mathbf{n},j)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(\mathbf{m},i)}}$ are maps $\phi\colon{{\mathbf{m}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{n}}}$ of $\Delta$ such that $j\leq\phi(i)$. This category comes equipped with a natural projection ${{\mathrm{M}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Delta^{{\mathrm{op}}}}}$.
It is easy to see that the projection ${{\mathrm{M}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Delta^{{\mathrm{op}}}}}$ is a Grothendieck fibration, and so the projection $\pi\colon{{N\mathrm{M}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}}}$ is a cartesian fibration. In fact, the category $\mathrm{M}$ is nothing more than the Grothendieck construction applied to the natural inclusion ${{(\Delta^{{\mathrm{op}}})^{{\mathrm{op}}}\cong\Delta}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}}}$. So the functor ${{\Delta}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}}}$ that classifies $\pi$ is given by the assignment ${{\mathbf{m}}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {\Delta^m}}$.
The nerve $N\mathrm{M}$ can be endowed with a pair structure by setting $$(N\mathrm{M})_{\dag}{\mathrel{\mathop:}=}N\mathrm{M}\times_{N\Delta^{{\mathrm{op}}}}\iota N\Delta^{{\mathrm{op}}}.$$ Put differently, an edge of $\mathrm{M}$ is ingressive just in case it covers an equivalence of $\Delta^{{\mathrm{op}}}$. Consequently, $\pi$ is automatically a pair cartesian fibration (Df. \[dfn:paircartfib\]); the functor ${{N\Delta}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}_{\infty}}}$ classified by $\pi$ is given by the assignment ${{\mathbf{m}}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {(\Delta^m)^{\sharp}}}$.
Now it is no problem to use the technology from Pr. \[prp:htt32213\] to define the cocartesian fibration ${{\mathscr{F}(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}\times S}}$ that we seek.
For any pair cocartesian fibration ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$, define a map ${{\mathscr{F}(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}\times S}}$, using the notation of Pr. \[prp:htt32213\] and Ex. \[exm:minpairmaxpair\], as $$\mathscr{F}(\mathscr{X}/S){\mathrel{\mathop:}=}T_{\pi\times\operatorname{id}_S}((N\Delta^{{\mathrm{op}}})^{\flat}\times\mathscr{X}).$$ Equivalently, we require, for any simplicial set $K$ and any map $\sigma\colon{{K}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}\times S}}$, a bijection between the set $\operatorname{Mor}_{N\Delta^{{\mathrm{op}}}\times S}(K,\mathscr{F}(\mathscr{X}/S))$ and the set $$\operatorname{Mor}_{s{\mathbf{Set}}(2)/(S,\iota S)}((K\times_{N\Delta^{{\mathrm{op}}}}N\mathrm{M},K\times_{N\Delta^{{\mathrm{op}}}}(N\mathrm{M})_{\dag}),(\mathscr{X},\mathscr{X}_{\dag}))$$ (Nt. \[ntn:ordcatofpairs\]), functorial in $\sigma$.
With this definition, Pr. \[prp:htt32213\] now implies the following.
Suppose $p\colon{{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ a pair cocartesian fibration. Then the functor $${{\mathscr{F}(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}\times S}}$$ is a cocartesian fibration.
Furthermore, the functor ${{N\Delta^{{\mathrm{op}}}\times S}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}}}$ that classifies the cocartesian fibration ${{\mathscr{F}(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}\times S}}$ is indeed the functor $${{(\mathbf{m},s)}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}((\Delta^m)^{\sharp},{\mathbf{X}}(s))}},$$ where ${\mathbf{X}}\colon{{S}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}_{\infty}}}$ is the functor that classifies $p$.
When $S=\Delta^0$, write $\mathscr{F}(\mathscr{C})$ for $\mathscr{F}(\mathscr{C}/S)$, and for any integer $m\geq 0$, write $\mathscr{F}_m(\mathscr{C})$ for the fiber $\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}((\Delta^m)^{\sharp},\mathscr{C})$ of the cocartesian fibration ${{\mathscr{F}(\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}}}$ over $\mathbf{m}$.
Hence for any Waldhausen cocartesian fibration ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$, the fiber of the cocartesian fibration ${{\mathscr{F}(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}\times S}}$ over a vertex $(\mathbf{m},s)$ is the Waldhausen $\infty$-category $\mathscr{F}_m(\mathscr{X}_s)$.
A Waldhausen structure on filtered objects of a Waldhausen $\infty$-category {#a-waldhausen-structure-on-filtered-objects-of-a-waldhausen-infty-category .unnumbered}
----------------------------------------------------------------------------
We may endow the $\infty$-categories $\mathscr{F}(\mathscr{X}/S)$ of filtered objects with a pair structure in a variety of ways, but we wish to focus on one pair structure that will retain good formal properties when we pass to quotients.
More specifically, suppose $\mathscr{C}$ a Waldhausen $\infty$-category. A morphism $f\colon{{X}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y}}$ of $\mathscr{F}_m(\mathscr{C})$ can be represented as a diagram $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{X_0&X_1&\cdots&X_m\\
Y_0&Y_1&\cdots&Y_m.\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge[>->] (m-1-2)
edge (m-2-1)
(m-1-2) edge[>->] (m-1-3)
edge (m-2-2)
(m-1-3) edge[>->] (m-1-4)
(m-1-4) edge (m-2-4)
(m-2-1) edge[>->] (m-2-2)
(m-2-2) edge[>->] (m-2-3)
(m-2-3) edge[>->] (m-2-4);
\end{tikzpicture}$$ What should it mean to say that $f$ is ingressive? It is natural to demand, first and foremost, that each morphism $f_i\colon{{X_i}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y_i}}$ is ingressive, but this will not be enough to ensure that the morphisms ${{X_j/X_i}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y_j/Y_i}}$ are all ingressive. Guaranteeing this turns out to be equivalent to the claim that in each of the squares $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{X_i&X_j\\
Y_i&Y_j,\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge[>->] (m-1-2)
edge[>->] (m-2-1)
(m-1-2) edge[>->] (m-2-2)
(m-2-1) edge[>->] (m-2-2);
\end{tikzpicture}$$ the morphism from the pushout $X_j\cup^{X_i}Y_i$ to $Y_j$ is a cofibration as well. This was noted by Waldhausen [@MR86m:18011 Lm. 1.1.2].
Our approach is thus to define a pair structure in such a concrete manner on $\mathscr{F}_1(\mathscr{C})$, and then to declare that a morphism $f$ of $\mathscr{F}_m(\mathscr{C})$ is ingressive just in case $\eta^{\star}(f)$ is so for any $\eta\colon{{\Delta^1}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Delta^m}}$.
\[dfn:injpairstruct\] Suppose $\mathscr{C}$ a Waldhausen $\infty$-category. We now endow the $\infty$-category $\mathscr{F}_1(\mathscr{C})$ with a pair structure by letting $\mathscr{F}_1(\mathscr{C})_{\dag}\subset\mathscr{F}_1(\mathscr{C})$ be the smallest subcategory containing the following classes of edges of $\mathscr{C}$:
1. \[item:degensourceingresstarg\] any edge ${{X}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y}}$ represented as a square $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{X_0&X_1\\
Y_0&Y_1\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge[>->] (m-1-2)
edge[inner sep=0.5pt] node[left]{$\sim$} (m-2-1)
(m-1-2) edge[>->] (m-2-2)
(m-2-1) edge[>->] (m-2-2);
\end{tikzpicture}$$ in which ${{X_0}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {Y_0}}$ is an equivalence and ${{X_1}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ {Y_1}}$ is ingressive, and
2. \[item:cocartedgeovercof\] any edge ${{X}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y}}$ represented as a pushout square $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{X_0&X_1\\
Y_0&Y_1\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge[>->] (m-1-2)
edge[>->] (m-2-1)
(m-1-2) edge[>->] (m-2-2)
(m-2-1) edge[>->] (m-2-2);
\end{tikzpicture}$$ in which ${{X_0}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ {Y_0}}$ and thus also ${{X_1}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ {Y_1}}$ are ingressive.
Let’s compare this definition to our more concrete one outlined above it. To this end, we need a bit of notation.
Let us denote by $\mathscr{R}$ the pair of $\infty$-categories whose underlying $\infty$-category is $$(\Delta^1\times(\Lambda^2_0)^{\rhd})/(\Delta^1\times\Lambda^2_0),$$ which may be drawn $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=5ex, column sep=5ex,
text height=1.5ex, text depth=0.25ex]
{0&1&[-3ex]\\
2&\infty'&\\
[-3ex]&&\infty,\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
edge (m-3-3)
(m-2-1) edge (m-2-2)
edge (m-3-3)
(m-2-2) edge (m-3-3);
\end{tikzpicture}$$ in which only the edges ${{0}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ {1}}$, ${{2}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ {\infty}}$ and ${{2}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ {\infty'}}$ are ingressive. In the notation of \[exm:LambdaDelta\], there is an obvious strict inclusion of pairs ${{\Lambda_0\mathscr{Q}^2}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{R}}}$, and there are two strict inclusions of pairs $${{\mathscr{Q}^2\cong\mathscr{Q}^2\times\Delta^{\{0\}}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{R}}}\textrm{\quad and\quad}{{\mathscr{Q}^2\cong\mathscr{Q}^2\times\Delta^{\{1\}}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{R}}}.$$
\[lem:concreteingressiveF1C\] Suppose $\mathscr{C}$ a Waldhausen $\infty$-category. Then a morphism $f\colon{{X}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y}}$ of $\mathscr{F}_1(\mathscr{C})$ is ingressive just in case the morphism ${{X_0}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y_0}}$ is ingressive and the corresponding square $$F\colon{{\mathscr{Q}^2\cong(\Delta^1)^{\flat}\times(\Delta^1)^{\sharp}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$$ has the property that for any diagram $\overline{F}\colon{{\mathscr{R}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$ such that $\overline{F}|_{\mathscr{Q}^2\times\Delta^{\{0\}}}$ is a pushout square, and $F=\overline{F}|_{\mathscr{Q}^2\times\Delta^{\{1\}}}$, the edge ${{\overline{F}(\infty')}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\overline{F}(\infty)}}$ is ingressive.
An easy argument shows that morphisms with this property form a subcategory of $\mathscr{F}_1(\mathscr{C})$, and it is clear that morphisms either of type (\[dfn:injpairstruct\].\[item:degensourceingresstarg\]) or of type (\[dfn:injpairstruct\].\[item:cocartedgeovercof\]) enjoy this property. Consequently, every ingressive morphism enjoys this property. On the other hand, a morphism ${{X}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y}}$ that enjoys this property can clearly be factored as $X\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ Y'\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ Y$, where ${{X}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y'}}$ is of type (\[dfn:injpairstruct\].\[item:cocartedgeovercof\]), and ${{Y'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y}}$ is of type (\[dfn:injpairstruct\].\[item:degensourceingresstarg\]), viz.: $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{X_0&X_1\\
Y_0&Y_{01}\\
Y_0&Y_1,\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge[>->] (m-1-2)
edge[>->] (m-2-1)
(m-1-2) edge[>->] (m-2-2)
(m-2-1) edge[>->] (m-2-2)
edge[-,double distance=1.5pt] (m-3-1)
(m-2-2) edge[>->] (m-3-2)
(m-3-1) edge[>->] (m-3-2);
\end{tikzpicture}$$ where the top square is a pushout square and ${{Y_{01}}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ {Y_1}}$ is ingressive.
\[dfn:ingressinFmC\] Now suppose ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ a Waldhausen cocartesian fibration. We endow the $\infty$-category $\mathscr{F}(\mathscr{X}/S)$ with the following pair structure. Let $\mathscr{F}(\mathscr{X}/S)_{\dag}\subset\mathscr{F}(\mathscr{X}/S)$ be the smallest pair structure containing any edge $f\colon{{\Delta^1}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{F}(\mathscr{X}/S)}}$ covering a degenerate edge $\operatorname{id}_{(\mathbf{m},s)}$ of $N\Delta^{{\mathrm{op}}}\times S$ such that for any edge $\eta\colon{{\Delta^1}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Delta^m}}$, the edge $$\Delta^1\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$f$}(0.5,0.5ex);\mathscr{F}_m(\mathscr{X}_s)\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$\eta^{\star}$}(0.5,0.5ex);\ \mathscr{F}_1(\mathscr{X}_s)$$ is ingressive in the sense of Df. \[dfn:injpairstruct\].
\[lem:checkingressonneighbors\] Suppose $\mathscr{C}$ a Waldhausen $\infty$-category. Then a morphism $f\colon{{X}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y}}$ of $\mathscr{F}_m(\mathscr{C})$ is ingressive just in case, for any integer $1\leq i\leq m$, the restricted morphism ${{X|_{(\Delta^{\{i-1,i\}})^{\sharp}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y|_{(\Delta^{\{i-1,i\}})^{\sharp}}}}$ is ingressive in $\mathscr{F}_{\{i-1,i\}}(\mathscr{X}_s)$.
Suppose $f$ satisfies this condition. It is immediate that every morphism ${{X_i}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ {Y_i}}$ is ingressive, so we can regard $f$ as an $m$-simplex $\sigma\colon{{\Delta^m}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{F}_1(\mathscr{C})}}$. By Lm. \[lem:concreteingressiveF1C\], this condition is equivalent to the condition that each edge $\sigma|_{\Delta^{i-1,i}}$ is ingressive, and since ingressive edges are closed under composition, it follows that every edge $\sigma|_{\Delta^{\{i,j\}}}$ is ingressive.
\[prp:FXSisWaldcocart\] Suppose $p\colon{{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ a Waldhausen cocartesian fibration. Then with the pair structure of , the functor $${{\mathscr{F}(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}\times S}}$$ is a Waldhausen cocartesian fibration.
It is easy to see that ${{\mathscr{F}(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}\times S}}$ a pair cocartesian fibration.
We claim that for any vertex $(\mathbf{m},s)\in N\Delta^{{\mathrm{op}}}\times S$, the pair $\mathscr{F}_m(\mathscr{X}_s)$ is a Waldhausen $\infty$-category. Note that since $\mathscr{X}_s$ admits a zero object, so does $\mathscr{F}_m(\mathscr{X}_s)$. For the remaining two axioms, one reduces immediately to the case where $m=1$. Then (\[dfn:preWald\].\[item:0toxingressive\]) follows from the presence of (\[dfn:injpairstruct\].\[item:degensourceingresstarg\]) among cofibrations. To prove (\[dfn:preWald\].\[item:pushcof\]), one may note that cofibations of $\mathscr{F}_1(\mathscr{X}_s)$ are in particular cofibrations of $\mathscr{O}(\mathscr{C})$, for which the existence of pushouts is clear. Finally, to prove (\[dfn:preWald\].\[item:pushcofcof\]), it suffices to see that a pushout of any edge of either of the classes (\[dfn:injpairstruct\].\[item:degensourceingresstarg\]) or (\[dfn:injpairstruct\].\[item:cocartedgeovercof\]) is of the same class. For the class (\[dfn:injpairstruct\].\[item:degensourceingresstarg\]), this follows from the fact that pushouts in $\mathscr{F}_1(\mathscr{X}_s)$ are computed pointwise. A pushout of a morphism of the class (\[dfn:injpairstruct\].\[item:cocartedgeovercof\]) is a cube $$X\colon{{(\Delta^1)^{\flat}\times(\Delta^1)^{\sharp}\times(\Delta^1)^{\sharp}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{X}_s}}$$ in which the faces $$X|_{\Delta^{\{0\}}\times(\Delta^1)^{\sharp}\times(\Delta^1)^{\sharp}},\quad X|_{(\Delta^1)^{\flat}\times\Delta^{\{0\}}\times(\Delta^1)^{\sharp}},\textrm{\quad and\quad}X|_{(\Delta^1)^{\flat}\times\Delta^{\{1\}}\times(\Delta^1)^{\sharp}}$$ are all pushouts. If $X$ is represented by the commutative diagram $$\begin{tikzpicture}[cross line/.style={preaction={draw=white, -,
line width=6pt}}]
\matrix(m)[matrix of math nodes,
row sep=2ex, column sep=0.75ex,
text height=1.5ex, text depth=0.25ex]
{&X_{100}&&X_{101}\\
X_{000}&&X_{001}&\\
&X_{110}&&X_{111}\\
X_{010}&&X_{011},&\\
};
\path[>=stealth,->,font=\scriptsize]
(m-1-2) edge[<-] (m-2-1)
edge[>->] (m-3-2)
edge[>->] (m-1-4)
(m-3-2) edge[<-] (m-4-1)
edge[>->] (m-3-4)
(m-2-1) edge[cross line,>->] (m-2-3)
edge[>->] (m-4-1)
(m-1-4) edge[<-] (m-2-3)
edge[>->](m-3-4)
(m-4-1) edge[>->] (m-4-3)
(m-3-4) edge[<-] (m-4-3)
(m-2-3) edge[cross line,>->] (m-4-3);
\end{tikzpicture}$$ then the front face, the top face, and the bottom face are all pushouts. By Quetzalcoatl (e.g., by [@HTT Lm. 4.4.2.1]), the back face $X|_{\Delta^{\{1\}}\times(\Delta^1)^{\sharp}\times(\Delta^1)^{\sharp}}$ must be a pushout as well; this is precisely the claim that the pushout is of the class (\[dfn:injpairstruct\].\[item:cocartedgeovercof\]).
For any $\mathbf{m}\in\Delta$ and any edge $f\colon{{s}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {t}}$ of $S$, since the functor $f_{\mathscr{X},!}\colon{{\mathscr{X}_s}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{X}_t}}$ is exact, it follows directly that the functor $$f_{\mathscr{F},!}\colon{{\mathscr{F}_m(\mathscr{X}_s)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{F}_m(\mathscr{X}_t)}}$$ is exact as well. Now for any fixed vertex $s\in S_0$ and any simplicial operator $\phi\colon{{\mathbf{n}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{m}}}$ of $\Delta$, the functor $$\phi_{\mathscr{F},!}\colon{{\mathscr{F}_m(\mathscr{X}_s)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{F}_n(\mathscr{X}_s)}}$$ visibly carries cofibrations to cofibrations, and it preserves zero objects as well as any pushouts that exist, since limits and colimits are formed pointwise.
Thanks to \[nul:Tpafunctor\], we have:
The assignment ${{(\mathscr{X}/S)}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{F}(\mathscr{X}/S)}}$ defines a functor $$\mathscr{F}\colon{{{\mathbf{Wald}_{\infty}}^{{\mathrm{cocart}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}^{{\mathrm{cocart}}}}}$$ covering the endofunctor ${{S}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}\times S}}$ of ${\mathbf{Cat}}_{\infty}$.
\[prp:Fisacategory\] Suppose ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ a Waldhausen cocartesian fibration, and suppose $${\mathbf{F}}_{\ast}(\mathscr{X}/S)\colon{{N\Delta^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}(S,{\mathbf{Wald}_{\infty}})}}$$ a functor that classifies the Waldhausen cocartesian fibration $${{\mathscr{F}(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}\times S}}.$$ Then ${\mathbf{F}}_{\ast}(\mathscr{X}/S)$ is a ***category object*** [@G Df. 1.1.1]; that is, the morphisms of $\Delta$ of the form ${{\{i-1,i\}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{m}}}$ induce morphisms that exhibit ${\mathbf{F}}_m(\mathscr{X}/S)$ as the limit in $\operatorname{Fun}(S,{\mathbf{Wald}_{\infty}})$ of the diagram $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=-6ex,
text height=1.5ex, text depth=0.25ex]
{{\mathbf{F}}_{\{0,1\}}(\mathscr{X}/S)&&{\mathbf{F}}_{\{1,2\}}(\mathscr{X}/S)&[6ex]&[6ex]{\mathbf{F}}_{\{m-2,m-1\}}(\mathscr{X}/S)&&{\mathbf{F}}_{\{m-1,m\}}(\mathscr{X}/S).\\
&{\mathbf{F}}_{\{1\}}(\mathscr{X}/S)&&\cdots&&{\mathbf{F}}_{\{m-1\}}(\mathscr{X}/S)&\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-2-2)
(m-1-3) edge (m-2-2)
edge (m-2-4)
(m-1-5) edge (m-2-4)
edge (m-2-6)
(m-1-7) edge (m-2-6);
\end{tikzpicture}$$
Since limits in $\operatorname{Fun}(S,{\mathbf{Wald}_{\infty}})$ are computed objectwise, it suffices to assume that $S=\Delta^0$. It is easy to see that $(\Delta^m)^{\sharp}$ decomposes in ${\mathbf{Pair}}_{\infty}$ as the pushout of the diagram $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=0ex,
text height=1.5ex, text depth=0.25ex]
{&(\Delta^{\{1\}})^{\sharp}&&[2ex]\cdots&&(\Delta^{\{m-1\}})^{\sharp}&&\\
(\Delta^{\{0,1\}})^{\sharp}&&(\Delta^{\{1,2\}})^{\sharp}&&(\Delta^{\{m-2,m-1\}})^{\sharp}&&(\Delta^{\{m-1,m\}})^{\sharp},\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-2) edge (m-2-1)
edge (m-2-3)
(m-1-4) edge (m-2-3)
edge (m-2-5)
(m-1-6) edge (m-2-5)
edge (m-2-7);
\end{tikzpicture}$$ since the analogous statement is true in ${\mathbf{Cat}}_{\infty}$. Thus ${\mathbf{F}}_m(\mathscr{X})$ is the desired limit in ${\mathbf{Cat}}_{\infty}$, and it follows immediately from Lm. \[lem:checkingressonneighbors\] that ${\mathbf{F}}_m(\mathscr{X})$ is the desired limit in the $\infty$-category ${\mathbf{Pair}}_{\infty}$ and thus also in the $\infty$-category ${\mathbf{Wald}_{\infty}}$.
Totally filtered objects {#totally-filtered-objects .unnumbered}
------------------------
Now we are in a good position to study the functoriality of filtered objects $X$ that are *separated* in the sense that $X_0$ is a zero object. We call these *totally filtered* objects.
\[dfn:totallyfilteredobject\] Suppose $\mathscr{C}$ a Waldhausen $\infty$-category. Then a filtered object $X\colon{{(\Delta^m)^{\sharp}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$ will be said to be ***totally filtered*** if $X_0$ is a zero object.
Suppose $p\colon{{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ a Waldhausen cocartesian fibration. Denote by $\mathscr{S}(\mathscr{X}/S)$ the full subpair (\[dfn:pair\].\[item:subpair\]) of $\mathscr{F}(\mathscr{X}/S)$ spanned by those filtered objects $X$ such that $X$ is totally filtered in $\mathscr{X}_{p(X)}$. When $S=\Delta^0$, write $\mathscr{S}(\mathscr{X})$ for $\mathscr{S}(\mathscr{X}/S)$, and for any integer $m\geq 0$, write $\mathscr{S}_m(\mathscr{X})$ for the fiber of ${{\mathscr{S}(\mathscr{X})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}}}$ over the object $\mathbf{m}\in N\Delta^{{\mathrm{op}}}$.
\[prp:FmisS1plusm\] Suppose $\mathscr{C}$ a Waldhausen $\infty$-category. For any integer $m\geq0$, the $0$-th face map defines an equivalence of $\infty$-categories $${{\mathscr{S}_{1+m}(\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\mathscr{F}_m(\mathscr{C})}},$$ and the map ${{\mathscr{S}_0(\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Delta^0}}$ is an equivalence.
It follows from Joyal’s theorem [@HTT Pr. 1.2.12.9] that the natural functor ${{\mathscr{S}_{1+m}(\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{F}_m(\mathscr{C})}}$ is a left fibration whose fibers are contractible Kan complexes — hence a trivial fibration.
As $\mathbf{m}$ varies, the functoriality of $\mathscr{S}_m(\mathscr{X})$ is, as we have observed, traditionally a matter of some consternation, as the functors involve various (homotopy) quotients, which are not uniquely defined on the nose. We all share the intuition that the uniqueness of these quotients is good enough for all practical purposes and that the coherence issues that appear to arise are mere technical issues. The theory of cocartesian fibrations allows us to make this intuition honest.
Below (Th. \[thm:SXiscocartfib\]), we’ll show that for any Waldhausen cocartesian fibration ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$, the functor ${{\mathscr{S}(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}\times S}}$ is a Waldhausen cocartesian fibration. Let’s reflect on what this means when $S=\Delta^0$; in this case, $\mathscr{X}$ is just a Waldhausen $\infty$-category. An edge ${{X}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y}}$ of $\mathscr{S}(\mathscr{X})$ that covers an edge given by a morphism $\eta\colon{{\mathbf{m}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{n}}}$ of $\Delta$ is by definition a commutative diagram $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{X_{\eta(0)}&X_{\eta(1)}&\cdots&X_{\eta(m)}\\
0&Y_{1}&\cdots&Y_{m}.\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge[>->] (m-1-2)
edge (m-2-1)
(m-1-2) edge[>->] (m-1-3)
edge (m-2-2)
(m-1-3) edge[>->] (m-1-4)
(m-1-4) edge (m-2-4)
(m-2-1) edge (m-2-2)
(m-2-2) edge (m-2-3)
(m-2-3) edge (m-2-4);
\end{tikzpicture}$$ To say that ${{X}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y}}$ is a cocartesian edge over $\eta$ is to say that $Y$ is initial among totally filtered objects under $\eta^{\star}X$. This is equivalent to the demand that each of the squares above must be pushout squares, i.e., that $Y_k\simeq X_{\eta(k)}/X_{\eta(0)}$. So if ${{\mathscr{S}(\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}}}$ is a Waldhausen cocartesian fibration, then the functor ${\mathbf{S}}_{\ast}\colon{{N\Delta^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}}}$ that classifies it works exactly as Waldhausen’s $S_{\bullet}$ construction: it carries an object $\mathbf{m}\in N\Delta^{{\mathrm{op}}}$ to the Waldhausen $\infty$-category ${\mathbf{S}}_m(\mathscr{C})$ of totally filtered objects of length, and it carries a morphism $\eta\colon{{\mathbf{m}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{n}}}$ of $\Delta$ to the exact functor ${{{\mathbf{S}}_n(\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{S}}_m(\mathscr{C})}}$ given by $${{[X_0\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X_1\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X_n]}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {[0\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X_{\eta(1)}/X_{\eta(0)}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X_{\eta(m)}/X_{\eta(0)}].}}$$ In other words, the *data* of the $\infty$-categorical $S_{\bullet}$ construction is already before us; we just need to confirm that it works as desired.
To prove Th. \[thm:SXiscocartfib\], it turns out to be convenient to study the “mapping cylinder” $\mathscr{M}(\mathscr{X}/S)$ of the inclusion functor ${{\mathscr{S}(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{F}(\mathscr{X}/S)}}$. We will discover that this inclusion admits a left adjoint, and then we will use this left adjoint to complete the proof of Th. \[thm:SXiscocartfib\].
For any Waldhausen cocartesian fibration ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$, let us write $\mathscr{M}(\mathscr{X}/S)$ for the full subcategory of $\Delta^1\times\mathscr{F}(\mathscr{X}/S)$ spanned by those pairs $(i,X)$ such that $X$ is totally filtered if $i=1$. This $\infty$-category comes equipped with an inner fibration $${{\mathscr{M}(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Delta^1\times N\Delta^{{\mathrm{op}}}\times S}}.$$ Define a pair structure on $\mathscr{M}(\mathscr{X}/S)$ so that it is a subpair of $(\Delta^1)^{\flat}\times\mathscr{F}(\mathscr{X}/S)$; that is, let $\mathscr{M}(\mathscr{X}/S)_{\dag}\subset\mathscr{M}(\mathscr{X}/S)$ be the subcategory whose edges are maps ${{(i,X)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(j,Y)}}$ such that $i=j$ and ${{X}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y}}$ is a cofibration of $\mathscr{F}(\mathscr{X}/S)$.
Our first lemma is obvious by construction.
For any Waldhausen cocartesian fibration ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$, the natural projection ${{\mathscr{M}(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Delta^1}}$ is a pair cartesian fibration.
Our next lemma, however, is subtler.
\[lem:MXScocartfib\] For any Waldhausen cocartesian fibration ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$, the natural projection ${{\mathscr{M}(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Delta^1}}$ is a pair cocartesian fibration.
By [@HTT 2.4.1.3(3)], it suffices to show that for any vertex $(\mathbf{m},s)\in(N\Delta^{{\mathrm{op}}}\times S)_0$, the inner fibration $$q\colon{{\mathscr{M}_m(\mathscr{X}_s)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Delta^1}}$$ is a pair cocartesian fibration. Note that an edge ${{X}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y}}$ of $\mathscr{M}_m(\mathscr{X}_s)$ covering the nondegenerate edge $\sigma$ of $\Delta^1$ is $q$-cocartesian if and only if it is an initial object of the fiber $\mathscr{M}_m(\mathscr{X}_s)_{X/}\times_{\Delta^1_{0/}}\{\sigma\}$. If $m=0$, then the map $${{\mathscr{M}_0(\mathscr{X}_s)_{X/}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Delta^1_{0/}}}$$ is a trivial fibration [@HTT Pr. 1.2.12.9], so the fiber over $\sigma$ is a contractible Kan complex. Let us now induct on $m$; assume that $m>0$ and that the functor $p\colon{{\mathscr{M}_{m-1}(\mathscr{X}_s)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Delta^1}}$ is a cocartesian fibration. It is easy to see that the inclusion ${{\{0,1,\dots,m-1\}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{m}}}$ induces an inner fibration $\phi\colon{{\mathscr{M}_m(\mathscr{X}_s)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{M}_{m-1}(\mathscr{X}_s)}}$ such that $q=p\circ\phi$. Again by [@HTT 2.4.1.3(3)], it suffices to show that for any object $X$ of $\mathscr{M}_m(\mathscr{X}_s)$ and any $p$-cocartesian edge $\eta\colon{{\phi(X)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y'}}$ covering $\sigma$, there exists a $\phi$-cocartesian edge ${{X}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y}}$ of $\mathscr{M}_m(\mathscr{X}_s)$ covering $\eta$. But this follows directly from (\[dfn:preWald\].\[item:pushcof\]).
We now show that $q$ is a *pair* cocartesian fibration. Suppose $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{X'&X\\
Y'&Y\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge (m-2-2);
\end{tikzpicture}$$ is a square of $\mathscr{M}_m(\mathscr{X}_s)$ in which ${{X'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X}}$ and ${{Y'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y}}$ are $q$-cocartesian morphisms and ${{X}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y}}$ is a cofibration. We aim to show that for any edge $\eta\colon{{\Delta^{\{p,q\}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Delta^m}}$, the morphism ${{X'|\Delta^{\{p,q\}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y'|\Delta^{\{p,q\}}}}$ is a cofibration. For this, we may factor ${{X}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y}}$ as $$X\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ Z\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ Y,$$ where $Z|\Delta^{\{0,\dots,p\}}=Y|\Delta^{\{0,\dots,p\}}$, and for any $r>p$, the edge ${{X|\Delta^{\{p,r\}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Z|\Delta^{\{p,r\}}}}$ is cocartesian. Now choose a cocartesian morphism ${{Z'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Z}}$ as well. The proof is now completed by the following observations.
1. Since the morphism ${{X|\Delta^{\{p,q\}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Z|\Delta^{\{p,q\}}}}$ is of type (\[dfn:injpairstruct\].\[item:cocartedgeovercof\]), it follows by Quetzalcoatl that the morphism ${{X'|\Delta^{\{p,q\}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Z'|\Delta^{\{p,q\}}}}$ is of type (\[dfn:injpairstruct\].\[item:cocartedgeovercof\]) as well.
2. The morphism ${{Z|\Delta^{\{p,q\}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y|\Delta^{\{p,q\}}}}$ is of type (\[dfn:injpairstruct\].\[item:degensourceingresstarg\]) and the morphism ${{Z'_p}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X'_p}}$ is an equivalence; so again by Quetzalcoatl, the morphism ${{Z'|\Delta^{\{p,q\}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {Y'|\Delta^{\{p,q\}}}}$ is of type (\[dfn:injpairstruct\].\[item:degensourceingresstarg\]).
\[ntn:JandF\] Together, these lemmas state that for any Waldhausen cocartesian fibration ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$, the functor ${{\mathscr{M}(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Delta^1}}$ exhibits an adjunction of $\infty$-categories [@HTT Df. 5.2.2.1] $${{F}\colon{\mathscr{F}(\mathscr{X}/S)}\ \begin{tikzpicture}[baseline] \draw[>=stealth,->] (0,1ex) -- (0.75,1ex); \draw[>=stealth,->] (0.75,0.25ex) -- (0,0.25ex); \end{tikzpicture}\ {\mathscr{S}(\mathscr{X}/S)}\colon{J}}$$ over $N\Delta^{{\mathrm{op}}}\times S$ in which both $F$ and $J$ are functors of pairs. In particular, for any integer $m\geq 0$ and any vertex $s\in S_0$, the fiber ${{\mathscr{M}_m(\mathscr{X}_s)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Delta^1}}$ over $(\mathbf{m},s)$ also exhibits an adjunction $${{F_m}\colon{\mathscr{F}_m(\mathscr{X}_{s})}\ \begin{tikzpicture}[baseline] \draw[>=stealth,->] (0,1ex) -- (0.75,1ex); \draw[>=stealth,->] (0.75,0.25ex) -- (0,0.25ex); \end{tikzpicture}\ {\mathscr{S}(\mathscr{X}_{s})}\colon{J_m}}.$$
Let’s unravel this a bit. Assume $S=\Delta^0$. The functor $J$ is the functor of pairs specified by the edge ${{\Delta^1}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}_{\infty}}}$ that classifies the *cartesian* fibration ${{\mathscr{M}(\mathscr{X})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Delta^1}}$. By construction, this is a forgetful functor: it carries a totally filtered object of $\mathscr{X}$ to its underlying filtered object. The functor $F$ is the functor of pairs specified by the edge ${{\Delta^1}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}_{\infty}}}$ that classifies the *cocartesian* fibration ${{\mathscr{M}(\mathscr{X})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Delta^1}}$, and it is much more interesting: it carries a filtered object $X$ represented as $$X_0\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X_1\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X_m$$ to the totally filtered object $FX$ that is initial among all totally filtered objects under $X$; in other words, $FX$ is the quotient of $X$ by $X_0$: $$0\simeq X_0/X_0\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X_1/X_0\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X_m/X_0.$$
This functor $F$ is a cornerstone for the following result:
\[thm:SXiscocartfib\] Suppose ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$ a Waldhausen cocartesian fibration. Then the functor $${{\mathscr{S}(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}\times S}}$$ is a Waldhausen cocartesian fibration.
We first show that the functor ${{\mathscr{S}(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}\times S}}$ is a cocartesian fibration by proving the stronger assertion that the inner fibration $$p\colon{{\mathscr{M}(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Delta^1\times N\Delta^{{\mathrm{op}}}\times S}}$$ is a cocartesian fibration. By \[prp:FXSisWaldcocart\], the map $$\label{eqn:MXSover0}
{{\Delta^{\{0\}}\times_{\Delta^1}\mathscr{M}(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Delta^{\{0\}}\times N\Delta^{{\mathrm{op}}}\times S}}$$ is a cocartesian fibration. By \[lem:MXScocartfib\], for any vertex $(\mathbf{m},s)\in(N\Delta^{{\mathrm{op}}}\times S)_0$, the map $$\label{eqn:MXSoverms}
{{\mathscr{M}(\mathscr{X}/S)\times_{N\Delta^{{\mathrm{op}}}\times S}\{(\mathbf{m},s)\}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Delta^1\times\{(\mathbf{m},s)\}}}$$ is a cocartesian fibration. Finally, for any $\mathbf{m}\in\Delta$ and any edge $f\colon{{s}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {t}}$ of $S$, the functor $f_!\colon{{\mathscr{X}_s}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{X}_t}}$ carries zero objects to zero objects; consequently, any cocartesian edge of $\mathscr{F}(\mathscr{X}/S)$ that covers $(\operatorname{id}_{\mathbf{m}},f)$ lies in $\mathscr{S}(\mathscr{X}/S)$ if and only if its source does. Thus the map $${{(\Delta^{\{1\}}\times\{\mathbf{m}\})\times_{\Delta^1\times N\Delta^{{\mathrm{op}}}}\mathscr{M}(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Delta^{\{1\}}\times \{\mathbf{m}\}\times S}}$$ is a cocartesian fibration.
Now to complete the proof that $p$ is a cocartesian fibration, thanks to [@HTT 2.4.1.3(3)] it remains to show that for any vertex $s\in S_0$, any simplicial operator $\phi\colon{{\mathbf{n}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{m}}}$, and any totally $m$-filtered object $X$ of $\mathscr{X}_s$, there exists a $p$-cartesian morphism ${{(1,X)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(1,Y)}}$ of $\mathscr{F}(\mathscr{X}/S)$ covering $(\operatorname{id}_1,\phi,\operatorname{id}_s)$. Write $\sigma$ for the nondegenerate edge of $\Delta^1$. The $p$-cartesian edge $e\colon{{(0,X)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(1,X)}}$ covering $(\sigma,\operatorname{id}_{\mathbf{m}},\operatorname{id}_s)$ is also $p$-cocartesian. Since is a cocartesian fibration, there exists a $p$-cocartesian edge $\eta'\colon{{(0,X)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(0,Y')}}$ covering $(\operatorname{id}_0,\phi,\operatorname{id}_s)$. Since is a cocartesian fibration, there exists a $p$-cocartesian edge $e'\colon{{(0,Y')}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(1,Y)}}$ covering $(\sigma,\operatorname{id}_{\mathbf{n}},\operatorname{id}_s)$. Since $e$ is $p$-cocartesian, we have a diagram $${{\Delta^1\times\Delta^1}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{M}(\mathscr{X}/S)\times_S\{s\}}}$$ of the form $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{(0,X)&(0,Y')\\
(1,X)&(1,Y).\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge node[above]{$\eta'$} (m-1-2)
edge node[left]{$e$} (m-2-1)
(m-1-2) edge node[right]{$e'$} (m-2-2)
(m-2-1) edge node[below]{$\eta$} (m-2-2);
\end{tikzpicture}$$ It follows from [@HTT 2.4.1.7] that $\eta$ is $p$-cocartesian.
From \[prp:FmisS1plusm\] and \[prp:FXSisWaldcocart\] it follows that the fibers of ${{\mathscr{S}(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}\times S}}$ are all Waldhausen $\infty$-categories. For any $\mathbf{m}\in\Delta$ and any edge $f\colon{{s}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {t}}$ of $S$, the functor $f_{\mathscr{X},!}\colon{{\mathscr{X}_s}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{X}_t}}$ is exact, whence it follows by \[prp:FmisS1plusm\] that the functor $$f_{\mathscr{S},!}\colon{{\mathscr{S}_m(\mathscr{X}_s)\simeq\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}((\Delta^{m-1})^{\sharp},\mathscr{X}_s)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}((\Delta^{m-1})^{\sharp},\mathscr{X}_t)\simeq\mathscr{S}_m(\mathscr{X}_t)}}$$ is exact, just as in the proof of \[prp:FXSisWaldcocart\]. Now for any fixed vertex $s\in S_0$ and any simplicial operator $\phi\colon{{\mathbf{n}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{m}}}$ of $\Delta$, the functor $\phi_{\mathscr{S},!}\colon{{\mathscr{S}_m(\mathscr{X}_s)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{S}_n(\mathscr{X}_s)}}$ is by construction the composite $$\mathscr{S}_m(\mathscr{X}_s)\ \tikz[baseline]\draw[>=stealth,right hook->,font=\scriptsize](0,0.5ex)--node[above]{$J_{m,s}$}(0.75,0.5ex);\ \mathscr{F}_m(\mathscr{X}_s)\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$\phi_{\mathscr{F},!}$}(0.5,0.5ex);\ \mathscr{F}_n(\mathscr{X}_s)\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$F_{n,s}$}(0.75,0.5ex);\ \mathscr{S}_n(\mathscr{X}_s),$$ and as $\phi_{\mathscr{F},!}$ is an exact functor (\[prp:FXSisWaldcocart\]), we are reduced to checking that the functors of pairs $J_{m,s}$ and $F_{n,s}$ are each exact functors.
For this, it is clear that $J_{m,s}$ and $F_{n,s}$ each carry zero objects to zero objects, and as $F_{n,s}$ is a left adjoint, it preserves any pushout squares that exist in $\mathscr{F}_n(\mathscr{X}_s)$. Moreover, a pushout square in $\mathscr{S}_m(\mathscr{X}_s)$ is nothing more than a pushout square in $\mathscr{F}_m(\mathscr{X}_s)$ of totally $m$-filtered objects; hence $J_{m,s}$ preserves pushouts along cofibrations.
For any Waldhausen cocartesian fibration ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$, write $$\mathbf{S}_{\ast}(\mathscr{X}/S)\colon{{N\Delta^{{\mathrm{op}}}\times S}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}}}$$ for the diagram of Waldhausen $\infty$-categories that classifies the Waldhausen cocartesian fibration ${{\mathscr{S}(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}\times S}}$, and, similarly, write $${\mathbf{F}}_{\ast}(\mathscr{X}/S)\colon{{N\Delta^{{\mathrm{op}}}\times S}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}}}$$ for the diagram of Waldhausen $\infty$-categories that classifies the Waldhausen cocartesian fibration ${{\mathscr{F}(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}\times S}}$. An instant consequence of the construction of the functoriality of $\mathscr{S}$ in the proof above is the following.
\[cor:Jisanattrans\] The functors $F_m\colon{{\mathscr{F}_m(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{S}_m(\mathscr{X}/S)}}$ assemble to a morphism $F\colon{{\mathscr{F}(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{S}(\mathscr{X}/S)}}$ of $\mathbf{Wald}_{\infty/N\Delta^{{\mathrm{op}}}\times S}^{{\mathrm{cocart}}}$, or, equivalently, a natural transformation $$F\colon{{{\mathbf{F}}_{\ast}(\mathscr{X}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{S}}_{\ast}(\mathscr{X}/S)}}.$$
Note, however, that it is *not* the case that the functors $J_m$ assemble to a natural transformation of this kind.
Virtual Waldhausen $\infty$-categories of filtered objects {#virtual-waldhausen-infty-categories-of-filtered-objects .unnumbered}
----------------------------------------------------------
Thanks to \[nul:Tpafunctor\], the assignments $${{(\mathscr{X}/S)}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {(\mathscr{F}(\mathscr{X}/S)/(N\Delta^{{\mathrm{op}}}\times S))}}\textrm{\quad and\quad}{{(\mathscr{X}/S)}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {(\mathscr{S}(\mathscr{X}/S)/(N\Delta^{{\mathrm{op}}}\times S))}}$$ define endofunctors of ${\mathbf{Wald}_{\infty}}^{{\mathrm{cocart}}}$ over the endofunctor ${{S}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}\times S}}$ of ${\mathbf{Cat}}_{\infty}$. We now aim to descend these functors to endofunctors of the $\infty$-category of virtual Waldhausen $\infty$-categories.
\[lem:FandScontinuous\] The functors ${{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{Wald}_{\infty/N\Delta^{{\mathrm{op}}}}^{{\mathrm{cocart}}}}}$ given by $${{\mathscr{C}}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {(\mathscr{F}(\mathscr{C})/N\Delta^{{\mathrm{op}}})}}\textrm{\qquad and\qquad}{{\mathscr{C}}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {(\mathscr{S}(\mathscr{C})/N\Delta^{{\mathrm{op}}})}}$$ each preserve filtered colimits.
By Cor. \[cor:colimsinWaldScocart\], it is enough to check the claim fiberwise. The assignment ${{\mathscr{C}}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{S}_0(\mathscr{C})}}$ is an essentially constant functor whose values are all terminal objects; hence since filtered simplicial sets are weakly contractible, this functor preserves filtered colimits. We are now reduced to the claim that for any natural number $m$, the assignment ${{\mathscr{C}}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{F}_m(\mathscr{C})}}$ defines a functor ${{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}}}$ that preserves filtered colimits.
Suppose now that $\Lambda$ is a filtered simplicial set; by [@HTT Pr. 5.3.1.16], we may assume that $\Lambda$ is the nerve of a filtered poset. Suppose $\mathscr{C}\colon{{\Lambda^{\rhd}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}}}$ a colimit digram of Waldhausen $\infty$-categories. Suppose $\widetilde{\mathscr{F}}_m(\mathscr{C})\colon{{\Lambda^{\rhd}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Pair}}_{\infty}}}$ be a colimit diagram such that $\widetilde{\mathscr{F}}_m(\mathscr{C})|\Lambda=\mathscr{F}_m(\mathscr{C}|\Lambda)$. By \[thm:Waldfiltcolims\], we are reduced to showing that the natural functor of pairs $$\nu\colon{{\widetilde{\mathscr{F}}_m(\mathscr{C})_{\infty}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{F}_m(\mathscr{C}_{\infty})}}$$ is an equivalence. Indeed, $\nu$ induces an equivalence of the underlying $\infty$-categories, since $(\Delta^m)^{\sharp}\times(\Delta^n)^{\flat}$ is a compact object of ${\mathbf{Pair}}_{\infty}$; hence it remains to show that $\nu$ is a strict functor of pairs. For this it suffices to show that for any cofibration $\psi\colon{{X}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ {Y}}$ of $\mathscr{F}(\mathscr{C}_{\infty})$, there exists a vertex $\alpha\in\Lambda$ and an edge $\overline{\psi}\colon{{\overline{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\overline{Y}}}$ of $\mathscr{F}(\mathscr{C}_{\alpha})$ lifting $\psi$. It is enough to assume that $m=1$ and to show that $\psi$ is either of type (\[dfn:injpairstruct\].\[item:degensourceingresstarg\]) or of type (\[dfn:injpairstruct\].\[item:cocartedgeovercof\]). That is, we may assume that $\psi$ is represented by a square $$\label{eqn:morphinF1C}
\begin{tikzpicture}[baseline]
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{X&Y\\
X'&Y'\\};
\path[>=stealth,>->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge (m-2-2);
\end{tikzpicture}$$ of cofibrations such that either ${{X}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ {X'}}$ is an equivalence or else the square is a pushout. Since $\Delta^1\times\Delta^1$ is a compact $\infty$-category, a square of cofibrations of the form must lift to a square of cofibrations $$\label{eqn:morphinF1Calpha}
\begin{tikzpicture}[baseline]
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\overline{X}&\overline{Y}\\
\overline{X}'&\overline{Y}'\\};
\path[>=stealth,>->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge (m-2-2);
\end{tikzpicture}$$ of $\mathscr{C}_{\alpha}$ for some vertex $\alpha\in\Lambda$. Now the argument is completed by the following brace of observations.
1. If ${{X}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ {X'}}$ is an equivalence, then, increasing $\alpha$ if necessary, we may assume that its lift ${{\overline{X}}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ {\overline{X}'}}$ in $\mathscr{C}_{\alpha}$ is an equivalence as well, since for example the pushout $$\Delta^3\cup^{(\Delta^{\{0,2\}}\sqcup\Delta^{\{1,3\}})}(\Delta^0\sqcup\Delta^0)$$ is compact in the Joyal model structure; hence it represents a cofibration of type (\[dfn:injpairstruct\].\[item:degensourceingresstarg\]) of $\mathscr{F}_1(\mathscr{C}_{\alpha})$.
2. If is a pushout, then one may form the pushout of $\overline{X}'\ \tikz[baseline]\draw[>=stealth,<-<](0,0.5ex)--(0.5,0.5ex);\ \overline{X}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ \overline{Y}$ in $\mathscr{C}_{\alpha}$. Since ${{\mathscr{C}_{\alpha}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}_{\infty}}}$ preserves such pushouts, we may assume that is a pushout square in $\mathscr{C}_{\alpha}$; hence it represents a cofibration of type (\[dfn:injpairstruct\].\[item:cocartedgeovercof\]) of $\mathscr{F}_1(\mathscr{C}_{\alpha})$.
One may compose the functors $$\mathscr{F}\colon{{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{Wald}_{\infty,/N\Delta^{{\mathrm{op}}}}}}\textrm{\qquad and\qquad}\mathscr{S}\colon{{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{Wald}_{\infty,/N\Delta^{{\mathrm{op}}}}}}$$ with the realization functor $|\cdot|_{N\Delta^{{\mathrm{op}}}}$ of Df. \[dfn:realization\]; the results are models for the functors ${{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathrm{D}(\mathbf{Wald}_{\infty})}}}$ that assign to any Waldhausen $\infty$-category $\mathscr{C}$ the formal geometric realizations of the simplicial Waldhausen $\infty$-categories ${\mathbf{F}}_{\ast}(\mathscr{C})$ and ${\mathbf{S}}_{\ast}(\mathscr{C})$ that classify $\mathscr{F}(\mathscr{C})$ and $\mathscr{S}(\mathscr{C})$. In particular, these composites $$|{\mathbf{F}}_{\ast}|_{N\Delta^{{\mathrm{op}}}},|{\mathbf{S}}_{\ast}|_{N\Delta^{{\mathrm{op}}}}\colon{{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathrm{D}(\mathbf{Wald}_{\infty})}}}$$ each preserve filtered colimits, whence one may form their left derived functors (Df. \[dfn:lderivWald\]), which we will abusively also denote $\mathscr{F}$ and $\mathscr{S}$. These are the essentially unique endofunctors of ${\mathrm{D}(\mathbf{Wald}_{\infty})}$ that preserve sifted colimits such that the squares $$\begin{tikzpicture}[baseline]
\matrix(m)[matrix of math nodes,
row sep=6ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{{\mathbf{Wald}_{\infty}}&\mathbf{Wald}_{\infty,/N\Delta^{{\mathrm{op}}}}^{{\mathrm{cocart}}}\\
{\mathrm{D}(\mathbf{Wald}_{\infty})}&{\mathrm{D}(\mathbf{Wald}_{\infty})}\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge node[above]{$\mathscr{F}$} (m-1-2)
edge node[left]{$j$} (m-2-1)
(m-1-2) edge node[right]{$|\cdot|_{N\Delta^{{\mathrm{op}}}}$} (m-2-2)
(m-2-1) edge node[below]{$\mathscr{F}$} (m-2-2);
\end{tikzpicture}
\textrm{\qquad and\qquad}
\begin{tikzpicture}[baseline]
\matrix(m)[matrix of math nodes,
row sep=6ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{{\mathbf{Wald}_{\infty}}&\mathbf{Wald}_{\infty,/N\Delta^{{\mathrm{op}}}}^{{\mathrm{cocart}}}\\
{\mathrm{D}(\mathbf{Wald}_{\infty})}&{\mathrm{D}(\mathbf{Wald}_{\infty})}\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge node[above]{$\mathscr{S}$} (m-1-2)
edge node[left]{$j$} (m-2-1)
(m-1-2) edge node[right]{$|\cdot|_{N\Delta^{{\mathrm{op}}}}$} (m-2-2)
(m-2-1) edge node[below]{$\mathscr{S}$} (m-2-2);
\end{tikzpicture}$$ commute via a specified homotopy.
Also note that the natural transformation $F$ from Cor. \[cor:Jisanattrans\] descends further to a natural transformation $F\colon{{\mathscr{F}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{S}}}$ of endofunctors of ${\mathrm{D}(\mathbf{Wald}_{\infty})}$.
As it happens, the functor $\mathscr{F}\colon{{{\mathrm{D}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathrm{D}(\mathbf{Wald}_{\infty})}}}$ is not particularly exciting:
\[prp:FCisacone\] For any virtual Waldhausen $\infty$-category $\mathscr{X}$, the virtual Waldhausen $\infty$-category $\mathscr{F}(\mathscr{X})$ is the zero object.
For any Waldhausen $\infty$-category $\mathscr{C}$, the virtual Waldhausen $\infty$-category $|\mathscr{F}(\mathscr{C})|_{N\Delta^{{\mathrm{op}}}}$ is by definition a functor ${{{\mathbf{Wald}_{\infty}}^{\omega}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$ that assigns to any compact Waldhausen $\infty$-category $\mathscr{Y}$ the geometric realization of the simplicial space $${{\mathbf{m}}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}^{\Delta}(\mathscr{Y},\mathscr{F}_m(\mathscr{C}))}}.$$ By Pr. \[prp:FmisS1plusm\], this simplicial space is the path space of the simplicial space $${{\mathbf{m}}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}^{\Delta}(\mathscr{Y},\mathscr{S}_m(\mathscr{C}))}}.\qedhere$$
For any Waldhausen $\infty$-category $\mathscr{C}$, we have a natural morphism ${{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{F}(\mathscr{C})}}$ in ${\mathrm{D}(\mathbf{Wald}_{\infty})}$, which is induced by the inclusion of the fiber over $0$. The previous result now entitles us to regard the virtual Waldhausen $\infty$-category $\mathscr{F}(\mathscr{C})$ as a *cone* on $\mathscr{C}$. With this perspective, in the next section we will end up thinking of the induced morphism $F\colon{{\mathscr{F}(\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{S}(\mathscr{C})}}$ induced by the functor $F$ as the quotient of $\mathscr{F}(\mathscr{C})$ by $\mathscr{C}$, thereby identifying $\mathscr{S}(\mathscr{C})$ as a *suspension* of $\mathscr{C}$ in a suitable localization of ${\mathrm{D}(\mathbf{Wald}_{\infty})}$.
The essential uniqueness of the extensions $\mathscr{F}$ and $\mathscr{S}$ to ${\mathrm{D}(\mathbf{Wald}_{\infty})}$ now implies the following.
If $S$ is a small sifted $\infty$-category, then the squares $$\begin{tikzpicture}[baseline]
\matrix(m)[matrix of math nodes,
row sep=6ex, column sep=3ex,
text height=1.5ex, text depth=0.25ex]
{\mathbf{Wald}_{\infty,/S}^{{\mathrm{cocart}}}&\mathbf{Wald}_{\infty,/N\Delta^{{\mathrm{op}}}\times S}^{{\mathrm{cocart}}}\\
{\mathrm{D}(\mathbf{Wald}_{\infty})}&{\mathrm{D}(\mathbf{Wald}_{\infty})}\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge node[above]{$\mathscr{F}$} (m-1-2)
edge node[left]{$|\cdot|_{S}$} (m-2-1)
(m-1-2) edge node[right]{$|\cdot|_{N\Delta^{{\mathrm{op}}}\times S}$} (m-2-2)
(m-2-1) edge node[below]{$\mathscr{F}$} (m-2-2);
\end{tikzpicture}
\textrm{\quad and\quad}
\begin{tikzpicture}[baseline]
\matrix(m)[matrix of math nodes,
row sep=6ex, column sep=3ex,
text height=1.5ex, text depth=0.25ex]
{\mathbf{Wald}_{\infty,/S}^{{\mathrm{cocart}}}&\mathbf{Wald}_{\infty,/N\Delta^{{\mathrm{op}}}\times S}^{{\mathrm{cocart}}}\\
{\mathrm{D}(\mathbf{Wald}_{\infty})}&{\mathrm{D}(\mathbf{Wald}_{\infty})}\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge node[above]{$\mathscr{S}$} (m-1-2)
edge node[left]{$|\cdot|_{S}$} (m-2-1)
(m-1-2) edge node[right]{$|\cdot|_{N\Delta^{{\mathrm{op}}}\times S}$} (m-2-2)
(m-2-1) edge node[below]{$\mathscr{S}$} (m-2-2);
\end{tikzpicture}$$ commute via a specified homotopy.
Of course this is no surprise for $\mathscr{F}\colon{{{\mathrm{D}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathrm{D}(\mathbf{Wald}_{\infty})}}}$, as we have already seen that $\mathscr{F}$ is constant at zero.
The fissile derived $\infty$-category of Waldhausen $\infty$-categories
=======================================================================
A functor $\phi\colon{{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$ may be described and studied through its left derived functor (Df. \[dfn:lderivWald\]) $$\Phi\colon{{{\mathrm{D}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}.$$ In this section, we construct a somewhat peculiar localization ${\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}$ of the $\infty$-category ${\mathrm{D}(\mathbf{Wald}_{\infty})}$ on which the functor $\mathscr{S}\colon{{{\mathrm{D}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathrm{D}(\mathbf{Wald}_{\infty})}}}$ constructied in the previous section can be identified as the suspension (Cor. \[cor:Sdotisreallysuspension\]). In the next section we will use this to show that $\phi$ is *additive* in the sense of Waldhausen just in case $\Phi$ factors through an excisive functor on ${\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}$ (Th. \[thm:additiveequiv\]).
Fissile virtual Waldhausen $\infty$-categories {#fissile-virtual-waldhausen-infty-categories .unnumbered}
----------------------------------------------
In Df. \[dfn:virtWald\], we defined a virtual Waldhausen $\infty$-category as a presheaf $\mathscr{X}\colon{{{\mathbf{Wald}_{\infty}}^{\omega,{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$ such that the natural maps $${{\mathscr{X}(\mathscr{C}\oplus\mathscr{D})}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\mathscr{X}(\mathscr{C})\times\mathscr{X}(\mathscr{D})}}$$ are equivalences. This condition implies in particular that the value of $\mathscr{X}$ on the Waldhausen $\infty$-category of *split* cofiber sequences in an $\infty$-category $\mathscr{C}$ agrees with the product $\mathscr{X}(\mathscr{C})\times\mathscr{X}(\mathscr{C})$. We can ask for more: we can demand that $\mathscr{X}$ be able split even those cofiber sequences that are not already split. That is, we can ask that $\mathscr{X}$ regard the Waldhausen $\infty$-categories of split exact sequences and that of all exact sequences in $\mathscr{C}$ as indistinguishable. This is obviously very closely related to Waldhausen’s additivity, and it is what we will mean by a *fissile* virtual Waldhausen $\infty$-category, and the $\infty$-category of these will be called the *fissile derived $\infty$-category* of Waldhausen $\infty$-categories. (The word “fissile” in geology and nuclear physics means, in essence, “easily split.” The intuition is that when we pass to the fissile derived $\infty$-category, filtered objects can be identified with the sum of their layers.)
But this is asking a lot of our presheaf $\mathscr{X}$. For example, while Waldhausen $\infty$-categories always represent virtual Waldhausen $\infty$-categories, they are almost never fissile. Nevertheless, any virtual Waldhausen $\infty$-category has a best fissile approximation. In other words, the inclusion of fissile virtual Waldhausen $\infty$-categories into virtual Waldhausen $\infty$-categories actually admits a left adjoint, which exhibits the $\infty$-category of fissile virtual Waldhausen $\infty$-categories as a localization of the $\infty$-category of all virtual Waldhausen $\infty$-categories.
\[cnstr:Em\] Suppose $\mathscr{C}$ a Waldhausen $\infty$-category. Then for any integer $m\geq 0$, we may define a fully faithful functor $$E_m\colon{{\mathscr{C}\simeq\mathscr{F}_0(\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{F}_m(\mathscr{C})}}$$ that carries an object $X$ of $\mathscr{C}$ to the constant filtration of length $m$: $$X\ \tikz[baseline]\draw[>=stealth,-,double distance=1.5pt](0,0.5ex)--(0.5,0.5ex);\ X\ \tikz[baseline]\draw[>=stealth,-,double distance=1.5pt](0,0.5ex)--(0.5,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,-,double distance=1.5pt](0,0.5ex)--(0.5,0.5ex);\ X.$$ This is the functor induced by the simplicial operator ${{\mathbf{0}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{m}}}$. One has a similiar functor $$E_m'\colon{{\Delta^0\simeq\mathscr{S}_0(\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{S}_m(\mathscr{C})}},$$ which is of course just the inclusion of a contractible Kan complex of zero objects into $\mathscr{S}_m(\mathscr{C})$.
We will also need to have a complete picture of how these functors transform as $\mathbf{m}$ and $\mathscr{C}$ each vary, so we give the following abstract description of them. Since there is an equivalence of $\infty$-categories $$\mathbf{Wald}_{\infty/N\Delta^{{\mathrm{op}}}}^{{\mathrm{cocart}}}\simeq\operatorname{Fun}(N\Delta^{{\mathrm{op}}},{\mathbf{Wald}_{\infty}})$$ (Pr. \[prp:Waldstraightening\]), and since $\mathbf{0}$ is an initial object of $N\Delta^{{\mathrm{op}}}$, it is easy to see that there is an adjunction $${{C}\colon{{\mathbf{Wald}_{\infty}}}\ \begin{tikzpicture}[baseline] \draw[>=stealth,->] (0,1ex) -- (0.75,1ex); \draw[>=stealth,->] (0.75,0.25ex) -- (0,0.25ex); \end{tikzpicture}\ {\mathbf{Wald}_{\infty/N\Delta^{{\mathrm{op}}}}^{{\mathrm{cocart}}}}\colon{R}},$$ where $C$ is the functor ${{\mathscr{C}}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}\times N\Delta^{{\mathrm{op}}}}}$, which represents the contstant functor ${{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}(N\Delta^{{\mathrm{op}}},{\mathbf{Wald}_{\infty}})}}$, and $R$ is the functor ${{(\mathscr{X}/N\Delta^{{\mathrm{op}}})}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{X}_0}}$, which represents evaluation at zero ${{\operatorname{Fun}(N\Delta^{{\mathrm{op}}},{\mathbf{Wald}_{\infty}})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}}}$. The counit ${{CR}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{id}}}$ of this adjunction can now be composed with the the natural transformation $F\colon{{\mathscr{F}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{S}}}$ (which we regard as a morphism of $\operatorname{Fun}({\mathbf{Wald}_{\infty}},\mathbf{Wald}_{\infty/N\Delta^{{\mathrm{op}}}}^{{\mathrm{cocart}}})$) to give a commutative square $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=8ex,
text height=1.5ex, text depth=0.25ex]
{CR\circ\mathscr{F}\times N\Delta^{{\mathrm{op}}}&CR\circ\mathscr{S}\\
\mathscr{F}&\mathscr{S}\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge node[above]{$CR\circ F$} (m-1-2)
edge node[left]{$E$} (m-2-1)
(m-1-2) edge node[right]{$E'$} (m-2-2)
(m-2-1) edge node[below]{$F$} (m-2-2);
\end{tikzpicture}$$ in the $\infty$-category $\operatorname{Fun}({\mathbf{Wald}_{\infty}},\mathbf{Wald}_{\infty/N\Delta^{{\mathrm{op}}}}^{{\mathrm{cocart}}})$.
Forming the fiber over an object $\mathbf{m}\in N\Delta^{{\mathrm{op}}}$, we obtain a commutative square $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\mathscr{F}_0&\mathscr{S}_0\\
\mathscr{F}_m&\mathscr{S}_m\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge node[above]{$F_0$} (m-1-2)
edge node[left]{$E_m$} (m-2-1)
(m-1-2) edge node[right]{$E'_m$} (m-2-2)
(m-2-1) edge node[below]{$F_m$} (m-2-2);
\end{tikzpicture}$$ in the $\infty$-category $\operatorname{Fun}({\mathbf{Wald}_{\infty}},{\mathbf{Wald}_{\infty}})$. We see that $E_m$ and $E_m'$ are the functors we identified above.
On the other hand, applying the realization functor $|\cdot|_{N\Delta^{{\mathrm{op}}}}$ (Df. \[dfn:realization\]), and noting that $$|CR\circ\mathscr{F}|_{N\Delta^{{\mathrm{op}}}}\simeq|\mathscr{F}_0\times N\Delta^{{\mathrm{op}}}|_{N\Delta^{{\mathrm{op}}}}\simeq\operatorname{id}$$ and $$|CR\circ\mathscr{S}|_{N\Delta^{{\mathrm{op}}}}\simeq|\mathscr{S}_0\times N\Delta^{{\mathrm{op}}}|_{N\Delta^{{\mathrm{op}}}}\simeq\Delta^0,$$ we obtain a commutative square $$\label{eqn:Sdotissuspension}
\begin{tikzpicture}[baseline]
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\operatorname{id}&0\\
\mathscr{F}&\mathscr{S},\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge node[left]{$E$} (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge node[below]{$F$} (m-2-2);
\end{tikzpicture}$$ in the $\infty$-category $\operatorname{Fun}({\mathrm{D}(\mathbf{Wald}_{\infty})},{\mathrm{D}(\mathbf{Wald}_{\infty})})$. When we pass to the fissile derived $\infty$-category, we will actually force this square to be pushout. Since $\mathscr{F}$ is the zero functor (Pr. \[prp:FCisacone\]), this will exhibit $\mathscr{S}$ as a suspension.
Before we give our definition of *fissibility*, we need a spot of abusive notation.
Recall (Cor. \[cor:WaldisInd\]) that we have an equivalence of $\infty$-categories ${\mathbf{Wald}_{\infty}}\simeq\operatorname{Ind}({\mathbf{Wald}_{\infty}}^{\omega})$. Consequently, we may use the transitivity result of [@HTT Pr. 5.3.6.11] to conclude that, in the notation of \[rec:PABofC\], we also have an equivalence $\mathscr{P}({\mathbf{Wald}_{\infty}}^{\omega})\simeq\mathscr{P}_{\mathscr{I}}^{\mathscr{K}}({\mathbf{Wald}_{\infty}})$, where $\mathscr{I}$ is the class of all small filtered simplicial sets, and $\mathscr{K}$ is the class of all small simplicial sets.
In particular, every presheaf $\mathscr{X}\colon{{{\mathbf{Wald}_{\infty}}^{\omega,{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$ extends to an essentially unique presheaf ${{{\mathbf{Wald}_{\infty}}^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$ with the property that it carries filtered colimits in ${\mathbf{Wald}_{\infty}}$ to the corresponding limits in ${\mathbf{Kan}}$. We will abuse notation by denoting this extended functor by $\mathscr{X}$ as well. This entitles us to speak of the value of a presheaf $\mathscr{X}\colon{{{\mathbf{Wald}_{\infty}}^{\omega,{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$ even on Waldhausen $\infty$-categories that may not be compact.
\[dfn:fissilepresheaf\] A presheaf $\mathscr{X}\colon{{{\mathbf{Wald}_{\infty}}^{\omega,{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$ will be said to be ***fissile*** if for every Waldhausen $\infty$-category $\mathscr{C}$ and every integer $m\geq0$, the exact functors $E_m$ and $J_m$ (Cnstr. \[cnstr:Em\] and Nt. \[ntn:JandF\]) induce functors $${{\mathscr{X}(\mathscr{F}_m(\mathscr{C}))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{X}(\mathscr{C})}}\textrm{\qquad and\qquad}{{\mathscr{X}(\mathscr{F}_m(\mathscr{C}))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{X}(\mathscr{S}_m(\mathscr{C}))}}$$ that together exhibit $\mathscr{X}(\mathscr{F}_m(\mathscr{C}))$ as the product of $\mathscr{X}(\mathscr{C})$ and $\mathscr{X}(\mathscr{S}_m(\mathscr{C}))$: $$(E_m^{\star},J_m^{\star})\colon{{\mathscr{X}(\mathscr{F}_m(\mathscr{C}))}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\mathscr{X}(\mathscr{C})\times\mathscr{X}(\mathscr{S}_m(\mathscr{C}))}}.$$
An induction using Pr. \[prp:FmisS1plusm\] demonstrates that the value of a fissile presheaf $\mathscr{X}\colon{{{\mathbf{Wald}_{\infty}}^{\omega,{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$ on the Waldhausen $\infty$-category of filtered objects $\mathscr{F}_m(\mathscr{C})$ of length $m$ is split into $1+m$ copies of $\mathscr{X}(\mathscr{C})$. That is, the $1+m$ different functors ${{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{F}_m(\mathscr{C})}}$ of the form $${{X}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {[0\ \tikz[baseline]\draw[>=stealth,-,double distance=1.5pt](0,0.5ex)--(0.5,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,-,double distance=1.5pt](0,0.5ex)--(0.5,0.5ex);\ 0\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X\ \tikz[baseline]\draw[>=stealth,-,double distance=1.5pt](0,0.5ex)--(0.5,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,-,double distance=1.5pt](0,0.5ex)--(0.5,0.5ex);\ X]}}$$ induce an equivalence $${{\mathscr{X}(\mathscr{F}_m(\mathscr{C}))}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\mathscr{X}(\mathscr{C})^{1+m}}}.$$
We began our discussion of fissile presheaves by thinking of them as special examples of virtual Waldhausen $\infty$-categories. That wasn’t wrong:
A presheaf $\mathscr{X}\in\mathscr{P}({\mathbf{Wald}_{\infty}}^{\omega})$ is fissile only if $\mathscr{X}$ carries direct sums in ${\mathbf{Wald}_{\infty}}^{\omega}$ to products — that is, only if $\mathscr{X}$ is a virtual Waldhausen $\infty$-category.
Suppose $\mathscr{C}$ and $\mathscr{D}$ two compact Waldhausen $\infty$-categories. Consider the retract diagrams $$\begin{tikzpicture}[baseline]
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=6ex,
text height=1.5ex, text depth=0.25ex]
{\mathscr{C}&\mathscr{C}\oplus\mathscr{D}&\mathscr{C}\\
\mathscr{C}\oplus\mathscr{D}&\mathscr{F}_1(\mathscr{C}\oplus\mathscr{D})&\mathscr{C}\oplus\mathscr{D}\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-1-3)
edge node[right]{$E_1$} (m-2-2)
(m-1-3) edge (m-2-3)
(m-2-1) edge node[below]{$E_1\oplus J_1$} (m-2-2)
(m-2-2) edge node[below]{$I_{1,0}\oplus F_1$} (m-2-3);
\end{tikzpicture}$$ and $$\begin{tikzpicture}[baseline]
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=6ex,
text height=1.5ex, text depth=0.25ex]
{\mathscr{D}&\mathscr{C}\oplus\mathscr{D}&\mathscr{D}\\
\mathscr{C}\oplus\mathscr{D}&\mathscr{F}_1(\mathscr{C}\oplus\mathscr{D})&\mathscr{C}\oplus\mathscr{D}.\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-1-3)
edge node[right]{$J_1$} (m-2-2)
(m-1-3) edge (m-2-3)
(m-2-1) edge node[below]{$E_1\oplus J_1$} (m-2-2)
(m-2-2) edge node[below]{$I_{1,0}\oplus F_1$} (m-2-3);
\end{tikzpicture}$$ Here $I_{1,0}$ is the functor induced by the morphism ${{0}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {0}}$. For any fissile virtual Waldhausen $\infty$-category $\mathscr{X}$, we have an induced retract diagram $$\label{eqn:retfordecompsarevirts}
\begin{tikzpicture}[baseline]
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\mathscr{X}(\mathscr{C}\oplus\mathscr{D})&\mathscr{X}(\mathscr{F}_1(\mathscr{C}\oplus\mathscr{D}))&\mathscr{X}(\mathscr{C}\oplus\mathscr{D})\\
\mathscr{X}(\mathscr{C})\times\mathscr{X}(\mathscr{D})&\mathscr{X}(\mathscr{C}\oplus\mathscr{D})\times\mathscr{X}(\mathscr{C}\oplus\mathscr{D})&\mathscr{X}(\mathscr{C})\times\mathscr{X}(\mathscr{D}).\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-1-3)
edge (m-2-2)
(m-1-3) edge (m-2-3)
(m-2-1) edge (m-2-2)
(m-2-2) edge (m-2-3);
\end{tikzpicture}$$ Since the center vertical map is an equivalence, and since equivalences are closed under retracts, so are the outer vertical maps.
Denote by $${\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}\subset{\mathrm{D}(\mathbf{Wald}_{\infty})}$$ the full subcategory spanned by the fissile functors. We’ll call this the ***fissile derived $\infty$-category*** of Waldhausen $\infty$-categories.
Since sifted colimits in ${\mathrm{D}(\mathbf{Wald}_{\infty})}$ commute with products [@HTT Lm. 5.5.8.11], we deduce the following.
\[cor:VaddWaldstableundersiftedcolims\] The subcategory ${\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}\subset{\mathrm{D}(\mathbf{Wald}_{\infty})}$ is stable under sifted colimits.
Fissile approximations to virtual Waldhausen $\infty$-categories {#fissile-approximations-to-virtual-waldhausen-infty-categories .unnumbered}
----------------------------------------------------------------
Note that representable presheaves are typically *not* fissile. Consequently, the obvious fully faithful inclusion ${{{\mathbf{Wald}_{\infty}}^{\omega}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {{\mathrm{D}(\mathbf{Wald}_{\infty})}}}$ does not factor through ${\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}\subset{\mathrm{D}(\mathbf{Wald}_{\infty})}$. Instead, in order to make a representable presheaf fissile, we’ll have to form a fissile approximation to it. Fortunately, there’s a universal way to do that.
\[prp:Laddexists\] The inclusion functor admits a left adjoint $$L^{{\mathrm{fiss}}}\colon{{{\mathrm{D}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}}},$$ which exhibits ${\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}$ as an accessible localization of ${\mathrm{D}(\mathbf{Wald}_{\infty})}$.
For any compact Waldhausen $\infty$-category $\mathscr{C}$ and every integer $m\geq 0$, consider the exact functor $$E_m\oplus J_m\colon{{\mathscr{C}\oplus\mathscr{S}_m(\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{F}_m(\mathscr{C})}};$$ let $S$ be the set of morphisms of ${\mathrm{D}(\mathbf{Wald}_{\infty})}$ of this form; let $\overline{S}$ be the strongly saturated class it generates. Since ${\mathbf{Wald}_{\infty}}^{\omega}$ is essentially small, the class $\overline{S}$ is of small generation. Hence we may form the accessible localization $S^{-1}{\mathrm{D}(\mathbf{Wald}_{\infty})}$. Since virtual Waldhausen $\infty$-categories are functors $\mathscr{X}\colon{{{\mathbf{Wald}_{\infty}}^{\omega,{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$ that preserve products, one sees that $S^{-1}{\mathrm{D}(\mathbf{Wald}_{\infty})}$ coincides with the full subcategory ${\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}\subset{\mathrm{D}(\mathbf{Wald}_{\infty})}$.
The fully faithful inclusion ${{{\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {{\mathrm{D}(\mathbf{Wald}_{\infty})}}}$ preserve finite products, and its left adjoint $L^{{\mathrm{fiss}}}$ preserve finite coproducts, whence we deduce the following.
The $\infty$-category ${\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}$ is compactly generated and admits finite direct sums, which are preserved by the inclusion $${{{\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {{\mathrm{D}(\mathbf{Wald}_{\infty})}}}.$$
Combining this with Lm. \[cor:VaddWaldstableundersiftedcolims\] and [@HA Lm. 1.3.2.9], we deduce the following somewhat surprising fact.
The subcategory ${\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}\subset{\mathrm{D}(\mathbf{Wald}_{\infty})}$ is stable under all small colimits.
Suspension of fissile virtual Waldhausen $\infty$-categories {#suspension-of-fissile-virtual-waldhausen-infty-categories .unnumbered}
------------------------------------------------------------
We now show that the suspension in the fissile derved $\infty$-category is essentially given by the functor $\mathscr{S}$. This is the key to showing that Waldhausen’s additivity is essentially equivalent to excision on the fissile derived $\infty$-category (Th. \[thm:additiveequiv\]). As a first step, we have the following observation.
\[prp:Sdotissuspension\] The diagram $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=5ex, column sep=5ex,
text height=1.5ex, text depth=0.25ex]
{{\mathrm{D}(\mathbf{Wald}_{\infty})}&{\mathrm{D}(\mathbf{Wald}_{\infty})}\\
{\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}&{\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge node[above]{$\mathscr{S}$} (m-1-2)
edge node[left]{$L^{{\mathrm{fiss}}}$} (m-2-1)
(m-1-2) edge node[right]{$L^{{\mathrm{fiss}}}$} (m-2-2)
(m-2-1) edge node[below]{$\Sigma$} (m-2-2);
\end{tikzpicture}$$ commutes (up to homotopy), where $\Sigma$ is the suspension endofunctor on the fissile derived $\infty$-category ${\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}$.
Apply $L^{{\mathrm{fiss}}}$ to the square to obtain a square $$\label{eqn:Laddpushoutsquare}
\begin{tikzpicture}[baseline]
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{L^{{\mathrm{fiss}}}&0\\
L^{{\mathrm{fiss}}}\circ\mathscr{F}&L^{{\mathrm{fiss}}}\circ\mathscr{S}\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge node[below]{$F$} (m-2-2);
\end{tikzpicture}$$ of natural transformations between functors ${{{\mathrm{D}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}}}$. Since $\mathscr{F}$ is essentially constant with value the zero object, this gives rise to a natural transformation ${{\Sigma\circ L^{{\mathrm{fiss}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {L^{{\mathrm{fiss}}}\circ\mathscr{S}}}$. To see that this natural transformation is an equivalence, it suffices to consider its value on a compact Waldhausen $\infty$-category $\mathscr{C}$. Now for any $\mathbf{m}\in N\Delta^{op}$, we have a diagram $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{L^{{\mathrm{fiss}}}\mathscr{S}_0(\mathscr{C})&L^{{\mathrm{fiss}}}\mathscr{F}_0(\mathscr{C})&L^{{\mathrm{fiss}}}\mathscr{S}_0(\mathscr{C})\\
L^{{\mathrm{fiss}}}\mathscr{S}_m(\mathscr{C})&L^{{\mathrm{fiss}}}\mathscr{F}_m(\mathscr{C})&L^{{\mathrm{fiss}}}\mathscr{S}_m(\mathscr{C})\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge node[above]{$J_0$} (m-1-2)
edge node[left]{$E'_m$} (m-2-1)
(m-2-1) edge node[below]{$J_m$} (m-2-2)
(m-1-2) edge node[above]{$F_0$} (m-1-3)
edge node[left]{$E_m$} (m-2-2)
(m-1-3) edge node[right]{$E'_m$} (m-2-3)
(m-2-2) edge node[below]{$F_m$} (m-2-3);
\end{tikzpicture}$$ of Waldhausen $\infty$-categories in which the horizontal composites are equivalences. Since $\mathscr{S}_0(\mathscr{C})$ is a zero object, the left-hand square is a pushout by definition; hence the right-hand square is as well. The geometric realization of the right-hand square is precisely the value of the square on $\mathscr{C}$.
The observation that $\Sigma\circ L^{{\mathrm{fiss}}}\simeq L^{{\mathrm{fiss}}}\circ\mathscr{S}$, nice though it is, doesn’t quite cut it: we want an even closer relationship between $\mathscr{S}$ and the suspension in the fissile derived $\infty$-category. More precisely, we’d like to know that it isn’t necessary to apply $L^{{\mathrm{fiss}}}$ to $\mathscr{S}(\mathscr{C})$ in order to get $\Sigma L^{{\mathrm{fiss}}}\mathscr{C}$. So we conclude this section with a proof that the functor $\mathscr{S}\colon{{{\mathrm{D}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathrm{D}(\mathbf{Wald}_{\infty})}}}$ already takes values in the fissile derived $\infty$-category ${\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}$.
\[prp:SXisdistrib\] For any virtual Waldhausen $\infty$-category $\mathscr{X}$, the virtual Waldhausen $\infty$-category $\mathscr{S}\mathscr{X}$ is fissile.
We may write $\mathscr{X}$ as a geometric realization of a simplicial diagram $\mathscr{Y}_{\ast}$ of Waldhausen $\infty$-categories. So our claim is that for any compact Waldhausen $\infty$-category $\mathscr{C}$ and any integer $m\geq 0$, the map $${{(\operatorname{colim}\mathscr{S}(\mathscr{Y}_{\ast}))(\mathscr{F}_m(\mathscr{C}))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(\operatorname{colim}\mathscr{S}(\mathscr{Y}_{\ast}))(\mathscr{C})\times(\operatorname{colim}\mathscr{S}(\mathscr{Y}_{\ast}))(\mathscr{S}_m(\mathscr{C}))}}$$ induced by $(E_m,J_m)$ is an equivalence. Since geometric realization commutes with products, we reduce to the case in which $\mathscr{Y}_{\ast}$ is constantat a Waldhausen $\infty$-category $\mathscr{Y}$. Now our claim is that for any compact Waldhausen $\infty$-category $\mathscr{C}$ and any integer $m\geq 0$, the map $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\mathrm{H}(\mathscr{F}_m(\mathscr{C}),(\mathscr{S}(\mathscr{Y})/N\Delta^{{\mathrm{op}}}))\\
\mathrm{H}(\mathscr{C},(\mathscr{S}(\mathscr{Y})/N\Delta^{{\mathrm{op}}}))\times\mathrm{H}(\mathscr{S}_m(\mathscr{C}),(\mathscr{S}(\mathscr{Y})/N\Delta^{{\mathrm{op}}}))\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-2-1);
\end{tikzpicture}$$ (Cnstr. \[ntn:geomrealinWald\]) induced by $(E_m,J_m)$ is a weak homotopy equivalence. To simplify notation, we write $\mathrm{H}(-,\mathscr{S}(\mathscr{Y}))$ for $\mathrm{H}(-,(\mathscr{S}(\mathscr{Y})/N\Delta^{{\mathrm{op}}}))$ in what follows.
Let’s use Joyal’s $\infty$-categorical variant of Quillen’s Theorem A [@HTT Th. 4.1.3.1]. Fix an object $$((\mathbf{p},\alpha),(\mathbf{q},\beta))\in\mathrm{H}(\mathscr{C},\mathscr{S}(\mathscr{Y}))\times\mathrm{H}(\mathscr{S}_m(\mathscr{C}),\mathscr{S}(\mathscr{Y})).$$ So $\mathbf{p}$ and $\mathbf{q}$ are objects of $N\Delta^{op}$, $\alpha$ is an exact functor ${{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{S}_p(\mathscr{Y})}}$, and $\beta$ is an exact functor ${{\mathscr{S}_m(\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{S}_p(\mathscr{Y})}}$. Write $J((\mathbf{p},\alpha),(\mathbf{q},\beta))$ for the pullback $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{J((\mathbf{p},\alpha),(\mathbf{q},\beta))&\mathrm{H}(\mathscr{C},\mathscr{S}(\mathscr{Y}))\times\mathrm{H}(\mathscr{S}_m(\mathscr{C}),\mathscr{S}(\mathscr{Y}))\\
\mathrm{H}(\mathscr{F}_m(\mathscr{C}),\mathscr{S}(\mathscr{Y}))&(\mathrm{H}(\mathscr{C},\mathscr{S}(\mathscr{Y}))\times\mathrm{H}(\mathscr{S}_m(\mathscr{C}),\mathscr{S}(\mathscr{Y})))_{((\mathbf{p},\alpha),(\mathbf{q},\beta))/}\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge (m-2-2);
\end{tikzpicture}$$ We may identify $J((\mathbf{p},\alpha),(\mathbf{q},\beta))$ with a quasicategory whose objects are tuples $(\mathbf{r},\gamma,\mu,\nu,\sigma,\tau)$ consisting of:
- $\mathbf{r}$ is an object of $\Delta$,
- $\gamma\colon{{\mathscr{S}_m(\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{S}_r\mathscr{Y}}}$ is an exact functor,
- $\mu\colon{{\mathbf{r}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{p}}}$ and $\nu\colon{{\mathbf{r}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{q}}}$ are morphisms of $\Delta$, and
- $\sigma\colon{{\mu^{\ast}\alpha}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\gamma|_{\mathscr{C}}}}$ and $\tau\colon{{\nu^{\ast}\beta}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\gamma|_{\mathscr{S}_m(\mathscr{C})}}}$ are equivalences of exact functors.
Denote by $\kappa$ the constant functor ${{J((\mathbf{p},\alpha),(\mathbf{q},\beta))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {J((\mathbf{p},\alpha),(\mathbf{q},\beta))}}$ at the object $$\left(0,0,{{\{0\}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{p}}},{{\{0\}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{q}}},0,0\right).$$ To prove that $J((\mathbf{p},\alpha),(\mathbf{q},\beta))$ is contractible, we construct an endofunctor $\lambda$ and natural transformations $$\operatorname{id}\ \tikz[baseline]\draw[>=stealth,<-](0,0.5ex)--(0.5,0.5ex);\ \lambda\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ \kappa.$$ We define the functor $\lambda$ by $$\lambda(\mathbf{r},\gamma,\mu,\nu,\sigma,\tau){\mathrel{\mathop:}=}(\mathbf{r}^{\lhd},s_0\circ\gamma,\mu',\nu',\sigma',\tau'),$$ where $\mu'|_{\mathbf{r}}=\mu$ and $\mu'(-\infty)=0$, $\nu'|_{\mathbf{r}}=\nu$ and $\nu'(-\infty)=0$, and $\sigma'$ and $\tau'$ are the obvious extensions of $\sigma$ and $\tau$. The inclusion ${{\mathbf{r}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{r}^{\lhd}}}$ induces a natural transformation ${{\lambda}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{id}}}$, and the inclusion ${{\{-\infty\}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{r}^{\lhd}}}$ induces a natural transformation ${{\lambda}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\kappa}}$.
We thus have the following enhancement of Pr. \[prp:Sdotissuspension\].
\[cor:Sdotisreallysuspension\] The diagram $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{{\mathrm{D}(\mathbf{Wald}_{\infty})}&\\
&{\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}\\
{\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}&\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge node[above right]{$\mathscr{S}$} (m-2-2)
edge node[left]{$L^{{\mathrm{fiss}}}$} (m-3-1)
(m-3-1) edge node[below right]{$\Sigma$} (m-2-2);
\end{tikzpicture}$$ commutes (up to homotopy), where $\Sigma$ is the suspension endofunctor on the fissile derived $\infty$-category ${\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}$.
Additive theories
=================
In this section we introduce the $\infty$-categorical analogue of Waldhausen’s notion of additivity, and we prove our Structure Theorem (Th. \[thm:additiveequiv\]), which identifies the homotopy theory of additive functors ${{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$ with the homotopy theory of certain excisive functors ${{{\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$ on the fissile derived $\infty$-category of the previous section. Using this, we can find the best additive approximation to any functor $\phi\colon{{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$ as a Goodwillie differential. Since suspension in this $\infty$-category is given by the functor $\mathscr{S}$, this best excisive approximation $D\phi$ can be exhibited by a formula $${{\mathscr{C}}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{colim}_n\Omega^n|\phi(S_{\ast}^n(\mathscr{C}))|}}.$$ If $\phi$ preserves finite products, the colimit turns out to be unnecessary, and $D\phi$ can be given by an even simpler formula: $${{\mathscr{C}}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {\Omega|\phi(S_{\ast}(\mathscr{C}))|}}.$$
In the next section, we’ll use this perspective on additivity to prove some fundamental things, such as the Eilenberg Swindle and Waldhausen’s Fibration Theorem, for general additive functors. In §\[sect:univpropKthy\], we’ll apply our additive approximation to the “moduli space of objects” functor $\iota$ to give a universal description of algebraic $K$-theory of Waldhausen $\infty$-categories, and the formula above shows that our algebraic $K$-theory extends Waldhausen’s.
Theories and additive theories {#theories-and-additive-theories .unnumbered}
------------------------------
The kinds of functors we’re going to be thinking about are called *theories*. What we’ll show is that among theories, one can isolate the class of *additive theories*, which split all exact sequences.
\[dfn:reducedtheories\] Suppose $C$ and $D$ $\infty$-categories, and suppose that $C$ is pointed. Recall ([@MR1076523 p. 1] or [@HA Df. 1.4.2.1(ii)]) that a functor ${{C}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {D}}$ is ***reduced*** if it carries the zero object of $C$ to the terminal object of $D$. We write $\operatorname{Fun}^{\ast}(C,D)\subset\operatorname{Fun}(C,D)$ for the full subcategory spanned by the reduced functors, and if $\mathscr{A}$ is a collection of simplicial sets, then we write $\operatorname{Fun}_{\mathscr{A}}^{\ast}(C,D)\subset\operatorname{Fun}(C,D)$ for the full subcategory spanned by the reduced functors that preserve $\mathscr{A}$-shaped colimits (\[rec:Ashapedcolim\]).
Suppose $\mathscr{E}$ an $\infty$-topos. By an ***$\mathscr{E}$-valued theory***, we shall here mean a reduced functor ${{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}}}$ that preserves filtered colimits. We write $\operatorname{Thy}(\mathscr{E})$ for the full subcategory of $\operatorname{Fun}({\mathbf{Wald}_{\infty}},\mathscr{E})$ spanned by $\mathscr{E}$-valued theories.
Those who grimace the prospect of contemplating general $\infty$-topoi can enjoy a complete picture of what’s going on by thinking only of examples of the form $\mathscr{E}=\operatorname{Fun}(S,{\mathbf{Kan}})$. The extra generality comes at no added expense, but we won’t get around to using it here.
Note that a theory $\phi\colon{{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}}}$ may be uniquely identified in different ways. On one hand, $\phi$ is (Cor. \[cor:WaldisInd\]) the left Kan extension of its restriction $$\phi|_{{\mathbf{Wald}_{\infty}}^{\omega}}\colon{{{\mathbf{Wald}_{\infty}}^{\omega}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}}};$$ on the other, we can extend $\phi$ to its left derived functor (Df. \[dfn:lderivWald\]) $$\Phi\colon{{{\mathrm{D}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}}},$$ which is the unique extension of $\phi$ that preserves all sifted colimits.
Many examples of theories that arise in practice have the property that the natural morphism ${{\phi(\mathscr{C}\oplus\mathscr{D})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\phi(\mathscr{C})\times\phi(\mathscr{D})}}$ is an equivalence. We’ll look at these theories more closely below (Df. \[dfn:preadditive\]). In any case, when this happens, the sum functor ${{\mathscr{C}\oplus\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$ defines a monoid structure on $\pi_0\phi(\mathscr{C})$. For invariants like $K$-theory, we’ll want to demand that this monoid actually be a group. We thus make the following definition, which is sensible for *any* theory.
A theory $\phi\in\operatorname{Thy}(\mathscr{E})$ will be said to be ***grouplike*** if, for any Waldhausen $\infty$-category $\mathscr{C}$, the *shear functor* ${{\mathscr{C}\oplus\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}\oplus\mathscr{C}}}$ defined by the assignment ${{(X,Y)}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {(X,X\vee Y)}}$ induces an equivalence ${{\pi_0\phi(\mathscr{C}\oplus\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\pi_0\phi(\mathscr{C}\oplus\mathscr{C})}}$.
To formulate our Structure Theorem, we need to stare at a few functors between various Waldhausen $\infty$-categories of filtered objects.
Suppose $m\geq 0$ an integer, and suppose $0\leq k\leq m$. We consider the morphism $i_k\colon{{\mathbf{0}\cong\{k\}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{m}}}$ of $\Delta$. For any Waldhausen $\infty$-category $\mathscr{C}$, write $I_{m,k}$ for the induced functor ${{\mathscr{F}_m(\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{F}_0(\mathscr{C})}}$, and write $I'_{m,k}$ for the induced functor ${{\mathscr{S}_m(\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{S}_0(\mathscr{C})}}$. Of course $\mathscr{F}_0(\mathscr{C})\simeq\mathscr{C}$ and $\mathscr{S}_0(\mathscr{C})\simeq0$. So the functor $I_{m,k}$ extracts from a filtered object $$X_0\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X_1\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X_m$$ its $k$-th filtered piece $X_k$, and the functor $I'_{m,k}$ is, by necessity, the trivial functor.
We may now contemplate a square of retract diagrams $${{(\Delta^2/\Delta^{\{0,2\}})\times(\Delta^2/\Delta^{\{0,2\}})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}}}$$ given by $$\label{eqn:pushpullsquares}
\begin{tikzpicture}[baseline]
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\mathscr{S}_0(\mathscr{C})&\mathscr{F}_0(\mathscr{C})&\mathscr{S}_0(\mathscr{C})\\
\mathscr{S}_m(\mathscr{C})&\mathscr{F}_m(\mathscr{C})&\mathscr{S}_m(\mathscr{C})\\
\mathscr{S}_0(\mathscr{C})&\mathscr{F}_0(\mathscr{C})&\mathscr{S}_0(\mathscr{C}).\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge node[above]{$J_0$} (m-1-2)
edge node[left]{$E'_m$} (m-2-1)
(m-1-2) edge node[above]{$F_0$} (m-1-3)
edge node[left]{$E_m$} (m-2-2)
(m-1-3) edge node[right]{$E'_m$} (m-2-3)
(m-2-1) edge node[below]{$J_m$} (m-2-2)
edge node[left]{$I'_{m,k}$} (m-3-1)
(m-2-2) edge node[above]{$F_m$} (m-2-3)
edge node[right]{$I_{m,k}$} (m-3-2)
(m-2-3) edge node[right]{$I_{m,k}$} (m-3-3)
(m-3-1) edge node[below]{$J_0$} (m-3-2)
(m-3-2) edge node[below]{$F_0$} (m-3-3);
\end{tikzpicture}$$ Only the upper right square of is (by Cnstr. \[cnstr:Em\]) functorial in $\mathbf{m}$.
We may now apply the localization functor $L^{{\mathrm{fiss}}}$ to . In the resulting diagram $$\label{eqn:Laddpushpullsquares}
\begin{tikzpicture}[baseline]
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{L^{{\mathrm{fiss}}}\mathscr{S}_0(\mathscr{C})&L^{{\mathrm{fiss}}}\mathscr{F}_0(\mathscr{C})&L^{{\mathrm{fiss}}}\mathscr{S}_0(\mathscr{C})\\
L^{{\mathrm{fiss}}}\mathscr{S}_m(\mathscr{C})&L^{{\mathrm{fiss}}}\mathscr{F}_m(\mathscr{C})&L^{{\mathrm{fiss}}}\mathscr{S}_m(\mathscr{C})\\
L^{{\mathrm{fiss}}}\mathscr{S}_0(\mathscr{C})&L^{{\mathrm{fiss}}}\mathscr{F}_0(\mathscr{C})&L^{{\mathrm{fiss}}}\mathscr{S}_0(\mathscr{C}),\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge node[above]{$J_0$} (m-1-2)
edge node[left]{$E'_m$} (m-2-1)
(m-1-2) edge node[above]{$F_0$} (m-1-3)
edge node[left]{$E_m$} (m-2-2)
(m-1-3) edge node[right]{$E'_m$} (m-2-3)
(m-2-1) edge node[below]{$J_m$} (m-2-2)
edge node[left]{$I'_{m,k}$} (m-3-1)
(m-2-2) edge node[above]{$F_m$} (m-2-3)
edge node[right]{$I_{m,k}$} (m-3-2)
(m-2-3) edge node[right]{$I_{m,k}$} (m-3-3)
(m-3-1) edge node[below]{$J_0$} (m-3-2)
(m-3-2) edge node[below]{$F_0$} (m-3-3);
\end{tikzpicture}$$ the square in the upper left corner is a pushout, whence every square is a pushout.
Now we are ready to state the Structure Theorem.
\[thm:additiveequiv\] Suppose $\mathscr{E}$ an $\infty$-topos. Suppose $\phi$ an $\mathscr{E}$-valued theory. Then the following are equivalent.
1. \[item:add\] For any Waldhausen $\infty$-category $\mathscr{C}$, any integer $m\geq1$, and any integer $0\leq k\leq m$, the functors $$\phi(F_m)\colon{{\phi(\mathscr{F}_m(\mathscr{C}))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\phi(\mathscr{S}_m(\mathscr{C}))}}\textrm{\quad and\quad}\phi(I_{m,k})\colon{{\phi(\mathscr{F}_m(\mathscr{C}))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\phi(\mathscr{F}_0(\mathscr{C}))}}$$ exhibit $\phi(\mathscr{F}_m(\mathscr{C}))$ as a product of $\phi(\mathscr{S}_m(\mathscr{C}))$ and $\phi(\mathscr{F}_0(\mathscr{C}))$.
2. \[item:group\] For any Waldhausen $\infty$-category $\mathscr{C}$ and for any functor $${\mathbf{S}}_{\ast}(\mathscr{C})\colon{{N\Delta^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}}}$$ that classifies the Waldhausen cocartesian fibration ${{\mathscr{S}(\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}}}$, the induced functor $\phi\circ{\mathbf{S}}_{\ast}(\mathscr{C})\colon{{N\Delta^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}_{\ast}}}$ is a group object [@HTT Df. 7.2.2.1].
3. \[item:stableadd\] The theory $\phi$ is grouplike, and for any Waldhausen $\infty$-category $\mathscr{C}$ and any integer $m\geq1$, the functors $$\phi(F_m)\colon{{\phi(\mathscr{F}_m(\mathscr{C}))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\phi(\mathscr{S}_m(\mathscr{C}))}}\textrm{\quad and\quad}\phi(I_{m,0})\colon{{\phi(\mathscr{F}_m(\mathscr{C}))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\phi(\mathscr{F}_0(\mathscr{C}))}}$$ exhibit $\phi(\mathscr{F}_m(\mathscr{C}))$ as a product of $\phi(\mathscr{S}_m(\mathscr{C}))$ and $\phi(\mathscr{F}_0(\mathscr{C}))$.
4. \[item:stableaddmisone\] The theory $\phi$ is grouplike, and for any Waldhausen $\infty$-category $\mathscr{C}$, the functors $$\phi(F_1)\colon{{\phi(\mathscr{F}_1(\mathscr{C}))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\phi(\mathscr{S}_1(\mathscr{C}))}}\textrm{\quad and\quad}\phi(I_{1,0})\colon{{\phi(\mathscr{F}_1(\mathscr{C}))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\phi(\mathscr{F}_0(\mathscr{C}))}}$$ exhibit $\phi(\mathscr{F}_1(\mathscr{C}))$ as a product of $\phi(\mathscr{S}_1(\mathscr{C}))$ and $\phi(\mathscr{F}_0(\mathscr{C}))$.
5. \[item:condonhocat\] The theory $\phi$ is grouplike, it carries direct sums to products, and, for any Waldhausen $\infty$-category $\mathscr{C}$, the images of $\phi(I_{1,1})$ and $\phi(I_{1,0}\oplus F_1)$ in the set $\operatorname{Mor}_{h\mathscr{E}_{\ast}}(\mathscr{F}_1(\mathscr{C}),\mathscr{C})$ are equal.
6. \[item:stablecat\] The theory $\phi$ is grouplike, and for any Waldhausen $\infty$-category $\mathscr{C}$ and any functor ${\mathbf{S}}_{\ast}(\mathscr{C})\colon{{N\Delta^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}}}$ that classifies the Waldhausen cocartesian fibration ${{\mathscr{S}(\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}}}$, the induced functor $\phi\circ{\mathbf{S}}_{\ast}(\mathscr{C})\colon{{N\Delta^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}_{\ast}}}$ is a category object see or [@G Df. 1.1.1].
7. \[item:excisive\] The left derived functor $\Phi\colon{{{\mathrm{D}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}}}$ of $\phi$ factors through an excisive functor $$\Phi_{{\mathrm{add}}}\colon{{{\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}}}.$$
The equivalence of conditions (\[thm:additiveequiv\].\[item:add\]) and (\[thm:additiveequiv\].\[item:group\]) follows from Pr. \[prp:FmisS1plusm\] and the proof of [@HTT Pr. 6.1.2.6]. (Also see [@HTT Rk. 6.1.2.8].) Conditions (\[thm:additiveequiv\].\[item:stableadd\]) and (\[thm:additiveequiv\].\[item:stablecat\]) are clearly special cases of (\[thm:additiveequiv\].\[item:add\]) and (\[thm:additiveequiv\].\[item:group\]), respectively, and condition (\[thm:additiveequiv\].\[item:stableaddmisone\]) is a special case of (\[thm:additiveequiv\].\[item:stableadd\]). The equivalence of (\[thm:additiveequiv\].\[item:stableadd\]) and (\[thm:additiveequiv\].\[item:stablecat\]) also follows directly from Pr. \[prp:FmisS1plusm\].
Let us show that (\[thm:additiveequiv\].\[item:stableaddmisone\]) implies (\[thm:additiveequiv\].\[item:condonhocat\]). We begin by noting that we have an analogue of the commutative diagram : $$\begin{tikzpicture}[baseline]
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\phi(\mathscr{C}\oplus\mathscr{D})&\phi(\mathscr{F}_1(\mathscr{C}\oplus\mathscr{D}))&\phi(\mathscr{C}\oplus\mathscr{D})\\
\phi(\mathscr{C})\times\phi(\mathscr{D})&\phi(\mathscr{C}\oplus\mathscr{D})\times\phi(\mathscr{C}\oplus\mathscr{D})&\phi(\mathscr{C})\times\phi(\mathscr{D}),\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-1-3)
edge (m-2-2)
(m-1-3) edge (m-2-3)
(m-2-1) edge (m-2-2)
(m-2-2) edge (m-2-3);
\end{tikzpicture}$$ and once again it is a retract diagram in $\mathscr{E}$. Since $\mathscr{E}$ admits filtered colimits, equivalences therein are closed under retracts, so since the center vertical morphism is an equivalence, the outer vertical morphisms are as well. Hence $\phi $ carries direct sums to products. Now the exact functor $I_{1,0}\oplus F_1$ admits a (homotopy) section $\sigma\colon{{\mathscr{C}\oplus\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{F}_1(\mathscr{C})}}$ such that $I_{1,1}\circ\sigma\simeq\nabla$. Hence if $\phi$ satisfies (\[thm:additiveequiv\].\[item:stableaddmisone\]), then $\phi(I_{1,0}\oplus F_1)$ is an equivalence with homotopy inverse $\phi(\sigma)$, whence $\phi(I_{1,1})$ and $\phi(I_{1,0}\oplus F_1)$ are equal in $\operatorname{Mor}_{h\mathscr{E}}(\mathscr{F}_1(\mathscr{C}),\mathscr{C})$.
It is now easy to see that (\[thm:additiveequiv\].\[item:stablecat\]) implies (\[thm:additiveequiv\].\[item:group\]).
We now show that (\[thm:additiveequiv\].\[item:condonhocat\]) implies (\[thm:additiveequiv\].\[item:stableadd\]). For any natural number $m$, suppose the images of $\phi(I_{1,1})$ and $\phi(I_{1,0}\oplus F_1)$ in $\operatorname{Mor}_{h\mathscr{E}}(\mathscr{F}_1(\mathscr{F}_m(\mathscr{C})),\mathscr{F}_m(\mathscr{C}))$ are equal; we must show that $\phi(I_{m,0}\oplus F_m)$ is an equivalence. Compose $I_{1,1}$ and $I_{1,0}\oplus F_1$ with the exact functor ${{\mathscr{F}_m(\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{F}_1(\mathscr{F}_m(\mathscr{C}))}}$ that sends a filtered object $$X_0\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.75,0.5ex);\ X_1\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.75,0.5ex);\ X_2\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.75,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.75,0.5ex);\ X_m$$ to the cofibration of filtered objects given by the diagram $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{X_0&X_0&X_0&\cdots&X_0\\
X_0&X_1&X_2&\cdots&X_m;\\};
\path[>=stealth,font=\scriptsize]
(m-1-1) edge[-,double distance=1.5pt] (m-1-2)
edge[-,double distance=1.5pt] (m-2-1)
(m-1-2) edge[-,double distance=1.5pt] (m-1-3)
edge[>->] (m-2-2)
(m-1-3) edge[-,double distance=1.5pt] (m-1-4)
edge[>->] (m-2-3)
(m-1-4) edge[-,double distance=1.5pt] (m-1-5)
(m-1-5) edge[>->] (m-2-5)
(m-2-1) edge[>->] (m-2-2)
(m-2-2) edge[>->] (m-2-3)
(m-2-3) edge[>->] (m-2-4)
(m-2-4) edge[>->] (m-2-5);
\end{tikzpicture}$$ the exact functor $I_{m,0}\oplus F_m$ also admits a section $\sigma\colon{{\mathscr{C}\oplus\mathscr{S}_m(\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{F}_m(\mathscr{C})}}$ (up to homotopy) such that $I_{m,1}\circ\sigma\simeq\nabla$, and applying our condition on $\phi$, we find that $\phi(\sigma\circ(I_{m,0}\oplus F_m))\simeq\phi(\operatorname{id})$.
We now set about showing that (\[thm:additiveequiv\].\[item:stableadd\]) implies (\[thm:additiveequiv\].\[item:excisive\]). First, we show that $\Phi$ factors through a functor $$\Phi_{{\mathrm{add}}}\colon{{{\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}}}.$$ As above, we find that $\Phi$ carries direct sums to products, and from this we deduce that $\Phi$ carries morphisms of the class $S$ described in Pr. \[prp:Laddexists\] to equivalences. We further claim that the family $T$ of those morphisms of ${\mathrm{D}(\mathbf{Wald}_{\infty})}$ that are carried to equivalences by $\Phi$ is a strongly saturated class. Since $\Phi$ sends direct sums to products, it carries any finite coproduct of elements of $T$ to equivalences. Moreover, since $\Phi$ preserves sifted colimits, it preserves any morphism that can be exhibited as a small sifted colimit of elements of $T$. Hence the full subcategory of $\mathscr{O}({\mathrm{D}(\mathbf{Wald}_{\infty})})$ spanned by the elements of $T$ is closed under all small colimits. Finally, to prove that any pushout $\psi'\colon{{\mathscr{X}'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{Y}'}}$ of an element $\psi\colon{{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{Y}}}$ of $T$ (along any morphism ${{\mathscr{X}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{X}'}}$), we note that we may exhibit $\psi'$ as the natural morphism of geometric realizations[^1] $${{|B_{\ast}(\mathscr{X}',\mathscr{X},\mathscr{X})|}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {|B_{\ast}(\mathscr{X}',\mathscr{X},\mathscr{Y})|}},$$ where the simplicial objects $B_{\ast}(\mathscr{X}',\mathscr{X},\mathscr{X})$ and $B_{\ast}(\mathscr{X}',\mathscr{X},\mathscr{Y})$ are two-sided bar constructions defined by $$B_n(\mathscr{X}',\mathscr{X},\mathscr{X}){\mathrel{\mathop:}=}\mathscr{X}'\oplus\mathscr{X}^{\oplus n}\oplus\mathscr{X}\textrm{\quad and\quad}B_n(\mathscr{X}',\mathscr{X},\mathscr{Y}){\mathrel{\mathop:}=}\mathscr{X}'\oplus\mathscr{X}^{\oplus n}\oplus\mathscr{Y}.$$ Since $T$ is closed under formation of products, each map $${{B_n(\mathscr{X}',\mathscr{X},\mathscr{X})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {B_n(\mathscr{X}',\mathscr{X},\mathscr{Y})}}$$ is an element of $T$, and since $T$ is closed under geometric realizations, the morphism ${{\mathscr{X}'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{Y}'}}$ is an element of $T$. Hence $T$ is strongly saturated and therefore contains $\overline{S}$; thus $\Phi$ factors through a functor $\Phi_{{\mathrm{add}}}\colon{{{\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}}}$.
We now show that $\Phi_{{\mathrm{add}}}$ is excisive. For any nonnegative integer $m$, apply $\phi$ to the diagram with $k=0$. The lower right corner of the resulting diagram is a pullback. Hence the upper right corner of the diagram resulting from applying $\phi$ to the diagram is also a pullback. Now we may form the geometric realization of this simplicial diagram of squares to obtain a square $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\Phi(\mathscr{F}_0(\mathscr{C}))&\Phi(\mathscr{S}_0(\mathscr{C}))\\
\Phi(\mathscr{F}(\mathscr{C}))&\Phi(\mathscr{S}(\mathscr{C})).\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge (m-2-2);
\end{tikzpicture}$$ It follows from the Segal delooping machine ([@MR50:5782] and [@HTT Lm. 7.2.2.11]) that this square is a pullback as well, since for any functor $S_{\ast}(\mathscr{C})\colon{{N\Delta^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}}}$ classified by the Waldhausen cocartesian fibration ${{\mathscr{S}(\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}}}$, the simplicial object $\Phi\circ S_{\ast}(\mathscr{C})$ is a group object, and $\mathscr{F}(\mathscr{C})$ and $\mathscr{S}_0(\mathscr{C})$ are zero objects. Since $\mathscr{S}$ is a suspension functor in ${\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}$, we find that the natural transformation ${{\Phi_{{\mathrm{add}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Omega_{\mathscr{E}}\circ\Phi_{{\mathrm{add}}}\circ\Sigma}}$ is an equivalence, whence $F_{{\mathrm{add}}}$ is excisive [@HA Pr. 1.4.2.13].
To complete the proof, it remains to show that (\[thm:additiveequiv\].\[item:excisive\]) implies (\[thm:additiveequiv\].\[item:add\]). It follows from (\[thm:additiveequiv\].\[item:excisive\]) that for any nonnegative integer $m$ and any integer $0\leq k\leq m$, applying $\Phi$ to yields the same result as applying $\Phi_{{\mathrm{add}}}$ to . Since the lower right square of the latter diagram is a pushout in ${\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}$, the excisive functor $F_{{\mathrm{add}}}$ carries it to a pullback square in $\mathscr{E}$, whence we obtain the first condition.
\[dfn:additivefunctor\] Suppose $\mathscr{E}$ an $\infty$-topos. An $\mathscr{E}$-valued theory $\phi$ will be said to be ***additive*** just in case it satisfies any of the equivalent conditions of Th. \[thm:additiveequiv\]. We denote by ${\mathrm{Add}}(\mathscr{E})$ the full subcategory of $\operatorname{Thy}(\mathscr{E})$ spanned by the additive theories.
Our Structure Theorem (Th. \[thm:additiveequiv\]) yields an identification of additive theories and excisive functors on distributive virtual Waldhausen $\infty$-categories.
\[thm:addfuncsareexc\] Suppose $\mathscr{E}$ an $\infty$-topos. The functor $L^{{\mathrm{fiss}}}\circ j$ induces an equivalence of $\infty$-categories $${{\operatorname{Exc}_{\mathscr{G}}({\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})},\mathscr{E})}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {{\mathrm{Add}}(\mathscr{E})}},$$ where $\operatorname{Exc}_{\mathscr{G}}({\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})},\mathscr{E})\subset\operatorname{Fun}^{\ast}({\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})},\mathscr{E})$ is the full subcategory spanned by the reduced excisive functors that preserve small sifted colimits.
It follows from Th. \[thm:additiveequiv\] that composition with $L^{{\mathrm{fiss}}}\circ j$ defines an essentially surjective functor $${{\operatorname{Exc}_{\mathscr{G}}({\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})},\mathscr{E})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathrm{Add}}(\mathscr{E})}}.$$ To see that this functor is fully faithful, it suffices to note that we have a commutative diagram $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\operatorname{Exc}_{\mathscr{G}}({\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})},\mathscr{E})&{\mathrm{Add}}(\mathscr{E})\\
\operatorname{Fun}({\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})},\mathscr{E})&\operatorname{Fun}({\mathrm{D}(\mathbf{Wald}_{\infty})},\mathscr{E})\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge[right hook->] (m-2-1)
(m-1-2) edge[right hook->] (m-2-2)
(m-2-1) edge[right hook->] (m-2-2);
\end{tikzpicture}$$ in which the vertical functors are fully faithful by definition, and the bottom functor is fully faithful because the $\infty$-category ${\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}$ is a localization of ${\mathrm{D}(\mathbf{Wald}_{\infty})}$ (Pr. \[prp:Laddexists\]).
By virtue of [@HA Pr. 1.4.4.10], this result now yields a *canonical* delooping of any additive functor.
\[cor:additivevaluedinstab\] Suppose $\mathscr{X}$ an $\infty$-topos. Then composition with the canonical functor $\Omega^{\infty}\colon{{\operatorname{Stab}(\mathscr{E})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}_{\ast}}}$ induces an equivalence of $\infty$-categories $${{\operatorname{Fun}^{\mathrm{rex}}_{\mathscr{G}}({\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})},\operatorname{Stab}(\mathscr{E}))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathrm{Add}}(\mathscr{E})}},$$ where $\operatorname{Fun}^{\mathrm{rex}}_{\mathscr{G}}({\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})},\operatorname{Stab}(\mathscr{E}))\subset\operatorname{Fun}({\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})},\operatorname{Stab}(\mathscr{E}))$ denotes the full subcategory spanned by the right exact functors $\mathbf{\Phi}\colon{{{\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Stab}(\mathscr{E})}}$ such that $\Omega^{\infty}\circ\mathbf{\Phi}\colon{{{\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}}}$ preserves sifted colimits.
Additivization {#additivization .unnumbered}
--------------
We now find that any theory admits an additive approximation given by a Goodwillie differential. The nature of colimits computed in ${\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}$ will then permit us to describe this additive approximation as an $\infty$-categorical $S_{\bullet}$ construction. As a result, we find that any such theory deloops to a *connective* spectrum.
We first need the following well-known lemma, which follows from [@HA Lm. 5.3.6.17] or, alternately, from a suitable generalization of [@HA Cor. 5.1.3.7].
\[lem:loopspreservesiftedcolims\] For any $\infty$-topos $\mathscr{E}$, the loop functor $\Omega_{\mathscr{E}}\colon{{\mathscr{E}_{\ast}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}_{\ast}}}$ preserves sifted colimits of connected objects.
\[thm:additivizationexists\] Suppose $\mathscr{E}$ an $\infty$-topos. The inclusion functor $${{{\mathrm{Add}}(\mathscr{E})}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Thy}(\mathscr{E})}}$$ admits a left adjoint $D$ given by a Goodwillie differential [@MR1076523; @MR1162445; @MR2026544] $$D\phi\simeq\underset{n\to\infty}{\operatorname{colim}}\ \Omega_{\mathscr{E}}^n\circ\Phi\circ\mathscr{S}^n\circ j,$$ where $\Phi\colon{{{\mathrm{D}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}}}$ is the left derived functor of $\phi$.
Let us write $\mathscr{F}$ for the class of small filtered colimits. By [@MR2026544 Th. 1.8] or [@HA Cor. 7.1.1.10], the inclusion $${{\operatorname{Exc}_{\mathscr{F}}({\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})},\mathscr{E})}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}^{\ast}_{\mathscr{F}}({\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})},\mathscr{E})}}$$ (Df. \[dfn:reducedtheories\]) admits a left adjoint given by the assignment $${{\Phi}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {\underset{n\to\infty}{\operatorname{colim}}\ \Omega_{\mathscr{E}}^n\circ\Phi\circ\Sigma_{{\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}}^n}}.$$ Now the inclusion $i\colon{{{\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {{\mathrm{D}(\mathbf{Wald}_{\infty})}}}$ induces a left adjoint $${{\operatorname{Fun}^{\ast}_{\mathscr{F}}({\mathrm{D}(\mathbf{Wald}_{\infty})},\mathscr{E})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}^{\ast}_{\mathscr{F}}({\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})},\mathscr{E})}}$$ to the forgetful functor induced by $L^{{\mathrm{fiss}}}$. By composing these adjoints, we thus obtain a left adjoint ${\mathbf{D}}$ to the forgetful functor $${{\operatorname{Exc}_{\mathscr{F}}({\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})},\mathscr{E})}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}^{\ast}_{\mathscr{F}}({\mathrm{D}(\mathbf{Wald}_{\infty})},\mathscr{E})}}.$$ The left adjoint ${\mathbf{D}}$ is given by the assignment $${{\Phi}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {\underset{n\to\infty}{\operatorname{colim}}\ \Omega_{\mathscr{E}}^n\circ\Phi\circ i\circ\Sigma_{{\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}}^n}}.$$ By Cor. \[cor:Sdotisreallysuspension\], if $n\geq 1$, then one may rewrite the functor $\Omega_{\mathscr{E}}^n\circ F\circ i\circ\Sigma_{{\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}}^n$ as $$\Omega_{\mathscr{E}}^n\circ\Phi\circ i\circ\Sigma_{{\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}}^n\circ L^{{\mathrm{fiss}}}\circ i\simeq\Omega_{\mathscr{E}}^n\circ\Phi\circ i\circ\mathscr{S}^n.$$ Now if $\Phi\colon{{{\mathrm{D}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}}}$ is the left derived functor of a theory, then for any virtual Waldhausen $\infty$-category $\mathscr{Y}$, since $\Phi$ is pointed, and since $\mathscr{S}(\mathscr{Y})$ is the colimit of a simplicial virtual Waldhausen $\infty$-category ${\mathbf{S}}_{\ast}(\mathscr{Y})$ with ${\mathbf{S}}_0(\mathscr{Y})\simeq 0$, the object $\Phi(\mathscr{S}(\mathscr{Y}))$ is connected as well. By Lm \[lem:loopspreservesiftedcolims\], $\Omega_{\mathscr{E}}$ commutes with sifted colimits of connected objects of $\mathscr{E}$, whence it follows that the restriction of ${\mathbf{D}}\colon{{\operatorname{Fun}_{\mathscr{F}}^{\star}({\mathrm{D}(\mathbf{Wald}_{\infty})},\mathscr{E})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Exc}_{\mathscr{F}}({\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})},\mathscr{E})}}$ to $$\operatorname{Thy}(\mathscr{E})\simeq\operatorname{Fun}_{\mathscr{G}}^{\star}({\mathrm{D}(\mathbf{Wald}_{\infty})},\mathscr{E})\subset\operatorname{Fun}_{\mathscr{F}}^{\star}({\mathrm{D}(\mathbf{Wald}_{\infty})},\mathscr{E})$$ in fact factors through the full subcategory $$\operatorname{Exc}_{\mathscr{G}}({\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})},\mathscr{E})\subset\operatorname{Exc}_{\mathscr{F}}({\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})},\mathscr{E}).$$ Thanks to Th. \[thm:addfuncsareexc\], the functor ${\mathbf{D}}$ consequently descends to a functor $$D\colon{{\operatorname{Thy}(\mathscr{E})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathrm{Add}}(\mathscr{E})}}$$ given by the assignment $${{\Phi}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {\underset{n\to\infty}{\operatorname{colim}}\ \Omega_{\mathscr{E}}^n\circ\Phi\circ\Sigma_{{\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}}^n\circ L^{{\mathrm{fiss}}}\circ j}}.$$ Now another application of Cor. \[cor:Sdotisreallysuspension\] completes the proof.
The left adjoint $$D\colon{{\operatorname{Thy}(\mathscr{E})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathrm{Add}}(\mathscr{E})}}$$ of the previous corollary will be called the *additivization*.
Suppose $\phi\colon{{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}}}$ a theory; denote by $\Phi$ its left derived functor. For any virtual Waldhausen $\infty$-category $\mathscr{Y}$ and any natural number $n$, since the virtual Waldhausen $\infty$-category $\mathscr{S}^n(\mathscr{Y})$ is the colimit of a reduced $n$-simplicial diagram ${\mathbf{S}}_{\ast}({\mathbf{S}}_{\ast}(\cdots{\mathbf{S}}_{\ast}(\mathscr{Y})\cdots))$, it follows that the object $\Phi(\mathscr{S}^n(\mathscr{Y}))$ is $n$-connected. This proves the following.
\[prp:deloopingofdfisconnective\] The canonical delooping of the additivization $D\phi$ of a theory $\phi\colon{{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}_{\ast}}}$ is valued in connective spectra: $${{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Stab}(\mathscr{E})_{\geq0}}}.$$
Pre-additive theories {#pre-additive-theories .unnumbered}
---------------------
We have already mentioned that many of the theories that arise in practice have the property that they carry direct sums of Waldhausen $\infty$-categories to products. What’s really useful about theories $\phi$ that enjoy this property is that the colimit $$\operatorname{colim}[\phi\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ \Omega\circ\Phi\circ\mathscr{S}\circ j\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ \ \Omega^{2}\circ\Phi\circ\mathscr{S}^{2}\circ j\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ \cdots]$$ that appears in the formula for the additivization (Th. \[thm:additivizationexists\]) stabilizes after the first term; that is, only one loop is necessary to get an additive theory.
\[dfn:preadditive\] Suppose $\mathscr{E}$ an $\infty$-topos. Then a theory $\phi\in\operatorname{Thy}(\mathscr{E})$ is said to be ***pre-additive*** if it carries direct sums of Waldhausen $\infty$-categories to products in $\mathscr{E}$.
Suppose $\mathscr{E}$ an $\infty$-topos, and suppose $\phi\in\operatorname{Thy}(\mathscr{E})$ a pre-additive theory with left derived functor $\Phi$. Then the morphisms $${{\Phi(\mathscr{S}(\mathscr{F}_m(\mathscr{C})))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Phi(\mathscr{S}(\mathscr{C}))}}\textrm{\quad and\quad}{{\Phi(\mathscr{S}(\mathscr{F}_m(\mathscr{C})))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Phi(\mathscr{S}(\mathscr{S}_m(\mathscr{C})))}}$$ induced by $I_{m,0}$ and $F_m$ together exhibit $\Phi(\mathscr{S}(\mathscr{F}_m(\mathscr{C})))$ as a product of $\Phi(\mathscr{S}(\mathscr{C}))$ and $\Phi(\mathscr{S}(\mathscr{S}_m(\mathscr{C})))$.
Since $\phi$ is pre-additive, the morphism from $\Phi(\mathscr{S}(\mathscr{F}_m(\mathscr{C})))$ to the desired product may be identified with the morphism $${{\Phi(\mathscr{S}(\mathscr{F}_m(\mathscr{C})))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Phi(\mathscr{S}(\mathscr{C})\oplus\mathscr{S}(\mathscr{S}_m(\mathscr{C})))}},$$ which can in turn be identified with the natural morphism $${{\Phi(i\circ\Sigma_{{\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}}\circ L^{{\mathrm{fiss}}}(\mathscr{F}_m(\mathscr{C})))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Phi(i\circ\Sigma_{{\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}}\circ L^{{\mathrm{fiss}}}(\mathscr{C}\oplus\mathscr{S}_m(\mathscr{C})))}}$$ by Cor. \[cor:Sdotisreallysuspension\]. The upper right corner of is a pushout, and since $E_m\oplus J_m$ is a section of $I_{m,0}\oplus F_m$, the natural morphism ${{L^{{\mathrm{fiss}}}(\mathscr{F}_m(\mathscr{C}))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {L^{{\mathrm{fiss}}}(\mathscr{C}\oplus\mathscr{S}_m(\mathscr{C}))}}$ is an equivalence.
By Th. \[thm:additiveequiv\], we obtain the following repackaging of Waldhausen’s Additivity Theorem.
\[for:preaddsareeasy\] Suppose $\mathscr{E}$ an $\infty$-topos, and suppose $\phi\in\operatorname{Thy}(\mathscr{E})$ a pre-additive theory with left derived functor $\Phi$. Then the additivization is given by $$D\phi\simeq\Omega\circ\Phi\circ\mathscr{S}\circ j.$$
Suppose $\mathscr{E}$ an $\infty$-topos, and suppose $\phi\in\operatorname{Thy}(\mathscr{E})$ a pre-additive theory. Then the counit ${{\phi}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {D\phi}}$ is the initial object of the $\infty$-category ${\mathrm{Add}}(\mathscr{E})\times_{\operatorname{Thy}(\mathscr{E})}\operatorname{Thy}(\mathscr{E})_{\phi/}$. By Th. \[thm:additiveequiv\], this means that $D\phi$ is the initial object of the full subcategory of $\operatorname{Thy}(\mathscr{E})_{\phi/}$ spanned by those natural transformations ${{\phi}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\phi'}}$ such that for any Waldhausen $\infty$-category $\mathscr{C}$ and for any functor ${\mathbf{S}}_{\ast}(\mathscr{C})\colon{{N\Delta^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}}}$ that classifies the Waldhausen cocartesian fibration ${{\mathscr{S}(\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}}}$, the induced functor $\phi'\circ{\mathbf{S}}_{\ast}(\mathscr{C})\colon{{N\Delta^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}_{\ast}}}$ is a group object.
Motivated by this, we may now note that the inclusion of the full subcategory ${\mathbf{Grp}}(\mathscr{E})$ of $\operatorname{Fun}(N\Delta^{{\mathrm{op}}},\mathscr{E})$ spanned by the group objects admits a left adjoint $L$. (It is an straightforward matter to note that ${\mathbf{Grp}}(\mathscr{E})\subset\operatorname{Fun}(N\Delta^{{\mathrm{op}}},\mathscr{E})$ is stable under arbitrary limits and filtered colimits; alternatively, one may find a small set $S$ of morphisms of $\operatorname{Fun}(N\Delta^{{\mathrm{op}}},\mathscr{E})$ such that a simplicial object $X$ of $\mathscr{E}$ is a group object if and only if $X$ is $S$-local.) Hence one may consider the following composite functor $L^{\phi}_{\ast}$: $${\mathbf{Wald}_{\infty}}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{${\mathbf{S}}_{\ast}$}(0.75,0.5ex);\ \operatorname{Fun}(N\Delta^{{\mathrm{op}}},{\mathbf{Wald}_{\infty}})\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$\phi$}(0.75,0.5ex);\ \operatorname{Fun}(N\Delta^{{\mathrm{op}}},\mathscr{E})\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$L$}(0.75,0.5ex);\ {\mathbf{Grp}}(\mathscr{E}).$$ If $\operatorname{ev}_1\colon{{{\mathbf{Grp}}(\mathscr{E})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}}}$ is the functor given by evaluation at $1$, then the functor $\operatorname{ev}_1\circ L$ may be identified with the functor $\Omega_{\mathscr{E}}\circ\operatorname{colim}_{N\Delta^{{\mathrm{op}}}}$. It therefore follows from the previous corollary that the functor $L^{\phi}_1=\operatorname{ev}_1\circ L^{\phi}_{\ast}$ can be identified with the additivization of $\phi$. This provides us with a *local* recognition principle for $D\phi$.
\[prp:localrecogofDf\] Suppose $\mathscr{E}$ an $\infty$-topos, suppose $\phi\in\operatorname{Thy}(\mathscr{E})$ a pre-additive theory, and suppose $\mathscr{C}$ a Waldhausen $\infty$-category. Write $${\mathbf{S}}_{\ast}(\mathscr{C})\colon{{N\Delta^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}}}$$ for the functor that classifies the Waldhausen cocartesian fibration ${{\mathscr{S}(\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}}}$. Then the object $D\phi(\mathscr{C})$ is canonically equivalent to underlying object of the group object that is initial in the $\infty$-category $${\mathbf{Grp}}(\mathscr{E})\times_{\operatorname{Fun}(N\Delta^{{\mathrm{op}}},\mathscr{E})}\operatorname{Fun}(N\Delta^{{\mathrm{op}}},\mathscr{E})_{\phi\circ{\mathbf{S}}_{\ast}(\mathscr{C})/}.$$
One may hope to study the rest of the Taylor tower of a theory. In particular, for any positive integer $n$ and any theory $\phi\in\operatorname{Thy}(\mathscr{E})$, one may define a symmetric “multi-additive” theory $D^{(n)}\phi$ via a formula $$D^{(n)}\phi(\mathscr{C}_1,\dots,\mathscr{C}_n)=\underset{(j_1,\dots,j_n)}{\operatorname{colim}}\ \Omega_{\mathscr{E}}^{j_1+\cdots+j_n}\operatorname{cr}_n\Phi(\mathscr{S}^{j_1}\mathscr{C}_1,\dots,\mathscr{S}^{j_n}\mathscr{C}_n),$$ where $\Phi$ is the left derived functor of $\phi$, and $\operatorname{cr}_n\Phi$ is the $n$-th cross-effect functor of the restriction of $\Phi$ to ${\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}$. However, if $\phi$ is pre-additive, then for $n\geq 2$, the cross-effect functor $\operatorname{cr}_n\Phi$ vanishes, whence $D^{(n)}\phi$ vanishes as well. As a result, the Taylor tower for $\Phi$ is constant above the first level. More informally, the best polynomial approximation to $\Phi$ is linear. Consequently, if $\phi\colon{{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}_{\ast}}}$ is pre-additive, then $\Phi$ factors through an $n$-excisive functor ${{{\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}_{\ast}}}$ for some $n\geq 1$ if and only if $\phi$ is an additive theory, in which case $n$ may be allowed to be $1$. This seems to suggest a rather peculiar dichotomy: a pre-additive theory is either additive or staunchly non-analytic.
Easy consequences of additivity
===============================
Additivity theories, which we introduced in the last section, are quite special. In this section, we’ll prove some simple results that will illustrate just how special they really are. We’ll show that additive theories vanish on any Waldhausen $\infty$-categories that are “too large” (Pr. \[prp:Eilenbergswindle\]), and we’ll show that additive functors do not distinguish between Waldhausen $\infty$-categories whose pair structure is maximal and suitable stable $\infty$-categories extracted from them. As a side note, we’ll remark that, rather curiously, the fissile derived $\infty$-category is only one loop away from being stable. Finally, and most importantly, we’ll prove our $\infty$-categorical variant of Waldhausen’s Fibration Theorem. In the next section, we’ll introduce a richer structure into this story, to prove a more useful variant of this result.
The Eilenberg Swindle {#the-eilenberg-swindle .unnumbered}
---------------------
We now show that Waldhausen $\infty$-categories with “too many” coproducts are invisible to additive theories.
\[prp:Eilenbergswindle\] Suppose $\mathscr{E}$ an $\infty$-topos, and suppose $\phi\in{\mathrm{Add}}(\mathscr{E})$. Then for any Waldhausen $\infty$-category $\mathscr{C}$ that admits countable coproducts, $\phi(\mathscr{C})$ is terminal in $\mathscr{E}$.
Denote by $I$ the set of natural numbers, regarded as a discrete $\infty$-category, and denote by $\psi\colon{{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$ the composite of the constant functor ${{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}(I,\mathscr{C})}}$ followed by its left adjoint ${{\operatorname{Fun}(I,\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$. The inclusion ${{\{0\}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {I}}$ and the successor bijection $\sigma\colon{{I}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {I-\{0\}}}$ together specify a natural cofibration ${{\operatorname{id}}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ {\psi}}$. This defines an exact functor ${{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{F}_1(\mathscr{C})}}$. Applying $I_{1,1}$ and $I_{1,0}\oplus F_1$ to this functor, we find that $\phi(\psi)=\phi(\operatorname{id})+\phi(\psi)$, whence $\phi(\operatorname{id})=0$.
Stabilization and approximation {#stabilization-and-approximation .unnumbered}
-------------------------------
We prove that the value of an additive theory on a Waldhausen $\infty$-category whose pair structure is maximal agrees with its value on a certain stable $\infty$-category. Using this, we show that for these Waldhausen $\infty$-categories, equivalences on the homotopy category suffice to give equivalences under any additive theory.
\[prp:suspension\] Suppose $\mathscr{A}$ a Waldhausen $\infty$-category whose pair structure is maximal. Then for any additive theory $\phi\in{\mathrm{Add}}(\mathscr{E})$, the suspension functor $\Sigma\colon{{\mathscr{A}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{A}}}$ induces multiplication by $-1$ on the group object $\phi(\mathscr{A})$.
This follows directly from the existence of the pushout square of endofunctors of $\mathscr{A}$ $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\operatorname{id}&0\\
0&\Sigma.\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge (m-2-2);
\end{tikzpicture}\qedhere$$
\[cor:stabilization\] Suppose $\mathscr{A}$ a Waldhausen $\infty$-category whose pair structure is maximal. Write $\widetilde{{\mathbf{Sp}}}(\mathscr{A})$ for the colimit $$\mathscr{A}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$\Sigma$}(0.5,0.5ex);\ \mathscr{A}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$\Sigma$}(0.5,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$\Sigma$}(0.5,0.5ex);\ \mathscr{A}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$\Sigma$}(0.5,0.5ex);\ \cdots$$ in ${\mathbf{Wald}_{\infty}}$. Then for any additive theory $\phi\in{\mathrm{Add}}(\mathscr{E})$, the canonical functor $$\Sigma^{\infty}\colon{{\mathscr{A}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\widetilde{{\mathbf{Sp}}}(\mathscr{A})}}$$ induces an equivalence ${{\phi(\mathscr{A})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\phi(\widetilde{{\mathbf{Sp}}}(\mathscr{A}))}}$.
We now obtain the following corolary, which we can regard as a version of Waldhausen’s Approximation Theorem. Very similar results appear in work of Cisinski [@MR2746284 Th. 2.15] and Blumberg–Mandell [@MR2764905 Th. 1.3], and an interesting generalization has recently appeared in a preprint of Fiore [@fiore].
\[prp:approx\] Suppose $\mathscr{C}$ and $\mathscr{D}$ two $\infty$-categories that each contain zero objects and all finite colimits, and regard them as Waldhausen $\infty$-categories equipped with the maximal pair structure . Then any exact functor $\psi\colon{{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{D}}}$ that induces an equivalence of homotopy categories ${{h\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {h\mathscr{D}}}$ also induces an equivalence ${{\psi(\mathscr{A})}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\psi(\mathscr{B})}}$ for any additive theory $\phi\in{\mathrm{Add}}(\mathscr{E})$.
We note that since the homotopy category functor ${{\mathscr{C}}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {h\mathscr{C}}}$ preserves colimits, the induced functor ${{h\widetilde{{\mathbf{Sp}}}(\mathscr{A})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {h\widetilde{{\mathbf{Sp}}}(\mathscr{B})}}$ is an equivalence. Now we combine Prs. \[cor:stabilization\] and \[prp:preapprox\].
The $\infty$-category $\widetilde{{\mathbf{Sp}}}(\mathscr{A})$ is not always the stabilization of $\mathscr{A}$, but when $\mathscr{A}$ is idempotent complete, it is.
\[prp:costabisstab\] Suppose $\mathscr{A}$ an idempotent complete $\infty$-category that contains a zero object and all finite colimits. Regard $\mathscr{A}$ as a Waldhausen $\infty$-category with its maximal pair structure. Then $\widetilde{{\mathbf{Sp}}}(\mathscr{A})$ is equivalent to the stabilization ${\mathbf{Sp}}(\mathscr{A})$ of $\mathscr{A}$.
The colimit of the sequence $$\mathscr{A}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$\Sigma$}(0.5,0.5ex);\ \mathscr{A}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$\Sigma$}(0.5,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$\Sigma$}(0.5,0.5ex);\ \mathscr{A}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$\Sigma$}(0.5,0.5ex);\ \cdots$$ in ${\mathbf{Wald}_{\infty}}$ agrees with the same colimit taken in ${\mathbf{Cat}}_{\infty}(\kappa_1)^{\mathrm{Rex}}$ by [@HTT Pr. 5.5.7.11] and Pr. \[thm:Waldfiltcolims\]. Since $\operatorname{Ind}$ is a left adjoint [@HTT Pr. 5.5.7.10], the colimit of the sequence $$\operatorname{Ind}\mathscr{A}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$\Sigma$}(0.5,0.5ex);\ \operatorname{Ind}\mathscr{A}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$\Sigma$}(0.5,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$\Sigma$}(0.5,0.5ex);\ \operatorname{Ind}\mathscr{A}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$\Sigma$}(0.5,0.5ex);\ \cdots$$ in ${\mathbf{Pr}}^{\mathrm{L}}_{\omega}$ is $\operatorname{Ind}(\widetilde{{\mathbf{Sp}}}(\mathscr{A}))$. By [@HTT Nt. 5.5.7.7], there is an equivalence between ${\mathbf{Pr}}^{\mathrm{L}}_{\omega}$ and $({\mathbf{Pr}}^{\mathrm{R}}_{\omega})^{{\mathrm{op}}}$, whence $\operatorname{Ind}(\widetilde{{\mathbf{Sp}}}(\mathscr{A}))$ can be identified with the limit of the sequence $$\cdots\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$\Omega$}(0.5,0.5ex);\ \operatorname{Ind}\mathscr{A}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$\Omega$}(0.5,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$\Omega$}(0.5,0.5ex);\ \operatorname{Ind}\mathscr{A}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$\Omega$}(0.5,0.5ex);\ \operatorname{Ind}\mathscr{A}$$ in ${\mathbf{Pr}}^{\mathrm{R}}_{\omega}$. Since the inclusion ${{{\mathbf{Pr}}^{\mathrm{R}}_{\omega}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}(\kappa_1)}}$ preserves limits [@HTT Pr. 5.5.7.6], it follows that $\operatorname{Ind}(\widetilde{{\mathbf{Sp}}}(\mathscr{A}))\simeq{\mathbf{Sp}}(\operatorname{Ind}(\mathscr{A}))$. Now the functor ${{C}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C^{\omega}}}$ is an equivalence of $\infty$-categories between ${\mathbf{Pr}}^{\mathrm{R}}_{\omega}$ and the full subcategory of ${\mathbf{Cat}}_{\infty}(\kappa_1)^{\mathrm{Lex}}$ spanned by the essentially small, idempotent complete $\infty$-categories, whence it follows that $$\widetilde{{\mathbf{Sp}}}(\mathscr{A})\simeq\operatorname{Ind}(\widetilde{{\mathbf{Sp}}}(\mathscr{A}))^{\omega}\simeq{\mathbf{Sp}}(\operatorname{Ind}(\mathscr{A}))^{\omega}\simeq{\mathbf{Sp}}(\mathscr{A}).\qedhere$$
\[exm:stabinftytopoi\] Suppose $\mathscr{E}$ an $\infty$-topos. (One may, again, think of $\operatorname{Fun}(X,{\mathbf{Kan}})$ for a simplicial set $X$.) For any additive theory $\phi$, the results above show that one has an equivalence $$\phi(\mathscr{E}^{\omega}_{\ast})\simeq\phi({\mathbf{Sp}}(\mathscr{E}^{\omega})).$$
Digression: the near-stability of the fissile derived $\infty$-category {#digression-the-near-stability-of-the-fissile-derived-infty-category .unnumbered}
-----------------------------------------------------------------------
By analyzing the additivization of the Yoneda embedding, we now find that a every distributive virtual Waldhausen $\infty$-category one step away from being an infinite loop object. This implies that the $\infty$-category ${\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}$ can be said to admit a much stronger form of the Blakers–Massey excision theorem than the $\infty$-category of spaces. Armed with this, we give an easy necessary and sufficient criterion for a morphism of virtual Waldhausen $\infty$-categories to induce an equivalence on every additive theory.
We shall call a theory $\phi\in\operatorname{Thy}(\mathscr{E})$ ***left exact*** just in case its left derived functor $\Phi$ preserves finite limits.
Clearly every left exact theory is pre-additive. Moreover, the best excisive approximation $P_1(G\circ F)$ to the composite $G\circ F$ of a suitable functor $F\colon{{C}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {D}}$ with a functor $G\colon{{D}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {D'}}$ that preserves finite limits is simply the composite $G\circ P_1(F)$. Accordingly, we have the following.
\[prp:OmegaSigmaisadditive\] Suppose $\phi\in\operatorname{Thy}(\mathscr{E})$ a left exact theory. Then $$D\phi\simeq\Phi\circ\Omega_{{\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}}\circ\mathscr{S}.$$
The Yoneda embedding $y\colon{{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{P}({\mathbf{Wald}_{\infty}}^{\omega})}}$ is a left exact theory; its left derived functor $Y\colon{{{\mathrm{D}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{P}({\mathbf{Wald}_{\infty}}^{\omega})}}$ is simply the canonical inclusion. Consequently, thanks to Cor. \[for:preaddsareeasy\], the additivization of $y$ is now given by the formula $$Dy\simeq\Omega\circ\mathscr{S}\circ j.$$
Let’s give some equivalent descriptions of the functor $Dy$. Since $\mathscr{F}(\mathscr{C})$ is contractible, one may write $$Dy(\mathscr{C})\simeq\mathscr{F}(\mathscr{C})\times_{\mathscr{S}(\mathscr{C})}\mathscr{F}(\mathscr{C}).$$ Alternately, since suspension in ${\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}$ is given by $\mathscr{S}$, the functor $$Dy(\mathscr{C})\colon{{{\mathbf{Wald}_{\infty}}^{\omega,{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$$ can be described by the formula $$Dy(\mathscr{C})(\mathscr{D})\simeq\operatorname{Map}_{{\mathrm{D}(\mathbf{Wald}_{\infty})}}(\mathscr{S}(\mathscr{D}),\mathscr{S}(\mathscr{C})).$$
In other words, $\Omega\Sigma\simeq\Omega\mathscr{S}$ is the Goodwillie differential of the identity on ${\mathrm{D}_{\mathrm{fiss}}(\mathbf{Wald}_{\infty})}$.
Waldhausen’s Generic Fibration Theorem {#waldhausens-generic-fibration-theorem .unnumbered}
--------------------------------------
Let’s now examine the circumstances under which a sequence of virtual Waldhausen $\infty$-categories gives rise to a fiber sequence under any additive functor. In this direction we have Pr. \[prp:fibthmi\] below, which is an analogue of Waldhausen’s [@MR86m:18011 Pr. 1.5.5 and Cor. 1.5.7]. We will deduce from this a necessary and sufficient condition for an exact functor to induce an equivalence under *every* additive theory (Pr. \[cor:invertedbyeveryaddthy\]).
Suppose $\psi\colon{{\mathscr{B}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{A}}}$ an exact functor of Waldhausen $\infty$-categories. Write $$\mathscr{K}(\psi){\mathrel{\mathop:}=}|\mathscr{F}(\mathscr{A})\times_{\mathscr{S}(\mathscr{A})}\mathscr{S}(\mathscr{B})|_{N\Delta^{{\mathrm{op}}}}$$ for the realization (Df. \[dfn:realization\]) of the Waldhausen cocartesian fibration $${{\mathscr{F}(\mathscr{A})\times_{\mathscr{S}(\mathscr{A})}\mathscr{S}(\mathscr{B})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}}}.$$
In other words, the virtual Waldhausen $\infty$-category $\mathscr{K}(\psi)$ is the geometric realization of the simplicial Waldhausen $\infty$-category whose $m$-simplices consist of a totally filtered object $$0\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.75,0.5ex);\ U_1\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.75,0.5ex);\ U_2\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.75,0.5ex);\ \dots\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.75,0.5ex);\ U_m$$ of $\mathscr{B}$, a filtered object $$X_0\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.75,0.5ex);\ X_1\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.75,0.5ex);\ X_2\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.75,0.5ex);\ \dots\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.75,0.5ex);\ X_m$$ of $\mathscr{A}$, and a diagram $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{X_0&X_1&X_2&\dots&X_m\\
0&\psi(U_1)&\psi(U_2)&\dots&\psi(U_m)\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge[>->] (m-1-2)
edge (m-2-1)
(m-1-2) edge[>->] (m-1-3)
edge (m-2-2)
(m-1-3) edge[>->] (m-1-4)
edge (m-2-3)
(m-1-4) edge[>->] (m-1-5)
(m-1-5) edge (m-2-5)
(m-2-1) edge[>->] (m-2-2)
(m-2-2) edge[>->] (m-2-3)
(m-2-3) edge[>->] (m-2-4)
(m-2-4) edge[>->] (m-2-5);
\end{tikzpicture}$$ of $\mathscr{A}$ in which every square is a pushout.
The object $\mathscr{K}(\psi)$ is not itself the corresponding fiber product of virtual Waldhausen $\infty$-categories; however, for any additive functor $\phi\colon{{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}}}$ with left derived functor $\Phi$, we shall now show that $\Phi(\mathscr{K}(\psi))$ is in fact the fiber of the induced morphism ${{\Phi(\mathscr{S}(\mathscr{A}))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Phi(\mathscr{S}(\mathscr{B}))}}$.
\[prp:fibthmi\] Suppose $\psi\colon{{\mathscr{B}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{A}}}$ an exact functor of Waldhausen $\infty$-categories. Then for any additive theory $\phi\colon{{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}}}$ with left derived functor $\Phi$, there is a diagram $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\phi(\mathscr{B})&\phi(\mathscr{A})&\ast\\
\ast&\Phi(\mathscr{K}(\psi))&\Phi(\mathscr{S}(\mathscr{B})).\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
edge (m-1-3)
(m-1-3) edge (m-2-3)
(m-2-1) edge (m-2-2)
(m-2-2) edge (m-2-3);
\end{tikzpicture}$$ of $\mathscr{E}$ in which each square is a pullback.
For any vertex $\mathbf{m}\in N\Delta^{{\mathrm{op}}}$, there exist functors $$s{\mathrel{\mathop:}=}(E_m\oplus\mathscr{S}_m(\psi),\operatorname{pr}_2)\colon{{\mathscr{F}_0(\mathscr{A})\oplus\mathscr{S}_m(\mathscr{B})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{F}_m(\mathscr{A})\times_{\mathscr{S}_m(\mathscr{A})}\mathscr{S}_m(\mathscr{B})}}$$ and $$p{\mathrel{\mathop:}=}(I_{m,0}\circ\operatorname{pr}_1)\oplus\operatorname{pr}_2\colon{{\mathscr{F}_m(\mathscr{A})\times_{\mathscr{S}_m(\mathscr{A})}\mathscr{S}_m(\mathscr{B})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{F}_0(\mathscr{A})\oplus\mathscr{S}_m(\mathscr{B})}}.$$ Clearly $p\circ s\simeq\operatorname{id}$; we claim that $\phi(s\circ p)\simeq\phi(\operatorname{id})$ in $\mathscr{E}_{\ast}$. This follows from additivity applied to the functor $${{\mathscr{F}_m(\mathscr{A})\times_{\mathscr{S}_m(\mathscr{A})}\mathscr{S}_m(\mathscr{B})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{F}_1(\mathscr{F}_m(\mathscr{A})\times_{\mathscr{S}_m(\mathscr{A})}\mathscr{S}_m(\mathscr{B}))}}$$ given by the cofibration of functors ${{(E_m\circ I_{m,0}\circ\operatorname{pr}_1,0)}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{id}}}$. Thus the value $\phi(\mathscr{F}_m(\mathscr{A})\times_{\mathscr{S}_m(\mathscr{A})}\mathscr{S}_m(\mathscr{B}))$ is exhibited as the product $\phi(\mathscr{F}_0(\mathscr{A}))\times\phi(\mathscr{S}_m(\mathscr{B}))$.
We may therefore consider the following commutative diagram of $\mathscr{E}_{\ast}$: $$\begin{tikzpicture}[baseline]
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=5ex,
text height=1.5ex, text depth=0.25ex]
{\phi(\mathscr{F}_0(\mathscr{B}))&\phi(\mathscr{F}_0(\mathscr{A}))&\phi(\mathscr{S}_0(\mathscr{B}))\\
\phi(\mathscr{F}_m(\mathscr{B}))&\phi(\mathscr{F}_m(\mathscr{A})\times_{\mathscr{S}_m(\mathscr{A})}\mathscr{S}_m(\mathscr{B}))&\phi(\mathscr{S}_m(\mathscr{B}))\\
&\phi(\mathscr{F}_m(\mathscr{A}))&\phi(\mathscr{S}_m(\mathscr{A}))\\
&\phi(\mathscr{F}_0(\mathscr{A}))&\phi(\mathscr{S}_0(\mathscr{A})).\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge node[above]{$\mathscr{F}_0(\psi)$} (m-1-2)
edge node[left]{$E_m$} (m-2-1)
(m-1-2) edge node[above]{$F_0$} (m-1-3)
edge (m-2-2)
(m-1-3) edge node[right]{$E'_m$} (m-2-3)
(m-2-1) edge node[below]{$(0,F_m)$} (m-2-2)
(m-2-2) edge node[above]{$\operatorname{pr}_1$} (m-2-3)
edge node[left]{$\operatorname{pr}_2$} (m-3-2)
(m-2-3) edge node[right]{$\mathscr{S}_m(\psi)$} (m-3-3)
(m-3-2) edge node[below]{$F_m$} (m-3-3)
edge node[left]{$I_{m,0}$} (m-4-2)
(m-3-3) edge node[right]{$I'_{m,0}$} (m-4-3)
(m-4-2) edge node[below]{$F_0$} (m-4-3);
\end{tikzpicture}$$ The lower right-hand square is a pullback square by additivity; hence, in light of the identification above, all the squares on the right hand side are pullbacks as well. Again by additivity the wide rectangle of the top row is carried to a pullback square under $\phi$, whence all the squares of this diagram are carried to pullback squares.
Since $\phi$ is additive, so is $\Phi\circ\mathscr{S}$. Hence we obtain a commutative diagram in $\mathscr{E}_{\ast}$: $$\begin{tikzpicture}[baseline]
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\Phi(\mathscr{S}(\mathscr{F}_0(\mathscr{B})))&\Phi(\mathscr{S}(\mathscr{F}_0(\mathscr{A})))&\Phi(\mathscr{S}(\mathscr{S}_0(\mathscr{B})))\\
\Phi(\mathscr{S}(\mathscr{F}_m(\mathscr{B})))&\Phi(\mathscr{S}(\mathscr{F}_m(\mathscr{A})\times_{\mathscr{S}_m(\mathscr{A})}\mathscr{S}_m(\mathscr{B})))&\Phi(\mathscr{S}(\mathscr{S}_m(\mathscr{B})))\\
&\Phi(\mathscr{S}(\mathscr{F}_m(\mathscr{A})))&\Phi(\mathscr{S}(\mathscr{S}_m(\mathscr{A}))),\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-1-3)
edge (m-2-2)
(m-1-3) edge (m-2-3)
(m-2-1) edge (m-2-2)
(m-2-2) edge (m-2-3)
edge (m-3-2)
(m-2-3) edge (m-3-3)
(m-3-2) edge (m-3-3);
\end{tikzpicture}$$ in which every square is a pullback. All the squares in this diagram are functorial in $\mathbf{m}$, and since the objects that appear are all connected, it follows from [@HA Lm. 5.3.6.17] that the squares of the colimit diagram $$\begin{tikzpicture}[baseline]
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\Phi(\mathscr{S}(\mathscr{F}_0(\mathscr{B})))&\Phi(\mathscr{S}(\mathscr{F}_0(\mathscr{A})))&\Phi(\mathscr{S}(\mathscr{S}_0(\mathscr{B})))\\
\Phi(\mathscr{S}(\mathscr{F}(\mathscr{B})))&\Phi(\mathscr{S}(\mathscr{K}(\psi)))&\Phi(\mathscr{S}(\mathscr{S}(\mathscr{B})))\\
&\Phi(\mathscr{S}(\mathscr{F}(\mathscr{A})))&\Phi(\mathscr{S}(\mathscr{S}(\mathscr{A}))),\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-1-3)
edge (m-2-2)
(m-1-3) edge (m-2-3)
(m-2-1) edge (m-2-2)
(m-2-2) edge (m-2-3)
edge (m-3-2)
(m-2-3) edge (m-3-3)
(m-3-2) edge (m-3-3);
\end{tikzpicture}$$ are all pullbacks. Applying the loopspace functor $\Omega_{\mathscr{E}}$ to this diagram now produces a diagram equivalent to the diagram $$\begin{tikzpicture}[baseline]
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\phi(\mathscr{F}_0(\mathscr{B}))&\phi(\mathscr{F}_0(\mathscr{A}))&\phi(\mathscr{S}_0(\mathscr{B}))\\
\Phi(\mathscr{F}(\mathscr{B}))&\Phi(\mathscr{K}(\psi))&\Phi(\mathscr{S}(\mathscr{B}))\\
&\Phi(\mathscr{F}(\mathscr{A}))&\Phi(\mathscr{S}(\mathscr{A})),\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-1-3)
edge (m-2-2)
(m-1-3) edge (m-2-3)
(m-2-1) edge (m-2-2)
(m-2-2) edge (m-2-3)
edge (m-3-2)
(m-2-3) edge (m-3-3)
(m-3-2) edge (m-3-3);
\end{tikzpicture}$$ in which every square again is a pullback.
\[cor:invertedbyeveryaddthy\] The following are equivalent for an exact functor $\psi\colon{{\mathscr{B}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{A}}}$ of Waldhausen $\infty$-categories.
1. \[item:addequiv\] For any $\infty$-topos $\mathscr{E}$ and any $\phi\in{\mathrm{Add}}(\mathscr{E})$ with left derived functor $$\Phi\colon{{{\mathrm{D}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}}},$$ the induced morphism $\Phi(\psi)\colon{{\Phi(\mathscr{B})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Phi(\mathscr{A})}}$ is an equivalence of $\mathscr{E}$.
2. \[item:gpfiber\] For any $\infty$-topos $\mathscr{E}$ and any $\phi\in{\mathrm{Add}}(\mathscr{E})$ with left derived functor $$\Phi\colon{{{\mathrm{D}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}}},$$ the object $\Phi(\mathscr{K}(\psi))$ is contractible.
3. \[item:SKpsiiszero\] The virtual Waldhausen $\infty$-category $\mathscr{S}(\mathscr{K}(\psi))$ is contractible.
In light of Pr. \[prp:OmegaSigmaisadditive\], if (\[cor:invertedbyeveryaddthy\].\[item:addequiv\]) holds, then the induced morphism $$\Omega\mathscr{S}(\psi)\colon{{\Omega\mathscr{S}(\mathscr{B})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Omega\mathscr{S}(\mathscr{A})}}$$ is an equivalence of virtual Waldhausen $\infty$-categories. Since $\mathscr{S}(\mathscr{B})$ and $\mathscr{S}(\mathscr{A})$ are connected objects of $\mathscr{P}({\mathbf{Wald}_{\infty}}^{\omega})$, this in turn implies (using, say, [@HA Cor. 5.1.3.7]) that the induced morphism of virtual Waldhausen $\infty$-categories $$\mathscr{S}(\psi)\colon{{\mathscr{S}(\mathscr{B})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{S}(\mathscr{A})}}$$ is an equivalence and therefore by Pr. \[prp:fibthmi\] that (\[cor:invertedbyeveryaddthy\].\[item:gpfiber\]) holds.
Now if (\[cor:invertedbyeveryaddthy\].\[item:gpfiber\]) holds, then in particular, $\Omega\mathscr{S}(\mathscr{K}(\psi))$ is contractible. Since the virtual Waldhausen $\infty$-category $\mathscr{S}(\mathscr{K}(\psi))$ is connected, it is contractible, yielding (\[cor:invertedbyeveryaddthy\].\[item:SKpsiiszero\]).
That the last condition implies the first now follows immediately from Pr. \[prp:fibthmi\].
Labeled Waldhausen $\infty$-categories and Waldhausen’s Fibration Theorem {#subsect:labeledWald}
=========================================================================
We have remarked (Ex. \[exm:catwithcofibsisWaldinftycat\]) that nerves of Waldhausen’s categories with cofibrations are natural examples of Waldhausen $\infty$-categories. But Waldhausen’s categories with cofibrations and weak equivalences don’t fit so easily into this story. One may attempt to form the relative nerve (Df. \[dfn:relnerve\]) of the underlying relative category and to endow the resulting $\infty$-category with a suitable pair structure, but part of the point of Waldhausen’s set-up was precisely that one didn’t need to assume things such as the two-out-of-three axiom. For example, Waldhausen considers ([@MR86m:18011 Part 3] or [@WJR:stabparhcob]) categories of spaces in which the weak equivalences are chosen to be the simple maps. In these situations the $K$-theory of the relative nerve will not correctly encode the Waldhausen $K$-theory.
The time has come to address this issue. Fortunately, the machinery we have developed provides a useful alternative. Namely, we introduce the notion of a *labeled Waldhausen $\infty$-category* (Df. \[dfn:labeledWaldcats\]), which is a Waldhausen $\infty$-category equipped with a subcategory of *labeled edges* that satisfy the analogue of Waldhausen’s axioms for a category with cofibrations and weak equivalences. There is a relative form of this, too, as an example, we show how to label Waldhausen cocartusian fibrations of filtered objects.
It is possible to extract from these categories with cofibrations and weak equivalences useful *virtual Waldhausen $\infty$-categories* (Nt. \[ntn:BasavirtWaldcat\]). These virtual Waldhausen $\infty$-categories are constructed as realizations of certain Waldhausen cocartesian fibrations over $N\Delta^{{\mathrm{op}}}$; they are not Waldhausen $\infty$-categories, but they are “close” (Pr. \[prp:BCwCiswFunblankC\]). We also discuss the relationship between the virtual Waldhausen $\infty$-categories attached to a labeled Waldhausen $\infty$-category and the result from formally inverting (in the $\infty$-categorical sense, of course) the labeled edges (Nt. \[ntn:gammaC\]).
The main result of this section is a familiar case of the Generic Fibration Theorem I (Th. \[prp:fibthmi\]). This result (Th. \[thm:fibration\]) gives, for any labeled Waldhausen $\infty$-category $(\mathscr{A},w\mathscr{A})$ satisfying a certain compatibility between the cofibrations and the labeled edges (Df. \[dfn:enoughcofibrations\]) a fiber sequence $$\phi(\mathscr{A}^w)\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ \phi(\mathscr{A})\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ \Phi(\mathscr{B}(\mathscr{A},w\mathscr{A}))$$ for any additive theory $\phi$ with left derived functor $\Phi$. This result is the foundation of virtually all fiber sequences that arise in $K$-theory.
Labeled Waldhausen $\infty$-categories {#labeled-waldhausen-infty-categories .unnumbered}
--------------------------------------
In analogy with Waldhausen’s theory of categories with cofibrations and weak equivalences, we study here Waldhausen $\infty$-categories with certain compatible classes of *labeled morphisms*.
\[dfn:labeledWaldcats\] Suppose $\mathscr{C}$ a Waldhausen $\infty$-category. Then a ***gluing diagram*** in $\mathscr{C}$ is a functor of pairs $$X\colon{{\mathscr{Q}^2\times(\Delta^1)^{\flat}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$$ (Ex. \[exm:minpairmaxpair\] and \[exm:LambdaDelta\]) such that the squares $X|_{(\mathscr{Q}^2\times\Delta^{\{0\}})}$ and $X|_{(\mathscr{Q}^2\times\Delta^{\{1\}})}$ are pushouts. We may depict such gluing diagrams as cubes $$\begin{tikzpicture}[cross line/.style={preaction={draw=white, -,
line width=6pt}}]
\matrix(m)[matrix of math nodes,
row sep=2ex, column sep=1.5ex,
text height=1.5ex, text depth=0.25ex]
{&X_{00}&&X_{10}\\
X_{20}&&X_{\infty0}&\\
&X_{01}&&X_{11}\\
X_{21}&&X_{\infty1}&\\
};
\path[>=stealth,->,font=\scriptsize]
(m-1-2) edge (m-2-1)
edge (m-3-2)
edge[>->] (m-1-4)
(m-3-2) edge (m-4-1)
edge[>->] (m-3-4)
(m-2-1) edge[cross line,>->] (m-2-3)
edge (m-4-1)
(m-1-4) edge (m-2-3)
edge (m-3-4)
(m-4-1) edge[>->] (m-4-3)
(m-3-4) edge (m-4-3)
(m-2-3) edge[cross line] (m-4-3);
\end{tikzpicture}$$ in which the top and bottom faces are pushout squares.
\[dfn:labeling\] A ***labeling*** of a Waldhausen $\infty$-category is a $w\mathscr{C}$ of $\mathscr{C}$ that contains $\iota\mathscr{C}$ (i.e., a a pair structure on $\mathscr{C}$) such that for any gluing diagram $X$ of $\mathscr{C}$ in which the morphisms $${{X_{00}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X_{01}}}\textrm{,\qquad}{{X_{10}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X_{11}}}\textrm{,\quad and\quad}{{X_{20}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X_{21}}}$$ lie in $w\mathscr{C}$, the morphism ${{X_{\infty0}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X_{\infty1}}}$ lies in $w\mathscr{C}$ as well. In this case, the edges of $w\mathscr{C}$ will be called ***labeled edges***, and the pair $(\mathscr{C},w\mathscr{C})$ is called a ***labeled Waldhausen $\infty$-category***.
A ***labeled exact functor*** between two labeled Waldhausen $\infty$-categories $\mathscr{C}$ and $\mathscr{D}$ is an exact functor ${{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{D}}}$ that carries labeled edges to labeled edges.
Note that a labeled Waldhausen $\infty$-category has two pair structures: the cofibrations and the labeled edges.
\[exm:catwithcofibsandwesislabWaldinftycat\] We have remarked (Ex. \[exm:catwithcofibsisWaldinftycat\]) that the nerve of an ordinary *category with cofibrations* in the sense of Waldhausen is a Waldhausen $\infty$-category. Similarly, if $(C,\operatorname{cof}C,wC)$ is a *category with cofibrations and weak equivalences* in the sense of Waldhausen [@MR86m:18011 §1.2], then $(NC,N\operatorname{cof}C,NwC)$ is a labeled Waldhausen $\infty$-category.
Suppose $(\mathscr{C},w\mathscr{C})$ a labeled Waldhausen $\infty$-category. For gluing diagrams $X$ of $\mathscr{C}$ in which the edges $$\begin{aligned}
{{X_{00}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X_{20}}},&&{{X_{00}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X_{01}}},\nonumber\\
{{X_{10}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X_{\infty0}}},&&{{X_{10}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X_{11}}}\nonumber\end{aligned}$$ are all degenerate, the condition above reduces to a guarantee that pushouts of labeled morphism along cofibrations are labeled. For gluing diagrams $X$ of $\mathscr{C}$ in which the edges $$\begin{aligned}
{{X_{00}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X_{10}}},&&{{X_{00}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X_{01}}},\nonumber\\
{{X_{20}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X_{\infty0}}},&&{{X_{20}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X_{21}}}\nonumber\end{aligned}$$ are all degenerate, the condition above reduces to a guarantee that the pushout of any labeled cofibration along any morphism exists and is again a labeled cofibration.
Denote by $${\ell\mathbf{Wald}}\subset{\mathbf{Wald}_{\infty}}\times_{{\mathbf{Cat}}_{\infty}}{\mathbf{Pair}}_{\infty}$$ the full subcategory spanned by the labeled Waldhausen $\infty$-categories.
\[prp:lWaldpres\] The $\infty$-category ${\ell\mathbf{Wald}}$ is presentable.
The inclusion $${{{\ell\mathbf{Wald}}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}\times_{{\mathbf{Cat}}_{\infty}}{\mathbf{Pair}}_{\infty}}}$$ admits a left adjoint, which assigns to any object $(\mathscr{C},\mathscr{C}_{\dag},w\mathscr{C})$ the labeled Waldhausen $\infty$-category $(\mathscr{C},\mathscr{C}_{\dag},\overline{w}\mathscr{C})$, where $\overline{w}\mathscr{C}$ is the smallest labeling containing $w\mathscr{C}$. It is easy to see that ${\ell\mathbf{Wald}}$ is stable under filtered colimits in ${\mathbf{Wald}_{\infty}}\times_{{\mathbf{Cat}}_{\infty}}{\mathbf{Pair}}_{\infty}$; hence ${\ell\mathbf{Wald}}$ is an accessible localization of ${\mathbf{Wald}_{\infty}}\times_{{\mathbf{Cat}}_{\infty}}{\mathbf{Pair}}_{\infty}$. Since the latter $\infty$-category is locally presentable by [@HTT Pr. 5.5.7.6], the proof is complete.
The Waldhausen cocartesian fibration attached to a labeled Waldhausen $\infty$-category {#the-waldhausen-cocartesian-fibration-attached-to-a-labeled-waldhausen-infty-category .unnumbered}
---------------------------------------------------------------------------------------
In §\[sect:filt\], we defined the virtual Waldhausen $\infty$-category of filtered objects of a Waldhausen $\infty$-category $\mathscr{C}$. We did this by first using Pr. \[prp:htt32213\] to write down a cocartesian fibration that is classified by the simplicial $\infty$-category $${\mathbf{F}}_{\ast}(\mathscr{C})\colon{{N\Delta^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}}}$$ such that for any integer $m\geq 0$, the $\infty$-category ${\mathbf{F}}_m(\mathscr{C})$ has as objects sequences of cofibrations $$X_0\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X_1\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X_m.$$ Then we defined the virtual Waldhausen $\infty$-category we were after by forming the formal geometric realization of the diagram ${\mathbf{F}}_{\ast}(\mathscr{C})$.
Here, we introduce an analogous construction when $\mathscr{C}$ admits a labeling, in which the role of the cofibrations is played instead by the labeled edges. That is, we will define a cocartesian fibration ${{\mathscr{B}(\mathscr{C},w\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}}}$ that is classified by the simplicial $\infty$-category $${\mathbf{B}}_{\ast}(\mathscr{C},w\mathscr{C})\colon{{N\Delta^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}}}$$ such that for any integer $m\geq 0$, the $\infty$-category ${\mathbf{B}}_m(\mathscr{C},w\mathscr{C})$ has as objects sequences of labeled edges $$X_0\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--(0.5,0.5ex);\ X_1\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--(0.5,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--(0.5,0.5ex);\ X_m.$$ The pair structure will be simpler than in §\[sect:filt\], but once again we will define the virtual Waldhausen $\infty$-category we’re after by forming the formal geometric realization of the diagram ${\mathbf{B}}_{\ast}(\mathscr{C},w\mathscr{C})$.
\[cnstr:BXSwXs\] Suppose $\mathscr{C}$ a Waldhausen $\infty$-category, and suppose $w\mathscr{C}\subset\mathscr{C}$ a labeling thereof. Define a map ${{\mathscr{B}(\mathscr{C},w\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}}}$, using the notation of Pr. \[prp:htt32213\], Ex. \[exm:minpairmaxpair\], and Nt. \[ntn:M\], as $$\mathscr{B}(\mathscr{C},w\mathscr{C}){\mathrel{\mathop:}=}T_{\pi}((N\Delta^{{\mathrm{op}}})^{\flat}\times(\mathscr{C},w\mathscr{C})).$$ Equivalently, we require, for any simplicial set $K$ and any map $\sigma\colon{{K}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}\times S}}$, a bijection between the set $\operatorname{Mor}_{N\Delta^{{\mathrm{op}}}\times S}(K,\mathscr{B}(\mathscr{C},w\mathscr{C}))$ and the set $$\operatorname{Mor}_{s{\mathbf{Set}}(2)}((K\times_{N\Delta^{{\mathrm{op}}}}N\mathrm{M},K\times_{N\Delta^{{\mathrm{op}}}}(N\mathrm{M})_{\dag}),(\mathscr{C},w\mathscr{C}))$$ (Nt. \[ntn:ordcatofpairs\]), functorial in $\sigma$.
In other words, $\mathscr{B}(\mathscr{C},w\mathscr{C})$ is the simplicial set $\mathscr{F}(\mathscr{C},w\mathscr{C})$, where $\mathscr{C}$ is regarded as a pair with its subcategory of *labeled edges*, rather than is subcategory of cofibrations.
It follows from \[prp:htt32213\] that ${{\mathscr{B}(\mathscr{C},w\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}}}$ is a cocartesian fibration. We endow the $\infty$-category $\mathscr{B}(\mathscr{C},w\mathscr{C})$ with a pair structure in the following manner.
For any Waldhausen $\infty$-category $\mathscr{C}$ and any labeling $w\mathscr{C}\subset\mathscr{C}$ thereof, we endow the $\infty$-category $\mathscr{B}(\mathscr{C},w\mathscr{C})$ with a pair structure in the following manner. We let $\mathscr{B}_{\dag}(\mathscr{C},w\mathscr{C})$ be the smallest pair structure containing morphisms of the form $(\operatorname{id},\operatorname{id},\psi)\colon{{(\mathbf{m},s,Y)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(\mathbf{m},s,X)}}$, where for any integer $0\leq k\leq m$, the induced morphism ${{Y_k}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X_k}}$ is ingressive.
For any Waldhausen $\infty$-category $\mathscr{C}$ and any labeling $w\mathscr{C}\subset\mathscr{C}$ thereof, the cocartesian fibration $p\colon{{\mathscr{B}(\mathscr{C},w\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}}}$ is a Waldhausen cocartesian fibration.
It is plain to see that $p$ is a pair cocartesian fibration.
Now suppose $m\geq0$ an integer. Since limits and colimits of the $\infty$-category $\operatorname{Fun}(\Delta^m,\mathscr{C})$ are computed pointwise, a zero object in $\operatorname{Fun}(\Delta^m,\mathscr{C})$ is an essentially constant functor whose value at any point of $\Delta^m$ is a zero object. Since any equivalence of $\mathscr{C}$ is contained in $w\mathscr{C}$, this zero object is contained in $\mathscr{B}(\mathscr{C},w\mathscr{C})_m$ as well. Again since pushouts in $\operatorname{Fun}(\Delta^m,\mathscr{C})$ are formed objectwise, a pushout square in $\operatorname{Fun}(\Delta^m,\mathscr{C})$ is a functor $$X\colon{{\Delta^1\times\Delta^1\times\Delta^{\{0,k\}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$$ such that for any integer $0\leq k\leq m$, the restriction $X|_{(\Delta^1\times\Delta^1\times\Delta^{\{0,k\}})}$ is a pushout square; now if $X$ is in addition a functor of pairs ${{\mathscr{Q}^2\times(\Delta^m)^{\flat}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$, then it follows from the gluing axiom that if $X|_{(\{0\}\times\Delta^{\{0,k\}})}$, $X|_{(\{1\}\times\Delta^{\{0,k\}})}$, and $X|_{(\{2\}\times\Delta^{\{0,k\}})}$ all factor through $w\mathscr{C}\subset\mathscr{C}$, then so does $X|_{(\{\infty\}\times\Delta^{\{0,k\}})}$. Hence the fibers $\mathscr{B}_m(\mathscr{C},w\mathscr{C})$ of $p$ are Waldhausen $\infty$-categories, and, again using the fact that colimits and limits are computed objectwise, we conclude that $p$ is a Waldhausen cocartesian fibration.
The virtual Waldhausen $\infty$-category attached to a labeled Waldhausen $\infty$-category {#the-virtual-waldhausen-infty-category-attached-to-a-labeled-waldhausen-infty-category .unnumbered}
-------------------------------------------------------------------------------------------
It follows from \[nul:Tpafunctor\] that the assignment $${{(\mathscr{C},w\mathscr{C})}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{B}(\mathscr{C},w\mathscr{C})}}$$ defines a functor $$\mathscr{B}\colon{{{\ell\mathbf{Wald}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{Wald}_{\infty/N\Delta^{{\mathrm{op}}}}^{{\mathrm{cocart}}}}}.$$ By composing with the realization functor (Df. \[dfn:realization\]), we find a functorial construction of virtual Waldhausen $\infty$-categories from labeled Waldhausen $\infty$-categories:
\[ntn:BasavirtWaldcat\] By a small abuse of notation, we denote also as $\mathscr{B}$ the composite functor $${\ell\mathbf{Wald}}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$\mathscr{B}$}(0.5,0.5ex);\ \mathbf{Wald}_{\infty/N\Delta^{{\mathrm{op}}}}^{{\mathrm{cocart}}}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$|\cdot|_{N\Delta^{{\mathrm{op}}}}$}(1.5,0.5ex);\ {\mathrm{D}(\mathbf{Wald}_{\infty})}.$$
One deduces from Ex. \[exm:catwithcofibsandwesislabWaldinftycat\] that a category $(C,\operatorname{cof}C,wC)$ with cofibrations and weak equivalences gives rise to a virtual Waldhausen $\infty$-category $\mathscr{B}(NC,N\operatorname{cof}C,NwC)$.
\[ntn:sigmastar\] Note that the pair cartesian fibration $\pi\colon{{N\mathrm{M}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}}}$ of Nt. \[ntn:M\] admits a section $\sigma$ that assigns to any object $\mathbf{m}\in\Delta$ the pair $(\mathbf{m},0)\in\mathrm{M}$. For any labeled Waldhausen $\infty$-category $(\mathscr{C},w\mathscr{C})$, this section induces a functor of pairs over $N\Delta^{{\mathrm{op}}}$ $$\sigma^{\star}_{(\mathscr{C},w\mathscr{C})}\colon{{\mathscr{B}(\mathscr{C},w\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(N\Delta^{{\mathrm{op}}})^{\flat}\times\mathscr{C}}},$$ which carries any object $(\mathbf{m},X)$ of $\mathscr{F}(\mathscr{C},w\mathscr{C})$ to the pair $(\mathbf{m},X_0)$ and any morphism $(\phi,\psi)\colon{{(\mathbf{n},Y)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(\mathbf{m},X)}}$ to the composite $$Y_0\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--(0.5,0.5ex);\ Y_{\phi(0)}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--node[above]{$\psi_0$}(0.5,0.5ex);\ X_0.$$
The section $\sigma$ induces a map of simplicial sets $${{\mathrm{H}(\mathscr{D},\mathscr{B}(\mathscr{C},w\mathscr{C}))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {w\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{C})}},$$ natural in $\mathscr{D}$, where $w\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{C})\subset\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{C})$ denotes the subcategory containing all exact functors ${{\mathscr{D}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$ and those natural transformations that are pointwise labeled.
\[lem:iotaBCwCiswC\] For any labeled Waldhausen $\infty$-category $(\mathscr{C},w\mathscr{C})$ and any compact Waldhausen $\infty$-category $\mathscr{D}$, the map ${{\mathrm{H}(\mathscr{D},\mathscr{B}(\mathscr{C},w\mathscr{C}))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {w\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{C})}}$ induced by $\sigma$ is a weak homotopy equivalence.
Using (the dual of) Joyal’s $\infty$-categorical version of Quillen’s Theorem A [@HTT Th. 4.1.3.1], we are reduced to showing that for any exact functor $X\colon{{\mathscr{D}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$, the simplicial set $$\mathrm{H}(\mathscr{D},\mathscr{B}(\mathscr{C},w\mathscr{C}))\times_{w\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{C})}w\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{C})_{X/}$$ is weakly contractible. This simplicial set is the geometric realization of the simplicial space $${{\mathbf{n}}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {\mathrm{H}_{1+n}(\mathscr{D},\mathscr{B}(\mathscr{C},w\mathscr{C}))\times_{w\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{C})}\{X\}}};$$ in particular, it may be identified with the path space of the fiber of the map $${{\mathrm{H}(\mathscr{D},\mathscr{B}(\mathscr{C},w\mathscr{C}))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {w\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{C})}}$$ over the vertex $X$.
Consider the ordinary category $\Delta_{w\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}^{\flat}(\mathscr{D},\mathscr{C})}$ of simplices of the simplicial set $w\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{C})$. Corresponding to the natural map $${{N(\Delta_{w\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{C})}^{{\mathrm{op}}}\times_{\Delta^{{\mathrm{op}}}}\mathrm{M}_{\dag})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{C})}}$$ is a map $${{N\Delta_{w\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{C})}^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathrm{H}(\mathscr{D},\mathscr{B}(\mathscr{C},w\mathscr{C}))}}.$$ This map identifies the nerve $N\Delta_{w\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{C})}^{{\mathrm{op}}}$ with the simplicial subset of $\mathrm{H}(\mathscr{D},\mathscr{B}(\mathscr{C},w\mathscr{C}))$ whose simplices correspond to maps $${{\Delta^n\times_{\Delta^{{\mathrm{op}}}}\mathrm{M}_{\dag}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{C})}}$$ that carry cocartesian edges (over $\Delta^n$) to degenerate edges. The composite $$N\Delta_{w\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{C})}^{{\mathrm{op}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ \mathrm{H}(\mathscr{D},\mathscr{B}(\mathscr{C},w\mathscr{C}))\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ w\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{C})$$ is the “initial vertex map,” which is a well-known weak equivalence. A simple argument now shows that the map ${{N\Delta_{w\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{C})}^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathrm{H}(\mathscr{D},\mathscr{B}(\mathscr{C},w\mathscr{C}))}}$ is also a weak equivalence.
In other words, the virtual Waldhausen $\infty$-category $\mathscr{B}(\mathscr{C},w\mathscr{C})$ attached to a labeled Waldhausen $\infty$-category $(\mathscr{C},w\mathscr{C})$ is not itself representable, but it’s close:
\[prp:BCwCiswFunblankC\] The virtual Waldhausen $\infty$-category $\mathscr{B}(\mathscr{C},w\mathscr{C})$ attached to a labeled Waldhausen $\infty$-category $(\mathscr{C},w\mathscr{C})$ is equivalent to the functor $${{\mathscr{D}}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {w\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{C})}}.$$
Inverting labeled edges {#inverting-labeled-edges .unnumbered}
-----------------------
Unfortunately, for a labeled Waldhausen $\infty$-category $(\mathscr{C},w\mathscr{C})$, the functor (Nt. \[ntn:sigmastar\]) $$\sigma^{\star}_{(\mathscr{C},w\mathscr{C})}\colon{{\mathscr{B}(\mathscr{C},w\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(N\Delta^{{\mathrm{op}}})^{\flat}\times\mathscr{C}}}$$ will typically fail to be a morphism of $\mathbf{Wald}_{\infty/N\Delta^{{\mathrm{op}}}}^{{\mathrm{cocart}}}$, because the cocartesian edges of $\mathscr{B}(\mathscr{C},w\mathscr{C})$ will be carried to labeled edges, but not necessarily to equivalences. Hence one may not regard $\sigma_{(\mathscr{C},w\mathscr{C})}^{\star}$ as a natural transformation of functors ${{N\Delta^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}}}$. To rectify this, we may formally invert the edges in $w\mathscr{C}$ in the $\infty$-categorical sense.
The inclusion functor ${{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {{\ell\mathbf{Wald}}}}$ defined by the assignment ${{(\mathscr{C},\mathscr{C}_{\dag})}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {(\mathscr{C},\mathscr{C}_{\dag},\iota\mathscr{C})}}$ admits a left adjoint ${{{\ell\mathbf{Wald}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}}}$.
The inclusion functor ${{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {{\ell\mathbf{Wald}}}}$ preserves all limits and all filtered colimits. Now the result follows from the Adjoint Functor Theorem [@HTT Cor. 5.5.2.9] along with Pr. \[prp:lWaldpres\].
Let us denote by $w\mathscr{C}^{-1}\mathscr{C}$ the image of a labeled Waldhausen $\infty$-category $(\mathscr{C},w\mathscr{C})$ under the left adjoint above. The canonical exact functor ${{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {w\mathscr{C}^{-1}\mathscr{C}}}$ is initial with the property that it carries labeled edges to equivalences. As an example, let us consider the case of an ordinary category with cofibrations and weak equivalences in the sense of Waldhausen [@MR86m:18011 §1.2].
\[thm:classWaldareWald\] If $(C,\operatorname{cof}C,wC)$ is a category with cofibrations and weak equivalences that is a *partial model category* [@BarKan1102] in the sense that: the weak equivalences satisfy the two-out-of-six axiom [@MR2102294 9.1], and the weak equivalences and trivial cofibrations are part of a three-arrow calculus of fractions [@MR2102294 11.1], then the Waldhausen $\infty$-category $(NwC)^{-1}(NC)$ is equivalent to the relative nerve $N(C,wC)$, equipped with the smallest pair structure containing the images of $\operatorname{cof}C$ .
We first claim that $N(C,wC)$ is a Waldhausen $\infty$-category. Indeed, the image of the zero object of $C$ is again a zero object of $N(C,wC)$ as well as an initial object of $N(C,wC)_{\dag}$. The $\infty$-category $\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\Lambda_0\mathscr{Q}^2,N(C,wC))$ is the relative nerve of the full subcategory $C^{\ulcorner}$ of $\operatorname{Fun}(\mathbf{1}\cup^{\{0\}}\mathbf{1},C)$ spanned by those functors that carry the first arrow ${{0}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {1}}$ to a cofibration, equipped with the objectwise weak equivalences. Similarly, $\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{Q}^2,N(C,wC))$ is the relative nerve of the full subcategory $C^{\Box}$ of $\operatorname{Fun}(\mathbf{1}\times\mathbf{1},C)$ spanned by those functors that carry the arrows ${{(0,0)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(0,1)}}$ and ${{(1,0)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(1,1)}}$ each to cofibrations, equipped with the objectwise weak equivalences. The forgetful functor $U\colon{{C^{\Box}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C^{\ulcorner}}}$ and its left adjoint $F\colon{{C^{\ulcorner}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C^{\Box}}}$ are each relative functors, whence they descend to an adjunction $${{F}\colon{\operatorname{Ho}(C^{\ulcorner})}\ \begin{tikzpicture}[baseline] \draw[>=stealth,->] (0,1ex) -- (0.75,1ex); \draw[>=stealth,->] (0.75,0.25ex) -- (0,0.25ex); \end{tikzpicture}\ {\operatorname{Ho}(C^{\Box})}\colon{U}}$$ on the $\operatorname{Ho}s{\mathbf{Set}}$-enriched homotopy categories, using [@MR2102294 36.3]. Thus the forgetful functor $${{\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{Q}^2,N(C,wC))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\Lambda_0\mathscr{Q}^2,N(C,wC))}}$$ admits a left adjoint. Thus $N(C,wC)$ is a Waldhausen $\infty$-category.
Moreover, if ${{X}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ {Y}}$ is a cofibration of $C$ and if ${{X}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X'}}$ is an arrow of $C$, it is easy to see that a square $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{X&Y\\
X'&Y'\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge[>->] (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge[>->] (m-2-2);
\end{tikzpicture}$$ in $N(C,wC)$ is a pushout just in case it is equivalent to the image of a pushout square in $C$.
Now suppose $\mathscr{D}$ a Waldhausen $\infty$-category. Since the canonical functor $${{NC}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N(C,wC)}}$$ is exact, there is an induced functor $$R\colon{{\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(N(C,wC),\mathscr{D})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}'_{{\mathbf{Wald}_{\infty}}}(NC,\mathscr{D})}},$$ where $\operatorname{Fun}'_{{\mathbf{Wald}_{\infty}}}(NC,\mathscr{D})\subset\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(NC,\mathscr{D})$ is the full subcategory spanned by those exact functors that carry arrows in $wC$ to equivalences in $D$. The universal property of $N(C,wC)$, combined with the definition of its pair structure, guarantees an equivalence $${{\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(N(C,wC),\mathscr{D})}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\operatorname{Fun}'_{{\mathbf{Pair}}_{\infty}}(NC,\mathscr{D})}},$$ where $\operatorname{Fun}'_{{\mathbf{Pair}}_{\infty}}(NC,\mathscr{D})\subset\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(NC,\mathscr{D})$ is the full subcategory spanned by those functors of pairs that carry arrows in $wC$ to equivalences in $\mathscr{D}$. Hence $R$ is fully faithful. Since an object (respectively, a morphism, a square) in $N(C,wC)$ is a zero object (resp., a cofibration, a pushout square along a cofibration) just in case it is equivalent to the image of one under the functor ${{NC}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N(C,wC)}}$, it follows that a functor of pairs ${{N(C,wC)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{D}}}$ that induces an exact functor ${{C}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{D}}}$ is itself exact. Thus $R$ is essentially surjective.
Let us give another example of a situation in which we can identify the Waldhausen $\infty$-category $w\mathscr{C}^{-1}\mathscr{C}$, up to splitting certain idempotents. We thank an anonymous referee and Peter Scholze for identifying an error in the original formulation of this result.
\[dfn:weakcofinal\] We say that a full Waldhausen subcategory $\mathscr{C}'\subset\mathscr{C}$ of a Waldhausen $\infty$-category is *weakly cofinal* if it satisfies the following conditions.
1. The subcategory $\mathscr{C}'$ is *closed under extensions* in the following sense. If $U\in\mathscr{C}'$ and ${{U}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ {V}}$ is an ingressive morphism such that the cofiber $V/U\in\mathscr{C}$, then $V\in\mathscr{C}$ as well.
2. For any object $X\in\mathscr{C}$, there exists an object $Y\in\mathscr{C}$ such that $X\vee Y\in\mathscr{C}'$.
We will show below in Th. \[thm:cofinality\] that a weakly cofinal subcategory $\mathscr{C}'\subset\mathscr{C}$ of a Waldhausen $\infty$-category has the same algebraic $K$-theory as $\mathscr{C}$ in positive degrees.
\[prp:localizationofperfwaldcat\] Suppose $C$ a compactly generated $\infty$-category containing a zero object, suppose $L\colon{{C}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {D}}$ an accessible localization of $C$, and suppose the inclusion ${{D}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {C}}$ preserves filtered colimits. Assume also that the class of all $L$-equivalences of $C$ is generated (as a strongly saturated class) by the $L$-equivalences between compact objects. Then if $wC^{\omega}\subset C^{\omega}$ is the subcategory consisting of $L$-equivalences between compact objects, then $D^{\omega}$ is the idempotent completion of $(wC^{\omega})^{-1}C^{\omega}$. In particular, with maximal pair structures, the inclusion ${{(wC^{\omega})^{-1}C^{\omega}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {D^{\omega}}}$ is weakly cofinal.
Let us show that the natural functor ${{(wC^{\omega})^{-1}C^{\omega}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {D^{\omega}}}$ has the universal property of an idempotent completion. For any Waldhausen $\infty$-category $\mathscr{A}$, let us write $\operatorname{Fun}'_{{\mathbf{Wald}_{\infty}}}(C^{\omega},\mathscr{A})\subset\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(C^{\omega},\mathscr{A})$ for the full subcategory spanned by those functors of pairs that carry arrows in $wC^{\omega}$ to equivalences in $\mathscr{A}$. Consider the subcategory $\mathscr{A}_{\dag}\subset\mathscr{A}$. Now $\mathscr{A}_{\dag}$ admits all finite colimits, and the inclusion ${{\mathscr{A}_{\dag}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{A}}}$ induces equivalences $${{\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(C^{\omega},\mathscr{A})}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\operatorname{Fun}_{\mathrm{Rex}}(C^{\omega},\mathscr{A}_{\dag})}}$$ and $${{\operatorname{Fun}'_{{\mathbf{Wald}_{\infty}}}(C^{\omega},\mathscr{A})}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\operatorname{Fun}'_{\mathrm{Rex}}(C^{\omega},\mathscr{A}_{\dag})}},$$ where $\operatorname{Fun}_{\mathrm{Rex}}(C^{\omega},\mathscr{A}_{\dag})$ is the $\infty$-category of right exact functors, and $$\operatorname{Fun}'_{\mathrm{Rex}}(C^{\omega},\mathscr{A}_{\dag})\subset\operatorname{Fun}_{\mathrm{Rex}}(C^{\omega},\mathscr{A}_{\dag})$$ is the full subcategory spanned by those functors that carry arrows in $wC$ to equivalences in $\mathscr{A}$.
On the other hand, if $\mathscr{A}_{\dag}$ is idempotent complete, note that a functor ${{C^{\omega}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{A}_{\dag}}}$ carries the morphisms of $wC^{\omega}$ to equivalences just in case the corresponding functor ${{C}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Ind}(\mathscr{A}_{\dag})}}$ carries $L$-local equivalences to equivalences. Hence the universal property of localizations guarantees that the functor $${{\operatorname{Fun}_{\mathrm{Rex}}(D^{\omega},\mathscr{A}_{\dag})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Fun}_{\mathrm{Rex}}(C^{\omega},\mathscr{A}_{\dag})}}$$ is fully faithful with essential image $\operatorname{Fun}'_{\mathrm{Rex}}(C^{\omega},\mathscr{A}_{\dag})$.
\[ntn:gammaC\] Composing the canonical exact functor ${{\mathscr{C}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {w\mathscr{C}^{-1}\mathscr{C}}}$ with the functor $${{\mathscr{B}(\mathscr{C},w\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(N\Delta^{{\mathrm{op}}})^{\flat}\times\mathscr{C}}},$$ we obtain a morphism of $\mathbf{Wald}_{\infty/N\Delta^{{\mathrm{op}}}}^{{\mathrm{cocart}}}$ $${{\mathscr{B}(\mathscr{C},w\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(N\Delta^{{\mathrm{op}}})^{\flat}\times w\mathscr{C}^{-1}\mathscr{C}}}$$ that carries cocartesian edges of $\mathscr{B}(\mathscr{C},w\mathscr{C})$ to equivalences. Applying the realization $|\cdot|_{N\Delta^{{\mathrm{op}}}}$ (Df. \[dfn:realization\]), we obtain a morphism of ${\mathrm{D}(\mathbf{Wald}_{\infty})}$ $$\gamma_{(\mathscr{C},w\mathscr{C})}\colon{{\mathscr{B}(\mathscr{C},w\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {w\mathscr{C}^{-1}\mathscr{C}}}.$$
We emphasize that for a general labeled Waldhausen $\infty$-category $(\mathscr{C},w\mathscr{C})$, the comparison morphism $\gamma_{(\mathscr{C},w\mathscr{C})}$ is not an equivalence of ${\mathrm{D}(\mathbf{Wald}_{\infty})}$; nevertheless, we will find (Pr. \[prp:gammaonKisequiv\]) that $\gamma_{(\mathscr{C},w\mathscr{C})}$ often induces an equivalence on $K$-theory.
Waldhausen’s Fibration Theorem, redux {#waldhausens-fibration-theorem-redux .unnumbered}
-------------------------------------
We now aim to prove an analogue of Waldhausen’s Generic Fibration Theorem [@MR86m:18011 Th. 1.6.4]. For this we require a suitable analogue of Waldhausen’s cylinder functor in the $\infty$-categorical context. This should reflect the idea that a labeled edge can, to some extent, be replaced by a labeled cofibration.
\[ntn:wdagFunDC\] To this end, for any labeled Waldhausen $\infty$-category $(\mathscr{A},\mathscr{A}_{\dag})$, write $w_{\dag}\mathscr{A}{\mathrel{\mathop:}=}w\mathscr{A}\cap\mathscr{A}_{\dag}$. The subcategory $w_{\dag}\mathscr{A}\subset\mathscr{A}$ defines a new pair structure, *but not a new labeling*, of $\mathscr{A}$. Nevertheless, we may consider the full subcategory $\mathscr{B}(\mathscr{A},w_{\dag}\mathscr{A})\subset\mathscr{F}(\mathscr{A})$ spanned by those filtered objects $$X_0\ \tikz[baseline]\draw[>=stealth,>->,font=\scriptsize](0,0.5ex)--(0.75,0.5ex);\ X_1\ \tikz[baseline]\draw[>=stealth,>->,font=\scriptsize](0,0.5ex)--(0.75,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,>->,font=\scriptsize](0,0.5ex)--(0.75,0.5ex);\ X_m$$ such that each cofibration ${{X_{i}}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ {X_{i+1}}}$ is labeled; we shall regard it as a subpair. One may verify that $\mathscr{B}_{m}(\mathscr{A},w_{\dag}\mathscr{A})\subset\mathscr{F}_m(\mathscr{A})$ is a Waldhausen subcategory, and ${{\mathscr{B}(\mathscr{A},w_{\dag}\mathscr{A})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}}}$ is a Waldhausen cocartesian fibration.
For any pair $\mathscr{D}$, write $w_{\dag}\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{D},\mathscr{A})\subset\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{D},\mathscr{A})$ for the following pair structure. A natural transformation $$\eta\colon{{\mathscr{D}\times\Delta^1}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{A}}}$$ lies in $w_{\dag}\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{D},\mathscr{A})$ if and only if the it satisfies the following two conditions.
1. For any object $X$ of $\mathscr{D}$, the edge ${{\Delta^1\cong\Delta^1\times\{X\}\subset\Delta^1\times\mathscr{D}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{A}}}$ is both ingressive and labeled.
2. For any cofibration $f\colon{{X}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ {Y}}$ of $\mathscr{D}$, the corresponding edge $${{\Delta^1}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{F}_1(\mathscr{A})}}$$ is ingressive in the sense of Df. \[dfn:injpairstruct\].
If $\mathscr{D}$ is a Waldhausen $\infty$-category, write $$w_{\dag}\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{A})\subset w_{\dag}\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{D},\mathscr{A})$$ for the full subcategory spanned by the exact functors.
Note that the proofs of Lm. \[lem:iotaBCwCiswC\] apply also to the pair $(\mathscr{A},w_{\dag}\mathscr{A})$ to guarantee that for any compact Waldhausen $\infty$-category $\mathscr{D}$, the natural map $${{\mathrm{H}(\mathscr{D},(\mathscr{B}(\mathscr{A},w_{\dag}\mathscr{A})/N\Delta^{{\mathrm{op}}}))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {w_{\dag}\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{A})}}$$ induced by $\sigma$ is a weak homotopy equivalence.
\[dfn:enoughcofibrations\] Suppose $(\mathscr{A},w\mathscr{A})$ a labeled Waldhausen $\infty$-category. We shall say that $(\mathscr{A},w\mathscr{A})$ ***has enough cofibrations*** if for any small pair of $\infty$-categories $\mathscr{D}$, the inclusion $${{w_{\dag}\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{D},\mathscr{A})}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {w\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{D},\mathscr{A})}}$$ is a weak homotopy equivalence.
In particular, if every labeled edge of $(\mathscr{A},w\mathscr{A})$ is a cofibration, then $(\mathscr{A},w\mathscr{A})$ has enough cofibrations. More generally, this may prove to be an extremely difficult condition to verify, but the following lemma simplifies matters somewhat.
A labeled Waldhausen $\infty$-category $(\mathscr{A},\mathscr{A}_{\dag})$ has enough cofibrations if and only if, for any of the pairs $\mathscr{D}\in\{\Delta^0,(\Delta^1)^{\flat},(\Delta^1)^{\sharp}\}$, the inclusion $${{w_{\dag}\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{D},\mathscr{A})}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {w\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{D},\mathscr{A})}}$$ is a weak homotopy equivalence.
Any pair of $\infty$-categories can be written as a suitable colimit of pairs of the form $\Delta^0$, $(\Delta^1)^{\flat}$, or $(\Delta^1)^{\sharp}$.
\[lem:wAhasenoughcofibs\] If a labeled Waldhausen $\infty$-category $(\mathscr{A},w\mathscr{A})$ has enough cofibrations, then for any Waldhausen $\infty$-category $\mathscr{D}$, the inclusion $${{w_{\dag}\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{A})}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {w\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{A})}}$$ is a weak homotopy equivalence.
For any Waldhausen $\infty$-category $\mathscr{B}$, the square $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{w_{\dag}\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{A})&w\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{A})\\
w_{\dag}\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{D},\mathscr{A})&w\operatorname{Fun}_{{\mathbf{Pair}}_{\infty}}(\mathscr{D},\mathscr{A})\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge (m-2-2);
\end{tikzpicture}$$ is a pullback, and the vertical maps are inclusions of connected components.
\[thm:fibration\] Suppose $(\mathscr{A},w\mathscr{A})$ a labeled Waldhausen $\infty$-category that has enough cofibrations. Suppose $\phi\colon{{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{E}}}$ an additive theory with left derived functor $\Phi$. Then the inclusion $i\colon{{\mathscr{A}^w}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{A}}}$ and the morphism of virtual Waldhausen $\infty$-categories $e\colon{{\mathscr{A}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{B}(\mathscr{A},w\mathscr{A})}}$ give rise to a fiber sequence $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\phi(\mathscr{A}^w)&\phi(\mathscr{A})\\
\ast&\Phi(\mathscr{B}(\mathscr{A},w\mathscr{A})).\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge (m-2-2);
\end{tikzpicture}$$
It follows from Pr. \[prp:fibthmi\] that it is enough to exhibit an equivalence between $\Phi(\mathscr{B}(\mathscr{A},w\mathscr{A}))$ and $\Phi(\mathscr{K}(i))$ as objects of $\mathscr{E}_{\phi(\mathscr{A})/}$.
The forgetful functor ${{\mathscr{K}(i)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{F}\mathscr{A}}}$ is fully faithful, and its essential image $\widetilde{\mathscr{F}}^w\mathscr{A}$ consists of those filtered objects $$X_0\ \tikz[baseline]\draw[>=stealth,>->,font=\scriptsize](0,0.5ex)--(0.75,0.5ex);\ X_1\ \tikz[baseline]\draw[>=stealth,>->,font=\scriptsize](0,0.5ex)--(0.75,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,>->,font=\scriptsize](0,0.5ex)--(0.75,0.5ex);\ X_m$$ such that the induced cofibration ${{X_{i}/X_0}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ {X_{i+1}/X_0}}$ is labeled; this contains the subcategory $\mathscr{B}(\mathscr{A},w_{\dag}\mathscr{A})$. We claim that for any $m\geq 0$, the induced morphism ${{\phi(\mathscr{B}_m(\mathscr{A},w_{\dag}\mathscr{A}))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\phi(\widetilde{\mathscr{F}}_m^w(\mathscr{A}))}}$ is an equivalence. Indeed, one may select an exact functor $p\colon{{\mathscr{K}_m(i)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{B}_m(\mathscr{A},w_{\dag}\mathscr{A})}}$ that carries an object $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{X_0&X_1&X_2&\dots&X_m\\
0&U_1&U_2&\dots&U_m\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge[>->] (m-1-2)
edge (m-2-1)
(m-1-2) edge[>->] (m-1-3)
edge (m-2-2)
(m-1-3) edge[>->] (m-1-4)
edge (m-2-3)
(m-1-4) edge[>->] (m-1-5)
(m-1-5) edge (m-2-5)
(m-2-1) edge[>->] (m-2-2)
(m-2-2) edge[>->] (m-2-3)
(m-2-3) edge[>->] (m-2-4)
(m-2-4) edge[>->] (m-2-5);
\end{tikzpicture}$$ to the filtered object $$X_0\ \tikz[baseline]\draw[>=stealth,>->,font=\scriptsize](0,0.5ex)--(0.75,0.5ex);\ X_0\vee U_1\ \tikz[baseline]\draw[>=stealth,>->,font=\scriptsize](0,0.5ex)--(0.75,0.5ex);\ X_0\vee U_2\ \tikz[baseline]\draw[>=stealth,>->,font=\scriptsize](0,0.5ex)--(0.75,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,>->,font=\scriptsize](0,0.5ex)--(0.75,0.5ex);\ X_0\vee U_m.$$ When $m=0$, this functor is compatible with the canonical equivalences from $\mathscr{A}$. Additivity now guarantees that $p$ defines a (homotopy) inverse to the morphism ${{\phi(\mathscr{B}_m(\mathscr{A},w_{\dag}\mathscr{A}))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\phi(\widetilde{\mathscr{F}}_m^w\mathscr{A})}}$.
One has an obvious forgetful functor ${{\mathscr{B}(\mathscr{A},w_{\dag}\mathscr{A})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{B}(\mathscr{A},w\mathscr{A})}}$ over $N\Delta^{{\mathrm{op}}}$. We claim that this induces an equivalence of virtual Waldhausen $\infty$-categories ${{|\mathscr{B}(\mathscr{A},w_{\dag}\mathscr{A})|_{N\Delta^{{\mathrm{op}}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {|\mathscr{B}(\mathscr{A},w\mathscr{A})|_{N\Delta^{{\mathrm{op}}}}}}$. So we wish to show that for any compact Waldhausen $\infty$-category $\mathscr{D}$, the morphism $${{\mathrm{H}(\mathscr{D},(\mathscr{B}(\mathscr{A},w_{\dag}\mathscr{A})/N\Delta^{{\mathrm{op}}}))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathrm{H}(\mathscr{D},(\mathscr{B}(\mathscr{A},w\mathscr{A})/N\Delta^{{\mathrm{op}}}))}}$$ of simplicial sets is a weak homotopy equivalence.
By Lm. \[lem:iotaBCwCiswC\] and its extension to the pair $(\mathscr{A},w_{\dag}\mathscr{A})$, we have a square $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\mathrm{H}(\mathscr{D},(\mathscr{B}(\mathscr{A},w_{\dag}\mathscr{A})/N\Delta^{{\mathrm{op}}}))&\mathrm{H}(\mathscr{D},(\mathscr{B}(\mathscr{A},w\mathscr{A})/N\Delta^{{\mathrm{op}}}))\\
w_{\dag}\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{C})&w\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(\mathscr{D},\mathscr{C})\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge (m-2-2);
\end{tikzpicture}$$ in which the vertical maps are weak homotopy equivalences. Since $(\mathscr{A},w\mathscr{A})$ has enough cofibrations, the horizontal map along the bottom is a weak homotopy equivalence as well by Lm. \[lem:wAhasenoughcofibs\].
We are finally prepared to describe the Waldhausen $K$-theory of $\infty$-categories. We define (Df. \[dfn:Ktheory\]) $K$-theory as the additivization of the the theory $\iota$ that assigns to any Waldhausen $\infty$-category the maximal $\infty$-groupoid (Nt. \[ntn:interior\]) contained therein. Since the theory $\iota$ is representable by the particularly simple Waldhausen $\infty$-category $N\Gamma^{{\mathrm{op}}}$ of pointed finite sets (Pr. \[prp:iotaisrepresentable\]), we obtain, for any additive theory $\phi$, a description of the space of natural transformations ${{K}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\phi}}$ as the value of $\phi$ on $N\Gamma^{{\mathrm{op}}}$.
Following this, we briefly describe two key examples that exploit certain features of the algebraic $K$-theory functor of which we are fond. The first of these (§\[sect:example1\]) lays the foundations for the algebraic $K$-theory of $E_1$-algebras in a variety of monoidal $\infty$-categories, and we prove a straightforward localization theorem. Second (§\[sect:example2\]), we extend algebraic $K$-theory to the context of spectral Deligne–Mumford stacks in the sense of Lurie, and we prove Thomason’s “proto-localization” theorem in this context.
The universal property of Waldhausen $K$-theory {#sect:univpropKthy}
===============================================
In this section, we *define* algberaic $K$-theory as the additivization of the functor that assigns to any Waldhausen $\infty$-category its moduli space of objects. More precisely, the functor $\iota\colon{{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$ that assigns to any Waldhausen $\infty$-category its interior $\infty$-groupoid (Nt. \[ntn:interior\]) is a theory.
\[dfn:Ktheory\] The ***algebraic $K$-theory functor*** $$K\colon{{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$$ is defined as the additivization $K{\mathrel{\mathop:}=}D\iota$ of the interior functor $\iota\colon{{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$. We denote by ${\mathbf{K}}\colon{{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Sp}}_{\geq 0}}}$ its canonical connective delooping, as guaranteed by Cor. \[cor:additivevaluedinstab\] and Pr. \[prp:deloopingofdfisconnective\].
Unpacking this definition, we obtain a global universal property of the natural morphism ${{\iota}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {K}}$.
\[prp:univpropofk\] For any additive theory $\phi$, the morphism ${{\iota}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {K}}$ induces a natural homotopy equivalence $${{\operatorname{Map}(K,\phi)}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\operatorname{Map}(\iota,\phi)}}.$$
We will prove in Cor. \[cor:easycompare\] and Cor. \[cor:hardcompare\] that our definition extends Waldhausen’s.
\[exm:Atheoryofinfintopoi\] For any $\infty$-topos $\mathscr{E}$, one may define the $A$-theory space $$A(\mathscr{E}){\mathrel{\mathop:}=}K(\mathscr{E}_{\ast}^{\omega})$$ (Ex. \[exm:inftytopoigiveWaldcats\]). In light of Ex. \[exm:stabinftytopoi\], we have $$A(\mathscr{E})\simeq K({\mathbf{Sp}}(\mathscr{E}^{\omega})).$$ For any Kan simplicial set $X$, if $$\mathscr{E}=\operatorname{Fun}(X,{\mathbf{Kan}})\simeq{\mathbf{Kan}}_{/X},$$ then it will follow from Cor. \[cor:fromWaldtoWald\] that $A(\mathscr{E})$ agrees with Waldhausen’s $A(X)$, where one defines the latter via the category $\mathscr{R}_{\mathrm{df}}(X)$ of finitely dominated retractive spaces over $X$ [@MR86m:18011 p. 389]. Then of course one has $A(\mathscr{E})\simeq K(\operatorname{Fun}(X,{\mathbf{Sp}}^{\omega}))$.
Representability of algebraic $K$-theory {#representability-of-algebraic-k-theory .unnumbered}
----------------------------------------
Algebraic $K$-theory is controlled, as an additive theory, by the theory $\iota$. It is therefore valuable to study this functor as a theory. As a first step, we find that it is corepresentable.
For any finite set $I$, write $I_+$ for the finite set $I\sqcup\{\infty\}$. Denote by $\Gamma^{{\mathrm{op}}}$ the ordinary category of pointed finite sets. Denote by $\Gamma^{{\mathrm{op}}}_{\dag}\subset\Gamma^{{\mathrm{op}}}$ the subcategory comprised of monomorphisms ${{J_+}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {I_+}}$.
\[prp:iotaisrepresentable\] For any Waldhausen $\infty$-category $\mathscr{C}$, the inclusion $${{\{\ast\}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {N\Gamma^{{\mathrm{op}}}}}$$ induces an equivalence of $\infty$-categories $${{\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(N\Gamma^{{\mathrm{op}}},\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {\mathscr{C}}}.$$ In particular, the functor $\iota\colon{{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$ is corepresented by the object $N\Gamma^{{\mathrm{op}}}$.
Write $N\Gamma^{{\mathrm{op}}}_{\leq 1}$ for the full subcategory of $N\Gamma^{{\mathrm{op}}}$ spanned by the objects $\varnothing$ and $\ast$. Then it follows from Joyal’s theorem [@HTT Pr. 1.2.12.9] that the inclusion ${{\{\ast\}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {N\Gamma^{{\mathrm{op}}}}}$ induces an equivalence between the full subcategory $\operatorname{Fun}^{\ast}(N\Gamma^{{\mathrm{op}}}_{\leq1},\mathscr{C})$ of $\operatorname{Fun}(N\Gamma^{{\mathrm{op}}}_{\leq1},\mathscr{C})$ spanned by functors $z\colon{{N\Gamma^{{\mathrm{op}}}_{\leq1}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$ such that $z(\varnothing)$ is a zero object. Now the result follows from the observation that $\operatorname{Fun}_{{\mathbf{Wald}_{\infty}}}(N\Gamma^{{\mathrm{op}}},\mathscr{C})$ can be identified as the full subcategory of $\operatorname{Fun}(N\Gamma^{{\mathrm{op}}},\mathscr{C})$ spanned by those functors $Z\colon{{N\Gamma^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$ such that (1) $Z(\varnothing)$ is a zero object, and (2) the identity exhibits $Z$ as a left Kan extension of $Z|_{(N\Gamma^{{\mathrm{op}}}_{\leq1})}$ along the inclusion ${{N\Gamma^{{\mathrm{op}}}_{\leq1}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {N\Gamma^{{\mathrm{op}}}}}$.
In the language of Cor. \[cor:Wconstruction\], we find that $W(\Delta^0)\simeq N\Gamma^{{\mathrm{op}}}$. Note also that it follows that the left derived functor $I\colon{{{\mathrm{D}(\mathbf{Wald}_{\infty})}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$ of $\iota$ is given by evaluation at $W(\Delta^0)\simeq N\Gamma^{{\mathrm{op}}}$. From this, the Yoneda lemma combines with Pr. \[prp:univpropofk\] to imply the following.
\[cor:tracesarephiofgamma\] For any additive theory $\phi\colon{{{\mathbf{Wald}_{\infty}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}_{\ast}}}$, there is a homotopy equivalence $$\operatorname{Map}(K,\phi)\simeq\phi(N\Gamma^{{\mathrm{op}}}),$$ natural in $\phi$.
In particular, the theorem of Barratt–Priddy–Quillen [@MR0358767] implies the following.
The space of endomorphisms of $K$-theory is given by $$\operatorname{End}(K)\simeq QS^0.$$
The local universal property of algebraic $K$-theory {#the-local-universal-property-of-algebraic-k-theory .unnumbered}
----------------------------------------------------
Though conceptually pleasant, the universal property of $K$-theory as an object of ${\mathrm{Add}}({\mathbf{Kan}})$ does not obviously provide an easy recognition principle for the $K$-theory of any *particular* Waldhausen $\infty$-category. For that, we note that $\iota$ is pre-additive, and we appeal to Cor. \[for:preaddsareeasy\] to obtain the following result.
For any virtual Waldhausen $\infty$-category $\mathscr{X}$, the $K$-theory space $K(\mathscr{X})$ is homotopy equivalent to the loop space $\Omega I(\mathscr{S}(\mathscr{X}))$, where $I$ is the left derived functor of $\iota$.
We observe that for any sifted $\infty$-category and any Waldhausen cocartesian fibration ${{\mathscr{Y}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$, the space $I(\mathscr{S}(|\mathscr{Y}|_S))$ may be computed as the underlying space of the subcategory $\iota_{N\Delta^{{\mathrm{op}}}\times S}\mathscr{S}(\mathscr{Y}/S)$ of the $\infty$-category $\mathscr{S}(\mathscr{Y})$ comprised of the cocartesian edges with respect to the cocartesian fibration ${{\mathscr{S}(\mathscr{Y}/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}\times S}}$ (Df. \[rec:leftfib\]). This provides us with a (singly delooped) model of the algebraic $K$-theory space $K(|\mathscr{Y}|_S)$ as the underlying simplicial set of an $\infty$-category.
For any sifted $\infty$-category $S$ and any Waldhausen cocartesian fibration ${{\mathscr{Y}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {S}}$, the $K$-theory space $K(|\mathscr{Y}|_S)$ is homotopy equivalent to the loop space $\Omega\iota_{(N\Delta^{{\mathrm{op}}}\times S)}\mathscr{S}(\mathscr{Y}/S)$.
The total space of a left fibration is weakly equivalent to the homotopy colimit of the functor that classifies it. So the $K$-theory space $K(\mathscr{C})$ of a Waldhausen $\infty$-category is given by $$K(\mathscr{C})\simeq\Omega(\operatorname{colim}\iota{\mathbf{S}}_{\ast}(\mathscr{C})),$$ where $${\mathbf{S}}_{\ast}(\mathscr{C})\colon{{N\Delta^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}}}$$ classifies the Waldhausen cocartesian fibration ${{\mathscr{S}(\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}}}$. Since this is precisely how Waldhausen’s $K$-theory is defined [@MR86m:18011 §1.3], we obtain a comparison between our $\infty$-categorical $K$-theory and Waldhausen $K$-theory.
\[cor:easycompare\] If $(C,\operatorname{cof}C)$ is an ordinary category with cofibrations in the sense of Waldhausen [@MR86m:18011 §1.1], then the algebraic $K$-theory of the Waldhausen $\infty$-category $(NC,N(\operatorname{cof}C))$ is naturally equivalent to Waldhausen’s algebraic $K$-theory of $(C,\operatorname{cof}C)$.
The fact that the algebraic $K$-theory space $K(\mathscr{X})$ of a virtual Waldhausen $\infty$-category $\mathscr{X}$ can be exhibited as the loop space of the underlying simplicial set of an $\infty$-category permits us to find the following sufficient condition that a morphism of Waldhausen cocartesian fibrations induce an equivalence on $K$-theory.
\[cor:ThmAKtheory\] For any sifted $\infty$-category $S$, a morphism ${{(\mathscr{Y}'/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(\mathscr{Y}'/S)}}$ of Waldhausen cocartesian fibrations induces an equivalence $${{K(|\mathscr{Y}'|_S)}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {K(|\mathscr{Y}|_S)}}$$ if the following two conditions are satisfied.
1. For any object $X\in\iota_S\mathscr{Y}$, the simplicial set $$\iota_S\mathscr{Y}'\times_{\iota_S\mathscr{Y}}(\iota_S\mathscr{Y})_{/X}$$ is weakly contractible.
2. For any object $Y\in\iota_S\mathscr{F}_1(\mathscr{Y}/S)$, the simplicial set $$\iota_S\mathscr{F}_1(\mathscr{Y}'/S)\times_{\iota_S\mathscr{F}_1(\mathscr{Y}/S)}\iota_S\mathscr{F}_1(\mathscr{Y}/S)_{/Y}$$ is weakly contractible.
We aim to show that the map ${{\iota_{N\Delta^{{\mathrm{op}}}\times S}\mathscr{S}(\mathscr{Y}'/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\iota_{N\Delta^{{\mathrm{op}}}\times S}\mathscr{S}(\mathscr{Y}/S)}}$ is a weak homotopy equivalence; it is enough to show that for any $\mathbf{n}\in\Delta$, the map ${{\iota_S\mathscr{F}_n(\mathscr{Y}'/S)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\iota_S\mathscr{F}_n(\mathscr{Y}/S)}}$ is a weak homotopy equivalence. Since $\mathscr{F}(\mathscr{Y}'/S)$ and $\mathscr{F}(\mathscr{Y}/S)$ are each category objects (Pr. \[prp:Fisacategory\]), it is enough to prove this claim for $n\in\{0,1\}$. The result now follows from Joyal’s $\infty$-categorical version of Quillen’s Theorem A [@HTT Th. 4.1.3.1].
Using Pr. \[prp:localrecogofDf\], we further deduce the following recognition principle for the $K$-theory of a Waldhausen $\infty$-category.
\[prp:univpropKofC\] For any Waldhausen $\infty$-category $\mathscr{C}$, and any functor $${\mathbf{S}}_{\ast}(\mathscr{C})\colon{{N\Delta^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}}}$$ that classifies the Waldhausen cocartesian fibration ${{\mathscr{S}(\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}}}$, the $K$-theory space $K(\mathscr{C})$ is the underlying space of the initial object of the $\infty$-category $${\mathbf{Grp}}({\mathbf{Kan}})\times_{\operatorname{Fun}(N\Delta^{{\mathrm{op}}},{\mathbf{Kan}})}\operatorname{Fun}(N\Delta^{{\mathrm{op}}},{\mathbf{Kan}})_{\iota{\mathbf{S}}_{\ast}(\mathscr{C})/}.$$
The algebraic $K$-theory of labeled Waldhausen $\infty$-category {#the-algebraic-k-theory-of-labeled-waldhausen-infty-category .unnumbered}
----------------------------------------------------------------
We now study the $K$-theory of labeled Waldhausen $\infty$-categories.
Suppose $(\mathscr{C},w\mathscr{C})$ a labeled Waldhausen $\infty$-category (Df. \[dfn:labeling\]). Then we define $K(\mathscr{C},w\mathscr{C})$ as the $K$-theory space $K(\mathscr{B}(\mathscr{C},w\mathscr{C}))$.
If $\mathscr{C}$ is a Waldhausen $\infty$-category, and if $w\mathscr{C}\subset\mathscr{C}$ is a labeling, then define $w_{N\Delta^{{\mathrm{op}}}}\mathscr{S}(\mathscr{C})\subset\mathscr{S}(\mathscr{C})$ as the smallest subcategory containing all cocartesian edges and all morphisms of the form $(\operatorname{id},\psi)\colon{{(\mathbf{m},Y)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(\mathbf{m},X)}}$, where for any integer $0\leq k\leq m$, the induced morphism ${{Y_k}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X_k}}$ is labeled.
In light of Lm. \[lem:iotaBCwCiswC\], we now immediately deduce the following.
For any labeled Waldhausen $\infty$-category $(\mathscr{C},w\mathscr{C})$, the $K$-theory space $K(\mathscr{C},w\mathscr{C})$ is weakly homotopy equivalent to the loopspace $$\Omega(w_{N\Delta^{{\mathrm{op}}}}\mathscr{S}(\mathscr{C})).$$
In other words, for any labeled Waldhausen $\infty$-category $(\mathscr{C},w\mathscr{C})$, the simplicial set $K(\mathscr{C},w\mathscr{C})$ is weakly homotopy equivalent to the loopspace $$\Omega\operatorname{colim}w{\mathbf{S}}_{\ast}(\mathscr{C}).$$ Since this again is precisely how Waldhausen’s $K$-theory is defined [@MR86m:18011 §1.3], we obtain a further comparison between our $\infty$-categorical $K$-theory for labeled Waldhausen $\infty$-categories and Waldhausen $K$-theory, analogous to Cor. \[cor:easycompare\].
\[cor:hardcompare\] If $(C,\operatorname{cof}C,wC)$ is an ordinary category with cofibrations and weak equivalences in the sense of Waldhausen [@MR86m:18011 §1.2], then the algebraic $K$-theory of the labeled Waldhausen $\infty$-category $(NC,N(\operatorname{cof}C),wC)$ is naturally equivalent to Waldhausen’s algebraic $K$-theory of $(C,\operatorname{cof}C,wC)$.
Using Cor. \[cor:ThmAKtheory\], we obtain the following.
\[prp:gammaonKisequiv\] Suppose $(\mathscr{C},w\mathscr{C})$ a labeled Waldhausen $\infty$-category. Then the comparison morphism $\gamma_{(\mathscr{C},w\mathscr{C})}$ induces an equivalence $${{K(\mathscr{C},w\mathscr{C})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {K(w\mathscr{C}^{-1}\mathscr{C})}}$$ of $K$-theory spaces if the following conditions are satisfied.
1. For any object $X$ of $w\mathscr{C}^{-1}\mathscr{C}$, the simplicial set $$w\mathscr{C}\times_{\iota(w\mathscr{C}^{-1}\mathscr{C})}\iota(w\mathscr{C}^{-1}\mathscr{C})_{/X}$$ is weakly contractible.
2. For any object $Y$ of $\mathscr{F}_1(w\mathscr{C}^{-1}\mathscr{C})$, the simplicial set $$w\mathscr{F}_1(\mathscr{C})\times_{\iota\mathscr{F}_1(w\mathscr{C}^{-1}\mathscr{C})}\iota\mathscr{F}_1(w\mathscr{C}^{-1}\mathscr{C})_{/Y}$$ is weakly contractible.
Pr. \[thm:classWaldareWald\], combined with Cor. \[prp:gammaonKisequiv\], yields a further corollary.
\[cor:fromWaldtoWald\] If $(C,wC)$ is a *partial model category* [@BarKan1102] in which the weak equivalences and trivial cofibrations are part of a three-arrow calculus of fractions, then the Waldhausen $K$-theory of $(C,\operatorname{cof}C,wC)$ is canonically equivalent to the $K$-theory of a relative nerve $N(C,wC)$, equipped with the smallest pair structure containing the image of $\operatorname{cof}C$ .
The only nontrivial point is to verify that the natural maps $${{NwC}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\iota N(C,wC)}}\textrm{\quad and\quad}{{Nw\operatorname{Fun}(\mathbf{1},C)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\iota\operatorname{Fun}(\Delta^1,N(C,wC))}}$$ are weak homotopy equivalences, but this is precisely the content of the main theorem of [@BarKan1102].
Cofinality and more fibration theorems {#cofinality-and-more-fibration-theorems .unnumbered}
--------------------------------------
We may also use Cor. \[prp:gammaonKisequiv\] in combination with Pr. \[prp:localizationofperfwaldcat\] to specialize the second Generic Fibration Theorem (Th. \[thm:fibration\]). We first prove a cofinality result, which states that weakly cofinal inclusions (Df. \[dfn:weakcofinal\]) of Waldhausen $\infty$-categories do not affect the $K$-theory in high degrees. We are thankful to Peter Scholze for noticing an error that necessitated the inclusion of this result. We follow closely the model of Staffeldt [@MR990574 Th. 2.1], which works in our setting with only superficial changes.
\[thm:cofinality\] The map on $K$-theory induced by the inclusion $i\colon{{\mathscr{C}'}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$ of a weakly cofinal subcategory fits into a fiber sequence $$K(\mathscr{C}')\to K(\mathscr{C})\to A,$$ where $A$ is the abelian group $K_0(\mathscr{C})/K_0(\mathscr{C}')$, regarded as a discrete simplicial set.
It is convenient to describe the classifying space $BA$ in the following manner. Denote by $BA$ the nerve of the following ordinary category. An object $(m,(x_{i}))$ consists of an integer $m\geq 0$ and a tuple $(x_{i})_{i\in \{1,\dots,m\}}$, and a morphism $${{(m,(x_{i}))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(n,(y_{j}))}}$$ is a morphism $\phi\colon{{\mathbf{n}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{m}}}$ of $\Delta$ such that for any $j\in\{1,\dots,n\}$, $$y_{j}=\prod_{\phi(j-1)\leq i-1\leq\phi(j)-1}x_{i}.$$ The projection ${{BA}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}}}$ clearly induces a left fibration, and the simplicial space ${{N\Delta^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{Kan}}}$ that classifies it visibly satisfies the Segal condition and thus exhibits $(BA)_1\cong A$ as the loop space $\Omega BA$.
We appeal to the Generic Fibration Theorem \[prp:fibthmi\]. Consider the left fibration $$p\colon{{\iota_{N(\Delta^{{\mathrm{op}}}\times\Delta^{{\mathrm{op}}})}\mathscr{S}\!\!\mathscr{K}\!(i)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N(\Delta^{{\mathrm{op}}}\times\Delta^{{\mathrm{op}}})}}$$ and more particularly its composite $q{\mathrel{\mathop:}=}\operatorname{pr}_2\circ p$ with the projection $$\operatorname{pr}_2\colon{{N(\Delta^{{\mathrm{op}}}\times\Delta^{{\mathrm{op}}})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}}}$$ (whose fiber over $\mathbf{n}\in\Delta$ is $\iota_{N(\Delta^{{\mathrm{op}}}\times\Delta^{{\mathrm{op}}})}\mathscr{S}_{\!n}\mathscr{K}\!(i)$). The Generic Fibration Theorem will imply the Cofinality Theorem once we have furnished an equivalence $\iota_{N(\Delta^{{\mathrm{op}}}\times\Delta^{{\mathrm{op}}})}\mathscr{S}\!\!\mathscr{K}\!(i)\simeq BA$ over $N\Delta^{{\mathrm{op}}}$.
Observe that an object $X$ of the $\infty$-category $\iota_{N(\Delta^{{\mathrm{op}}}\times\Delta^{{\mathrm{op}}})}\mathscr{S\!\!K}\!(i)$ consists of a diagram in $\mathscr{C}$ of the form $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{0&0&\cdots&0\\
X_{01}&X_{11}&\cdots&X_{m1}\\
\vdots&\vdots&&\vdots\\
X_{0n}&X_{1n}&\cdots&X_{mn},\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge[-,double distance=1.5pt] (m-1-2)
edge[>->] (m-2-1)
(m-1-2) edge[-,double distance=1.5pt] (m-1-3)
edge[>->] (m-2-2)
(m-1-3) edge[-,double distance=1.5pt] (m-1-4)
(m-1-4) edge[>->] (m-2-4)
(m-2-1) edge[>->] (m-2-2)
edge[>->] (m-3-1)
(m-3-1) edge[>->] (m-4-1)
(m-2-2) edge[>->] (m-2-3)
edge[>->] (m-3-2)
(m-3-2) edge[>->] (m-4-2)
(m-2-4) edge[>->] (m-3-4)
(m-3-4) edge[>->] (m-4-4)
(m-4-1) edge[>->] (m-4-2)
(m-4-2) edge[>->] (m-4-3)
(m-2-3) edge[>->] (m-2-4)
(m-4-3) edge[>->] (m-4-4);
\end{tikzpicture}$$ such that each $X_{k\ell}/X_{(k-1)\ell}\in\mathscr{C}'$ and the maps $${{X_{(k-1)\ell}\cup^{X_{(k-1)(\ell-1)}}X_{k(\ell-1)}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X_{k\ell}}}$$ are all cofibrations. Consequently, we may define a map $$\Phi\colon{{\iota_{N(\Delta^{{\mathrm{op}}}\times\Delta^{{\mathrm{op}}})}\mathscr{S\!\!K}\!(i)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {BA}}$$ that carries an $n$-simplex $$X(0)\to\cdots\to X(n)$$ of $\iota_{N(\Delta^{{\mathrm{op}}}\times\Delta^{{\mathrm{op}}})}\mathscr{S\!\!K}\!(i)$ to the obvious $n$-simplex whose $i$-th vertex is $$\left(q(X(i)),([X(i)_{0\ell}/X(i)_{0(\ell-1)}])_{\ell\in\{1,\dots,q(X(i))\}}\right)$$ of $BA$, where $[Z]$ denotes the image of any object $Z\in\mathscr{C}$ in $K_0(\mathscr{C})/K_0(\mathscr{C}')$. This is easily seen to be a map of simplicial sets over $N\Delta^{{\mathrm{op}}}$.
Our aim is now to show that $\Phi$ is a fiberwise equivalence. Note that the target satisfies the Segal condition by construction, and the source satisfies it thanks to the Additivity Theorem. Consequently, we are reduced to checking that the induced map $$\Phi_1\colon{{\iota_{N\Delta^{{\mathrm{op}}}}\mathscr{K}(i)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(BA)_1\cong A}}$$ is a weak equivalence.
One may check that $\Phi_1$ induces a bijection ${{\pi_0\iota_{N\Delta^{{\mathrm{op}}}}\mathscr{K}\!(i)}\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ {A}}$ exactly as in [@MR990574 p. 517].
Now fix an object $Z\in\mathscr{C}$, and write $\iota_{N\Delta^{{\mathrm{op}}}}\mathscr{K}\!(i)_{Z}\subset\iota_{N\Delta^{{\mathrm{op}}}}\mathscr{K}\!(i)$ for the connected component corresponding to the class $[Z]$. This is the full subcategory spanned by those objects $$X_0\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X_n$$ such that $[X_0]=[Z]$ in $A$. We may construct a functor $$T\colon{{\iota_{N\Delta^{{\mathrm{op}}}}\mathscr{F}(\mathscr{C}')}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\iota_{N\Delta^{{\mathrm{op}}}}\mathscr{K}\!(i)_{Z}}}$$ that carries an object $$Y_0\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ Y_n$$ to an object $$Y_0\vee Z\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ Y_n\vee Z.$$ In the other direction, choose an object $W\in\mathscr{C}$ such that $Z\vee W\in\mathscr{C}'$. Let $S\colon{{\iota_{N\Delta^{{\mathrm{op}}}}\mathscr{K}\!(i)_{Z}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\iota_{N\Delta^{{\mathrm{op}}}}\mathscr{F}(\mathscr{C}')}}$ be the obvious functor that carries an object $$X_0\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X_n$$ to an object $$X_0\vee W\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ \cdots\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ X_n\vee W.$$
Now for any finite simplicial set $K$ and any map $g\colon{{K}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\iota_{N\Delta^{{\mathrm{op}}}}\mathscr{F}(\mathscr{C}')}}$, we construct a map $$G\colon{{K\times\Delta^1}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\iota_{N\Delta^{{\mathrm{op}}}}\mathscr{F}(\mathscr{C}')}}$$ such that $$G|(K\times\Delta^{\{0\}})\cong g\textrm{\quad and\quad}G|(K\times\Delta^{\{1\}})\cong S\circ T\circ g$$ in the following manner. We let the map ${{K\times\Delta^1}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}}}$ induced by $G$ be the projection onto $K$ followed by the map ${{K}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}}}$ induced by $g$. The natural transformation from the identity on $\mathscr{C}'$ to the functor ${{X}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {X\vee Z\vee W}}$ now gives a map ${{(K\times\Delta^1)\times_{N\Delta^{{\mathrm{op}}}}N\mathrm{M}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{C}}}$, which by definition corresponds to the desired map $G$.
In almost exactly the same manner, for any map $f\colon{{K}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\iota_{N\Delta^{{\mathrm{op}}}}\mathscr{K}\!(i)_{Z}}}$, one may construct a map $$F\colon{{K\times\Delta^1}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\iota_{N\Delta^{{\mathrm{op}}}}\mathscr{K}\!(i)_{Z}}}$$ such that $$F|(K\times\Delta^{\{0\}})\cong f\textrm{\quad and\quad}F|(K\times\Delta^{\{1\}})\cong T\circ S\circ f.$$ We therefore conclude that for any simplicial set $K$, the functors $T$ and $S$ induce a bijection $$[K,\iota_{N\Delta^{{\mathrm{op}}}}\mathscr{F}(\mathscr{C}')]\cong[K,\iota_{N\Delta^{{\mathrm{op}}}}\mathscr{K}\!(i)_{Z}],$$ whence $S$ and $T$ are homotopy inverses. Now since $\iota_{N\Delta^{{\mathrm{op}}}}\mathscr{F}(\mathscr{C}')$ is contractible, it follows that $\iota_{N\Delta^{{\mathrm{op}}}}\mathscr{K}\!(i)_{Z}$ is as well. Thus $\iota_{N\Delta^{{\mathrm{op}}}}\mathscr{K}\!(i)$ is equivalent to the discrete simplicial set $A$, as desired.
In the situation of Pr. \[prp:localizationofperfwaldcat\], we find that the natural map $${{K((wC^{\omega})^{-1}C^{\omega})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {K(D^{\omega})}}$$ is a homotopy monomorphism; that is, it induces an inclusion on $\pi_0$ and an isomorphism on $\pi_k$ for $k\geq 1$. We therefore obtain the following.
\[cor:specialfibration\] Suppose $C$ a compactly generated $\infty$-category containing a zero object, suppose $L\colon{{C}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {D}}$ an accessible localization, and suppose the inclusion ${{D}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {C}}$ preserves filtered colimits. Assume also that the class of all $L$-equivalences of $C$ is generated (as a strongly saturated class) by the $L$-equivalences between compact objects. Then $L$ induces a pullback square of spaces $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{K(E^{\omega})&K(C^{\omega})\\
\ast&K(D^{\omega}),\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge (m-2-2);
\end{tikzpicture}$$ where $C^{\omega}$ and $D^{\omega}$ are equipped with the maximal pair structure, and $E^{\omega}\subset C^{\omega}$ is the full subcategory spanned by those objects $X$ such that $LX\simeq 0$.
A further specialization of this result is now possible. Suppose $C$ a compactly generated stable $\infty$-category. Then $C=\operatorname{Ind}(A)$ for some small $\infty$-category $A$, and so, since $\operatorname{Ind}(A)\subset\mathscr{P}(A)$ is closed under filtered colimits and finite limits, it follows that filtered colimits of $C$ are left exact [@HTT Df. 7.3.4.2]. Suppose also that $C$ is equipped with a t-structure such that $C_{\leq 0}\subset C$ is stable under filtered colimits. Then the localization $\tau_{\geq 1}\colon{{C}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}$, being the fiber of the natural transformation ${{\operatorname{id}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\tau_{\leq 0}}}$, preserves filtered colimits as well. Now by [@HA Pr. 1.2.1.16], the class $S$ of morphisms $f$ such that $\tau_{\leq 0}(f)$ is an equivalence is generated as a quasi-saturated class by the class $\{{{0}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X}}\ |\ X\in C_{\geq 1}\}$. But now writing $X$ as a filtered colimit of compact objects and applying $\tau_{\geq 1}$, we find that $S$ is generated under filtered colimits in $\operatorname{Fun}(\Delta^1,C)$ by the set $\{{{0}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {X}}\ |\ X\in C^{\omega}\cap C_{\geq 1}\}$. Hence the $\tau_{\leq 0}$-equivalences are generated by $\tau_{\leq 0}$-equivalences between compact objects, and we have the following.
\[cor:tstruct\] Suppose $C$ a compactly generated stable $\infty$-category. Suppose also that $C$ is equipped with a t-structure such that $C_{\leq 0}\subset C$ is stable under filtered colimits. Then the functor $\tau_{\leq 0}$ induces a pullback square $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{K(C^{\omega}\cap C_{\geq 1})&K(C^{\omega})\\
\ast&K(C^{\omega}\cap C_{\leq 0}),\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge (m-2-2);
\end{tikzpicture}$$ where the $\infty$-categories that appear are equipped with their maximal pair structure.
In particular, we can exploit the equivalence of [@HTT Pr. 5.5.7.8] to deduce the following.
Suppose $A$ a small stable $\infty$-category that is equipped with a t-structure. Then the functor $\tau_{\leq 0}$ induces a pullback square $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{K(A_{\geq 1})&K(A)\\
\ast&K(A_{\leq 0}),\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge (m-2-2);
\end{tikzpicture}$$ where the $\infty$-categories that appear are equipped with their maximal pair structure.
If $A$ is idempotent-complete, then we can appeal to Cor. \[cor:tstruct\] and [@HTT Pr. 5.5.7.8] directly. If not, then we may embed $A$ in its idempotent completion $A'$, and we extend the t-structure using the condition that any summand of an object $X\in A_{\leq 0}$ (respectively, $X\in A_{\geq 1}$) must lie in $A'_{\leq 0}$ (resp., $A'_{\geq 1}$). Now we appeal to the Cofinality Theorem \[thm:cofinality\] to complete the proof.
Example: Algebraic $K$-theory of $E_1$-algebras {#sect:example1}
===============================================
To any associative ring in any suitable monoidal $\infty$-category we can attach its $\infty$-category of modules. We may then impose suitable finiteness hypotheses on these modules and extract a $K$-theory spectrum. Here we identify some important examples of these $K$-theory spectra.
Suppose $\mathscr{A}$ a presentable, symmetric monoidal $\infty$-category [@HA Df. 2.0.0.7] with the property that the tensor product $\otimes\colon{{\mathscr{A}\times\mathscr{A}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{A}}}$ preserves (small) colimits separately in each variable; assume also that $\mathscr{A}$ is *additive* in the sense that it admits direct sums (Df. \[item:directsums\]), and the resulting commutative monoids $\operatorname{Mor}_{h\mathscr{A}}(X,Y)$ are all abelian groups. We denote by ${\mathbf{Alg}}(\mathscr{A})$ the $\infty$-category of $E_1$-algebras in $\mathscr{A}$, and we denote by ${\mathbf{Mod}}^{\ell}(\mathscr{A})$ the $\infty$-category $\mathrm{LMod}(\mathscr{A})$ defined in [@HA Df. 4.2.1.13]. We have the canonical presentable fibration $$\theta\colon{{{\mathbf{Mod}}^{\ell}(\mathscr{A})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Alg}}(\mathscr{A})}}$$ [@HA Cor. 4.2.3.7], whose fiber over any $E_1$-algebra $\Lambda$ is the stable $\infty$-category ${\mathbf{Mod}}^{\ell}_{\Lambda}$ of left $\Lambda$-modules. Informally, we describe the objects of ${\mathbf{Mod}}^{\ell}(\mathscr{A})$ as pairs $(\Lambda,E)$ consisting of an $E_1$-algebra $\Lambda$ in $\mathscr{A}$ and a left $\Lambda$-module $E$.
Our aim now is to impose hypotheses on the objects of $(\Lambda,E)$ and pair structures on the resulting full subcategories in order to ensure that the restriction of $\theta$ is a Waldhausen cocartesian fibration.
\[ntn:perfectmodule\] For any $E_1$-algebra $\Lambda$ in $\mathscr{A}$, a left $\Lambda$-module $E$ will be said to be ***perfect*** if it satisfies the following two conditions.
1. As an object of the $\infty$-category ${\mathbf{Mod}}^{\ell}_{\Lambda}$ of left $\Lambda$-modules, $E$ is compact.
2. The functor ${{{\mathbf{Mod}}^{\ell}_{\Lambda}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{A}}}$ corepresented by $E$ is exact.
Denote by ${\mathbf{Perf}}^{\ell}(\mathscr{A})\subset{\mathbf{Mod}}^{\ell}(\mathscr{A})$ the full subcategory spanned by those pairs $(\Lambda,E)$ in which $E$ is perfect.
These two conditions can be more efficiently expressed by saying that $E$ is perfect just in case the functor ${{{\mathbf{Mod}}^{\ell}_{\Lambda}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{A}}}$ corepresented by $E$ preserves all small colimits. Note that this is *not* the same as *complete compactness*, i.e., requiring that the functor ${{{\mathbf{Mod}}^{\ell}_{\Lambda}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$ corepresented by $E$ preserves all small colimits.
When $\mathscr{A}$ is the nerve of the ordinary category of abelian groups, ${\mathbf{Alg}}(\mathscr{A})$ is the category of associative rings, and ${\mathbf{Mod}}^{\ell}(\mathscr{A})$ is the nerve of the ordinary category of pairs $(\Lambda,E)$ consisting of an associative ring $\Lambda$ and a left $\Lambda$-module $E$. An $\Lambda$-module $E$ is perfect just in case it is (1) finitely presented and (2) projective. Thus ${\mathbf{Perf}}^{\ell}_{\Lambda}$ is the nerve of the ordinary category of finitely generated projective $\Lambda$-modules.
\[exm:summapyperfinconn\] When $\mathscr{A}$ is the $\infty$-category of connective spectra, ${\mathbf{Alg}}(\mathscr{A})$ can be identified with the $\infty$-category of connective $E_1$-rings, and ${\mathbf{Mod}}^{\ell}(\mathscr{A})$ can be identified with the $\infty$-category of pairs $(\Lambda,E)$ consisting of a connective $E_1$-ring $\Lambda$ and a connective left $\Lambda$-module $E$. Since the functor $\Omega^{\infty}\colon{{{\mathbf{Sp}}_{\geq 0}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$ is conservative [@HA Cor. 5.1.3.9] and preserves sifted colimits [@HA Pr. 1.4.3.9], it follows using [@HA Lm. 1.3.3.10] the second condition of Df. \[ntn:perfectmodule\] amounts to the requirement that $E$ be a projective object. Now [@HA Pr. 8.2.2.6 and Cor. 8.2.2.9] guarantees that the following are equivalent for a left $\Lambda$-module $E$.
1. The left $\Lambda$-module $E$ is perfect.
2. The left $\Lambda$-module $E$ is projective, and $\pi_0E$ is finitely generated as a $\pi_0\Lambda$-module.
3. The $\pi_0\Lambda$-module $\pi_0E$ is finitely generated, and for every $\pi_0A$-module $M$ and every integer $m\geq 1$, the abelian group $\operatorname{Ext}^m(E,M)$ vanishes.
4. There exists a finitely generated free $\Lambda$-module $F$ such that $E$ is a retract of $F$.
\[exm:summapyperfinsmod\] The situation for modules over simplicial associative rings is nearly identical. When $\mathscr{A}$ is the $\infty$-category of simplicial abelian groups, ${\mathbf{Alg}}(\mathscr{A})$ can be identified with the $\infty$-category of simplicial associative rings, and ${\mathbf{Mod}}^{\ell}(\mathscr{A})$ can be identified with the $\infty$-category of pairs $(\Lambda,E)$ consisting of a simplicial associative ring $\Lambda$ and a left $\Lambda$-module $E$. Since the forgetful functor ${{\mathscr{A}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$ is conservative and preserves sifted colimits, it follows that the second condition of Df. \[ntn:perfectmodule\] amounts to the requirement that $E$ be a projective object. One may show that the following are equivalent for a left $\Lambda$-module $E$.
1. The left $\Lambda$-module $E$ is perfect.
2. The left $\Lambda$-module $E$ is projective, and $\pi_0E$ is finitely generated as a $\pi_0A$-module.
3. The $\pi_0\Lambda$-module $\pi_0E$ is finitely generated, and for every $\pi_0\Lambda$-module $M$ and every integer $m\geq 1$, the abelian group $\operatorname{Ext}^m(E,M)$ vanishes.
4. There exists a finitely generated free $\Lambda$-module $F$ such that $E$ is a retract of $F$.
\[nul:summaryperfectmad\] When $\mathscr{A}$ is the $\infty$-category of *all* spectra, ${\mathbf{Alg}}(\mathscr{A})$ is the $\infty$-category of $E_1$-rings, and ${\mathbf{Mod}}^{\ell}(\mathscr{A})$ is the $\infty$-category of pairs $(\Lambda,E)$ consisting of an $E_1$-ring $\Lambda$ and a left $\Lambda$-module $E$. Suppose $\Lambda$ an $E_1$-ring. The second condition of Df. \[ntn:perfectmodule\] is vacuous since $\mathscr{A}$ is stable. Hence by [@HA Pr. 8.2.5.4], the following are equivalent for a left $\Lambda$-module $E$.
1. The left $\Lambda$-module $E$ is perfect.
2. The left $\Lambda$-module $E$ is contained in the smallest stable subcategory of the $\infty$-category ${\mathbf{Mod}}^{\ell}_{\Lambda}$ of left $\Lambda$-modules that contains $\Lambda$ itself and is closed under retracts.
3. The left $\Lambda$-module $E$ is compact as an object of ${\mathbf{Mod}}^{\ell}_{\Lambda}$.
4. There exists a right $\Lambda$-module $E^{\vee}$ such that the functor ${{{\mathbf{Mod}}^{\ell}_{\Lambda}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$ informally written as $\Omega^{\infty}(E^{\vee}\otimes_{\Lambda}-)$ is corepresented by $E$.
Now we wish to endow ${\mathbf{Perf}}^{\ell}(\mathscr{A})$ with a suitable pair structure. In general, this may not be possible, but we can isolate those situations in which it is possible.
\[cofsinperf\] Denote by $S$ the class of morphisms ${{(\Lambda',E')}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(\Lambda,E)}}$ of the $\infty$-category ${\mathbf{Perf}}^{\ell}(\mathscr{A})$ with the following two properties.
1. The morphism ${{\Lambda'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Lambda}}$ of ${\mathbf{Alg}}(\mathscr{A})$ is an equivalence.
2. Any pushout diagram $$\begin{tikzpicture}[baseline]
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{(\Lambda',E')&(\Lambda,E)\\
(\Lambda',0)&(\Lambda,E'')\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge (m-2-2);
\end{tikzpicture}$$ in ${\mathbf{Mod}}^{\ell}(\mathscr{A})$ in which $0\in{\mathbf{Mod}}^{\ell}_{\Lambda'}$ is a zero object is also a pullback diagram, and the $\Lambda$-module $E''$ is perfect.
We shall say that $\mathscr{A}$ is *admissible* if the class $S$ is stable under pushout in ${\mathbf{Perf}}^{\ell}(\mathscr{A})$ and composition.
When $\mathscr{A}$ is the nerve of the category of abelian groups, $S$ is the class of morphisms ${{(\Lambda',E')}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(\Lambda,E)}}$ such that ${{\Lambda'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Lambda}}$ is an isomorphism, and the induced map of $\Lambda'$-modules ${{E'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {E}}$ is an admissible monomorphism. It is a familiar fact that these are closed under pushout and composition, so that the nerve of the category of abelian groups is admissible.
When $\mathscr{A}$ is the $\infty$-category of connective spectra or the $\infty$-category of simplicial abelian groups, $S$ is the class of morphisms ${{(\Lambda',E')}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {(\Lambda,E)}}$ such that ${{\Lambda'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Lambda}}$ is an equivalence, and the induced homomorphism $${{\operatorname{Ext}^0(E,M)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Ext}^0(E',M)}}$$ is a surjection for every $\pi_0\Lambda'$-module $M$. This is visibly closed under composition. To see that these are closed under pushouts, let us proceed in two steps. First, for any morphism ${{\Lambda}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Lambda'}}$ of ${\mathbf{Alg}}(\mathscr{A})$, the functor informally described as ${{E}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {E\otimes_{\Lambda}\Lambda'}}$ clearly carries morphisms of ${\mathbf{Perf}}^{\ell}_{\Lambda}$ that lie in $S$ to morphisms of ${\mathbf{Perf}}^{\ell}_{\Lambda'}$ that lie in $S$. Now, for a fixed $E_1$-algebra $\Lambda$ in $\mathscr{A}$, suppose $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{E'&E\\
F'&F\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge (m-2-2);
\end{tikzpicture}$$ a pushout square in ${\mathbf{Perf}}^{\ell}_{\Lambda}$ in which ${{E'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {E}}$ lies in the class $S$, and suppose $M$ a $\pi_0\Lambda$-module $M$. For any morphism ${{F'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {M}}$, one may precompose to obtain a morphism ${{E'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {M}}$. Our criterion on the morphism ${{E'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {E}}$ now guarantees that there is a commutative square $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{E'&E\\
F'&M\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge (m-2-2);
\end{tikzpicture}$$ up to homotopy. Now the universal property of the pushout yields a morphism ${{F}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {M}}$ that extends the morphism ${{F'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {M}}$, up to homotopy. Thus both connective spectra and simplicial abelian groups are admissible $\infty$-categories.
When $\mathscr{A}$ is the $\infty$-category of all spectra, every morphism is contained in the class $S$. Hence the $\infty$-category of all spectra is an admissible $\infty$-category.
If $\mathscr{A}$ is admissible, denote by ${\mathbf{Perf}}^{\ell}_{\dag}(\mathscr{A})$ the subcategory of ${\mathbf{Perf}}^{\ell}(\mathscr{A})$ whose morphisms are those that lie in the class $S$. With this pair structure, the $\infty$-category ${\mathbf{Perf}}^{\ell}(\mathscr{A})$ is a Waldhausen $\infty$-category.
If $\mathscr{A}$ is admissible, then the functor ${{{\mathbf{Perf}}^{\ell}(\mathscr{A})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Alg}}(\mathscr{A})}}$ is a Waldhausen cocartesian fibration.
It is clear that the fibers of this cocartesian fibration are Waldhausen $\infty$-categories. We claim that for any morphism ${{\Lambda'}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Lambda}}$ of $E_1$-algebras, the corresponding functor $${{{\mathbf{Mod}}^{\ell}_{\Lambda'}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Mod}}^{\ell}_{\Lambda}}}$$ given informally by the assignment ${{E'}\ \tikz[baseline]\draw[|->](0,0.5ex)--(0.5,0.5ex);\ {\Lambda\otimes_{\Lambda'}E'}}$ carries perfect modules to perfect modules. Indeed, it is enough to show that the right adjoint functor $${{{\mathbf{Mod}}^{\ell}_{\Lambda}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Mod}}^{\ell}_{\Lambda'}}}$$ preserves small colimits. This is immediate, since colimits are computed in the underlying $\infty$-category $\mathscr{A}$ [@HA Pr. 3.2.3.1].
The induced functor ${{{\mathbf{Perf}}^{\ell}_{\Lambda'}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Perf}}^{\ell}_{\Lambda}}}$ carries a cofibration ${{F'}\ \tikz[baseline]\draw[>=stealth,>->](0,0.5ex)--(0.5,0.5ex);\ {E'}}$ to the morphism of left $\Lambda$-modules ${{F'\otimes_{\Lambda'}\Lambda}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {E'\otimes_{\Lambda'}\Lambda}}$, which fits into a pushout square $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{(\Lambda',F')&(\Lambda',E')\\
(\Lambda,F'\otimes_{\Lambda'}\Lambda)&(\Lambda,E'\otimes_{\Lambda'}\Lambda)\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge (m-2-2);
\end{tikzpicture}$$ in ${\mathbf{Perf}}^{\ell}(\mathscr{A})$; hence ${{F'\otimes_{\Lambda'}\Lambda}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {E'\otimes_{\Lambda'}\Lambda}}$ is a cofibration.
The ***algebraic $K$-theory of $E_1$-rings***, which we will abusively denote $${\mathbf{K}}\colon{{{\mathbf{Alg}}(\mathscr{A})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Sp}}_{\geq0}}},$$ is the composite functor ${\mathbf{K}}\circ P$, where $P\colon{{{\mathbf{Alg}}(\mathscr{A})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}}}$ is the functor classified by the Waldhausen cocartesian fibration ${{{\mathbf{Perf}}^{\ell}(\mathscr{A})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Alg}}(\mathscr{A})}}$.
The preceding definition ensures that $K$ is well-defined up to a contractible ambiguity. To obtain an explicit model of $K$, we proceed in the following manner. Apply $\mathscr{S}$ to ${{{\mathbf{Perf}}^{\ell}(\mathscr{A})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Alg}}(\mathscr{A})}}$ in order to obtain a Waldhausen cocartesian fibration ${{\mathscr{S}({\mathbf{Perf}}^{\ell}(\mathscr{A}))}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {N\Delta^{{\mathrm{op}}}\times{\mathbf{Alg}}(\mathscr{A})}}$. Now consider the subcategory $$\iota_{(N\Delta^{{\mathrm{op}}}\times{\mathbf{Alg}}(\mathscr{A}))}\mathscr{S}({\mathbf{Perf}}^{\ell}(\mathscr{A}))\subset\mathscr{S}({\mathbf{Perf}}^{\ell}(\mathscr{A}))$$ consisting of cocartesian edges. The composite $$\iota_{(N\Delta^{{\mathrm{op}}}\times{\mathbf{Alg}}(\mathscr{A}))}\mathscr{S}{\mathbf{Perf}}^{\ell}(\mathscr{A})\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--(0.75,0.5ex);\ N\Delta^{{\mathrm{op}}}\times{\mathbf{Alg}}(\mathscr{A})\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--(0.75,0.5ex);\ {\mathbf{Alg}}(\mathscr{A})$$ is now a left fibration with a contractible space of sections given by $${\mathbf{Alg}}(\mathscr{A})\cong\{0\}\times{\mathbf{Alg}}(\mathscr{A})\ \tikz[baseline]\draw[>=stealth,<-,font=\scriptsize,inner sep=0.5pt](0,0.5ex)--node[above]{$\sim$}(0.5,0.5ex);\ \iota\mathscr{S}_0{\mathbf{Perf}}^{\ell}(\mathscr{A})\ \tikz[baseline]\draw[>=stealth,right hook->,font=\scriptsize](0,0.5ex)--(0.75,0.5ex);\ \iota_{(N\Delta^{{\mathrm{op}}}\times{\mathbf{Alg}}(\mathscr{A}))}\mathscr{S}{\mathbf{Perf}}^{\ell}(\mathscr{A}).$$ It is clear by construction that this left fibration classifies a functor $$L\colon{{{\mathbf{Alg}}(\mathscr{A})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}}}$$ such that $K\simeq\Omega\circ L$.
Let us now concentrate on the case in which $\mathscr{A}$ is the $\infty$-category of spectra.
\[prp:protolocforE1rings\] Suppose $\Lambda$ an $E_1$ ring spectrum, and suppose $S\subset\pi_{\ast}\Lambda$ a collection of homogeneous elements satisfying the left Ore condition [@HA Df. 8.2.4.1]. Then the morphism ${{\Lambda}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\Lambda[S^{-1}]}}$ of ${\mathbf{Alg}}({\mathbf{Sp}})$ induces a fiber sequence of connective spectra $${\mathbf{K}}(\mathbf{Nil}_{(\Lambda,S)}^{\ell,\omega})\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{K}}(\Lambda)\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{K}}(\Lambda[S^{-1}]),$$ where $\mathbf{Nil}_{(\Lambda,S)}^{\ell,\omega}\subset{\mathbf{Perf}}^{\ell}_{\Lambda}$ is the full subcategory spanned by those perfect left $\Lambda$-modules that are $S$-nilpotent.
Consider the t-structure $$(\mathbf{Nil}_{(\Lambda,S)}^{\ell},\mathbf{Loc}_{(\Lambda,S)}^{\ell}),$$ where $\mathbf{Nil}_{(\Lambda,S)}^{\ell}\subset{\mathbf{Mod}}^{\ell}_{\Lambda}$ is the full subcategory spanned by the $S$-nilpotent left $\Lambda$-modules, and $\mathbf{Loc}_{(\Lambda,S)}^{\ell}\subset{\mathbf{Mod}}^{\ell}_A$ is the full subcategory spanned by the $S$-local left $\Lambda$-modules. We claim that this t-structure restricts to one on ${\mathbf{Perf}}_{\Lambda}^{\ell}$. To this end, we note that ${\mathbf{Mod}}^{\ell}_{\Lambda}$ is compactly generated, and $\mathbf{Loc}_{(\Lambda,S)}^{\ell}\subset{\mathbf{Mod}}^{\ell}_{\Lambda}$ is in fact stable under all colimits [@HA Rk. 8.2.4.16]. Now we apply Cor. \[cor:tstruct\], and our description of the cofiber term now follows from the discussion preceding [@HA Rk. 8.2.4.26].
Such a result is surely well-known among experts; see for example [@BM Pr. 1.4 and Pr. 1.5].
\[exm:ARconj\] For a prime $p$ (suppressed from the notation) and an integer $n\geq 0$, the truncated Brown–Peterson spectra $\mathrm{BP}\langle n\rangle$, with coefficient ring $$\pi_{\ast}\mathrm{BP}\langle n\rangle\cong{\mathbf{Z}}_{(p)}[v_1,v_2,\dots,v_n]$$ admit compatible $E_1$ structures [@MR1990937 p. 506]. We may consider the multiplicative system $S\subset\pi_{\ast}\mathrm{BP}\langle n\rangle$ of homogeneous elements generated by $v_n$. Then $\mathrm{BP}\langle n\rangle[v_n^{-1}]$ is an $E_1$-algebra equivalent to the Johnson–Wilson spectrum $E(n)$. The exact sequence above yields a fiber sequence of connective spectra $${\mathbf{K}}(\mathbf{Nil}_{(\mathrm{BP}\langle n\rangle,S)}^{\ell,\omega})\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{K}}(\mathrm{BP}\langle n\rangle)\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{K}}(E(n)).$$ The content of a well-known conjecture of Ausoni–Rognes [@MR1947457 (0.2)] identifies the fiber term (possibly after p-adic completion) as ${\mathbf{K}}(\mathrm{BP}\langle n-1\rangle)$. In light of results such as [@HA Lm. 8.4.2.13], such a result will follow from a suitable form of a *Dévissage Theorem* [@MR0338129 Th. 4]; we hope to return to such a result in later work (cf. [@MR92f:19001 1.11.1]).
Of course, when $n=1$, such a Dévissage Theorem has already been provided thanks to beautiful work of Andrew Blumberg and Mike Mandell [@BM]. They prove that the $K$-theory of the $\infty$-category of perfect, $\beta$-nilpotent modules over the $p$-local Adams summand can be identified with the $K$-theory of ${\mathbf{Z}}_{(p)}$. Consequently, they provide a fiber sequence $$K({\mathbf{Z}}_{(p)})\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ K(\ell)\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ K(L).$$
Example: Algebraic $K$-theory of derived stacks {#sect:example2}
===============================================
Here we introduce the algebraic $K$-theory of spectral Deligne–Mumford stacks in the sense of Lurie, and we prove an easy localization theorem (analogous to what Thomason called the “Proto-localization Theorem”) in this context.
We appeal here to the theory of nonconnective spectral Deligne–Mumford stacks and their module theory as exposed in [@DAGV; @DAGVIII]. Much of what we will say can probably be done in other contexts of derived algebraic geometry as well, such as [@MR2137288; @MR2394633]; we have opted to use Lurie’s approach only because that is the one with which we are least unfamiliar. We begin by summarizing some general facts about quasicoherent modules over nonconnective spectral Deligne–Mumford stacks. Since Lurie at times concentrates on connective Deligne–Mumford stacks, we will at some points comment on how to extend the relevant definitions and results to the nonconnective case.
\[nul:summary\] Recall from [@DAGVIII §2.3, Pr. 2.5.1] that the functor $${{{\mathbf{Sch}}(\mathscr{G}_{\textrm{\'et}}^{n\mathrm{M}})^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Stk}}^{\mathrm{nc}}}}$$ is a cocartesian fibration, and its fiber over a nonconnective spectral Deligne–Mumford stack $(\mathscr{E},\mathscr{O})$ is the stable, presentable $\infty$-category ${\mathbf{QCoh}}(\mathscr{E},\mathscr{O})$ of *quasicoherent* $\mathscr{O}$-modules.
For any nonconnective Deligne–Mumford stack $(\mathscr{E},\mathscr{O})$, the following are equivalent for an $\mathscr{O}$-module $\mathscr{M}$.
1. The $\mathscr{O}$-module $\mathscr{M}$ is quasicoherent.
2. For any morphism ${{U}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {V}}$ of $\mathscr{E}$ such that $(\mathscr{X}_{/U},\mathscr{O}_{|U})$ and $(\mathscr{X}_{/V},\mathscr{O}_{|V})$ are affine, the natural morphism ${{\mathscr{M}(V)\otimes_{\mathscr{O}(V)}\mathscr{O}(U)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{M}(U)}}$ is an equivalence.
3. \[item:simplecharqcoh\] The following conditions obtain.
1. For every integer $n$, the homotopy sheaf $\pi_n\mathscr{M}$ is a quasicoherent module on the underlying ordinary Deligne–Mumford stack of $(\mathscr{E},\mathscr{O})$
2. The object $\Omega^{\infty}\mathscr{M}$ is hypercomplete in the $\infty$-topos $\mathscr{E}$.
Using ideas from [@DAGVIII §2.7], we shall now make sense of the notion of quasicoherent module over any functor ${{\mathbf{CAlg}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}(\kappa_1)}}$. As suggested in [@DAGVIII Rk. 2.7.9], write $${\mathbf{QCoh}}\colon{{\operatorname{Fun}(\mathbf{CAlg},{\mathbf{Kan}}(\kappa_1))^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}(\kappa_1)}}$$ for the right Kan extension of the functor ${{\mathbf{CAlg}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Cat}}_{\infty}(\kappa_1)}}$ that classifies the cocartesian fibration ${{{\mathbf{Mod}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{CAlg}}}$. Then for any functor $X\colon{{\mathbf{CAlg}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}(\kappa_1)}}$, we obtain the $\infty$-category of *quasicoherent modules* ${\mathbf{QCoh}}(X)$ on the functor $X$. Many of the results of §2.7 of loc. cit. hold in this context with precisely the same proofs, including the following brace of results.
For any functor $X\colon{{\mathbf{CAlg}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}(\kappa_1)}}$, the $\infty$-category ${\mathbf{QCoh}}(X)$ is stable.
\[prp:nonconnversionof2718\] Suppose $(\mathscr{E},\mathscr{O})$ a nonconnective Deligne–Mumford stack representing a functor $X\colon{{\mathbf{CAlg}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}(\kappa_1)}}$. Then there is a canonical equivalence of $\infty$-categories $${\mathbf{QCoh}}(\mathscr{E},\mathscr{O})\simeq{\mathbf{QCoh}}(X).$$
Suppose $X\colon{{\mathbf{CAlg}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}(\kappa_1)}}$ a functor. We say that a quasicoherent module $\mathscr{M}$ on $X$ is ***perfect*** if for any $E_{\infty}$ ring $A$ and any point $x\in X(A)$, the $A$-module $\mathscr{M}(x)$ is perfect (Df. \[ntn:perfectmodule\]). Write ${\mathbf{Perf}}(X)\subset{\mathbf{QCoh}}(X)$ for the full subcategory spanned by the perfect modules.
In particular, we can now use Pr. \[prp:nonconnversionof2718\] to specialize the notion of perfect module to the setting of nonconnective Deligne–Mumford stacks.
Denote by ${\mathbf{Perf}}\subset{\mathbf{Sch}}(\mathscr{G}_{\textrm{\'et}}^{\mathrm{M}})^{{\mathrm{op}}}$ the full subcategory of those objects $(\mathscr{E},\mathscr{O},\mathscr{M})$ such that $\mathscr{M}$ is perfect.
For any functor $X\colon{{\mathbf{CAlg}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}(\kappa_1)}}$, the $\infty$-category ${\mathbf{QCoh}}(X)$ admits a symmetric monoidal structure [@DAGVIII Nt. 2.7.27]. Moreover, this is functorial, yielding a functor $${\mathbf{QCoh}}^{\otimes}\colon{{\operatorname{Fun}(\mathbf{CAlg},{\mathbf{Kan}}(\kappa_1))^{{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{CAlg}({\mathbf{Cat}}_{\infty}(\kappa_1))}}.$$
For any functor $X\colon{{\mathbf{CAlg}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Kan}}(\kappa_1)}}$ a quasicoherent module $\mathscr{M}$ on $X$ is perfect if and only if it is a dualizable object of ${\mathbf{QCoh}}(X)$.
Since the pullback functors are symmetric monoidal, they preserve dualizable objects. This proves the following.
The functor ${{{\mathbf{Perf}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Stk}}^{\mathrm{nc}}}}$ is a cocartesian fibration.
We endow ${\mathbf{Perf}}$ with a pair structure by ${\mathbf{Perf}}_{\dag}{\mathrel{\mathop:}=}{\mathbf{Perf}}\times_{{\mathbf{Stk}}^{\mathrm{nc}}}\iota{\mathbf{Stk}}^{\mathrm{nc}}$, so that the fibers are equipped with the maximal pair structure.
The functor ${{{\mathbf{Perf}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Stk}}^{\mathrm{nc}}}}$ is a Waldhausen cocartesian fibration.
In fact, the fiber over a nonconnective Deligne–Mumford stack $(\mathscr{E},\mathscr{O})$ is a stable $\infty$-category ${\mathbf{Perf}}(\mathscr{E},\mathscr{O})$.
\[dfn:algKthyDMstacks\] We now define the ***algebraic $K$-theory of nonconnective Deligne–Mumford stacks*** as a functor that we abusively denote $${\mathbf{K}}\colon{{{\mathbf{Stk}}^{\mathrm{nc}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Sp}}_{\geq 0}}}$$ given by the composite ${\mathbf{K}}\circ P$, where $P$ is the functor ${{{\mathbf{Stk}}^{\mathrm{nc},{\mathrm{op}}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Wald}_{\infty}}}}$ classified by the Waldhausen cocartesian fibration ${{{\mathbf{Perf}}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Stk}}^{\mathrm{nc}}}}$.
For any open immersion of quasicompact nonconnective spectral Deligne–Mumford stacks $j\colon{{\mathscr{U}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{X}}}$, the induced functor $$j_{\star}\colon{{{\mathbf{QCoh}}(\mathscr{U})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{QCoh}}(\mathscr{X})}}$$ is fully faithful.
When $\mathscr{X}$ is of the form $\operatorname{Spec}^{\textrm{\'et}}A$, this is proved in [@DAGVIII Cor. 2.4.6]. For any map $x\colon{{\operatorname{Spec}^{\textrm{\'et}}A}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{X}}}$, we have the open immersion $${{\mathscr{U}\times_{\mathscr{X}}\operatorname{Spec}^{\textrm{\'et}}A}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Spec}^{\textrm{\'et}}A}},$$ which induces a fully faithful functor $${{{\mathbf{QCoh}}(\mathscr{U}\times_{\mathscr{X}}\operatorname{Spec}^{\textrm{\'et}}A)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{QCoh}}(\operatorname{Spec}^{\textrm{\'et}}A)}}.$$ Now letting $A$ vary and applying [@DAGVIII Pr. 2.4.5(3)], we obtain a functor $${{\mathbf{CAlg}_{\mathscr{X}/}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{O}({\mathbf{Cat}}_{\infty}(\kappa_1))}}$$ whose values are all fully faithful functors. Thanks to Pr. \[prp:nonconnversionof2718\], the limit of this functor is then equivalent to a functor $$\alpha\colon{{\lim_{A\in\mathbf{CAlg}_{\mathscr{X}/}}{\mathbf{QCoh}}(\mathscr{U}\times_{\mathscr{X}}\operatorname{Spec}^{\textrm{\'et}}A)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{QCoh}}(\mathscr{X})}},$$ which is thus fully faithful. We aim to identify this functor with $j_{\star}$.
Since each of the $\infty$-categories ${\mathbf{QCoh}}(\mathscr{U}\times_{\mathscr{X}}\operatorname{Spec}^{\textrm{\'et}}A)$ can itself be described as the limit of the $\infty$-categories ${\mathbf{Mod}}_B$ for $B\in\mathbf{CAlg}_{\mathscr{U}\times_{\mathscr{X}}\operatorname{Spec}^{\textrm{\'et}}A/}$, it follows that the source of $\alpha$ can be expressed as the limit of the $\infty$-categories ${\mathbf{Mod}}_B$ over the $\infty$-category $C$ of squares of nonconnective Deligne–Mumford stacks of the form $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\operatorname{Spec}^{\textrm{\'et}}B&\operatorname{Spec}^{\textrm{\'et}}A\\
\mathscr{U}&\mathscr{X}.\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge (m-1-2)
edge (m-2-1)
(m-1-2) edge (m-2-2)
(m-2-1) edge node[below]{$j$} (m-2-2);
\end{tikzpicture}$$ Now there is a forgetful functor $g\colon{{C}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathbf{CAlg}_{\mathscr{U}/}}}$ that carries an object as above to the morphism ${{\operatorname{Spec}^{\textrm{\'et}}B}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{U}}}$. This is the functor that induces the canonical functor $${{\lim_{A\in\mathbf{CAlg}_{\mathscr{X}/}}{\mathbf{QCoh}}(\mathscr{U}\times_{\mathscr{X}}\operatorname{Spec}^{\textrm{\'et}}A)}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{QCoh}}(\mathscr{U})}};$$ hence it suffices to show that $g$ is right cofinal. This now follows from the fact that the functor $g$ admits a right adjoint ${{\mathbf{CAlg}_{\mathscr{U}/}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {C}}$, which carries a morphism $x\colon{{\operatorname{Spec}^{\textrm{\'et}}C}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{U}}}$ to the object $$\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=4ex, column sep=4ex,
text height=1.5ex, text depth=0.25ex]
{\operatorname{Spec}^{\textrm{\'et}}C&\operatorname{Spec}^{\textrm{\'et}}C\\
\mathscr{U}&\mathscr{X}.\\};
\path[>=stealth,->,font=\scriptsize]
(m-1-1) edge[-,double distance=1.5pt] (m-1-2)
edge node[left]{$x$} (m-2-1)
(m-1-2) edge node[right]{$j\circ x$} (m-2-2)
(m-2-1) edge node[below]{$j$} (m-2-2);
\end{tikzpicture}$$ The proof is complete.
For any open immersion $j\colon{{\mathscr{U}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{X}}}$ of quasicompact nonconnective spectral Deligne–Mumford stacks, let us write ${\mathbf{Perf}}(\mathscr{X},\mathscr{X}\setminus\mathscr{U})$ for the full subcategory of ${\mathbf{Perf}}(\mathscr{X})$ spanned by those perfect modules $\mathscr{M}$ on $\mathscr{X}$ such that $j^{\star}\mathscr{M}\simeq0$. Write $${\mathbf{K}}(\mathscr{X},\mathscr{X}\setminus\mathscr{U}){\mathrel{\mathop:}=}{\mathbf{K}}({\mathbf{Perf}}(\mathscr{X},\mathscr{X}\setminus\mathscr{U})).$$
\[prp:protolocforDMstacks\] For any quasicompact open immersion $j\colon{{\mathscr{U}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{X}}}$ of quasicompact, quasiseparated spectral algebraic spaces [@DAGVIII Df. 1.3.1] and [@DAGXII Df. 1.3.1], the functor $j^{\star}\colon{{{\mathbf{Perf}}(\mathscr{X})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Perf}}(\mathscr{U})}}$ induces a fiber sequence of connective spectra $${\mathbf{K}}(\mathscr{X},\mathscr{X}\setminus\mathscr{U})\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--(0.75,0.5ex);\ {\mathbf{K}}(\mathscr{X})\ \tikz[baseline]\draw[>=stealth,->,font=\scriptsize](0,0.5ex)--(0.75,0.5ex);\ {\mathbf{K}}(\mathscr{U}).$$
We wish to employ the Special Fibration Theorem \[cor:specialfibration\]. We note that by [@DAGXII Cor. 1.5.12], the $\infty$-category ${\mathbf{QCoh}}(\mathscr{X})$ is compactly generated, and one has ${\mathbf{Perf}}(\mathscr{X})={\mathbf{QCoh}}(\mathscr{X})^{\omega}$; the analogous claim holds for $\mathscr{U}$. It thus remains to show that $j^{\star}$-equivalences of ${\mathbf{QCoh}}(\mathscr{X})$ — i.e., the class of morphisms of ${\mathbf{QCoh}}(\mathscr{X})$ whose restriction to $\mathscr{U}$ is an equivalence — is generated (as a strongly saturated class) by $j^{\star}$-equivalences between compact objects. Since ${\mathbf{QCoh}}(\mathscr{X})$ is stable, we find that it suffices to show that the full subcategory ${\mathbf{QCoh}}(\mathscr{X},\mathscr{X}-\mathscr{U})$ of ${\mathbf{QCoh}}(\mathscr{X})$ spanned by the $j^{\star}$-acyclics — i.e., those quasicoherent modules $M$ such that $j^{\star}M\simeq 0$ — is generated by compact objects of ${\mathbf{QCoh}}(\mathscr{X})$. This will follow from [@DAGXII Th. 1.5.10] once we know that the quasicoherent stack $\Phi_{\mathscr{X}}({\mathbf{QCoh}}(\mathscr{X},\mathscr{X}-\mathscr{U}))$ of [@DAGXI Constr. 8.5] is locally compactly generated.
So suppose $R$ a connective $E_{\infty}$ ring spectrum, and suppose $\eta\in\mathscr{X}(R)$. We wish to show that the $\infty$-category $$\Phi_{\mathscr{X}}({\mathbf{QCoh}}(\mathscr{X},\mathscr{X}-\mathscr{U}))(\eta)\simeq{\mathbf{Mod}}_R\otimes_{{\mathbf{QCoh}}(\mathscr{X})}{\mathbf{QCoh}}(\mathscr{X},\mathscr{X}\setminus\mathscr{U})$$ is compactly generated. It easy to see that this $\infty$-category can be identified with the full subcategory of ${\mathbf{Mod}}_R$ spanned by those modules $M$ that are carried to zero by the functor $${{{\mathbf{Mod}}_R\simeq{\mathbf{Mod}}_R\otimes_{{\mathbf{QCoh}}(\mathscr{X})}{\mathbf{QCoh}}(\mathscr{X})}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {{\mathbf{Mod}}_R\otimes_{{\mathbf{QCoh}}(\mathscr{X})}{\mathbf{QCoh}}(\mathscr{U})}}.$$ By a theorem of Ben Zvi, Francis, and Nadler [@DAGXI Cor. 8.22], this functor may be identified with the restriction functor along the open embedding $$j'\colon{{\mathscr{U}'{\mathrel{\mathop:}=}\operatorname{Spec}^{\textrm{\'et}}R\times_{\mathscr{X}}\mathscr{U}}\ \tikz[baseline]\draw[>=stealth,right hook->](0,0.5ex)--(0.5,0.5ex);\ {\operatorname{Spec}^{\textrm{\'et}}R}}.$$ The open immersion $j'$ is determined by a quasicompact open $U\subset\operatorname{Spec}^{\mathrm{Z}}A$, which consists of those prime ideals of $\pi_0A$ that do not contain a finitely generated ideal $I$. The proof is now completed by [@DAGVIII Pr. 4.1.15 and Pr. 5.1.3].
When $j$ is the open complement of a closed immersion $i\colon{{\mathscr{Z}}\ \tikz[baseline]\draw[>=stealth,->](0,0.5ex)--(0.5,0.5ex);\ {\mathscr{X}}}$, one may ask whether ${\mathbf{K}}(\mathscr{X}\setminus\mathscr{U})$ can be identified with ${\mathbf{K}}(\mathscr{Z})$. In general, the answer is no, but in special situations, such an identification is possible. Classically, this is the result of a *Dévissage Theorem* [@MR0338129 Th. 4]; we hope to return to a higher categorical analogue of such a result in later work (cf. [@MR92f:19001 1.11.1])
[^1]: We are grateful to Jacob Lurie for this observation.
|
---
address:
- 'D. M.: Mathématiques - bât. M2, Université Lille 1, F-59655 Villeneuve d’Ascq Cedex, France'
- 'Yu. G.: Algebra Section, Dept. of Mathematics, Moscow State University, 117234 Moscow, Russia'
author:
- 'D. Markushevich'
- 'Yu. G. Prokhorov'
title: Exceptional quotient singularities
---
Introduction and statement of main results
==========================================
The notion of exceptional singularity was introduced by Shokurov [@Sh1]. A singularity $(X,P)$ is called [*exceptional*]{}, if for any log canonical boundary, there is at most one exceptional divisor of discrepancy $-1$ over $P$ (see Definition \[def-exc\]). The reason for distinguishing these singularities is that they have more complicated multiple anticanonical systems $|-nK_X|$, than the nonexceptional ones. However, Shokurov suggests, and proves in dimension $3$, that the exceptional singularities are in a sense bounded (loc. cit., Corollary 7.3). The search of “good" divisors in $|-nK_X|$, or so called $n$-[*complements*]{}, is an essential ingredient of Shokurov’s project of the inductive study of log flips, log contractions and of the classification of log canonical singularities [@Sh1 Sect. 7].
According to [@Sh 5.2.3, 5.6], [@Sh1 1.5], the exceptional log terminal singularities in dimension 2 are exactly the singularities of types $E_6,E_7,E_8$ in the sense of Brieskorn [@Br]. Shokurov’s approach gives a description of the dual graphs of their resolutions (cf [@I], [@Ut Chapter 3]). According to [@Br], the exceptional graphs correspond to finite subgroups of $GL_2({{\mathbb C}})$ of tetrahedral ($E_6$), octahedral ($E_7$) or icosahedral ($E_8$) types. We found a classification-free approach to the proof which works also in dimension 3. The exceptional groups of types $E_6$, $E_7$, $E_8$ are exactly those finite subgroups of $GL_2({{\mathbb C}})$ which have no semiinvariants of degree $\leq 2$. Hence the following proposition implies Shokurov’s statement.
\[Sh\_2\] A two-dimensional quotient singularity $X={{\mathbb C}}^2/G$ by a finite group $G$ without reflections is exceptional if and only if $G$ has no semiinvariants of degree $\le 2$.
In dimension 3, we started the study of exceptional quotient singularities in our previous publication [@MP], where we showed that the quotients of ${{\mathbb C}}^3$ by Klein’s simple group of order 168 and by its unique central extension contained in ${{SL}_3({{\mathbb C}})}$ (of order 504) are exceptional, in using the configuration of the action of Klein’s group on ${{\mathbb P}}^2$ [@Kl], [@W]. The following Theorem is the main result of the present paper.
\[main\] A three-dimensional quotient singularity $X={{\mathbb C}}^3/G$ by a finite group $G$ without reflections is exceptional if and only if $G$ has no semiinvariants of degree $\le 3$.
Using Miller–Blichfeldt–Dickson classification [@MBD] of finite subgroups of ${{GL}_3({{\mathbb C}})}$, we obtain a complete list of such subgroups yielding exceptional singularities (Theorem \[list-exc\]).
Section \[s-1\] contains basic definitions and preliminary results. In particular, we show that the exceptional divisor with discrepancy $-1$ for an exceptional singularity is birationally unique and that its image is a point, independently of the choice of the boundary and in any dimension. Section \[s-2\] contains the proofs of Proposition \[Sh\_2\], Theorem \[main\], and the list of finite subgroups of ${{GL}_3({{\mathbb C}})}$ with exceptional quotients.
[Acknowledgement.]{} The second author is grateful to V. V. Shokurov for discussions on complements and the exceptionality.
Basics on exceptional singularities {#s-1}
===================================
For reader’s convenience, we reproduce here some basic facts about log canonical singularities. We are using the terminology and notation of [@MP], [@Ut], [@Sh] and [@Sh1] (see also [@KoP] for a nice introduction to the subject).
Let $(X\ni P)$ be a normal singularity (not necessarily isolated) and let $D=\sum d_iD_i$ be a divisor on $X$ with real coefficients. $D$ is called a boundary if $0\leq d_i\leq 1$ for all $i$. It is called a subboundary, if it is majorated by a boundary. A proper birational morphism $f \colon Y{{\longrightarrow}}X$ is called a log resolution of $(X,D)$ at $P$, if $Y$ is nonsingular near $f^{-1}(P)$ and ${\operatorname{Supp}}(D)\cup E$ is a normal crossing divisor on $Y$ near $f^{-1}(P)$, where $D$ is used to denote both the subboundary on $X$ and its proper transform on $Y$, and $E=\cup E_i$ is the exceptional divisor of $f$. The pair $(X,D)$ or, by abuse of language, the divisor $K_X+D$ is called [*terminal, canonical, Kawamata log terminal, purely log terminal,*]{} and, respectively, [*log canonical*]{} near $P$, if the following conditions are verified:
\(i) $K_X+D$ is ${{\mathbb R}}$-Cartier.
\(ii) Let us write for any proper birational morphism $f \colon Y{{\longrightarrow}}X$ $$K_Y\equiv f^\ast (K_X+D)+\sum a(E,X,D)E ,$$ where $E$ runs over prime divisors on $Y$, $a(E,X,D)\in{{\mathbb R}}$, and $a(D_i,X,D)=-d_i$ for each component $D_i$ of $D$. Then, for some log resolution of $(X,D)$ at $P$ and for all prime divisors $E$ on $Y$ near $P$, we have: $a(E,X,D)>0$ (for terminal), $a(E,X,D)\geq 0$ (for canonical), $a(E,X,D)> -1$ and no $d_i=1$ (for Kawamata log terminal), $a(E,X,D)> -1$ (for purely log terminal, without any restriction on the subboundary $D$), and, respectively, $a(E,X,D)\geq -1$ (for log canonical).
The coefficients $a(E,X,D)$ are called [*discrepancies*]{} of $f$, or of $(X,D)$; they depend on the discrete valuations of the function field of $X$ associated to the prime divisors $E$ and on $D$, but not on the choice of $f$. We will identify prime divisors with corresponding discrete valuations, when speaking about ‘divisors $E$ over $X$’ without indicating, on which birational model $E$ is realized. The conditions given by inequalities in part (ii) of the above definition do not depend on the choice of a log resolution.
Let $V$ be a normal variety and let $D=\sum d_iD_i$ be a ${{\mathbb Q}}$-divisor on $V$ such that $K_V+D$ is ${{\mathbb Q}}$-Cartier. A subvariety $W\subset V$ is said to be a [*center of log canonical singularities*]{} for $(V,D)$ if there exists a birational morphism from a normal variety $g\colon \tilde{V}\to V$ and a prime divisor $E$ on $\tilde{V}$ with discrepancy $a(E,V,D)\le -1$ such that $g(E)=W$. (The case $d_i=1$, $W=D_i$ is also possible.) The union of all centers of log canonical singularities is called the [*locus of log canonical singularities*]{} and is denoted by $LCS(V,D)$ [@Sh 3.14].
The following statement is a weak form of [@Ut 17.4] (in dimension $2$ it was proved earlier by Shokurov [@Sh]).
\[conn-1\] Let $X$ be a normal projective variety and let $D=\sum d_iD_i$ be a ${{\mathbb Q}}$-divisor on $X$ such that $K_X+D$ is ${{\mathbb Q}}$-Cartier. If $-(K_X+D)$ is nef and big, then the locus of log canonical singularities is connected.
\[conn-2\] Let $X$ be a normal projective surface and let $D=\sum d_iD_i$ be a boundary on $X$ such that $K_X+D$ is ${{\mathbb Q}}$-Cartier. If $K_X+D\equiv 0$, then the locus of log canonical singularities has one or two connected components.
Shokurov informed us that the previous theorem is valid in any dimension modulo Log Minimal Model Program.
\[def-exc\] Let $(X\ni P)$ be a normal singularity and let $D=\sum d_iD_i$ be a boundary on $X$ such that $K_X+D$ is log canonical. The pair $(X,D)$ is said to be [*exceptional*]{} if there exists at most one exceptional divisor $E$ over $X$ with discrepancy $a(E,X,D)=-1$. The singularity $(X,P)$ is said to be [*exceptional*]{} if $(X,D)$ is exceptional for any $D$ whenever $K_X+D$ is log canonical.
\[red\] Assume that there exists a reduced divisor $S=\sum S_i$ passing through $P$ such that $K_X+S$ is log canonical. Then $(X\ni P)$ is nonexceptional.
For the proof, see Lemma 1.7 in [@MP]. This implies, in particular, that any three-dimensional cDV-singularity [@YPG], and any three-dimensional terminal singularity is nonexceptional (see [@MP 1.9]).
\[convex\] Let $(X,P)$ be an exceptional singularity. Then the divisor $E$ from Definition \[def-exc\] with discrepancy $a(E,X,D)=-1$ is birationally unique. This means, that there exists a unique discrete valuation $\nu$ of the field of rational functions $k(X)$, such that for any pair $(X,D)$, exceptional at $P$, and for any log resolution $f:\tilde{X}
{{\longrightarrow}}X$, if there is an exceptional divisor $E$ in $\tilde{X}$ with $a(E,X,D)=-1$, then the corresponding discrete valuation $\nu_E=\nu$.
Let $\{ D_1,\ldots ,D_r\}$ be a finite set of irreducible divisors on $X$, and $f:\tilde{X}{{\longrightarrow}}X$ a log resolution of $(X,\sum_{i=1}^r D_i)$ at $P$. We can represent the boundaries $D=\sum_i d_iD_i$ with components from $\{ D_1,\ldots ,D_r\}$ as the unit cube $I^r\subset {{\mathbb R}}^r$ of vectors $(d_i)$. Then the subset $\Lambda\subset I^r$ corresponding to log canonical pairs $(X,D)$ is given by a finite number of linear inequalities in $d_i$’s $$a(E,X,D)=a(E,X,0)-\sum_id_i{\operatorname{mult}}_E(D_i)\geq -1 ,$$ where $E$ runs over the exceptional divisors such that $f(E)\ni P$. Let $\partial_+\Lambda$ be the closure of $\{ (d_i)\in\partial\Lambda\mid
0<d_i<1\;\forall \; i=1,\ldots ,r\}$. Then the exceptionality of $(X,P)$ implies that there are no points in $\partial_+\Lambda$ satisfying the equality $a(E,X,0)-\sum_i{\operatorname{mult}}_E(D_i)= -1$ for two different $E$’s. So, $\partial_+\Lambda$ (if nonempty) is an open convex polyhedron lying in exactly one hyperplane $\Pi_E$ with equation $a(E,X,0)-\sum_id_i{\operatorname{mult}}_E(D_i)= -1$.
Now, let $(X,P)$ be an exceptional singularity, $(X,D^{(1)}),(X,D^{(2)})$ two exceptional pairs, $f_i:X_i{{\longrightarrow}}X$ two log resolutions with divisors $E_i$ such that $a(E_i,X,D^{(i)})=-1$. Then we can apply the previous argument, taking the union of all the components of $D^{(1)},D^{(2)}$ as $\{ D_1,\ldots ,D_r\}$, and some log resolution dominating $f_1,f_2$ as $f$. Then we will have : firstly, by Lemma \[red\], $D^{(1)},D^{(2)}$ are represented by two different points of $\partial_+\Lambda$, and secondly, $\Pi_{E_1}=\Pi_{E_2}$. But then, both of the $E_i$ are of discrepancy $-1$ with respect to any $D$ represented by a point of $\partial_+\Lambda$. By exceptionality, $E_1=E_2$. This proves the assertion of the proposition.
\[LCS-point\] Let $(X,P)$ be an exceptional singularity. Then for any pair $(X,D)$ exceptional at $P$ and such that $LCS(X,D)\neq\varnothing$, we have $LCS(X,D)=\{ P\}$.
By Proposition \[convex\], the exceptional divisor $E$ with discrepancy $-1$ is biratonally unique. Suppose, $LCS(X,D)\supsetneqq\{ P\}$. Then for a generic hyperplane section $H$ of $(X,P)$, we have ${\operatorname{mult}}_E(H)=0$. This implies that $a(E,X,D+\epsilon H)=a(E,X,D)=-1$, and, by exceptionality, $a(E',X,D+\epsilon H)>-1$ for any sufficiently small positive ${\epsilon}$ and any exceptional divisor $E'\neq E$. Let ${\epsilon}_\ast =
\sup\;\{{\epsilon}\;\mid\; (X,D+\epsilon H)\ \text{is log canonical}\ \}$. Then $(X,D+{\epsilon}_\ast H)$ has two different divisors with discrepancy $-1$: $E$ and either the one of the $E'\neq E$ with minimal discrepancy, or $H$ in the case when ${\epsilon}_\ast =1$. This contradicts to the exceptionality of $(X,P)$.
\[cartesian\] Let $(X_i,P_i)$, $i=1,2,$ be two germs of normal varieties of dimension $>0$. Then $(X_1\times X_2,(P_1,P_2))$ is nonexceptional.
If at least one of $X_i$ is not log canonical with zero boundary, we are done, for $X_1\times X_2$ is not log canonical with any boundary. If $(X_i,0)$ are log canonical, then, starting from any nonzero boundaries $D^{(i)}$ at $P_i$, we can find ${\epsilon}_i\geq 0$ such that both pairs $(X_i,{\epsilon}_iD^{(i)})$ are log canonical and possess at least one discrete valuation of discrepancy $-1$ for $i=1,2$. Let $W_i$ be their centers. Then $LCS(X_1\times X_2,{\epsilon}_1 D^{(1)}\times X_2)$ contains $W_1\times X_2$, and $LCS(X_1\times X_2, X_1\times {\epsilon}_2 D^{(2)})$ contains $X_1\times W_2$. Hence $(X_1\times X_2,(P_1,P_2))$ is nonexceptional by Proposition \[LCS-point\].
Quotients ${{\mathbb C}}^m/G$ {#s-2}
=============================
In the first part of this section we work in arbitrary dimension. The following Lemma was proved in [@MP 2.1] in the $3$-dimensional case; now we generalize it to arbitrary dimension.
\[quot-lem1\] Let $\pi\colon V\to X$ be a finite morphism of normal varieties, étale in codimension $1$. Let $D$ be a boundary on $X$, and $D'=\pi^\ast D$. Assume that $K_X+D$ is log canonical (and then automatically is $K_V+D'$). Then $(X,D)$ is exceptional if and only if $(V,D')$ is.
For every proper birational morphism $f\colon Y\to X$ consider the following commutative diagram $$\label{1}
\begin{CD}
W@>{\varphi}>>Y\\
@V{g}VV@VfVV\\
V@>{\pi}>>X.\\
\end{CD}$$ where $W$ is the normalization of the dominant component of $V\times_{X}Y$. By the ramification formula (cf. [@KoP proof of 3.16]) for every exceptional divisor $F$ on $Y$ and every exceptional divisor $E$ on $W$ which dominates $F$ we have $$\label{1.5}
a(E,V,D')+1=r(a(F,X,D)+1),$$ where $r$ is the ramification index at the generic point of $E$. Therefore $$\label{iff}
a(E,V,D')=-1\qquad\text{ if and only if}\qquad a(F,X,D)=-1.$$ Thus if $(X,D)$ is nonexceptional, then $(V,D')$ is so.
Conversely, assume that $(V,D')$ is nonexceptional. Then there are infinitely many exceptional divisors over $V$ with discrepancy $a(\phantom{E},V,D')=-1$. As in [@KoP proof of 3.16] we can see that all these divisors appear in some commutative square (\[1\]) (for suitable $f$), whence $(X,D)$ is also nonexceptional by (\[iff\]).
### Warning {#warning .unnumbered}
The Lemma does not imply that the singularity of $X$ is exceptional if and only if $V$ is. The assertion concerns [*pairs*]{} $(V,D')$ with $D'$ a pullback of a boundary from $X$.
{#V,G}
Let $\pi\colon V\to X$ be the quotient morphism, where $V={{\mathbb C}}^m$. Let $D$ be a boundary on $X$ and let $D':=\pi^*D$. By [@Sh 2.2], (see also [@Ut 20.3], [@KoP 3.16]) $K_X+D$ is log canonical (resp., purely log terminal, Kawamata log terminal) if and only if so is $K_{V}+D'$.
\[conj\] In the notation of \[V,G\], assume that $G$ has a semiinvariant of degree $\le m$. Does this imply that $(X\ni P)$ is nonexceptional?
Let $\psi$ be such a homogeneous semiinvariant of minimal degree $d\le m$ and let $D'$ be its zero locus. By Lemma \[quot-lem1\] it is sufficient, for the positive answer to 3.3, to prove that $K_V+D'$ is log canonical. It is clear that $D'$ is a cone. The following Proposition is an easy particular case of this question.
\[impr\] Let $G$ be maximally imprimitive, that is, let $G$ contain a normal abelian subgroup $A$ whose character subspaces $V_\chi$ are $1$-dimensional and form one orbit under the action of $G$. Then $V/G$ is nonexceptional.
$G/A$ acts by permutations of the $V_\chi$, so $G$ has a semiinvariant of the form $x_1\cdots x_m$, where $x_i$ are coordinate linear forms. Since $D'=\{ x_1\cdots x_m=0\}$ is a simple normal crossing divisor, $(V,D')$ is log canonical. Then $(X,D)$ is also log canonical, and the assertion follows from Lemma \[red\].
{#Vi}
We cannot treat in general the case of [*imprimitive*]{} groups, that is, groups $G$ which permute transitively factors of some direct sum decomposition $V=\oplus V_i$. [*Maximally imprimitive*]{} are those for which $\dim V_i =1$. But if the action of $G$ is not transitive for some direct sum decomposition, the singularity is also nonexceptional. This follows from the next proposition.
A group $G$ is called [*reducible*]{}, if $V=V_1\oplus V_2$ with $V_1,V_2$ invariant under $G$, $\dim V_i>0\; (i=1,2)$.
\[reducible\] If $G$ is reducible, then $V/G$ is nonexceptional.
The proof is similar to that of Proposition \[cartesian\], though the result is not a corollary of \[cartesian\]. Let $G_i$ denote the image of $G$ in $GL(V_i)$. Let $D_i\; (i=1,2)$ be any nonzero boundaries for $(V_i,0)$, given by $G_i$-semiinvariant polynomials from $k[V_i]$. We can find ${\epsilon}_i>0$ such that both pairs $(V_i,{\epsilon}_iD_i)$ possess at least one discrete valuation of discrepancy $-1$ for $i=1, 2$. Let $W_i$ be their centers. Define $D^{(1)}=D_1\times V_2,D^{(2)}=V_1\times D_2$ in $V$. Then $LCS((V,{\epsilon}_iD^{(i)})$ contains $W_1\times V_2$ or $V_1\times W_2$. Using a commutative square of the same type as (\[1\]), we conclude from (\[1.5\]) that the images of $W_1\times V_2$ and $V_1\times W_2$ in $V/G$ are also centers of log canonical singularities. Hence $V/G$ is nonexceptional by Proposition \[LCS-point\].
\[quot-lem2\] If $\dim X=m\leq 3$, the answer to \[conj\] is affirmative.
In [@MP Lemma 2.2], we proved the assertion in dimension $3$. The same and even easier argument works in dimension $2$.
This Lemma shows the “if" part of Proposition \[Sh\_2\] and Theorem \[main\].
{#logic}
Now we will explain the logic of our proof of the exceptionality of singularities in Proposition \[Sh\_2\] and Theorem \[main\]. Assume that $(X\ni P)$ is nonexceptional. Then there exists a nonexceptional log canonical $K_X+D$. Further, we will use notations of Lemma \[quot-lem1\]. Take the smallest $n\in{{\mathbb N}}$ such that $F:=nD'$ is an integer divisor. Then $F$ locally near $0$ can be defined by a semiinvariant function, say $\psi$. Denote $d:={\operatorname{mult}}_0(\psi)$. \[begin\] Let $\sigma\colon W\to V={{\mathbb C}}^m$ be the blow-up of the origin and let $S\simeq{{\mathbb P}}^{m-1}$ be the exceptional divisor. Then $K_W=\sigma^*K_V+(m-1)S$ and $\sigma^*F=R+dS$, where $R$ is the proper transform of $F$.
Further $$K_W+S+\frac{m}{d}R=\sigma^*(K_V+\frac{m}{d}F).$$ By [@KoP Lemma 3.10] $K_V+\frac{m}{d}F$ is log canonical if and only if so is $K_W+S+\frac{m}{d}R$. In this case the pair $(V,\alpha F)$ is exceptional for all $0\leq\alpha\leq \frac{m}{d}$ if and only if $K_W+S+\frac{m}{d}R$ is purely log terminal. By the inversion of adjunction (see [@Sh 5.13], [@Ut 17.6]) the purely log terminal condition for $K_W+S+\frac{m}{d}R$ is equivalent to that $K_S+\frac{m}{d}C$ is Kawamata log terminal, where $C=R\cap S$. It is clear that $C$ is given by the equation $\psi_{{\min}}=0$, where $\psi_{{\min}}$ is the homogeneous component of $\psi$ of minimal degree $d$. Therefore we have
\[check\] In the above notations, if $K_S+\frac{m}{d}C$ is Kawamata log terminal, then $(V,\alpha F)$ is exceptional for any $0\leq\alpha \leq\frac{m}{d}$.
\[remark\] In the above notations, if $G$ has no semiinvariants of degree $\le m$, then ${\lfloor \frac{m}{d}C\rfloor}=0$.
Two-dimensional case
--------------------
### Proof of Proposition \[Sh\_2\] {#proof-of-proposition-sh_2 .unnumbered}
We obtain Proposition \[Sh\_2\] as an easy corollary of \[check\]. Assume that $G$ has no invariants of degree $\le 2$. Recall that $S\simeq{{\mathbb P}}^1$ in our case, so $C$ is a finite set. By \[check\] it is sufficient to prove that $K_S+\frac{2}{d}C$ is Kawamata log terminal, and this is equivalent to that ${\lfloor \frac{2}{d}C\rfloor}=0$. The last assertion follows by \[remark\].
Three-dimensional case
----------------------
### Proof of Theorem \[main\] {#proof-of-theorem-main .unnumbered}
Assume that $G$ has no invariants of degree $\le 3$. We shall prove that ${{\mathbb C}}^3/G$ is exceptional. By Proposition \[check\] it is sufficient to prove that $K_S+\frac{3}{d}C$ is Kawamata log terminal. Take $c$ to be the log canonical threshold of $(S,C)$, that is, the maximal ${\alpha}$ such that $K_S+{\alpha}C$ is log canonical. If $K_S+\frac{3}{d}C$ is not Kawamata log terminal, then $c\le 3/d$. First we consider the case when $c<3/d$. Then $-(K_S+cC)$ is ample. By connectedness theorem \[conn-1\], the locus of log canonical singularities is connected. By \[remark\] ${\lfloor \frac{3}{d}C\rfloor}=0$, so any divisor of discrepancy $-1$ should be exceptional. Therefore the locus of log canonical singularities on $S$ is a unique point, which must be invariant under the action of $G$. The dual line gives us a semiinvariant of degree $1$, a contradiction with our assumptions. In the case when $c=3/d$, we can use Theorem \[conn-2\]. Similarly to the above, we see that the locus of log canonical singularities is one or two points. In both cases, there is an invariant line. This ends the proof of Theorem \[main\].
The finite subgroups of $G\subset{{GL}_3({{\mathbb C}})}$ were classified by Miller–Blichfeldt–Dickson [@MBD] modulo central extensions (compare with [@P]). There are 10 types of such groups, denoted by A,B, …,J in [@MBD]. A stands for abelian, B for reducible, and C, D are imprimitive. The groups of type C are called [*tetrahedral*]{}; their image in the symmetric group ${\frak S}_3$ permuting the $V_i,\; i=1,2,3$ (notations as in \[Vi\]) is cyclic of order 3. The groups of type D are called [*general monomial*]{}; their map to ${\frak S}_3$ is surjective. The primitive subgroups of ${{GL}_3({{\mathbb C}})}$ belong to the $6$ types E, F, G, H, I, J. The orders of the associated collineation groups $PG=G/(G\cap{{\mathbb C}}^\ast)\subset{{PGL}_3({{\mathbb C}})}$ are $36$, $72$, $216$, $60$, $360$, $168$; the first three are solvable, and the last three are simple. The collineation groups from G to J have their names: the Hessian group, the icosahedral one, the alternating group ${\frak A}_6$ of degree $6$, and, finally, Klein’s simple group. We have also $P$E$\triangleleft P$F$\triangleleft P$G.
The following assertion is a consequence of Theorem \[main\].
\[list-exc\] Let $G$ be a finite subgroup of ${{GL}_3({{\mathbb C}})}$. Then the quotient ${{\mathbb C}}^3/G$ is exceptional if and only if $G$ belongs to one of the $4$ types [*F, G, I, J*]{}.
We can eliminate the types A–D by Propositions \[impr\], \[reducible\]. Further, the icosahedral group in its $3$-dimensional representation has an invariant of degree two, which is a semiinvariant of any group of type H. This follows, for example, from the fact that the $3$-dimensional representation is the complexification of the standard real one, which has an invariant scalar product.
\[groupE\] Any group of type [*E*]{} has two nonproportional semiinvariants of degree $3$. Groups of types [*F, G*]{} have no semiinvariants of degree $\leq 3$.
According to [@MBD Sect. 115], a group $G$ of one of the types E, F, G is an extension of a group $H$ of type D which leaves invariant the set of four triangles $t_1,t_2,
t_3,t_4$ defined in appropriate coordinates by the equations $$x_1x_2x_2=0,\; (x_1+x_2+x_3)(x_1+{\omega}x_2+{\omega}^{2+i}x_3)
(x_1+{\omega}^{2}x_2+{\omega}^{1+i}x_3)=0 \;$$ $$\phantom{x_1x_2x_2=0,\; (x_1+x_2+x_3)(x_1+{\omega}x_2}
({\omega}=\exp \frac{\scriptstyle 2\pi\sqrt{-1}}
{\scriptstyle 3},
i=0,1,2),$$ and the associated collineation group $PG$ is completely characterized by its image in the group ${\frak S}_4$ of permutations on the set $\{ t_1,t_2,
t_3,t_4\}$. It is the subgroup of order $2$, conjugate to $\{ 1, (t_1t_2)(t_3t_4)\}$ for the type E, $\{ 1, (t_1t_2)(t_3t_4),
(t_1t_3)(t_2t_4),(t_1t_4)(t_2t_3)\}$ for the type F, and the full alternating group on four letters for the type G. Moreover, one can easily verify that the $t_i$ belong to one pencil $\Phi$ of plane cubics (see [@Sp Lemma 4.7.6]). So, a group of type E acts on the pencil $\Phi$ with image ${{\mathbb Z}}/2{{\mathbb Z}}$ in ${\operatorname{Aut}}({{\mathbb P}}^1)$. Any involution on ${{\mathbb P}}^1$ has two fixed points, which implies the result for type E.
Since the image of $G$ in ${\operatorname{Aut}}({{\mathbb P}}^1)$ has no fixed points for the groups of types F, G, they do not have semiinvariants in the pencil $\Phi$. However, any semiinvariant of $G$ should be also that of $H$. According to [@MBD Sect. 113], $PH$ is generated by the cycle $c=(x_1x_2x_3)$, transposition $\tau =(x_2x_3)$ and dilatation $\kappa =\:{\operatorname{diag}}(1,{\omega},{\omega}^2)$. Any semiinvariant of ${\frak S}_3=<c,\tau>$ is a polynomial in elementary symmetric functions $\sigma_i\; (i=1,2,3)$ in $x_1,x_2,x_3$ and in $\Delta =(x_1-x_2)(x_2-x_3)(x_1-x_3)$. A direct verification shows that in degrees $\leq 3$, only the linear combinations of $x_1^3+x_2^3+x_3^3=\sigma_1^3-3\sigma_1\sigma_2+3\sigma_3$ and of $\sigma_3$ are (skew-)symmetric semiinvariants under $\kappa$. This yields exactly the pencil $\Phi$.
Thus, by Lemma \[quot-lem2\], the quotients of types E, H are not exceptional. By Theorem \[main\], the quotients of types F, G are exceptional. It remains to verify that the groups of types I, J have no semiinvariants of degree $\leq 3$.
For the groups of type I, we can take a representative $I_0\subset{{SL}_3({{\mathbb C}})}$ of order $1080$. The homogeneous semiinvariants of any group $G$ of type I will coincide with those of $I_0$, because they are central extensions of the same group $PI_0\simeq {\frak A}_6$. But all the semiinvariants of the group $I_0$ are indeed invariants. This follows from the fact that it has no normal subgroups with abelian quotient: the only nontrivial normal subgroup of $I_0$ is its center, isomorphic to ${{\mathbb Z}}/3{{\mathbb Z}}$, and its quotient is the simple group ${\frak A}_6$. The algebra of invariants of $I_0$ was determined by Wiman [@Wi] (see also a modern account of invariants of finite subgroups of ${{SL}_3({{\mathbb C}})}$ in [@YY], where I of [@MBD] is denoted by L); it is generated by basic invariants of degrees $6$, $12$, $30$ and $45$ with one relation of weighted degree $90$ between them. Thus, there are no semiinvariants of degree $\leq 3$, and we are done.
Klein’s simple group has a representation $J_{168}\subset{{SL}_3({{\mathbb C}})}$. As above, the only its semiinvariants are invariants, and they were determined by Klein [@Kl], see also [@W] or [@YY]. The degrees of basic invariants here are $4$, $6$, $14$, $21$, and there is one relation of weihgted degree $42$ between them. Again there are no semiinvariants of degree $\leq 3$, and we are done. This ends the proof of Theorem \[list-exc\].
Using the classification of finite subgroups in ${{SL}_3({{\mathbb C}})}$ ([@P], [@YY]), one can get the following assertion.
Let $G$ be a finite subgroup of ${{GL}_3({{\mathbb C}})}$ such that ${{\mathbb C}}^3/G$ is an exceptional canonical singularity. Then $G$ is, up to conjugation, one of the following subgroups of ${{SL}_3({{\mathbb C}})}$:
1. Klein’s simple group $J_{168}\subset{{SL}_3({{\mathbb C}})}$,
2. the unique central extension $J'_{504}$ of $J_{168}$ contained in ${{SL}_3({{\mathbb C}})}$,
3. the Hessian group $G_{648}\subset{{SL}_3({{\mathbb C}})}$,
4. the normal subgroup $F_{216}$ of $G_{648}$,
5. a central extension $I_{1080}$ of $\mathfrak{A}_6$.
It suffices to show that $G\subset{{SL}_3({{\mathbb C}})}$. Let $r$ be the order of the center $Z$ of $G$. Then ${{\mathbb C}}^3/Z$ is resolved by a single blow up, giving an exceptional divisor with discrepancy $-1+\frac{3}{r}$. Construct, as in the proof of Lemma \[quot-lem1\], a diagram (\[1\]) with ${{\mathbb C}}^3/Z,\: {{\mathbb C}}^3/G$ in place of $V,X$ respectively. Then the formula (\[1.5\]) implies that the minimal discrepancy of exceptional divisors $E$ over $X\;$ $\;\; a_{\min}\leq -1+\frac{3}{r}$. For canonical singularities, $a_{\min}\geq 0$, hence $r\leq 3$. If $r=1$ or $3$, $\; G\subset{{SL}_3({{\mathbb C}})}$ and we are done.
Assume that $r=2$. Then $Z=\{ \pm 1\}$. Let $G_0=G\cap{{SL}_3({{\mathbb C}})}$. The orders of the subgroups of ${{SL}_3({{\mathbb C}})}$ of types F, G, I, J are even, hence $G_0$ contains an element $g$ of order 2. Then either $g$ or $-g$ is a reflection. This contradicts to our hypotheses. Hence $r=2$ is impossible.
Recall that a variety $X$ is said to have $\epsilon$-log terminal singularities if $a(E,X,0)>-1+\epsilon$ for any exceptional divisor $E$ over $X$. Adapting the proof of the previous corollary to the case $a_{\min} > -1+{\epsilon}$ instead of $a_{\min}\geq 0$, we obtain the following result:
Fix $\epsilon>0$. Then the set of subgroups $G\subset{{GL}_3({{\mathbb C}})}$ without reflections such that ${{\mathbb C}}^3/G$ is an exceptional $\epsilon$-log terminal singularity is finite up to conjugation.
A similar finiteness result was obtained by [@Bor] for abelian quotients of any dimension (which are never exceptional by Proposition \[reducible\]).
[000]{}
Borisov, A. [*On classification of toric singularities,*]{} e-print math.AG/9804137.
Brieskorn, E. [*Rationale Singularitäten komplexer Flächen,*]{} Invent. Math. [**4**]{} (1968), 336–358.
Iliev, A. [*Log-terminal singularities of algebraic surfaces,*]{} Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1986) [No. 3]{}, 38–43; English transl.: Mosc. Univ. Math. Bull. [**41**]{} (1986), 46-53.
Klein, F. [*Gesammelte Mathematische Abhandlungen, II.*]{} Springer: Berlin, 1973.
Markushevich, D., Prokhorov, Yu. G. [*Klein’s group defines an exceptional singularity of dimension $3$,*]{} e-print math.AG/9805004, to appear in J. Math. Sci., 1999.
Miller, G. A., Blichfeldt, H. F., Dickson, L. E. [*Theory and Applications of finite groups.*]{} Dover Publ. Inc.: New York, 1916.
Pouyanne, N.: [*Une résolution en singularités toriques simpliciales des singularités-quotient de dimension trois,*]{} Ann. Fac. Sci. Toulouse, VI, Ser. Math. [**1**]{}(1992), 363–398.
*Young persons guide to canonical singularities,* in *Algebraic Geometry, Bowdoin, 1985,* Proc. Symp. Pure Math. vol. 46, 1987, pp. 345–414.
Kollár J. et al. [*Flips and abundance for algebraic threefolds,*]{} A summer seminar at the University of Utah, Salt Lake City, 1991. Astérisque. [**211**]{} (1992).
Kollár J. [*Singularities of pairs,*]{} Proc. Symp. Pure Math. [**62**]{}, pt. 1 (1997), 221-287.
Shokurov V. V. [*3-fold log flips,*]{} Izv. AN SSSR Ser. mat. [**56**]{} (1992), 105–201 & [**57**]{} (1993), 141–175; English transl. Russian Acad. Sci. Izv. Math. [**40**]{} (1993), 93–202 & [**43**]{} (1994), 527–558.
Shokurov V. V. [*Complements on surfaces,*]{} preprint, alg-geom/9711024
Springer T. [*Invariant theory,*]{} Lect. Notes. Math. [**585**]{}, Springer, 1977.
Weber, H. [*Lehrbuch der Algebra. II.*]{} Chelsea: New York, 1961.
Wiman, A. [*Ueber eine enfache Gruppe von $360$ ebenen Collineationen,*]{} Math. Annalen [**47**]{}(1896), 531-556.
Yau, S. S.-T., Yu, Y. [*Gorenstein Quotient Singularities in Dimension Three,*]{} Memoirs AMS [**505**]{}. Providence, 1993.
|
---
abstract: 'Motivated by recent experiments, we present here a systematic ab-initio study of the length dependence of the thermal conductance of single-molecule junctions. We make use of a combination of density functional theory with non-equilibrium Green’s function techniques to investigate the length dependence of the phonon transport in single alkane chains, contacted with gold electrodes via both thiol and amine anchoring groups. Additionally, we study the effect of the substitution of the hydrogen atoms in the alkane chains by heavier fluorine atoms to form polytetrafluoroethylenes. Our results demonstrate that (i) the room-temperature thermal conductance is fairly length-independent for chains with more than 5 methylene units and (ii) the efficiency of the thermal transport is strongly influenced by the strength of the phononic metal-molecule coupling. Our study sheds new light onto the phonon transport in molecular junctions, and it provides clear guidelines for the design of molecular junctions for thermal management.'
author:
- 'J. C. Klöckner$^{1}$'
- 'M. Bürkle$^{2}$'
- 'J. C. Cuevas$^{3}$'
- 'F. Pauly$^{1}$'
title: 'Length dependence of the thermal conductance of alkane-based single-molecule junctions: An ab-initio study'
---
Introduction
============
Heat conduction by phonons is a fundamental physical process with great relevance in numerous problems in science and engineering as well as in technological applications [@Minnich2014]. In the context of macroscale systems, phonon conduction is, to some extent, well understood thanks to the use of semiclassical theoretical methods such as the Boltzmann transport equation [@Ziman-book] or molecular-dynamics-based approaches [@Minnich2014], which are often used in combination with ab-initio techniques for the determination of the phonon spectra and coupling constants. However, with the advent of novel nanoscale systems and devices it has become possible to study phonon conduction in a new regime, where the system size can be smaller than the inelastic mean free path for phonons (due to phonon-phonon interactions), and in some cases even smaller than the corresponding elastic mean free path leading to quasi-ballistic transport [@Cahill2002; @Pop2010; @Luo2013; @Cahill2014]. In this new regime, phonon conduction has to be described by purely quantum mechanical means, which poses a very interesting and challenging problem for the theory.
In this work we are interested in the heat conduction via phonons in single-molecule junctions, which constitute the ultimate limit of miniaturization of electronic devices. These nanojunctions have emerged in recent years as ideal platforms to study the quantum theories that describe both the charge and energy transport in nanoscale devices [@Cuevas2010]. In particular, recent experimental advances have made it possible to investigate different aspects of energy and heat conduction in molecular junctions such as thermoelectricity [@Reddy2007; @Kim2014; @Rincon-Garcia2016], Joule heating [@Lee2013], and thermal conductance [@Segal2016]. These advances stimulate the development of fully ab-initio methods, which in turn should elucidate the main aspects and physical mechanisms of phonon conduction in atomic-scale junctions. In this sense, the goal of this paper is to use an ab-initio approach, based on a combination of density functional theory (DFT) and non-equilibrium Green’s function (NEGF) techniques, to study one of the most fundamental questions in the context of phonon transport in atomic-scale systems, namely the length dependence of the thermal conductance of a molecular chain.
The theoretical discussion of the thermal conductance of a linear one-dimensional (1D) chain has a long history [@Lepri2003; @Dhar2008; @Li2012]. The general conclusion is that in long ideal chains (of more than 100 identical units) exhibiting nonlinear interactions, momentum conservation and no disorder, the thermal conductance decays algebraically as $\kappa_{\rm ph} \propto L^{\alpha-1}$, where $L$ is the chain length and the exponent $\alpha$ is found to be $\alpha = 1/3$. Although these results are of fundamental interest, it is not clear that they are very relevant for actual experiments on nanoscale systems due to the long chain lengths required.
In the context of molecular junctions, the issue of the length dependence of the thermal conductance has already been addressed both experimentally and theoretically. Often the molecules of choice have been alkane chains [@Wang2007; @Majumdar2015], which are saturated molecules, whose electrical properties have been widely studied in the context of molecular electronics [@Cui2001; @Wang2003; @Haiss2004; @Li2006; @Gonzalez2006; @Fujihira2006; @Jang2006; @Chen2006; @Venkataraman2006; @Park2007; @Li2008; @Gonzalez2008; @Martin2008; @Akkerman2008]. From the experimental side, the length dependence of the thermal conductance of alkane-based molecular junctions was investigated by Wang *et al.* [@Wang2006] making use of alkanedithiol self-assembled monolayers (SAMs) that were sandwiched between Au and GaAs electrodes. It was found that the thermal conductance of junctions with 8, 9, and 10 CH$_2$ units did not depend significantly on the molecule length. More recently, Meier *et al.* [@Meier2014] reported thermal conductance measurements of monothiolated alkane monolayers, self-assembled on Au(111) surfaces as a function of their length (ranging from 2 to 18 methylene units). Making use of a scanning thermal microscope with a Si tip, it was found that the thermal conductance first increases for short chains, reaching a maximum for 4 units, and then it exhibits an (arguable) slow decay for lengths above 8 units. From the theory side, Segal *et al.* [@Segal2003] made use of semi-empirical methods to predict that the thermal conductance of alkane chains approaches a constant value for more than 20 CH$_2$ units for a weakly coupled junction, while it decreases inversely proportional to the length for the strongly coupled case. On the other hand, Duda *et al.* [@Duda2011], using a diffusive transport model combined with Hartree-Fock calculations of the vibrational modes of alkane chains, suggested that the thermal conductance should be fairly length-independent for chains with more than 5 CH$_2$ units. In the only ab-initio study of this issue that we are aware of, Sadeghi *et al.* [@Sadeghi2015] explored the phonon transport in alkane-based single-molecule junctions with 2, 4, 8, and 16 CH$_2$ units and found a very pronounced decay of the thermal conductance for the longest molecular length. These seemingly contradictory theoretical results call for new inspections of the fundamental issue of the length dependence of the thermal conductance of molecular junctions.
In this work we present a systematic ab-initio study with no adjustable parameters of the phonon transport in alkane-based single-molecule junctions, but discuss also electronic contributions to the heat transport. Making use of a combination of DFT and NEGF techniques, we analyze the thermal conductance of Au-alkane-Au molecular junctions as a function of the length of the alkane chains ranging from 2 to 30 methylene units. Moreover, in order to elucidate the role of the metal-molecule interface, we investigate two different anchoring groups, namely thiol (SH) and amine (NH$_2$). Additionally, we determine the impact of the substitution of the H atoms in the alkane chains by F atoms for both types of anchoring groups. The ensemble of our results shows that the room-temperature phonon thermal conductance of all investigated junctions is fairly length-independent for chains with more than 5 CH$_2$ units. This statement also holds for the total thermal conductance that consists of phononic and electronic contributions, since we show that the electronic contribution plays a significant role only for the shortest chain lengths below 4 units. Moreover, we demonstrate that the relative efficiency of the phonon heat conduction is mainly, although not exclusively, dictated by the strength of the metal-molecule phononic coupling. Our results yield novel insights into the heat conduction in molecular wires and provide clear predictions that will surely be possible to test in the near future due to the recent rapid advances in the experimental techniques.
The rest of the manuscript is organized as follows. First, in section \[sec-Methods\] we briefly describe the theoretical approach employed to obtain the results presented in this work. Then, in section \[sec-Results\] we describe and analyze the main results of this manuscript on phonon heat transport. Section \[sec-Further\] is devoted to additional discussions of our results such as electronic contributions to the thermal conductance and to the comparison with published experimental and theoretical work. Finally, we present in section \[sec-Conclusions\] our conclusions.
Theoretical approach {#sec-Methods}
====================
Our primary goal is to compute the thermal conductance due to phonon transport in single-molecule junctions. For this purpose, we make use of the first-principles formalism developed by us and reported in Ref. \[\]. Our approach is based on a combination of DFT and NEGF that allows us to compute both the electronic and phononic contributions to all basic linear response transport properties of a nanoscale system. In what follows, we briefly describe the main features of our method and refer the reader to Ref. \[\] for further details.
Contact geometries, electronic structure, and vibrational properties
--------------------------------------------------------------------
The first step in our modeling is the construction of the molecular junction geometries. We use DFT to compute equilibrium geometries through total energy minimization and to describe their electronic structure. Vibrational properties of the optimized equilibrium contacts are subsequently obtained in the framework of density functional perturbation theory. We use both procedures as implemented in the quantum chemistry software package TURBOMOLE 6.5 [@TURBOMOLE; @Deglmann2002; @Deglmann2004], employ the PBE exchange-correlation functional [@Perdew1992; @Perdew1996], the basis set def2-SV(P) [@Weigend2005] and the corresponding Coulomb fitting basis [@Weigend2006]. To make sure that the vibrational properties, i.e. vibrational energies and force constants, are accurately determined, we use very stringent convergence criteria. Thus, total energies are converged to a precision of better than $10^{-9}$ a.u., while geometry optimizations are performed until the change of the maximum norm of the Cartesian gradient is below $10^{-5}$ a.u.
Phonon transport
----------------
We compute the phononic contribution to the heat conductance within the framework of the Landauer-Büttiker picture, i.e., we ignore inelastic and anharmonic effects that are expected to play a minor role in short molecular junctions. Within this picture, the heat current due to phonons can be expressed as [@Rego1998; @Mingo2003; @Yamamoto2006] $$J_{\rm ph}(T_{\rm L},T_{\rm R}) = \dfrac{1}{h} \int_{0}^{\infty} dE \, E \tau_{\rm ph}(E)
\left[ n(E,T_{\rm L}) - n(E,T_{\rm R})\right], \label{eq-Qph}$$ where $\tau_{\rm ph}(E)$ is the phonon transmission, $n(E,T)=[\exp(E/k_{\rm
B}T)-1]^{-1}$ the Bose function describing the phonon occupation in the left (L) or right (R) electrode, and $T_X$ is the temperature in electrode $X=\text{L},\text{R}$. In this work, we focus on the linear response regime in which $J_{\rm ph} = -\kappa_{\rm ph} \Delta T$ is proportional to the temperature difference $\Delta T=T_{L}-T_{R}$ and the phonon thermal conductance is given by $$\kappa_{\rm ph}(T) = \frac{1}{h} \int_{0}^{\infty} dE \,E \tau_{\rm ph}(E)
\frac{\partial n(E,T)}{\partial T}. \label{eq-kph}$$ The phonon transmission appearing in the previous equations can be determined with the help of Green’s function techniques [@Mingo2003; @Yamamoto2006; @Burkle2015]. Briefly, the starting point is the description of the phonons or vibrational modes of the junction within the harmonic approximation. In this approximation, the phonon Hamiltonian for small displacements $\{Q_{\xi}\}$ of the atoms around their equilibrium positions $\{R_{\xi}^{(0)}\}$ adopts the following form $$\hat H = \frac{1}{2} \sum_{\xi} \hat{p}_{\xi}^{2} + \frac{1}{2\hbar^{2}}
\sum_{\xi\chi} \hat{q}_{\xi} \hat{q}_{\chi} K_{\xi\chi},$$ where we have introduced mass-weighted displacement operators $\hat{q}_{\xi} =
\sqrt{M_{\xi}}\hat{Q}_{\xi}$ and mass-scaled momentum operators $\hat{p}_{\xi}=\hat{P}_{\xi}/\sqrt{M_{\xi}}$ as conjugate variables. These variables obey the following commutation relations: $[\hat{q}_{\xi},\hat{p}_{\chi}]=\mbox{i}\hbar\delta_{\xi\chi}$ and $[\hat{q}_{\xi},\hat{q}_{\chi}]=[\hat{p}_{\xi},\hat{p}_{\chi}]=0$, where $\xi=(i,c)$ denotes a Cartesian component $c=x,y,z$ of atom $i$ at position $\vec{R}_{i}=\vec{R}_{i}^{(0)}+\vec{Q}_{i}$. The phonon system is characterized by its dynamical matrix $K_{\xi\chi} = \hbar^{2}
\mathcal{H}_{\xi\chi} /\sqrt{M_{\xi}M_{\chi}}$, which is the mass-weighted Hessian of the DFT total ground state energy with respect to the Cartesian atomic coordinates, $\mathcal{H}_{\xi\chi} = \partial_{\xi\chi}^{2}E$. These harmonic force constants are computed within density functional perturbation theory.
The use of a local displacement basis enables the partitioning of the dynamical matrix into three parts, a central (C) scattering region, and the two semi-infinite L and R electrodes, i.e.$$\boldsymbol{K}=\left(\begin{array}{ccc}
\boldsymbol{K}_{\mathrm{LL}} & \boldsymbol{K}_{\mathrm{LC}} & \mathrm{\boldsymbol{0}}\\
\boldsymbol{K}_{\mathrm{CL}} & \boldsymbol{K}_{\mathrm{CC}} & \boldsymbol{K}_{\mathrm{CR}}\\
\boldsymbol{0} & \boldsymbol{K}_{\mathrm{RC}} & \boldsymbol{K}_{\mathrm{RR}}
\end{array}\right). \label{eq-Kmatrix}$$ Notice that there is no direct coupling between L and R. The phonon transmission can be expressed as [@Mingo2003; @Asai2008] $$\tau_{\rm ph}(E) = \mathrm{Tr} \left[ \boldsymbol{D}_{\mathrm{CC}}^{\textrm{r}}(E)
\boldsymbol{\Lambda}_{\textrm{L}}(E) \boldsymbol{D}_{\mathrm{CC}}^{\textrm{a}}(E)
\boldsymbol{\Lambda}_{\textrm{R}}(E)\right], \label{eq-tauph}$$ where $\boldsymbol{D}_{\mathrm{CC}}^{\textrm{r,a}}(E)$ are the retarded and advanced phonon Green’s functions of the central region that can be computed by solving the following Dyson equation $$\boldsymbol{D}_{\mathrm{CC}}^{\mathrm{r}}(E) = \left[\left(E+i\eta
\right)^{2}\boldsymbol{1} - \boldsymbol{K}_{\textrm{CC}} -
\boldsymbol{\Pi}_{\textrm{L}}^{\mathrm{r}}(E) -
\boldsymbol{\Pi}_{\textrm{R}}^{\mathrm{r}}(E) \right]^{-1}$$ with $\boldsymbol{D}_{\mathrm{CC}}^{\mathrm{a}}(E)=\boldsymbol{D}_{\mathrm{CC}}^{\mathrm{r}}(E)^{\dagger}$ and an infinitesimal quantity $\eta>0$. On the other hand, the linewidth-broadening matrices $$\boldsymbol{\Lambda}_{X}(E)= i
\left[\boldsymbol{\Pi}_{X}^{\mathrm{r}}(E)-\boldsymbol{\Pi}_{X}^{\mathrm{a}}(E)\right]$$ are related to the corresponding contact self-energies $$\boldsymbol{\Pi}_{X}^{\mathrm{r}}(E) = \boldsymbol{K}_{\textrm{C}X} \boldsymbol{d}_{XX}^{\mathrm{r}}(E)
\boldsymbol{K}_{X\mathrm{C}},$$ describing the coupling between the central region C and electrode $X$. In the expressions $\boldsymbol{d}_{XX}^{\mathrm{r}}(E)$ is the surface Green’s function of lead $X=\text{L},\text{R}$ and $\boldsymbol{\Pi}_{X}^{\textrm{a}}(E)=\boldsymbol{\Pi}_{X}^{\textrm{r}}(E)^{\dagger}$.
In analogy with the concept of conduction channels in coherent electron transport [@Cuevas2010], it is interesting to decompose the total phonon transmission in terms of individual transmission coefficients of the different phononic scattering eigenfunctions that contribute to the phonon conduction at each energy. The total phonon transmission in Eq. (\[eq-tauph\]) can be rewritten as a sum of independent contributions $$\tau_{\rm ph}(E) = \mbox{Tr} \{ \boldsymbol{t}_{\rm ph} (E)
\boldsymbol{t}_{\rm ph}^{\dagger} (E) \} = \sum_n \tau_{{\rm ph},n}(E) ,$$ where $\boldsymbol{t}_{\rm ph}(E) = \boldsymbol{\Lambda}^{1/2}_{\textrm{R}}(E)
\boldsymbol{D}_{\mathrm{CC}}^{\textrm{r}}(E) \boldsymbol{\Lambda}^{1/2}_{\textrm{L}}(E)$ is the phonon transmission matrix and $\tau_{{\rm ph},n}(E)$ are the energy-dependent eigenvalues of $\boldsymbol{t}_{\rm ph} \boldsymbol{t}_{\rm ph}^{\dagger}$.
To conclude this discussion, let us remark that to compute the different parts of the dynamical matrix in Eq. (\[eq-Kmatrix\]) we follow the strategy of our cluster-based approach for charge quantum transport [@Pauly2008]. In this approach, we first compute the dynamical matrix for an extended central cluster including the molecule and large parts of the leads. Subsequently, we extract from there the matrices $\boldsymbol{K}_{\mathrm{CC}}$, $\boldsymbol{K}_{\mathrm{XC}}$. On the other hand, the surface Green’s functions of the electrodes $\boldsymbol{d}_{XX}^{\mathrm{r}}(E)$ are obtained by extracting bulk force constants from a separate calculation of a big cluster of several hundred atoms and then using these extracted parameters in combination with a decimation technique to describe the surface of a semi-infinite perfect crystal (see Ref. \[\] for details). In this way, we achieve a consistent, parameter-free ab-initio description of the phonon system in nanoscale devices.
Electronic transport
--------------------
For completeness, in this work we also investigate the electronic contribution to the thermal conductance. In this way we can assess the relative importance of electrons and phonons in the studied molecular junctions. Similar to the phononic case, we assume that the electronic transport is dominated by elastic tunneling processes. Under these circumstances the electronic contribution to the thermal conductance in the linear response regime can be computed within the Landauer-Büttiker formalism. It is given by [@Sivan1986; @Cuevas2010] $$\kappa_{\rm el}(T) = \frac{2}{hT} \left( K_{2}(T)- \frac{K_{1}(T)^{2}}{K_{0}(T)}
\right), \label{eq-kel}$$ where the $K_n(T)$ coefficients are defined as $$K_n(T) = \int^{\infty}_{-\infty} dE \, \tau_{\rm el}(E) \left(-\tfrac{\partial
f(E)}{\partial E}\right)(E-\mu)^{n},
\label{eq-Kn}$$ $\tau_{\rm el}(E)$ is the energy-dependent electron transmission and $f(E,\mu,T) = \left\{ \exp[(E-\mu)/k_{\mathrm{B}}T]+1\right\} ^{-1}$ is the Fermi function. Here, the chemical potential $\mu\approx E_{\textnormal{F}}$ is approximately given by the Fermi energy $E_{\textnormal{F}}$ of the Au electrodes.
We have computed the electron transmission making use of our DFT-NEGF formalism implemented in TURBOMOLE and explained in detail in Ref. . It is worth stressing that in order to correct for the known inaccuracies in DFT related to quasiparticle energies, we have made use of the DFT+$\Sigma$ approach [@Quek2007], which was implemented in our method as explained in Ref. \[\].
Results {#sec-Results}
=======
To investigate the length dependence of the thermal conductance of single-molecule junctions, we study alkanes chains of different length with an even number $n$ of methylene (CH$_2$) units ranging between 2 and 30. These saturated molecules are electrically rather insulating, which ensures that the thermal transport is dominated by phonons (see below). To elucidate the role of the anchoring or terminal group in the length dependence of the phonon transport, we analyze two standard groups in molecular electronics, namely thiol (S) and amine (NH$_2$) groups. Moreover, we study the impact of the substitution of the H atoms in the alkane chains by heavier F atoms to investigate polytetrafluoroethylenes (PTFEs) with both thiol and amine groups. Thus, in an attempt to draw general conclusions, we investigate four different families of molecules attached in all cases to gold electrodes.
In what follows, we shall focus on the binding geometries illustrated in Fig. \[fig-geometries\], where the molecules are bonded to the electrodes in an atop position via the corresponding anchoring group. Let us stress that in all cases the molecular junctions were carefully optimized to find the minimum energy configuration. Moreover, in order to establish a meaningful comparison between the different compounds, special care was taken to avoid both strain effects and the appearance of defects in the molecular chains. In this sense, the alkane chains remain linear in our case, while the introduction of F atoms causes the carbon atoms to deviate from this linear structure, leading to more disordered PTFE chains. Independent of this, the distance between the carbon atoms stays nearly constant in all chains with a value of around 1.5 Å.
![(Color online) Examples of the four types of molecular junctions studied in this work. The four molecules have the structure $X$-(C$Y_2$)$_n$-$X$, where the terminal or anchoring group $X$ is either S or NH$_2$, while the $Y$ atom in the repeated unit is either H or F. The integer number $n$ denotes the number of C$Y_2$ segments in the molecules. In all cases, the electrodes are made of gold and the contact geometries are such that molecules are bonded to the electrodes in an atop position via the anchoring groups.[]{data-label="fig-geometries"}](Fig1.pdf){width="0.9\columnwidth"}
We summarize in Fig. \[fig-kph\] the main result of this work, namely the room-temperature phononic thermal conductance for the four types of molecular junctions as a function of the number of C$Y_2$ segments ($Y=\text{H},\text{F}$) ranging from 2 to 30, which corresponds to a maximum length of around 4.5 nm. The first thing to mention is that the conductance values range from 15 to 45 pW/K. Second, for chains with more than 5 segments the conductance exhibits small variations, but it is basically length-independent, irrespective of the molecular family. Third, the molecules with amine anchoring groups typically exhibit a lower conductance than the corresponding dithiolated ones. This is particularly evident in the case of the PTFE-diamine molecules, which exhibit thermal conductances that are about a factor of 2 smaller than those of the other families. Overall, these results show that the phononic thermal conductance of alkane-based single-molecule junctions is rather insensitive to the molecular length for up to 30 segments, which is a signature of ballistic phonon transport.
![(Color online) Room-temperature ($T=300$ K) phonon thermal conductance as a function of the number of C$Y_2$ units ($Y=\text{H},\text{F}$) in the molecule for both anchoring groups, thiol and amine. The distance between neighboring C atoms in the chains of around 1.5 Å allows to translate the number of units to a corresponding distance.[]{data-label="fig-kph"}](Fig2.pdf){width="\columnwidth"}
To understand these results, let us first analyze the energy dependence of the phonon transmission. An example for the four types of molecules with $n=10$ C$Y_2$ segments is shown in Fig. \[fig-Tph\]. First of all, notice that the transmission is different from zero only in an energy region between 0 and 20 meV, which is determined by the phonon density of states of the gold electrodes (see Ref. \[\]). Second, the most obvious feature in these results is the fact that for the molecules with the amine group the transmission spectra exhibit narrower peaks, which explains the lower conductance obtained for these molecules. (This is particularly well pronounced for the PTFE-diamine ones.) This strongly suggests that the phononic metal-molecule coupling for the amine group is weaker than for thiol. On the other hand, although the fluorinated molecules are expected to exhibit more vibrational modes in the transport window than the alkane chains due to the larger mass of the F atoms, one does not observe significant differences.
To further clarify how the elastic phonon transport takes place, we show in Fig. \[fig-taun\] the largest individual phonon transmission coefficients for molecular junctions with alkanedithiols of three different lengths ($n=10$, $20$, and $30$ methylene units). The first thing to notice is that the total transmission is dominated by a single or two phonon channels, with a third one giving only a small contribution. The number of channels is controlled in this case by the rather linear molecules, and the three channels correspond to the three polarizations of the vibrational modes. It is important to realize that this number of channels cannot be altered by the number of vibrational modes in the molecular chains, which only determine the actual values of the transmission coefficients. More important for our discussion of the length dependence is the fact that the dominant channel is fully open (transmission equal to 1) over a wide range of energies, irrespective of the length of the molecule. This is indeed the true signature of ballistic phonon transport, which is realized here with alkane and alkane-related molecular chains.
![(Color online) Phonon transmission as a function of energy for junctions containing molecules with 10 C$Y_2$ units.[]{data-label="fig-Tph"}](Fig3.pdf){width="\columnwidth"}
![(Color online) Individual phonon transmission coefficients as a function of the energy for three alkanedithiols with different lengths (10, 20, and 30 methylene units). We show the coefficients of the three most transmissive channels.[]{data-label="fig-taun"}](Fig4.pdf){width="\columnwidth"}
To gain further insight into our ab-initio results, we have developed a simple 1D model that is schematically represented in Fig. \[fig-1Dmodel\](a). In the following we will use the notation $k_{ij}$ for elements of the dynamical matrix coupling atoms $i$ and $j$ in the toy model as compared to $K_{ij}$ in the full ab-initio results. In the model we consider only nearest-neighbor couplings between segments in the molecular chain, $k_{\rm CC}$, and the leads are modeled as 1D Au chains with an analogous nearest-neighbor coupling, $k_{\rm Au Au}$. Finally, the metal-molecule coupling is described by a single constant, $k_{\rm Au C}=k_{{\rm Au} X}\sqrt{M_X/M_{\rm C}}$ with $X=\text{S},\text{N}$. We extracted these parameters from the DFT calculations as the highest eigenvalue of $K_{ij}$, see Eq. (\[eq-Kmatrix\]). These parameters depend on the molecular species, but not on the molecular length, and we summarize their values in Table \[table\].
$X$ $Y$ $k_{\mathrm{Au}X}$ $k_{\mathrm{CC}}$ $k_{\rm Au Au}$
----- ----- -------------------- ------------------- -----------------
S H 485 6150 100
S F 426 4850 100
N H 218 6150 100
N F 55 4850 100
: Parameters for the 1D model schematically represented in Fig. \[fig-1Dmodel\](a). All elements $k_{ij}$ of the dynamical matrix are given in units of meV$^2$. Notice that the same value of $k_{\rm Au Au}$ was used in all cases. \[table\]
Using these parameters, we computed the corresponding thermal conductance with the Green’s function method described in section \[sec-Methods\] and the results are displayed in Fig. \[fig-1Dmodel\](b). As one can see, this simple model is able to reproduce all the salient features of our ab-initio results in Fig. \[fig-kph\]. In particular, it nicely reproduces the fact that the amine-terminated molecules exhibit a lower thermal conductance, which is especially evident in the case of the PTFE-diamine chains. Now we can confirm that this lower thermal conductance is due to a weaker coupling to the leads (see values of $k_{\mathrm{Au}X}$ in Table \[table\]). The strongly reduced coupling for PTFE-diamine chains as compared to their alkane-diamine counterparts results in practice from the larger distance (about 0.15 Å) between the Au electrodes and the N atoms in these molecular junctions. Let us also mention that we attribute the higher conductance values obtained with the 1D model as compared to the ab-initio results to the fact that we extract the parameters $k_{ij}$ from the largest eigenvalues of the dynamical submatrices $K_{ij}$.
![(Color online) (a) Schematic representation of the 1D model used to understand our findings. The meaning of the different parameters is explained in the text. (b) Results obtained with the 1D model for the room-temperature phonon thermal conductance as a function of the number of C$Y_2$ units ($Y=\text{H},\text{F}$) in the molecule for both anchoring groups, thiol and amine.[]{data-label="fig-1Dmodel"}](Fig5.pdf){width="0.9\columnwidth"}
Further discussion {#sec-Further}
==================
So far we have focused on the phononic contribution to the thermal conductance and one may wonder whether the electrons play any role. Alkanes are known to be poor electrical conductors [@Cui2001; @Wang2003; @Haiss2004; @Li2006; @Gonzalez2006; @Fujihira2006; @Jang2006; @Chen2006; @Venkataraman2006; @Park2007; @Li2008; @Gonzalez2008; @Martin2008; @Akkerman2008]. Thus, on the basis of the Wiedemann-Franz law [@Cuevas2010; @Burkle2015], one therefore does not expect the electrons to give a significant contribution to the thermal conductance. To check this, we have studied the electronic contribution to the thermal conductance, $\kappa_{\rm el}$, in all of our molecular junctions using the ab-initio methodology briefly described in subsection \[sec-Methods\].C. Using $T=300$ K, we show in Fig. \[fig-ratio\] the results for the ratio between the electronic and phononic thermal conductances as a function of the molecular length for the four families of molecules investigated here. As one can see, when molecular chains have more than 4 segments, the electronic contribution is negligible. Notice that the exponential decay of the conductance ratio is a simple consequence of the exponential decay of the electrical conductance with length in the off-resonant transport situation. It follows from the proportionality of $\kappa_{\rm el}$ and the electrical conductance and a rather length-independent $\kappa_{\rm ph}$.
![(Color online) Ratio of the electronic ($\kappa_{\rm el}$) and phononic ($\kappa_{\rm ph}$) room-temperature thermal conductance as a function of the molecular length for the 4 different types of molecules studied in this work. In the inset the thermal conductance due to both electronic and phononic contributions is displayed for short chain lengths.[]{data-label="fig-ratio"}](Fig6.pdf){width="\columnwidth"}
In the previous section we discussed the results for the thermal conductance at room temperature. For completeness, we now briefly address the issue of the temperature dependence of the phonon transport. Fig. \[fig-Tdep\] displays the temperature dependence from 0 to 300 K of the phonon thermal conductance for junctions with the four molecular species and featuring $n=10$ C$Y_2$ segments. This temperature dependence is relatively insensitive to the molecular length and thus, the results of Fig. \[fig-Tdep\] are representative of the four molecular species studied in this work. As one can see, the thermal conductance raises abruptly at low temperatures, it tends to saturate, depending on the molecule, above approximately 100 to 200 K, and around room temperature it is fairly constant. This overall behavior simply reflects the fact that for temperatures above the Debye temperature of gold, all the phonons of the metal electrodes as well as the vibrational modes of the molecule with energies within the transport window are thermally occupied, while below this temperature the higher-lying modes are only partially occupied and the thermal conductance hence becomes sensitive to the temperature.
![(Color online) Phonon thermal conductance as a function of the temperature for junctions containing molecules with 10 C$Y_2$ units.[]{data-label="fig-Tdep"}](Fig7.pdf){width="0.9\columnwidth"}
Let us now discuss the comparison with existent results. From the theory side our results, showing a thermal conductance relatively insensitive of the molecular length, are qualitatively compatible with the findings of Segal *et al.* [@Segal2003] for weakly coupled alkane chains. This is also the case for the results of Duda *et al.* [@Duda2011]. Our results are also in agreement with the molecular dynamics calculations performed by Luo and Lloyd [@Luo2010], which studied octanedithiol SAMs sandwiched between gold electrodes and estimated a thermal conductance per molecule of 43 pW/K. However, our results are clearly at variance with the strong decay for the longest molecule reported by Sadeghi *et al.* [@Sadeghi2015], which indeed is the only ab-initio study published to date on the length dependence of the thermal conductance of alkane single-molecule junctions. The reason for the discrepancy is unclear to us, but it is worth pointing out that those authors only analyzed a very limited number of molecules (with 2, 4, 8, and 16 methylene units) and they used a different anchoring group (hydrobenzothiophene).
With respect to existent experiments on the length dependence, we cannot establish a direct comparison, but our length independence is compatible with the experiments of Wang *et al.* [@Wang2006] in junctions based on alkanedithiol SAMs sandwiched between Au and GaAs electrodes. On the other hand, the experiments of Meier *et al.* [@Meier2014], which represent the most systematic study of the length dependence of the thermal conductance of alkane junctions to date, exhibit some basic differences with the junctions studied in this work. First, the tip electrode was made of Si. Second, the alkanes were monothiolated. Third, the temperature difference used in the experiment was between 200 and 300 $^{\rm o}$C, which is most likely beyond the linear response regime addressed in our work. Finally, in this experiment the number of contacted molecules is not directly determined, and it is only inferred with the help of tip models. Having said all that, let us mention that the estimated conductance values per molecule lie in the range of our calculated values and a careful inspection of the experimental data shows that the conductance is indeed fairly independent of the molecular length for chains with more than 8 methylene units. Anyway, a rigorous comparison with experiment to settle the issue of the length dependence of the thermal conductance requires true single-molecule experimental techniques, and our results provide clear predictions that we hope will be tested experimentally in the near future.
Related to the role of the anchoring group, let us mention that Losego *et al.* [@Losego2012] investigated experimentally with the time-domain thermoreflectance technique its impact on heat transport in SAMs of alkanes contacted to quartz and gold films. They observed that the replacement of amines as the binding group to the Au film by thiols led to a 60% increase of the thermal conductance per unit area for chains with 11 methylene units. This is consistent with the general trend found here in the sense that the amine-terminated molecules exhibit in general a lower thermal conductance than their thiolated counterparts.
Let us conclude this section by stressing that in spite of the fact that we are using an approach, where anharmonic effects are not taken into account, the finding of a length-independent thermal conductance, as reported, here is by no means trivial. The intrinsic disorder in the molecular chains, which is present e.g. in the PTFE chains, can in principle lead to a diffusive transport regime, where the conductance decays linearly with length, or ultimately to an Anderson-localized regime, where the conductance is exponentially suppressed with length. In our study we see that for chains of up to 30 segments (with a length of up to 4.5 nm) the conductance exhibits a weak length dependence, indicating that the elastic mean free path is larger than the system size. It would be interesting to study the limits of this quasi-ballistic transport, and we hope to address this issue in the near future. Let us also emphasize that Segal *et al.* [@Segal2003] showed that anharmonicity in the alkane chains is rather weak and it is not expected to play a crucial role in the range of lengths studied in this work. Therefore, our approach is well justified.
Conclusions {#sec-Conclusions}
===========
In summary, we have presented a systematic ab-initio study of the length dependence of the thermal transport through alkane-based single-molecule junctions. In particular, we have investigated the role of the anchoring group in this length dependence and we have compared the results for alkane chains with those for PTFE chains, which are obtained from the alkanes by substituting the H atoms by F atoms. By investigating the phonon transport for chains with up to 30 segments, we find that the phonon thermal conductance is rather insensitive to the molecular length, irrespective of the molecule. These results strongly suggest that the phonon transport in these molecular wires is quasi-ballistic. On the other hand, our analysis of the role of the anchoring group shows that these groups ultimately determine the efficiency of the phonon transport and, in particular, we find that the thiol group leads to higher conductance values than the amine group. Electronic contributions to the thermal conductance do not modify these conclusions, since we demonstrate that they are negligible for alkanes with 4 or more segments. Overall, our results provide clear predictions that we hope will be tested experimentally in the near future given the rapid advances in the field of nanothermometry. Moreover, our work sheds new light on the thermal transport mechanisms in molecular wires and provides clear guidelines for the design of molecular junctions for thermal management.
Acknowledgments
===============
J.C.K. and F.P. gratefully acknowledge funding from the Carl Zeiss foundation and the Junior Professorship Program of the Ministry of Science, Research, and the Arts of the state of Baden-Württemberg. M.B. was supported by a Grant-in-Aid for Young Scientists (Start-up) (KAKENHI \#15H06889) from the Japan Society for the Promotion of Science and J.C.C. through the Spanish Ministry of Economy and Competitiveness (Contract No. FIS2014-53488-P) and thanks the German Research Foundation (DFG) and Collaborative Research Center (SFB) 767 for sponsoring his stay at the University of Konstanz as Mercator Fellow. An important part of the numerical modeling was carried out on the computational resources of the bwHPC program, namely the bwUniCluster and the JUSTUS HPC facility.
[00]{}
A. J. Minnich, J. Phys.: Condens. Matter [**27**]{}, 053202 (2015).
J. M. Ziman, *Electrons and Phonons: The Theory of Transport Phenomena in Solids* (Oxford University Press, Oxford, 2001).
D. G. Cahill, W. K. Ford, K. E. Goodson, G. D. Mahan, A. Majumdar, H. J. Maris, R. Merlin, and S. R. Phillpot, J. Appl. Phys. [**93**]{}, 793 (2003).
E. Pop, Nano Res. [**3**]{}, 147 (2010).
T. Luo and G. Chen, Phys. Chem. Chem. Phys. [**15**]{}, 3389 (2013).
D. G. Cahill, P. V. Braun, G. Chen, D. R. Clarke, S. Fan, K. E. Goodson, P. Keblinski, W. P. King, G. D. Mahan, A. Majumdar, H. J. Maris, S. R. Phillpot, E. Pop, and L. Shi, Appl. Phys. Rev. [**1**]{}, 011305 (2014).
J. C. Cuevas and E. Scheer, *Molecular Electronics: An Introduction to Theory and Experiment* (World Scientific, Singapore, 2010).
P. Reddy, S. Y. Jang, R. Segalman, A. Majumdar, Science [**315**]{}, 1568 (2007).
Y. Kim, W. Jeong, K. Kim, W. Lee, P. Reddy, Nat. Nanotech. [**9**]{}, 881 (2014).
L. Rincón-García, C. Evangeli, G. Rubio-Bollinger, and N. Agraït, Chem. Soc. Rev. [**45**]{}, 4285 (2016).
W. Lee, K. Kim, W. Jeong, L. A. Zotti, F. Pauly, J. C. Cuevas, and P. Reddy, Nature (London) [**498**]{}, 209 (2013).
D. Segal and B. K. Agarwalla, Annu. Rev. Phys. Chem. [**67**]{}, 185 (2016).
S. Lepri, R. Livi, and A. Politi, Phys. Rep. [**377**]{}, 1 (2003).
A. Dhar, Adv. Phys. [**57**]{}, 457 (2008).
N. Li, J. Ren, L. Wang, G. Zhang, P. Hänggi, and B. Li, Rev. Mod. Phys. [**84**]{}, 1045 (2012).
Z. Wang, J. A. Carter, A. Lagutchev, Y. K. Koh, N. H. Seong, D. G. Cahill, D. D. Dlott, Science [**317**]{}, 787 (2007).
S. Majumdar, J. A. Sierra-Suarez, S. N. Schiffres, W. L. Ong, C. F. Higgs, A. J. H. McGauney, J. A. Malen, Nano Lett. [**15**]{}, 2985 (2015).
X. D. Cui, A. Primak, X. Zarate, J. Tomfohr, O. F. Sankey, A. L. Moore, T. A. Moore, D. Gust, G. Harris, S. M. Lindsay, Science [**294**]{}, 571 (2001).
W. Wang, T. Lee, M. A. Reed, Phys. Rev. B [**68**]{}, 035416 (2003).
W. Haiss, R. J. Nichols, H. van Zalinge, S. J. Higgins, D. Bethell, D. J. Schiffrin, Phys. Chem. Chem. Phys. [**6**]{}, 4330 (2004).
X. Li, J. He, J. Hihath, B. Xu, S. M. Lindsay, N. Tao, J. Am. Chem. Soc. [**128**]{}, 2135 (2006).
M. T. González, R. Huber, S. J. van der Molen, C. Schönenberger, M. Calame, Nano Lett. [**6**]{}, 2238 (2006).
M. Fujihira, M. Suzuki, S. Fujii, A. Nishikawa, Phys. Chem. Chem. Phys. [**8**]{}, 3876 (2006).
S. Y. Jang, P. Reddy, A. Majumdar, R. A. Segalman, Nano Lett. [**6**]{}, 2362 (2006).
F. Chen, X. Li, J. Hihath, Z. Huang, N. J. Tao, J. Am. Chem. Soc. [**128**]{}, 15874 (2006).
L. Venkataraman, J. E. Klare, I. W. Tam, C. Nuckolls, M. S. Hybertsen, M. L. Steigerwald, Nano Lett. [**6**]{}, 458 (2006).
Y. S. Park, A. C. Whalley, M. Kamenetska, M. L. Steigerwald, M. S. Hybertsen, C. Nuckolls, L. Venkataraman, J. Am. Chem. Soc. [**129**]{}, 15768 (2007).
C. Li, I. Pobelov, T. Wandlowski, A. Bagrets, A. Arnold, F. Evers, J. Am. Chem. Soc. [**130**]{}, 318 (2008).
M. T. González, J. Brunner, R. Huber, S. Wu, C. Schönenberger, M. Calame, New J. Phys. [**10**]{}, 065018 (2008).
C. A. Martin, D. Ding, J. K. Sørensen, T. Bjørnholm, J. M. van Ruitenbeek, H. S. J. van der Zant, J. Am. Chem. Soc. [**130**]{}, 13198 (2008).
H. B. Akkerman and B. de Boer, J. Phys.: Condens. Matter [**20**]{}, 013001 (2008).
R. Y. Wang, R. A. Segalman, and A. Majumdar, Appl. Phys. Lett. [**89**]{}, 173113 (2006).
T. Meier, F. Menges, P. Nirmalraj, H. Hölscher, H. Riel, and B. Gotsmann, Phys. Rev. Lett. [**113**]{}, 060801 (2014).
D. Segal, A. Nitzan, and P. Hänggi, J. Chem. Phys. [**119**]{}, 6840 (2003).
J. C. Duda, C. B. Saltonstall, P. M. Norris, and P. E. Hopkins, J. Chem. Phys. [**134**]{}, 094704 (2011).
H. Sadeghi, S. Sangtarash, and C. J. Lambert, Nano Lett. [**15**]{}, 7467 (2015).
M. Bürkle, T. J. Hellmuth, F. Pauly, and Y. Asai, Phys. Rev. B [**91**]{}, 165419 (2015).
TURBOMOLE 6.5, TURBOMOLE GmbH Karlsruhe, http://www.turbomole.com. TURBOMOLE is a development of University of Karlsruhe and Forschungszentrum Karlsruhe 1989-2007, TURBOMOLE GmbH since 2007.
P. Deglmann, F. Furche, and R. Ahlrichs, Chem. Phys. Lett. [**362**]{}, 511 (2002).
P. Deglmann, K. May, F. Furche, and R. Ahlrichs, Chem. Phys. Lett. [**384**]{}, 103 (2004).
J. P. Perdew and Y. Wang, Phys. Rev. B [**45**]{}, 13244 (1992).
J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. [**77**]{}, 3865 (1996).
F. Weigend and R. Ahlrichs, Phys. Chem. Chem. Phys. [**7**]{}, 3297 (2005).
F. Weigend, Phys. Chem. Chem. Phys. [**8**]{}, 1057 (2006).
F. Pauly, J. K. Viljas, U. Huniar, M. Häfner, S. Wohlthat, M. Bürkle, J. C. Cuevas, and G. Schön, New J. Phys. [**10**]{}, 125019 (2008).
L. G. C. Rego and G. Kirczenow, Phys. Rev. Lett. [**81**]{}, 232 (1998).
N. Mingo and L. Yang, Phys. Rev. B [**68**]{}, 245406 (2003).
T. Yamamoto and K. Watanabe, Phys. Rev. Lett. [**96**]{}, 255503 (2006).
Y. Asai, Phys. Rev. B [**78**]{}, 045434 (2008).
U. Sivan and Y. Imry, Phys. Rev. B [**33**]{}, 551 (1986).
S. Y. Quek, L. Venkataraman, H. J. Choi, S. G. Louie, M. S. Hybertsen, and J. B. Neaton, Nano Lett. [**7**]{}, 3477 (2007).
L. A. Zotti, M. Bürkle, F. Pauly, W. Lee, K. Kim, W. Jeong, Y. Asai, P. Reddy, and J. C. Cuevas, New J. Phys. [**16**]{}, 015004 (2014).
T. Luo and J. R. Lloyd, Int. J. Heat Mass Transfer [**53**]{}, 1 (2010).
M. D. Losego, M. E. Grady, N. R. Sottos, D. G. Cahill, and P. V. Braun, Nat. Mater. [**11**]{}, 502–506 (2012).
|
---
abstract: 'We provide a complete set of linearizability conditions for nonlinear partial difference equations defined on four points and, using them, we classify all linearizable multilinear partial difference equations defined on four points up to a Möbious transformation.'
---
[**[Classification of discrete equations linearizable by point transformation on a square lattice]{}\
[ ]{}**]{}
[**Christian Scimiterna and Decio Levi**]{}
[Dipartimento di Ingegneria Elettronica\
Università degli Studi Roma Tre and INFN Sezione di Roma Tre\
Via della Vasca Navale 84, 00146 Roma, Italy]{}
[[**E-mail:**]{} scimiterna@fis.uniroma3.it; levi@roma3.infn.it ]{}
PACS numbers:
Mathematics Subject Classification:
Introduction
============
In a series of papers [@ls0; @ls; @ls1; @ls2] one has provided necessary conditions for the linearizability of real dispersive multilinear difference equations on a quad–graph (see Fig. \[fig1\]).
\[fig1\]
(200,140)(-50,-20) ( 0, 0)[(1,0)[100]{}]{} ( 100, 0)[(-1,0)[100]{}]{} ( 0,100)[(1,0)[100]{}]{} ( 100,100)[(-1,0)[100]{}]{} ( 0, 0)[(0,1)[100]{}]{} ( 0, 100)[(0,-1)[100]{}]{} (100, 0)[(0,1)[100]{}]{} (100, 100)[(0,-1)[100]{}]{} (97, -3)[$\bullet$]{} (-3, -3)[$\bullet$]{} (-3, 97)[$\bullet$]{} (97, 97)[$\bullet$]{} (-32,-13)[$u_{n,m}$]{} (103,-13)[$u_{n+1,m}$]{} (103,110)[$u_{n+1,m+1}$]{} (-32,110)[$u_{n,m+1}$]{}
and on three points (Fig. 2).
\[fig2\]
(100,51) (80,10) (50,10) (50,40) (95,11)[$r_1$]{} (51,46)[$r_2$]{} (41,51)[$r_3$]{} (49,07)[$u_{n,m}$]{} (79,07)[$u_{n+1,m}$]{} (51,39)[$u_{n,m+1}$]{} (40,50)[(1,-1)[55]{}]{} (40,10) [(1,0)[60]{}]{} (50,10) [(0,1)[50]{}]{}
These conditions, obtained by considering the existence of point transformations and symmetries have been sufficient to classify the multilinear equations defined on three points [@ls1] but not those defined on four points. In [@ls2] we considered the problem from the point of view of the symmetries, both point and nonlocal. In this way we get a different set of conditions with respect to those obtained before which, however, are not yet sufficient to classify the multilinear equations defined on four points. So here, using the experience of [@ls1] and [@ls2] we construct the largest possible set of linearizability conditions and, through them, we classify the multilinear equations on a square lattice. We assume a partial difference equation on a quad–graph to be given by \[1.1a\] E =E(u\_[n,m]{}, u\_[n,m+1]{}, u\_[n+1,m]{},u\_[n+1,m+1]{})=0, =0, i,j=0,1, for a field $u_{n,m}$ which linearizes into a linear autonomous equation for $\tilde u_{n,m}$ \[1.3a\] a[u]{}\_[n,m]{}+b[u]{}\_[n+1,m]{}+c[u]{}\_[n,m+1]{}+d[u]{}\_[n+1,m+1]{}+e = 0 with $a$, $b$, $c$, $d$ and $e$ being ($n,m$)–independent arbitrary non zero complex coefficients. The choice that (\[1.3a\]) be autonomous is a restriction but it is also a natural simplifying ansatz when one is dealing with autonomous equations. Moreover, as (\[1.1a\], \[1.3a\]) are taken to be autonomous equations, i.e. they have no $n,m$ dependent coefficients, they are translationally invariant under shifts in $n$ and $m$. So we can with no loss of generality choose as reference point $n=0$ and $m=0$. This will also be assumed to be true for the linearizing point tranformation.
By a [*linearizing point transformation*]{} we mean a transformation \[1.3b\] u\_[0,0]{} = f(u\_[0,0]{}) between (\[1.3a\]) and (\[1.1a\]) characterized by a function depending just from the function $u_{0,0}$ and on some constant parameters. It will be a [*Lie point transformation*]{} if $f=f_{0,0}$ satisfies all Lie group axioms and, in particular, the composition law. In the following we will require much less, i.e. we will only assume the differentiability of the function $f$ up to at least second order.
In Section 2 we discuss point transformations, present the linearizability conditions which ensure that the given equation is linearizable and the differential equations which define the transformation $f$. In Section 3 we classify all multilinear equations which belong to the class up to a Möbious transformation while in the final Section we present some conclusive remarks and open problems.
Discrete equations defined on a square linearizable by a point transformation.
==============================================================================
In the autonomous case a generic partial difference equation for the complex function $u_{n,m}\doteq u_{0,0}$ can be rewritten as E(u\_[0,0]{},u\_[1,0]{},u\_[0,1]{},u\_[1,1]{})=0, =0, i,j=0,1, \[Roma\] We will assume that we can solve (\[Roma\]) with respect to each one of the four variables in its argument
\[Zero\] \[Zeroa\] u\_[1,1]{}&=&F(u\_[0,0]{},u\_[1,0]{},u\_[0,1]{}), F\_[,u\_[0,0]{}]{}=0 F\_[,u\_[1,0]{}]{}=0, F\_[,u\_[0,1]{}]{}=0,\
\[Zerob\] u\_[1,0]{}&=&G(u\_[0,0]{},u\_[0,1]{},u\_[1,1]{}), G\_[,u\_[0,0]{}]{}=0, G\_[,u\_[0,1]{}]{}=0, G\_[,u\_[1,1]{}]{}=0,\
\[Zeroc\] u\_[0,1]{}&=&S(u\_[0,0]{},u\_[1,0]{},u\_[1,1]{}), S\_[,u\_[0,0]{}]{}=0, S\_[,u\_[1,0]{}]{}=0 S\_[,u\_[1,1]{}]{}=0,\
\[Zerod\] u\_[0,0]{}&=&T(u\_[1,0]{},u\_[0,1]{},u\_[1,1]{}), T\_[,u\_[1,0]{}]{}=0, T\_[,u\_[0,1]{}]{}=0 T\_[,u\_[1,1]{}]{}=0,
and that can be linearized by the linearizing autonomous point transformation into the linear equation for the complex function $u_{n,m}\doteq u_{0,0}$, i.e au\_[0,0]{}+bu\_[1,0]{}+cu\_[0,1]{}+du\_[1,1]{}+e=0.\[Janus\] Hence, assuming we can solve (\[Roma\]) with respect to $u_{1,1}$, we can choose as independent variables $u_{0,0}$, $u_{1,0}$ and $u_{0,1}$ and we will have that af\_[0,0]{}+bf\_[1,0]{}+cf\_[0,1]{}+df\_[1,1]{}\_[u\_[1,1]{}=F]{}+e=0,\[Amor\] must be identically satisfied for any $ u_{0,0}$, $u_{1,0}$ and $u_{0,1}$. Differentiating (\[Amor\]) with respect to $u_{0,0}$, $u_{1,0}$ or $u_{0,1}$, we obtain
\[Saturno\] a+d\_[u\_[1,1]{}=F]{}F\_[,u\_[0,0]{}]{}=0,\[Saturno1\]\
\[Saturno2\] b+d\_[u\_[1,1]{}=F]{}F\_[,u\_[1,0]{}]{}=0,\
\[Saturno3\] c+d\_[u\_[1,1]{}=F]{}F\_[,u\_[0,1]{}]{}=0,
which have to be identically satisfied for any $ u_{0,0}$, $u_{1,0}$ and $u_{0,1}$. From them, considering that $\frac{df\left(x\right)}{dx}\not=0$, we derive that $d\not=0$, otherwise $a=b=c=e=0$. As a consequence, considering that also $F_{,u_{0,0}}\not=0$, $F_{,u_{1,0}}\not=0$ and $F_{,u_{0,1}}\not=0$, we have $a\not=0$, $b\not=0$ and $c\not=0$. Then in all generality we can divide (\[Janus\]) by $d$ and, introducing the new parameters $\alpha\doteq a/d\not=0$, $\beta\doteq b/d\not=0$, $\gamma\doteq c/d\not=0$ and $\epsilon\doteq e/a$, (\[Janus\]) can be rewritten as u\_[0,0]{}+u\_[1,0]{}+u\_[0,1]{}+u\_[1,1]{}+=0. Defining $\frac{df\left(x\right)}{dx}\doteq H\left(x\right)$, from (\[Saturno\]) we obtain
\[Isis\] =,\
=.
From (\[Isis\]) we get the following linearizability conditions
\[Osiris\] A(x,u\_[0,1]{})\_[u\_[0,0]{}=u\_[1,0]{}=x]{}=, x, u\_[0,1]{},\[Osiris1\]\
B(x,u\_[1,0]{})\_[u\_[0,0]{}=u\_[0,1]{}=x]{}=, x, u\_[1,0]{},\[Osiris2\]\
C(x,u\_[0,0]{})\_[u\_[1,0]{}=u\_[0,1]{}=x]{}=, x, u\_[0,0]{},\[Osiris3\]\
=0, u\_[0,0]{}, u\_[1,0]{}, u\_[0,1]{},\[Osiris4\] \
=0, u\_[0,0]{}, u\_[1,0]{}, u\_[0,1]{},\[Osiris5\] \
=0, u\_[0,0]{}, u\_[1,0]{}, u\_[0,1]{}.\[Osiris6\]
Alternatively the conditions (\[Osiris1\]-\[Osiris3\]) can be substituted by the following ones
A(x,u\_[0,1]{})=0, x, u\_[0,1]{},\
B(x,u\_[1,0]{})=0, x, u\_[1,0]{},\
C(x,u\_[0,0]{})=0, x, u\_[0,0]{}.
Taking the (principal value of the) logarithm of (\[Saturno1\]), we have -\_[u\_[1,1]{}=F]{}=(-) (2).\[Sat\] Then, let us introduce the linear operator $\mathcal B$ \[lob\] B-, such that and $B\phi\left(F\left(u_{0,0},u_{1,0},u_{0,1}\right)\right)=0$, where $\phi$ is an arbitrary functions of its argument. When we apply to (\[Sat\]), we obtain an ordinary differential equation describing the evolution of the linearizing transformation =W\_[(u\_[0,0]{})]{},\[Romulius\] where $W_{(x)}\left[f;g\right]\doteq fg_{,x}-gf_{,x}$ stands for the Wronskian of the functions $f$ and $g$. Let’s remark that the linearizability conditions (\[Osiris4\]-\[Osiris6\]) imply that the right hand member of the equation (\[Romulius\]) does not depend on $u_{1,0}$ and $u_{0,1}$. These conditions were considered in [@ls] and had not been sufficient to classify . Other similar conditions can be obtained starting from or . The linearizability conditions presented here have been obtained starting from . Similar results could be obtained starting from , or . However these results would not have provided any really new linearizability condition. So, here, in the next Section we start the classifying process from the more basic equations as we did in the case of equations depending on just three points [@ls2].
Classification of complex autonomous multilinear partial difference equations defined on four points linearizable by a point transformation.
============================================================================================================================================
Let $\mathcal E\left(u_{0,0},u_{1,0},u_{0,1},u_{1,1}\right)=0$ be the complex multilinear equation \[Mars\] a\_[1]{}u\_[0,0]{}&+&a\_[2]{}u\_[1,0]{}+a\_[3]{}u\_[0,1]{}+a\_[4]{}u\_[1,1]{}+a\_[5]{}u\_[0,0]{}u\_[1,0]{}+a\_[6]{}u\_[0,0]{}u\_[0,1]{}+\
&+&a\_[7]{}u\_[0,0]{}u\_[1,1]{}+a\_[8]{}u\_[1,0]{}u\_[0,1]{}+a\_[9]{}u\_[1,0]{}u\_[1,1]{}+a\_[10]{}u\_[0,1]{}u\_[1,1]{}+\
&+&a\_[11]{}u\_[0,0]{}u\_[1,0]{}u\_[0,1]{}+a\_[12]{}u\_[0,0]{}u\_[1,0]{}u\_[1,1]{}+a\_[13]{}u\_[0,0]{}u\_[0,1]{}u\_[1,1]{}+\
&+&a\_[14]{}u\_[1,0]{}u\_[0,1]{}u\_[1,1]{}+a\_[15]{}u\_[0,0]{}u\_[1,0]{}u\_[0,1]{}u\_[1,1]{}+a\_[16]{}=0, where $a_{j}$, $j=1,\ldots$, $16$, are arbitrary complex free parameters. This equation is invariant under a Möbious transformation of the dependent variable u\_[0,0]{}, where $b_{k}$, $k=1,\ldots$, $4$ are four arbitrary complex parameters such that $b_{1}b_{4}-b_{2}b_{3}\not=0$. As we operate in the field of complex numbers and we classify up to Möbious transformations, using inversions, dilations and translations we can always simplify by setting either [**1)**]{} $a_{15}=a_{16}=0$ or [**2)**]{} $\sum_{k=1}^{4}a_{k}=\sum_{k=5}^{10}a_{k}=\sum_{k=11}^{14}a_{k}=a_{15}=0$, $a_{16}=1$. Let’s now apply to these two cases the six necessary linearizability conditions (\[Osiris\]). This amounts to solving a system of 96 algebraic in general nonlinear equations involving the coefficients $a_{j}$, $j=1,\ldots$, $14$. Their solution implies, through the integration of the differential equation (\[Romulius\]), that the function $f\left(x\right)$ appearing in the linearizing transformation (\[1.3a\]) can only be of the following two types:
1. The fractional linear function f(x)=, c\_[1]{}c\_[4]{}-c\_[2]{}c\_[3]{}=0, which, as the classification is up to Möbious transformations of $u_{0,0}$, can always be reduced to be $f\left(x\right)=1/x$;
2. The (principal branch) of the logarithmic function f(x)=d\_[1]{}()+d\_[6]{} d\_[1]{}=0, d\_[2]{}d\_[5]{}-d\_[3]{}d\_[4]{}=0, which, as the classification is up to Möbious transformations, can always be reduced to $f\left(x\right)=\log\left(x\right)$. Moreover in this case the ratios $\alpha/\beta$ and $\alpha/\gamma$ are always real and of modulus $1$, so that the possible linear equations can be only of the following four types:
(u\_[0,0]{}+u\_[1,0]{}+u\_[0,1]{})+u\_[1,1]{}+=0;\
(u\_[0,0]{}+u\_[1,0]{}-u\_[0,1]{})+u\_[1,1]{}+=0;\
(u\_[0,0]{}-u\_[1,0]{}+u\_[0,1]{})+u\_[1,1]{}+=0;\
(u\_[0,0]{}-u\_[1,0]{}-u\_[0,1]{})+u\_[1,1]{}+=0.
It is easy to prove that, if the transformation $\tilde u_{0,0}\doteq\log\left(u_{0,0}\right)$ has to produce a multilinear equation for $u_{0,0}$, we must have $\alpha=\pm 1$. In fact, as $F\left(u_{0,0},u_{1,0},u_{0,1}\right)$ should be a fractional linear function of $u_{0,0}$ with coefficients depending on $u_{1,0}$ and $u_{0,1}$, we have that the relations \[new\] u\_[0,0]{}\^(e\_[1]{}u\_[0,0]{}+e\_[2]{})=e\_[3]{}u\_[0,0]{}+e\_[4]{}, e\_[j]{}=e\_[j]{}(u\_[1,0]{},u\_[0,1]{}), j=1,…,4,where $e_{1}e_{4}-e_{2}e_{3}$ is not identically zero for all $ u_{1,0}$ and $u_{0,1}$, must be identically satisfied for all $ u_{0,0}$. Differentiating twice with respect to $u_{0,0}$, we get $e_{1}\left(\alpha+1\right)u_{0,0}+e_{2}\left(\alpha-1\right)=0$ identically for all $ u_{0,0}$, so that e\_[1]{}(+1)=0, e\_[2]{}(-1)=0.Considering that $e_{1}$ and $e_{2}$ cannot be simultaneously identically zero, it follows that $\alpha=\pm 1$. In this way we obtain a set of eight linear equations corresponding to eight linearizable nonlinear equations.
Hence we can summarize the results obtained in the following Theorem:
\[t1\] Apart from the class of equations linearizable by a Möbious transformation, which can be represented up to a Möbious transformation of the dependent variable (eventually composed with an exchange of the independent variables $n\leftrightarrow m$) by the equation w\_[0,1]{}w\_[1,1]{}(w\_[1,0]{}+w\_[0,0]{})+w\_[0,0]{}w\_[1,0]{}(w\_[1,1]{}+w\_[0,1]{})+w\_[0,0]{}w\_[1,0]{}w\_[0,1]{}w\_[1,1]{}=0, =0,1, ,,=0, linearizable by the inversion $\tilde u_{0,0}=1/w_{0,0}$ to the equation u\_[0,0]{}+u\_[1,0]{}+u\_[0,1]{}+u\_[1,1]{}+=0, the only other linearizable equations are up to a Möbious transformation of the dependent variable (eventually composed with an exchange of the independent variables $n\leftrightarrow m$), represented by the following six nonlinear equations
&w\_[0,0]{}w\_[1,0]{}w\_[0,1]{}w\_[1,1]{}-1&=0,\[Maro1\]\
&w\_[0,0]{}-w\_[1,0]{}w\_[0,1]{}w\_[1,1]{}&=0,\[Maro2\]\
&w\_[1,0]{}-w\_[0,0]{}w\_[0,1]{}w\_[1,1]{}&=0,\[Maro3\]\
&w\_[1,1]{}-w\_[0,0]{}w\_[1,0]{}w\_[0,1]{}&=0,\[Maro4\]\
&w\_[0,1]{}w\_[1,1]{}-w\_[0,0]{}w\_[1,0]{}&=0,\[Maro5\]\
&w\_[0,0]{}w\_[1,1]{}-w\_[1,0]{}w\_[0,1]{}&=0,\[Maro6\] where $\theta\not=0$ is an otherwise arbitrary complex parameter. They are linearizable by the transformation $\tilde u_{0,0}=\log w_{0,0}$ to the equations &u\_[0,0]{}+u\_[1,0]{}+u\_[0,1]{}+u\_[1,1]{}&=2z,\[Lati1\]\
&-u\_[0,0]{}+u\_[1,0]{}+u\_[0,1]{}+u\_[1,1]{}&=2z,\[Lati2\]\
&u\_[0,0]{}-u\_[1,0]{}+u\_[0,1]{}+u\_[1,1]{}&=2z,\[Lati3\]\
&-u\_[0,0]{}-u\_[1,0]{}-u\_[0,1]{}+u\_[1,1]{}&=2z,\[Lati4\]\
&-u\_[0,0]{}-u\_[1,0]{}+u\_[0,1]{}+u\_[1,1]{}&=+2z,\[Lati5\]\
&u\_[0,0]{}-u\_[1,0]{}-u\_[0,1]{}+u\_[1,1]{}&=+2z,\[Lati6\]
where $\log$ always stands for the principal branch of the complex logarithmic function and where $\log\theta$ stands for the principal branch of the complex logarithmic function of the parameter $\theta$.
Let’s note that, given a solution $w_{0,0}$ of eqs. (\[Maro1\]-\[Maro6\]), the choice of the principal branch of the complex logarithm function in the transformation $\tilde u_{0,0}=\log w_{0,0}$ reflects in the inhomogeneous terms of the linear eqs. (\[Lati1\]-\[Lati6\]) and is always $|z|\leq 2$. This can be easily seen considering that we must have $2\pi|z|\leq |Im\left(u_{0,0}\right)|+|Im\left(u_{1,0}\right)|+|Im\left(u_{0,1}\right)|+|Im\left(u_{1,1}\right)|\leq 4\pi$ (with a slight difference in and ). Through the translation $\tilde u_{0,0}=v_{0,0}+\pi\ri z/2$ the linear equation (\[Lati1\]) can be made homogeneous. As a consequence in (\[Maro1\]) $w_{0,0}=e^{v_{0,0}}e^{\pi\ri z/2}$, so that the integer parameter $z$ can take the values $z=-1$, $0$, $1$, $2$. Hence we conclude that the term $2\pi\ri z$ in (\[Lati1\]) takes into account the discrete symmetry of (\[Maro1\]) given by $w_{0,0}\rightarrow\zeta w_{0,0}$, where $\zeta$ stands for one of the four roots of unity $\zeta=1$, $-1$, $\ri$ and $-\ri$. The same happens when applying the translation $\tilde u_{0,0}=v_{0,0}+\pi\ri z$ to the linear equations (\[Lati2\], \[Lati3\]) and $\tilde u_{0,0}=v_{0,0}-\pi\ri z$ to . In (\[Maro2\], \[Maro3\], \[Maro4\]) we have $w_{0,0}=e^{v_{0,0}}e^{\pm\pi\ri z}$ so that, without any loss of generality, we can restrict the values of $z$ to $z=0$, $1$. The term $2\pi\ri z$ in (\[Lati2\], \[Lati3\], \[Lati4\]) takes into account the discrete symmetry of (\[Maro2\], \[Maro3\], \[Maro4\]) given by $w_{0,0}\rightarrow\zeta w_{0,0}$, where $\zeta$ stands for one of the two roots of unity $\zeta=1$ and $-1$. The same happens for the linear equations (\[Lati5\]) when we consider the non autonomous translation $\tilde u_{0,0}=v_{0,0}+\left(\log\theta +2\pi\ri z\right)m/2$. In (\[Maro5\]) $w_{0,0}=e^{v_{0,0}}e^{\pi\ri mz}\theta^{m/2}$ so that, without any loss of generality, we can also restrict the values $z$ to $z=0$, $1$. Hence we conclude that the term $2\pi\ri z$ in (\[Lati5\]) takes into account the discrete symmetry of (\[Maro5\]) given by $w_{0,0}\rightarrow\left(-1\right)^m w_{0,0}$. The same happens for the linear equation (\[Lati6\]) with the non autonomous translation $\tilde u_{0,0}=v_{0,0}+\left(\log\theta+2\pi\ri z\right)nm$. Now, as $e^{2\pi\ri nmz}=1$ for any $ z$, in (\[Maro6\]) $w_{0,0}=e^{v_{0,0}}\theta^{nm}$ so that, without any loss of generality, we can choose $z=0$.
Concluding remarks and outlook
==============================
In this paper we have classified all multilinear partial difference equations which can be linearized by a point transformation. The resulting linearizable equations are presented in Theorem \[t1\] together with their linearized counterparts.
It seems interesting at this point to try to analyze the case of more complicate classes of equations involving more lattice points and which could provide in the continuous case parabolic or elliptic partial differential equations. Work on this is in progress.
Acknowledgements {#acknowledgements .unnumbered}
================
DL and CS have been partly supported by the Italian Ministry of Education and Research, 2010 PRIN “Continuous and discrete nonlinear integrable evolutions: from water waves to symplectic maps".
[99]{} C. Scimiterna and D. Levi, C-Integrability Test for Discrete Equations via Multiple Scale Expansions, [*SIGMA*]{} [**6**]{} (2010), 070, 17 pages, arXiv:1005.5288, http://dx.doi.org/10.3842/SIGMA.2010.070.
D. Levi a and C. Scimiterna, Linearizability of Nonlinear Equations on a Quad-Graph by a Point, Two Points and Generalized Hopf-Cole Transformations, [*SIGMA*]{} [**7**]{} (2011), 079, 24 pages, arXiv:1108.3648, http://dx.doi.org/10.3842/SIGMA.2011.079.
D. Levi and C. Scimiterna, Linearization through symmetries for discrete equations, submitted to SIGMA 2012.
C. Scimiterna and D. Levi, Three points partial difference equations linearizable by local and nonlocal transformations, submitted to J. Phys. A, 2012.
|
---
abstract: 'We consider a non-isothermal multi-phase field model. We subsequently discretize implicitly in time and with linear finite elements. The arising algebraic problem is formulated in two variables where one is the multi-phase field, and the other contains the inverse temperature field. We solve this saddle point problem numerically by a non-smooth Schur–Newton approach using truncated non-smooth Newton multigrid methods. An application in grain growth as occurring in liquid phase crystallization of silicon is considered.'
address:
- |
Carsten Gräser\
Freie Universität Berlin\
Institut für Mathematik\
Arnimallee 6\
D - 14195 Berlin\
Germany
- |
Max Kahnt\
Freie Universität Berlin\
Institut für Mathematik\
Arnimallee 6\
D - 14195 Berlin\
Germany
- |
Ralf Kornhuber\
Freie Universität Berlin\
Institut für Mathematik\
Arnimallee 6\
D - 14195 Berlin\
Germany
author:
- Carsten Gräser
- Max Kahnt
- Ralf Kornhuber
bibliography:
- 'paper.bib'
title: 'Numerical approximation of multi-phase Penrose–Fife systems'
---
Introduction
============
The mathematical modelling of phase transitions has a long history and has stimulated new developments in the field of variational inequalities and free boundary value problems over more than three decades [@BaiocchiCapelo1984; @Crank1982; @kinderlehrer1980introduction]. Particular attention was paid to problems of Stefan-type [@Visintin2012] and their mathematical description by phase field models [@BrokateSprekels96]. In this approach, phase transitions are represented by an order parameter that is strongly varying across the (diffuse) interface. The evolution of the order parameter is typically obtained from some gradient flow of a suitable Ginzburg–Landau free energy that provides non-decreasing entropy (thermodynamical consistency) and could be mass conserving (phase separation) or non-conserving (phase transition). More recently Stinner et al. [@Stinner2004] extended well-established thermodynamically consistent, two-phase Penrose–Fife models [@BrokateSprekels96] to multiple phases (non-conserved) and components (conserved). Existence of solutions to the resulting balance equations for the energy, order parameters, and concentrations of components was studied in [@stinner2007weak].
While the numerical analysis of two-phase Penrose–Fife models was based on implicit time discretization [@OKlein97], previous numerical computations with multiple phases and components were typically based on an explicit approach [@nestler2008phase]. In this way, the solution of non-smooth, large-scale algebraic systems is avoided at the expense of severe stability constraints on the time step.
In this paper, we consider a multi-phase extension of the classical Penrose–Fife system [@BrokateSprekels96; @OKlein97; @penrose1990thermodynamically]. Following Stinner et al. [@Stinner2004], this system is derived from a general entropy functional that combines a Ginzburg–Landau energy with the thermodynamic entropy. We concentrate on a numerical approach based on semi-implicit time discretization (with explicit treatment of the concave terms [@deckelnick2005computation; @graser2013time]) and first-order Taylor approximation of nonlinearities associated with inverse temperature. Variational arguments are used to show the existence and uniqueness of solutions of the resulting spatial problems and the thermodynamic consistency of this time discretization. Spatial discretization is performed by piecewise linear finite elements with adaptive mesh refinement based on hierarchical a posteriori error estimation [@graeser:2014a; @GraeserKornhuberSack2010]. The resulting large-scale non-smooth algebraic systems are solved by non-smooth Schur–Newton multigrid (NSNMG) methods [@graeserthesis; @graeser2014; @graser2009truncated] exploiting again the saddle point structure of these problems. In our numerical experiments, we observe optimal order of convergence of the spatial discretization and mesh-independent, fast convergence speed of NSNMG with nested iteration. Furthermore, our computations suggest that non-decreasing entropy is preserved under the spatial discretization. Application to a liquid phase crystallization (LPC) process occurring in the fabrication of thin film silicon solar cells [@kuhnapfel2015towards] underline the potential of the presented solution approach.
Phase field modelling {#ss:assumptions}
=====================
Thermodynamical background
--------------------------
Let $\Omega \subset {\mathbb{R}}^d$, $d=1$, $2$, $3$, be a bounded domain with Lipschitz boundary $\Gamma= \partial \Omega$. Following [@Stinner2004], we consider the entropy functional $$\begin{aligned}
S(e,\phi) = \int_\Omega s(e,\phi) - (\tfrac{\varepsilon}2 \gamma^2(\phi, \nabla\phi) + \tfrac1{\varepsilon}\psi(\phi)) {\,\textnormal{d}}x,\end{aligned}$$ where the entropy density $s$ depends on the internal energy density $e$ and on the multi-phase field $\phi= (\phi_{{\alpha}})_{{\alpha}=1}^M$, $\gamma$ represents the surface gradient entropy, and $\psi$ a multi-well potential with $M$ distinct minima. The components of $\phi$ describe relative fractions of a given substance. Hence, it is natural to impose the algebraic constraint $$\label{eq:ONE}
\sum_{{\alpha}=1}^M \phi_{\alpha}=1.$$ We postulate the Gibbs relation [@AltPawlow; @Stinner2004] $$\begin{aligned}
\label{Gibbs-relation}
{\,\textnormal{d}}f &= -s {\,\textnormal{d}}T + \sum_{{\alpha}=1}^M f_{,\phi_{\alpha}} {\,\textnormal{d}}\phi_{\alpha}\end{aligned}$$ with absolute temperature $T>0$ and a Helmholtz free energy density $f=f(T,\phi)$ according to $$\label{energy-contributions}
e = f + sT.$$ As a consequence, we have $$\label{eq:entropytemp}
s=-f_{,T},\qquad
{\,\textnormal{d}}s = \tfrac{1}{T}({\,\textnormal{d}}e - {\,\textnormal{d}}f) = \displaystyle \tfrac{1}{T} {\,\textnormal{d}}e- \sum_{{\alpha}=1}^M \tfrac{1}{T} f_{, \phi_{{\alpha}}} {\,\textnormal{d}}\phi_{\alpha},$$ and therefore $$\label{eq:ENTROPDIFF}
s_{, e}= \tfrac{1}{T},\qquad s_{,\phi_{\alpha}}= - \tfrac{1}{T} f_{, \phi_{\alpha}} \quad \alpha =1, \dots, M.$$ We assume that the free energy density $f$ is obtained by interpolation of the individual bulk free energies $L_{\alpha}\tfrac{T-T_{\alpha}}{T_{\alpha}} - c_vT(\ln(T)-1)$ for each phase ${\alpha}$. Here, ${L}_{\alpha}\geq 0$ and $T_{\alpha}>0$ represent the latent heat and melting temperature of pure phase ${\alpha}$, respectively, and $c_v>0$ is the specific heat capacity. In the light of , this leads to $$\label{total-free-energy}
f(T,\phi) = \sum_{{\alpha}=1}^M L_{\alpha}\tfrac{T-T_{\alpha}}{T_{\alpha}}\phi_\alpha - c_vT(\ln(T)-1),$$ and we have $$\label{individual-free-energy}
f_{, \phi_{\alpha}}(T,\phi) = L_{\alpha}\tfrac{T-T_{\alpha}}{T_{\alpha}}.$$
Utilizing , , and the state equation , we can represent the entropy $s$ and the energy $e$ in terms of the temperature $T$ and the phase field $\phi$ according to $$\label{energy}
s(e,\phi) = \tilde{s}(T,\phi) = - \sum_{{\alpha}= 1}^M L_{\alpha}\tfrac{1}{T_{\alpha}}\phi_{\alpha}+ c_v \ln(T), \quad
e = \tilde{e}(T,\phi) = - \sum_{{\alpha}=1}^M L_{\alpha}\phi_{\alpha}+c_v T.$$ Though our approach could be extended to anisotropic interfacial energies [@graser2013time], we choose $$\label{eq:INTERFACE}
\gamma(\phi,\nabla\phi) = \vert \nabla\phi\vert$$ for simplicity. Finally, $\psi$ stands for the classical multi-obstacle potential [@blowey_elliot:ch_obst_math:1991; @barrett1997finite] $$\label{well-potential}
\psi(\phi) = \chi_G(\phi) + \tfrac12\phi^T K \phi,$$ with $\chi$ denoting the characteristic function $$\chi_A(x) = 0 \textnormal{ if } x \in A \textnormal{ and } \chi_A(x) = \infty \textnormal{ if } x \not\in A$$ and $G$ the Gibbs simplex $$\label{eq:GIBBSSIMP}
G=\{v=(v_{\alpha})\in {\mathbb{R}}^M {\,|\,}\sum_{{\alpha}=1}^M v_{\alpha}= 1 \text{ and }v_{\alpha}\geq 0, \; {\alpha}=1,\dots,M\} \subset {\mathbb{R}}^M.$$ We choose the negative definite interaction matrix $$K=- I \in {\mathbb{R}}^{M,M}$$ leading to the concave contribution $\phi^T K\phi = - |\phi|^2$ to the multi-obstacle potential.
With these specifications the entropy functional takes the form $$\begin{aligned}
\label{eq:entropy_with_obstacle}
S(e,\phi) =S_0(e,\phi) - \chi_{\mathcal{G}}(\phi), \quad S_0(e,\phi)= \int_\Omega s(e,\phi) - \tfrac{{\varepsilon}}{2}|\nabla\phi|^2 - \tfrac{1}{2{\varepsilon}} \phi^TK\phi {\,\textnormal{d}}x\end{aligned}$$ where $\chi_{\mathcal{G}}$ is the characteristic functional of $${\mathcal{G}}=\{v \in H^1(\Omega)^M {\,|\,}v(x)\in G\text{ a.e. in }\Omega\}.$$
A multi-phase Penrose–Fife system
---------------------------------
We postulate the continuity equation $$\label{eq:ECONT}
e_t=-\nabla \cdot J_0(e,\phi) + q(e,\phi)$$ with the flux $$J_0(e,\phi) =
\kappa \nabla (\delta_e S)(e, \phi),$$ mobility $\kappa>0$, variational derivative $\delta_e S$, and a source term $q(e,\phi)$ to obtain $$\label{energy-balance}
e_t= - \nabla \cdot \kappa \nabla ( \delta_e S ) (e,\phi) + q(e, \phi).$$
We assume that the outward energy flux is proportional to the difference of the temperature $T$ and a given boundary temperature $T_\Gamma$ or, more precisely, we prescribe $$\label{eq:ENGBND}
J_0 \cdot n = {h_c}(T -T_\Gamma)$$ with the convection coefficient ${h_c}>0$ and the outward normal $n$ to $\Omega$. It is also convenient to introduce the inverse temperature $$\label{eq:TTRANS}
\theta = \frac{1}{T}, \qquad \theta_\Gamma = \frac{1}{T_\Gamma}.$$ Note that for given $\phi$ the variables $e$, $T$, and $\theta$ can be transformed into each other due to the strictly monotone relationships and .
In order to provide a non-decreasing entropy $S(e,\phi)$ in the course of the phase evolution, we set $$\label{maximize-entropy}
{\varepsilon}{\beta}\phi _t \in \delta_\phi S_0(e,\phi) - \partial \chi_{\mathcal{G}}(\phi) ,$$ with a kinetic coefficient ${\beta}>0$, the variational derivative $\delta_\phi S_0(e,\phi)$, and the subdifferential $\partial \chi_{\mathcal{G}}(\phi)$ of the convex functional $\chi_{\mathcal{G}}$. For the phase field we impose homogeneous Neumann boundary conditions $$\label{eq:PHASEBND}
\frac{\partial}{\partial n}\phi_\alpha = 0, \quad \alpha =1, \dots, M.$$ Utilizing , , , and the transformation , a weak formulation of the differential equations and with boundary conditions and , respectively, reads as follows.
\
\[prob:PF\]Find the phase field $\phi \in L^2( (0, t^\ast), H^1(\Omega)^M) \cap H^1((0, t^\ast), L^2(\Omega)^M)$ and positive inverse temperature $\theta \in L^2((0, t^\ast), H^1(\Omega)) \cap H^1((0, t^\ast), L^2(\Omega))$ such that $$\label{eq:INIT}
\phi(\cdot,0) = \phi^0,\qquad \theta(\cdot,0) = \theta^0$$ holds with given initial conditions $\phi^0\in L^2(\Omega)^M$, $\theta^0\in L^2(\Omega)$, $\theta^0 \geq 0$ a.e. in $\Omega$ and
\[pvi\] $$\begin{aligned}
\displaystyle ({\varepsilon}{\beta}\phi_t + \widetilde{{L}} - \theta {L}+\tfrac1{\varepsilon}K\phi,v-\phi)
+ {\varepsilon}(\nabla \phi, \nabla (v-\phi)) + \chi_{\mathcal{G}}(v)-\chi_{\mathcal{G}}(\phi) &\geq 0 \label{pvi:1} \\
\left( -L^T\phi_t + c_v \tfrac{1}{\theta^2}\theta_t - q, w\right)
- (\kappa\nabla \theta , \nabla w) + \left({h_c}\left(\tfrac{1}{\theta} - \tfrac{1}{\theta_\Gamma}\right), w\right)_{\Gamma} &= 0
\label{pvi:2}\end{aligned}$$
holds for all $v\in H^1(\Omega)^M$ and $w \in H^1(\Omega)$.
Here, $(0,t^\ast)\in {\mathbb{R}}$ is the considered time interval, ${L}=({L}_\alpha)_{\alpha=1}^M$ and $\widetilde{{L}}=(\frac{{L}_\alpha}{T_{\alpha}})_{\alpha=1}^M$ are constant vectors used to simplify the notation of and , and $(\cdot, \cdot)$, $(\cdot, \cdot)_{\Gamma}$ stand for the scalar product in $L^2(\Omega)$, $L^2(\Gamma)$, respectively.
For existence and uniqueness results in the special case $M=2$, we refer to [@BrokateSprekels96 Section 7.2], [@OKlein97], and the references cited therein.
\[prop:THERMOCONSISTENCY\] The multi-phase Penrose–Fife system is thermodynamically consistent in the sense that $$\label{eq:TCC}
S(e(t), \phi(t))\geq S(e(t_0), \phi(t_0)) \quad \forall t \in [t_0,t^*]\subset (0,t^*]$$ holds for any solution $(\phi, \theta)$ of Problem \[prob:PF\] with $q=0$, ${h_c}=0$ satisfying $\phi \in C^1([t_0, t^*], H^1(\Omega)^M)$ for $[t_0,t^*]\subset (0,t^*]$.
Since $\phi \in {\mathcal{G}}$ for almost all $t$ and in view of and we can write $$\begin{aligned}
\label{eq:S_hat}
S(e,\phi) = \hat S(\theta,\phi) = \int_\Omega -\widetilde{{L}}^T \phi + c_v \ln(\tfrac1\theta) - \tfrac{\varepsilon}2 \lvert \nabla\phi \rvert^2 - \tfrac1{2{\varepsilon}} \phi^T K \phi {\,\textnormal{d}}x.
\end{aligned}$$ Testing with $v = \phi(t-\tau)$, dividing by $\tau>0$, and letting $\tau \to 0$ we get $$\begin{aligned}
0 \leq {\varepsilon}\Vert \sqrt{{\beta}}\phi_t \Vert^2
\leq ( - \widetilde{{L}} + \theta {L}- \tfrac1{\varepsilon}K\phi, \phi_t) - {\varepsilon}(\nabla\phi, \nabla\phi_t)
\end{aligned}$$ whereas testing with $w=\theta$ yields $$\begin{aligned}
0 \leq (\kappa \nabla\theta, \nabla \theta) =
(-{L}^T\phi_t + c_v \tfrac1{\theta^2}\theta_t, \theta).
\end{aligned}$$ Adding both we get $$\begin{aligned}
0
\leq ( - \widetilde{{L}}, \phi_t) - ( \tfrac{c_v}{\theta}, \theta_t) - {\varepsilon}(\nabla\phi, \nabla\phi_t) - \tfrac1{\varepsilon}(K\phi, \phi_t)
= \langle \nabla \hat S(\theta, \phi)), (\theta_t, \phi_t)\rangle.
\end{aligned}$$ Now integrating over $[t_0,t]$ provides the assertion.
Thin film approximation
-----------------------
We consider a domain of the form $\Omega = \Omega' \times (0,H) \subset {\mathbb{R}}^d$, $d = 2,3$, with a bounded Lipschitz domain $\Omega'\subset {\mathbb{R}}^{d-1}$ and $H>0$. We assume that $\Omega$ is “thin” in the sense that variations of $\phi$, $\phi^0$, $\theta$, $\theta^0$, and $q$ normal to $\Omega'$ as well as the flux $J_0$ across $\partial \Omega'\times (0,H)$ can be neglected and $J_0\cdot n(\cdot,0) = J_0\cdot n (\cdot,H)$ holds a.e. in $\Omega'$. Inserting these assumptions into , , we obtain the following thin film approximation of Problem \[prob:PF\].
\
\[prob:PFTHIN\]Find the phase field $\phi \in L^2( (0, t^\ast), H^1(\Omega')^M) \cap H^1((0, t^\ast), L^2(\Omega')^M)$ and positive inverse temperature $\theta \in L^2((0, t^\ast), H^1(\Omega')) \cap H^1((0, t^\ast), L^2(\Omega'))$ such that $$\phi(\cdot,0) = \phi^0 ,\qquad \theta(\cdot,0) = \theta^0$$ holds with given initial conditions $\phi^0 \in L^2(\Omega)^M$, $\theta^0 \in L^2(\Omega)$, $\theta^0 \geq 0$ a.e. in $\Omega$ and
\[pvi2D\] $$\begin{aligned}
\displaystyle ({\varepsilon}{\beta}\phi_t + \widetilde{{L}} - \theta {L}+\tfrac1{\varepsilon}K\phi,v-\phi)
+ {\varepsilon}(\nabla \phi, \nabla (v-\phi)) + \chi_{\mathcal{G}}(v)-\chi_{\mathcal{G}}(\phi) &\geq 0 \label{pv2Di:1} \\
\left( -L^T\phi_t + c_v \tfrac{1}{\theta^2}\theta_t +{h_c}'\tfrac1\theta - {h_c}'\tfrac1{\theta_\Gamma} -q, w\right)
- (\kappa\nabla \theta , \nabla w) &= 0
\label{pvi2D:2}\end{aligned}$$
and ${h_c}'=\tfrac{2{h_c}}{H}$ for all $v\in H^1(\Omega')^M$ and $w \in H^1(\Omega')$.
Here, $(\cdot, \cdot)=(\cdot, \cdot)_{\Omega'}$ stands for the scalar product in $L^2(\Omega')$ for ease of notation. Note that Problem \[prob:PFTHIN\] is essentially a $(d-1)$-dimensional analogue of Problem \[prob:PF\] with similar mathematical properties. For example, the thermodynamic consistency in the sense of Proposition \[prop:THERMOCONSISTENCY\] is still valid and all considerations concerning the discretization and algebraic solution of discretized problems to be reported below carry over from Problem \[prob:PF\] to its thin film approximation. For this reason, in the remainder we consider the more general equation $$\begin{aligned}
( -L^T\phi_t + c_v \tfrac{1}{\theta^2}\theta_t +h_{\Omega}\tfrac1\theta - q', w)
- (\kappa\nabla \theta , \nabla w) + (h_\Gamma(\tfrac1\theta - \tfrac1{\theta_\Gamma}), w)_\Gamma = 0
\label{eq:therm-general}\end{aligned}$$ with the coefficients $h_\Omega, h_\Gamma \geq 0$, of which exactly one is zero to recover Problems \[prob:PF\] resp. \[prob:PFTHIN\]. Specifically, choose $h_\Omega =0$, $h_\Gamma = {h_c}$, and $q'=q$ to obtain and $h_\Omega ={h_c}'$, $h_\Gamma = 0$, and $q'= q + {h_c}'\tfrac1{\theta_\Gamma}$ to obtain .
Discretization {#section:discretization}
==============
In this section we present a discretization of Problem \[prob:PF\] and Problem \[prob:PFTHIN\] using the general equation by Euler-type discretizations in time and finite elements in space. Since an efficient approximation of the phase field $\phi$ requires time-dependent, locally refined spatial grids, it is convenient to use Rothe’s method [@bornemann:adapt_multilevel_parabolic:1991], i.e., the variational problem is first discretized in time and the resulting spatial problems are then discretized in space, independently from each other.
Implicit time discretization
----------------------------
In light of the well-known stiffness of the non-linear parabolic system of equations, we use a semi-implicit Euler method. More precisely, after approximating the time derivatives $\phi_t, \theta_t$ by backward finite differences with step size $\tau>0$, the nonlinearities $1/\theta$, $1/\theta^2$ are approximated by first-order Taylor expansion (cf., e.g., [@BornemannDeuflhard2008 Section 6.4]) $$\tfrac1{\theta(t)}
\doteq \tfrac{2}{\theta(t-\tau)} - \tfrac{\theta(t)}{(\theta(t-\tau))^2},\qquad
\tfrac1{\theta(t)^2}
\doteq \tfrac{3}{\theta(t-\tau)^2} - \tfrac{2\theta(t)}{\theta(t-\tau)^3}.$$ In particular, this leads to $$\tfrac{1}{\theta(t)^2} \theta_t(t)
\doteq \tfrac{1}{\theta(t)^2 }\tfrac{\theta(t)-\theta(t-\tau)}{\tau}
=\tfrac{1}{\tau}\Bigl(\tfrac{1}{\theta(t)} - \tfrac{\theta(t-\tau)}{\theta(t)^2} \Bigr)
\doteq \tfrac{1}{\tau}
\Bigl(\tfrac{\theta(t)}{\theta(t-\tau)^2} - \tfrac{1}{\theta(t-\tau)} \Bigr).$$ Only the concave term $\frac{1}{{\varepsilon}}K\phi$ is taken explicitly [@deckelnick2005computation; @graser2013time], trading unconditional stability for a potential loss of accuracy, c.f. [@blank2013primal] or [@bartels2015numerical section 6.3.1].
For simplicity, we utilize the uniform time step size $\tau= t^*/n^*$ with given $n^*\in {\mathbb{N}}$, and denote the approximations of $\phi(t_n)$, $\theta(t_n)$ at $t_n=n\tau$, $n=1,\dots,n^*$ by $\phi^n$, $\theta^n$, respectively. The spatial problem to be solved in the $n$-th time step then reads as follows.
\
\[prob:PFSPAT\]Find the phase field $\phi^n \in H^1(\Omega)^M$ and positive inverse temperature $\theta^n \in H^1(\Omega)$ such that
\[pviTD\] $$\begin{aligned}
a(\phi^n,v-\phi^n) + \chi_{\mathcal{G}}(v) - \chi_{\mathcal{G}}(\phi^n)
+ b(v-\phi^n,\theta^n) &\geq \ell_1^n(v-\phi^n) \label{pviTD1}\\
b(\phi^n,w) - c^n(\theta^n,w) &= \ell_2^n(w) \label{pviTD2}
\end{aligned}$$
holds with the bilinear forms
\[eq:BIFO\] $$\begin{aligned}
a(v,v') &= {\varepsilon}({\beta}v,v') + {\varepsilon}\tau\left(\nabla v,\nabla v'\right), \\
b(v,w) &= -\tau(L^T v,w),\\
c^n(w,w') &= \tau(\tfrac{c_v + \tau h_\Omega}{(\theta^{n-1})^2} w,w' )
+ \tau^{2} (\tfrac{h_\Gamma}{(\theta^{n-1})^2} w,w' )_\Gamma
+ \tau^{2} (\kappa\nabla w, \nabla w' )
\intertext{and the linear functionals}
\ell_1^n(v)&= ({\varepsilon}{\beta}\phi^{n-1} - \tau \widetilde{{L}} - \tfrac{\tau}{{\varepsilon}} K\phi^{n-1},v), \label{eq:ell1}\\
\ell_2^n(w)&= (\tau^{2} q' - \tau L^T \phi^{n-1} - \tau\tfrac{c_v + 2\tau h_\Omega}{\theta^{n-1}},w) - \tau^{2} h_\Gamma ( \tfrac{2}{\theta^{n-1}} - \tfrac{1}{\theta_\Gamma} ,w )_\Gamma
\end{aligned}$$
defined for all $v, v'\in H^1(\Omega)^M$ and $w,w' \in H^1(\Omega)$.
\[prop:THERMOCONSISTENCY-disc\] The time-discrete multi-phase Penrose–Fife system is thermodynamically consistent in the sense that $$\label{eq:TCC-disc}
S(e^n, \phi^n)\geq S(e^{n-1}, \phi^{n-1})$$ holds for any solution $(\phi^n, \theta^n)$ of Problem \[prob:PFSPAT\] with $q=0$, ${h_c}=0$. Here, $e^n = \tilde{e}(\tfrac{1}{\theta^n},\phi^n)$ is defined according to .
Testing equation (\[pviTD1\]) with $v=\phi^{n-1}$ yields $$\begin{aligned}
\tfrac{\varepsilon}2\vert\nabla\phi^n\vert^2 - \tfrac{\varepsilon}2\vert\nabla\phi^{n-1}\vert^2 &\leq ( \widetilde{L}- {L}\theta^n + \tfrac1{\varepsilon}K\phi^{n-1}, \phi^{n-1}-\phi^n)
\end{aligned}$$ and testing equation (\[pviTD2\]) with $w=\theta^n$ for $q=0, {h_c}=0$ yields $$\begin{aligned}
0 \leq \tau \kappa\vert \nabla \theta^n \vert^2 &= ({L}\theta^n, \phi^{n-1}-\phi^n) + c_v\int_\Omega \tfrac{\theta^n}{\theta^{n-1}}(1-\tfrac{\theta^n}{\theta^{n-1}}){\,\textnormal{d}}x.\end{aligned}$$ Adding both inequalities, we obtain $$\begin{gathered}
- ( \widetilde{L}, \phi^{n-1}-\phi^n) - \tfrac{\varepsilon}2(\vert\nabla\phi^{n-1}\vert^2 - \vert\nabla\phi^n\vert^2) \\
\leq \,\, c_v\int_\Omega \tfrac{\theta^n}{\theta^{n-1}}(1-\tfrac{\theta^n}{\theta^{n-1}}){\,\textnormal{d}}x + \tfrac1{\varepsilon}(K\phi^{n-1},\phi^{n-1} - \phi^n)\end{gathered}$$ which, using the representation , finally can be used to show $$\begin{aligned}
S(e^{n-1},\phi^{n-1}) - S(e^n,\phi^n)
\leq
\int_\Omega c_v (\ln(\tfrac{\theta^{n}}{\theta^{n-1}})+\tfrac{\theta^n}{\theta^{n-1}}(1-\tfrac{\theta^n}{\theta^{n-1}})){\,\textnormal{d}}x.\end{aligned}$$ Since the right hand side is non-positive, this provides the assertion.
It turns out that the system can be regarded as optimality conditions for a Lagrange-functional.
\[prop:timediscrete\_saddle\] Problem \[prob:PFSPAT\] is equivalent to find $\phi^n \in H^1(\Omega)^M$ and $\theta^n \in H^1(\Omega)$ such that $${\mathcal{L}}^n(\phi^n,w)
\leq {\mathcal{L}}^n(\phi^n, \theta^n)
\leq {\mathcal{L}}^n(v, \theta^n)
\qquad \forall \; v \in H^1(\Omega)^M,\; w \in H^1(\Omega),$$ with the Lagrangian ${\mathcal{L}}^n$ given by $$\begin{aligned}
\label{eq:lagrange_functional}
{\mathcal{L}}^n(v,w) = {\mathcal{J}}^n(v) - \ell_1^n(v) + b(v,w) -\ell_2^n(w)
- \tfrac{1}{2}c^n(w,w)
\end{aligned}$$ denoting ${\mathcal{J}}^n(v)=\tfrac{1}{2}a(v,v) + \chi_{\mathcal{G}}(v)$.
\[thm:timediscrete\_existence\] Let $\phi^{n-1} \in {\mathcal{G}}$, $\theta^{n-1}\in H^1(\Omega)$ and $\theta^{n-1} \geq c$ a.e. in $\Omega$ with a positive constant $c$. Then the spatial Problem \[prob:PFSPAT\] admits a unique solution.
Under the given assumptions, $\ell_1^n$ and $\ell_2^n$ are bounded linear functionals and $a(\cdot,\cdot)$ is symmetric and coercive. Since ${\mathcal{G}}$ is closed and convex the functional ${\mathcal{L}}^n(\cdot,w)$ is strictly convex, coercive, and lower semi-continuous for all fixed $w \in H^1(\Omega)$. Thus, we can define the dual functional $$\begin{aligned}
h(w) = - \inf_{v \in H^1(\Omega)^M} {\mathcal{L}}^n(v,w) = ({\mathcal{J}}^n)^*(\ell_1^n - b(\cdot,w)) + \tfrac{1}{2}c^n(w,w) + \ell_2^n(w).
\end{aligned}$$ Here, $({\mathcal{J}}^n)^*$ is the convex and continuous polar of ${\mathcal{J}}^n$. Notice that $h$ is also convex and continuous because $c^n(\cdot,\cdot)$ is symmetric and positive-definite.
By integrability of $(\theta^{n-1})^2$ there must be a subset $\Omega' \subset \Omega$ with positive measure and a constant $c_2>0$ such that $\theta^{n-1}\leq c_2$ on $\Omega'$. As a consequence we get $$\begin{aligned}
c^n(w,w) \geq \tfrac{c_v \tau}{c_2^2}\|w\|_{L^2(\Omega')}^2 + \tau^2 \kappa \|\nabla w\|^2
\end{aligned}$$ and therefore coercivity of $c^n(\cdot,\cdot)$. Hence, $h$ is coercive, continuous, and strictly convex and thus has a unique minimizer. Existence and uniqueness now follows from the fact that $(\phi^n,\theta^n)$ is a solution of Problem \[prob:PFSPAT\], if and only if $\theta^n$ is a minimizer of $h$ and $\phi^n = {{\operatorname{arg\,min\,}}_{v \in H^1(\Omega)^M}} {\mathcal{L}}^n(v, \theta^n)$.
In general it is not clear if the uniform positivity of the inverse temperature is preserved by solving Problem \[prob:PFSPAT\]. We refer, however, to [@OKlein97] for such kind of results in the scalar case $M=2$.
Adaptive finite element discretization {#subsec:fe-discretization}
--------------------------------------
We will now consider the adaptive finite element discretization of the spatial Problem \[prob:PFSPAT\] for an individual fixed time stepIn order to improve readability we will from now on drop all superscripts $(\cdot)^n$ that identify the current time step. We will designate quantities from the previous time step by the superscript $(\cdot)^{\textnormal{old}}$ whenever necessary.
### Finite element discretization
In the following, we assume that ${\mathcal{T}}$ is a simplicial grid that is either conforming or obtained via local hanging node refinement of a conforming initial grid. We will discretize the spatial Problem \[prob:PFSPAT\] with respect to the conforming first order finite element space $$\begin{aligned}
\label{eq:fe_space}
{\mathcal{S}}= {\mathcal{S}}({\mathcal{T}}) = \Bigl\{ v \in C(\overline{\Omega}) {\,\Big|\,}v|_{\tau}\text{ is affine } \forall {\tau}\in {\mathcal{T}}\Bigr\}
\subset H^1(\Omega).\end{aligned}$$ Notice that ${\mathcal{S}}$ has a uniquely defined nodal basis $\{\lambda_p {\,|\,}p \in {\mathcal{N}}\}$ satisfying $\lambda_p(q) = \delta_{pq}$ for all $p,q \in {\mathcal{N}}$ where ${\mathcal{N}}$ is the set of non-hanging nodes of ${\mathcal{T}}$. If the grid is obtained by uniform or local hanging node refinement, the non-conforming mesh hierarchy induces a natural hierarchy of subspaces of ${\mathcal{S}}$ that can be used in geometric multigrid methods. For a detailed discussion of finite element spaces on hierarchies of non-conforming, locally refined grids we refer to [@graeserthesis; @graeser:2014a].
Note that linearity implies that the Gibbs constraint can be evaluated node-wise, i.e., $${\mathcal{G}}\cap {\mathcal{S}}^M = \{v \in {\mathcal{S}}^M {\,|\,}v(p)\in G\;\forall p \in {\mathcal{N}}\}.$$
\
\[prob:PFSPATDISC\] Find the phase field $\phi_{\mathcal{T}}\in {\mathcal{S}}^M$ and positive inverse temperature $\theta_{\mathcal{T}}\in {\mathcal{S}}$ such that
$$\begin{aligned}
a(\phi_{\mathcal{T}},v-\phi_{\mathcal{T}}) + \chi_{\mathcal{G}}(v) - \chi_{\mathcal{G}}(\phi_{\mathcal{T}}) + b(v-\phi_{\mathcal{T}},\theta_{\mathcal{T}}) &\geq \ell_1(v-\phi_{\mathcal{T}}),
\label{pvi:PFSPATDISC1} \\
b(\phi_{\mathcal{T}},w) - c(\theta_{\mathcal{T}},w) &= \ell_2(w)
\label{pvi:PFSPATDISC2}
\end{aligned}$$
holds for all $v\in {\mathcal{S}}^M$ and $w \in {\mathcal{S}}$.
Here, $\ell_1,\ell_2$, and $c(\cdot,\cdot)$ are defined as in Problem \[prob:PFSPAT\] but with $\phi^{n-1}$ and $\theta^{n-1}$ replaced by their finite element approximations $\phi^{\textnormal{old}}$ and $\theta^{\textnormal{old}}$. To avoid negative values for $\theta_{\mathcal{T}}$ due to overshooting when $q'$ is not resolved by the grid, we replace the latter by its interpolation in ${\mathcal{S}}$. Notice that $\phi^{\textnormal{old}}$ and $\theta^{\textnormal{old}}$ are finite element functions on a grid ${\mathcal{T}^\textnormal{old}}$. In case of adaptive refinement, ${\mathcal{T}^\textnormal{old}}$ is usually different from ${\mathcal{T}}$. See [@graeser:2014a] for a detailed discussion.
In analogy to its continuous counterpart, Problem \[prob:PFSPATDISC\] can be written as a non-linear, non-smooth saddle point problem.
\[prop:SADDLE\] Problem \[prob:PFSPATDISC\] is equivalent to finding $\phi_{\mathcal{T}}\in {\mathcal{S}}^M$ and $\theta_{\mathcal{T}}\in {\mathcal{S}}$ such that $${\mathcal{L}}(\phi_{\mathcal{T}},w)
\leq {\mathcal{L}}(\phi_{\mathcal{T}}, \theta_{\mathcal{T}})
\leq {\mathcal{L}}(v, \theta_{\mathcal{T}})
\qquad \forall \; v \in {\mathcal{S}}^M,\; w \in {\mathcal{S}},$$ with the Lagrangian ${\mathcal{L}}$ given according to .
Existence and uniqueness are also inherited from the continuous case.
\[thm:discrete:existence\] Let $\phi^{\textnormal{old}}\in {\mathcal{G}}$, $\theta^{\textnormal{old}}\in H^1(\Omega)$ and $\theta^{\textnormal{old}}\geq c$ a.e. in $\Omega$ with a positive constant $c$. Then the spatial Problem \[prob:PFSPATDISC\] admits a unique solution.
The proof can be carried out using the same arguments as in the proof of Theorem \[thm:timediscrete\_existence\].
### Hierarchical a posteriori error estimation {#subsubsection:APOSTERIORI}
As the phase field $\phi$ is expected to strongly vary across the phase boundaries, spatial adaptivity based on a posteriori error estimates is mandatory. Similarly, the consumption of heat by phase changes may lead to strongly varying $\theta$. Hierarchical error estimates rely on the solution of local defect problems. While originally introduced for linear elliptic problems [@FABornemann_BErdmann_RKornhuber_1993b; @PDeuflhard_PLeinen_HYserentant_1989a; @HolstOvallSzypowski:2011; @OCZienkiewicz_JPSRGago_DWKelly_1983a] this technique was successfully extended to non-linear problems [@REBank_RKSmith_1993a], constrained minimization [@RHWHoppe_RKornhuber_1994a; @RKornhuber_1995b; @RKornhuberQZou_2008; @SiebertVeeser2007; @graeser_kornhuber_veeser_zou:hest_energy_obstacle:2011] and non-smooth saddle point problems [@GraeserKornhuberSack2010; @graeser:2014a; @graeserthesis].
Following [@graeserthesis; @GraeserKornhuberSack2010; @graeser:2014a], we now derive an a posteriori error estimate by a suitable approximation of the defect problem associated with the defect Lagrangian $$\begin{aligned}
{\mathcal{D}}(e_\phi,e_\theta) = {\mathcal{L}}(\phi_{\mathcal{T}}+ e_\phi, \theta_{\mathcal{T}}+ e_\theta).\end{aligned}$$ In the first step the defect problem is discretized with respect to a larger finite element space ${\mathcal{Q}}^M\times {\mathcal{Q}}$, where ${\mathcal{Q}}= {\mathcal{S}}({\mathcal{T}}')$ is defined analogously to for the grid ${\mathcal{T}}'$ obtained by uniform refinement of ${\mathcal{T}}$. Note that we have ${\mathcal{Q}}={\mathcal{S}}\oplus {\mathcal{V}}$ with ${\mathcal{V}}$ denoting the incremental space $$\begin{aligned}
{\mathcal{V}}= {\operatorname{span}}\{\lambda_p' {\,|\,}p \in {\mathcal{E}}\}
\subset {\operatorname{span}}\{\lambda_p' {\,|\,}p \in {\mathcal{N}}' \} = {\mathcal{S}}'.\end{aligned}$$ Here, ${\mathcal{N}}'$ denotes the set of non-hanging nodes in ${\mathcal{T}}'$, $\{\lambda_p' {\,|\,}p \in {\mathcal{N}}'\}$ the nodal basis of ${\mathcal{S}}'$, and ${\mathcal{E}}={\mathcal{N}}' \setminus {\mathcal{N}}$ is the set of all edge mid points in ${\mathcal{T}}$ that are non-hanging in ${\mathcal{T}}'$.
In the second step, the discrete defect problem is localized by ignoring the coupling between ${\mathcal{S}}$ and ${\mathcal{V}}$ and also the coupling between $\lambda_p'$ for all $p \in {\mathcal{E}}$. Denoting ${\mathcal{D}}_p(r,s) = {\mathcal{D}}(r\lambda_p',s\lambda_p')$, this results in the local saddle point problems $$\begin{aligned}
(e_{\phi,p},e_{\theta,p}) \in {\mathbb{R}}^{M}\times {\mathbb{R}}: \;\;
{\mathcal{D}}_p(e_{\phi,p},s)
\leq
{\mathcal{D}}_p(e_{\phi,p},e_{\theta,p})
\leq
{\mathcal{D}}_p(r,e_{\theta,p})
\;\;\forall (r,s) \in {\mathbb{R}}^M\times {\mathbb{R}}\end{aligned}$$ for all $p \in {\mathcal{E}}$ that give rise to the hierarchical a posteriori error estimate $$\eta=\Bigl(\sum_{p\in{\mathcal{E}}} \eta_p^2\Bigr)^{\frac12}, \qquad
\eta_p^2=
{{\left\lVerte_{\phi,p}\lambda_p'\right\rVert_{\mathrm{\phi}}}}^2 +
{{\left\lVerte_{\theta,p}\lambda_p'\right\rVert_{{\mathrm{\theta}}}}}^2, \qquad
p \in {\mathcal{E}}\label{eq:APOST}$$ with the problem-dependent norms $$\begin{aligned}
{{\left\lVertv\right\rVert_{\mathrm{\phi}}}}^2 &=
a(v,v), &
{{\left\lVertw\right\rVert_{{\mathrm{\theta}}}}}^2 &=
c(w,w)\end{aligned}$$ on ${\mathcal{V}}^M$, ${\mathcal{V}}$, respectively.
### Adaptive mesh refinement {#subsubsec:adaptivity}
The initial grid for the adaptive refinement should be sufficiently fine to detect basic features of the unknown spatial approximation and sufficiently coarse for efficiency of the overall adaptive procedure. The construction of such a grid starts with the grid ${\mathcal{T}^\textnormal{old}}$ from the preceding time step. In the first time step, we select a suitable, uniformly refined grid ${\mathcal{T}^\textnormal{old}}$.
We begin by coarsening ${\mathcal{T}^\textnormal{old}}$. To this end, we keep all simplices from the grid ${\mathcal{T}^\textnormal{old}}$ from the preceding time step that were obtained by at most $j_{{\operatorname{min}}}$ refinements. In addition, we keep all simplices ${\tau}$ such that $\phi^{{\textnormal{old}}}$ exhibits a strong local variation on ${\tau}$ that is not visible after coarsening, i.e., such that $$\begin{aligned}
\| |\nabla (I_{{\tau}} \phi^{{\textnormal{old}}}) | \|_{L^\infty({\tau})}\geq{{\operatorname{Tol}}_{\mathrm{derefine}}}\qquad \textnormal{ and } \qquad
\| |\nabla (I_{{\tau}'} \phi^{{\textnormal{old}}} ) | \|_{L^\infty({\tau}')}<{{\operatorname{Tol}}_{\mathrm{derefine}}}\end{aligned}$$ holds with ${\tau}'$ denoting the simplex resulting from coarsening of ${\tau}$. $I_{{\tau}}$ and $I_{{\tau}'}$ are the linear interpolation operators to ${\tau}$ and ${\tau}'$, respectively. This set of simplices is completed by additional local refinements. Possible additional refinement is used to uniformly bound the ratio of diameters of adjacent simplices.
The adaptive mesh refinement of the resulting initial grid ${\mathcal{T}}$ is based on the local error indicators $\eta_p$ defined in . In each step, the indicators $\eta_{p_i}$, $i=1,\dots,|{\mathcal{E}}|$, are arranged with decreasing value, to determine the minimal number $i_0$ of indicators such that $$\label{eq:REFTHRES}
\sum_{i=1}^{i_0} \eta_{p_i}^2 \geqslant \rho\eta^2$$ holds with a given parameter $\rho\in [0,1]$. Then all simplices ${\tau}\in {\mathcal{T}}$ with the property $p_i\in {\tau}$ for some $p_i$ with $i\leq i_0$ are marked for refinement [@Doerfler1996]. Each marked simplex is partitioned by (red) refinement [@Bey2000; @bornemann:1993]. Again, possible additional refinement is used to uniformly bound the ratio of diameters of adjacent simplices. The refinement process is stopped, when the estimated relative error is less than a given tolerance ${{\operatorname{Tol}}_{\mathrm{adapt}}}>0$, i.e., if $$\label{eq:STOPCRIT}
\eta < {{\operatorname{Tol}}_{\mathrm{adapt}}}\cdot
\biggl( {{\left\lVert\phi_{\mathcal{T}}\right\rVert_{\mathrm{\phi}}}}^2 + {{\left\lVert\theta_{\mathcal{T}}\right\rVert_{{\mathrm{\theta}}}}}^2\biggr)^{\frac 1 2}.$$
Algebraic solution {#section:solver}
==================
Several methods have been proposed for the algebraic solution of discretized multi-phase field equations. The solution of multi-phase Allen–Cahn-type equations via primal–dual active set methods was discussed in [@BlankGarkeSarbuStyles2013] and multigrid methods for such problems where proposed in [@GraeserSander2014_preprint; @kornhuber_krause:mg_vector_ac:2006]. In contrast to the second order problems with minimization structure considered there, Problem \[prob:PFSPATDISC\] is a saddle point problem for a discretized fourth order equation. For similar problems resulting from multi-component Cahn–Hilliard systems block Gauß–Seidel-type algorithms with component-wise and vertex-wise blocking where proposed in [@BloweyCopettiElliott1996] and [@Nuernberg2009], respectively. To overcome the mesh-dependence of the Gauß–Seidel approach, a nonsmooth Schur–Newton method was proposed in [@graeser2014].
While, for Cahn–Hilliard-type systems, the local sum constraint in $G$ can be enforced using the chemical potential as a natural Lagrange multiplier (cf. [@graeser2014]), Problem \[prob:PFSPATDISC\] is structurally different and the introduction of such a multiplier would change the structure of the problem. Therefore, we now introduce a nonsmooth Schur–Newton method that does not involve such a multiplier.
Matrix notation
---------------
For the presentation of the algebraic solver for the iterative solution of Problem \[prob:PFSPATDISC\] we first formulate this problem in terms of coefficient vectors and matrices. To this end let $N={\operatorname{dim}}{\mathcal{S}}$ and introduce an enumeration of the nodes ${\mathcal{N}}=\{p_1,\dots, p_N\}$. To simplify the presentation, we use the abbreviated notation $\lambda_k = \lambda_{p_k}$ for the nodal basis $\lambda_1,\dots,\lambda_N$ of ${\mathcal{S}}$ and introduce the basis $\lambda^1,\dots,\lambda^{MN}$ of ${\mathcal{S}}^M$ where $\lambda^{\pi(i,k)} = b^i \lambda_{k}$, $b^i \in {\mathbb{R}}^M$ is the $i$-th Euclidean basis vector, and $\pi : \lbrace 1,\ldots, N\rbrace \times \{ 1,\dots, M\}$ is the bijective index map given by $$\begin{aligned}
\pi(k,i) = i+M(k-1).\end{aligned}$$
For $v\in {\mathcal{S}}^M, w\in {\mathcal{S}}$ we then get the associated coefficient vectors $V \in {\mathbb{R}}^{MN}, W \in {\mathbb{R}}^N$ $$\begin{aligned}
v &= \textstyle\sum_{i=1}^{MN} V_i \lambda^i,
&
w &= \textstyle\sum_{i=1}^N W_i \lambda_i.\end{aligned}$$ Using the matrices $A\in{\mathbb{R}}^{MN,MN}$, $B\in{\mathbb{R}}^{N,MN}$, $C\in {\mathbb{R}}^{N,N}$ and vectors $F\in {\mathbb{R}}^{MN}, G\in {\mathbb{R}}^{N}$ given by $$\begin{aligned}
A_{ij} &= a(\lambda^j, \lambda^i), &
B_{ij} &= b(\lambda^j, \lambda_i), &
C_{ij} &= c(\lambda_j, \lambda_i), &
F_{i} &= \ell_1(\lambda^i), &
G_{i} &= \ell_2(\lambda_i),\end{aligned}$$ and the characteristic functional $\chi_{G_N}: {\mathbb{R}}^{MN} \to {\mathbb{R}\cup \{\infty\}}$ of $$\begin{aligned}
G_N = \{V \in {\mathbb{R}}^{MN} {\,|\,}(V_{\pi(k,i)})_{i=1,\dots,M} \in G \quad \forall k \}
= \{V \in {\mathbb{R}}^{MN} {\,|\,}\textstyle\sum_{i=1}^{MN} V_i \lambda^i \in {\mathcal{G}}\}\end{aligned}$$ Problem \[prob:PFSPATDISC\] can be written as:
\
\[prob:algebraic\_vi\] Find the coefficient vectors $\Phi \in {\mathbb{R}}^{MN}$ and $\Theta \in {\mathbb{R}}^N$ of $\phi_{\mathcal{T}}$ and $\theta_{\mathcal{T}}$, respectively, such that
$$\begin{aligned}
{\langle A \Phi, V-\Phi \rangle} + \chi_{G_N}(V) - \chi_{G_N}(\Phi) + {\langle B^T \Theta, V-\Phi \rangle} &\geq {\langle F, V-\Phi \rangle},
\label{eq:algebraic_vi_1} \\
B \Phi - C\Theta &= G
\label{eq:algebraic_vi_2}
\end{aligned}$$
holds for all $V \in {\mathbb{R}}^{MN}$.
Problem \[prob:algebraic\_vi\] can equivalently be written in operator notation as a non-linear saddle point problem using in turn the subdifferential of $\chi_{G_N}$.
\
\[prob:discrete\_spp\] Find the coefficient vectors $\Phi \in {\mathbb{R}}^{MN}$ and $\Theta \in {\mathbb{R}}^N$ of $\phi_{\mathcal{T}}$ and $\theta_{\mathcal{T}}$, respectively, such that $$\begin{aligned}
\begin{pmatrix}
A+\partial \chi_{G_N} & B^T\\
B & -C
\end{pmatrix}
\begin{pmatrix}
\Phi\\
\Theta
\end{pmatrix}
\ni
\begin{pmatrix}
F\\
G
\end{pmatrix}.
\end{aligned}$$
For later reference we note that the Lagrangian for this saddle point problem is given by the following discrete analogue $$\begin{aligned}
L(V,W) = J(V) - {\langle F, V \rangle} + {\langle BV - G, W \rangle} - \tfrac{1}{2} {\langle CW, W \rangle}\end{aligned}$$ of ${\mathcal{L}}^n$ where $J(V)=\tfrac{1}{2}{\langle AV, V \rangle} + \chi_{G_N}(V)$ is the analogue of ${\mathcal{J}}^n$.
Non-smooth Schur–Newton multigrid methods
-----------------------------------------
In the context of non-smooth Schur–Newton methods as introduced in [@graeserthesis; @graeser2014], it is shown that problems of the form of Problem \[prob:discrete\_spp\] can equivalently be formulated as the following dual minimization problem.
\
\[prob:discrete\_dual\] Find $\Theta \in {\mathbb{R}}^N$ such that $$\begin{aligned}
h(\Theta) \leq h(W) \qquad \forall W\in {\mathbb{R}}^N
\end{aligned}$$ where $h:{\mathbb{R}}^N \to {\mathbb{R}}$ is the dual functional $$\begin{aligned}
h(W) &= - \inf_{V \in {\mathbb{R}}^{MN}} L(V,W) = -L(\Phi(W),W)
\end{aligned}$$ and $\Phi(W) = (A+\partial \chi_{G_N})^{-1}(F-B^T W)$.
\[prop:discrete\_dual\_problem\] Problems \[prob:discrete\_spp\] and \[prob:discrete\_dual\] are equivalent in the sense that $(\Phi,\Theta)$ solves Problem \[prob:discrete\_spp\] if and only if $\Theta$ solves Problem \[prob:discrete\_dual\] and $\Phi = \Phi(\Theta)$. Furthermore, the dual functional $h:{\mathbb{R}}^N \to {\mathbb{R}}$ is convex and continuously differentiable with the Lipschitz-continuous derivative $$\begin{aligned}
\nabla h(W)
&= -B\Phi(W) + CW + G \\
&= -B(A+\partial \chi_{G_N})^{-1}(F- B^TW) + CW + G.
\end{aligned}$$
The proof of Proposition \[prop:discrete\_dual\_problem\] can be done analogously to the proof of [@GraeserKornhuber2009a Theorem 2.1]. This proof also shows that $h$ can be written as $$\begin{aligned}
h(W) = J^*(F-B^T W) + \tfrac{1}{2}{\langle CW, W \rangle} + {\langle G, W \rangle}\end{aligned}$$ where $J^* : {\mathbb{R}}^{MN} \to {\mathbb{R}}$ is the polar (or conjugate) functional of $J$ which is convex itself. This especially shows that $h$ is a strongly convex functional because $C$ is positive definite due to coercivity of the associated bilinear form $c(\cdot,\cdot)$ (cf. proof of Theorem \[thm:timediscrete\_existence\]).
As a consequence of Proposition \[prop:discrete\_dual\_problem\] we can apply gradient-related descent methods of the form $$\begin{aligned}
\label{eq:descent_method}
\Theta^{\nu+1} = \Theta^\nu + \rho_\nu D^\nu\end{aligned}$$ where $D^\nu \in {\mathbb{R}}^N$ is a decent direction and $\rho_\nu$ a step size. The non-smooth Schur–Newton method as introduced in [@GraeserKornhuber2009a; @graeserthesis; @graeser2014] is such a descent method where $D^\nu$ is taken to be $$\begin{aligned}
\label{eq:schur_newton_direction}
D^\nu = - S_\nu^{-1} \nabla h(\Theta^\nu)\end{aligned}$$ and $S_\nu \in {\mathbb{R}}^{N,N}$ is a generalized linearization of the non-smooth, non-linear but Lipschitz continuous Schur complement operator $-\nabla h$ at $\Theta^\nu$.
The derivation of $S_\nu$ essentially amounts to deriving a generalized linearization of the operator $(A+\partial \chi_{G_N})^{-1}$ at $Y=F- B^T \Theta^\nu$. For simple component-wise bound constraints is has been shown in [@GraeserKornhuber2009a] that such a linearization is given by $(A_{{\mathcal{W}}(X)})^+$. Here, $A_{{\mathcal{W}}(X)}$ is the restriction of $A$ to ${\mathcal{W}}(X) \times {\mathcal{W}}(X)$, ${\mathcal{W}}(X)$ is the maximal subspace such that $J$ is locally smooth in $$\begin{aligned}
(X + {\mathcal{W}}(X)) \cap U_X\end{aligned}$$ for some neighborhood $U_X$ of $X = (A+\partial\chi_{G_N})^{-1}(Y)$, and $(\cdot)^+$ is the Moore–Penrose pseudoinverse or, equivalently, the inverse of $A_{{\mathcal{W}}(X)}: {\mathcal{W}}(X) \to {\mathcal{W}}(X)$.
For the simplex constraints in the present problem we will use exactly the same approach. In the following we will outline the construction of $A_{{\mathcal{W}}(X)}$ for local simplex constraints following [@GraeserSander2014_preprint]. To this end we identify vectors $V \in {\mathbb{R}}^{MN}$ with block-structured vectors $\hat{V} \in ({\mathbb{R}}^M)^N$ such that $V_{\pi(k,i)} = (\hat{V}_k)_i$.
Due to the product structure $$\begin{aligned}
G_N = \{V \in {\mathbb{R}}^{MN} {\,|\,}\hat{V} \in G^N\},\end{aligned}$$ of the feasible set $G_N$ we can determine the subspace ${\mathcal{W}}(X)$ in each block individually. Hence ${\mathcal{W}}(X)$ takes the form $$\begin{aligned}
{\mathcal{W}}(X) = \bigl\{ V \in {\mathbb{R}}^{MN} {\,|\,}\hat{V} \in \prod_{k=1}^N {\mathbb{W}}(\hat{X}_k)\bigr\}\end{aligned}$$ where ${\mathbb{W}}(\hat{X}_k)$ is the maximal subspace where $\chi_G$ is locally smooth near $\hat{X}_k$. As outlined in [@GraeserSander2014_preprint] the local subspace ${\mathbb{W}}(\xi)$ is given by $$\begin{aligned}
{\mathbb{W}}(\xi) = {\operatorname{span}} \{ b^i - b^j \in {\mathbb{R}}^M {\,|\,}1 \leq i < j \leq M, \,\xi_i >0, \,\xi_j>0 \}\end{aligned}$$ for $\xi \in {\mathbb{R}}^M$. Since ${\mathcal{W}}(X)$ is a product space the orthogonal projection ${\mathcal{P}}_{{\mathcal{W}}(X)} : {\mathbb{R}}^{MN} \to {\mathcal{W}}(X)$ is given by a block diagonal matrix where the $k$-th diagonal block is the orthogonal projection ${\mathcal{P}}_{{\mathbb{W}}(\hat{X}_k)} : {\mathbb{R}}^M \to {\mathbb{W}}(\hat{X}_k)$. For an explicit representation of ${\mathcal{P}}_{{\mathbb{W}}(\hat{X}_k)} \in {\mathbb{R}}^{M,M}$ we refer to [@GraeserSander2014_preprint]. Using ${\mathcal{P}}_{{\mathcal{W}}(X)}$ we now get $$\begin{aligned}
A_{{\mathcal{W}}(X)} = {\mathcal{P}}_{{\mathcal{W}}(X)} A {\mathcal{P}}_{{\mathcal{W}}(X)}.\end{aligned}$$ Although the chain rule does in general not hold true for generalized Jacobians in the sense of Clarke (see, e.g. [@graeserthesis]), we define a generalized linearization of the non-linear Schur complement operator $-\nabla h$ at $\Theta^\nu$ in an analogous manner by $$\begin{aligned}
\label{eq:linearized_schur_complement}
S_\nu = B \Bigl(A_{{\mathcal{W}}(\Phi(\Theta^\nu))}\Bigr)^+ B^T + C.\end{aligned}$$
As a consequence of the convexity of $h$ we can show global convergence.
\[thm:schur\_newton\_convergence\] Assume that the step sizes $\rho_\nu$ are efficient (cf. [@ortega_rheinboldt:iterative_solution:1970; @GraeserKornhuber2009a]), then the iterates produced by the descent method with Schur–Newton directions for $S_\nu$ given by converge to the solution $\Theta$ of Problem \[prob:discrete\_dual\] for any initial guess $\Theta^0 \in {\mathbb{R}}^N$.
Notice that the $S_\nu$ are uniformly bounded from above and below. Hence global convergence follows from [@GraeserKornhuber2009a Theorem 4.2].
Efficient step sizes $\rho_{\nu}$ as required in Theorem \[thm:schur\_newton\_convergence\] can be obtained by classical step size rules like, e.g., the Armijo rule or bisection. Notice that it is not necessary to evaluate $S_\nu^{-1}$ exactly in because global convergence is preserved as long as the approximation of $S_\nu^{-1}$ is sufficiently accurate. Since the dual functional $h$ is strongly convex, one can also show global linear convergence with a rate depending on the bounds for $S_\nu$ and the step size rule. For further details we refer to [@GraeserKornhuber2009a].
During each iteration of the algorithm two types of subproblems have to be solved. The evaluation of $-\nabla h(\Theta^\nu)$ requires to compute $\Phi(\Theta^\nu) = (A+\partial \chi_{G_N})^{-1}(F-B^T \Theta^\nu)$. This is equivalent to minimizing $J(\cdot) + {\langle B^T \Theta^\nu, \cdot \rangle}$, i.e., a convex minimization problem for a quadratic functional with local simplex constraints. If the step size rule requires several trial steps further problems of this type have to be solved for each evaluation of $h$ or $\nabla h$. These convex minimization problems can efficiently be solved using non-linear multigrid methods [@kornhuber_krause:mg_vector_ac:2006; @GraeserSander2014_preprint]. More precisely the TNNMG method for simplex-constrained problems as proposed in [@GraeserSander2014_preprint] allows to solve these problems with an effective complexity of $O(M^2 N)$. This method was used in all numerical examples presented below.
The second type of subproblems are the linear problems for the symmetric positive definite operators $S_\nu$. Since each $S_\nu$ is a linear Schur complement this is equivalent to solving the linear saddle point problem $$\begin{aligned}
\begin{pmatrix}
A_{{\mathcal{W}}(\Phi(\Theta^\nu))} & (B{\mathcal{P}}_{{\mathcal{W}}(\Phi(\Theta^\nu))})^T \\
B {\mathcal{P}}_{{\mathcal{W}}(\Phi(\Theta^\nu))} & -C
\end{pmatrix}
\begin{pmatrix}
\tilde{V}^\nu \\
D^\nu
\end{pmatrix}
=
\begin{pmatrix}
0 \\
\nabla h ( \Theta^\nu)
\end{pmatrix}\end{aligned}$$ whose solution is unique in $({\operatorname{ker}} {\mathcal{P}}_{{\mathcal{W}}(\Phi(\Theta^\nu))})^\perp \times {\mathbb{R}}^N$. In the numerical examples presented below we used a linear multigrid method with a Vanka-type smoother to solve these problems. Notice that there is no convergence proof for this linear iterative method. To increase its robustness it can be used as preconditioner for a GMRes iteration.
Numerical experiments
=====================
All our computations are based on a non-dimensionalized version of the Penrose–Fife systems stated in Problem \[prob:PF\] and Problem \[prob:PFTHIN\], respectively, as obtained by setting $$\theta= \frac{T_{\rm ref}}{T}$$ instead of . While the order parameter $\phi_1$ is representing the liquid fraction, the order parameters $\phi_2,\dots, \phi_M$ are associated with certain solid states, as, e.g., crystalline structures, of the given material. With this in mind, the positive reference value $T_{\rm ref}\in {\mathbb{R}}$ is chosen to be the melting temperature and we set $T_{\rm ref} = T_1 = \cdots = T_M = 1$. Accordingly we set $L_1=0$ and $L_2=\cdots = L_M > 0$ in all our computational examples.
Efficient step sizes $\rho_\nu$ for the Schur–Newton iteration as required in Theorem \[thm:schur\_newton\_convergence\] are determined by bisection. The iteration is stopped, once the criterion $$\begin{aligned}
\label{eq:nsn_termination}
\tfrac{\Vert \theta^{\nu+1} - \theta^{\nu} \Vert_\theta}{\Vert \theta^{\nu}\Vert_\theta} \leq {{\operatorname{Tol}}_{\mathrm{correction}}}\end{aligned}$$ is satisfied. We use ${{\operatorname{Tol}}_{\mathrm{correction}}}= 10^{-11}$ in all our computations.
For each time step a grid hierarchy is obtained either by uniform refinement or according to the adaptive coarsening and refinement strategy described in Subsection \[subsubsec:adaptivity\].
The initial iterate for the algebraic Schur–Newton solver is derived by nested iteration, i.e., on each refinement level an initial iterate is obtained by nodal interpolation of the final iterate from the preceding one. On the first refinement level, the initial iterate is obtained by nodal interpolation of the final approximation in the preceding time step. For the first time step, the continuous initial conditions are interpolated to the initial grid ${\mathcal{T}^\textnormal{old}}$.
All numerical experiments were conducted using the DUNE (Distributed and Unified Numerics Environment) framework and the DUNE-modules *dune-subgrid* and *dune-tnnmg* (cf. [@bastian_et_al:dune2:2008; @bastian_et_al:dune1:2008; @graser2009dune]).
Experimental order of convergence {#subsec:EOC}
---------------------------------
In order to numerically assess the spatial discretization error of the finite element discretization stated in Problem \[prob:PFSPATDISC\], we consider the multi-phase Penrose–Fife Problem \[prob:PF\], with $\Omega=(0,2)^2\subset {\mathbb{R}}^2$, $M=5$ phases of which only the liquid and one solid phase is present, and the following parameters $$\begin{aligned}
{\varepsilon}&= 6\cdot 10^{-2}, &
c_v &= 1, &
q &= 0, &
{h_c}&= 0, \\
\kappa &= 1, &
{\beta}&= 1, & L_1 &= 0, & L_{\alpha}&= 2, \; \alpha =2,\dots,M.\end{aligned}$$ We select the initial temperature $\theta^0=0.5$. The initial phase field $\phi^0$ is given by $$\begin{aligned}
\phi^0_2(x) &=
\begin{cases}
1 & \textrm{ if } d(x) < 0.5 \\
\vert\tfrac12\cos(5\pi(d(x)-0.5))+0.5\vert & \textrm{ if } 0.5 \leq d(x) < 0.7 \\
0 & \textrm{ else }
\end{cases}\end{aligned}$$ where $d(x)$ stands for the Euclidean distance from $x$ to $(1,1)$, $\phi^0_1(x) = 1 - \phi^0_2(x,t)$, and $\phi^0_\alpha = 0$ for $\alpha = 3, 4, 5$, as depicted in Figure \[fig:initialvalue\]. We select the uniform time step size $\tau = 5\cdot 10^{-4}$. A sequence ${\mathcal{T}}_0,\dots, {\mathcal{T}}_{10}$ of grids is obtained by uniform refinement of ${\mathcal{T}}_0$ consisting of a partition of $\Omega$ into two triangles.
![Initial phase field: A circular solid phase (orange) in a liquid environment (teal).[]{data-label="fig:initialvalue"}](img/disc-err-init.png){width="31.00000%"}
Figure \[fig:disc-err\] shows the approximate discretization error in the first time step plotted over the mesh size $h_j$, $j=2,\dots,9$. The exact error is approximated by $e_\phi = \phi_{\mathcal{T}}- \phi^*$ and $e_\theta = \theta_{\mathcal{T}}- \theta^*$, with approximations $\phi^*$ and $\theta^*$ obtained from ${\mathcal{T}}_{10}$. Our results suggest optimal order $O(h)$ of the discretization error $\|e\| = \|e_\phi\|_\phi + \|e_\theta\|_\theta$.
We next investigate the convergence properties of the non-smooth Schur–Newton method as applied to the discrete saddle point problem in the first time step. Figure \[fig:rateItSteps\] depicts the number $\nu_{\textrm{max}}$ of iteration steps needed until the stopping criterion is satisfied plotted over $N={\operatorname{dim}}{\mathcal{S}}_j$, where ${\mathcal{S}}_j$ is the finite element space associated with ${\mathcal{T}}_j$. The results indicate mesh independence of the Schur–Newton iteration. While up to $\nu_{\textrm{max}}=17$ iteration steps are required on coarser levels, only $\nu_{\textrm{max}}\leq 7$ steps are needed once the diffuse interface is properly resolved by sufficiently fine grids. This is in accordance with previous computations with multi-component Cahn–Hilliard systems [@graeser2014].
We also computed the approximate solution for the first 500 time steps utilizing the grid ${\mathcal{T}}_7$ obtained by seven uniform refinements to illustrate the evolution of the approximate entropy as depicted in Figure \[fig:circleEntropy\].
Evolution of energy and entropy
-------------------------------
In order to illustrate the equilibration of energy in terms of latent heat and temperature and the evolution of entropy, we consider the multi-phase Penrose–Fife Problem \[prob:PF\] on the unit square $\Omega=(0,1)^2\subset {\mathbb{R}}^2$ with $M=5$ phases of which only the liquid and one solid phase are present, and the parameters $$\begin{aligned}
{\varepsilon}&= 8\cdot 10^{-2}, &
c_v &= 1, &
q &= 0, &
{h_c}&= 0, \\
\kappa &= 1, &
{\beta}&=1, & L_1 &= 0, & L_{\alpha}&= 2,\; {\alpha}=2,\dots,5.
\end{aligned}$$ We choose an initial configuration with two phases (liquid and solid) and a planar interface according to $$\begin{aligned}
\phi^0_2 (x_1, x_2) =
\begin{cases}
1 & \textrm{ if } x_1 > x^\star + 0.1, \\
10(x_1 - x^\star) & \textrm{ if } x^\star +0.1 \geq x_1 > x^\star \\
0 & \textrm{ if } x^\star \geq x_1,
\end{cases} ,
\end{aligned}$$ $\phi^0_1 = 1- \phi^0_2$, and $\phi^0_\alpha = 0$, $\alpha = 3,4,5$. The parameter $x^\star$ and constant initial temperature $\theta^0 = \theta^\star$ will be fixed later.
We select the time step size $\tau = 5\cdot 10^{-3}$. The grid ${\mathcal{T}}$ is obtained by eight uniform refinements of an initial partition of $\Omega$ into two triangles.
The evolution of temperature is illustrated in terms of its maximal variation $$\begin{aligned}
\theta_d^n = \max_{x\in\Omega}\theta^n(x) - \min_{x\in\Omega}\theta^n(x),
\qquad n = 1,\dots, 500,\end{aligned}$$ and the average $$\begin{aligned}
\theta_m^n = \tfrac12 (\max_{x\in\Omega}\theta^n(x) + \min_{x\in\Omega}\theta^n(x)),
\qquad n = 1,\dots, 500,\end{aligned}$$ of its extremal values. The parameter $x^\star$ and constant initial temperature $\theta^0 = \theta^\star$ are set to $x^\star=0.8$ and $\theta^\star = 0.2^{-1}$ in our first experiment and to $x^\star=0.2$ and $\theta^\star = 1.5^{-1}$ in our second experiment. The corresponding two evolutions are illustrated in Figure \[fig:feedback01\] and Figure \[fig:feedback02\], respectively. Both figures show several time steps of the phase field in the left picture. As the solution is constant in vertical direction only a cut-out is shown. The picture on the right shows the evolution of temperature in terms of $\theta_m^n$ and $\theta_d^n$ and of the entropy $S=\hat{S}(\theta^n, \phi^n)$ (cf. ) approximated by numerical quadrature.
In the first experiment we observe a growth of the initial solid grain that slows down continuously due to intrinsic specimen heating by solidification. Conversely, the shrinking of the initial grain observed in the second experiment slows down due to intrinsic cooling by melting. In both experiments, absorption or release of latent heat is driving the approximate temperature $\tfrac1{\theta^n}$ towards the melting temperature $T=1$ at equilibrium. Both cases exhibit a monotonically increasing entropy.
A liquid phase crystallization process {#s:application}
======================================
Liquid phase crystallization (LPC) is an emerging technology to produce silicon thin film solar cells with advanced photoelectronic properties that enable high efficiency devices. In an LPC process, silicon is deposited on a substrate and then is swept over with a heat source for local melting and subsequent recrystallization to coarser, photoelectronically beneficial structures. Optimization of parameters like speed, shape or intensity of the heat source for various semi-conducting materials is the subject of current experimental research, cf., e.g., [@amkreutz2011electron; @eggleston2012large; @kuhnapfel2015towards].
Mathematical modelling of LPC processes can be performed in the framework of multi-phase field models presented in Section \[ss:assumptions\]. To this end, we consider the thin-film approximation Problem \[prob:PFTHIN\] on $\Omega = (0,2)^2$ with $M=5$ phases with $\phi_{\alpha}$, ${\alpha}=2,\dots M$, representing different crystal structures and the parameters $$\begin{aligned}
{\varepsilon}&= 5\cdot 10^{-2}, &
c_v &= 1, &
T_\Gamma &= 0.1, &
T_{\alpha}&= 1, \; {\alpha}=2,\dots, M,
\\
{h_c}' &= 5\cdot 10^2, &
\kappa &= 1, &
L_1 &= 0, &
L_{\alpha}&= 1, \; {\alpha}=2,\dots, M .\end{aligned}$$
In order to prescribe a slower solid–solid interface evolution in comparison the to solid–liquid interfaces, we now choose a solution-dependent kinetic coefficient ${\beta}= ({\beta}_\alpha(\phi, \nabla\phi))_{\alpha=1}^N$ according to $$\begin{aligned}
{\beta}_\alpha(\phi_1, \dots, \phi_M, \nabla\phi_1, \dots, \nabla\phi_M) &= \begin{cases}
100 & \vert\phi_1\nabla\phi_\alpha - \phi_\alpha\nabla\phi_1\vert < 10^{-5},\\ 1 & \textrm{else}.
\end{cases}\end{aligned}$$
The heat source is represented by $$\begin{aligned}
q(\theta, x, t) &= q_{{\operatorname{max}}}(\theta) \cdot\exp\left(-\tfrac{\lvert x_1 - q_p(t)\rvert}{2q_w^2}\right), \\
q_{{\operatorname{max}}}(\theta) &= \tfrac1\theta h_q (1-\tfrac{\theta_q}{\theta}) + {h_c}'(\tfrac1{\theta} - T_\Gamma)\end{aligned}$$ and we select the parameters $$\begin{aligned}
q_p(t) &= 0.9 + 1.5t, &
q_w &= 0.2, &
\theta_q &= \tfrac18 &
h_q &= 5\cdot 10^2.\end{aligned}$$ Notice that the consistency statements of Proposition \[prop:THERMOCONSISTENCY\] and \[prop:THERMOCONSISTENCY-disc\] cannot be applied in this case, because $q\neq 0$ and because the coefficient ${\beta}$ depends on $\phi$. It is unclear if similar results can be shown for solution dependent coefficients.
Observe that the heat source peaks at $x_1=q_p(t)$ and is moving across the device from left to right with constant speed. The initial temperature is given by $\theta^0=10=\tfrac1{T^0} = \tfrac1{0.1}$ and the initial phase configuration $\phi^0$ is depicted in the upper left picture of Figure \[fig:lpc01\] with teal color representing the liquid phase. We choose an initial value that already prescribes a local liquid phase, because this simplifies to exclude superheating effects such as instantaneous global melting of the whole material.
We select the time step size $\tau = 2\cdot 10^{-3}$. In each time step, we construct a sequence of locally refined meshes ${\mathcal{T}}_0, \dots, {\mathcal{T}}_J$ using the adaptive refinement algorithm described in Subsection \[subsubsec:adaptivity\]. The derefinement and refinement parameters are selected as follows $$\begin{aligned}
j_{{\operatorname{min}}}&= 2, &
{{\operatorname{Tol}}_{\mathrm{derefine}}}&= 10^{-6}, &
\rho &= 0.9, &
{{\operatorname{Tol}}_{\mathrm{adapt}}}&= 8\cdot10^{-3}.\end{aligned}$$ In the first time step, we start with an initial grid ${\mathcal{T}^\textnormal{old}}$ obtained by eight uniform refinements of an initial partition of $\Omega$ into two triangles. The final mesh ${\mathcal{T}}$ for time steps 5 and 100 is depicted in Figure \[fig:lpc\_mesh\]. In both cases, the mesh is obtained by 6 adaptive refinement steps after coarsening.
Figure \[fig:lpc01\] shows the (approximate) evolution of phases and temperature. For each of the time steps 0, 5, 10, 20, 100, 250 the left picture depicts the liquid and the different crystalline phases while the right picture shows the temperature distribution. The liquid phase adapts to the shape of the heat source in course of the evolution. As the heat source travels on, the right hand crystalline phases start melting, while recrystallization occurs on the left solid–liquid interfaces, because the local temperature drops below melting temperature. Note that recrystallization leads to coarser grain structures which is a characteristic feature of LPC.
To briefly highlight the efficiency of the Schur–Newton method with nested iteration, the number of Schur–Newton steps $\nu_{\textrm{max}}$ required on the finest mesh is plotted over the time step in the left of Figure \[fig:rateLPC\]. We observe that this number does not exceed 3 in any time step. Mesh independence is illustrated by the number of Schur–Newton steps over $N={\operatorname{dim}}{\mathcal{S}}_j$ in time step $5$.
Acknowledgment {#acknowledgment .unnumbered}
==============
The work was supported by the Helmholtz Virtual Institute HVI-520 “Microstructure Control for Thin-Film Solar Cells”, the Einstein Foundation Berlin via the research center [Matheon]{}, and by the Sino-German Science Center (grant id 1228) on the occasion of the Chinese-German Workshop on Computational and Applied Mathematics in Augsburg 2015.
|
---
abstract: 'In quasars which are lensed by galaxies, the point-like images sometimes show sharp and uncorrelated brightness variations (microlensing). These brightness changes are associated with the innermost region of the quasar passing through a complicated pattern of caustics produced by the stars in the lensing galaxy. In this paper, we study whether the universal properties of optical caustics could enable extraction of shape information about the central engine of quasars. We present a toy model with a crescent-shaped source crossing a fold caustic. The silhouette of a black hole over an accretion disk tends to produce roughly crescent sources. When a crescent-shaped source crosses a fold caustic, the resulting light curve is noticeably different from the case of a circular luminosity profile or Gaussian source. With good enough monitoring data, the crescent parameters, apart from one degeneracy, can be recovered.'
author:
- |
Mihai Tomozeiu[^1],$^{1,2}$ Irshad Mohammed,$^{1,2,3}$ Manuel Rabold,$^{1,2}$ Prasenjit Saha,$^{1,2}$ and Joachim Wambsganss$^{1,4}$\
$^1$[Physik-Institut, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland]{}\
$^2$[Institute for Computational Science, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland]{}\
$^3$[Theoretical Astrophysics Group, Fermi National Accelerator Laboratory, Batavia, IL 60510, USA]{}\
$^4$[Zentrum für Astronomie der Universität Heidelberg, Mönchhofstrasse 12–14, 69120 Heidelberg, Germany]{}
bibliography:
- 'heap.bib'
title: 'Microlensing as a possible probe of event-horizon structure in quasars'
---
Supermassive black holes, microlensing, quasars.
introduction
============
Active galactic nuclei are thought to be powered by the accretion of matter from the proximal environment into a supermassive black hole. The radiation emitted excites the surrounding medium which becomes detectable as narrow line regions, broad line regions and optical continuum. Moreover, in the direction perpendicular to the accretion disc, where the medium is more transparent, jets will appear. If a jet is oriented towards the Earth, a quasar is observed [e.g., @1984RvMP...56..255B]. While the basic mechanism [originating in the work of @1964ApJ...140..796S; @1964SPhD....9..246Z; @1969Natur.223..690L] is not in doubt, the central engines, near the event horizons of the black holes, remain to be probed.
For the black holes in the Galactic centre and the centre of M87 —the two nearest objects that barely qualify as AGN— the central engines (which are $<0.1\rm\,mas$ on the sky) are close to being resolved through very long baseline interferometry, which shows preliminary indications of the jet-launching structures [@2008JPhCS.131a2055D; @2012Sci...338..355D; @2013MNRAS.434..765K; @2016arXiv160205527F]. Current data do not deliver images but require fitting to predefined models for the images. A whole range of models have been applied, starting from simple geometric models to more complex physical models [@2008Natur.455...78D; @2011ApJ...738...38B; @2009ApJ...706..497M; @2010ApJ...717.1092D]. The more complicated models nonetheless tend to predict a crescent shaped silhouette of the black hole. This motivated [@2013MNRAS.434..765K] to use a simple geometric crescent model to fit the data, and argued that the crescent is nothing but the silhouette of the event horizon.
The great majority of quasars, however, lie at redshifts beyond 2 and their central engines would be orders of magnitude smaller on the sky. The direct observations of the black hole silhouettes of quasars are far beyond foreseeable instrumentation. In the present paper, we consider a possible indirect method, related to [@1999ApJ...524...49A], through which gravitational microlensing could probe the black hole shadow and its proximal quasar environment.
Gravitational microlensing of quasars reviewed in Section \[sec:microlensing\] below, refers to sharp changes in the observed brightness of quasars that have been lensed by an intervening galaxies, without any changes in the intrinsic luminosity. Microlensing affects only the light from the innermost part of the quasar, such as the optical continuum and is a consequence of two things: the very small size of the central engine, and granularity of the mass distribution of a lensing galaxy due to stars. The latter means that the local magnification is not a smooth function of source position. It contains a complicated network of singular curves, known as caustics. Figure \[fig:magnification\_map\] shows part of a magnification map with a few caustics. The lensed brightness would be given by placing the source on such a magnification map and integrating the surface brightness weighted by the magnification. Most astrophysical sources straddle several caustics, and hence, their net brightness varies smoothly with location. The central engine of quasars, however, is smaller than the typical spacing between caustics. As a result, the lensed brightness undergoes sudden changes as a quasar crosses a caustic. The effect supplies an upper limit on the size of the central engine, and can also be used to study the mass distribution and kinematics of stars in the lensing galaxy as well [e.g., @2012ApJ...744..111P]. Caustics have an additional remarkable feature: though they can be very complicated, they have some universal properties well-known from catastrophe theory. In particular, very close to the simplest caustics (known as folds), the magnification is approximately constant on one side and $\propto1/\sqrt p$ where $p$ is the transverse distance of the source from the caustic. This property will be exploited later.
In Section \[sec:source-models\] we introduce the three source profiles used in our subsequent models and simulations: a constant-brightness disc, a circular Gaussian, and the crescent source introduced by [@2013MNRAS.434..765K]. The latter is simply a constant-brightness disc with a smaller, non-concentric disc cut out of it. We also derive the half-light radius for a crescent. The half-light radius can characterise the source size for all three types of source.
Section \[sec:fold-crossing\] shows the light curves that result when each of the model sources crosses an ideal fold. This would apply in Figure \[fig:magnification\_map\] to sources along the path AB or BC, for sources small enough that the curvature of the caustics is negligible. With this assumption one can imagine the caustic as an infinite wall to be crossed by the source as presented in Figure \[fig:infinite\_fold\]. The source brightness distribution parallel to the caustic naturally makes no difference to the observable brightness; each source can be replaced by an effective one-dimensional source profile, by flattening the source so it becomes perpendicular to the caustic. In principle, the effective one-dimensional brightness profile could be recovered from the light curve by deconvolution. [@1999ApJ...524...49A] modelled this profile as the result of a circular accretion disc seen through the spacetime around a Kerr black hole, and [@2012MNRAS.423..676A] have applied the idea to observed light curves to infer properties of quasar accretion discs. In this work, we take a simpler but arguably more robust approach: we study features in the light curves characteristic of a crescent-like source which in turn would indicate a black-hole silhouette. Figure \[fig:char\_points\] shows the qualitative features: there is a period during which the dark cutout disc is crossing the caustic, and before and after there are periods when the only the bright parts of the crescent are in transit across the caustic. The details depend on the orientation of the crescent, but basically the dark disc causes a rising light curve to plateau or dip. These features are still present, albeit faintly, if the simple crescent is replaced by a source based on an accretion-disc simulation of a black-hole environment (Figures \[fig:M87\_image\]–\[fig:M87\_plots\]).
In Section \[sec:numerics\] we carry out source fitting to lightcurves, with both noise and systematic errors are present. We generate three lightcurves by taking a uniform disc, a circular Gaussian, and a crescent across the path AB in Figure \[fig:magnification\_map\], and then adding noise. The path simulates crossing a clean but not ideal fold. In addition, we generate templates lightcurves by running the three source types, with various parameter values, across the path CB. That is, the templates come from a similar but not identical caustic, thus deliberately generating a systematic error. We then fit the noisy lightcurves to the templates using Markov chain Monte-Carlo. We find that the correct source type can be inferred from the $\chi^2$ values. The parameter values can also be inferred. The fitting errors are larger than the formal uncertainties, which is expected in the presence of systematic errors, but still appear acceptable.
Finally in Section \[sec:discussion\] we discuss in more depth the implications of the results presented in the previously mentioned sections. One of the most interesting implication is the possibility to estimate the black hole mass from the reconstructed parameters. This further requires good approximations of the relative transversal velocity between the quasar and the lens. Proper constraints can be set with independent observations of the stellar structure that contain the gravitational lens.
The thorough study of the possibility to reconstruct the quasar’s structural parameters from light-curves containing multiple microlensing events represents the target of future work. This will most probably require the use of powerful statistical tools. Another path for future work is the designing of an observation regime best suited for acquiring the necessary microlensed light-curve.
Microlensing {#sec:microlensing}
============
We start by introducing in a succinct manner the gravitational lensing theory that is relevant for microlensing in general and for the scope of the present paper in particular. More detailed presentation of the theory can be found in several references, such as [@2001stgl.book.....P].
Magnification
-------------
The gravitational lens equation $$\vec\beta = \vec\theta - \vec\alpha(\vec\theta)
\label{eqn:lens}$$ relates the apparent sky position $\vec\theta$ of a light source to its true but unobservable sky position $\vec\beta$ through the bending angle $\vec\alpha$. The latter is an integral $$\vec\alpha = (1+z_{_L})\frac{D_S-D_L}{D_SD_L} \frac{4G}{c^2}
\int \Sigma(\vec\theta')
\frac{\vec\theta-\vec\theta'}{|\vec\theta-\vec\theta'|^2}\,
d^2\vec\theta'
\label{eqn:alpha}$$ depending on the projected density $\Sigma(\vec\theta)$ (kg/steradian) of the lens, the lens redshift $z_{_L}$ and the comoving distances $D_L$ and $D_S$ to the lens and source. The derivative of the apparent position with respect to the source position $$M(\vec\theta) =
\left(\frac{\partial\beta}{\partial\theta}\right)^{-1}
\label{eqn:magnif-matrix}$$ is known as the magnification matrix, and its determinant $$\mu(\vec\theta) = \det|M(\vec\theta)|$$ is the brightness amplification of an image of a point source. In other words, the source will brighten or dim according to whether $\mu(\vec\theta)$ is more or less than unity. If there are multiple mages at distinct but not observationally resolved $\vec\theta_i$ from the same $\vec\beta$, a total brightness amplification of$$\mu_{\rm total} = \sum_{i} \mu(\vec\theta_i)$$ applies. It is possible for the magnification to become formally infinite, as a result of an eigenvalues of the magnification matrix (\[eqn:magnif-matrix\]) becoming infinite. This generically happens on curves on the $\vec\theta$ plane, known as [*critical curves*]{}. Mapping a critical curve to the source plane, through the lens equation (\[eqn:lens\]), to the source plane $\vec\beta$ gives the so-called caustics. Caustics can appear in the optical system, not just gravitational lensing. For a point source, caustics are singularities of the magnification; for finite-size sources caustics correspond to high and sharply changing magnification.
Caustics are important in all forms of lensing with multiple images, but they have a special significance for lens quasars, first pointed out by [@1979Natur.282..561C]. The granularity of the mass distribution $\Sigma(\vec\theta)$ due to individual stars produces a caustic network on the scale of $\sqrt{GM_\odot D_L/c^2}$ or $10^{-6}
\sqrt{M/M_\odot}\rm\,arcsec$ for typical lens and source redshift [@2001PASA...18..207W]. Extended sources wash out this micro-caustic structure, but the optical-continuum source of quasars is even smaller.
![\[fig:magnification\_map\] Magnification map used later in this paper (Section \[sec:numerics\]). The white line marks the trajectory of the center of the sources for which the lightcurves presented in figure 11 are generated. Note that this represents an atypically simple region of any realistic magnification map.](figures/IRIS567_path.eps){width="0.9\hsize"}
Magnification near a fold caustic
---------------------------------
A simple example of a caustic network is shown in Figure \[fig:magnification\_map\]. There are two general categories of caustics in gravitational lensing, cusps and fold, and examples of both kinds can be seen in this figure. Magnification near a caustic has universal properties, independent of the system and has been extensively studied . In particular, at distance $p$ from a fold caustic $$\mu(p) = \mu_0 + C_0 \frac{1}{\sqrt{p}} \Theta(p).$$ Here the magnification of a point source near a caustic is equal to the sum of the magnification due to other reasons $\mu_0$, assumed to be locally constant, and a decrease with the square root of the distance from the fold. The latter term becomes activated only after the source enters the region interior to the caustic curves when the values of the step function $\Theta(p)$ become unity. The proportionality constant $C_0$ depends on the local conditions in the vicinity of the caustic.
A source of arbitrary shape can be described by a two-dimensional brightness function $S_{2D}(p - p_s, q - q_s)$ defined for a coordinate system $p,q$ where $p_s, q_s$ denote the coordinates of the center of a source.
For a microlensing event the lightcurve can be written for an undefined source shape as: $$F(t) = \int_{-\infty}^\infty \int_{-\infty}^\infty S_{2D}(p-p_s(t), q-q_s(t)) \mu_t(p) \mathrm{d}q \mathrm{d}p
\label{eqn:ft2d}$$ In order to build the previous equation we have considered that the time dependency of the flux $F$ is given only by the motion of the source with respect to a fixed caustic. Therefore the only time dependent quantities in the right hand side of the equation are the coordinates of the center of the source $p_s,q_s$ and by construct $S_{2D}$.
Due to the choice of the coordinate system the amplification factor has no dependence on the $q$ coordinate. The previous equation can be rewritten as:
$$F(t)
= \int_{-\infty}^\infty \mu_t(p) S_{1D}\left(p-p_s(t)\right) \mathrm{d}p,
\label{eqn:ft}$$
\
where we have defined the one dimensional flux function as: $$S_{1D}(p-p_s(t)) = \int_{-\infty}^\infty S_{2D}(p-p_s(t), q-q_s(t)) \mathrm{d}q$$
This representation is a valid approximation only when the apparent size of the source is much smaller than the corresponding Einstein angle of the lens. In this context all the information about the source shape and brightness that can be contained in the lightcurve is exhaustively given by the 1D flux function. In other words, if two sources with different $S_{2D}$ have the same $S_{1D}$ they cannot be distinguished by studying their lightcurves.
Models for extended sources {#sec:source-models}
===========================
In the present study we are analysing three types of sources with different surface brightness:
1. a rotationally symmetric source with a bivariate gaussian surface brightness distribution,
2. a disk source with constant surface brightness distribution,
3. a crescent shaped source with constant surface brightness distribution.
The first two sources are the typical choices used in the literature to describe the luminous parts of a quasar (Prasenjit should give some citations here). The third one is a recently proposed variant [@2013MNRAS.434..765K].
Rotationally symmetric source with a bivariate gaussian surface brightness distribution
---------------------------------------------------------------------------------------
A symmetric 2D gaussian can be described mathematically as:
$$S_{2D}^G(p-p_s, q-q_s) = \frac{S_0^G}{2 \pi \sigma^2} e^{-\frac{(p-p_s)^2}{2 \sigma^2}} e^{-\frac{(q-q_s)^2}{2 \sigma^2}}.$$
\
The corresponding 1D brightness is:
$$S_{1D}^G(p-p_s) = \frac{S_0^G}{\sqrt{2 \pi} \sigma} e^{-\frac{(p-p_s)^2}{2 \sigma^2}}.$$
\
Other parameters of the model are the total flux $S_0^G$ and $\sigma$.
Although such a definition for a source would have non-zero surface/linear brightness for any coordinate $p,q$, the amount of light received by a detector from outside a $3 \sigma$ disk centered at $p_s, q_s$ would be insignificant. For a gaussian distributed surface brightness source the half-light radius is directly proportional to the parameter $\sigma$ according to the equation: $$r_{1/2} = \sqrt{ln(4)} \sigma.$$
Disk source with constant surface brightness distribution
---------------------------------------------------------
One can construct mathematically a disk source with constant surface brightness and radius $R$ using a stepfunction: $$S_{2D}^D(p-p_s, q-q_s) = \frac{S_0^D}{\pi R^2} \Theta \left( R^2 - \left( p-p_s \right)^2 - \left( q-q_s \right)^2 \right).$$ By integrating over $q$ coordinate the linear brightness function is obtained:
$$S_{1D}^D(p-p_s) = \frac{2 S_0^D}{\pi R} \sqrt{1 - \frac{(p-p_s)^2}{R^2} } \Theta \left( R^2 - \left( p-p_s \right)^2 \right).$$
\
The half-light radius of a uniform disk source is $R/\sqrt{2}$.
Crescent source with constant surface brightness distribution {#subsec:crescent}
-------------------------------------------------------------
The surface brightness distribution of a geometric crescent can be built by considering two disk sources of constant brightness. One larger disk will contribute positively to the total flux while one smaller disk that is interior to the large one will contribute negatively. This superposition can be written for 2D as:\
$$S_{2D}^C = S_{2D}^{Dp} - S_{2D}^{Dn}
\label{eqn:s2d}$$ with\
$$S_{2D}^{Dp}(p-p_{sn}, q-q_{sn}) = \frac{S_0^{Dp}}{\pi R_p^2} \Theta \left( R_p^2 - \left( p-p_{sp} \right)^2 - \left( q-q_{sp} \right)^2 \right)$$\
and $$S_{2D}^{Dn}(p-p_{sn}, q-q_{sn}) = \frac{S_0^{Dn}}{\pi R_n^2} \Theta \left( R_n^2 - \left( p-p_{sn} \right)^2 - \left( q-q_{sn} \right)^2 \right).$$\
The following notations were used: $R_p, (p_{sp}, q_{sp}), R_n, (p_{sn},q_{sn})$ are the radii and coordinate of the center for the larger positive disk and smaller negative disk respectively. $S_0^{Dp},S_0^{Dn}$ represent the total flux of radiation received from the large and small disk. From this point forward we will not use the total flux from each source. Instead we will use the difference which in this case is th total flux from the crescent-shaped source $S_0^C$.\
Equation \[eqn:s2d\] can be written as:\
$$\begin{aligned}
S_{2D}^C &= \frac{S_0^C}{\pi \left(R_p^2-R_n^2 \right)} \left\{ \Theta \left[ R_p^2 - \left( p-p_{sp} \right)^2 - \left( q-q_{sp} \right)^2 \right] \right.\nonumber\\
&\qquad \left. {} - \Theta \left[ R_n^2 - \left( p-p_{sn} \right)^2 - \left( q-q_{sn} \right)^2 \right] \right\}.\end{aligned}$$\
Analogous for the linear brightness function:
$$\begin{aligned}
S_{1D}^C &= \frac{2 S_0^C}{\pi \left(R_p^2-R_n^2 \right)} \left\{ \sqrt{R_p^2 - (p-p_{sp})^2} \Theta \left[ R_p^2 - \left( p-p_{sp} \right)^2 \right] \right.\nonumber\\
&\qquad \left. {} - \sqrt{R_n^2 - (p-p_{sn})^2 } \Theta \left[ R_n^2 - \left( p-p_{sn} \right)^2 \right] \right\}.
\label{eqn:s1_d}\end{aligned}$$
There are some constraints on the parameters used to define a crescent in the previously presented manner that need to be stated. First, we must impose the obvious $R_p > R_n$ relation. Secondly, the small disk must always be interior to the large disk: $$R_p \ge R_n + \sqrt{\left(p_{sp} - p_{sn} \right)^2 + \left(q_{sp} - q_{sn} \right)^2}$$
For the distances between the centers of the two disks we will use the same notations as the one found in the paper [@2013MNRAS.434..765K], $a \equiv p_{sn} - p_{sp}$ and $b \equiv q_{sn} - q_{sp}$.
The half-light radius of any source is invariant to any rotational transformation. In the present case of a crescent source the effective radius is dependent on the parameters $R_p$, $R_n$ and $\sqrt{a^2+b^2}\equiv c $ exclusively. From symmetry considerations the centroid of the source is collinear with the centers of the two disks and it is situated at a distance $d_c$ from the center of the bright disk. $d_c$ can be computed numerically with the use of a variation of equation \[eqn:s1\_d\]:
$$\begin{aligned}
\frac{S_0^C}{2} & = \frac{2 S_0^C}{\pi \left(R_p^2-R_n^2 \right)} \int_{d_c}^{R_p} \bigg[ \sqrt{R_p^2 - p^2} \\
& - \sqrt{R_p^2 - \left(p-c\right)^2} \Theta \left(R_p^2 - \left(p-c\right)^2 \right) \bigg] dp.
\end{aligned}$$
With the position of the centroid determined, the half-light radius can be also be computed numerically:
$$\begin{aligned}
S_0^C &= \frac{4S_0^C}{\pi \left(R_p^2-R_n^2 \right)} \int_{0}^{r_{1/2}} \left[ \Theta_2 - \Theta_1 \right] p dp; \\
\Theta_1 &= \bigg[ \pi - \arccos \frac{R_p^2 + p^2 -d_c^2}{2 R_p p} \\
& \phantom{= \bigg[ \pi} - \arccos \frac{R_p^2 - p^2 + d_c^2}{2 R_p d_c}
\bigg] \Theta \left( p - R_p + d_c \right); \\
\Theta_2 &= \pi - \arccos \frac{ p^2 + (c+d_c)^2 - R_n^2}
{2 p \left(c + d_c \right)} \\
& \phantom{= \pi - } \Theta \left( p + R_n - d_c -c \right).
\end{aligned}$$
Lightcurves of the extended sources during fold crossing {#sec:fold-crossing}
========================================================
Using equation \[eqn:ft\] and the one-dimensional flux function presented in the previous section one can compute numerically the lightcurves of the three extended sources for the simplified infinite-wall-caustics model.
Lightcurve of the gaussian source
---------------------------------
The amount of light received by an observer from a source with a gaussian distributed brightness with $\sigma$ and total flux $S_0^G$ in the absence of any gravitational lensing is: $$F^G(t) = \int_{-\infty}^\infty \left( \mu_0 + \frac{C_0}{\sqrt{p}} \Theta \left( p \right) \right) \left( \frac{S_0^G}{\sqrt{2 \pi} \sigma} e^{-\frac{(p-p_s(t))^2}{2 \sigma^2}} \right) \mathrm{d}p.$$
which can be simplified to: $$F^G(t) = \mu_0 S_0^G + \frac{C_0 S_0^G}{\sqrt{2\pi} \sigma} \int_{0}^\infty \frac{e^{-\frac{(p-p_s(t))^2}{2 \sigma^2}}}{\sqrt{p}} \mathrm{d}p.$$
Lightcurve of the disk shaped source
------------------------------------
Analogous to the gaussian shaped source, the disk source with uniform brightness, radius $R$ and unmagnified flux $S_0^D$ has a lightcurve described by the equation:
$$\begin{aligned}
F^D(t) &= \int_{-\infty}^\infty \left( \mu_0 + \frac{C_0}{\sqrt{p}} \Theta \left( p \right) \right) \\
& \bigg[ \frac{2 S_0^D}{ \pi R} \sqrt{1 - \frac{\left( p-p_s(t) \right)^2}{R^2}} \Theta \left(R^2 - \left(p-p_s(t) \right)^2 \right) \bigg] \mathrm{d}p.
\end{aligned}$$
which is equivalent to: $$F^D(t) = \mu_0 S_0^D + \frac{2 C_0 S_0^D}{\pi R} \int_{max(0, p_s(t) - R)}^{max(0, p_s(t) + R)} \frac{1}{\sqrt{p}} \sqrt{1 - \frac{\left( p-p_s(t) \right)^2}{R^2}} \mathrm{d}p.$$
Lightcurve of the crescent shaped source
----------------------------------------
The lightcurve of a crescent shaped source with unamplified flux $S_0^C$, radii $R_p$, $R_n$ and center displacement $a(t)$ is:
$$\begin{aligned}
F^c(t) &= \mu_0 S_0^C + C_0 \frac{2 S_0^C}{\pi \left( R_p^2 -R_n^2 \right) } \\
&\bigg[ \int_{max(0, p_s(t) - R)}^{max(0, p_s(t) + R)} \sqrt{\frac{R_p^2 - \left( p-p_s(t) \right)^2 }{p}} \mathrm{d}p \\
& - \int_{max(0, p_s(t) - a(t) - R)}^{max(0, p_s(t) -a(t) + R)} \sqrt{\frac{R_p^2 - \left( p-p_s(t) +a(t) \right)^2 }{p}} \mathrm{d}p \bigg] .
\end{aligned}$$
The function $p_s(t)$ can be chosen to be equal to $v_p(t-t_0) + p_{s0}$. Where $p_{s0}$ is the coordinate $p$ of the source at the initial time, and $v_p$ is the component of the velocity along the $p$ axis. Such a modelling of the motion of the object in the source plane describes a linear motion with constant velocity. Furthermore, we reduce the complexity of the model by choosing the function $a(t)$ to be constant in time.
There are four characteristic points visible on the resulting one-dimensional light profile of the crescent shaped source. Two of the points, p1 and p4, mark the outer boundaries of the luminous disc component. The other two, p2 and p3, mark the boundaries of the dark disc component. Since $S_{1D}$ is a projection of $S_{2D}$ on a line perpendicular to the caustic, the following relation holds: $$p_2-p_1 = R_p -R_n - a.$$ In addition there are two other obvious relations the characteristic points which are independent of the projection: $$p_4 -p_1 = 2 R_p,$$
$$p_3 -p_2 = 2 R_n.$$
All four points mark the positions where derivative $\frac{dS_{1D}}{dp}$ is discontinuous. The points can be used to define three regions: $p_1 - p_2$ where $S_1D$ is convex, $p_2 - p_3$ where $S_1D$ is concave, and $p_3 - p_4$ where $S_1D$ is convex again.\
Due to the nature of the caustic and the monotonic behaviour of the magnification map on both sides of the caustic. The previously mentioned characteristic points are inherited by the microlensing lightcurve. The points on the temporal dimension $t_1, t_2, t_3$ and $t_4$ correspond to instances in time when the fold is aligned with $p_1, p_2, p_3$ and $p_4$, respectively. For a constant relative velocity $v_p$ between the source and the caustic there is a simple relation between the points $p_i$ and instances $t_i$:
$$t_j - t_i = \frac{p_j - p_i}{v_p}.$$
With the use of the previous four equations the following identities can be written:
$$R_p = \frac{v_p \left( t_4 -t_1 \right)}{2},$$
$$R_n = \frac{v_p \left( t_3 -t_2 \right)}{2},$$
$$a = \frac{v_p \left( t_4 +t_1 - t_3 - t_2 \right)}{2}.$$
Figure 4 reveals that the lightcurve of a crescent source has more visible features than the other two light-curves corresponding to the disc and gaussian shape. There are three regions where $S_{1D}$ and by inheritance, the lightcurve has distinguishable behaviour. The first region that would be recorded on a lightcurve plot represents the period of time when the bright disk begins to be overlapped by the caustic and stops when just before the dark disk reaches the caustic. During this period of time, the flux of light from the source is increasingly magnified. The second period starts and ends with the overlapping of the dark disc. As a boundary of the two regions, there is a distinguishable point where the slope of the magnification is drastically changed. This apparent discontinuity in the first derivative of the magnification function is caused by the caustic amplification of the sudden drop in the $S_{1d}$ function. During the respective period, the magnification growth slows or even reverses during the first part of the period and starts to grow faster again as the dark disk ends its overlap with the caustic. At the point where the dark disc clears the caustic, the growth of the magnification is infinite, which appears as a saddle point on the lightcurve. Next, the final period corresponds to the case when the dark disc has cleared the caustic and the bright disc continues to overlap with the caustic. During this period, the lightcurve reaches a peak that for most of the parameter space is global and for the rest of the parameter space local. At the end of the period, the growth of the magnification is negative infinite. the respective point appears as a second saddle point on the lightcurve. Past this point the magnification of any finite source will decay in roughly the same manner.\
The impact of parameter $a$ on the shape of the lightcurve can be observed in figure 5 (a-var). Sources where the center of the dark disc reaches the caustic before the center of the bright disc are characterized by a smoother broad peak in contrast to the cases where the center of the dark disc reaches the caustic after the bright disk. In the case of the latter the instance when the dark disc reaches the caustic corresponds to larger and larger magnifications until it becomes a local and even global peak. The effect of the $R_n$ parameter on the light curve is presented in figures 6 and 7. For the particular set of parameters where $R_p = R_n +a$ the third period of time discussed previously does not exist. Particularizing further, if the value of the radius of the dark disc is comparable to the value of the radius of the bright one then the position and shape of the maximum magnification are strikingly different. In case the crescent reaches the caustic with the bright region first, the peak magnification happens when the dark region reaches the caustic and it is characterized by a sharp variation in magnification growth, In the opposite case, the peak appears before the end of the bright disc reaches the caustic and shape is smoother.\
Microlensing a simulated image of M87
-------------------------------------
[@2012MNRAS.421.1517D] have created a radiative image of M87 based on the GRMHD simulations presented in [@2009MNRAS.394L.126M]. The top right-most image in figure 5 of [@2012MNRAS.421.1517D] has been projected to a 1D profile associated to the perpendicular direction to a fold caustic approaching the image from the right. The projection is presented in the upper panel of figure 9. The amplification values of the flux of light corresponding to a microlensing event are presented in the lower panel of figure 9. In general, the behaviour of the lightcurve is similar to the geometric crescent source with the caveat that the outer regions surrounding the luminous parts of the image have non-zero flux and thus are more extended than the simplified source model.
Fitting mock data {#sec:numerics}
=================
Having used the simple model of an ideal fold to gain insight, we now move to the numerical computation of microlensing light curves with realistic mass distributions, and then fit source models to lightcurves in the presence of random and systematic errors.
First we generate microlensing magnification maps using the microlensing code created by Joachim Wambsganss . We then generate mock light curves for three source models — a crescent, a uniform disc and a Gaussian disc — and then fit each mock light curve to all three source models. Parameter fitting and marginalising was done by Markov-chain Monte Carlo (MCMC).
Numerical microlensing magnification maps
-----------------------------------------
The code uses a ray-shooting technique to compute the gravitational lensing effect of a mass distribution consisting of (a) a smooth component and (b) a random distribution of point masses representing stars. The ray shooting maps a grid of $\vec\theta$ to $\vec\beta$ using the lens equation (\[eqn:lens\]). That is, the rays are shot from the observer back to the source.
For the computation of the individual deflection angles $\vec\alpha(\vec\theta)$ a hierarchical or tree method is used. The positions of all lensing masses are put into a grid of $\vec\theta$. Each grid cell is subdivided into four smaller squares recursively until every cell contains only one mass. Nearby masses are added individually while distance masses are clumped into larger grid cells whose net contribution is approximated by its first few multipole moments. Scaled units are used, with the constant pre-factor in the deflection angle (\[eqn:alpha\]) separated out.
The result of ray shooting is a pixel map on the $\vec\beta$ plane of the number of lightrays which arrive at the source plane from a particular observer. This intermediate result is effectively a magnification map on the source plane. Once the map is created, the lightcurve can be obtained by specifying the transit path of the source across the map. At each point on this transit line, the code computes equation (\[eqn:ft2d\]) convolving the brightness distribution of the source with the magnification pattern of the map. In real life, not only are both lens and source moving but the lens lens configuration, and with it the magnification pattern, is also changing with time. While the first subtlety is taken care of by a coordinate transformation in this analysis, for the second one the lens configuration is assumed to be constant in time.
When generating the magnification map depicted in Figure \[fig:magnification\_map\], which is used in our analysis below, only two point masses were included. This was done in order to have clean fold caustics. For the computation of the actual lightcurve, the code was modified to also allow for crescent shaped images specified through the parameter set $R_p,R_n,a,b$. Here $R_p$ denotes the outer radius of the crescent and $R_n$ the inner one. The orientation of the source image with respect to the magnification map is specified by the parameters $a$ and $b$ as the shift of the center of the inner disk from the center of the outer disk in $x$- and $y$-direction respectively. In the original version of the code, gaussian and disk-shaped images were already implemented. Those are completely characterised by the single parameter $R_p$. The values of the parameters are specified in pixel units corresponding to the magnification map. Further, one needs to specify the start and end point coordinates of the path, which the center of the source image follows through the magnification map (see the depiction in figure \[fig:magnification\_map\]). Hence, the points along this path are specified through the number of time steps for which the computation of the brightness is to be carried out. Those points correspond to the actual measurement of the brightness of an object in the observational case. For each timestep the two-dimensional convolution of equation \[eqn:ft2d\] is carried out numerically for the position of the source on the magnification map.
For the purpose of this analysis, it was desirable to mimic the analytical behaviour of a simple fold as much as possible, for comparing the numerical result with the analytical one, therefore, the path of the source was chosen, so that it intersects the border of the caustic perpendicularly, and on a point where the border is a fold caustic.
![\[fig:mockdata\] Three different mock datasets with three different parametrizations fitted to each — crescent (red), uniform-disk (blue) and Gaussian disk (green).](figures/data_cc.eps "fig:"){width="0.9\hsize"} ![\[fig:mockdata\] Three different mock datasets with three different parametrizations fitted to each — crescent (red), uniform-disk (blue) and Gaussian disk (green).](figures/data_dd.eps "fig:"){width="0.9\hsize"} ![\[fig:mockdata\] Three different mock datasets with three different parametrizations fitted to each — crescent (red), uniform-disk (blue) and Gaussian disk (green).](figures/data_gg.eps "fig:"){width="0.9\hsize"}
Mock light curves
-----------------
Mock light curves were generated by running a source across a caustic, specifically along the line from point C to point B in Figure \[fig:magnification\_map\]. The model light curves, to which these are fitted, are made by running a source across a different caustic, along the line from point A to point B in the same figure. That is, the mock data are generated with a clean but not ideal fold caustic and then fitted with another such caustic. This mimics the unavoidable systematic error of not knowing the caustic exactly.
The crescent source had parameters (as explained in §\[subsec:crescent\]) $R_p$ and $R_n$ being the radii of the outer and inner circles and $(\alpha,\beta)$ being the coordinates of the inner circle with respect to the centre of the outer circle in the coordinate system associated with the image in Figure 1 and not associated with the caustic surface as defined in the previous sections. The parameter values were $$(R_p, R_n, \alpha, \beta) = (50.0, 30.0, 15.0, 10.0).
\label{eqn:cp}$$ The uniform disc had the same (outer) radius as the crescent. In the Gaussian source, we set $3\sigma=50$. For convenience, below we will refer to $R_p$ of the Gaussian source, by which we actually mean $3\sigma$.
Each mock light curve also has three nuisance parameters, namely the beginning and end of the event and the brightness normalisation. These are to be marginalised out by the MCMC.
Figure \[fig:mockdata\] shows the three light curves. Each has 250 points regularly spaced in time, with Gaussian noise at the level of 10% of the current brightness. This amounts formally to a single summary data point with a signal to noise of $10\times\sqrt{250}\simeq160$.
Model fitting and likelihood analysis
-------------------------------------
Each of the three mock light curves was fitted to all three models. We denote the nine possible cases with letter pairs, with the first letter denoting the assumed model for the fitting procedure and the second letter representing the source: thus CG means a [*C*]{}rescent model was fitted to mock data from a [*G*]{}aussian source, DC stands for a uniform [ *D*]{}isc model fitted to mock data from a [*C*]{}rescent source, and so on.
![\[fig:mcmc\] Posterior probability distribution for the source size $R_p$. The vertical line is the correct value. Same color represents the same dataset whereas same line style corresponds to same model fit. The legend gives the reduced $\chi^2$ of the best fit in each case.](figures/Rp4all.eps){width="0.9\hsize"}
Figure \[fig:mcmc\] shows the posterior probability distribution of $R_p$ for all nine cases, along with the minimum reduced $\chi^2$ for each case. The area under each curve is unity. Note that the height of the curves are not the likelihood, they are probability densities in parameter space.
![\[fig:crescentfit\] The 2$\sigma$ contours (or error ellipses) of the crescent model when fitting with the three different datasets.](figures/Rhalf_RnRp.eps "fig:"){width="0.9\hsize"} ![\[fig:crescentfit\] The 2$\sigma$ contours (or error ellipses) of the crescent model when fitting with the three different datasets.](figures/aRp_bRp.eps "fig:"){width="0.9\hsize"}
Let us first consider the three cases where a crescent model was fitted. These are the solid curves in Figure \[fig:mcmc\], with the colours of the curves indicating the source. Meanwhile, Figure \[fig:crescentfit\] shows $2\sigma$ of the inferred parameter values.
1. solid red curve in Figure \[fig:mcmc\] and red ellipses in Figure \[fig:crescentfit\]. In this case, a light curve from a crescent source was being fitted to a crescent model. The fit gives reduced $\chi^2$ close to unity, as expected. The recovered parameter values are near or slightly outside the $2\sigma$ ellipses. This is expected since we added a small systematic error, by using different caustics (though both clean folds) for the mock data and the model fit. The apparent degeneracy between $a$ and $b$, seen in Figure \[fig:crescentfit\], is also expected, since only the distance of the crescent’s small circle from the caustic influences the magnification.
2. solid blue curve and blue ellipses. Here a crescent model was fitted to a light curve from a uniform disc. The situation is formally a crescent with $R_n=0$ and $a,b$ arbitrary, and this shows in the blue ellipses in the recovered parameter values. The redundant parameters $\alpha$ and $\beta$, in effect, allow the model to partly fit the noise, and hence, the $\chi^2$ is somewhat lower than in the previous case.
![\[fig:burnin\] The $\chi^2$ of each step of the full MCMC for one case.](figures/burnin_cg.eps){width="0.9\hsize"}
3. Solid green curve and green ellipses. Additionally, Figure \[fig:burnin\] shows the progress of reduction of $\chi^2$ in this case, where a crescent-source model is used to fit data from a Gaussian source. The parameter values are seemingly tightly constrained, but the recovered $R_p$ is completely wrong since the correct value is far from the green curve and green ellipses. The reduced $\chi^2$ is only 0.85, indicative of over-fitting. We can see what has happened from the red curve in the bottom panel of Figure \[fig:mockdata\]. The best-fit model covers only a small part of the light curve. That is, the fitting procedure has exploited the nuisance parameters to find a good fit to mainly noise.
Next we consider the three cases where a disc model is fitted. These correspond to the dashed curves in Figure \[fig:mcmc\]. There is only one interesting parameter to fit, the disc radius $R_p$, and no equivalent of Figure \[fig:crescentfit\] is needed.
1. red dashed curve. On fitting a disc model to a crescent-source source, the recovered $R_p$ is incorrect, but the reduced $\chi^2$ of 1.29 signals that the data reject the model.
2. blue dashed curve. When the correct model is fitted to a disc-source, the best fit $\chi^2=0.91$ is good and $R_p$ is recovered within the uncertainty estimate.
3. green dashed curve. On fitting a disc model to a Gaussian-source model, the recovered $R_p$ is incorrect, but the reduced $\chi^2$ gives no signal that something is amiss. It appears that a Gaussian source could mimic a uniform disc.
Finally, we consider the three cases where a Gaussian model is fitted. These correspond to the dotted curves in Figure \[fig:mcmc\]. Again, there is only one interesting parameter to fit, the Gaussian $3\sigma$-radius, which we have called $R_p$.
1. red dotted curve. When a Gaussian-source model is fitted to data from a crescent source, the recovered $R_p$ is wrong but the reduced $\chi^2=1.29$ shows the data rejecting the model.
2. blue dotted curve. When a Gaussian-source model is fitted to data from a uniform disc, the recovered $R_p$ is wrong but the model is rejected anyway.
3. green dotted curve. When data from a Gaussian is fitted to the correct model, $R_p$ is recovered within it’s estimated uncertainty and the best fit reduced $\chi^2$ is close to unity.
The above results suggest the following strategy for fitting a lightcurve from an unknown source profile: first try a Gaussian-source model; if the data reject that model, try a uniform disc; if the uniform disc is also rejected by the data, try a crescent model. Our numerical experiments indicate — assuming one of the three sources models are correct — that the reduced $\chi^2$ would unmask the correct model, and its parameters would be correctly recovered.
Discussion {#sec:discussion}
==========
In the current paper, we simulate and study the resulting microlensing lightcurves of geometric crescent-shaped sources and compare them with the microlensing lightcurves of other simple mathematically describable source profiles. In order to mimic the behaviour of the flux of light from the source in the proximity of a fold caustic, we make use of the simple approximation described in equation (5). The equation would exhaustively describe the magnification map and offer a good universal approximation for the particular microlensing regime that we consider. Namely, the shape of the caustic boundary in the proximity of the source in the respective plane can be approximated by a line due to reason that the local radius of curvature of caustic is orders of magnitude greater than the half-light radius of the studied source. In particular cases in which the previously mentioned approximation loses its validity, the impact on the quality of the lightcurves is not evenly distributed. The shape of the lightcurve will be maintained. The data points corresponding to the source position before and during the overlapping of the caustic will be affected by smaller errors than the data points corresponding to later times.\
The first two source profiles that we consider are the uniform disk and symmetric Gaussian source. Both of them can be described by a half-light radius $r_{1/2}$ and a total unlensed light flux $S_0$. With the two parameters constrained the one-dimensional profiles, as well as the lightcurves of the two source, are completely determined, since no free parameter remains.
The previous statement does not hold for a crescent source. In the case of the crescent source there are in total five parameters: the integrated flux of the source $S_0$, the radii of the bright/dark disk $R_p$/$R_n$ and the displacement of the centers of the two disks on the axes perpendicular and parallel to the caustic $a$ and $b$.
Two of the parameters can be reduced by expressing the results in terms of $S_0$ and $r_{1/2}$. The later being determinable for any set of parameters $R_p, R_n$ and $a^2+b^2$. Moreover, one of the displacement parameters $b$ has no impact on the one-dimensional profile of the source that results from the projection of the source image on an axis perpendicular to the caustic.
Since the one-dimensional source profile that corresponds to an axis perpendicular to the caustic contains exhaustively all the information regarding the source that can be revealed by the lightcurve, the value of the parameter $b$ does not have an effect on the shape of the lightcurve.
Nevertheless, the $b$ parameter is relevant for the calculation of $r_{1/2}$. It’s qualitative effect is to decrease the value of the half-light radius when the absolute value of the parameter is increased. With two parameters constrained and another irrelevant to the shape of the lightcurve, two free parameters remain $R_n$ and $a$. Figure 4 reveals that the lightcurve of a crescent source has more visible features than the other two light-curves corresponding to the disc and Gaussian shape. The parameters $R_n$ and $a$ have strong influences on the shape of the microlensed lightcurve as can be seen in figures 5, 6 and 7. Moreover, the one-dimensional source profile corresponding to the direction perpendicular to the caustic reveal four characteristic points. The overlap of each of these points with the caustic leaves visible features on the lightcurve at the corresponding instances of time. In timely order, the instances correspond to the start of the overlap between the caustic and the bright disk, the start of the overlap between the caustic and the dark disk, the end of the overlap between the caustic and dark disk and finally the end of the overlap between the caustic and the bright disk.
With the different source profiles and their corresponding lightcurves studied we can change our point of view of the system to that of an observer. The observer would basically detect only the lightcurve of such a source. As described in section 4.3 the timing of the onset and offset of the previously described periods can be used to estimate the values of the radii and one of the displacement parameters when assuming a geometric crescent. All quantities can be estimated in terms of the relative velocity of the source in a direction perpendicular to the caustic.
Furthermore, a simulated image of M87 presented in [@2012MNRAS.421.1517D] has been microlensed (figure 9). On the resulting lightcurve the instances corresponding to the start and end of the black hole shadow and caustic overlap were distinguishable.
In the case of a high-quality lightcurve with insignificant noise and measurement errors, the parameters can be obtained by simply identifying the characteristic instances of time without making use of the actual values of the magnification map. If the effect of the errors and the noise distorts the magnification time function enough so that the characteristic epochs are not identifiable with the characteristic periods still visible, the boundaries of the periods can be roughly estimated. Furthermore, if direct estimates of the parameters cannot be obtained we propose the use of a strong statistical tool such as Markov-Chain Monte Carlo. In Section 6 we have studied the possibility of identifying a crescent source and the possibility of recovering the respective parameters. As magnification map, we have used a complex numerical one generated with the microlensing code by . In addition, we have added to the signal a Gaussian noise with an SNR of 1.6. In the experiment, we have considered all nine combinations of original source profiles and assumed fitting source models. Effectively we have fitted using MCMC all three sources with all three assumed fitting models. The results of the experiment allowed us to build a procedure for distinguishing the shape of the source assuming that one of the three models we have considered is a good approximation. The procedure would be useful for observers that endeavour to gain more information about an unresolved source for which they can study the microlensing lightcurve. As a first step in the procedure, one should first attempt to use MCMC with a Gaussian model assumption. If either the value of the $\chi^2$ or the number of rejected datapoints is large then the next step is to change the assumed source model to a uniform disc and redo the fitting. Finally, if the uniform disc assumption is rejected as well then the fitting should be done with a crescent source model assumption. If at each of the three steps the data rejects the model then the source cannot be approximated by any of the three models. Otherwise, if at one of the steps the data does not reject the model then the respective model is a good approximation.
The previously mentioned abstract parameters can be related to physical quantities specific to the central region of a quasar. As such the luminous region would correspond to the bright accretion disc that surrounds the black hole. The later’s gravity would cause a shadow in the bright region limited by the extent of the event horizon of the black hole. Therefore, the radius of the bright disk would provide an estimate of the size of the accretion disk and the radius of the dark disc would provide and estimate of the gravitationally magnified Schwarzschild radius of the black hole $R_{S}^{magnified} =
\Delta t_{dark} \cdot v_p$. By $R_{S}^{magnified}$ we refer to the apparent Schwarzschild radius which is larger than the real value at large distances due to the black hole’s own gravity. Moreover, the gravitationally magnified value of the Schwarzschild radius is a monotonic function of the black hole’s mass. Therefore, it can be used to estimate the mass of the black hole if it was not rotating. In the previous expression, the $\Delta t_{dark}$ denotes the period of overlap between the black hole shadow and fold caustic. $v_p$ denotes the component of the relative velocity of the source and fold which is perpendicular to the caustic. The respective velocity is an unknown, though it can be constrained on a case by case basis to an order of magnitude or even better. This would require the study of the dynamics of the stellar structure which contains the gravitational lens. A better estimate of the relative velocity would facilitate a better estimate of the effective non-rotating black hole mass associated with the black hole shadow.
The parameters whose values cannot be determined due to the loss of information from the directions parallel to the caustic could be obtained in the eventuality in which the same source crosses multiple caustics that are not parallel. Multiple crossing of caustics can reveal details of the one-dimensional flux profile corresponding to multiple distinct directions which would allow the reconstruction of the two-dimensional profile analogous to the process through which an image of a CT scan is obtained.
*Author contributions*\
Prasenjit Saha provided the original idea and plan for the research project as well as multiple contributions to the analysis and manuscript preparation. Mihai Tomozeiu simulated and studied the ideal behaviour of the microlensing lightcurves for the different source profiles discussed and prepared the manuscript. Joachim Wambsganss contributed to the research planning and provided the numerical code used by Manuel Rabold to create the magnification map and corresponding lightcurves used in the MCMC analysis performed by Irshad Mohammed in the last part of the presented work. Both Manuel Rabold and Irshad Mohammed had large contributions in writing the “Fitting Mock Data” section.
Acknowledgement
===============
IM is supported by Fermi Research Alliance, LLC under Contract No. De-AC02-07CH11359 with the United States Department of Energy. J.W. would like to acknowledge and thank the Pauli Center for Theoretical Studies of ETH Zurich and University of Zurich for generous support during the Schrödinger visiting professorship in 2013.
[^1]: mihai@physik.uzh.ch
|
[**Turing pattern with proportion preservation**]{}
Shuji Ishihara$^{1}$ and Kunihiko Kaneko$^{1,2}$
$^1$Department of Pure and Applied Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8902, Japan\
$^2$ ERATO Complex Systems Biology Project, JST\
Correspondence should be addressed to S. Ishihara\
E-mail : shuji@complex.c.u-tokyo.ac.jp\
Department of Pure and Applied Sciences, College of Arts and Sciences,\
The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan.\
Tel/Fax : +81-3-5454-6731\
Although Turing pattern is one of the most universal mechanisms for pattern formation, in its standard model the number of stripes changes with the system size, since the wavelength of the pattern is invariant: It fails to preserve the proportionality of the pattern, i.e., the ratio of the wavelength to the size, that is often required in biological morphogeneis. To get over this problem, we show that the Turing pattern can preserve proportionality by introducing a catalytic chemical whose concentration depends on the system size. Several plausible mechanisms for such size dependence of the concentration are discussed. Following this general discussion, two models are studied in which arising Turing patterns indeed preserve the proportionality. Relevance of the present mechanism to biological morphogenesis is discussed from the viewpoint of its generality, robustness, and evolutionary accessibility.
[**Keywords**]{} : Turing pattern; Morphogenesis; size-invariance
=3.5mm
Introduction \[sec:intro\]
==========================
Since the seminal paper by Turing (1952), a large variety of pattern formation phenomena in nature has been explained by his theory. The original motivation of Turing himself lied in the explanation of biological morphogenesis, as was succeeded to Gierer and Meinhardt (Gierer and Meinhardt, 1972), and others over a half century (Gray and Scott, 1984; Murray 1993; Pearson, 1993; Kondo and Asai, 1995; Meinhardt and Gierer, 2000). Although the Turing pattern is one of the most beautiful and ubiquitous mechanisms for morphogenesis, frequent criticism raised to it is non-adjustability of the characteristic wave length of the pattern against the system size. Because the generation of the Turing pattern comes from instability of an uniform state over a certain range of wavelengths, the possible range of the wavelengths is pre-fixed, and is invariant against the change of the system size. Hence the number of segments or stripes is proportional to the system size as shown in Fig. \[fig:Prop\] (a). In contrast, the number of segments or stripes, rather than the wavelength, is often invariant against the change of the size in many biological systems as shown in Fig. \[fig:Prop\] (b). In a biological system, the scale of pattern is often proportional to the system size, and also it is desirable to have such proportionality in many situations. However, neither the Turing pattern nor the positional information theory by Wolpert (1969) which assumes the morphogen gradient to convey positional information satisfies the scale-invariance.
For example, it is observed that the patterns in *Hydra* and *Dictyostelium discoideum* slugs have proportionality with the size. In *Dictyostelium discoideum* the ratio of two different cell types is almost fixed, independent of the size. Transgenic mice indeed preserve the body proportion despite their larger size in phenotype (Palmiter et al., 1982). In the development of *Drosophila Melanogaster*, it was reported that the expression of gap gene *hb* is robust and preserves the proportion over different sizes of individual eggs (Houchmandzadeh et al., 2002). The proportion preservation is important for robust morphogenesis in general.
In the present paper we discuss a general mechanism which enables the proportionality of wavelength with the size in Turing patterns as in Fig. \[fig:Prop\] (b).
To explain the proportion regulation of different cell types for *Dictyostelium discoideum* slug, Meinhardt (1982) proposed an activator-inhibitor model in which the ratio of two cell types is preserved. The mechanism works well for a pattern with only a single boundary between two types, but is not valid for multiple stripes pattern. Theory based on globally coupled dynamical systems also provides a regulation mechanism for the ratio of different cell types (Kaneko and Yomo, 1994, 1999; Mizuguchi and Sano, 1995; Furusawa and Kaneko, 2001), but a proportion preservation of a pattern with multiple stripes is not discussed yet.
As an extension of Turing pattern, Othmer and Pate (1980) showed that if diffusion constants depend on the concentration of auxiliary chemical factor which has size-dependence, then size-invariant pattern formation is possible, where the size-dependence of the auxiliary chemical concentration is provided by choosing a proper boundary condition (see also Pate and Othmer, 1984; Dillon et al., 1994). Hunding and Sørensen (1988) also discussed a simple mechanism to explain such concentration-dependent diffusion by an auxiliary chemical factor. In these models, it is necessary that all the diffusion coefficients are regulated in the same manner. Apart from Turing pattern, a model for the proportion regulation in *Drosophila Melanogaster* was proposed by Aegerter-Wilmsen et al. (2005), also by assuming effectively concentration-dependent diffusion. It is not yet sure if such regulation of diffusion is really adopted to control the proportionality.
In this paper we discuss mechanisms of scale-invariant Turing pattern without considering any variations of diffusion constants. Instead we seek for a possibility that concentration of some chemical changes with some power of the system size, which influences the rate of reaction for the Turing pattern so that the size-invariance is generated. We introduce a chemical component whose concentration depends on the size of the system. In Section \[sec:sdc\] we discuss several possibilities for such size-dependent concentration of a chemical. Following this general discussion, we give two specific examples leading to the Turing pattern whose wavelength is proportional to the system size. The first example introduced in Section \[sec:Model1\] adopts a size-dependent auxiliary chemical component, in the same way as Othmer and Pate (1980) or Hunding and Sørensen (1988), while we believe it is simpler than the earlier models, and is also plausible biologically because change between active and inactive forms adopted therein is ubiquitous in a biochemical process. The second example introduced in Section \[sec:Model2\] contains only two chemical components, which is the minimum number for the Turing instability. The model provides a novel mechanism for the proportionality preservation based on the conservation of some quantity. Due to the size dependence of the conserved quantity, the scale-invariance in Turing pattern is resulted. In Section \[sec:discussion\] we summarize the mechanisms for the proportion preservation and discuss possible relevance of them to biological morphogenesis.
Before discussing the mechanisms, we make one remark what we have in mind with the term “system” in the present paper. In a class of examples, the system refers to a cell, in which case the boundary of the system is a membrane, while in some other cases, the system refers to cell aggregates (or tissue). As a mathematical expression of reaction-diffusion equation the two cases are treated in the same way, and thus we discuss the two cases together by adopting the term “system”.
Size dependent concentration \[sec:sdc\]
========================================
Following Section \[sec:intro\], we seek for a mechanism in which concentration of some chemical components changes with the system size. Let us consider the case in which a single component W satisfies the size-dependence in concentration. Here we will discuss several possible mechanisms in which the concentration of W indeed changes with some power of the system size. We take a three-dimensional system with a size scale $\sim L$, and assume that the size of the system (e.g., a cell) varies keeping conformity in shape, so that the volume of the system is proportional to $L^3$, while the area of the boundary increases with $L^2$. We further assume that the diffusion of W is so rapid as $\sqrt{D_w/\gamma_w} >
L$ where $D_w$ and $\gamma_w$ are the diffusion coefficient and the degradation rate of W respectively, so that W is distributed almost homogeneously. This condition, however, is not so restrictive to realize the size-dependence, and the argument below can be generalized even by relaxing the condition.
1. Consider the case in which the total quantity of W is conserved against the change of the system size (or through the growth), as shown in Fig. \[fig:turingcase\] (A). In this case the concentration of W is proportional to $L^{-3}$ due to the conservation of the total amount and the dilution by the increase of the size. Here the case with only a single chemical component (W) is discussed, but $W$ can be a sum of multiple components if the total sum of them is conserved. In Section \[sec:Model2\], we discuss an example of such case, which results in the size-invariant Turing pattern.
2. Consider the case in which W is generated whole through the system at a rate $g$, while W escapes out of the system only through the boundary, as in Fig. \[fig:turingcase\] (B-(i)). As another possibility, consider the case in which W is decomposed by enzymes bounded on the membrane (if a system is a cell) or by specific cells that are located at the boundary of a tissue as in Fig. \[fig:turingcase\] (B-(ii)).
The case (i) is the same as that discussed by Othmer and Pate (1980). In this case, the boundary condition is represented by the following equation $$\begin{aligned}
-\vec{n}\cdot D_w \nabla w = b w \label{eq:bound}\end{aligned}$$ where $\vec{n}$ is an unit vector perpendicular to the boundary, and $b$ is the mass transfer coefficient of W at the surface of the system. In the cases (ii), degradation of W is catalyzed only at the boundary. Then $w(\vec{r},t)$ follows the equation $$\begin{aligned}
\frac{\partial w}{\partial t} = D_w \triangle w + g - \gamma w
\delta(\vec{r}-\vec{r}_s), \label{eq:wB}\end{aligned}$$ where $\vec{r}_s$ denotes the coordinate of the boundary.
In both cases, W is distributed almost homogeneously in the system if the diffusion coefficient is sufficiently large. In the steady state, concentration $w$ in the system is evaluated by the integration, where W is synthesized with the rate proportional to $L^3$ and is decomposed in proportion to $L^2$. Thus the abundances of W are proportional to $L$.
It is often the case that the generation of some chemical factors are limited at a localized region in a system. Bcd-protein in the embryo of *Drosophila* is an example, where Bcd-mRNA is fixed in the anterior of the cell. In such case, the rate of synthesis of W is independent of the system size ($L^0$), and thus the concentration of W is proportional to $L^{-2}$.
3. As is shown in Fig. \[fig:turingcase\] (C), W flows into a system from (or is synthesized by a chemical factor from) the environment of the system, and is decomposed within the system. Then the former rate is proportional to $L^2$, and the latter to $L^3$, so that the concentration of W is proportional to $L^{-1}$. This situation is typical for morphogenesis where each part of the embryo transmits and receives signals with each other. As another example, cAMP in a cell is synthesized by the membrane-bound enzyme Adenylcyclase, and thus the concentration of cAMP follows the above scaling relation.
4. Consider a chemical factor Z that is synthesized whole through the system while it is degraded on the boundary. Then the concentration of Z, $z$, is proportional to the system size ($z \propto L$). Also consider a chemical W that is synthesized whole through the system while it is degraded, catalyzed by two molecules of Z, such as $2Z + W \to 2Z +
G$, as shown in Fig. \[fig:turingcase\] (D). Then the concentration of W is proportional to $ L^{-2}$. In general, cooperative reactions as in this example can induce various $L$ dependence.
Of course, some other situations are possible in which $w$ depends on the size of a system. Next we give specific examples of reaction-diffusion equations that leads to the scale-invariant Turing pattern, based on this scaling behavior of a chemical W. In these models, chemical factors U and V regulate each other, which, we assume, are impenetrable through the boundary (membrane).
Model I : Turing model with a size regulator\[sec:Model1\]
==========================================================
Controlling proportionality
---------------------------
Here we give an example of size-invariant Turing pattern, based on the chemical $W$ with the size-dependent concentration in Section \[sec:sdc\]. Consider a reaction-diffusion system composed of three chemical components U, V, W. The concentrations of U and V at time $t$ and at position $\vec{r}$, $u(\vec{r},t)$ and $v(\vec{r},t)$, obey the following equations
$$\begin{aligned}
\frac{\partial u}{\partial t} & = & D_u \triangle u +f(u,v;w) \label{eq:Turinga}\\
\frac{\partial v}{\partial t} & = & D_v \triangle v +g(u,v;w). \label{eq:Turingb}\end{aligned}$$
W is a factor controlling the reactions, which is a size-dependent component at the same time. The reaction terms are represented by $f(u,v;w)$ and $g(u,v;w)$, and the wavelength of U-V pattern is controlled by the concentration of W ($w$). In general, the change of $w$ is accompanied with the change of the homogeneous steady state itself, and as a result the characteristic wave length at the unstable uniform state may change in a complicated manner. Here, we just give two simple classes of reaction equations that satisfy the scale-invariant pattern formation.
In the first case, all the reactions for U and V are homogeneously regulated by W, i.e. $f(u,v) \propto w^{\mu}$ and $g(u,v) \propto
w^{\mu}$. Here, by the spatial scale transformation $x \to
x/w^{\mu/2}$, the $w$-independent differential equations are obtained for a steady-state pattern. Such regulation was also assumed in earlier study by Saunders and Ho (1995), but it might not be so natural, as W regulates all the reaction process in the same manner.
As far as we know, the second case has been slipped over, in which the reactions for U and V have the functional forms $$\begin{aligned}
f(u,v;w) = F(w^{\nu}u,w^{\nu}v), ~~~~g(u,v;w) =G(w^{\nu}u,w^{\nu}v). \label{eq:exp}\end{aligned}$$ In this case a homogeneous fixed point given by the conditions $f(u,v;w)=g(u,v;w)=0$ is realized at $\hat{u} \equiv
w^{\nu}u=\hat{u}^0$ and $\hat{v} \equiv w^{\nu}v=\hat{v}^0$, where $(\hat{u}^0,\hat{v}^0)$ is the solution of $F(u,v)=G(u,v)=0$, so that the linearized partial differentiations around the fixed point are given by $f_u=w^{\nu} F_{\hat{u}}(\hat{u}^0,\hat{v}^0)$ and so forth. By the transformation of $(u,v)$ to $(\hat{u},\hat{v})$, the equations are invariant under the spatial scale transformation $x \to
x/w^{\nu/2}$,
In the above two cases, the characteristic wavelength for the unstable uniform steady state of U, V is controlled by the concentration of W. In the next subsection, by taking a simple specific reaction-diffusion model we show that this latter case arises rather naturally.
A reaction-diffusion model
--------------------------
Here we study the following reaction-diffusion model based on Brusselator (Prigogine and Lefever, 1968; Nicolis and Prigogine, 1977), in addition to the size-regulator W;
----------- ----------------------------------------------------------- --- -------------------------------------------------------------- -- --
**(I)** U is generated by A at a constant rate : A $\stackrel{k_A}{\to}$ U
**(II)** U is activated into U$^*$ by W with a reversible reaction : U $+$ W $\mathop{\rightleftharpoons}_{k^{-1}_U}^{k_U}$ U$^*$
**(III)** V is activated into V$^*$ by W with a reversible reaction : V $+$ W $\mathop{\rightleftharpoons}_{k^{-1}_V}^{k_V}$ V$^*$
**(IV)** U$^*$ changes to V$^*$ at a constant rate : U$^*$ $\stackrel{k_b}{\to}$ V$^*$
**(V)** Dimer of U$^*$ catalyzes V$^*$ into U$^*$ : 2U$^*$ + V$^*$ $\stackrel{k_a}{\to}$ 3U$^*$
**(VI)** U$^*$ is degraded at a constant rate : U$^*$ $\stackrel{k_G}{\to}$ G
----------- ----------------------------------------------------------- --- -------------------------------------------------------------- -- --
The model is illustrated in Fig. \[fig:Brusselator\]. In the model, U and V have active and inactive states, and can react only in its active state “\*”. At the same time, U and V are activated by W.
Under a proper rescaling and redefinition of the parameters, the rate equations for the system are given by
$$\begin{aligned}
\dot{u}_i &=& A-k_U\,w u_i + k_U^{-1} u_a \\
\dot{u}_a &=& k_U \,w u_i -k_U^{-1} u_a - u_a - B u_a + u_a^2 v_a\\
\dot{v}_i &=& - k_V w v_i + k_V^{-1} v_a\\
\dot{v}_a &=& k_V w v_i - k_V^{-1} v_a + B u_a - u_a^2 v_a\end{aligned}$$
where $u_i,~u_a,~v_i,~v_a$, and $w$ are concentrations of U, U$^*$, V, V$^*$, and W respectively. Let us assume that the reversible reactions between active and inactive states (II, III) are sufficiently rapid and in equilibrium. Then, the ratio of U to U$^*$ (V to V$^*$) is given by a constant $u_a = \left(k_U\,w/ k_U^{-1} \right) u_i$ ($v_a =
\left(k_Vb/ k_V^{-1} \right) v_i $), by which their terms with $u_a$ and $v_a$ are replaced. As a result, we obtain the equations of $u \equiv u_a+u_i$ and $v \equiv v_a+v_i$ as:
$$\begin{aligned}
\dot{u} & = & A -m(w)\, u + m(w)^2 n(w) \, u^2 v- B m(w)\, u \\
\dot{v} & = & -m(w)^2 n(w) \,u^2 v + B m(w)\, u
\end{aligned}$$
where $m(w) = k_U\,w/(k_U\,w + k_U^{-1}) $ and $n(w) = k_V\,w/( k_V\,w
+ k_V^{-1})$. Note that $u_a = m(w) u $ and $v_a = n(w) v$. In the case $k_U^{-1} \gg k_U $ and $k_V^{-1} \gg k_V $, where inactive states are dominant, $m(w) \sim (k_U / k_U^{-1}) w$ and $n(w) \sim
(k_V / k_V^{-1}) w$ approximately. Accordingly, the conditions of Eq. (\[eq:exp\]) are satisfied with $\nu=1$. Notice that this satisfaction of Eq. (\[eq:exp\]) is not specific to this model, but is general when the chemicals have active and inactive states and only the former participates in the reaction.
Now we consider the situation given by **(B)** in Section \[sec:sdc\], where W is synthesized at a limited domain in the system. We just consider an one-dimensional pattern, and study a model represented by the following partial differential equations;
$$\begin{aligned}
\frac{\partial u}{\partial t } &= & D_U \frac{\partial^2 u}{\partial x^2} + A -m(w)\, u + m(w)^2 n(w) \, u^2 v- B m(w)\, u \\
\frac{\partial v}{\partial t } &= & D_V \frac{\partial^2 v}{\partial x^2} - m(w)^2 n(w) \,u^2 v + B m(w)\, u \\
\frac{\partial w}{\partial t } &= & D_W \frac{\partial^2 w}{\partial x^2} + H(x) - \gamma w \label{eq:PB3}
\end{aligned}$$
\[eq:simueq\]
where $H(x)=1$ for $0<x<x_0$ and 0 otherwise. To represent the synthesis of W in some definite area of the system irrespective of the system size, the constant $x_0$ is independent of $L$ and is set at $1.0$ in the simulation. The last term $-\gamma w$ represents the escape (or decomposition) of W through the surface. We have carried out simulations on the temporal evolution of $u_n, v_n ~ (n=0 \sim
N-1)$ under Dirichlet boundary condition $U(0)=U(L)=V(0)=V(L)=0.0$. Since we consider a one-dimensional direction of a three dimensional system with the size $L$, the flow-out of the escaping chemicals at the boundary should be proportional to $L^2$, so that the $\gamma$ term in the above model equation should be scaled by $L^{2}$. Under these assumptions on $H(x)$ and $\gamma$, the discussion in **(B)** of Section \[sec:sdc\] is valid. Indeed, we have confirmed that the concentration $w$ is proportional to $L^{-2}$, in the simulation for large $D_W$. Thus we take $w=W_0 L^{-2}$ for the simulations below.
In Fig. \[fig:model1x-L\], we plot the wavelength $\xi$ corresponding to the wavenumber that leads to the largest eigenvalue in the linear stability analysis, for a given system size $L$. $\xi$ increases in proportion to $L$, and thus the generated pattern from this instability is expected to preserve the proportion.
The results of the simulation are shown in Fig. \[fig:SimulationTuring\], which clearly show that the number of stripes does not change against the change of system size $L$ as long as it is sufficiently large. For small size, because $w$ is large, the saturation in the terms in $m(w)$ (or $n(w)$) is not negligible, so that the number of stripes is decreased.
Simplicity of the mechanism \[sec:CondTuring\]
----------------------------------------------
The above example gives us a simple but plausible model for the regulation of the wave length, which leads to the proportion preservation for a pattern generated by Turing instability. As shown in Fig.\[fig:TuringMech\], conditions for this proportion preservation are summarized as follows:
1. Each chemical component has active and inactive states. The activation is reversible and is catalyzed by a chemical factor W (whose concentration changes with the size). Only chemicals in the active state can participate in reactions.
2. The reaction-diffusion system shows Turing instability.
3. W is generated at a localized region in the system (cell), diffuses rapidly, and goes out of, or is degraded on, the surface (membrane).
These are the only conditions for the proportion preservation of a Turing pattern, which works regardless of the specific choice of a model. The condition **** can be replaced by some other conditions in which the concentration of the factor is scaled as $w \propto L^{-2}$. Notice that the above conditions are independent of each other so that they would be easily satisfied by combining each process that satisfies each condition. Thus, a size-invariant Turing pattern by the above conditions may be achieved easily through the evolution.
Model II : Model with a conserved quantity\[sec:Model2\]
========================================================
Size invariant Turing instability by the conserved quantity
-----------------------------------------------------------
Here we give another model with two chemical components, which are regarded as two states of a single chemical species. At the same time both U and V are not synthesized or decomposed, so that the total quantity of the chemical components is conserved. According to the mechanism **(A)** discussed in Section \[sec:sdc\], we seek for the possibility of the proportion preservation in this model. For simplicity, we assume one-dimensional system in this section.
Consider two components U and V that regulate the concentration of each other through a reaction, as shown in Fig. \[Fig:Model2\]. Then the reaction-diffusion equations are represented by
$$\begin{aligned}
\frac{\partial u}{\partial t} &=& D_u \frac{\partial^2 u}{\partial x^2} + F(u,v) \\
\frac{\partial v}{\partial t} &=& D_v \frac{\partial^2 v}{\partial x^2} - F(u,v)
\end{aligned}$$
\[eqn:rd\]
Hence the total quantity of U and V is conserved; $$\begin{aligned}
S \equiv \int dx \left( u+v \right) = \mbox{constant}.\end{aligned}$$ As discussed in the case **(A)** in Section \[sec:sdc\], the increase in the system size leads to the dilution of the concentration of $S$.
In the steady homogeneous state, the Jacobian matrix for the reaction terms is given by $$\begin{aligned}
J=\left(
\begin{array}{cc}
F_u & F_v \\
-F_u & -F_v
\end{array}
\right)\end{aligned}$$ where $F_u$ denotes the partial derivative of $F$ by $u$ at a homogeneous steady state of Eq. (\[eqn:rd\]), and so forth. Through the stability analysis of the Fourier transform of the linearized equation around the homogeneous state by using $J$, the wave number that has the largest eigenvalue is obtained, which gives the wavenumber that grows most rapidly from this unstable homogeneous state. This wavenumber is given by $$\begin{aligned}
k_m^2 = \frac{-\hat{D}(F_u+F_v)+ (1+\hat{D}) \sqrt{F_u F_v \hat{D}}}{\hat{D} (\hat{D}-1)}
\label{eq:Awacond}\end{aligned}$$ with $\hat{D}\equiv D_v/D_u$, where $\hat{D}$ is larger than $1$ for Turing instability to occur (Turing, 1952). To preserve the proportional pattern by increasing the length of the system $L$, it is necessary that both $F_u$ and $F_v$ behave so as to $k_m$ scales as $L^{-1}$ in Eq. (\[eq:Awacond\]), at least approximately. Below, we give an explicit example corresponding to this case.
An explicit reaction-diffusion model with a conserved quantity
--------------------------------------------------------------
Consider the following reaction diffusion system corresponding to the reactions shown in Fig. \[Fig:Model2\].
$$\begin{aligned}
\frac{\partial u}{\partial t} &=& D_u \frac{\partial^2 u}{\partial x^2} + u^3v - Bu^2 \\
\frac{\partial v}{\partial t} &=& D_v \frac{\partial^2 v}{\partial x^2} - u^3v + Bu^2 \end{aligned}$$
\[eq:simueq2\]
This system is a modified version of the Brusselator, so that supplies and degradations of substances are excluded (Awazu and Kaneko, 2004). Although the reaction term $u^3v$ is higher than the original Brusselator, indeed, the reaction with a lower order cannot satisfy the requirement of the last subsection for the proportion preservation. As far as we have examined, this choice is one of the simplest to satisfy the requirement (See Appendix A). Also, in a biological system, such high order catalysis is not so uncommon. Hence we adopt this reaction model.
In this system, the corresponding homogeneous fixed point $(u_0,v_0)$ is given by $u_0v_0=B$ and $u_0+v_0=S/L$. Note that $u_0$ is almost proportional to $S$ as long as $S$ is sufficiently large. Jacobian around the uniform steady state is given by $$\begin{aligned}
J=\left( \begin{array}{cc} Bu_0& {u_0}^3 \\ -Bu_0& -{u_0}^3
\end{array}
\right)\end{aligned}$$ If $\hat{D}$ is large enough, the dominant term in Eq. (\[eq:Awacond\]) is the one containing $\sqrt{F_u F_v}$, because other terms are lower order with regards to $\hat{D}$. Thus, $k_m^2
\sim \sqrt{{u_0}^4} \sim S^2$ holds in the model, which results in the proportion preservation.
We plot the characteristic wavelength $\xi=2\pi/k_m$, corresponding to the most unstable mode given by the linear analysis of $J$, for the system size $L$ in Fig. \[fig:model2anal\]. $\xi$ increases in proportion to $L$ over a wide range of $L$ for sufficiently large $S$, thus leading to a size-invariant pattern. We have carried out numerical simulation of Eq. (\[eq:simueq2\]). The results are shown in Fig. \[fig:SimulationTuring2\], which support the above estimation to realize the size-invariant Turing pattern formation.
The model we give here is one of the simplest, in the sense that it contains the lowest order polynomial reaction term among such equations, as is also explained in the Appendix A, where more detailed estimations as well as some other equations leading to the size-invariant pattern formation.
Summary and Discussion \[sec:discussion\]
=========================================
In this paper, we have discussed a possible mechanism of proportion regulation based on the control of the reaction rate in reaction-diffusion systems. We have introduced a morphogen W which itself does not convey positional information (Wolpert, 1969), but works as a carrier of information on the size of the system. It is important to recognize that the proportion preservation is possible by such simple mechanism. We have discussed several possible schemes that can naturally realize the proposed mechanism in a biochemical system.
In some earlier studies and in our model, it is assumed that there are chemical factors whose concentration depends on the system size. As discussed by Hunding and Sørensen (1988), a candidate of such chemical component is cAMP, which is synthesized by the membrane-bound enzyme Adenylcyclase. Indeed, cAMP is involved in a number of important biological processes.
Some proteins can also fit as the size regulator W. A candidate is the product of the gene *staufen* (*stau*) working in the early development of *Drosophila*. Houchmandzadeh et al. (2002) reported that in the *Drosophila* embryo, the domain boundary of zygotic gene *hunchback* (*hb*) expression is tuned precisely at a half of the embryo, despite individually fluctuating embryo size and expression of its direct regulator Bicoid. It is discussed that maternal gene *stau* may play a major role to control such positioning of *hb* expression, as mutants lacking *stau* lose precise expression boundary of *hb* at the half of the embryo. Although a model based on the effective change of diffusion constant was proposed by Aegerter-Wilmsen et al. (2005) recently, it may be interesting to seek for the possibility that *stau* may work as a size-regulator W with the mechanism (A) or (B) in Section \[sec:sdc\]. In general, it will be interesting to search for some molecules that work as size regulators, or carry out a knock-out experiment on a candidate molecule of such regulator.
In Section \[sec:Model1\], we have introduced a simple model in which the Turing pattern is size-invariant. The proposed scheme for the proportion preservation there is rather general and robust, and at the same time is naturally realized in a biological system. We give three conditions for the scheme, which are rather simple and plausible to be achieved in biological morphogenesis. Additionally these conditions are independent of each other, which is a good feature from an evolutionary viewpoint, because they can be established one by one through evolution, without any influence with each other. Consider two neighbor species with similar proportional organization, but with different sizes. Most of the genes are common between the two, and they may share the same diffusion coefficients for most of their products. Under these conditions, ordinary Turing pattern or other mechanisms cannot explain conformity in their morphology for two species with different sizes. On the other hand, in the mechanism we proposed in Sec. \[sec:Model1\], control of solely a single chemical W can lead to proper adaptive patterning. This is one of evolutionary advantages of the present mechanism.
Independence of each condition is also a good feature for an experimetal reailzation of the present mechansim. One can realize the present proportion-prserving Turing pattern based on the established experimental examples (Castets et al., 1990; Ouyang and Swinney 1990). Because ordinary Turing pattern has just one definite wavenumber, it is interesting experimetally to construct a pattern whose intrinsic scale is changed flexibly according to environmental conditions, using the mechanism discussed in this paper.
In Section \[sec:Model2\], we have given an example for the size-invariant Turing pattern realized by chemical reactions with a conserved quantity as proposed in (**(A)** in Section \[sec:sdc\]). Existence of such conserved quantity in morphogenesis may be a rather natural assumption. However, in contrast to the models (conditions) given in Section \[sec:Model1\], the model here may lack generality, since the condition for the scale-invariance, i.e., the combination of exponents, may be rather specific. Also, accurate control for the initial value of the conserved quantity $S$ (the total quantity of U and V) may be required. In this sense, search for the mechanism in Section \[sec:Model1\] may be more important in a biological context.
As another possible explanation for the size-invarint pattern, one could assume that Turing mechanism works only in certain period of early development, leading to a pattern with differentiated cell types, and then the cells grow at the same rate, keeping the proportionality. In general, this mechanism has a low tolerance for the individual fluctuation of body size at the earlier stage of development (Houchmandzadeh et al., 2002). Also the growth process keeping the proportionality is required, and the mechanism is vulnerable by disturbance through the development. On the other hand, in our mechanism the proportionality is tolerant against size fluctuations, and is recoverable again such disturbances or external manipulation.
At last we give a speculation on a relationship between the size regulation and pattern formation. In general the size of an organ is much flexibly regulated by the mutual compensation between cell size and cell number (Frabkenhauser, 1945; Potter and Xu, 2001). However, the mechanism of size control in development is not so clear yet. Potter and Xu (2001) discuss the relationship between size regulation and pattern formation, in which mutations in genes regulating pattern result in the changes in total tissue mass. If the size regulator W discussed in this paper also concerns with size control through its concentration, the pattern formation process is tightly coupled with the organ size. It may be interesting to seek for this possibility, since the present mechanism then allows for adaptive control of the pattern scale as well as the organ size.
In conclusion, we have shown that morphogenesis with proportion preservation is possible under Turing instability, by simply utilizing a catalytic molecule whose concentration is properly scaled with the system size.
Appendix A : On the Equations (\[eq:simueq2\]) {#appendix-a-on-the-equations-eqsimueq2 .unnumbered}
==============================================
Here we explain why we choose Eq. (\[eq:simueq2\]) as a model to satisfy $k_m \sim L^{-1}$. First consider the reaction-diffusion equation in a polynomial form
$$\begin{aligned}
\frac{\partial u}{\partial t} &=& D_u \frac{\partial^2 u}{\partial x^2} + u^mv^n - Bu^l \\
\frac{\partial v}{\partial t} &=& D_v \frac{\partial^2 v}{\partial x^2} - u^mv^n + Bu^l
\end{aligned}$$
Then the steady uniform solution $(u_0,v_0)$ is given by
$$\begin{aligned}
u_0^{m-l}v_0^n&=&B \label{eq:uvst1}\\
u_0+v_0&=&S/L \label{eq:uvst2}\end{aligned}$$
\[eq:uvsteady\]
and the Jacobian at this solution is given by $$\begin{aligned}
J=\left(
\begin{array}{cc}
(m-l) B u_0^{l-1} & n Bu_0^l v_0^{-1}\\
-(m-l) B u_0^{l-1} & -n Bu_0^l v_0^{-1}
\end{array}
\right) \end{aligned}$$ The steady state solution $(u_0,v_0)$ is represented as crossing points of Eq. (\[eq:uvst1\]) and (\[eq:uvst2\]). When $m>l$, the relationship is represented as in Fig. \[fig:uvApend1\] on the $u$-$v$ plane. Then the crossing point satisfies $u_0 \propto S/L$ approximately for large $S$. On the other hand, if $m<l$, $v_0$ is an increasing function of $u_0$ in the relationship Eq. (\[eq:uvst1\]). Then the crossing point does not satisfy $u_0
\propto S/L$. Hence we assume $m>l$ here. Recall that if $D_v/D_u$ is sufficiently large, the leading term determining the characteristic scale length $\xi$ is given by $\sqrt{-F_vG_u}$ (Eq. (\[eq:Awacond\])), $$\begin{aligned}
\xi^{-2} = k_m^{2} \sim \left( u_0^{2l-1} v_0^{-1} \right)^{1/2} \sim
u_0^{\frac{2l-1}{2}+\frac{m-l}{2n}} \sim
L^{-\frac{2l-1}{2}-\frac{m-l}{2n}}\end{aligned}$$ (note $v_0 \sim u_0^{(l-m)/n}$ from Eq. (\[eq:uvst1\])). Since the exponent has to be -2 to sustain the proportionality, we get the condition $$\begin{aligned}
n=\frac{m-l}{5-2l}\end{aligned}$$ Suppose that $n,~m,~l$ are positive integers. Then because $m-l>0$, $l$ can be only $1$ or $2$. Now by choosing $m=3, n=1, l=2$ Eq. (\[eq:simueq2\]) (Model II) is derived, which is the system with the lowest degree of exponent. Another choice will be $m=4, n=1, l=1$ which leads to the following equation:
$$\begin{aligned}
\frac{\partial u}{\partial t} &=& D_u \frac{\partial^2 u}{\partial x^2} + u^4v - Bu \\
\frac{\partial v}{\partial t} &=& D_v \frac{\partial^2 v}{\partial x^2} - u^4v + Bu
\end{aligned}$$
\[eq:ModelIIb\]
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors are grateful to A. Awazu, K. Fujimoto, T. Shibata, and H. Takagi for discussions.
References {#references .unnumbered}
==========
Aegerter-Wilmsen, T., Aegerter, C. M., Bisseling, T., 2005. Model for the robust establishment of precise proportions in the early Drosophila embryo. J. Theor. Biol. 234, 13-19
Awazu, A., Kaneko, K., 2004. Is relaxation to equilibrium hindered by transient dissipative structures in closed systems? Phys. Rev. Lett. 92, 258302
Castets, V., Dulos, E., Boissonade, J., and De Kepper, P., 1990. Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern. Phys. Rev. Lett. 64, 2953-2965
Dillon, R., Maini, P. K., Othmer, H. G., 1994. Pattern formation in generalized Turing systems I. Steady-state patterns in systems with boundary conditions. J. Math. Biol. 32, 345-393.
Frankenhauser, G., 1945. The effects of changes in chromosome number on amphibian development. Q. Rev. Biol. 20, 20-78
Furusawa, C., Kaneko, K., 2001. Theory of Robustness of Irreversible Differentiation in a Stem Cell System: Chaos Hypothesis. J. Theor. Biol. 209, 395-416
Gierer, A., Meinhert, H., 1972. A theory of biological pattern formation. Kybernet 12, 30-39
Gray, P., Scott, S. K., 1984. Autocatalytic reactions in the isothermal continuous stirred tank reactor: oscillations and instabilities in the system a + 2b $\to$ 3b; b $\to$ c. Chem. Eng. Sci. 39,1087-1097.
Houchmandzadeh, B., Wieschais, E., Leibler, S., 2002. Establishment of developmental precision and proportions in the early Drosophila embryo. Nature 415, 798-802
Hunding, A., Sørensen, P. G., 1988. Size adaptation of Turing prepatterns. J. Math. Biol. 26, 27-39
Kaneko, K., Yomo, Y., 1994. Cell Division, Differentiation, and Dynamic Clustering. Physica D 75, 89-102
Kaneko, K., Yomo, T., 1999. Isologous Diversification for Robust Development of Cell Society. J. Theor. Biol. 199, 243-256
Kondo, S., Asai, R., 1995. A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus. Nature 376, 765-768.
Meinhardt, H., 1982. Models of Biological Pattern Formation. Academic Press New York
Meinhardt, H., Gierer, A., 2000. Pattern formation by local self-activation and lateral inhibition BioEssays 22, 753-760.
Mizuguchi, T., Sano, M., 1995. Proportion regulation of Biological Cells in Globally Coupled Nonlinear Systems. Phys. Rev. Lett. 75, 966-969.
Murray, J. D., 1993. Mathematical Biology (2nd ed.) Spriger-Verlag
Nicolis, G., Prigogine, I., 1977. Self Organization in Non-Equilibrium Systems. J. Wiley and Sons, New York
Othmer, H. G., Pate, E., 1980. Scale-invariance in reaction-diffusion models of spatial pattern formation. Proc. Natl. Acad. Sci. 77, 4180-4184
Ouyang, Q. and Swinney, H. L., 1991 Transition from a uniform state to hexagonal and striped Turing patterns. Nature 352, 610-611
Potter, C. J. and Xu, T., 2001. Mechanism of size control. Curr. Opin. Gen. Dev. 11. 279-286
Palmiter, R.D., Brinster, R. L., Hammer, R.E., Trumbauer, M.E., Rosenfeld, M.G., Birnberg, N.C., Evans. R.M., 1982. Dramatic growth of mice that develop from eggs microinjected with metallothionein-growth hormone fusion genes. Nature 300,611-615.
Pate, E., Othmer, H. G., 1984. Application of a model for scale-invariant pattern formation on developing systems. Differentiation 28, 1-8.
Pearson, J. E., 1993. Complex Patterns in a Simple System. Science 261, 189-192
Prigogine, I., Lefever, R., 1968. Symmetry Breaking Instabilities in Dissipative Systems II. J. Chem. Phys. 48, 1695-1700.
Saunders, P. T., Ho, M. W., 1995. Reliable segmentation by successive bifurcation. Bull. Math. Biol. 57, 539-556.
Turing, A. M., 1952. The chemical basis of Morphogenesis. Philos. Trans. Roy. Soc. Lond. B. 237, 37-72.
Wolpert, L., 1969. Positional information and the spatial pattern of cellar differentiation. J. Theor. Biol. 25, 1-47.
Figure \[fig:Prop\] : Pattern formation (a) with a fixed wave length and (b) with a fixed proportion. Ordinary Turing pattern belongs to the class (a), as the pattern arises by instability of a certain range of wave lengths.
Figure \[fig:turingcase\] : Several cases for the scaling behavior of the concentration of W against the system size, as discussed in the text.
Figure \[fig:Brusselator\] : Two-state Brusselator model. Only chemical components in the active state can react in the model.
Figure \[fig:model1x-L\] : Characteristic length $\xi$ of the pattern in the two-state Brusselator model (Eq. (\[eq:simueq\])) plotted against the system size $L$. The parameters are $k_U=0.5,
k_U^{-1}=10.0, k_V=0.1, k_V^{-1}=2.0, A=2.0, B=4.0, W_0=5.0 \times
10^4$. We plot for $\hat{D} \equiv D_v/D_u = 4.0, 6.0 8.0,
10.0$. Normalized wavelengths by the system size ($\xi/L$) are plotted in the inset.
Figure \[fig:SimulationTuring\] : Pattern of the two-state Brusselator model in Eq. (\[eq:simueq\]) obtained numerically, for the system size $L=$ 64, 128, 256, 512, and 1024. The parameters are $k_U=0.5, k_U^{-1}=10.0, k_V=0.1, k_V^{-1}=2.0, A=2.0, B=4.0, D_U=0.5,
D_V=3.0, W_0=5.0 \times 10^4$. The number of stripes are invariant over a wide range of system size, while for too small system size, the proportion is no longer sustained due to the nonlinearity (saturation) in $m(w)$ and $n(w)$. Simulations are carried out with the grid size 1.
Figure \[fig:TuringMech\] : A scenario for proportion preservation of a Turing pattern.
Figure \[Fig:Model2\] : A model with two states of morphogen U and V whose sum is conserved, where reaction between them to regulate the concentration brings about Turing instability.
Figure \[fig:model2anal\] : Characteristic length $\xi$ of the pattern by Eq. \[eq:simueq2\] plotted against the system size $L$, for various values of the total quantity $S\,(= \int
u(x)+v(x)dx)$. For large $S$, $\xi$ increase in proportion to the system size $L$. $B=0.5, \hat{D}=5.0\times 10^2$. Normalized wavelengths by the system size ($\xi/L$) are plotted in the inset.
Figure \[fig:SimulationTuring2\] : Simulation results of the modified Brusselator model with a conserved quantity in Eq. (\[eq:simueq2\]) for various size $L=$64, 96, 128, 160, 192, 224 and 256. The number of stripes are $6$ and invariant for $L \ge 128$. The parameters are $D_u=1.0 \times 10^2 , D_v=1.0 \times 10^4, B=0.5$, $S=2560.0$.
Figure \[fig:uvApend1\] : Steady state solution plotted in $u$-$v$ plane. The crossing points of the line and the curve give homogeneous steady solutions of reaction equations Eq. (\[eq:uvsteady\]).
![Ishihara and Kaneko[]{data-label="fig:Prop"}](Fig1.eps){width=".8\textwidth"}
![Ishihara and Kaneko[]{data-label="fig:turingcase"}](Fig2.eps){width="1.\textwidth"}
![Ishihara and Kaneko[]{data-label="fig:Brusselator"}](Fig3.eps){width="1.\textwidth"}
![Ishihara and Kaneko[]{data-label="fig:model1x-L"}](Fig4.eps){width="1.\textwidth"}
![Ishihara and Kaneko[]{data-label="fig:SimulationTuring"}](Fig5.eps){width="1.\textwidth"}
![Ishihara and Kaneko[]{data-label="fig:TuringMech"}](Fig6.eps){width="1.\textwidth"}
![Ishihara and Kaneko[]{data-label="Fig:Model2"}](Fig7.eps){width=".6\textwidth"}
![Ishihara and Kaneko[]{data-label="fig:model2anal"}](Fig8.eps){width=".9\textwidth"}
![Ishihara and Kaneko[]{data-label="fig:SimulationTuring2"}](Fig9.eps){width="1.\textwidth"}
![Ishihara and Kaneko[]{data-label="fig:uvApend1"}](Fig10.eps){width=".6\textwidth"}
|
---
abstract: 'We prove that the fully asynchronous dynamics of a Boolean network $f:\{0,1\}^n\to\{0,1\}^n$ without negative loop can be simulated, in a very specific way, by a monotone Boolean network with $2n$ components. We then use this result to prove that, for every even $n$, there exists a monotone Boolean network $f:\{0,1\}^n\to\{0,1\}^n$, an initial configuration $x$ and a fixed point $y$ of $f$ such that: (i) $y$ can be reached from $x$ with a fully asynchronous updating strategy, and (ii) all such strategies contains at least $2^{\frac{n}{2}}$ updates. This contrasts with the following known property: if $f:\{0,1\}^n\to\{0,1\}^n$ is monotone, then, for every initial configuration $x$, there exists a fixed point $y$ such that $y$ can be reached from $x$ with a fully asynchronous strategy that contains at most $n$ updates.'
author:
- Tarek Melliti
- Damien Regnault
- 'Adrien Richard[^1]'
- Sylvain Sené
bibliography:
- 'bib.bib'
title: Asynchronous simulation of Boolean networks by monotone Boolean networks
---
Boolean networks, monotone networks, asynchronous updates.
Introduction
============
A [*Boolean network*]{} with $n$ components is a discrete dynamical system usually defined by a global transition function $$f:\B^n\to\B^n,\qquad x=(x_1,\dots,x_n)\mapsto f(x)=(f_1(x),\dots,f_n(x)).$$ Boolean networks have many applications. In particular, since the seminal papers of McCulloch and Pitts [@MP43], Hopfield [@H82], Kauffman [@K69; @K93] and Thomas [@T73; @TA90], they are omnipresent in the modeling of neural and gene networks (see [@B08; @N15] for reviews). They are also essential tools in Information Theory, for the network coding problem [@ANLY00; @GRF14].
The structure of a Boolean network $f$ is usually represented via its [**interaction graph**]{}, which is the signed digraph $G(f)$ defined as follows: the vertex set is $[n]:=\{1,\dots,n\}$ and, for all $i,j\in [n]$, there exists a positive (resp. negative) arc from $j$ to $i$ is there exists $x\in\B^n$ such that $$f_i(x_1,\dots,x_{j-1},1,x_{j+1},\dots,x_n)-f_i(x_1,\dots,x_{j-1},0,x_{j+1},\dots,x_n)$$ is positive (resp. negative). Note that $G(f)$ may have both a positive and a negative arc from one vertex to another. Note also that $G(f)$ may have [*loops*]{}, that is, arcs from a vertex to itself. The sign of a cycle of $G(f)$ is, as usual, the product of the signs of its arcs (cycles are always directed and without “repeated” vertices).
From a dynamical point of view, there are several ways to derive a dynamics from $f$, depending on the chosen [*updating strategy*]{}. With the so-called [*synchronous*]{} or [*parallel strategy*]{}, each component is updated at each step: if $x^t$ is the configuration of the system at time $t$, then $f(x^t)$ is the configuration of the system at time $t+1$. Hence, the dynamics is just given by the successive iterations of $f$. On the opposite way, with the so-called [*(fully) asynchronous strategy*]{}, exactly one component is updated at each time. This strategy is very often used in practice, in particular in the context of gene networks [@TA90]. More formally, given an infinite sequence $i_0i_1i_2\dots$ of indices taken in $[n]$, the dynamics of $f$ resulting from an initial configuration $x^0$ and the asynchronous strategy $i_0i_1i_2\dots$ is given by the following recurrence: for all $t\in{\mathbb{N}}$ and $i\in [n]$, $x^{t+1}_i=f_i(x^t)$ if $i=i_t$ and $x^{t+1}_i=x^t$ otherwise.
All the possible asynchronous dynamics can be represented in a compact way by the so-called [**asynchronous graph $\Gamma(f)$**]{}, defined as follows: the vertex set is $\B^n$ and, for all $x,y\in\B^n$, there is an arc from $x$ to $y$, called [*transition*]{}, if there exists $i\in [n]$ such that $f_i(x)=y_i\neq x_i$ and $y_j=x_j$ for all $j\neq i$. Note that $f$ and $\Gamma(f)$ share the same information. The [*distance*]{} between two configurations $x$ and $y$ in $\Gamma(f)$, denoted $d_{\Gamma(f)}(x,y)$, is the minimal length of a path of $\Gamma(f)$ from $x$ to $y$, with the convention that the distance is $\infty$ if no such paths exist. Note that $d_{\Gamma(f)}(x,y)$ is at least the Hamming distance $d_H(x,y)$ between $x$ and $y$. A path from $x$ to $y$ in $\Gamma(f)$ is then called a [**geodesic**]{} if its length is exactly $d_H(x,y)$. In other words, a geodesic is a path along which each component is updated at most one. The [**diameter**]{} of $\Gamma(f)$ is $${\mathrm{diam}}(\Gamma(f)):=\max\{d_{\Gamma(f)}(x,y):x,y\in\B^n,d_{\Gamma(f)}(x,y)<\infty\}.$$
In many contexts, as in molecular biology, the first reliable information are represented under the form of an interaction graph, while the actual dynamics are very difficult to observe [@N15; @TK01]. A natural question is then the following: [*What can be said about $\Gamma(f)$ according to $G(f)$ only?*]{}
Robert proved the following partial answer [@R86; @R95].
\[thm:robert\] If $G(f)$ is acyclic then $f$ has a unique fixed point $y$. Furthermore, $\Gamma(f)$ is acyclic and, for every configuration $x$, $\Gamma(f)$ has a geodesic from $x$ to $y$.
In other words, $d_{\Gamma(f)}(x,y)=d_H(x,y)$ for every $x\in\B^n$. However, the acyclicity of $G(f)$ is not sufficient for $\Gamma(f)$ to have a short diameter. Indeed, in a rather different setting, Domshlak [@D02] proved (a slightly stronger version of) the following result.
\[thm:diam\_acyclic\] For every $n\geq 8$ there exists $f:\B^n\to\B^n$ such that $G(f)$ is acyclic and ${\mathrm{diam}}(\Gamma(f))\geq 1.5^{\frac{n}{2}}$.
Now, what can be said if $G(f)$ contains cycles ? Thomas highlighted the fact that the distinction between positive and negative cycles is highly relevant (see [@TA90; @TK01] for instance). The subtlety and versatility of the influences of interactions between positive and negative cycles lead researchers to first focus on networks with only positive cycles or only negative cycles. In particular, the following basic properties was proved in [@A08; @RRT08; @R10]: [*If $G(f)$ has no positive (resp. negative) cycles, then $f$ has at most (resp. at least) one fixed point.*]{} This gives a nice proof by dichotomy of the first assertion in Theorem \[thm:robert\].
In [@MRRS13], the authors showed that the absence of negative cycles essentially corresponds to the study of [**monotone networks**]{}, that is, Boolean networks $f:\B^n\to\B^n$ such that $$x\leq y~\Rightarrow~f(x)\leq f(y)$$ where $\leq$ is the usual partial order ($x\leq y$ if and only if $x_i\leq y_i$ for all $i\in[n]$). More precisely, they proved the following: [*If $G(f)$ is strongly connected and without negative cycles, then there exists a monotone network $f':\B^n\to\B^n$ such that: $G(f)$ and $G(f')$ have the same underlying unsigned digraph, and $\Gamma(f)$ and $\Gamma(f')$ are isomorphic*]{}. Furthermore, they proved the following reachability result, that shares some similarities with Theorem \[thm:robert\].
If $f:\B^n\to\B^n$ is monotone, then, for every configuration $x$, $\Gamma(f)$ has a geodesic from $x$ to a fixed point $y$ of $f$.
Here, we prove the following theorem, that shows that there may exist, under the same hypothesis, a configuration $x$ and a fixed point $y$ such that $y$ is reachable from $x$ with paths of exponential length only. This result contrasts with the previous one, and may be seen as an adaptation of Theorem \[thm:diam\_acyclic\] for monotone networks.
\[thm:diam\_monotone\] For every even $n$, there exists a monotone network $f:\B^n\to\B^n$, two configurations $x$ and $y$ such that $y$ is a fixed point of $f$ and $${\mathrm{diam}}(\Gamma(f))\geq d_{\Gamma(f)}(x,y)\geq 2^{\frac{n}{2}}.$$
The proof is by construction, and the idea for the construction is rather simple. Let $A$, $B$ and $C$ be the sets of configurations that contains $n/2-1$, $n/2$ and $n/2+1$ ones. Clearly, $A$, $B$ and $C$ are antichains of exponential size, and, in these antichains, obviously, the monotonicity of $f$ doesn’t apply. This leaves enough freedom to defined $f$ on $A\cup B\cup C$ in such a way that subgraph $\Gamma(f)$ induced by $A\cup B\cup C$ contains a configuration $x$ and fixed point $y$ reachable from $x$ with paths of exponential length only. To obtain a network as in the theorem, it is then sufficient to extend $f$ on the whole space $\B^n$ by keeping $f$ monotone and without creating shortcuts from $x$ to $y$ in the asynchronous graph. This idea, that consists in using large antichains to construct special monotone functions, is also present in [@GRR15] and [@ADG04c] for instance.
Let $f:\B^n\to\B^n$ be any Boolean network such that $G(f)$ has no negative loops. With the technic described above, we can go further and prove that $\Gamma(f)$ can be embedded in the asynchronous graph $\Gamma(f')$ of a monotone network $f':\B^{2n}\to\B^{2n}$ in such a way that fixed points and distances between configurations are preserved. The formal statement follows. If $x,y\in\B^n$, then the concatenation $(x,y)$ is seen as a configuration of $\B^{2n}$ and, conversely, each configuration in $\B^{2n}$ is seen as the concatenation of two configurations in $\B^n$. As usual, we denote by ${\overline{x}}$ the configuration obtained from $x$ by switching every component.
\[thm:embedding\] Let $f:\B^n\to\B^n$. If $G(f)$ has no negative loops, then there exists a monotone network $f':\B^{2n}\to\B^{2n}$ such that the following two properties holds. First, $x$ is a fixed point of $f$ if and only if $(x,{\overline{x}})$ is a fixed point of $f'$. Second, for all $x,y\in\B^n$, $\Gamma(f)$ has a path from $x$ to $y$ of length $\ell$ if and only if $\Gamma(f')$ has a path from $(x,\bar x)$ to $(y,\bar y)$ of length $2\ell$.
Theorem \[thm:diam\_monotone\] is now an easy corollary of Theorem \[thm:embedding\].
Let $r=2^n$, and let $x^1,x^2,\dots,x^r$ be any enumeration of the elements of $\B^n$ such that $d_H(x^k,x^{k+1})=1$ for all $1\leq k<r$ (take the Gray code for instance). Let $f:\B^n\to\B^n$ be defined by $f(x^k)=x^{k+1}$ for all $1\leq k<r$ and $f(x^r)=x^r$. Let $x=x^0$ and $y=x^r$. Then $y$ is the unique fixed point of $f$. Furthermore, since the set of transitions of $\Gamma(f)$ is $\{x^k\to x^{k+1}:1\leq k<r\}$, we deduce that $d_{\Gamma(f)}(x,y)=2^n-1$. We also deduce that $G(f)$ has no negative loops (this is an easy exercise to prove that $G(f)$ has a negative loop if and only if $\Gamma(f)$ has a cycle of length two). Hence, by Theorem \[thm:embedding\], there exists a monotone network $f':\B^{2n}\to\B^{2n}$ such that $(y,{\overline{y}})$ is a fixed point and $$d_{\Gamma(f')}((x,{\overline{x}}),(y,{\overline{y}}))=2d_{\Gamma(f)}(x,y)=2^{n+1}-2\geq 2^n.$$
The paper is organized as follows. The proof of Theorem \[thm:embedding\] is given in Section \[sec:proof\]. A conclusion and some open questions are then given in Section \[sec:conclusion\].
Proof of Theorem \[thm:embedding\] {#sec:proof}
==================================
We first fix some notations: $$\begin{aligned}
{\overline{x}}^i&:=(x_1,\dots,{\overline{x_i}},\dots,x_n)&(\textrm{$x\in\B^n$ and $i\in [n]$}),\\
w(x)&:=|\{i\in[n]:x_i=1\}|&(\textrm{$x\in\B^n$}),\\
w(x,y)&:=w(x)+w(y)&(\textrm{$x,y\in\B^n$}),\\
\Omega&:=\{(x,{\overline{x}}):x\in\B^n\}. \end{aligned}$$
The function $f'$ in Theorem \[thm:embedding\] is defined as follows from $f$.
Given $f:\B^n\to\B^n$, we define $f':\B^{2n}\to\B^{2n}$ by: for all $i\in [n]$ and $x,y\in\B^n$, $$f'_i(x,y)=
\left\{
\begin{array}{ll}
f_i(x)&\text{ if $y={\overline{x}}$ or ${\overline{y}}^i={\overline{x}}$}\\[2mm]
{\overline{x_i}}&\text{ if $w(x,y)=n$ and $y\neq {\overline{x}}$}\\[2mm]
1&\text{ if $w(x,y)=n+1$ and ${\overline{y}}^i\neq{\overline{x}}$}\\[2mm]
0&\text{ if $w(x,y)=n-1$ and ${\overline{y}}^i\neq{\overline{x}}$}\\[2mm]
1&\text{ if $w(x,y)\geq n+2$}\\[2mm]
0&\text{ if $w(x,y)\leq n-2$}
\end{array}
\right.
\quad\text{and}\quad
f'_{n+i}(x,y)={\overline{f'_i({\overline{y}},{\overline{x}})}}.$$
$f'_i(x,y)={\overline{f'_{n+i}({\overline{y}},{\overline{x}})}}$.
Let $A$, $B$ and $C$ be sets of configurations $(x,y)\in\B^{2n}$ such that $w(x,y)$ is $n-1$, $n$ and $n+1$, respectively (these are the three sets discussed in the introduction) (we have $\Omega\subseteq B$). One can see that $f'_i$ behave as $f_i$ when $x$ and $y$ are mirroring each other ($y={\overline{x}}$) or almost mirroring each other (${\overline{y}}^i={\overline{x}}$); and in both cases, $(x,y)$ lies in $A\cup B\cup C$. One can also see that $f'_i$ equals $0$ below the layer $A$ and equals $1$ above the layer $C$. The same remarks apply on $f'_{n+i}$, excepted that $f'_{n+1}$ behaves as the negation ${\overline{f_i}}$ in $A\cup B\cup C$. Hence, roughly speaking, $f$ behaves as $(f,{\overline{f}})$ in the middle layer $A\cup B\cup C$, and it converges toward the all-zeroes or all-ones configuration outside this layer.
\[lem:monotone\] If $G(f)$ has no negative loops, then $f'$ is monotone.
Suppose, for a contradiction, that there exists $a,b,c,d\in\B^n$ and $i\in [n]$ such that $$(a,b)<(c,d)
\text{ and }
f'_i(a,b)>f'_i(c,d).$$ Then we have $$n-1\leq w(a,b)<w(c,d)\leq n+1.$$ This leaves three possibilities.
[*Case 1: $w(a,b)=n-1$ and $w(c,d)=n+1$*]{}.
: Since $f'_i(a,b)=1$, we fall in the first case of the definition of $f'_i$, that is, $$f'_i(a,b)=f_i(a)=1\text{ and }{\overline{b}}^i={\overline{a}}.$$ Similarly $$f'_i(c,d)=f_i(c)=0\text{ and }{\overline{d}}^i={\overline{c}}.$$ Thus $$(a,b)=(a,{\overline{{\overline{a}}}}^i)<(c,d)=(c,{\overline{{\overline{c}}}}^i).$$ So for all $j\neq i$, we have $a_j\leq c_j$ and ${\overline{a_j}}=({\overline{{\overline{a}}}}^i)_j\leq ({\overline{{\overline{c}}}}^i)_j={\overline{c_j}}$ thus $c_j\leq a_j$. So $a_j=c_j$ for all $j\neq i$, that is, $c\in\{a,{\overline{a}}^i\}$. Since $f_i(a)<f_i(c)$ we have $c={\overline{a}}^i$, and since $a\leq c$ we deduce that $a_i=0$. Thus $G(f)$ has a negative arc from $i$ to $i$, a contradiction.
[*Case 2: $w(a,b)=n-1$ and $w(c,d)=n$*]{}.
: As in Case 1, we have $$f'_i(a,b)=f_i(a)=1\text{ and }{\overline{b}}^i={\overline{a}}.$$ For $f'_i(c,d)$ we have two cases. Suppose first that $$f'_i(c,d)=f_i(c)=0\text{ and }d={\overline{c}}.$$ Then $$(a,b)=(a,{\overline{{\overline{a}}}}^i)<(c,d)=(c,{\overline{c}}).$$ So for all $j\neq i$, we have $a_j\leq c_j$ and ${\overline{a_j}}=({\overline{{\overline{a}}}}^i)_j\leq {\overline{c_j}}$ thus $c_j\leq a_j$. So $a_j=c_j$ for all $j\neq i$, that is, $c\in\{a,{\overline{a}}^i\}$. Since $f_i(a)<f_i(c)$ we have $c={\overline{a}}^i$, and since $a\leq c$ we deduce that $a_i=0$. Thus $G(f)$ has a negative arc from $i$ to $i$, a contradiction. The other case is $$f'_i(c,d)={\overline{c_i}}=0\text{ and }d\neq {\overline{c}}.$$ First, observe that for all $j\neq i$, if $c_j=0$ then $a_j=0$ thus $1=({\overline{{\overline{a}}}}^i)_j\leq d_j$. Since $c_i=1$ we deduce that ${\overline{c}}\leq d$. Now, suppose that $c_j=d_j=1$ for some $j\in [n]$. Since $w(c,d)=n$, we deduce that there exists $k\neq j$ such that $c_k=d_k=0$, and this contradicts ${\overline{c}}\leq d$. Thus, for all $j\in [n]$, either $d_j=0$ or $d_j>c_j$, that is, $d\leq {\overline{c}}$. Thus $c={\overline{d}}$, a contradiction.
[*Case 3: $w(a,b)=n$ and $w(c,d)=n+1$*]{}.
: We obtain a contradiction as in Case 2.
So we have proven that $f'_i$ is monotone for all $i\in [n]$. It remains to prove that $f'_{n+i}$ is monotone. Using the monotony of $f'_i$ for the implication we get: $$\begin{array}{rcl}
(a,b)\leq (c,d)
&\iff& ({\overline{c}},{\overline{d}})\leq ({\overline{a}},{\overline{b}})\\[2mm]
&\iff& ({\overline{d}},{\overline{c}})\leq ({\overline{b}},{\overline{a}})\\[2mm]
&\Longrightarrow& f'_i({\overline{d}},{\overline{c}})\leq f'_i({\overline{b}},{\overline{a}})\\[2mm]
&\iff& {\overline{f'_i({\overline{b}},{\overline{a}})}}\leq{\overline{f'_i({\overline{d}},{\overline{c}})}}\\[2mm]
&\iff& f'_{i+n}(a,b)\leq f'_{i+n}(c,d).
\end{array}$$
\[lem:fixed\] For all $x\in\B^n$ we have $f(x)=x$ if and only if $f'(x,{\overline{x}})=(x,{\overline{x}})$.
By definition we have $$f(x)=x\quad\iff\quad f'_i(x,{\overline{x}})=x_i~\forall i\in[n].$$ So it is sufficient to prove that $$f'(x,{\overline{x}})=(x,{\overline{x}})\quad\iff\quad f'_i(x,{\overline{x}})=x_i~\forall i\in[n].$$ The direction $\Rightarrow$ is obvious, and $\Leftarrow$ is a consequence of the following equivalences: $$\begin{aligned}
f'_i(x,{\overline{x}})=x_i
&\iff {\overline{f'_{n+i}({\overline{{\overline{x}}}},{\overline{x}})}}=x_i\\
&\iff f'_{n+i}(x,{\overline{x}})={\overline{x_i}}\\
&\iff f'_{n+i}(x,{\overline{x}})=(x,{\overline{x}})_{n+i}.\end{aligned}$$
\[lem:01\] For all $x,y\in\B^n$, if $\Gamma(f')$ has a path from $(x,y)$ to $\Omega$ then $n-1\leq w(x,y)\leq n+1$.
It is sufficient to prove that, $$w(x,y)\leq n-2~\Rightarrow f'(x,y)=0
\quad\text{and}\quad
w(x,y)\geq n+2~\Rightarrow f'(x,y)=1.$$ Let $i\in[n]$. If $w(x,y)\leq n-2$ (resp. $w(x,y)\geq n+2$) then $f'_i(x,y)=0$ (resp. $f'_i(x,y)=1$) by definition. Now, if $w(x,y)\leq n-2$ then $w(\bar y,\bar x)\geq n+2$ thus $$f'_{n+i}(x,y)={\overline{f'_i(\bar y,\bar x)}}={\overline{1}}=0,$$ and if $w(x,y)\geq n+2$ then $w(\bar y,\bar x)\leq n-2$ thus $$f'_{n+i}(x,y)={\overline{f'_i(\bar y,\bar x)}}={\overline{0}}=1.$$
\[lem:transition\] If $G(f)$ has no negative loops, then, for all $x,y\in\B^n$, the following assertions are equivalent:
1. $x\to y$ is a transition of $\Gamma(f)$.
2. $(x,{\overline{x}})\to(y,{\overline{x}})\to (y,{\overline{y}})$ is a path of $\Gamma(f')$.
3. $(x,{\overline{x}})\to(x,{\overline{y}})\to (y,{\overline{y}})$ is a path of $\Gamma(f')$.
4. $\Gamma(f')$ has a path from $(x,{\overline{x}})$ to $(y,{\overline{y}})$ without internal vertex in $\Omega$.
Furthermore, the only possible paths of $\Gamma(f')$ from $(x,{\overline{x}})$ to $(y,{\overline{y}})$ without internal vertex in $\Omega$ are precisely the ones in [*(2)*]{} and [*(3)*]{}.
Suppose that $\Gamma(f)$ has a transition $x\to y$, and let $i\in[n]$ be such that $y={\overline{x}}^i$. We have $f'_i(x,{\overline{x}})=f_i(x)\neq x_i$ thus $\Gamma(f')$ has a transition from $(x,{\overline{x}})$ to $({\overline{x}}^i,{\overline{x}})=(y,{\overline{x}})$. Since $$f'_{n+i}({\overline{x}}^i,{\overline{x}})={\overline{f_i({\overline{{\overline{x}}}},{\overline{{\overline{x}}^i}})}}={\overline{f_i(x,{\overline{{\overline{x}}^i}})}}={\overline{f_i(x)}}=x_i\neq ({\overline{x}}^i,{\overline{x}})_{n+i},$$ $\Gamma(f')$ has a transition from $({\overline{x}}^i,{\overline{x}})$ to $${\overline{({\overline{x}}^i,{\overline{x}})}}^{n+i}=({\overline{x}}^i,{\overline{{\overline{x}}}}^i)=(y,{\overline{y}}).$$ This proves the implication $(1)\Rightarrow (2)$. Now, if $\Gamma(f')$ contains the transition $(x,{\overline{x}})\to (y,{\overline{x}})$ then there exists $i\in [n]$ such that $y={\overline{x}}^i$ and $y_i=f'_i(x,{\overline{x}})=f_i(x)$. Thus $x\to y$ is a transition of $\Gamma(f)$. So we have $(1)\iff (2)$ and we prove similarly that $(1)\iff (3)$.
Since $[(2)\text{ or }(3)]\Rightarrow (4)$ is obvious, to complete the proof it is sufficient to prove that if $\Gamma(f')$ has a path $P$ from $(x,{\overline{x}})$ to $(y,{\overline{y}})$ without internal vertex in $\Omega$ then either $P=(x,{\overline{x}})\to(y,{\overline{x}})\to (y,{\overline{y}})$ or $P=(x,{\overline{x}})\to(x,{\overline{y}})\to (y,{\overline{y}})$. Let $a$ be the configuration following $(x,{\overline{x}})$ in $P$, and let $b$ be the configuration following $a$ in $P$. We will prove that $b=(y,{\overline{y}})$ and $a=(x,{\overline{y}})$ or $a=(y,{\overline{x}})$. We have $w(a)=n\pm 1$ and thus $w(b)\in\{n-2,n,n+2\}$, but if $w(b)=n\pm 2$ then we deduce from Lemma \[lem:01\] that $\Gamma(f')$ has no paths from $b$ to a configuration in $\Omega$, a contradiction. Thus $w(b)=n$. Let $i\in [n]$ be such that $a=({\overline{x}}^i,{\overline{x}})$ or $a=(x,{\overline{{\overline{x}}}}^i)$. We have four cases.
[*Case 1: $a=({\overline{x}}^i,{\overline{x}})$ and $w(a)=n-1$*]{}.
: Since $w(a)=n-1$ we have $x_i=1$, and thus $f'_i(x,{\overline{x}})=f_i(x)=0$. Also $f'_i(a)=f'_i({\overline{x}}^i,{\overline{x}})=f_i({\overline{x}}^i)=0$ since otherwise $G(f)$ has a negative loop on vertex $i$. Let $1\leq j\leq 2n$ be such that $b={\overline{a}}^j$. Since $w(a)<w(b)=n$, we have $a_j=0$ and $f'_j(a)=1$. If $1\leq j\leq n$ then $j\neq i$ (since $f'_i(a)=0$) so ${\overline{{\overline{x}}}}^j\neq{\overline{{\overline{x}}^i}}$ and since $w({\overline{x}}^i,{\overline{x}})=n-1$, we deduce from the definition of $f'$ that $f'_j(a)=f'_j({\overline{x}}^i,{\overline{x}})=0$, a contradiction. So $n< j\leq 2n$. Let $k=j-n$. We have $$f'_j(a)=f'_{n+k}(a)=f'_{n+k}({\overline{x}}^i,{\overline{x}})=\overline{f'_k({\overline{{\overline{x}}}},{\overline{{\overline{x}}^i}})}=\overline{f'_k(x,{\overline{{\overline{x}}^i}})}.$$ Since $w({\overline{x}}^i,{\overline{x}})=n-1$ we have $w(x,{\overline{{\overline{x}}}}^i)=n+1$. So if $k\neq i$ we have ${\overline{{\overline{{\overline{x}}}}^i}}^k\neq{\overline{x}}$. Thus by the definition of $f'$ we have $f'_k(x,{\overline{{\overline{x}}^i}})=1$ thus $f'_j(a)=0$, a contradiction. We deduce that $k=i$, that is, $j=n+i$. Thus $b={\overline{a}}^{n+i}=({\overline{x}}^i,{\overline{{\overline{x}}}}^i)\in\Omega$, and we deduce that $$P=(x,{\overline{x}})\to (y,{\overline{x}})\to (y,{\overline{y}}).$$
[*Case 2: $a=({\overline{x}}^i,{\overline{x}})$ and $w(a)=n+1$*]{}.
: We prove with similar arguments that $$P=(x,{\overline{x}})\to (y,{\overline{x}})\to (y,{\overline{y}}).$$
[*Case 3: $a=(x,{\overline{{\overline{x}}}}^i)$ and $w(a)=n-1$*]{}.
: We prove with similar arguments that $$P=(x,{\overline{x}})\to (x,{\overline{y}})\to (y,{\overline{y}}).$$
[*Case 4: $a=(x,{\overline{{\overline{x}}}}^i)$ and $w(a)=n+1$*]{}.
: We prove with similar arguments that $$P=(x,{\overline{x}})\to (x,{\overline{y}})\to (y,{\overline{y}}).$$
\[lem:path\] If $G(f)$ has no negative loops, then for all $x,y\in\B^n$, the following two assertions are equivalent:
1. $\Gamma(f)$ has a path from $x$ to $y$ of length $\ell$.
2. $\Gamma(f')$ has a path from $(x,\bar x)$ to $(y,\bar y)$ of length $2\ell$.
According to Lemma \[lem:transition\], $x^0\to x^1\to x^2\to \cdots\to x^\ell$ is a path of $\Gamma(f)$ if and only if $$(x^0,{\overline{x^0}})\to(x^0,{\overline{x^1}})\to (x^1,{\overline{x^1}})\to(x^1,{\overline{x^2}})\to (x^2,{\overline{x^2}}) \cdots\to (x^\ell,{\overline{x^\ell}})$$ is a path of $\Gamma(f')$. This proves $(1)\Rightarrow (2)$. To prove $(2)\Rightarrow(1)$ suppose that $\Gamma(f')$ has a path $P$ from $(x,\bar x)$ to $(y,\bar y)$ of length $2\ell$. Let $(a^0,{\overline{a^0}}),(a^1{\overline{a^1}}),\dots,(a^p,{\overline{a^p}})$ be the configurations of $P$ that belongs to $\Omega$, given in the order (so $a^0=x$ and $a^p=y$). According to Lemma \[lem:transition\], there exists $b^1,b^2,\dots,b^p$ with $b^q\in\{(a^{q-1},{\overline{a^q}}),(a^q,{\overline{a^{q-1}}})\}$ for all $1\leq q\leq p$ such that $$P=(a^0,{\overline{a^0}})\to b^1 \to (a^1,{\overline{a^1}})\to b^2\to\cdots \to b^p\to (a^p,{\overline{a^p}}).$$ Thus $p=\ell$, and again by Lemma \[lem:transition\], $x^0\to x^1\to\cdots \to x^\ell$ is a path of $\Gamma(f)$.
Theorem \[thm:embedding\] result from Lemmas \[lem:monotone\], \[lem:fixed\] and \[lem:path\].
Conclusion and open questions {#sec:conclusion}
=============================
In this paper we have proved that the asynchronous graph of every $n$-component Boolean network without negative loop can be embedded in the asynchronous graph of a $2n$-component [*monotone*]{} Boolean network, in such a way that fixed points and distances between configurations are preserved. A consequence of this result, which was our initial goal, is that the asynchronous graph of a monotone network may have an exponential diameter. More precisely, it may exist a configuration $x$ and a fixed point $y$ reachable from $x$ such that the distance between $x$ and $y$ is at least $2^{\frac{n}{2}}$. This contrasts with the fact that for every configuration $x$ there exists a fixed point $y$ such that the distance between $x$ and $y$ is at most $n$.
These results raise several questions. Could it be possible to embed, in a similar way, a $n$-component network [*with*]{} negative loops into a $m$-component monotone network? Maybe this would require $m$ to be even larger than $2n$. Besides, the embedding we propose is based on the injection $x\mapsto (x,\overline{x})$ from $\B^n$ to the balanced words of length $2n$. The well-known Knuth’s balanced coding scheme [@K86] provides a rather simple injection from $\B^n$ to the balanced words of length $n+2\log_2n$ only. Could this technique be used to decrease the number of components in the host monotone network from $2n$ to $n+2\log_2n$? Finally, it could be interesting to study the interaction graph of monotone networks with large diameter. Does it necessarily contain long cycles, or many disjoint cycles?
#### Acknowledgment
This work has been partially supported by the project PACA APEX FRI. We wish also to thank Pierre-Etienne Meunier, Maximilien Gadouleau and an anonymous reviewer for stimulating discussions and interesting remarks.
[^1]: Corresponding author
|
---
abstract: 'We demonstrate a simple and quick method for the measurement of the in-plane thermal conductance of thin films via steady-state IR thermography. The films are suspended above a hole in an opaque substrate and heated by a homogeneous visible light source. The temperature distribution of the thin films is captured via infrared microscopy and fitted to the analytical expression obtained for the specific hole geometry in order to obtain the in-plane thermal conductivity. For thin films of poly(3,4-ethylenedioxythiophene):polystyrene sulfonate post-treated with ethylene glycol and of polyimide we find conductivities of and at room temperature, respectively. These results are in very good agreement with literature values, validating the method developed.'
author:
- Anton Greppmair
- Benedikt Stoib
- Nitin Saxena
- Caroline Gerstberger
- 'Peter M[ü]{}ller-Buschbaum'
- Martin Stutzmann
- 'Martin S. Brandt'
bibliography:
- 'ThermalCond\_AntonGreppmair.bib'
title: 'Measurement of the in-plane thermal conductivity by steady-state infrared thermography'
---
Introduction: Thermal conductivity of thin films
================================================
With a rising interest in the thermoelectric properties of meso-structured[@Isa14; @Li13; @Wan08; @Dre07] and organic materials[@Sta12; @Zhe14], the full electric and thermal characterization of thin films becomes ever more relevant. Though the electrical parameters are often well studied, there is still little data on the thermal conductivity of novel thin film materials like organic semiconductors. Even for more standardized materials information about the dependence of the thermal conductivity on measurement orientation, material type, and film fabrication is scarce.[@Ben99; @Yok95; @Cho87] Therefore, a quick and versatile method to measure the thermal conductivity of films thinner than is needed, which is challenging for established methods.
Problematic for the measurement of the thermal conductivity of thin films can be the substrates often used to support the film under investigation. These substrates usually are orders of magnitude thicker than the film and therefore easily have a higher thermal conductance than the thin film itself. Similarly, heat transport through wires for local heaters or thermometers in contact with the films can disturb thermal conductivity measurements. Thus substrate- and contact-free methods are generally preferred. The Raman-shift method [@Bal11; @Sto14b; @Rep14; @Sto14] is such a possible technique, but, while it has successfully been applied to well-understood materials such as SiGe alloys or graphene, the necessary complex calibration procedure somewhat restricts the use of this method for other materials. Furthermore, to ensure reasonable measurement durations and sensitivity, a high illumination power is typically applied in Raman experiments which generates high temperature differences and further limits this technique. Time-resolved optical pump-and-probe measurements[@Dal02] are an alternative, but also depend on calibration and smooth film surfaces without diffuse reflection. Transient 3$\omega$ on-chip measurements[@Koj15; @Vol13] circumvent the need for calibration and are less restricted in terms of the type of film investigated, but sample preparation is very demanding. Since either sample mounting or measurement are time consuming in all of these techniques, they cannot easily provide information about the homogeneity of a film as a whole.
A method which is quicker and allows easy on-sample calibration is IR thermography. For the determination of the in-plane thermal conductivity with thermography, the [Å]{}ngstr[ø]{}m method, which is a dynamic measurement technique, is traditionally used.[@Wol06; @Hor03] It, however, relies on well-defined illumination patterns generated, e.g., by focused laser beams. Here we demonstrate an IR thermography method for the measurement of the in-plane thermal conductivity of thin films where a steady-state temperature gradient is generated via homogeneous heating by strongly absorbed visible light and a substrate serving as a lateral heat sink. Below, we will introduce the measurement principle and present the necessary experimental setup and the calibration steps. We apply the method to thin films of poly(3,4-ethylenedioxythiophene):polystyrene sulfonate () and of polyimide (PI). Finally, we discuss the range of conductances which we expect can be measured with this technique.
Measurement Principle
=====================
In order to measure a thermal conductivity, a temperature gradient needs to be generated by a known heat flux. Figure \[fig:method\] shows a schematic diagram of the method used here to generate such a temperature gradient, where thin films are placed over a cavity in a substrate. The sample is illuminated from below, by which the required known heat flux is generated. The opaque substrate serves as a shadow mask for the illumination and as a heat sink, generating the necessary temperature gradient. The resulting temperature distribution of the free-standing part of the film depends on the form of the cavity. Using analytical or numerical models the thermal conductivity can then be obtained from the magnitude of the temperature increase taking into account the illumination power density, the absorptance of the film, and its thickness. These are the only quantities that need to be measured in addition to the temperature distribution for the determination of the thermal conductivity via this method.
![\[fig:method\] A film suspended over a hole in an opaque substrate. Illumination in the visible range (straight red arrows) from below heats the free-standing film homogeneously. Heat (jagged black arrows) is transferred along the film into the substrate acting as a heat sink. In steady state a maximum temperature of the free-standing film and a maximum thermal radiation (wavelike orange arrows) is reached in the center of the hole.](Fig1_MethodModel.eps)
In this work we concentrate on circular cavities. For free-standing thin films suspended over circular cavities with radius $R$ (and also assuming a homogeneous absorption of the illumination over the depth of the film), rotational invariance allows an analytical solution of the temperature distribution. In cylindrical coordinates the resulting one-dimensional problem can be written as $$q_r (r)=-\kappa_{||} \frac{\partial T(r)}{\partial r}
\label{eq:fourier}$$ with the radial component of the heat flux vector $q_r$, the in-plane thermal conductivity $\kappa_{||}$, and the temperature distribution $T(r)$ at a distance $r$ from the cavity center.[@Ser14] For a homogeneously absorbed areal power density $p_\mathrm{abs}$, symmetry leads to a vanishing heat flux at the center of the hole ($r=0$). At steady-state conditions the power absorbed by a circle area $r^2 \pi$ needs to be equal to the power flowing through the circumference of said circle. This leads to $$r^2 \pi \cdot p_\mathrm{abs}=2 r \pi \cdot d \cdot q_r (r)
\label{eq:EnergyConservation}$$ where $d$ is the film thickness.
Merging (\[eq:fourier\]) and (\[eq:EnergyConservation\]), integrating, and applying the boundary condition $T(r=R)=T_\mathrm{substrate}$, i.e. the substrate temperature, leads to the temperature distribution $$T(r)=-\frac{p_\mathrm{abs}}{4\kappa_{||} d}r^2+\frac{p_\mathrm{abs}}{4\kappa_{||} d}R^2+T_\mathrm{substrate},
\label{eq:TDistribution}$$ expected for the experimental geometry investigated here.
The simple boundary condition used does not take into account the exact transition from the parabolic behavior in the middle of the free-standing film to the substrate-supported film. However, this transition, influenced by thermal contact resistance and shadowing at the border of the hole, will also be rotationally invariant leading to an additional constant offset to the temperature distribution. This offset and the constant term $\frac{p_\mathrm{abs}}{4\kappa_{||} d}R^2+T_\mathrm{substrate}$ in (\[eq:TDistribution\]) can be incorporated into an effective substrate temperature $T_\mathrm{substrate, eff}$. Therefore, $\kappa_{||}$ can be directly determined from the curvature of the temperature distribution near the center of the hole, $p_\mathrm{abs}$, and $d$ only. Since the curvature can be obtained from a measurement of local temperature differences the demand for the absolute temperature accuracy is minimal.
Furthermore, the above derivation of (\[eq:TDistribution\]) assumes vanishing losses, e.g. from convection and thermal radiation, generating additional cooling processes. We will discuss thermal radiation losses towards the end of the paper. Losses through air are reduced by a vacuum chamber, described below.
Setup details {#sec:setup}
=============
Integral parts of the setup are the IR camera for the spatially resolved temperature measurement and the vacuum chamber, shown in Fig. \[fig:setup\], for the reduction of the losses by air.
![\[fig:setup\] Exploded view of the measurement setup. IR thermography of the sample in vacuum is performed from above through a sapphire window. The sample holder allows to heat the sample and to measure its temperature directly via thermocouples. The heating of the thin film is achieved optically via illumination from below through a glass window. A translation stage allows lateral positioning.](Fig2_Setup.eps)
The IR camera used was an InSb 640 SM high speed camera from DCG systems equipped with a Micro 1x lens, that projects the sample image with a 1:1 scale onto the detector of the camera. The resulting distance on the sample between two pixels in the camera image $l$ was determined to be and the resolution of the camera is 640x480 pixels, giving a total field of measurement of . The noise equivalent differential temperature (NEDT) is below and for each measurement at least 128 images were averaged at a frame rate of .
The IR camera observes the sample from above. For the window material to the vacuum chamber we use sapphire to allow the transmission of the IR radiation. The sample consists of sheets of aluminum with a thickness of and holes ranging from to in diameter as the opaque substrate with the film under investigation on top. The aluminum substrate allows simple manufacturing and offers a large thermal conductivity for fast removal of excess heat. It also reflects most of the incident light, thus the increase of the substrate temperature was always below , even for the highest illumination powers.
The thickness of the films was determined by placing the films on a glass substrate and using profilometry. The results were confirmed with scanning electron microscopy and atomic force microscopy. Since the thickness $d$ of the films investigated here is orders of magnitude below the hole radius $R$, the slight inhomogeneity in absorption with respect to the depth in the film can be neglected.
The whole sample is suspended underneath the sapphire window and is illuminated from below via an LED. The LED is placed outside of the vacuum chamber and irradiates the sample through a glass window, which blocks the IR radiation of the hot LED due to its low IR transmittance. The visible light of the LED itself is not detected by the IR camera. The sample holder has an integrated heater and a feedthrough for a thermocouple in contact with the substrate. Heater and thermocouple are used for the calibration summarized below. In order to reduce background radiation, thermal insulation assures that the chamber temperature remains constant while the sample and holder are heated. The setup is mounted on a translation stage for sample positioning and measurement of the illumination power density.
For the latter, a circular silicon photo-diode with a diameter of is fixed above the film, pointing in the direction of the LED, as shown in Fig. \[fig:Scan\](a). The photo-diode was cross-calibrated with a NIST-traceable power sensor. The chamber with the LED and the sample are then scanned underneath the photo-diode, yielding the map of the holes in Fig. \[fig:Scan\](b). In a second step the sample, consisting of the film and the supporting substrate, is removed and the scan repeated for all LED intensities used for the thermal conductivity measurement. This allows the acquisition of the spatially resolved illumination power density in Fig. \[fig:Scan\](c). This power density includes the effects of the glass window and allows verification of the lateral homogeneity. The hole position from Fig. \[fig:Scan\](b) is than correlated to the power density in Fig. \[fig:Scan\](c) and the power density for each hole is determined by averaging the values within each red circle.
The absorptance $A$ in the range of the optical wavelengths used for heating and the IR transmittance and reflectance of the films are determined via UV-Vis and FTIR transmission and reflection measurements, respectively. For UV-Vis spectroscopy an integrating sphere was used to include diffuse scattering. For these experiments the thin films were again placed on sheets of aluminum with a hole. This time the holes had a diameter of in order to increase the signal intensities.
Once the absorptance $A$ of the film under investigation as well as the illumination power density is known the areal power density $p_{abs}$ can be calculated providing the desired known heat flux.
![\[fig:Scan\] (a) Scanning the sample with a calibrated photo-diode (black) with a diameter of by moving the chamber and a blue LED underneath the fixed diode by steps. (b) and (c) show the measured power density with and without a PI sample on a supporting substrate, respectively. The red circles illustrate the position of the holes. The difference in absolute intensity between (b) and (c) is due to the low transmittance of PI at the used illumination wavelength.](Fig3_Scan.eps)
![\[fig:TempDisp\] (a) and (b) IR images of a film over a cavity with a diameter of in the dark and under illumination, respectively. (c) False color temperature map derived from (a) and (b) after calibration. (d) Temperature distribution of the same film over a hole with a diameter of under illumination (blue dots). The gray surface shows a fitted paraboloid. The sample in images (b) through (d) was illuminated with a power density of .](Fig4_TemperatureDistribution.eps)
Temperature Measurement
=======================
We calibrate the IR camera, the second important step in the method presented here, directly with the films investigated, and thereby also account for the emissivity of the film.
The signal detected by the IR camera is proportional to the IR emission of the film in the detection wavelength range of the camera. For the temperature calibration of the camera two images of the sample at different temperatures, as determined by the thermocouple, are taken. An example of such a calibration image without illumination can be seen in Fig. \[fig:TempDisp\](a). For small temperature differences the calibration can be approximated by a linear relation between the temperature and the camera signal. More accurate results, however, are obtained when the nonlinear dependence of the number of photons $N_\mathrm{photon}$, emitted by the film at a temperature $T$, is taken into account. This correlation is given by the photon radiance $$N_\mathrm{photon}(\lambda) \propto \frac{1}{\lambda^4 \left[\exp{\left(\frac{hc}{\lambda k_\mathrm{B} T}\right)}-1 \right]}~,
\label{eq:planck}$$ where $\lambda$ is the wavelength, $k_\mathrm{B}$ the Boltzmann constant, $h$ the Planck constant, and $c$ the speed of light. This dependence is treated numerically. The dependences of the lens, the camera, and the film emissivity on $\lambda$ are effectively removed by using a narrow band IR filter in front of the camera that is transparent between and only. While the application of this correction in principle requires knowledge of the absolute temperature, simulations have shown that the effect of the exact absolute temperature on the $\kappa_{||}$ determined is minimal.
After calibration, an IR image of the film under illumination can be taken, shown in Fig. \[fig:TempDisp\](b). From this the temperature distribution in Fig. \[fig:TempDisp\](c) can be calculated, including the correction for the slight non-linearity in (\[eq:planck\]), which is then used to deduce the in-plane thermal conductivity. For fitting, only the center of the film is used, as discussed above in conjunction with (\[eq:TDistribution\]). In Fig. \[fig:TempDisp\](d) the central temperature distribution of the same film over a smaller hole is compared to a paraboloid fitted to this data. Clearly, the temperature distribution in blue is described well by the first term in (\[eq:TDistribution\]), indicated by the gray surface, which allows the direct determination of $\frac{p_\mathrm{abs}}{4 \kappa_{||} d}$ and, together with the experiments described in Sec. \[sec:setup\], the measurement of $\kappa_{||}$.
Sample Preparation
==================
The method presented here requires a sample that has a sufficient emissivity in the detection range of the camera and a sufficient absorptance at the wavelength of the illumination used for heating. For this work we have studied two different thin film materials that comply with these requirements, and PI. While is an example of a novel organic semiconductor, PI is a rather standardized material, but nonetheless offers an interesting study case as varying values for the thermal conductivity $\kappa_{||}$ are found in the literature, which we will discuss later.
Free-floating thin films of PEDOT:PSS on water were produced by the method described in Ref. . The solution, containing Zonyl, was spin-coated thrice onto a glass substrate covered with a thin layer of poly(dimethylsiloxane) (PDMS). Such a thin layer of PDMS reduces adhesion of the film to the substrate, allowing for easier lift-off later. After each spin coating the samples were annealed for at . Post-treatment was carried out by drop-casting ethylene glycol (EG), letting it soak in for , spinning it off, and annealing again for at . This yielded an electrical conductivity $\sigma=\SI{720\pm30}{\siemens\per\centi\meter}$ at a typical film thickness of .[@Pal14] Poly(vinylalcohol) (PVA) was drop-cast onto the sample and after drying, the PVA/ film could easily be removed from the PDMS surface. The PVA was then dissolved in water and the remaining free-floating film was transferred from the water onto the desired substrate. The adhesion of the PEDOT:PSS to the final substrate was very good, ensuring sufficient thermal contact with the support.
We also investigated commercial type HN polyimide films with a thickness of . Transfer was much simpler for the PI, however, the adhesion and thus the thermal contact to the substrate were inferior. To improve adhesion and thermal contact, thermal grease was applied between the aluminum and the PI film.
Results and Discussion
======================
We start with the optical characterization of the thin films in the visible spectral range used for heating. The transmittance $T$ and reflectance $R$ are plotted in Fig. \[fig:UVVis\] for the and PI films. The absorptance $A$ for each film was then calculated according to $A+T+R=1$. For direct comparison, the spectra emitted by the two LEDs used with mean wavelengths of and are also plotted in Fig. \[fig:UVVis\] in arbitrary units in red and blue, respectively. The highest absorptance was achieved by using the red LED for the film and the blue LED for the PI film. Weighting the absorptance spectra with the LED spectra results in effective absorptances of and , respectively.
Additionally, the IR transmittance of the films was measured via FTIR spectroscopy to determine the influence of the increase in background radiation during calibration, caused by an unavoidable slight increase of the chamber temperature. For the film, the IR transmittance is below while the increase in background radiation during calibration is less than of the increase in film radiation from holes covered by the film. Therefore the total background influence on the calibration is below and can be neglected. The IR transmittance of the PI film studied is significantly higher, ranging between and . At the same time the emissivity of the PI film is significantly lower than that of the PEDOT:PSS film and thus the increase in background radiation is about of the increase in film radiation. This is corrected for in the calibration.
![\[fig:UVVis\] UV-Vis measurements of the investigated PI and films, shown as dashed and solid lines, respectively. An integrating sphere was used to measure the transmittance (black) and reflectance (violet) of the films and the absorptance (green) was calculated with $A+T+R=1$. The spectral data of the blue and red LED are shown in arbitrary units to indicate the absorptance of the PI and PEDOT:PSS films, respectively.](Fig5_UVVisLED.eps)
Together with the film thicknesses, we now have all data available to determine $\kappa_{||}$ via steady-state thermography. The results for $\kappa_{||}$ are shown in Fig. \[fig:cond\] as a function of the maximum temperature increase, measuring films on holes with different diameters and with different illumination power densities. As expected, a larger hole diameter and a higher illumination power lead to a larger temperature increase. For an increase below the results for show a stable value of $\kappa_{||}=\SI{1.00\pm0.04}{\watt\per\meter\per\kelvin}$ for all hole diameters with the peak-to-peak distribution of values given in parenthesis. For a higher temperature increase the measured values of $\kappa_{||}$ increase. This is most likely due to increased thermal radiation losses already mentioned earlier, proportional to $T^4$, which were not corrected.
For PI we obtain $\kappa_{||}=\SI{0.42\pm0.02}{\watt\per\meter\per\kelvin}$. Because of a larger film thickness the PI does not heat up as much as the PEDOT:PSS for the same power densities. Thus the radiation losses are smaller and the apparent increase in $\kappa_{||}$ for large temperature increases is less pronounced.
![\[fig:cond\] The in-plane thermal conductivity of films with a thickness of and PI films with a thickness of as a function of the maximum temperature increase resulting from the LED illumination. The illumination power density was varied between and .](Fig6_ThermalConductivity.eps)
We now estimate the uncertainty in the determination of $\kappa_{||}$. The fit error and the influence of the camera noise depend on the maximum temperature increase. For increases above , they are typically below and can be neglected. Uncertainties for the absorptance measured by UV-Vis spectroscopy, for the thickness determination, and for the illumination power were all estimated at below and uncertainties for the temperatures determined by the thermocouples at below . For the PI film measurements an additional error of was estimated, accounting for the calibration correction due to background radiation.
The influence of thermal radiation losses of the thin films on the calibration and the actual measurements is more challenging to estimate. Simulations have shown that radiation losses during calibration and LED illumination cancel each other out for small illumination powers and hole diameters. This is finally supported by the data in Fig. \[fig:cond\], where the values for $\kappa_{||}$ are independent of the hole diameter for low illumination powers. We conservatively account for the effect of radiation losses with another uncertainty. This leads to values for $\kappa_{||}$ of and for the investigated PEDOT:PSS and PI films, respectively, with the errors now accounting for all parts of the experimental method.
Our results are in very good agreement with literature values of the in-plane thermal conductivity of after post-treatment. The results for dimethyl sulfoxide- (DMSO) and EG-doped in Ref. and Ref. , respectively, are within our uncertainties. Those measurements were also performed on PEDOT:PSS that was prepared on PDMS and eventually removed from the substrate, as was done here.[@Gre11] Reference and Ref. reported lower values for $\kappa_{||}$, however, these samples were not prepared on PDMS and the results were deduced from cross-plane thermal conductivities.
As mentioned above varying values for the thermal conductivity $\kappa$ of PI can be found depending on PI type, film thickness, and measurement orientation.[@Ben99; @Yok95; @Cho87] The manufacturer of the PI film studied here, DuPont, and the reseller, GoodFellow, however, state a thermal conductivity $\kappa$ of and , respectively, for bulk and thin films alike. The measurement geometry used to determine $\kappa$ is not given. A more in-depth study by Rule *et al.*, using an unguarded steady-state parallel-plate apparatus and stacking 33 layers of PI film, found a thermal conductivity of $\kappa_{||}=\SI{0.38}{\watt\per\meter\per\kelvin}$ for the in-plane thermal conductivity of thin films of type HN PI, the type of PI studied here, again well within our uncertainties.[@Rul96]
Finally we briefly address the necessity to perform these measurements in vacuum. Figure \[fig:vac\] shows the dependence of the maximum temperature increase of illuminated PEDOT:PSS films on the chamber pressure.
![\[fig:vac\] Normalized maximum increase of the thermography signal of optically heated PEDOT:PSS films for varying environmental pressure. Solid symbols and open symbols represent illumination power densities of and , respectively. Thinner films and larger hole diameters show a stronger pressure dependence.](Fig7_Vacuum.eps)
It becomes clear that the signal detected is significantly lower at atmospheric pressure, where the surrounding gas acts as an additional heat sink. The thermal conductivity of the air then follows the characteristics of the Pirani pressure gauge[@Woo61; @Shi95] before saturating for a pressure below where the influence of the remaining gas can be neglected. Worth mentioning is the independence of this effect on the LED power, evident from comparing the open and solid black squares in Fig. \[fig:vac\], as well as the fact that thicker films show less influence because of a reduced surface-to-volume ratio.
Range of thermal conductivities accessible
==========================================
Finally we want to estimate the range of thermal conductivities which might be measured with the method presented here. Large thermal conductivities will result in a small temperature increase. To be able to detect this, high optical heating powers and large cavity diameters will be necessary. To estimate the maximum thermal conductivity accessible, we conservatively assume that the minimum signal that can be achieved by the highest LED power over the largest hole diameter should be equivalent to the maximum temperature increase of for our film, which has an emissivity of $\epsilon_\mathrm{ir}=0.3$ as estimated by FTIR. This would still allow a measurement series with varying illumination power and hole diameters to confirm the independence of the value of $\kappa_{||}$ on power and diameter as in Fig. \[fig:cond\], while offering temperature differences well above the NETD of the IR camera. When further assuming an ideal emissivity of $\epsilon_\mathrm{ir}=1$ of the film investigated, a temperature increase $\Delta T=\SI{0.3}{\kelvin}$ should be detectable. Equation (\[eq:TDistribution\]) can now be used to link this temperature increase, the hole radius $R$, the optical heating power density $p_\mathrm{abs}$, and the film thickness $d$ via $$\Delta T=-\frac{p_\mathrm{abs}}{4\kappa_{||} d}R^2.
\label{eq:tempinc}$$ The largest hole radius that can be captured by the camera using the 1x microscope objective with the 1:1 image scaling is $R\approx\SI{5}{\milli\meter}$. With an illumination power density of this leads to the maximum thermal conductivity detectable of $$\kappa_\mathrm{max}=\frac{\epsilon_\mathrm{ir}A}{d} \cdot \SI{0.02}{\watt\per\kelvin},
\label{eq:kappamax}$$ where A is the absorptance of the thin film.
For low thermal conductivities radiation losses are the limiting factor. The radiation losses per area for both sides of the film can be expressed as $$p_\mathrm{rad}=2 \epsilon_\mathrm{tot}\sigma\left(\left(\Delta T+T_\mathrm{amb}\right)^4-T_\mathrm{amb}^4\right),
\label{eq:radloss}$$ with the total emissivity of the film $\epsilon_\mathrm{tot}$, the Stefan-Boltzmann constant $\sigma$, and the ambient temperature $T_\mathrm{amb}$. First order approximation and division by $p_\mathrm{abs}$ from (\[eq:tempinc\]) leads to $$\frac{p_\mathrm{rad}}{p_\mathrm{abs}}=2 R^2 \frac{\epsilon_\mathrm{tot}\sigma T_\mathrm{amb}^3}{\kappa_{||} d}.
\label{eq:radfigure}$$ This can be used as a figure of merit for the influence of the radiation losses on the measurement. As an upper limit for this figure we choose the measurement of over a hole with a diameter of , since for low illumination power densities again no dependence of $\kappa_{||}$ on the variation of the experimental conditions is observed. With a total emissivity of the film of $\epsilon_\mathrm{ir}=0.3$, again estimated by FTIR, the ratio (\[eq:radfigure\]) should thus be below 0.2. A simple way to realize such a value of $\frac{p_\mathrm{rad}}{p_\mathrm{abs}}$ is the reduction of $R$, however, this is limited by the resolution of the IR camera. Assuming that 15 pixels should be the minimum for the image of a hole diameter to assure a satisfying fit (cf. Fig. \[fig:TempDisp\]), we find again a minimum $R$ of , using the 1x microscope objective. With (\[eq:radfigure\]) the minimum thermal conductivity detectable by this setup is thus $$\kappa_\mathrm{min}=\frac{\epsilon_\mathrm{tot}}{d} \cdot \SI{2e-7}{\watt\per\kelvin}.
\label{eq:kappamin}$$
The limits for $\kappa_{||} \cdot d$, that we expect should be detectable with the setup developed here, can thus be written as $$\epsilon_\mathrm{ir}A \cdot \SI{0.02}{\watt\per\kelvin} \gtrsim \kappa_{||}d \gtrsim \epsilon_\mathrm{tot} \cdot \SI{2e-7}{\watt\per\kelvin}.
\label{eq:limitsetup}$$ This means, e.g, that a black sheet of aluminum with a thermal conductivity of and assuming $A=1$ and $\epsilon_\mathrm{ir}=1$ should have a maximum thickness of to be within this thermal conductivity range.[@Ho72] A film on the other hand should have a minimum thickness of , assuming that the optical properties remain as they were observed in this work.
Using a 10x microscope objective and thereby reducing the distance on the sample between two pixels in the camera image to $l=\SI{1.5}{\micro\meter}$ would potentially reduce the lower limit by a factor of 100, but at the cost of more noise. Generalizing our findings to a different camera setup we get $$0.016 \cdot \frac{\epsilon_\mathrm{ir} p_\mathrm{max} A R_\mathrm{max}^2}{NEDT} \gtrsim \kappa_{||}d \gtrsim \epsilon_\mathrm{tot} l \cdot \SI{0.013}{\watt\per\meter\per\kelvin},
\label{eq:limitany}$$ with the maximum illumination power density $p_\mathrm{max}$ and the maximum possible radius $R_\mathrm{max}$.
Conclusion
==========
In summary, we have presented a fast contact-free method for the determination of the in-plane thermal conductivity of free-standing thin films and outlined the method’s possible range of application. After calibration the measurement of samples with sizes between and , depending on the microscope objective, with multiple measurement cavities can be performed in a few seconds, providing statistically relevant results and allowing to judge the homogeneity of the samples investigated. The results obtained on the low conductivity test materials used agree quantitatively with the existing literature. With on-sample calibration, no additional knowledge for the temperature calibration is necessary, thus the technique could be applied to novel materials such as highly porous SiGe films, as well as composite and organic-inorganic hybrid materials.[@Sto15; @Sch16; @Car12; @See10b] The setup presented here furthermore allows to measure $\kappa_{||}$ as a function of temperature (by additionally heating the substrate in the vacuum chamber) and the application to brittle materials (due to the small hole diameters possible). While for large temperature differences generated by the optical heating losses due to thermal radiation were identified as an issue, for low temperature differences consistent results were observed. Should this requirement lead to signal amplitudes too small to be measured accurately with the steady-state thermography presented here, the application of quasi-steady-state lock-in methods could be helpful. Eventually leaving the steady-state regime, an investigation of the heating dynamics should even yield the heat capacity of the film under investigation with the measurement geometry presented here.
Acknowledgments
===============
We gratefully acknowledge funding by the Bavarian State Ministry of the Environment and Consumer Protection via the project UMWELTnanoTECH, by the Bavarian State Ministry of Education, Science, and the Arts via a project from Energy Valley Bavaria at the Munich School of Engineering, and by the Nanosystems Initiative Munich (NIM).
|
---
abstract: 'Traditional tree search algorithms supply a blueprint for modeling problem solving behaviour. A diverse spectrum of problems can be formulated in terms of tree search. Quantum computation, in particular Grover’s algorithm, has aroused a great deal of interest since it allows for a quadratic speedup to be obtained in search procedures. In this work we consider the impact of incorporating classical search concepts alongside Grover’s algorithm into a hybrid quantum search system. Some of the crucial points examined include: (1) the reverberations of contemplating the use of non-constant branching factors; (2) determining the consequences of incorporating an heuristic perspective into a quantum tree search model.'
author:
- 'Lu’s Tarrataca[^1]'
- Andreas Wichert
title: Tree Search and Quantum Computation
---
Introduction \[sec:introduction\]
=================================
The concepts of knowledge representation and the reasoning processes that support knowledge application have long been a key interest area in the field of artificial intelligence. Knowledge enables problem-solving agents to determine an appropriate action in order to better deal with complex environments. Reasoning allows an agent to perform complex decisions by employing a finite amount of knowledge [@luger1993]. One such form of reasoning consists of classical tree searching. Tree search is employed whenever decisions must be made that are based on complex knowledge. Tree search algorithms play a crucial role in many applications, e.g. production systems [@post1943] [@newell1959] [@newell1963] [@ernst1969] [@anderson1983] [@laird1986] [@laird1987], game playing programs [@feldmann1993] [@hsu1999] [@hsu2002] [@campbell2002] and robot control systems [@coelho2009a] [@coelho2009b].
The next few section are organized as follows: Section \[sec:classicalTreeSearch\] reviews some of the major tree search algorithms; Section \[sec:quantumSearch\] present an alternative search method based on quantum computation; Section \[sec:hybridSystem\] proposed an hybrid search system combining classical tree search algorithms alongside quantum search. Section \[sec:objectivesAndProblems\] presents the objectives and associated problems.
Classical tree search \[sec:classicalTreeSearch\]
-------------------------------------------------
The simplest tree search algorithms rely on a “brute-force” approach. These methods perform an exhaustive examination of all possible sequences of moves until goal states are reached. The search through the state space systematically checks if the current state is a goal state. If a non-goal state is discovered then the current state is expanded by applying a successor function, generating a new set of states. The choice of which state to expand is determined by a search strategy. In a great deal of occasions, an artificial intelligence application does not possess an adequate level of knowledge enabling the choice of the most promising state. Strategies that can only distinguish between between goal states and non-goal states, without being able to determine if one state is more promising than another, are referred to as uninformed search strategies. Examples of uninformed search strategies include the well known breadth first search [@moore1959], depth-first search [@hopcroft1973], and also the iterative deepening search [@slate1977]. Uninformed strategies are only successful for small problem instances. Typically, most problems search space is characterized by an exponential growth [@garey1979]. Due to the mammoth dimensions of the search space it becomes impractical, both time- and space-wise, to perform an exhaustive examination.
Alternatively, it is possible to employ additional insights, that arise beyond the definition of the problem. The use of this information, thus the term informed search strategies, allows for solutions to be found more efficiently. Typically, informed search strategies employ an evaluation function $f(n)$ which considers a cost function $g(n)$ alongside a heuristic function $h(n)$. Function $g(n)$ can be interpreted as representing the cost to reach node $n$ whilst $h(n)$ represents an estimate on the cost to reach a leaf node from node $n$. Traditionally, the node with the lowest evaluation value is selected for expansion. Examples of some of the best known informed search strategies include greedy search [@newell1965] and A$^{*}$ search [@hart1968]. More recent advances on informed strategies include IDA$^{*}$ [@korf1985] and RBFS [@korf1991], [@korf1993]. A time complexity comparative assessment between the various algorithms is presented in Table \[table:treeSearchAlgorithmComparison\].
Quantum Search \[sec:quantumSearch\]
------------------------------------
Grover’s algorithm performs a generic search for a solution using the principles of quantum computation and mechanics [@grover1996] [@grover1998a]. Suppose we wish to search through a problem’s search space of dimension $N$. Also, consider that we are also capable of efficiently perceiving a solution to our problem. This is similar to the NP class of problems whose solutions are verifiable in polynomial time $O(n^{k})$ for some constant $k$, where $n$ is the size of the input to the problem [@edmonds1965]. Grover’s search algorithm employs quantum superposition and reversible computation in order to query many elements of the search space simultaneously.
Grover’s algorithm was later experimentally demonstrated in [@chuang1998]. The algorithm provides a polynomial speed-up when compared with the best-performing classical search algorithms. As previously mentioned, any such classical algorithm requires $O(N)$ time in order to search $N$ elements. Grover’s algorithm requires $O(\sqrt{N})$ time, providing a quadratic speedup, which is considerable when $N$ is large.
### Oracle unitary operator \[sec:oracleUnitaryOperator\]
In quantum computation mathematical objects known as unitary operators are responsible for the time-evolution of the state of a close quantum system. This is one of the foundational principles behind quantum computation, and is also known as the evolution postulate [@kaye2007a]. A black box, also referred to as an oracle [@nielsen2000], representing a unitary operator $U$, is employed in order to indicate, through a reverse of the associated amplitude, which of the values present in an amplitude register corresponds to the searched ones. This process can be performed by adding an additional input bit $c$ to the original $n$-bit input register $x$ and performing a XOR operation. This behaviour is illustrated in Expression \[eq:oracle\] which employs the ket notation introduced by Paul Dirac [@dirac1939] [@dirac1981]. Classical reversible circuitry can also be described through such a formulation, albeit without employing the ket notation.
$$U : {| x \rangle} {| c \rangle} \mapsto {| x \rangle} {| c \oplus g( x ) \rangle}
\label{eq:oracle}$$
Grover’s algorithm employs a process of amplitude amplification, known as Grover’s iterate, in order to amplify the amplitudes of the solutions and in the process diminish those of the non-solutions. This process is performed by setting the control register $c$ to a specified eigenvector, which, when combined with Grover’s iterate can be mathematically proven to perform an inversion about the mean of the amplitudes [@kaye2007a]. As a direct result of Grover’s iterate, the probability of an answer bearing state increases. However, the amplitude of the solution value is amplified only in a linear way. If the function $f$ is provided as a black box, then $O(\sqrt{N})$ applications of the black box are necessary in order to solve the search problem with high probability for any input [@nielsen2000]. [^2]
Traditionally, classical computation is seen as an irreversible process, a direct consequence from the use of many-to-one binary gates. A logical gate is a function $f: \{0,1\}^{k} \rightarrow \{0,1\}^{l}$ from some fixed number $k$ of input bits to some fixed number $l$ of outputs bits [@mano2002]. A computation is said to be reversible if given the outputs we can recover the inputs [@toffoli1980a] [@toffoli1980b]. Mathematically, a reversible computation corresponds to the concept of a bijective function. It turns out that there is a general mechanism for converting irreversible computations into reversible ones. Each irreversible gate can be made reversible by adding some additional input and output wires [@kaye2007a]. This conversion introduces a certain number of inputs and outputs to each irreversible gate. It is this additional information that provides for reversible computation.
Intuitively, it should come as no surprise that an irreversible circuit can be made reversible by substituting each irreversible gate by an equivalent reversible gate [@toffoli1980a]. By substituting each element of the circuit with its respective inverse we are able to perform the inverse operation of the original circuit. In practice, this means that if we run the reversed circuit with an output, we will obtain the originating input bit register.
Emil Post’s research into the principals of mathematical logic and description of the complete sets of truth functions [@post1941] implied that all functions $f: \{ 0, 1 \}^{k} \rightarrow \{ 0, 1\}^{l}$ could be computed by binary circuits employing logical gates $\neg , \lor$ and $\land$. Since it is always possible to build a reversible version of an irreversible circuit, reversible computation can also compute all functions $f: \{ 0, 1 \}^{k} \rightarrow \{ 0, 1\}^{l}$.
### Quantum superpositions \[sec:quantumSuperpositions\]
In order to gain a “quantum advantage”, Grover assembles a quantum superposition containing all possible values that should be presented as input to the unitary operator. At its core, the superposition principle simply conveys the notion that multiple quantum states exist at the same time. Let ${| \psi \rangle}$ denote the $n$-bit input register presented to unitary operator $U$, then ${|\psi\rangle}$ takes the form illustrated in Expression \[eq:superpositionInputRegisterV1\].
$${| \psi \rangle} = \frac{1}{\sqrt{2^{n}}} \sum_{ x = 0}^{2^{n} - 1 } {| x \rangle}
\label{eq:superpositionInputRegisterV1}$$
Since unitary operators obey linearity principles we are now in a position to apply unitary operator $U$ to the superposition register. In practice this process means that all values present in the superposition register are processed simultaneously. This operation is illustrated in Expression \[eq:unitaryOperatorAppliedToSuperpositionInputRegister\]. Additionally, unitary operators as well as input registers can be described by matrices. Accordingly, in light of Expression \[eq:unitaryOperatorAppliedToSuperpositionInputRegister\], applying unitary operator $U$ to input register ${|\psi\rangle}$, can be understood as performing matrix multiplication $U {| \psi \rangle}$.
$$U {| \psi \rangle} = \frac{1}{\sqrt{2^{n}}} \sum_{ x = 0}^{2^{n} - 1 } U {| x \rangle}
\label{eq:unitaryOperatorAppliedToSuperpositionInputRegister}$$
Hybrid system \[sec:hybridSystem\]
----------------------------------
As previously mentioned, Grover’s algorithm performs a generic search by employing a unitary operator $U$. It was designed with the purpose of searching an unstructured collection of registers. I.e. the algorithm’s purpose can be understood as looking for binary strings within a collection. Albeit, since binary strings can be used to encode mathematical abstractions, can we build upon this behaviour in order to develop a hybrid system performing classical hierarchical search using Grover’s algorithm? This hybrid approach would combine traditional tree search mechanisms with the efficiency gains promised by Grover’s algorithm.
Assume that the unitary operator $U$ employed during Grover’s iterate is developed in such a way as to recognize potential solutions. Intuitively, it is fairly easy to see that $U$ will need to contemplate the sequence of steps taken in each path of a tree in order to determine if it leads to a solution. However, it needs to do so by restricting itself to a binary representation. This coding strategy serves two direct purposes, namely (i) provide a numerical basis on which the development of $U$ can be built-upon; (ii) being able to translate Grover’s output into a set of actions leading to a solution. These actions are to be interpreted as the set of decisions executed at each step of the tree search, leading from the root node to a goal state.
In order to formally introduce the coding mechanism lets start by considering the binary tree presented in Figure \[fig:constantBranchingFactor\]. The illustrated binary tree has a root node $A$. Also, has it is possible to see each layer of depth $d$ provides an additional $2^{d}$ nodes to the tree. At each node it is always possible to apply two actions, i.e. we have a branching factor $b = 2$. Each possible path leading to a leaf node can thus be perceived as a concatenation of the binary strings encoding the associated set of performed actions. Using this approach we are able to build a superposition ${| \psi \rangle}$ containing all the possible paths. Superposition ${| \psi \rangle}$ and a unitary operator $U$ can then be employed by Grover’s algorithm to determine the paths leading to solutions.
![A search tree with a constant branching factor $b = 2$, depth $d = 3$ for a total of $b^d = 8$ leaf nodes\[fig:constantBranchingFactor\].](binaryTreeExample1.eps){width="8cm"}
Objectives and Problems \[sec:objectivesAndProblems\]
-----------------------------------------------------
As previously mentioned, unitary operators can be derived by transforming irreversible circuits into reversible ones. Accordingly, it is always possible to build a unitary operator $U$ which checks if a set of actions produces a solution. In this work we will not be concerned with the actual implementation details of $U$ but rather on how such a problem could potentially be approached. Additionally, we will also be interested in determining how such an hybrid system would be affected by traditional search concepts. For instance, what are the ramifications of a variable such as the branching factor? Would the system require the use of a constant branching factor? Clearly, this is not always the case for the complete set of problems that can potentially be addressed by search algorithms. When considering a non-constant branching factor, what would be the associated impacts in overall system performance? Additionally, traditional search strategies typically employ some kind of information to determine which states are more promising than others when trying to reach a goal state. Can an heuristic bounded procedure be incorporated into such an approach? If so, do we stand to gain any significant advantage?
These and other questions will be addressed in the remaining sections of this work. In Section \[sec:branchingFactorRamifications\] we will focus on assessing how such an hybrid system would perform from a branching factor perspective. In Section \[sec:heuristicPerspective\] we devote our efforts to researching on the impacts of incorporating heuristic concepts into the hybrid proposal. Section \[sec:finalConsiderations\] presents and discusses the parallels, alongside the differences, between our proposal and another well-known kind of graph inspection tool, respectively the quantum random walk. We present the conclusions of this work in Section \[sec:conclusions\]. With these sections we hope to lend some intuition into the advantages, and disadvantages, of quantum computation and a possible model for hybrid hierarchical quantum search. We will strive for presenting a accessible mathematical analysis of our hybrid approach which takes into account the referred objectives.
Branching factor ramifications \[sec:branchingFactorRamifications\]
===================================================================
In theoretical computer science one possible way to measure a problem’s complexity consists in assessing how long a given algorithm takes to find a solution. However, time performance is dependent on a multitude of hardware related factors. Accordingly, it is often more suitable to take appropriate steps to determine the total number of items that are to be evaluated. In the case of a classical tree search this equates to the number of nodes to take into account. From a classical tree search perspective complexity is expressed in terms of $b$, the branching factor or maximum number of successors of any node; $d$, the depth of the shallowest goal node; $m$, the maximum length of any path in the state space [@russell2003]. Section \[sec:constantBranchingFactor\] focuses on the aspects surrounding a constant branching factor. The requirements for a non-constant branching factor are presented in Section \[sec:nonConstantBranchingFactor\]. The impact of using a non-constant branching factor is presented in Section \[sec:mathematicalAnalysis\].
Constant branching factor \[sec:constantBranchingFactor\]
---------------------------------------------------------
As previously stated, our proposal for a hybrid quantum tree search system only relied on a constant branching factor $b$. This was mostly due to simplification reasons. However, a constant branching factor requirement is not feasible when considering potential applications of search algorithms. Since, at its essence, our system can be perceived as evaluating a superposition of all possible paths up to a depth level $d$, it is pivotal to determine the impact of a non-constant branching factor in our approach. Lets proceed by examining a couple of examples in order to have a clear understanding of the search process.
Figure \[fig:constantBranchingFactor\] illustrates a search tree where at any given node it is always possible to apply two actions (respectively labeled as $a_{0}$ and $a_{1}$), i.e. the branching factor for this search tree is $2$. Recall from Section \[sec:hybridSystem\] that we need to encode these actions in a binary fashion. In order to do so we need to determine how many bits $n$ do we require. For an even number of actions we can simply calculate the base-$2$ logarithm. However, we need to take into account that an odd number of actions might be required. In this case we need to map the value into the next largest integer, which can be done through the ceiling function, i.e. $n = \left \lceil log_{2}{|a|} \right \rceil$ bits, where $|a|$ denotes the cardinality of the action set. Notice that the complete range of values allowed with $n$ bits might not be used, i.e. $|a| < 2^{n}$. From our point of view, it is the unitary operator’s responsibility to validate whether a binary string is an admissible action.
In the case of the search tree illustrated in Figure \[fig:constantBranchingFactor\] which possesses a branching factor $b = 2$ actions we need $\lceil log_{2}{2} \rceil = 1$ bit. Accordingly, let value $0$ denote $a_{0}$ and value $1$ represent $a_{1}$. The binary strings encoding the paths leading to each leaf nodes of the search tree illustrated in Figure \[fig:constantBranchingFactor\] are presented in Table \[table:binaryEncondingsConstantBranchingFactor\].
-- ----- ----- -----
$0$ $0$ $0$
$0$ $0$ $1$
$0$ $1$ $0$
$0$ $1$ $1$
$1$ $0$ $0$
$1$ $0$ $1$
$1$ $1$ $0$
$1$ $1$ $1$
-- ----- ----- -----
: Binary encoding for each possible path of the search tree illustrated in Figure \[fig:constantBranchingFactor\]\[table:binaryEncondingsConstantBranchingFactor\]
Suppose we wish to perform a search up to depth level $d$. We can easily build a string of $d$ elements, one for each possible depth, with each element requiring a binary representation using $n$ bits. In total, our binary string will employ $n \times d$ bits. We are also able to construct a quantum superposition ${| \psi \rangle}$ encompassing all the actions to be applied up to depth level $d$ as illustrated by Expression \[eq:superpositionInputRegisterV2\]. This superposition ${| \psi \rangle}$ can then be employed alongside our unitary operator $U$ and Grover’s algorithm.
$${| \psi \rangle} = \frac{1}{\sqrt{2^{n \times d}}}\sum_{ x = 0}^{2^{n \times d} - 1 } {| x \rangle}
\label{eq:superpositionInputRegisterV2}$$
Non-constant branching factor \[sec:nonConstantBranchingFactor\]
----------------------------------------------------------------
Now consider the tree presented in Figure \[fig:nonConstantBranchingFactor\] with an action set $a = \{a_{0}, a_{1}, a_{2}, a_{3},a_{4}\}$. The first thing one notices is that we no longer have a constant branching factor at each node. In fact for this particular case it is convenient to distinguish between two types of branching factor, namely, the theoretical maximum branching factor $b_{max} = |a| = 5$, and the average branching factor of each node $b_{avg} = 2$, not including the leafs. In order to encode each of the possible actions we require $n = \left \lceil \log_{2}{|a|} \right \rceil = 3$ bits. Let the encodings of each action be those presented in Table \[table:binaryEncondingsProductionNonConstantBranchingFactor\]. Accordingly, the binary strings leading to each leaf node are illustrated in Table \[table:binaryEncondingsNonConstantBranchingFactor\].
![A search tree with a maximum branching factor $b_{max} = 5$ and an average branching factor $b_{avg} = \frac{2 + 1 + 2 + 2 + 1 + 4}{6} = 2$. \[fig:nonConstantBranchingFactor\]](nonBinaryTreeExample.eps){width="8cm"}
action $b_{0}$ $b_{1}$ $b_{2}$
----------- --------- --------- ---------
$a_{0}$ $0$ $0$ $0$
$a_{1}$ $0$ $0$ $1$
$a_{2}$ $0$ $1$ $0$
$a_{3}$ $0$ $1$ $1$
$a_{4}$ $1$ $0$ $0$
undefined $1$ $0$ $1$
undefined $1$ $1$ $0$
undefined $1$ $1$ $1$
: Binary encoding for each possible path of the search tree illustrated in Figure \[fig:nonConstantBranchingFactor\]\[table:binaryEncondingsProductionNonConstantBranchingFactor\]
-- ------- ------- -------
$000$ $000$ $000$
$000$ $000$ $001$
$001$ $010$ $011$
$001$ $100$ $000$
$001$ $100$ $001$
$001$ $100$ $010$
$001$ $100$ $100$
-- ------- ------- -------
: Binary encoding for each possible path of the search tree illustrated in Figure \[fig:constantBranchingFactor\]\[table:binaryEncondingsNonConstantBranchingFactor\]
In order to employ Grover’s algorithm we would need to build a quantum superposition state ${| \psi^{\prime}\rangle}$ similar to that presented in Expression \[eq:superpositionInputRegisterV2\]. Again, our superposition would consist of those states with $d$ elements, each of which with $n$ bits. However, there is a crucial difference between both tree searches. Consider the case illustrated in Figure \[fig:nonConstantBranchingFactor\] where a search to depth level $d = 3$ alongside $b_{max} = 5$ is performed. Superposition ${| \psi^{\prime}\rangle}$ would contain all the quantum states belonging to the range $[0, 2^{d \times n} - 1] = [0, 2^{9} - 1] = [0, 511 ]$. I.e. we would be able encode 512 possible paths when in reality we would only need to encode the states presented in Table \[table:binaryEncondingsNonConstantBranchingFactor\] [^3]. In reality, the vast majority of the states present in the superposition would contain inadmissible configurations of actions.
It is important to draw attention to the fact that Grover’s algorithm provides a quadratic speedup $O( \sqrt{N} )$ where $N$ is the number of elements present in the superposition. By employing $n = \left \lceil \log_{2}{b_{max}} \right \rceil$ bits, when in practice $n = \left \lceil \log_{2}{b_{avg}} \right \rceil$ bits might have sufficed, we are extending the search space and in the process loosing some of the speedup provided by Grover’s algorithm.
Naturally, the question arises: Considering the above encoding mechanism, and a $b_{avg} < b_{max}$ when does Grover stop providing a speedup over classical approaches?
Analysis \[sec:mathematicalAnalysis\]
-------------------------------------
So, how can we proceed in order to analyze the problem depicted in the previous section? As with so many other fields it is usually easier to start out with an example and extrapolate from that. Accordingly, lets consider the following scenario: we want to perform a tree search using our hybrid quantum search system up to depth level $10$; the maximum branching factor, $b_{max}$ is $5$, however on average we are only able to perform three actions, i.e. $b_{avg} = 3$. Does our approach still provides an advantage over classical search strategies?
In order to answer this question we need to consider the complexities of classical search algorithms. Traditionally, these methods experience some type of exponential growth in the number of leaf nodes that need to be assessed. Since the number of elements to be evaluated is a function of the branching factor $b$ and the depth of the search $d$ the associated complexity is typically of the form $O(b^{d})$. Clearly, in the case of our scenario we need to differentiate between the two branching factors, respectively $b_{max}$ and $b_{avg}$. Accordingly, if we consider $b_{max}$ then a total of $ O( b_{max}^{d} ) = O( 5^{10} ) = 9765625$ nodes might potentially need to be evaluated. On the other hand, by employing $b_{avg}$ a total of $O( b_{avg}^{d} ) = O( 3^{10} ) = 59049$ nodes may be considered. These values differ by a factor of $\approx 165$ which is considerable when in practice it is acceptable to consider $b_{avg}$ as the *de facto* branching factor.
Now lets reflect on the number of times one would need to apply Grover’s iterate. In this case there is no distinction to be made between $b_{max}$ and $b_{avg}$ since we would still need to build a quantum superposition state encoding all of the $b_{max}$ actions up to a depth level $d$. As previously stated this would result in a total of $n \times d = 3 \times 10 = 30$ bits being required, where $n = \left \lceil \log_{2}{b_{max}} \right \rceil = \left \lceil \log_{2}{5} \right \rceil = 3$. Accordingly, for the above scenario we would need to apply Grover’s iterate a total of $O( \sqrt{N} ) = O( \sqrt{2^{30}} ) = 32768$ times in order to obtain a solution. By comparing the number of states classically evaluated by using $b_{avg}$ against the total number of iterations required by Grover’s algorithm we see that both values differ by a factor of $\approx 1.8$. Not surprisingly, although we still obtain a speedup over the classical approach it is severely lessened. Intuitively, it should be clear that this behaviour can be perceived in the following manner:
- As $b_{avg}$ grows closer to $b_{max}$ the number of times to apply Grover’s iterate will be optimal relatively to the classical approaches;
- As $b_{avg}$ grows more distant to $b_{max}$ the number of Grover iterations to apply will be closer to $b_{avg}^{d}$.
From the above reasoning we are now able to successfully determine the number of times that Grover’s iterate should be applied, respectively $|G|$, as illustrated by Expression \[eq:numberOfTimesToApplyGrover\].
$$\begin{aligned}
|G| &= \sqrt{N} \nonumber \\
&= \sqrt{2^{n \times d}} \nonumber \\
&= \sqrt{2^{\left \lceil log_{2}{b_{max}} \right \rceil \times d}} \nonumber \\
&= 2^{ \frac{ \left \lceil log_{2}{b_{max}} \right \rceil \times d }{2} }
\label{eq:numberOfTimesToApplyGrover}\end{aligned}$$
Keep in mind that we wish to determine where the threshold lies between the number of elements of a classical search, respectively $b_{avg}^{d}$ and the total number of times to apply Grover’s iterate $|G|$. This process can be formulated as presented in Expression \[eq:result1\] which when solved results in Expression \[eq:result2\].
$$\begin{aligned}
b_{avg}^{d} &= |G| \label{eq:result1} \\
\Leftrightarrow b_{avg}^{d} &= 2^{ \frac{ \left \lceil log_{2}{b_{max}} \right \rceil }{2}^{d} } \nonumber \\
\Leftrightarrow b_{avg} &= 2^{ \frac{ \left \lceil log_{2}{b_{max}} \right \rceil }{2} } \label{eq:result2}\end{aligned}$$
Accordingly, when $b_{avg} < 2^{ \frac{ \left \lceil log_{2}{b_{max}} \right \rceil }{2} }$ then the total number of nodes evaluated in classical search will be less than the number of times to apply Grover’s iterate, i.e. $b_{avg}^{d} < |G|$. Appropriately, when $b_{avg} > 2^{ \frac{ \left \lceil log_{2}{b_{max}} \right \rceil }{2} }$ then our hybrid system will yield a speedup over classical search algorithms. The plot of Expression \[eq:result2\], for $b_{max} \in [2,128]$, is presented in Figure \[fig:plotAverageBranchingFactorVsGroverIterations\]. The shaded area indicates those values of $b_{avg}$ that will produce better performance results classically over the proposed hybrid search.
![The area plot of $b_{avg} \leq 2^{ \frac{ \left \lceil log_{2}{b_{max}} \right \rceil }{2} } $ for $b_{max} \in [2,128]$. The shaded area indicates those values of $b_{avg}$ that will produce better performance results over our hybrid quantum production system. \[fig:plotAverageBranchingFactorVsGroverIterations\]](plotAverageBranchingFactorVsGroverIterations.eps){width="10cm"}
Figure \[fig:plotAverageBranchingFactorVsGroverIterations\] also showcases the characteristic ladder effect of functions employing the ceiling function required by the branching factor binary coding mechanism. Notice that in practice this means that there will always be ranges of values for $b_{max}$ where the number of iterations, $|G|$, will remain the same. Consequently, the average branching factor, $b_{avg}$, for the boundary condition presented in Expression \[eq:result1\] will also remain constant in those intervals. This happens despite the fact that $b_{max}$ is growing in the associated range.
Lets proceed by elaborating a bit more on each $b_{max}$ and the associated range of $b_{avg}$ values. In order to encode in a binary fashion any $b_{max}$ value we require a total of $n = \lceil log_{2}{b_{max}} \rceil$ bits. The use of the ceiling function effectively forces certain ranges of $b_{max}$ values to require the same number of $n$ bits. As we have seen previously, the number of Grover iterations is a function of the total number of bits required. We can see how this influences $b_{avg}$ by taking Expression \[eq:result2\] into account. Clearly, for those $b_{max}$ values requiring the same number of bits, the associated $b_{avg}$ values will remain the same. It is only when the required number of bits for $b_{max}$ changes that an impact will be felt regarding $b_{avg}$. Accordingly, we are interested in studying what happens when a transition if performed from, for example, $n$ to $n + 1$ bits. In this case the $b_{avg}$ value grows from $ 2^{\frac{n}{2}}$ to $2^{\frac{n + 1}{2}}$ which differ by a factor of $\sqrt{2}$. Let $b_{avg}^{n}$ denote the average branching factor $b_{avg}$ associated with an interval requiring $n$ bits. Then, it is possible to express $b_{avg}^{n + 1}$ as a function of $b_{avg}^{n}$. This recurrence behaviour is illustrated in Expression \[eq:relationshipBetweenIntervals\].
$$b_{avg}^{n + 1} = \sqrt{2} b_{avg}^{n}
\label{eq:relationshipBetweenIntervals}$$
Expression \[eq:relationshipBetweenIntervals\] can be improved if we allow ourselves to leave behind the use of the ceiling function employed in Expression \[eq:result2\]. In doing so, we are deliberately abdicating of the ladder effect presented in Figure \[fig:plotAverageBranchingFactorVsGroverIterations\] and giving the $b_{avg}$ function a quadratic form. This new formulation is presented in Expression \[eq:functionWithoutCeiling\].
$$b_{avg} = 2^{ \frac{ log_{2}{ b_{max} } }{2} } = \sqrt{b_{max}}
\label{eq:functionWithoutCeiling}$$
Since we no longer have a “range-bounded” function, a direct mapping of Expression \[eq:relationshipBetweenIntervals\] is impossible. Instead, we can simply define a function, $b^{\prime}_{avg}$, depicting the new upper-range values of $b_{avg}$, as illustrated in Expression \[eq:relationshipBetweenFunctions\]. This $\sqrt{2}$-constant growth factor is depicted in Figure \[fig:plotAverageBranchingFactorVsGroverIterationsConstantGrowth\].
$$b^{\prime}_{avg} = \sqrt{2} b_{avg} = \sqrt{2 b_{max}}
\label{eq:relationshipBetweenFunctions}$$
![The $\sqrt{2}$-constant growth between $b_{avg}^{\prime}$ and $b_{avg}$ superimposed on the original $b_{avg} = 2^{ \frac{ \left \lceil log_{2}{b_{max}} \right \rceil }{2} } $ function for $b_{max} \in [2,128]$. \[fig:plotAverageBranchingFactorVsGroverIterationsConstantGrowth\]](plotAverageBranchingFactorVsGroverIterationsConstantGrowth.eps){width="10cm"}
Heuristic Perspective \[sec:heuristicPerspective\]
==================================================
Determining which production rule should be applied to the working memory is an important part of the control strategy of production system theory. In a great deal of occasions an artificial intelligence application does not possess the adequate level of knowledge allowing for full differentiation amongst the production rules [@nilsson1982].
At any given point in time during the search process it would be useful to somehow know which production might produce a state which is closer to a goal state. Intuitively, we may describe this process as trying to determine the quality of a path of actions with an optimal solution having the lowest path cost among all solutions [@russell2003]. A key component of these systems, and many other algorithms in artificial intelligence, consists in an heuristic function $h(n)$ responsible for presenting an estimate of the distance that a given state $n$ is relatively to a goal state. Function $h(n)$ is typically employed alongside a function $g(n)$ which reflects the search cost incurred to reach state $n$. Traditionally, the conjunction of both these functions is incorporated within a single evaluation function $f( n )$ as illustrated by Expression \[eq:evaluationFunction\].
$$f(n) = g(n) + h(n)
\label{eq:evaluationFunction}$$
Not surprisingly, function $h(n)$ can have a number of different definitions depending on the specifics of the problem at hand. In a sense, heuristic functions are responsible for providing extra-information about how-well a search is performing. Production systems need not depend on the use of heuristics. Systems which do not take into consideration any kind of auxiliary information are referred to as uninformed strategies. In the case of “uninformed” production systems the choice of which rule to apply is performed at random. On the other hand, “informed” strategies enable a production system to select an appropriate rule.
Our quantum production system model can be perceived as systematically trying to apply actions in an uninformed fashion. This operation is performed until a goal state is reached. Such behaviour resembles that of a standard blind depth-limited search process. Intuitively, the heuristic concepts incorporated into informed search strategies appear to be an adequate extension idea to our quantum production system. However, it remains to be seen if these concepts can be adequately mapped into our approach and if doing so provides an advantage. The remainder of this section is organized as follows: Section \[sec:theQuantumHeuristic\]) considers how to incorporate the use of an heuristic the unitary operator; Section \[sec:interpretation\]) provides an analysis of the quantum computation procedure incorporating the use of an heuristic. Section \[sec:extentingTheQuantumHeuristic\]) builds on these results to present an extended quantum heuristic mechanism.
The Quantum Heuristic \[sec:theQuantumHeuristic\]
-------------------------------------------------
As previously mentioned, at its core, our system can be perceived as evaluating a superposition of all possible paths. Expression \[eq:superpositionInputRegisterV2\] illustrated this perspective, where a quantum superposition state is composed by an amplitude value and a multitude of sub-states, i.e. the computational basis. From a quantum mechanics perspective it is important to reinforce the idea that the mechanisms allowing quantum states of a superposition to communicate with the other states are complex and restricted in what they achieve. Those algorithms that are able to provide a meaningful speedup do so by employing, and determining, some type of global property. E.g. Grover’s algorithm is able to determine the mean amplitude $A$ of a quantum superposition and perform an inversion about the mean [@grover1996]. Also Shor’s algorithm for factoring numbers is able to efficiently determine the period of periodic quantum states [@shor1994]. Accordingly, there is no clear method to communicate between states.
This fact immediately poses a problem: traditionally, the use of heuristics is employed to choose among possible tree paths, ideally producing an optimal sequence of actions. However, with our system, we are unable to quantum mechanically perform such a comparison. Ergo, is there any way to incorporate any heuristic concepts into our hybrid approach?
From a simplified point of view, an heuristic function outputs a value estimating the distance to a goal. We can therefore opt to consider only those states whose $f$ value is below a certain threshold $T$. Notice that in doing so we are deliberately ignoring the fact that these same states may later on provide an optimal solution path. In order to for the search process to incorporate the heuristic function we need to reflect it upon the unitary operator’s design. Recall that a unitary operator $U$ in order to be employed by Grover’s algorithm is only required to flip the amplitudes of the solutions states. Expression \[eq:unitaryOperatorWithHeuristic\] reflects both of the previous requirements, where ${| n \rangle}$ represents the current node being processed and $a_{i}$ an action taken at depth $i$.
$$U {| b \rangle} {| a_{1} a_{2} \cdots a_{d} \rangle} =
\left\{
\begin{array}{lll}
-&{| b \rangle} {| a_{1} a_{2} \cdots a_{d} \rangle} & \text{if} f(b, a_{1}, a_{2}, \cdots, a_{d}) \leq T \\
&{| b \rangle} {| a_{1} a_{2} \cdots a_{d}\rangle} & \text{otherwise}
\end{array} \right.
\label{eq:unitaryOperatorWithHeuristic}$$
In a similar way as to deciding which of two possible heuristics to employ, the matter of deciding the threshold value $T$ could be left to statistical studies, or even informed intuition based on hands-on experience [@nilsson1982]. Take notice that in no way did we circumvent the problem of comparing multiple paths. In conclusion, we are able to incorporate some of the concepts surrounding the use of heuristics, but not all of them.
Interpretation \[sec:interpretation\]
-------------------------------------
What happens when we employ a unitary operator $U$ with the form presented in Expression \[eq:unitaryOperatorWithHeuristic\] alongside Grover’s algorithm? I.e. do we stand to gain anything by incorporating heuristic concepts into our hybrid search system? Section \[sec:decomposingTheSuperpositionState\] illustrates how superposition ${| \psi \rangle}$ can be decomposed into two components. Section \[eq:heuristicImpact\]) provides a graphical illustration of the impacts of performing search up to different depth levels.
### Decomposing the superposition state \[sec:decomposingTheSuperpositionState\]
Answering this question requires extending our analysis of what happens when unitary operator $U$ is applied to superposition ${| \psi \rangle}$, i.e. $ U {| \psi \rangle}$. Suppose superposition ${| \psi \rangle}$ takes the form illustrated in Expression \[eq:standardQuantumSuperposition\], where $n$ represents the number of bits.
$${| \psi \rangle} = \sqrt{ \frac{1}{2^{n}} } \sum_{x = 0}^{2^{n} - 1} {|x\rangle}
\label{eq:standardQuantumSuperposition}$$
Before advancing any further, notice that in an uniform superposition, associated with each state ${|x\rangle}$ there is an amplitude $\alpha \in \mathbb{C}$, which in this case is $\sqrt{ \frac{1}{2^{n}}}$ for all states. Let $\alpha_{i}$ denote the amplitude associated with state ${|i\rangle}$. Quantum computation requires the norm of amplitudes to be unit-length, i.e. $\sum_{x = 0}^{2^{n} - 1} |\alpha_{x}|^{2} = 1$, at all times. Not surprisingly any unitary operator must also preserve the norm.
Now note that we can decompose our initial quantum superposition $ {| \psi \rangle} $ into two parts [@kaye2007a]. One part will contain all those states that are solutions, which we will respectively label as set $X_{good}$. Another part will include the non-solutions states, respectively labeled as set $X_{bad}$. Also, assume that the problem we are interested in solving contains $k$ solutions. This implies that $|X_{good}| = k$ and $|X_{bad}| = 2^{n} - k$. Accordingly, an uniform superposition of the states belonging to $X_{good}$ would set an equal amplitude among its $k$ elements, i.e. $\sqrt{ \frac{1}{k} }$. By the same reasoning, an uniform superposition of the states in $X_{bad}$ would impose an amplitude values of $\sqrt{ \frac{1}{2^{n} - k} }$. Accordingly, we can define the superposition states ${| \psi_{good} \rangle}$ and ${| \psi_{bad} \rangle}$, respectively presented in Expression \[eq:psiGood\] and Expression \[eq:psiBad\].
$${| \psi_{good} \rangle} = \sqrt{ \frac{1}{k} } \sum_{x \in X_{good}} {|x\rangle}
\label{eq:psiGood}$$
$${| \psi_{bad} \rangle} = \sqrt{ \frac{1}{2^{n} - k} } \sum_{x \in X_{bad}} {|x\rangle}
\label{eq:psiBad}$$
We are now able to express superposition ${| \psi \rangle}$ in terms of the subspaces ${| \psi_{good} \rangle}$ and ${| \psi_{bad} \rangle}$. This process is illustrated in Expression \[eq:superpositionReformulated\] which is indispensable to the rest of our analysis.
$${| \psi \rangle} = \sqrt{ \frac{k}{2^{n}} } {| \psi_{good} \rangle} + \sqrt{ \frac{2^{n} - k}{2^{n}}} {| \psi_{bad}\rangle}
\label{eq:superpositionReformulated}$$
The requirement that the norm must be preserved at all times induces a probabilistic behaviour. Consider state ${| \psi \rangle}$ whose norm is $\left | \sqrt{ \frac{k}{2^{n}} } \right |^{2} + \left | \sqrt{ \frac{2^{n} - k}{2^{n}}} \right |^{2} = 1$. I.e. we have a sum of values which sum up to $1$ similarly to a probability distribution. Accordingly, the probability of obtaining state ${| \psi_{good}\rangle} = \left |\sqrt{ \frac{k}{2^{n}} } \right |^{2}$ and state $ {| \psi_{bad}\rangle} = \left | \sqrt{ \frac{2^{n} - k}{2^{n}}} \right |^{2}$. Applying Grover’s iterate effectively changes the amplitudes, maximizing the probability of obtaining a solution contained in the subspace spawned by ${| \psi_{good} \rangle}$. As a result, the amplitude associated with state ${| \psi_{good} \rangle}$ will increase. Since the norm of state ${| \psi \rangle}$ must be preserved, employing Grover also implies that the amplitude of ${| \psi_{bad} \rangle}$ will be decreased.
### Heuristic Impact \[eq:heuristicImpact\]
In order to continue with our analysis lets assume we possess an admissable heuristic which always eliminates candidate states with each additional level of depth. Although this is a rather optimistic strategy the heuristic’s behaviour can be viewed as ideal. We will use this best case scenario to demonstrate the system’s potential performance.
Given a sufficiently high depth level $d$ the heuristic will have to eventually produce the exact number of solutions $k$. Let $k_{d}$ denote the number of solutions at depth level $d$. Then, with such an heuristic we would have $k_{1} < k_{2} < \cdots < k_{d} \leq k$. However, we can never produce fewer than $k$ solutions, since that would violate basic assumptions about the search space of the problem.
The above process can be visualized geometrically. Notice that our initial superposition ${| \psi \rangle}$ containing $k$ solutions can be understood as vector with two components $( {| \psi_{good}\rangle}, {| \psi_{bad}\rangle} ) = ( \sqrt{ \frac{k}{2^{n}} }, \sqrt{ \frac{2^{n} - k}{2^{n}}})$. Therefore, we are able to map it into a two dimensional plane with axis $ {| \psi_{good}\rangle}$ and ${| \psi_{bad}\rangle}$. This mapping process is illustrated in Figure \[fig:geometricVisualization\]. From a quantum computation perspective, we can envisage the use of different states ${| \psi_{k_{d}} \rangle}$ reflecting a search up to depth level $d$. State ${| \psi_{k_{d}} \rangle}$ is a simple reformulation in terms of $k_{d}$ of state ${|\psi\rangle}$, as presented in Expression \[eq:superpositionReformulatedInTermsOfSolutionsPerDepth\].
$${| \psi_{k_{d}} \rangle} = \sqrt{ \frac{k_{d}}{2^{n}} } {| \psi_{good} \rangle} + \sqrt{ \frac{2^{n} - k_{d}}{2^{n}}} {| \psi_{bad}\rangle}
\label{eq:superpositionReformulatedInTermsOfSolutionsPerDepth}$$
Clearly, as the number of solutions $k_{d}$ grows, to the allowable maximum of $k$, the ${|\psi_{good}\rangle}$ and ${|\psi_{bad}\rangle}$ components will tend towards the values of the original state vector ${|\psi\rangle}$. This behaviour is illustrated in Expression \[eq:limitsStudy\].
$$\lim_{k_{d} \rightarrow k } \left( \sqrt{ \frac{k_{d}}{2^{n}} }, \sqrt{ \frac{2^{n} - k_{d}}{2^{n}}} \right) = \left( \sqrt{ \frac{k}{2^{n}} }, \sqrt{ \frac{2^{n} - k}{2^{n}}} \right)
\label{eq:limitsStudy}$$
Not surprisingly, as the depth of the search increases the corresponding state vector ${| \psi_{k_{d}} \rangle}$ gets closer to the original ${| \psi \rangle}$, as can be perceived from Figure \[fig:geometricVisualization\]. Additionaly, each state ${| \psi_{k_{d}} \rangle}$ can be understood as forming an angle $\theta$ whose tangent is shown in Expression \[eq:angleTangent\].
$$\tan \theta_{k_{d}} = \frac{ \sqrt{ \frac{k_{d}}{2^{n}} } }{ \sqrt{ \frac{2^{n} - k_{d}}{2^{n}}} } = \sqrt{\frac{k_{d}}{2^{n} - k_{d}}}
\label{eq:angleTangent}$$
Using this approach we can perform a comparison between the angles of different search depth levels. Lets say we wish to compare how the search advanced between depth levels $d_{1}$ and $d_{2}$. We can define an operator $\Delta \theta_{k_{d_{1}}, k_{d_{2}}}$ with the behaviour defined in Expression \[eq:angleOperator\].
$$\Delta \theta_{k_{d_{1}}, k_{d_{2}}} := \underbrace{\arctan{\sqrt{\frac{k_{d_{2}}}{2^{n} - k_{d_{2}}}}}}_{\theta_{k_{d_{2}}}} - \underbrace{\arctan{\sqrt{\frac{k_{d_{1}}}{2^{n} - k_{d_{1}}}}}}_{\theta_{k_{d_{1}}}}
\label{eq:angleOperator}$$
Operator $\Delta \theta_{k_{d_{1}}, k_{d_{2}}}$ provides a mechanism for determining the operational success of adding $(d_{2} - d_{1})$ extra levels of depth relatively to $d_{1}$. Accordingly, small $\Delta \theta_{k_{d_{1}}, k_{d_{2}}}$ values can be understood as not contributing in a significant manner to changing the system’s overall state. Conversely, high $\Delta \theta_{k_{d_{1}}, k_{d_{2}}}$ values reveal that the system’s state significantly shifted towards ${|\psi\rangle}$.
![The geometric interpretation of the original state vector ${| \psi \rangle}$ alongside different state vectors ${| \psi_{i}\rangle}$ reflecting the state attained by performing a search to depth-level $i$. Deeper searches will be closer to the original ${|\psi\rangle}$ state. All states are unit-length vectors. \[fig:geometricVisualization\]](geometricVisualization.eps){width="8cm"}
Ultimately, we can only expect to get as far as state ${|\psi\rangle}$. From this point on Grover’s usual $O(\sqrt{N})$ iterations will still be required in order to perform a rotation of ${| \psi \rangle}$ towards ${| \psi_{good}\rangle} $, leaving the algorithms overall complexity unchanged.
Extending the quantum heuristic \[sec:extentingTheQuantumHeuristic\]
--------------------------------------------------------------------
Can the concepts of the quantum heuristic be extended in such a way as to perform some kind of useful task? In order to answer this question we will assume that the heuristic function employed has some kind of probabilistic distribution. Accordingly, we will start by reviewing some principles surrounding heuristic distributions. We will then show how to incorporate these concepts alongside our hybrid quantum search proposal.
### Heuristic distributions \[eq:heuristicDistributions\]
Lets assume that the heuristic function employed has a probabilistic distribution, i.e. the probability of each output value is between $0$ and $1$. This behaviour can be written as $ 0 \leq P( X = x ) \leq 1$ where $X$ is a random variable representing an output event, and $x$ is a possible value of $X$. Additionally, a random variable may be either discrete or continuous depending on the values that it can assume. Random variables whose set of possible values belong to $\mathbb{Z}$ are said to be discrete. On the other other hand, continuous variables map events to an uncountable set such as $\mathbb{R}$. For a discrete random variable $X$, the sum of the set containing all possible probability values is $1$ as illustrated by Expression \[eq:discreteRandomVariable\].
$$\sum_{i = 1}^{n} P( X = x_{i} ) = 1
\label{eq:discreteRandomVariable}$$
As a concrete example of a discrete heuristic we can consider the sliding block puzzle, also known as the $n$-puzzle, search problem. A sliding block puzzle challenges a player to shift pieces around on a board without lifting them to establish a certain end-configuration, as illustrated in Figure \[fig:3x3slidingBlockPuzzle\]. This non-lifting property makes finding moves, and the paths opened up by each move important parts of solving sliding block puzzles [@hordern1987]. Accordingly, we can define a heuristic function $h_{1}$ which simply calculates the number of misplaced tiles between a board and a target board configuration. Function $h_{1}$ outputs values belonging to the range $[0, 9]$ and consequently can be classified as discrete. The discrete probability distribution for function $h_{1}$ regarding the $8$-puzzle is presented in Figure \[fig:discreteHeuristic8Puzzle\].
![Discrete probability distribution for $h_{1}$ when applied to the $8$-puzzle \[fig:discreteHeuristic8Puzzle\]](discreteHeuristic8Puzzle.eps){width="10cm"}
If $X$ is a continuous random variable then there exists a nonnegative function $P(X)$, the probability density function of the random variable $X$. The density function $P(X=c)$ is defined as the ratio of the probability that $X$ falls into an interval around $c$, divided by the width of the interval, as the interval width goes to zero [@degroot2002]. This process is illustrated in Expression \[eq:probabilityDensityFunction\]. Additionally, the density function must be nonnegative for all arguments and must obey the behaviour presented in Expression \[eq:probabilityDensityFunctionMustHaveIntegralEqualToOne\] [@russell2003].
$$P( X = c ) = \lim_{dx \to 0} P( c \leq X \leq c + dx ) / dx
\label{eq:probabilityDensityFunction}$$
$$\int_{-\infty}^{+\infty} P(X) dx = 1
\label{eq:probabilityDensityFunctionMustHaveIntegralEqualToOne}$$
As an example of a continuous heuristic function we can again consider the $n$-puzzle search problem. This time we need only to determine an heuristic function $h_{2}$ mapping values into a real codomain. This can be done by employing a metric such as the euclidean distance. For instance, we can define $h_{2}$ in such a fashion as to calculate the euclidean distance for all the corresponding elements of a board and a target board configuration. I.e., $ h_{2} = \sum_{ \forall{ \text{ tiles}} } d( l_{\text{tile}}, l_{\text{tile}}^{\prime})$, where $d$ is the euclidean distance, $l_{\text{tile}}$ the location of a tile in a board configuration and $l_{\text{tile}}^{\prime}$ the location of the corresponding tile in the target configuration. The probability density function for $h_{2}$ regarding the $8$-puzzle is presented in Figure \[fig:continuousHeuristic8Puzzle\].
![Continuous probability density function for $h_{2}$ when applied to the $8$-puzzle \[fig:continuousHeuristic8Puzzle\]](continuousHeuristic8Puzzle.eps){width="10cm"}
The probability that a random variable $X$ takes on a value that is less than or equal to $x$ is referred to as the cumulative distributive function $F$ which has the form shown in Expression \[eq:cumulativeDistributiveFunction\].
$$F( x ) = P( X \leq x )
\label{eq:cumulativeDistributiveFunction}$$
The cumulative distribution function $F(a)$ for a discrete random variable is a simple sum of the values up to element $a$. This behaviour is illustrated in Expression \[eq:cumulativeDistributionFunctionDiscreteRandomVariable\]. Similarly, a cumulative probability density function $F(a)$ can be defined for a continuous random variable as shown in Expression \[eq:cumulativeDistributionFunctionContinuousRandomVariable\].
$$F(a) = \sum_{\text{all } x \leq a} P( X = x )
\label{eq:cumulativeDistributionFunctionDiscreteRandomVariable}$$
$$F(a) = \int_{-\infty }^{a} P(x) dx
\label{eq:cumulativeDistributionFunctionContinuousRandomVariable}$$
Now suppose we wish to determine when, that is for which values, the cumulative distribution function equals a certain probability $p$, i.e. $F( x ) = p$. In probability theory, this mapping is known as the quantile function of a cumulative distribution function $F(x)$ and is expressed as $F^{-1}(p) = x$. Once more we need to be careful in order to differentiate between discrete and continuous random variables. In the former, there may exist gaps between values in the domain of the cumulative distribution function, accordingly $F^{-1}$ is defined as presented in Expression \[eq:quantileFunctionDiscreteRandomVariable\] where $inf$ denotes the infimum operator [@gilchrist2000].
$$F^{-1}(p) = inf\{ x \in \mathbb{R} : p \leq F(x) \}
\label{eq:quantileFunctionDiscreteRandomVariable}$$
In the case of a continuous random variable obtaining a clear expression for the quantile function is not so trivial since it requires determining the inverse of an integral. Notwithstanding, it is still possible to determine such expressions, for instance, the quantile function of a normal distribution with mean $\mu$ and standard deviation $\sigma$ is illustrated in Expression \[eq:quantileFunctionNormalDistribution\], where $erf$ is the Gauss error function [@gilchrist2000].
$$F^{-1}( p, \mu, \sigma^{2} ) = \mu + \sqrt{2} \sigma erf^{-1}( 2p - 1), \quad p \in [0, 1]
\label{eq:quantileFunctionNormalDistribution}$$
### Extended quantum heuristic \[extendedQuantumHeuristic\]
Naturally, the question arises, how can this process be combined with our hybrid quantum search approach? It turns out that when a fourth of all possible states analyzed by Grover’s algorithm are marked as solutions, i.e. $k = \frac{1}{4} . N$ then a single iteration is required in order to obtain with certainty one of the solution states [@hirvensalo2004]. Accordingly, we can try to fine tune the behaviour of our unitary operator such that it outputs a solution for a fourth of all cases. This procedure can be performed with the assistance of the quantile function concept introduced in the previous section.
Lets say we have an heuristic function $f: X \to Y$ and are interested in obtaining the states which are closest to a solution. Accordingly, we are interested in marking as a solution those states that produce the smallest values of codomain $Y$. Lets assume that the states which are closer to a goal node are those who tend to happen less times (as exemplified by Figure \[fig:discreteHeuristic8Puzzle\] and Figure \[fig:continuousHeuristic8Puzzle\]). From a probabilistic point-of-view it is possible to check if the heuristic value is less than or equal to the quantile function output for a probability of $25\%$. This behaviour is illustrated in Expression \[eq:unitaryOperatorWithExtendedHeuristicV1\].
$$U {| b \rangle} {| a_{1} a_{2} \cdots a_{d} \rangle} =
\left\{
\begin{array}{lll}
-&{| b \rangle} {| a_{1} a_{2} \cdots a_{d} \rangle} & \text{if} f(b, a_{1}, a_{2}, \cdots, a_{d}) \leq F^{-1}(0.25) \\
&{| b \rangle} {| a_{1} a_{2} \cdots a_{d}\rangle} & \text{otherwise}
\end{array} \right.
\label{eq:unitaryOperatorWithExtendedHeuristicV1}$$
Ideally, by using this approach it is possible to obtain a superposition containing one fourth of the “closest” states to a goal configuration by applying a single iteration of Grover’s algorithm. Additionally, we can further expand on the results of Expression \[eq:unitaryOperatorWithExtendedHeuristicV1\] in order to contemplate different sections of a probability distribution. For instance, we can choose to obtain in a single iteration of Grover’s algorithm the states which lie between heuristic values $(F^{-1}(0.5) - F^{-1}(0.25))$. Similarly, we could also choose to obtain the states belonging to $(F^{-1}(0.75) - F^{-1}(0.5))$ or $(F^{-1}(1) - F^{-1}(0.75))$. More generally, let $a$ and $b$ denote two probabilities values such that $b - a = 0.25$, then it is possible to determine if a given heuristic value belongs to the range $[F^{-1}(a), F^{-1}(b)]$. This strategy for selecting $25\%$ of the search space is presented in the unitary operator shown in Expression \[eq:unitaryOperatorWithExtendedHeuristicV2\],
$$U {| b \rangle} {| a_{1} a_{2} \cdots a_{d} \rangle} =
\left\{
\begin{array}{lll}
-&{| b \rangle} {| a_{1} a_{2} \cdots a_{d} \rangle} & \text{if} f(b, a_{1}, a_{2}, \cdots, a_{d}) \in [F^{-1}(a), F^{-1}(b)] \\
&{| b \rangle} {| a_{1} a_{2} \cdots a_{d}\rangle} & \text{otherwise}
\end{array} \right.
\label{eq:unitaryOperatorWithExtendedHeuristicV2}$$
In a certain sense, employing this type of procedure allows for a kind of partial selection of the search space to be performed using a single Grover iteration. The selection is only partial because upon measurement a random collapse amongst the marked states is obtained.
Final considerations \[sec:finalConsiderations\]
================================================
It is important to mention that some of the graph dynamics considered in this work relate directly to the well-studied quantum random walks on graphs (for an introduction to this research area please refer to [@ambainis2003], [@kempe2003b] and [@ambainis2004]. Quantum random walks are the quantum equivalents of their classical counterparts (we refer the reader to [@hughes1995] [@woess2000] for basic facts regarding random walks). Quantum random walks were initially approached in [@aharonov1993], [@meyer1996], [@nayak2000] and [@ambainis2001] in one-dimensional terms, i.e. walk on a line. The system is described in terms of a position $n$ on the line and a direction $d$, i.e. ${|n\rangle}{|d\rangle}$ in the Hilbert space $\mathcal{H} = \mathcal{H}_{n} \otimes \mathcal{H}_{d}$ where $\mathcal{H}_{n}$ is the Hilbert space spanned by the basis vectors encoding the position and $\mathcal{H}_{d}$ the Hilbert space spanned by the vectors of the direction. The direction register, sometimes referred to as the coin space, is initialized to a superposition of the possible direction, in the case of a walk on a line, either left or right, and the position $n$ updated based on the direction of the walk. The choice of which superposition to apply is also a matter investigated. Surprisingly, if the system is executed for $t$ steps it behaves rather differently than its classical random walks. Specifically, the authors found that the system spreads quadratically faster over the line than its classical equivalent.
Some of the first approaches proposing quantum random walking on graphs can be found in [@farhi1997], [@hogg1998], [@aharonov2001] and [@childs2002a]. Let $G(V,E)$ represent a $d$-regular graph, and let $\mathcal{H}_{V}$ be the Hilbert space spanned by states ${|v\rangle}$ where $v \in V$, and $\mathcal{H}_{E}$ be an Hilbert space of dimension $d$ spanned by basis states ${{|1\rangle}, \cdots, {|d\rangle}}$. Overall, the system can be described through basis states ${|v\rangle}{|e\rangle}$ for all $v \in V$ and $e \in E$. The common approach is to randomly select one of the edges $e$ adjacent to $v$ and the update the current position of the graph, i.e. $U {|v\rangle} {|e\rangle} \rightarrow {|v^{\prime}\rangle} {|e\rangle}$, if $e$ has edge points $v$ and $v^{\prime}$. The random selection may also be performed using a $d$-dimensional coin space. Perhaps more interesting is the hitting time, i.e. the time it takes to reach a certain vertex $B$ starting from a vertex $A$. In [@childs2002a] and [@childs2002b] a graph example is presented where a classical random walk would take $\Omega(2^{d})$ steps to reach $B$, where $d$ is the depth of the graph. However, the quantum equivalent walk can reach $B$ in $O(d^{2})$ computational steps, providing for an exponential speedup! Other examples of quantum random walks include how to adapt the models to perform a search [@shenvi2003], [@ambainis2004], [@ambainis2005], and [@ambainis2007]
Not surprisingly, due to the hard computational problems being asked, the vast majority of these approaches put a strong emphasis on actually determining if a node can be reached, and if so how fast. Consequently, questions about the path leading to a pre-specified node have not been properly addressed. For instance, this fact is explicitly pointed out in [@childs2002b], namely: “Note that although our algorithm Þnds the name of the exit, it does not Þnd a particular path from entrance to exit”. For many artificial intelligence applications the ability to answer this question is a crucial one as it provides the basis for powerful inference mechanisms capable of knowledge deduction. The ability to obtain the full path open measurement is a key feature of the production system proposed in this work. Additionally, quantum random walks incorporate previous knowledge of a graph in the form of the relationship $G(V,E)$. This kind of knowledge may not always be available from start, e.g. systems where new nodes are generated based on the information accessible to a system at any given point in time.
The quantum random walk approach differs drastically from the model presented in the previous sections which focused on detailing some of the key notions supporting an hybrid quantum search system. Such a system incorporated classical search concepts, expressed through unitary operators, with Grover’s quantum search algorithm. From a classical point of view, applying a search procedure can be understood as partitioning the search space into blocks. Each block is then examined in order to determine if a solution is present. However, such an approach did not in any way performed a kind of partition of the “quantum search space”. Instead, it executed a sequence of actions in order to determine if a goal state was reached. So the question naturally arises: Is there any procedure to perform a hierarchical quantum search based solely on decomposition of the quantum search space?
Grover and Radhakrishnan were some of the first to purpose a possible approach to this problem in [@grover2004a]. The main motivation of the work was the following: consider a quantum search space containing a single solution is divided into $l$-blocks of equal size, suppose that we wish to determine in which of the $l$-blocks the solution is, can this process be performed with fewer queries than the original Grover algorithm? In practice, this problem reduces to the one of determining the first $m$ bits of the $n$ bit computational basis containing the solution, with $m \leq n$. The authors proceed by analysing what happens when a variation of Grover’s iterate for amplitude amplification is applied to each block. They conclude that it is indeed easier to determine the initial $m$ bits. However, as $m$ grows closer to $n$ the computational gains obtained disappear [@grover2004a]. Later, an optimization to this approach was proposed in [@korepin2007], albeit only marginally improving the lower bound on the number of oracle queries.
A number of authors have also focused on extending Grover’s original search algorithm. Namely, in [@zalka1999a] the quantum search algorithm was shown to be optimal in the sense that it gives the maximal possible probability of finding a solution. Other examples of possible extensions to Grover’s original work include returning a solution with certainty (please refer to [@zalka1999b], [@brassard2000], [@long2001] and [@hu2002]) and assessing the impact of having multiple solutions [@boyer1998]. Additionally, it should be mentioned that no quantum black-box search algorithm can solve the search problem by making fewer than $\Omega( \sqrt{N} )$ iterations. This result was first shown in [@bennett1997] and later revised on [@boyer1998].
Conclusions \[sec:conclusions\]
===============================
In this work we examined the ramifications of an hybrid quantum search system. We chose to focus on two key aspects: the use of a non-constant branching factor and the adoption of an heuristic point of view. Clearly, both of these cases are not without its flaws. However, we consider the additional insight provided to be valuable. In the case of the non-constant branching factor we were able to verify that deciding whether or not to Grover, or to proceed classically, should take into account the maximum and the average branching factor. Additionally, by evaluating the states that are within a given threshold we are able to incorporate some classical heuristic concepts into our approach. These concepts were further extended with the use of probabilistic distribution functions allowing for a selection mechanism to be obtained. This mechanism enables specific ranges of quantum states to be obtained in an efficient manner.
Acknowledgements
================
This work was supported by FCT (INESC-ID multiannual funding) through the PIDDAC Program funds and FCT grant DFRH - SFRH/BD/61846/2009.
Aharonov, D., Ambainis, A., Kempe, J., and Vazirani, U. (2001). Quantum walks on graphs. In [*Proceedings of ACM Symposium on Theory of Computation (STOC’01)*]{}, pages 50–59.
Aharonov, Y., Davidovich, L., and Zagury, N. (1993). Quantum random walks. , 48(2):1687–1690.
Ambainis, A. (2003). Quantum walks and their algorithmic applications. , 1:507.
Ambainis, A. (2004). Quantum search algorithms. , 35(2):22–35.
Ambainis, A. (2007). Quantum walk algorithm for element distinctness. , 37:210.
Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., and Watrous, J. (2001). One-dimensional quantum walks. In [*[ACM]{} Symposium on Theory of Computing*]{}, pages 37–49.
Ambainis, A., Kempe, J., and Rivosh, A. (2005). Coins make quantum walks faster.
Anderson, J. R. (1983). . Harvard University Press, Cambridge, Massachusetts, USA.
, C. H., [Bernstein]{}, E., [Brassard]{}, G., and [Vazirani]{}, U. (1997). . .
Boyer, M., Brassard, G., Hoeyer, P., and Tapp, A. (1998). Tight bounds on quantum searching. , 46:493.
, G., [Hoyer]{}, P., [Mosca]{}, M., and [Tapp]{}, A. (2000). . .
Campbell, M., Hoane Jr., A. J., and Hsu, F.-h. (2002). Deep blue. , 134:57–83.
Childs, A. M., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., and Spielman, D. (2003). Exponential algorithmic speedup by quantum walk. In [*Proceedings of the 35th ACM Symposium on Theory of Computing (STOC 2003)*]{}, pages 59–68.
Childs, A. M., Farhi, E., and Gutmann, S. (2002). An example of the difference between quantum and classical random walks. , 1(1):35–43.
Chuang, I. L., Gershenfeld, N., and Kubinec, M. (1998). Experimental implementation of fast quantum searching. , 80(15):3408–3411.
DeGroot, M. H. and Schervish, M. J. (2002). . Addison-Wesley, 3 edition.
Dirac, P. A. M. (1939). A new notation for quantum mechanics. In [*Proceedings of the Cambridge Philosophical Society*]{}, volume 35, pages 416–418.
Dirac, P. A. M. (1981). . Oxford University Press.
Edmonds, J. (1965). Paths, trees, and flowers. , 17:449–467.
Ernst, G. and Newell, A. (1969). . Academic Press, New York, NY, USA.
Farhi, E. and Gutmann, S. (1998). Quantum computation and decision trees. , 58(2):915–928.
Feldmann, R. (1993). . PhD thesis, University of Paderborn.
Garey, M. and Johnson, D. (1979). . Series of Books in the Mathematical Sciences. W. H. Freeman.
Gilchrist, W. (2000). . Chapman and Hall/CRC.
Grover, L. K. (1996). A fast quantum mechanical algorithm for database search. In [*STOC ’96: Proceedings of the twenty-eighth annual ACM symposium on Theory of computing*]{}, pages 212–219, New York, NY, USA. ACM.
Grover, L. K. (1998a). A framework for fast quantum mechanical algorithms. In [*STOC ’98: Proceedings of the thirtieth annual ACM symposium on Theory of computing*]{}, pages 53–62, New York, NY, USA. ACM.
Grover, L. K. (1998b). Quantum computers can search rapidly by using almost any transformation. , 80(19):4329–4332.
Grover, L. K. (1999). Quantum search on structured problems. , 10(10):1695 – 1705.
Grover, L. K. (2002). Trade-offs in the quantum search algorithm. , 66(5):052314.
Grover, L. K. (2005). Fixed-point quantum search. , 95(15):150501.
, L. K. and [Radhakrishnan]{}, J. (2004). .
Hart, P., Nilsson, N., and Raphael, B. (1968). A formal basis for the heuristic determination of minimum cost paths. , 4(2):100–107.
Hirvensalo, M. (2004). . Springer-Verlag, Berlin Heidelberg.
Hogg, T. (1998). A framework for structured quantum search. , 120:102.
Hopcroft, J. and Tarjan, R. (1973). Algorithm 447: efficient algorithms for graph manipulation. , 16(6):372–378.
Hordern, E. (1987). . Recreations in Mathematics, No 4. Oxford University Press, USA.
Hsu, F.-h. (1999). Ibm’s deep blue chess grandmaster chips. , 19(2):70–82.
Hsu, F.-h. (2002). . Princeton University Press.
Hu, C.-R. (2002). A family of sure-success quantum algorithms for solving a generalized grover search problem.
Hughes, B. (1995). . Random Walks and Random Environments. Oxford University Press, USA.
Kaye, P. R., Laflamme, R., and Mosca, M. (2007). . Oxford University Press, USA.
Kempe, J. (2003). Quantum random walks - an introductory overview. , 44:307.
, V. E. and [Xu]{}, Y. (2007). . , 21:5187–5205.
Korf, R. E. (1985). Depth-first iterative-deepening : An optimal admissible tree search. , 27(1):97 – 109.
Korf, R. E. (1991). Best-first search with limited memory. .
Korf, R. E. (1993). Linear-space best-first search. , 62(1):41–78.
Laird, J. E., Newell, A., and Rosenbloom, P. S. (1987). Soar: An architecture for general intelligence. , 33(1):1–64.
Laird, J. E., Rosenbloom, P. S., and Newell, A. (1986). Chunking in soar: The anatomy of a general learning mechanism. , 1(1):11–46.
Long, G. L. (2001). Grover algorithm with zero theoretical failure rate. , 64(2):022307.
Luger, G. F. and Stubblefield, W. A. (1993). . The Benjamin/Cummings Publishing Company, Inc, Menlo Park, CA, USA.
Mano, M. and Kime, C. R. (2002). . Prentice Hall, Englewood Cliffs, NJ, USA.
Meyer, D. (1996). From quantum cellular automata to quantum lattice gases. , 85(5):551–574.
Moore, E. (1959). The shortest path through a maze. In [*Proceeding of an International Symposium on the Theory of Switching, Part II*]{}, pages 285–292, Cambridge, Massachusetts. Harvard University Press.
Nayak, A. and Vishwanath, A. (2000). Quantum walk on the line. Technical report, DIMACS Technical Report.
Newell, A. (1963). A guide to the general problem-solver program gps-2-2. Technical Report RM-3337-PR, RAND Corporation, Santa Monica, CA, USA.
Newell, A. and G, E. (1965). The search for generality. In [*Information Processing 1965: Proceeding of IFIP Congress*]{}, volume 1, pages 17–24, Chicago. Spartan.
Newell, A., Shaw, J., and Simon, H. A. (1959). Report on a general problem-solving program. In [*Proceedings of the International Conference on Information Processing*]{}, pages 256–264.
Nielsen, M. A. and Chuang, I. L. (2000). . Cambridge University Press, Cambridge, MA, USA.
Nilsson, N. J. (1982). . Morgan Kaufmann Publishers, Inc.
Post, E. (1941). . Princeton University Press.
Post, E. (1943). Formal reductions of the general combinatorial problem. , 65:197–268.
Russell, S. J., Norvig, P., Canny, J. F., Edwards, D. D., Malik, J. M., and Thrun, S. (2003). . Prentice Hall.
Santos, A. C., Tarrataca, L., and Jo[ã]{}o, C. (2009a). An analysis of navigation algorithms for smartphones using j2me. In [*In Proceedings of the Second International ICST Conference on MOBILe Wireless MiddleWARE, Operating Systems, and Applications (Mobilware’09, Berlin-Germany, April 28-29), LNICST*]{}, volume 7, pages 266–279. Springer.
Santos, A. C., Tarrataca, L., and Jo[ã]{}o, C. (2009b). Context inference for mobile applications in the upcase project. In [*In Proceedings of the Second International ICST Conference on MOBILe Wireless MiddleWARE, Operating Systems, and Applications (Mobilware’09, Berlin-Germany, April 28-29), LNICST*]{}, volume 7, pages 352–365. Springer.
Shenvi, N., Kempe, J., and Whaley, K. B. (2003). Quantum random-walk search algorithm. , 67(5):052307.
Shor, P. (1994). Algorithms for quantum computation: discrete logarithms and factoring. In [*Proceedings 35th Annual Symposium on Foundations of Computer Science*]{}, pages 124–134.
Slate, D. and Atkin, L. R. (1977). Chess 4.5 - northwestern university chess program. In [*Chess Skill in Man and Machine*]{}, pages 82–118, Berlin. Springer-Verlag.
Toffoli, T. (1980a). Reversible computing. In [*Proceedings of the 7th Colloquium on Automata, Languages and Programming*]{}, pages 632–644, London, UK. Springer-Verlag.
Toffoli, T. (1980b). Reversible computing. Technical report, Massschusetts Institute of Technology, Laboratory for Computer Science, Massachusetts, MA, USA.
Woess, W. (2000). . Number 138 in Cambridge Tracts in Mathematics. Cambridge University Press.
Zalka, C. (1999a). A grover-based quantum search of optimal order for an unknown number of marked elements.
Zalka, C. (1999b). Grover’s quantum searching algorithm is optimal. , 60:2746.
[^1]: Lu’s Tarrataca was supported by FCT (INESC-ID multiannual funding) through the PIDDAC Program funds and FCT grant DFRH - SFRH/BD/61846/2009.
[^2]: A number of improvements have been purposed since Grover’s original work [@grover2002] [@grover2005]. These improvements essentially targeted reduced time complexity bounds for non-query operations and overall robustness. For a number of several novel search related applications please refer to [@grover1998a] [@grover1998b] [@grover1999].
[^3]: An alternative approach would consist in encoding each possible state, instead of encoding in an binary fashion the sequence of actions. However, this would have a meaningful impact on the complexity and design of unitary operator $U$ since each admissible input string would have to be mapped onto a predefined sequence of actions.
|
---
abstract: |
We report the serendipitous discovery of a ring nebula around a candidate Wolf-Rayet (WR) star, HBHA4202-22, in Cygnus using the [*Spitzer Space Telescope*]{} archival data. Our spectroscopic follow-up observations confirmed the WR nature of this star (we named it WR138a) and showed that it belongs to the WN8-9h subtype. We thereby add a new example to the known sample of late WN stars with circumstellar nebulae. We analyzed the spectrum of WR138a by using the Potsdam Wolf-Rayet (PoWR) model atmospheres, obtaining a stellar temperature of 40kK. The stellar wind composition is dominated by helium with 20 per cent of hydrogen. The stellar spectrum is highly reddened and absorbed ($E_{B-V} = 2.4$mag, $A_V
= 7.4$mag). Adopting a stellar luminosity of $\log L/L_{\odot} =
5.3$, the star has a mass-loss rate of $10^{-4.7} \, {\rm\,M_\odot}\, {\rm
yr}^{-1}$, and resides in a distance of 4.2 kpc. We measured the proper motion for WR138a and found that it is a runaway star with a peculiar velocity of $\simeq 50 \, {{\rm\,km\,s^{-1}}}$. Implications of the runaway nature of WR138a for constraining the mass of its progenitor star and understanding the origin of its ring nebula are discussed.
date: 'Accepted 2009 August 3, Received 2009 July 20; in original form 2009 June 3'
title: 'Discovery of a new Wolf-Rayet star and its ring nebula in Cygnus'
---
\[firstpage\]
line: identification – circumstellar matter – stars: individual: HBHA4202-22 – stars: Wolf-Rayet
Introduction {#sec:intro}
============
It is believed that single stars of solar metallicity with an initial mass of $\ga 20-25 \, {\rm\,M_\odot}$ end their lives as Wolf-Rayet (WR) stars (Vanbeveren, De Loore & Van Rensbergen 1998; Meynet & Maeder 2003). The relatively small number of massive stars and the short duration of the WR phase ($\la 0.5$ Myr) makes the WR stars rare objects. The present total number of WR stars in the Galaxy does not exceed several thousands (e.g. Shara et al. 1999; van der Hucht 2001). The known population of WR stars is, however, an order of magnitude less numerous (van der Hucht 2006). The large disparity between the expected and the observed number of WR stars is mostly caused by the huge interstellar extinction in the Galactic plane, which strongly limits the traditional method for searching for WR stars using optical surveys. In this respect, the infrared (IR) observations are of high importance since the interstellar medium (ISM) is much more transparent in the IR compared with the optical.
The effectiveness of IR observations in identifying new WR stars was demonstrated by Shara et al. (2009) who carried out a near-IR narrow-band imaging survey of the inner Galactic plane to select candidate WR stars and discovered 41 new WR stars in spectroscopic follow-ups (cf. Homeier et al. 2003; Hadfield et al. 2007; Mauerhan, Van Dyk & Morris 2009). Another possible way to search for new WR (and other evolved massive) stars is through detection of their (IR) circumstellar nebulae, created after a massive star leaves the main sequence (e.g. Marston et al. 1999; Egan et al. 2002; Clark et al. 2003).
In this paper, we report the discovery of a new WN8-9h star in Cygnus via the detection of its IR nebula. We thereby add a new example to the known sample of late WN (WNL) stars with circumstellar nebulae (e.g. Esteban et al. 1993). Inspired by this discovery, we undertook an extensive search for similar objects using the archival data of the Spitzer Legacy Programs[^1] and discovered many dozens of ring-like and bipolar IR shells (Gvaramadze, Kniazev & Fabrika 2009; cf. Carey et al. 2009). Spectroscopic follow-ups of central stars associated with two dozens of the shells showed that they are either candidate Luminous Blue Variables or related (WNL, blue supergiant) stars (Gvaramadze et al. 2009, in preparation; Kniazev et al. 2009, in preparation), which confirmed that the IR imaging provides a powerful tool for revealing evolved massive stars via detection of their circumstellar nebulae.
A new ring nebula and its central star {#sec:neb}
======================================
The nebula, which is the subject of this paper, was discovered serendipitously during our search for bow shocks around OB stars running away from star clusters in the Cygnus X region (for motivation of this search see Gvaramadze & Bomans 2008a). The nebula was detected in the archival data of the [*Spitzer Space Telescope*]{}, obtained with the Infrared Array Camera (IRAC) and the Multiband Imaging Photometer for [*Spitzer*]{} (MIPS) (Werner et al. 2004) within the framework of the Cygnus-X Spitzer Legacy Survey[^2].
![ [*Spitzer*]{} MIPS $24 \, \mu$m image of a new ring nebula in Cygnus and its central star HBHA4202-22 (WR138a). The orientation of the spectrograph slit in our optical observation is indicated by a line. []{data-label="fig:neb"}](Fig1.ps){width="1.0\columnwidth"}
Fig.\[fig:neb\] shows the MIPS $24 \, \mu$m image of the nebula and its central star. The nebula has a clear ring-like structure with a diameter of $\simeq 2.3$ arcmin and enhanced brightness along the west rim. The central star is offset for $\sim 0.2$ arcmin from the geometric centre of the nebula, being closer to the brightest portion of the nebula (a possible origin of the nebula and its brightness asymmetry are discussed in Section\[sec:nebula\]). The Cygnus-X Spitzer Legacy Survey also provides IRAC $3.6, 5.8 \,
\mu$m and MIPS $70 \, \mu$m images of the field containing the nebula; none of them shows its signatures.
The optical counterpart to the central star was identified by Dolidze (1971) as a possible WR star due to the presence in its spectrum of an emission band around $\lambda 6750$ Å. This star was included in the Catalogue of H-alpha emission stars in the northern Milky Way (Kohoutek & Wehmeyer 1999) as a candidate WR star (named in the SIMBAD database as \[KW97\]44-47 or HBHA4202-22), but was not listed in the VIIth Catalogue of Galactic Wolf-Rayet Stars (van der Hucht 2001) nor in the annex to this catalogue (van der Hucht 2006). Recently, Corradi et al. (2008) put this star in the list of candidate symbiotic stars on the basis of its IPHAS and 2MASS colours.
Observations {#sec:obs}
============
Spectroscopy
------------
To verify the nature of HBHA4202-22, we had performed spectroscopic observations with the Russian 6-m telescope using the SCORPIO[^3] focal reducer (Afanasiev & Moiseev 2005) in a long-slit mode with a slit width of 1 arcsec, providing a spectral resolution of 5 Å. The orientation of the (6 arcmin long) slit is shown in Fig.\[fig:neb\]. The spectra were taken on two occasions: 2008 October 10 (spectral range $\lambda\lambda 4000-5700$ Å) and 2008 November 11 (spectral ranges $\lambda\lambda 4000-5700$ Å and $\lambda\lambda 5700-7500$ Å), and reduced using standard procedures.
Photometry
----------
Using secondary photometric standards in the regions of symbiotic variable stars V407 Cyg and AS 323 published by Henden & Honeycutt (1997), we determined accurate magnitudes of four comparison stars near HBHA4202-22. The observations were carried out with the Special Astrophysical Observatory (SAO) 1-m Zeiss reflector and $UBVR_{\rm C}I_{\rm C}$ photometer in the good photometric night on 2009 May 29. The photometry accuracy is better than 0.02 mag. We have made expanded CCD standard star set of 18 stars to measure a DSS POSS-1 blue image. HBHA4202-22 was measured using POSS-O plate taken in the epoch of 1954.57, CCD frames taken with the Russian 6-m telescope in 2008.90 and CCD frames taken with SAO 1-m Zeiss telescope in 2009. The results are given in Table\[tab:res\].
JD $B$ $V$ $R_{\rm C}$ Source
--------- ---------------- ---------------- ---------------- -------------
2434951 $17.15\pm0.06$ – – POSS-1
2454796 $17.25\pm0.01$ $15.44\pm0.01$ – 6-m/SCORPIO
2454981 $17.34\pm0.02$ $15.57\pm0.02$ $13.90\pm0.02$ 1-m Zeiss
2455008 $17.27\pm0.02$ $15.53\pm0.02$ $13.85\pm0.02$ 1-m Zeiss
The uniform 1-m Zeiss photometry shows that the object is variable with amplitudes between 0.04 and 0.07 in different filters in the time scale of a month. A total variability range in B filter between 1955 and 2009 is 0.19 mag. This level of variability is typical of WNL stars (e.g. Marchenko et al. 1998).
To measure the [*Spizer*]{}’s IRAC and MIPS fluxes from HBHA4202-22, we performed aperture photometry of the star using the [MOPEX/APEX]{} source extraction package and applied aperture correction, obtained from the nearby bright point sources. The fluxes are listed in Table\[tab:obs\]; the estimated errors are $\sim 2-3$ per cent. For the flux at $70 \, \mu$m, we give a $3\sigma$ upper limit since HBHA4202-22 was not detected at this wavelength.
The details of HBHA4202-22 are summarized in Table\[tab:obs\]. The coordinates and the $J,H,K_s$ magnitudes are taken from the 2MASS All-Sky Catalog of Point Sources (Skrutskie et al. 2006).
--------------- --------------------------------- -- -- --
RA(J2000) $20^{\rm h} 17^{\rm m} 08\fs12$
Dec.(J2000) $41\degr 07\arcmin 27\farcs0$
$l,b$ 78.3203, 3.1536
$B$ (mag) $17.25\pm0.01$
$V$ (mag) $15.44\pm0.01$
$J$ (mag) $10.15\pm0.02$
$H$ (mag) $9.27\pm0.02$
$K_s$ (mag) $8.65\pm0.02$
$[3.6]$ (mJy) $168.0\pm5.0$
$[5.8]$ (mJy) $154.0\pm4.6$
$[24]$ (mJy) $45.7\pm1.4$
$[70]$ (mJy) $< 28.1$
--------------- --------------------------------- -- -- --
: Details of HBHA4202-22 (WR138a)[]{data-label="tab:obs"}
Proper motion {#sec:obs-prop}
-------------
We measured the proper motion for HBHA4202-22 using four epochs in the POSS-1 and POSS-2 Digitized Sky Surveys and the epoch of our 6-m/SCORPIO frames. The full epoch range is 54.3 years. 21 reference stars located within an angular radius of 2.5 arcmin from the object position were used. Typical rms deviations of reference stars in these frames are between 0.21 and 0.56 arcsec. Linear trends are well seen in both coordinates. To evaluate the stellar proper motion and its errors, we got a linear approximation of these trends using the method of least squires and derived mean rms deviations of stellar coordinates from these linear approximations. The proper motion errors were estimated by dividing these mean deviations by the epoch difference. The result is $\mu _\alpha = -4.9\pm 1.2 \,
{\rm mas} \, {\rm yr}^{-1}$, $\mu _\delta =-4.7\pm 1.2 \, {\rm mas}
\, {\rm yr}^{-1}$.
To convert the observed proper motion into the true tangential velocity of the star, we use the Galactic constants $R_0 =8$ kpc and $\Theta =200 \, {{\rm\,km\,s^{-1}}}$ (e.g. Reid 1993; Avedisova 2005) and the solar peculiar motion ($U_{\odot} , V_{\odot} , W_{\odot}) = (10.00, 5.25,
7.17) \, {{\rm\,km\,s^{-1}}}$ (Dehnen & Binney 1998), and adopt a distance $d=4.2$ kpc (see Section\[sec:dist\]). We found that the star is moving in the west-southwest direction with a peculiar (transverse) velocity of $\simeq 50\pm 24 {{\rm\,km\,s^{-1}}}$ (here we assume that the errors are mainly due to the errors in our proper motion measurements). Note that the vector of stellar peculiar velocity is oriented by chance almost exactly along the spectrograph slit (see Fig.\[fig:neb\]).
A new Wolf-Rayet star – WR138a
==============================
Spectral type {#sec:type}
-------------
![Blue (top) and red (bottom) spectra of HBHA4202-22 (WR138a) with some lines indicated (upper row). In the blue spectrum, the bottom row of vertical bars marks numerous He[ii]{} lines while the middle one marks He[i]{} lines. In the red spectrum, the vertical bars mark He[ii]{} lines, whereas very strong He[i]{} lines are not marked. H[i]{} emissions are not marked on both spectra as well. CCD fringes are notable at wavelengths longer than $\sim 6900$Å.[]{data-label="fig:spec"}](Fig2a_spectrum.ps "fig:"){width="0.7\columnwidth"} ![Blue (top) and red (bottom) spectra of HBHA4202-22 (WR138a) with some lines indicated (upper row). In the blue spectrum, the bottom row of vertical bars marks numerous He[ii]{} lines while the middle one marks He[i]{} lines. In the red spectrum, the vertical bars mark He[ii]{} lines, whereas very strong He[i]{} lines are not marked. H[i]{} emissions are not marked on both spectra as well. CCD fringes are notable at wavelengths longer than $\sim 6900$Å.[]{data-label="fig:spec"}](Fig2b_spectrum.ps "fig:"){width="0.7\columnwidth"}
In Fig.\[fig:spec\] we present the spectrum of HBHA4202-22 in the blue and red regions. The spectrum shows strong emission lines of hydrogen, He[i]{} and He[ii]{}. We detect emissions of N[ii]{}, N[iii]{}, C[iii]{}, N[iv]{}, Si[iv]{}. No forbidden lines can be seen in the spectrum. Many of the weaker lines in the spectrum show PCygni profiles, while the strongest lines are entirely in emission. The strong absorptions visible in the red spectrum are telluric. Numerous diffuse interstellar bands (DIBs) are observed, in the blue region they are $\lambda\lambda
4428, 4726, 4762$, in the red the strongest one is at 6280 Å.
Our spectra also show nebular emission lines along the whole length of the slit. Radial velocity in H$\alpha$ line is constant along the slit and equal to $-21\pm5 \, {{\rm\,km\,s^{-1}}}$. From the Balmer decrement, we estimated the interstellar extinction and obtained $A_V =
1.9\pm0.1$mag (also constant along the slit), which is much less than $A_V = 7.4$mag derived from the stellar spectrum (see Section 4.4). Hence, we conclude that the nebular line emission originates from the foreground and is not related to HBHA4202-22 and its ring nebula.
The dominance of helium and nitrogen emission lines indicates that HBHA4202-22 belongs to the nitrogen (WN) sequence of WR stars. The presence of hydrogen emission lines is typical for WNL subtypes (Hamann et al. 1991). We named this star WR138a, in accordance with the numbering system of the VIIth Catalogue of Galactic Wolf-Rayet Stars by van der Hucht (2001). To determine the subtype of WR138a more precisely, we use the classification scheme proposed by Smith (1968) and updated for WNL stars by Crowther, Hillier & Smith (1995) and Smith, Shara & Moffat (1996). The relative strengths of N[iii]{} $\lambda \, 4640$, N[iv]{} $\lambda \, 4058$ and He[ii]{} $\lambda \, 4686$ lines and the PCygni profiles of He[i]{} lines suggest that the star belongs to the WN8-9 subtype. The WN8-9 classification for WR138a also follows from the position of this star on the diagrams showing a comparison of the emission equivalent widths (EW) of He[i]{} $\lambda 5876$ versus He[ii]{} $\lambda 4686$ lines and He[ii]{} $\lambda 4686$ EW versus FWHM for WNL stars in the Milky Way and the Large Magellanic Cloud (LMC) (see Fig.1 of Crowther & Smith 1997). With EW(5876)=40.0 Å, EW(4686)=19.5 Å and FWHM(4686)=7.7 Å, WR138a lies almost exactly on the line separating the WN8 and WN9 stars. The further evidence for this classification is given in Section\[sec:PoWR\]. Following the three-dimensional classification for WN stars by Smith et al. (1996), we add a ‘h’ suffix to WN8-9 to indicate the presence of hydrogen emission lines, so that WR138a is a WN8-9h star.
Radial velocity {#sec:rad}
---------------
We have measured the radial velocities of the main emission lines in the spectrum and the line widths corrected for the spectral resolution. We used the Gaussian analysis for the measurements. The accuracy of radial velocity measurements is $\la 10 \, {{\rm\,km\,s^{-1}}}$, while the relative velocities are measured with a notably better precision.
For hydrogen lines H$\gamma$, H$\beta$ and H$\alpha$, we measured heliocentric radial velocities $v_{\rm r} \simeq -10 \, {{\rm\,km\,s^{-1}}}$ (FWHM $\simeq 610 \pm 30 \, {{\rm\,km\,s^{-1}}}$). The strongest He[ii]{} $\lambda
4686$ line shows $v_{\rm r} \simeq -37 \, {{\rm\,km\,s^{-1}}}$ (FWHM $\simeq 490 \,
{{\rm\,km\,s^{-1}}}$) and it does not have the PCyg profile. All He[ii]{} lines with the PCyg profile have $v_{\rm r} \simeq 49 \, {{\rm\,km\,s^{-1}}}$ (FWHM $\simeq 170 \, {{\rm\,km\,s^{-1}}}$) and the radial velocity measured for the absorption peak $v_{\rm a} \simeq -380 \, {{\rm\,km\,s^{-1}}}$. Blue shifted absorption obviously shifts emission to the red side and makes it narrower. He[i]{} lines show the same behaviour as He[ii]{} lines. The mean radial velocity for all ‘non-PCyg’ He[i]{} lines is $\simeq -11 \, {{\rm\,km\,s^{-1}}}$ (FWHM $\simeq 590 \, {{\rm\,km\,s^{-1}}}$).
We may adopt for the star’s radial velocity $v_{\rm r} \sim -20 \,
{{\rm\,km\,s^{-1}}}$, which is an average over the hydrogen lines, He[ii]{} $\lambda 4686$ and the ‘non-PCyg’ He[i]{} lines. It is necessary to note, however, that the radial velocity of the star cannot be determined with good accuracy, because all the spectral lines are formed in the wind. Resulting from a complicate formation process in the expanding atmosphere, the emergent line profiles are not exactly symmetric. Due to the high wind speed ($700 \, {{\rm\,km\,s^{-1}}}$, as found in our spectral analysis, see below), these effects are large compared to possible radial velocity uncertainties.
Spectral analysis and stellar parameters {#sec:PoWR}
----------------------------------------
{width="\textwidth"}
To analyze the stellar spectrum and to derive the fundamental parameters of WR138a, we use the Potsdam Wolf-Rayet (PoWR) models for expanding stellar atmospheres. These models account for complex model atoms including iron-line blanketing in non-LTE (for a detailed description see Hamann & Gräfener 2004). For abundances of trace elements, we adopt values which are typical for Galactic WN stars – N: 0.015, C: 0.0001, Fe: 0.0014 (Hamann & Gräfener 2004).
The main parameters of a WR-type atmospheres are the stellar temperature, $T_\ast$, and the so-called transformed radius, $R_{\rm
t}$. The stellar temperature $T_*$ denotes the effective temperature related to the radius $R_\ast$, i.e. $L = \sigma T_\ast^4 4 \pi
R_\ast^2$, where $\sigma$ is the Stefan-Boltzmann constant and $R_\ast$ is by definition at a Rosseland optical depth of 20. $R_{\rm t}$ is related to the mass-loss rate $\dot{M}$ and defined by $$\begin{aligned}
R_{\rm t} = R_*
\left[\frac{v_\infty}{2500 \, {{\rm\,km\,s^{-1}}}} \left/
\frac{\sqrt{D} \dot M}
{10^{-4} \, {\rm\,M_\odot}\, {\rm yr^{-1}}}\right]^{2/3} \right. ~, \nonumber\end{aligned}$$ where $D$ is the clumping contrast (adopted here to be 4 throughout the wind as a typical value for WN stars; see Hamann & Koesterke 1998), and $v_\infty$ is the terminal velocity of the wind.
These basic stellar parameters $T_\ast$ and $R_{\rm t}$ are determined from fitting the lines in the normalized spectrum (see Fig.\[fig:linefit\]). The normalization is in fact achieved by dividing the absolutely calibrated observed spectrum by the theoretical continuum, which makes the total procedure described in the following an iterative process.
A first orientation about the proper choice of these parameters can be obtained by comparing the observed spectrum to the published grids of WN models (Hamann & Gräfener 2004). For a detailed fit, we calculate individual models for this star with the adequate terminal wind velocity, which is $v_\infty = 700\,{{\rm\,km\,s^{-1}}}$ as inferred from fitting the width of the emission line profiles.
The stellar temperature is adjusted such that the balance between the lines from He[i]{} versus He[ii]{} is reproduced. At the same time, $R_t$ is adjusted, which influences the strength of the emission lines in general. The parameters of the best-fitting model are compiled in Table\[tab:model\] (where for the completeness we also give the hydrogen ionising luminosity, $\Phi _{\rm i}$). Note that according to its location in the $T_\ast - R_{\rm t}$ plane (Hamann & Gräfener 2004), WR138a belongs to a spectral subtype later than WN8 (cf. Section\[sec:type\]).
Fig.\[fig:linefit\] shows that the Balmer lines of hydrogen dominate their blend with lines of the He[ii]{} Pickering series. Radial velocities of the Balmer lines confirm this. These blends are nicely matched by models with a hydrogen mass fraction of 20 per cent, leaving about 80 per cent of the mass for helium which is a typical composition for WNL stars.
The spectral analysis alone cannot tell the absolute dimensions of the star. These are related to the distance of the object, and will be discussed in the following subsection.
Luminosity and distance of WR138a {#sec:dist}
---------------------------------
The lowest plausible luminosity for a massive WNL star is about $\log L/L_{\odot} = 5.3$ (Hamann, Gräfener & Liermann 2006). Let us adopt this value for a moment. This choice implies the values for the stellar radius and mass-loss rate as given in Table\[tab:model\]. Note that, in fair approximation, WR models can be scaled to different luminosities, with $R_\ast \propto
L^{1/2}$, $\dot{M} \propto L^{3/4}$, and $d \propto L^{1/2}$ (Schmutz, Hamann & Wessolowski 1989; Hamann & Koesterke 1998).
To estimate the corresponding distance to WR138a, we now fit its observed spectral energy distribution (SED) with the model SED (Fig.\[fig:SED\]). Two parameters can be adjusted for the fit, the distance $d$ and the reddening $E_{B-V}$. The reddening law we adopt from Seaton (1979) in the optical and from Moneti et al. (2001) in the IR. A perfect fit to the whole SED is achieved with $E_{B-V} = 2.4\,{\rm mag}$, implying an extinction in the visual of $A_V = 7.4$mag and $d=4.2$ kpc. The strong interstellar absorption also shows up in the observed spectrum by pronounced DIBs (see Fig.\[fig:spec\]).
The line-of-sight towards WR138a is nearly tangential to the local (Orion) spiral arm, whose extent in this direction is $\simeq 4-6$ kpc (McCutcheon & Shuter 1970; Russeil 2003). A distance of about 4kpc would imply that this star is located at the far end of the Orion arm, which is compatible with the high reddening.
Alternatively, one might consider if our object is in fact not a massive star with ring nebula, but a planetary nebula (PN). PN central stars with WN-type spectra are very rare; only two of them are known in our Galaxy \[PMR5, Morgan, Parker & Cohen (2003) and PB8, Todt et al. (2009)\] and an eruptive one in the LMC \[N66; Hamann et al. (2003)\]. With a typical luminosity of a PN central star, 6000$L_\odot$, the photometric distance would become only 0.5kpc, placing the object into the foreground of the Orion arm. In this case, the strong reddening would have to be of circumstellar origin. Although we cannot strictly rule out this scenario, it appears very artificial and unlikely.
The distance of $d = 4.2$ kpc, which relies on the adopted stellar luminosity of $\log L/L_\odot = 5.3$, places WR138a at a height of 230pc above the Galactic plane. As we will discuss in the next section, this is just compatible with the inference that WR138a is a runaway star that has been ejected from the Galactic plane. Many WNL stars have a much larger luminosity than $\log L/L_\odot = 5.3$ (Hamann et al. 2006). A higher luminosity, however, would increase the implied distance ($d \propto L^{1/2}$) and hence lead to an even larger height above the Galactic plane, while on the other hand the evolutionary lifetime decreases with higher luminosity. Both effects together tend to make the runaway scenario impossible for much larger $d$ and $L$ (see Sect.\[sec:run\]). Hence, we think that our choice of $\log L/L_\odot = 5.3$ is the most plausible one.
-------------------------------------------------- --------
Spectral type WN8-9h
$T_\ast$ \[kK\] 40
$\log R_{\rm t} \, [R_\odot ]$ 0.9
$v_\infty \, [{{\rm\,km\,s^{-1}}}]$ 700
$\log L \, [L_\odot ]$ 5.3
$R_\ast \, [R_\odot ]$ 9.4
$\log \dot{M} \, [{\rm\,M_\odot}{\rm yr}^{-1} ]$ -4.7
$E_{B-V}$ \[mag\] 2.4
$d$ \[kpc\] 4.2
$\log \Phi _{\rm i} \, [{\rm s}^{-1}]$ 48.9
-------------------------------------------------- --------
: Stellar parameters for WR138a[]{data-label="tab:model"}
![Observed flux distribution of WR138a (blue/noisy) in absolute units, including the calibrated spectrum and the photometric measurements compiled in Table\[tab:obs\], compared to the emergent flux of the model continuum (red/smooth line), in the optical also shown with lines. The model flux has been reddened and scaled to the distance according to the parameters given in Table\[tab:model\]. []{data-label="fig:SED"}](sed.ps){width="1.0\columnwidth"}
Discussion {#sec:dis}
==========
WR138a as a runaway star {#sec:run}
------------------------
Our proper motion measurements for WR138a (Section\[sec:obs-prop\]) showed that the star is moving in the west-southwest direction (i.e. away from the Galactic plane) with a peculiar (transverse) velocity of $\simeq 50 \, {{\rm\,km\,s^{-1}}}$ (for $d=4.2$ kpc), which is typical of runaway stars. In the Galactic coordinates, the components of the peculiar velocity are almost equal to each other: $v_l \simeq -36 \, {{\rm\,km\,s^{-1}}}$, $v_b \simeq 34 \,
{{\rm\,km\,s^{-1}}}$.
Assuming that the progenitor star of WR138a was born near the Galactic plane and ejected from the parent star cluster soon after the birth, one has that the time it needs to reach the present height above the Galactic plane, $t_\ast \simeq d\sin b /v_b$, should be $\leq t_{\rm H}$, where $t_{\rm H}$ is the H-burning lifetime of the star. Adopting the minimum plausible luminosity for WNL stars of $\log (L/L_{\odot})=5.3$, one finds the current mass of WR138a of $\simeq 13 \, {\rm\,M_\odot}$, which in turn implies the initial mass of the star of $\simeq 30 \, {\rm\,M_\odot}$ and $t_{\rm H} \simeq
6.0-7.0$ Myr (e.g. Meynet & Maeder 2003). Thus, one has that $t_\ast (\simeq 6.7 $ Myr) $\la t_{\rm H}$, so that the star had enough time to reach its present location.
Placing WR138a at, for example, $d=6$ kpc would result in a twofold increase of the stellar luminosity and in a increase of the present and initial mass of the star to, respectively, $\simeq 17 \,
{\rm\,M_\odot}$ and $\simeq 40 \, {\rm\,M_\odot}$. In this case, $t_\ast \simeq 7.0 $ Myr $> t_{\rm H} \simeq 4.5-5.5$ Myr. Further increase of $d$ makes the discrepancy between $t_\ast$ and $t_{\rm H}$ more severe. The discrepancy would be even more sever if the star spent several Myr in the parent cluster before it was ejected (Gvaramadze & Bomans 2008b; Gvaramadze, Gualandris & Portegies Zwart 2009). From this follows that the most likely initial mass of the progenitor star of WR138a was $\simeq 25-30 \, {\rm\,M_\odot}$. One cannot, however, exclude a possibility that the progenitor star was a blue straggler formed via merging of two less massive stars in the course of close binary-binary encounter, resulting in ejection of the merger product from the parent cluster (e.g. Gvaramadze & Bomans 2008b). In this case, the distance to and the mass of WR138a could be larger.
The inference that WR138a is a runaway star could also be used to understand the origin of its ring nebula.
Origin of the nebula around WR138a {#sec:nebula}
----------------------------------
A runaway wind-blowing (e.g. WR) star moving through the ISM creates a highly elongated bubble (Weaver et al. 1977; Brighenti & D’Ercole 1994), whose shape in the upstream direction is determined by a ram pressure balance between the stellar wind and the ISM (Baranov, Krasnobaev & Kulikovskii 1971). The characteristic scale of the leading edge of the bubble is $R_{\rm WR} \simeq (\dot{M} _{\rm WR}
v_{\rm WR} /5.6\pi m_{\rm H} n_0 v_{\ast} ^2 )^{1/2}$, where $\dot{M} _{\rm WR}$ and $v_{\rm WR}$ are the stellar mass-loss rate and the wind velocity, $n_0$ is the number density of the ambient medium, $v_{\ast}$ is the stellar peculiar velocity and $m_{\rm H}$ is the mass of a hydrogen atom. For $\dot{M} _{\rm WR} = 10^{-4.7}
\, {\rm\,M_\odot}{\rm yr}^{-1}$ and $v_{\rm WR} = 700 \, {{\rm\,km\,s^{-1}}}$ (see Table\[tab:model\]), $v_{\ast} = 50 \, {{\rm\,km\,s^{-1}}}$ and $n_0 \simeq 0.1
\, {\rm cm}^{-3}$ (typical of the ISM at $z=230$ pc; Dickey & Lockman 1990), one finds $R_{\rm WR} \simeq 20$ pc. The small size and the nearly circular shape of the nebula around WR138a (at $d=4.2$ kpc, the linear radius of the nebula $r_{\rm neb} \simeq
1.4$ pc) imply that the stellar wind interacts with the dense ambient medium comoving with the star (cf. Lozinskaya 1992). This consideration suggests that the immediate precursor of WR138a was a red supergiant (RSG) star (i.e. the initial mass of the progenitor star was $\leq 40\, {\rm\,M_\odot}$; cf. Section\[sec:run\]) and that the WR wind still propagates through the region occupied by the dense material shed during the RSG phase.
According to stellar evolutionary models, a massive star with the initial mass in the range from $25$ to $40 \, {\rm\,M_\odot}$ evolves through the sequence O $\rightarrow$ RSG $\rightarrow$ WN (e.g. Meynet & Maeder 2003). During the RSG phase, the star loses a considerable fraction of its initial mass in the form of slow, dense wind. The subsequent fast WR wind sweeps up the RSG wind and creates a circumstellar shell. For a spherically symmetric RSG wind, the shell driven by the WR wind expands with a constant velocity (e.g. Chevalier & Imamura 1983) $v_{\rm sh} \simeq (\dot{M} _{\rm WR}
v_{\rm WR} ^2 v_{\rm RSG} /3\dot{M} _{\rm RSG} )^{1/3}$, where $\dot{M} _{\rm RSG}$ and $v_{\rm RSG}$ are, respectively, the mass-loss rate and the wind velocity during the RSG phase. Adopting $\dot{M} _{\rm RSG} =3\times 10^{-5} {\rm\,M_\odot}{\rm yr}^{-1}$ and $v_{\rm RSG} =10 \, {{\rm\,km\,s^{-1}}}$ (the figures typical of RSGs), one has $v_{\rm sh} \simeq 100 \, {{\rm\,km\,s^{-1}}}$, which in turn gives us the dynamical age of the nebula $r_{\rm neb} /v_{\rm sh} \simeq 1.4
\times 10^4$ yr. The latter estimate suggests that WR138a only recently entered the WR phase.
The runaway nature of WR138a allows us to propose a natural explanation of the brightness asymmetry of the nebula and the offset of the star towards the brightest portion of the nebula. Namely, we suggest that both are caused by the effect of the ram pressure of the ISM, which affects the zone occupied by the RSG wind by making it denser and more compact ahead of the star. This suggestion could be supported by an estimate of the upstream (minimum) size of the zone occupied by the RSG wind, $R_{\rm RSG} \simeq (\dot{M} _{\rm
RSG} v_{\rm RSG} /5.6\pi m_{\rm H} n_0 v_{\ast} ^2 )^{1/2}$ ($\simeq
1.6$ pc for the parameters adopted above) $\sim r_{\rm neb}$, which shows that although the WR wind still propagates through the zone occupied by the RSG wind, it is already close to the edge of this zone to fill the density enhancement caused by the stellar motion. An additional support for our suggestion comes from the proper motion measurements for WR138a, which show that the star is moving in the direction implied by the above consideration.
Our interpretation of the nebula around WR138a as a circumstellar one (i.e. created via the wind-wind interaction) is consistent with the observational fact that small-scale (circumstellar) nebulae are predominantly associated with WNL stars (e.g. Lozinskaya & Tutukov 1981; Esteban et al. 1993; Barniske, Oskinova & Hamann 2008), i.e. with young WR stars, whose winds interact with the material lost during the preceding evolutionary phases rather than with the ambient ISM. The scarcity of circumstellar nebulae around WR stars suggests that they are visible on a time-scale much shorter than the duration of the WR phase. It is conceivable to associate this time-scale with the crossing-time of the region occupied by the RSG wind, i.e. $\simeq r_{\rm RSG} /v_{\rm sh} \la 2-5\times 10^4$ yr, where $r_{\rm RSG} = v_{\rm RSG} t_{\rm RSG} \simeq 2-5$ pc and $t_{\rm RSG} \simeq 0.2-0.5$ Myr is the duration of the RSG phase.
Summary
=======
We have serendipitously discovered a ring nebula around a candidate WR star, HBHA4202-22, in Cygnus using the MIPS $24 \, \mu$m data from the [*Spitzer Space Telescope*]{} archive. Our spectroscopic follow-up confirmed the WR nature of this star and showed that it belongs to the WN8-9h subtype (we named the star WR138a). We analyzed the spectrum of WR138a by using the Potsdam Wolf-Rayet (PoWR) model atmospheres, obtaining a stellar temperature of 40kK. The stellar wind composition is dominated by helium with 20 per cent of hydrogen. The stellar spectrum is highly reddened and absorbed ($E_{B-V} = 2.4$mag, $A_V = 7.4$mag). Adopting a stellar luminosity of $\log L/L_{\odot} = 5.3$, the star has a mass-loss rate of $10^{-4.7} \, {\rm\,M_\odot}\, {\rm yr}^{-1}$, and resides in a distance of 4.2 kpc. We measured the proper motion for WR138a and found that it is a runaway star with a peculiar velocity of $\simeq
50 \, {{\rm\,km\,s^{-1}}}$. The runaway nature of WR138a was used to constrain the initial mass of its progenitor star and to propose an explanation of the origin of its circular nebula. We found that the most likely initial mass of the progenitor star was $25-30 \, {\rm\,M_\odot}$ (i.e. the immediate precursor of WR138a was a RSG star) and suggested that the nebula around WR138a is a circumstellar one, created via the interaction between the WR wind and the dense material shed during the preceding RSG phase.
Acknowledgements
================
We are grateful to A.Y.Kniazev for critically reading the manuscript and to the anonymous referee for useful suggestions. VVG and DJB acknowledge financial support from the Deutsche Forschungsgemeinschaft (grants 436 RUS 17/104/06 and BO 1642/14-1) for research visits of VVG at the Astronomical Institute of the Ruhr-University Bochum. SF, OS and AFV acknowledge support from the RFBR grants N07-02-00909 and 09-02-00163. AMC acknowledges support from the RFBR grant N08-02-01220 and the State Program of Support for Leading Scientific Schools of the Russian Federation (grant NSh-1685.2008.2). This work is based in part on archival data obtained with the [*Spitzer Space Telescope*]{}, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA, and has made use of the NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration, the SIMBAD database and the VizieR catalogue access tool, both operated at CDS, Strasbourg, France.
Afanasiev V.L., Moiseev A.V., 2005, AstL, 31, 194 Avedisova V.S., 2005, ARep, 49, 435 Baranov, V.B., Krasnobaev K.V., Kulikovskii A.G., 1971, SPhD, 15, 791 Barniske A., Oskinova L.M., Hamann W.-R., 2008, A&A 486, 971 Brighenti F., D’Ercole A., 1994, MNRAS, 270, 65 Carey S.J. et al., 2009, PASP, 121, 76 Chevalier R.A., Imamura J.N., 1983, ApJ, 270, 554 Clark J.S., Egan M.P., Crowther P.A., Mizuno D.R., Larionov V.M., Arkharov A., 2003, A&A, 412, 185 Corradi R.L.M. et al., 2008, A&A, 480, 409 Crowther P.A., Smith L.J., 1997, A&A, 320, 500 Crowther P.A., Hillier D.J., Smith L.J., 1995, A&A, 293, 172 Dehnen W., Binney J.J., 1998, MNRAS, 298, 387 Dickey J.M., Lockman F.J., 1990, ARA&A, 28, 215 Dolidze M.V., 1971, Astron. Tsirk., 629, 6 Egan M.P., Clark J.S., Mizuno D.R., Carey S.J., Steele I.A., Price S.D., 2002, ApJ, 572, 288 Esteban C., Smith L.J., Vilchez J.M., Clegg R.E.S., 1993, A&A, 272, 299 Gvaramadze V.V., Bomans D.J., 2008a, A&A, 485, L29 Gvaramadze V.V., Bomans D.J., 2008b, A&A, 490, 1071 Gvaramadze V.V., Gualandris A., Portegies Zwart S., 2009, MNRAS, 396, 570 Gvaramadze V.V., Kniazev A.Y., Fabrika S., 2009, preprint (astro-ph/0909.0458) Hadfield L.J., van Dyk S.D., Morris P.W., Smith J.D., Marston A.P., Peterson D.E., 2007, MNRAS, 376, 248 Hamann W.-R., Gräfener G., 2004, A&A, 427, 697 Hamann W.-R., Koesterke, L., 1998, A&A, 335, 1003 Hamann W.-R., Gräfener G., Liermann A., 2006, A&A, 457, 1015 Hamann W.-R., Peña M., Gräfener G., Ruiz M.T., 2003, A&A, 409, 969 Hamann W.-R., Duennebeil G., Koesterke L., Wessolowski U., Schmutz W., 1991,A&A, 249, 443 Henden A.A., Honeycutt R.K., 1997, PASP, 109, 441 Homeier N.L., Blum R.D., Pasquali A., Conti P.S., Damineli A., 2003, A&A, 408, 153 Kohoutek L., Wehmeyer R., 1999, A&AS, 134, 255 Lozinskaya T.A., 1992, Supernovae and Stellar Wind in the Interstellar Medium. Am. Inst. Phys., New York Lozinskaya T.A., Tutukov A.V., 1981, NInfo, 49, 21 Marchenko S.V., Moffat A.F.J., Eversberg T., Morel T., Hill G.M., Tovmassian G.H., Seggewiss W., 1998, MNRAS, 294, 642 Marston A.P., Welzmiller J., Bransford M.A., Black J.H., Bergman P., 1999, ApJ, 518, 769 Mauerhan J., Van Dyk S., Morris P., 2009, PASP, 121, 591 McCutcheon W.H., Shuter W.L.H., 1970, AJ, 75, 910 Meynet G., Maeder A., 2003, A&A, 404, 975 Moneti A., Stolovy S., Blommaert J.A.D.L., Figer D.F., Najarro F., 2001, A&A, 366, 106 Morgan D.H., Parker Q.A., Cohen M., 2003, MNRAS, 346, 719 Reid M.J., 1993, ARA&A, 31, 345 Russeil D., 2003, A&A, 397, 133 Schmutz W., Hamann W.-R., Wessolowski U., 1989, A&A, 210, 236 Seaton M.J., 1979, MNRAS, 187, 73 Shara M.M., Moffat A.F.J., Smith L.F., Niemela V.S., Potter M., Lamontagne R., 1999, AJ, 118, 390 Shara M.M. et al., 2009, AJ, 138, 402 Skrutskie M.F. et al., 2006, AJ, 131, 1163 Smith L.F., 1968, MNRAS, 138, 109 Smith L.F., Shara M.M., Moffat A.F.J., 1996, MNRAS, 281, 163 Todt H., Peña M., Hamann W.-R., Gräfener, 2009, A&A, submitted van der Hucht, K.A., 2001, New Astr. Rev., 45, 135 van der Hucht, K.A., 2006, A&A, 458, 453 Vanbeveren D., De Loore C., Van Rensbergen W., 1998, A&AR, 9, 63 Weaver R., McCray R., Castor J., Shapiro P., Moore R., 1977, ApJ, 218, 377 Werner M.W. et al., 2004, ApJS, 154, 1
[^1]: http://irsa.ipac.caltech.edu/Missions/spitzer.html
[^2]: http://www.cfa.harvard.edu/cygnusX/index.html
[^3]: Spectral Camera with Optical Reducer for Photometrical and Interferometrical Observations; http://www.sao.ru/hq/lsfvo/devices/scorpio/scorpio.html
|
---
abstract: 'Additive tree functionals represent the cost of many divide-and-conquer algorithms. We derive the limiting distribution of the additive functionals induced by toll functions of the form (a) $n^\alpha$ when $\alpha > 0$ and (b) $\log{n}$ (the so-called shape functional) on uniformly distributed binary trees, sometimes called Catalan trees. The Gaussian law obtained in the latter case complements the central limit theorem for the shape functional under the random permutation model. Our results give rise to an apparently new family of distributions containing the Airy distribution ($\alpha = 1$) and the normal distribution \[case (b), and case (a) as $\alpha \downarrow 0$\]. The main theoretical tools employed are recent results relating asymptotics of the generating functions of sequences to those of their Hadamard product, and the method of moments.'
address:
- 'Department of Applied Mathematics and Statistics, The Johns Hopkins University, 3400 N. Charles St., Baltimore MD 21218, USA'
- 'Department of Computer Science, California Institute of Technology, MC 256-80, 1200 E. California Blvd., Pasadena CA 91125, USA'
author:
- James Allen Fill
- Nevin Kapur
bibliography:
- 'msn.bib'
- 'leftovers.bib'
date: 'June 4, 2003. Revised April 1, 2004.'
title: Limiting distributions for additive functionals on Catalan trees
---
[^1]
Introduction {#sec:introduction}
============
Binary trees are fundamental data structures in computer science, with primary application in searching and sorting. For background we refer the reader to Chapter 2 of the excellent book [@MR93f:68045]. In this article we consider additive functionals defined on uniformly distributed binary trees (sometimes called Catalan trees) induced by two types of toll sequences \[$(n^{\alpha})$ and $(\log n)$\]. (See the simple Definition \[def:additive-functional\].) Our main results, Theorems \[thm:limit-dist\] and \[thm:shape\_clt\], establish the limiting distribution for these induced functionals.
A competing model of randomness for binary trees—one used for binary search trees—is the *random permutation model* (RPM); see Section 2.3 of [@MR93f:68045]. While there has been much study of additive functionals under the RPM (see, for example, [@MR93f:68045 Section 3.3] and [@MR92f:68028; @MR97f:68021; @01808498; @MR1871558]), little attention has been paid to the distribution of functionals defined on binary trees under the uniform (Catalan) model of randomness. Fill [@MR97f:68021] argued that the functional corresponding to the toll sequence $ (\log{n}) $ serves as a crude measure of the “shape” of a binary tree, and explained how this functional arises in connection with the move-to-root self-organizing scheme for dynamic maintenance of binary search trees. He derived a central limit theorem under the RPM, but obtained only asymptotic information about the mean and variance under the Catalan model. (The latter results were rederived in the extension [@MR99j:05171] from binary trees to simply generated rooted trees.) In this paper (Theorem \[thm:shape\_clt\]) we show that there is again asymptotic normality under the Catalan model.
In [@MR88h:68033 Prop. 2] Flajolet and Steyaert gave order-of-growth information about the mean of functionals induced by tolls of the form $ n^\alpha $. (The motivation is to build a “repertoire” of tolls from which the behavior of more complicated tolls can be deduced by combining elements from the repertoire. The corresponding results under the random permutation model were derived by Neininger [@MR1910527].) Tak[á]{}cs established the limiting (Airy) distribution of path length in Catalan trees [@MR92m:60057; @MR93e:60175; @MR92k:60164], which is the additive functional for the toll $n-1$. The additive functional for the toll $n^2$ arises in the study of the Wiener index of the tree and has been analyzed by Janson [@janson:_wiener]. In this paper (Theorem \[thm:limit-dist\]) we obtain the limiting distribution for Catalan trees for toll $n^{\alpha}$ for any $\alpha > 0$. The family of limiting distributions appears to be new. In most cases we have a description of the distribution only in terms of its moments, although other descriptions in terms of Brownian excursion, as for the Airy distribution and the limiting distribution for the Wiener index, may be possible. This is currently under investigation by the authors in collaboration with others.
The uniform model on binary trees has also been used recently by Janson [@janson02:_ideal] in the analysis of an algorithm of Koda and Ruskey [@MR94f:94015] for listing ideals in a forest poset.
This paper serves as the first example of the application of recent results [@FFK], extending singularity analysis [@MR90m:05012], to obtain limiting distributions. In [@FFK], it is shown how the asymptotics of generating functions of sequences relate to those of their Hadamard product. First moments for our problems were treated in [@FFK] and a sketch of the technique we employ was presented there. (Our approach to obtaining asymptotics of Hadamard products of generating functions differs only marginally from the Zigzag Algorithm as presented in [@FFK].) As will be evident soon, Hadamard products occur naturally when one is analyzing moments of additive tree functionals. The program we carry out allows a fairly mechanical derivation of the asymptotics of moments of each order, thereby facilitating application of the method of moments. Indeed, preliminary investigations suggest that the techniques we develop are likewise applicable to the wider class of simply generated trees; this is work in progress.
The organization of this paper is as follows. Section \[sec:preliminaries\] establishes notation and states certain preliminaries that will be used in the subsequent proofs. In Section \[sec:toll-sequence-nalpha\] we consider the toll sequence $(n^\alpha)$ for general $\alpha > 0$. In Section \[sec:asympotics-mean\] we compute the asymptotics of the mean of the corresponding additive functional. In Section \[sec:higher-moments\] the analysis diverges slightly as the nature of asymptotics of the higher moments differs depending on the value of $\alpha$. Section \[sec:asymptotics-moments\] employs singularity analysis [@MR90m:05012] to derive the asymptotics of moments of each order. In Section \[sec:limit-distr\] we use the results of Section \[sec:asymptotics-moments\] and the method of moments to derive the limiting distribution of the additive tree functional. In Section \[sec:shape-functional\] we employ the approach again to obtain a normal limit theorem for the shape functional. Finally, in Section \[sec:suff-cond-asympt\], we present heuristic arguments that may lead to the identification of toll sequences giving rise to a normal limit.
Notation and Preliminaries {#sec:preliminaries}
==========================
Additive tree functionals {#sec:addit-tree-funct}
-------------------------
We first establish some notation. Let $T$ be a binary tree. We use $|T|$ to denote the number of nodes in $T$. Let $L(T)$ and $R(T)$ denote, respectively, the left and right subtrees rooted at the children of the root of $T$.
\[def:additive-functional\] A functional $ f $ on binary trees is called an *additive tree functional* if it satisfies the recurrence $$\label{eq:2.1}
f(T) = f( L(T) ) + f( R(T) )+ b_{|T|},$$ for any tree $ T $ with $ |T| \geq 1 $. Here $ (b_n)_{n \geq
1} $ is a given sequence, henceforth called the *toll function*.
We analyze additive functionals defined on binary trees uniformly distributed over $\{ T\!:\, |T|=n \}$ for given $n$. Let $X_n$ be such an additive functional induced by the toll sequence $(b_n)$. It is well known that the number of binary trees on $n$ nodes is counted by the $n$th Catalan number $$\beta_n := \frac1{n+1}\binom{2n}{n},$$ with generating function $$\label{eq:36}
\operatorname{CAT}(z) := \sum_{n=0}^\infty \beta_n z^n = \frac1{2z}(1 - \sqrt{1-4z}).$$ In our subsequent analysis we will make use of the identity $$\label{eq:50}
z \operatorname{CAT}^2(z) = \operatorname{CAT}(z) - 1.$$ The mean of the cost function $a_n := {\mathbf{E}\,X_n} $ can be obtained recursively by conditioning on the size of $L(T)$ as $$a_n = \sum_{j=1}^n \frac{\beta_{j-1}\beta_{n-j}}{\beta_n}(a_{j-1} +
a_{n-j}) + b_n, \qquad n \geq 1.$$ This recurrence can be rewritten as $$\label{eq:35}
(\beta_n a_n) = 2 \sum_{j=1}^n (\beta_{j-1} a_{j-1}) \beta_{n-j} +
(\beta_n b_n), \qquad n \geq 1.$$ Recall that the *Hadamard product* of two power series $F$ and $G$, denoted by $F(z) \odot G(z)$, is the power series defined by $$( F \odot G)(z) \equiv F(z) \odot G(z) := \sum_{n} f_n g_n z^n,$$ where $$F(z) = \sum_{n} f_n z^n \qquad \text{ and } \qquad G(z) = \sum_{n}
g_n z^n.$$ Multiplying (\[eq:35\]) by $z^n/4^n$ and summing over $n \geq 1$ we get $$\label{eq:1}
A(z)\odot\operatorname{CAT}(z/4) = \frac{B(z)\odot\operatorname{CAT}(z/4)}{\sqrt{1-z}}, $$ where $A(z)$ and $B(z)$ are the ordinary generating functions of $(a_n)$ and $(b_n)$ respectively.
\[rem:Catalan\] Catalan numbers are ubiquitous in combinatorial applications; see [@MR2000k:05026] for a list of 66 instances and <http://www-math.mit.edu/~rstan/ec/> for more.
In the sequel the notation \[$\cdots$\] is used both for Iverson’s convention [@knuth97 1.2.3(16)] and for the coefficient of certain terms in the succeeding expression. The interpretation will be clear from the context. For example, $ [ \alpha > 0 ] $ has the value 1 when $ \alpha > 0 $ and the value 0 otherwise. In contrast, $ [z^n]
F(z) $ denotes the coefficient of $ z^n $ in the series expansion of $ F(z) $. Throughout this paper $\Gamma$ and $\zeta$ denote Euler’s gamma function and the Riemann zeta function, respectively.
Singularity analysis {#sec:singularity-analysis}
--------------------
*Singularity analysis* is a systematic complex-analytic technique that relates asymptotics of sequences to singularities of their generating functions. The applicability of singularity analysis rests on the technical condition of *$\Delta$-regularity*. Here is the definition. See [@FFK] or [@MR90m:05012] for further background.
\[def:delta-regular\] A function defined by a Taylor series about the origin with radius of convergence equal to $1$ is *$\Delta$-regular* if it can be analytically continued in a domain $$\Delta(\phi,\eta) := \{z: |z| < 1 + \eta, |\arg(z-1)| > \phi\},$$ for some $\eta > 0$ and $0 < \phi < \pi/2$. A function $f$ is said to admit a *singular expansion* at $z=1$ if it is $\Delta$-regular and $$f(z) = \sum_{j=0}^J c_j(1-z)^{\alpha_j} + O(|1-z|^A)$$ uniformly in $z \in \Delta(\phi,\eta)$, for a sequence of complex numbers $(c_j)_{0 \leq j \leq J}$ and an increasing sequence of real numbers $(\alpha_j)_{0 \leq j \leq J}$ satisfying $\alpha_j < A$. It is said to satisfy a singular expansion *“with logarithmic terms”* if, similarly, $$f(z) = \sum_{j=0}^J c_j\left(L(z)\right)(1-z)^{\alpha_j} + O(|1-z|^A),
\qquad
L(z):=\log\frac{1}{1-z},$$ where each $c_j(\cdot)$ is a polynomial.
Following established terminology, when a function has a singular expansion with logarithmic terms we shall say that it is *amenable* to singularity analysis.
Recall the definition of the *generalized polylogarithm*:
\[def:li\] For $\alpha$ an arbitrary complex number and $r$ a nonnegative integer, the *generalized polylogarithm* function $\operatorname{Li}_{\alpha,r}$ is defined for $|z| < 1$ by $$\label{eq:4.1.50}
\operatorname{Li}_{\alpha,r}(z) := \sum_{n=1}^\infty \frac{(\log{n})^r}{n^\alpha} z^n.$$
The key property of the generalized polylogarithm that we will employ is $$\operatorname{Li}_{\alpha,r} \odot \operatorname{Li}_{\beta,s} = \operatorname{Li}_{\alpha+\beta,r+s}.$$ We will also make extensive use of the following consequences of the singular expansion of the generalized polylogarithm. Neither this lemma nor the ones following make any claims about uniformity in $\alpha$ or $r$. Note that $\operatorname{Li}_{1,0}(z) = L(z) = \log\bigl((1-z)^{-1}\bigr)$.
\[lem:litoomz\] For any real $\alpha < 1$ and nonnegative integer $r$, we have the singular expansion $$\operatorname{Li}_{\alpha,r}(z) = \sum_{k=0}^r \lambda_k^{(\alpha,r)}
(1-z)^{\alpha-1} L^{r-k}(z) +
O(|1-z|^{\alpha-\epsilon}) + (-1)^r \zeta^{(r)}(\alpha)[\alpha > 0],$$ where $\lambda_k^{(\alpha,r)} \equiv \binom{r}{k}
\Gamma^{(k)}(1-\alpha)$ and $\epsilon > 0$ is arbitrarily small.
By Theorem 1 in [@MR2000a:05015], $$\label{eq:37}
\operatorname{Li}_{\alpha,0}(z) \sim \Gamma(1-\alpha) t^{\alpha-1} + \sum_{j
\geq 0} \frac{(-1)^j}{j!} \zeta(\alpha-j) t^j, \qquad t = -\log z
= \sum_{l=1}^\infty \frac{(1-z)^l}{l},$$ and for any positive integer $r$, $$\operatorname{Li}_{\alpha,r}(z) = (-1)^r \frac{\partial^r}{\partial\alpha^r}
\operatorname{Li}_{\alpha,0}(z).$$ Moreover, as also shown in [@MR2000a:05015], the singular expansion for $\operatorname{Li}_{\alpha,r}$ is obtained by performing the indicated differentiation of (\[eq:37\]) term-by-term. To establish the claim we set $ f = \Gamma(1-\alpha) $ and $ g =
t^{\alpha-1} $ in the general formula for the $r$th derivative of a product: $$(fg)^{(r)} = \sum_{k=0}^r \binom{r}{k} f^{(k)} g^{(r-k)}$$ to first obtain $$(-1)^r \frac{\partial^r}{\partial\alpha^r}
[\Gamma(1-\alpha)t^{\alpha-1}] = (-1)^r \sum_{k=0}^r \binom{r}{k}
(-1)^k\Gamma^{(k)}(1-\alpha) t^{\alpha-1} (\log t)^{r-k}$$ The claim then follows easily.
The following “inverse” of Lemma \[lem:litoomz\] is very useful for computing with Hadamard products.
\[lem:omztoli\] For any real $\alpha < 1$ and nonnegative integer $r$, there exists a region $\Delta(\phi,\eta)$ as in Defintion \[def:delta-regular\] such that $$(1-z)^{\alpha-1}L^r(z) = \sum_{k=0}^r
\mu_k^{(\alpha,r)} \operatorname{Li}_{\alpha,r-k}(z) + O(|1-z|^{\alpha-\epsilon})
+ c_r(\alpha)[\alpha > 0]$$ holds uniformly in $z \in \Delta(\phi,\eta)$, where $\mu_0^{(\alpha,r)}=1/\Gamma(1-\alpha)$, $ c_r(\alpha) $ is a constant, and $\epsilon > 0$ is arbitrarily small.
We use induction on $r$. For $r=0$ we have $$\operatorname{Li}_{\alpha,0}(z) = \Gamma(1-\alpha)(1-z)^{\alpha-1} +
O(|1-z|^{\alpha-\epsilon}) + \zeta(\alpha) [ \alpha > 0 ]$$ and the claim is verified with $$\mu_0^{(\alpha,0)} = \frac{1}{\Gamma(1-\alpha)} \qquad \text{ and
} \qquad c_0(\alpha) = -\frac{\zeta(\alpha)}{\Gamma(1-\alpha)}.$$ Let $r \geq 1$. Then using Lemma \[lem:litoomz\] and the induction hypothesis we get $$\begin{aligned}
&\operatorname{Li}_{\alpha,r}(z) \\&=
\Gamma(1-\alpha)(1-z)^{\alpha-1}
L^r(z) \\
& \quad + \sum_{k=1}^r \lambda_{k}^{(\alpha,r)} \left[
\sum_{l=0}^{r-k}\mu_l^{(\alpha,r-k)} \operatorname{Li}_{\alpha,r-k-l}(z) +
O(|1-z|^{\alpha-\epsilon}) +
c_{r-k}(\alpha) [\alpha > 0 ] \right]\\
& \quad + O(|1-z|^{\alpha-\epsilon})
+ (-1)^r\zeta^{(r)}(\alpha)[\alpha > 0]\\
&=
\Gamma(1-\alpha)(1-z)^{\alpha-1}L^r(z)
+ \sum_{k=1}^r \lambda_{k}^{(\alpha,r)} \sum_{s=0}^{r-k}
\mu_{r-k-s}^{(\alpha,r-k)} \operatorname{Li}_{\alpha,s}(z) \\
& \quad {}+ O(|1-z|^{\alpha-\epsilon}) +
\left(
\sum_{k=1}^r \lambda_k^{(\alpha,r)} c_{r-k}(\alpha)
+ (-1)^r\zeta^{(r)}(\alpha)
\right)[\alpha > 0]\\
&=
\Gamma(1-\alpha)(1-z)^{\alpha-1}
L^r(z)
+ \sum_{s=0}^{r-1} \nu_s^{(\alpha,r)} \operatorname{Li}_{\alpha,s}(z)\\
& \quad + O(|1-z|^{\alpha-\epsilon}) + \gamma_r(\alpha)[\alpha >
0],
\end{aligned}$$ where, for $0 \leq s \leq r-1$, $$\nu_s^{(\alpha,r)} := \sum_{k=1}^{r-s} \lambda_k^{(\alpha,r)}
\mu_{r-s-k}^{(\alpha,r-k)},$$ and where $$\gamma_r(\alpha) := \sum_{k=1}^r
\lambda_k^{(\alpha,r)}c_{r-k}(\alpha) + (-1)^r \zeta^{(r)}(\alpha).$$ Setting $$\mu_0^{(\alpha,r)} = \frac{1}{\Gamma(1-\alpha)},\qquad
\mu_k^{(\alpha,r)} =
-\frac{\nu_{r-k}^{(\alpha,r)}}{\Gamma(1-\alpha)}, \quad 1 \leq k
\leq r,$$ and $$c_r(\alpha) = - \frac{\gamma_r(\alpha)}{\Gamma(1-\alpha)},$$ the result follows.
For the calculation of the mean, the following refinement of a special case of Lemma \[lem:litoomz\] is required. It is a simple consequence of Theorem 1 of [@MR2000a:05015].
\[lem:li0omz\] When $\alpha < 0$, we have the singular expansion $$\operatorname{Li}_{\alpha,0}(z) = \Gamma(1-\alpha)(1-z)^{\alpha-1} -
\Gamma(1-\alpha)\frac{1-\alpha}2 (1-z)^{\alpha} +
O(|1-z|^{\alpha+1}) + \zeta(\alpha)[\alpha > -1] .$$
For the sake of completeness, we state a result of particular relevance from [@FFK].
\[thm:hadamard\] If $f$ and $g$ are amenable to singularity analysis and $$f(z) = O(|1-z|^a) \qquad\text{ and }\qquad g(z) = O(|1-z|^b)$$ as $z \to 1$, then $f \odot g$ is also amenable to singularity analysis. Furthermore
(a) If $a+b+1 < 0$ then $$f(z) \odot g(z) = O(|1-z|^{a+b+1}).$$\[item:hadamard1\]
(b) If $k < a+b+1 < k+1$ for some integer $-1 \leq k < \infty$, then $$f(z) \odot g(z) = \sum_{j=0}^k \frac{(-1)^j}{j!} (f \odot
g)^{(j)}(1) (1-z)^j + O(|1-z|^{a+b+1}).$$ \[item:hadamard2\]
(c) If $a+b+1$ is a nonnegative integer then $$f(z) \odot g(z) = \sum_{j=0}^{a+b} \frac{(-1)^j}{j!} (f \odot
g)^{(j)}(1) (1-z)^j + O(|1-z|^{a+b+1}|L(z)|).$$ \[item:hadamard3\]
The toll sequence n\^ {#sec:toll-sequence-nalpha}
=====================
In this section we consider additive functionals when the toll function $b_n$ is $n^\alpha$ with $\alpha > 0$.
Asympotics of the mean {#sec:asympotics-mean}
----------------------
The main result of this Section \[sec:asympotics-mean\] is a singular expansion for $A(z) \odot \operatorname{CAT}(z/4)$. The result is , , or according as $\alpha <
1/2$, $\alpha = 1/2$, or $\alpha > 1/2$.
Since $b_n = n^\alpha$, by definition $B=\operatorname{Li}_{-\alpha,0}$. Thus, by Lemma \[lem:li0omz\], $$B(z) = \Gamma(1+\alpha)(1-z)^{-\alpha-1} -
\Gamma(1+\alpha)\frac{\alpha+1}2(1-z)^{-\alpha} +
O(|1-z|^{-\alpha+1}) + \zeta(-\alpha)[\alpha < 1].$$ We will now use (\[eq:1\]) to obtain the asymptotics of the mean.
First we treat the case $\alpha < 1/2$. From the singular expansion $\operatorname{CAT}(z/4) = 2 + O(|1-z|^{1/2})$ as $z
\to 1$, we have, by part (\[item:hadamard2\]) of Theorem \[thm:hadamard\], $$B(z)\odot\operatorname{CAT}(z/4) = C_0 + O(|1-z|^{-\alpha+\tfrac12}),$$ where $$\label{eq:20}
C_0 := B(z)\odot\operatorname{CAT}(z/4) \Bigr\rvert_{z=1} = \sum_{n=1}^\infty n^\alpha
\frac{\beta_n}{4^n}.$$ We now already know the constant term in the singular expansion of $B(z)\odot\operatorname{CAT}(z/4)$ at $z=1$ and henceforth we need only compute lower-order terms. The constant $\bar{c}$ is used in the sequel to denote an unspecified (possibly 0) constant, possibly different at each appearance.
Let’s write $B(z) = L_1(z) + R_1(z)$, and $\operatorname{CAT}(z/4) = L_2(z) +
R_2(z)$, where $$\begin{aligned}
L_1(z) &:= \Gamma(1+\alpha)(1-z)^{-\alpha-1} -
\Gamma(1+\alpha)\frac{\alpha+1}2(1-z)^{-\alpha} + \zeta(-\alpha),\\
R_1(z) &:= B(z) - L_1(z) = O(|1-z|^{1-\alpha}),\\
L_2(z) &:= 2(1 - (1-z)^{1/2}),\\
R_2(z) &:= \operatorname{CAT}(z/4) - L_2(z) = O(|1-z|).\end{aligned}$$ We will analyze each of the four Hadamard products separately. First, $$\begin{aligned}
L_1(z)\odot{}L_2(z) &= -
2\Gamma(1+\alpha)\bigl[(1-z)^{-\alpha-1}\odot(1-z)^{1/2}\bigr]\\
& \quad + 2\Gamma(1+\alpha)\frac{\alpha+1}2
\bigl[(1-z)^{-\alpha}\odot(1-z)^{1/2}\bigr] + \bar{c}.\end{aligned}$$ By Theorem 4.1 of [@FFK], $$(1-z)^{-\alpha-1}\odot(1-z)^{1/2} = \bar{c} +
\frac{\Gamma(\alpha-\tfrac12)}{\Gamma(\alpha+1)\Gamma(-1/2)}
(1-z)^{-\alpha+\tfrac12} + O(|1-z|),$$ and $$(1-z)^{-\alpha}\odot(1-z)^{1/2} = \bar{c} + O(|1-z|)$$ by another application of part (\[item:hadamard2\]) of Theorem \[thm:hadamard\], this time with $k=1$. Hence $$L_1(z)\odot{}L_2(z) = \bigl[L_1(z)\odot{}L_2(z)\bigr]\Bigr\rvert_{z=1} +
\frac{\Gamma(\alpha-\tfrac12)}{\sqrt\pi}
(1-z)^{-\alpha+\tfrac12} + O(|1-z|).$$ The other three Hadamard products are easily handled as $$\begin{aligned}
L_1(z)\odot{}R_2(z) &= \bigl[L_1(z)\odot{}R_2(z)\bigr]\Bigr\rvert_{z=1} +
O(|1-z|^{-\alpha+1}), \\
L_2(z)\odot{}R_1(z) &= \bigl[L_2(z)\odot{}R_1(z)\bigr]\Bigr\rvert_{z=1} +
O(|1-z|), \\
R_1(z)\odot{}R_2(z) &= \bigl[R_1(z)\odot{}R_2(z)\bigr]\Bigr\rvert_{z=1} +
O(|1-z|).\end{aligned}$$ Putting everything together, we get $$B(z)\odot\operatorname{CAT}(z/4) = C_0 +
\frac{\Gamma(\alpha-\tfrac12)}{\sqrt\pi}
(1-z)^{-\alpha+\tfrac12} + O(|1-z|^{-\alpha+1}).$$ Using this in , we get $$\label{eq:9}
A(z)\odot\operatorname{CAT}(z/4) = C_0(1-z)^{-1/2} +
\frac{\Gamma(\alpha-\tfrac12)}{\sqrt\pi} (1-z)^{-\alpha} +
O(|1-z|^{-\alpha+\tfrac12}).$$ To treat the case $\alpha \geq 1/2$ we make use of the estimate $$\label{eq:21}
(1-z)^{1/2} = \frac1{\Gamma(-1/2)}[ \operatorname{Li}_{3/2,0}(z) - \zeta(3/2) ]
+ O(|1-z|),$$ a consequence of Theorem 1 of [@MR2000a:05015], so that $$B(z) \odot (1-z)^{1/2} = \operatorname{Li}_{-\alpha,0}(z) \odot (1-z)^{1/2} =
\frac{1}{\Gamma(-1/2)} \operatorname{Li}_{\tfrac32-\alpha,0}(z) + R(z),$$ where $$\label{eq:55}
R(z) =
\begin{cases}
\bar{c} + O(|1-z|^{1-\alpha}) & 1/2 \leq \alpha < 1 \\
O(|L(z)|) & \alpha=1 \\
O(|1-z|^{1-\alpha}) & \alpha > 1.
\end{cases}$$ Hence $$B(z)\odot\operatorname{CAT}(z/4) = -\frac2{\Gamma(-1/2)}\operatorname{Li}_{\tfrac32-\alpha,0}(z) +
\widetilde{R}(z),$$ where $\widetilde{R}$, like $R$, satisfies (with a possibly different $\bar{c}$). When $\alpha=1/2$, this gives us $$B(z)\odot\operatorname{CAT}(z/4) = -\frac2{\Gamma(-1/2)}L(z) +
\bar{c} + O(|1-z|^{1/2}),$$ so that $$\label{eq:7}
A(z)\odot\operatorname{CAT}(z/4) =
\frac1{\sqrt\pi}(1-z)^{-1/2}L(z) +
\bar{c}(1-z)^{-1/2} + O(1).$$ For $\alpha > 1/2$ another singular expansion leads to the conclusion that $$\label{eq:8}
A(z)\odot\operatorname{CAT}(z/4) =
\frac{\Gamma(\alpha-\tfrac12)}{\sqrt\pi}(1-z)^{-\alpha} + \widehat{R}(z),$$ where $$\widehat{R}(z) =
\begin{cases}
O(|1-z|^{-\tfrac12}) & 1/2 < \alpha < 1 \\
O(|1-z|^{-\tfrac12}|L(z)|) & \alpha = 1 \\
O(|1-z|^{-\alpha+\tfrac12}) & \alpha > 1.
\end{cases}$$
We defer deriving the asymptotics of $a_n$ until Sections \[sec:higher-moments\]–\[sec:asymptotics-moments\].
Higher moments {#sec:higher-moments}
--------------
We will analyze separately the cases $0 < \alpha < 1/2$, , and . The reason for this will become evident soon; though the technique used to derive the asymptotics is induction in each case, the induction hypothesis is different for each of these cases.
### Small toll functions {#sec:small-toll-functions}
We start by restricting ourselves to tolls of the form $n^\alpha$ where $0 < \alpha < 1/2$. In this case we observe that by singularity analysis applied to (\[eq:9\]), $$\frac{a_n \beta_n}{4^n} = \frac{C_0}{\sqrt\pi} n^{-1/2} + O(n^{-3/2}) +
O(n^{\alpha-1}) = \frac{C_0}{\sqrt\pi} n^{-1/2} +
O(n^{\alpha-1}),$$ so $$a_n = n^{\tfrac32}[1 + O(n^{-1})][ C_0n^{-\tfrac12} +
O(n^{\alpha-1})] = C_0 n + O(n^{\alpha+\tfrac12}) = (C_0 + o(1)) (n+1).$$ The lead-order term of the mean $a_n = {\mathbf{E}\,X_n}$ is thus linear, irrespective of the value of $0 < \alpha < 1/2$ (though the coefficient $C_0$ does depend on $\alpha$). We next perform an approximate centering to get to further dependence on $\alpha$.
Define $\widetilde{X}_n := X_n -
C_0(n+1)$, with $X_0 := 0$; $\tilde{\mu}_n(k) :=
{\mathbf{E}\,\widetilde{X}_n^k}$, with $\tilde{\mu}_n(0) = 1$ for all $n \geq
0$; and $ \hat{\mu}_n(k) :=
\beta_n\tilde{\mu}_n(k)/4^n $. Let $\widehat{M}_k(z)$ denote the ordinary generating function of $\hat{\mu}_n(k)$ in the argument $n$.
By an argument similar to the one that led to (\[eq:35\]), we get, for $k \geq 2$, $$\hat{\mu}_n(k) = \frac12 \sum_{j=1}^n \frac{\beta_{n-j}}{4^{n-j}}
\hat{\mu}_{j-1}(k) + \hat{r}_n(k), \qquad n \geq 1,$$ where $$\begin{aligned}
\hat{r}_n(k) &:= \frac14\sum_{j=1}^n \sum_{\substack{k_1+k_2+k_3=k\\
k_1,k_2 < k}} \binom{k}{k_1,k_2,k_3}
\hat{\mu}_{j-1}(k_1)
\hat{\mu}_{n-j}(k_2) b_n^{k_3}\\
&= \frac14 \sum_{\substack{k_1+k_2+k_3=k\\
k_1,k_2 < k}} \binom{k}{k_1,k_2,k_3} b_n^{k_3} \sum_{j=1}^n
\hat{\mu}_{j-1}(k_1) \hat{\mu}_{n-j}(k_2),\end{aligned}$$ for $n \geq 1$ and $\hat{r}_0(k) := \hat{\mu}_0(k) = \tilde{\mu}_0(k) =
(-1)^kC_0^k$. Let $\widehat{R}_k(z)$ denote the ordinary generating function of $\hat{r}_n(k)$ in the argument $n$. Then, mimicking (\[eq:1\]), $$\label{eq:4}
\widehat{M}_k(z) = \frac{\widehat{R}_k(z)}{\sqrt{1-z}}$$ with $$\label{eq:3}
\widehat{R}_k(z) = (-1)^kC_0^k + \sum_{\substack{k_1+k_2+k_3=k\\
k_1,k_2<k}} \binom{k}{k_1,k_2,k_3} \bigl( B(z)^{\odot k_3}
\bigr) \odot
\bigl[\frac{z}4 \widehat{M}_{k_1}(z)\widehat{M}_{k_2}(z)\bigr],$$ where for $k$ a nonnegative integer $$B(z)^{\odot k} := \underbrace{B(z)\odot\cdots\odot B(z)}_{k}.$$ Note that $\widehat{M}_0(z) = \operatorname{CAT}(z/4)$.
\[thm:0alpha14\] Let $\epsilon > 0$ be arbitrary, and define $$c :=
\begin{cases}
2\alpha - \epsilon & 0 < \alpha \leq 1/4 \\
1/2 & 1/4 < \alpha < 1/2.
\end{cases}$$ Then we have the singular expansion $$\widehat{M}_k(z) = C_k(1-z)^{-k(\alpha+\tfrac12) +\tfrac12} +
O(|1-z|^{-k(\alpha+\tfrac{1}{2}) + \tfrac12 + c}),$$ The $C_k$’s here are defined by the recurrence $$\label{eq:10}
C_k = \frac14 \sum_{j=1}^{k-1} \binom{k}{j} C_{j} C_{k-j} +
kC_{k-1} \frac{ \Gamma(k\alpha+\tfrac{k}2-1)}{\Gamma((k-1)\alpha +
\tfrac{k}2 -1)}, \quad k \geq 2; \quad C_1 =
\frac{\Gamma(\alpha-\tfrac12)}{\sqrt\pi}.$$
For $k=1$ the claim is true as shown in with $C_1$ as defined in . We will now analyze each term in for $k \geq 2$.
One can analyze separately the cases $0 < \alpha \leq 1/4$ and $1/4
< \alpha < 1/2$. The proof technique in either case is induction. We shall treat here the case $0 < \alpha \leq 1/4$; the details in the other case can be found in [@FK-catalan-arXiv].
For notational convenience, define $\alpha' := \alpha+\tfrac12$. Also, observe that $$B(z)^{\odot{}k} = \operatorname{Li}_{-k\alpha,0}(z)
= \Gamma(1+k\alpha)(1-z)^{-k\alpha-1} + O(|1-z|^{-k\alpha-\epsilon})$$ by Lemma \[lem:litoomz\]. We shall find that the dominant terms in the sum in are those with (i) $k_3 = 0$, (ii) $(k_1,
k_2, k_3) = (k-1, 1, 0)$, and (iii) $(k_1, k_2, k_3) = (0, k-1,
1)$.
For this paragraph, consider the case that $k_1$ and $k_2$ are both nonzero. It follows from the induction hypothesis that $$\begin{aligned}
\frac{z}4 \widehat{M}_{k_1}(z)\widehat{M}_{k_2}(z) &= \frac14(1-(1-z))
\bigl[ C_{k_1}(1-z)^{-k_1\alpha'+\tfrac12} +
O(|1-z|^{-k_1\alpha'+\tfrac12+(2\alpha-\epsilon)})
\bigr]\\
&\times
\bigl[ C_{k_2}(1-z)^{-k_2\alpha'+\tfrac12} +
O(|1-z|^{-k_2\alpha'+\tfrac12+(2\alpha-\epsilon)})
\bigr]\\
&= \frac14C_{k_1}C_{k_2} (1-z)^{-(k_1+k_2)\alpha' + 1}
+ O(|1-z|^{-(k_1+k_2)\alpha'+1+(2\alpha-\epsilon)}).
\end{aligned}$$ If $k_3=0$ then the corresponding contribution to $\widehat{R}_k(z)$ is $$\frac14\binom{k}{k_1}
C_{k_1}C_{k_2} (1-z)^{-k\alpha' + 1}
+ O(|1-z|^{-k\alpha'+1+(2\alpha-\epsilon)}).$$ If $k_3 \ne 0$ we use Lemma \[lem:omztoli\] to express $$\begin{gathered}
\frac{z}4 \widehat{M}_{k_1}(z)\widehat{M}_{k_2}(z) =
\frac{C_{k_1}C_{k_2}}{4\Gamma((k_1+k_2)\alpha'-1)}
\operatorname{Li}_{-(k_1+k_2)\alpha'+2,0}(z)\\
+
O(|1-z|^{-(k_1+k_2)\alpha'+1+(2\alpha-\epsilon)}) -
\frac{C_{k_1}C_{k_2}}{4}
[(k_1+k_2)\alpha' <
2]\frac{\zeta(-(k_1+k_2)\alpha'+2)}
{\Gamma((k_1+k_2)\alpha'-1)}.
\end{gathered}$$ The corresponding contribution to $\widehat{R}_{k}(z)$ is then $\binom{k}{k_1,k_2,k_3}$ times: $$\frac{C_{k_1}C_{k_2}}{4\Gamma((k_1+k_2)\alpha'-1)}
\operatorname{Li}_{-k\alpha'+\tfrac{k_3}2+2,0}(z)
+ \operatorname{Li}_{-k_3\alpha,0}(z)
\odot O(|1-z|^{-(k_1+k_2)\alpha'+1+(2\alpha-\epsilon)}).$$ Now $k_3 \leq k-2$ so $-k\alpha'+\tfrac{k_3}2 + 2 < 1$. Hence the contribution when $k_3 \ne 0$ is $$O(|1-z|^{-k\alpha'+\tfrac{k_3}2+1}) =
O(|1-z|^{-k\alpha'+\tfrac32}) =
O(|1-z|^{-k\alpha'+1+(2\alpha-\epsilon)}).$$
Next we consider the case when $k_1$ is nonzero but $k_2=0$. In this case using the induction hypothesis we see that $$\begin{aligned}
\frac{z}4 \widehat{M}_{k_1}(z)\widehat{M}_{k_2}(z) &=
\frac{z}4 \operatorname{CAT}(z/4)\widehat{M}_{k_1}(z)\\
&= \frac{1 - (1-z)^{1/2}}{2} \bigl[
C_{k_1}(1-z)^{-k_1\alpha' + \tfrac12}\bigr]
+
O(|1-z|^{-k_1\alpha' + \tfrac12 +
(2\alpha-\epsilon)})\\
&= \frac{C_{k_1}}2 (1-z)^{-k_1\alpha'+\tfrac12} +
O(|1-z|^{-k_1\alpha' + \tfrac12 + (2\alpha-\epsilon)}).
\end{aligned}$$ Applying Lemma \[lem:omztoli\] to the last expression we get $$\begin{gathered}
\frac{z}4 \widehat{M}_{k_1}(z)\widehat{M}_{k_2}(z) =
\frac{C_{k_1}}{2\Gamma(k_1\alpha'-\tfrac12)}
\operatorname{Li}_{-k_1\alpha'+\tfrac32,0}(z) \\
+ O(|1-z|^{-k_1\alpha' + \tfrac12 + (2\alpha-\epsilon)})
- \frac{C_{k_1}}{2}[k_1\alpha'-\tfrac12 < 1]
\frac{\zeta(-k_1\alpha'+\tfrac32)}
{\Gamma(k_1\alpha'-\tfrac12)}.
\end{gathered}$$ The contribution to $\widehat{R}_{k}(z)$ is hence $\binom{k}{k_1}$ times: $$\frac{C_{k_1}}{2\Gamma(k_1\alpha'-\tfrac12)}
\operatorname{Li}_{-k\alpha'+\tfrac{k_3}2+\tfrac32,0}(z) +
\operatorname{Li}_{-k_3\alpha,0}(z) \odot O(|1-z|^{-k_1\alpha' +
\tfrac12 + (2\alpha-\epsilon)}).$$ Using the fact that $\alpha > 0$ and $k_3 \leq k-1$, we conclude that $-k\alpha'+\tfrac{k_3}2+\tfrac32 < 1$ so that, by Lemma \[lem:litoomz\] and part (\[item:hadamard1\]) of Theorem \[thm:hadamard\], the contribution is $$O(|1-z|^{-k\alpha'+\tfrac{k_3}2+\tfrac12}) =
O(|1-z|^{-k\alpha'+\tfrac32})$$ where the displayed equality holds unless $k_3=1$. When $k_3=1$ we get a corresponding contribution to $\widehat{R}_k(z)$ of $\binom{k}{k-1}$ times: $$\frac{C_{k-1}\Gamma(k\alpha'-1)}
{2\Gamma((k-1)\alpha'-\tfrac12)}
(1-z)^{-k\alpha'+1} +
O(|1-z|^{-k\alpha'+1+(2\alpha-\epsilon)}),$$ since for $k \geq 2$ we have $k\alpha' > 1 +
(2\alpha-\epsilon)$. The introduction of $\epsilon$ handles the case when $k\alpha' = 1 + 2\alpha$, which would have otherwise, according to part (\[item:hadamard3\]) of Thoerem \[thm:hadamard\], introduced a logarithmic remainder. In either case the remainder is $ O(|1-z|^{-k\alpha'+1+(2\alpha-\epsilon)})$. The case when $k_2$ is nonzero but $k_1=0$ is handled similarly by exchanging the roles of $k_1$ and $k_2$.
The final contribution comes from the single term where both $k_1$ and $k_2$ are zero. In this case the contribution to $\widehat{R}_k(z)$ is, recalling , $$\label{eq:5}
\operatorname{Li}_{-k\alpha,0}(z) \odot [\frac{z}4 \operatorname{CAT}^2(z/4)] =
\operatorname{Li}_{-k\alpha,0}(z) \odot
(\operatorname{CAT}(z/4)-1)= \operatorname{Li}_{-k\alpha,0}(z) \odot \operatorname{CAT}(z/4).$$ Now, using Theorem 1 of [@MR2000a:05015], $$\begin{aligned}
\operatorname{CAT}(z/4) & = 2-2(1-z)^{1/2} + O(|1-z|) \\
& = 2 + 2\frac{\zeta(3/2)}{\Gamma(-1/2)}
- \frac{2}{\Gamma(-1/2)}\operatorname{Li}_{3/2,0}(z) +
O(|1-z|),
\end{aligned}$$ so that is $$-\frac2{\Gamma(-1/2)} \operatorname{Li}_{\tfrac32-k\alpha,0}(z) +
O(|1-z|^{1-k\alpha}) +
\begin{cases}
0 & 1-k\alpha < 0, \\
O(|1-z|^{-\epsilon}) & 1-k\alpha=0, \\
O(1) & 1-k\alpha > 0.
\end{cases}$$ When $\tfrac32-k\alpha < 1$ this is $O(|1-z|^{-k\alpha+\tfrac12})$; when $\tfrac32-k\alpha \geq 1$, it is $O(1)$. In either case we get a contribution which is $O(|1-z|^{-k\alpha'+1+(2\alpha-\epsilon)})$.
Hence $$\begin{aligned}
\widehat{R}_k(z) &= \Biggl[ \sum_{\substack{k_1+k_2=k\\k_1,k_2<k}}
\binom{k}{k_1} \frac{C_{k_1}C_{k_2}}4 + 2k \frac{C_{k-1}}2
\frac{\Gamma(k\alpha+\tfrac{k}2-1)}{\Gamma((k-1)\alpha+\tfrac{k}2-1)}
\Biggr] (1-z)^{-k\alpha'+1}\\
& \qquad\qquad + O(|1-z|^{-k\alpha'+1+(2\alpha-\epsilon)})\\
&= C_k(1-z)^{-k\alpha'+1} + O(|1-z|^{-k\alpha'
+ 1 + (2\alpha-\epsilon)}),
\end{aligned}$$ with the $C_k$’s defined by the recurrence . Now using , the claim follows.
### Large toll functions {#sec:large-toll-functions}
When $\alpha \geq 1/2$ there is no need to apply the centering techinques. Define ${\mu}_n(k) := {\mathbf{E}\,{X}_n^k}$ and $ \bar{\mu}_n(k) := \beta_n{\mu}_n(k)/4^n $. Let $\overline{M}_k(z)$ denote the ordinary generating function of $\bar{\mu}_n(k)$ in $n$. Observe that $\overline{M}_0(z) = \operatorname{CAT}(z/4)$. As earlier, conditioning on the key stored at the root, we get, for $k \geq 2$, $$\bar{\mu}_n(k) = \frac12 \sum_{j=1}^n \frac{\beta_{n-j}}{4^{n-j}}
\bar{\mu}_{j-1}(k) + \bar{r}_n(k),
\qquad n \geq 1,$$ where $$\bar{r}_n(k) :=
\frac14 \sum_{\substack{k_1+k_2+k_3=k\\
k_1,k_2 < k}} \binom{k}{k_1,k_2,k_3} b_n^{k_3} \sum_{j=1}^n
\bar{\mu}_{j-1}(k_1) \bar{\mu}_{n-j}(k_2),$$ for $n \geq 1$ and $\bar{r}_0(k) := \bar{\mu}_0(k) = \mu_0(k) = 0$. Let $\overline{R}_k(z)$ denote the ordinary generating function of $\bar{r}_n(k)$ in $n$. Then $$\overline{M}_k(z) = \frac{\overline{R}_k(z)}{\sqrt{1-z}}$$ and $$\label{eq:12}
\overline{R}_k(z) = \sum_{\substack{k_1+k_2+k_3=k\\
k_1,k_2<k}} \binom{k}{k_1,k_2,k_3} \bigl( B(z)^{\odot k_3}
\bigr) \odot
\bigl[ \frac{z}4 \overline{M}_{k_1}(z)\overline{M}_{k_2}(z)\bigr].$$
We can now state the result about the asymptotics of the generating function $\overline{M}_k$ when $\alpha > 1/2$. The case $\alpha=1/2$ will be handled subsequently, in Proposition \[thm:alpha=12\].
\[thm:12alpha\] Let $\epsilon > 0$ be arbitrary, and define $$\label{eq:56}
c :=
\begin{cases}
\alpha - \tfrac12 & \tfrac12 < \alpha < 1 \\
\tfrac12 - \epsilon & \alpha = 1 \\
\tfrac12 & \alpha > 1.
\end{cases}$$ Then the generating function $\overline{M}_k(z)$ of $\bar{\mu}_n(k)$ has the singular expansion $$\overline{M}_k(z) = C_k(1-z)^{-k(\alpha+\tfrac12)+\tfrac12} +
O(|1-z|^{-k(\alpha+\tfrac12)+\tfrac12+c})$$ for $k \geq 1$, where the $C_k$’s are defined by the recurrence .
The proof is very similar to that of Proposition \[thm:0alpha14\]. We present a sketch. The reader is invited to compare the cases enumerated below to those in the earlier proof.
When $k=1$ the claim is true by . We analyze the various terms in for $k \geq 2$, employing the notational convenience $\alpha' := \alpha+\tfrac12$.
When both $k_1$ and $k_2$ are nonzero then the contribution to $\overline{R}_k(z)$ is $$\frac14\binom{k}{k_1} C_{k_1}C_{k_2} (1-z)^{-k\alpha'+1} +
O(|1-z|^{-k\alpha'+c+1})$$ when $k_3=0$ and is $O(|1-z|^{-k\alpha'+c+1})$ otherwise.
When $k_1$ is nonzero and $k_2=0$ the contribution to $\overline{R}_k(z)$ is $$k \frac{C_{k-1} \Gamma(k\alpha'-1)}{2\Gamma((k-1)\alpha'-\tfrac12)}
(1-z)^{-k\alpha'+1} + O(|1-z|^{-k\alpha'+c+1})$$ when $k_3=1$ and $O(|1-z|^{-k\alpha'+c+1})$ otherwise. The case when $k_2$ is nonzero and $k_1=0$ is identical.
The final contribution comes from the single term when both $k_1$ and $k_2$ are zero. In this case we get a contribution of $O(|1-z|^{-k\alpha+\tfrac12})$ which is $O(|1-z|^{-k\alpha'+c+1})$. Adding all these contributions yields the desired result.
The result when $\alpha=1/2$ is as follows. Recall that $L(z) :=
\log((1-z)^{-1})$.
\[thm:alpha=12\] Let $\alpha=1/2$. In the notation of Proposition \[thm:12alpha\], $$\overline{M}_k(z) = (1-z)^{-k+\tfrac12} \sum_{l=0}^k C_{k,l}
L^{k-l}(z) + O(|1-z|^{-k+1-\epsilon})$$ for $k \geq 1$ and any $\epsilon > 0$, where the $C_{k,l}$’s are constants. The constant multiplying the lead-order term is given by $$\label{eq:11}
C_{k,0} = \frac{(2k-2)!}{2^{2k-2}(k-1)!\pi^{k/2}}.$$
We omit the proof, referring the interested reader to [@FK-catalan-arXiv].
Asymptotics of moments {#sec:asymptotics-moments}
----------------------
For $0 < \alpha< 1/2 $, we have seen in Proposition \[thm:0alpha14\] that the generating function $\widehat{M}_k(z)$ of $ \hat{\mu}_n(k) = \beta_n\tilde{\mu}_n(k)/4^n $ has the singular expansion $$\widehat{M}_k(z) = C_k(1-z)^{-k(\alpha+\tfrac{1}2)+\tfrac12} +
O(|1-z|^{-k(\alpha+\tfrac{1}{2})+\tfrac12 + c}),$$ where $c := \min\{2\alpha-\epsilon,1/2\}$. By singularity analysis [@MR90m:05012], $$\frac{\beta_n\tilde{\mu}_n(k)}{4^n} =
C_k
\frac{n^{k(\alpha+\tfrac12)-\tfrac32}}{\Gamma(k(\alpha+\tfrac12)-\tfrac12)}
+ O(n^{k(\alpha+\tfrac{1}{2})-\tfrac32-c}).$$ Recall that $$\beta_n =
\frac{4^n}{\sqrt{\pi}n^{3/2}} \left(1 + O(\tfrac1n)\right),$$ so that $$\label{eq:15}
\tilde{\mu}_n(k)
= \frac{C_k\sqrt{\pi}}{\Gamma(k(\alpha+\tfrac{1}2)-\tfrac12)}
n^{k(\alpha+\tfrac{1}2)} +
O(n^{k(\alpha+\tfrac{1}2) - c}).$$
For $\alpha > 1/2$ a similar analysis using Proposition \[thm:12alpha\] yields $$\label{eq:16}
\mu_n(k) = \frac{C_k\sqrt{\pi}}{\Gamma(k(\alpha+\tfrac{1}2)-\tfrac12)}
n^{k(\alpha+\tfrac{1}2)} +
O(n^{k(\alpha+\tfrac{1}2) - c}),$$ with now $c$ as defined at . Finally, when $\alpha=1/2$ the asymptotics of the moments are given by $$\label{eq:17}
\mu_n(k) =
\left( \frac{1}{\sqrt\pi} \right)^k(n\log{n})^k +
O(n^k(\log{n})^{k-1}).$$
The limiting distributions {#sec:limit-distr}
--------------------------
In Section \[sec:alpha-ne-12\] we will use our moment estimates and with the method of moments to derive limiting distributions for our additive functions. The case $\alpha=1/2$ requires a somewhat delicate analysis, which we will present separately in Section \[sec:alpha=12\].
### alpha not 1/2 {#sec:alpha-ne-12}
We first handle the case $0 < \alpha < 1/2$. (We assume this restriction until just before Proposition \[thm:limit\_law\_alpha\_ne\_12\].) We have $$\label{eq:39}
\tilde\mu_n(1) = {\mathbf{E}\,\widetilde}{X}_n = {\mathbf{E}\,[}X_n - C_0(n+1)]
= \frac{C_1\sqrt\pi}{\Gamma(\alpha)} n^{\alpha+\tfrac12} +
O(n^{\alpha+\tfrac12-c})$$ with $c := \min\{2\alpha-\epsilon,1/2\}$ and $$\tilde\mu_n(2) = {\mathbf{E}\,\widetilde}{X}_n^2 =
\frac{C_2\sqrt\pi}{\Gamma(2\alpha+\tfrac12)} n^{2\alpha+1} +
O(n^{2\alpha+1-c}).$$ So $$\label{eq:40}
{\mathbf{Var}\,X_n} = {\mathbf{Var}\,\widetilde{X}_n} = \tilde\mu_n(2) -
[\tilde\mu_n(1)]^2 = \sigma^2 n^{2\alpha+1} + O(n^{2\alpha+1-c}),$$ where $$\label{eq:46}
\sigma^2 := \frac{C_2\sqrt\pi}{\Gamma(2\alpha+\tfrac12)} -
\frac{C_1^2\pi}{\Gamma^2(\alpha)}.$$ We also have, for $k \geq 1$, $$\label{eq:41}
{\mathbf{E}\,\left}[ \frac{\widetilde{X}_n}{n^{\alpha+\tfrac12}} \right]^k
= \frac{\tilde{\mu}_n(k)}{n^{k(\alpha+\tfrac12)}}
= \frac{C_k\sqrt\pi}{\Gamma(k(\alpha+\tfrac12)-\tfrac12)} + O(n^{-c}).$$
The following lemma provides a sufficient bound on the moments facilitating the use of the method of moments.
\[lem:Ckbound\] Define $\alpha' := \alpha+\tfrac12$. There exists a constant $A
< \infty $ depending only on $\alpha$ such that $$\left|\frac{C_k}{k!}\right| \leq A^k k^{\alpha'k}$$ for all $k \geq 1$.
The proof is fairly similar to those of Propositions \[thm:0alpha14\], \[thm:12alpha\] and Proposition \[thm:shape\_moments\]. We omit the details, referring the reader to [@FK-catalan-arXiv].
It follows from Lemma \[lem:Ckbound\] and Stirling’s approximation that $$\label{eq:19}
\left|\frac{C_k\sqrt\pi}{k! \Gamma(k(\alpha+\tfrac12)-\tfrac12)}\right|
\leq B^k$$ for large enough $B$ depending only on $\alpha$. Using standard arguments [@MR95k:60001 Theorem 30.1] it follows that $X_n$ suitably normalized has a limiting distribution that is characterized by its moments. Before we state the result, we observe that the argument presented above can be adapted with minor modifications to treat the case $\alpha > 1/2$, with $\widetilde{X}_n$ replaced by $X_n$. We can now state a result for $\alpha \ne 1/2$. We will use the notation $\stackrel{\mathcal{L}}{\to}$ to denote convergence in law (or distribution).
\[thm:limit\_law\_alpha\_ne\_12\] Let $X_n$ denote the additive functional on Catalan trees induced by the toll sequence $(n^\alpha)_{n \geq 0}$. Define the random variable $Y_n$ as follows: $$Y_n :=
\begin{cases}
\displaystyle\frac{X_n-C_0(n+1)}{n^{\alpha+\tfrac12}} & 0 <
\alpha < 1/2,\\
\displaystyle\frac{X_n}{n^{\alpha+\tfrac12}} & \alpha > 1/2,
\end{cases}$$ where $$C_0 := \sum_{n=0}^\infty n^\alpha \frac{\beta_n}{4^n}, \qquad
\beta_n = \frac{1}{n+1} \binom{2n}{n}.$$ Then $$Y_n \stackrel{\mathcal{L}}{\to} Y;$$ here $Y$ is a random variable with the unique distribution whose moments are $$\label{eq:49}
{\mathbf{E}\,Y}^k = \frac{C_k
\sqrt\pi}{\Gamma(k(\alpha+\tfrac12)-\tfrac12)},$$ where the $C_k$’s satisfy the recurrence $$C_k = \frac{1}{4} \sum_{j=1}^{k-1} \binom{k}{j} C_j C_{k-j} + k
\frac{\Gamma(k\alpha + \tfrac{k}2 -1)}{\Gamma((k-1)\alpha +
\tfrac{k}2 - 1)} C_{k-1}, k \geq 2; \quad C_1 =
\frac{\Gamma(\alpha-\tfrac12)}{\sqrt\pi}.$$
The case $\alpha=1/2$ is handled in Section \[sec:alpha=12\], leading to Proposition \[prop\_alpha\_12\], and a unified result for all cases is stated as Theorem \[thm:limit-dist\].
\[remark\_y\_alpha\_properties\] We now consider some properties of the limiting random variable $Y
\equiv Y(\alpha)$ defined by its moments at for $\alpha
\ne 1/2$.
(a) When $ \alpha=1 $, setting $\Omega_k := C_k/2$ we see immediately that $${\mathbf{E}\,Y}^k = \frac{-\Gamma(-1/2)}{\Gamma((3k-1)/2)}\Omega_k,$$ where $$2\Omega_k = \sum_{j=1}^{k-1} \binom{k}{j} \Omega_j \Omega_{k-j} + k(3k - 4)
\Omega_{k-1}, \qquad \Omega_1 = \frac12.$$ Thus $Y$ has the ubiquitous Airy distribution and we have recovered the limiting distribution of path length in Catalan trees [@MR92m:60057; @MR92k:60164]. The Airy distribution arises in many contexts including parking allocations, hashing tables, trees, discrete random walks, mergesorting, etc.—see, for example, the introduction of [@MR2002j:68115] which contains numerous references to the Airy distribution.
(b) When $\alpha = 2$, setting $\eta := Y/\sqrt2$ and $a_{0,l} :=
2^{2l-1} C_l$, we see that $${\mathbf{E}\,\eta^l} = \frac{\sqrt\pi}{2^{(5l-2)/2} \Gamma((5l-1)/2)} a_{0,l},$$ where $$a_{0,l} = \frac{1}{2} \sum_{j=1}^{l-1} \binom{l}{j} a_{0,j}
a_{0,l-j} + l(5l-4)(5l-6), \qquad a_{0,1} = 1.$$ We have thus recovered the recurrence for the moments of the distribution $\mathcal{L}({\eta})$, which arises in the study of the Wiener index of Catalan trees [@janson:_wiener proof of Theorem 3.3 in Section 5].
(c) Consider the variance $\sigma^2$ defined at .
(i) Figure \[fig:variance\], plotted using `Mathematica`, suggests that $\sigma^2$ is positive for all $\alpha > 0$. We will prove this fact in Theorem \[thm:limit-dist\].
![$\sigma^2$ of as a function of $\alpha$.[]{data-label="fig:variance"}](variance)
There is also numerical evidence that $\sigma^2$ is unimodal with $\max_{\alpha} \sigma^2(\alpha) \doteq 0.198946$ achieved at $\alpha \doteq
0.682607$. (Here $\doteq$ denotes approximate equality.)
(ii) As $\alpha \to \infty$, using Stirling’s approximation one can show that $\sigma^2 \sim (\sqrt{2}-1)\alpha^{-1}$.
(iii) As $\alpha \downarrow 0$, using a Laurent series expansion of $\Gamma(\alpha)$ we see that $\sigma^2 \sim 4(1-\log2) \alpha$.
(iv) Though the random variable $Y(\alpha)$ has been defined only for $\alpha \ne 1/2$, the variance $\sigma^2$ has a limit at $\alpha=1/2$: $$\label{eq:51}
\lim_{\alpha \to 1/2} \sigma^2(\alpha) = \frac{8 \log 2}{\pi} -
\frac{\pi}2.$$
(d) Figure \[fig:mc3\] shows the third central moment ${\mathbf{E}\,[}Y - {\mathbf{E}\,Y}]^3$ as a function of $\alpha$. The plot suggests that the third central moment is positive for each $\alpha
> 0$, which would also establish that $Y(\alpha)$ is not normal for any $\alpha > 0$. However we do not know a proof of this positive skewness. \[Of course, the law of $Y(\alpha)$ is not normal for any $\alpha > 1/2$, since its support is a subset of $[0,\infty)$.\]
![${\mathbf{E}\,[}Y - {\mathbf{E}\,Y}]^3$ of Proposition \[thm:limit\_law\_alpha\_ne\_12\] as a function of $\alpha$.[]{data-label="fig:mc3"}](mc3)
\[item:1\]
(e) When $\alpha=0$, the additive functional with toll sequence $(n^\alpha=1)_{n \geq 1}$ is $n$ for all trees with $n$ nodes. However, if one considers the random variable $\alpha^{-1/2}Y(\alpha)$ as $\alpha \downarrow 0$, using and induction one can show that $\alpha^{-1/2}Y(\alpha)$ converges in distribution to the normal distribution with mean 0 and variance $4(1-\log2)$.
(f) Finally, if one considers the random variable $\alpha^{1/2}Y(\alpha)$ as $\alpha \to \infty$, again using and induction we find that $\alpha^{1/2}Y(\alpha)$ converges in distribution to the unique distribution with $k$th moment $\sqrt{k!}$ for $k=1,2,\ldots$. In Remark \[remark:sqrkfact\] next, we will show that the limiting distribution has a bounded, infinitely smooth density on $(0,
\infty)$.
\[remark:sqrkfact\] Let $Y$ be the unique distribution whose $k$th moment is $\sqrt{k!}$ for $k=1,2,\ldots$. Taking $Y^*$ to be an independent copy of $Y$ and defining $X := Y Y^*$, we see immediately that $X$ is Exponential with unit mean. It follows by taking logarithms that the distribution of $\log Y$ is a convolution square root of the distribution of $\log X$. In particular, the characteristic function $\phi$ of $\log Y$ has square equal to $\Gamma(1 + it)$ at $t \in (-\infty, \infty)$; we note in passing that $\Gamma(1 + it)$ is the characteristic function of $-G$, where $G$ has the Gumbel distribution. By exponential decay of $\Gamma(1 + it)$ as $t \to
\pm\infty$ and standard theory (see, e.g., [@MR42:5292 Chapter XV]), $\log Y$ has an infinitely smooth density on $(-\infty, \infty)$, and the density and each of its derivatives are bounded.
So $Y$ has an infinitely smooth density on $(0, \infty)$. By change of variables, the density $f_Y$ of $Y$ satisfies $$f_Y(y) = \frac{f_{\log Y}(\log y)}{y}.$$ Clearly $f_Y(y)$ is bounded for $y$ *not* near 0. (We shall drop further consideration of derivatives.) To determine the behavior near 0, we need to know the behavior of $f_{\log Y}(\log y)/y$ as $y \to 0$. Using the Fourier inversion formula, we may equivalently study $$e^x f_{\log{Y}}(-x) = \frac{1}{2\pi} \int_{-\infty}^\infty
e^{(1+it)x} \phi(t)\, dt,$$ as $x \to \infty$. By an application of the method of steepest descents \[(7.2.11) in [@MR89d:41049], with $g_0=1$, $\beta =
1/2$, $w$ the identity map, $z_0=0$, and $\alpha=0$\], we get $$f_Y(y) \sim \frac{1}{\sqrt{\pi\log{(1/y)} }} \quad \text{as $y
\downarrow 0$}.$$ Hence $f_Y$ is bounded everywhere.
Using the Cauchy integral formula and simple estimates, it is easy to show that $$f_Y(y) = o( e^{-My} ) \quad \text{as $y \to \infty$}$$ for any $M < \infty$. Computations using the <span style="font-variant:small-caps;">WKB</span> method [@MR30:3694] suggest $$\label{eq:57}
f_Y(y) \sim (2/\pi)^{1/4} y^{1/2} \exp(-y^2/2) \quad \text{as $y
\to \infty$},$$ in agreement with numerical calculations using `Mathematica`. \[In fact, the right-side of appears to be a highly accurate approximation to $f_Y(y)$ for all $y
\geq 1$.\] Figure \[fig:sqrtfactdensity\] depicts the salient features of $f_Y$. In particular, note the steep descent of $f_Y(y)$ to 0 as $y \downarrow 0$ and the quasi-Gaussian tail.
![$f_Y$ of Remark \[remark:sqrkfact\].[]{data-label="fig:sqrtfactdensity"}](little1 "fig:"){width="2.4in"} ![$f_Y$ of Remark \[remark:sqrkfact\].[]{data-label="fig:sqrtfactdensity"}](little2 "fig:"){width="2.4in"}
### alpha = 1/2 {#sec:alpha=12}
For $\alpha=1/2$, from (\[eq:17\]) we see immediately that $${\mathbf{E}\,\left}[ \frac{X_n}{n\log{n}} \right]^k = \left(
\frac1{\sqrt\pi} \right)^k + O\left( \frac{1}{\log n} \right).$$ Thus the random variable $X_n/(n\log n)$ converges in distribution to the degenerate random variable $1/\sqrt\pi$. To get a nondegenerate distribution, we carry out an analysis similar to the one that led to (\[eq:7\]), getting more precise asymptotics for the mean of $X_n$. The refinement of (\[eq:7\]) that we need is the following, whose proof we omit: $$A(z) \odot \operatorname{CAT}(z/4) = \frac{1}{\sqrt\pi} (1-z)^{-1/2} L(z) +
D_0(1-z)^{-1/2} + O(|1-z|^{\tfrac12-\epsilon}),$$ where $$\label{eq:58}
D_0 = \sum_{n=1}^\infty n^{1/2} [ 4^{-n} \beta_n - \frac{1}{\sqrt{\pi}}
n^{-3/2} ].$$ By singularity analysis this leads to $$\label{eq:43}
{\mathbf{E}\,X_n} = \frac{1}{\sqrt\pi} n \log n + D_1 n + O(n^\epsilon),$$ where $$\label{eq:59}
D_1 = \frac{1}{\sqrt{\pi}} ( 2 \log2 + \gamma + \sqrt{\pi} D_0 ).$$ Now analyzing the random variable $X_n -
\pi^{-1/2} n \log n$ in a manner similar to that of Section \[sec:small-toll-functions\] we obtain $$\label{eq:44}
{\mathbf{Var}\,[ X_n - \pi^{-1/2} n \log n ]} =
\left(
\frac{8}{\pi} \log 2 - \frac{\pi}{2}
\right)
n^2 + O(n^{\tfrac32 + \epsilon}).$$ Using (\[eq:43\]) and (\[eq:44\]) we conclude that $${\mathbf{E}\,\left}[
\frac{X_n - \pi^{-1/2} n \log n - D_1 n}{n}
\right]
= o(1)$$ and $$\label{eq:45}
{\mathbf{Var}\,
\left[
\frac{X_n - \pi^{-1/2} n \log n - D_1 n}{n}
\right]}
\longrightarrow \frac{8}{\pi} \log 2 - \frac{\pi}{2} = \lim_{\alpha \to 1/2}
\sigma^2(\alpha),$$ where $\sigma^2 \equiv \sigma^2(\alpha)$ is defined at (\[eq:46\]) for $\alpha \ne 1/2$. \[Recall of Remark \[remark\_y\_alpha\_properties\].\]
It is possible to carry out a program similar to that of Section \[sec:higher-moments\] to derive asymptotics of higher order moments using singularity analysis. However we choose to sidestep this arduous, albeit mechanical, computation. Instead we will derive the asymptotics of higher moments using a somewhat more direct approach akin to the one employed in [@MR97f:68021]. The approach involves approximation of sums by Riemann integrals. To that end, define $$\label{eq:47}
\widetilde{X}_n := X_n - \pi^{-1/2}(n+1) \log (n+1) -
D_1(n+1), \quad\text{ and }\quad \hat\mu_n(k) := \frac{\beta_n}{4^{n+1}}
{\mathbf{E}\,\widetilde}{X}_n^k.$$ Note that $\widetilde{X}_0 = -D_1$, $\hat\mu_n(0) = \beta_n/4^{n+1}$, and $\hat\mu_0(k) = (-D_1)^k/4$. Then, in a now familiar manner, for $n
\geq 1$ we find $$\hat\mu_n(k) = 2 \sum_{j=1}^n \frac{\beta_{j-1}}{4^j}
\hat\mu_{n-j}(k) + \hat{r}_n(k),$$ where now we define $$\begin{gathered}
\hat{r}_n(k) := \sum_{\substack{k_1+k_2+k_3=k\\k_1,k_2 < k}}
\binom{k}{k_1,k_2,k_3} \sum_{j=1}^n \hat\mu_{j-1}(k_1)
\hat{\mu}_{n-j}(k_2)\\
\times \left[
\frac{1}{\sqrt\pi} ( j\log j + (n+1-j) \log (n+1-j) - (n+1) \log
(n+1) + \sqrt\pi n^{1/2})
\right]^{k_3}\end{gathered}$$ Passing to generating functions and then back to sequences one gets, for $n \geq 0$, $$\hat{\mu}_n(k) = \sum_{j=0}^n (j+1) \frac{\beta_j}{4^j}
\hat{r}_{n-j}(k).$$ Using induction on $k$, we can approximate $\hat{r}_n(k)$ and $\hat{\mu}_n(k)$ above by integrals and obtain the following result. We omit the proof, leaving it as an exercise for the ambitious reader.
\[prop\_alpha\_12\] Let $X_n$ be the additive functional induced by the toll sequence $(n^{1/2})_{n \geq 1}$ on Catalan trees. Define $\widetilde{X}_n$ as in [(\[eq:47\])]{}, with $D_1$ defined at and $D_0$ at . Then $${\mathbf{E}\,[}
{\widetilde{X}_n}/{n}
]^k
= m_k + o(1) \text{ as $n \to \infty$},$$ where $m_0=1$, $m_1=0$, and, for $k \geq 2$, $$\begin{gathered}
\label{eq:48}
m_k = \frac{1}{4\sqrt\pi} \frac{\Gamma(k-1)}{\Gamma(k-\tfrac12)}\\
\times \left[
\sum_{\substack{k_1+k_2+k_3=k\\k_1,k_2 < k}}
\binom{k}{k_1,k_2,k_3} m_{k_1} m_{k_2}
\left(
\frac{1}{\sqrt\pi}
\right)^{k_3}
J_{k_1,k_2,k_3}
+ 4 \sqrt\pi k m_{k-1}
\right],
\end{gathered}$$ where $$J_{k_1,k_2,k_3} := \int_{0}^1 x^{k_1-\tfrac32}
(1-x)^{k_2-\tfrac32} [ x \log x + (1-x) \log (1-x) ]^{k_3} \, dx.$$ Furthermore $\widetilde{X}_n/(n+1) \stackrel{\mathcal{L}}{\to} Y$, where $Y$ is a random variable with the unique distribution whose moments are ${\mathbf{E}\,Y^k} = m_k$, $k \geq 0$.
### A unified result {#sec:unified-result}
The approach outlined in the preceding section can also be used for the case $\alpha \ne 1/2$. For completeness, we state the result for that case here (without proof).
\[prop:alpha\_ne\_12\_riemann\] Let $X_n$ be the additive functional induced by the toll sequence $(n^{\alpha})_{n \geq 1}$ on Catalan trees. Let $\alpha' := \alpha
+ \tfrac12$. Define $\widetilde{X}_n$ as $$\label{eq:52}
\widetilde{X}_n :=
\begin{cases}
X_n - C_0(n+1) -
\displaystyle\frac{ \Gamma(\alpha-\tfrac12
)}{\Gamma(\alpha)}(n+1)^{\alpha'}
& 0 < \alpha <
1/2, \\
X_n - \displaystyle\frac{ \Gamma(\alpha-\tfrac12
)}{\Gamma(\alpha)}(n+1)^{\alpha'} & \alpha > 1/2,
\end{cases}$$ where $$C_0 := \sum_{n=1}^\infty n^\alpha \frac{\beta_n}{4^n}.$$ Then, for $k=0,1,2,\ldots$, $${\mathbf{E}\,\left}[ {\widetilde{X}_n}/{n^{\alpha'}} \right]^k =
m_k + o(1) \text{ as $n \to \infty$},$$ where $m_0=1$, $m_1=0$, and, for $k \geq 2$, $$\begin{gathered}
\label{eq:42}
m_k = \frac{1}{4\sqrt\pi}
\frac{\Gamma(k\alpha'-1)}{\Gamma(k\alpha'-\tfrac12)}\\
\times \left[
\sum_{\substack{k_1+k_2+k_3=k\\k_1,k_2 < k}}
\binom{k}{k_1,k_2,k_3} m_{k_1} m_{k_2}
\left( \frac{\Gamma(\alpha-\tfrac12)}{\Gamma(\alpha)}\right)^{k_3}
J_{k_1,k_2,k_3}
+ 4 \sqrt\pi k m_{k-1}
\right],
\end{gathered}$$ with $$J_{k_1,k_2,k_3} := \int_{0}^1 x^{k_1\alpha'-\tfrac32}
(1-x)^{k_2\alpha'-\tfrac32} [ x^{\alpha'} + (1-x)^{\alpha'} - 1
]^{k_3} \, dx.$$ Furthermore, $\widetilde{X}_n/n^{\alpha'}
\stackrel{\mathcal{L}}{\to} Y$, where $Y$ is a random variable with the unique distribution whose moments are ${\mathbf{E}\,Y^k} = m_k$.
\[The reader may wonder as to why we have chosen to state Proposition \[prop:alpha\_ne\_12\_riemann\] using several instances of $n+1$, rather than $n$, in . The reason is that use of $n+1$ is somewhat more natural in the calculations that establish the proposition.\]
In light of Propositions \[thm:limit\_law\_alpha\_ne\_12\], \[prop\_alpha\_12\], and \[prop:alpha\_ne\_12\_riemann\], there are a variety of ways to state a unified result. We state one such version here.
\[thm:limit-dist\] Let $X_n$ denote the additive functional induced by the toll sequence $(n^\alpha)_{n \geq 1}$ on Catalan trees. Then $$\frac{X_n - {\mathbf{E}\,X_n}}{\sqrt{{\mathbf{Var}\,X_n}}}
\stackrel{\mathcal{L}}{\to} W,$$ where the distribution of $W$ is described as follows:
\(a) For $\alpha \ne 1/2$, $$W = \frac{1}{\sigma}
\left(
Y - \frac{C_1\sqrt\pi}{\Gamma(\alpha)}
\right), \quad \text{ with } \quad \sigma^2 :=
\frac{C_2\sqrt\pi}{\Gamma(2\alpha+\tfrac12)} -
\frac{C_1^2\pi}{\Gamma^2(\alpha)} > 0,$$ where $Y$ is a random variable with the unique distribution whose moments are $${\mathbf{E}\,Y^k} =
\frac{C_k\sqrt\pi}{\Gamma(k(\alpha+\tfrac12)-\tfrac12)},$$ and the $C_k$’s satisfy the recurrence [(\[eq:10\])]{}.
\(b) For $\alpha=1/2$, $$W = \frac{Y}{\sigma}, \quad \text{ with } \quad \sigma^2 :=
\frac{8}{\pi} \log 2 - \frac{\pi}{2},$$ where $Y$ is a random variable with the unique distribution whose moments $m_k = {\mathbf{E}\,Y^k}$ are given by [(\[eq:48\])]{}.
Define $$W_n := \frac{X_n - {\mathbf{E}\,X_n}}{\sqrt{{\mathbf{Var}\,X_n}}}$$
\(a) Consider first the case $\alpha < 1/2$ and let $\alpha' := \alpha +
\tfrac12$. By , $$\label{eq:53}
{\mathbf{E}\,X_n} = C_0(n+1) + \frac{C_1\sqrt\pi}{\Gamma(\alpha)}
n^{\alpha'} + o(n^{\alpha'}).$$ Since $\widetilde{X}_n$ defined at and $X_n$ differ by a deterministic amount, ${\mathbf{Var}\,X_n}={\mathbf{Var}\,\widetilde{X}_n}$. Now by Proposition \[prop:alpha\_ne\_12\_riemann\], $$\label{eq:54}
{\mathbf{Var}\,\widetilde{X}_n} = {\mathbf{E}\,\widetilde{X}_n^2} -
({\mathbf{E}\,\widetilde{X}_n})^2 = (m_2 + o(1))n^{2\alpha'} - (m_1^2 +
o(1))n^{2\alpha'} = (m_2 + o(1))n^{2\alpha'}.$$ So $\sigma^2$ equals $m_2$ defined at (\[eq:42\]), namely, $$\frac{1}{4\sqrt\pi}\frac{\Gamma(2\alpha'-1)}{\Gamma(2\alpha'-\tfrac12)}
\left(\frac{\Gamma(\alpha-\tfrac12)}{\Gamma(\alpha)}\right)^2 J_{0,0,2}.$$ Thus to show $\sigma^2 > 0$ it is enough to show that $J_{0,0,2} > 0$. But $$J_{0,0,2} = \int_{0}^1 x^{-3/2} (1-x)^{-3/2} [ x^{\alpha'} +
(1-x)^{\alpha'} - 1]^2 \,dx,$$ which is clearly positive. Using and , $$W_n = \frac{X_n - C_0(n+1) - \frac{C_1\sqrt\pi}{\Gamma(\alpha)}
n^{\alpha'} + o(n^{\alpha'})}{ (1 + o(1))\sigma
n^{\alpha'}},$$ so, by Proposition \[thm:limit\_law\_alpha\_ne\_12\] and Slutsky’s theorem [@MR95k:60001 Theorem 25.4], the claim follows.
The case $\alpha > 1/2$ follows similarly.
\(b) When $\alpha=1/2$, $${\mathbf{E}\,X_n} = \frac{1}{\sqrt\pi} n\log{n} + D_1 n + o(n)$$ by and $${\mathbf{Var}\,X_n} =
\left(
\frac{8}{\pi}\log2 - \frac{\pi}{2} + o(1)
\right)n^2$$ by . The claim then follows easily from Proposition \[prop\_alpha\_12\] and Slutsky’s theorem.
The shape functional {#sec:shape-functional}
====================
We now turn our attention to the shape functional for Catalan trees. The shape functional is the cost induced by the toll function $b_n \equiv \log{n}$, $n \geq 1$. For background and results on the shape functional, we refer the reader to [@MR97f:68021] and [@MR99j:05171].
In the sequel we will improve on the mean and variance estimates obtained in [@MR97f:68021] and derive a central limit theorem for the shape functional for Catalan trees. The technique employed is singularity analysis followed by the method of moments.
Mean {#sec:shape_function_mean}
----
We use the notation and techniques of Section \[sec:asympotics-mean\] again. Observe that now $B(z) =
\operatorname{Li}_{0,1}(z)$ and gives the singular expansion $$\begin{gathered}
\operatorname{CAT}(z/4) = 2 - \frac2{\Gamma(-1/2)}[ \operatorname{Li}_{3/2,0}(z) -
\zeta(3/2)]\\
+ 2\left(1-\frac{\zeta(1/2)}{\Gamma(-1/2)} \right)(1-z) +
O(|1-z|^{3/2}).\end{gathered}$$ So $$B(z)\odot\operatorname{CAT}(z/4) = -\frac2{\Gamma(-1/2)} \operatorname{Li}_{3/2,1}(z)
+ \bar{c} + \bar{\bar{c}}(1-z) + O(|1-z|^{\tfrac32-\epsilon}),$$ where $ \bar{c} $ and $ \bar{\bar{c}} $ denote unspecified (possibly 0) constants. The constant term in the singular expansion of $ B(z)\odot\operatorname{CAT}(z/4) $ is already known to be $$C_0 = B(z)\odot\operatorname{CAT}(z/4) \Bigr\rvert_{z=1} = \sum_{n=1}^\infty
(\log{n}) \frac{\beta_n}{4^n}.$$ Now using the singular expansion of $ \operatorname{Li}_{3/2,1}(z) $, we get $$B(z) \odot \operatorname{CAT}(z/4) = C_0 - 2(1-z)^{1/2}L(z) -
2(2(1-\log(2))-\gamma)(1-z)^{1/2} + O(|1-z|),$$ so that $$\label{eq:22}
A(z)\odot\operatorname{CAT}(z/4) = C_0(1-z)^{-1/2} - 2L(z) -
2(2(1-\log2)-\gamma) + O(|1-z|^{1/2}).$$ Using singularity analysis and the asymptotics of the Catalan numbers we get that the mean $ a_n $ of the shape functional is given by $$\label{eq:24}
a_n = C_0(n+1) - 2\sqrt\pi n^{1/2} + O(1),$$ which agrees with the estimate in Theorem 3.1 of [@MR97f:68021] and improves the remainder estimate.
Second moment and variance {#sec:shape_functional_variance}
--------------------------
We now derive the asymptotics of the approximately centered second moment and the variance of the shape functional. These estimates will serve as the basis for the induction to follow. We will use the notation of Section \[sec:small-toll-functions\], centering the cost function as before by $C_0(n+1)$.
It is clear from that $$\label{eq:38}
\widehat{M}_1(z) = -2L(z) - 2(2(1-\log2)-\gamma) +
O(|1-z|^{1/2}),$$ and with $ k=2 $ gives us, recalling , $$\label{eq:34}
\widehat{R}_2(z) = C_0^2 + \operatorname{CAT}(z/4)\odot\operatorname{Li}_{0,2}(z) + 4\operatorname{Li}_{0,1}(z)
\odot [\frac{z}4 \operatorname{CAT}(z/4)
\widehat{M}_1(z)] + \frac{z}2\widehat{M}_1^2(z).$$ We analyze each of the terms in this sum. For the last term, observe that $z/2 \to 1/2$ as $z \to 1$, so that $$\frac{z}2 \widehat{M}_1^2(z) = 2L^2(z) + 4(2(1-\log2)-\gamma)L(z) +
2(2(1-\log2)-\gamma)^2 + O(|1-z|^{\tfrac12-\epsilon}),$$ the $ \epsilon $ introduced to avoid logarithmic remainders. The first term is easily seen to be $$\operatorname{CAT}(z/4) \odot \operatorname{Li}_{0,2}(z) = K + O(|1-z|^{\tfrac12-\epsilon}),$$ where $$K := \sum_{n=1}^\infty (\log{n})^2 \frac{\beta_n}{4^n}.$$ For the middle term, first observe that $$\frac{z}4 \operatorname{CAT}(z/4)\widehat{M}_1(z) = -L(z) - (2(1-\log2)-\gamma) +
(1-z)^{1/2}L(z) + O(|1-z|^{1/2})$$ and that $ L(z) = \operatorname{Li}_{1,0}(z) $. Thus the third term on the right in is 4 times: $$-\operatorname{Li}_{1,1}(z) + \bar{c} + O(|1-z|^{\tfrac12-2\epsilon}) = -\frac12
L^2(z) + \gamma L(z) + \bar{c} + O(|1-z|^{\tfrac12-\epsilon}).$$ \[The singular expansion for $ \operatorname{Li}_{1,1}(z) $ was obtained using the results at the bottom of p. 379 in [@MR2000a:05015]. We state it here for the reader’s convenience: $$\operatorname{Li}_{1,1}(z) = \frac12 L^2(z) - \gamma L(z) + \bar{c} + O(|1-z|),$$ where $\bar{c}$ is again an unspecified constant.\] Hence $$\widehat{R}_2(z) = 8(1-\log2)L(z) + \bar{c} +
O(|1-z|^{\tfrac12-\epsilon}),$$ which leads to $$\label{eq:23}
\widehat{M}_2(z) = 8(1-\log2)(1-z)^{-1/2}L(z) +
\bar{c}(1-z)^{-1/2} + O(|1-z|^{-\epsilon}).$$ We draw the attention of the reader to the cancellation of the ostensible lead-order term $ L^2(z) $. This kind of cancellation will appear again in the next section when we deal with higher moments.
Now using singularity analysis and estimates for the Catalan numbers we get $$\label{eq:25}
\tilde\mu_n(2) = 8(1-\log2)n\log{n} + \bar{c}n + O(n^{\tfrac12+\epsilon}).$$ Using , $${\mathbf{Var}\,X_n} = \tilde\mu_n(2) - \tilde\mu_n(1)^2 =
8(1-\log2)n\log{n} + \bar{c}n + O(n^{\tfrac12+\epsilon}),$$ which agrees with Theorem 3.1 of [@MR97f:68021] (after a correction pointed out in [@MR99j:05171]) and improves the remainder estimate. In our subsequent analysis we will not need to evaluate the unspecified constant $\bar{c}$.
Higher moments {#sec:shape_function_higher-moments}
--------------
We now turn our attention to deriving the asymptotics of higher moments of the shape functional. The main result is as follows.
\[thm:shape\_moments\] Define $\widetilde{X}_n := X_n - C_0(n+1)$, with $X_0 := 0$; $\tilde{\mu}_n(k) := {\mathbf{E}\,\widetilde{X}_n^k} $, with $\tilde{\mu}_n(0) = 1$ for all $n \geq 0$; and $\hat{\mu}_n(k)
:= \beta_n\tilde{\mu}_n(k)/4^n $. Let $\widehat{M}_k(z)$ denote the ordinary generating function of $\hat{\mu}_n(k)$ in the argument $n$. For $ k \geq 2 $, $ \widehat{M}_k(z) $ has the singular expansion $$\widehat{M}_k(z) = (1-z)^{-\tfrac{k-1}2}
\sum_{j=0}^{\lfloor {k}/2 \rfloor} C_{k,j}
L^{\lfloor {k}/2 \rfloor-j}(z) +
O(|1-z|^{-\tfrac{k}2+1-\epsilon}),$$ with $$C_{2l,0} = \frac14 \sum_{j=1}^{l-1} \binom{2l}{2j}
C_{2j,0}C_{2l-2j,0}, \qquad C_{2,0} = 8(1-\log2).$$
The proof is by induction. For $ k=2 $ the claim is true by . We note that the claim is *not* true for $ k=1 $. Instead, recalling (\[eq:38\]), $$\label{eq:28}
\widehat{M}_1(z) = -2L(z) -
2(2(1-\log2)-\gamma) + O(|1-z|^{1/2}).$$ For the induction step, let $ k \geq 3 $. We will first get the asymptotics of $ \widehat{R}_k(z) $ defined at with $B(z) = \operatorname{Li}_{0,1}(z)$. In order to do that we will obtain the asymptotics of each term in the defining sum. We remind the reader that we are only interested in the form of the asymptotic expansion of $ \widehat{R}_k(z) $ and the coefficient of the lead-order term when $ k $ is even. This allows us to “define away” all other constants, their determination delayed to the time when the need arises.
For this paragraph suppose that $ k_1 \geq2 $ and $ k_2 \geq 2 $. Then by the induction hypothesis $$\begin{gathered}
\label{eq:26}
\frac{z}4 \widehat{M}_{k_1}(z)\widehat{M}_{k_2}(z) = \frac14
(1-z)^{-\tfrac{k_1+k_2}2+1}
\sum_{l=0}^{\lfloor {k_1}/2 \rfloor +
\lfloor {k_2}/2 \rfloor} A_{k_1,k_2,l}
L^{\lfloor {k_1}/2 \rfloor + \lfloor {k_2}/2
\rfloor - l}(z)\\
{}+ O(|1-z|^{-\tfrac{k_1+k_2}2+\tfrac32-\epsilon}),
\end{gathered}$$ where $ A_{k_1,k_2,0}= C_{k_1,0}C_{k_2,0} $. (a) If $ k_3=0 $ then $ k_1+k_2=k $ and the corresponding contribution to $
\widehat{R}_k(z) $ is given by $$\begin{gathered}
\label{eq:27}
\frac14 \binom{k}{k_1} (1-z)^{-\tfrac{k}2+1}\\
\times
\sum_{l=0}^{\lfloor {k_1}/2 \rfloor +
\lfloor ({k-k_1})/2 \rfloor} A_{k_1,k-k_1,l}
L^{\lfloor {k_1}/2 \rfloor + \lfloor ({k-k_1})/2 \rfloor -
l}(z) + O(|1-z|^{-\tfrac{k}2+\tfrac32-\epsilon}).
\end{gathered}$$ Observe that if $ k $ is even and $ k_1 $ is odd the highest power of $ L(z) $ in is $ \lfloor {k}/2 \rfloor-1 $. In all other cases the the highest power of $ L(z) $ in is $ \lfloor {k}/2 \rfloor $. (b) If $ k_3 \ne 0 $ then we use Lemma \[lem:omztoli\] to express as a linear combination of $$\left\{
\operatorname{Li}_{-\tfrac{k_1+k_2}2+2,l}(z) \right\}_{l=0}^{\lfloor {k_1}/2 \rfloor
+ \lfloor {k_2}/2 \rfloor}$$ with a remainder that is $O(|1-z|^{-\tfrac{k_1+k_2}2+\tfrac32-\epsilon})$. When we take the Hadamard product of such a term with $ \operatorname{Li}_{0,k_3}(z) $ we will get a linear combination of $$\left\{
\operatorname{Li}_{-\tfrac{k_1+k_2}2+2,l+k_3}(z) \right\}_{l=0}^{\lfloor
{k_1}/2 \rfloor + \lfloor {k_2}/2 \rfloor}$$ and a smaller remainder. Such terms are all $ O(|1-z|^{-\tfrac{k_1+k_2}2+1-\epsilon}) $, so that the contribution is $ O(|1-z|^{-\tfrac{k}2+\tfrac32-\epsilon}) $.
Next, consider the case when $ k_1=1 $ and $ k_2 \geq 2 $. Using the induction hypothesis and we get $$\label{eq:29}
\begin{split}
\frac{z}4 \widehat{M}_{k_1}(z) \widehat{M}_{k_2}(z) = -\frac12
(1-z)^{-\tfrac{k_2-1}2} \sum_{j=0}^{\lfloor {k_2}/2 \rfloor+1}
B_{k_2,j} L^{\left\lfloor \tfrac{k_2}2 \right\rfloor + 1 - j}(z) \\
{}+ O(|1-z|^{-\tfrac{k_2}2+1-2\epsilon}),
\end{split}$$ with $ B_{k_2,0} = C_{k_2,0} $. (a) If $ k_3=0 $ then $
k_2=k-1 $ and the corresponding contribution to $ \widehat{R}_k(z) $ is given by $$\label{eq:30}
-\frac{k}{2} (1-z)^{-\tfrac{k}2+1}
\sum_{j=0}^{\lfloor ({k-1})/{2} \rfloor+1} B_{k-1,j}
L^{\left\lfloor \tfrac{k-1}2 \right\rfloor+1-j}(z) +
O(|1-z|^{-\tfrac{k}2+\tfrac32-2\epsilon}).$$ (b) If $ k_3 \ne 0 $ then Lemma \[lem:omztoli\] can be used once again to express in terms of generalized polylogarithms, whence an argument similar to that at the end of the preceding paragraph yields that the contributions to $\widehat{R}(z)$ from such terms is $ O(|1-z|^{-\tfrac{k_2-1}2-\epsilon}) $, which is $ O(|1-z|^{-\tfrac{k}2+\tfrac32-\epsilon}) $. The case when $k_1
\geq 2$ and $k_2 =1$ is handled symmetrically.
When $ k_1=k_2=1 $ then $
(z/4)\widehat{M}_{k_1}(z)\widehat{M}_{k_2}(z) $ is $ O(|1-z|^{-\epsilon}) $ and when one takes the Hadamard product of this term with $ \operatorname{Li}_{0,k_3}(z) $ the contribution will be $ O(|1-z|^{-2\epsilon}) $.
Now consider the case when $ k_1=0 $ and $ k_2 \geq 2 $. Since $ \widehat{M}_0(z) = \operatorname{CAT}(z/4) $, we have $$\label{eq:31}
\frac{z}4 \widehat{M}_{k_1}(z)\widehat{M}_{k_2}(z) = \frac12
(1-z)^{-\tfrac{k_2-1}2} \sum_{j=0}^{\lfloor {k_2}/2 \rfloor}
C_{k_2,j} L^{\lfloor {k_2}/2 \rfloor-j}(z) +
O(|1-z|^{-\tfrac{k_2}2+1-\epsilon}).$$ By Lemma \[lem:omztoli\] this can be expressed as a linear combination of $$\left\{ \operatorname{Li}_{-\tfrac{k_2-1}2+1,j}(z)
\right\}_{j=0}^{\lfloor {k_2}/2 \rfloor}$$ with a $ O(|1-z|^{-\tfrac{k_2}2+1-\epsilon}) $ remainder. When we take the Hadamard product of such a term with $ \operatorname{Li}_{0,k_3}(z) $ we will get a linear combination, call it $S(z)$, of $$\left\{ \operatorname{Li}_{-\tfrac{k_2-1}2+1,j+k_3}(z)
\right\}_{j=0}^{\lfloor {k_2}/2 \rfloor}$$ with a remainder of $ O(|1-z|^{-\tfrac{k_2}2 + 1 - 2\epsilon})
$, which is $ O(|1-z|^{-\tfrac{k}2 + \tfrac32 - 2\epsilon}) $ unless $ k_2=k-1 $. When $ k_2 = k-1 $, by Lemma \[lem:omztoli\] the constant multiplying the lead-order term $ \operatorname{Li}_{-\tfrac{k}{2}+2,\lfloor\tfrac{k-1}{2} \rfloor + 1}(z) $ in $S(z)$ is $\frac{C_{k-1,0}}2
\mu_0^{(-\tfrac{k}{2}+2,\lfloor\tfrac{k-1}{2}\rfloor)}$. When we take the Hadamard product of this term with $ \operatorname{Li}_{0,k_3}(z) $ we get a lead-order term of $$\frac{C_{k-1,0}}2
\mu_0^{(-\tfrac{k}{2}+2,\lfloor\tfrac{k-1}{2}\rfloor)}
\operatorname{Li}_{-\tfrac{k}{2}+2,\lfloor\tfrac{k-1}{2}\rfloor+1}(z).$$ Now we use Lemma \[lem:litoomz\] and the observation that $ \lambda_0^{(\alpha,r)}\mu_0^{(\alpha,s)}=1 $ to conclude that the contribution to $\widehat{R}_k(z)$ from the term with $ k_1=0 $ and $ k_2=k-1 $ is $$\label{eq:32}
\frac{k}2 (1-z)^{-\tfrac{k}2+1}
\sum_{j=0}^{\lfloor\frac{k-1}{2}\rfloor+1} D_{k,j}
L^{\lfloor\frac{k-1}{2}\rfloor+1-j}(z)
+ O(|1-z|^{-\tfrac{k}2+\tfrac{3}2-\epsilon}),$$ with $ D_{k,0}=C_{k-1,0} $. Notice that the lead order from this contribution is precisely that from but with opposite sign; thus the two contributions cancel each other to lead order. The case $k_2 = 0$ and $k_1 \geq 2$ is handled symmetrically.
The last two cases are $ k_1=0 $, $ k_2=1 $ (or vice-versa) and $ k_1=k_2=0 $. The contribution from these cases can be easily seen to be $ O(|1-z|^{-\tfrac{k}2+\tfrac32-2\epsilon}) $.
We can now deduce the asymptotic behavior of $ \widehat{R}_k(z)
$. The three contributions are , , and , with only (in net) contributing a term of the form $ (1-z)^{-\tfrac{k}{2}+1} L^{\lfloor {k}/{2}
\rfloor}(z) $ when $k$ is even. The coefficient of this term when $ k $ is even is given by $$\frac14 \sum_{\substack{0 < k_1 < k\\k_1\text{ even}}}
\binom{k}{k_1} C_{k_1,0}C_{k_2,0}.$$ Finally we can sum up the rest of the contribution, define $ C_{k,j} $ appropriately and use to claim the result.
A central limit theorem {#sec:shape_functional_centr-limit-theor}
-----------------------
Proposition \[thm:shape\_moments\] and singularity analysis allows us to get the asymptotics of the moments of the “approximately centered” shape functional. Using arguments identical to those in Section \[sec:asymptotics-moments\] it is clear that for $ k \geq 2 $ $$\label{eq:33}
\tilde{\mu}_n(k) = \frac{C_{k,0}\sqrt\pi}{\Gamma(\tfrac{k-1}2)}
n^{{k}/{2}} [\log{n}]^{\lfloor{k}/{2}\rfloor} +
O(n^{{k}/{2}} [\log{n}]^{\lfloor{k}/{2}\rfloor-1}).$$ This and the asymptotics of the mean derived in Section \[sec:shape\_function\_mean\] give us, for $ k \geq 1 $, $$E\left[ \frac{\tilde{X}_n}{\sqrt{n\log{n}}} \right]^{2k} \to
\frac{C_{2k,0}\sqrt\pi}{\Gamma(k-\tfrac12)}, \qquad E\left[
\frac{\tilde{X}_n}{\sqrt{n\log{n}}} \right]^{2k-1} = o(1)$$ as $n \to \infty$. The recurrence for $ C_{2k,0} $ can be solved easily to yield, for $k
\geq 1$, $$C_{2k,0} = \frac{(2k)!(2k-2)!}{2^k2^{2k-2}k!(k-1)!} \sigma^{2k},$$ where $ \sigma^2 := 8(1-\log2) $. Then using the identity $$\frac{\Gamma(k-\tfrac12)}{\sqrt\pi} =
\left[2^{2k-2} \frac{(k-1)!}{(2k-2)!}\right]^{-1}$$ we get $$\frac{C_{2k,0}\sqrt\pi}{\Gamma(k-\tfrac12)} = \frac{(2k)!}{2^k
k!}\sigma^{2k}.$$ It is clear now that both the “approximately centered” and the normalized shape functional are asymptotically normal.
\[thm:shape\_clt\] Let $ X_n $ denote the shape functional, induced by the toll sequence $(\log{n})_{n \geq 1}$, for Catalan trees. Then $$\frac{X_n-C_0(n+1)}{\sqrt{n\log{n}}} \stackrel{\mathcal{L}}{\to}
\mathcal{N}(0,\sigma^2) \quad\text{ and }\quad
\frac{X_n - {\mathbf{E}\,X_n} }{\sqrt{ {\mathbf{Var}\,X_n} }}
\stackrel{\mathcal{L}}{\to} \mathcal{N}(0,1),$$ where $$C_0 := \sum_{n=1}^\infty (\log{n}) \frac{\beta_n}{4^n}, \qquad
\beta_n = \frac1{n+1}\binom{2n}{n},$$ and $ \sigma^2 := 8(1-\log2) $.
Concerning numerical evaluation of the constant $C_0$, see the end of Section 5.2 in [@FFK].
Sufficient conditions for asymptotic normality {#sec:suff-cond-asympt}
==============================================
In this speculative final section we briefly examine the behavior of a general additive functional $X_n$ induced by a given “small” toll sequence $(b_n)$. We have seen evidence \[Remark \[remark\_y\_alpha\_properties\](\[item:1\])\] that if $(b_n)$ is the “large” toll sequence $n^{\alpha}$ for any fixed $\alpha
> 0$, then the limiting behavior is non-normal. When $b_n = \log n$ (or $b_n = n^\alpha$ and $\alpha \downarrow 0$), the (limiting) random variable is normal. Where is the interface between normal and non-normal asymptotics? We have carried out arguments similar to those leading to Propositions \[prop\_alpha\_12\] and \[prop:alpha\_ne\_12\_riemann\] (see also [@MR97f:68021]) that suggest a sufficient condition for asymptotic normality, but our “proof” is somewhat heuristic, and further technical conditions on $(b_n)$ may be required. Nevertheless, to inspire further work, we present our preliminary indications.
We assume that $b_n \equiv b(n)$, where $b(\cdot)$ is a function of a nonnegative real argument. Suppose that $x^{-3/2} b(x)$ is (ultimately) nonincreasing and that $x b'(x)$ is slowly varying at infinity. Then $${\mathbf{E}\,X_n} = C_0 (n+1) - (1 + o(1)) 2 \sqrt{\pi} n^{3/2} b'(n),$$ where $$C_0 = \sum_{n=1}^\infty b_n \frac{\beta_n}{4^n}.$$ Furthermore, $${\mathbf{Var}\,X_n} \sim 8 (1 - \log 2) [n b'(n)]^2 n\log n,$$ and $$\frac{X_n - C_0 (n + 1)}{n b'(n) \sqrt{n \log n}}
\stackrel{\mathcal{L}}{\to} \mathcal{N}(0,\sigma^2), \text{ where }
\sigma^2 = 8(1 - \log 2).$$ This asymptotic normality can also be stated in the form $$\frac{X_n - {\mathbf{E}\,X_n} }{\sqrt{{\mathbf{Var}\,X_n}}}
\stackrel{\mathcal{L}}{\to} \mathcal{N}(0,1).$$
**Acknowledgments.** We thank two anonymous referees for helpful comments.
[^1]: Research for both authors supported by NSF grants DMS-9803780 and DMS-0104167, and by The Johns Hopkins University’s Acheson J. Duncan Fund for the Advancement of Research in Statistics. Research for the second author supported by NSF grant 0049092 and carried out primarily while this author was affiliated with what is now the Department of Applied Mathematics and Statistics at The Johns Hopkins University.
|
---
abstract: 'Given a sufficient statistic for a parametric family of distributions, one can estimate the parameter without access to the data. However, the memory or code size for storing the sufficient statistic may nonetheless still be prohibitive. Indeed, for $n$ independent samples drawn from a $k$-nomial distribution with $d=k-1$ degrees of freedom, the length of the code scales as $d\log n+O(1)$. In many applications, we may not have a useful notion of sufficient statistics (e.g., when the parametric family is not an exponential family) and we also may not need to reconstruct the generating distribution exactly. By adopting a Shannon-theoretic approach in which we allow a small error in estimating the generating distribution, we construct various [*approximate sufficient statistics*]{} and show that the code length can be reduced to $\frac{d}{2}\log n+O(1)$. We consider errors measured according to the relative entropy and variational distance criteria. For the code constructions, we leverage Rissanen’s minimum description length principle, which yields a non-vanishing error measured according to the relative entropy. For the converse parts, we use Clarke and Barron’s formula for the relative entropy of a parametrized distribution and the corresponding mixture distribution. However, this method only yields a weak converse for the variational distance. We develop new techniques to achieve vanishing errors and we also prove strong converses. The latter means that even if the code is allowed to have a non-vanishing error, its length must still be at least $\frac{d}{2}\log n$.'
author:
- 'Masahito Hayashi,$^\dagger$ [*Fellow, IEEE*]{}, $\,\, \,$ Vincent Y. F. Tan,$^\ddagger$ [*Senior Member, IEEE*]{} [^1] [^2] [^3] [^4][^5]'
bibliography:
- 'isitbib.bib'
title: Minimum Rates of Approximate Sufficient Statistics
---
Approximate sufficient statistics, Minimum rates, Memory size reduction, Minimum description length, Exponential families, Pythagorean theorem, Strong converse
Acknowledgements {#acknowledgements .unnumbered}
----------------
The authors thank the Associate Editor Dr. Peter Harremoës and the two reviewers for their insightful comments that helped to improve the paper significantly.
[Masahito Hayashi]{}(M’06–SM’13–F’17) was born in Japan in 1971. He received the B.S. degree from the Faculty of Sciences in Kyoto University, Japan, in 1994 and the M.S. and Ph.D. degrees in Mathematics from Kyoto University, Japan, in 1996 and 1999, respectively. He worked in Kyoto University as a Research Fellow of the Japan Society of the Promotion of Science (JSPS) from 1998 to 2000, and worked in the Laboratory for Mathematical Neuroscience, Brain Science Institute, RIKEN from 2000 to 2003, and worked in ERATO Quantum Computation and Information Project, Japan Science and Technology Agency (JST) as the Research Head from 2000 to 2006. He also worked in the Superrobust Computation Project Information Science and Technology Strategic Core (21st Century COE by MEXT) Graduate School of Information Science and Technology, The University of Tokyo as Adjunct Associate Professor from 2004 to 2007. He worked in the Graduate School of Information Sciences, Tohoku University as Associate Professor from 2007 to 2012. In 2012, he joined the Graduate School of Mathematics, Nagoya University as Professor. He also worked in Centre for Quantum Technologies, National University of Singapore as Visiting Research Associate Professor from 2009 to 2012 and as Visiting Research Professor from 2012 to now. He also worked in Center for Advanced Intelligence Project, RIKEN as a Visiting Scientist from 2017. In 2011, he received Information Theory Society Paper Award (2011) for “Information-Spectrum Approach to Second-Order Coding Rate in Channel Coding”. In 2016, he received the Japan Academy Medal from the Japan Academy and the JSPS Prize from Japan Society for the Promotion of Science.
In 2006, he published the book “Quantum Information: An Introduction” from Springer, whose revised version was published as “Quantum Information Theory: Mathematical Foundation” from Graduate Texts in Physics, Springer in 2016. In 2016, he published other two books “Group Representation for Quantum Theory” and “A Group Theoretic Approach to Quantum Information” from Springer. He is on the Editorial Board of [*International Journal of Quantum Information*]{} and [*International Journal On Advances in Security*]{}. His research interests include classical and quantum information theory and classical and quantum statistical inference.
[Vincent Y. F. Tan]{} (S’07-M’11-SM’15) was born in Singapore in 1981. He is currently an Assistant Professor in the Department of Electrical and Computer Engineering (ECE) and the Department of Mathematics at the National University of Singapore (NUS). He received the B.A. and M.Eng. degrees in Electrical and Information Sciences from Cambridge University in 2005 and the Ph.D. degree in Electrical Engineering and Computer Science (EECS) from the Massachusetts Institute of Technology in 2011. He was a postdoctoral researcher in the Department of ECE at the University of Wisconsin-Madison and a research scientist at the Institute for Infocomm (I$^2$R) Research, A\*STAR, Singapore. His research interests include network information theory, machine learning, and statistical signal processing.
Dr. Tan received the MIT EECS Jin-Au Kong outstanding doctoral thesis prize in 2011, the NUS Young Investigator Award in 2014, the Engineering Young Researcher Award in the Faculty of Engineering, NUS in 2018, and the Singapore National Research Foundation (NRF) Fellowship (Class of 2018). He has authored a research monograph on [*“Asymptotic Estimates in Information Theory with Non-Vanishing Error Probabilities”*]{} in the Foundations and Trends in Communications and Information Theory Series (NOW Publishers). He is currently an Editor of the IEEE Transactions on Communications and the IEEE Transactions on Green Communications and Networking.
[^1]: $^\dagger$Masahito Hayashi is with the Graduate School of Mathematics, Nagoya University, and the Center for Quantum Technologies (CQT), National University of Singapore (Email: masahito@math.nagoya-u.ac.jp).
[^2]: $\ddagger$Vincent Y. F. Tan is with the Department of Electrical and Computer Engineering and the Department of Mathematics, National University of Singapore (Email: vtan@nus.edu.sg).
[^3]: This paper was presented in part [@HayashiTan17] at the 2017 International Symposium on Information Theory (ISIT) in Aachen, Germany.
[^4]: MH is partially supported by a MEXT Grant-in-Aid for Scientific Research (B) No. 16KT0017. MH is also partially supported by the Okawa Research Grant and Kayamori Foundation of Informational Science Advancement. The Centre for Quantum Technologies is funded by the Singapore Ministry of Education and the National Research Foundation as part of the Research Centres of Excellence programme.
[^5]: VYFT is partially supported an NUS Young Investigator Award (R-263-000-B37-133) and a Singapore Ministry of Education Tier 2 grant “Network Communication with Synchronization Errors: Fundamental Limits and Codes” (R-263-000-B61-112).
|
---
abstract: 'Space-based high contrast imaging mission concepts for studying rocky exoplanets in reflected light are currently under community study. We develop an inverse modeling framework to estimate the science return of such missions given different instrument design considerations. By combining an exoplanet albedo model, an instrument noise model, and an ensemble Markov chain Monte Carlo sampler, we explore retrievals of atmospheric and planetary properties for Earth twins as a function of signal-to-noise ratio (SNR) and resolution ($R$). Our forward model includes Rayleigh scattering, single-layer water clouds with patchy coverage, and pressure-dependent absorption due to water vapor, oxygen, and ozone. We simulate data at $R = 70$ and $R = 140$ from 0.4–1.0 $\mu$m with SNR $ = 5, 10, 15, 20$ at 550 nm (i.e., for HabEx/LUVOIR-type instruments). At these same SNR, we simulate data for [*WFIRST*]{} paired with a starshade, which includes two photometric points between 0.48–0.6 $\mu$m and $R = 50$ spectroscopy from 0.6–0.97 $\mu$m. Given our noise model for [*WFIRST*]{}-type detectors, we find that weak detections of water vapor, ozone, and oxygen can be achieved with observations with at least $R = 70$ / SNR$\ = 15$, or $R = 140$ / SNR$\ = 10$ for improved detections. Meaningful constraints are only achieved with $R = 140$ / SNR$\ = 20$ data. The [*WFIRST*]{} data offer limited diagnostic information, needing at least SNR = 20 to weakly detect gases. Most scenarios place limits on planetary radius, but cannot constrain surface gravity and, thus, planetary mass.'
author:
- '<span style="font-variant:small-caps;">Y. Katherina Feng, Tyler D. Robinson</span>, Jonathan J. Fortney, Roxana E. Lupu, Mark S. Marley, Nikole K. Lewis, Bruce Macintosh, Michael R. Line'
---
Introduction
============
The scientific field of exoplanets has been rapidly advancing since the hallmark discovery of the first planet orbiting a Sun-like star [@mayor1995]. Following the launch of NASA’s [*Kepler*]{} mission [@borucki2003; @Borucki11], the field has seen the discovery of thousands of transiting exoplanets and the exciting result that planets with radii between $0.75$–$2.5\ R_{\oplus}$ are common around solar-type stars [@burke2015]. Only within the last decade have observational studies for exoplanet atmospheric characterization seen substantial development, starting with the first detection of an exoplanet’s atmosphere by @Charb02.
To date, the majority of exoplanet atmospheric characterization investigations have focused on transiting worlds. Hot Jupiters, owing to their large sizes and short orbital periods, are typically emphasized as targets for these studies. Characterization of small, potentially rocky exoplanets is limited to worlds with cool stellar hosts (K and M dwarfs), which offer favorable planet-to-star size ratios. Recently, @dewit2016 studied the combined transmission spectra of two transiting Earth-sized planets orbiting the ultracool dwarf TRAPPIST-1 using the *Hubble Space Telescope*. While no gas absorption features were detected by @dewit2016, this work highlights the improvements in signal size when terrestrial-sized transiting planets are studied around low-mass stars. Additionally, since the Habitable Zone [@kasting1993] around a low-mass star is relatively close-in, characterization studies of potentially habitable exoplanets around cool stars can benefit from the frequency of transit events. However, for Sun-like hosts, the planet-to-star size ratio is much less favorable and the Habitable Zone is located far from the star, thus severely limiting the potential for atmospheric characterization.
Direct, high-contrast imaging has now emerged as an essential technique for studying the atmospheres of planets at larger orbital separations from their host star (i.e., at orbital distances $\gtrsim 1$ au). Thus far, high-contrast imaging has been proven successful in studying atmospheres of young, self-luminous gas giants in the near-infrared and mid-infrared [e.g., @barman2011; @Skemer14; @macintosh2015]. These worlds, owing to their intrinsic brightness, have typical contrast ratios of $10^{-4}$ with respect to their hosts. A true Jupiter analog at visible wavelengths, by comparison, would have a contrast ratio of $10^{-9}$, while an Earth analog would have a contrast ratio of order $10^{-10}$. Reflected light in the visible probes to atmospheric depths of up to $\sim 10$ bar for giant planets [@marley2014], which is complimentary to the relatively low pressures probed in transit observations (typically less than 10–100 mbar). Additionally, the wavelength range of 0.4–1.0 holds rich information about a planet’s atmosphere, including signatures of methane, water vapor, and haze [@marley2014; @burrows2014].
In spite of the incredible technological challenges, there are multiple planned or in-development space-based missions that would be capable of high-contrast imaging of exoplanets in reflected light. First among these will be NASA’s Wide-Field InfraRed Survey Telescope [[*WFIRST*]{}, @spergel2013], which was identified as the top priority space mission in the 2010 National Academy of Sciences Decadal Survey of Astronomy and Astrophysics[^1]. The [*WFIRST*]{} mission will carry a Coronagraphic Instrument (CGI) with imaging capability and a visible-light integral field spectrograph of wavelength resolution $\sim 50$ [@noecker2016; @trauger2016; @seo2016; @cady2016; @balasubramanian2016; @groff2018]. Although envisioned primarily as a technology demonstrator, it may study the atmospheres of relatively cool gas giant exoplanets that have been previously detected using the radial velocity technique [@Traub2016].
While *WFIRST* could also have some capability to survey stars in the solar neighborhood for lower-mass planetary companions , it is anticipated that the core optical throughput of the [*WFIRST*]{} CGI will be low for planetary signals. This stems primarily from the complexities of accommodating for [*WFIRST*]{}’s on-axis secondary mirror and support structures within the high-contrast instruments [@Traub2016; @krist2016]. Low throughput drives long requisite integration times, thereby likely making spectroscopic observations of smaller, less-bright worlds (such as super-Earth exoplanets) unfeasible except around the very closest stars [@robinson2016]. However, if the [*WFIRST*]{} spacecraft were to be paired with an external starshade [@cash2006; @Kasdin2012], the CGI can be operated in a direct mode without coronagraphic masks, substantially increasing throughput. High-contrast imaging of sub-Neptune and terrestrial-sized exoplanets may then become possible. The feasibility of a starshade “rendezvous” with the *WFIRST* spacecraft is under active investigation [@seager2015; @crill2017].
In advance of the 2020 astronomy and astrophysics decadal survey, several large-scale space-based mission concepts are being studied[^2]. Of these, two have a strong focus on the characterization of rocky exoplanets with direct imaging: the Habitable Exoplanet Imaging Mission [HabEx; @habexRef] and the Large Ultra-Violet/Optical/InfraRed Surveyor [LUVOIR; @luvoirRef]. HabEx and LUVOIR are incorporating aspects of design that would allow the detection of water vapor and biosignatures on planets in the Habitable Zones of nearby Sun-like stars. It is therefore timely and critical that we explore observational approaches that maximize science yield during the development of these large-scale mission concepts as well as the *WFIRST* rendezvous concept. To accomplish this, we must perform atmospheric and instrument modeling to simulate the types of spectra we can expect to measure, and we must develop tools to infer planetary properties from these simulated observations.
Traditionally, the comparison to a limited range of forward models has been used to infer atmospheric properties (such as temperature structure and gas abundances) from spectral observations. This involves iterating to a radiative-convective solution for a given set of planetary parameters (e.g., gravity, metallicity, equilibrium abundances, incident flux), and can include detailed treatment of aerosols, chemistry, and dynamics within the model atmosphere [@marley2015]. The goal is to generate a spectrum that matches available data and, thus, offers one potential explanation for the world’s atmospheric state [e.g., @Konopacky13; @macintosh2015; @barman2015]. A more data-driven interpretation of atmospheric observations is accomplished through inverse modeling, or retrievals. Developed for Solar System studies and remote sensing [e.g., @rodgers76; @irwin2008], retrievals have become a valuable tool in constraining our understanding of the atmospheres of transiting exoplanets. Early exoplanet retrieval work invoked grid-based optimization schemes [@madhu2009], while subsequent works have taken advantage of Bayesian inference with methods such as optimal estimation and Markov chain Monte Carlo (MCMC) [e.g., @lee2012; @Benneke12; @Line2013].
Several studies have examined the hypothetical yield from characterizing giant exoplanets observed with a space-based coronagraph (such as *WFIRST*) with retrieval techniques. @marley2014, for example, modeled spectra we could expect from known radial velocity gas giants if observed by the *WFIRST* CGI. Given the diversity of cool giant planets, the model spectra have a variety of input assumptions for clouds, surface gravity, and atmospheric metallicity. @marley2014 then applied retrieval methods to these synthetic spectra, enabling the exploration of how well atmospheric parameters are constrained under varying quality of data. @lupu2016 further investigated the feasibility of characterizing cool giant planet atmospheres through retrieval, focusing on the ability to constrain the [$\rm CH_4$]{} abundance and cloud properties. The systematic study of the impact of conditions like signal-to-noise ratios or wavelength resolution is essential to quantifying the scientific return of these reflected-light observations. @nayak2017 considered the impact of an unknown phase angle on the inference of properties such as planet radius and gravity. In all of these studies, the signal-to-noise ratio (SNR) of the data has a significant influence on the constraints of atmospheric properties.
Previous work on smaller planets in the context of possible future space missions includes @vonparis13, who synthesized infrared emission observations of a cloud-free, directly-imaged Earth-twin, and employed a least-squares approach and $\chi^2$ maps to perform retrievals and explore parameter space (considering the effects of instrument resolution and SNRs). A collection of recent studies [@wang2017a; @mawet2017; @wang2017b] examined atmospheric species detection using “High Dispersion Coronagraphy”, which couples starlight suppression technologies with high resolution spectroscopy. In these studies, simulated observations (typically at spectral resolutions, $R=\lambda/\Delta\lambda$, of many hundreds to tens of thousands) are cross-correlated with template molecular opacity spectra to explore the feasibility of species detection. While this novel approach can yield detections of key atmospheric constituents, the abundance of these these atmospheric species cannot be robustly constrained.
To date, there still does not exist a systematic study of atmospheric characterization of small exoplanets using retrieval techniques on reflected light observations at spectral resolutions relevant to [*WFIRST*]{} rendezvous, HabEx, and LUVOIR. Motivated by this need, we present here our extension of Bayesian retrieval techniques into the terrestrial regime. We construct a forward model suitable for simulating reflectance spectra of Earth-like planets in the visible wavelength range of 0.4 to 1.0 . We explore retrievals of planetary and atmospheric properties from simulated data sets at varying spectral resolutions and SNRs. A retrieval framework such as this allows us to quantify uncertainties we expect for key planetary parameters given certain observing scenarios. Thus, our approach enables us to search for the minimal observing conditions that achieve the scientific goal of identifying traits associated with habitability and life. In particular, we are interested in our ability to detect and constrain abundances of molecules such as water, oxygen, and ozone, characterize basic properties of a cloud layer, and measure bulk parameters such as radius.
In section \[methods\], we describe our forward model and construction of simulated data. In section \[buildup\], we validate our forward model by building up retrieval complexity (i.e., number of retrieved parameters). We perform a study of retrieval performance with respect to spectral resolution and SNR in section \[snr\_study\], with implications for HabEx/LUVOIR. We also study the retrieval performance for data sets expected from a [*WFIRST*]{} rendezvous scenario, where the CGI would provide modest-resolution spectroscopy in the red (600–970 nm) and photometry in the blue (480–600 nm). We present our discussion and conclusions in sections \[discussion\] and \[conclusion\], respectively.
Methods
=======
The observed quantity for a directly imaged exoplanet in reflected light at a given phase (i.e., planet-star-observer) angle, $\alpha$, is the wavelength dependent planet-to-star flux ratio, $$\frac{F_p}{F_s} = A_g \Phi (\alpha)
\Big(\frac{R_p}{r}\Big)^2, \label{fpfs}$$ where $A_g$ is the geometric albedo, $\Phi(\alpha)$ is the phase function, $R_p$ is the radius of the planet, and $r$ is the orbital separation. The phase function (which depends on wavelength) translates the planetary brightness at full phase (i.e., where $\alpha=0^{\circ}$) to its brightness at different phase angles. The wavelength dependent geometric albedo is the ratio between the measured flux from the planet at full phase to that from a perfectly reflecting Lambert (i.e., isotropically-reflecting) disk with the size of the planet. We denote the product of the geometric albedo and the phase function as the phase dependent “reflectance” of the planet. In general, the geometric albedo encodes information about the composition and structure (i.e., “state”) of an atmosphere, while the phase function is strongly related to the scattering properties of an atmosphere [e.g., @Marley99; @burrows2014].
To understand the information contained in direct imaging observations of exoplanets in reflected light, we employ a retrieval (or inverse analysis) framework that consists of several linked simulation tools and models. Of central importance is a well-tested three-dimensional albedo model—described in greater depth below—that computes a reflectance spectrum at high resolution for a planet given a description of its atmospheric state [@Mckay89; @Marley99; @cahoy2010; @lupu2016; @nayak2017]. When coupled with a simulator for degrading a high resolution spectrum to match the resolution of an instrument, we refer to these two tools as the “forward model.” By adding simulated noise to forward model spectra, we generate faux “observations” of worlds as would be studied by future high-contrast imaging missions. To create “observed” spectra, we adopt a widely-used direct imaging instrument simulator [@robinson2016] that generates synthetic observations given an input, noise-free spectrum.
Given an “observed” planet-to-star flux ratio spectrum, our inverse analyses use a Bayesian inference tool that compares the observation to forward model outputs to sample the posterior probability distributions for a collection of atmospheric state parameters. In other words, our inverse analyses indicate what range of atmospheric state parameters (e.g., gas abundances) adequately describe a direct imaging observation. Our Bayesian parameter estimations use an open-source affine invariant Markov-Chain Monte Carlo (MCMC) ensemble sampler—[emcee]{} [@goodman2010; @foreman2013].
In this work, retrieval analyses generally proceed by first simulating a noise-free spectrum of a world with a known atmospheric state (e.g., Earth). We then add simulated observational noise to this spectrum to create a synthetic observation. Following Bayesian parameter estimation on this synthetic observation, we can compare a retrieved atmospheric state to the original, known atmospheric state, thereby allowing us to understand how observational noise affects our ability to deduce the true nature of an exoplanetary atmosphere.
Albedo Model {#fm}
------------
Our three-dimensional albedo model [see also @cahoy2010] divides a world into a number of plane-parallel facets with coordinates of longitude ($\zeta$) and co-latitude ($\eta$), with the former referenced from the sub-observer location and the latter ranging from 0 at the northern pole to $\pi$ at the southern pole. A single facet has downwelling incident stellar radiation from a zenith angle $\mu_{\rm s} = \cos{\theta_{\rm s}} = \sin{\eta}\cos{(\zeta - \alpha)}$, where, as earlier, $\alpha$ is the phase angle. The facet reflects to the observer in a direction whose zenith angle is given by $\mu_{\rm o} = \cos{\theta_{\rm o}} = \sin{\eta} \cos{\zeta}$. Note that, at full phase (where the geometric albedo is defined) the observer and the source are colinear such that $\mu_{\rm o} = \mu_{\rm s}$ for all facets.
The atmosphere above each facet is divided into a set of pressure levels, and we perform a radiative transfer calculation to determine the emergent intensity. With the intensities calculated for an entire visible hemisphere, we follow the methods outlined by @horak1950 and @horak1965 to perform integration using Chebychev-Gauss quadrature, thus producing the reflectance value at a given wavelength. We repeat this procedure at each of the wavelength points within a specified range to build up a reflectance spectrum.
Taking $I(\tau,\mu,\phi)$ to be the wavelength-dependent intensity at optical depth $\tau$ in a direction determined by the zenith and azimuth angles $\mu$ and $\phi$, we ultimately need to determine the emergent intensity from each facet in the direction of the observer, $I(\tau=0,\mu_{\rm o},\phi_{\rm o})$. Thus, for each facet we must solve the one-dimensional, plane-parallel radiative transfer equation, $$\mu\frac{dI}{d\tau} = I(\tau,\mu,\phi) - S(\tau,\mu,\phi),$$ where $S$ is the wavelength dependent source function. Following @meador1980, @Toon89, and @marley2015, the source function is $$\begin{aligned}
S(\tau,\mu,\phi) = \frac{\bar{\omega}}{4\pi}F_{\rm s} \cdot p(\tau,\mu,\phi,-\mu_{\rm s},\phi_{\rm s}) \cdot \mathrm{e}^{-\tau/\mu_{\odot}} \\
+\, \bar{\omega}\int_0^{2\pi} d\phi' \int_{-1}^{1}\frac{d\mu'}{4\pi} \cdot I(\tau,\mu',\phi') \cdot p(\tau,\mu,\phi,\mu',\phi'),
\end{aligned}
\label{eq:sourcefunction}$$ where $\bar{\omega}$ is the single scattering albedo, $F_{\rm s}$ is the incoming stellar flux at the top of the atmosphere (which we normalize to unity so that emergent intensities correspond to reflectivities), $\phi_{\rm s}$ is the stellar azimuth angle, and $p$ is the scattering phase function. Note that our source function does not include an emission term since we are not computing thermal spectra. Recall that the first term in Equation \[eq:sourcefunction\] describes directly scattered radiation from the direct solar beam while the final term describes diffusely scattered radiation from the ($\mu',\phi'$) direction scattering into the ($\mu,\phi$) direction.
Like most standard tools for solving the radiative transfer equation, we separate treatments of directly-scattered radiation from diffusely-scattered radiation, and, for both, it is convenient to express the scattering phase function in terms of a unique scattering angle, $\Theta$. As single scattered radiation typically has more distinct forward and backward scattering features, we choose to represent the scattering phase function for the direct beam with a two-term Henyey-Greenstein (TTHG) phase function [@kattawar1975], $$p_{\rm TTHG}(\Theta) = fp_{\rm HG}(g_{\rm f},\Theta) + (1-f)p_{\rm HG}(g_{\rm b},\Theta) \,$$ where $p_{\rm HG}$ is the Henyey-Greenstein (HG) phase function with, $$p_{\rm HG} = \frac{1}{4\pi}\frac{1 - \bar{g}^2}{(1 + \bar{g}^2 - 2\bar{g}\cos{\Theta})^{3/2}} \ .$$ Recall that a TTHG phase function can represent both forward and backward scattering peaks, while the (one term) Henyey-Greenstein phase function only has one peak (typically in the forward direction). In the previous expressions, $\bar{g}$ is the asymmetry parameter, $f$ is the forward/backward scatter fraction, $g_{\rm f}$ is the asymmetry parameter for the forward-scattered portion of the TTHG, and $g_{\rm b}$ is the asymmetry parameter for the backward-scattered portion of the TTHG. For the forward and backward scattering portions of the TTHG phase function, we use $g_{\rm f} = \bar{g}$, $g_{\rm b} = -\bar{g}/2$, and $f = 1 - g_{\rm b}^2$. Substituting these into the TTHG phase function expression yields, $$p_{\rm TTHG} = \frac{\bar{g}^2}{4}p_{\rm HG}(-\bar{g}/2,\Theta) + (1-\frac{\bar{g}^2}{4})p_{\rm HG}(\bar{g},\Theta).$$ For radiation that is single-scattered from the solar beam to the observer, the scattering geometry is fixed by the planetary phase angle such that $\Theta = \pi-\alpha$. Our choice of parameters, and their relation to $\bar{g}$, in the TTHG was designed by @cahoy2010 to roughly reproduce the phase function of liquid water clouds. This parameterization, however, is different from that proposed by @kattawar1975. We do not expect our results to be sensitive to the details of a particular phase function treatment as @lupu2016 showed that scattered-light retrievals struggle to constrain phase function parameters.
We adopt a standard two-stream approach to solving the radiative transfer equation [@meador1980]. In this case, the diffusely-scattered component of the source function is azimuthally averaged. Combined with our representation of the directly-scattered component, we have, $$\begin{aligned}
S(\tau,\mu, \mu_{\rm s}, \alpha) = {}& \frac{\bar{\omega}}{4\pi}F_{\rm s} \cdot p_{\rm TTHG}(\mu,-\mu_{\rm s}) \cdot \mathrm{e}^{-\tau/\mu_{\rm s}} \\
& + \frac{\bar{\omega}}{4\pi}\int_{-1}^{1} I(\tau,\mu')p(\mu,\mu') d\mu',
\end{aligned}
\label{eq:directdiffuse}$$ where the azimuth-averaged phase functions are given by, $$p(\mu,\mu') = \frac{1}{2\pi} \int_{0}^{2\pi} p(\mu,\phi,\mu',\phi) d\phi \ .$$ We represent the azimuth-averaged scattering phase functions as a series of Legendre polynomials, $P_l(\mu)$, expanded to order $M$ with, $$p(\mu,\mu') = \sum_{l = 0}^{M}g_l P_l(\mu) P_l(\mu') \ ,$$ where the phase function moments, $g_l$, are defined according to, $$g_l = \frac{2l+1}{2}\int_{-1}^1 p(\cos{\Theta}) P_l(\cos{\Theta}) d\cos{\Theta} \ .$$ The first moment of the phase function is related to the asymmetry parameter, with $\bar{g} = g_1/3$. We use a second order expansion of the phase function, giving, $$p(\mu,\mu') = 1 + 3\bar{g}\mu\mu' + \frac{g_2}{2}(3\mu^2 - 1)(3\mu'^2 - 1) \ .$$
In a given atmospheric layer of our albedo model, the optical depth is the sum of the scattering optical depth and the absorption optical depth, $\tau = \tau_{\rm scat} + \tau_{\rm abs}$. The scattering optical depth has contributions from Rayleigh scattering and clouds, so that $\tau_{\rm scat} = \tau_{\rm Ray} + \bar{\omega}_{\rm cld}\tau_{\rm cld}$, where $\bar{\omega}_{\rm cld}$ is the cloud single scattering albedo. The single scattering albedo for a layer is then $\bar{\omega}=\tau_{\rm scat}/\tau$. We determine the asymmetry parameter, $\bar{g}$, with an optical depth weighting on the Rayleigh scattering asymmetry parameter (which is zero) and the cloud scattering asymmetry parameter, yielding $\bar{g} = \bar{g}_{\rm cld} (\tau_{\rm cld}/\tau_{\rm scat})$. When representing the second moment of the phase function, we use $g_2 = \frac{1}{2}(\tau_{\rm Ray}/\tau_{\rm scat})$ so that $g_2$ tends towards the appropriate value for Rayleigh scattering [i.e., 1/2; @hansen1974] when the Rayleigh scattering optical depth dominates the scattering optical depth.
Model Upgrades {#subsec:upgrades}
--------------
As compared to prior investigations that have used the @cahoy2010 albedo tool [e.g., @lupu2016; @nayak2017], we have updated the model to include an optional isotropically-reflecting (Lambertian) lower boundary (mimicking a planetary surface), and have added pressure-dependent absorption due to [$\rm H_2O$]{}, [$\rm O_3$]{}, [$\rm O_2$]{}, and [$\rm CO_2$]{}. [$\rm CH_4$]{} remains a radiatively active species in the model, as in previous studies. We also include Rayleigh scattering from [$\rm H_2O$]{}, [$\rm O_2$]{}, [$\rm CO_2$]{}, and N$_2$ (in addition to H$_2$ and He from previous studies). As in @lupu2016, we allow for an extended gray-scattering cloud in our atmospheres.
In @lupu2016, their two-layer cloud model atmosphere includes a deeper, optically thick cloud deck that essentially acts as a reflective surface. Unlike the gas giants within that study, though, terrestrial planets have a solid surface we can probe. We characterize our isotropically-reflecting lower boundary using a spherical albedo for the planetary surface, $A_{\rm s}$, which represents the specific power in scattered, outgoing radiation compared to that in incident radiation. For this study, we simply adopt gray surface albedo values, which reduces complexity and computation time. For the inhomogeneous surface of a realistic Earth, featuring oceans and continents, the surface albedo is wavelength-dependent, and we hope to investigate the significance of such surfaces in future work.
We undertook a test to check our reflective lower boundary condition in the limit of a transparent atmosphere. Without atmospheric absorption or scattering, our assumption of a Lambertian surface would imply that the reflectivity (or phase function) determined by our albedo code should follow the analytic Lambert phase function, $$\Phi_{\rm{L}}(\alpha) = \frac{\sin{\alpha} + (\pi - \alpha)\cos{\alpha}}{\pi}.
\label{lamb_phase}$$
Figure \[lambert\_test\] compares the model phase function with the analytic phase function and shows complete agreement, confirming that our treatment of the surface is correct.
![Comparing our model phase function to the analytic Lambertian phase function (Equation \[lamb\_phase\]). No atmospheric absorption or scattering is present in the forward model. []{data-label="lambert_test"}](phase_function.pdf){width="47.00000%"}
Previous work featuring the albedo model adopted here used a pre-defined atmospheric pressure grid. To accommodate the finite surface pressures of rocky planets as well as the various combinations of cloud parameters our retrievals will explore, we instead establish an adaptive method of determining the pressure grid. Here, we divide the atmosphere into a pressure grid of $N_{\rm level}$, bounded by $P = P_{\rm top}$ at the top of the atmosphere and $P = P_0$ at the surface. In a cloud-free scenario, we simply divide the atmosphere evenly in log-$P$ space.
For our simulations that include a single cloud deck, we adaptively determine the pressure value at each level depending on the location, thickness, and optical depth of the cloud. The quantities that define the cloud deck are $p_{\rm t}$, the cloud-top pressure, $dp$, the atmospheric pressure across the cloud, and $\tau$, the cloud optical depth. We begin by assigning a number of layers to the cloud, imposing two conditions: (1) there should be at least three model pressure layers to each atmospheric pressure scale height (${\tt perH} =3$), and (2) the cloud optical depth in a layer must remain below at most 5 (${\tt maxtau} = 5$). This allows us to avoid any one layer from spanning a large extent within the atmosphere, and also avoids cloud layers that have extremely large scattering optical depths.
When beginning our gridding process, we propose an initial number of cloud layers, $N_{\rm{c}} = {\tt
perH} \times {\tt numH}$, where $ {\tt numH} =
\ln{\frac{p_{\rm t}+dp}{p_{\rm t}}}$ is the number of e-folding distances through the cloud (serving as a proxy for scale height). The aerosol optical depth for each pressure layer within the cloud would then simply be $\Delta\tau = \frac{\tau}{N_{\rm c}}$. However, if $\Delta \tau > {\tt maxtau}$, we adjust the cloud resolution by increasing $N_{\rm{c}}$ by a factor of $\frac{\Delta\tau}{{\tt maxtau}}$ and then round up to the nearest integer. In other words, we increase the resolution of the pressure grid through the cloud until the layer optical depth is under [maxtau]{}. We determine successive pressure level values through the cloud with $p[i] = p[i-1] + \Delta\ln{p}$, where $\Delta\ln{p}
= \frac{\ln{(p_{\rm t} + dp)} - \ln{p_{\rm t}}}{N_{\rm c}}$, starting from the top of the cloud. We divide the remaining $N_{\rm level} - N_{\rm c}$ levels in uniform $\ln{p}$ space on either side of the cloud, weighted by the number of pressure scale heights above ($N_{\rm t}$) and below ($N_{\rm b}$) the cloud. Figure \[fig:atmscheme\] visualizes the three portions of the atmosphere.
For simplicity, we assume an isothermal atmosphere (at $T=250$ K), as temperature has little effect on the reflected-light spectrum [@robinson2017]. Pressure, however, has a strong impact on molecular opacities, as seen in Figure \[fig:pdep\]. We incorporated high-resolution pressure-dependent opacities for all molecules in our atmosphere. The absorption opacities are generated line-by-line from the HITRAN2012 line list [@rothman2013] for seven orders of magnitude in pressure ($10^{-5} - 10^{2}$ bar) at $T$, spanning our entire wavelength range at $<1$ cm$^{-1}$ resolution. Figure \[fig:pdep\] also illustrates how absorption features of [$\rm H_2O$]{}, [$\rm O_2$]{}, and [$\rm O_3$]{} change when in an atmosphere of 1 bar versus one of 10 bar.
We interpolate our high-resolution opacity tables to the slightly lower resolution of the forward model in order to maintain short model runtimes while not affecting the accuracy of the output spectra. For each model layer, we interpolate over the opacities from our table given the pressure. The chemical abundances in our forward model atmosphere are constant as a function of pressure, and we also adopt a uniform acceleration due to gravity.
![Illustrative schematic of our model atmosphere’s structure. The atmosphere has $N_t + N_c + N_b$ layers. Table \[tab:params\] lists the definitions, fiducial values, and priors of the presented parameters.[]{data-label="fig:atmscheme"}](atmosphere_schematic.pdf){width="41.00000%"}
We have also added an option to include partial cloudiness across a planetary disk, whose fractional coverage is described by $f_{c}$. To mimic partial cloudiness as we see on Earth, we call the forward model twice. We use the same set of atmospheric and planetary parameters for both calls, except for the cloud optical depth. “Cloudy” is the call that has a non-zero cloud optical depth, while “cloud-free” is the call where we set cloud optical depth to zero. Each call returns a geometric albedo spectrum, and we combine the two sets with the fractional cloudiness parameter such that the combined spectrum follows $f_c \times {\rm cloudy} + (1 - f_c) \times$ [cloud-free]{}.
Albedo Model Fiducial Values and Validation {#subsec:fiducialvals}
-------------------------------------------
The generalized three-dimensional albedo model described above can simulate reflected-light spectra of a large diversity of planet types, spanning solid-surfaced worlds to gas giants with a variety of prescribed atmospheric compositions. For the present study, however, we choose to focus on Earth-like worlds, which are described in detail below. Thus, we define a set of fiducial model input parameters that are designed to mimic Earth and thereby enable us to generate simulated observational datasets for an Earth twin.
Table \[tab:params\] summarizes the fiducial model parameter values adopted for our Earth twin. Also shown are the priors for these parameters, which we use when performing retrieval analyses. For an Earth-like setup, the surface atmospheric pressure is $P_0 = 1$ bar and we adopt a surface albedo of $A_{\rm s} = 0.05$, which is representative of mostly ocean-covered surface. We adopt a uniform acceleration due to gravity of $g = 9.8$ m s$^{-2}$ and set the planetary radius to $R_{\oplus}$. For convenience, we sometime refer to these four variables ($P_0$, $A_{\rm s}$, $g$, and $R_{\rm p}$) as the bulk planetary and atmospheric parameters.
{width="0.49\linewidth"} {width="0.49\linewidth"}
We focus on molecular absorption due to [$\rm H_2O$]{}, [$\rm O_3$]{}, and [$\rm O_2$]{}. While our albedo model includes opacities from [$\rm CH_4$]{} and [$\rm CO_2$]{} as well, we omit these two species as the reflected-light spectrum of Earth in the visible contains no strong features for these molecules. The input values for the molecular abundances (or volume mixing ratios) are [$\rm H_2O$]{} = $3\times10^{-3}$, [$\rm O_3$]{} = $7\times10^{-7}$, and [$\rm O_2$]{}= 0.21. These abundance values are based on column weighted averages from a standard Earth model atmosphere with vertically-varying gas mixing ratios [@mcclatchey1972]. The primary Rayleigh scatterer and background gas in our fiducial model is [$\rm N_2$]{}, whose abundance makes up the remainder of the atmosphere after all other gases are accounted for (i.e., roughly 0.79). Rayleigh scattering is treated according to @hansen1974 with constants to describe the scattering properties of [$\rm N_2$]{}, [$\rm O_2$]{}, and [$\rm H_2O$]{} from @allen2000. We do not include polarization or Raman scattering effects.
Our cloud model was designed to be minimally parametric while still enabling us to sufficiently reproduce realistic spectra of Earth. Our single-layer gray [$\rm H_2O$]{} cloud has $\bar{\omega} = 1$ and $\bar{g} = 0.85$, which are characteristic of water clouds across the visible range. These two parameters were fixed to minimize retrieval model complexity, as we believe that water is the most likely condensate for worlds in the Habitable Zone. Nevertheless, future studies may not wish to assume values of $\bar{\omega}$ and $\bar{g}$ [*a priori*]{}. Cloud top pressure ($p_{\rm t}$) and fractional coverage ($f_{c}$) are set at 0.6 bar and 50%, respectively, which are roughly consistent with observations of optically thick cloud coverage on Earth [@stubenrauch13]. Cloud thickness ($dp$) and optical depth ($\tau$) were set to 0.1 bar and 10, respectively, based on results from the MODIS instrument (http://modis-atmos.gsfc.nasa.gov) used in @robinson2011.
With fiducial values chosen, we validate our forward model against a simulated high-resolution disk-integrated spectrum of Earth at full phase, as shown in Figure \[fig:3dcompare\]. The comparison spectrum is produced by the NASA Astrobiology Institute’s Virtual Planetary Laboratory (VPL) sophisticated 3D line-by-line, multiple scattering spectral Earth model [@robinson2011]. The @robinson2011 tool can simulate images and disk-integrated spectra of Earth from the ultraviolet to the infrared. It has been validated against observations at visible wavelengths taken by NASA’s [*EPOXI*]{} mission [@robinson2011] and NASA’s [*LCROSS*]{} mission [@robinsonetal2014].
Features of the @robinson2011 model include Rayleigh scattering due to air molecules, realistic patchy clouds, and gas absorption from a variety of molecules, including [$\rm H_2O$]{}, [$\rm CO_2$]{}, [$\rm O_2$]{}, [$\rm O_3$]{}, and [$\rm CH_4$]{}. Surface coverage of different land types (e.g., forest, desert) is informed by satellite data, and water surfaces incorporate specular reflectance of sunlight. A grid of thousands of surface pixels are nested beneath a grid of 48 independent atmospheric pixels, all of equal area. For each surface pixel, properties from the overlying atmospheric pixels are used as inputs to a full-physics, plane-parallel radiative transfer solver— the Spectral Mapping Atmospheric Radiative Transfer (SMART) model [@meadows1996]. Intensities from this solver are integrated over the pixels with respect to solid angle, thereby returning a disk-integrated spectrum.
The sophistication of the @robinson2011 model makes it unsuitable to retrieval studies, however, as model runtimes are measured in weeks for the highest-complexity scenarios. This, in part, justifies our adoption of a minimally-parameteric albedo model (whose runtime is measured in seconds). Furthermore, as in Figure \[fig:3dcompare\], our efficient albedo model reproduces all of the key features of the @robinson2011 model. The most notable differences—that the efficient model, as compared to the @robinson2011 model, is more reflective in the blue and less reflective in the red—are simply due to our adoption of a gray surface albedo. Land and plants, which cover roughly 29% of Earth’s surface, are generally more reflective in the red than in the blue. Figure \[fig:3dcompare\] also compares a spectrum from our forward model against a spectrum of a partially clouded ocean planet generated with the @robinson2011 model. This ocean world is identical to Earth except for the fact that its surface is covered entirely by an ocean, with no land present. The surface albedo in the ocean model is gray beyond 500 nm; shortward of this the reflectivity increases, likely leading to the discrepancy in our comparison at the bluest wavelengths. Still, with a more accurate match to a planet that has a nearly gray albedo through the visible, we consider our assumption of gray surface albedo to be the main reason for the discrepancies when compared to the @robinson2011 realistic model.
Finally, in our albedo model we set 100 facets for the visible hemisphere and calculate a high-resolution geometric albedo spectrum at 1000 wavelength points between 0.35$\mu$m and 1.05$\mu$m. Like @lupu2016, we only consider a planet at full phase ($\alpha = 0^{\circ}$). While direct imaging missions will not obtain observations of exoplanets at full phase, this assumption makes little difference for our results as we are not computing integration times and only work in SNR space. Additionally, as @nayak2017 followed up @lupu2016 by retrieving phase information from giant planets in reflected light, we anticipate performing a similar expansion in the future. Our forward model has 61 pressure levels in an isothermal atmosphere of 250 K, bounded below by a reflective surface. The top of the atmosphere is set at $P_{\rm top} = 10^{-4}$ bar.
[llll]{} $\log{P_0}$ (bar) & Surface pressure & $\log{(1)}$ & \[-2,2\]\
$\log{\rm H_2 O}$ & Water vapor mixing ratio & $\log{(3\times10^{-3})}$ & \[-8,-1\]\
$\log{\rm O_3}$ & Ozone mixing ratio & $\log{(7\times10^{-7})}$ & \[-10,-1\]\
$\log{\rm O_2}$ & Molecular oxygen mixing ratio & $\log{(0.21)}$ & \[-10,0\]\
$R_{p}$ (R$_{\oplus}$) & Planet radius & $1$ & \[0.5, 12\]\
$\log{g}$ (m s$^{-2}$) & Surface gravity & $\log{(9.8)}$ & \[0,2\]\
$\log{A_s}$ & Surface albedo & $\log{(0.05)}$ & \[-2, 0\]\
$\log{p_t}$ (bar) & Cloud top pressure & $\log{(0.6)}$ & \[-2,2\]\
$\log{dp}$ (bar) & Cloud thickness & $\log{(0.1)}$ & \[-3,2\]\
$\log{\tau}$ & Cloud optical depth & $\log{(10)}$ & \[-2,2\]\
$\log{f_c}$ & Cloudiness fraction & $\log{(0.5)}$ & \[-3,0\]\
[l|l|l]{} Wavelength ($\mu$m) & 0.4 – 1.0 & 0.506, 0.575, $R = 50$: 0.6 – 0.97\
Data quality & SNR$_{\rm 550nm} = 5,\ 10,\ 15,\ 20$ & SNR$_{\rm 600nm} = 5,\ 10,\ 15,\ 20$\
[llll]{} I & Surface conditions & $P_0$, $A_{\rm s}$ & 2\
II & $+$ Bulk properties & $P_0$, $A_{\rm s}$, $g$, $R_{\rm p}$ & 4\
III & $+$ Gas mixing ratios & $P_0$, $A_{\rm s}$, $g$, $R_{\rm p}$ & 7\
& & [$\rm H_2O$]{}, [$\rm O_2$]{}, [$\rm O_3$]{}&\
IV & $+$ Cloud properties & $P_0$, $A_{\rm s}$, $g$, $R_{\rm p}$ & 11\
& & [$\rm H_2O$]{}, [$\rm O_2$]{}, [$\rm O_3$]{}&\
& & $p_{\rm t}$, $dp$, $\tau$, $f_{\rm c}$&\
Retrieval Setup and Noise Model {#subsec:retrieval}
-------------------------------
We convert a high resolution geometric albedo spectrum to a synthetic planet-to-star flux ratio spectrum given the resolution of an instrument and a noise model. We then apply a Bayesian inference tool on the synthetic data set to sample the posterior probability distributions of the forward model input parameters. To perform Bayesian parameter estimation, we utilize the open-source affine invariant Markov-Chain Monte Carlo (MCMC) ensemble sampler [emcee]{} [@goodman2010; @foreman2013]. Ensemble refers to the use of many chains, or walkers, to traverse parameter space; as a massively parallelized algorithm, it is computationally efficient. Affine-invariance refers to the invariant performance under linear transformations of parameter space, enabling the algorithm to be insensitive to parameter covariances [@foreman2013]. With a cloudy retrieval, we can expect complex correlations that a sampler should be able to reveal. As it is more agnostic to the shape of the posterior, we choose [emcee]{} following @nayak2017 over [Multinest]{}, another sampler @lupu2016 considered that yielded consistent results. The albedo model is coded in Fortran; we convert it into a Python-callable library with the F2PY package. Each call to the forward model takes approximately 10 seconds of clock time on an 8-core processor. To visualize the MCMC results, we utilize the [corner]{} plotting package developed by @corner.
Table \[tab:params\] lists the priors for our parameters. We offer a generous range on the molecular abundances; we allow [$\rm O_2$]{} in particular to be able to dominate the atmospheric composition. Our choice of radius range (0.5–12 $\rm R_\oplus$) reflects the range of of planetary sizes from Mars to Jupiter. Also, when performing retrievals, we impose two limiting conditions to maintain physical scenarios. First, we limit the mixing ratio of [$\rm N_2$]{}, $f_{\rm N_2} = 1 - \sum (\rm gas\ abundances)$, to be between 0 and 1. Second, for the cloud pressure terms, we reject any drawn value that does not satisfy $10^{p_{\rm t}} + 10^{dp} < 10^{P_0}$ (i.e., that the cloud base cannot extend below the bottom of the atmosphere). Note that for the purposes of the retrieval, we consider pressures in log space.
We simulate noise in our observations following the expressions given in @robinson2016. For simplicity, we include only read noise and dark current, as [@robinson2016] showed that detector noise will be the dominant noise source in *WFIRST*-type spectral observations of exoplanets. Detector and instrument parameters for the HabEx and LUVOIR concepts are only loosely defined, and advances in detector technologies for these missions may move observations out of the detector-noise dominated regime. In the detector-noise dominated regime the signal-to-noise ratio is simply, $${\rm SNR} = \frac{c_p \times t_{\rm int}}{\sqrt{(c_d + c_r) \times t_{\rm int}}} \,
\label{snr}$$ where $t_{\rm int}$ is the integration time, $c_p$ is planet count rate, and $c_d$ is the dark noise count rate, and $c_r$ is the read noise count rate. More rigorously, it can be shown that, at constant spectral resolution, ${\rm SNR} \propto q \mathcal{T} A_g \Phi(\alpha) B_\lambda \lambda$, where $q$ is the wavelength-dependent detector quantum efficiency, $\mathcal{T}$ is throughput, and $B_\lambda$ is the host stellar specific intensity (taken here as a Planck function at the stellar effective temperature). We use a stellar temperature of 5780 K for the blackbody. When the SNR at one wavelength is specified, this scaling implies that the the calculation of the signal-to-noise ratio at other wavelengths is independent of the imaging raw contrast of the instrument. We can expect the noise at the redder end of our range to be large, as the detector quantum efficiency (taken to be appropriate for the [*WFIRST*]{}/CGI) rapidly decreases. Since we treat only SNR rather than modeling exposure times, the exact mix of noise sources is not relevant (so that, e.g., dark current and readnoise are indistinguishable). The key relevant properties of the noise model is that it is uncorrelated between spectral channels and its magnitude only depends on wavelength via a dependence on point spread function area [@robinson2016 their Equation 26], which will be true for detector-limited cases but may not be true for large-aperture instruments limited by speckle noise.
![The high resolution (1000 wavelength points from $0.35 - 1.05\ \mu$m) forward model spectrum, overplotted with simulated *WFIRST* rendezvous, $R = 70$, $R = 140$ data, from top panel to bottom. Key spectral features for atmospheric gases in our model are labeled. In the top panel, “1” and “2” mark the span of the *WFIRST* Design Cycle 7 filters (see Table \[tab:runs\]). []{data-label="fig:labeledSpec"}](labeledDataOnSpec.pdf){width="45.00000%"}
For our study, we will consider multiple wavelength resolutions, $R$, and SNRs. Working in SNRs (instead of integration times) makes our investigations independent of telescope diameter, target distance, and other system-specific or observing parameters. Because the SNR is dependent on wavelength, we reference our values to be at V-band (550 nm) for all resolutions for HabEx/LUVOIR. Since the [*WFIRST*]{}/CGI spectrograph is currently planned to only extend to 600 nm at the blue end, we opt to reference our [*WFIRST*]{} SNRs to this wavelength. Unlike previous studies [@lupu2016; @nayak2017], our simulated *WFIRST* rendezvous data include two photometric points in the blue, which is consistent with current CGI designs. We set the SNR in the [*WFIRST*]{} filters to be equal to that at 600 nm.
Our simulation grid setup is shown in Table \[tab:runs\], where the spectral resolutions and SNRs assumed for different observing scenarios are indicated. Figure \[fig:labeledSpec\] demonstrates the *WFIRST* rendezvous scenario data along with $R = 70$ and $R = 140$ data points (for HabEx/LUVOIR) plotted over the forward model spectrum before noise is added. The scaling of SNR with wavelength for [*WFIRST*]{} rendezvous (normalized to unity at 600 nm) as well as our $R = 70$ and $R = 140$ cases (normalized to unity at 550 nm) is shown in Figure \[fig:snrscaling\]. The impact of the host stellar SED sets the overall shape of the SNR scaling, with additional influence from atmospheric absorption bands detector as well as quantum efficiency effects (that have strong impacts at red wavelengths). Thus, Figure \[fig:snrscaling\] can be used to translate our stated SNR to the SNR at any other wavelength (e.g., a SNR$=10$ simulation has a SNR in the continuum shortward of the 950 nm water vapor band of roughly $0.3\times10=3$).
![Scaling of SNR with wavelength for [*WFIRST*]{} rendezvous, $R = 70$, and $R = 140$ cases. The [*WFIRST*]{} curve is normalized to unity at 600 nm while the $R=70$ and $R=140$ curves are normalized to unity at 550 nm, following our definite of simulation SNR at these respective wavelengths. Also shown is the wavelength-dependent detector quantum efficiency (QE) that we adopt.[]{data-label="fig:snrscaling"}](snr_scaling_v2.pdf){width="47.00000%"}
When generating simulated data with a noise model, there are several options for handling the placement/sampling of the mock observational data points. Previous studies [@lupu2016; @nayak2017] have generated a single, randomized dataset for a given SNR. The placement of a single spectral data point is determined by randomly sampling a Gaussian distribution whose width is determined by the wavelength-dependent SNR. While this treatment can accurately simulate a single observational instance, it also runs the risk (especially at lower spectral resolution and SNR) of biasing retrieval results, as the random placement of only a small handful of spectral data points can significantly impact the outcome. Given this, it is ideal to retrieve on a large number ($\gtrsim 10$) of simulated data sets at a given spectral resolution and SNR, where a comprehensive view of all the posteriors from the collection of instances will indicate expected telescope/instrument performance. Unfortunately, given the large number of $R$/SNR pairs in our study (10) and the long runtime of an individual retrieval (of order 1 week on a cluster), running $\sim$10 noise instances for each of our $R$/SNR pairs is computationally unfeasible (requiring $\sim$100 weeks of cluster time). Thus, we opt for an intermediate approach that maintains computational feasibility and avoids potential biases from individual noise instances. Here, we run only a single noise instance at a given $R$/SNR pair, but we do not randomize the placement of the individual spectral points. In other words, the individual simulated spectral points are placed on the “true” planet-to-star flux ratio point and are assigned error bars according to the SNR and noise model. While this approach prevents having a small handful of randomized data points from biasing retrieval results, it does lead to likely optimistic results, especially at modest SNR (i.e., SNR$\sim$10), since data point randomization is, in effect, an additional “noise” source that we are omitting. This means that the posterior distributions will usually be centered on the true values in an unrealistic fashion. However, the width and shape of the posterior covariances will be representative of ‘real’ observations, so the fidelity of retrievals can be assessed. We keep this optimism in mind when discussing results in later sections; in particular, we compare the performance of retrievals on multiple noise instances of a subset of the cases we consider to the non-randomized case in Section \[subsec:multiple\].
Retrieval Validation {#buildup}
====================
Before using our framework on simulated data, we validate its accuracy and examine its performance. For this initial validation, we use non-randomized, wavelength-independent noise at a signal-to-noise ratio of 20 for a spectrum at a resolution of $R = 140$. Table \[tab:buildup\] lists our four validation model variants, each increasing in complexity as we systematically explore how the addition of retrieved parameters influences the posterior distributions and correlations. In Model I, we fix all parameters except $P_0$ and $A_{\rm s}$. In Model II, we add $g$ and $R_p$; in Model III, we then add in gases as retrieved parameters ([$\rm H_2O$]{}, [$\rm O_3$]{}, [$\rm O_2$]{}); and in Model IV, we add all cloud parameters. Incrementally increasing the number of free parameters (from 2 to 11) allows us to see the interconnections between them, and helps us understand how clouds can obscure our inferences.
![Posterior distributions of Model I from Table \[tab:buildup\], where we fix all parameters but $P_0$ and $A_{\rm s}$. We retrieve on $R = 140$, SNR$\ = 20$ data with wavelength-independent noise. Overplotted in solid light-blue color are the fiducial parameter values. The 2D marginalized posterior distribution, used in interpreting correlations, is overplotted with the 1-, 2-, and 3-$\sigma$ contours. Above the 1D marginalized posterior for each parameter, we list the median retrieved value with uncertainties that indicate the 68% confidence interval. Dashed lines (left to right) mark the 16%, 50%, and 84% quantiles.[]{data-label="fig:b1stairs"}](b1_cloudy_stairs.pdf){width="45.00000%"}
{width="70.00000%"}
{width="\textwidth"}
{width="\textwidth"}
In Figure \[fig:b1stairs\], we present the posterior distributions for Model I. In the two-dimensional correlation histogram, a higher probability corresponds to a darker shade. With all else held constant, we see narrow posterior distributions and a slight correlation between $P_0$ and $A_s$. For lower values of surface pressure, which controls the turn off of the Rayleigh scattering slope, we need a brighter surface to maintain the measured brightness, especially in the red end of the spectrum, and vice versa. We mark the 16%, 50%, and 84% quantiles in the marginalized one-dimensional posterior distributions. The posterior distributions for Model II are shown in Figure \[fig:b2stairs\], and are generally narrow (as only four parameters are being retrieved). There are two key correlations, one between $g$ and $P_0$, and one between $R_{\rm p}$ and $A_{\rm s}$. Both gravity and surface pressure influence the column mass, so that, when attempting to fit a spectrum, we can trade a larger gravity with a larger surface pressure and maintain a similar column mass (which controls, e.g., the Rayleigh scattering feature). Additionally, we can trade off a larger reflecting surface area (i.e., larger $R_{\rm p}$) with a darker surface (lower $A_{\rm s}$), which is a statement of the typical “radius/albedo degeneracy” problem. The posterior for surface albedo is now an upper limit instead of a constraint. As a result, the radius posterior distribution appears truncated at larger values given the tight correlation between these two parameters. The correlation seen originally in Model I, between $P_0$ and $A_{\rm s}$, then acts as a chain between the other two, more prominent, correlations to induce correlations between parameters such as $A_{\rm s}$ and $g$ or $R_{\rm p}$ and $P_0$.
Once we allow gases to be free parameters in Model III (Figure \[fig:b3stairs\]), the $P_0$ and $A_{\rm s}$ correlation becomes diminished as [$\rm H_2O$]{}, due to its numerous bands across the spectral range, becomes a primary control of brightness. The significant impact of [$\rm H_2O$]{} on the spectrum leads to a strong, positive correlation between [$\rm H_2O$]{} abundance and planetary size, as additional water vapor absorption can be compensated by a larger planetary size to maintain fixed brightness. We now see gravity linked to the molecular abundances, which is expected as surface gravity directly influences the column abundance of a species. This key correlation also causes the individual gas abundances to be correlated with each other. The main correlations from Model II are still present. We note once more that we do not have constraint on the surface albedo, again leading to an asymmetric distribution for radius. Thus, from the strong correlation of [$\rm H_2O$]{} with with $R_p$, and the fundamental correlation between $R_{\rm p}$ and $A_{\rm s}$, we see correlations between planetary radius, surface albedo, and all gas abundances. Weak correlations between surface pressure and the gas abundances are due to column abundance effects.
Finally, as shown by Figure \[fig:b4stairs\], we retrieve on the data with the full forward model, adding in the cloud parameters $p_{\rm t}$, $dp$, $\tau$, and $f_{\rm c}$. This version of the model is what we apply when simulating direct-imaging data in the upcoming sections, and represents our most realistic (i.e., true to the actual Earth) scenario. The optical depth is shown to only have a lower limit constraint. Thus, the retrieval detects a cloud but cannot constrain the optical depth beyond showing that the cloud is optically thick. There is an expected correlation between $\tau$ and $f_{\rm c}$; a higher cloudiness fraction can complement a less optically thick cloud, and vice versa. There is only an upper limit to $dp$, which is a result of the lack of vertical sensitivity given the constant-with-pressure abundance distributions. The posterior distribution for [$\rm O_2$]{} becomes a lower limit instead of a constraint as in Model III. Surface gravity is less precisely and less accurately constrained compared to the previous, less complex renditions of the model.
For optically thin clouds, we expect to better constrain surface albedo; however, we do not consider this scenario in our study. We examined instead the performance of a completely cloud-free model on data generated with our cloudy model. We find that while the model can fit the data and return accurate estimates of e.g., the mixing ratios, we get inaccurate estimates of the surface albedo and the surface pressure. These two parameters are biased, with lower surface pressure paired with higher surface albedo as the preferred configuration in the cloud-free case. As a result, we move forward with utilizing our cloudy forward model on our simulated data. However, we note that in realistic cases where we do not know the true state of a planet’s atmosphere, we could obtain complementary information relating to the presence of clouds (e.g., variability) such that we may choose the most appropriate forward model.
Results {#snr_study}
=======
We generate data sets for HabEx and LUVOIR-like missions ($0.4 - 1.0\ \mu$m at $R=70$, $R = 140$) at SNR$= 5,\ 10,\ 15,\ 20$, and for the [*WFIRST*]{} rendezvous scenario (two photometric points within $0.4 - 0.6\ \mu$m plus a spectrum of $R = 50$ for $0.6 - 0.96\ \mu$m) also at SNR$=5,\ 10,\ 15,\ 20$. In all cases, we used the noise model to generate uncertainties expected for high-contrast imaging instead of the wavelength-independent noise for the validations in the previous section. As Section \[subsec:retrieval\] described, the SNR refers to the value at 0.55 $\mu$m for $R = 70,\ 140$, and at 0.6 $\mu$m for [*WFIRST*]{}. We record the specific runs in Table \[tab:runs\]. In place of showing the correlations for all parameters for all cases, we refer to Figure \[fig:b4stairs\], which represents the ideal case correlations among the parameter posteriors. We only show the posterior probability distributions themselves to better highlight any trends with respect to SNR and/or $R$. We grouped the posteriors in terms of bulk atmospheric and planetary parameters ($P_0$, $R_{\rm p}$, $g$, $A_{\rm s}$), then cloud parameters ($p_{\rm t}$, $dp$, $\tau$, $f_{\rm c}$), and finally gases ([$\rm H_2O$]{}, [$\rm O_3$]{}, [$\rm O_2$]{}). For each case, [emcee]{} was run with 16 MCMC chains (walkers) per parameter for at least 12000 steps, the last 5000 of which are used to determine the posterior distributions. From those 5000 steps, we randomly selected 1000 sets of parameters to calculate their corresponding high resolution spectra. These spectra are plotted with the data to show the 1-$\sigma$, 2-$\sigma$, and median fits.
Results for $R = 70$, $R = 140$ simulated data
----------------------------------------------
For both $R = 70$ and $R = 140$, we simulated data sets at SNR = 5, 10, 15, 20. Table \[tab:r70results\] lists the median and 1-$\sigma$ values of all retrieved parameters for each SNR at $R = 70$. Figure \[fig:compareR70\] shows the marginalized posterior distributions for the model parameters for all SNR cases for $R = 70$, plotted with the fiducial or “truth” values. Table \[tab:r140results\] lists the median and 1-$\sigma$ values of all retrieved parameters for each SNR at $R = 140$. Figure \[fig:compareR140\] shows the posterior distributions for $R = 140$ for the model parameters for all SNR cases compared against their input values. Figure \[fig:yarr\] shows the corresponding spread in fits and the median fit to the data for each SNR for both resolutions.
Results for *WFIRST* rendezvous simulated data
----------------------------------------------
For the *WFIRST* rendezvous scenario, we utilized the Design Cycle 7 instrument parameters to set the locations of the two photometric points and the range and resolution of the spectrometer ($R=50$; see Table \[tab:runs\]). Because of this particular set-up, we reference the SNRs in our grid (5, 10, 15, 20) at 600 nm, and assign the photometric points the same SNR as at 600 nm. Table \[tab:wfirstresults\] lists the median and 1-$\sigma$ values of all retrieved parameters for each SNR variant. Figure \[fig:compareWfirst\] presents the posterior distributions for the four *WFIRST* rendezvous variants with respect to the input values. Figure \[fig:yarr\] shows the spread in fits and median fit to the data for each variant.
[lrrrrr]{} $\log \, \rm H_2 O$ & $-2.52$ & ${-5.07}_{-1.92}^{+2.34}$ & ${-3.85}_{-2.60}^{+1.77}$ & ${-3.12}_{-1.71}^{+0.97}$ & ${-2.76}_{-0.88}^{+0.62}$\
$\log \, \rm O_3$ & $-6.15$ & ${-7.55}_{-1.46}^{+1.49}$ & ${-6.79}_{-1.81}^{+0.93}$ & ${-6.37}_{-0.84}^{+0.55}$ & ${-6.24}_{-0.60}^{+0.47}$\
$\log \, \rm O_2$ & $-0.68$ & ${-5.12}_{-3.23}^{+3.25}$ & ${-4.51}_{-3.61}^{+3.24}$ & ${-1.86}_{-3.99}^{+1.29}$ & ${-1.00}_{-1.01}^{+0.66}$\
$\log \, \rm P_0$ & $0.0$ & ${0.02}_{-0.84}^{+1.35}$ & ${-0.03}_{-0.70}^{+0.87}$ & ${0.28}_{-0.56}^{+0.85}$ & ${0.25}_{-0.49}^{+0.56}$\
R$_{\rm p}$ & $1.0$ & ${1.23}_{-0.58}^{+1.54}$ & ${1.33}_{-0.52}^{+1.23}$ & ${0.97}_{-0.27}^{+0.68}$ & ${0.98}_{-0.25}^{+0.44}$\
$\log \, \rm g$ & $0.99$ & ${1.33}_{-0.77}^{+0.48}$ & ${1.48}_{-0.68}^{+0.38}$ & ${1.28}_{-0.66}^{+0.51}$ & ${1.24}_{-0.69}^{+0.55}$\
$\log \, \rm A_s$ & $-1.3$ & ${-0.96}_{-0.74}^{+0.58}$ & ${-1.05}_{-0.59}^{+0.55}$ & ${-0.70}_{-0.62}^{+0.37}$ & ${-0.63}_{-0.46}^{+0.29}$\
$\log \, \rm p_t$ & $-0.22$ & ${-1.14}_{-0.61}^{+0.97}$ & ${-1.19}_{-0.56}^{+0.93}$ & ${-0.92}_{-0.71}^{+0.86}$ & ${-0.94}_{-0.73}^{+0.84}$\
$\log \, \rm dp$ & $-1.0$ & ${-1.67}_{-0.92}^{+1.24}$ & ${-1.71}_{-0.91}^{+1.18}$ & ${-1.35}_{-1.14}^{+1.17}$ & ${-1.43}_{-1.06}^{+1.11}$\
$\log \, \tau$ & $1.0$ & ${0.10}_{-1.43}^{+1.30}$ & ${0.21}_{-1.48}^{+1.23}$ & ${0.49}_{-1.66}^{+1.03}$ & ${0.61}_{-1.66}^{+0.93}$\
$\log \, \rm f_c$ & $-0.3$ & ${-1.43}_{-1.07}^{+0.99}$ & ${-1.33}_{-1.12}^{+0.94}$ & ${-0.93}_{-1.32}^{+0.71}$ & ${-1.05}_{-1.27}^{+0.80}$\
[lrrrrr]{} $\log \, \rm H_2 O$ & $-2.52$ & ${-4.56}_{-2.35}^{+2.14}$ & ${-2.74}_{-1.07}^{+0.69}$ & ${-2.61}_{-0.65}^{+0.47}$ & ${-2.43}_{-0.56}^{+0.39}$\
$\log \, \rm O_3$ & $-6.15$ & ${-7.36}_{-1.65}^{+1.26}$ & ${-6.26}_{-0.68}^{+0.53}$ & ${-6.18}_{-0.48}^{+0.42}$ & ${-6.03}_{-0.48}^{+0.34}$\
$\log \, \rm O_2$ & $-0.68$ & ${-4.45}_{-3.69}^{+3.08}$ & ${-1.06}_{-1.43}^{+0.76}$ & ${-0.76}_{-0.79}^{+0.51}$ & ${-0.60}_{-0.59}^{+0.43}$\
$\log \, \rm P_0$ & $0.0$ & ${0.07}_{-0.84}^{+1.01}$ & ${0.20}_{-0.49}^{+0.72}$ & ${0.12}_{-0.36}^{+0.49}$ & ${0.07}_{-0.31}^{+0.39}$\
R$_{\rm p}$ & $1.0$ & ${1.25}_{-0.52}^{+1.16}$ & ${1.01}_{-0.28}^{+0.60}$ & ${0.99}_{-0.23}^{+0.42}$ & ${1.05}_{-0.27}^{+0.42}$\
$\log \, \rm g$ & $0.99$ & ${1.36}_{-0.74}^{+0.46}$ & ${1.31}_{-0.77}^{+0.49}$ & ${1.14}_{-0.65}^{+0.56}$ & ${1.20}_{-0.64}^{+0.50}$\
$\log \, \rm A_s$ & $-1.3$ & ${-0.98}_{-0.60}^{+0.54}$ & ${-0.67}_{-0.50}^{+0.32}$ & ${-0.68}_{-0.44}^{+0.29}$ & ${-0.79}_{-0.69}^{+0.34}$\
$\log \, \rm p_t$ & $-0.22$ & ${-1.23}_{-0.55}^{+1.03}$ & ${-0.96}_{-0.71}^{+0.80}$ & ${-0.79}_{-0.82}^{+0.70}$ & ${-0.66}_{-0.85}^{+0.53}$\
$\log \, \rm dp$ & $-1.0$ & ${-1.72}_{-0.91}^{+1.25}$ & ${-1.43}_{-1.09}^{+1.13}$ & ${-1.55}_{-1.00}^{+1.10}$ & ${-1.49}_{-0.98}^{+1.00}$\
$\log \, \tau$ & $1.0$ & ${0.18}_{-1.49}^{+1.31}$ & ${0.50}_{-1.66}^{+1.09}$ & ${0.61}_{-1.61}^{+0.98}$ & ${0.79}_{-1.40}^{+0.87}$\
$\log \, \rm f_c$ & $-0.3$ & ${-1.30}_{-1.13}^{+0.93}$ & ${-1.31}_{-1.21}^{+0.94}$ & ${-0.99}_{-1.27}^{+0.76}$ & ${-0.76}_{-1.26}^{+0.54}$\
[lrrrrr]{} $\log \, \rm H_2 O$ & $-2.52$ & ${-4.94}_{-2.05}^{+2.35}$ & ${-4.89}_{-2.11}^{+2.48}$ & ${-4.03}_{-2.52}^{+1.87}$ & ${-3.11}_{-1.71}^{+1.17}$\
$\log \, \rm P_0$ & $0.0$ & ${-0.16}_{-0.80}^{+1.32}$ & ${-0.19}_{-0.71}^{+1.03}$ & ${0.03}_{-0.74}^{+1.16}$ & ${0.45}_{-0.85}^{+1.01}$\
$\log \, \rm O_3$ & $-6.15$ & ${-7.66}_{-1.59}^{+1.65}$ & ${-7.53}_{-1.66}^{+1.54}$ & ${-7.16}_{-1.66}^{+1.19}$ & ${-6.80}_{-1.30}^{+0.94}$\
$\log \, \rm O_2$ & $-0.68$ & ${-5.05}_{-3.39}^{+3.26}$ & ${-4.89}_{-3.54}^{+3.43}$ & ${-3.43}_{-4.41}^{+2.50}$ & ${-2.26}_{-3.88}^{+1.71}$\
$\log \, \rm P_0$ & $0.0$ & ${-0.16}_{-0.80}^{+1.32}$ & ${-0.19}_{-0.71}^{+1.03}$ & ${0.03}_{-0.74}^{+1.16}$ & ${0.45}_{-0.85}^{+1.01}$\
R$_{\rm p}$ & $1.0$ & ${1.13}_{-0.50}^{+1.60}$ & ${1.13}_{-0.48}^{+1.27}$ & ${1.02}_{-0.38}^{+1.10}$ & ${0.80}_{-0.19}^{+0.81}$\
$\log \, \rm g$ & $0.99$ & ${1.42}_{-0.82}^{+0.42}$ & ${1.45}_{-0.75}^{+0.41}$ & ${1.41}_{-0.83}^{+0.43}$ & ${1.26}_{-0.83}^{+0.52}$\
$\log \, \rm A_s$ & $-1.3$ & ${-0.89}_{-0.74}^{+0.63}$ & ${-0.84}_{-0.70}^{+0.56}$ & ${-0.95}_{-0.68}^{+0.64}$ & ${-0.76}_{-0.79}^{+0.53}$\
$\log \, \rm p_t$ & $-0.22$ & ${-1.26}_{-0.52}^{+0.98}$ & ${-1.24}_{-0.55}^{+0.83}$ & ${-1.23}_{-0.57}^{+0.84}$ & ${-0.84}_{-0.73}^{+0.89}$\
$\log \, \rm dp$ & $-1.0$ & ${-1.75}_{-0.87}^{+1.16}$ & ${-1.70}_{-0.87}^{+1.12}$ & ${-1.46}_{-1.06}^{+1.14}$ & ${-1.49}_{-1.03}^{+1.49}$\
$\log \, \tau$ & $1.0$ & ${0.03}_{-1.39}^{+1.33}$ & ${0.05}_{-1.40}^{+1.42}$ & ${0.61}_{-1.55}^{+0.94}$ & ${0.99}_{-1.44}^{+0.73}$\
$\log \, \rm f_c$ & $-0.3$ & ${-1.41}_{-1.07}^{+0.96}$ & ${-1.42}_{-1.07}^{+1.00}$ & ${-0.82}_{-1.28}^{+0.60}$ & ${-0.58}_{-0.97}^{+0.41}$\
Discussion
==========
The results from our retrieval analyses enable us to identify the SNR required, at a given spectral resolution, to constrain key planetary and atmospheric quantities. These findings have important implications for the development of future space-based direct imaging missions. We discuss these ideas below, and also touch on impacts of certain model assumptions and ideas for future research directions.
In what follows, we define a “weak detection” for a given parameter as having a posterior distribution that has a marked peak but which also has a substantial tail towards extreme values (indicating that, e.g., for a gas we could not definitively state that the gas is present in the atmosphere). A “detection” implies a peaked posterior without tails towards extreme values but whose 1-$\sigma$ width is larger than an order of magnitude. We use the term “constraint” to indicate a detection whose posterior distribution has 1-$\sigma$ width smaller than an order of magnitude. A non-detection would be indicated by a flat posterior distribution across the entire (or near-entire) prior range. For planetary radius, which is not retrieved in logarithmic space, we distinguish between a “detection” and a “constraint” when the 1-$\sigma$ uncertainties are small enough to firmly place the planet in the Earth/super-Earth regime [i.e., with a radius below $1.5R_{\oplus}$, @rogers2016; @chen2017]. A visual summary of weak detections, detections, and constraints as a function of SNR for our different observing scenarios and for a selection of key parameters are given in Tables \[tab:r70detect\], \[tab:r140detect\], and \[tab:wfirstdetect\].
Influence of SNR on Inferred Properties
---------------------------------------
For $R = 70$ at SNR = 5, Figure \[fig:compareR70\] shows there is only a weak detection of $P_0$ and a detection of $R_{\rm p}$, which merely suggests the planet has an atmosphere and is not a giant planet. As SNR increases to 10, the [$\rm O_3$]{} posterior distribution has a weak peak near the fiducial value, and the gas is only weakly detected. Once the SNR is equal to 15, we weakly detect [$\rm H_2O$]{}, [$\rm O_3$]{}, and [$\rm O_2$]{}. At a SNR of 20, it is possible to detect each of [$\rm H_2O$]{}, [$\rm O_3$]{}, and [$\rm O_2$]{}. At this SNR, the oxygen mixing ratio is estimated to be above roughly $10^{-3}$, indicating that we are unable to determine if [$\rm O_2$]{} is a major atmospheric constituent (i.e., present at the 1% level or more). Gravity (and, thus, planetary mass) remain undetected at all SNRs, similar to the findings of @lupu2016. The surface albedo is unconstrained (or worse) at all SNRs, but shows a weak bias toward a higher value of $A_s \approx 0.3$ ($\log{A_{\rm s}} \approx -0.5$) at the highest SNRs, which is likely due to the relatively large error bars at red wavelengths (driven primarily by low detector quantum efficiency) where we have the most sensitivity to the surface. We are able to get weak detections of $\tau$ and $f_{\rm c}$, which are shown in Figure \[fig:b4stairs\] to be correlated. Yet, with these posteriors, we cannot rule out scenarios without cloud cover. We note the drop-off in the posteriors of $p_{\rm t}$ and $dp$ at higher pressure values likely result from the limiting conditions that the cloud base cannot extend below the surface pressure and the upper limit of the $P_0$ prior. The improved signal-to-noise ratio leads to a posterior more concentrated around the true value for $p_{\rm t}$, $dp$, and $R_{\rm p}$. For improved constraints on cloud properties, it may be beneficial to observe time variability with photometry [e.g., @ford2001] or use polarimetry [e.g., @rossi2017].
At a higher spectral resolution ($R = 140$), the improvement in detections and constraints begin at a lower SNR, as illustrated by Figure \[fig:compareR140\]. Gravity remains undetected for all SNRs. At a SNR equal to 5, $P_0$ and $R_{\rm p}$ have a weak detection and a detection, respectively. At SNR = 10, it is possible to detect [$\rm H_2O$]{}, [$\rm O_3$]{}, and [$\rm O_2$]{}. As with the $R=70$ case, surface albedo is unconstrained (or worse) at all SNRs, and, at the highest SNRs, the model is biased towards $A_{\rm s} \approx 0.3$ (as with $R = 70$). Moving to SNR = 15 adds a constraint to $R_{\rm p}$, $P_0$, and [$\rm O_3$]{}, as well as weak detections of cloud parameters. Increasing the SNR to 20 does not dramatically change the posterior distributions, although the posteriors for [$\rm H_2O$]{} and [$\rm O_2$]{} become narrow enough to offer constraints. Here, the constraint on [$\rm O_2$]{} suggests it is a major constituent in the atmosphere. In spite of the generous SNR, though, the 1-$\sigma$ uncertainties on the gas mixing ratios are not more precise than roughly an order of magnitude (see Table \[tab:r140results\]).
Considering both $R = 140$ and $R = 70$, we see that SNR = 5 data offer very little information about the planetary atmosphere. In the case of $R = 140$, SNR = 10 data offer detections but no constraints, and SNR = 20 data are required to constrain all included gas species. In other words, the conclusions we would draw about the planet (e.g., the amount of gases, the bulk and cloud properties) improve significantly between SNR = 10 and SNR = 20. With $R = 70$, the boost from SNR = 10 to SNR = 15 provides weak detections of key atmospheric and surface parameters, and SNR = 20 data offer detections but few constraints (i.e., except on planetary radius).
For the *WFIRST* rendezvous data sets, we are able to infer very little information at a SNR of 5 or 10 except for weak detection of surface pressure and a detection of the planetary radius. All gases remain undetected at these SNRs. The posterior distributions for most parameters do not vary much as SNR improves, although there are weak detections of cloud optical depth and fractional coverage at the highest SNRs. Like all previous cases, we do not detect the surface gravity. At SNR$\ = 15,\ 20$, the detection of $f_c$ is unable to rule out scenarios with little cloud cover. To obtain weak detections of the atmospheric gases we require a SNR of 20, but, even here, the posteriors have tails that extend to near-zero mixing ratios.
To compare the performance of a *WFIRST* rendezvous scenario against HabEx/LUVOIR scenarios at $R = 70$ and $R = 140$, we plot together the posterior distributions of the parameters for the SNR$\ = 10$ results from *WFIRST* rendezvous, $R = 70$, and $R = 140$ in Figure \[fig:compareMixed\]. While this comparison sheds light on the corresponding trade-off in terms of parameter estimation for the same SNR, these cases do not represent equal integration times, which scales with resolution and SNR. If the dominant noise source does not depend on resolution (e.g., detector noise), the cases of $R = 140$ at SNR = 10, $R = 70$ at SNR = 20, and $R = 50$ at SNR = 28 would be roughly equal in integration times. However, if the dominant noise source does depend on resolution (e.g., exozodiacal dust), the cases of $R = 140$ at SNR = 10, $R = 70$ at SNR = 14, and $R = 50$ at SNR = 17 would roughly have equivalent integration times. Tables \[tab:r70detect\] through \[tab:wfirstdetect\] allow approximate comparisons of these different scenarios, excluding a *WFIRST* rendezvous scenario at high SNR = 28 that we have not considered.
[lcccc]{} [$\rm H_2O$]{}& $-$ & $-$ & W & D\
[$\rm O_3$]{}& $-$ &W & W&D\
[$\rm O_2$]{}&$-$ & $-$ & W &D\
$P_0$ &W &W &W &D\
$R_p$ &D &D &D & C\
[lcccc]{} [$\rm H_2O$]{}& $-$ &D &D &C\
[$\rm O_3$]{}& $-$ &D & C&C\
[$\rm O_2$]{}&$-$ &D &D &C\
$P_0$ &W &D & C &C\
$R_p$ &D &D & C& C\
[lcccc]{} [$\rm H_2O$]{}& $-$ &$-$ &$-$ &W\
[$\rm O_3$]{}& $-$ &$-$ & $-$ &W\
[$\rm O_2$]{}&$-$ &$-$ &W &W\
$P_0$ & W& W& W&W\
$R_p$ &D &D &D & D\
From Figure \[fig:compareMixed\], we see that the performance of the *WFIRST* rendezvous retrieval is similar to that of $R = 70$ at SNR = 10. The noticeable difference is a weak detection of [$\rm O_3$]{} with $R = 70$. Because we adopt the photometric setup from *WFIRST* Design Cycle 7 through the shorter wavelengths, the data do not provide complete spectroscopic coverage across the significant [$\rm O_3$]{} feature from $0.5 - 0.7\ \mu$m, as in the case of HabEx/LUVOIR simulated data. Figure \[fig:labeledSpec\] shows the sampling of the forward model spectrum for the three types of data sets we considered. We compare the spectral fits in Figure \[fig:yarr\] and note the much wider spread in the possible fits for wavelengths shorter than $0.6\ \mu$m for *WFIRST* rendezvous versus $R = 70$ or $R = 140$, which have continuous coverage in the full range. The $R = 140$, SNR = 10 data set was able to offer [*detections*]{} of all atmospheric gases, setting it apart from the other two. We stress, however, that [*constraints*]{} were only found at SNR = 20 and $R = 140$.
Considering Multiple Noise Instances {#subsec:multiple}
------------------------------------
Our parameter estimations are likely to be optimistic as a consequence of our adoption of non-randomized spectral data points in our faux observations. Thus, the requisite SNRs for detection detailed above should be seen as lower limits. Ultimately, our decision to use non-randomized data points stemmed from computational limitations (preventing us from running large numbers of randomized faux observations for each of our $R$/SNR pairs) and from a desire to avoid the biases that can occur from attempting to make inferences from retrievals performed on a single, randomized faux observation [@lupu2016].
However, we deemed it necessary to investigate the consistency of our findings with respect to different noise instances. To work within our computational restrictions, we realized that cases such as $R=70$ with SNR $=5$ yielded little detection information for any parameter even in the ideal scenario of non-randomized data. We then decided to focus on two “threshold” cases based on the results from the non-randomized data: $R=140$ with SNR $=10$ and $R=70$ with SNR $=15$. We ran 10 noise instances of these two cases where it is likely the optimistic non-randomized data makes the difference between detection and constraint for several parameters (see Tables \[tab:r70detect\] and \[tab:r140detect\]).
Each noise instance is run for at least 10000 steps in [emcee]{}. Figure \[fig:demoOutlier\] shows all the individual posteriors for the gas mixing ratios from each noise instance for $R=70$, SNR $=15$. We highlight the posteriors from one “outlier” case where there is no oxygen detection. The corresponding set of data points are shown as well. This highlights the fact that single noise instances can mislead our interpretation and the benefit of having many noise instances run to obtain a more comprehensive understanding of the state of an atmosphere.
{width="0.8\linewidth"}
To summarize the noise instance results, we concatenate samples from the last 1000 steps in each noise instance and construct an averaged set of posteriors. We are able to do this because the noise instances are equally likely, having been drawn in the same manner from a Gaussian with set parameters (i.e., the same SNR as the standard deviation). In Figure \[fig:noiseR70\], we plot up the combined posteriors of the 10 noise instances of $R=70$, SNR $=15$ and compared them to the posterior from the last 5000 steps of the non-randomized data case. We illustrate the same comparison for $R=140$, SNR $=10$ in Figure \[fig:noiseR140\]. We overplot the truth values as well as the 68% confidence interval and median value for each parameter from the combined noise-instances posterior and the non-randomized data posterior.
For all parameters in both the $R=70$ and $R=140$ cases, we find that the average posterior from the 10 noise instances agree with the posterior from the non-randomized data set qualitatively. Their medians and 68% confidence interval ranges are also similar with significant overlap. The overall conclusions we can draw from the average posteriors do not appear to differ much from those using the non-randomized data set posteriors.
Implications for Future Direct Imaging Missions
-----------------------------------------------
Future space-based direct imaging missions will have a diversity of goals for exoplanet studies, and will likely emphasize the detection and characterization of Earth-like exoplanets. For the detection of oxygen and ozone—which are key biosignature gases—in the atmospheres of Earth twins, our results indicate that spectra at a minimum characteristic SNR of 10 will suffice if at $R=140$, while data at SNR of at least 15–20 would be needed at $R=70$. For a [*WFIRST*]{} rendezvous-like observing setup, these gases would only be weakly detected even at a SNR of 20. Methane, which is another important biosignature gas, has no strong signatures in the visible wavelength range for the modern Earth, so we did not consider detection of this gas. Thus, we could not use our simulated data and retrievals to argue for detections of atmospheric chemical disequilibrium [@sagan1993; @krissansen2016].
Key habitability indicators include atmospheric water vapor and surface pressure. Detecting the former requires a SNR of 15–20 at $R=70$, but only a SNR of 10 at $R=140$. Surface pressure can be constrained to within an order of magnitude for SNR $\gtrsim 15$ at $R=140$, although the overall lack of temperature information in these reflected-light spectra would make it impossible to use pressure/temperature data to argue for habitability [@robinson2017]. Surface temperature information may then need to come from climate modeling investigations that are constrained by retrieved gas mixing ratios.
For all of our observing setups, the data yield detections of, and in some cases constraints on, the planetary radius. Except at SNR of 20 for $R=70$ or SNR $>15$ for $R=140$, the posterior distributions are not well-enough constrained to distinguish a Earth/super-Earth ($R_p < 1.5\,R_{\oplus}$) from a mini-Neptune based on size alone, although the data do rule out planetary sizes larger than Neptune. Additional atmospheric information (e.g., composition) could potentially be used to help distinguish between terrestrial planets and mini-Neptunes. These findings are consistent with the gas giant-focused work of @nayak2017, who note that observations at multiple phase angles can also help to better constrain planetary size. Our overall lack of surface gravity constraints, paired with the weak constraints on planet size, implies that we do not have a constraint on the planetary mass. Follow-up (or precursor) radial velocity observations (or, potentially, astrometric observations) could offer additional constraints on planet mass.
We can make rough comparisons of our $R$/SNR results to those of @brandt2014, who used minimally parametric models to investigate detections of [$\rm O_2$]{} and [$\rm H_2O$]{} for Earth twins. These comparisons are not direct, however, as @brandt2014 were fitting for fewer parameters (8 versus our 11) and also only assumed that SNR was proportional to planetary reflectance (versus our more complicated scaling, as shown in Figure \[fig:snrscaling\]). For [$\rm O_2$]{}, @brandt2014 find $R=150$ and SNR $=6$ for a 90% detection probability, which is consistent with our $R=140$ posteriors moving from a non-detection at SNR $=5$ to a detection at SNR $=10$. When investigating [$\rm H_2O$]{}, @brandt2014 find $R=40$ and SNR $=7.5$ or $R=150$ and SNR $=3.3$ for a 90% detection probability. Using Figure \[fig:snrscaling\] to scale our SNRs to 890 nm (i.e., to the continuum just shortward of 950 nm water vapor band), at $R=50$ we only find a weak detection of [$\rm H_2O$]{} for SNR$_{\rm 890~nm}$ $=10$, and at $R=140$ we transition from a water vapor non-detection to detection between a SNR$_{\rm 890~nm}$ of 2.5–5. Taken altogether, these comparisons indicate that we agree with @brandt2014 at higher spectral resolution ($R=140$–150), but that detection of [$\rm H_2O$]{} at lower spectral resolution ($R=50$) will likely require higher SNRs than originally indicated.
The discussion above emphasizes mere detections, not constraints (which, again, we define as having peaked posterior distributions with 1-$\sigma$ widths less than an order of magnitude). While uncertain, we anticipate that characterization of climate, habitability, and life likely require constraints, not simple detections. Here, as is shown in Table \[tab:r140detect\], only $R=140$ and SNR = 20 observations offer the appropriate constraints. Thus, future space-based high contrast imaging missions with goals of characterizing Earth-like planetary environments are likely to need to achieve $R=140$ and SNR = 20 observations (or better). Of course, combining near-infrared capabilities, which would provide access to additional gas absorption bands, may help loosen these requirements.
Impacts of Model Assumptions
----------------------------
Several key assumptions adopted in this study warrant further comment. First, as noted earlier, we do not retrieve on planetary phase angle and planet-star distance, both of which influence the planet-to-star flux ratio. Thus, in effect, we are assuming that the planetary system has been revisited multiple times for photometric and astrometric measurements, such that the planetary orbit is reasonably well-constrained (i.e., that the orbital distance and phase angle are not the dominant sources of uncertainty when interpreting the observed planet-to-star flux ratio spectrum). If the orbit is not well-constrained, @nayak2017 showed that strong correlations can exist between the retrieved phase angle and the planet radius.
Second, we have assumed detector-dominated noise and a quantum efficiency appropriate for the [*WFIRST*]{}/CGI for all of our observational setups. While this is likely a fair assumption for our [*WFIRST*]{} rendezvous studies, it is likely that detector development will lead to major improvements in instrumentation for a HabEx/LUVOIR-like mission. Here, the rapid decrease into the red due to detector quantum efficiency may not be as dramatic, implying that spectra would have relatively more information content at red wavelengths as compared to the present study. Furthermore, a HabEx/LUVOIR-like mission may no longer be in the detector-dominated noise regime. In the limit of noise dominated by astrophysical sources (e.g., exo-zodiacal light or stellar leakage), the SNR only varies as $\sqrt{q\mathcal{T}B_{\lambda}}$.
Finally, we adopt a relatively simple parameterization of cloud three-dimensional structure. Specifically, we allow for only a single cloud deck in the atmosphere, and we then permit these clouds to have some fractional coverage over the entire planet. This parameterization of fractional cloudiness implies uniform latitudinal and longitudinal distribution of patchy clouds. In reality, clouds on Earth have a complex distribution in altitude, latitude, and longitude [@stubenrauch13], and variations in time also have an observational impact [@cowan2009; @cowanfujii2017]. However, given the overall inability of our retrievals to constrain cloud parameters [at least at the SNRs investigated here; see also @lupu2016; @nayak2017], it seems challenging for future space-based exoplanet characterization missions to detect (or constrain) more complex cloud distributions with the types of observations studied here and data of similar quality.
Future Work
-----------
Our current forward model is able to include both CO$_2$ and CH$_4$, although we did not retrieve on these gases in the current study due to their overall lack of strong features in the visible wavelength range for modern Earth. However, these species do have stronger features in the near-infrared wavelength range. As both of the HabEx and LUVOIR concepts are considering near-infrared capabilities, it will be essential to extend our current studies to longer wavelengths and to investigate whether or not constraints on additional gases (i.e., beyond water, oxygen, and ozone) can be achieved at these wavelengths.
Additionally, given the likely huge diversity of exoplanets that will be discovered by future missions (and that have already been identified and studied by [*Kepler*]{}, [*Hubble*]{}, and [*Spitzer*]{}), it will be necessary to extend our parameter estimation studies to include a wider range of worlds. Both super-Earths and mini-Neptunes are more-favorable targets for a *WFIRST* rendezvous mission, and may also be easier targets for HabEx/LUVOIR-like missions. Our forward model is already capable of simulating these types of worlds, and we anticipate emphasizing a variety of exoplanet types in future studies. Such future studies may also include retrievals on planetary phase angle, which would be relevant to observing scenarios where the planetary orbit is poorly constrained.
Summary {#conclusion}
=======
We have developed a retrieval framework for constraining atmospheric properties of an Earth-like exoplanet observed with reflected light spectroscopy spanning the visible range ($0.4 - 1.0\mu$m). We have upgraded an existing, well-tested albedo model to generate high-resolution geometric albedo spectra used to simulate data at resolutions and quality relevant to future telescopes, such as the HabEx and LUVOIR mission concepts. We combined our albedo model with Bayesian inference techniques and applied MCMC sampling to perform parameter estimation. The data we considered were for *WFIRST* paired with a starshade (i.e., the rendezvous scenario), $R = 70$, and $R = 140$ at SNR$ = 5, 10, 15, 20$. We validated our forward model, and we demonstrated the successful application of our retrieval approach by gradually adding complexity to our inverse analyses.
Following work by @lupu2016 and @nayak2017, who have constructed a retrieval framework for gas giants in refleted light, we made several modifications to the albedo model featured in these previous studies. Our model has a reflective surface, absorption due to water vapor, oxygen, and ozone, Rayleigh scattering from nitrogen and other key gases, pressure-dependent opacities, an adaptive pressure grid, and a single-layer water vapor cloud layer with fractional cloudiness. We performed our retrievals with the goal of estimating our ability to detect and constrain the atmosphere of an Earth twin. We found that $R = 70$, SNR $=15$ data allowed us to weakly detect surface pressure as well as water vapor, ozone, and oxygen. At $R = 140$, we found that SNR $=10$ was needed to more firmly detect these parameters. At $R = 140$, a SNR of 20 was needed to constrain key planetary parameters, and $R = 70$ data at this SNR offered extremely few constraints. A *WFIRST* rendezvous scenario, with its photometric points and lower resolution spectrum ($R = 50$), is only able to offer limited diagnostic information. For example, at SNR $=10$, we only weakly detect and detect surface pressure and planetary radius, respectively. To weakly detect the gases, *WFIRST* rendezvous data needed to be at least SNR $= 20$. Throughout our runs, we found that we are unable to accurately constrain surface albedo or place estimates on the surface gravity, although we can straightforwardly rule out planetary sizes above roughly the radius of Uranus or Neptune.
Our findings demonstrate that direct imaging of Earth-like exoplanets in reflected light offers a promising path forward for detecting and constraining atmospheric biosignature gases. Instrument spectral resolution for future missions strongly impacts requisite SNRs for detection and characterization, and this must be taken into consideration during mission design. Thus, the scientific yield of future space-based exoplanet direct imaging missions can only be maximized by simultaneously considering mission characterization goals, integration time constraints, and instrument spectral performance.
YKF is supported by the National Science Foundation Graduate Research Fellowship under Grant DGE1339067. TR gratefully acknowledges support from NASA through the Sagan Fellowship Program executed by the NASA Exoplanet Science Institute. The authors thank the 2016 Kavli Summer Program in Astrophysics, its Scientific and Local Organizing Committees, the program founder, Pascale Garaud, and the Kavli Foundation for supporting the genesis of this work. This work was made possible by support from the UCSC Other Worlds Laboratory and the [*WFIRST*]{} Science Investigation Team program. The results reported herein benefited from collaborations and/or information exchange within NASA’s Nexus for Exoplanet System Science (NExSS) research coordination network sponsored by NASA’s Science Mission Directorate. Certain essential tools used in this work were developed by the NASA Astrobiology Institute’s Virtual Planetary Laboratory, supported by NASA under Cooperative Agreement No. NNA13AA93A. Computation for this research was performed by the UCSC Hyades supercomputer, which is supported by National Science Foundation (award number AST-1229745) and UCSC. We thank Cecilia Leung, Asher Wasserman, Daniel Thorngren, Eric Gentry, and Chris Stark for stimulating discussions and essential guidance.
natexlab\#1[\#1]{}
Allen, C., & Cox, A. 2000, Allen’s Astrophysical Quantities, Allen’s Astrophysical Quantities (Springer)
, K., [White]{}, V., [Yee]{}, K., [et al.]{} 2016, Journal of Astronomical Telescopes, Instruments, and Systems, 2, 011005
, T. S., [Konopacky]{}, Q. M., [Macintosh]{}, B., & [Marois]{}, C. 2015, , 804, 61
, T. S., [Macintosh]{}, B., [Konopacky]{}, Q. M., & [Marois]{}, C. 2011, , 733, 65
, B., & [Seager]{}, S. 2012, , 753, 100
, W. J., [Koch]{}, D. G., [Lissauer]{}, J. J., [et al.]{} 2003, in , Vol. 4854, Future EUV/UV and Visible Space Astrophysics Missions and Instrumentation., ed. J. C. [Blades]{} & O. H. W. [Siegmund]{}, 129–140
, W. J., [Koch]{}, D. G., [Basri]{}, G., [et al.]{} 2011, , 728, 117
, T. D., & [Spiegel]{}, D. S. 2014, Proceedings of the National Academy of Science, 111, 13278
Burke, C. J., Christiansen, J. L., Mullally, F., [et al.]{} 2015, The Astrophysical Journal, 809, 8
, A. 2014, ArXiv e-prints, arXiv:1412.6097
, E., [Prada]{}, C. M., [An]{}, X., [et al.]{} 2016, Journal of Astronomical Telescopes, Instruments, and Systems, 2, 011004
, K. L., [Marley]{}, M. S., & [Fortney]{}, J. J. 2010, , 724, 189
, W. 2006, , 442, 51
, D., [Brown]{}, T. M., [Noyes]{}, R. W., & [Gilliland]{}, R. L. 2002, , 568, 377
, J., & [Kipping]{}, D. 2017, , 834, 17
, N. B., & [Fujii]{}, Y. 2017, ArXiv e-prints, arXiv:1704.07832
, N. B., [Agol]{}, E., [Meadows]{}, V. S., [et al.]{} 2009, , 700, 915
, B., & [Siegler]{}, N. 2017, ArXiv e-prints, arXiv:1709.06660
de Wit, J., Wakeford, H. R., Gillon, M., [et al.]{} 2016, Nature, advance online publication,
, E. B., [Seager]{}, S., & [Turner]{}, E. L. 2001, , 412, 885
Foreman-Mackey, D. 2016, The Journal of Open Source Software, 24, doi:10.21105/joss.00024
, D., [Hogg]{}, D. W., [Lang]{}, D., & [Goodman]{}, J. 2013, , 125, 306
, J., & [Weare]{}, J. 2010, Commun. Appl. Math. Comput. Sci., 5, 65
, J. P., & [Burrows]{}, A. 2015, , 808, 172
, T., [Gong]{}, Q., [Mandell]{}, A. M., [et al.]{} 2018, in American Astronomical Society Meeting Abstracts, Vol. 231, American Astronomical Society Meeting Abstracts, \#355.47
, J. E., & [Travis]{}, L. D. 1974, , 16, 527
, H. G. 1950, , 112, 445
, H. G., & [Little]{}, S. J. 1965, , 11, 373
, P. G. J., [Teanby]{}, N. A., [de Kok]{}, R., [et al.]{} 2008, , 109, 1136
, N. J., [Lisman]{}, D., [Shaklan]{}, S., [et al.]{} 2012, in , Vol. 8442, Space Telescopes and Instrumentation 2012: Optical, Infrared, and Millimeter Wave, 84420A
, J. F., [Whitmire]{}, D. P., & [Reynolds]{}, R. T. 1993, , 101, 108
, G. W. 1975, , 15, 839
, Q. M., [Barman]{}, T. S., [Macintosh]{}, B. A., & [Marois]{}, C. 2013, Science, 339, 1398
, J., [Bergsman]{}, D. S., & [Catling]{}, D. C. 2016, Astrobiology, 16, 39
, J., [Nemati]{}, B., & [Mennesson]{}, B. 2016, Journal of Astronomical Telescopes, Instruments, and Systems, 2, 011003
, J.-M., [Fletcher]{}, L. N., & [Irwin]{}, P. G. J. 2012, , 420, 170
Line, M. R., Wolf, A. S., Zhang, X., [et al.]{} 2013, ApJ, 775, 137
, R. E., [Marley]{}, M. S., [Lewis]{}, N., [et al.]{} 2016, ArXiv e-prints, arXiv:1604.05370
, B., [Graham]{}, J. R., [Barman]{}, T., [et al.]{} 2015, Science, 350, 64
, N., & [Seager]{}, S. 2009, , 707, 24
, M., [Lupu]{}, R., [Lewis]{}, N., [et al.]{} 2014, ArXiv e-prints, arXiv:1412.8440
, M. S., [Gelino]{}, C., [Stephens]{}, D., [Lunine]{}, J. I., & [Freedman]{}, R. 1999, , 513, 879
, M. S., & [Robinson]{}, T. D. 2015, , 53, 279
, D., [Ruane]{}, G., [Xuan]{}, W., [et al.]{} 2017, , 838, 92
, M., & [Queloz]{}, D. 1995, , 378, 355
, R. A., [Fenn]{}, R. W., [Selby]{}, J. E. A., [Volz]{}, F. E., & [Garing]{}, J. S. 1972, [Optical Properties of the Atmosphere 3rd edn]{}
, C. P., [Pollack]{}, J. B., & [Courtin]{}, R. 1989, Icarus, 80, 23
, W. E., & [Weaver]{}, W. R. 1980, Journal of Atmospheric Sciences, 37, 630
, V. S., & [Crisp]{}, D. 1996, , 101, 4595
, B., [Gaudi]{}, S., [Seager]{}, S., [et al.]{} 2016, in , Vol. 9904, Space Telescopes and Instrumentation 2016: Optical, Infrared, and Millimeter Wave, 99040L
, M., [Lupu]{}, R., [Marley]{}, M. S., [et al.]{} 2017, , 129, 034401
, M. C., [Zhao]{}, F., [Demers]{}, R., [et al.]{} 2016, Journal of Astronomical Telescopes, Instruments, and Systems, 2, 011001
, B. M., [Fischer]{}, D., & [LUVOIR Science and Technology Definition Team]{}. 2017, in American Astronomical Society Meeting Abstracts, Vol. 229, American Astronomical Society Meeting Abstracts, 405.04
, T. D. 2017, ArXiv e-prints, arXiv:1701.05205
, T. D., [Ennico]{}, K., [Meadows]{}, V. S., [et al.]{} 2014, , 787, 171
, T. D., [Stapelfeldt]{}, K. R., & [Marley]{}, M. S. 2016, , 128, 025003
, T. D., [Meadows]{}, V. S., [Crisp]{}, D., [et al.]{} 2011, Astrobiology, 11, 393
, C. D. 1976, Reviews of Geophysics and Space Physics, 14, 609
, L. A. 2015, , 801, 41
, L., & [Stam]{}, D. M. 2017, , 607, A57
, L. S., [Gordon]{}, I. E., [Babikov]{}, Y., [et al.]{} 2013, , 130, 4
, C., [Thompson]{}, W. R., [Carlson]{}, R., [Gurnett]{}, D., & [Hord]{}, C. 1993, , 365, 715
, D., & [Garrett]{}, D. 2016, Journal of Astronomical Telescopes, Instruments, and Systems, 2, 011006
, S., [Turnbull]{}, M., [Sparks]{}, W., [et al.]{} 2015, in , Vol. 9605, Techniques and Instrumentation for Detection of Exoplanets VII, 96050W
, B.-J., [Gordon]{}, B., [Kern]{}, B., [et al.]{} 2016, Journal of Astronomical Telescopes, Instruments, and Systems, 2, 011019
, A. J., [Marley]{}, M. S., [Hinz]{}, P. M., [et al.]{} 2014, , 792, 17
, D., [Gehrels]{}, N., [Breckinridge]{}, J., [et al.]{} 2013, ArXiv e-prints, arXiv:1305.5422
, D., [Gehrels]{}, N., [Baltay]{}, C., [et al.]{} 2015, ArXiv e-prints, arXiv:1503.03757
Stubenrauch, C. J., Rossow, W. B., Kinne, S., [et al.]{} 2013, Bulletin of the American Meteorological Society, 94, 1031
, O. B., [McKay]{}, C. P., [Ackerman]{}, T. P., & [Santhanam]{}, K. 1989, Journal of Geophysical Research, 94, 16287
, W. A., [Breckinridge]{}, J., [Greene]{}, T. P., [et al.]{} 2016, Journal of Astronomical Telescopes, Instruments, and Systems, 2, 011020
, J., [Moody]{}, D., [Krist]{}, J., & [Gordon]{}, B. 2016, Journal of Astronomical Telescopes, Instruments, and Systems, 2, 011013
, P., [Hedelt]{}, P., [Selsis]{}, F., [Schreier]{}, F., & [Trautmann]{}, T. 2013, , 551, A120
, J., [Mawet]{}, D., [Ruane]{}, G., [et al.]{} 2017, Proc.SPIE, 10400, 10400
, J., [Mawet]{}, D., [Ruane]{}, G., [Hu]{}, R., & [Benneke]{}, B. 2017, , 153, 183
[^1]: [http://sites.nationalacademies.org/bpa/bpa\\textunderscore049810](http://sites.nationalacademies.org/bpa/bpa\textunderscore049810)
[^2]: <https://science.nasa.gov/astrophysics/2020-decadal-survey-planning>
|
---
title: '[Reduced Effective Lagrangians]{}'
---
phyzzx
=cmbx10 scaled3 =cmssq8 scaled1 \#1[\_[[\#1 ]{}]{}]{}
and
[Introduction]{}
================
Within the framework of accelerator physics, there are two ways we may obtain information on physics above the weak scale. By building higher and higher energy machines, we hope to directly observe new particles as the energy of the machine passes the threshold of particle production. At the SSC, for example, we hope to find some direct evidence of a mass-generating mechanism or other new phenomena. We may also hope that high-precision measurements at lower energies will provide indirect evidence of high-energy physics. In these measurements we search for deviations from the Standard Model; any such deviation is either evidence that the Standard Model is incorrect or a signal of some new physics on a higher energy scale.
If we accept the validity of the Standard Model, it becomes important to ask how to characterize any possible deviations from it. Although we do not know the structure of physics beyond the Standard Model, we know that the low-energy effects of the full higher-energy theory (valid at energies above the mass $\Lambda$ of some new particle) can be incorporated into an effective Lagrangian \[\]. For any given extension of the Standard Model we can write an effective Lagrangian composed of only low-energy (Standard-Model) fields in a series of terms of higher and higher dimension: $$\lef = \sum_{n=-2}^{\infty} {1\over{{\Lambda}^n}}\thinspace
\alpha_{\ocal}\thinspace \ocal^{(n+4)}
\eqn\eflag$$ The operators $\ocal^{(n+4)}$ have dimension $[\hbox{mass}]^{(n+4)}$ and contain only derivatives and fields with masses below $\Lambda$. If the high-energy physics decouples \[\], then the sum starts at $n=0$, and $\ocal^{(4)}$ is the Standard Model. All operators (not including those related by the equations of motion) up to $\ocal^{(6)}$ have been listed by Buchmüller and Wyler \[\]. If the physics above scale $\Lambda$ does not decouple from the low energy physics, then we can write our effective theory as a gauged chiral model \[,,\]. In this case $\Lambda \sim 4\pi v$ \[\], and there exist terms $\ocal^{(2)}$ of chiral dimension 2. The (renormalization-scale-dependant) constants $\alpha_{\ocal}$ determine the strength of the contribution of $\ocal$; they are calculated by matching the Green’s functions (or S-matrix elements) of the effective Lagrangian to those of the full high-energy theory (for an explanation see, for example, \[\]).
The situation at hand is somewhat different. We do not know the high-energy theory, and so we cannot perform the matching to find the values for the $\alpha$’s. Whatever the high-energy theory is, though, it will generate in the effective Lagrangian a tower of terms $\ocal^{(n)}$ each obeying the symmetries of the theory. So we may parametrize all possible forms of new high-energy physics by writing down an effective Lagrangian containing all operators $\ocal$ which respect the symmetries of the theory. Since higher order terms are all suppressed by higher powers of ${ 1 \over \Lambda}$, we can terminate the series at some point with negligible effects.
This still leaves a large number of terms to be considered. The number can of course be reduced by integration by parts; the action S is usually unchanged by such a manipulation. Further, the classical equations of motion can clearly be used to remove terms when the effective Lagrangian is only to be used at tree level \[,,,\]. Recently Georgi has shown that in certain cases the classical equations of motion can be used even in the quantum theory (in loop diagrams) \[\]. Explicitly exempted from this in \[\] are terms quadratic in the fields. Also, gauge theories and spontaneously broken theories are not fully considered. These limitations preclude the use of this simplification in the Standard Model. In this paper we will show that, quite generally, terms in an effective Lagrangian that are connected by the equations of motion are redundant. They may be dropped from the effective Lagrangian without changing observables. A generalization of the equivalence theorem ß[A. Salam and J. Strathdee, [*Phys. Rev.*]{} [**D2**]{} (1970) 2869.]{} \[,ß,\] shows that this is also true for quadratic terms. Spontaneously broken gauge theories present no new problems. The result is that any effective Lagrangian can be brought into a canonical form consisting of a reduced set of operators that are gauge invariant and unrelated by the classical equations of motion. We may choose such a set to minimize the number of higher derivative terms; this is often (but not always - for example see ) the most useful form.
[proof]{}
=========
The purpose of this section is to show that we may use the equations of motion on any terms of dimension d $\geq 5$ in the effective Lagrangian. In particular, we show that any such term which contains $D^2 \phi$, $D^\mu
F^I_{\mu\nu}$, or $\Dslash\psi$ gives contributions to the S matrix identical to those from a term with fewer derivatives, with which we may therefore replace it. (Here $\phi$ is a scalar, $F^I_{\mu\nu}$ is a gauge field strength, $\psi$ is a fermion, and $D_\mu$ is a covariant derivative.) The generalization will be clear.
Let $\lcal$ be an effective Lagrangian valid for energies below $\Lambda$. It can be written as $$\lef = \sum_{n=-2}^{\infty} {1\over{{\Lambda}^n}}\thinspace
\alpha_{\ocal}\thinspace \ocal^{(n+4)}
\equiv \sum_{n=0}^{\infty} \eta^n \lcal_n
\eqn\eq$$ where $\eta$ is a small parameter (such as $1 / \Lambda$) and $\lcal_0$ includes any terms with negative powers of $\eta$. The operators $\ocal$ can all be chosen to be local; in the following we assume this choice has been made.
As an example, consider $\lef = \half (\partial_\mu \phi)^2 - \half m^2 \phi^2
- \lambda \phi^4 + \eta g_1 \phi^6 + \eta g_2 \phi^3 \partial^2 \phi$. By making the shift of variables $\phi \rightarrow \phi + \eta g_2 \phi^3$ we induce $\lef \rightarrow \half (\partial_\mu \phi)^2 - \half m^2 \phi^2 -
\lambda' \phi^4 + \eta g_1' \phi^6 + \ocal(\eta^2)$. By the equivalence theorem \[,ß,\] the S matrix is unaffected by the change in variables, and so (to first order in $\eta$) we may choose to use the new effective Lagrangian in place of the original one. In the following, we will generalize the equivalence theorem, showing that a similar procedure is possible for any effective Lagrangian.
In the following $\varphi_i$ stands for any of the various fields in the theory. First consider an effective Lagrangian containing a term like $\eta
T[\varphi] D^2 \phi$ where $T[\varphi]$ is any local function of any of the fields and their derivatives ($\eta$ is some appropriate power of $1 /
\Lambda$). Let $Z'[j_i]$ be the generating functional for the Green’s functions with the $j_i$ sources for each of the fields. Then (working for now only to first order in $\eta$) $$Z'[j_i] = \int {\prod_l \dcal\varphi'_l\
\exp\thinspace i\int{d^4x\left[\lcal_0' + (\lcal_1' - \eta T D^2 \phi')
+ \eta T D^2 \phi' + \sum_i j_i \varphi'_i \right]}}
+ \ocal(\eta^2).
\eqn\genfun$$ The term in $\lcal_1'$ to be removed (that is, $\eta T D^2 \phi'$) has been written explicitly. We can now change variables so that $(\phi')^\dagger = \phi^\dagger + \eta T$. (If the scalar is real, then we let $\phi' = \phi + \eta T$.) This change is only a redefinition of a variable of integration, so we expect $Z'$ (and therefore all Green’s functions) to be unchanged.
Written in the shifted variable $$\eqalign{
Z'[j_i] = \int \prod_l \dcal\varphi_l
\left|{\delta(\phi')^\dagger \over \delta\phi^\dagger}\right|
\exp\thinspace i\int d^4x\Bigg[\lcal_0' &+
\eta T ({\delta\lcal_0' \over \delta\phi^\dagger} -
\partial_\mu {\delta\lcal_0' \over \delta \partial_\mu\phi^\dagger})
+ (\lcal_1' - \eta T D^2 \phi) \cr
&+ \eta T D^2 \phi
+ \sum_{i} j_i \varphi_i
+ j_{\phi^\dagger} \eta T \Bigg]
+ \ocal(\eta^2). \cr }
\eqn\genfunii$$ In this equation we have expanded $\lcal\left(\phi^\dagger + \eta T\right)$ in a Taylor series about $\phi^\dagger$. The shifted $Z'$ differs from in three ways: there is a new Lagrangian, a Jacobian of the transformation, and a new coupling to the source $j_{\phi^\dagger}$.
The change of the variable of integration we have performed in presumably has no effect. It is commonly known as the equivalence theorem \[,\] that in many cases we may make the change of variables in [*only*]{} the Lagrangian without changing the S matrix. In other words, we may remove the Jacobian of the transformation and the additional coupling to $j_{\phi^\dagger}$ in without changing the S matrix. The only effect on mass shell is to renormalize the Green’s functions (though in general the off-shell Green’s functions are changed). Statements of the equivalence theorem for $\phi\rightarrow F(\phi)$ usually require $F$ to be a point transformation, or $\phi\rightarrow \phi + F(\phi)$, where the expansion of $F(\phi)$ begins with the term second order in $\phi$. For applications to spontaneously broken theories, this would require the use of the shifted field, destroying the (broken) symmetry. These requirements are in fact too demanding; we will see that the transformation $(\phi')^\dagger
= \phi^\dagger + \eta T$ leaves the S matrix unchanged for any function $T$ to any order in $\eta$. In the next three paragraphs we consider the three differences between equations and , respectively: the change in the Lagrangian, the Jacobian of the transformation, and the new coupling to the source $j_{\phi^\dagger}$.
The new Lagrangian is just the original Lagrangian plus $\eta T$ times the classical equation of motion for $\phi^\dagger$. The variable shift we have performed respects the symmetries of the theory; since $\phi^\dagger \phi$ and $T\phi$ are both invariant under symmetry operations, $\phi^\dagger +
\eta T$ transforms as $\phi^\dagger$ does. Because of this the new Lagrangian explicitly retains all the symmetries of the original. If $\lcal_0$ has the usual quadratic terms, then the new Lagrangian is $\lef + \eta T (-D^2\phi -
m^2 \phi + \hbox{terms with two or more fields)}$. The first term cancels $\eta T D^2\phi$ in $\lcal_1$. Georgi \[\] has pointed out that since the effective Lagrangian contains [*all*]{} terms allowed by the symmetries of the theory, each of remaining terms is of a form already present. We can absorb them by changing the coefficients $\alpha_\ocal$ of some terms already present in $\lef$ (this is true to all orders in $\eta$).
Regardless of the the structure of $T[\varphi]$ (but assuming it is local) the presence of the Jacobian has no effect on the theory. It can be written as a ghost coupling $\bar c c + \eta \bar c {\delta T \over \delta \phi} c$. We can see that in any diagram containing ghosts, there will always be at least one loop containing only ghost propagators. Assume without loss of generality that $T$ has only one term. The ghost Lagrangian from the shift will have exactly two terms; one is $\bar c c$, and the other will be a kinetic term only if $T = \partial^m \phi^\dagger$ for any m. So there can be [*either*]{} a kinetic term for c (in which case the ghosts will not couple to physical fields) [*or*]{} the ghosts will couple to physical fields (in which case there will be no kinetic term for the ghosts, which can therefore be consistently disregarded when dimensional regularization is employed), but not both. Another way to see the point is to note that the effective theory is valid only up to energies of order $\Lambda = 1 / \sqrt{\eta}$. Let $T =
(\partial^2\phi + \lambda \phi^3)$, so the ghost Lagrangian is $\bar c( 1 +
\eta\partial^2 + 3\eta\lambda\phi^2) c$. Now rescale $c\rightarrow c/
\sqrt{\eta}$. Even though the ghosts propagate, their mass is on the order of the cutoff! In any loop we can therefore expand the ghost propagators into the numerator, and so loops consisting purely of ghosts will contribute only quadratic (or more highly divergent) terms. Any diagrams containing ghosts will therefore not contribute to the S matrix. Note that for a more general field transformation, the equivalence theorem does not hold because the ghosts do not decouple \[ß\]. For instance, for $\phi\rightarrow \partial^2\phi +
\lambda\phi^3$ the ghost Lagrangian will be $\bar c \partial^2 c + 3 \lambda
\bar c c \phi^2$, which will have physical effects. The transformation necessary for the case at hand avoids this pitfall.
Again, whether or not $T$ is linear in $\phi$, the term $\eta
j_{\phi^\dagger}T$ has no effect on the S matrix. Instead of $${G^{(n)}}' = \bra{0} T\left[\phi(x_1)...\phi(x_n)\right] \ket{0}$$ we have $$G^{(n)} =\bra{0} T\left[(\phi(x_1) + \eta T(x_1))...(\phi(x_n) + \eta
T(x_n))\right]\ket{0}.$$ It can be seen diagrammatically that $G^{(n)} =
f^n(p) {G^{(n)}}' +$ (terms with fewer than n poles). The term $\eta
j_{\phi^\dagger} T$ has only the effect (on-shell) of multiplying each n-point Green’s function $G^{(n)}$ by $f^n(p)$, the nth power of some function of momentum (see \[\] for a full explanation in a slightly more restricted context). Indeed, if $T$ is linear in $\phi$, even the off-shell Green’s functions are related to the original ones in this way. In any case, this multiplicative factor cancels out in the definition of the S matrix and leaves all S-matrix elements unaffected.
The result is that $Z'$ gives the same S matrix as the generating functional $$Z[j_i] = \int {\prod_l \dcal\varphi_l\
\exp\thinspace i\int{d^4x \left[\lcal_0
+ (\lcal_1 - \eta T D^2 \phi)
+ \sum_i j_i \varphi_i \right]}}
+ \ocal(\eta^2).
\eqn\genfuniii$$ The term $\eta T D^2 \phi$ has been removed, the on-shell Green’s functions are the same up to a renormalization, and the (unknown) values of some $\alpha_\ocal$’s have changed to linear combinations of the original $\alpha_\ocal$’s (changing $\lcal'$ to $\lcal$). This equivalence is true regardless of the structure of the local function $T$.
The preceding comments are true also for a term $\eta T\Dslash\psi$. In this case one makes the change of variables $\bar\psi \rightarrow \bar\psi + \eta
T$. The equations of motion contain $\Dslash\psi$ rather than $D^2\phi$, and the rest follows similarly.
Finally, a term like $\eta T^\nu_a D^\mu F_{\mu\nu}^a$ is also redundant. Here we must make the change of variables $A^\nu_a \rightarrow A^\nu_a + \eta
T^\nu_a$ (where A is any abelian or non-abelian Yang-Mills gauge field, $\mu$ and $\nu$ are Lorentz indices, and a,b, and c are symmetry-group indices). Note that this change respects the local gauge symmetries; since the term $\eta
T^\nu_a D^\mu F_{\mu\nu}^a$ is gauge invariant, $T^\nu_a$ transforms like $F^{\mu\nu}_a$. Under a gauge transformation $$\eqalign{
A_a^\nu &\rightarrow A_a^\nu + \partial^\nu \Lambda_a
+ gf_{abc}\Lambda_b A_c^\nu \cr
T_a^\nu &\rightarrow T_a^\nu + gf_{abc}\Lambda_b T_c^\nu \cr}
\eqn\gt$$ ($f_{abc}$ are the structure constants of the symmetry group), so $A^\nu_a + \eta T^\nu_a$ transforms just like $A_\nu^a$: $$(A + \eta T)_a^\mu \rightarrow (A + \eta T)_a^\mu + \partial^\mu \Lambda_a
+ gf_{abc}\Lambda_b (A + \eta T)_c^\mu.
\eqn\gt1$$ The proof follows through unchanged from above, but now the action also contains pieces whose job it is to fix the gauge. The variable change $A^\nu_a \rightarrow (A+\eta T)^\nu_a$ which takes us from to produces the following change in the gauge-fixing term (using a simple choice as an example): $$\eqalign{
\lcal_{GF} &= - {1 \over 2\xi} [f]^2 \cr
f =\partial_\mu A^\mu_a
\ \ \ &\rightarrow \ \ \
f=\partial_\mu (A+\eta T)^\mu_a, \cr}
\eqn\gf$$ and the following change in the Faddeev-Popov ghost term: $$\lcal_{FP} = \partial_\mu\omega_a^*(\partial^\mu\omega_a
+ gf_{abc}\omega_b A^\mu_c)
\ \ \ \rightarrow \ \ \
\partial_\mu\omega_a^*(\partial^\mu\omega_a
+ gf_{abc}\omega_b (A+\eta T)^\mu_c).
\eqn\fp$$ The new FP ghost term is exactly that needed $(\omega^*_a {\delta f \over
\delta \Lambda} \omega_a)$ for the new gauge-fixing term, and so the symmetry is consistently fixed. After making the change of variables we can choose instead to gauge-fix with the original gauge-fixing term, with the original FP ghost term - the net effect is that our change of variables hasn’t changed $\lcal_{GF} + \lcal_{FP}$.
By repeatedly shifting variables, we can continue the process outlined above for all redundant terms to all orders. First we remove all order $\eta$ terms containing the appropriate derivative forms. Once we have removed all possible derivative terms to order $\eta$, we can continue the process with derivative terms of order $\eta^2$; since the change of variables is $\phi
\rightarrow \phi +\eta^2 T$, all order $\eta$ terms will be unaffected. In this way we can successively remove derivative terms of the given form order-by-order in $\eta$.
For a spontaneously broken theory the proof above carries through unchanged. We may write the Lagrangian in terms of the shifted field $\phi$ (for example $\phi= { \phi_1 + v \choose \phi_2 }$) so that the (broken) symmetries of the theory are still apparent. Then a term $\eta T[\varphi] D^2 \phi$ is redundant, with the required shift in fields again being $\phi^\dagger \rightarrow \phi^\dagger + \eta T$ (in the example this would mean $\phi_1 +v \rightarrow
\phi_1^* + v^* + \eta T$ and $\phi_2^* \rightarrow \phi_2^* + \eta T$). In this case it is crucial that $T[\varphi]$ linear in the fields not be disallowed, since upon expanding any shifted scalars $\phi$, $T[\varphi]$ may in general be a sum of terms, some of which are linear in the fields $\phi_1$ or $\phi_2$. That $T[\varphi]$ may be linear means all of these terms in $\eta T[\varphi] D^2 \phi$ are redundant, not just those with two or more fields.
[technical considerations]{}
============================
The preceeding manipulations are only formal, for two reasons. Firstly, we have treated the path integral as if it were a simple integral in the variable $\varphi$. A rigorous treatment would discretize the variables $\varphi(x)$ and write the path integral as an infinite product of integrals over the discrete variables $\varphi_x$. We would then perform the shift in variables, and rewrite the result as a path integral over continuous variables. Alternately, we could change variables in the canonical operator formalism. In this way it is found \[,\] that, in general, the formal manipulations above are actually incorrect; a discrepancy arises at the two-loop level. It will turn out, however, that this complication can be dealt with.
In the operator formalism, the operator Hamiltonian $\hat H$ contains non-commuting factors $\hat Q$ and $\hat P$. It is related to the classical Hamiltonian $H$ by $$\bra{q} \hat H(\hat Q,\hat P) \ket{p}_{QP}
= { e^{ipq/\hbar} \over \sqrt{2 \pi \hbar}} H(p,q)
\eqn\heqh$$ The subscript QP indicates that H is “QP ordered”, so that all factors of $\hat Q$ are placed to the left of all factors of $\hat P$. When we make a change of variables, the new Hamiltonian $ \hat H' = \hat H(f(\hat Q,\hat P),g(\hat Q,\hat P))$ is no longer QP ordered. Because of the non-commuting factors, $$\bra{q} \hat H \ket{p}_{QP} =
\bra{q} \hat H' \ket{p} =
\bra{q} (\hat H' + \hat H'_{new}) \ket{p}_{QP}
= { e^{ipq/\hbar} \over \sqrt{2 \pi \hbar}}
( H' + H'_{new})
\eqn\eqn$$ where the term after the first equals sign is not QP ordered, and $\hat
H'_{new}$ arises because $\hat P$ and $\hat Q$ do not commute. We see that the Hamiltonian in the new variables is not just $H'(f(Q,P),g(Q,P))$; there is an additional term.
If, rather than the operator form, we consider a path integral over classical fields, we see the same effect differently expressed. Because of the stochastic nature of the path integral, terms of order $\epsilon$ which we have naively neglected must be considered \[\]. This is important, because the chain rule is not valid to order $\epsilon$; this can be inferred from ${dq \over dt} = \lim_{\epsilon\rightarrow 0}
{ q(t+\epsilon) - q(t) \over \epsilon} =
\lim_{\epsilon\rightarrow 0}
[{dq \over dt} + \half\epsilon {d^2 q \over dt^2} +\ocal(\epsilon^2)]$. When we change variables in the continuous Lagrangian, we ignore the $\ocal(\epsilon)$ term. But when we work carefully, with discrete variables, these order $\epsilon$ terms in the integral are seen to contribute an additional potential term in the limit $\epsilon \rightarrow 0$. This is identical to $H'_{new}$.
In fact, this extra potential term is regulator dependant. Salomonson \[\] noted this by explicitly comparing Feynman graphs calculated before and after a change of variables. When using a cutoff, the extra potential was needed, but when using dimensional regularization, no new term appeared. It can be shown for a local field theory that the additional potential term is proportional to $\delta^n(0)$, where $n+1$ is the number of space-time dimensions. (In operator language, this factor comes from the commutator of $\hat P$ and $\hat
Q$. In the path integral version, it enters through additional volume elements connected with the space integrals.) The quantum mechanical path integral, equivalent to a 0+1 dimensional field theory, has no such delta functions factors. If the theory is regulated dimensionally, it would appear that these terms leave the S matrix unchanged, and so the extra potential can be disregarded.
Another possible source of error in the last section is the manipulation of divergent integrals. The conclusions reached above are strictly true only if we assume the theory to have been regularized beforehand. Let us assume the theory in has been rendered finite by dimensional regularization; as $\epsilon$ approaches zero, the regulation is removed. In this case the generating functionals in and give identical S-matrix elements with identical $\epsilon$ dependence. We can write the effective Lagrangian in as $\lef' + \lct'$ (a counter-term Lagrangian $\lct'$ has been extracted), so that the S-matrix elements from $Z'$ are UV finite (, they have no $\epsilon$ dependence). We can then use the results of the last section to show that all terms in $\lct'$ of dimension d $\geq 5$ which contain $D^2 \phi$, $D_\mu F^I_{\mu\nu}$, or $\Dslash\psi$ can be dropped in favor of terms with fewer derivatives. The reduced form of the full Lagrangian will contain no terms with these derivatives, either in $\lef$ or $\lct$, but the S matrix will be the same as the that from $\lef'$ and $\lct'$, and so it will be finite ($\epsilon$-independent). The reduced set of effective operators is therefore renormalizable, in the sense that counter terms of the form already present in the reduced Lagrangian are sufficient to renormalize all S-matrix elements.
In practice this type of renormalization may be cumbersome, as Green’s functions are divergent, and it is not until S-matrix elements are calculated that the coefficients of the counter terms are apparent. In some cases it may be easier to imagine employing all possible counter terms, including those in the form of terms not in the reduced $\lef$. In this way we can clearly make all Green’s functions finite. This approach has the disadvantage that terms removed at one scale may reappear at another. When running the couplings, some that we removed will be reintroduced. Of course, we may remove them again, at the new scale, in the same way they were removed originally. The relationships among the $\alpha's$ will therefore be different at different scales, in just such a way as to give the same $\alpha(\mu)'s$ in this method of renormalization as in the other.
[Conclusions]{}
===============
Not all terms in an effective Lagrangian contribute independently to the S matrix. By using the classical equations of motion in the Lagrangian, the number of terms which must be considered can be reduced while maintaining all symmetries of the effective Lagrangian (including any broken symmetries). This is true for all terms in an effective Lagrangian (including those quadratic in fields) and for gauged and spontaneously broken theories.
A result of the discussion above is that, in the full effective Lagrangian, some of the parameters $\alpha$ are redundant; it is only the the coefficients of the terms in the maximally reduced effective Lagrangian that are completely determined by the high-energy theory. For each equation of motion, there is one arbitrary parameter, and we may exploit this ambiguity to choose values as we see fit. Additionally, it is clear that that the value of some $\alpha$ may not be the same in two effective Lagrangians with differing sets of terms, and so care must be taken when comparing estimates of an $\alpha$ to also compare the forms of the Lagrangians used in calculations.
In practice this result is quite useful. It is of great utility to be able to work with as few terms in $\lef$ as possible when calculating loop diagrams. That the results apply in spontaneously broken theories, without destroying the broken symmetry, makes them useful for calculating in the Standard Model \[\], and these results have been assumed in, for example, \[\]. Generally, it is easiest to calculate with an effective Lagrangian in which the number of derivatives is minimized, though other choices are possible. Indeed other choices of operators are sometimes more useful, for example when the derivative terms are more tightly constrained by experiment than are those with which we would replace them \[\], or when the derivative terms in the effective Lagrangian are expected, based on some knowledge of the high-energy theory, to be small \[\].
[Acknowledgements]{}
====================
I am very grateful to M.B. Einhorn for many valuable discussions, helpful suggestions, and critical readings of the manuscript. I also thank J. Wudka for useful discussions of the subject. Portions of this work were supported by the Department of Energy and by the National Science Foundation (Grant No. PHY89-0435).
|
---
abstract: 'We present a stepwise approach to estimate high dimensional Gaussian graphical models . We exploit the relation between the partial correlation coefficients and the distribution of the prediction errors, and parametrize the model in terms of the Pearson correlation coefficients between the prediction errors of the nodes’ best linear predictors. We propose a novel stepwise algorithm for detecting pairs of conditionally dependent variables. We show that the proposed algorithm outperforms existing methods such as the graphical lasso and CLIME in simulation studies and real life applications. In our comparison we report different performance measures that look at different desirable features of the recovered graph and consider several model settings.'
author:
- 'Ginette LAFIT, Francisco J. NOGALES, Marcelo RUIZ and Ruben H. ZAMAR'
bibliography:
- 'bibliography.bib'
title: '**A Stepwise Approach for High-Dimensional Gaussian Graphical Models**'
---
\#1
[*Keywords:*]{} Covariance Selection; Gaussian Graphical Model; Forward and Backward Selection; Partial Correlation Coefficient.
Introduction {#sec_1}
============
High-dimensional [*Gaussian graphical models*]{} (GGM) are widely used in practice to represent the linear dependency between variables. The underlying idea in GGM is to measure linear dependencies by estimating partial correlations to infer whether there is an association between a given pair of variables, conditionally on the remaining ones. Moreover, there is a close relation between the nonzero partial correlation coefficients and the nonzero entries in the inverse of the covariance matrix. Covariance selection procedures take advantage of this fact to estimate the GGM conditional dependence structure given a sample [@dempster1972covariance; @lauritzen1996graphical; @edwards2000introduction].
When the dimension $p$ is larger than the number $n$ of observations, the sample covariance matrix $S$ is not invertible and the maximum likehood estimate (MLE) of $\boldsymbol{\Sigma}$ does not exist. When $p/n \leq 1$, but close to $1$, $S$ is invertible but ill-conditioned, increasing the estimation error [@ledoit2004well]. To deal with this problem, several covariance selection procedures have been proposed based on the assumption that the inverse of the covariance matrix, $\Omega$, called [*precision matrix*]{}, is sparse.
We present an approach to perform covariance selection in a high dimensional GGM based on a forward-backward algorithm called *graphical stepwise* (GS). Our procedure takes advantage of the relation between the partial correlation and the Pearson correlation coefficient of the residuals.
Existing methods to estimate the GGM can be classified in three classes: nodewise regression methods, maximum likelihood methods and limited order partial correlations methods. The nodewise regression method was proposed by [@meinshausen2006high]. This method estimates a lasso regression for each node in the graph. See for example [@peng2009partial], [@yuan2010high], [@liu2012tiger], [@zhou2011high] and [@ren2015asymptotic]. Penalized likelihood methods include [@yuan2007model], [@banerjee2008model], [@friedman2008sparse], [@johnson2011high] and [@ravikumar2011high] among others. [@cai2011constrained] propose an estimator called CLIME that estimates precision matrices by solving the dual of an $\ell_1$ penalized maximum likelihood problem. Limited order partial correlation procedures use lower order partial correlations to test for conditional independence relations. See [@spirtes2000causation], [@kalisch2007estimating], [@rutimann2009high], [@liang2015equivalent] and [@huang2016].
The rest of the article is organized as follows. Section \[section\_2\] introduces the stepwise approach along with some notation. Section \[section\_3\] gives simulations results and a real data example. Section \[section\_4\] presents some concluding remarks. The Appendix shows a detailed description of the crossvalidation procedure used to determine the required parameters in our stepwise algorithm and gives some additional results from our simulation study.
Stepwise Approach to Covariance Selection {#section_2}
=========================================
Definitions and Notation {#sec_1a}
------------------------
In this section we review some definitions and technical concepts needed later on. Let $\mathcal{G}=(V,E)$ be a graph where $V\neq \emptyset$ is the set of nodes or vertices and $E\subseteq V\times V= V^2$ is the set of edges. For simplicity we assume that $V=\{1,\ldots,p \}$. We assume that the graph $\mathcal{G}$ is undirected, that is, $ (i,j)\in E$ if and only if $(j,i)\in E$. Two nodes $i$ and $j$ are called connected, adjacent or neighbors if $(i,j) \in E$.
A [*graphical model*]{} (GM) is a graph such that $V$ indexes a set of variables $\{ X_1,\ldots,X_p\}$ and $E$ is defined by: $$\label{pmp}
(i,j) \notin E \text{ if and only if } X_i {
\mathrel{\text{
{\rotatebox[origin=c]{90}{\resizebox{2.25ex}{1.65ex}{$\vDash$}}}
}}
}X_j \mid X_{V \setminus \{i,j\}.
}$$ Here $ {
\mathrel{\text{
{\rotatebox[origin=c]{90}{\resizebox{2.25ex}{1.65ex}{$\vDash$}}}
}}
}$ denotes [*conditional independence*]{}.
Given a node $i\in V$, its neighborhood $\mathcal{A}_i$ is defined as $$\label{eq2-1}
\mathcal{A}_i = \{l \in V \setminus \{i\}: (i,l) \in E \}.$$ Notice that $\mathcal{A}_i$ gives the nodes directly connected with $i$ and therefore a GM can be effectively described by giving the system of neighborhoods $\displaystyle \left\{\mathcal{A}_i\right\}_{i = 1}^{p}$.
We further assume that $\displaystyle (X_1,\ldots,X_p)^{\top} \sim \text{N}(\boldsymbol{0}, \boldsymbol{\Sigma})$, where $\boldsymbol{\Sigma}=(\sigma_{ij})_{i,j=1 \ldots,p}$ is a positive-definite covariance matrix. In this case the graph is called a [*Gaussian graphical model*]{} (GGM). The matrix $\boldsymbol{\Omega}=(\omega_{ij})_{i,j=1 \ldots,p}=\boldsymbol{\Sigma}^{-1}$ is called [*precision matrix*]{}.
There exists an extensive literature on GM and GGM. For a detailed treatment of the theory see for instance [@lauritzen1996graphical], [@edwards2000introduction], and [@buhlmann2011statistics].
Conditional dependence in a GGM {#cdep}
-------------------------------
In a GGM the set of edges $E$ represents the conditional dependence structure of the vector $(X_1,\ldots, X_p)$. To represent this dependence structure as a statistical model it is convenient to find a parametrization for $E$.
In this subsection we introduce a convenient parametrization of $E$ using well known results from classical multivariate analysis. For an exhaustive treatment of these results see, for instance, @andersonmult, @cramer, @lauritzen1996graphical and @eaton2007.
Given a subset $\mathcal{A}$ of $V$, $\mathbf{X}_{\mathcal{A}}$ denotes the vector of variables with subscripts in $\mathcal{A}$ in increasing order. For a given pair of nodes $(i,l)$, set $\mathbf{X}_{1}^{\top}=\left( X_{i},X_{l}\right) $, $\mathbf{X}_{2}^{\top}=\mathbf{X}_{V\backslash\left\{ i,l\right\} } $ and $ \mathbf{X}=\left(\mathbf{X}_{1}^{\top},\mathbf{X}_{2}^{\top} \right)^{\top}$. Note that $ \mathbf{X}$ has multivariate normal distribution with mean $ \mathbf{0}$ and covariance matrix $$\label{matpart}
\begin{pmatrix}
\Sigma_{11} & \Sigma_{12}\\
\Sigma_{21} & \Sigma_{22}
\end{pmatrix}$$ such that $\Sigma_{11}$ has dimension $2\times 2$, $\Sigma_{12}$ has dimension $2\times (p-2)$ and so on. The matrix in is a partition of a permutation of the original covariance matrix $\Sigma$, and will be also denoted by $\Sigma$, after a small abuse of notation.
Moreover, we set $$\begin{aligned}
\Omega =
\begin{pmatrix}
\Sigma_{11} & \Sigma_{12}\\
\Sigma_{21} & \Sigma_{22}
\end{pmatrix}^{-1} =
\begin{pmatrix}
\Omega_{11} & \Omega_{12}\\
\Omega_{21} & \Omega_{22}
\end{pmatrix}.\end{aligned}$$ Then, by (B.2) of @lauritzen1996graphical, the blocks $\Omega_{i,j}$ can be written explicitly in terms of $\Sigma_{i,j}$ and $\Sigma_{i,j}^{-1}$. In particular
$\Omega_{11} =\left( \Sigma_{11}-\Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21}\right) ^{-1}$ where $$\Omega_{11}=
\begin{pmatrix}
\omega_{ii} & \omega_{il}\\
\omega_{li} & \omega_{ll}
\end{pmatrix}$$ is the submatrix of $\Omega$ (with rows $i$ and $l$ and columns $i$ and $l$). Hence, $$\begin{aligned}
\label{alfin}
{\mbox{\sc cov}}\left( \mathbf{X}_{1}|\mathbf{X}_{2}\right) &=& \Sigma_{11}-\Sigma
_{12}\Sigma_{22}^{-1}\Sigma_{21}\\ \nonumber
&=& \Omega_{11}^{-1} \\ \nonumber
&=& \frac{1}{ \omega_{ii} \omega_{ll}- \omega_{il} \omega_{li} }
\begin{pmatrix}
\omega_{ll} & -\omega_{il}\\
- \omega_{li} & \omega_{ii}
\end{pmatrix}\end{aligned}$$ and, in consequence, the partial correlation between $X_i$ and $X_l$ can be expressed as $$\label{eq2-3}
{\mbox{\sc corr}}\left( X_i,X_l |\mathbf{X}_{V\backslash\left\{ i,l\right\} } \right) = - \frac{\omega_{il}}{ \sqrt{\omega_{ii}\omega_{ll} }}.$$ This gives the standard parametrization of $E$ in terms of the support of the precision matrix $$\begin{aligned}
\label{suppomega}
{\mbox{\sc supp}}\left( \Omega \right)= \{(i,l) \in V^2:\, i\neq l, \omega_{i,l} \neq 0 \}.\end{aligned}$$
We now introduce another parametrization of $E$, which we need to define and implement our proposed method. We consider the regression error for the regression of $\mathbf{X}_{1}$ on $\mathbf{X}_{2}$, $$\displaystyle \boldsymbol{\varepsilon} = \mathbf{X}_{1} - \widehat{\mathbf{X}}_{1}=\mathbf{X}_{1} - \boldsymbol{\beta}^{\top} \mathbf{X}_{2}$$ and let $\varepsilon_i$ and $\varepsilon_l$ denote the entries of $\boldsymbol{\varepsilon}$ (i.e. $\boldsymbol{\varepsilon}^{\top}=(\varepsilon_i, \varepsilon_l)$). The regression error $\boldsymbol{\varepsilon}$ is independent of $ \widehat{\mathbf{X}}_{1}$ and has normal distribution with mean $\boldsymbol{0}$ and covariance matrix $\Psi_{11} $ with elements denoted by $$\begin{aligned}
\Psi_{11} & =
\begin{pmatrix}
\psi_{ii} & \psi_{il}\\
\psi_{li} & \psi_{ll}
\end{pmatrix}.\end{aligned}$$ A straightforward calculation shows that
$$\begin{aligned}
\Psi_{11} & = {\mbox{\sc cov}}\left( \mathbf{X}_{1}\right) +{\mbox{\sc cov}}\left( \widehat{\mathbf{X}}
_{1}\right) -2 {\mbox{\sc cov}}\left( \mathbf{X}_{1},\widehat{\mathbf{X}}_{1}\right)
\nonumber\\
& \nonumber\\
& =\Sigma_{11}+\Sigma_{12}\Sigma_{22}^{-1}\Sigma_{22}\Sigma_{22}^{-1}
\Sigma_{21}-2\Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21}\nonumber\\
& \nonumber\\
& =\Sigma_{11}-\Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21}=\Omega_{11}^{-1}.
\label{eq4}\end{aligned}$$
See @cramer [ Section 23.4].
Therefore, by this equality, and , the partial correlation coefficient and the conditional correlation are equal $$\begin{aligned}
\rho_{il\cdot V\backslash\left\{ i,l\right\} } =
{\mbox{\sc corr}}\left( X_i,X_l | \mathbf{X}_{V\backslash\left\{ i,l\right\} } \right) \label{eq2-3bis}
= \frac{\psi_{il}}{\sqrt{\psi_{ii}\psi_{ll}}}. \nonumber\end{aligned}$$ Summarizing, the problem of determining the conditional dependence structure in a GGM (represented by $E$) is equivalent to finding the pairs of nodes of $V$ that belong to the set $$\begin{aligned}
\label{supppartial}
\{(i,l) \in V^2:\, i\neq l, \psi_{i,l} \neq 0 \}\end{aligned}$$ which is equal to the support of the precision matrix, ${\mbox{\sc supp}}\left( \Omega \right)$, defined by .
As noticed above, under normality, partial and conditional correlation are the same. However, in general they are different concepts [@lawrance].
\[Rem2\] Let $ \beta_{i,l}$ be the regression coefficient of $X_l$ in the regression of $X_i$ versus $\mathbf{X}_{V\backslash\left\{ i\right\} }$ and, similarly let $\beta_{l,i}$ be the regression coefficient of $X_i$ in the regression of $X_l$ versus $\mathbf{X}_{V\backslash\left\{ i\right\} }$. Then it follows that $\displaystyle \rho_{il\cdot V\backslash\left\{ i,l\right\} }= \text{sign} \left( \beta_{l,i} \right) \sqrt{ \beta_{l,i}\beta_{i,l} }$. This allows for another popular parametrization for $E$. Moreover, let $\epsilon_i$ be the error term in the regression of the $i^{\text{th}}$ variable on the remaining ones. Then by Lemma 1 in [@peng2009partial] we have that ${\mbox{\sc cov}}( \epsilon_i,\epsilon_l)=\omega_{il}/\omega_{ii}\omega_{ll}$ and ${\mbox{\sc var}}( \epsilon_i)=1/\omega_{ii}$.
The Stepwise Algorithm
-----------------------
Conditionally on its neighbors, $X_i$ is independent of all the other variables. Formally, for all $i$, $$\label{pairneigh}
\text{if } l \notin {\cal A}_i\ \text{and } l\neq i \text{ then } X_{i} {
\mathrel{\text{
{\rotatebox[origin=c]{90}{\resizebox{2.25ex}{1.65ex}{$\vDash$}}}
}}
}X_{l} | \mathbf {X}_{{ \mathcal A_i}}.$$ Therefore, given a system of neighborhoods $\left\{
\mathcal{A}_{i}\right\} _{i=1}^{p}$ and $l\notin\mathcal{A}_{i}$ (and so $i\notin\mathcal{A}_{l}$), the partial correlation between $X_{i}$ and $X_{l}$ can be obtained by the following procedure: (i) regress $X_{i}$ on $\mathbf{X}_{\mathcal{A}_{i}}$ and compute the regression residual $\varepsilon_{i}$; regress $X_{l}$ on $\mathbf{X}_{\mathcal{A}_{l}}$ and compute the regression residual $\varepsilon_{l}$; (ii) calculate the Pearson correlation between $\varepsilon_{i}$ and $\varepsilon
_{l}.$
This reasoning motivates the graphical stepwise algorithm (GSA). It begins with the family of empty neighborhoods, $\hat{\mathcal{A}}_j^{(0)} = \emptyset$ for each $j \in V$. There are two basic steps, the forward and the backward steps. In the forward step, the algorithm adds a new edge $(j_{0},l_{0})$ if the largest absolute empirical partial correlation between the variables $X_{j_{0}}, X_{l_{0}}$ is above the given threshold $\alpha_{f}$. In the backward step the algorithm deletes an edge $(j_{0},l_{0})$ if the smallestt absolute empirical partial correlation between the variables $X_{j_{0}}, X_{l_{0}}$ is below the given threshold $\alpha_{b}$. A step by step description of GSA is as follows:
**Graphical Stepwise Algorithm**
- **Input:** the (centered) data $\left\{ \mathbf{x}_{1},...,\mathbf{x}_{n}\right\}
, $ and the forward and backward thresholds $\alpha_{f}$ and $\alpha_{b}.$
- **Initialization**. $k=0$: set $\widehat{\mathcal{A}}_{1}^{0}=\widehat{\mathcal{A}}_{2}
^{0}=\cdots=\widehat{\mathcal{A}}_{p}^{0}=\phi$.
- **Iteration Step.** Given $\widehat{\mathcal{A}}_{1}^{k},\widehat{\mathcal{A}}_{2}^{k},...,\widehat{\mathcal{A}}_{p}^{k}$ we compute $\widehat{\mathcal{A}}_{1}^{k+1},\mathcal{A}_{2}^{k+1},...,\widehat{\mathcal{A}}_{p}
^{k+1}$ as follows.
1. **Forward.** For each $j=1,...,p$ do the following.
For each $l\notin \widehat{\mathcal{A}}_{j}^{k}$ calculate the partial correlations $f_{jl}^{k}$ as follows.
1. Regress the $j^{th}$ variable on the variables with subscript in the set $\widehat{\mathcal{A}}_{j}^{k}$ and compute the regression residuals $\mathbf{e}_{j}^{k}=\left( e_{1j}^{k},e_{2j}^{k},...,e_{nj}^{k}\right).$
2. Regress the $l^{th}$ variables on the variables with subscript in the set $\widehat{\mathcal{A}}_{l}^{k}$ and compute the regression residuals $\mathbf{e}_{l}^{k}=\left( e_{1l}^{k},e_{2l}^{k},...,e_{nl}^{k}\right).$
3. Obtain the partial correlation $f_{jl}^{k}$ by calculating the Pearson correlation between $\mathbf{e}_{j}
^{k}$ and $\mathbf{e}_{l}^{k}.$
If $$\max_{l\notin\widehat{\mathcal{A}}_{j}^{k},j\in V}\left\vert f_{jl}^{k}\right\vert
=\left\vert f_{j_{0}l_{0}}^{k}\right\vert \geq\alpha_{f}$$ set $\widehat{\mathcal{A}}_{j_{0}}^{k+1}=\widehat{\mathcal{A}}_{j_{0}}^{k}\cup\left\{
l_{0}\right\} ,$ $\widehat{\mathcal{A}}_{l_{0}}^{k+1}=\widehat{\mathcal{A}}_{l_{0}}^{k}
\cup\left\{ j_{0}\right\} ,$ $\widehat{\mathcal{A}}_{l}^{k+1}=\widehat{\mathcal{A}}_{l}^{k}$ for $l\neq j_{0},l_{0}$ If $$\max\left\vert f_{jl}^{k}\right\vert =\left\vert f_{j_{0}l_{0}}^{k}\right\vert
<\alpha_{f}, \text{ stop}.$$
2. **Backward.** For each $j=1,...,p$ do the following.
For each $l\in\widehat{\mathcal{A}}_{j}^{k+1}$ calculate the partial correlation $b_{jl}^{k}$ as follows.
1. Regress the $j^{th}$ variables on the variables with subscript in the set $\widehat{\mathcal{A}}_{j}^{k+1}\backslash\left\{ l\right\} $ and compute the regression residuals $\mathbf{r}_{j}^{k}=\left( r_{1j}^{k},r_{2j}
^{k},...,r_{nj}^{k}\right) .$
2. Regress the $l^{th}$ variable on the variables with subscript in the set $\widehat{\mathcal{A}}_{l}^{k+1}\backslash\left\{ j\right\} $ and compute the regression residuals $\mathbf{r}_{l}^{k}=\left( r_{1l}^{k},r_{2l}
^{k},...,r_{nl}^{k}\right) .$
3. Compute the partial correlation $b_{jl}^{k}$ by calculating the Pearson correlation between $\mathbf{r}_{j}
^{k}$ and $\mathbf{r}_{l}^{k}.$
If $$\min_{l\in\widehat{\mathcal{A}}_{j}^{k},j\in V}\left\vert b_{jl}^{k}\right\vert
=\left\vert b_{j_{0}l_{0}}^{k}\right\vert \leq\alpha_{b}$$ set $\widehat{\mathcal{A}}_{j_{0}}^{k+1}\rightarrow\widehat{\mathcal{A}}_{j_{0}}^{k+1}
\backslash\left\{ l_{0}\right\} ,$ $\widehat{\mathcal{A}}_{l_{0}}^{k+1}\rightarrow
\widehat{\mathcal{A}}_{l_{0}}^{k+1}\backslash\left\{ j_{0}\right\}$.
- [**Output**]{}
1. A collection of estimated neighborhoods $\widehat{\mathcal{A}}_{j}$, $j=1,\ldots,p$.
2. The set of estimated edges $\widehat{E}=\left\{(i,l)\in V^2: i\in \widehat{\mathcal{A}}_{l} \right\}$.
3. An estimate of $\boldsymbol{\Omega }$, $\widehat{\boldsymbol{\Omega }}=\left( \widehat{\omega }
_{il}\right) _{i,l=1}^{p}$ with $\widehat{\omega }_{il}$ defined as follow: in the case $i=l,$ $\widehat{\omega }_{ii}=n/(\boldsymbol{e}_{i}^{T}\boldsymbol{e}_{i})$ for $i=1,...,p,$ where $\boldsymbol{e}_{i}$ is the vector of the prediction errors in the regression of the $i^{\text{th}}$ variable on $\boldsymbol{X}_{\widehat{\mathcal{A}}_{i}}.$ In the case $i\neq
l$ we must distinguish two cases, if $l\notin \widehat{\mathcal{A}}_{i}$ then $\widehat{\omega }_{il}=0,$ otherwise $\widehat{\omega }_{il}=n\left(
\boldsymbol{e}_{i}^{T}\boldsymbol{e}_{l}\right) /\left[ \left( \boldsymbol{e}_{i}^{T}\boldsymbol{e}_{i}\right) \left( \boldsymbol{e}_{l}^{T}\boldsymbol{e}_{l}\right) \right]$ (see Remark \[Rem2\]).
Thresholds selection by cross-validation {#cross}
-----------------------------------------
Let $\boldsymbol{X}$ be the $n\times p$ matrix with rows $\mathbf{x}_{i}=\left(x_{i1},\ldots,x_{ip} \right)$, $i=1,\ldots,n$, corresponding to $n$ observations. We randomly partition the dataset $\{\mathbf{x}_{i}\}_{1\leq i\leq n}$ into $K$ disjoint subsets of approximately equal sizes, the $t^{th}$ subset being of size $n_{t}\geq 2$ and $\displaystyle \sum_{t=1}^{K}n_{t}=n$. For every $t$, let $\displaystyle \{ \mathbf{x}_{i}^{(t)}\}_{1\leq i\leq n_{t}}$ be the $t^{th}$ *validation subset*, and its complement $\displaystyle \{ \widetilde{\mathbf{x}}_{i}^{(t)} \}_{1\leq i\leq n-n_{t}}$, the $t^{th}$ *training subset*. For every $t$ and for every pair $(\alpha_f, \alpha_b)$ of threshold parameters let $ \widehat{\mathcal{A}}_1^{(t)}, \ldots, \widehat{\mathcal{A}}_{p}^{( t)} $ be the estimated neighborhoods given by GSA using the $t^{th}$ training subset. For every $j=1,\ldots,p$ let $ \widehat{\beta}_{ \widehat{\mathcal{A}}_j^{(t)}}$ be the estimated coefficient of the regression of the variable $X_j$ on the neighborhood $\widehat{ \mathcal{A} }^{(t)}_j $.
Consider now the $t^{th}$ validation subset. So, for every $j$, using $\widehat{\beta}_{\mathcal{A}_j^{(t)}}^{(t)}$, we obtain the vector of predicted values $\widehat{\mathbf{X}}_j^{(t)} \left( \alpha_f, \alpha_b \right)$. If $\mathcal{A}_j^{(t)}=\emptyset$ we predict each observation of $X_j$ by the sample mean of the observations in the $t^{th}$ dataset of this variable.
Then, we define the $K$–fold cross–validation function as $$CV\left( \alpha_f, \alpha_b \right)=\frac{1}{n}\sum_{t=1}^{K}\sum_{j=1}^{p_j} \left\| \mathbf{X}_j^{(t)} - \widehat{\mathbf{X}}_j^{(t)}\left( \alpha_f, \alpha_b \right) \right\|^2$$ where $\left\|\cdot \right\|$ the L2-norm or euclidean distance in $\mathbb{R}^p$. Hence the $K$–fold cross–validation forward–backward thresholds $\widehat{\alpha}_f$, $\widehat{ \alpha}_b$ is $$\left( \widehat{\alpha}_f, \widehat{ \alpha}_b \right)=:\mathop{\rm argmin}_{\left( \alpha_f, \alpha_b \right)\in \mathcal{H}}CV\left( \alpha_f, \alpha_b \right)$$ where $\mathcal{H}$ is a grid of ordered pairs $\left( \alpha_f, \alpha_b \right)$ in $[0,1]\times [0,1]$ over which we perform the search. For a detail description see the Appendix.
Example {#algor_example}
-------
To illustrate the algorithm we consider the GGM with 16 edges given in the first panel of Figure \[GGM\_Gstepwise\_Example\]. We draw $n=1000$ independent observations from this model (see the next section for details). The values for the threshold parameters $\alpha_f=0.17$ and $\alpha_b=0.09$ are determined by $5$-fold cross-validation. The figure also displays the selected pairs of edges at each step in a sequence of successive updates of $\widehat{\mathcal{A}}_{j}^{k}$, for $k=1,4, 9, 12$ and the final step $k=16$, showing that the estimated graph is identical to the true graph.
\[H\]
[3]{} ![ True graph and sequence of successive updates of $\widehat{\mathcal{A}}_{j}^{k}$, for $k=1,4, 9, 12, 16$ of the GSA.[]{data-label="GGM_Gstepwise_Example"}](Model_NN_p20_n1000_Graph "fig:"){width="\linewidth"}
![ True graph and sequence of successive updates of $\widehat{\mathcal{A}}_{j}^{k}$, for $k=1,4, 9, 12, 16$ of the GSA.[]{data-label="GGM_Gstepwise_Example"}](GStepwise_NN_p20_n1000_Graph_Step_1){width="\linewidth"}
![ True graph and sequence of successive updates of $\widehat{\mathcal{A}}_{j}^{k}$, for $k=1,4, 9, 12, 16$ of the GSA.[]{data-label="GGM_Gstepwise_Example"}](GStepwise_NN_p20_n1000_Graph_Step_4){width="\linewidth"}
[3]{} ![ True graph and sequence of successive updates of $\widehat{\mathcal{A}}_{j}^{k}$, for $k=1,4, 9, 12, 16$ of the GSA.[]{data-label="GGM_Gstepwise_Example"}](GStepwise_NN_p20_n1000_Graph_Step_9 "fig:"){width="\linewidth"}
![ True graph and sequence of successive updates of $\widehat{\mathcal{A}}_{j}^{k}$, for $k=1,4, 9, 12, 16$ of the GSA.[]{data-label="GGM_Gstepwise_Example"}](GStepwise_NN_p20_n1000_Graph_Step_12){width="\linewidth"}
![ True graph and sequence of successive updates of $\widehat{\mathcal{A}}_{j}^{k}$, for $k=1,4, 9, 12, 16$ of the GSA.[]{data-label="GGM_Gstepwise_Example"}](GStepwise_NN_p20_n1000_Graph_Step_16){width="\linewidth"}
Numerical results and real data example {#section_3}
========================================
We conducted extensive Monte Carlo simulations to investigate the performance of GS. In this section we report some results from this study and a numerical experiment using real data.
Monte Carlo simulation study {#section_3.1}
----------------------------
**Simulated Models**
We consider three dimension values $p = 50, 100, 150$ and three different models for $\boldsymbol{\Omega}$:
- [**Model 1.**]{} Autoregressive model of orden $1$, denoted $\text{AR}(1)$. In this case $\Sigma_{ij}=0.4^{|i-j|}$ for $i,j=1,\ldots p$.
- [**Model 2.**]{} Nearest neighbors model of order 2, denoted $\text{NN}(2)$. For each node we randomly select two neighbors and choose a pair of symmetric entries of $\boldsymbol{\Omega}$ using the NeighborOmega function of the R package Tlasso.
- [**Model 3.**]{} Block diagonal matrix model with $q$ blocks of size $p/q$, denoted BG. For $p=50,100$ and $150$, we use $q=10, 20$ and $30$ blocks, respectively. Each block, of size $p/q=5$, has diagonal elements equal to $1$ and off-diagonal elements equal to $0.5$.
For each $p$ and each model we generate $R=50$ random samples of size $n=100$. These graph models are widely used in the genetic literature to model gene expression data. See for example [@Lee] and [@lighi]. Figure \[adjmatrices\] displays graphs from Models 1-3 with $p=100$ nodes.
\[H\]
[3]{} ![ Graphs of $\text{AR}(1)$, $\text{NN}(2)$ and BG graphical models for $p=100$ nodes.[]{data-label="adjmatrices"}](adjAR1_p100 "fig:"){width="\linewidth"}
![ Graphs of $\text{AR}(1)$, $\text{NN}(2)$ and BG graphical models for $p=100$ nodes.[]{data-label="adjmatrices"}](adjNN2_p100){width="\linewidth"}
![ Graphs of $\text{AR}(1)$, $\text{NN}(2)$ and BG graphical models for $p=100$ nodes.[]{data-label="adjmatrices"}](adjBG_p100){width="\linewidth"}
**Methods**
We compare the performance of GS with Graphical lasso (Glasso) and Constrained $l_1$-minimization for inverse matrix estimation (CLIME) proposed by [@friedman2008sparse] and [@cai2011constrained] respectively. Therefore, the methods compared in our simulation study are:
- The proposed method GS with the forward and backward thresholds, $\left( {\alpha}_f, { \alpha}_b \right)$, estimated by $5$-fold crossvalidation on a grid of $20 $ values in $[0,1]\times[0,1] $, as described in Subsection \[cross\]. The computing algorithm is available by request.
- The Glasso estimate obtained by solving the $\ell_1$ penalized-likelihood problem: $$\label{Glasso}
\min_{\boldsymbol{\Omega} \succ 0} \ \left( -\text{log}\{ \text{det}[\boldsymbol{\Omega}]\} + \text{tr}\{\boldsymbol{\Omega}\textbf{X}^{\top}\textbf{X}\} + \lambda \parallel \boldsymbol{\Omega} \parallel_1 \right).$$ In our simulations and examples we use the R-package <span style="font-variant:small-caps;">CVglasso</span> with the tuning parameter $\lambda$ selected by $5-$fold crossvalidation (the package default).
- The CLIME estimate obtained by symmetrization of the solution of $$\label{Clime}
\min \{ \parallel \boldsymbol{\Omega} \parallel_1 \text{ subject to } \left| S \boldsymbol{\Omega} -\boldsymbol{I}\right|_{\infty} \leq \lambda \},$$ where $S$ is the sample covariance, $\boldsymbol{I}$ is the identity matrix, $\left| \cdot \right|_{\infty}$ is the elementwise $l_{\infty}$ norm, and $\lambda$ is a tuning parameter. For computations, we use the R-package <span style="font-variant:small-caps;">clime</span> with the tuning parameter $\lambda$ selected by $5-$fold crossvalidation (the package default).
To evaluate the ability of the methods for finding the pairs of edges, for each replicate, we compute the Matthews correlation coefficient [@matt] $$\text{MCC}= \frac{\text{TP} \times \text{TN} - \text{FP} \times \text{FN}}{\sqrt{(\text{TP} + \text{FP})(\text{TP} + \text{FN})(\text{TN} + \text{FP})(\text{TN} + \text{FN})}},$$ the $\text{Specificity} = \text{TN}/(\text{TN}+\text{FP})$ and the $\text{Sensitivity} = \text{TP}/(\text{TP}+\text{FN})$, where $\text{TP}$, $\text{TN}$, $\text{FP}$ and $\text{FN}$ are, in this order, the number of true positives, true negatives, false positives and false negatives, regarding the identification of the nonzero off-diagonal elements of $\boldsymbol{\Omega}$. Larger values of MCC, Sensitivity and Specificity indicate a better performance [@fan2009network; @baldi2000].
For every replicate, the performance of $\widehat{\boldsymbol{\Omega}}$ as an estimate for ${\boldsymbol{\Omega}}$ is measured by $m_F=|| \widehat{\boldsymbol{\Omega}}-\boldsymbol{\Omega} ||_{F}$ (where $|| \cdot||_{F}$ denotes the Frobenius norm) and by the normalized Kullback-Leibler divergence defined by $m_{NKL}=D_{KL}/(1+D_{KL})$ where $$\begin{aligned}
D_{KL} =\frac12 \left(\text{tr} \left\{\widehat{\boldsymbol{\Omega}}\boldsymbol{\Omega}^{-1}\right\} - \text{log}\left\{\text{det}\left[\widehat{\boldsymbol{\Omega}}\boldsymbol{\Omega}^{-1}\right]\right\}-p\right)\end{aligned}$$ is the the Kullback-Leibler divergence between $\widehat{\boldsymbol{\Omega}}$ and $\boldsymbol{\Omega}$.
**Results**
Table \[clasperf\] shows the MCC performance for the three methods under Models 1-3. GS clearly outperforms the other two methods while CLIME just slightly outperforms Glasso. [@cai2011constrained] pointed out that a procedure yielding a more sparse $\widehat{\boldsymbol{\Omega}}$ is preferable because this facilitates interpretation of the data. The sensitivity and specificity results, reported in Table \[clasperfapp\] in Appendix, show that in general GS is more sparse than the CLIME and Glasso, yielding fewer false positives (more specificity) but a few more false negatives (less sensitivity). Table \[Numerical\_Performance\] shows that under models $\text{AR}(1)$ and $\text{NN}(2)$ the three methods achieve fairly similar performances for estimating $\Omega$. However, under model BG, GS clearly outperforms the other two.
Figure \[Heatmaps\_1\] display the heat-maps of the number of non-zero links identified in the $50$ replications under model $\text{AR}(1)$. Notice that among the three compared methods, the GS sparsity patterns best match those of the true model. Figures \[Heatmaps\_2\] and \[Heatmaps\_3\] in the Appendix lead to similar conclusions for models $\text{NN}(2)$ and BG.
Model $p$
---------------- ----- --------------- --------------- ---------------
50 0.741 (0.009) 0.419 (0.016) 0.492 (0.006)
$\text{AR}(1)$ 100 0.751 (0.004) 0.433 (0.020) 0.464 (0.004)
150 0.730 (0.004) 0.474 (0.017) 0.499 (0.003)
50 0.751 (0.004) 0.404 (0.014) 0.401 (0.007)
$\text{NN}(2)$ 100 0.802 (0.005) 0.382 (0.006) 0.407 (0.005)
150 0.695 (0.007) 0.337 (0.008) 0.425 (0.003)
50 0.898 (0.005) 0.356 (0.009) 0.482 (0.005)
BG 100 0.857 (0.005) 0.348 (0.004) 0.461 (0.002)
150 0.780 (0.008) 0.314 (0.003) 0.408 (0.003)
: Comparison of means and standard deviations (in brackets) of MCC over $R=50$ replicates.[]{data-label="clasperf"}
\
\
---------------- ----- -------- -------- -------- --------- --------- ---------
Model $p$
50 0.70 3.82 0.64 3.90 0.63 3.91
(0.00) (0.00) (0.00) ( 0.02) (0.00) (0.01)
$\text{AR}(1)$ 100 0.83 5.73 0.80 5.72 0.79 5.75
(0.00) (0.00) (0.00) (0.02) (0.00) (0.01)
150 1.25 7.16 1.17 7.21 1.17 7.25
(0.00) (0.00) (0.00) (0.02) (0.00) (0.01)
50 0.99 6.98 0.99 6.65 0.99 6.64
(0.00) (0.00) (0.00) (0.01) (0.00) (0.00)
$\text{NN}(2)$ 100 0.10 10.11 1.00 9.64 1.00 9.601
(0.00) (0.00) (0.00) (0.009) (0.000) (0.005)
150 1.00 12.37 1.00 11.90 1.00 11.79
(0.00) (0.00) (0.00) (0.01) (0.00) (0.00)
BG 50 0.46 1.44 0.85 5.45 0.82 5.03
(0.00) (0.00) (0.00) (0.10) (0.00) (0.05)
100 0.71 2.94 0.93 9.16 0.92 8.71
(0.00) (0.00) (0.00) (0.07) (0.00) (0.02)
150 0.88 6.10 0.96 11.59 0.96 11.42
(0.00) (0.00) (0.00) (0.06) (0.00) (0.02)
---------------- ----- -------- -------- -------- --------- --------- ---------
: Comparison of means and standard deviations (in brackets) of $\text{m}_F$ and $\text{m}_{NKL}$ over $R=50$ replicates.[]{data-label="Numerical_Performance"}
Analysis of Breast Cancer Data {#subsection_5_2}
------------------------------
In preoperative chemoterapy, the complete eradication of all invasive cancer cells is referred to as [*pathological complete response*]{}, abbreviated as pCR. It is known in medicine that pCR is associated with the long-term cancer-free survival of a patient. Gene expression profiling (GEP) – the measurement of the activity (expression level) of genes in a patient – could in principle be a useful predictor for the patient’s pCR.
Using normalized gene expression data of patients in stages I-III of breast cancer, [@hess2006pharmacogenomic] aim to identify patients that may achieve pCR under [*sequential anthracycline paclitaxel*]{} preoperative chemotherapy. When a patient does not achieve pCR state, he is classified in the group of residual disease (RD), indicating that cancer still remains. Their data consist of 22283 gene expression levels for 133 patients, with 34 pCR and 99 RD. Following [@fan2009network] and [@cai2011constrained] we randomly split the data into a training set and a testing set. The testing set is formed by randomly selecting 5 pCR patients and 16 RD patients (roughly $1/6$ of the subjects) and the remaining patients form the training set. From the training set, a two sample t-test is performed to select the 50 most significant genes. The data is then standardized using the standard deviation estimated from the training set.
We apply a linear discriminant analysis (LDA) to predict whether a patient may achieve pathological complete response (pCR), based on the estimated inverse covariance matrix of the gene expression levels. We label with $r=1$ the pCR group and $r=2$ the RD group and assume that data are normally distributed, with common covariance matrix $\boldsymbol{\Sigma}$ and different means $\boldsymbol{\mu}_r$. From the training set, we obtain $\widehat{\boldsymbol{\mu}}_r$, $\widehat{\boldsymbol{\Omega}}$ and for the test data compute the linear discriminant score as follows $$\label{LDA-score}
\delta_r(\textbf{x}) = \textbf{x}^{\top}\widehat{\boldsymbol{\Omega}} \widehat{\boldsymbol{\mu}}_r - \frac{1}{2} \boldsymbol{\mu}_r^{\top}\widehat{\boldsymbol{\Omega}} \boldsymbol{\mu}_r + \text{log} \widehat{\pi}_r \quad \text{for}\; i=1,\ldots,n,$$ where $\widehat{\pi}_r$ is the proportion of group $r$ subjects in the training set. The classification rule is $$\label{LD-rule}
\widehat{r}(\textbf{x}) = \operatorname*{argmax}\delta_r (\textbf{x}) \;\;\; \text{for} \; r=1,2.$$ For every method we use 5-fold cross validation on the training data to select the tuning constants. We repeat this scheme 100 times.
Table \[Breast\_Cancer\_Table\] displays the means and standard errors (in brackets) of Sensitivity, Specificity, MCC and Number of selected Edges using $\widehat{\boldsymbol{\Omega}}$ over the 100 replications. Considering the MCC, GS is slightly better than CLIME and CLIME than Glasso. While the three methods give similar performance considering the Specificity, GS and CLIME improve over Glasso in terms of Sensitivity.
----------------- -------------- -------------- -------------- --
Sensitivity 0.798 (0.02) 0.786 (0.02) 0.602 (0.02)
Specificity 0.784 (0.01) 0.788 (0.01) 0.767 (0.01)
MCC 0.520 (0.02) 0.516 (0.02) 0.334 (0.02)
Number of Edges 54 (2) 4823 (8) 2103 (76)
----------------- -------------- -------------- -------------- --
: Comparison of means and standard deviations (in brackets) of Sensitivity, Specificity, MCC and Number of selected edges over 100 replications.[]{data-label="Breast_Cancer_Table"}
Concluding remarks {#section_4}
==================
This paper introduces a stepwise procedure, called GS, to perform covariance selection in high dimensional Gaussian graphical models. Our method uses a different parametrization of the Gaussian graphical model based on Pearson correlations between the best-linear-predictors prediction errors. The GS algorithm begins with a family of empty neighborhoods and using basic steps, forward and backward, adds or delete edges until appropriate thresholds for each step are reached. These thresholds are automatically determined by cross–validation.
GS is compared with Glasso and CLIME under different Gaussian graphical models ($\text{AR}(1)$, $\text{NN}(2)$ and BG) and using different performance measures regarding network recovery and sparse estimation of the precision matrix $\Omega$. GS is shown to have good support recovery performance and to produce simpler models than the other two methods (i.e. GS is a parsimonious estimation procedure).
We use GS for the analysis of breast cancer data and show that this method may be a useful tool for applications in medicine and other fields.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thanks the generous support of NSERC, Canada, the Institute of Financial Big Data, University Carlos III of Madrid and the CSIC, Spain.
Appendix
==========
Selection of the thresholds parameters by cross-validation
----------------------------------------------------------
Let $\boldsymbol{X}$ be the $n\times p$ matrix with rows $\mathbf{x}_{i}=\left(x_{i1},\ldots,x_{ip} \right)$, $i=1,\ldots,n$, corresponding to $n$ observations. For each $j=1,\ldots, p$, let $\mathbf{X}_j=\left( x_{1j}, \ldots, x_{nj} \right)^{\top}$ denote the jth–column of the matrix $\mathbf{X}$.
We randomly partition the dataset $\{\mathbf{x}_{i}\}_{1\leq i\leq n}$ into $K$ disjoint subsets of approximately equal size, the $t^{th}$ subset being of size $n_{t}\geq 2$ and $\displaystyle \sum_{t=1}^{K}n_{t}=n$. For every $t$, let $\displaystyle \{ \mathbf{x}_{i}^{(t)}\}_{1\leq i\leq n_{t}}$ be the $t^{th}$ *validation subset*, and its complement $\displaystyle \{ \widetilde{\mathbf{x}}_{i}^{(t)} \}_{1\leq i\leq n-n_{t}}$, the $t^{th}$ *training subset*.
For every $t=1,\ldots, K$ and threshold parameters $(\alpha_f, \alpha_b)\in [0,1]\times [0,1]$ let $ \widehat{\mathcal{A}}_1^{(t)}, \ldots, \widehat{\mathcal{A}}_{p}^{(t)} $ be the estimated neighborhoods given by GSA using the $t^{th}$ training subset $\displaystyle \{ \widetilde{\mathbf{x}}_{i}^{(t)} \}_{1\leq i\leq n-n_{t}}$ with $ \widetilde{\mathbf{x}}_{i}^{(t)} = (\widetilde{x}_{i1}^{(t)},\ldots, \widetilde{x}_{ip}^{(t)}),$ $ 1\leq i \leq n-n_{t}$. Consider for every node $j$ the estimated neighborhood $\widehat{ \mathcal{A} }^{(t)}_j=\left\{l_1,\ldots, l_q\right\}$ and let $ \widehat{\beta}_{ \widehat{\mathcal{A}}_j^{(t)}}$ be the estimated coefficient of the regression of $\widetilde{\mathbf{X}}_j=(\widetilde{x}_{1j}^{(t)},\ldots, \widetilde{x}_{n-n_tj}^{(t)})^{\top}$ on $X_{l_1},\ldots, X_{l_q}$, represented in (red colour).
Consider the $t^{th}$ validation subset $\displaystyle \{ \mathbf{x}_{i}^{(t)}\}_{1\leq i\leq n_{t}}$ with $ \mathbf{x}_{i}^{(t)} =( x_{i1}^{(t)},\ldots, x_{ip}^{(t)})$, $ 1 \leq i \leq n_t $ and for every $j$ let $\mathbf{X}_j^{(t)}=\left( x_{1j}^{(t)}, \ldots, x_{n_{t}j}^{(t)} \right)^{\top}$ and define the vector of predicted values $$\begin{aligned}
\widehat{\mathbf{X}}_j^{(t)} \left( \alpha_f, \alpha_b \right) = \mathbf{X}_{ \widehat{\mathcal{A}}_j^{ (t) }} \widehat{\beta}_{ \mathcal{A}_j^{(t)}}^{(t)},\end{aligned}$$ where $\mathbf{X}_{ \widehat{\mathcal{A}}_j^{ (t) }}$ is the matrix with rows $( x_{il_1}^{(t)},\ldots, x_{il_q}^{(t)} )$, $ 1 \leq i \leq n_t $ represented in (in blue colour). If the neighborhood $\mathcal{A}_j^{(t)}=\emptyset$ we define $$\begin{aligned}
\widehat{\mathbf{X}}_j^{(t)}\left( \alpha_f, \alpha_b \right)=(\bar{x}_{j}^{(t)},\ldots, \bar{x}_{j}^{(t)})^{\top}\end{aligned}$$ where $\bar{x}_{j}^{(t)}$ is the mean of the sample of observations $x_{1j}^{(t)}, \ldots, x_{n_{t}j}^{(t)} $.
We define the $K$–fold cross–validation function as $$CV\left( \alpha_f, \alpha_b \right)=\frac{1}{n}\sum_{t=1}^{K}\sum_{j=1}^{p} \left\| \mathbf{X}_j^{(t)} - \widehat{\mathbf{X}}_j^{(t)}\left( \alpha_f, \alpha_b \right) \right\|^2$$ where $\left\|\cdot \right\|$ the L2-norm or euclidean distance in $\mathbb{R}^p$. Hence the $K$–fold cross–validation forward–backward thresholds $\widehat{\alpha}_f$, $\widehat{ \alpha}_b$ is $$\label{threscross}
\left( \widehat{\alpha}_f, \widehat{ \alpha}_b \right)=:\mathop{\rm argmin}_{\left( \alpha_f, \alpha_b \right)\in \mathcal{H}}CV\left( \alpha_f, \alpha_b \right)$$ where $\mathcal{H}$ is a grid of ordered pairs $\left( \alpha_f, \alpha_b \right)$ in $[0,1]\times [0,1]$ over which we perform the search.
$$\begin{aligned}
\label{matrixfig}
\left( \begin{array}{c c ccccccccc}
t^{th} & \text{training} & \text{subset } \\
\cdots & \textcolor[rgb]{1,0,0}{\widetilde{x}_{1j}^{(t)}} & \cdots & \textcolor[rgb]{1,0,0}{\widetilde{x}_{1l_1}^{(t)}} & \cdots & \textcolor[rgb]{1,0,0}{\widetilde{x}_{1l_q}^{(t)}} & \cdots \\
\vspace{0.5cm}
\vdots & \vdots & \vdots & \vdots & \vdots& \vdots & \vdots\\
\vspace{0.5cm}
\cdots & \textcolor[rgb]{1,0,0}{\widetilde{x}_{n-n_tj}^{(t)}} & \cdots & \textcolor[rgb]{1,0,0}{\widetilde{x}_{n-n_t l_1}^{(t)}} & \cdots & \textcolor[rgb]{1,0,0}{\widetilde{x}_{n-n_tl_q}^{(t)}} & \cdots \\
\midrule
t^{th} & \text{validation} & \text{subset } \\
\cdots & \textcolor[rgb]{0,0,1}{x_{1j}^{(t)}} & \cdots & \textcolor[rgb]{0,0,1}{x_{1l_1}^{(t)}} & \cdots & \textcolor[rgb]{0,0,1}{x_{1l_q}^{(t)}} & \cdots \\
\vdots & \vdots & \vdots & \vdots & \vdots& \vdots & \vdots\\
\cdots & \textcolor[rgb]{0,0,1}{x_{n_tj}^{(t)}} & \cdots & \textcolor[rgb]{0,0,1}{x_{n_t l_1}^{(t)}} & \cdots & \textcolor[rgb]{0,0,1}{x_{n_tl_q}^{(t)}} & \cdots \\
\end{array}\right)\end{aligned}$$
Matrix represents, for every node $j$ the comparison between estimated and predicted values for cross-validation. $\widehat{\beta}_{ \widehat{\mathcal{A}}_j^{(t)}}$ is computed using the observations $\widetilde{\mathbf{X}}_j=(\widetilde{x}_{1j}^{(t)},\ldots, \widetilde{x}_{n-n_tj}^{(t)})^{\top}$ and the matrix $\widetilde{ \mathbf{X}}_{ \widehat{\mathcal{A}}^{(t)}_j } $ with rows $(\widetilde{x}_{il_1}^{(t)},\ldots, \widetilde{x}_{il_q}^{(t)})$, $i=1,\ldots,n-n_t$ in the $t^{th}$ training subset (red colour). Based on the $t^{th}$ validation set $\widehat{\mathbf{X}}_j^{(t)}$ is computed using $\mathbf{X}_{ \widehat{\mathcal{A}}_j^{ (t) }}$ and compared with $\mathbf{X}_j$ (in blue color).
Complementary simulation results
----------------------------------
---------------- ----- --------- --------- ---------- --------- --------- ---------- ---------- --------- ---------
Model $p$
50 0.756 0.988 0.741 0.994 0.823 0.419 0.988 0.891 0.492
(0.015) (0.002) (0.009) (0.002) (0.012) (0.016) (0.002) (0.003) (0.006)
$\text{AR}(1)$ 100 0.632 0.999 0.751 0.989 0.897 0.433 0.983 0.934 0.464
(0.007) (0.000) (0.004) (0.002) (0.009) (0.020) (0.002) (0.001) (0.004)
150 0.607 0.999 0.730 0.981 0.943 0.474 0.972 0.964 0.499
(0.006) (0.000) (0.004) (0.002) (0.007) (0.017) (0.002) (0.001) (0.003)
50 0.632 0.999 0.751 0.971 0.864 0.404 0.984 0.875 0.401
(0.007) (0.000) (0.004 ) (0.004) (0.010) (0.014) (0.003) (0.004) (0.007)
$\text{NN}(2)$ 100 0.730 0.999 0.802 0.987 0.924 0.382 0.985 0.937 0.407
(0.008) (0.000) (0.005) (0.002) (0.004) (0.006) (0.002) (0.001) (0.005)
150 0.555 0.999 0.695 0.952 0.936 0.337 0.934 0.965 0.425
(0.017) (0.000) (0.007) (0.004) (0.002) (0.008) ( 0.003) (0.001) (0.003)
50 0.994 0.981 0.898 0.867 0.697 0.356 0.962 0.807 0.482
(0.002) (0.001) (0.005) (0.032) (0.021) (0.009) (0.004) (0.005) (0.005)
BG 100 0.949 0.989 0.857 0.569 0.908 0.348 0.818 0.920 0.4615
(0.007) (0.000) (0.005) (0.039) (0.011) ( 0.004) (0.005) (0.005) (0.002)
150 0.782 0.994 0.780 0.426 0.952 0.314 0.626 0.959 0.408
(0.021) (0.000) (0.008) (0.035) (0.006) (0.003) (0.006) (0.001) (0.003)
---------------- ----- --------- --------- ---------- --------- --------- ---------- ---------- --------- ---------
: Comparison of means and standard deviations (in brackets) of Specificity, Sensitivity and MCC over $R=50$ replicates. []{data-label="clasperfapp"}
\
\
\
\
\
\
|
---
abstract: 'We provide a purely quantum version of polar codes, achieving the coherent information of any quantum channel. Our scheme relies on a recursive channel combining and splitting construction, where a two-qubit gate randomly chosen from the Clifford group is used to combine two single-qubit channels. The inputs to the synthesized bad channels are frozen by preshared EPR pairs between the sender and the receiver, so our scheme is entanglement assisted. We further show that quantum polarization can be achieved by choosing the channel combining Clifford operator randomly, from a much smaller subset of only $9$ two-qubit Clifford gates. Subsequently, we show that a Pauli channel polarizes if and only if a specific classical channel over four symbol input set polarizes. We exploit this equivalence to prove fast polarization for Pauli channels, and to devise an efficient successive cancellation based decoding algorithm for such channels. Finally, we present a code construction based on chaining several quantum polar codes, which is shown to require a rate of preshared entanglement that vanishes asymptotically.'
author:
- |
to\
[${}^{1\text{a}}$ Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France]{}\
[${}^{1\text{b}}$ Université de Montréal, Québec, Canada]{}\
[${}^2$ Univ. Grenoble Alpes, Grenoble INP, LIG, F-38000 Grenoble, France]{}\
[${}^3$ Univ. Grenoble Alpes, CNRS, Grenoble INP, LIG, F-38000 Grenoble, France]{}\
[${}^4$ CEA-LETI, MINATEC, F-38054 Grenoble, France]{}
bibliography:
- 'big.bib'
title: 'Polarization of Quantum Channels using Clifford-based Channel Combining'
---
Introduction
============
Polar codes, proposed by Arikan [@arikan09], are the first explicit construction of a family of codes that provably achieve the channel capacity for any binary-input, symmetric, memoryless channel. His construction relies on a channel combining and splitting procedure, where a CNOT gate is used to combine two instances of the transmission channel. Applied recursively, this procedure allows synthesizing a set of so-called virtual channels from several instances of the transmission channel. When the code length goes to infinity, the synthesized channels tend to become either noiseless (good channels) or completely noisy (bad channels), a phenomenon which is known as “channel polarization”. Channel polarization can effectively be exploited by transmitting messages via the good channels, while freezing the inputs to the bad channels to values known to the both encoder and decoder. Polar codes have been generalized for the transmission of classical information over quantum channels in [@wg13-2], and for transmitting quantum information in [@rdr11; @wg13; @rw12]. It was shown in [@rdr11] that the recursive construction of polar codes using a CNOT polarizes in both amplitude and phase bases for Pauli and erasure channels, and [@rw12] extended this to general channels. Then, a CSS-like construction was used to generalize polar codes for transmitting quantum information. This construction requires a small number of EPR pairs to be shared between the sender and the receiver, in order to deal with virtual channels that are bad in both amplitude and phase bases, thus making the resulting code entanglement-assisted in the sense of [@hsieh-devetak-brun2]. This construction was further refined in [@srdr13], where preshared entanglement is completely suppressed at the cost of a more complicated multilevel coding scheme, in which polar coding is employed separately at each level. However, all of these quantum channel coding schemes essentially exploit classical polarization, in either amplitude or phase basis. In this paper, we give a purely quantum version of polar codes , [*i.e.*]{}, a family of polar codes where the good channels are good as quantum channels, and not merely in one basis. Our construction uses a two-qubit gate chosen randomly from the Clifford group to combine two single-qubit channels, which bears similarities to the randomized channel combining/splitting operation proposed in [@sta09], for the polarization of classical channels with input alphabet of arbitrary size. We show that the synthesized quantum channels tend to become either noiseless or completely noisy as quantum channels, and not merely in one basis. Similar to the classical case, information qubits are transmitted through good (almost noiseless) channels, while the inputs to the bad (noisy) channels are “frozen” by sharing EPR pairs between the sender and the receiver. We show that the proposed scheme achieves the coherent information of the quantum channel, for a uniform input distribution. Further, we show that polarization can be achieved while reducing the set of two-qubit Clifford gates, used to randomize the channel combining operation, to a subset of $9$ Clifford gates only. We also present an efficient decoding algorithm for the proposed quantum polar codes for the particular case of Pauli channels. To a Pauli channel, we associate a classical symmetric channel, with both input and output alphabets given by the quotient of the $1$-qubit Pauli group by its centralizer, and show that the former polarizes quantumly if and only if the latter polarizes classically. This equivalence provides an alternative proof of the quantum polarization for a Pauli channel and, more importantly, an effective way to decode the quantum polar code for such channels, by decoding its classical counterpart. Fast polarization properties [@sta09; @at09] are also proven for Pauli channels, by using techniques similar to those in [@sta09]. Finally, we present a code construction based on chaining several quantum polar codes, which is shown to require a rate of preshared entanglement that vanishes asymptotically.
Preliminaries
=============
Here are some basic definitions that we will need to prove the quantum polarization. First, we will need the conditional sandwiched Rényi entropy of order 2, as defined by Renner [@renner-phd]:
Let $\rho_{AB}$ be a quantum state. Then, $${\tilde{H}^{\downarrow}_2}(A|B)_{\rho} := -\log \operatorname{Tr}\left[ \rho_B^{-\frac{1}{2}} \rho_{AB} \rho_B^{-\frac{1}{2}} \rho_{AB} \right].$$ \[def:renyi-2-entropy\]
We will also need the conditional Petz Rényi entropy of order $\frac{1}{2}$:
Let $\rho_{AB}$ be a quantum state. Then, $${H^{\uparrow}_{\frac{1}{2}}}(A|B)_{\rho} := 2 \log \sup_{\sigma_B} \operatorname{Tr}\left[ \rho_{AB}^{\frac{1}{2}} \sigma^{\frac{1}{2}}_B \right].$$ \[def:renyi-half-entropy\]
As shown in [@tbh14 Theorem 2], those two quantities satisfy a duality relation: given a pure tripartite state $\rho_{ABC}$, ${\tilde{H}^{\downarrow}_2}(A|B)_{\rho} = -{H^{\uparrow}_{\frac{1}{2}}}(A|C)_{\rho}$.
We will also need the concept of the complementary channel:
Let $\mathcal{N}_{A' \rightarrow B}$ be a channel with a binary input and output of arbitrary dimension, and let $U_{A' \rightarrow BE}$ be a Stinespring dilation of $\mathcal{N}$ (i.e. a partial isometry such that $\mathcal{N}(\cdot) = \operatorname{Tr}_E[U (\cdot) U^{\dagger}]$). The complementary channel of $\mathcal{N}$ is then $\mathcal{N}^c_{A' \rightarrow E}$ is then given by $\mathcal{N}^c(\cdot) := \operatorname{Tr}_B[U (\cdot) U^{\dagger}]$. \[def:complementary-channel\]
Technically this depends on the choice of the Stinespring dilation, so the complementary channel is only unique up to an isometry on the output system. However, this will not matter for any of what we do here.
Finally, we need the following lemma, providing necessary conditions for the convergence of a stochastic process. The lemma below is a slightly modified version of [@sta09 Lemma 2], so as to meet our specific needs. The proof is omitted, since it is essentially the same as the one in [*loc. cit.*]{} (see also [@sta09 Remark 1]).
\[lemma:stochastic\_proc\_convergence\] Suppose $B_i$, $i=1,2,\dots$ are i.i.d., $\{0,1\}$-valued random variables with $P(B_1 = 0) = P(B_1 = 1) = 1/2$, defined on a probability space $(\Omega, {\cal F}, P)$. Set ${\cal F}_0 = \{ \phi, \Omega\}$ as the trivial $\sigma$-algebra and set ${\cal F}_n$, $n\geq 1$, to be the $\sigma$-field generated by $(B_1,\dots, B_n)$. Suppose further that two stochastic processes $\{I_n : n\geq 0\}$ and $\{T_n : n\geq 0\}$ are defined on this probability space with the following properties:
[*(i.1)*]{}
: $I_n$ takes values in $[\iota_0, \iota_1]$ and is measurable with respect to ${\cal F}_n$. That is, $I_0$ is a constant, and $I_n$ is a function of $B_1,\dots, B_n$.
[*(i.2)*]{}
: $\{ (I_n, {\cal F}_n) : n \geq 0\}$ is a martingale.
[*(t.1)*]{}
: $T_n$ takes values in the interval $[\theta_0, \theta_1]$ and is measurable with respect to ${\cal F}_n$.
[*(t.2)*]{}
: $T_{n+1} \leq f(T_n)$ when $B_{n+1} = 1$, where $f: [\theta_0, \theta_1] \rightarrow [\theta_0, \theta_1]$ is a continuous function, such that $f(\theta) < \theta, \forall \theta \in (\theta_0, \theta_1)$.
[*(i&t.1)*]{}
: For any $\epsilon > 0$ there exists $\delta > 0$, such that $I_n \in (\iota_0+\epsilon, \iota_1-\epsilon)$ implies $T_n \in (\theta_0+\delta, \theta_1-\delta)$.
Then, $I_\infty {:=}\lim_{n\rightarrow\infty} I_n$ exists with probability 1, $I_\infty$ takes values in $\{\iota_0, \iota_1\}$, and ${\mathbb{E}}(I_\infty){:=}\iota_0 P(I_\infty = \iota_0) + \iota_1 P(I_\infty = \iota_1) = I_0$.
Purely Quantum Polarization {#sec:purely_quantum_polarization}
===========================
In this section, we introduce our purely quantum version of polar codes, which is based on the channel combining and slitting operations depicted in Figure \[fig:transform\] and Figure \[fig:good-bad-channel\]. For the channel combining operation (Figure \[fig:transform\]), we consider a randomly chosen two-qubit Clifford unitary, to combine two independent copies of a quantum channel ${\cal W}$. The combined channel is then split, with the corresponding bad and good channels shown in Figure \[fig:good-bad-channel\].
(0,0) node\[draw\] (canal1) [$\mathcal{W}$]{} (canal1) ++(0, -1) node\[draw\] (canal2) [$\mathcal{W}$]{} ($.5*(canal1)+.5*(canal2)$) ++(-1.5, 0) node\[draw, minimum height=1.8cm\] (C) [$C$]{} ;
(canal1 -| C.east) to node\[above\] [$X_1$]{} (canal1) (canal2 -| C.east) to node\[above\] [$X_2$]{} (canal2) (canal1 -| C.west) to ++(-1, 0) node\[left\] [$U_1$]{} (canal2 -| C.west) to ++(-1, 0) node\[left\] [$U_2$]{} (canal1.east) to ++(1, 0) node\[right\] [$Y_1$]{} (canal2.east) to ++(1, 0) node\[right\] [$Y_2$]{} ;
(0,0) node\[draw\] (canal1) [$\mathcal{W}$]{} (canal1) ++(0, -1) node\[draw\] (canal2) [$\mathcal{W}$]{} ($.5*(canal1)+.5*(canal2)$) ++(-1.5, 0) node\[draw, minimum height=1.8cm\] (C) [$C$]{} ;
(canal1 -| C.east) to node\[above\] [$X_1$]{} (canal1) (canal2 -| C.east) to node\[above\] [$X_2$]{} (canal2) (canal1 -| C.west) to ++(-1.5, 0) node\[left\] [$U_1$]{} (canal2 -| C.west) to ++(-0.5, 0) node\[left\] (ground) [$\frac{{\mathbbm{1}}}{2}$]{} (canal1.east) to ++(1, 0) node\[right\] (y1) [$Y_1$]{} (canal2.east) to ++(1, 0) node\[right\] (y2) [$Y_2$]{} ;
;
(0,0) node\[draw\] (canal1) [$\mathcal{W}$]{} (canal1) ++(0, -1) node\[draw\] (canal2) [$\mathcal{W}$]{} ($.5*(canal1)+.5*(canal2)$) ++(-1.5, 0) node\[draw, minimum height=1.8cm\] (C) [$C$]{} ;
(canal1 -| C.east) to node\[above\] [$X_1$]{} (canal1) (canal2 -| C.east) to node\[above\] [$X_2$]{} (canal2) (canal2 -| C.west) to ++(-1.5, 0) node\[left\] [$U_2$]{} (canal1.east) to ++(1, 0) node\[right\] (y1) [$Y_1$]{} (canal2.east) to ++(1, 0) node\[right\] [$Y_2$]{} (canal1 -| C.west) to ++(-0.5, 0) to ++(-.5, .5) coordinate (coude) to ++(.5, .5) coordinate (topline) to (topline -| y1.west) node\[right\] (R) [$R_1$]{} ;
;
\(a) $\mathcal{W} \boxast_C \mathcal{W}$ (b) $\mathcal{W} \varoast_C \mathcal{W}$
In other words, the bad channel $\mathcal{W} \boxast_C \mathcal{W}$ is a channel from $U_1$ to $Y_1 Y_2$ that acts as $(\mathcal{W} \boxast_C \mathcal{W})(\rho) = \mathcal{W}^{{\varotimes}2}\left( C (\rho {\varotimes}\frac{{\mathbbm{1}}}{2}) C^{\dagger} \right)$. Likewise, the good channel $\mathcal{W} \varoast_C \mathcal{W}$ is a channel from $U_2$ to $R_1 Y_1 Y_2$ that acts as $(\mathcal{W} \varoast_C \mathcal{W})(\rho) = \mathcal{W}^{{\varotimes}2}\left( C(\Phi_{R_1 U_1} {\varotimes}\rho) C^{\dagger} \right)$, where $\Phi_{R_1 U_1}$ is an EPR pair. The polarization construction is obtained by recursively applying the above channel combining and spiting operations. Let us denote ${\cal W}_C^{(0)} {:=}{\cal W}\boxast_C{\cal W}$, ${\cal W}_C^{(1)} {:=}{\cal W}\varoast_C{\cal W}$, where index $C$ in the above notation indicates the Clifford unitary used for the channel combining operation. To accommodate a random choice of $C$, a classical description of $C$ must be included as part of the output of the bad/good channels at each step of the transformation. To do so, for $i = 0, 1$, we define $${\cal W}^{(i)}(\rho) = \frac{1}{|C_2|} \sum_{C \in C_2} {\mathchoice{{{\ensuremath{\left\lvertC\right\rangle}}}}{{{\ensuremath{\lvertC\rangle}}}}{{{\ensuremath{\lvertC\rangle}}}}{{{\ensuremath{\lvertC\rangle}}}}}{\mathchoice{{{\ensuremath{\left\langleC\right\rvert}}}}{{{\ensuremath{\langleC\rvert}}}}{{{\ensuremath{\langleC\rvert}}}}{{{\ensuremath{\langleC\rvert}}}}} {\varotimes}{\cal W}_C^{(i)}(\rho) $$ where $C_2$ denotes the Clifford group on two qubits, and $\{ {\mathchoice{{{\ensuremath{\left\lvertC\right\rangle}}}}{{{\ensuremath{\lvertC\rangle}}}}{{{\ensuremath{\lvertC\rangle}}}}{{{\ensuremath{\lvertC\rangle}}}}} \}_{C \in C_2}$ denotes an orthogonal basis of some auxiliary system. Now, applying twice the operation ${\cal W} \mapsto \left({\cal W}^{(0)},{\cal W}^{(1)}\right)$, we get channels ${\cal W}^{(i_1 i_2)} {:=}\left( {\cal W}^{(i_1)} \right)\,\!^{(i_{2})}$, where $(i_1i_{2}) \in \{00, 01, 10, 11\}$. In general, after $n$ levels or recursion, we obtain $2^n$ channels: $$\label{eq:recursive_construction}
{\cal W}^{(i_1\dots i_{n})} {:=}\left( {\cal W}^{(i_1\dots i_{n-1})} \right)\,\!^{(i_{n})}, \mbox{ where } (i_1\dots i_{n}) \in \{0,1\}^{n}$$ Our main theorem below states that as $n$ goes to infinity, the symmetric coherent information of the synthesized channels ${\cal W}^{(i_1\dots i_{n})}$ polarizes, meaning that it goes to either $-1$ or $+1$, except possibly for a vanishing fraction of channels. We recall that the symmetric coherent information of a quantum channel $\mathcal{N}_{A' \rightarrow B}$ is defined as the coherent information of the channel for a uniformly distributed input, that is $$I(\mathcal{N}) := -H(A|B)_{\mathcal{N}(\Phi_{A'A})} \in [-1, 1].$$ To prove the polarization theorem, we will utilize Lemma \[lemma:stochastic\_proc\_convergence\]. This basically requires us to find two quantities $I$ and $T$ that respectively play the role of the symmetric mutual information of the channel and of the Bhattacharyya parameter from the classical case. As mentioned above, for $I$ we shall consider the symmetric coherent information of the quantum channel. For $T$, we will need to be slightly more creative. For any channel $\mathcal{N}_{A' \rightarrow B}$, let us define $R(\mathcal{N})$ as $$R(\mathcal{N}) := 2^{{H^{\uparrow}_{\frac{1}{2}}}(A|B)_{\mathcal{N}(\Phi_{AA'})}} = 2^{-{\tilde{H}^{\downarrow}_2}(A|E)_{\mathcal{N}^c(\Phi_{AA'})}} \in \left[\tfrac{1}{2}, 2\right]$$ This quantity will be our $T$ and we will call it the “Réyni-Bhattacharyya” parameter. We can see from the expression of ${H^{\uparrow}_{\frac{1}{2}}}$ that this indeed looks vaguely like the Bhattacharyya parameter; however we will work mostly with the second form involving the complementary channel as this will be more mathematically convenient for us.
Before stating the main theorem, we first provide the following lemma on the symmetric coherent information $I$ and the Réyni-Bhattacharyya parameter $R$ of a classical mixture of quantum channels. It will allow us to derive the main steps in the proof of the polarization theorem, by conveniently working with the ${\cal W}_C^{(0)}(\rho)$/ ${\cal W}_C^{(1)}(\rho)$ construction, rather than the ${\cal W}^{(0)}(\rho)$/ ${\cal W}^{(1)}(\rho)$ mixture (in which a classical description of $C$ is included in the output). The proof is omitted, since part (a) is trivial, and part (b) follows easily from [@mdsft13 Section B.2].
\[lem:i-and-r-for-mixture\] Let ${\cal N}(\rho) = \sum_{x\in X} \lambda_x {\mathchoice{{{\ensuremath{\left\lvertx\right\rangle}}}}{{{\ensuremath{\lvertx\rangle}}}}{{{\ensuremath{\lvertx\rangle}}}}{{{\ensuremath{\lvertx\rangle}}}}}{\mathchoice{{{\ensuremath{\left\langlex\right\rvert}}}}{{{\ensuremath{\langlex\rvert}}}}{{{\ensuremath{\langlex\rvert}}}}{{{\ensuremath{\langlex\rvert}}}}} {\varotimes}{\cal N}_x(\rho)$, be a classical mixture of quantum channels ${\cal N}_x$, where $\{{\mathchoice{{{\ensuremath{\left\lvertx\right\rangle}}}}{{{\ensuremath{\lvertx\rangle}}}}{{{\ensuremath{\lvertx\rangle}}}}{{{\ensuremath{\lvertx\rangle}}}}}\}_{x\in X}$ is some orthonormal basis of an auxiliary system, and $\sum_{x\in X} \lambda_x = 1$. Then
- $I({\cal N}) = {\mathbb{E}}_X I({\cal N}_x) {:=}\sum_{x\in X} \lambda_x I({\cal N}_x)$
- $R({\cal N}) = {\mathbb{E}}_X R({\cal N}_x) {:=}\sum_{x\in X} \lambda_x R({\cal N}_x)$
We can now state the polarization theorem.
\[thm:quantum\_polarization\] For any $\delta > 0$, $$\lim_{n\rightarrow\infty} \frac{\#\{(i_1\dots i_{n}) \in \{0,1\}^{n} : I\left( {\cal W}^{(i_1\dots i_{n})} \right) \in (-1+\delta, 1-\delta) \}}{2^n} = 0$$ and furthermore, $$\lim_{n \rightarrow \infty} \frac{\#\left\{ (i_1,\dots,i_n) \in \{0,1\}^n : I(\mathcal{W}^{(i_1,\dots,i_n)}) \geqslant 1-\delta\right\} }{2^n} = \frac{I(\mathcal{N}) + 1}{2}$$
Let $\{B_n : n\geq 1\}$ be a sequence of i.i.d., $\{0,1\}$-valued random variables with $P(B_n = 0) = P(B_n = 1) = 1/2$, as in Lemma \[lemma:stochastic\_proc\_convergence\]. Let $\{I_n : n\geq 0\}$ and $\{R_n : n\geq 0\}$ be the stochastic processes defined by $I_n {:=}I\left( {\cal W}^{(B_1\dots B_{n})} \right)$ and $R_n{:=}R\left( {\cal W}^{(B_1\dots B_{n})} \right)$. By convention, ${\cal W}^{(\varnothing)} {:=}{\cal W}$, thus $I_0 = I({\cal W})$ and $R_0 = R({\cal W})$. We prove that all the conditions of Lemma \[lemma:stochastic\_proc\_convergence\] hold for $I_n$ and $T_n{:=}R_n$.
(i.1)
: Straightforward (with $[\iota_0, \iota_1] = [-1,1]$)
(i.2)
: We must show that $I_n$ forms a martingale. In other words, that the channel combining and splitting transformation doesn’t change the total coherent information, [*i.e.*]{}, $I\left({\cal W}^{(0)}\right) + I\left({\cal W}^{(1)}\right) = 2I\left({\cal W}\right)$. This follows from Lemma \[lem:i-is-a-martingale\] below, and Lemma \[lem:i-and-r-for-mixture\] (a).
(t.1)
: Straightforward (with $[\theta_0, \theta_1] = [\frac{1}{2}, 2]$).
(t.2)
: Here, we will show that $R_{n+1} = \frac{2}{5} + \frac{2}{5} R_n^2$, when $B_{n+1}=1$. It is enough to prove it for $n=0$ ([*i.e.*]{}, the first step of recursion), since in the general case the proof is obtained simply by replacing ${\cal W}$ with ${\cal W}^{(B_1\dots B_n)}$. First, by using Lemma \[lem:i-and-r-for-mixture\] (b), and assuming $B_{1}=1$, we get $R_{1} {:=}R\left( {\cal W}^{(1)} \right) = {\mathbb{E}}_C R\left( {\cal W}_C^{(1)} \right) = {\mathbb{E}}_C R\left( \mathcal{W} \varoast_C \mathcal{W} \right)$, where the last equality is simply a reminder of our notation ${\cal W}_C^{(1)} {:=}\mathcal{W} \varoast_C \mathcal{W}$. We then prove that $ {\mathbb{E}}_C R\left(\mathcal{W} \varoast_C \mathcal{W}\right) = \frac{2}{5} + \frac{2}{5} R(\mathcal{W})^2$. This is where most of the action happens, and the proof is in Lemma \[lem:d-good-channel\].
(i&t.1)
: For any $\varepsilon > 0$, there exists a $\delta > 0$ such that $I_n \in (-1+\varepsilon, 1-\varepsilon)$ implies that $R_n \in (\frac{1}{2}+\delta, 2-\delta)$. In other words, we need to show that if $R$ polarizes, then so does $I$. This holds for any choice of the Clifford unitary in the channel combining operation, and is proven in Lemma \[lem:i-and-t-1\].
We now proceed with the lemmas. The following lemmas are stated in slightly more general settings, with the channel combining construction applied to two quantum channels ${\cal N}$ and ${\cal M}$, rather than to two copies of the same quantum channel ${\cal W}$.
\[lem:i-is-a-martingale\] Given two channels $\mathcal{N}_{A'_1 \rightarrow B_1}$ and $\mathcal{M}_{A'_2 \rightarrow B_2}$ with qubit inputs, then $$I(\mathcal{N} \varoast_C \mathcal{M}) + I(\mathcal{N} \boxast_C \mathcal{M}) = I(\mathcal{N}) + I(\mathcal{M}),$$ and this holds for all choices of $C$.
Consider the state $\rho = (\mathcal{N} {\varotimes}\mathcal{M})( C( \Phi_{A_1 A'_1} {\varotimes}\Phi_{A_2 A'_2} ) C^{\dagger} )$ on systems $A_1 A_2 B_1 B_2$. We have that $I(\mathcal{N} \boxast_C \mathcal{M}) = -H(A_1|B_1 B_2)_{\rho}$ and $I(\mathcal{N} \varoast_C \mathcal{M}) = -H(A_2|A_1 B_1 B_2)_{\rho}$. Therefore, by the chain rule, $$\begin{aligned}
I(\mathcal{N} \boxast_C \mathcal{M}) + I(\mathcal{N} \varoast_C \mathcal{M}) &= -H(A_1|B_1 B_2)_{\rho} - H(A_2|A_1 B_1 B_2)_{\rho}\\
&= -H(A_1 A_2|B_1 B_2)_{\rho}. \end{aligned}$$ Now, recall that the EPR pair has the property that $(Z {\varotimes}{\mathbbm{1}}) {\mathchoice{{{\ensuremath{\left\lvert\Phi\right\rangle}}}}{{{\ensuremath{\lvert\Phi\rangle}}}}{{{\ensuremath{\lvert\Phi\rangle}}}}{{{\ensuremath{\lvert\Phi\rangle}}}}} = ({\mathbbm{1}}{\varotimes}Z^{\top}) {\mathchoice{{{\ensuremath{\left\lvert\Phi\right\rangle}}}}{{{\ensuremath{\lvert\Phi\rangle}}}}{{{\ensuremath{\lvert\Phi\rangle}}}}{{{\ensuremath{\lvert\Phi\rangle}}}}}$ for any matrix $Z$. Using this, we can move $C$ from the input systems $A'_1$ and $A'_2$ to the purifying systems $A_1 A_2$: $\rho = C^{\top}(\mathcal{N} {\varotimes}\mathcal{M})(\Phi_{A_1 A'_1} {\varotimes}\Phi_{A_2 A'_2}) \bar{C}$. Hence, we have $$\begin{aligned}
-H(A_1 A_2 | B_1 B_2)_{\rho} &= -H(A_1 A_2 | B_1 B_2)_{(\mathcal{N} {\varotimes}\mathcal{M})(\Phi)}\\
&= -H(A_1|B_1)_{\mathcal{N}(\Phi)} - H(A_2 | B_2)_{\mathcal{M}(\Phi)}\\
&= I(\mathcal{N}) + I(\mathcal{M}).\end{aligned}$$
\[lem:d-good-channel\] Given two channels $\mathcal{N}_{A'_1 \rightarrow B_1}$ and $\mathcal{M}_{A'_2 \rightarrow B_2}$ with qubit inputs, then $${\mathbb{E}}_C R(\mathcal{N} \varoast_C \mathcal{M}) = \frac{2}{5} + \frac{2}{5} R(\mathcal{N}) R(\mathcal{M}),$$ where $C$ is the channel combining Clifford operator and is chosen uniformly at random over the Clifford group.
Let $\mathcal{N}^c_{A'_1 \rightarrow E_1}$ and $\mathcal{M}^c_{A'_2 \rightarrow E_2}$ be the complementary channels of $\mathcal{N}$ and $\mathcal{M}$ respectively. It’s not too hard to show that $(\mathcal{N} \varoast_C \mathcal{M})^c(\rho) = (\mathcal{N}^c {\varotimes}\mathcal{M}^c)\left( C \left( \frac{{\mathbbm{1}}_{A'_1}}{2} {\varotimes}\rho \right) C^{\dagger} \right)$, and therefore $R(\mathcal{N} \varoast_C \mathcal{M}) = 2^{-{\tilde{H}^{\downarrow}_2}(A_2|E_1 E_2)_{\rho}}$, where $\rho_{A_2 E_1 E_2} = (\mathcal{N} \varoast \mathcal{M})^c(\Phi_{A_2 A'_2})$. Note that $\rho_{E_1 E_2} = \mathcal{N}^c\left(\frac{{\mathbbm{1}}}{2} \right)_{E_1} {\varotimes}\mathcal{M}^c\left(\frac{{\mathbbm{1}}}{2} \right)_{E_2}$, which is independent of $C$. Now, to compute the expected value of this for a random choice of $C$, we proceed as follows:
$$\begin{aligned}
{\mathbb{E}}_C 2^{-{\tilde{H}^{\downarrow}_2}(A_2|E_1 E_2)_{\rho}} & = {\mathbb{E}}_C \operatorname{Tr}\left[ \left( \rho_{E_1 E_2}^{-\frac{1}{4}} \rho_{A_2 E_1 E_2} \rho_{E_1 E_2}^{-\frac{1}{4}} \right)^2 \right]\\
&= {\mathbb{E}}_C \operatorname{Tr}\left[ \left( \rho_{E_1 E_2}^{-\frac{1}{4}} (\mathcal{N}^c {\varotimes}\mathcal{M}^c)\left( C \left( \frac{{\mathbbm{1}}_{A'_1}}{2} {\varotimes}\Phi_{A_2 A'_2} \right) C^{\dagger} \right) \rho_{E_1 E_2}^{-\frac{1}{4}} \right)^2 \right].
\end{aligned}$$
Now, note that this is basically the same calculation as in [@fred-these], at Equation (3.32) (there, $U$ is chosen according to the Haar measure over the full unitary group, but all that is required is a 2-design, and hence choosing a random Clifford yields the same result). However, since here we are dealing with small systems, we will not make the simplifications after (3.44) and (3.45) in [@fred-these] but will instead keep all the terms. We therefore get ${\mathbb{E}}_C 2^{-{\tilde{H}^{\downarrow}_2}(A_2|E_1 E_2)_{\rho}} = \alpha \operatorname{Tr}\left[ \pi_{A_2}^2 \right] + \beta \operatorname{Tr}\left[ \pi_{A'_1}^2 {\varotimes}\Phi_{A_2 A'_2} \right]
= \frac{1}{2} \alpha + \frac{1}{2} \beta$, where $\alpha = \frac{16}{15} - \frac{4}{15} 2^{-{\tilde{H}^{\downarrow}_2}(A_1 A_2|E_1 E_2)_{\omega}}$, $\beta = \frac{16}{15} 2^{-{\tilde{H}^{\downarrow}_2}(A_1 A_2|E_1 E_2)_{\omega}} - \frac{4}{15}$, and $\omega_{A_1 A_2 E_1 E_2} := (\mathcal{N}^c {\varotimes}\mathcal{M}^c)(\Phi_{A_1 A'_1} {\varotimes}\Phi_{A_2 A'_2})$. Hence, $$\begin{aligned}
{\mathbb{E}}_C 2^{-{\tilde{H}^{\downarrow}_2}(A_2|E_1 E_2)_{\rho}} &= \frac{6}{15} + \frac{6}{15} 2^{-{\tilde{H}^{\downarrow}_2}(A_1 A_2|E_1 E_2)_{\omega}}\\
&= \frac{2}{5} + \frac{2}{5} R(\mathcal{N}) R(\mathcal{M}).
\end{aligned}$$
Let $\mathcal{N}_{A' \rightarrow B}$ be a channel with qubit input. Then,
- $R(\mathcal{N}) \leqslant \frac{1}{2}+\delta \Rightarrow I(\mathcal{N}) \geqslant 1 - \log(1+2\delta)$.
- $R(\mathcal{N}) \geqslant 2 - \delta \Rightarrow I(\mathcal{N}) \leqslant -1 + 4\sqrt{2 \delta} + 2h(\sqrt{2\delta}),$
where $h(\cdot)$ denotes the binary entropy function. \[lem:i-and-t-1\]
We first prove point 1. Observe that for any state $\sigma_{AB}$, the inequality $H(A|B)_{\sigma} \leqslant {H^{\uparrow}_{\frac{1}{2}}}(A|B)_{\sigma}$ holds. Now, for $\rho_{AB} = \mathcal{N}(\Phi_{AA'})$, we have that $$\begin{aligned}
\frac{1}{2} + \delta &\geqslant R(\mathcal{N})\\
&= 2^{{H^{\uparrow}_{\frac{1}{2}}}(A|B)_{\rho}}\\
&\geqslant 2^{H(A|B)_{\rho}}\\
&= 2^{-I(\mathcal{N})},
\end{aligned}$$ and hence $I(\mathcal{N}) \geqslant 1 - \log(1+2\delta)$.
We now turn to the second point. We have that $$\begin{aligned}
2 - \delta &\leqslant R(\mathcal{N})\\
&= \max_{\sigma_B} \operatorname{Tr}\left[ \rho^{\frac{1}{2}}_{AB} \sigma^{\frac{1}{2}}_B \right]^2\\
&= 2 \max_{\sigma_B} \operatorname{Tr}\left[ \sqrt{\rho_{AB}} \sqrt{\frac{{\mathbbm{1}}_A}{2} {\varotimes}\sigma_B} \right]^2\\
&\leqslant 2 \max_{\sigma_B} \left\| \sqrt{\rho_{AB}} \sqrt{\frac{{\mathbbm{1}}_A}{2} {\varotimes}\sigma_B} \right\|_1^2\\
&= 2 \max_{\sigma_B} F\left( \rho_{AB}, \frac{{\mathbbm{1}}_A}{2} {\varotimes}\sigma_B \right)^2.
\end{aligned}$$ Now, using the Fuchs-van de Graaf inequalities, we get that there exists a $\sigma_B$ such that $$\left\| \rho_{AB} - \frac{{\mathbbm{1}}_A}{2} {\varotimes}\sigma_B \right\|_1 \leqslant \sqrt{2\delta}.$$ We are now in a position to use the Alicki-Fannes [@AF03] inequality, which states that $$\begin{aligned}
\left| H(A|B)_{\rho} - 1 \right| \leqslant 4 \sqrt{2 \delta} + 2 h(\sqrt{2\delta}).
\end{aligned}$$ This concludes the proof of the lemma.
Quantum Polarization Using Only $9$ Clifford Gates {#sec:polarization_9_cliffords}
==================================================
In this section, we prove that quantum polarization can be achieved while reducing the set of two-qubit Clifford gates used to randomize the channel combining operation, to a subset of $9$ Clifford gates only. To do so, we need to find a subset of Clifford gates such that the condition $(t.2)$ from Lemma \[lemma:stochastic\_proc\_convergence\] is still fulfilled.
Let $\mathcal{C}_n$ denote the $n$-qubit Clifford group. Clearly $\mathcal{C}_1 {\varotimes}\mathcal{C}_1 \leq \mathcal{C}_2$, and we may define an equivalence relation on $\mathcal{C}_2$, whose equivalence classes are the left cosets of $\mathcal{C}_1 {\varotimes}\mathcal{C}_1$.
We say that $C'$ and $C'' \in \mathcal{C}_2$ are equivalent, and denote it by $C'\sim C''$, if there exist $C_1, C_2 \in \mathcal{C}_1$ such that $C'' = C'(C_1 {\varotimes}C_2)$ (see also Figure \[fig:equiv\_cliffords\]).
(0,0) node\[left\] [$U_2$]{} – +(0.75,0); (0,1) node\[left\] [$U_1$]{} – +(0.75,0); (0.75,-0.25) rectangle (1.25,0.25); (0.75,0.75) rectangle (1.25,1.25); (1,0) node [$C_2$]{}; (1,1) node [$C_1$]{}; (1.25,0) – +(0.5,0); (1.25,1) – +(0.5,0); (1.75,-0.25) rectangle (2.25,1.25); (2,0.5) node [$C'$]{}; (2.25,0) – +(0.75,0) node\[right\] [$X_2$]{}; (2.25,1) – +(0.75,0) node\[right\] [$X_1$]{}; (0.5,-0.5) rectangle (2.5,1.5); (1.5,-0.5) node\[below\] [$C''$]{};
Now, the main observation is that two equivalent Clifford gates used to combine any two quantum channels with qubit inputs, yield the same Réyni-Bhattacharyya parameter of the bad/good channels. This is stated in the following lemma, whose proof is provided in Appendix \[sec:proof-quantum-red-clifford\].
\[lemma:equiv\_conditions\] Let $C', C'' \in \mathcal{C}_2$. If $C'\sim C''$, then for any two quantum channels $\mathcal{M}$ and $\mathcal{N}$ with qubit inputs, we have: $$R(\mathcal{M} \boxast_{C'} \mathcal{N}) = R( \mathcal{M} \boxast_{C''} \mathcal{N}) \text{ and }
R(\mathcal{M} \varoast_{C'}\mathcal{N}) = R( \mathcal{M} \varoast_{C''} \mathcal{N})$$
As a consequence, one may ensure polarization while restricting the set of Clifford gates to any set of representatives of the equivalence classes of the above equivalence relation (since such a restriction will not affect the ${\mathbb{E}}_C R(\mathcal{N} \varoast_C \mathcal{M})$ value, for any two quantum channels $\mathcal{M}$ and $\mathcal{N}$ with qubit inputs). Since $|\mathcal{C}_1| = 24$ and $|\mathcal{C}_2| = 11520$, it follows that there are exactly $11520 / (24\times 24) = 20$ equivalence classes. A set of representatives of these $20$ equivalence classes can be chosen as follows[^1]:
- For 2 of these equivalence classes, one may choose the identity gate $I$ and swap gate $S$, as representatives.
- For 9 out of remaining 18 equivalence classes, one may find representatives of form $(C_1 {\varotimes}C_2) \text{\sc cnot}_{21}$, where $\text{\sc cnot}_{21}$ denotes the controlled-NOT gate with control on the second qubit and target on the first qubit, $C_1 \in \{I, \sqrt{Z}, \sqrt{Y}\}$, $C_2 \in \{I, \sqrt{X}, \sqrt{Y}\}$, and $\sqrt{P} = \frac{(1-i)({\mathbbm{1}}+ i P)}{2}$, for any Pauli matrix $P \in \{X, Y, Z\}$. We denote this set by $\mathcal{L}$, which is further depicted in Figure \[fig:cliff\_generators\]. $$\mathcal{L} := \left\{ (C_1 {\varotimes}C_2) \text{\sc cnot}_{21} \mid C_1 \in \{I, \sqrt{Z}, \sqrt{Y}\}, C_2 \in \{I, \sqrt{X}, \sqrt{Y}\} \right\}$$
- For the remaining 9 equivalence classes, one may find representatives of form $S L$, $S$ is the swap gate and $L\in \mathcal{L}$. We denote this set by $ \mathcal{R}$. $$\mathcal{R} := \left\{ SL \mid L\in \mathcal{L} \right\}$$
(0, 0) node\[draw, minimum width=, minimum height=\](C1)[$I$]{} (C1) ++(0, -1) node\[draw, minimum width=, minimum height=\](C2)[$I$]{} (C1) ++(-1.25, 0) node\[not\](C3) (C2) ++(-1.25, 0) node\[phase\](C4) ; (C3) to (C1) (C4) to (C2) (C3) to (C4) (C3) to ++(-0.8, 0) (C4) to ++(-0.8, 0) (C2) to ++(+1, 0) (C1) to ++(+1, 0) (C1) ++(-2.8, -.5) node\[\]()[$L_{1,1}:$]{} ; (0, -2.5) node\[draw, minimum width=, minimum height=\](C1)[$\sqrt{Z}$]{} (C1) ++(0, -1) node\[draw, minimum width=, minimum height=\](C2)[$I$]{} (C1) ++(-1.25, 0) node\[not\](C3) (C2) ++(-1.25, 0) node\[phase\](C4) ; (C3) to (C1) (C4) to (C2) (C3) to (C4) (C3) to ++(-0.8, 0) (C4) to ++(-0.8, 0) (C2) to ++(+1, 0) (C1) to ++(+1, 0) (C1) ++(-2.8, -.5) node\[\]()[$L_{2,1}:$]{} ; (0, -5.0) node\[draw, minimum width=, minimum height=\](C1)[$\sqrt{Y}$]{} (C1) ++(0, -1) node\[draw, minimum width=, minimum height=\](C2)[$I$]{} (C1) ++(-1.25, 0) node\[not\](C3) (C2) ++(-1.25, 0) node\[phase\](C4) ; (C3) to (C1) (C4) to (C2) (C3) to (C4) (C3) to ++(-0.8, 0) (C4) to ++(-0.8, 0) (C2) to ++(+1, 0) (C1) to ++(+1, 0) (C1) ++(-2.8, -0.5) node\[\]()[$L_{3,1}:$]{} ;
(0, 0) node\[draw, minimum width=, minimum height=\](C1)[$I$]{} (C1) ++(0, -1) node\[draw, minimum width=, minimum height=\](C2)[$\sqrt{X}$]{} (C1) ++(-1.25, 0) node\[not\](C3) (C2) ++(-1.25, 0) node\[phase\](C4) ; (C3) to (C1) (C4) to (C2) (C3) to (C4) (C3) to ++(-0.8, 0) (C4) to ++(-0.8, 0) (C2) to ++(+1, 0) (C1) to ++(+1, 0) (C1) ++(-2.8, -0.5) node\[\]()[$L_{1,2}:$]{} ; (0, -2.5) node\[draw, minimum width=, minimum height=\](C1)[$\sqrt{Z}$]{} (C1) ++(0, -1) node\[draw, minimum width=, minimum height=\](C2)[$\sqrt{X}$]{} (C1) ++(-1.25, 0) node\[not\](C3) (C2) ++(-1.25, 0) node\[phase\](C4) ; (C3) to (C1) (C4) to (C2) (C3) to (C4) (C3) to ++(-0.8, 0) (C4) to ++(-0.8, 0) (C2) to ++(+1, 0) (C1) to ++(+1, 0) (C1) ++(-2.8, -0.5) node\[\]()[$L_{2,2}:$]{} ; (0, -5) node\[draw, minimum width=, minimum height=\](C1)[$\sqrt{Y}$]{} (C1) ++(0, -1) node\[draw, minimum width=, minimum height=\](C2)[$\sqrt{X}$]{} (C1) ++(-1.25, 0) node\[not\](C3) (C2) ++(-1.25, 0) node\[phase\](C4) ; (C3) to (C1) (C4) to (C2) (C3) to (C4) (C3) to ++(-0.8, 0) (C4) to ++(-0.8, 0) (C2) to ++(+1, 0) (C1) to ++(+1, 0) (C1) ++(-2.8, -.5) node\[\]()[$L_{3,2}:$]{} ;
(0, 0) node\[draw, minimum width=, minimum height=\](C1)[$I$]{} (C1) ++(0, -1) node\[draw, minimum width=, minimum height=\](C2)[$\sqrt{Y}$]{} (C1) ++(-1.25, 0) node\[not\](C3) (C2) ++(-1.25, 0) node\[phase\](C4) ; (C3) to (C1) (C4) to (C2) (C3) to (C4) (C3) to ++(-0.8, 0) (C4) to ++(-0.8, 0) (C2) to ++(+1, 0) (C1) to ++(+1, 0) (C1) ++(-2.8, -0.5) node\[\]()[$L_{1,3}:$]{} ; (0, -2.5) node\[draw, minimum width=, minimum height=\](C1)[$\sqrt{Z}$]{} (C1) ++(0, -1) node\[draw, minimum width=, minimum height=\](C2)[$\sqrt{Y}$]{} (C1) ++(-1.25, 0) node\[not\](C3) (C2) ++(-1.25, 0) node\[phase\](C4) ; (C3) to (C1) (C4) to (C2) (C3) to (C4) (C3) to ++(-0.8, 0) (C4) to ++(-0.8, 0) (C2) to ++(+1, 0) (C1) to ++(+1, 0) (C1) ++(-2.8, -.5) node\[\]()[$L_{2,3}:$]{} ; (0, -5) node\[draw, minimum width=, minimum height=\](C1)[$\sqrt{Y}$]{} (C1) ++(0, -1) node\[draw, minimum width=, minimum height=\](C2)[$\sqrt{Y}$]{} (C1) ++(-1.25, 0) node\[not\](C3) (C2) ++(-1.25, 0) node\[phase\](C4) ; (C3) to (C1) (C4) to (C2) (C3) to (C4) (C3) to ++(-0.8, 0) (C4) to ++(-0.8, 0) (C2) to ++(+1, 0) (C1) to ++(+1, 0) (C1) ++(-2.8, -.5) node\[\]()[$L_{3,3}:$]{} ;
Now, we prove that two Clifford gates $C'$ and $C''$, such that $C'' = SC'$, used to combine two copies of a quantum channel $\mathcal{W}$ with qubit input, yield the same Réyni-Bhattacharyya parameter of the bad/good channels. Although this property is weaker that the one in Lemma \[lemma:equiv\_conditions\], which holds for any two quantum channels $\mathcal{M}$ and $\mathcal{N}$, it is sufficient for whatever we need here.
\[lemma:swap\_eq\] Let $C', C'' \in \mathcal{C}_2$, such that $C'' = SC'$, where $S$ is the swap gate. Then, for two copies of a quantum channel $\mathcal{W}$ with qubit input, $$R(\mathcal{W} \boxast_{C'} \mathcal{W}) = R( \mathcal{W} \boxast_{C''} \mathcal{W}) \text{ and }
R(\mathcal{W} \varoast_{C'}\mathcal{W}) = R( \mathcal{W} \varoast_{C''} \mathcal{W})$$
First, we note that by applying a unitary on the output of any quantum channel does not change the Réyni-Bhattacharyya parameter. Precisely, let $\mathcal{N}_{A \rightarrow B}$ be any quantum channel, and $U\mathcal{N} U^\dagger_{A \rightarrow B}$ be the quantum channel[^2] obtained by applying the unitary $U$ on the output system $B$, that is, $U\mathcal{N} U^\dagger_{A \rightarrow B}(\rho_A) := U\mathcal{N}_{A \rightarrow B}(\rho_A) U^\dagger$. Then, $$\label{eq:u_same_r}
R\left( U\mathcal{N}_{A \rightarrow B} U^\dagger \right) = R\left( \mathcal{N}_{A \rightarrow B} \right)$$
Going back to the proof of our Lemma, by the definition of $ \mathcal{W} \boxast_{C} \mathcal{W}$ and using that $S^\dagger = S$, we may write: $$\begin{aligned}
(\mathcal{W} \boxast_{C''} \mathcal{W}) (\rho) &= (\mathcal{W} {\varotimes}\mathcal{W}) \left(C''(\rho {\varotimes}\frac{{\mathbbm{1}}}{2})C''^\dagger\right)
= (\mathcal{W} {\varotimes}\mathcal{W}) \left(SC'(\rho {\varotimes}\frac{{\mathbbm{1}}}{2})C'^\dagger S\right) \end{aligned}$$ Now, it is easily seen that applying the swap gate on either the input or the output system of the $\mathcal{W} {\varotimes}\mathcal{W}$ channel yields the same quantum channel, hence we may further write: $$\begin{aligned}
(\mathcal{W} \boxast_{C''} \mathcal{W}) (\rho) &= S(\mathcal{W} {\varotimes}\mathcal{W}) \left(C'(\rho {\varotimes}\frac{{\mathbbm{1}}}{2})C'^\dagger \right) S \\
&= S (\mathcal{W} \boxast_{C'} \mathcal{W}) (\rho) S = \left(S (\mathcal{W} \boxast_{C'} \mathcal{W}) S\right) (\rho) \end{aligned}$$ Hence, $\mathcal{W} \boxast_{C''} \mathcal{W} = S (\mathcal{W} \boxast_{C'} \mathcal{W}) S$, and using (\[eq:u\_same\_r\]), with $\mathcal{N}:= \mathcal{W} \boxast_{C'} \mathcal{W}$ and $U:=S$, we get $$R\left( \mathcal{W} \boxast_{C'} \mathcal{W} \right) = R\left( \mathcal{W} \boxast_{C''} \mathcal{W} \right),$$ as desired. The equality $R(\mathcal{W} \varoast_{C''} \mathcal{W}) = R(\mathcal{W} \boxast_{C'} \mathcal{W})$ may be proven in a similar way.
The following lemma implies that polarization can be achieved by choosing the channel combining Clifford operator randomly from either $\mathcal{L}$ or $\mathcal{R}$. It is the analogue of the Lemma \[lem:d-good-channel\] used to check the $(t.2)$ condition in the proof of the polarization Theorem \[thm:quantum\_polarization\].
\[lemma:quantum\_pol9\] Given two copies of a quantum channel $\mathcal{W}_{A'_1 \rightarrow B_1}$ with qubit input, we have $${\mathbb{E}}_{C \in \mathcal{L}} R(\mathcal{W} \varoast_C \mathcal{W}) = {\mathbb{E}}_{C \in \mathcal{R}} R(\mathcal{W} \varoast_C \mathcal{W}) = \frac{4}{9} - \frac{1}{9}R(\mathcal{W}) + \frac{4}{9}R(\mathcal{W})^2,$$ where $C$ is the channel combining Clifford operator and is chosen uniformly either from the set $\mathcal{L}$ or from the set $\mathcal{R}$, each containing $9$ Clifford gates.
Since $\mathcal{S}:=\{I, S\} \cup \mathcal{L} \cup \mathcal{R}$ is a set of representatives of the $20$ equivalence classes partitioning the Clifford group $\mathcal{C}_2$, we have: $${\mathbb{E}}_{C \in \mathcal{S}} R(\mathcal{W} \varoast_C \mathcal{W}) = {\mathbb{E}}_{C \in \mathcal{C}_2} R(\mathcal{W} \varoast_C \mathcal{W}) = \frac{2}{5} + \frac{2}{5} R(\mathcal{W})^2,$$ where the first equality follows from Lemma \[lemma:equiv\_conditions\], and the second from Lemma \[lem:d-good-channel\]. Now, using Lemma \[lemma:swap\_eq\], we have $R(\mathcal{W} \varoast_S \mathcal{W}) = R(\mathcal{W} \varoast_I \mathcal{W}) = R(\mathcal{W})$ and ${\mathbb{E}}_{C \in \mathcal{L}} R(\mathcal{W} \varoast_C \mathcal{W}) = {\mathbb{E}}_{C \in \mathcal{R}} R(\mathcal{W} \varoast_C \mathcal{W})$. Hence, $${\mathbb{E}}_{C \in \mathcal{S}} R(\mathcal{W} \varoast_C \mathcal{W}) = \frac{2R(\mathcal{W}) + 9{\mathbb{E}}_{C \in \mathcal{L}} R(\mathcal{W} \varoast_C \mathcal{W}) + 9{\mathbb{E}}_{C \in \mathcal{R}} R(\mathcal{W} \varoast_C \mathcal{W})}{20},$$ and therefore $${\mathbb{E}}_{C \in \mathcal{L}} R(\mathcal{W} \varoast_C \mathcal{W}) = {\mathbb{E}}_{C \in \mathcal{R}} R(\mathcal{W} \varoast_C \mathcal{W}) = \frac{4}{9} - \frac{1}{9}R(\mathcal{W}) + \frac{4}{9}R(\mathcal{W})^2$$ Finally, we also note that the above expected value is less than the one in Lemma \[lem:d-good-channel\], namely ${\mathbb{E}}_{C \in \mathcal{C}_2} R(\mathcal{W} \varoast_C \mathcal{W}) = \frac{2}{5} + \frac{2}{5} R(\mathcal{W})^2$, since the expected value can only decrease by taking out the identity and swap gate from the set of representatives.
Quantum Polar Coding {#sec:quantum_polar_coding}
====================
Polar coding is a coding method that take advantage of the channel polarization phenomenon [@arikan09]. To construct a quantum polar code of length $N=2^n$, $n>0$, we start with $N$ copies of the quantum channel $\mathcal{W}$, pair them in $N/2$ pairs, and apply the channel combining and splitting operation on each pair. The same channel combining Clifford gate is used for each of the $N/2$ pairs, which will be denoted by $C$. By doing so, we generate $N/2$ copies of the channel $\mathcal{W}^{(0)} := \mathcal{W} \boxast_{C} \mathcal{W}$ and $N/2$ copies of the channel $\mathcal{W}^{(1)} := \mathcal{W} \varoast_{C} \mathcal{W}$. Hence, for each $i_1=0,1$, we group together the $N/2$ copies of the $\mathcal{W}^{(i_1)}$ channel, pair them in $N/4$ pairs, and apply the channel combining and splitting operation on each pair, by using some channel combining Clifford gate denoted by $C_{i_1}$. By performing $n$ [*polarization steps*]{}, we generate quantum channels $\mathcal{W}^{(i_1\dots i_n)}$, which can be recursively defined for $n > 0$, as follows: $$\label{eq:polar_coding_recursion}
\mathcal{W}^{(i_1\dots i_n)} := \left\{ \begin{array}{ll}
\mathcal{W}^{(i_1\dots i_{n-1})} \boxast_{C_{i_1\dots i_{n-1}}} \mathcal{W}^{(i_1\dots i_{n-1})}, & \text{if } i_n = 0 \\
\mathcal{W}^{(i_1\dots i_{n-1})} \varoast_{C_{i_1\dots i_{n-1}}} \mathcal{W}^{(i_1\dots i_{n-1})}, & \text{if } i_n = 1
\end{array} \right.$$ where, for $n=1$, in the right hand side term of the above equality, we set by convention $\mathcal{W}^{(\varnothing)} := \mathcal{W}$ and $C_\varnothing := C$. Note that, for the sake of simplicity, we have dropped the channel combining Clifford gate from the $\mathcal{W}^{(i_1\dots i_n)}$ notation. The construction is illustrated in Figure \[fig:polar\_code\_N8\], for $N=8$. Horizontal “wires” represent qubits, and for each polarization step, we have indicated on each wire the virtual channel $\mathcal{W}^{(i_1i_2\dots)}$ “seen” by the corresponding qubit state.
![Quantum polar code of length $N=8$[]{data-label="fig:polar_code_N8"}](polar_code_N8.pdf)
The above construction synthesizes a set of $N$ channels and, for any $i=0,\dots,N-1$, we shall further denote $\mathcal{W}^{(i)} := \mathcal{W}^{(i_1\dots i_n)}$, where $i_1\dots i_n$ is the binary decomposition of $i$. Let $\mathcal{I} \subset\{0,1,\dots,N-1 \}$ denote the set of good channels ([*i.e.*]{}, with coherent information close to $1$, or equivalently, Réyni-Bhattacharyya parameter close to $1/2$), and let $\mathcal{J} := \{0,1,\dots,N-1 \} \setminus \mathcal{I}$. With a slight abuse of notation, we shall also denote by ${\cal I}$ and ${\cal J}$ two qudit systems, of dimension $2^{|{\cal I}|}$ and $2^{|{\cal J}|}$, respectively (it will be clear from the context whether the notation is meant to indicate a set of indexes or a quantum system).
A quantum state $\rho_{\cal I}$ on system ${\cal I}$ is encoded by supplying it as input to channels $i\in\mathcal{I}$, while supplying each channel $j\in\mathcal{J}$ with half of an EPR pair, shared between the sender and the receiver. Precisely, let $\Phi_{\cal JJ'}$ be a maximally entangled state, defined by $$\label{eq:max_entangled_state}
\Phi_{\cal JJ'} = {\varotimes}_{j\in \mathcal{J}} \Phi_{jj'},$$ where indexes $j$ and $j'$ indicate the $j$-th qubits of $\mathcal{J}$ and $\mathcal{J}'$ systems, respectively, and $\Phi_{jj'}$ is an EPR pair. Let also $G_q$ denote the quantum polar transform, that is the unitary operator defined by the applying the Clifford gates corresponding to the $n$ polarization steps. The encode state, denoted $\varphi_{{\cal IJJ}'}$, is obtained by applying the $G_q{\varotimes}I_{{\cal J}'}$ unitary on the $\mathcal{I}\mathcal{J}\mathcal{J}'$ system, hence: $$\varphi_{{\cal IJJ}'} {:=}(G_q{\varotimes}I_{{\cal J}'}) (\rho_{\cal I} {\varotimes}\Phi_{\cal JJ'}) (G_q^\dagger {\varotimes}I_{{\cal J}'})$$ Since no errors occur on the $\mathcal{J}'$ system, the channel output state is given by: $$\psi_{{\cal IJJ}'} {:=}({\cal W}^{{\varotimes}N} {\varotimes}I_{{\cal J}'}) (\varphi_{{\cal IJJ}'})$$
It is worth noticing that randomness is used only at the code construction stage (since Clifford gates used in the $n$ polarization steps are randomly chosen from some predetermined set of gates), but not at the encoding stage. Decoding for general quantum channels is an open problem. However, for Pauli channels, an efficient decoding algorithm will be introduced in Section \[sec:decod\_pauli\_channel\], below.
Polarization of Pauli Channels {#sec:polarization_of_pauli_channels}
==============================
This section further investigates the quantum polarization of Pauli channels. First, to a Pauli channel ${\cal N}$ we associate a classical symmetric channel ${\cal N}^\#$, with both input and output alphabets given by the quotient of the $1$-qubit Pauli group by its centralizer. We then show that the former polarizes quantumly if and only if the latter polarizes classically. We use this equivalence to provide an alternative proof of the quantum polarization for a Pauli channel, as well as fast polarization properties. We then devise an effective way to decode a quantum polar code on a Pauli channel, by decoding its classical counterpart.
Let $P_n$ denote the Pauli group on $n$ qubits, and $\bar{P}_n = P_n/\{\pm 1, \pm i\}$ the Abelian group obtained by taking the quotient of $P_n$ by its centralizer. We write $\bar{P}_1 = \{\sigma_i \mid i=0,\dots,3\}$, with $\sigma_0 = I$, $\sigma_1 = X$, $\sigma_2 = Y$, $\sigma_3 = Z$, and $\bar{P}_2 = \{\sigma_{i,j}{:=}\sigma_i {\varotimes}\sigma_j \mid i,j=0,\dots,3\} \simeq \bar{P}_1\times\bar{P}_1$. For any two-qubit Clifford unitary $C$, we denote by $\Gamma(C)$, or simply $\Gamma$ when no confusion is possible, the conjugate action of $C$ of $\bar{P}_2$. Hence, $\Gamma$ is the automorphism of $\bar{P}_2$ (or equivalently $\bar{P}_1\times\bar{P}_1$), defined by $\Gamma(\sigma_{i,j}) = C \sigma_{i,j} C^\dagger$. Let ${\cal N}$ be a Pauli channel defined by[^3] ${\cal N}(\rho) = \sum_{i=0}^3 p_i \sigma_i \rho \sigma_i^{\dagger}$, with $\sum_{i=0}^3 p_i = 1$. Its coherent information for a uniformly distributed input is given by $I({\cal N}) = 1 - h(\mathbf{p})$, where $h(\mathbf{p}) = -\sum_{i=0}^3p_i\log(p_i)$ denotes the entropy of the probability vector $\mathbf{p} = (p_0,p_1,p_2,p_3)$.
\[def:classical\_channel\] Let ${\cal N}$ be a Pauli channel. The classical counterpart of ${\cal N}$, denoted by ${\cal N}^\#$, is the classical channel with input and output alphabet $\bar{P}_1$, and transition probabilities ${\cal N}^\#(\sigma_i \mid \sigma_j) = p_k$, where $k$ is such that $\sigma_i \sigma_j = \sigma_k$ [^4].
Hence, ${\cal N}^\#$ is a memoryless symmetric channel, whose capacity is given by the mutual information for uniformly distributed input $\mathtt{I}({\cal N}^\#) = \frac{1}{2}(2 - h(\mathbf{p})) \in [0,1]$. It follows that $$\mathtt{I}({\cal N}^\#) = \displaystyle \frac{1 + I({\cal N})}{2}$$ Note that the right hand side term in the above equation is half the mutual information of the Pauli channel ${\cal N}$, for a uniformly distributed input.
It is worth noticing that the quantum channels synthesized during the quantum polarization of a Pauli channel are [*identifiable*]{} (see below) to classical mixtures of Pauli channels (this will be proved in Proposition \[prop:cq\_equiv\]). A Classical Mixture of Pauli (CMP) channels is a quantum channel ${\cal N}(\rho) = \sum_{x\in X} \lambda_x {\mathchoice{{{\ensuremath{\left\lvertx\right\rangle}}}}{{{\ensuremath{\lvertx\rangle}}}}{{{\ensuremath{\lvertx\rangle}}}}{{{\ensuremath{\lvertx\rangle}}}}}{\mathchoice{{{\ensuremath{\left\langlex\right\rvert}}}}{{{\ensuremath{\langlex\rvert}}}}{{{\ensuremath{\langlex\rvert}}}}{{{\ensuremath{\langlex\rvert}}}}} {\varotimes}{\cal N}_x(\rho)$, where $\{{\mathchoice{{{\ensuremath{\left\lvertx\right\rangle}}}}{{{\ensuremath{\lvertx\rangle}}}}{{{\ensuremath{\lvertx\rangle}}}}{{{\ensuremath{\lvertx\rangle}}}}}\}_{x\in X}$ is some orthonormal basis of an auxiliary system, ${\cal N}_x$ are Pauli channels, and $\sum_{x\in X} \lambda_x = 1$. We further extend Definition \[def:classical\_channel\] to the case of CMP channels, by defining the classical channel ${\cal N}^\#$ as the mixture of the channels ${\cal N}_x^\#$, where channel ${\cal N}_x^\#$ is used with probability $\lambda_x$. Hence, input and output alphabets of ${\cal N}^\#$ are $\bar{P}_1$ and $X \times \bar{P}_1$, respectively, with channel transition probabilities defined by ${\cal N}^\#(x, \sigma_i \mid \sigma_j) = \lambda_x\,{\cal N}_x(\sigma_i \mid \sigma_j)$. It also follows that: $$\displaystyle \mathtt{I}({\cal N}^\#) = \sum_x \lambda_x \mathtt{I}({\cal N}_x^\#) =\sum_x \lambda_x \frac{1 + I({\cal N}_x)}{2} = \frac{1 + I({\cal N})}{2}$$
Given two classical channels ${\cal U}$ and ${\cal V}$, we say they are equivalent, and denote it by ${\cal U} \equiv {\cal V}$, if they are defined by the same transition probability matrix, modulo a permutation of rows and columns. The following lemma states that the classical channel associated with a CMP channel does not depend on the basis.
\[lem:nclassical-indep-of-basis\]Let ${\cal N}(\rho) = \sum_{x\in X} \lambda_x {\mathchoice{{{\ensuremath{\left\lvertx\right\rangle}}}}{{{\ensuremath{\lvertx\rangle}}}}{{{\ensuremath{\lvertx\rangle}}}}{{{\ensuremath{\lvertx\rangle}}}}}{\mathchoice{{{\ensuremath{\left\langlex\right\rvert}}}}{{{\ensuremath{\langlex\rvert}}}}{{{\ensuremath{\langlex\rvert}}}}{{{\ensuremath{\langlex\rvert}}}}} {\varotimes}{\cal N}_x(\rho)$ and ${\cal M}(\rho) = \sum_{y\in Y} \tau_y {\mathchoice{{{\ensuremath{\left\lverty\right\rangle}}}}{{{\ensuremath{\lverty\rangle}}}}{{{\ensuremath{\lverty\rangle}}}}{{{\ensuremath{\lverty\rangle}}}}}{\mathchoice{{{\ensuremath{\left\langley\right\rvert}}}}{{{\ensuremath{\langley\rvert}}}}{{{\ensuremath{\langley\rvert}}}}{{{\ensuremath{\langley\rvert}}}}} {\varotimes}{\cal M}_y(\rho)$ be two CMP channels, where $\{{\mathchoice{{{\ensuremath{\left\lvertx\right\rangle}}}}{{{\ensuremath{\lvertx\rangle}}}}{{{\ensuremath{\lvertx\rangle}}}}{{{\ensuremath{\lvertx\rangle}}}}}\}_{x\in X}$ and $\{{\mathchoice{{{\ensuremath{\left\lverty\right\rangle}}}}{{{\ensuremath{\lverty\rangle}}}}{{{\ensuremath{\lverty\rangle}}}}{{{\ensuremath{\lverty\rangle}}}}}\}_{y\in Y}$ are orthonormal bases of the same auxiliary system. If ${\cal N} = {\cal M}$, then there exists a bijective mapping $\pi:X\rightarrow Y$, such that $\lambda_x = \tau_{\pi(x)}$ and ${\cal N}_x = {\cal M}_{\pi(x)}$. In particular, ${\cal N}^\# \equiv {\cal M}^\#$.
Finally, we say that a quantum channel ${\cal N}_{U\rightarrow AX}$ is [*identifiable*]{} to a channel ${\cal N}'_{U\rightarrow A}$ if, for some unitary operator $C$ on the $AX$ system, we have that ${\cal N}(\rho) = C\left({\cal N}'(\rho) {\varotimes}\frac{I_X}{|X|}\right)C^\dagger$, where $|X|$ denotes the dimension of the $X$ system. In other words, ${\cal N}$ and ${\cal N}'$ are equal modulo the conjugate action of an unitary operator $C$, and possibly after discarding a “useless” output system $X$. If ${\cal N}_{U\rightarrow AX}$ is identifiable to a CMP channel ${\cal N}'_{U\rightarrow A}$, we shall define ${\cal N}^\# {:=}\left({\cal N}'\right)^\#$. It can be seen that ${\cal N}^\#$ is well defined up to equivalence of classical channels, that is, if ${\cal N}_{U\rightarrow AX}$ is identifiable to another CMP channel ${\cal N}''_{U\rightarrow A}$, then $\left({\cal N}'\right)^\# \equiv \left({\cal N}''\right)^\#$. This follows from the following lemma, proven in Appendix \[sec:proof-nclassical-well-defined\].
\[lem:nclassical-well-defined\] Let ${\cal N}'$ and ${\cal N}''$ be two CMP channels, such that ${\cal N}'(\rho) {\varotimes}\frac{I_X}{|X|} = C\left({\cal N}''(\rho) {\varotimes}\frac{I_X}{|X|}\right)C^\dagger$, for some unitary $C$. Then $\left({\cal N}'\right)^\# \equiv \left({\cal N}''\right)^\#$.
Classical Channel Combining and Splitting Operations {#subsec:classical_combining_splitting}
----------------------------------------------------
[**Simplified notation:**]{} To simplify notation, we shall identify $\left(\bar{P}_1, \times\right) \cong \left(\{0,1,2,3\}, \oplus\right)$, by identifying $\sigma_u \cong u$, $\forall u=0,\dots,3$, where the additive group operation $u\oplus v$ is given by the bitwise exclusive [or]{} ([xor]{}) between the binary representations of integers $u,v$. The classical counterpart ${\cal N}^\#$ of a Pauli channel ${\cal N}(\rho) = \sum_{u=0}^3 p_u \sigma_u \rho \sigma_u^{\dagger}$ (Definition \[def:classical\_channel\]), is therefore identified to a channel with input and output alphabet $\bar{P}_1 \cong \{0,1,2,3\}$, and transition probabilities ${\cal N}^\#(u \mid v) = p_{u\oplus v}$.
Let $N$ and $M$ be two classical channels, both with input alphabet $\bar{P}_1 \cong \{0,1,2,3\}$, and output alphabets $A$ and $B$, respectively. Channel transition probabilities are denoted by $N(a\mid u)$ and $M(b \mid v)$, for $u,v\in \bar{P}_1$, $a\in A$, and $b \in B$. Let $\Gamma : \bar{P}_1\times\bar{P}_1 \rightarrow \bar{P}_1\times\bar{P}_1$ be any permutation, and write $\Gamma = (\Gamma_1, \Gamma_2)$, with $\Gamma_i: \bar{P}_1\times\bar{P}_1 \rightarrow \bar{P}_1$, $i=1,2$. The [*combined channel*]{} $N\bowtie_\Gamma M$ is defined by: $$(N\bowtie_\Gamma M) (a,b \mid u,v) = N(a \mid \Gamma_1(u,v))M(b \mid \Gamma_2(u,v))$$ It is further [*split*]{} into two channels $N \boxast_\Gamma M$ and $N \varoast_\Gamma M$, defined by: $$\begin{aligned}
(N \boxast_\Gamma M)(a, b \mid u) & = & \frac{1}{4} \sum_v (N \bowtie_\Gamma M) (a,b \mid u,v) \\
(N \varoast_\Gamma M)(a, b, u \mid v) & = & \frac{1}{4} (N \bowtie_\Gamma M) (a,b \mid u,v),\end{aligned}$$
Applying the above construction to classical counterparts of CMP channels, we have the following proposition, proven in Appendix \[sec:proof\_cq\_equiv\].
\[prop:cq\_equiv\] Let ${\cal N}_{U\rightarrow A}$ and ${\cal M}_{V\rightarrow B}$ be two CMP channels, and $C$ be any two-qubit Clifford unitary, acting on the two qubit system $UV$. Let ${\cal N}^\#$ and ${\cal M}^\#$ denote the two classical counterparts of the above CMP channels, and $\Gamma {:=}\Gamma(C)$ be the permutation induced by the conjugate action of $C$ on $\bar{P}_1 \times \bar{P}_1$. Then ${\cal N}\boxast_C{\cal M}$ and ${\cal N}\varoast_C{\cal M}$ are identifiable to CMP channels, thus $({\cal N}\boxast_C{\cal M})^\#$ and $({\cal N}\varoast_C{\cal M})^\#$ are well defined, and the following properties hold:
- $({\cal N}\boxast_C{\cal M})^\# \equiv {\cal N}^\#\boxast_\Gamma{\cal M}^\#$
- $({\cal N}\varoast_C{\cal M})^\# \equiv {\cal N}^\#\varoast_\Gamma{\cal M}^\#$
A consequence of the above proposition is that a CMP channel polarizes under the recursive application of the channel combining and splitting rules, if and only if its classical counterpart does so. Moreover, processes of both quantum and classical polarization yield the same set of indexes for the good/bad channels. More precisely, we have the following:
Let ${\cal W}$ be a CMP channel, and ${\cal W}^{(i_1\dots i_{n})}$ be defined recursively as in (\[eq:polar\_coding\_recursion\]), $\forall n > 0$, $\forall i_1,\dots,i_n\in\{0,1\}$. Let ${\cal W}^\#$ be the classical counterpart of ${\cal W}$, and $({\cal W}^\#)^{(i_1\dots i_{n})}$ be defined recursively, similar to (\[eq:polar\_coding\_recursion\]), while replacing ${\cal W}$ by ${\cal W}^\#$, and Clifford unitaries $C_{i_1\dots i_n}$ by the corresponding permutations $\Gamma_{i_1\dots i_n} = \Gamma(C_{i_1\dots i_n})$. Then $ \left({\cal W}^{(i_1\dots i_{n})}\right)^\# \equiv ({\cal W}^\#)^{(i_1\dots i_{n})}$, $\forall n, \forall i_1,\dots, i_{n}$. In particular: $$\displaystyle \mathtt{I}\left(({\cal W}^\#)^{(i_1\dots i_{n})}\right) = \frac{1 + I\left({\cal W}^{(i_1\dots i_{n})}\right)}{2}$$
As we already know that the quantum transform polarizes, it follows that the classical transform does also polarize. Moreover, a direct proof of the classical polarization can be derived by verifying the conditions from Lemma \[lemma:stochastic\_proc\_convergence\], with stochastic process $\{T_n : n\geq 0\}$ given by Bhattacharyya parameter $Z$ of the classical channels synthesized during the recursive construction. We recall below the definition of the Bhattacharyya parameter for a classical channel $W$, as defined in [@sta09]. We shall restrict out attention to classical channels with input alphabet $\bar{P}_1$.
\[def:bhattacharyya\] Let $W$ be a classical channel, with input alphabet $\bar{P}_1 \cong (\{0,1,2,3\}, \oplus)$ and output alphabet $Y$. For $u,u',d\in \bar{P}_1$, we define $$\begin{aligned}
Z(W_{u, u^\prime}) &:= \sum_{y\in Y} \sqrt{W(y|u) W(y|u^\prime)} \label{eq:bhattacharyya1} \\
Z_d(W) &:= \frac{1}{4}\sum_{u \in \bar{P}_1} Z(W_{u, u\oplus d}) \label{eq:bhattacharyya_d}\end{aligned}$$ In particular, note that $Z(W_{u, u}) = 1, \forall u\in \bar{P}_1$, and $Z_0(W) = 1$. The Bhattacharyya parameter of $W$, denoted $Z(W)$, is then defined as $$\label{eq:bhattacharyya}
Z(W) := \frac{1}{3} \sum_{d\neq 0} Z_d(W) = \frac{1}{12} \sum_{u \neq u^\prime} Z(W_{u, u^\prime})$$
Polarization of the classical channel ${\cal W}^\#$ follows then from the lemma below, whose proof is provided in Appendix \[apnd:pol9\].
\[lem:z-good-classical-channel\] Let ${\cal W}$ be a CMP channel and ${\cal W}^\#$ its classical counterpart. Given two instances of the channel ${\cal W}^\#$, then $$\label{eq:pauli_pol_9_clif}
{\mathbb{E}}_{\Gamma \in \Gamma(\mathcal{L})} Z\left({\cal W}^\# \varoast_{\Gamma} {\cal W}^\# \right) = {\mathbb{E}}_{\Gamma \in \Gamma(\mathcal{R})} Z\left({\cal W}^\# \varoast_{\Gamma} {\cal W}^\#\right) = \frac{1}{3}Z({\cal W}^\#) + \frac{2}{3}Z({\cal W}^\#)^2,$$ where $\Gamma(\mathcal{L})$ and $\Gamma(\mathcal{R})$ denote the set of permutations generated on $\bar{P}_2$ by the conjugate action of Cliffords in $\mathcal{L}$ and $\mathcal{R}$, respectively.
Polarization Using Only 3 Clifford Gates {#subsec:polarization_3_cliffords}
----------------------------------------
In this section, we show that for Pauli channels the set of channel combining Clifford gates can be reduced to 3 gates only, while still ensuring polarization. Let ${\cal S}$ denote the set containing the Clifford gates $L_{1,3}$, $L_{2,2}$, and $L_{3,1}$ from Figure \[fig:cliff\_generators\], and $\Gamma(\mathcal{S})$ denote the corresponding set of permutations, namely $\Gamma(L_{1,3})$, $\Gamma(L_{2,2})$ and $\Gamma(L_{3,1})$, generated by the conjugate actions of $L_{1,3}$, $L_{2,2}$, and $L_{3,1}$ on $\bar{P}_1 \times \bar{P}_1$.
\[lem:pol\_3\_clif\] Let ${\cal W}$ be a CMP channel and ${\cal W}^\#$ its classical counterpart. Given two instances of the channel ${\cal W}^\#$, then $$\label{eq:pauli_pol_3_clif}
{\mathbb{E}}_{\Gamma \in \Gamma(\mathcal{S})} Z\left(W^\# \varoast_{\Gamma} W^\# \right) \leq \frac{1}{3}Z(W^\#) + \frac{2}{3}Z(W^\#)^2$$
The proof is given in appendix \[apnd:pol3\].
Speed of Polarization
---------------------
Before discussing decoding of quantum polar codes over Pauli channels (Section \[sec:decod\_pauli\_channel\]), it is worth noticing that classical polar codes come equipped with a decoding algorithm, known as successive cancellation (SC) [@arikan09]. However, the effectiveness of the classical SC decoding, [*i.e.*]{}, its capability of successfully decoding at rates close to the capacity, depends on the speed of polarization. The Bhattacharyya parameter of the synthesized channels plays an important role in determining the speed at which polarization takes place. First, we note that for a classical channel $W$, the Bhattacharyya parameter upper bounds the error probability of uncoded transmission. Precisely, given a classical channel $W$ with input alphabet $X$, the error probability of the maximum-likelihood decoder for a single channel use, denoted $P_e$, is upper-bounded as follows ([@sta09 Proposition 2]): $$P_e \leq (|X|-1)Z(W)$$ Now, consider a polar code defined by the recursive application of $n$ polarization steps to the classical channel $W:=\mathcal{W}^\#$ (the input alphabet is $X:=\bar{P}_1$, of size $|\bar{P}_1|=4$). The construction is the same as the one in Section \[sec:quantum\_polar\_coding\], while replacing the quantum channel $\mathcal{W}$ by its classical counterpart $W$, and channel combining Clifford gates $C_{i_1 i_2\dots}$ by the corresponding permutations $\Gamma_{i_1 i_2\dots} := \Gamma(C_{i_1 i_2\dots})$. For any $i=0,\dots,N-1$, let $W^{(i)} := (\mathcal{W}^\#)^{(i_1\dots i_n)}$, where $i_1\dots i_n$ is the binary decomposition of $i$. For the sake of simplicity, we drop the channel combining permutations $\Gamma$’s from the above notation. Let $\mathcal{I} \subset\{0,1,\dots,N-1 \}$ denote the set of good channels ([*i.e.*]{}, channels used to transmit information symbols, as opposed to bad channels, which are frozen to symbol values known to the both encoder and decoder). Since the SC decoding proceeds by decoding successively the synthesized good channels[^5], it can be easily seen that the block error probability of the SC decoder, denoted by $P_e(N, \mathcal{I})$, is upper-bounded by (see also [@arikan09 Proposition 2]): $$\label{eq:pe_sc_upperbound}
P_e(N, \mathcal{I}) \leq 3\sum_{i \in\mathcal{I}}Z(W^{(i)})$$ If the the Bhattacharyya parameters of the $W^{(i)}$ channels, with $i \in\mathcal{I}$, converge sufficient fast to zero, one can use (\[eq:pe\_sc\_upperbound\]) to ensure that $P_e(N, \mathcal{I})$ goes to zero. Since the number of terms in the right hand side of (\[eq:pe\_sc\_upperbound\]) is linear in $N$, it is actually enough to prove that $Z(W^{(i)}) \leq O(N^{-(1+\theta)}), \forall i\in \mathcal{I}$, for some $\theta > 0$.
The proof of fast polarization properties in [@sta09 Lemma 3], for channels with non-binary input alphabets, exploits two main ingredients:
- The quadratic improvement of the Bhattacharyya parameter, when taking the good channel, [*i.e.*]{}, $Z\left(W^{(i_1\dots i_{n-1}i_n)}\right) \leq Z\left(W^{(i_1\dots i_{n-1})}\right)^2$, $\forall i_1\dots i_{n-1}i_n\in\{0,1\}^n$, such that $i_n=1$.
- The linearly upper-bounded degradation of the Bhattacharyya parameter, when taking the bad channel, [*i.e.*]{}, $Z\left(W^{(i_1\dots i_{n-1}i_n)}\right) \leq \kappa Z\left(W^{(i_1\dots i_{n-1})}\right)$, $\forall i_1\dots i_{n-1}i_n\in\{0,1\}^n$, such that $i_n=0$, for some constant $\kappa>0$.
Regarding the second condition, in our case we have the following lemma, where for a classical channel $W$ with input alphabet $\bar{P}_1 \cong \{0,1,2,3\}$, we define $$\bar{Z}(W) := \max_{d=1,2,3} Z_d(W)$$
\[lem:Fast\_Polarization\] For any classical channel $W$ with input alphabet $\bar{P}_1$, and any linear permutation $\Gamma : \bar{P}_1\times\bar{P}_1 \rightarrow \bar{P}_1\times\bar{P}_1$, the following inequalities hold: $$\begin{aligned}
\bar{Z}(W \boxast_\Gamma W) &\leq 4 \bar{Z}(W) \\
Z(W \boxast_\Gamma W) &\leq 12 Z(W)\end{aligned}$$
The proof is given in Appendix \[sec:proof\_lemma\_fast\_polarization\].
Condition (1) above – quadratic improvement of the Bhattacharyya parameter, when taking the good channel – is more problematic, due to the linear term in the right hand side of equations (\[eq:pauli\_pol\_9\_clif\]) and (\[eq:pauli\_pol\_3\_clif\]). In particular, we can not apply [@sta09 Lemma 3] to derive fast polarization properties in our case. Instead, we will prove fast polarization properties by drawing upon arguments similar to those in the proof of [@arikan09 Theorem 2]. First, we need the following definition.
Let $W$ be a classical channel with input alphabet $\bar{P}_1$, and $\bm{\Gamma} = \{ \Gamma, \Gamma_{i_1\dots i_n} \mid n > 0, \break i_1\dots i_n \in\{0,1\}^n \}$ be an infinite sequence of permutations. For $n > 0$, let $$\label{eq:c_polar_recursion}
W^{(i_1\dots i_n)} := \left\{ \begin{array}{ll}
W^{(i_1\dots i_{n-1})} \boxast_{\Gamma_{i_1\dots i_{n-1}}} W^{(i_1\dots i_{n-1})}, & \text{if } i_n = 0 \\
W^{(i_1\dots i_{n-1})} \varoast_{\Gamma_{i_1\dots i_{n-1}}} W^{(i_1\dots i_{n-1})}, & \text{if } i_n = 1
\end{array} \right.$$ where, for $n=1$, in the right hand side term of the above equality, we set by convention $W^{(\varnothing)} := W$ and $\Gamma_\varnothing := \Gamma$. We say that $\bm{\Gamma}$ is a [*polarizing sequence*]{} (or that polarization happens for $\bm{\Gamma}$), if for any $\delta > 0$, $$\lim_{n\rightarrow\infty} \frac{\#\{(i_1\dots i_{n}) \in \{0,1\}^{n} : \mathtt{I}\left( {\cal W}^{(i_1\dots i_{n})} \right) \in (\delta, 1-\delta) \}}{2^n} = 0$$
Note that different from (the classical counterpart of) Theorem \[thm:quantum\_polarization\], we consider here a given sequence of permutations, instead of averaging over some set of sequences. If $W=\mathcal{W}^\#$ is the classical counterpart of a CMP channel $\mathcal{W}$, by Lemma \[lem:pol\_3\_clif\], we know that polarization happens when averaging over all the sequences $\bm{\Gamma} \in \Gamma(\mathcal{S})^\infty$. As a consequence, there exists a subset $\Gamma(\mathcal{S})^\infty_{\text{pol}} \subset \Gamma(\mathcal{S})^\infty$ of positive probability[^6], such that polarization happens for any $\bm{\Gamma} \in \Gamma(\mathcal{S})^\infty_{\text{pol}}$.
\[prop:fast\_polarization\] Let $\mathcal{W}$ be a CMP channel, $W:=\mathcal{W}^\#$ its classical counterpart, and $\mathcal{S}$ the set of three Clifford gates from Section \[subsec:polarization\_3\_cliffords\]. Then the following [*fast polarization property*]{} holds for almost all $\bm{\Gamma}$ sequences in $\Gamma(\mathcal{S})^\infty_{\text{pol}}$:
For any $\theta > 0$ and $R < \mathtt{I}(W)$, there exists a sequence of sets $\mathcal{I}_N \subset \{0,\dots,N-1\}$, $N\in\{1, 2, \dots,2^n,\dots\}$, such that $|\mathcal{I}_N| \geq NR$ and $Z\left( W^{(i)} \right) \leq O\left(N^{-(1+\theta)}\right)$, $\forall i\in \mathcal{I}_N$. In particular, the block error probability of polar coding under SC decoding satisfies $$P_e(N, \mathcal{I}_N) \leq O\left(N^{-\theta}\right)$$
Decoding the Quantum Polar Code by Using its Classical Counterpart {#sec:decod_pauli_channel}
------------------------------------------------------------------
Let ${\cal W}$ be a CMP channel and ${\cal W}^\#$ its classical counterpart. Let $G_q$ denote the unitary operator corresponding to the quantum polar code (defined by the recursive application of $n$ polarization steps, see Section \[sec:quantum\_polar\_coding\]), and $G_c$ denote the linear transformation corresponding to the classical polar code. Let ${\cal I}$ and ${\cal J}$ be the set of indexes corresponding to the good and bad channels, respectively, with $|{\cal I}| + |{\cal J}| = N {:=}2^n$. We shall use the following notation from Section \[sec:quantum\_polar\_coding\]:
- $\rho_{\cal I}$ denotes the original state of system ${\cal I}$,
- $\varphi_{{\cal IJJ}'} {:=}(G_q{\varotimes}I_{{\cal J}'}) (\rho_{\cal I} {\varotimes}\Phi_{\cal JJ'}) (G_q^\dagger {\varotimes}I_{{\cal J}'})$ denotes the [*encoded state*]{}, where $\Phi_{\cal JJ'}$ is a maximally entangled state, as defined in (\[eq:max\_entangled\_state\]).
- $\psi_{{\cal IJJ}'} {:=}({\cal W}^{{\varotimes}N} {\varotimes}I_{{\cal J}'}) (\varphi_{{\cal IJJ}'})$ denotes the [*channel output state*]{}.
Since ${\cal W}$ is a CMP channel, it follows that:
$$\psi_{{\cal IJJ}'} = (E_{{\cal I}{\cal J}}G_q{\varotimes}I_{{\cal J}'}) (\rho_{\cal I} {\varotimes}\Phi_{\mathcal{J}\mathcal{J'}}) (G_q^\dagger E_{{\cal I}{\cal J}}^\dagger {\varotimes}I_{{\cal J}'})$$
for some [*error*]{} $E_{{\cal I}{\cal J}} \in P_N$. Hence, quantum polar code decoding can be performed in the 4 steps described below.
[**Step 1: Apply the inverse quantum polar transform on the channel output state.**]{} Applying $G_q^\dagger$ on the output state $\psi_{{\cal IJJ}'}$, leaves the ${\cal IJJ}'$ system in the following state: $$\begin{aligned}
\psi'_{{\cal IJJ}'} &= (G_q^\dagger E_{{\cal I}{\cal J}}G_q{\varotimes}I_{{\cal J}'}) (\rho_{\cal I} {\varotimes}\Phi_{\mathcal{J}\mathcal{J'}}) (G_q^\dagger E_{{\cal I}{\cal J}}^\dagger G_q {\varotimes}I_{{\cal J}'}) \nonumber \\
&= (E'_{{\cal I}{\cal J}}{\varotimes}I_{{\cal J}'}) (\rho_{\cal I} {\varotimes}\Phi_{\mathcal{J}\mathcal{J'}}) (E{'}_{{\cal I}{\cal J}}^{\,\dagger} {\varotimes}I_{{\cal J}'})\end{aligned}$$ where $E'_{{\cal I}{\cal J}} {:=}G_q^\dagger E_{{\cal I}{\cal J}}G_q$. Since we only need to correct up to a global phase, we may assume that $E'_{{\cal I}{\cal J}}, E_{{\cal I}{\cal J}} \in {P}_N / \{\pm 1, \pm i\} \simeq {\bar P}_1^{N}$, and thus write $E'_{{\cal I}{\cal J}} = G_c^{-1} E_{{\cal I}{\cal J}}$, or equivalently: $$E_{{\cal I}{\cal J}} = G_c E'_{{\cal I}{\cal J}}$$ Put differently, $E_{{\cal I}{\cal J}}$ is the classical polar encoded version of $E'_{{\cal I}{\cal J}}$.
[**Step 2: Quantum measurement.**]{} Let $E'_{{\cal I}{\cal J}} = \displaystyle \mathop{{\varotimes}}_{i\in {\cal I}} E'_i \mathop{{\varotimes}}_{j\in {\cal J}} E'_j$, with $E'_i, E'_j \in \bar{P}_1$. Measuring $X_jX_{j'}$ and $Z_jZ_{j'}$ observables[^7], allows determining the value of $E'_j$, for any $j \in {\cal J}$, since no errors occurred on the ${\cal J}'$ system.
[**Step 3: Decode the classical polar code counterpart.**]{} We note that the error $E_{{\cal I}{\cal J}}$ can be seen as the output of the classical vector channel $({\cal W}^\#)^{N}$, when the [*“all-identity vector”*]{} $\sigma_0^N \in \bar{P}_1^N$ is applied at the channel input. However, by the definition of the classical channel ${\cal W}^\#$, we have $({\cal W}^\#)^{N}(E_{{\cal I}{\cal J}} \mid \sigma_0^N) = ({\cal W}^\#)^{N}(\sigma_0^N \mid E_{{\cal I}{\cal J}})$, meaning that we can equivalently consider $\sigma_0^N$ as being the observed channel output, and $E_{{\cal I}{\cal J}}$ the (unknown) channel input. Hence, we have given $(i)$ the value of $E'_{\cal J} {:=}\mathop{{\varotimes}}_{j\in {\cal J}} E'_j$, and $(ii)$ a noisy observation (namely $\sigma_0^N$) of $E_{{\cal I}{\cal J}} = G_c E'_{{\cal I}{\cal J}}$. We can then use classical polar code decoding to recover the value of $E'_{\cal I} {:=}\mathop{{\varotimes}}_{i\in {\cal I}} E'_i$. [**Step 4: Error correction.**]{} Once we have recovered the $E'_{\cal J}$ (step 2) and $E'_{\cal I}$ (step 3) values, we can apply the $E'_{{\cal I}{\cal J}} {\varotimes}I_{{\cal J}'}$ operator on $\psi'_{{\cal IJJ}'}$, thus leaving the ${\cal IJJ}'$ system in the state $\rho_{\cal I} {\varotimes}\Phi_{\mathcal{J}\mathcal{J'}}$.
Polarization with Vanishing Rate of Preshared Entanglement
==========================================================
In this section we present a code construction using an asymptotically vanishing rate or preshared entanglement, while achieving a transmission rate equal to the coherent information of the channel. In particular, we shall assume that the coherent information of the channel is positive, $I({\cal W}) > 0$. The proposed construction bears similarities to the universal polar code construction in [@hassani2014universal Section V], capable of achieving the compound capacity of a finite set of classical channels.
Let $P_q(N, \mathcal{J}, \mathcal{I})$ denote a quantum polar code of length $N=2^n$, for some $n > 0$, where $\mathcal{I}$ and $\mathcal{J}$ denote the sets of good and bad channels respectively. By Theorem \[thm:quantum\_polarization\], as $n$ goes to infinity, $|{\cal I}|$ approaches $\frac{1+I(\mathcal{W})}{2}N$, and thus $|{\cal J}|$ approaches $\frac{1-I(\mathcal{W})}{2}N$. Since $I(\mathcal{W}) > 0$, it follows that $|{\cal J}| < |{\cal I}|$, provided that $n$ is large enough. Therefore, we may find a subset of good channels $\mathcal{I}' \subset \mathcal{I}$, such that $|\mathcal{I}'| = |\mathcal{J}|$. In the sequel, we shall extend the definition of a polar code to include such a subset ${\cal I}'$, and denote it by $P_q(N, \mathcal{J}, \mathcal{I}, \mathcal{I}')$.
Let us now consider $k$ copies of a quantum polar code $P_q(N, \mathcal{J}, \mathcal{I}, \mathcal{I}')$, denoted by $P_q^l(N, \mathcal{J}_l, \mathcal{I}_l, \mathcal{I}'_l)$ or simply by $P_q^l$, for any $ l \in \{0, 1, \dots k-1 \}$. We define a quantum code $C_q^k$ of codelength $|C_q^k| = kN$, by [*chaining*]{} them in the following way (see also Figure \[fig:three\_polar\_Catal\]):
- For system $\mathcal{J}_0$, the input quantum state before encoding is half of a maximally entangled state $\Phi_{\mathcal{J}_0\mathcal{J}'_0}$, where system $\mathcal{J}'_0$ is part of channel output. This is the only preshared entanglement between the sender and the receiver.
- For systems $\mathcal{I}'_{l-1}$ and $\mathcal{J}_{l}$, with $l \neq 0$, the input quantum state before encoding is a maximally entangled state $\Phi_{\mathcal{I}'_{l-1}\mathcal{J}_{l}}$.
- Systems $ \mathcal{I}_l \setminus \mathcal{I}'_l$, for $l \neq k - 1$, and $\mathcal{I}_{k-1}$ are information systems, meaning that the corresponding quantum state is the one that has to be transmitted from the sender to the receiver.
(0, 0) to node\[above\](c) ++(1,0) to++(0,-0.1) (0,0)++(1, 0) to ++(0,0.1) (0,0)++(1, 0) to node\[above\](c) ++(.75, 0) to ++(0,0.1) (0,0)++(1, 0) to ++(.75, 0) to ++(0,-0.1) (0,0)++(1, 0) to ++(.75, 0) to ++(1, 0) ; (0,0) – (1,0) node \[black,midway,yshift=.5cm\] [$\mathcal{J}_0$]{} ; (1,0) – (2.75,0) node \[black,midway,yshift=-.65cm\] [$\mathcal{I}_0$]{} ; (1.75,0) – (2.75,0) node \[black,midway,yshift=.5cm\] [$\mathcal{I}'_0$]{} ; (0, 0) ++(1.37, 0) ++ (0, -1.1) node\[\][$P_q^0$]{} ; (4,0) to node\[above\](c) ++(1,0) to++(0,-0.1) (4,0)++(1, 0) to ++(0,0.1) (4,0)++(1, 0) to node\[above\](c) ++(.75, 0) to ++(0,0.1) (4,0)++(1, 0) to ++(.75, 0) to ++(0,-0.1) (4,0)++(1, 0) to ++(.75, 0) to ++(1, 0) ; (4,0) – (5,0) node \[black,midway,yshift=.5cm\] [$\mathcal{J}_1$]{} ; (5,0) – (6.75,0) node \[black,midway,yshift=-.65cm\] [$\mathcal{I}_1$]{} ; (5.75,0) – (6.75,0) node \[black,midway,yshift=.5cm\] [$\mathcal{I}'_1$]{} ; (4, 0) ++(1.37, 0) ++ (0, -1.1) node\[\][$P_q^1$]{} ; (8, 0) to node\[above\](c) ++(1,0) to++(0,-0.1) (8,0)++(1, 0) to ++(0,0.1) (8,0)++(1, 0) to node\[above\](c) ++(.75, 0) to ++(0,0.1) (8,0)++(1, 0) to ++(.75, 0) to ++(0,-0.1) (8,0)++(1, 0) to ++(.75, 0) to ++(1, 0) ; (8,0) – (9,0) node \[black,midway,yshift=.5cm\] [$\mathcal{J}_2$]{} ; (9,0) – (10.75,0) node \[black,midway,yshift=-.65cm\] [$\mathcal{I}_2$]{} ; (9.75,0) – (10.75,0) node \[black,midway,yshift=.5cm\] [$\mathcal{I}'_2$]{} ; (2.25, 0) ++(0, 0.8) to ++(0, .3) to ++(1.1, .75) to ++(1.1, -.75) to ++(0, -.3) (6.25, 0) ++(0, 0.8) to ++(0, .3) to ++(1.1, .75) to ++(1.1, -.75) to ++(0, -.3) ; (3.35,2.1) node[$\Phi_{\mathcal{I}'_{0}\mathcal{J}_{1}}$]{} (7.35,2.1) node[$\Phi_{\mathcal{I}'_{1}\mathcal{J}_{2}}$]{} ; (8, 0) ++(1.37, 0) ++ (0, -1.1) node\[\][$P_q^2$]{} ; (-2.25,0) – (-1.25,0) node \[black,midway,yshift=.5cm\] [$\mathcal{J}'_0$]{} ; (-2.25,0) to (-1.25,0) ; (-2.25,-0.1) to (-2.25,0.1) (-1.25,-0.1) to (-1.25,0.1) ; (-1.75, 0) ++(0, 0.8) to ++(0, .3) to ++(1.1, .75) ; (-0.65, 1.85) to ++(1.1, -.75) to ++(0, -.3) ; (-0.65,2.1) node[$\Phi_{\mathcal{J}'_{0}\mathcal{J}_{0}}$]{} ; (-0.65,1) to (-0.65,-1.5) ; (-0.65,-1.1) node\[left\][Receiver]{} (-0.65,-1.1) node\[right\][Sender]{} ;
It can be easily seen that the transmission (coding) rate of the proposed scheme is given by $$R := \frac{\sum_{l=0}^{k-2} |\mathcal{I}_l \setminus \mathcal{I}'_l| + |\mathcal{I}_{k-1}|}{kN} \xrightarrow[n\rightarrow \infty]{}
\frac{(k-1)I(\mathcal{W}) + \frac{1+I(\mathcal{W})}{2}}{k} \xrightarrow[k\rightarrow \infty]{} I(\mathcal{W}),$$ while the rate of preshared entanglement is given by $$E := \frac{|\mathcal{J}_0|}{kN} \xrightarrow[n\rightarrow \infty]{} \frac{1-I(\mathcal{W})}{2k} \xrightarrow[k\rightarrow \infty]{} 0.$$
**Decoding $C_q^k$:** We shall assume that we have given an effective algorithm capable of decoding the quantum polar code $P_q$. We note that this is indeed the case for Pauli channels (Section \[sec:decod\_pauli\_channel\]), but it is an open problem for general quantum channels. In this case, $C_q^k$ can be decoded sequentially, by decoding first $P_q^0$, then $P_q^1$, $P_q^2$, and so on. Indeed, after decoding $P_q^0$, thus in particular correcting the state of the $\mathcal{I}'_0$ system, the EPR pairs $\Phi_{\mathcal{I}'_0\mathcal{J}_1}$ will play the role of the preshared entanglement required to decode $P_q^1$. Therefore, $P_q^1$ can be decoded once $P_q^0$ has been decoded, and similarly, $P_q^l$ can be decoded after $P_q^{l - 1}$ has been decoded, for any $ l \in \{2, \dots k - 1 \}$.
[**Entanglement as a catalyst:**]{} Finally, the above coding scheme can be slightly modified, such that preshared entanglement between the sender and the receiver is not consumed. In the above construction, we have considered that for the last $P_q^{k-1}$ polar code, the $\mathcal{I}'_{k-1}$ system is an information system, [*i.e.*]{}, used to transmit quantum information from the sender to the receiver (system $\mathcal{I}'_{2}$ in Figure \[fig:three\_polar\_Catal\]). Let us now assume that the input quantum state to the $\mathcal{I}'_{k-1}$ system is half of a maximally entangled state $\Phi_{\mathcal{I}'_{k-1} \mathcal{J}_{k}}$, where quantum system $\mathcal{J}_{k}$ is held by the sender. When the receiver completes decoding of the $C_q^k$ code, it restores the initial state of the $\mathcal{I}'_{k-1}$, thus resulting in a maximally entangled state $\Phi_{\mathcal{I}'_{k-1} \mathcal{J}_{k}}$ shared between the sender ($\mathcal{J}_{k}$ system) and the receiver ($\mathcal{I}'_{k-1}$ system). Hence, the initial preshared entanglement $\Phi_{\mathcal{J}_0\mathcal{J}'_0}$ acts as a catalyst, in that it produces a new state $\Phi_{\mathcal{I}'_{k-1} \mathcal{J}_{k}}$ shared between the sender and the receiver, which can be used for the next transmission.
Conclusion and Perspectives
===========================
In this paper, we have shown that, with entanglement assistance, the polarization phenomenon appears at the quantum level with a construction using randomized two-qubit Clifford gates instead of the CNOT gate. In the case of Pauli channels, we have proven that the quantum polarization is equivalent to a classical polarization for an associated non-binary channel which allows us to have an efficient decoding scheme. We also proved a fast polarization property in this case. Finally, we presented a quantum polar code chaining construction, for which the required entanglement assistance is negligible with respect to the code length.
A natural further direction would be to see whether it is possible to achieve quantum polarization without entanglement assistance and also to find an efficient decoding scheme for general quantum channels.
Acknowledgements {#acknowledgements .unnumbered}
================
This research was supported in part by the “Investissements d’avenir” (ANR-15-IDEX-02) program of the French National Research Agency. AG acknowledges the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 754303.
Proof of Lemma \[lemma:equiv\_conditions\] {#sec:proof-quantum-red-clifford}
==========================================
We need to prove that $C'$ and $C''$ yield bad/good quantum channels with the same Réyni-Bhattacharyya parameter, when used for combining (then splitting) two quantum channels $\mathcal{N}$ and $\mathcal{M}$. In this section, we use notation $ U\Big[ \rho \Big] {:=}U \rho U^\dagger $.
### Bad Channel {#subsec:red_cliff_bad}
We have following equalities for the complementary of bad channel:
$$\begin{aligned}
\rho_{A_1E_1R_2E_2}'' = (\mathcal{N} \boxast_{C''} \mathcal{M})^c(\Phi_{A_1 A_1'}) = \mathcal{N}^c {\varotimes}\mathcal{M}^c \Big( C_{A_1'A_2'}''\Big[\Phi_{A_1A_1'} {\varotimes}\Phi_{R_2 A_2'}\Big]\Big)\end{aligned}$$
$$\begin{aligned}
\rho_{A_1E_1R_2E_2}' = (\mathcal{N} \boxast_{C'} \mathcal{M})^c(\Phi_{A_1 A_1'}) = \mathcal{N}^c {\varotimes}\mathcal{M}^c \Big( C_{A_1'A_2'}'\Big[\Phi_{A_1A_1'} {\varotimes}\Phi_{R_2 A_2'}\Big]\Big),\end{aligned}$$
where $ C_{A_1'A_2'}'' = C_{A_1'A_2'}' (C_{A_1'}^1 {\varotimes}C_{A_2'}^2)$.
\[prop:equi\_cliff\_bad\] $$\rho_{A_1E_1R_2E_2}'' = C_{A_1}^{1^{\top}} {\varotimes}C_{R_2}^{2^{\top}} \Big[\rho_{A_1E_1R_2E_2}' \Big] \nonumber$$
$$\begin{aligned}
\rho_{A_1E_1R_2E_2}'' &= \mathcal{N}_{A_1' \rightarrow E_1}^c {\varotimes}\mathcal{M}_{A_2' \rightarrow E_2}^c \big( C_{A_1'A_2'}' \cdot C_{A_1'}^1 {\varotimes}C_{A_2'}^2 \big[\Phi_{A_1A_1'} {\varotimes}\Phi_{R_2 A_2'})\big] \big) \\
& = \mathcal{N}_{A_1' \rightarrow E_1}^c {\varotimes}\mathcal{M}_{A_2' \rightarrow E_2}^c \Big( C_{A_1'A_2'}' \cdot C_{A_1}^{1^\top}{\varotimes}C_{R_2}^{2^\top} \Big[\Phi_{A_1A_1'} {\varotimes}\Phi_{R_2 A_2'} \Big] \Big)\\
&= C_{A_1}^{1^\top} {\varotimes}C_{R_2}^{2^\top} \Big[\rho_{A_1E_1R_2E_2}' \Big],\end{aligned}$$
where second equality follows from the relation $({\mathbbm{1}}{\varotimes}Z) \Big[\Phi\Big] = (Z^\top {\varotimes}{\mathbbm{1}}) \Big[\Phi\Big]$, for any matrix $Z$.
**Proof of $R( \mathcal{N} \boxast_{C''} \mathcal{M} ) = R( \mathcal{N} \boxast_{C'} \mathcal{M} )$:** By definition [@tbh14], $$-\Tilde{H}_2^{\downarrow}(A|B)_{\rho} = \Tilde{D}_2(\rho_{AB} || {\mathbbm{1}}{\varotimes}\rho_B ),$$ where $\Tilde{D}_2(\rho || \sigma)$ is quantum Rényi divergence of order 2 defined in [@mdsft13], and it satisfies following unitary equivalence: $$\label{eq:uni_equiv}
\Tilde{D}_2(\rho || \sigma) = \Tilde{D}_2(U \rho U^\dagger || U \sigma U^\dagger )$$ Now, $$\begin{aligned}
-\Tilde{H}_2^{\downarrow}(A_1|E_1R_2E_2)_{\rho''} & = \Tilde{D}_2(\rho_{A_1E_1R_2E_2}''|| {\mathbbm{1}}{\varotimes}\rho_{E_1R_2E_2}'')\nonumber \\
& =\Tilde{D}_2\Big(C_{A_1}^{1^\top} {\varotimes}C_{R_2}^{2^\top} \big[\rho_{A_1E_1R_2E_2}'\big]
|| {\mathbbm{1}}{\varotimes}(C_{R_2}^{2^\top} \Big[\rho_{E_1R_2E_2}'\big])\Big)\nonumber \\
& = \Tilde{D}_2(\rho_{A_1E_1R_2E_2}' || {\mathbbm{1}}{\varotimes}\rho_{E_1R_2E_2}') \nonumber \\
& = -\Tilde{H}_2(A_1|E_1R_2E_2)_{\rho'} \nonumber \\
\implies R( \mathcal{N} \boxast_{C''} \mathcal{M} ) & = R( \mathcal{N} \boxast_{C'} \mathcal{M} ),\end{aligned}$$
where the second equality follows from Proposition \[prop:equi\_cliff\_bad\] and $\rho_{E_1R_2E_2}''= \text{tr}_{A_1}(C_{A_1}^{1^{\top}} {\varotimes}C_{R_2}^{2^{\top}} \Big[\rho_{A_1E_1R_2E_2}' \Big]) = C_{R_2}^{2^\top} \big[\rho_{E_1R_2E_2}'\big]$, and the third equality follows from equation (\[eq:uni\_equiv\]). $\blacksquare$
### Good Channel {#subsec:red_cliff_good}
We have following equalities for the complementary of good channel:
$$\rho_{A_2E_1E_2}'' = (\mathcal{N} \varoast_{C''} \mathcal{M})^c (\Phi_{A_2A_2'}) =\mathcal{N}^c {\varotimes}\mathcal{M}^c \Big( C_{A_1'A_2'}'' \Big[ \frac{ {\mathbbm{1}}_{A_1'}}{2} {\varotimes}\Phi_{A_2A_2'} ) \Big] \Big) \nonumber$$
$$\begin{aligned}
\rho_{A_2E_1E_2}' = (\mathcal{N} \varoast_{C'} \mathcal{M})^c (\Phi_{A_2A_2'}) =\mathcal{N}^c {\varotimes}\mathcal{M}^c \Big( C_{A_1'A_2'}' \Big[ \frac{ {\mathbbm{1}}_{A_1'}}{2} {\varotimes}\Phi_{A_2A_2'} ) \Big] \Big) \nonumber \end{aligned}$$
\[prop:equi\_cliff\_good\] $$\rho_{A_2E_1E_2}'' = C_{A_2}^{2^\top} \Big[\rho_{A_2E_1E_2}'\Big]$$
Proof is similar to the proof of proposition \[prop:equi\_cliff\_bad\]
**Proof of $R( \mathcal{N} \varoast_{C''} \mathcal{M} ) = R( \mathcal{N} \varoast_{C'} \mathcal{M} )$:** Using proposition \[prop:equi\_cliff\_good\], it can be proved similar to the proof for bad channel in subsection \[subsec:red\_cliff\_bad\] that:
$$R( \mathcal{N} \varoast_{C''} \mathcal{M} ) = R( \mathcal{N} \varoast_{C'} \mathcal{M} )$$
$\blacksquare$
Proof of Lemma \[lem:nclassical-well-defined\] {#sec:proof-nclassical-well-defined}
==============================================
We have to prove that if ${\cal N}'$ and ${\cal N}''$ are CMP channels, such that $$\label{eq:identifiable_channels}
{\cal N}'(\rho) {\varotimes}\frac{I_X}{|X|} = C \left({\cal N}''(\rho) {\varotimes}\frac{I_X}{|X|}\right)C^\dagger,$$ for some unitary $C$, then $\left({\cal N}'\right)^\# \equiv \left({\cal N}''\right)^\#$. We restrict ourselves to the case when ${\cal N}'$ and ${\cal N}''$ are Pauli channels, since the case of CMP channels follows in a similar manner, by introducing an auxiliary system providing a classical description of the Pauli channel being used. Hence, we may write ${\cal N}'(\rho) = \sum_{i=0}^3 p'_i \sigma_i \rho \sigma_i^{\dagger}$ and ${\cal N}''(\rho) = \sum_{i=0}^3 p''_i \sigma_i \rho \sigma_i^{\dagger}$, with $\sum_{i=0}^{3}p'_i = \sum_{i=0}^{3}p''_i = 1$. It follows that ${\cal N}'(\sigma_k) = \alpha'_k \sigma_k$ and ${\cal N}''(\sigma_k) = \alpha''_k \sigma_k$, where $\alpha'_0 = \alpha''_0 = 1$, and for $k=1,2,3$, $\alpha'_k = p'_0+p'_k - p'_{k_1} - p'_{k_2}$, $\alpha''_k = p''_0+p''_k - p''_{k_1} - p''_{k_2}$, with $\{k_1, k_2\} = \{1,2,3\}\setminus \{k\}$. Using bold notation for vectors $\bm{p}' {:=}(p'_0,p'_1, p'_2,p'_3)$, and similarly $\bm{p}'', \bm{\alpha}', \bm{\alpha}''$, the above equalities rewrite as $$\label{eq:a-matrix}
\bm{\alpha}' = A \bm{p}' \mbox{ and } \bm{\alpha}'' = A \bm{p}'',$$ $$\text{ where } A {:=}\left(\begin{array}{rrrr}
1 & 1 & 1 & 1 \\
1 & 1 & -1 & -1 \\
1 & -1 & 1 & -1 \\
1 & -1 & -1 & 1
\end{array}\right)$$ Now, replacing $\rho$ by $\sigma_k$ in (\[eq:identifiable\_channels\]), we have that $$\alpha'_k \sigma_k {\varotimes}I_X = C \left( \alpha''_k \sigma_k {\varotimes}I_X \right) C^\dagger.$$ Since the conjugate action of the unitary $C$ preserves the Hilbert–Schmidt norm of an operator, it follows that $\left\| \alpha'_k \sigma_k {\varotimes}I_X \right\|_{\text{HS}} = \left\| \alpha''_k \sigma_k {\varotimes}I_X \right\|_{\text{HS}}$, and therefore $|\alpha'_k| = |\alpha''_k|$.
[**Case 1:**]{} We first assume that $\alpha'_k = \alpha''_k, \forall k=1,2,3$. In this case, using (\[eq:a-matrix\]), it follows that $\bm{p}' = \bm{p}''$, and therefore $\left({\cal N}'\right)^\# = \left({\cal N}''\right)^\#$.
[**Case 2:**]{} We consider now the case when $\alpha'_k \neq \alpha''_k$, for some $k=1,2,3$. To address this case, we start by writing $C = \sum_{i=0}^3 \sigma_i {\varotimes}C_i$, where $C_i$ are linear operators on the system $X$. Hence, equation (\[eq:identifiable\_channels\]) rewrites as $${\cal N}'(\rho) {\varotimes}\frac{I_X}{|X|} = \sum_{i,j}\left(\sigma_i {\cal N}''(\rho)\sigma_j^\dagger\right) {\varotimes}\frac{C_i C_j^\dagger}{|X|}.$$ Tracing out the $X$ system, we have $$\label{eq:nprime}
{\cal N}'(\rho) = \sum_{i,j} \gamma_{i,j} \sigma_i {\cal N}''(\rho)\sigma_j^\dagger, \mbox{ where } \gamma_{i,j} = \frac{1}{|X|} \operatorname{Tr}(C_i C_j^\dagger).$$ We define $\gamma_i {:=}\gamma_{i,i}$, and from (\[eq:nprime\]) it follows that $\gamma_i {:=}\gamma_{i,i} \in \mathbb{R}_+$. Replacing $\rho = \sigma_k$ in (\[eq:nprime\]), we have that for all $k=0,\dots,3$, $$\alpha'_k \sigma_k = \alpha''_k\sum_{i }\gamma_{i}\sigma_i \sigma_k \sigma_i^\dagger + \alpha''_k\sum_{i,j, i \neq j}\gamma_{i,j}\sigma_i \sigma_k \sigma_j^\dagger$$ The left hand side of the above equation has only $\sigma_k$ term, so only $\sigma_k$ on the right hand side should survive as Pauli matrices form an orthogonal basis. It follows that either $\alpha'_k = \alpha''_k = 0$, or the terms of the second sum in the right hand side of the above equation necessarily cancel each other. In both cases, we have that $$\begin{aligned}
\alpha'_k \sigma_k = &\ \alpha''_k\sum_{i }\gamma_{i}\sigma_i \sigma_k \sigma_i^\dagger = \alpha''_k \lambda_k \sigma_k, \\
\mbox{and thus, } \qquad \alpha'_k = &\ \lambda_k\alpha''_k, \label{eq:lambda}\\
\mbox{where, } \qquad \lambda_0 {:=}&\ \gamma_0 + \gamma_1 + \gamma_2 + \gamma_3 \label{eq:lambda_0}\\
\lambda_1 {:=}&\ \gamma_0 + \gamma_1 - \gamma_2 - \gamma_3 \label{eq:lambda_1}\\
\lambda_2 {:=}&\ \gamma_0 - \gamma_1 + \gamma_2 - \gamma_3 \label{eq:lambda_2}\\
\lambda_3 {:=}&\ \gamma_0 - \gamma_1 - \gamma_2 + \gamma_3 \label{eq:lambda_3}
\end{aligned}$$ We also note that $\lambda_0 = 1$, since $\alpha'_0 = \alpha''_0 = 1$. We further rewrite equation (\[eq:lambda\]) as $$\label{eq:lambda-rewrite}
\bm{\alpha}' = \Lambda \bm{\alpha}''$$ where $\Lambda = \mbox{diag}(\lambda_0, \lambda_1, \lambda_2, \lambda_3)$ is the square diagonal matrix with $\lambda_i$’s on the main diagonal. Plugging equation (\[eq:a-matrix\]) into equation (\[eq:lambda-rewrite\]), and using $A^2 = 4I$, we get $$\bm{p}' = \frac{1}{4}A \Lambda A \bm{p}'' = \Gamma \bm{p}'', \label{eq:Gamma_matrix}$$
$$\text{ where } \Gamma {:=}\frac{1}{4}A \Lambda A =
\left( \begin{array}{rrrr}
\gamma_0 & \gamma_1 & \gamma_2 & \gamma_3 \\
\gamma_1 & \gamma_0 & \gamma_3 & \gamma_2 \\
\gamma_2 & \gamma_3 & \gamma_0 & \gamma_1 \\
\gamma_3 & \gamma_2 & \gamma_1 & \gamma_0
\end{array}\right)$$
We now come back to our assumption, namely $\alpha'_k \neq \alpha''_k$, for some $k=1,2,3$. Without loss of generality, we may assume that $\alpha'_1 \neq \alpha''_1$. Since $|\alpha'_1| = |\alpha''_1|$ and $\alpha'_1 = \lambda_1 \alpha''_1$, it follows that $\lambda_1 = -1$. Then, using (\[eq:lambda\_0\]) and (\[eq:lambda\_1\]), we have that $2(\gamma_0 + \gamma_1) = \lambda_0 + \lambda_1 = 0$, which implies $$\label{eq:zero-gammas}
\gamma_0 = \gamma_1 = 0,$$ since they are non-negative. We proceed now with several sub-cases:
- either $\alpha'_2 \neq \alpha''_2$ or $\alpha'_3 \neq \alpha''_3$. Similarly to the derivation of equation (\[eq:zero-gammas\]), we get either $\gamma_2 = 0$ (in which case $\gamma_3 = 1$) or $\gamma_3 = 0$ (in which case $\gamma_2 = 1$). In either case $\Lambda$ is a permutation matrix, which implies that $\left({\cal N}'\right)^\# \equiv \left({\cal N}''\right)^\#$, as desired.
- $\alpha'_2 = \alpha''_2$ and $\alpha'_3 = \alpha''_3$, and either $\alpha'_2 = \alpha''_2 \neq 0$ or $\alpha'_3 = \alpha''_3 \neq 0$. Let us assume that $\alpha'_2 = \alpha''_2 \neq 0$. In this case, using (\[eq:lambda\]), we have that $\lambda_2 = 1$, and from (\[eq:lambda\_2\]) it follows that $\gamma_2 - \gamma_3 = 1$. This implies $\gamma_2 = 1$ and $\gamma_3 = 0$, therefore $\Lambda$ is a permutation matrix, and thus $\left({\cal N}'\right)^\# \equiv \left({\cal N}''\right)^\#$, as desired.
- $\alpha'_2 = \alpha''_2 = 0$ and $\alpha'_3 = \alpha''_3 = 0$. Using $\alpha'_k = 2(p'_0 + p'_k) - 1, \forall k\neq 0$, we get $p'_2 = p'_3 = \frac{1}{2} - p'_0$, and similarly $p''_2 = p''_3 = \frac{1}{2} - p''_0$. Moreover, using (\[eq:Gamma\_matrix\]) and the fact that $\gamma_2 + \gamma_3 = 1$, we get $p'_0 = p'_1 = p''_2 = p''_3$ and $p'_2 = p'_3 = p''_0 = p''_1$. This implies that $\left({\cal N}'\right)^\# \equiv \left({\cal N}''\right)^\#$, as desired.
This concludes the second case, and finishes the proof. $\blacksquare$
Proof of Proposition \[prop:cq\_equiv\] {#sec:proof_cq_equiv}
=======================================
Using the notation from Section \[subsec:classical\_combining\_splitting\], we shall identify $\left(\bar{P}_1, \times\right) \cong \left(\{0,1,2,3\}, \oplus\right)$, where $\sigma_i\cong i, \forall i =0,\dots,3$, and thus assume that the classical channel ${\cal N}^\#$ – associated with a Pauli channel ${\cal N}(\rho) = \sum_{i=0}^3 p_i \sigma_i \rho \sigma_i^{\dagger}$ – has alphabet $\{0,1,2,3\}$, with transition probabilities defined by ${\cal N}^\#(i \mid j) = p_{i\oplus j}$. Moreover, the automorphism $\Gamma = \Gamma(C)$ induced by the conjugate action of a two-qubit Clifford unitary $C$ on $\bar{P}_1 \times \bar{P}_1$, is identified to a linear permutation $\Gamma: \{0,1,2,3\}^2 \rightarrow \{0,1,2,3\}^2$, such that $C\sigma_{i,j}C^\dagger = \sigma_{\Gamma(i,j)}$. We shall also write $\Gamma = (\Gamma_1, \Gamma_2)$, with $\Gamma_i : \{0,1,2,3\}^2 \rightarrow \{0,1,2,3\}$, $i=1,2$.
It can be easily seen that it is enough to prove the statement of Proposition \[prop:cq\_equiv\] for the case when ${\cal N}$ and ${\cal M}$ are Pauli channels. Let ${\cal N}(\rho) = \sum_{i=0}^3 p_i \sigma_i \rho \sigma_i^{\dagger}$ and ${\cal M}(\rho) = \sum_{j=0}^3 q_j \sigma_j \rho \sigma_j^{\dagger}$.
We start by proving $(i)$. $$\begin{aligned}
({\cal N}\boxast{\cal M})(\rho_U) & = & ({\cal N}{\varotimes}{\cal M})\left( C \left( \rho_U {\varotimes}\frac{I_V}{2} \right) C^\dagger \right) \\
& = & \sum_{i,j} p_i q_j \sigma_{i,j} C \left( \rho_U {\varotimes}\frac{I_V}{2} \right) C^\dagger \sigma_{i,j}^\dagger \\
& = & \sum_{i,j}r_{i,j} C \sigma_{\Gamma^{-1}(i,j)} \left( \rho_U {\varotimes}\frac{I_V}{2} \right) \sigma_{\Gamma^{-1}(i,j)}^\dagger C^\dagger, \mbox{ where } r_{i,j} {:=}p_i q_j\\
& = & C \left( \sum_{i,j}r_{\Gamma(i,j)} \sigma_{i,j} \left( \rho_U {\varotimes}\frac{I_V}{2} \right) \sigma_{i,j}^\dagger \right) C^\dagger \label{eq:var_change}\\
& = & C \left( \sum_{i,j}r_{\Gamma(i,j)} \sigma_{i} \rho_U \sigma_{i}^\dagger {\varotimes}\frac{I_V}{2} \right) C^\dagger \\
& = & C \left( \sum_{i} s_i\, \sigma_{i} \rho_U \sigma_{i}^\dagger {\varotimes}\frac{I_V}{2} \right) C^\dagger, \mbox{ where } s_i {:=}\sum_j r_{\Gamma(i,j)}\end{aligned}$$ where Eq. (\[eq:var\_change\]) follows from variable change $(i,j) \mapsto \Gamma(i,j)$. Omitting the conjugate action of the unitary $C$ and discarding the $V$ system, we may further identify: $$({\cal N}\boxast{\cal M})(\rho_U) = \sum_{i} s_i \sigma_{i} \rho_U \sigma_{i}^\dagger$$ Hence, the associated classical channel $({\cal N}\boxast{\cal M})^\#$ is defined by the probability vector $\mathbf{s} = (s_0,s_1,s_2,s_3)$, meaning that $$\label{eq:quantum_bad_channel}
({\cal N}\boxast{\cal M})^\#(i \mid j) = s_{i\oplus j}$$
On the other hand, we have: $$\begin{aligned}
({\cal N}^\#\boxast{\cal M}^\#)(a, b \mid u) & = & \frac{1}{4} \sum_v {\cal N}^\#(a \mid {\Gamma_1(u,v)}) {\cal M}^\#(b \mid {\Gamma_2(u,v)})\\
& = & \frac{1}{4} \sum_v p_{a \oplus \Gamma_1(u,v)} q_{b \oplus \Gamma_2(u,v)} \end{aligned}$$ Applying $\Gamma^{-1}$ on the channel output, we may identify ${\cal N}^\#\boxast{\cal M}^\#$ to a channel with output $(a',b') = \Gamma^{-1}(a,b)$, and transition probabilities given by: $$\begin{aligned}
({\cal N}^\#\boxast{\cal M}^\#)(a', b' \mid u) & = & \frac{1}{4} \sum_v p_{\Gamma_1(a',b') \oplus \Gamma_1(u,v)} q_{\Gamma_2(a',b') \oplus \Gamma_2(u,v)} \\
& = & \frac{1}{4} \sum_v p_{\Gamma_1((a',b') \oplus (u,v))} q_{\Gamma_2((a',b') \oplus (u,v))} \\
& = & \frac{1}{4} \sum_v p_{\Gamma_1(a'\oplus u, b'\oplus v)} q_{\Gamma_2(a' \oplus u, b' \oplus v)} \\
& = & \frac{1}{4} \sum_v p_{\Gamma_1(a'\oplus u, v)} q_{\Gamma_2(a' \oplus u, v)} \\
& = & \frac{1}{4} \sum_v r_{\Gamma(a'\oplus u, v)} \\
& = & \frac{1}{4} s_{a'\oplus u}\end{aligned}$$ We can then discard the $b'$ output, since the channel transition probabilities do not depend on it, which gives a channel defined by transition probabilities: $$\label{eq:classical_bad_channel}
({\cal N}^\#\boxast{\cal M}^\#)(a' \mid u) = s_{a'\oplus u}$$ Finally, using Eq. (\[eq:quantum\_bad\_channel\]) and Eq. (\[eq:classical\_bad\_channel\]), and noticing that omitting the conjugate action of the unitary $C$ and discarding the $V$ system in the derivation of Eq. (\[eq:quantum\_bad\_channel\]) is equivalent to applying $\Gamma^{-1}$ on the channel output and discarding the $b'$ output in the derivation of Eq. (\[eq:classical\_bad\_channel\]), we conclude that $({\cal N}\boxast{\cal M})^\# \equiv {\cal N}^\#\boxast{\cal M}^\#$
We prove now the $(ii)$ statement. Similar to the derivations used for $(i)$, we get: $$\begin{aligned}
({\cal N}\varoast{\cal M})(\rho_V) & = C \left( \sum_{i,j}r_{\Gamma(i,j)} \sigma_{i,j} \left( \Phi_{U'U} {\varotimes}\rho_V \right) \sigma_{i,j}^\dagger \right) C^\dagger \\
& = C \left( \sum_{i,j}r_{\Gamma(i,j)} \left( (I_{U'}{\varotimes}\sigma_i)(\Phi_{U'U})(I_{U'}{\varotimes}\sigma_i^\dagger)\right) {\varotimes}(\sigma_j \rho_V \sigma_j^\dagger) \right) C^\dagger \label{eq:xxx}\end{aligned}$$ Omitting the conjugate action of the unitary $C$, and expressing $(I_{U'}{\varotimes}\sigma_i)(\Phi_{U'U})(I_{U'}{\varotimes}\sigma_i^\dagger)$ in the Bell basis, $\{{\mathchoice{{{\ensuremath{\left\lverti\right\rangle}}}}{{{\ensuremath{\lverti\rangle}}}}{{{\ensuremath{\lverti\rangle}}}}{{{\ensuremath{\lverti\rangle}}}}}\}_{i=0,\dots,3} {:=}\{ \frac{{\mathchoice{{{\ensuremath{\left\lvert00\right\rangle}}}}{{{\ensuremath{\lvert00\rangle}}}}{{{\ensuremath{\lvert00\rangle}}}}{{{\ensuremath{\lvert00\rangle}}}}}+{\mathchoice{{{\ensuremath{\left\lvert11\right\rangle}}}}{{{\ensuremath{\lvert11\rangle}}}}{{{\ensuremath{\lvert11\rangle}}}}{{{\ensuremath{\lvert11\rangle}}}}}}{\sqrt{2}}, \frac{{\mathchoice{{{\ensuremath{\left\lvert01\right\rangle}}}}{{{\ensuremath{\lvert01\rangle}}}}{{{\ensuremath{\lvert01\rangle}}}}{{{\ensuremath{\lvert01\rangle}}}}}+{\mathchoice{{{\ensuremath{\left\lvert10\right\rangle}}}}{{{\ensuremath{\lvert10\rangle}}}}{{{\ensuremath{\lvert10\rangle}}}}{{{\ensuremath{\lvert10\rangle}}}}}}{\sqrt{2}}, \frac{{\mathchoice{{{\ensuremath{\left\lvert01\right\rangle}}}}{{{\ensuremath{\lvert01\rangle}}}}{{{\ensuremath{\lvert01\rangle}}}}{{{\ensuremath{\lvert01\rangle}}}}}-{\mathchoice{{{\ensuremath{\left\lvert10\right\rangle}}}}{{{\ensuremath{\lvert10\rangle}}}}{{{\ensuremath{\lvert10\rangle}}}}{{{\ensuremath{\lvert10\rangle}}}}}}{\sqrt{2}}, \frac{{\mathchoice{{{\ensuremath{\left\lvert00\right\rangle}}}}{{{\ensuremath{\lvert00\rangle}}}}{{{\ensuremath{\lvert00\rangle}}}}{{{\ensuremath{\lvert00\rangle}}}}}-{\mathchoice{{{\ensuremath{\left\lvert11\right\rangle}}}}{{{\ensuremath{\lvert11\rangle}}}}{{{\ensuremath{\lvert11\rangle}}}}{{{\ensuremath{\lvert11\rangle}}}}}}{\sqrt{2}} \}$, we get: $$\begin{aligned}
({\cal N}\varoast{\cal M})(\rho_V) & = & \sum_{i,j}r_{\Gamma(i,j)} {\mathchoice{{{\ensuremath{\left\lverti\right\rangle}}}}{{{\ensuremath{\lverti\rangle}}}}{{{\ensuremath{\lverti\rangle}}}}{{{\ensuremath{\lverti\rangle}}}}}{\mathchoice{{{\ensuremath{\left\langlei\right\rvert}}}}{{{\ensuremath{\langlei\rvert}}}}{{{\ensuremath{\langlei\rvert}}}}{{{\ensuremath{\langlei\rvert}}}}} {\varotimes}(\sigma_j \rho_V \sigma_j^\dagger) \end{aligned}$$ Let $\lambda_i {:=}\sum_j r_{\Gamma(i,j)}$ and $s_{i,j} {:=}r_{\Gamma(i,j)} / \lambda_i$ (with $s_{i,j} {:=}0$ if $\lambda_i=0$). Denoting by ${\cal S}_i$ the Pauli channel defined by ${\cal S}(\rho)_i = \sum_j s_{i,j} \sigma_j \rho_V \sigma_j^\dagger$, we may rewrite: $$\begin{aligned}
({\cal N}\varoast{\cal M})(\rho_V) & = & \sum_{i,j} \lambda_{i} {\mathchoice{{{\ensuremath{\left\lverti\right\rangle}}}}{{{\ensuremath{\lverti\rangle}}}}{{{\ensuremath{\lverti\rangle}}}}{{{\ensuremath{\lverti\rangle}}}}}{\mathchoice{{{\ensuremath{\left\langlei\right\rvert}}}}{{{\ensuremath{\langlei\rvert}}}}{{{\ensuremath{\langlei\rvert}}}}{{{\ensuremath{\langlei\rvert}}}}} {\varotimes}{\cal S}_i(\rho_V) \end{aligned}$$ Hence, $({\cal N}\varoast{\cal M})^\#$ is the mixture of the channels ${\cal S}_i^\#$, with ${\cal S}_i^\#$ being used with probability $\lambda_i$, whose transition probabilities are given by: $$\label{eq:quantum_good_channel}
({\cal N}\varoast{\cal M})^\#(i, j\mid k) = \lambda_i s_{i, j \oplus k} = r_{\Gamma(i, j\oplus k)}$$
On the other hand, we have: $$\begin{aligned}
({\cal N}^\#\varoast{\cal M}^\#)(a, b, u \mid v) & = & \frac{1}{4} {\cal N}^\#(a \mid {\Gamma_1(u,v)}) {\cal M}^\#(b \mid {\Gamma_2(u,v)})\\
& = & \frac{1}{4} p_{a \oplus \Gamma_1(u,v)} q_{b \oplus \Gamma_2(u,v)} \end{aligned}$$ We apply $\Gamma^{-1}$ on the $(a,b)$ output of the channel, which is equivalent to omitting the conjugate action of the unitary $C$ in Eq. (\[eq:xxx\]), and then identify ${\cal N}^\#\varoast{\cal M}^\#$ to a channel with output $(a',b', u)$, where $(a',b') = \Gamma^{-1}(a,b)$, and transition probabilities: $$\begin{aligned}
({\cal N}^\#\varoast{\cal M}^\#)(a', b', u \mid v) & = & \frac{1}{4} p_{\Gamma_1(a',b') \oplus \Gamma_1(u,v)} q_{\Gamma_2(a',b') \oplus \Gamma_2(u,v)} \\
& = & \frac{1}{4} p_{\Gamma_1(a'\oplus u, b'\oplus v)} q_{\Gamma_2(a' \oplus u, b' \oplus v)} \\
& = & \frac{1}{4} r_{\Gamma(a'\oplus u, b'\oplus v)} \end{aligned}$$ We further perform a change of variable, replacing $(a', u)$ by $(a'\oplus u, u)$, which makes the above transition probability independent of $u$. We may then discard the $u$ output, and thus identify ${\cal N}^\#\varoast{\cal M}^\#$ to a channel with output $(a',b')$ and transition probabilities: $$\begin{aligned}
\label{eq:classical_good_channel}
({\cal N}^\#\varoast{\cal M}^\#)(a', b' \mid v) & = & r_{\Gamma(a', b'\oplus v)} \end{aligned}$$ Finally, using Eq. (\[eq:quantum\_good\_channel\]) and Eq. (\[eq:classical\_good\_channel\]), we conclude that $({\cal N}\varoast{\cal M})^\# \equiv {\cal N}^\#\varoast{\cal M}^\#$
$\blacksquare$
Proof of Lemma \[lem:z-good-classical-channel\] {#apnd:pol9}
===============================================
We prove first the following lemma.
\[lemma:Zd\_good\_channels\] For any classical channels $N, M$, with input alphabet $\bar{P}_1 \cong \left( \{0,1,2,3\}, \oplus \right)$, and any linear permutation $\Gamma = (A, B) : \bar{P}_1\times\bar{P}_1 \rightarrow \bar{P}_1\times\bar{P}_1$, the following equality holds for any $d \in \bar{P}_1$: $$\begin{aligned}
Z_d(N \varoast_\Gamma M) &= Z_{A(0,d)}(N) Z_{B(0,d)}(M)\end{aligned}$$
According to Definition \[def:bhattacharyya\], for the channel $ N \varoast_{\Gamma} M$, we have: $$\begin{aligned}
Z\left((N \varoast_{\Gamma} M)_{v,\, v'}\right) &= \sum_{u, y_1, y_2} \sqrt{(N \varoast_{\Gamma} M)(y_1, y_2, u \mid v) \times (N \varoast_{\Gamma} M)(y_1, y_2, u \mid v') } \\
&= \frac{1}{4} \sum_{u, y_1, y_2} \Big[ \sqrt{N(y_1 \mid A(u,v)) M(y_2 \mid B(u,v))} \ \times \nonumber \\
& \qquad \qquad \qquad \qquad \qquad \qquad \sqrt{N(y_1 \mid A(u,v')) M(y_2 \mid B(u,v'))} \Big] \\
&= \frac{1}{4} \sum_{u, y_1, y_2} \Big[ \sqrt{N(y_1 \mid A(u,v)) N(y_1 \mid A(u,v')) } \ \times \nonumber \\
& \qquad \qquad \qquad \qquad \qquad \qquad \sqrt{M(y_2 \mid B(u,v)) M(y_2 \mid B(u,v'))} \Big] \\
&= \frac{1}{4} \sum_{u} Z\left(N_{A(u,v),\, A(u,v')}\right) Z\left(M_{B(u,v),\, B(u,v')}\right) \end{aligned}$$ Therefore, $$\begin{aligned}
Z_d\left(N \varoast_{\Gamma} M\right) &= \frac{1}{4}\sum_{v} Z\left((N \varoast_{\Gamma} M)_{v,\, v\oplus d}\right) \\
&= \frac{1}{16} \sum_{u,v} Z\left(N_{A(u,v),\, A(u,v\oplus d)}\right) Z\left(M_{B(u,v),\, B(u,v\oplus d)}\right) \\
&= \frac{1}{16} \sum_{u,v} Z\left(N_{A(u,v),\, A(u,v)\oplus A(0, d)}\right) Z\left(M_{B(u,v),\, B(u,v)\oplus B(0, d)}\right) \label{eq:D:92}\\
&= \frac{1}{16} \sum_{a} Z\left(N_{a,\, a\oplus A(0, d)}\right)\sum_{b} Z\left(M_{b,\, b\oplus B(0, d)}\right) \label{eq:D:93} \\
&= Z_{A(0,d)}(N) Z_{B(0,d)}(M),\end{aligned}$$ where (\[eq:D:92\]) follows from the linearity of the permutation $\Gamma = (A,B)$, and (\[eq:D:93\]) follows from the change of basis for summation from $(u,v)$ to $(a,b) := (A(u,v), B(u,v))$.
Throughout the remaining of this section, we shall denote by $u:=[u_1, u_2]$ the binary representation of a given $u\in \bar{P}_1 \cong \{0,1,2,3\}$, where $u_1, u_2 \in \{0,1\}$ and $u_2$ is the least significant bit.
\[lemma:Ai\_Bj\_permutations\] Let $\Gamma_{i,j} := \Gamma(L_{i,j}) : \bar{P}_1\times\bar{P}_1 \rightarrow \bar{P}_1\times\bar{P}_1$ be the permutation defined by the conjugate action of $L_{i,j} \in {\cal L}$, where ${\cal L}$ is the set of two-qubit Clifford gates defined in Section \[sec:polarization\_9\_cliffords\] (Figure \[fig:cliff\_generators\]). Then $\Gamma_{i,j} = (A_i, B_j), \forall 1 \leq i,j \leq 3$, with $A_i, B_j : \bar{P}_1\times\bar{P}_1 \rightarrow \bar{P}_1$ given by: $$\begin{aligned}
A_1(u,v) &= [u_1, u_2\oplus v_1\oplus v_2], & B_1(u,v) &= [u_1 \oplus v_1, u_1 \oplus v_2] \nonumber \\
A_2(u,v) &= [u_2\oplus v_1\oplus v_2, u_1 ], & B_2(u,v) &= [u_1 \oplus v_1, v_1 \oplus v_2] \nonumber \\
A_3(u,v) &= [u_1 \oplus u_2\oplus v_1\oplus v_2, u_2\oplus v_1\oplus v_2] & B_3(u,v) &= [v_1\oplus v_2, u_1 \oplus v_2] \nonumber\end{aligned}$$ where $u$ and $v$ inputs are represented in binary form, $u:=[u_1, u_2]$ and $v := [v_1, v_2]$, with $u_1, u_2, v_1, v_2 \in\{0,1\}$ ($\Gamma_{i,j}$ permutations are also depicted in Figure \[fig:perm\_left\_set\]).
Recall from Section \[sec:polarization\_9\_cliffords\], that $L_{i,j} = (C' {\varotimes}C'')\text{\sc cnot}_{21}$, where $C' \in \{I, \sqrt{Z}, \sqrt{Y}\}$, and $C'' \in \{I, \sqrt{X}, \sqrt{Y}\}$. Recall also that by identifying $\bar{P}_1 \cong \{0,1,2,3\}$, we have $I = \sigma_0 \cong 0$, $X = \sigma_1 \cong 1$, $Y = \sigma_2 \cong 2$, $Z = \sigma_3 \cong 3$. The conjugate action of $\sqrt{X}$ on $\bar{P}_1$, fixes $I$ and $X$, and permutes $Y$ and $Z$. Hence, the corresponding permutation on $\bar{P}_1 \cong \{0,1,2,3\}$, can be written as $(0, 1, 3, 2)$. Similarly, the conjugate action of $\sqrt{Y}$ and $\sqrt{Z}$ induces the permutations $(0, 3, 2, 1)$ and $(0, 2, 1, 3)$, respectively. Replacing $u\in \{0,1,2,3\}$ by its binary representation $[u_1, u_2]$, we may write: $$\label{eq:sqrt_sigma_perm}
\hspace*{-2mm}\sqrt{X}: [u_1,u_2] \mapsto [u_1, u_1\oplus u_2],\ \sqrt{Y}: [u_1,u_2] \mapsto [u_1\oplus u_2, u_2],\ \sqrt{Z}: [u_1,u_2] \mapsto [u_2, u_1]$$ Moreover, the permutation induced by the conjugate action of the $\text{\sc cnot}_{21}$ gate is the linear permutation on $\bar{P}_1 \times \bar{P}_1$ such that: $$\begin{aligned}
\text{\sc cnot}_{21}&: \ (X, I) \mapsto (X, I),\ \ (Z, I)\mapsto (Z, Z),\ \ (I, X) \mapsto (X, X),\ \ (I, Z) \mapsto (I, Z) \\
\Rightarrow \text{\sc cnot}_{21}&: \left([u_1,u_2], [v_1,v_2]\right) \mapsto \left( [u_1, u_2 \oplus v_1 \oplus v_2], [u_1 \oplus v_1, u_1\oplus v_2] \right) \label{eq:cnot21_perm}\end{aligned}$$ Finally, using (\[eq:sqrt\_sigma\_perm\]) and (\[eq:cnot21\_perm\]), it can be easily verified that $\Gamma_{i,j} = (A_i, B_j), \forall 1 \leq i,j \leq 3$, with $A_i$ and $B_j$ as given in the lemma.
[**Proof of Lemma \[lem:z-good-classical-channel\].**]{} To simplify notation, let $W {:=}{\cal W}^\#$ be the classical counterpart of the CMP channel $\mathcal{W}$ from Lemma \[lem:z-good-classical-channel\]. Applying Lemma \[lemma:Zd\_good\_channels\] and Lemma \[lemma:Ai\_Bj\_permutations\], we may express $Z_d(W\varoast_{\Gamma_{i,j}} W)$ as a function of $(Z_1(W), Z_2(W), Z_3(W))$, for any $\Gamma_{i,j} \in\Gamma(\mathcal{L})$ and any $d=1,2,3$ (recall that $Z_0(W) = 1$). The corresponding expressions are given in Table \[table:Zd\_good\_channel\].
$$\begin{array}{c|c|c|c}
\hline
(i,j) & Z_1(W\varoast_{\Gamma_{i,j}} W) & Z_2(W\varoast_{\Gamma_{i,j}} W) & Z_3(W\varoast_{\Gamma_{i,j}} W) \\
\hline \hline
(1, 1) & Z_1(W)^2 & Z_1(W)Z_2(W) & Z_3(W) \\
(1, 2) & Z_1(W)^2 & Z_1(W)Z_3(W) & Z_2(W) \\
(1, 3) & Z_1(W)Z_3(W) & Z_1(W)Z_2(W) & Z_1(W) \\
(2, 1) & Z_1(W)Z_2(W) & Z_2(W)^2 & Z_3(W) \\
(2, 2) & Z_1(W)Z_2(W) & Z_2(W)Z_3(W) & Z_2(W) \\
(2, 3) & Z_2(W)Z_3(W) & Z_2(W)^2 & Z_1(W) \\
(3, 1) & Z_1(W)Z_3(W) & Z_2(W)Z_3(W) & Z_3(W) \\
(3, 2) & Z_1(W)Z_3(W) & Z_3(W)^2 & Z_2(W) \\
(3, 3) & Z_3(W)^2 & Z_2(W)Z_3(W) & Z_1(W) \\
\hline
\end{array}$$
Hence, $$\begin{aligned}
\sum_{\Gamma \in \Gamma(\cal L)} Z\left(W \varoast_{\Gamma} W\right) &= \frac{1}{3} \sum_{\Gamma \in \Gamma(\cal L)}
\left( Z_1\left(W \varoast_{\Gamma} W\right) + Z_2\left(W \varoast_{\Gamma} W\right) + Z_3\left(W \varoast_{\Gamma} W\right) \right) \\
&= \frac{3(Z_1(W) + Z_2(W) + Z_3(W)) + 2 (Z_1(W) + Z_2(W) + Z_3(W))^2}{3} \\
& = 3Z(W) + 6Z(W)^2,\end{aligned}$$ and therefore, $$\begin{aligned}
{\mathbb{E}}_{\Gamma \in \Gamma(\cal L)} Z\left(W \varoast_{\Gamma} W\right) &= \frac{1}{9} \sum_{\Gamma \in \Gamma^{\cal L}} Z\left(W \varoast_{\Gamma} W\right) = \frac{1}{3}Z(W) + \frac{2}{3}Z(W)^2\end{aligned}$$
The case $\Gamma\in\mathcal{R}$ can be derived in a similar way. Alternatively, similarly to the proof of Lemma \[lemma:swap\_eq\] in the quantum case, it can be directly verified that ${\mathbb{E}}_{\Gamma \in \Gamma(\cal L)} Z\left(W \varoast_{\Gamma} W\right) = {\mathbb{E}}_{\Gamma \in \Gamma(\cal R)} Z\left(W \varoast_{\Gamma} W\right)$.
$\blacksquare$
Proof of Lemma \[lem:pol\_3\_clif\] {#apnd:pol3}
===================================
Using Table \[table:Zd\_good\_channel\] from Appendix \[apnd:pol9\], for $\Gamma \in \Gamma({\cal S}) = \{\Gamma_{1,3}, \Gamma_{2,2}, \Gamma_{3,1}\}$, we get $$\begin{aligned}
{\mathbb{E}}_{\Gamma \in \Gamma({\cal S})} Z\left(W \varoast_{\Gamma} W\right) &= \frac{1}{3} \sum_{\Gamma \in \Gamma({\cal S})} Z\left(W \varoast_{\Gamma} W\right) =
\frac{1}{9} \sum_{\Gamma \in \Gamma({\cal S})} \sum_{d=1,2,3} Z_d\left(W \varoast_{\Gamma} W\right) \\
&= \frac{1}{9}\left( Z_1(W) + Z_2(W) + Z_3(W) \right) + \nonumber \\
&\qquad \qquad \frac{2}{9}\left( Z_1(W)Z_2(W) + Z_1(W)Z_3(W) + Z_2(W) Z_3(W) \right) \\
&\leq \frac{1}{3}Z(W) + \frac{2}{9}\left(Z_1(W)^2 + Z_2(W)^2 + Z_3(W)^2 \right) \label{eq:E:116} \\
&\leq \frac{1}{3}Z(W) + \frac{2}{3} Z(W)^2,\end{aligned}$$ where (\[eq:E:116\]) follows from $Z_i(W)Z_j(W) \leq (Z_i(W)^2 + Z_j(W)^2)/2$ and the last inequality follows from the Cauchy-Schwarz inequality.
$\blacksquare$
Proof of Lemma \[lem:Fast\_Polarization\] {#sec:proof_lemma_fast_polarization}
=========================================
We prove first the following lemma.
\[lemma:Zd\_bad\_channels\] For any classical channels $N, M$, with input alphabet $\bar{P}_1 \cong \left( \{0,1,2,3\}, \oplus \right)$, and any linear permutation $\Gamma = (A, B) : \bar{P}_1\times\bar{P}_1 \rightarrow \bar{P}_1\times\bar{P}_1$, the following inequality holds for any $d \in \bar{P}_1$: $$\begin{aligned}
Z_d(N \boxast_\Gamma M) &\leq \sum_{d' \in \bar{P}_1} Z_{A(d,d')}(N) Z_{B(d,d')}(M)\end{aligned}$$
According to Definition \[def:bhattacharyya\], for the channel $ N \boxast_{\Gamma} M$, we have: $$\begin{aligned}
Z\left((N \boxast_{\Gamma} M)_{u,\, u'}\right) &= \sum_{y_1, y_2} \sqrt{(N \boxast_{\Gamma} M)(y_1, y_2 \mid u) \times (N \boxast_{\Gamma} M)(y_1, y_2 \mid u') } \\
&= \frac{1}{4} \sum_{y_1, y_2} \left[ \sqrt{ \sum_v N(y_1 \mid A(u,v)) M(y_2 \mid B(u,v))} \right. \times \nonumber \\
& \qquad \qquad \qquad \qquad \qquad \left. \sqrt{ \sum_{v'} N(y_1 \mid A(u',v')) M(y_2 \mid B(u',v'))} \right] \\
&\leq \frac{1}{4} \sum_{v,v'} \sum_{y1,y2} \left[ \sqrt{ N(y_1 \mid A(u,v)) M(y_2 \mid B(u,v))} \right. \times \nonumber \\
& \qquad \qquad \qquad \qquad \qquad \left. \sqrt{ N(y_1 \mid A(u',v')) M(y_2 \mid B(u',v'))} \right] \label{eq:F:98} \\
&= \frac{1}{4} \sum_{v,v'} \sum_{y1,y2} \Big[ \sqrt{N(y_1 \mid A(u,v)) N(y_1 \mid A(u',v')) } \ \times \nonumber \\
& \qquad \qquad \qquad \qquad \qquad \qquad \sqrt{M(y_2 \mid B(u,v)) M(y_2 \mid B(u',v'))} \Big] \\
&= \frac{1}{4} \sum_{v,v'} Z\left(N_{A(u,v),\, A(u',v')}\right) Z\left(M_{B(u,v),\, B(u',v')}\right),\end{aligned}$$ where (\[eq:F:98\]) follows from $\sqrt{\sum_v x_v} \leq \sum_v \sqrt{x_v}$.
Therefore, $$\begin{aligned}
Z_d\left(N \boxast_{\Gamma} M\right) &= \frac{1}{4}\sum_{u} Z\left((N \boxast_{\Gamma} M)_{u,\, u\oplus d}\right) \\
&\leq \frac{1}{16} \sum_{u,v,v'} Z\left(N_{A(u,v),\, A(u\oplus d,v')}\right) Z\left(M_{B(u,v),\, B(u\oplus d,v')}\right) \\
&= \frac{1}{16} \sum_{u,v,d' \text{\makebox[0mm][l]{\ ($d' := v\oplus v'$)}}} Z\left(N_{A(u,v),\, A(u\oplus d,v\oplus d')}\right) Z\left(M_{B(u,v),\, B(u\oplus d,v\oplus d')}\right) \\
&= \frac{1}{16} \sum_{u,v,d'} Z\left(N_{A(u,v),\, A(u,v)\oplus A(d, d')}\right) Z\left(M_{B(u,v),\, B(u,v)\oplus B(d, d')}\right) \label{eq:F:104}\\
&= \frac{1}{16} \sum_{d'} \sum_{a} Z\left(N_{a,\, a\oplus A(d, d')}\right)\sum_{b} Z\left(M_{b,\, b\oplus B(d, d')}\right) \label{eq:F:105}\\
&= \sum_{d'} Z_{A(d,d')}(N) Z_{B(d,d')}(M),\end{aligned}$$ where (\[eq:F:104\]) follows from the linearity of the permutation $\Gamma = (A,B)$, and (\[eq:F:105\]) follows from the change of basis for summation from $(u,v)$ to $(a,b) := (A(u,v), B(u,v))$.
[**Proof of Lemma \[lem:Fast\_Polarization\].**]{} To simplify notation, let $W {:=}{\cal W}^\#$ be the classical counterpart of the CMP channel $\mathcal{W}$ from Lemma \[lem:Fast\_Polarization\]. Using Lemma \[lemma:Zd\_bad\_channels\], we have $Z_d(W \boxast_\Gamma W) \leq \sum_{d' \in \bar{P}_1} Z_{A(d,d')}(W) Z_{B(d,d')}(W)$. For $d\neq 0$, $A(d,d')$ and $B(d, d')$ cannot be simultaneously zero (recall that $Z_0(W) = 1$), and therefore we get $Z_{A(d,d')}(W) Z_{B(d,d')}(W) \leq \bar{Z}(W)$. Hence, $Z_d(W \boxast_\Gamma W) \leq 4\bar{Z}(W), \forall d=1,2,3$, which implies $\bar{Z}(W \boxast_\Gamma W) \leq 4\bar{Z}(W)$, as desired. Finally, we have $$Z(W \boxast_\Gamma W) \leq \bar{Z}(W \boxast_\Gamma W) \leq 4\bar{Z}(W) \leq 12 Z(W)$$ $\blacksquare$
Proof of Proposition \[prop:fast\_polarization\] {#sec:proof_prop_fast_polarization}
================================================
We proceed first with several lemmas. In the following, the notation $x = x(\cdot)$ means that the value of $x$ depends only on the list of variables $(\cdot)$ enclosed between parentheses.
\[lemma:dprime\_good\_channel\] $(i)$ For any permutation $\Gamma \in \Gamma(\mathcal{S})$, there exist $\delta_1 = \delta_1(\Gamma)$, $\delta_2 = \delta_2(\Gamma)$, $\delta_3 = \delta_3(\Gamma)$, such that $\{\delta_1, \delta_2, \delta_3\} = \{1, 2, 3\}$, and $$\begin{aligned}
Z_{3} (W \varoast_{\Gamma} W) &= Z_{\delta_3}(W)\\
Z_{2} (W \varoast_{\Gamma} W) &= Z_{\delta_3}(W) Z_{\delta_2}(W) \\
Z_{1} (W \varoast_{\Gamma} W) &= Z_{\delta_3}(W) Z_{\delta_1}(W) \end{aligned}$$ and the above equalities hold for any $W$ channel. $(ii)$ For any $d \in \{1,2,3\}$, there exists exactly one permutation $\Gamma \in \Gamma(\mathcal{S})$, such that $\delta_3(\Gamma) = d$.
Follows from Table \[table:Zd\_good\_channel\] in Appendix \[apnd:pol9\], wherein $\Gamma(\mathcal{S}) = \{\Gamma_{1,3}, \Gamma_{2,2},\Gamma_{3,1}\}$. Precisely, we have $\delta_3(\Gamma_{1,3}) = 1, \delta_3(\Gamma_{2,2}) = 2, \delta_3(\Gamma_{3,1}) = 3$.
\[lemma:dprime\_bad\_channel\] There exist a constant $\kappa > 1$ and $\bm{\delta} = \bm{\delta}(W) \in\{1, 2, 3\}$, such that for any $\Gamma \in \Gamma(\mathcal{S})$ and any $d\in\{1,2,3\}$, the following equality holds $$Z_{d} (W \boxast_{\Gamma} W) \leq \kappa Z_{\bm{\delta}}(W)$$
Follows from Lemma \[lem:Fast\_Polarization\], for $\kappa = 4$ and $\displaystyle \bm{\delta} = \bm{\delta}(W) := \operatorname{argmax}_{d=1,2,3} Z_{d}(W)$.
We shall also use the following lemma (known as Hoeffding’s inequality) providing an upper bound for the probability that the mean of $n$ independent random variables falls below its expected value mean by a positive number.
\[lemma:mean\_independent\_rv\] Let $X_1,X_2,\dots,X_n$ be independent random variables such that $0 \leq X_i\leq 1$, for any $i=1\dots,n$. Let $\bar{X} := \frac{1}{n}\sum_{i=1}^n X_i$, and $\mu = {\mathbb{E}}(\bar{X})$. Then, for any $0< t < \mu$, $$\Pr\left\{ \bar{X} \leq \mu - t \right\} \leq e^{-2nt^2}$$
Now, let $\Gamma(\mathcal{S})^{\infty}$ be the infinite Cartesian product of countable many copies of $\Gamma(\mathcal{S})$. It is endowed with an infinite product probability measure [@hewitt2013real], denoted by $P$, where the uniform probability measure is taken on each copy of $\Gamma(\mathcal{S})$. For our purposes, an infinite sequence $\bm{\Gamma} \in\Gamma(\mathcal{S})^{\infty}$ should be written as $\bm{\Gamma} := \left\{ \Gamma, \Gamma_{i_1\dots i_n} \mid n > 0, i_1,\dots,i_n \in\{0,1\} \right\}$ (this is always possible, since the set of indexes is countable). We further define a sequence of independent and identically distributed (i.i.d) Bernoulli random variables on $\Gamma(\mathcal{S})^{\infty}$, denoted $\Delta^{i_1\dots i_n}$, $n \geq 0$, $i_1,\dots,i_n\in\{0,1\}^n$, $$\Delta^{i_1\dots i_n}(\bm{\Gamma}) := \mathbf{1}_{\{\delta_3(\Gamma_{i_1\dots i_n}) \in \{1,2\}\}},$$ that is, $\Delta^{i_1\dots i_n}(\bm{\Gamma})$ is equal to $1$, if $\delta_3(\Gamma_{i_1\dots i_n}) \in\{1,2\}$, and equal to $0$, if $\delta_3(\Gamma_{i_1\dots i_n}) = 3$. From Lemma \[lemma:dprime\_good\_channel\] $(ii)$, it follows that ${\mathbb{E}}(\Delta^{i_1\dots i_n}) = 2/3$, $\forall n \geq 0$, $\forall i_1,\dots,i_n\in\{0,1\}^n$.
For $0 < \gamma < 2/3$ and $m > 0$, we define $$\begin{aligned}
\Pi_m(\gamma) &= \left\{ \bm{\Gamma} \in \Gamma(\mathcal{S})^{\infty} \Bigm| \sum_{i_1\dots i_{m-1}} \Delta^{i_1\dots i_{m-1} 1}(\bm{\Gamma}) \geq \left( \frac{2}{3}-\gamma\right) 2^{m-1} \right\} \label{eq:G:128}\\
\overline{\Pi}_m(\gamma) &= \bigcap_{n \geq m} \Pi_n(\gamma)\label{eq:G:129}\end{aligned}$$ Note that in (\[eq:G:128\]), $\Pi_m(\gamma)$ is defined by requiring that at least a fraction of $(2/3-\gamma)$ of $\Delta^{i_1\dots i_{m-1} i_m}$ variables are equal to $1$, where $i_m=1$. In (\[eq:G:129\]), the above condition must hold for any $n\geq m$.
For any $0 < \gamma < 2/3$ and $m > 0$, $$\label{eq:Pi_m_lowerbound}
P\left( \overline{\Pi}_m(\gamma) \right) \geq 2 - \frac{1}{1 - e^{-\gamma^2 2^{m}}}$$
By Lemma \[lemma:mean\_independent\_rv\], $P\left( \Pi_m(\gamma) \right) \geq 1 - e^{-\gamma^2 2^{m}}$. Therefore, we have $$\begin{aligned}
P\left( \overline{\Pi}_m(\gamma) \right) &\geq 1 - \sum_{n\geq m} e^{-\gamma^2 2^{n}} \\
&= 1 - \sum_{n\geq 0} \left(e^{-\gamma^2 2^{m}}\right)^{2^n} \\
&\geq 1 - \sum_{n\geq 1} \left(e^{-\gamma^2 2^{m}}\right)^{n} \\
&= 1 - \left(\frac{1}{1 - e^{-\gamma^2 2^{m}}} - 1 \right) \\
&= 2 - \frac{1}{1 - e^{-\gamma^2 2^{m}}}\end{aligned}$$
Note that the right hand side term in (\[eq:Pi\_m\_lowerbound\]) converges to $1$ as $m$ goes to infinity. Hence, for $\varepsilon > 0$, we denote by $m(\gamma,\epsilon)$ the smallest $m$ value, such that $2 - \frac{1}{1 - e^{-\gamma^2 2^{m}}} \geq 1-\epsilon$. It follows that $P\left( \overline{\Pi}_{m(\gamma,\varepsilon)}(\gamma) \right) \geq 1 -\varepsilon$.
In the following, we fix once for all some $\gamma$ value, such that $0 < \gamma < 2/3$. The value of $\gamma$ will no matter for any of what we do here, we only need $(2/3-\gamma)$ to be positive. We proceed now with the proof of Proposition \[prop:fast\_polarization\].
[**Proof of Proposition \[prop:fast\_polarization\].**]{} Let $\Omega := \{0,1\}^\infty$ denote the set of all binary sequences $\omega := (\omega_1,\omega_2,\dots) \in \{0,1\}^\infty$. Hence, $\Omega$ can be endowed with an infinite product probability measure, by taking the uniform probability measure on each $\omega_n$ component. We denote this probability measure by $P$ (the notation is the same as for the probability measure on $\Gamma(\mathcal{S})^{\infty}$, but no confusion should arise, since the sample spaces are different).
Let $\varepsilon > 0$ and fix any $\bm{\Gamma} \in \Gamma(\mathcal{S})^\infty_{\text{pol}} \cap \overline{\Pi}_{m(\gamma,\varepsilon)}(\gamma) $. Given $\bm{\Gamma}$, the polarization process can be formally described as a random process on the probability space $\Omega$ [@arikan09]. Precisely, for any $\omega = (\omega_1,\omega_2,\dots) \in \Omega$ and $n > 0$, we define $$\begin{aligned}
Z^{[n]}(\omega) &:= Z\left(W^{(\omega_1\dots \omega_n)}\right) \\
Z_{d}^{[n]}(\omega) &:= Z_d\left(W^{(\omega_1\dots \omega_n)}\right), \forall d\in \{1,2,3\}\end{aligned}$$ Note that $W^{(\omega_1\dots \omega_n)}$ is recursively defined as in (\[eq:c\_polar\_recursion\]), through the implicit assumption of using the channel combining permutations in the given sequence $\bm{\Gamma}$. For $n=0$, we set $Z^{[0]}(\omega) := Z(W)$ and $Z_{d}^{[0]}(\omega) := Z_d(W)$.
For $\zeta > 0$ and $m \geq 0$, we define $$T_m(\zeta) = \left\{ \omega \in \Omega \mid Z_{d}^{[n]}(\omega) \leq \zeta, \forall d=1,2,3, \forall n \geq m \right\}$$
Hence, for $\omega \in T_m(\zeta)$, $d\in \{1,2,3\}$, and $n > m$, we may write $$\label{eq:zd_decomp}
Z_{d}^{[n]}(\omega) = \frac{Z_{d_{n}}^{[n]}(\omega)}{Z_{d_{n-1}}^{[n-1]}(\omega)} \frac{Z_{d_{n-1}}^{[n-1]}(\omega)}{Z_{d_{n-2}}^{[n-2]}(\omega)} \cdots
\frac{Z_{d_{m+1}}^{[m+1]}(\omega)}{Z_{d_{m}}^{[m]}(\omega)}Z_{d_{m}}^{[m]}(\omega),$$ where $d_n := d$, and $d_{n-1},\dots, d_m$ are defined as explained below. Recall that $Z_{d}^{[k]}(\omega) := Z_d(W^{(\omega_1\dots \omega_k)})$, and for $k=n,n-1,\dots,m+1$, we have $$W^{(\omega_1\dots \omega_k)} = \left\{ \begin{array}{ll}
W^{(\omega_1\dots \omega_{k-1})} \boxast_{\Gamma_{\omega_1\dots \omega_{k-1}}} W^{(\omega_1\dots \omega_{k-1})}, & \text{if } \omega_k = 0 \\
W^{(\omega_1\dots \omega_{k-1})} \varoast_{\Gamma_{\omega_1\dots \omega_{k-1}}} W^{(\omega_1\dots \omega_{k-1})}, & \text{if } \omega_k = 1
\end{array} \right.$$ Hence, if $\omega_k = 0$, we set $d_{k-1} := \bm{\delta}\left(W^{(\omega_1\dots \omega_{k-1})}\right)$ from Lemma \[lemma:dprime\_bad\_channel\], such that we have $$\label{eq:bad_ubound}
\frac{Z_{d_{k}}^{[k]}(\omega)}{Z_{d_{k-1}}^{[k-1]}(\omega)} \leq \kappa, \ \text{ if } \omega_k = 0$$ If $\omega_k = 1$, we set $d_{k-1} := \delta_3\left(\Gamma_{\omega_1\dots \omega_{k-1}}\right)$ from Lemma \[lemma:dprime\_good\_channel\], such that we have $$\begin{aligned}
\frac{Z_{d_{k}}^{[k]}(\omega)}{Z_{d_{k-1}}^{[k-1]}(\omega)} &= 1, \text{ if } \omega_k = 1 \text{ and } d_k = 3 \\
\frac{Z_{d_{k}}^{[k]}(\omega)}{Z_{d_{k-1}}^{[k-1]}(\omega)} &\leq \zeta, \text{ if } \omega_k = 1 \text{ and } d_k \in \{1,2\} \label{eq:good_ubound}\end{aligned}$$ Let $A_{m,n}(\omega) := \{ k = m+1,\dots,n \mid \omega_k = 1 \}$, and $B_{m,n}(\omega) := \{ k = m+1,\dots,n \mid \omega_k = 1 \text{ and } d_k \in \{1,2\} \}$. Using (\[eq:zd\_decomp\]), (\[eq:bad\_ubound\])–(\[eq:good\_ubound\]), for $\omega \in T_m(\zeta)$ and $n > m$, we get: $$\label{eq:Zdn_upper_bound_1}
Z_{d}^{[n]}(\omega) \leq \kappa^{(n-m)-|A_{m,n}(\omega)|}\zeta^{|B_{m,n}(\omega)|}\zeta$$ Now, we want to upper-bound the right hand side term of the above inequality, by providing lower-bounds for the $|A_{m,n}(\omega)|$ and $|B_{m,n}(\omega)|$ values.
[*$|A_{m,n}(\omega)|$ lower-bound:*]{} Let $A^{[k]}(\omega) := \omega_k$, hence $|A_{m,n}(\omega)| = \sum_{k=m+1}^n A^{[k]}(\omega)$. Fix any $\alpha \in (0, 1/2)$, and let $$\mathcal{A}_{m,n}(\alpha) := \left\{ \omega \in \Omega \Bigm| \sum_{m=m+1}^n A^{[m]}(\omega) \geq \left(\frac{1}{2} - \alpha\right) (n-m) \right\}$$ Hence, for any $\omega\in \mathcal{A}_{m,n}(\alpha)$, $$\label{eq:Amn_lower_bound}
|A_{m,n}(\omega)| \geq (1/2 - \alpha)(n-m)$$ Moreover, by Lemma \[lemma:mean\_independent\_rv\], $P\left( \mathcal{A}_{m,n}(\alpha) \right) \geq 1 - e^{-2\alpha^2(n-m)}$.
[*$|B_{m,n}(\omega)|$ lower-bound:*]{} First, note that $d_k$ is defined depending on $\omega_{k+1}$ value. Hence, we may write $$\begin{aligned}
B_{m,n}(\omega) &= \left\{ k = m+1,\dots,n \mid \omega_k = 1 \text{ and } d_k \in \{1,2\} \right\} \\
&\supseteq \left\{ k = m+1,\dots,n-1 \mid \omega_k = 1, \omega_{k+1} = 1, \text{ and } d_k \in \{1,2\} \right\} \\
&= \left\{ k = m+1,\dots,n-1 \mid \omega_k = 1, \omega_{k+1} = 1, \text{ and } \delta_3\left(\Gamma_{\omega_1\dots \omega_{k}}\right) \in \{1,2\} \right\}\end{aligned}$$ Let $B^{[k]}$ be the Bernoulli random variable on $\Omega$, defined by $$B^{[k]}(\omega) := \mathbf{1}_{\{\omega_{k+1} = 1\}} \mathbf{1}_{\{\omega_{k} = 1\}} \mathbf{1}_{\{\delta_3(\Gamma_{\omega_1\dots \omega_k}) \in \{1,2\}\}}$$ The expected value of $B^{[k]}$ is given by $$\begin{aligned}
{\mathbb{E}}B^{[k]} &= \frac{1}{2^{k+1}} \sum_{i_1\dots i_k i_{k+1}} \mathbf{1}_{\{i_{k+1} = 1\}} \mathbf{1}_{\{i_{k} = 1\}} \mathbf{1}_{\{\delta_3(\Gamma_{i_1\dots i_k}) \in \{1,2\}\}} \\
&= \frac{1}{2^{k+1}} \sum_{i_1\dots i_{k-1}} \mathbf{1}_{\{\delta_3(\Gamma_{i_1\dots i_{k-1} 1}) \in \{1,2\}\}} \\
&= \frac{1}{2^{k+1}} \sum_{i_1\dots i_{k-1}} \Delta^{i_1\dots i_{k-1} 1}(\bm{\Gamma})\end{aligned}$$ Since $\bm{\Gamma} \in \overline{\Pi}_{m(\gamma,\varepsilon)}(\gamma)$, for $k > m \geq m(\gamma,\epsilon)$, we get $${\mathbb{E}}B^{[k]} \geq \gamma_0 := \frac{1}{4}\left(\frac{2}{3} - \gamma \right)$$ Let $\mathcal{K}(m,n) := \{ k = m+1,\dots, n-1 \mid k = m+1 \mod 2\}$, the set of integers $m+1,m+3,\dots$ comprised between $m+1$ and $n-1$. Random variables $B^{[k]}$, $k\in \mathcal{K}(m,n)$, are independent, and the expected value of their mean, denoted ${\mathbb{E}}B_{\mathcal{K}(m,n)} := \frac{1}{|\mathcal{K}(m,n)|}{\mathbb{E}}B^{[k]} $, satisfies ${\mathbb{E}}B_{\mathcal{K}(m,n)} \geq \gamma_0$. Fix any $\beta \in (0, \gamma_0)$, and let $$\mathcal{B}_{m,n}(\beta) := \left\{ \omega \in \Omega \Bigm| \sum_{k\in \mathcal{K}(m,n)} B^{[k]}(\omega) \geq (\gamma_0 - \beta) |\mathcal{K}(m,n)| \right\}$$ Hence, for $m \geq m(\gamma,\epsilon)$ and $\omega \in \mathcal{B}_{m,n}(\beta)$, we have[^8] $$\label{eq:Bmn_lower_bound}
|B_{m,n}(\omega)| \geq \sum_{k=m+1}^{n-1} B^{[k]}(\omega) \geq \sum_{k\in \mathcal{K}(m,n)} B^{[k]}(\omega) \geq (\gamma_0 - \beta) |\mathcal{K}(m,n)| \geq (\gamma_0 - \beta)\frac{n-m}{3}$$ Moreover, by applying Lemma \[lemma:mean\_independent\_rv\], we have $$\begin{aligned}
P\left( \mathcal{B}_{m,n}(\alpha) \right) &\geq P\left( \sum_{k\in \mathcal{K}(m,n)} B^{[k]}(\omega) \geq ({\mathbb{E}}B_{\mathcal{K}(m,n)} - \beta) |\mathcal{K}(m,n)| \right) \\
&\geq 1 - e^{-2\beta^2|\mathcal{K}(m,n)|} \\
&\geq 1 - e^{-2\beta^2 \frac{n-m}{3}} \end{aligned}$$ We define $\mathcal{U}_{m,n}(\zeta, \alpha, \beta) := T_m(\zeta) \cap \mathcal{A}_{m,n}(\alpha) \cap \mathcal{B}_{m,n}(\beta)$. Using (\[eq:Zdn\_upper\_bound\_1\]), (\[eq:Amn\_lower\_bound\]), and (\[eq:Bmn\_lower\_bound\]), for $n > m \geq m(\gamma,\epsilon)$ and $\omega \in \mathcal{U}_{m,n}(\zeta, \alpha, \beta)$, we have $$\label{eq:Zdn_upper_bound_2}
Z_{d}^{[n]}(\omega) \leq \kappa^{(\alpha+\frac{1}{2})(n-m)}\zeta^{\frac{\gamma_0 - \beta}{3}(n-m)}\zeta = \left( \kappa^{\alpha + \frac{1}{2}} \zeta^{\frac{\gamma_0 - \beta}{3}} \right)^{n-m} \zeta$$ Note that $\alpha, \beta$, and $\gamma$ (thus, $\gamma_0$) are some fixed constants. Hence, for any $\theta > 0$ (as in the fast polarization property), we may choose $\zeta > 0$, such that $\kappa^{\alpha + \frac{1}{2}} \zeta^{\frac{\gamma_0 - \beta}{3}} \leq 2^{-(1+\theta)}$. Using $Z^{[n]}(\omega) \leq \max_{d=1,2,3} Z_{d}^{[n]}(\omega)$, we get the following inequality, that holds for any $n > m \geq m(\gamma,\epsilon)$ and any $\omega \in \mathcal{U}_{m,n}(\zeta, \alpha, \beta)$: $$Z^{[n]}(\omega) \leq c 2^{-n(1+\theta)} = cN^{-(1+\theta)}$$ where $c = c(m,\alpha,\beta,\gamma,\zeta) := \left( \kappa^{\alpha + \frac{1}{2}} \zeta^{\frac{\gamma_0 - \beta}{3}} \right)^{-m} \zeta$, and $N = 2^n$. Note that $\alpha,\beta,\gamma$, and $\zeta$ have been fixed at this point, and only the value of $m$ can still be varied.
To complete the proof, we need to show that $\mathcal{U}_{m,n}(\zeta, \alpha, \beta)$ is sufficiently large (for some $m$, and large enough $n>m$), so that we may find information sets $\mathcal{I}_N$ of size $|\mathcal{I}_N| \geq RN$, for $R < \mathtt{I}(W)$. For this, we need the following lemma, which is essentially the same as Lemma 1 in [@arikan09], and the proof follows using exactly the same arguments as in [*loc. cit.*]{} (and also using the fact that $\bm{\Gamma}$ is a polarizing sequence).
\[lemma:Tm\_zeta\_large\_enough\] For any fixed $\zeta > 0$ and any $0 \leq \delta < \mathtt{I}(W)$, there exists an integer $m_0(\zeta, \delta)$, such that $$P\left( T_{m_0}(\zeta) \right) \geq \mathtt{I}(W) - \delta$$
Therefore, $P\left( T_{m}(\zeta) \right)$ can be made arbitrarily close to $\mathtt{I}(W)$, by taking $m$ large enough, and once we have made $P\left( T_{m}(\zeta) \right)$ as close as desired to $\mathtt{I}(W)$, we can make $P\left( \mathcal{A}_{m,n}(\alpha) \right)$ and $P\left( \mathcal{B}_{m,n}(\alpha) \right)$ arbitrarily close to $1$, by taking $n > m$ large enough. Hence, for any $R < \mathtt{I}(W)$, we may find $m_0 = m_0(\zeta, R)$ and $n_0 = n_0(m_0, \alpha, \beta, \gamma) > m_0$, such that $$P\left( \mathcal{U}_{m_0,n}(\zeta, \alpha, \beta) \right) > R, \ \ \forall n \geq n_0,$$ and since we may assume that $m_0 \geq m(\gamma,\varepsilon)$, we also have $$\label{eq:znomega_ubound}
Z^{[n]}(\omega) \leq c_0 N^{-(1+\theta)}, \ \ \forall n \geq n_0, \ \forall \omega \in \mathcal{U}_{m_0,n}(\zeta, \alpha, \beta)$$ where $c_0 := c(m_0,\alpha,\beta,\gamma,\zeta)$.
Now, for $n > 0$, let $\mathcal{V}_n := \{ \omega \in \Omega \mid Z^{[n]}(\omega) \leq c_0 N^{-(1+\theta)} \}$. Using (\[eq:znomega\_ubound\]), we have that $\mathcal{U}_{m_0,n}(\zeta, \alpha, \beta) \subseteq \mathcal{V}_n$, for any $n \geq n_0$, and therefore $P\left[ \mathcal{V}_n \right] \geq R$. On the other hand, $$P\left[ \mathcal{V}_n \right] = \sum_{i_1\dots i_n \in \{0,1\}^n} \frac{1}{2^n} \mathbf{1}\left\{ Z(W^{(i_1\dots i_n)}) \leq c_0 N^{-(1+\theta)} \right\} = \frac{1}{N}|\mathcal{I}_N|,$$ where $\mathcal{I}_N := \left\{ i\in \{0,\dots,N-1\} \mid Z(W^{(i)}) \leq cN^{-(1+\theta)} \right\}$. It follows that $|\mathcal{I}_N| \geq RN$, for $n\geq n_0$.
We have shown that, given $\varepsilon > 0$, the fast polarization property holds for any $\bm{\Gamma} \in \Gamma(\mathcal{S})^\infty_{\text{pol}} \cap \overline{\Pi}_{m(\gamma,\varepsilon)}(\gamma)$, with $P\left( \overline{\Pi}_{m(\gamma,\varepsilon)}(\gamma) \right) \geq 1 -\varepsilon$. We therefore conclude that it holds for any $\bm{\Gamma} \in \Gamma(\mathcal{S})^\infty_{\text{pol}} \bigcap \left(\bigcup_{\epsilon > 0} \overline{\Pi}_{m(\gamma,\varepsilon)}(\gamma)\right)$, which is a measurable subset of $\Gamma(\mathcal{S})^\infty_\text{pol}$, of same probability.
$\blacksquare$
[^1]: We have used a computer program to determine such a set of representatives
[^2]: To see that $U\mathcal{N} U^\dagger_{A \rightarrow B}$ is a quantum channel, it is enough to notice that if $\mathcal{N}_{A \rightarrow B}$ is defined by Kraus operators $\{E_k\}$, then $U\mathcal{N} U^\dagger_{A \rightarrow B}$ is defined by Kraus operators $\{ U E_k U^\dagger\}$.
[^3]: We use $\sigma_i^\dagger$ in the definition of the Pauli channel, to explicitly indicate that the definition does not depend on the representative of the equivalence class.
[^4]: Here, equality is understood as equivalence classes in $\bar{P}_1$
[^5]: Each good channel is decoded by taking a maximum-likelihood decision, according to the observed channel output and the previously decoded channels.
[^6]: Note that $\Gamma(\mathcal{S})^{\infty}$ is the infinite product space of countable many copies of $\Gamma(\mathcal{S})$, and it is endowed with the infinite product probability measure, taking the uniform probability measure on each copy of $\Gamma(\mathcal{S})$. See [@hewitt2013real] for infinite product probability measures.
[^7]: Here, indexes $j$ and $j'$ indicate the $j$-th qubits of ${\cal J}$ and ${\cal J}'$ systems
[^8]: The last inequality could be tighten, but we only need a non-zero fraction of $n-m$.
|
---
abstract: '[ We discuss the search for the heavy charged Higgs bosons $H^\pm$, implicitly of the , in the $\tau\nu$ decay mode at the LHC. Compared to the dominant decay mode $H^\pm\to bt$, the channel suffers from suppression due to the branching ratio and the lack of direct mass reconstruction, but the reduced background makes it a feasible channel especially in the large region. We study the production in ‘$bt$ fusion’ via $bg\to tH^-$, and the leading irreducible background $bg\to tW^-$. Our results indicate that for the $H^\pm$ mass of greater than 200 GeV and up to 1 TeV and higher, they can be discovered in this channel for a vast range of the parameter space, down to at least $\tan\beta\sim3$ and potentially the whole range of $\tan\beta$ down to 1.5 if the signal selection efficiency could be improved fourfold. Our analysis is sensitive to top quark identification at large rapidity, and should be supplemented with a full study including jet showering and detector effects. ]{}'
---
[RAL–TR–1999–012]{}\
[January 1999]{}\
[ **Searching for heavy charged Higgs bosons\
in the neutrino–tau decay mode at LHC**]{}\
[Kosuke Odagiri]{}\
[*Rutherford Appleton Laboratory,*]{}\
[*Chilton, Didcot, Oxon OX11 0QX, UK.*]{}\
Introduction
============
The charged Higgs bosons $H^\pm$ are a central prediction of the Two Higgs Doublet Model which is necessitated by supersymmetry. Given the results of recent experimental analyses, the neutral Higgs exclusion at LEP2 [@lep2search] and the constraint on the rare bottom quark decay $b\to s\gamma$ [@bdecay], both of which favour the elusive heavy mass region for $H^\pm$, at least in the , it is important to establish the parameter regions in which $H^\pm$ can be discovered at present and future colliders.
It is believed [@mo_lep2] that LEP2 can not discover the charged Higgs bosons in the mass regions indicated by the () and the current bound [@lep2search] on neutral Higgs bosons, viz the mass relation $M^2_{H^\pm}=M^2_W+M^2_{A^0}\gtrsim110$ GeV. The search for such a particle needs greater centre-of-mass energy, notably at LHC and, to a more limited extent, Tevatron [@spira]. The purpose of this paper is to establish the significance of the secondary $H^\pm$ decay channels, mainly at LHC.
If $H^\pm$ are ‘heavy’, which is taken from here on to imply $M_{H^\pm}\gtrsim m_t$, they can not be produced in top decay and the main production mode for them at hadron colliders is ‘$bt$ fusion’, viz: $$\label{gbth} gb\to tH^-.$$ We have taken $b$ as a sea quark, as is appropriate at such high energy collisions, whereas $t$ is created via $g\to t{\bar t}^*$. Of course the charge conjugate process $g\bar b\to\bar tH^+$ must also be included, and this behaves in exactly the same way since there is no intrinsic $b$ component of the proton.
The above production channel has been studied extensively in the literature [@heavysearches]. Recent analyses have focussed on the dominant decay mode $H^+\to t\bar b$ and the reduction of the background associated with it. The utilisation of the less dominant decay modes was discussed in the past [@hunters]. Here we focus on the decay mode $H^+\to\tau^+\nu_\tau$, improving on previous analyses by adopting the latest numbers for parameters and parton distribution functions as well as using kinematic selection techniques and the tau polarisation analysis [@bkm].
The result of our analysis indicates that the discovery region for $H^\pm$ is far greater than has previously been thought, covering the whole of the $(M_{H^\pm}, \tan\beta)$ parameter space up to $M_{H^\pm}\sim1$ TeV and higher, and down to at least $\tan\beta\sim3$. Optimisation in the analysis procedure and parameters can potentially extend the lower limit in $\tan\beta$ to 1.5, below which is excluded by LEP2 measurements so far in the .
Calculation
===========
In order to minimise the parameter dependence of our calculations and to make them applicable to as wide a range of parameters as possible, we adopt $M_{H^\pm}$ and $\tan\beta$ as the two Higgs sector parameters. When considering the $H^\pm$ decays, let us furthermore assume that the neutral Higgs mixing angle $\alpha$ is given by $\beta-\alpha=\pi/2$ which holds in at tree level in the heavy mass limit of the charged Higgs bosons, explicitly, $M^2_{H^\pm}\gg M^2_Z$. In this limit, practically the only decay modes are $tb$ and $\tau\nu$, and we assume superpartners too heavy to be produced in $H^\pm$ decays.
We restrict our discussions to the tree level, since our primary purpose is to establish the plausibility of signal detection, rather than to propose precision measurements. We adopt MRS 1998 leading order parton distributions [@mrs98], and calculate the strong coupling at one loop using the default MRS parameter values, neglecting virtual top contributions in both cases.
We assume zero kinematic bottom quark mass to be consistent with the sea quark picture we are adopting, whereas we retain finite bottom quark pole mass $m_b$ for the Yukawa coupling, which we take to be 4.25 GeV. Similarly, we take the tau kinematic mass to be zero and set the Yukawa coupling mass to 1.78 GeV. Both kinematic and Yukawa masses of the top quark are taken to be 175 GeV. The top quark, $H^\pm$ and $W^\pm$ widths are calculated at tree level, although we adopt the narrow width approximation for the top quark and $H^\pm$. We also note that the quark masses implicit in the parton distribution functions are independent parameters which are in general distinct from the kinematic and Yukawa masses we choose to adopt.
The electroweak parameters are $\aem=1/128$, $\sin^2\theta_W=0.2315$, $M_Z=91.187$ GeV, $M_W=M_Z\cos\theta_W\approx79.94$ GeV. The Cabbibo–Kobayashi–Maskawa matrix element $\ckm[bt]$ is taken to be 1 (it is practically 1. See [@pdg]).
As for the LHC collider parameters [@lhc], the centre-of-mass energy for $pp$ collision is 14 TeV. Integrated luminosities of 10 fb$^{-1}$ and 100 fb$^{-1}$ per year are expected in the low luminosity and high luminosity options, respectively.
The squared matrix elements are calculated by hand with the usual trace method, convoluted with the MRS parton distribution functions, and integrated using [@vegas]. The results are tested against the literature [@hunters] where they are available, and tests of gauge independence were carried out by hand.
When computing decay level distributions, the top quark is assumed to be on the mass shell using the so-called ‘narrow width approximation’, in order to minimise the number of ‘unphysical’ Feynman graphs we must consider without violating gauge invariance. We similarly take the $W^\pm$ boson arising from top quark decay to be on shell.
For the signal events, we assume that the top quark decays independently of production, by neglecting the spin correlation effects which are anyhow small for our purposes, in order to maximise the parameter space that can be covered in our analysis. Since we take top quark to be on the mass shell, this procedure is exact in the limit of identical left and right $H^\pm$ coupling, at $\tan\beta=\sqrt{m_t/m_b}$. We also take $H^\pm$ to be on the mass shell since we do not consider the finite width effect to be important at this stage.
For the background we incorporate the exact decay distribution of the top quark, and the finite constant width of $W^\pm$, both of which are potentially important when considering kinematic cuts to reduce the background. The calculation was tested using [@form].
The theoretical uncertainties associated with the above procedure are expected to be dominated by two sources. Firstly, the higher order corrections which are of the order $\as(M_{H^\pm})\sim10\%$. To be more precise, our procedure will need to be tested for jet and detector effects, which must be carried out using simulations where the leading corrections are resummed and factorised. Secondly, as we discuss in the next chapter, the possibility of the other decay modes, such as the $H^\pm\to hW^\pm$ decay mode for relatively light $H^\pm$ at low $\tan\beta$.
Analysis strategy
=================
The tree level Feynman diagrams for the signal process are shown in figure \[feyn\]. The spin- and colour-averaged matrix element squared for the process simplifies as follows: $$\begin{aligned}
\label{mgbth}\MSQ(g_{(1)}b_{(2)}\to t_{(3)}H^-_{(4)})&=&
\left(\frac{g^2_s}{2\nc}\right)\left(\frac{e^2}{2\sin^2\theta_W}\right)
\left(\frac{m_b^2\tan^2\beta+m_t^2\cot^2\beta}{2M_W^2}\right)
\times\nonumber\\&&\times\left(-\frac{u^2_4}{st_3}\right)
\left[1+2\frac{m_4^2-m_3^2}{u_4}
\left(1+\frac{m_3^2}{t_3}+\frac{m_4^2}{u_4}\right)\right].
\end{aligned}$$ Here $g_s$ is the strong coupling, $g_s^2/4\pi=\as$, and as a first approximation we evaluate it at scale $Q=(m_3+m_4)$. $N_c=3$ for . $s$, $t$ and $u$ are the usual Mandelstam variables, defined with the partonic momenta. $t_3=t-m_3^2=t-m_t^2$ and $u_4=u-m_4^2=u-M^2_{H^\pm}$. $\tan\beta$ is the ratio of the two expectation values of the two Higgs doublets, and we assume $1<\tan\beta<m_t/m_b$.
We do not tag the spectator bottom quark. However, in order to reduce the background, for example, from top pair, $W^\pm Z^0$ and single $W^\pm$ production, and to have a grip on the irreducible background, we do make use of the ‘spectator’ top quark, $t_{(3)}$. In [@hunters; @stifflepton], the following ‘stiff lepton’ procedure was proposed: the top quark moves nearly parallel to the original $g_{(1)}$ parton direction, and will be lost in the beam jets. However, if it decays leptonically, the electron or muon will be kicked out with $p_T$ of the order $m_t/2$ so that this can be used to trigger the signal events. Potentially the bottom quark can also be tagged, especially when it decays leptonically.
In this study we consider both the leptonic and hadronic decays. Although the leptonic mode is more inclusive, the hadronic branching fraction is larger ($BR[{\rm hadronic}]=2/3$), and the lack of the missing momentum from leptonic decay makes the handling of the kinematic distributions more transparent.
In the leptonic decay mode, we consider the inclusive sample of top-like events, wherein we assume the top quark identification to be carried out with a universal efficiency of $\varepsilon_t$, with $\varepsilon_t\sim
BR[{\rm leptonic}] = 2/9$.
In the hadronic decay mode, we require the inclusive multi-jet final state to have a reconstructed mass near the top mass. If bottom jet tagging can be utilised, this will aid the procedure by allowing the $W^\pm$ mass to be reconstructed also. We can not make a full simulation of the top quark identification efficiency, but as a first approximation we require all parton level decay products to be reasonably away from the beam directions, at pseudorapidities of less than 1.5 in order to reduce the noise. We do not impose cuts on separation between top decay products.
It has been shown [@stifflepton], and we will assume, that the tagging of the top quark is very effective in reducing the background, eliminating all but the irreducible background: $$\label{gbtw} gb\to tW^-,$$ whose presence is the main obstacle to the observation of the signal (\[gbth\]).
Let us assume the leptonic decay mode $H^-\to\tau\bar\nu$. In the limit of heavy $H^\pm$, $M^2_{H^\pm}\gg m^2_t$, the branching ratio is easily seen to be given by: $$\label{bratios}
BR[\tau\nu]\approx\frac{\Gamma[\tau\nu]}{\Gamma[bt]}=
\frac{m_\tau^2\tan^2\beta}{\nc(m_t^2\cot^2\beta+m_b^2\tan^2\beta)}.$$ For finite $M_{H^\pm}$ the branching ratio is always greater than this limit, as there is kinematic suppression on top quark for the $H^\pm\to
bt$ decay mode, which goes as $(1-m_t^2/M^2_{H^\pm})$. Thus if there are no other decay modes: $$\label{exactbratios}
BR[\tau\nu]=\left[1+\frac{m^2_b\nc}{m^2_\tau}
\left(1+\frac{m^2_t}{m^2_b\tan^4\beta}\right)
\left(1-\frac{m^2_t}{M^2_{H^\pm}}\right) \right]^{-1}.$$
Figure \[brplot\]a shows the $M^2_{H^\pm}\gg m^2_t$ limit of equation (\[exactbratios\]), as a function of $\tan\beta$. By requiring the other decay modes to vanish, we have implicitly assumed $\cos^2(\beta-\alpha)=0$ which holds in the heavy mass limit in the at tree level. The $H^\pm\to W^\pm h^0$ decay rate is proportional to this. The ratio of this mode to the $tb$ mode is given by: $$\label{whmode}
\frac{\Gamma[W^\pm h^0]}{\Gamma[bt]}=
\frac{\lambda^{3/2}[M_{H^\pm},M_{W^\pm},M_{h^0}]\cos^2(\beta-\alpha)}
{2\nc\lambda[M_{H^\pm},m_t,0](m_b^2\tan^2\beta+m_t^2\cot^2\beta)}$$ where $\lambda[m_1,m_2,m_3]=(m_1^2-(m_2+m_3)^2)(m_1^2-(m_2-m_3)^2)$ and is symmetric under the permutations of the three masses. At tree level in , it can be shown that $\cos^2(\beta-\alpha)$ is given in terms of the pseudoscalar Higgs mass $M_A$ and $\tan\beta$ by: $$\cos^2(\beta-\alpha)=\frac{1}{2}\left[1-\left[1+
\left(\frac{M_Z^2\sin4\beta}{M_A^2-M_Z^2\cos4\beta}\right)^2
\right]^{-1/2}\right].$$ Combined with equation (\[whmode\]), the approximation is valid if $M^2_A\gg M^2_Z$ which is implicit in $M^2_{H^\pm}\gg m^2_t$ since $M_{H^\pm}>M_A$ in . However, if $m_t(+m_b)>M_{W^\pm}+M_{h^0}$, the $W^\pm h^0$ decay could dominate in a small window, noting $M_{h^0}\gtrsim
80$ GeV, just below the threshold for the $bt$ mode.
The factor $(m_t^2\cot^2\beta+m_b^2\tan^2\beta)$ cancels between (\[mgbth\]) and (\[bratios\]), and we see that the cross section relevant to us goes as: $$\label{brnorm}
\MSQ(gb\to tH^-\to t\tau\bar\nu_\tau)\propto
\frac{m_\tau^2\tan^2\beta}{2\nc M_W^2}.$$ In figure \[brplot\](b) we plot this quantity, which represents the overall $\tan\beta$ dependence of production and decay in the $\tau\nu$ mode. The $1/BR[\tau\nu]$ contribution, neglected in (\[bratios\]) and (\[brnorm\]), is also included in this plot. In addition, in our forthcoming simulations in the next chapter, we also include the kinematic $(1-m_t^2/M^2_{H^\pm})$ suppression on the $bt$ decay mode.
The complete matrix element squared for the background process (\[gbtw\]), including the $t$ and $W^\pm$ decay, is available in [@ellisparke]. However, in order to gain insight into the structure of the background process let us present the spin- and colour- averaged matrix element squared for (\[gbtw\]). The Feynman graphs are identical to figure \[feyn\] with the substitution $H^-\to W^-$, in the following simple form[^1]: $$\begin{aligned}
\label{mgbtw}\MSQ(g_{(1)}b_{(2)}&\to& t_{(3)}W^-_{(4)})=
\left(\frac{g^2_s}{2\nc}\right)\left(\frac{e^2}{2\sin^2\theta_W}\right)
\times\nonumber\\&&\times\left[2+\left(1+\frac{m_3^2}{2m_4^2}\right)
\left(-\frac{u^2_4}{st_3}\right) \left[1+2\frac{m_4^2-m_3^2}{u_4}
\left(1+\frac{m_3^2}{t_3}+\frac{m_4^2}{u_4}\right)\right]\right].
\end{aligned}$$ This contains an ‘isotropic’ part, the 2 inside the first square brackets, and a ‘scalar’ part whose kinematics are exactly proportional to the signal cross section (\[mgbth\]) in the limit of degenerate $H^\pm$ and $W^\pm$ masses. The ratio of the two is around 1:1 (see forthcoming table \[tablea\]). As for the $W^\pm$ branching ratio into $\tau\nu_\tau$, this equals $1/9\approx11\%$. Thus in this limit, the ratio of signal to non-isotropic, ‘scalar’ background, assuming $H^\pm$ branching ratio of equation (\[bratios\]), is: $$\frac{\sigma(gb\to tH^-\to t\tau\nu_\tau)}
{\sigma(gb\to tW^-\to t\tau\nu_\tau)_{\rm scalar}}
= \frac{9}{2\nc}\left(\frac{m_\tau\tan\beta}{M_W}\right)^2
\frac{2M_W^2}{m_t^2+2M_W^2}.$$ $M_W/m_\tau\sim40$, so this amounts to some excess of signal over background for large values of $\tan\beta$. In reality, this will be reduced further by the phase space suppression on $H^\pm$ production, but to compensate for this, the mass difference between $H^\pm$ and $W^\pm$ can be exploited in kinematic cuts. A more subtle point is that $H^\pm$ are scalar whereas $W^\pm$ are vector, leading to different distributions of the decay products which may be utilised once a significant excess of signal over background is observed through the kinematic cuts. As a final stage of the signal–background analysis, we can use tau polarisation as described in [@bkm], noting that their helicities are predominantly left in $W^\pm$ decay, whereas they are predominantly right in $H^\pm$ decay.
To summarise, our strategy is as follows. Firstly, we select events with top quarks, possibly identified using the stiff lepton trigger, of the forms $t+\tau+\etmiss+X$ in the hadronic top quark decay mode, or $\ell+\tau+\etmiss+X$ in the leptonic decay mode, where $\ell$ is $e$ or $\mu$, is the typically large missing transverse energy, and $X$ contains jets near the beam directions. Secondly, we utilise kinematic cuts to screen out $W^\pm$ events. At this stage, it is desirable to be able to reconstruct the $H^\pm$ mass from the transverse momentum distribution of tau. In experimental analyses, the kinematic selection cuts will be replaced by the fitting of the distributions with signal and background expectations, which is a more elegant and effective procedure. Thirdly, we apply tau polarisation and spin correlation analyses in order to confirm the presence of $H^\pm$.
Results
=======
The total signal cross section, corresponding to (\[gbth\]) but excluding the $\tan\beta$ dependence of equation (\[brnorm\]), the kinematic suppression on $tb$ mode of equation (\[bratios\]), and excluding the $bg$ initial state and the charge conjugate processes, is shown in figure \[figa\]. It can be seen that the cross section falls approximately exponentially with increasing $M_{H^\pm}$, but even for large masses the cross section is significant, bearing in mind the integrated luminosity of 100 fb$^{-1}$ per year in the high luminosity option at LHC.
In table \[tablea\] we present the typical signal and background cross sections including the $\tan\beta$ dependence, the kinematic suppression on $bt$ mode, factor 4 for the $bg$ initial state and charge conjugation, and the $W^\pm\to\tau\nu$ branching ratio of 1/9. It is clear that the signal suffers from large background, against which we must find powerful cuts or distributions in order to make the signal visible.
In order to proceed further, we need to examine the distributions of decay products. In figures \[figb\] and \[figc\] we present the rapidity and transverse momentum distributions. Since all particles in our final state are treated as being massless, the rapidity can be replaced by pseudorapidity. We have plotted the leptonic top decay modes only. For the hadronic top decay, discarding the small effects of spin correlation, the light quark jet distributions are as the lepton distributions, and the missing momentum distribution is identical to the tau transverse momentum distribution.
We have also computed the rapidity difference between tau and the lepton, and found that the distribution does not behave significantly differently from the rapidity distribution of the lepton alone.
At first sight there seems little hope for signal detection. The rapidity distributions are similar, the small difference being partly due to spin correlation and partly due to the different resonance mass. The spin correlation can not be used to reduce the background since we have made the assumption that the top quark decays independently of production when generating the signal plots. Anyhow, there will be a small spin correlation effect in kinematic distributions involving the top quark decay, which can be used to confirm the coupling structure of signal and background events if we acquire a sufficient sample.
The tail of transverse momentum distribution for the background does not fall sufficiently to allow us to extract a convincing signal, especially if $\tan\beta$ is not high enough. This is as remarked in [@hunters; @stifflepton].
On a closer examination of the the tail of tau transverse momenta for the background events, we note that this is not primarily due to off-shell $W^\pm$ production, but due to $W^\pm$ being produced back-to-back with the top quark at large transverse momenta. Indeed, the author has made a similar plot using the ‘narrow width approximation’ for the $W^\pm$ and the difference between this and figure \[figc\]c is small. At a more analytical level too, it is easy to see that the off-shell effect is smaller than the kinematic effect of back-to-back production.
As noted earlier, there is an ‘isotropic’ component in the background production cross section (\[mgbtw\]) which makes the background even more prone to this back-to-back production than the signal. Having said that, we see that at large transverse momenta, the signal distributions of figures \[figc\]a, b also exhibit ‘dips’ in missing transverse momenta at high energies corresponding to the back-to-back production.
Such events are simpler to deal with than the ‘down-the-beam-pipe’ type events, since in this case the hadronic top quark decay can be used to reconstruct the top quark mass, giving us more control over the signal distributions.
Let us proceed as follows. Firstly, we remove events where tau and missing transverse momenta are not back-to-back, viz $\phi_{\tau-{\rm miss}}<\phi^{\rm cut}=\pi/2$ where $\phi$ is the azimuthal angle. Secondly, out of the events which survive the above requirement, we remove events where, in the leptonic case, a top quark remnant, either the bottom quark or the lepton, is harder than either the tau or the missing transverse momentum. In the hadronic case, we can proceed similarly, imposing some requirement on the reconstructed top quark (or a remnant) transverse momentum against either the tau or the missing transverse momentum. As default, we adopt $p_T(t)$ as the momentum cut-off. Thirdly, we make cuts on the tau and the missing transverse momenta, either by an explicit cut or by a plot of the cross section against $p_T^{\rm cut}$.
As for the detector cuts, for the leptonic decay mode of the top quark, the only requirement is on the tau and the missing transverse momenta. This will also help eliminate the reducible background, for example, from top pair production. We tentatively set this at 50 GeV. The final transverse momentum cut mentioned above will alter this minimum value at a later stage in the analysis. We also require the tau rapidity to be between $-2.5$ and $+2.5$, although this may be redundant after the transverse momentum cuts.
For the hadronic decay mode of the top quark we require all top decay products to have rapidity between $-1.5$ and $+1.5$, in addition to the transverse momentum requirement above.
To summarise, the cuts are: $$\begin{aligned}
\label{leptoniccuts}
\phi_{\tau-{\rm miss}}&>&\phi^{\rm cut}=\pi/2; \\
\min(p_T(\tau),\etmiss) &>& \max(50\ {\rm GeV},p_T(b),p_T(\ell)); \\
|\eta(\tau)| &<& 2.5
\end{aligned}$$ for the leptonic decay of the top quark, and $$\begin{aligned}
\label{hadroniccuts}
\phi_{\tau-{\rm miss}}&>&\phi^{\rm cut}=\pi/2; \\
\min(p_T(\tau),\etmiss) &>& \max(50\ {\rm GeV},p_T(t)); \\
|\eta(\tau)| &<& 2.5; \\
\max(|\eta(b)|,|\eta(q)|,|\eta(q')|)&<&1.5
\end{aligned}$$ for the hadronic decay of the top quark.
The resulting $p_T(\tau)$ and distributions are used to determine the optimum transverse momentum cut procedure.
The result is shown in tables \[tableb\] and \[tablec\].
The cuts are not optimised, and it is possible that the signal against background ratio can be improved without reducing the absolute rate too much. In any case, it is likely that the hadronic decay of the top quark offers a far greater chance of signal resolution, even though there is a significant loss of top remnants near the beam pipe. The improved background reduction is because of the absence of the neutrino from the decay of the top quark, allowing us to know the transverse momentum of the tau neutrino. Let us concentrate on the hadronic decay from here on.
In figure \[figd\] we show the tau transverse momentum distribution for the background events, after cuts, before the final transverse momentum cut. The missing transverse momentum , which we do not plot here, should have the same profile neglecting the spin correlation effects, but in practice the large integration error prevents this coincidence to materialise in our simulation.
Comparing figures \[figc\] and \[figd\] we see that rather than to keep $p_T^{\rm cut}$ as a free parameter as one would do in an experimental analysis, an easy way to simulate the effect of transverse momenta cuts is to impose cuts at 100 GeV on both tau and missing transverse momenta. The numerical integration for the background becomes very unstable at this value and renders higher cuts impractical in our simulation.
The optimisation of cuts can be achieved by balancing between keeping the production of the bosons $H^\pm$ and $W^\pm$ at low transverse momenta via the cuts described above for example, and the final cuts on, or the distribution of, the tau and missing transverse momenta. As we lower the transverse momenta of the bosons we also lower the transverse momentum of the top quark. Since we can not present a full simulation of top quark identification near the beam directions we do not consider it worthwhile speculating further into this, until we can utilise the full simulation.
Table \[tabled\] shows the result after the final cuts.
The cut is very effective at large $M_{H^\pm}$, where it is clear that larger values of $p_T^{\rm cut}$ would allow lower values of $\tan\beta$ to be probed. This can be easily achieved in practice by plotting the cross section after cuts against $p_T^{\rm cut}$. The signal will show a ‘bump’ in the distribution, which can be confirmed as due to $H^\pm$ rather than statistical fluctuation by tau polarisation and spin correlation analyses.
For the intermediate values of $\tan\beta$ we can extrapolate the figures by the known dependence of the cross section on $\tan\beta$. At $M_{H^\pm}=200$ GeV, where the transverse momentum cut is least effective, the signal-to-background ratio becomes 1:1 at $\tan\beta\sim2.3$. Given the LHC integrated luminosity of 100 fb$^{-1}$ for the high luminosity option, this corresponds to observing 3 events per year.
For large values of $M_{H^\pm}$, say masses higher than 500 GeV, the transverse momentum cut becomes very effective at reducing the background, but we must deal with the exponentially falling cross section (see figure \[figa\]). As an example, at $M_{H^\pm}=1$ TeV, applying the same cuts as above but setting $p_T^{\rm cut}=300$ GeV, we obtain a cross section of 0.00168 fb at $\tan\beta=1.5$. The numerical integration for the background is very unstable, but we consider it negligible. We find that in order to observe one event per year at the integrated luminosity of 100 fb$^{-1}$, $\tan\beta$ must be greater than 3.7.
We note that at such high values of $M_{H^\pm}$, the $bt$ decay mode becomes relatively clean, as what we observe will be $b$ and $t$ jets back-to-back with transverse momenta of about 500 GeV each. In the $\tau\nu_\tau$ decay mode too, the signal will be spectacular, with the hard tau and the missing momentum pointed back to back. Presumably the requirement of top identification becomes less important here, and the minimum $\tan\beta$ can be lowered significantly. As a conservative estimate, $\tan\beta>3$ is accessible in the $\tau\nu_\tau$ decay mode of $H^\pm$ for $M_{H^\pm}$ between 200 GeV and 1 TeV.
For smaller values of $M_{H^\pm}$, the transverse momentum cut becomes less effective, we must impose stricter preliminary cuts, and the analysis becomes complex. On the other hand, this range involves kinematic suppression on the $tb$ decay mode of $H^\pm$, and the branching ratio into $\tau\nu_\tau$ becomes greater. As an added bonus, the $W^\pm h^0$ mode kicks in at low $\tan\beta$ where the $\tau\nu_\tau$ channel becomes the least effective, which is easy to handle by imposing $b$ vertex tagging on $h^0\to b\bar b$ since we will hopefully know the $h^0$ mass at an early stage in LHC operation.
Conclusions
===========
We have discussed the search for the charged Higgs bosons $H^\pm$ of the Two Higgs Doublet Model, inherent for example in , through the $\tau\nu_\tau$ decay mode at LHC through the ‘$tb$ fusion’ production channel.
We have found that after a combination of kinematic cuts, there is a vast parameter space in which $H^\pm$ can be discovered, which have previously been thought inaccessible at LHC.
For $H^\pm$ masses heavier than 200 GeV, up to say 1 TeV, they can be discovered for $\tan\beta\gtrsim3$. This is without the optimisation of cuts.
Provided we neglect the other decay modes which are model dependent, the cross section is roughly proportional to $\tan^2\beta$, which implies that if we improve the procedure merely by fourfold, which is not unrealistic, the minimum $\tan\beta$ that can be probed falls to 1.5, which is below the maximum $\tan\beta$ probed so far at LEP2 assuming .
The signal thus discovered can be confirmed by analysing the tau polarisation and spin correlations.
The plots of the tau and the missing transverse momenta indicate that the $H^\pm$ mass can be evaluated to some degree from the signal distribution provided there is large enough sample. At least before cuts, the $p_T$ distributions of the tau and the missing transverse momenta are independent of $\tan\beta$, allowing one to reconstruct $M_{H^\pm}$. The total cross section, being proportional to $\tan^2\beta$, is then a good measure of $\tan\beta$. The spin correlations involving the top quark decay products can be used to confirm this measurement.
The model dependent region near and below the $tb$ decay threshold, at $M_{H^\pm}<200$ GeV, has not been treated in detail. A full study of this region is desired, assuming , or otherwise.
We only considered the irreducible background to this process. The reducible background includes top pair production and $W^\pm+$jets production. The second one is presumably removable by imposing the top selection and as long as we impose some jet profile criterion in the selection (so as to filter out soft multi-jets), it is at higher order in anyhow. The top pair background is only significant if one of the bottom quarks from top decay escapes in the beampipe. Although the cross section for this may not be negligible, this is hardly likely to lead to large tau and missing transverse momenta, as is the $W+$jets background, and should be negligible in our analysis.
We did not consider the supersymmetric background, notably the $2\to2$ sbottom–neutralino production from $gb$ fusion, with the sbottom decaying into top quark, $W^\pm$ and neutralino. In some regions of the parameter space where the sbottom decays into a top quark and a chargino and the chargino decays into a $W^\pm$ and a neutralino, such background will be large.
Once $H^\pm$ has been discovered in this channel, the mass and the $\tan\beta$ thus obtained can be used to optimise the signal extraction procedure for other modes, notably the dominant $tb$ decay, leading to a better understanding of the two-doublet Higgs sector.
Our analysis is sensitive to top quark identification near the beam direction, especially in the hadronic channel. This needs to be studied with full jet and detector simulations, leading to the optimisation of the signal extraction procedure, before the exact discovery contour can be elucidated. We note that as long as top quark is identified as such, we do not require much more information, other than the guarantee that there is no other source of missing transverse momentum, for our analysis. In fact, this guarantee may be sufficient on its own when using our procedure of selecting high transverse momentum decay products from charged bosons produced at low transverse momentum. The effect on reducible background of relaxing the top quark identification requirement may thus be worth analysing.
Acknowledgements {#acknowledgements .unnumbered}
----------------
I thank Stefano Moretti for advice, discussions and for reading the manuscript, and Bryan Webber for his guidance through my Ph.D. at Cambridge.
[99]{}
F. Richard, LAL 98–74, hep-ex/9810045, talk given at Zuoz Summer School on Hidden Symmetries and Higgs Phenomena, Zuoz, Switzerland, 16–22 August 1998.
The ALEPH collaboration, B429 1998 169.
See, for example, S. Moretti and K. Odagiri, G23 1997 537 and references therein.
M. Spira, talk presented at “Physics at Run II: Workshop on Supersymmetry”, Batavia, IL, 19–21 November 1998, DESY–98-159.
A.Krause, T. Plehn, M. Spira and P.M. Zerwas, B519 1988 85;\
V. Barger, R.J.N. Phillips and D.P. Roy, B324 1994 236;\
M. Guchait and D.P. Roy, D55 1997 7263.
J.F. Gunion, H.E. Haber, G.L. Kane and S. Dawson, [*‘The Higgs Hunter’s Guide’*]{} (Addison-Wesley, Reading MA, 1990).
B.K. Bullock, K. Hagiwara and A.D. Martin, 67 1991 3055.
A.D. Martin, R.G. Roberts, W.J. Stirling, R.S. Thorne, B443 1998 301.
Particle Data Group, ‘Review of particle physics’, C3 1998 1.
CMS Technical Proposal, CERN/LHC/94-43 LHCC/P1 (December 1994);\
ATLAS Technical Proposal, CERN/LHC/94-43 LHCC/P2 (December 1994).
G.P. Lepage, [*Jour. Comp. Phys.*]{} 27 1978 192.
J.A.M. Vermaseren, The Symbolic Manipulation Program FORM, Computer Algebra Netherland, Amsterdam 1991.
J.F. Gunion, H.E. Haber, F.E. Paige, W.K. Tung and S.S.D. Willenbrock, B294 1987 621.
R.K. Ellis and S. Parke, D46 1992 3785.
Table captions {#table-captions .unnumbered}
--------------
Figure captions {#figure-captions .unnumbered}
---------------
[|c|c|c|]{} $M_{H^\pm}$ (GeV) & $\tan\beta$ & $\sigma$ (fb)\
\
200 & 1.5 & 4.78\
200 & 30 & 1530\
500 & 1.5 & 0.139\
500 & 30 & 52.1\
\
& 13260\
& 6650\
& 6610\
\
Table 1
[|c|c|c|]{} $M_{H^\pm}$ (GeV) & $\tan\beta$ & $\sigma$ (fb)\
\
200 & 1.5 & 0.252\
200 & 30 & 81.0\
500 & 1.5 & 0.0205\
500 & 30 & 7.69\
\
& 0.86\
\
Table 2
[|c|c|c|]{} $M_{H^\pm}$ (GeV) & $\tan\beta$ & $\sigma$ (fb)\
\
200 & 1.5 & 0.388\
200 & 30 & 124\
500 & 1.5 & 0.0240\
500 & 30 & 9.00\
\
& 371\
\
Table 3
[|c|c|c|]{} $M_{H^\pm}$ (GeV) & $\tan\beta$ & $\sigma$ (fb)\
\
200 & 1.5 & 0.0123\
200 & 30 & 3.94\
500 & 1.5 & 0.0177\
500 & 30 & 6.659\
\
& $0.030\pm0.009$\
\
Table 4
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
[^1]: In our decay level simulations we supplement (\[mgbtw\]) by including the top quark and $W^\pm$ decay, with the top quark being on shell, as well as the $W^\pm$ from top quark decay. For the signal, (\[mgbth\]) is supplemented similarly, but we set the $H^\pm$ on shell, and assume the top quark to decay independently of production by averaging over the top quark helicities, as mentioned in the previous chapter. Neither of these approximations are expected to have a significant effect on our results.
|
---
abstract: 'We investigate the dynamics of pairs of drops in microfluidic ladder networks with slanted bypasses, which break the fore-aft structural symmetry. Our analytical results indicate that unlike symmetric ladder networks, structural asymmetry introduced by a single slanted bypass can be used to modulate the relative drop spacing, enabling them to contract, synchronize, expand, or even flip at the ladder exit. Our experiments confirm all these behaviors predicted by theory. Numerical analysis further shows that while ladder networks containing several identical bypasses are limited to nearly linear transformation of input delay between drops, combination of forward and backward slant bypasses can cause significant non-linear transformation enabling coding and decoding of input delays.'
author:
- 'Jeevan Maddala, William S. Wang, Siva A. Vanapalli and Raghunathan Rengaswamy'
bibliography:
- 'apstemplate.bib'
title: 'Drop Traffic in Microfluidic Ladder Networks with Fore-Aft Structural Asymmetry'
---
Understanding the spatiotemporal dynamics of confined immiscible plugs in interconnected fluidic paths is essential for applications ranging from lab-on-chip technologies [@review_angewad; @song_ismagilov] to physiological flows [@vascular_flow_review] to porous media flows [@prs_media]. The traffic of drops or bubbles in even simple networks such as bifurcating channels can be astonishingly complex due to collective hydrodynamic resistive interactions in the branches [@spack; @drp_trfc_engl; @fuerstman]. Although such intricate dynamics, in the case of lab-on-chip applications, make device design challenging, the collective behaviors can be harnessed to perform useful tasks such as droplet sorting [@cristobalsortin] and storage [@swastika; @rails_anchors_stroing].
Recently, the collective dynamics between pairs of drops have been harnessed in the so-called microfluidic ladder networks (MLNs) to control their relative drop spacing [@manu; @*manu_conf]. In MLNs, two droplet-carrying parallel channels are connected by narrow bypass channels through which the motion of drops is forbidden but the carrier fluid can leak. Current versions of ladders have fore-aft structural symmetry due to equally-spaced vertical bypasses. Such symmetric ladders are limited in functionality because the distance between pairs of drops have been shown to decrease at the exit only for constant inlet flow [@rail_road; @manu; @ajdari]. Since flexible manipulation of drop spacing in networks is crucial for passively regulating a variety of tasks including drop coalescence [@ladder_sync_drp_gen], detection and storage, there is a need to design microfluidic ladders with multiple functionalities.
From a fundamental perspective, the dynamics of drops in MLNs is distinct compared to the widely-studied microfluidic loops [@spack; @ajdari; @fuerstman; @cmptn_locl_flws]. In loops, drops at junctions choose a given branch. These discrete choices make such systems non-linear, enabling coding and decoding of input signals. Since drops do not typically make decisions at the bypass junctions in ladders, an open question is: is it possible to design microlfuidic ladders that yield significant non-linear transformation of input signal?
In this Letter, we study the dynamics of spacing between drop pairs in MLNs with slanted bypasses. We find that because the slant breaks the fore-aft symmetry, it provides significantly more control over drop spacing than symmetric MLNs. We also discover that inclusion of slanted bypasses in ladders can non-linearly transform the initial delay between drops. These advanced capabilities arise because slanted bypasses flexibly manipulate *(i)* the locations in the channels where drop velocity changes occur and *(ii)* the time drops spend with bypasses between them.
As shown in Fig. \[fig:confgr\], the key framework for quantifying drop spacing comes from understanding the variation in relative velocity ($u$) between drops in the top and bottom channel as they cross nodes in the network. Consider the simple case of a symmetric MLN with one vertical bypass. As shown in Fig. \[fig:confgr\](b) left, when two drops driven by a constant flow rate enter the ladder with an initial separation $(\Delta x_{in})$, they maintain the same separation, as $u=0$. When the leading drop crosses the node, a new configuration (see Fig. \[fig:confgr\](b) center) is reached, and the relative velocity changes as fluid leaks into the bypass, i.e. $u < 0$, causing a contraction in inter-drop distance. When both drops cross the bypass, $u=0$ again as the pressure drop across the bypass is zero due to equal downstream branch resistances (see Fig. \[fig:confgr\](b) right). Thus, the outlet drop spacing, $(\Delta x_{out})$, is less than the initial separation and is given by $\Delta x_{out}$ = $\Delta x_{in}$+u$\Delta T$, where $\Delta T$ is the duration drops remain in the particular configuration of Fig. \[fig:confgr\](b) center.
In networks with many bypasses, more drop configurations and relative velocity changes are possible than that of Fig. \[fig:confgr\](b) as drops traverse through multiple nodes. In general, we find that $${\Delta}x_{out} = {\Delta}x_{in} + \displaystyle\sum_{j=1}^p u_j\Delta T_j
\label{eqn:delxt}$$ where $p$ is the number of distinct configurations of droplets occurring in the network, $u_j$ and $\Delta T_j$ are the associated relative velocities and time periods.
The degree of separation achieved at the ladder exit depends on the contribution of the summation term in Eqn (\[eqn:delxt\]) both with respect to magnitude and sign. The strength of this contribution is modulated by the specific architecture of the ladder network. To compute this contribution, we use resistive network modeling [@ajdari], where we assume each drop is a point object with the same hydrodynamic resistance $(R_d)$. Since the drop velocity $(V)$ is linearly dependent on liquid flow rate $(Q)$ [@garstecki_linear_flow], we have $\mathfrak{V = \beta Q/S}$ , where $S$ is the channel cross-sectional area and $\mathfrak{0 < \beta < 2}$. To preserve the relative separation when drops leave the ladder network, we choose the downstream channel resistances to be equal.
We begin by discussing the effect of a single slanted bypass in regulating the dynamics of drop spacing. Fig. \[fig:basicunit\] ($a$, $b$) shows a representative ladder network with a slanted bypass. In contrast to the vertical bypass, a new control parameter, $\Delta L$ is needed to describe two possible structural configurations—backward slant for $\Delta L < 0$ and forward slant for $\Delta L > 0$.
To determine the drop spacing at the exit due to the slanted bypass, we identify the relative velocities that are non-zero and the corresponding durations as prescribed by Eqn (\[eqn:delxt\]). Similar to the vertical bypass in Fig. \[fig:confgr\], non-zero $u$ occurs only after one of the droplets crosses a node. Solving the hydrodynamic circuit for this drop configuration analytically, we obtain $\mathfrak{{u} = \beta Q/S \cdot (R_d/(R_{b}+2R_e+R_d))}$ and $\mathfrak{\Delta T = (|{\Delta}x_{in}-\Delta L|)S/({\beta}Q})$, where $R_b$ and $R_e$ are the bypass and exit channel resistance respectively. Thus, Eqn (\[eqn:delxt\]) for the case of an MLN with a slanted bypass transforms to $$\begin{aligned}
{\Delta}x_{out} = {\Delta}x_{in}-M ({\Delta}x_{in}-\Delta L)
\label{eqn:comprhnsve}\end{aligned}$$ where $\mathfrak{M = {R_d}/{(R_b+2R_e+R_d)}}$ and 0 $<$ M $<$ 1. Note in Eqn (\[eqn:comprhnsve\]), $\Delta x_{in} > 0$ corresponds to the top drop leading over the bottom drop in the ladder.
Remarkably, Eqn (\[eqn:comprhnsve\]) captures several dynamical regimes emerging from structural asymmetry due to a slanted bypass as illustrated in Fig. \[fig:basicunit\](c). For the particular case of a vertical bypass ($\Delta L=0$), Eqn (\[eqn:comprhnsve\]) reveals that drops can only undergo contraction (see Fig. \[fig:confgr\](a)). Perfect synchronization of drop pairs, i.e. $\Delta x_{out}$=0 is difficult to achieve with a single vertical bypass, as it requires $M$ to be unity.
In contrast to the vertical bypass, we find that the backward slant, where $\Delta L < 0$, yields flexible control over drop spacing as illustrated in Fig. \[fig:basicunit\](ii). For large input drop spacing, $\mathfrak{{\Delta}x_{in} > \frac{M\Delta L}{M-1}}$, the pairs at the exit undergo contraction for the same reason as in the vertical bypass. Perfect synchronization can also be realized with just a single backward slant when $\mathfrak{{\Delta}x_{in} = \frac{M\Delta L}{M-1}}$. Moreover, when $\mathfrak{{\Delta}x_{in} < \frac{M\Delta L}{M-1}}$, a new regime emerges that we refer to as $\textit flipping$. We observe that the leading droplet is initially ahead of the lagging droplet. However, when the leading drop crosses the bypass first, its velocity is reduced and the lagging drop has sufficient duration to catch up and overtake it. Thus, the flipping behavior yields $\Delta x_{out} < 0$ as shown in Fig. \[fig:basicunit\](ii).
Similar to the backward slant, the forward slant (where $\Delta L > 0$) provides additional means of control. Interestingly, in contrast to the backward slant, the behavioral transitions depend only on the value of input drop spacing relative to $\Delta L$. When $\mathfrak{{\Delta}x_{in} < \Delta L}$, the lagging drop crosses the bypass first, resulting in expansion as highlighted in Fig. \[fig:basicunit\](iii). Alternatively, if $\mathfrak{{\Delta}x_{in} > \Delta L}$, the leading droplet crosses the bypass first, resulting in contraction. If $\mathfrak{{\Delta}x_{in} = \Delta L}$, then both drops cross the bypass simultaneously, and the input spacing is preserved. Thus forward slant allows drop pairs to expand, contract or remain unchanged.
It would be misleading to visualize the slanted ladder network as being equivalent to a vertical bypass network with the addition of $\Delta L$ to the inlet spacing as we have already shown that vertical bypasses can only reduce the drop spacing for a fixed inlet flow. We assumed the exit channels to have equal lengths to simplify the analysis; however, the analysis presented in this letter extends trivially to unequal exit lengths also.
To confirm the different behaviors predicted by our theory, we sought to construct an MLN with a single slant. However, the design space is large, requiring optimization of upstream and downstream transport channel resistance, slant resistance, slant slope, and hydrodynamic resistance of drops, which itself is a complex function of the flow conditions and fluid properties [@siva_drop_res; @labrot]. Fortunately, insights from Eqn (\[eqn:comprhnsve\]) reduce the search space. According to Fig. \[fig:experimental\], the backward slant is the best candidate to achieve maximum contraction, perfect synchronization and flipping. Moreover, Eqn (\[eqn:comprhnsve\]) reveals that if $\Delta x_{in} < 0$ ($.i.e.,$ the top drop is lagging behind the bottom drop in the ladder), then the forward slant becomes a backward slant, allowing access to the expansion regime as well as the condition where the input separation does not change. Thus, we chose the backward slant to test our predictions.
We incorporated two flow-focusing drop generators in polydimethyl(siloxane) devices to introduce drops at a constant flow rate into the ladder. To amplify the effects produced by the backward slant, we maximized the value of $M$ by minimizing $R_b$ and $R_e$. For example, as shown in Fig. \[fig:experimental\](a), the bypass has an enlarged mid-section to minimize $R_b$. We also ensured $R_b$ and $R_e$ to be $\sim$ O($R_d$). In Fig. \[fig:experimental\](b), we show experimental curves corresponding to each of the dynamical regimes predicted by our theory. We find the drop spacing to be relatively unchanged when the drop pair is before or after the bypass. However, as pairs of drops cross the bypass section, their spacing may expand, contract, flip, synchronize or remain unchanged depending on the initial separation. By comparing the data of Fig. \[fig:experimental\] with the expression for $u$, we estimate $R_d \approx 1.2kg/mm^{4}s$; $M \approx 0.2$. Taken together, our analytical results and experiments suggest that MLNs with slanted bypasses provide greater flexibility than ladders with vertical bypasses.
To understand the flexibility due to additional bypasses, we investigated ladders containing several identical bypasses. In contrast to the single bypass case, in ladders with $n$ bypasses, the maximum number of configurations where $u$ is non-zero is $n(n+1)/2$. However, all these configurations need not be realized for a given $\Delta x_{in}$, which makes the theoretical analysis complex. We therefore used resistive network based simulations [@ajdari] to fully quantify the exit drop spacing for arbitrary input delay. As shown in Fig. \[fig:slopeslant\](a), we find that additional vertical bypasses simply amplify the contraction effect due to a single bypass, *i.e.,* slope decreases with increasing number of bypasses. A similar outcome also holds, as shown in Fig. \[fig:slopeslant\](b), for the particular case of an MLN with forward slants. We also find that the largest change in drop spacing occurs in the first few bypasses. Ladders with multi-bypasses could therefore be useful to maintain the same behavior as their single bypass counterparts, while dampening the effect of small fluctuations in input drop spacing.
Our analysis of MLN designs with identical bypasses has shown that at small input delay, drop spacing may either contract or expand, while it always contracts at large input delay (c.f. Fig. \[fig:slopeslant\]). The contraction at large input delay is expected because the leading drop is the first to cross the bypass and slow down. Initially, it appears that expansion is not possible at large input delays. To further probe this notion, we developed an evolutionary algorithm [@Genetic_algo] to search for ladder designs containing any combination of slant and/or vertical bypasses that might be capable of contraction at low input spacing and expansion at large input spacing. Our search strategy revealed that such networks do exist, and an example is shown in Fig. \[fig:nonlinear\](a). The dynamics of drop spacing in this network cannot be rationalized from mere addition of functionalities of the single bypasses shown in Fig. \[fig:slopeslant\]. Instead, we find that the first five bypasses collectively cause contraction of drop spacing, while the last two bypasses cause drops to expand. However, the relative magnitudes of contraction and expansion from these sets of bypasses depends on the input drop spacing. We observe that at small input delays, the first five bypasses dominate, resulting in contraction behavior (see Fig. \[fig:nonlinear\](a)), whereas at large input delays, the last two bypasses dominate, yielding expansion.
A striking observation from Fig. \[fig:nonlinear\](a) is that the curve is significantly nonlinear compared to the almost linear dependence observed in ladders with identical bypasses. This result is significant because it implies nonlinear transformation of input delay without any droplet decision-making at bypass junctions. A unique consequence of this nonlinear transformation is the capability to encode and decode input delays as shown in Fig. \[fig:nonlinear\](b) where the entrance delay between pairs of drops represents the input signal and the system of bypasses represents the encoder and decoder. First consider the curves in Fig. \[fig:experimental\] where the input signal is ‘scrambled’ in the bypass section, but does not revert to its original value and therefore is not decoded. In striking contrast, we find in Fig. \[fig:nonlinear\](a), that input signal gets encoded and decoded at two different values of $\mathfrak{\Delta x_{in} = 4.5, 7(\Delta L)}$, where $\mathfrak{\Delta x_{in} = \Delta x_{out}}$. At these input delays, we find that the above-discussed contraction and expansion effects introduced by sets of bypasses negate each other. Thus, our results highlight a new route to code and decode signals compared to earlier studies that use drop decision making events in networks [@fuerstman]. The encoder/decoder illustration in this letter is just an exemplar demonstration of the potential of such networks in realizing complex behavior without any active components.
Given that ladders with mixed combinations of bypasses can display nonlinear behavior, we ask what degree of nonlinearity can be achieved in ladders. A qualitative indication can be obtained by considering reversibility in ladder networks. Reversibility implies that the original input delay is recovered when the flow is reversed. Ladder networks are reversible because of the absence of decision-making events [@ajdari]. This reversibility criterion demands that the relationship between input and output delays remain bijective. In addition, this functional relationship cannot have maxima, minima, or saddle points because it has to be strictly monotonic. Thus, we believe reversibility imposes bounds on the degree of nonlinearity that can be achieved with microfluidic ladders.
In summary, we observe that MLNs with mixed bypasses display rich dynamics in drop spacing as well as nonlinear behavior. Such ladder networks in fact also exist in natural systems including leaf venation [@leaf_ven_reveiw], microvasculature [@vascular_flow_review; @vascular_flow_bmfc] and neural systems [@rat_ladder_neurons]. Interestingly, the nonlinear shape of Fig. \[fig:nonlinear\](a), resembles that of the widely observed sigmoid function, which is essential for coding/decoding neural signals [@single_neuron], as it produces an invertible map. Finally, the framework described here can be expanded to explore not only pairs of drops, but also trains of drops to further probe collective hydrodynamics in drop-based microfluidic networks.\
We acknowledge National Science Foundation for partial financial support (Grant No. CDI-1124814).
|
---
abstract: 'An overview is given of recent developments in the field of Dirac equations generalized to curved space-times. An illustrative discussion is provided. We conclude with a variation of Dirac’s large-number hypothesis which relates a number of physical quantities on cosmological scales.'
author:
- 'U. D. Jentschura'
title: |
From Dirac theories in curved space-times to a\
variation of Dirac’s large-number hypothesis
---
Some physicists, even today, may think that it is impossible to connect relativistic quantum mechanics and general relativity [@Ei1915; @Hi1915; @Ei1916]. This impression, however, is false, as demonstrated, among many other recent works, in an article published not long ago in this journal [@GoNe2014]. In order to understand the subtlety, let us remember the meaning of first [@CTDiLa1978vol1; @CTDiLa1978vol2], second [@ItZu1980; @AbHa1986], and third quantization (see Refs. [@PiMo2000; @Ki2014; @Bl2014]). In first quantization, what was previously a well-defined particle trajectory now becomes smeared and defines a probability density of the quantum mechanical particle. In second quantization, what was previously a well-defined functional value of a physical field at a given space-time point now becomes a field operator, which is subject to quantum fluctuations. E.g., quantum fluctuations about the flat space-time metric can be expressed in terms of the graviton field operator, for which an explicit expression can be found in Eq. (5) of Ref. [@AbHa1986].
In third quantization, what was previously a well-defined space-time point now becomes an operator, defining a conceivably noncommutative entity, which describes space-time quantization. E.g., the emergence of the noncommutative Moyal product in quantized string theory is discussed in a particularly clear exposition in Ref. [@Bl2014], in the derivation leading to Eq. (32) therein. Speculations about conceivable effects, caused by the noncommutativity of space-time, in hydrogen and other high-precision experiments, have been recorded in the literature [@ChSJTu2001] (these rely on noncommuting coordinates which define a Moyal–Weyl plane). It is instructive to realize that relativistic quantum mechanics can easily be combined with general relativity if the latter is formulated classically, while the former relies on wave functions expressed in terms of the space-time coordinates.
We thus observe that space-time quantization is not necessary, a priori, in order to combine general relativity and quantum mechanics. Indeed, before we could ever conceive to observe effects due to space-time quantization, we should first consider the leading-order coupling of the Dirac particle to a curved space-time. Space-time curvature is visible on the classical level, and can be treated on the classical level [@Ei1916]. Deviations from perfect Lorentz symmetry caused by effects other than space-time curvature may result in conceivable anisotropies of space-time; they have been investigated by Kostelecky [*et al.*]{} in a series of papers (see Refs. [@CoKo1997; @DiKoMe2009; @KoMe2012] and references therein) and would be visible in tiny deviations of the dispersion relations of the relativistic particles from the predictions of Dirac theory. Measurements on neutrinos [@KoMe2012] can be used in order to constrain the Lorentz-violating parameters. Other recent studies concern the modification of gravitational effects under slight global violations of Lorentz symmetry [@KoTa2011].
Brill and Wheeler [@BrWh1957] were among the first to study the gravitational interactions of Dirac particles, and they put a special emphasis on neutrinos. The motivation can easily be guessed: Neutrinos, at the time, were thought to be massless and transform according to the fundamental $(\tfrac12,0)$ and $(0,\tfrac12)$ representations of the Lorentz group. Yet, their interactions have a profound impact on cosmology [@Wh1960]. Being (almost) massless, their gravitational interactions can, in principle, only be formulated on a fully relativistic footing. This situation illustrates a pertinent dilemma: namely, the gravitational potential, unlike the the Coulomb potential, cannot simply be inserted into the Dirac Hamiltonian based on the correspondence principle of classical and quantum physics [@CTDiLa1978vol1; @CTDiLa1978vol2; @SoMuGr1977; @Pa1980; @GiLi2013].
For illustration, the free Dirac Hamiltonian is $\vec\alpha \cdot \vec p + \beta \, m$, the Dirac–Coulomb Hamiltonian is $\vec\alpha \cdot \vec p + \beta \, m - Z\alpha/r$, but the Dirac–Schwarzschild [@DoHo1986; @Ob2001; @SiTe2005; @JeNo2013pra] Hamiltonian is [*not*]{} simply given as $\vec\alpha \cdot \vec p + \beta \, m - G \, m \, M/r$. Here, $Z$ is the nuclear charge number, $\alpha$ is the fine-structure constant, $G$ is Newton’s gravitational constant, $m$ and $M$ are the masses of the two involved particles, and $r$ is the distance from the gravitational center. We use the Dirac $\vec\alpha$ and $\beta$ matrices in the standard representation [@JeNo2013pra]. Natural units with $\hbar = c = \epsilon_0 = 1$ are employed.
Key to the calculation of the gravitational coupling of Dirac particles is the observation that the Dirac–Clifford algebra needs to be augmented to include the local character of the space-time metric, $$\label{flatcurved}
\{ \gamma^\mu, \gamma^\nu \} = 2 \, g^{\mu\nu} \to
\{ \gamma^\mu(x), \gamma^\nu(x) \} = 2 \, g^{\mu\nu}(x) \,,$$ and yet, the local Dirac matrices $\gamma^\mu(x)$ have to be in full consistency with the requirement that the covariant derivative of the metric vanishes (axiom of local Lorentz frames in general relativity). Curved and flat space-time matrices are denoted with and without the argument $x$, respectively. This leads to the following ansatz for the covariant derivative of a Dirac spinor, $$\partial_\mu \psi(x) \to \nabla_\mu \psi(x) =
\left[ \partial_\mu -\Gamma_\mu(x) \right] \, \psi(x) \,,$$ and the spin-connection matrices $\Gamma^\mu(x)$ are expressed in terms of the $\gamma^\mu(x)$ as follows [@BrWh1957; @Je2013pra], $$\Gamma_\rho = - \frac{{\mathrm i}}{4} \, g_{\mu\alpha}(x) \,
\left( \frac{\partial {b_\nu}^\beta(x)}{\partial x^\rho} \,
{a^\alpha}_\beta(x) - {\Gamma^\alpha}_{\nu \rho} \right) \,
\sigma^{\mu\nu}(x),$$ where $\sigma^{\mu\nu}(x) =
\frac{{\mathrm i}}{2} [\gamma^\mu(x), \gamma^\nu(x)]$ is a local spin matrix and the $a$ and $b$ coefficients belong to the square root of the metric, $\gamma_\rho(x) = {b_\rho}^\alpha(x) \, \gamma_\alpha$, and $\gamma^\alpha(x) = {a^\alpha}_\rho(x) \, \gamma^\rho$, while the Christoffel symbols are ${\Gamma^\alpha}_{\nu \rho} \equiv {\Gamma^\alpha}_{\nu \rho}(x)$. This formalism was later shown to be compatible with the exponentiation of the local generators [@Bo1975prd; @SoMuGr1977; @GiLi2013] of the Lorentz group in the spinor representation, as explained for flat space-time in detail in Ref. [@ItZu1980]. Contrary to wide-spread belief, it is indeed possible to combine relativistic quantum mechanics and general relativity, at least on the level of quantum mechanics and quantum fields, in curved space-times, which are defined by a nontrivial (but classical, non-quantized) metric $g^{\mu\nu}(x)$. This formalism is free from ambiguities and has been used in order to evaluate perturbative corrections to the hydrogen spectrum in curved space-times with a nontrivial metric [@Pa1980prd; @PaPi1982].
Recent developments in this field include a series of papers [@HeNi1990; @Ob2001; @SiTe2005; @SiTe2007; @ObSiTe2009; @ObSiTe2011; @Je2013pra; @JeNo2013pra; @JeNo2014jpa; @Je2014pra], where the gravitationally coupled Dirac particle is formulated first in a fully relativistic setting, and then, a generalized Foldy—Wouthuysen transformation is applied in order to isolate the nonrelativistic limit, plus correction terms. For the photon emission in curved space-time, we refer to the result for the vector transition current in Ref. [@JeNo2013pra]. A certain pitfall, hard to spot at first glance because a deceptively elegant Eriksen–Kolsrud [@ErKo1958] variant of the standard Foldy–Wouthuysen transformation exists, leads to the rather unfortunate appearance of spurious, parity-violating terms. In Ref. [@JeNo2014jpa], the conclusion has been reached that, in a typical, nontrivial space-time geometry, the parity-breaking terms in the in the “chiral” modification [@ErKo1958] of the Foldy—Wouthuysen transformation change the interpretation of the wave functions and of the transformed operators to the extent that the transformed entities cannot be associated any more with their “usual” physical interpretation after the transformation, questioning the usefulness of the “chiral” transformation (for more details, see Sec. 4 of Ref. [@JeNo2014jpa]). We here take the opportunity to discuss the Foldy–Wouthuysen transformation of gravitationally coupled Dirac particles in some further detail. Of particular importance is the Dirac–Schwarzschild Hamiltonian [@Ob2001; @SiTe2005; @JeNo2013pra], which describes the most basic central-field problem in gravitation and probably replaces the Dirac–Coulomb Hamiltonian [@SwDr1991a; @SwDr1991b; @SwDr1991c] as the paradigmatic bound-state problem in gravitation.
Inherently, the Foldy–Wouthuysen program is perturbative; for nontrivial geometries, one tries to disentangle the particles and the antiparticle degrees of freedom which are otherwise [*simultaneously*]{} described by the Dirac theory [@Je2013pra]. A Hamiltonian, which preserves the full parity symmetry of the Schwarzschild geometry, and which allows for a clear identification of the relativistic corrections terms, has recently been presented in Ref. [@JeNo2013pra]. It reads $$\begin{aligned}
\label{HFW}
& H_{\rm FW} =
\beta \, \left( m + \frac{\vec p^{\,2}}{2 m} -
\frac{\vec p^{\,4}}{8 m^3} \right)
- \beta \, \frac{m \, r_s}{2 \, r}
\\[1.0ex]
& \; + \beta \, \left(
- \frac{3 r_s}{8 m} \,
\left\{ \vec p^{\,2}, \frac{1}{r} \right\}
+ \frac{3 \pi r_s}{4 m} \delta^{(3)}(\vec r) \,
+ \frac{3 r_s}{8 m} \, \frac{\vec\Sigma \cdot \vec L}{r^3} \right) \,.
\nonumber\end{aligned}$$ Here, the Schwarzschild radius is given as $r_s = 2 \, G \, M$. This Hamiltonian is equivalent to the one presented in Eq. (26) of the recent paper published in this journal [@GoNe2014], provided the $V$ and $W$ functions are expanded to first order in the gravitational constant $G$ (first order in the Schwarzschild radius $r_S$). The subject is not without pitfalls: We note that Donoghue and Holstein [@DoHo1986], two authors otherwise known for their extremely careful analysis of a number of physical problems, obtained spurious parity-violating terms after the Foldy–Wouthuysen transformation \[see the text surrounding Eq. (46) of Ref. [@DoHo1986]\]; the community seems to have converged on the result that these terms actually are absent. The manifestly and fully parity-conserving Hermitian form allows for an immediate answer to a pressing question: If the electrodynamic interaction were absent, would the quantum mechanical Schrödinger problem of proton and electron still have a solution and feature bound states? The answer is yes. The leading nonrelativistic terms from Eq. exactly have the structure expected from a naive insertion of the gravitational potential into the Schrödinger Hamiltonian, namely $$\begin{aligned}
\label{HS_grav}
H_S =& \;
\frac{\vec p^{\,2}}{2 m} - \frac{m \, r_s}{2 \, r}
= \alpha_G^2 m_e c^2 \, \left( -\frac12 \, \vec\nabla^2_\rho -
\frac{1}{\rho} \right) \,,\end{aligned}$$ where the gravitational fine-structure constant is $\alpha_G = G \, m_e \, m_p/(\hbar c)$ and $\rho = G m_e m_p r/\hbar^2$. The relativistic corrections are described by the expectation value of the remaining terms in Eq. on Schrödinger–Pauli wave functions and lead to the gravitational fine-structure formula [@Je2014pra] $$\label{mainres}
E_{n \varkappa} =
-\frac{\alpha_G^2 m_e c^2}{2 n^2} +
\frac{\alpha^4_G m_e c^2}{n^3} \, \left( \frac{15}{8 n} -
\frac{14 \, \varkappa + 3}{2 \,|\varkappa| \, (2 \varkappa + 1)} \right)\,,$$ where $n$ is the principal quantum number, $\varkappa$ is the Dirac angular quantum number, which summarizes both the orbital angular momentum $\ell$ as well as the total angular momentum $j$ of the bound particle into a single integer, according to the formula $\varkappa = (-1)^{j+\ell+1/2} \, \left(j + \frac12 \right)$. Initial steps toward the calculation of the fine-structure formula had been taken in Refs. [@GaPoCh1983; @VrEtAl2013]; the formula is in agreement with the form of the gravitational Hamiltonian given in Eq. (26) of Ref. [@GoNe2014] and with Refs. [@JeNo2013pra; @JeNo2014jpa]. The energy levels lift the $(n,j)$ degeneracy known from the Dirac–Coulomb problem. For electron-proton interactions, the gravitational fine-structure constant $\alpha_G \approx 3.2164 \times 10^{-42} $ is a lot smaller than the electromagnetic fine-structure constant $\alpha_{\rm QED} \approx 1/137.036$, but $\alpha_G$ can be a lot larger for particles bound to black holes even if the “tiny” black holes are lighter than the Earth. A “scatter plot” of a gravitational bound state (circular Rydberg state) which illustrates the quantum-classical correspondence is given in Fig. \[fig1\].
![\[fig1\] Probability density of a neutron in a quantum Rydberg state with $n=19$ and maximum orbital angular momentum quantum number $\ell=m=18$, gravitationally bound to a (tiny) black hole located at the origin. The scaled coordinates $\vec \rho$ is defined in the text immediately following Eq. . For a neutron bound to a black hole of mass equal to $10^{-15}$ times the mass of the Earth, the gravitational fine-structure constant has the numerical $\alpha_G = 0.0211$. The points are distributed with a probability density proportional to $|\psi|^2$, i.e., proportional to the probability of finding the electron at a particular coordinate in the immediate vicinity of the plotted coordinate. The circular Rydberg state approximates a classical circular trajectory with $\sqrt{ \langle \vec\rho^2 \rangle} = 375.22$.](fig1.eps){width="0.81\linewidth"}
Dirac’s large-number hypothesis [@Di1938LNH] is based on the observation that the ratio $$\frac{\alpha_{\rm QED}}{\alpha_G} =
\frac{e^2}{4 \pi \epsilon_0 G \, m_e \, m_p} \approx 2.3 \times 10^{39}
\approx \frac{c \, T}{r_e}$$ is approximately equal to the ratio of the age of the Universe, $T$, to the time it takes light to travel a distance equal to the classical electron radius, $r_e/c$, where $r_e$ is the classical electron radius, $r_e = \alpha \hbar/(m_e c)$. Eddington [@Ed1931] observed that the ratio of $\alpha_{\rm QED}$ to the gravitational fine-structure constant $\alpha^{(ee)}_G$ for two gravitating electrons is approximately equal to the square root of the number $N$ of charged particles in the Universe, $\alpha_{\rm QED}/\alpha^{(ee)}_G =
e^2/(4 \pi \epsilon_0 G \, m^2_e) \approx 4.2 \times 10^{42}
\approx \sqrt{ \, N \,} $. One may also curiously observe, as a variation of the other observations recorded in the literature, that $$\label{spec}
\frac{1}{\sqrt{\alpha_G}} \,
\exp\left( -\sqrt{\frac{m_p}{m_e}} \right)
\approx 137 \approx \frac{1}{\alpha_{\rm QED}} \,.$$ With the most recent [CODATA]{} value [@MoTaNe2012] for $G$ plugged into the left-hand side of Eq. , the left-hand side and right-hand sides of Eq. are in agreement to better than half a permille. It is unclear at present if the emergence of the famous factor “$137$” from an equation whose only adjustable prefactor is the number one \[see Eq. \] constitutes a pure accident, or if there is a deeper physical interpretation available. Variants of unified electromagnetic and gravitational theories, with Kaluza–Klein compactification, may lead to massless, or mass-protected spinors coupled to the gauge fields [@MBNi2006]. Possible connections of gravitational and electromagnetic interactions have intrigued physicists ever since Gertsenshtein [@Ge1962], as well as Zeldovich and Novikov [@ZeNo1983], discovered the possibility of graviton-photon conversion.
Work on this project has been supported by the National Science Foundation (Grant PHY–1403973) and by the National Institute of Standards and Technology (precision measurement grant).
[10]{}
A. Einstein, Königlich Preussische Akademie der Wissenschaften (Berlin), Sitzungsberichte, 799–801 (1915).
D. Hilbert, Nachrichten der Königlichen Gesellschaft der Wissenschaften (Göttingen), Math.-phys. Klasse, 395–407 (1915).
A. Einstein, Ann. Phys. (Leipzig) [**49**]{}, 796 (1916).
M. V. Gorbatenko and V. P. Neznamov, Ann. Phys. (Berlin) [**526**]{}, 195 (2014).
C. Cohen-Tannoudji, B. Diu, and F. Lalo$\ddot{e}$, [**]{}, 1 ed. (J. Wiley & Sons, New York, 1978).
C. Cohen-Tannoudji, B. Diu, and F. Lalo$\ddot{e}$, [**]{}, 1 ed. (J. Wiley & Sons, New York, 1978).
C. Itzykson and J. B. Zuber, [**]{} (McGraw-Hill, New York, 1980).
L. F. Abbott and D. D. Harari, Nucl. Phys. B [**264**]{}, 487 (1986).
L. O. Pimentel and C. Mora, Phys. Lett. A [**280**]{}, 191 (2000).
S. P. Kim, Nucl. Phys. Proc. Suppl. [**246**]{}, 68 (2014).
R. Blumenhagen, [*A Course on Noncommutative Geometry in String Theory*]{}, e-print 1403.4805 \[hep-th\].
M. Chaichian, M. M. Sheikh-Jabbari, and A. Tureanu, Phys. Rev. Lett. [**86**]{}, 2716 (2001).
D. Colladay and V. A. Kostelecky, Phys. Rev. D [**55**]{}, 6760 (1997).
J. S. Diaz, V. A. Kostelecky, and M. Mewes, Phys. Rev. D [**80**]{}, 076007 (2009).
V. A. Kostelecky and M. Mewes, Phys. Rev. D [**85**]{}, 096005 (2012).
V. A. Kostelecky and J. D. Tasson, Phys. Rev. D [**83**]{}, 016013 (2011).
D. R. Brill and J. A. Wheeler, Rev. Mod. Phys. [**29**]{}, 465 (1957).
J. A. Wheeler, [**]{} (Tipografia Compositori, Bologna, 1960).
M. Soffel, B. Müller, and W. Greiner, J. Phys. A [**10**]{}, 551 (1977).
G. Parisi, J. Stat. Phys. [**23**]{}, 49 (1980).
H. Gies and S. Lippoldt, , arXiv:1310.2509 \[hep-th\].
J. F. Donoghue and B. R. Holstein, Am. J. Phys. [**54**]{}, 827 (1986).
Y. N. Obukhov, Phys. Rev. Lett. [**86**]{}, 192 (2001).
A. J. Silenko and O. V. Teryaev, Phys. Rev. D [**71**]{}, 064016 (2005).
U. D. Jentschura and J. H. Noble, Phys. Rev. A [**88**]{}, 022121 (2013).
U. D. Jentschura, Phys. Rev. A [**88**]{}, 062514 (2013).
D. G. Boulware, Phys. Rev. D [**12**]{}, 350 (1975).
L. Parker, Phys. Rev. D [**22**]{}, 1922 (1980).
L. Parker and L. O. Pimentel, Phys. Rev. D [**25**]{}, 3180 (1982).
F. W. Hehl and W.-T. Ni, Phys. Rev. D [**42**]{}, 2045 (1990).
A. J. Silenko and O. V. Teryaev, Phys. Rev. D [**76**]{}, 061101(R) (2007).
Y. N. Obukhov, A. J. Silenko, and O. V. Teryaev, Phys. Rev. D [**80**]{}, 064044 (2009).
Y. N. Obukhov, A. J. Silenko, and O. V. Teryaev, Phys. Rev. D [**84**]{}, 024025 (2011).
U. D. Jentschura and J. H. Noble, J. Phys. A [**47**]{}, 045402 (2014).
U. D. Jentschura, , e-print 1404.1944 \[gr-qc\].
E. Eriksen and M. Kolsrud, Supplemento al Nuovo Cimento [**18**]{}, 1 (1958).
R. A. Swainson and G. W. F. Drake, J. Phys. A [**24**]{}, 79 (1991).
R. A. Swainson and G. W. F. Drake, J. Phys. A [**24**]{}, 95 (1991).
R. A. Swainson and G. W. F. Drake, J. Phys. A [**24**]{}, 1801 (1991).
D. V. Galtsov, G. V. Pomerantseva, and G. A. Chizhov, Sov. Phys. Journal [ **26**]{}, 743 (1983).
M. A. Vronsky, M. V. Gorbatenko, N. S. Kolesnikov, V. P. Neznamov, E. Y. Popov, and I. I. Safronov, [*Stationary Bound States of Dirac Particles in the Schwarzschild Gravitational Field*]{}, e-print 1301.7595 \[gr-qc\].
P. A. M. Dirac, Proc. Roy. Soc. London, Ser. A [**165**]{}, 199 (1938).
A. S. Eddington, Proc. Camb. Phil. Soc. [**27**]{}, 15 (1931).
P. J. Mohr, B. N. Taylor, and D. B. Newell, Rev. Mod. Phys. [**84**]{}, 1527 (2012).
N. Mankoc Borstnik and H. B. Nielsen, Phys. Lett. B [**633**]{}, 771 (2006).
M. E. Gertsenshtein, Sov. Phys. JETP [**14**]{}, 84 (1962).
Y. B. Zeldovich and I. D. Novikov, [*The structure and evolution of the universe*]{}, Rel. Astrophys. Vol. 2 (1983), Chicago University Press.
|
---
author:
- Julien Lesgourgues
- and Thomas Tram
bibliography:
- 'superhyperpaperreferences.bib'
title: 'Fast and accurate CMB computations in non-flat FLRW universes'
---
CERN-PH-TH/2013-298, LAPTH-071/13
Introduction
============
The large amount of cosmological information stored in the statistical properties of Cosmic Microwave Background (CMB) anisotropies and of the large scale structure of the universe has motivated several ambitious space-based and ground-based experiments. The interpretation of the data relies on a comparison with theoretical predictions, computed in the linear regime by sophisticated Bolztmann codes. In order to trust this interpretation, one should keep asking two questions: (i) are we postulating the correct model to describe the evolution of the universe? (ii) for a given model, are theoretical predictions calculated with sufficient accuracy, given observational errors? While the first question can be addressed by a continuous theoretical effort and by evaluating the goodness of fit of different models given the data, the second question calls for several independent tests of the robustness of Boltzmann codes. On top of that, progress in these codes are triggered by the fact that their speed is crucial, given that cosmological parameter extraction relies on Monte Carlo methods. These require the evaluation of tens of thousands of models each time that one cosmology is compared to one data set. In the global analysis of a data set like Planck [@Ade:2013ktc], many cosmologies and combinations of data need to be considered, leading to millions of Boltzmann code runs.
Current codes and data
----------------------
Several Boltzmann codes have been made public and compared with each other, including [[cmbfast]{}]{}, [@Seljak:1996is; @Zaldarriaga:1996xe; @Zaldarriaga:1997va], implementing for the first time the line-of-sight method, and later [[camb]{}]{} [@Lewis:1999bs], [[cmbeasy]{}]{} [@Doran:2003sy] and [[class]{}]{} [@Lesgourgues:2011re; @Blas:2011rf]. Both [[camb]{}]{} and [[class]{}]{} are being maintained and pushed to ever higher precision, and both were used in different parts of the Planck data analysis.
All current cosmological data can be well fitted with a flat FLRW model. Still, it is important to have efficient Boltzmann codes which also cover non-flat models, in order to check for small deviations from spatial flatness in future data. Currently, the bounds read $100\Omega_k=-1^{+1.8}_{-1.9}$ at the 95% confidence level using Planck alone and $100\Omega_k=-0.10^{+0.62}_{-0.65}$ when combining Planck with BAO [@Ade:2013zuv]. The effect of curvature on the CMB consists mainly in a shift of the angular scale of the acoustic peaks, due to a modification of the angular diameter distance to recombination. It also impacts the primordial spectrum and the evolution of very large wavelengths comparable to the curvature radius of the universe, especially through the late integrated Sachs-Wolfe effect (see Figure \[fig:decomposition\], left panel).
The computation of temperature and polarisation power spectra in non-flat FLRW universes requires the calculation of hyperspherical Bessel functions when employing the line-of-sight method. Because of memory restrictions, these functions must be computed on the fly, and this imposes an execution penalty when analysing non-flat models. Spatial curvature was successfully implemented in both [[cmbfast]{}]{} and [[camb]{}]{}, although both codes run significantly slower than in flat models. Their accuracy is also not well tested, due to the lack of a robust and accurate way of calculating hyperspherical Bessel functions without making any approximation.
To this end, we implement a new and improved method for computing the hyperspherical Bessel functions which is fast and accurate. We also found a new approximation scheme which gradually takes over in the flat limit $K\rightarrow 0$. This scheme results in a significant speed-up for nearly flat models, and it ensures continuity in the point $K=0$ when doing parameter estimation. Our approach is implemented in the release 2.0 of the [[class]{}]{} code[^1].
\[fig:decomposition\]![[*(Left panel)*]{} Temperature anisotropy spectrum for some closed, flat and open models ($\Omega_k=-0.05, 0, 0.05$, with all other cosmological parameters kept fixed). In each case, we show the auto-correlation spectrum of the total temperature anisotropy [*(solid)*]{}, of the late Integrated Sachs-Wolfe (ISW) contribution alone [*(dotted)*]{}, and of the total minus this contribution [*(dashed)*]{}. [*(Right panel)*]{} For a flat model, auto-correlation spectrum of the total temperature anisotropy and of different contributions (intrinsic temperature corrected by the Sachs-Wolfe term (SW), early plus late ISW term, Doppler term, polarisation contribution enhanced by a factor $10^2$). Such a decomposition can be obtained using the latest [[class]{}]{} version without any modifications (see Appendix A). Note that the individual auto-correlation spectra do not sum up to the total one, due to the existence of cross-correlation spectra.](split/decomp_curved "fig:"){width="0.5\columnwidth"}![[*(Left panel)*]{} Temperature anisotropy spectrum for some closed, flat and open models ($\Omega_k=-0.05, 0, 0.05$, with all other cosmological parameters kept fixed). In each case, we show the auto-correlation spectrum of the total temperature anisotropy [*(solid)*]{}, of the late Integrated Sachs-Wolfe (ISW) contribution alone [*(dotted)*]{}, and of the total minus this contribution [*(dashed)*]{}. [*(Right panel)*]{} For a flat model, auto-correlation spectrum of the total temperature anisotropy and of different contributions (intrinsic temperature corrected by the Sachs-Wolfe term (SW), early plus late ISW term, Doppler term, polarisation contribution enhanced by a factor $10^2$). Such a decomposition can be obtained using the latest [[class]{}]{} version without any modifications (see Appendix A). Note that the individual auto-correlation spectra do not sum up to the total one, due to the existence of cross-correlation spectra.](split/decomp "fig:"){width="0.5\columnwidth"}
Outline of the paper
--------------------
In section 2, we describe our approach for calculating hyperspherical Bessel functions. We also show a comparison with other methods, in order to estimate its accuracy on robust ground. In section 3, we describe other aspects of our implementation of curvature in a Boltzmann code. We explain how and why we simplified the temperature and polarisation source terms. We write the photon transfer functions, featuring source functions and radial functions, in a way which unifies flat and non-flat calculations. We clarify the issue of how to define primordial spectra and to integrate over wavenumbers in non-flat space. In section 4, we describe our new approximation scheme for speeding up calculations in curved space, the flat rescaling approximation. In section 5, we evaluate the accuracy of our implementation for “reference precision” and “default precision” setting. We perform a detailed comparison of our results with those of [[camb]{}]{} (version of November 2013) and compare the performances of the two codes. We present our conclusions in section 6. In appendix A, we show how to use [[class]{}]{} to split the temperature power spectrum into different contributions as in figure \[fig:decomposition\]. Finally, in appendix B and C, we summarise two other approximation schemes which were also introduced in the new version of [[class]{}]{}.
Hyperspherical Bessel functions\[sec:hyper\]
============================================
Here we give a brief overview of how we compute the hyperspherical Bessel functions in [[class]{}]{} [@Blas:2011rf; @Lesgourgues:2011re]. The algorithms are given in much more detail in the paper [@Tram:2013xpa].
Definition and notation
-----------------------
In order for us to give unified formulae for any $K$, it is convenient to introduce the notation $$\begin{aligned}
{\sin _K}(\chi) &\equiv \frac{1}{{\,\text{csc} _K}\chi} \equiv \left\{
\begin{array}{ll}
\sinh \chi & {\hat{K}}=-1 \\
\chi & {\hat{K}}=0\\
\sin \chi & {\hat{K}}=1
\end{array}
\right. , &
{\cot _K}(\chi) &\equiv \left \{
\begin{array}{ll}
\coth \chi & {\hat{K}}=-1 \\
\frac{1}{\chi}& {\hat{K}}=0\\
\cot \chi & {\hat{K}}=1
\end{array}
\right. .\end{aligned}$$ We use ${\hat{K}}\equiv K/|K|$ for $K\neq 0$, otherwise $\hat{K}=0$. The three cases ${\hat{K}}=0,1,-1$ refer respectively to a universe with null, elliptic or hyperbolic curvature, or in more common words, to a flat, closed or open Universe. The hyperspherical Bessel functions $\Phi^\nu_\ell (\chi)$ are given by $\Phi^\nu_\ell (\chi) = {\,\text{csc} _K}(\chi) u^\nu_\ell (\chi)$, where $u^\nu_\ell (\chi)$ is the solution of $${\frac{{\text{d}}^2 u^\nu_\ell }{{\text{d}}\chi^2}} = {\left[}\ell (\ell +1){\,\text{csc} _K}^2(\chi) - \nu^2{\right]}u^\nu_\ell (\chi),
\label{eq:u_equation}$$ which is regular at the origin. For ${\hat{K}}=0$ the hyperspherical Bessel functions are just the usual spherical Bessel functions, $\Phi^\nu_\ell (\chi) = j_\ell (\nu \chi)$, while for ${\hat{K}}\pm 1$, they can be expressed in terms of Legendre functions [@Harrison:1967; @Tram:2013xpa]: $$\label{eq:phiLegendre}
\Phi^\nu_\ell (\chi) = \left\{
\begin{array}{ll}
{\left(}\prod\limits_{n=1}^\ell {\sqrt{\nu^2+n^2}} {\right)}\sqrt{\frac{\pi }{2\sinh \chi}} P^{-1/2-\ell }_{-1/2+{\mathbbm{i}}\nu}(\cosh\chi) &{\hat{K}}=-1\\
{\left(}\prod\limits_{n=1}^\ell {\sqrt{\nu^2-n^2}} {\right)}\sqrt{\frac{\pi}{2\sin \chi}} P^{-1/2-\ell }_{-1/2+\nu}(\cos \chi) &{\hat{K}}=1\\
\end{array}
\right. .$$ When ${\hat{K}}=1$ we must also require the solution to be regular at the second boundary $\chi=\pi$. This leads to the requirement that $\nu$ must be an integer, so the eigenmode spectrum becomes discrete [@Tram:2013xpa] which is the familiar energy quantisation for a particle trapped in a potential. It also has the consequence that the solutions become symmetric (anti-symmetric) around $\chi=\frac{\pi}{2}$ for $\nu-\ell -1$ even (odd), so we only need to compute the solution in the range $[0; \pi/2]$. However we found that this has one more important consequence, namely, that the ${\hat{K}}=1$ hyperspherical Bessel functions can be re-expressed in terms of Gegenbauer polynomials $C_\alpha^{(\beta)}(x)$ [@Tram:2013xpa]: $$\label{eq:Gegenbauer_identity}
\Phi^\nu_\ell (\chi) = 2^\ell \ell ! \sqrt{\frac{(\nu-\ell -1)!}{\nu(\nu+\ell )!}} \sin^\ell (\chi) C_{\nu-\ell -1}^{(\ell +1)}(\cos \chi).$$ This identity can be used directly in a compact, stable method for ${\hat{K}}=1$ hyperspherical Bessel functions, or it can be used for solving the long-standing issue [@Kosowsky:1998nc] of using backward recurrence for ${\hat{K}}=1$.
Establishing a reference method
-------------------------------
The first step in building the hyperspherical Bessel module was to find a reliable reference implementation to which we could compare our methods. From equation we see that they are given in terms of Legendre functions and also hypergeometric functions using well-known identites. But no general and stable numerical algorithm exist for the computation of Legendre functions or hypergeometric functions of large order, not even in commercial software. So for establishing a set of trusted methods, we implemented many different methods in MATLAB. Whenever possible we coded the routines such that they would work also for ${\hat{K}}=0$: the first check would then be to test the routine against MATLAB’s built-in Bessel function routine which is known to be precise. We implemented the following 5 schemes in MATLAB:
1. [**Direct evaluation of equation .**]{} The Legendre functions in equation can be rewritten as hypergeometric functions and evaluated using the hypergeometric series. While initially promising, the series will not converge numerically for large $\ell $ since the positive and negative part of the series become large and nearly equal.
2. [**Direct integration of the differential equation .**]{} Such an implementation starts with an arbitrary, finite initial condition deep inside the dissipative region where the solution is heavily damped. It then evolves into the dispersive regime where the solution oscillates, and the solution will be the regular one up to an overall normalisation factor. This factor is then typically fixed by a forward recurrence. The standard ODE-methods works well for low to medium accuracy, but for high accuracy we implemented the modified Magnus method [@Iserles:2002:GED].
3. [**WKB approximation.**]{} Equation lends itself to a WKB approximation [@Kosowsky:1998nc; @Tram:2013xpa]. While this approximation is not sufficiently accurate as a [[class]{}]{} reference method, it is an important cross check. We also expected this method to be faster than the recurrence method for standard precision, but that turned out not to be the case[^2].
4. [**Using recurrence relations.**]{} Forward recurrence is only stable inside the dispersive region, so in the dissipative region one must use backward recurrence instead. It was thought for a long time [@Kosowsky:1998nc] that backward recurrence could not be used for ${\hat{K}}=1$ due to the restriction $\ell <\nu$, but this problem is eliminated by the identity we found, equation .
5. [**Using the Gegenbauer identity directly.**]{} Equation gives the ${\hat{K}}=1$ solutions in terms of Gegenbauer polynomials which we compute using (stable) forward recurrence.[^3]
We compared all the methods on the parameter space consisting of $\ell $, $\nu$ and $\chi$. For instance, we selected $N$ $\chi$-samples in a prescribed way and generated a figure of $N$ subplots, each containing a 2D image of the logarithm of the relative error between two methods. Similarly, we kept $\nu$ fixed and plotted the logarithm of the relative difference between methods in the $(x,\ell )$-plane, and we show such a plot for ${\hat{K}}=1$ for a fixed $\nu=2500$ in figure \[fig:reldif\]. The left panel compares method 2 and 5, while the right panel compares the WKB method 3 to method 5. This is probably the first time that the WKB approximation to the hyperspherical Bessel function has been properly quantified, and we can see that the error is usually of the order $10^{-4}$. The dashed black line shows the position of the classical turning point, so the solutions decay exponentially to the left of this line.
\[fig:reldif\]{width="0.5\columnwidth"}{width="0.5\columnwidth"}
The method implemented in [[camb]{}]{} is the following: the differential equation is integrated from an initial condition computed primarily using the WKB approximation and sometimes from the recurrence relations. The integration method used is the fourth order explicit Runge-Kutta method, RK4. Unlike the usual embedded methods, RK4 has no built-in error estimate of the local truncation error for adaptive step size, so the step sizes are taken from the time sampling of sources but with a hard coded maximum step. The differential equation is solved in tandem with the source convolution which makes it difficult to judge the accuracy of the actual spherical Bessel functions. It is also clear that this method will usually be limited by the accuracy limit of the WKB approximation.
Some but not all of the hyperspherical Bessel implementation in [[camb]{}]{} is inherited from [[cmbfast]{}]{} [@Zaldarriaga:1997va]. This includes the idea of using RK4 and summing up the source convolutions simultaneously. However, [[cmbfast]{}]{} found initial values from interpolation in a precomputed table, while [[camb]{}]{} is using recurrence and WKB.
Choosing a scheme for CLASS {#sec:CLASS_scheme}
---------------------------
The differential equation method has the advantage that one gets all $\chi$-values with a single call to the method, while the recurrence method will generate all $\ell $-values with a single call. Both methods will generate the derivative ${\Phi^\nu_\ell }^\prime(\chi)$ for free, which will be important later for interpolation. One can argue for both methods, but we chose to implement the recurrence method because it is more standard, and it would be needed anyway for normalising the ODE-method. It also has the advantage of requiring a negligible amount of library function calls such as trigonometric functions and square roots, so it would possibly be easier to vectorise.
![Hermite interpolation of ${\hat{K}}=-1$, $\nu=16$ hyperspherical Bessel functions for 3 values of $\ell $. Only $7$ points are used, corresponding to a sampling parameter of $1.75$. The top plot shows the exact functions in solid lines and the interpolated functions in black dashed lines. The bottom plot shows the relative error of the interpolated function, which is of the order $10^{-2}$ despite the very low value of the sampling parameter.[]{data-label="fig:interpolation"}](interpolation_figure){width="0.8\columnwidth"}
We found the recurrence method to be accurate at the level of $\sim 10^{-10}$, which is of course much better than what is required for standard precision in [[class]{}]{}. To extend the dynamic range of the method (i.e. increasing speed by reducing accuracy), we coded an interpolation method based on Hermite interpolation. (See e.g. [@burden:2004] and the help entry for the Mathematica function `InterpolatingPolynomial`.) By storing just $\Phi^\nu_\ell $ and ${\Phi^\nu_\ell }^\prime$, all higher derivatives can be found from the differential equation and its derivatives. We then construct the unique order 5 polynomial which matches $\Phi^\nu_\ell $, ${\Phi^\nu_\ell }^\prime$ and ${\Phi^\nu_\ell }^{\prime\prime}$ inside each interval for interpolating $\Phi^\nu_\ell $. However, since we sometimes also need to interpolate ${\Phi^\nu_\ell }^\prime$ and ${\Phi^\nu_\ell }^{\prime\prime}$, we need formulae for ${\Phi^\nu_\ell }^{\prime\prime\prime}$ and ${\Phi^\nu_\ell }^{\prime\prime\prime\prime}$ as well. We refer the interested reader to [@Tram:2013xpa] for more details on the implementation.
The interpolation method is illustrated in figure \[fig:interpolation\] for a sampling parameter of $1.75$. This parameter is defined as the number of points per approximate wavelength $\frac{2\pi}{\nu}$, so with 1.75 we compute less than two points per oscillation. Still the relative error for such a sampling parameter is as low as $\sim 10^{-2}$. Increasing the sampling gradually reduces the relative error all the way to a sampling of $\sim 30$ where the relative error due to the interpolation becomes $10^{-10}$, i.e. comparable to the accuracy of the underlying method. A typical sampling value of $5.0$ yields a relative error of $\sim 10^{-5}$. We will see at the end of section \[sec:transfer\] how this interpolation scheme is implemented in [[class]{}]{}.
Efficient CMB spectrum calculations in curved space
===================================================
In this paper and inside the code, we follow the notations of Ma & Berstchinger [@Ma:1995ey] for metric perturbations, matter perturbations and Bolztmann hierarchies (see [@Tram:2013ima] for the generalisation of these hierarchies to curved space, and for vector/tensor mode notations). In this paper, we write equations only in the Newtonian gauge, although in [[class]{}]{} they are implemented in both the Newtonian and synchronous (comoving with CDM) gauges. It is trivial to infer the synchronous gauge formulas using the gauge transformation rules presented in [@Ma:1995ey].
Source functions\[sec:sources\]
-------------------------------
It is well-known since the work of [@Seljak:1996is] that photon transfer functions can be conveniently obtained by a time integral of a set of source functions multiplied by certain radial functions. The total angular momentum method introduced in [@Hu:1997hp] (and generalised to curved space in [@Hu:1997mn]) leads to a unified set of simple integrals, accounting for temperature $T$ (decomposed in three contributions $T_j$ with $j=0,1,2$), E-type polarisation $E$, and B-type polarisation $B$: $$\begin{aligned}
{\Delta_\ell^{T_j}}^{(m)}(q) &=& \int_{\tau_\mathrm{ini}}^{\tau_0} d\tau S_{T_j}^{(m)}(k,\tau) \phi^{jm}_{\ell}(\nu,\chi)~, \label{eq:transT}\\
{\Delta_\ell^{E}}^{(m)}(q) &=& \int_{\tau_\mathrm{ini}}^{\tau_0} d\tau S_{P}^{(m)}(k,\tau) \epsilon^{m}_{\ell}(\nu,\chi)~, \\
{\Delta_\ell^{B}}^{(m)}(q) &=& \int_{\tau_\mathrm{ini}}^{\tau_0} d\tau S_{P}^{(m)}(k,\tau) \beta^{m}_{\ell}(\nu,\chi)~. \label{eq:transB}\end{aligned}$$ The $\Delta$’s are the photon transfer functions, depending on the mode index $m$ ($m=0,1,2$ for scalars, vectors, tensors), on the multipole $\ell$ and on the generalised wavenumber $$q=\sqrt{k^2 + (1+m)K}~, \qquad K=-H_0^2(1-\Omega_\mathrm{tot})~.$$ The radial functions $\{ \phi^{jm}_{\ell}, \epsilon^{m}_{\ell}, \beta^{m}_{\ell} \}$ are linear combinations of (hyper-)spherical Bessel functions and their derivatives. They depend on the rescaled generalised wavenumber $\nu=q/\sqrt{|K|}$ and the rescaled radial coordinate $\chi\equiv \sqrt{|K|} (\tau_0-\tau)$. This remains true in the flat limit as well, since the hyperspherical Bessel functions becomes $\Phi^\nu_\ell (\chi) \rightarrow j_\ell (\nu \chi) = j_\ell (k (\tau_0-\tau)$ so the two $\sqrt{|K|}$’s cancels out. All the relevant radial functions in flat space may be found in [@Hu:1997hp], while the ${\hat{K}}=-1$ radial functions can be found in [@Hu:1997mn]. In section \[sec:radial\] we write all the radial functions in a unified form where the flat limit is transparent.
The $S$’s are the source functions which depends on the Fourier wavenumber $k(q)$ and on conformal time $\tau$. In the Newtonian gauge, and omitting vector modes, the source functions derived in [@Hu:1997hp] read $$\begin{aligned}
S_{T0}^{(0)} &=\frac{g}{4}\delta_\gamma^{(0)} + e^{-\kappa} \phi'~, &
S_{T1}^{(0)} &= \frac{g}{k} \theta_b^{(0)} + e^{-\kappa} k \psi ~, &
S_{T2}^{(0)} &= g P^{(0)}~, \label{eq:sourceT}\\
S_P^{(0)} &= \sqrt{6} g P^{(0)} ~, &
S_{T2}^{(2)} &= g P^{(2)} -e^{-\kappa}h' ~, &
S_P^{(2)} &= \sqrt{6} g P^{(2)}~,\end{aligned}$$ while $S_{T0}^{(2)}=S_{T1}^{(2)}=0$. The polarisation source terms $P^{(0)}$ and $P^{(2)}$ can be expressed in terms of the coefficients of the optimal Boltzmann hierarchies presented in [@Tram:2013ima]. They are given in equation (2.16) of that reference but we will reprint them here for convenience: $$\begin{aligned}
P^{(0)} &= \frac{1}{8}\left[F_{\gamma 2}^{(0)}+G_{\gamma 0}^{(0)}+G_{\gamma 2}^{(0)}\right]~,\\
P^{(2)} &= -\frac{1}{\sqrt{6}} \left[ \frac{1}{10} F_{\gamma 0}^{(2)} + \frac{1}{7} F_{\gamma 2}^{(2)} + \frac{3}{70} F_4^{(2)}
- \frac{3}{5} G_{\gamma 0}^{(2)} + \frac{6}{7} G_{\gamma 2}^{(2)} - \frac{3}{70} G_{\gamma 4}^{(2)}\right]~.\end{aligned}$$ Note that we extend the naming scheme of [@Ma:1995ey] for the lowest multipoles such that $F_{\gamma 2}^{(0)} = 2 s_2 \sigma_{\gamma 2}^{(2)}$, $F_{\gamma 2}^{(2)} = 2 \sigma_\gamma^{(2)}$ and $F_{\gamma 0}^{(m)} = \delta_\gamma^{(m)}$.
In previous codes ([[cmbfast]{}]{} [@Seljak:1996is], [[camb]{}]{} [@Lewis:1999bs], [[cmbeasy]{}]{} [@Doran:2003sy] and [[class]{}]{} up to version 1.7), the temperature source functions were not implemented as such. They were integrated by part (once for the $T_1$ and twice for $T_2$), in order to reduce the problem to the integration of a single source $S_T$ multiplied by a single Bessel function. This approach is not very practical, because terms like e.g. $\psi'$ (or its counterpart in the synchronous gauge[^4]) are not directly given by Einstein equations. One needs to evaluate the time derivative of these equations, which involves the time derivative of the pressure $\bar{p}_i$ and anisotropic stress $\sigma_i$ of photons and baryons. This can be done in two ways:
- in [[cmbfast]{}]{}, [[camb]{}]{} and [[cmbeasy]{}]{}, all these derivatives are calculated exactly using the equations of motion. As a result, the evaluation of $S_T$ requires several extra equations (e.g. for computing the derivative of massive neutrino pressure). This causes the final expression of the temperature source function to be very lengthy and difficult to understand. For instance, in this scheme it is non-trivial to split the source term into the different contributions: Sachs-Wolfe (SW), Integrated Sachs-Wolfe (ISW), Doppler, etc.
- in [[class]{}]{} up to version 1.7, the code computed $S_{T_0}$, $S_{T_1}^\prime$ and $S_{T_2}^\prime$ exactly, while $S_{T_2}^{\prime \prime}$ was inferred from $S_{T_2}^\prime$ using a numerical derivative. This scheme enabled relatively simple and compact formulas without derivatives of Einstein equations, but some fine-tuning of precision parameters was required for computing the derivative with sufficient accuracy.
In [[class]{}]{} version 2.0 and higher, the strategy consists in keeping all three terms $S_{T_0}$, $S_{T_1}$, $S_{T_2}$ for the scalar temperature source functions. They lead to the three transfer functions ${\Delta_{\ell}^{T_j}}^{(m)}(q)$ introduced in equation (\[eq:transT\]), which are summed up in the module called “spectra”, to obtain a single temperature transfer function ${\Delta^T_{\ell}}^{(m)}(q)$. The advantage of this method is that we maintain high-accuracy while sticking to compact and simple source functions. The drawback is that the number of transfer functions increases from 1 to 3 when CMB temperature is required, but this actually has a minor impact on the performance of the code since 2 of the 3 contributions can be computed quickly using the approximations in appendix B and C.
We can still perform some integrations by part if we want to distribute the various terms differently between $S_{T_0}$, $S_{T_1}$, $S_{T_2}$. A good policy consists in using this freedom for optimising the speed and numerical stability of the code, up to the extent to which the formulas remain simple, and do not require derivatives of Einstein equations. This can be achieved by reorganising the terms in such a way that $S_{T_0}$, $S_{T_1}$, $S_{T_2}$ all become very small between recombination and reionisation. If this is the case, over this interval of time, we will not need a very accurate time-sampling of Bessel functions. This does not happen when one sticks to the original expressions (\[eq:sourceT\]), in which $S_{T0}^{(0)}$ and $S_{T_1}^{(0)}$ both remain large after recombination, but nearly cancelling each other. To avoid this, we can introduce a new set of source functions: $$\tilde{S}_{T_0}^{(0)}=g\left( \frac{1}{4}\delta_\gamma+\phi\right)+e^{-\kappa} 2 \phi' + k^{-2}(g \theta_b'+g' \theta_b)~,~~~~
\tilde{S}_{T_1}^{(0)}=e^{-\kappa} k (\psi-\phi)~,~~~~
\tilde{S}_{T_2}^{(0)}=S_{T2}^{(0)}~.
\label{eq:sourceTbis}$$ Using integrations by part, one can easily show that this choice is exactly equivalent to (\[eq:sourceT\]). However $\tilde{S}_{T_1}^{(0)}$ is proportional to $(\phi-\psi)$, which is very small at any time inside the Hubble radius, due to the decay of all anisotropic stresses. Similarly, $\tilde{S}_{T_0}^{(0)}$ remains very small between reionisation and recombination because $g$ and $g'$ are small, and $\phi$ is nearly constant. This version of the source functions is both compact and numerically efficient, and is the one implemented in [[class]{}]{}[^5]. Its counterpart in the synchronous gauge can be read directly in the code. In appendix \[app:sources\], we show how to break this expression into SW, ISW, Doppler and polarisation contributions.
Computing the transfer functions \[sec:transfer\]
-------------------------------------------------
In order to compute the functions $\phi_\ell^{j(m)}(\nu, \chi)$, $\epsilon_\ell^{(m)}(\nu, \chi)$ and $\beta_\ell^{(m)}(\nu, \chi)$, we need to know the hyperspherical Bessel functions $\Phi_\ell^\nu(\chi)$ and their first and second derivatives for several values of $\ell$, $\chi$ and $\nu$. For a given rescaled wavenumber $\nu$, a single run of the recurrence method summarised in section \[sec:hyper\] gives $\Phi_\ell^\nu$ and $\Phi_\ell^{\nu \prime}$ for all $\ell$, and the interpolation routine will compute the higher derivatives as they become necessary. In the “transfer” module of the code, i.e. the module in charge of performing the integrals of equations (\[eq:transT\]-\[eq:transB\]), we wish to avoid any redundancy in the calculation of hyperspherical Bessel functions. Since we can not store them in memory, we are left with only one possible strategy:
- implement the loop over $q$ values (or equivalently over $\nu$ values) as the outermost loop in the “transfer” module.
- for each new value of $q$ and $\nu$, compute all hyperspherical Bessel functions for a discrete set of arguments $\{ \chi_i\}$, and store the result in a form that will allow fast interpolation at any arbitrary $\chi$.
- inside the $q$-loop, loop over modes (scalar, vectors, tensors), over initial conditions (adiabatic, isocurvature), over types of transfer functions (temperature $T_0$, $T_1$, $T_2$, polarisation $E$, $B$, and other types related to weak lensing and density power spectra), and finally over multipoles $\ell $.
Since the calculation of Bessel functions is the most time-consuming part of the code for large curvature, it is crucial to run the recurrence for as few values of $\{ \chi_i\}$ as possible. This is where the interpolation scheme of section \[sec:CLASS\_scheme\] comes into play. As explained in that section, we checked explicitly that by increasing the sampling parameter, we can obtain at least $10^{-10}$ precision on the hypersperical Bessel function, i.e. much more than will ever be needed by Boltzmann codes. We will show in section \[sec:accuracy\] how to obtain “reference” and “default” precision parameter settings, and which accuracy they imply for the $C_\ell$’s. We found that a sufficient choice for the sampling parameter in the default case is $6$ for $\nu<1000$, and $3$ for $\nu>1000$. In the high-precision settings stored in the file [cl\_ref.pre]{}, $N=10$ for any value of $\nu$. These numerical values can of course be changed by the user[^6].
Radial functions\[sec:radial\]
------------------------------
The radial functions $\phi_\ell^{j(m)}$, $\epsilon_\ell^{(m)}$, $\beta_\ell^{(m)}$ are given in Ref. [@Hu:1997mn] in terms of the hyperspherical functions $\Phi_\ell ^\nu(\chi)$. In the code, we wrote these functions in such a way that the flat-space limit is completely transparent. We define five functions with simple flat-space limits: $$\begin{aligned}
{\mathfrak{j}_\ell ^\nu}(\chi) &\equiv \Phi_\ell ^\nu(\chi), &{\mathfrak{j}_\ell ^\nu}(\chi) &\xrightarrow[K\rightarrow0]{} j_\ell (k (\tau_0-\tau)), \\
{\mathfrak{j}_\ell ^{\nu\prime}}(\chi) &\equiv \frac{\sqrt{|K|}}{k} \Phi_\ell ^{\nu\prime} (\chi), &{\mathfrak{j}_\ell ^{\nu\prime}}(\chi) &\xrightarrow[K\rightarrow0]{} j_\ell '(k (\tau_0-\tau)), \\
{\mathfrak{j}_\ell ^{\nu\prime\prime}}(\chi) &\equiv \frac{|K|}{k^2} \Phi_\ell ^{\nu\prime\prime} (\chi), &{\mathfrak{j}_\ell ^{\nu\prime\prime}}(\chi) &\xrightarrow[K\rightarrow0]{} j_\ell ''(k (\tau_0-\tau)), \\
{\mathfrak{csc}_K}(\chi) &\equiv \frac{\sqrt{|K|}}{k} {\,\text{csc} _K}(\chi), &{\mathfrak{csc}_K}(\chi) &\xrightarrow[K\rightarrow0]{} \frac{1}{k (\tau_0-\tau)}, \\
{\mathfrak{cot}_K}(\chi) &\equiv \frac{\sqrt{|K|}}{k} {\cot _K}(\chi), &{\mathfrak{cot}_K}(\chi) &\xrightarrow[K\rightarrow0]{} \frac{1}{k (\tau_0-\tau)}~.\end{aligned}$$ Note that the last two functions are related to each other through $${\cot _K}^2(\chi) + {\hat{K}}= {\,\text{csc} _K}^2(\chi) ~~~~\Rightarrow~~~~ {\mathfrak{cot}_K}^2(\chi) + \frac{K}{k^2} = {\mathfrak{csc}_K}^2(\chi).$$ In Ref. [@Tram:2013ima], we defined a set of numbers $s_\ell $ which enter in the coefficients of the Boltzmann hierarchy: $$s_\ell = \sqrt{1-\frac{\ell ^2-1}{k^2}K},$$ where $\ell $ is a positive integer. However, for the purpose of writing the radial functions in a simple form, we will extend this definition to $\ell \in \mathbb{C}$. Specifically, we need $$k^2+pK =k^2{\left(}1-\frac{-p}{k^2}K{\right)}= k^2 s_{\sqrt{1-p}}^2,$$ where $p$ is a positive or negative integer. We find explicitly $$\begin{aligned}
\sqrt{k^2-K} &=k s_{\sqrt{2}}, &
\sqrt{k^2-2K} &= k s_{\sqrt{3}}, &
\sqrt{k^2-3K} &= k s_2, &
\sqrt{k^2-4K} &= k s_{\sqrt{5}}, \nonumber \\
\sqrt{k^2+K} &=k s_0, &
\sqrt{k^2+2K} &=k s_{{\mathbbm{i}}}, &
\sqrt{k^2+3K} &=k s_{\sqrt{2}{\mathbbm{i}}}, &
\sqrt{k^2+4K} &=k s_{\sqrt{3}{\mathbbm{i}}}. \label{eq:explicitsl}\end{aligned}$$ We can rewrite the radial functions of [@Hu:1997hp] and [@Hu:1997mn] in a compact and unified way,
#### Scalar modes:
\_\^[(00)]{} &= [\_\^]{}(), \_\^[(10)]{} = [\_\^]{}(), & \_\^[(20)]{} &= $$3 {\mathfrak{j}_\ell ^{\nu\prime\prime}}(\chi) + {\mathfrak{j}_\ell ^\nu}(\chi)$$, &\
\_\^[(0)]{} &= [\_K]{}\^2() [\_\^]{}(), & \_\^[(0)]{} &=0. &
#### Vector modes:
\_\^[(11)]{} &= [\_K]{}() [\_\^]{}(), &\
\_\^[(21)]{} &= [\_K]{}() $${\mathfrak{j}_\ell ^{\nu\prime}}(\chi) -{\mathfrak{cot}_K}(\chi) {\mathfrak{j}_\ell ^\nu}(\chi)$$, &\
\_\^[(1)]{} &= [\_K]{}() $ {\mathfrak{cot}_K}(\chi) {\mathfrak{j}_\ell ^\nu}(\chi) + {\mathfrak{j}_\ell ^{\nu\prime}}(\chi) $ , &\
\_\^[(1)]{} &= [\_K]{}() [\_\^]{}().
#### Tensor modes:
\_\^[(22)]{} &= [\_K]{}\^2() [\_\^]{}(), &\
\_\^[(2)]{} &= $${\mathfrak{j}_\ell ^{\nu\prime\prime}}(\chi) + 4 {\mathfrak{cot}_K}(\chi) {\mathfrak{j}_\ell ^{\nu\prime}}(\chi) -{\left(}s_{\sqrt{3}{\mathbbm{i}}}^2 -2{\mathfrak{cot}_K}^2(\chi) {\right)}{\mathfrak{j}_\ell ^\nu}(\chi)$$, &\
\_\^[(2)]{} &= $${\mathfrak{j}_\ell ^{\nu\prime}}(\chi) +2{\mathfrak{cot}_K}(\chi) {\mathfrak{j}_\ell ^\nu}(\chi)$$.
In [[class]{}]{}, radial functions are found in exactly that form in the function [transfer\_radial\_function()]{} inside the “transfer” module.
Harmonic power spectra\[sec:spectra\]
-------------------------------------
In flat or open space, the integral leading to the harmonic power spectra ${C_\ell^{XY}}^{(m)}$ (with $X$ referring to $T$, $E$, $B$, or the density or lensing potential of sources in a given bins, and $m=0,1,2$ for scalars, vectors, tensors) reads [@Hu:1997hp; @Hu:1997mn] $${C_\ell^{XY}}^{(m)} = 4 \pi \int \frac{dk}{k}\,\, {\Delta_\ell^X}^{(m)}(q,\tau_0) \,\, {\Delta_\ell^Y}^{(m)}(q,\tau_0) \,\, {\cal P}^{(m)}(k)~, \label{eq:spec1}$$ where $q$ is seen as a function of $k$, and ${\cal P}^{(m)}(k)$ is the primordial spectrum of the mode $m$. In closed space, the integral is replaced by a sum over discrete wavenumbers $k_n=\sqrt{(n^2 - 1-m)K}$, corresponding to $\nu_n=q_n/\sqrt{K}=n$, with $n=3,4,5,...,\infty$.
For vector modes, there is no significant generation of CMB anisotropies through a passive mechanism, i.e. from initial conditions in the very early universe, so the vector primordial spectrum is not a relevant quantity. For scalar and tensor modes, the definition and normalisation of the primordial spectrum in a non-flat universe is presented in the literature under different forms, sometimes confusing, and usually with scarce explanations. In this section we wish to clarify this issue, to introduce the conventions used in both [[camb]{}]{} and [[class]{}]{}, and to compare them with expressions appearing in the literature.
In general, each primordial spectrum refers to a given perturbation $A$ (metric, curvature, density, etc.) which can be chosen almost arbitrarily. For consistency, the transfer functions $\Delta_\ell^X(q)$ must be computed starting from a set of initial conditions for the system of cosmological perturbations such that the quantity $A$ has initially a unit value. At initial time the power spectrum $P_A(k)$ of $A$ is defined as $$\langle A(\vec{k}) A^*(\vec{k}') \rangle = P_A(k) \delta(\vec{k}-\vec{k}') ~,$$ and the dimensionless power spectrum ${\cal P}_A(k)$ follows from ${\cal P}_A(k)=\frac{k^3}{2 \pi^2} P_A(k)$. This definition is designed in such a way that the dimensionless power spectrum represents the contribution of a given logarithmic interval to any integral in $k$ space: $$\forall f, \qquad \int \frac{d^3 \vec{k}}{(2\pi)^3} P_A(k) f(k) = \int \frac{dk}{k} {\cal P}_A(k) f(k)~.$$
#### Scalar modes.
For adiabatic initial conditions, in [[camb]{}]{} and [[class]{}]{}, ${\cal P}^{(0)}(k)$ is assumed to be the curvature power spectrum ${\cal P}_{\cal R}(k)$, where ${\cal R}$ stands for the comoving curvature perturbation in the comoving gauge. Then the adiabatic transfer function must be computed starting from the initial condition ${\cal R} \longrightarrow 1$ when $k \tau \longrightarrow 0$. This corresponds to $\eta \longrightarrow 1$ in the synchronous gauge, still using the notations of Ref. [@Ma:1995ey] for synchronous metric perturbations. The equivalent initial condition in the Newtonian gauge is obtained from a simple gauge transformation. Hence the scalar harmonic spectrum is given by $${C_\ell^{XY}}^{(0)} = 4 \pi \int \frac{dk}{k} \,\, {\Delta_\ell^{X}}^{(0)}(q,\tau_0) \,\, {\Delta_\ell^{Y}}^{(0)}(q,\tau_0) \,\, {\cal P}_{\cal R}(k)~,
\label{eq:spec2}$$ or by the corresponding sum in a closed universe. In the case of mixed initial condition (adiabatic plus isocurvature), equation (\[eq:spec2\]) needs to be replaced by a sum over each pair of initial conditions, involving auto- and cross-correlation primordial spectra ${\cal P}_{\cal R \cal R}(k)$, ${\cal P}_{\cal R \cal I}(k)$, ${\cal P}_{\cal I \cal I}(k)$.
Lyth & Stewart [@Lyth:1990dh] showed that the dimensionless scalar spectrum of primordial metric fluctuations ${\cal P}_{\phi}(k)$ generated by single-field inflation in an open universe is scale-invariant in the limit of negligible slow-roll parameters, i.e. for $n_s=1$. This result also holds in the closed case [@Starobinsky:1996ek]. Since ${\cal R}= \frac{3}{2} \phi$ on super-Hubble scale and during radiation domination, this corresponds to a constant curvature spectrum, ${\cal P}_{\cal R}\equiv A_s$. To account for deviations from this limit, it is customary to introduce a power-law primordial spectrum $${\cal P}_{\cal R}(k) = A_s k^{n_s-1},
\label{eq:scalprispec}$$ where the scalar tilt $n_s$ can be related to the first-order slow-roll parameters. In the literature, the integral of equation (\[eq:spec2\]) (or the corresponding sum in a closed universe) is often written in terms of $q$ or $\nu$: $$\begin{aligned}
{C_l^{XY}}^{(0)}
&=& 4 \pi \int dq \, q^ 2 {\Delta_\ell^X}^{(0)}(q,\tau_0) {\Delta_\ell^Y}^{(0)}(q,\tau_0) \frac{{\cal P}_{\cal R}(k)}{q\left(q^2- K\right)} \label{eq:spec3}\\
&=& 4 \pi \int d\nu \, \nu^ 2 {\Delta_\ell^X}^{(0)}(q,\tau_0) {\Delta_\ell^Y}^{(0)}(q,\tau_0) \frac{{\cal P}_{\cal R}(k)}{\nu\left(\nu^2- \hat{K} \right)}~,\label{eq:spec4}\end{aligned}$$ where we used $q^2 = k^2+K$ and $k\,dk = q\,dq$. The literature also often refers to a dimensionless primordial spectrum in $q$-space that we will denote $\widetilde{\cal P}_{\cal R}(q)$. Since the dimensionless spectrum refers to the contribution of a given logarithmic interval, it is natural to define $\widetilde{\cal P}_{\cal R}(q)$ as: $$\frac{dk}{k} {\cal P}_{\cal R}(k) \equiv \frac{dq}{q} \widetilde{\cal P}_{\cal R}(q)
\qquad \Longrightarrow \qquad
\widetilde{\cal P}_{\cal R}(q) = \frac{q^2}{\left(q^2- K\right)}{\cal P}_{\cal R}(k)~. \label{eq:corpkpq}$$
#### Tensor modes.
The perturbed FLRW metric $g_{\mu \nu} = a^2 \left( \gamma_{\mu \nu} + h_{\mu \nu}\right)$ (with $\gamma_{00}=-1$) features gravitational waves in the traceless transverse part of $h_{ij}$. For each wavelength and direction of propagation of the waves, $h_{ij}$ contains two independent degrees of freedom (called the two polarisation states). In a Friedmann-Lemaître universe, the evolution equation and initial power spectrum of these two states are identical, and independent of the direction of propagation. Hence the statistical properties of gravitational waves can be inferred from a unique power spectrum ${\cal P}_H(k)$ and transfer function $H(\tau, k)$. In standard notations, $H$ is normalized in such a way that it obeys the evolution equation (see e.g. [@Hu:1997mn]) $$H'' + 2\frac{a^\prime}{a} H + \left(k^2+2K\right) H = 8 \pi G a^2 \sum_{i=\gamma,\nu}
p_i \, \pi^{(2)}_i ~,$$ where the anisotropic stress $\pi^{(2)}_i$ is related to the coefficients of the optimal Bolztmann hierarchy of Ref. [@Tram:2013ima] through $$\pi^{(2)}_i = -4 \sqrt{6} \left(\frac{1}{15} F^{(2)}_{i 0} + \frac{2}{21} F^{(2)}_{i 2} + \frac{1}{35} F^{(2)}_{i 4}\right)~,$$ and the index $i$ runs over relativistic photons and neutrinos[^7] (non-relativistic particles have a negligible anisotropic stress, and do not contribute to the source term). The tensor harmonic spectra follows from $$\begin{aligned}
{C_\ell^{XY}}^{(2)} &=& 4 \pi \int \frac{dk}{k} \,\, {\Delta_\ell^X}^{(2)}(q,\tau_0) \,\, {\Delta_\ell^Y}^{(2)}(q,\tau_0) \,\, {\cal P}_{H}(k) \\
&=& 4 \pi \int \frac{dq}{q} \,\, {\Delta_\ell^X}^{(2)}(q,\tau_0) \,\, {\Delta_l^Y}^{(2)}(q,\tau_0) \,\, \widetilde{\cal P}_{H}(q)~,\end{aligned}$$ or the equivalent sum of discrete $k$ modes in the case of a closed universe. The relation between ${\cal P}_{H}(k)$ and $\widetilde{\cal P}_{H}(q)$ is now given by $$\frac{dk}{k} {\cal P}_H(k) \equiv \frac{dq}{q} \widetilde{\cal P}_H(q)
\qquad \Longrightarrow \qquad
\widetilde{\cal P}_H(q) = \frac{q^2}{\left(q^2- 3 K\right)}{\cal P}_H(k)~. \label{eq:corpkhpq}$$ It was shown by [@Bucher:1997xs] (see also [@Hu:1997ws]) that during slow-roll inflation in a bubble with negative curvature, and in the limit of negligible slow-roll parameters, the minimal primordial spectrum of gravitational waves is proportional to $${\cal P}_{H}(k) \propto \frac{k^2(k^2-K)}{(k^2+3K)(k^2+2K)} \, \mathrm{tanh}
\left( \frac{\pi}{2} \sqrt{\frac{k^2+3K}{K}}\right)~, \qquad (K < 0).$$ The hyperbolic tangent cut-off accounts for the suppression of wavelengths which are large compared to the radius of curvature of the universe. In a closed universe the same result applies but without the cut-off. Hence it is convenient to define $$t_K(k) = \left\{
\begin{tabular}{ll}
$\mathrm{tanh} \left( \frac{\pi}{2} \sqrt{\frac{k^2+3K}{K}} \right) = \mathrm{tanh} \left( \frac{\pi \nu}{2} \right)$ & $\mathrm{if}~K < 0$~,\\
$1$&$\mathrm{if}~K\geq 0$~.
\end{tabular}
\right.$$ The normalisation of the spectrum is set by a parameter $A_t$, and deviations from the slow-roll limit are encoded at first order in a power-law $k^{n_t}$, where the tensor tilt $n_t$ relates to the first slow-roll parameter. The usual normalisation convention for $A_t$ is such that for single-field inflation and in the slow-roll limit, the tensor-to-scalar ratio $r \equiv A_t/A_s$ relates to the tensor index through $$r \equiv \frac{A_t}{A_s} = 16 \epsilon = -8 n_t~.$$ A careful computation shows that this definition is compatible with $${\cal P}_H(k) = \frac{A_t}{6} k^{n_t} \frac{k^2(k^2-K)}{(k^2+3K)(k^2+2K)} t_K(k)~,$$ which corresponds, using (\[eq:corpkhpq\]), to $$\widetilde{\cal P}_H(q) = \frac{A_t}{6} k^{n_t} \frac{(k^2-K)}{(k^2+2K)} t_K(k) = \frac{A_t}{6} k^{n_t} \frac{(q^2-4K)}{(q^2-K)} t_K(q) = \frac{A_t}{6} k^{n_t} \frac{(\nu^2-4\hat{K})}{(\nu^2-\hat{K})} t_K(\nu)~.$$
#### Comparison with other references.
Ref. [@Hu:1997mn] refers to the primodial spectra of metric fluctuations $\phi$ and tensor fluctuations $H$. Following our notations, we infer from previous results that in the slow-roll limit $n_s=1$ and $n_t=0$, these spectra should be given by $$\begin{aligned}
P_\phi(q) &=& \frac{2 \pi^2}{q^3} \widetilde{\cal P}_\phi (q) \propto \frac{1}{q(q^2-K)} \propto \frac{1}{\nu\left(\nu^2- \hat{K} \right)}~,\\
P_H(q) &=& \frac{2 \pi^2}{q^3} \widetilde{\cal P}_H (q) \propto \frac{q^2-4K}{q^3(q^2-K)} \propto \frac{(\nu^2-4\hat{K})}{\nu^3(\nu^2-\hat{K})}~t_K(\nu)~.\end{aligned}$$ This coincides exactly with equation (44) of [@Hu:1997mn] for $\hat{K}=-1$. In Lyth & Stewart [@Lyth:1990dh], the final result for scalar modes is expressed in terms of the dimensionless density power spectrum $P_\delta$. Using the Poisson equation $$4 \pi G a^2 \bar{\rho}_\mathrm{tot} \delta_\mathrm{tot} = - s_2^2 k^2 \phi = - (k^2-3K) \phi = - (q^2-4K) \phi~,$$ we see that in the slow-roll limit where $n_s=1$, $$P_\delta(q) \propto (q^2-4K)^2 P_\phi(q) \propto \frac{(q^2-4K)^2}{q(q^2-K)} \propto \frac{(\nu^2-4\hat{K})^2}{\nu(\nu^2-\hat{K})} ~. \label{eq:densprispec2}$$ This result coincides with [@Lyth:1990dh] for $\hat{K}=-1$. Seljak et al. [@Zaldarriaga:1997va] use equation (\[eq:densprispec2\]) to define their scalar primordial spectrum. By doing so, they implicitly assume that the transfer functions ${\Delta_\ell^X}^{(0)}(k,\tau_\mathrm{ini})$ are normalised at initial time to $\delta_\mathrm{tot}=1$ instead of the more conventional choices $\phi=1$, or ${\cal R}=1$ like in [[camb]{}]{} and [[class]{}]{}.
Flat rescaling approximation\[sec:approx\]
==========================================
In a curved universe, the comoving angular diameter distance to the last scattering surface reads $$r_A^\mathrm{rec} = \frac{d_A^\mathrm{rec}}{a^\mathrm{rec}} = \frac{1}{\sqrt{|K|}} \sin_K \left(\sqrt{|K|} (\tau_0-\tau_\mathrm{rec})\right)~,$$ where $\tau_\mathrm{rec}$ is the conformal time at recombination. This means that features on the last scattering surface are shifted in angular space by $$\alpha = \frac{r_A^\mathrm{rec}}{(\tau_0-\tau_\mathrm{rec})} \label{eq:alpha}$$ with respect to a universe with the same conformal age and recombination time[^8], as clearly seen in the left panel of figure \[fig:decomposition\]. This suggests various possible approximation schemes, like computing the CMB spectra $C_l^{XY}$ in flat space and rescaling them horizontally by $\alpha$, or replacing the hyperspherical Bessel functions $\Phi_l^\nu(\chi)$ by $j_l(\alpha \nu \chi)$. Such schemes were already attempted by [@Zaldarriaga:1997va] and found to be very inaccurate. Still, some good approximation must exist, to account for the fact that when the curvature is small, the CMB physics is almost the same as in flat space, except for a small impact of the curvature $K$ on the evolution of the smallest wavenumbers $k \sim K$, and for a small change in the angular diameter distance to recombination. The modes with $k \gg K$, i.e. $\nu \gg1$, only experience the latter effect.
Approximating hyperspherical Bessel functions \[sec:ahbf\]
----------------------------------------------------------
We investigate an approximation scheme which is supposed to be valid above a critical value $\nu_*$. For large $\nu$’s, one notices that the hyperspherical bessel functions are very close to their flat spherical counterpart with a linearly rescaled argument. Our goal is then to use an approximation of the type $$\Phi_l^\nu(\chi) \simeq A_\nu^l(\chi) j_l(\gamma \nu \chi) \qquad \mathrm{if}~\nu\geq\nu_* ~,\label{eq:flat-approx}$$ where the oscillatory part is described by the flat Bessel functions, while $A$ is a smooth function of $\chi$.
Ideally, we would like our approximation scheme to reach its maximum precision for $\chi$ values close to the first peak of the Bessel function, since this is the region contributing most to the integrals of equations (\[eq:transT\], \[eq:transB\]). The position and amplitude of the first peak is difficult to estimate analytically. It is more convenient to match the exact and approximate functions at the turning point, i.e. at the first point where the second derivative vanishes, located just before the first peak. Its position is given exactly by the equation $${\sin _K}\chi_\mathrm{tp} = \frac{\sqrt{l(l+1)}}{\nu} ~.$$ Hence the rescaling factor $\gamma$ can be defined as the ratio of the turning points for the hyperspherical and spherical Bessel functions: $$\gamma = \frac{\sqrt{l(l+1)}}{\nu \,\, \mathrm{arcsin}_K\left( \frac{\sqrt{l(l+1)}}{\nu}\right)}~. \label{rescaling_argument}$$ The amplitude of the hyperspherical Bessel function compared to the spherical one around the turning point can be estimated from the ratio of their respective WKB approximations [@Kosowsky:1998nc; @Tram:2013xpa] in the limit $\chi \rightarrow \chi
_\text{tp}$. By defining $Q(\chi)\equiv {\,\text{csc} _K}^2 \chi - \frac{\nu^2}{l(l+1)}$ and expanding $Q(\chi) \simeq Q'(\chi_\text{tp}) (\chi - \chi_\text{tp}
)$ around $\chi_\text{tp}$, equation becomes the Airy equation. Evaluating the solutions at $\chi = \chi_\text{tp}$ we find $$\Phi_l^\nu (\chi_\text{tp}) \sim Q'(\chi_\text{tp}) ^{-\frac{1}{6}} = {\left(}\frac{-2 {\sin _K}\chi_\text{tp} {\cot _K}\chi_\text{tp}}{{\sin _K}^3 \chi_\text{tp}} {\right)}^{-\frac{1}{6}} = {\left(}\frac{-2 {\sin _K}\chi_\text{tp} {\cot _K}\chi_\text{tp}}{{\left[}\frac{l(l+1)}{\nu^2}{\right]}^{\frac{3}{2}}} {\right)}^{-\frac{1}{
6}},$$ so the difference in amplitude at the turning point is due to the product: ${\sin _K}\chi_\text{tp} {\cot _K}\chi_\text{tp}$. Since ${\sin _K}\chi {\cot _K}\chi = 1$ for ${\hat{K}}=0$, the ratio of amplitudes at the turning point is exactly $$\begin{aligned}
A_\nu^l(\chi_\text{tp}) \equiv \frac{\Phi_l^{\nu,\text{ \text{curved}}} (\chi_\text{tp})}{\Phi_l^{\nu,\text{ \text{flat}}} (\chi_\text{tp})}
&= {\left(}{\sin _K}\chi_\text{tp} {\cot _K}\chi_\text{tp} {\right)}^{-\frac{1}{6}} = \left\{
\begin{array}{ll}
{\left(}\cosh {\,\text{asinh}}{\left[}\frac{\sqrt{l(l+1)}}{\nu} {\right]}{\right)}^{-\frac{1}{6}}, & {\hat{K}}=-1, \\
{\left(}\cos {\,\text{asin}}{\left(}\frac{\sqrt{l(l+1)}}{\nu} {\right]}{\right)}^{-\frac{1}{6}}, & {\hat{K}}=1,
\end{array}
\right. \nonumber \\
&= {\left[}1 -{\hat{K}}\frac{l(l+1)}{\nu^2} {\right]}^{-\frac{1}{12}}.\end{aligned}$$ Using constant rescaling factors $\gamma$ and $A_\nu^l(\chi_\text{tp})$ in equation (\[eq:flat-approx\]) already provides a good approximation, but not good enough for our purpose, since such a scheme leads to percent level errors in the $C_\ell$’s. The reason is that the ratio $A_\nu^l(\chi_\text{tp}) $ defined above gives a very good approximation of the amplitude of the first peak, but overestimates (underestimates) the amplitude of the subsequent peaks for $K<0$ ($K>0$). We know from the WKB approximation that in the limit $\chi \longrightarrow \infty$, the hyperspherical to spherical peak ratio approaches $\chi / \sin_K(\chi)$, but the CMB is mainly sensitive to the first hundred peaks or so, which are still far from this asymptotic regime. We find numerically that a quadratic rescaling formula of the type[^9] $$A_\nu^l(\chi) = A_\nu^l(\chi_\text{tp}) \left[ 1 + a \,\, {\,\text{atan}}\left({l}/{\nu}\right) \,\, (\chi-\chi_\text{tp}) + b \,\, \left({\,\text{atan}}\left({l}/{\nu}\right)\right)^2 \,\, (\chi-\chi_\text{tp})^2 \right]~, \label{rescale_function}$$ with $$\begin{aligned}
a=0.34, & b= 2.0 & (K>0)~,\nonumber \\
a=-0.38, & b=0.4 & (K<0)~,\end{aligned}$$ leads to 0.1% precision on the $C_\ell$’s with $\nu_*=4000$, as discussed later and shown in Figure \[fig:flatapprox\]. In the flat limit, $K\rightarrow 0$, we have $\nu \rightarrow \infty$ and consequently ${\,\text{atan}}(l/\nu)\rightarrow 0$. So $A_\nu^l(\chi)\rightarrow 1$ in the flat limit as it should.
At this point we would like to emphasise that we are not trying to interpolate transfer functions or power spectra as a function of the cosmological parameter $K$. Such a scheme would be most effective when implemented at the level of a parameter extraction code such as Pico [@Fendt:2006uh]. What we are doing is just approximating one special function with another which is easy to compute in a region of parameters where the two must be similar. This is in fact similar to the uniform WKB approximation where the hyperspherical Bessel functions are related to the easily computable Airy function.
Implementation of the flat approximation
----------------------------------------
In [[class]{}]{} v2.0, the [transfer]{} module first computes the flat spherical Bessel functions and stores them in memory. It then loops over growing values of $\nu$. For $\nu<\nu_*$, Bessel functions $\Phi_l^\nu(\chi)$ are recomputed at the start of each new loop. For $\nu>\nu_*$ this computation is switched off. The code interpolates from the array of flat Bessel function with a rescaled argument, using the rescaling factor of equation (\[rescaling\_argument\]), and multiplies the result by the rescaling function of equation (\[rescale\_function\]). For simplicity, $n$-th derivatives of hyperspherical Bessel functions are approximated as $${\Phi_l^{\nu}}^{(n)}(\chi) \simeq A_\nu^l(\chi)\, (\gamma \nu)^n \, j_l^{(n)}(\gamma \nu \chi) \qquad \mathrm{if}~\nu\geq\nu_* ~,\label{eq:flat-approx-derivs}$$ neglecting derivatives of the smoothly varying $A_\nu^l$ function. Models with a smaller curvature have a larger fraction of their $\nu$ values above $\nu_*$, so the execution time of the code tends towards that of flat models in the limit $K\longrightarrow0$. For very small $K$, all Bessel functions are replaced by their flat rescaled approximation, with rescaling factors $\gamma$ and $A_\nu^l(\chi)$ tending towards one. This ensures that $C_\ll^{XY}$ is perfectly continuous with respect to $K$ across the special point $K=0$. In the default version of [[class]{}]{} v2.0, the approximation is switched on at $\nu_*=4000$[^10]. In the high-precision settings stored in the file [cl\_ref.pre]{}, this number is pushed up to $10^6$. In Figure \[fig:flatapprox\], we show the ratio of $C_\ell^{TT}$’s and $C_\ell^{EE}$’s computed with and without this approximation.
The error introduced by the approximation goes to 0 when $|K| \longrightarrow \infty$, because it is never used (all $\nu$ values being below $\nu_*$), and we checked that the same is true in the limit $|K| \longrightarrow 0$, in which the approximate $\Phi_l^\nu$ tends towards the real ones. For intermediate values of $|K|$, the approximation affects $C_\ell^{XY}$’s above some value $l_*(\nu_*)$ growing with $\nu_*$. Above this multipole, the amplitude of the error also grows with $\nu_*$.
\[fig:flatapprox\]![Relative difference between scalar unlensed $C_\ell$’s computed with and without the flat rescaling approximation for $\nu_*=4000$. Temperature $C_\ell$’s are displayed in solid lines and E-type polarisation is shown in dashed lines. *Left panel:* Positive curvature. *Right panel:* Negative curvature. Different colours correspond to different values of $\Omega_K$ picked up in the range where the approximation is the most harmful: for smaller $|\Omega_K|$’s the approximation is very accurate at every $l$, while for larger $|\Omega_K|$’s it introduces a significant error only for multipoles $l > 2500$. All calculations have been performed using the reference settings of the file [cl\_ref.pre]{} for all precision parameters but $\nu_*$.](fig_flatapprox/fig_posK "fig:"){width="0.5\columnwidth"}![Relative difference between scalar unlensed $C_\ell$’s computed with and without the flat rescaling approximation for $\nu_*=4000$. Temperature $C_\ell$’s are displayed in solid lines and E-type polarisation is shown in dashed lines. *Left panel:* Positive curvature. *Right panel:* Negative curvature. Different colours correspond to different values of $\Omega_K$ picked up in the range where the approximation is the most harmful: for smaller $|\Omega_K|$’s the approximation is very accurate at every $l$, while for larger $|\Omega_K|$’s it introduces a significant error only for multipoles $l > 2500$. All calculations have been performed using the reference settings of the file [cl\_ref.pre]{} for all precision parameters but $\nu_*$.](fig_flatapprox/fig_negK "fig:"){width="0.5\columnwidth"}
For the choice $\nu_*=4000$ and positive curvature, models with $|\Omega_K| < 0.1$ are affected at almost any $l$, but by a negligible amount. For $|\Omega_K|>0.06$, the approximation is harmless because values with $2<l<2500$ are almost unaffected by the approximation (used only in the tail of the integrals over time). Larger $l$’s are more affected, but they are not constrained very accurately by CMB data. In the intermediate range $0.01<|\Omega_K|<0.06$, the error in the range $2<l<2500$ is maximal. Figure \[fig:flatapprox\] shows that this error peaks around 0.1% for temperature and E-type polarisation $C_\ell$’s. This is true for scalar spectra: tensor spectra are affected roughly at the same level, and no high-precision measurements of tensor $C_\ell$’s is foreseeable at the moment.
We conclude that the setting $\nu_*=4000$ is sufficient for fitting Planck data, but the user is free to increase $\nu_*$ slightly in order to decrease the maximal error. If necessary, it would also be possible to increase the accuracy of the rescaling function of equation (\[rescale\_function\]).
Additional approximations and performance
-----------------------------------------
In this section we have described the most important new approximation which was introduced for non-zero curvature. Two additional approximations have been introduced in [[class]{}]{} v2.0 and they are described in appendix \[sec:time\_cut\] and \[sec:multipole\_cut\]. In Table \[tab:performance\], we show the impact of all three approximations on the performance of the code. In previous figures, for all precision parameters not related to these three approximations, we adopted reference settings; on the contrary, for this performance test, we adopted default precision settings. All input parameters were the same as in the input file [explanatory.ini]{} distributed with [[class]{}]{}, except $\Omega_k$ (switched to non-zero values in the last six models) and ${\tt modes}$ (switched to “scalars plus tensors” in the second model). The reported numbers account for the number of CPU cycles, rescaled in such way to give approximately the real time in seconds for a single-core run on a 2.3GHz processor. On the laptop used to perform this test, the actual execution time was approximately eight times smaller than these numbers, due to a combination of multi-threading and turbo-boost. We see that the time cut and multipole cut approximations are both important, especially in the flat case. As expected, in the non-flat case, the execution time is much larger in absence of any approximation, due to the computation of hyperspherical Bessel function. This time can be considerably reduced thanks to the flat rescaling approximation, especially for small $|\Omega_k|$.
\[tab:performance\]
------------------ ------ ------ ----------- ----------- ----------- ------------ ------------ ------------
$\Omega_k$ 0 0 $10^{-3}$ $10^{-2}$ $10^{-1}$ $-10^{-3}$ $-10^{-2}$ $-10^{-1}$
modes s s+t s s s s s s
no approx. 18.0 28.3 76.6 78.7 63.8 95.1 96.7 163.7
time cut 14.2 24.3 71.3 71.8 59.5 86.0 89.6 153.2
+ $l$ cut 5.7 10.9 64.2 65.5 54.6 78.8 78.4 128.3
+ flat rescaling 5.7 10.9 18.9 17.6 32.5 20.8 21.2 45.4
------------------ ------ ------ ----------- ----------- ----------- ------------ ------------ ------------
: Execution time of [[class]{}]{}, for different input value of $\Omega_k$. The numbers actually reflect the number of CPU cycles, rescaled in such way to give approximately the real time in seconds for a single-core run on a 2.3GHz processor. As explained in the text, all input parameters were fixed by the file [explanatory.ini]{}, except for $\Omega_k$. Tensor spectra were requested only in the second column. In the first line, the three approximation schemes discussed in this section were switched off. In the next three lines, they were progressively restored. Hence the last line corresponds to the default performances of the code.
Accuracy and comparison with other codes\[sec:accuracy\]
========================================================
Establishing reference precision settings and power spectra\[sec:precision flat\]
---------------------------------------------------------------------------------
In Ref. [@Lesgourgues:2011rg], we compared reference precision settings for [[class]{}]{} v1.0 and [[camb]{}]{} (version of January 2011) for a flat minimal $\Lambda$CDM model, and we found that the temperature spectra agreed at the 0.01% level (0.02% for polarisation). This result extends to all “reasonable” $\Lambda$CDM models: there is nothing in the codes that could cause a jump in the precision when the cosmological parameters are varied over the range allowed by current cosmological data. This good agreement strongly suggests that modern Boltzmann codes are accurate at such a level. Both codes could of course be wrong if the underlying model was not correct. For instance, one could imagine that the recombination model was wrong. However, the good agreement between [RECFAST]{}, [HyRec]{} and [CosmoRec]{}, all embedded within [[class]{}]{} and/or [[camb]{}]{}, suggests that standard recombination physics is indeed well understood, and modelled at the accuracy level required by cosmological data. What the comparison really shows is that if the underlying physical model is correct, the codes do not introduce a numerical error (related to numerical methods, discretisation, approximations, etc.) larger than about 0.01%[^11]. This result is not trivial, given the high complexity of these codes.
In [@Lesgourgues:2011rg], the reference settings of [[class]{}]{} v1.0 were obtained by varying [*all*]{} precision parameters[^12], up to the point at which a typical variation of these parameters by a factor two induces less than $10^{-3}$% variations in the $C_\ell$’s in the range $2 \leq \ell \leq 2500$. Since then, [[class]{}]{} has evolved, even in the flat case. Versions v1.1 to 1.7 introduced very minor changes in the $C_\ell$ computation, and we checked that their respective reference spectra remained stable at the $10^{-3}$% level. However version 2.0 introduces a new scheme for the decomposition of sources and radial functions (already described in section \[sec:sources\] and \[sec:radial\]), a new algorithm for the computation of all spherical and hyperspherical Bessel functions (described in section \[sec:hyper\]), and finally a new scheme for the sampling in wavenumber space (documented within the code). Hence it is important to check whether the previous 0.01% level agreement with [[camb]{}]{} still holds. To do so, we established a new set of reference accuracy parameters, identical to the ones of previous versions, except for the new parameters introduced in version 2.0. For these new parameters, we again established the convergence of the code at the $10^{-3}$% level. The resulting set of parameters is contained in the file [cl\_ref.pre]{}, delivered together with the code.
In figure \[fig:accuracy\_flat\], we show the difference between the old and new reference spectra, and conclude that the new scheme impacts the reference model at the 0.01% level for temperature, and $0.02$% for polarisation. Since this matches the maximum accuracy claimed for the [[class]{}]{} and [[camb]{}]{} codes, we conclude that [[class]{}]{} v2.0 achieves the same precision as earlier versions. This is important, because the source decomposition and sampling schemes are so different in [[class]{}]{} v1.x and v2.0 that, together with [[camb]{}]{}, they can almost be seen as three independent Boltzmann codes. Hence the present comparison makes our claim of a 0.01% maximum accuracy even stronger than before.
Accuracy of [[class]{}]{} and [[camb]{}]{} in flat $\Lambda$CDM models
----------------------------------------------------------------------
Having a reference model in the flat case, we degrade the precision of the new precision parameters introduced in [[class]{}]{} v2.0, in order to speed up the code while still achieving 0.1% precision on scalar temperature and 0.2% on scalar polarisation. These default settings are used when one runs the code without passing any precision parameters in input. Figure \[fig:accuracy\_flat\] shows the resulting error for a particular minimal flat $\Lambda$CDM model.
In Ref. [@Lesgourgues:2011rg], the reference precision settings of [[camb]{}]{} (version of January 2011) were obtained, first, by pushing the accuracy boost parameter up to very large values, and second, by decreasing by hand the sampling step size of a few functions ($j_\ell(x)$ as a function of $x$, and ionisation fraction as a functions of $z$). Since then, the accuracy of [[camb]{}]{} has increased: these two samplings have been improved, and the code now comes with a “high accuracy” flag, aimed roughly at 0.1% precision on scalar $C_\ell$’s. Hence it is interesting to compare the spectra obtained with the latest version of [[camb]{}]{} (from November 2013) with our reference model[^13]. Figure \[fig:accuracy\_flat\] shows the accuracy achieved with default [[camb]{}]{} precision, when using the “high accuracy” flag, and by using the “high accuracy” while also increasing the three “accuracy boost parameters” from 1 to 2. In the first case, the error can be as large as 0.3% for scalar temperature, or 0.8% for scalar polarisation. The “high accuracy” setting does not reduce the error for $\ell < 20$[^14], but this is harmless, due to cosmic variance. More importantly, it does keep the error at the 0.1% level for larger $\ell$’s, making it comparable to [[class]{}]{} with default precision. Finally, with much higher precision settings (all “accuracy boost parameters” set to 2) the error decreases to about 0.05%. We did not push the comparison further, relying on the conclusion of [@Lesgourgues:2011rg] that in the flat model, the agreement can be pushed to the 0.01% level.
\[fig:accuracy\_flat\]![Accuracy of [[class]{}]{} and [[camb]{}]{} in a minimal flat $\Lambda$CDM model. All spectra for unlensed temperature (left) and E-type polarisation (right) are compared to the [[class]{}]{} reference spectra, argued to be accurate at the 0.01% level (see text). For [[class]{}]{}, we show the reference settings of v1.0, known to agree at the 0.01% level with the high-precision limit of [[camb]{}]{}, and the default settings of v2.0. For [[camb]{}]{}, we show three precision settings: default, “high accuracy” (HA), and “high accuracy” plus accuracy boost parameters set to 2 (HA+).](hyperfigs/Ok0_CLASS_vs_CAMB_TTv1 "fig:"){width="0.5\columnwidth"}![Accuracy of [[class]{}]{} and [[camb]{}]{} in a minimal flat $\Lambda$CDM model. All spectra for unlensed temperature (left) and E-type polarisation (right) are compared to the [[class]{}]{} reference spectra, argued to be accurate at the 0.01% level (see text). For [[class]{}]{}, we show the reference settings of v1.0, known to agree at the 0.01% level with the high-precision limit of [[camb]{}]{}, and the default settings of v2.0. For [[camb]{}]{}, we show three precision settings: default, “high accuracy” (HA), and “high accuracy” plus accuracy boost parameters set to 2 (HA+).](hyperfigs/Ok0_CLASS_vs_CAMB_EEv1 "fig:"){width="0.5\columnwidth"}
\[fig:accuracy\_closed\]![Accuracy of [[class]{}]{} and [[camb]{}]{} in closed $\Lambda$CDM models with, from top to bottom, $\Omega_k=-0.1, -0.01, -0.001$. All spectra for unlensed temperature (left) and E-type polarisation (right) are compared to the [[class]{}]{} reference settings, argued to be accurate at the 0.01% level (see text). For [[class]{}]{}, we show the default settings of v2.0. For [[camb]{}]{}, we show three precision settings: default, “high accuracy” (HA), and “high accuracy” plus accuracy boost parameters set to 2 (HA+).](hyperfigs/Ok-0d1_CLASS_vs_CAMB_TT "fig:"){width="0.5\columnwidth"} ![Accuracy of [[class]{}]{} and [[camb]{}]{} in closed $\Lambda$CDM models with, from top to bottom, $\Omega_k=-0.1, -0.01, -0.001$. All spectra for unlensed temperature (left) and E-type polarisation (right) are compared to the [[class]{}]{} reference settings, argued to be accurate at the 0.01% level (see text). For [[class]{}]{}, we show the default settings of v2.0. For [[camb]{}]{}, we show three precision settings: default, “high accuracy” (HA), and “high accuracy” plus accuracy boost parameters set to 2 (HA+).](hyperfigs/Ok-0d1_CLASS_vs_CAMB_EE "fig:"){width="0.5\columnwidth"}\
![Accuracy of [[class]{}]{} and [[camb]{}]{} in closed $\Lambda$CDM models with, from top to bottom, $\Omega_k=-0.1, -0.01, -0.001$. All spectra for unlensed temperature (left) and E-type polarisation (right) are compared to the [[class]{}]{} reference settings, argued to be accurate at the 0.01% level (see text). For [[class]{}]{}, we show the default settings of v2.0. For [[camb]{}]{}, we show three precision settings: default, “high accuracy” (HA), and “high accuracy” plus accuracy boost parameters set to 2 (HA+).](hyperfigs/Ok-0d01_CLASS_vs_CAMB_TT "fig:"){width="0.5\columnwidth"} ![Accuracy of [[class]{}]{} and [[camb]{}]{} in closed $\Lambda$CDM models with, from top to bottom, $\Omega_k=-0.1, -0.01, -0.001$. All spectra for unlensed temperature (left) and E-type polarisation (right) are compared to the [[class]{}]{} reference settings, argued to be accurate at the 0.01% level (see text). For [[class]{}]{}, we show the default settings of v2.0. For [[camb]{}]{}, we show three precision settings: default, “high accuracy” (HA), and “high accuracy” plus accuracy boost parameters set to 2 (HA+).](hyperfigs/Ok-0d01_CLASS_vs_CAMB_EE "fig:"){width="0.5\columnwidth"}\
![Accuracy of [[class]{}]{} and [[camb]{}]{} in closed $\Lambda$CDM models with, from top to bottom, $\Omega_k=-0.1, -0.01, -0.001$. All spectra for unlensed temperature (left) and E-type polarisation (right) are compared to the [[class]{}]{} reference settings, argued to be accurate at the 0.01% level (see text). For [[class]{}]{}, we show the default settings of v2.0. For [[camb]{}]{}, we show three precision settings: default, “high accuracy” (HA), and “high accuracy” plus accuracy boost parameters set to 2 (HA+).](hyperfigs/Ok-0d001_CLASS_vs_CAMB_TT "fig:"){width="0.5\columnwidth"} ![Accuracy of [[class]{}]{} and [[camb]{}]{} in closed $\Lambda$CDM models with, from top to bottom, $\Omega_k=-0.1, -0.01, -0.001$. All spectra for unlensed temperature (left) and E-type polarisation (right) are compared to the [[class]{}]{} reference settings, argued to be accurate at the 0.01% level (see text). For [[class]{}]{}, we show the default settings of v2.0. For [[camb]{}]{}, we show three precision settings: default, “high accuracy” (HA), and “high accuracy” plus accuracy boost parameters set to 2 (HA+).](hyperfigs/Ok-0d001_CLASS_vs_CAMB_EE "fig:"){width="0.5\columnwidth"}
\[fig:accuracy\_open\]![Accuracy of [[class]{}]{} and [[camb]{}]{} in open $\Lambda$CDM models with, from top to bottom, $\Omega_k=0.1, 0.01, 0.001$. All spectra for unlensed temperature (left) and E-type polarisation (right) are compared to the [[class]{}]{} reference settings, argued to be accurate at the 0.01% level (see text). For [[class]{}]{}, we show the default settings of v2.0. For [[camb]{}]{}, we show three precision settings: default, “high accuracy” (HA), and “high accuracy” plus accuracy boost parameters set to 2 (HA+).](hyperfigs/Ok0d1_CLASS_vs_CAMB_TT "fig:"){width="0.5\columnwidth"} ![Accuracy of [[class]{}]{} and [[camb]{}]{} in open $\Lambda$CDM models with, from top to bottom, $\Omega_k=0.1, 0.01, 0.001$. All spectra for unlensed temperature (left) and E-type polarisation (right) are compared to the [[class]{}]{} reference settings, argued to be accurate at the 0.01% level (see text). For [[class]{}]{}, we show the default settings of v2.0. For [[camb]{}]{}, we show three precision settings: default, “high accuracy” (HA), and “high accuracy” plus accuracy boost parameters set to 2 (HA+).](hyperfigs/Ok0d1_CLASS_vs_CAMB_EE "fig:"){width="0.5\columnwidth"}\
![Accuracy of [[class]{}]{} and [[camb]{}]{} in open $\Lambda$CDM models with, from top to bottom, $\Omega_k=0.1, 0.01, 0.001$. All spectra for unlensed temperature (left) and E-type polarisation (right) are compared to the [[class]{}]{} reference settings, argued to be accurate at the 0.01% level (see text). For [[class]{}]{}, we show the default settings of v2.0. For [[camb]{}]{}, we show three precision settings: default, “high accuracy” (HA), and “high accuracy” plus accuracy boost parameters set to 2 (HA+).](hyperfigs/Ok0d01_CLASS_vs_CAMB_TT "fig:"){width="0.5\columnwidth"} ![Accuracy of [[class]{}]{} and [[camb]{}]{} in open $\Lambda$CDM models with, from top to bottom, $\Omega_k=0.1, 0.01, 0.001$. All spectra for unlensed temperature (left) and E-type polarisation (right) are compared to the [[class]{}]{} reference settings, argued to be accurate at the 0.01% level (see text). For [[class]{}]{}, we show the default settings of v2.0. For [[camb]{}]{}, we show three precision settings: default, “high accuracy” (HA), and “high accuracy” plus accuracy boost parameters set to 2 (HA+).](hyperfigs/Ok0d01_CLASS_vs_CAMB_EE "fig:"){width="0.5\columnwidth"}\
![Accuracy of [[class]{}]{} and [[camb]{}]{} in open $\Lambda$CDM models with, from top to bottom, $\Omega_k=0.1, 0.01, 0.001$. All spectra for unlensed temperature (left) and E-type polarisation (right) are compared to the [[class]{}]{} reference settings, argued to be accurate at the 0.01% level (see text). For [[class]{}]{}, we show the default settings of v2.0. For [[camb]{}]{}, we show three precision settings: default, “high accuracy” (HA), and “high accuracy” plus accuracy boost parameters set to 2 (HA+).](hyperfigs/Ok0d001_CLASS_vs_CAMB_TT "fig:"){width="0.5\columnwidth"} ![Accuracy of [[class]{}]{} and [[camb]{}]{} in open $\Lambda$CDM models with, from top to bottom, $\Omega_k=0.1, 0.01, 0.001$. All spectra for unlensed temperature (left) and E-type polarisation (right) are compared to the [[class]{}]{} reference settings, argued to be accurate at the 0.01% level (see text). For [[class]{}]{}, we show the default settings of v2.0. For [[camb]{}]{}, we show three precision settings: default, “high accuracy” (HA), and “high accuracy” plus accuracy boost parameters set to 2 (HA+).](hyperfigs/Ok0d001_CLASS_vs_CAMB_EE "fig:"){width="0.5\columnwidth"}
Accuracy of [[class]{}]{} and [[camb]{}]{} in curved $\Lambda$CDM models
------------------------------------------------------------------------
For several values of $\Omega_k$ in the range ${\left[}-0.1:0.1{\right]}$, we tuned the parameters governing the computation of hyperpsherical Bessel function and the sampling of sources and transfer functions in such a way that, like in the flat case, temperature and polarisation scalar spectra converge at the $10^{-3}$% level (we also checked the convergence of tensor spectra, but at a lower level, given the limited prospects for high-accuracy tensor observations in the future). The reference accuracy settings of the file [cl\_ref.pre]{} guarantee this level of convergence for $\ell \leq 2500$. In section \[sec:precision flat\], we explained why we believe that in the flat case, this leads to robust “reference spectra”, precise at the 0.01% level for scalar temperature. In non-flat models, we expect that reference settings obtained with the same method lead to the same precision. This might not be the case if the code was switching between different algorithms or sampling methods, depending on the value of curvature. The only place in the code where this could be true would be the calculation of Bessel functions, since the starting value for the recurrence method in the closed case is sometimes based on equation . However, in the $K\rightarrow 0$, our hyperspherical Bessel algorithm is the same for closed and open models.
In the rest of the code, no difference is made between the flat and non-flat case: all physical equations are continuous in the variable $K$, we use the same routines and the same sampling strategies everywhere for flat and non-flat models, and we switch off the flat rescaling approximation, which is specific to non-flat models, when we establish reference settings. Hence, we expect our non-flat reference setting to provide the same accuracy as in the flat case, namely 0.01% for scalar temperature and 0.02% for scalar polarisation.
Like in the flat case, we degraded the precision parameters in such a way that, by default, the code achieves roughly 0.1% precision on scalar temperature spectra, or 0.2% on scalar polarisation, throughout the range $-0.1<\Omega_k<0.1$ and for $\ell \leq 2500$. The resulting error is shown in figures \[fig:accuracy\_closed\] and \[fig:accuracy\_open\] for several values of $\Omega_k$. In the same figures, we show [[camb]{}]{} errors for the same precision choices as before: default, with “high accuracy”, and finally with further increasing the three “accuracy boost parameters” from 1 to 2. The fact that the last setting converges towards the [[class]{}]{} reference model at the level of 0.03% (except for $\Omega_k=-0.1$) is important: it brings the first independent test of the fact that the method implemented in [[camb]{}]{} for hyperspherical Bessel functions can reach higher accuracy than what is requested by current and future CMB data.
On the other hand, the default precision settings in [[camb]{}]{} turn out to produce rather large errors in the large curvature limit: up to 0.5% for temperature and 1% for polarisation. With “high accuracy” settings, the low multipoles remain equally inaccurate, but in the region where high precision is really needed, i.e. for $500 < \ell < 2500$, the “high accuracy” settings lead to roughly 0.1% accuracy on temperature for all values of $\Omega_k$ with the exception of $\Omega_k=-0.1$. We conclude that with the current versions of the two codes, [[camb]{}]{} with “high accuracy” is nearly as precise as [[class]{}]{} with default settings, even for non-flat models. For these settings, a comparison of running time (see in Table \[tab:performance2\]) shows that [[class]{}]{} is typically 2 to 3 times faster than [[camb]{}]{} for non-flat models. This can be attributed to the efficient algorithmic implementation described in section \[sec:hyper\] as well as the flat rescaling approximation.
\[tab:performance2\]
$\Omega_k$ 0 $10^{-3}$ $10^{-2}$ $10^{-1}$ $-10^{-3}$ $-10^{-2}$ $-10^{-1}$
--------------- ----- ----------- ----------- ----------- ------------ ------------ ------------
[[class]{}]{} 9.7 29.7 21.7 43.4 24.5 27.7 79.4
[[camb]{}]{} 6.4 94.7 89.9 87.2 68.4 72.3 163.7
: Execution time of [[class]{}]{} (default precision) and [[camb]{}]{} (with “high accuracy” settings) for different values of $\Omega_k$. In these settings, [[class]{}]{} is slightly more accurate, especially for low $\ell$’s and large positive curvature. We asked the codes to compute the lensed temperature and polarisation scalar spectra up to the same $l_\mathrm{max}=3000$. For a reliable comparison, we switched off [OpenMP]{}, and compiled the codes with recent versions of [gcc]{} or [gfortran]{}, using exactly the same optimisation flags, and running on the same processor. These times are expressed in seconds, but they are only useful for the sake of comparison: on modern processors with many cores the execution time will of course be much smaller.
Continuity across $\Omega_k=0$
------------------------------
\[fig:accuracy\_continuity\]![Test of the continuity of the spectra across $\Omega_k=0$. The results for $\Omega_k=10^{-5}$ (solid curves) and $\Omega_k=-10^{-5}$ (dashed curves) are compared to the [[class]{}]{} reference *flat* model, for unlensed temperature (*left panel*) and E-type polarisation (*right panel*). For [[class]{}]{}, we show the reference settings and default settings of v2.0. For [[camb]{}]{}, we show three precision settings: default, “high accuracy” (HA), and “high accuracy” plus accuracy boost parameters set to 2 (HA+).](hyperfigs/continuity_TT "fig:"){width="0.5\columnwidth"}![Test of the continuity of the spectra across $\Omega_k=0$. The results for $\Omega_k=10^{-5}$ (solid curves) and $\Omega_k=-10^{-5}$ (dashed curves) are compared to the [[class]{}]{} reference *flat* model, for unlensed temperature (*left panel*) and E-type polarisation (*right panel*). For [[class]{}]{}, we show the reference settings and default settings of v2.0. For [[camb]{}]{}, we show three precision settings: default, “high accuracy” (HA), and “high accuracy” plus accuracy boost parameters set to 2 (HA+).](hyperfigs/continuity_EE "fig:"){width="0.5\columnwidth"}
Checking the continuity of the $C_\ell$’s across the special value $\Omega_k=0$ can be seen as a further test of precision. One might fear that, when all cosmological parameters but $\Omega_k$ are fixed, a discontinuity could be observed in $\Omega_k=0$, either due to insufficient accuracy settings or a real issue with the algorithms used in the code (for instance, in the calculation of closed or open hyperspherical Bessel functions, or due to the fact that $q$ takes discrete values in a closed universe or arbitrary values in an open one). In a Monte Carlo run for parameter extraction, such a discontinuity could cause a significant problem and bias the final results.
In figure \[fig:accuracy\_continuity\], we show the ratio of temperature and polarisation spectra computed for $\Omega_k=\pm10^{-5}$ compared to the [[class]{}]{} reference spectra for $\Omega_k=0$. We expect the true difference between the spectra to be also of the order $10^{-5}$. The precision settings to be used in a parameter extraction code should be such that these models do not differ by more than 0.1%, meaning that they could not be discriminated from each other, given the accuracy of current CMB data.
We check that with [[class]{}]{} and reference precision settings, the spectra indeed agrees at the $10^{-5}$ level. With default settings, [[class]{}]{} is supposed to produce an error of the order of 0.1% at most. This is consistent with the difference observed in figure \[fig:accuracy\_continuity\] between the $\Omega_k=0$ and $\Omega_k=\pm10^{-5}$ spectra. Actually, in this case continuity across $\Omega_k=0$ is achieved automatically due to the flat rescaling approximation. For [[camb]{}]{}, the differences are at the expected level: 0.3% for temperature with default settings, 0.1% with “high accuracy”, and even less with all boost parameters set to 2. We conclude that the two codes successfully deal with the $\Omega_k \rightarrow 0$ limit, and do not suffer from artificial step effects between closed and open models.
Lensing spectrum and lensed CMB spectra
---------------------------------------
The previous sections refer to the accuracy of unlensed temperature/polarisation spectra. The lensed spectra are obtained as a function of the unlensed ones and of the CMB lensing potential spectrum $C_\ell^{\phi \phi}$. In this section, we will test the accuracy with which this quantity is computed by [[class]{}]{} in curved space. This is crucial to establish the accuracy of lensed spectra. Instead, the step leading from unlensed to lensed CMB spectra, coded in the [lensing]{} module of CLASS, is exactly the same in flat and curved space; its accuracy has been extensively tested in [@Lesgourgues:2011rg], and there is no need to present new tests in curved space.
The CMB lensing spectrum $C_\ell^{\phi \phi}$ follows from the CMB lensing transfer function ${\Delta_\ell^\phi}(q,\tau_0)$ through Eq. (\[eq:spec1\]). For low $\ell$’s, this transfer function is computed by integrating over time, $${\Delta_\ell^\phi}(q,\tau_0) = \int_{\tau_\mathrm{rec}}^{\tau_0} d\tau S_{\phi}(k,\tau) \Phi^{\nu}_{\ell}(\chi)~, \label{eq:delta_phi}$$ where the relations between $k$, $q$, $\nu$, $\chi$ are the same as in the rest of this paper. This integral is of the same form as the one leading to the first temperature transfer function ${\Delta_\ell^{T_0}}(q,\tau_0)$, except for the shape of the source function: for CMB lensing, the source is very smooth and very broad in the range $\tau_\mathrm{rec} < \tau < {\tau_0}$. Previous tests presented in this section show that the [[class]{}]{} default precision settings are sufficient for getting an accurate and well-sampled temperature transfer function. Nevertheless, given the very different shape of the CMB lensing source function, similar tests should be repeated for the lensing potential.
The smoothness of $S_{\phi}(k,\tau)$ allows to use the Limber approximation for large $\ell$’s, i.e. to replace the integral by a simple evaluation of the source function at a given time. Details on this approximation (in flat and curved space) are discussed in Appendix \[sec:limber\].
To check the accuracy of $C_\ell^{\phi \phi}$ in the range which is most relevant for CMB lensing, we computed some reference spectra for various values of $\Omega_k$, ensuring that:
- the Limber approximation was never used in the range $2 \leq \ell \leq 1000$,
- all precision parameters governing the $\tau$-sampling in the integral, the $q$-sampling of transfer functions, and the accuracy with which hyperspherical bessel functions $\Phi^{\nu}_{\ell}(\chi)$ are computed were pushed to extreme values such that the $C_\ell^{\phi \phi}$’s are fully converged and stable (up to the level of $10^{-5}$).
Then, we computed the same $C_\ell^{\phi \phi}$ with default accuracy settings (i.e. with poor precision settings for the integral at low $\ell$, and using the Limber approximation for $\ell \geq {\tt l\_switch\_limber}=10$). We show the ratio of the default over reference spectra in figure \[fig:accuracy\_lensing\] (left plot).
\[fig:accuracy\_lensing\]![Test of the accuracy of the CMB lensing potential $C_\ell^{\phi \phi}$ (left plot), and of the propagation of the error in $C_\ell^{\phi \phi}$ to the lensed power spectra (right plot). For $\Omega_k=0$, $0.1$, $-0.1$, we ran [[class]{}]{} with reference settings for the parameters governing the accuracy of $C_\ell^{\phi \phi}$ (in particular, not using the Limber approximation), and then with default precision. We show the ratio of CMB lensing potential spectra on the left, and the corresponding ratio of lensed temperature spectra on the right. At low-$\ell$, before using the Limber approximation, the default $C_\ell^{\phi \phi}$ are not precise, but the impact of this error on lensed spectra is negligible; precise $C_\ell^{\phi \phi}$ at low $\ell$ can be obtained by just halfing the precision parameter [source\_sampling\_step]{} (leading to the dashed curves). These tests were performed later than the rest of this paper, with [[class]{}]{} v2.3.3.](fig_limber/fig_limber_phi "fig:"){width="0.5\columnwidth"}![Test of the accuracy of the CMB lensing potential $C_\ell^{\phi \phi}$ (left plot), and of the propagation of the error in $C_\ell^{\phi \phi}$ to the lensed power spectra (right plot). For $\Omega_k=0$, $0.1$, $-0.1$, we ran [[class]{}]{} with reference settings for the parameters governing the accuracy of $C_\ell^{\phi \phi}$ (in particular, not using the Limber approximation), and then with default precision. We show the ratio of CMB lensing potential spectra on the left, and the corresponding ratio of lensed temperature spectra on the right. At low-$\ell$, before using the Limber approximation, the default $C_\ell^{\phi \phi}$ are not precise, but the impact of this error on lensed spectra is negligible; precise $C_\ell^{\phi \phi}$ at low $\ell$ can be obtained by just halfing the precision parameter [source\_sampling\_step]{} (leading to the dashed curves). These tests were performed later than the rest of this paper, with [[class]{}]{} v2.3.3.](fig_limber/fig_limber_lensed "fig:"){width="0.5\columnwidth"}
For $l<10$, the error is large, up to 2%. The settings leading to accurate small-scale temperature spectra do not lead to accurate CMB lensing spectra. This can easily be fixed by just increasing the time sampling in the integral (\[eq:delta\_phi\]): by halfing the precision parameter [source\_sampling\_step]{}, we reach 0.5% precision (dashed curves in figure \[fig:accuracy\_lensing\]). However, there is no need to change the default precision settings just for improving low-$l$ $C_\ell^{\phi \phi}$’s, because on very large angular scales the lensing spectrum is impossible to measure accurately, and impacts the final lensed $C_\ell$’s by a negligible amount, as we shall see in the next paragraph. For $l\geq 10$, we see the power of the Limber approximation: it induces an error of at most $0.5\%$ near $l=10$, and it becomes increasingly good at large $\ell$’s.
To check how errors in the lensing spectrum propagate to lensed $C_\ell$’s, we computed the ratio of lensed temperature spectra for the same pair of models. We find an accuracy level better than 0.1% (except for $l=3$ in the open model), clearly sufficient for fitting current and forthcoming CMB experimental data. Hence, the 2% error on the very low $\ell$ CMB lensing spectrum has a negligible impact on lensed temperature spectra, which are mainly sensitive to larger $\ell$’s. For default accuracy, we found in figures \[fig:accuracy\_flat\], \[fig:accuracy\_closed\] , \[fig:accuracy\_open\] that the error in the unlensed spectra is of the order of 0.2%; figure \[fig:accuracy\_lensing\] shows that in the lensing spectra the error may increase at most to the level of 0.3%.
Discussion
==========
Having efficient Boltzmann codes for non-flat models is still useful, despite the fact that the data is compatible with spatial flatness. Indeed we will have soon more accurate Planck temperature data, Planck polarisation data and accurate polarisation data coming from other ground-based experiments. In addition, there will be new generations of large scale structure experiments in the future and maybe also another CMB experiment. Each time that new data arrives, we will need to check that the flat model is still preferred, and to derive bounds on curved models.
Previous implementations of curvature in Boltzmann codes were not thoroughly tested due to the lack of independent methods. Moreover, the method encoded in [[camb]{}]{} for computing hyperspherical Bessel function is difficult to test because it is built into the computation of the integrals. Here we have presented a new method for calculating CMB anisotropies in a non-flat FLRW universe, relying on a very stable algorithm for the calculation of hyperspherical Bessel functions, that can be pushed to arbitrary precision levels. We also introduce a new approximation scheme which gradually takes over in the flat space limit, and significant speeds up calculations. We described several aspects of our implementation of the equations in the code (sources, radial functions, transfer functions, primordial spectra) aimed at simplicity and unification of flat and non-flat computations.
We used our code to benchmark the accuracy of the [[camb]{}]{} code in curved space: by default [[camb]{}]{} achieves roughly 0.3% on scalar temperature, or 0.1% with the “high accuracy” flag tuned on. This is roughly what was claimed before, and it is similar to what one gets in the flat case. The exception is the limit of large positive curvature, since our comparison reveals slightly larger errors for $\Omega_k=-0.1$. We will check in the future that we obtain the same lower bound on $\Omega_k$ when using [[class]{}]{} instead of CAMB to analyse Planck data, but this is likely to be the case up to insignificant differences.
However we find that for the same precision level, [[class]{}]{} is significantly faster, usually by a factor 3 for non-flat models. This is mainly due to the flat rescaling approximation, which could in principle be easily implemented in other codes.
Acknowledgements {#acknowledgements .unnumbered}
================
This project was supported by the Swiss National Foundation.
Appendix {#appendix .unnumbered}
========
The appendix describe 4 approximation schemes used by [[class]{}]{}. The splitting of the temperature source function described in appendix \[app:sources\] is unique to [[class]{}]{}, but the time cut approximation and the multipole cut approximation of appendix \[sec:time\_cut\] and \[sec:multipole\_cut\] may have been used in previous Boltzmann codes in some form. (They are rather straightforward.) However, this is the first time the physical motivations for these approximations have been published. The Limber approximation of appendix \[sec:limber\] is more standard.
Splitting the temperature source functions \[app:sources\]
==========================================================
There are several ways to split the temperature source functions in a set of physical contributions: this is just a matter of convention. Here we refer to the most common splitting. To make it more readable, we reorganize the contributions to $S_{T_0}$, $S_{T_1}$, $S_{T_2}$ in a way which differs from both (\[eq:sourceT\]) and (\[eq:sourceTbis\]), but is still fully equivalent after some integrations by part: $$S_{T0}^{(0)}= g \left( \frac{1}{4}\delta_\gamma + \psi \right) + e^{-\kappa} (\phi^\prime+\psi^\prime)~,~~~~
S_{T1}^{(0)}= \frac{g}{k} \theta_b ~,~~~~
S_{T2}^{(0)}=g P^{(0)}~. \label{eq:sourceTter}$$ The first term in $S_{T0}^{(0)}$ contains the intrinsic temperature fluctuation ($\frac{1}{4}\delta_\gamma$) and the gravitational redshift term $\psi$. The words “Sachs-Wolfe term" sometimes refer to the latter, or to the sum of the two: here we call the whole term proportional to $g$ “Sachs-Wolfe". The second term proportional to $e^{-\kappa}$ is the Integrated Sachs-Wolfe term. $S_{T1}^{(0)}$ contains the Doppler term, and $S_{T2}^{(0)}$ conatins some polarisation-related contributions (that would vanish if the Thomson scattering term was averaged over directions). We can now write the source terms (\[eq:sourceTbis\]) implemented in [[class]{}]{} with a set of switching coefficients $\{ s_\mathrm{SW}, s_\mathrm{ISW}, s_\mathrm{Dop.}, s_\mathrm{Pol.}\}$ which should all be set to 1 in order to recover the full temperature spectrum, or some of them can be set to 0 in order to kill each of the four physical contributions: $$\begin{aligned}
\tilde{S}_{T_0}^{(0)} &=& s_\mathrm{SW} \left[ g\left( \frac{1}{4}\delta_\gamma+\psi \right)\right] + s_\mathrm{ISW} \left[ g(\phi-\psi) +e^{-\kappa} 2 \phi' \right]
+ s_\mathrm{Dop.} \left[ k^{-2}(g \theta_b'+g' \theta_b)\right]~, \nonumber\\
\tilde{S}_{T_1}^{(0)} &=& s_\mathrm{ISW} \left[ e^{-\kappa} k (\psi-\phi)\right]~,~~~~
\tilde{S}_{T_2}^{(0)} = s_\mathrm{Pol.} \left[ g P^{(0)} \right]~.\end{aligned}$$ In version 2.0 of [[class]{}]{}, we decomposed the expression (\[eq:sourceTbis\]) of the source functions in this way, to allow users to switch off some terms when necessary (e.g, to study ISW correlations with large scale stucture, or to understand the physically impact of some cosmological ingredient). We have shown this feature in the right panel of figure \[fig:decomposition\]. As explained in the file [explanatory.ini]{}, the [[class]{}]{} user can specify a list of terms to be included in the temperature calculation. In this list, the ISW terms has been further split into early and late ISW[^15]
Time cut approximation \[sec:time\_cut\]
========================================
The lower boundary of the conformal time integral in equations (\[eq:transT\]–\[eq:transB\]) is found automatically by [[class]{}]{}: it is the time at which the Thomson scattering rate $\kappa'=an_e\sigma_T$ exceeds a given fraction of the Hubble rate[^16]. The upper boundary is by default $\tau_0$, the time today. However it gets automatically reduced by the code for large $\ell$’s, using the fact that for large $\ell$ and small $\chi$, $\Phi_\ell^\nu(\chi)$ (or $j_l(\nu \chi)$) is negligible. All Bessel functions are approximated by zero when they are below a given threshold $\Phi_*$[^17], and this translates into an upper bound of the conformal time integral.
In [[class]{}]{} version v2.0, we introduced an additional approximation, allowing to further reduce the upper bound. The source terms $\tilde{S}_{T_2}^{(0)}$ and $S_P^{(m)}$ are proportional to the visibility function $g$. This function has a large peak near the recombination time, and a much smaller peak near the reionisation time. Small angular scales are not sensitive to the effect of rescattering at reionisation, and feel the imprint of the source terms $\tilde{S}_{T_2}^{(0)}$ and $S_P^{(m)}$ only around the time of recombination. For these scales, we can cut the integral over time soon after recombination, when the visibility function goes below a threshold $g_c$. This approximation should not be used on large angular scales, which are sensitive to the small peak of the visibility function near the reionisation time. In conclusion, the time cut approximation consists in cutting the time integral for $\tilde{S}_{T_2}^{(0)}$ and $S_P^{(m)}$ at a time $\tau_c$ such that $g(\tau_c) = g_c$, but only for $l < \alpha \, l_c$ ($\alpha$ is the angular rescaling factor defined in equation (\[eq:alpha\]), equal to one in flat space). In the default version of [[class]{}]{}, these parameters[^18] are fixed to $(g_c, l_c)=(10^{-3}, 400)$, while in the reference settings of the file [cl\_ref.pre]{} they are pushed to $(10^{-30}, 3000)$, such that the approximation is never used.
In the case of $\tilde{S}_{T_1}^{(0)}$, we can use the same approximation in order to save extra time, because this term accounts for one part of the late ISW effect, and only contributes to small multipoles. With default settings, the time cut does not alter any calculation for $l<400 \alpha$. This covers the whole region in which the ${T_1}$ term contributes to the total ${C_\ell^{TT}}^{(0)}$’s at more than 0.01%, and to the late ISW ${C_\ell^{TT}}^{(0)}$’s at more than one per cent[^19]. Hence the time cut approximation can be safely used for $T_1$.
\[fig:timecut\]![Relative difference between unlensed $C_\ell$’s computed using the time cut approximation, with $(g_c, l_c)=(10^{-3}, 400)$, over $C_\ell$’s without this approximation, for scalar temperature (left, solid), scalar E-type polarisation (left, dotted), tensor E-type polarisation (right, solid) and tensor B-type polarisation (right, dotted), each time for three values of spatial curvature. All calculations have been performed using the reference settings of the file [cl\_ref.pre]{} for all precision parameters but $(g_c, l_c)$. The vertical scale is the same as in the previous figure.](fig_timecut/fig_scal "fig:"){width="0.5\columnwidth"}![Relative difference between unlensed $C_\ell$’s computed using the time cut approximation, with $(g_c, l_c)=(10^{-3}, 400)$, over $C_\ell$’s without this approximation, for scalar temperature (left, solid), scalar E-type polarisation (left, dotted), tensor E-type polarisation (right, solid) and tensor B-type polarisation (right, dotted), each time for three values of spatial curvature. All calculations have been performed using the reference settings of the file [cl\_ref.pre]{} for all precision parameters but $(g_c, l_c)$. The vertical scale is the same as in the previous figure.](fig_timecut/fig_tens "fig:"){width="0.5\columnwidth"}
In Figure \[fig:timecut\], we show that the impact of the time cut approximation on scalar and tensor $C_\ell$’s is well below the 0.1% level for default settings. The advantage of this approximation is that the time spent by the code in calculating the transfer functions for $T_1^{(0)}$, $T_2^{(0)}$, $E^{(m)}$, $B^{(2)}$ is significantly smaller than the time spent in calculating $T_0^{(0)}$ and $T_2^{(2)}$. In the new approach of [[class]{}]{} v2.0 for source functions, described in section \[sec:sources\], we have increased the number of temperature source term from one to three for scalar modes; thanks to the time cut approximation, this extension is done at reduced cost and does not compromise the speed of the code.
Multipole cut approximation \[sec:multipole\_cut\]
==================================================
The execution time of Boltzmann codes depends crucially on the number of time integrals that must be performed in order to compute each transfer function ${\Delta_l^X}^{(m)}(q)$. Usually, a comparable amount of time is spent in integrating the system of cosmological perturbations, and in evaluating the transfer functions[^20]. Hence, limiting the number of discrete values $l_i$ and $q_j$ for which transfer functions are calculated has a great potential in speeding up the codes.
The CMB transfer functions ${\Delta_l^X}^{(m)}(q)$ (with $X \in \{T_0, T_1, T_2, E, B\}$) peak close to the (generalised) wavenumber $q$ corresponding to Fourier modes seen under an angle $\pi/l$ when they propagate orthogonally to the line-of-sight on the last scattering surface. For a given $l$, these modes are given by $q = q(l) \equiv l/r_A^\mathrm{rec}$. For $q\ll q(l)$, the $\Delta$’s vanish exponentially, because larger wavelengths cannot project under the angle $\pi/l$. For $q\gg q(l)$, the transfer function exhibits damped oscillations, because smaller wavelength can be seen under the same angle if they propagate with an appropriate angle with respect to the line-of-sight. Depending on the type $X$ and mode $(m)$, the transfer function decreases faster or slower with $q$, because some transfer functions can also receive a physical contribution from much smaller wavelengths than $q(l)$ at smaller redshifts, due to the ISW effect or to reionisation.
For the reasons mentioned in the previous paragraph, the ${\Delta_l^X}^{(m)}(q)$ should not be computed over a rectangular shape in $(q,l)$ space, but around an oblique band encompassing the line $l = q r_A^\mathrm{rec}$. In [[class]{}]{} v2.0 the most exterior loop is over $q$, so we must express this condition in terms of minimal and maximal values of $l$ for which ${\Delta_l^X}^{(m)}(q)$ should be computed, given the wavenumber $q$, the type $X$ and the mode $m$.
The minimal $l$ values are easy to find, and are defined in the same way in all versions of [[class]{}]{}. In section \[sec:time\_cut\], we have seen that for each integral over time, the lower boundary is fixed by a threshold value of the Thomson scattering rate, and the upper boundary by a threshold value of the Bessel function (unless the time cut approximation imposes a stronger condition). For $l$ slightly bigger than $q r_A^\mathrm{rec}$, these two boundaries cross each other, because the support of the Bessel function and of the source function do not overlap. In this situation, the code does not perform any integration over time, and simply assigns zero to the transfer function.
\[tab:lcut\]
---------------------- ------------- ------------- ------------- ----------- ------------- ----------- -----------
$X^{(m)}$ $T_0^{(0)}$ $T_1^{(0)}$ $T_2^{(0)}$ $E^{(0)}$ $T_2^{(2)}$ $E^{(2)}$ $B^{(2)}$
$(\Delta q)_X^{(m)}$ 0.15 0.04 0.15 0.11 0.20 0.25 0.10
---------------------- ------------- ------------- ------------- ----------- ------------- ----------- -----------
: Default setting for the multiple cut approximation, for scalar and vector modes.
\[fig:lcut\]![Relative difference between unlensed $C_\ell$’s computed using the multipole cut approximation, with the settings of Table 1, over $C_\ell$’s without this approximation, for scalar temperature (left, solid), scalar E-type polarisation (left, dashed), tensor temperature (right, solid), tensor E-type polarisation (right, dashed) and tensor B-type polarisation (right, dotted), each time for three values of spatial curvature. All calculations have been performed using the reference settings of the file [cl\_ref.pre]{} for all precision parameters but the $(\Delta q)_X^{(m)}$’s. For scalars, the vertical scale is the same as in previous figures, while for tensors (requiring less precision) the scale is multiplied by ten.](fig_lcut/fig_l_scal "fig:"){width="0.5\columnwidth"}![Relative difference between unlensed $C_\ell$’s computed using the multipole cut approximation, with the settings of Table 1, over $C_\ell$’s without this approximation, for scalar temperature (left, solid), scalar E-type polarisation (left, dashed), tensor temperature (right, solid), tensor E-type polarisation (right, dashed) and tensor B-type polarisation (right, dotted), each time for three values of spatial curvature. All calculations have been performed using the reference settings of the file [cl\_ref.pre]{} for all precision parameters but the $(\Delta q)_X^{(m)}$’s. For scalars, the vertical scale is the same as in previous figures, while for tensors (requiring less precision) the scale is multiplied by ten.](fig_lcut/fig_l_tens "fig:"){width="0.5\columnwidth"}
The scheme for the maximal $l$ has been simplified in [[class]{}]{} v2.0 with respect to versions 1.x. We simply assume a linear boundary for the region in $(q,l)$ space where ${\Delta_l^X}^{(m)}(q)$ is not negligible. In other words, we set the transfer function to zero whenever the condition $l < \left(q-(\Delta q)_X^{(m)}\right) r_A^\mathrm{rec}$ is fulfilled, where the $(\Delta q)_X^{(m)}$’s are precision parameters tuned to achieve a given precision[^21]. Table \[tab:lcut\] shows the default settings of [[class]{}]{} v2.0, which are sufficient for fitting Planck data. Indeed, Figure \[fig:lcut\] shows that these settings introduce an error well below the 0.1% level for all scalar ${C_\ell^{XX}}^{(0)}$’s. Primordial tensor spectra are not yet observed and require less precision, especially on small angular scales. For all ${C_\ell^{XX}}^{(2)}$’s, Figure \[fig:lcut\] shows that the error is below 0.1% for $l<500$, and increases to the level of 1% for $l\sim 1000$.
The user is free to avoid ever using the multiple cut approximation by setting all precision parameters $(\Delta q)_X^{(m)}$ to very large values, as done in the reference settings of the file [cl\_ref.pre]{}.
Limber approximation \[sec:limber\]
===================================
In both the flat and curved space, we use the Limber approximation for speeding up the calculation of the lensing potential transfer function $\Delta_l^\phi(q, \tau_0)$ for large $l$ (by default, $l>10$). This approximation is very useful for lensing, because of the slow variation of the source function with respect to (hyper)spherical Bessel functions. It is never used for CMB transfer functions, due to the quickly oscillating behaviour of the underlying source functions.
In flat space, the Limber approximation can be derived by Taylor expansion of the function that we are convolving with the spherical Bessel function: $$\begin{aligned}
\int_0^\infty dx j_l(x) f(x) &\simeq \int_0^\infty dx j_l(x) \left( f(x_0) + f'(x_0) (x-x_0) + \cdots \right) \\
&= f(x_0) \int_0^\infty dx j_l(x) + f'(x_0) \int_0^\infty dx j_l(x) (x-x_0) + \cdots\end{aligned}$$ The second integral vanishes if we take $$\begin{aligned}
x_0 &= (l+1)\frac{\Gamma\left( \frac{l+2}{2} \right)^2}{\Gamma\left( \frac{l+1}{2} \right) \Gamma\left( \frac{l+3}{2} \right)} \\
&\simeq \frac{1}{2} + l + \frac{1}{8l} - \frac{1}{16 l^2} + \cdots\end{aligned}$$ where the last expansion is valid for large $l$. The Limber approximation now becomes $$\begin{aligned}
\int_0^\infty dx j_l(x) f(x) &\simeq f(x_0) \frac{\sqrt{\pi}}{2} \frac{\Gamma\left( \frac{l+1}{2} \right)}{\Gamma\left( \frac{l+2}{2} \right)} \equiv f(x_0) I_l^\text{flat} \\
&\simeq f(x_0) \sqrt{\frac{\pi}{2l}} \left\{ 1 - \frac{1}{4l} + \frac{1}{32l^2} + \cdots \right\}\end{aligned}$$ We should note that one can also derive a flat space Limber approximation based on a Taylor expansion of $f(x)/\sqrt{x}$. The advantage is that the spherical Bessel function becomes an ordinary Bessel function of order $l+1/2$ and the Limber approximation can be derived to all orders using the Laplace transform [@LoVerde:2008re]. This leads to the Limber approximation $$\begin{aligned}
\int_0^\infty dx j_l(x) f(x) &\simeq \frac{f(\tilde{x}_0)}{\sqrt{\tilde{x}_0}} \sqrt{\frac{\pi}{2}} = f(l+1/2) \sqrt{\frac{\pi}{2l+1}}.\end{aligned}$$ By comparing the two formulae, it is clear that they are equivalent for large $l$. However, the first approach is more easily generalised to curved space. First note that the classical turning point is a good approximation to $x_0$ as is the case in flat space where $x_\text{tp}=\sqrt{l(l+1)} \simeq l + \frac{1}{2} - \frac{1}{8l} + \cdots$. We found numerically that the integrals over the hyperspherical Bessel functions could be expressed in terms of the flat integrals $I_l^\text{flat}$ to a good approximation as $$\begin{aligned}
\int_0^\infty d\chi \Phi_l^\nu(\chi) &\simeq \left[1 - {\hat{K}}\frac{l^2}{\nu^2} \right]^{-1/4} \frac{1}{\nu} I_l^\text{flat},\end{aligned}$$ where $\infty$ in the integrals should be understood as the equivalent point $\pi/2$ for ${\hat{K}}=1$. The Limber approximation in curved space now becomes $$\int_0^\infty d\chi \Phi_l^\nu (\chi) f(x) \simeq f(\chi_0) \frac{1}{\nu} \left(1 - \hat{K} \frac{l^2}{\nu^2} \right)^{-1/4} I_l^\text{flat}$$ Like in flat space, we choose $\chi_0= \mathrm{Arcsin}_{\hat{K}} \left(\frac{l + \frac{1}{2}}{\nu}\right)$, very close to the turning point $\chi_\mathrm{tp}$ defined in section \[sec:ahbf\]. Then we can derive: $$\begin{aligned}
\Delta_l(q) &=& \int d \tau S(k,\tau) \Phi_l^{\nu=q/\sqrt{|K|}}\left(\chi=\sqrt{|K|}(\tau_0-\tau) \right)\\
&=& \int d \chi \frac{1}{\sqrt{|K|}} S\left(k,\tau=\tau_0-\frac{\chi}{\sqrt{|K|}}\right) \Phi_l^{\nu=q/\sqrt{|K|}}(\chi) \\
&\simeq & I_l^\text{flat} \left(1 - \hat{K} \frac{l^2}{\nu^2} \right)^{-1/4} \frac{1}{q} S\left(k,\tau=\tau_0-\frac{\chi_0}{\sqrt{|K|}}\right) \end{aligned}$$ This approximation is nearly the same as the one implemented in [camb]{}, up to tiny corrections (terms $+1$ or $+\frac{1}{2}$). Note that in the code, the function that we want to interpolate with respect to time with is not $S$ but the product $(\tau_0-\tau) S$: this product is better behaved in $(\tau_0-\tau) \longrightarrow 0$. Like in flat space, we push the calculation one step further, because we want to interpolate $(\tau_0-\tau) S$ instead of $S$: $$\begin{aligned}
\Delta_l(q)
&\simeq&
I_l^\text{flat} \left(1 - \hat{K} \frac{l^2}{\nu^2} \right)^{-1/4} \frac{1}{q} \frac{\sqrt{|K|}}{\chi_0} \left[ (\tau_0-\tau) S(k,\tau) \right]_{\tau =\tau_0-\frac{\chi_0}{\sqrt{|K|}}} \\
&=&
I_l^\text{flat} \left(1 - \hat{K} \frac{l^2}{\nu^2} \right)^{-1/4} \frac{1}{\nu \chi_0} \left[ (\tau_0-\tau) S(k,\tau) \right]_{\tau =\tau_0-\frac{\chi_0}{\sqrt{|K|}}} ~.\end{aligned}$$ This version of the Limber approximation in non-flat space has been implemented in [[class]{}]{} v2.3.3.
[^1]: To be precise, this paper always refer to the version with release number 2.0.4. The first [[class]{}]{} release including spatial curvature was 2.0.0. It was quickly followed by some minor revisions fixing small bugs, and at the time of submitting this work, by the revision 2.0.4, in which we improved the sampling scheme in wavenumber space and the tuning of some accuracy parameters, corresponding exactly to the results presented in this paper.
[^2]: Each evaluation of the WKB approximation requires two calls to library functions and a Chebyshev approximation of the Airy function. While it is faster than the recurrence method for a single $\ell $-value, it becomes slower when we need a few hundred $\ell 's$ for each $\nu$ which is usually the case.
[^3]: Note that this is different from using the Gegenbauer identity to set initial conditions for the backward recurrence, since the Gegenbauer recurrences and hyperspherical recurrences moves along different lines in the $(\ell,\nu)$-plane.
[^4]: In the synchronous gauge, $\psi$ corresponds to $\alpha'+\frac{a'}{a} \alpha$, with $\alpha \equiv (h'+6\eta')/(2k^2)$. Hence $\psi'$ involves $\alpha''$.
[^5]: in the function [perturb\_sources()]{} of the module [perturbations.c]{}.
[^6]: In [[class]{}]{} v2.0, the sampling parameter is fixed by the precision parameters [hyper\_sampling\_curved\_high\_nu]{}, [hyper\_sampling\_curved\_low\_nu]{}, [hyper\_nu\_sampling\_step]{}, which by default are equal to 3, 6, 1000.
[^7]: The contribution from massive neutrinos has not yet been implemented.
[^8]: This ratio $\alpha$ is computed by [[class]{}]{} in the [thermodynamics]{} module and called [angular\_rescaling]{}. It is used in many places in the code since all characteristic quantities in multipole space (sampling step sizes, values at which a given approximation should be switched on) are systematically scaled by $\alpha$.
[^9]: We deduced the functional dependence on $l$ and $\nu$ by fitting second order polynomials to the ratios of peaks on a $(l,\nu)$-grid. We then observed that the coefficients seemed to depend mostly on the angle $\beta = {\,\text{atan}}(l/\nu)$ in this plane.
[^10]: In the code, $\nu_*$ is called [hyper\_flat\_approximation\_nu]{}.
[^11]: Since the two codes are fully independent (apart from the underlying equations and the recombination modules), a very unusual coincidence would be needed to hide any larger error.
[^12]: This is made easier by the fact that these parameters are all grouped inside a single structure, and set in the same place. No numbers referring to a choice of precision ever appear in the bulk of the code. Hence it is not conceivable that some precision parameters have been forgotten when doing the tuning.
[^13]: All these test presented in this work rely on this version of [[camb]{}]{}.
[^14]: The spike at $l=16$ had already been observed in [@Lesgourgues:2011rg] and has not disappeared. It might be due to a jump in the precision between $\ell=16$ and $17$, since the value [llmax]{} is fixed to 17 in [[camb]{}]{}’s [DoSourceIntegration]{} routine (A. Lewis, 2011, private communication).
[^15]: For instance, to include only the late ISW term, the user should write [temperature contributions = lisw]{} in the input file. For all contributions except the late ISW, the syntax would be [temperature contributions = tsw, eisw, dop,pol]{}. The splitting between early and late ISW occurs at an arbitrary redshift which can be adjusted by passing a value for the input parameter [early/late isw redshift]{} (set by default to 120).
[^16]: Defining the characteristic times $\tau_c=1/\kappa^\prime$ and $\tau_H=a/a^\prime$, the lower boundary is the time at which $\tau_c/\tau_H = 0.008$ (default setting), or $\tau_c/\tau_H = 0.006$ (high precision setting).
[^17]: In [[class]{}]{} v1.x, this value was called [bessel\_j\_cut]{} and fixed by default to $10^{-5}$. In [[class]{}]{} v2.0, since the whole calculation of Bessel functions was revised as described in Section \[sec:hyper\], this parameter has changed: it is called [hyper\_phi\_min\_abs]{} and fixed to $10^{-10}$
[^18]: Inside the code, $g_c$ and $l_c$ are called respectively [neglect\_CMB\_sources\_below\_visibility]{} and [transfer\_neglect\_late\_source]{}.
[^19]: Moreover, on small angular scales with $l>400 \alpha$, the late ISW signal is very difficult to observe, even when cross-correlating temperature and large scale structure maps.
[^20]: in [[class]{}]{}, these two tasks are distributed respectively to the [perturbation]{} and [transfer]{} modules. Depending on the input cosmological model and on the requested output, most of the time can be spent in one or the other module. In non-flat space, for large enough $|\Omega_k|$, most of the time is actually spent in computing hyperspherical Bessel functions, but the next longest tasks remain the two previous ones.
[^21]: in the code, the parameters $(\Delta q)_X^{(m)}$ are called, e.g., [transfer\_neglect\_delta\_k\_S\_t0]{} (for scalar modes and type $T_0$).
|
---
author:
- 'Hsuan-Hao Lu'
- 'Joseph M. Lukens'
- 'Nicholas A. Peters'
- 'Brian P. Williams'
- 'Andrew M. Weiner'
- Pavel Lougovski
title: 'Controllable two-photon interference with versatile quantum frequency processor'
---
[^1]
[^2]
[13.45]{}(1.35,12) This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan. (http://energy.gov/downloads/doe-public-access-plan).
**Quantum information is the next frontier in information science, promising unconditionally secure communications, enhanced channel capacities, and computing capabilities far beyond their classical counterparts [@Gisin2002; @Ladd2010]. And as quantum information processing devices continue to transition from the lab to the field, the demand for the foundational infrastructure connecting them with each other and their users—the quantum internet [@Kimble2008; @Pirandola2016]—will only increase. Due to the remarkable success of frequency multiplexing and control in the *classical* internet, quantum information encoding in optical frequency offers an intriguing synergy with state-of-the-art fiber-optic networks. Yet coherent quantum frequency operations prove extremely challenging, due to the difficulties in mixing frequencies efficiently, arbitrarily, in parallel, and with low noise. Here we implement an original approach based on a reconfigurable quantum frequency processor, designed to perform arbitrary manipulations of spectrally encoded qubits. This processor’s unique tunability allows us to demonstrate frequency-bin Hong-Ou-Mandel interference with record-high 94% visibility. Furthermore, by incorporating such tunability with our method’s natural parallelizability, we synthesize independent quantum frequency gates in the same device, realizing the first high-fidelity flip of spectral correlations on two entangled photons. Compared to quantum frequency mixing approaches based on nonlinear optics [@Kobayashi2016; @Clemmen2016], our linear method removes the need for additional pump fields and significantly reduces background noise. Our results demonstrate multiple functionalities in parallel in a single platform, representing a huge step forward for the frequency-multiplexed quantum internet.**
In the classical domain, the ultrabroad bandwidth supported by optical fiber has proven crucial in solidifying fiber optics in the digital communications revolution. Wavelength-division multiplexing (WDM)—either in its standard embodiment with independent frequency channels, or more complex versions with channels comprising interleaved bands [@Hillerkuss2011]—forms an essential component, and will continue to do so even as novel data formats and multiplexing techniques are incorporated [@Agrell2016]. Such success has naturally brought WDM approaches to the forefront for the *quantum* internet as well. In particular, potentially large amounts of information can be stored in single photons encoded in spectro-temporal modes [@Lukens2014; @Brecht2015; @Zhong2015; @Islam2017], and frequency multiplexing is essential for scaling up quantum memories [@Sinclair2014].
To that end, considerable progress has been made in generating multiphoton entanglement across a comb of narrowband frequency modes, or bins, including optical parametric oscillators below threshold [@Lu2003], straightforward filtering of broadband parametric downconversion [@Xie2015], and, recently, on-chip production of quantum frequency combs using microring resonators [@Reimer2016; @Jaramillo2017; @Kues2017; @Imany2018]. Likewise, an explosion of research in quantum frequency conversion has showcased coherent translation of single-photon states across both wide [@Huang1992; @Tanzilli2005] and narrow [@Wright2017] bandwidths. But the step from viewing frequency as a channel—wherein quantum information is carried by some other parameter, such as time or polarization—to encoding the information itself in frequency is significant, requiring markedly more complex operations: i.e., universal gate sets in frequency modes. As important milestones in that direction, frequency beamsplitters [@Kobayashi2016; @Clemmen2016; @Joshi2017b] and quantum pulse gates [@Brecht2015] based on optical nonlinearities have shown coherent interference and mode selection of frequency-encoded photons. Yet the need for powerful optical control fields makes nonlinear approaches challenging, given the potential for extra noise photons from Raman scattering and imperfect isolation.
Recently, we proposed a general framework [@Lukens2017] for spectrally encoded photonic state control, based on electro-optic phase modulators (EOMs) and Fourier-transform pulse shapers. Enabling universal quantum information processing in a scalable fashion, our approach is also optically linear, obviating the need for additional pump fields. Figure \[fig1\] sketches an example of such a quantum frequency processor, with the particular operations chosen to match the ensuing experiments. In general, an input quantum state consisting of a superposition of photons spread over discrete frequency bins is manipulated by the designed network of EOMs and pulse shapers, which applies various unitary operations to combinations of frequency bins. After each step, some of the frequency bins can be detected, with the newly available bandwidth re-provisioned with freshly encoded photons. Note that, although we draw each frequency bin as a separate “rail” for conceptual purposes, the physical encoding occurs within a single fiber-optic spatial mode, thereby enabling natural phase stability and providing compatibility with current fiber networks. This paradigm has allowed us to experimentally demonstrate frequency beamsplitters and tritters with ultrahigh operation fidelity and parallelizability across a 40-nm optical bandwidth [@Lu2018]. While then validated with weak coherent states, a quantum frequency network will require multiphoton and nonclassical interference phenomena as well. In this work, we experimentally show that not only does our operation indeed realize unrivaled interference of fully quantum frequency states, but it also enables independent simultaneous operations *in the same device*, via its unique tunability and parallelizability.
Two nonclassical phenomena of particular significance in quantum photonics are Hong-Ou-Mandel (HOM) interference [@Hong1987] and the Einstein-Podolsky-Rosen (EPR) paradox [@Einstein1935]. In the conventional HOM interferometer, two photons mixed on a 50/50 spatial beamsplitter bunch, never exiting in different output ports; a general feature of bosons, HOM interference forms the basis of essentially all multiqubit gates in linear optics [@Kok2007]. The EPR paradox notes the strange behavior in quantum mechanics that two spatially distant particles can share strong correlations in noncommuting observables (such as position and momentum). This highlights a fundamental inconsistency between the completeness of quantum mechanics and local realism, ultimately underpinning Bell tests of nonlocality [@Bell1964] and security in quantum key distribution [@Gisin2002]. Both phenomena are properly foundational in that they shed light on fundamental interpretations of quantum mechanics and enable practical quantum information applications.
Figure \[fig2\]a shows our setup for processing quantum information encoded in frequency. Our test source of entanglement is a biphoton frequency comb (BFC) generated by pumping a periodically poled lithium niobate (PPLN) waveguide with a continuous-wave Ti:sapphire laser, filtering the broadband emission with an etalon to produce 25-GHz-spaced frequency bins, and setting the relative amplitude and phase of each frequency bin with a pulse shaper. In Fig. \[fig2\]b, we plot the measured frequency correlations of this source, obtained by bypassing the quantum frequency processor (QFP), scanning the filters of the output wavelength-selective switch, and counting coincidences between two detectors. Each frequency-bin index $n$ corresponds to the filter centered at $\omega_n = \omega_0 + n\Delta\omega$, where $\omega_0/2\pi = 193.6000$ THz (ITU channel 36 at 1548.51 nm) and $\Delta\omega/2\pi = 25$ GHz. Over this $50\times 50$ mode grid, we observe high coincidence counts only for frequency-bin pairs satisfying $n_A+n_B=1$, as expected by energy conservation. The processor itself consists of a pulse shaper sandwiched between two EOMs. Each EOM is driven by a 25-GHz sinusoidal voltage, while the pulse shaper imparts a user-defined phase to each spectral bin; this combination was shown to enable a frequency Hadamard gate $H$ with 99.998% fidelity and only 2.61% photon leakage into neighboring modes [@Lu2018].
![**Quantum frequency processor setup.** **a**, Experimental configuration. Parametric downconversion followed by an etalon generates a biphoton frequency comb with 25-GHz spacing, which is spectrally filtered and sent through the quantum frequency processor, after which a wavelength-selective switch is used to measure photon coincidences between frequency bins. **b**, Joint frequency spectrum of source, measured with etalon output connected directly to coincidence detection setup. Frequency bins satisfying $n_A+n_B=1$ show strong correlations, whereas all other combinations are at the expected accidentals level. Coincidences are counted over 5 seconds.[]{data-label="fig2"}](Fig2.pdf){width="4.5in"}
In the case of HOM interference, we seek to apply such a gate to a pair of photons located in adjacent frequency bins, which we obtain directly by filtering out all but bins 0 and 1 of the source (Fig. \[fig2\]b); the $H$ gate applied to 0 and 1 should cause both photons to bunch in either bin 0 or 1, with no coincidences between the two bins. To measure the strength of quantum interference, one must scan some parameter which controls the distinguishability of the two-photon probability amplitudes leading to clicks on both output detectors; a visibility exceeding $50\%$ indicates nonclassicality [@Hong1987]. In the case of frequency mixers, one can introduce a temporal delay between the two modes [@Kobayashi2016] or scan the photon frequency spacing relative to that of the frequency beamsplitter [@Joshi2017b]. In our case, we adjust the mixing probability of the operation itself, analogous to varying the reflectivity $\mathcal{R}$ of a spatial beampslitter. To do so, we note a valuable feature of the our spectral beamsplitter: tunability. For by simply changing the depth of the phase shift imparted by the pulse shaper between frequency bins 0 and 1, the spectral reflectivity $\mathcal{R}$ can be tuned smoothly from 0 to $\sim$0.5 and back to 0 (see Methods). Figure \[fig3\]a plots the theoretically predicted (curves) and experimentally measured (symbols) beamsplitter transmission and reflection coefficients between bins 0 and 1, when scanning the pulse shaper phase. A phase setting of $\pi$ results in an $H$ gate; $0$ and $2\pi$ phase shifts yield an identity operation. It is important to note that both EOMs remain fixed throughout the scan, so that the tunability is effected only by adjusting the phase applied by the pulse shaper.
Sending in the photon pair $|1_{\omega_0}\rangle_A |1_{\omega_1}\rangle_B$ (i.e., one photon in frequency-bin 0, assigned to party $A$, and one photon in frequency-bin 1, assigned to party $B$) and scanning the pulse shaper phase, we measure the coincidence counts between output bins 0 and 1 shown in Fig. \[fig3\]b. The solid curve is the theoretical prediction, scaled and vertically offset to match the data points via linear least squares; the visibility obtained from this fit is $0.94 \pm 0.01$, with the reduction from unity completely consistent with the accidentals level expected for our measured counts and timing resolution. This visibility far exceeds the previous values measured for frequency-domain HOM interference—namely, $0.71 \pm 0.04$ [@Kobayashi2016] and $0.68\pm 0.03$ [@Joshi2017b]—and is a consequence of both the reduced optical noise and fine controllability of the operation. We also record the singles counts for bins $0$ and $1$, as well as the adjacent sidebands ($-1$ and $2$). As shown in Fig. \[fig3\]c, the two central modes retain nearly constant flux across the full scan, showing that the dip in coincidence counts results from truly quantum HOM interference as opposed to photon loss (see Methods for detailed verification). Moreover, the small reduction in singles counts around $\pi$—accompanied by the increase in singles counts for bins $-1$ and $2$—also qualitatively matches expectations, given the fact that the full $H$ gate scatters $2.61\%$ of the input photons out of the computational space into adjacent sidebands. We note that even this scattering could be removed by driving the EOMs with more complicated waveforms [@Lukens2017].
The quantum frequency processor’s tunability, invoked in the above realization of HOM interference, relies only on modifying the spectral phase, which suggests the ability to perform independent operations by setting different phase shifts on appropriate subbands in the pulse shaper’s bandwidth. Accordingly, this form of parallelizability is even stronger than previously shown, where the *same* operation was replicated across the bandwidth [@Lu2018]. To demonstrate this, we set the BFC shaper to pass modes $\{-4, -3, 4, 5\}$ (cf. Fig. \[fig2\]b), preparing the input entangled state $|\Psi\rangle \propto |1_{\omega_{-4}}\rangle_A |1_{\omega_5}\rangle_B + |1_{\omega_{-3}}\rangle_A |1_{\omega_4}\rangle_B$. On each pair of frequency bins—$\{-4,-3\}$ and $\{4,5\}$—we set the spectral phase to apply either the identity $\mathbbm{1}$ or Hadamard $H$ gates, and then measure coincidence counts between the frequency bins at the output. Figure \[fig4\] furnishes the results for all four combinations of $\mathbbm{1}$ and $H$; when the two gates match, near-perfect spectral correlations result (a and d), whereas mismatched cases produce uniform population of the two-qubit space (b and c). By measuring correlations in adjacent bins as well, we confirm the self-contained nature of our operation; even in the worst case (Fig. \[fig4\]d), less than $6\%$ of the total coincidences lie outside of the $2\times 2$ subspace, whereas similar state manipulation with only one EOM suffers from high probability of qubit scattering [@Kues2017; @Imany2018]. Importantly, the transition from $\mathbbm{1}_A \otimes \mathbbm{1}_B$ to $H_A \otimes H_B$ actually flips the correlations entirely, eliminating the negative frequency dependence resulting from pump energy conservation in favor of a positive dependence. Our demonstration is the first sign flip of entangled-photon frequency correlations *post-generation* and the first electro-optic-based two-qubit rotation that is closed in the $2\times 2$ computational space—essential for the ideal two-level logic underpinning qubit-based forms of quantum information processing.
![ **Two-photon state manipulation.** Coincidences between output frequency bins after application of the following gates: **a**, identity on both photons; **b**, Hadamard on photon $A$; **c**, Hadamard on photon $B$; **d**, Hadamard on both photons. Coincidences are collected over 120 s.[]{data-label="fig4"}](Fig4.pdf){width="4.5in"}
Using the strong correlations in two mutually unbiased measurements ($\mathbbm{1}$ and $H$), we can quantify directly the EPR-like nature of our entangled state. We define the conditional entropies $\mathcal{H}(\mathbbm{1}_A|\mathbbm{1}_B)$ and $\mathcal{H}(H_A|H_B)$ as the uncertainty of the measured frequency mode of $A$ {$-4,-3$} given knowledge of $B$’s result {$4,5$}, for the two cases of matched transformations (Fig. \[fig4\]a and d). Retrieving the probabilities from the raw counts via Bayesian mean error estimation (with no accidental subtraction), we recover $\mathcal{H}(\mathbbm{1}_A|\mathbbm{1}_B)= 0.19 \pm 0.03$ and $\mathcal{H}(H_A|H_B) = 0.29 \pm 0.04$, violating the Maassen-Uffink bound $q_{MU}$ for separable states \[$\mathcal{H}(\mathbbm{1}_A|\mathbbm{1}_B) + \mathcal{H}(H_A|H_B) \geq q_{MU}$\] [@Coles2017] by nearly 10 standard deviations (see Methods). Moreover, despite the fact we have implemented a tomographically incomplete set of measurements in Fig. \[fig4\], Bayesian methods allow us to estimate the full density matrix; any lack of information is reflected directly in the retrieved uncertainty. Doing so, we obtain the density matrix $\hat{\rho}$ with fidelity $\mathcal{F}=\langle\Psi|\hat{\rho}|\Psi\rangle = 0.92\pm0.01$, where $|\Psi\rangle$ is the maximally entangled ideal state (see Methods and Extended Data Fig. 1). Such findings demonstrate the utility of our quantum frequency processor for manipulating joint quantum systems coherently and independently, preserving a state’s built-in entanglement in the process—an entirely new functionality in frequency-bin qubit control.
**Acknowledgements.** We thank A. Sproles for the graphics in Fig. \[fig1\], D. E. Leaird for technical assistance, and B. Qi for valuable discussions. This work was performed in part at Oak Ridge National Laboratory, operated by UT-Battelle for the U.S. Department of Energy under Contract No. DEAC05-00OR22725. Funding was provided by ORNL’s Laboratory Directed Research and Development Program.
**Author Contributions.** H-H.L. led the experiments and data synthesis, and contributed to the theoretical analysis. J.M.L. initiated the concepts, assisted in the laboratory, and led writing of the paper. N.A.P. assisted with experiments, data interpretation, and conceptual design. B.P.W. set up the time-tagging electronics and performed the Bayesian estimation. A.M.W. supervised the work at Purdue and assisted with system design. P.L. initiated the concepts, managed the theoretical analysis, and supervised the project. All authors reviewed the results and contributed to the manuscript.
**Author Information.** The authors declare no competing financial interests. Correspondence and requests for materials should be addressed to J.M.L. (lukensjm@ornl.gov) or P.L. (lougovskip@ornl.gov).
METHODS {#methods .unnumbered}
=======
**Experimental details.** We couple a continuous-wave Ti:sapphire laser (M Squared) into a fiber-pigtailed periodically poled lithium niobite (PPLN; SRICO) waveguide, temperature controlled at $\sim$85$^{\circ}$C for spontaneous parametric down-conversion under type-0 phase matching. Spectrally entangled photon pairs spanning $>$2.5 THz are subsequently filtered by a Fabry-Perot etalon (Optoplex) with 25-GHz mode spacing (matched to the ITU grid) to produce a biphoton frequency comb (BFC), with each comb line possessing a full-width at half-maximum linewidth of 1.8 GHz. The center frequency of the pump laser is carefully locked to align the generated signal-idler pairs with etalon peaks, i.e., to maximize coincidences between the spectrally filtered modes. We utilize a pulse shaper (BFC shaper; Finisar) to perform amplitude and phase filtering to prepare particular input states for quantum frequency processing. Our quantum frequency processor (QFP) consists of two 40-Gb/s EOMs (EOSpace) with a pulse shaper [@Weiner2011] (QFP shaper; Finisar) sandwiched between them, with a total insertion loss of 12.5 dB [@Lu2018]. We note that synchronizing the biphoton emission time to the EOM phase is not necessary when the two-photon coherence time is much larger than the modulation period. This condition is guaranteed, e.g., by a pump laser with narrow linewidth, as in the current experiments. To implement a Hadamard operation, i.e., frequency-bin beamsplitter, we drive the two EOMs with 25-GHz $\pi$-phase-shifted sinewaves, and apply a step function with $\pi$-phase jump between the two computational modes on the shaper. The specific phase patterns are obtained in advance from optimization program in refs. , which achieve fidelity $\mathcal{F} =0.9999$ and success probability $\mathcal{P} =0.9760$ numerically for the Hadamard operation. The output photons are frequency-demultiplexed by an amplitude-only wavelength selective switch (WSS; Finisar) having 12.5-GHz channel specificity. Each time we route two different spectral modes (each takes up two pixels on the WSS) to two superconducting nanowire single-photon detectors (SNSPD; Quantum Opus) to record single counts as well as the coincidences within 1.5-ns bins.
**Quantum operations.** The specific configuration for our Hadamard gate (cf. supplement of ref. ) relies on the temporal phase modulation $\varphi(t) = \pm \Theta \sin\Delta\omega t$ ($\Theta = 0.8169$ rad) on the first and second EOMs, respectively. And for a gate operating on bins 0 and 1, the discrete pulse shaper phases can be written as $$\label{E0}
\nonumber
\phi_n =
\begin{cases}
\phi_0 & ; n \leq 0 \\
\phi_0 + \alpha & ; n \geq 1.
\end{cases}$$ Here $\phi_0$ is an offset with no physical significance, while $\alpha=\pi$ for the ideal Hadamard. Yet $\alpha$ can be tuned as well; doing so actually permits tunable reflectivity.
Specifically, if we write out the $2\times 2$ transformation matrix on modes 0 and 1 as a function of this phase, $$\label{E3}
\nonumber
V =
\begin{bmatrix}
V_{00}(\alpha) & V_{01}(\alpha) \\
V_{10}(\alpha) & V_{11}(\alpha)
\end{bmatrix},$$ we can define the variable reflectivities (i.e., mode-hopping probabilities) and transmissivities (probabilities of preserving frequency) as $$\nonumber
%\begin{aligned}
%\mathcal{R}_{0\rightarrow 1} = |V_{10}(\alpha)|^2 = 4 \left[ \sum_{k=1}^\infty J_k(\Theta) J_{k-1}(\Theta) \right]^2 \sin^2\frac{\alpha}{2}\\
%\mathcal{R}_{1\rightarrow 0} = |V_{01}(\alpha)|^2 = 4\left[ \sum_{k=1}^\infty J_k(\Theta) J_{k-1}(\Theta) \right]^2 \sin^2\frac{\alpha}{2} \\
%\mathcal{T}_{0\rightarrow 0} = |V_{00}(\alpha)|^2 = J_0^4(\Theta) \sin^2\frac{\alpha}{2} + \cos^2\frac{\alpha}{2} \\
%\mathcal{T}_{1\rightarrow 1} = |V_{11}(\alpha)|^2 = J_0^4(\Theta) \sin^2\frac{\alpha}{2} + \cos^2\frac{\alpha}{2},
%\end{aligned}
\begin{aligned}
\mathcal{R}_{0\rightarrow 1} = |V_{10}(\alpha)|^2 = \left| (1-e^{i\alpha}) \sum_{k=1}^\infty J_k(\Theta) J_{k-1}(\Theta) \right|^2 \\
\mathcal{R}_{1\rightarrow 0} = |V_{01}(\alpha)|^2 = \left| (1-e^{i\alpha}) \sum_{k=1}^\infty J_k(\Theta) J_{k-1}(\Theta) \right|^2 \\
\mathcal{T}_{0\rightarrow 0} = |V_{00}(\alpha)|^2 = \left| J_0^2(\Theta) + (1+e^{i\alpha})\frac{1-J_0^2(\Theta)}{2} \right|^2 \\
\mathcal{T}_{1\rightarrow 1} = |V_{11}(\alpha)|^2 = \left| e^{i\alpha} J_0^2(\Theta) + (1+e^{i\alpha})\frac{1-J_0^2(\Theta)}{2} \right|^2,
\end{aligned}$$ where $J_k(\Theta)$ is the Bessel function of the first kind. We note that, when $\alpha=\pi$, the elements $\{V_{00},V_{01},V_{10}\}$ are all real and positive, while $V_{11}$ is real and negative—in accord with the ideal Hadamard and leading to destructive HOM interference between the reflect/reflect and transmit/transmit two-photon probability amplitudes. Additionally, these expressions satisfy $\mathcal{R}_{0\rightarrow 1}=\mathcal{R}_{1\rightarrow 0}\equiv \mathcal{R}$ and $\mathcal{T}_{0\rightarrow 0} = \mathcal{T}_{1\rightarrow 1} \equiv \mathcal{T}$. As $\alpha$ is tuned over $0\rightarrow\pi\rightarrow2\pi$, $\mathcal{R}$ follows from 0 to a peak of 0.4781 and back to 0, while $\mathcal{T}$ starts at 1, drops to 0.4979, and returns to 1. The sum $\mathcal{R+T}$ defines the gate success probability, which drops slightly at $\alpha=\pi$ due to the use of single-frequency electro-optic modulation. These particular values are confirmed experimentally in Fig. \[fig2\]a with coherent state measurements [@Lu2018].
**Hong-Ou-Mandel interference.** The generated biphoton frequency comb can be described as a state of the form $$\label{e1}
|\Psi\rangle = \sum_{n=1}^N c_n |1_{\omega_{1-n}}\rangle_A |1_{\omega_{n}}\rangle_B,$$ or in terms of bosonic mode operators, $$\label{e1-1}
\nonumber
|\Psi\rangle = \sum_{n=1}^N c_n \hat{a}_{1-n}^\dagger \hat{a}_{n}^\dagger |\mathrm{vac}\rangle_A |\mathrm{vac}\rangle_B,$$ where $\hat{a}_{n}$ ($\hat{a}_{n}^\dagger$) annihilates (creates) one photon in the frequency bin centered at $\omega_n$. The $A$ and $B$ nomenclature defines the modes held by each of two parties: $A$ consists all $\omega_n$ such that $n\leq 0$, $B$ everything with $n\geq1$. We favor this notation over the more traditional “signal” and “idler” classification because (i) our frequency operations can move photons between $A$ and $B$ mode sets—and indeed *does* in the case of HOM—and (ii) there are no other distinguishing degrees of freedom to label the photons.
Our quantum frequency processor transforms these bins into outputs $\hat{b}_m$ (at frequencies $\omega_m$) via $$\label{e2}
\nonumber
\hat{b}_m = \sum_{n=-\infty}^\infty V_{mn} \hat{a}_n.$$ The matrix $V$ describes the entire operation over all modes. Then at the output we measure the spectrally resolved coincidences between bins $n_A$ and $n_B$, i.e., $$\label{e3}
\nonumber
C_{n_A n_B} = \langle\Psi |\hat{b}_{n_A}^\dagger \hat{b}_{n_B}^\dagger \hat{b}_{n_B} \hat{b}_{n_A}| \Psi\rangle,$$ as well as the singles $$\label{e4}
\nonumber
S_n = \langle\Psi |\hat{b}_{n_A}^\dagger \hat{b}_{n_A}| \Psi\rangle$$
In the case of HOM interference, we filter out all photon pairs except $c_1$ \[Eq. (\[e1\])\], so the input state is $|\Psi\rangle = |1_{\omega_0}\rangle_A |1_{\omega_1}\rangle_B$, which gives $C_{01} = |V_{00}V_{11}+V_{01}V_{10} |^2$ and $S_n= |V_{n0}|^2+|V_{n1}|^2$. In light of the previous discussion on beamsplitter tunability, we thus predict: $$\label{e5}
\nonumber
\begin{aligned}
C_{01}=|\mathcal{R}(\alpha)-\mathcal{T}(\alpha)|^2 \\
S_0 = S_1 = \mathcal{R}(\alpha) + \mathcal{T}(\alpha) \\
S_{-1} = S_2 \approx 1 - \mathcal{R}(\alpha) - \mathcal{T}(\alpha),
\end{aligned}$$ where the nonunity success probability \[$\mathcal{R}(\pi)+\mathcal{T}(\pi)=0.976$\] results in some photons scattering into bins $-1$ and $2$. (Scattering beyond these modes is not observable in experiment, consistent with the theoretical prediction of only $\sim$10$^{-4}$ probability to leave the center four bins.) Invoking the theoretically predicted values for $\mathcal{R}$ and $\mathcal{T}$, we use weighted least-squares to fit the function $f(\alpha)=K_0+K_1 C_{01}(\alpha)$ to the data in Fig. \[fig3\]b and extract the visibility $$\label{e51}
\nonumber
\mathcal{V}=\frac{K_1 \left[C_{01}(0)-C_{01}(\pi) \right]}{2K_0+K_1 \left[C_{01}(0)+C_{01}(\pi) \right]} = 0.94\pm 0.01.$$ Now, because the singles $S_0$ and $S_1$ drop slightly at $\alpha=\pi$ (cf. Fig. \[fig3\]b)—which is not the case in a traditional HOM experiment—we also look at the visibility of the normalized cross-correlation function, $g_{01}^{(2)} = \frac{C_{01}}{S_0 S_1}.$ For in the most pathological case, a reduction in the unnormalized coincidences $C_{01}$ could in principle be due to dropping singles $S_0$ or $S_1$, which would not be surprising from a classical view: if one detector rarely clicks, of course its coincidences with another detector will drop as well. On the other hand, the normalized $g_{01}^{(2)}$ does not suffer from this issue, by accounting for singles counts directly. Accordingly, we repeat the least-squares fit using the theoretically predicted $g_{01}^{(2)}(\alpha)$, along with the measured coincidences (Fig. \[fig3\]b) divided by the product of mode 0 and 1 single counts (Fig. \[fig3\]c). In this more conservative case, we still retrieve $\mathcal{V}=0.94 \pm 0.01$, fully confirming the nonclassicality of our HOM interference.
**Quantum state manipulation.** For the state rotation experiments, we filter out all modes except four, leaving the entangled qubits \[$n=4,5$ in Eq. (\[e1\])\]: $$\label{e6}
|\Psi\rangle = \frac{1}{\sqrt{2}} \left( |1_{\omega_{-3}}\rangle_A |1_{\omega_{4}}\rangle_B + |1_{\omega_{-4}}\rangle_A |1_{\omega_{5}}\rangle_B \right).$$ Ideally, parametric downconversion and filtering should produce this relative phase automatically; but in order to compensate any residual dispersion, we also fine-tune the phase with the BFC pulse shaper, experimentally maximizing spectral correlations in the $H_A \otimes H_B$ measurement case (see below). We have six initially empty modes between those populated in $A$ and $B$, allowing us to apply combinations of Hadamard operations and the identity to each pair of modes—$\{-4,-3\}$ and $\{4,5\}$—without any fear of the photon in $A$ jumping over to $B$’s modes, and vice versa (cf. guardband discussion in ref. ). Accordingly, after the frequency-bin transformation $V$ (chosen to apply the desired joint operation), the coincidence probability for any ($n_A\leq0, n_B\geq 1$) is given by $$\label{e7}
\nonumber
C_{n_A n_B} = \left|V_{n_A,-3} V_{n_B,4} + V_{n_A,-4} V_{n_B,5} \right|^2.$$ This expression accounts for all aspects of the potentially nonideal mode transformation. Focusing on the qubit modes ($n_A\in\{-4,3\}, n_B\in\{4,5\}$), we have the ideal coincidences under all four cases of Fig. \[fig4\] as: $$\label{e8}
\begin{aligned}
V(\mathbbm{1}_A \otimes \mathbbm{1}_B) & \Longrightarrow & C_{n_A n_B}^{\mathbbm{1}_A \otimes \mathbbm{1}_B} & = & \frac{1}{2} & \left( \delta_{n_A,-3}\delta_{n_B,4} + \delta_{n_A,-4}\delta_{n_B,5} \right) \\
V(\mathbbm{1}_A \otimes H_B) & \Longrightarrow & C_{n_A n_B}^{\mathbbm{1}_A \otimes H_B} & = & \frac{1}{4} & \left( \delta_{n_A,-3}\delta_{n_B,4} + \delta_{n_A,-3}\delta_{n_B,5}+ \delta_{n_A,-4}\delta_{n_B,4} + \delta_{n_A,-4}\delta_{n_B,5} \right) \\
V(H_A \otimes \mathbbm{1}_B) & \Longrightarrow & C_{n_A n_B}^{H_A \otimes \mathbbm{1}_B} & = & \frac{1}{4} & \left( \delta_{n_A,-3}\delta_{n_B,4} + \delta_{n_A,-3}\delta_{n_B,5} + \delta_{n_A,-4}\delta_{n_B,4} + \delta_{n_A,-4}\delta_{n_B,5} \right) \\
V(H_A \otimes H_B) & \Longrightarrow & C_{n_A n_B}^{H_A \otimes H_B} & = & \frac{1}{2} & \left( \delta_{n_A,-4}\delta_{n_B,4} + \delta_{n_A,-3}\delta_{n_B,5} \right),
\end{aligned}$$ where $\delta_{nm} = \{1 \text{ if } n=m, 0 \text{ if } n\neq m\}$. These expressions predict perfect negative correlations for the case $\mathbbm{1}_A \otimes \mathbbm{1}_B$—i.e., detecting the low frequency of $A$ occurs in coincidence with the high frequency of $B$, and vice versa—while positive correlations result for $H_A \otimes H_B$. For the other two cases, no frequency correlations are present, with all four combinations equally likely.
The predictions of Eq. (\[e8\]) are precisely those of the EPR paradox [@Einstein1935], particularly the discrete version formulated by Bohm [@Bohm1951]. Applying unitaries $\mathbbm{1}$ and $H$ followed by frequency detection produces measurements of the Pauli $Z$ and $X$ bases, respectively, so that our simultaneous correlations in two mutually unbiased bases reveals the strange characteristics of quantum mechanics, although admittedly without verifying, in our experiments, its nonlocal character. Finally, while we formulate the theory here viewing the operations as *measurements*—as this is most direct in explaining our results—they can equivalently be considered as *gates* on the input state, a useful distinction in considering more complex frequency networks where the photons are processed further before detection.
**Entanglement witness.** The EPR paradox exemplified by the theory of Eq. (\[e8\]) and results in Fig. \[fig4\] can be quantified in terms of an entropic entanglement witness between party $A$ and $B$. Adapting Eq. (329) of ref. to our experiments gives following inequality, satisfied by all separable states: $$\label{e9}
\nonumber
\mathcal{H}(\mathbbm{1}_A|\mathbbm{1}_B) + \mathcal{H}(H_A|H_B) \geq q_{MU},$$ where $\mathcal{H}(U_A|U_B^\prime)$ is the entropy of $A$’s detected frequency bin, given knowledge of $B$’s result, when unitary operations $U$ and $U^\prime$ are applied. (We use the transformations as arguments, rather than the observables per se, for maximal clarity with our experimental configuration.) The Maassen-Uffink bound $q_{MU}$ [@Maassen1988] depends on the overlap between the frequency basis vectors in the rows of $\mathbbm{1_A}$ and $H_A$, which can be written in terms of the beamsplitter reflectivity $\mathcal{R}$ and transmissivity $\mathcal{T}$ (defined above) as $$\label{e10}
\nonumber
q_{MU} = -\log_2 \max \left( \left\{ \frac{\mathcal{R}}{\mathcal{R+T}}, \frac{\mathcal{T}}{\mathcal{R+T}} \right\}\right),$$ where $\{\mathcal{R},\mathcal{T}\}$ are evaluated at $\alpha=\pi$. Because we postselect on coincidences in the two-qubit Hilbert space, we have normalized $\{\mathcal{R},\mathcal{T}\}$ by the success probability; plugging in theoretical values, we obtain $q_{MU}=0.9710$—close to the maximum of 1 for perfect mutually unbiased bases in $d=2$ dimensions ($\log_2 d$).
To calculate the conditional entropies corresponding to the measurements in Fig. \[fig4\], we employ Bayesian mean estimation (BME) on the raw count data [@Blume2010; @Williams2017]. Unlike alternative approaches, such as maximum likelihood estimation, BME yields error bars directly for any computed function and incorporates prior knowledge into the calculation naturally. To produce as conservative an estimate as possible, we make no specifying assumptions about the underlying state. For each situation in Fig. \[fig4\], we posit a three-parameter multinomial likelihood function (four probabilities minus normalization), with counts taken directly from the raw data; we take the prior as uniform. The estimated means and standard deviations of the conditional entropies are then $$\nonumber
\begin{aligned}
\mathcal{H}(\mathbbm{1}_A|\mathbbm{1}_B) & = & &0.19\pm 0.03 \\
\mathcal{H}(H_A|\mathbbm{1}_B) & = & &0.997\pm 0.003 \\
\mathcal{H}(\mathbbm{1}_A|H_B) & = & &0.993\pm 0.005 \\
\mathcal{H}(H_A|H_B) & = & &0.29\pm 0.04. \\
\end{aligned}$$ As expected, the mismatched bases have near-maximal entropy (1 bit), while matched cases are much lower. We emphasize that the full effect of accidentals are included in these numbers; appreciably lower matched entropies may be possible in a model incorporating dark counts as well. Nonetheless, summing these entropies directly gives $\mathcal{H}(\mathbbm{1}_A|\mathbbm{1}_B) + \mathcal{H}(H_A|H_B) = 0.48 \pm 0.05$—violating the bound $q_{MU}$ by 9.8 standard deviations, and thereby confirming the nonseparability of our quantum state with high confidence.
**State reconstruction.** To estimate the complete two-qubit density matrix, we again employ BME [@Blume2010; @Williams2017] but now with the assumption of a single quantum state underlying all four measurements in Fig. \[fig4\]. As noted above, these four combinations are equivalent to joint measurements of the two-qubit observables $\{Z_A \otimes Z_B, X_A\otimes Z_B, Z_A\otimes X_B, X_A\otimes X_B \}$, where the identity $\mathbbm{1}$ permits measurement of $Z$, and the unitary $H$ allows measurement of $X$. As we have no information from the Pauli $Y$ basis, our set of projections is tomographically incomplete. We are nevertheless able to use the information from the measured bases to infer a complete state estimate, with appropriately higher uncertainties in the unmeasured bases. Finally, we emphasize that experimentally we only have access to the detector click (or no-click) events that are more naturally described in terms of positive-operator valued measures (POVMs) rather than von Neumann type projectors on the eigenvectors of Pauli $X$ and $Z$ operators.
For a specific two-qubit observable and chosen pair of frequency bins, we have the POVMs $\Lambda^{(A)}=\{\hat{\Pi}^{(A)}, \mathbbm{1}-\hat{\Pi}^{(A)}\}$ for subsystem $A$, and $\Lambda^{(B)}=\{\hat{\Pi}^{(B)}, \mathbbm{1}-\hat{\Pi}^{(B)}\}$ for subsystem $B$, where $\hat{\Pi}^{(A,B)}$ correspond to photon clicks, $\mathbbm{1}-\hat{\Pi}^{(A,B)}$ to the absence of a click. Absence of a click can be due to detection inefficiency or the photon being in an unmonitored mode. An outcome of a two-qubit POVM $\Lambda^{(A)} \otimes \Lambda^{(B)}$ will fall into one of the three experimentally recorded numbers: coincidence counts ($C_{AB}$), singles counts on detector $A$ ($S_A$), and singles counts on detector $B$ ($S_B$). These form our specific data set $\mathcal{D}=\{C_{AB}, S_A, S_B \}$. In our model, we assume fixed channel efficiencies for $A$ and $B$ propagation and detection ($\eta_A$ and $\eta_B$), and the following normalized probabilities under no loss and perfect detection: $p_{AB}$ (coincidence, one photon in mode $A$ and one photon in mode $B$), $p_{A0}$ (one photon in mode $A$ and no photon in mode $B$), $p_{0B}$ (one photon in mode $B$ and no photon in mode $A$), $p_{00}$ (no photon in mode $A$ or $B$).
Letting $N$ denote the number of photon pairs generated in the measured time interval, we can enumerate the following four experimental possibilities, formed by the products of all operators from this POVM pair. (i) $\hat{\Pi}^{(A)}\otimes \hat{\Pi}^{(B)}$: coincidence between detectors $A$ and $B$. This occurs with probability $\eta_A \eta_B p_{AB}$ and is observed $C_{AB}$ times. (ii) $\hat{\Pi}^{(A)}\otimes [\mathbbm{1}-\hat{\Pi}^{(B)}]$: click on detector $A$, no click on $B$. This occurs with probability $\eta_A[p_{AB}(1-\eta_B) + p_{A0}]$ and is observed $S_A-C_{AB}$ times. (iii) $[\mathbbm{1}-\hat{\Pi}^{(A)}]\otimes \hat{\Pi}^{(B)}$: no click on detector $A$, click on detector $B$. This has probability $\eta_B [p_{AB}(1-\eta_A) + p_{0B}]$ and is observed $S_B-C_{AB}$ times. (iv) $[\mathbbm{1}-\hat{\Pi}^{(A)}]\otimes [\mathbbm{1}-\hat{\Pi}^{(B)}]$: no click on either detector. This occurs with probability $p_{AB}(1-\eta_A)(1-\eta_B) + p_{A0}(1-\eta_A) + p_{0B}(1-\eta_B) + p_{00}$ and is counted $N-S_A-S_B+C_{AB}$ times. Our likelihood function, $P(\mathcal{D}|\alpha)$, is then a multinomial distribution over the aforementioned probabilities and outcomes, where $\alpha=\{\hat{\rho},\eta_A,\eta_B,N\}$ is the underlying parameter set. The idealized probabilities $\{p_{AB}, p_{A0}, p_{0B}, p_{00} \}$ are all functions of the density matrix $\hat{\rho}$, which we limit to physically allowable states [@Williams2017].
Up to this point, we have focused on a *specific* choice of POVMs, $\Lambda^{(A)} \otimes \Lambda^{(B)}$. To account for all 16 POVM combinations (basis pairs and frequency-bin pairs) in the two-qubit space of Fig. \[fig4\], we form the product over all settings, leaving the complete posterior distribution $$\label{b2}
P(\alpha|\bm{\mathcal{D}}) = \frac{\left[ \prod_j P(\mathcal{D}_j|\alpha) \right] P(\alpha)}{P(\bm{\mathcal{D}})},$$ where the bolded $\bm{\mathcal{D}}$ represents the union of the respective results $\mathcal{D}_j$ from each particular setting ($j=1,2,...,16$). Our prior $P(\alpha)$ is taken to be uniform in a Haar-invariant sense, and the marginal $P(\bm{\mathcal{D}})$ is found by integrating the numerator in Eq. (\[b2\]). With this posterior distribution, we can estimate any parameter of interest via integration, such as the mean density matrix $$\label{b3}
\nonumber
\hat{\rho}_\mathrm{BME} = \int d\alpha \; P(\alpha|\bm{\mathcal{D}}) \hat{\rho}.$$ Due to the complexity of integrals of this form, we employ numerical slice sampling for their evaluation [@Neal2003; @Williams2017]. The density matrix estimate is plotted in Extended Data Fig. 1. To highlight BME’s natural treatment of error, we include the standard deviation for each matrix element as well. The error is extremely low for the real elements, due to our complete coverage of the $X$ and $Z$ bases, whereas several of the imaginary components (particularly the center two of the antidiagonal) have significantly higher error; this is a natural consequence of our lack of any measurements in the Pauli $Y$ basis. Nevertheless, physical requirements do bound this error, such that our Bayesian estimate of the fidelity, $\mathcal{F}=\langle\Psi|\hat{\rho}|\Psi\rangle$—with $|\Psi\rangle$ as defined in Eq. (\[e6\])—has extremely low uncertainty: $\mathcal{F}=0.92\pm0.01$. Such high fidelity provides positive corroboration of our frequency-bin control, and is fairly conservative, given that: (i) dark counts are not removed, and thus can degrade the state; and (ii) we lump any imperfections of our $H$ gate rotation onto the state itself, so that impurities in either the input state or quantum frequency processor will contribute to lower $\mathcal{F}$.
![**Reconstructed density matrix.** **a**, Real part of average density matrix. **b**, Imaginary part. **c**, Standard deviations of the real density matrix elements. **d**, Standard deviations of the imaginary elements. Label definitions: $00 \equiv |1_{\omega_{-4}}\rangle_A |1_{\omega_{4}}\rangle_B$, $01 \equiv |1_{\omega_{-4}}\rangle_A |1_{\omega_{5}}\rangle_B$, $10 \equiv |1_{\omega_{-3}}\rangle_A |1_{\omega_{4}}\rangle_B$, $11 \equiv |1_{\omega_{-3}}\rangle_A |1_{\omega_{5}}\rangle_B$.[]{data-label="figE1"}](state_estimation_2.pdf){width="4.5in"}
[36]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****, ()](\doibase
10.1103/RevModPhys.74.145) @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} [****, ()](\doibase 10.1103/PhysRevLett.117.223601) @noop [****, ()]{} [****, ()](http://stacks.iop.org/2040-8986/18/i=6/a=063002) @noop [****, ()]{} [****, ()](\doibase
10.1103/PhysRevX.5.041017) @noop [****, ()]{} [****, ()](\doibase
10.1126/sciadv.1701491) [****, ()](\doibase
10.1103/PhysRevLett.113.053603) [****, ()](\doibase 10.1103/PhysRevLett.91.163602) @noop [****, ()]{} [****, ()](\doibase 10.1126/science.aad8532) [****, ()](\doibase 10.1364/OPTICA.4.000655) [****, ()](http://dx.doi.org/10.1038/nature22986) [****, ()](\doibase 10.1364/OE.26.001825) [****, ()](\doibase 10.1103/PhysRevLett.68.2153) [****, ()](\doibase
10.1038/nature04009) [****, ()](\doibase 10.1103/PhysRevLett.118.023601) in @noop [**]{} (, ) p. [****, ()](\doibase 10.1364/OPTICA.4.000008) [****, ()](\doibase
10.1103/PhysRevLett.120.030502) [****, ()](\doibase
10.1103/PhysRevLett.59.2044) [****, ()](\doibase
10.1103/PhysRev.47.777) [****, ()](\doibase 10.1103/RevModPhys.79.135) @noop [****, ()]{} [****, ()](\doibase 10.1103/RevModPhys.89.015002) [****, ()](\doibase
http://dx.doi.org/10.1016/j.optcom.2011.03.084) @noop [**]{} (, ) [****, ()](\doibase
10.1103/PhysRevLett.60.1103) [****, ()](http://stacks.iop.org/1367-2630/12/i=4/a=043034) [****, ()](http://stacks.iop.org/1367-2630/19/i=4/a=043003) [****, ()](http://www.jstor.org/stable/3448413)
[^1]: These authors contributed equally to this work.
[^2]: These authors contributed equally to this work.
|
---
abstract: 'The structure of large networks can be revealed by partitioning them to smaller parts, which are easier to handle. One of such decompositions is based on $k$–cores, proposed in 1983 by Seidman. In the paper an efficient, $O(m)$, $m$ is the number of lines, algorithm for determining the cores decomposition of a given network is presented.'
author:
- |
Vladimir Batagelj, Matjaž Zaveršnik\
University of Ljubljana, FMF, Department of Mathematics,\
and IMFM Ljubljana, Department of TCS,\
Jadranska ulica 19, 1000 Ljubljana, Slovenia\
`vladimir.batagelj@uni-lj.si`\
`matjaz.zaversnik@fmf.uni-lj.si`
date:
title: |
**An $O(m)$ Algorithm for\
Cores Decomposition of Networks**
---
Introduction
============
“One of the major concerns of social network analysis is identification of cohesive subgroups of actors within a network. Cohesive subgroups are subsets of actors among whom there are relatively strong, direct, intense, frequent, or positive ties” ([@WaF94], p. 249). Several notions were introduced to formally describe cohesive groups: cliques, $n$–cliques, $n$–clans, $n$–clubs, $k$–plexes, $k$–cores, lambda sets, …For most of them it turns out that they are algorithmically difficult (NP hard [@GJ] or at least quadratic), but for cores a very efficient algorithm exists. We describe it in details in this paper.
Cores
=====
The notion of core was introduced by Seidman in 1983 [@SEIDMAN].
Let $G=(V,L)$ be a graph. $V$ is the set of [*vertices*]{} and $L$ is the set of [*lines*]{} ([*edges*]{} or [*arcs*]{}). We will denote $n = |V|$ and $m = |L|$. A subgraph $H=(W,L|W)$ induced by the set $W$ is a $k$-*core* or a *core of order* $k$ iff $\forall v\in W: \deg_H(v)\geq k$ and $H$ is a maximum subgraph with this property. The core of maximum order is also called the *main* core. The *core number* of vertex $v$ is the highest order of a core that contains this vertex.
The degree $\deg(v)$ can be: in-degree, out-degree, in-degree $+$ out-degree, …determining different types of cores.
In figure \[cores\] an example of cores decomposition of a given graph is presented. From this figure we can see the following properties of cores:
![*0, 1, 2 and 3 core*[]{data-label="cores"}](pics/cores.eps){width="80mm"}
- The cores are nested: $i<j \ \ \Longrightarrow \ \ H_j\subseteq H_i$
- Cores are not necessarily connected subgraphs.
Algorithm
=========
Our algorithm for determining the cores hierarchy is based on the following property [@BMCON]:
> If from a given graph $G=(V,L)$ we recursively delete all vertices, and lines incident with them, of degree less than $k$, the remaining graph is the $k$-core.
The outline of the algorithm is as follows:
xxxxxxxxxxxxxxxxxxxx=\
INPUT: graph $G=(V,L)$ represented by lists of neighbors\
OUTPUT: table $core$ with core number for each vertex\
\
1.1compute the degrees of vertices;\
1.2order the set of vertices $V$ in increasing order of their degrees;\
2[**for each**]{} $v\in V$ in the order [**do begin**]{}\
2.1$core\lbrack v\rbrack := degree\lbrack v\rbrack$;\
2.2[**for each**]{} $u \in Neighbors(v)$ [**do**]{}\
2.2.1[**if**]{} $degree\lbrack u\rbrack > degree\lbrack v\rbrack$ [**then begin**]{}\
2.2.1.1$degree\lbrack u\rbrack := degree\lbrack u\rbrack - 1$;\
2.2.1.2reorder $V$ accordingly\
[**end**]{}\
[**end**]{};
In the refinements of the algorithm we have to provide efficient implementations of steps 1.2 and 2.2.1.2.
Detailed Algorithm
==================
We describe an implementation of the algorithm in a Pascal like language.
Structure [`graph`]{} is used to represent a given graph $G=(V,L)$. We will not describe the structure into details, because there are several possibilities, how to do this. We assume that the vertices of $G$ are numbered from 1 to $n$. The user has also to provide functions [`size`]{} and [`in Neighbors`]{}, described in the table:
[`name(parameters)`]{} returned value
--------------------------- --------------------------------------------------------------
[`size(G)`]{} number of vertices in graph $G$
[`u in Neighbors(G,v)`]{} $u$ is a not yet visited neighbor of vertex $v$ in graph $G$
Using an adequate representation of graph $G$ (lists of neighbors) we can implement both functions to run in constant time.
![*Arrays*[]{data-label="arrays"}](pics/arrays.eps){width="12cm"}
Two types of integer arrays ([`tableVert`]{} and [`tableDeg`]{}) are also introduced. Both of them must be of length at least $n$. The only difference is how we index their elements. We start with index 1 in [`tableVert`]{} and with index 0 in [`tableDeg`]{}.
The algorithm is implemented by procedure [`cores`]{}. The input is graph $G$, represented by variable [`g`]{} of type [`graph`]{}, the output is array [`deg`]{} of type [`tableVert`]{} containing core number for each vertex of graph $G$.
01 [**procedure**]{} cores([**var**]{} g: graph; [**var**]{} deg: tableVert);\
02 [**var**]{}\
03 n, d, md, i, start, num: integer;\
04 v, u, w, du, pu, pw: integer;\
05 vert, pos: tableVert;\
06 bin: tableDeg;\
07 [**begin**]{}\
08 n := size(g); md := 0;\
09 [**for**]{} v := 1 [**to**]{} n [**do**]{} [**begin**]{}\
10 d := 0; [**for**]{} u in Neighbors(g, v) [**do**]{} inc(d);\
11 deg\[v\] := d; [**if**]{} d > md [**then**]{} md := d;\
12 [**end**]{};\
13 [**for**]{} d := 0 [**to**]{} md [**do**]{} bin\[d\] := 0;\
14 [**for**]{} v := 1 [**to**]{} n [**do**]{} inc(bin\[deg\[v\]\]);\
15 start := 1;\
16 [**for**]{} d := 0 [**to**]{} md [**do**]{} [**begin**]{}\
17 num := bin\[d\];\
18 bin\[d\] := start;\
19 inc(start, num);\
20 [**end**]{};\
21 [**for**]{} v := 1 [**to**]{} n [**do**]{} [**begin**]{}\
22 pos\[v\] := bin\[deg\[v\]\];\
23 vert\[pos\[v\]\] := v;\
24 inc(bin\[deg\[v\]\]);\
25 [**end**]{};\
26 [**for**]{} d := md [**downto**]{} 1 [**do**]{} bin\[d\] := bin\[d-1\];\
27 bin\[0\] := 1;\
28 [**for**]{} i := 1 [**to**]{} n [**do**]{} [**begin**]{}\
29 v := vert\[i\];\
30 [**for**]{} u in Neighbors(g, v) [**do**]{} [**begin**]{}\
31 [**if**]{} deg\[u\] > deg\[v\] [**then**]{} [**begin**]{}\
32 du := deg\[u\]; pu := pos\[u\];\
33 pw := bin\[du\]; w := vert\[pw\];\
34 [**if**]{} u <> w [**then**]{} [**begin**]{}\
35 pos\[u\] := pw; vert\[pu\] := w;\
36 pos\[w\] := pu; vert\[pw\] := u;\
37 [**end**]{};\
38 inc(bin\[du\]); dec(deg\[u\]);\
39 [**end**]{};\
40 [**end**]{};\
41 [**end**]{};\
42 [**end**]{};
We need (03-06) some integer variables and three additional arrays. Array [`vert`]{} contains the set of vertices, sorted by their degrees. Positions of vertices in array [`vert`]{} are stored in array [`pos`]{}. Array [`bin`]{} contains for each possible degree the position of the first vertex of that degree in array [`vert`]{}. See also Figure \[arrays\] in which a Pascal implementation of our algorithm for the case of simple undirected graph $G=(V,E)$, $E$ is the set of edges, is presented.
In a real implementation of the proposed algorithm dynamically allocated arrays should be used. To simplify our description of the algorithm we replaced them by static.
At the beginning we have to initialize some local variables and arrays (08-12). First we determine [`n`]{}, the number of vertices of graph [`g`]{}. Then [**[*we compute degree for each vertex*]{}**]{} [`v`]{} [**[*in graph*]{}**]{} [`g`]{} and store it into array [`deg`]{}. Simultaneously we also compute the maximum degree [`md`]{}.
After that we [**[*sort the vertices in increasing order of their degrees*]{}**]{} using bin-sort (13-25). First we count (13-14) how many vertices will be in each bin (bin consists of vertices with the same degree). Bins are numbered from 0 to [`md`]{}.
From bin sizes we can determine (15-20) starting positions of bins in array [`vert`]{}. Bin 0 starts at position 1, while other bins start at position, equal to the sum of starting position and size of the previous bin. To avoid additional array we used the same array ([`bin`]{}) to store starting positions of bins. Now we can put (21-25) vertices of graph $G$ into array [`vert`]{}. For each vertex we know to which bin it belongs and what is the starting position of that bin. So we can put vertex to the proper place, remember its position in table [`pos`]{}, and increase the starting position of the bin we used. The vertices are now sorted by their degrees.
In the final step of initialization phase we have to [**[*recover starting positions of the bins*]{}**]{} (26-27). We increased them several times in previous step, when we put vertices into corresponding bins. It is obvious, that the changed starting position is the original starting position of the next bin. To restore the right starting positions we have to move the values in array [`bin`]{} for one position to the right. We also have to reset starting position of bin 0 to value 1.
The [**[*cores decomposition*]{}**]{}, implementing the [**for each**]{} loop from the algorithm described in section 3, is done in the main loop (28-41) that runs over all vertices [`v`]{} of graph [`g`]{} in the order, determined by table [`vert`]{}. The core number of current vertex [`v`]{} is the current degree of that vertex. This number is already stored in table [`deg`]{}. For each neighbor [`u`]{} of vertex [`v`]{} with higher degree we have to decrease its degree and move it for one bin to the left. Moving vertex [`u`]{} for one bin to the left is operation, which can be done in constant time. First we have to swap vertex [`u`]{} and the first vertex in the same bin. In array [`pos`]{} we also have to swap their positions. Finally we increase starting position of the bin (we increase previous and reduce current bin for one element).
Time complexity
---------------
We shall show that the described algorithm runs in time $O(\max(m,n))$.
To compute (08-12) the degrees of all vertices we need time $O(\max(m,n))$ since we have to consider each line at most twice. The [*bin sort*]{} (13-27) consists of five loops of size at most $n$ with constant time $O(1)$ bodies – therefore it runs in time $O(n)$.
The statement (29) requires a constant time and therefore contributes $O(n)$ to the algorithm. The conditional statement (31-39) also runs in constant time. Since it is executed for each edge of $G$ at most twice the contribution of (30-40) in all repetitions of (28-41) is $O(\max(m,n))$.
Summing up — the total time complexity of the algorithm is $O(\max(m,n))$. Note that in a connected network $m \geq n-1$ and therefore $O(\max(m,n)) = O(m)$.
Adaption of the algorithm for directed graphs
---------------------------------------------
For directed simple graphs without loops only few changes in the implementation of the algorithm are needed depending on the interpretation of the *degree*. In the case of in-degree and out-degree the function [`in Neighbors`]{} must return next not yet visited in-neighbor and out-neighbor respectively. If degree is defined as in-degree $+$ out-degree, the maximum degree can be at most $2n-2$. In this case we must provide enough space for table [`bin`]{} ($2n-1$ elements). Function [`in Neighbors`]{} must return next not yet visited in-neighbor or out-neighbor.
Example
=======
We applied the described algorithm for cores decomposition on a network based on the Knuth’s English dictionary [@DICK]. This network has 52652 vertices (English words having 2 to 8 characters) and 89038 edges (two vertices are adjacent, if we can get one word from another by changing, removing or inserting a letter). The obtained network is sparse: density is 0.0000642. The program took on PC only 0.01 sec to compute the core numbers. In the table below the summary results are presented.
----- ------- -------- ------- ---------
$k$ \# % \# %
25 26 0.049 26 0.049
16 34 0.065 60 0.114
15 16 0.030 76 0.144
14 59 0.112 135 0.257
13 82 0.156 217 0.412
12 200 0.380 417 0.792
11 202 0.384 619 1.176
10 465 0.883 1084 2.059
9 504 0.957 1588 3.016
8 923 1.753 2511 4.769
7 1114 2.116 3625 6.885
6 1590 3.020 5215 9.905
5 2423 4.602 7638 14.507
4 3859 7.329 11497 21.836
3 5900 11.206 17397 33.042
2 8391 15.937 25788 48.978
1 13539 25.714 39327 74.693
0 13325 25.308 52652 100.000
----- ------- -------- ------- ---------
Vertices with core number 0 are isolated vertices. Vertices with core number 1 have only one neighbor in the network. The 25-core (main core) consists of 26 vertices, where each vertex has at least 25 neighbors inside the core (obviously this is a clique). The corresponding words are [`a’s`]{}, [`b’s`]{}, [`c’s`]{}, …, [`y’s`]{}, [`z’s`]{}.
The 16-core has additional 34 vertices ([`an`]{}, [`on`]{}, [`ban`]{}, [`bon`]{}, [`can`]{}, [`con`]{}, [`Dan`]{}, [`don`]{}, [`eon`]{}, [`fan`]{}, [`gon`]{}, [`Han`]{}, [`hon`]{}, [`Ian`]{}, [`ion`]{}, [`Jan`]{}, [`Jon`]{}, [`man`]{}, [`Nan`]{}, [`non`]{}, [`pan`]{}, [`pon`]{}, [`ran`]{}, [`Ron`]{}, [`San`]{}, [`son`]{}, [`tan`]{}, [`ton`]{}, [`van`]{}, [`von`]{}, [`wan`]{}, [`won`]{}, [`yon`]{}, [`Zan`]{}). There are no edges between vertices with core number 25 and vertices with core number 16. The adjacency matrix of the subgraph induced by these 34 vertices is presented on figure \[core16\]. In this matrix we can see two 17-cliques and some additional edges.
![*Adjacency matrix of 16-core without 25-core*[]{data-label="core16"}](pics/core16.eps){width="100mm"}
The 15-core has additional 16 vertices ([`ow`]{}, [`bow`]{}, [`cow`]{}, [`Dow`]{}, [`how`]{}, [`jow`]{}, [`low`]{}, [`mow`]{}, [`now`]{}, [`pow`]{}, [`row`]{}, [`sow`]{}, [`tow`]{}, [`vow`]{}, [`wow`]{}, [`yow`]{}). This is a clique again, because only the first letters of the words are different.
Conclusion
==========
The cores, because they can be efficiently determined, are one among few concepts that provide us with meaningful decompositions of large networks. We expect that different approaches to the analysis of large networks can be built on this basis. For example, the sequence of vertices in sequential coloring can be determined by their core numbers (combined with their degrees). Cores can also be used to reveal interesting subnetworks in large networks [@BMZ; @ERDOS].
The described algorithm is implemented in program for large networks analysis [`Pajek`]{} (Slovene word for Spider) for Windows (32 bit) [@BMCON]. It is freely available, for noncommercial use, at its homepage:
[`http://vlado.fmf.uni-lj.si/pub/networks/pajek/`]{}
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was supported by the Ministry of Education, Science and Sport of Slovenia, Project J1-8532. It is a detailed version of the part of the talk presented at *Recent Trends in Graph Theory, Algebraic Combinatorics, and Graph Algorithms*, September 24–27, 2001, Bled, Slovenia,
[99]{}
BATAGELJ, V. & MRVAR, A. (1998). Pajek – A Program for Large Network Analysis. *Connections* **21 (2)**, 47–57.
BATAGELJ, V. & MRVAR, A. (2000). Some Analyses of Erdős Collaboration Graph. *Social Networks* **22**, 173–186.
BATAGELJ, V., MRVAR, A. & ZAVERŠNIK, M. (1999). Partitioning approach to visualization of large graphs. In KRATOCHVÍL, Jan (ed.). Proceedings of 7th International Symposium on Graph Drawing, September 15-19, 1999, Štiřín Castle, Czech Republic. (Lecture notes in computer science, 1731). Berlin \[etc.\]: Springer, 90–97.
GAREY, M. R. & JOHNSON, D. S. (1979). *Computer and intractability*. San Francisco: Freeman.
KNUTH, D. E. (1992). Dictionaries of English words.\
[`ftp://labrea.stanford.edu/pub/dict/`]{} .
SEIDMAN, S. B. (1983). Network structure and minimum degree. *Social Networks* **5**, 269–287.
WASSERMAN, S. & FAUST, K. (1994). *Social Network Analysis: Methods and Applications*. Cambridge: Cambridge University Press.
|
---
abstract: 'The efficient certification of nonclassical effects of light forms the basis for applications in optical quantum technologies. We derive general correlation conditions for the verification of nonclassical light based on multiplexed detection. The obtained nonclassicality criteria are valid for imperfectly balanced multiplexing scenarios with on-off detectors and do not require any knowledge about the detector system. In this sense, they are fully independent of the detector system. In our experiment, we study light emitted by clusters of single-photon emitters, whose photon number may exceed the number of detection channels. Even under such conditions, our criteria certify nonclassicality with high statistical significance.'
author:
- Martin Bohmann
- Luo Qi
- Werner Vogel
- Maria Chekhova
title: 'Detection-device-independent verification of nonclassical light'
---
Introduction
============
The verification of quantum correlations in optical systems is a key task in quantum optics. Besides its fundamental significance for the understanding of radiation fields, the identification of genuine quantum features is becoming ever more important as they can be used for applications in quantum technologies [@Braunstein05; @Kok07; @Gisin07]. A major goal, in this context, is to develop robust methods which ideally do not rely on any knowledge or assumptions about the studied system, leading to the concept of device-independent quantum characterization [@Branciard13; @Lim14; @Gheorghiu15; @Altorio16; @Sperling17].
An important task is the characterization of light in the few-photon regime. For the analysis of quantum light in this regime, so-called multiplexing strategies [@Paul96; @Kok01; @Banaszek03; @Rehavcek03; @Fitch03; @Achilles03; @Castelletto07; @Schettini07; @Lundeen09; @Matthews16] have been developed as a way of gaining insights in the measured quantum state even when a photon-number-resolving measurement is not accessible. Such strategies do not provide a direct access to the photon-number distribution and consequently the interpretation of the measurement statistics as the photon-number statistics can lead to a false certification of nonclassicality [@Sperling12]. However, nonclassicality criteria which can be directly applied to the recorded click-counting statistics have been formulated [@Sperling12a; @Sperling13; @Luis15; @Filip13; @Lachman16; @Lee16]. In particular, such criteria are very efficient and successful in certifying nonclassicality from experimental data [@Bartley13; @Sperling15; @Heilmann16; @Sperling16; @Bohmann17b; @Bohmann18; @Obsil18; @Tiedau18].
One common assumption for such conditions is that the incoming light is equally split and detected in each detection channel. Recently, the detector-independent verification of quantum light for such equal splitting has been reported [@Sperling17; @Sperling17b]. In some cases, however, an equal splitting ratio might be hard to realize and requires the careful characterization of the optical elements. Furthermore, other multiplexing strategies such as fiber-loop detectors [@Banaszek03; @Tiedau18] by design do not provide an equal splitting. For such unequal-splitting scenarios, a condition based on second-order moments [@Lee16] has been derived as a generalization of the corresponding equal-splitting condition [@Sperling12a]. More general higher order criteria have, however, not yet been reported for such unequal-splitting detections.
In this paper, we introduce detector-independent general (higher order) conditions for the certification of nonclassical light measured with unbalanced multiplexing schemes and on-off detectors. The presented conditions are fully independent of the properties of the used detection scheme. Based on Chebyshev’s integral inequality, we derive a family of inequality conditions for the no-click events at the output channels which have to be fulfilled for any classical radiation fields; their violation verifies nonclassicality. The so-obtained inequalities include simple covariance conditions between two detection channels and more general higher order correlation conditions. Our approach is based on minimal assumptions and requirements which guarantees the applicability to any multiplexing setup, even without the knowledge about the used detectors and the splitting ratios. We demonstrate the strength of the obtained criteria by certifying nonclassicality of light from clusters of single-photon emitters with high statistical significance. The relations of the presented nonclassicality certifiers to other nonclassicality criteria based on the Mandel $Q$ parameter and the matrix of moments approach are discussed.
Multiplexing detection
======================
We are interested in the certification of quantum correlations based on general multiplexing scenarios, as schematically sketched in Fig. \[fig:Multiplexing\]. The incident quantum state of light is (unequally) split into $N$ output channels. Each of these channels is then measured by a single on-off detector, which is the standard working principle of multiplexing detectors [@Paul96; @Kok01; @Banaszek03; @Rehavcek03; @Fitch03; @Achilles03; @Castelletto07; @Schettini07; @Lundeen09; @Matthews16].
![ Working principle of a multiplexing device. The input quantum state $\hat \rho$ is split at an unbalanced multiport splitter and each output channel is measured with an on-off detector. []{data-label="fig:Multiplexing"}](Multiplexing.pdf){width="0.75\columnwidth"}
Let us describe this multiplexing step formally. We express the input quantum state in terms of the Glauber Sudarshan $P$ representation [@Glauber63; @Sudarshan63], $$\begin{aligned}
\hat \rho_{\mathrm{in}}=\int d^2\alpha P(\alpha)|\alpha\rangle\langle\alpha|,
\end{aligned}$$ where $|\alpha\rangle$ is a coherent state. A quantum state is called classical if and only if (iff) its $P$ function is non-negative [@Titulaer65; @Mandel86]. Multiplexing devices act as $N{\times} N$ multimode splitter and can be described via the unitary operation $\mathbf{U}(N)= (u_{i,j})_{i,j=1}^{N}$ which relates the input to the output field operators via $\hat{\mathbf{a}}_{\mathrm{out}}{=}\mathbf{U}(N)\hat{\mathbf{a}}_{\mathrm{in}}$ and $\hat{\mathbf{a}}_{\mathrm{in(out)}}{=}(\hat a_{1\mathrm{in(out)}},\dots,\hat a_{N\mathrm{in(out)}})$. In the multiplexing case, only the first mode is occupied and the other ones are in the vacuum state. Consequently, the output quantum state can be written as $$\begin{aligned}
\label{eq:Pout}
\hat \rho_{\mathrm{out}}{=}\int d^2\alpha P(\alpha)|u_{1,1}\alpha,\dots,u_{1,N}\alpha\rangle\langle u_{1,1}\alpha,\dots,u_{1,N}\alpha|,
\end{aligned}$$ with $\sum_k |u_{1,k}|^2{=}1$ [@Note1]. We explicitly do not restrict our consideration to the case of uniform splitting, i.e., $\forall k:\quad|u_{1,k}|^2=1/N$.
We are now interested in the detector-click probability in each channel. The probability of detecting no click in the $k$-th channel is given by the expectation value $$\begin{aligned}
\langle{:} \hat m_k {:}\rangle=\int d^2\alpha P(\alpha)\langle u_{1,k}\alpha|{:}\hat m_k{:}|u_{1,k}\alpha\rangle,
\end{aligned}$$ where ${:}\dots {:}$ denotes the normal-ordering prescription; cf. [@Vogel06]. The corresponding operator is defined as $\hat m_{k}{=}e^{-\hat \Gamma_k(\hat n_k)}$, the subscript $k$ indicates the detection channel and $\hat n_k$ is the photon-number operator in the corresponding channel. The probability of obtaining a click in this channel is given by $1{-}\langle{:}\hat m_k{:}\rangle$. The detector response function $\hat \Gamma_k(\hat n_k)$ is a function of $\hat n_k$ and describes the connection between the electromagnetic field and the generation of a click [@Kelly64; @Vogel06]. The detector response function can be determined via direct calibration techniques; see, e.g., [@Bohmann17a].
Conditions for quantum correlations
===================================
We aim at formulating nonclassicality conditions based on the correlations between the no-click events of the different output channels which do not depend on the characteristics of the used multiplexing architecture and detectors. The simplest case we can consider in this context is the correlation between the no-click events of two output channels [@Note2]. In fact, we can use Chebyshev’s integral inequality (see, e.g., [@Mitrinovic70]) to derive the simple condition, $$\begin{aligned}
\label{eq:covar}
\langle{:}\mathrm{Cov}(\hat m_i, \hat m_j){:}\rangle=\langle {:} \hat m_i \hat m_j {:} \rangle - \langle {:} \hat m_i {:}\rangle \langle {:} \hat m_j {:}\rangle \stackrel{\mathrm{cl}}\geq 0,
\end{aligned}$$ which must hold for any classical input state, i.e., a quantum state with a non-negative $P$ function. Details on the derivation are provided in the Appendix \[sec:derivation\]. The violation of this inequality is a direct and experimentally easily accessible signature of nonclassicality and is directly related to the negativities of the $P$ function of the studied state. In particular, only one multiplexing step and on-off detection is sufficient for the application of condition .
Condition has a clear physical interpretation. Non-negative normal-ordered covariances can be explained in terms of a classical description of the measured radiation field, i.e., by a classical $P$ function. In particular, for an input coherent state, the no-click events are uncorrelated \[$\langle{:}\mathrm{Cov}(\hat m_i, \hat m_j){:}\rangle{=}0$\], which represents the boundary between classical and nonclassical radiation fields. On the other hand, an anticorrelation, i.e., a negative covariance of the no-click events, can only arise from negativities in the $P$ function of the considered state. For example, a single-photon input state leads to an anticorrelation of the no-click events which is revealed by a negative covariance.
We can further generalize condition to multimode higher order moment conditions. Again by making use of Chebyshev’s integral inequality [@Mitrinovic70], we formulate the family of correlation conditions (see Appendix \[sec:derivation\] for details) $$\begin{aligned}
\label{eq:partitions}
\langle {:} \hat m_{{\mathcal{I}}_1} \dots \hat m_{{\mathcal{I}}_K} {:}\rangle - \langle {:} \hat m_{{\mathcal{I}}_1} {:}\rangle \cdots \langle {:} \hat m_{{\mathcal{I}}_K} {:}\rangle \stackrel{\mathrm{cl}}\geq 0,
\end{aligned}$$ where $\mathcal{I}_1\dots\mathcal{I}_{K}$ are mutually disjoint subsets (partitions) of $\mathcal{I}=\{1,\dots,N\}$ and $\hat m_J$ is the no-click operator for all detection channels in $\mathcal{I}_J$, $\hat m_{\mathcal{I}_J}{=}\prod_{j\in \mathcal{I}_J}\hat m_j$. These general conditions also include asymmetric partitions and the clustering of channels. Note that Eq. generalizes the approach in [@Lachman16] to unequal splitting, uncharacterized detectors, and arbitrary partitions. In Appendix \[sec:derivation\], we show that the conditions and are not affected by dark counts or other uncorrelated noise.
Example
=======
Before turning to the experiment and the data analysis, let us consider an example. As an input state, we choose an $n$-photon-added thermal state $\mathcal{N}_n\left(\hat a^\dagger\right)^n\hat\rho_{\mathrm{th}}\hat a^n$, where the thermal state is $\hat\rho_{\mathrm{th}}=1/(\overline{n}+1)\sum_{k=0}^\infty \left(\overline{n}/(\overline{n}+1)\right)^k |k\rangle\langle k|$ with $\mathcal{N}_n$ being the normalization constant [@Agarwal92]. Such states have been realized in experiments [@Zavatta04; @Kiesel08]. We consider a single unbalanced multiplexing step (beam splitter) with an intensity splitting of $70{:}30$ and a detection efficiency of $0.7$. By violating the covariance inequality , we can certify nonclassicality. We compare this nonideal detection scheme with a classicality condition based on the photon-number covariance, $$\begin{aligned}
\label{eq:ncovar}
\langle \hat n_i\hat n_j \rangle-\langle \hat n_i \rangle\langle \hat n_j \rangle\stackrel{\mathrm{cl}}\geq 0,
\end{aligned}$$ where $\hat n_j$ is the photon-number operator in the $j$th detection channel. The application of this condition would require experimental access to the second-order moments of the photon-number operator. The violation of this condition corresponds to the nonclassicality condition in terms of the second-order intensity correlation function, $g^{(2)}(0)<1$, which is closely related to the sub-Poissonian photon statistics and the Mandel $Q$ parameter [@Mandel79].
![ The no-click-covariance condition (solid lines; scaled by a factor of $5$) and the photon-number covariance \[Eq. \] (dashed) are shown for one- and two-photon-added thermal states in dependence on the mean thermal photon number $\overline{n}$. []{data-label="fig:mpats"}](PlotTheory.pdf){width="0.8\columnwidth"}
In Fig. \[fig:mpats\], the behavior of the two conditions is shown for one and two-photon-added thermal states in dependence on the thermal photon number. The violation of the no-click inequality detects nonclassicality in a wider range of $\overline{n}$ than corresponding violation of the photon-number condition . Similar behaviors have been observed for the sampling of phase-space distributions from multiplexing detection data [@Luis15; @Bohmann18].
An explanation for this behavior can be found when one considers the measurement operator of the click detection, i.e., the no-click operator $\hat m_k$. This operator is an exponential function of the photon-number operator and, thus, higher order moments of the photon-number operator contribute to the condition . Therefore, the click detection may be more sensitive toward nonclassical effects than the detection of the first two moments of the photon-number operator in Eq. .
Experiment
==========
In our experimental setup, we study light from clusters of single-photon emitters which we detect with the help of time-bin multiplexed click detection. We used multiphoton light emitted by clusters of colloidal CdSe/CdS quantum ”dot in rods” (DRs) [@Carbone2007; @Pisanello2010; @Manceau2014; @Vezzoli2015] coated on a fused silica substrate and excited by picosecond pulses at $355$ nm. For a detailed description of the experiment, see Ref. [@Qi2018]. To get rid of the pump radiation, the emitted light was filtered using a long-pass filter and a band-pass filter (center wavelength $607$ nm, bandwidth $42$ nm). The size of the cluster was determined by assuming that the mean number of photons emitted per pulse scales with the number of emitters in the cluster. In this way, we obtained clusters with an effective size between $2$ and $14$ emitters. Each DR in such a cluster, provided that it is excited, emits a quantum state that is close to a single-photon one, with an extremely small admixture of a two-photon component. Taking into account the $25\%$ excitation probability per excitation pulse, the state emitted by a single DR can be written as $\hat \rho_{\mathrm{DR}}{=}p_0|0\rangle\langle0|+p_1|1\rangle\langle1|+p_2|2\rangle\langle2|$, where $p_0{\approx}0.9$, $p_1{\approx}0.1$, and $p_2{\approx}10^{-4}$. The low probability of two-photon emission leads to strongly nonclassical $g^{(2)}(0){\le}0.05$. Although different DRs in a cluster emit incoherently, the resulting state manifests nonclassicality because the total number of emitted photons is restricted according to the size of the cluster [@Shcherbina2014; @Qi2018].
The radiation emitted by a single cluster was collected with an efficiency of $44\%$, which takes into account the losses at all optical elements, and sent to a fiber multiplexed detection setup, cf. Fig. \[fig:setup\], comprising two click detectors based on avalanche photodiodes, with the quantum efficiency $60\%$. The use of fiber loops provided two time bins for each detector, and therefore the setup was equivalent to the one shown in Fig. \[fig:Multiplexing\], with $N{=}4$ channels. For each of the studied clusters, we collected a dataset containing between $10^7$ and $10^8$ pulses, and for each pulse, the number of click counts in each channel was registered. Depending on the size of the cluster, the mean click number per pulse was between 0.05 and 0.1 due to the low excitation and detection efficiency.
![ Principal scheme of the setup. The pump radiation ($355$ nm) is focused into a cluster of DRs through objective lens O1 and then cut off by the filters F. The radiation emitted by the cluster is coupled into an objective lens O2 and, after filtering, is sent into the fiber-assisted multiplexed detection setup where two detectors D1, D2 and two different path lengths create four detection channels, C1, $\dots$, C4. []{data-label="fig:setup"}](setup.pdf){width="0.9\columnwidth"}
Results
=======
We apply the nonclassicality criteria based on the inequalities and to the multiplexing data obtained for the different cluster sizes. Our approach can be directly applied to the measured data without the knowledge of the detection system or data post-processing. We analyze the correlation condition between two of the detection channels (first and third) and the condition corresponding to the full partition of all four detection channels \[cf. Eq. \] for the different cluster sizes. The no-click moments and their statistical errors can be directly sampled from the measurement statistics [@Sperling15].
![ a) The covariance condition between two detection channels (upper, orange markers) and the condition based on the full partition of all channels, cf. Eq. (lower, green markers), scaled by a factor of $10^4$, are shown together with their sampling errors (gray bars) for different cluster sizes. The dashed and dotted lines represent the corresponding model, Eqs. and with $\eta=0.009$ and different $M$ (the lines should guide the eye). b) The behavior for higher numbers of emitters following the theoretical model. The nonclassicality conditions only approach the classical limit for high number of emitters due to saturation of all detectors. []{data-label="fig:data1"}](DataPlot2.pdf){width="0.8\columnwidth"}
The results are shown in Fig. \[fig:data1\] a). Even the simple covariance condition between two of the output channels is capable of certifying nonclassicality with statistical significances (estimated value divided by its statistical sampling error) above three standard derivations for all cluster sizes. Furthermore, we identify that the condition based on the full partition of the no-click operator into four parts \[cf. Eq. \] even yields verifications with higher statistical significances of up to $29.3$ standard deviations. Thus, the used conditions provide robust and easily applicable approaches for the faithful detector-independent verification of nonclassical light. Importantly, we can certify nonclassicality from the measurement outcomes of an uncharacterized detection device even if no assumption on the quantum state of the measured light is used.
Let us discuss the dependence on the cluster size. We observe that for both conditions the distance to the classical limit (zero) increases with the number of emitters. At first glance, this is a surprising observation as the nonclassicality of the recorded light stems from the single-photon characteristics (”antibunching”) of the light from individual emitters. Therefore, one would expect that increasing the number of incoherently emitting quantum dots would suppress the nonclassical character and that the nonclassicality conditions eventually approach the classical limit. However, the relatively low efficiency (excitation, collection, detection) of the whole experiment leads to an initial increase of the distance to the classical limit with increasing numbers of emitters.
The following model can explain this. We assume that each excited quantum dot emits a single photon and all losses and the non-unit excitation probability can be summarized by an overall quantum efficiency $\eta$. The light emitted by a cluster of $M$ incoherently emitting quantum dots is an $M$-mode tensor-product state of single-photon states. Furthermore, we assume for simplicity symmetric multiplexing, and that the $M$-mode state is mode-insensitively recorded by the detectors. This yield the expressions of the conditions and , $$\begin{aligned}
&\langle{:}\mathrm{Cov}(\hat m_1, \hat m_3){:}\rangle=\left(1-\frac{\eta}{2}\right)^M-\left(1-\frac{\eta}{4}\right)^{2M}\quad \text{and}\label{eq:model1}
\\
&\langle {:} \hat m_{1} \dots \hat m_{4} {:}\rangle - \prod_{i=1}^{4} \langle {:} \hat m_{i} {:}\rangle= \left(1-\eta\right)^M-\left(1-\frac{\eta}{4}\right)^{4M}\label{eq:model2}
\end{aligned}$$ respectively; cf. Appendix \[sec:model\] for more details. This simple model captures the dependence on the cluster size \[Fig. \[fig:data1\] a)\] and explains that for the relatively low efficiency of the experiment ($\eta\approx0.009$) both conditions become more negative with increasing cluster size in the considered interval. Additionally, the model can be used to predict the further behavior for higher numbers of emitters \[see Fig. \[fig:data1\] b)\], where we in fact observe that for a large number of emitters the conditions approach the classical limit. The overall efficiency $\eta$ determines how fast the classical limit is reached. Examples with other efficiencies are provided in Appendix \[sec:model\]. Furthermore, we can conclude that with the proposed criteria it would be possible to certify nonclassicality of the light emitted by clusters of several hundreds of emitters with the used experimental setup.
Discussion
==========
The obtained covariance condition is closely connected to the Mandel $Q$ parameter [@Mandel79] and related second-order moment conditions such as the sub-binomial $Q_{\mathrm{B}}$ [@Sperling12a] and sub-Poisson-binomial $Q_{\mathrm{PB}}$ [@Lee16] parameters. In fact, we show in Appendix \[sec:relation\] that these criteria can be traced back to condition . Thus, we could identify the covariance condition as the fundamental building block of these other nonclassicality criteria.
Let us now consider conditions including higher order moments, such as in Eq. . This includes also asymmetric partitions of the corresponding no-click operators, i.e., partitions in which different parts may have different number of elements, such as, e.g., the partition $\langle {:} \hat m_{1}\hat m_{2} {:}\rangle \langle {:} \hat m_{3} {:}\rangle$. Other approaches [@Sperling13; @Sperling15; @Heilmann16; @Sperling17] are based on the matrix of moments which, by construction, cannot involve asymmetric-partition conditions. In Appendix \[sec:relation\], a comparison of these methods with the derived conditions is presented. Hence, the here derived conditions are by construction different from already existing approaches and provide a wider applicability [@Note3].
A detector-independent method for the certification of quantum light through multiplexing was already introduced [@Sperling17]. In that work, the detector independence refers solely to the detectors and not to the whole detection system, including the multiplexing, as an equal splitting into the detection channels is required. With our approach, we can relax the latter requirement and, hence, have a condition which is fully independent of the whole detection scheme.
Furthermore, we would like to discuss the relation of the introduced nonclassicality conditions to entanglement. The derived conditions can certify nonclassicality of the input quantum state of a multiplexing device. In Sec. IV of Ref. [@Vogel14] it was undoubtedly demonstrated that the multiplexing of a nonclassical state of light yields multipartite entangled states. This is just what happens in our experiment. However, the direct certification of the entanglement requires an extension of the measurement setup, which is beyond the scope of this paper. Although we are testing nonclassicality of the input state, our conditions in fact reveal quantum correlations between the different detection channels. Hence, the obtained nonclassicality conditions are closely related to multimode entanglement. This opens possibilities for future applications in quantum technologies.
Conclusions
===========
We formulated conditions for the verification of nonclassical light detected by general multiplexing setups. The obtained criteria do not require any knowledge about splitting ratios and used detectors. Hence, our methods are fully independent of the detection scheme. The obtained correlation criteria include conditions based on second-order and higher order moments of the no-click events. We demonstrated the strength of our approach by certifying nonclassicality of light emitted by clusters of single-photon emitters. Importantly, the presented criteria are capable of detecting quantum light even if the number of photon emitters is higher than the number of used detection channels.
We could show that our conditions based on on-off detectors can provide even more insight into the nonclassical character of the recorded light than comparable approaches based on ideal photon-number-resolving measurements. Furthermore, we discussed the relation to established nonclassicality indicators and showed that our approach provides new forms of nonclassicality conditions which cannot be deduced from other existing criteria. The present results provide useful and simple tools for the detector-independent verification of quantum light, applicable to many experimental scenarios.
This work has been supported by Deutsche Forschungsgemeinschaft through Grant No. VO 501/22-2. MB acknowledges financial support by the Leopoldina Fellowship Programme of the German National Academy of Science (LPDS 2019-01) and thanks Valentin Gebhart and Elizabeth Agudelo for helpful comments.
Derivation of the nonclassicality conditions {#sec:derivation}
============================================
Here we will show how the introduced nonclassicality conditions can be derived by using Chebyshev’s integral inequality. Furthermore, we demonstrate that the results of the obtained conditions are not influenced by uncorrelated noise sources.
Chebyshev’s integral inequality
-------------------------------
In order to construct the nonclassicality conditions we will make use of Chebyshev’s integral inequality; see, e.g., [@Mitrinovic70]. Let $f$ and $g$ be two functions which are integrable and monotone in the same sense on ($a,b$) and let $p$ be a positive and integrable function on the same interval. Then the Chebyshev’s integral inequality $$\begin{aligned}
\label{eq:Chebyshev}
\int_a^b p(x) f(x) g(x)dx \int_a^b p(x) dx \geq
\int_a^b p(x) f(x)dx \int_a^b p(x) g(x)dx,
\end{aligned}$$ holds. In the following, we will see that we can make use of this inequality for the derivation of the nonclassicality conditions. In our case, the $p(x)$ will be the (phase-averaged) $P$ function of a classical quantum state and $f$, $g$ will be the expectation values of the normal-ordered no-click operators with coherent states.
Two-channel no-click correlation
--------------------------------
The starting point of this consideration is the multimode state after the multiplexing step, $$\begin{aligned}
\hat \rho_{\mathrm{out}}=\int d^2\alpha P(\alpha)|u_{1,1}\alpha,\dots,u_{1,N}\alpha\rangle\langle u_{1,1}\alpha,\dots,u_{1,N}\alpha|,
\end{aligned}$$ with $\sum_k |u_{1,k}|^2=1$. We consider the no-click operators $\hat m_i=e^{-\Gamma(\hat n_i)}$ whose normally ordered expectation values with coherent states, $\langle \beta| {:}\hat m_i|\beta{:}\rangle=\langle \beta| {:}e^{-\Gamma(\hat n_i)} {:}|\beta\rangle=e^{-\Gamma(|\beta|^2)}$, are monotonically decreasing functions of $|\beta|$. A typical example of a detector response function is a linear response function $ \Gamma=\eta_i|\beta|^2+\nu_i$, where $\eta_i$ and $\nu_i$ are the quantum efficiency and the dark-count rate in the $i$th mode, respectively.
Then, the normal-ordered expectation value of the product of the two no-click operators $\hat m_i$ and $\hat m_j$ reads as $$\begin{aligned}
\langle{:}\hat m_i \hat m_j{:}\rangle=\mathrm{Tr}[\hat \rho_{\mathrm{out}}{:}\hat m_i \hat m_j{:}]=\int d^2\alpha P(\alpha) e^{-\Gamma_i(|u_{1,i}|^2|\alpha|^2)}e^{-\Gamma_j(|u_{1,j}|^2|\alpha|^2)}.
\end{aligned}$$ As both functions $e^{-\Gamma(|u_{1,i}|^2|\alpha|^2)}$ and $e^{-\Gamma(|u_{1,i}|^2|\alpha|^2)}$ are monotonically decreasing functions of $|\alpha|^2$ and we assume a non-negative (classical) $P$ distribution, we can apply Chebyshev’s integral inequality and obtain $$\begin{aligned}
\int d^2\alpha P(\alpha) e^{-\Gamma_i(|u_i|^2|\alpha|^2)}e^{-\Gamma_j(|u_j|^2|\alpha|^2)} \geq \int d^2\alpha P(\alpha) e^{-\Gamma_i(|u_i|^2|\alpha|^2)} \int d^2\alpha P(\alpha) e^{-\Gamma_j(|u_j|^2|\alpha|^2)},
\end{aligned}$$ which can be written in terms of the covariance, $$\begin{aligned}
\langle{:}\mathrm{Cov}(\hat m_i, \hat m_j){:}\rangle=\langle{:}\hat m_i \hat m_j{:}\rangle - \langle{:}\hat m_i {:}\rangle\langle{:} \hat m_j{:}\rangle \stackrel{\mathrm{cl}}\geq 0.
\end{aligned}$$ This inequality has to hold for any non-negative (classical) $P$ and, thus, a violation of the inequality immediately uncovers nonclassicality of the considered state. Let us stress that the derivation does not rely on the knowledge of the splitting ratios and the properties of the detectors.
Multimode generalization
------------------------
Here, we will show how the multimode generalizations of the two-mode covariance condition can be obtained by applying Chebyshev’s integral inequality several times. To derive the multimode conditions for $N$ detection modes, we make several times use of Chebyshev’s integral inequality. We start from the expectation value $$\begin{aligned}
\langle{:}\prod_{i=1}^N\hat m_i {:}\rangle=\int d^2\alpha P(\alpha) \prod_{i=1}^N e^{-\Gamma_i(|u_{1,i}|^2|\alpha|^2)},
\end{aligned}$$ which can be written as $$\begin{aligned}
\langle{:}\prod_{i=1}^N\hat m_i {:}\rangle=\int d^2\alpha P(\alpha) e^{-\sum_{i=1}^N\Gamma_i(|u_{1,i}|^2|\alpha|^2)}=\int d^2\alpha P(\alpha) e^{-\sum_{i\in \mathcal{I}}\Gamma_i(|u_{1,i}|^2|\alpha|^2)}e^{-\sum_{i\in \mathcal{\overline{I}}}\Gamma_i(|u_{1,i}|^2|\alpha|^2)}
\end{aligned}$$ where $\mathcal{I}$ and $ \mathcal{\overline{I}}$ are two bipartitions of the considered operator functions. Note that both $e^{-\sum_{i\in \mathcal{\overline{I}}}\Gamma_i(|u_{1,i}|^2|\alpha|^2)}$ and $e^{-\sum_{i\in \mathcal{I}}\Gamma_i(|u_{1,i}|^2|\alpha|^2)}$ are monotonically decreasing functions. As in the case above, we can now apply Chebyshev’s integral inequality which yields $$\begin{aligned}
\int d^2\alpha P(\alpha) e^{-\sum_{i=1}^N\Gamma_i(|u_{1,i}|^2|\alpha|^2)}\geq \int d^2\alpha P(\alpha) e^{-\sum_{i\in \mathcal{I}}\Gamma_i(|u_{1,i}|^2|\alpha|^2)} \int d^2\alpha P(\alpha) e^{-\sum_{i\in \mathcal{\overline{I}}}\Gamma_i(|u_{1,i}|^2|\alpha|^2)},
\end{aligned}$$ which holds for any non-negative $P$ function. This may also be written as $\langle{:}\prod_{i=1}^N\hat m_i {:}\rangle\geq \langle{:}\prod_{i\in \mathcal{J}}\hat m_i{:}\rangle\langle {:}\prod_{j\in \mathcal{\overline{J}}}\hat m_j{:}\rangle$. This procedure can be repeated several times which leads to the general form $$\begin{aligned}
\langle {:} \hat m_{{\mathcal{I}}_1} \dots \hat m_{{\mathcal{I}}_K} {:}\rangle - \langle {:} \hat m_{{\mathcal{I}}_1} {:}\rangle \cdots \langle {:} \hat m_{{\mathcal{I}}_K} {:}\rangle \stackrel{\mathrm{cl}}\geq 0,
\end{aligned}$$ where $\mathcal{I}_1\dots\mathcal{I}_{K}$ are mutually disjoint subsets (partitions) of $\mathcal{I}=\{1,\dots,N\}$ and $\hat m_J$ is the no-click operator for all detection channels in $\mathcal{I}_J$, $\hat m_{\mathcal{I}_{J}}{=}\prod_{j\in \mathcal{I}_{J}}\hat m_j$. A full partition of the non-click operators yields the condition $$\begin{aligned}
\label{eq:full}
\langle {:} \hat m_1 \dots \hat m_N {:}\rangle - \langle {:} \hat m_1 {:}\rangle \cdots \langle {:} \hat m_N {:}\rangle \stackrel{\mathrm{cl}}\geq 0.
\end{aligned}$$
Independence of uncorrelated noise contributions
------------------------------------------------
Furthermore, we can show that the derived covariance conditions do not depend on uncorrelated noise contributions such as detector dark counts. Let us assume that two detectors have linear detector responses which can be described by the functions $\hat\Gamma_{i(j)}(\hat n_{i(j)})=\eta_{i(j)}\hat n_{i(j)}+\nu_{i(j)}$, where $\eta_{i(j)}$ and $\nu_{i(j)}$ are the quantum efficiencies and the uncorrelated noise-count rates, respectively. In this case, the normal-ordered expectation values of the no-click operators are $\langle{:}\hat m_i \hat m_j{:}\rangle=e^{-(\nu_i+\nu_j)}\int d^2\alpha P(\alpha) e^{-\eta_i|u_i|^2|\alpha|^2}e^{-\eta_j|u_j|^2|\alpha|^2}$ and $\langle{:}\hat m_{i(j)} {:}\rangle=e^{-\nu_{i(j)}}\int d^2\alpha P(\alpha) e^{-\eta_{i(j)}|u_{i(j)}|^2|\alpha|^2}$. As above, we can now derive an inequality which can be written in the form $$\begin{aligned}
e^{-(\nu_i+\nu_j)}\left[\int d^2\alpha P(\alpha) e^{-\eta_i|u_i|^2|\alpha|^2}e^{-\eta_j|u_j|^2|\alpha|^2}-\int d^2\alpha P(\alpha) e^{-\eta_{i}|u_{i}|^2|\alpha|^2}\int d^2\alpha P(\alpha) e^{-\eta_{j}|u_{j}|^2|\alpha|^2}\right]\stackrel{\mathrm{cl}}\geq0
\end{aligned}$$ This can be further simplified to $$\begin{aligned}
e^{-(\nu_i+\nu_j)}\langle{:}\mathrm{Cov}(\hat m_i, \hat m_j){:}\rangle_{\mathrm{FUN}}\stackrel{\mathrm{cl}}\geq0,
\end{aligned}$$ where $\langle{:}\mathrm{Cov}(\hat m_i, \hat m_j){:}\rangle_{\mathrm{FUN}}$ corresponds to the covariance in the case when the detected light field would be free of any uncorrelated noise (FUN). We see that the uncorrelated noise contributions only result in an overall scaling of the inequality but do not influence the sign of the inequality. Therefore, the uncorrelated noise contributions do not alter the corresponding nonclassicality verification. This consideration can be straightforwardly generalized to the multimode conditions. Note that a similar independence of dark-count contributions has been reported in the context of phase-sensitive measurements with multiplexing detectors [@Lipfert15].
Radiation state and detection model for clusters of single-photon emitters {#sec:model}
==========================================================================
We derive the model for the multiplexed detection of light from a cluster of incoherently emitting single-photon emitters. This model is used to explain and interpret the experimental results for the different cluster sizes presented in Fig. \[fig:data1\] of the main text. We consider clusters consisting of $M$ quantum-dot emitters and we assume that all quantum dots have equal properties and that each quantum dot emits a single photon. Thus, the light emitted by the cluster can be expressed by a tensor-product state of single-photon states, i.e., $|\chi\rangle=\prod_{i=1}^M\otimes|1\rangle^i$ where the upper index indicates the different single-photon modes. Note that the no-click operators of the different detection channels are labeled with a lower index. In the experiment each quantum dot is excited with a certain finite probability, so that unexcited quantum dots do not contribute to the single photon emission. We note that a quantum dot which does not emit a photon is equivalent to one emitting a photon but the photon gets lost, i.e., it is not recorded. Therefore, we consider that each quantum dot emits a single photon but we assign a quantum efficiency $\eta_{\mathrm{ex}}$ in the detection of the light which accounts for the finite probability of exciting each quantum dot. Furthermore, not all light emitted by the clusters is collected which can be modeled by the collection efficiency $\eta_{\mathrm{col}}$. We assume that the multiplexed detection is symmetric and that the on-off detectors have a linear detector response which is characterized through the detector efficiency $\eta_{\mathrm{det}}$. All the different efficiencies can be summarized in the total efficiency of the experiment, $\eta=\eta_{\mathrm{ex}}\eta_{\mathrm{coll}}\eta_{\mathrm{det}}$. Furthermore, the detection in each channel is not mode sensitive as all impinging photons can lead to a detection click. Therefore, we can write the no-click operator of the $j$-th detection channel as $\hat m_j=\exp [\eta(\sum_{i=1}^M \hat n_j^i)]$.
Let us now consider the multiplexing of the tensor product single-photon states. The multiplexing device acts equally on each single-photon state and we assume a symmetric splitting into the four detection channels which yields $$\begin{aligned}
\prod_{i=1}^M\otimes|1\rangle^i\to
\prod_{i=1}^M\otimes\left[\frac{1}{\sqrt{4}}\left(|1,0,0,0\rangle^i+|0,1,0,0\rangle^i+|0,0,1,0\rangle^i+|0,0,0,1\rangle^i\right)\right].
\end{aligned}$$ Now we calculate the expectation value of the no-click operator in the $j$-th detection channel to be $\langle \hat m_j \rangle=(1-\frac{\eta}{4})^M$. Similarly, the expectation value of the joint no-click event for all four detection channels is given by $\langle {:} \hat m_{1} \dots \hat m_{4} {:}\rangle = (1-\eta)^M$. This allows us to evaluate the expressions of the condition and $(N=4)$ to be $$\begin{aligned}
\langle {:} \hat m_{1} \hat m_{3} {:}\rangle - \langle {:} \hat m_{1} {:}\rangle \langle {:} \hat m_{3} {:}\rangle=(1-\frac{\eta}{2})^M-[(1-\frac{\eta}{4})^M]^2
\\
\langle {:} \hat m_{1} \dots \hat m_{4} {:}\rangle - \prod_{i=1}^{4} \langle {:} \hat m_{i} {:}\rangle= (1-\eta)^M-[(1-\frac{\eta}{4})^M]^4,
\end{aligned}$$ respectively, which are given as as Eqs. and in the main text. These results are used in Fig. \[fig:data1\] of the main paper to explain the dependence of the nonclassicality conditions on the cluster size.
![ The values of the two nonclassicality conditions scaled by a factor of $10^4$ for the state and detection model, Eq. (dotted) and Eq. (dashed), are shown in dependence on the number of emitters for $\eta=0.05$ (a) and $\eta=0.2$ (b). []{data-label="fig:eta"}](PlotAppendix.pdf){width="0.9\columnwidth"}
Let us note that the only parameter in this model is the overall efficiency $\eta$ which accounts for all inefficiencies of the whole setup. For the analysis of our experimental data we estimate this efficiency to be $\eta\approx 0.009$. This value is in good agreement with the estimated losses and efficiencies of the experiment, and the corresponding model fits well the experimental results (cf. Fig. \[fig:data1\]). The behavior of the calculated nonclassicality conditions for two other values of $\eta$ is exemplarily shown in Fig. \[fig:eta\]. We observe that they all follow the same behavior, i.e., that the nonclassicality condition is at first getting more negative with increasing numbers of emitters, reaches its minimum for a certain number of emitters $M_{\mathrm{min}(\eta)}$, and eventually approaches the classical limit of zero for large numbers of emitters. The position of its minimal value is determined by the overall efficiency $\eta$. In the limit of unity efficiency $(\eta\to1)$, the minimal value is reached by one emitter ($M=1$) which is the manifestation of the single-photon character and the resulting anticorrelations in the click events (”antibunching”). However, we see that for lower efficiencies $\eta$ we can still certify nonclassicality even for large numbers of single-photon emitters. Remarkably, this is possible even if the number of detection channels ($N=4$) is orders of magnitude smaller than the numbers of emitters; cf. Fig \[fig:eta\] and Fig. \[fig:data1\] in the main text. We note that the presented model is rather simple and that it is based on several assumptions. This explains deviations of the model from some of the experimental data points while it describes well the overall dependence on the cluster size.
Relation to other nonclassicality conditions {#sec:relation}
============================================
In this section, we show some relations and differences to established methods for the verification of nonclassicality from multiplexing measurements. First, we will discuss the relation to conditions of the form of the Mandel $Q$ parameter. Second, we show that our approach delivers criteria which cannot be obtained via the matrix of moments approach.
Relation to conditions of the form of the Mandel $Q$ parameter
--------------------------------------------------------------
Here, we will show how the obtained covariance condition relates to established nonclassicality parameters of the form of the Mandel $Q$ parameter [@Mandel79]. Besides the Mandel parameter itself, we will also consider the recently introduced sub-binomial $Q_{\mathrm{B}}$ [@Sperling12a] and sub-Poisson-binomial $Q_{\mathrm{PB}}$ [@Lee16] parameters. The latter two were introduced for the verification of nonclassical light in symmetric and general (asymmetric) multiplexing schemes, respectively. All three parameter have in common that—whenever they attain negative values—they uncover nonclassicality of the considered quantum state.
### Sub-Poisson-Binomial parameter
Let us start by considering the sub-Poisson-Binomial parameter. The sub-Poisson-binomial parameter [@Lee16] may be written as $$\begin{aligned}
\label{eq:QPB}
Q_{\mathrm{PB}}=\frac{\sum_{i\neq j}^N \langle{:}\mathrm{Cov}(\hat m_i \hat m_j){:}\rangle}{\sum_{i}^N \langle {:}\hat m_i{:}\rangle (1-\langle{:}\hat m_i{:}\rangle)},
\end{aligned}$$ where $N$ is the number of detection channels and $\langle{:}\mathrm{Cov}(\hat m_i \hat m_j){:}\rangle$ is the normal-ordered expectation value of the covariance defined in Eq. . $Q_{\mathrm{PB}}<0$ reveals nonclassicality and its sign solely depends on the numerator in Eq. as the denominator is always positive. Hence, $Q_{\mathrm{PB}}$ characterizes nonclassicality if the sum over all possible $\langle{:}\mathrm{Cov}(\hat m_i \hat m_j){:}\rangle$ is negative and, thus, it is based on the violation of the simple classical covariance condition . However, it is a more complex condition as it requires evaluation of all possible correlations between the different detection channels.
### Sub-Binomial parameter
If we now assume that the multiplexing is performed with an equal splitting into the channels and each channel is detected with detectors which have equal properties, i.e., having the same detector response (quantum efficiency and dark-count rate), the $Q_{\mathrm{B}}$ parameter [@Sperling12a], $$\begin{aligned}
\label{eq:QB}
Q_{\mathrm{B}}=\frac{(N-1)\langle{:}\mathrm{Var}(\hat m){:}\rangle}{\langle {:}\hat m{:}\rangle (1-\langle{:}\hat m{:}\rangle)},
\end{aligned}$$ may be applied. A negative value of $Q_{\mathrm{B}}$ uncovers nonclassicality. Note that $Q_{\mathrm{B}}$ is the special form of $Q_{\mathrm{PB}}$ in the case of equal splitting and detection. In this case, all no-click operators $\hat m_i$ are equal and we can replace them by $\hat m$. Then, the covariance in Eq. reduces to the variance in Eq. . Still the negativities of $Q_{\mathrm{B}}$ arise from a negative variance, which is nothing else as the violation of the classical covariance condition for the case in which all $\hat m_i$ are the same. The negativity of the variance is also a direct indication that the corresponding quantum state cannot be described by a classical (non-negative) $P$ function. Note, however, that $Q_{\mathrm{B}}$ is only applicable if the assumptions of equal splitting and detection are fulfilled.
### Mandel parameter
In [@Sperling12a], it has been shown that $Q_{\mathrm{B}}$ approaches the Mandel $Q$ parameter $$\begin{aligned}
\label{eq:Q}
Q=\frac{\langle{:}\mathrm{Var}(\hat n){:}\rangle}{\langle{:} \hat n{:}\rangle},
\end{aligned}$$ in the case of an infinite number of detection channels, i.e., $\lim_{N\to\infty} Q_{\mathrm{B}}=Q$. In other words, for an equal splitting into an infinite number of on-off detectors the variance of the no-click operator yields the normal-ordered expectation value of the variance of the photon-number operator, $\langle{:}\mathrm{Var}(\hat n){:}\rangle$. Hence, $Q$ can be seen as the limiting case of recording light with an infinite number of equal on-off detectors. This finding agrees with a derivation of the photon-counting formula in the 1960s [@Scully69], where a bulk material described by an infinite number of single atoms (acting in the same way as on-off detectors) detects the light.
We can summarize that criteria of the form of the parameters $Q$, $Q_{\mathrm{B}}$, and $Q_{\mathrm{PB}}$ are in the end based on the covariance condition in Eq. . Importantly, the condition in Eq. is the most simple form of such conditions and provides the essential building block for the other criteria. Moreover, the simple covariance condition using only two on-off detectors can be more sensitive than the condition provided by the $Q$ parameter, as we show in Fig. , for the example of a photon-added thermal state.
Relation to criteria based on the matrix of moments approach
------------------------------------------------------------
Here, we compare the obtained conditions with the matrix of moments approach for multiplexing detection [@Sperling13]. For classical states, the matrix of moments $\boldsymbol M$ is positive semidefinite, with $$\begin{aligned}
0\stackrel{\mathrm{cl}}\leq& \boldsymbol M=\left(\langle{:} \hat m^{l+l'}{:}\rangle\right)_{l,l'=0}^{\lfloor N/2\rfloor}.
\end{aligned}$$ where the floor function yields $\lfloor N/2 \rfloor=N/2$ for even $N$ and $\lfloor N/2 \rfloor=(N-1)/2$ for odd $N$. A violation of this positive semidefiniteness would imply that the measured quantum state is a nonclassical one. As an example we can consider the $2\times2$ matrix with moments up to the $2l$th order $(l\geq1)$ $$\begin{aligned}
\det \begin{pmatrix}
\langle{:} \hat m^{0}{:}\rangle & \langle{:} \hat m^{l}{:}\rangle\\
\langle{:} \hat m^{l}{:}\rangle & \langle{:} \hat m^{2l}{:}\rangle
\end{pmatrix}
=\langle{:}\mathrm{Var}(\hat m^l){:}\rangle \stackrel{\mathrm{cl}}\geq 0.
\end{aligned}$$ This, in fact, corresponds to the correlation conditions derived here if all channels are recorded equally. It is, however, important to mention that, contrary to the conditions presented here, the matrix of moments approach can only be applied if equal splitting and detection are considered.
We have seen that for some cases (equal splitting and detection, and $2\times2$ matrices) the matrix of moments methods yield the same conditions as our approach based on Chebyshev’s integral inequality. This analogy, however, does not hold in general. In particular, our most general conditions in Eq. include arbitrary partitions which cannot be obtained with the matrix of moments approach. Examples are asymmetric partitions of the form $$\begin{aligned}
\langle {:} \hat m^k {:}\rangle - \langle {:} \hat m {:}\rangle \langle {:} \hat m^{k-1} {:}\rangle \stackrel{\mathrm{cl}}\geq 0\quad \text{with}\quad k>2
\end{aligned}$$ or multi-partitions $$\begin{aligned}
\langle {:} \hat m^k {:}\rangle - \langle {:} \hat m {:}\rangle \langle {:} \hat m^{k-2} {:}\rangle\langle {:} \hat m {:}\rangle \stackrel{\mathrm{cl}}\geq 0\quad \text{with}\quad k>2.
\end{aligned}$$ Hence, the approach presented here provides new conditions which are not covered by the matrix of moments approach. Therefore, these more general conditions might be able to certify nonclassicality in cases where other methods fail to do so.
[99]{}
S. L. Braunstein and P. van Loock, Quantum information with continuous variables, [Rev. Mod. Phys. **77**, 513 (2005)](https://doi.org/10.1103/RevModPhys.77.513). P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn Linear optical quantum computing with photonic qubits [Rev. Mod. Phys. **79**, 135 (2007)](https://doi.org/10.1103/RevModPhys.79.135). Quantum communication N. Gisin and R. Thew [Nature Photonics **1**, 165 (2007)](https://doi.org/10.1038/nphoton.2007.22). C. Branciard, D. Rosset, Y.-C. Liang, and N. Gisin, Measurement-Device-Independent Entanglement Witnesses for All Entangled Quantum States, [Phys. Rev. Lett. **110**, 060405 (2013)](https://doi.org/10.1103/PhysRevLett.110.060405). C. C. W. Lim, B. Korzh, A. Martin, F. Bussières, R. Thew, and H. Zbinden, Detector-device-independent quantum key distribution, [Appl. Phys. Lett. **105**, 221112 (2014)](https://doi.org/10.1063/1.4903350). A. Gheorghiu, E. Kashefi, and P. Wallden, Robustness and device independence of verifiable blind quantum computing, [New J. Phys. **17**, 083040 (2015).](https://doi.org/10.1088/1367-2630/17/8/083040) M. Altorio, M. G. Genoni, F. Somma, and M. Barbieri, Metrology with Unknown Detectors, [Phys. Rev. Lett. **116**, 100802 (2016)](https://doi.org/10.1103/PhysRevLett.116.100802). J. Sperling, W. R Clements., A. Eckstein, M. Moore, J. J. Renema, W. S. Kolthammer, S. W. Nam, A. Lita, T. Gerrits, W. Vogel, G. S. Agarwal, I. A. Walmsley, Detector-Independent Verification of Quantum Light, [Phys. Rev. Lett. **118**, 163602 (2017)](https://doi.org/10.1103/PhysRevLett.118.163602). H. Paul, P. Törmä, T. Kiss, and I. Jex, Photon Chopping: New Way to Measure the Quantum State of Light, [Phys. Rev. Lett. **76**, 2464 (1996)](http://dx.doi.org/10.1103/PhysRevLett.76.2464). P. Kok and S. L. Braunstein, Detection devices in entanglement-based optical state preparation, [Phys. Rev. A **63**, 033812 (2001)](http://dx.doi.org/10.1103/PhysRevA.63.033812). K. Banaszek and I. A. Walmsley, Photon counting with a loop detector, [Opt. Lett. **28**, 52 (2003)](https://doi.org/10.1364/OL.28.000052). J. Řehàček, Z. Hradil, O. Haderka, J. Peřina Jr., and M. Hamar, Multiple-photon resolving fiber-loop detector, [Phys. Rev. A **67**, 061801(R) (2003)](https://doi.org/10.1103/PhysRevA.67.061801). M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, Photon-number resolution using time-multiplexed single-photon detectors, [Phys. Rev. A **68**, 043814 (2003)](https://doi.org/10.1103/PhysRevA.68.043814). D. Achilles, C. Silberhorn, C. Śliwa, K. Banaszek, and I. A. Walmsley, Fiber-assisted detection with photon number resolution, [Opt. Lett. **28**, 2387 (2003)](https://doi.org/10.1364/OL.28.002387). S. A. Castelletto, I. P. Degiovanni, V. Schettini, and A. L. Migdall, Reduced deadtime and higher rate photon-counting detection using a multiplexed detector array, [J. Mod. Opt. **54**, 337 (2007)](https://doi.org/10.1080/09500340600779579). V. Schettini, S. V. Polyakov, I. P. Degiovanni, G. Brida, S. Castelletto, and A. L. Migdall, Implementing a multiplexed system of detectors for higher photon counting rates, [IEEE J. Sel. Top. Quantum Electron. **13**, 978 (2007)](https://doi.org/10.1109/MODTQE.2007.902846). J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnell, C. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, Tomography of quantum detectors, [Nat. Phys. **5**, 27 (2009)](https://doi.org/10.1038/nphys1133). J. C. F. Matthews, X.-Q. Zhou, H. Cable, P. J. Shadbolt, D. J. Saunders, G. A. Durkin, G. J. Pryde, and J. L. O’Brien, Towards practical quantum metrology with photon counting, [npj Quantum Inf. **2**, 16023 (2016)](https://doi.org/10.1038/npjqi.2016.23). J. Sperling, W. Vogel, and G. S. Agarwal, True photocounting statistics of multiple on-off detectors, [Phys. Rev. A **85**, 023820 (2012)](https://doi.org/10.1103/PhysRevA.85.023820). J. Sperling, W. Vogel, and G. S. Agarwal, Sub-Binomial Light, [Phys. Rev. Lett. **109**, 093601 (2012)](https://doi.org/10.1103/PhysRevLett.109.093601). J. Sperling, W. Vogel, and G. S. Agarwal, Correlation measurements with on-off detectors, [Phys. Rev. A **88**, 043821 (2013)](http://dx.doi.org/10.1103/PhysRevA.88.043821). R. Filip and L. Lachman, Hierarchy of feasible nonclassicality criteria for sources of photons, [Phys. Rev. A **88**, 043827 (2013)](https://doi.org/10.1103/PhysRevA.88.043827). L. Lachman, L. Slodicka, and R. Filip, Nonclassical light from a large number of independent single-photon emitters, [Sci. Rep. **6** 19760 (2016)](https://doi.org/10.1038/srep19760). A. Luis, J. Sperling, and W. Vogel, Nonclassicality Phase-Space Functions: More Insight with Fewer Detectors, [Phys. Rev. Lett. **114**, 103602 (2015)](https://doi.org/10.1103/PhysRevLett.114.103602). C. Lee, S. Ferrari, W. H. P. Pernice, and C. Rockstuhl, Sub-Poisson-Binomial Light, [Phys. Rev. A **94**, 053844 (2016)](https://doi.org/10.1103/PhysRevA.94.053844). T. J. Bartley, G. Donati, X.-M. Jin, A. Datta, M. Barbieri, and I. A. Walmsley, Direct Observation of Sub-Binomial Light, [Phys. Rev. Lett. **110**, 173602 (2013)](https://doi.org/10.1103/PhysRevLett.110.173602). J. Sperling, M. Bohmann, W. Vogel, G. Harder, B. Brecht, V. Ansari, and C. Silberhorn, Uncovering Quantum Correlations with Time-Multiplexed Click Detection, [Phys. Rev. Lett. **115**, 023601 (2015)](https://doi.org/10.1103/PhysRevLett.115.023601). R. Heilmann, J. Sperling, A. Perez-Leija, M. Gräfe, M. Heinrich, S. Nolte, W. Vogel, and A. Szameit, Harnessing click detectors for the genuine characterization of light states, [Sci. Rep. **6**, 19489 (2016)](http://dx.doi.org/10.1038/srep19489). J. Sperling, T. J. Bartley, G. Donati, M. Barbieri, X.-M. Jin, A. Datta, W. Vogel, and I. A. Walmsley, Quantum Correlations from the Conditional Statistics of Incomplete Data, [Phys. Rev. Lett. **117**, 083601 (2016)](https://doi.org/10.1103/PhysRevLett.117.083601). M. Bohmann, R. Kruse, J. Sperling, C. Silberhorn, and W. Vogel, Probing free-space quantum channels with laboratory-based experiments, [Phys. Rev. A [**95**]{}, 063801 (2017)](https://doi.org/10.1103/PhysRevA.95.063801). M. Bohmann, J. Tiedau, T. Bartley, J. Sperling, C. Silberhorn, and W. Vogel, Incomplete Detection of Nonclassical Phase-Space Distributions, [Phys. Rev. Lett. **120**, 063607 (2018)](https://doi.org/10.1103/PhysRevLett.120.063607). P. Obšil, L. Lachman, T. Pham, A. Lešundák, V. Hucl, M. Čížek, J. Hrabina, O. Číp, L. Slodička, and R. Filip, Nonclassical Light from Large Ensembles of Trapped Ions, [Phys. Rev. Lett. **120**, 253602 (2018)](https://doi.org/10.1103/PhysRevLett.120.253602). J. Tiedau, E. Meyer-Scott, T. Nitsche, S. Barkhofen, T. J. Bartley, C. Silberhorn, A high dynamic range optical detector for measuring single photons and bright light, [Opt. Express **27**, 1 (2019)](https://doi.org/10.1364/OE.27.000001). J. Sperling, A. Eckstein, W. R. Clements, M. Moore, J. J. Renema, W. S. Kolthammer, S. W. Nam, A. Lita, T. Gerrits, I. A. Walmsley, G. S. Agarwal, and W. Vogel, Identification of nonclassical properties of light with multiplexing layouts, [Phys. Rev. A **96**, 013804 (2017)](https://doi.org/10.1103/PhysRevA.96.013804). R. J. Glauber, Coherent and Incoherent States of the Radiation Field, [Phys. Rev. **131**, 2766 (1963)](https://doi.org/10.1103/PhysRev.131.2766). E. C. G. Sudarshan, Equivalence of Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams, [Phys. Rev. Lett. **10**, 277 (1963)](https://doi.org/10.1103/PhysRevLett.10.277). U. M. Titulaer and R. J. Glauber, Correlation functions for coherent fields, [Phys. Rev. **140**, B676 (1965)](https://doi.org/10.1103/PhysRev.140.B676). L. Mandel, Non-classical states of the electromagnetic field, [Phys.Scr.T **12**, 34, (1986)](https://doi.org/10.1088/0031-8949/1986/T12/005). Here, we consider loss-less multi-splitter which can be described by a unitary operation. Note that this approach is also applicable to lossy splitters if one accounts for the overall (or mode individual) losses in the detection process by an effective quantum efficiency including the additional losses in the multiplexing step. W. Vogel and D.-G. Welsch, Quantum Optics (Wiley-VCH, Weinheim, Germany, 2006). P. L. Kelley and W. H. Kleiner, Theory of Electromagnetic Field Measurement and Photoelectron Counting, [Phys. Rev. **136**, A316 (1964)](https://doi.org/10.1103/PhysRev.136.A316). M. Bohmann, R. Kruse, J. Sperling, C. Silberhorn, and W. Vogel, Direct calibration of click-counting detectors, [Phys. Rev. A **95**, 033806 (2017)](https://doi.org/10.1103/PhysRevA.95.033806). As the no-click event is complementary to the click event, the formulation in terms of the no-click operator yields the same results as for the click operator. Due to notational convenience, we formulate all conditions in terms of no-click events. D.S. Mitrinović, Analytic Inequalities, (Springer-Verlag, Berlin, Germany, 1970). G. S. Agarwal and K. Tara, Nonclassical character of states exhibiting no squeezing or sub-Poissonian statistics, [Phys. Rev. A **46**, 485 (1992)](https://doi.org/10.1103/PhysRevA.46.485). A. Zavatta, S. Viciani, and M. Bellini, Quantum-to-Classical Transition with Single-Photon-Added Coherent States of Light, [Science **306**, 660 (2004).](https://doi.org/10.1126/science.1103190) T. Kiesel, W. Vogel, V. Parigi, A. Zavatta, and M. Bellini, Experimental determination of a nonclassical Glauber-Sudarshan P function, [Phys. Rev. A **78**, 021804(R) (2008).](https://doi.org/10.1103/PhysRevA.78.021804) L. Mandel, Sub-Poissonian photon statistics in resonance fluorescence, [Opt. Lett. **4**, 205 (1979)](https://doi.org/10.1364/OL.4.000205). L. Carbone, C. Nobile, M. D. Giorgi, F. D. Sala, G. Morello, P. Pompa, M. Hytch, E. Snoeck, A. Fiore, I. R. Franchini, M. Nadasan, A. F. Silvestre, L. Chiodo, S. Kudera, R. Cingolani, R. Krahne, and L. Manna, Synthesis and micrometer-scale assembly of colloidal cdse/cds nanorods prepared by a seeded growth approach, [Nano Letters **7**, 2942 (2007)](https://doi.org/10.1021/nl0717661). F. Pisanello, L. Martiradonna, G. Lemenager, P. Spinicelli, F. Fiore, L. Manna, J. Hermier, R. Cingolani, E. Giacobino, M. De Vittorio, and A. Bramati, Room temperature-dipolelike single photon source with a colloidal dot-in-rod, [Appl. Phys. Lett. **96**, 033101 (2010)](https://doi.org/10.1063/1.3291849). M. Manceau, S. Vezzoli, Q. Glorieux, F. Pisanello, E. Giacobino, L. Carbone, M. De Vittorio, and A. Bramati, Effect of charging on CdSe/CdS dot-in-rods single-photon emission, [Phys. Rev. B **90**, 035311 (2014)](https://doi.org/10.1103/PhysRevB.90.035311). S. Vezzoli, M. Manceau, G. Lemenager, Q. Glorieux, E. Giacobino, L. Carbone, M. De Vittorio, and A. Bramati, Exciton Fine Structure of CdSe/CdS Nanocrystals Determined by Polarization Microscopy at Room Temperature, [ACS Nano **9**, 7992 (2015)](https://doi.org/10.1021/acsnano.5b01354). L. Qi, M. Manceau, A. Cavanna, F. Gumpert, L. Carbone, M. de Vittorio, A. Bramati, E. Giacobino, L. Lachman, R. Filip, and M. Chekhova, Multiphoton nonclassical light from clusters of single-photon emitters, [New J. Phys. **20**, 073013 (2018)](https://doi.org/10.1088/1367-2630/aacf21). O. A. Shcherbina, G. A. Shcherbina, M. Manceau, S. Vezzoli, L. Carbone, M. De Vittorio, A. Bramati, E. Giacobino, M. V. Chekhova, and G. Leuchs, Photon correlations for colloidal nanocrystals and their clusters, [Opt. Lett. **39**, 1791 (2014)](https://doi.org/10.1364/OL.39.001791). Note that the application of the matrix of moments approach requires an equal splitting into the detection bins and is therefore limited to experiments fulfilling this condition. W. Vogel and J. Sperling, Unified quantification of nonclassicality and entanglement, [Phys. Rev. A **89**, 052302 (2014)](https://doi.org/10.1103/PhysRevA.89.052302). T. Lipfert, J. Sperling, and W. Vogel, Homodyne detection with on-off detector systems, [Phys. Rev. A 92, 053835 (2015)](https://doi.org/10.1103/PhysRevA.92.053835). M. O. Scully and W. E. Lamb, Jr., Quantum Theory of an Optical Maser. III. Theory of Photoelectron Counting Statistics, [Phys. Rev. **179**, 368 (1969)](https://doi.org/10.1103/PhysRev.179.368).
|
---
abstract: 'Temperature and doping dependent tunneling spectroscopy on $Bi_2Sr_2CaCu_2O_{8+\delta}$ single crystals has been investigated by using break-junction technique. The results provide direct evidence that the pseudogap persists into heavily overdoped regime, possibly in whole doping range of the superconducting phase. The gap value determined from the peak-to-peak of the conductance, $\Delta_{p-p}$, exhibits a jump-like increase as $T$ is raised across the critical temperature. The magnitude of the jump, the pseudogap opening temperature and superconducting gap decrease with raising doping. The $\Delta_{p-p}$ below $T_c$ can be naturally treated as the vector sum of two competing orders: a BCS-like superconducting gap and a residual pseudogap.'
address: |
$^1$National Laboratory for Superconductivity, Institute of Physics and Center for Condensed Matter Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing 100080, China\
$^2$Max-Planck-Institut für Festkörperforschung, 70569 Stuttgart, Germany
author:
- 'Y. Xuan$^1$, H. J. Tao$^1$[@responce], Z. Z. Li$^1$, C. T. Lin$^2$, Y. M. Ni$^1$, B. R. Zhao$^1$, Z. X. Zhao$^1$'
title: ' Superconducting Gap, Pseudogap and Their Relationship Investigated by Break-junction Tunneling Spectroscopy on $Bi_2Sr_2CaCu_2O_{8+\delta}$ Single Crystals '
---
It is generally agreed that for underdoped cuprate high-$T_c$ superconductors a generic phenomenon is there is the pseudogap (PG) in their excitation spectrum for temperature ($T$) above the critical temperature $T_c$. The PG has been observed with various probes both spin and charge sensitive[@Timusk]. Among the probes tunneling spectroscopy, one of the direct measures of quasiparticle excitations in a superconductor has attracted great deal of attention. However, thus far the results of tunneling, are still controversial. Different types of relation between the SG and the PG have been reported, such as smooth evolution[@Renner], competing[@Ekino], coexistence of the two gaps[@Krasnov], and distinct two gaps[@Suzuki], with scanning tunneling microscope (STM)[@Renner], break-junction[@Ekino], and c-axis intrinsic junction[@Krasnov; @Suzuki]. There is also the question that whether the PG exists in overdoped regime, especially in heavily overdoped region? Apparently, experimentally clarifying the above discrepancies is a pressing task and will set constraint to any theoretical models on the origin of the pseudogap and on the mechanism of high-$T_c$ superconductivity.
In this paper we present systematic investigation on the detailed evolution behavior of the gap $\Delta_{p-p}$, determined by a quarter of the span from the peak-to-peak in the tunneling conductance, on $Bi_2Sr_2CaCu_2O_{8+\delta}$ (Bi2212) single crystals with break-junction technique, specifically focusing on the overdoped regime. For all samples studied, including the highly overdoped ($T_c$ = 72 K), our low-noise tunneling conductance clearly demonstrates the existence of the pseudogap. Detailed $T$-dependent measurements show a jump-like increase of $\Delta_{p-p}$ as $T$ is raised across $T_c$ within a very narrow temperature range, indicating an unsmooth transformation from the superconducting state to the PG state. We show that the measured gap for $T$ below $T_c$ can be naturally treated as a vector sum of two competing components: the superconducting gap (SG) and the residual PG. The PG opening temperature, $T^*$, the magnitude of the jump of $\Delta_{p-p}$ at $T_c$, $\Lambda$, and the value of the SG, and PG decrease with increase of doping. The doping dependence of $T^*$ implies that for Bi2212 the PG may exist in the whole overdoped regime provided there is the superconductivity. To our knowledge, this doping dependence of $\Lambda$, as a characteristic parameter for the unsmooth transformation is newly observed by tunneling, and the present work is the first directly tunneling observation of the doping dependence of the $T^*$.
We chose break-junction to carry out our tunneling investigation because it provides a clean and fresh interface easily, which avoids surface degradation and diminishes the barrier scattering. In addition, the break-junction is of the superconductor-insulator-superconductor (SIS) type junction, which enhances the gap feature in the tunneling conductance due to its unique convolution relationship from the electronic density of states (DOS) of two superconductors on both sides of tunnel barrier. Therefore, the peak position is less affected by thermal smearing and can be taken as a characteristic parameter[@Wolf]. More importantly, previous studies at 4.2 K have provided reproducible and low-noise tunneling conductance consisting of sharp coherence peaks, a close to zero zero-bias conductance, and a significant peak-dip-hump structure[@Mandrus; @Miyakawa; @Zasadzinski]. Those facts demonstrate that the break-junction technique is a reliable tool with high-energy resolution to investigate the tunneling spectroscopy.
The Bi2212 single crystals used in this study were of three doping levels. The near-optimally doped samples were the as-grown crystals grown by traveling-solvent-floating-zone method. The medium-overdoped samples were achieved by annealing the as-grown crystals ($T_c$ = 89 K), synthesized by directional solidification method, in flowing O$_2$ at 500$^\circ$C for 120 hours. And the heavily overdoped samples, also from the as-grown Bi2212 single crystals grown by directional solidification, were obtained by annealing them under a pressure of 140 atm of O$_2$ at 500$^\circ$C for 50 hours. The on-set $T_c$ of the three batches of samples was 90 K, 82 K and 72 K with transition width of 1 K, 1 K and 3 K, respectively, measured by AC susceptibility using Lakeshore AC Susceptometer-7000. Hereafter we label the samples as OP90, OV82 and OV72. All the characterizations were carried out on segments of the very single crystals used to measure tunneling spectroscopy. The tunneling conductance curve, $dI/dV - V$, was measured by using a standard low-frequency ac lock-in technique with the four-probe method and a computer data acquisition system. To achieve low-resistive contacts ($<$ 2 $\Omega$) between the crystals and the leads, gold film electrodes were sputtered onto each sample.
The break-junctions were formed in situ at 4.2 K[@Mandrus]. To realize it, thin (less than 0.1 mm) single crystal was glued onto phosphor-copper substrate with epoxy, and broken by applying a bending force perpendicular to the CuO$_2$ plane with a precise differential-screw. After a break-junction was formed, a key problem left was to keep it from being unstable as temperature varied. Because there is mismatch between the thermal expansion coefficients among the Bi2212 crystal, epoxy and phosphor-copper substrate, which often causes the contact of two pieces of the broken off crystal become loosen or tighten, and leads to the junction conductance changed drastically when the temperature varies. It even breaks down the junction at high temperature. Through practice we found that by adjusting the thickness of epoxy, which depends on the dimensions of the substrate, we were able to overcome this problem and perform stable $T$-dependent investigation.
To diminish the complexity bringing about by Josephson current and to facilitate the analysis of experimental data, each time we carefully adjusted the differential screw to increase the junction resistance and depress the Cooper pair tunneling to undetectable small. Tunneling measurements were carried out at temperatures ranging from near 4.2 K to where the PG disappears.
Figure 1, 2 and 3 show the normalized tunneling conductance of different temperatures measured on OP90, OV82 and OV72, respectively. The normalization was done by divided each measured conductance by its conductance of high bias voltage. For clarity, the curves of $T \leq T_c$ are sequentially shifted up with a unit value (in (a) panel of each figure), whilst those of $T \geq T_c$ are plotted with same coordinate scale as they were measured (in (b) panel of each figure).
In Fig. 1 to 3 several points are notable. (1) At low $T$, the conductance for OP90 and OV82 exhibit the significant dip-hump structure, which has been observed in ARPES[@Ding] and tunneling measurements and is related to the ($\pi$, $\pi$) resonance in inelastic neutron scattering[@Zasadzinski]. The dip feature for OV72 is not as significant as of OP90 and OV82, but it is still visible in Fig. (3.c) by comparing the low-$T$ conductance to the ones of higher temperature. Those facts demonstrate the high quality of our junctions. (2) For $T \ll T_c$, such as $T$ = 4.2 K, it is generally safe to take $\Delta_{p-p}$ as the SG, $\Delta$(0). The low-T parameters, including $\Delta$(0), and the dip position $E_{dip}$ in tunneling conductance for the three samples are listed in Table I. These values are consistent well with the results of Ref. [@Zasadzinski], in which a detailed examination over a wide range of oxygen doping at 4.2 K was performed. On the other hand, the tunneling curve calculated from a weak-coupling d-wave DOS has significant deviation from the experimental date in the shape of conductance peak[@Zasadzinski; @Coffey]. This brings about difficulty to model fit the experimental curves even in the low $T$. (3) In Fig. 1, 2 and 3, the conductance curves at $T > T_c$ clearly demonstrate the existence of PG even in heavily overdoping level with the presence of peaks, and the low-noise background enables us to get the value of $\Delta_{p-p}(T)$, a quarter of the peak-to-peak span. (4) The opening temperature of the PG, $T^*$, can be determined as the temperature at which the conductance peaks vanish, meanwhile the central dip in the conductance is filled-in. By so doing, the obtained $T^*$ is 190, 150 and 120 K for sample OP90, OV82 and OV72, respectively, and we estimate the error bars are $\pm$15, $\pm$15, $\pm$10 K, respectively.
To gain more insights into the physics, we depict in Fig. 4 the $\Delta_{p-p}(T)$ taken from Fig. 1, 2 and 3, respectively, as a function of the normalized temperature $T/T_c$,. The filled squares, deltas and circles is corresponding to sample OP90, OV82 and OV72, respectively. The solid lines are the BCS $T$-dependent gaps corresponding to measured $\Delta$(0) of OP90, OV82 and OV72, respectively. A general feature for the three junctions in Fig. 4 is that for $2/3T_c < T < T_c$, the $\Delta_{p-p}(T)$ decreases with increasing $T$ much slower than the BCS behavior. There are at least two factors that may cause this slow decrease. First, the life-time effect of the quasuiparticles, which smears the tunneling conductance and gives rise to a larger $\Delta_{p-p}(T)$ than the SG, $\Delta_s(T)$. Second, the measured single gap is a vector sum of the SG and the residual PG for $T$ below $T_c$: $\Delta_{p-p}(T) = (\Delta_s^2(T) + \Delta_p^2(T))^{1/2}$. For simplicity, if we do not take into account the life-time broadening effect, and subtract the BCS gap from the $\Delta_{p-p}(T)$ according to the above formula we get the residual PG for $T < T_c$, which is shown in the inset of Fig. 4. The data for $T \geq T_c$ in the inset are the same as those in the main part of Fig. 4. The vector sum scenario is suggested by a number of theoretical models[@Chakravarty] stating that in superconducting state there exist two competing orders: the SG and the PG. As temperature is raised across $T_c$, $\Delta_{p-p}$ undergoes an abrupt jump within a very narrow temperature range ($\sim$2 K in OP90) and the $T_c$ can be regarded as where the maximum of $d\Delta_{p-p}(T)/dT$ is, or the middle of the jump. This behavior demonstrates that the maximum of DOS moves to higher energy unsmoothly as soon as long-range phase coherence disappears. To characterize this unsmooth behavior, we introduce $\Lambda$, the jump in $\Delta_{p-p}$ within the measured temperature interval in the vicinity of $T_c$. It can be seen that $\Lambda$ has the same trend with doping as $T^*$ (see also Tab. I).
Here one point we would like to emphasis is that the measured gap $\Delta_{p-p}$ in the [*normal*]{} state is not the true magnitude of the PG, because apparently the value of the PG should decrease or keep as constant when $T$ is raised. In general, high fluctuation[@Dynes] or strong pair-breaking[@Burstein] can bring out broadening conductance peak and widen separation between the peaks in the conductance. As to how to extract the true value of the PG and its temperature behavior from tunneling date is an open issue remaining to be address. The fact that for OP90 $\Delta_{p-p}$ keeps increasing as $T$ is raised reflects that in the normal state there is high fluctuation or strong pair-breaking which enhance as $T$ is raised. Moreover, the trend of doping dependence of $\Delta_{p-p}(T)$ demonstrates that the fluctuation or pair-breaking effect become weak in high doping region. Those features are consistent with the picture of precursor paring[@Emery] in electronic or spin channel[@Anderson] in the normal state but we emphasize it exists in the overdoped regime, and directly show that the difference in electronic excitation between the superconducting state and the normal state near $T_c$ increases with lowering doping.
To further illustrate the present observation we plot the doping dependence of the $\Delta(0)$ and $T^*$ in Fig. 5, in which the dotted line is the BCS mean-field gap, $\Delta = 2.14k_BT_c$, the hole concentration $p$ is obtained from the empirical relation $T_c/T_{c,max} = 1- 82.6 (p - 0.16)^2$[@Tallon1], with $T_{c,max} = 92 K$. The filled circles are measured $\Delta$(0) and the filled squares are the measured $2.14k_BT^*$. The thick straight line is a guide to the eye. It is surprising that the extrapolation of the thick line just crosses to the abscissa at $p$ = 0.27, where the superconductivity disappears. That implies so long as it is a superconductor, there may be the PG state. A similar trend for Bi2212 can be found in Ref.[@Oda]. This phase diagram of Bi2212 is qualitatively different from that of YBa$_2$Cu$_3$O$_{7-\delta}$, to which it is shown the PG state only exists in the region where the hole concentration $p$ is less than a certain value, 0.19[@Tallon2]. Obviously this is an issue that needs to be further addressed.
To conclude, we have performed investigations on temperature and doping dependent tunneling spectroscopy of Bi2212 single crystals with break-junction technique, focusing on overdoped regime. The results provide direct evidence that for Bi2212 the PG persists into heavily overdoped regime, possibly in whole doping range accompanying the SG. Detailed measurements near $T_c$ have showed the measured $\Delta_{p-p}$ undergoes a jump-like increase within a very narrow temperature range as $T$ is raised across $T_c$, indicating an unsmooth transformation from the superconducting state to the PG state. The magnitude of the jump of $\Delta_{p-p}$, $T^*$, the SG, and PG decrease with raising doping. We show the measure gap $\Delta_{p-p}(T)$ for $T$ below $T_c$ can be naturally treated as the vector sum of two competing orders: a BCS-like SG and a residual PG.
This work was supported by a grant for State Key Program for Basic Research of China, and the National Center for Research and Development on Superconductivity.
Email address: hjtao@aphy.iphy.ac.cn T. Timusk and B. Statt, Rep. Prog. Phys. [**62**]{}, 61 (1999). C. Renner et al., Phys. Rev. Lett. [**80**]{}, 149 (1998). T. Ekino, Y. Sezaki, and H. Fujii, Phys. Rev. B [**60**]{}, 6916 (1999). V. M. Krasnov et al., Phys. Rev. Lett. [**84**]{}, 5860 (2000); V. M. Krasnov, A. E. Kovalev, A. Yurgens, and D. Winkler, Phys. Rev. Lett. [**86**]{}, 2657 (2001). M. Suzuki and T. Watanabe, Phys. Rev. Lett. [**85**]{}, 4787 (2000). E. L. Wolf, [*Principles of Electron Tunneling Spectroscopy*]{}, (Oxford University, New York, 1985). D. Mandrus, L. Forro, D. Koller, and L. Mihaly, Nature [**351**]{}, 460 (1991) N. Miyakawa et al., Phys. Rev. Lett. [**80**]{}, 157 (1998); N. Miyakawa et al., Phys. Rev. Lett. [**83**]{}, 1018 (1999). J. F. Zasadzinski et al., Cond-mat/0102475. H. Ding et al., Phys. Rev. Lett. [bf 76]{}, 1533 (1996). L.Coffey, Cond-mat/0103518. S. Chakravarty, R. B. Laughlin, D. K. Morr, and C. Nayak, Phys. Rev. B [**63**]{}, 094503 (2001); I. Kosztin, Q. J. Chen, Y. J. Kao, and K. Levin, Phys. Rev. B [**61**]{}, 11662 (2000); J. W. Loram et al., J. Supercond. [**7**]{}, 243 (1994); R. S. Markiewicz, C. Kusko, and V. Kidambi, Phys. Rev. B [**60**]{}, 627 (1999). R. C. Dynes, V. Narayanamurti, and J. P. Garno, Phy. Rev. Lett. [**41**]{}, 1509 (1978); A. A. Varlamov, G. Balestrino, E. Milani, and D. V. Livanov, Adv. Phys. [**48**]{}, 655, (1999). E. Burstein and S. Lundqvist, [*Tunneling Phenomena in Solids*]{}, (Plenum, New York, 1969), Chapter 29, P. Fulde; Chapter 30, T. Claeson. V. J. Emery and S. A. Kivelson, Nature [**374**]{}, 434 (1995). P. W. Anderson, Science [**235**]{}, 1196 (1987). J. L. Tallon et al., Phys. Rev. B [**51**]{}, 12911 (1995). M. Oda, T. Matsuzaki, N. Momono, and M. Ido, Physica C [**341-348**]{}, 847 (2000). J. T. Tallon and J. W. Loram, Physica C [**349**]{}, 53 (2001).
|
---
abstract: 'The “new fields" or “superconformal functions" on $N=1$ super Riemann surfaces introduced recently by Rogers and Langer are shown to coincide with the Abelian differentials (plus constants), viewed as a subset of the functions on the associated $N=2$ super Riemann surface. We confirm that, as originally defined, they do not form a super vector space.'
author:
- |
Jeffrey M. Rabin\
Dept. of Mathematics, U.C.S.D.\
La Jolla, CA 92093\
jrabin@ucsd.edu
title: Old and New Fields on Super Riemann Surfaces
---
It has been known for some time that the globally defined holomorphic functions on a generic super Riemann surface (SRS) with odd spin structure do not form a super vector space [@nelson]. Neither do the holomorphic superconformal tensors of weights $-1$ through $2$ [@luke]. That is, one cannot uniquely express an arbitrary tensor as a linear combination of some small basis set. The proposed “basis" inevitably contains nilpotent tensors which are annihilated by some elements of the set of constant Grassmann parameters (exterior algebra) $\Lambda$. Therefore linear combinations with coefficients from $\Lambda$ cannot be unique. This is a special case of the general fact that sheaf cohomology groups $H^i(M,\cal F)$ of complex supermanifolds over $\Lambda$ are $\Lambda$-modules, but these modules are not necessarily free. In comparison with the theory of Riemann surfaces used in bosonic string theory, this complicates the discussion of period matrices, Jacobian varieties, functional determinants, and other ingredients in the construction of superstring amplitudes [@dhoker].
Recently, Rogers and Langer introduced a notion of “superconformal functions" on a SRS with the intent of overcoming these difficulties [@rogers]. These are a certain subset ${{\cal SC}}$ of the holomorphic sections of a rank $(1,1)$ super vector bundle $E$ over the SRS. On the basis of a direct but very complicated calculation, the module of global superconformal functions $H^0(M,{{\cal SC}})$ was claimed to be free, that is, a super vector space. Subsequently, an error in the calculation was corrected, with the result that $H^0(M,{{\cal SC}})$ is not free but a free submodule can be picked out by imposing an additional condition on the sections [@corrig]. In this paper we will give a geometric interpretation of ${{\cal SC}}$ leading to a simpler verification of these facts.
One can associate to every $N=1$ SRS, indeed to every complex supermanifold $M$ of dimension $(1,1)$, or “supercurve", an $N=2$ (untwisted) SRS $M_2$ [@schwarz]. This correspondence is generically two-to-one, so that each supercurve $M$ has a dual supercurve $\hat{M}$ with the same associated $M_2$. SRSs can be characterized by the property of self-duality: $M=\hat{M}$ iff $M$ is a SRS [@berg]. We will see that contour integration theory on $N=2$ SRSs [@cohn] makes it possible to integrate Abelian differentials (tensors of weight $1$) on $M$ to obtain functions on $M_2$. The super vector bundle $E$ will be shown to be precisely the sheaf of functions ${{\cal O}_2}$ on $M_2$, and ${{\cal SC}}$ will be that subsheaf obtained by integrating Abelian differentials on $M$. In fact, ${{\cal O}_2}$ splits as a sheaf into the direct sum of the Abelian differentials ${{\cal K}}$ and the functions $\cal O$ on $M$. In terms of this splitting, ${{\cal SC}}$ consists of ${{\cal K}}$ plus integration constants from $\cal O$. Since the global Abelian differentials on $M$ do not form a super vector space [@luke; @dhoker], we confirm that the global superconformal functions (as originally defined) do not either.
We will be rather casual about supermanifold theory in this paper. The reader can think of Rogers [@oldrogers] or DeWitt [@dewitt] supermanifolds (topologically vector bundles over their bodies) over a finite- or infinite-dimensional complex Grassmann algebra $\Lambda$ with generators $\eta_i$, $i = 1, 2, \ldots, L$ in the finite-dimensional case, or of families of sheaf-theoretic supermanifolds [@sheafrefs] over the parameter superspace $({\rm point},\Lambda)$. All that matters is that the various sheaves and cohomology groups considered are $\Lambda$-modules; linear combinations are taken with $\Lambda$ coefficients.
Let $M$ be a supercurve, covered by charts $U_\alpha$, with transition functions $$z_\beta = F_{\beta\alpha}(z_\alpha,\theta_\alpha),\;\;\;\;
\theta_\beta = \Psi_{\beta\alpha}(z_\alpha,\theta_\alpha) \label{Mtrans}$$ in $U_\alpha \cap U_\beta$. The associated split supercurve $M_{\rm split}$ is obtained by setting to zero the generators $\eta_i$ of $\Lambda$ in the transition functions. The body or reduced space $M_{\rm red}$ is obtained by dropping $\theta$ as well. We denote by ${{\cal K}}$ the canonical bundle of $M$, the line bundle having transition functions $k^{-1}_{\beta\alpha}$, where $$k_{\beta\alpha} =
{\rm sdet}\left[\begin{array}{cc}
\partial_z F_{\beta\alpha} & \partial_z \Psi_{\beta\alpha} \\
\partial_\theta F_{\beta\alpha} & \partial_\theta \Psi_{\beta\alpha}
\end{array}\right].$$ Its sections will be called Abelian differentials. The associated $N=2$ SRS $M_2$ has coordinates $(z,\theta,\rho)$ with the transition functions [@schwarz; @berg] $$z_\beta = F_{\beta\alpha}(z_\alpha,\theta_\alpha),\;\;\;\;
\theta_\beta = \Psi_{\beta\alpha}(z_\alpha,\theta_\alpha),\;\;\;\;
\rho_\beta = \rho_\alpha k_{\beta\alpha}(z_\alpha,\theta_\alpha) +
\phi_{\beta\alpha}(z_\alpha), \label{N=2transfuncts}$$ where $$\phi_{\beta\alpha} = \frac{\partial_\theta F_{\beta\alpha}}
{\partial_\theta \Psi_{\beta\alpha}}.$$ The sheaf of functions ${{\cal O}_2}$ on $M_2$ is then an extension of ${{\cal K}}$ by the sheaf $\cal O$ of functions on $M$; there is an exact sequence $$0 \longrightarrow {\cal O} \stackrel{\rm inc}{\longrightarrow} {{\cal O}_2}\stackrel{D^+}{\longrightarrow} {{\cal K}}\longrightarrow 0, \label{exact}$$ where $D^+ = \partial_\rho$. That is, the (local) functions on $M_2$ include the functions on $M$ as those which do not depend on $\rho$, whereas the coefficient of $\rho$ in a function on $M_2$ transforms as a section of ${{\cal K}}$. The compatibility of the transition functions imposes the following cocycle relation on $\phi_{\beta\alpha}$: $$\phi_{\gamma\alpha}(z_\alpha) =
\phi_{\gamma\beta}(z_\beta) +
k_{\gamma\beta}(z_\beta,\theta_\beta) \phi_{\beta\alpha}(z_\alpha).$$ This is the cocycle in $H^1(M,{{\cal K}}^{-1})$ which classifies extensions of the form (\[exact\]). If a function $G(z,\theta,\rho)$ on $M_2$ is written in the form $g(z,\theta) - \rho \gamma(z,\theta)$, then the pair $(g,\gamma)$ transforms as $$\left[\begin{array}{c}g_\beta \\ \gamma_\beta\end{array}\right] =
\left[\begin{array}{cc}1 & k^{-1}_{\beta\alpha}\phi_{\beta\alpha} \\
0 & k^{-1}_{\beta\alpha}\end{array}\right]
\left[\begin{array}{c}g_\alpha \\ \gamma_\alpha\end{array}\right],
\label{vectorbundle}$$ i.e., as a section of a rank $(1,1)$ vector bundle $E$ over $M$. (A.S. Schwarz has informed me that the equivalence between ${{\cal O}_2}$ and $E$ has been observed previously [@icmschwarz].) If the extension cocycle $\phi_{\beta\alpha}$ vanishes, this reduces to the direct sum of line bundles ${\cal O} \oplus {{\cal K}}$. If the cocycle is cohomologically trivial, meaning $$\label{cohomtriv}
\phi_{\beta\alpha} = \sigma_\beta - k_{\beta\alpha}\sigma_\alpha$$ for some functions $\sigma_\alpha(z_\alpha,\theta_\alpha)$ on $U_\alpha$, then the change of coordinates $$\label{coordchange}
\tilde{\rho}_\alpha = \rho_\alpha - \sigma_\alpha(z_\alpha,\theta_\alpha)$$ induces a change of basis $(g,\gamma) \rightarrow (g-\sigma\gamma,\gamma)$ in the fibers of $E$ which again reduces it to $\cal O \oplus {{\cal K}}$. We will show below that this cocycle is always trivial when $M$ is a SRS.
The correspondence between supercurves $M$ and $N=2$ SRSs is generically two-to-one; there is a dual curve $\hat{M}$ yielding the same $M_2$. To construct it, introduce in each chart the new coordinate $u = z - \theta\rho$. One finds that $(u,\rho)$ then transform independently of $\theta$, so they patch together as coordinates for another supercurve, $\hat{M}$. Its transition functions are $$u_\beta = \hat{F}_{\beta\alpha}(u_\alpha,\rho_\alpha),\;\;\;\;\;\;\;\;
\rho_\beta = \hat{\Psi}_{\beta\alpha}(u_\alpha,\rho_\alpha),$$ where $$\hat{F} = F + \frac{DF}{D\Psi} \Psi,\;\;\;\;\;\;\;
\hat{\Psi} = \frac{DF}{D\Psi},$$ and as usual $D = \partial_\theta + \theta \partial_z$. The exact sequence analogous to (\[exact\]) is $$0 \longrightarrow \hat{\cal O} \stackrel{\rm inc}{\longrightarrow} {{\cal O}_2}\stackrel{D^-}{\longrightarrow} \hat{{{\cal K}}} \longrightarrow 0, \label{hatexact}$$ where $D^- = \partial_\theta + \rho \partial_z$. These operators $D^\pm$ are the usual superconformal derivatives on an $N=2$ SRS, satisfying $(D^\pm)^2 = 0,\;\{D^+,D^-\} = \partial_z$. However, $(z,\theta,\rho)$ are coordinates adapted to the projection $M_2 \rightarrow M$ rather than the customary superconformal coordinates \[which are $(z-\frac{1}{2}\theta\rho,\theta,\rho)$\].
Contour integration theory on an $N=2$ SRS takes the following form [@cohn]. The derivative of a function $G$ is defined to be the pair $D^+G \oplus D^-G$, which is a section of ${{\cal K}}\oplus \hat{{{\cal K}}}$. Conversely, an antiderivative $G$ of such a section $\omega \oplus \hat{\omega}$ always exists locally and is unique up to a constant. In coordinates, if $G = a + \theta \alpha - \rho (\beta + \theta b)$, then $$\begin{aligned}
D^+G \oplus D^-G & = & (-\beta - \theta b) \oplus
[\alpha + \rho (a'+b + \theta \alpha')]. \nonumber \\
& = & [-\beta(z) - \theta b(z)] \oplus
[\alpha(u) + \rho a'(u) + \rho b(u)]. \label{derivs}\end{aligned}$$ In finding an antiderivative, $\alpha,\beta,$ and $b$ can be read off directly, and the only actual integration required is in finding $a$ from $a'$. The definite (contour) integral $\int_{P_0}^P \omega \oplus \hat{\omega}$, with $P_0$ a fixed $\Lambda$-valued basepoint and $P$ a variable point of $M_2$, is defined directly as $G(P) - G(P_0)$ when the contour lies in a single contractible chart, and in general by subdividing any contour into such pieces. It is generally a multivalued function of $P$, changing by periods when $P$ is taken around homology cycles. This integration theory on $M_2$ is consistent with the corresponding integration theories on $M$ and $\hat{M}$ [@berg]. If $c_2$ is a contour on $M_2$, it projects down to contours $c$ and $\hat{c}$ on $M$ and $\hat{M}$ respectively, and we have the relation, $$\int_{c_2} \omega \oplus \hat{\omega} = \int_{\hat{c}} \omega +
\int_c \hat{\omega}.$$ (Note that differentials on a supercurve are invariantly integrated over contours on the dual curve.) In particular, for closed contours, $c_2,c,\hat{c}$ all define the same homology class and the periods of $\omega,\hat{\omega}$ sum to the periods of the pair.
Now let us specialize to the case that $M$ is itself an $N=1$ SRS (with odd spin structure) [@crane]. Then $M$ is self-dual; $M=\hat{M}$ [@schwarz] (and in fact [*only*]{} SRSs are self-dual [@berg]). Its transition functions are superconformal, and conventionally parametrized as $$z_\beta = F_{\beta\alpha}(z_\alpha,\theta_\alpha) =
f_{\beta\alpha}(z_\alpha) + \theta_\alpha \psi_{\beta\alpha}(z_\alpha)
\sqrt{f'_{\beta\alpha}(z_\alpha)},
\label{srstrans}$$ $$\label{oddtrans}
\theta_\beta = \Psi_{\beta\alpha}(z_\alpha,\theta_\alpha) =
\psi_{\beta\alpha}(z_\alpha) + \theta_\alpha
\sqrt{f'_{\beta\alpha}(z_\alpha) +
\psi_{\beta\alpha}(z_\alpha)\psi'_{\beta\alpha}(z_\alpha)}.$$ The canonical bundle ${{\cal K}}$ has the transition function $$k^{-1}_{\beta\alpha} = (D_\alpha \Psi_{\beta\alpha})^{-1},$$ satisfying $D_\beta = k^{-1}_{\beta\alpha} D_\alpha.$ This is the usual bundle of superconformal tensors of weight $1$. $M_2$ has the additional transition function $$\begin{aligned}
\rho_\beta & = & \psi_{\beta\alpha}(z_\alpha) + \rho_\alpha
[\sqrt{f'_{\beta\alpha}(z_\alpha) +
\psi_{\beta\alpha}(z_\alpha)\psi'_{\beta\alpha}(z_\alpha)} +
\theta_\alpha \psi'_{\beta\alpha}] \nonumber \\
& = & \psi_{\beta\alpha}(z_\alpha) + \rho_\alpha k_{\beta\alpha}.\end{aligned}$$ The cocycle $\phi_{\beta\alpha}$ characterizing ${{\cal O}_2}$ as an extension of ${{\cal K}}$ is simply $\psi_{\beta\alpha}$. Then the vector bundle description (\[vectorbundle\]) of ${{\cal O}_2}$ precisely reduces to the rank $(1,1)$ vector bundle $E$ on $M$ defined by Rogers and Langer, cf. Eq. (11) of [@rogers].
Because $M$ is self-dual, any Abelian differential $\omega$ on $M$ is also a differential on $\hat{M}$, so it gives rise to a pair $\omega \oplus -\omega$ which can be integrated on $M_2$. Furthermore, all periods of this pair vanish: $$\oint_{c_2} \omega \oplus -\omega = \oint_c \omega - \oint_c \omega = 0.$$ Thus, once we choose a basepoint $P_0$, each section $\omega$ of ${{\cal K}}$ maps to a single-valued global function on $M_2$ by $$\omega \mapsto \int_{P_0}^P \omega \oplus -\omega,$$ and a change of basepoint alters this function by a constant. In coordinates, $$\begin{aligned}
\omega \oplus -\omega & = & [-\beta(z) - \theta b(z)] \oplus
[\beta(u) + \rho b(u)] \nonumber \\
& = & (-\beta - \theta b) \oplus [\beta + \rho (b + \theta \beta')].\end{aligned}$$ Compared to the general case (\[derivs\]) we have $\alpha = \beta$ and $a'=0$. The antiderivative is trivially $$G(z,\theta,\rho) = a + \theta \beta(z) - \rho[\beta(z) + \theta b(z)],$$ with $a$ constant.
A superconformal function is defined by Rogers and Langer as a (global) section of $E$ locally represented by pairs $(g,\gamma)$ satisfying $$\gamma(z,\theta) = D[g(z,\theta) + r(z)],$$ for some $r(z)$. This is equivalent to the constraint $\alpha = \beta$ on the corresponding section $G$ of ${{\cal O}_2}$. The global superconformal functions automatically satisfy the constraint $a'=0$ as well [@rogers], so we have identified them as the subset of the functions on $M_2$ obtained by integrating global sections of ${{\cal K}}$. They are in one-to-one correspondence with such sections, apart from an integration constant.
The local situation becomes clear after a change of coordinates (\[coordchange\]) that reduces ${{\cal O}_2}$ to a sum of line bundles. Indeed, condition (\[cohomtriv\]) for the cohomological triviality of $\psi_{\beta\alpha}$ is satisfied with $\sigma_\alpha = \theta_\alpha$ as an immediate consequence of (\[oddtrans\]). The transition function for the new coordinate $\tilde{\rho} = \rho - \theta$ is then $$\tilde{\rho}_\beta = \tilde{\rho}_\alpha k_{\beta\alpha}.$$ Writing functions on $M_2$ in the form $a + \theta \tilde{\alpha} - \tilde{\rho} (\beta + \theta b)$, where $\tilde{\alpha} = \alpha - \beta$, explicitly shows the splitting ${{\cal O}_2}= {\cal O} \oplus {{\cal K}}$. Although Rogers and Langer consider superconformal functions only as global objects, we can also view them locally as sections of the subsheaf ${{\cal SC}}$ of ${{\cal O}_2}$ defined by $\tilde{\alpha}=0$ and $a=$ constant in every chart. Indeed, the second condition follows from the first for infinite-dimensional $\Lambda$, since the preservation of $\tilde{\alpha}=0$ under arbitrary coordinate changes (\[srstrans\]) requires $\psi a' = 0$ for all odd $\psi$, which makes $a$ constant in this case (for finite-dimensional $\Lambda$ we can add functions proportional to $\eta_1 \eta_2 \cdots \eta_L$). In terms of the splitting ${{\cal O}_2}= {\cal O} \oplus {{\cal K}}$, ${{\cal SC}}$ consists of ${{\cal K}}$ plus constants from $\cal O$.
The global sections of ${{\cal K}}$ have been discussed in [@luke] and completely described in [@dhoker]. For the split SRS $M_{\rm split}$, sections have the form $\omega = \beta + \theta b$, with $\beta$ a holomorphic spinor on the body $M_{\rm red}$ and $b$ an Abelian differential on the body. One can choose a basis of such spinors and differentials and try to modify the resulting split sections $\omega$ order by order in the generators of $\Lambda$ to obtain a basis of sections on $M$. In [@dhoker] this problem is analyzed in the 2d supergravity picture with the $\Lambda$-dependence of the transition functions of $M$ (i.e., the supermoduli) described by a super Beltrami differential $\chi$, with the result that $\beta$ can always be extended to a section, but $\theta b$ extends iff $b$ obeys a nilpotent constraint (Eq. (6.23) of [@dhoker]) $\int d^2z\; h(z) \chi(z) b(z) = 0$ for every holomorphic spinor $h$ (generically there is exactly one up to normalization) on $M_{\rm red}$. The usual Abelian differentials $b$ will generally not satisfy this constraint, but since $\chi$ is odd, multiples by sufficiently nilpotent elements of $\Lambda$ will, showing that the global sections $H^0(M,{{\cal K}})$ are not freely generated. As a result, neither are the superconformal functions $H^0(M,{{\cal SC}})$. The extra condition imposed on superconformal functions in [@corrig] has the effect of restricting them to the free submodule generated by the extensions of the spinors $\beta$ only.
We have seen that, for any SRS $M$, ${{\cal O}_2}$ splits as ${\cal O} \oplus {{\cal K}}$ because the extension cocycle $\psi_{\beta\alpha}$ is trivial in $H^1(M,{{\cal K}}^{-1})$. The reader may still wonder how this is possible without the supermoduli which distinguish $M$ from $M_{\rm split}$ (more precisely, since the bosonic moduli in $f_{\beta\alpha}$ may also contain nilpotents, from a projected SRS having $\psi_{\beta\alpha}=0$) being trivial as well. The two issues are conceptually distinct: the splitting of ${{\cal O}_2}$ means that a change of the $\rho$ coordinate can remove the term $\psi$ from its transition functions, while the projectedness of $M$ would mean that a superconformal change of the $(z,\theta)$ coordinates could remove $\psi$ from their transition functions. The obstructions to the latter lie in the tangent space to supermoduli space, $H^1(M,{{\cal K}}^{-2})$ [@lebrun]. To avoid discussion of the global structure of supermoduli space, consider the issue at the linearized level only. To lowest order in the generators $\eta_i$ of $\Lambda$, sections of ${{\cal K}}^{-1}$ and ${{\cal K}}^{-2}$ split into components as $${{\cal K}}^{-1} = {{\cal K}}^{-1}_{\rm red} \oplus \theta {{\cal K}}^0_{\rm red},\;\;\;\;
{{\cal K}}^{-2} = {{\cal K}}^{-2}_{\rm red} \oplus \theta {{\cal K}}^{-1}_{\rm red},$$ where ${{\cal K}}^{-1}_{\rm red}$ is the square root of the holomorphic tangent bundle to $M_{\rm red}$. To this order, $\psi = \eta_i \psi_i$ with $\psi_i$ belonging to $H^1(M_{\rm red},{{\cal K}}_{\rm red}^{-1})$, so that as we know $\psi$ is a cocycle in $H^1(M,{{\cal K}}^{-1})$ (in fact, to all orders in the $\eta_i$), while $\theta \psi$ is a cocycle in $H^1(M,{{\cal K}}^{-2})$ (to lowest order only). However, $H^1(M,{{\cal K}}^{-2})$, Serre dual to $H^0(M,{{\cal K}}^3)$, is freely generated while $H^1(M,{{\cal K}}^{-1})$, dual to $H^0(M,{{\cal K}}^2)$, is not [@luke]. This means precisely that a nilpotent combination of nontrivial classes $\psi_i$ can be trivial in $H^1(M,{{\cal K}}^{-1})$ but not in $H^1(M,{{\cal K}}^{-2})$.
The question of which classes of supercurves enjoy the splitting ${{\cal O}_2}= {\cal O} \oplus {{\cal K}}$ is discussed further in [@berg], particularly for the “generic SKP curves" which yield algebro-geometric solutions to the super KP hierarchies. In contrast to the situation just discussed for SRSs, for such curves this splitting is equivalent to projectedness.
Acknowledgements {#acknowledgements .unnumbered}
================
This work grew out of a collaboration with Maarten Bergvelt [@berg], and I have enjoyed discussions with him and with Mitchell Rothstein.
[99]{}
S. B. Giddings and P. Nelson, “Line bundles on super Riemann surfaces," Commun. Math. Phys. [**118**]{} (1988) 289–302.
L. Hodgkin, “Problems of fields on super Riemann surfaces," J. Geom. and Phys. [**6**]{} (1989) 333–348.
E. D’Hoker and D. H. Phong, “Conformal scalar fields and chiral splitting on super Riemann surfaces," Commun. Math. Phys. [**125**]{} (1989) 469–513.
A. Rogers and M. Langer, “New fields on super Riemann surfaces," Classical and Quantum Gravity [**11**]{} (1994) 2619–2626.
A. Rogers and M. Langer, Corrigendum to “New fields on super Riemann surfaces," Classical and Quantum Gravity [**12**]{} (1995) 2619–2620.
S. N. Dolgikh, A. A. Rosly and A. S. Schwarz, “Supermoduli spaces," Commun. Math. Phys. [**135**]{} (1990) 91–100.
M. J. Bergvelt and J. M. Rabin, “Supercurves, their Jacobians, and super KP equations," preprint, January 1996, alg-geom/9601012.
J. Cohn, “$N=2$ super Riemann surfaces," Nucl. Phys. [**B284**]{} (1987) 349–364.
A. Rogers, “A global theory of supermanifolds," J. Math. Phys. [**21**]{} (1980) 1352–65.
B. S. DeWitt, [*Supermanifolds,*]{} second edition, Cambridge University Press, Cambridge, New York 1992.
P. Nelson, “Lectures on supermanifolds and strings," in [*Particles, Strings, and Supernovae*]{}, ed. A. Jevicki and C.-I. Tan, World Scientific, Singapore 1989; Yu. I. Manin, [*Topics in Noncommutative Geometry,*]{} Princeton University Press, Princeton, 1991.
A. S. Schwarz, “Geometry of fermionic string," in [*Proceedings of the International Congress of Mathematicians, Kyoto 1990*]{}, Vol. II, ed. I. Satake, Springer-Verlag, Tokyo, New York 1991. See esp. p. 1383.
L. Crane and J. M. Rabin, “Super Riemann surfaces: uniformization and Teichmüller theory," Commun. Math. Phys. [**113**]{} (1988) 601–623.
C. LeBrun and M. Rothstein, “Moduli of super Riemann surfaces," Commun. Math. Phys. [**117**]{} (1988) 159–176.
|
---
abstract: |
Dark matter cores within galaxy haloes can be formed by energy feedback from star forming regions: an energy balance suggests that the maximum *core formation efficiency* arises in galaxies with [M$_{\rm star}$]{}$\sim10^{8.5}$[M$_\odot$]{}. We show that a model population of galaxies, in which the density profile has been modified by such baryonic feedback, is able to explain the observed galaxy velocity function and Tully-Fisher relations significantly better than a model in which a universal cuspy density profile is assumed.
Alternative models, namely warm or self-interacting dark matter, also provide a better match to these observed relations than a universal profile model does, but make different predictions for how halo density profiles vary with mass compared to the baryonic feedback case. We propose that different core formation mechanisms may be distinguished based on the imprint they leave on galaxy populations over a wide range of mass.
Within the current observational data we find evidence of the expected signatures of the mass dependence of core formation generated by baryonic feedback.
bibliography:
- 'archive.bib'
date: 'Accepted XXXX . Received XXXX; in original form XXXX'
title: Signatures of Dark Matter Halo Expansion in Galaxy Populations
---
\[firstpage\]
galaxies: evolution - formation - haloes cosmology: theory - dark matter
Introduction {#sec:introduction}
============
The rotational velocity of gas and stars in galaxy discs implies that galaxies are embedded within dark matter ‘haloes’. A central tenet of the canonical cold dark matter ([$\Lambda$CDM]{}) cosmological model is that such haloes have dense central regions with steep, “cuspy" inner density profiles (@navarro96, hereafter NFW profile). However, observed rotation curves imply that dwarf galaxies have flat “cores" at their centre [@moore94]. This “cusp-core" crisis is a major challenge to the [$\Lambda$CDM]{} paradigm, heightened by studies which demonstrate that some non-standard dark matter particles form cored haloes [@maccio12; @vogelsberger12].
On the other hand, astrophysical processes [@navarro96b; @pontzen12] within a cold dark matter framework may modify halo density profiles and create cored haloes at specific halo masses [@DiCintio2014a]. While observed rotation curves of individual galaxies provide compelling evidence for the existence of cores in low surface brightness and dwarf galaxies [@oh11], they are not able to distinguish between the various proposed origins of cores. Therefore, it is essential to establish whether the “cusp-core" crisis is pointing to a new cosmological paradigm, or is instead the result of astrophysical processes within the standard [$\Lambda$CDM]{} model.
In the [$\Lambda$CDM]{} paradigm, cores may result from the non-adiabatic impact of gas outflows on dark matter haloes. There is significant evidence [@weiner09; @martin12] that energy feedback from star-formation activity drives gas out of galaxies, and that processes such as radiation energy from massive stars, stellar winds and supernova explosions are important in galaxy formation [e.g. @brook11]. The degree to which such energetic processes flatten the inner density profiles of haloes depends on the ratio of their stellar to halo masses [@DiCintio2014a see Section 2 for an analytic argument].
Low mass galaxies are dark matter dominated: those with [M$_{\rm star}$]{}${\lower.5ex\hbox{{$\; \buildrel < \over \sim \;$}}}$3$\times$10$^6$M$_\odot$ do not produce enough energy to flatten their halo’s inner density profile, which remains steep [@penarrubia12; @governato12]. As stellar mass increases relative to the dark matter mass, the inner density profile becomes increasingly flat [@governato12]. The flattening is greatest when [M$_{\rm star}$]{}$\sim$3$\times$10$^8$[M$_\odot$]{}, after which the increasingly deep potential well is able to oppose halo expansion resulting in less flattening for more massive galaxies [@DiCintio2014a].
One approach to test core formation by baryonic processes has been to study low mass dwarf galaxies, where the theory predicts cuspy, NFW profiles. A multitude of studies have attempted to determine whether galaxies such as Sculptor have a core [@Battaglia08; @walker09; @evans09; @walker11; @agnello12; @amorisco12; @laporte13; @adams14; @strigari14], without a clear consensus emerging.
In this study, we therefore take another approach and explore the properties of an ensemble of galaxies. We show that the mass dependence of cores formed through baryonic processes should leave an imprint on galaxy populations, and that such imprint may be used to distinguish between different core formation mechanisms. Indeed, while alternative dark matter particles could also be able to create cores, they would result in a different dependence of core sizes versus galaxy mass.
In particular, we study the velocity function of galaxies [@zavala09; @zwaan10; @tg11; @papastergis11; @obreschkow13; @klypin14], as well as the relation between velocity and stellar mass [@tully77], over a wide mass range in which core sizes are expected to vary in different ways according to different models.
To model haloes that are flattened by energetic feedback processes, we use the mass dependent density profile [@DiCintio2014b], hereafter DC14 profile, in which galaxies with [M$_{\rm star}$]{} between $\sim$3$\times$$10^6$ and $\sim$$10^{10}$ [M$_\odot$]{} have central densities shallower than the NFW profile [@DiCintio2014a]. The DC14 model has already been used to study the variation in core sizes for Local Group dwarf galaxies, in which their observed kinematics has been explained in terms of a density profile dependent on [M$_{\rm star}$]{}/[M$_{\rm halo}$]{} [@brook15].
Standard alternative dark matter models, in particular a warm dark matter (WDM) and a self interacting dark matter (SIDM) one, are also explored and the expected properties of galaxy populations within such models are compared with the fiducial NFW model. We highlight the manner in which the alternative dark matter models presented can be generalized to a wide range of parameterizations.
In Section 2 we present the ingredients of our models, including an analytic argument for our expectation of how core sizes vary with mass in the baryonic feedback scenario. In Section 3 we create velocity distribution functions of galaxies and [M$_{\rm star}$]{}-velocity relations, and we compare the predictions coming from the DC14 model to the results obtained by using a standard NFW model (Sections 3.1 and 3.2). We also compare velocity functions and [M$_{\rm star}$]{}-velocity relations for the alternative dark matter models, namely WDM and SIDM (Section 3.3). We discuss possible uncertainties and caveats in Section 3.4 and 3.5. In Section 4 we show that the different models result in different observational trends with mass within galaxy populations, and that such trends may be distinguishable with careful observations.
![ Left Panel:The ratio of supernova energy from stars within the central 1$\,$kpc, E$_{SN}$(r$<$1$\,$kpc), and the energy required to transform an NFW profile to a profile with a 1$\,$kpc core ($\Delta$W), plotted as a function of [M$_{\rm star}$]{}. Only galaxies in the mass range, $3\times$10$^6$${\lower.5ex\hbox{{$\; \buildrel < \over \sim \;$}}}$ [M$_{\rm star}$]{}${\lower.5ex\hbox{{$\; \buildrel < \over \sim \;$}}}$10$^{10}$[M$_\odot$]{} have sufficient energy to form cores. Right Panel: The ratio of supernova energy from stars within the central 1$\,$kpc, and the potential of an NFW profile with embedded baryonic disk](plots/F1.png){width=".5\textwidth"}
as a function of [M$_{\rm star}$]{}. This ratio gives an indication of the efficiency of core formation: one expects the largest cores at [M$_{\rm star}$]{}$\sim$3$\times$10$^8$[M$_\odot$]{}.
\[fig:analytic\]
**Ingredients of our models**
=============================
Our models consist of a mass function for dark matter haloes with appropriate concentrations, a relation between stellar and halo mass, and exponential stellar and gas discs. Dark matter haloes are modelled by different density profiles, as explained below. Halo mass distributions for [$\Lambda$CDM]{} and WDM assume Planck cosmological parameters and are created with [HMFcalc]{} [@HMFcalc13], with appropriate corrections to the WDM mass function [@schneider12]. The WDM model has a particle mass of 2$\,$KeV, with the effect of using 1, 2 & 3$\,$KeV particles shown in section \[wdmvary\].
We use the concentration-mass relation from Planck [$\Lambda$CDM]{} [@Dutton14], and adjust the WDM halo concentration [@schneider12]. An adiabatic contraction correction to the dark matter haloes is applied to all models, as found in cosmological simulations for galaxies as massive as the Milky Way [@DiCintio2014b]. Such contraction alters the concentration (i.e. the scale radius) rather than the inner slope, allowing core sizes to be set. Specifically we use $${\rm C}=(1.0+ 0.00001e^{3.4{\rm X}})\times {\rm C}_{\rm NFW}\\
\label{c}$$ where X$=\log_{10}$([M$_{\rm star}$]{}/[M$_{\rm halo}$]{}) + 4.5, and C$_{\rm NFW}$ is the concentration given from the adopted [@Dutton14] relation.\
Haloes are matched to galaxy stellar masses using the empirical abundance matching relation of @guo10. We explore the effects of different abundance matching relations in Section \[section:ab\].
Once a relation between dark matter halo mass and galaxy stellar mass has been set, we add a stellar exponential disc with scale-length h$_{s}$ taken from the observed [M$_{\rm star}$]{}-h$_s$ relation, log$_{10}$h$_{\rm s}$=$-$2.462+0.281log$_{10}$[M$_{\rm star}$]{}(equation 7 in @dutton11b). We further add a gas disk with mass M$_{\rm gas}$$=$1.3M$_{\rm HI}$, with log$_{\rm 10}$(M$_{\rm HI}$/[M$_{\rm star}$]{})$=$$-0.43$log$_{10}$[M$_{\rm star}$]{}$+3.75$ [@papastergis12] and disc scale length h$_{g}$=3h$_{s}$.
The circular velocity of a thin exponential disk of mass M$_{\rm d}$ and scale length h$_{\rm i}$ is given by:
$$\rm{V}_{disk}^2(r)=\frac{GM_d}{h_i}2y^2[I_0(y)K_0(y) - I_1(y)K_1(y)]$$
where $y=r/(2h_i)$ and $I_n,K_n$ are modified Bessel functions [@dutton11].
In all models we include the universal dark matter fraction of mass ($\Omega_{\rm DM}$/$\Omega_{m}$=0.85) in dark matter haloes. The density profile of dark matter haloes is set to be either the universal, NFW one [@navarro96], or the mass-dependent DC14 density profile described in the next section [@DiCintio2014b].
WDM halos are known to have cored profiles [@tremaine79; @bode01]. However, [@maccio12] find that WDM particles with the masses considered in our study, 1-3KeV, result in very small cores of order 10pc for halos of mass 10$^{10}$[M$_\odot$]{}, and that it requires much lower mass ($\sim$0.1KeV) WDM particles to create cores of 1kpc. In this study, we approximate the very small cores of WDM halos in our model by retaining the NFW profile.
In the adopted SIDM model, core sizes are set to zero for Milky Way mass galaxies ([M$_{\rm star}$]{}=6$\times$10$^{10}$[M$_\odot$]{}) and increases as the scale-length decreases, until reaching a core size of 1$\,$kpc for [M$_{\rm star}$]{}$=$10$^8$[M$_\odot$]{} [@vogelsberger12; @vogelsberger14]. Lower mass haloes all have core sizes of 1$\,$kpc. Other parameterizations for varying core sizes are explored in section \[corevary\]. Of course, they do not cover all SIDM models and it is possible to fine tune SIDM parameters in order to have a mass variation of halo profile similar to the one of baryonic outflow models. On the other hand, the parameterisations that we do explore here can be generalized, since they are representative of a range of alternative dark matter candidates that predict similar behaviour of core size with galaxy mass.
Cored profiles of dark matter for the SIDM case take the form:
$$\rho(r)=\frac{\rho_0 r_{scale}^3}{(r_c+r)(r_{scale}+r)^2}$$
where $\rho_0$ is a characteristic halo density, $r_{scale}$ is a scale radius and $r_c$ is the core size [@penarrubia12]. The velocity profile of dark matter haloes, $\rm{V}_{DM}$, will thus depend on the type of density profile chosen. The theoretical, total circular velocity profile of a galaxy at any radius, defined as V$_{\rm rot}$, is therefore given by the quadratic sum of the circular velocities of the various components:
$$\rm{V}_{rot}(r)=\sqrt{\rm{V}_{star}^2(r) + \rm{V}_{gas}^2(r) + \rm{V}_{DM}^2(r)}$$
The mass dependent ‘DC14’ profile {#DC14}
---------------------------------
The analytic work in this section follows the work of [@penarrubia12] who model the balance between the gravitational potential of galaxies and the energy from central star formation regions as the main determinant of core formation in the case of cores created by astrophysical processes [@penarrubia12]. Once an exponential stellar and gas discs have been set as explained above, using observed scaling relations [@dutton11b; @fathi10; @papastergis12], it is possible to compute the supernova energy from stars within the central 1$\,$kpc of each galaxy and compare with the depth of its potential well.
The left panel of Figure \[fig:analytic\] shows the ratio of such supernova energy, E$_{SN}$(r$<$1$\,$kpc), divided by the gravitational potential energy $\Delta$W required to transform an initially NFW halo to a halo whose profile has a 1$\,$kpc core, plotted as a function of [M$_{\rm star}$]{}.
$\Delta$W is defined as $\Delta$W=(W$_{cusp}$-W$_{core}$)/2, where: $$W=-4\pi G\int_0^{rvir}\rho(r)M(r)rdr$$ The supernova energy within 1 Kpc is easily computed by: $$E_{SN}=\rm M_{\rm star}(<1kpc)\bar{f}_{SN}\epsilon10^{51}erg$$
where $\bar{f}_{SN}$ is the fraction of stars that will explode as supernovae divided by the mean stellar mass according to the initial mass function selected ($\bar{f}_{SN}=0.00925$ for a @kroupa02 IMF), $\epsilon$ is the fraction of energy that will be transferred into the dark matter halo and $10^{51}$erg is the energy released during a single supernova event.
Figure \[fig:analytic\] assumes that galaxies are connected to dark matter haloes via the abundance matching relation of [@guo10]. It is clear that only galaxies in the mass range 3$\times$10$^6$${\lower.5ex\hbox{{$\; \buildrel < \over \sim \;$}}}$ [M$_{\rm star}$]{}${\lower.5ex\hbox{{$\; \buildrel < \over \sim \;$}}}$$\times$10$^{10}$[M$_\odot$]{} have sufficient energy to form cores.
The right panel of Figure \[fig:analytic\] shows the ratio of supernova energy from stars within the central 1$\,$kpc, and the gravitational potential energy of an NFW halo plus a baryonic disk as a function of [M$_{\rm star}$]{}. This ratio, namely the balance between SNe energy vs potential of the galaxy, gives an indication of the efficiency of core formation: the largest cores are expected at the peak of this relation, i.e. at [M$_{\rm star}$]{}$\sim$3$\times$10$^8$[M$_\odot$]{}.
In Figure \[fig:analytic\] we used $\epsilon=0.4$, and we note that using different values will simply shift the plots up or down, changing the minimum mass for core formation but without modifying the position of the peak. Whilst we have included only energy from SNe in this calculation, it is likely that ionising radiation will help drive outflows [@murray10]: again our analytic argument of where we expect the peak of core formation will not be affected, as long as this feedback scales with star formation rate.
We will explore the effects of AGN, which do not scale with star formation rate, later in the paper; for now we just note that the stellar mass where core formation is most efficient is below the mass range where we expect AGN feedback to be important. Bulges have not been included in our models because at the core formation peak mass in Figure \[fig:analytic\] most galaxies do not have bulges. Nevertheless, we tested the results of adding empirically motived bulges, following the Hernquist formalism [@hernquist90]: we found that the only difference is the appearance of a “shoulder" on the right side of the the plots, since more SNe energy is stored within the inner 1kpc of a galaxy with a bulge, while the peak of core formation does not get affected.
Of course, our model is very simplistic. In particular, [@penarrubia12] showed that the energy requirements for core formation are significantly lower at higher redshift. And further, [@maxwell15] recently showed that the radius to which dark matter is moved when it is expelled from the cusp region, will also have an effect on the energy required to form a core. Yet a very similar relation between core formation and galaxy mass has been found in cosmological simulations [@DiCintio2014a] of galaxies that match a wide range of galaxy scaling relations [@brook12b; @stinson13; @kannan14; @obreja14]: within these simulations, the flattest profiles are also found at [M$_{\rm star}$]{}$\sim$3$\times$10$^{8}$[M$_\odot$]{}.
The mass dependent DC14 density profile is based on such cosmological simulations, and accounts for the expansion of dark matter haloes due to the effects of feedback from star forming regions [@DiCintio2014b]. These profiles self consistently account for the distance to which dark matter is moved, addressing the concerns of [@maxwell15], and take the form: $$\rho(r)=\frac{\rho_s}{\left(\frac{r}{r_s}\right)^{\gamma}\left[1 + \left(\frac{r}{r_s}\right)^{\alpha}\right] ^{(\beta-\gamma)/\alpha}}
\label{five}$$ where ($\alpha,\beta,\gamma$) indicate the sharpness of the transition, the outer and the inner slope, respectively [@merritt06].
The DC14 model describes profiles that have a range of inner slopes, with the parameters of the double power law model being dependent on the stellar-to-halo mass ratio of a galaxy in the following way [@DiCintio2014b]:
$$\begin{aligned}
&\alpha= 2.94 - \log_{10}[(10^{X+2.33})^{-1.08} + (10^{X+2.33})^{2.29}]\\
&\beta=4.23+1.34X+0.26X^2\\
&\gamma= -0.06 + \log_{10}[(10^{X+2.56})^{-0.68} + (10^{X+2.56})]
\end{aligned}
\label{abg}$$
where $X=\log_{10}($[M$_{\rm star}$]{}/[M$_{\rm halo}$]{}). The scale radius, $r_s$, is connected to the concentration of the halo, which varies with mass.
As long as the flattening of the central dark matter density is mass dependent, and there is a maximum flattening at a similar stellar mass as found in both the analytic model and the simulations, the exact form of the profile will not affect the arguments presented in this study.
{width="1.\textwidth"}
Results
=======
Velocity Function
-----------------
When testing theoretical cosmological models, dark matter haloes can be matched to observed galaxies within a given volume [e.g. @guo10; @moster10; @papastergis14]. The density profiles of dark matter haloes, together with the above mentioned disk distributions of stars and gas, can be used to determine the theoretical velocity profile of a galaxy at a given radius, $\rm{V}_{rot}$, according to Eq.3: such velocity is determined by the total mass of dark matter, gas and stars contained within that radius.
The standard [$\Lambda$CDM]{} model, in which haloes have cuspy, NFW density profiles [@navarro96], predicts a far greater number of galaxies than are observed with line-of-sight velocity ([V$_{\rm los}$]{}) less than $\sim$60 km$\,$s$^{-1}$ [@zavala09; @zwaan10; @tg11; @papastergis11; @klypin14].
This can be seen by comparing the [$\Lambda$CDM]{} prediction, purple line, to the observational data points [@klypin14] in the left panel of Figure \[fig:CDM\]. The observed [V$_{\rm los}$]{} distributions are taken from compiled data of the of HI line-widths of Local Volume galaxies [@karachentsev13] measured at 50% of the maximum intensity, $W50$, with a further accounting for the number of early type galaxies which do not have significant amounts of HI [@klypin14].
The comparison is made assuming that the observed velocities of gas within galaxies track the maximum possible velocity associated with the corresponding dark matter halo, [V$_{\rm max}$]{}. The assumption made in @klypin14 that observed velocities relate directly to [V$_{\rm max}$]{} may be justified in high mass galaxies, but is not necessarily correct in less massive galaxies, where the rotation curves are often still increasing at their outermost measured point [@catinella06; @swaters09; @deblok08; @oh11]. For example, in a dwarf galaxy with stellar mass [M$_{\rm star}$]{}=$10^{7-8}$[M$_\odot$]{}, observed velocity measurements often come from stars within 1-2$\,$kpc of the galaxy centre, while the [V$_{\rm max}$]{} of the corresponding NFW halo is at $\sim$7-10$\,$kpc. If haloes are cored or expanded, then the [V$_{\rm max}$]{} may be even further from the centre of the galaxy: indeed, the maximum observed rotation velocity may be a poor estimate of [V$_{\rm max}$]{} in the case of a cored profile.
In @papastergis14 this issue is addressed by attempting to place observed galaxies on the V$_{\rm rot}$-[V$_{\rm max}$]{} diagram expected from abundance matching, by fitting several dark matter haloes to observed rotation curves and considering only the last measured point of such velocity curves in order to determine the maximum velocity of the most massive halo that fits such point (see also @ferrero12). In this sense, this is a more conservative approach than the one followed by @klypin14. However such methodology still assumes that dark matter haloes follow the universal NFW profile, but for some of the observed rotation curves in low mass galaxies this is not the case (see for example UGC7577 in their Fig.5).
Debating whether rotation curves [*appear flat*]{} may not be the optimal manner to address the issue of how to best compare theory and observation. Models provide predictions for velocities at all radii, so one can compare observations with theoretical predictions for velocities at any radius, as long as the two are done consistently. This is particularly important when comparing models of different density profiles.
We therefore suggest a different approach, which is to estimate the physical radius at which measurements are made, and then compare models and observations at that same radius. In order to determine the radial region where [V$_{\rm los}$]{} is measured in HI surveys, we find the radius, R$_{\rm m}$, at which observed rotation curves reach the inclination and velocity dispersion adjusted value V$_{\rm rot}$.
In Figure \[rmeasure\] we plot, as black squares, R$_{\rm m}$ as a function of [M$_{\rm star}$]{} for 26 observed galaxies which have [M$_{\rm star}$]{}${\lower.5ex\hbox{{$\; \buildrel < \over \sim \;$}}}$10$^{10}$[M$_\odot$]{}, high resolution rotation curves [@oh08; @deblok08; @oh11; @cannon11; @oh15] and corresponding $W50$ values [@walter08]. The orange line shows a fit to the data, $$\label{rm}
\rm log(R_{\rm m})=-2.75+0.37\times \rm log(M_{\rm star}),$$ with the shaded region indicating R$_{\rm m}$$\pm$20%. The purple line is the radius R$_{\rm max}$ at which a halo of a specific stellar mass, according to [M$_{\rm star}$]{}-[M$_{\rm halo}$]{} relations, reaches its maximum velocity [$V_{\rm max}$]{}. It is clear how moving toward low mass galaxies the gas is tracking an increasingly inner region of the halo rather than extending to R$_{\rm max}$. The lower is the mass of a galaxy, the higher is the separation between R$_{\rm m}$ and R$_{\rm max}$. We therefore decided to use the radius R$_{\rm m}$ as the radius where we measure velocities in our theoretical models, to be consistent with observations.
In the left panel of Figure \[fig:CDM\], we derived theoretical velocity functions by first assuming a mass function for dark matter haloes, with appropriate halo concentration-mass dependence, then using the density profiles of different models to derive galaxies’ circular velocities at each radius. We then measure circular velocities at the radius $\rm{R_m}$ to mimic the region where observations are made. We remark that before comparing the observed line-of-sight velocity, [V$_{\rm los}$]{}, with the theoretically predicted rotation curve, $\rm{V}_{rot}$, a correction must be applied to account for the pressure support of gas in real galaxies. When using HI line widths, [V$_{\rm los}$]{}$=$$W50/2$, where $W50$ is the width of the HI line profile measured at 50 per cent of the peak value, and the rotational velocity is then: $$\rm{V}_{rot}=\sqrt{\left(\frac{\rm V_{los}}{sin(i)}\right)^2-\sigma_v^2},
\label{eq:inc}$$ where $i$ is the inclination of the galaxy and $\sigma_v$=8 km/s is the typical amplitude of gas turbulent motion, in addition to rotation [@papastergis14]. In our theoretical velocity function, we average over all possible inclinations. We use equation \[eq:inc\] to transform from circular velocities measured at R$_{\rm m}$ to [V$_{\rm los}$]{} velocities, and do this self-consistently for all the models, i.e. NFW, DC14, WDM and SIDM.
Using the standard [$\Lambda$CDM]{} model with associated cuspy, NFW halo density profiles, the model still overpredicts the number of galaxies with [V$_{\rm los}$]{}${\lower.5ex\hbox{{$\; \buildrel < \over \sim \;$}}}$60$\,$km$\,$s$^{-1}$: the theoretical NFW velocity function lies significantly above observations as can be seen from the blue shaded line in Figure \[fig:CDM\].
This discrepancy motivates the exploration of other models which may lead to different predictions for the distribution of [V$_{\rm los}$]{} of galaxies. We first explore the mass dependent DC14 density profile, based on simulations where cores are formed through outflows driven by star formation activity, and fully described in Section \[DC14\].
![The radius, R$_{\rm m}$, at which the observed rotation curves of 26 observed galaxies taken from the literature reach the inclination and velocity dispersion adjusted value V$_{\rm rot}$$=$$\sqrt{((W50/2/sin(i))^2-\sigma_v^2)}$, plotted against stellar mass (black squares). The orange line is a fit to this data, with the shaded region indicating R$_{\rm m}$$\pm$20%. The purple line is the radius R$_{\rm max}$ at which an halo of a specific stellar mass [@guo10] reaches its maximum velocity [$V_{\rm max}$]{}. The discrepancy between R$_{\rm max}$ and R$_{\rm m}$ is more severe moving toward low mass galaxies.[]{data-label="rmeasure"}](plots/F3.png){width=".45\textwidth"}
The theoretical velocity function based on a DC14 profile is shown as a red shaded region in Figure \[fig:CDM\] and its agreement with observations is notable. As we move from massive to intermediate mass galaxies, the DC14 profile describes haloes which become increasingly flat in their inner regions, resulting in fewer galaxies with 30$\,{\lower.5ex\hbox{{$\; \buildrel < \over \sim \;$}}}\,$[V$_{\rm los}$]{}$\,{\lower.5ex\hbox{{$\; \buildrel < \over \sim \;$}}}\,$60$\,$km$\,$s$^{-1}$, in line with observations. Below [M$_{\rm star}$]{}$\sim$3$\times$10$^8$[M$_\odot$]{}, the inner region of DC14 profiles become increasingly steep. This results in a slight upward bend at [V$_{\rm los}$]{}${\lower.5ex\hbox{{$\; \buildrel < \over \sim \;$}}}$20$\,$km$s^{-1}$ with the [V$_{\rm los}$]{} distribution eventually joining the standard [$\Lambda$CDM]{} model for [V$_{\rm los}$]{}${\lower.5ex\hbox{{$\; \buildrel < \over \sim \;$}}}$10$\,$km$s^{-1}$, where the haloes retain a NFW profile given the insufficient energy from SNe to create a core.
This upward trend in the velocity function at [V$_{\rm los}$]{}${\lower.5ex\hbox{{$\; \buildrel < \over \sim \;$}}}$20$\,$km$s^{-1}$ is a prediction of models in which cores are formed by baryonic outflows, and in which density profiles remain steep in the lowest mass galaxies. Interestingly, the low velocity end of observations also show an upturn, as derived in @klypin14 and precisely as predicted by the DC14 model.
Of course, the upturn in the observations needs to be taken with a grain of salt. Indeed, moving to lower masses, observations are less well constrained for two reasons. Firstly, below stellar mass of $\sim$10$^9$[M$_\odot$]{} the HI disks become on average thicker [@brinks02] with the velocity dispersions being a larger fraction of the total rotation velocity, making the interpretation of $W50$ increasingly difficult and the approximation of where to measure V$_{\rm rot}$ in the models less well defined. Secondly, at these masses a correction for completeness is required to account for the increasingly significant number of galaxies that have no measured HI [@klypin14]. Future surveys with larger samples, more accurate velocity measurements, detailed accounting for completeness and precise understanding of the radius at which velocities are measured, are required in order to determine whether there is a statistically solid upturn in the [V$_{\rm los}$]{} distribution at [V$_{\rm los}$]{}$<$20$\,$km$\,$s$^{-1}$.
{width=".8\textwidth"}
Bearing these caveats in mind, the comparison between the theoretical and observed velocity function highlights three points:\
(i) the cold dark matter model where haloes follow the widely used NFW density profile is ruled out: even by including a correction for the radius at which velocities are measured, the data could not be reconciled with a universal NFW model,\
(ii) the discrepancy between observations and theory can be easily explained as a manifestation of the cusp/core problem, with the mass dependent DC14 profile capturing the essence of it: a notable agreement between theory and observations is found if such a model is applied, and\
(iii) the dark matter cores in galaxies formed by energetic baryonic outflows predict an upturn in the velocity function at low velocities, corresponding to the low mass galaxies regime where the energy form supernovae is less efficient at expanding haloes: such a feature may be tested with a careful comparison between theory and future updated observations.
We note that the results at the low mass end are not particularly sensitive to our disc scale-lengths, as stars become a decreasing fraction of total mass. At the high mass end, the position where the the model [V$_{\rm los}$]{} intersects with the dark matter only [V$_{\rm max}$]{}, is sensitive to the adopted disc scale-lengths, as well as to the assumed adiabatic contraction, but this does not affect our main results.
[M$_{\rm star}$]{}-Velocity Relation
------------------------------------
In this section, we take another perspective of the problem, exploring the relation between the velocity and stellar mass of galaxies [@tully77]. Observational and systematic biases in this process will differ from biases that may exist in the velocity distribution comparison made above.
Galaxy stellar masses from large surveys [@baldry12] can be matched to dark matter haloes [@guo10], allowing us to derive theoretical rotation velocity for galaxies of given stellar mass at any radius. We then obtain high resolution velocity information for a sample of observed galaxies [@pizagno07; @kirby14; @klypin14; @oh08; @walter08; @deblok08; @oh11] and compare observed and theoretical values of [V$_{\rm los}$]{} between 2.2 and 3.5 scale lengths. To derive this region, we use the empirical relation between disc scale-lengths and [M$_{\rm star}$]{} [@dutton11b; @fathi10].
In the right panel of Figure \[fig:CDM\] we show results of this comparison. THINGS galaxies are plotted at both 2.2 (upward pointing triangles) and 3.5 scale-lengths (downward pointing triangles) [@walter08; @oh08; @deblok08; @oh11]. At the high mass galaxies end [@pizagno07], there is very little difference between measuring [V$_{\rm los}$]{} at 2.2 and 3.5 scale-lengths. At the low mass end, the region of 2.2-3.5 disc scale-lengths is a reasonable proxy for the radial range at which kinematics of dwarf spheroidal galaxies are measured [@kirby14].
As shown in the right panel of Figure \[fig:CDM\], the [$\Lambda$CDM]{} model with NFW profile (blue shaded line) predicts that dwarf galaxies with [M$_{\rm star}$]{}$\sim$5$\times$10$^6$$-$5$\times$10$^8\,$M$_\odot$, should have significantly higher [V$_{\rm los}$]{} than is observed, a well documented issue within the standard [$\Lambda$CDM]{} model [e.g. @read06; @oh11; @tollerud14]. The DC14 model (red shaded line), instead, predicts that the dark matter density profile is flattest for massive dwarf galaxies, such that the relation between [V$_{\rm los}$]{}-[M$_{\rm star}$]{} becomes lower than the NFW expectation at masses $10^7$$<$[M$_{\rm star}$]{}/M$_\odot$$<$$10^9$. Moving to even lower stellar masses, the steepening of the halo profile causes the [V$_{\rm los}$]{}-[M$_{\rm star}$]{} relation to flatten, eventually re-joining the NFW model for galaxies with stellar masses of [M$_{\rm star}$]{}$\sim$10$^6$[M$_\odot$]{}.
The well known independence of velocity from stellar mass observed in low mass dwarf galaxies, such as satellites of the Milky Way, Andromeda and ultrafaint dwarfs (@strigari08 [@gk14], and references therein), is therefore a feature naturally reproduced by haloes that are flattened by astrophysical processes, such as supernovae-energy-driven outflows of gas, and that retain a cuspy profile at the smallest scales, [M$_{\rm star}$]{}$<3\times10^6$[M$_\odot$]{} [@governato12; @brooks14; @DiCintio2014a]. The overall result is that the DC14 model can explain the observed [V$_{\rm los}$]{} of dwarf galaxies over a wide range in stellar mass.
A significant caveat must be mentioned here, regarding the observed measures of galaxy velocities. For the dwarf spheroidal galaxies, the circular velocity has been approximated using V$(r_{1/2})=\sqrt{3\langle\sigma^2\rangle}$, which provides a reasonable estimate at the half light radius of non-rotating, dispersion supported galaxies, minimizing the errors introduced by uncertainties of the anisotropy parameter [@walker09; @wolf10]. This methodology has been shown to work also in non-spherically symmetric systems [@thomas11]. Systematic differences exist between these estimates of circular velocity, and those coming from rotation curve data. We therefore do not want to overstate any comparison between the predictions and observations. The purpose of the paper is to highlight ways that competing theories may be contrasted: it remains observationally challenging to test for the differences that we highlight, over the full mass range that may be required.
Alternative Dark Matter Models
-------------------------------
Alternative dark matter particles have also been suggested as solutions to the velocity function problem [@zavala09; @vogelsberger12; @schneider14; @klypin14; @papastergis14]. Such models make different predictions for how halo density profiles vary with galaxy mass.
Here, we explore predictions for the warm dark matter (WDM) and self interacting dark matter (SIDM) models introduced in Section [2]{}. In the top left panel of Figure \[fig:altDM\] we plot the velocity function derived assuming a 2$\,$KeV WDM particle, which results in a different halo mass function when compared to CDM predictions. We show this effect measuring velocities at [V$_{\rm max}$]{} and using a CDM concentration-mass relation (dashed purple line). We then make adjustments for WDM concentrations [@schneider12] including adiabatic contraction (dot-dashed purple line). Clearly, modification to the halo mass function driven by a 2$\,$KeV particle is not enough to reconcile theory and observations, if haloes’ velocities are measured at [V$_{\rm max}$]{}.
Therefore we procede to analyze a WDM model which includes the universal dark matter fraction plus a stellar and a gas discs, as presented in Section [2]{}, and in which halo velocities are measured according to Equation \[rm\]. We show the resulting velocity function in light blue in the top left panel of Figure \[fig:altDM\]. In the top right panel we show the [M$_{\rm star}$]{}-[V$_{\rm los}$]{} relation for WDM where the [V$_{\rm los}$]{} has been measured between 2.2 and 3.5 disc scale-lengths.
The bottom left panel shows the [V$_{\rm los}$]{} distribution for cored profiles (green) representative of the adopted SIDM models [@vogelsberger12]. Core sizes in the model increase in proportion to the decrease in disc scale length from zero at [M$_{\rm star}$]{}=6$\times$10$^{10}$[M$_\odot$]{} to a maximum core size of 1$\,$kpc at [M$_{\rm star}$]{}$=$10$^8$[M$_\odot$]{}. The [M$_{\rm star}$]{}-[V$_{\rm los}$]{} relation for the SIDM model is shown in the bottom right panel.
![Left Panel: The velocity function for WDM particles of 1,2 & 3 KeV is shown as dot dashed, solid and dashed line respectively. Right Panel: The [V$_{\rm los}-$M$_{\rm star}$]{} relation for the different WDM particle masses. The trends are the same in all cases, i.e. the [V$_{\rm los}$]{} distribution bends down compared to the [$\Lambda$CDM]{} model as [V$_{\rm los}$]{} become smaller, while the [V$_{\rm los}-$M$_{\rm star}$]{} relation is steeper than the [$\Lambda$CDM]{} relation. []{data-label="fig:wdm"}](plots/F5.png){width=".53\textwidth"}
Both these alternative dark matter theories result in a flattening [@zavala09] of the [V$_{\rm los}$]{} distribution in the region 30${\lower.5ex\hbox{{$\; \buildrel < \over \sim \;$}}}$[V$_{\rm los}$]{}${\lower.5ex\hbox{{$\; \buildrel < \over \sim \;$}}}$60 km$\,$s$^{-1}$, as seen in Figure \[fig:altDM\]. As velocities become even smaller, however, the [V$_{\rm los}$]{} distribution of these models increasingly diverge from the [$\Lambda$CDM]{} distribution, contrary to what occurs in the DC14 model. This finding is not specific to the WDM and SIDM models we show in Figure \[fig:altDM\], but is general to any alternative DM model where core size increases relative to the virial radius, are constant at all masses, or scale with the virial radius of galaxies. No such model is able to match observations over the entire velocity range.
Alternative DM models that predict a decrease in the number of low mass haloes, and/or flattened halo density profiles even in the lowest mass galaxies, result in the velocity-[M$_{\rm star}$]{} relation bending down compared to the standard [$\Lambda$CDM]{} case, better matching the velocities of observed high mass dwarf galaxies (right panels in Figure \[fig:altDM\]). However, as we move to even lower mass, such models continue to bend downward and do not match the velocity values observed for dwarf galaxies with low stellar mass. The better such models are able to account for the velocity of high mass dwarfs, the worse they fare with respect to low mass dwarfs. Our results can be generalized to rule out any model where core size increases monotonically as halo mass decreases.
### Different WDM particle masses {#wdmvary}
We show the [V$_{\rm los}$]{} distribution and [V$_{\rm los}-$M$_{\rm star}$]{} relations for WDM particles of 1, 2 & 3 KeV as dashed, solid and dot dashed lines in Figure \[fig:wdm\]. The trends are the same in all cases, i.e. the [V$_{\rm los}$]{} distribution bends down compared to the [$\Lambda$CDM]{} model as [V$_{\rm los}$]{} become smaller, while the [V$_{\rm los}-$M$_{\rm star}$]{} relation is steeper than the [$\Lambda$CDM]{} relation. None of the particle masses considered is able to reproduce observations, consistently with what has been shown previously [@klypin14; @schneider14]. Moreover, we note that constraints from the Lyman-alpha forest indicate that WDM particles should have mass ${\lower.5ex\hbox{{$\; \buildrel > \over \sim \;$}}}$3$\,$KeV [@viel13].
![ The [V$_{\rm los}$]{} distribution (left panel) and [V$_{\rm los}-$M$_{\rm star}$]{} relation (right panel) for various core sizes and scalings with mass. Core sizes increase linearly in proportion to the decrease in disc scale length from zero core at [M$_{\rm star}$]{}=6$\times$10$^{10}$[M$_\odot$]{} to a maximum core size of 1 (solid green line) and 2 (dashed green line)$\,$kpc at [M$_{\rm star}$]{}$=$10$^8$[M$_\odot$]{}. The purple dashed line shows cores which scale in size with virial radius, core size = R$_{\rm vir}$/100. []{data-label="fig:cores"}](plots/F6.png){width=".53\textwidth"}
### Different core sizes {#corevary}
The parameterization of core sizes with mass, used within the adopted SIDM model in Section 3.3, mimics the main features of core dependence on halo mass for velocity dependent SIDM particles [@vogelsberger12]. In Figure \[fig:cores\] we show the results for different parameterizations of core sizes with mass. Core sizes increase linearly in proportion to the decrease in disc scale length from zero core at [M$_{\rm star}$]{}=6$\times$10$^{10}$[M$_\odot$]{} to a maximum core size of 1$\,$kpc (solid green line) and 2$\,$kpc (dashed green line) at [M$_{\rm star}$]{}$=$10$^8$[M$_\odot$]{}.
Keeping a constant 1 or 2$\,$kpc core at high masses makes very little difference (making the fit to observations at the high mass end slightly worse), so is not shown. We also show a core which scales in size with virial radius, such that core size = R$_{\rm vir}$/100, which results in less flattening of the [V$_{\rm los}$]{} distribution. None of the used parameterizations is able to reproduce the [V$_{\rm los}$]{} distribution and the [V$_{\rm los}-$M$_{\rm star}$]{} relation over the entire mass ranges.
AGN and cores in massive disc galaxies.
---------------------------------------
The large amounts of energy from AGN in the inner region of galaxies could lead to a further transformation of the inner dark matter profile [@martizzi13]. We did not include AGN in our study. Firstly, AGN activity is not likely to be significant in the mass range that is relevant to this study.
Secondly, baryons make an increasingly significant contribution to the inner density profiles at high masses: the differences in the dark matter density profile are therefore less important in high mass galaxies. We verified that a model which follows the mass dependence of the DC14 density profile at low mass, but has cored profiles for all galaxies with [M$_{\rm star}$]{}$>$3$\times$10$^8$[M$_\odot$]{}, in order to resemble a possible AGN-driven core, does not change our results.
Effects of Abundance Matching Uncertainties {#section:ab}
-------------------------------------------
![The [M$_{\rm star}$]{} versus [M$_{\rm halo}$]{} relations [@guo10; @gk14; @moster10; @sawala14] that are explored. These relations cover a wide range of normalisations and slopes in the region relevant for our study. []{data-label="fig:AM"}](plots/F7.png){width=".45\textwidth"}
In this study we relate stellar masses and halo masses using an empirical abundance matching relation [@guo10]. Several other studies provide different formulations for such relation. Here we show that the particular form of the abundance matching relation that we employ is not driving our results and conclusions.
Abundance matching relations are generally complete down to a stellar mass of few [M$_{\rm star}$]{}$\sim10^8$[M$_\odot$]{}, corresponding to the lower limit of large scale surveys such as SDSS [@baldry08] and GAMA [@baldry12]. Above this mass, there is relatively little difference in the abundance matching studies [@guo10; @moster10; @behroozi13], and such differences are insignificant in terms of our study. However for stellar masses below $\sim$10$^8$[M$_\odot$]{} observational uncertainties exists and, as this is a crucial mass range in our study, we checked if using different abundance matching prescriptions affects the results at low stellar masses.
There are two recent studies that have extended the abundance matching relation to masses lower than [M$_{\rm star}$]{}$\sim10^8$[M$_\odot$]{}, by using the observed stellar mass function of the Local Group [@brook14; @gk14]. In the first study, @brook14, it was shown that using the average halo mass function of simulated local groups, which is well described by a power law, implies a steep relation between [M$_{\rm star}$]{}-[M$_{\rm halo}$]{} in the region 10$^{6.5}$$<$[M$_{\rm star}$]{}/[M$_\odot$]{}$<$10$^8$. The empirically extended relation found in @brook14 matches well the extrapolated abundance matching relation used in this paper [@guo10].
The second study, @gk14, choses a particular collisionless simulation of the local group which has a flatter than average halo mass function, allowing the use of a flatter [M$_{\rm star}$]{}-[M$_{\rm halo}$]{} relation when matching haloes to the Local Group stellar mass function. It also showed that, using an even flatter [M$_{\rm star}$]{}-[M$_{\rm halo}$]{} relation as in @behroozi13, the results is a severe over-estimate of the number of local group galaxies with stellar masses 10$^{6.5}$${\lower.5ex\hbox{{$\; \buildrel < \over \sim \;$}}}$[M$_{\rm star}$]{}/[M$_\odot$]{}${\lower.5ex\hbox{{$\; \buildrel < \over \sim \;$}}}$10$^8$.
Considering these empirical constraints, we explore the effects of choosing flatter abundance matching relations than the one used in the paper. The three tested relations [@guo10; @gk14; @moster10], shown in Figure \[fig:AM\], cover a very broad range in slope and normalisation, particularly at low masses. For a halo mass of 10$^{10}$[M$_\odot$]{}, there is approximately an order of magnitude difference in the matched stellar mass according to the different relations used. The velocity function relation of the DC14, NFW, SIDM and WDM models will only suffer from a negligible change when different abundance matching are applied, and therefore it is not shown here.
![The [M$_{\rm star}$]{}-[V$_{\rm los}$]{} relations for the NFW and DC14 models, using different abundance matching relations [@guo10; @gk14; @moster10], which span a wide range of normalisation and slope. The conclusions of the paper are not affected: the DC14 profile still provides a very good fit to the data, for all the abundance matching relations, while the NFW profile does not match observations when the full range of dwarfs masses is considered.[]{data-label="fig:AMrel"}](plots/F8.png){width=".47\textwidth"}
We plot [M$_{\rm star}$]{}-[V$_{\rm los}$]{} using the different abundance matching relations for the NFW and DC14 models in Figure \[fig:AMrel\], and for the alternative dark matter models in Figure \[fig:S9\].
The most important general observation is that altering the abundance matching will alter each of the models in a consistent manner. To be specific, choosing an abundance matching relation that is flatter than the fiducial one used in the paper [@guo10], steepens the [M$_{\rm star}$]{}-[V$_{\rm los}$]{}relation in all models. Thus, all models that were already too steep in the region 5$\times$10$^6$$<$[M$_{\rm star}$]{}/[M$_\odot$]{}$<$3$\times$10$^8$, (i.e. all but the DC14 model), worsen. The DC14 model still does very well at matching the observations in this important mass range, and the advantage over the other models is clearly maintained in full. The NFW, SIDM, and WDM models all fail at reproducing the observed [M$_{\rm star}$]{}-[V$_{\rm los}$]{} relation, predicting dwarf galaxies with an increasingly low [V$_{\rm los}$]{} when a flatter abundance matching relation is assumed.
It may seem a little surprising that a large change in the abundance matching relation results in a relatively small change in the [M$_{\rm star}$]{}-[V$_{\rm los}$]{} relation. This is in part due to lower mass haloes having higher concentrations. We remind that the fact that the mass-concentration relation results in similar velocity measurements, for low mass galaxies over a range of halo masses, has been shown previously [@kravtsov10]. This study concerns the effects of baryons in altering the haloes in which galaxies are hosted, which we implement using the DC14 model. A different proposal is that baryonic processes effectively “adjust" the halo mass function [@sawala14], flattening the [M$_{\rm star}$]{}-[M$_{\rm halo}$]{} relation. In the most important region of our study, 5$\times$10$^6$$<$[M$_{\rm star}$]{}/[M$_\odot$]{}$<$3$\times$10$^8$, galaxy stellar masses are matched to haloes that are well above the mass where re-ionization may render some haloes dark, [M$_{\rm halo}$]{}$\sim$5$\times$10$^{9}$[M$_\odot$]{} [@bullock00], regardless of which abundance matching relation is used. The “bend" in the [M$_{\rm star}$]{}-[M$_{\rm halo}$]{}relation due to re-ionisation, as proposed by [@sawala14], occurs at lower masses than what is considered in this study, [M$_{\rm star}$]{}$<$10$^6$[M$_\odot$]{}.
Conclusions
===========
The *cusp-core* discrepancy is a challenge to the paradigm, which is heightened by studies showing that non-standard dark matter particles can result in cored dark matter haloes. In this paper we have highlighted how it is possible, from a theoretical point of view, to distinguish between astrophysical processes and alternative dark matter scenarios that are able to create cores in galaxies.
The key is that different models make different predictions for how core formation depends on mass. We show the signatures of these differences in galaxy populations, in particular by studying the velocity function and the Tully-Fisher relation.
Using analytic models, we have shown that the density profiles, DC14, resulting from energetic feedback processes like supernova explosions, have a particular mass dependence with a maximum core formation efficiency at [M$_{\rm star}$]{}$\sim$10$^{8.5}$[M$_\odot$]{} [@DiCintio2014a; @DiCintio2014b]. This value matches the peak mass of core formation occurring in hydrodynamical simulations [@stinson13; @brook12]. These findings allow us to create model populations of galaxies within dark matter haloes that have been affected by such energetic processes, and to compare them with models that assume a universal, steep, NFW density profile.
![The [M$_{\rm star}$]{}-[V$_{\rm los}$]{} relations for the WDM and and SIDM models, using different abundance matching relations [@guo10; @gk14; @moster10]. A flatter relation between [M$_{\rm star}$]{} and [M$_{\rm halo}$]{} results in a steeper [M$_{\rm star}$]{}-[V$_{\rm los}$]{} relation, worsening the mismatch with observations. []{data-label="fig:S9"}](plots/F9.png){width=".55\textwidth"}
In agreement with previous studies, we showed that it is difficult to reconcile a model galaxy population that assumes an NFW profile for dark matter haloes with the observed velocity function of galaxies with line-of-sight velocity [V$_{\rm los}$]{}$<$60 km/s [@zavala09; @papastergis11; @tg11; @klypin14; @papastergis14], and with the observed Tully-Fisher relation of galaxies with stellar mass $\sim$10$^7$-10$^9$[M$_\odot$]{}. The mismatch holds without assuming a priori that galaxies rotation curves are tracking the maximum velocity of dark matter haloes.
By contrast, a model in which the density profile varies according to the ratio [M$_{\rm star}$]{}/[M$_{\rm halo}$]{}, as expected when the halo profile is affected by baryonic outflows of gas, is able to reproduce both the velocity function and the [M$_{\rm star}$]{}-velocity relation of galaxies, under plausible assumptions regarding the manner in which observations and theory are compared. Further, we show that the particular mass dependence of cores formed by baryonic outflows will leave signatures within both the velocity function and the [M$_{\rm star}$]{}-velocity relation of galaxies. Specifically, the steepening of the profiles as galaxy mass decreases below [M$_{\rm star}$]{}$\sim$10$^{8.5}$[M$_\odot$]{} results in an upturn in the velocity function for [V$_{\rm los}$]{}${\lower.5ex\hbox{{$\; \buildrel < \over \sim \;$}}}$20$\,$ and in a flattening of the [M$_{\rm star}$]{}-velocity relation in the region 10$^{6}$${\lower.5ex\hbox{{$\; \buildrel < \over \sim \;$}}}$[M$_{\rm star}$]{}/[M$_\odot$]{}${\lower.5ex\hbox{{$\; \buildrel < \over \sim \;$}}}$10$^{8}$, both features in agreement with current data [@klypin14].
The independence of velocity from stellar mass observed in low mass dwarf galaxies (@strigari08 [@gk14], and references therein) is thus a feature naturally reproduced by haloes that are flattened by astrophysical processes. Evidence for an observed upturn in the velocity function is more tenuous, relying on the manner in which galaxies that are devoid of gas are added to results coming from large scale surveys of HI line-widths. Further to this, an increased ratio of velocity dispersion to rotation velocity means that HI line-widths of low mass galaxies may be more difficult to interpret than in more massive galaxies.
Alternative dark matter models also make predictions for how halo density profiles vary with galaxy mass. We explored predictions from WDM and SIDM models, finding that both result in a flattening of the [V$_{\rm los}$]{} distribution in the region 30${\lower.5ex\hbox{{$\; \buildrel < \over \sim \;$}}}$[V$_{\rm los}$]{}${\lower.5ex\hbox{{$\; \buildrel < \over \sim \;$}}}$60 km$\,$s$^{-1}$; as velocities become even smaller, however, the [V$_{\rm los}$]{} distribution of these models increasingly diverge from the [$\Lambda$CDM]{} distribution, contrary to what occurs in the DC14 model. Any model that predicts a decrease in the number of low mass haloes (such as WDM) and/or cores in the lowest mass galaxies (such as SIDM), results in the [M$_{\rm star}$]{}-velocity relation better matching observations of high mass dwarf galaxies than the NFW model, but as we move to the lowest masses, such models continue to bend downward and do not match the observed velocities of the faintest dwarf galaxies. The better such models are able to account for the velocity of high mass dwarfs, the worse they fare with respect to low mass dwarfs.
These findings can be generalized to any alternative dark matter model in which core size increases monotonically as halo mass decreases, are constant at all masses, or scale with the virial radius of galaxies.
It must be noted that in this paper we did not use the distribution of gas and stars from hydrodynamical simulations, but we rather coupled the mass dependent DC14 profile with empirical galaxy scaling relations to make model galaxy populations.
In a future study we will compare HI line-widths measured directly from our simulations to the empirical $\rm V_{max}-V_{rot}$ abundance matching relation found in @papastergis14. We note that the hydrodynamical simulations of @governato12 and @brooks14 included in @papastergis14 did not match the $\rm V_{max}-V_{rot}$ abundance matching at the low mass end, and we will explore whether this holds in our simulations and whether there are differences between simulated galaxies and empirical model for galaxy properties adopted here.
It remains a theoretical and observational challenge to compare measured velocities of observed dwarf galaxies to model predictions, particularly as the ratio of velocity dispersion to rotational velocity increases for low mass galaxies. Uncertainties associated with stellar velocity dispersions in local dSphs should be reduced thanks to GAIA satellite data [@perryman01], and possibly also next generation 40m telescopes such as the European ELT, working with multi adaptive optic cameras [@davies10]. Meanwhile, the next generation of HI surveys, as we enter the era of SKA, promise high spatial resolution HI gas kinematics in a large sample of galaxies, with increased fidelity in the derivation of the velocity function [@lister15].
More sophisticated comparisons between observations and theory are certainly required, as are further models exploring the nature of density profiles that form under the influence of baryonic outflows; different feedback implementations, higher resolution simulations and more diverse initial conditions will likely result in an improvement of our understanding of how density profiles are modified by baryonic physics.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Manolis Papastergis and Simon White for very helpful comments on an earlier draft. CB thanks the MICINN (Spain) for the financial support through the MINECO grant AYA2012-31101 and the Ramon y Cajal program. He further thanks the DARK cosmology centre for the kind hospitality. ADC is supported by the DARK independent fellowship program.
\[lastpage\]
|
---
address: 'Institute of Solid State Physics, Russian Acaddemy of Sciences, 142432 Chernogolovka, Russian Federation'
author:
- 'V.S. Khrapai, D.V. Shovkun'
title: 'Shot noise measurements in a wide-channel transistor near pinch-off'
---
A current $I$ flowing in a two-terminal conductor placed in external electric circuit exhibits fluctuations around it’s mean value $\overline{I}$. The second moment of the fluctuations is related to the noise spectral density $\left.\overline{(I-\overline{I})^2}\right|_{\Delta f}\equiv
S_I\Delta f$, where $\Delta f$ is a measurement bandwidth. In the absence of current ($\overline{I}=0$) the noise is related to thermal fluctuations of the occupation number of the electronic states, known Johnson-Nyquist noise (JN-noise). In this case $S_I=4k_BTR^{-1}$, where $k_B,T$ and $R$ are, respectively, the Boltzman constant, the temperature and the resistance of the conductor. Away from the equilibrium, when the voltage drop across the conductor is high enough $|eV|\gg k_BT$, and in the absence of dissipation inside the conductor and spurious noises, the current fluctuations are caused by the discreteness of the elementary charge $e$ [@schottky]. This noise is referred to as shot noise and for a voltage-biased conductor has a spectral density of $S_I=2F|e\overline{I}|$, where $F$ is called a Fano-factor.
In a non-interacting system in the linear transport regime, the shot noise is caused by a partition of incident carriers, which can be viewed as quantum effect [@blanterreview]. In this case, the Fano-factor is determined by the distribution of the eigen-channel transparencies ($T_n$) of the conductor ${F=
\overline{T_n(1-T_n)}/\overline{T_n}}\leq1$ [@blanterreview]. The noise is strongest ($F=1$) in the Poissonian regime, which is obtained when all $T_n\ll1$. In a quasi one dimensional metallic diffusive conductor the distribution of $T_n$ is universal [@beenaker] and $F=1/3$, which has been experimentally confirmed [@henny]. The universality of the value $F=1/3$ in metallic conductors has been proven to be independent of geometry [@nazarov]. Near the transition from the metallic to the insulating state one expects an increase of the partition noise to the Poissonian value [@blanterreview], although this regime haven’t been studied experimentally.
We study the shot noise in a gated wide channel transistor near pinch-off, where the transition to the insulating state is expected to occur. The dependence of $S_I$ on current is slightly non-linear in the shot noise regime. The differential Fano-factor $F_D=(2|e|)^{-1}|dS_I/dI|$ is enhanced above the universal metallic value $1/3<F_D\leq0.5$ for low $|I|$ and is close to this value $F_D\approx1/3$ for higher $|I|$. We find that the enhancement of $F_D$ is not necessarily related to the quantum partition noise. In contrast, it can be explained by a classical effect of thermal fluctuations in a strongly nonlinear transport regime near pinch-off.
The skecth of the measurement is shown in fig. \[fig1\]. The sample and the cryogenic amplifier (LTAmp) are placed in a $^4$He gas chamber with the walls maintained at 4.2K. The sample is below the LTAmp and is connected to it with a 20 cm cable. A heat sink connects the LTAmp to a liquid $^4$He bath. The actual temperature of the LTAmp ($T_{LTA}$) measured with a thermometer is about 5.3K. The actual temperature of the sample is taken to be 4.2K, consistent with the JN-noise measurement (see below). At the input of the setup (CAL IN) a 50 $\Omega$ cable is connected to the transistor source (S) via a divider. This input is used both for driving a current $I$ and external rf calibration. A transistor drain (D) is followed by an $L-C$ resonator, which is connected to a cryogenic amplifier (LTAmp) ($\approx20$dB gain). The resonator serves to match a high-impedance of the sample and a low input impedance $Z_0=50~\Omega$ of the LTAmp. The output of the LTAmp is connected via a second 50 $\Omega$ cable to the input stage of the room-temperature low-noise amplifiers (total gain of 3$\times$20 dB). Finally, the amplified signal is filtered with a $\sim$30 MHz bandpass filter at the resonance frequency $f_0\approx125$ MHz and rectified by a detector (8473C by Agilent Technologies). The ac modulation of the rectified signal thanks to a current modulation, gate voltage chopping or amplitude modulation of the external rf is measured with the lock-in. Alternatively, a spectrum analyzer can be used to analyze the frequency spectra at the output of the room-temperature amplifiers.
![Fig. \[fig1\]: The sketch of the setup. The low temperature parts are shown by dashed boxes. The parameters of the resonant $LC$ circuit and the undesired stray capacitances $C_{SD}$ and $C_D$ (shown by dashed lines inside the 4.2K box) are given in the text. The signal from the room temperature amplifiers (Room T Amps) is either sent to the filter and detector for a Lock-in measurement or to the spectrum analyzer (dashed line with an arrow) for wide range spectra acquisition.[]{data-label="fig1"}](newsetup.eps){width="0.9\columnwidth"}
We study shot noise of a commercial pseudomorphic AlGaAs/InGaAs/GaAs pHEMT ATF-35143 by Agilent with a gate length (width) of 0.5 (400) $\rm \mu m$. This transistor is known for low noise at room temperature and is used as an active element in cryogenic LTAmp’s by us and other authors [@RoschierCryogenics]. We preformed the resistance measurements in two such transistors (samples 1 and 2), and noise measurements and calibration only in sample 2.
![Fig. \[fig2\]. (a) Differential resistance $R_{diff}$ as a function of current for three values of the gate voltage in sample 2. Inset – linear response S-D resistance as a function of gate voltage for sample 1 (solid line) and three different states of sample 2 (symbols). The $V_g$ axis for different sample states was compensated for random shifts on the order of 50 mV. (b) Experimental $I$-$V$ curves calculated from the data of (a) shown by the same symbols as the corresponding data for $R_{diff}$ in (a). Model fits (see text) are shown by dashed lines.[]{data-label="fig2"}](fig2new.eps){width="0.7\columnwidth"}
Negative gate voltage $V_g<0$ is used to deplete the channel which results in increase of the linear-response S-D resistance $R\equiv
dV/dI|_{I\rightarrow0}$. The huge aspect ratio of the gate electrode allows to work near the pinch-off with high channel resistivity in the range of M$\Omega/\Box$ while keeping a reasonable $R$. The dependence $R(V_g)$ at 4.2 K is shown in the inset of fig. \[fig2\] for sample 1 (solid line) and for several states of sample 2 (symbols). In all cases, the behavior $R(V_g)$ is roughly exponential and is reproducible up to insignificant random threshold voltage shifts (see caption). Hence, most likely, the current is homogeneously distributed across the transistor channel near pinch-off. Such a strong dependence $R(V_g)$ might indicate that we enter the insulating phase near the pinch-off [@ando]. The measured differential resistance $R_{diff}\equiv dV/dI$ is shown as a function of current in fig. \[fig2\]a for three values of the gate voltage in sample 2. These data are taken simultaneously with the shot noise measurements presented below. $R_{diff}$ is maximum in the linear response ($I=0$) and falls down at finite current. The reduction is most pronounced at low $|I|$ and for more depleted channel. $R_{diff}$ is an asymmetric function of current, which is related to the capacitive population/depopulation of the channel at negative/positive bias. The nonlinear $I$-$V$ curves numerically calculated from these data are plotted in fig. \[fig2\]b. The nonlinearity is somewhat similar to the behavior of the $I$-$V$ curves in the insulator breakdown regime [@shashkinPRL], although much less pronounced, possibly because of the much higher temperature.
The main ingredient of the shot noise measurements is the calibration of the setup gain and bandwidth described below. A rectified voltage $V_{\rm DET}$ at the output of the detector is proportional to the power $P$ incident on the detector. The power $P$ is the sum of contributions $P_T$ proportional to the transistor noise $S_I^{T}$, $P_{Z_0}$ proportional to the input current noise of the LTAmp $S_I^{Z_0}$ and a constant contribution coming from the input voltage noise of the LTAmp and noises of all other amplifiers. The LTAmp’s input current noise is dominated by the Johnson-Nyquist noise of the input resistor $S_I^{Z_0}\approx4k_BT_{LTA}/Z_0$. The actual noise temperature can be somewhat higher owing to extra current noise from the active parts of the LTAmp. Small noises from the two resistors at the input divider (fig. \[fig1\]) are neglected. Below we are interested only in a differential part of $V_{\rm DET}$ measured with a lock-in, which depends on the transistor S-D current $I$ and/or its (linear or differential) S-D resistance. Hence, up to an unimportant constant one gets: $$\begin{split}
V_{DET}^{noise} = D(P_T+P_{Z_0}) = S_I^{T}R^2\cdot DG\int |k_T|^2df+\\
+S_I^{Z_0}Z_0^2\cdot DG\int |k_{Z_0}(f,R)|^2df, \label{eq1} \end{split}$$ where $D$ is the detector power to voltage conversion coefficient, $G$ – total power gain of the amplifiers divided by the input resistance $Z_0$. $k_T$ and $k_{Z_0}(f,R)$ denote the voltage transfer functions of the corresponding noise sources to the input of the LTAmp, which both depend on $f$ and $R$. The transfer functions are set by the parameters of the circuit in fig. \[fig1\], which we determine via the following calibration procedure. We apply an rf signal of amplitude $U_{CAL}$ at a frequency $f$ to the input CAL IN (see fig. \[fig1\]) and measure the contribution $V_{DET}^{CAL}$ to the output detector voltage. The detector signal is proportional to power $P_{CAL}$ incident on the detector: $$V_{DET}^{CAL}=DP_{CAL} =D|k^\prime_T|^2\cdot |k_{GEN}|^2G\cdot
U_{CAL}^2, \label{eq2}$$ where $k_{GEN}$ is the voltage divider coefficient at the CAL IN input of the circuit (fig. \[fig1\]). $k^\prime_T$ is the rf-voltage transfer function, which is related to the noise transfer function $k_T$ from eq. (\[eq1\]) as $|k^\prime_T|^2=\alpha |k_T|^2$. Factor $\alpha=1+4\pi^2f^2R^2C_{SD}^2$ accounts for the suppression of the transistor voltage noise caused by a stray S-D capacitance $C_{SD}$. The power $P_{CAL}$ measured with a spectrum analyzer is shown in fig. \[fig3\]a as a function of $f$ for a set of gate voltages (symbols). The quality factor of the $LC$-resonator in fig. \[fig1\] increases with $R$, which results in narrower peak for more depleted channel in fig. \[fig3\]a. Solid lines represent the best fits to the data used to accurately determine the values of $L\approx 300$ nH, $C\approx1.5~$pF, the drain-ground stray capacitance $C_D\approx3.9$ pF and $C_{SD}\approx0.2~$pF. We find that $C_D$ is dominated by a stray capacitance of the hand-made inductor ($L$ in fig. \[fig1\]) and the value of $C_{SD}$ is close to an intrinsic parameter of the transistor. The quality of the fits is almost perfect, apart from small oscillations presumably caused by resonances in the rf-tract. These discrepancies are not important as they occur beyond the bandwidth used for noise measurements.
![Fig. \[fig3\]. (a) Frequency response spectra of the setup at $I=0$ acquired with a spectrum analyzer for a set of linear response resistances $R$ indicated in the legend. Experimental data and fits are shown by symbols and lines, respectively. (b) Calibration of the setup via equilibrium noise measurement (see text). Symbols and line: measured chop amplitude of the detector voltage and fit, respectively.[]{data-label="fig3"}](fig3.eps){width="0.7\columnwidth"}
Fitting the data of fig. \[fig3\]a with eq. (\[eq2\]) returns the value of the product $|k_{GEN}|^2G$, whereas a separate knowledge of $G$ is required for noise measurements. This is achieved via a measurement of the equilibrium noise of the setup, which depends on $R$ (see eq. (\[eq1\])). We chop the transistor gate voltage between the two values, corresponding to nearly zero ($\sim10\Omega$) and finite $R$ and measure the first harmonic ac component of the detector voltage $V_{DET}^{noise}$ with a lock-in. The result is plotted in fig. \[fig3\]b as a function of $R$ (symbols). The noise signal $V_{DET}^{noise}$ increases as the transistor is depleting, which reflects the increase of the JN voltage noise of the transistor. The overall dependence is caused by the interplay of the $R$-dependent bandwidth of the resonant circuit and a (negative) contribution from the chopped LTAmp’s input current noise. The experimental behavior is well captured by the dashed line fit in fig. \[fig3\]b. The fitting parameters include $G$, the LTAmp’s input noise temperature of $T_{LTA}\approx4.7$K and the sample temperature of $T=4.2$K. Consistently, an independent thermometry returned the value of $\approx$5.3K for the temperature of the resistor $Z_0$ (fig. \[fig1\]). Under assumption of $T_{LTA}=5.3$K the best fit to the data of fig \[fig3\]b would be obtained for a sample temperature of $\approx4.8$K. This discrepancy represents a possible systematic error in our calibration and shot noise measurements, so that the Fano factor values given below may actually be within 10% higher.
![Fig. \[fig6\]. Integrated detector voltage $V_{DET}^{noise}=V_{DET}^T+V_{DET}^{Z_0}$ as a function of S-D current (dashed line). Contributions from the transistor noise ($V_{DET}^T$) and input current noise of the LTAmp ($V_{DET}^{Z_0}$) are shown by solid and dotted lines, respectively. The data are taken for the same gate voltage as the traces shown by squares in fig .\[fig2\] (linear-response resistance indicated in the figure)[]{data-label="fig6"}](fig4.eps){width="0.7\columnwidth"}
As follows from fig. \[fig3\], at $I=0$ the rf-response of the transistor and its equilibrium noise are successfully described by a single stray capacitance parameter $C_{SD}$. We find that this is not the case under non-linear transport conditions. Presumably, the reason is the inhomogeneous electron density distribution below the gate at $I\neq0$, which can change, e.g., distributed gate-drain and gate-source capacitances. Instead of introducing more fitting parameters at $I\neq0$, we calibrate the noise transfer function $k_T$ in-situ. According to eq. (\[eq2\]) integration of the frequency response of the setup to the external rf-signal gives: $$DG\int|k_T|^2df=(\alpha |k_{GEN}|^2U_{CAL}^2)^{-1}\int
V_{DET}^{CAL}df \label{eq3}$$ The quantity $K_{CAL}\equiv DG\int|k_T|^2df$ obtained in this way accounts for the $I$-dependent gain and bandwidth of the shot noise measurement in the nonlinear regime.
Using equations (\[eq1\]) and (\[eq3\]) the measurement of the shot noise spectral density $S_I^T$ is straightforward. In the nonlinear regime the voltage noise of the transistor is determined by the differential resistance $R_{diff}$ which substitutes $R$ in eq. (\[eq1\]). The LTAmp’s noise transfer function $k_{Z_0}(f,R)$ is evaluated with the known $I=0$ circuit parameters and $R=R_{diff}$. We measure the dc contribution to the detector voltage caused by finite S-D current $V_{DET}^{noise}$. This is achieved via a lock-in measurement of the derivative $dV_{DET}/dI$ and subsequent numeric integration. The integration constant is obtained via the equilibrium noise measurement (fig. \[fig3\]b). The (arbitrarily offset) result is shown in fig. \[fig6\] for one value of $V_g$ (see caption). Here, the dashed line is the experimental $V_{DET}^{noise}$. The evaluated contribution of the LTAmp’s input current noise $V_{DET}^{Z_0}$ is shown by dots. $V_{DET}^{Z_0}$ is not a constant, thanks to the $R_{diff}$ dependence on $I$ in the nonlinear regime, which slightly modifies the impedance connected to the input of the LTAmp. As seen from fig. \[fig2\], $R_{diff}$ changes stronger for $I<0$, which results in a corresponding asymmetry of $V_{DET}^{Z_0}$ as a function of $I$ in fig. \[fig6\]. The difference $V_{DET}^T=V_{DET}^{noise}-V_{DET}^{Z_0}$ is the contribution thanks to transistor shot noise shown by solid line. The functional dependence of $V_{DET}^T$ on $I$ is related to that of the noise spectral density as $S_I^{T}=V_{DET}^T/(R_{diff}^2K_{CAL})$.
In figs. \[fig7\]a,\[fig7\]b and \[fig7\]c the noise spectral density $S_I^{T}$ is plotted as a function of $I$ for three values of the gate voltage (symbols). At $I=0$ the noise spectral density is minimum and equals the JN value. At $I\neq0$, $S_I^{T}$ increases as a function of $|I|$ and demonstrates a nearly linear behavior at high enough currents. We find that for all experimental traces, in the limit of high currents, the differential Fano-factor is close to the universal value $F_D\approx1/3$. For comparison, we plot $S_I$ expected for a metallic diffusive conductor in the linear regime by dashed lines in fig. \[fig7\]. These lines are drawn according to the standard formula $S_I=\frac{2}{3}R^{-1}\left[4k_BT+|eV|\coth(|eV|/2k_BT)\right]$, where $R$ is the experimental linear response resistance and $V$ is the associated voltage drop $V=IR$ [@henny]. The symbols in figs. \[fig7\]a and \[fig7\]b (obtained for a less depleted transistor) are systematically above the dashed lines, i.e. for the same $I$ the noise spectral density exceeds the one obtained with the above formula. Hence, at low currents the differential Fano-factor is enhanced compared to 1/3, which is most pronounced for the data in fig. \[fig7\]b with $F_D\approx0.5$ for $|I|\sim1\mu$A. This discrepancy is beyond the experimental uncertainty and can be explained by the effect of thermal fluctuations in the nonlinear transport regime, as we propose below.
![Fig. \[fig7\]: Shot noise spectral density $S_I^T$ as a function of S-D current for three values of the gate voltage (linear-response resistance $R$ shown in the figures). Experimental data in (a),(b),(c) are measured simultaneously with the data of fig. \[fig2\] and are shown by the respective symbols. Dashed lines are fits to the standard linear response shot noise formula [@henny]. Solid line are fits according to eq. (\[eq4\]) and a model of nonlinear transport described in the text.[]{data-label="fig7"}](fig5.eps){width="0.7\columnwidth"}
The general result for shot noise spectral density of a two-terminal conductor is usually express in terms of the energy-dependent 1D eigen-channel transparencies $T_n(E)$ [@blanterreview]. For the case of wide channel transistor, it is convenient to express the same result in terms of the energy-dependent conductance $\sigma(E)=e^2/h\sum T_n(E)$, where $h$ is the Planck’s constant, and the Fano-factor of the partition noise ${F= \overline{T_n(1-T_n)}/\overline{T_n}}$: $$\begin{split}
S_I=2\int \sigma(E)dE\{
f_L(1-f_L)+\\+f_R(1-f_R)+F(f_L-f_R)^2\}\label{eq4}
\end{split}$$ Here, $f_i=(1+\exp[(E-\mu_i)/k_BT])^{-1}$ are the Fermi distributions of the left ($i=L$) and right ($i=R$) reservoirs, with respective electrochemical potentials of $\mu_i=\pm |e|V/2$. The first two terms in the integrand of eq. (\[eq4\]) represent the thermal noise of the reservoirs, while the last term stands for the shot noise. Eq. (\[eq4\]) provides a phenomenological description of the shot noise behavior in the nonlinear regime. We find that the experimental enhancement of the differential Fano-factor $F_D>1/3$ (fig. \[fig7\]a and \[fig7\]b) is not necessarily related to energy dependence of $F$. Below we assume that the partition noise Fano-factor in the last term of eq. (\[eq4\]) has a universal value $F=1/3$ for metallic diffusive conductors independent of energy.
The energy dependent conductance $\sigma(E)$ is directly related to the transport current as ${I=|e|^{-1}\int\sigma(E)(f_L-f_R)dE}$ and can be obtained by fitting the experimental $I$-$V$ curves (fig. \[fig2\]b). We model $\sigma(E)$ by a step-like functional dependence on energy $\sigma(E)=\sigma_0(1+\tanh[(E-E_0-\lambda
|e|V)/\Delta])$. Parameters $\sigma_0\sim10^{-2}\Omega^{-1}$ and $E_0,\Delta\sim1$ meV define the shape of the conductance step, whereas the parameter $\lambda\sim0.1$ accounts for the asymmetry of the $I$-$V$ curves (fig. \[fig2\]b) thanks to capacitive effects of finite bias. This conductance model was chosen for its analogy to the step-like behavior of the density of states near the metal-insulator transition in two dimensions [@ando]. Fig. \[fig2\]b demonstrates that the model provides good fits (dashed lines) to the experimental nonlinear $I$-$V$ curves (symbols). The so-obtained $\sigma(E)$ and eq. (\[eq4\]) predict the behavior of the noise spectral density, which is shown by solid lines in fig. \[fig7\].
The enhancement of the differential Fano-factor in a less depleted channel at small $I$ (symbols in figs. \[fig7\]a and \[fig7\]b) is qualitatively captured by the fits according to eq. (\[eq4\]) (solid lines). Note, that thanks to assumption of $F=1/3$ the eq. (\[eq4\]) reduces to $S_I=2/3|eI|$ at $T=0$. In other words, within our model the enhancement of $F_D>F=1/3$ is a finite temperature effect. The finite temperature determines the thermal fluctuations in the electron flow incident on the conductor. In the nonlinear regime, the contribution of thermal fluctuations to the current noise increases as a function of $I$, thanks to the energy dependence of the conductance $\sigma(E)$. This results in enhanced differential Fano-factor $F_D>F=1/3$. Note, that this nonlinear effect is not related to a thermalization of the non-equilibrium carriers inside the conductor [@nagaev] or in the reservoirs [@henny].
The importance of the thermal fluctuations is best illustrated in the ultimate case of thermally activated conductance in the insulating phase, which in our model is achieved for $E_0,\Delta\gg T$. This is expected to occur in a strongly depleted transistor. Here, the eq. (\[eq4\]) predicts $F_D=1$ at low $|I|$. In this case, the Poissonian value of the Fano-factor is caused by a classical reason that the occupation number of the current carrying states is small $f_i(E\sim E_0)\ll1$ ($i=L,R$). This is fully analogous to the case of the shot noise for thermionic emission in a vacuum tube considered by Schottky [@schottky], and is not related to particular properties of the model we used for $\sigma(E)$. In the limit of high currents, the eq. (\[eq4\]) predicts a crossover to the partition noise in the insulator breakdown regime with $F_D\rightarrow F$. Unfortunately, we could not observe such a behavior experimentally. Fig. \[fig7\]c shows the noise spectral density for the lowest gate voltage (symbols), where the non-linearities are most pronounced (same symbols in fig. \[fig2\]). Unlike the prediction of eq. (\[eq4\]) (solid line), $F_D\approx1/3$ and the data falls close to the standard metallic linear-response result (dashed line). Note, however, that at low $|I|$ this behavior might be an artefact caused by the uncertainty in the voltage noise suppression factor $\alpha$, which is most crucial for measurements at high $R_{diff}\gtrsim3k\Omega$.
In summary, we performed the shot noise measurements in a commercial high-frequency transistor near pinch-off. The dependence of the shot noise on current is slightly nonlinear. The differential Fano-factor is about $F_D\approx1/3$ in the limit of high currents, and somewhat enhanced above the universal metallic value at lower currents $1/3<F_D<0.5$. The model of nonlinear transport near the pinch-off is suggested, which allows to explain the results in terms of classical effect of thermal fluctuation, without assuming the enhancement of the partition noise.
We acknowledge the discussions with A.A. Shashkin, V.T. Dolgopolov and V.F. Gantmakher. Financial support by RFBR and the grant MK-3470.2009.2 is gratefully acknowledged. VSK acknowledges support from the Russian Science Support Foundation.
[16]{} W. Schottky, Ann. Phys. (Leipzig) [**57**]{}, 541 (1918) Ya.M. Blanter, M. B$\rm \ddot{u}$ttiker Phys. Rep. [**336**]{}, 1 (2000) C.W.J. Beenakker, Rev. Mod. Phys. [**69**]{}, 731 (1997) M. Henny, S. Oberholzer, C. Strunk, C. Sch$\rm \ddot{o}$nenberger, Phys. Rev. B [**59**]{}, 2871 (1999) Yu.V. Nazarov, Phys. Rev. Lett. [**73**]{}, 134 (1994) L. Roschier, P. Hakonen, Cryogenics [**44**]{}, 783 (2004) T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. [**54**]{}, 437 (1982). A.A. Shashkin, V.T. Dolgopolov, G.V. Kravchenko, M. Wendel, R. Schuster et al., Phys. Rev. Lett. [**73**]{}, 3141 (1994) K.E. Nagaev, Phys. Rev. B [**52**]{}, 4740 (1995)
|
---
abstract: 'Training artificial neural networks requires a tedious empirical evaluation to determine a suitable neural network architecture. To avoid this empirical process several techniques have been proposed to automatise the architecture selection process. In this paper, we propose a method to perform parameter and architecture selection for a quantum weightless neural network (qWNN). The architecture selection is performed through the learning procedure of a qWNN with a learning algorithm that uses the principle of quantum superposition and a non-linear quantum operator. The main advantage of the proposed method is that it performs a global search in the space of qWNN architecture and parameters rather than a local search.'
address:
- |
Departamento de Estatística e Informática\
Universidade Federal Rural de Pernambuco, Brazil
- |
Centro de Informática\
Universidade Federal de Pernambuco, Brazil
author:
- 'Adenilton J. da Silva$^\dag$'
- 'Wilson R. de Oliveira'
- 'Teresa B. Ludermir'
bibliography:
- 'bibliografia1.bib'
title: Weightless neural network parameters and architecture selection in a quantum computer
---
Quantum neural networks ,quantum weightless neural networks ,quantum learning ,architecture selection
Introduction
============
The exponential reduction of computers’ components known as Moore’s law took computation from the classical physical domain to the quantum physics. The idea of quantum computation was initially proposed in [@Feynman], where Feynman states that quantum computers can simulate quantum physical systems exponentially faster than classical computers. Some quantum algorithms also overcome the best knew classical algorithms; the most famous examples being the Shor’s factoring algorithm [@shor:97] that is exponentially faster than the best know classical algorithm and the Grover’s search algorithm [@grover:96] with quadratic gain in relation to the best classical algorithm for unordered search. It is true that quantum computers are not yet a reality, but there has been an explosion of investment in quantum computing, the result of which are numerous proposals for quantum computers and the general belief which soon they will be realised. The use of an adiabatic quantum system with 84 quantum bits is reported in [@Bian2013] and in [@Monz2011] is reported the creation of a quantum system with 14 quantum bits. Empirical evaluations of ideas presented in this work for real problems require a quantum computer with capacity to manipulate hundreds of qubits which is impossible with current technology.
One of the main characteristics of quantum computation is the quantum parallelism that for some problems allows quantum algorithms to have a speedup in relation to the classical algorithms. With quantum parallelism is possible to calculate all possible $2^n$ values of a $n-$ary Boolean function in a single query. However, we cannot visualise these outputs directly. A quantum measurement is necessary and it returns probabilistically only a more restrict value. The quantum algorithm design problem is then to perform quantum operations to increase the probability of the desired output.
Designing quantum algorithms is not an intuitive task. Attempts to bring quantum computing power to a greater range of problems are the development of quantum machine-learning algorithms as decision trees [@Farhi1998], evolutionary algorithms [@Malossini2008] and artificial neural networks [@panella:09; @Altaisky; @oliveira:08; @ventura:04; @Behrman; @daSilva:12; @Narayanan; @Oliveira2009; @Liu2013]. In this paper, we are concerned in the field of quantum weightless neural networks.
Weightless neural networks (WNN) are not the most used model of artificial neural networks. WNN have been proposed by Aleksander [@Aleksander1966] as engineering tools to perform pattern classification. Applications of WNN are described in several works [@Staffa2014a; @Carvalho2014; @Cardoso2014; @esann:2014:tutorial] and quantum versions of WNN have been proposed in [@oliveira:08; @Oliveira2009; @daSilva:12].
The idea of quantum neural computation has been proposed in the nineties [@kak:95], since then several models of quantum neural networks have been proposed. For instance, quantum weightless neural networks [@daSilva:12], neural networks with quantum architecture [@panella:09] and a simple quantum neural network [@ventura:04]. In all these works [@daSilva:12; @panella:09; @ventura:04] a quantum neural network configuration is represented by a string of qubits and quantum learning algorithms are proposed within a common framework. The main idea of the learning algorithms in [@daSilva:12; @panella:09; @ventura:04] is to present input data to all possible neural networks for a given architecture in superposition and perform a quantum search in the resulting superposition. The objective of this paper is to generalise this idea to allow architecture selection through the training of a quantum weightless neural network. To achieve this objective we use a quantum weightless neural network that stores representations of weightless neural networks with different architectures in its memory positions and we define a quantum learning algorithm using the non-linear operator proposed in [@PhysRevLett.81.3992] and the measurement and feedback strategy [@Gammelmark:09].
Selection of a neural network architecture is an important task in neural networks applications. Normally this task requires a lot of empirical evaluation performed by an expert. To avoid the tedious empirical evaluation process and help inexperienced users some algorithms have been proposed to perform automatic selection of neural networks architecture. Techniques such as meta-learning [@Abraham20041] and evolutionary computation [@Almeida2010] have been used for architecture selection.
In this paper, we show how to use a quantum weightless neural network with a non-linear quantum-learning algorithm to find a quantum neural network architecture and parameters with a desired performance. The proposed algorithm uses quantum superposition principle and a non-linear quantum operator. The proposed algorithm performs a global search in architecture and parameters space and its computational time is polynomial in relation to the number of training patterns, architectures and quantum weightless network memory size.
The rest of the paper is organised as follows. Section 2 presents basics concepts on quantum computation such as quantum bits, operators, measure and parallelism. Section 3 presents the concept of weightless neural networks, quantum neural networks and quantum weightless neural networks. Section 4 describes a quantum learning algorithm for weightless neural networks and how to apply this learning algorithm to perform architecture selection. Finally, Section 5 is the conclusion.
Quantum computing
=================
Deep knowledge of classical physics is not required for designing classical algorithms. In the same vein, the development of quantum algorithms does not require a deep knowledge of quantum physics and there are several books [@nielsen:00; @Hirvensalo2003; @mermin2007quantum] that follow this approach by introducing only the strictly necessary knowledge of quantum physics for the understanding of quantum computing. In order to create a self-contained text a brief introduction to quantum computing is presented.
The state of a quantum computer with $n$ quantum bits (or *qubits*) can be mathematically represented by a unit vector of an $2^n$-dimensional complex vector space with inner product. For instance, one single qubit can be represented in the vector space $\mathbb{C}^2$ as described in Equation , $$\ket{\psi}=\alpha\ket{0}+\beta\ket{1}
\label{eq:qubit}$$ where $\alpha, \beta \in \mathbb{C}$, $\left|\alpha\right|^2+\left|\beta\right|^2 = 1$ and $\ket{0}$ and $\ket{1}$ are the vectors described in Equation [^1]. $$\ket{0} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \mbox{ and } \ket{1} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}
\label{eq:canbasis}$$ One qubit in a $n$-dimensional quantum system is represented by the $2^n$-dimensional vector space as described in Equation , $$\sum_{i=0}^{2^n-1}\alpha_i \ket{\psi_i}
\label{eq:qubits}$$ where the sum of the squared modulus of the amplitude $\sum_i |\alpha_i|^2$ is equal to one and the set $\{\ket{\psi_0}, \ket{\psi_1}, \cdots, \ket{\psi_{2^n-1}} \}$ is an orthonormal basis of $\mathbb{C}^{2^n}$.
A *Quantum operator* in a quantum system with $n$ qubits is an unitary operator[^2] in the vector space $\mathbb{C}^{2^n}$. Let $U$ be an unitary operator over $\mathbb{C}^{2^n}$ and $\ket{\psi_{t_1}}$ the state of the quantum system. After applying the quantum operator $U$ the system will be in the state $\ket{\psi_{t_2}}$ described in Equation . $$\ket{\psi_{t_2}} = U\ket{\psi_{t_1}}
\label{eq:evolution}$$ In the computational basis, the matrix representation of the quantum operators the *not operator* $X$ and the *Hadamard operator* $H$ over one qubit are described in Equation . $$X = \begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix} \mbox{ and }
H =\frac{1}{\sqrt{2}} \begin{bmatrix}
1 & 1 \\
1 & -1
\end{bmatrix}
\label{eq:exop}$$ $X$ acts on the computational basis vectors as a not operator ($X\ket{0}=\ket{1}$ and $X\ket{1}=\ket{0}$) and $H$ applied to a state in the computational basis creates a “uniform" superposition (or linear combination) of the two basis: $$\begin{split}
H\ket{0} = \frac{1}{\sqrt{2}}(\ket{0} + \ket{1})\\
H\ket{1} = \frac{1}{\sqrt{2}}(\ket{0} - \ket{1})
\end{split}
\label{eq:exH1}$$ both represent a state which is $\ket{0}$ with probability $\frac{1}{2}$ and $\ket{1}$ with probability $\frac{1}{2}$, and can be thought of a state which is both $\ket{0}$ and $\ket{1}$. That is why one says that a qubit is able to “store" two classical bits simultaneously. This scale up exponentially with the number of qubits $n$, $H\ket{0}\otimes \cdots \otimes H\ket{0} = H^{\otimes n}\ket{0\cdots 0}$, with $0\cdots 0$ being a sequence of $n$ $0$’s, is the superposition of all $2^n$ possibles $n$-qubits. Equation shows the result for $n=2$, where $H^{\otimes 2} = H\otimes H$: $$\begin{split}
H^{\otimes 2} \ket{0}\ket{0} = \frac{1}{2}\left(\ket{0} + \ket{1}\right)\otimes \left(\ket{0} + \ket{1}\right)
= \\ \frac{1}{2}(\ket{0}\ket{0} + \ket{0}\ket{1} + \ket{1}\ket{0} + \ket{1}\ket{1})
\end{split}
\label{eq:exH}$$
*Quantum parallelism* is one of the main properties of quantum computation and it is used in the majority of quantum algorithms. Let $U_f$ be a quantum operator with action described in Equation , $$U_f\ket{x,c} = \ket{x, c \oplus f(x)}
\label{eq:uf}$$ where $f:B^m \rightarrow B^n$ is a Boolean function. Applying this operator in a state in superposition $\sum_i\ket{x_i,0}$ the value of $x_i$ will be calculated for all $i$ in a single quantum operation, as described in Equation . $$U_f\left(\sum_{i}\ket{x_i,0}\right) = \sum_{i}U_f \ket{x_i,0} = \sum_{i}\ket{x_i,f(x_i)}
\label{eq:parallelism}$$ Despite the possibility of obtaining all possible outputs of a Boolean function in a single query, quantum parallelism cannot be used directly.
Results in quantum computation are obtained via *measurement* which returns only a limited information about the system. For instance, if a measurement is performed in a quantum state $\ket{\psi} = \alpha_i\ket{\psi_i}$ the result will be $\ket{\psi_i}$ with probability $|\alpha_i|^2$. After measurement state $\ket{\psi}$ collapses to the output obtained and new measurements will result in the same output.
With the definition given above, also adopted by the mainstream quantum literature as [@nielsen:00], quantum operators are linear operators. In this paper, we suppose the viability of a nonlinear quantum operator $Q$ proposed in [@PhysRevLett.81.3992] whose action is described in Equation if at least one $\ket{c_i}$ is equal to $\ket{1}$ otherwise its action is described in Equation . $$Q \left(\sum_i \ket{\psi_i}\ket{c_i} \right)= \left(\sum_i \ket{\psi_i}\right)\ket{1}
\label{eq:nonlin}$$ $$Q \left(\sum_i \ket{\psi_i}\ket{c_i}\right) = \left(\sum_i \ket{\psi_i}\right)\ket{0}
\label{eq:nonlin2}$$ The speedup obtained by the application of non-linear operators have been associated with unphysical effects, however in [@czachor1998remarks; @czachor1998notes] it is presented a version of this non linear quantum operator free of unphysical influences.
Classical and quantum weightless neural networks
================================================
This work deals with quantum weightless neural networks. Weightless Neural Networks (WNN) are neural networks without weights associated in their connections where the information is stored in a look up table. The first model of WNN named RAM has been proposed in [@Aleksander1966], since then several neural networks models have been proposed, for instance the Probabilistic Logic Neuron (PLN), the Multi-valued Probabilistic Logic Neuron (MPLN), Goal Seeking Neuron (GSN) and the quantum RAM neuron (qRAM).
A weightless neuron with $n$ input values has a memory with $2^n$ addressable positions. The learning procedure of a WNN does not require differential calculus or any complex mathematical calculations. The learning procedure is performed by writing in the look up table. This learning strategy is faster than techniques based in gradient descendant methods and are suitable to implementation in conventional digital hardware.
Several models of weightless neural networks are described on [@Ludermir1999]. In this paper we deal with the qRAM neural network. The qRAM neuron is based in the simplest weightless model, the RAM neuron. Besides its simplicity RAM neurons can be trained very rapidly. Some applications of RAM and RAM based neurons in real world problems are described e.g. in [@Staffa2014; @Cardoso2014; @Carvalho2014; @DeSouza2009]. For a recent review see [@esann:2014:tutorial]. In [@Staffa2014; @Carvalho2014] a WiSARD system is used to track moving objects or human beings, in [@Cardoso2014] a WiSARD clustering version is proposed to perform credit analysis and in [@DeSouza2009] a VG-RAM weightless neural network is used to perform multi-label text categorisation.
RAM Node
--------
A RAM neuron with $n$ inputs has a memory $C$ with $2^n$ addressable positions. Each memory position of a RAM neuron stores a Boolean value and its address is a Boolean string in $\{0,1\}^n$ also called Boolean vector. When a RAM neuron receives a Boolean vector $x=x_1\cdots x_n$ as input it will produce the output $C[x]$. Learning in the qRAM node is very simple and can be accomplished updating the bits in memory positions for each one of the patterns in the training set.
Architectures of weightless neural networks are weekly connected as a consequence of the limited number of neurons inputs. Two common architectures are pyramidal where the output of a neuron in one layer is connected with a single neuron in the next layer or with only one layer where the neural network output is the sum of each neuron output.
Quantum neural networks
-----------------------
The notion of quantum neural networks has been proposed on several occasions [@panella:09; @ventura:04; @Behrman]. In [@panella:09; @ventura:04] quantum neural models are pure abstract mathematical devices and in [@Behrman] quantum neural networks are described as a physical device. In this paper we follow the first approach where a neural network is a mathematical model. It is also possible to classify quantum neural networks models as either quantum neural model [@Andrecut2002; @panellaneurofuzzy; @panella:09; @ventura:04; @Behrman; @daSilva:12] or quantum inspired model [@Li2013; @kouda:05]. Quantum inspired models are classical models of computation that uses ideas from quantum computing. Implementation of the quantum weightless neural network mathematically described in this paper requires a real quantum computer. Recent reviews on quantum neural networks can be found in [@Schuldquest; @Altaiskycurrent].
Models of quantum weightless neural networks are proposed or analysed in [@oliveira:08; @Oliveira2009; @daSilva:12; @DaSilva2012]. Quantum weightless neural networks’ models are first proposed in [@oliveira:08], in [@daSilva:10a] a quantum version of the RAM neuron based on an associative quantum memory is presented and in [@daSilva:12] the qRAM neuron and a learning algorithm for quantum weightless neural networks are presented.
Learning algorithms for quantum neural networks are also proposed in [@panella:09; @ventura:04] where a superposition of neural networks with a fixed architecture is created and a quantum search is performed to recover the best neural network architecture. In this paper we propose a variation of this methodology to train quantum weightless neural networks. In our training strategy, weightless neural networks with different architectures are in a superposition. The neural network model used in this learning methodology is the qRAM neural network.
qRAM Node
---------
The qRAM neuron is a quantum version of the RAM neuron. As in classical case, a $n$ input qRAM neuron has a quantum memory with $2^n$ memory positions. The content of the qRAM memory cannot be directly stored because a measurement of the output could destroy the information stored in the qRAM memory. We store quantum bits in the computational basis named selectors and apply one quantum operator $A$ to obtain the stored qubit.
The $A$ operator used in the qRAM is the control $X$ operator described in Equation \[eq:CNOT\]. With this operator a quantum RAM neuron can be described as in Definition \[def:qram\], where memory contents are stored in quantum register selectors.
$$\begin{array}{lr}
A = \begin{pmatrix}
I & 0 \\
0 & X
\end{pmatrix}
&
\begin{array}{l}
\mbox{where}\\
A\ket{00} = \ket{0}I\ket{0}\\
A\ket{10} = \ket{1}X\ket{0}\\
\end{array}
\end{array}
\label{eq:CNOT}$$
A qRAM node with $n$ inputs is represented by the operator $\textsf{N}$ described in Equation . The inputs, selectors and outputs of $\textsf{N}$ are organised in three quantum registers $\ket{i}$ with $n$ qubits, $\ket{s}$ with $2^n$ qubits and $\ket{o}$ with 1 qubit. The quantum state $\ket{i}$ describe qRAM input, and quantum state $\ket{s}\ket{o}$ describes qRAM state. \[def:qram\]
$$\textsf{N} = \sum_{i=0}^{2^n-1} \ket{i}_n \bra{i}_n A_{s_i,o}
\label{eq:N}$$
The qRAM neural network functions exactly as a RAM neural network when the selectors are in the computational basis. For instance, when the quantum register selectors of a qRAM neuron is in the state $\ket{c_{00}c_{01}c_{10}c_{11}}$ and the input $\ket{xy}$ is presented its output is $\ket{c_{xy}}$. The difference between the qRAM and RAM neurons can be observed when the selectors are initialised with a state in superposition. Suppose an initialisation of the quantum register selector with a state in the superposition $\frac{1}{\sqrt{2}}\left( \ket{c_{00}c_{01}c_{10}c_{11}} + \ket{c'_{00}c'_{01}c'_{10}c'_{11}}\right)$. When the neuron receives an input $\ket{xy}$, the output for each configuration in the superposition will be calculated and the quantum register output will be in the state $\frac{1}{\sqrt{2}}\left(\ket{ c_{xy} } + \ket{ c'_{xy} }\right)$, a sort of parallel execution of the network.
Classical and quantum weightless neurons require a memory (in the classical case) and a number of selectors (in the quantum case) exponential in relation to the number of inputs. To avoid exponential memory requirements, classical and quantum weightless neural networks use a feed-forward, low connected, pyramidal architecture. A pyramidal, feed-forward neural network with three two inputs qRAM Nodes is shown in Figure \[fig:qRAMNet1\]. A pyramidal qRAM network with $n$ inputs and composed of neurons with two inputs will have $2^{log_2(n)}-1 = n-1$ neurons. Each two input neuron has a memory with 4 selectors than network memory will need of $4 \cdot (n-1)$ selectors (linear memory size instead of an exponential memory size).
Configuration of a qRAM neural network is realised by the neuron selectors. For instance the configuration of the qRAM neural network in Figure \[fig:qRAMNet1\] is the state of quantum registers $\ket{s_1,s_2,s_3}$. For instance, a qRAM network with architecture displayed in Figure \[fig:qRAMNet1\] with configuration $\ket{s_1}=\ket{0110}$, $\ket{s_2}=\ket{0110}$ and $\ket{s_3}=\ket{0110}$ can solve the 4 bit parity problem. Superposition of qRAM neural networks with a given architecture can be obtained with the initialisation of qRAM neural configuration with a state in superposition. In Section 5, we explore superposition of qRAM networks in the learning procedure to allow neural network architecture selection.
(85,50)(0,0) (20,24)[(15,15)]{} (26,31)[$\textsf{N}_1$]{} (0,35)[$i_1$]{} (0,30)[$i_2$]{} (5,25)[$s_1$]{} (5,35.5)[(1,0)[10]{}]{} (5,30.5)[(1,0)[10]{}]{} (10,25.5)[(1,0)[5]{}]{} (20,5)[(15,15)]{} (26,11)[$\textsf{N}_2$]{} (0,15)[$i_3$]{} (0,10)[$i_4$]{} (5,5)[$s_2$]{} (5,15.5)[(1,0)[10]{}]{} (5,10.5)[(1,0)[10]{}]{} (10,5.5)[(1,0)[5]{}]{} (35,32.5)[(1,0)[10]{}]{} (45,32.5)[(0,-1)[5]{}]{} (45,27.5)[(1,0)[18]{}]{} (35,12.5)[(1,0)[10]{}]{} (45,12.5)[(0,1)[10]{}]{} (45,22.5)[(1,0)[18]{}]{} (65,15)[(15,15)]{} (71,21)[$\textsf{N}_3$]{} (53,15)[$s_3$]{} (58,15.5)[(1,0)[5]{}]{} (80,22.5)[(1,0)[10]{}]{}
Non linear quantum learning
===========================
Nonlinear quantum operators have been used previously [@panella:09; @zhou:12]. In this section we show how to train a weightless neural network with a nonlinear quantum algorithm. The proposed algorithm is based on a strategy proposed in [@Gammelmark:09], where the learning procedure is performed by measurement and feedback. Figure \[fig:mf\] illustrates how the measurement and feedback strategy works. The input is presented to a controlled quantum operator named quantum processor, and the result of a measurement performed in the output registers is used to update qubits in the control quantum register. The procedure is repeated until the control qubits $\ket{s}$ are set to some desired value.
![Measurement and feedback methodology[]{data-label="fig:mf"}](mf){width="0.7\columnwidth"}
The quantum processor in our learning strategy will be a qRAM weightless neural network with a fixed architecture. This quantum weightless neural network can have any number of layers and neurons and must have a feed-forward architecture. Patterns selected from a training set will be presented to several neural networks in parallel. This step cannot be efficiently performed in a classical computer, but it can be performed in a quantum computer using quantum parallelism. Operation performed by the quantum processor is described in Figure \[fig:qp\] where each pattern $x$ is presented to all qRAM network configurations represented in the quantum register $\ket{s}$ and the performance quantum register is updated to indicate if the neural network output is equal to the desired output $d(x)$. After the presentation of all patterns in the training set all pairs of neural network configuration and its respective performance will be in superposition.
Control qubits of the quantum processor in the measurement and feedback strategy are selectors of the qRAM neural network. In the $k$th iteration of the measurement and feedback methodology a non-linear quantum operator and a measurement are performed to determine the $k$th quantum bit of the selectors quantum register $\ket{s}$. After all iterations, the quantum register $\ket{s}$ will hold a qRAM configuration with performance greater than or equal to a given threshold $\theta$ for given training set (if exists).
Algorithm \[alg:la\] presents the proposed learning strategy. It requires six quantum registers. Input quantum register $\ket{i}$ used to present patterns from the training set to the qRAM network. The free parameters or selectors quantum register $\ket{s}$ used to store qRAM neural network configuration. The output quantum register $\ket{o}$ used to store the qRAM neural network output, the desired output quantum register $\ket{d}$, performance quantum register $\ket{{\mathit{perf}}}$ used to store the performance of each classifier in the superposition. And the objective quantum register $\ket{obj}$ used to mark configurations with desired performance. A configuration of the weightless neuron during the execution of Algorithm \[alg:la\] will be represented using the quantum state $\ket{\psi}$ described in Equation . $$\ket{\psi} = \ket{i}\ket{s}\ket{o}\ket{d}\ket{{\mathit{perf}}}\ket{obj}
\label{eq:qr}$$
The for loop starting in line \[line:for1\] will be repeated $n_s$ times, where $n_s$ is the number of quantum bits in quantum register $\ket{s}$. At the end of the $k$th iteration a non-linear quantum operator is performed to determine the $k$th bit $l_k$ of the quantum register $\ket{s}$.
Steps \[line:initinput\], \[line:initoutput\], \[line:initper\], \[line:initobj\] initialise quantum registers input, output, performance and objective. Step \[line:setselect\] of Algorithm \[alg:la\] initialises selector quantum register. After this step, the state of quantum registers $\ket{s}$ is described in Equation , where the value of first $k$ qubits $l_i$ were determined in $i$th iteration of the for loop and the last $n_s-k$ qubits are initialised with $H\ket{0}$ state. $$\ket{s} = \left(\frac{1}{\sqrt{2}}\right)^{n_s-k+1}\ket{l_1 \cdots l_{k-1}} \left(\ket{0}+\ket{1}\right)^{\otimes(n_s-k+1)}
\label{eq:init}$$
The for loop starting in line \[line:for2\] performs the quantum processor operation. It calculates the performance of all configurations in the superposition for the given architecture simultaneously due to principle of quantum parallelism. Step \[line:loadpattern\] initialises quantum register input with a pattern $x$ from the data-set, and desired output quantum register with the desired output of $x$ named $d(x)$. These initialisation steps can be performed by unitary operators controlled by a classical system using the classical representation of $x$ and $d(x)$. Step \[line:run\] runs the qRAM neural network and its output quantum register is set to the calculated output $y(x,s)$ for pattern $x$ with neural network configuration $s$. Steps \[line:if\] to \[line:endif\] adds 1 to quantum register performance if $y(x,s)$ is equal to $d(x)$. After these steps, description of state $\ket{\psi}$ is presented in Equation , where state $\ket{s}$ is described in Equation and $\ket{{\mathit{perf}}(x,s)}$ is the performance of the neural network with selectors $s$ after reading the input $x$. $$\ket{\psi} = \ket{x}\ket{s}\ket{y(x,s)}\ket{d(x)}\ket{{\mathit{perf}}(x,s)}\ket{0}
\label{eq:step4}$$
Step \[line:remove\] removes $\ket{x}$ and $\ket{d(x)}$ of quantum registers $\ket{i}$ and $\ket{d}$ performing the inverse operation of Step \[line:loadpattern\]. After the execution of the for loop starting in line \[line:for2\] the performance of each classifier $\ket{{\mathit{perf}}(s)}$ will be in superposition with its representation $\ket{s}$.
The for loop starting in line \[line:for3\] performs the measurement and feedback. An exhaustive non-linear quantum search is performed to determine the value of the $k$th bit in quantum state $\ket{s}$. Step 16 sets the quantum register $\ket{obj}$ to $\ket{1}$ if ${\mathit{perf}}(s) =\theta$ and $k=l$. This step can be performed by a unitary controlled operator $U_g$ that flips objective quantum register if and only if ${\mathit{perf}}(x,s) \geq \theta$ and $k=l$. After Step 16 the state of quantum registers $\ket{s}$, $\ket{{\mathit{perf}}}$ and $\ket{obj}$ is described in Equation , where $\delta_{ms,l,{\mathit{perf}}(ms)}$ is equal to 1 if ${\mathit{perf}}(s) \geq \theta$ and the $k$th quantum bit in $\ket{s}$ is equal to $l$. $$\ket{s,{\mathit{perf}},obj}=\ket{s,{\mathit{perf}}(s),\delta_{s,l,{\mathit{perf}}(s)}}
\label{eq:f}$$
All previous steps can be performed utilising only linear quantum operators. Step \[line:nonlin\] applies the non-linear quantum operator proposed in [@PhysRevLett.81.3992] to the objective quantum register. The objective quantum register will be changed to the basis state $\ket{1}$ if there is at least one configuration in the superposition with objective equal to one. In this case, Steps \[line:if2\] to \[line:endif2\] performs a measurement in state $\ket{\psi}$ and changes the $k$th quantum bit in quantum register $\ket{s}$ to $l$.
The computational cost of Algorithm 1 depends on the number of patterns in the training set $n_t$ and on the number of qubits used in selector quantum register $n_s$. The for loop starting in line 1 will be repeated $n_s$ times. Steps \[line:initinput\] to \[line:initobj\] have constant computational time. For loop in lines \[line:for2\] to 13 will be repeated $n_t$ times and each inner line has constant computational cost. For loop in lines \[line:for3\] to \[line:endfor3\] does not depend on $n_t$ and $n_s$ and it has a constant computational cost. In this way the overall cost of the Algorithm 1 is $O(n_t \cdot n_s )$. Then Algorithm 1 has polynomial time in relation to the number of qubits used to represent the qRAM neural network selectors and the number of patterns in the training set.
A concrete example of Algorithm 1 execution is presented to illustrate its functionality. Without loss of generality we use a qRAM neural network composed by only one neuron with two inputs to learn the 2-bit XOR toy problem described in Equation . For this problem, quantum register input needs two qubits, quantum register selectors has 4 qubits, quantum register output needs 1 qubit, quantum register performance has 3 qubits and quantum register objective has 1 qubit.
$$T=\left\{\left(\ket{00},\ket{0}\right),\left(\ket{01},\ket{1}\right),\left(\ket{10},\ket{1}\right),\left(\ket{11},\ket{0}\right)\right\}
\label{eq:xor}$$
In Steps 2, 4, 5 and 6 bits in quantum registers input, output, performance and objective are initialised with the quantum state $\ket{0}$. The number of quantum bits in $\ket{s}$ quantum register is equal to 4 and in the first iteration $n_s-k+1$ is also equal to 4, then all four qubits in quantum register $\ket{s}$ are initialised with the state $H\ket{0}$. After these initialisation steps, neural network configuration $\ket{\psi}$ is described in Equation . $$\begin{split}
\ket{\psi} = \frac{1}{4}\ket{00}\left(\ket{0}+\ket{1}\right)^{\otimes 4}\ket{0}\ket{0}\ket{000}\ket{0} = \\
\frac{1}{4}\sum_{j\in \left\{0,1\right\}^4}\ket{00}\ket{j}\ket{0}\ket{0}\ket{000}\ket{0}
\end{split}
\label{eq:6}$$
Suppose that in the first iteration of the for loop starting in line \[line:for1\] $x$ assumes value $\ket{01}$ and $d(x)$ is $\ket{1}$. Step \[line:loadpattern\] initialises pattern and desired output quantum register respectively to $\ket{01}$ and $\ket{1}$. This initialisation can be performed through CNOT operators applied to $\ket{\psi}$ resulting in state $\ket{\psi_1}$ described in Equation .
$$\frac{1}{4}\sum_{j\in \left\{0,1\right\}^4}\ket{01}\ket{j}\ket{0}\ket{1}\ket{000}\ket{0}
\label{eq:7}$$
Step \[line:run\] runs the neural network and this output is calculated in quantum register $\ket{o}$. After this step we obtain the state $\ket{\psi_2}$ described in Equation , where $j_1$ is the qubit in memory position 01 and $\delta_{j_1,1}=1$ if and only if $j_1=1$. $$\frac{1}{4}\sum_{j\in \left\{0,1\right\}^4}\ket{01}\ket{j}\ket{\delta_{j_1,1}}\ket{1}\ket{000}\ket{0}
\label{eq:8}$$
Step \[line:if\] to \[line:endif\] check if desired output is equal to the calculated output, adding one to the performance quantum register if they are equal. The resulting state after Step \[line:endif\] $\ket{\psi_3}$ is described in Equation . These steps can be performed using a unitary operator describing the qRAM neural network and a quantum operator that adds one to the quantum register performance with controls $\ket{o}$ and $\ket{d}$. $$\begin{split}
\ket{\psi_3} = \frac{1}{4}\left(\ket{01}\ket{0000}\ket{0}\ket{1}\ket{000}\ket{0} \right. \\
+ \ket{01}\ket{0001}\ket{0}\ket{1}\ket{000}\ket{0}
+ \ket{01}\ket{0001}\ket{0}\ket{1}\ket{000}\ket{0} \\
+ \ket{01}\ket{0001}\ket{0}\ket{1}\ket{000}\ket{0}
+ \ket{01}\ket{0001}\ket{0}\ket{1}\ket{000}\ket{0} \\
+ \ket{01}\ket{0001}\ket{0}\ket{1}\ket{000}\ket{0}
+ \ket{01}\ket{0001}\ket{0}\ket{1}\ket{000}\ket{0} \\
+ \ket{01}\ket{0001}\ket{0}\ket{1}\ket{000}\ket{0}
+ \ket{01}\ket{0001}\ket{0}\ket{1}\ket{000}\ket{0} \\
+ \ket{01}\ket{0001}\ket{0}\ket{1}\ket{000}\ket{0}
+ \ket{01}\ket{0001}\ket{0}\ket{1}\ket{000}\ket{0} \\
+ \ket{01}\ket{0001}\ket{0}\ket{1}\ket{000}\ket{0}
+ \ket{01}\ket{0001}\ket{0}\ket{1}\ket{000}\ket{0} \\
+ \ket{01}\ket{0001}\ket{0}\ket{1}\ket{000}\ket{0}
+ \ket{01}\ket{0001}\ket{0}\ket{1}\ket{000}\ket{0} \\
+ \ket{01}\ket{0001}\ket{0}\ket{1}\ket{000}\ket{0}
\end{split}
\label{eq:9}$$
Step \[line:remove\] removes the values of $\ket{x}$ and $\ket{d(x)}$ from quantum registers $\ket{i}$ and $\ket{d}$ allowing the initialisation of the next for loop iteration. After the for loop last execution only one configuration in superposition, with $\ket{s}=\ket{0110}$, has performance 100% and the selectors and performance quantum registers are described by quantum state in Equation , where ${\mathit{perf}}(j)<4$ for all $j \neq 0110$. $$\ket{s,{\mathit{perf}}} = \frac{1}{4}\left(\ket{0110}\ket{4}_3 + \sum_{j\in \{0,1\}^4, j\neq 0110}\ket{j}\ket{{\mathit{perf}}(j)}\right)
\label{eq:10}$$
Setting $\theta$ to 100%, in the first iteration of the for loop ($l=0$) in line \[line:for3\], Step \[line:obj\] changes objective register to $\ket{1}$ when the $k$th qubit of $\ket{s}$ is $\ket{0}$ and performance is $\theta$. After Step 15 selectors, performance and objective quantum registers are described in Equation (11). $$\ket{s,{\mathit{perf}},obj} = \frac{1}{4}\left(\ket{0110}\ket{4}_3\ket{1} + \sum_{j\in \{0,1\}^4, j\neq 0110}\ket{j}\ket{{\mathit{perf}}(j)}\ket{0}\right)
\label{eq:11}$$
Step \[line:nonlin\] applies the nonlinear quantum operator in objective quantum register and the state of selectors, performance and objective quantum registers are described in Equation . The nonlinear quantum operator sets the quantum register objective to $\ket{1}$. $$\ket{s,{\mathit{perf}},obj} = \frac{1}{4}\left(\ket{0110}\ket{4}_3 + \sum_{j\in \{0,1\}^4, j\neq 0110}\ket{j}\ket{{\mathit{perf}}(j)}\right)\ket{1}
\label{eq:12}$$
Since the objective quantum register is in a base state we can check whether $\ket{obj} = \ket{1}$ with no information loss. In Steps \[line:if2\] to \[line:endif2\] a measurement is performed in quantum register $\ket{s}$ and the first qubit of $\ket{s}$ is set to $\ket{l_1} = \ket{0}$. This qubit will not be changed in the next iterations.
At the end of the main for loop the selector quantum register $\ket{s}$ will be in the state $\ket{0110}$ and the desired configuration was found. Next section shows how to perform a search in the architecture space of a quantum weightless neural network.
Architecture learning
=====================
The operator $A$ in a qRAM neural network is known as controlled not operator. In other models of quantum weightless neural networks this operator can assume different forms. For instance, in [@oliveira:08] the $A$ operators of qPLN are represented in computational basis by the quantum operator $A_{qPLN}$ described in Equation , where $\textsf{U}$ is an arbitrary quantum operator $$\begin{split}
A_{qPLN} =\ket{00}\bra{00}\otimes \textsf{I} + \ket{01}\bra{01}
\otimes \textsf{X} + \\ \ket{10}\bra{10}\otimes\textsf{H} +
\ket{11}\bra{11}\otimes\textsf{U}
\end{split}
\label{eq:aqpln}$$ and the $A$ operators of a qMPLN with $n$ qubits in each memory position are represented by the matrix described in Equation , where ${\textsf{U}}_{p_k}$ is a rotation operator with angle $p_k$. $$A_{qMPLN} = \sum_{k=0}^{n-1}\ket{k}\bra{k}\otimes {\textsf{U}}_{p_k}
\label{eq:aqmpln}$$
These $A$ operators are used to generate the values stored in a specific memory position. For instance in the qPLN, instead of storing the qubit $\frac{1}{\sqrt{2}}\left(\ket{0} + \ket{1}\right)$, we store the qubits in the computational basis $\ket{10}$ and uses the operator ${\textsf{A}}_{qPLN}$ to generate the content $\frac{1}{\sqrt{2}}\left(\ket{0} + \ket{1}\right)$.
![Quantum neuron representing a weightless neural networks with four different architectures[]{data-label="fig:qn"}](architectureLearning){width="\columnwidth"}
The main idea in this Section is to allow a weightless neural network to store the output of a weightless neural network with a given input $x$ and selectors $s$. In this case, the quantum version of this weightless neural network will need a matrix $A$ representing the weightless neural network to generate the output of the weightless neural network. Then the $A$ operators are replaced by operators representing weightless neural networks and selectors are replaced by the neural network inputs and selectors. Figure \[fig:qn\] illustrates this weightless neuron with two inputs, where $\ket{a_1 a_2}$ are architecture selectors, input pattern $x$ and selectors are combined in one single quantum register and acts as the free parameters of the neuron, and quantum register output is shared by all weightless networks $N_0,N_1,N_2,N_3$.
With this quantum neuron the action of the quantum processor in Figure \[fig:mf\] can be described by Figure \[fig:fram2\]. Initialisation of the architecture selector quantum register with a quantum state in superposition will put different architectures, represented by the doted boxes, into superposition. And the initialisation of selectors quantum registers puts different configurations of each architecture into superposition. Problem of architecture selection is reduced to the problem of training the weightless neuron in Figure \[fig:qn\] where the input is represented by quantum register $\ket{x}$ and selectors are represented by quantum registers $\ket{a,s}$. In this way, Algorithm 1 can be used to learning parameters and architecture simultaneously.
Architecture selection computational time is directly related to computational time of the Algorithm \[alg:la\]. Due to the linearity of quantum operators, neurons can share selectors and under supposition that all architectures are pyramidal and low connected then network memory size (or the necessary number of selectors) will be polynomial in relation to the number of neural network inputs. The cost of architecture selection will be $O\left(n_a+n_s+n_t\right)$, where $n_a$ is the number of architectures, $n_s$ is the number of selectors in the most complex (with more selectors) architecture and $n_t$ is the number of training patterns.
Architecture selection with SAL algorithm
-----------------------------------------
Quantum computers are not yet a reality and we cannot evaluate SAL algorithm in real problems. In this Section we present a concrete example (with low dimensionality) of the SAL algorithm in architecture selection. Weuse the artificial dataset described in Table \[tab:artificialDataSet\]obtained in the following way. Two weightless neural network architectures were defined and an exhaustive search was performed to find a dataset in each one architecture can learn the dataset and the other architecture cannot learn the dataset using selectors in the computational basis.
Class
--- ------- --- --- ---
0 0 0 0 1
0 0 0 1 1
0 0 1 0 0
0 0 1 1 1
0 1 0 0 1
0 1 0 1 1
0 1 1 0 0
0 1 1 1 1
1 0 0 0 1
1 0 0 1 1
1 0 1 0 0
1 0 1 1 1
1 1 0 0 0
1 1 0 1 1
1 1 1 0 0
1 1 1 1 1
: Simple artificial data set[]{data-label="tab:artificialDataSet"}
The architectures used in the experiment are two layers, pyramidal qRAM weightless neural networks. The first architecture ${\textsf{N}}_0$ has two qRAM neurons each with two inputs in the first layer and one qRAM neuron with two inputs in the second layer. Figure \[fig:qRAMNet1\] displays architecture ${\textsf{N}}_0$. The second architecture ${\textsf{N}}_1$ has two qRAM neurons in the first layer where the first neuron has three inputs and the second neuron has one input and the second layer has one qRAM neuron with two inputs.
The first architecture needs of 12 quantum bits for representing selector quantum register, 4 quantum bits for representing input of the first layer, 2 quantum bits to represent the second layer input, and 1 quantum bit to representing the neural network output. In this way, the first architecture representation needs of 19 quantum bits. The second architecture needs of 14 quantum bits for representing selector quantum register and the same number of quantum bits used by the first architecture to represent neurons inputs and network output than the second architecture representation requires 21 quantum bits.
These two qRAM neural networks are represented in a single circuit with six quantum registers. Neurons inputs quantum register $\ket{i}$ with 6 quantum bits, selectors quantum register $\ket{s}$ with 14 quantum bits, output quantum register $\ket{o}$ with one qubit and architecture selector quantum register $\ket{a}$ with 1 qubit. Performance quantum register $\ket{{\mathit{perf}}}$ with 5 quantum bits. Output quantum register with 1 quantum bit.
The qRAM neural network with architecture ${\textsf{N}}_0$ uses all qubits in quantum registers selectors, input and output. The qRAM neural network with architecture ${\textsf{N}}_1$ uses all qubits in inputs and output quantum register and uses only 12 qubits in selectors quantum register. The architecture quantum register $\ket{a}$ is used to select the architecture. If $\ket{a}$ is equal to 0 the architecture 1 is used. If $\ket{a}$ is equal to 1 the architecture 2 is used.
After the initialization steps of Algorithm \[alg:la\], the state of quantum registers $\ket{a}\ket{s}\ket{{\mathit{perf}}}$ is described in Equation , where $\ket{a}$ and $\ket{s}$ are in a superposition with all possible values and the quantum bits in performance quantum register are initialized with $\ket{0}$. $$\ket{a}\ket{s}\ket{{\mathit{perf}}} = (\ket{0}+\ket{1})\sum_{k\in\{0,1\}^{14}}\ket{k}\ket{00000}
\label{eq:aftinit}$$ After the datased presetation to the neural network performed in Steps 7 to 14 of Algorithm \[alg:la\], the state of quantum registers $\ket{a}\ket{s}\ket{{\mathit{perf}}}$ is described in Equation , where ${\mathit{perf}}(k,N_i)$ is the performance of qRAM neural network with architecture $N_i$ and selectors $\ket{k}$.
$$\begin{split}
\ket{a}\ket{s}\ket{{\mathit{perf}}} = \\ \ket{0}\left(\sum_{k\in\{0,1\}^{12}}\ket{k}{\textsf{H}}^{\otimes 2}\ket{00}\right)\ket{{\mathit{perf}}(k,N_0)} \\ +\ket{1}\sum_{k\in\{0,1\}^{14}}\ket{k}\ket{{\mathit{perf}}(k,N_1)}
\end{split}
\label{eq:aftpresentation}$$
${\textsf{N}}_0$ architecture cannot learn the dataset with 100% of accuracy and ${\textsf{N}}_1$ can learn the dataset with 100% of accuracy when its selectors are in the set $$\begin{split}
T = \{\ket{0 1 0 1 0 1 1 1, 0 1, 1 1 0 1},
\ket{0 1 0 1 0 1 1 1, 1 0, 1 1 1 0}, \\
\ket{1 0 1 0 1 0 0 0, 0 1, 0 1 1 1},
\ket{1 0 1 0 1 0 0 0, 1 0, 1 0 1 1} \}.\\
\end{split}$$ In the second iteration of for loop starting in line 15, the quantum register objective is set to $\ket{1}$ if and only if the performance is greather than a given threshold $\theta$. Here we use $\theta$ equal to 16 (100% of accuracy), after this operation the state of quantum registers $\ket{a}\ket{s}\ket{{\mathit{perf}}}\ket{obj}$ is described in Equation .
$$\begin{split}
\ket{a}\ket{s}\ket{{\mathit{perf}}}\ket{obj} =\\ \ket{0}\left(\sum_{k\in\{0,1\}^{12}}\ket{k}{\textsf{H}}^{\otimes 2}\ket{00}\right)\ket{{\mathit{perf}}(k,N_0)}\ket{0} \\ +\ket{1}\sum_{k\in\{0,1\}^{14}, k\notin T}\ket{k}\ket{{\mathit{perf}}(k,N_1)}\ket{0} \\ +\ket{1}\sum_{k\in T}\ket{k}\ket{{\mathit{perf}}(k,N_1)}\ket{1}
\end{split}
\label{eq:afttheta}$$
Step 17 applies the nonlinear quantum operator and the resultant state of quantum registers $\ket{a}\ket{s}\ket{{\mathit{perf}}}\ket{obj}$ is described in Equation , where a measurement can be performed and the architecture register will be in state $\ket{1}$ and the architecture ${\textsf{N}}_1$ was chosen.
$$\begin{split}
\ket{a}\ket{s}\ket{{\mathit{perf}}}\ket{obj} = \ket{1}\sum_{k\in T}\ket{k}\ket{{\mathit{perf}}(k,N_1)}\ket{1}
\end{split}
\label{eq:aftnonlinear}$$
Discussion
----------
We proposed a methodology to select quantum neural network parameters and architecture using a quantum weightless neural networks in polynomial time in relation to the number of training patterns, architectures and neural network free parameters. The proposed algorithm, named Superposition based Architecture Learning (SAL), performs a non-linear global search in the space of weightless neural networks parameters and for a given data set returns a classifier with a desired performance $\theta$ or returns that there is no classifier otherwise.
A classical polynomial time algorithm to perform neural network architecture selection is not known. Classical techniques used to perform architecture selection are heuristics that do not guarantee to find an exact solution. Some strategies used to find near optimal neural networks architectures or parameters are evolutionary algorithms [@Almeida2010] and meta-learning [@Miranda201427]. Running time of evolutionary algorithms used in architecture selection are displayed in [@Miranda201427] and even in benchmark problems the running time of these classical strategies can vary from 3 to 400 minutes.
In the application of the SAL algorithm to perform architecture selection, if there is a solution in the space search then the solution will be found in polynomial time. SAL algorithm puts all neural network configurations with some architectures in superposition, the performance is calculated and a nonlinear operator is used to recover the configuration and architecture with desired performance. SAL algorithm is the first algorithm to perform quantum weightless neural network architecture selection in polynomial time in relation to the number of patterns, architectures.
Superposition principle allows the evaluation of neural networks architectures in a way that is not possible in classical neural networks. In a classical neural network the architecture evaluation is biased by a choice of neural network parameters. In SAL algorithm all neural network parameters are initialized with all parameters in superposition allowing the evaluation of neural network architecture without the bias of a given set of parameters.
The gain in computational time of the proposed strategy is a result of the use of non-linear quantum operator proposed in [@PhysRevLett.81.3992]. Despite non-linear quantum computing has been used in several works, there still remains some controversy with some authors claiming that non linear quantum operators are not physically realisable [@PhysRevLett.81.3992] while other researchers claiming otherwise [@czachor1998remarks].
Even if non-linear quantum operators do not become a reality, the proposed learning algorithm furnishes a framework for the development of linear quantum algorithms to perform neural network architecture selection. The proposed idea is to define a quantum weightless neural network such that its memory positions store configurations of neural networks with different architectures.
Conclusion
==========
For some problems there are quantum algorithms which are asymptotically faster than the known classical algorithms [@grover:96; @shor:97; @trugenberger:02]. In this paper, we defined a quantum Superposition based Architecture Learning algorithm for weightless neural networks that finds architecture and parameters with polynomial time in relation to the number of training patterns, architectures and the size of the selectors quantum register. The proposed algorithm used the quantum superposition principle and a nonlinear quantum operator.
A linear version of the proposed algorithm is challenging research topic which is the subject of on going work. This linear version should be a quantum probabilistic algorithm, because the problem of training a weightless neural networks is a NP-complete problem. One could use the quantum processor to create a superposition of weightless neural networks with different architectures and to perform classical learning steps in these neural networks in superposition before performing the measurement and feedback.
Quantum weightless neural networks proposed in [@oliveira:08] are generalisation of the classical models based on a classical RAM memory. Another possible future work is the analysis of quantum memories [@altaiskymemory; @ventura:98] for the development of weightless neural networks models. These quantum memories has an exponential gain in memory capacity when compared with classical memories.
Acknowledgements {#acknowledgements .unnumbered}
================
This work is supported by research grants from CNPq, CAPES and FACEPE (Brazilian research agencies).
[^1]: We could have used any other orthonormal basis but in quantum computing the canonical basis also called *computational basis* is the most employed.
[^2]: An operator (or matrix, for the finite dimensional case once fixed a basis) $A$ is *unitary* if $AA^\dagger=A^\dagger A=I$ where $A^\dagger$ is the complex conjugate of the transpose of $A$
|
---
abstract: 'We propose a modified theory of gravitation constructed by the addition of the term $f(T_{\mu\nu}T^{\mu\nu})$ to the Einstein-Hilbert action, and elaborate a particular case $f(T_{\mu\nu}T^{\mu\nu})=\alpha(T_{\mu\nu}T^{\mu\nu})^{\eta}$, where $\alpha$ and $\eta$ are real constants, dubbed as energy-momentum powered gravity (EMPG). We search for viable cosmologies arising from EMPG especially in the context of the late-time accelerated expansion of the Universe. We investigate the ranges of the EMPG parameters $(\alpha,\eta)$ on theoretical as well as observational grounds leading to the late-time acceleration of the Universe with pressureless matter only, while keeping the successes of standard general relativity at early times. We find that $\eta=0$ corresponds to the $\Lambda$CDM model, whereas $\eta\neq 0$ leads to a $w$CDM-type model. However, the underlying physics of the EMPG model is entirely different in the sense that the energy in the EMPG Universe is sourced by pressureless matter only. Moreover, the energy of the pressureless matter is not conserved, namely, in general it does not dilute as $\rho\propto a^{-3}$ with the expansion of the Universe. Finally, we constrain the parameters of an EMPG-based cosmology with a recent compilation of 28 Hubble parameter measurements, and find that this model describes an evolution of the Universe similar to that in the $\Lambda$CDM model. We briefly discuss that EMPG can be unified with Starobinsky gravity to describe the complete history of the Universe including the inflationary era.'
author:
- Özgür Akarsu
- 'Nihan Kat[i]{}rc[i]{}'
- Suresh Kumar
title: 'Cosmic acceleration in a dust only universe via energy-momentum powered gravity'
---
Introduction
============
Inflation is a theory of exponential expansion of space in the early Universe corresponding to energy scales $\sim10^{16}\,{\rm GeV}$ [@Starobinsky:1980te; @Guth:1980zm; @Linde:1981mu; @Albrecht:1982wi; @Baumann:2009ds; @Linde:2014nna; @Ade:2015lrj]. Besides, today it is confirmed by means of several independent observations that the Universe again started to expand at an accelerated rate approximately $5$ Gyr ago, and it has happened at low energy scales $\sim 10^{-4}\, {\rm eV}$ [@Komatsu:2010fb; @Aubourg:2014yra; @Ade:2015xua]. The totally different energy scales show that the physics behind these two phenomena should be different. Inflation requires a modification of gravitation that plays a role at high energy densities such as the Starobinsky model [@Starobinsky:1980te], while the late-time acceleration of the Universe would be related with a similar mechanism but working at very low energy scales or a modification to the general theory of relativity (GR) that plays a role at sufficiently low energies. The approaches accommodating the current accelerated expansion of the Universe are classified into two main categories, subject to imminent observational discrimination. First is the introduction of a source in GR with a large negative pressure, which is called “dark energy” (DE) and is described most frequently by a scalar field or conventional vacuum energy which is mathematically equivalent to the cosmological constant $\Lambda$ [@Sahni:1999gb; @Peebles:2002gy; @Copeland:2006wr; @Bamba:2012cp]. The most successful cosmological model capable of predicting the observed pattern of the expansion of the Universe so far is the six-parameter base $\Lambda$CDM model that is simple and in reasonably good agreement with the currently available high-precision data [@Komatsu:2010fb; @Aubourg:2014yra; @Ade:2015xua]. On the other hand, this model suffers from profound theoretical difficulties, such as the cosmological constant and coincidence problems [@Weinberg:1988cp; @Sahni:1999gb; @Carroll:2000fy; @Peebles:2002gy; @Sahni:2004ai; @Bull:2015stt]. There are also observations suggesting small deviations from $\Lambda$CDM in order to describe the current Universe [@Aubourg:2014yra; @Ade:2015xua; @Zhao:2017cud]. On the other hand, we do not have a promising and concrete fundamental theory giving rise to dark energy models more general than the $\Lambda$CDM model that would account for small deviations from $\Lambda$CDM. The second approach is the modification of gravity at large distances, rather than imposing an unknown kind of source, such as in $f(R)$, scalar-tensor theories, Brans-Dicke theories of gravity, etc. [@DeFelice:2010aj; @Clifton:2011jh; @Capozziello:2011et; @Nojiri:2017ncd]. The essence of this approach is to modify the form of the coupling between the source described by the energy-momentum tensor (EMT) $T_{\mu\nu}$ and the spacetime geometry described by the metric tensor $g_{\mu\nu}$. Depending on the modifications in the action, this approach can lead to modifications on either the left- or right-hand side of Einstein’s field equations (EFE), $R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\kappa T_{\mu\nu}$, where the terms have the usual meaning. In this approach where one avoids introducing a new kind of source, it may be possible to define the corrections that appear on the left-hand side of the EFE, as in the $f(R)$ theories [@Nojiri:2010wj], as a separate effective source by moving them to the right-hand side of the EFE. However, this might not always be trivial or possible, as in, for example, the generalization of $f(R)$ theories via an explicit coupling of an arbitrary function of the Ricci scalar $R$ with the matter Lagrangian density $\mathcal{L}_{\rm m}$ given in Ref. [@Harko:2010mv]. Tests of the modified gravity models including nonminimal coupling between matter and geometry, using direct astronomical and astrophysical observations at the galactic or extragalactic scale, are also possible [@Harko:2010vs]. The modifications in the action that lead to the modifications on the left-hand side of the EFE are much more commonly studied in the literature compared to the ones that lead to the modification on the right-hand side of the EFE, namely, in the form of how $T_{\mu\nu}$ appears in the field equations. Of course, it could be possible to find a corresponding modification on the left-hand side of the EFE for a modification appearing on the right-hand side of the EFE though this also might not be trivial or possible. Hence, confining ourselves to the modifications in the action leading to modifications on the left-hand side of the EFE may result in missing a large class of successful modified gravity laws that can be obtained from the EMT-type modifications.
In the following, we briefly describe the EMT type of modifications considered/studied in the literature. In most of the works, each side $-$ both $R$ and $T$ (the trace of the EMT)$-$ were modified and the resulting cosmological implications were studied with the motivation to explain problems or shortcomings of the standard big bang model and Einstein’s standard GR, such as inflation, late-time acceleration etc. $\Lambda(T)$ gravity relates cosmic acceleration via the most general form of the EMT, and yields a relativistically covariant model of interacting DE, based on the principle of least action [@Poplawski:2006ey], where $\Lambda$ in the gravitational Lagrangian is a function of the trace of the EMT. The $f(R,T)$ model was proposed and studied in Ref. [@Harko:2011kv] with some specific choices of the function $f(R, T)$. The authors of Ref. [@Moraes:2016gpe] discussed a complete cosmological scenario from $f(R,T^{\phi})$ gravity theory proposed in Ref. [@Harko:2011kv]. In Refs. [@Odintsov:2013iba; @Haghani:2013oma; @Haghani:2014ina], more generalized modified theories of gravity with $f(R,T,R_{\mu\nu}T^{\mu\nu})$ gravity were studied. One of us has proposed a Lorentz invariant and covariant generalization of GR in Ref. [@Arik:2013sti] considering $f(R,T_{\mu\nu}T^{\mu\nu})$. The last two studies are different in the sense that $T_{\mu\nu}T^{\mu\nu}$- or $R_{\mu\nu}T^{\mu\nu}$-like terms still survive for the $T=0$ case, and do not simply reduce to $f(R)$ theories. For instance, for the electromagnetic field, $f(R,T,R_{\mu\nu}T^{\mu\nu})$ and $f(R,T_{\mu\nu}T^{\mu\nu})$ theories differ, whereas the predictions of $f(R,T)$ and $f(R)$ theories may be the same. Rather than GR, $f(\mathcal{T})$ gravity$-$ called the teleparallel equivalent of GR$-$ was extended via a nonminimal torsion-matter coupling in the action in Ref. [@Harko:2014sja]. Going one step further, the coupling of the torsion scalar with the trace of the EMT was studied in $f(\mathcal{T},T)$ gravity [@Harko:2014aja].
As stated above, the self-contraction of the EMT was first proposed in Ref. [@Arik:2013sti]. Here we extend it to a more general form $f(R,T_{\mu\nu}T^{\mu\nu})=\alpha(T_{\mu\nu}T^{\mu\nu})^{\eta}$, whereas in Ref. [@Arik:2013sti] $-$particular values of its power$-$ $\eta$ were studied. For instance, it was found that the relation between the Hubble parameter and energy density is of a form familiar from the Cardassian expansion studied in the context of late-time cosmic acceleration, or that $\eta=\frac{1}{2}$ leads to a slight deviation from the standard pressureless matter and radiation behavior, violating energy conservation. The higher-order matter terms are reminiscent of the terms (corrections) that arise naturally in loop quantum gravity [@Ashtekar:2006wn; @Ashtekar:2011ni], and those in the brane world models [@Brax:2003fv]. In Ref. [@Roshan:2016mbt], the model was analyzed with the $\eta=1$ case, which corresponds to the EMT squared contribution, dubbed energy-momentum squared gravity (EMSG). For this power of the EMT, the correction terms are important only at sufficiently early times, and therefore this model does not give accelerated expansion without any contribution from other extra fields such as scalar fields which can enter the matter Lagrangian. Since the difference appears in the high energy density regime, the charged black hole solution in energy-momentum squared gravity is different from the standard Reissner-Nordstr" om spacetime [@Roshan:2016mbt]. Here, we present a detailed theoretical and observational analysis of energy-momentum powered gravity (EMPG) provided by the general case $f(R,T_{\mu\nu}T^{\mu\nu})=\alpha(T_{\mu\nu}T^{\mu\nu})^{\eta}$ in the context of the late-time accelerated expansion of the Universe. We investigate the ranges of the EMPG model parameters $(\alpha,\eta)$ for viable cosmologies leading to the late-time acceleration of the Universe with pressureless matter only, while keeping the successes of standard general relativity at early times. We demonstrate that only the dust content is sufficient to explain the observed cosmic acceleration. Moreover, the sequence of the matter-dominated phase, deceleration-acceleration transition, and acceleration is obtained similar to the $\Lambda$CDM model in the presence of dust alone. After the first appearance of our study on the arXiv, the $(T_{\mu\nu}T^{\mu\nu})^{\eta}$-type modification to GR was also studied in Ref. [@Board:2017ign] where the authors presented a range of exact solutions for isotropic universes, and discussed their behaviors with reference to the early- and late-time evolution, accelerated expansion, and the occurrence or avoidance of singularities.
The paper is structured as follows. In the following section, we present the detailed framework of EMPG. In Sec. III, we demonstrate the viable cosmologies arising from EMPG that lead to the late-time acceleration of the Universe. In Sec. IV, we constrain the EMPG model parameters with a recent compilation of 28 Hubble parameter measurements, and discuss the evolution of the EMPG model in contrast with the $\Lambda$CDM model. In Sec. V, we give our concluding remarks and discuss some future perspectives of the study.
Energy-Momentum Powered Gravity
===============================
We start with the action constructed by the addition of the term $f(T_{\mu\nu}T^{\mu\nu})$ [@Arik:2013sti] to the Einstein-Hilbert (EH) action[^1] as follows: $$\begin{aligned}
S=\int \left(\frac{1}{2\kappa}R+f(T_{\mu\nu}T^{\mu\nu})\right)\sqrt{-g}\,{\rm d}^4x+\int \mathcal{L}_{\rm m}\sqrt{-g}\,{\rm d}^4x,
\label{action}\end{aligned}$$ where $\kappa$ is Newton’s constant, $R$ is the Ricci scalar, $g$ is the determinant of the metric, and $\mathcal{L}_{\rm m}$ is the Lagrangian density corresponding to the source that would be described by the energy-momentum tensor $T_{\mu\nu}$. We vary the action with respect to the inverse metric as $$\begin{aligned}
\delta S=\int\, {\rm d}^4 x \sqrt{-g} \bigg[\frac{1}{2\kappa}\delta R+\frac{\partial f}{\partial(T_{\mu\nu}T^{\mu\nu})}\frac{\delta(T_{\sigma\epsilon}T^{\sigma\epsilon})}{\delta g^{\mu\nu}}\delta g^{\mu\nu} \nonumber \\
-\frac{1}{2}g_{\mu\nu}\left(R+f(T_{\sigma\epsilon}T^{\sigma\epsilon})\right)\delta g^{\mu\nu} +\frac{1}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{\rm m})}{\delta g^{\mu\nu}}\bigg],
\end{aligned}$$ and, as usual, we define the EMT as $$\begin{aligned}
\label{tmunudef}
T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{\rm m})}{\delta g^{\mu\nu}}=g_{\mu\nu}\mathcal{L}_{\rm m}-2\frac{\partial \mathcal{L}_{\rm m}}{\partial g^{\mu\nu}},
\end{aligned}$$ which depends only on the metric tensor components, and not on its derivatives. Consequently, the field equations read as follows: $$\begin{aligned}
G_{\mu\nu} =\kappa T_{\mu\nu}+\kappa\left(fg_{\mu\nu}-2\frac{\partial f}{\partial(T_{\mu\nu}T^{\mu\nu})}\theta_{\mu\nu}\right),
\label{genfieldeq}\end{aligned}$$ where $G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}$ is the Einstein tensor and $\theta_{\mu\nu}$ is the new tensor defined as $$\begin{aligned}
\theta_{\mu\nu}&= T^{\sigma\epsilon}\frac{\delta T_{\sigma\epsilon}}{\delta g^{\mu\nu}}+T_{\sigma\epsilon}\frac{\delta T^{\sigma\epsilon}}{\delta g^{\mu\nu}} \\
&=-2\mathcal{L}_{\rm m}\left(T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T\right)-TT_{\mu\nu}\\
&\quad\quad+2T_{\mu}^{\gamma}T_{\nu\gamma}-4T^{\sigma\epsilon}\frac{\partial^2 \mathcal{L}_{\rm m}}{\partial g^{\mu\nu} \partial g^{\sigma\epsilon}}.
\label{theta}
\end{aligned}$$ We note that the EMT given in Eq. does not include the second variation of $\mathcal{L}_{\rm m}$, and hence the last term of Eq. is null. As the definition of the matter Lagrangian that gives the perfect-fluid EMT is not unique, one could choose either $\mathcal{L}_{\rm m}=p$ or $\mathcal{L}_{\rm m}=-\rho$, which provide the same EMT (see [@Bertolami:2008ab; @Faraoni:2009rk] for a detailed discussion). In the present study, we consider $\mathcal{L}_{\rm m}=p$.
We proceed with a particular form of the model as $$f(T_{\mu\nu}T^{\mu\nu})=\alpha(T_{\mu\nu}T^{\mu\nu})^{\eta},$$ which we shall refer to as the EMPG model. The action now reads as $$S=\int \left[\frac{1}{2\kappa}R+\alpha(T_{\mu\nu}T^{\mu\nu})^{\eta}+\mathcal{L}_{\rm m}\right]\sqrt{-g}\,{\rm d}^4x,
\label{eq:action}$$ where $\eta$ is the power of the self-contraction of the EMT, and $\alpha$ is a constant that would take part in determining the coupling strength of the EMT-powered modification to gravity. The Einstein field equations for this action become $$G_{\mu\nu}=\kappa T_{\mu\nu}+\kappa \alpha (T_{\sigma\epsilon}T^{\sigma\epsilon})^{\eta}\left[g_{\mu\nu}-2\eta\frac{\theta_{\mu\nu}}{T_{\sigma\epsilon}T^{\sigma\epsilon}}\right].
\label{fieldeq}$$ Using Eq. , the covariant divergence of the EMT reads as $$\begin{aligned}
\label{nonconservedenergy}
\nabla^{\mu}T_{\mu\nu}=&-\alpha g_{\mu\nu}\nabla^{\mu}(T_{\sigma\epsilon}T^{\sigma\epsilon})^{\eta}\\
&+2\alpha\eta\nabla^{\mu}\left(\frac{\theta_{\mu\nu}}{(T_{\sigma\epsilon}T^{\sigma\epsilon})^{1-\eta}}\right).
\end{aligned}$$ We notice that, in our model, the EMT is not conserved in general since the right-hand side of this equation vanishes only for some particular values of the constants.
In this paper, we shall study the gravity model under consideration in the context of cosmology. Therefore, we proceed by considering the spatially maximally symmetric spacetime metric, i.e., the Robertson-Walker metric, with flat space-like sections $$\begin{aligned}
\label{RW}
ds^2=-dt^2+a^2\,({\rm d}x^2+{\rm d}y^2+{\rm d}z^2),\end{aligned}$$ where the scale factor $a=a(t)$ is a function of cosmic time $t$ only, and the perfect fluid form of the EMT is given by $$\begin{aligned}
\label{em}
T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+p g_{\mu\nu},\end{aligned}$$ where $\rho$ is the energy density, $p$ is the thermodynamic pressure, and $u_{\mu}$ is the four-velocity satisfying the conditions $u_{\mu}u^{\mu}=-1$ and $\nabla_{\nu}u^{\mu}u_{\mu}=0$. We assume a barotropic equation-of-state (EoS), $$\begin{aligned}
\label{eos}
\frac{p}{\rho}=w={\rm const.}\end{aligned}$$ to describe the physical ingredient of the Universe. Using Eqs. and , we find $\theta_{\mu\nu}$ given in Eq. and the self-contraction of the EMT as $$\begin{aligned}
\label{thetafrw}
\theta_{\mu\nu}&=-\rho^2(3w+1)(w+1)u_{\mu}u_{\nu},\\
T_{\mu\nu}T^{\mu\nu}&=\rho^2(3w^2+1),
\label{trace}\end{aligned}$$ respectively. Then, using Eqs. and as well as the metric in the field equations , we obtain the following set of two linearly independent differential equations in two unknown functions $H$ and $\rho$: $$\begin{aligned}
3 H^2=&\kappa\rho+\kappa' \rho_0\left(\frac{\rho}{\rho_0}\right)^{2\eta} , \label{eq:rhoprime}\\
-2\dot{H}-3H^2=&\kappa w\rho+ \frac{\kappa' \rho_0}{2\eta-1+\frac{8w\eta}{3w^2+1}} \left(\frac{\rho}{\rho_0}\right)^{2\eta}, \label{eq:presprime}\end{aligned}$$ where $H=\dot{a}/a$ is the Hubble parameter and the subscript $0$ refers to the present-day values of the parameters. The constant $\kappa'$ is the gravitational coupling of the EMT-powered modification, and is given by $$\kappa'=\alpha' \kappa= \alpha\,\kappa \rho_{0}^{2\eta-1}(3w^2+1)^{\eta}\left[2\eta-1 +\frac{8w\eta}{3w^2+1}\right].
\label{eq:alphaprime}$$
We note that the source (i.e., $T^{\mu\nu}$) that we considered in the EMT-powered term is the same as the one obtained from $\mathcal{L}_{\rm m}$, but the terms that appear in the field equations due to the EMT-powered term in the action couple to gravity with a different strength as $\kappa'=\alpha'\kappa$, where $\alpha'$ is the ratio of this coupling with respect to the conventional Newtonian coupling $\kappa$. We further note that $\alpha'=\alpha'(\alpha,\eta,\rho_{0},w)$, namely, $\alpha'$ depends not only on $\alpha$ but also on the current energy density $\rho_0$, and the type of source described by EoS parameter $w$ provided that $\eta\neq0$. This implies that the EMT-powered term would lead to a violation of the equivalence principle, which is intimately connected with some of the basic aspects of the unification of gravity with particle physics such as string theories (see Ref. [@Uzan:2010pm] and references therein). We will not elaborate the implications of this property of our model in this paper, since here we shall discuss the dynamics of the Universe in the presence of only a pressureless fluid, i.e., a monofluid Universe, with the purpose of describing the late-time acceleration of the Universe (where the radiation is negligible) without invoking a cosmological constant or any dark energy source.
Next, we note that the first Friedmann equation is in the form of the well-known Cardassian expansion ($H^2=A\rho+B\rho^n$, with $A$, $B$, and $n$ being constants) [@Freese:2002sq], which was motivated by the fact that the term of the form $\rho^n$ can generically appear as a consequence of embedding the observable Universe as a brane in extra dimensions [@Freese:2002sq; @Chung:1999zs]. On the other hand, in the second Friedmann equation , we see that the additional pressure term (the latter term that appears due to the EMPG) is in the form of the pressure of the generalized Chaplygin gas ($p=-A/\rho^{\alpha}$, where $A$ is a positive constant) [@Kamenshchik:2001cp; @Bento:2002ps]. Also, for the special case $\eta=1$, the total pressure \[the whole right-hand side of Eq. \] is similar to the quadratic equation of state ($p=p_0+\alpha\rho+\beta\rho^2$, where $p_0$, $\alpha$, and $\beta$ are constants) of dark energy [@Ananda:2005xp]. However, one may check that our model in fact does not correspond to any of them, namely, the modified Friedmann equations of our model and do not simultaneously match the Friedmann equations of each of these models. The main reason behind this is the violation of the local/covariant energy-momentum conservation in our model. The corresponding energy conservation equation is $$\begin{aligned}
\label{noncons}
&\dot \rho+ 3H(1+w)\rho=\nonumber \\
&-2\alpha'\eta\, \left[ \dot \rho +3H \rho\left(\frac{1+\frac{4w}{3w^2+1}}{2\eta-1+\frac{8w\eta}{3w^2+1}}\right) \right]\,\left(\frac{\rho}{\rho_0}\right)^{2\eta-1}.\end{aligned}$$ Here we can see that the local/covariant energy-momentum conservation $\nabla^{\mu}T_{\mu\nu}=0$, which would lead to $\rho\propto a^{-3(1+w)}$, is not satisfied for $\alpha'\neq0$ in general. Some particular cases, in which the right-hand side of the equation vanishes are as follows: (i) the case $\eta=0$, which is trivial; (ii) the case $w=-1$, i.e., the conventional vacuum energy; and (iii) the case $\eta=\frac{3w^2+3w+2}{2(3w+1)(w+1)}$, which gives $\eta=1$ for $w=0$ (dust) and $\eta=\frac{5}{8}$ for $w=\frac{1}{3}$ (radiation). One may also check that the standard energy conservation is not satisfied for the range $\frac{24-9\sqrt{6}}{48-20\sqrt{6}}<\eta<\frac{1}{2}$. In the context of late-time acceleration, the violation of energy conservation is not uncommon in the literature [@Josset:2016vrq; @Shabani:2017vns].
Late-time acceleration in a dust-only Universe
==============================================
The late-time acceleration of the Universe takes place at relatively low energies, and hence it would be wise to search for suitable ranges of the model parameters $\alpha'$ and $\eta$ in which our modification is effective at sufficiently low energy densities, but negligible at high energy densities, namely, at energies higher than that of recombination. In this way, the successes of the standard cosmology would be untouched, and we would be able to derive accelerated expansion. Knowing that the energy density of the matter source $\rho$ should be positive, we would like to ensure that the EMT-powered contribution to the Hubble parameter \[the latter term in Eq. \] is positive as well. So we set $\kappa'>0$, implying that $\alpha'>0$. In the $\alpha'=0$ case, the EMT-powered modification vanishes, and our model reduces to the standard Friedmann model as the standard EH action is recovered. In this case, we get a matter (e.g., dust) dominated Universe in GR, and to get accelerated expansion of the Universe we need a cosmological constant or DE source. Therefore, in what follows, we consider $\alpha'>0$, focus our attention on the free parameter $\eta$ and investigate the model.
$\Lambda$CDM-type behavior ($\eta=0$) without $\Lambda$ {#subsec:A}
-------------------------------------------------------
Let us start with the special case $\eta=0$. In this case, in the presence of pressureless matter $\rho_{\rm m}$ with $w=w_{\rm m}=0$, from Eqs. and we obtain $3H^2=\kappa\rho_{{\rm m},0}\left( \frac{a}{a_0}\right)^{-3}+\kappa' \rho_{{\rm m},0}$. Hence, the model yields the same mathematical structure as the $\Lambda$CDM model $3H^2=\kappa\rho_{{\rm m},0}\left( \frac{a}{a_0}\right)^{-3}+\kappa\rho_{\Lambda}$, where $\rho_{\Lambda}=\rm const$. But we stress that the underlying physics of these two models are completely different. Namely, the only source in our model is pressureless matter, which is not the case in the $\Lambda$CDM model. For instance, if we consider an empty Universe ($\rho_{{\rm m},0}=0$), we find $H=0$ (a static Universe) in our model, whereas we find $H=\sqrt{\frac{\rho_{\Lambda}}{3}}=\rm const$ (the de Sitter solution) in the $\Lambda$CDM model. On the other hand, we can make use of this correspondence to estimate the value of $\alpha'$. Setting $a=a_0$ in both models, we obtain $3H_0^2=(\kappa+\kappa')\rho_{{\rm m},0}$ for our model, and $3H_0^2=\kappa(\rho_{{\rm m},0}+\rho_{\Lambda})$ for the $\Lambda$CDM model. Then, using these we obtain the following correspondence between the parameters of these two models: $\alpha'=\rho_{\Lambda}/\rho_{{\rm m},0}$. Hence, from the recent Planck results [@Ade:2015xua] giving $\Omega_{\Lambda,0}=\frac{\rho_{\Lambda}}{\rho_{{\rm m},0}+\rho_{\Lambda}}\sim 0.69$ for the current Universe, we estimate that $\alpha' \sim 2.2$. Knowing that $\Lambda$CDM is very successful in describing the observed Universe, it would not be wrong to conclude from this simple investigation that our model with $\eta\sim0$ and $\alpha'\sim2$ would successfully describe the background dynamics of the observed Universe. We note that the role of the cosmological constant in $\Lambda$CDM has been taken over by the energy density of the pressureless matter itself due to its new form of coupling to gravity which is about $2$ times stronger than that of the conventional coupling. On the other hand, we could also obtain accelerated expansion by including a cosmological constant in our model from the beginning (which is obvious), or by considering vacuum energy for the EMT. In the latter case$-$namely, when we consider the conventional vacuum energy described by the EoS parameter $w=w_{\rm vac}=-1$$-$irrespective of the value of $\eta$, our model gives de sitter expansion, as in standard GR, but with an enhanced Hubble parameter $H=\sqrt{\frac{1}{3}\rho_{\rm vac,0}(\kappa+\kappa')}={\rm const.}$ due to the coupling $\kappa'$ of the EMT-powered modification. However, obtaining accelerated expansion without invoking a cosmological constant or vacuum energy $-$ in contrast to $\Lambda$CDM $-$ would render our model immune to the well-known cosmological constant problem, namely, the extreme fine-tuning of the value of $\Lambda$. Although we know of no special symmetry that could enforce a vanishing vacuum energy while remaining consistent with the known laws of physics, it is usually thought to be easier to imagine an unknown mechanism that would set $\Lambda$ precisely to zero than one that would suppress it by just the right amount $\rho_{\Lambda}^{\rm (observation)}/\rho_{\Lambda}^{(\rm theory)}\sim10^{-120}$ to yield an observationally accessible cosmological constant (see Refs. [@Weinberg:1988cp; @Sahni:1999gb; @Carroll:2000fy; @Peebles:2002gy; @Sahni:2004ai; @Bull:2015stt] and references therein). In our model, on the other hand, in the presence of only a pressureless fluid the observations would favor $\kappa' \sim 2\kappa$, which means that the coupling strength of the corresponding energy density of the EMT-powered modification to gravity has the same order of magnitude as the conventional coupling of the energy density to gravity.
It may be noted that we have avoided the introduction of a nonzero cosmological constant in the action , though according to Lovelock’s theorem[^2] the cosmological constant $\Lambda$ arises as a constant of nature like Newton’s gravitational constant $G=\frac{\kappa}{8\pi}$. Indeed, if we stick to the usual interpretation of $\Lambda$ as the term representing the vacuum energy $\rho_{\rm vac}$, one then gets $R=4\Lambda=4\kappa \rho_{\rm vac}$ in GR. However, then the fine-tuning problem arises, as we discussed above, as well as some other problems as well inherited from the standard $\Lambda$CDM model due to the presence of $\Lambda$ (see Ref. [@Bull:2015stt] for a recent review). Similarly, in our model, as mentioned above, in the presence of only pure vacuum $w_{\rm vac}=-1$, we obtain $R=4({\kappa+\kappa'}) \rho_{\rm vac}$. Therefore, considering a dust-only ($w=0$) Universe implies that a constant of nature, which would correspond to $\Lambda=({\kappa+\kappa'}) \rho_{\rm vac}$, is set to zero in the construction of our model. On the other hand, as discussed in the previous paragraph, setting $\Lambda$ to zero (i.e., considering only dust without vacuum energy) alleviates the fine-tuning problem in our model. However, there are several other reasons for such a setting within the scope of the current study. If we consider a nonzero $\Lambda$, then we can see from the field equations and that, for the case $\eta=0$, the constant contribution from dust itself, i.e., $\kappa'\rho_0$, and $\Lambda$ would be degenerate at least at the background level. In addition, it is conceivable that such a degeneracy would also appear for the case $\eta\sim0$. Besides, for the case $\eta>\frac{1}{2}$, the new terms that arise due to EMPG with respect to GR would be suppressed at relatively low energy densities, and then one can recover the $\Lambda$CDM model in the late Universe and let EMPG alter the early Universe. For instance, in [@Roshan:2016mbt] the Universe filled with dust approached the $\Lambda$CDM model at late times in EMSG, i.e., the special case ${\rm EMPG}_{\eta=1}$ of our model. Indeed, in Ref. [@ACEK], the parameter $\alpha$ was well constrained from neutron stars and it was found that EMSG can lead to a significant modification in the dynamics of the Universe only when the age of the Universe is $t\lesssim10^{-4}\,{\rm s}$, long before the physical processes relevant to big bang nucleosynthesis that take place when $t\sim 10^{-2}\,-10^{2}\,{\rm s}$, and leaves the standard cosmology unaltered [@ACEK]. It is clear from these studies that our model in the presence of $\Lambda$ would create a degeneracy between the cases $\eta\sim0$ and $\eta>\frac{1}{2}$ when subjected to the data from cosmological observations. Moreover, when we carry out our observational analysis, the case with $\eta\sim0$, $\alpha'\sim 2.2$, and $\Lambda\sim0$ would be degenerate with the case $\eta>\frac{1}{2}$, $\frac{\rho_{{\rm m},0}}{\rho_{\Lambda}+\rho_{{\rm m},0}}\sim 0.7$ for a range of $\alpha'$ depending on the value of $\eta$. However, it is possible to overcome some of the above-mentioned degeneracies, e.g., by considering compact astrophysical objects such as neutron stars for the case $\eta>\frac{1}{2}$ (e.g., in Ref. [@ACEK] EMSG (the case ${\rm EMPG}_{\eta=1}$) was studied) , although such an analysis is beyond the scope of the current study. Finally, we would like to mention that the inclusion of $\Lambda$ in the action may provide a much richer theory and we postpone this idea to future work.
Viable cosmologies more general than $\Lambda$CDM without $\Lambda$
-------------------------------------------------------------------
In the above, we have demonstrated that our model for the case $\eta=0$ provides not only the same background dynamics as the $\Lambda$CDM model but also some additional promising features. Let us now elaborate our discussion by considering the $\eta\neq0$ cases. We first explore the range for the value of $\eta$ to get accelerated expansion in the relatively late Universe. We note from Eq. that the parameter $\eta$ determines whether the EMPG modification would come into play at large or small values of $\rho$. In the case $\eta=\frac{1}{2}$, the terms from the standard EH and the EMT-powered modification in Eq. have the same power, and they track each other, i.e., their relative contributions to $H$ do not depend on the value of the energy density, but only on $\alpha'$ only. In the case of $\eta>\frac{1}{2}$, the EMT-powered term manifests itself at larger values of $\rho$, namely, in the relatively early Universe, say, at a time before big bang nucleosynthesis (BBN) processes took place; in this case, one could discuss alternative scenarios for the beginning of Universe. For instance, in the model introduced in Ref. [@Roshan:2016mbt], which corresponds to the case $\eta=1$ of our model in the presence of $\Lambda$, the Universe may not reach an initial singularity and bounce when $\rho$ increases to a certain value in the early Universe, and at sufficiently low values of $\rho$ the model becomes indistinguishable from GR and hence it is $\Lambda$ that leads to a late-time accelerated expansion. In the case $\eta<\frac{1}{2}$, the EMT$-$powered term manifests itself at lower values of $\rho$ (say, in the present Universe $\frac{\rho_{\rm m}}{\rho_{{\rm m},0}} \sim 1$), where it may play a role in the accelerated expansion of the Universe, thereby providing an alternative to $\Lambda$ or dark energy sources while keeping the successes of the standard cosmology based on GR at earlier times of the Universe. For instance, the particular case $\eta=\frac{1}{4}$ of our model may lead to a late-time accelerated expansion as discussed in Ref. [@Arik:2013sti].
### Accelerated expansion at low energies
In light of the above discussion, we see that we should consider the case $\alpha'>0$ with $\eta<\frac{1}{2}$ for the EMT-powered term to be effective at lower values of the energy density. As a further step, we write the deceleration parameter $q=-1-\frac{\dot{H}}{H^2}$ to check whether the EMT-powered modification would give rise to the accelerated expansion, $$\begin{aligned}
\label{decpar}
q=\frac{1+3w+\alpha'\left(\frac{\rho}{\rho_0}\right)^{2\eta-1}\left[1+\frac{3}{2\eta-1+\frac{8w\eta}{3w^2+1}}\right]}
{2+2\alpha'\left(\frac{\rho}{\rho_0}\right)^{2\eta-1}}.\end{aligned}$$ As we discussed above, the EMPG modification becomes negligible at larger values of $\rho$ for $\eta<\frac{1}{2}$. It is possible to set the values of $\alpha'$ and $\eta$ such that at energy scales larger than that of recombination, our model would be indistinguishable from GR, i.e., the terms with $\alpha'$ in Eqs. and would be negligible and lead to $q=\frac{1+3w}{2}$ and $\dot \rho+ 3H(1+w)\rho=0$, respectively. Hence, the standard cosmology would be left unaltered for times before recombination that took place at $z\approx1100$. Because the Universe should always be matter dominated (hence $w=0$, implying $q=\frac{1}{2}$) at recombination [@Barrow:1976rda] and we introduce no sources (such as $\Lambda$ or DE) that would dominate over pressureless matter after recombination, we can assume that the Universe is only filled with pressureless matter $\rho_{\rm m}$. Hence, we use the deceleration parameter $$\begin{aligned}
\label{qdef}
q=\frac{1+2\alpha'\left(\frac{\eta+1}{2\eta-1}\right)\left(\frac{\rho_{\rm m}}{\rho_{{\rm m},0}}\right)^{2\eta-1}}
{2+2\alpha'\left(\frac{\rho_{\rm m}}{\rho_{{\rm m},0}}\right)^{2\eta-1}},\end{aligned}$$ obtained by substituting $w=0$ in Eq. , to investigate the evolution the Universe at $z\lesssim1100$ as well as the late-time accelerated expansion, which starts at $z\sim 0.6$. Because the condition $\alpha'>0$ makes the denominator always positive, $q$ can take negative values if $$\label{cond1}
-1<\eta<\frac{1}{2},$$ due to the term $\frac{\eta+1}{2\eta-1}$ in the numerator. We note that the upper limit coincides with the one we obtained in order for the EMPG modification to be effective at lower energy density values. Thus, the condition guarantees standard GR at earlier times, and cosmic acceleration at later times by means of the EMPG modification. Finally, we get accelerated expansion in the present Universe at $z=0$, implying $\rho_{\rm m}=\rho_{{\rm m},0}$, provided that $$\label{cond2}
-1+\frac{3}{2\alpha'+2}<\eta<\frac{1}{2},$$ which is a stronger condition than the one given in Eq. for all positive values of $\alpha'$. We next see from Eq. that $$q\approx\frac{\eta+1}{2\eta-1}\quad\textnormal{for}\quad \rho_{\rm m}\ll\rho_{{\rm m},0},$$ provided that $\rho_{\rm m}$ can decrease to sufficiently small values under the condition . (We shall discuss the minimum $\rho_{\rm m}$ values that can be achieved at the end of this section.) According to this, the Universe evolves toward $-1<q<0$ for $-1<\eta<0$, toward $q=-1$ for $\eta=0$, and toward $q<-1$ for $0<\eta<\frac{1}{2}$. These are in line with the effective energy density and pressure $\rho'= \rho_{{\rm m},0}\left(\frac{\rho_{\rm m}}{\rho_{{\rm m},0}}\right)^{2\eta}$ and $p'= \frac{\rho_{{\rm m},0}}{2\eta-1}\left(\frac{\rho_{\rm m}}{\rho_{{\rm m},0}}\right)^{2\eta}$ from the EMPG modification in the modified Friedmann equations $-$, respectively. In the presence of only pressureless matter, we find that the effective EoS parameter of the EMPG modification is $w'=\frac{p'}{\rho'}=\frac{1}{2\eta-1}$, which gives a quintessence-like EoS parameter $-1<w'<-\frac{1}{3}$ for $-1<\eta<0$, a vacuum-like EoS parameter $w'=-1$ for $\eta=0$, and a phantom-like EoS parameter $w'<-1$ for $0<\eta<\frac{1}{2}$. These signal that our model would lead to a Universe that exhibits a similar (but not the same, since covariant energy-momentum conservation is not always satisfied in our model) evolution as that of the well-known $w$CDM model, which is the simplest (phenomenological) and most widely considered extension of the $\Lambda$CDM model, in observational cosmology. Based on GR, in the $w$CDM model (where $w$ represents the EoS of the DE source as $w_{\rm DE}={\rm constant}$) the energy densities of pressureless matter and DE evolve as $\rho_{\rm m} \propto a^{-3}$ and $\rho_{\rm DE} \propto a^{-3(1+w_{\rm DE})}$, respectively, due to the covariant conservation of the EMT tensors of these sources. These imply that $\rho_{\rm DE} \propto \rho_{\rm m}^{(1+w_{\rm DE})}$, which leads to the following deceleration parameter for the spatially flat $w$CDM model: $$\begin{aligned}
\label{qdefw1}
q_{w{\rm CDM}}=\frac{1+(3{w_{\rm DE}+1)\,\frac{1-\Omega_{{\rm m},0}}{\Omega_{{\rm m},0}}\left(\frac{\rho_{\rm m}}{\rho_{{\rm m},0}}\right)^{w_{\rm DE}}}}{2+2\,\frac{1-\Omega_{{\rm m},0}}{\Omega_{{\rm m},0}}\left(\frac{\rho_{\rm m}}{\rho_{{\rm m},0}}\right)^{w_{\rm DE}}}.\end{aligned}$$ Its mathematical form is very similar to our model’s deceleration parameter . Indeed, under the transformations $w_{\rm DE} \rightarrow 2\eta-1$ and $\frac{1-\Omega_{{\rm m},0}}{\Omega_{{\rm m},0}}\rightarrow\alpha'$, this can be recast as follows: $$\begin{aligned}
\label{qdefw2}
q_{w{\rm CDM}}=\frac{1+2\alpha' (3\eta-1)\left(\frac{\rho_{\rm m}}{\rho_{{\rm m},0}}\right)^{2\eta-1}}
{2+2\alpha'\left(\frac{\rho_{\rm m}}{\rho_{{\rm m},0}}\right)^{2\eta-1}}.\end{aligned}$$ However, we note that the coefficients in front of the latter terms in the numerators in Eqs. and are not the same, except for the cases $\eta=0$ corresponding to $w'=w_{\rm DE}=-1$ ($\Lambda$/conventional vacuum) and $\eta=1$ corresponding to $w'=w_{\rm DE}=1$, i.e., a stiff fluid that does not lead to accelerated expansion and hence cannot be a DE candidate.
![$q$ versus $\frac{\rho_{\rm m}}{\rho_{{\rm m},0}}$ curve of the $w$CDM and EMPG models. The figure is plotted by using $\Omega_{{\rm m},0}=0.305\pm 0.010$ and $w_{\rm DE}=-0.97\pm0.05$, corresponding to $\alpha'=2.28\pm0.11$ and $\eta=0.015\pm0.025$ according to the transformations between the two models, from the constraints for the $w$CDM model from Planck+BAO+SN data presented in Ref. [@Aubourg:2014yra].[]{data-label="qvsrho"}](q_rho_plot_suresh1.pdf){width="50.00000%"}
In Fig. \[qvsrho\], we show that the EMPG \[Eq. \] and $w$CDM \[ Eq. \] models differ only slightly by depicting the deceleration parameter versus the energy density of the pressureless matter, using $\Omega_{{\rm m},0}=0.305\pm 0.010$ and $w_{\rm DE}=-0.97\pm0.05$, corresponding to $\alpha'=2.28\pm0.11$ and $\eta=0.015\pm0.025$ according to the above transformations, from the constraints for the $w$CDM model from Planck+BAO+SN data presented in Ref. [@Aubourg:2014yra]. We see that our model would lead to exactly the same behavior as in $\Lambda$CDM model when $\eta=0$, though the physics is different, and would also lead to a behavior similar to that in the $w$CDM model when $\eta\neq0$. The DE source of the $w$CDM model is phenomenological rather than derived from a fundamental theory, and may be described by a scalar field yielding a suitable potential, e.g., as it was reconstructed analytically to give $\rho_{\rm DE}\propto a^{-3(1+w_{\rm DE})}$ in two fluid cosmological models in Ref. [@Rubano:2001xi]. The case $w_{\rm DE}<-1$ for the $w$CDM model corresponds to phantom DE, which would be described by a noncanonical scalar field with a negative kinetic term, which would eventually lead to $q<-1$, i.e., the Universe would end in a big rip [@Caldwell:1999ew; @Caldwell:2003vq]. On the other hand, the behavior we would get in the range $0<\eta<\frac{1}{2}$ in the presence of ordinary matter only does the job of the scalar field with a negative kinetic term, namely, the phantom field. We know that models with a negative kinetic term suffer from the problem of quantum instability due to an unbounded vacuum state from below in the matter sector, and hence they are problematic [@carroll03; @cline04]. It is interesting that we obtained a $w$CDM-like behavior without invoking a scalar field with a particular kind of potential that is not from a fundamental theory, but rather reconstructed and requires a negative kinetic term for the case $w_{\rm DE}<-1$.
### Evolution of the energy density
The energy conservation equation for $\rho=\rho_{\rm m}$ with $w=0$ reads as $$\begin{aligned}
\label{nonconser}
&\dot \rho_{\rm m}+ 3H\rho_{\rm m}=-2\alpha'\eta\, \left[ \dot \rho_{\rm m} +\frac{3H \rho_{\rm m}}{2\eta-1} \right]\,\left(\frac{\rho_{\rm m}}{\rho_{{\rm m},0}}\right)^{2\eta-1},\end{aligned}$$ and it gives the time rate of change of the matter energy density as $$\begin{aligned}
\label{eqn:conh}
\frac{\dot{\rho}_{\rm m}}{\rho_{\rm m}} = -3 H \, \frac{1+\frac{2\alpha'\eta}{2\eta-1}\left(\frac{\rho_{\rm m}}{\rho_{{\rm m},0}}\right)^{2\eta-1}}{1+2\alpha'\eta \left(\frac{\rho_{\rm m}}{\rho_{{\rm m},0}}\right)^{2\eta-1}}.\end{aligned}$$ Taking the derivative of Eq. with respect to time $t$, and then using the result together with Eq. , we find $$\label{eqn:hdot}
\dot{H}=-\frac{\kappa}{2}\rho_{\rm m} \left[1+\frac{2\alpha' \eta}{2\eta-1} \left(\frac{\rho_{\rm m}}{\rho_{{\rm m},0}}\right)^{2\eta-1} \right].$$ Among many other features that are required for a successful cosmological model, a viable cosmological model would be expected to satisfy the conditions $H>0$, $\dot{H}<0$, $\rho_{\rm m}>0$, and $\dot \rho_{\rm m}<0$ when $\rho_{\rm m}>\rho_{{\rm m},0}$, at least since after the matter-radiation equality that took place about $z\sim3600$ until the present time $z\sim0$ ($\rho_{\rm m}\sim\rho_{{\rm m},0}$), in line with the standard cosmological model. Accordingly, using Eqs. , , and , we find that these conditions imply the following relations between $\alpha'$ and $\eta$: $$\begin{aligned}
\label{eqn:fcon}
\alpha'<-\frac{1}{2\eta}&\quad\textnormal{for}\quad -1<\eta<0,\\
\alpha'<-1+\frac{1}{\eta}&\quad \textnormal{for}\quad 0<\eta<\frac{1}{2},
\end{aligned}$$ These are in addition to the condition that guarantees that EMPG gravity approaches GR at high energy densities, and leads to accelerated expansion at sufficiently low energy densities in the presence of only pressureless matter.
We next show that under the above conditions , the matter energy density $\rho_{\rm m}$ cannot decrease to zero as the Universe expands, but it can decrease to a nonzero minimum value which in turn also determines the minimum value of the deceleration parameter that the Universe could achieve from Eq. . We see from Eq. that $H$ would never become null in the case $-1<\eta<0$ and $\alpha'>0$, and we also see from Eq. that in an expanding Universe, i.e., $H>0$, the matter energy density decreases with time, $\dot{\rho}_{\rm m}<0$, for $\rho_{\rm m}>\rho_{{\rm m},0} (-2\alpha'\eta)^{\frac{1}{1-2\eta}}$ and increases with time, $\dot{\rho}_{\rm m}>0$, for $\rho_{\rm m}<\rho_{{\rm m},0} (-2\alpha'\eta)^{\frac{1}{1-2\eta}}$. It follows that $$\label{eqn:rmin1}
\rho_{\rm m,min}=\rho_{{\rm m},0}\left(\frac{1}{-2\alpha'\eta}\right)^{\frac{1}{2\eta-1}} \quad\textnormal{for}\quad -1<\eta<0.$$ Substituting this into , we find the minimum value of the deceleration parameter as $$q_{\rm min}=\frac{2\eta^2-2\eta-1}{(2\eta-1)^2}\quad\textnormal{for}\quad -1<\eta<0.$$ Next, we see from Eq. that $H$ would become null for $0<\eta<\frac{1}{2}$ and $\alpha'>0$ if $\rho_{m}$ could be null. However, we see from Eqs. and that as the Universe expands ($H>0$), in the case $0<\eta<\frac{1}{2}$ and $\alpha'>0$ the matter energy density decreases with time as long as $\rho_{\rm m}<\rho_{{\rm m},0} \left(\frac{1-2\eta}{2\alpha'\eta}\right)^{\frac{1}{2\eta-1}}$, and eventually $\dot{\rho}_{\rm m}\rightarrow 0$ and $\dot{H}\rightarrow 0$ as $\rho_{\rm m}\rightarrow \rho_{{\rm m},0} \left(\frac{1-2\eta}{2\alpha'\eta}\right)^{\frac{1}{2\eta-1}}$. These signal that the matter energy density asymptotically approaches its minimum value $$\label{eqn:rmin2}
\rho_{\rm m,min}=\rho_{{\rm m},0} \left(\frac{1-2\eta}{2\alpha'\eta}\right)^{\frac{1}{2\eta-1}}\quad\textnormal{for}\quad 0<\eta<\frac{1}{2},$$ which gives \[when substituted into Eq. \] the minimum value of the deceleration parameter that the Universe asymptotically approaches, as $$\label{eqn:asym}
q_{\rm min}=-1 \quad\textnormal{for}\quad 0<\eta<\frac{1}{2}.$$ According to this, although the EMPG modification would effectively behave like a phantom field $w'<-1$ for $0<\eta<\frac{1}{2}$, it would not lead to a big rip since the matter energy density does not diverge during the evolution of the Universe but asymptotically approaches a positive constant, which implies that the Hubble parameter also asymptotically approaches a positive constant, namely, the Universe would asymptotically approach exponential expansion with $q_{\rm min}=-1$ as given in the above equation. Finally, the fact that $a\rightarrow\infty$ as $\rho_{\rm m}\rightarrow \rho_{{\rm m},0} \left(\frac{1-2\eta}{2\alpha'\eta}\right)^{\frac{1}{2\eta-1}}$ in Eq. \[which is the solution we give below for $a(\rho_{\rm m})$\] confirms our conclusions on the asymptotic behavior of our model for the case $0<\eta<\frac{1}{2}$.
In this paper, we do not intend to investigate the all possible solutions of the model, but rather those that satisfy the conditions we have discussed in this section for obtaining a viable cosmology. One may note that each different value of the parameter $\eta$ $-$which determines the power of the self-contraction of the energy-momentum tensor $-$would correspond to different theories of gravity, and hence a full mathematical analysis of the model for arbitrary values of $\eta$ or even for some particular values (such as that done, e.g., in Ref. [@Roshan:2016mbt] that corresponds to the case $\eta=1$ of our model in the presence of $\Lambda$) is beyond the scope of this paper. Here, our main purpose is to study whether viable cosmologies can be obtained from our model for certain ranges of the parameters $\alpha'$ (as well as $\alpha$) and $\eta$ estimated from observational data. Therefore, we continue here with the following solution for $a(\rho_{\rm m})$, satisfying the conditions given in Eqs. and and valid for $\rho_{\rm m}>\rho_{\rm m,min}$ given in Eqs. and : $$\begin{aligned}
\frac{\rho_{\rm m}}{\rho_{{\rm m},0}}\left[\frac{1+\frac{2\eta\alpha'}{2\eta-1}\left(\frac{\rho_{\rm m}}{\rho_{{\rm m},0}}\right)^{2\eta-1}}{1+\frac{2\eta\alpha'}{2\eta-1}}\right]^{\frac{2\eta-2}{2\eta-1}} =a^{-3}.
\label{eq:rh}\end{aligned}$$ It should be noted here that we are not able to give an explicit solution for $\rho_{\rm m}(a)$ since it is not possible to isolate $\rho_{\rm m}$ except for a couple of particular cases of $\eta$ in this equation, which stands as one of the difficulties in the investigation of the model for arbitrary values of $\eta$. However, it might be immediately seen from this equation that pressureless matter in our model does not evolve as $a^{-3}$ except for $\eta=0,1$, and that $a\rightarrow\infty$ as $\rho_{\rm m}\rightarrow \rho_{{\rm m},0} \left(\frac{1-2\eta}{2\alpha'\eta}\right)^{\frac{1}{2\eta-1}}$ for $0<\eta<\frac{1}{2}$, in line with our expectation that the model would asymptotically approach exponential expansion in this case, in accordance with our discussion regarding Eq. .
OBSERVATIONAL CONSTRAINTS {#obsanalysis}
=========================
In the previous section, we discussed in detail under what conditions our model could give viable cosmologies; we showed that, in the presence of only a matter source, our model can not only mimic the evolution of the Universe precisely as in the $\Lambda$CDM model at $\eta=0$, but it can also lead to an evolution similar to $w$CDM-like cosmology for $\eta\sim0$. However, in both cases the underlying physics of the EMPG model is entirely different than what we have in the $\Lambda$CDM and $w$CDM models. In this section, we investigate the observational constraints on the parameters $\alpha'$ (as well as $\alpha$) and $\eta$ by writing an approximated function for $a(\rho_{\rm m})$ \[given in Eq. \] for $a\lesssim1$ (or $z\gtrsim0$) that would allow us to isolate $\rho_{\rm m}$, which we can then substitute into $H(\rho_{\rm m})$ \[given in Eq. \] to obtain $H(z)$ for the purpose of observational data analysis. In order to investigate the observational constraints on the parameters of the EMPG model, first we need to determine the matter energy density $\rho_{\rm m}$ of the model explicitly in terms of the cosmological redshift $z=-1+\frac{1}{a}$. However, $\rho_{\rm m}$ cannot be isolated explicitly in terms of $z$ from Eq. except for a limited number of particular values of $\eta$. On the other hand, we find that the following explicit expression for the matter energy density $\rho_{\rm m}$ in terms of $z$ is a very good approximation to Eq. for $\rho_{\rm m}/\rho_{{\rm m},0}\gtrsim1$ and $\eta\sim0$ (see Appendix \[app\] for details): $$\begin{aligned}
\label{eq:rhoo}
\rho_{\rm m}={\rho_{{\rm m},0}} [\beta(1+z)^3+1-\beta],\end{aligned}$$ where $\beta=\left(1+\frac{2\eta\alpha'}{2\eta-1}\right)^{\frac{2\eta-2}{2\eta-1}}$. Finally, substituting Eq. and the Hubble constant $H_0^2=\frac{\kappa}{3}(1+\alpha'){\rho_{{\rm m},0}}$ into Eq. , the approximated modified Friedmann equation in terms of redshift $z$ is $$\begin{aligned}
\frac{H^2}{H_0^2}=&\frac{1}{1+\alpha'}\left[\beta (1+z)^3+1 - \beta\right] \\
&+\frac{\alpha'}{1+\alpha'}\left[\beta(1+z)^3+1 - \beta\right]^{2\eta},
\label{oo}
\end{aligned}$$ subject to the conditions $\rho_{\rm m}\gtrsim{\rho_{{\rm m},0}}>\rho_{\rm m,min}$ (or $z\gtrsim0$) and the ones given in Eqs. and .
Having determined the evolution of the Hubble parameter in terms of $z$ in Eq. , now we can constrain the EMPG model parameters $\eta$, $\alpha'$ and $H_0$ with the observational data. For this purpose, we use the compilation of 28 Hubble parameter measurements spanning the redshift range $0.07 \leq z \leq 2.3$, as displayed in Table \[tab:Hz\]. The $28$ $H(z)$ data points were compiled in Ref. [@Farooq:2013hq] to determine constraints on the parameters of various dark energy models. In a very recent paper [@Chen:2016uno], the compilation of the 28 $H(z)$ points was utilized to determine the Hubble constant $H_0$ in four different cosmological models, including the $\Lambda$CDM and $w$CDM models. Following the same methodology, here we constrain the parameters $(H_0,\alpha',\eta)$ of the EMPG model by minimizing $$\label{eq:chi2Hz} \chi_{H}^2 (H_0,\alpha',\eta) =
\sum_{i=1}^{28}\frac{[H^{\rm th} (z_i; H_0,\alpha',\eta)-H^{\rm
obs}(z_i)]^2}{\sigma^2_{{\rm H},i}}$$ for $28$ measured $H^{\rm obs} (z_i)$’s with variance $\sigma^2_{{\rm H},i}$ at redshift $z_i$ (as displayed in Table \[tab:Hz\]), whereas $H^{\rm th}$ is the predicted value of $H(z)$ in the EMPG model. The parameter space $(H_0,\alpha',\eta)$ of the model is explored by using the Markov chain Monte Carlo method coded in the publicly available package COSMOMC [@Lewis:2002ah].
[cccc]{} $z$ & $H(z)$ &$\sigma_{H}$ &Reference\
& (km s$^{-1}$ Mpc $^{-1}$) & (km s$^{-1}$ Mpc $^{-1}$)&\
\
0.070& 69& 19.6& [@Zhangetal2014]\
0.090& 69& 12& [@Simon:2004tf]\
0.120& 68.6& 26.2& [@Zhangetal2014]\
0.170& 83& 8& [@Simon:2004tf]\
0.179& 75& 4& [@Morescoetal2012]\
0.199& 75& 5& [@Morescoetal2012]\
0.200& 72.9& 29.6& [@Zhangetal2014]\
0.270& 77& 14& [@Simon:2004tf]\
0.280& 88.8& 36.6& [@Zhangetal2014]\
0.350& 76.3& 5.6& [@ChuangWang2013]\
0.352& 83& 14& [@Morescoetal2012]\
0.400& 95& 17& [@Simon:2004tf]\
0.440& 82.6& 7.8& [@Blakeetal2012]\
0.480& 97& 62& [@Sternetal2010]\
0.593& 104& 13& [@Morescoetal2012]\
0.600& 87.9& 6.1& [@Blakeetal2012]\
0.680& 92& 8& [@Morescoetal2012]\
0.730& 97.3& 7.0& [@Blakeetal2012]\
0.781& 105&12& [@Morescoetal2012]\
0.875& 125& 17& [@Morescoetal2012]\
0.880& 90&40& [@Sternetal2010]\
0.900& 117& 23& [@Simon:2004tf]\
1.037& 154& 20& [@Morescoetal2012]\
1.300&168& 17& [@Simon:2004tf]\
1.430& 177& 18& [@Simon:2004tf]\
1.530& 140& 14& [@Simon:2004tf]\
1.750& 202& 40& [@Simon:2004tf]\
2.300& 224& 8& [@Buscaetal2013]\
Our results are presented in Table \[tab:results\] and Fig. \[contour\].
Parameter Mean 68% C.L. 95% C.L.
------------------- ---------- ------------------- ------------------
$H_0$ $68.7$ $[66.5, 70.9]$ $[64.2, 73.2]$
\[6pt\] $\alpha'$ $2.80$ $[2.54,3.05]$ $[2.42,3.17]$
\[6pt\] $\eta$ $-0.003$ $ [-0.014,0.009]$ $[-0.026,0.020]$
\[6pt\]
: Mean values of EMPG model parameters are displayed with 68% and 95% confidence levels (C.L.). We find $\chi^2_{\rm min}=17$.[]{data-label="tab:results"}
![1$\sigma$ and 2$\sigma$ confidence contours of the EMPG model parameters. Marginalized probability distributions of the individual parameters are also displayed.[]{data-label="contour"}](contour1.pdf){width="48.00000%"}
We find that the values of the EMPG model parameters are $H_0=68.7^{+2.2+4.5}_{-2.2-4.5}$ km s$^{-1}$ Mpc $^{-1}$, $\alpha'=2.80^{+0.25+0.37}_{-0.26-0.38}$ and $\eta=-0.003^{+0.012+0.023}_{-0.011-0.023}$, where the error limits are at $1\sigma$ and $2\sigma$ confidence levels with $\chi^2_{\rm min}=17$. We first note that the values of $\alpha'$ and $\eta$ satisfy all of the conditions , , and as described in the previous section for a viable cosmology. We note that the mean value of $\eta$ is almost equal to zero, signaling that our model$-$in light of observational data$-$predicts a $\Lambda$CDM-type background evolution at least up to the present time. The constraints on the$\Lambda$CDM model parameters read as $H_0=68.3^{+2.7+5.2}_{-2.6-5.1}$ km s$^{-1}$ Mpc $^{-1}$ and $\Omega_{\rm m,0}=0.276^{+0.032+0.072}_{-0.039-0.068}$, with error limits at the $1\sigma$ and $2\sigma$ confidence levels and $\chi^2_{\rm min}=17$ (see Table I of Ref. [@Chen:2016uno]). Recalling that the case $\eta=0$ corresponds to $\Lambda$CDM-type expansion with the transformation $\alpha'=\rho_{\Lambda}/\rho_{{\rm m},0}$ in our model, it is easy to deduce $H_0=68.3^{+2.7+5.2}_{-2.6-5.1}$ km s$^{-1}$ Mpc $^{-1}$, $\alpha'=2.623^{+0.596+1.185}_{-0.376-0.750}$ for $\eta=0$ (fixed). It is noteworthy that the errors of $\alpha'$ become larger when $\eta$ is fixed to zero, which may be interpreted as the data suggesting that fixing $\eta$ to zero is not preferable. On the other hand, we see that the observational data of the Hubble parameter measurements fit equally well to the approximated EMPG model, in contrast with the $\Lambda$CDM model, and predict a similar value of $H_0$ in both models.
In Fig. \[Hzfit\], we show the mean value $H(z)/(1+z)$ curves with the 1$\sigma$ error region of the EMPG and $\Lambda$CDM models in the redshift range $0\leq z\leq 2.5$ along with the data points from Table \[tab:Hz\]. We see that the $H(z)/(1+z)$ curves of the two models overlap, indicating a similar evolution of the Universe in the two models.
![$H(z)/(1+z)$ curves of the EMPG and $\Lambda$CDM models are plotted with the mean values and 1$\sigma$ error regions of the model parameters. The $28$ $H(z)$ data points of Table 1 are also displayed.[]{data-label="Hzfit"}](Hz_plot_suresh.pdf){width="50.00000%"}
The mean value evolution trajectories of the deceleration parameter $q=-1+(1+z)\frac{H'}{H}$ and jerk parameter $j=1-2(1+z)\frac{H'}{H}+(1+z)^2\left(\frac{H'^2}{H^2}+\frac{H''}{H}\right)$ with 1$\sigma$ error regions of the EMPG model and $\Lambda$CDM model (yielding a constant jerk parameter equal to unity, $j_{\Lambda{\rm CDM}}=1$) in the redshift range $0\leq z\leq 1100$ (log scale on the $z$ axis), are shown in Figs. \[qz\] and \[jz\], respectively.
![The mean value deceleration parameter curves with 1$\sigma$ error regions of the EMPG and $\Lambda$CDM models are plotted in the redshift range $(0,1100)$ (log scale on the $z$-axis) with the mean values of the model parameters. []{data-label="qz"}](qz_plot.pdf){width="50.00000%"}
![The mean value jerk parameter curves with 1$\sigma$ error regions of the EMPG and $\Lambda$CDM models are plotted in the redshift range $(0,1100)$ (log scale on $z$-axis) with the mean values of the model parameters.[]{data-label="jz"}](jerk_plot.pdf){width="50.00000%"}
Again, we observe a similar evolution of the Universe in the two models in the whole redshift range $0\leq z\leq 1100$. Thus, the EMPG and $\Lambda$CDM models provide similar descriptions of the dynamics and kinematics of the Universe up to the present time, in light of observational data from Hubble parameter measurements, despite the fact that the underlying physics of the EMPG model is completely different from that of the $\Lambda$CDM model. Thus, the observed Hubble data supports the new physics employed in this work to develop the EMPG model, which would deviate in various ways from the $\Lambda$CDM model depending on the sign of the $\eta$ (as the data allow both negative and positive values of $\eta$). However, we do not discuss the future dynamics of the Universe in the EMPG model in this paper.
Finally, to find the constraints on the parameter $\alpha$$-$which like $\eta$, is a true constant of the model that appears in the action of the EMPG model$-$ we first obtain the equation $$\label{eqn:Aprime}
\alpha=\frac{\alpha'}{2\eta-1}\left[\frac{3H_0^2}{\kappa(\alpha'+1)}\right]^{1-2\eta}$$ in the presence of only a matter source upon using Eq. with Eq. by substituting $w=0$. Hence, using the constraints on the parameters given in Table \[tab:results\] in Eq. we obtain the constraint on the parameter $\alpha$ as $$\alpha = - 0.60^{+0.50}_{-0.69}\times10^{-8} (\rm erg/cm^3)^{1-2\eta}\quad (95\%\, \rm C.L.).
\label{calpha}$$ One may note that the units of $\alpha$ depend on $\eta$, which indicates that each different value of $\eta$ should be considered as another gravity theory. In Fig. \[alpahaeta\] we present the three dimensional constraints in the $\eta$-$\alpha$ plane from $H(z)$ data, where the samples are colored by the parameter $\alpha$.
![Constraints in the $\eta$-$\alpha$ plane with 1$\sigma$ and 2$\sigma$ confidence level contours from $H(z)$ data in the EMPG model where the samples are colored by the parameter $\alpha'$.[]{data-label="alpahaeta"}](alphap_vs_alpha_eta.pdf){width="49.00000%"}
It may be noted that we did not take the parameter $\alpha$ into account in the cosmological model that we studied in the presence of only pressureless matter, but rather the more useful parameter $\alpha'$ \[defined in Eq. \], particularly for a monofluid cosmological solution of the model. However, the true constants of the EMPG model are the parameters $\alpha$ and $\eta$ that appear in the action \[eq:action\] of the EMPG model. In other words, the constraints we give on $H_0$, $\eta$, and $\alpha'$ above constitute the constraints on the cosmological model in the presence of only a pressureless source, while the constraints on $\eta$ and $\alpha$ constitute the constraints on the parameters of the EMPG model by means of the cosmological model under consideration. One may try to obtain constraints on $\eta$ and $\alpha$ from noncosmological physics, e.g., neutron stars, parametrized post-Newtonian parameters etc. In this case it might not be possible to properly define a parameter $\alpha'$ corresponding to the one we defined in this paper. On the other hand, one may try to obtain constraints on $\eta$ and $\alpha$ from, for instance, big bang nucleosynthesis, during which the Universe would be considered to be filled with only radiation ($w=1/3$). In this case one can properly use the $\alpha'$ given in Eq. by substituting $w=1/3$. However, even in this case, one cannot compare the constraint that would be found for $\alpha'$ for radiation with the one we give here for pressureless fluid in Table \[tab:results\]; rather, one should deduce the constraint on $\alpha$ and then compare that with the one we give in Eq. . A good demonstration of this point can be given by the following example. We know from Eq. that it is not necessary for $\alpha'$ to be the same for different fluids since it depends on the EoS parameter $w$ of the considered fluid. We found here that $\alpha'>0$ for pressureless matter ($w=0$) and then deduced that $\alpha<0$. However, if we use the above constraints we obtained for $\eta$ and $\alpha$, we see from that $\alpha'<0$ for radiation ($w=1/3$). Hence, the constraints we obtained on $\eta$ and $\alpha$ using data obtained from the late Universe predict that the EMPG modification would lower the value of the Hubble parameter in the radiation-dominated Universe$-$for instance, when BBN took place$-$though the EMPG modification would be completely negligible at the energy scales of BBN.
Concluding remarks and future perspectives
==========================================
In this study, we have proposed a modified theory of gravitation constructed by the addition of the term $f(T_{\mu\nu}T^{\mu\nu})$ to the EH action of GR, and elaborated a particular case $f(T_{\mu\nu}T^{\mu\nu})=\alpha(T_{\mu\nu}T^{\mu\nu})^{\eta}$ (where $\alpha$ and $\eta$ are real constants dubbed EMPG. We have further investigated the EMPG model on theoretical and observational grounds by considering the cosmological evolution that it predicts. We have discussed the conditions under which it leads to viable cosmologies, and showed that there are ranges of the parameters of the EMPG model in which it approaches standard GR at high energy densities and the accelerated expansion at sufficiently low energy densities (namely, in the present Universe) without invoking a cosmological constant or any other dark energy source. We have shown that, in the presence of only a matter source, EMPG can give rise to not only precisely the same background evolution as the $\Lambda$CDM model when $\eta=0$, but also an evolution similar to that of the $w$CDM model when $\eta\sim0$. However, in both cases the underlying physics of the EMPG-driven cosmologies are entirely different than what we have in the $\Lambda$CDM and $w$CDM models. For instance, we introduced neither $\Lambda$ nor a dark energy source to drive the late-time acceleration of the Universe. Our model alleviates the cosmological constant problem that arises when a cosmological constant is introduced as in the $\Lambda$CDM model, and the issue of introducing an *ad hoc* scalar field that can lead to quintessence and/or phantom (leading additional severe problems) dark energy source with a constant or slightly varying equation-of-state parameter as in the $w$CDM-like models. The constraints on the EMPG model parameters from the recent compilation of $28$ Hubble parameter measurements reveal that the EMPG model describes an evolution of the Universe similar to that in the $\Lambda$CDM model.
We have shown that our modification of the EH action that leads to the EMPG action becomes negligible at sufficiently high energy densities leaving the standard cosmology based on GR untouched for times earlier than the time of onset of the late-time acceleration of the Universe. This means that if we had added the term $f(T_{\mu\nu}T^{\mu\nu})=\alpha(T_{\mu\nu}T^{\mu\nu})^{\eta}$ to the Starobinsky action [@Starobinsky:1980te] rather than the EH action, then this term would be even more negligible at inflationary energy scales and hence the Starobinsky inflation model [@Starobinsky:1980te] (which is the leading inflationary model supported by the most recent cosmological observations [@Ade:2015lrj]) would also be left untouched. Hence, a modified gravity model such as the Starobinsky action$+\alpha(T_{\mu\nu}T^{\mu\nu})^{\eta}$ with $\eta\sim0$ and $\alpha\sim2.5$ would be able to successfully describe the complete history of the Universe (see Appendix \[MGR\] for more details).
Our findings in this paper are promising enough to justify further study of EMPG, as it is apparently a very rich topic. In our detailed discussions here we have mainly considered a pressureless matter source and its evolution throughout the history of Universe, and discussed the late-time acceleration of the Universe. Hence, one direction may be to extend our analysis by considering effects of other known cosmological fluids on the evolution of the Universe. For instance, radiation could be studied in order to obtain constraints on the parameters of the EMPG from BBN processes that took place when the Universe was radiation dominated. Of course, the inclusion of a positive $\Lambda$ in our model is also of further interest to us since in this case the model would provide much richer dynamics (e.g. Ref. [@Board:2017ign]). In particular, if we consider the case $\eta>\frac{1}{2}$, then EMPG would alter the dynamics of the early Universe, but lead to an evolution of the Universe like that in the $\Lambda$CDM model after a certain amount of time (see Refs. for EMSG, i.e., ${\rm EMPG}_{\eta=1}$). In this case, one may need to study not only the cosmological dynamics but also the physical processes/astrophysical objects relevant to high energy densities such as BBN/neutron stars, which in turn can be used jointly to constrain the parameters of the model without suffering from degeneracy issues. Another direction may be to consider functions other than $f(T_{\mu\nu}T^{\mu\nu})=\alpha(T_{\mu\nu}T^{\mu\nu})^{\eta}$ that lead to the EMPG we studied here; for instance, $f(T_{\mu\nu}T^{\mu\nu})=\alpha\log(T_{\mu\nu}T^{\mu\nu})$ would give $\log(\rho^2)$-type modifications to Einstein’s field equations, as in Ref. [@Chavanis:2016pcp]. The essence of constructing modified gravity models is to redefine the coupling of the energy-momentum tensor to the spacetime geometry. However, this is usually done by modifying the action that leads to modifications on the left-hand side of the Einstein’s field equations, in contrast to what we have done here. Manipulating the introduction of the energy-momentum tensor into the action would lead to modified Einstein’s field equations that might not be possible to obtain or to easily find from the Einstein’s field equations with a modification on the left-hand side while the energy density and pressure of a fluid appear on the right-hand side of the equations as usual. Hence, it would be interesting to investigate the $f(R)$ correspondence of $T_{\mu\nu}T^{\mu\nu}$-type modifications in the metric and Palatini formalisms, similarly to, for instance, how the Ricci-tensor-powered gravity was studied in the Palatini variational approach regarding the powered form of the Ricci tensor contraction in Ref. [@Li:2007xw]. Finally, the geometrical counterpart of $T_{\mu\nu}T^{\mu\nu}$-type modifications can also be studied, such as the powered $G_{\mu\nu}G^{\mu\nu}$ term, and even the geometry and source can be mixed by considering terms like $G_{\mu\nu}T^{\mu\nu}$. For instance, if the scalar field EMT is considered as the source, the model should correspond to a very popular subclass of Horndeski scalar-tensor theories [@Horndeski:1974wa], where the scalar field is nonminimally coupled with gravity through the Einstein tensor such that $G_{\mu\nu}\partial^{\mu}\phi\partial^{\nu}\phi$.
We thank to R.C. Nunes for constructive discussions. Ö.A. acknowledges the support by the Science Academy in scheme of the Distinguished Young Scientist Award (BAGEP). N.K. acknowledges the post-doctoral research support she is receiving from the [İ]{}stanbul Technical University. S.K. gratefully acknowledges the support from SERB-DST project No. EMR/2016/000258.
Approximation procedure {#app}
=======================
The matter energy density $\rho_{\rm m}$ appears implicitly in Eq. , $$\begin{aligned}
\label{eqn:apx1}
\frac{\rho_{\rm m}}{\rho_{{\rm m},0}}\left[\frac{1+\frac{2\eta\alpha'}{2\eta-1}\left(\frac{\rho_{\rm m}}{\rho_{{\rm m},0}}\right)^{2\eta-1}}{1+\frac{2\eta\alpha'}{2\eta-1}}\right]^{\frac{2\eta-2}{2\eta-1}} =(1+z)^{3},\end{aligned}$$ and it cannot be isolated in terms of redshift $z$. Therefore, it is not possible to explicitly write the exact function for the Hubble parameter $H(z)$ by substituting $\rho_{\rm m}(z)$ in Eq. , but a good approximation can be found. To do so, we first note that Eq. can be written as $$\begin{aligned}
\label{eqn:fs}
\left({1+\frac{2\alpha'\eta}{2\eta-1}}\right)^{-\frac{2\eta-2}{2\eta-1}}\rho_{\rm m}\approx \rho_{{\rm m},0}(1+z)^{3}\;\;\textnormal{for}\;\; \rho_{\rm m}\gg\rho_{{\rm m},0},\end{aligned}$$ provided that $\eta<\frac{1}{2}$ in accordance with the conditions given in Eq. . Of course, this approximation is not good in the vicinity of $z= 0$, and we need to improve it. To do so, we first note that the approximation error (the deviation from the true value) at $z=0$ is $\left({1+\frac{2\alpha'\eta}{2\eta-1}}\right)^{-\frac{2\eta-2}{2\eta-1}}\rho_{{\rm m},0}-\rho_{{\rm m},0}$. Notice that, in this case, we do not get $\rho_{\rm m}=\rho_{{\rm m},0}$ at $z=0$, but $\rho_{\rm m}=\left({1+\frac{2\alpha'\eta}{2\eta-1}} \right)^{-\frac{2\eta-2}{2\eta-1}}\rho_{{\rm m},0}$. Therefore, subtracting this approximation error from the left-hand side of Eq. would not only give the actual value of $\rho_{\rm m}$ when $z=0$, but also decrease the approximation error when $\rho_{\rm m}\sim\rho_{{\rm m},0}$, viz., when $z\sim0$. Accordingly, compensating the approximation error at $z=0$ in Eq. , we reach the following improved approximation: $$\begin{aligned}
&\left({1+\frac{2\alpha'\eta}{2\eta-1}}\right)^{-\frac{2\eta-2}{2\eta-1}}\rho_{\rm m}\\
&-\left[\left({1+\frac{2\alpha'\eta}{2\eta-1}}\right)^{-\frac{2\eta-2}{2\eta-1}}\rho_{{\rm m},0}-\rho_{{\rm m},0}\right]\approx\rho_{{\rm m},0}(1+z)^3,
\end{aligned}$$ which is an approximation both for $\rho_{\rm m}\gg\rho_{{\rm m},0}$ and $\rho_{\rm m}\sim\rho_{{\rm m},0}$. In addition, it allows us to isolate $\rho_{\rm m}$ in terms of $z$ as $$\begin{aligned}
\label{eqn:apxr}
\rho_{\rm m}\approx\rho_{\rm m,approx}=&\rho_{{\rm m},0} \left[1-\left({1+\frac{2\alpha'\eta}{2\eta-1}}\right)^{\frac{2\eta-2}{2\eta-1}}\right]\\
&+\rho_{{\rm m},0} \left({1+\frac{2\alpha'\eta}{2\eta-1}}\right)^{\frac{2\eta-2}{2\eta-1}} (1+z)^3.
\end{aligned}$$ In Fig. \[rhofit\], we depict the relative error between the true value of the matter energy density $\rho_{\rm m}$ from Eq. \[or Eq. \] and the approximated value of the matter energy density $\rho_{\rm m,approx}$ from Eq. $-$namely, $\frac{\delta\rho_{\rm m}}{\rho_{\rm m}}=\frac{\rho_{\rm m,approx}-\rho_{\rm m}}{\rho_{\rm m}}$ versus $\rho_{\rm m}/\rho_{{\rm m},0}$ from $\rho_{\rm m}/\rho_{{\rm m},0}=1\; (z=0)$ to $\rho_{\rm m}/\rho_{{\rm m},0}=10^9\; (z\sim1100)$ $-$while using the mean values $\alpha'=2.8$, $\eta=-0.003$ from Table \[tab:results\]. We see that the relative error is indeed negligible, as the maximum relative error is only $0.01$ percent while it is much smaller for $\rho_{\rm m}\sim\rho_{{\rm m},0}$ and $\rho\gg\rho_{{\rm m},0}$.
![The relative error $\delta\rho_{\rm m}/\rho_{\rm m}$ versus $\rho_{\rm m}/\rho_{{\rm m},0}$ from $\rho_{\rm m}/\rho_{{\rm m},0}=1$ $(z=0)$ to $\rho_{\rm m}/\rho_{{\rm m},0}=10^9$ $(z\sim1100)$ for the mean values $\alpha'=2.8$, $\eta=-0.003$ given in Table \[tab:results\].[]{data-label="rhofit"}](rhodelfit.pdf){width="49.00000%"}
The complete history of the Universe via the Starobinsky action extension {#MGR}
=========================================================================
EMPG is studied on theoretical as well as observational grounds leading to the late-time acceleration of the Universe with only pressureless matter while keeping the successes of standard general relativity at earlier times; however, inflation is not considered. For completeness, we may consider the modification that gives EMPG in the Starobinsky gravity action [@Starobinsky:1980te] instead of that in the Einstein-Hilbert action as follows: $$\begin{aligned}
S=\int \left[\frac{1}{2\kappa}\left(R+\frac{R^2}{6M_{\rm p}^2}\right)+\alpha(T_{\mu\nu}T^{\mu\nu})^{\eta}+\mathcal{L}_{\rm m}\right]\sqrt{-g}d^4x,
\label{eq:staroaction}\end{aligned}$$ The field equations become $$\begin{aligned}
\left(1+\frac{R}{3M_{\rm p}^2}\right)G_{\mu\nu}+\left(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu}\right)\left(1+\frac{R}{3M_{\rm p}^2}\right)
\\
=\kappa T_{\mu\nu} +\kappa \alpha (T_{\sigma\epsilon}T^{\sigma\epsilon})^{\eta}\left[g_{\mu\nu}-2\eta\frac{\theta_{\mu\nu}}{T_{\sigma\epsilon}T^{\sigma\epsilon}}\right].
\label{starofieldeq}
\end{aligned}$$ Then, using Eqs. and as well as the metric in the field equations , we get the following set of two independent differential equations: $$\begin{aligned}
3\left(1+\frac{R}{3M_{\rm p}^2}\right)H^2+\frac{H\dot{R}}{M_{\rm p}^2}-\frac{R^2}{12M_{\rm p}^2}=\kappa\rho+\kappa' \rho_0\left(\frac{\rho}{\rho_0}\right)^{2\eta} ,
\label{eq:starhoprime}
\end{aligned}$$ $$\begin{aligned}
\left(-2\dot{H}-3H^2\right)\left(1+\frac{R}{3M_{\rm p}^2}\right)-\frac{H\dot{R}+\ddot{R}}{3M_{\rm p}^2}+\frac{R^2}{12M_{\rm p}^2}
\\=\kappa w\rho+ \frac{\kappa' \rho_0}{2\eta-1+\frac{8w\eta}{3w^2+1}} \left(\frac{\rho}{\rho_0}\right)^{2\eta}.\\
\label{eq:stapresprime}
\end{aligned}$$ We note that the terms that appear in the field equations given in Eqs. and due to the $R^2$ term in the action become effective for large values of $H$, namely, in the very early Universe at very high energy densities where the terms that appear in the field equations due to the modification $\alpha(T_{\mu\nu}T^{\mu\nu})^{\eta}$ in the action, are completely negligible for $\eta<\frac{1}{2}$. It is easy to see in this case that, for $\eta\sim0$ and $\alpha \sim-10^{-8} (\rm erg/cm^3)^{1-2\eta}$ (or $\kappa'\sim2.5\kappa$ in the presence of only dust) as found in this study, at inflationary energy scales the terms that appear in the field equations due to $R^2$ and the usual matter Lagrangian $\mathcal{L}_{\rm m}$ in the action would be effective, while the terms due to $R$ and the EMP modification would be negligible, so that $R^2$ inflation$-$which is the preferred model in light of the latest observational data by the Planck Collaboration [@Ade:2015lrj]$-$would take place during the early Universe in our model. Then, at intermediate energy scales (at the time of BBN, recombination, etc.), the terms that appear in the field equations due to the terms $R$ and $\mathcal{L}_{\rm m}$ in the action would be effective, but the terms due to the terms $R^2$ and $\alpha(T_{\mu\nu}T^{\mu\nu})^{\eta}$ would be negligible so that the standard cosmology would work. Finally, at sufficiently low energy density scales the terms that appear in the field equations due to the terms $R$ and $\alpha(T_{\mu\nu}T^{\mu\nu})^{\eta}$ in addition to the $\mathcal{L}_{\rm m}$ in the action would be effective, while the terms in the field equations due to $R^2$ in the action would be negligible so that the Universe would start to accelerate at later times. Thus, the unification of EMPG with Starobinsky gravity would provide us with a complete history of the Universe. For instance, the Starobinsky model within the $f(R,T)$ formalism was studied in Ref. [@Moraes:2016zdw], where the radiation$-$, matter$-$ and dark energy-dominated eras as well as the transition between these phases were studied.
[99]{} A. A. Starobinsky, A new type of isotropic cosmological models without singularity, Phys. Lett. B [**91**]{}, 99 (1980).
A. H. Guth, The inflationary universe: A possible solution to the horizon and flatness problems, Phys. Rev. D [**23**]{}, 347 (1981). A. D. Linde, A new inflationary universe scenario: A possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems, Phys. Lett. B [**108**]{}, 389 (1982). A. Albrecht and P. J. Steinhardt, Cosmology for grand unified theories with radiatively induced symmetry breaking, Phys. Rev. Lett. [**48**]{}, 1220 (1982). D. Baumann, Inflation, in TASI 2009: Physics of the Large and the Small, edited by C. Csaki and S. Dodelson (World Scientific, Singapore, 2011), p. 523, \[arXiv:0907.5424 \[hep-th\]\].
A. Linde, Inflationary cosmology after Planck 2013, arXiv:1402.0526. P. A. R. Ade [*et al.*]{} \[Planck Collaboration\], Planck 2015 results. XX. Constraints on inflation, Astron. Astrophys. [**594**]{}, A20 (2016), \[arXiv:1502.02114 \[astro-ph.CO\]\].
E. Komatsu [*et al.*]{} \[WMAP Collaboration\], Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Cosmological interpretation, Astrophys. J. Suppl. Ser. [**192**]{}, 18 (2011), \[arXiv:1001.4538 \[astro-ph.CO\]\]..
E. Aubourg [*et al.*]{}, Cosmological implications of baryon acoustic oscillation measurements, Phys. Rev. D [**92**]{}, 123516 (2015), \[arXiv:1411.1074 \[astro-ph.CO\]\].
P. A. R. Ade [*et al.*]{} \[Planck Collaboration\], Planck 2015 results. XIII. Cosmological parameters, Astron. Astrophys. [**594**]{}, A13 (2016), \[arXiv:1502.01589 \[astro-ph.CO\]\]..
V. Sahni and A. A. Starobinsky, The case for a positive cosmological $\Lambda$-term, Int. J. Mod. Phys. D [**9**]{}, 373 (2000), \[astro-ph/9904398\].
P. J. E. Peebles and B. Ratra, The cosmological constant and dark energy, Rev. Mod. Phys. [**75**]{}, 559 (2003), \[astro-ph/0207347\].
E. J. Copeland, M. Sami and S. Tsujikawa, Dynamics of dark energy, Int. J. Mod. Phys. D [**15**]{}, 1753 (2006), \[hep-th/0603057\].
K. Bamba, S. Capozziello, S. Nojiri and S. D. Odintsov, Dark energy cosmology: The equivalent description via different theoretical models and cosmography tests, Astrophys. Space Sci. [**342**]{}, 155 (2012), \[arXiv:1205.3421 \[gr-qc\]\].
S. Weinberg, The cosmological constant problem, Rev. Mod. Phys. [**61**]{}, 1 (1989).
S. M. Carroll, The cosmological constant, Living Rev. Relativity [**4**]{}, 1 (2001), \[astro-ph/0004075\].
V. Sahni, Dark matter and dark energy, Lect. Notes Phys. [**653**]{}, 141 (2004), \[astro-ph/0403324\]..
P. Bull [*et al.*]{}, Beyond $\Lambda$CDM: Problems, solutions, and the road ahead, Phys. Dark Univ. [**12**]{}, 56 (2016), \[arXiv:1512.05356 \[astro-ph.CO\]\].
G. B. Zhao [*et al.*]{}, Dynamical dark energy in light of the latest observations, Nature Astron. [**1**]{}, 627 (2017), \[arXiv:1701.08165 \[astro-ph.CO\]\]..
T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis, Modified gravity and cosmology, Phys. Rep. [**513**]{}, 1 (2012), \[arXiv:1106.2476 \[astro-ph.CO\]\]..
A. De Felice and S. Tsujikawa, $f(R)$ theories, Living Rev. Relativity [**13**]{}, 3 (2010), \[arXiv:1002.4928 \[gr-qc\]\].
S. Capozziello and M. De Laurentis, Extended theories of gravity, Phys. Rep. [**509**]{}, 167 (2011), \[arXiv:1108.6266 \[gr-qc\]\].
S. Nojiri, S. D. Odintsov and V. K. Oikonomou, Modified gravity theories on a nutshell: Inflation, bounce and late-time evolution, Phys. Rep. [**692**]{}, 1 (2017) \[arXiv:1705.11098 \[gr-qc\]\].
S. Nojiri and S. D. Odintsov, Unified cosmic history in modified gravity: From $F(R)$ theory to Lorentz non-invariant models, Phys. Rep. [**505**]{}, 59 (2011), \[arXiv:1011.0544 \[gr-qc\]\].
T. Harko and F. S. N. Lobo, $f(R, \mathcal{L}_{\rm m}$) gravity, Eur. Phys. J. C [**70**]{}, 373 (2010), \[arXiv:1008.4193 \[gr-qc\]\].
T. Harko, Galactic rotation curves in modified gravity with nonminimal coupling between matter and geometry, Phys. Rev. D [**81**]{}, 084050 (2010), \[arXiv:1004.0576 \[gr-qc\]\].
N. J. Poplawski, A Lagrangian description of interacting dark energy, arXiv: gr-qc/0608031. T. Harko, F. S. N. Lobo, S. Nojiri and S. D. Odintsov, $f(R,T)$ gravity, Phys. Rev. D [**84**]{}, 024020 (2011), \[arXiv:1104.2669 \[gr-qc\]\].
P. H. R. S. Moraes and J. R. L. Santos, A complete cosmological scenario from $f(R,T^{\phi })$ gravity theory, Eur. Phys. J. C [**76**]{}, 60 (2016), \[arXiv:1601.02811 \[gr-qc\]\].
S. D. Odintsov and D. Sáez-Gómez, $f(R, T, R_{\mu\nu} T^{\mu\nu})$ gravity phenomenology and $\Lambda$CDM universe, Phys. Lett. B [**725**]{}, 437 (2013), \[arXiv:1304.5411 \[gr-qc\]\].
Z. Haghani, T. Harko, F. S. N. Lobo, H. R. Sepangi, and S. Shahidi, Further matters in space-time geometry: $f(R,T,R_{\mu\nu}T^{\mu\nu})$ gravity, Phys. Rev. D [**88**]{}, 044023 (2013), \[arXiv:1304.5957 \[gr-qc\]\].
Z. Haghani, T. Harko, H. R. Sepangi, and S. Shahidi, Matter may matter, Int. J. Mod. Phys. D [**23**]{}, 12 (2014), \[arXiv:1304.5957 \[gr-qc\]\].
N. Kat[i]{}rc[i]{} and M. Kavuk, $ f(R,T_{\mu\nu}T^{\mu\nu})$ gravity and Cardassian-like expansion as one of its consequences, Eur. Phys. J. Plus [**129**]{}, 163 (2014), \[arXiv:1302.4300 \[gr-qc\]\].
T. Harko, F. S. N. Lobo, G. Otalora and E. N. Saridakis, Nonminimal torsion-matter coupling extension of $f(T)$ gravity, Phys. Rev. D [**89**]{}, 124036 (2014), \[arXiv:1404.6212 \[gr-qc\]\].
T. Harko, F. S. N. Lobo, G. Otalora and E. N. Saridakis, $f(T,\mathcal{T})$ gravity and cosmology, J. Cosmol. Astropart. Phys. 12 (2014) 021, \[arXiv:1405.0519 \[gr-qc\]\].
A. Ashtekar, T. Pawlowski and P. Singh, Quantum nature of the big bang: Improved dynamics, Phys. Rev. D [**74**]{}, 084003 (2006), \[gr-qc/0607039\].
A. Ashtekar and P. Singh, Loop quantum cosmology: A status report, Classical Quantum Gravity [**28**]{}, 213001 (2011), \[arXiv:1108.0893 \[gr-qc\]\]..
P. Brax and C. van de Bruck, Cosmology and brane worlds: A review, Classical Quantum Gravity [**20**]{}, R201 (2003), \[hep-th/0303095\].
M. Roshan and F. Shojai, Energy-momentum squared gravity, Phys. Rev. D [**94**]{}, 4 044002 (2016), \[arXiv:1607.06049 \[gr-qc\]\].
C. V. R. Board and J. D. Barrow, Cosmological models in energy-momentum squared gravity, Phys. Rev. D **96**, 123517 (2017), arXiv:1709.09501 \[gr-qc\].
O. Bertolami, F. S. N. Lobo and J. Paramos, Non-minimum coupling of perfect fluids to curvature, Phys. Rev. D [**78**]{}, 064036 (2008), \[arXiv:0806.4434 \[gr-qc\]\].
V. Faraoni, The Lagrangian description of perfect fluids and modified gravity with an extra force, Phys. Rev. D [**80**]{}, 124040 (2009), \[arXiv:0912.1249 \[astro-ph.GA\]\].
J. P. Uzan, Varying constants, gravitation and cosmology, Living Rev. Relativity [**14**]{}, 2 (2011), \[arXiv:1009.5514 \[astro-ph.CO\]\]..
K. Freese and M. Lewis, Cardassian expansion: A model in which the universe is flat, matter dominated, and accelerating, Phys. Lett. B [**540**]{}, 1 (2002), \[astro-ph/0201229\].
D. J. H. Chung and K. Freese, Cosmological challenges in theories with extra dimensions and remarks on the horizon problem, Phys. Rev. D [**61**]{}, 023511 (2000), \[hep-ph/9906542\].
A. Y. Kamenshchik, U. Moschella and V. Pasquier, An alternative to quintessence, Phys. Lett. B [**511**]{}, 265 (2001), \[gr-qc/0103004\].
M. C. Bento, O. Bertolami and A. A. Sen, Generalized Chaplygin gas, accelerated expansion and dark energy matter unification, Phys. Rev. D [**66**]{}, 043507 (2002), \[gr-qc/0202064\].
K. N. Ananda and M. Bruni, Cosmo-dynamics and dark energy with non-linear equation of state: a quadratic model, Phys. Rev. D [**74**]{}, 023523 (2006), \[astro-ph/0512224\].
T. Josset, A. Perez and D. Sudarsky, Dark energy from violation of energy conservation, Phys. Rev. Lett. [**118**]{}, 021102 (2017), \[arXiv:1604.04183 \[gr-qc\]\].
H. Shabani and A. H. Ziaie, Consequences of energy conservation violation: Late-time solutions of $\Lambda (\mathsf{T}) \mathsf{CDM}$ subclass of $f(\mathsf{R},\mathsf{T})$ gravity using dynamical system approach, Eur. Phys. J. C [**77**]{}, 282 (2017), \[arXiv:1702.07380 \[gr-qc\]\].
D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. [**12**]{}, 498 (1971).
D. Lovelock, The four-dimensionality of space and the Einstein tensor, J. Math. Phys. [**13**]{}, 874 (1972).
O. Akarsu, J. D. Barrow, S. Ç[i]{}k[i]{}ntoğlu, K. Y. Ekşi and N. Katirci, Constraints on energy-momentum squared gravity from neutron stars and its cosmological implications, arXiv:1802.02093.
N. Straumann, General Relativity: With Applications to Astrophysics. (Springer-Verlag, 2004).
J. Barrow, Light elements and the isotropy of the universe, Mon. Not. R. Astron. Soc. [**175**]{}, 359 (1976).
C. Rubano and J. D. Barrow, Scaling solutions and reconstruction of scalar field potentials, Phys. Rev. D [**64**]{}, 127301 (2001), \[gr-qc/0105037\].
R. R. Caldwell, A phantom menace?, Phys. Lett. B [**545**]{}, 23 (2002), \[astro-ph/9908168\].
R. R. Caldwell, M. Kamionkowski and N. N. Weinberg, Phantom energy and cosmic doomsday, Phys. Rev. Lett. [**91**]{}, 071301 (2003), \[astro-ph/0302506\].
S. M. Carroll, M. Hoffman, and M. Trodden, Can the dark energy equation-of-state parameter w be less than $-1$?, Phys. Rev. D [**68**]{}, 023509 (2003), \[arXiv:astro-ph/0301273\].
J. M. Cline, S. Jeon, and G. D. Moore, The phantom menaced: Constraints on low-energy effective ghosts, Phys. Rev. D [**70**]{}, 043543, (2004), \[arXiv:hep-ph/0311312\].
O. Farooq and B. Ratra, Hubble parameter measurement constraints on the cosmological deceleration-acceleration transition redshift, Astrophys. J. [**766**]{}, L7 (2013), \[arXiv:1301.5243 \[astro-ph.CO\]\].
Y. Chen, S. Kumar and B. Ratra, Determining the Hubble constant from Hubble parameter measurements, Astrophys. J. [**835**]{}, 1 86 (2017), \[arXiv:1606.07316 \[astro-ph.CO\]\].
A. Lewis and S. Bridle, Cosmological parameters from CMB and other data: A Monte Carlo approach, Phys. Rev. D [**66**]{}, 103511 (2002), \[astro-ph/0205436\].
J. Simon, L. Verde and R. Jimenez, Constraints on the redshift dependence of the dark energy potential, Phys. Rev. D [**71**]{}, 123001 (2005), \[astro-ph/0412269\].
, [*et* al.]{}, Cosmic chronometers: Constraining the equation-of-state of dark energy. II. A spectroscopic catalog of red galaxies in galaxy clusters, J. Cosmol. Astropart. Phys. 02 (2010) 008, \[arXiv:0907.3152\].
, Improved constraints on the expansion rate of the Universe up to $z\sim1.1$ from the spectroscopic evolution of cosmic chronometers, J. Cosmol. Astropart. Phys. 08, (2012) 006, \[arXiv:1201.3609\].
, Baryon acoustic oscillations in the Ly-$\alpha$ forest of BOSS quasars, Adyton. Astrophys. [**552**]{}, A96 (2013), \[arXiv:1211.2616\].
, Four New Observational $H(z)$ Data From Luminous Red Galaxies of Sloan Digital Sky Survey Data Release Seven, Res. Astron. Astrophys., [**14**]{}, 1221 (2014), \[arXiv:1207.4541\].
, The WiggleZ Dark Energy Survey: Joint measurements of the expansion and growth history at $z <1$, Mon. Not. R. Astron. Soc. [**425**]{}, 405 (2012), \[arXiv:1204.3674\].
and [Y. Wang]{}, Modeling the anisotropic two-point galaxy correlation function on small scales and single-probe measurements of $H(z)$, $D_A(z)$, and $f(z)\sigma_8(z)$ from the Sloan Digital Sky Survey DR7 luminous red galaxies", Mon. Not. R. Astron. Soc., [**435**]{}, 255 (2013), \[arXiv:1209.0210\].
P. H. Chavanis and S. Kumar, Comparison between the Logotropic and $\Lambda$CDM models at the cosmological scale, J. Cosmol. Astropart. Phys. 05 (2017) 018, \[arXiv:1612.01081 \[astro-ph.CO\]\].
B. Li, J. D. Barrow and D. F. Mota, The cosmology of Ricci-tensor-squared gravity in the Palatini variational approach, Phys. Rev. D [**76**]{}, 104047 (2007), \[arXiv:0707.2664 \[gr-qc\]\].
G. W. Horndeski, Second-order scalar-tensor field equations in a four-dimensional space, Int. J. Theor. Phys. [**10**]{}, 363 (1974).
P. H. R. S. Moraes, R. A. C. Correa and G. Ribeiro, The Starobinsky model within the $f(R,T)$ formalism as a cosmological model, arXiv:1701.01027, arXiv:1701.01027 \[gr-qc\].
[^1]: According to Lovelock’s theorem, the cosmological constant $\Lambda$ arises as a constant of nature, which we have set to zero in Eq. for the reasons discussed in detail in Sec. \[subsec:A\].
[^2]: Lovelock’s theorem [@Lovelock:1971yv; @Lovelock:1972vz] states that the only possible second-order Euler-Lagrange expression obtainable in a four-dimensional space from a scalar density of the form $\mathcal{L}= \mathcal{L}(g_{\mu\nu})$ is $E_{\mu\nu}=\sqrt{-g}\left(\lambda_1 G_{\mu\nu}+\lambda_2 g_{\mu\nu}\right)$, where $\lambda_1$ and $\lambda_2$ are constants, leading to Newton’s gravitational constant $G$ and cosmological constant $\Lambda$ in Einstein’s field equations $G_{\mu\nu}+\Lambda g_{\mu\nu}=\kappa T_{\mu\nu}$ (see [@Bull:2015stt; @Clifton:2011jh; @Straumann] for further reading).
|
---
abstract: 'We study the ground state phase diagram of a nonlinear two-photon Rabi-Hubbard (RH) model in one dimension using quantum Monte Carlo (QMC) simulations and density matrix renormalization group (DMRG) calculations. Our model includes a nonlinear photon-photon interaction term. Absent this term, the RH model has only one phase, the normal disordered phase, and suffers from spectral collapse at larger values of the photon-qubit interaction or inter-cavity photon hopping. The photon-photon interaction, no matter how small, stabilizes the system which now exhibits [*two*]{} quantum phase transitions: Normal phase to [*photon pair*]{} superfluid (PSF) transition and PSF to single particle superfluid (SPSF). The discrete $Z_4$ symmetry of the Hamiltonian spontaneously breaks in two stages: First it breaks partially as the system enters the PSF and then completely breaks when the system finally enters the SPSF phase. We show detailed numerical results supporting this, and map out the ground state phase diagram.'
author:
- Shifeng Cui
- 'B. Grémaud'
- Wenan Guo
- 'G. G. Batrouni'
title: 'Nonlinear two-photon Rabi-Hubbard model: superradiance and photon/photon-pair Bose-Einstein condensate'
---
Introduction
============
Light-matter interaction is ubiquitous in nature and is, therefore, the focus of much theoretical and experimental work. The relatively simple Rabi model[@rabi36; @rabi37] gives a good description of the interaction of photons with a two level system (qubit) and recent advances in manipulating the interactions of photons with single atoms have resulted in its experimental realization using two-level atoms in cavities[@haroche06] (cavity quantum electrodynamics, QED) or Josephson junctions on solid state chips[@wallraff04; @chen07; @lambert09; @nataf10; @viehmann11] (circuit QED). In the strong coupling limit, $g/\omega \lesssim 0.1$, one can exploit the random wave approximation (RWA) to justify ignoring the counter-rotating (CR) terms resulting in a Hamiltonian which conserves the exciton number (number of photons plus excited qubits) due to its $U(1)$ symmetry. This results in the Jaynes-Cummings (JC) model[@jaynes63] which can be solved exactly. Such cavities can be arranged in a chain where the coupling between near neighbors can be controlled via a tunable tunneling of the photon mode thus resulting in the Jaynes-Cummings-Hubbard model[@schiro13] which describes itinerant bosons (the photons) interacting with localized qubits. This model has been shown to behave like the one dimensional Bose-Hubbard model (BHM)[@batrouni90] with a superfluid phase (of excitons) and incompressible Mott insulator (MI) lobes[@greentree06; @hartmann06; @rossini07; @koch08; @hohenadler08; @hartmann08; @schmidt09; @zhao08]. The MI is approximately a product of single site states obtained from a superposition of photons and excited atoms[@schmidt09] and exhibits behavior similar to photon blockade[@birnbaum05] where there is a finite energy gap opposing the addition of a photon.
In the experimentally attainable[@niemczyk10; @forndiaz10; @chen17; @forndiaz17; @yoshihara17; @kockum17], ultra-strong coupling regime, $g/\omega\sim 1$, the CR terms can no longer be ignored. Their restoration to the Hamiltonian reduces the $U(1)$ symmetry to $Z_2$. Consequently, the number of excitons is no longer conserved thus removing the possibility of a MI phase. The phase diagram now consists of a disordered phase and an ordered (coherent) one separated by a quantum phase transition in the universality class of the two-dimensional Ising model[@schiro13; @zheng11; @kumar13; @flottat16]. The ordered phase, therefore, exhibits a Bose-Einstein condensate (BEC) of photon. This transition resembles the incoherent/coherent (normal/superradiant) phase transition in the Dicke model[@hepp73; @rotondo15].
The two-photon Rabi model has also attracted much attention as new systems are realized where multi-photon processes come into play. For example, it has been used to describe second order processes in Rydberg atoms in cavities[@bertet02] and quantum dots[@stufler06; @delvalle10], and mechanisms have been proposed to realize it in circuit QED[@felicetti18]. The two-photon model undergoes spectral collapse where the Hamiltonian is no longer bounded when the coupling exceeds a certain value[@emary02; @dolya09; @travenec12; @maciejewski15; @travenec15; @peng17; @chen12; @felicetti15]. In the strong coupling regime, the CR terms can be ignored, as in the one-photon case, and result in the two-photon JC model with $U(1)$ symmetry and conserved number of excitons. In the ultra-strong regime, the CR terms are restored and the symmetry is reduced to $Z_4$. The ground state of the many-body two-photon model was studied in the context of the Dicke model[@dicke54] using mean field[@garbe17] and QMC[@cui19]. Excellent agreement was found between the two methods which showed the system to exhibit a quantum phase transition between a normal (disordered) and superradiant phase. Here, however, the term superradiant is used to indicate a macroscopic change in the average number of photons but which remains relatively small. This is in contrast with the one-photon case where there is a very large number of photons in the superradiant phase which form a BEC[@schiro13; @zheng11; @kumar13; @flottat16]. QMC simulations were also used to examine the two-photon JCH and Rabi Hubbard (RH) models[@cui19]. It was found that the JCH model has only one MI lobe with two excitons/cavity, unlike the one-photon case where there is a succession of MI lobes[@greentree06; @hartmann06; @rossini07; @koch08; @hohenadler08; @hartmann08; @schmidt09; @zhao08]. Furthermore, it was found that there are two superfluid (SF) phases; a single photon SF phase (SPSF) and a photon pair SF phase (PSF). In the former, the single particle Green function decays as a power indicating the quasi-long range order for the photons. In the latter case, however, the single particle Green function decays exponentially while the photon pair Green function decays as a power indicating quasi long range order for bound photon pairs but not for simgle photons. When the CR terms are restored, the symmetry of the Hamiltonian is reduced from $U(1)$ to $Z_4$ and it was found that the two-photon RH model does not exhibit any quantum phase transitions; it has only the disordered phase and the spectral collapse region[@cui19].
In this paper, we examine the possibility that non-linear photon terms in the Hamiltonian could stabilize the system and allow the appearance of quantum phase transitions in the RH model. Non-linear terms, [ *i.e.*]{} photon-photon interactions, have attracted experimental and theoretical interest as a means to generate topological photon pairs[@gorlach17; @mittal18; @olekhno19] which have robust transport properties. To this end we exploit the density matrix renormalization group[@white92; @white93] (DMRG) with open boundary conditions (OBC) using the ALPS library[@alps], and the stochastic Green function[@rousseau08; @rousseau08b] (SGF) quantum Monte Carlo method, with periodic boundary conditions (PBC), to study the phase diagram of the two-photon RH model with a nonlinear photon term. We show that the photon-photon interaction term, stabilizes the system, eliminating spectral collapse, and leads to the appearance of two quantum phase transitions, the first is the transition from the disordered phase to PSF phase where the $Z_4$ symmetry patially breaks, the second is the transition between the PSF and the SPSF phases where the $Z_4$ symmetry is completely broken. We show that in the PSF phase, we have a BEC of photon pairs whereas in the SPSF phase we have a BEC of photons.
The paper is organized as follows. In section [**II**]{} we present the model and discuss briefly the methods used to perform the numerical calculations and simulations. In section [**III**]{} we present and sicuss our results, and in section [**IV**]{} we discuss our conclusions.
Model and method
================
The one-dimensional nonlinear two-photon Rabi-Hubbard (RH) model we study is governed by the Hamiltonian, $$\begin{aligned}
\nonumber
H_{RH}&=& -J \sum_{i=1}^N \left ( {\hat a}^\dagger_i {\hat
a}^{\phantom\dagger}_{i+1} + h.c. \right ) +
\sum_{i=1}^N
\left (\omega {\hat a}^\dagger_i {\hat a}^{\phantom\dagger}_i +
\omega_q \sigma^+_i \sigma^-_i \right )\\
&&+ g\sum_{i=1}^N \left ( \sigma^+_i + \sigma^-_i \right ) \left
({\hat a}^2_i+{\hat a}^{\dagger 2}_i \right )+U\sum_i ({\hat
a}^\dagger_i {\hat a}^{\phantom\dagger}_i)^2,
\label{rabihub}\end{aligned}$$ where $N$ is the number of sites (or cavities) and ${\hat
a}^{\phantom\dagger}_i$ (${\hat a}^\dagger_i$) is the photon destruction (creation) operator in the $i$th cavity. Photon tunneling between sites is governed by the hopping parameter $J$. The photon frequency is $\omega$ and the qubit energy spacing is $\omega_q$; $\sigma^z_i$ and $\sigma^x_i = \sigma^+_i + \sigma^-_i$ are the Pauli matrices acting on the $i$th qubit, $\sigma^+_i$ ($\sigma^-_i$) is the corresponding raising (lowering) operator. The last term in Eq.(\[rabihub\]) describes the onsite photon-photon interaction. Ignoring the CR terms in Eq. (\[rabihub\]), $(\sigma^+_i{\hat a}^{\dagger 2}_i + \sigma^-_i {\hat a}^2_i$), leads to the Jaynes-Cummings-Hubbard (JCH) model where the system is invariant under the generalized rotation operator, $${\cal R}(\theta) = {\rm exp} \left (i \theta {\hat a}^\dagger_j
{\hat a}^{\phantom\dagger}_j
+ i2 \theta \sigma^+_j \sigma^-_j \right ),
\label{rotop}$$ with ${\cal R}(\theta)^\dagger {\hat a}_j {\cal
R}(\theta)^{\phantom\dagger}={\rm e}^{i\theta} {\hat a}_j$, and ${\cal R}(\theta)^\dagger \sigma^-_j {\cal
R}(\theta)^{\phantom\dagger}={\rm e}^{i2\theta} \sigma^-_j$ for any $\theta$, thus exhibiting $U(1)$ symmetry and conservation of the number of excitons $N_{exc} = N_{ph} + 2N_{q}$ where $N_q$ is the number of qubits in the excited state and $N_{ph}$ is the number pf photons. However, the CR terms break the $U(1)$ symmetry, they pick up a phase ${\rm exp}(i4\theta)$ due to the action of the operator ${\cal
R}(\theta)$; this reduces the symmetry to $Z_4$. Now the system is left invariant by this rotation only for $\theta=n2\pi/4$, $n=0,1,2,3$ and $N_{exc}$ is no longer conserved. The discrete $Z_4$ symmetry can break spontaneously in the ground state leading to an ordered BEC phase.
To characterize the various possible phases, we calculate several Green functions, $$G_{\alpha,\beta}(r) \equiv \frac{1}{2N} \sum_i \langle \alpha_i \beta_{i+r} +
{\rm h.c.} \rangle,
\label{greens}$$ where $\alpha$ and $\beta$ denote creation and annihilation operators of the photons (${\hat a}^\dagger_i$ and ${\hat
a}^{\phantom\dagger}_i$) or the qubits ($\sigma^+_j$ and $\sigma^-_j$). $\langle {\cal O}\rangle$ denotes the ground state expectation value, $\langle GS |{\cal O} | GS \rangle$, for DMRG, and the statistical average for QMC, $$\langle {\cal O}\rangle \equiv \frac{1}{Z(\beta)}{\rm Tr} \left
[{\rm e}^{-\beta H}{\cal O}\right ],$$ where $Z={\rm Tr} \left [{\rm e}^{-\beta H}{\cal O}\right ]$ and $\beta = 1/T$. For example, the one-body photon Green function at equal time is given by, $$G_{a^\dagger, a^{\phantom\dagger}}(r) = \frac{1}{2N} \sum_i \langle
{\hat a}^\dagger_i {\hat a}^{\phantom\dagger}_{i+r} + {\hat a}^\dagger_{i+r}
{\hat a}^{\phantom\dagger}_{i} \rangle.
\label{photongreen}$$ The qubit Green function is, $$G_{\sigma^-, \sigma^+}(r)=\frac{1}{2N}\sum_i \langle \sigma^-_i
\sigma^+_{i+r} + \sigma^-_{i+r}\sigma^+_i \rangle,
\label{qubitgreen}$$ and the following two functions will be particularly useful: $$G_{\sigma^+, a}(r)=\frac{1}{2N}\sum_i \langle {\hat a}^{\phantom\dagger}_i
\sigma^+_{i+r}
+ \sigma^-_{i+r}{\hat a}^\dagger_i \rangle,
\label{qubitphoton}$$ and $$G_{\sigma^+, a^2}(r)=\frac{1}{2N}\sum_i \langle {\hat a}^{2}_i \sigma^+_{i+r}
+ \sigma^-_{i+r}{\hat a}^{\dagger 2}_i \rangle.
\label{qubitphoton2}$$ The photon pair Green function is given by, $$G_{a^{\dagger\,2},a^2}(r)=\frac{1}{N}\sum_i\langle {\hat
a}^{\dagger\, 2}_{i}a^2_{i+r}+ H.c.\rangle.
\label{gpair}$$ The Green functions described by Eqs. (\[greens\]-\[gpair\]) are defined on a lattice with perdiodic boundary conditions used in the QMC simulations. When using DMRG with open boundaries, we choose the origin of the correlation function in the center of the lattice and calculate the correlations from that point. Power law decay of one of these Green functions would indicate quasi-long range order for the corresponding quantity. If the Green function decays to a finite constant, it signals long range order and the spontaneous breaking of the corresponding symmetry. The Fourier transform of the single particle Green function, Eq.(\[photongreen\]), gives the photon momentum distribution, $n_{ph}(k)$; the Fourier transform of the pair Green function gives the pair momentum distribution, $n_{pair}(k)$; the Fourier transform of the qubit Green function, Eq.(\[qubitgreen\]), gives the qubit momentum distribution, $n_{q}(k)$ . These quantities will indicate whether a condensate ([ *i.e.*]{} long range order) is present. We also measure the average number of excitons, $$N_{exc} = \sum_i \langle {\hat a}^\dagger_i{\hat a}^{\phantom\dagger}_i +
2\sigma_i^+\sigma_i^- \rangle.
\label{nexc}$$ We calculate these quantities using the ALPS DMRG package[@alps] and the SGF QMC algorithm[@rousseau08; @rousseau08b]. For DMRG, we verified that, in all cases, the number of states we kept and sweeps we performed were sufficient for proper convergence (up to $240$ states and $300$ sweeps for large systems).
Results
=======
The results we present here were obtained using DMRG and SGF QMC depending on the physical quantities being studied and the boundary conditions: Open boundary conditions were typically used with DMRG and periodic ones with SGF. We begin, therefore, by comparing the two methods to ensure that results yielded by one can be reliably compared with the other. Figure \[dmrgsgfcompare\] compares DMRG and SGF results, both with periodic boundary conditions, for several Green functions in the PSF phase (see below) versus distance. The finite temperature SGF QMC simulations were performed at very low temperature, $\beta=360$, and show excellent agreement with the zero-temperature DMRG calculations.
\
-0.5cm
In what follows, we take $U=1$ to fix the energy scale. We also choose the photon frequency $\omega=1$ and the qubit energy spacing $\omega_q=0.1$ and determine the phase diagram of the system governed by Eq.(\[rabihub\]) in the $(J,g)$ plane. It was seen in Refs.[@garbe17; @cui19] that the average number of excited qubits per site, $n_q=N_q/L$, is a good probe for the phase transition between the normal (incoherent) phase and the superradiant phase. Here, we will use both $n_q$ and $n_{ph}=N_{ph}/L$ (the average number of photons per site) as probes for possible phase transitions. We use Padé approximants[@cui19] to fit our numerical results for $n_q$ and $n_{ph}$ as functions of $g$ or $J$. Differentiating the Padé forms, and also numerically differentiating the DMRG data, reveal the presence of peaks indicating the location of the phase transitions. The analytical Padé and the numerical derivatives are in excellent agreement. Figure \[nphnqvsgJ0.4\] shows DMRG results for $n_{ph}$ and $n_q$ and the corresponding numerical and Padé derivatives, ${\rm d}n_{ph}/{\rm
d}g$ and ${\rm d}n_{q}/{\rm d}g$, versus $g$ for constant $J=0.4$ and for $L=12,\,16,\,20,\,24$. In Fig.\[nphnqvsgJ0.4\](a) we show $n_{ph}$ versus $g$ at fixed $J=0.4$, and in (b) we show ${\rm
d}n_{ph}/{\rm d}g$ which, remarkably, displays two peaks indicating the presence of two quantum phase transitions. $n_q$ and its derivative are shown in Fig.\[nphnqvsgJ0.4\](c) and (d) respectively. It is seen that ${\rm d}n_{q}/{\rm d}g$ exhibits only one peak which matches the first peak observed for ${\rm d}n_{ph}/{\rm
d}g$. We conclude, therefore, that the qubits have one type of potential order while the photons have two. This will be examined below. It is seen that the positions of the peaks shift to smaller $g$ values as $L$ increases; this shift is used to extrapolate the location of the transition to the thermodynamic limit and construct the phase diagram.
Similar behavior is observed when $g$ is fixed and and $J$ is changed, as shown in Figs.\[nphnqvsJg2.0\].
\
-0.5cm
We conclude, therefore, that typically, for this system, two quantum phase transitions are encountered as $g$ ($J$) is fixed and $J$ ($g$) is varied. We recall that for the two-photon Rabi model in the absence of the quartic term, no phase transition is present[@cui19]. The addition of the quartic term, $U\sum_i (a^\dagger_i
a^{\phantom\dagger}_i)^2$, stabilized the system and allowed the appearance of quantum phase transitions.
To determine the nature of the three phases present, we study the behavior of the Green functions, Eqs.(\[greens\]-\[gpair\]). Figure \[greensnormalphase\] shows four of these functions, obtained with DMRG, on log-log scale clearly exhibiting decay faster than power (exponential) with distance and, therefore, corresponding to the normal (incoherent) phase. This behavior is characteristic of the region in parameter space, ($J,g$), where $J$ and/or $g$ are small, [*i.e.*]{} before the first peak in Figs.\[nphnqvsgJ0.4\] and \[nphnqvsJg2.0\] leading to the conclusion that this region corresponds to a disordered (incoherent) phase.
Choosing, for example, $(J,g)=(0.6,1.02)$ puts the system between the two peaks observed in Figs.\[nphnqvsgJ0.4\] or \[nphnqvsJg2.0\]. We see in Fig.\[greensPSF\] that, in this case, the one-body photon Green function, Eq.(\[photongreen\]), still decays exponentially whereas all other Green functions shown, rapidly saturate to constant values. The exponential decay of $G_{a^\dagger,a}(r)$ indicates the absence of off-diagonal long range or quasi-long range photon order and, consequently, no photon SF, and $\langle \hat a\rangle=0$. However, the saturation (constant value) of the other Green functions as the distance increases, indicates the establishment of true off-diagonal long range order (ODLRO) for [ *photon pairs*]{}. The saturation value is the square of the order parameter, [*e.g.*]{} $G_{a^{\dagger\,2},a^2}^{saturated} \sim
|\langle a^2\rangle|^2$. Therefore, between the two peaks of Figs.\[nphnqvsgJ0.4\] and \[nphnqvsJg2.0\], the system is in a photon pair SF (PSF) phase with a pair BEC. In terms of symmetry breaking, this means that the $Z_4$ symmetry of the Hamiltonian, Eq.(\[rabihub\]), generated by the operator Eq.(\[rotop\]), is only partially broken: $\langle {\hat a}^2\rangle \neq 0$, and $\langle {\hat a}\rangle =0$.
Increasing $g$ and/or $J$ further, puts the system on the right of the second peaks in Figs.\[nphnqvsgJ0.4\] and \[nphnqvsJg2.0\]. We see in Fig.\[greensSPSF\] that now all Green functions saturate to constant values as the distance increases. This shows that now the system is in a phase with true off-diagonal long range photon order indicating the complete breaking of the $Z_4$ symmetry and the establishment of a single photon SF.
To elucidate further the nature of the two SF phases, we examine the behavior of the qubit, the one-body and the pair momentum distributions. In Fig.\[momdistSP\] we show the normalized photon momentum distribution, $n_{ph}(k)$, in the three phases we have identified and for several system sizes. We see that, as the system size increases, $n_{ph}(k=0)$ decreases when the system is in the normal or PSF phases indicating the absence of a photon BEC and, therefore, the absence of (ODLRO), $\langle {\hat a}_i\rangle =
0$. However, $n_{ph}(k=0)$ remains constant in the SPSF phase indicating that here the photons have formed a BEC, $\langle {\hat
a}_i \rangle \neq 0$, and the $Z_4$ symmetry is broken.
On the other hand, Fig.\[momdistPair\] shows the pair momentum distribution, $n_{pair}(k)$, for the same parameters as in Fig.\[momdistSP\]. Here we see that in the normal phase, $n_{pair}(k=0)$ decreases as $L$ increases but remains constant in the PSF phase. This means that in this phase, [*bound photon pairs*]{} have formed a BEC even though the photons themselves have not, and, therefore, $\langle {\hat a}^2_i\rangle \neq0$ but $\langle {\hat
a}_i\rangle =0$. In the SPSF phase, we already saw, Fig.\[momdistSP\], that $n_{ph}(k=0)\neq 0$, and, therefore, it is not surprising that $n_{pair}(k=0)$ is also (trivially) nonzero in this phase.
Figure \[momdistQ\] shows the momentum distribution of the qubits. In the disordered phase, $n_q(k=0)$ decreases as $L$ increases showing that there is no order. In both the PSF and SPSF, $n_q(k)$ behaves in the same way: $n_q(k=0)$ remains constant as $L$ increases showing that the qubits are ordered. In fact, when the qubits order at the transition between the normal and PSF phases, they remain ordered in the same manner as the system transitions from PSF to SPSF. This is manifested by appearance of only one peak in the derivatives of $n_{q}$ as seen above.
To map out the phase diagram, we perform simulations along several lines of constant $g$ while varying $J$ and also constant $J$ while varying $g$. As shown above, the two peaks exhibited by ${\rm
d}n_{ph}/{\rm d}g$ and by ${\rm d}n_{ph}/{\rm d}J$ indicate the location of the quantum phase transition for the system size being studied. We do the calculations for several system sizes ($L=12,\,16,\,20,\,24$) and extrapolate the critical values to the thermodynamic limit. The separation between the two peaks in the derivatives of $n_{ph}$ is not constant; it decreases as $g$ or $J$ get larger. Figure \[peaksapproch\] illustrates this for several values of $J$. The inset shows the separation between the two peaks, $\delta g\equiv g_c^{(2)}-g_c^{(1)}$, where $g_c^{(1)}$ is the critical $g$ for the transition between the disordered and PSF phases, and $g_c^{(2)}$ is the critical $g$ for the transition between the PSF and SPSF. We see that $\delta g$ gets smaller as $J$ increases. We observe similar behavior for ${\rm d}n_{ph}/{\rm d}J$ as $J$ increases.
Putting all these results together leads to the phase diagram we show in Fig.\[phasediag\]. The figure shows the three phases discussed above: Normal incoherent phase, superradiant photon condensate phase (single particle SF, SPSF) and sandwiched in between is the pair SF phase (PSF). As the two transitions approach each other, for example at large $J$, they become hard to distinguish and appear to merge eventually into one single transition leading to a direct passage from the normal phase to the SPSF phase without passing first through the PSF phase. We also show in Fig.\[phasediag\] the Gutzwiller mean field result for the boundary between the SPSF and the other phases (see appendix for details).
Conclusions
===========
A lot of attention has been given to the two-photon Rabi model, its applications [@bertet02; @stufler06; @delvalle10; @felicetti18] and the spectral collapse [@emary02; @dolya09; @travenec12; @maciejewski15; @travenec15; @peng17; @chen12; @felicetti15] it undergoes in certain regions of its parameter space. Interestingly, in contradistinction with the one-photon Rabi-Hubbard model which, in its ground state, exhibits a quantum phase transtion from a disordered phase to a superradiant one, chracterized by a spontaneously broken symmetry ($Z_2$) and a photon BEC, the two-photon RH model only exhibits the disordered phase. In this work, we used exact computational methods (DMRG and QMC) and showed that the system can be stabilized by a nonlinear (quartic) term which models effective photon-photon interactions [@gorlach17; @mittal18; @olekhno19]. This stabilization eliminates the spectral collapse of the model and, in fact, exposes two quantum phase transtions in the ground state. The first transition, a consquence of partial spontaneous symmetry breaking, takes the system from the disordered phase to the pair superfluid (PSF) phase which is characterized by a nonvanishing pair condensate order parameter, $\langle a^2\rangle\neq 0$, while at the same time $\langle a\rangle =0$. The second transition is from the PSF to the single particle SF (SPSF) phase and completes the symmetry breaking with the photon condensate order parameterm acquiring a nonvanishing expectation value, $\langle a\rangle \neq 0$.
An interesting question to ask is what happens in higher order photon processes, for example the three-photon model? Without stabilization, the three-photon model undergoes spectral collapse but, with a nonlinear photon-photon term like we used here, it will be stabilized. It would be interesting to study this model both in its Jaynes-Cummings limit (ignoring the counter-rotating terms) where the symmetry is $U(1)$ and will not break in one dimension, and also in its full Rabi form where the CR terms are kept and where the symmetry is now $Z_6$. Specifically, will there be two or more kinds of SF phases? For example, will there be photon triplet SF and BEC where three photons act as a single boson that condenses? Will the $Z_6$ symmetry break down in more stages than the smaller $Z_4$ and, consequently, result in more phases?
S. C. and W. G. are supported by the NSFC under Grant No. 11775021 and No. 11734002; S. C. is also supported under NSAF Grant No. U1930402 at CSRC.
Gutzwiller mean field
=====================
We used Gutzwiller mean field to determine the phase boundary of the single particle superfluid (SPSF), in other words where the order parameter $\langle \hat{a}_i\rangle=\langle \hat{a}^+_i\rangle=\psi$ acquires a nonzero value, $\psi \neq 0$. Outside the SPSF, $\psi = 0$ both in the pair superfluid phase (PSF) and the disordered phase. We do not determine the boundary between the disordered phase and PSF where $\langle a^2_i\rangle $ acquires a nonzero value while $\psi =
0$.
We start by writing $$\begin{aligned}
{\hat a}^\dagger_i {\hat a}^{\phantom\dagger}_{i+1} \approx\langle
{\hat a}^\dagger_i\rangle {\hat
a}^{\phantom\dagger}_{i+1}+\langle{\hat
a}^{\phantom\dagger}_{i+1}\rangle{\hat a}^\dagger_i -\langle {\hat
a}^\dagger_i\rangle\langle {\hat a}^{\phantom\dagger}_{i+1}\rangle,\end{aligned}$$ where we ignore fluctuations. This decouples the sites of the lattice and leads to the single-site Hamiltonian $$\begin{aligned}
H_i&=&-2J(\psi{\hat a}^\dagger_i+\psi{\hat
a}^{\phantom\dagger}_i)+\omega {\hat a}^\dagger_i {\hat
a}^{\phantom\dagger}_i+\omega_q \sigma^+_i \sigma^-_i \nonumber
\\ &+&g( \sigma^+_i + \sigma^-_i) ({\hat a}^2_i+{\hat a}^{\dagger
2}_i)+U({\hat a}^\dagger_i {\hat a}^{\phantom\dagger}_i)^2.\end{aligned}$$
Now we define the wavefunction basis: $|n_{ph},g\rangle$ and $|n_{ph},e\rangle$ where $n_{ph}=0,1,2,\dots, n_{max}$ is the number of photons in the cavity; $g$ ($e$) denotes a qubit in the ground (exicted) state. The wave function can then be written in terms of this basis, $$|\Psi\rangle=\sum_{i=0,\,k=(g,e)}^{n_{max}}c_{k,i}|i,k\rangle.$$ The order parameter, $\psi = \langle \Psi | \ a |\Psi \rangle$ becomes, $$\begin{aligned}
\nonumber
\psi&=&c_{g,0}*c_{g,1} + \sqrt{2}c_{g,1}*c_{g,2} \\
&& \ldots+\sqrt{n_{max}}c_{g,n_{max}-1}*c_{g,n_{max}}
\nonumber \\ &+&c_{e,0}*c_{e,1}+\sqrt{2}c_{e,1}*c_{e,2}+\ldots
\nonumber \\ &+&\sqrt{n_{max}}c_{e,n_{max}-1}*c_{e,n_{max}}\end{aligned}$$ The coefficients are first chosen randomly, the Hamiltonian matrix is calculated and diagonalized. This gives a new estimate for the ground state wavefunction which is then used iteratively to calculate an improved ground state wavefunction and so on until the process converges. This way, we calculate $\psi$ for a chosen fixed value of $U$ and many values of $\omega,\, \omega_q,\, J,\, g$, and determine the region in phase space where $\psi\neq 0$. For example, for $U=1$, we obtain the dashed red line in Fig.\[phasediag\].
As an example, we show the Hamiltonian matrix for $n_{max}=2$ with basis order:$|0,g\rangle$, $|1,g\rangle$, $|2,g\rangle$, $|0,e\rangle$, $|1,e\rangle$, $|2,e\rangle$: $$\left(\begin{array}{cccccc}
0 & -2J\psi & 0 & 0 & 0 & \sqrt{2}g \\
-2J\psi & \omega+U & -2\sqrt{2}J\psi & 0 & 0 & 0 \\
0 & -2\sqrt{2}J\psi & 2\omega+4U & \sqrt{2}g & 0 & 0 \\
0 & 0 & \sqrt{2}g & \omega_q & -2J\psi & 0 \\
0 & 0 & 0 & -2J\psi & \omega +\omega_q+U & -2\sqrt{2}J\psi \\
\sqrt{2}g & 0 & 0 & 0 & -2\sqrt{2}J\psi & 2\omega+\omega_q+4U
\end{array}\right)$$
[99]{}
I.I. Rabi, Phys. Rev. [**49**]{}, 324 (1936).
I.I. Rabi, Phys. Rev. [**51**]{}, 652 (1937).
S. Haroche and J.M. Raymond, Exploring the Quantum: Atoms, Cavities and Photons (Oxford Univ. Press, 2006).
A. Wallraff, D.I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S.M. Girvin, and R.J. Schoelkopf, Nature [**431**]{}, 162 (2004).
G. Chen, Z. Chen, and J. Liang, Phys. Rev. A [**76**]{}, 055803 (2007).
N. Lambert, Y.-n. Chen, R. Johansson, and F. Nori, Phys. Rev. B [**80**]{}, 165308 (2009).
P. Nataf and C. Ciuti, Nat. Commun. [**1**]{}, 72 (2010).
O. Viehmann, J. von Delft, and F. Marquardt, Phys. Rev. Lett. [**107**]{}, 113602 (2011).
E.T. Jaynes and F.W. Cummings, Proc. IEEE [**51**]{}, 89109 (1963).
M. Schiró, M. Bordyuh, B. Öztop, H.E. Türeci, J. Phys. B [**46**]{}, 224021 (2013).
G. G. Batrouni, R. T. Scalettar, and G. T. Zimanyi, Phys. Rev. Lett. [**65**]{}, 1765 (1990).
A.D. Greentree, C. Tahan, J.H. Cole, and L.C.L. Hollenberg, Nat. Phys. [**2**]{}, 856 (2006).
M. Hartmann, F. Brandao, and M.B. Plenio, Nat. Phys. [**2**]{}, 849 (2006).
D. Rossini and R. Fazio, Phys. Rev. Lett. [**99**]{}, 186401 (2007).
J. Koch and K. LeHur, Phys. Rev. A [**80**]{}, 023811 (2009).
M. Aichhorn, M. Hohenadler, C. Tahan, P. B. Littlewood, Phys. Rev. Lett. [**100**]{}, 216401 (2008).
M. Hartmann, F. Brandao, and M.B. Plenio, Laser Photon. Rev. [**2**]{}, 527 (2008).
S. Schmidt and G. Blatter, Phys. Rev. Lett. [**103**]{}, 086403, (2009).
J. Zhao, A.W. Sandvik, and K. Ueda, arXiv:0806.3603 (2008).
K.M. Birnbaum, A. Boca, R. Miller, A.D. Boozer, T.E. Northup, and H.J. Kimble, Nature [**436**]{}, 87 (2005).
T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco, T. Hümmer, E. Solano, A. Marx, and R. Gross, Nat. Phys. [**6**]{}, 772 (2010).
P. Forn-Diaz, J. Lisenfeld, D. Marcos, J. J. García-Ripoll, E. Solano, C. J. P. M. Harmans, and J. E. Mooij, Phys. Rev. Lett. [**105**]{}, 237001 (2010).
Z. Chen, Y. Wang, T. Li, L. Tian, Y. Qiu, K. Inomata, F. Yoshihara, S. Han, F. Nori, J. S. Tsai, and J. Q. You, Phys. Rev. A [**96**]{}, 012325 (2017).
P. Forn-Diaz, J. J. García-Ripoll, B. Peropadre, M. A. Yurtalan, J.-L. Orgiazzi, R. Belyansky, C. M. Wilson, and A. Lupascu, Nat. Phys. [**13**]{}, 39 (2017).
F. Yoshihara, T. Fuse, S. Ashhab, K. Kakuyanagi, S. Saito, and K. Semba, Nat. Phys. [**13**]{}, 44 (2017).
A.F. Kockum, A. Miranowicz, S.D. Liberato, S. Savasta, F. Nori, Nature Reviews Physics 1, 19 (2019).
H. Zheng and Y. Takada, Phys. Rev. A [**84**]{}, 043819 (2011).
B. Kumar, S. Jalal, Phys. Rev. A [**88**]{}, 011802(R) (2013).
T. Flottat, F. Hébert, V. G. Rousseau, and G. G. Batrouni, Eur. Phys. J. D [**70**]{}, 213 (2016).
K. Hepp and E.H. Lieb, Ann. Phys. [**76**]{}, 360 (1973).
P. Rotondo, M.C. Lagomarsino, and G. Viola, Phys. Rev. Lett. [**114**]{}, 143601 (2015).
P. Bertet, S. Osnaghi,P. Milman, A. Auffeves, P. Maioli, M. Brune, J. M. Raimond, and S. Haroche, Phys. Rev. Lett. [**88**]{}, 143601 (2002).
S. Stufler, P. Machnikowski, P. Ester, M. Bichler, V. M. Axt, T. Kuhn, and A. Zrenner, Phys. Rev. B [**73**]{}, 125304 (2006).
E. del Valle, S. Zippilli, F. P. Laussy, A. Gonzalez-Tudela, G. Morigi, and C. Tejedor, Phys. Rev. B [**81**]{}, 035302 (2010).
S. Felicetti, D. Z. Rossatto, E. Rico, E. Solano, and P. Forn-Díaz Phys. Rev. [**A97**]{}, 013851 (2018), S. Felicetti, M-J. Hwang, and A. Le Boité, Phys. Rev. [**A98**]{}, 053859 (2018).
C. Emary and R. F. Bishop, J. Math. Phys. (NY) [**43**]{}, 3916 (2002).
S. N. Dolya, J. Math. Phys. [**50**]{}, 033512 (2009).
I. Travěnec, Phys. Rev. A [**85**]{}, 043805 (2012).
A.J. Maciejewski, M. Przybylska, and T. Stachowiak, Phys. Rev. A [**91**]{}, 037801 (2015).
I. Travěnec, Phys. Rev. A [**91**]{}, 037802 (2015).
J. Peng, C. Zheng, G. Guo, X. Guo, X. Zhang, C. Deng, G. Ju, Z. Ren, L. Lamata, and E. Solano, J. Phys. A: Math. Theor. [**50**]{}, 174003 (2017).
Q.-H. Chen, C.Wang, S. He, T. Liu, and K.-L. Wang, Phys. Rev. A [**86**]{}, 023822 (2012).
S. Felicetti, J. S. Pedernales, I. L. Egusquiza, G. Romero, L. Lamata, D. Braak, and E. Solano, Phys. Rev. A [**92**]{}, 033817 (2015).
R. H. Dicke, Phys. Rev. [**93**]{}, 99 (1954).
L. Garbe, I. L. Egusquiza, E. Solano, C. Ciuti, T. Coudreau, P. Milman, and S. Felicetti, Phys. Rev. A [**95**]{}, 053854 (2017)
S. Cui, F. Hébert, B. Grémaud, W. Guo, and G. G. Batrouni, Phys. Rev. [**A100**]{}, 033608 (2019).
M. A. Gorlach and A. N. Poddubny, Phys. Rev. [ **A95**]{}, 053866 (2017).
S. Mittal, E. A. Goldschmidt and Mohammad Hafezi, Nature 561, 502 (2018).
N. A. Olekhno, E. I. Kretov, A. A. Stepanenko, D. S. Filonov, V. V. Yaroshenko, B. Cappello, L. Matekovits, and M. A. Gorlach, arXiv:1907.01016.
S. R. White, Phys. Rev. Lett. [**69**]{}, 2863 (1992).
S. R. White, Phys. Rev. B [**48**]{}, 10345 (1993).
B. Bauer [*et al.*]{}, J. Stat. Mech. P05001 (2011).
V.G. Rousseau, Phys. Rev. E [**77**]{}, 056705 (2008).
V.G. Rousseau, Phys. Rev. E [**78**]{}, 056707 (2008).
|
---
abstract: 'Region of interest (ROI) alignment in medical images plays a crucial role in diagnostics, procedure planning, treatment, and follow-up. Frequently, a model is represented as triangulated mesh while the patient data is provided from CAT scanners as pixel or voxel data. Previously, we presented a 2D method for curve-to-pixel registration. This paper contributes (i) a general mesh-to-raster (M2R) framework to register ROIs in multi-modal images; (ii) a 3D surface-to-voxel application, and (iii) a comprehensive quantitative evaluation in 2D using ground truth provided by the simultaneous truth and performance level estimation (STAPLE) method. The registration is formulated as a minimization problem where the objective consists of a data term, which involves the signed distance function of the ROI from the reference image, and a higher order elastic regularizer for the deformation. The evaluation is based on quantitative light-induced fluoroscopy (QLF) and digital photography (DP) of decalcified teeth. STAPLE is computed on 150 image pairs from 32 subjects, each showing one corresponding tooth in both modalities. The ROI in each image is manually marked by three experts (900 curves in total). In the QLF-DP setting, our approach significantly outperforms the mutual information-based registration algorithm implemented with the Insight Segmentation and Registration Toolkit (ITK) and Elastix.'
author:
- Rosalia Tatano
- Benjamin Berkels
- 'Thomas M. Deserno'
bibliography:
- 'paperRevision.bib'
title: 'Mesh-to-raster region-of-interest-based non-rigid registration of multi-modal images'
---
[**\***Rosalia Tatano, ]{}
Introduction {#sect:intro}
============
Two-dimensional (2D) as well as three-dimensional (3D) images play a crucial role in diagnosis, treatment planning, treatment, and the assessment of progression and/or regression of a condition or disease in a patient. In this context, comparisons of subject data acquired using different imaging modalities or of subject and model data are often necessary. Hence, multi-modal image registration methods aim at aligning images (2D) or volumes (3D) acquired with different devices, thus integrating the information provided by this data. The goal is to find the optimal transform that best aligns structures in two input 2D or 3D images [@Keszei16].
Mutual information (MI) has been widely used in multi-modal image registration [@viola1997; @pluim2003]. The idea is to maximize iteratively the MI between the two images globally with respect to the transformation. This is equivalent to minimizing the joint entropy of the two data sets, which occurs when the two images or the two volumes are correctly registered.
However, in several cases, there is a particular region of interest (ROI) predefined in the medical recording, such as a tumorous region, a lesion, or some other pathologies. Therefore, accurate ROI alignment is of primary importance. Moreover, in some applications, the ROI might change, e.g., due to pathology or tumor growth. Using MI to register these ROIs might lead to inaccuracies, since MI is computed on the entire image disregarding the corresponding and overlapping parts of the images and hence, is sensitive to the ROI’s size and content [@pluim2003].
Therefore, ROI-based registration has been addressed in several works already. Wilkie et al.[@wilkie2005], proposed a modification of MI registration that takes into account information from the ROIs, using a weighted combination of the statistics of the entire images and of the ROIs. However, the registration is affected by the value of the weighting parameter, which is difficult to determine, and the ROIs probability distributions. Yavariabdi et al. presented a registration method for magnetic resonance imaging (MRI) and transvaginal ultrasound (TUS) that matches manually marked contours of the ROIs in the two modalities through a one-step deformable iterative closest point (ICP) method[@yavariabdi2013]. Gu et al.[@gu2013] have proposed a contour-guided deformable image registration (DIR) algorithm for adaptive radiotherapy that deforms images using a demons registration algorithm with an additional regularization term based on modified image intensities inside manually marked ROIs. These modified images differ from the original image only if the intensities of the ROIs differs from the surrounding. Finite element model (FEM) -based deformable registration has been employed in Penjweini et al.[@Penjweini2016] to match the contours of the ROIs in a series of computed tomography (CT) scans of the lungs acquired pre-operatively with intra-operative images acquired using an infrared camera-based navigation system during the surgery stage of pleural photodynamic therapy. Zhong et al. [@zhong2016] proposed a method for a dental implant guide that uses ICP on teeth contours point sets extracted from CT scans and from 3D patient model’s cross sections to retrieve the rigid transformation between two images. A registration method for dynamic contrast-enhanced MRI (DCE-MRI) images of the liver is proposed by Zhang et al.,[@zhang2016] but the energy to be minimized is evaluated just on the segmented liver, disregarding the other image areas completely.
Whenever considering a ROI, automatic segmentation of such is important. Obviously, the accuracy of the registration depends on the accuracy of segmentation. ROI segmentation is frequently based on the pixel or contour levels, and several approaches have been proposed in the literature. By way of example, the following approaches are taken from dentistry, where digital photography (DP) and quantitative light-induced fluoroscopy (QLF) are common 2D imaging modalities:
- [ manual draw]{}: Hope et al. [@hope2014] manually draw the contour around the boundary of the tooth;
- [ gray scale]{}: A threshold technique based on the intensities of the tooth region and of the background is used in Yan et al. [@yan2011] to determine the tooth region in fluorescence and white light images. The underlying assumption is that the tooth region has higher intensities than the background;
- [ color]{}: A color based segmentation technique combined with morphological operations is used in Datta et al. [@datta2015] to segment a tooth from gum, lips and neighbors teeth in RGB images;
- [ statistical model]{}: In Mansoor et al. [@mansoor2015], an initial segmentation of the tooth was achieved using a Gauss-Markov random field statistical model and then refined by the practitioner;
- [ active contour]{}: In Shah et al. [@shah2006], the contour of teeth from dental images is extracted using an active contour algorithm that depends on the intensity of the tooth region. However, the algorithm is sensitive to poor contrast in image intensities and the presence of neighboring teeth;
Another issue in registration is the multi-modality. Usually, multi-modal means that data from the same subject is taken with different imaging modalities, such as, for example, computed tomography (CT) and magnetic resonance imaging (MRI). Furthermore, atlas registration is required frequently in modern medicine to transfer knowledge coded in a general model (i.e., the atlas) to a specific subject (e.g., the patient in a diagnostic or therapeutic process). Such comparison of virtual physiological human (VPH) models[@viceconti2016] with subject-specific scan data bears another challenge for registration: the VPH models are usually in triangulated mesh-based coding, while patient measurements are obtained with computed axial tomography (CAT) scanners and stored as pixel or voxel data [@oliveira2016]. This yields curve-to-pixel and surface-to-voxel registration problems in 2D and 3D, respectively, disregarding whether registration is considered as global or local (i.e., ROI-based) problem.
In our previous work, a curve-to-pixel-based registration method has been presented [@berkelsSPIE2016; @berkelsBVM2016] and used to align the ROIs of 2D images acquired with QLF and DP. The ROIs are segmented using a color space transform into grayscale, which were adapted to the imaging modality, and thresholded. Registration is based on aligning the ROI’s contours, i.e., the tooth region. Our curve-to-pixel method allows superimposing DP with QLF and thus, a direct comparison of the detected demineralized areas, an undesirable side effect of orthodontic treatment with fixed appliances [@Gorelick82]. To the best of our knowledge, registration of QLF and DP for direct comparison of these two image modalities had not been explicitly addressed before.
In our study on the comparison of the demineralized areas in DP and QLF images of a tooth, the demineralized area on the tooth surface often is more evident in one modality than in the other, or is even indistinguishable in one of the modalities. Thus, the correlation between the demineralized areas shown in the two modalities needs to be investigated. Relying on ROI-features can induce a correlation bias in the registration results and therefore, using a registration method that does not use this information is desirable. In addition, while it is relatively easy to segment the tooth region in QLF images, this segmentation is more challenging in the photo due to the poor contrast between tooth and gum colors and the lack of separation of neighboring teeth.
Motivated by the setting outlined above, the aim of the present work is to provide a general registration methodology that is applicable in 2D and 3D, copes with different imaging modalities as well as types of data representations (mesh and raster-scan), and handles global and local (ROI-based) problems. The key features of the proposed method, whose combination sets it apart from the existing methods, are as follows:
- a contour-to-image approach: the proposed method is able to directly link data types of different dimension, e.g. it can align a surface to a volumetric image. Most existing methods can only align data of the same dimension and type, e.g., surface-to-surface or image-to-image;
- only ROI shape required: previous works rely on information about the interior of the entire image or the ROI (e.g., internal features, intensities, statistics...). The proposed method does not make any assumption on the features inside the ROIs. This is particularly suitable for studies where the correlation of the modalities inside the ROIs is unknown and supposed to be determined. In fact, the use of feature-based registration algorithms is bound to induce a bias towards the correlation assumed by the registration;
- no strict requirement on the ROI segmentation in the reference image: unlike previous works, the proposed method only requires an accurate contour of the ROI in the template. The classification of the reference image ROI only needs to provide sufficient information about the ROI’s shape (see below for an example);
- topology preserving ROI segmentation: the method can also be used to segment the ROI in the reference image. In this case, the contour of the template image serves as initial guess for the segmentation of the ROI in the other image modality. This is helpful in case the reference image modality is difficult to segment but the topology of the segment is known;
- nonlinear least-square problem: the proposed method leads to a nonlinear least-square problem, which can be solved efficiently using the Gauss-Newton algorithm. Many multi-modal registration approached like Mutual Information lead to much more involved optimization problems with fewer (or no) guarantees on optimality of minimization algorithms.
Due to the properties listed above, the proposed method is particularly suited for the QLF/DP registration problem the method was originally designed for. Indeed, the proposed method allows to align the tooth areas in the two modalities without relying on ROI-features (cf. ii). Due to the challenges posed by the photo classification into tooth area and background, neighboring teeth are still present. However, due to the fact that the bottom and the upper part the tooth region is clearly delineated, the proposed method is able to extract the contour of a target tooth (cf. iii), disregarding the neighboring teeth on the side of it, and align the corresponding tooth region to the one shown in the QLF, since the algorithm preserves the ROI topology (cf. i, iv). Nevertheless, the method can be applied to a much larger class of registration problems.
Furthermore, we aim to comprehensively and reliably evaluate the general registration approach using sufficiently large and reliable data sets.
Materials and methods
=====================
In this section, we present a general mesh-to-raster (M2R) registration framework, its application, and the methodology used for quantitative evaluation.
M2R registration method
-----------------------
The M2R method is constructed in the continuous setting as minimization problem and then discretized. After the discretization, we propose a numerical minimization strategy. Moreover, a parametric registration is performed to provide a first alignment of the two input images.
### Continuous approach {#subsec:MatMed}
Let us assume that we are given two data sets, named $f$ and $g$, of the same anatomical structure acquired with different modalities and that the image $f$ is given as mapping $f:\Omega\to\mathbb{R}^n$, $n\in\{2,3\}$, on the unit cube $\Omega=[0,1]^n$. Let $\mathcal{C}$ indicate a hypersurface, i.e. a curve for $n=2$ and a surface for $n=3$, representing the contour of the region of interest (ROI), which needs to be extracted from $g$, in case $g$ is also an image or may be identical to $g$, if $g$ was directly acquired as hypersurface, for instance, by a laser scan.
We denote the region of interest in the image $f$ by $S_f\subset\Omega$ and now want to find a non-rigid deformation $\phi:\Omega\to\mathbb{R}^n$ that matches $\mathcal{C}$ to the boundary of the set $S_f\subset\Omega$. To this end, let $d$ be the signed distance function of $S_f$, i.e. $d(c,S_f)=\pm \text{dist}(c,\partial S_f)$. Thus, $d$ is the Euclidean distance of the point $c$ to the boundary of $S_f$, where the sign is positive if $c$ is outside of $S_f$ and negative otherwise [@bauer]. Then, the desired alignment of $S_f$ and $\mathcal{C}$ is attained by minimizing the energy $$E[\phi] = E_\text{match}[\phi] + E_\text{reg}[\phi] = \frac{1}{2}\int_{\mathcal{C}}w_c\left(d(\phi(c), S_f)\right)^2 \mathrm{d}\mathcal{H}^{n-1}(c)+ \frac{\lambda^2}{2}\int_\Omega \Vert \Delta \phi(x)\Vert^2\mathrm{d}x,\label{eq:En}$$ where $\Delta \phi = (\Delta\phi_1, \dots, \Delta\phi_n)$ is the vector of the Laplacian of the components of $\phi$ and $\mathrm{d}\mathcal{H}^{n-1}$ denotes the $n-1$ dimensional Hausdorff measure. Thus, the first term is a hypersurface integral. In particular, it is a curve integral in case $n=2$ and a surface integral in case $n=3$.
The energy measures the distance of the deformed hypersurface to the boundary of the ROI in the image $f$ and the smoothness of the deformation. The parameter $\lambda>0$ controls the smoothness of $\phi$. Here, $w_c>0$ are application dependent weights defined to control the influence of the hypersurface points. Since the data term is using the integral over the hypersurface, it involves only the deformation of $\mathcal{C}$. However, the use of a higher order regularizer extends the non-rigid deformation to the whole domain $\Omega$.
In contrast to our previous implementation [@berkelsSPIE2016; @berkelsBVM2016], here the data term is formulated as the integral over the hypersurface. Previously, the data term was defined using the sum over the hypersurface points of the weighted signed distance function calculated at the deformed hypersurface points. However, this approach might lead to problems if the hypersurface points are not approximatively equidistant. Using the integral instead, the distance of a hypersurface point with respect to its neighbors is taken into account.
### Discretization
The deformation $\phi$ is expressed as displacement $u:\Omega\to\mathbb{R}^n$ via $\phi(x)=x+u(x)$, for $x\in\Omega$, noting that $\Delta\phi=\Delta u$. For the spatial discretization of $u$, we use multilinear Finite Elements on a uniform rectangular grid on the image domain $\Omega$ [@braess]. Let $\{\psi_j\}_{j\in J}$ denote the FE basis functions, with nodal index set $J$. Let $M$ and $L$ denote the lumped mass matrix and the stiffness matrix respectively, i.e. $M_{i,j}=\int_{\Omega}\mathcal{I}(\psi_i\psi_j)\mathrm{d}x\text{ and }L_{i,j}=\int_\Omega \nabla\psi_i\nabla\psi_j\mathrm{d}x,$ where $\mathcal{I}$ is the bilinear Lagrangian interpolation. Although the chosen regularizer involves second derivatives, it can be approximated as $E_\text{reg}[u]=\frac{\lambda^2}{2}\sum_{i=1}^n\Vert M^{\frac{1}{2}}LU_i\Vert^2$ [@berkels], using the lumped mass and stiffness matrices. Here, $U_i$ denotes the vector of nodal values that uniquely represents the scalar FE function $u_i$.
### Minimization {#sec:GenMinimization}
In contrast to our previous implementation [@berkelsBVM2016], the data term now is evaluated using simplicial finite elements on the hypersurface using a quadrature rule for the numerical evaluation of the integral. The minimization of $E$ is formulated as nonlinear least squares problem, $E[u]=\frac{1}{2}\Vert F[u] \Vert^2$, where $$\label{eq:VecFGeneral}F[u]= \left[\left\{\sqrt{w_c}\sqrt{m_k^q}d(x_k^q+u(x_k^q), S_f)\right\}_{\substack{k=1,\dots,N\\q=1,\dots,Q}}, \lambda M^{-\frac{1}{2}}LU_1, \dots,\lambda M^{-\frac{1}{2}}LU_n\right]^T.$$ Here, $m_k^q$ are the weights corresponding to the $q$-th quadrature point $x_k^q$ in the $k$-th simplex describing the hypersurface. $N$ and $Q$ denote the total number of simplices and of quadrature points in a simplex, respectively. The minimization is efficiently solved using the Gauss-Newton method [@Gratton07p106]. To avoid the minimization from getting stuck in local minima, the non-rigid registration problem is solved for decreasing values of the parameter $\lambda$. For the sake of simplicity, we use this strategy to avoid a a multi level approach, which would also require to create a multi level representation of the unstructured simplicial grid of the hypersurface.
Parametric registration algorithm
---------------------------------
To provide a reasonable initial guess for the Gauss-Newton algorithm above, a regularized parametric registration [@chum] is performed. The aim is to find an affine deformation $\varphi( c) = Ac+t $, where $A$ is a $n\times n$ matrix and $t$ a translation vector, that minimizes the energy $E[\varphi] = E_\text{match}[\varphi] + E_\text{reg}^\text{par}[\varphi]$, with $$\begin{aligned} E_\text{match}[\varphi] ={}& \frac{1}{2}\int_\mathcal{C}w_c\left(d(A c+t, S_f)\right)^2\mathrm{d}\mathcal{H}^{n-1}(c),\\ E_\text{reg}^\text{par}[\varphi]:={}&\frac{\alpha^2}{2}\int_\Omega \Vert J(\varphi(x)-x)\Vert_F^2\;\mathrm{d}x+ \frac{\mu^2}{2}\int_\Omega\sum_{i=1}^n \left\Vert I-\frac{1}{\partial_{x_i}(\varphi_i(x)-x_i)}D_{J(\varphi(x))}\right\Vert_F^2\mathrm{d}x\\={}& \frac{\alpha^2}{2}\sum_{i=1}^n\int_\Omega \Vert \partial_{x_i}((A-I)x)\Vert^2\mathrm{d}x + \frac{\mu^2}{2}\sum_{i=1}^n \left\Vert I-\frac{1}{a_{ii}}D_A\right\Vert_F^2, \end{aligned}\label{eq:regPar}$$ where $J(\varphi(x))$ indicates the Jacobian of $\varphi$ with respect to $x$ and $\varphi_i(x)$ represents the $i$-th component of the vector $\varphi(x)$. Here, the data term is the same as the non-rigid model. In contrast to our previous formulation [@berkelsBVM2016], here as prior for the deformation the sum of two terms is used, with positive scalars $\alpha$ and $\mu$ that weight the contribution of these terms to the value of the energy. The first term in the regularizer is the Dirichlet energy of the displacement. Noting that $J(\varphi(x))=A$, the second term is the squared Frobenius norm of the matrix $I-\frac{1}{a_{ii}}D_A$, where $I$ is the identity matrix and $D_A$ is a diagonal matrix whose entries are the diagonal entries of the matrix $A$. Using this term, the difference of all the possible ratios of the diagonal entries of $A$ from 1 is penalized, thus isotropic scalings are preferred to anisotropic ones.
Similarly to the non-rigid case, the parametric registration is formulated as a least squares problem by defining the vector $F[\varphi]$ and solved using Gauss-Newton.
Applications
------------
In this section, we demonstrate how to apply the method for solving registration problems in 2D and 3D.
### 2D Example {#sec:2D}
The selected 2D application aims at registering the tooth as ROI in DP and QLF for demineralization assessment. In contrast to the state of the art in ROI segmentation, we apply the proposed curve-to-image registration method for both the extraction of the tooth contour from the QLF (ROI segmentation) and for its alignment to the tooth region shown in the DP (Fig. \[fig:Method\]).
- [*Segmentation*]{}: In the QLF segmentation step, the tooth contour is represented as a small circle in the center of the image, since a tooth is topologically equivalent to a circle. The size of the circle was chosen to ensure that its entire boundary is within the tooth. Then, the registration algorithm aligns the circle to the boundary of the tooth region shown in the QLF, thus, yielding the shape of the tooth[@berkelsBVM2016]. For the alignment of DP and QLF, the proposed algorithm is applied to the extracted QLF tooth contour and the tooth region shown in the DP.
- [*Registration*]{}: We assume that the curve is discretized as set of line segments between points $\{c_i\}_{i=1}^N$, where $N$ is the number of points. The data term is discretized using the midpoint rule over each line segment $[c_i,c_{i+1}]$ as $$E_\text{match}[\phi]= \frac{1}{2}\int_{\mathcal{C}}w_c\left(d(\phi(c), S_f)\right)^2 dc \approx \sum_{i=1}^Nw_{c_i}l_{c_i}\left(d\left(\phi\left(c_{i+\frac{1}{2}}\right), S_f\right)\right)^2$$ where $c_{i+\frac{1}{2}}$ indicates the midpoint of the line segment with endpoints $c_i$ and $c_{i+1}$ and $l_{c_i}$ its the length, i.e. $l_{c_i}=\Vert c_{i+1}-c_i\Vert$. The weights $w_{c_i}$ of the data term in Eq. (\[eq:En\]) are used to get a proper alignment of the curve points on the vertical boundary of the tooth. In fact, these points may have no counterpart in the boundary of $S_f$, since often the thresholded images do not exhibit a clear separation between a tooth and its neighbors. Thus, the weights $w_{c_i}$ are defined using the vector $\vec{v}_{c_i}=\frac{1}{2}(c_{i+1}-c_i)$ that characterizes the orientation of the curve. The bigger the absolute value of the $x$-component of $\frac{\vec{v}_{c_i}}{|\vec{v}_{c_i}|}$, the less vertical is $C$ at $c_i$. Hence, $w_{c_i}$ it set to this value.
Thus, the vector $F$ that encodes the non-linear least squares problem in the 2D setting is $$F[u]= \left[\left\{\sqrt{w_{c_i}l_{c_i}}d\left(c_{i+\frac{1}{2}}+u\left(c_{i+\frac{1}{2}}\right), S_f\right)\right\}_{i=1,\dots,N}, \lambda M^{-\frac{1}{2}}LU_1,\lambda M^{-\frac{1}{2}}LU_2\right]^T.$$ The vector $F$ for the parametric registration step is defined similarly (see Appendix \[app-1\]).
At each iteration of the Gauss-Newton algorithm, the resulting linear system is solved using a sparse Cholesky factorization of the matrix $J_F^TJ_F$, where $J_F$ denotes the Jacobian of $F$. To this end, we use CHOLMOD from the SuiteSparse[@cholmod2008] library.
### 3D Example {#sec:3D}
The 3D use-case is taken from the Regional Anaesthesia Simulator and Assistance (RASimAs) project [@rasimas], where subject-specific MRI need to be registered to a VPH model composed of mesh-based surfaces for skin, fascia, muscle, bone, vessels, and nerves.
Hence, we assume that $f$ is obtained from a volumetric scan, the surface $\mathcal{C}$ represents the ROI contour of $g$, and that $g$ is given as a triangle mesh. Then, the data term is discretized using linear FE on the triangle mesh as $$E_\text{match}[\phi]= \frac{1}{2}\int_{\mathcal{C}}w_c\left(d(\phi(c), S_f)\right)^2 dc \approx \sum_{T\in \mathcal{T}}m_T^q\left(d(\phi(x_T^q), S_f)\right)^2 ,$$ where the weights $w_c$ are chosen to be equal to 1, $T$ is a triangle in the set of triangles $\mathcal{T}$ defining the triangle mesh of the ROI, $x_T^q$ is the barycenter of the triangle $T$ and $m_T^q$ is the area of $T$. Thus, the vector $F$ that encodes the non-linear least squares problem in the 3D setting is $$F[u]= \left[\left\{\sqrt{m_T^q}d(x_T^q+u(x_T^q), S_f)\right\}_{T\in\mathcal{T}}, \lambda M^{-\frac{1}{2}}LU_1,\lambda M^{-\frac{1}{2}}LU_2, \lambda M^{-\frac{1}{2}}LU_3\right]^T.$$
The definition of the vector $F$ for the parametric registration is similar to the non-rigid case. The details are presented in Appendix \[app-2\].
As in the 2D case, Gauss-Newton requires the solution of a linear system in each iteration of the algorithm. Unlike in the 2D case, it is not feasible though to assemble the system matrix $J_F^TJ_F$ due to memory requirements. $J_F$ does not only have more rows and columns, but also considerably more non-zero entries in each row. Thus, it is crucial not to assemble this product matrix. Instead, we solve the linear system using the LSMR[@lsmr2011] algorithm, where it is sufficient to compute and store the matrix $J_F$ and its transposed.
Figure \[fig:results3Dcolored\] illustrates the effect of the minimization strategy proposed in Section \[sec:GenMinimization\]. First the parametric registration is performed with the empirically determined parameters $\alpha=1$ and $\mu=1$. Estimating the parameters is rather straightforward since their value mainly depends on the order of magnitude of the initial energy. Then, the non-rigid registration is performed iteratively for decreasing values of the parameter $\lambda$. In this case, the chosen values of the parameters were $\lambda=10^{-i},\;i=\{0,1,2,3,4\}$. This allows to get more accurate registration results as $\lambda$ decreases. For the different settings of $\lambda$, Figure \[fig:results3Dcolored\] visualizes the distances $d(\phi(c), S_f)$, for $c\in\mathcal{C}$, on the registered template mesh, $\phi(\mathcal{C}):=\{\phi(c):c\in\mathcal{C}\}$, for human hips.
Evaluation {#sec:Eval}
----------
Reliable valuation of non-rigid registration is difficult, since large databases with reliable ground truth (GT) annotations are not available. Nonetheless, we aim at quantitatively evaluating our approach.
### Aim
Our evaluation aims at determining the absolute error (accuracy) of the method. To define accuracy, we need to rely on a ROI-based registration problem. This requires automatic segmentation, and the accuracy of the segmentation deeply impacts the overall performance. In addition, we aim at comparing the results to a state-of-the-art method for multi-modal registration, which is considered to be based on mutual information (MI)[@oliveira2014].
Metrics
-------
The alignment accuracy is assessed by the Dice Coefficient[@dice] (DC) and the symmetric Hausdorff distance[@huttenlocher1993] (HD), which quantify the agreement of two segmentation and the accuracy of the contour alignment, respectively. The larger the DC and the smaller the HD the better the two ROIs correspond.
### Database
Quantitative measures on ROI alignment are image specific. In order to obtain statistically significant results, a sufficiently large number of images shall be processed. Hence, large databases of images annotated with reliable GT are needed. Therefore, we selected the 2D application of QLF / DP registration, where the ROI is defined as tooth contour. In total, 150 pairs of QLF and DP of upper and lower incisors and canines have been acquired from 32 subjects. All subjects were exhibiting white spot lesions after orthodontic treatment with a fixed appliance.
### Ground truth
Manually references are unreliable, since they cannot be reproduced exactly, even with the same rater[@SPIE-gold-silver]. However, based on several manual markings, a gold standard can be estimated using the simultaneous truth and performance level estimation (STAPLE) algorithm[@staple2004]. The core idea of STAPLE is to iteratively (i) compute the observer-weighted mean of a binary segmentation and (ii) adjust the weights of the observers with respect to the similarity to that mean segmentation. In other words, if a observer has large discrepancies to the estimated GT, the corresponding weights are lowered in the next iteration.
To generate the ground truth with STAPLE, three trained raters (R1, R2, R3) manually marked the tooth contours on both of the image modalities. All the raters were presented the 300 images in random order.
### Assessment of segmentation
The accuracy of the automatic ROI segmentation is assessed by calculating the DCs and HDs of the automatic segmentation and the ground truth estimated using the STAPLE algorithm. Including the automatic segmentation, in total four observers are available. The performance of the raters is compared for both modalities, QLF and DP. The contour extraction from the QLF images is done with the parameters $\alpha=10^{-3}$, $\mu=10^{-3}$ and $\lambda=10^{-i},\;i=\{4,5,6,7\}$ for the parametric registration step and the non-rigid step, respectively, while for the DP, $\alpha=10^{-3}$, $\mu=10^{-1}$ and $\lambda=10^{-i},\;i=\{4,5,6\}$ were used.
A repeated measures analysis of variance (ANOVA) is applied to assess statistical significances between the automatic segmentation and the human raters, and between the performance of the algorithm in QLF and DP modalities. The significance level is $\alpha = 0.05$.
### Assessment of registration
The accuracy of the DP/QLF ROI alignment is measured again by DCs and HDs of the photo ground truth deformed using the deformation fields obtained from the proposed registration method and the QLF ground truth. For the DP/QLF alignment, $\alpha=10^{-3}$, $\mu=10^{-1}$ and $\lambda=10^{-i},\;i=\{4,5,6\}$ were used.
To compare our approach with the state-of-the-art, the same analysis is done using the deformation fields obtained from the Insight Segmentation and Registration Toolkit (ITK)[@johnson2015itk] MI registration and those obtained from the MI registration implemented with Elastix[@elastix2010], which is an established registration method.[@monti2017; @broggi2017; @malinsky2013] In both cases, the registration is performed on the same grayscale version of the DP/QLF pairs, which are used for the proposed automatic segmentation. The parameters used for the registrations are specified in Table \[tab:paramMI\]. The listed parameters are the default values suggested by the original authors, except for the “Final Grid spacing" parameter for Elastix. The latter has been increased from its default value of 16 pixels to 48 pixels, as using the default value was resulting in obviously unrealistic transformations. In the ITK implementation, the MI is optimized using the Limited-memory Broyden Fletcher Goldfarb Shannon[@lbfgsb] (L-BFGS-B) algorithm.
Based on DC and HD, a one-way repeated measures ANOVA is used to compare the proposed method with the mutual information-based ITK implementation that is considered as gold standard.
------------------- ------------------------------------------------------------------------ --
[**Parameter**]{} **[ITK]{}&**[Elastix]{}\
Metric & Mattes MI & Advanced Mattes MI\
Number of histogram bin & 50 & 32\
Transformation & Third-order B-spline & Affine + Third-order B-spline\
Final Grid spacing & \# pixels in the input image & 48 pixels\
Optimization algorithm & L-BFGS-B & Adaptive stochastic gradient\
& & descent\
Maximum number of iterations & 1000 & 200 (Affine) - 500 (B-spline)\
Number of multiresolution levels & - & 4\
****
------------------- ------------------------------------------------------------------------ --
: \[tab:paramMI\] Parameters used for ITK and Elastix. Here, MI indicates mutual information and L-BFGS-B refers to Limited-memory Broyden Fletcher Goldfarb Shannon algorithm.
Results {#sec:Res}
=======
In this section, we present the results of qualitative registration in 2D and 3D, as well as the accuracy determined in the 2D evaluation for segmentation and registration.
2D application
--------------
Figure \[fig:figRes\] shows qualitative results obtained in the exemplary 2D application. After non-rigid deformation, the DP matches the QLF, as the QLF-based contour (red) matches the tooth in the DP, as depicted in Panel (d).
3D application {#subsec:3DMethod}
--------------
The M2R registration method has been applied in 3D, too. For the use case from the RASimAs project, the resolution of the corresponding 3D grid is $129\times 129\times129$, with pixel dimensions (3.73, 3.73, 3.73) mm. Figure \[fig:results3D\] depicts the results and suggests an appropriate surface-to-voxel alignment. The Dice coefficient of the two meshes after the alignment is 0.9959. As measure of the distance between the contours, the Hausdorff distance is calculated. The Hausdorff distance of the reference mesh from the template mesh is 90.60 mm (24.28 pixel units) before performing the registration and 15.73 mm (4.22 pixel units) after. The 99th percentile of the Hausdorff distance of the reference from the template is 78.67 mm (21.14 pixel units) before and 2.12 mm (0.57 pixel units) after the registration. Note that the 99th percentile after the registration is much lower than the Hausdorff distance after the registration. This indicates the presence of just very few outlier points (Fig. \[fig:results3Dcolored\], blue/red dots in the last row) that make the Hausdorff distance appear relatively large even though the alignment is very accurate. Furthermore, note that we are not considering the symmetric Hausdorff distance here since one surface only contains part of the other surface. The proposed method, in fact, has the advantage that it can cope with non-symmetric relations between the two input meshes.
Quantitative evaluation of segmentation performance
---------------------------------------------------
Based on the 150 QLF images, the mean of the DC between the segments obtained from the automatic procedure and the STAPLE-based GT that is calculated from the three human observers is 0.978 (range 0.9420-0.9940). The mean HD is 0.032 (range 0.0102-0.1060).
Based on the 150 DP images, these values turn to 0.981 (range 0.9440-0.9940) and 0.030 (range 0.0102-0.0792) for DC and HD, respectively (Table \[tab:compGTQLFDP\]).
For both QLF and DP, a pairwise comparison of the means of the DC as well as of the HD using a one-way repeated measure ANOVA was performed between the automatic segmentation and each rater, determining a statistical significant difference between these means (Table \[tab:ANOVA\]).
The one-way repeated measures ANOVA revealed a statistical significant difference between the means of the DC ($F(1,149)=8.68$, $p=0.0037$) between DP and QLF modalities, but no significant difference between the means of the HD ($F(1,149)=1.86$, $p=0.1742$).
[**Modality**]{}
------------------ --------------- -------- ------- -------- --------------- -------- ------- --------
[**Metrics**]{}
Mean SD Mean SD Mean SD Mean SD
[**Method**]{}
Rater R1 0.990 0.0098 0.015 0.0117 0.984 0.0076 0.021 0.0087
Rater R2 0.980 0.0068 0.023 0.0067 0.991 0.0066 0.015 0.0094
Rater R3 [**0.992**]{} 0.0082 0.012 0.0116 [**0.996**]{} 0.0049 0.008 0.0079
M2R 0.978 0.0101 0.032 0.0185 0.981 0.0083 0.030 0.0129
: \[tab:compGTQLFDP\] Mean and standard deviation (SD) of Dice coefficient (DC) and symmetric Hausdorff distance (HD) of the ground truth and manual markings and of ground truth and the proposed automatic segmentation (M2R) for QLF and DP.
Quantitative evaluation of registration performance
---------------------------------------------------
The descriptive statistics (mean and standard deviation) of DCs and HDs measuring the accuracy of the alignment for the ITK-MI and Elastix state-of-the-art methods and our mesh-to-raster (M2R) approach are reported in Table \[tab:compGTdef\]. Based on the 150 QLF/DP image pairs, the means of the DC are 0.940 (range 0.7600-0.9900), 0.959 (range 0.874-0.991) and 0.971 (range 0.8910-0.9930) for ITK-MI, Elastix and M2R, respectively. The means of the HD are 0.072 (range 0.0181-0.2630), 0.055 (range 0.014-0.153) and 0.041 (range 0.0102-0.1050), respectively.
The one-way repeated measures ANOVA revealed a statistical significant difference between the means of the DC ($F(1,149)=71.96$, $p<0.001$) as well as the means of the HD ($F(1,149)=61.99$, $p<0.001$) for the registration with ITK and the proposed method. Similar results were given by ANOVA for the means of the DC ($F(1,149)= 62.96$, $p<0.001$) and for the means of the HD ($F(1,149)=38.14$, $p<0.001$) for M2R and Elastix. The statistical significance is emphasized by the box plots in Figure \[fig:BoxplotQLFVI\], which are visualizing the data from Table \[tab:compGTdef\]. Both the DCs and the HDs exhibit more variability in the case of ITK-MI and Elastix rather than for our method.
----------------- --------------- -------- --------------- --------
[**Metrics**]{}
Mean SD Mean SD
[**Method**]{}
**M2R** [**0.971**]{} 0.0129 [**0.041**]{} 0.0180
ITK-MI 0.940 0.0471 0.072 0.0494
Elastix 0.959 0.0219 0.055 0.0279
----------------- --------------- -------- --------------- --------
: \[tab:compGTdef\] Mean and standard deviation (SD) of Dice coefficient (DC) and symmetric Hausdorff distance (HD) of the deformed DP ground truth and the QLF ground truth for the proposed mesh-to-raster (M2R), the ITK-MI and Elastix reference methods.
![\[fig:BoxplotQLFVI\] Boxplots of the Dice coefficients and the symmetric Hausdorff distance of the QLF ground truth and the DP ground truth deformed using the proposed algorithm (M2R), the ITK mutual information (ITK-MI) and the registration implemented with Elastix.](Fig5.pdf){height="5cm"}
Discussion
==========
The alignment of ROIs in multi-modal medical images is important for many applications. Here, a mesh-to-raster (M2R)-based method is described, which aligns the contour of the ROI of the reference image to the target image with a deformation field that is determined on the whole image domain. Although this paper provides examples in 2D and 3D only, the method is applicable in general in any dimension.
The quantitative evaluation is based on multi-modal 2D data from dentistry. Our methods was applied to QLF and DP images, including an evaluation of the automatic segmentation that is needed to determine the tooth region, which acts as ROI.
With respect to the evaluation of the ROI segmentation, our M2R approach is not as accurate as a human observer (a significant difference among the means of each human rater and the automatic segmentation was determined by ANOVA). In absolute numbers for QLF, the distance in terms of mean DC between M2R ($ {\rm DC} = 0.978$) and the best-performing human observer R3 ($ {\rm DC} = 0.992$) is only about $1.5 \%$, while the distance between the M2R and the worst-performing human observer R2 is negligible (about $0.2\%$). Similar results hold for the DP. Also, the mean of the HD for the automatic segmentation is slightly bigger than those of the manual markings.
The observed deviation from the GT in a QLF is due to the extended and smooth transition zone between tooth and background, leading to an imprecise classification of the image into tooth and non-tooth regions and thus to a less accurate contour extraction than in cases where the images exhibit clear distinction between tooth and background. Similarly for the DP images, small inaccuracies in tooth and non-tooth regions classification are caused mostly by the low contrast between the colors of tooth and gum or adjacent teeth. However, the segmentation step may be replaced easily by another algorithm.
Contour-based registration methods have been used in several works for ROI extraction and alignment. Chen et al.[@chen2005] used an automatic method to extract and align teeth contours to register dental radiographs. A deformable image registration method that used ROI’s contour propagation was proposed by Wu et al.[@wu2015] in radiotherapy. Here, the proposed method is used for both ROI extraction and matching. The method is correctly aligning the contour of the ROI in the target image to the boundary of the ROI in the reference image. However, if the boundary of ROI in the reference image is not accurately delineated, it might lead to inaccuracies in the registration step, as already pointed out in the case of QLF and DP segmentation.
The best and worst DCs and HDs for our method, the ITK-MI registration and the MI-based registration implemented with Elastix are depicted in Fig. \[fig:CIR\_MI\]. The MI-registration implemented with Elastix yields better results than the ITK-MI registration. However, in some cases, the registration implemented with Elastix returned unrealistic deformations, as shown for example in the last row of Figure \[fig:CIR\_MI\]. These results clearly illustrate the improved accuracy of the proposed method over both the ITK-MI and Elastix that was shown quantitatively but in an abstract manner by the DC and HD means. Disregarding that we have used a non-optimal ROI extractor, we have been able to outperform both the ITK-MI registration and the MI-registration implemented with Elastix significantly. This is due to the ROI-based vs. global approach, respectively. It emphasizes the need of ROI-based registration, which is in line with Yi et al.[@yi2006].
Using an optimal ROI segmentation improves the accuracy of the registration with the proposed algorithm. In the case of the DP/QLF alignment, we performed also the registration using the segmentation of the ROIs provided from the ground truth estimated from the manual markings instead of automatically computing the segmentation. This eliminates the segmentation error that is implicitly included in the results shown above. We calculated also the Dice coefficients and the Hausdorff measure for all 150 image pairs. The mean and standard deviation of the DC for the M2R registration with optimal segmentation (ground truth estimated from manual segmentation) are 0.9913 and 0.0018 respectively. The mean and the standard deviation for the HD are 0.0126 and 0.0048. The registration results are therefore even more accurate if an optimal segmentation of the ROI (or classification of the images to be registered into ROI and background) is available.
For both 2D and 3D, the objects of interest presented in our applications are relatively simple in shape. However, the proposed method is capable of handling more complex shapes. As an example in 2D, our method was applied to images of a hand and a bone taken from the 1070-Shape Dataset of the Laboratory for Engineering Man/Machine Systems (LEMS) [@LEMS]. The registration results, with values of the parameters $\alpha=1$, $\mu=1$ and $\lambda=10^{-i}$, $i=\{2,3,4,5,6,7\}$, are given in Figure \[fig:figRes2dShapes\]. The Dice coefficient and the symmetric Hausdorff distance are 0.9819 and 0.0102 for the bone example, 0.9899 and 0.0488 for the first hand example (middle row in Figure \[fig:figRes2dShapes\]) and 0.9902 and 0.0191 for the second hand example (last row in Figure \[fig:figRes2dShapes\]).
As an example of application of the proposed method to more complicated shapes in 3D, a brain CT scan is used. The template has been obtained by applying a synthetic deformation to the reference mesh. The considered deformation was defined as $u_1(x) = -\beta x_1(x_2-\frac{1}{2})\allowbreak+\allowbreak\beta (1-x_1)(x_1-\frac{1}{2})\allowbreak+\beta (1-x_1)(x_3-\frac{1}{2})$, $u_2(x)=\beta x_1(x_2-\frac{1}{2})+\beta(1-x_1)(x_2-\frac{1}{2})-\beta(1-x_1)(x_3-\frac{1}{2})$, $u_3(x)=\allowbreak -\beta x_1(x_3-\frac{1}{2})\allowbreak +\allowbreak \beta(1-x_1)(x_3-\frac{1}{2})$, with $\beta=0.3$. After applying the defined nonlinear deformation to the reference mesh, a parametric rigid body transformation was applied to the resulting mesh. The chosen parameters were $t=(t_1,t_2,t_3)=(0.,0.,0.1)$ as translation vector and $\theta=(\theta_1, \theta_2,\theta_3)=(0.1,\allowbreak 0.5,-0.03)$ radians for the rotation angles. Figure \[fig:results3Dbrain\] shows the results of the proposed registration method for the brain example, using $\alpha=10^2$, $\mu=10^2$ and $\lambda=10^{-i},\;i=\{-1,0,1,2,3,4,5\}$ as parameters values. As in the hips example, the resolution of the 3D grid is $129\times 129\times129$, with pixels dimensions (1.9380, 1.9380, 1.9380) mm. Figure \[fig:results3DcoloredBrain\] displays the distances $d(\phi(c), S_f)$. The Dice coefficient after the registration is 0.9993. The two one-sided Hausdorff distances before the registration are 100.72 mm (51.97 pixel units) for the reference mesh from the template and 45.30 mm (23.38 pixel units) for the other direction. The 99th percentiles of the Hausdorff distance of the reference mesh from the template are 94.82 mm (48.93 pixel units) and 39.50 mm (20.38 pixel units) for the other direction. After the registration, the one-sided Hausdorff distance of the reference mesh from the registered template is 1.35 mm (0.0054 pixel units) and 6.27 mm (0.0251) for the other direction, while the 99th percentiles are 1 mm (0.52 pixel units) and 2.65 mm (1.37 pixel units) for the Hausdorff distance of the reference mesh from the registered template and the other direction respectively. The noticeable difference between the Hausdorff distance and its 99th percentile is due to the presence of a few outliers, displayed as colored (red or blue) dots in Figure \[fig:results3DcoloredBrain\].
A limitation of our work is that the extensive quantitative evaluation is performed only in 2D. In future, we plan to obtain extensive quantitative assessment in 3D, too. Here data from the Evaluation of Methods for Pulmonary Image Registration 2010 (EMPIRE10) or other challenges might be helpful [@murphy2011]. In this context, the proposed method could be used to perform lung registration by aligning the lung boundaries. However, to align the major fissures inside the lung, an additional data term would need to be included in the proposed method, which we plan to exploit in a future work.
Since the proposed method can also be used for segmentation, even though it is performed as a pre-processing step, it would be interesting to explore the possibility of extending the method to an integrated segmentation/registration scheme in the future.
Conclusions
===========
A ROI-based registration method for multi-modal images was presented. It uses a curve-to-pixel or surface-to-voxel approach to align the ROIs from the reference and the target images in 2D or 3D, respectively. Qualitative examples in 2D and 3D were presented. The accuracy of the alignment was tested on multi-modal 2D images, as well as in 3D examples, by calculating the Dice coefficient and the Hausdorff distance between the registered images. For the quantitative evaluation in 2D, the ground truth was established by applying the STAPLE algorithm to manually marked images. In a comprehensive analysis based on 150 pairs of images, the proposed method statistical significantly outperforms mutual information (MI)-based global registration implemented using both ITK and Elastix, which is considered as state-of-the-art method.
details of the parametric registration in 2D {#app-1}
============================================
For the parametric registration in 2D, if $A=\left(\begin{matrix}a_{11} & a_{12}\\ a_{21} & a_{22}\end{matrix} \right)$, $D_A=\left(\begin{matrix}a_{11} & 0\\ 0 & a_{22}\end{matrix} \right)$, the regularizer can be expressed as $$E_\text{reg}[\varphi] = \frac{\alpha^2}{2}\left((a_{11}-1)^2+a_{12}^2+a_{21}^2+(a_{22}-1)^2\right)+\frac{\mu^2}{2}\left(\left(1-\frac{a_{11}}{a_{22}}\right)^2+\left(1-\frac{a_{22}}{a_{11}}\right)^2\right)_.$$ Therefore, the vector $F$ for the parametric registration is $$F[\varphi]= \left[\begin{array}{c}\left\{\sqrt{w_{c_i}l_{c_i}}d\left(\varphi\left(c_{i+\frac{1}{2}}\right), S_f\right)\right\}_{i=1,\dots,N}\\ \alpha (1-a_{11})\\ \alpha a_{12}\\ \alpha a_{21}\\ \alpha (1-a_{22})\\ \mu \left(1-\frac{a_{11}}{a_{22}}\right)\\ \mu \left(1-\frac{a_{22}}{a_{11}}\right)\end{array}\right].$$
details of the parametric registration in 3D {#app-2}
============================================
For the parametric registration in 3D, if $A=\left(\begin{smallmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{smallmatrix} \right)$, $D_A=\left(\begin{smallmatrix}a_{11} & 0 &0\\ 0 & a_{22}&0\\ 0&0&a_{33}\end{smallmatrix} \right)$, the regularizer for the parametric registration, (\[eq:regPar\]), can be expressed as Therefore, the vector $F$ for the parametric registration is $$F[\varphi]= \left[\begin{array}{c}\left\{\sqrt{m_T^q}d(x_T^q+u(x_T^q), S_f)\right\}_{T\in\mathcal{T}}\\ \alpha (1-a_{11})\\ \alpha a_{12}\\ \alpha a_{13}\\ \alpha a_{21}\\ \alpha (1-a_{22})\\ \alpha a_{23}\\ \alpha a_{31}\\ \alpha a_{32}\\ \alpha (1-a_{33})\\ \mu \left(1-\frac{a_{11}}{a_{22}}\right)\\ \mu \left(1-\frac{a_{11}}{a_{33}}\right)\\ \mu \left(1-\frac{a_{22}}{a_{11}}\right)\\ \mu \left(1-\frac{a_{22}}{a_{33}}\right)\\ \mu \left(1-\frac{a_{33}}{a_{11}}\right)\\ \mu \left(1-\frac{a_{33}}{a_{22}}\right)\\ \end{array}\right].$$
details of the ANOVA for the segmentation evaluation {#app-3}
====================================================
This section presents the results ($F$ and $p-$value) of the one-way repeated measures ANOVA for the comparison of the means of the Dice coefficient (DC) as well as of the symmetric Hausdorff distance (HD) of each human rater and the automatic segmentation for both the QLF and the DP.
Informed consent {#informed-consent .unnumbered}
================
Informed consent was obtained from all individuals participating in the trial. The study was approved by the IRB no IORG0006299 (Ethics Committee, Uniklinik RWTH Aachen).
Disclosures {#disclosures .unnumbered}
===========
No conflicts of interest, financial or otherwise, are declared by the authors.
The authors would like to thank Eva E. Ehrlich and Ekaterina Sirazitdinova, Uniklinik RWTH Aachen, for their assistance in the preparation of the 2D data. The authors at AICES RWTH Aachen were funded in part by the Excellence Initiative of the German Federal and State Governments. The RASimAs project has received funding from the European Union’s Seventh Framework Programme for research, technological development and demonstration under grant agreement no 610425. The authors declare no conflict of interest and have nothing to disclose.
|
[**Effective Theory of Squeezed Correlation Functions**]{}
[ and [**Marko Simonović**]{}]{}
0.5cm
------------------------------------------------------------------------
------------------------------------------------------------------------
Introduction
============
One of the main goals of modern cosmology is to understand the origin of primordial fluctuations. We are fairly confident, given the in-phase acoustic oscillations, that the fluctuations have been super-horizon for a while. Inflation seems to be the simplest, and most widely accepted, explanation for generating the observed slightly red-tilted and almost Gaussian spectrum. However, there are many different inflationary scenarios, and there are alternatives to inflation, which we would like, and potentially be able to distinguish using data from observations in the foreseeable future. This program requires an understanding of the cosmological observables, and a classification of different models in terms of their distinctive observational signatures.
Perhaps the most decisive of these signatures whose detection rules out non-inflationary alternatives is the $B$-type polarization of the cosmic microwave background caused by primordial tensor modes. However, depending on the energy scale of inflation, the amplitude of tensor modes may be too small for the $B$-modes to be observable, and even if observed, there is still much to learn about inflation. In particular, understanding what other degrees of freedom besides the inflaton are active during inflation and what is their spectrum of masses and couplings can teach us about the underlying ultraviolet physics.
It is well known through several examples that the information about these degrees of freedom is encoded in the squeezed limit of correlation functions in momentum space, when one or a partial sum of momenta becomes “soft”, i.e. much smaller in magnitude than the others. Single-field models of inflation satisfy a set of identities relating the squeezed limit correlation functions to lower order statistics [@Maldacena; @Creminelli:2004yq]. In particular, denoting the soft momentum by $q$ and factoring out the power spectrum of the soft mode $P(q)$, the zeroth and first order terms in $q$ are fully fixed by these identities [@Creminelli_sct; @Hinterbichler]. The model dependent self-interactions (for instance, the equilateral type 3-point correlators) start to contribute starting from second order. That is, neglecting the tensorial structure, \[equil\] q\^2. On the other hand additional light degrees of freedom during inflation leave characteristically different imprints in the squeezed limit correlation functions whose scaling in the ratio of the small momentum to the large ones $q/k$ is determined by the ratio of the mass to the expansion rate during inflation ($m/H$) [@Chen; @Baumann]. The purpose of this note is to give a more unified treatment of this limit.
We take an approach similar in spirit to the effective field theory (EFT) of single-field inflation [@Cheung:2007st]. In the simplest single-field, slow-roll models of inflation the background expansion as well as the dynamics of scalar fluctuations are fully described in terms of one fundamental scalar field, the inflaton. However, there exist several models where the background is not driven by a weakly coupled fundamental field, such as DBI and $k$-inflation, but the fluctuations have a richer phenomenology (e.g. non-Gaussianity). These models are generally characterized by having a lower strong-coupling scale compared to the energy scale at which de Sitter isometries are broken [@Baumann_equil]. The advantage of the EFT of inflation is to provide a unifying framework to study the perturbations in these models by means of separating the discussion of background evolution from the fluctuations and parameterizing the former in a set of time-dependent coefficients. This allows a much more efficient and systematic treatment of the fluctuations, and in terms of parameters that are directly determined by observations.
Similarly, the simplest inflationary models with more than one degree of freedom are usually described by the addition of new fields that are weakly coupled up to an energy scale $\Lambda \gg H$. They can be included in the EFT framework as in [@Baumann; @Senatore_multi], and their coupling to the inflaton, or their subsequent conversion to adiabatic modes modifies the phenomenology of inflation. However, distinct features generally appear in the behavior of the correlation functions in the squeezed limit. Away from this limit the shapes of correlation functions are expected to be degenerate with those resulting from self-interactions in a single-field model. Hence, although the EFT of multi-field inflation allows us to calculate the full shape of non-Gaussianity, only a part of it is of special interest. We therefore look for an effective description of this part by working directly at the level of the correlation functions rather than the Lagrangian. We will argue that such description exists, it is independent of details of interactions and is parameterized in terms of quantities that are directly related to observations.
One can also imagine scenarios in which the additional degrees of freedom are strongly coupled below and around the Hubble scale, i.e. $\Lambda \sim H$. The EFT approach can fail in this case, while as we will argue, at super-horizon scales one seems to be able to always talk about weakly coupled degrees of freedom. This is a consequence of locality and the slow-roll assumption, namely that the time-variation of the parameters of the underlying theory is much slower than the expansion rate $H$. Therefore, one advantage of our approach is that it proves many of the known results about the squeezed limit behavior of inflationary correlation functions hold more universally, beyond the explicit models in which they were originally derived. In particular, the squeezed limit behavior is describable in terms of a discrete sum of power laws with either real positive or pairs of complex conjugate powers whose real parts are multiples of $3/2$. Any deviation (such as negative powers or a continuous spectrum) seems to indicate a radically different alternative to inflation.[^1]
In what follows, we first investigate the evolution of local operators at super-horizon scales in more detail, and argue that they have simple scaling behavior as a function of the conformal time $\eta$, up to slow-roll corrections. In many cases of interest they become classical and their time and space dependence factorizes: \[O\] (,x)(x) \^[\_]{}. We then show that to describe the squeezed limit behavior of the observed correlation functions, all that is needed is the spectrum of $\{\Delta_\sigma\}$, plus a set of coefficients parameterizing the degree of coupling between adiabatic fluctuations and each local operator, e.g.
[()() ]{}\_=&\_0()+() \_() +\_i()\^i\_() +\_i\_j()\^[ij]{}\_()+\
&+\^2()\_[\^2]{}()+\_i()\^[i]{}\_()+
At this order the $\xi$ functions are fixed up to numerical coefficients which are the theoretical data directly related to the observations. This applies to quasi-single field and multi-field models but also when $\zeta$ is directly coupled to a composite operator like $\sigma^2$. The contribution of an operator of dimension $\Delta_O\geq 0$ to the squeezed limit scales as q\^[\_O]{}. The special case in which the perturbations of the intermediate operator do not decay significantly by expansion ($\Delta_O \simeq 0$) deserves a more detailed discussion. These fluctuations are called entropic perturbations, which can convert to adiabatic perturbations at a later time during or after the end of inflation. The squeezed limit behavior is again simply related to a set of mixing and coupling coefficients. In the case where there is only one entropy mode, or there are several identical ones, we can determine the degree to which the observed adiabatic fluctuations are coming from the initial fluctuations of the inflaton, and derive a new consistency condition satisfied by ratios of squeezed limit correlation functions. In the more general case, unitarity imposes an inequality condition on these ratios [@Suyama; @Smith].
Exceptions to occur when there are massive particles of $m>3H/2$. This case is discussed separately since the corresponding super-horizon modes never become classical; they are described by two complex conjugate power laws and the notion of particle remains applicable. However, there is still a sense in which the correlation functions factorize in the squeezed limit which is analogous to that of scattering amplitudes on single particle poles. The squeezed limit of the bispectrum now scales as =Ø(q\^[3/2]{}), but it also oscillates as a function of the ratio and the angle between the short and long momenta. This case is particularly interesting since the squeezed limit correlation functions, which in real space describe the correlation between short distance $\sim 1/k$ measurements performed at large separation $\sim 1/q$, have quantum mechanical origin in contrast to all previous cases. The oscillations arise from interference between two branches of the wavefunction [@Noumi; @Arkani-Hamed].
Finally, we discuss the observational prospects of these squeezed limit behaviors. It is known that local non-Gaussianity (a non-Gaussianity type that can be generated by entropy modes) has a higher chance of being detected by observation of “non-local biasing” in large-scale structure surveys [@Dalal]. This is because the squeezed limit behavior of local non-Gaussianity is distinct from what can be caused by gravitational dynamics in the formation of structures, which scales as $q^2$ in the soft momentum. In particular, by going to larger scales, or equivalently sending $q\to 0$ the contribution of local non-Gaussianity rises compared to the gravitational one. A milder enhancement can arise in all of the above scenarios. However, due to the finite size of the universe the amount of signal in the long wavelength modes is limited. Through an estimate of the signal-to-noise ratio we will argue that only those non-Gaussian 3-point functions that have the squeezed limit scaling below $\O(q)$ are better detectable in this limit.
Super-horizon degrees of freedom
================================
Let us begin by considering an interacting theory of fields of non-zero mass during inflation. We denote them by $\{\sigma_a\}$, and order them with respect to their mass: $m_a\leq m_{a+1}$. As long as the time variation of coupling coefficients in the interaction Hamiltonian $\mathcal H$ is slower than the Hubble rate (the slow-roll assumption), it is easy to see that interactions become irrelevant for super-horizon modes. Hence, one can choose a basis of free fields.
Note first that if one ignores the interactions among the sub-horizon and super-horizon modes and focus only on the latter, the terms with spatial derivative are negligible. Thus the equations of motion form a system of ODEs in time. Take the equation(s) for the lightest field(s). It is of the general form \[eom1\] a\^[-3]{}\_t(a\^3 \_1)+m\_1\^2\_1=F\_1({\_a},t), where $F_1$ is at least of second order in fields or their time-derivatives and it can explicitly depend on time through time-dependent coupling coefficients. Over-dot denotes $\d_t$. To argue that $F_1$ becomes negligible outside of the horizon we need to analyze the linear solutions. Up to slow-roll suppressed corrections, the super-horizon free-field solutions in a near de Sitter space-time of expansion rate $H$ are of the form \[sigmapm\] \^(,), \_\^= , where $\eta = -1/aH$ is the conformal time and the $q$ dependence is fixed by the canonical quantization condition. It follows from the solution that for masses below $3H/2$ the mode functions divide into growing and decaying modes. Therefore, after a few Hubble times the field fluctuations are dominated by a single growing mode and the spatial and temporal dependence of the perturbations factorize, giving (,t)-\_\^[-]{} H (,t), up to corrections that decay with a negative power of $a$. It follows that the field has become classical: by locality the equal-time real space commutator $[\sigma(\x,t),\dot\sigma(\x',t)]$ is proportional to $a^{-3}\delta^3(\x-\x')$ which implies that the momentum space commutator \[commut\] \[(,t),([’]{},t)\]a\^[-3]{} \^3(+’) decays much faster than the product $\sigma({\q},t)\dot\sigma({\q'},t)\propto a^{-2\Delta_\sigma^-}$. Hence the field and its time derivative can be simultaneously known with negligible uncertainty. For heavier fields with $m>3H/2$ this conclusion does not hold.
Returning to the main problem, the slow-roll assumption about the explicit time dependence of $F_1$ together with $m_a>0$ then imply that the interactions on the right-hand side of decay by a negative power of $a$, and hence become negligible at late times. That is $\sigma_1$ becomes free.
We can now proceed to the next field, say $\sigma_2$ with equation of motion given by \[eom2\] a\^[-3]{}\_t(a\^3 \_2)+m\_2\^2\_2=F\_2({\_a},t). Let us focus on one example, for instance, $F_2= \lambda \sigma_1^2$ (the generalization to arbitrary interaction is straightforward). If $\Delta_{\sigma_2}^- < 2\Delta_{\sigma_1}^-$, then the above argument can be repeated to show that $F_2$ becomes irrelevant at late times. However, in the opposite regime there can be interactions made of lighter fields which are not negligible. Here we can use the factorization property of the solutions for light fields to diagonalize the equation . Note that $\Delta_{\sigma_2}^- \geq 2\Delta_{\sigma_1}^-$ can happen only if $\Delta_{\sigma_1}^- \leq 3/4$ for which the above-mentioned dominance of the growing mode and factorization of super-horizon solution is valid. We can therefore redefine the heavier field by adding to it a particular solution of the equation made of the lighter fields. In the above example, take \_2 = \_2 +\_1\^2, where $\Delta_1\equiv\Delta_{\sigma_1}^-$. The field $\tilde\sigma_2$ now satisfies a free field equation of mass $m_2$, plus corrections that are negligible outside of the horizon. This procedure can be continued to obtain a complete basis of free fields $\{\tilde \sigma_a\}$. The tildes will be dropped in the following.
So far the interactions with modes of the horizon size or shorter have been ignored. Once they are included and averaged over two types of corrections arise. First, their back-reaction to the long modes renormalizes the mass and coupling coefficients of the theory (or introduces new interactions), and secondly, the short modes can stochastically combine to make up a long wavelength mode. The first correction does not introduce any qualitative difference in the above argument except that the masses must be replaced by the renormalized ones. On the other hand, the stochastic contribution of horizon size modes on a super-horizon mode $q$ is suppressed because it requires $N\sim (aH/q)^3$ independent Hubble patches to conspire. The contribution to the variance of $\sigma$ is therefore expected to be of order $(q/aH)^3$ and hence negligible [@IR].
Since every perturbative theory of massive fields approaches the free theory outside of the horizon, we expect any (possibly strongly coupled) theory which can be continuously connected to a perturbative theory by dialing its coupling coefficients to also satisfy this property. If there are massless fields that are not protected by a symmetry, the interactions do not necessarily decay with $a$. However, keeping these scalar fields massless seems to require considerable fine-tuning. We discard this possibility in what follows. Note that the metric fluctuations (both tensor and scalar modes) are massless, but their interactions contain sufficient number of derivatives and die off outside the horizon.
It is important to stress that the above argument does not ensure that the super-horizon degrees of freedom continue to exist as weakly coupled fields at horizon and sub-horizon scales. However some general statements about correlation functions can be made even in these cases. For example, one can imagine that the physics at the sub-horizon scales is described by a strongly coupled conformal field theory (CFT).[^2] Or the compactification scale of the string theory may happen to be close to $H$ in which case a weakly coupled 4d description might not exist. The other interesting example is models with dissipation [@LopezNacir:2011kk]. In all these and similar scenarios it is hard or impossible to calculate the correlation functions. Nevertheless, the results of the following sections hold regardless of that. They are based just on the existence of free super-horizon degrees of freedom.
Squeezed limit in a classical background field
==============================================
The result of the previous section has important consequences for the squeezed limit correlation functions. In order to see that let us consider a long-wavelength perturbation of a light field whose space and time dependence factorizes as argued above, $\sigma(\x,\eta)\simeq \hat\sigma(\x)g(\eta)$, where $g(\eta)$ is the growing solution. This long wavelength mode slightly modifies the background seen by the short wavelength fluctuations. Hence, in the squeezed limit of cosmological correlation functions, the contribution of $\sigma$ is through the response of shorter wavelength modes to this background. More specifically, this happens when the short modes start interacting with the long mode after its horizon crossing, or if their memory of earlier interactions is erased due to adiabatic evolution. In these cases an effective description can be employed. First, note that the short scale physics can only depend on local observables made of the long-wavelength mode which can be organized in a derivative expansion. For instance, the 2-point correlation functions of shorter wavelength metric fluctuations $\zeta$ on the background can be written as \[xi\]
[()() ]{}\_=&\_0()+() \_() +\_i()\^i\_() +\_i\_j()\^[ij]{}\_()+\
&+\^2()\_[\^2]{}()+\_i()\^[i]{}\_()+
where $\xi_0$ is the correlation function in the absence of the particular long wavelength mode. This expansion in terms of local operators is not necessarily valid to all orders in the soft momentum $q$. For now, we are mainly interested in the leading contribution, however this point will be discussed in more detail in section \[sec:ope\].
Equation implies that \[z3xi\] [\_\_[\_1]{}\_[\_2]{} ]{}=P\_(q)\_(\_1)+where $P_{\zeta\hat\sigma}(q)$ is the cross correlation $P_{\zeta\hat\sigma}(q)={\left\langle \zeta_\q\hat\sigma_{-\q} \right\rangle}$ and dots represent other contributions to the squeezed limit 3-point function. If the mixing $r\equiv P_{\zeta\hat\sigma}/\sqrt{P_\zeta(q)P_{\hat\sigma}(q)}\ll 1$, then to leading order in $r$ we can write \[z3fac0\] [\_\_[\_1]{}\_[\_2]{} ]{}\_ , where $\eta_0$ is a time late enough so that all of the modes are super-horizon. The same formula can also be used to find the squeezed limit of a four-point function when the sum of two momenta is much less than individual momenta (this is sometimes called the counter-collinear limit). We get (for $\k_1\approx -\k_2$ and $\k_3\approx -\k_4$) \[z4fac\] [\_[\_1]{}\_[\_2]{}\_[\_3]{}\_[\_4]{} ]{}=\_[\_1,\_2{,\_a}]{} P\_[\_1\_2]{}(q)\_[\_1]{}(\_1)\_[\_2]{}(\_3)+
Note that the background long wavelength mode can be a fluctuation in $\zeta$ (or the tensor modes $\gamma_{ij}$). However, unlike the generic case, now there are consistency conditions on how the short modes respond to a long-wavelength metric fluctuation. Some linear superpositions of these fluctuations are locally indistinguishable from coordinate transformations and the response of the short modes is completely fixed by symmetries [@Hinterbichler]. Model dependence starts to show up at order $\d_i\d_j\zeta$ which is the reason for the squeezed limit behavior of equilateral-type shapes . In the absence of other super-horizon degrees of freedom the above constraints lead to consistency conditions on the full squeezed limit correlation functions, which hold regardless of the details of sub-horizon physics. For instance, an explicit derivation when the sub-horizon dynamics is effectively dissipative can be found in [@Nacir].
In the presence of other degrees of freedom, two cases are of special interest and will be discussed separately in the following sections. First, when $\sigma$ has mass comparable to $H$ (or more precisely $m^2/H^2\gg 1-n_s$, the scalar tilt). In this case $\sigma$ fluctuations gradually decay outside the horizon, and the long-short correlation forms around the horizon crossing time of the short modes during inflation. The second case is when $m/H\ll 1$. In this case can form simultaneously for all $k$, long after they have all crossed the horizon. We denote these nearly massless fluctuations as entropy modes.
Massive fields with $1-n_s\ll m^2/H^2 < 9/4$ {#sec:quasi}
--------------------------------------------
Neglecting slow-roll corrections, the growing mode of a field with mass $m<3H/2$ at super-horizon scales behaves as \[sigq\] \_()\_\^[\_\^-]{}. Since the correlation forms around the horizon crossing time $\eta\sim -1/k$, the factor $\eta^{\Delta_\sigma^{-}}$ translates into $k^{-\Delta_\sigma^{-}}$. On the other hand, the scale invariance of de Sitter (i.e. the symmetry under $(\eta,\x)\to (\lambda\eta,\lambda\x)$) implies that [\_\_[-]{}(\_0) ]{}’. From we see that $P_\sigma(q,\eta_0)\propto (q\eta_0)^{2\Delta_\sigma^{-}}/q^3$ up to normalization constants and ${\left\langle \sigma_\q(\eta_0)\hat\sigma_{-\q} \right\rangle}=P_\sigma(q,\eta_0)/\eta_0^{\Delta_\sigma^{-}}$. Hence, the $q$ dependence of and are fully fixed and using the scale invariance we can write \[quasi\] [\_\_[\_1]{}\_[\_2]{} ]{}\_P\_(q)P\_(k) ()\^[\_\^[-]{}]{}, where in this approximation $P_\zeta(q)\propto q^{-3}$. It is seen that a wide range of squeezed limit behaviors with $0<\Delta_\sigma^{-} <3/2$ result depending on the mass of $\sigma$. The quasi-single field scenario [@Chen:2009zp] is the perturbative realization of this possibility. A review of the perturbative calculation is provided in appendix \[funda\]. Note that there can also be a mixing between $\zeta$ and a composite operator made of $\sigma$. For instance, if P\_[\^2]{}(q)[\_\^2() ]{}’0 then there will be a contribution like but with $\Delta_\sigma^{-}\to 2\Delta_\sigma^{-}$. More generally for any local operator $O$ of dimension $\Delta_O\leq 2$, we have the squeezed limit behavior $(q/k)^{\Delta_O}$ that decays slower than the equilateral-type contributions. This is explicitly demonstrated in a perturbative example in appendix \[operator\]. Note that the whole range between the local and equilateral type non-Gaussianity is covered in these models. This type of correlation between the metric fluctuation $\zeta$ and operators constructed out of super-horizon degrees of freedom can naturally arise, for instance via a coupling $\dot\zeta \sigma^2$.
### Squeezed limit and operator product expansion\[sec:ope\]
Before discussing the case of entropic fluctuations, let us make some comments which cast doubt on any general connection between squeezed limit correlation functions during inflation (or a near de Sitter phase) and operator product expansion (OPE) in a hypothetical conformal field theory in one lower dimension (hints that such a connection might exist can be found e.g. in [@Arkani-Hamed; @Assassi]). The presence of such a connection is motivated by the fact that the action of de Sitter isometries on super-horizon fields can be identified with the action of conformal group on some dual operators in a CFT living on a spatial slice. This can best be seen at the level of the wavefunction of the universe. The wavefunction is most naturally expressed for fields of dimension lower than $3/2$ in terms of sufficiently smooth (super-horizon) field configurations as $\Psi[\{\vphi\},\eta]$. This is the analog of coordinate basis for particle quantum mechanics $\psi(x,t)$, but now with one real variable for each $\vphi(\x)$, or for each $\vphi(\q)$ with $q\ll -\eta^{-1}$. The expectation values are calculated by integrating over all of these fields [F\[{()}\] ]{}=D||\^2 F\[{}\]. Note that $\vphi$ on the r.h.s. is an integration variable and the time-dependence is fully encoded in $\Psi$.
The coefficient of the term $\vphi_1\cdots \vphi_n$ in $\log \Psi$ have the same structure as an $n$-point correlation function of CFT operators ${\left\langle O_1\cdots O_n \right\rangle}$ with dimensions $\Delta_{O_i}=\Delta^{+}_{\vphi_i}=3-\Delta^{-}_{\vphi_i}$ times $\eta^{\Delta_1^{-}}\cdots \eta^{\Delta_n^{-}}$. (For scalar fields heavier than $3H/2$ the mapping is less clear since both asymptotic solutions, $\Delta^{\pm}_\vphi$, decay the same way.) Now one may expect that the dominant squeezed limit behavior of correlation functions should be related to the OPE limit of the CFT operators, where two operators dual to the short modes are brought close to each other in position space, and their product is expanded as a sum over operators. We will show that this is not necessarily the case.
As a concrete example consider $\phi(\x_1)\phi(\x_2)\sigma(\x_3)$ where $\phi$ is a dimension zero field (an approximation to the almost massless fluctuations of the inflaton). The coefficient of this term has the structure of ${\left\langle \Phi(\x_1)\Phi(\x_2)\Sigma(\x_3) \right\rangle}$, where $\Delta_\Phi = 3$ and $\Delta_\Sigma = \Delta_\sigma^+$. In the limit $|\x_{12}|=|\x_1-\x_2|\to 0$, one expects O\_1(\_1)O\_2(\_2)\~\_[Q]{} c\_[12]{}\^[\_Q]{} x\_[12]{}\^[\_Q-\_1-\_2]{} Q(\_1), where the constants $c_{12}^{\Delta_Q}$ are called OPE coefficients. Using ${\left\langle Q(\x_1)Q(\x_2) \right\rangle}\propto x_{12}^{-2\Delta_Q}$ gives a term in $\log \Psi$ of the form \_i d\^3\_i(\_1)(\_2)(\_3) x\_[12]{}\^[\^[+]{}\_-6]{}x\_[13]{}\^[-2\_\^[+]{}]{}. Taking the Fourier transform, we obtain \[f(k,q)\] d\^3d\^3()(--)() f(k,q,), f(k,q,)k\^[3-\^[+]{}\_]{} q\^[2\^+\_-3]{} \^[-\_\^-]{}. The 3-point correlation function can be computed via [\_[\_1]{}()\_[\_2]{}()\_[\_3]{}() ]{}= DD(\_1)(\_2)(\_3)||\^2. For small non-Gaussianities, $|\Psi|^2$ is dominated by the Gaussian part ||\^2=, and the non-Gaussian correlation functions can be calculated perturbatively. The cubic term gives the squeezed limit contribution \[OPE\] [\_()\_[--]{}()\_() ]{}\_[OPE]{} =f(k,q,)P\_\^2 (k)P\_(q,)\^[\_\^-]{} k\^[3-\^[+]{}\_]{}P\_\^2(k), where we have used the fact that P\_(q,) = \^[2\_\^-]{} q\^[2\_\^+ -3]{}. However, one can show that is a subleading contribution in the squeezed limit. The leading contribution comes from correlating with $\sigma_\q(\eta)$ a similar expression as with the substitution $\zeta\to \phi$. Up to an overall constant, this is fixed by the scale invariance \[phiphisigma\] [\_\_[--]{}\_() ]{} \^[-\_\^-]{} k\^[-\^[-]{}\_]{}P\_(k)P\_(q,). (Note that super-horizon $\phi_\k$ is time-independent in our approximation.) Hence \[ope/squeezed\] \~()\^[3-2\_\^[-]{}]{}1, where we used the fact that $\Delta_\sigma^{-}<3/2$. Note that on the level of in-in diagrams the OPE result corresponds to having both power spectra on the short modes, while the diagrams for expression have one of the power spectra on the long mode. These different diagrams are shown in Fig. \[fig:diagrams\].
![(a) The diagram that corresponds to the squeezed limit bispectrum calculated using OPE (see equation ). Both power spectra are on the short modes. (b) Diagrams for the leading part of the squeezed limit bispectrum given by . One of the power spectra is on the long mode. []{data-label="fig:diagrams"}](ope.pdf){width="99.00000%"}
The dominant squeezed limit contribution arises from the following cubic term in $\log \Psi$: d\^3d\^3()(--)() k\^[-\^[-]{}\_]{} P\_\^[-1]{}(k) \^[-\_\^-]{}. This is a local function of $x_{13}$ when transformed to the position space. Interpreted as a correlation function of dual operators, it is a contact term ${\left\langle \Phi(\x_1)\Phi(\x_2)\Sigma(\x_3) \right\rangle}_{\rm contact} \propto \delta^3(\x_1-\x_3)$. However, this does not make it any less relevant for cosmological correlation functions. We can translate to correlation functions of $\zeta$ by using -, where $\dot\phi_{\rm s.r.}$ is the background slow-roll rate of the change of $\phi$. The field $\sigma$ would then contribute to the 3-point function if there is a mixing $P_{\zeta\sigma}$, and, even if not, its exchange contributes to the squeezed 4-point function ($|\q|=|\k_1+\k_2|\ll k_i$): [\_[\_1]{}\_[\_2]{}\_[\_3]{}\_[\_4]{} ]{}\_=()\^4 . The resulting three and four point functions are non-analytic functions of $q$ which lead to long-range correlations in position space.
Our expansion keeps only these local terms in $\log\Psi$, which as argued above play the dominant role in the squeezed limit. The corrections generically start at order estimated in .
Entropic perturbations $m/H \ll 1$
----------------------------------
The fluctuations of a very light field $\sigma$ remain almost constant during inflation and the power in fluctuations is nearly scale invariant. As a consequence they can affect cosmological observables at a later epoch of cosmic evolution, e.g. during reheating or when the $\sigma$ fluctuations become time-dependent again (for instance when the expansion rate drops below their mass). In the latter case the mode functions are not given by with $\Delta^-\simeq 0$ anymore, however we will argue that the modes remain classical and to study the affect of super-horizon perturbations of $\sigma$ on the observables one can still use an expansion in terms of local observables .
Suppose at some time the solutions to the linear equation of motion for $\sigma$ become oscillatory with a (time-dependent) physical frequency $\omega$. Now both mode functions have to be kept but the reason why the field can still be treated classically is that the fluctuations have very large occupation number. The amplitude of the inflationary fluctuations are ${\left\langle \sigma_k^2 \right\rangle}'\sim H_{\rm inf}^2/k^3$ while the typical size of the vacuum fluctuations for a fluctuating field of frequency $\omega$ is ${\left\langle \sigma_k^2 \right\rangle}_{\rm vac}\sim 1/a^3\omega$. The ratio of the two is a measure of the occupation number which is N \~\~. This is much larger than unity if the mode starts oscillating while it is still super-horizon, in which case $k/a\ll \omega\ll H_{\rm inf}$, and also when it enters the (sound) horizon long after the end of inflation, when $\omega = k/a\sim H_{\rm reentry} \ll H_{\rm inf}$. This implies that $\sigma$ and $\dot\sigma$ can both be known with great accuracy and the evolution of the field is determined in terms of its freeze-out value during inflation $\hat \sigma$. As long as the $\sigma$ mode is super-horizon this evolution is also local and therefore an expansion like can be used to relate observables to the locally measurable operators made of $\sigma$ which can all be related to the local value of $\hat \sigma$ and its derivatives. Moreover, since the interactions occur when all modes are already super-horizon they should be independent of $k$, therefore the squeezed limit behavior is the same as local non-Gaussianity. Note that, as before this local non-Gaussianity can be generated via a mixing between $\zeta$ and $\sigma^2$ or other composite operators made of $\sigma$—energy density is perhaps the most natural candidate—while $P_{\zeta\sigma}$ can in principle vanish.
These models are known to violate standard single-field consistency conditions since unlike long wavelength $\zeta$ perturbations those of $\sigma$ are not locally equivalent to a coordinate transformation. That is, while in the squeezed limit \[single\_cons\] \_[\_]{}(\_1,,\_N)[\_[\_1]{}\_[\_N]{} ]{}, where $\D$ is the dilatation operator, $\xi_{\sigma_\q}(\k_1,\cdots,\k_N)$ is completely model dependent. However, we can still find weaker conditions which are satisfied by models in which the violation of the single-field relations is triggered by mixing with other fields, thereby distinguishing them from models in which $\zeta$ by itself evolves at super-horizon scales.[^3]
The idea is to use the knowledge about the vertices with soft $\zeta$ modes \[like \], and the factorization to eliminate the ignorance about interactions of soft $\sigma$ modes. We first consider only a single entropic mode, which is easily generalizable to the case of several identical ones. Here, one can fully determine the degree of mixing between $\sigma$ and $\zeta$ from combinations of observable squeezed-limit correlation functions. The mixing cannot exceed $1$ by unitarity, otherwise the two-by-two correlation matrix will have negative eigenvalues. Moreover, different ways of determining it from different combinations of correlation functions must agree. This gives a consistency condition for these models. In the more general case, the unitarity requirement still puts a weaker constraint on the ratio of squeezed limit correlation functions. Most notably, on $f_{NL}$ and $\tau_{NL}$, which characterize respectively the 3-point and the 4-point correlators when the sum of two momenta becomes soft. However, these weaker constraints on observables must be obeyed as a consequence of the axioms of probability [@Smith], and therefore any consistent theory of fluctuations would automatically satisfy them.
### Consistency conditions on two-field scenarios
Consider having only one entropy mode. In this case, we have [\_\_[\_1]{}\_[\_2]{} ]{}’=P\_(q)\_(\_1)+P\_(q)\_(\_1), and
[\_[\_1]{}\_[\_2]{}\_[\_3]{}\_[\_4]{} ]{}’=& \_(\_1)\_(\_3)P\_(q) +\_(\_1) \_(\_3)P\_(q)\
&+\[\_(\_1)\_(\_3)+\_(\_1)\_(\_3)\]P\_(q),
for the counter-collinear limit of the 4-point function $\k_1+\k_2=\q\to0$. Solving for $r = P_{\zeta\hat\sigma}/\sqrt{P_\zeta(q)P_{\hat\sigma}(q)}$, we obtain[^4] \[eq:r2\_nonpert\] r\^2 =. This must hold for all $\k_1$ and $\k_3$ as long as they are much larger than $\q$.[^5] Using the standard consistency condition , and assuming that $r\ll 1$ simplifies the above equation to \[z3/z4\] r\^2=.
The above equation tells us that, even though the $\sigma$ field itself is unobservable, the mixing parameter $r$ can still be measured just from the observable correlation functions of $\zeta$. Moreover, $r$ can also be solved in terms of the counter-collinear limit of the 5-point function and the squeezed limit of 3- and 4-point functions, and so on. Then, given that the two expressions for $r$ must agree, one can write down an equality that involves just the correlators of $\zeta$. These relations are new consistency conditions for the two-field inflation that hold irrespectively of the details of $\sigma$ interactions.
### Unitarity constraints on multi-field models
In the more general case where a set of local operators $\{O_a\}$ contribute to the squeezed limits, the constraints are weaker, since in general one cannot solve for the individual mixings $r_a$. However, there are inequalities that must be satisfied for the theory to be unitary.[^6] We derive the simplest one: The squeezed limit of the 3-point function and the counter-collinear limit of 4-point function now read [\_\_[\_1]{}\_[\_2]{} ]{}’=\_a P\_[a]{}(q)\_a(\_1)+P\_(q)\_(\_1), and
[\_[\_1]{}\_[\_2]{}\_[\_3]{}\_[\_4]{} ]{}’=& \_(\_1)\_(\_3)P\_(q) +\_[a,b]{} \_[a]{}(\_1) \_[b]{}(\_3)P\_[ab]{}(q)\
&+\_a \[\_(\_1)\_[a]{}(\_3)+\_[a]{}(\_1)\_(\_3)\]P\_[a]{}(q),
where $P_{ab}={\left\langle \hat O_a\hat O_b \right\rangle}$. Choosing an orthogonal basis for the operators such that $P_{ab}\propto\delta_{ab}$, and assuming for simplicity that $\k_1\simeq \k_3$, we arrive at the following constraint (using the Cauchy-Schwartz theorem) \^2(\_a r\_a\^2) P\_(q)\[[\_[\_1]{}\_[\_2]{}\_[\_3]{}\_[\_4]{} ]{}’ - 2[\_[\_1]{}\_[\_2]{}\_ ]{}’ \_(\_1) +P\_(q) \_(\_1)\_(\_1)\]. In the extremely squeezed limit where only local shape survives, this simplifies to \[tildfNL\] (5 f\_[NL]{})\^2(\_a r\_a\^2), where $\tilde f_{NL}$ is the observable part of the local non-Gaussianity: the difference between total $f_{NL}$ and the single-field part. A stronger bound can be obtained by subtracting the contribution of tensor mode exchange from the 4-point function since $P_{\zeta\gamma}=0$.
On the other hand, $1-\sum_a r_a^2$ is proportional to the determinant of the total 2-point correlation matrix $\det P_{\varphi\varphi'}$, where $\varphi$ indicates all local fields and operators including $\zeta$. The positivity of this determinant together with then implies (65 f\_[NL]{})\^2 \_[NL]{}.
\* \* \*
Let us close this section by two general remarks. First, the underlying assumption for the expansion was that the short modes are in vacuum until the long mode has crossed the horizon and become classical. This assumption can be broken for instance if the short wavelength modes get excited inside the horizon. An explicit example of this sort is axion models of inflation [@McAllister; @Flauger] where there is another physical frequency $\omega>H$ in the problem at which the modes get excited. This case has been studied in appendix \[resonant\]. As seen after sufficient squeezing (here $q/k \ll H/\omega$) our formalism will be applicable again. In particular, in the absence of other degrees of freedom the single-field consistency conditions hold for sufficiently squeezed correlators [@Flauger_squeezed].
The second remark is regarding the resemblance between the expansion for classical long wavelength modes and the bias expansion for the distribution of halos and galaxies. In the latter case the dynamics of the short scale modes is influenced by the local observables of the long wavelength perturbations of gravitational potential (made of $\d_i\d_j\phi$) along the trajectory of the short modes. However unlike a super-horizon $\sigma$ field whose time-evolution is local and fully determined in terms of the field $\hat\sigma$ and its spatial derivatives at a single time-slice, the sub-horizon density fluctuations have long range gravitational interactions. Therefore the bias expansion won’t be local if written just in terms of the initial field and its spatial derivatives (see e.g. [@bias]).
Heavy fields $m/H>3/2$
======================
In this case the two solutions of the free-field equation decay with the same rate and oscillate, scaling as $\eta^{3/2\pm i\mu}$ where $\mu = \sqrt{m^2/H^2-9/4}$. Hence, the fields are not classical and we cannot expand as in . However, there is still a sense in which the correlation functions arising from the exchange of heavy fields factorize in the squeezed limit. This is because in this limit the dominant contribution arises from the production of on-shell heavy particles, whose amplitude is suppressed by a Boltzmann factor $\exp(-\pi\mu)$ for each pair of particles. In the weakly coupled case the other contributions in which no on-shell particle is created are captured by a set of effective derivative self-couplings of $\zeta$ arising from integrating out the heavy particle. These self couplings are not suppressed by any Boltzmann factor, but they lead to equilateral type non-Gaussianities. For instance, the squeezed limit bispectrum scales as $q^2$, which decays by a factor of $(q/k)^{1/2}$ faster compared to the former contribution from production of on-shell particles which goes as $(q/k)^{3/2}$. For higher statistics these on-shell particles give a sub-dominant piece but they still have a characteristically different functional form. This problem has been thoroughly studied in [@Noumi; @Arkani-Hamed] and we will just touch upon the salient features of the squeezed limit correlators.[^7]
Instead of correlation functions in a classical background, here we should talk about the wavefunction of universe to keep track of the phase of the $\sigma$ field. Since the concept of particle is well-defined for late-time heavy fields, it is more natural to expand the wavefunction in terms of eigenstates of particle number operator as in the scattering theory. Let us choose an intermediate time $\tilde\eta$ satisfying $k^{-1}\ll -\tilde\eta\ll q^{-1}$, so that the super-horizon field $\sigma_\q$ is well-defined but the hard $k$ modes are in vacuum. The wavefunction at $\eta_0$, when all correlations are formed, can be written as \[Psi\] |(\_0)= \_N\_[n=1]{}\^N D U(\_0,) |{},N , where $U(\eta_0,\tilde\eta)$ is the time-evolution operator and the sum is only over $N$-particle states of $\sigma$ with super-horizon wavelengths. Other possible degrees of freedom have been omitted. The wavefunction $|\Psi(\tilde\eta)\rangle$ depends on the sub-horizon dynamics, and if $\sigma$ exists there as a weakly coupled degree of freedom can be perturbatively calculated. However, on general grounds we expect that e\^[-N/2]{}, since we assume that the theory is in the adiabatic vacuum deep inside the horizon, and hence particle production is a result of evolution in the expanding background. Secondly, all modes that are still deeply sub-horizon must be in vacuum: (\_), -k1. (The $i\ep$ prescription evolves such a wavefunction into the true vacuum.) On the other hand, the soft $\zeta_q$ modes are constant up to corrections of order $(q\tilde \eta)^2$, therefore \_|U(\_0,)|\_. The correlation functions are obtained by taking averages using the wavefunction . Keeping the minimum number of Boltzmann factors, we get: [\_\_[\_1]{}\_[\_2]{} ]{}\_[\_[\_1]{}\_[\_2]{}|1\_]{} d\_ \_ \^+ The second factor on the r.h.s. depends on the sub-horizon mixing of $\zeta_\q$ and $\sigma_\q$. We expect from the perturbative calculation that this correlation should form at $-q \eta \sim \mu\gg 1$ and to introduce another factor of $\exp(-\mu\pi/2)$. The first factor on the r.h.s. corresponds to a massive particle decaying into a pair of $\zeta_\k$ modes when their total energy red-shifts to values around the mass, i.e. $\eta\sim -\mu/k$. The super-horizon wavefunction $\sigma_q \propto (q\eta)^{3/2+i\mu}/q^{3/2}+\text{c.c.}$ then leads to an oscillating piece $\cos(\mu\log k+\phi_q)$ and a dilution factor $(q/k)^{3/2}$. The phase $\phi_q$ is fixed if we require that $\sigma$ should be in the adiabatic vacuum at very early times. As emphasized in [@Arkani-Hamed] the oscillations result from interference: the expectation value is calculated between the Gaussian part of the wavefunction, and the part with production of a pair of entangled $\sigma$ particles, one of them oscillating into a $\zeta_\q$ mode, and the other decaying at a much later time into a pair of short wavelength $\zeta_{\pm \k}$ modes. Fixing the other factors by approximate scale invariance, we obtain \[massive\] [\_\_[\_1]{}\_[\_2]{} ]{}\_ P\_(q)P\_(k) ()\^[3/2]{}(-\_0). This is enhanced in the squeezed limit by $(k/q)^{1/2}$ compared to equilateral type shapes. Moreover, the oscillating component makes this shape distinct from other contributions. We could also consider the $\sigma$-exchange contribution to the 4-point correlator: [\_[\_1]{}\_[\_2]{}\_[\_3]{}\_[\_4]{} ]{}\_e\^[2]{} [\_[\_1]{}\_[\_2]{}|1\_]{} [\_[\_3]{}\_[\_4]{}|1\_[-]{} ]{}+[c.c.]{} Again this is an interference between the Gaussian piece, and a piece which describes the following process: (i) An entangled pair of massive particles is produced in the time-dependent background. (ii) The particles move out of causal contact. (iii) They subsequently decay into several $\zeta$ fields, when the total energy of the red-shifting modes coincides with $m$.
Finally, if the exchanged field has spin $s>0$, the squeezed limit result would be proportional to $\ep^{i_1\cdots i_s}(\q)\hat \k_{i_1}\cdots \hat \k_{i_s}$. Only the longitudinal polarization of a massive tensor field can mix with a scalar $\zeta$, e.g. via $\dot\zeta \d_{i_1}\cdots\d_{i_s}\sigma^{i_1\cdots i_s}$. Therefore, \[spin\] [\_\_[\_1]{}\_[\_2]{} ]{}\_ P\_(q)P\_(k) ()\^[3/2]{}(-\_0)¶\_s(), where $\P_s$ is the Legendre polynomial.
Quantum versus classical correlation
------------------------------------
We have seen that the squeezed limit correlation functions have an expansion in terms of a discrete set of power-laws
[\_\_[\_1]{}\_[\_2]{} ]{}&\~P\_(q)P\_(k)\_a\_()\^,\
[\_[\_1]{}\_[\_2]{}\_[\_3]{}\_[\_4]{} ]{}&\~P\_(k\_1)P\_(k\_3)\_b\_()\^ ,
with $\Delta$ integers $\geq 2$ for equilateral type non-Gaussianities, real numbers corresponding to the light ($m/H<3/2$) degrees of freedom and their derivatives and products, or pairs of complex numbers corresponding to heavy degrees of freedom. This expansion is unambiguous and non-trivial and seems to be dictated just by the isometries of a quasi-de Sitter space-time. As emphasized in [@Arkani-Hamed] the case of heavy field exchange is fundamentally different as it corresponds to a quantum mechanical interference effect.[^8] In fact one can see that the fundamental difference between the case of heavy intermediate fields and the other cases is the difference between quantum versus classical correlations. The outcome of two short distance measurements in two far separated laboratories are correlated with each other in all cases, but in one case they are deterministic while in the other case the outcome of one measurement influences the result of the other.
This difference has a distinct imprint in the squeezed limit correlators: When the long wavelength mode becomes classical, the $\sigma$ field works as a hidden variable and the expansion is possible. As a result the dependence on the momenta on two sides of the intermediate long wavelength mode factorizes (see for instance and ), or more generally it can be written as a sum of factorized terms weighted by the correlations among various derivatives and products of $\sigma$. Different short distance measurements have no influence on one another; they are correlated because they are both influenced by a single long mode. On the other hand, the heavy field contribution is not factorizable because of the dependence of the squeezed 3-point and 4-point functions, respectively, on $\log (q/k)$ and $\log(q^2/k_{12}k_{34})$ (where $k_{ij}=|\k_i-\k_j|$) [@Arkani-Hamed].
Prospects in large-scale structure surveys
==========================================
In this section we will discuss the observability of various squeezed limit behaviors, when contrasted with the contribution from the gravitational interactions during the structure formation (see [@Flauger_squeezed] for a related analysis in the context of single-field inflation). Consider a non-Gaussian 3-point correlator whose squeezed limit behavior is \[fnl\] [\_\_\_[’]{} ]{}=f\_[NL]{} P(k)P(q)()\^(2)\^3 \^3(++’). The signal to noise ratio in a 3d survey of volume $V$ can be estimated as (S/N)\^2=d\^3\_1 d\^3’ d\^3 . Using , approximating the denominator by the Gaussian contribution, and replacing $(2\pi)^3\delta(\bsb 0)\to V$ (appropriate for finite volume surveys), we get (S/N)\^2 = f\_[NL]{}\^2d\^3d\^3P(q) ()\^[2]{}, where we are calculating the signal in the squeezed limit configurations $q\ll k$. The $\k$ integral is dominated by the highest values of $k$. This is generally the case since the number of modes grows as $k^3$. For biased tracers, $k$ really corresponds to the modes that have become nonlinear. The dependence on $k$ would then be different. However, we are interested in the dependence on $q$ as it is made small so that $\zeta_\q$ mode is linear and $P(q)\propto 1/q^3$. Limiting the inner integral to $q<q_{\rm max}$ we get a contribution which is proportional to \[S/N\] (S/N)q\_[max]{}\^. That is, the error with which this type of non-Gaussianity can be measured by considering modes longer than $\qmax^{-1}$ scales as $\qmax^{-\Delta}$. This suggests that unless $\Delta =0$, there is more signal in larger values of $q$.
However, there is a guaranteed level of non-Gaussianity due to the subsequent gravitational interaction of the modes that also increases as $q$ becomes bigger and competes with primordial signal. For very large values of the hard momenta $k$, this is not accurately calculable. However, in the squeezed limit its $q$ dependence must scale as \[gr\] [\_\_\_[’]{} ]{}\_[gr]{}Ø(q\^2) P(q), since the short scale dynamics can only depend on the local observables made of the long mode. As $k$ is kept fixed around the largest available values and $q$ is made smaller, this gravitational contribution decays faster than for all $\Delta<2$. Therefore, the relevant question is how the signal in modes with $q<\qmax$, i.e. equation , compares to the ratio of to , which scales as $\qmax ^{\Delta-2}$. One finds that for $\Delta <1$ the primordial shape is better distinguishable in the squeezed limit, while for higher values of $\Delta$ analyzing the equilateral configurations significantly improves the chance of detectability. However, unlike the squeezed limit the equilateral primordial shapes are not expected to be very distinct from gravitational contributions, hence the theoretical uncertainty in determining the latter close to the nonlinear scale poses a challenge to increasing $k$.
Conclusions
===========
We argued that several squeezed limit properties of non-single-field inflationary models hold more generically, and can be studied in a unified fashion. The fields in a quasi-de Sitter space-time become free at super-horizon scales and follow power-law time evolution. The light fields become classical and the leading squeezed limit behaviors can be obtained from an effective parameterization of short-distance correlations in terms of a local expansion in derivatives and powers of the background long wavelength mode. We showed how the known results, such as the squeezed limit behavior in quasi-single-field models, and the unitarity constrains on multi-field models, can be derived in a more model independent way from this formalism. Heavy fields, on the other hand, have complex wavefunction at late times and are naturally described in terms of many particle states. The squeezed limit correlation functions due to the production of these heavy particles by the time-dependent background is dominated by interference effects and hence they oscillate as a function of the ratio of $k_{\rm short}/k_{\rm long}$. All these features have not much to do with the details of sub-horizon dynamics, but they essentially depend on quasi-de Sitter evolution and the existence of (not too heavy) super-horizon degrees of freedom other than metric and inflaton fluctuations. The form can be used as a general test of a quasi-de Sitter cosmic evolution. Finally, we discussed the observability of various squeezed limit scalings of the bispectrum.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Nima Arkani-Hamed, Paolo Creminelli, Juan Maldacena, and Matias Zaldarriaga for stimulating discussions. M.M. is supported by NSF Grants No. PHY-1314311 and No. PHY-0855425. M.S. is supported by the Institute for Advanced Study.
Purturbative examples of factorization
======================================
Fundamental fields {#funda}
------------------
Suppose $\zeta$ is coupled to a light field $\sigma$ via \[H0\] = \_1+\_2 = \^2+ . We are interested in the leading contribution of a $\sigma$ exchange to the squeezed limit of the 3-point function ${\left\langle \zeta_\q\zeta_{\k_1}\zeta_{\k_2} \right\rangle}_\sigma$: \[z30\]
[\_\_[\_1]{}\_[\_2]{} ]{}\_[\^2]{}= -2[\_(\_0)\_[\_1]{}(\_0)\_[\_2]{}(\_0)\^[\_0]{}\_[-]{} \^[\_1]{}\_[-]{}\_1(\_1)\_2(\_2) ]{}\
+[\^[\_0]{}\_[-]{}\_1(\_1) \_(\_0)\_[\_1]{}(\_0)\_[\_2]{}(\_0) \^[\_1]{}\_[-]{}\_2(\_2) ]{}+ \_1(\_1)\_2(\_2) ,
where we introduced an infrared regulator $\eta_0$ since the field $\sigma$ is generically time-dependent at super-horizon scales. We will often drop the time argument of $\zeta$ if it is $\eta_0$. The main contribution to the squeezed limit is when the mixing $\H_2$ between the soft $\zeta$ and $\sigma$ fields occurs at a time around $\eta_q \sim -1/q$ much earlier than the $\H_1$ interaction at $\eta_k\sim -1/k$. Substituting in , expanding in terms of mode functions and using this approximation, we obtain \[z320\]
&- \_(\_0)\_[\_1]{}(\_0)\_[\_2]{}(\_0) \^[\_0]{}\_[-]{} \^[\_1]{}\_[-]{} \^\*\_[\_1]{}(\_1)\^\*\_[\_2]{}(\_1)\_q(\_1) \^\*\_q(\_2)\^\*\_(\_2)+[c.c.]{}\
&+ \_(\_0)\^\*\_[\_1]{}(\_0)\^\*\_[\_2]{}(\_0) \^[\_0]{}\_[-]{} \_[\_1]{}(\_1)\_[\_2]{}(\_1)\_q(\_1) \^[\_0]{}\_[-]{}\^\*\_q(\_2)\^\*\_(\_2)+[c.c.]{}
We next argue that this expression factorizes into \[z3fac\] [\_\_[\_1]{}\_[\_2]{} ]{}\_ P(q)P(k)()\^[\_]{}, where $\Delta_\sigma=\frac{3}{2}-\sqrt{\frac{9}{4}-\frac{m^2}{H^2}}$ is the scaling of the growing mode and $P_\sigma(q,\eta_0)=|\sigma_q(\eta_0)|^2$. The two mixed correlators in the numerator are given by \[zs\] [\_\_[-]{}(\_0) ]{}=-i \_(\_0)\_[-]{}(\_0) \^[\_0]{}\_[-]{}\^\*\_[-]{}()\^\*\_()+[c.c.]{} and \[zzs\] [\_[\_1]{}\_[\_2]{}\_(\_0) ]{}=-i\_[\_1]{}(\_0)\_[\_2]{}(\_0)\_(\_0) \^[\_0]{}\_[-]{} \^\*\_[\_1]{}(\_1)\^\*\_[\_2]{}(\_1)\^\*\_(\_1)+[c.c.]{} The relation is then verified by noticing that: (a) The integral in is independent of $\eta_0$ as long as $-q\eta_0\ll 1$. Therefore, since the $\eta_1$ integral on the first line of is dominated by $-\eta_1\sim 1/k \ll 1/q$, we can replace the upper-bound of the $\eta_2$ integral with $\eta_0$. (b) At late times the time-dependence of the mode functions factorizes and we can replace: \_q(\_2)\_q\^\*(\_2).
### Light fields {#light}
For fields of dimension $\Delta_\sigma\leq 1$, the integral in is naively dominated by late times, and the approximation (a) seems to be invalid. However, this late-time dominance is spurious since it is pure imaginary and cancels in the final result. This independence from the late-time cutoff $\eta_0$ (apart from the trivial scaling in $\sigma_q(\eta_0)$) becomes manifest by reorganizing the perturbation theory as time-evolution problem [@Musso]. In this language there are two contributions to ${\left\langle \zeta_\q\sigma_{-\q}(\eta_0) \right\rangle}$: \[zsig\] [\_\_[-]{}(\_0) ]{}=\^\*\_[-]{}(\_0) \^[\_0]{}\_[-]{} G\^R\_[\_]{}(\_0,)\_[-]{}() +\_(\_0) \^[\_0]{}\_[-]{} G\^R\_[\_[-]{}]{}(\_0,)\^\*\_(), where G\_[f\_]{}\^R(\_0,)=(f\_(\_0)f\^\*\_()-[c.c.]{}) is the retarded propagator of the indicated field. At late times it decays as $\eta^3$, while $\dot\zeta_\q(\eta) \propto \eta^2$ and $\sigma_\q(\eta)\propto \eta^{\Delta_\sigma}$. Hence, for all $\Delta_\sigma>0$ each term in is manifestly IR-safe, i.e. it does not depend on the upper limit of the integral as long as $-q\eta_0\ll 1$, but only the sum of the two terms is real. The case of an unprotected massless field $\Delta_\sigma =0$ is special. Here, the dependence on $\eta_0$ is real and corresponds to the possibility of the generation of a mass for $\sigma$ via its mixing with $\zeta$.
In this formalism, the leading contribution of $\sigma$ exchange to the squeezed limit comes from two diagrams
[\_\_[\_1]{}\_[\_2]{} ]{}= \^[\_0]{}\_[-]{} G\^R\_[\_]{}(\_0,\_2)\_[-]{}(\_2) \^[\_0]{}\_[-]{}\^\*\_(\_1)\
+\_(\_0) \^[\_0]{}\_[-]{}\^[\_0]{}\_[-]{} G\^R\_[\_[-]{}]{}(\_1,\_2)\^\*\_(\_2),
where dots represent the part of the diagrams corresponding to the interaction among the high momentum modes at $\eta_1\sim -1/k$. The approximation (b) can now be rephrased as G\^R\_[\_]{}(\_1,\_2)G\^R\_[\_]{}(\_0,\_2), -q\_11, and leads to the same factorization formula .
Local operator exchange {#operator}
-----------------------
Suppose $\zeta$ is coupled to a light field $\sigma$ of mass $m<3H/2$ via \[H\] = \_1+\_2 = \^2\^2 + \^2. We are interested in the leading contribution of a $\sigma$ loop to the squeezed limit of the 3-point function ${\left\langle \zeta_\q\zeta_{\k_1}\zeta_{\k_2} \right\rangle}_\sigma$: \[z3\]
[\_\_[\_1]{}\_[\_2]{} ]{}\_[\^2]{}= -2[\_(\_0)\_[\_1]{}(\_0)\_[\_2]{}(\_0)\^[\_0]{}\_[-]{} \^[\_1]{}\_[-]{}\_1(\_1)\_2(\_2) ]{}\
+[\^[\_0]{}\_[-]{}\_1(\_1) \_(\_0)\_[\_1]{}(\_0)\_[\_2]{}(\_0) \^[\_1]{}\_[-]{}\_2(\_2) ]{}+ \_1(\_1)\_2(\_2) ,
where we introduced an infrared regulator $\eta_0$ since the field $\sigma$ is generically time-dependent at super-horizon scales. We will often drop the time argument of $\zeta$ if it is $\eta_0$. The contribution to the squeezed limit is insignificant unless the momenta of the virtual pair $\q_1$ and $\q_2$ are of the same order as the soft momentum $\q$, when an $\H_2$ interaction among the three soft modes occur around $\eta_q \sim -1/q$ much earlier than the $\H_1$ interaction at $\eta_k\sim -1/k$. Substituting in , expanding in terms of mode functions and using this approximation, we obtain \[z32\]
-2\^[\_0]{}\_[-]{} \^[\_0]{}\_[-]{} [\^2\_(\_1)\^2\_[-]{}(\_2) ]{},
where \[sig2\] [\^2\_(\_1)\^2\_[-]{}(\_2) ]{}=\_[\_1+\_2 =]{} \_[\_1]{}(\_1)\_[\_2]{}(\_1)\^\*\_[\_1]{}(\_2)\^2\_[\_2]{}(\_2). Using the late time behavior: \[late\] ()\^[\_]{} \_=- in and substituting in , it is easy to see that it factorizes: [\_\_[\_1]{}\_[\_2]{} ]{}\_\_[\_1+\_2 =]{} . However, one can show that the above squeezed limit 3-point function fully factorizes. Note first that because of the late time behavior \[zzss\] [\_[\_1]{}\_[\_2]{}\_[\_1]{}(\_0)\_[\_2]{}(\_0) ]{} =f(k\_1,\_0)P\_(q\_1,\_0)P\_(q\_2,\_0) , which implies \[z33\] [\_\_[\_1]{}\_[\_2]{} ]{}\_[\^2]{}f(k\_1,\_0) [\_\^2\_[-]{}(\_0) ]{}. Next, integrate over $\q_1$ and $\q_2$, with $\q_1+\q_2=\q$, to obtain \[zzss2\] [\_[\_1]{}\_[\_2]{}\^2\_(\_0) ]{} =f(k\_1,\_0)P\_[\^2]{}(q,\_0). Combining and we finally obtain [\_\_[\_1]{}\_[\_2]{} ]{}\_[\^2]{} P(q)P(k)()\^[2\_]{}. Let us make a few remarks:
- The above derivation holds only if $\Delta_\sigma\leq 1$. Otherwise, the $\eta_1$ integral in is dominated by $\eta_1\sim \eta_2$ and not by $\eta_1\sim -1/ q$: the UV divergence in the $\sigma$ loop wins over slow variation of $\zeta_q(\eta_1)$ at super-horizon scales. Nevertheless, by renormalizing the composite operator one expects to obtain the same factorized contribution.
- Although the above formula was derived for the operator $\sigma^2$, it is expected to hold more generally for any intermediate operator $O$ [\_\_[\_1]{}\_[\_2]{} ]{}\_[O]{} P(q)P(k)()\^[\_O]{}, and the last scaling to be valid even for large mixing.
Soft internal lines {#internal}
-------------------
The approximation (b) of section \[funda\] can be used to also show that the contribution of soft internal lines (or operators in general) to the correlation functions factorizes. For instance, a 4-point function in the counter-collinear limit $\k_1+\k_2=\q\to 0$ can be written as \[cc\] [\_[\_1]{}\_[\_2]{}\_[\_3]{}\_[\_4]{} ]{}\_[O]{} . The intermediate operator can now be a soft $\zeta_q$ mode.
Resonant non-Gaussianity {#resonant}
------------------------
The above predictions always require sufficient amount of squeezing $q/k\ll 1$. This is needed to ensure that the short modes are in vacuum before the long mode crosses the horizon. An illustrative example is resonant non-Gaussianity where the factorization on soft internal lines, e.g. , can fail due to insufficient squeezing. Here the modes get excited at physical frequencies of order the resonant frequency $\omega\gg H$ and the interactions start to be important when the short modes cross this threshold. If $q/k > H/\omega$ the soft intermediate mode is not yet super-horizon when the interactions of the short modes become important. Thus, the approximation (b) cannot be made. More explicitly, the leading contribution to a 4-point function in the counter-collinear limit contains a term
-\_(\_0)&\_[\_1]{}(\_0)\_[\_2]{}(\_0) \^[\_0]{}\_[-]{} V\^[(3)]{}(\_1) \^[\_1]{}\_[-]{} V\^[(3)]{}(\_2)\
&\^\*\_[\_1]{}(\_1)\^\*\_[\_2]{}(\_1)\_(\_1) \^\*\_[\_3]{}(\_2)\^\*\_[\_4]{}(\_2)\^\*\_(\_2) +{\_1,\_2}{\_3,\_4}+[c.c.]{}
where $V^{(n)}(\eta)\propto \cos \omega t$ is the leading cubic vertex. The integrals are dominated when $-k\eta\sim \omega/H$ around the saddle point. Suppose $q\ll k$ but $q/k\gg H/\omega$ then we can neglect the factor $\exp(-iq\eta)$ in $\zeta_q(\eta)$ and replace \_q()=\_q(\_0) (1+iq)e\^[-iq]{}iq\_q(\_0), which gives an opposite contribution compared to $\zeta^*_q(\eta)\zeta_q(\eta_0)/\zeta_q^*(\eta_0)$ in the same approximation. Hence the approximation (b) does not hold in this regime.
[99]{}
J. M. Maldacena, “Non-Gaussian features of primordial fluctuations in single field inflationary models,” JHEP [**0305**]{} (2003) 013 \[arXiv:astro-ph/0210603\]. P. Creminelli and M. Zaldarriaga, “Single field consistency relation for the 3-point function,” JCAP [**0410**]{}, 006 (2004) \[astro-ph/0407059\].
P. Creminelli, J. Norena and M. Simonovic, “Conformal consistency relations for single-field inflation,” JCAP [**1207**]{}, 052 (2012) \[arXiv:1203.4595 \[hep-th\]\].
K. Hinterbichler, L. Hui and J. Khoury, “An Infinite Set of Ward Identities for Adiabatic Modes in Cosmology,” JCAP [**1401**]{}, 039 (2014) \[arXiv:1304.5527 \[hep-th\]\].
X. Chen and Y. Wang, “Large non-Gaussianities with Intermediate Shapes from Quasi-Single Field Inflation,” Phys. Rev. D [**81**]{}, 063511 (2010) \[arXiv:0909.0496 \[astro-ph.CO\]\]. D. Baumann and D. Green, “Signatures of Supersymmetry from the Early Universe,” Phys. Rev. D [**85**]{}, 103520 (2012) \[arXiv:1109.0292 \[hep-th\]\]. C. Cheung, P. Creminelli, A. L. Fitzpatrick, J. Kaplan and L. Senatore, “The Effective Field Theory of Inflation,” JHEP [**0803**]{}, 014 (2008) \[arXiv:0709.0293 \[hep-th\]\]. D. Baumann and D. Green, “Equilateral Non-Gaussianity and New Physics on the Horizon,” JCAP [**1109**]{}, 014 (2011) \[arXiv:1102.5343 \[hep-th\]\]. L. Senatore and M. Zaldarriaga, “The Effective Field Theory of Multifield Inflation,” JHEP [**1204**]{}, 024 (2012) \[arXiv:1009.2093 \[hep-th\]\]. K. Hinterbichler and J. Khoury, “The Pseudo-Conformal Universe: Scale Invariance from Spontaneous Breaking of Conformal Symmetry,” JCAP [**1204**]{}, 023 (2012) \[arXiv:1106.1428 \[hep-th\]\]. T. Suyama and M. Yamaguchi, “Non-Gaussianity in the modulated reheating scenario,” Phys. Rev. D [**77**]{}, 023505 (2008) \[arXiv:0709.2545 \[astro-ph\]\]. K. M. Smith, M. LoVerde and M. Zaldarriaga, “A universal bound on N-point correlations from inflation,” Phys. Rev. Lett. [**107**]{}, 191301 (2011) \[arXiv:1108.1805 \[astro-ph.CO\]\].
T. Noumi, M. Yamaguchi and D. Yokoyama, “Effective field theory approach to quasi-single field inflation and effects of heavy fields,” JHEP [**1306**]{}, 051 (2013) \[arXiv:1211.1624 \[hep-th\]\]. N. Arkani-Hamed and J. Maldacena, “Cosmological Collider Physics,” arXiv:1503.08043 \[hep-th\]. N. Dalal, O. Dore, D. Huterer and A. Shirokov, “The imprints of primordial non-gaussianities on large-scale structure: scale dependent bias and abundance of virialized objects,” Phys. Rev. D [**77**]{}, 123514 (2008) \[arXiv:0710.4560 \[astro-ph\]\]. L. Senatore and M. Zaldarriaga, “On Loops in Inflation II: IR Effects in Single Clock Inflation,” JHEP [**1301**]{}, 109 (2013) \[arXiv:1203.6354 \[hep-th\]\]. D. Green, M. Lewandowski, L. Senatore, E. Silverstein and M. Zaldarriaga, “Anomalous Dimensions and Non-Gaussianity,” JHEP [**1310**]{}, 171 (2013) \[arXiv:1301.2630\]. D. Lopez Nacir, R. A. Porto, L. Senatore and M. Zaldarriaga, “Dissipative effects in the Effective Field Theory of Inflation,” JHEP [**1201**]{}, 075 (2012) \[arXiv:1109.4192 \[hep-th\]\]. D. Lopez Nacir, R. A. Porto and M. Zaldarriaga, “The consistency condition for the three-point function in dissipative single-clock inflation,” JCAP [**1209**]{}, 004 (2012) \[arXiv:1206.7083 \[hep-th\]\]. X. Chen and Y. Wang, “Quasi-Single Field Inflation and Non-Gaussianities,” JCAP [**1004**]{}, 027 (2010) \[arXiv:0911.3380 \[hep-th\]\]. V. Assassi, D. Baumann and D. Green, “Symmetries and Loops in Inflation,” JHEP [**1302**]{}, 151 (2013) \[arXiv:1210.7792 \[hep-th\]\]. S. Weinberg, “Adiabatic modes in cosmology,” Phys. Rev. D [**67**]{}, 123504 (2003) \[astro-ph/0302326\]. D. Baumann, D. Green, H. Lee and R. A. Porto, “Signs of Analyticity in Single-Field Inflation,” arXiv:1502.07304 \[hep-th\].
L. McAllister, E. Silverstein and A. Westphal, “Gravity Waves and Linear Inflation from Axion Monodromy,” Phys. Rev. D [**82**]{}, 046003 (2010) \[arXiv:0808.0706 \[hep-th\]\]. R. Flauger, L. McAllister, E. Pajer, A. Westphal and G. Xu, “Oscillations in the CMB from Axion Monodromy Inflation,” JCAP [**1006**]{}, 009 (2010) \[arXiv:0907.2916 \[hep-th\]\]. R. Flauger, D. Green and R. A. Porto, “On squeezed limits in single-field inflation. Part I,” JCAP [**1308**]{}, 032 (2013) \[arXiv:1303.1430 \[hep-th\]\].
M. Mirbabayi, F. Schmidt and M. Zaldarriaga, “Biased Tracers and Time Evolution,” arXiv:1412.5169 \[astro-ph.CO\]. E. Dimastrogiovanni, M. Fasiello and M. Kamionkowski, “Imprints of Massive Primordial Fields on Large-Scale Structure,” arXiv:1504.05993 \[astro-ph.CO\]. M. Musso, “A new diagrammatic representation for correlation functions in the in-in formalism,” JHEP [**1311**]{}, 184 (2013) \[hep-th/0611258\].
[^1]: In many of the existing alternatives to inflation the fluctuating fields see an effective de Sitter background metric (see [@Hinterbichler:2011qk] and references therein), and hence our conclusions remain unmodified.
[^2]: One such example was proposed in [@cft]. There it was assumed that a sector of theory is described both at super- and sub-horizon scales by a CFT that is conformally coupled to the metric. We think one can consider a more general case where the coupling of the ultraviolet CFT to the metric is not conformal and hence it flows to a QFT with a mass gap of order $H$ on curved background. The above arguments would then apply in this case.
[^3]: The latter case can be characterized by the condition that the constant adiabatic mode of Weinberg [@Weinberg_adia] is a sub-dominant solution to the linearized field equations.
[^4]: Notice that in this equation we treat $r$ non-perturbatively.
[^5]: One may wonder if the denominator can be equal to zero for some momentum configurations. It is equal to $P_{\hat\sigma}(q) P_{\zeta}(q) \xi_{\sigma}({\bf k}_1) \xi_{\sigma}({\bf k}_3)$ and therefore it can vanish only when the numerator vanishes as well. Note that in this case $r$ cannot be determined using .
[^6]: For a discussion of unitarity constraints on single field inflation see [@anal].
[^7]: For a study of the impact of heavy fields on correlators involving tensor modes see [@Dimastrogiovanni:2015pla].
[^8]: It was also argued that they are distinguishable due to the non-locality of the real space correlator. However, almost all terms in the above expansion lead to large distance (non-local) correlations when Fourier transformed with respect to $q$ (see the discussion of section \[sec:ope\]).
|
---
abstract: 'Anticipating the intentions of vulnerable road users (VRUs) such as pedestrians and cyclists is critical for performing safe and comfortable driving maneuvers. This is the case for human driving and, thus, should be taken into account by systems providing any level of driving assistance, from advanced driver assistant systems (ADAS) to fully autonomous vehicles (AVs). In this paper, we show how the latest advances on monocular vision-based human pose estimation, [[[*i.e.*]{}]{}]{} those relying on deep Convolutional Neural Networks (CNNs), enable to recognize the intentions of such VRUs. In the case of cyclists, we assume that they follow traffic rules to indicate future maneuvers with arm signals. In the case of pedestrians, no indications can be assumed. Instead, we hypothesize that the walking pattern of a pedestrian allows to determine if he/she has the intention of crossing the road in the path of the ego-vehicle, so that the ego-vehicle must maneuver accordingly ([[[*e.g.*]{}]{}]{} slowing down or stopping). In this paper, we show how the same methodology can be used for recognizing pedestrians and cyclists’ intentions. For pedestrians, we perform experiments on the JAAD dataset. For cyclists, we did not found an analogous dataset, thus, we created our own one by acquiring and annotating videos which we share with the research community. Overall, the proposed pipeline provides new state-of-the-art results on the intention recognition of VRUs.'
author:
- 'Zhijie Fang, Antonio M. López[^1].[^2]'
bibliography:
- 'IEEEabrv.bib'
- 'root.bib'
title: '**Intention Recognition of Pedestrians and Cyclists by 2D Pose Estimation**'
---
Autonomous Driving, ADAS, Computer Vision, Pedestrians Intention Recognition, Cyclists Intention Recognition
[Zhijie Fang]{} received his B.S. degree in Automation from South China Agricultural University, Guangzhou, China, in 2011. He received his M.S. degree in Control Science and Control Engineering from South China University of Technology, Guangzhou, China, in 2014. And he is currently a Ph.D. Candidate in the Department of Computer Science at the Universitat Autònoma de Barcelona (UAB). His research interests are in the areas of computer vision and pedestrian behavior analyses.
[Antonio M. López]{} is the PI of the Autonomous Driving lab of the Computer Vision Center (CVC) at the UAB, where he has also a tenure position as associated professor of the Comp. Sci. Dpt. Antonio has a long research trajectory at the intersection of computer vision, computer graphics, machine learning and autonomous driving. Antonio has been deeply involved in the creation of the SYNTHIA dataset and the CARLA open-source simulator. He is actively working with industry partners to bring state-of-the-art techniques to the field of autonomous driving. Antonio is granted by the Catalan ICREA Academia program.
[^1]: Zhijie and Antonio are with the Dpt. of Computer Science at the Universitat Autònoma de Barcelona (UAB), and with the Computer Vision Center (CVC) at the UAB. e-mail: {zfang, antonio}@cvc.uab.es
[^2]: We acknowledge the financial support by the Spanish project TIN2017-88709-R (MINECO/AEI/FEDER, UE). Antonio thanks the financial support by ICREA under the ICREA Academia Program. Zhijie thanks the Chinese Scholarship Council, Grant 201406150062. We thank the Generalitat de Catalunya CERCA Program and its ACCIO agency. We thank the student Arjun Gupta who annotated part of CASR during an internship at CVC.
|
---
abstract: 'We use the stability conditions of the Landau parameters for the symmetric nuclear matter and pure neutron matter to calculate the critical densities for the Skyrme type effective nucleon-nucleon interactions. We find that the critical density can be maximized by adjusting appropriately the values of the enhancement factor $\kappa$ associated with isovector giant dipole resonance, the quantity $L$ which is directly related to the slope of the symmetry energy and the Landau parameter $G_0^\prime$. However, restricting $\kappa$, $L$ and $G_0^\prime$ to vary within acceptable limits reduces the maximum value for the critical density $\tilde\rho_{cr}$ by $\sim 25\%$. We also show that among the various quantities characterizing the symmetric nuclear matter, $\tilde\rho_{cr}$ depends strongly on the isoscalar effective mass $m^*/m$ and surface energy coefficient $E_s$. For realistic values of $m^*/m$ and $E_s$ we get $\tilde\rho_{cr} = 2\rho_0$ to $ 3\rho_0$ ($\rho_0 = 0.16$fm$^{-3}$).'
author:
- 'B. K. Agrawal, S. Shlomo, and V. Kim Au'
title: Critical densities for the Skyrme type effective interactions
---
There are many different parameterizations of the Skyrme type effective nucleon-nucleon interaction as recently reviewed in Ref. [@Bender03]. Values of the Skyrme parameters are usually obtained by fitting the results of the Hartree-Fock calculations for the binding energies, charge radii and spin-orbit splittings of a few closed shell nuclei to the observed ones using a least square procedure. The Skyrme parameters are also constrained to yield appropriate values for some properties of the infinite symmetric nuclear matter at the saturation density $\rho_{nm}$. Over time, several improvements were made by appropriately treating the center of mass correction to the binding energy, modifying the spin-orbit part of the interaction and including in the least square fit the equation of state (EOS) for the pure neutron matter obtained from a realistic interaction [@Chabanat97; @Chabanat98]. Nevertheless, not all the Skyrme parameters are well determined in this way.
Recently, it has been shown in Ref. [@Margueron02] that a more realistic parameterization of the Skyrme interaction can be obtained by subjecting it to stability requirements of the EOS defined by the inequality conditions for the Landau parameters for symmetric nuclear matter and pure neutron matter. In other words, few of the Skyrme parameters not well determined by the experimental data used in the least square procedure can be restricted by requiring that the inequality conditions are satisfied up to a maximum value for a nuclear matter density, also referred to as the critical density $\rho_{cr}$. A very recent systematic study carried out in Ref. [@Stone03] using several Skyrme interactions indicates that the density dependence of the symmetry energy coefficient $J$ plays a critical role in determining the properties of neutron star. Out of 87 different parameterizations for the Skyrme interaction considered in Ref. [@Stone03] only 27 of them, having a positive slope for the symmetry energy coefficient at nuclear matter densities $\rho$ up to $3\rho_0$ ($\rho_0 = 0.16$fm$^{-3}$), are found to be suitable for the neutron star model. Thus, it appears that the parameters of the Skyrme interactions can be better constrained by combining the findings of Refs. [@Margueron02] and [@Stone03].
In the present work we study the dependence of $\rho_{cr}$ on the nuclear matter saturation density $\rho_{nm}$, binding energy per nucleon $B/A$, isoscalar effective mass $m^*/m$, incompressibility coefficient $K_{nm}$, surface energy $E_s$, and symmetry energy coefficient $J$, as also considered in Ref. [@Margueron02]. It must be emphasized that we determine $\rho_{cr}$ in terms of the enhancement factor $\kappa$, the coefficient $L = 3\rho dJ/d\rho$ and the Landau parameter $G_0^\prime$ (at $\rho_{nm}$) instead of the combinations $t_ix_i$ ($i=1, 2, $ and 3) of the Skyrme parameters $t_i$ and $x_i$ as used in Ref. [@Margueron02]. Unlike the combinations $t_ix_i$, the quantities $\kappa$, $L$ and $G_0^\prime$, which can be expressed in terms of the Skyrme parameters, are related to some physical processes. The enhancement factor $\kappa$ accounts for the deviations from the Thomas-Reiche-Kuhn (TRK) sum rule in the case of the isovector giant dipole resonance. The value of $\kappa $ at the $\rho_{nm}$ is expected to be $\sim 0.5$ [@Chabanat97; @Keh91]. The slope of the symmetry energy coefficient must be positive for $\rho $ up to $ 3\rho_0$, so that resulting Skyrme interaction can be suitable for the study of the properties of neutron stars [@Stone03]. The Landau parameter $G_0^\prime$ must be positive at $\rho \le \rho_{nm}$ in order to appropriately describe the position of the isovector $M1$ and Gamow-Teller states [@Keh91]. We find that by restricting the values of $\kappa$, $L$ and $G_0^\prime$ within the acceptable limits, the maximum critical densities are lowered by about $25\%$ compared to the ones obtained without such restrictions [@Margueron02]. The constrains proposed in the present work not only maximizes the value of the critical density, but also ensures that the resulting Skyrme interaction can be used to study the bulk properties of finite nuclei as well as those of neutron stars.
Recently [@Bender03; @Bender02] a generalized form for the Skyrme energy density functional has been obtained using the Hohenberg-Kohn-Sham approach. However, in the present work we restrict ourselves to the energy density functional associated with the commonly used Skyrme interaction [@Chabanat97; @Vautherin72], $$\begin{aligned}
\label{v12}
&V_{12}&= t_0\left (1+x_0 P_{12}^\sigma\right )\delta({\bf r}_1-{\bf
r}_2)\nonumber\\
&&+\frac{1}{2}t_1\left (1+x_1 P_{12}^\sigma\right )
\times \left[\overleftarrow{k}_{12}^2\delta({\bf r}_1-{\bf r}_2)+\delta({\bf
r}_1-{\bf r}_2)\overrightarrow{k}_{12}^2\right] \nonumber\\
&&+t_2\left (1+x_2 P_{12}^\sigma\right )\overleftarrow{k}_{12}\delta({\bf r}_1-{\bf r}_2)\overrightarrow{k}_{12} \nonumber\\
&&+ \frac{1}{6}t_3\left (1+x_3 P_{12}^\sigma\right )\rho^\alpha\left(\frac{{\bf r}_1+{\bf r}_2}{2}\right )
\delta({\bf r}_1-{\bf r}_2) \nonumber\\
&&+iW_0\overleftarrow{k}_{12} \delta({\bf r}_1-{\bf
r}_2)(\overrightarrow{\sigma_1}+
\overrightarrow{\sigma_2})\times \overrightarrow{k}_{12}\end{aligned}$$ where, $P_{12}^\sigma$ is the spin exchange operator, $\overrightarrow{\sigma}_i$ is the Pauli spin operator, $\overrightarrow{k}_{12} = -i(
\overrightarrow{\nabla}_1-\overrightarrow{\nabla}_2)/2$ and $\overleftarrow{k}_{12} =
-i(\overleftarrow{\nabla}_1-\overleftarrow{\nabla}_2)/2\, .$ Here, the right and left arrows indicate that the momentum operators act on the right and on the left, respectively. The Skyrme parameters $t_i$, $x_i$ and $\alpha$ for a fixed value of $W_0$ can be expressed in terms of the quantities associated with the symmetric nuclear matter as follows [@Chabanat97; @Margueron02; @Gomez92]. $$\begin{aligned}
t_0&=&\frac{8}{\rho_{nm}}\left [\frac{\left (-{B/A}+\left (2m/m^*-3\right )\left (\hbar^2/10m\right )k_f^2\right )\left (\frac{1}{27}K_{nm}-\left (1-6m^*/5m\right )\left (\hbar^2/9m^*\right )
k_f^2\right )}{-{B/A}+\frac{1}{9}K_{nm}-\left (4m/3m^*-1\right )\left
(\hbar^2/10m\right )k_f^2}\right . \nonumber\\
&+&\left . \left (1-\frac{5m}{3m^*}\right )\frac{\hbar^2}{10m}k_f^2\right ],
\label{t0}\end{aligned}$$ $$t_1=\frac{2}{3}\left [T_0 + T_s\right ],$$ $$t_2=t_1+\frac{8}{3}\left[\left (
\frac{1}{4}t_0+\frac{1}{24}t_3\rho_{nm}^\alpha\right )
\frac{2m^*}{\hbar^2}\frac{k_f}{\pi^2}+G_0^\prime\right ]
\frac{\hbar^2}{m^*\rho_{nm}},
\label{t2}$$ $$\label{t3}
t_3=\frac{16}{\rho_{nm}^{\alpha+1}}\frac{\left (-{B/A}+\left (2m/m^*-3\right )\left (\hbar^2/10m\right )k_f^2\right )^2}
{-{B/A}+\frac{1}{9}K_{nm}-\left (4m/3m^*-1\right )\left (\hbar^2/10m\right )k_f^2},$$ $$x_0=\frac{4}{t_0\rho_{nm}}\left [\frac{\hbar^2}{6m}k_f^2 -\frac{1}{24}t_3
(x_3+\frac{1}{2})\rho_{nm}^{\alpha+1} + \frac{1}{24}\left (t_2\left
(4+5x_2\right )-3t_1x_1\right )\rho_{nm} k_f^2 -J \right]- \frac{1}{2},$$ $$x_1=\frac{1}{t_1}\left [4\frac{\hbar^2 \kappa}{m\rho_{nm}}-t_2(2+x_2)\right ] -2,
\label{x1}$$ $$x_2=\frac{1}{4t_2}\left [8T_0-3t_1-5t_2\right ],$$ $$x_3=-\frac{8}{\alpha t_3\rho_{nm}^{\alpha+1}}\left
[\frac{\hbar^2}{6m}k_f^2-\frac{1}{12}\left ((4+5 x_2)t_2-3t_1 x_1\right
)\rho_{nm} k_f^2-3 J +L\right ]-\frac{1}{2},
\label{x3}$$ $$\alpha=\frac{{B/A}-\frac{1}{9}K_{nm}+\left (4m/3m^*-1\right )\left (\hbar^2/10m\right )k_f^2}
{-{B/A}+\left (2m/m^*-3\right )\left (\hbar^2/10m\right )k_f^2},
\label{alpha}$$ where,
$$T_0=\frac{1}{8}\left (3t_1+(5+4 x_2)t_2\right )=
\frac{\hbar^2}{m\rho_{nm}}\left (\frac{m}{m^*}-1\right ),
\label{T0}$$
$$T_s=\frac{1}{8}\left [9t_1-(5+4 x_2)t_2\right ],
\label{TS}$$
and $$k_{f}= \left (\frac {3\pi^2}{2}\rho_{nm} \right)^{1/3}.$$ In Eqs. (\[t0\])-(\[alpha\]), the various quantities characterizing the nuclear matter are the binding energy per nucleon $B/A$, isoscalar effective mass $m^*/m$, nuclear matter incompressibility coefficient $K_{nm}$, symmetry energy coefficient $J$, the coefficient $L$ which is directly related to the slope of the symmetry energy coefficient ($L=3\rho dJ/d\rho$), enhancement factor $\kappa$ and Landau parameter $G_0^\prime$. All these quantities are taken at the saturation density $\rho_{nm}$. It must be pointed out that we include consistently all terms in the energy density functional. In particular, the expression for the parameter $G_0^\prime$ used in the Eq. (\[t2\]) is consistent with the energy density functional obtained using Eq. (\[v12\]). For a more generalized Skyrme energy density functional, the expression for $G_0^\prime$ should be appropriately modified following Ref. [@Bender02]. Once, $T_0$ is known, $T_s$ can be calculated for a given value of the surface energy $E_s$ as [@Margueron02], $$\label{Es}
E_s=8\pi r_0^2\int_0^{\rho_{nm}} d\rho
\left [\frac{\hbar^2}{36m}-\frac{5}{36}T_0\rho+\frac{1}{8}T_s\rho
-\frac{m^*}{\hbar^2}V_{\rm so}\rho^2\right ]^{1/2}
\left [B(\rho_{nm})/A-B(\rho)/A\right ]^{1/2}$$ where, $B(\rho)/A$ is the binding energy per nucleon given by, $$\frac{B(\rho)}{A}=-\left [\frac{3\hbar^2}{10m^*}k_f^2+\frac{3}{8}t_0\rho+\frac{1}{16}t_3\rho^{\alpha+1} \right ]$$ and, $$\begin{aligned}
r_0=\left [ \frac{3}{4\pi\rho_{nm}}\right ]^{1/3}, \quad
V_{\rm so}=\frac{9}{16}{W_0}^2.\end{aligned}$$ The manner in which Eqs. (\[t0\]) - (\[alpha\]) can be used to evaluate the Skyrme parameters $t_i$, $x_i$ and $\alpha$ is as follows. First, the parameters $t_0$ and $\alpha$ can be obtained from Eqs. (\[t0\]) and (\[alpha\]), respectively. Then, the parameter $t_3$ can be calculated using Eq. (\[t3\]). Next, $T_0$ and $T_s$ can be calculated using Eqs. (\[T0\]) and (\[Es\]), respectively. Once, the combinations $T_0$ and $T_s$ of the Skyrme parameters are known, one can calculate the remaining parameters in the following sequence, $t_1$, $t_2$, $x_2$, $x_1$, $x_3$ and $x_0$.
The stability criteria requires that [@Migdal67], $$\label{ic}
{\cal X}_l > -(2l + 1),$$ where, ${\cal X}_l$ stands for the Landau parameters $F_l$, $F_l^\prime$, $G_l$ and $G_l^\prime$ for a given multipolarity $l$. The Skyrme interaction only contains monopolar and dipolar contributions to the particle-hole interaction so that all Landau parameters are zero for $l > 1$. Thus, there are 12 different Landau parameters, i.e., $F_l$, $F_l^\prime$, $G_l$ and $G_l^\prime$ ($l = 0, 1$) for the symmetric nuclear matter and $F_l^{(n)}$, $G_l^{(n)}$ ($l= 0, 1$) for the pure neutron matter. Each of these Landau parameters must satisfy the inequality condition given by Eq. (\[ic\]). Using the expressions for the Landau parameters in terms of the Skyrme parameters, given in Refs. [@Margueron02], one can obtain the values of the Landau parameters at any density for a given set of the Skyrme parameters. Thus, the critical density which is nothing but the maximum density up to which all the inequality conditions are met can be easily determined.
We have compiled the values of $\kappa$, $L$ and $G_0^\prime$ for several parameterization of the Skyrme interaction presented in Refs. \[2,3,11-16\]. We find that the values of $\kappa$, $L$ and $G_0^\prime$ vary over a wide ranges $0 - 2$, $40 - 160$ MeV and $-0.15 - 1.0$, respectively. This is due to the fact that the experimental data used in the least-square procedure to fit the values of the Skyrme parameters can not constrain well the values of these quantities. These quantities can be constrained by requiring a reasonable value for the critical density. In what follows, we shall refer to the set of standard values for six quantities $\rho_{nm}= 0.16 $fm$^{-3}$, ${B/A} = 16$ MeV, $K_{nm}
=230 $ MeV, $m^*/m =0.7$, $E_s =18$ MeV and $J =32 $MeV in short as the STD values for the nuclear matter input as used in Ref. [@Margueron02].
We now study the dependence of $\rho_{cr}$ on the $\rho_{nm}$, $B/A $, $m^*/m$, $K_{nm}$, $E_s$ and $J$. For a given set of values for these quantities, we calculate the maximum value of $\rho_{cr}$ by varying $\kappa$, $L$ and $G_0^\prime$ at the $\rho_{nm}$ within acceptable limits. We denote the maximum value of the critical density by $\tilde\rho_{cr}$. It must be emphasized here that the present work differs from the one performed in Ref. [@Margueron02] by the fact that we constrain the values of $\kappa$, $L$ and $G_0^\prime$, whereas, in Ref. [@Margueron02], the value of $\tilde\rho_{cr}$ was calculated by varying the combinations $t_ix_i$ ($i=1, 2,$ and 3) of Skyrme parameters with no restrictions. We find that if the space of the parameters $t_ix_i$ is not restricted, it may lead to an unreasonable values of $\kappa$ and $L$. As an example, for the STD values of nuclear matter input, $\tilde\rho_{cr}$ becomes $3.5\rho_0$ for $\kappa = 1.0$, $L = 36$ MeV and $G_0^\prime = 0.20$ (at $\rho_0$). The value of $G_0^\prime$ seems reasonable, but the value of $\kappa =
1.0$ is a little too large [@Chabanat97; @Keh91]. Further, we find that for $\rho > \rho_0$ the value of $L$ decreases with increasing $\rho$ and it becomes negative for $\rho > 1.6\rho_0$, which makes the interaction not favorable for the neutron star model.
In Fig. \[rhocr\_std\] we have displayed the results for $\tilde\rho_{cr}$ obtained by varying the various quantities associated with the nuclear matter around their standard values. To calculate $\tilde\rho_{cr}$ we allow for $0.25 \le \kappa \le 0.5 $, $0 \le L\le
100 $ MeV and $0 \le G_0^\prime\le 0.5 $ at the $\rho_{nm}$. We further demand that $L > 0$ at $3\rho_0$. Fig. \[rhocr\_std\] shows that $\tilde\rho_{cr}$ depends strongly on $m^*/m$ and $E_s$. Whereas, $\tilde\rho_{cr}$ depends weakly on $\rho_{nm}$, ${B/A}$ and $K_{nm}$ and it is almost independent of $J$. These features for $\tilde\rho_{cr}$ are qualitatively similar to the ones presented in Ref. [@Margueron02]. However, due to the restrictions imposed on the values of $\kappa$, $L$ and $G_0^\prime$, the values of $\tilde\rho_{cr}$ becomes smaller than that obtained in Ref. [@Margueron02] by up to $25\%$. For $m^*/m = 0.6$ (0.7) and keeping all the other nuclear matter quantities equal to their standard values, we get $\tilde\rho_{cr} =
4.5\rho_0$ ($2.8\rho_0$) compared to $6\rho_0$ ($3.5\rho_0$) obtained in Ref. [@Margueron02].
It may be instructive to present the values of $\kappa$, $L$ and $G_0^\prime$ required to obtained $\tilde\rho_{cr}$ for a given set of values for the nuclear matter input. We find that $\kappa$ lies in the range of $0.45 - 0.5$ for variations in the nuclear matter input by up to $\pm 15\%$ relative to their standard values. These values of $\kappa$ clearly reflects the fact that restricting $\kappa$ to take values in the range of $0.25 - 0.5$ delimits the $\tilde\rho_{cr}$ to a lower value. In Figs. \[L-nm\]a and \[L-nm\]b we have plotted the values of $L$ and $G_0^\prime$ (at $\rho_{nm}$), respectively, which are needed to yield the $\tilde\rho_{cr}$. We see that $L$ varies from $20$ to $ 60$ MeV for different values of the nuclear matter input. For the STD values of nuclear matter input we find that $L = 47$ MeV at $\rho = \rho_0$. This value is quite large compared to the values of $L = 35, 27$ and 16 MeV associated with the the Skz0, Skz1 and Skz2 interactions [@Margueron02], respectively, which were obtained for the same standard values of the nuclear matter input, but by varying the combinations $t_1 x_1$ and $t_2 x_2$ with no restrictions and keeping $t_3 x_3$ fixed to some arbitrary values. We see from Fig. \[L-nm\]b that, except for $J$, the value of $G_0^\prime$ (at $\rho_{nm}$) depends strongly on the values of the various quantities associated with nuclear matter. The dependence of $G_0^\prime$ on the surface energy coefficient $E_s$ is the most pronounced one. We note that $E_s$ is mainly determined by the ground state properties of light nuclei. Thus, the center of mass correction to the binding energy and charge radii must be appropriately taken into account as they are very important for light nuclei and may affect the values obtained for $E_s$. We see from Fig. \[L-nm\]b that $G_0^\prime$ tends to vanish rapidly with increasing $E_s$.
As pointed out earlier (see from Fig. \[rhocr\_std\]), the dependence of $\tilde\rho_{cr}$ on $\rho_{nm}$, $J$, and ${B/A}$ is quite weak. Thus, it may be sufficient to calculate $\tilde\rho_{cr}$ as a function of $m^*/m$, $E_s$ and $K_{nm}$ only. In Fig. \[efm\_es\_k230\] we display our results for the variation of $E_s$ versus $m^*/m$, obtained for fixed values of $\tilde\rho_{cr}$ with the remaining nuclear matter quantities kept equal to their standard values. It can be seen from Fig. \[efm\_es\_k230\] that for a fixed value of $\tilde\rho_{cr}$, $E_s$ decreases with the increase in $m^*/m$. It is quite interesting to note that for $E_s = 18 \pm 1$ MeV (as most of the Skyrme interactions yield), $\tilde\rho_{cr} = 2\rho_0$ and $3\rho_0$ for $m^*/m = 0.72 -
0.85$ and $0.63 - 0.73$, respectively. For $\tilde\rho_{cr} = 4\rho_0$ one must have $m^*/m \sim 0.65$ for not too low value of $E_s$. The value of $m^*/m$ is also constrained by the centroid energy of the isoscalar giant quadrupole resonance [@Reinhard99_1] which favors $m^*/m \ge 0.7$. Thus, for reasonable values of $E_s$ and $m^*/m$, one may obtain a Skyrme interaction with $\tilde\rho_{cr} = 2\rho_0$ to $3\rho_0$.
In summary, we have used the stability conditions of the Landau parameters for the symmetric nuclear matter and pure neutron matter to calculate the critical densities for the Skyrme type effective nucleon-nucleon interactions. We find that the critical density can be maximized by appropriately adjusting the values of the enhancement factor $\kappa$, coefficient $L$ and the Landau parameter $G_0^\prime$ as these quantities are not well determined by the Skyrme parameters, conventionally obtained by fitting the experimental data for the ground state properties of finite nuclei. We exploit the fact that i) The value of $\kappa$ should be in the range of $0.25 - 0.5$, needed to describe the TRK sum rule for the isovector giant dipole resonance [@Chabanat97; @Keh91], ii) $L > 0$ for $ 0 \le \rho \le 3\rho_0$; a condition necessary for a Skyrme interaction to be suitable for studying the properties of neutron star [@Stone03]. and iii) $G_0^\prime > 0$ in order to reproduce the energies of the isovector $M1$ and Gamow-Teller states [@Keh91]. The maximum value of the critical density so obtained is lower by up to $25\%$ compared to the ones obtained without any such restrictions [@Margueron02]. We show that the critical density obtained for realistic values of the surface energy coefficient ($E_s = 18 \pm 1$ MeV) and isoscalar effective mass ($m^*/m = 0.7 \pm 0.1$) lie in the range of $2\rho_0$ to $3\rho_0$. Finally, we would like to remark that the pairing correlations were not included in the present work. It may be worthwhile to investigate the influence of the pairing correlations on the various quantities associated with the nuclear matter which may alter the value of the critical density.
This work was supported in part by the US Department of Energy under Grant No. DOE-FG03-93ER40773 and by the National Science Foundation under the Grant No. PHY-0355200.
[99]{} M. Bender, P-H Heenen and P-G Reinhard, Rev. Mod. Phys. [**75**]{}, 121 (2003). E. Chabanat, P. Bonche, P. Haensel, J. Meyer and R. Schaeffer, Nucl. Phys. [**A627**]{}, 710 (1997).
E. Chabanat, P. Bonche, P. Haensel, J. Meyer and R. Schaeffer, Nucl. Phys. [**A635**]{}, 231 (1998).
J. Margueron, J. Navarro and Nguyen Van Giai, Phys. Rev. C [**66**]{}, 014303 (2002).
J. Rikovska Stone, J. C. Miller, R. Koncewicz, P. D. Stevenson and M. R. Strayer, Phys. Rev. C [**68**]{}, 034324 (2003).
Keh-Fie Liu, Zhonga Ma, Qingbiao Shen and S. A. Mozkowski Nucl. Phys. [**A534**]{}, 1 (1991).
M. Bender, J. Dobaczewski, J. Engel and W. Nazarewicz, Phys. Rev. C[**65**]{}, 054322 (2002). D. Vautherin and D. M. Brink, Phys. Rev. C [**5**]{}, 626 (1972). J. M. G. Gomez, C. Prieto and J. Navarro, Nucl. Phys. [**A 549**]{}, 125 (1992). A. B. Migdal, [*Theory of finite Fermi system and application to atomic nuclei*]{}, London, 1967. J. Friedrich and P.-G. Reinhard, Phys. Rev. C [**33**]{}, 335 (1986). Nguyen Van Giai and H. Sagawa, Phys. Lett. [**106B**]{}, 379 (1981). J. Mayer, P.-G. Reinhard, M. Bender, and W. Nazarewicz(private communication). P.-G. Reinhard, D. J. Dean, W. Nazarewicz, J. Dobaczewski, J. A. Maruhnand and M. R. Strayer, Phys. Rev. C [**60**]{}, 014316 (1999). F. Tondeur, M. Brack, M. Farine and J. M. Pearson, Nucl. Phys. [**A420**]{}, 297 (1984). P.-G. Reinhard and H. Flocard, Nucl. Phys. [**A584**]{}, 467 (1995). P.-G. Reinhard, Nucl. Phys. [**A649**]{}, 305c (1999).
|
---
abstract: 'A comparison theorem for the isoperimetric profile on the universal cover of surfaces evolving by normalised Ricci flow is proven. For any initial metric, a model comparison is constructed that initially lies below the profile of the initial metric and which converges to the profile of the constant curvature metric. The comparison theorem implies that the evolving metric is bounded below by the model comparison for all time and hence converges to the constant curvature profile. This yields a curvature bound and a bound on the isoperimetric constant, leading to a direct proof that the metric converges to the constant curvature metric.'
address: 'Department of Mathematics, University of California San Diego'
author:
- Paul Bryan
bibliography:
- 'refs.bib'
title: Curvature bounds via an isoperimetric comparison for Ricci flow on surfaces
---
Introduction {#sec-1}
============
The Ricci flow is the nonlinear geometric parabolic evolution equation $$\label{eq:RF}
\begin{cases}
\pd{t} \metric &= -2\ricci(t) \\
\metric(0) &= \metric_0
\end{cases}$$ for a smooth family of Riemannian metrics $\metric(t)$ on a smooth manifold $M$ with Ricci curvature $\ricci(t)$ and an arbitrary smooth initial metric $\metric_0$. Here we are interested in the case of closed surfaces, that is, $2$ dimensional, compact manifolds $M$ without boundary. The results here pertain to the *normalised* flow, preserving the $2$-dimensional volume of $M$. After rescaling the initial metric to have volume $4\pi$ and applying the Gauss-Bonnet formula, the normalised flow on surfaces takes the form $$\label{eq:NRF}
\pd{t}\metric = -2(\gausscurv - \avg{\gausscurv})\metric$$ where $\gausscurv$ is the Gaussian curvature and $\avg{\gausscurv} = \frac{1}{4\pi} \int_M \gausscurv \metricmeasure{\metric}$ is the average Gauss curvature on $M$ [@MR2729306]. An important consequence of writing the equation in this way is that it may be lifted to the universal cover $\unicover{M} \to M$. That is, the pullback metric $\unicover{\metric}(t) = \pullback{\uniproj} \metric (t)$ also evolves according to equation .
The main theorem of this paper is a comparison theorem for the isoperimetric profile of a surface with metric evolving by the normalised Ricci flow, generalising the comparison theory in [@MR2729306] for $M=\sphere^2$ to arbitrary closed surfaces. Recall, the isoperimetric profile is the least boundary area enclosing a given volume (see section \[sec:isoprofile\] for a precise definition).
\[Main Theorem \[thm:comparison\]\] Let $(M, \metric(t))$ be a Ricci flow of a closed surface and $(\unicover{M}, \unicover{\metric}(t))$ be the lift to the universal cover. Let $\phi: (0, \abs{\unicover{M}}) \times [0, T) \to \RR$ be a smooth, strictly positive, strictly concave function satisfying $$\pd[\phi]{t} \leq \phi'' \phi^2 - (\phi')^2 \phi + \phi'\left(4\pi - 2(1-{\operatorname{\lambda}})a_0\right) + (1-{\operatorname{\lambda}})\phi.$$ along with the asymptotic behaviour $$\limsup_{a\to 0} \frac{\phi(a, t)}{\sqrt{4\pi a}} \leq 1$$ and $$\limsup_{a\to \infty} \left(\isoprofile(a, t) - \phi(a,t)\right) \geq 0.$$
If the initial inequality, $\phi(a, 0) < \isoprofile_{\unicover{\metric}(0)} (a)$ for all $a \in (0, \abs{\unicover{M}})$ holds, then $\phi(a, t) \leq \isoprofile_{\unicover{\metric}(t)} (a)$ for all $a, t$ with strict inequality if the inequality in is strict.
As an application, by a standard bootstrapping argument, a proof of the Hamilton-Chow theorem (Theorem \[thm:convergence\]) is obtained directly as a corollary of Theorem \[thm:comparison\]. This achieved by suitable choices of comparison functions, leading to explicit curvature bounds and bounds on the isoperimetric constant as described in sections \[sec:models\] and \[sec:convergence\].
\[[@MR954419; @MR1094458]\] \[thm:convergence\] Given any initial metric $\metric_0$, there exists a unique solution to the normalised Ricci flow existing for all time $t \in [0,\infty)$ and such that $\metric(t) \to_{C^{\infty}} \metric_{{\operatorname{\lambda}}}$, the metric of constant curvature $\gausscurv = 1-{\operatorname{\lambda}}$ as $t\to\infty$.
The use of the isoperimetric profile here is an extension from the 2-sphere to arbitrary surfaces of the results in [@MR2729306], which in turn are based on the isoperimetric estimates in [@MR1369139]. We begin in section \[sec:isoprofile\] with a treatment of the isoperimetric profile of surfaces, deriving a viscosity equation via variational techniques which forms the heart of the comparison theorem. The comparison theorem is proven in section \[sec:comparison\] by coupling the time-variation of the isoperimetric profile under the normalised Ricci flow with the spatial viscosity equation, yielding the parabolic version of the viscosity equation. Section \[sec:models\] is devoted to constructing suitable model comparisons. The construction on the $2$-sphere was given in [@MR2729306] and is briefly described. Curiously, the most difficult case to deal with is for surfaces of genus ${\operatorname{\lambda}}> 1$, which historically was perhaps the easiest case by applying the maximum principle and introducing a potential function [@MR954419]. For initially negatively curved surfaces however, the model given here is quite appealing. Finally in section \[sec:convergence\], the boostrapping convergence argument is briefly described.
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to thank Professor Ben Andrews for his generous guidance and support supervising my Ph.D. thesis on which this paper is based. I would also like to thank Professor Gang Tian and BICMR for sponsoring a very enjoyable stay in Beijing where my thoughts on variational techniques were greatly clarified on the banks of WeiMing Lake. Finally, this paper was written during my time as SEW Assistant Professor at UCSD.
The Isoperimetric Profile {#sec-2}
=========================
\[0b158ef8-2a10-4c0e-9d45-0de44bbddf06\]
\[sec:isoprofile\]
Definition and Basic Properties {#sec-2-1}
-------------------------------
The *isoperimetric profile*, $\isoprofile_M : (0, \abs{M}) \to \RR_+$ of $M$ is defined by $$\isoprofile_M(a) = \inf \left\{\abs{\bdry{\Omega}} : \abs{\Omega} = a \right\}$$ where the infimum is taken over all relatively compact open sets $\Omega$ with smooth boundary. Such $\Omega$ are said to be *admissible regions*. If $\Omega$ is an admissible region such that $\isoprofile_M(\abs{\Omega}) = \abs{\bdry{\Omega}}$, we will call $\Omega$ an *isoperimetric region*.
A basic theorem we will assume here is that for every $a\in(0,\abs{M})$, there exists a corresponding isoperimetric region (with smooth boundary apart from a set of Hausdorff dimension at most $n-7$ on an $n$ dimensional manifold) provided $M$ is either compact or co-compact. In particular, smooth isoperimetric regions exist on a closed surface and its universal cover equipped with the pull-back metric. The proof of this fact is a standard result of geometric measure theory [@MR2455580 pp. 128-129]. A simplified proof in the case of surfaces is given in [@MR1661278] using regularity techniques developed in [@MR1417428].
It will be important for us to understand the behaviour of the isoperimetric profile near the end points $\{0, \abs{M}\}$. In the situation where $M$ is compact, then the complement of an isoperimetric region is again an isoperimetric region, so the isoperimetric profile is symmetric about $\abs{M}/2$ and it suffices to consider only the behaviour near $0$. In the non-compact case, the behaviour near $0$ is the same as for the compact case, so let us begin with the behaviour near $0$.
\[thm:bdrybehaviour\] Let $M$ be a smooth Riemannian surface without boundary and such that $\sup_M \gausscurv < \infty$. Then the isoperimetric profile satisfies $$\isoprofile(a)= \sqrt{4\pi a} - \frac{\sup_M \gausscurv}{4\sqrt{\pi}} a^{3/2} + O(a^{5/2}) \quad \text{as} \quad a\to 0.$$
Small geodesic balls about any point $p$ are admissible regions. The result of [@MR0339002 Theorem 3.1] gives $\abs{B_r(p)} = \pi r^2 \left(1 - \frac{\gausscurv(p)}{12} r^2 + O(r^4)\right)$ and $\abs{\bdry{B_r(p)}} = 2\pi r \left(1 - \frac{\gausscurv(p)}{6} r^2 +O(r^4)\right)$. The upper bound follows since $\abs{\bdry{B_r(p)}} \geq \isoprofile(\abs{B_r(p)})$.
To prove the lower bound, first choose $a_0$ sufficiently small to ensure that $\isoprofile(a_0)$ is much smaller than the injectivity radius of $M$. Then an isoperimetric region $\Omega_0$ corresponding to $a_0$ lies inside a geodesic ball about some point $p$ (width is bounded above by perimeter for surfaces). Since geodesic balls are simply connected and $\gausscurv \leq \gausscurv_0 = \sup_M\gausscurv$, the Bol-Fiala inequality (see [@MR0500557]) then gives $$\begin{aligned}
\isoprofile(a_0) \geq \sqrt{4\pi a_0 - \gausscurv_0 a_0^2} = \sqrt{4\pi a_0} - \frac{\gausscurv_0}{4\sqrt{\pi}} a_0^{3/2} + O(a_0^2).\end{aligned}$$
Next, we have the asymptotics of the isoperimetric profile near $\infty$ for non-compact $\unicover{M}$.
\[thm:asymptotics\_large\] Let $M$ be a closed, genus ${\operatorname{\lambda}}\geq 1$ surface with metric $\metric$, normalised to have $\abs{M} = 4\pi$ and let $\uniproj: \unicover{M} \to M$ be the universal cover of $M$ equipped with the pull-back metric $\unicover{\metric} = \uniproj^{\star}\metric$. Then $$\isoprofile_{\unicover{\metric}}(a) \to C \sqrt{4\pi a - (1-{\operatorname{\lambda}}) a^2}$$ as $a\to\infty$ for some $C>0$.
By the uniformisation theorem, $\unicover{\metric}$ is conformal to a metric of constant curvature so that $$\unicover{\metric} = \phi\metric_{1-{\operatorname{\lambda}}}$$ with $\metric_{1-{\operatorname{\lambda}}}$ the metric of constant curvature $1-{\operatorname{\lambda}}$ and $\phi$ a positive function $\phi: \unicover{M} \to \RR$ invariant under the deck transformation group of $\unicover{M}$. Thus $\phi$ is uniformly bounded above and below.
The isoperimetric inequality for simply connected Riemannian surfaces of constant curvature $1-{\operatorname{\lambda}}$ implies that the isoperimetric profile $I_{1-{\operatorname{\lambda}}}$ of the constant curvature metric $\metric_{1-{\operatorname{\lambda}}}$ is given by $$\isoprofile_{1-{\operatorname{\lambda}}}(a) = \sqrt{4\pi a - (1-{\operatorname{\lambda}})a^2}.$$
Since $\unicover{\metric}$ is conformal to $\metric_{1-{\operatorname{\lambda}}}$ with conformal factor $\phi$ uniformly bounded, we have $$\begin{aligned}
\frac{1}{C_1} \abs{\bdry{\Omega}}_{\metric_{1-{\operatorname{\lambda}}}} &\leq \abs{\bdry{\Omega}}_{\unicover{\metric}} \leq C_1 \abs{\bdry{\Omega}}_{\metric_{1-{\operatorname{\lambda}}}} \\
\frac{1}{C_2} \abs{\Omega}_{\metric_{1-{\operatorname{\lambda}}}} &\leq \abs{\Omega}_{\unicover{\metric}} \leq C_2 \abs{\Omega}_{\metric_{1-{\operatorname{\lambda}}}}\end{aligned}$$ for constants $C_1,C_2>0$ which gives the result.
It would be preferable if we didn’t have to refer to the uniformisation theorem, as then the results of this paper provide a proof of the uniformisation theorem. In the case ${\operatorname{\lambda}}=0$, we have such a proof since $M$ is compact. In the case ${\operatorname{\lambda}}=1$, the result of [@MR1354290] implies that $$\isoprofile(a) \to C\sqrt{a}$$ as $a\to \infty$ for $0 < C \leq 4\pi$ with $C=4\pi$ if and only if $M$ is flat. This is precisely the required asymptotics in the theorem for ${\operatorname{\lambda}}=1$ surfaces obtained without requiring the use of the uniformisation theorem. The only problem here then is for ${\operatorname{\lambda}}>1$ surfaces. The volume growth of $\unicover{M}$ is controlled by the number of generators for the fundamental group, but controlling the perimeter is rather more difficult. I am not aware of an applicable result for ${\operatorname{\lambda}}>1$ surfaces, though such a result would be interesting.
Variational Formulae and Consequences {#sec-2-2}
-------------------------------------
Our techniques are based on applying the standard variational formula for isoperimetric regions, with a slight change in the second variation, obtained by applying the Gauss-Bonnet theorem. Let us briefly recall the applicable variational formulae and describe the approach used here in obtaining the second variation.
Let $\Omega_0$ be an isoperimetric region and $\Omega_{\epsilon}$ a smooth normal variation with variational vector field $\eta \nor$ for a smooth function $\eta : \bdry{\Omega_0} \to \RR$. The first variation formulae are $$\begin{aligned}
\label{eq:firstvar_bdry}
\pd{\epsilon} \abs{\bdry{\Omega_{\epsilon}}} &= \int_{\bdry{\Omega_{\epsilon}}} \eta \curvecurv \\
\intertext{and}
\label{eq:firstvar_vol}
\pd{\epsilon} \abs{\Omega_{\epsilon}} &= \int_{\bdry{\Omega_{\epsilon}}} \eta .\end{aligned}$$ where $\curvecurv$ is the geodesic curvature of $\bdry{\Omega_{\epsilon}}$. In particular, the vanishing of the first variation for all functions $\eta$ such that $\int_{\bdry{\Omega_{\epsilon}}} \eta = 0$ (area preserving variations) implies that $\curvecurv$ is constant.
For the second variation, we need only consider unit-speed variations ($\eta \equiv 1$) and so immediately conclude the second variation for area, $$\label{eq:secondvar_vol}
\pdd{\epsilon} \abs{\Omega_{\epsilon}} = \int_{\bdry{\Omega_{\epsilon}}} \curvecurv.$$
For the second variation of boundary length, it suits our purposes to first apply the Gauss-Bonnet formula and then differentiate equation . Thus $$\begin{split}
\pdd{\epsilon} \abs{\bdry{\Omega_{\epsilon}}} &= \pd{\epsilon} \int_{\bdry{\Omega_{\epsilon}}} \curvecurv \\
&= \pd{\epsilon} \left(2\pi\eulerchar{\Omega(\epsilon)} - \int_{\Omega(\epsilon)}\gausscurv_M\right)
\end{split}$$ where $\eulerchar{\Omega{\epsilon}}$ is the Euler characteristic of $\Omega_{\epsilon}$ which is independent of $\epsilon$ since each $\phi_{\epsilon}$ is a diffeomorphism and $\gausscurv_M$ is the Gauss curvature of $M$. The latter has no explicit dependence on $\epsilon$ and so the Reynold’s Transport Theorem (or differentiating under the integral sign) yields $$\label{eq:secondvar_bdry}
\pdd{\epsilon} \abs{\bdry{\Omega_{\epsilon}}} = - \int_{\bdry{\Omega_{\epsilon}}} \gausscurv_M.$$
Our approach is based on weak differential inequalities for the isoperimetric profile arising from the variational formulae.
A function $f: (a,b) \to \RR$ has weak derivatives satisfying $$\pd[f^-]{x} \leq C_1 \leq \pd[f^+]{x} \quad \text{and} \quad \pdd[f]{x} \leq C_2$$ in the *support* (or sometimes *Calabi*) sense at $x_0$ if $f$ supports a smooth function $\phi$ at $x_0$ ($f(x_0)=\phi(x_0)$ and $f(x) \leq x_0$ for $x$ near $x_0$) such that $$\pd[\phi]{x} (x_0) = C_1 \quad \text{and} \quad \pdd[\phi]{x} (x_0) = C_2.$$
\[[@MR875084] (see also [@MR2229062] pp. 249-251)\] \[prop:support\_iso\_nobdry\] For each $a_0 \in (0,\abs{M})$, let $\Omega_0$ be a corresponding isoperimetric region with constant curvature $\curvecurv(a_0)$ along the boundary. Then the isoperimetric profile satisfies $$\pd[\isoprofile^-]{a} \leq \curvecurv(a_0) \leq \pd[\isoprofile^+]{a} \quad \text{and} \quad \pdd[\isoprofile]{a} \leq \frac{-1}{\isoprofile^2}\left(\curvecurv(a_0)^2\isoprofile + \int_{\bdry{\Omega_0}} \gausscurv_M\right).$$ in the support sense. Moreover, if $\gausscurv_M \geq \gausscurv_0$, the function $$a \mapsto \isoprofile(a)^2 + \gausscurv_0 a^2$$ is concave, hence $\isoprofile^2$ is locally Lipschitz and in particular $\isoprofile$ is continuous.
Since we are assuming $M$ is compact or co-compact, $\gausscurv_M$ is bounded hence $\isoprofile$ is continuous. Note also that since $\sqrt{-}$ is smooth away from $0$, by Rademacher’s theorem, $\isoprofile$ is differentiable almost everywhere.
\[cor:iso\_concave\_nobdry\] With the notation of the proposition, if $\gausscurv_0\geq0$ then $\isoprofile$ is concave and so too is $\isoprofile^2$. If the inequality is strict, then $\isoprofile$ and $\isoprofile^2$ are strictly concave.
The last results of this section concern the topology of isoperimetric regions. We generally don’t have a priori control over the topology of isoperimetric regions and so we don’t know the precise form of the differential inequality for $\isoprofile$ because of the integral over the unknown regions $\Omega_0$. However, there is a useful sufficient condition for obtaining control of the topology of isoperimetric regions. The idea comes from [@MR1674097].
\[thm:curvetopology\] Let $a_0\in (0,\abs{M})$ and $\Omega_0$ a corresponding isoperimetric region. If there exists a strictly positive, strictly concave function $\phi: (0,\abs{M}) \to \RR$ supporting $I$ at $a_0$ ($\phi(a_0) = \isoprofile(a_0)$ and $\phi(a) \leq \isoprofile(a)$ for all $a\in(0,\abs{M})$), then $\Omega_0$ is connected. If $M$ is compact then $\Omega_0$ has connected complement.
It is worth pointing out that while the conclusion of the lemma is local, pertaining to a particular value of $a_0$ and corresponding isoperimetric region, the hypotheses are global in nature in that we need a *globally defined* supporting function $\phi$ (not just in a neighbourhood of $a_0$).
First note that since $\phi > 0$ on $(0,\abs{M})$, $\phi \leq \isoprofile$, and $\isoprofile(0) = 0$, we have $\phi(0) = 0$. Thus since $\phi$ is strictly concave, $\phi$ is strictly subadditive.
Now suppose $\Omega_0$ is not connected. Then we can write $\Omega_0 = \Omega_1\cup\Omega_2$ with $\Omega_1 \cap \Omega_2 = \emptyset$. Since $\bdry{\Omega_0}$ is smooth we must have $\bdry{\Omega_0} = \bdry{\Omega_1} \cup \bdry{\Omega_2}$ and $\bdry{\Omega_1} \cap \bdry{\Omega_2} = \emptyset$. Thus we have $\abs{\Omega_0} = \abs{\Omega_1} + \abs{\Omega_2}$ and $\abs{\bdry{\Omega_0}} = \abs{\bdry{\Omega_1}} + \abs{\bdry{\Omega_2}}$, and all of these are non-zero. But then we get $$\begin{aligned}
\phi\left(\abs{\Omega_1}\right) + \phi\left(\abs{\Omega_2}\right) &\leq \abs{\bdry{\Omega_1}} + \abs{\bdry{\Omega_2}} \\
&= \abs{\bdry{\Omega_0}} \\
&= \phi\left(\abs{\Omega_0}\right) \\
&= \phi\left(\abs{\Omega_1} + \abs{\Omega_2}\right) \\
&< \phi\left(\abs{\Omega_1}\right) + \phi\left(\abs{\Omega_2}\right).\end{aligned}$$ This is a contradiction, so $\Omega_0$ is connected.
If $M$ is compact, then $M\setminus\Omega_0$ is also an isoperimetric region with $\abs{M\setminus\Omega_0} = \abs{M}-\abs{\Omega_0} = \abs{M} - a_0$ and $\isoprofile(\abs{M} - a_0) = \abs{\bdry{M\setminus\Omega_0}} = \abs{\bdry{\Omega_0}} = \isoprofile(a_0)$. Reflecting $\phi$ about $a=\abs{M}/2$ gives a function satisfying the hypothesis of the proposition at $\abs{M}-a_0$ hence $M\setminus\Omega_0$ is also connected.
\[cor:sphere\_iso\_conn\] With the hypothesis of lemma \[thm:curvetopology\], if $M$ is diffeomorphic to $\sphere^2$ then $\Omega_0$ is simply connected.
Follows from the Jordan curve theorem for $\sphere^2$.
\[cor:plane\_iso\_conn\] With the hypothesis of lemma \[thm:curvetopology\], if $M$ is diffeomorphic to $=\RR^2$ then $\Omega_0$ is simply connected.
Since $\RR^2$ is not compact, we cannot immediately conclude that $\Omega_0$ has connected complement as before. To achieve this result, first note that if $M$ is $\RR^2$, then $\phi$ is a (strictly) positive concave function on $(0,\infty)$ and hence is strictly increasing. Since $\Omega_0$ is connected, topologically it is a disc with finitely many discs removed. Let $\Omega_1$ denote the interior of the external boundary of $\Omega_0$, i.e. $\Omega_1$ is equal to $\Omega_0$ with the “holes” filled in. Then $\Omega_1$ has strictly larger area than $\Omega_0$ and strictly smaller boundary length. But then $$\phi(\abs{\Omega_0}) = \isoprofile(\abs{\Omega_0}) = \abs{\bdry{\Omega_0}} > \abs{\bdry{\Omega_1}} \geq \isoprofile(\abs{\Omega_1}) \geq \phi(\abs{\Omega_1})$$ contradicting that $\phi$ is increasing. Therefore $\RR^2 \setdiff \Omega_0$ is connected and now the Jordan curve theorem implies $\Omega_0$ is simply connected.
Let us finish by noting that in positive curvature, we have complete knowledge of the topology of isoperimetric regions.
\[cor:isoregions\_positive\_curvature\] If $M$ is diffeomorphic to either $\sphere^2$ or $\RR^2$ (for instance if $M$ is the universal cover of a closed surface) equipped with any metric (not necessarily the pull-back from a compact surface) and $\gausscurv_0 > 0$, then all isoperimetric regions are simply connected.
By Corollary \[cor:iso\_concave\_nobdry\], $\isoprofile$ is strictly concave so the hypotheses of Corollaries \[cor:sphere\_iso\_conn\] and \[cor:plane\_iso\_conn\] are satisfied at any $a_0\in (0,\abs{M})$ by choosing $\phi=\isoprofile$ itself.
I don’t know if this result can be extended to $\gausscurv_0 = 0$ since in this case $\isoprofile$ is not necessarily strictly concave.
A viscosity equation for the isoperimetric profile {#sec-2-3}
--------------------------------------------------
The results in this section formalise some of the ideas used in [@MR2729306]. We obtain a differential inequality in the viscosity sense for the isoperimetric profile of a surface. This is somewhat dual to the results in the previous section and those in [@MR2041647] in that we assert conditions which *lower* supporting functions must satisfy as opposed to the aforementioned results which assert the existence of an *upper* supporting function with bounds on the derivatives. The methods however, are essentially the same and the support inequality implies the viscosity inequality. As the isoperimetric profile is defined as an extrema, viscosity equations turn out to be well suited to this situation. Indeed, viscosity equations were introduced in [@MR690039] to study Hamilton-Jacobi equations, also arising from optimisation problems. A central feature of viscosity equations, that forms the basis of the comparison theorem \[thm:comparison\], is that they enjoy a maximum principle. See [@MR1351007] for more on viscosity equations.
A lower semi-continuous function $f: (a,b) \to \RR$ is a viscosity super-solution of the 2nd order differential equation $$A(x, f, f', f'') = 0$$ if for every $x_0\in (a,b)$ and every $C^2$ function $\phi$ such that $\phi(x_0) = f(x_0)$ and $\phi(x) \leq f(x)$ in a neighbourhood of $x_0$, we have $A(x_0, \phi(x_0), \phi'(x_0), \phi''(x_0)) \geq 0$. An upper semi-continuous function is a viscosity sub-solution if the same statements hold with all the inequalities reversed.
For $f$ a viscosity super(sub)-solution of $A(x, f, f', f'') = 0$, we will abuse notation slightly and write $A(x, f, f', f'') \geq 0 (\leq 0)$ (in the viscosity sense).
In the definition, the existence of a lower (upper) supporting function at a point is not required, rather we assert that if such a supporting function exists, it must satisfy the appropriate differential inequality. For instance, the absolute value function, $x\mapsto \abs{x}$ is a viscosity sub-solution of $f'=1$ even though no $C^1$ upper support function exists at $x=0$.
\[thm:viscosity\_nobdry\] The isoperimetric profile is a viscosity super-solution of $$\label{eq:spatial_viscosity}
-\left(\isoprofile'' \isoprofile^2 + (\isoprofile')^2 \isoprofile + \int_{\bdry{\Omega_0}} \gausscurv_M\right) = 0$$ where $\Omega_0$ is any isoperimetric region corresponding to $a_0$ ($\abs{\Omega_0} = a_0$ and $\isoprofile(a_0) = \abs{\bdry{\Omega_0}}$) and $\gausscurv_M$ is the gauss curvature of $M$.
In particular, if $\gausscurv_M \geq \gausscurv_0$ is bounded below on $M$, then $$-\left(\isoprofile'' \isoprofile^2 + (\isoprofile')^2 \isoprofile + \gausscurv_0 \isoprofile\right) \geq 0$$ in the viscosity sense.
The integral term in the first equation is difficult to deal with; even though the Gauss curvature $\gausscurv$ is a given function on the ambient space $M$, we don’t have any a-priori knowledge of $\Omega_0$. Nevertheless, the first form will be the most useful to us when considering the Ricci flow, since the integral term will also appear in the time variation of isoperimetric regions under the Ricci flow allowing us to connect the spatial variational formulae with the time variational formulae.
The isoperimetric profile is continuous by proposition \[prop:support\_iso\_nobdry\] and the remark following it.
Let $\phi$ be a smooth function defined on a neighbourhood of $a_0 \in (0, \abs{M})$ such that $\phi\leq\isoprofile$ and $\phi(a_0) = \isoprofile(a_0)$. Let $\Omega_0$ be an isoperimetric region corresponding to $a_0$. Choose a unit speed normal variation of $\bdry{\Omega_0}$ and define $$f(\epsilon) = \abs{\bdry{\Omega_{\epsilon}}} - \phi(\abs{\Omega_{\epsilon}}).$$ Then we have $$f(\epsilon) \geq \isoprofile(\abs{\Omega_{\epsilon}}) - \phi(\abs{\Omega_{\epsilon}}) \geq 0$$ and $$f(0) = \abs{\bdry{\Omega_0}} - \phi(\abs{\Omega_0}) = \isoprofile(\abs{\Omega_0}) - \phi(\abs{\Omega_0}) = 0.$$ Thus $0$ is a minima of $f$ so that $\inpd[f]{\epsilon}(0) = 0$ and $\inpdd[f]{\epsilon}(0) \geq 0$. Now we use the first variation formula to compute $$\pd[f]{\epsilon} = \int_{\bdry{\Omega_{\epsilon}}} \curvecurv - \phi' \abs{\bdry{\Omega_{\epsilon}}}$$ which at $\epsilon = 0$ gives $$0 = \int_{\bdry{\Omega_0}} \curvecurv - \phi'(a_0) \abs{\bdry{\Omega_0}} = (\curvecurv - \phi'(a_0)) \abs{\bdry{\Omega_0}}$$ since $\curvecurv$ is constant along $\bdry{\Omega_0}$. Thus $\curvecurv = \phi'(a_0)$ along $\bdry{\Omega_0}$.
The second variation gives $$\begin{split}
0 \leq \pdd[f]{\epsilon} &= \pdd{\epsilon} \abs{\bdry{\Omega_{\epsilon}}} - \phi'' (\pd{\epsilon} \abs{\Omega_{\epsilon}})^2 - \phi' \pdd{\epsilon} \abs{\Omega_{\epsilon}} \\
&= - \int_{\bdry{\Omega_{\epsilon}}} \gausscurv - \phi'' (\abs{\bdry{\Omega_{\epsilon}}})^2 - \phi' \int_{\bdry{\Omega_{\epsilon}}} \curvecurv \\
&= - \int_{\bdry{\Omega_0}} \gausscurv - \phi''(a_0) \phi^2(a_0) - (\phi')^2(a_0) \phi(a_0).
\end{split}$$ recalling that $\phi(a_0) = \abs{\Omega_0}$ and using that $\curvecurv = \phi'(a_0)$ along $\bdry{\Omega_0}$ as just obtained from the first variation.
A comparison theorem {#sec-3}
====================
\[9b6175ce-34f4-40c4-9b36-e7588e0a53d7\]
\[sec:comparison\]
Comparison equation under the Ricci flow {#sec-3-1}
----------------------------------------
Let us now couple the spatial viscosity equation with the Ricci flow. For this we need to know the time-variation of isoperimetric regions under the Ricci flow. It is quite remarkable that this is possible at all and heavily relies on the fact that $M$ is $2$ dimensional. It would be interesting to see if similar results hold in higher dimensions, though this seems unlikely unless some topological and/or curvature restrictions are imposed.
We first need the parabolic version of viscosity equations.
A lower semi-continuous function $f: (a,b) \times [0, T) \to \RR$ is a viscosity super-solution of the 2nd order parabolic equation $$\pd[f]{t} + A(x, t, f, f', f'') = 0$$ if for every $(x_0, t_0) \in (a,b) \times [0, T)$ and every $C^2$ function $\phi$ such that $\phi(x_0, t_0) = f(x_0, t_0)$ and $\phi(x, t) \leq f(x, t)$ for $x$ in a neighbourhood of $x_0$ and $t\leq t_0$ near $t_0$, we have $\pd[\phi]{t} (x_0, t_0) + A(x_0, t_0, \phi, \phi' , \phi'') \geq 0$. An upper semi-continuous function is a viscosity sub-solution if the same statements hold with the inequalities reversed.
\[thm:ricci\_iso\_viscosity\] Let $M$ be a closed surface of genus ${\operatorname{\lambda}}$, $\metric(t)$ a solution of the normalised Ricci flow on $M$ and $\unicover{\metric}(t) = \uniproj^{\ast}\metric (t)$ the corresponding solution on the universal cover $\uniproj: \unicover{M} \to M$. For any $a_0$, let $\chi_0$ be the Euler characteristic of $\Omega_0$ an isoperimetric region corresponding to $a_0$. Then the isoperimetric profile, $\isoprofile_{\unicover{\metric}(t)}$ satisfies $$\label{eq:temporal_viscosity}
\pd{t}\isoprofile - \left[\isoprofile'' \isoprofile^2 + (\isoprofile')^2 \isoprofile + (4\pi\chi_0 - 2(1-{\operatorname{\lambda}})a)\isoprofile' + (1-{\operatorname{\lambda}}) \isoprofile\right] \geq 0$$ in the viscosity sense.
For convenience sake, let us write $\abs{\cdot}_t = \abs{\cdot}_{\unicover{\metric}(t)}$ and $\isoprofile_t = \isoprofile_{\unicover{\metric}(t)}$. Let $\phi$ be a $C^2$ function such that $\phi(a_0, t_0) = \isoprofile_{t_0}(a_0)$ and $\phi \leq \isoprofile$ for $a$ near $a_0$ and $t\leq t_0$ near $t_0$. We need to show that $\phi$ satisfies the differential inequality .
We compute the time variation of isoperimetric regions. Given $a_0$, let $\Omega_0\subset \unicover{M}$ be an isoperimetric region in $\unicover{M}$ with respect to the metric $\unicover{\metric}(t_0)$. That is $\abs{\Omega_0}_{t_0} = a_0$ and $\abs{\bdry{\Omega_0}}_{t_0} = \phi(a_0, t_0)$. Since $\isoprofile_t(a) \geq \phi(a,t)$ for $t\leq t_0$ and $a$ near $a_0$, we have $$\abs{\bdry{\Omega_0}}_t \geq \phi\left(\abs{\Omega_0}_t, t\right)$$ for $t\leq t_0$, and equality holds when $t=t_0$. Since both sides of this equation are differentiable in $t$, it follows that under the normalised Ricci flow, $$\label{eq:timeineq1}
\frac{\partial}{\partial t}\Big|_{t=t_0} \abs{\bdry{\Omega_0}}_t \leq \pd[\phi]{t} (a_0,t_0) + \phi'(a_0,t_0) \pd{t}\Big|_{t=t_0} \abs{\Omega_0}_{t}.$$ The time derivative on the left can be computed as follows: Parametrise $\bdry{\Omega_0}$ by $\gamma: u \in \sphere^1 \mapsto M$ and write $\gamma_u = \gamma_{\ast} \pd{u}$. Then recalling that the metric evolves by the normalised Ricci flow, $\pd{t}\unicover{\metric} = -2(\gausscurv - (1-{\operatorname{\lambda}}))\unicover{\metric}$, we obtain $$\begin{aligned}
\pd{t}\Big|_{t=t_0} \abs{\bdry{\Omega_0}} &= \pd{t} \int_{\bdry{\Omega_0}} \sqrt{\unicover{\metric}_t(\gamma_u,\gamma_u)}\,du = -\int_{\bdry{\Omega_0}}(\gausscurv_M - (1 - {\operatorname{\lambda}}))ds \\
& = -\int_{\bdry{\Omega_0}} \gausscurv_M\,ds + (1-{\operatorname{\lambda}})\phi (a_0, t_0),\end{aligned}$$ where $ds$ is the arc-length element along $\bdry{\Omega_0}$.
For the right hand side, by differentiating the determinant and using the normalised Ricci flow equation again, we have $\pd{t} \measure_{\unicover{\metric}} = -2(\gausscurv_M - (1-{\operatorname{\lambda}}))\measure_{\unicover{\metric}}$ where $\measure_{\unicover{\metric}}$ is the measure on $\unicover{M}$ induced by the metric $\unicover{\metric}$. Thus, $$\pd{t} \Big|_{t=t_0} \abs{\Omega_0}_t = -2\int_{\Omega_0}(\gausscurv_M - (1-{\operatorname{\lambda}}) d\measure_{\unicover{\metric}(t_0)}.$$ Writing $\chi_0 = \chi(\Omega_0)$ the Euler characteristic of $\Omega_0$ and applying the Gauss-Bonnet theorem yields $$\pd{t}\Big|_{t=t_0} \abs{\Omega_0}_t = 2(1-{\operatorname{\lambda}})\abs{\Omega_0} - 2\left(2\pi\chi_0 - \int_{\bdry{\Omega_0}} \curvecurv\,ds\right) = 2(1-{\operatorname{\lambda}})a_0 -4\pi\chi_0 +2 \int_{\bdry{\Omega_0}}\curvecurv\,ds,$$ were $\curvecurv$ is the geodesic curvature of the curve $\bdry{\Omega_0}$. Thus the inequality becomes $$\label{eq:timeineq}
-\int_{\bdry{\Omega_0}}\gausscurv_M\,ds + (1-{\operatorname{\lambda}})\phi \leq \pd{t}\phi + \phi'\left(2(1-{\operatorname{\lambda}})a_0 -4\pi\chi_0 + 2 \int_{\bdry{\Omega_0}}\curvecurv\,ds\right).$$
Now recall that theorem \[thm:viscosity\_nobdry\] states that for each time $t$, the isoperimetric profile $\isoprofile_t$ satisfies $$-\left(\isoprofile'' \isoprofile^2 + (\isoprofile')^2 \isoprofile + \int_{\bdry{\Omega_0}} \gausscurv_M\right) \geq 0$$ in the viscosity sense. Since at $a_0$, $\phi(-, t_0)$ is a supporting function for $\isoprofile_{t_0}(-)$ we also have $$\label{eq:spaceineq}
\phi'' \phi^2 + (\phi')^2 \phi \leq -\int_{\bdry{\Omega_0}} \gausscurv_M.$$ Also, the vanishing of the first spatial variation gives $\curvecurv = \phi'(a_0)$ is constant along $\bdry{\Omega_0}$ and so $$\label{eq:curvfirstvar}
\int_{\bdry{\Omega_0}} \curvecurv \, ds = \phi(a_0) \phi' (a_0).$$
Putting together the inequalities and and using we obtain $$\begin{split}
\pd[\phi]{t} &\geq -\int_{\bdry{\Omega_0}}\gausscurv_M\,ds + (1-{\operatorname{\lambda}})\phi - \phi'\left(2(1-{\operatorname{\lambda}})a_0 - 4\pi\chi_0 + 2\phi\phi'\right) \\
&\geq \phi'' \phi^2 + (\phi')^2 \phi + (1-{\operatorname{\lambda}})\phi + \phi'\left(4\pi\chi_0 - 2(1-{\operatorname{\lambda}})a_0\right) - 2\phi(\phi')^2 \\
&= \phi'' \phi^2 - (\phi')^2 \phi + (1-{\operatorname{\lambda}})\phi + \phi'\left(4\pi\chi_0 - 2(1-{\operatorname{\lambda}})a_0\right)
\end{split}$$ which is the required inequality.
The viscosity equation includes the $\chi_0$ term which, without any topological knowledge of isoperimetric regions is essentially unknown and could a priori, take on any possible value. By Corollary \[cor:isoregions\_positive\_curvature\], in the particular case that $\gausscurv_M > 0$, we may conclude that $\chi_0 = 1$ for all $a_0$. In general however, we need not expect any particular bound on Euler characteristic from a curvature bound alone.
Even though the topological uncertainty is a real problem, for our purposes we may avoid it entirely by appealing to the underlying concavity of the isoperimetric profile. This is exploited in the next theorem, the comparison theorem, which is the central result of this paper.
\[thm:comparison\] Let $(M, \metric(t))$, $(\unicover{M}, \unicover{\metric}(t))$ be as in the previous theorem. Let $\phi: (0, \abs{\unicover{M}}) \times [0, T) \to \RR$ be a smooth, strictly positive, strictly concave function satisfying $$\label{eq:ricci_iso_diff_inequal}
\pd[\phi]{t} \leq \phi'' \phi^2 - (\phi')^2 \phi + \phi'\left(4\pi - 2(1-{\operatorname{\lambda}})a_0\right) + (1-{\operatorname{\lambda}})\phi.$$ along with the asymptotic behaviour $$\limsup_{a\to 0} \frac{\phi(a, t)}{\sqrt{4\pi a}} \leq 1$$ and $$\limsup_{a\to \infty} \left(\isoprofile(a, t) - \phi(a,t)\right) \geq 0$$
Then if the initial inequality, $\phi(a, 0) < \isoprofile_{\unicover{\metric}(0)} (a)$ for all $a \in (0, \abs{\unicover{M}})$ holds, the inequality $\phi(a, t) \leq \isoprofile_{\unicover{\metric}(t)} (a)$ holds for all $a, t$ with strict inequality if the inequality in is strict.
The large scale asymptotic requirements are rather imprecise because we don’t have a priori control over the constant $C$ in Theorem \[thm:asymptotics\_large\]. However, this will not prove problematic for us by Proposition \[prop:comparison\] below.
First suppose that we have strict inequality in the differential inequality and in the asymptotic inequalities. We argue by contradiction. The conditions $\phi(a, 0) < \isoprofile_{\unicover{\metric}(0)} (a)$ and $\phi(a, t) < \isoprofile_{\unicover{\metric}(t)} (a)$ for $a$ sufficiently close to $\{0, \abs{\unicover{M}}\}$ imply that if the theorem is false, there is a first time $t_0>0$ and an $a_0 \in (0, \abs{\unicover{M}})$ such that $\phi(a_0, t_0) = \isoprofile_{t_0} (a_0)$. Thus $\phi(a, t) \leq \isoprofile_t(a)$ for $t\leq t_0$ with equality at $(a_0, t_0)$. Since $\phi$ is strictly concave, the hypotheses of Lemma \[thm:curvetopology\] are satisfied, so $\Omega_0$ is simply connected and $\chi_0 = 1$.
But now observe that $\phi$ is a lower supporting function for $\isoprofile_t$ at $a_0$ and by theorem \[thm:ricci\_iso\_viscosity\], $$\pd[\phi]{t} \geq \phi'' \phi^2 - (\phi')^2 \phi + \phi'\left(4\pi - 2(1-{\operatorname{\lambda}})a_0\right) + (1-{\operatorname{\lambda}})\phi$$ a contradiction, hence the theorem is true when the inequalities are strict.
If any of the inequalities are not strict, define $$\phi_{\epsilon} = (1-\epsilon) \phi$$ for any $\epsilon$ with $0<\epsilon<1$. Then we have $\phi_{\epsilon} < \phi$ giving strict inequality for the asymptotics. We also have $$\begin{split}
&\pd[\phi_{\epsilon}]{t} - (\phi_{\epsilon}'' \phi_{\epsilon}^2 - (\phi_{\epsilon}')^2 \phi_{\epsilon}) - \phi_{\epsilon}'\left(4\pi - 2(1-{\operatorname{\lambda}})a_0\right) - (1-{\operatorname{\lambda}})\phi_{\epsilon} \\
&= (1-\epsilon)\left(\pd[\phi]{t} - (1-\epsilon)^2(\phi'' \phi^2 - (\phi')^2 \phi) - \phi'\left(4\pi - 2(1-{\operatorname{\lambda}})a_0\right) - (1-{\operatorname{\lambda}})\phi\right) \\
&\leq \epsilon(1-\epsilon)(2-\epsilon)(\phi^2\phi'' - (\phi')^2\phi) \\
&< 0
\end{split}$$ since $\phi''< 0$.
Thus $\phi_{\epsilon} (a,t) < \isoprofile (a,t)$ by the result for strict inequalities and the result follows by letting $\epsilon \to 0$.
It’s not entirely clear whether strict concavity may be relaxed to merely concavity. A strictly concave approximation to $\phi$ may increase $\phi$ violating the inequality $\phi \leq \isoprofile$.
Using the theorem, and the asymptotics of $\isoprofile$ from Theorem \[thm:bdrybehaviour\], $$\isoprofile(a) = \sqrt{4 \pi a}(1 - \frac{\sup_M\gausscurv}{8\pi} a + O(a^2))$$ as $a\to 0$, we may now obtain a curvature bound for $\unicover{\metric}(t)$ and hence for $\metric(t)$.
\[cor:ricci\_comp\_curv\_bnd\] With the notation of the previous theorem, $\phi$ satisfying the hypothesis of the theorem and such that $$\phi (a, t) = \sqrt{4\pi a}(1 - \frac{\gausscurv_0(t)}{8\pi} a + O(a^2)),$$ we have $$\sup_M \gausscurv_M(t) \leq \gausscurv_0(t).$$
The isoperimetric constant of a non-compact surface is defined to be $$\operatorname{\mathcal{I}}= \inf\left\{ \frac{\abs{\bdry{\Omega}}^2}{\abs{\Omega}} : \Omega \quad \text{admissible} \right\} = \inf \left\{\frac{\isoprofile(a)^2}{a} : 0 < a < \infty \right\}.$$ For a compact surface, the (modified) isoperimetric constant is defined by $$\operatorname{\mathcal{I}}= \inf\left\{ \frac{\abs{\bdry{\Omega}}^2}{\min(\abs{\Omega}, \abs{M \setdiff \Omega})} : \Omega \quad \text{admissible} \right\} = \inf \left\{ \frac{\isoprofile(a)^2}{a} : 0 < a < \frac{\abs{M}}{2} \right\}.$$
\[cor:isoconst\_bdd\] With the notation of the previous theorem and, $\phi$ satisfying the hypothesis of the theorem we have $$\operatorname{\mathcal{I}}_{\unicover{M}}(t) \geq \inf\left\{\frac{\phi(a, t)^2}{a} : 0 < a < \frac{\abs{\tilde{M}}}{2}\right\}.$$
\[rem:isoconst\_bdd\] Note that area (2 dimensional volume) on $\unicover{M}$ equipped with the pull-back metric $\pullback{\uniproj} \metric$ grows like the growth of the fundamental group, but boundary length can’t be controlled so easily. For instance, the torus with arbitrarily small ratio of principal radii may be equipped with the flat metric, giving control of the isoperimetric constant on $\unicover{M}$, but with arbitrarily small isoperimetric constant on $M$. Note however, that by normalising the area of $M$ to $4\pi$, we avoid this issue, and I conjecture that the isoperimetric constant of $M$ may be bounded below by that of $\unicover{M}$ for any ${\operatorname{\lambda}}\geq 1$. For the matter at hand, when ${\operatorname{\lambda}}>0$ (so that $\unicover{M}$ is not compact), we can’t immediately transfer control of the isoperimetric constant on $\unicover{M}$ to control of the isoperimetric constant on $M$.
Let us finish this section by recording a useful result for surfaces of genus ${\operatorname{\lambda}}\geq 1$ that shows the large scale asymptotics of $\phi$ are superfluous.
\[prop:comparison\] Let $M$ be a closed surface of genus $\geq 1$ (so that $\unicover{M}$ is not compact). Let $\phi$ be a strictly positive, strictly concave function satisfying the differential inequality and the small scale asymptotics from the comparison theorem \[thm:comparison\]. Then if $\phi(a, 0) < \isoprofile_{\unicover{\metric}(0)}(a)$ for all $a\in (0,\infty)$, then $\phi(a, t) \leq \isoprofile_{\unicover{\metric}(t)} (a)$ for all $a,t$.
The only thing missing from Theorem \[thm:comparison\] is the large scale asymptotics. It is convenient to work with the function $v=\phi^2$. This satisfies $$\label{eq:ricci_isosq_diff_inequal}
\pd[v]{t} \leq v^2 \laplace \ln v + (4\pi - 2(1-{\operatorname{\lambda}})a) v' + 2(1-{\operatorname{\lambda}}) v.$$ For any $C>0$, define $$u_C(a, t) = C e^{2(1-{\operatorname{\lambda}})t}.$$ Then $u_C$ satisfies equality in equation . Since $u_C$ is constant for each fixed $t$ and $\isoprofile$ grows at least linearly as $a\to\infty$ by Theorem \[thm:asymptotics\_large\], we have also have $u_C(a, t) < \isoprofile_{\unicover{\metric}(t)}(a)$ for all $a$ large enough. Now take the harmonic mean, $$\operatorname{H}(a, t) = \left(\frac{1}{v(a,t)} + \frac{1}{u_C(a,t)}\right)^{-1}.$$ This has the property that for any $(a,t)$ we have $$v(a, t) = \lim_{C\to\infty} \frac{v(a, t) u_C(a, t)}{v(a, t) + u_C(a, t)} = \lim_{C\to\infty} \operatorname{H}(a, t).$$ Therefore to prove the result, we need to show $\operatorname{H}$ satisfies the hypotheses of theorem \[thm:comparison\] since this will give the inequality for $\operatorname{H}$ for every $C>0$ and so too for $v$ being the limit $C\to\infty$ of $\operatorname{H}$.
First, since $0 < \operatorname{H}\leq v, u_C$, the initial inequality $\operatorname{H}< \isoprofile_0$ is satisfied along with the necessary small and large scale asymptotics.
For strict concavity of $\operatorname{H}$, we use $$\operatorname{H}= \frac{u_C v}{u_C + v}$$ and $(u_C)' = 0$ to compute $$\begin{aligned}
\operatorname{H}' &= \frac{u_Cv'}{u_C + v} - \frac{u_C v v'}{(u_C + v)^2} \\
&= \frac{u_C^2 v'}{(u_C + v)^2}\end{aligned}$$ and so $$\operatorname{H}'' = \frac{u_C^2 v''}{(u_C + v)^2} - \frac{2u_C^2(v')^2}{(u_C + v)^3} < 0$$ by strict concavity of $v$ and positivity of $v, u_C$. Thus $H$ is strictly concave.
Now let us consider the differential inequality. Define $$L_{\pm} = \left(4\pi - 2(1-{\operatorname{\lambda}})a\right)\pd{a} \pm 2(1-{\operatorname{\lambda}}).$$ The differential inequality then reads $$\left(\pd{t} - L_-\right) v \leq v^2 \laplace \ln v.$$ For any function $f$ we have $$\label{eq:linopf}
\left(\pd{t} - L_{\pm}\right) \frac{1}{f} = -\frac{1}{f^2} \left(\pd{t} - L_{\mp}\right) f.$$ Applying equation to $\operatorname{H}= 1/f$ with $f = v^{-1} + u_C^{-1}$ gives $$\begin{aligned}
\left(\pd{t} - L_-\right) \operatorname{H}&= -\operatorname{H}^2 \left((\pd{t} - L_+) \frac{1}{v} + (\pd{t} - L_+) \frac{1}{u_C}\right) \\
&= -\operatorname{H}^2 \left((\pd{t} - L_+) \frac{1}{v} - 2(1-{\operatorname{\lambda}})\frac{1}{u_C}\right)\end{aligned}$$ since $L_{\pm} (u_C) = 0$ and $u_C$ satisfies equality in .
Next applying equation to $v^{-1}$ we get $$\begin{aligned}
\left(\pd{t} - L_-\right) \operatorname{H}&= \frac{\operatorname{H}^2}{v^2} (\pd{t} - L_-) v + 2\operatorname{H}^2 (1-{\operatorname{\lambda}})\frac{1}{u_C} \\
&\leq \frac{\operatorname{H}^2}{v^2} v^2 \laplace \ln v \\
&= \operatorname{H}^2 \laplace \ln v.\end{aligned}$$ since $(1-{\operatorname{\lambda}}) \leq 0$. Here the inequality is strict if $v$ (or equivalently $u$) satisfies strict inequality in the differential inequality. We want to show the right hand side is less than or equal to $\operatorname{H}^2 \laplace \ln \operatorname{H}$. We compute $$\begin{aligned}
\operatorname{H}^2 \laplace \ln \operatorname{H}&= \operatorname{H}\operatorname{H}'' - (\operatorname{H}')^2 \\
&= \frac{v u_C}{v+u_C} \left[\left(\frac{u_C}{v+u_C}\right)^2 v'' - \frac{2u_C^2}{(v+u_C)^3} (v')^2\right] - \left(\frac{u_C}{v+u_C}\right)^4 (v')^2 \\
&= \left(\frac{v u_C}{v + u_C}\right)^2 \left[\left(\frac{u_C}{v+u_C}\right) \frac{v''}{v} - \frac{2u_C v}{(v+u_C)^2} \frac{(v')^2}{v^2} - \frac{u_C^2}{(v+u_C)^2} \frac{(v')^2}{v^2}\right] \\
&= \operatorname{H}^2 \left[\left(\frac{u_C}{v+u_C}\right) \frac{v''}{v} - \left(\frac{(v+u_C)^2 - v^2}{(v+u_C)^2}\right) \frac{(v')^2}{v^2}\right] \\
&\geq \operatorname{H}^2 \left[ \frac{v''}{v} - \frac{(v')^2}{v^2}\right] = \operatorname{H}^2 \laplace \ln v\end{aligned}$$ where the inequality follows from the concavity of $v$ and the positivity of $v$ and $u_C$.
A connection with logarithmic porous media {#sec-3-2}
------------------------------------------
For positive functions $\phi$, the differential inequality $$\pd[\phi]{t} < \phi^2\phi'' - \phi(\phi')^2 + (4\pi - 2(1-{\operatorname{\lambda}})a)\phi' + (1-{\operatorname{\lambda}})\phi$$ is equivalent to the logarithmic porous media inequality $$\pd[u]{t} > \laplace \ln u.$$ To see this, observe that $$\phi^3 \laplace \ln \phi = \phi^2\phi'' - \phi(\phi')^2.$$ Letting $u = \phi^{-2}$ we have $\laplace \ln u = -2 \laplace \ln \phi$ and so $$\begin{split}
\pd[u]{t} &= \frac{-2}{\phi^3} \pd[\phi]{t} > \frac{-2}{\phi^3} \left[\phi^3 \laplace \ln \phi + (4\pi - 2(1-{\operatorname{\lambda}})a)\phi' + (1-{\operatorname{\lambda}})\phi\right] \\
&= \laplace \ln u + (4\pi - 2(1-{\operatorname{\lambda}})a)u' -2(1-{\operatorname{\lambda}})u.
\end{split}$$ A change of the independent variables $(a,t)$ can now be made to get rid of the lower order terms. This point of view may prove useful since the logarithmic porous media equation has been extensively studied, but we do not use it here.
Model solutions {#sec-4}
===============
\[396a40e0-d4d8-422d-a69b-4e25da6de9c3\]
\[sec:models\]
This section is devoted to exhibiting suitable comparison functions $\phi$ and hence curvature and isoperimetric bounds for metrics evolving by the normalised Ricci flow via Corollaries \[cor:ricci\_comp\_curv\_bnd\] \[cor:isoconst\_bdd\]. We will need to treat the cases ${\operatorname{\lambda}}=0, {\operatorname{\lambda}}=1, {\operatorname{\lambda}}>1$ separately. The next and final section briefly outlines how such bounds lead, via standard arguments, to the convergence results described in Theorem \[thm:convergence\].
genus 0 {#sec-4-1}
-------
In [@MR2729306], we showed that the isoperimetric profile of the Rosenau solution provided a suitable comparison solution. Let us briefly recall the result. The Rosenau solution is an explicit axially symmetric solution of the normalized Ricci flow on the two-sphere. The metric is given by $\bar g(t) = u(x,t)(dx^2+dy^2)$, where $(x,y)\in\RR\times[0,4\pi]$, and $$u(x,t) = \frac{\sinh(e^{-2t})}{2e^{-2t}\left(\cosh(x)+\cosh(e^{-2t})\right)}.$$ This extends to a smooth metric on the two-sphere at each time with area $4\pi$, and which evolves according to the normalized Ricci flow equation . A direct computation gives the isoperimetric profile, $$\label{eq:Rosenauprofile}
\varphi(a,t) = \sqrt{4\pi}\sqrt{\frac{\sinh(a e^{-2t})\sinh((1-a)e^{-2t})}{\sinh(e^{-2t})e^{-2t}}}.$$
By translating $t \mapsto t - t_0$ with $t_0$ chosen so that initial inequality of the isoperimetric profile holds, the comparison theorem leads to the following bounds for solutions of the normalised Ricci flow on the $2$-sphere:
\[thm:sphere\_bounds\] Let $\metric$ be a solution of the normalised Ricci flow on $\sphere^2$. Then there exists constants $A,C>0$ depending only on the metric at the initial time such that $$\sup_{\sphere^2} \gausscurv (t) \leq C e^{-At}.$$ There also exists a constant $\operatorname{\mathcal{I}}_0 > 0$, depending only on the initial metric $\metric_0$, such that $$\operatorname{\mathcal{I}}(t) > \operatorname{\mathcal{I}}_0$$ where $\operatorname{\mathcal{I}}(t)$ is the isoperimetric constant of $(\sphere^2, \metric(t))$.
genus 1 {#sec-4-2}
-------
Next, let us describe a comparison solution for the universal cover of surfaces of genus ${\operatorname{\lambda}}= 1$, i.e. for $\RR^2$.
Recall, we need to find a function satisfying the differential inequality $$\phi_t\geq \phi^2\phi''-\phi(\phi')^2+4\pi \phi'.$$ We look for solutions with equality. First, to simply matters, let $v=\phi^2$ which satisfies the equation $$v_t = vv''-(v')^2+4\pi v' = v^2\left(\frac{v'}{v} - \frac{4\pi}{v}\right)'.$$ Taking the Ansatz $v(a,t) = tV(a/t)$, we obtain an integrable equation, which adding in the limiting behaviour $V(0) = 0$, has the family of solutions $$V_C(z) = \frac{1}{C}\left(4\pi - \frac{1}{C}\right) \left(1 - e^{-Cz}\right) + \frac{z}{C}.$$ That is, we have $$\label{eq:ricci_plane_comparison}
v_C(a,t) = \frac{a}{C} + \frac{t}{C}\left(4\pi - \frac{1}{C}\right) \left(1 - e^{-\tfrac{Ca}{t}}\right).$$
We can now use $v_C$ as a comparison for ${\operatorname{\lambda}}= 1$ surfaces, as in the following:
\[thm:plane\_comparison\] Let $\metric(t)$ be any solution of the normalised Ricci flow on $M$ a closed, genus $1$ surface and let $\unicover{\metric} = \uniproj^{\ast}\metric$ be the pull back metric to the universal cover $\unicover{M} = \RR^2$. Then there exists a $C>0$ such that the function $\phi = \sqrt{v_c}$ where $v_c$ is defined by satisfies $\phi(a, t) < \isoprofile_{\unicover{\metric}} (a, t)$ for all $a \in (0,\infty)$ and $t \in [0, T)$. Therefore the Gauss curvature $\gausscurv$ of $M$ satisfies the bound $$\sup_M \gausscurv \leq \frac{A}{t}$$ for a constant $A>0$ depending only on the initial metric $\metric_0$.
We know that $v_C$ satisfies the differential inequality and it’s easy to see that $v_C$ is strictly concave, so we need to show that $v_C$ meets the other requirements for the comparison theorem in the form of Proposition \[prop:comparison\]. At $t=0$, we have $v_C(a,0) = \tfrac{a}{C}$ so by choosing $C$ large enough, we have the initial comparison since $\isoprofile \simeq \sqrt{C_1 a + C_2 a^2}$ as $a\to \infty$.
On the small scale we have $$\phi(a, t) = \sqrt{4\pi a} \left(1 - (\frac{C}{4} - \frac{1}{16\pi})\frac{1}{t} a + O(a^2)\right)$$ as required for the small scale asymptotics and also providing the stated curvature bound with $A = 2\pi C - 1/2$ (which is positive for $C > 1/4\pi$) by Corollary \[cor:ricci\_comp\_curv\_bnd\].
genus $>1$ {#sec-4-3}
----------
In this section, we construct the model comparison solution for the final case, ${\operatorname{\lambda}}> 1$. When $\sup_{M_0} \gausscurv > 0$, the construction is a little involved.
### $K < 0$ case {#sec-4-3-1}
First let us consider the case where $\sup_{M_0} \gausscurv \leq 0$, since it admits a simple, appealing comparison solution.
For any $A,C>0$, let $$\label{eq:ricci_hyperbolic_comparison}
v(a, t) = 4\pi a + B(t) a^2$$ with $$B(t) = ({\operatorname{\lambda}}- 1) - \frac{C}{1 + A e^{({\operatorname{\lambda}}- 1)t}}.$$ Direct computation shows that $v_C$ is a solution of the differential equation $$v_t = vv''-(v')^2 + (4\pi - (1-{\operatorname{\lambda}})a) v' + 2(1-{\operatorname{\lambda}})v,$$ which is the required equation for $v = \phi^2$ as in the genus $1$ case above.
\[thm:ricci\_hyperbolic\_curvature\_bound\_negative\] Let $\metric(t)$ be any solution of the normalised Ricci flow on $M$ a closed, genus $>1$ surface with $\sup_{M_0} \gausscurv \leq 0$ and let $\unicover{\metric} = \uniproj^{\ast}\metric$ the pull back to $\HH^2$ with $\uniproj: \HH^2 \to M$ the universal cover. Then for $\phi = \sqrt{v}$ where $v$ is defined by , there exists $A,C>0$ such that $\phi(a, t) < \isoprofile_{\unicover{\metric}} (a, t)$ for all $a \in (0,\infty)$. Therefore, the Gauss curvature $\gausscurv_M$ satisfies the bound $$\sup_{M_t} \gausscurv \leq C_1 e^{-C_2t}$$ for positive constants $C_1,C_2$.
Since the comparison function is a quadratic with zero constant term and linear coefficient equal to $4\pi$, the small scale asymptotics are satisfied providing that $B(t) \leq - \text{const} \sup_M\gausscurv$, by the asymptotics of the isoperimetric profile given in theorem \[thm:bdrybehaviour\]. Since we require $B(t)\geq 0$, this can only be achieved in the case $\sup_{M_t} \gausscurv \leq 0$ which is true by the maximum principle under the assumption $\sup_{M_0} \gausscurv \leq 0$. In this case, we choose $A,C$ large enough so that the initial comparison holds. Concavity is easily checked. Proposition \[prop:comparison\] completes the proof that $\phi(a, t) < \isoprofile_{\unicover{\metric}} (a, t)$ for all $a \in (0,\infty)$. The curvature bound now follows directly from Corollary \[cor:ricci\_comp\_curv\_bnd\].
### General case {#sec-4-3-2}
#### Stationary solution {#sec-4-3-2-1}
Recall we have the equation, $$\pd{t} v - \left\{vv'' - (v')^2 + [4\pi - 2(1-{\operatorname{\lambda}})a]v' + 2(1-{\operatorname{\lambda}})v\right\} \leq 0.$$ We can write this as $$\label{eq:hyperbolic_squared_diff_ineq}
\pd{t} v \leq v^2 \left(\frac{v'}{v} - \frac{4\pi - 2(1-{\operatorname{\lambda}})a}{v}\right)'.$$ Stationary solutions (with equality) to this equation that satisfy the conditions $v(0)=0$ and $\limsup_{x\to\infty} \tfrac{v(x)}{x^2} < \infty$ are given by $$\label{eq:hyperbolic_stationary}
\begin{split}
v_C(x) &= \frac{1}{C}[4\pi + \frac{2(1 - {\operatorname{\lambda}})}{C}][1 - e^{-Cx}] - \frac{2(1-{\operatorname{\lambda}})}{C^2} (Cx) \\
&= 4\pi x + \frac{1}{C}[4\pi + \frac{2(1 - {\operatorname{\lambda}})}{C}][1 - Cx - e^{-Cx}] \\
&= 4\pi x - \frac{1}{C}[4\pi + \frac{2(1 - {\operatorname{\lambda}})}{C}]\frac{(Cx)^2}{2} \\
& \quad + \frac{1}{C}[4\pi + \frac{2(1 - {\operatorname{\lambda}})}{C}][1 - Cx + \frac{(Cx)^2}{2} - e^{-Cx}]
\end{split}$$ for any $C\geq 0$. The last line is obtained from the Taylor expansion for $e^{-Cx}$. Each of the three expressions illustrates different properties of $v_C$. For instance, the first line shows that $v_C$ grows at most linearly. The second and third lines give the first and second order Taylor expansions with explicit remainders.
For later use, the first and second derivatives of $v_C$ are $$\begin{aligned}
\label{eq:hyperbolic_stationary_1st}
v_C' &= 4\pi + [4\pi + \frac{2(1 - {\operatorname{\lambda}})}{C}][-1 + e^{-Cx}] \\
\label{eq:hyperbolic_stationary_2nd}
v_C'' &= -C[4\pi + \frac{2(1 - {\operatorname{\lambda}})}{C}]e^{-Cx}.\end{aligned}$$ In particular, provided that $$C \geq C_{\text{crit}} = - \frac{1-{\operatorname{\lambda}}}{2\pi}$$ we have $[4\pi + \frac{2(1 - {\operatorname{\lambda}})}{C}] \geq 0$ and so $V_C$ is concave, strictly so when $C>C_{\text{crit}}$.
Such functions prove useful, but are not quite sufficient for our purposes. The comparison is constructed from the function, $$\label{eq:f_hyperbolic}
f(x, t) = \sqrt{v_{C} (x) + b x^2}$$ with $b\geq 0$.
\[lem:hyperbolic\_stationary\_concave\] Let $f$ be defined as in equation with $C\geq C_{\text{crit}}$. Then $f$ is concave, if and only if $$b \leq b_{\text{crit}} = \frac{(1-{\operatorname{\lambda}})^2}{\frac{1}{C}[4\pi + \frac{2(1-{\operatorname{\lambda}})}{C^2}]},$$ with strict concavity corresponding to strict inequality.
We have $$f'' = \frac{1}{2f^3}[v_Cv_C'' - \frac{1}{2}(v_C')^2 + b(2v_c - 2xv_C' + x^2 v_C'')]$$ so that $f$ is concave if and only if $$b \leq \frac{\frac{1}{2}(v_C')^2 - v_Cv_C''}{(2v_c - 2xv_C' + x^2 v_C'')}$$ since $b$ is non-negative.
First, consider the numerator. Since $v_C\geq 0$ and $v_C''\leq 0$ we have $$\frac{1}{2}(v_C')^2 - v_C v_C'' \geq \frac{1}{2}(v_C')^2.$$ Again using $v_C''\leq 0$ we have $$v_C'(x) \geq \lim_{x\to\infty} v_C' = \frac{-2(1-{\operatorname{\lambda}})}{C}$$ Also $\lim_{x\to\infty} v_Cv_C'' = 0$ so that in fact, $$\inf_{x}\left(\frac{1}{2}[(v_C')^2 - v_Cv_C'']\right) = \frac{2(1-{\operatorname{\lambda}})^2}{C^2}.$$
For the denominator, we have $$\begin{aligned}
2v_C - 2xv_C' + x^2v_C &= \frac{2}{C}[4\pi + \frac{2(1 - {\operatorname{\lambda}})}{C}][1 - e^{-Cx}(1 + Cx + (Cx)^2/2)] \\
&\leq \frac{2}{C}[4\pi + \frac{2(1 - {\operatorname{\lambda}})}{C}]\end{aligned}$$ with equality at $x=0$ so that $$\sup_{x} \left(2v_C - 2xv_C' + x^2v_C\right) = \frac{2}{C}[4\pi + \frac{2(1 - {\operatorname{\lambda}})}{C}].$$ Therefore $f$ is concave if and only if $$b \leq \frac{\frac{2(1-{\operatorname{\lambda}})^2}{C^2}}{\frac{2}{C}[4\pi + \frac{2(1 - {\operatorname{\lambda}})}{C}]} = \frac{(1-{\operatorname{\lambda}})^2}{\frac{1}{C}[4\pi + \frac{2(1-{\operatorname{\lambda}})}{C^2}]}.$$
\[rem:hyperbolic\_stationary\_concave\] Observe that the denominator in $b_{\text{crit}}$ is zero for $C=C_{\text{crit}}$, is positive for $C>C_{\text{crit}}$ and, approaches $0$ as $C\to\infty$. Thus for $C_0 \geq C_{\text{crit}}$, $b_{\text{crit}} ([C_{\text{crit}}, C_0])$ is bounded below away from $0$. This will prove useful later.
Let us also record the small and large scale asymptotics of $f$ in a lemma for later reference.
\[lem:hyperbolic\_asymptotics\] The function $v_C$ satisfies the asymptotic behaviour $$v_C (x) = 4\pi x - \frac{1}{C}\left[4\pi + \frac{2(1-{\operatorname{\lambda}})}{C}\right] \frac{(Cx)^2}{2} + \bigo((Cx)^3))$$ as $Cx\to 0$. Moreover, $$\limsup_{x\to\infty} v_C(x) = -\frac{2(1-{\operatorname{\lambda}})}{C^2} (Cx).$$
Therefore, $f$ satisfies the asymptotic behaviour $$f^2(x) = 4\pi x +\left(\frac{2b}{C^2} - \frac{1}{C}\left[4\pi + \frac{2(1-{\operatorname{\lambda}})}{C}\right]\right) \frac{(Cx)^2}{2} + \bigo((Cx)^3))$$ as $Cx\to 0$. Moreover $$\limsup_{x\to \infty} f^2 = b x^2.$$
\[rem:hyperbolic\_asymptotics\] In particular notice that the coefficient of $x^2/2$ (rather than $(Cx)^2/2$) from the small-scale asymptotics of $f$ is $$2b - 4\pi C - 2(1-{\operatorname{\lambda}})$$ and this can be made arbitrarily large negative by choosing $0 \leq b \ll 1-{\operatorname{\lambda}}$ and $C \gg C_{\text{crit}}$.
#### Construction of the comparison function {#sec-4-3-2-2}
The comparison is built from the function $f$ defined in equation by letting $C=C(t), b=b(t)$. If $C(t) \searrow C_{\text{crit}}$ and $b(t) \nearrow -(1-{\operatorname{\lambda}})$ as $t\to \infty$, with $b(t) \leq b_{\text{crit}}(C(t))$ (which choice is possible by remark \[rem:hyperbolic\_stationary\_concave\]), then $$\label{eq:hyperbolic_comparison}
f(x,t) = \sqrt{v_{C(t)} + b(t)x^2}$$ is a concave function with $$\lim_{t\to\infty} f(x,t) = \sqrt{4\pi x + (1-{\operatorname{\lambda}})x^2}$$ the isoperimetric profile of the metric of constant curvature $1-{\operatorname{\lambda}}$ (which is the curvature of the metric lifted from the constant curvature surface with area $4\pi$ and genus ${\operatorname{\lambda}}$ by the Gauss-Bonnet theorem).
By choosing $C(0)>C_{\text{crit}}$ sufficiently large and $0\leq b(0) < \max\{b_{\text{crit}}, 1-{\operatorname{\lambda}}\}$, sufficiently small, lemma \[lem:hyperbolic\_asymptotics\] and remark \[rem:hyperbolic\_asymptotics\] imply that for any initial metric, initial inequality is satisfied along with the asymptotic behaviour required by comparison theorem in the form of Proposition \[prop:comparison\].
Thus for $f$ to be a suitable comparison function, we need to choose $C(t)$ and $b(t)$ so that the differential inequality is satisfied. As before, it is more convenient to work with $v=f^2 = v_c + bx^2$
\[lem:hyperbolic\_diff\_ineq\] Let $$\begin{aligned}
b(t) &= \left[\left(\frac{1}{b_0} + \frac{1}{1-{\operatorname{\lambda}}}\right) e^{4(1-{\operatorname{\lambda}})t} - \frac{1}{1-{\operatorname{\lambda}}}\right]^{-1} \\
C(t) &= (C_0 - C_{\text{crit}})\sqrt{b_0} e^{2(1-{\operatorname{\lambda}})t} \left[\left(\frac{1}{b_0} + \frac{1}{1-{\operatorname{\lambda}}}\right) e^{4(1-{\operatorname{\lambda}})t} - \frac{1}{1-{\operatorname{\lambda}}}\right]^{-1/2} + C_{\text{crit}}\end{aligned}$$ Then $v = f^2$ satisfies the differential inequality with $f$ defined by .
First, for the time derivative we have $$\label{eq:dt_hyperbolic_stationary}
\begin{split}
\pd[v]{t} &= \dd[C]{t} \pd[v_C]{C} + \dd[b]{t} x^2 \\
&= -\dd[C]{t} \frac{1}{C^2}[4\pi + \frac{2(1-{\operatorname{\lambda}})}{C}] \frac{(Cx)^2}{2} \\
&\quad - \dd[C]{t} \left(\frac{1}{C^2}[4\pi + \frac{4(1-{\operatorname{\lambda}})}{C}] \right) \left(1 - Cx + \frac{(Cx)^2}{2} - e^{-Cx}\right) \\
&\quad - \dd[C]{t} \left(\frac{1}{C^2}[4\pi + \frac{2(1-{\operatorname{\lambda}})}{C}]\right)\left(Cx\right)\left(1 - Cx - e^{-Cx}\right) \\
&\quad + \frac{2}{C^2} \dd[b]{t} \frac{(Cx)^2}{2} \\
&< \frac{1}{C^2} \left(2\dd[b]{t} - \dd[C]{t} [4\pi + \frac{2(1-{\operatorname{\lambda}})}{C}]\right) \frac{(Cx)^2}{2} \\
&\quad - \dd[C]{t} \left(\frac{1}{C^2}[4\pi + \frac{2(1-{\operatorname{\lambda}})}{C}] \right) \left(1 - Cx + \frac{(Cx)^2}{2} - e^{-Cx}\right) \\
&\quad - \dd[C]{t} \left(\frac{1}{C^2}[4\pi + \frac{2(1-{\operatorname{\lambda}})}{C}\right)\left(Cx\right)\left(1 - Cx - e^{-Cx}\right) \\
\end{split}$$ The inequality occurs in the second line after the inequality by replacing the $4$ with a $2$ using the fact that $C$ is decreasing and $(1-{\operatorname{\lambda}})<0$.
For the spatial part, we first use the fact that for two functions $g,h$ we have $$(g+h)^2 \laplace \ln (g+h) = (g+h)(g+h)'' - (g+h)' = g^2\laplace \ln g + h^2 \laplace \ln h + gh'' + hg'' - 2g'h'$$ Thus with $g=v_C$ and $h=bx^2$ we get $$v^2 \laplace \ln v = v_C^2 \laplace \ln v_C - 2b^2x^2 + b (2v_C - 4 x v_C' + x^2 v_C'')$$ so that $$\begin{split}
v^2 \laplace \ln v + L[v] &= v_C^2 \laplace\ln v_C + L[v_C] + 8\pi b x - 2(b^2 + (1-{\operatorname{\lambda}})b)x^2 \\
&\quad + b (2v_C - 4 x v_C' + x^2 v_C'') \\
&= 8\pi b x - 2(b^2 + (1-{\operatorname{\lambda}})b)x^2 + b (2v_C - 4 x v_C' + x^2 v_C'')
\end{split}$$ since $v_C$ satisfies $v_C^2 \laplace\ln v_C + L[v_C] = 0$. Expand the last term in parenthesis in a Taylor series using equations , and to get $$\label{eq:spatial_hyperbolic_stationary}
\begin{split}
v^2 \laplace \ln v + L[v] &= \left(\frac{4b}{C}[4\pi + \frac{2(1-{\operatorname{\lambda}})}{C}] - \frac{4(b^2 + (1-{\operatorname{\lambda}})b)}{C^2} \right) \frac{(Cx)^2}{2} \\
&\quad + \frac{2b}{C}[4\pi + \frac{2(1-{\operatorname{\lambda}})}{C}] \left(1 - Cx + \frac{(Cx)^2}{2} - e^{-Cx}\right) \\
&\quad + \frac{4b}{C}[4\pi + \frac{2(1-{\operatorname{\lambda}})}{C}] \left(Cx\right) \left(1 - Cx - e^{-Cx}\right) \\
&\quad + \frac{2b}{C}[4\pi + \frac{2(1-{\operatorname{\lambda}})}{C}] \frac{(Cx)^2}{2} \left(1 - e^{-Cx}\right).
\end{split}$$
Now we compare the terms from equation with those of equation to obtain the following necessary inequalities:
- $(Cx)^2/2$: $$\begin{gathered}
\frac{1}{C^2} \left(2\dd[b]{t} - \dd[C]{t} [4\pi + \frac{2(1-{\operatorname{\lambda}})}{C}]\right) \\
< \left(\frac{4b}{C}[4\pi + \frac{2(1-{\operatorname{\lambda}})}{C}] - \frac{4(b^2 + (1-{\operatorname{\lambda}})b)}{C^2} \right)
\end{gathered}$$
- $1 - Cx + \frac{(Cx)^2}{2} - e^{-Cx}$: $$-\dd[C]{t} \left(\frac{1}{C^2}[4\pi + \frac{2(1-{\operatorname{\lambda}})}{C}] \right)
< \frac{2b}{C}[4\pi + \frac{2(1-{\operatorname{\lambda}})}{C}]$$
- $Cx \left(1 - Cx - e^{-Cx}\right)$: $$-\dd[C]{t} \left(\frac{1}{C^2}[4\pi + \frac{2(1-{\operatorname{\lambda}})}{C}\right)
< \frac{4b}{C}[4\pi + \frac{2(1-{\operatorname{\lambda}})}{C}]$$
- $\frac{(Cx)^2}{2} \left(1 - e^{-Cx}\right)$: $$0 < \frac{2b}{C}[4\pi + \frac{2(1-{\operatorname{\lambda}})}{C}]$$
All the above inequalities are satisfied if $$\begin{aligned}
\label{eq:b_bernoulli_hyperbolic_stationary}
\dd[b]{t} &< -4\left(b^2 + (1-{\operatorname{\lambda}})b\right) \\
\label{eq:c_hyperbolic_stationary}
\dd{t} \ln C &> - 2b\end{aligned}$$ and $C \geq C_{\text{crit}}$ which ensures that $4\pi + \tfrac{2(1-{\operatorname{\lambda}})}{C} \geq 0$.
It is now a simple matter, left to the reader, to check that $b(t)$ as given in the statement of the lemma satisfies equality in the Bernoulli equation and that equality in is satisfied by $$\tilde{C} = (C_0 - C_{\text{crit}})\sqrt{b_0} e^{2(1-{\operatorname{\lambda}})t} \left[\left(\frac{1}{b_0} + \frac{1}{1-{\operatorname{\lambda}}}\right) e^{4(1-{\operatorname{\lambda}})t} - \frac{1}{1-{\operatorname{\lambda}}}\right]^{-1/2}.$$ Since $C(t) = \tilde{C} + C_{\text{crit}} > \tilde{C}$ and $\dd{t} \tilde{C} < 0$, we then have $$\dd{t} \ln C = \frac{\dd{t}\tilde{C}}{\tilde{C} + C_{\text{crit}}} > \frac{\dd{t}\tilde{C}}{\tilde{C}} = -2b$$ competing the proof.
Observe that with $b(t), C(t)$ as given in lemma \[lem:hyperbolic\_diff\_ineq\], $b(t)$ monotonically increases from $b_0$ to $-(1-{\operatorname{\lambda}})$ and $C(t)$ monotonically decreases from $C_0$ to $C_{\text{crit}}$ so that $f = \sqrt{v}$ converges to the constant curvature $1-{\operatorname{\lambda}}$ isoperimetric profile.
Finally, applying corollary \[cor:ricci\_comp\_curv\_bnd\] we obtain
\[thm:ricci\_hyperbolic\_curvature\_bound\_general\] Let $\metric(t)$ be any solution of the normalised Ricci flow on $M$ a closed, genus $>1$ surface and let $\unicover{\metric} = \uniproj^{\ast}\metric$ the pull back to $\HH^2$ with $\uniproj: \HH^2 \to M$ the universal cover. Then for $\phi = \sqrt{v_{C(t)} + b(t)a^2}$ where $C(t), b(t)$ are defined as in Lemma \[lem:hyperbolic\_diff\_ineq\] there exists $C_0, b_0>0$ such that $\phi(a, t) < \isoprofile_{\unicover{\metric}} (a, t)$ for all $a \in (0,\infty)$. Therefore, the Gauss curvature $\gausscurv_M$ satisfies the bound $$\sup_{M} \gausscurv_{M} (t) \leq \left(\frac{C(t)}{2}\left[4\pi + \frac{2(1-{\operatorname{\lambda}})}{C(t)}\right] - b(t)\right)$$ which decays exponentially fast to $(1-{\operatorname{\lambda}})$ as $t \to \infty$.
The exponential decay in the theorem follows from the fact that $b(t) \to -(1-g)$ exponentially fast, $C(t) \to C_{\text{crit}}$ exponentially fast and hence $\left[4\pi + \frac{2(1-{\operatorname{\lambda}})}{C(t)}\right] \to 0$ exponentially fast.
Convergence {#sec-5}
===========
\[50880488-de51-46b5-9883-fc520d40172c\]
\[sec:convergence\]
In this last section, let us briefly discuss the proof of Theorem \[thm:convergence\]. The argument is very standard, following from bootstrapping the curvature bounds to higher derivative bounds. Here we will only outline the steps, indicating how the results here may be applied.
\[Proof of Theorem \[thm:convergence\]\] First, observe that by Theorems \[thm:sphere\_bounds\], \[thm:plane\_comparison\] and, \[thm:ricci\_hyperbolic\_curvature\_bound\_negative\] and, \[thm:ricci\_hyperbolic\_curvature\_bound\_general\] and the fact that $\uniproj: \unicover{M} \to M$ is a local isoemetry, we have uniform upper bounds $\gausscurv_M \leq \gausscurv_0(t) + (1-{\operatorname{\lambda}})$ with $\gausscurv_0(t)$ uniformly bounded and such that $\lim_{t\to\infty} \gausscurv_0(t) = 0$. Since the Gauss curvature evolves according to $\pd{t} \gausscurv = \laplace \gausscurv + \gausscurv(\gausscurv - ({\operatorname{\lambda}}- 1))$, by an ODE comparison we also have uniform lower bounds converging to $0$ as $t\to\infty$. Thus $\abs{\gausscurv(t)}$ is uniformly bounded for all $t>0$ and hence the solution exists for all time.
$L^1$ convergence of the curvature to $1-{\operatorname{\lambda}}$ now follows easily. By Gauss-Bonnet, $\gausscurv \leq \gausscurv_0(t) + (1-{\operatorname{\lambda}})$ and, the fact that $\abs{M} = 4\pi$, $$0 = \int_M \gausscurv - (1-{\operatorname{\lambda}}) d\mu \leq - \int_{\gausscurv \leq (1-{\operatorname{\lambda}})} \abs{\gausscurv - (1-{\operatorname{\lambda}})} d\mu + 4\pi \gausscurv_0(t),$$ which rearranges to give $\int_{\gausscurv \leq (1-{\operatorname{\lambda}})} \abs{\gausscurv - (1-{\operatorname{\lambda}})} d\mu \leq 4\pi \gausscurv_0(t)$. Therefore we get $$\int_M \abs{\gausscurv - (1-{\operatorname{\lambda}})} d\mu \leq 8\pi \gausscurv_0(t)$$ which converges to $0$ as $t\to\infty$.
Next we bound the higher derivatives of $\gausscurv$. By the bootstrapping argument described in [@MR1375255 Section 7], and from the uniform curvature bounds we obtain $$\abs{\nabla^{(j)} \gausscurv}^2 \leq C_j ((1-{\operatorname{\lambda}}) + t^{-j})$$ for constants $C_j>0$.
In the genus ${\operatorname{\lambda}}=0$ case, the lower bound on the isoperimetric constant affords very strong analytic control allowing us to apply Gagliardo-Nirenberg inequalities to deduce that $\gausscurv \to_{C^{\infty}} (1-{\operatorname{\lambda}})$ uniformly as $t \to \infty$. See [@MR2729306].
For higher genus surfaces, we don’t have such control as noted in remark \[rem:isoconst\_bdd\]. Thus instead, for ${\operatorname{\lambda}}>0$ surfaces, with a little more work, another bootstrapping argument gives $\gausscurv \to_{C^{\infty}} (1-{\operatorname{\lambda}})$ uniformly as $t \to \infty$ [@MR1375255 Section 7].
Finally, in the cases ${\operatorname{\lambda}}\ne 1$, we have $\gausscurv_0(t) = Ce^{-at}$ which gives for any non-zero $v\in TM$, $$\abs{\pd{t} \ln \metric(t) (v,v)} = 2\abs{\gausscurv - (1-{\operatorname{\lambda}})} \leq C^{-at}$$ which is integrable in $t$ on $[0,\infty)$. Smooth convergence of the metric now follows by the argument in [@MR664497 Section 17].
For the case ${\operatorname{\lambda}}= 0$ we only have $\gausscurv_0(t) = C/t$ which is not integrable. However, we may use the fact that $\abs{\nabla \gausscurv_0} \leq C/t^{3/2}$ which is integrable to again deduce smooth convergence [@MR2061425].
Note that we have control of the isoperimetric constant on $\unicover{M}$ and a curvature bound. We cannot however use these to obtain the simpler convergence proof using Gagliardo-Nirenberg inequalities since these rely on $L^1$ convergence of the curvature. But this is invalid for ${\operatorname{\lambda}}> 1$ since $\unicover{M}$ is not compact. Perhaps one might deduce $L^1_{\text{loc}}$ convergence, but note that the above $L^1$ convergence argument uses Gauss-Bonnet. In the $L^1_{\text{loc}}$ case, we would need to deal with boundary terms arising from Gauss-Bonnet and I don’t know how to control these. This is perhaps related to transferring isoperimetric control from $\unicover{M}$ to $M$.
|
---
abstract: 'We consider the issue of multiple agents learning to communicate through reinforcement learning within partially observable environments, with a focus on information asymmetry in the second part of our work. We provide a review of the recent algorithms developed to improve the agents’ policy by allowing the sharing of information between agents and the learning of communication strategies, with a focus on Deep Recurrent Q-Network-based models. We also describe recent efforts to interpret the languages generated by these agents and study their properties in an attempt to generate human-language-like sentences. We discuss the metrics used to evaluate the generated communication strategies and propose a novel entropy-based evaluation metric. Finally, we address the issue of the cost of communication and introduce the idea of an experimental setup to expose this cost in cooperative-competitive game.'
author:
- |
Mohamed Salah Zaiem[^1]\
Ecole polytechnique, Paris\
Etienne Bennequin\
Ecole polytechnique, Paris\
bibliography:
- 'mybib.bib'
title: 'Learning to Communicate in Multi-Agent Reinforcement Learning : A Review'
---
#### Disclaimer
This review has been written in December 2018 as coursework. In a very dynamic field, interesting results may have been found meanwhile, and a few assumptions here may have become outdated. However, we believe that this remains a good start for people discovering this field or trying to get an idea on what has been done about communication in multi-agent reinforcement learning settings.
Introduction
============
In recent years, Multi-Agent Reinforcement Learning has received a lot of interest, with more and more complex algorithms and structures improving the policy of multiple agents in more and more complex environments [@comprehensive:survey]. In Multi-agent settings, agents take actions and learn from their rewards simultaneously in the same environment. These interactions could be either competitive [@gosilver] or cooperative or a mix of the two. This review mainly tackles the case of partially observable multi-agent environments. In real world environments it is rare that the full state of the system can be provided to the agent or even be determined. In this kind of environments, agents may hold very valuable information for the decision making of other agents. In the case of common reward, agents would need to communicate seamlessly with the other agents in order to pass that information, coordinate their behaviours and increase the common reward [@Littman:1994].
Classic independent Q-learning or DQN learning have showed poor performance for this kind of setups [@matignon:2012]. Since no communication is involved, other agents learning and acting accordingly are considered part of the environment in these models. The environment is therefore non stationary, and agents fail to converge to an optimal policy. Allowing the agents to share messages and learn what to communicate can significantly improve their adaptability to a given situation.
To allow this information to circulate between agents, there has been a lot of research work on creating communication channels and protocols. These channels can be either continuous, therefore allowing easy derivation and optimization, or discrete with a set of symbols/letters copying human messages’ generation. These messages can either be determined in advance by a human operator or be learned by the agents, using the reward as the only learning signal. In this work, we will focus on the latter case.
We will separate two different uses of this framework. They can be roughly explained this way : the first tries to improve agent’s ability in a task through communication between agents, while the second one tries to improve the quality of the generated “language” (or communication) and its closeness to natural language through mastering a task.
The first use does not give a primary importance to the format of communication. Information could circulate through handmade communication channels and protocols as explained above. But it also could be latent and implied, through sharing the Deep Q-Networks weights.
The second use is motivated by the work of linguists on the emergence of language. Word2Vec embeddings arose from applying to huge textual datasets a famous quote of John Rupert Firth “ You shall know a word by the company it keeps.” Concerning the rise of language, Wittgenstein [@wittgenstein] said that “Language derives its meaning from use”. From this quote, a lot of works have been trying to create a language relying on its functionality, and Reinforcement Learning represents a perfect framework for such creation. In other words, language is a way to coordinate and achieve common objectives and cannot be represented by simple statistical associations like Language Models. Moreover, cognitive studies tend to show that feedback is paramount for the language learning process [@sachs].
Emergent communication and language is not a new field. Wagner *et al.* [@wagner] have already proposed a thorough review on the progress of the research in this domain. Since the majority of the works presented by them are not Reinforcement Learning related, and since the Deep Q-Networks offered a lot of new possibilities for researchers, we will not present these works in details.
We will start by defining the useful tools used in the different setups and models described in the review, then we will see the effect of introducing communication channels and algorithms on the performance of agents in multi-agent partially observable settings. We will then inspect the link between the generated languages and natural ones, studying the attempts of researchers to ground the communication sequences into human language. In the final part, we will discuss the metrics used in the experiments proposing a new entropy-based one, and we will propose an new multi-agent environment that could expose a cost in communication between agents.
Useful tools
============
#### Deep Q-Networks
Mnih *et al.* proposed in 2015 a representation of the Q-function of the policy of an agent using deep neural networks [@dqn].
Assume a single-agent observing at each time step $t$ the current state $s_t$ (fully observable) and taking an action $a_t$ following a policy $\pi$ with the objective of maximizing an expectation of the sum of future $\gamma$-discounted rewards $R_t = r_t + \gamma r_{t+1} + \dots$. In Q-learning, this expectation is represented by the value function of the policy $Q^{\pi}(s, a) = \mathbf{E}(R_t|s_t=s, a_t=a)$, with the optimal value function obeying the Bellman equation : $$Q^*(s,a) = \mathbf{E}\left[ r + \gamma \max_{a'}Q^*(s',a')|s,a \right]$$ In DQN, a neural network is used to represent the value function $Q(s,a,\theta)$ where $\theta$ is the parameters of the network. The training of the network is done using *experience replay*, *i.e.* a dataset of past experiences $e_t = (s_t, a_t, r_t, s_{t+1})$ sampled at random to constitute mini-batches, and the loss function w.r.t. the target $y = r + \gamma \max_{a'}Q(s',a', \theta^-)$ ($\theta^-$ being the parameters of a copy of the network, frozen at the beginning of each update iteration) : $$L(\theta) = \mathbf{E}_{s,a,r,s'} \left[ ( y - Q(s,a,\theta) ) ^2 \right]$$ In the case of multiple agents, this can be extended to *Independent DQN*, in which the Q-network of each agent is updated independently from the others. It has to be noted that this algorithm can lead to convergence problems, since the environment can appear non-stationary to one agent because of the learning of the other agents. Also, this assumes full observability of the state of the environment by all agents, which is a strong assumption in practice.
#### Deep Recurrent Q-Networks
Hausknecht & Stone proposed an adaptation of the DQN able to address the issue of a partially observable environment using Recurrent Neural Networks [@drqn]. Instead of approximating a value function $Q(s,a)$, this approximates $Q(o,a)$, where $o$ is the observation made by the agent. Here, $Q$ is computed using a RNN (usually, an LSTM layer), *i.e.* we have at each time step $Q(o_t, h_{t-1}, a, \theta )$. The observations can then be aggregated over time.
Multi-agent Cooperation
=======================
Cooperation & communication
---------------------------
Most real life situations can be modeled as several or many agents interacting in an environment. These agents can cooperate, compete, or some combination of the two. Tampuu *et al.*’s work presents all of these three possibilities in a game of Pong [@pong]. In their experiment, two agents play the famous Atari game with each other. Each of them can fully observe the state of the environment and tries to find its optimal policy using the independent DQN algorithm. No reward is given during the point. At the end of each point, the player who scored receives a reward $\rho$, and the other one receives a reward $-1$. Then the authors make $\rho$ vary between +1 and -1. When $\rho = 1$, the game is fully competitive, since each player has a full incentive to score. When $\rho=-1$, the game is fully cooperative : the two players are encouraged to keep the ball alive. Their experiment showed that with a competitive rewarding scheme, the number of paddle-bounces between the two players is about a hundred times lower than it is with a cooperative rewarding scheme, which means that in the latter, both players learned how to cooperate in order not to drop the ball.
This experiment shows that the notion of cooperation in a multi-agent environment is close to the notion of shared reward between the agent. If several agents evolving in the same environment share the same common reward, then they have full incentive to cooperate.
However, this does not mean that agents sharing the same reward will have an incentive to communicate. According to most papers in this area, the incentive to communicate actually comes from a partially observable environment. Take for instance the switch riddle : one hundred prisoners arrive in a new prison. The warden tells them that each day, one of them will be chosen at random (among all prisoners, including those having already been picked) and placed in a room containing only a light bulb. This prisoner can observe the state of the light bulb (*on* or *off*) and choose whether to change its state or not. He can also announce that he believes that all prisoners have visited the room. If he is right, all prisoners will be set free, but if he is wrong, they will all be executed. The prisoners can agree in advance on a strategy but as soon as the challenge begin, they will not be able to communicate with each other, neither will they know who is picked to go to the light bulb room.
In this riddle, prisoner-agents cannot observe the full state. They cannot observe anything, unless they are picked to go in the room, in which case they only know that they are the one in the room, along with the state of the switch. However, the common knowledge of all agents covers the full state and allows to solve the riddle. Therefore, the agents have an incentive to communicate, through the light bulb.
The problem of a partially observable multi-agent environment was already addressed in 2004 by Ghavamzadeh & Mahadevan [@hrl], who used Hierarchical Reinforcement Learning to allow multiple taxi drivers to learn to pick-up as many customers as possible in a finite time. After executing an action, a taxi driver is able to receive the state of the other cabs. However, communication in this case is not free : there is a arbitrarily predefined cost for getting the information from the other taxi drivers. Thus each agent had to compare the value function without communication with the sum of the value function with communication and the cost of communication. The authors did a comparison between a “selfish multi-agent” baseline, in which the agents do not communicate with each other, and their algorithm for different values of the communication cost. They found that even though the results were always better with their algorithm, the number of satisfied customers decreased when the communication cost increased, tending close to the baseline results. These experiments have to their credit that they managed to quantify the trade-off between partial observation and communication cost, and therefore to quantify the incentive to communicate.
Centralised learning and Deep Reccurent Q-Networks
--------------------------------------------------
Some constraints on the nature of the communication can make it non-differentiable, thus invalidating a lot of learning algorithms. A widely used solution to avoid this issue is centralised training for decentralised execution. Agents have a learning period in which they develop their communication schemes along with their policies, which they apply afterwards, at test time, possibly with more constraints on the message.
#### Learning communication with backpropagation
Sukhbaatar *et al.* [@backprop] developed a “Communication Network”, that takes as entries all the partial observations of the $M$ agents and outputs the actions. Each layer $i$ of this $N$-layered network has $M$ cells $\lbrace f^i_m \rbrace _{m=1,\dots,M}$. Each cell takes as input $(c_m^i, h_m^i)$ where $h_m^i$ is the output of $f_m^{i-1}$ (with $h_m^0 = g(s_m)$) and $c_m^i = \operatorname{avg}\limits_{m'\neq m}h_{m'}^{i-1}$ is the message coming from other agents (it can also be computed only on a subset of the set of other agents), and outputs : $$h_m^{i+1} = \sigma(C^i c_m^i + H^i h_m^i)$$ where $\sigma$ is some non-linearity (as an alternative, $f_m^i$ can also be an LSTM). Finally $f^N_m(c_m^N, h_m^N) = a_m$. The weights of the network are trained all-together using backpropagation from a loss related to the reward on a batch of experiences. Their model proved to heavily outperform a baseline model where communication inside the network occurs in discrete symbol.
After some promising results, the research on centralised learning of communication schemes took a turn with the utilisation of Deep Recurrent Q-Networks. Unlike the independent Deep Q-Network used in Tampuu *et al.*, Deep Recurrent Q-Networks do not assume full observability and do not lead to convergence problems (in independent DQN, one agent’s learning makes the environment appear non stationnary to the others).
Foerster *et al.* proposed the two following algorithms deriving from DRQN in which multiple agents learn to communicate in a centralized fashion and sharing the weights of their network. Both make the assumption of a partially observable environment in which agents share the same reward.
#### Deep Distributed Recurrent Q-Network [@DDRQN]
This algorithm derives from a combination of DRQN and independent Q-learning, with three modification : the experience replay is forgotten (as said before, the experience of one agent is made obsolete by the learning of the others), the last action is added as input, and most importantly the weights of the network are shared between all the agents : there is not one Q-network for each agent, but one Q-network in total, allowing more efficient (and way less costly) common learning. Each agent can still behave differently from the others since it gets its own individual inputs. The index of the agent is also given as input, in order to allow each agent to specialise. The Q-function to be learned by the RNN becomes of the form $Q(o_t^m, h^m_{t-1}, m, a_{t-1}^m, a_t^m, \theta)$, with $o_t^m$ and $a_t^m$ respectively the observation and the action of agent $m$ at time step $t$, and $h^m_{t-1}$ the hidden state of agent $m$ after time step $t-1$. The network is trained in the following way : 1) the multiple agents evolve during one episode with an $\epsilon$-greedy policy following the Q-function given by the RNN. 2) The weights of the network are updated following Bellman equations.
#### Differentiable Inter-Agent Learning [@DIAL]
At each time-step, in addition to an environment action $a_t^m$, each agent $m$ takes a communication action $c_t^m$. This message is not restricted during the centralised learning but is restricted to a limited-bandwidth channel during the decentralised execution.\
Here, the Q-network does not only output the Q-function, but also the real-valued (thus differentiable) message $c_t^m$ which is fed at time-step $t+1$ to the Q-networks of the other agents (that still share the same weights). Then two separate gradients back-propagate through the Q-network : 1) the one associated to the reward, coming from the environment. 2) the one coming from the error backpropagated by the recipients of the message in the next turn. Figure \[fig:DIAL\] shows the DIAL architecture with two agents. With this model, messages can be updated more quickly to minimize future DQN loss.
Note that in the first model, there is no explicit message fluctuating between the agents. The communication lies in the actions that are taken by the multiple agents and can influence the observations of other agents in the future. But in both algorithms, there is a big deal of implicit communication during the learning, when all agents actually come together and agree on a common strategy, just like the prisoners in the switch riddle.
Both algorithms having been tested on their performance on this switch riddle, we can compare their results. Both algorithms usually converge to the same rewards : $100 \%$ of the Oracle (maximal possible reward) for $n=3$, and $90 \%$ for $n=4$. It is also interesting to see that the same communication protocol is used by both algorithms after convergence for $n=3$ (this protocol is shown in Figure \[fig:switch\]). However, DIAL converges about 20 times faster than DDRQN in both cases (after one thousand episodes for DIAL, but 20 thousands for DDRQN). Plus, in the case of $n=4$, even though DDRQN converges in most instances to $90 \%$ of the Oracle, there are a significant number of instances in which we cannot see any convergence after 500 thousands episodes. This is a good argument in favor of differentiable message directly adjusted following downstream DQN loss.
[.45]{} ![\[fig:DIAL\]Figure \[fig:DIAL\] : Dial architecture with two agents : red arrows represent backpropagation. Different Q-Net cells are copies of the same shared network.](switch_decision_tree.png "fig:"){width=".99\linewidth"}
[.45]{} ![\[fig:DIAL\]Figure \[fig:DIAL\] : Dial architecture with two agents : red arrows represent backpropagation. Different Q-Net cells are copies of the same shared network.](DIAL.png "fig:"){width=".99\linewidth"}
Cooperation through public belief
---------------------------------
However, the DIAL method assume the existence of a channel on which agents can communicate freely without affecting the environment. In practice, this is not always the case. Foerster *et al.* [@BAD] introduce the notion of *public belief*, which can be defined as the estimation of the probability of possible states (*i.e.* observable or not) knowing all public observations (*i.e.* observations available to all agents) : assuming that at time step $t$, $f_t$ is the discrete set of features of the environment that compose the state $s_t$, and that $f_t^{pub}$ is the subset of features that are commonly known by all agents, we define the public belief $\mathcal{B}_t = P(f_t | f^{pub}_{1,\dots,t})$.
#### Bayesian Action Decoder [@BAD]
The authors proposed the *BAD* algorithm to use the notion of public belief in order to allow agents to learn a cooperative policy.\
They assume a partially observable environment with no communication channel. At each time step $t$, agent $m$ takes an action $a_t^m$ following a policy $\pi^m$ knowing all its past observations and actions. All agents share a common reward. The training is centralised, which means that each agent knows the policies of all the other agents.\
The main idea is to generate a virtual third-party agent with *public belief policy* $\pi_{\mathcal{B}_t}$ that conditions on public information and public belief, and therefore can be computed by each agent through a common algorithm. At each time step, this virtual third-party outputs a policy $\hat \pi_t$ that maps all possible private observations made by the agent $m$ acting at this time step to the set of its possible actions. Then, at time step $t+1$, since all agents know the policy $\hat \pi_t$, the action that agent $m$ has taken following this policy gives public information about the private observation he made : $$P(f_t^m | a_t^m, \mathcal{B}_t, f_t^{pub}, \hat \pi_t) \propto \mathbf{1}(\hat \pi_t(f^m_t) = a_t^m)P(f_t^m|\mathcal{B}_t, f^{pub}_t)$$ with $f^m_t$ being the state features observed by agent $m$ at time $t$. This allowing to update the public belief.
The authors chose to apply their algorithm to the cooperative firework-themed playing card game Hanabi. In this game, each player can see the other players’ cards but not its own, and can chose to hint a teammate about one of this teammate’s cards, or to play one of its own cards, using the hints given by the other players. In the case of a two players game, the score converged after ten thousands games, achieving state of the art results. What is even more interesting is that agents developed their own communication conventions through their actions. For instance, the authors found that an agent hinting “red” or “yellow” is a huge incentive for the other agent to use its newest card at the next action. In this way, the agents learned a crucial ability in the interactions between humans, which is the ability to deduce, from the action of someone, latent information about their situation and observations.
Learning a language
===================
Main models : Referential Games
--------------------------------
As we mentioned earlier, emergent languages and communication is not a new research topic, but the development of Deep Recurrent Q-Networks methods offered a whole new lot of possibilities to language researchers.
A majority of the reinforcement learning setups we will describe in this part rely on referential games. Referential games are a variant of the Lewis signaling game [@lewis] and have been often used in linguistics and cognitive sciences. They start by presenting a target object to a the first agent (the speaker). The speaker is therefore allowed to send a message describing the object to a second agent (the listener). Finally, the listener tries to guess the target shown to the speaker among a list of possible candidates. The communication is successful, and thus the reward positive, if the listener picks the correct candidate.
Let us describe more precisely a specific model, the one developed in [@lazaridou:17]. Target objects and candidates are sampled pictures from ImageNet. They represent basic concepts like cats, cars... classified among 20 general categories (animal, vehicle...). As it receives raw pixel input, the speaker agent has to detect by itself the features it could send to the listener. Other papers like [@lazaridou:18] tried to use disentangled inputs with attribute-based object vectors.
Two images $i_R $ and $i_L$ are drawn randomly from the set. One of them is the target $t$. The sender knows which one is the target, and generates a message following the policy $s(\theta_s (i_L, i_R, t )) $ from a vocabulary V of size K. The listener tries to decode the message and guess the target image following a policy $r(i_L, i_R, s(\theta_s (i_L, i_R, t ))) \in \{L,R\} $. If $r$, the guess of the listener, is equal to $t$, they both receive a positive reward, otherwise, they receive 0 as a reward.
### Vocabulary
The size of the vocabulary has a huge influence on the performance of communicating agents. Foerster *et al.* [@DIAL] started with sending simple 1-bit message, with their environments needing “small” information (for instance, the switch riddle or finding the parity of a number), but the more complex the problems get, the more vocabulary is needed to ensure informative messages. Lazaridou *et al.* [@lazaridou:17] tried single-symboled messages with a vocabulary size going from 0 to 100. Later research, inspired by natural language, suggested the use of LSTM based encoder-decoder architectures to generate sequences [@havrylov] [@lazaridou:18]. An important point is that these symbols have no *a priori* meaning. They do get linked to visual or physical attributes only during learning.
One of the main benefits of generating sequences is allowing the speakers to generate languages ensuring compositionnality. With a compositional language, the speaker would be able to describe a red box, if he had prior suffixes or prefixes for “red” and “box” given the training set, even if no “red box” is seen during training.
### Q-Models
Each agent (speaker and listener) has an input network that takes the incoming inputs (images, messages, previous actions...) then embeds them using Recurrent Neural Networks or simple shallow Feedforward networks. Lazaridou [@lazaridou:17] observed that producing these features with Convolutional Neural Networks in the case of images input yielded better features and therefore better results in the guessing phase.
To generate a sequence of symbols, Lazaridou *et al.* [@lazaridou:18] chose a recurrent policy for the decoder [@drqn]. The purpose being that the RNN will be able to retain information from past states and use them to perform better on long sequences generation. Evtimova *et al.* [@evtimova] added an attention mechanism during the generation of messages, reporting that it improved the communication success with unseen objects, as the embedding layers focus on recognizable objects.
To make it clearer, from the images $i_L$ and $i_R$, a convolutional network outputs a hidden state $h_{LR}$ that is input to a single-layer LSTM decoder (the speaker’s recurrent policy). This layer generates a message $m = g(h_{LR}, \theta_g ) $. The listener applies the single-layer LSTM encoding for the message (since it is also dealing with sequence input) producing an encoding $z = h(m, \theta_h)$. Coupled with encoding for the candidate images, it predicts a target $t$ following similarity measures between candidates $u \in U$ and encoded vector $v$.
All the weights of both the speaker and listener agents are jointly optimized using the choice of the agents as the only learning signal. Since their task is different, no weights are shared between the trained agents.
### Variants
We described above a popular model in language emergence, but since the emergent languages literature has become consistent, there is a few interesting variants which we will discuss here.\
A first interesting one is proposed in the work of [@jorge:16] and [@cao]. To get closer to human-like interactions, they implemented multi-step communication between the agents. In [@jorge:16], the task is very similar, recognizing target images (celebrities in this case), but both agents generate messages. The guessing agent starts the communication given the candidates by asking “questions”. The answering agent, who knows the target, answers these questions by yes or no, like in the “Guess Who” game. The best results were obtained using two rounds of question-answer, and their analysis shows that the questioning agent targets different features in Question 1 and Question 2, getting more information.
Cao *et al.* [@cao] also tested multi-step communication but with important changes. It is no longer about referential games, but a negotiation to trade items having different utilities for the two parties. Two channels are implemented, the first one is made to communicate information (like the channels seen earlier), the second one is a concrete trade offer that can be either accepted or rejected by the other agent. When the agents cooperate, *i.e.* the reward is common, the messages indicate to the other agent which items are most valuable and their value. The results show that the “linguistic” channel helps reach a Nash equilibrium and reduces the variance in joint optimality, making the trade system more robust.
Results
-------
The first question this part should answer is whether the communication is successful, *i.e.* whether the agents complete their task. In almost all the papers we have read, agents learn to coordinate almost perfectly and reach very high accuracies.
For instance, in the work of Lazaridou *et al.* [@lazaridou:17], depending on the model, agents reach a $99\%$ success rate, while the best ones do not fail at choosing the right target. The main differences concern the number of episodes needed before convergence. Best models are the ones that encode best the input images, thus facilitating message generation by a better representation/classification of the described objects.
In [@lazaridou:18], the higher number of distractors (*i.e.* non target inputs to the listener) slightly reduces the accuracy ($98.5\%$). We have to note that success rate increases highly with the size of the alphabet at the beginning, before stabilizing when the vocabulary is large enough to answer the task properly. Table \[tab:lazaridou-results\] shows that influence. Lexicon size denotes the effective number of unique messages used by the speaker agent.
Max length Alphabet Size Lexicon size Training accuracy
------------ --------------- -------------- -------------------
2 10 31 92.0%
5 17 293 98.2%
10 40 355 98.5%
: \[tab:lazaridou-results\]Results of Lazaridou *et al.* 2018 [@lazaridou:18]
Link with Natural Language
--------------------------
As we said in the introduction, research work in multi-agent communication and language emergence did not stick to improving the behaviour of agents facing partially-observable environments. Apart from getting closer to optimal policies, emergent communication has been seen as a way to tackle Natural Language Processing issues. Roughly put, if machines are able to generate a language conveying the same basic functions as human language, an eventual translator from “machine” to human could solve a lot of unsolved NLP tasks.
Therefore, two challenges arise. The first is interpreting the generated messages. The second is one grounding the emergent language into natural language. Some works have been trying to link the generated messages to human communication, going until twisting the tasks to force that closeness.
### Interpretability
Starting from rough and noisy messages, interpretation is a rough task. In a lot of cases, researchers fail to understand completely which information the agents are passing and stick to formulating vague hypotheses about meanings. Figure \[fig:conversations\] shows the attemps of Jorge *et al.* [@jorge:16] to understand the meaning of Question A and B (questions asked by the questioning agent). Looking at the table, it seems that A checks whether the top of the photo is coloured or not(dark or white hair). The operation is even harder with sequences, Table 3 shows the top messages for each object(an object is identified by its shape and color) in the work of Choi *et al.* (ICLR).
![\[fig:conversations\]Figure \[fig:conversations\] : Samples presented according to the answer they triggered to two questions [@jorge:16]](partition_guess.png){width="50.00000%"}
![\[fig:conversations\]Table 3 : Most common sequences describing objects. (The letters between brackets represent common alternatives for the last letter) [@choi]](tablechoi.png){width="100.00000%"}
Lazaridou *et al.* [@lazaridou:17] tried to estimate closeness to human representations. They used the purity index to assess the quality of “message clusters”, *i.e.* the images represented by the same message by the speaker agent. We compare these clusters with the initial ones (the groups englobing the concepts depicted in the images, Dog $\in$ Animals for instance). The purity of a clustering is the proportion of category labels in the clusters that agree with the respective cluster majority category. In other terms, for a given message, we take the label having a majority of elements represented by that message, and check how many images do not pertain to that label but are still represented by the same message. With CNN encoding, and 100 symbols in the vocabulary (43 of them are effectively used) their model reaches $41\%$ purity.
Choi *et al.* [@choi] evaluate closeness to human language by assessing the ability of agents to compose meanings. In this purpose, they use zero-shot evaluation. In their setup, the listener has to guess an image among a set of 4 candidates. Each image is generated randomly given its shape (cylinder, cube...) and color. Zero-shot evaluation consists in showing the trained speaker an unseen combination of shape/color, with previously (multiple times) seen color and shape separately. The point is to evaluate his ability to convey the message of an unseen element composing the knowledge he had on its elements. Communication accuracies do not vary much between seen and unseen objects and stay very close to 1, around 0.97. The emerged language seems able to compose since it succeeds in describing unseen objects.
### Grounding it into Natural Language
In order to get easy interpretable languages, researchers have tried to compel the agents to generate messages closer to human representations.In [@lazaridou:17], they introduced a few changes to the referential game. Instead of presenting the same image to the speaker and the listener (among other candidates for the latter), they presented an image pertaining to the same concept. For instance, during training, they showed a dog to the speaker, and then the listener had to pick the picture of another dog among a list of non-dog candidates. The point of this manipulation is to force the speaker to send the message “dog”, like a human would do, and not send messages on the color of the background, or the luminosity, for instance. This change induced a slight increase in purity reaching $45\%$ . Furthermore, to force even more the speaker to classify “humanly” the images, they alternate its learning between playing the game (generating messages to the listener) and a classic image classification task.
Discussion and propositions
===========================
Lack of evaluation metrics
--------------------------
The accuracy of the choice is a natural and relevant metric to evaluate the communication’s quality, and thus the quality of the language generated. But the metrics used to assess the link with natural language are completely biased by the human macro-representation of raw images. To be clearer, the purity index introduced above is not a valid evaluation of interpretability. We believe that forcing the agent to describe macro-representations(dog, car..) instead of raw image descriptions ( color, background, luminosity..), especially given the lack of prior information on these macro-representations, is not a natural way to solve the task for the agents, and it may harm their communication success.
However, we believe that it would be interesting to develop an evaluation metric for a learned communication protocol that would be independent from human assumptions as well as from its effectiveness in the execution of a singular task. Zero-shot evaluation for compositionnality is indeed relevant in this perspective, but it cannot capture all the features that make a language relevant.
In 2003, Prokopenko & Wang studied the entropy on the public belief in a setting with multi-agent coordination [@entropy]. We can put that in relation with the results of the Bayesian Action Decoder on the Hanabi game, which shows that the entropy of the public belief (*i.e.* the posterior on all possible states) decreases as the algorithm develops useful communication conventions. In the same spirit, we introduce here the idea of an evaluation metric on a language or communication protocol, independently from the task to be completed by the agents.
#### Language-Entropy Evolution
Let us assume a two-agent environment, Agent *Smith* being the speaker and Agent *Lee* being the listener. At each time step $t$, *Smith* observes the full state $s_t \in \mathcal{S}$ and sends to *Lee* a message $m_t \in \mathcal{M} $. *Lee* then computes the probability $p_t(s)$ of each possible state $s$ given all messages $m_{1,\dots,t}$. We evaluate the entropy of this probability distribution : $$H(m_1, \dots, m_t) = - \sum\limits_{s\in \mathcal{S}}p_t(s) \log(p_t(s))$$ The comparison between $H(\dots, m_{t-1})$ and $H(\dots, m_t)$ gives a measure of the uncertainty that has been lifted by message $m_t$, in other words how discriminant message $m_t$ is in this context.
Note that the probabilities computed by *Lee* can be on the action set instead of the state set. Then the entropy measures the uncertainty about which action to take instead of the uncertainty about which state the agents are in. In some applications where it is uneasy to compute a probability distribution, this alternative might be more relevant.
We believe that this metric can be useful in the evaluation of learned languages and communication protocol.
EDIT : This idea has been explored in the work of Lowe *et al.* [@lowe] , along with other interesting evaluation metrics.
Cost of communication
---------------------
![\[fig:conversations\]Figure \[fig:conversations\] : 4 Players 2 vs 2 Pong Game](doublepong.png){width="35.00000%"}
We also noticed that an interesting study could take place on the possible cost of communication in these systems. Apart from the obvious computational cost, we were thinking about environments with competitive groups of cooperative agents.
In this context, we have been thinking about a game setting where we would observe a cost on communication in a cooperation/competition setup. The game is represented in Figure \[fig:conversations\], the goal would be to make teams of two agents play against each other a 4-players Pong setting. Multiple cases could be tested :
- No communication channel.
- The communication between members of the same team is not heard by others.
- Communication is public, or communication is public for one team and secret for the other.
We believe that these experiments could lead to a better understanding of the cost of communication in an environment where sending a message has an indirect influence on the state of the environment, through the reaction of the opposite team.
Conclusion
==========
We reviewed the main recent algorithms addressing the issue of communication and cooperation between multiple agents in a Reinforcement Learning environment. We also presented the current efforts in interpreting the language learned by these algorithms, and proposed a novel metric to evaluate the relevance of a generated communication strategy.
In the future, we want to implement the 4-players pong environment with the multiple cases described above in a personal project to see how the policies of the teams evolve depending on the constraints on their communication protocol.
[^1]: Both authors contributed equally
|
---
abstract: 'Parity constraints, common in application domains such as circuit verification, bounded model checking, and logical cryptanalysis, are not necessarily most efficiently solved if translated into conjunctive normal form. Thus, specialized parity reasoning techniques have been developed in the past for propagating parity constraints. This paper studies the questions of deciding whether unit propagation or equivalence reasoning is enough to achieve full propagation in a given parity constraint set. Efficient approximating tests for answering these questions are developed. It is also shown that equivalence reasoning can be simulated by unit propagation by adding a polynomial amount of redundant parity constraints to the problem. It is proven that without using additional variables, an exponential number of new parity constraints would be needed in the worst case. The presented classification and propagation methods are evaluated experimentally.'
author:
- 'Tero Laitinen, Tommi Junttila, and Ilkka Niemelä'
bibliography:
- 'paper.bib'
title: 'Classifying and Propagating Parity Constraints (extended version)[^1]'
---
[^1]: The original version of the paper [@LJN:CP2012] has been presented in the 18th International Conference on Principle and Practice of Constraint Programming, CP 2012. An earlier version of the extended version has been presented for CP 2012 reviewers. This revised version uses proof techniques from [@LJN:ICTAI2012full].
|
---
abstract: 'The main result of this paper is the following: if $F$ is any field and $R$ any $F$-subalgebra of the algebra $\mathbb{M}_n(F)$ of $n\times n$ matrices over $F$ with Lie nilpotence index $m$, then $$\mathrm{dim}_{F}R \leqslant M(m+1,n)$$ where $M(m+1,n)$ is the maximum of $\frac{1}{2}\!\left(n^{2}-\sum_{i=1}^{m+1}k_{i}^{2}\right)+1$ subject to the constraint $\sum_{i=1}^{m+1}k_{i}=n$ and $k_{1},k_{2},\ldots,k_{m+1}$ nonnegative integers. This answers in the affirmative a conjecture by the first and third authors. The case $m=1$ reduces to a classical theorem of Schur (1905), later generalized by Jacobson (1944) to all fields, which asserts that if $F$ is an algebraically closed field of characteristic zero, and $R$ is any commutative $F$-subalgebra of $\mathbb{M}_{n}(F)$, then $\mathrm{dim}_{F}R \leqslant \left\lfloor\frac{n^{2}}{4}\right\rfloor+1$. Examples constructed from block upper triangular matrices show that the upper bound of $M(m+1,n)$ cannot be lowered for any choice of $m$ and $n$. An explicit formula for $M(m+1,n)$ is also derived.'
address:
- 'Institute of Mathematics, University of Miskolc, 3515 Miskolc-Egyetemváros, Hungary'
- 'Department of Mathematics and Applied Mathematics, University of Pretoria, Private Bag X20, Hatfield, Pretoria 0028, South Africa'
- 'Department of Mathematical Sciences, Stellenbosch University, Private Bag X1, Matieland 7602, Stellenbosch, South Africa'
- 'Faculty of Mathematics and Information Science, Technical University of Warsaw, 00-661 Warsaw, Poland'
author:
- 'J. Szigeti'
- 'J. van den Berg\*'
- 'L. van Wyk'
- 'M. Ziembowski'
title: 'The maximum dimension of a Lie nilpotent subalgebra of $\boldsymbol{\mathbb{M}_n(F)}$ of index $\boldsymbol{m}$'
---
[^1] [^2] [^3] [^4] [^5]
Contents {#contents .unnumbered}
========
[[.]{}]{}
Introduction
Preliminaries
The passage to local algebras over an algebraically closed field
Simultaneous triangularization and the passage to upper triangular matrix algebras
Subalgebras of ${\mathbb{U}}_{n}^{\ast}(F)$
Lie nilpotent subalgebras of ${\mathbb{U}}_{n}^{\ast}(F)$: the main theorem
The function $M(\ell,n)$
An illustrative example
Open questions
Introduction
============
In 1905 Schur [@Sc05 Satz I, p. 67] proved that the dimension over the field of complex numbers $\mathbb{C}$ of any commutative subalgebra of $\mathbb{M}_{n}(\mathbb{C})$ is at most $\left\lfloor\frac{n^{2}}{4}\right\rfloor+1$, where $\lfloor\hspace*{0.3em}\rfloor$ denotes the integer floor function. Some forty years later, Jacobson [@Ja44 Theorems 1 and 2, p. 434] extended Schur’s result by showing that the upper bound holds for commutative subalgebras of $\mathbb{M}_{n}(F)$ for all fields $F$.
In a subsequent further improvement, Gustafson [@Gu76 Section 2, p. 558] showed that Schur’s theorem in its most general form could be proved with much greater efficiency using module theoretic methods. We record here that Gustafson’s elegant arguments are the inspiration for a key proposition in this paper.
There have also appeared in the literature a number of papers offering alternative proofs of Schur’s theorem and its subsequent extensions. In this regard, we refer the reader to [@ZG64], [@Mi98] and [@Ka10].
In response to a question posed in [Gu76]{} Cowsik [@Co93] has proved a version of Schur’s theorem for artinian rings that are not algebras, in which the module length of a faithful module substitutes for the dimension of the $F$-space on which the matrices act.
The common approach to establishing Schur’s upper bound has been to show that if $F$ is a field and $R$ a commutative $F$-subalgebra of $\mathbb{M}_{n}(F)$, then there exist positive integers $k_{1}$ and $k_{2}$ such that $k_{1}+k_{2}=n$ and
$$\mathrm{dim}_{F}R \leqslant k_{1}k_{2}+1.$$
An application of rudimentary Calculus then shows that
$\mathrm{max}\hspace*{0.1em}\{k_{1}k_{2}+1:(k_{1},k_{2})\in\mathbb{N}\times\mathbb{N}$ and $k_{1}+k_{2}=n\}=\left\lfloor\frac{n^{2}}{4}\right\rfloor+1$,
whence $\mathrm{dim}_{F}R \leqslant \left\lfloor\frac{n^{2}}{4}\right\rfloor+1$.
The upper bound of \[1.3\]\[0.7\]$\left\lfloor\frac{n^{2}}{4}\right\rfloor+1$ is, moreover, easily seen to be optimal. Indeed, let $F$ be any field and $(k_{1},k_{2})$ any pair of positive integers satisfying $k_{1}+k_{2}=n$. Define rectangular array $B$ by
$$B\,\overset{\mathrm{def}}{=}\{(i,j)\in\mathbb{N}\times\mathbb{N}:1\leqslant
i\leqslant k_{1}< j\leqslant n\},$$
and subset $J$ of $\mathbb{M}_{n}(F)$ by
$$\begin{aligned}
J\,\overset{\mathrm{def}}{=}\left\{
\raisebox{0.7ex}[3.2ex][0ex]{
$\displaystyle\sum\limits_{(i,j)\in B}b_{ij}E_{(i,j)}\::\:b_{ij}\in F \hspace*{0.6em}\forall(i,j)\in B$}\right\},\end{aligned}$$
where $E_{(i,j)}$ denotes the matrix unit in $\mathbb{M}_{n}(F)$ associated with position $(i,j)$. Observe that $J$ comprises the set of all block upper triangular matrices that correspond with $B$; it has the following illuminating pictorial representation (the unshaded region in the picture below corresponds with zero entries):
; (0,5)–(5,0); (5,5) rectangle (2,3);
at (0,2.5) [$J\,=\,$]{};
(5,3.05) – (5,4.95) node \[black,midway,xshift=0.6cm\][$k_{1}$]{};
(5,0.05) – (5,2.95) node \[black,midway,xshift=0.6cm\][$k_{2}$]{};
Denote by
$$FI_{n}\,\overset{\mathrm{def}}{=}\{aI_{n}:a\in F\} \,\makebox{ {\rm ($I_{n}$
is the $n\times n$ identity matrix)}}$$
the set of all $n\times n$ scalar matrices over $F$, and define
$$\begin{aligned}
R\,\overset{\mathrm{def}}{=} FI_{n}+J.\end{aligned}$$
It is easily seen that $R$ is a local $F$-subalgebra of $\mathbb{M}_{n}(F)$ with (Jacobson) radical $J(R)=J$ such that $J^{2}=0$. This entails $R$ is commutative. It is clear too, that
$$\mathrm{dim}_{F}R=k_{1}k_{2}+1.$$
The above simple construction shows that the upper bound $\left\lfloor\frac{n^{2}}{4}\right\rfloor+1=\mathrm{max}\hspace*{0.1em}\{k_{1}k_{2}+1:(k_{1},k_{2})\in\mathbb{N}\times\mathbb{N}$ and $k_{1}+k_{2}=n\}$ cannot be lowered for any $n\geqslant 2$, and is thus optimal, as claimed.
We construct now an $F$-subalgebra $R$ of $\mathbb{M}_{n}(F)$ similar to the one constructed above, but whose radical $J$ comprises $m$ blocks rather than a single block. We require first a compact notation for the description of such rings. To this end, let $k_{1},k_{2},\ldots,k_{m+1}$ be a sequence of positive integers such that $k_{1}+k_{2}+\cdots+k_{m+1}=n$. For each $p\in\{1,2,\ldots,m\}$, define rectangular array
$$B_{p}\,\overset{\mathrm{def}}{=}
\begin{cases}
\{(i,j)\in\mathbb{N}\times\mathbb{N}:1\leqslant i\leqslant k_{1}<j\leqslant
n\},\makebox{ {\rm if} }p=1, \\[2ex]
\{(i,j)\in\mathbb{N}\times\mathbb{N}:\parbox[t]{20em}{$k_{1}+k_{2}+\cdots+k_{p-1}<i\leqslant k_{1}+k_{2}+\cdots+k_{p}$\\[0.2\baselineskip]
$<j\leqslant n\}$, if $p>1$.}\end{cases}$$
Put
$$\begin{aligned}
B\,\overset{\mathrm{def}}{=}\bigcup\limits_{p=1}^{m}B_{p}.\end{aligned}$$
Define $J$ as in (1) but with $B$ defined as in (3) above. The following pictorial representation of $J$ reveals a stack of $m$ blocks
(0,5)–(5,0);
;
(5,5) rectangle (1,4); (5,4) rectangle (1.7,3.3); (5,1.8) rectangle (4.3,0.7);
at (4.5,2.05) [$\cdot$]{}; at (4.5,2.25) [$\cdot$]{}; at (4.5,2.45) [$\cdot$]{}; at (4.5,2.65) [$\cdot$]{}; at (4.5,2.85) [$\cdot$]{}; at (4.5,3.05) [$\cdot$]{};
at (5.32,1.65) [$\cdot$]{}; at (5.32,1.85) [$\cdot$]{}; at (5.32,2.05) [$\cdot$]{}; at (5.32,2.25) [$\cdot$]{}; at (5.32,2.45) [$\cdot$]{}; at (5.32,2.65) [$\cdot$]{}; at (5.32,2.85) [$\cdot$]{}; at (5.32,3.05) [$\cdot$]{}; at (5.32,3.25) [$\cdot$]{};
at (0,2.5) [$J\,=\,$]{};
(5,4.05) – (5,4.95) node \[black,midway,xshift=0.6cm\][$k_{1}$]{};
(5,3.35) – (5,3.95) node \[black,midway,xshift=0.6cm\][$k_{2}$]{};
(5,0.75) – (5,1.75) node \[black,midway,xshift=0.65cm\][$k_{m}$]{};
(5,0.05) – (5,0.65) node \[black,midway,xshift=0.8cm\][$k_{m+1}$]{};
We shall call the $F$-algebra $R$ defined as in (2), the *algebra of $n\times n$ matrices over $F$ of type $(k_{1},k_{2},\ldots,k_{m+1})$.* We see that $R$ is again a local $F$-subalgebra of $\mathbb{M}_{n}(F)$ with radical $J(R)=J$ such that $J^{m+1}=0$ and
$$\begin{aligned}
\mathrm{dim}_{F}R &=k_{1}(n-k_{1})+k_{2}(n-k_{1}-k_{2})+\cdots \notag \\
& \hspace*{1.1em}+k_{m}(n-k_{1}-k_{2}-\cdots -k_{m})+1. \notag \\
&= \sum_{j=1}^{m}k_{j}\left(n-\sum_{i=1}^{j}k_{i}\right)+1.\end{aligned}$$
A routine inductive argument shows that the expression (less $1$) appearing on the right-hand-side of (4), simplifies as
$$\sum_{j=1}^{m}k_{j}\left(n-\sum_{i=1}^{j}k_{i}\right)=\frac{1}{2}\left(n^{2}-\sum_{i=1}^{m+1}k_{i}^{2}\right)=\sum_{i,j=1,\,i<j}^{m+1}k_{i}k_{j},$$
so that (4) becomes
$$\begin{aligned}
\mathrm{dim}_{F}R=\frac{1}{2}\left(n^{2}-\sum_{i=1}^{m+1}k_{i}^{2}\right)+1=\sum_{i,j=1,\,i<j}^{m+1}k_{i}k_{j}+1.\end{aligned}$$
The algebra of $n\times n$ matrices over $F$ of type $(k_{1},k_{2},\ldots,k_{m+1})$ is clearly not commutative (unless $m=1$), but it does satisfy a weak form of commutativity called Lie nilpotence. To put this notion in context, we first recall some basic facts about Lie algebras.
Let $\mathfrak{g}$ be a Lie algebra[^6] and $x_{1},x_{2},\ldots,x_{m}$ a finite sequence of elements in $\mathfrak{g}$. We define element $[x_{1},x_{2},\ldots,x_{m}]^{\ast}$ of $\mathfrak{g}$ recursively as follows
$$\begin{aligned}
[x_{1}]^{\ast} &\overset{\mathrm{def}}{=} x_{1},\hspace*{0.3em}\makebox{\rm
and} \\[0.5ex]
[x_{1},x_{2},\ldots,x_{m}]^{\ast} &\overset{\mathrm{def}}{=}
[[x_{1},x_{2},\ldots,x_{m-1}]^{\ast},x_{m}],\hspace*{0.3em}\makebox{\rm for
$m>1$.}\end{aligned}$$
Recall that if ${\mathfrak{h}}$ is any ideal of ${\mathfrak{g}}$, then the *Lower Central Series* $\{{\mathfrak{h}}_{[m]}\}_{m\in\mathbb{N}}$ of ${\mathfrak{h}}$ is defined by
$${\mathfrak{h}}_{[m]}\overset{\mathrm{def}}{=}\{[x_{1},x_{2},\ldots,x_{m}]^{\ast}: x_{i}\in{\mathfrak{h}} \,\makebox{ {\rm for $1\leqslant i\leqslant
m$}}\}.$$
We say ${\mathfrak{g}}$ is *nilpotent* if ${\mathfrak{g}}_{[m]}=0$ for some $m\in\mathbb{N}$, $m>1$, and more specifically, *nilpotent of index $m$*, if ${\mathfrak{g}}_{[m+1]}=0$.
Every ring $R$ may be endowed with the structure of a Lie algebra (over the centre of $R$), by choosing as bracket the *commutator* defined by
$$\forall r,s\in R, \,[r,s]\,\overset{\mathrm{def}}{=} rs-sr.$$
Following [@SvW15 p. 4785], we call a ring $R$ *Lie nilpotent* \[resp. *Lie nilpotent of index $m$*\] if $R$, considered as a Lie algebra via the commutator, is nilpotent \[resp. nilpotent of index $m$\]. The reader will observe that the commutative rings are precisely the rings that are Lie nilpotent of index $1$.
A ring $R$ is said to satisfy the *Engel condition of index $m$* if the identity
$$[x,\overbrace{y,\ldots,y}^{m\mathrm{\ times}}]^{\ast}=0,$$
holds in $R$. A ring is said to satisfy the *Engel condition* if it satisfies the Engel condition of index $m$ for some $m\in{\mathbb{N}}$. Clearly a ring that is Lie nilpotent of index $m$ satisfies the Engel condition of index $m$. The following result of Riley and Wilson [@RW99 p. 974] establishes a partial converse.
If $F$ is any field and $R$ an $F$-algebra that is generated by a finite number $d$ of elements, and $R$ satisfies the Engel condition of index $m$, then $R$ is Lie nilpotent of index $f(d,m)\geqslant m$, where the index $f(d,m)$ depends only on $d$ and $m$.
Lie nilpotent rings have been shown to play an important role in the proofs of certain classical results about polynomial and trace identities in the $F$-algebra $\mathbb{M}_{n}(F)$ (see [@Dr00] and [@DrFu04]). For fields $F$ of characteristic zero, Kemer’s [@Ke91] pioneering work on the T-ideals of associative algebras has revealed the importance of identities satisfied by $n\times n$ matrices over the Grassmann (exterior) algebra $E=F\langle\{x_{i}:i\in{\mathbb{N}}\}:x_{i}x_{j}+x_{j}x_{i}=0$ whenever $1\leqslant i\leqslant j\rangle$ generated by an infinite family $\{x_{i}:i\in{\mathbb{N}}\}$ of anticommutative indeterminates. For $n\times n$ matrices over a Lie nilpotent ring of index $m$, a Cayley-Hamilton identity of degree $n^{m}$ (with left- or right-sided scalar coefficients) was found in [Sz97]{}. Since the Grassmann algebra $E$ is Lie nilpotent of index $m=2$, the aforementioned Cayley-Hamilton identity for matrices in $\mathbb{M}_{n}(E)$ is of degree $n^{2}$. In [@Do98], Domokos presents a slightly modified version of this identity in which the coefficients are invariant under the conjugation action of $\mathrm{GL}_{n}(F)$.
This paper is an attempt to answer a conjecture posed in [SvW15]{}. The statement of this conjecture is rendered less cumbersome if expressed in terms of a function $M(\ell,n)$ of positive integer arguments $\ell$ and $n$, defined as follows
$$\begin{aligned}
M(\ell,n)\,\overset{\mathrm{def}}{=} & \hspace*{0.4em}\mathrm{max}\left\{\frac{1}{2}\left(n^{2}-\sum_{i=1}^{\ell}k_{i}^{2}\right)+1:k_{1},k_{2},\ldots,k_{\ell} \makebox{ {\rm are}}\right. \notag \\
& \left.\,\makebox{{\rm nonnegative integers such that} }\sum_{i=1}^{\ell}k_{i}=n\right\}.\end{aligned}$$
> **Conjecture.** *Let $F$ be any field, $m$ and $n$ positive integers, and $R$ an $F$-subalgebra of $\mathbb{M}_{n}(F)$ with Lie nilpotence index $m$. Then*
*$$\begin{aligned}
\mathrm{dim}_{F}R \leqslant M(m+1,n).\end{aligned}$$*
We shall henceforth refer to the above as *‘the Conjecture’*. More specifically, if $F$ is any fixed field, we shall say that *‘the Conjecture holds in respect of $F$’*, if (7) holds for all positive integers $m$ and $n$, and $F$-subalgebras $R$ of $\mathbb{M}_{n}(F)$ with Lie nilpotence index $m$.
If $R$ is any algebra over a field $F$, then a module $V$ over $R$ is precisely a representation of $R$ via action on the underlying $F$-space structure on $V$. If the module is faithful, then this representation is faithful thus yielding an embedding of $R$ into $\mathrm{End}_{F}V$, the $F$-algebra of $F$-space endomorphisms on $V$. If $V$ is also finite dimensional over $F$, say $\mathrm{dim}_{F}V=n$, then $\mathrm{End}_{F}V$ is isomorphic to $\mathbb{M}_{n}(F)$ and so we have an $F$-algebra embedding of $R$ into $\mathbb{M}_{n}(F)$. (We point out that such a finite dimensional $V
$ is certain to exist if $R$ is finite dimensional, for $V$ can always be chosen to be $R$ itself.) Thus, seen through a representation theoretic lens, inequality (7) sheds light on a possible *lower* bound for the dimension of a faithful module over a given Lie nilpotent algebra.
In the same spirit, Domokos [@Do03 Theorem 1, p. 156] derives a lower bound for the dimension of a faithful module over a finite dimension algebra satisfying the polynomial identity $[x_{1},y_{1}][x_{2},y_{2}]\ldots[x_{m},y_{m}]=0$, in terms of $m$.
Our initial task, which is easily accomplished, shall be to argue that the upper bound (7) is optimal for all choices of $m$ and $n$.
Suppose first that $m+1\leqslant n$. It is proven in Corollary 27(a) that for such $m$ and $n$, $M(m+1,n)=\frac{1}{2}\left(n^{2}-\sum_{i=1}^{m+1}k_{i}^{2}\right)+1$ for some sequence of *positive* integers $k_{1},k_{2},\ldots,k_{m+1}$ satisfying $\sum_{i=1}^{m+1}k_{i}=n$. Let $F$ be any field and $R$ the algebra of $n\times n$ matrices over $F$ of type $(k_{1},k_{2},\ldots,k_{m+1})$. As noted earlier, $R$ has the form $R=FI_{n}+J$ with radical $J$ satisfying $J^{m+1}=0$. Since the set $FI_{n}$ of scalar matrices is central in $R$, it can be shown that the $k$th terms of the Lower Central Series for $R$ (interpreted as a Lie algebra via the commutator) and $J$ coincide, that is to say, $R_{[k]}=J_{[k]}$, for $k>1$. It is also evident that $J_{[k]}\subseteq J^{k}$ for every $k\in\mathbb{N}$. Thus $R_{[m+1]}=J_{[m+1]}\subseteq J^{m+1}=0$, so $R$ is Lie nilpotent of index $m$. It follows from (5) that $\mathrm{dim}_{F}R=\frac{1}{2}\left[n^{2}-\sum_{i=1}^{m+1}k_{i}^{2}\right]+1=M(m+1,n)$.
Now suppose $m+1>n$. No generality is lost if we suppose $n>1$. It is proven in Corollary 27(b) that for such $m$ and $n$, $M(m+1,n)=M(n,n)=\frac{1}{2}(n^{2}-n)+1$, and this, by (5), is equal to $\mathrm{dim}_{F}R$ where $R$ is the algebra of $n\times n$ matrices over field $F$ of type $(k_{1},k_{2},\ldots,k_{n})$ with $k_{1}=k_{2}=\cdots=k_{n}=1$. (The reader will see that in this instance, $R$ is just the algebra of all upper triangular matrices over $F$ with constant main diagonal.) As shown in the previous paragraph, such an algebra $R$ is Lie nilpotent of index $n-1$ and thus Lie nilpotent of index $m$, since $m\geqslant n-1$.
The theorem below collects together the conclusions drawn above.
Let $F$ be any field, and $m$ and $n$ arbitrary positive integers. Then there exists an $F$-subalgebra $R$ of $\mathbb{M}_{n}(F)$ with Lie nilpotence index $m$ such that
$$\mathrm{dim}_{F}R=M(m+1,n).$$
The main body of theory in this paper is developed in Sections 5 and 6 with module theoretic methods our primary tools. Sections 3 and 4 show that the Conjecture reduces to a consideration of local subalgebras of upper triangular matrix rings over an algebraically closed field. Section 7, which can be read independently of earlier sections, establishes important properties of the function $M(\ell,n)$ required in earlier theory. An explicit formula for $M(\ell,n)$ is also derived which is then shown to have a more simplified form for small values of $\ell$. In Section 8 the algebra of $n\times n$ matrices of type $(d_{1},d_{2},\ldots,d_{\ell})$ is used to provide a pictorial representation of the objects introduced in earlier theory. The content of Section 9, which is titled Open questions, is self-evident.
Preliminaries
=============
The symbol $\subseteq$ denotes containment and $\subset$ proper containment for sets.
If $X$ is any set, then $X^{n}$ denotes the cartesian product of $n$ copies of $X$.
$\mathbb{N}$ and $\mathbb{N}_{0}$ will denote the sets of positive integers, and nonnegative integers, respectively.
All rings are associative and possess identity, and all modules are unital.
Let $R$ be a ring and $V$ a right $R$-module. We write $W\leqslant V$ to indicate that $W$ is a submodule of $V$. If $X$ is a nonempty subset of $V$ and $I$ is a right ideal of $R$, then
$$(0:^{I}X)\,\overset{\mathrm{def}}{=}\{a\in I:Xa=0\}=I\cap(0:^{R}X).$$
Observe that $(0:^{I}X)$ is always a right ideal of $R$.
Let $F$ be a field. For each $n\in\mathbb{N}$, $\mathbb{M}_{n}(F)$ \[resp. $\mathbb{U}_{n}(F)$\] \[resp. $\mathbb{U}_{n}^{\ast}(F)$\] shall denote the $F$-algebra of all $n\times n$ matrices over $F$ \[resp. upper triangular $n\times n$ matrices over $F$\] \[resp. upper triangular $n\times n$ matrices over $F$ with constant main diagonal\].
The passage to local algebras over an algebraically closed field
================================================================
In this section we show that the Conjecture reduces to a consideration of local algebras over an algebraically closed field.
Let $F$ be a subfield of field $K$ and $R$ an $F$-algebra. Let $r_{1}\otimes
b_{1},r_{2}\otimes b_{2},\ldots,r_{m}\otimes b_{m}\in R\otimes_{F}K$ with $r_{i}\in R$ and $b_{i}\in K$ for $i\in\{1,2,\ldots,m\}$. Then
$$[r_{1}\otimes b_{1},r_{2}\otimes b_{2},\ldots,r_{m}\otimes
b_{m}]^{\ast}=[r_{1},r_{2},\ldots,r_{m}]^{\ast}\otimes(b_{1}b_{2}\ldots
b_{m}).$$
We provide only a proof of the inductive step.
Putting $r=[r_{1},r_{2},\ldots,r_{m}]^{\ast}$ and $b=b_{1}b_{2}\ldots b_{m}$ we see that
$$\begin{aligned}
&[r_{1}\otimes b_{1},r_{2}\otimes b_{2},\ldots,r_{m+1}\otimes b_{m+1}]^{\ast}
\\[0.5ex]
&=[r\otimes b,r_{m+1}\otimes b_{m+1}]\hspace*{0.6em}\makebox{\rm [by the
inductive hypothesis]} \\[0.5ex]
&=(r\otimes b)(r_{m+1}\otimes b_{m+1})-(r_{m+1}\otimes b_{m+1})(r\otimes b)
\\[0.5ex]
&=(rr_{m+1})\otimes (bb_{m+1})-(r_{m+1}r)\otimes (b_{m+1}b) \\[0.5ex]
&=\parbox[t]{26em}{$(rr_{m+1})\otimes (bb_{m+1})-(r_{m+1}r)\otimes (bb_{m+1})$\hspace*{0.6em}[because $K$ is a field\\[0.5ex]
\,so $b_{m+1}b=bb_{m+1}$]} \\[0.5ex]
&=(rr_{m+1}-r_{m+1}r)\otimes(bb_{m+1}) \\[0.5ex]
&=[r_{1},r_{2},\ldots,r_{m+1}]^{\ast}\otimes(b_{1}b_{2}\ldots b_{m+1}).\end{aligned}$$
Let $F$ be a subfield of field $K$ and $R$ an $F$-subalgebra of $\mathbb{M}_{n}(F)$. Then:
[[()]{}]{}
${\rm dim}_{F}R=\,{\rm dim}_{K}(R\otimes_{F}K)$.
$R\otimes_{F}K$ is isomorphic to a $K$-subalgebra of ${\mathbb{M}}_{n}(K)$.
If $R$ is Lie nilpotent of index $m$, then so is $R\otimes_{F}K$.
\(a) is standard theory - see for example [@AF92 Exercise 19.3, p. 231].
\(b) The hypothesis entails $R\otimes_{F}K$ is a $K$-subalgebra of $\mathbb{M}_{n}(F)\otimes_{F}K$. The result follows noting that $\mathbb{M}_{n}(F)\otimes_{F}K\cong\mathbb{M}_{n}(K)$ as $K$-algebras (see [@Pa91 Chapter 9, Exercise 10, p. 94]).
\(c) Suppose $R$ is Lie nilpotent of index $m$. Take $x_{1},x_{2},\ldots,x_{m+1}\in R\otimes_{F}K$. Since the expression $[x_{1},x_{2},\ldots,x_{m+1}]^{\ast}$ is additive in each of its $m+1$ arguments, $[x_{1},x_{2},\ldots,x_{m+1}]^{\ast}$ is expressible as a sum of elements of the form $[r_{1}\otimes b_{1},r_{2}\otimes b_{2},\ldots,r_{m+1}\otimes
b_{m+1}]^{\ast}$ where $r_{i}\in R$ and $b_{i}\in K$ for $i\in\{1,2,\ldots,m+1\}$. By Lemma 3
$$\begin{aligned}
&[r_{1}\otimes b_{1},r_{2}\otimes b_{2},\ldots,r_{m+1}\otimes b_{m+1}]^{\ast}
\\[0.5ex]
&=[r_{1},r_{2},\ldots,r_{m+1}]^{\ast}\otimes(b_{1}b_{2}\ldots b_{m+1}) \\[0.5ex]
&=0\otimes(b_{1}b_{2}\ldots b_{m+1})\hspace*{0.6em}\makebox{\rm [because $R$
is Lie nilpotent of index $m$]} \\[0.5ex]
&=0.\end{aligned}$$
It follows that $[x_{1},x_{2},\ldots,x_{m+1}]^{\ast}=0$, so $R\otimes_{F}K$ is Lie nilpotent of index $m$.
Let ${\mathcal{C}}$ be a nonempty class of fields and $\overline{\mathcal{C}}
$ the class of all subfields of fields in ${\mathcal{C}}$. The following statements are equivalent:
[[()]{}]{}
The Conjecture holds in respect of all fields in ${\mathcal C}$;
The Conjecture holds in respect of all fields in $\overline{\mathcal C}$.
(b)$\Rightarrow$(a) is obvious since ${\mathcal{C}}\subseteq\overline{\mathcal{C}}$.
(a)$\Rightarrow$(b) Let $m$ and $n$ be positive integers, $F\in\overline{\mathcal{C}}$, and $R$ an $F$-subalgebra of $\mathbb{M}_{n}(F)$ with Lie nilpotence index $m$. We must show that $\mathrm{dim}_{F}R\leqslant M(m+1,n)$.
Choose field extension $K$ of $F$ such that $K\in{\mathcal{C}}$. By Proposition 4 ((b) and (c)), the $K$-algebra $R\otimes_{F}K$ is Lie nilpotent of index $m$ and is isomorphic to a $K$-subalgebra of $\mathbb{M}_{n}(K)$. By part (a) of this theorem, $\mathrm{dim}_{K}(R\otimes_{F}K)\leqslant M(m+1,n)$. Hence by Proposition 4(a),
$$\mathrm{dim}_{F}R=\mathrm{dim}_{K}(R\otimes_{F}K)\leqslant M(m+1,n),$$
as required.
It follows from Theorem 5 that the Conjecture will hold for a given field $F$, if it can be shown to hold for the algebraic closure of $F$. We shall exploit this fact in the next section.
Every idempotent in a ring satisfying the Engel condition is central.
If $R$ is an arbitrary ring and $e=e^{2}\in R$, then a routine calculation shows that for each $a\in R$,
$$[[1-e,(1-e)a],e]=(1-e)ae.$$
Putting $\alpha=(1-e)ae$ we see that $\alpha e=\alpha$ and $e\alpha=0$ from which it follows that $[\alpha,e]=\alpha$. Iterating, we obtain
$$\begin{aligned}
&[[\alpha,e],e]=\alpha, \\[0.5ex]
&[[[\alpha,e],e],e]=\alpha,\hspace*{0.6em}\makebox{\rm and in general} \\[0.5ex]
&[\alpha,\overset{m}{\overbrace{e,e,\ldots,e}}]^{\ast}=\alpha,\hspace*{0.6em}\makebox{\rm for each $m\in{\mathbb{N}}$.}\end{aligned}$$
If $R$ satisfies the Engel condition of index $m$, then we have
$$\alpha=[\alpha,\overset{m}{\overbrace{e,e,\ldots,e}}]^{\ast}=0,$$
and so
$$\begin{aligned}
(1-e)ae=0.\end{aligned}$$
Interchanging the roles of $e$ and $1-e$ in the above argument yields
$$\begin{aligned}
ea(1-e)=0.\end{aligned}$$
Equations (8) and (9) imply $ae-eae=0$ and $ea-eae=0$ whence $ea=ae
$. We conclude that $e$ is central.
Every right artinian ring satisfying the Engel condition is isomorphic to a finite direct product of local rings.
It is known (see [@AF92 Theorem 27.6, p. 304] or [@Pa91 Theorem 5.9, p. 49]) that every right artinian ring $R$ contains a complete set of primitive orthogonal idempotents $\{e_{1},e_{2},\ldots,e_{k}\}$ such that $R$ decomposes as
$$R_{R}\cong e_{1}R\oplus e_{2}R\oplus \cdots \oplus e_{k}R,$$
where each $e_{i}R$ has unique maximal proper submodule $e_{i}J(R)$. If $R$ satisfies the Engel condition, then each idempotent $e_{i}$ is central by Proposition 6, so the above decomposition is a decomposition of (two-sided) ideals with each $e_{i}R=e_{i}Re_{i}$ a local ring.
Let $F$ be a field and $e$ an idempotent of $\mathbb{M}_{n}(F)$. If $\mathrm{rank}\,e=r$, then $e\mathbb{M}_{n}(F)e\cong\mathbb{M}_{r}(F)$ as $F$-algebras.
Since $\mathrm{rank}\,e=r$, $F^{(n)}e$ has dimension $r$ as an $F$-space, so $F^{(n)}e\cong F^{(r)}$ as $F$-spaces. Then
$$\begin{aligned}
e\mathbb{M}_{n}(F)e & \cong\mathrm{End}_{F}\left(F^{(n)}e\right) \\[0.5ex]
& \cong\mathrm{End}_{F}\left(F^{(r)}\right) \\[0.5ex]
& \cong \mathbb{M}_{r}(F).\end{aligned}$$
The following theorem tells us that for a given field $F$, the Conjecture will hold for all $F$-subalgebras of $\mathbb{M}_{n}(F)$, if it can be shown to hold for all local $F$-subalgebras of $\mathbb{M}_{n}(F)$.
The following statements are equivalent for a field $F$:
[[()]{}]{}
The Conjecture holds in respect of $F$;
For all positive integers $m$ and $n$, if $R$ is any [*local*]{} $F$-subalgebra of ${\mathbb{M}}_{n}(F)$ with Lie nilpotence index $m$, then
$${\rm dim}_{F}R\leqslant M(m+1,n).$$
(a)$\Rightarrow$(b) is obvious.
(b)$\Rightarrow$(a) Let $m$ and $n$ be positive integers and $R$ an $F$-subalgebra of $\mathbb{M}_{n}(F)$ with Lie nilpotence index $m$. Note that $R$ satisfies the Engel condition of index $m$. Since $R$ is a finite dimensional $F$-algebra, it is right (and left) artinian, and so by Proposition 7, $R\cong R_{1}\times R_{2}\times\cdots\times R_{k}$ where each $R_{i}$ is a local right artinian ring. This entails the existence of a complete set of central primitive orthogonal idempotents $\{e_{1},e_{2},\ldots,e_{k}\}$ in $R$ such that
$$\begin{aligned}
R_{R}\cong e_{1}R\oplus e_{2}R\oplus\cdots\oplus e_{k}R\end{aligned}$$
with $e_{i}R=e_{i}Re_{i}\cong R_{i}$ for each $i\in\{1,2,\ldots,k\}
$. For each $i\in\{1,2,\ldots,k\}$ put
$$\begin{aligned}
r_{i}\,\overset{\mathrm{def}}{=}\mathrm{rank}\,e_{i}.\end{aligned}$$
The equation $1_{R}=I_{n}=e_{1}+e_{2}+\cdots+e_{k}$ induces the $F$-space decomposition
$$F^{(n)}=F^{(n)}e_{1}\oplus F^{(n)}e_{2}\oplus \cdots \oplus F^{(n)}e_{k}.$$
Thus
$$\begin{aligned}
n & = \mathrm{dim}_{F}F^{(n)} \notag \\[0.5ex]
& = \mathrm{dim}_{F}(F^{(n)}e_{1})+\mathrm{dim}_{F}(F^{(n)}e_{2})+\cdots+\mathrm{dim}_{F}(F^{(n)}e_{k}) \notag \\[0.5ex]
& = r_{1}+r_{2}+\cdots+r_{k}\hspace*{0.6em}\makebox{\rm [by (11)].}\end{aligned}$$
Observe that each local ring $e_{i}R$ is an $F$-subalgebra of $e_{i}\mathbb{M}_{n}(F)e_{i}$, and that $e_{i}\mathbb{M}_{n}(F)e_{i}\cong
\mathbb{M}_{r_{i}}(F)$ for each $i\in\{1,2,\ldots,k\}$, by Lemma 8. It is clear too that each $e_{i}R$ must be Lie nilpotent of index $m$, since $R$ has the same property and $e_{i}R\subseteq R$.
The aforementioned facts, together with (b), imply that $\mathrm{dim}_{F}(e_{i}R)\leqslant M(m+1,r_{i})$ for each $i\in\{1,2,\ldots, k\}$. Then
$$\begin{aligned}
\mathrm{dim}_{F}R & = \sum_{i=1}^{k}\mathrm{dim}_{F}(e_{i}R)\hspace*{0.6em}\makebox{\rm [by (10)]} \notag \\[0.5ex]
& \leqslant\sum_{i=1}^{k}M(m+1,r_{i}) \notag \\[0.5ex]
& \leqslant M\left(m+1,\sum_{i=1}^{k}r_{i}\right)\hspace*{0.6em}\makebox{\rm
[by Proposition 28]} \notag \\[0.5ex]
& = M(m+1,n)\hspace*{0.6em}\makebox{\rm [by (12)].}\end{aligned}$$
Simultaneous triangularization and the passage to upper triangular matrix rings
===============================================================================
The main result of this section (Theorem 12) shows that for algebraically closed fields $F$, the Conjecture reduces to a consideration of $F$-subalgebras of $\mathbb{U}_{n}^{\ast}(F)$, the algebra of upper triangular matrices over $F$ with constant main diagonal.
Recall that an $F$-subalgebra $R$ of $\mathbb{M}_{n}(F)$ is said to be *simultaneously upper triangularizable* in $\mathbb{M}_{n}(F)$ if there exists an invertible $U\in\mathbb{M}_{n}(F)$ such that $U^{-1}RU\subseteq
\mathbb{U}_{n}(F)$.
A key result is the following. Although implicit in [Ja44]{} we shall provide a proof in the absence of a suitable reference.
Let $F$ be an algebraically closed field.
[[()]{}]{}
If $R$ is a finite dimensional local $F$-algebra, then $R$ has $F$-space decomposition $R=F\cdot 1_{R}\oplus J(R)$.
If $R$ is a local $F$-subalgebra of ${\mathbb{M}}_{n}(F)$, then there exists an invertible $U\in{\mathbb{M}}_{n}(F)$ such that $U^{-1}RU\subseteq {\mathbb{U}}_{n}^{\ast}(F)$. Thus, $R$ is isomorphic to an $F$-subalgebra of ${\mathbb{U}}_{n}^{\ast}(F)$.
\(a) Since $R$ is local, it follows that $R/J(R)$ is a division algebra that is finite dimensional over $F$. Since $F$ is algebraically closed this implies $R/J(R)\cong F$. Inasmuch as $F\cdot 1_{R}\cap J(R)=0$, the equation
$$\begin{aligned}
\mathrm{dim}_{F}(F\cdot 1_{R}+J(R)) &=1+\mathrm{dim}_{F}J(R) \\[0.5ex]
&=\mathrm{dim}_{F}(R/J(R))+\mathrm{dim}_{F}J(R) \\[0.5ex]
&=\mathrm{dim}_{F}R\end{aligned}$$
entails $R=F\cdot 1_{R}\oplus J(R)$.
\(b) It is known (see [@RR12 Theorem 1.4.6, p. 12]) that for an algebraically closed field $F$, a necessary and sufficient condition for an $F$-subalgebra $R$ of $\mathbb{M}_{n}(F)$ to be simultaneously upper triangularizable in $\mathbb{M}_{n}(F)$ is that $R/J(R)$ is commutative, a condition that is clearly met in our case. Hence $U^{-1}RU\subseteq \mathbb{U}_{n}(F)$ for some invertible $U\in\mathbb{M}_{n}(F)$. Putting $S=U^{-1}RU$ we note that since $S$ is local, $S=FI_{n}\oplus J(S)$ by (a). Since every element of $J(S)$ is a nilpotent matrix in $\mathbb{U}_{n}(F)$, and a nilpotent upper triangular matrix is strictly upper triangular, we have
$$U^{-1}RU=S=FI_{n}\oplus J(S)\subseteq \mathbb{U}_{n}^{\ast}(F).$$
\(a) The observation that the factor ring $R/J(R)$ is commutative, is key in the proof of Proposition 10(b). We point out that this property is possessed by all Lie nilpotent rings. Indeed, [@SvW15 Proposition 3.1(3), p. 4790] asserts that if $\mathrm{rad}(R)$ denotes the prime radical of a Lie nilpotent ring $R$, then $R/\mathrm{rad}(R)$ is commutative. Since $\mathrm{rad}(R)\subseteq J(R)$, the commutativity of $R/J(R)$ follows.
\(b) If $F$ is an algebraically closed field of characteristic zero (the latter assumption is not made in Proposition 10) and $R$ is any Lie nilpotent $F$-subalgebra of $\mathbb{M}_{n}(F)$, then $R$ can be shown to be simultaneously upper triangularizable in $\mathbb{M}_{n}(F)$ as a consequence of Lie’s Theorem which asserts that if ${\mathfrak{g}}$ is a finite dimensional solvable Lie algebra with representation $\mathbb{M}_{n}(F)$, then ${\mathfrak{g}}$ is simultaneously upper triangularizable in $\mathbb{M}_{n}(F)$. Lie’s Theorem applies inasmuch as every Lie nilpotent ring is a nilpotent Lie algebra with respect to the commutator, and nilpotent Lie algebras are solvable. (This latter fact is explained in the second open question of Section 9.)
The following statements are equivalent for an algebraically closed field $F$:
[[()]{}]{}
The Conjecture holds in respect of $F$;
For all positive integers $m$ and $n$, if $R$ is any [*local*]{} $F$-subalgebra of ${\mathbb{M}}_{n}(F)$ with Lie nilpotence index $m$, then $${\rm dim}_{F}R\leqslant M(m+1,n);$$
For all positive integers $m$ and $n$, if $R$ is any $F$-subalgebra of ${\mathbb{U}}_{n}^{\ast}(F)$ with Lie nilpotence index $m$, then $${\rm dim}_{F}R\leqslant M(m+1,n).$$
\(a) and (b) are equivalent by Theorem 9 without any restriction on the field $F$.
The equivalence of (b) and (c) is a consequence of Proposition 10(b) which tells us that up to isomorphism, the local $F$-subalgebras of $\mathbb{M}_{n}(F)$ are precisely the $F$-subalgebras of $\mathbb{U}_{n}^{\ast}(F)$.
Subalgebras of $\mathbb{U}_{n}^{\ast}(F)$
=========================================
The main body of theory is developed in this section.
Throughout this section and unless otherwise stated, $F$ shall denote a field and $R$ an $F$-subalgebra of $\mathbb{U}_{n}^{\ast}(F)$.
Let $V$ be a faithful right $R$-module. We define a sequence $\{R_{k}\}_{k\in\mathbb{N}}$ of $F$-subalgebras of $R$, a sequence $\{J_{k}\}_{k\in\mathbb{N}}$ where each $J_{k}$ is an ideal of $R_{k}$, and a sequence $\{U_{k}\}_{k\in\mathbb{N}}$ of $F$-subspaces of $V$ as follows
$$\begin{cases}
R_{1}\overset{\mathrm{def}}{=} R, \\[1ex]
J_{1}\overset{\mathrm{def}}{=} J(R_{1}),\hspace*{0.3em}\makebox{\rm and} \\[1ex]
U_{1}\overset{\mathrm{def}}{=}\hspace*{0.2em}\makebox{\rm any $F$-subspace
complement of $VJ_{1}$ in $V$.}\end{cases}$$
For $k\in\mathbb{N}$, $k\geqslant 2$, define
$$\begin{aligned}
\begin{cases}
R_{k}\overset{\mathrm{def}}{=} FI_{n} + (0:^{R_{k-1}}U_{k-1}), \\[1ex]
J_{k}\overset{\mathrm{def}}{=} J(R_{k}),\hspace*{0.3em}\makebox{\rm and} \\[1ex]
U_{k}\overset{\mathrm{def}}{=}\hspace*{0.2em}\parbox[t]{20em}{any $F$-subspace complement of
$VJ_{1}J_{2}\ldots J_{k}$ in\\[0.2\baselineskip]
$VJ_{1}J_{2}\ldots J_{k-1}$.}\end{cases}$$
It follows from the definition of $U_{k}$ that
$$\begin{aligned}
VJ_{1}J_{2} \ldots J_{k-1} = U_{k} \oplus VJ_{1}J_{2} \ldots J_{k}\end{aligned}$$
as $F$-spaces.
For convenience we put $J_{0}=R$.
Since $(0:^{R_{k-1}} U_{k-1})\subseteq R_{k-1}$ and since every $F$-subalgebra of $\mathbb{U}_{n}^{\ast}(F)$ contains $FI_{n}$, it is clear from the definition of $R_{k}$ in (14) that $R_{k-1} \supseteq R_{k}$ for every $k\in\mathbb{N}$, $k\geqslant 2$. We thus have
$$\begin{aligned}
R_{1}\supseteq R_{2}\supseteq\cdots\end{aligned}$$
It is easily shown that if $S$ and $T$ are any $F$-subalgebras of $\mathbb{U}_{n}^{\ast}(F)$, then $S \subseteq T$ if and only if $J(S)
\subseteq J(T)$. In the light of this observation, (16) implies that
$$\begin{aligned}
J_{1}\supseteq J_{2}\supseteq\cdots\end{aligned}$$
Since $J_{k} \subseteq J_{1}$ for all $k\in \mathbb{N}$, and $J_{1}
$ is nilpotent, we must have $J_{0}J_{1} \ldots J_{k}=0$ for $k$ sufficiently large. Define
$$\begin{aligned}
\ell\overset{\mathrm{def}}{=}\min\hspace*{0.1em}\{k\in\mathbb{N}:
J_{0}J_{1}\ldots J_{k}=0\}.\end{aligned}$$
It follows from (17) that $J_{0}J_{1}\ldots J_{k-1}\supseteq
J_{0}J_{1}\ldots J_{k}$ for each $k\in\mathbb{N}$. We thus have the descending chain
$$R=J_{0}\supseteq J_{0}J_{1}\supseteq\cdots\supseteq J_{0}J_{1}\ldots
J_{\ell-1}\supseteq J_{0}J_{1}\ldots J_{\ell}=0.$$
This, in turn, induces a descending chain
$$\begin{aligned}
V = VJ_{0}\supseteq V J_{0}J_{1}\supseteq\cdots\supseteq VJ_{0}J_{1}\ldots
J_{\ell-1}\supseteq 0.\end{aligned}$$
Note that $VJ_{0}J_{1} \ldots J_{\ell-1} \neq 0$ since $J_{0}J_{1}
\ldots J_{\ell-1} \neq 0$ and $V$ is a faithful right $R$-module.
Recall that if $R$ is an arbitrary ring, then a submodule $N$ of a right $R$-module $M$ is said to be *superfluous* if
$$\forall L\leqslant M, \,N+L=M\Rightarrow L=M.$$
If $I$ is a nilpotent ideal of an arbitrary ring $R$ and $M$ is any right $R$-module, then $MI$ is a superfluous submodule of $M$.
Suppose $MI+L=M$ with $L\leqslant M$. Multiplying by $I$ we obtain $MI^{2}+LI=MI$, so $MI^{2}+LI+L=MI+L=M$. Continuing in this way, we obtain $MI^{k}+L=M$ for all $k\in\mathbb{N}$. Since $I$ is nilpotent this yields, for $k$ sufficiently large, the equation $MI^{k}+L=M\cdot 0+L=L=M$.
Important properties of the chain (19) are established in the next lemma.
Let the sequences $\{R_{k}\}_{k\in\mathbb{N}}$, $\{J_{k}\}_{k\in\mathbb{N}}$ and $\{U_{k}\}_{k\in\mathbb{N}}$ be defined as in (14), and positive integer $\ell$ defined as in (18). Let $k \in \{1,2,\ldots,\ell\}$. Then:
[[()]{}]{}
$VJ_{0}J_{1} \ldots J_{k}$ is a superfluous $R_{k}$-submodule of $VJ_{0}J_{1} \ldots J_{k-1}$;
$U_{k}R_{k}=VJ_{0}J_{1} \ldots J_{k-1}=U_{k}\oplus \cdots \oplus U_{\ell}$;
$J_{k+1}=(0:^{R_{k}}U_{k})$;
$VJ_{0}J_{1} \ldots J_{k-1}$ is a faithful right $R_{k}$-module.
\(a) That $VJ_{0}J_{1}\ldots J_{k}$ is a right $R_{k}$-module is a consequence of the fact that $J_{0}J_{1}\ldots J_{k}$ is an $F$-subspace of $R$ that is closed under right multiplication by elements from $R_{k}$.
Since $R_{k}\subseteq R_{k-1}$, every right $R_{k-1}$-module is canonically a right $R_{k}$-module. In particular, $VJ_{0}J_{1}\ldots J_{k-1}$ is a right $R_{k}$-module.
It remains to show that $VJ_{0}J_{1}\ldots J_{k}$ is superfluous in $VJ_{0}J_{1}\ldots J_{k-1}$. Put $U=VJ_{0}J_{1}\ldots J_{k-1}$. Since $J_{k}\subseteq J_{1}$ and $J_{1}$ is nilpotent, $J_{k}$ must also be nilpotent. It follows from Lemma 13 that $UJ_{k}$ is a superfluous submodule of $U$, as required.
\(b) Since $U_{k}R_{k}\supseteq U_{k}$, it follows from (15) that
$$VJ_{0}J_{1}\ldots J_{k-1}=U_{k}R_{k}+VJ_{0}J_{1}\ldots J_{k}$$
where the right-hand-side of the above equation is a sum of $R_{k}$-submodules of $VJ_{0}J_{1}\ldots J_{k-1}$. Since $VJ_{0}J_{1}\ldots J_{k}$ is a superfluous $R_{k}$-submodule of $VJ_{0}J_{1}\ldots J_{k-1}$ by (a), we must have $U_{k}R_{k}=VJ_{0}J_{1}\ldots J_{k-1}$.
To establish the equation $VJ_{0}J_{1}\ldots J_{k-1}=U_{k}\oplus\cdots\oplus
U_{\ell}$, we note first that the $U_{i}$ constitute an independent family of $F$-subspaces of $V$. This is clear from the definition of the $U_{i}$ in (14). This means that the sum $U_{k}\oplus\cdots\oplus U_{\ell}$ is indeed a direct sum of $F$-subspaces. It remains to establish equality.
Since, by (14), $U_{\ell}$ is an $F$-subspace complement of $VJ_{0}J_{1}\ldots J_{\ell}$ in $VJ_{0}J_{1}\ldots J_{\ell-1}$, and since $VJ_{0}J_{1}\ldots J_{\ell}=0$ by definition of $\ell$, we must have
$$VJ_{0}J_{1}\ldots J_{\ell-1}=U_{\ell}.$$
Repeated application of the formula for $U_{k}$ in (14) shows that $$VJ_{0}J_{1}\ldots J_{\ell-2}=U_{\ell-1}\oplus U_{\ell},$$
and, more generally, that
$$VJ_{0}J_{1}\ldots J_{k-1}=U_{k}\oplus\cdots\oplus U_{\ell},$$
as required.
\(c) Since $k-1<\ell$, it follows from (b) and the minimality of $\ell$ that $U_{k}R_{k}=VJ_{0}J_{1}\ldots J_{k-1}\neq 0$, whence $U_{k}\neq 0$. This means that $(0:^{R_{k}}U_{k})$ must be a proper right ideal of $R_{k}$ and so cannot contain any units of $R_{k}$. Inasmuch as $R_{k}$ is an $F$-subalgebra of $\mathbb{U}_{n}^{\ast}(F)$, $(0:^{R_{k}}U_{k})$ must therefore comprise strictly upper triangular matrices. Since, by (14), $R_{k+1}=FI_{n}+(0:^{R_{k}}U_{k})$, we must have $J_{k+1}=J(R_{k+1})=(0:^{R_{k}}U_{k})$.
\(d) We use induction on $k$. Take $k=1$. Then $VJ_{0}J_{1}\ldots
J_{k-1}=VJ_{0}=V$, which is a faithful $R_{1}$-module by hypothesis. This establishes the base case.
To establish the inductive step, take $t\in R_{k}$ with $k\geqslant 2$ and suppose
$$\begin{aligned}
(VJ_{0}J_{1}\ldots J_{k-1})t=0.\end{aligned}$$
Since $VJ_{0}J_{1}\ldots J_{k-1}\neq 0$, $t$ cannot be a unit of $R_{k}$, and since $R_{k}$ is local, we must have $t\in J_{k}$. By (c), $J_{k}=(0:^{R_{k-1}}U_{k-1})$, so
$$\begin{aligned}
U_{k-1}t=0.\end{aligned}$$
We thus have
$$\begin{aligned}
(VJ_{0}J_{1}\ldots J_{k-2})t &=(U_{k-1}+VJ_{0}J_{1}\ldots J_{k-1})t\hspace*{0.6em}\makebox{\rm [by (15)]} \\[0.5ex]
&=0\hspace*{0.6em}\makebox{\rm [by (20) and (21)].}\end{aligned}$$
By the inductive hypothesis, $VJ_{0}J_{1}\ldots J_{k-2}$ is a faithful right $R_{k-1}$-module. Since $t\in J_{k}\subseteq R_{k}\subseteq
R_{k-1}$, the above equation entails $t=0$. We conclude that $VJ_{0}J_{1}\ldots J_{k-1}$ is a faithful $R_{k}$-module.
\(a) Taking $k=1$ in Lemma 14(b) yields the $F$-subspace decomposition
$$\begin{aligned}
V=U_{1}\oplus U_{2}\oplus\cdots\oplus U_{\ell}.\end{aligned}$$
Substituting the equation $VJ_{0}J_{1}\ldots
J_{k-1}=U_{k}\oplus\cdots\oplus U_{\ell}$ of Lemma 14(b) into (22) yields
$$\begin{aligned}
V=U_{1}\oplus\cdots\oplus U_{k-1}\oplus VJ_{0}J_{1}\ldots J_{k-1}.\end{aligned}$$
\(b) The faithfulness of $VJ_{0}J_{1}\ldots J_{k-1}$ proved in Lemma 14(d) means that $(VJ_{0}J_{1}\ldots J_{k-1})J_{k}=0$ if and only if $J_{k}=0$. Moreover, since $V$ is faithful as a right $R$-module, we have that
$$VJ_{0}J_{1}\ldots J_{k}=0 \,\Leftrightarrow\, J_{0}J_{1}\ldots J_{k}=0.$$
It follows that $$J_{0}J_{1}\ldots J_{k}=0 \,\Leftrightarrow\, J_{k}=0.$$
This has the consequence that
$$\ell=\min\hspace*{0.1em}\{k\in\mathbb{N}: J_{0}J_{1}\ldots J_{k}=0\}=\min\hspace*{0.1em}\{k\in\mathbb{N}: J_{k}=0\}.$$
Let the sequences $\{R_{k}\}_{k\in\mathbb{N}}$, $\{J_{k}\}_{k\in\mathbb{N}}$ and $\{U_{k}\}_{k\in\mathbb{N}}$ be defined as in (14), and positive integer $\ell$ defined as in (18). Then
$$R=R_{1}\supset R_{2}\supset\cdots\supset R_{\ell}=R_{\ell+1}=\cdots$$
is a strictly descending chain of $F$-subalgebras of $\mathbb{U}_{n}^{\ast}(F)$ that stabilizes at $R_{\ell}$. Moreover, $J_{\ell}=0$, so that $R_{\ell}=FI_{n}$.
Suppose $R_{k}=R_{k+1}$ for some $k\leqslant\ell$. Note that we cannot have $U_{k}=0$ since this would imply, by Lemma 14(b), that $VJ_{0}J_{1}\ldots
J_{k-1}=0$, which contradicts the fact that $VJ_{0}J_{1}\ldots
J_{k-1}\supseteq VJ_{0}J_{1}\ldots J_{\ell-1}\neq 0$. Now
$$\begin{aligned}
0 &=U_{k}J_{k+1}\hspace*{0.6em}\makebox{\rm [because
$J_{k+1}=(0:^{R_{k}}U_{k})$ by Lemma 14(c)]} \\[1ex]
&=U_{k}J_{k}\hspace*{0.6em}\makebox{\rm [because
$J_{k}=J(R_{k})=J(R_{k+1})=J_{k+1}$ by hypothesis]} \\[1ex]
&=(U_{k}R_{k})J_{k}\hspace*{0.6em}\makebox{\rm [because $J_{k}$ is an ideal
of $R_{k}$]} \\[1ex]
&=(VJ_{0}J_{1}\ldots J_{k-1})J_{k}\hspace*{0.6em}\makebox{\rm [by Lemma
14(b)].}\end{aligned}$$
Since $k\leqslant\ell$ it follows from the minimality of $\ell$ that $k=\ell$. We have thus proven that $R_{k}\supset R_{k+1}$ for $k\in\{1,2,\ldots,\ell-1\}$.
In Remark 15(b) we noted that $J_{\ell}=0$. Since $R_{\ell}\subseteq\mathbb{U}_{n}^{\ast}(F)$, this entails $R_{\ell}=FI_{n}$. However, since every $F$-subalgebra of $\mathbb{U}_{n}^{\ast}(F)$ contains $FI_{n}$, the descending chain of $F$-subalgebras must stabilize at $R_{\ell}$.
Let the sequences $\{R_{k}\}_{k\in\mathbb{N}}$, $\{J_{k}\}_{k\in\mathbb{N}}$ and $\{U_{k}\}_{k\in\mathbb{N}}$ be defined as in (14), and positive integer $\ell$ defined as in (18). For each $k\in\{1,2,\ldots,\ell\}$ define
$$\begin{aligned}
d_{k}\overset{\mathrm{def}}{=}\mathrm{dim}_{F}U_{k}.\end{aligned}$$
A key step in the proof of Theorem 17(c) below is inspired by [@Gu76 2. Proof of Schur’s Inequality, p. 558].
Let the sequences $\{R_{k}\}_{k\in\mathbb{N}}$, $\{J_{k}\}_{k\in\mathbb{N}}$ and $\{U_{k}\}_{k\in\mathbb{N}}$ be defined as in (14), positive integer $\ell$ defined as in (18), and $\{d_{k}:1\leqslant k\leqslant \ell\}$ defined as in (24). Then:
[[()]{}]{}
${\displaystyle{\rm dim}_{F}(U_{k}J_{k})={\rm dim}_{F}V-\sum_{i=1}^{k}d_{i}}$ for each $k\in\{1,2,\ldots,\ell\}$.
${\displaystyle{\rm dim}_{F}V=\sum_{i=1}^{\ell}d_{i}}$.\
${\rm dim}_{F}R \leqslant M(\ell,{\rm dim}_{F}V)$.
\(a) Inasmuch as
$$\begin{aligned}
V &=U_{1}\oplus\cdots\oplus U_{k}\oplus VJ_{0}J_{1}\ldots J_{k}\hspace*{0.6em}\makebox{\rm [by (23)]} \\[0.5ex]
&=U_{1}\oplus\cdots\oplus U_{k}\oplus (VJ_{0}J_{1}\ldots J_{k-1})J_{k} \\[0.5ex]
&=U_{1}\oplus\cdots\oplus U_{k}\oplus (U_{k}R_{k})J_{k}\hspace*{0.6em}\makebox{\rm [by Lemma 14(b)]} \\[0.5ex]
&=U_{1}\oplus\cdots\oplus U_{k}\oplus U_{k}J_{k}\hspace*{0.6em}\makebox{\rm
[because $J_{k}$ is an ideal of $R_{k}$],}\end{aligned}$$
we have $\mathrm{dim}_{F}V=d_{1}+\cdots+d_{k}+\mathrm{dim}_{F}(U_{k}J_{k})$, from which (a) follows.
\(b) is an immediate consequence of (22) and (24).
\(c) If $\ell=1$, then $J=J_{\ell}=0$, so $\mathrm{dim}_{F}R=1=M(1,\mathrm{dim}_{F}V)$ and there is nothing further to prove. Suppose $\ell\geqslant 2$.
We next derive the recursive formula
$$\begin{aligned}
\mathrm{dim}_{F}J_{k}\leqslant d_{k}\left(\mathrm{dim}_{F}V-\sum_{i=1}^{k}d_{i}\right)+\mathrm{dim}_{F}J_{k+1}\hspace*{0.6em}(1\leqslant
k\leqslant\ell).\end{aligned}$$
To this end, take $k\in\{1,2,\ldots,\ell\}$, $X\in J_{k}$ and let $\rho_{X}:U_{k}\rightarrow U_{k}J_{k}$ be the right multiplication by $X$ map. Observe that $\rho_{X}$ is an $F$-linear map and thus a member of $\mathrm{Hom}_{F}(U_{k},U_{k}J_{k})$.
Define the map $\Theta: J_{k} \rightarrow \mathrm{Hom}_{F}(U_{k},U_{k}J_{k})$ by $\Theta(X)=\rho_{X}$. It is also easily seen that $\Theta$ is an $F$-linear map. Note that
$$\begin{aligned}
\mathrm{Ker}\,\Theta & = \{X\in J_{k}:\rho_{X}=0\} \notag \\[0.5ex]
& = \{X\in J_{k}:U_{k}X=0\} \notag \\[0.5ex]
& = (0:^{J_{k}}U_{k}) \notag \\[0.5ex]
& = J_{k}\cap (0:^{R_{k}}U_{k}) \notag \\[0.5ex]
& =J_{k}\cap J_{k+1}\hspace*{0.6em}\makebox{\rm [by Lemma 14(c)]} \notag \\[0.5ex]
& =J_{k+1}\hspace*{0.6em}\makebox{\rm [because $J_{k}\supseteq J_{k+1}$].}\end{aligned}$$
We thus have
$$\begin{aligned}
\mathrm{dim}_{F}J_{k} &=\mathrm{rank}\,\Theta+\mathrm{nullity}\,\Theta \\[0.5ex]
&\leqslant \mathrm{dim}_{F}\left(\mathrm{Hom}_{F}(U_{k},U_{k}J_{k})\right)+\mathrm{dim}_{F}J_{k+1}\hspace*{0.6em}\makebox{\rm [by (26)]} \\[0.5ex]
&=\mathrm{dim}_{F}U_{k}\cdot\mathrm{dim}_{F}(U_{k}J_{k})+\mathrm{dim}_{F}J_{k+1} \\[0.5ex]
&=d_{k}\left(\mathrm{dim}_{F}V-{\displaystyle\sum_{i=1}^{k}d_{i}}\right)+\mathrm{dim}_{F}J_{k+1}\hspace*{0.6em}\makebox{\rm [by (a)],}\end{aligned}$$
which is (25).
Letting $k$ take on the values from $1$ to $\ell-1$ in (25), we see that
$$\begin{aligned}
\mathrm{dim}_{F}J_{1} &\leqslant\sum_{j=1}^{\ell-1}d_{j}\left(\mathrm{dim}_{F}V-\sum_{i=1}^{j}d_{i}\right)+\mathrm{dim}_{F}J_{\ell} \notag \\[0.5ex]
&=\sum_{j=1}^{\ell-1}d_{j}\left(\mathrm{dim}_{F}V-\sum_{i=1}^{j}d_{i}\right)\hspace*{0.6em}\makebox{\rm [because $J_{\ell}=0$]} \notag \\[0.5ex]
&=\frac{1}{2}\left((\mathrm{dim}_{F}V)^{2}-\sum_{i=1}^{\ell}d_{i}^{2}\right)\hspace*{0.6em}\makebox{\rm [because ${\rm dim}_{F}V=\sum_{i=1}^{\ell}d_{i}$
by (b)]} \notag \\[0.5ex]
&\leqslant
\parbox[t]{24em}{$M(\ell,{\rm dim}_{F}V)-1$\hspace*{0.6em}[by the definition of $M(\ell,{\rm dim}_{F}V)$ \\[0.5ex]
\,noting that ${\rm dim}_{F}V=\sum_{i=1}^{\ell}d_{i}$].}\end{aligned}$$
Since $R$ has $F$-space decomposition $R=FI_{n}\oplus J$, we have
$$\begin{aligned}
\mathrm{dim}_{F}R &=1+\mathrm{dim}_{F}J \\[0.5ex]
&=1+\mathrm{dim}_{F}J_{1}\hspace*{0.6em}\makebox{\rm [because $J=J_{1}$]} \\[0.5ex]
&\leqslant 1+M(\ell,\mathrm{dim}_{F}V)-1\hspace*{0.6em}\makebox{\rm [by
(27)]} \\[0.5ex]
&=M(\ell,\mathrm{dim}_{F}V).\end{aligned}$$
In Proposition 29 it is shown that $M(\ell,n)$ is an increasing function in both arguments. This means, with reference to Theorem 17(c), that the smaller the value of $\ell$, the lower the upper bound $M(\ell,\mathrm{dim}_{F}V)$ for $\mathrm{dim}_{F}R$.
We shall show presently that if the $F$-subalgebra $R$ of $\mathbb{U}_{n}^{\ast}(F)$ has radical $J$ satisfying $J^{m}=0$ for some $m\in\mathbb{N}
$, then the value of $\ell$ cannot exceed $m$, and so
$$\mathrm{dim}_{F}R\leqslant M(m,\mathrm{dim}_{F}V).$$
In the next section we shall strengthen the above by proving that if $R$ has Lie nilpotence index $m$ (this is the case if $J^{m+1}=0$), then the value of $\ell$ cannot exceed $m+1$, from which we may deduce
$$\mathrm{dim}_{F}R\leqslant M(m+1,\mathrm{dim}_{F}V).$$
Since the $d_{i}$ are positive in Theorem 17(b), we must have $\ell\leqslant\mathrm{dim}_{F}V$. A combination of Theorem 17(c), the fact that $M(\ell,n)$ is increasing in its first argument (Proposition 29), and the formula for $M(n,n)$ derived in Corollary 27(a), yields:
If $R$ is an $F$-subalgebra of $\mathbb{U}_{n}^{\ast}(F)$ and $V$ any faithful right $R$-module, then
$$\mathrm{dim}_{F}R\leqslant M(\mathrm{dim}_{F}V,\mathrm{dim}_{F}V)={\textstyle \frac{1}{2}}\left((\mathrm{dim}_{F}V)^{2}-\mathrm{dim}_{F}V\right)+1.$$
If $V=F^{n}=\overbrace{F\times F\times\cdots\times F}^{n\hspace*{0.2em}\mathrm{times}}$ is interpreted as a $1\times n$ matrix over $F$, then it has the canonical structure of a faithful right module with respect to any $F
$-subalgebra of the matrix algebra $\mathbb{M}_{n}(F)$. For such a module $V$, we have
$$\mathrm{dim}_{F}V=n.$$
This allows us to replace $\mathrm{dim}_{F}V$ with $n$ in each of the results in this, and subsequent, sections. In particular, taking $\mathrm{dim}_{F}V=n$ in the previous corollary yields the upper bound
$$\mathrm{dim}_{F}R\leqslant{\textstyle \frac{1}{2}}(n^{2}-n)+1,$$
an observation that has little value, since the expression $\frac{1}{2}(n^{2}-n)+1$ coincides with the dimension of the overlying $F$-algebra $\mathbb{U}_{n}^{\ast}(F)$.
Let the sequences $\{R_{k}\}_{k\in\mathbb{N}}$, $\{J_{k}\}_{k\in\mathbb{N}}$ and $\{U_{k}\}_{k\in\mathbb{N}}$ be defined as in (14), and positive integer $\ell$ defined as in (18). If $J^{m}=0$ for some $m\in\mathbb{N}$, then $\ell\leqslant m$.
Inasmuch as $J_{0}J_{1}\ldots J_{m}\subseteq J^{m}=0$, it follows from the definition of $\ell$ in (18) that $\ell\leqslant m$.
If $R$ is an $F$-subalgebra of $\mathbb{U}_{n}^{\ast}(F)$ satisfying $J^{m}=0
$, and $V$ is any faithful right $R$-module, then
$$\mathrm{dim}_{F}R\leqslant M(m,\mathrm{dim}_{F}V).$$
It follows from Theorem 17(c) and Proposition 20 that there exists a positive integer $\ell\leqslant m$ such that $\mathrm{dim}_{F}R\leqslant
M(\ell,\mathrm{dim}_{F}V)$. By Proposition 29, $M(\ell,\mathrm{dim}_{F}V)\leqslant M(m,\mathrm{dim}_{F}V)$, whence $\mathrm{dim}_{F}R\leqslant
M(m,\mathrm{dim}_{F}V)$.
Lie nilpotent subalgebras of $\mathbb{U}_{n}^{\ast}(F)$: the main theorem
=========================================================================
A routine inductive argument establishes the following.
Let $R$ be an arbitrary ring and $\{r_{i}:1\leqslant i\leqslant m\}\subseteq
R$. Then
$$[r_{1},r_{2},\ldots,r_{m}]^{\ast}=\sum_{\sigma\in
S_{m}}c_{\sigma}r_{\sigma(1)}r_{\sigma(2)}\ldots r_{\sigma(m)}$$
where $c_{\sigma}\in\{-1,0,1\}$ for all $\sigma\in S_{m}$, and $\{\sigma\in S_{m}: c_{\sigma}\neq 0$ and $\sigma(1)=1\}$ is a singleton comprising the identity permutation.
Let the sequences $\{R_{k}\}_{k\in\mathbb{N}}$, $\{J_{k}\}_{k\in\mathbb{N}}$ and $\{U_{k}\}_{k\in\mathbb{N}}$ be defined as in (14), and positive integer $\ell$ defined as in (18). If $R$ is Lie nilpotent of index $m$, then $\ell\leqslant m+1$.
By Lemma 14(c), we have $J_{2}=(0:^{R}U_{1})$. Pick arbitrary $r\in R$ and $b_{k}\in J_{k}$ for each $k\in\{2,\ldots,m+1\}$. Since $U_{1}J_{2}=0$ and $J_{2}\supseteq J_{3}\supseteq\cdots\supseteq J_{m+1}$, we have $U_{1}b_{k}=0$ for all $k\in\{2,\ldots,m+1\}$. Thus, using Lemma 22, we see that $U_{1}[r,b_{2},\ldots,b_{m+1}]^{\ast}=U_{1}rb_{2}\ldots b_{m+1}$. But $R$ is Lie nilpotent of index $m$, so $[r,b_{2},\ldots,b_{m+1}]^{\ast}=0$ whence $U_{1}rb_{2}\ldots b_{m+1}=0$. Since $r$ is arbitrary, we get
$$\begin{aligned}
0 &=(U_{1}R)b_{2}\ldots b_{m+1} \\[0.5ex]
&=V b_{2}\ldots b_{m+1}\hspace*{0.6em}\makebox{\rm [because $U_{1}R=V$ by
Lemma 14(b)],} \\[0.5ex]\end{aligned}$$
from which we infer $b_{2}\ldots b_{m+1}=0$ since $V$ is faithful. It follows that $J_{2}\ldots J_{m+1}=0$, so $\ell\leqslant m+1$ by definition of $\ell$.
For all positive integers $m$ and $n$, and fields $F$, if $R$ is any $F$-subalgebra of $\mathbb{U}_{n}^{\ast}(F)$ with Lie nilpotence index $m$, then
$$\mathrm{dim}_{F}R\leqslant M(m+1,n).$$
Let $m$ and $n$ be arbitrary positive integers, and $F$ an arbitrary field. Let $R$ be an $F$-subalgebra of $\mathbb{U}_{n}^{\ast}(F)$ with Lie nilpotence index $m$. If sequences $\{R_{k}\}_{k\in\mathbb{N}}$, $\{J_{k}\}_{k\in\mathbb{N}}$ and $\{U_{k}\}_{k\in\mathbb{N}}$ are defined as in (14), and positive integer $\ell$ defined as in (18), then it follows from Theorem 17(c) that
$$\mathrm{dim}_{F}R\leqslant M(\ell,\mathrm{dim}_{F}V).$$
Choose $V$ to be $F^{n}$, so that $\mathrm{dim}_{F}V=n$ (see Remark 19). By Proposition 23, $\ell\leqslant m+1$. Since $M(\ell,n)$ is increasing in its first argument by Proposition 29, we have
$$\mathrm{dim}_{F}R\leqslant M(\ell,\mathrm{dim}_{F}V)=M(\ell,n)\leqslant
M(m+1,n).$$
Let $R$ be any $F$-subalgebra of $\mathbb{U}_{n}^{\ast}(F)$ satisfying the polynomial identity
$$f(x_{1},x_{2},\ldots,x_{m})=\sum_{\sigma\in
S_{m}}c_{\sigma}x_{\sigma(1)}x_{\sigma(2)}\ldots x_{\sigma(m)}=0$$
where $c_{\sigma}\in F$ for all $\sigma\in S_{m}$, and $\{\sigma\in S_{m}: c_{\sigma}\neq 0$ and $\sigma(1)=1\}$ is a singleton comprising the identity permutation.
Arguments similar to those used earlier in this section show that
$$\mathrm{dim}_{F}R\leqslant M(m+1,n).$$
We are finally in a position to complete the proof of the Conjecture.
> **Proof of Conjecture.** Let $F$ be any field with algebraic closure $K$. Taking the field $F$ of Theorems 12 and 24 to be $K$, we see that the latter is just Statement (c) of the former. It thus follows from Theorem 12 ((c)$\Rightarrow$(a)) that the Conjecture holds in respect of field $K$.
>
> Taking the class of fields ${\mathcal{C}}$ in Theorem 5 to be the singleton ${\mathcal{C}}=\{K\}$ and noting that $F$ is a subfield of $K$, we conclude that the Conjecture holds in respect of field $F$. Since $F$ was chosen arbitrarily, the proof is complete.
The function $M(\ell,n)$
========================
The purposes of this section are twofold. First, to establish a number of important properties of the function $M(\ell,n)$ that are required in earlier theory, and second to obtain an explicit description of $M(\ell,n)$; without such a description, the important results of this paper remain somewhat opaque. This task will involve the solution of an integer-variable optimization problem. Our methods, however, are first principled and require no background knowledge of integer optimization techniques.
We shall make use of the following notation: if ${\mathbf{k}}=(k_{1},k_{2},\ldots,k_{\ell})\in\mathbb{N}_{0}^{\ell}$, then:
[[()]{}]{}
${\rm supp}\,{\mathbf k}\,{\stackrel{\rm def}{=}}\{i\in\{1,2,\ldots,\ell\}:k_{i}>0\}$; and
${\displaystyle
|{\mathbf k}|\,{\stackrel{\rm def}{=}}\left(\sum_{i=1}^{\ell}k_{i}^{2}\right)^{1/2}}$ so that ${\displaystyle |{\mathbf k}|^{2}=\sum_{i=1}^{\ell}k_{i}^{2}}$.
Let $\ell$ and $n$ be positive integers. Then the following statements are equivalent for ${\mathbf{k}}=(k_{1},k_{2},\ldots,k_{\ell})\in\mathbb{N}_{0}^{\ell}$ such that $\sum_{i=1}^{\ell}k_{i}=n$:
[[()]{}]{}
$M(\ell,n)=\frac{1}{2}\left(n^{2}-|{\mathbf k}|^{2}\right)+1$;
$|k_{i}-k_{j}|\leqslant 1$ for all $i,j\in\{1,2,\ldots,\ell\}$.
(a)$\Rightarrow$(b) Suppose (a) holds but $|k_{p}-k_{q}|\geqslant 2$ for some $p,q\in\{1,2,\ldots,\ell\}$. Without loss of generality, we may suppose that $k_{p}\geqslant k_{q}+2$. Define ${\mathbf{k}}^{\prime}=(k_{1}^{\prime},k_{2}^{\prime},\ldots,k_{\ell}^{\prime})\in\mathbb{N}_{0}^{\ell}$ by:
$$k_{i}^{\prime}\overset{\mathrm{def}}{=}
\begin{cases}
k_{i},\hspace*{0.4em}\makebox{\rm if $i\notin\{p,q\}$} \\[1ex]
k_{p}-1,\hspace*{0.4em}\makebox{\rm if $i=p$} \\[1ex]
k_{q}+1,\hspace*{0.4em}\makebox{\rm if $i=q$.}\end{cases}$$
Note that $\sum_{i=1}^{\ell}k_{i}^{\prime}=\sum_{i=1}^{\ell}k_{i}=n
$. Then:
$$\begin{aligned}
&{\textstyle\frac{1}{2}}\left(n^{2}-|{\mathbf{k}}^{\prime}|^{2}\right)+1-M(\ell,n) \\[1ex]
&= {\textstyle \frac{1}{2}}\left(n^{2}-|{\mathbf{k}}^{\prime}|^{2}\right)+1-\left({\textstyle \frac{1}{2}}\left(n^{2}-|{\mathbf{k}}|^{2}\right)+1\right)
\\
&= \frac{1}{2}\left(\sum_{i=1}^{\ell}\left(k_{i}^{2}-(k_{i}^{\prime})^{2}\right)\right) \\[0.5ex]
&= {\textstyle \frac{1}{2}}\left(k_{p}^{2}+k_{q}^{2}-(k_{p}^{\prime})^{2}-(k_{q}^{\prime})^{2}\right) \\[1ex]
&= {\textstyle \frac{1}{2}}\left(k_{p}^{2}+k_{q}^{2}-(k_{p}-1)^{2}-(k_{q}+1)^{2}\right) \\[1ex]
&= {\textstyle \frac{1}{2}}\left(2k_{p}-2k_{q}-2\right) \\[1ex]
&= k_{p}-k_{q}-1>0\hspace*{0.6em}\makebox{\rm [because $k_{p}\geqslant
k_{q}+2$].}\end{aligned}$$
This implies that $\frac{1}{2}\left(n^{2}-|{\mathbf{k}}^{\prime}|^{2}\right)+1>M(\ell,n)$, a contradiction.
(b)$\Rightarrow$(a) Suppose ${\mathbf{k}}=(k_{1},k_{2},\ldots,k_{\ell})\in\mathbb{N}_{0}^{\ell}$ is such that $\sum_{i=1}^{\ell}k_{i}=n$ and $|k_{i}-k_{j}|\leqslant 1$ for all $i,j\in\{1,2,\ldots,\ell\}$. Inasmuch as each $k_{i}$ is nonnegative this implies the existence of some $r\in\mathbb{N}$ such that
$$\begin{aligned}
k_{i}\in\{r-1,r\}\hspace*{0.5em}\forall i\in\{1,2,\ldots,\ell\}.\end{aligned}$$
Now suppose $M(\ell,n)=\frac{1}{2}\left(n^{2}-|{\mathbf{k}}^{\prime}|^{2}\right)+1$ with ${\mathbf{k}}^{\prime}=(k_{1}^{\prime},k_{2}^{\prime},\ldots,k_{\ell}^{\prime})\in\mathbb{N}_{0}^{\ell}$ such that $\sum_{i=1}^{\ell}k_{i}^{\prime}=n$. It follows from implication (a)$\Rightarrow$(b) that $|k_{i}^{\prime}-k_{j}^{\prime}|\leqslant 1$ for all $i,j\in\{1,2,\ldots,\ell\}$, so there must exist some $s\in\mathbb{N}$ such that
$$\begin{aligned}
k_{i}^{\prime}\in\{s-1,s\}\hspace*{0.5em}\forall i\in\{1,2,\ldots,\ell\}.\end{aligned}$$
If $r<s$, then it follows from (28) and (29) that
$$k_{i}\leqslant r\leqslant s-1\leqslant k_{i}^{\prime}\hspace*{0.5em}\forall
i\in\{1,2,\ldots,\ell\}.$$
Since $\sum_{i=1}^{\ell}k_{i}=\sum_{i=1}^{\ell}k_{i}^{\prime}$, the above inequalities can only be satisfied if $k_{i}=k_{i}^{\prime}$ for all $i\in\{1,2,\ldots,\ell\}$, whence ${\mathbf{k}}={\mathbf{k}}^{\prime}$.
A similar argument shows that ${\mathbf{k}}={\mathbf{k}}^{\prime}$ whenever $r>s$. Thus if $r\neq s$, then ${\mathbf{k}}={\mathbf{k}}^{\prime}$, whence $\frac{1}{2}\left(n^{2}-|{\mathbf{k}}|^{2}\right)+1=\frac{1}{2}\left(n^{2}-|{\mathbf{k}}^{\prime}|^{2}\right)+1=M(\ell,n)$ and the proof is complete.
Now suppose $r=s$. Since $k_{i},k_{i}^{\prime}\in\{r,r-1\}$ for each $i\in\{1,2,\ldots,\ell\}$ and since $\sum_{i=1}^{\ell}k_{i}=\sum_{i=1}^{\ell}k_{i}^{\prime}$, it is easily seen that ${\mathbf{k}}$ and ${\mathbf{k}}^{\prime}$ are equal to within permutation of their coordinates, that is to say, there exists a permutation $\sigma\in S_{\ell}$ such that $k_{i}^{\prime}=k_{\sigma(i)}$ for all $i\in\{1,2,\ldots,\ell\}$. Clearly, in such a situation $|{\mathbf{k}}|=|{\mathbf{k}}^{\prime}|$ and $\frac{1}{2}\left(n^{2}-|{\mathbf{k}}|^{2}\right)+1=\frac{1}{2}\left(n^{2}-|{\mathbf{k}}^{\prime}|^{2}\right)+1=M(\ell,n)$.
Let $\ell$ and $n$ be positive integers. Then:
[[()]{}]{}
If $\ell\leqslant n$, then $M(\ell,n)=\frac{1}{2}\left(n^{2}-|{\mathbf k}|^{2}\right)+1$ for some ${\mathbf k}=(k_{1},k_{2},\ldots,k_{\ell})\in{\mathbb{N}}_{0}^{\ell}$ with $k_{i}\geqslant 1$ for all $i\in\{1,2,\ldots,\ell\}$. In particular, $M(n,n)=\frac{1}{2}\left(n^{2}-n\right)+1$.
If $\ell>n$, then $M(\ell,n)=M(n,n)=\frac{1}{2}\left(n^{2}-n\right)+1$.
By Proposition 26, we can choose ${\mathbf{k}}=(k_{1},k_{2},\ldots,k_{\ell})\in\mathbb{N}_{0}^{\ell}$ such that $\sum_{i=1}^{\ell}k_{i}=n$, $M(\ell,n)=\frac{1}{2}\left(n^{2}-|{\mathbf{k}}|^{2}\right)+1$ and $|k_{i}-k_{j}|\leqslant 1$ for all $i,j\in\{1,2,\ldots,\ell\}$.
\(a) Suppose $\ell\leqslant n$. If $k_{j}=0$ for some $j\in\{1,2,\ldots,\ell\}
$, then $k_{i}\in\{0,1\}$ for all $i\in\{1,2,\ldots,\ell\}$, whence $n=\sum_{i=1}^{\ell}k_{i}<\ell\leqslant n$, a contradiction.
If $\ell=n$, then clearly $k_{i}=1$ for all $i\in\{1,2,\ldots,\ell\}$, so $|{\mathbf{k}}|^{2}=n$ and $M(\ell,n)=M(n,n)=\frac{1}{2}\left(n^{2}-n\right)+1$.
\(b) Suppose $\ell> n$. Since $n=\sum_{i=1}^{\ell}k_{i}$, we must have $k_{j}=0$ for some $j\in\{1,2,\ldots,\ell\}$. Thus $k_{i}\in\{0,1\}$ for all $i\in\{1,2,\ldots,\ell\}$, so $|{\mathbf{k}}|^{2}=n$ and $M(\ell,n)=M(n,n)$.
Let $\ell$ be an integer satisfying $\ell\geqslant 2$ and $n_{1},n_{2},\ldots,n_{k}$ any sequence of positive integers. Then $$M\!\left(\ell,\sum_{i=1}^{k}n_{i}\right)\geqslant
\sum_{i=1}^{k}M(\ell,n_{i}).$$
We provide a proof in the case $k=2$; the arguments used can be applied mutatis-mutandis to establish the inductive step in a proof by induction on $k$. Choose ${\mathbf{k}}=(k_{1},k_{2},\ldots,k_{\ell})\in\mathbb{N}_{0}^{\ell}$ such that
$$\begin{aligned}
\sum_{i=1}^{\ell}k_{i}=n_{1}\end{aligned}$$
and
$$\begin{aligned}
M(\ell,n_{1})={\textstyle \frac{1}{2}}\left(n_{1}^{2}-|{\mathbf{k}}|^{2}\right)+1,\end{aligned}$$
and choose $\overline{\mathbf{k}}=(\bar{k}_{1},\bar{k}_{2},\ldots,\bar{k}_{\ell})\in\mathbb{N}_{0}^{\ell}$ such that
$$\begin{aligned}
\sum_{i=1}^{\ell}\bar{k}_{i}=n_{2}\end{aligned}$$
and
$$\begin{aligned}
M(\ell,n_{2})={\textstyle \frac{1}{2}}\left(n_{2}^{2}-|\overline{\mathbf{k}}|^{2}\right)+1.\end{aligned}$$
If $|\mathrm{supp}\,{\mathbf{k}}|=|\mathrm{supp}\,{\overline{\mathbf{k}}}|=1$, then it follows from (30) that $M(\ell,n_{1})=\frac{1}{2}\left(n_{1}^{2}-n_{1}^{2}\right)+1=1$, and from (32) that $M(\ell,n_{2})=\frac{1}{2}\left(n_{2}^{2}-n_{2}^{2}\right)+1=1$. Since $\ell$, $n_{1}+n_{2}\geqslant 2$, it is clear that we can choose ${\mathbf{k}}^{\ast}=(k_{1}^{\ast},k_{2}^{\ast},\ldots,k_{\ell}^{\ast})\in\mathbb{N}_{0}^{\ell}$ such that $|\mathrm{supp}\,{\mathbf{k}}^{\ast}|\geqslant 2$ and $\sum_{i=1}^{\ell}k_{i}^{\ast}=n_{1}+n_{2}$. Then
$$\begin{aligned}
M(\ell,n_{1}+n_{2}) &\geqslant {\textstyle\frac{1}{2}\left((n_{1}+n_{2})^{2}-|{\mathbf{k}}^{\ast}|^{2}\right)+1} \\[0.5ex]
&= \frac{1}{2}\left(\left(\sum_{i=1}^{\ell}k_{i}^{\ast}\right)^{2}-\sum_{i=1}^{\ell}(k_{i}^{\ast})^{2}\right)+1 \\[0.5ex]
&= \sum_{i,j=1,\,i<j}^{\ell}k_{i}^{\ast}k_{j}^{\ast}+1 \\[0.5ex]
&\geqslant 2=M(\ell,n_{1})+M(\ell,n_{2}),\end{aligned}$$
as required.
Now suppose $|\mathrm{supp}\,{\mathbf{k}}|\geqslant 2$ or $|\mathrm{supp}\,{\overline{\mathbf{k}}}|\geqslant 2$.
Put ${\overline{\overline{\mathbf{k}}}}=(\bar{\bar{k}}_{1},\bar{\bar{k}}_{2},\ldots,\bar{\bar{k}}_{\ell})={\mathbf{k}}+{\overline{\mathbf{k}}}$. By (30) and (32)
$$\begin{aligned}
\sum_{i=1}^{\ell}\bar{\bar{k}}_{i}=n_{1}+n_{2}.\end{aligned}$$
Then
$$\begin{aligned}
M(\ell,n_{1}+n_{2}) &\geqslant
\parbox[t]{22em}{
$\frac{1}{2}\left((n_{1}+n_{2})^{2}-|{\overline{\overline{\mathbf k}}}|^{2}\right)+1$\hspace*{0.6em}[by (34) and the defini-\\[0.2\baselineskip]
tion of $M(\ell,n_{1}+n_{2})$]} \notag \\[0.5ex]
& = {\textstyle\frac{1}{2}}\left(n_{1}^{2}+n_{2}^{2}-|{\mathbf{k}}|^{2}-|{\overline{\mathbf{k}}}|^{2}\right)+ n_{1}n_{2}-\sum_{i=1}^{\ell}k_{i}\bar{k}_{i}+1 \notag \\[1ex]
& =
\parbox[t]{22em}{
$M(\ell,n_{1})+M(\ell,n_{2})+n_{1}n_{2}-{\displaystyle\sum_{i=1}^{\ell}}k_{i}\bar{k}_{i}-1$\hspace*{0.6em}[by (31)\\
and (33)]
} \notag \\[0.5ex]
& =
M(\ell,n_{1})+M(\ell,n_{2})+\left(\sum_{i=1}^{\ell}k_{i}\right)\left(\sum_{i=1}^{\ell}\bar{k}_{i}\right)-\sum_{i=1}^{\ell}k_{i}\bar{k}_{i}-1
\notag \\
&\hspace*{1.3em}{[}\parbox[t]{15em}{by (30) and (32)]} \notag \\[0.5ex]
& = M(\ell,n_{1})+M(\ell,n_{2})+\sum_{i,j=1,\,i\neq j}^{\ell}k_{i}\bar{k}_{j}-1.\end{aligned}$$
Since, by hypothesis, $|\mathrm{supp}\,{\mathbf{k}}|\geqslant 2$ or $|\mathrm{supp}\,{\overline{\mathbf{k}}}|\geqslant 2$, we must have $\sum_{i,j=1,\,i\neq j}^{\ell}k_{i}\bar{k}_{j}\geqslant 1$, hence by (35), $M(\ell,n_{1}+n_{2})\geqslant M(\ell,n_{1})+M(\ell,n_{2})$, as required.
The function $M(\ell,n)$ is increasing in both its arguments.
That $M(\ell,n)$ is increasing in its second argument is an immediate consequence of Proposition 28.
To show that $M(\ell,n)$ is increasing in its first argument, it suffices to show that $M(\ell,n)\leqslant M(\ell+1,n)$. Choose ${\mathbf{k}}=(k_{1},k_{2},\ldots,k_{\ell})\in\mathbb{N}_{0}^{\ell}$ such that $\sum_{i=1}^{\ell}k_{i}=n$ and $M(\ell,n)=\frac{1}{2}\left(n^{2}-|{\mathbf{k}}|^{2}\right)+1$. Putting ${\mathbf{k}}^{\prime}=(k_{1},k_{2},\ldots,k_{\ell},0)\in\mathbb{N}_{0}^{\ell+1}$, we see that $M(\ell,n)=\frac{1}{2}\left(n^{2}-|{\mathbf{k}}|^{2}\right)+1=\frac{1}{2}\left(n^{2}-|{\mathbf{k}}^{\prime}|^{2}\right)+1\leqslant M(\ell+1,n)$, as required.
We attempt now an explicit description of the function $M(\ell,n)$. This is achieved in Theorem 31. If $\ell$ and $n$ are positive integers with $\ell\geqslant n$, then Corollary 27 exhibits the simple formula $M(\ell,n)=\frac{1}{2}\left(n^{2}-n\right)+1$. We shall therefore restrict our attention to the case $\ell\leqslant n$. For such integers $\ell $ and $n$ we denote by $n\,(\mathrm{mod}\,\ell)$ the nonnegative remainder on dividing $n$ by $\ell$, that is, the unique integer $r<\ell$ that satisfies
$$n=\left\lfloor\frac{n}{\ell}\right\rfloor\ell +r.$$
Let $r=n\,(\mathrm{mod}\,\ell)$ and define ${\mathbf{d}}=(d_{1},d_{2},\ldots,d_{\ell})\in\mathbb{N}_{0}^{\ell}$ by
$$\begin{aligned}
d_{i}\overset{\mathrm{def}}{=}
\begin{cases}
\left\lfloor\frac{n}{\ell}\right\rfloor,\hspace*{0.4em}\makebox{\rm for
$1\leqslant i\leqslant \ell-r$} \\[2ex]
\left\lfloor\frac{n}{\ell}\right\rfloor+1,\hspace*{0.4em}\makebox{\rm for
$\ell-r<i\leqslant\ell$}.\end{cases}$$
We omit the proof of the following routine lemma.
Let $\ell$ and $n$ be positive integers with $\ell\leqslant n$ and $r=n\,(\mathrm{mod}\,\ell)$. If ${\mathbf{d}}$ is defined as in (36), then
$$|{\mathbf{d}}|^{2}=(\ell-r)\left\lfloor\frac{n}{\ell}\right\rfloor^2+r\left(\left\lfloor\frac{n}{\ell}\right\rfloor+1\right)^2= \frac{n^{2}-r^{2}}{\ell}+r.$$
Let $\ell$ and $n$ be positive integers with $\ell\leqslant n$ and $r=n\,(\mathrm{mod}\,\ell)$. If ${\mathbf{d}}$ is defined as in (36), then
$$\begin{aligned}
M(\ell,n) & ={\textstyle\frac{1}{2}}\left(n^{2}-|{\mathbf{d}}|^{2}\right)+1
\\[1ex]
& =\frac{1}{2}\left(n^2 - (\ell-r)\left\lfloor\frac{n}{\ell}\right\rfloor^2
- r\left(\left\lfloor\frac{n}{\ell}\right\rfloor + 1\right)^2\right) + 1 \\[1ex]
& = \frac{n^{2}(\ell-1)}{2\ell}+\frac{1}{2}\left(\frac{r^{2}}{\ell}-r\right)+1.\end{aligned}$$
It is clear from the definition of ${\mathbf{d}}$ in (36) that $\sum_{i=1}^{\ell}d_{i}=n$ and $|d_{i}-d_{j}|\leqslant 1$ for all $i,j\in\{1,2,\ldots,\ell\}$. Hence, by Proposition 26 ((b)$\Rightarrow$(a)),
$$\begin{aligned}
M(\ell,n) &= {\textstyle\frac{1}{2}}\left(n^{2}-|{\mathbf{d}}|^{2}\right)+1
\notag \\[1ex]
& = \frac{1}{2}\left(n^2 - (\ell-r)\left\lfloor\frac{n}{\ell}\right\rfloor^2
- r\left(\left\lfloor\frac{n}{\ell}\right\rfloor + 1\right)^2\right) + 1\hspace*{0.6em}\makebox{\rm [by Lemma 30]} \\[1ex]
& = \frac{1}{2}\left(n^{2}-\left(\frac{n^{2}-r^{2}}{\ell}+r\right)\right)+1\hspace*{0.6em}\makebox{\rm [by Lemma 30]} \\[1ex]
& = \frac{1}{2}\left(n^{2}-\frac{n^{2}}{\ell}+\frac{r^{2}}{\ell}-r\right)+1
\\[1ex]
& = \frac{n^{2}(\ell-1)}{2\ell}+\frac{1}{2}\left(\frac{r^{2}}{\ell}-r\right)+1.\end{aligned}$$
Suppose $F$ is any field and $R$ the algebra of $n\times n$ matrices over $F$ of type
$${\mathbf{d}}=(\overbrace{ {\textstyle\left\lfloor\frac{n}{\ell}\right\rfloor}, {\textstyle\left\lfloor\frac{n}{\ell}\right\rfloor},\ldots,{\textstyle\left\lfloor\frac{n}{\ell}\right\rfloor}}^{(\ell-r)\mathrm{\ times}},\overbrace{{\textstyle\left\lfloor\frac{n}{\ell}\right\rfloor+1},{\textstyle\left\lfloor\frac{n}{\ell}\right\rfloor+1},\ldots, {\textstyle\left\lfloor\frac{n}{\ell}\right\rfloor +1}}^{r\mathrm{\ times}}),$$
with $n\geqslant\ell\geqslant 2$.
Figure 1 is a pictorial representation of the radical $J$ of $R$.
(0,9)–(9,0);
;
(9,9) rectangle (0.5,8.5); (9,8.5) rectangle (1,8); (9,7) rectangle (2.5,6.5); (9,6.5) rectangle (3.5,5.5); (9,5.5) rectangle (4.5,4.5); (9,2) rectangle (8,1);
at (8.5,7.25) [$\cdot$]{}; at (8.5,7.5) [$\cdot$]{}; at (8.5,7.75) [$\cdot$]{};
at (8.5,2.25) [$\cdot$]{}; at (8.5,2.5) [$\cdot$]{}; at (8.5,2.75) [$\cdot$]{}; at (8.5,3) [$\cdot$]{}; at (8.5,3.25) [$\cdot$]{}; at (8.5,3.5) [$\cdot$]{}; at (8.5,3.75) [$\cdot$]{}; at (8.5,4) [$\cdot$]{}; at (8.5,4.25) [$\cdot$]{};
at (9.32,7.25) [$\cdot$]{}; at (9.32,7.5) [$\cdot$]{}; at (9.32,7.75) [$\cdot$]{}; at (9.32,2) [$\cdot$]{}; at (9.32,2.25) [$\cdot$]{}; at (9.32,2.5) [$\cdot$]{}; at (9.32,2.75) [$\cdot$]{}; at (9.32,3) [$\cdot$]{}; at (9.32,3.25) [$\cdot$]{}; at (9.32,3.5) [$\cdot$]{}; at (9.32,3.75) [$\cdot$]{}; at (9.32,4) [$\cdot$]{}; at (9.32,4.25) [$\cdot$]{}; at (9.32,4.5) [$\cdot$]{};
at (0,4.5) [$J\,=\,$]{};
(9,8.55) – (9,8.95) node \[black,midway,xshift=1.1cm\][$d_{1}=\left\lfloor\frac{n}{\ell}\right\rfloor$]{};
(9,8.05) – (9,8.45) node \[black,midway,xshift=1.1cm\][$d_{2}=\left\lfloor\frac{n}{\ell}\right\rfloor$]{};
(9,6.55) – (9,6.95) node \[black,midway,xshift=1.3cm\][$d_{\ell-r}=\left\lfloor\frac{n}{\ell}\right\rfloor$]{};
(9,5.55) – (9,6.45) node \[black,midway,xshift=1.8cm\][$d_{\ell-r+1}=\left\lfloor\frac{n}{\ell}\right\rfloor+1$]{};
(9,4.55) – (9,5.45) node \[black,midway,xshift=1.8cm\][$d_{\ell-r+2}=\left\lfloor\frac{n}{\ell}\right\rfloor+1$]{};
(9,1.05) – (9,1.95) node \[black,midway,xshift=1.6cm\][$d_{\ell-1}=\left\lfloor\frac{n}{\ell}\right\rfloor+1$]{};
(9,0.05) – (9,0.95) node \[black,midway,xshift=1.4cm\][$d_{\ell}=\left\lfloor\frac{n}{\ell}\right\rfloor+1$]{};
Inasmuch as $R$ has the form $R=FI_{n}+J$ with $J$ satisfying $J^{\ell}=0$, it follows that $R$ is Lie nilpotent of index $\ell-1$. (This assertion is explained in more detail in the discussion following the statement of the Conjecture (7).) Moreover,
$$\begin{aligned}
\mathrm{dim}_{F}R &=
\sum_{j=1}^{\ell-1}d_{j}\left(n-\sum_{i=1}^{j}d_{i}\right)+1 \\[0.5ex]
&= {\textstyle\frac{1}{2}}\left(n^{2}-|{\mathbf{d}}|^{2}\right)+1 \\[0.5ex]
&= M(\ell,n)\hspace*{0.6em}\makebox{\rm [by Theorem 31].}\end{aligned}$$
Thus $R$ is an $F$-subalgebra of $\mathbb{M}_{n}(F)$ whose dimension is maximal amongst $F$-subalgebras of $\mathbb{M}_{n}(F)$ with Lie nilpotence index $\ell-1$.
If $\frac{1}{2}\left(n^{2}-|{\mathbf{k}}|^{2}\right)+1$ is interpreted as a real-valued function of real variables ${\mathbf{k}}=(k_{1},k_{2},\ldots,k_{\ell})\in{\mathbb{R}}^{\ell}$, the methods of multivariable Calculus show that the function $\frac{1}{2}\left(n^{2}-|{\mathbf{k}}|^{2}\right)+1$, subject to the constraint $\sum_{i=1}^{\ell}k_{i}=n$, attains a maximum of $\frac{n^{2}(\ell-1)}{2\ell}+1$ at ${\mathbf{k}}=(\frac{n}{\ell},\frac{n}{\ell},\ldots,\frac{n}{\ell})\in{\mathbb{R}}^{\ell}$. Thus
$$\begin{aligned}
\left\lfloor\frac{n^{2}(\ell-1)}{2\ell}\right\rfloor+1\geq M(\ell,n).\end{aligned}$$
We explore now instances in which (37) is an equation, a situation that arises precisely when $D<1$, where
$$D\,\overset{\mathrm{def}}{=}\frac{n^{2}(\ell-1)}{2\ell}+1-M(\ell,n).$$
It follows from Theorem 31 that
$$\begin{aligned}
D=\frac{1}{2}\left(r-\frac{r^{2}}{\ell}\right)\end{aligned}$$
where $r=n\,(\mathrm{mod}\,\ell)$. Observe that $D=D(r,\ell)$ is a function only of $r$ and $\ell$.
Figure 2 is a sketch of the level curve $D(r,\ell)=1$ in the $r\ell$-plane, interpreting $r$ and $\ell$ as real-valued variables. A simple calculation shows that the curve has equation
$$\ell=\frac{r^{2}}{r-2}.$$
Its essential features are obtained using elementary Calculus.
(axis cs:0,16) – (axis cs:16,16); (axis cs:0,0) – (axis cs:16,0);
coordinates [(2,0) (2,16)]{}; coordinates [(0,8) (4,8)]{}; coordinates [(4,0) (4,8)]{};
(axis cs:2.34314575,16)–(axis cs:13.6568542,16); ;
; ;
(axis cs:4,8) \[fill\] circle \[blue, radius=1\];
at (axis cs:3,6)[${\mathcal S}$]{};
at (axis cs:9,10);
at (axis cs:10,12);
The shaded region is
$${\mathcal{S}}\,\overset{\mathrm{def}}{=}\{(r,\ell)\in{\mathbb{R}}^{2}:0\leqslant r\leqslant\ell-1\makebox{ {\rm and} }D(r,\ell)<1\}.$$
The content of Theorem 32 below is easily gleaned from Figure 2 by assembling together points $(r,\ell)$ belonging to ${\mathcal{S}}$ that have integral coordinates.
Let $\ell$ and $n$ be positive integers with $\ell\leqslant n$ and $r=n\,(\mathrm{mod}\,\ell)$. Then the following statements are equivalent:
[[()]{}]{}
${\displaystyle M(\ell,n)=\left\lfloor\frac{n^{2}(\ell-1)}{2\ell}\right\rfloor+1}$;\
$(r,\ell)$ belongs to one of the following (disjoint) sets:
[[()]{}]{}
$\{(r,\ell):0\leqslant r\leqslant \ell-1$ and $1\leqslant\ell\leqslant 7\}$;
$\{(r,\ell):0\leqslant r\leqslant 2$ and $\ell\geqslant 8\}$;
$\{(r,r+1):r\geqslant 7\}\cup\{(r,r+2):r\geqslant 7\}$;
$\{(3,8),(5,8)\}$.
The reader will observe with reference to Theorem 32(b)(i), that if, amongst others, $1\leqslant\ell\leqslant 7$, we have the simplified formula
$$M(\ell,n)=\left\lfloor\frac{n^{2}(\ell-1)}{2\ell}\right\rfloor+1.$$
In particular, if $\ell=2$, then
$$M(2,n)=\left\lfloor\frac{n^{2}}{4}\right\rfloor+1,$$
which corresponds with the upper bound in Schur’s classical result.
An illustrative example
=======================
The main body of theory developed in Section 5 is based on the triple of sequences $\{R_{k}\}_{k\in\mathbb{N}}$, $\{J_{k}\}_{k\in\mathbb{N}}$ and $\{U_{k}\}_{k\in\mathbb{N}}$ defined in (14). In this section we show that the terms in these sequences are easily visualized in the case where $R$ is the algebra of $n\times n$ matrices over field $F$ of type $(d_{1},d_{2},\ldots,d_{\ell})$. Indeed, this special case provides the germ for our proof strategy.
Let $F$ be any field and $(d_{1},d_{2},\ldots,d_{\ell})$ any sequence of positive integers satisfying $\sum_{i=1}^{\ell}d_{i}=n$ with $\ell\geqslant 2
$. Let $R$ be the algebra of $n\times n$ matrices over $F$ of type $(d_{1},d_{2},\ldots,d_{\ell})$. We saw in Section 1 that the radical $J$ of $R$ has pictorial representation
(0,5)–(5,0);
;
(5,5) rectangle (1,4); (5,4) rectangle (1.7,3.3); (5,1.8) rectangle (4.3,0.7);
at (4.5,2.05) [$\cdot$]{}; at (4.5,2.25) [$\cdot$]{}; at (4.5,2.45) [$\cdot$]{}; at (4.5,2.65) [$\cdot$]{}; at (4.5,2.85) [$\cdot$]{}; at (4.5,3.05) [$\cdot$]{};
at (5.32,1.65) [$\cdot$]{}; at (5.32,1.85) [$\cdot$]{}; at (5.32,2.05) [$\cdot$]{}; at (5.32,2.25) [$\cdot$]{}; at (5.32,2.45) [$\cdot$]{}; at (5.32,2.65) [$\cdot$]{}; at (5.32,2.85) [$\cdot$]{}; at (5.32,3.05) [$\cdot$]{}; at (5.32,3.25) [$\cdot$]{};
at (0,2.5) [$J_{1}=J=\,$]{};
(5,4.05) – (5,4.95) node \[black,midway,xshift=0.6cm\][$d_{1}$]{};
(5,3.35) – (5,3.95) node \[black,midway,xshift=0.6cm\][$d_{2}$]{};
(5,0.75) – (5,1.75) node \[black,midway,xshift=0.8cm\][$d_{\ell-1}$]{};
(5,0.05) – (5,0.65) node \[black,midway,xshift=0.6cm\][$d_{\ell}$]{};
Observe that $\mathrm{dim}_{F}J_{1}$ corresponds with the sum of the dimensions (to be visualized as areas) of each of the $\ell -1$ blocks that make up $J_{1}$. With this perspective we see that
$$\mathrm{dim}_{F}J_{1}=\overbrace{d_{1}(n-d_{1})}^{\mathrm{1st \ block}} +
\overbrace{d_{2}(n-d_{1}-d_{2})}^{\mathrm{2nd \ block}}+ \cdots+\overbrace{d_{\ell-1}(n-d_{1}-\cdots-d_{\ell-1})}^{(\ell-1)\mathrm{th \ block}}.$$
Note also that
$$J_{1}^{\ell}=0,$$
from which we infer that $R_{1}$ is Lie nilpotent of index $\ell-1$. (This inference is explained in the discussion following the statement of the Conjecture (7).)
Take $V=F^{n}$, which in this context is to be visualized as a $1\times n$ block thus
at (0,0) [$V=\,$]{}; (0,-0.15) rectangle (5,0.15);
Given the above pictorial representations of $V$ and $J_{1}$, we see that
at (0,0) [$VJ_{1}\,=\,$]{}; (0,-0.15) rectangle (1,0.15); (1,-0.15) rectangle (5,0.15); (0.05,0) – (0.95,0) node \[black,above,midway,xshift=0cm\];
Choosing
at (0,0) [$U_{1}\,=\,$]{}; (0,-0.15) rectangle (1,0.15); (1,-0.15) rectangle (5,0.15); (0.05,0) – (0.95,0) node \[black,above,midway,xshift=0cm\];
we see that
(0,5)–(5,0);
;
(5,4) rectangle (1.7,3.3); (5,1.8) rectangle (4.3,0.7);
at (4.5,2.05) [$\cdot$]{}; at (4.5,2.25) [$\cdot$]{}; at (4.5,2.45) [$\cdot$]{}; at (4.5,2.65) [$\cdot$]{}; at (4.5,2.85) [$\cdot$]{}; at (4.5,3.05) [$\cdot$]{};
at (5.32,1.65) [$\cdot$]{}; at (5.32,1.85) [$\cdot$]{}; at (5.32,2.05) [$\cdot$]{}; at (5.32,2.25) [$\cdot$]{}; at (5.32,2.45) [$\cdot$]{}; at (5.32,2.65) [$\cdot$]{}; at (5.32,2.85) [$\cdot$]{}; at (5.32,3.05) [$\cdot$]{}; at (5.32,3.25) [$\cdot$]{};
at (0,2.5) [$J_{2}=(0:^{R_{1}}U_{1})=\,$]{};
(5,4.05) – (5,4.95) node \[black,midway,xshift=0.6cm\];
(5,3.35) – (5,3.95) node \[black,midway,xshift=0.6cm\][$d_{2}$]{};
(5,0.75) – (5,1.75) node \[black,midway,xshift=0.8cm\][$d_{\ell-1}$]{};
(5,0.05) – (5,0.65) node \[black,midway,xshift=0.6cm\][$d_{\ell}$]{};
Here:
[[()]{}]{}
${\rm dim}_{F}J_{2}=\overbrace{d_{2}(n-d_{1}-d_{2})}^{2{\rm nd \ block}}+\cdots+
\overbrace{d_{\ell-1}(n-d_{1}-\cdots-d_{\ell-1})}^{(\ell-1){\rm th \ block}}$;
$J^{\ell-1}_{2}=0$;
$R_{2}$ is Lie nilpotent of index $\ell-2$;
;
.
Continuing in this manner, we arrive at a smallest $F$-subalgebra of $R$ properly containing $FI_{n}$, namely $R_{\ell-1}$, and this has radical comprising a single block
(0,5)–(5,0);
;
(5,1.8) rectangle (4.3,0.7);
at (5.32,1.65) [$\cdot$]{}; at (5.32,1.85) [$\cdot$]{}; at (5.32,2.05) [$\cdot$]{}; at (5.32,2.25) [$\cdot$]{}; at (5.32,2.45) [$\cdot$]{}; at (5.32,2.65) [$\cdot$]{}; at (5.32,2.85) [$\cdot$]{}; at (5.32,3.05) [$\cdot$]{}; at (5.32,3.25) [$\cdot$]{};
at (0,2.5) [$J_{\ell-1}=\,$]{};
(5,4.05) – (5,4.95) node \[black,midway,xshift=0.6cm\];
(5,3.35) – (5,3.95) node \[black,midway,xshift=0.6cm\];
(5,0.75) – (5,1.75) node \[black,midway,xshift=0.8cm\][$d_{\ell-1}$]{};
(5,0.05) – (5,0.65) node \[black,midway,xshift=0.6cm\][$d_{\ell}$]{};
Here:
[[()]{}]{}
${\rm dim}_{F}J_{\ell-1}=\overbrace{d_{\ell-1}(n-d_{1}-\cdots-d_{\ell-1})}^{(\ell-1){\rm th \ block}}$;
$J^{2}_{\ell-1}=0$;
$R_{\ell-1}$ is Lie nilpotent of index $1$ and is thus commutative;
;
.
Open questions
==============
[[()]{}]{}
The sequence $\{U_k\}_{k \in {\mathbb{N}}}$ of $F$-subspace complements defined in (14) is not unique. This has the consequence that the sequence $\{R_{k}\}_{k\in{\mathbb{N}}}$ of $F$-subalgebras of $R$ is not uniquely determined by $R$. Are the $R_{k}$ unique to within isomorphism perhaps? Or failing this, are the dimensions (over $F$) of the $R_{k}$ unique?
Recall that if ${\mathfrak g}$ is a Lie algebra, then the [*Derived Series*]{} $\{{\mathfrak g}^{[m]}\}_{m\in{\mathbb{N}}}$ for ${\mathfrak g}$ is defined recursively as follows
$$\begin{aligned}
{\mathfrak g}^{[1]} &{\stackrel{\rm def}{=}}{\mathfrak g},\hspace*{0.3em}\makebox{\rm and}\\[0.5ex]
{\mathfrak g}^{[m]} &{\stackrel{\rm def}{=}}[{\mathfrak g}^{[m-1]},{\mathfrak g}^{[m-1]}],\hspace*{0.3em}\makebox{\rm for $m>1$.}\end{aligned}$$
We say ${\mathfrak g}$ is [*solvable*]{} if ${\mathfrak g}^{[m]}=0$ for some $m\in{\mathbb{N}}$, $m>1$, and more specifically, [*solvable of index $m$*]{}, if ${\mathfrak g}^{[m+1]}=0$. We call a ring $R$ [*Lie solvable*]{} \[resp. [*Lie solvable of index $m$*]{}\] if $R$, considered as a Lie algebra via the commutator, is solvable \[resp. solvable of index $m$\]. If $\{{\mathfrak g}_{[m]}\}_{m\in{\mathbb{N}}}$ denotes the Lower Central Series for ${\mathfrak g}$, it is easily seen that ${\mathfrak g}^{[m]}\subseteq{\mathfrak g}_{[m]}$ for all $m\in{\mathbb{N}}$, from which it follows that every ring $R$ that is Lie nilpotent of index $m$, is also Lie solvable of index $m$. This being so, it is natural to ask whether the main theorems of this paper remain valid if the condition ‘Lie nilpotent of index $m$’ is substituted with the weaker ‘Lie solvable of index $m$’.
Expressed in terms that make no explicit reference to the overlying matrix ring, a key result in this paper asserts that if $R$ is an $F$-algebra with Lie nilpotence index $m$, and $V$ is any faithful right $R$-module, then ${\rm dim}_{F}R\leqslant M(m+1,{\rm dim}_{F}V)$. (This is Theorem 24 with ${\rm dim}_{F}V$ in place of $n$.) We ask whether the same inequality holds if the requirement that $R$ be a finite dimensional $F$-algebra is weakened to $R$ being merely a (two-sided) artinian ring. In such a situation, ‘$R$-module length’ takes the place of ‘$F$-dimension’ thus yielding the conjecture
> *If $R$ is a (two-sided) artinian ring with Lie nilpotence index $m$ and $V$ is any faithful right $R$-module with finite composition length, then*
>
> $${\rm length}\,R_{R}\leqslant M(m+1,{\rm length}\,V_{R}).$$
In the case where $m=1$, the above reduces to the question [@Gu76 Section 5, Open problem (a), p. 562] that is answered in [@Co93][^7].
[99]{} F. W. Anderson and K. R. Fuller, *Rings and Categories of Modules*, Graduate Texts in Mathematics, Series No. 13, 2nd Edition, Springer-Verlag, Berlin, Heidelberg, New York, 1992.
R. C. Cowsik, A short note on the Schur-Jacobson Theorem, Proc. Amer. Math. Soc. **118**(2) (1993), 675-676.
M. Domokos, Cayley-Hamilton theorem for $2\times2$ matrices over the Grassmann algebra, J. Pure Appl. Algebra **133** (1998), 69-81.
M. Domokos, On the dimension of faithful modules over finite dimensional basic algebras, Linear Algebra Appl. **365** (2003), 155-157.
V. Drensky, *Free Algebras and PI-Algebras,* Springer-Verlag, 2000.
V. Drensky and E. Formanek, *Polynomial Identity Rings,* Birkhäuser-Verlag, 2004.
W. H. Gustafson, On Maximal Commutative Algebras of Linear Transformations, J. Algebra, **42** (1976), 557-563.
N. Jacobson, Schur’s theorems on commutative matrices, Bull. Amer. Math. Soc. **50** (1944), 431-436.
N. S. Karamzadeh, Schur’s inequality for the dimension of commuting families of matrices, Math. Inequal. Appl. **13**(3) (2010), 625-628.
A. R. Kemer, *Ideals of Identities of Associative Algebras*, Translations of Math. Monographs, Vol. 87 (1991), AMS, Providence, Rhode Island.
M. Mirzakhani, A simple proof of a theorem of Schur, Amer. Math. Monthly **105**(3) (1998), 260-262.
D. S. Passman, *A Course in Ring Theory*, Wadsworth & Brooks/Cole Advanced Books & Software, 1991.
H. Radjavi and P. Rosenthal, *Simultaneous Triangularization*, Springer-Verlag, New York, Inc., 2012.
D. M. Riley and M. C. Wilson, Associative rings satisfying the Engel condition, Proc. Amer. Math. Soc. **127**(4) (1999), 973-976.
J. Schur, Zur Theorie der vertauschbaren Matrizen, J. Reine Angew. Math. **130** (1905), 66-76.
J. Szigeti, New determinants and the Cayley-Hamilton theorem for matrices over Lie nilpotent rings, Proc. Amer. Math. Soc. **125**(8) (1997), 2245-2254.
J. Szigeti and L. van Wyk, On Lie nilpotent rings and Cohen’s Theorem, Comm. Algebra **43**(11) (2015), 4783-4796.
Z. Wan and G. Li, The two theorems of Schur on commutative matrices, Acta Math. Sinica **14**, 143-150 (Chinese); translated as Chinese Math. **5** (1964), 156-164.
[^1]: \*Corresponding author
[^2]: With reference to the first author, this research was carried out as part of the TAMOP-4.2.1.B-10/2/KONV-2010-0001 project, supported by the European Union and co-financed by the European Social Fund.
[^3]: The second and third authors were supported by the National Research Foundation of South Africa under Grant Numbers UID 85784 and UID 72375, respectively. All opinions, findings and conclusions or recommendations expressed in this publication are those of the authors and therefore the National Research Foundation does not accept any liability in regard thereto.
[^4]: The fourth author was supported by the Polish National Science Centre grant UMO-2013/09/D/ST1/03669.
[^5]: The authors thank K. C. Smith for fruitful discussions on the topic of this paper.
[^6]: Our Lie algebras are over a commutative ring that is not necessarily a field. No harm shall come of this more general interpretation since, in the few instances where results about standard Lie algebras are used, the underlying commutative ring is a field.
[^7]: Although the ring $R$ is assumed to be local in Gustafson’s formulation, Cowsik’s proof does not assume localness.
|
---
abstract: 'For a connected smooth projective curve $X$ of genus $g$, global sections of any line bundle $L$ with $\deg(L) \geq 2g+ 1$ give an embedding of the curve into projective space. We consider an analogous statement for a Berkovich skeleton in nonarchimedean geometry, in which projective space is replaced by tropical projective space, and an embedding is replaced by a homeomorphism onto its image preserving integral structures (called a faithful tropicalization). Let $K$ be an algebraically closed field which is complete with respect to a non-trivial nonarchimedean value. Suppose that $X$ is defined over $K$ and has genus $g \geq 2$ and that $\Gamma$ is a skeleton (that is allowed to have ends) of the analytification $X^{\mathrm{an}}$ of $X$ in the sense of Berkovich. We show that if $\deg(L) \geq 3g-1$, then global sections of $L$ give a faithful tropicalization of $\Gamma$ into tropical projective space. As an application, when $Y$ is a suitable affine curve, we describe the analytification $Y^{\mathrm{an}}$ as the limit of tropicalizations of an effectively bounded degree.'
address:
- 'Department of Mathematical Sciences, Doshisha University, Kyoto 610-0394, Japan'
- 'Institute for Liberal Arts and Sciences, Kyoto University, Kyoto, 606-8501, Japan'
author:
- Shu Kawaguchi
- Kazuhiko Yamaki
title: Effective faithful tropicalizations associated to linear systems on curves
---
[^1]
Introduction {#sec:intro}
============
Let $X$ be a connected smooth projective curve over a field $K$ of genus $g$, and let $L$ be an ample line bundle over $X$. If $\deg(L) \geq 2g+1$, then there exist nonzero global sections $s_0, \ldots, s_N \in H^0(X, L)$ such that the associated morphism $$X \longrightarrow {{\mathbb{P}}}^N, \quad
p \mapsto (s_0(p): \cdots: s_N(p))$$ is an embedding into projective space. In this paper, we consider an analogous statement for a Berkovich skeleton in nonarchimedean geometry, in which projective space is replaced by tropical projective space, and an embedding is replaced by a homeomorphism onto its image preserving integral structures, called a faithful tropicalization.
In the following, we assume that $K$ is an algebraically closed field which is complete with respect to a non-trivial nonarchimedean absolute value. Let $X^{{\operatorname{an}}}$ denote the analytification of $X$ in the sense of Berkovich, and let $\Gamma$ be a skeleton (which is allowed to have ends) of $X^{{{\operatorname{an}}}}$; $\Gamma$ is a closed subset of $X^{{\operatorname{an}}}\setminus X(K)$ and has an integral structure (see §\[subsec:Berko:skeleta\] for details). The integral structures play important roles in many aspects. For example, they are used to describe measures on the Berkovich spaces (cf. [@gubler1; @gubler4; @yamaki5; @yamaki6]); they are also of interest in mirror symmetry through the work of Kontsevich–Soibelman [@KS].
We set ${{\mathbb{T}}}:= {{\mathbb{R}}}\cup \{+\infty\}$. The $N$-dimensional tropical projective space is defined to be ${{\mathbb{T}}}{{\mathbb{P}}}^N = ({{\mathbb{T}}}^{N+1}\setminus\{(+\infty, \ldots, +\infty)\})/\sim$, where $(x_0, \ldots, x_N) \sim (y_0, \ldots, y_N)$ if there exists $c \in {{\mathbb{R}}}$ such that $y_i = x_i + c$ for any $i$ (see [@MZ]). We remark that ${{\mathbb{T}}}{{\mathbb{P}}}^{N}$ has a natural integral structure; see §\[subsec:tropical:geometry\].
A faithful tropicalization is a tropicalization that is faithful with respect to the integral structures. Nonzero global sections $s_0, \ldots, s_N \in H^0(X, L)$ define a map $$\label{eqn:trop:map:L}
\varphi: X^{{\operatorname{an}}}\longrightarrow {{\mathbb{T}}}{{\mathbb{P}}}^N,
\quad p = (p, |\cdot|) \mapsto \left(-\log|s_0(p)|: \cdots: -\log|s_N(p)|\right),$$ called a tropicalization map associated to $L$. We say that *$\varphi$ is a faithful tropicalization of $\Gamma$* if its restriction to $\Gamma$ is a homeomorphism onto its image preserving integral structures. We say that *$L$ admits a faithful tropicalization of $\Gamma$* or that $\Gamma$ has a [*faithful tropicalization associated to the linear system $|L|$*]{} if for some nonzero $s_0, \ldots, s_N \in H^0(X, L)$, $\varphi$ is a faithful tropicalization of $\Gamma$. See §\[subsec:unimodular:faithful\] for details.
Then, in view of the classical result of embedding of connected smooth projective curves by linear systems, it will be natural to ask the following question.
\[question:first\] Does there exist a constant $d(g)$ depending only on $g \geq 0$ with the following property? For any connected smooth projective curve $X$ over $K$ of genus $g$, for any skeleton $\Gamma$ of $X^{{{\operatorname{an}}}}$, and for any line bundle $L$ over $X$, if $\deg (L) \geq d(g)$, then $L$ admits a faithful tropicalization of $\Gamma$. Furthermore, if there exists such one, can we give a concrete $d(g)$?
The following is our main result, which answers Question \[question:first\].
\[thm:main:unimodular:faithful\] Let $K$ be an algebraically closed field that is complete with respect to a non-trivial nonarchimedean absolute value. Set $$t (g) :=
\begin{cases}
1 & \text{if $g=0$,}
\\
3 & \text{if $g=1$,}
\\
3g-1 & \text{if $g\geq 2$.}
\end{cases}$$ Let $X$ be a connected smooth projective curve over $K$ of genus $g$, let $\Gamma$ be a skeleton of $X^{{{\operatorname{an}}}}$, and let $L$ be a line bundle over $X$. Then if $\deg (L) \geq t (g) $, then $L$ admits a faithful tropicalization of $\Gamma$.
As a consequence of Theorem \[thm:main:unimodular:faithful\], we obtain the following result on tropicalization associated to the pluri-canonical systems, because $g \geq 2$ implies $3\deg(\omega_X)
\geq
t (g)$.
\[cor:faithful:canonical\] Let $K$, $X$, and $\Gamma$ be as in Theorem \[thm:main:unimodular:faithful\]. Suppose that $g \geq 2$. Then $\Gamma$ has a faithful tropicalization associated to the pluri-canonical linear system $|\omega_X^{\otimes m}|$ for any $m \geq 3$.
Once one obtains an affirmative answer to Question \[question:first\], the next natural question would be the following.
\[Question:optimalbound\] What is the best lower bound $d_{{\rm best}}(g)$, which depends only on $g \geq 0$, with the following property? For any connected smooth projective curve $X$ over $K$ of genus $g$, for any skeleton $\Gamma$ of $X^{{{\operatorname{an}}}}$, and for any line bundle $L$ over $X$, if $\deg (L) \geq d_{{\rm best}}(g)$, then $L$ admits a faithful tropicalization of $\Gamma$.
By Theorem \[thm:main:unimodular:faithful\], $d_{{\rm best}}(g) \leq t (g)$. Although we do not have the complete answer to Question \[Question:optimalbound\], we have the answer for small $g$; for $g = 0,1,2$, the bound in Theorem \[thm:main:unimodular:faithful\] is optimal, i.e., $d_{{\rm best}}(g) = t (g)$.
\[thm:best:possible\] Let $K$ be as in Theorem \[thm:main:unimodular:faithful\]. Then there exist a connected smooth projective curve $X$ over $K$ of genus $2$ resp. $1$, $0$, a skeleton $\Gamma$ of $X^{{{\operatorname{an}}}}$, and a line bundle $L$ of degree $4$ resp. $2$, $0$ such that $\Gamma$ does [*not*]{} have a faithful tropicalization associated to $|L|$.
As one sees from the proofs of Propositions \[prop:complement:0\], \[prop:complement:1\], and \[prop:complement:2\], line bundles that we consider for the proof of Theorem \[thm:best:possible\] are not very ample. One might expect that if $L$ is very ample, then it should always admit a faithful tropicalization of a given skeleton. In fact, this expectation is not true, which makes Question \[Question:optimalbound\] more interesting. We show the existence of a connected smooth projective curve with a [*very ample*]{} line bundle that does not admit a faithful tropicalization of any skeleton of the curve.
\[prop:countereg:very:ample\] Let $K$ be as in Theorem \[thm:main:unimodular:faithful\]. Let $d \geq 4$ be an integer. Then there exists a connected smooth projective plane curve $X$ of ${{\mathbb{P}}}^2$ of degree $d$ over $K$ such that no skeleton of $X^{{{\operatorname{an}}}}$ has a faithful tropicalization associated to $|{{\mathcal{O}}}_X(1)|$, where ${{\mathcal{O}}}_X(1)$ is the very ample line bundle over $X$ given by restriction of ${{\mathcal{O}}}_{{{\mathbb{P}}}^2}(1)$.
Let us explain the relation to previous works on faithful tropicalization of skeleta. Katz–Markwig–Markwig [@KMM; @KMM2], Baker–Payne–Rabinoff [@BPR], Gubler–Rabinoff–Werner [@GRW], Baker–Rabinoff [@BR] showed that for a smooth projective variety $X$ and any skeleton $\Gamma$ of $X^{{{\operatorname{an}}}}$, there exist nonzero rational functions $f_1, \ldots, f_N$ of $X$ such that $$\psi: X^{{{\operatorname{an}}}} \dasharrow {{\mathbb{R}}}^N, \quad p = (p, |\cdot|)\mapsto (-\log|f_1(p)|, \ldots, -\log|f_N(p)|)$$ gives a faithful tropicalization of $\Gamma$. Note that there exists a positive integer $d$ such that for any line bundle $L$ over $X$ with $\deg (L) \geq d$, $1, f_1, \ldots, f_N$ can be regarded as elements of $H^0(X, L)$. This means that for any $X$ and $\Gamma$ as above, there exists a positive integer $d (X, \Gamma)$, which may depend on $X$ and $\Gamma$, such that any line bundle $L$ with $\deg (L) \geq d ( X, \Gamma)$ admits a faithful tropicalization of $\Gamma$. Theorem \[thm:main:unimodular:faithful\], on the other hand, tells us that once $g$ is fixed, then one has a uniform and effective lower bound for the degree of a line bundle such that the line bundle admits a faithful tropicalization of any skeleton of $X^{{{\operatorname{an}}}}$, as far as $X$ is a connected smooth projective curve of genus $g$.
We remark that the tropicalization map is not injective on the whole $X^{{{\operatorname{an}}}}$. We also remark that, from a different perspective, Hrushovski–Loeser–Poonen [@HLP] studies if $X^{{{\operatorname{an}}}}$ is embedded in the Euclidean space ${{\mathbb{R}}}^N$ by a homeomorphism onto its image (where one does not consider integral structures). In another direction, Haase–Musiker–Yu [@HMY] and Amini [@Amini] study embeddings of [*tropical curves*]{} associated to linear systems. See also Cueto–Markwig [@CM] for an algorithmic side of tropicalizations of plane curves using tropical modifications.
After we wrote up the first draft of this paper, through communications with Sam Payne, we realize that Theorem \[thm:main:unimodular:faithful\] has an unexpected application to limit tropicalizations of affine space curves. We begin by recalling Payne’s result [@payne] that the Berkovich analytification is the limit of the tropicalizations. Let $Y$ be an affine variety over $K$. We call a closed embedding of $Y$ into affine space an *affine embedding of $Y$*. Any affine embedding $\iota: Y \hookrightarrow {\mathbb{A}}^n$ induces a standard tropicalization map $Y (K) \to {{\mathbb{T}}}^n$ given by $p \mapsto (-\log |x_1 (\iota (p) )|_K, \ldots, -\log |x_n (\iota (p))|_K)$, where $x_1 , \ldots , x_n$ are the standard coordinates of ${\mathbb{A}}^n$ and $|\cdot|_K$ is the absolute value of $K$. The *tropicalization ${\operatorname{Trop}}(Y, \iota)$ of $Y$ with respect to $\iota$* is defined to be the closure of the image of the standard tropicalization map. By [@payne Proposition 2.2], this map extends to a unique continuous map $\pi_\iota: Y^{{{\operatorname{an}}}} \to {\operatorname{Trop}}(Y, \iota)$. Let $I$ be the set of all affine embeddings of $Y$. One makes $I$ a directed set by declaring $\iota_1 \leq \iota_2$ for two elements $\iota_1: Y \hookrightarrow {\mathbb{A}}^{n_1}$ and $\iota_2: Y \hookrightarrow {\mathbb{A}}^{n_2}$ of $I$ if there exists a torus-equivariant morphism $\varphi: {\mathbb{A}}^{n_2} \to {\mathbb{A}}^{n_1}$ with $\varphi\circ \iota_2 = \iota_1$ (cf. §\[sec:limit:trop\]). If $\iota_1 \leq \iota_2$, then one has a natural continuous map ${\operatorname{Trop}}(Y, \iota_2) \to {\operatorname{Trop}}(Y, \iota_1)$, and thus $\left({\operatorname{Trop}}(Y, \iota)\right)_{\iota \in I}$ constitutes an inverse system in the category of topological spaces. Then Payne’s theorem [@payne Theorem 1.1] asserts that the map $${\rm LTrop}: Y^{{{\operatorname{an}}}} \to \varprojlim_{\iota \in I} {\operatorname{Trop}}(Y, \iota)$$ induced from $( \pi_{\iota} )_{\iota \in I}$ is a homeomorphism. This homeomorphism says that various tropicalizations of $Y$ give rise to the Berkovich space $Y^{{{\operatorname{an}}}}$ associated to $Y$.
Now, fix a distinguished closed embedding $Y \subset {\mathbb{A}}^N = {{\operatorname{Spec}}}(K[z_1, \ldots, z_N])$. Then for any positive integer $D$, we have the notion of affine embeddings of degree at most $D$ with respect to $z_1, \ldots, z_N$, so that we consider a bounded degree version of Payne’s construction. To be precise, we say that an affine embedding $\iota : Y \hookrightarrow \mathbb{A}^n$ has *degree at most $D$* if there exist $f_1 , \ldots , f_n \in K [z_1 , \ldots , z_N]$ such that $\deg (f_j) \leq D$ for $j = 1, \ldots, n$ and $\iota = (f_1, \ldots, f_n)$. Let $I_{\leq D}$ denote the set of all affine embeddings of $Y$ whose degree is at most $D$. One sees that $I_{\leq D}$ is a directed set, and thus $\left( {\operatorname{Trop}}(Y, \iota ) \right)_{\iota \in I_{\leq D}}$ again constitutes an inverse system. We denote by $$\label{eqn:limit:tropical:map:d:intro}
\mathrm{LTrop}_{\leq D} : Y^{{{\operatorname{an}}}} \to \varprojlim_{\iota \in I_{\leq D}} {\operatorname{Trop}}(Y, \iota)$$ the map induced from $( \pi_{\iota} )_{\iota \in I_{\leq D}}$.
As an application of Theorem \[thm:main:unimodular:faithful\], we show that when $Y$ is a suitable affine curve, $Y^{{{\operatorname{an}}}}$ is homeomorphic to the inverse limit of tropicalizations of an [*effectively bounded degree*]{}, i.e., $\mathrm{LTrop}_{\leq D}$ is a homeomorphism for some effectively computed $D$. Let $Y \subset {\mathbb{A}}^N = {{\operatorname{Spec}}}(K[z_1, \ldots, z_N])$ be a connected smooth affine curve. We say that *$Y$ has smooth compactification in ${{\mathbb{P}}}^N$* if the closure of $Y$ in ${{\mathbb{P}}}^N$ is smooth, where we regard ${\mathbb{A}}^N$ as an open subset of ${{\mathbb{P}}}^N$. The *degree $d$ of $Y$* is defined as the degree of $X$. For a real number $x$, $\lceil x \rceil$ denotes the smallest integer with $\lceil x \rceil \geq x$. We set $$\label{eqn:D:def}
D :=
\begin{cases}
\max
\left\{
\left\lceil
\frac{3d^2 - 9 d + 4}{2d}
\right\rceil
,
1
\right\}
&
\parbox{8cm}{\flushleft if
$N \leq 2$, or
if
$N \geq 3$ and
$Y$ is contained in
some affine plane in ${\mathbb{A}}^N$,}
\\
\max \{ d-2 , 1 \}
& \text{otherwise.}
\end{cases}$$ Then we have the following; see also Remark \[remark:morethanD\].
\[thm:limit:trop:d:intro\] Let $Y \subset{\mathbb{A}}^N$ be a connected smooth affine curve of degree $d \geq 1$. Assume that $Y$ has smooth compactification in ${{\mathbb{P}}}^N$. Let $D$ be as in . Then the map $\mathrm{LTrop}_{\leq D}$ in is a homeomorphism.
As sample examples, let $Y_3$ be a connected affine plane curve of degree $3$ that has smooth compactification in ${{\mathbb{P}}}^2$ (an affine part of an elliptic curve). Then $D = 1$, and Theorem \[thm:limit:trop:d:intro\] says that $Y_3^{{{\operatorname{an}}}}$ is the limit of the tropicalizations of embeddings of degree at most $1$, namely, the limit of [*linear*]{} tropicalizations. Next, let $Y_4$ be a connected affine plane curve of degree $4$ that has smooth compactification in ${{\mathbb{P}}}^2$ (an affine part of a connected smooth non-hyperelliptic curve of genus $3$). Then $D = 2$, and Theorem \[thm:limit:trop:d:intro\] says that $Y_4^{{{\operatorname{an}}}}$ is the limit of tropicalization of embeddings of degree $2$, namely, the limit of [*quadratic*]{} tropicalizations. We note that in general, $Y_4^{{{\operatorname{an}}}}$ is not the limit of linear tropicalizations. For details, see Examples \[eg:limit:trop:linear\], \[eg:limit:trop:linear:impossible\], and \[eg:limit:trop:quad\].
Let us explain the ideas of our proof of Theorem \[thm:main:unimodular:faithful\]. For simplicity, assume here that $X$ has genus $g \geq 2$ and the given skeleton is the minimal skeleton $\Gamma_{\min}$, which is associated to a Deligne–Mumford strictly semistable curve with generic fiber $X$. It is well-defined for $X$ and does not depend on the choice of such a Deligne–Mumford semistable model. It is known that $\Gamma_{\min}$ has a canonical structure of finite graph such that the set $V$ of vertices corresponds to the set of the irreducible components of the special fiber of the stable model of $X$. With this finite graph structure, the set $E$ of edges corresponds to the set of nodes of the special fiber of the stable model.
To realize a faithful tropicalization of $\Gamma_{\min}$ by $L$, we construct global sections of $L$ according to the following aims: for a given $e \in E$, to give a unimodular tropicalization of $e$, where a map is said to be unimodular if it is a piecewise linear map preserving the integral structures, and to separate points in the relative interior $\mathrm{relin} (e)$ of $e$; for given distinct $e , f \in E$, to separate two points $x$ and $y$ with $x \in \mathrm{relin} (e)$ and $y \in f$; for given two distinct points of $V$, to separate the two points. Then the collection of those global sections will give a faithful tropicalization of $\Gamma_{\min}$.
Global sections as above are constructed as global sections of models. Here, a model $(\mathscr{X} , \mathscr{L})$ means a pair consisting of a semistable curve over the ring of integers $R$ of $K$ equipped with an isomorphism ${{\mathscr{X}}}\times_{{{\operatorname{Spec}}}(R)}{{\operatorname{Spec}}}(K) \cong X$ and a line bundle $\mathscr{L}$ over $\mathscr{X}$ with ${{\mathscr{L}}}\otimes_R K \cong L$. For each aim as above, we will define “suitable” global sections of a model.
Now the issue is divided into two parts: the first part is to construct a “good” model; the second part is to construct “suitable” global sections of the model.
In the first part, we use the theory of $\Lambda$-divisors on $\Gamma_{\min}$, which has been recently developed (see, for example, [@AC; @BN; @GK; @MZ]), together with Raynaud’s type theorem on the surjectivity of the principal divisors under the specialization map (see, for example, [@AC; @B; @BR; @Ra]). We take a divisor $\tilde{D}$ on $X$ such that $L \otimes \omega_X^{\otimes -1} \cong {{\mathcal{O}}}_X(\tilde{D})$. The retraction map $X^{{{\operatorname{an}}}} \to \Gamma_{\min}$ induces the specialization map $
\tau_*: {\operatorname{Div}}(X) \to {\operatorname{Div}}_\Lambda(\Gamma_{\min})
$ from the divisor group on $X$ to the $\Lambda$-divisor group on $\Gamma_{\min}$. Using Riemann’s inequality, the notion of reduced divisors on metric graphs, and so forth, we construct, according to each aim, a $\Lambda$-divisor $D$ on $\Gamma_{\min}$ that are linearly equivalent to $\tau_*(\tilde{D})$. Then using a Raynaud’s type theorem, we construct a “good” Deligne–Mumford model $({{\mathscr{X}}}, {{\mathscr{L}}})$ of $(X, L)$ such that $\sum_{C} \deg
\left({\left.{{{\mathscr{L}}}\otimes \omega_{\mathscr{X}/R}^{\otimes -1}}\right\vert_{{C}}}
\right) [C] = D$, where $C$ runs through all the irreducible component of the special fiber ${{\mathscr{X}}}_s$ and $[C] \in \Gamma_{\min}$ is the point in the Berkovich space $X^{{{\operatorname{an}}}}$ associated to $C$, called the Shilov point.
In the second part, we do algebraic geometry on semistable curves. We construct (1) global sections of the restriction of (a modification of) ${{\mathscr{L}}}$ to the special fiber $\mathscr{X}_s$, and then (2) we lift those sections to global sections of ${{\mathscr{L}}}$. In (1), we use some basepoint freeness over nodal curves, and in (2), we use base-change theorem. In those arguments, the key is vanishing of cohomologies of line bundles over the special fiber. By the choice of the above $D \in {\operatorname{Div}}_{\Lambda} ( \Gamma_{\min} )$, we can show desired vanishing of cohomologies. Thus we find desired global sections and obtain a faithful tropicalization of the minimal skeleton.
Let us explain here why $3g-1$ appears when $g \geq 2$, even in a very rough form. In the second part above, to show vanishing of cohomologies, we need ${{\mathscr{L}}}\otimes \omega_{{{\mathscr{X}}}/R}^{\otimes -1}$ to be nef over the special fiber, which requires $\deg (L) \geq 2g-2$, at least. Further, depending on our each aim, we often want ${{\mathscr{L}}}\otimes \omega_{{{\mathscr{X}}}/R}^{\otimes -1}$ to have positive degree over a specified irreducible component $C$. To obtain such positivity, we want a $\Lambda$-divisor $D$ in the first part above to be effective and positive at $[C] \in \Gamma_{\min}$. To find such a $D$, we use Riemann’s inequality ($|E| \neq \emptyset$ if $\deg(E) \geq g$) on $\Gamma_{\min} \subset X^{{{\operatorname{an}}}}$, and this requires $\deg \left( \tau_{\ast} ( \tilde{D}) - [v] \right) \geq g$ and hence $\deg \left( \tilde{D} \right) \geq g+1$. Since $\deg
\left( \tau_{\ast} ( \tilde{D})
\right) = \deg \left( L \otimes \omega_{X}^{\otimes -1} \right)$, this requires that $\deg (L)$ should be at least $g + 1+ (2g-2) = 3g -1$.
The above description how $3g-1$ appears indicates that by our approach, it is difficult to reduce $3g-1$ further. Indeed, when $g=2$, $t(2)$ is optimal by Theorem \[thm:best:possible\]. However, a different approach may give a better lower bound for a faithful tropicalization and may contribute to Question \[Question:optimalbound\].
As explained as above, for the proof of Theorem \[thm:main:unimodular:faithful\], we show the existence of “good” divisors on a metric graph, and then we lift them to divisors on a curve under the specialization map. Albeit from a different perspective, this type of lifting argument is sometimes quite successful in tropical geometry. For example, Cools, Draisma, Payne and Robeva [@CDPR] gave a tropical proof of the Brill-Noether theorem. See also [@CLM; @CJP; @KY0; @KY1], for example.
In [@KY], we consider faithful tropicalizations associated to line bundles for projective varieties of any dimension. We recall what the main result of [@KY] asserts for curves: Let $X$ be a connected smooth projective curve over $K$ of genus $g$, and let $L$ be an ample line bundle over $X$; suppose that there exist a discrete valuation ring $R_0$ dominated by $R$, a strictly semistable regular scheme $\mathscr{X} \to {{\operatorname{Spec}}}(R_0)$ over $R_0$ with $\mathscr{X} \otimes_{R_0} K = X$, a *relatively ample* line bundle $\mathscr{N}$ over $\mathscr{X}$, and an integer $m \geq 2$ such that ${\left.{\mathscr{N}^{\otimes m} \otimes \omega_{\mathscr{X}/R_0}}\right\vert_{{X}}} = L$; then $L$ admits a faithful tropicalization of the skeleton associated to ${{\mathscr{X}}}$.
For curves, Theorem \[thm:main:unimodular:faithful\] in this paper has an advantage over [@KY]. For example, the result in [@KY] cannot contribute to Question \[question:first\], even if we work under the condition that $X$ can be defined over a discrete valued field and we consider only a faithful tropicalization of the minimal skeleton $\Gamma_{\min}$ of $X^{{{\operatorname{an}}}}$; see §\[subsection:comparison\] for details.
In [@KY], we work with a given model $(\mathscr{X} , \mathscr{N})$. On the other hand, to establish Theorem \[thm:main:unimodular:faithful\] in this paper, we need much finer analysis for constructing various carefully chosen models $({{\mathscr{X}}}, {{\mathscr{L}}})$ of $(X, L)$ as we do in this paper. That is one of the main reasons why this paper has much more pages than [@KY].
Theorem \[thm:limit:trop:d:intro\] does not follow automatically from Theorem \[thm:main:unimodular:faithful\], because we need to separates type IV points of $Y^{{{\operatorname{an}}}}$ (cf. Remark \[rmk:four:types\]) and they are outside skeleta of $Y^{{{\operatorname{an}}}}$. To separate a type IV point from another point, we will use global sections constructed in §\[sec:ft:general:model\] originally for the proof of Theorem \[thm:main:unimodular:faithful\]. Other ingredients of the proof of Theorem \[thm:limit:trop:d:intro\] are Castelnuovo’s bound for a curve in projective space and a criterion for a complete linear system to be traced out by hypersurfaces on the projective space. See §\[sec:limit:trop\] for details.
The organization of this paper is as follows. Section \[sec:preliminaries\] is preliminary. We recall semistable models and semistable pairs, some known facts on Berkovich spaces and tropical geometry. In particular, we recall a minimal skeleton $\Gamma_{\min}$ for a connected smooth projective curve. We also give the definition of a faithful tropicalization. In §§\[sec:models\]-\[section:FTcan\], we generally assume that $g \geq 2$. In §\[sec:models\], we introduce the notion of a good model. Further, for a given effective divisor on the minimal skeleton of $X^{{{\operatorname{an}}}}$, we construct a good model with a line bundle which is associated to a divisor that is a “lifting” of the divisor on the skeleton. In §\[sec:usuful:lemmas\], we show several lemmas that are useful for constructing required sections. Then we show that the minimal skeleton $\Gamma_{\min}$ can be embedded into ${{\mathbb{T}}}{{\mathbb{P}}}^N$ piecewisely isometrically (viz. a unimodular tropicalization). In §\[section:FTcan\], we construct sections that separate points in $\Gamma_{\min}$, and establish a faithful tropicalization for $\Gamma_{\min}$. In §\[section:FT:lowgenus\], we discuss the tropicalization when $g = 0$ or $g=1$. In §\[sec:ft:general:model\], we establish a faithful tropicalization for an arbitrary skeleton possibly with ends, thus completing the proof of Theorem \[thm:main:unimodular:faithful\]. In §\[sec:complementary\], we give proofs of complementary results such as Theorem \[thm:best:possible\] and Proposition \[prop:countereg:very:ample\]. We also compare Theorem \[thm:main:unimodular:faithful\] with [@KY Theorem 1.1]. In §\[sec:limit:trop\], we give a proof of Theorem \[thm:limit:trop:d:intro\].
Originally, we essentially worked only over the completion of the algebraic closure of a discretely valued field, and the bound of the degree was $5g-4$. While we were making gradual improvement of our results, we had opportunities to give talks in several conferences and have fruitful discussions. We especially thank Professor Antoine Ducros for helpful discussions, which lead us to work over any algebraically closed field that is complete with respect to a non-trivial nonarchimedean absolute value. Our deep thanks go to Professor Sam Payne for helpful communications, which has lead us to Proposition \[prop:countereg:very:ample\] and Theorem \[thm:limit:trop:d:intro\]. Our deep thanks go to Professor Matthew Baker for helpful comments, which improve the exposition.
Notation and Conventions {#notation-and-conventions .unnumbered}
------------------------
By the convention of this paper, an inclusion $A \subset B$ of sets allows the case $A =B$.
A [*variety*]{} over a field $k$ means a reduced separated scheme of finite type over $k$. (We allow a variety to be reducible unless otherwise specified.) A *curve* is a variety of pure dimension $1$. For a curve $Y$ over $k$, let ${\operatorname{\operatorname{Irr}}}(Y)$ denote the set of irreducible components of $Y$ and let ${\operatorname{Sing}}(Y)$ denote the set of non-smooth points of $Y \to {{\operatorname{Spec}}}(k)$.
Let $Y$ be a curve over $k$. A [*curve in $Y$*]{} is a closed subscheme of $Y$ that is a curve over $k$ in the above sense. If there is no danger of confusion, we identify a curve in $Y$ with a union of irreducible components of $Y$. A *connected curve in $Y$* is a curve in $Y$ that is connected. For a curve $D$ in $Y$, we denote by $Y - D$ the union of irreducible components of $Y$ other than those in $D$, i.e., $Y - D := \bigcup_{C \in {\operatorname{\operatorname{Irr}}}(Y) \setminus {\operatorname{\operatorname{Irr}}}(D)} C$. By convention, if $D = Y$, then $Y-D = \emptyset$. Notice that $Y - D$ is not equal to the (set-theoretic) complement $Y \setminus D$ of $D$ unless $D$ is a union of connected components of $Y$. Let $D_1 , \ldots , D_r$ be curves in $Y$. Assuming that ${\operatorname{\operatorname{Irr}}}(D_i) \cap {\operatorname{\operatorname{Irr}}}(D_j) = \emptyset$ for $i \neq j$, we denote by $D_1 + \cdots + D_r = \sum_{i=1}^r D_i$ the reduced curve $\bigcup_{i = 1}^r D_i$ in $Y$. The notation $\sum_{i=1}^r D_i$ is used only under this assumption.
Throughout this paper, $K$ denotes an algebraically closed field that is complete with respect to a non-trivial nonarchimedean absolute value $|\cdot|_K$. We denote by $R$ the ring of integers of $K$, by ${{\mathfrak{m}}}$ the maximal ideal of $R$, and by $k$ the residue field $R/{{\mathfrak{m}}}$. Let $v_K : K^{\times} \to {{\mathbb{R}}}$ be the valuation map given by $v_K := - \log | \cdot |_K$, and let $\Lambda := \{v_K(x) \mid x \in K^\times\} \subset {{\mathbb{R}}}$ be the value group of $K$.
We use notions on graphs. A [*finite graph*]{} $G$ means a finite connected graph (that is allowed to have loops and multiple edges), and we denote by $V(G)$ and $E(G)$ the set of vertices and the set of edges, respectively. We assign each edge $e \in E(G)$ length in $\Lambda$. Then the graph $G$ assigned with edge lengths gives a $\Lambda$-metric graph $\Gamma$ together with the set of vertices $V(G)$ and the set of edges $E(G)$. A [*$\Lambda$-metric graph*]{} $\Gamma$ is a connected metric graph such that there exists a finite graph $G$ assigned with edge lengths in $\Lambda$ that induces $\Gamma$ as above. A point $x$ in a $\Lambda$-metric graph is a [*$\Lambda$-valued point*]{} if the distance from $x$ to one (hence any) vertex $v$ of $G$ lies in $\Lambda$. We denote by $\Gamma_{\Lambda}$ the set of $\Lambda$-valued points of $\Gamma$. Notice that the notion of $\Lambda$-valued points is independent of the choice of a finite graph $G$ as above.
Let $\Gamma$ be a $\Lambda$-metric graph with a set $V$ of vertices and a set $E$ of edges. We naturally regard $V$ as a finite subset of $\Gamma$ and each $e \in E$ as a closed subset of $\Gamma$. We take any $e \in E$ and set $\partial e := e \cap V$. We call a point in $\partial e$ an *end vertex* of $e$. If $e$ is isometric to a closed interval, then $\# \partial e = 2$; if $e$ is a loop, then $\# \partial e = 1$. We define the *relative interior of $e$* to be $
\mathrm{relin}(e) := e \setminus \partial e
$.
Preliminaries {#sec:preliminaries}
=============
In this section, after recalling semistable models and semistable pairs, we briefly review some facts on Berkovich spaces and tropical geometry, which we use later. Then we define a unimodular and a faithful tropicalization associated to a linear system. Our basic references for Berkovich spaces and skeleta are [@BPR2], [@gubler4 §5], and [@GRW §3]. We use the language of schemes (rather than formal schemes), which might be more familiar to readers in algebraic geometry, even though actual proofs use formal geometry.
Semistable models and semistable pairs {#subsec:semistable:models:pairs}
--------------------------------------
This subsection is mainly to fix the notation. We recall semistable curves, semistable models, and semistable pairs.
### Semistable curves {#semistable-curves .unnumbered}
We first recall the notion of semistable curves over schemes. Let $S$ be a scheme. A [*semistable curve*]{} over $S$ of genus $g \geq 0$ is a proper flat morphism $\pi: {{\mathscr{X}}}\to S$ whose geometric fibers ${{\mathscr{X}}}_{\bar{s}}$ are reduced, separated, connected, $1$-dimensional schemes that has only ordinary double points (called nodes) as singularities and satisfies $h^1({{\mathcal{O}}}_{{{\mathscr{X}}}_{\bar{s}}}) = g$. A semistable curve ${{\mathscr{X}}}$ over a scheme $S$ is said to be [*strictly semistable*]{} if for any scheme point $s$ of $S$, any singular point of ${{\mathscr{X}}}_s$ is defined over the residue field $\kappa(s)$ and any irreducible component of ${{\mathscr{X}}}_s$ is geometrically irreducible and smooth.
Let ${{\mathscr{X}}}$ be a strictly semistable curve over a scheme $S$ of genus $g$. For $p \in {\operatorname{Sing}}(\mathscr{X}_s)$, we say that $p$ is of *connected type* if the partial normalization $\widetilde{\mathscr{X}_s}$ of $\mathscr{X}_s$ at $p$ is connected; we say that $p$ is of *disconnected type* otherwise.
A line bundle ${{\mathscr{L}}}$ over ${{\mathscr{X}}}$ is [*vertically nef*]{} if ${\left.{{{\mathscr{L}}}}\right\vert_{{{{\mathscr{X}}}_{\bar{s}}}}}$ is nef for all geometric fibers ${{\mathscr{X}}}_{\bar{s}}$. A semistable curve is called a [*Deligne–Mumford semistable curve*]{} if the dualizing sheaf $\omega_{{{\mathscr{X}}}/S}$ is vertically nef. A Deligne–Mumford semistable curve is called a Deligne–Mumford [*stable*]{} curve (or simply a [*stable*]{} curve) if $\omega_{{{\mathscr{X}}}/S}$ is ample.
Let ${{\mathscr{X}}}$ be a semistable curve over $S$. An irreducible component $C$ of ${{\mathscr{X}}}_{\bar{s}}$ is called a [*$(-1)$-curve*]{} if $C \cong {{\mathbb{P}}}^1$ and $\#(C \cap ({{\mathscr{X}}}_{\bar{s}}-C)) = 1$, and is called a [*$(-2)$-curve*]{} if $C \cong {{\mathbb{P}}}^1$ and $\#(C \cap ({{\mathscr{X}}}_{\bar{s}}-C)) = 2$. By the adjunction formula, an irreducible component $C$ of ${{\mathscr{X}}}_{\bar{s}}$ is a $(-1)$-curve if and only if $\deg({\left.{\omega_{{{\mathscr{X}}}/S}}\right\vert_{{C}}}) = -1$, and $C$ is a $(-2)$-curve if and only if $\deg({\left.{\omega_{{{\mathscr{X}}}/S}}\right\vert_{{C}}}) = 0$. Suppose that ${{\mathscr{X}}}$ is a Deligne–Mumford semistable curve of genus $g \geq 1$. Let $E$ be a connected curve in ${{\mathscr{X}}}_{\bar{s}}$. Then $E$ consists of $(-2)$-curves if and only if $\deg({\left.{\omega_{{{\mathscr{X}}}/S}}\right\vert_{{E}}}) = 0$. If this is the case, then $E$ is a cycle of $(-2)$-curves or a chain of $(-2)$-curves. Furthermore, if $g \geq 2$, then $\mathscr{X}_{\bar{s}}$ does not contain a cycle of $(-2)$-curves, and hence $E$ is a chain.
A chain consisting of $(-2)$-curves are called a *$(-2)$-chain*. Let $E$ be a $(-2)$-chain in ${{\mathscr{X}}}_{\bar{s}}$. We say that $E$ is [*of connected type*]{} if ${{\mathscr{X}}}_{\bar{s}} - E$ is connected, and $E$ is [*of disconnected type*]{} if ${{\mathscr{X}}}_{\bar{s}} - E$ is disconnected. Note that if $\mathscr{X}$ is a Deligne–Mumford semistable curve and $E$ is of disconnected type, then ${{\mathscr{X}}}_{\bar{s}} - E$ consists of two connected components each of which has positive arithmetic genus.
A $(-2)$-chain $E$ is called a [*maximal*]{} $(-2)$-chain if $F$ is a $(-2)$-chain with $E \subset F$, then $E = F$. When $g \geq 2$, any connected curve in $\mathscr{X}_{\bar{s}}$ consisting of $(-2)$-curves is a chain, and hence any two maximal $(-2)$-chains are disjoint.
### Semistable models {#subsec:semistabe:models .unnumbered}
Let $X$ be a variety over $K$. A *model of $X$ over $R$* is a flat morphism $
\pi : \mathscr{X} \to {{\operatorname{Spec}}}(R)$ of finite type equipped with an identification of the generic fiber of $\pi$ with $X$. When $X$ is proper over $K$, we require that a model be also proper over $R$. The special fiber of ${{\mathscr{X}}}$ is denoted by ${{\mathscr{X}}}_s$.
For two models $\mathscr{X}'$ and $\mathscr{X}$ of $X$ over $R$, we say that *$\mathscr{X}'$ dominates $\mathscr{X}$* if the identity morphism on $X$ extends to a morphism $\mathscr{X}' \to \mathscr{X}$.
Now, let $X$ be a connected smooth projective curve over $K$ of genus $g$. A [*semistable model*]{} of $X$ over $R$ is a model $\pi : \mathscr{X} \to {{\operatorname{Spec}}}(R)$ that is a semistable curve. If a semistable model ${{\mathscr{X}}}$ is strictly semistable, then ${{\mathscr{X}}}$ is called a [*strictly*]{} semistable model. If ${{\mathscr{X}}}$ is Deligne–Mumford semistable (resp. stable), then ${{\mathscr{X}}}$ is called a [*Deligne–Mumford semistable*]{} (resp. [*stable*]{}) model of $X$. By [@BL1 Theorem 7.1] or [@BPR2 Corollary 4.23 and Remark 4.24], there exists a Deligne–Mumford semistable model of $X$ if $g \geq 1$, and there exists a unique stable model of $X$ if $g \geq 2$. When $g \geq 2$ and we denote a model of $X$ by $\mathscr{X}^{{\operatorname{st}}} \to {{\operatorname{Spec}}}(R)$, this stands for the stable model. It is known that the stable model is “minimal” in the sense that if $\mathscr{X}$ is any semistable model of $X$, then $\mathscr{X}$ dominates $\mathscr{X}^{{\operatorname{st}}}$.
Let $L$ be a line bundle over $X$. Let $\mathscr{X}$ be a model of $X$ over $R$. We call a line bundle $\mathscr{L}$ over $\mathscr{X}$ a [*model*]{} of $L$ if ${\left.{\mathscr{L}}\right\vert_{{X}}} = L$. In this case, we call the pair $(\mathscr{X} , \mathscr{L})$ a *model* of $(X, L)$. If ${{\mathscr{X}}}$ is semistable (resp. strictly semistable, Deligne–Mumford semistable, stable), we say that $(\mathscr{X} , \mathscr{L})$ is a [*semistable*]{} (resp. [*strictly semistable*]{}, [*Deligne–Mumford semistable*]{}, [*stable*]{}) model of $(X, L)$.
### Strictly semistable pairs {#subsec:semistabe:pairs .unnumbered}
Let $X$ be a connected smooth projective curve over $K$. Let $\pi: \mathscr{X} \to {{\operatorname{Spec}}}(R)$ be a strictly semistable model of $X$. For a section $\sigma: {{\operatorname{Spec}}}(R) \to \mathscr{X}$ of $\pi$, we denote the image of $\sigma$ by $\sigma (R)$, the image of the special (resp. generic) point of ${{\operatorname{Spec}}}(R)$ by $\sigma (k)$ (resp. $\sigma(K)$).
Let $\sigma_1 , \ldots , \sigma_r$ be sections of $\pi$. The pair $( \mathscr{X} ; \sigma_1 , \ldots , \sigma_r)$ is called a *strictly semistable pair* if $\sigma_i (R)$ is contained in the smooth locus of $\pi$ and $\sigma_i (k) \neq \sigma_j (k)$ for $i \neq j$. We note that the definition of a strictly semistable pair is the same as in [@GRW]. Indeed, since $\sigma_i(R)$ is contained in the smooth locus of $\pi$, $\sigma_1 (R) , \ldots , \sigma_r (R)$ are prime Cartier divisors on $\mathscr{X}$, and by the condition $\sigma_i (k) \neq \sigma_j (k)$ for $i \neq j$, these Cartier divisors have disjoint supports.
Berkovich spaces {#subsec:Berko:skeleta}
----------------
Let $X$ be a variety over $K$. For each $x \in X$, let $\kappa(x)$ denote the residue field. Let $X^{{{\operatorname{an}}}}$ be the [*Berkovich (analytic) space*]{} associated to $X$. (For details, see fundamental papers by Berkovich [@Be1; @Be2; @Be3; @Be4].) Here we recall $X^{{{\operatorname{an}}}}$ as a set and the underlying topology of $X^{{{\operatorname{an}}}}$. As a set, $X^{{{\operatorname{an}}}}$ is defined by $$X^{{{\operatorname{an}}}} :=
\left\{
(x, |\cdot|) \mid \text{$x \in X$ and
$|\cdot|$ is an absolute value of $\kappa(x)$ extending $|\cdot|_{K}$}
\right\}.$$ The *Berkovich topology* on $X^{{{\operatorname{an}}}}$ is the weakest topology such that for any Zariski open subset $U$ of $X$ and for any regular function $g \in {{\mathcal{O}}}_X(U)$, the map $\iota^{-1}(U) \to {{\mathbb{R}}}, (x, |\cdot|) \mapsto |g(x)|$ is continuous, where $\iota: X^{{{\operatorname{an}}}} \to X$ is the natural map given by $(x, |\cdot|) \mapsto x$.
The set $X(K)$ of $K$-valued points are naturally regarded as points of $X^{{{\operatorname{an}}}}$ via $x \mapsto (x, |\cdot|_K)$ for $x \in X(K)$, where $| \cdot |_{K}$ is the original absolute value on $K = \kappa (x)$. Those points are called *classical points*. It is known that $X(K)$ is dense in $X^{{\operatorname{an}}}$.
There is another class of points of $X^{{{\operatorname{an}}}}$, called the *Shilov points*. Let $\mathscr{X} \to {{\operatorname{Spec}}}(R)$ be a model of $X$. Assume that any irreducible component of the special fiber ${{\mathscr{X}}}_s$ is reduced. Let $C$ be an irreducible component of $\mathscr{X}_s$ and let $\xi$ be the generic point of $C$. Then one associates a unique absolute value $|\cdot|_{C}$ on the function field $K ( X )$ of $X$, which is characterized by the property that for any $f \in K ( X )$, $$| f |_{C}
=
\inf
\{
|a| \in {{\mathbb{R}}}\mid
a \in K^{\times} ,
a^{-1} f \in {{\mathcal{O}}}_{\mathscr{X},\xi}
\}
.$$ Thus $C$ gives a point $[C]$ of $X^{{{\operatorname{an}}}}$, which we call the *Shilov point* associated to $(\mathscr{X} , C)$ or simply to $C$. A Shilov point associated to $(\mathscr{X} , C)$ for some irreducible component $C$ of $\mathscr{X}_s$ is called a *Shilov point with respect to $\mathscr{X}$*. In this paper, let $V( \mathscr{X} )$ denote the set of Shilov points with respect to $\mathscr{X}$. If $\mathscr{X}^\prime \to {{\operatorname{Spec}}}(R)$ is another model of $X$ with reduced special fiber and with a surjective morphism $\mathscr{X}^\prime \to \mathscr{X}$, then we have $V( \mathscr{X} ) \subset V( \mathscr{X}^\prime )$. A point which is a Shilov point with respect to some model is called a *Shilov point*.
\[rem:type2points\] Let $x \in X^{{{\operatorname{an}}}}$ be a Shilov point. Then for any $f \in K(X)^{\times}$, we have $- \log |f ( x )| \in \Lambda$. One sees this fact from the maximum modulus principle [@BGR Proposition 6.2.1/4] with the description [@BPR2 §2.3] or from the description of $\mathrm{Val}$ in [@gubler4 §5.3].
### Reduction map {#reduction-map .unnumbered}
We recall the reduction map associated to a model ${{\mathscr{X}}}$ of $X$. Here, we assume that $X$ is proper over $K$. Let $(x, |\cdot|)$ be a point of $X^{{\operatorname{an}}}$. Then we have a natural morphism ${{\operatorname{Spec}}}( \kappa (x) ) \to \mathscr{X}$. Let $V$ be the ring of integers of $\kappa (x)$ with respect to $| \cdot |$. Since $V$ is a valuation ring and $\mathscr{X} \to {{\operatorname{Spec}}}(R)$ is proper, the morphism ${{\operatorname{Spec}}}( \kappa (x) ) \to \mathscr{X}$ extends to a unique morphism ${{\operatorname{Spec}}}(V) \to \mathscr{X}$. Let ${\operatorname{red}}_{\mathscr{X}} ( x, |\cdot| ) \in \mathscr{X}_s$ be the image of the special point of ${{\operatorname{Spec}}}(V)$ by this morphism. Then the assignment $( x, |\cdot| ) \to {\operatorname{red}}_{\mathscr{X}} ( x, |\cdot| )$ defines a map ${\operatorname{red}}_{\mathscr{X}} : X^{{{\operatorname{an}}}} \to \mathscr{X}_s$. We call this map the *reduction map associated to $\mathscr{X}$*.
If $x = (x, |\cdot|_K) \in X^{{{\operatorname{an}}}}$ is a classical point and if $\sigma$ is the corresponding section of the model $\mathscr{X} \to {{\operatorname{Spec}}}(R)$ guaranteed by the valuative criterion, then ${\operatorname{red}}_{\mathscr{X}} (x) = \sigma (k)$.
Skeleta associated to strictly semistable models {#subsection:skeleta}
------------------------------------------------
We recall the notion of skeleta. For simplicity, in the sequel, we assume that $X$ is a connected smooth projective curve, although skeleta can be defined in a more general setting.
### Standard model and its skeleton {#standard-model-and-its-skeleton .unnumbered}
Let $\varpi \in R$ be an element with $0 < |\varpi| < 1$. We set $
\mathscr{S} :=
{{\operatorname{Spec}}}( R
[x,y]/(xy - \varpi))
$, and call ${{\mathscr{S}}}$ a *standard model*.
Let $\mathscr{S}_K = {{\operatorname{Spec}}}(K[x, y]/(xy-\varpi))$ be the generic fiber of $\mathscr{S} \to {{\operatorname{Spec}}}(R)$. We first define the skeleton $S ( \mathscr{S}) \subset \mathscr{S}_K^{{{\operatorname{an}}}}$. Since $K [x,y]/(x y - \varpi)
\cong K [y^{\pm}]$, we regard an element of $K [x,y]/(x y - \varpi)$ as a Laurent polynomial on $y$. For any $v \in [ 0 , - \log |\varpi| ]$, one constructs a unique absolute value $|\cdot|_{v}$ on the function field of $K [x,y]/(x y - \varpi)$ such that for any Laurent polynomial $f= \sum_{m} a_{m} y^m
$, we have $
|f|_{v} =
\max_{m}
\left\{
|a_{m}|
\exp(- vm
)
\right\}
.
$
Let $\eta$ be the generic point of $\mathscr{S}$. The assignment $v \to (\eta, |\cdot|_{v})$ gives a map $[ 0 , - \log |\varpi| ] \to \mathscr{S}_K^{{{\operatorname{an}}}}$. It is known that this map is continuous and injective. The [*skeleton*]{} $S({{\mathscr{S}}})$ of ${{\mathscr{S}}}$ is defined to be the image of this map. Then the map $[ 0 , - \log |\varpi| ] \to S({{\mathscr{S}}})$ is a homeomorphism with inverse
$$\begin{aligned}
\label{eq:inverseisometry}
(\eta, | \cdot |) \mapsto - \log |y|
.\end{aligned}$$
Thus $S ( \mathscr{S} )$ has a structure of $1$-simplex via the homeomorphism $[ 0 , - \log |\varpi| ] \cong S ( \mathscr{S} )$. One sees that the end points of $S ( \mathscr{S} )$ are the Shilov points corresponding to the irreducible components of $\mathscr{S}_s$.
### Local étale atlas that distinguishes a node {#local-étale-atlas-that-distinguishes-a-node .unnumbered}
Let $\mathscr{X} \to {{\operatorname{Spec}}}(R)$ be a strictly semistable model of $X$. We take any $p \in {\operatorname{Sing}}( \mathscr{X}_s )$. Then there exist $C, C' \in
{\operatorname{\operatorname{Irr}}}( \mathscr{X}_s)$ such that $C \neq C'$ and $p \in C \cap C'$. By the same argument as in the proof of [@BPR2 Proposition 4.3], one can take an open neighborhood of $p$ in ${{\mathscr{X}}}$ and an étale morphism from this open subset to a standard model. Furthermore, replacing the open neighborhood by a smaller one, we can take an affine open neighborhood $\mathscr{U}_p
\subset \mathscr{X}$ of $p$, an element $\varpi_p \in R$ with $0 < |\varpi_p| < 1$, and an étale morphism $\psi_p : \mathscr{U}_p \to \mathscr{S}:={{\operatorname{Spec}}}( R [x,y]/(x y - \varpi_p))$ such that the special fiber $\mathscr{U}_p \cap {{\mathscr{X}}}_s$ consists of exactly two irreducible components and $p$ is a unique node of $\mathscr{U}_p \cap {{\mathscr{X}}}_s$ (cf. [@gubler4 Proposition 5.2]). We call $\psi_p$ a *local étale atlas that distinguishes $p$*.
Note that, with the above notation, $v_K(\varpi_p)$ depends only on $p \in \mathscr{X}$ and is independent of the choice of $\mathscr{U}_p$. Thus the following definition makes sense.
\[def:multiplicityatanodes\] We call $v_K (\varpi_p)$ the *multiplicity* (of $\mathscr{X}$) at $p \in {\operatorname{Sing}}( \mathscr{X}_s)$.
### Skeleton $S ( \mathscr{X} )$ associated to ${{\mathscr{X}}}$ {#skeleton-s-mathscrx-associated-to-mathscrx .unnumbered}
Let $\mathscr{X} \to {{\operatorname{Spec}}}(R)$ be a strictly semistable model of $X$, and let $p \in {\operatorname{Sing}}({{\mathscr{X}}}_s)$.
\[def:canonical:1:simplex:node\] The *canonical $1$-simplex corresponding to $p$* is a closed subset $\Delta_p \subset X^{{{\operatorname{an}}}}$ characterized by the following conditions.
1. For any local étale atlas $\psi_p : \mathscr{U}_p \to \mathscr{S}:={{\operatorname{Spec}}}( R [x,y]/(x y - \varpi_p))$ that distinguishes $p$, we have $\Delta_p \subset U_p^{{{\operatorname{an}}}}$, where $U_p$ is the generic fiber of $\mathscr{U}_p \to {{\operatorname{Spec}}}(R)$.
2. The restriction to $\Delta_p$ of the induced map $\psi_p^{{{\operatorname{an}}}} : U^{{{\operatorname{an}}}} \to \mathscr{S}_K^{{{\operatorname{an}}}}$ gives a homeomorphism $\Delta_p \to S ( \mathscr{S} )$.
Note that the end points $\Delta_p$ are the Shilov points which correspond to the irreducible components of the special fiber of $\mathscr{U}$.
\[def:skeleton:model\] Let ${{\mathscr{X}}}$ be a strictly semistable model of $X$. The [*skeleton*]{} $S({{\mathscr{X}}})$ associated to ${{\mathscr{X}}}$ is defined by $$S({{\mathscr{X}}}) = \bigcup_{p \in
{\operatorname{Sing}}( \mathscr{X}_s )} \Delta_{p},$$ which is a compact subset of $X^{{{\operatorname{an}}}}$.
One sees that for any $p_1 , p_2 \in {\operatorname{Sing}}( \mathscr{X}_s )$, $\Delta_{p_1} = \Delta_{p_2}$ if and only if $p_1 = p_2$; if $p_1 \neq p_2$, then $\mathrm{relin} ( \Delta_{p_1}) \cap \mathrm{relin} ( \Delta_{p_2}) = \emptyset$, where $\mathrm{relin}$ means the relative interior, and the points in $\Delta_{p_1} \cap \Delta_{p_2}$ are the Shilov points associated to irreducible components of $\mathscr{X}_s$ that contain $p_1$ and $p_2$. Thus $S ( \mathscr{X} )$ has a canonical structure of simplicial set and is regarded as the dual graph of $\mathscr{X}_s$. Note that the set $V (\mathscr{X}) =
\{ [C] \mid C \in {\operatorname{\operatorname{Irr}}}(\mathscr{X}_a )\ \}$ of Shilov points with respect to $\mathscr{X}$ is actually a set of vertices of this simplicial set. Each canonical $1$-simplex is called an edge, and we denote by $E ( \mathscr{X} )$ the set of edges.
It is known that $X^{{{\operatorname{an}}}}\setminus X(K)$ has a canonical metric structure (cf. [@BPR2 Corollary 5.7]). We describe this canonical metric structure on $S({{\mathscr{X}}})$. We take any $p \in {\operatorname{Sing}}( \mathscr{X}_s )$ and denote by $\lambda_p$ the multiplicity at $p$ (cf. Definition \[def:multiplicityatanodes\]). The composition of the above homeomorphisms $\Delta_p \to S({{\mathscr{S}}})$ and $S({{\mathscr{S}}}) \to [0, \lambda_p]$ (see ) gives a homeomorphism $\Delta_p \to [0, \lambda_p]$. Then, with respect to the canonical metric structure on $\Delta_p$ induced from $X^{{{\operatorname{an}}}}\setminus X(K)$, this homeomorphism is an isometry ([@BPR2 Theorem 5.15] and [@GRW §4]). In particular, if $C$ and $C'$ are the distinct irreducible components with $p
\in C \cap C'$ and if $v$ and $v'$ are the Shilov points of $C$ and $C'$, then the distance between $v$ and $v'$ in $\Delta_p$ equals the multiplicity at $p$.
Thus the skeleton $S ( \mathscr{X} )$ has a structure of a $\Lambda$-metric graph. For a node $p \in \mathscr{X}_s$, a point of $\Delta_p$ that lies in $[0 , \lambda_p] \cap \Lambda$ via the isometry $\Delta_p \to [0 , \lambda_p]$ is a $\Lambda$-rational point of $S ( \mathscr{X} )$. We denote the set of $\Lambda$-rational points by $S_{\Lambda} (\mathscr{X} )$. When we write the skeleton as $\Gamma$, we write the subset of those points as $\Gamma_\Lambda$, which is compatible with “Notation and convention” in the introduction.
We end this subsection by defining a canonical $1$-simplex corresponding to a connected curve.
\[def:canonical:1:simplex:curve\] Let ${{\mathscr{X}}}$ be a strictly semistable model of $X$. Let $E$ be a connected curve in $\mathscr{X}_s$. We set $$\Delta_E := \bigcup_{p \in E \cap {\operatorname{Sing}}( \mathscr{X}_s)} \Delta_p,$$ which we call the *canonical $1$-simplex corresponding to $E$*.
Note that if $E$ is a chain in $\mathscr{X}_s$, then $\Delta_E$ is isometric to a closed interval of length $\sum_{p \in E \cap {\operatorname{Sing}}( \mathscr{X}_s)} \lambda_p$, where $\lambda_p$ denotes the multiplicity at $p$.
Skeleta associated to strictly semistable pairs {#subsec:skeleta:ass:ssp}
-----------------------------------------------
We recall the notion of skeleta associated to strictly semistable pairs, which generalize skeleta associated to strictly semistable models in §\[subsection:skeleta\]. See also [@Tyo].
### Standard pair for ends and its skeleton {#standard-pair-for-ends-and-its-skeleton .unnumbered}
We set $\mathscr{T} := {{\operatorname{Spec}}}( R [y] )$ and denote by $\mathscr{T}_K = {{\operatorname{Spec}}}(K [y] )$ the generic fiber of $\mathscr{T} \to {{\operatorname{Spec}}}(R)$. Let $\tau$ be a section of $\mathscr{T} \to {{\operatorname{Spec}}}(R)$ given by $R[y] \to R [y]/(y) = R$. We call $(\mathscr{T} ; \tau )$ the *standard pair for an end*.
One defines the skeleton $S(\mathscr{T} ; \tau ) \subset \mathscr{T}_K^{{{\operatorname{an}}}}$ as follows. For any $v \in {{\mathbb{R}}}_{\geq 0}$, one constructs a unique absolute value $|\cdot|_v$ on $K (y)$ such that for any $f = \sum_{m} a_m y^m \in K [y]$, we have $
|f|_{v} := \max_{m}
\{ |a_m|
\exp (-vm)
\}
.
$
Let $\eta$ be the generic point of $\mathscr{T}$. The assignment $v \to (\eta, | \cdot |_v)$ gives a map ${{\mathbb{R}}}_{\geq 0} \to \mathscr{T}_K^{{{\operatorname{an}}}}$. It is known that this map is continuous and injective. The [*skeleton*]{} $S(\mathscr{T} ; \tau )$ of $(\mathscr{T} ; \tau )$ is defined to be the image of this map. The map ${{\mathbb{R}}}_{\geq 0} \to S ( \mathscr{T} ; \tau )$ is a homeomorphism with inverse given by $(\eta, | \cdot |) \to - \log |y|$.
The point $\xi$ in $S ( \mathscr{T} ; \tau )$ corresponding to $0 \in {{\mathbb{R}}}_{\geq 0}$ is the Shilov point associated to the special fiber of $\mathscr{T}$. We write $\mathrm{relin} (S ( \mathscr{T} ; \tau ))$ for the relative interior of $S ( \mathscr{T} ; \tau )$. We remark that $\mathrm{relin} ( S ( \mathscr{T} ; \tau )) =
S ( \mathscr{T} ; \tau )\setminus \{ \xi \}$.
### Local étale atlas that distinguishes a section {#local-étale-atlas-that-distinguishes-a-section .unnumbered}
Let $\mathscr{X} \to {{\operatorname{Spec}}}(R)$ be a strictly semistable curve and let $\sigma$ be a section. Assume that $(\mathscr{X} ; \sigma)$ is a strictly semistable pair. Then there exist an affine open neighborhood $\mathscr{U}_{\sigma}$ of $\sigma$ and an étale morphism $$\psi_{\sigma} : \mathscr{U}_{\sigma} \to
{{\operatorname{Spec}}}( R [ y ] ) = \mathscr{T}$$ such that the special fiber of $\mathscr{U}$ is irreducible and ${\operatorname{div}}( \psi^{\ast} (y)) = \sigma (R)$. We call such an étale morphism a *local étale atlas that distinguishes $\sigma$*.
### Skeleton with ends {#skeleton-with-ends .unnumbered}
Let $\mathscr{X} \to {{\operatorname{Spec}}}(R)$ be a strictly semistable model of $X$, and let $(\mathscr{X} ; \sigma)$ be a strictly semistable pair.
The [*canonical end*]{} $\Delta ( \sigma)$ associated to $ \sigma $ is a closed subspace of $X^{{{\operatorname{an}}}} \setminus X (K)$ which is characterized by the following conditions.
1. For any local étale atlas $\psi_{\sigma} : \mathscr{U}_{\sigma} \to
{{\operatorname{Spec}}}( R [ y ] ) = \mathscr{T}$ that distinguishes $\sigma$, we have $\Delta ( \sigma ) \subset U_{\sigma}^{{{\operatorname{an}}}}$, where $U_{\sigma}$ is the generic fiber of $\mathscr{U}_{\sigma}$.
2. The restriction to $\Delta ( \sigma )$ of the induced map $\psi_{\sigma}^{{{\operatorname{an}}}} : U_{\sigma}^{{{\operatorname{an}}}} \to \mathscr{T}_K^{an}$ gives a homeomorphism $\Delta ( \sigma ) \to S ( \mathscr{T} ; \tau )$.
The canonical metric structure on $X^{{{\operatorname{an}}}}\setminus X(K)$ gives a metric on $\Delta ( \sigma )$, and the map $\Delta ( \sigma ) \to {{\mathbb{R}}}_{\geq 0}$ given by $(\eta' , | \cdot |) \mapsto - \log |\psi_p^{\ast} (y)|$ is an isometry, where $\eta'$ is the generic point of $X$ and $\psi_p^{\ast} (y)$ is regarded as a rational function on $X$. The endpoint of $\Delta ( \sigma)$, i.e, the point corresponding to $0 \in {{\mathbb{R}}}_{\geq 0}$ via the isometry, is the Shilov point associated to the irreducible component of $\mathscr{X}_s$ that contains the point $\sigma (k)$.
\[def:definitionofskeleton\] Let $\mathscr{X}$ be a strictly semistable model of $X$, and let $( \mathscr{X} ; \sigma_1 , \ldots , \sigma_r)$ be a strictly semistable pair. The *skeleton* $S (\mathscr{X} ; \sigma_1 , \ldots , \sigma_r)$ associated to $( \mathscr{X} ; \sigma_1 , \ldots , \sigma_r)$ is defined by $$\begin{aligned}
\label{align:definitionofskeleton}
S (\mathscr{X} ; \sigma_1 , \ldots , \sigma_r)
:= S ( \mathscr{X} ) \cup \bigcup_{i = 1}^{r} \Delta ( \sigma_i).\end{aligned}$$
One shows that $\mathrm{relin} ( \Delta ( \sigma_i) )
\cap \mathrm{relin} ( \Delta ( \sigma_j)) = \emptyset$ for $i \neq j$ and that $\mathrm{relin} ( \Delta ( \sigma_i) )
\cap S ( \mathscr{X} ) = \emptyset$ for any $i$. On the other hand, for each $i$, since the boundary point $\xi_i$ of $\Delta ( \sigma_i)$ is the Shilov point associated to the irreducible component of $\mathscr{X}_s$ that meets $\sigma_i$, $\xi_i$ belongs to $S ( \mathscr{X} )$ as well. Thus $S (\mathscr{X} ; \sigma_1 , \ldots , \sigma_r)$ is the space obtained by successive one-point sum of $S (\mathscr{X} )$ with $\Delta (\sigma_1) , \ldots , \Delta (\sigma_r)$ at Shilov points associated to the irreducible components that contain $\sigma_1 (k) ,
\ldots , \sigma_r (k)$, respectively.
Since $X^{{{\operatorname{an}}}} \setminus X(K)$ has a canonical metric, the skeleton $S (\mathscr{X} ; \sigma_1 , \ldots , \sigma_r)$ is equipped with a metric. For each $\sigma_i$, we have a unique isometry $\Delta ( \sigma_i) \cong {{\mathbb{R}}}_{\geq 0}$. We also consider $\Lambda$-rational points of $\Delta ( \sigma_i)$ via the isometry. Thus one has the notion of $\Lambda$-rational points of $S (\mathscr{X} ; \sigma_1 , \ldots , \sigma_r)$, and we denote the union of the set of those points and $S_{\Lambda} ( \mathscr{X})$ by $S_{\Lambda} (\mathscr{X} ; \sigma_1 , \ldots , \sigma_r)$. When we write $\Gamma$ for the skeleton, we write $\Gamma_\Lambda$ for the subset of $\Lambda$-rational points.
Note that Definition \[def:definitionofskeleton\] makes sense even if $r = 0$, which is the skeleton $S({{\mathscr{X}}})$ associated to the model ${{\mathscr{X}}}$ in the sense of Definition \[def:skeleton:model\]. Then we make the following definition.
\[def:skeletongeneral\] A subset $\Gamma$ of $X^{{{\operatorname{an}}}}$ is called a *skeleton of $X^{{\operatorname{an}}}$* if there exists a strictly semistable pair $(\mathscr{X} ; \sigma_1 , \ldots , \sigma_r)$ (possibly $r=0$) such that $\Gamma = S(\mathscr{X} ; \sigma_1 , \ldots , \sigma_r)$. A skeleton $\Gamma$ of $X^{{\operatorname{an}}}$ is called a [*compact skeleton*]{} if $\Gamma$ is compact.
Remark that a skeleton $\Gamma$ is a compact skeleton if and only if $\Gamma = S({{\mathscr{X}}})$ for some strictly semistable model of $X$. We remark also that the definition of skeleta in this paper and that in [@BPR2] are the same by [@BPR2 Theorem 4.11].
Some properties of skeleta {#subsec:properties:skeleta}
--------------------------
In this subsection, we recall some properties of skeleta. Basic references are [@BPR2].
### Subdivision of a skeleton {#subdivision-of-a-skeleton .unnumbered}
As we have seen, a skeleton associated to a strictly semistable model has a structure of a simplicial set such that the set of irreducible components of the special fiber of the model coincides with the set of vertices. Here, we give remarks on the relationship between the subdivision of this simplicial structure and strictly semistable models dominating the given model.
The following propositions are essentially parts of [@BPR2 Theorem 4.11]. Indeed, they follow from [@BPR2 Theorem 4.11] and the fact that an admissible formal model of a smooth projective curve is algebraizable (cf. [@FK Proposition 10.3.2]).
\[prop:subdivision1\] Let $X$ be a connected smooth projective curve over $K$, and let $\pi : \mathscr{X} \to {{\operatorname{Spec}}}(R)$ be a strictly semistable model of $X$. Let $\pi' : \mathscr{X}' \to {{\operatorname{Spec}}}(R)$ be a strictly semistable model of $X$ that dominates $\mathscr{X}$, and let $\mu : \mathscr{X}' \to \mathscr{X}$ be the morphism extending the identity morphism on $X$. Let $\{E_1 , \ldots , E_m\}$ be the set of irreducible components of $\mathscr{X}'_s$ such that each $\mu (E_i)$ is a singleton of $\mathscr{X}_s$, and we assume that $\{E_1 , \ldots , E_m\}$ consists of $(-2)$-curves. Namely, we assume that $\mu$ is given by contracting only $(-2)$-curves. For $i = 1 , \ldots , m$, let $[E_i]$ be the Shilov point associated to $E_i$.
1. Then $S({{\mathscr{X}}}') = S({{\mathscr{X}}})$ as subsets of $X^{{{\operatorname{an}}}}$, and $V ( \mathscr{X}) = V ( \mathscr{X}') \setminus
\{ [E_1] , \ldots , [E_m] \}$.
2. Let $p \in {\operatorname{Sing}}( \mathscr{X}_s )$, and we set $\{q_0 , \ldots , q_r\} := \{q \in {\operatorname{Sing}}(\mathscr{X}'_s) \mid \mu(q)= p\}$. Then $\Delta_{p} = \bigcup_{i=1}^{r} \Delta_{q_i}$.
The next proposition is a converse of Proposition \[prop:subdivision1\].
\[prop:subdivision1:b\] Let $X$ be a connected smooth projective curve over $K$ and let $\pi : \mathscr{X} \to {{\operatorname{Spec}}}(R)$ be a strictly semistable model of $X$. Let $V'$ be a finite subset of $S_{\Lambda} (\mathscr{X})$ with $V' \supset V (\mathscr{X})$. Then there exists a unique up to a canonical isomorphism strictly semistable model $\pi' : \mathscr{X}' \to {{\operatorname{Spec}}}(R)$ of $X$ such that $V ( \mathscr{X}') = V'$. Further, $\mathscr{X}'$ has the following properties.
1. There exists a unique morphism $\mu : \mathscr{X}' \to \mathscr{X}$ that extends the identity morphism on $X$.
2. Any $E \in {\operatorname{\operatorname{Irr}}}( \mathscr{X}'_s)$ such that $\mu (E)$ is a singleton is a $(-2)$-curve.
3. We have $S ( \mathscr{X}') = S (\mathscr{X})$ as subsets of $X^{{\operatorname{an}}}$.
### Retraction map {#subsect:retraction .unnumbered}
Let $X$ be a connected smooth projective curve over $K$. Let $\mathscr{X} \to {{\operatorname{Spec}}}(R)$ be a strictly semistable model of $X$, and set $\Gamma : = S ( \mathscr{X} )$. Then we have a unique strongly deformation retraction $\tau : X^{{{\operatorname{an}}}} \to \Gamma$ that is the identity on $\Gamma$. Indeed, one shows that for each connected component $B$ of $X^{{{\operatorname{an}}}} \setminus \Gamma$, the boundary $\partial B$ consists of a unique point $v$ in $\Gamma_{\Lambda}$ and the closure $B \cup \{ v \}$ of $B$ is retracted to $v$. We call $\tau$ the *retraction map* with respect to $\Gamma$. The retraction map is also described in terms of valuation map; see [@gubler4 §5.3].
\[lemma:retraction:rational\] Let $\Gamma$ be a compact skeleton of $X^{{{\operatorname{an}}}}$, and let $\tau_{\Gamma} : X^{{{\operatorname{an}}}} \to \Gamma$ be the retraction map. Let $A$ be a connected component of $X^{{{\operatorname{an}}}} \setminus \Gamma$. Then the following hold.
1. There exists $v \in \Gamma$ such that $\tau_{\Gamma} (A) = \{ v \}$. Further, $v \in \Gamma_\Lambda$.
2. Let $v$ be as in . Let $\mathscr{X}$ be a strictly semistable model of $X$ such that $S( \mathscr{X} ) = \Gamma$ and $v \in V( \mathscr{X})$. Let $C \in {\operatorname{\operatorname{Irr}}}( \mathscr{X}_s )$ with $[C] = v$. Then there exists a unique $q \in C(k) \setminus {\operatorname{Sing}}( \mathscr{X}_s )$ such that ${\operatorname{red}}_{\mathscr{X}} (A) = \{ q \}$.
[[*Proof.*]{}]{}The first assertion in (1) follows from the definition of the retraction map. We show the second assertion, i.e., that $v$ is a $\Lambda$-rational point. Since $\Gamma$ is a compact skeleton, there exists a strictly semistable model $\mathscr{X}$ of $X$ such that $S(\mathscr{X}) = \Gamma$. We take $p \in {\operatorname{Sing}}( \mathscr{X}_s )$ such that $v \in \Delta_p$. Let $\psi_p : \mathscr{U}_p \to \mathscr{S}:={{\operatorname{Spec}}}( R [x,y]/(x y - \varpi_p))$ be a local étale atlas that distinguishes $p$. Then $\varphi := - \log \left| {\left.{\psi_p^{\ast} (y)}\right\vert_{{X}}} \right| $ gives an isometry $\Delta_p \to [0 , - \log | \varpi_p |_K]$. Therefore, (1) is reduced to showing that $\varphi (v) \in \Lambda$. We see from [@gubler4 §5.3] or [@BPR2 Lemma 3.8] that $$\begin{aligned}
\label{align:retraction:val}
\varphi \circ {\left.{\tau_{\Gamma}}\right\vert_{{A}}} = {\left.{\varphi}\right\vert_{{A}}}.\end{aligned}$$ Since $X(K)$ is dense in $X^{{{\operatorname{an}}}}$, there exists $P \in A \cap X(K)$. Since $\tau_{\Gamma} (A) = \{ v \}$, we have $v = \tau_{\Gamma} (P)$. By (\[align:retraction:val\]), $\varphi (v) = \varphi (\tau_{\Gamma} (P) ) = \varphi (P)$, and this belongs to $\Lambda$ by the definition of $\varphi$. Thus we obtain (1).
Let us prove (2). Set $C^{\circ} := C \setminus {\operatorname{Sing}}( \mathscr{X}_s )$. By [@gubler4 Proposition 5.7], ${\operatorname{red}}_{\mathscr{X}} ( \tau_{\Gamma}^{-1} (v) ) = C^{\circ}$. Recall that $v$ is the unique point in $X^{{{\operatorname{an}}}}$ such that ${\operatorname{red}}_{\mathscr{X}} (v)$ is the generic point of $C$. Then we have $\tau_{\Gamma}^{-1} (v) \setminus \{ v \} \subset
{\operatorname{red}}_{\mathscr{X}}^{-1} (C^{\circ} (k) )$, so that $A \subset {\operatorname{red}}_{\mathscr{X}}^{-1} (C^{\circ} (k) )$. Note that $
{\operatorname{red}}_{\mathscr{X}}^{-1} (C^{\circ} (k) ) = \bigcup_{q \in C^{\circ} (k)}
{\operatorname{red}}_{\mathscr{X}}^{-1} ( q )$, Since ${\operatorname{red}}_{\mathscr{X}}$ is anti-continuous, each ${\operatorname{red}}_{\mathscr{X}}^{-1} ( q )$ is an open subset of $X^{{{\operatorname{an}}}}$, and thus $\bigcup_{q \in C^{\circ} (k)}
{\operatorname{red}}_{\mathscr{X}}^{-1} ( q )$ a disjoint union of open subsets of ${\operatorname{red}}_{\mathscr{X}}^{-1} (C^{\circ} (k) )$. Since $A$ is a connected open subset of ${\operatorname{red}}_{\mathscr{X}}^{-1} (C^{\circ} (k) )$, it follows that there exists a unique $q \in C^{\circ} (k)$ such that $A \subset {\operatorname{red}}_{\mathscr{X}}^{-1} (q)$. This proves (2).
500
### Minimal skeleton and Deligne–Mumford stable model {#minimal-skeleton-and-delignemumford-stable-model .unnumbered}
Let $X$ be a connected smooth projective curve of genus $g$. Let $\mathscr{X}$ be a strictly semistable model of $X$. We say that $\mathscr{X}$ is *minimal* if $\mathscr{X}_s$ does not have a $(-1)$-curve. Remark that in the case where $g \geq 1$, $\mathscr{X}$ is minimal if and only if $\mathscr{X}$ is Deligne–Mumford semistable.
\[def:minimal:skeleton\] We say that a skeleton $\Gamma$ of $X^{{{\operatorname{an}}}}$ is a *minimal skeleton* if there exists a minimal model $\mathscr{X}$ of $X$ with $\Gamma = S ( \mathscr{X} )$.
We note that the notion of a minimal skeleton is the same as that in [@BPR2]. Remark that if $\Gamma$ is minimal, then any model $\mathscr{X}'$ with $\Gamma = S ( \mathscr{X}')$ is minimal.
Assume that $g \geq 1$. Then the minimal skeleton is unique. Indeed, let $\mathscr{X} \to {{\operatorname{Spec}}}(R)$ be any Deligne–Mumford strictly semistable model of $X$; then the minimal skeleton equals $S ( \mathscr{X} )$ as a subspace of $X^{{{\operatorname{an}}}}$, and it does not depend on the choice of ${{\mathscr{X}}}$. We denote by $\Gamma_{\min}$ the minimal skeleton.
Furthermore, assume that $g \geq 2$. Then the minimal skeleton $\Gamma_{\min}$ of $X^{{{\operatorname{an}}}}$ has a [*canonical*]{} finite graph structure determined by the [*stable*]{} model. Let $\mathscr{X}^{{\operatorname{st}}}$ be the stable model of $X$. As is noted in §\[subsec:semistabe:models\], there exists a Deligne–Mumford strictly semistable model $\mathscr{X}$ that dominates $\mathscr{X}^{{\operatorname{st}}}$. It follows that the set of Shilov points $V (\mathscr{X}^{{\operatorname{st}}})$ with respect to $\mathscr{X}^{{\operatorname{st}}}$ is a subset of $V ( \mathscr{X})$ (in fact, $V (\mathscr{X}^{{\operatorname{st}}})$ equals to the subset of $V ( \mathscr{X}) $ consisting of the Shilov points associated to non-$(-2)$-curves), and hence $V (\mathscr{X}^{{\operatorname{st}}})$ is also a subset of $\Gamma_{\min}$. We give $\Gamma_{\min}$ a finite graph structure so that $V (\mathscr{X}^{{\operatorname{st}}})$ is the set of vertices. Let $E ( \mathscr{X}^{{\operatorname{st}}})$ denote the set of edges with respect to this graph structure. Note that this finite graph may have self-loops in general.
We remark that for any $p \in {\operatorname{Sing}}(\mathscr{X}^{{\operatorname{st}}}_s)$, $\Delta_p$ makes sense. Indeed, let $\mathscr{X}$ be a Deligne–Mumford strictly semistable model of $X$ and let $\mu : \mathscr{X} \to \mathscr{X}^{{\operatorname{st}}}_s$ be the morphism extending the identity morphism on $X$. Let $\{q_1 , \ldots , q_r\}$ be the set of nodes of $\mathscr{X}_s$ such that $\mu (q_1) = \cdots = \mu (q_r) = p$. Then $\bigcup_{i = 1}^{r} \Delta_{q_i}$ is a subset of $\Gamma_{\min}$ and does not depend on the choice of $\mathscr{X}$. We set $\Delta_p := \bigcup_{i = 1}^{r} \Delta_{q_i}$. Note that $\Delta_p \in E(\mathscr{X}^{{\operatorname{st}}})$.
When $g \geq 2$, we use the following convention unless otherwise specified: When we say that $v$ is a vertex of $\Gamma_{\min}$, this means that $v \in V ( \mathscr{X}^{{\operatorname{st}}})$; when we say that $e$ is an edge, this means that $e \in E (\mathscr{X}^{{\operatorname{st}}})$.
Tropical geometry {#subsec:tropical:geometry}
-----------------
Let ${{\mathbb{G}}}_{m}^N$ be the algebraic torus of dimension $N$ over $K$ with coordinates $z_1 , \ldots , z_N$ and let ${{\mathbb{G}}}_m^{N , {{\operatorname{an}}}}$ be the associated analytic space. The map $$\begin{aligned}
\label{eqn:trop:map:torus}
{\operatorname{trop}}_{{{\mathbb{G}}}_m^N} : {{\mathbb{G}}}_{m}^{N ,{{\operatorname{an}}}} \to {{\mathbb{R}}}^N
,
\quad
p = (p , | \cdot |)
\mapsto
( - \log |z_1 (p) | , \ldots , - \log |z_N (p)|)\end{aligned}$$ is called the *tropicalization map of algebraic torus*. We extend this tropicalization map from the analytic space associated to projective space to tropical projective space. As in the introduction, we set ${{\mathbb{T}}}:= {{\mathbb{R}}}\cup\{+ \infty\}$. The *$N$-dimensional tropical projective space* is defined to be $${{\mathbb{T}}}{{\mathbb{P}}}^N :=
({{\mathbb{T}}}^{N+1}\setminus\{(+ \infty, \ldots, +\infty)\})/\sim,$$ where $x := (x_0, \ldots, x_N), y := (y_0, \ldots, y_N) \in {{\mathbb{T}}}^{N+1}\setminus\{(+\infty, \ldots, +\infty)\}$ satisfy $x \sim y$ if there exists $c \in {{\mathbb{R}}}$ such that $y_i = x_i + c$ for all $i = 0, \ldots, N$ (see [@MZ]). The equivalence class of $x$ in ${{\mathbb{T}}}{{\mathbb{P}}}^N$ is written as $(x_0: \cdots: x_N)$. Now, let ${{\mathbb{P}}}^N$ be the $N$-dimensional projective space over $K$ with homogeneous coordinates $X_0, \ldots, X_N$ and let ${{\mathbb{P}}}^{N, {{\operatorname{an}}}}$ denote the Berkovich analytification of ${{\mathbb{P}}}^N$. Then we define the tropicalization map to be $$\label{eqn:trop:map}
{\operatorname{trop}}: {{\mathbb{P}}}^{N, {{\operatorname{an}}}} \to {{\mathbb{T}}}{{\mathbb{P}}}^N, \qquad p = (p, |\cdot|) \mapsto (-\log|X_0(p)|
: \cdots: -\log|X_N(p)|).$$
The tropicalization map on (\[eqn:trop:map\]) extends that on (\[eqn:trop:map:torus\]), which we make precise now. The tropical projective space is equipped with $(N+1)$ charts $U_i := \{x = (x_0: \cdots: x_N) \in {{\mathbb{T}}}{{\mathbb{P}}}^N\mid x_i \neq + \infty\}$. We set $E:=\bigcap_{i = 0}^N U_i$. For each $i= 0 , \ldots , N$, we have a natural homeomorphism $$\phi_i : {{\mathbb{R}}}^{N} \to E,
\quad
(u_1 , \ldots , u_N) \mapsto ( u_1 : \cdots : u_{i-1} : 0 : u_{i}
: \cdots : u_N ).$$ Thus $E$ is an $N$-dimensional Euclidean space embedded in ${{\mathbb{T}}}{{\mathbb{P}}}^N$. Further, for each $i=0,\ldots ,N$, let $$\psi_i : {{\mathbb{G}}}_m^N \hookrightarrow
{{\mathbb{P}}}^N, \quad
(z_1 , \ldots , z_N)
\mapsto (z_1 : \cdots : z_{i-1} : 1 : z_{i} \cdots : z_N )$$ denote the open immersion. Then we have ${\operatorname{trop}}\circ \psi_i = \phi_i \circ {\operatorname{trop}}_{{{\mathbb{G}}}_m^N}$. Remark that ${\operatorname{trop}}^{-1} ( E ) = \psi_i ( {{\mathbb{G}}}_m^N)$ for any $i=0 , \ldots , N$.
\[remark:integralstructure\] Let $i , j \in \{ 0 , 1 , \ldots , N \}$. Then we have an homeomorphism $\phi_j^{-1} \circ \phi_i : {{\mathbb{R}}}^N \to {{\mathbb{R}}}^N$. It is clear from the definition of $\phi_i$ and $\phi_j$ that this restricts an isomorphism from ${{\mathbb{Z}}}^N$ to ${{\mathbb{Z}}}^N$ as ${{\mathbb{Z}}}$-modules, i.e., $\phi_j^{-1} \circ \phi_i$ is given by the multiplication of an element of ${\operatorname{GL}}_N ({{\mathbb{Z}}})$.
Let $Y$ be an irreducible closed subvariety of ${{\mathbb{P}}}^N$. Fix an $i = 0 , \ldots , N$. Then $Y_i^{\circ} := \psi_i^{-1} (Y)$ is an irreducible closed subvariety of ${{\mathbb{G}}}_m^N$. The results of Bieri–Groves and Speyer–Sturmfels (cf. [@MS] and [@GuTool Theorem 3.3]) describe the polytopal structure of ${\operatorname{trop}}_{{{\mathbb{G}}}_m^N}\left(Y_i^{\circ, {{\operatorname{an}}}}\right)$. Although we do not recall their results in full generality, we describe the polytopal structure when $\dim(Y_i^\circ) = 1$, because that is the case we are concerned with. In this case, there exists a finite subset $V \subset
{\operatorname{trop}}_{{{\mathbb{G}}}_m^N}\left(Y_i^{\circ, {{\operatorname{an}}}}\right) \cap \Lambda^N$ such that if $\Sigma$ is the set of closures of the connected components of of ${\operatorname{trop}}_{{{\mathbb{G}}}_m^N}\left(Y_i^{\circ, {{\operatorname{an}}}}\right) \setminus V$, then each $\Delta \in \Sigma$ is of form
1. $\Delta = \{y + t z \mid t \in [0, \ell]\}$ or
2. $\Delta = \{y + t z \mid t \in [0, +\infty)\}$,
where $y \in \Lambda^N$, $z = (z_1, \ldots, z_N) \in {{\mathbb{Z}}}^N$ with ${\rm GCD}(z_1, \ldots, z_N) = 1$, and $\ell \in \Lambda \cap {{\mathbb{R}}}_{> 0}$.
Using the above illustration, we endow ${\operatorname{trop}}_{{{\mathbb{G}}}_m^N}\left(Y_i^{\circ, {{\operatorname{an}}}}\right)$ with a metric structure. Indeed, we put a metric by lattice length: If $\Delta \in \Sigma$ is as in (1) (resp. (2)), then we identify $\Delta$ with $[0, \ell]$ (resp. $[0, + \infty)$) via $y + t z \mapsto t$, and with this identification, $\Delta$ is a metric space; since ${\operatorname{trop}}_{{{\mathbb{G}}}_m^N}\left(Y_i^{\circ, {{\operatorname{an}}}}\right) = \bigcup_{\Delta} \Delta$, this metric structure does not depend on the choice of $\Sigma$ and is well-defined for ${\operatorname{trop}}_{{{\mathbb{G}}}_m^N}\left(Y_i^{\circ, {{\operatorname{an}}}}\right)$. Thus ${\operatorname{trop}}_{{{\mathbb{G}}}_m^N}\left(Y_i^{\circ, {{\operatorname{an}}}}\right)$ is a metric space.
Furthermore, the above metric structure on each ${\operatorname{trop}}_{{{\mathbb{G}}}_m^N}\left(Y_i^{\circ, {{\operatorname{an}}}}\right)$ gives a well-defined metric on ${\operatorname{trop}}(Y^{{{\operatorname{an}}}}) \cap E$. Indeed, for each $i = 0 , \ldots , N$, since ${\operatorname{trop}}^{-1} ( E ) = \psi_i ( {{\mathbb{G}}}_m^N)$, we have a homeomorphism ${\left.{\phi_i}\right\vert_{{{\operatorname{trop}}_{{{\mathbb{G}}}_m^N}\left(Y_i^{\circ, {{\operatorname{an}}}}\right)}}} :
{\operatorname{trop}}_{{{\mathbb{G}}}_m^N}\left(Y_i^{\circ, {{\operatorname{an}}}}\right) \to {\operatorname{trop}}(Y^{{{\operatorname{an}}}}) \cap E$, and since ${\operatorname{trop}}_{{{\mathbb{G}}}_m^N}\left(Y_i^{\circ, {{\operatorname{an}}}}\right)$ has a metric structure, this homeomorphism induces a metric structure on ${\operatorname{trop}}(Y^{{{\operatorname{an}}}}) \cap E$. Since the metric on ${\operatorname{trop}}_{{{\mathbb{G}}}_m^N}\left(Y_i^{\circ, {{\operatorname{an}}}}\right)$ is given by the lattice length, one sees from Remark \[remark:integralstructure\] that this metric structure on ${\operatorname{trop}}(Y^{{{\operatorname{an}}}}) \cap E$ does not depend on $i$. Thus ${\operatorname{trop}}(Y^{{{\operatorname{an}}}}) \cap E$ is a metric space.
Remark that tropical geometry near the boundary ${{\mathbb{T}}}{{\mathbb{P}}}^N \setminus
E$ is rather subtle (see [@MZ]). However, for our purposes, namely, for faithful tropicalizations, we will not need detailed analysis on the boundary because the tropicalization of a skeleton is contained in $E$, as we will see in the next subsection.
Faithful tropicalization {#subsec:unimodular:faithful}
------------------------
Let $X$ be a connected smooth projective curve over $K$. Let $L$ be a line bundle over $X$. Suppose that we are given nonzero global sections $s_0, s_1, \ldots, s_N \in H^0(X, L)$. Associated to those sections, we define a map $\varphi : X^{{\operatorname{an}}}\to {{\mathbb{T}}}{{\mathbb{P}}}^N$ as follows: Let $\varphi^\prime: X^{{\operatorname{an}}}\longrightarrow {{\mathbb{P}}}^{N, {{\operatorname{an}}}}$ be the morphism induced by $p \mapsto \left(s_0(p): \cdots: s_N(p)\right)$; we define $\varphi : X^{{\operatorname{an}}}\to {{\mathbb{T}}}{{\mathbb{P}}}^N$ to be ${\operatorname{trop}}\circ \varphi^\prime$, where ${\operatorname{trop}}$ is the map in (\[eqn:trop:map\]). We write $$\label{eqn:map:tropical}
\varphi: X^{{\operatorname{an}}}\longrightarrow {{\mathbb{T}}}{{\mathbb{P}}}^N, \quad p = (p, |\cdot|) \mapsto \left(-\log|s_0(p)|: \cdots: -\log|s_N(p)|\right).$$
Let $E$ be the $N$-dimensional Euclidean space embedded in ${{\mathbb{T}}}{{\mathbb{R}}}^N$ as in the previous subsection. Note that $\varphi ( X^{{{\operatorname{an}}}} \setminus X (K) )
\subset \varphi (X^{{{\operatorname{an}}}} ) \cap E$. Indeed, since each $s_i$ has zero only at points in $X (K)$, $- \log |s_i (p)| \neq + \infty$ for any $p \in X^{{\operatorname{an}}}\setminus X (K)$, and hence $\varphi ( X^{{\operatorname{an}}}\setminus X (K) ) \subset E$.
We describe ${\left.{\varphi}\right\vert_{{X^{{\operatorname{an}}}\setminus X (K) }}}$ in terms of the coordinates given by the homeomorphism $\phi_i : {{\mathbb{R}}}^N \to E$ for $i = 0, \ldots, N$. We explain only the case $i = 0$, since the other cases are the same. We identify $E = {{\mathbb{R}}}^N$ via $\phi_0$. Then $$\label{eq:trop:affine}
{\left.{\varphi}\right\vert_{{X^{{\operatorname{an}}}\setminus X (K) }}} =
\left(
-
\log \left| \frac{s_1}{s_0}
\right|
,
\ldots
,
- \log \left| \frac{s_N}{s_0}
\right|
\right)
.$$
Let $\Gamma = S({{\mathscr{X}}}; \sigma_1, \ldots, \sigma_r)$ be a skeleton (possibly with ends) and write $\Gamma = \bigcup_{q \in {\operatorname{Sing}}( \mathscr{X}_s)} \Delta_q \cup \bigcup_{\sigma_i} \Delta(\sigma_i)$ be the decomposition of $\Gamma$ as in . Recall that $\Gamma$ is a metric space; $\Delta_q$ is canonically identified with $[0 , \lambda_q]$, where $\lambda_q$ is the multiplicity at $q$, and $\Delta (\sigma_i)$ is canonically identified with ${{\mathbb{R}}}_{\geq 0}$. By the definition of $\Gamma$, we have $\Gamma \subset X^{{\operatorname{an}}}\setminus X (K)$. Since $\varphi ( X^{{{\operatorname{an}}}} \setminus X (K) )
\subset \varphi (X^{{{\operatorname{an}}}} ) \cap E$, we have a map $\Gamma \to \varphi (X^{{{\operatorname{an}}}} ) \cap E$ between metric spaces by restriction. Thus the following definition makes sense.
Let $X$, $L$, and $\Gamma$ be as above. Let $s_0, s_1, \ldots, s_N$ be nonzero global sections of $L$, and let $\varphi: X^{{\operatorname{an}}}\longrightarrow {{\mathbb{T}}}{{\mathbb{P}}}^N$ denote the associated morphism . We call $\varphi$ a [*unimodular tropicalization*]{} of $\Gamma$ if there exists a finite subset $V \subset \Gamma_\Lambda$ such that for the closure $\Delta$ of any connected component of $\Gamma \setminus V$, the restriction map ${\left.{\varphi}\right\vert_{{\Delta}}}: \Delta\to\varphi(\Delta)$ is an isometry.
Namely, a unimodular tropicalization is a piecewise isometry.
\[remark:unimodular\] Let $X$, $\Gamma$, and $L$ be as above. Let $s_0 , \ldots , s_M , s_{M+1} , \ldots , s_N$ be global sections of $L$ ($N \geq M+1$). Let $\varphi_1 : X^{{{\operatorname{an}}}} \to {{\mathbb{T}}}{{\mathbb{P}}}^M$ be the associated morphism to $s_0 , \ldots , s_M$, and let $\varphi_2 : X^{{{\operatorname{an}}}} \to {{\mathbb{T}}}{{\mathbb{P}}}^N$ be the associated morphism to $s_0 , \ldots , s_N$. Let $e$ be a connected subspace of $\Gamma$. Suppose that ${\left.{\varphi_1}\right\vert_{{e}}}$ is an isometry. Then ${\left.{\varphi_2}\right\vert_{{e}}}$ is also an isometry. Indeed, with the description on (\[eq:trop:affine\]), we have ${\left.{\varphi_1}\right\vert_{{e}}} =
\left(
-
\log \left| \frac{s_1}{s_0}
\right|
,
\ldots
,
- \log \left| \frac{s_M}{s_0}
\right|
\right)$ and ${\left.{\varphi_2}\right\vert_{{e }}} =
\left(
{\left.{\varphi_1}\right\vert_{{e}}}
,
-
\log \left| \frac{s_{M+1}}{s_0}
\right|
,
\ldots
,
- \log \left| \frac{s_N}{s_0}
\right|
\right)$. Since the metric on the tropicalization is given by the lattice length, one then sees that if ${\left.{\varphi_1}\right\vert_{{e}}}$ is an isometry, then so is ${\left.{\varphi_2}\right\vert_{{e}}}$.
Let $X$, $L$, and $\Gamma$ be as above. Let $s_0, s_1, \ldots, s_N$ be nonzero global sections of $L$ and let $\varphi: X^{{\operatorname{an}}}\longrightarrow {{\mathbb{T}}}{{\mathbb{P}}}^N$ denote the associated morphism . We call $\varphi$ a [*faithful tropicalization*]{} of $\Gamma$ if it is an injective unimodular tropicalization; in other words, the restriction of $\varphi$ to $\Gamma$ is a homeomorphism of $\Gamma$ onto its image preserving the metric.
In relation to faithful tropicalizations, we recall the notion of $\Lambda$-rational polyhedral complexes and their integral structures, and remark that faithful tropicalizations are exactly homeomorphisms preserving the integral structures.
Here we describe $\Lambda$-rational polyhedral complexes of dimension at most one. A $\Lambda$-rational polyhedron $\sigma$ in ${{\mathbb{R}}}$ of dimension at most one is a subset defined by $\sigma = \{x \in {{\mathbb{R}}}\mid A x \leq b\}$ for some $A \in M_{r, 1}({{\mathbb{Z}}})$ and $b \in \Lambda^r$. A $\Lambda$-rational polyhedral complex $\Sigma$ of dimension at most one on a Hausdorff topological space $Y$ is a finite collection of closed topological spaces $\Delta \subset Y$ such that $\Delta$ is homeomorphic to a $\Lambda$-rational polyhedron $\sigma$ in ${{\mathbb{R}}}$ of dimension at most one with the following properties: Any face of $\Delta \in \Sigma$ belongs to $\Sigma$; if $\Delta$ and $\Delta'$ are in $\Sigma$, then $\Delta \cap \Delta'$ (if non-empty) are in $\Sigma$. Such a polyhedral complex $\Sigma$ is said to be of pure dimension one if $|\Sigma| := \{y \in Y \mid \text{$y \in \Delta$ for some $\Delta \in \Sigma$} \}$ is connected and not a singleton. A [*refinement*]{} of $\Sigma$ is a $\Lambda$-rational polyhedral complex $\widetilde{\Sigma}$ with $|\widetilde{\Sigma}| = |\Sigma|$ such that for any $\widetilde{\Delta} \in \widetilde{\Sigma}$, there exists $\Delta \in \Sigma$ with $\widetilde{\Delta} \subset \Delta$.
Let $\Sigma$ be a $\Lambda$-rational polyhedral complex of pure dimension one on $Y$. A member $\Delta \in \Sigma$ is said to be zero-dimensional (resp. one-dimensional) if the corresponding $\Lambda$-rational polyhedron $\sigma$ in ${{\mathbb{R}}}$ is zero-dimensional (resp. one-dimensional). Let $|\Sigma|_0$ be the union of all $\Delta \in \Sigma$ such that $\Delta$ is zero-dimensional, and let $|\Sigma|_1$ be the union of all $\Delta \in \Sigma$ such that $\Delta$ is one-dimensional. Then $|\Sigma|_0$ (resp. $|\Sigma|_1$) is naturally a manifold of dimension $0$ (resp. $1$) with an integral structure. Namely, for $n = 0$, this means nothing but that each member of $|\Sigma|_0$ is identified with a point in $\Lambda \subset {{\mathbb{R}}}$. For $n=1$, this means that there exist an open covering $\{U_i\}$ of $|\Sigma|_1$ and homeomorphisms $\phi_i: U_i \to V_i \subset {{\mathbb{R}}}$ such that for each $i$ and $j$, the transition map ${\left.{\phi_i \circ \phi_j^{-1}}\right\vert_{{\phi_j(U_i \cap U_j)}}}: \phi_j(U_i \cap U_j) \to
\phi_i(U_i \cap U_j)$ is the restriction of the map $x \mapsto c x + d$, where $c = \pm1 $ and $d \in \Lambda$.
Let $\Sigma$ (resp. $\Sigma'$) be a $\Lambda$-rational polyhedral complex of pure dimension one on $Y$ (resp. $Y'$). Let $f:Y \to Y'$ be a continuous map. We say that the restriction of $f$ to $|\Sigma|$ is a homeomorphism onto $|\Sigma'|$ preserving the integral structures if there exist refinements $\widetilde{\Sigma}$ of $\Sigma$ and $\widetilde{\Sigma'}$ of $\Sigma'$ with the following properties: $|\widetilde{\Sigma}|_0$ and $|\widetilde{\Sigma'}|_0$ are bijective under $f$; there exist open coverings $\{U_i\}$ of $|\widetilde{\Sigma}|_1$, $\{U'_i\}$ of $|\widetilde{\Sigma'}|_1$, and homeomorphisms $\phi_i: U_i \to V_i \subset {{\mathbb{R}}}$ and $\phi'_i: U'_i \to V'_i \subset {{\mathbb{R}}}$ that give the integral structures of $|\widetilde{\Sigma}|_1$ and $|\widetilde{\Sigma'}|_1$ such that $\phi'_i \circ f \circ\phi_i^{-1}: V_i \to V'_i$ is a homeomorphism of the form $x \mapsto c x + d$, where $c = \pm 1$ and $d \in \Lambda$.
Let $X$, $L$, and $\Gamma$ be as above. Let $s_0, s_1, \ldots, s_N$ be nonzero global sections of $L$ and let $\varphi: X^{{\operatorname{an}}}\longrightarrow {{\mathbb{T}}}{{\mathbb{P}}}^N$ denote the associated morphism . Then each of $\Gamma$ and $\varphi(\Gamma)$ is equipped with a natural structure of a $\Lambda$-rational polyhedral complex of dimension at most one, and $\varphi$ is a faithful tropicalization of $\Gamma$ in the above definition if and only if ${\left.{\varphi}\right\vert_{{\Gamma}}}: \Gamma \to \varphi(\Gamma)$ is a homeomorphism of $\Gamma$ onto its image preserving the integral structures. See [@MZ §3], [@BPR2], and [@GRW §§2.2–2.3] for more details.
Good models {#sec:models}
===========
Let $X$ be a connected smooth projective curve over $K$ of genus $g$, and let $L$ be a line bundle over $X$. In the proof of the main Theorem \[thm:main:unimodular:faithful\], we need to construct many suitable global sections of a line bundle $L$ over $X$ which will give a faithful tropicalization for a skeleton. In our strategy, we will construct those global sections as global sections of $\mathscr{L}$ for some models $( \mathscr{X} , \mathscr{L} )$ of $(X,L)$. Thus we will need to construct various models of $(X,L)$, and to do that, we use the theory of divisors on graphs.
In this section, we firstly define the notion of good models $\mathscr{X}$ of $X$. In subsequent sections, over a good model ${{\mathscr{X}}}$ of $X$, we will construct a line bundle ${{\mathscr{L}}}$ with ${{\mathscr{L}}}\otimes_R K \cong L$ and suitable global sections of ${{\mathscr{L}}}$. We secondly recall some recent development on the divisor theory on $\Lambda$-metric graphs. We thirdly introduce the notion of [*islands*]{} of weighted $\Lambda$-metric graphs and prove a positivity result on divisors on $\Lambda$-metric graphs. In the last subsection of this section, we introduce the machinery that produces various models $( \mathscr{X} , \mathscr{L} )$ which will be crucially used in the subsequent arguments. Further, we use it to construct a model of $(X,L)$ which will be frequently used.
Good models of $X$ {#subsec:def:good:model}
------------------
In this subsection, we define the notion that a model ${{\mathscr{X}}}$ of $X$ is good. Let $\pi : \mathscr{X} \to {{\operatorname{Spec}}}(R)$ be a strictly semistable model of $X$. Recall from §\[subsec:semistable:models:pairs\] that a $(-2)$-chain is a connected curve in $\mathscr{X}_s$ consisting of $(-2)$-curves, and it is a [*maximal*]{} $(-2)$-chain if it is maximal with respect to the inclusion among $(-2)$-chains. Recall also from §\[subsec:semistable:models:pairs\] that a $(-2)$-chain $E$ is of connected type if $\mathscr{X}_s - E$ is connected, and is of disconnected type otherwise.
\[def:symmetricmultiplicity\] Let $E$ be a maximal $(-2)$-chain of connected type in $\mathscr{X}_s$. We say that $E$ *has symmetric multiplicities* if it satisfies the following conditions.
1. The number $\# {\operatorname{\operatorname{Irr}}}(E)$ is odd and at least $3$. We write $\# {\operatorname{\operatorname{Irr}}}(E) = 2 \ell - 1$ for $\ell \geq 2$.
2. Put $D := \mathscr{X}_s - E$. Let $E_1 , \ldots , E_{2 \ell - 1}$ be the irreducible components of $E$ such that $E_{i} \cap E_{i+1} \neq \emptyset$ for $i = 1 , \ldots , 2 \ell -2$. Let $p_0 , \ldots , p_{2 \ell -1}$ be the nodes of $\mathscr{X}_s$ such that $\{ p_i \} = E_i \cap E_{i+1}$ for $i = 1 , \ldots , 2 \ell -2$, $\{ p_0 \} = D \cap E_1$, and $\{ p_{2 \ell-1} \} = D \cap E_{2 \ell -1}$. For $i = 0 , \ldots , 2 \ell - 1$, let $\lambda_i$ be the multiplicity at $p_i$ (cf. Definition \[def:multiplicityatanodes\]). Then $\lambda_j = \lambda_{2\ell - 1 -j}$ for $j = 0 , \ldots , \ell -1$.
3. We have $\lambda_0 = \lambda_{\ell-1}$. (Thus $\lambda_0 = \lambda_{\ell-1}
= \lambda_{\ell} = \lambda_{2\ell-1}$.)
From here on to the end of this subsection, we assume that $g \geq 2$. Let $\mathscr{X}$ be a strictly semistable model of $X$. Then two distinct maximal $(-2)$-chains in $\mathscr{X}_s$ are disjoint.
\[def:goodmodel:new\] Let $X$ be a connected smooth projective curve of genus $g \geq 2$, and let $\mathscr{X}$ be a model of $X$. We say that $\mathscr{X}$ is a *good model* if it satisfies the following conditions.
1. The model $\mathscr{X}$ is Deligne–Mumford strictly semistable.
2. For any $p \in {\operatorname{Sing}}( \mathscr{X}_s)$, there exists a $(-2)$-chain $E$ such that $p \in E$.
3. Any maximal $(-2)$-chain $E$ in $\mathscr{X}_s$ of connected type has symmetric multiplicities.
We make clear the relationship between a good model $\mathscr{X}$ and the canonical finite graph structure of the minimal skeleton $\Gamma_{\min}$ with the set $V({{\mathscr{X}}}^{{\operatorname{st}}})$ of vertices and the set $E({{\mathscr{X}}}^{{\operatorname{st}}})$ of edges. Recall that $V({{\mathscr{X}}}^{{\operatorname{st}}}) =
\{ [C] \mid C \in {\operatorname{\operatorname{Irr}}}(\mathscr{X}^{{\operatorname{st}}}_s) \}$ by definition, where $[C]$ is the Shilov point associated to $C$, and an element of $E({{\mathscr{X}}}^{{\operatorname{st}}})$ is characterized by the closure of a connected component of $\Gamma_{\min} \setminus V({{\mathscr{X}}}^{{\operatorname{st}}})$. We say that $e \in E({{\mathscr{X}}}^{{\operatorname{st}}})$ is [*of connected type*]{} if $\Gamma_{\min} \setminus
\mathrm{relin}(e)$ is connected, and $e \in E({{\mathscr{X}}}^{{\operatorname{st}}})$ is [*of disconnected type*]{} otherwise.
\[lemma:make:clear\] Let $\mathscr{X} \to {{\operatorname{Spec}}}(R)$ be a good model of $X$, and let $\mu: \mathscr{X} \to {{\mathscr{X}}}^{{\operatorname{st}}}$ be the morphism extending the identity morphism on $X$.
1. There exist natural one-to-one correspondences between the following three sets: $${\operatorname{Sing}}({{\mathscr{X}}}_s^{{\operatorname{st}}}) \longleftrightarrow E({{\mathscr{X}}}^{{\operatorname{st}}})
\longleftrightarrow \{\text{maximal $(-2)$-chain in ${{\mathscr{X}}}_s$}\}.$$
2. Let $e \in E({{\mathscr{X}}}^{{\operatorname{st}}})$ correspond to a maximal $(-2)$-chain $E$ in ${{\mathscr{X}}}_s$ under the natural correspondence in . Then $e$ is of connected type resp. of disconnected type if and only if $E$ is of connected type resp. of disconnected type.
[[*Proof.*]{}]{}(1) Since ${{\mathscr{X}}}$ is Deligne–Mumford semistable, the irreducible components of ${{\mathscr{X}}}_s$ contracted by $\mu$ are the $(-2)$-curves. For $p \in {\operatorname{Sing}}({{\mathscr{X}}}_s^{{\operatorname{st}}})$, we set $E := \mu^{-1}(p)$. Since ${{\mathscr{X}}}$ satisfies condition (ii) of Definition \[def:goodmodel:new\], $E$ is not a singleton, and thus is a maximal $(-2)$-chain in ${{\mathscr{X}}}_s$. This gives a natural one-to-one correspondence between the first and the third sets. For a maximal $(-2)$-chain $E$ in ${{\mathscr{X}}}_s$, we set $\Delta_E:= \bigcup_{q \in E \cap {\operatorname{Sing}}( \mathscr{X}_s)} \Delta_q$ as in Definition \[def:canonical:1:simplex:curve\]. Then $\Delta_E \in E({{\mathscr{X}}}^{{\operatorname{st}}})$, and this gives a natural one-to-one correspondence between the third and the second sets. Finally, we remark that $\Delta_p$ at the end of §\[subsec:properties:skeleta\] is exactly $\Delta_E$. We have thus natural correspondences: $$\label{eqn:correspondences}
p = \mu(E) \quad\longleftrightarrow\quad \Delta_p = \Delta_E
\quad\longleftrightarrow\quad E:= \mu^{-1}(p).$$
\(2) The assertion follows from the definitions of being of connected type and of disconnected type.
500
Let $E$ be a maximal $(-2)$-chain in $\mathscr{X}_s$. Then $\Delta_{E}$ is a circle or a closed line segment in $S ( \mathscr{X}_s)$. The irreducible components $C \in {\operatorname{\operatorname{Irr}}}({{\mathscr{X}}}_s)$ having the properties that $C \notin {\operatorname{\operatorname{Irr}}}(E)$ and $C \cap E \neq \emptyset$ give points in $\Delta_E$, and those points are in $V ( \mathscr{X}^{{\operatorname{st}}})$.
We finish this subsection by introducing the notation, which will be frequently used. Let $\mathscr{X}$ be a good model of $X$ and let $\mu : \mathscr{X} \to \mathscr{X}^{{\operatorname{st}}}$ be the morphism extending the identity morphism on $X$. For each $q \in {\operatorname{Sing}}( \mathscr{X}_s )$, let $\Delta_q$ denote the corresponding $1$-simplex in the minimal skeleton $\Gamma_{\min}$. Remark that if $E$ is the maximal $(-2)$-chain with $q \in E$, then $\Delta_q \subset \Delta_E = \Delta_{\mu (q)}$ with the notation in (\[eqn:correspondences\]). For a given compact subset $\Gamma_1 \subset \Gamma_{\min}$, we set $$\begin{aligned}
\label{align:nodes:subset}
{\operatorname{Sing}}(\mathscr{X}_s )_{\subset \Gamma_1} :=
\left\{
q \in {\operatorname{Sing}}(\mathscr{X}_s )
\mid
\Delta_q \subset \Gamma_1
\right\}
.\end{aligned}$$ Remark that for each $e \in E ( \mathscr{X}^{{\operatorname{st}}})$, $$\begin{aligned}
{\operatorname{Sing}}(\mathscr{X}_s )_{\subset e} =
\left\{
q \in {\operatorname{Sing}}(\mathscr{X}_s )
\mid
\Delta_{\mu (q)} = e
\right\}
=
{\operatorname{Sing}}(\mathscr{X}_s ) \cap E
,\end{aligned}$$ where $E$ is the maximal $(-2)$-chain with $\Delta_E = e$.
Theory of divisors on $\Lambda$-metric graphs {#subsec:theory:divisors}
---------------------------------------------
As in “Notation and conventions,” let $\Lambda \subset {{\mathbb{R}}}$ denote the value group of $K$. We review the theory of divisors on $\Lambda$-metric graphs, which we use later. Our basic references are Amini, Baker, Brugallé, and Rabinoff [@ABBR], Baker and Norine [@BN], Amini and Caporaso [@AC], and Baker and Rabinoff [@BR]. See also [@KY0; @KY1].
Let $\Gamma$ be a $\Lambda$-metric graph (see “Notation and conventions”). Let ${\operatorname{Div}}(\Gamma)$ be the free abelian group generated by the points in $\Gamma$. An element of ${\operatorname{Div}}(\Gamma)$ is called a [*divisor*]{} on $\Gamma$. Any divisor $D \in {\operatorname{Div}}(\Gamma)$ is uniquely written as a finite sum $D = \sum_{x \in \Gamma} n_x [x]$ for $n_x \in {{\mathbb{Z}}}$. We set $D(x) := n_x$. Remark that $D = \sum_{x \in \Gamma} D(x) [x]$. The [*support*]{} ${\operatorname{Supp}}(D)$ of $D$ is the set of points with $D(x) \neq 0$. The [*degree*]{} of $D$ is defined to be $\deg(D) =
\sum_{x \in \Gamma} D(x)$. We say that a divisor $D$ is [*effective*]{} and write $D \geq 0$ if $D(x) \geq 0$ for any $x \in \Gamma$. For a subset $S$ of $\Gamma$, we set ${\left.{D}\right\vert_{{S}}} := \sum_{x \in S} D(x)[x]$ and $$\label{eqn:deg:extended}
\deg\left({\left.{D}\right\vert_{{S}}}\right)
= \sum_{x \in S} D(x).$$
We recall the notion of linear equivalence of divisors on a metric graph. For a point $x \in \Gamma$, ${\operatorname{val}}(x)$ denotes the valency at $x$, i.e., the number of branches emanating from $x$. A [*rational function*]{} on $\Gamma$ is a continuous function $f: \Gamma \to {{\mathbb{R}}}$ such that there exists a finite subset $\{x_1, \ldots, x_n\}$ of $\Gamma$ containing $\{x \in \Gamma \mid {\operatorname{val}}(x) \neq 2\}$ such that $f$ is an affine map with integer slopes on $\Gamma \setminus \{x_1, \ldots, x_n\}$. We denote by ${\operatorname{Rat}}(\Gamma)$ the set of rational functions on $\Gamma$. For $f \in {\operatorname{Rat}}(\Gamma)$ and $x \in \Gamma$, we define ${\operatorname{ord}}_x(f)$ to be the sum of outgoing slopes of $f$ at $x$. Then ${\operatorname{div}}(f) := \sum_{x \in \Gamma} {\operatorname{ord}}_x(f) [x]$ is a divisor on $\Gamma$, which is called the *principal divisor* of $f$. We set $ {\operatorname{Prin}}(\Gamma) := \{{\operatorname{div}}(f) \mid f \in {\operatorname{Rat}}(\Gamma)\}$, the set of [*principal divisors*]{} on $\Gamma$. For $D_1 , D_2 \in {\operatorname{Div}}(\Gamma)$, we say that $D_1$ is *linearly equivalent to $D_2$*, denoted by $D_1 \sim D_2$, if $D_1 - D_2 \in {\operatorname{Prin}}(\Gamma)$. Linear equivalence is an equivalence relation.
In studying divisors on graphs, it is often convenient to consider reduced divisors, which we are recalling now. (The notion of reduced divisors was considered in [@BN] to prove the Riemann–Roch formula for a finite graph.) For any closed subset $A$ of $\Gamma$ and $v \in \Gamma$, the [*out-degree*]{} of $v$ from $A$, denoted by $\mathrm{outdeg}^\Gamma_A(v)$, is defined to be the maximum number of internally disjoint segments of $\Gamma\setminus A$ with an open end $v$. For $D \in {\operatorname{Div}}(\Gamma)$, a point $v \in \partial A$ is [*saturated*]{} for $D$ with respect to $A$ if $D(v) \geq \mathrm{outdeg}^\Gamma_A(v)$, and [*non-saturated*]{} otherwise. We fix a point $v_0\in\Gamma$. A divisor $E \in {\operatorname{Div}}(\Gamma)$ is said to be [*$v_0$-reduced*]{} if $E(x) \geq 0$ for any $x \in \Gamma\setminus\{v_0\}$ and every connected compact subset $A$ of $\Gamma\setminus\{v_0\}$ contains a non-saturated point $v \in \partial A$ for $E$ with respect to $A$. It is known that for any $D \in {\operatorname{Div}}( \Gamma )$, there exists a unique $v_0$-reduced $D_{v_0} \in {\operatorname{Div}}( \Gamma )$ that is linearly equivalent to $D$ (cf. [@MZ], [@Luo Theorem 2.3]).
In this paper, we mainly consider divisors supported in $\Gamma_{\Lambda}$. We set ${\operatorname{Div}}_{\Lambda}(\Gamma) :=
\{ D \in {\operatorname{Div}}( \Gamma) \mid {\operatorname{Supp}}(D) \subset \Gamma_{\Lambda} \}$. The elements of ${\operatorname{Div}}_{\Lambda}(\Gamma)$ are called [*$\Lambda$-divisors*]{} on $\Gamma$. For a $\Lambda$-divisor $D$, the complete linear system $|D|$ is defined by $$|D| := \{D^\prime \in {\operatorname{Div}}_{\Lambda}(\Gamma) \mid D^\prime \geq 0, D^\prime - D \in {\operatorname{Prin}}(\Gamma)\}.$$
\[prop:reduced:Lambda-div\] For any $D \in {\operatorname{Div}}_{\Lambda} ( \Gamma )$ and for any $v_0 \in \Gamma_\Lambda$, the $v_0$-reduced divisor $D_{v_0}$ that is linearly equivalent to $D$ is a $\Lambda$-divisor. Further, $D_{v_0} \geq 0$ if and only if $|D| \neq \emptyset$.
[[*Proof.*]{}]{}We take any $D \in {\operatorname{Div}}_\Lambda(\Gamma)$ and $v_0 \in \Gamma_\Lambda$. Let $D_{v_0} \in {\operatorname{Div}}( \Gamma )$ be the $v_0$-reduced divisor that is linearly equivalent to $D$. The first assertion $D_{v_0} \in {\operatorname{Div}}_{\Lambda} ( \Gamma )$ follows from, for example, Luo’s algorithm to construct $v_0$-reduced divisors in [@Luo].
For the second assertion, it is obvious that if $D_{v_0} \geq 0$, then $|D| \neq \emptyset$. The other implication follows from [@Luo Corollary 2.18].
500
Let $g(\Gamma)$ denote the first Betti number of $\Gamma$.
\[Riemann’s inequality on a metric graph\] \[prop:RIforMG\] If $\deg(D) \geq g(\Gamma)$, then $|D| \neq \emptyset$.
[[*Proof.*]{}]{}Suppose that $\deg(D) \geq g(\Gamma)$. Take any $v_0 \in \Gamma_{\Lambda}$, and let $D_{v_0}$ be the $v_0$-reduced divisor with $D_{v_0} \sim D$. Since $\deg(D) \geq g(\Gamma)$, the Riemann–Roch formula on metric graphs (see [@GK Proposition 3.1], [@MZ Theorem 7.4], or [@HKN Theorem 1.2]) shows that there exists an effective divisor that is linearly equivalent to $D$. By [@Luo Corollary 2.18], it follows that $D_{v_0} \geq 0$. Further, by Proposition \[prop:reduced:Lambda-div\], we have $D_{v_0} \in {\operatorname{Div}}_{\Lambda} ( \Gamma )$. Thus $D_{v_0} \in |D|$, which shows the proposition.
500
\[rmk:on:reduced:divisors\] Let $D \in {\operatorname{Div}}_\Lambda(\Gamma)$ and $v_0 \in \Gamma_\Lambda$. Let $D_{v_0}$ be the $v_0$-reduced divisor that is linearly equivalent to $D$. Then $D_{v_0}(v_0) \geq \deg(D) - g(\Gamma)$. Indeed, we set $E := D - (\deg(D) - g(\Gamma)[v_0]$. Since $D$ is a $v_0$-reduced divisor, it follows from the definition of $v_0$-reduced divisor that $E$ is also a $v_0$-reduced divisor. Since $\deg(E) \geq g(\Gamma)$, Propositions \[prop:reduced:Lambda-div\] and \[prop:RIforMG\] show that $E \geq 0$. Thus $D_{v_0}(v_0) \geq \deg(D) - g(\Gamma)$.
Weighted $\Lambda$-metric graphs {#subsec:weighted:Lambda:metric:graph}
--------------------------------
In this subsection, we show in Proposition \[prop:good:eff:div\] the existence of a certain effective divisor on a $\Lambda$-metric graph, which will be used to construct models $({{\mathscr{X}}}, {{\mathscr{L}}})$ of $(X, L)$. Here we use the theory of [*weighted*]{} $\Lambda$-metric graphs. For a general account of the theory of weighted metric graphs, we refer to Amini and Caporaso [@AC].
A [*weighted*]{} $\Lambda$-metric graph $\bar{\Gamma} = (\Gamma, \omega)$ is a pair of a $\Lambda$-metric graph $\Gamma$ and a function $\omega: \Gamma \to {{\mathbb{Z}}}_{\geq 0}$ such that the set $\{x \in \Gamma \mid \omega(x) \neq 0\}$ is finite and contained in $\Gamma_\Lambda$. The function $\omega$ is called a weight function. The [*genus*]{} of $\bar{\Gamma}$ is defined to be $g(\bar{\Gamma}) := g(\Gamma) + \sum_{x \in \Gamma_\Lambda} \omega(x)$.
Let $\bar{\Gamma} = (\Gamma, \omega)$ be a weighted $\Lambda$-metric graph. Since $g(\bar{\Gamma}) \geq g(\Gamma)$, the following follows from Proposition \[prop:RIforMG\].
\[prop:riemann:inequality\] For $D \in {\operatorname{Div}}_\Lambda(\Gamma )$, if $\deg(D) \geq g(\bar{\Gamma})$, then $|D| \neq \emptyset$.
Let $\bar{\Gamma} = (\Gamma, \omega)$ be a weighted $\Lambda$-metric graph. Assume now that $g(\bar{\Gamma}) \geq 2$ and that there does not exist $x \in \Gamma_\Lambda$ with ${\operatorname{val}}(x) = 1$ and $\omega(x) = 0$. Then $\Gamma$ has a finite graph structure with the set of vertices $V(\bar{\Gamma}):= \{x \in \Gamma_\Lambda \mid {\operatorname{val}}(x) \neq 2\}
\cup \{x \in \Gamma_\Lambda \mid \omega(x) \neq 0\}$. Let $E(\bar{\Gamma})$ denote the set of edges.
An edge $e \in E(\bar{\Gamma})$ is called an edge of connected type if $\Gamma \setminus {\mathrm{relin}(e)}$ is disconnected; $e$ is called an edge of disconnected type otherwise. (An edge of disconnected type is also called a bridge, but in this paper, we do not use the terminology “bridge.”) Let $\{e_1, \ldots, e_r\}$ be the set of the edges of disconnected type, and we decompose $\Gamma \setminus \left({\mathrm{relin}(e_1)}\cup\cdots\cup {\mathrm{relin}(e_r)}\right)$ as $$\Gamma \setminus \left({\mathrm{relin}(e_1)}\cup\cdots\cup {\mathrm{relin}(e_r)}\right)
= \Gamma_1 \amalg \cdots \amalg \Gamma_{r+1},$$ where each $\Gamma_i$ is a connected component and thus is a $\Lambda$-metric graph. We set $\omega_i := {\left.{\omega}\right\vert_{{\Gamma_i}}}: \Gamma_i \to {{\mathbb{Z}}}_{\geq 0}$. Then $\bar{\Gamma}_i := (\Gamma_i, \omega_i)$ is a weighted $\Lambda$-metric graph, and $g(\bar{\Gamma}) = \sum_{i=1}^{r+1} g(\bar{\Gamma}_i)$. Since we assume that there does not exist $x \in \Gamma_\Lambda$ such that ${\operatorname{val}}(x) = 1$ and $\omega(x) = 0$, we have $g(\bar{\Gamma}_i) \geq 1$ for each $i$.
\[def:island\] We call each $\Gamma_i$ an [*island*]{} of a weighted $\Lambda$-metric graph $\bar{\Gamma} = (\Gamma, \omega)$. By slight abuse of terminology, we also call $\Gamma_i$ an island of $\Gamma$ if there is no confusion for the choice of a weight function $\omega$.
The following proposition shows the existence of an effective divisor on $\Gamma$ whose restriction to each island is not trivial.
\[prop:good:eff:div\] Let $\bar{\Gamma} = (\Gamma, \omega)$ be a weighted $\Lambda$-metric graph. Assume that $g(\bar{\Gamma}) \geq 2$ and that there does not exist $x \in \Gamma_\Lambda$ such that ${\operatorname{val}}(x) = 1$ and $\omega(x) = 0$. Let $D \in {\operatorname{Div}}_{\Lambda}(\Gamma)$. If $\deg(D) \geq g(\bar{\Gamma})$, then there exists $E \in {\operatorname{Div}}_{\Lambda}(\Gamma)$ with the following properties
1. $E \in |D|$, i.e., $E$ is effective and linearly equivalent to $D$.
2. For any island $\Gamma_i$ of $\Gamma$, we have $\deg\left({\left.{E}\right\vert_{{\Gamma_i}}}\right) \geq 1$.
3. For any edge $e \in E(\bar{\Gamma})$ of disconnected type, we have $\deg\left({\left.{E}\right\vert_{{{\mathrm{relin}(e)}}}}\right) =0$. Further, for any edge $e \in E(\bar{\Gamma})$ that is of connected type, we have $\deg\left({\left.{E}\right\vert_{{{\mathrm{relin}(e)}}}}\right) \leq 1$.
[[*Proof.*]{}]{}Suppose that $\deg(D) \geq g(\bar{\Gamma})$. We construct such $E \in {\operatorname{Div}}_\Lambda(\Gamma)$ step by step. Let $\{e_1, \ldots, e_r\} \subset E(\bar{\Gamma})$ be the set of edges of disconnected type.
By Riemann’s inequality (Proposition \[prop:riemann:inequality\]), one has $|D| \neq \emptyset$, and thus there exists an effective divisor $E_1$ with $E_1 \sim D$. The divisor $E_1$ satisfies condition (i).
In this step, we replace $E_1$ with another $E_2 \in |D|$ so that $E_2$ satisfies conditions (i) and (ii) and the first condition in (iii). We are going to show a stronger statement that for any effective divisor $E_1 \in {\operatorname{Div}}_\Lambda(\Gamma)$ with $\deg(E_1) \geq g(\bar{\Gamma})$, there exists an effective divisor $E_2 \in {\operatorname{Div}}_\Lambda(\Gamma)$ such that $E_1 \sim E_2$ and $\deg\left({\left.{E_2}\right\vert_{{\Gamma_i}}}\right) \geq g(\bar{\Gamma}_i)$ for any island $\Gamma_i$ of $\Gamma$. Since any two points on an edge of disconnected type are linearly equivalent to each other, we may assume that $\deg\left({\left.{E_1}\right\vert_{{{\mathrm{relin}(e_i)}}}}\right) = 0$ for any $1 \leq i \leq r$.
Suppose that $r = 0$, i.e., $\Gamma$ has no edge of disconnected type. In this case, $\Gamma$ is the only island, and we have $\deg\left({\left.{E_1}\right\vert_{{\Gamma}}}\right)
= \deg(E_1) \geq g(\bar{\Gamma})$. Thus it suffices to take $E_2 := E_1$.
Suppose that $r \geq 1$. Let $v^\prime$ and $ v^{\prime\prime}$ be the end vertices of $e_1$. Let $\Gamma^\prime$ and $\Gamma^{\prime\prime}$ be the connected components of $\Gamma\setminus{\mathrm{relin}(e_1)}$ with $v^\prime \in \Gamma^\prime$ and $v^{\prime\prime} \in \Gamma^{\prime\prime}$. We set $\bar{\Gamma}^\prime = (\Gamma^\prime, \omega^\prime)$ and $\bar{\Gamma}^{\prime\prime} = (\Gamma^{\prime\prime}, \omega^{\prime\prime})$, where $\omega^\prime := {\left.{\omega}\right\vert_{{\Gamma^\prime}}}$ and $\omega^{\prime\prime} := {\left.{\omega}\right\vert_{{\Gamma^{\prime\prime}}}}$. Since $$\deg({\left.{E_1}\right\vert_{{\Gamma^\prime}}}) + \deg({\left.{E_1}\right\vert_{{\Gamma^{\prime\prime}}}}) = \deg(E_1)
\geq g(\bar{\Gamma}) = g(\bar{\Gamma}^\prime) + g(\bar{\Gamma}^{\prime\prime}),$$ we may assume without loss of generality that $\deg({\left.{E_1}\right\vert_{{\Gamma^{\prime\prime}}}}) \geq g(\bar{\Gamma}^{\prime\prime})$. On $\Gamma'$, set $$E^{\prime}_1 := {\left.{E_1}\right\vert_{{\Gamma^{\prime}}}}
+ \left(\deg({\left.{E_1}\right\vert_{{\Gamma^{\prime\prime}}}}) - g(\bar{\Gamma}^{\prime\prime})\right) [v^{\prime}]
\in {\operatorname{Div}}_\Lambda(\Gamma^{\prime})
.$$ Then $E^{\prime}_1 \geq 0$. Further, $
\deg(E^{\prime}_1) = \deg(E_1) - g(\bar{\Gamma}^{\prime\prime})
\geq g(\bar{\Gamma}) - g(\bar{\Gamma}^{\prime\prime}) = g(\bar{\Gamma}^{\prime})
$. On $\Gamma''$, since $$\deg
\left( {\left.{E_1}\right\vert_{{\Gamma^{\prime\prime}}}} -
(\deg({\left.{E_1}\right\vert_{{\Gamma^{\prime\prime}}}}) - g(\bar{\Gamma}^{\prime\prime}))[v''] \right) =
g ( \bar{\Gamma}^{\prime\prime}),$$ Riemann’s inequality (cf. Proposition \[prop:riemann:inequality\]) gives an effective divisor $E_1^{\prime\prime} \in {\operatorname{Div}}_{\Lambda} ( \Gamma^{\prime\prime})$ such that $$E_1^{\prime\prime}
\sim {\left.{E_1}\right\vert_{{\Gamma^{\prime\prime}}}}
- (\deg({\left.{E_1}\right\vert_{{\Gamma^{\prime\prime}}}}) - g(\bar{\Gamma}^{\prime\prime}))[v''].$$ Since $\Gamma^\prime$ and $\Gamma^{\prime\prime}$ are subgraphs of $\Gamma$, we naturally regard $E^\prime_1$ and $E^{\prime\prime}_1$ as $\Lambda$-divisors on $\Gamma$. Since $v'$ and $v''$ are the end vertices of an edge of disconnected type, we have $[v^{\prime}] \sim [v^{\prime\prime}] $, and hence $E^{\prime}_1 + E^{\prime\prime}_1 \in |E_1| = |D|$. Further, $
\deg \left( {\left.{(E^{\prime}_1 + E^{\prime\prime}_1)}\right\vert_{{\Gamma'}}}\right)
=
\deg(E^{\prime}_1) \geq g(\bar{\Gamma}^{\prime})
$ and $
\deg \left( {\left.{(E^{\prime}_1 + E^{\prime\prime}_1)}\right\vert_{{\Gamma''}}}\right)
=
\deg(E^{\prime\prime}_1) = g(\bar{\Gamma}^{\prime\prime})
$.
If $g(\bar{\Gamma}^\prime) =1$ (resp. $g(\bar{\Gamma}^{\prime\prime}) =1$), then $\Gamma^{\prime}$ (resp. $\Gamma^{\prime\prime}$) is an island of $\Gamma$, and we have $\deg\left({\left.{
E^{\prime}_1 + E^{\prime\prime}_1
}\right\vert_{{\Gamma^\prime}}}\right) \geq g(\Gamma^\prime)$ (resp. $\deg\left({\left.{
E^{\prime}_1 + E^{\prime\prime}_1
}\right\vert_{{\Gamma^{\prime\prime}}}}\right) \geq g(\Gamma^{\prime\prime})$). If $g(\bar{\Gamma}^\prime) \geq 2$ (resp. $g(\bar{\Gamma}^\prime) \geq 2$), we repeat this argument to $\bar{\Gamma}^\prime$ and $E^\prime_1$ (resp. $\bar{\Gamma}^{\prime\prime}$ and $E^{\prime\prime}_1$). Then we obtain $E_2 \in {\operatorname{Div}}_\Lambda(\Gamma)$ that satisfies conditions (i) and (ii). Further, we may assume that $\deg\left({\left.{E_2}\right\vert_{{{\mathrm{relin}(e_i)}}}}\right) = 0$ for any $1 \leq i \leq r$.
In the final step, we replace $E_2$ with $E_3$ so that $E_3$ satisfies conditions (i)–(iii). Suppose that $e$ is an edge that is of connected type and that $\deg\left({\left.{E_2}\right\vert_{{{\mathrm{relin}(e)}}}}\right) \geq 2$. Let $\gamma : [0,\ell] \to e$ be a parameterization that restricts to an isometry ${\left.{\gamma}\right\vert_{{(0,\ell)}}} : (0,\ell) \to \mathrm{relin} (e)$. Then there exist $t_1$ and $t_2$ with $0 < t_1 \leq t_2 < \ell$ such that $E_2 - [\gamma(t_1)] -
[\gamma(t_2)] \geq 0$. We set $m = \min\{t_1, \ell- t_2\}$ and $E_2^\prime:= E_2 - [t_1] - [t_2] +
[\gamma(t_1 -m)] + [\gamma(t_2 + m)] \geq 0$. Then we have $\deg\left({\left.{E_2^\prime}\right\vert_{{{\mathrm{relin}(e)}}}}\right) < \deg\left({\left.{E_2}\right\vert_{{{\mathrm{relin}(e)}}}}\right)$ and $\deg\left({\left.{E_2}\right\vert_{{\Gamma_i}}}\right) = \deg\left({\left.{E_2^\prime}\right\vert_{{\Gamma_i}}}\right)$ for any island $\Gamma_i$, and $E_2^\prime$ still satisfies conditions (i)(ii) and the first condition of (iii). Applying this argument for all such $e$ repeatedly, we obtain $E_3 \in {\operatorname{Div}}_\Lambda(\Gamma)$ that satisfies conditions (i)–(iii).
500
Skeleton as a weighted $\Lambda$-metric graph (with a finite graph structure) {#subsec:skeleton:weighted:metric:graph}
-----------------------------------------------------------------------------
Let $X$ be a connected smooth projective curve over $K$ of genus $g \geq 0$. Let $\mathscr{X}$ be a strictly semistable model of $X$. Then the skeleton $\Gamma := S(\mathscr{X})$ is equipped with a natural weight function $\omega$, and thus $(\Gamma, \omega)$ becomes a weighted $\Lambda$-metric graph. Indeed, we recall that we put $V({{\mathscr{X}}}) := \{[C] \mid C \in {\operatorname{\operatorname{Irr}}}({{\mathscr{X}}}_s)\}$, which is the set of Shilov points with respect to $\mathscr{X}$. We define a weight function $\omega: \Gamma \to {{\mathbb{Z}}}_{\geq 0}$ by letting $\omega([C])$ be the geometric genus of $C \in {\operatorname{\operatorname{Irr}}}({{\mathscr{X}}}_s)$ and letting $\omega(x) = 0$ for any $x \not\in V({{\mathscr{X}}})$. Then $\bar{\Gamma} = (\Gamma, \omega)$ is a weighted $\Lambda$-metric graph. One sees that $g (\bar{\Gamma}) = g$. Further, $g ( \Gamma ) = g$ if and only if $\omega$ is trivial.
### Minimal weighted skeleton {#minimal-weighted-skeleton .unnumbered}
Assume that $g \geq 1$. Then we have a unique minimal skeleton $\Gamma_{\min}$ of $X^{{{\operatorname{an}}}}$ (cf. the paragraph after Definition \[def:minimal:skeleton\]). We take any Deligne–Mumford semistable model ${{\mathscr{X}}}$ of $X$. Remark that $\Gamma_{\min} = S ( \mathscr{X} )$, and we define a canonical weight function $\omega$ from $\mathscr{X}$ as above. Note that $\omega$ does not depend on the choice of a Deligne–Mumford semistable model ${{\mathscr{X}}}$ of $X$. Thus we obtain a weighted $\Lambda$-metric graph $$\bar{\Gamma}_{\min} := (\Gamma_{\min} , \omega)$$ for $X$. We call $\bar{\Gamma}_{\min}$ the *minimal weighted skeleton of $X^{{{\operatorname{an}}}}$*. Remark that since $\mathscr{X}$ is Deligne–Mumford semistable, one sees that there does not exist $x \in \Gamma_{\min , \Lambda}$ such that ${\operatorname{val}}(x) = 1$ and $\omega(x) = 0$.
### Minimal weighted skeleton with the canonical finite graph structure {#minimal-weighted-skeleton-with-the-canonical-finite-graph-structure .unnumbered}
Assume that $g \geq 2$. Recall that we have given the minimal skeleton $\Gamma_{\min}$ with the canonical finite graph structure (cf. the second paragraph after Definition \[def:minimal:skeleton\]) arising from the stable model: the set $V({{\mathscr{X}}}^{{\operatorname{st}}})$ of vertices is given by the set of Shilov points with respect to the stable model ${{\mathscr{X}}}^{{\operatorname{st}}}$ of $X$, and the set $E({{\mathscr{X}}}^{{\operatorname{st}}})$ of edges is determined by $V({{\mathscr{X}}}^{{\operatorname{st}}})$. On the other hand, since $\Gamma_{\min}$ has a natural weight function $\omega$, this gives a finite graph structure to $\Gamma_{\min}$ as in §\[subsec:weighted:Lambda:metric:graph\]: the set $V(\bar{\Gamma}_{\min})$ of vertices is given by $$V(\bar{\Gamma}_{\min}):=\{x \in \Gamma_{\min , \Lambda} \mid {\operatorname{val}}(x) \neq 2\}
\cup \{x \in \Gamma_{\min , \Lambda} \mid \omega(x) \neq 0\},$$ and the set $E(\bar{\Gamma}_{\min})$ of edges is determined by $V(\bar{\Gamma}_{\min})$. Then we have $$V({{\mathscr{X}}}^{{\operatorname{st}}}) = V(\bar{\Gamma}_{\min})
\quad\text{and}\quad
E({{\mathscr{X}}}^{{\operatorname{st}}}) = E(\bar{\Gamma}_{\min}).$$ This says that the finite graph structures on $\Gamma_{\min }$ that are determined by two ways are in fact the same, and we have a canonical finite graph structure on $\Gamma_{\min }$. The minimal weighted skeleta $\bar{\Gamma}_{\min}$ for genus $g \geq 2$ with this canonical finite graph structures are important examples on which one can apply Proposition \[prop:good:eff:div\].
\[remark:valence:stable\] Assume that $g \geq 2$. With the notation above, suppose that $g ( \Gamma_{\min} ) = g$. Then $\omega$ is trivial. Since there does not exist $x \in \Gamma_{\min, \Lambda}$ such that ${\operatorname{val}}(x) = 1$ and $\omega(x) = 0$, it follows that $V ( \mathscr{X}^{{\operatorname{st}}} ) =
\{ x \in \Gamma_{\min , \Lambda} \mid {\operatorname{val}}(x) \geq 3 \}$.
Construction of a model of $(X, L)$ {#subsec:starting:point}
-----------------------------------
Let $X$ be a connected smooth projective curve over $K$. Let $\mathscr{X}$ be a strictly semistable model of $X$, and let $\Gamma = S ( \mathscr{X} )$ be the associated skeleton. Let $\tau : X^{{{\operatorname{an}}}} \to \Gamma$ be the retraction map. Since $X(K) \subset X^{{{\operatorname{an}}}}$, the restriction of $\tau$ to $X(K)$ gives the map $\tau: X(K) \to \Gamma$ (still denoted by $\tau$). Since $K$ is algebraically closed, we have ${\operatorname{Div}}(X) = {\operatorname{Div}}(X(K))$. By linearity and Lemma \[lemma:retraction:rational\](1), we extend $\tau$ to $$\label{eqn:specialization:map}
\tau_*: {\operatorname{Div}}(X) \to {\operatorname{Div}}_\Lambda(\Gamma).$$ This map $\tau_*$ is called the [*specialization map*]{}. Note that $\tau_*$ preserves degrees and linearly equivalent classes (cf. [@B Lemma 2.1])
Let $N$ be a line bundle over $X$. We take a divisor $\tilde{D} \in {\operatorname{Div}}(X)$ with $N \cong {{\mathcal{O}}}_X(\tilde{D})$, and we consider $\tau_*(\tilde{D}) \in {\operatorname{Div}}_\Lambda(\Gamma)$. Since $\tau_*$ preserves linear equivalent classes, the linear system $|\tau_*(\tilde{D})|$ does not depend on the choice of $\tilde{D}$. We denote $|\tau_*(\tilde{D})|$ by $|\tau_*(N)|$.
For a line bundle ${{\mathscr{N}}}$ over ${{\mathscr{X}}}$, we set $$\label{eqn:def:DM}
D_{{{\mathscr{N}}}} := \sum_{C \in {\operatorname{\operatorname{Irr}}}( \mathscr{X}_s )}
\deg
({\left.{\mathscr{N}}\right\vert_{{C}}}) [C]
\in {\operatorname{Div}}_{\Lambda} (\Gamma)
.$$ We show the following proposition, which will be the starting point to construct various good strictly semistable models $({{\mathscr{X}}}, {{\mathscr{L}}})$.
\[prop:pregoodmodel:pre\] Let $X$ be a connected smooth projective curve over $K$ of genus $g \geq 2$. Let $N$ be a line bundle over $X$ and let $D \in |\tau_*(N)|$. Then there exists a model $({{\mathscr{X}}}, {{\mathscr{N}}})$ of $(X, N)$ with the following properties
1. $\mathscr{X}$ is a good model cf. Definition
2. $D_{{{\mathscr{N}}}} = D$.
Further, if ${{\mathscr{X}}}^0$ is a Deligne–Mumford strictly semistable model of $X$, then we may take ${{\mathscr{X}}}$ so that ${{\mathscr{X}}}$ dominates ${{\mathscr{X}}}^0$, i.e., there exists a morphism $\mu: {{\mathscr{X}}}\to {{\mathscr{X}}}^0$ extending the identity morphism on $X$.
[[*Proof.*]{}]{}[**Step 1.**]{} We take $\tilde{D}^\prime \in {\operatorname{Div}}(X)$ such that $N \cong {{\mathcal{O}}}_X(\tilde{D}^\prime)$. Let $\Gamma_{\min}$ be the minimal skeleton of $X^{{{\operatorname{an}}}}$, and set $$D^\prime := \tau_{\ast} (\tilde{D}^\prime) \in {\operatorname{Div}}_\Lambda(\Gamma_{\min})
.$$ Then $D \in |\tau_*(N)| = | \tau_{\ast} (\tilde{D}^\prime)|$. We take a $P \in {\operatorname{Prin}}_\Lambda(\Gamma_{\min})$ such that $D =D^\prime + P$. Remark that our ${\operatorname{Prin}}_\Lambda(\Gamma_{\min})$ coincides with the group of $\Lambda$-rational principal divisors in [@BR]. Then by [@BR Theorem 1.1], which proves Raynaud’s type theorem on the surjectivity between the groups of principal divisors under the specialization map, there exists a principal divisor $\tilde{P}$ on $X$ such that $\tau_{\ast} (\tilde{P}) = P$. We have $D = \tau_*(\tilde{D}^\prime + \tilde{P})$. Since $K$ is algebraically closed, we write $\tilde{D}^\prime + \tilde{P} = \sum_{i= 1}^{m} P_i$ with $P_i \in X(K)$.
Let $\mathscr{X}^0$ be a Deligne–Mumford strictly semistable model of $X$. Then Proposition \[prop:subdivision1:b\] gives a strictly semistable model $
\pi^{1} : \mathscr{X}^{1} \to
{{\operatorname{Spec}}}(R)$ such that $V ( \mathscr{X}^{1} ) = V ( \mathscr{X}^0) \cup \{ \tau (P_1) ,
\ldots , \tau (P_m )\}$. Since $S(\mathscr{X}^{1}) = S(\mathscr{X}^{0}) = \Gamma_{\min}$, $\mathscr{X}^{1}$ is Deligne–Mumford semistable.
We modify ${{\mathscr{X}}}^1$ to obtain a good model. Let $p_1 , \ldots , p_\alpha \in {\operatorname{Sing}}( \mathscr{X}^1_s)$ be the nodes such that there does not exist $(-2)$-chain $E$ with $p_i \in E$. Let $\Delta_{p_i}$ be the canonical $1$-simplex corresponding to $p_i$. Take a $\Lambda$-valued points $v_i$ in $\mathrm{relin}( \Delta_{p_i})$ and put $M_1 := \{ v_1 , , \ldots , v_\alpha \}$. Then by Proposition \[prop:subdivision1:b\], there exists a strictly semistable model $\mathscr{X}^2 \to {{\operatorname{Spec}}}(R)$ of $X$ such that $V \left( \mathscr{X}^2 \right) =
V \left( \mathscr{X}^1 \right) \cup M_1$. By the construction, $\mathscr{X}^2$ satisfies condition (ii) in Definition \[def:goodmodel:new\]. Since $M_1 \subset S({{\mathscr{X}}}^1) = \Gamma_{\min}$, $\mathscr{X}^2$ is Deligne–Mumford semistable. Further, $\mathscr{X}^2 $ dominates $\mathscr{X}^1$.
Next, we modify $\mathscr{X}^2$ to make a model that satisfies condition (iii) in Definition \[def:goodmodel:new\]. Let $E$ be a maximal $(-2)$-chain of connected type. Put $D := {{\mathscr{X}}}^2 - E$. Let $E_1, \ldots, E_{r-1}$ be the irreducible components of $E$ with a numbering such that $\# (E_i \cap E_{i+1}) = 1$ for $i = 1 , \ldots r-2$. Let $p_0 , \ldots , p_{r-1}$ be the distinct nodes such that $\{ p_0 , \ldots , p_{r-1} \} = {\operatorname{Sing}}( {{\mathscr{X}}}^2_s ) \cap E
$, where the numbering is given in such a way that $\{p_i\} = E_i \cap E_{i+1}$ for $i = 1, \ldots, r-2$, $p_0 \in D \cap E_1$, and $p_{r-1} \in D \cap E_{r-1}$. Take $E_0, E_r \in {\operatorname{\operatorname{Irr}}}(D)$ such that $p_0 \in E_0$ and $p_{r-1} \in E_r$ (we may have $E_0 = E_r$). We set $\Delta_{E} := \bigcup_{i=0}^{r-1} \Delta_{p_i}$ in the skeleton $\Gamma_{\min} = S \left( \mathscr{X}^2 \right)$. There exists a unique nontrivial isometry $\iota : \Delta_E \to \Delta_E$ such that $\iota ( \partial \Delta_E) = \partial \Delta_E$ and $\iota^2 = {\operatorname{id}}$, where $\partial \Delta_E$ is the set of end vertices of $\Delta_E$, i.e., $\partial \Delta_E = \Delta_E \cap V(\mathscr{X}^{{\operatorname{st}}})$. There exists a unique point $w \in \Delta_E \setminus \partial \Delta_E$ with $\iota (w) = w$. We can take a finite set $M_{E}$ of $\Lambda$-valued points of $\Delta_{E} \subset S({{\mathscr{X}}}^2)$ such that $[E_0], [E_1], \ldots, [E_{r-1}], [E_{r}], w $ are all in $M_E$ and such that $\iota (M_E ) = M_{E}$. Note that $\# \left( M_{E} \setminus \{ [E_0] , [E_r]\} \right)$ is odd, and we write this number by $2 \ell - 1$. Furthermore, adding $\Lambda$-valued points to $M_E$ if necessary, we may and do assume that $\ell \geq 2$ and $M_E$ satisfies the following condition: Let $w_1 (\neq [E_0]) \in M_E$ be a point that is the nearest to $[E_0]$ and let $w_{\ell -1} (\neq w) \in M_E$ be a point that is the nearest to $w$; then the distance between $[E_0]$ and $w_1$ equals the distance between $w$ and $w_{\ell - 1}$.
For all maximal $(-2)$-chains $E$ of connected type, we take $M_{E}$ as above, and we set $M' := \bigcup_{E} M_E$. By Proposition \[prop:subdivision1:b\], there exists a strictly semistable model $\mathscr{X} \to {{\operatorname{Spec}}}(R)$ such that $V \left( \mathscr{X} \right) =
V \left( \mathscr{X}^2 \right) \cup M'$ and $\mathscr{X}$ dominates ${{\mathscr{X}}}^2$. Since $M' \subset S({{\mathscr{X}}}^2) = \Gamma_{\min}$, $\mathscr{X}$ is Deligne–Mumford semistable. By the construction, $\mathscr{X}$ satisfies condition (iii) in Definition \[def:goodmodel:new\]. Thus ${{\mathscr{X}}}$ is a good model in Definition \[def:goodmodel:new\]. Further, $\mathscr{X}$ dominates $\mathscr{X}^2$ and hence dominates $\mathscr{X}^0$.
[**Step 3.**]{} Finally, we construct a line bundle $\mathscr{N}$ over $\mathscr{X}$ that satisfies condition (ii) of the proposition. Let $P_1, \ldots , P_m \in X(K)$ be as in Step 1. We note that $\tau_{\ast} ( P_i ) \in V ( \mathscr{X}^1)
\subset V ( \mathscr{X})$. Let $\sigma_i: {{\operatorname{Spec}}}(R) \to {{\mathscr{X}}}$ be the corresponding section to $P_i$ guaranteed by the valuative criterion of properness. By Lemma \[lemma:retraction:rational\](2), there exists, for any $i = 1 , \ldots , m$, a unique $F_i \in {\operatorname{\operatorname{Irr}}}\left( \mathscr{X}_s \right)$ such that $\sigma_i (k) \in F_i \setminus {\operatorname{Sing}}\left( \mathscr{X}_s \right)$, where $\sigma_i (k) = {\operatorname{red}}_{\mathscr{X}} (P_i)$ by the definition of the reduction map ${\operatorname{red}}_{\mathscr{X}}$. Since $\sigma_i (k) \notin {\operatorname{Sing}}\left( \mathscr{X}_s \right)$, $\sigma_i (R)$ is a Cartier divisor on $\mathscr{X}$. We define a line bundle ${{\mathscr{N}}}$ over ${{\mathscr{X}}}$ to be $
{{\mathcal{O}}}_{{{\mathscr{X}}}}
\left( \sum_{i=1}^{m} \sigma_i (R)
\right)$. We have ${\left.{{{\mathscr{N}}}}\right\vert_{{X}}} = {{\mathcal{O}}}_X \left(\sum_{i=1}^{m}P_i\right) = {{\mathcal{O}}}_{X} ( \tilde{D}) \cong N$. For any $C \in {\operatorname{\operatorname{Irr}}}(\mathscr{X}_s)$ and $i= 1 ,\ldots , m$, we have $$\deg ( {\left.{{{\mathcal{O}}}_{\mathscr{X}} (\sigma_i (R) )}\right\vert_{{C}}}) =
\begin{cases}
1
& \text{if $C = F_i$,}
\\
0
&
\text{otherwise.}
\end{cases}$$ By Lemma \[lemma:retraction:rational\], $\tau_{\ast} (P_i) = [F_i]$. It follows that for any $C \in {\operatorname{\operatorname{Irr}}}(\mathscr{X}_s)$, $$\deg \left({\left.{\mathscr{N}}\right\vert_{{C}}}\right) =
\tau_* \left(\tilde{D}^\prime + \tilde{P}\right)([C])
= D([C]).$$ This shows that $D_{{{\mathscr{N}}}} = D$. Thus the model $({{\mathscr{X}}}, {{\mathscr{N}}})$ satisfies condition (ii). This completes the proof of the proposition.
500
Now, via Proposition \[prop:pregoodmodel:pre\], we construct a model $({{\mathscr{X}}}, {{\mathscr{L}}})$ of $(X, L)$, which will be frequently used to construct various global sections of $L$. Recall from the previous subsection that we have the minimal weighted skeleton $\bar{\Gamma}_{\min} = (\Gamma_{\min}, \omega)$ of $X^{{{\operatorname{an}}}}$ for a connected smooth projective curve $X$ of genus $g \geq 2$.
\[prop:pregoodmodel\] Let $X$ be a connected smooth projective curve over $K$ of genus $g \geq 2$, and let $L$ be a line bundle over $X$. Suppose that $\deg (L) \geq 3g-1$. Let $x \in \Gamma_{\min}$. Then there exist a model $(\mathscr{X},\mathscr{L})$ such that $\mathscr{X}$ is a good model cf. Definition and such that as a divisor on the minimal skeleton $\Gamma_{\min}$, $$D_{{{\mathscr{M}}}} - [x] \geq 0,$$ where $\mathscr{M} := \mathscr{L} \otimes \omega_{\mathscr{X}/R}^{\otimes -1}$ and $D_{{{\mathscr{M}}}}$ is the divisor on $\Gamma_{\min}$ defined in . Further, we may take ${{\mathscr{L}}}$ so that the following properties are also satisfied
1. For any island $\Gamma_i$ of $\Gamma_{\min}$, we have $\deg\left({\left.{(D_{{{\mathscr{M}}}} - [x])}\right\vert_{{\Gamma_i}}}\right) \geq 1$.
2. For any edge $e \in E({{\mathscr{X}}}^{{\operatorname{st}}})$ of $\Gamma_{\min}$, we have $\deg\left({\left.{(D_{{{\mathscr{M}}}} - [x])}\right\vert_{{{\mathrm{relin}(e)}}}}\right) \leq 1$.
Furthermore, if ${{\mathscr{X}}}^0$ is a Deligne–Mumford strictly semistable model of $X$, then we may take ${{\mathscr{X}}}$ so that ${{\mathscr{X}}}$ dominates ${{\mathscr{X}}}^0$.
[[*Proof.*]{}]{}Set $M := L \otimes \omega_X^{\otimes -1}$. We take $\tilde{D} \in {\operatorname{Div}}(X)$ with $M \cong {{\mathcal{O}}}_X(\tilde{D})$. Since $\deg(M) \geq 3g-1 - (2g-2) = g+1$, we have $$\deg \left(\tau_*(\tilde{D}) -
[x]
\right) \geq
g = g(\bar{\Gamma}_{\min}).$$ Then by Proposition \[prop:good:eff:div\], we take $E \in \left\vert \tau_{\ast} (\tilde{D}) - [x]\right\vert$ satisfying conditions (i)–(iii) of that proposition. Now, apply Proposition \[prop:pregoodmodel:pre\] to $
E + [x] \in \left|\tau_{\ast} (\tilde{D})\right| = \left|\tau_{\ast} (M)\right| $, and we obtain a good model ${{\mathscr{X}}}$ of $X$ and a line bundle ${{\mathscr{M}}}$ with ${\left.{{{\mathscr{M}}}}\right\vert_{{X}}} \cong M$ such that $
D_{{{\mathscr{M}}}} = E + [x]
$. Further, if ${{\mathscr{X}}}^0$ is a Deligne–Mumford strictly semistable model of $X$, then we may take ${{\mathscr{X}}}$ so that ${{\mathscr{X}}}$ dominates ${{\mathscr{X}}}^0$. Set $\mathscr{L} := \mathscr{M} \otimes \omega_{\mathscr{X}/R}$. Then, since $E$ satisfies conditions (ii) and (iii) in Proposition \[prop:good:eff:div\], it is straightforward to see that $\mathscr{L}$ has the required properties.
500
Unimodular tropicalization of minimal skeleta for $g \geq 2$ {#sec:usuful:lemmas}
============================================================
Let $X$ be a connected smooth projective curve over $K$ of genus $g$, and let $L$ be a line bundle over $X$. In this section, we prove that if $g \geq 2$ and $\deg (L) \geq 3 g - 1$, then the minimal skeleton $\Gamma_{\min}$ of $X^{{{\operatorname{an}}}}$ has a unimodular tropicalization associated to $|L|$.
Useful lemmas
-------------
We begin by showing some lemmas which will be frequently used not only in this section but also in the following sections.
We say that a line bundle $L$ is [*free*]{} at a point $p$ if $p$ is not a base point of $L$. We would often like to show that a line bundle is free at a point. The following simple lemma serves for this purpose. Recall that $k$ denotes the residue field of the ring of integers $R$ of $K$.
\[lemma:nonbasepoint\] Let $D$ be a semistable curve over $k$, let $L$ be a line bundle over $D$, and let $p \in D (k)$. We set $M := L \otimes \omega_D^{\otimes -1}$.
1. Assume that $p$ is a regular point of $D$. If $h^0\left((M(-p))^{\otimes -1}\right) = 0$, then $L$ is free at $p$.
2. Assume that $p$ is a node of $D$. Let $\nu : \widetilde{D} \to D$ be the partial normalization of $D$ at $p$. If $h^0\left(\nu^*\left(
M^{\otimes -1}\right)\right) = 0$, then $L$ is free at $p$.
[[*Proof.*]{}]{}(1) By the Serre duality, $h^0\left((M(-p))^{\otimes -1}\right) = 0$ is equivalent to $h^{1} ( L ( -p ) ) = 0$. From the exact sequence $0 \to L(-p) \to L \to k_p \to 0$, we then have $h^{0} ( L ( -p ) ) = h^{0} ( L ) - 1$. Thus $L$ is free at $p$.
\(2) Let $\frak{m}_p$ be the maximal ideal of ${{\mathcal{O}}}_D$ at $p$. We claim that if ${\operatorname{\operatorname{Hom}}}\left(\frak{m}_p,
\left(L \otimes \omega_D^{\otimes -1} \right)^{\otimes -1}\right) = 0$, then $L$ is free at $p$. Indeed, by the Serre duality, ${\operatorname{\operatorname{Hom}}}\left(\frak{m}_p,
\left(L \otimes \omega_D^{\otimes -1} \right)^{\otimes -1}\right) = 0$ is equivalent to $h^1({{\mathfrak{m}}}_p L) = 0$ (cf. [@Ha Theorem III.7.6]). By the same argument as (1) using the exact sequence $0 \to {{\mathfrak{m}}}_p L \to L \to k_p \to 0$, we obtain the claim.
As is noted in the proof of [@DM Theorem 1.2], we have $$\label{eqn:basepointfreeforpositive-ft:2}
{\operatorname{\operatorname{Hom}}}\left( \mathfrak{m}_p , \left(L \otimes \omega_D^{\otimes -1} \right)^{\otimes -1}\right)
\cong
H^{0}
\left(
\widetilde{D} ,
\nu^*\left((L \otimes \omega_D^{\otimes -1})^{\otimes -1}\right)
\right).$$ Thus we get the assertion.
500
In view of Lemma \[lemma:nonbasepoint\], we would often like to show the vanishing of global sections of a certain line bundle. The following lemmas serve for this purpose.
\[lemma:vanishing0\] Let $D$ be a connected proper reduced curve over $k$ and let $M$ be a line bundle over $D$. If $M$ is nef and $\deg(M) > 0$, then $h^0(M^{\otimes -1}) = 0$.
[[*Proof.*]{}]{}Let $C \in {\operatorname{\operatorname{Irr}}}(D)$. Since $\deg({\left.{M}\right\vert_{{C}}}) \geq 0$, we see that $H^0({\left.{M^{\otimes -1}}\right\vert_{{C}}}) \neq 0$ if and only if ${\left.{M}\right\vert_{{C}}} \cong {{\mathcal{O}}}_C$. We take any $\sigma \in H^0(M^{\otimes -1})$. Then ${\left.{\sigma}\right\vert_{{C}}} \in H^0({\left.{M^{\otimes -1}}\right\vert_{{C}}})$ is either zero or nowhere vanishing. On the other hand, there exists $C_1 \in {\operatorname{\operatorname{Irr}}}(D)$ with $\deg({\left.{M}\right\vert_{{C_1}}}) > 0$, and we have ${\left.{\sigma}\right\vert_{{C_1}}} = 0$. Since $D$ is connected, we have $\sigma = 0$.
500
\[lemma:vanishing:a\] Let $D$ be a connected proper reduced curve over $k$ and let $M$ be a line bundle over $D$. Let $E$ be a curve in $D$ and set $F := D - E$. Assume that $F \neq \emptyset$, ${\left.{M}\right\vert_{{E}}}$ is nef and $h^0({\left.{M^{\otimes -1}}\right\vert_{{F}}}) = 0$. Then $h^0(M^{\otimes -1}) = 0$.
[[*Proof.*]{}]{}Our argument is similar to the proof of Lemma \[lemma:vanishing0\]. We take any $\sigma \in H^0(M^{\otimes -1})$. For any $C \in {\operatorname{\operatorname{Irr}}}(E)$, since $\deg({\left.{M}\right\vert_{{C}}}) \geq 0$, ${\left.{\sigma}\right\vert_{{C}}} \in H^0({\left.{M^{\otimes -1}}\right\vert_{{C}}})$ is either zero or nowhere vanishing. On the other hand, ${\left.{\sigma}\right\vert_{{F}}} = 0$. Since $D$ is connected, we have $\sigma = 0$.
500
\[lemma:vanishing7\] Let $D$ be a proper reduced curve over $k$ with at most nodes as singularities, and let $M$ be a line bundle over $D$. Let $E$ be a curve in $D$ and set $F := D - E$. Regard $\Sigma := F \cap E$ as a Cartier divisor on $F$. Suppose that $h^0 \left( {\left.{M^{\otimes -1}}\right\vert_{{E}}} \right) = 0$ and $h^0 \left(
\left( {\left.{M}\right\vert_{{F}}} ( \Sigma )
\right)^{\otimes -1}
\right) = 0$. Then $h^0(M^{\otimes -1}) = 0$.
[[*Proof.*]{}]{}Take any $\eta \in H^0 \left( M^{\otimes -1}
\right)$. Since $h^0 \left( {\left.{M^{\otimes -1}}\right\vert_{{E}}} \right) = 0$, we have ${\left.{\eta}\right\vert_{{E}}} = 0$, and thus ${\left.{\eta}\right\vert_{{E \cap F}}} = 0$. This means that via the natural inclusion ${\left.{M^{\otimes -1}}\right\vert_{{F}}}
(- \Sigma) \hookrightarrow
{\left.{M^{\otimes -1}}\right\vert_{{F}}}$, ${\left.{\eta}\right\vert_{{F}}} \in H^0
\left(
M^{\otimes -1} ( - \Sigma )
\right)$. Since $h^0 \left(
{\left.{M^{\otimes -1}}\right\vert_{{F}}} ( - \Sigma)
\right) = 0$, ${\left.{\eta}\right\vert_{{F}}} = 0$. It follows that $\eta = 0$. This proves $h^0(M^{\otimes -1}) = 0$.
500
Now, let $X$ be a connected smooth projective curve over $K$, and let ${{\mathscr{X}}}$ be a strictly semistable model of $X$. Using the above lemmas, we give a sufficient condition in terms of $\Lambda$-metric graphs for a line bundle over ${{\mathscr{X}}}$ to be free at a node $p \in {{\mathscr{X}}}_s$.
\[lemma:freeatnode:graphversion:emptyset\] Let $X$ be a connected smooth projective curve over $K$, and let ${{\mathscr{X}}}$ be a strictly semistable model of $X$. Let $\Gamma := S({{\mathscr{X}}})$ be the associated skeleton. Let $\mathscr{L}$ be a line bundle over ${{\mathscr{X}}}$, and we set ${{\mathscr{M}}}:= {{\mathscr{L}}}\otimes \omega_{{{\mathscr{X}}}/R}^{\otimes -1}$. Let $p \in {\operatorname{Sing}}( \mathscr{X}_s)$, and denote by $\Delta_p$ the corresponding $1$-simplex in $\Gamma$ cf. Definition . Assume that that $D_{\mathscr{M}} \geq 0$ cf. . Suppose that for any connected component $\Gamma^\prime$ of $\Gamma \setminus \mathrm{relin}(\Delta_p)$, we have $\deg
\left(
{\left.{D_{\mathscr{M}}}\right\vert_{{\Gamma^\prime}}} \right) \geq 1$. Then $\mathscr{L}$ is free at $p$.
[[*Proof.*]{}]{}To ease notation, we set $M := {\left.{{{\mathscr{M}}}}\right\vert_{{{{\mathscr{X}}}_s}}}$. Note that $D_{\mathscr{M}} \geq 0$ means that $M$ is nef. Let $\nu : \widetilde{\mathscr{X}_s} \to \mathscr{X}_s$ be the partial normalization at $p$. Let $F$ be any connected component of $\widetilde{\mathscr{X}_s}$. Then by the assumptions, ${\left.{\nu^*\left(M\right)}\right\vert_{{F}}}$ is nef and $\deg\left({\left.{\nu^*\left(M\right)}\right\vert_{{F}}}\right) >0$. By Lemma \[lemma:vanishing0\], we have $h^{0}
\left(
\nu^*\left(M^{\otimes -1}\right)
\right) = 0$. By Lemma \[lemma:nonbasepoint\](2), ${\left.{\mathscr{L}}\right\vert_{{\mathscr{X}_s}}}$ is free at $p$.
Since $H^0 \left(
M^{\otimes -1}
\right) \subset H^0
\left(
\nu^\ast \left(
M^{\otimes -1}
\right)
\right)$, we have $h^{0}
\left(
M^{\otimes -1}
\right) = 0$. By the Serre duality, $h^1
\left(
{\left.{\mathscr{L}}\right\vert_{{\mathscr{X}_s}}}
\right) = 0$. By the base-change theorem, it follows that the natural homomorphism $H^0 ( \mathscr{L}) \to H^{0}
\left(
{\left.{\mathscr{L}}\right\vert_{{\mathscr{X}_s}}}
\right)$ is surjective. Since ${\left.{\mathscr{L}}\right\vert_{{\mathscr{X}_s}}}$ is free at $p$, this concludes that $\mathscr{L}$ is free at $p$.
500
In the next lemma, we consider a case where $D_{{{\mathscr{M}}}}$ is not effective.
\[lemma:freeatnode:graphversion:nonemptyset\] Let $X$, $\mathscr{X}$, $\Gamma$, $\mathscr{L}$, $\mathscr{M}$, $p$, and $\Delta_p$ be as in Lemma \[lemma:freeatnode:graphversion:emptyset\]. Let $\Gamma_1$ be a connected component of $\Gamma \setminus \mathrm{relin}(\Delta_p)$. Set $
V := \{
v \in V({{\mathscr{X}}}) \mid
D_{\mathscr{M}}(v) <0\}
$. Assume that $V \neq \emptyset$ and $V \subset \Gamma_1$. Suppose the following.
1. For any connected component $\Gamma^\prime$ of $\Gamma \setminus \mathrm{relin}(\Delta_p)$ with $\Gamma^\prime \neq \Gamma_1$, $\deg
\left({\left.{D_{\mathscr{M}}}\right\vert_{{\Gamma^\prime}}} \right) \geq 1$.
2. There exist $s \geq 1$ and distinct connected components $\Gamma_{11}^\circ, \ldots, \Gamma_{1s}^\circ$ of $\Gamma_1 \setminus V$ such that
1. $\deg\left({\left.{D_{\mathscr{M}}}\right\vert_{{\Gamma_{1j}^\circ}}} \right) \geq 1$ for any $1 \leq j \leq s$, and
2. for any $v \in V$, the valence of $\Gamma_{11} \cup \cdots \cup \Gamma_{1s}$ at $v$ is at least $-D_{{{\mathscr{M}}}}(v) + 1$, where $\Gamma_{1j}$ is the closure of $\Gamma_{1j}^\circ$ in $\Gamma_1$.
Then $\mathscr{L}$ is free at $p$.
[[*Proof.*]{}]{}Let $\nu : \widetilde{\mathscr{X}_s} \to \mathscr{X}_s$ be the partial normalization at $p$. Via $\nu$ we identify ${\operatorname{\operatorname{Irr}}}\left( \widetilde{\mathscr{X}_s} \right)$ with ${\operatorname{\operatorname{Irr}}}\left( \mathscr{X}_s \right)$. For $C \in {\operatorname{\operatorname{Irr}}}\left( \widetilde{\mathscr{X}_s} \right)$, let $[C]$ denote the corresponding point in $\Gamma \setminus {\rm relin}(\Delta_p)$. To ease notation, we set $M := {\left.{{{\mathscr{M}}}}\right\vert_{{{{\mathscr{X}}}_s}}}$.
\[claim:in:lemma:freeatnode:graphversion:nonemptyset\] We have $h^{0}\left(\nu^*(M)^{\otimes -1}\right) = 0$.
Indeed, we set $F_1 :=
\sum_{C \in {\operatorname{\operatorname{Irr}}}\left(\widetilde{\mathscr{X}_s} \right) , [C] \in \Gamma_1} C$ and set $F_{1j} := \sum_{C \in {\operatorname{\operatorname{Irr}}}\left(\widetilde{\mathscr{X}_s} \right) , [C] \in \Gamma^{\circ}_{1j}}
C$ for $1 \leq j \leq s$. We take $C_1 , \ldots , C_{r} \in \widetilde{\mathscr{X}_s}$ such that $V' = \{[C_1], \ldots, [C_r]\}$. By the assumption, $F_1$ is the connected component of ${{\mathscr{X}}}_s \setminus \{p\}$ such that $\{C_1 , \ldots , C_{r}\} \subset F_1$. Further, by (ii)(a) in the lemma, each $F_{1j}$ is a connected component of $F_1 - (C_1 + \cdots + C_{r})$, ${\left.{\nu^*(M)}\right\vert_{{F_{1j}}}}$ is nef, and $\deg\left({\left.{\nu^*(M)}\right\vert_{{F_{1j}}}}\right) > 0$. Thus Lemma \[lemma:vanishing0\] gives $$\label{eqn:freeatnode:graphversion:nonemptyset}
h^0\left(
{\left.{\nu^*(M)^{\otimes -1}}\right\vert_{{\sum_{j=1}^s F_{1j}}}}
\right) = 0.$$
Take any $i = 1 , \ldots , r$. Set $\Sigma_i := C_i \cap (\sum_{j=1}^s F_{1j})$, which we regard as a Cartier divisor on $C_i$. Then, by condition (ii)(b) in the lemma, $\deg \left(
{\left.{\nu^*(M)}\right\vert_{{C_i}}} (\Sigma_i)
\right)
\geq 1$. Since $C_i$ is irreducible, Lemma \[lemma:vanishing0\] tells us that $$\label{eqn:freeatnode:graphversion:nonemptyset:2}
h^0
\left(
\left(
{\left.{\nu^*(M)}\right\vert_{{C_i}}} (\Sigma_i)
\right)^{\otimes -1}
\right) = 0.$$
Note that $h^{0}
\left(
{\left.{\nu^*(M)^{\otimes -1}}\right\vert_{{F_1}}}
\right) = 0$. Indeed, we take any $\sigma \in H^{0}
\left({\left.{\nu^*(M)^{\otimes -1}}\right\vert_{{\sum_{i=1}^r C_i + \sum_{j=1}^s F_{1j}}}}
\right)$. By and , Lemma \[lemma:vanishing7\] gives $h^{0}
\left(
{\left.{\nu^*(M)^{\otimes -1}}\right\vert_{{C_i + \sum_{j=1}^s F_{1j}}}}
\right) = 0$. Thus ${\left.{\sigma}\right\vert_{{C_i + \sum_{j=1}^s F_{1j}}}} = 0$ for each $i=1, \ldots , r$. Since the union of $\left\{ C_i + \sum_{j=1}^s F_{1j} \right\}_{i=1}^r$ equals $\sum_{i=1}^r C_i + \sum_{j=1}^s F_{1j}$, this proves $\sigma = 0$. Thus $h^0
\left(
{\left.{\nu^*(M)^{\otimes -1}}\right\vert_{{\sum_{i=1}^r C_i + \sum_{j=1}^s F_{1j}}}}
\right) = 0$. Since ${\left.{\nu^*(M)}\right\vert_{{F_1 - (\sum_{i=1}^r C_i + \sum_{j=1}^s F_{1j})}}}$ is nef, Lemma \[lemma:vanishing:a\] concludes that $h^{0}
\left(
{\left.{\nu^*(M)^{\otimes -1}}\right\vert_{{F_1}}}
\right) = 0$.
Let $\Gamma^\prime$ be any connected component of $\Gamma \setminus \mathrm{relin}(\Delta_p)$ with $\Gamma^\prime \neq \Gamma_1$, and we set $
F^\prime :=
\sum_{C \in {\operatorname{\operatorname{Irr}}}\left(\widetilde{\mathscr{X}_s} \right) , [C] \in \Gamma^\prime} C$. It follows from $V \subset \Gamma_1$ that ${\left.{\nu^*(M)}\right\vert_{{F^\prime}}}$ is nef. It also follows from condition (i) in the lemma that $\deg({\left.{\nu^*(M)}\right\vert_{{F^\prime}}}) \geq 1$. By Lemma \[lemma:vanishing0\], we have $h^{0}
\left(
{\left.{\nu^*(M)}\right\vert_{{F^\prime}}}^{\otimes -1}
\right) = 0$. Thus $h^{0}\left(
\nu^*(M)^{\otimes -1}\right) = 0$. This completes the proof of Claim \[claim:in:lemma:freeatnode:graphversion:nonemptyset\].
By Claim \[claim:in:lemma:freeatnode:graphversion:nonemptyset\] and Lemma \[lemma:nonbasepoint\], it follows that ${\left.{{{\mathscr{L}}}}\right\vert_{{{{\mathscr{X}}}_s}}}$ is free at $p$. Then the same argument as in the proof of Lemma \[lemma:freeatnode:graphversion:emptyset\] concludes that ${{\mathscr{L}}}$ is free at $p$.
500
While Lemmas \[lemma:freeatnode:graphversion:emptyset\] and \[lemma:freeatnode:graphversion:nonemptyset\] consider freeness of ${{\mathscr{L}}}$ at a node, the following two lemmas consider freeness of ${{\mathscr{L}}}$ at a regular point. The reader may skip these two lemmas and return when reading §\[section:FTcan\], because we will use them not in this section but in §\[section:FTcan\].
\[lemma:freeatregular:graphversion:effective\] Let $X$ be a connected smooth projective curve over $K$, and let ${{\mathscr{X}}}$ be a strictly semistable model of $X$. Let $\Gamma := S({{\mathscr{X}}})$ be the associated skeleton. Let $\mathscr{L}$ be a line bundle over ${{\mathscr{X}}}$, and we set ${{\mathscr{M}}}:= {{\mathscr{L}}}\otimes \omega_{{{\mathscr{X}}}/R}^{\otimes -1}$. Let $C$ be an irreducible component of $\mathscr{X}_s$, and let $v := [C] \in V ( \mathscr{X} )$ be the Shilov point in $\Gamma$ corresponding to $C$. Suppose that $D_{{{\mathscr{M}}}} - [v]$ is effective on $\Gamma$ and $\deg(D_{{{\mathscr{M}}}} - [v]) \geq 1$ . Then $\mathscr{L}$ is free at any $p \in C(k)
\setminus {\operatorname{Sing}}( \mathscr{X}_s)$.
[[*Proof.*]{}]{}We take any $p \in C(k)
\setminus {\operatorname{Sing}}( \mathscr{X}_s)$. To ease notation, we set $M := {\left.{\mathscr{M}}\right\vert_{{\mathscr{X}_s}}}$. Since $D_{{{\mathscr{M}}}} - [v]$ is effective on $\Gamma$ and has positive degree, $M (-p)$ is nef and has positive degree. By Lemma \[lemma:vanishing0\] we have $h^0
\left(
M (-p)^{\otimes -1}
\right) = 0$, and then Lemma \[lemma:nonbasepoint\](1) implies that ${\left.{\mathscr{L}}\right\vert_{{\mathscr{X}_s}}}$ is free at $p$. Since $h^0
\left(
M (-p)^{\otimes -1}
\right) = 0$, we have $h^0
\left(
M^{\otimes -1}
\right) = 0$. By the Serre duality, we have $
h^1
\left(
{\left.{\mathscr{L}}\right\vert_{{\mathscr{X}_s}}}
\right)
=
h^0
\left(
M^{\otimes -1}
\right) = 0$. By the base-change theorem, $H^0 ( \mathscr{L} ) \to H^0 \left(
{\left.{\mathscr{L}}\right\vert_{{\mathscr{X}_s}}} \right)$ is surjective. Thus $\mathscr{L}$ is free at $p$.
500
In the next lemma, we consider a case where $D_{{\mathscr{M}}}- [v]$ is not effective.
\[lemma:freeatregular:graphversion\] Let $X$, ${{\mathscr{X}}}$, $\Gamma$, $\mathscr{L}$, ${{\mathscr{M}}}$, $C$, and $v$ be as in Lemma . Suppose that $D_{\mathscr{M}}$ is effective and that $D_{{{\mathscr{M}}}} (v) = 0$. Further, suppose that there exist connected components $\Gamma_1^\circ$ and $\Gamma_2^\circ$ of $\Gamma \setminus \{ v \}$ possibly $\Gamma_1^\circ = \Gamma_2^\circ$ such that
1. $\deg ({\left.{D_{\mathscr{M}}}\right\vert_{{\Gamma_j^\circ}}}) \geq 1$ for any $j = 1 , 2$
2. the valence of $\Gamma_1 \cup \Gamma_2$ at $v$ is at least $2$, where $\Gamma_j:= \Gamma_j^\circ \cup \{v\}$ is the closure of $ \Gamma_j^\circ$ in $\Gamma$.
Then $\mathscr{L}$ is free at any $p \in C(k) \setminus {\operatorname{Sing}}( \mathscr{X}_s)$.
[[*Proof.*]{}]{}To ease notation, we set $M := {\left.{\mathscr{M}}\right\vert_{{\mathscr{X}_s}}}$. We take any $p \in C(k)
\setminus {\operatorname{Sing}}( \mathscr{X}_s)$. We prove that $h^{0}
\left(
M(-p)^{\otimes -1}
\right) = 0$. Put $F_j := \sum_{C^\prime \in {\operatorname{\operatorname{Irr}}}\left(
\mathscr{X}_s \right) , [C^\prime] \in \Gamma_j^\circ} C^\prime$ for $j=1,2$. Then each $F_j$ ($j=1,2$) is a connected component of $\mathscr{X}_s - C_1
$. By condition (i), ${\left.{M (-p)}\right\vert_{{F_j}}}$ is nef and has positive degree for $j =
1 ,2$. By Lemma \[lemma:vanishing0\], we have $h^0
\left(
{\left.{M(-p)}\right\vert_{{F_j}}}^{\otimes -1}
\right) = 0$. We set $$F_{3} :=
\begin{cases}
F_1
& \text{if $F_1 = F_2$}
\\
F_1 + F_2
&
\text{if $F_1 \neq F_2$.}
\end{cases}$$ Then we have $h^0({\left.{M(-p)}\right\vert_{{F_3}}}^{\otimes -1}) = 0$. We set $\Sigma :=
C \cap F_3$, which we regard as a Cartier divisor on $C$. Then by condition (ii) and $D_{\mathscr{M}} (v) = 0$, we have $\deg \left(
{\left.{M(-p)}\right\vert_{{C}}} ( \Sigma )
\right)
\geq 1$, and then Lemma \[lemma:vanishing0\] gives $h^0
\left(
\left(
{\left.{M(-p)}\right\vert_{{C}}} ( \Sigma )
\right)^{\otimes -1}
\right) = 0$. By Lemma \[lemma:vanishing7\], we have $h^{0}
\left(
{\left.{M(-p)}\right\vert_{{C + F_3}}}^{\otimes -1}
\right) = 0$. Since $D_{\mathscr{M}}$ is effective, ${\left.{M(-p)}\right\vert_{{\mathscr{X}_s - (C + F_3)}}}$ is nef. Thus Lemma \[lemma:vanishing:a\] gives $h^{0}
\left(
M(-p)^{\otimes -1}
\right) = 0$.
We have shown that $h^{0}
\left(M(-p)^{\otimes -1}\right) = 0$. Then Lemma \[lemma:nonbasepoint\](1) implies that ${\left.{\mathscr{L}}\right\vert_{{\mathscr{X}_s}}}$ is free at $p$. By the same argument as in the proof of Lemma \[lemma:freeatregular:graphversion:effective\], we obtain that $\mathscr{L}$ is free at $p$.
500
The following remark will not be used in this section or in §\[section:FTcan\] but in §\[subsec:good:model:classical:point\].
\[remark:proofs:lemma:freeatnode:graphversion:emptyset\] In the proofs of Lemma \[lemma:freeatnode:graphversion:emptyset\] and Lemma \[lemma:freeatregular:graphversion\], we have actually shown the following statements.
1. Let $X, \mathscr{X}, \Gamma$, $p$ and $\Delta_p$ be as in Lemma \[lemma:freeatnode:graphversion:emptyset\]. Let $M$ be a line bundle over $\mathscr{X}_s$, and we set $D_{M} := \sum_{C \in {\operatorname{\operatorname{Irr}}}( \mathscr{X}_s )}
\deg ({\left.{M}\right\vert_{{C}}}) [C]
\in {\operatorname{Div}}_{\Lambda} (\Gamma)$. Assume that $D_M \geq 0$ and that for any connected component $\Gamma'$ of $\Gamma \setminus
\mathrm{relin} ( \Delta_p )$, we have $\deg ( {\left.{D_M}\right\vert_{{\Gamma'}}}) \geq 1$. Then $h^{0}\left(
\nu^*(M)^{\otimes -1}\right) = 0$, where $\nu$ is the partial normalization of ${{\mathscr{X}}}_s$ at $p$.
2. Let $X$, ${{\mathscr{X}}}$, $\Gamma$, $C$, and $v$ be as in Lemma \[lemma:freeatregular:graphversion:effective\]. Let $M$ be a line bundle over $\mathscr{X}_s$. Set $D_{M}
:= \sum_{C \in {\operatorname{\operatorname{Irr}}}( \mathscr{X}_s )}
\deg ({\left.{M}\right\vert_{{C}}}) [C]
\in {\operatorname{Div}}_{\Lambda} (\Gamma)$. Suppose that $D_{M}$ is effective and $D_{M} (v) = 0$. Further, suppose that there exist connected components $\Gamma_1^\circ$ and $\Gamma_2^\circ$ of $\Gamma \setminus \{ v \}$ possibly $\Gamma_1^\circ = \Gamma_2^\circ$ such that conditions (i) and (ii) of Lemma are satisfied with $D_{{{\mathscr{M}}}}$ replaced by $D_M$. Then $h^{0}
\left(
M(-p)^{\otimes -1}
\right) = 0$ for any $p \in C(k) \setminus {\operatorname{Sing}}( \mathscr{X}_s )$.
Fundamental vertical divisors {#subsec:fvd}
-----------------------------
Let $X$ be a connected smooth projective curve over $K$, and let ${{\mathscr{X}}}\to {{\operatorname{Spec}}}(R)$ be a strictly semistable model of $X$. Since $\mathscr{X}$ is a local-ringed space, we have the notion of Cartier divisors on $\mathscr{X}$. In this paper, divisors on $\mathscr{X}$ always mean Cartier divisors. Let ${\operatorname{Div}}( \mathscr{X})$ denote the group of divisors on $\mathscr{X}$.
Let $C \in {\operatorname{\operatorname{Irr}}}(\mathscr{X}_s)$, and let $|\cdot|_C$ denote the absolute value on the function field $K (X)$ that gives the Shilov point associated to $C$. For any $f \in K (X)^{\times}$, ${\operatorname{ord}}_C (f) := - \log |f|_C$ is called the *order of $f$ along $C$.* Let ${\mathscr{D}}$ be a divisor of ${{\mathscr{X}}}$. Let $\xi \in \mathscr{X}$ be the generic point of $C$, and let $g$ be a local equation of ${\mathscr{D}}$ at $\xi$, i.e., a rational function on ${{\mathscr{X}}}$ such that ${\operatorname{div}}(g) = {\mathscr{D}}$ on some open neighborhood of $\xi$ in $\mathscr{X}$. We define *order of $\mathscr{D}$ along $C$* to be ${\operatorname{ord}}_C ({\mathscr{D}}) := - \log |g|_C$. Remark that ${\operatorname{ord}}_C ({\mathscr{D}})$ does not depend on the choice of $g$ and that ${\operatorname{ord}}_C ({\mathscr{D}}) \in \Lambda$ (cf. Remark \[rem:type2points\]).
For a divisor ${\mathscr{D}}$, its support ${\operatorname{Supp}}({\mathscr{D}})$ is the locus on $\mathscr{X}$ at which the local equation of ${\mathscr{D}}$ is not a locally invertible regular function. We remark that ${\operatorname{Supp}}({\mathscr{D}})$ is a closed subset of $\mathscr{X}$. A divisor ${\mathscr{D}}$ on $\mathscr{X}$ is said to be *vertical* if the restriction of ${\mathscr{D}}$ to $X$ is trivial, which is equivalent to ${\operatorname{Supp}}({\mathscr{D}}) \subset \mathscr{X}_s$.
We recall from Definition \[def:multiplicityatanodes\] the multiplicity at a node. Let $F$ be a connected curve in $\mathscr{X}_s$. For each $p \in F \cap ( \mathscr{X}_s - F)$, take a local étale atlas $$\label{eqn:local:atlas:psi:p}
\psi_p :
\mathscr{U}_p \to
{{\operatorname{Spec}}}( R[x, y]/(xy - \varpi_p))$$ that distinguishes $p$, where $\varpi_p \in R$. Then $v_K ( \varpi_p)$ is called the multiplicity of ${{\mathscr{X}}}$ at $p$.
Assume that $F$ satisfies the following conditions (F):
- There exists a $\lambda \in \Lambda$ such that for any $p \in F \cap ( \mathscr{X}_s - F)$, the multiplicity of $\mathscr{X}$ at $p$ equals $\lambda$.
Then we define a vertical divisor ${{\mathscr{F}}}$ as follows. For each $p \in F \cap ( \mathscr{X}_s - F)$, we take a local étale atlas such that $\psi_{p}^{\ast} (x)$ defines the branch $C$ at $p$ with $C \subset F$. Then there exists a unique divisor ${{\mathscr{F}}}$ characterized by the properties that
1. for any $p \in F \cap ( \mathscr{X}_s - F)$, ${{\mathscr{F}}}$ has a local equation $\psi_p^{\ast} (x)$ on $\mathscr{U}_p$;
2. ${{\mathscr{F}}}= {\operatorname{div}}( \varpi_p )$ on some open neighborhood of $F \setminus ( \mathscr{X}_s - F)$;
3. ${{\mathscr{F}}}$ is trivial on some open neighborhood of $( \mathscr{X}_s - F) \setminus F$.
One sees that ${{\mathscr{F}}}$ does not dependent on the choice of a local atlas $\psi_p$ at $p$ nor of $\varpi_p$. Further, ${{\mathscr{F}}}$ has support $F$. We call ${{\mathscr{F}}}$ the *fundamental vertical divisor with support $F$*. The number $\lambda$ is called the *multiplicity of ${{\mathscr{F}}}$*. The fundamental vertical divisor with support $F$ for some connected curve $F$ is simply called a *fundamental vertical divisor*.
\[rem:restrictionofcartierdivisor\] Let ${{\mathscr{F}}}$ be a fundamental vertical divisor on $\mathscr{X}$. Let $D$ be a connected curve with ${\operatorname{\operatorname{Irr}}}(D) \cap {\operatorname{\operatorname{Irr}}}( {\operatorname{Supp}}({{\mathscr{F}}})) = \emptyset$. Then ${{\mathscr{F}}}|_{D}$ is a reduced Cartier divisor on $D$ with support ${\operatorname{Supp}}( {{\mathscr{F}}}) \cap D$. This can be checked by using a local equation of ${{\mathscr{F}}}$ at each point of ${\operatorname{Supp}}( {{\mathscr{F}}}) \cap D$.
Let $X$ be a connected smooth projective curve over $K$, and let $\pi : \mathscr{X} \to {{\operatorname{Spec}}}(R)$ be a strictly semistable model of $X$.
\[lem:forinjectivity\] Let $X$ and $\pi : \mathscr{X} \to {{\operatorname{Spec}}}(R)$ be as above. Let $g$ be a non-zero rational function on $\mathscr{X}$, and let $h:= {\left.{g}\right\vert_{{X}}}$ denote the restriction of $g$ to the generic fiber $X$. Let $\mathscr{W}
\in {\operatorname{Div}}(\mathscr{X})
$ be a ${{\mathbb{Z}}}$-linear combination of fundamental vertical divisors. Then the following hold, where for any $C \in {\operatorname{\operatorname{Irr}}}(\mathscr{X}_s)$, $[C]$ denotes the Shilov point associated to $C$.
1. Let $C \in {\operatorname{\operatorname{Irr}}}( \mathscr{X}_s)$. Suppose that there exists an open neighborhood $\mathscr{U} \subset
\mathscr{X}$ of the generic point $\xi_C$ such that $
{\left.{
\left(
{\operatorname{div}}(g) - \mathscr{W}
\right)}\right\vert_{{\mathscr{U}}}}
$ is effective. Then $- \log |h ([C])| \geq {\operatorname{ord}}_C (\mathscr{W})$. Furthermore, if ${\left.{
\left(
{\operatorname{div}}(g) - \mathscr{W}
\right)}\right\vert_{{\mathscr{U}}}}$ is trivial, then $- \log |h ( [C] )| = {\operatorname{ord}}_C (\mathscr{W})$.
2. Let $p \in {\operatorname{Sing}}( \mathscr{X}_s )$, and let $C_1, C_2 \in {\operatorname{\operatorname{Irr}}}(\mathscr{X}_s)$ be distinct irreducible components with $p \in C_{1} \cap C_{2}$. Let $\varphi$ be the affine function on $\Delta_p$ with $\varphi ([C_i]) = {\operatorname{ord}}_{C_i} (\mathscr{W})$ for $i=1,2$. Suppose that there exists an open neighborhood $\mathscr{U} \subset
\mathscr{X}$ of $p$ such that $
{\left.{
\left(
{\operatorname{div}}(g) - \mathscr{W}
\right)}\right\vert_{{\mathscr{U}}}}
$ is effective. Then $- \log |h (u) | \geq \varphi (u)$ for any $u \in \Delta_p$. Furthermore, if ${\left.{
\left(
{\operatorname{div}}(g) - \mathscr{W}
\right)}\right\vert_{{\mathscr{U}}}}$ is trivial, then $- \log |h (u) | = \varphi (u)$ for any $u \in \Delta_p$.
[[*Proof.*]{}]{}Assertion (1) follows from the definition of ${\operatorname{ord}}$. Let us prove (2). We take a distinguished local étale morphism $\psi_p : \mathscr{U}_p \to {{\operatorname{Spec}}}(R [x,y]/(xy - \varpi_p))$ such that $\mathscr{U}_p \subset \mathscr{U}$. We may assume that, in the special fiber, $C_1$ is defined by $\psi_p^{\ast} (x)$ and $C_2$ by $\psi_p^{\ast} (y)$. Since $\mathscr{W}$ is a ${{\mathbb{Z}}}$-linear combination of fundamental vertical divisors, we have a local equation of $\mathscr{W}$ of the form $\varpi \psi_p^{\ast} (x)^{m_1} \psi_p^{\ast} (y)^{m_2}$ for some $m_1 , m_2 \in {{\mathbb{Z}}}$ and $\varpi \in K^{\times}$, and then $\varphi = - \log \left|{\left.{\varpi \psi_p^{\ast} (x)^{m_1} \psi_p^{\ast} (y)^{m_2}}\right\vert_{{X}}}\right|$. Put $g_1:= g \cdot
\left(\varpi \psi_p^{\ast} (x)^{m_1} \psi_p^{\ast} (y)^{m_2} \right)^{-1}$. Since $
{\left.{
\left(
{\operatorname{div}}(g) - \mathscr{W}
\right)}\right\vert_{{\mathscr{U}}}}
$ is effective, $g_1$ is regular on $\mathscr{U}_p$ and thus $- \log \left|{\left.{g_1}\right\vert_{{X}}}\right| \geq 0$ on $\Delta_p$. Thus we obtain $- \log \left|{\left.{g}\right\vert_{{X}}}\right| \geq \varphi$. If $
{\left.{
\left(
{\operatorname{div}}(g) - \mathscr{W}
\right)}\right\vert_{{\mathscr{U}}}}
$ is trivial on some open neighborhood of $p$, then $g_1$ is an invertible regular function around $p$, and hence $- \log \left|{\left.{g_1}\right\vert_{{X}}}\right| = 0$. In this case, we have $- \log \left|{\left.{g}\right\vert_{{X}}}\right| = \varphi$.
500
Stepwise vertical divisors {#subsection:stepwise}
--------------------------
Let $X$ be a connected smooth projective curve over $K$ of genus $g$. In this subsection, we assume that $g \geq 2$. Let $\mathscr{X}$ be a strictly semistable model of $X$.
In this subsection, we define the stepwise vertical divisors on ${{\mathscr{X}}}$ associated to an oriented edge of disconnected type and those associated to an edge of connected type. These divisors is defined as the sum of suitable fundamental vertical divisors. They will be mainly used to construct a tropicalization map that is unimodular over the edge.
Let $e$ be an edge of $\Gamma_{\min}$, i.e., $e \in E ( \mathscr{X}^{{\operatorname{st}}} )$, where $\mathscr{X}^{{\operatorname{st}}}$ is the stable model of $X$; see the last paragraph in §\[subsec:properties:skeleta\]. Let $E$ be the corresponding maximal $(-2)$-chain in $\mathscr{X}_s$ (cf. Lemma \[lemma:make:clear\]).
### Stepwise vertical divisor for an edge of disconnected type {#stepwise-vertical-divisor-for-an-edge-of-disconnected-type .unnumbered}
We consider the case where $e \in E ( \mathscr{X}^{{\operatorname{st}}} )$ is of disconnected type.
Since we use orientations on $e$, we first fix the notation. Let $\lambda_e$ be the length of $e$. Then there are exactly two isometries from $[0 , \lambda_e]$ to $e$, and an *orientation* means one of the choices of an isometry. Once an orientation on $e$ is fixed, we usually denote by $\vec{e}$ the edge $e$ with the orientation, and we call $\vec{e}$ an *oriented edge*. For an oriented edge $\vec{e}$ with orientation $\gamma : [0 , \lambda_e]
\to e$, we call $\gamma (0)$ the *head of $\vec{e}$* and $\gamma (\lambda_e )$ the *tail of $\vec{e}$*. We remark that giving an orientation on $e$ is equivalent to specifying an end vertex of $e$ to be the head.
Fix an orientation on $e$, and let $\vec{e}$ denote the oriented edge. Let $v_0$ be the head of $\vec{e}$. Let $E_0$ be the irreducible component of $\mathscr{X}_s$ with $[E_0] = v_0$. Let $E_{1} , \ldots , E_{r-1}$ be the irreducible components of $E$, where we put the numbering in such a way that $\# (E_i \cap E_{i+1} ) = 1$ for $i=0 , \ldots , r-2$. We remark that this numbering is unique for $\vec{e}$. Since $e$ is of disconnected type, $\mathscr{X}_s - E$ consists of two connected components; let $D$ be the connected component with $E_{0} \subset D$ and let $T$ be the other connected component. Let $p_i$ denote the node with $\{ p_i \} = E_i \cap E_{i+1}$ for $i = 0
, \ldots , r-2$ and let $p_{r-1}$ be the node with $\{ p_{r-1 }\} = E_{r-1} \cap T$. Let $E_r \in {\operatorname{\operatorname{Irr}}}(T)$ with $p_{r-1} \in E_r$. See Figure \[figure:disconn:type\].
$$\setlength\unitlength{0.08truecm}
\begin{picture}(100, 65)(0,0)
\qbezier(0, 5)(12.5, 15)(25, 5)
\put(-20,7){\line(1,0){25}}
\put(10,7){\line(3,4){17}}
\put(15,20){\line(1,0){25}}
\put(30,17){\line(3,4){14}}
\put(55,28){\line(1,0){25}}
\put(65,25){\line(3,4){17}}
\qbezier(70, 35)(82.5, 45)(95, 35)
\put(85,38){\line(1,0){25}}
\put(45, 27){$\cdots$}
\put(0, 10){$E_0$}
\put(40, 18){$E_2$}
\put(18, 27){$E_1$}
\put(70, 47){$E_{r-1}$}
\put(86, 42){$E_r$}
\put(81, 26){$E_{r-2}$}
\put(12.2,10){\circle*{1.5}}
\put(20,20){\circle*{1.5}}
\put(67.2,28){\circle*{1.5}}
\put(75, 38){\circle*{1.5}}
\put(12, 5){$p_0$}
\put(20, 15){$p_1$}
\put(67, 23){$p_{r-2}$}
\put(75, 33){$p_{r-1}$}
\put(-22, 12){$\overbrace{\phantom{AAAAAAAAA}}$}
\put(-10, 20){$D$}
\put(85,45){$\overbrace{\phantom{AAAAAAAA}}$}
\put(97,53){$T$}
\put(15, 50){$\overbrace{\phantom{AAAAAAAAAAAAAAAA}}$}
\put(42,60){$E$}
\end{picture}$$
For each $j =0, \ldots , r-1$, the connected curve $\sum_{i = j+1}^{r-1} E_i + T$ satisfies condition (F) in §\[subsec:fvd\], and let $\mathscr{F}_j$ be the fundamental vertical divisor associated to $\sum_{i = j+1}^{r-1} E_i + T$. (Note that $\mathscr{F}_{r-1}$ is the fundamental vertical divisor associated to $T$.) Further, we set $$\mathscr{V}_{\vec{e}} :=
\sum_{i=0}^{r-1} \mathscr{F}_i,$$ which we call the *stepwise vertical divisor associated to $\vec{e}$.* For the opposite orientation $\vec{e}^{\, opp}$, we also have the stepwise vertical divisor $\mathscr{V}_{\vec{e}^{\, opp}}$. When we say a *stepwise vertical divisor for $e$*, we mean $\mathscr{V}_{\vec{e}}$ or $\mathscr{V}_{\vec{e}^{\, opp}}$.
We compute the degree of the stepwise vertical divisor over each $C \in {\operatorname{\operatorname{Irr}}}( \mathscr{X}_s)$. Noting Remark \[rem:restrictionofcartierdivisor\], we obtain $$\label{eqn:degreepositivetype}
\deg
\left(
{\left.{{{\mathcal{O}}}_{\mathscr{X}} ( \mathscr{V}_{\vec{e}})}\right\vert_{{C}}}
\right)
=
\begin{cases}
1
& \text{if $C = E_0$}
\\
-1
& \text{if $C = E_{r}$}
\\
0
&
\text{otherwise.}
\end{cases}$$
For the order at any $C \in {\operatorname{\operatorname{Irr}}}( \mathscr{X}_s )$, we have $$\label{eqn:ordVdisconnected}
{\operatorname{ord}}_{C} \left( \mathscr{V}_{\vec{e}} \right)
=
\begin{cases}
0
&
\text{if $C \in {\operatorname{\operatorname{Irr}}}(D)$,}
\\
\sum_{j=0}^{i-1} \lambda_{j} &
\text{if $C = E_{i}$ ($i = 1 , \ldots , r-1$ ),}
\\
\sum_{j=0}^{r-1} \lambda_{j}
&
\text{if $C \in {\operatorname{\operatorname{Irr}}}(T)$,}
\end{cases}$$ where $\lambda_j$ is the multiplicity at $p_j$, which also equals the length of the canonical $1$-simplex $\Delta_{p_j}$ corresponding to $p_j$ (cf. §\[subsection:skeleta\]).
We explain how a stepwise vertical divisor looks by pretending that $R$ is a discrete valuation ring and $\mathscr{X}$ is a regular scheme. (This is not rigorous and only intuitive because we are working over a non-Noetherian ring $R$.) In this situation, the stepwise vertical divisor $\mathscr{V}_{\vec{e}}$ for an oriented edge of disconnected type is $$\text{
``$E_1 + 2 E_2 + \cdots + (r-1) E_{r-1} + r T$.''}$$ Similarly, for an edge $e$ of connected type, the stepwise vertical divisor $\mathscr{V}_e$ defined below is $$\text{
``$E_1 + 2 E_2 + \cdots + (\ell -1) E_{\ell -1} + \ell E_{\ell}
+ (\ell -1) E_{\ell +1} + \cdots + 2 E_{2\ell-2} + E_{2\ell-1}$.''}$$
### Stepwise vertical divisor for an edge of connected type {#stepwise-vertical-divisor-for-an-edge-of-connected-type .unnumbered}
Next, we consider the case where $e$ is of connected type. Here we assume that $E$ has symmetric multiplicities (cf. Definition \[def:symmetricmultiplicity\]), and let $E_{1} , \ldots , E_{2 \ell - 1}$ be the irreducible components of $E$ with the numbering such that $\# (E_{i} \cap E_{i+1}) = 1$ for $i=1 , \ldots , 2 \ell -2$. Let $E_0$ be the irreducible component of $\mathscr{X}_s - E$ with $E_{0} \cap E_1 \neq \emptyset$ and let $E_{2 \ell}$ be the irreducible component of $\mathscr{X}_s - E$ with $E_{2 \ell} \cap E_{2 \ell - 1} \neq \emptyset$. We remark that $E_0 = E_{2 \ell}$ if and only if $e$ is a loop. Let $p_i$ denote the node with $p_i \in E_{i} \cap E_{i+1}$ for $i=0 ,
\ldots , 2 \ell -1$. See Figure \[figure:conn:type\].
$$\setlength\unitlength{0.08truecm}
\begin{picture}(180, 70)(0,0)
\put(57,48){\line(4,1){16}}
\put(102,48){\line(-4,1){16}}
\put(50, 40){$\cdots$}
\put(100, 40){$\cdots$}
\put(64,51){\line(1,0){30}}
\put(76, 53){$E_\ell$}
\put(45, 48){$E_{\ell-1}$}
\put(104, 48){$E_{\ell+1}$}
\put(69,51){\circle*{1.5}}
\put(90,51){\circle*{1.5}}
\put(68, 46){$p_{\ell-1}$}
\put(90, 46){$p_{\ell}$}
\qbezier(0, 5)(17.5, 15)(35, 5)
\put(25,7){\line(1,6){3}}
\put(25,18){\line(1,1){16}}
\put(32,28){\line(2,1){16}}
\put(27,20){\circle*{1.5}}
\put(133,20){\circle*{1.5}}
\put(30, 19){$p_1$}
\put(120, 19){$p_{2\ell-2}$}
\put(25.5,9){\circle*{1.5}}
\put(27, 11){$p_0$}
\put(134.5,9){\circle*{1.5}}
\put(123, 11){$p_{2 \ell-1}$}
\qbezier(125, 5)(142.5, 15)(160, 5)
\put(135,7){\line(-1,6){3}}
\put(135,18){\line(-1,1){16}}
\put(128,28){\line(-2,1){16}}
\put(3, 10){$E_0$}
\put(20,23){$E_1$}
\put(37,36){$E_2$}
\put(118,36){$E_{2\ell-2}$}
\put(135,23){$E_{2\ell-1}$}
\put(152, 10){$E_{2 \ell}$}
\put(28, 55){$\overbrace{\phantom{AAAAAAAAAAAAAAAAAAAAAAAAAA}}$}
\put(76, 65){$E$}
\end{picture}$$
Since $E$ has symmetric multiplicities, the connected curve $\sum_{i = j+1}^{2\ell - 1 - j} E_i$ satisfies condition (F) in §\[subsec:fvd\] for each $j = 0 , \ldots \ell-1$, and hence we have the fundamental vertical divisor $\mathscr{F}_j$ with support $\sum_{i = j+1}^{2\ell - 1 - j} E_i$. We remark that the divisor $\mathscr{F}_i$ does not depend on such numbering of the irreducible components of $E$, and hence well-defined for $e$. We set $$\mathscr{V}_{e} :=
\sum_{i = 0}^{\ell - 1}
\mathscr{F}_i,$$ which we call the *stepwise vertical divisor associated to $e$.*
We compute the degree of the stepwise vertical divisor over each $C \in {\operatorname{\operatorname{Irr}}}( \mathscr{X}_s)$. If $e$ is a loop, then $E_0 = E_{2\ell}$, and noting Remark \[rem:restrictionofcartierdivisor\], we obtain $$\label{eqn:degreetype0:pre}
\deg
\left(
{\left.{{{\mathcal{O}}}_{\mathscr{X}} ( \mathscr{V}_{e})}\right\vert_{{C}}}
\right)
=
\begin{cases}
2
& \text{if $C = E_0 \;(=E_{2 \ell})$,}
\\
-2
& \text{if $C = E_{\ell}$,}
\\
0
&
\text{otherwise.}
\end{cases}$$ If $e$ is not a loop, then $E_0 \neq E_{2\ell}$, and we have $$\label{eqn:degreetype0}
\deg
\left(
{\left.{{{\mathcal{O}}}_{\mathscr{X}} ( \mathscr{V}_{e})}\right\vert_{{C}}}
\right)
=
\begin{cases}
1
& \text{if $C = E_0$ or $C = E_{2 \ell}$,}
\\
-2
& \text{if $C = E_{\ell}$,}
\\
0
&
\text{otherwise.}
\end{cases}$$
For the orders, regardless $e$ being a loop or not, we have $${\operatorname{ord}}_{E_i} \left( \mathscr{V}_{e} \right)
=
{\operatorname{ord}}_{E_{2 \ell - i}} \left( \mathscr{V}_{e} \right)
=
\sum_{j=0}^{i-1} \lambda_{j}$$ for $i = 1, \ldots, \ell$, where $\lambda_j$ is the multiplicity at the node $p_j$, which equals the length of the canonical $1$-simplex $\Delta_{p_j}$ corresponding to $p_j$ (cf. §\[subsection:skeleta\]). Further, ${\operatorname{ord}}_{C} \left( \mathscr{V}_{e} \right) = 0$ for any $C \in {\operatorname{\operatorname{Irr}}}(\mathscr{X}_s) \setminus {\operatorname{\operatorname{Irr}}}( E)$.
Edge-base sections and edge-unimodularity sections
--------------------------------------------------
Let $X$ be a connected smooth projective curve over $K$ of genus $g$. From here on to the end of this section, assume that $g \geq 2$. Let $L$ be a line bundle over $X$. In this subsection, we fix an edge $e \in E(\mathscr{X}^{{\operatorname{st}}})$. Let $({{\mathscr{X}}}, {{\mathscr{L}}})$ be a model of $(X, L)$ such that ${{\mathscr{X}}}$ is a good model in Definition \[def:goodmodel:new\]. Recall that ${\operatorname{Sing}}( \mathscr{X}_s)_{\subset e} :=
\{ p \in {\operatorname{Sing}}( \mathscr{X}_s ) \mid
\Delta_p \subset e \}$ (cf. (\[align:nodes:subset\])), where $\Delta_p$ denotes the canonical $1$-simplex corresponding to $p$ in $S({{\mathscr{X}}}) = \Gamma_{\min}$.
\[def:edge-base:section\] A nonzero global section $\widetilde{s} \in H^0(\mathscr{L})$ is called an *$e$-base section* if $\widetilde{s} (p) \neq 0$ for any $p \in {\operatorname{Sing}}( \mathscr{X}_s)_{\subset e}$.
In the following, we define the notion of edge-unimodularity sections for an edge according to the type of the edge.
### Edge of disconnected type {#edge-of-disconnected-type .unnumbered}
Assume that $e \in E({{\mathscr{X}}}^{{\operatorname{st}}})$ is of disconnected type.
A nonzero global section $\widetilde{s} \in H^0(\mathscr{L})$ is called an *$e$-unimodularity section* if there exists an orientation on $e$ such that ${\operatorname{div}}(\widetilde{s}) - \mathscr{V}_{\vec{e}}$ is effective on ${{\mathscr{X}}}$ and is trivial on some open neighborhood of ${\operatorname{Sing}}( \mathscr{X}_s)_{\subset e}$, where $\vec{e}$ denotes the oriented edge.
The next lemma asserts that an $e$-base section and an $e$-unimodularity section give a tropicalization map which is unimodular over the edge $e$.
\[lemma:actuallyunimodular:discon\] Let $e \in E({{\mathscr{X}}}^{{\operatorname{st}}})$ be an edge of disconnected type. Suppose that there exists a model $({{\mathscr{X}}}, {{\mathscr{L}}})$ of $(X, L)$ such that ${{\mathscr{X}}}$ is a good model cf. Definition and such that there exist an $e$-base section $\widetilde{s}^{(e)}_{0} \in H^0({{\mathscr{L}}})$ and an $e$-unimodularity section $\widetilde{s}^{(e)}_{1} \in H^0({{\mathscr{L}}})$. Set $s^{(e)}_0 := {\left.{\widetilde{s}^{(e)}_0}\right\vert_{{X}}}$ and $s^{(e)}_{1} := {\left.{\widetilde{s}^{(e)}_{1}}\right\vert_{{X}}}$, which are non-zero global sections of $L$. We define a function $\varphi^{(e)} :
X^{{\operatorname{an}}}\setminus X(K) \to {{\mathbb{R}}}$ by $\varphi^{(e)} := - \log \left| s^{(e)}_{1}
/ s^{(e)}_{0} \right|$. Let $\lambda_e$ denote the length of $e$. Then $\varphi^{(e)}$ gives an isometry $e \to [0 , \lambda_e ]$.
[[*Proof.*]{}]{}To ease notation, we omit the superscript $(e)$ in the proof. We define the rational function $g$ on $\mathscr{X}$ by $
g:=\widetilde{s}_{1} / \widetilde{s}_0
$. Since $\widetilde{s}_{1}$ is an $e$-unimodularity section, there exists an oriented edge $\vec{e}$ such that ${\operatorname{div}}\left(
\widetilde{s}_{1}
\right) - \mathscr{V}_{\vec{e}}$ is effective on ${{\mathscr{X}}}$ and is trivial on some open neighborhood of ${\operatorname{Sing}}( \mathscr{X}_s )_{\subset e}$. Since $\widetilde{s}_0$ does not have zero on some open neighborhood of ${\operatorname{Sing}}( \mathscr{X}_s )_{\subset e}$, there exists an open neighborhood $\mathscr{U}$ of ${\operatorname{Sing}}( \mathscr{X}_s )_{\subset e}$ such that ${\left.{{\operatorname{div}}\left( g \right)
- \mathscr{V}_{\vec{e}}}\right\vert_{{\mathscr{U}}}}$ is trivial.
We use the notation used in §\[subsection:stepwise\]; in particular, we have $E = \sum_{i=1}^{r-1} E_i$, $[E_0]$ is the head of $\vec{e}$, $E_0 \subset D$, $E_r \subset T$, and $\{ p_i \} = E_i \cap E_{i+1}$ for $i = 0, \ldots r-1$. For each $i = 0 , \ldots , r-1$, we take a local étale atlas $\psi_{i} : \mathscr{U}_{i} \to {{\operatorname{Spec}}}( R [x,y]/ (xy - \varpi_{i}))$ that distinguishes $p_i$ such that $\mathscr{U}_i
\subset \mathscr{U}$. Exchanging $x$ and $y$ if necessary, we may assume that in the special fiber $\left(
\mathscr{U}_{i}
\right)_s$ of $\mathscr{U}_i$, the branch $E_{i+1}$ at $p_i$ is defined by $\psi_i^\ast (y)$. Since ${\left.{{\operatorname{div}}\left( g \right)
- \mathscr{V}_{\vec{e}}}\right\vert_{{\mathscr{U}}}}$ is trivial, $g$ is a local equation of $\mathscr{V}_{\vec{e}}$ at each $p_i$. By (\[eqn:ordVdisconnected\]), it follows that for any $i = 0 , \ldots , r-1$, there exists a unit $u_i \in {{\mathcal{O}}}_{\mathscr{X}} ( \mathscr{U}_i)^{\times}$ such that $g = u_i \prod_{j=0}^{i-1} \varpi_{j}
\psi_i^\ast (y)$ over $\mathscr{U}_i$; indeed, see the properties (i)–(iii) in the definition of fundamental vertical divisor and the definition of $\mathscr{V}_{\vec{e}}$ in §\[subsec:fvd\]. Note that $v_K
\left(
\prod_{j=0}^{i-1} \varpi_{j}
\right) = \sum_{j = 0}^{i-1} \lambda_j$, where $v_K = - \log |\cdot|_{K}$. Since $\varphi = - \log \left| {\left.{g}\right\vert_{{X^{{{\operatorname{an}}}}}}} \right|$, $\varphi =
\sum_{j = 0}^{i-1} \lambda_j
- \log \left| {\left.{\psi_1^{\ast} (y)}\right\vert_{{X^{{{\operatorname{an}}}}}}} \right|$. It follows from Lemma \[lem:forinjectivity\](1) that $\varphi$ maps $\Delta_{p_i}$ to $\left[ \sum_{j = 0}^{i-1} \lambda_j ,
\sum_{j = 0}^{i} \lambda_j
\right]$ isometrically. Since $\lambda_e = \sum_{j = 0}^{r-1} \lambda_j$, this shows that $\varphi$ maps $e = \Delta_{E} = \bigcup_{i=0}^{r-1} \Delta_{p_i}$ to $\left[ 0,
\lambda_e \right]$ isometrically.
500
### Edge of connected type {#edge-of-connected-type .unnumbered}
Assume that $e \in E({{\mathscr{X}}}^{{\operatorname{st}}})$ is of connected type. In this case, we divide $e$ at the middle point of $e$ into two $1$-simplices, and let $e^\prime$ denote one of them.
A nonzero global section $\widetilde{s} \in H^0(\mathscr{L})$ is called an *$e^\prime$-unimodularity section* if ${\operatorname{div}}(\widetilde{s}) - \mathscr{V}_{e}$ is effective on ${{\mathscr{X}}}$ and is trivial on some open neighborhood of ${\operatorname{Sing}}( \mathscr{X}_s)_{\subset e^\prime}$.
\[lemma:actuallyunimodular:conn\] Let $e \in E({{\mathscr{X}}}^{{\operatorname{st}}})$ be of connected type and let $e^\prime$ be as above. Suppose that there exists a model $({{\mathscr{X}}}, {{\mathscr{L}}})$ of $(X, L)$ such that ${{\mathscr{X}}}$ is a good model and that there exist an $e$-base section $\widetilde{s}^{(e^\prime)}_{0} \in H^0({{\mathscr{L}}})$ and an $e^\prime$-unimodularity section $\widetilde{s}^{(e^\prime)}_1 \in H^0({{\mathscr{L}}})$. Set $s^{(e^\prime)}_0 := {\left.{\widetilde{s}^{(e^\prime)}_0}\right\vert_{{X}}}$ and $s^{(e^\prime)}_{1} := {\left.{\widetilde{s}^{(e^\prime)}_{1}}\right\vert_{{X}}}$, which are non-zero global sections of $L$. We define a function $\varphi^{(e^\prime)} :
X^{{\operatorname{an}}}\setminus X(K) \to {{\mathbb{R}}}$ by $\varphi^{(e^\prime)} := - \log \left| s^{(e^\prime)}_{1}
/ s^{(e^\prime)}_{0} \right|$. Let $\lambda_{e^\prime}$ denote the length of $e^\prime$. Then $\varphi^{(e^\prime)}$ gives an isometry $e^\prime \to [0 , \lambda_{e^\prime} ]$.
Here we place a superscript $(e^\prime)$ to an edge-base section $\widetilde{s}^{(e^\prime)}_{0}$ in (1), because a model $({{\mathscr{X}}}, {{\mathscr{L}}})$ depends on $e^\prime$.
[[*Proof.*]{}]{}We can prove the lemma by almost the same idea as in Lemma \[lemma:actuallyunimodular:discon\], so that we omit the details.
500
Unimodular tropicalization {#subsec:edge:good:model:disconnected}
--------------------------
In view of Lemmas \[lemma:actuallyunimodular:discon\] and \[lemma:actuallyunimodular:conn\], we would like to prove the following Proposition \[prop:existence:sections:stronger\] to complete the proof of a unimodular tropicalization of the minimal skeleton for $g \geq 2$. Recall that a good model is defined in Definition \[def:goodmodel:new\] and an edge-base section is defined in Definition \[def:edge-base:section\]. Given a model $({{\mathscr{X}}}, {{\mathscr{L}}})$ of $(X, L)$ such that ${{\mathscr{X}}}$ is a good model, we put ${{\mathscr{M}}}:= {{\mathscr{L}}}\otimes \omega_{{{\mathscr{X}}}/R}^{\otimes -1}$ as before, and for a line bundle ${{\mathscr{N}}}$ over ${{\mathscr{X}}}$, let $D_{{{\mathscr{N}}}}$ be the divisor on $\Gamma_{\min}$ defined in .
\[prop:existence:sections:stronger\] Let $X$ be a connected smooth projective curve over $K$ of genus $g \geq 2$, and let $L$ be a line bundle over $X$. Suppose that $\deg(L) \geq 3g-1$. Let $e \in E({{\mathscr{X}}}^{{\operatorname{st}}})$.
1. Suppose that $e$ is of disconnected type. Fix an orientation on $e$ and let $\vec{e}$ denote the oriented edge. Then there exists a model $({{\mathscr{X}}}, {{\mathscr{L}}})$ of $(X, L)$ such that $\mathscr{X}$ is a good model with the following properties there exist sections $\widetilde{s}^{(e)}_{0} , \widetilde{s}^{(e)}_{1} \in H^0({{\mathscr{L}}})$ such that $\widetilde{s}^{(e)}_{0}(p) \neq 0$ for any $p \in {\operatorname{Sing}}({{\mathscr{X}}}_s)$ and such that ${\operatorname{div}}\left( \widetilde{s}^{(e)}_{1} \right)
- \mathscr{V}_{\vec{e}}$ is effective on ${{\mathscr{X}}}$ and is trivial on some neighborhood of ${\operatorname{Sing}}( \mathscr{X}_s )$. In particular, $\widetilde{s}^{(e)}_{0}$ is an $e$-base section, and $\widetilde{s}^{(e)}_{1}$ is an $e$-unimodularity section.
2. Suppose that $e$ is of connected type. Let $e^\prime$ be a $1$-simplex that appears by subdividing $e$ at the middle point, i.e., let $e^\prime$ be a half of $e$. Then there exist a model $({{\mathscr{X}}}, {{\mathscr{L}}})$ of $(X, L)$ such that ${{\mathscr{X}}}$ is a good model and such that there exist an $e$-base section $\widetilde{s}^{(e^\prime)}_{0} \in H^0({{\mathscr{L}}})$ and an $e^\prime$-unimodularity section $\widetilde{s}^{(e^\prime)} \in H^0({{\mathscr{L}}})$.
To prove Proposition \[prop:existence:sections:stronger\], we show the following lemma based on Proposition \[prop:pregoodmodel\].
\[lemma:forbasesectionUT:dc\] Let $X$ be a connected smooth projective curve over $K$ of genus $g \geq 2$, and let $L$ be a line bundle over $X$. Assume that $\deg (L) \geq 3g-1$. Take an edge $e \in E({{\mathscr{X}}}^{{\operatorname{st}}})$. Let $v$ be an end vertex of $e$. Let $({{\mathscr{X}}}, {{\mathscr{L}}})$ be a model of $(X,L)$. Assume that $\mathscr{X}$ is a good model, $D_{{{\mathscr{M}}}} - [v] \geq 0$ on the minimal skeleton $\Gamma_{\min}$, and $({{\mathscr{X}}}, {{\mathscr{L}}})$ satisfies conditions and in Proposition . Then the following hold.
1. The line bundle $\mathscr{L}$ is free at any $q \in {\operatorname{Sing}}( \mathscr{X}_s )$.
2. Suppose that $e$ is of disconnected type. Put the orientation on $e$ with head $v$, and let $\vec{e}$ denote the oriented edge. Then $\mathscr{L} ( - \mathscr{V}_{\vec{e}})$ is free at any $q \in {\operatorname{Sing}}( \mathscr{X}_s )$.
3. Suppose that $e$ is of connected type. Divide $e$ into two $1$-simplices at the middle point of $e$, and let $e^\prime$ be one of them with $v \in e^\prime$. Then $\mathscr{L} ( - \mathscr{V}_{e})$ is free at any $q \in {\operatorname{Sing}}( \mathscr{X}_s )_{\subset e^\prime}$.
[[*Proof.*]{}]{}To show (1), we use Lemma \[lemma:freeatnode:graphversion:emptyset\]. Note that $\deg ( D_{{{\mathscr{M}}}} ) \geq g+1$ and $D_{{{\mathscr{M}}}} \geq 0$; indeed, the inequality on the degree follows from $\deg (L) \geq 3g-1$, and the effectivity follows from $D_{{{\mathscr{M}}}} - [v] \geq 0$. We take any $q \in {\operatorname{Sing}}( \mathscr{X}_s )$. If $q$ is of connected type, then $\Gamma_{\min}\setminus \mathrm{relin}(\Delta_q)$ is connected and $\deg({\left.{D_{{{\mathscr{M}}}}}\right\vert_{{\Gamma_{\min}\setminus \mathrm{relin}(\Delta_q)}}})
\geq (g+1) - \deg({\left.{D_{{{\mathscr{M}}}}}\right\vert_{{\mathrm{relin}(\Delta_q)}}})
\geq g > 1$, and thus Lemma \[lemma:freeatnode:graphversion:emptyset\] concludes that ${{\mathscr{L}}}$ is free at $q$. Suppose that $q$ is of disconnected type. Then $\Delta_q$ is a part of an edge of disconnected type. Then any connected component $\Gamma^\prime$ of $\Gamma_{\min}\setminus \mathrm{relin}(\Delta_q)$ contains an island $\Gamma_i$ of $\Gamma_{\min}$, and Proposition \[prop:pregoodmodel\](i) shows that $\deg({\left.{D_{{{\mathscr{M}}}}}\right\vert_{{\Gamma^\prime}}}) \geq 1$. It follows from Lemma \[lemma:freeatnode:graphversion:emptyset\] that ${{\mathscr{L}}}$ is free at $q$. Thus we have (1).
Assertion (2) can be shown similarly. Let $v^\prime$ the end vertex of $e$ with $v \neq v^\prime$. By , we have $
D_{\mathscr{M} ( - \mathscr{V}_{\vec{e}})}
= D_{\mathscr{M}} - [v] + [v^\prime]
$. Further, since $D_{{{\mathscr{M}}}} - [v] \geq 0$, we have $D_{\mathscr{M} ( - \mathscr{V}_{\vec{e}})} \geq 0$. Then the same argument as in (1) proves that ${{\mathscr{L}}}( - \mathscr{V}_{\vec{e}})$ is free at any $q \in {\operatorname{Sing}}( \mathscr{X}_s )$. Thus we have (2).
Let us prove (3). If $e$ is not a loop, then let $v^\prime$ be the end vertex of $e$ other than $v$; if $e$ is a loop, then set $v = v^\prime$. Let $w$ denote the middle point of $e$. By and , we have $
D_{\mathscr{M} ( - \mathscr{V}_{e})}
= D_{\mathscr{M}} - [v] - [v^\prime] + 2[w]
$. We take any $q \in {\operatorname{Sing}}( \mathscr{X}_s )_{\subset e^\prime}$.
Suppose that $D_{\mathscr{M} ( - \mathscr{V}_{e})}(v^\prime) \geq 0$. Then, since $D_{{{\mathscr{M}}}} - [v] \geq 0$, we have $D_{\mathscr{M} ( - \mathscr{V}_{e})}
\geq 0$. Since $\Delta_q$ is a part of the edge $e$ of connected type, $\Gamma_{\min}\setminus \mathrm{relin}(\Delta_q)$ is connected, and Proposition \[prop:pregoodmodel\](i) shows that $$\deg({\left.{D_{{{\mathscr{M}}}( - \mathscr{V}_{e})}}\right\vert_{{\Gamma_{\min}\setminus \mathrm{relin}(\Delta_q)}}})
= \deg({\left.{D_{{{\mathscr{M}}}}}\right\vert_{{\Gamma_{\min}\setminus \mathrm{relin}(\Delta_q)}}})
\geq (g+1) - \deg\left({\left.{D_{{{\mathscr{M}}}}}\right\vert_{{ \mathrm{relin}(\Delta_q)}}}\right)
\geq g \geq 1.$$ It follows from Lemma \[lemma:freeatnode:graphversion:emptyset\] that ${{\mathscr{L}}}( - \mathscr{V}_{e})$ is free at $q$.
$$\setlength\unitlength{0.08truecm}
\begin{picture}(100, 65)(0,0)
\put(5, 25){\line(1,0){25}}
\qbezier(30, 25)(45, 65)(60, 25)
\put(60, 25){\line(1,0){25}}
\put(60, 25){\line(0,-1){15}}
\put(30, 25){\line(0,-1){15}}
\put(30, 25){\circle*{1.5}}
\put(60, 25){\circle*{1.5}}
\put(45, 45){\circle*{1.5}}
\put(75, 25){\circle*{1.5}}
\put(33, 32){\circle*{1.5}}
\put(36, 38){\circle*{1.5}}
\multiput(33.5, 10)(2,0){12}{\line(1,0){1}}
\put(25, 0){$\Gamma_1 := \Gamma_{\min}\setminus{\operatorname{relin}}(\Delta_q)$}
\put(26, 20){$v$}
\put(43, 49){$w$}
\put(61, 19){$v^\prime$}
\put(74, 20){$u$}
\put(35, 32){$\Delta_q$}
\put(28, 35){$e^\prime$}
\put(57, 35){$e^{\prime\prime}$}
\put(44, 35){$e$}
\put(45, 60){$\Gamma_{11}^\circ$}
\put(72, 36){$\Gamma_{12}^\circ$}
\put(35.5, 50){$\overbrace{\phantom{AAAAAA}}$}
\put(60.5, 26){$\overbrace{\phantom{AAAAAAA}}$}
\thicklines
\put(33, 32){\line(1,2){3}}
\end{picture}$$ \[figure:for:lemma:forbasesectionUT:dc\]
Suppose that $D_{\mathscr{M} ( - \mathscr{V}_{e})}(v^\prime) < 0$. Since $D_{{{\mathscr{M}}}} - [v] \geq 0$, this means that $D_{\mathscr{M} ( - \mathscr{V}_{e})} (v') =-1$. We use Lemma \[lemma:freeatnode:graphversion:nonemptyset\]. Set $V := \{ x \in \Gamma_{\min} \mid D_{\mathscr{M} ( - \mathscr{V}_{e})} (x) < 0 \}$. In this case, we have $V= \{v^\prime\}$. Set $\Gamma_1:= \Gamma_{\min}\setminus \mathrm{relin}(\Delta_q)$. Since $( \mathscr{X} , \mathscr{L})$ satisfies condition (ii) in Proposition \[prop:pregoodmodel\], we have $\deg\left({\left.{(D_{\mathscr{M}}-[v])}\right\vert_{{\mathrm{relin}(e)}}}\right) \leq 1$. It follows that $$\deg\left({\left.{(D_{\mathscr{M}}-[v])}\right\vert_{{\Gamma_{\min}\setminus\mathrm{relin}(e)}}}\right)
= \deg\left(D_{\mathscr{M}}-[v]\right) -
\deg\left({\left.{(D_{\mathscr{M}}-[v])}\right\vert_{{\mathrm{relin}(e)}}}\right)
\geq g - 1 \geq 1.$$ Since $D_{\mathscr{M} ( - \mathscr{V}_{e})}(v^\prime) = -1$, we have $(D_{\mathscr{M}} - [v])(v') = -1$. Further, we have $v' \in
\Gamma_{\min}\setminus\mathrm{relin}(e)$. It follows form the above inequality that there exists $u \in \Gamma_{\min}\setminus\mathrm{relin}(e)$ such that $u \neq v^\prime$ and $(D_{\mathscr{M}}-[v])(u) \geq 1$.
Let $\Gamma_{11}^\circ$ be the connected component of $\Gamma_1 \setminus \{v^\prime\}$ that contains $w$, and let $\Gamma_{12}^\circ$ be the connected component of $\Gamma_1 \setminus \{v^\prime\}$ that contains $u$. Note that $\Gamma_{11}^\circ \neq \Gamma_{12}^\circ$. Indeed, since $\Delta_q \subset e^\prime$, we have $\Gamma_{11}^\circ \subset \mathrm{relin}(e)$; on the other hand, since $u \notin \mathrm{relin}(e)$, $\Gamma_{12}^\circ \not\subset \mathrm{relin}(e)$. Since the coefficients of $D_{\mathscr{M}(- \mathscr{V}_{e})}$ are nonnegative except at $v'$, we have $$\deg\left({\left.{D_{\mathscr{M}(- \mathscr{V}_{e})}}\right\vert_{{\Gamma_{11}^\circ}}}\right)
\geq \deg(2 [w]) = 2 \geq 1, \quad
\deg\left({\left.{D_{\mathscr{M}(- \mathscr{V}_{e})}}\right\vert_{{\Gamma_{12}^\circ}}}\right)
\geq \deg([u]) = 1.$$ Further, if we set $\Gamma_{1j} := \Gamma_{1j}^\circ \cup \{v^\prime\}$ for $j = 1, 2$, then the valence of $\Gamma_{11} \cup \Gamma_{12}$ at $v^\prime$ is at least $2 = - D_{\mathscr{M} ( - \mathscr{V}_{e})}(v^\prime) + 1$. Thus by Lemma \[lemma:freeatnode:graphversion:nonemptyset\], ${{\mathscr{L}}}( - \mathscr{V}_{e})$ is free at $q$.
500
[*Proof of Proposition \[prop:existence:sections:stronger\].*]{} We construct a model $( \mathscr{X} , \mathscr{L} )$. We take an end-vertex $v$ of $e$ as follows: if $e$ is of disconnected type, we take $v$ to be the head of $\vec{e}$ (for (1)); if $e$ is of connected type, we take $v$ such that $v \in e^\prime$ (for (2)). We apply Proposition \[prop:pregoodmodel\], where we take $v$ as $x$. Then we get a model $({{\mathscr{X}}}, {{\mathscr{L}}})$ such that: ${{\mathscr{X}}}$ is a good model; $D_{{{\mathscr{M}}}} - [v] \geq 0$ on $\Gamma_{\min}$; $({{\mathscr{X}}}, {{\mathscr{L}}})$ satisfies conditions (i) and (ii) in Proposition \[prop:pregoodmodel\]. By Lemma \[lemma:forbasesectionUT:dc\](1), ${{\mathscr{L}}}$ is free at any $p \in {\operatorname{Sing}}( \mathscr{X}_s )$, which implies that there exists a global section of $\widetilde{s}_{0} \in H^0({{\mathscr{L}}})$ that does not vanish at any $p \in {\operatorname{Sing}}( \mathscr{X}_s )$. Thus in assertion (1), setting $\widetilde{s}_{0}^{(e)}$ to be $\widetilde{s}_{0}$ proves the existence of $\widetilde{s}_{0}^{(e)}$; in assertion (2), setting $\widetilde{s}_{0}^{(e')}$ to be $\widetilde{s}_{0}$ proves that of $\widetilde{s}_{0}^{(e')}$.
Suppose that $e$ is of disconnected type. We show (1). By Lemma \[lemma:forbasesectionUT:dc\](2), ${{\mathscr{L}}}( - \mathscr{V}_{\vec{e}})$ is free at any $p \in
{\operatorname{Sing}}( \mathscr{X}_s )$, which implies that there exists a global section of $\widetilde{s}^{(e)}_{1,-} \in H^0( {{\mathscr{L}}}( - \mathscr{V}_{\vec{e}}))$ that does not vanish at any $p \in {\operatorname{Sing}}( \mathscr{X}_s )$. Let $\widetilde{s}^{(e)}_{1}$ be the image of $\widetilde{s}^{(e)}_{1,-}$ by the natural inclusion ${{\mathscr{L}}}( - \mathscr{V}_{\vec{e}})
\hookrightarrow {{\mathscr{L}}}$. Then $\widetilde{s}^{(e)}_{1}$ satisfies the required condition. This completes the proof of (1).
To show (2), suppose that $e$ is of connected type. Then the same argument as above which uses Lemma \[lemma:forbasesectionUT:dc\](3) instead of Lemma \[lemma:forbasesectionUT:dc\](2) shows the existence of an $e^\prime$-unimodularity section.
500
We are ready to construct a unimodular tropicalization of the minimal skeleton $\Gamma_{\min}$. In fact, we construct a tropicalization map that is not only unimodular but also separate two points which are in some special positions. To do this we prove one more lemma.
\[lemma:ES:d-c\] Let $e$ be an edge of disconnected type. Put an orientation on $e$, and let $\vec{e}$ denote the oriented edge. Let $\Gamma_1$ be the connected component of $\Gamma \setminus
\mathrm{relin}(e)$ containing the head of $\vec{e}$, and let $\Gamma_2$ be the other connected component. Let $(\mathscr{X} , \mathscr{L})$ be a model of $(X,L)$ such that $\mathscr{X}$ is a good model. Let $\widetilde{s}_0$ and $\widetilde{s}_1$ be global sections of $\mathscr{L}$ such that
1. $\widetilde{s}_0(p) \neq 0$ for any $p \in {\operatorname{Sing}}({{\mathscr{X}}}_s)$
2. ${\operatorname{div}}(\widetilde{s}_1) - \mathscr{V}_{\vec{e}}$ is effective on ${{\mathscr{X}}}$ and is trivial on some open neighborhood of ${\operatorname{Sing}}( \mathscr{X}_s )$.
Let $h$ denote the rational function that is the restriction to $X$ of the rational function $\widetilde{s}_1 / \widetilde{s}_0$ on $\mathscr{X}$. Set $\varphi := - \log |h|$. Let $\lambda_e$ denote the length of $e$. Then $\varphi ( \Gamma_1 ) = \{ 0 \}$ and $\varphi ( \Gamma_2 ) = \{ \lambda_e \}$.
[[*Proof.*]{}]{}We define the rational function $g$ on ${{\mathscr{X}}}$ by $g := \widetilde{s}_1 / \widetilde{s}_0$, so that $h = {\left.{g}\right\vert_{{X}}}$. By conditions (i) and (ii), there exists an open neighborhood $\mathscr{U}$ of ${\operatorname{Sing}}( \mathscr{X}_s )$ such that ${\operatorname{div}}(g) - \mathscr{V}_{\vec{e}} = \left({\operatorname{div}}(\widetilde{s}_1) - \mathscr{V}_{\vec{e}}\right)- {\operatorname{div}}(\widetilde{s}_0)$ is trivial on $\mathscr{U}$.
First, we take any $p \in {\operatorname{Sing}}( \mathscr{X}_s)_{\subset \Gamma_1}$. Let $C_1$ and $C_2$ be the irreducible components of $\mathscr{X}_s$ such that $p \in C_1 \cap C_2$. By the definition of $\mathscr{V}_{\vec{e}}$, we have ${\operatorname{ord}}_{C_i} ( \mathscr{V}_{\vec{e}}) = 0$ for $i=1,2$. By Lemma \[lem:forinjectivity\], it follows that $\varphi ( \Delta_p ) = 0$. Since $\Gamma_1 = \bigcup_{p \in {\operatorname{Sing}}( \mathscr{X}_s)_{\subset \Gamma_1}} \Delta_p$, this proves $\varphi ( \Gamma_1 ) = \{ 0 \}$.
Next, we take any $p' \in {\operatorname{Sing}}( \mathscr{X}_s)_{\subset \Gamma_2}$. Let $C_1'$ and $C_2'$ be the irreducible components of $\mathscr{X}_s$ such that $p' \in C_1' \cap C_2'$. By the definition of $\mathscr{V}_{\vec{e}}$, we have ${\operatorname{ord}}_{C_i'} ( \mathscr{V}_{\vec{e}}) = \lambda_e$ for $i=1,2$. By Lemma \[lem:forinjectivity\], it follows that $\varphi ( \Delta_p ) = \lambda_e$. Since $\Gamma_2 = \bigcup_{p' \in {\operatorname{Sing}}( \mathscr{X}_s)_{\subset \Gamma_2}} \Delta_{p'}$, this proves $\varphi ( \Gamma_2 ) = \{ \lambda_e \}$.
500
\[theorem:UT:canonical\] Let $X$ be a connected smooth projective curve over $K$ of genus $g \geq 2$, and let $L$ be a line bundle over $X$. Suppose that $\deg (L) \geq 3g-1$. Then there exist nonzero global sections $s_0 , \ldots , s_N \in H^{0} (L)$ such that the map $\varphi : X^{{\operatorname{an}}}\to {{\mathbb{T}}}{{\mathbb{P}}}^N$ defined by $$\varphi :=
( - \log |s_0| : \cdots :
- \log |s_N| )$$ gives a unimodular tropicalization of the minimal skeleton $\Gamma_{\min}$. Furthermore, we can take $s_0 , \ldots , s_N$ in such a way that $\varphi$ has the following properties.
1. Let $e \in E({{\mathscr{X}}}^{st})$ be of disconnected type. Then ${\left.{\varphi}\right\vert_{{e}}}$ is an isometry.
2. Let $e \in E({{\mathscr{X}}}^{st})$ be of connected type. Let $e^{\prime}$ and $e^{\prime\prime}$ be the $1$-simplices arising by dividing $e$ at the middle point. Then ${\left.{\varphi}\right\vert_{{e^{\prime}}}}$ and ${\left.{\varphi}\right\vert_{{e^{\prime\prime}}}}$ are isometries.
3. Let $e \in E({{\mathscr{X}}}^{st})$ be of disconnected type. Let $x , y \in \Gamma_{\min}$ be points that do not belong to the same connected component of $\Gamma_{\min} \setminus \mathrm{relin}(e)$. Then $\varphi (x) \neq \varphi (y)$ and $\varphi (x) , \varphi (y) \notin \varphi ( \mathrm{relin}(e) )$.
[[*Proof.*]{}]{}We write $E({{\mathscr{X}}}^{{\operatorname{st}}}) = \{e_i\}_{i=1}^{m_1} \amalg \{e_i\}_{i=m_1+1}^{m_1 + m_2}$, where $e_i$ ($1 \leq i \leq m_1$) are the edges of disconnected type and $e_i$ ($m_1 + 1 \leq i \leq m_1 + m_2$) are the edges of connected type. For $1 \leq i \leq m_1$, we give an orientation to $e_i$ and let $\vec{e_i}$ denote the oriented edge. Then Proposition \[prop:existence:sections:stronger\](1) gives a model $({{\mathscr{X}}}_i, {{\mathscr{L}}}_i)$ of $(X, L)$, a section $\widetilde{s}^{(e_i)}_{0} \in H^0({{\mathscr{L}}}_{i})$ such that $\widetilde{s}^{(e_i)}_{0}(p) \neq 0$ for any $p \in {\operatorname{Sing}}({{\mathscr{X}}}_s)$, and a section $\widetilde{s}^{(e_i)}_{1} \in H^0({{\mathscr{L}}}_{i})$ such that ${\operatorname{div}}\left(\widetilde{s}^{(e_i)}_{1} \right) - \mathscr{V}_{\vec{e_i}}$ is effective on ${{\mathscr{X}}}$ and is trivial on some open neighborhood of ${\operatorname{Sing}}( \mathscr{X}_s )$. Note in particular that $\widetilde{s}^{(e_i)}_{0}$ is an $e_i$-base section, and $\widetilde{s}^{(e_i)}_{1}$ is an $e_i$-unimodularity section.
For $m_1 + 1 \leq i \leq m_1+ m_2$, we divide $e_i$ into two $1$-simplices at the middle point and let $e_{i}^\prime$ and $e_{i}^{\prime\prime}$ be the $1$-simplices arising from this subdivision. Proposition \[prop:existence:sections:stronger\](2) gives a model $({{\mathscr{X}}}_{i}^\prime, {{\mathscr{L}}}_{i}^\prime)$ (resp. $({{\mathscr{X}}}_{i}^{\prime\prime}, {{\mathscr{L}}}_{i}^{\prime\prime})$) of $(X, L)$, an $e_{i}$-base section $\widetilde{s}^{(e_{i}^\prime)}_{0} \in H^0({{\mathscr{L}}}_{i}^\prime)$ (resp. $\widetilde{s}^{(e_{i}^{\prime\prime})}_{0} \in H^0({{\mathscr{L}}}_{i}^{\prime\prime})$), and an $e_{i}^\prime$-unimodularity section $\widetilde{s}^{(e_{i}^\prime)}_{1} \in H^0({{\mathscr{L}}}_{i}^\prime)$ (resp. an $e_{i}^{\prime\prime}$-unimodularity section $\widetilde{s}^{(e_{i}^{\prime\prime})}_{1} \in H^0({{\mathscr{L}}}_{i}^{\prime\prime})$).
Set $N := 2m_1 + 4 m_2 - 1$. We consider ${\left.{\widetilde{s}^{(e_i)}_0}\right\vert_{{X}}}, {\left.{\widetilde{s}^{(e_i)}_{1}}\right\vert_{{X}}} \in H^{0} (L)$ for $1 \leq i \leq m_1$ and ${\left.{\widetilde{s}^{(e_{i}^\prime)}_0}\right\vert_{{X}}}, {\left.{\widetilde{s}^{(e_{i}^\prime)}_{1}}\right\vert_{{X}}},
{\left.{\widetilde{s}^{(e_{i}^{\prime\prime})}_0}\right\vert_{{X}}},
{\left.{\widetilde{s}^{(e_{i}^{\prime\prime})}_{1}}\right\vert_{{X}}} \in H^{0} (L)
$ for $m_1 + 1 \leq i \leq m_1+ m_2$, and we denote those $(N+1)$ global sections of $L$ by $s_0, \ldots, s_{N}$. Then it is straightforward from Lemmas \[lemma:actuallyunimodular:discon\] and \[lemma:actuallyunimodular:conn\] that the map $\varphi$ determined by $s_0 , \ldots , s_N$ gives a unimodular tropicalization of $\Gamma_{\min}$ having properties (i) and (ii).
Furthermore, this tropicalization also has property (iii). Indeed, let $e$, $x$, and $y$ be as in (iii), and take $i = 1 , \ldots , m_1$ with $e = e_i$. By the definition of $s_0 , \ldots , s_N$, there exist $a , b = 1 ,\ldots , N$ such that $s_a = {\left.{\widetilde{s}^{(e_i)}_0}\right\vert_{{X}}}$ and $s_b = {\left.{\widetilde{s}^{(e_i)}_{1}}\right\vert_{{X}}}$. Set $h := - \log |s_b/ s_a|$. Then by Lemma \[lemma:ES:d-c\], $h (x) \neq h (y)$. Further, by Lemma \[lemma:actuallyunimodular:discon\] and Lemma \[lemma:ES:d-c\], we have $h (x) , h (y) \notin h ( \mathrm{relin} (e_i))$. This shows that $\varphi$ has property (iii) of the theorem.
500
Faithful tropicalization of minimal skeleta for $g \geq 2$ {#section:FTcan}
==========================================================
Let $X$ be a connected smooth projective curve over $K$ of genus $g$. In this section, we assume that $g \geq 2$. Let $\Gamma_{\min}$ be the minimal skeleton of $X^{{{\operatorname{an}}}}$. As explained in §\[subsec:skeleton:weighted:metric:graph\], we endow $\Gamma_{\min}$ with the canonical weight function $\omega$, so that $\bar{\Gamma}_{\min} = (\Gamma_{\min}, \omega)$ is the minimal weighted skeleton. Let $\mathscr{X}^{{\operatorname{st}}}$ denote the stable model of $X$ over $R$. Also, as explained in §\[subsec:skeleton:weighted:metric:graph\], we endow $\Gamma_{\min}$ with the canonical finite metric structure with the set of vertices $V({{\mathscr{X}}}^{{\operatorname{st}}}) = V(\bar{\Gamma}_{\min})$ and the set of edges $E({{\mathscr{X}}}^{{\operatorname{st}}}) = E(\bar{\Gamma}_{\min})$. We note that $g = g(\bar{\Gamma}_{\min}) \geq g(\Gamma_{\min})$.
Let $L$ be a line bundle over $X$. If $\deg (L) \geq 3g-1$, then we have already constructed enough global sections of $L$ which give a unimodular tropicalization of $\Gamma_{\min}$. However, we have not yet obtained a faithful tropicalization of $\Gamma_{\min}$. In view of Theorem \[theorem:UT:canonical\](i)(ii)(iii), we still need to construct global sections of $L$ that separate distinct points $x ,y$ of $\Gamma_{\min}$ as below:
1. $e \in E ( \mathscr{X}^{{\operatorname{st}}})$ is an edge of connected type with middle point $w$, and $x , y \in \mathrm{relin}(e)$ belong to the different connected components of $\mathrm{relin}(e) \setminus \{w\} $;
2. $e, f\in E ( \mathscr{X}^{{\operatorname{st}}})$ are distinct edges, and $x \in \mathrm{relin}(e)$ and $y \in f$;
3. $x , y \in V ( \mathscr{X}^{{\operatorname{st}}})$ are distinct vertices.
We will construct global sections which take care of (i) in §\[subsection:AS\], (ii) in §\[subsection:ES\], and (iii) in §\[subsection:VS\], and then we will show in §\[subsec:ft:min:skeleton\] that those sections give a faithful tropicalization of $\Gamma_{\min}$. This section will be technically the most difficult in this paper.
Notation and terminology of §\[section:FTcan\] {#notation-and-terminology-of-sectionftcan .unnumbered}
----------------------------------------------
We put together the notation and terminology that are used throughout this section. Let $X$ be a connected smooth projective curve over $K$ of genus $g \geq 2$, and let $ L$ be a line bundle over $X$. The minimal skeleton $\Gamma_{\min}$ of $X^{{{\operatorname{an}}}}$ is endowed with the canonical weight function and the canonical finite graph structure as above. A good model of $X$ always means a model defined in Definition \[def:goodmodel:new\]. Given a model $({{\mathscr{X}}}, {{\mathscr{L}}})$ of $(X, L)$ such that ${{\mathscr{X}}}$ is a good model, we put ${{\mathscr{M}}}:= {{\mathscr{L}}}\otimes \omega_{{{\mathscr{X}}}/R}^{\otimes -1}$ as before, and for a line bundle ${{\mathscr{N}}}$ over ${{\mathscr{X}}}$, let $D_{{{\mathscr{N}}}}$ be the divisor on $\Gamma_{\min}$ defined in . For a node $p \in {\operatorname{Sing}}({{\mathscr{X}}}_s)$, let $\Delta_p$ denote the canonical $1$-simplex corresponding to $p$ in $\Gamma_{\min}$ (cf. Definition \[def:canonical:1:simplex:node\]).
Separating points on an edge of connected type {#subsection:AS}
----------------------------------------------
In this subsection, we construct global sections which will be used to separate points in an edge of connected type.
Let $e \in E ( \mathscr{X}^{{\operatorname{st}}})$ be en edge of connected type with middle point $w$. Let $\mathscr{X}$ be a good model of $X$. By Lemma \[lemma:make:clear\], we take the maximal $(-2)$-chains $E$ of $\mathscr{X}_s$ with $\Delta_E = e$. Let $E_1 , \ldots , E_{2 \ell - 1}$ be the irreducible components of $E$. Since $\mathscr{X}$ is a good model, we have $2 \ell - 1 \geq 3$. We give the numbering for $E_1 , \ldots , E_{2 \ell - 1}$ as is illustrated in Figure \[figure:conn:type\] in §\[subsection:stepwise\]. Note that $w = [E_\ell]$. As in Figure \[figure:conn:type\] in §\[subsection:stepwise\], let $E_0$ and $E_{2 \ell}$ denote the irreducible components of $\mathscr{X}_s - E$ with $E _0 \cap E_1 \neq \emptyset$ and $E _{2 \ell -1} \cap E_{2 \ell} \neq \emptyset$, and we set $\{ p_i \} := E_{i} \cap E_{i+1}$ for $i= 0 , \ldots , 2 \ell - 1$.
Let $\lambda_i$ denote the multiplicity of $\mathscr{X}$ at $p_i$. Since $\mathscr{X}$ is a good model, we have $\lambda_{0} = \lambda_{\ell - 1}
= \lambda_{\ell} = \lambda_{2 \ell - 1}$. It follows that there exist unique fundamental vertical divisors $\mathscr{A}_{1}$ and $\mathscr{A}_{2}$ with $${\operatorname{Supp}}(\mathscr{A}_{1}) = E_1 + \cdots + E_{\ell - 1}
\quad\text{and}\quad
{\operatorname{Supp}}(\mathscr{A}_{2}) = E_{\ell + 1} + \cdots +
E_{2 \ell - 1}.$$ Then $\Delta_{{\operatorname{Supp}}( \mathscr{A}_1 )}$ and $\Delta_{{\operatorname{Supp}}( \mathscr{A}_2 )}$ are the two $1$-simplices in $e$ that arise by dividing $e$ at $w$.
\[separating divisor for an edge of connected type, separating divisor for $e$\] \[def:assymetric:divisor\] Let $e \in E ( \mathscr{X}^{{\operatorname{st}}})$ be an edge of connected type, and let $\mathscr{X}$ be a good model of $X$. We call a divisor $\mathscr{A}$ on ${{\mathscr{X}}}$ a *separating divisor for $e$* if $\mathscr{A} = \mathscr{A}_1$ or $\mathscr{A} = \mathscr{A}_2$.
We remark that the notion of a separating divisor is well-defined for $e$, and it does not depend on the two choices of the numbering $E_1 , \ldots , E_{2 \ell - 1}$ for the irreducible components of $E$.
\[separating section for an edge of connected type, separating section for $e$\] \[def:asymmetric:section:along:e\] Let $e \in E ( \mathscr{X}^{{\operatorname{st}}})$ be an edge of connected type. Let $( \mathscr{X} , \mathscr{L})$ be a model of $(X,L)$ such that $\mathscr{X}$ is a good model. Let $\widetilde{s}$ be a nonzero global section of $\mathscr{L}$. We call $\widetilde{s}$ a *separating section for $e$* if there exists a separating divisor $\mathscr{A}$ for $e$ such that ${\operatorname{div}}\left( \widetilde{s} \right) -
\mathscr{A}$ is effective on ${{\mathscr{X}}}$ and is trivial on some open neighborhood of ${\operatorname{Sing}}( \mathscr{X}_s )_{\subset e}
\setminus {\operatorname{Supp}}( \mathscr{A})$.
Let $e^\prime$ and $e^{\prime\prime}$ be the two $1$-simplices that arise by dividing $e$ at $w$. We take the separating divisor $\mathscr{A}$ with $\Delta_{{\operatorname{Supp}}( \mathscr{A})} = e^{\prime\prime}$. Then ${\operatorname{Sing}}( \mathscr{X}_s )_{\subset e}
\setminus {\operatorname{Supp}}( \mathscr{A}) = {\operatorname{Sing}}({\operatorname{Supp}}( \mathscr{A}))_{\subset e^\prime}$. (For example, if $e^{\prime\prime} =
\Delta_{E_{\ell+1} + \cdots + E_{2\ell-1}}$, then $\mathscr{A} = \mathscr{A}_2$ and ${\operatorname{Sing}}( \mathscr{X}_s )_{\subset e}
\setminus {\operatorname{Supp}}( \mathscr{A}) =\{p_{0}, \ldots, p_{\ell-1}\}$.)
We use the following lemma to separate points in $\mathrm{relin}(e)$ in §\[subsec:ft:min:skeleton\]. Recall that an edge-base section is defined in Definition \[def:edge-base:section\].
\[lemma:asymmetric:section:separate:new\] Let $e \in E({{\mathscr{X}}}^{{\operatorname{st}}})$ be an edge of connected type with middle point $w$. Suppose that there exists a model $({{\mathscr{X}}}, {{\mathscr{L}}})$ of $(X, L)$ such that ${{\mathscr{X}}}$ is a good model and that there exist an $e$-base section $\widetilde{s}^{(e)}_0$ and a separating section $\widetilde{s}^{(e)}_{1}$ for $e$. We set $g := \widetilde{s}^{(e)}_1/\widetilde{s}^{(e)}_{0}$ and define a nonzero rational function $h$ on $X$ by $h:= {\left.{g}\right\vert_{{X}}}$. Let $e^\prime$ and $e^{\prime\prime}$ be the two $1$-simplices that arise by dividing $e$ at $w$. Then for any $x \in \mathrm{relin}(e^\prime)$ and $y \in \mathrm{relin}(e^{\prime\prime})$, we have $- \log |h (x)| \neq - \log |h (y)|$.
[[*Proof.*]{}]{}Since $\widetilde{s}^{(e)}_0(p) \neq 0$ for any $p \in {\operatorname{Sing}}({{\mathscr{X}}}_s)_{\subset e}$, we take an open neighborhood ${{\mathscr{U}}}$ of ${\operatorname{Sing}}({{\mathscr{X}}}_s)_{\subset e}$ such that ${\left.{\widetilde{s}^{(e)}_0}\right\vert_{{{{\mathscr{U}}}}}}$ is nowhere vanishing over ${{\mathscr{U}}}$. Then ${\operatorname{div}}( {\left.{\widetilde{s}^{(e)}_0}\right\vert_{{{{\mathscr{U}}}}}} ) = 0$. By the definition of a separating section $\widetilde{s}^{(e)}_{1}$ for $e$, there exists a separating divisor $\mathscr{A}$ for $e$ as in Definition \[def:asymmetric:section:along:e\]. Set $g := \widetilde{s}^{(e)}_{1} / \widetilde{s}^{(e)}_0$. Then ${\operatorname{div}}(g) - \mathscr{A}$ is effective over ${{\mathscr{U}}}$.
Note that $\Delta_{{\operatorname{Supp}}(\mathscr{A} )}$ equals $e^{\prime}$ or $e^{\prime\prime}$. Without loss of generality, we assume that $\Delta_{{\operatorname{Supp}}(\mathscr{A} )}
= e^{\prime\prime}$. We set $\varphi := - \log |h| = \log \left| {\left.{g}\right\vert_{{X}}}\right|$. Let $C \in {\operatorname{\operatorname{Irr}}}({{\mathscr{X}}}_s)$ with $[C] \in {\operatorname{relin}}(e^{\prime\prime})$. Then we have ${\operatorname{ord}}_{C}(\mathscr{A}) > 0$. Further, since $e^{\prime\prime} \subset e$, we have $C \cap {{\mathscr{U}}}\neq \emptyset$, so that the generic point of $C$ belongs to ${{\mathscr{U}}}$. Then it follows from Lemma \[lem:forinjectivity\](1) that $\varphi ([C]) \geq {\operatorname{ord}}_{C}(\mathscr{A}) > 0$. By Lemma \[lem:forinjectivity\](2), we have $\varphi (y) > 0$ for any $y \in \mathrm{relin}(e^{\prime\prime})$. On the other hand, since $\widetilde{s}^{(e)}_{1}$ does not vanish at any $p \in {\operatorname{Sing}}( \mathscr{X}_s )_{\subset e^\prime} =
{\operatorname{Sing}}( \mathscr{X}_s )_{\subset e} \setminus {\operatorname{Supp}}(\mathscr{A})$, $g^{-1}$ is regular on some neighborhood of ${\operatorname{Sing}}( \mathscr{X}_s )_{\subset e^\prime}$. By Lemma \[lem:forinjectivity\], we see that $\varphi (x) \leq 0$ for any $x \in e^\prime$. This proves the lemma.
500
In view of Lemma \[lemma:asymmetric:section:separate:new\], our next task is to show the existence of a model of $(X,L)$ that has an $e$-base section and a separating section for $e$.
\[prop:model:AS\] Suppose that $\deg (L) \geq 3 g -1$. Then for any edge $e \in E ( \mathscr{X}^{{\operatorname{st}}})$ of connected type, there exists a model $({{\mathscr{X}}}, {{\mathscr{L}}})$ of $(X,L)$ such that ${{\mathscr{X}}}$ is a good model and such that there exist an $e$-base section and a separating section for $e$ of $\mathscr{L}$.
[[*Proof.*]{}]{}We take an edge $e \in E ( \mathscr{X}^{{\operatorname{st}}})$ of connected type. Let $w$ denote the middle point of $e$. Applying Proposition \[prop:pregoodmodel\] for $x = w$, we obtain a model $( \mathscr{X} , \mathscr{L} )$ of $(X,L)$ with the following properties:
1. $\mathscr{X}$ is a good model of $X$;
2. $\mathscr{M}$ is vertically nef (i.e., $D_{\mathscr{M}} \geq 0$), $D_{\mathscr{M}}(w) \geq 1$, and $\deg ({\left.{D_{\mathscr{M}}}\right\vert_{{\mathrm{relin}(e)}}}) \leq 2$.
We are going to show that there exist an $e$-base section and a separating section for $e$ of ${{\mathscr{L}}}$.
We consider the existence of an $e$-base section of $\mathscr{L}$. Take any $p \in {\operatorname{Sing}}( \mathscr{X}_s)_{\subset e}$ and put $\widetilde{\Gamma} := \Gamma_{\min} \setminus \mathrm{relin} (\Delta_p)$. Since $p$ is of connected type, $\widetilde{\Gamma}$ is connected. By condition (ii) above, ${\left.{D_{\mathscr{M}}}\right\vert_{{\widetilde{\Gamma}}}}$ is effective and $\deg\left({\left.{D_{\mathscr{M}}}\right\vert_{{\widetilde{\Gamma}}}}\right)
\geq \deg \left(D_{\mathscr{M}}\right) - \deg({\left.{D_{\mathscr{M}}}\right\vert_{{\mathrm{relin} (e)}}}) \geq (g+1) -2 \geq 1$. By Lemma \[lemma:freeatnode:graphversion:emptyset\], $\mathscr{L}$ is free at $p$, which shows the existence of an $e$-base section.
We show that $\mathscr{L}$ has a separating section for $e$. The proof will be somewhat similar to that of Lemma \[lemma:forbasesectionUT:dc\](3). By Lemma \[lemma:make:clear\], we take the maximal $(-2)$-chains of ${{\mathscr{X}}}_s$ with $\Delta_E = e$. We use the notation as in the beginning of this subsection. In particular, $E_1, \ldots, E_{2\ell-1}$ are the irreducible components of $E$, where numbering is given as in Figure \[figure:conn:type\]. We denote by $e^\prime$ and $e^{\prime\prime}$ the $1$-simplices arising by dividing $e$ at $w$ such that $e^\prime
= \Delta_{E_1 + \cdots + E_{\ell}}$ and $e^{\prime\prime} = \Delta_{E_{\ell+1} + \cdots + E_{2\ell-1}}$. We consider the separating divisor $\mathscr{A}_2$ for $e$. Recall that ${\operatorname{Sing}}( \mathscr{X}_s )_{\subset e}
\setminus {\operatorname{Supp}}( \mathscr{A}_2 ) =
{\operatorname{Sing}}( \mathscr{X}_s )_{\subset e^\prime} = \{p_0, \ldots, p_\ell\}$.
We set $\mathscr{L}^\prime := \mathscr{L} ( - \mathscr{A}_2)$. We show that $\mathscr{L}^\prime$ is free at any $q \in {\operatorname{Sing}}( \mathscr{X}_s )_{\subset e^\prime}$ by using Lemma \[lemma:freeatnode:graphversion:emptyset\] and Lemma \[lemma:freeatnode:graphversion:nonemptyset\]. Take any $q \in {\operatorname{Sing}}( \mathscr{X}_s )_{\subset e^\prime}$ and set $\Gamma_1 := \Gamma_{\min} \setminus \mathrm{relin} ( \Delta_q )$. Since $q$ is of connected type, $\Gamma_1$ is connected. We set $\mathscr{M}^\prime := \mathscr{L}^\prime \otimes \omega_{\mathscr{X}/R}^{\otimes -1}
= \mathscr{M} ( - \mathscr{A}_2 )$. We set $w_i = [E_i] \in \Gamma_{\min}$ for $i = 0, \ldots, 2 \ell$. Note that $w = w_\ell$. Then we have $$\label{eqn:D:M:prime}
D_{\mathscr{M}^\prime}
= D_{\mathscr{M}} -D_{{{\mathcal{O}}}_{\mathscr{X}} (\mathscr{A}_2)}
= (D_{\mathscr{M}} - [w]) + [w_{\ell + 1}] +
[w_{2 \ell - 1}] - [w_{2 \ell}].$$ By condition (ii), we have $D_{\mathscr{M}} - [w] \geq 0$, and thus $V := \{v \in V({{\mathscr{X}}}^{{\operatorname{st}}}) \mid D_{\mathscr{M}^\prime} (v) <0\}$ is either the empty set or equal to the one point set $\{w_{2 \ell}\}$. We argue it case by case.
Suppose that $V = \emptyset$, i.e., $D_{\mathscr{M}^\prime} \geq 0$. Since $$\deg\left({\left.{D_{\mathscr{M}^\prime}}\right\vert_{{\Gamma_1}}}\right)
= \deg\left({\left.{D_{\mathscr{M}}}\right\vert_{{\Gamma_1}}}\right)
\geq
\deg \left(D_{\mathscr{M}}\right) -\deg({\left.{D_{\mathscr{M}}}\right\vert_{{\mathrm{relin} (e)}}}) \geq (g+1) -2 \geq 1,$$ Lemma \[lemma:freeatnode:graphversion:emptyset\] tells us that $\mathscr{L}^\prime$ is free at $q$.
Suppose that $V = \{w_{2 \ell}\}$. In this case, ${\left.{D_{\mathscr{M}^\prime}}\right\vert_{{\Gamma_{\min} \setminus {\{w_{2 \ell}}\}}}} \geq 0$ and $D_{\mathscr{M}^\prime} (w_{2 \ell}) = -1$ by . Let $\Gamma_{11}^{\circ}$ be the connected component of $\Gamma_1 \setminus \{w_{2 \ell}\}$ such that $w_{2 \ell - 1} \in \Gamma_{11}^\circ$. Then $\deg
\left( {\left.{D_{\mathscr{M}^\prime}}\right\vert_{{\Gamma_{11}^{\circ}}}} \right) \geq
D_{\mathscr{M}^\prime} (w_{2 \ell - 1}) \geq 1$. Since $\deg \left(
{\left.{D_{\mathscr{M}}}\right\vert_{{\mathrm{relin} (e)}}}
\right) \leq 2$, we get from that $$\deg \left(
{\left.{D_{\mathscr{M}^\prime}}\right\vert_{{\mathrm{relin} (e)}}}
\right)
=
\deg \left(
{\left.{D_{\mathscr{M}}}\right\vert_{{\mathrm{relin} (e)}}}\right)
+ \deg(- [w] + [w_{\ell + 1}] + [w_{2 \ell - 1}] )
\leq 2 + 1 = 3.$$ Since $\deg (D_{\mathscr{M}^\prime}) =
\deg (D_{\mathscr{M}}) \geq g+ 1 \geq 3$ and $D_{\mathscr{M}^\prime} (w_{2\ell}) = -1$, we have $$\deg \left( {\left.{D_{\mathscr{M}^\prime}}\right\vert_{{
\Gamma_{\min}\setminus ( \mathrm{relin} (e) \cup \{w_{2\ell}\})}}}
\right)
=
\deg (D_{\mathscr{M}^\prime}) -
\deg \left(
{\left.{D_{\mathscr{M}^\prime}}\right\vert_{{\mathrm{relin} (e)}}}
\right)
- D_{\mathscr{M}^\prime} (w_{2\ell})
\geq 1.$$ Since $\Gamma_{\min}\setminus ( \mathrm{relin} (e) \cup \{w_{2\ell}\}) \subset \Gamma_1 \setminus \{{w_{2\ell}\}}$ and $\Gamma_{11}^\circ \subset \mathrm{relin} (e)$, we see that there exists a connected component $\Gamma_{12}^{\circ}$ of $\Gamma_1 \setminus \{w_{2 \ell}\}$ such that $\Gamma_{12}^{\circ} \neq \Gamma_{11}^{\circ}$ and $\deg
\left( {\left.{D_{\mathscr{M}^\prime}}\right\vert_{{\Gamma_{12}^{\circ}}}} \right) \geq 1$.
$$\setlength\unitlength{0.08truecm}
\begin{picture}(100, 65)(0,0)
\put(5, 25){\line(1,0){25}}
\qbezier(30, 25)(45, 65)(60, 25)
\put(60, 25){\line(1,0){25}}
\put(60, 25){\line(0,-1){15}}
\put(30, 25){\line(0,-1){15}}
\put(60, 25){\circle*{1.5}}
\put(45, 45){\circle*{1.5}}
\put(33, 32){\circle*{1.5}}
\put(36, 38){\circle*{1.5}}
\multiput(33.5, 10)(2,0){12}{\line(1,0){1}}
\put(25, 0){$\Gamma_1 := \Gamma_{\min}\setminus{\operatorname{relin}}(\Delta_q)$}
\put(58, 30){\circle*{1.5}}
\put(50, 42.5){\circle*{1.5}}
\put(53, 42){$w_{\ell+1}$}
\put(61, 20){$w_{2\ell}$}
\put(35, 32){$\Delta_q$}
\put(28, 35){$e^\prime$}
\put(57, 35){$e^{\prime\prime}$}
\put(44, 35){$e$}
\put(60, 30){$w_{2\ell-1}$}
\put(38, 48){$w = w_{\ell}$}
\put(45, 60){$\Gamma_{11}^\circ$}
\put(35.5, 50){$\overbrace{\phantom{AAAAAA}}$}
\thicklines
\put(33, 32){\line(1,2){3}}
\end{picture}$$ \[figure:for:prop:model:AS\]
Let $\Gamma_{11} := \Gamma_{11}^{\circ} \cup \{w_{2\ell}\}$ be the closure of $\Gamma_{11}^{\circ}$ in $\Gamma_1$, and let $\Gamma_{12} := \Gamma_{12}^{\circ} \cup \{w_{2\ell}\}$ be the closure of $\Gamma_{12}^{\circ}$ in $\Gamma_1$. Since the valence of $\Gamma_{11} \cup
\Gamma_{12}$ at $w_{2\ell}$ is at least $2
= - D_{\mathscr{M}^\prime} (w_{2\ell}) + 1$, one can apply Lemma \[lemma:freeatnode:graphversion:nonemptyset\]. (With the notation of Lemma \[lemma:freeatnode:graphversion:nonemptyset\], condition (i) of Lemma \[lemma:freeatnode:graphversion:nonemptyset\] is vacuous, and for condition (ii), we take $s=2$ and consider $\Gamma_{11} $ and $\Gamma_{12}$.) This proves that $\mathscr{L}^\prime$ is free at $q$ also in this case.
Thus $\mathscr{L}^\prime$ is free at $q$ in any case. It follows that there exists a global section $s^\prime $ of $\mathscr{L}^\prime
:= \mathscr{L} ( - \mathscr{A}_2 ) $ such that $s^\prime (q) \neq 0$ for any $q \in
{\operatorname{Sing}}( \mathscr{X}_s )_{\subset e^\prime} =
{\operatorname{Sing}}( \mathscr{X}_s )_{\subset e}
\setminus {\operatorname{Supp}}( \mathscr{A}_2 )$. Let $s$ be the image of $s^\prime$ by the natural homomorphism $\mathscr{L} ( - \mathscr{A}_2 )
\hookrightarrow \mathscr{L}$. Then $s$ is a separating section for $e$.
500
Separating points in different edges {#subsection:ES}
------------------------------------
Let $e, f \in E (\mathscr{X}^{{\operatorname{st}}})$ be distinct edges of $\Gamma_{\min}$. In this subsection, we construct global sections that separate a point in $\mathrm{relin}(e)$ and a point in $f$.
\[remark:disconnect-other\] If $e$ is an edge of disconnected type, then we have already constructed such global sections. Indeed, let $\varphi$ be the tropicalization map as in Theorem \[theorem:UT:canonical\]. If $e$ is of disconnected type, then $f$ is contained in a connected component of $\Gamma
\setminus \mathrm{relin}(e)$. By property (iii) of Theorem \[theorem:UT:canonical\], we obtain $\varphi (f) \cap \varphi ( \mathrm{relin}(e)) =
\emptyset$, and thus $\varphi$ separates a point in $\mathrm{relin}(e)$ and a point in $f$.
With Remark \[remark:disconnect-other\], in the rest of this subsection, we assume that $e$ is an edge of connected type. Suppose that $\mathscr{X}$ is a good model of $X$. By Lemma \[lemma:make:clear\], we take the maximal $(-2)$-chain $E$ of ${{\mathscr{X}}}_s$ with $\Delta_E = e$. By the definition of a good model, $E$ has symmetric multiplicities. It follows that there exists a unique fundamental vertical divisor with support $E$ (cf. §\[subsec:fvd\]), which we denote by $\mathscr{F}_e$.
### Edges-separating sections
Let $e, f \in E({{\mathscr{X}}}^{{\operatorname{st}}})$ be distinct edges such that $e$ is of connected type. Let $(\mathscr{X} , \mathscr{L})$ be a model of $(X,L)$ such that $\mathscr{X}$ is a good model. Let $\widetilde{s}$ be a nonzero global section of $\mathscr{L}$. We call $\widetilde{s}$ an *$(e,f)$-separating section* if $
{\operatorname{div}}\left(
\widetilde{s}
\right)
-
\mathscr{F}_e
$ is effective on ${{\mathscr{X}}}$ and is trivial on some open neighborhood of ${\operatorname{Sing}}( \mathscr{X}_s)_{\subset f}
$.
We use in §\[subsec:ft:min:skeleton\] the following lemma to separate a point in ${\operatorname{relin}}(e)$ and a point in $f$.
\[lemma:ES:separate:edges\] Let $e, f \in E({{\mathscr{X}}}^{{\operatorname{st}}})$ be distinct edges such that $e$ is of connected type. Assume that there exists a model $(\mathscr{X} , \mathscr{L})$ such that $\mathscr{X}$ is a good model and such that there exist an $e$-base section $\widetilde{s}_0$ cf. Definition and an $(e,f)$-separating section $\widetilde{s}_1$ of $\mathscr{L}$. We set $g := \widetilde{s}_1 / \widetilde{s}_0$, and define a nonzero rational function $h$ on $X$ by $h:= {\left.{g}\right\vert_{{X}}}$. Then for any $x \in \mathrm{relin}(e)$ and any $y \in f$, we have $-\log|h(x)| > 0$ and $-\log|h(y)| \leq 0$, and in particular, $-\log|h(x)| \neq -\log|h(x)|$.
[[*Proof.*]{}]{}Since $\widetilde{s}_0$ is an $e$-base section, ${\operatorname{div}}( \widetilde{s}_0)$ is trivial over some open neighborhood $\mathscr{U}_0$ of ${\operatorname{Sing}}( \mathscr{X}_s )_{\subset e}$. On the other hand, since $\widetilde{s}_1$ is an $(e,f)$-separating section, ${\operatorname{div}}( \widetilde{s}_1) - \mathscr{F}_e$ is effective on $\mathscr{X}$ and is trivial over some neighborhood $\mathscr{U}_1$ of ${\operatorname{Sing}}( \mathscr{X}_s )_{\subset f}$. Since $e \neq f$, we have ${\operatorname{Sing}}( \mathscr{X}_s )_{\subset f} \cap {\operatorname{Supp}}( \mathscr{F}_e ) =
\emptyset$, so that shrinking $\mathscr{U}_1$ if necessary, we may furthermore assume that $\mathscr{U}_1 \cap {\operatorname{Supp}}( \mathscr{F}_e ) = \emptyset$. Then ${\operatorname{div}}( \widetilde{s}_1)$ is trivial on $\mathscr{U}_1$. By the definition of $g$, ${\operatorname{div}}(g) - \mathscr{F}_s = ({\operatorname{div}}( \widetilde{s}_1) - \mathscr{F}_s) - {\operatorname{div}}( \widetilde{s}_0)$ is effective on $\mathscr{U}_0$, and $- ( {\operatorname{div}}(g) ) = {\operatorname{div}}(g^{-1})$ is effective over $\mathscr{U}_1$.
Take any $C \in {\operatorname{\operatorname{Irr}}}( \mathscr{X}_s)$ with $[C] \in \mathrm{relin} (e)$. Since ${\operatorname{div}}(g) - \mathscr{F}_e$ is effective on $\mathscr{U}_0$, it follows from Lemma \[lem:forinjectivity\](1) that $ - \log |h ([C]) | \geq {\operatorname{ord}}_C(\mathscr{F}_e) > 0$. By Lemma \[lem:forinjectivity\](2), we see that $- \log |h (x)| > 0$ for any $x \in \mathrm{relin} (e)$. Similarly, take any $C' \in {\operatorname{\operatorname{Irr}}}( \mathscr{X}_s)$ with $[C'] \in f$. Since $- ( {\operatorname{div}}(g) ) = {\operatorname{div}}(g^{-1})$ is effective on $\mathscr{U}_1$, it follows from Lemma \[lem:forinjectivity\](1) that $ - \log |h ([C]) | \leq 0$. By Lemma \[lem:forinjectivity\](2), we see that $- \log |h (y)| \leq 0$ for any $y \in f$. This concludes the lemma.
500
Our task is then to show the existence of a model of $(X, L)$ that has an $e$-base section and an $(e,f)$-separating section. The goal of this subsection is to prove the following proposition.
\[prop:model:ES\] Suppose that $\deg (L) \geq 3g -1$. Let $e, f \in E({{\mathscr{X}}}^{{\operatorname{st}}})$ be distinct edges such that $e$ is of connected type. Then there exists a model $( \mathscr{X} , \mathscr{L} )$ of $(X,L)$ such that $\mathscr{X}$ is a good model and such that $\mathscr{L}$ has an $e$-base section and an $(e , f)$-separating section.
The above proposition follows the proposition below.
\[prop:sep:edges\] Suppose that $\deg (L) \geq 3g -1$, and let $e$ and $f$ be as in Proposition \[prop:model:ES\]. Then there exists a model $( \mathscr{X} , \mathscr{L})$ of $(X,L)$ with the following properties.
1. $\mathscr{X}$ is a good model.
2. For any $p \in {\operatorname{Sing}}( \mathscr{X}_s )_{\subset e}$, $\mathscr{L}$ is free at $p$.
3. Let $\mathscr{V}_e$ denote the stepwise vertical divisor for $e$. Set $\mathscr{L}_1 := \mathscr{L} ( - \mathscr{V}_e)$. Then for any $q \in {\operatorname{Sing}}( \mathscr{X}_s )_{\subset f}$, $\mathscr{L}_1$ is free at $q$.
We prove that Proposition \[prop:sep:edges\] implies Proposition \[prop:model:ES\]. Let $( \mathscr{X} , \mathscr{L} )$ be as in Proposition \[prop:sep:edges\]. Then since $\mathscr{L}$ is free at any $p \in {\operatorname{Sing}}(\mathscr{X}_s)_{\subset e}$, there exists a global section $\widetilde{s}_0$ of $\mathscr{L}$ such that $\widetilde{s}_0 (p) \neq 0$ for any such $p$. Thus we have an $e$-base section. Since $\mathscr{L}_1 = \mathscr{L} ( - \mathscr{V}_e)
$ is free at any $q \in {\operatorname{Sing}}(\mathscr{X}_s)_{\subset f}$, there exists a global section $\widetilde{s}_1^-$ of $\mathscr{L}_1$ such that $\widetilde{s}_1^- (q) \neq 0$ for any such $q$. Let $\widetilde{s}_1$ be the image of $\widetilde{s}_1^-$ by the canonical injection $\mathscr{L}_1 \hookrightarrow \mathscr{L}$. Then ${\operatorname{div}}(\widetilde{s}_1) - \mathscr{V}_e$ is effective and is trivial on some open neighborhood of ${\operatorname{Sing}}(\mathscr{X}_s)_{\subset f}$. Since $\mathscr{V}_e - \mathscr{F}_e$ is effective and is trivial on some open neighborhood of ${\operatorname{Sing}}(\mathscr{X}_s)_{\subset f}$, it follows that ${\operatorname{div}}(\widetilde{s}_1) - \mathscr{F}_e = ({\operatorname{div}}(\widetilde{s}_1) - \mathscr{V}_e)
+ (\mathscr{V}_e - \mathscr{F}_e)$ is effective and is trivial on some open neighborhood of ${\operatorname{Sing}}(\mathscr{X}_s)_{\subset f}$. Thus $\widetilde{s}_1$ is an $(e , f)$-separating section, and we obtain Proposition \[prop:model:ES\].
Thus our goal is to show Proposition \[prop:sep:edges\]. We will prove it according to the position of $e$ and $f$ in $\Gamma_{\min}$. The construction of a model will be done separately according to the following two cases:
1. $f$ is of disconnected type, or $f$ is of connected type and $\Gamma_{\min} \setminus ( \mathrm{relin} (e) \cup \mathrm{relin} (f) )$ is connected;
2. $f$ is of connected type and $\Gamma_{\min} \setminus ( \mathrm{relin} (e) \cup \mathrm{relin} (f) )$ is not connected.
For case (A), we may take the same kind of models obtained by Proposition \[prop:pregoodmodel\] as before, but for case (B), we need to construct another kind of model. The construction will be done below.
### Model for case
We construct a suitable model $({{\mathscr{X}}}, {{\mathscr{L}}})$ for case (B). Under the condition of (B), we remark that $e$ nor $f$ is a loop and that $\Gamma_{\min} \setminus ( \mathrm{relin} (e) \cup \mathrm{relin} (f) )$ has exactly two connected components. Indeed, since $e$ is of connected type, $\Gamma_{\min} \setminus \mathrm{relin} (e)$ is connected; since $
\Gamma_{\min} \setminus ( \mathrm{relin} (e) \cup \mathrm{relin} (f) )
=
( \Gamma_{\min} \setminus \mathrm{relin} (e) ) \setminus \mathrm{relin} (f)$ is not connected, it follows that this has exactly two connected components.
\[prop:separation:special2\] Suppose that $\deg (L) \geq 3g -1$, and let $e$ and $f$ be as in Proposition . Assume that Case (B) applies for $e, f$. Let $\Gamma_1$ and $\Gamma_2$ be the connected components of $\Gamma_{\min}\setminus (\mathrm{relin} (e) \cup \mathrm{relin}(f))$. Then there exists a model $(\mathscr{X} , \mathscr{L})$ of $(X,L)$ such that $\mathscr{X}$ is a good model and $\deg \left(
{\left.{D_{\mathscr{M}}}\right\vert_{{\Gamma_i}}}
\right) \geq 1$ for $i= 1, 2$.
To construct a model as in Proposition \[prop:separation:special2\], we will apply Proposition \[prop:pregoodmodel:pre\] together with the following lemma on the $\Lambda$-metric graph $\Gamma_{\min}$.
\[lemma:separation:special1\] Let $e$ and $f$ be edges of $\Gamma_{\min}$ such that case applies for $e$ and $f$. Let $\Gamma_1$ and $\Gamma_2$ be as in Proposition \[prop:separation:special2\]. Let $D \in {\operatorname{Div}}_{\Lambda} ( \Gamma_{\min} )$ be a divisor with $\deg (D) \geq
g + 1$. Then there exists an effective divisor $E \in {\operatorname{Div}}_{\Lambda} ( \Gamma_{\min} )$ such that $E \sim D$ and $\deg ({\left.{E}\right\vert_{{\Gamma_i}}}) \geq 1$ for both $i =1,2$.
[[*Proof.*]{}]{}Let $w_1$ and $w_2$ be the end vertices of $f$ such that $w_i \in \Gamma_i$.
$$\setlength\unitlength{0.08truecm}
\begin{picture}(60, 50)(0,0)
\put(5, 15){\line(0,1){20}}
\put(50, 15){\line(0,1){20}}
\put(5, 35){\line(2,1){10}}
\put(5, 15){\line(2,-1){10}}
\put(50, 35){\line(-2,1){10}}
\put(50, 15){\line(-2,-1){10}}
\multiput(18, 40)(2,0){10}{\line(1,0){1}}
\multiput(18, 9)(2,0){10}{\line(1,0){1}}
\put(26, 45){$\Gamma_1$}
\put(26, 2){$\Gamma_2$}
\put(5, 15){\circle*{1.5}}
\put(5, 35){\circle*{1.5}}
\put(50, 15){\circle*{1.5}}
\put(50, 35){\circle*{1.5}}
\put(52, 13){$w_2$}
\put(52, 33){$w_1$}
\put(46, 23){$f$}
\put(7, 23){$e$}
\end{picture}$$ \[figure:for:lemma:separation:special1:1\]
First we suppose that $\deg (D) \geq g (\Gamma_{\min}) + 2$. Then using Riemann’s inequality (Proposition \[prop:RIforMG\]), we have an effective divisor $E^\prime$ with $E^\prime \sim D - [w_1] - [w_2]$, and $E := E^\prime + [w_1] + [w_2] \in {\operatorname{Div}}_{\Lambda} ( \Gamma_{\min} )$ is a desired effective divisor. Thus we are done.
In the remaining of the proof, we assume that $\deg (D) \leq g (\Gamma_{\min}) + 1$. Since $g = g ( \bar{\Gamma}_{\min})\geq g ( \Gamma_{\min})$ and we assume that $\deg (D) \geq g + 1$, we have $\deg (D) = g (\Gamma_{\min}) + 1$ and $g ( \Gamma_{\min}) = g$. Then the following claim follows from Remark \[remark:valence:stable\].
\[claim:lemma:separation:special1:valence:3\] Every vertex of $V({{\mathscr{X}}}^{{\operatorname{st}}})$ has valency at least $3$.
We note the following claim.
We have $g (\Gamma_{i} ) \geq 1$ for any $i = 1,2$.
To prove the above claim by contradiction, we assume that $g (\Gamma_{i} ) = 0$ for some $i$. Without loss of generality, we assume that $g (\Gamma_{1} ) = 0$. Let $v_1$ be the end point of $e$ with $v_1 \in \Gamma_1$.
Suppose that $\Gamma_{1} $ is a singleton. Then $\Gamma_{1} = \{ w_1 \}$ and $w_1 = v_1$. Since $\Gamma_{1}$ is a connected component of $\Gamma_{\min} \setminus ( \mathrm{relin} (e) \cup \mathrm{relin} (f) )$, $e$ and $f$ are the edges of $\Gamma_{\min}$ emanating from $w_1$. Since $e$ nor $f$ is a loop, this means that the valence of $\Gamma_{\min}$ at $w_1$ equals $2$. However, that contradicts to Claim \[claim:lemma:separation:special1:valence:3\].
Suppose that $\Gamma_{1} $ is a chain. Then $\Gamma_1$ is a segment connecting $w_1$ and $v_1$. Let $f'$ be an edge of $\Gamma_{\min}$ emanating from $w_1$ with $f' \neq f$. Since $\Gamma_{1}$ is a connected component of $\Gamma_{\min} \setminus ( \mathrm{relin} (e) \cup \mathrm{relin} (f) )$, we have $f' \subset \Gamma_1$. It follows that the valence of $\Gamma_{\min}$ at $w_1$ equals $2$. However, that contradicts to Claim \[claim:lemma:separation:special1:valence:3\].
Thus $\Gamma_1$ is a tree with at least three vertices at which $\Gamma_1$ has valence $1$. Take a valence $1$ point $u$ of $\Gamma_1$ with $u \neq v_1$ and $u \neq w_1$. Then $u$ is a point at which $\Gamma$ also has valence $1$. However, that is impossible by Claim \[claim:lemma:separation:special1:valence:3\]. This concludes that $g (\Gamma_{1} ) \geq 1$, and thus we have shown the claim.
Let $w$ be the middle point of $f$. By Proposition \[prop:reduced:Lambda-div\], we take the $w$-reduced divisor $D_{w} \in {\operatorname{Div}}_{\Lambda} ( \Gamma )$ with $D_{w} \sim D$. Since $\deg (D) = g (\Gamma_{\min}) + 1$, $D_{w}$ is effective and $D_{w} (w) \geq 1$ (cf. Remark \[rmk:on:reduced:divisors\]).
\[claim.reduced:divisor:new3\] If ${\operatorname{Supp}}(D_{w}) \cap \Gamma_i = \emptyset$ for some $i = 1,2$, then $D_{w} (w) \geq 2$.
We prove the above claim. Without loss of generality, we assume that ${\operatorname{Supp}}(D_{w}) \cap \Gamma_2 = \emptyset$. Let $\Gamma'$ be the topological space obtained from $\Gamma$ by contracting $\Gamma_2$ to a singleton, and let $\alpha : \Gamma \to \Gamma'$ be the natural surjective map; we remark that $\{ \alpha (w_2) \} = \alpha ( \Gamma_2 )$. Let $\beta : \Gamma \setminus \Gamma_2 \to
\Gamma' \setminus \alpha ( \Gamma_2)$ denote the restriction of $\alpha$. Since $\beta$ is a homeomorphism and since $\alpha ( \Gamma_2)$ is a singleton, we endow $\Gamma'$ with a $\Lambda$-metric graph structure such that $\beta$ is an isometry.
$$\setlength\unitlength{0.08truecm}
\begin{picture}(60, 40)(0,0)
\qbezier(10, 25)(12, 10)(32.5, 5)
\qbezier(55, 25)(45, 10)(32.5, 5)
\put(10, 25){\line(2,1){10}}
\put(55, 25){\line(-2,1){10}}
\put(0, 15){$\Gamma^\prime$}
\multiput(23,30)(2,0){10}{\line(1,0){1}}
\put(30, 35){$\Gamma_1$}
\put(55, 25){\circle*{1.5}}
\put(32.5, 5){\circle*{1.5}}
\put(10, 25){\circle*{1.5}}
\put(45.5, 13.5){\circle*{1.5}}
\put(57, 23){$w_1$}
\put(47, 11){$w$}
\put(19, 13){$e$}
\put(40, 13){$f$}
\put(30, 0){$\alpha(w_2)$}
\end{picture}$$ \[figure:for:lemma:separation:special1:2\]
Let $ D_{w}'$ be the pushout of $D_{w}$ by $\alpha$; in this situation, since ${\operatorname{Supp}}(D_{w}) \cap \Gamma_2 = \emptyset$, it is the divisor supported on $\Gamma' \setminus \{ \alpha (w_2)\}$ that coincides with $D_{w} $ via the isometry $\beta$. We show that $ D_{w} '$ is an $\alpha (w)$-reduced divisor. To argue it by contradiction, suppose that $ D_{w} '$ is not an $\alpha (w)$-reduced divisor. Then there exists a connected compact subset $A$ of $\Gamma' \setminus \alpha (w)$ such that any point in $\partial A$ is saturated for $D_{w} '$. The set $\alpha^{-1} ( A )$ is a connected compact subset of $\Gamma \setminus \{ w \}$. Since $D_{w}' (\alpha (w_2)) = 0$, $\alpha (w_2) \notin \partial A$. It follows that $\alpha$ is a homeomorphism over $\partial A$, and hence $\alpha^{-1} ( \partial A ) = \partial
\alpha^{-1} ( A )$. Since $\alpha$ is a homeomorphism over $\partial A$ and over ${\operatorname{Supp}}( D_{w} ')$, it follows that any point in $\partial
\alpha^{-1} ( A )$ is saturated for $D_{w}$. However, this contradicts that $D_w$ is $w$-reduced. This shows that $D_{w}'$ is $\alpha (w)$-reduced.
Since $g ( \Gamma_2) \geq 1$, $g ( \Gamma' ) =
g (\Gamma ) - g ( \Gamma_2)
\leq g (\Gamma ) - 1$, and hence $\deg ( D_{w}' ) \geq g ( \Gamma' ) + 2$. By Remark \[rmk:on:reduced:divisors\], it follows that $D_{w}' (\alpha (w) ) \geq 2$. Thus $D_{w} (w) \geq 2$. We have shown the claim.
Let us finish the proof of the lemma. If ${\operatorname{Supp}}(D_{w}) \cap \Gamma_i \neq \emptyset$ for any $i = 1,2$, then $D_{w}$ is a desired effective divisor, since $D_{w} \in |D|$. Suppose that ${\operatorname{Supp}}( D_{w} ) \cap \Gamma_i = \emptyset$ for some $i = 1, 2$. Then by Claim \[claim.reduced:divisor:new3\], $D_{w} (w) \geq 2$. Then $E^\prime := D_{w} - 2 [w]$ is an effective divisor. Since $w$ is the middle point of $f$, $2 [w] \sim [w_1] + [w_2]$, so that $D \sim D_{w} \sim E^\prime + [w_1] + [w_2]$. Then $E:= E^\prime + [w_1] + [w_2]$ is a desired effective divisor. Thus the lemma holds.
500
*Proof of Proposition \[prop:separation:special2\].*We take a divisor $\tilde{D}$ on $X$ such that $L \otimes \omega_{X}^{\otimes -1} \cong {{\mathcal{O}}}_{X} ( \tilde{D} )$. Let $\tau : X^{{{\operatorname{an}}}} \to \Gamma_{\min}$ be the retraction map. Then $\deg
\left(
\tau_{\ast} ( \tilde{D})
\right) \geq (3g-1) - (2g-2) = g+1$, and by Lemma \[lemma:separation:special1\], there exists a $D \in
\left|
\tau_{\ast} ( \tilde{D})
\right|$ such that $\deg
\left(
{\left.{D}\right\vert_{{\Gamma_1}}}
\right)
\geq
1$ and $\deg
\left(
{\left.{D}\right\vert_{{\Gamma_2}}}
\right)
\geq
1$. By Proposition \[prop:pregoodmodel:pre\], there exist a good model $\mathscr{X}$ of $X$ and a line bundle $\mathscr{M}$ over $\mathscr{X}$ such that $D_{\mathscr{M}} = D$. Set $\mathscr{L} := \mathscr{M} \otimes \omega_{\mathscr{X}/R}$. Then by construction, $(\mathscr{X} , \mathscr{L})$ is a model that has the required properties in Proposition \[prop:separation:special2\].
500
### Proof of Proposition \[prop:sep:edges\]
Let $v_1$ and $v_2$ denote the end vertices of $e$ and let $v$ be the middle point of $e$. We remark that if $e$ is a loop, then $v_1 = v_2$ and $v$ is the point antipodal to $v_1 (= v_2)$.
Our argument goes according to cases (A) or (B) above.
Suppose that $f$ is of disconnected type, or $f$ is of connected type and $\Gamma_{\min} \setminus ( \mathrm{relin} (e) \cup \mathrm{relin} (f) )$ is connected.
Applying Proposition \[prop:pregoodmodel\], we obtain a model $( \mathscr{X} , \mathscr{L})$ with the following properties:
1. $\mathscr{X}$ is a good model;
2. $D_{\mathscr{M}} - [v_1]$ is effective;
3. $D_{\mathscr{M}} - [v_1]$ has positive degree over any island of $\Gamma_{\min}$.
4. $\deg({\left.{D_{\mathscr{M}}}\right\vert_{{{\operatorname{relin}}(e)}}}) \leq 1$ and $\deg({\left.{D_{\mathscr{M}}}\right\vert_{{{\operatorname{relin}}(f)}}}) \leq 1$. (With comparison with Proposition \[prop:pregoodmodel\](ii), we note that $v_1$ does not lie in the relative interior of $e$ nor $f$.)
We are going to show that $( \mathscr{X} , \mathscr{L})$ enjoys the required properties.
We take any $p \in {\operatorname{Sing}}( \mathscr{X}_s )_{\subset e}$. Since $e$ is of connected type and $\Delta_p \subset e$ , we see that $\Gamma_{\min} \setminus
{\operatorname{relin}}(\Delta_p)$ is connected. Since $\deg({\left.{\mathscr{M}}\right\vert_{{\Gamma_{\min} \setminus
{\operatorname{relin}}(\Delta_p)}}}) \geq \deg({\left.{\mathscr{M}}\right\vert_{{\Gamma_{\min} \setminus
{\operatorname{relin}}(e)}}}) \geq (g+1) - \deg({\left.{D_{\mathscr{M}}}\right\vert_{{{\operatorname{relin}}(e)}}}) \geq g > 0$ by condition (iv), if follows from Lemma \[lemma:freeatnode:graphversion:emptyset\] that ${{\mathscr{L}}}$ is free at $p$. Thus property (ii) of Proposition \[prop:sep:edges\] is verified.
It remains to verify property (iii) of Proposition \[prop:sep:edges\], i.e., $\mathscr{L}_1 = \mathscr{L} (-\mathscr{V}_e)$ is free at any $q \in {\operatorname{Sing}}( \mathscr{X}_s )_{\subset f}$. Fixing any $q \in {\operatorname{Sing}}( \mathscr{X}_s )_{\subset f}$, we set $\widetilde{\Gamma} := \Gamma_{\min} \setminus \mathrm{relin}(\Delta_q)$. Note that $\widetilde{\Gamma}$ is connected if and only if $f$ is of connected type. If $\widetilde{\Gamma}$ is not connected, it has exactly two connected components, each of which contains some island of $\Gamma_{\min}$.
We set $\mathscr{M}_1 :=
\mathscr{L}_1 \otimes \omega_{\mathscr{X}/R}^{\otimes -1}
=
\mathscr{M}
( - \mathscr{V}_e )$. Then $$D_{\mathscr{M}_1} = (D_{\mathscr{M}} - [v_1] ) - [v_2] +2 [v].$$ Remark that $D_{\mathscr{M}_1}$ is effective over $\Gamma_{\min} \setminus
\{ v_2 \}$.
Suppose that $D_{\mathscr{M}_1} (v_2) \geq 0$. Then $D_{\mathscr{M}_1}$ is effective. If $\widetilde{\Gamma}$ is connected, then we note from condition (iv) that $\deg({\left.{D_{\mathscr{M}_1}}\right\vert_{{\widetilde{\Gamma}}}}) \geq
(g+1) - \deg({\left.{D_{\mathscr{M}}}\right\vert_{{{\operatorname{relin}}(f)}}}) \geq g > 0$. If $\widetilde{\Gamma}$ is not connected, then by condition (iii) above, $D_{\mathscr{M}_1}$ has positive degree on any connected component of $\widetilde{\Gamma}$. Thus Lemma \[lemma:freeatnode:graphversion:emptyset\] concludes that $\mathscr{L}_1$ is free at $q$.
Suppose that $D_{\mathscr{M}_1} (v_2) \leq -1$. To show that $\mathscr{L}_1$ is free at $q$, we use Lemma \[lemma:freeatnode:graphversion:nonemptyset\]. Since $D_{\mathscr{M}} - [v_1]$ is effective, we then have $D_{\mathscr{M}_1} (v_2) = -1$ and $D_{\mathscr{M}_1} (x) \geq 0$ for any other $x \in \Gamma$. If $\widetilde{\Gamma}'$ is a connected component of $\widetilde{\Gamma}$ with $v_2 \notin \widetilde{\Gamma}'$, then ${\left.{D_{\mathscr{M}_1}}\right\vert_{{\widetilde{\Gamma}'}}}$ has positive degree by condition (iii) above.
Let $\widetilde{\Gamma}_2$ be the connected component of $\widetilde{\Gamma}$ with $v_2 \in \widetilde{\Gamma}_2$. Since $e$ is a connected subspace of $\widetilde{\Gamma}$ and has a common point $v_2$ with $\widetilde{\Gamma}_2$, we have $e \subset \widetilde{\Gamma}_2$. Let $\widetilde{\Gamma}_3^{\circ}$ be the connected component of $\widetilde{\Gamma}_2 \setminus \{ v_2 \}$ containing $\mathrm{relin} (e)$. Since $\deg
\left(
{\left.{D_{\mathscr{M}_1}}\right\vert_{{\mathrm{relin} (e)}}}
\right)
\geq
D_{\mathscr{M}_1} (v)
\geq 2$, we have $\deg \left(
{\left.{D_{\mathscr{M}_1}}\right\vert_{{\widetilde{\Gamma}_3^{\circ}}}}
\right) \geq 2$.
To apply Lemma \[lemma:freeatnode:graphversion:nonemptyset\], we need to show that the valence of $\Gamma_3 := \widetilde{\Gamma}_3^{\circ} \cup \{ v_2 \}$ at $v_2$ is at least $2$. If $f$ is of disconnected type, then since $e$ is of connected type, we see that $\widetilde{\Gamma}_2 \setminus \mathrm{relin}(e)$ is still connected. Suppose that $f$ is of connected type. Then $\widetilde{\Gamma}$ is connected, and thus $\widetilde{\Gamma}_2 = \widetilde{\Gamma}$. By the condition of (A), $\widetilde{\Gamma} \setminus \mathrm{relin} (e)
= \widetilde{\Gamma}_2 \setminus \mathrm{relin} (e) $ is connected. Thus for Case (A), $\widetilde{\Gamma}_2 \setminus \mathrm{relin}(e)$ is connected. It follows that the valence of $\widetilde{\Gamma}_3$ at $v_2$ is at least $2$; otherwise, it equals $1$, and since $\widetilde{\Gamma}_3^{\circ}$ is a connected component of $\widetilde{\Gamma} \setminus \{ v_2 \}$, $\widetilde{\Gamma}_2 \setminus \mathrm{relin}(e)$ should not be connected. Thus $\widetilde{\Gamma}_3$ at $v_2$ is at least $2$.
Now, using Lemma \[lemma:freeatnode:graphversion:nonemptyset\], we obtain that $\mathscr{L}_1$ is free at $q$ for Case (A).
Suppose that $f$ is of connected type and $\Gamma_{\min} \setminus ( \mathrm{relin} (e) \cup \mathrm{relin} (f) )$ is not connected.
In this case, since $e$ is not a loop, $v_1 \neq v_2$. Further, $\Gamma_{\min} \setminus ( \mathrm{relin} (e) \cup \mathrm{relin} (f) )$ has exactly two connected components. We denote these connected components by $\Gamma_1$ and $\Gamma_2$, where $v_1 \in \Gamma_1$ and $v_2 \in \Gamma_2$. Then by Proposition \[prop:separation:special2\], there exists a model $( \mathscr{X} , \mathscr{L})$ of $(X,L)$ with the following properties:
1. $\mathscr{X}$ is a good model;
2. $D_{\mathscr{M}}$ is effective, and ${\left.{D_{\mathscr{M}}}\right\vert_{{\Gamma_1}}}$ and ${\left.{D_{\mathscr{M}}}\right\vert_{{\Gamma_2}}}$ have positive degree.
We prove that $( \mathscr{X} , \mathscr{L})$ enjoys the required properties. Take any $p \in {\operatorname{Sing}}( \mathscr{X}_s )_{\subset e}$. Since $e$ is of connected type and $\Delta_p \subset e$ , we see that $\Gamma_{\min} \setminus
{\operatorname{relin}}(\Delta_p)$ is connected. Since $\Gamma_{\min} \setminus
{\operatorname{relin}}(\Delta_p)$ contains $\Gamma_1$ and $\Gamma_2$, we have $\deg({\left.{\mathscr{M}}\right\vert_{{\Gamma_{\min} \setminus
{\operatorname{relin}}(\Delta_p)}}}) =
\deg\left({\left.{D_{\mathscr{M}}}\right\vert_{{\Gamma_1}}}\right)
+ \deg\left({\left.{D_{\mathscr{M}}}\right\vert_{{\Gamma_2}}}\right)
\geq 2 > 0$ by condition (b) above. Then Lemma \[lemma:freeatnode:graphversion:emptyset\] shows that ${{\mathscr{L}}}$ is free at $p$. Thus property (ii) of Proposition \[prop:sep:edges\] is verified.
Let us verify property (iii) of Proposition \[prop:sep:edges\]. We fix any $q \in {\operatorname{Sing}}({{\mathscr{X}}}_s)_{\subset f}$. Our goal is to show that $\mathscr{L}_1 := \mathscr{L}(- \mathscr{V}_e)$ is free at $q$. As in Case (A), we set $\widetilde{\Gamma} := \Gamma_{\min}\setminus {\operatorname{relin}}(\Delta_q)$. Since $f$ is of connected type, $\widetilde{\Gamma}$ is connected. By the condition of Case (B), $\widetilde{\Gamma} \setminus
\mathrm{relin} (e)
$ is not connected. Let $\widetilde{\Gamma}_1$ and $\widetilde{\Gamma}_2$ be the connected components of $\widetilde{\Gamma} \setminus
\mathrm{relin} (e)$ with $v_1 \in \widetilde{\Gamma}_1$ and $v_2 \in \widetilde{\Gamma}_2$. Note that $\Gamma_1 \subset \widetilde{\Gamma}_1$ and $\Gamma_2 \subset \widetilde{\Gamma}_2$.
Set $ \mathscr{M}_1 :=
\mathscr{L}_1 \otimes \omega_{\mathscr{X}/R}^{\otimes -1}
= \mathscr{M}(- \mathscr{V}_e )$ as in Case (A), so that $D_{\mathscr{M}_1} = D_{\mathscr{M}} - [v_1] - [v_2] + 2 [v]$. We note that $$V_1 :=
\{ v \in V ( \mathscr{X} ) \mid
D_{\mathscr{M}_1} (v) \leq -1 \} \subseteq \{ v_1 , v_2 \}
.$$ If $V_1 = \emptyset$, then ${\left.{D_{\mathscr{M}_1}}\right\vert_{{\widetilde{\Gamma}}}}$ is effective and $\deg\left({\left.{D_{\mathscr{M}_1}}\right\vert_{{\widetilde{\Gamma}}}}\right) \geq
\deg\left({\left.{D_{\mathscr{M}}}\right\vert_{{\Gamma_1}}}\right)
+ \deg\left({\left.{D_{\mathscr{M}}}\right\vert_{{\Gamma_2}}}\right)
\geq 2 > 0$, and hence Lemma \[lemma:freeatnode:graphversion:emptyset\] concludes that $\mathscr{L}_1$ is free at $q$.
We suppose that $V_1 \neq \emptyset$. We want to use Lemma \[lemma:freeatnode:graphversion:nonemptyset\] to prove that $\mathscr{L}_1$ is free at $q$. First, suppose moreover that $V_1 = \{ v_1 \}$ or $V_1 = \{ v_2 \}$. By symmetry, we may and do assume that $V_1 = \{ v_2 \}$. Then it follows from $D_{{{\mathscr{M}}}} \geq 0$ that $D_{\mathscr{M}_1} (v_2) = -1$. We set $\widetilde{\Gamma}_1^{\prime\circ} :=
\widetilde{\Gamma}_1 \cup \mathrm{relin} (e) $ and $\widetilde{\Gamma}_2^{\prime\circ} :=\widetilde{\Gamma}_2 \setminus \{ v_2 \}$. We have $\deg
\left(
{\left.{D_{\mathscr{M}_1}}\right\vert_{{\widetilde{\Gamma}_1^{\prime\circ}}}}
\right) \geq
\deg
\left(
{\left.{D_{\mathscr{M}_1}}\right\vert_{{\mathrm{relin}(e)}}}
\right) \geq D_{\mathscr{M}_1} (v) = 2 > 0$. By property (b) above, $\deg
\left(
{\left.{D_{\mathscr{M}}}\right\vert_{{\widetilde{\Gamma}_2}}}
\right) \geq 1$. Since $D_{\mathscr{M}_1} (v_2) = -1$, we have $D_{\mathscr{M}} (v_2) = 0$. It follows that $\deg
\left(
{\left.{D_{\mathscr{M}}}\right\vert_{{\widetilde{\Gamma}_2^{\prime\circ}}}}
\right) \geq 1$. Since $D_{\mathscr{M}_1}$ coincides with $D_{\mathscr{M}}$ on $\widetilde{\Gamma}_2^{\prime\circ}$, we also have $\deg
\left(
{\left.{D_{\mathscr{M}_1}}\right\vert_{{\widetilde{\Gamma}_2^{\prime\circ}}}}
\right) \geq 1$. Then there exists a connected component $\widetilde{\Gamma}_2^{\prime\prime\circ}$ of $\widetilde{\Gamma}_2^{\prime\circ}$ such that $\deg
\left(
{\left.{D_{\mathscr{M}_1}}\right\vert_{{\widetilde{\Gamma}_2^{\prime\prime\circ}}}}
\right) \geq 1$. We see that the valence of $\widetilde{\Gamma}_1^{\prime\circ} \cup \widetilde{\Gamma}_2^{\prime\prime\circ} \cup \{ v_2 \}$ at $v_2$ is at least $2$. Then Lemma \[lemma:freeatnode:graphversion:nonemptyset\] concludes that $\mathscr{L}_1$ is free at $q$.
Finally, we consider the case where $V_1 = \{ v_1 , v_2 \}$. In this case, $\widetilde{\Gamma} \setminus V_1$ is the disjoint union of $\widetilde{\Gamma}_1 \setminus \{ v_1 \}$, $\widetilde{\Gamma}_2 \setminus \{ v_2 \}$, and $\mathrm{relin} (e)$. Since $D_{\mathscr{M}_1} (v_1) = D_{\mathscr{M}_1} (v_2) = -1$, we have $D_{\mathscr{M}} (v_1) =D_{\mathscr{M}} (v_2) = 0$. Since $\deg
\left( {\left.{D_{\mathscr{M}}}\right\vert_{{\widetilde{\Gamma}_1}}}
\right) \geq 1$ and $\deg
\left( {\left.{D_{\mathscr{M}}}\right\vert_{{\widetilde{\Gamma}_2}}}
\right) \geq 1$, it follows that $\deg
\left( {\left.{D_{\mathscr{M}}}\right\vert_{{\widetilde{\Gamma}_1 \setminus \{ v_1 \}}}}
\right) \geq 1$ and $\deg
\left( {\left.{D_{\mathscr{M}}}\right\vert_{{\widetilde{\Gamma}_2 \setminus \{ v_2 \}}}}
\right) \geq 1$. Thus $\deg
\left( {\left.{D_{\mathscr{M}_1}}\right\vert_{{\widetilde{\Gamma}_1 \setminus \{ v_1 \}}}}
\right) \geq 1$ and $\deg
\left( {\left.{D_{\mathscr{M}_1}}\right\vert_{{\widetilde{\Gamma}_2 \setminus \{ v_2 \}}}}
\right) \geq 1$. It follows that there exist connected components $\widetilde{\Gamma}_1^{\circ}$ and $\widetilde{\Gamma}_2^{\circ}$ of $\widetilde{\Gamma}_1 \setminus \{ v_1 \}$ and $\widetilde{\Gamma}_2 \setminus \{ v_2 \}$, respectively, such that $\deg
\left( {\left.{D_{\mathscr{M}_1}}\right\vert_{{\widetilde{\Gamma}_1^{\circ}}}}
\right) \geq 1$ and $\deg
\left( {\left.{D_{\mathscr{M}_1}}\right\vert_{{\widetilde{\Gamma}_2^{\circ}}}}
\right) \geq 1$. We also have $\deg
\left( {\left.{D_{\mathscr{M}_1}}\right\vert_{{\mathrm{relin} (e)}}}
\right) \geq D_{\mathscr{M}_1} (v) = 2$. Furthermore, for each $i=1, 2$, the valence of $\widetilde{\Gamma}_1^{\circ}
\cup
\widetilde{\Gamma}_2^{\circ}
\cup
\mathrm{relin}(e)
\cup
V_1$ at $v_i$ is at least $2$. (Indeed, in this subgraph, $e$ is an edge emanating from $v_i$, and there exists in $\widetilde{\Gamma}_{i}^{\circ} \cup \{ v_i \}$ another edge emanating from $v_i$.) Thus Lemma \[lemma:freeatnode:graphversion:nonemptyset\] concludes that $\mathscr{L}_1$ is free at $q$. This completes the proof of Proposition \[prop:sep:edges\].
500
Separating vertices {#subsection:VS}
-------------------
In this subsection, we consider the separation of vertices. Recall that if ${{\mathscr{X}}}$ is a Deligne–Mumford strictly semistable model of $X$ (for example, if ${{\mathscr{X}}}$ is a good model), then the associated skeleton $S({{\mathscr{X}}})$ is equal to the minimal skeleton $\Gamma_{\min}$.
Let $v_1 , v_2 \in V ( \mathscr{X}^{{\operatorname{st}}})$ be vertices. Let $( \mathscr{X} , \mathscr{L} )$ be a model of $(X,L)$ such that ${{\mathscr{X}}}$ is a good model. Let $C_1$ and $C_2$ denote the irreducible components of $\mathscr{X}_s$ with $[C_1] = v_1$ and $[C_2] = v_2$.
1. We call $\widetilde{s} \in H^0 ( \mathscr{L} )$ a *$v_1$-base section* if there exists $p \in C_1$ with $\widetilde{s} (p) \neq 0$.
2. Suppose that $v_1 \neq v_2$. We call $\widetilde{s}
\in H^0 ( \mathscr{L})$ a *$(v_1 , v_2)$-separating section* if ${\left.{\widetilde{s}}\right\vert_{{C_1}}} = 0$ in $H^0({\left.{\mathscr{L}}\right\vert_{{C_1}}})$ and and ${\left.{\widetilde{s}}\right\vert_{{C_2}}} \neq 0$ in $H^0({\left.{\mathscr{L}}\right\vert_{{C_2}}})$.
The following lemma shows that once one has a model with a vertex-base section and a vertices-separating section, then one can separate vertices.
\[lemma:vertices-separating\] Let $v_1 , v_2 \in V ( \mathscr{X}^{{\operatorname{st}}})$ be distinct vertices. Suppose that there exists a model $( \mathscr{X} , \mathscr{L} )$ of $(X,L)$ such that ${{\mathscr{X}}}$ is a good model and that there exists a $v_1$-base section $\widetilde{s}_0$ and a $(v_1 , v_2)$-separating section $\widetilde{s}_1$. We set $g := \widetilde{s}_1 / \widetilde{s}_0$ and define a nonzero rational function $h$ on $X$ by $h := {\left.{g}\right\vert_{{X}}}$. Then $- \log |h (v_1)| > 0$ and $ - \log | h (v_2) | \leq 0$. In particular, we have $- \log |h (v_1)| \neq
- \log | h (v_2) |$.
[[*Proof.*]{}]{}Let $C_1$ and $C_2$ be the irreducible components of $\mathscr{X}_s$ with $[C_1] = v_1$ and $[C_2] = v_2$. Then ${\operatorname{ord}}_{C_1} (g) > 0$ and ${\operatorname{ord}}_{C_2}( g ) \leq 0$. Thus the lemma follows.
500
Our task is then to show the existence of a model of $(X,L)$ that has an vertex-base section and a vertices-separating section. The key is the following proposition.
\[prop:existence:sep:sections\] Let $v_1 , v_2 \in V ( \mathscr{X}^{{\operatorname{st}}})$ be distinct vertices. Assume that $v_1$ and $v_2$ do not belong to the same island of $\Gamma_{\min}$ (cf. Definition \[def:island\]). Further, assume that $\deg (L) \geq 3g-1$. Then there exists a model $( \mathscr{X} , \mathscr{L} )$ of $(X,L)$ such that ${{\mathscr{X}}}$ is a good model and such that or holds.
1. $\mathscr{L}$ has a $v_1$-base section and a $(v_1 , v_2)$-separating section.
2. $\mathscr{L}$ has a $v_2$-base section and a $(v_2 , v_1)$-separating section.
The proof of Proposition \[prop:existence:sep:sections\] will require a careful analysis of $\Gamma_{\min}$. Before moving on to the proof, we show the following lemma, which gives a sufficient condition in terms of $\Lambda$-metric graphs for a model to have a vertices-separated section.
\[lemma:exist:vert:sep:sections\] Let $v_1 , v_2 \in V ( \mathscr{X}^{{\operatorname{st}}})$ be distinct vertices. Let $( \mathscr{X} , \mathscr{L} )$ be model of $(X,L)$ such that ${{\mathscr{X}}}$ is a good model. Assume that there exists a $v_1' \in \Gamma_{\min} \setminus \{ v_2 \}$ such that $v_1$ and $v_1'$ belong to the same connected component of $\Gamma_{\min} \setminus \{ v_2 \}$ and such that $D_{\mathscr{M}} - [v_1 ] - [v_1']$ is effective. Then there exists a $(v_2 , v_1)$-separating section of $\mathscr{L}$.
[[*Proof.*]{}]{}Since $\mathscr{X}$ dominates $\mathscr{X}^{{\operatorname{st}}}$, we note that $v_1 , v_2 \in V( \mathscr{X} )$. Further, since $D_{\mathscr{M}} - [v_1 ] - [v_1']$ is effective, we note that $v_1' \in V( \mathscr{X} )$. Let $C_1$, $C_2$, and $C_1'$ be the irreducible components of $\mathscr{X}_s$ with $[C_1] = v_1$, $[C_2] = v_2$, and $[C_1'] = v_1'$. Let $\Gamma_1^\circ$ be the connected component of $\Gamma \setminus \{ v_2 \}$ on which $v_1$ and $v_1'$ lie. Set $F_1 := \sum_{C \in {\operatorname{\operatorname{Irr}}}( \mathscr{X}_s ), [C] \in \Gamma_1^\circ} C $. Fix a point $p_1 \in C_1 (k) \setminus {\operatorname{Sing}}( \mathscr{X}_s)$. Set $\Sigma := F_1 \cap C_2$, which we regard as a nontrivial effective Cartier divisor on $C_2$. Since $D_{\mathscr{M}}$ is effective, ${\left.{\mathscr{M}}\right\vert_{{\mathscr{X}_s}}}$ is nef.
$$\setlength\unitlength{0.08truecm}
\begin{picture}(200, 60)(0,0)
\put(5, 20){\line(1,0){80}}
\put(70, 20){\line(1,1){15}}
\put(70, 20){\line(-1,1){15}}
\put(10, 20){\circle*{1.5}}
\put(50, 20){\circle*{1.5}}
\put(70, 20){\circle*{1.5}}
\put(40, 40){$\Gamma_{\min}$}
\put(8, 14){$v_1^\prime$}
\put(48, 14){$v_1$}
\put(68, 14){$v_2$}
\put(5, 12){$\underbrace{\phantom{AAAAAAAAAAAAAAAaa}}$}
\put(36, 3){$\Gamma_1^\circ$}
\put(100, 10){\line(1,0){10}}
\multiput(110,10)(1,0){10}{\line(1,0){0.5}}
\put(120, 10){\line(1,0){10}}
\put(135, 25){\line(1,0){10}}
\multiput(145,25)(1,0){10}{\line(1,0){0.5}}
\put(155, 25){\line(1,0){10}}
\put(163, 33){\line(1,0){15}}
\put(168, 38){\line(1,0){15}}
\put(173, 43){\line(1,0){15}}
\put(133, 18){\circle*{1.5}}
\put(160, 25){\circle*{1.5}}
\put(115, 23){$C_1^\prime$}
\put(143, 31){$C_1$}
\put(183, 50){$C_2$}
\put(135, 15){$p_1$}
\put(160, 19){$\Sigma$}
\put(100, 32){$\overbrace{\phantom{AAAAAAAAAAAAAAA}}$}
\put(127, 42){$F_1$}
\thicklines
\put(100, 5){\line(1,1){15}}
\put(123, 8){\line(1,1){20}}
\put(158, 23){\line(1,1){25}}
\end{picture}$$ \[figure:for:lemma:exist:vert:sep:sections\]
We claim that ${\left.{\mathscr{L}}\right\vert_{{F_1}}} ( - \Sigma )$ is free at $p_1$. Indeed, since ${\left.{\mathscr{M}}\right\vert_{{F_1}}}$ is nef and since $\deg
\left(
{\left.{{{\mathscr{M}}}}\right\vert_{{C_1}}}
\right) \geq 1$ and $
\deg
\left(
{\left.{\mathscr{M}}\right\vert_{{C_1'}}}
\right)
\geq 1$, it follows that ${\left.{\mathscr{M}}\right\vert_{{F_1}}} ( -p_1 )$ is nef and $\deg\left({\left.{\mathscr{M}}\right\vert_{{F_1}}} ( -p_1 )\right) > 0$. Since $F_1$ is connected, Lemma \[lemma:vanishing0\] gives $h^0 \left(
\left(
{\left.{\mathscr{M}}\right\vert_{{F_1}}} ( -p_1 )
\right)^{\otimes -1}
\right) = 0$. By the adjunction formula, we have $${\left.{\mathscr{L} }\right\vert_{{F_1}}} (- \Sigma -p_1) \otimes \omega_{F_1}^{\otimes -1}
=
{\left.{\mathscr{L} \otimes \omega_{\mathscr{X}/R}^{\otimes -1}}\right\vert_{{F_1}}} (-p_1)
=
{\left.{\mathscr{M}}\right\vert_{{F_1}}} ( -p_1 )$$ By the Serre duality, we obtain $h^1 \left( {\left.{\mathscr{L} }\right\vert_{{F_1}}} (- \Sigma -p_1) \right) =
h^0 \left(
\left(
{\left.{\mathscr{M}}\right\vert_{{F_1}}} ( -p_1 )
\right)^{\otimes -1}
\right) = 0$, and then Lemma \[lemma:nonbasepoint\] shows that ${\left.{\mathscr{L}}\right\vert_{{F_1}}} (- \Sigma) $ is free at $p_1$.
It follows that there exists a global section $\eta$ of ${\left.{\mathscr{L} }\right\vert_{{F_1}}} (- \Sigma) $ such that $\eta (p_1) \neq 0$. Note that ${\left.{\mathscr{L} }\right\vert_{{F_1}}} (- \Sigma)
\subset {\left.{\mathscr{L} }\right\vert_{{F_1}}} $. Extending $\eta$ to be zero outside of $F_1$, we obtain a global section $\eta_1$ of ${\left.{\mathscr{L}}\right\vert_{{\mathscr{X}_s}}}$ such that ${\left.{\eta_1}\right\vert_{{F_1}}} = \eta$ as a global section of ${\left.{\mathscr{L}}\right\vert_{{F_1}}}$. Since $C_2 \notin {\operatorname{\operatorname{Irr}}}(F_1)$, we have ${\left.{\eta_1}\right\vert_{{C_2}}} = 0$.
Since ${\left.{\mathscr{M}}\right\vert_{{\mathscr{X}_s}}}$ is nef and has positive degree, $h^0
\left( {\left.{\mathscr{M}}\right\vert_{{\mathscr{X}_s}}}^{\otimes -1} \right) = 0$, and thus $h^1 \left( {\left.{\mathscr{L}}\right\vert_{{\mathscr{X}_s}}} \right) = 0$ by the Serre duality. By the base-change theorem, the natural map $H^0 ( \mathscr{L} ) \to
H^0 \left( {\left.{\mathscr{L}}\right\vert_{{\mathscr{X}_s}}} \right)$ is surjective. It follows that there exists a global section $\widetilde{s}_1 \in H^0 ( \mathscr{L} )$ such that ${\left.{\widetilde{s}_1}\right\vert_{{\mathscr{X}_s}}} = \eta_1$. Then ${\left.{\widetilde{s}_1}\right\vert_{{C_1}}} \neq 0$ and ${\left.{\widetilde{s}_1}\right\vert_{{C_2}}} = 0$, so that $\widetilde{s}_1$ is a $(v_2 , v_1)$-separating section of $\mathscr{L}$.
500
We start the proof of Proposition \[prop:existence:sep:sections\].
*Proof of Proposition \[prop:existence:sep:sections\].*Let $e \in E({{\mathscr{X}}}^{{\operatorname{st}}})$ be any edge of disconnected type. For a good model ${{\mathscr{X}}}$ of $X$, we denote by $C_1$ and $C_2$ the irreducible components of ${{\mathscr{X}}}_s$ with $v_1 = [C_1]$ and $v_2 = [C_2]$.
[**Case 1.**]{} Suppose that $\Gamma_{\min} \setminus \{ v_1 \}$ is connected or $\Gamma_{\min} \setminus \{ v_2 \}$ is connected. Without loss of generality, we may assume that $\Gamma_{\min} \setminus \{ v_2 \}$ is connected. By Proposition \[prop:pregoodmodel\], there exists a model $( \mathscr{X} , \mathscr{L})$ of $(X,L)$ with the following properties:
1. $\mathscr{X}$ is a good model;
2. $D_{\mathscr{M}} - [v_1]$ is effective.
First, suppose that $D_{\mathscr{M}} (v_2) \geq 2$. Then applying Lemma \[lemma:exist:vert:sep:sections\] with $v_2$, $ v_1 $, and $ v_2$ in place of $v_1$, $v_2 $, and $ v_1'$, respectively, we see that $\mathscr{L}$ has a $(v_1 , v_2)$-separating section. Further, since $D_{\mathscr{M}} - [v_1]$ is effective, Lemma \[lemma:freeatregular:graphversion:effective\] shows that $\mathscr{L}$ is free at any point of $C_1(k) \setminus {\operatorname{Sing}}( \mathscr{X}_s )$. Thus $\mathscr{L}$ has a $v_1$-base section, which proves the proposition.
Next, suppose that $D_{\mathscr{M}} (v_2) \leq 1$. Then, since $\deg ( D_{\mathscr{M}} - [v_1] ) \geq (g+1) - 1\geq 2$, we have $\deg \left(
{\left.{(D_{\mathscr{M}} - [v_1])}\right\vert_{{\Gamma_{\min} \setminus \{ v_2 \}}}}
\right) \geq 1$. It follows that there exists a $v_1' \in \Gamma_{\min}
\setminus \{ v_2 \}$ such that $D_{\mathscr{M}} - [v_1] - [v_1']$ is effective. Since $\Gamma_{\min} \setminus \{ v_2 \}$ is assumed to be connected, Lemma \[lemma:exist:vert:sep:sections\] gives that $\mathscr{L}$ has a $(v_2 , v_1)$-separating section.
We need to prove that $\mathscr{L}$ has a $v_2$-base section. Since $D_{\mathscr{M}} - [v_1]$ is effective and we are assuming that $D_{\mathscr{M}} (v_2) \leq 1$, we have $0 \leq D_{\mathscr{M}} (v_2) \leq 1$. If $D_{\mathscr{M}} (v_2) = 1$, then $D_{\mathscr{M}} - [v_2]$ is effective and $\deg ( D_{\mathscr{M}} - [v_2] ) \geq (g+1) - 1\geq 2 > 0$, so that Lemma \[lemma:freeatregular:graphversion:effective\] gives that $\mathscr{L}$ is free at any $q \in C_{2}(k) \setminus {\operatorname{Sing}}( \mathscr{X}_s)$; thus $\mathscr{L}$ has a $v_2$-base section.
Suppose that $D_{\mathscr{M}} (v_2) = 0$. If the valence of $v_2$ is equal to $1$, then the edge $e$ with an end vertex $v_2$ is of disconnected type and $v_1$ and $v_2$ belong to different connected components of $\Gamma_{\min} \setminus \mathrm{relin}(e)$, which contradicts the assumption that $v_1$ and $v_2$ belong to the same island of $\Gamma_{\min}$. Thus the valence of $v_2$ is at least $2$. Further, $\Gamma_{\min} \setminus \{ v_2 \}$ is assumed to be connected. By Lemma \[lemma:freeatregular:graphversion\], $\mathscr{L}$ is free at any point of $C_2(k) \setminus {\operatorname{Sing}}(\mathscr{X}_s)$. Thus $\mathscr{L}$ has a $v_2$-base section.
To sum up, we have shown that there exists a model $( \mathscr{X} , \mathscr{L})$ such that (i) or (ii) of Proposition \[prop:existence:sep:sections\] holds in Case 1.
Suppose that $\Gamma_{\min} \setminus \{ v_1 \}$ is not connected and $\Gamma_{\min} \setminus \{ v_2 \}$ is not connected.
Suppose that $\Gamma_{\min}$ has an edge $e$ of disconnected type. Then one of the following holds:
1. $v_1$ and $\mathrm{relin}(e)$ are contained in the same connected component of $\Gamma_{\min} \setminus \{ v_2 \}$;
2. $v_2$ and $\mathrm{relin}(e)$ are contained in the same connected component of $\Gamma_{\min} \setminus \{ v_1 \}$
Indeed, suppose that (II) does not hold. We take any point $w \in \mathrm{relin}(e)$. Then there exists a continuous map $\gamma: [0, 1] \to \Gamma_{\min}$ such that $\gamma(0) = v_2$, $\gamma(1/2) = v_1$ and $\gamma(1) = w$ and such that $\gamma(t) \neq v_2$ for any $t \ne 0$ and $\gamma(t) \neq v_1$ for any $t \neq 1/2$. The half path $\gamma: [1/2, 1] \to \Gamma_{\min}$ connects $v_1$ and $w$ in $\Gamma_{\min} \setminus \{ v_2 \}$, and thus (I) holds.
Without loss of generality, we may and do assume that (I) holds. Let $\Gamma_1^{\circ}$ be the connected component of $\Gamma_{\min}
\setminus \{ v_2 \}$ with $v_1 \in \Gamma_1^{\circ}$, and set $\Gamma_1 := \Gamma_1^{\circ} \cup \{ v_2 \}$, which is the closure of $\Gamma_1^{\circ}$ in $\Gamma_{\min}$. Then $e \subset \Gamma_1$.
By Proposition \[prop:pregoodmodel\], there exists a model $( \mathscr{X} , \mathscr{L})$ of $(X,L)$ such that ${{\mathscr{X}}}$ is a good model with the following properties.
1. $D_{\mathscr{M}} - [v_1]$ is effective;
2. $\deg
\left(
{\left.{\left(
D_{\mathscr{M}} - [v_1]
\right)}\right\vert_{{\Gamma'}}}
\right) \geq 1$ for any island $\Gamma'$ of $\Gamma_{\min}$.
Since $e \subset \Gamma_1$, there exists an island $\Gamma_1'$ of $\Gamma_{\min}$ satisfying $\Gamma_1' \subset \Gamma_1 \setminus \{ v_2 \}$. By conditions (i) and (ii) above, there exists a $v_1' \in \Gamma_1'$ such that $D_{\mathscr{M}} - [v_1] - [v_1']$ is effective. Note that $\Gamma_1 \setminus \{ v_2 \} = \Gamma_1^\circ$ and that $v_1, v_1^\prime \in \Gamma_1^\circ$. Since $v_1$ and $v_1'$ belong to the same connected component $\Gamma_1^\circ$ of $\Gamma_{\min} \setminus \{ v_2 \}$, Lemma \[lemma:exist:vert:sep:sections\] concludes that $\mathscr{L}$ has a $(v_2 , v_1)$-separating section.
$$\setlength\unitlength{0.08truecm}
\begin{picture}(90, 40)(0,0)
\put(5, 20){\line(1,0){80}}
\put(10, 20){\circle*{1.5}}
\put(50, 20){\circle*{1.5}}
\put(70, 20){\circle*{1.5}}
\put(20, 20){\circle*{1.5}}
\put(40, 20){\circle*{1.5}}
\put(40, 30){$\Gamma_{\min}$}
\put(8, 14){$v_1^\prime$}
\put(48, 14){$v_1$}
\put(68, 14){$v_2$}
\put(29, 22){$e$}
\put(36, 3){$\Gamma_1$}
\put(10, 30){$\Gamma_1^\prime$}
\thicklines
\put(20, 19.9){\line(1,0){20}}
\put(20, 20.1){\line(1,0){20}}
\put(20, 20){\line(1,0){20}}
\put(5, 12){$\underbrace{\phantom{AAAAAAAAAAAAAAAAa}}$}
\put(5, 20){$\overbrace{\phantom{AAAa}}$}
\end{picture}$$ \[figure:for:prop:existence:sep:sections:case2\]
In the rest of this subcase, we construct a $v_2$-base section of $\mathscr{L}$. Since $D_{\mathscr{M}} - [v_1]$ is effective, we have $D_{\mathscr{M}} (v_2) \geq 0$.
Suppose that $D_{\mathscr{M}} (v_2) \geq 1$. Then $D_{\mathscr{M}} - [v_2] $ is effective and $\deg ( D_{\mathscr{M}} - [v_2] ) \geq (g+1) - 1\geq 2 > 0$. By Lemma \[lemma:freeatregular:graphversion:effective\], $\mathscr{L}$ is free at any $q \in C_{2}(k) \setminus {\operatorname{Sing}}( \mathscr{X}_s)$, and $\mathscr{L}$ has a $v_2$-base section.
Suppose that $D_{\mathscr{M}} (v_2) = 0$. Let $\Gamma_2'$ be the island of $\Gamma_{\min}$ with $v_2 \in \Gamma_2'$. Since $D_{\mathscr{M}} (v_2) = 0$, it follows from condition (ii) above that there exists a connected component $\Gamma_2^{\prime\prime\circ}$ of $\Gamma_2' \setminus \{ v_2 \}$ such that $\deg
\left(
{\left.{D_{\mathscr{M}}}\right\vert_{{\Gamma_2^{\prime\prime\circ}}}}
\right) \geq 1$. Let $\Gamma_2^{\prime\prime} := \Gamma_2^{\prime\prime\circ} \cup \{v_2\}$ be the closure of $\Gamma_2^{\prime\prime}$ in $\Gamma_2'$ (and thus the closure of $\Gamma_2^{\prime\prime}$ in $\Gamma_{\min}$).
We claim that the valence of $\Gamma_2^{\prime\prime}$ at $v_2$ is at least $2$. To derive a contradiction, assume that the valence is $1$. Then there exists a unique edge $e_2$ with end vertex $v_2$ such that $e_2 \subset \Gamma_2''$. Since the valence is $1$, $e_2$ is not a loop. Further, $v_2$ and $\Gamma_2^{\prime\prime\circ} \setminus {\operatorname{relin}}(e_2)$ do not belong to the same connected component of $\Gamma_2' \setminus \mathrm{relin}(e_2)$. It follows that $\Gamma_2' \setminus \mathrm{relin}(e_2)$ is not connected. On the other hand, since $e_2 \subset \Gamma_2'$, this contradicts $\Gamma_2'$ being an island of $\Gamma_{\min}$.
Now, wince the valence of $\Gamma_2^{\prime\prime}$ at $v_2$ is at least $2$, Lemma \[lemma:freeatregular:graphversion\] shows that $\mathscr{L}$ is free at any $q \in C_2(k) \setminus
{\operatorname{Sing}}( \mathscr{X}_s)$. Thus $\mathscr{L}$ has a $v_2$-base section.
We consider the case where $\Gamma_{\min}$ does not have an edge of disconnected type.
In this case, by Lemma \[lemma:goodmodel:VS\], which we prove below, we take a model $(\mathscr{X} , \mathscr{L})$ of $(X, L)$ satisfying conditions (i) or (ii) in Lemma \[lemma:goodmodel:VS\].
Suppose that this model satisfies condition (i). Then by Lemma \[lemma:exist:vert:sep:sections\], $\mathscr{L}$ has a $(v_2 , v_1)$-separating section. Further, by the same argument using Lemma \[lemma:freeatregular:graphversion\] as in Case 1, $\mathscr{L}$ has a $v_2$-base section. Thus Proposition \[prop:existence:sep:sections\] holds in this case.
Suppose that $(\mathscr{X} , \mathscr{L})$ satisfies (ii) in Lemma \[lemma:goodmodel:VS\]. Then by Lemma \[lemma:exist:vert:sep:sections\], $\mathscr{L}$ has a $(v_1 , v_2)$-separating section. Further, since $D_{\mathscr{M}} (v_1) \geq 1$, the same argument using Lemma \[lemma:freeatregular:graphversion:effective\] as in Case 1 shows that $\mathscr{L}$ has a $v_1$-base section. This completes the proof of Proposition \[prop:existence:sep:sections\].
500
\[lemma:goodmodel:VS\] Let $v_1, v_2 \in V({{\mathscr{X}}}^{{\operatorname{st}}})$ be distinct vertices of $
\Gamma_{\min}$. Assume that $\deg(L) \geq 3g-1$. Suppose that $\Gamma_{\min}\setminus\{v_1\}$ and $\Gamma_{\min}\setminus\{v_2\}$ are not connected and that $\Gamma_{\min}$ does not have an edge of disconnected type. Let $\Gamma_1^\circ$ be the connected component of $\Gamma_{\min}\setminus\{v_2\}$ with $v_1 \in \Gamma_1^\circ$, and set $\Gamma_1 := \Gamma_1^\circ \cup\{v_2\}$, which is the closure of $\Gamma_1^\circ$ in $\Gamma_{\min}$. Then there exists a model $(\mathscr{X} , \mathscr{L})$ of $(X,L)$ such that $D_{\mathscr{M}} - [v_1]$ is effective and such that one of the following holds
1. there exists a $v_1' \in \Gamma_1^{\circ}$ such that $
D_{\mathscr{M}} - [v_1] - [v_1']
$ is effective;
2. $D_{\mathscr{M}} (v_2) \geq 2$.
[[*Proof.*]{}]{}Since $\Gamma_1$ contains $v_1$ and $v_2$, $\Gamma_1$ is not a singleton. We set $\Gamma_2 := \Gamma_{\min} \setminus \Gamma_1^\circ$. Since $\Gamma_{\min} \setminus \{ v_2 \}$ is not connected, $\Gamma_2$ is not a singleton. We have $\Gamma_1 \cap \Gamma_2 = \{v_2\}$ and $\Gamma_1 \cup \Gamma_2 = \Gamma_{\min}$, and $\Gamma_1$ and $\Gamma_2$ are connected. We endow $\Gamma_1$ and $\Gamma_2$ with a finite graph structure by restricting that of $\Gamma_{\min}$.
We use the following claim.
\[claim:goodmodel:VS:divisorongraph\] Let $E \in {\operatorname{Div}}_{\Lambda} ( \Gamma_{\min} )$ with $\deg (E) \geq g$. Suppose that for any $E' \in |E|$, ${\left.{E'}\right\vert_{{\Gamma_1^{\circ}}}}$ is trivial. Then there exists an $E'' \in |E|$ such that $E'' (v_2) \geq 2$.
Let us show the claim in three steps.
We prove that $\Gamma_1$ does not have an edge of disconnected type. To argue by contradiction, assume that $\Gamma_1$ has an edge $e$ of disconnected type. Then $\Gamma_1
\setminus \mathrm{relin}(e) $ has two connected component. Let $\Gamma_1'$ be the connected component of $\Gamma_1
\setminus \mathrm{relin}(e) $ with $v_2 \in \Gamma_1'$, and let $\Gamma_1''$ be the other. Then $\Gamma_1'' \cap \Gamma_2 = \emptyset$ and $\Gamma_1' \cap \Gamma_2
= \{ v_2 \}$, and we see that $\Gamma_1''$ and $\Gamma_1' \cup \Gamma_2$ are connected components of $\Gamma_{\min} \setminus \mathrm{relin}(e)$. This means that $e$, as an edge of $\Gamma_{\min}$, is an edge of disconnected type. However, this contradicts the assumption of $\Gamma_{\min}$ in the lemma. Thus $\Gamma_1$ does not have an edge of disconnected type.
$$\setlength\unitlength{0.08truecm}
\begin{picture}(90, 45)(0,0)
\put(5, 20){\line(1,0){80}}
\put(70, 20){\circle*{1.5}}
\put(20, 20){\circle*{1.5}}
\put(40, 20){\circle*{1.5}}
\put(35, 35){$\Gamma_{\min}$}
\put(68, 14){$v_2$}
\put(29, 22){$e$}
\put(36, 3){$\Gamma_1$}
\put(79, 3){$\Gamma_2$}
\put(10, 30){$\Gamma_1^{\prime\prime}$}
\put(53, 30){$\Gamma_1^{\prime}$}
\thicklines
\put(20, 19.9){\line(1,0){20}}
\put(20, 20.1){\line(1,0){20}}
\put(20, 20){\line(1,0){20}}
\put(5, 12){$\underbrace{\phantom{AAAAAAAAAAAAAAAAa}}$}
\put(5, 20){$\overbrace{\phantom{AAAa}}$}
\put(41, 20){$\overbrace{\phantom{AAAAAAaa}}$}
\put(71, 12){$\underbrace{\phantom{AAAAA}}$}
\end{picture}$$ \[figure:for:lemma:goodmodel:VS\]
[**Step 2.**]{} We prove $g ( \Gamma_1 ) \geq 2$. To argue by contradiction, suppose that $g ( \Gamma_1 ) \leq 1$. Since $\Gamma_1$ is not a singleton and since $\Gamma_1$ does not have an edge of disconnected type by Step 1, it follows that $g(\Gamma_1) \neq 0$. Then $g ( \Gamma_1 ) = 1$. Since $\Gamma_1$ does not have an edge of disconnected type, we see that $\Gamma_1$ is a circle. Thus $\Gamma_1 \setminus \{ v_1 \}$ is connected. Since $\Gamma_2$ is connected and $( \Gamma_1 \setminus \{ v_1 \}) \cap \Gamma_2 = \{ v_2 \}$, it follows that $\Gamma_{\min} \setminus \{ v_1 \}
= (\Gamma_1 \setminus \{ v_1 \}) \cup \Gamma_2$ is connected. However, this contradicts to the assumption that $\Gamma \setminus \{ v_1 \}$ is not connected. Thus $g ( \Gamma_1 ) \geq 2$.
Since $\deg (E) \geq g$, there exists an $E' \in |E|$ by Riemann’s inequality (cf. Proposition \[prop:riemann:inequality\]). By the assumption of the claim, ${\operatorname{Supp}}(E') \subset \Gamma_2$, and hence we regard $E'$ as a divisor on $\Gamma_2$. Since $g ( \Gamma_1) + g (\Gamma_2 ) = g(\Gamma_{\min}) \leq g(\bar{\Gamma}_{\min}) = g$, Step 2 gives $g ( \Gamma_2 ) \leq g-2$. Thus $\deg (E' - 2[v_2]) \geq g (\Gamma_2)$. It follows from Proposition \[prop:RIforMG\] that there exists an effective divisor $E'_2 \in {\operatorname{Div}}_{\Lambda} ( \Gamma_2)$ such that $E'_2 + 2[v_2] \sim E'$ as divisors on $\Gamma_2$. Regard $E'_2 + 2[v_2]$ as a $\Lambda$-divisor on $\Gamma$, which we denote by $E''$. Then, since $\Gamma_1 \cup \Gamma_2 = \Gamma_{\min}$ and $\Gamma_1 \cap \Gamma_2
= \{ v_2 \}$, we have $E'' \sim E$ on $\Gamma$ as well, and $E'' (v_2) \geq 2$. This completes the proof of the claim.
We resume the proof of Lemma \[lemma:goodmodel:VS\]. Take a divisor $\tilde{D}_1 \in {\operatorname{Div}}( X )$ such that $L \otimes \omega_{X}^{\otimes -1} \cong {{\mathcal{O}}}_X ( \tilde{D}_1 )$. Recall from that $\tau_*: {\operatorname{Div}}(X) \to {\operatorname{Div}}_{\Lambda} ( \Gamma_{\min} )$ denotes the specialization map. Set $E := \tau_{\ast} ( \tilde{D}_1) - [v_1]$. Then $\deg (E) = \deg( \tilde{D}_1) - 1 \geq (g+1) - 1 = g$. By Proposition \[prop:riemann:inequality\], $|E| \neq \emptyset$.
We divide our argument into two cases. First, suppose that there exists an $E' \in |E|$ such that $\deg \left( {\left.{E'}\right\vert_{{\Gamma_1^{\circ}}}} \right) \geq 1$. Then there exists a $v_1' \in \Gamma_{1}^{\circ}$ such that $E' (v_1') \geq 1$. We set $D := E' + [v_1]$. Remark that $D \sim \tau_{\ast} ( \tilde{D}_1)$. By Proposition \[prop:pregoodmodel:pre\], there exists a good model $\mathscr{X}$ and a line bundle $\mathscr{M}$ over $\mathscr{X}$ such that $D_{\mathscr{M}} = D$. Set $\mathscr{L} := \mathscr{M} \otimes \omega_{\mathscr{X}/R}$. Then $D_{\mathscr{M}} - [v_1] = D - [v_1] = E'$, which is effective. Further, the above construction immediately shows that $( \mathscr{X} , \mathscr{L} )$ is a model of $(X , L)$ which satisfies condition (i) of Lemma \[lemma:goodmodel:VS\]. This completes the proof of the lemma in this case.
Next, suppose that there does not exist an $E' \in |E|$ such that $\deg \left( {\left.{E'}\right\vert_{{\Gamma_1^{\circ}}}} \right) \geq 1$. Then by Claim \[claim:goodmodel:VS:divisorongraph\], there exists an $E'' \in |E|$ such that $E'' (v_2) \geq 2$. Set $D := E'' + [v_1]$. By the same way as above, we construct a model $(\mathscr{X} , \mathscr{L})$ such that $D_{\mathscr{M}} = D$, where $\mathscr{M} = \mathscr{L} \otimes \omega_{\mathscr{X}/R}^{\otimes -1}$. Note that $D_{\mathscr{M}} - [v_1] = E''$ is effective. Further, it is straightforward to see that this model satisfies (ii) of Lemma \[lemma:goodmodel:VS\]. Thus we obtain the lemma.
500
Faithful tropicalization of the minimal skeleton {#subsec:ft:min:skeleton}
------------------------------------------------
We construct a faithful tropicalization of the minimal skeleton $\Gamma_{\min}$.
\[theorem:FT:canonical\] Let $X$ be a connected smooth projective curve over $K$ of genus $g \geq 2$, and let $L$ be a line bundle over $X$. Suppose that $\deg (L) \geq 3g-1$. Then there exist $s_0 , \ldots , s_N \in H^0 ( X,L)$ such that the map $\varphi : X^{{\operatorname{an}}}\to {{\mathbb{T}}}{{\mathbb{P}}}^N$ defined by $\varphi := ( - \log |s_0| : \cdots : - \log |s_N|)$ gives a faithful tropicalization of $\Gamma_{\min}$.
[[*Proof.*]{}]{}[**Step 1.**]{} By Theorem \[theorem:UT:canonical\], we take $s_0 , \ldots , s_{N_1} \in H^0 ( X,L)$ such that the map $\varphi_1 : X^{{\operatorname{an}}}\to {{\mathbb{T}}}{{\mathbb{P}}}^{N_1}$ defined by $\varphi_1 := ( - \log |s_0| : \cdots : - \log |s_{N_1}|)$ satisfies conditions (i)–(iii) in Theorem \[theorem:UT:canonical\] for $\varphi = \varphi_1$. In particular, $\varphi_1$ is unimodular.
We take care of the separation of points in the interior of an edge. We set $\{e_{1} , \ldots , e_{\alpha}\} := \{ e \in E ( \mathscr{X}^{{\operatorname{st}}}_s ) \mid
\text{$e$ is of connected type}\}$. For each $i = 1 ,\ldots , \alpha$, Proposition \[prop:model:AS\] gives a model $( \mathscr{X}_i , \mathscr{L}_i )$ of $(X,L)$ such that $\mathscr{L}_i$ has an $e_i$-base section $\widetilde{s}^{(e_i)}_{0}$ and a separating section $\widetilde{s}^{(e_i)}_{1}$ for $e_i$. Set $s_{N_1+2i-1} := {\left.{\widetilde{s}^{(e_i)}_{0}}\right\vert_{{X}}}$ and $s_{N_1+2i} := {\left.{\widetilde{s}^{(e_i)}_{1}}\right\vert_{{X}}}$ for $i = 1 , \ldots , \alpha$, and set $N_2 := N_1 + 2 \alpha$. Define $\varphi_2 : X^{{{\operatorname{an}}}} \to {{\mathbb{T}}}{{\mathbb{R}}}^{N_2}$ by $\varphi_2 :=
( - \log |s_0| : \cdots : - \log |s_{N_1}| :
- \log |s_{N_1+1}| : \cdots : - \log |s_{N_2}|)$. Since $\varphi_1$ is unimodular, so is $\varphi_2$ (cf. Remark \[remark:unimodular\]).
Let $x, y \in \Gamma$ be distinct points that belong to the interior of some common edge $e$. If $e$ is of disconnected type, then $\varphi_1 (x) \neq \varphi_1 (y)$ by condition (i) in Theorem \[theorem:UT:canonical\], and hence $\varphi_2 (x) \neq \varphi_2 (y)$. Suppose that $e$ is of connected type. We divide $e$ at the middle point and let $e^\prime$ and $e^{\prime\prime}$ be the $1$-simplices arising from this division. If $x , y \in e^\prime$ or $x , y \in e^{\prime\prime}$, then by condition (ii) in Theorem \[theorem:UT:canonical\], we have $\varphi_1 (x) \neq \varphi_1 (y)$ and hence $\varphi_2 (x) \neq \varphi_2 (y)$. If $x \in \mathrm{relin}(e^{\prime})$ and $y \in \mathrm{relin}(e^{\prime\prime})$, then by and Lemma \[lemma:asymmetric:section:separate:new\], we obtain $\varphi_2 (x) \neq \varphi_2 (y)$. Thus $\varphi_2$ separate two distinct points that belong to the interior of the same edge.
We take care of separation of two distinct points one of which is not a vertex. We set $$\{(e_1, f_1), \ldots, (e_\beta, f_\beta)\}
:=
\{
(e, f) \in E ( \mathscr{X}^{{\operatorname{st}}}) \times E ( \mathscr{X}^{{\operatorname{st}}})
\mid
\text{$e$ is of connected type, $e \neq f$}.
\}$$ For each $i$, Proposition \[prop:model:ES\] gives a model a model $( \mathscr{X}_i , \mathscr{L}_i )$ of $(X,L)$ such that $\mathscr{L}_i$ has an $e_i$-base section $\widetilde{s}^{(e_i)}_{0}$ and an $(e_i, f_i)$-separating section $\widetilde{s}^{(e_i , f_i)}_{1}$. Set $s_{N_2+2i-1} := {\left.{\widetilde{s}^{(e_i)}_{0}}\right\vert_{{X}}}$ and $s_{N_2+2i} := {\left.{\widetilde{s}^{(e_i , f_i)}_{1}}\right\vert_{{X}}}$ for $i = 1 , \ldots , \beta$, and set $N_3 := N_2 + 2 \beta$. Define $\varphi_3 : X^{{{\operatorname{an}}}} \to {{\mathbb{T}}}{{\mathbb{R}}}^{N_3}$ by $\varphi_3 :=
( - \log |s_0| : \cdots : - \log |s_{N_2}| :
- \log |s_{N_2+1}| : \cdots : - \log |s_{N_3}|)$. Since $\varphi_2$ is unimodular, so is $\varphi_3$ (cf. Remark \[remark:unimodular\]).
Let $x , y \in \Gamma$ be distinct points. Assuming that not both of $x ,y$ are vertices, let us show that $\varphi_3 (x) \neq \varphi_3 (y)$. Without loss of generality, we may and do assume that there exists an edge $e$ with $x \in \mathrm{relin}(e)$. If $y \in \mathrm{relin}(e)$, then $\varphi_2 (x) \neq \varphi_2 (y)$ by Step 2, and hence $\varphi_3 (x) \neq \varphi_3 (y)$. Suppose that $y \notin \mathrm{relin}(e)$. Then there exists an edge $f$ with $f \neq e$ such that $y \in f$. If $e$ is of disconnected type, then $\varphi_1 (x) \neq \varphi_1 (y)$ as noted in Remark \[remark:disconnect-other\], and hence $\varphi_3 (x) \neq \varphi_3 (y)$. Suppose that $e$ is of connected type. Then we take $1 \leq i \leq \beta$ with $(e,f) = (e_i , f_i)$. By Lemma \[lemma:ES:separate:edges\], $- \log |s_{N_2+2i} / s_{N_2+2i-1} (x)|
\neq - \log |s_{N_2+2i} / s_{N_2+2i-1} (y)|$, and hence $\varphi_3 (x) \neq \varphi_3 (y)$. Thus $\varphi_3 (x) \neq \varphi_3 (y)$ if $x \neq y$ and not both of $x$ and $y$ are vertices.
We take care of separation of two distinct vertices. Set $$W := \{ (v,w) \in V ( \mathscr{X}^{{\operatorname{st}}})
\times V ( \mathscr{X}^{{\operatorname{st}}})
\mid
\text{$v \neq w$, and $v$ and $w$ belong to the same island of $\Gamma_{\min}$}
\}.$$ For any $(v,w) \in W$, Proposition \[prop:existence:sep:sections\] gives a model $( \mathscr{X}^{(v,w)} , \mathscr{L}^{(v,w)})$ such that $\mathscr{L}^{(v,w)}$ has a $v$-separating section and a $(v,w)$-separating section or has a $w$-separating section and a $(w,v)$-separating section. We take an (indexed) subset $\{(v_i, w_i)\}_{i=1}^{\gamma}$ such that
- the model $\mathscr{L}^{(v_i,w_i)}$ has a $v_i$-base section $\widetilde{s}^{(v_i,w_i)}_0$ and a $(v_i, w_i)$-separating section $\widetilde{s}^{(v_i , w_i)}_{1}$;
- for any $(v , w) \in W$, there exists a unique $i = 1 ,\ldots , \gamma$ such that $\{ v ,w \} = \{ v_i, w_i \}$.
Set $s_{N_3+2i-1} := {\left.{\widetilde{s}^{(v_i,w_i)}_{0}}\right\vert_{{X}}}$ and $s_{N_3+2i} := {\left.{\widetilde{s}^{(v_i , w_i)}_{1}}\right\vert_{{X}}}$ for $i = 1 , \ldots , \gamma$, and set $N := N_3 + 2 \gamma$. We define $\varphi : X^{{{\operatorname{an}}}} \to {{\mathbb{T}}}{{\mathbb{R}}}^{N}$ by $\varphi
:= ( - \log |s_0| : \cdots : - \log |s_{N_3}| :
- \log |s_{N_3+1}| : \cdots : - \log |s_{N}|)$. Since $\varphi_3$ is unimodular, so is $\varphi$ (cf. Remark \[remark:unimodular\]).
Take any distinct $v, w \in V (\mathscr{X}^{{\operatorname{st}}})$. If $v$ and $w$ do not belong to some common island, then by condition (iii) in Theorem \[theorem:UT:canonical\] we have $\varphi_1 (v) \neq \varphi_1 (w)$, and hence $\varphi (v) \neq \varphi (w)$. Suppose that $v$ and $w$ belong to the same island. Then there exists $1 \leq i \leq \gamma$ with $(v,w) = (v_i , w_i)$ or $(w,v) = (v_i , w_i)$. By Lemma \[lemma:vertices-separating\], it follows that $- \log |s_{N_3+2i} / s_{N_3 +2i -1} (v)|
\neq - \log |s_{N_3+2i} / s_{N_3 +2i -1} (w)|$, and hence $\varphi (v) \neq \varphi (w)$. This completes the proof of the theorem.
500
Faithful tropicalization of minimal skeleta in low genera {#section:FT:lowgenus}
=========================================================
Let $X$ be a connected smooth projective curve of genus $g \geq 0$. In the previous section, we have shown the existence of a faithful tropicalization of the minimal skeleton of $X^{{{\operatorname{an}}}}$ when $g \geq 2$. In this section, we prove the existence when $g = 0,1$.
\[thm:FT:low:genus\] Let $X$ be a connected smooth projective curve of genus $g = 0,1$, and let $\Gamma_{\min}$ be a minimal skeleton of $X^{{{\operatorname{an}}}}$. Let $L$ be a line bundle over $X$. Suppose that $$\deg (L)
\geq
\begin{cases}
1
&
\text{if $g=0$,}
\\
3
&
\text{if $g=1$.}
\end{cases}$$ Then there exist $s_0 , \ldots , s_N \in H^{0} (X,L)$ such that the tropicalization map $\varphi : X^{{{\operatorname{an}}}} \to {{\mathbb{T}}}{{\mathbb{P}}}^N$ defined by $\varphi =
(- \log |s_0| : \cdots : - \log |s_N|)$ gives a faithful tropicalization of $\Gamma$.
Genus $0$ case {#subsection:case:singleton:genus:0}
--------------
Assume that $g = 0$. Then the existence of a faithful tropicalization of a minimal skeleton is trivial. Indeed, a minimal skeleton is a point. Thus if $\deg (L) \geq 1$, then there exists two non-zero sections $s_0 , s_1 \in H^0 (L)$, and the tropicalization with respect to those sections gives a faithful tropicalization of a minimal skeleton.
Genus $1$ case {#subsection:case:genus:1}
--------------
Assume that $g = 1$. Then there exists a unique minimal skeleton of $X^{{{\operatorname{an}}}}$, which we denote by $\Gamma_{\min}$ as before (cf. §\[subsec:properties:skeleta\]). We have two possibilities of $\Gamma_{\min}$: a singleton or a circle. If $\Gamma_{\min}$ is a singleton, then the existence of faithful tropicalization is again trivial. Indeed, if $\deg (L) \geq 3$, then there exist two nontrivial sections $s_0 , s_1 \in H^0 (L)$, and the tropicalization with respect to those sections gives a faithful tropicalization of $\Gamma_{\min}$.
In the remaining of this subsection, we assume that $\Gamma_{\min}$ is a circle. We fix a point $v_0 \in \Gamma_{\min, \Lambda}$, and we let $e$ denote the loop edge of $\Gamma_{\min}$ such that $\mathrm{relin}
(e) = \Gamma_{\min} \setminus \{ v_0 \}$. We are going to show the existence of a faithful tropicalization of $\Gamma_{\min}$ with a similar strategy to that we used for curves of genus at least $2$.
\[def:goodmodel:g=1\] Let $\mathscr{X}$ be a model of $X$. We say that $\mathscr{X}$ is a *good model* if it satisfies the following conditions.
1. The model ${{\mathscr{X}}}$ is Deligne–Mumford strictly semistable.
2. There exists a $C_0 \in {\operatorname{\operatorname{Irr}}}(\mathscr{X}_s )$ such that $[C_0] = v_0$ and such that $E := \mathscr{X}_s - C_0$ is a $(-2)$-chain with symmetric multiplicities.
\[lemma:pregoodmodel:g=1\] Let $X$ be a connected smooth projective curve of genus $1$, and let $L$ be a line bundle over $X$ with $\deg (L) \geq 3$. Then for any $x,y \in \Gamma_{\min, \Lambda}$, there exists a model $( \mathscr{X} , \mathscr{L})$ of $(X,L)$ such that $\mathscr{X}$ is a good model in Definition and such that $D_{\mathscr{L}} - [x] - [y]$ is effective.
[[*Proof.*]{}]{}Recall from that $\tau_*: {\operatorname{Div}}(X) \to {\operatorname{Div}}_{\Lambda} ( \Gamma_{\min} )$ denotes the specialization map. Take a divisor $\tilde{D}$ on $X$ with $L \cong {{\mathcal{O}}}_X ( \tilde{D} )$. Then $\tau_{\ast} ( \tilde{D} ) \in {\operatorname{Div}}_{\Lambda} ( \Gamma_{\min} )$ and the linear system $\left| \tau_{\ast} ( \tilde{D} ) \right|$ does not depend on the choice of $\tilde{D}$ (cf. the proof of Proposition \[prop:pregoodmodel:pre\]). Since $\deg ( \tau_{\ast} ( \tilde{D} ) - [x] - [y] ) \geq 1$, $\left| \tau_{\ast} ( \tilde{D} ) \right| \neq \emptyset$ by Proposition \[prop:riemann:inequality\]. Thus there exists an effective $E \in {\operatorname{Div}}_{\Lambda} (\Gamma_{\min})$ such that $E + [x] + [ y] \sim \tau_{\ast} ( \tilde{D} )$. By the same argument as in the proof of Proposition \[prop:pregoodmodel:pre\], we construct a model $( \mathscr{X} , \mathscr{L} )$ of $(X,L)$ such that $\mathscr{X}$ is good in the above sense and such that $D_{\mathscr{L}} = E + [x] + [y]$.
500
Let $(\mathscr{X} , \mathscr{L} )$ be a model of $(X, L)$ such that ${{\mathscr{X}}}$ is a good model in Definition \[def:goodmodel:g=1\]. Let $\widetilde{s}$ be a nonzero global section of $\mathscr{L}$. We call $\widetilde{s}$ a *base section* if $\widetilde{s} (p) \neq 0$ for any $p \in {\operatorname{Sing}}( \mathscr{X}_s )$. Further, let $C_0$ and $E$ be as in Definition \[def:goodmodel:g=1\](ii). Since $E$ is a maximal $(-2)$-chain with symmetric multiplicities and $\Delta_E = e$, we define the stepwise vertical divisor $\mathscr{V}_e$ by the same way as in §\[subsection:stepwise\]. We call $\widetilde{s}$ a *unimodularity section* if ${\operatorname{div}}(\widetilde{s}) - \mathscr{V}_e$ is effective on ${{\mathscr{X}}}$ and is trivial on some open neighborhood of ${\operatorname{Sing}}( \mathscr{X}_s )$.
\[prop:model:US:g=1\] Let $X$ and $ L$ be as in Lemma . Then there exists a model $(\mathscr{X} , \mathscr{L} )$ such that $\mathscr{X}$ is a good model in Definition \[def:goodmodel:g=1\] and such that $\mathscr{L}$ has a base section and a unimodularity section.
[[*Proof.*]{}]{}Applying Lemma \[lemma:pregoodmodel:g=1\] to $x = y = v_0$, we obtain a model $(\mathscr{X} , \mathscr{L})$ of $(X,L)$ such that $\mathscr{X}$ is a good model and $D_{\mathscr{L}} - 2 [v_0]$ is effective. By the same argument using Lemma \[lemma:freeatnode:graphversion:emptyset\] as in the proof of Lemma \[lemma:forbasesectionUT:dc\](1), we see that $\mathscr{L}$ is free at any $p \in {\operatorname{Sing}}( \mathscr{X}_s )$, and then by the same argument as the proof of Proposition \[prop:existence:sections:stronger\], we obtain a base section.
We set $\mathscr{L}_1 := \mathscr{L} ( - \mathscr{V}_e )$. By the same argument as in the proof of Lemma \[lemma:forbasesectionUT:dc\](3), we see that $\mathscr{L}_1$ is free at any $p \in {\operatorname{Sing}}( \mathscr{X}_s )$. By the same argument as the proof of Proposition \[prop:existence:sections:stronger\], we obtain a unimodularity section of $\mathscr{L}$.
500
Let $\mathscr{X}$ be a good model in Definition \[def:goodmodel:g=1\]. Since the maximal $(-2)$-chain $E$ has symmetric multiplicity, we can consider a separating divisor $\mathscr{A}$ for $e$ as in Definition \[def:assymetric:divisor\] also in this setting. Let $\mathscr{L}$ be a model of $L$. We call a nonzero global section $s$ of $\mathscr{L}$ an *separating section* if there exists a separating divisor $\mathscr{A}$ (for $e$) such that ${\operatorname{div}}(s ) - \mathscr{A}$ is trivial on some open neighborhood of ${\operatorname{Sing}}( \mathscr{X}_s ) \setminus {\operatorname{Supp}}( \mathscr{A})$ (Here we remark that ${\operatorname{Sing}}( \mathscr{X}_s )_{\subset e} = {\operatorname{Sing}}( \mathscr{X}_s )$.)
Let $X$ and $ L$ be as in Lemma . Then there exists a model $(\mathscr{X} , \mathscr{L} )$ of $(X,L)$ such that $\mathscr{X}$ is a good model in Definition and such that $\mathscr{L}$ has a base section and a separating section.
[[*Proof.*]{}]{}Let $v_1 \in \Gamma_{\min}$ be the point antipodal to $v_0$. In other words, let $v_1$ be the middle point of $e$. Applying Lemma \[lemma:pregoodmodel:g=1\] to $x = v_0$ and $y = v_1$, we obtain a model $(\mathscr{X} , \mathscr{L})$ of $(X,L)$ such that $\mathscr{X}$ is a good model and $D_{\mathscr{L}} - [v_0] - [v_1]$ is effective. By the same argument as in Step 1 of the proof of Proposition \[prop:model:AS\], one sees that $\mathscr{L}$ has a base section.
We set $\mathscr{L}_1 :=
\mathscr{L} ( - \mathscr{A})$. By the same argument as in Step 2 of the proof of Proposition \[prop:model:AS\], one sees that $\mathscr{L}_1$ is free at any $q \in
{\operatorname{Sing}}( \mathscr{X}_s ) \setminus {\operatorname{Supp}}( \mathscr{A})$, and hence there exists a section $\widetilde{s}_1 \in H^0 ( \mathscr{L}_1 )$ such that $\widetilde{s}_1 (q) \neq 0$ for any $q \in
{\operatorname{Sing}}( \mathscr{X}_s ) \setminus {\operatorname{Supp}}( \mathscr{A})$. The same argument as in the last paragraph of the proof of Proposition \[prop:model:AS\] gives the existence of a separating section of $\mathscr{L}$.
500
Since we have a model that has a base section and a unimodularity section, the same argument as the proof of Theorem \[theorem:UT:canonical\] gives us a unimodular tropicalization of $\Gamma_{\min}$. Furthermore, since we have a model that has a base section and a separating section, the same argument as the proof of Theorem \[theorem:FT:canonical\] gives us a faithful tropicalization of $\Gamma_{\min}$.
Together with §\[subsection:case:singleton:genus:0\], we have shown Theorem \[thm:FT:low:genus\].
Faithful tropicalization of an arbitrary skeleton {#sec:ft:general:model}
=================================================
In this section, we prove the main Theorem \[thm:main:unimodular:faithful\].
Notation and terminology of §\[sec:ft:general:model\] {#notation-and-terminology-of-secftgeneralmodel .unnumbered}
-----------------------------------------------------
Throughout this section, we use the following notation and terminology. Let $X$ be a connected smooth projective over $K$ of genus $g \geq 0$, and let $L$ be a line bundle over $X$. For any compact skeleton $\Gamma$ of $X^{{{\operatorname{an}}}}$, let $\tau_{\Gamma}: X^{{{\operatorname{an}}}} \to \Gamma$ denote the retraction map with respect to $\Gamma$.
Geodesic paths {#subsection:geodesic:line}
--------------
Recall that $X^{{{\operatorname{an}}}}\setminus X(K)$ has a canonical metric structure (cf. [@BPR2 Corollary 5.7]). In this subsection, we briefly review basic properties of geodesic paths in $X^{{{\operatorname{an}}}} \setminus X(K)$. We refer to [@BPR2 §5] for details.
Let $x , y \in X^{{{\operatorname{an}}}} \setminus X (K)$, and let $\gamma$ be a path in $X^{{{\operatorname{an}}}} \setminus X (K)$ from $x$ to $y$, which we mean a continuous map from a closed interval $[a , b] \subset {{\mathbb{R}}}$ to $X^{{{\operatorname{an}}}} \setminus X (K)$ such that $\gamma (a) = x$ and $\gamma (b) = y$. We call $\gamma$ a *geodesic path connecting $x$ and $y$* if $\gamma$ is an isometry from $[a,b]$ to its image (cf. [@BPR2 §5]). In what follows, we identify a geodesic path with its image in $X^{{{\operatorname{an}}}} \setminus X (K)$. Remark that two geodesic paths with the same image differ only by parameterizations: if $\gamma_1 : [a_1 , b_1] \to X^{{{\operatorname{an}}}} \setminus X (K)$ and $\gamma_2 : [a_2 , b_2] \to X^{{{\operatorname{an}}}} \setminus X (K)$ are geodesic paths connecting $x$ and $y$ such that $\gamma_1 ([a_1 , b_1]) = \gamma_2 ([a_2 , b_2])$, then there exists a unique isometry $\phi : [a_1 , b_1] \to
[a_2 , b_2]$ such that $\gamma_1 = \gamma_2 \circ \phi$.
Before discussing geodesic paths, we show the following lemma on the retraction map, which will be frequently used.
\[lemma:connectedcomponents\] Let $\Gamma$ be a compact skeleton of $X^{{{\operatorname{an}}}}$, and let $x$ be a point in $X^{{{\operatorname{an}}}} \setminus \Gamma$. Let $A$ be the connected component of $X^{{{\operatorname{an}}}} \setminus \Gamma$ with $x \in A$. Then $A$ is the connected component of $X^{{{\operatorname{an}}}}
\setminus \{ \tau_{\Gamma} (x) \}$ with $x \in A$.
[[*Proof.*]{}]{}By the definition of $\tau_{\Gamma}$, $\overline{A} = A \cup \{ \tau_{\Gamma} (x) \}$ and $\tau_{\Gamma} (A) = \{ \tau_{\Gamma} (x) \}$. Let $B$ be the connected component of $X^{{{\operatorname{an}}}} \setminus \{ \tau_{\Gamma} (x) \}$ with $B \cap A \neq \emptyset$. Since $A \subset X^{{{\operatorname{an}}}} \setminus \{ \tau_{\Gamma} (x) \}$ and $A$ is connected, we have $A \subset B$. Set $C := B \setminus A$. Then we have $B = A \cup C$ and $A \cap C = \emptyset$. Since $ \tau_{\Gamma} (x) \notin B$, we have $C = B \setminus A = B \setminus (A \cup \{ \tau_{\Gamma} (x) \})
= B \setminus \overline{A}$, which is an open subset of $B$. On the other hand, since $X^{{{\operatorname{an}}}}$ is locally connected, $A$ is an open subset of $X^{{{\operatorname{an}}}}$ and hence of $B$. Since $B$ is connected and $A \neq \emptyset$, we have $C = \emptyset$, and thus $A = B$. This shows the lemma.
500
The following lemma shows basic properties of geodesic paths, which are essentially treated in [@BPR2].
\[lemma:geodesic:path\] Let $x , y \in X^{{{\operatorname{an}}}}$. Suppose that there exists a compact skeleton $\Gamma$ of $X^{{{\operatorname{an}}}}$ such that $x,y \in \Gamma$. We fix a minimal skeleton $\Gamma_{\min}$.
1. Then there exists a geodesic path connecting $x$ and $y$.
2. Furthermore, suppose that $\tau_{\Gamma_{\min}} (x) = \tau_{\Gamma_{\min}} (y)$. Then a geodesic path connecting $x$ and $y$ as the image in $X^{{{\operatorname{an}}}}$ is unique. In particular, if there exists a compact skeleton $\Gamma_1$ of $X^{{{\operatorname{an}}}}$ such that $\tau_{\Gamma_1} (x) = \tau_{\Gamma_1} (y)$, then we have the uniqueness of the geodesic path connecting $x$ and $y$.
[[*Proof.*]{}]{}Assertion (1) is a direct consequence of the first paragraph of [@BPR2 §5.3].
For assertion (2), put $z := \tau_{\Gamma_{\min}} (x) = \tau_{\Gamma_{\min}} (y)$. First, we prove that the uniqueness of a geodesic path connecting $x$ and $z$. If $x = z$, then it is trivial, so that we assume that $x \neq z$. Then $x \notin \Gamma_{\min}$. Let $\Gamma_x^\circ$ be the connected component of $\Gamma \setminus \Gamma_{\min}$ containing $x$, and let $\Gamma_x$ be the closure of $\Gamma_x^{\circ}$ in $X^{{{\operatorname{an}}}}$. Then we claim that $\Gamma_x$ is tree. Indeed, let $A_x$ be the connected component of $X^{{{\operatorname{an}}}} \setminus \Gamma_{\min}$ that contains $x$. Since $\Gamma_x^\circ \subset X^{{{\operatorname{an}}}} \setminus \Gamma_{\min}$ and $\Gamma_x^\circ$ is connected, we then have $\Gamma_x^\circ \subset A_x$. By the definition of $\tau_{\Gamma_{\min}}$, we have $\overline{A_x} = A_x \cup \{ z \}$. Note that, since $\Gamma$ is connected and $\Gamma_{\min} \subset \Gamma$ is closed, $\Gamma_x^{\circ}$ is not closed. This means that the closure $\Gamma_x$ of $\Gamma_x^\circ$ in $\Gamma$ is equal to $\Gamma_x = \Gamma_x^\circ\cup \{z\}$. On the other hand, by [@BPR2 Lemma 3.4(3)], $A_x$ is an open ball. Then the argument in [@BPR2 §5.8] shows that $\Gamma_x$ is a tree.
Let $\gamma$ be a geodesic path connecting $x$ and $z$. By Lemma \[lemma:connectedcomponents\], $A_x$ is the connected component of $X^{{{\operatorname{an}}}} \setminus \{ z \}$ with $x \in A_x$. Since $\gamma \setminus \{ z \}$ is a connected subspace in $X^{{{\operatorname{an}}}} \setminus \{ z \}$ and $x \in \gamma \setminus \{ z \}$, it follows that $\gamma \setminus \{ z \} \subset A_x$. Further, by [@BPR2 Corollary 5.10], $\gamma \subset \Gamma$. It follows that $\gamma \setminus \{ z \} \subset \Gamma \cap A_x = \Gamma_x^{\circ}$, and hence $\gamma \subset \Gamma_x$. Thus $\gamma$ is a geodesic path in $\Gamma_x$. Since $\Gamma_x$ is a tree, this proves that $\gamma$ is unique. This shows that the geodesic path connecting $x$ and $z$ is unique.
If $y \neq z$, then we define $A_y$, $\Gamma_y^{\circ}$, and $\Gamma_y$ with $x$ replaced with $y$. Further, the same argument as above also shows that a geodesic path connecting $y$ and $z$ is unique.
Next, let us show that the uniqueness of a geodesic path connecting $x$ and $y$. We may and do assume that $x \neq z$ and $y \neq z$, since otherwise we have already shown the assertion. We remark that in this situation, $A_x$, $\Gamma_x^{\circ}$, $\Gamma_x$, $A_y$, $\Gamma_y^{\circ}$, and $\Gamma_y$ are defined above. Let $\gamma$ be a geodesic path connecting $x$ and $y$. By [@BPR2 Corollary 5.10], $\gamma \subset \Gamma$.
Suppose that $A_x = A_y$. Then $\Gamma_x = \Gamma_y$. We remark that $\Gamma_x \cap \Gamma_{\min} = \{ z \}$. Then, since $\gamma$ is geodesic in $\Gamma$, one sees that $\gamma \subset \Gamma_x$. Since $\Gamma_x$ is a tree, $\gamma$ is unique.
Suppose that $A_x \neq A_y$. Then $A_x \cap A_y = \emptyset$. It follows that $\Gamma_x^\circ \cap \Gamma_y^\circ = \emptyset$, and thus $\Gamma_x \cap \Gamma_y = \{ z \}$. This means that $\Gamma_x \cup \Gamma_y$ is a one point sum at $z$ of two trees, so that it is also a tree. Further, $(\Gamma_x \cap \Gamma_y) \cap \Gamma_{\min} = \{ z \}$. Since $\gamma$ is geodesic in $\Gamma$, it follows that $\gamma \subset \Gamma_x \cup \Gamma_y$. Since $\Gamma_x \cup \Gamma_y$ is a tree, this shows that $\gamma$ is unique. This completes the proof of the first assertion of (2).
Since $\Gamma_x$ and $\Gamma_y$ are trees, we see that $\gamma$ is a unique geodesic path connecting $x$ and $y$ lying in $\tilde{\Gamma}$; in fact, $\gamma$ is the union of a unique geodesic path connecting $x$ and $z$ in $\Gamma_x$ and a unique geodesic path connecting $z$ and $y$ in $\Gamma_y$. We have shown that a geodesic path connecting $x$ and $y$ that lie in $\tilde{\Gamma}$ is unique.
Suppose that $\gamma_1$ and $\gamma_2$ are geodesic paths connecting $x$ and $y$. Then we take a compact skeleton $\gamma_1 , \gamma_2\Gamma$ by [@BPR2 Corollary 5.10]. Then the above argument gives that $\gamma_1 = \gamma_2$. This completes the proof of the former assertion of (2).
For the second assertion of (2), if $g \geq 1$, then $\Gamma_{\min}$ is unique and is contained in $\Gamma_1$. It follows from $\tau_{\Gamma_1} (x) = \tau_{\Gamma_1} (y)$ that $\tau_{\Gamma_{\min}} (x) = \tau_{\Gamma_{\min}} (y)$. If $g = 0$, then $\Gamma_{\min}$ is a singleton, and we always have $\tau_{\Gamma_{\min}} (x) = \tau_{\Gamma_{\min}} (y)$. Thus the second assertion follows from the first assertion.
500
Let $x $ and $ y$ be points in $X^{{{\operatorname{an}}}}$ as in Lemma \[lemma:geodesic:path\]. We assume that $\tau_{\Gamma_{\min}}(x) = \tau_{\Gamma_{\min}}(y)$, so that there exists a unique geodesic path $\gamma$ connecting $x$ and $y$. We call the image of $\gamma$ in $X^{{{\operatorname{an}}}}$ the *geodesic segment connecting $x$ and $y$*. We use the notation $[x,y]$ for the geodesic segment connecting $x$ and $y$. Further, if $x \neq y$, then we set $$[x , y) := [x,y] \setminus \{ y \}, \quad
(x , y] := [x,y] \setminus \{ x \}, \quad\text{and}\quad
(x , y) := [x,y] \setminus \{ x, y \}.$$
We give some properties of geodesic segments. The following two lemmas are also essentially treated in [@BPR2].
\[lemma:to:be:used\] Let $\Gamma$ and $\Gamma^\prime$ be compact skeleta with $\Gamma \subset \Gamma^\prime$ and let $x \in \Gamma^\prime_\Lambda \setminus \Gamma_\Lambda$. Then we have the following.
1. \[lemma:to:be:used:addingsegments\] The union $\Gamma \cup [ \tau_{\Gamma} (x) , x ]$ is a compact skeleton of $X^{{{\operatorname{an}}}}$.
2. \[lemma:to:be:used:3\] We have $[\tau_{\Gamma}(x), x] \subset \Gamma^\prime$. Further, we have $\Gamma \cap [\tau_{\Gamma}(x), x] = \{\tau_{\Gamma}(x)\}$.
3. \[lemma:to:be:used:4\] For any compact skeleton $\Gamma_1$ with $\Gamma \subset \Gamma_1 \subset \Gamma^\prime$, we have $\tau_{\Gamma_1}(x) \in [\tau_{\Gamma}(x), x]$.
[[*Proof.*]{}]{}We first note that, since $x$ and $\tau_{\Gamma} (x)$ is contained in a compact skeleton and $\tau_{\Gamma}(x) = \tau_\Gamma(\tau_{\Gamma}(x))$, $[x, \tau_\Gamma(x)]$ makes sense by Lemma \[lemma:geodesic:path\].
(\[lemma:to:be:used:addingsegments\]) We write $\Gamma^\prime = S( \mathscr{X}^\prime)$ for some strictly semistable model ${{\mathscr{X}}}^\prime$ of $X$. By [@BPR2 Corollary 5.10], we have $[\tau_{\Gamma} (x) , x ] \subset S( \mathscr{X}^\prime)$. By successively contracting the irreducible components $E$ such that $[E] \notin \Gamma \cup [\tau_{\Gamma} (x) , x ]$, we obtain a strictly semistable model $\mathscr{X}^{\prime\prime}$ such that $\Gamma \cup [\tau_{\Gamma} (x) , x ] = S( \mathscr{X}^{\prime\prime})$. This shows that $\Gamma \cup [\tau_{\Gamma} (x) , x ]$ is a compact skeleton.
(\[lemma:to:be:used:3\]) Since $x , \tau_{\Gamma} (x) \in \Gamma'$, $[\tau_{\Gamma} (x) , x] \subset \Gamma'$ by [@BPR2 Corollary 5.10]. For the second assertion, let $A$ be the connected component of $X^{{{\operatorname{an}}}} \setminus \Gamma$ with $x \in A$. By Lemma \[lemma:connectedcomponents\], $A$ is the connected component of $X^{{{\operatorname{an}}}} \setminus \{
\tau_{\Gamma} (x) \}$ with $x \in A$. Since $(\tau_{\Gamma} (x) , x]$ is a connected subspace of $X^{{{\operatorname{an}}}} \setminus \{ \tau_{\Gamma} (x) \}$ containing $x$, we get $(\tau_{\Gamma} (x) , x]
\subset A$. This proves that $\Gamma \cap [\tau_{\Gamma}(x), x] = \{\tau_{\Gamma}(x)\}$.
(\[lemma:to:be:used:4\]) By the first assertion in (\[lemma:to:be:used:3\]) applied to $\Gamma \subset \Gamma_1$ and $\tau_{\Gamma_1}(x)$, we have $[\tau_{\Gamma} (x) , \tau_{\Gamma_1} (x)] \subset \Gamma_1$. By the second assertion in (\[lemma:to:be:used:3\]), we then have $[\tau_{\Gamma} (x) , \tau_{\Gamma_1} (x)]
\cap [\tau_{\Gamma_1} (x) , x] =
\{ \tau_{\Gamma_1} (x) \}
$, which means that $[\tau_{\Gamma} (x) , \tau_{\Gamma_1} (x)]
\cup [\tau_{\Gamma_1} (x) , x]$ is a geodesic segment connecting $\tau_{\Gamma} (x)$ and $x$. By the uniqueness of a geodesic segment of Lemma \[lemma:geodesic:path\](2), $[\tau_{\Gamma} (x) , \tau_{\Gamma_1} (x)]
\cup [\tau_{\Gamma_1} (x) , x]
= [\tau_{\Gamma} (x) , x]$. Thus $\tau_{\Gamma_1} (x) \in [\tau_{\Gamma} (x) , x]$.
500
Lemma \[lemma:to:be:used\](\[lemma:to:be:used:addingsegments\]) can be also stated as follows.
\[lemma:addingsegments\] Let $\Gamma$ be a compact skeleton of $X^{{{\operatorname{an}}}}$. Let $x$ be a point of $X^{{{\operatorname{an}}}} \setminus \Gamma$. Assume that there exists a compact skeleton $\Gamma'$ such that $x \in \Gamma'_{\Lambda}$. Then $\Gamma \cup [ \tau_{\Gamma} (x) , x ]$ is a compact skeleton of $X^{{{\operatorname{an}}}}$.
[[*Proof.*]{}]{}By [@BPR2 Proposition 3.13(3)], there exists a compact skeleton $\Gamma^{\prime\prime}$ such that $\Gamma \cup \{ x \} \subset \Gamma^{\prime\prime}$. Then the assertion is reduced to Lemma \[lemma:to:be:used\](\[lemma:to:be:used:addingsegments\]) applied to $\Gamma^{\prime\prime}$.
500
Next, we would like to discuss a path one of whose endpoint is a point in $X(K)$. Here, we write $[0 , + \infty] = {{\mathbb{T}}}$ by notation.
\[lemma:end=canonialend\] Let $\Gamma$ be a compact skeleton, and let $P$ be any point in $X(K)$. Then the following hold.
1. There exists a unique continuous map $\gamma: [0, + \infty] \to X^{{{\operatorname{an}}}}$ with $\gamma(0) = \tau_{\Gamma} (P)$ and $\gamma(1) = P$ such that $\gamma$ restricts to an isometry from $[0 , + \infty )$ to its image.
2. There exists a strictly semistable pair $( \mathscr{X} ; \sigma )$ such that $S( \mathscr{X}) = \Gamma$ and $\sigma (K) = P$.
3. We take $\gamma$ as in and $( \mathscr{X} ; \sigma )$ as in . We set $$[\tau_{\Gamma} (P) , P] := \gamma\left([0, + \infty ]\right)
\quad\text{and}\quad
[\tau_{\Gamma} (P) , P) := [\tau_{\Gamma} (P) , P] \setminus \{ P \} = \gamma\left([0, + \infty)\right).$$ Then $\Delta (\sigma) = [\tau_{\Gamma} (P) , P)$ and $\overline{\Delta (\sigma)} = [\tau_{\Gamma} (P) , P]$.
[[*Proof.*]{}]{}First, we prove the uniqueness property in (1). Let $\gamma^\prime: [0, + \infty] \to X^{{{\operatorname{an}}}}$ be a continuous map that satisfies the same property as $\gamma$. We are going to show that $\gamma = \gamma^\prime$.
We set $I := \gamma([0, + \infty])$ and $J:= \gamma^\prime([0, + \infty])$. It suffices to show that $I = J$. To argue by contradiction, assume that $I \neq J$. Then one easily sees that $I \nsubseteq J$ and $J \nsubseteq I$, and there exist $x \in I \setminus J$ and $y \in J \setminus I$ such that the distance $d(\tau_{\Gamma} (P) , x)$ from $\tau_{\Gamma} (P)$ to $x$ and the distance $d(\tau_{\Gamma} (P) , y)$ from $\tau_{\Gamma} (P)$ to $y$ are in $\Lambda$. By the uniqueness of geodesics (cf. Lemma \[lemma:geodesic:path\](2)), we have $[\tau_{\Gamma} (P) , x] \subset I$, $[\tau_{\Gamma} (P) , y] \subset J$. Further, we see that there exists $z \in [\tau_{\Gamma} (P) , x] \cap
[\tau_{\Gamma} (P) , y]$ such that $[\tau_{\Gamma} (P) , x] \cap
[\tau_{\Gamma} (P) , y] = [ \tau_{\Gamma} (P) , z]$. By Lemma \[lemma:addingsegments\], $\Gamma_0 := \Gamma \cup [\tau_{\Gamma} (P) , x]$ is a compact skeleton. By the choice of $z$, one sees that $z = \tau_{\Gamma_0} (y)$, which belongs to $\Gamma_{0,\Lambda}$ by Lemma \[lemma:retraction:rational\](1). Using Lemma \[lemma:addingsegments\] again, we see that $\Gamma_1 := \Gamma_0 \cup [z , y] =
\Gamma \cup
[\tau_{\Gamma} (P) , x] \cup [\tau_{\Gamma} (P) , y]$ is a compact skeleton.
Let $A$ be the connected component of $X^{{{\operatorname{an}}}} \setminus \Gamma_1$ with $P \in A$. By Lemma \[lemma:connectedcomponents\], $A$ is the connected component of $X^{{{\operatorname{an}}}} \setminus \{\tau_{\Gamma_1}(P)\}$, and we have $\tau_{\Gamma}(\overline{A}) = \{\tau_{\Gamma_1}(P)\}$ (cf. Lemma \[lemma:retraction:rational\]). Since $I \setminus [\tau_{\Gamma} (P) , x]$ is a connected subset with $I \setminus [\tau_{\Gamma} (P) , x]\subset X^{{{\operatorname{an}}}} \setminus \Gamma_1$ and $P \in I \setminus [\tau_{\Gamma} (P) , x] $, we have $I \setminus [\tau_{\Gamma} (P) , x] \subset A$. Since $x \in \overline{I \setminus [\tau_{\Gamma} (P) , x]}$, we have $x \in \overline{A}$. It follows that $\tau_{\Gamma_1} (P) = \tau_{\Gamma_1} (x) = x$. On the other hand, the same argument shows that $\tau_{\Gamma_1} (P) = y$. This contradicts with $x \neq y$. Thus we obtain the uniqueness property in (1) .
Next we prove (2). By Lemma \[lemma:retraction:rational\](2), we have $\tau_{\Gamma} (P) \in \Gamma_{\Lambda}$. It follows from Proposition \[prop:subdivision1:b\] that there exists a strictly semistable model $\mathscr{X}$ of $X$ such that $S ( \mathscr{X} ) = \Gamma$ and $\tau_{\Gamma} (P) \in V ( \mathscr{X})$. We take $C_{\tau_{\Gamma} (P)} \in {\operatorname{\operatorname{Irr}}}( \mathscr{X}_s)$ such that $\tau_{\Gamma} (P) = [ C_{\tau_{\Gamma} (P)} ]$. By the valuative criterion of properness, we take a section $\sigma$ of $\mathscr{X} \to {{\operatorname{Spec}}}(R)$ such that $\sigma (K) = P$. Since $\tau_{\Gamma} (P) = [ C_{\tau_{\Gamma} (P)} ]$, it follows from Lemma \[lemma:retraction:rational\] that ${\operatorname{red}}_{\mathscr{X}} (P) = \sigma (k) \in
C_{\tau_{\Gamma} (P)} (k) \setminus {\operatorname{Sing}}( \mathscr{X}_s )$. Thus $(\mathscr{X} ; \sigma )$ is a strictly semistable pair as desired.
Finally, we prove the existence of $\gamma$ in (1) and (3). Since we have a unique isometry $\Delta(\sigma) \cong {{\mathbb{R}}}_{\geq 0}$ (cf. §\[subsec:skeleta:ass:ssp\]), we have an isometry $\gamma: [0, + \infty) \to \Delta(\sigma)$. To complete the proof of the lemma, we have only to show that $\overline{\Delta(\sigma)} = \Delta(\sigma) \cup \{P\}$ and that $\gamma$ extends to a homomorphism $\gamma: [0, + \infty] \to \overline{\Delta(\sigma)}$. However, this is a fact mentioned in [@GRW Remark 4.12] and in (the proof of) [@BPR2 Lemma 3.4(2)]. Thus there exists $\gamma$ with property described in (1), and we have the equalities described in (3).
500
Let $\Gamma$ be a compact skeleton of $X^{{{\operatorname{an}}}}$, and let $P$ be a point in $X(K)$. By slight abuse of terminology, we call $[\tau_{\Gamma} (P) , P]$ a [*path connecting $\tau_{\Gamma} (P)$ and $P$*]{}. By Lemma \[lemma:end=canonialend\], $[\tau_{\Gamma} (P) , P]$ exists and is determined only by $\Gamma$ and $P$. We also write $$(\tau_{\Gamma} (P) , P) := [\tau_{\Gamma} (P) , P]
\setminus \{ \tau_{\Gamma} (P) , P \}.$$
Stepwise vertical divisor associated to a point in $X(K)$ {#subsection:setting1}
---------------------------------------------------------
Let $P$ be a point in $X(K)$, and let $\Gamma$ be a compact skeleton of $X^{{{\operatorname{an}}}}$. By Lemma \[lemma:end=canonialend\], there exists a strictly semistable pair $(\mathscr{X} ; \sigma)$ such that $S(\mathscr{X}) = \Gamma$ and $\sigma (K) = P$, and we have $[ \tau_{\Gamma} (P) , P ) = \Delta(\sigma)$ and $S( \mathscr{X} ; \sigma ) = \Gamma
\cup [ \tau_{\Gamma} (P) , P )$. Fix a minimal skeleton $\Gamma_{\min} \subset \Gamma$. Then $$\begin{aligned}
& [ \tau_{\Gamma_{\min}} (P) , P ) = [\tau_{\Gamma_{\min}} (P) ,
\tau_{\Gamma} (P)] \cup
[ \tau_{\Gamma} (P) , P ),
\\
& [\tau_{\Gamma_{\min}} (P) ,
\tau_{\Gamma} (P)] \cap
[ \tau_{\Gamma} (P) , P ) = \{ \tau_{\Gamma} (P) \}.\end{aligned}$$
For $P$, $\mathscr{X}$, and $\Gamma_{\min}$ above, we define a Cartier divisor $\mathscr{W}_P$ on $\mathscr{X}$, which is a sum of fundamental divisors, as follows. Let $E$ be the chain of curves in $\mathscr{X}_s$ characterized by $ \Delta_E
=
[\tau_{\Gamma_{\min}} (P) ,
\tau_{\Gamma} (P)]
$. We write the irreducible components of $E$ for $E_0 , \ldots , E_{r}$ in such a way that $[E_0] = \tau_{\min} (P)$ and $E_{\alpha} \cap E_{\alpha+1} \neq \emptyset$ for $\alpha = 0 , \ldots , r - 1$; then $[E_r] = \tau_{\Gamma} (P)$. Note that $\sigma (k) \in E_{r} \setminus {\operatorname{Sing}}( \mathscr{X}_s )$. Let $p_{\alpha}$ denote the node in $\mathscr{X}_s$ with $\{ p_{\alpha} \} = E_{\alpha} \cap E_{\alpha+1}$ for $\alpha = 0 , \ldots , r - 1$.
$$\setlength\unitlength{0.07truecm}
\begin{picture}(180, 85)(0,0)
\qbezier(0, 20)(15, 50)(0, 80)
\thicklines
\put(0,60){\line(4,1){30}}
\put(17,66){\line(1,0){40}}
\put(45,65){\line(4,1){30}}
\put(100,70){\line(1,0){30}}
\put(118,69){\line(4,1){30}}
\multiput(138,75)(2, 0){20}{\line(1,0){1}}
\thinlines
\put(35,70){\line(1,-4){7}}
\put(35,50){\line(4,-1){30}}
\put(60,48){\line(1,-4){6}}
\put(108,74){\line(1,-4){7}}
\put(108,54){\line(4,-1){30}}
\put(133,52){\line(1,-4){6}}
\put(85, 70){$\cdots$}
\put(0, 82){$E_0$}
\put(39, 68){$E_2$}
\put(10, 66){$E_1$}
\put(63, 74){$E_{3}$}
\put(127, 74){$E_r$}
\put(154, 77){$\sigma(R)$}
\put(6.5, 61.5){\circle*{1.5}}
\put(24, 66){\circle*{1.5}}
\put(49, 66){\circle*{1.5}}
\put(123, 70){\circle*{1.5}}
\put(142.5, 75){\circle*{1.5}}
\put(8, 58){$p_0$}
\put(23.5, 62){$p_1$}
\put(48.5, 62){$p_{2}$}
\put(122.5, 66){$p_{r-1}$}
\put(140, 69){$\sigma(k)$}
\put(180,73){$P$}
\put(23, 20){$\underbrace{\phantom{AAAAAAAAAAAAAAAAAAAAAAAAAAAAA}}$}
\put(84, 11){$F_1$}
\put(10, 9){$\underbrace{\phantom{AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA}}$}
\put(78, 0){$F_0$}
\end{picture}$$
If $r = 0$, then we set $\mathscr{W}_{P} = 0$. Assume that $r \geq 1$. For each $0 \leq \alpha \leq r -1$, the partial normalization at $p_{\alpha}$ has exactly two connected components. Let $F_{\alpha}$ be the connected component that contains $E_{r}$. We regard $F_{\alpha}$ as a curve in $\mathscr{X}_s$. Then $F_{\alpha}$ satisfies condition (F) in §\[subsec:fvd\]. Let $\mathscr{F}_{\alpha}$ be the fundamental vertical divisor with support $F_{\alpha}$. We define a vertical effective divisor on $\mathscr{X}$ for $P$ to be $$\begin{aligned}
\label{align:stepwise:for:ends}
\mathscr{W}_{P}
:=
\sum_{\alpha = 0}^{r - 1} \mathscr{F}_{\alpha}
.\end{aligned}$$ We call this divisor the *stepwise vertical divisor associated to $P$.*
The divisor $\mathscr{W}_P$ depends on the choice of $\Gamma_{\min}$. However, if $g \geq 1$, then $\Gamma_{\min}$ is unique and hence $\mathscr{W}_P$ is determined by $P$ and $\mathscr{X}$.
We compute the order of $\mathscr{W}_P$ at each irreducible component of $\mathscr{X}_s$ when $r \geq 1$. By the definition of $\mathscr{W}_P$, we have ${\operatorname{Supp}}( \mathscr{W}_P) = F_0$, which is a tree of irreducible components. For any $C \in {\operatorname{\operatorname{Irr}}}({{\mathscr{X}}}_s - F_0)$, we have ${\operatorname{ord}}_{C} ( \mathscr{W}_{P} ) = 0$. For each $j = 0 , \ldots , r-1$, let $\lambda_{j}$ denote the multiplicity at $p_{j}$, which also equals the length of the canonical $1$-simplex $\Delta_{p_j}$ corresponding to $p_{j}$ (cf. §\[subsection:skeleta\]); then for any $\alpha = 1, \ldots, r$, we have ${\operatorname{ord}}_{E_{\alpha}} ( \mathscr{W}_{P} ) = \sum_{j=0}^{\alpha-1} \lambda_{j}$, which also equals to the distance between $[E_0]$ and $[E_{\alpha}]$. For any $C \in {\operatorname{\operatorname{Irr}}}( F_0) \setminus {\operatorname{\operatorname{Irr}}}( E)$, since $F_0$ is a tree of irreducible components, there exists a unique $\alpha_C = 1 , \ldots , r$ such that $[E_{\alpha_C}]$ is the point in $\{ [E_\alpha ] \mid 1 \leq \alpha \leq r \}$ that is the nearest to $[C]$; then we have ${\operatorname{ord}}_{C} ( \mathscr{W}_P ) =
{\operatorname{ord}}_{E_{\alpha_C}} ( \mathscr{W}_P )
= \sum_{j=0}^{\alpha_C-1} \lambda_{j}$. To put together, we have the following. $$\label{eqn:remark:order:stepwise:leaf}
{\operatorname{ord}}_{C} \left( \mathscr{W}_P \right)
=
\begin{cases}
0
&
\text{if $C \in {\operatorname{\operatorname{Irr}}}({{\mathscr{X}}}_s - F_0)$,}
\\
\sum_{j=0}^{\alpha-1} \lambda_{j} &
\text{if $C = E_{\alpha}$ ($\alpha = 1 , \ldots , r$),}
\\
\sum_{j=0}^{\alpha_C-1} \lambda_{j}
&
\text{if $C \in {\operatorname{\operatorname{Irr}}}( F_0) \setminus {\operatorname{\operatorname{Irr}}}( E)$}.
\end{cases}$$
We also compute the degree of the stepwise vertical divisor $\mathscr{W}_P$ over each $C \in {\operatorname{\operatorname{Irr}}}({{\mathscr{X}}}_s)$. $$\label{eqn:remark:degree:stepwise:leaf}
\deg \left({\left.{{{\mathcal{O}}}(\mathscr{W}_P}\right\vert_{{C}}}) \right)
=
\begin{cases}
1
&
\text{if $C = E_0$,}
\\
-1 &
\text{if $C = E_r$,}
\\
0
&
\text{otherwise.}
\end{cases}$$
Base sections and $P$-unimodularity sections {#subsec:base:sections:and:P:unimodulatity:sections}
--------------------------------------------
We define key notions of global sections of a model, which will give global sections of $L$ used for a faithful tropicalization of an arbitrary skeleton.
Let $\Gamma$ be a compact skeleton of $X^{{{\operatorname{an}}}}$. We fix a minimal skeleton $\Gamma_{\min} \subset \Gamma$. Let $P$ be a point in $X(K)$. Let $(\mathscr{X} ; \sigma)$ be a strictly semistable pair such that $S( \mathscr{X} ) = \Gamma$ and $\sigma (K) = P$. Let $\mathscr{L}$ be a line bundle over $\mathscr{X}$ such that $( \mathscr{X} , \mathscr{L})$ is a model of $(X,L)$. Let $B$ be a finite subset of $\mathscr{X}_s (k) \setminus
( {\operatorname{Sing}}( \mathscr{X}_s) \cup \{ \sigma (k) \})$ (We allow $B = \emptyset$).
\[def:basesection:WBsection:leaf\] Let $\Gamma$, $\Gamma_{\min}$, $P$, $(\mathscr{X} ; \sigma)$, ${{\mathscr{L}}}$, and $B$ be as above.
1. A nonzero global section $\widetilde{s}_0$ of $\mathscr{L}$ is called a *base section with respect to $B$* if $\widetilde{s} (p) \neq 0$ for any $p \in
B \cup {\operatorname{Sing}}( \mathscr{X}_s ) \cup \{ \sigma (k) \}$.
2. A nonzero global section $\widetilde{s}_1$ of $\mathscr{L}$ is called a *$P$-unimodularity section with respect to $B$* if ${\operatorname{div}}(\widetilde{s}) - \mathscr{W}_{P} - \sigma (R)$ is trivial on some open neighborhood of $B \cup {\operatorname{Sing}}( \mathscr{X}_s ) \cup \{ \sigma (k) \}$.
The following lemma shows us importance of a base section and a $P$-unimodularity section.
\[lemma:faithful:ends2\] Let $\Gamma$, $\Gamma_{\min}$, $P$, $(\mathscr{X} ; \sigma)$, ${{\mathscr{L}}}$, and $B$ be as above. Suppose that there exist a base section $\widetilde{s}_{0}$ of $\mathscr{L}$ with respect to $B$ and a $P$-unimodularity section $\widetilde{s}_{1}$ of $\mathscr{L}$ with respect to $B$. Set $s_0 := {\left.{\widetilde{s}_{0}}\right\vert_{{X}}}$ and $s_{1} := {\left.{\widetilde{s}_{1}}\right\vert_{{X}}}$. Further, set $h := s_{1} / s_0$, which is a non-zero rational function on $X$. Define $\varphi :
X^{{\operatorname{an}}}\setminus X (K) \to {{\mathbb{R}}}$ by $\varphi := - \log |h|$. Then the following hold.
1. The function $\varphi$ gives an isometry from $[\tau_{\Gamma_{\min}} (P) , P)
$ to $[ 0 , + \infty )$.
2. The restriction of $\varphi$ to $\Gamma \setminus
(\tau_{\Gamma_{\min}} (P) , P)$ is locally constant.
3. Let $P_1$ be a point in $X(K)$. Suppose that ${\operatorname{red}}_{\mathscr{X}} (P_1) \in B$. Then the restriction of $\varphi$ to $[\tau_{\Gamma} (P_1) , P_1 )$ is constant.
[[*Proof.*]{}]{}Set $g := \widetilde{s}_{1}
/\widetilde{s}_{0}$, which is a non-zero rational function on $\mathscr{X}$. Note that ${\left.{g}\right\vert_{{X}}} = h$. By the definitions of a base section with respect to $B$ and a $P$-unimodularity section with respect to $B$, there exists an open neighborhood $\mathscr{U}$ of $B \cup {\operatorname{Sing}}( \mathscr{X}_s ) \cup \{ \sigma (k) \}
$ such that ${\operatorname{div}}(g) - \mathscr{W}_{P} - \sigma (R)$ is trivial on $\mathscr{U}$.
Let us prove (1). Let $E$ be the chain in $\mathscr{X}_s$ such that $\Delta_E = [\tau_{\Gamma_{\min}} (P) , \tau_{\Gamma} (P)] $. Let $\lambda_{E}$ denote the length of $\Delta_{E} \subset \Gamma$. (With the notation in §\[subsection:setting1\], $\lambda_{E}$ equals the distance between $[E_0]$ and $[E_{r}]$ in $\Gamma$, which is equal to $\sum_{j=0}^{r-1} \lambda_{j}$.) Since ${\operatorname{div}}(g) - \mathscr{W}_{P} - \sigma (R)$ is trivial on $\mathscr{U}$, the equality shows that $\varphi$ gives an isometry from $\Delta_{E}$ to $[0 , \lambda_{E}]$.
Let $\ell$ be a local equation of the Cartier divisor $\sigma (R)$ and we take $\varpi \in K^{\times}$ such that $v_K ( \varpi ) =
\lambda_{E}$. Then there exists a rational function $u$ on $\mathscr{X}$ such that $u$ is a unit regular function on some open neighborhood of $\sigma (R)$ and satisfies $g = \varpi u \ell$. Since $- \log
\left|
{\left.{\ell}\right\vert_{{X}}}
\right|$ gives an isometry from $\Delta (\sigma)= [ \tau_{\Gamma} (P) , P)$ to $[0 , + \infty )$, $\varphi = - \log |h|$ gives an isometry from $\Delta (\sigma) $ to $[ \lambda_{E} , + \infty)$. Since we have the simplicial decomposition $[
\tau_{\Gamma_{\min}} (P) , P ) = \Delta_{E} \cup \Delta ( \sigma)$, this concludes that $\varphi$ gives an isometry from $[
\tau_{\Gamma_{\min}} (P) , P )$ to $[ 0 , + \infty)$. Thus we have (1).
Let us prove (2). Let $\Gamma^{\circ}$ be a connected component of $\Gamma \setminus (\tau_{\Gamma_{\min}} (P) , P)$. Let $D$ be the connected curve in $\mathscr{X}_s$ such that $\Delta_D$ equals the closure of $\Gamma^{\circ}$ (cf. Definition \[def:canonical:1:simplex:curve\]). Let $F_0 , \ldots , F_{r-1}$ be the connected curves in $\mathscr{X}_s$ as in the paragraph where we define $\mathscr{W}_P$. Then one sees that for each $\alpha = 0 , \ldots r-1$, we have ${\operatorname{\operatorname{Irr}}}(D) \subset {\operatorname{\operatorname{Irr}}}(F_{\alpha})$ or ${\operatorname{\operatorname{Irr}}}(D) \cap {\operatorname{\operatorname{Irr}}}(F_{\alpha}) = \emptyset$. It follows from the definition of $\mathscr{W}_P$ that there exists $\varpi' \in K^{\times}$ such that for any $C \in {\operatorname{\operatorname{Irr}}}(D)$, ${\operatorname{ord}}_{C} (\mathscr{W}_P) = - \log | \varpi' | = : \lambda'$. Since ${\operatorname{div}}(g) - \mathscr{W}_{P} - \sigma (R)$ is trivial on $\mathscr{U}$, this shows that on some neighborhood of $D \cap \mathscr{U}$, ${\operatorname{div}}(g) = {\operatorname{div}}( \varpi' )$. Since $D \cap {\operatorname{Sing}}( \mathscr{X}_s ) \subset \mathscr{U}$, it follows that the restriction of $\varphi$ to $\Delta_D$ equals the constant $\lambda'$. Thus $\varphi$ is constant over $\Gamma^{\circ}$, and we have (2).
Let us prove (3). Let $\sigma_1$ be the section of $\mathscr{X} \to {{\operatorname{Spec}}}(R)$ such that $\sigma_1 (K) = P_1$. Since $\sigma_1 (k) = {\operatorname{red}}_{\mathscr{X}} (P_1) \in
B \subset
\mathscr{X}_s (k) \setminus
{\operatorname{Sing}}( \mathscr{X}_s ) $, $(\mathscr{X} , \sigma_1 )$ is a strictly semistable pair, and $\Delta ( \sigma_1 ) = [ \tau_{\Gamma} (P_1) , P_1)$ by Lemma \[lemma:end=canonialend\]. Recall that ${\operatorname{div}}(g) - \mathscr{W}_{P} - \sigma (R)$ is trivial on the neighborhood $\mathscr{U} \supset B$. Then, since $\sigma_1 (k) \in B$ and $B \cap ( {\operatorname{Sing}}(\mathscr{X}_s) \cup \{ \sigma (k) \}) = \emptyset$, there exist $\varpi_1 \in R \setminus \{ 0 \}$ and a rational function $u$ that is a unit regular function at $\sigma_1 (k)$ such that $g = \varpi_1 u$ on some neighborhood of $\sigma_1 (k)$. Since $\varphi = - \log |{\left.{g}\right\vert_{{X}}}|$, that shows that $\varphi$ equals the constant $- \log |\varpi_1|_K$ on $\Delta ( \sigma_1)$. This proves (3).
500
We want to show the following proposition, which assures the existence of a model $({{\mathscr{X}}}, {{\mathscr{L}}})$ that one can apply Lemma \[lemma:faithful:ends2\]. Here $t(g)$ is the quantity in Theorem \[thm:main:unimodular:faithful\].
\[prop:main:FT:ends\] Let $\Gamma$ be a compact skeleton of $X^{{{\operatorname{an}}}}$. We fix a minimal skeleton $\Gamma_{\min} \subset \Gamma$. Let $P$ be a point in $X(K)$. Assume that $\deg (L) \geq t(g)$. Then there exists a model $(\mathscr{X}, \mathscr{L})$ of $(X,L)$ with the following properties.
1. There exists a section $\sigma$ of $\mathscr{X} \to {{\operatorname{Spec}}}(R)$ such that $( \mathscr{X} ; \sigma )$ is a strictly semistable pair with $\Gamma
= S ( \mathscr{X} )$ and $\sigma (K) = P$.
2. For any finite subset $B \subset \mathscr{X}_s (k)
\setminus ( {\operatorname{Sing}}(\mathscr{X}_s) \cup \{ \sigma(k) \})$, $\mathscr{L}$ has a base section with respect to $B$ and a $P$-unimodularity section with respect to $B$ cf. Definition .
Further, if ${{\mathscr{X}}}^0$ is a strictly semistable model of $X$ with $S({{\mathscr{X}}}^0) = \Gamma$, then we may take ${{\mathscr{X}}}$ so that ${{\mathscr{X}}}$ dominates ${{\mathscr{X}}}^0$.
The proof of Proposition \[prop:main:FT:ends\] will be given in §\[subsection:proof:prop:main:FT:ends\].
Good model {#subsec:good:model:classical:point}
----------
In this subsection, we fix a minimal skeleton $\Gamma_{\min}$ of $X^{{{\operatorname{an}}}}$. When $g \geq 2$, we endow $\Gamma_{\min}$ with a canonical weight function and a canonical finite graph structure in §\[subsec:skeleton:weighted:metric:graph\].
\[def:goodDMmodel:end\] Let $P$ be a point in $X(K)$. Let $( \mathscr{X} , \mathscr{L})$ be a model of $(X,L)$. We call $( \mathscr{X} , \mathscr{L})$ a *good model for $P$* if it satisfies the following conditions, where $\mathscr{M}:= \mathscr{L} \otimes \omega_{\mathscr{X}/R}^{\otimes -1}$.
1. The model $\mathscr{X}$ is a strictly semistable model of $X$ such that $S ( \mathscr{X})
= \Gamma_{\min}$.
2. Let $C_P$ be the irreducible component of $\mathscr{X}_s$ with ${\operatorname{red}}_{\mathscr{X}} (P) \in C_P$ and set $v_P := [ C_P ]$. Then $
D_{\mathscr{M}} - [v_P]$ is effective on $\Gamma_{\min}$.
3. We have $\deg (D_{\mathscr{M}}) \geq 3$. Further, if $g \geq 2$, then for any island $\Gamma_i$ of $\Gamma_{\min}$, we have $\deg
\left(
{\left.{(D_{\mathscr{M}}
- [v_P])}\right\vert_{{\Gamma_i}}}
\right) \geq 1$.
If $g \geq 1$, then condition (i) in Definition \[def:goodDMmodel:end\] is equivalent to $\mathscr{X}$ being a Deligne–Mumford strictly semistable model of $X$.
We prove a couple of lemmas that will be used in constructing a base section and a $P$-unimodularity section.
\[lemma:vanishing:for:ends:new-1\] Let $( \mathscr{X} , \mathscr{L})$ be a good model of $(X,L)$ for $P$, and set $\mathscr{M} := \mathscr{L} \otimes
\omega_{\mathscr{X}/R}^{\otimes -1}$ as before. Let $q \in \mathscr{X}_s ( k )$ be a point.
1. Suppose that $q \notin {\operatorname{Sing}}( \mathscr{X}_s)$. Then we have $h^{0}
\left(
\left(
{\left.{\mathscr{M}}\right\vert_{{\mathscr{X}_s}}}
\left( - q - {\operatorname{red}}_{\mathscr{X}} (P)
\right)
\right)^{\otimes -1}
\right) = 0$ and $
h^{0}
\left(
{\left.{\mathscr{M}}\right\vert_{{\mathscr{X}_s}}} ( - {\operatorname{red}}_{\mathscr{X}} (P)
)^{\otimes -1}
\right) = 0$.
2. Suppose that $q \in {\operatorname{Sing}}( \mathscr{X}_s)$. Let $\nu : \widetilde{\mathscr{X}_s} \to \mathscr{X}_s$ be the partial normalization at $q$. Then $
h^{0}
\left(
\nu^{\ast}
\left(
{\left.{\mathscr{M}}\right\vert_{{\mathscr{X}_s}}} ( - {\operatorname{red}}_{\mathscr{X}} (P) )
\right)^{\otimes -1}
\right) = 0$.
[[*Proof.*]{}]{}We set $M := {\left.{\mathscr{M}}\right\vert_{{\mathscr{X}_s}}}
(- {\operatorname{red}}_{\mathscr{X}} (P))$ and $D_{M} :=
\sum_{C \in {\operatorname{\operatorname{Irr}}}( \mathscr{X}_s)} \deg
\left(
{\left.{M}\right\vert_{{C}}}
\right) [C]$. Then $D_{M} = D_{\mathscr{M}} - [v_P]$. Thus $D_{M}$ is effective by Definition \[def:goodDMmodel:end\](ii) and $\deg(D_M) \geq 2$ by Definition \[def:goodDMmodel:end\](iii).
\(1) Suppose that $q \notin {\operatorname{Sing}}( \mathscr{X}_s )$. We take a unique $C_1 \in {\operatorname{\operatorname{Irr}}}( \mathscr{X}_s )$ such that $q \in C_1$, and we set $v_1 := [C_1] \in V ( \mathscr{X} )$. By Definition \[def:goodDMmodel:end\](iii), we have $\deg
\left(
D_{M} - [v_1]
\right) \geq 1$.
Let us show the first equality $h^0 \left(
(M(-q))^{\otimes -1}
\right) = 0$. If $D_{M} - [v_1]$ is effective, then $M (-q)$ is nef and $\deg(M (-q)) > 0$. By Lemma \[lemma:vanishing0\], we get $h^0 \left(
(M(-q))^{\otimes -1}
\right) = 0$. Suppose that $D_{M} - [v_1]$ is not effective. Then $(D_{M} - [v_1]) (v_1) = -1$, and $v_1$ is the only point at which $D_{M} - [v_1]$ is negative. Further, we note that the valence of $\Gamma_{\min}$ at $v_1$ is at least $2$. Indeed, suppose that this is not the case. Then $\{ v_1 \}$ is an island, and hence by condition (iii) of Definition \[def:goodDMmodel:end\], $D_{M} (v_1) \geq 1$. This means that $D_{M} - [v_1]$ is effective, which is a contradiction. This proves that the valence of $\Gamma_{\min}$ at $v_1$ is greater than or equal to $2$. Thus we see that the assumptions described in Remark \[remark:proofs:lemma:freeatnode:graphversion:emptyset\](2) are fulfilled, and hence by this remark, we obtain $h^{0}
\left(
\left(
M (-q)
\right)^{\otimes -1}
\right) = 0$, which is the first equality.
Since $M^{\otimes -1} \subset M (-q)^{\otimes -1}$ the second equality follows from the first one.
\(2) Suppose that $q \in {\operatorname{Sing}}( \mathscr{X}_s )$. Set $\widetilde{\Gamma} := \Gamma_{\min} \setminus \mathrm{relin} (\Delta_q)$. Then $D_M$ is effective by Definition \[def:goodDMmodel:end\](ii) and has positive degree on any connected component of $\widetilde{\Gamma}$ by Definition \[def:goodDMmodel:end\](iii). Thus, as noted in Remark \[remark:proofs:lemma:freeatnode:graphversion:emptyset\](1), we obtain $
h^{0}
\left(
\nu^{\ast}(M)^{\otimes -1}
\right) = 0$.
500
\[lemma:vanishing:for:ends:new\] With notation and assumptions being in Lemma , we have the following:
1. ${\left.{\mathscr{L}}\right\vert_{{\mathscr{X}_s}}} \left(
- {\operatorname{red}}_{\mathscr{X}} (P)
\right)$ is free at any $q \in \mathscr{X}_s(k)$.
2. $\mathscr{L}$ is free at any $q \in \mathscr{X}_s(k)$.
[[*Proof.*]{}]{}Assertion (1) follows from Lemma \[lemma:nonbasepoint\] and Lemma \[lemma:vanishing:for:ends:new-1\]. We prove (2). For any $q \in \mathscr{X}_s (k) \setminus {\operatorname{Sing}}( \mathscr{X}_s )$, we have $h^{0}
\left(
\left(
{\left.{\mathscr{M}}\right\vert_{{\mathscr{X}_s}}}
\left( - q - {\operatorname{red}}_{\mathscr{X}} (P)
\right)
\right)^{\otimes -1}
\right) = 0$ by Lemma \[lemma:vanishing:for:ends:new-1\](1), and hence $h^{0}
\left(
{\left.{\mathscr{M}}\right\vert_{{\mathscr{X}_s}}} ( - q
)^{\otimes -1}
\right) = 0$. For any $q \in {\operatorname{Sing}}( \mathscr{X}_s )$, if $\nu : \widetilde{\mathscr{X}_s} \to
\mathscr{X}_s$ denotes the partial normalization at $q$, then $
h^{0}
\left(
\nu^{\ast}
\left(
{\left.{\mathscr{M}}\right\vert_{{\mathscr{X}_s}}} ( - {\operatorname{red}}_{\mathscr{X}} (P) )
\right)^{\otimes -1}
\right) = 0$ by Lemma \[lemma:vanishing:for:ends:new-1\](2), and hence $h^{0}
\left(
\nu^{\ast}
\left(
{\left.{\mathscr{M}}\right\vert_{{\mathscr{X}_s}}}
\right)^{\otimes -1}
\right) = 0$. By Lemma \[lemma:nonbasepoint\], it follows that ${\left.{\mathscr{L}}\right\vert_{{\mathscr{X}_s}}}$ is base-point free. The rest of the argument is the same as the proof of Lemma \[lemma:freeatregular:graphversion:effective\] when $q \in \mathscr{X}_s (k) \setminus {\operatorname{Sing}}( \mathscr{X}_s )$ and that of Lemma \[lemma:freeatnode:graphversion:emptyset\] when $q \in {\operatorname{Sing}}( \mathscr{X}_s )$.
500
Proof of Proposition \[prop:main:FT:ends\] {#subsection:proof:prop:main:FT:ends}
------------------------------------------
Let $\Gamma$ be a compact skeleton of $X^{{{\operatorname{an}}}}$. We fix a minimal skeleton $\Gamma_{\min} \subset \Gamma$. For any model $\mathscr{X}$ of $X$ with $S (\mathscr{X}) = \Gamma$, by [@BPR2 Theorem 4.11], there exist a unique model $\mathscr{X}^{\min}$ of $X$ and a homomorphism $\mu : \mathscr{X} \to \mathscr{X}^{\min}$ extending the identity on $X$ such that $S ( \mathscr{X}^{\min} ) = \Gamma_{\min}$ (namely, $\mathscr{X}^{\min}$ is minimal) and $V( \mathscr{X}^{\min}) =
V ( \mathscr{X} ) \cap \Gamma_{\min}$. We use this notation in the sequel.
In this subsection, we prove Proposition \[prop:main:FT:ends\]. First, we construct a model that dominates a good model for a point in $X(K)$.
\[prop:good:Gamma-model:ends\] Let $P$ be a point in $X(K)$. Assume that $\deg (L) \geq t(g)$. Then there exists a model $(\mathscr{X}, \mathscr{L})$ of $(X,L)$ with the following properties.
1. There exists a section $\sigma$ of $\mathscr{X} \to {{\operatorname{Spec}}}(R)$ such that $( \mathscr{X} ; \sigma )$ is a strictly semistable pair with $\Gamma
= S ( \mathscr{X} )$ and and $\sigma (K) = P$.
2. Let $\mathscr{X}^{\min}$ and $\mu: \mathscr{X}\to \mathscr{X}^{\min}$ be as above. Then there exists a line bundle $\mathscr{L}^{\min}$ over $\mathscr{X}^{\min}$ such that $( \mathscr{X}^{\min} , \mathscr{L}^{\min})$ is a good model for $P$ cf. Definition and $\mathscr{L} = \mu^{\ast} (\mathscr{L}^{\min})$.
Further, if ${{\mathscr{X}}}^0$ is a strictly semistable model of $X$ with $S({{\mathscr{X}}}^0) = \Gamma$, then we may take ${{\mathscr{X}}}$ so that ${{\mathscr{X}}}$ dominates ${{\mathscr{X}}}^0$.
[[*Proof.*]{}]{}By Lemma \[lemma:retraction:rational\](1), we have $\tau_{\Gamma} (P) \in \Gamma_{\Lambda}$. Let ${{\mathscr{X}}}^0$ be a strictly semistable model of $X$ with $S({{\mathscr{X}}}^0) = \Gamma$. By Lemma \[prop:subdivision1:b\], there exists a strictly semistable model $\mathscr{X}^1$ of $X$ such that $\Gamma = S( \mathscr{X}^1)$ and $V(\mathscr{X}^1) = V( \mathscr{X}^0) \cup \{ \tau_{\Gamma} (P) \}$. We take a minimal model $\mathscr{X}^{1,\min}$ of $X$ such that $S(\mathscr{X}^{1 , \min}) = \Gamma_{\min}$ and $V(\mathscr{X}^{1,\min})
= V(\mathscr{X}^{1}) \cap \Gamma_{\min}$. We note that $\mathscr{X}^{1,\min}$ is dominated by $\mathscr{X}^1$. Let $C_P$ be the irreducible component of $\mathscr{X}^{1,\min}_s$ with ${\operatorname{red}}_{\mathscr{X}^{1,\min}}(P) \in C_P$, and we set $v_P := [C_P]$.
Suppose that $g \geq 2$. Then $\mathscr{X}^{1,\min}$ is a Deligne–Mumford strictly semistable model of $X$. Since $\deg (L) \geq t(g) = 3g-1$, we use Proposition \[prop:pregoodmodel\] with $v_P$ in place of $x$ to obtain a model $( \mathscr{X}^2 , \mathscr{L}^2)$ with properties in Proposition \[prop:pregoodmodel\] such that the identity morphism on $X$ extends to a morphism $\mathscr{X}^2
\to \mathscr{X}^{1,\min}$. We remark that $S(\mathscr{X}^2) = \Gamma_{\min}$. We set $\mathscr{X} :=
\mathscr{X}^1 \times_{\mathscr{X}^{1 , \min}} \mathscr{X}^2$ and let $\mathscr{L}$ be the pullback of $\mathscr{L}^2$ to $\mathscr{X}$. Then by the construction of $\mathscr{X}^2$ and $\mathscr{X}$, we have $\mathscr{X}^{\min} = \mathscr{X}^2$. Set $\mathscr{L}^{\min} := \mathscr{L}^2$. Since $\left(\mathscr{X}^2, {{\mathscr{L}}}^2\right)$ has the properties in Proposition \[prop:pregoodmodel\], we see that $\left(\mathscr{X}^{\min}, \mathscr{L}^{\min} \right)$ is a good model for $P$ and thus has property (ii) of Proposition \[prop:good:Gamma-model:ends\]. Since $\deg\left( \mathscr{L}^{\min}\otimes \omega_{\mathscr{X}^{\min}/R}^{\otimes -1} \right)
\geq (3g-1) - (2g-2) = g + 1 \geq 3$, the first condition of Definition \[def:goodDMmodel:end\](iii) is satisfied.
By the valuative criterion of properness, we take a section $\sigma$ of $\mathscr{X} \to {{\operatorname{Spec}}}(R)$ such that $\sigma (K) = P$. By Lemma \[lemma:retraction:rational\](2), $\sigma (k) \in \mathscr{X}_s (k) \setminus {\operatorname{Sing}}(\mathscr{X}_s)$, and thus $( \mathscr{X} ; \sigma)$ is a strictly semistable pair. By the construction of ${{\mathscr{X}}}$, we have $S({{\mathscr{X}}}) = \Gamma$, and thus $\mathscr{X}$ has property (i) in the proposition.
We consider the case where $g = 0$ or $g = 1$. First, suppose that $g = 1$ and that the minimal skeleton is not a singleton. Then a similar argument using Lemma \[lemma:pregoodmodel:g=1\] instead of Proposition \[prop:pregoodmodel\] gives a model with the required properties. Here, we remark that $ \deg
\left({\left.{{{\mathscr{L}}}\otimes\omega_{\mathscr{X}^{\min}/R}^{\otimes -1}}\right\vert_{{\mathscr{X}^{\min}_s}}}\right) = \deg(L) \geq t(1) = 3$, since $\omega_{X}$ is trivial.
Suppose that $g = 1$ and that the minimal skeleton is a singleton, or suppose that $g = 0$. In this case, $\mathscr{X}^{1 , \min}$ is a smooth proper model, and there exists a line bundle $\mathscr{L}^{1 , \min}$ over $\mathscr{X}^{1,\min}$ such that ${\left.{\mathscr{L}^{1,\min}}\right\vert_{{X}}} = L$. Set $\mathscr{X} := \mathscr{X}^1$ and $\mathscr{L} := \mu^{\ast} ( \mathscr{L}^{1,\min})$, where $\mu : \mathscr{X}
\to \mathscr{X}^{1,\min}$ is the morphism extending the identity morphism on $X$. Then it is straightforward to check that $( \mathscr{X}, \mathscr{L})$ has the required properties.
500
\[lemma:proof:key:ends\] Let $P$ be a point in $X(K)$. Let $( \mathscr{X} ; \sigma )$ be a strictly semistable pair such that $S ( \mathscr{X} ) = \Gamma$ and $\sigma (K) = P$. Let $\mathscr{L}$ be a line bundle over $\mathscr{X}$. Assume that there exists a line bundle $\mathscr{L}^{\min}$ over $\mathscr{X}^{\min}$ such that $( \mathscr{X}^{\min} , \mathscr{L}^{\min})$ is a good model for $P$ and $\mathscr{L} = \mu^{\ast} ( \mathscr{L}^{\min})$. Then for any finite subset $B \subset \mathscr{X}_s (k) \setminus
( {\operatorname{Sing}}( \mathscr{X}_s) \cup \{ \sigma(k) \})$, $\mathscr{L}$ has a base section with respect to $B$ and a $P$-unimodularity section with respect to $B$.
[[*Proof.*]{}]{}Let $B$ be any finite subset of $\mathscr{X}_s (k) \setminus
( {\operatorname{Sing}}( \mathscr{X}_s) \cup \{ \sigma(k) \})
$. First, we construct a base section with respect to $B$. By Lemma \[lemma:vanishing:for:ends:new\](2), there exists an $\widetilde{s}_0^{\,\min}
\in H^0
( \mathscr{L}^{\min} )$ such that $\widetilde{s}_0^{\,\min} ( p ) \neq 0$ for any $p \in \mu \left(B \cup {\operatorname{Sing}}( \mathscr{X}_s \right)
\cup \{ \sigma (k) \})$. Set $\widetilde{s}_0 := \mu^{\ast} ( \widetilde{s}_0^{\,\min} ) \in H^0 \left( \mathscr{L} \right)$. Then we have $\widetilde{s}_0 (p) \neq 0$ for any $p \in B \cup {\operatorname{Sing}}( \mathscr{X}_s) \cup \{ \sigma (k) \}$. This shows that $\widetilde{s}_0$ is a base section with respect to $B$.
In the following, we construct a $P$-unimodularity section with respect to $B$. We use the notation in §\[subsection:setting1\]. In particular, $\mathscr{W}_P$ is the stepwise vertical divisor associated to $P$ defined in , and $F_0$ is the support of $\mathscr{W}_P$. We take a unique connected curve $D$ in $\mathscr{X}_s$ such that $\mu$ restricts to an isomorphism $\phi : D \to \mathscr{X}^{\min}_s$. We remark that $D \cap \sigma (R) \neq \emptyset$ if and only if $\mathscr{W}_P = 0$, and if this is the case, then $D \cap \sigma (R)$ is a singleton. Let $p_0$ be the point in ${{\mathscr{X}}}_s (k)$ such that $$\{ p_0 \}
=
\begin{cases}
D \cap F_0
& \text{if $\mathscr{W}_P \neq 0$,}
\\
D \cap \sigma (R)
& \text{otherwise.}
\end{cases}$$ We set $
\mathscr{L}_-
:=
\mathscr{L}
\left(
-
\left(
\mathscr{W}_{P}
+
\sigma (R)
\right)
\right)
$. Then ${\left.{{{\mathcal{O}}}_{\mathscr{X}}
\left( \mathscr{W}_{P} + \sigma (R)
\right)}\right\vert_{{D}}} = {{\mathcal{O}}}_{D} (p_0)$. Since $\phi ( p_0) = {\operatorname{red}}_{\mathscr{X}^{\min}} ( P )$, we have ${{\mathcal{O}}}_{D} ( p_0) = \phi^\ast \left( {{\mathcal{O}}}_{\mathscr{X}^{\min}_s} (
{\operatorname{red}}_{\mathscr{X}^{\min}} ( P ) ) \right)$. It follows that [$$\begin{aligned}
{\left.{\mathscr{L}_-}\right\vert_{{D}}}
&=
{\left.{\mathscr{L}
\left(
-
\left(
\mathscr{W}_{P}
+
\sigma (R)
\right)
\right)}\right\vert_{{D}}}
\\
&
=
{\left.{
\mu^{\ast}
\left(
\mathscr{L}^{\min}
\right)}\right\vert_{{D}}}
\otimes
\phi^\ast \left( {{\mathcal{O}}}_{\mathscr{X}^{\min}_s} (-
{\operatorname{red}}_{\mathscr{X}^{\min}} ( P )
) \right)
\\
&=
\phi^\ast
\left(
{\left.{\mathscr{L}^{\min}}\right\vert_{{\mathscr{X}^{\min}_s}}}
(-
{\operatorname{red}}_{\mathscr{X}^{\min}} ( P )
)
\right)
.\end{aligned}$$]{} Since $\phi$ is an isomorphism, it follows from Lemma \[lemma:vanishing:for:ends:new\] that $
{\left.{\mathscr{L}_-}\right\vert_{{D}}}
$ is basepoint free. Thus there exists $\xi_- \in H^0
\left(
{\left.{\mathscr{L}_-}\right\vert_{{D}}}
\right)$ such that $\xi_- (q) \neq 0$ for any $
q \in ( B \cup {\operatorname{Sing}}( \mathscr{X}_s ) \cup \{ \sigma (k) \})\cap D
$.
Let us define $\eta_- \in H^0\left({\left.{\mathscr{L}_-}\right\vert_{{\mathscr{X}_s}}}\right)$ such that $\eta_- (q) \neq 0$ for any $q \in
B \cup {\operatorname{Sing}}( \mathscr{X}_s ) \cup \{ \sigma (k) \}
$. If $\mathscr{X}_s = D$, then we set $\eta_- := \xi_-$, which satisfies the required condition. Suppose that $\mathscr{X}_s \neq D$. Since $\mu ( \mathscr{X}_s - D )$ is a finite set of points, ${\left.{\mathscr{L}
}\right\vert_{{\mathscr{X}_s - D}}} \cong {{\mathcal{O}}}_{\mathscr{X}_s - D}$. Further, it follows from that ${\left.{{{\mathcal{O}}}_{\mathscr{X}} \left( \mathscr{W}_{P}
+
\sigma(R) \right)}\right\vert_{{\mathscr{X}_s - D}}} \cong {{\mathcal{O}}}_{\mathscr{X}_s - D}$. Thus ${\left.{\mathscr{L}_-
}\right\vert_{{\mathscr{X}_s - D}}} \cong {{\mathcal{O}}}_{\mathscr{X}_s - D}$. It follows that there exists a (unique) section $\eta_-
\in H^0 \left(
{\left.{\mathscr{L}_-
}\right\vert_{{\mathscr{X}_s}}}
\right)$ such that ${\left.{\eta_-}\right\vert_{{D}}} = \xi_-$. We remark that $\phi^{-1}\left(\mu ( \mathscr{X}_s - D )\right)
\subset {\operatorname{Sing}}( \mathscr{X}_s ) \cap D$. Since $\xi_- (q) \neq 0$ for any $
q \in ( B \cup {\operatorname{Sing}}( \mathscr{X}_s ) \cup \{ \sigma (k) \}) \cap D
$, it follows that $\eta_-$ is nowhere vanishing over $\mathscr{X}_s - D$. This shows that $\eta_- (q) \neq 0$ for any $q \in B \cup {\operatorname{Sing}}( \mathscr{X}_s ) \cup \{ \sigma (k) \}
$.
Put $M := {\left.{\mathscr{L}_- \otimes \omega_{\mathscr{X}/R}^{\otimes -1}}\right\vert_{{\mathscr{X}_s}}}$. Put $\Sigma := D \cap \left( \mathscr{X}_s -D \right)$. By the adjunction formula, we see that [$$\begin{aligned}
{\left.{M}\right\vert_{{D}}} &=
{\left.{\mathscr{L}_-}\right\vert_{{D}}} \otimes
\omega_{D} ^{\otimes -1}
( -\Sigma )
\\
&=
\phi^{\ast}
\left(
{\left.{\mathscr{L}^{\min} \otimes
\omega_{\mathscr{X}^{\min}/R} ^{\otimes -1}}\right\vert_{{\mathscr{X}^{\min}_s}}}
(- {\operatorname{red}}_{\mathscr{X}^{\min}} (P))
\right)
( - \Sigma )
\\
&=
\phi^\ast
\left(
{\left.{\mathscr{M}^{\min}}\right\vert_{{\mathscr{X}^{\min}_s}}}
(- {\operatorname{red}}_{\mathscr{X}^{\min}} (P))
\right)
( - \Sigma ),\end{aligned}$$ ]{}where $\mathscr{M}^{\min} :=
\mathscr{L}^{\min} \otimes
\omega_{\mathscr{X}^{\min}/R} ^{\otimes -1}$. Since $({{\mathscr{X}}}^{\min}, {{\mathscr{L}}}^{\min})$ is a good model for $P$, Lemma \[lemma:vanishing:for:ends:new-1\](1) gives that $h^0
\left(\left(
{\left.{\mathscr{M}^{\min}}\right\vert_{{\mathscr{X}^{\min}_s}}}
(- {\operatorname{red}}_{\mathscr{X}^{\min}} (P))
\right)^{\otimes -1}
\right) = 0$. Since $\phi$ is an isomorphism, we get $$h^0\left(({\left.{M}\right\vert_{{D}}} (\Sigma))^{\otimes -1} \right)
= h^{0}
\left(
\phi^{\ast}
\left(
{\left.{\mathscr{M}^{\min}}\right\vert_{{\mathscr{X}^{\min}_s}}}
(- {\operatorname{red}}_{\mathscr{X}^{\min}} (P))
\right)^{\otimes -1}
\right) = 0.$$
Let us prove $h^0 \left( M^{\otimes -1} \right) = 0$. If $\mathscr{X}_s = D$, then $h^0 \left( M^{\otimes -1} \right) = h^0\left(({\left.{M}\right\vert_{{D}}} (\Sigma))^{\otimes -1} \right) = 0$. Suppose that $\mathscr{X}_s \neq D$. We take any connected component $F'$ of $\mathscr{X}_s - D$. We compute ${\left.{M}\right\vert_{{F'}}}$ by using the adjunction formula. Since ${\left.{\mathscr{L}_-}\right\vert_{{F'}}} \cong {{\mathcal{O}}}_{F'}$, as noted above, we have ${\left.{M}\right\vert_{{F'}}} \cong
{\left.{\omega_{\mathscr{X}/R}}\right\vert_{{F'}}}^{\otimes -1}$. Let $p'$ be the point with $\{ p' \} = D \cap F'$. Then we have ${\left.{\omega_{\mathscr{X}/R}}\right\vert_{{F'}}} = \omega_{F'} (p')$, and thus ${\left.{M}\right\vert_{{F'}}}^{\otimes -1} \cong
\omega_{F'} (-p')$. Applying the Riemann–Roch formula on $F'$ for ${{\mathcal{O}}}_{F'} (p')$, we see that $h^0 \left( {{\mathcal{O}}}_{F'} (p') \right) -
h^{0} \left(
\omega_{F'} (-p')
\right) = 2$. Since $h^0 \left( {{\mathcal{O}}}_{F'} (p') \right) =2$, it follows that $
h^0
\left(
{\left.{M}\right\vert_{{F'}}}^{\otimes -1}
\right)
=
h^{0} \left(
\omega_{F'} (-p')
\right) = 0$. By Lemma \[lemma:vanishing7\], we obtain $h^0
\left(
M^{\otimes -1}
\right) = 0$.
By the Serre duality, $
h^1
\left(
{\left.{\mathscr{L}_-}\right\vert_{{\mathscr{X}_s}}}
\right) =
h^0
\left(
M^{\otimes -1}
\right) = 0
$. By the base-change theorem, the restriction map $H^0 \left(
\mathscr{L}_-\right)
\to
H^0 \left(
{\left.{\mathscr{L}_-}\right\vert_{{\mathscr{X}_s}}} \right)
$ is surjective. Thus there exists a global section $\widetilde{s}_-
\in H^0
\left(
\mathscr{L}_-
\right)$ such that ${\left.{\widetilde{s}_-}\right\vert_{{\mathscr{X}_s}}} = \eta_-$. For any $
q \in B \cup {\operatorname{Sing}}( \mathscr{X}_s ) \cup \{ \sigma (k) \}
$, since $\eta_- (q) \neq 0$, we have $\widetilde{s}_- (q) \neq 0$. Let $\widetilde{s}$ be the image of $\widetilde{s}_-$ by the natural inclusion $\mathscr{L}_- \hookrightarrow \mathscr{L}$. Then we see that $\widetilde{s}$ is a $P$-unimodularity section with respect to $B$. Thus we complete the proof of the lemma.
500
*Proof of Proposition \[prop:main:FT:ends\].*Let us complete the proof of Proposition \[prop:main:FT:ends\]. We take a model $( \mathscr{X}, \mathscr{L})$ as in Proposition \[prop:good:Gamma-model:ends\]. By condition (i) in Proposition \[prop:good:Gamma-model:ends\], there exists a section $\sigma$ of $\mathscr{X} \to {{\operatorname{Spec}}}(R)$ such that $( \mathscr{X} ; \sigma)$ is a strictly semistable pair such that $\Gamma =
S ( \mathscr{X})$ and $\sigma(K) = P$. Thus $( \mathscr{X}, \mathscr{L})$ has property (i) in Proposition \[prop:main:FT:ends\]. By condition (ii) in Proposition \[prop:good:Gamma-model:ends\], $( \mathscr{X} , \mathscr{L})$ satisfies the condition of Lemma \[lemma:proof:key:ends\]. Then by this lemma, for any finite subset $B \subset \mathscr{X}_s (k)
\setminus ( {\operatorname{Sing}}(\mathscr{X}_s) \cup \{ \sigma (k) \})$, $\mathscr{L}$ has a base section with respect to $B$ and a $P$-unimodularity section with respect to $B$. Thus $(\mathscr{X} , \mathscr{L})$ has also property (ii) in Proposition \[prop:main:FT:ends\].
500
Proof of Theorem \[thm:main:unimodular:faithful\]
-------------------------------------------------
In this section, we complete the proof of the main Theorem \[thm:main:unimodular:faithful\], which is restated below.
Let $X$ be a connected smooth projective curve over $K$ of genus $g \geq 0$. Let $\Gamma$ be a skeleton of $X^{{{\operatorname{an}}}}$. Let $L$ be a line bundle over $X$. Suppose that $$\deg (L)
\geq
t(g) :=
\begin{cases}
1 & \text{if $g=0$,}
\\
3 & \text{if $g=1$,}
\\
3g-1 & \text{if $g\geq 2$.}
\end{cases}$$ Then there exist $s_0 , \ldots , s_N \in H^0 ( X,L)$ such that the map $\varphi : X^{{\operatorname{an}}}\to {{\mathbb{T}}}{{\mathbb{R}}}^N$ defined by $\varphi := ( - \log |s_0| : \cdots : - \log |s_N|)$ gives a faithful tropicalization of $\Gamma$.
The following two lemmas will be used to separate points of a skeleton with ends.
\[lemma:faithful:ends\] Let $\Gamma$ be a skeleton. We write $\Gamma = S ( \mathscr{X} ;
\sigma_1 , \ldots , \sigma_m)$ for some strictly semistable pair $( \mathscr{X} ;
\sigma_1 , \ldots , \sigma_m)$. Let $\Gamma_{\min} \subset S( \mathscr{X})$ be a minimal skeleton. Set $P_i := \sigma_i (K)$ for $i = 1, \ldots , m$. Assume that $\deg (L) \geq t (g)$. Then for any $i = 1 , \ldots , m$, there exist nonzero global sections $s_0^{(i)}$ and $s_1^{(i)}$ of $L$ such that the function $\varphi^{(i)} : \Gamma \to {{\mathbb{R}}}$ defined by $\varphi^{(i)} := - \log |s_1^{(i)} / s_0^{(i)}|$ has the following properties
1. $\varphi^{(i)}$ gives an isometry $[\tau_{\Gamma_{\min}} (P_i) , P_i ) \to [0 , + \infty)$
2. for $j \neq i$, the restriction of $\varphi^{(i)}$ to $[\tau_{\Gamma_{\min}} (P_j) , P_j )
\setminus [\tau_{\Gamma_{\min}} (P_i) , P_i )$ is a constant function, and if $(\tau_{\Gamma_{\min}} (P_j) , P_j )
\cap (\tau_{\Gamma_{\min}} (P_i) , P_i ) = \emptyset$, then the constant is $0$
3. $\varphi^{(i)} ( \Gamma_{\min} ) = \{ 0 \}$.
[[*Proof.*]{}]{}We fix any $i = 1 , \ldots , m$. Since $\deg ( L ) \geq t (g)$, we apply Proposition \[prop:main:FT:ends\] in place of $P$, $\Gamma$, and $\mathscr{X}^0$ with $P_i$, $S(\mathscr{X})$, and $\mathscr{X}$ respectively. Then we obtain a model $(\mathscr{X}^i , \mathscr{L}^i)$ of $(X,L)$ that satisfies the following conditions.
1. there exists a section $\sigma^{P_i}$ of $\mathscr{X}^i
\to {{\operatorname{Spec}}}(R)$ such that $(\mathscr{X}^i ; \sigma^{P_i})$ is a strictly semistable pair and such that $S (\mathscr{X}^i ) = S( \mathscr{X})$ and $\sigma^{P_i} (K) = P_i$;
2. the identity on $X$ extends to a morphism $\mu^i : \mathscr{X}^i \to \mathscr{X}$;
3. for any finite subset $B \subset
\mathscr{X}^i_s (k) \setminus
({\operatorname{Sing}}( \mathscr{X}^i_s) \cup \{ \sigma^{P_i} (k) \} )$, $\mathscr{L}^i$ has a base section with respect to $B$ (for $(\mathscr{X}^i ; \sigma^{P_i})$ in (a) above) and a $P_i$-unimodularity section with respect to $B$.
We set $\mathcal{L} ( \Gamma )
= \{ P_1, \ldots , P_m \}$. Note that $\sigma^{P_i}(k) = {\operatorname{red}}_{\mathscr{X}^i} (P_i)
\in {\operatorname{red}}_{\mathscr{X}^i} ( \mathcal{L} (\Gamma))$, and we set $B^i := {\operatorname{red}}_{\mathscr{X}^i} ( \mathcal{L} (\Gamma))
\setminus \{ \sigma^{P_i}(k)\}$. We claim the following.
\[claim:Bi\]
1. Any point in $B^{i}$ is a non-singular point of ${{\mathscr{X}}}^i_s$.
2. For any $j = 1 , \ldots , m$ with $j \neq i$, we have ${\operatorname{red}}_{\mathscr{X}^i} (P_j) \in B^i$.
We prove the claim. Note that ${\operatorname{red}}_{\mathscr{X}} ( \mathcal{L} (\Gamma) ) =
\{ \sigma_1(k) , \ldots , \sigma_m(k) \}$. Since $({{\mathscr{X}}}; \sigma_1, \ldots, \sigma_m)$ is a strictly semistable pair, ${\operatorname{red}}_{\mathscr{X}} ( \mathcal{L} (\Gamma) )$ is contained in the set of non-singular points of $\mathscr{X}_s$. Since ${\left.{\mu^i}\right\vert_{{\mathscr{X}^i_s}}} : \mathscr{X}^i_s \to \mathscr{X}_s$ is a contraction of $(-2)$-curves and ${\operatorname{red}}_{\mathscr{X}} = {\left.{\mu^i}\right\vert_{{\mathscr{X}^i_s}}}
\circ {\operatorname{red}}_{\mathscr{X}^i}$, we see that any point in ${\operatorname{red}}_{\mathscr{X}^i} ( \mathcal{L} (\Gamma) )$ and hence any point in $B^i$ is a non-singular point of ${{\mathscr{X}}}^i_s$. Thus we have (1).
Since $( \mathscr{X} ; \sigma_1 , \ldots , \sigma_m)$ is a strictly semistable pair, ${\left.{{\operatorname{red}}_{\mathscr{X}}}\right\vert_{{\mathcal{L}(\Gamma)}}}$ is injective. Since ${\operatorname{red}}_{\mathscr{X}} = {\left.{\mu^i}\right\vert_{{\mathscr{X}^i_s}}}
\circ {\operatorname{red}}_{\mathscr{X}^i}$, it follows that ${\left.{{\operatorname{red}}_{\mathscr{X}^i}}\right\vert_{{\mathcal{L}(\Gamma)}}}$ is injective. Thus $\# {\left.{{\operatorname{red}}_{\mathscr{X}^i}}\right\vert_{{\mathcal{L}(\Gamma)}}} = m$, which shows that ${\operatorname{red}}_{\mathscr{X}^i} (P_1) ,
\ldots , {\operatorname{red}}_{\mathscr{X}^i} (P_m)$ are all distinct. Since $B^i = {\operatorname{red}}_{\mathscr{X}^i} ( \mathcal{L}(\Gamma) ) \setminus
\{ {\operatorname{red}}_{\mathscr{X}^i} (P_i) \}$, (2) in the claim holds.
Since we have Claim \[claim:Bi\](1), property (c) allows us to take a base section $\widetilde{s}_0^{(i)}$ of $\mathscr{L}^{i}$ with respect to $B^i$ and a $P_i$-unimodularity section $\widetilde{s}_1^{(i)}$ of $\mathscr{L}^{i}$ with respect to $B^i$. We set $s_0^{(i)} := {\left.{\widetilde{s}_0^{(i)}}\right\vert_{{X}}}$ and $s_1^{(i)} := {\left.{\widetilde{s}_1^{(i)}}\right\vert_{{X}}}$.
Let us show that $\varphi^{(i)} = - \log |s_1^{(i)} / s_0^{(i)}|$ has the required properties. It follows from Lemma \[lemma:faithful:ends2\](1) that it has property (i). By Lemma \[lemma:faithful:ends2\](2), $\varphi^{(i)}$ is locally constant on $\Gamma \setminus
(\tau_{\Gamma_{\min}} (P_i) , P_i )$. Take any $j \neq i$. Since we have Claim \[claim:Bi\](2), Lemma \[lemma:faithful:ends2\](3) implies that $\varphi^{(i)}$ is constant on $[ \tau_{\Gamma} (P_j), P_j)$. Since $[\tau_{\Gamma_{\min}} (P_j) , P_j)
= [\tau_{\Gamma_{\min}} (P_j) , \tau_{\Gamma} (P_j)] \cup
[\tau_{\Gamma} (P_j) , P_j) \subseteq \Gamma \cup [\tau_{\Gamma} (P_j) , P_j)$, it follows that $\varphi^{(i)}$ is locally constant on $[\tau_{\Gamma_{\min}} (P_j) , P_j) \setminus ( \tau_{\Gamma_{\min}} (P_i) , P_i )$. Since $[\tau_{\Gamma_{\min}} (P_j) , P_j) \setminus ( \tau_{\Gamma_{\min}} (P_i) , P_i )$ is connected, this proves that $\varphi^{(i)}$ has the former property in (ii). Since $\tau_{\Gamma_{\min}} (P_i) \in \Gamma_{\min}$ and $\varphi^{(i)} (\tau_{\Gamma_{\min}} (P_i)) = 0$ by property (i), Lemma \[lemma:faithful:ends2\](2) implies that $\varphi^{(i)} ( \Gamma_{\min}) = \{ 0 \}$. Thus it has property (iii). Finally, suppose that $(\tau_{\Gamma_{\min}} (P_j) , P_j )
\cap
(\tau_{\Gamma_{\min}} (P_i) , P_i ) = \emptyset$. Then $[\tau_{\Gamma_{\min}} (P_j) , P_j )$ is contained in the connected component of $\Gamma \setminus (\tau_{\Gamma_{\min}} (P_i) , P_i )$ that contains $\Gamma_{\min}$. Since $\varphi^{(i)}$ is locally constant on $\Gamma \setminus (\tau_{\Gamma_{\min}} (P_i) , P_i )$ and $\varphi^{(i)} ( \Gamma_{\min}) = \{ 0 \}$, this shows that $\varphi^{(i)}$ equals $0$ on $[\tau_{\Gamma_{\min}} (P_j) , P_j )$. Thus it also has the latter property in (ii).
500
\[lemma:remark:toseparate:ends\] We keep the notation in Lemma . Let $x$ and $y$ be distinct points in $\Gamma$.
1. If $x , y \in [\tau_{\Gamma_{\min}} (P_i) , P_i)$ for some $i$, then $\varphi^{(i)} (x) \neq \varphi^{(i)} (y)$.
2. If $x \in (\tau_{\Gamma_{\min}} (P_i) , P_i) \setminus
(\tau_{\Gamma_{\min}} (P_j) , P_j)$ and $y \in (\tau_{\Gamma_{\min}} (P_j) , P_j)
\setminus (\tau_{\Gamma_{\min}} (P_i) , P_i)$ for some $i \neq j$, then $\varphi^{(i)} (x) \neq \varphi^{(i)} (y)$.
3. If $x \in (\tau_{\Gamma_{\min}} (P_i) , P_i)$ and $y \in \Gamma_{\min}$, then $\varphi^{(i)} (x) \neq \varphi^{(i)} (y)$.
[[*Proof.*]{}]{}Assertion (1) follows from Lemma \[lemma:faithful:ends\](i).
Assertion (3) follows from Lemma \[lemma:faithful:ends\](i)(iii). Indeed, we have $\varphi^{(i)} (x) > 0$ by Lemma \[lemma:faithful:ends\](i), and $\varphi^{(i)} (y) = 0$ by Lemma \[lemma:faithful:ends\](iii).
We show (2). Suppose that $x \in (\tau_{\Gamma_{\min}} (P_i) , P_i) \setminus
(\tau_{\Gamma_{\min}} (P_j) , P_j)$ and $y \in (\tau_{\Gamma_{\min}} (P_j) , P_j)
\setminus (\tau_{\Gamma_{\min}} (P_i) , P_i)$ for some $i \neq j$. If $\tau_{\Gamma_{\min}} (P_j) \neq \tau_{\Gamma_{\min}} (P_i)$, then Lemma \[lemma:faithful:ends\](i)(ii) gives $\varphi^{(i)} (x) > 0 = \varphi (y)$. Suppose that $\tau_{\Gamma_{\min}} (P_j) = \tau_{\Gamma_{\min}} (P_i)$. Then there exists $z \in
[\tau_{\Gamma_{\min}} (P_i) , P_i) \cap
[\tau_{\Gamma_{\min}} (P_j) , P_j)$ such that $[\tau_{\Gamma_{\min}} (P_i) , z ] =
[\tau_{\Gamma_{\min}} (P_i) , P_i) \cap
[\tau_{\Gamma_{\min}} (P_j) , P_j)$. Note that $x \in
(\tau_{\Gamma_{\min}} (P_i) , P_i) \setminus
(\tau_{\Gamma_{\min}} (P_j) , P_j)
=
[\tau_{\Gamma_{\min}} (P_i) , P_i) \setminus
[\tau_{\Gamma_{\min}} (P_i) , z]
$. Then by Lemma \[lemma:faithful:ends\](i)(ii), we see that $\varphi^{(i)} (y) = \varphi^{(i)} (z) < \varphi^{(i)} (x)$.
500
*Proof of the main Theorem \[thm:main:unimodular:faithful\].*If $\Gamma$ is minimal, then we have already shown a faithful tropicalization of $\Gamma$ in Theorem \[theorem:FT:canonical\] and Theorem \[thm:FT:low:genus\], so that we may assume that it is not minimal. We write $\Gamma = S( \mathscr{X} ; \sigma_1 , \ldots , \sigma_r )$ for some strictly semistable pair, where we allow $r = 0$. We add sections $\sigma_{r+1} , \ldots , \sigma_{m}$ with $m > r$ so that any $(-1)$-curve in $\mathscr{X}_s$ meets at least one of $\sigma_1 (k) , \ldots , \sigma_m (k)$. Since $\Gamma = S(\mathscr{X} ; \sigma_1 , \ldots , \sigma_{r})
\subset S(\mathscr{X} ; \sigma_1 , \ldots , \sigma_{m})$, it suffices to prove the theorem for $S(\mathscr{X} ; \sigma_1 , \ldots , \sigma_{m})$. Thus replacing $\Gamma$ if necessary, we assume that $\Gamma = S(\mathscr{X} ; \sigma_1 , \ldots , \sigma_{m})$.
Take any minimal skeleton $\Gamma_{\min} \subset \Gamma$. We set $P_i := \sigma_i (K)$ for $i=1 , \ldots , m$. We claim that $$\label{eqn:completeskeleton}
S(\mathscr{X} ; \sigma_1 , \ldots , \sigma_{m})
=
\Gamma_{\min} \cup \bigcup_{i = 1}^{m} [ \tau_{\Gamma_{\min}} (P_i),
P_i).$$ Indeed, if $S(\mathscr{X} ; \sigma_1 , \ldots , \sigma_{m}) \supsetneq \Gamma_{\min} \cup \bigcup_{i = 1}^{m} [ \tau_{\Gamma_{\min}} (P_i),
P_i)$, then there exists $E \in {\operatorname{\operatorname{Irr}}}(\mathscr{X}_{s})$ such that $[E]$ has valence $1$ in $S( \mathscr{X} )$ and $[E] \in S (\mathscr{X} ; \sigma_1 , \ldots , \sigma_{m})
\setminus \Gamma_{\min} \cup \bigcup_{i = 1}^{m} [ \tau_{\Gamma_{\min}} (P_i),
P_i)$. Then $E$ is a $(-1)$-curve in $\mathscr{X}_s$ with $\sigma_j (k) \notin E$ for any $j = 1 , \ldots , m$, which contradicts our assumption of (a newly replaced) $\Gamma$. Thus holds.
By Theorem \[theorem:FT:canonical\] (when $g \geq 2$) and by Theorem \[thm:FT:low:genus\] (when $g =0,1$), there exist $s_0 , \ldots , s_{N'} \in H^0 ( X,L)$ such that the map $\varphi' : X^{{\operatorname{an}}}\to {{\mathbb{T}}}{{\mathbb{R}}}^{N'}$ defined by $\varphi' := ( - \log |s_0| : \cdots : - \log |s_{N'}|)$ gives a faithful tropicalization of $\Gamma_{\min}$.
Note that $\mathcal{L} (\Gamma) = \{ P_1 , \ldots , P_m\}$. For each $i = 1 , \ldots , m$, we use Lemma \[lemma:faithful:ends\] to obtain global sections $s_0^{(i)}$ and $s_1^{(i)}$ such that $\varphi^{(i)} := - \log |s_1^{(i)} / s_0^{(i)}|$ enjoys properties (i)–(iii) of Lemma \[lemma:faithful:ends\]. Set $s_{N' + 2i-1} := s^{(i)}_0$ and $s_{N' + 2i} := s^{(i)}_{1}$ for $i = 1 , \ldots , m$, and set $N := N' + 2 m$. Further, we set $\varphi :=
( - \log |s_0| : \cdots : - \log |s_{N'}| :
- \log |s_{N'+1}| : \cdots : - \log |s_{N}|)$.
We show that $\varphi$ gives a faithful tropicalization of $\Gamma$. Since $\varphi'$ is gives a faithful tropicalization of $\Gamma_{\min}$, $\varphi$ also gives a faithful tropicalization of $\Gamma_{\min}$. By Lemma \[lemma:faithful:ends\](i), we see that $\varphi$ gives a unimodular tropicalization over $[\tau_{\Gamma_{\min}} (P_i) , P_i)$ for each $i=1, \ldots , m$. Thus by , $\varphi$ is a unimodular tropicalization of $\Gamma$.
To conclude that $\varphi$ is a faithful tropicalization of $\Gamma$, we need to show that $\varphi$ is injective on $\Gamma$. Let $x , y \in \Gamma$ be distinct points. If $x, y \in \Gamma_{\min}$, then we have shown that $\varphi (x) \neq \varphi (y)$ by Theorem \[theorem:FT:canonical\] and Theorem \[thm:FT:low:genus\]. Without loss of generality, we may assume that $x \notin
\Gamma_{\min}$. By (\[eqn:completeskeleton\]), we take an $i=1, \ldots , m$ with $x \in (\tau_{\Gamma_{\min}} (P_i) , P_i)$.
- If $y \in \Gamma_{\min}$, then Lemma \[lemma:remark:toseparate:ends\](3) shows that $\varphi (x) \neq \varphi (y)$.
- If $y \in (\tau_{\Gamma_{\min}} (P_i) , P_i)$, then Lemma \[lemma:remark:toseparate:ends\](1) shows that $\varphi (x) \neq \varphi (y)$.
- Suppose that $y \notin (\tau_{\Gamma_{\min}} (P_i) , P_i)$ and $y \notin \Gamma_{\min}$. Then by , there exists $j \neq i$ with $y \in (\tau_{\Gamma_{\min}} (P_j) , P_j ) \setminus (\tau_{\Gamma_{\min}} (P_i) , P_i)$. If $x \not\in (\tau_{\Gamma_{\min}} (P_j) , P_j )$, then Lemma \[lemma:remark:toseparate:ends\](2) shows that $\varphi (x) \neq \varphi (y)$. If $x \in (\tau_{\Gamma_{\min}} (P_j) , P_j )$, then Lemma \[lemma:remark:toseparate:ends\](1) applied to $\varphi^{(j)}$ shows that $\varphi (x) \neq \varphi (y)$.
Thus $\varphi$ is injective on $\Gamma$, so that $\varphi$ is a faithful tropicalization of $\Gamma$. This completes the proof of Theorem \[thm:main:unimodular:faithful\].
500
Complementary results {#sec:complementary}
=====================
Theorem \[thm:main:unimodular:faithful\] is optimal for curves in low genera
----------------------------------------------------------------------------
In this subsection, we prove Theorem \[thm:best:possible\], which shows that the bounds in Theorem \[thm:main:unimodular:faithful\] for $g=0,1,2$ are optimal.
We begin with the case of genus $0$. In this case, we have the following obvious result.
\[prop:complement:0\] Let ${{\mathbb{P}}}^1$ be the projective line over $K$. Let $L$ be a line bundle over ${{\mathbb{P}}}^1$ of degree $0$, i.e., $L \cong {{\mathcal{O}}}_{{{\mathbb{P}}}^1}$. Then for any non-zero global sections $s_0 , \ldots , s_N$ of $L$, the image of the tropicalization map $\varphi : {{\mathbb{P}}}^{1,{{\operatorname{an}}}} \to {{\mathbb{T}}}{{\mathbb{P}}}^N$ defined by $\varphi = ( - \log |s_0| :
\cdots : - \log |s_N| )$ is a singleton.
In particular, if $\Gamma$ is a skeleton of ${{\mathbb{P}}}^{1, {{\operatorname{an}}}}$ that is not a singleton, then ${{\mathcal{O}}}_{{{\mathbb{P}}}^1}$ does not admit a faithful tropicalization of $\Gamma$.
Next we consider the case of genus $1$. Recall that a smooth projective curve $X$ over $K$ of genus $g \geq 1$ is called a *Mumford–Tate curve* if its minimal skeleton (as a $\Lambda$-metric graph) has genus $g$.
\[prop:complement:1\] Let $X$ be a Mumford–Tate smooth projective curve over $K$ of genus $1$. Let $L$ be a line bundle over $X$ of degree $2$. Then $L$ does not admit a faithful tropicalization of the minimal skeleton $\Gamma_{\min}$.
[[*Proof.*]{}]{}Let $s_0, \ldots, s_N \in H^0(L)$ be any non-zero global sections. It is classically known that the associated morphism $X \to {{\mathbb{P}}}^N$ decomposes into the composite of a double covering $X \to {{\mathbb{P}}}^{1}$ and a morphism ${{\mathbb{P}}}^1 \to {{\mathbb{P}}}^N$. This means that the tropicalization $\varphi :
X^{{{\operatorname{an}}}} \to {{\mathbb{T}}}{{\mathbb{P}}}^N$ factors though ${{\mathbb{P}}}^{1, {{\operatorname{an}}}}$. Since the image $\varphi ( \Gamma_{\min} )$ is a skeleton of ${{\mathbb{P}}}^{1,{{\operatorname{an}}}}$, we see that the first Betti number of $\varphi ( \Gamma_{\min} )$ is $0$. On the other hand, we have $g ( \Gamma_{\min}) = 1$. This shows that $\Gamma_{\min}$ is not homeomorphic to $\varphi ( \Gamma_{\min} )$. Thus $\varphi$ is not a faithful tropicalization of $\Gamma_{\min}$.
500
Finally we show that a bicanonical system does not necessarily give a faithful tropicalization of a skeleton when $g=2$. We remark that the bicanonical divisor has degree $4 = t (2) - 1$ with the notation in Theorem \[thm:main:unimodular:faithful\].
\[prop:complement:2\] Let $X$ be a Mumford–Tate smooth projective curve over $K$ of genus $2$. Then the bicanonical line bundle $\omega_X^{\otimes 2}$ does not admit a faithful tropicalization of the minimal skeleton $\Gamma_{\min}$.
[[*Proof.*]{}]{}For any global sections $s_0, \ldots, s_N \in H^0(\omega_X^{\otimes 2})$, the associated morphism $
X \to {{\mathbb{P}}}^N
$ decomposes into the composite of the natural double covering $X \to {{\mathbb{P}}}^{1}$ given by the quotient of the hyperelliptic involution and a morphism ${{\mathbb{P}}}^1 \to {{\mathbb{P}}}^N$. This means that the tropicalization $\varphi :
X^{{{\operatorname{an}}}} \to {{\mathbb{T}}}{{\mathbb{P}}}^N$ factors through ${{\mathbb{P}}}^{1, {{\operatorname{an}}}}$. Then, by the same argument as in the proof of Proposition \[prop:complement:1\], we see that $\varphi$ is not a faithful tropicalization of $\Gamma_{\min}$.
500
A very ample line bundle that does not admits a faithful tropicalization {#subsec:very:ample:no:ft}
------------------------------------------------------------------------
In this subsection, we prove Proposition \[prop:countereg:very:ample\], which shows that a [*very ample*]{} line bundle does not necessarily admit a faithful tropicalization. Here we recall the assertion.
Let $d \geq 4$ be any integer. Then there exists a smooth projective plane curve $X$ of ${{\mathbb{P}}}^2$ of degree $d$ over $K$ such that no skeleton $\Gamma$ of $X^{{{\operatorname{an}}}}$ has a faithful tropicalization associated to $|{{\mathcal{O}}}_X(1)|$, where ${{\mathcal{O}}}_X(1)$ is the very ample line bundle over $X$ given by the restriction of ${{\mathcal{O}}}_{{{\mathbb{P}}}^2}(1)$.
[[*Proof.*]{}]{}We take any positive integers $d_1\geq 2$ and $d_2 \geq 2$ with $d_1 + d_2 = d$. For $i = 1, 2$, we take a homogeneous polynomial $f_{i} \in k[T_0, T_1, T_2]$ of degree $d_i$ such that the curve $C_i$ defined by $f_i$ in ${{\mathbb{P}}}^2_k$ is smooth and such that $C_1 \cup C_2$ is a strictly semistable curve over $k$.
First, we construct a plane curve $X$. We take any $\widetilde{F}_i \in R[T_0, T_1, T_2]$ which maps to $f_i$ under the natural surjection $R [T_0, T_1, T_2] \to k [T_0, T_1, T_2]$ arising from $R \to R/\mathfrak{m} = k$. We take a homogeneous polynomial $\widetilde{F} \in R[T_0, T_1, T_2]$ of degree $d$ and a nonzero element $\varpi \in R$ with $|\varpi|_K < 1$, and we set $$\widetilde{G} := \widetilde{F}_1 \widetilde{F}_2 + \varpi \widetilde{F} \in R[T_0, T_1, T_2].$$ When we regard $\widetilde{G}$ as an element of $K[T_0, T_1, T_2]$, we denote it by $G$. We set $$X := {\operatorname{\operatorname{Proj}}}(K[T_0, T_1, T_2]/(G)).$$ By taking $\widetilde{F}$ generally, we may and do assume that $X$ is smooth.
Let $\Gamma$ be any skeleton of $X^{{{\operatorname{an}}}}$. To ease notation, we set $L := {\left.{{{\mathcal{O}}}_{{{\mathbb{P}}}^2}(1)}\right\vert_{{X}}}$. We take any $N \geq 1$ and any nonzero global sections $s_0, \ldots, s_N \in H^0(L)$, and we consider the associated map $$\varphi: X^{{{\operatorname{an}}}} \to {{\mathbb{T}}}{{\mathbb{P}}}^N, \quad
p = (p, |\cdot|) \mapsto \left(-\log|s_0(p)|: \cdots: -\log|s_N(p)|\right).$$ Then our goal is to show that $\varphi$ is not a faithful tropicalization of $\Gamma$.
We set $${{\mathscr{X}}}:= {\operatorname{\operatorname{Proj}}}(R[T_0, T_1, T_2]/(\widetilde{G}))
,$$ which is naturally a model of $X$. By the definition of $\widetilde{G}$, the image of $\widetilde{G}$ by the natural homomorphism $R [ T_0 , T_1 , T_2] \to k [ T_0 , T_1 , T_2]$ equals $f_1 f_2$. This means that the special fiber $\mathscr{X}_s$ equals the strictly semistable semistable curve $C_1 \cup C_2$, and thus $\mathscr{X}$ is a strictly semistable model of $X$. For each $i = 1, 2$, since $\#(C_1 \cap C_2) = d_1 d_2 > 2$, $C_i$ is not a $(-1)$-curve nor a $(-2)$-curve in $\mathscr{X}_s$. Thus $\mathscr{X}$ is a Deligne–Mumford strictly semistable model of $X$. We remark that $X$ has genus $\frac{(d-1)(d-2)}{2} \geq 3$. Let $\Gamma_{\min}$ be the minimal skeleton of $X^{{{\operatorname{an}}}}$. Then for each $i=1,2$, the Shilov point $[C_i] \in X^{{{\operatorname{an}}}}$ associated to $C_i$ belongs to $\Gamma_{\min}$. Since $\Gamma_{\min} \subset \Gamma$, we note that $[C_1] , [C_2] \in \Gamma$.
Now, it suffices to show that $\varphi([C_1]) = \varphi([C_2])$. We set ${{\mathscr{L}}}:= {\left.{{{\mathcal{O}}}_{{{\mathbb{P}}}_R^2}(1)}\right\vert_{{{{\mathscr{X}}}}}}$. Then $({{\mathscr{X}}}, {{\mathscr{L}}})$ is a model of $(X, L)$. By adding the global section ${\left.{T_0}\right\vert_{{X}}} \in H^0(L)$, we may and do assume that $s_0 = {\left.{T_0}\right\vert_{{X}}}$. We set $\widetilde{s}_0 := {\left.{T_0}\right\vert_{{{{\mathscr{X}}}}}} \in H^0({{\mathscr{L}}})$. It follows from the exact sequence $$0 \to {{\mathcal{O}}}_{{{\mathbb{P}}}^2}(-d+1) \to {{\mathcal{O}}}_{{{\mathbb{P}}}^2}(1) \to L \to 0$$ and $H^1({{\mathcal{O}}}_{{{\mathbb{P}}}^2}(-d+1)) = 0$ that the restriction map $H^0({{\mathcal{O}}}_{{{\mathbb{P}}}^2}(1)) \to H^0(L)$ is surjective. Thus for any $1 \leq \ell \leq N$, there exists $(a_{\ell}, b_{\ell}, c_{\ell}) \neq (0, 0, 0) \in K^3$ such that $s_{\ell} = {\left.{\left(a_{\ell} T_0 + b_{\ell} T_1 + c_{\ell} T_2\right)}\right\vert_{{X}}}$. We take $\alpha_{\ell} \in K$ such that $a_{\ell}^\prime := a_{\ell}/\alpha_{\ell}, b_{\ell}^\prime := b_{\ell}/\alpha_{\ell}, c_{\ell}^\prime := c_{\ell}/\alpha_{\ell}$ belong to $R$ and such that one of $a_{\ell}^\prime, b_{\ell}^\prime, c_{\ell}^\prime$ is a unit of $R$. We set $\widetilde{s}_{\ell} = {\left.{\left(a_{\ell}^\prime T_0 + b_{\ell}^\prime T_1 + c_{\ell}^\prime T_2\right)}\right\vert_{{{{\mathscr{X}}}}}} \in H^0({{\mathscr{L}}})$ and $g_{\ell}:= \widetilde{s}_{\ell}/\widetilde{s}_0$. Then $g_{\ell}$ is a nonzero rational function on ${{\mathscr{X}}}$.
Recall that $C_1$ is a curve in ${{\mathbb{P}}}^2_k$ of degree $d_1 \geq 2$, and in particular it is not linear. Since ${\left.{(a_{\ell}^\prime T_0 + b_{\ell}^\prime T_1 + c_{\ell}^\prime T_2)}\right\vert_{{{{\mathbb{P}}}^2_k}}} \neq 0$, it follows that ${\left.{(a_{\ell}^\prime T_0 + b_{\ell}^\prime T_1 + c_{\ell}^\prime T_2)}\right\vert_{{C_1}}} \not\equiv 0$. This means that $\widetilde{s}_{\ell}$ does not vanish at the generic point $\xi_1$ of $C_1$. By the same argument, $\widetilde{s}_{0}$ does not vanish at $\xi_1$. Thus $g_{\ell}$ is a unit regular function of ${{\mathcal{O}}}_{{{\mathscr{X}}}, \xi_1}$, and we have $-\log|g_{\ell}|([C_1]) = 0$. It follows that $$\varphi([C_1]) = (0: -\log |\alpha_{1}|_K: \cdots: -\log |\alpha_{N}|_K).$$ By the same argument, we obtain $\varphi([C_2]) = (0: -\log |\alpha_{1}|_K: \cdots: -\log |\alpha_{N}|_K)$. This proves that $\varphi([C_1]) = \varphi([C_2])$, and thus this completes the proof.
500
Comparison with [@KY] {#subsection:comparison}
---------------------
In this subsection, we clarify the difference between the work in [@KY] and the current work with an emphasis of an advantage of Theorem \[thm:main:unimodular:faithful\].
The paper [@KY] discusses faithful tropicalization of projective varieties under some conditions of definability over discrete valuation rings (of equicharacteristic zero, cf. Remark \[remark:equicharacteristic\]). It has an obvious advantage of treating smooth projective varieties of arbitrary dimension. However, we will find that the current paper has an advantage when we restrict our attentions to curves, besides the definability assumption over discrete valuation rings.
First, let us recall what the result in [@KY] says for curves. The following assertion has essentially been proved in [@KY].
\[thm:KY\] Assume that $R$ has equicharacteristic zero cf. Remark . Let $X$ be a smooth projective curve over $K$, and let $\Gamma$ be a skeleton of $X^{{{\operatorname{an}}}}$. Let $L$ be a line bundle over $X$. Assume that there exists a discrete valuation ring $R'$ dominated by $R$ and a strictly semistable curve $\mathscr{X}' \to {{\operatorname{Spec}}}(R')$ such that $\Gamma$ is associated to the model $\mathscr{X}' \otimes_{R'} R$. Suppose that there exist a relatively ample line bundle $\mathscr{N}$ over $\mathscr{X}'$ and an integer $m \geq 2$ such that ${\left.{\mathscr{N}^{\otimes m} \otimes \omega_{\mathscr{X}^\prime/R^\prime}}\right\vert_{{X}}}
\cong L$. Then $L$ admits a faithful tropicalization of $\Gamma$.
[[*Proof.*]{}]{}We briefly explain how [@KY Theorem 1.1] leads to Theorem \[thm:KY\]. Let $K'$ be the fraction field of $R'$. We put an absolute value $|\cdot|_{K'}$ on $K'$ by restricting the absolute value $|\cdot|_K$ on $K$. Set $X' := \mathscr{X}' \otimes_{R'} K'$. Then the Berkovich analytic space $(X')^{{{\operatorname{an}}}}$ over $K'$ and the skeleton $S(\mathscr{X}')$ associated to a strictly semistable model $\mathscr{X}' \to {{\operatorname{Spec}}}(R^\prime)$ are defined similarly. Let $\alpha : X \to X'$ be the natural morphism, and the canonical map $\alpha^{{{\operatorname{an}}}} : X^{{{\operatorname{an}}}} \to (X')^{{{\operatorname{an}}}}$ induces an isometry $\Gamma = S ( \mathscr{X}' \otimes_{R'} R ) \cong S ( \mathscr{X}' )$.
Set $L' := {\left.{\mathscr{N}^{\otimes m} \otimes \omega_{\mathscr{X}'/R'}}\right\vert_{{X'}}}$. By [@KY Theorem 1.1], there exist global sections $s_0 , \ldots , s_N$ of $L'$ such that the associated morphism $\psi : (X')^{{{\operatorname{an}}}} \to {{\mathbb{T}}}{{\mathbb{P}}}^N$ gives a faithful tropicalization of $S ( \mathscr{X}')$. Since $\alpha^{\ast} \left(
L'
\right) = L$, we regard $s_0 , \ldots , s_N$ as global sections of $L$, and we also have the associated tropicalization map $\varphi : X^{{{\operatorname{an}}}} \to {{\mathbb{T}}}{{\mathbb{P}}}^N$. Further, we have $\varphi = \psi \circ \alpha^{{{\operatorname{an}}}}$. Since $\alpha^{{{\operatorname{an}}}}$ gives the isometry between the skeleta, this proves that $\varphi$ also gives a faithful tropicalization of $\Gamma$.
500
\[remark:equicharacteristic\] In fact, the assumption in Theorem \[thm:KY\] that $R$ has equicharacteristic zero is unnecessary, as we now explain. Suppose that $R$ has positive residue characteristic, that is, $\mathrm{char} (k) > 0$. Note that for any smooth projective curve $Z$ over any field $k$ and for any ample line bundle $N$ over $Z$, $N^{\otimes m} \otimes \omega_Z$ is basepoint free for any $m \geq 2$. This means the assumption “$m \geq \phi (d)$" in [@KY Theorem 1.1] is fulfilled. Further, for curves, the assertions of vanishing of cohomologies and basepoint-freeness as in [@KY §3], which are the technical keys to prove [@KY Theorem 1.1], hold true even in positive characteristic. Then one can construct global sections of ${\left.{\mathscr{N}^{\otimes m} \otimes \omega_{\mathscr{X}'/R'}}\right\vert_{{X}}}$ that give a faithful tropicalization of $\Gamma$ by essentially the same arguments in [@KY]. Thus the same conclusion of Theorem \[thm:KY\] holds true without the assumption that $R$ has equicharacteristic zero.
To clarify the crucial difference between the results of the two papers, we revisit Question \[question:first\]. Recall that Theorem \[thm:main:unimodular:faithful\] gives us a concrete answer to Question \[question:first\] that $d (g) = t (g)$, where $t (g)$ is given in Theorem \[thm:main:unimodular:faithful\]. Now, we want to ask whether or not Theorem \[thm:KY\] makes some contribution to Question \[question:first\]. The following proposition is crucial to this issue.
For a smooth projective curve $X$ over $K$ and a complete discrete valuation ring $R^\prime$ dominated by $R$, we say that $X$ is *definable over $R'$* if there exists a proper curve $\mathscr{X} \to {{\operatorname{Spec}}}(R')$ over $R'$ such that $\mathscr{X} \otimes_{R'} R$ is a model of $X$.
\[prop:cannotbecoverd\] Let $g$ and $d$ be any positive integers with $g \geq 2$. Then there exists a smooth projective curve $X$ over $K$ of genus $g$ that is definable over some complete discrete valuation ring dominated by $R$ and has the following property for any complete discrete valuation ring $R'$ dominated by $R$, for any Deligne–Mumford strictly semistable model $\mathscr{X}^\prime \to {{\operatorname{Spec}}}(R')$ of $X$ such that $\mathscr{X}^\prime$ is regular, and for any line bundle $\mathscr{N}$ over $\mathscr{X}^\prime$, if $\mathscr{N}$ is relatively ample, then $\deg \left( {\left.{\mathscr{N}}\right\vert_{{X}}} \right) \geq d$.
[[*Proof.*]{}]{}Note that for each prime number $p$, [@matsumura Theorem 29.1] gives a complete discrete valuation ring $W_p$ with residue field $k$ such that $p$ is a uniformizer in $W_p$. Set $$\begin{aligned}
V :=
\begin{cases}
k [[ t ]] &
\text{if $R$ has equicharacteristic,}
\\
W_p
&
\text{if $R$ has mixed characteristic and $\mathrm{char} (k) = p$.}
\end{cases}\end{aligned}$$ Then $V$ is a discrete valuation ring with residue field $k$. Let $\varpi$ be a uniformizer of $V$. Then there exists an injective homomorphism $V \to R$ of local rings that induces the identity between the residue fields $k$; see [@matsumura Theorem 29.2] when $R$ has mixed characteristic.
Since $g \geq 2$, there exists a stable curve $D$ over $V / ( \varpi) = k$ of genus $g$ that has at least two nodes, which we denote by $q_1$ and $q_2$. Using the deformation theory of stable curves, we obtain a generically smooth, strictly semistable and stable curve $\mathscr{X}^0 \to {{\operatorname{Spec}}}(V)$ of genus $g$ with special fiber $D$ such that the completion of the local ring of $\mathscr{X}^0$ at $q_1$ is isomorphic to $V [[x,y]] / (xy - \varpi)$ and that at $q_2$ is isomorphic to $V [[x,y]]/ (xy - \varpi^{d})$. We set $X := \mathscr{X}^0 \otimes_{V} K$. Then $X$ is a smooth projective curve over $K$ of genus $g$. We remark that $\mathscr{X}^{{\operatorname{st}}} := \mathscr{X}^0 \otimes_{V} R$ is the stable model of $X$ over $R$. Let $\Gamma_{\min}$ denote the minimal skeleton of $X^{{{\operatorname{an}}}}$. For $i=1,2$, let $e_i \in E({{\mathscr{X}}}^{{\operatorname{st}}})$ be the edge of $\Gamma_{\min}$ corresponding to $q_i$, and let $\lambda_i$ denote the length of $e_i$. Then $\lambda_2 = d \lambda_1$.
Let us prove that $X$ has the required properties. We take any complete discrete valuation ring $R'$ with a Deligne–Mumford strictly semistable model $\mathscr{X}^\prime \to {{\operatorname{Spec}}}(R')$ of $X$ such that $\mathscr{X}^\prime$ is regular. Let $\mathscr{X}^{\prime} \to \mathscr{X}^{\prime {\operatorname{st}}}$ be the contraction of $(-2)$-curves, where $\mathscr{X}^{\prime {\operatorname{st}}}$ is the stable model of $\mathscr{X}^{\prime}$. Since $\mathscr{X}^{\prime {\operatorname{st}}} \otimes_{R'} R$ is a stable curve with geometric generic fiber $X$, we have $\mathscr{X}^{\prime {\operatorname{st}}} \otimes_{R'} R = \mathscr{X}^{{\operatorname{st}}}$ by the uniqueness of the stable model. In particular, $\mathscr{X}^{\prime {\operatorname{st}}} \otimes_{R'} k = D$, where $k$ is regarded as an $R'$-algebra by the composite $R' \to R \to R/ \mathfrak{m} = k$. The morphism $\mathscr{X}^{\prime} \to \mathscr{X}^{\prime {\operatorname{st}}}$ induces a morphism $\alpha :
\mathscr{X}^{\prime} \otimes_{R'} k \to D$, which is the contraction of $(-2)$-curves in the Deligne–Mumford semistable curve $\mathscr{X}^{\prime} \otimes_{R'} k$ over $k$. For $i= 1,2$, let $n_i$ denote the number of nodes that $\alpha$ maps to $q_i$. Let $\varpi'$ denote the uniformizer of $R'$. Since $\mathscr{X}^\prime$ is strictly semistable and regular, we note that $n_i ( - \log | \varpi' |_K) =
\lambda_i$ for $i = 1,2$. Since $\lambda_2 = d \lambda_1$, it follows that $n_2 = d n_1 \geq d$.
We claim that $\# {\operatorname{\operatorname{Irr}}}( \mathscr{X}^{\prime} \otimes_{R'} k ) \geq d$. Since the assertion is trivial if $d = 1$, we may and do assume that $d \geq 2$. Then we have $n_2 \geq 2$, and hence there exists a maximal $(-2)$-chain $E$ in $\mathscr{X}^{\prime} \otimes_{R'} k$ such that $\Delta_{E} = e_2$. Note that $\# {\operatorname{\operatorname{Irr}}}(E) = n_2 -1 \geq d -1$. Since $\mathscr{X}^{\prime} \otimes_{R'} k$ has an irreducible components that is not a $(-2)$-curve, it follows that $
\# {\operatorname{\operatorname{Irr}}}( \mathscr{X}^{\prime} \otimes_{R'} k )
\geq
\# {\operatorname{\operatorname{Irr}}}( E ) + 1
\geq
d
$, as desired.
Now let $\mathscr{N}$ be any line bundle over $\mathscr{X}^\prime$ that is relatively ample. Then, since $\mathscr{X}^{\prime} \otimes_{R'} k$ has at least $d$ irreducible components, we have $\deg
\left(
{\left.{\mathscr{N}}\right\vert_{{X}}}
\right)
=
\deg \left(
{\left.{\mathscr{N}}\right\vert_{{\mathscr{X}^{\prime} \otimes_{R'} k}}}
\right) \geq d$. Thus the proposition holds.
500
Suppose that a smooth projective curve $X$ over $K$ is definable over a complete discrete valuation ring $R^\prime$ dominated by $R$, and we take a proper curve $\mathscr{X} \to {{\operatorname{Spec}}}(R')$ over $R'$ such that $\mathscr{X} \otimes_{R'} R$ is a model of $X$. Let $L$ be a line bundle over $X$, and let $\Gamma$ be a skeleton of $X^{{{\operatorname{an}}}}$. We say that $L$ is *definable over $R'$* if there exists a line bundle $\mathscr{L}$ over $\mathscr{X}$ such that $\mathscr{L} \otimes_{R'} R$ is a model of $L$. Further, we say that $\Gamma$ is *definable over $R'$* if there exists a strictly semistable curve $\mathscr{X} \to {{\operatorname{Spec}}}(R')$ over $R'$ such that $\mathscr{X} \otimes_{R'} R$ is a model of $X$ and such that $\Gamma$ is the skeleton associated to this model.
We fix any integer $g \geq 2$. Take any integer $s \geq 2g - 2$. We are going to show that there exist a smooth projective curve $X$ over $K$ of genus $g$ and a line bundle $L$ over $X$ with $\deg (L) = s$ such that $X$, $L$, and the minimal skeleton of $X$ are definable over some complete discrete valuation ring dominated by $R$, but such that there do [*not*]{} exist a complete discrete valuation ring $R'$ dominated by $R$, a Deligne–Mumford strictly semistable curve $\mathscr{X}' \to {{\operatorname{Spec}}}(R')$ with $\mathscr{X}' \otimes_{R'} R = X$, a relatively ample line bundle over $\mathscr{N}$ over $\mathscr{X}'$, and an integer $m \geq 2$ such that ${\left.{\mathscr{N}^{\otimes m} \otimes \omega_{\mathscr{X}'/R'}}\right\vert_{{X}}}
\cong L$. Indeed, suppose that this is proved. Then, the assumptions of Theorem \[thm:KY\] are not fulfilled. Since $s \geq 2g-2$ is taken arbitrarily, this suggests that the existence of a universal bound $d (g)$ as in Question \[question:first\] cannot be deduced from Theorem \[thm:KY\], so that Theorem \[thm:KY\] cannot contribute to Question \[question:first\].
The construction is done by Proposition \[prop:cannotbecoverd\]. Indeed, applying this proposition for $d := \lceil
(s- 2g + 2)/2
\rceil + 1$, where for a real number $x$, $\lceil x \rceil$ denotes the smallest integer with $\lceil x \rceil \geq x$, we get a smooth projective curve $X$ over $K$ of genus $g$ that is definable over some complete discrete valuation ring dominated by $R$ and enjoys the properties in the proposition. We note that the minimal skeleton of $X$ is also definable over the complete valuation ring. Since $X$ is definable over some complete discrete valuation ring dominated by $R$, there exists a line bundle $L$ over $X$ with $\deg (L) = s$ that is definable over some complete discrete valuation ring dominated by $R$.
However, for any complete discrete valuation ring $R'$ dominated by $R$, for any Deligne–Mumford strictly semistable curve $\mathscr{X}' \to {{\operatorname{Spec}}}(R')$ with $\mathscr{X}'
\otimes_{R'} R = X$, for any relatively ample line bundle $\mathscr{N}$ over $\mathscr{X}'$, and for any integer $m \geq 2$, we have ${\left.{\mathscr{N}^{\otimes m} \otimes \omega_{\mathscr{X}'/R'}}\right\vert_{{X}}}
\ncong L$. Indeed, for relatively ample $\mathscr{N}$, since $X$ enjoys the properties of Proposition \[prop:cannotbecoverd\], we have $$\deg ( {\left.{\mathscr{N}}\right\vert_{{X}}})
\geq d > (s- 2g + 2)/2
,$$ and hence $$\deg \left(
{\left.{\mathscr{N}^{\otimes m} \otimes \omega_{\mathscr{X}'/R'}}\right\vert_{{X}}}
\right)
>
m(s- 2g + 2)/2 + 2g - 2
\geq s
= \deg (L)
.$$ Thus we get the conclusion.
Limit of tropicalizations by polynomials of a bounded degree {#sec:limit:trop}
============================================================
Let $\mathbb{A}^N $ be the $N$-dimensional affine space with affine coordinate functions $z_1 , \ldots , z_N$, and let $Y \subset \mathbb{A}^N$ be a closed subvariety. In [@payne], Payne considers the inverse system consisting of all affine embeddings of $Y$ given by polynomials in $z_1, \ldots, z_N$ and shows that the inverse limit is homeomorphic to $Y^{{{\operatorname{an}}}}$. In his construction, there are no restrictions for affine embeddings of $Y$.
In this section, we consider a question whether $Y^{{{\operatorname{an}}}}$ is homeomorphic to the inverse limit of the affine embeddings given by polynomials of degree at most some effective bound. When $Y$ is a suitable affine curve, we will answer this question in the affirmative, as an application of Theorem \[thm:main:unimodular:faithful\].
Statement of the result {#subsec:limit:trop:statement}
-----------------------
Let us briefly recall Payne’s construction of limit tropicalizations. For any $n \geq 1$, let $\mathbb{A}^n := {{\operatorname{Spec}}}( K [x_1 , \ldots , x_n] )$ denote the $n$-dimensional affine space with affine coordinate functions $x_1 , \ldots , x_n$, on which the algebraic torus ${{\mathbb{G}}}_m^n :=
{{\operatorname{Spec}}}\left(
K [x_1^{\pm 1} , \ldots , x_n^{\pm 1}]
\right)$ acts naturally. Using the standard coordinates of $\mathbb{A}^n$, we have a map $$\mathbb{A}^{n} (K) \to {{\mathbb{T}}}^n, \quad
p \mapsto (- \log |x_1 (p)|_K , \ldots , - \log |x_n (p)|_K ),$$ which extends to a map $${\operatorname{Trop}}\colon
\mathbb{A}^{n,{{\operatorname{an}}}} \to {{\mathbb{T}}}^n, \quad
p \mapsto (- \log |x_1 (p)| , \ldots , - \log |x_n (p)| )
,$$ called the standard tropicalization map. On ${{\mathbb{T}}}^n$ acts ${{\mathbb{R}}}^n = {\operatorname{Trop}}({{\mathbb{G}}}_m^{n, {{\operatorname{an}}}})$ naturally, which is compatible with the map ${\operatorname{Trop}}$.
By an [*equivariant morphism*]{} $\varphi : \mathbb{A}^{m}
\to \mathbb{A}^{n}$, we mean a morphism that is equivariant by the torus action with respect to some group homomorphism ${{\mathbb{G}}}_m^m \to {{\mathbb{G}}}_m^n$. Then $\varphi$ induces a well-defined continuous map ${\operatorname{Trop}}(\varphi)\colon {{\mathbb{T}}}^m \to {{\mathbb{T}}}^n$ that is compatible with the tropicalization maps, i.e., ${\operatorname{Trop}}\circ \varphi = {\operatorname{Trop}}(\varphi) \circ {\operatorname{Trop}}$. We call ${\operatorname{Trop}}(\varphi)$ the tropicalization of the map $\varphi$.
A closed embedding $\iota: Y \hookrightarrow \mathbb{A}^{n}$ is called an *affine embedding*. Let $I$ denote the set of all affine embeddings of $Y$. Then $I$ has a structure of directed set as follows. Let $\iota_1 : Y \hookrightarrow \mathbb{A}^{n_1}$ and $\iota_2 : Y \hookrightarrow \mathbb{A}^{n_2}$ belong to $I$. Then we declare that $\iota_1 \leq \iota_2$ if there exists an equivariant morphism $\varphi : \mathbb{A}^{n_2} \to \mathbb{A}^{n_1}$ with $\varphi \circ \iota_2 = \iota_1$. Furthermore, for $\iota_1 : Y \hookrightarrow \mathbb{A}^{n_1}$ and $\iota_2 : Y \hookrightarrow \mathbb{A}^{n_2}$, we set $\iota_3 = (\iota_1, \iota_2): Y \hookrightarrow \mathbb{A}^{n_1+n_2}$. Since the natural projections ${\rm pr}_1\colon \mathbb{A}^{n_1+n_2} \to \mathbb{A}^{n_1}$ and ${\rm pr}_2\colon \mathbb{A}^{n_1+n_2} \to \mathbb{A}^{n_2}$ satisfy ${\rm pr}_1 \circ \iota_3 = \iota_1$ and ${\rm pr}_2 \circ \iota_3 = \iota_2$, we have $\iota_1 \leq \iota_3$ and $\iota_2 \leq \iota_3$.
For an affine embedding $\iota: Y \hookrightarrow \mathbb{A}^{n}$, we define the *tropicalization ${\operatorname{Trop}}(Y,\iota)$ of $Y$ with respect to $\iota$* to be the closure $\overline{
\{{\operatorname{Trop}}(\iota(y)) \mid y \in Y (K) \}}
$ of $\iota (Y (K) )$ in ${{\mathbb{T}}}^n$. By [@payne Proposition 2.2], $
{\operatorname{Trop}}(Y,\iota) =
{\operatorname{Trop}}(\iota^{{{\operatorname{an}}}} ( Y^{{{\operatorname{an}}}} ))$, where $\iota^{{{\operatorname{an}}}} : Y^{{{\operatorname{an}}}} \hookrightarrow \mathbb{A}^{n, {{\operatorname{an}}}}$ is the map between the Berkovich spaces associated to $\iota$. Let $$\pi_\iota : Y^{{{\operatorname{an}}}} \to {\operatorname{Trop}}(Y,\iota)$$ denote the restriction of ${\operatorname{Trop}}\circ \iota^{{{\operatorname{an}}}}$, which is surjective and continuous. Let $\iota_1 : Y \hookrightarrow \mathbb{A}^{n_1}$ and $\iota_2 : Y \hookrightarrow \mathbb{A}^{n_2} $ be elements of $I$. Suppose that $\iota_1 \geq \iota_2$. Then by the definition, there exists an equivariant morphism $\varphi : \mathbb{A}^{n_2} \to \mathbb{A}^{n_1}$ such that $\iota_1 = \varphi \circ \iota_2$. The map ${\operatorname{Trop}}(\varphi): {{\mathbb{T}}}^{n_2} \to {{\mathbb{T}}}^{n_1}$ induces the map ${\operatorname{Trop}}(Y,\iota_2) \to {\operatorname{Trop}}(Y,\iota_1)$ by restriction. Remark that ${\operatorname{Trop}}(Y,\iota_2) \to {\operatorname{Trop}}(Y,\iota_1)$ is determined only by $\iota_1$ and $\iota_2$, and is independent of the choice of $\varphi$.
To sum up, $\left( {\operatorname{Trop}}(Y, \iota ) \right)_{\iota \in I}$ constitutes an inverse system in the category of topological spaces, and we have the inverse limit $\varprojlim_{\iota \in I} {\operatorname{Trop}}(Y, \iota)$. Let $$\label{eqn:limit:tropical:map}
\mathrm{LTrop} : Y^{{{\operatorname{an}}}} \to \varprojlim_{\iota \in I} {\operatorname{Trop}}(Y, \iota)$$ to be the continuous map induced from $( \pi_{\iota} )_{\iota \in I}$ by the universal property of the inverse limit. Then Payne proves the following theorem.
\[thm:limit:trop\] Analytification is the limit of all tropicalizations, i.e., the map $\mathrm{LTrop}$ in is a homeomorphism.
Since ${\operatorname{Trop}}(Y,\iota)$ equals the closure $\overline{
\{{\operatorname{Trop}}(\iota(y)) \mid y \in Y (K) \}}
$ of $\iota (Y (K) )$ in ${{\mathbb{T}}}^n$, Theorem \[thm:limit:trop\] suggests that one can describe $Y^{{{\operatorname{an}}}}$ in terms of tropical geometry even without knowing the definition of the Berkovich space $Y^{{{\operatorname{an}}}}$ associated to $Y$.
Our aim here is to prove that $Y^{{{\operatorname{an}}}}$ is still homeomorphic to the inverse limit of tropicalizations of an [*effectively bounded degree*]{}. Recall that $Y$ is a closed subvariety of $\mathbb{A}^N = {{\operatorname{Spec}}}(K[z_1, \ldots, z_N])$. We say that an affine embedding $\iota : Y \hookrightarrow \mathbb{A}^n$ has *degree at most $D$* if there exist $f_1 , \ldots , f_n \in K [z_1 , \ldots , z_N]$ such that $\deg (f_j) \leq D$ for $j = 1, \ldots, n$ and $\iota = \left(
{\left.{f_1}\right\vert_{{Y}}}, \ldots, {\left.{f_n}\right\vert_{{Y}}} \right)$. Let $I_{\leq D}$ denote the set of all affine embeddings of $Y$ with degree at most $D$. Then since $I_{\leq D} \subset I$, it is an ordered set, and it is actually a directed set. Indeed, for $\iota_1 : Y \hookrightarrow \mathbb{A}^{n_1}$ and $\iota_2 : Y \hookrightarrow \mathbb{A}^{n_2}$, the map $(\iota_1, \iota_2): Y \hookrightarrow \mathbb{A}^{n_1 + n_2}$ belongs $I_{\leq D}$ and satisfies $\iota_1 \leq (\iota_1, \iota_2)$ and $\iota_2 \leq (\iota_1, \iota_2)$. Thus $\left( {\operatorname{Trop}}(Y, \iota ) \right)_{\iota \in I_{\leq D}}$ again constitutes an inverse system in the category of topological spaces, and we have the inverse limit $\varprojlim_{\iota \in I_{\leq D}} {\operatorname{Trop}}(Y, \iota)$. We denote by $$\label{eqn:limit:tropical:map:d}
\mathrm{LTrop}_{\leq D} : Y^{{{\operatorname{an}}}} \to \varprojlim_{\iota \in I_{\leq D}} {\operatorname{Trop}}(Y, \iota)$$ the continuous map induced from $( \pi_{\iota} )_{\iota \in I_{\leq D}}$ by the universality of the inverse limit.
We regard $\mathbb{A}^N$ as an open subset of ${{\mathbb{P}}}^N$. Let $X$ be the closure of $Y$ in ${{\mathbb{P}}}^N$. Then the degree of $X$ in ${{\mathbb{P}}}^N$ depend only on $Y$ and not on the choice of the open embedding $\mathbb{A}^N \hookrightarrow \mathbb{P}^N$. We call $d$ the *degree of $Y$*. Further, we say that *$Y$ has smooth compactification in ${{\mathbb{P}}}^N$* if $X$ is smooth. This notion is also well defined for $Y$.
The following is the main result of this section.
Let $Y \subset \mathbb{A}^N$ be a closed connected subvariety. Assume that $\dim (Y) = 1$ and $Y$ has smooth compactification. Let $d$ be the degree of $Y$. Set $$\label{eqn:thm:limit:trop:d}
D :=
\begin{cases}
\max
\left\{
\left\lceil
\frac{3d^2 - 9 d + 4}{2d}
\right\rceil
,
1
\right\}
&
\parbox{8cm}{\flushleft if
$N \leq 2$, or
if
$N \geq 3$ and
$Y$ is contained in
some affine plane in ${\mathbb{A}}^N$,}
\\
\max \{ d-2 , 1 \}
& \text{otherwise,}
\end{cases}$$ where for a real number $x$, $\lceil x \rceil$ denotes the smallest integer with $\lceil x \rceil \geq x$. Then the map $\mathrm{LTrop}_{\leq D}$ in is a homeomorphism.
In Theorem \[thm:limit:trop\], we take many polynomials $f_1, \ldots, f_n \in K[z_1, \ldots, z_N]$ (including a generating set for the coordinate ring of $Y$) and consider the embedding $\iota\colon Y \hookrightarrow {\mathbb{A}}^n$. Intuitively and roughly speaking, Theorem \[thm:limit:trop\] says that ${\operatorname{Trop}}(Y, \iota)$ approximates $Y^{{{\operatorname{an}}}}$ well, and if we take more and more polynomials $f_i$, then ${\operatorname{Trop}}(Y, \iota)$ approximates $Y^{{{\operatorname{an}}}}$ more and more. Theorem \[thm:limit:trop:d:intro\] asserts that under suitable conditions, we can do the same thing with only affine embeddings of degree at most $D$, which is effectively given in terms of the degree of $X$.
In the proof of the injectivity of $\mathrm{LTrop}_{\leq D}$, we will use Theorem \[thm:main:unimodular:faithful\] together with some results in §\[sec:ft:general:model\]. The proof will be given in the subsequent subsections.
\[eg:limit:trop:linear\] Suppose that $F \in K[Z_0, Z_1, Z_2]$ is a nonsingular irreducible homogeneous polynomial of degree $d = 3$. We put $f(z_1, z_2) := F(1, z_1, z_2) \in K[z_1, z_2]$ and let $Y$ be the curve in $\mathbb{A}^2 =
{{\operatorname{Spec}}}( K[z_1, z_2] )$ defined by $f$. Thus $Y$ is (an affine part of) an elliptic curve. In this case, $D :=
\max
\left\{
\left\lceil
\frac{3d^2 - 9 d + 4}{2d}
\right\rceil
,
1
\right\} = 1$, and Theorem \[thm:limit:trop:d:intro\] says that the map $$\label{eqn:limit:tropical:map:1}
\mathrm{LTrop}_{\leq 1}
\colon\;
Y^{{{\operatorname{an}}}} \to \varprojlim_{\iota \in I_{\leq 1}} {\operatorname{Trop}}(Y, \iota).$$ is a homeomorphism, namely, the analytification $Y^{{{\operatorname{an}}}}$ of $Y$ is the inverse limit of tropicalizations ${\operatorname{Trop}}(Y, \iota)$ of linear embeddings.
\[eg:limit:trop:linear:impossible\] In general, it is false that the analytification of an affine curve is the inverse limit of tropicalizations of [*linear*]{} embeddings, i.e., the map is not a homeomorphism in general. To see this, we take any $d \geq 4$ and take a homogeneous polynomial $F \in K[Z_0, Z_1, Z_2]$ as in the proof of Proposition \[prop:countereg:very:ample\] in §\[subsec:very:ample:no:ft\]. (Here we use $Z_0, Z_1, Z_2$ for the coordinate functions in place of $T_0, T_1, T_2$.) Let $X \subset {{\mathbb{P}}}^2$ be the connected smooth projective curve defined by $F$, and let $Y \subset {\mathbb{A}}^2$ be the connected smooth affine curve defined by $f(z_1, z_2) := F(1, z_1, z_2)$. We take $[C_1], [C_2] \in X^{{{\operatorname{an}}}}$ as in the proof of Proposition \[prop:countereg:very:ample\] in §\[subsec:very:ample:no:ft\]. Since $Y^{{{\operatorname{an}}}} \setminus Y(K) = X^{{{\operatorname{an}}}} \setminus X(K)$, we have $[C_1], [C_2] \in Y^{{{\operatorname{an}}}}$. For any $\iota \in I_{\leq 1}$, the proof of Proposition \[prop:countereg:very:ample\] shows that $\iota([C_1]) = \iota([C_2])$. Thus the images of $[C_1]$ and $[C_2]$ in $\varprojlim_{\iota \in I_{\leq 1}} {\operatorname{Trop}}(Y, \iota)$ coincide with each other, whence $\mathrm{LTrop}_{\leq 1}$ is not a homeomorphism.
\[eg:limit:trop:quad\] Let $X$ be a connected smooth non-hyperelliptic curve of genus $3$. Then the canonical map embeds $X$ as a degree $d = 4$ curve in ${{\mathbb{P}}}^2$. Let $Y$ be the restriction of $X$ to ${\mathbb{A}}^2$. Since $\lceil \frac{3d^2 - 9 d + 4}{2d} \rceil = 2$ when $d=4$, Theorem \[thm:limit:trop:d:intro\] says that the analytification $Y^{{{\operatorname{an}}}}$ is the limit of tropicalizations of quadratic embeddings.
It may be interesting to compare Examples \[eg:limit:trop:linear\], \[eg:limit:trop:linear:impossible\] and \[eg:limit:trop:quad\] with Cueto–Markwig [@CM], in which they study an algorithmic side of tropicalizations of plane curves, and with Wagner [@Wa].
Polynomial of bounded degree that separates two points
------------------------------------------------------
In order to show that $\mathrm{LTrop}_{\leq D}$ is injective, we need to find, for any distinct $x , y \in
Y^{{{\operatorname{an}}}}$, a polynomial $f (z_1 , \ldots , z_N)$ of degree at most $D$ such that $- \log |f (x)| \neq - \log |f (y)|$. In this section, we construct such an $f$. The key to the construction is Proposition \[prop:key:injective\]. In the proof of this proposition, we use not only Theorem \[thm:main:unimodular:faithful\] but also some global sections constructed in §\[sec:ft:general:model\] together with the lemma below. Here, we recall that for any compact skeleton $\Gamma$ of $X^{{{\operatorname{an}}}}$, $\tau_{\Gamma}: X^{{{\operatorname{an}}}} \to \Gamma$ denotes the retraction map with respect to $\Gamma$.
\[lemma:construction:Gamma1\] Let $x$ and $y$ be distinct points in $X^{{{\operatorname{an}}}}$ such that $x$ does not belong to any compact skeleton. We fix a minimal skeleton $\Gamma_{\min}$ of $X^{{{\operatorname{an}}}}$. Then there exists a compact skeleton $\Gamma$ that contains $\Gamma_{\min}$ and satisfies the following conditions $$\begin{aligned}
& \Gamma = \Gamma_{\min} \cup
[ \tau_{\Gamma_{\min}} (x) , \tau_{\Gamma} (x) ], \quad
\Gamma_{\min} \cap [\tau_{\Gamma_{\min}} (x) , \tau_{\Gamma} (x)]
= \{ \tau_{\Gamma_{\min}} (x) \} \\
&
\tau_{\Gamma} (x) \not\in \Gamma_{\min}, \quad
\tau_{\Gamma} (x) \neq \tau_{\Gamma} (y).\end{aligned}$$
[[*Proof.*]{}]{}By [@BPR2 Theorem 5.2], it follows from $x \neq y$ that there exists a compact skeleton $\Gamma_0$ containing $\Gamma_{\min}$ such that $\tau_{\Gamma_0}(x) \neq \tau_{\Gamma_0}(y)$. Since $x$ does not belong to any compact skeleton, we have $\tau_{\Gamma_{0}} (x) \neq x$, and by [@BPR2 Theorem 5.2] there exists a compact skeleton $\Gamma_1$ with $\Gamma_{0} \subset \Gamma_{1}$ such that $\tau_{\Gamma_0} (x) \neq \tau_{\Gamma_{1}} (x)$. Since $\Gamma_{0} \subset \Gamma_{1}$, we have $\tau_{\Gamma_{0}} (\tau_{\Gamma_1} (x)) = \tau_{\Gamma_{0}} (x) \neq \tau_{\Gamma_1} (x)$, so that $\tau_{\Gamma_1} (x) \not\in \Gamma_{0}$, and in particular, $\tau_{\Gamma_1} (x) \not\in \Gamma_{\min}$.
By Lemma \[lemma:retraction:rational\], we have $\tau_{\Gamma_1} (x) \in \Gamma_{1 , \Lambda}$. We set $\Gamma := \Gamma_{\min} \cup [ \tau_{\Gamma_{\min}} (x) , \tau_{\Gamma_1} (x) ]$. By Lemma \[lemma:addingsegments\], $\Gamma$ is a compact skeleton. Since $\tau_{\Gamma_{\min}} (x) , \tau_{\Gamma_1} (x) \in \Gamma_1$, Lemma \[lemma:to:be:used\](\[lemma:to:be:used:3\]) gives us $[ \tau_{\Gamma_{\min}} (x) , \tau_{\Gamma_1} (x) ] \subset \Gamma_1$. Thus $\Gamma \subset \Gamma_1$.
We have so far taken a compact skeleton $\Gamma_0$ with $\tau_{\Gamma_0}(x) \neq \tau_{\Gamma_0}(y)$, then we have taken a bigger compact skeleton $\Gamma_1$ with $\tau_{\Gamma_0}(x) \neq \tau_{\Gamma_1}(x)$, and then we have defined a compact skeleton $\Gamma$ as a part of this bigger skeleton $\Gamma_1$. We are going to show that $\Gamma$ has required properties.
We show $\tau_{\Gamma} (x) \not\in \Gamma_{\min}$ and $\Gamma = \Gamma_{\min} \cup
[\tau_{\Gamma_{\min}} (x) , \tau_{\Gamma} (x) ]$. It follows from $ \tau_{\Gamma_1} (x) \in \Gamma$ and $\Gamma \subset \Gamma_1$ that $\tau_{\Gamma} (x) = \tau_{\Gamma} ( \tau_{\Gamma_1} (x) )
= \tau_{\Gamma_1} (x)$. Since $\tau_{\Gamma_1} (x) \not\in \Gamma_{\min}$, we have $\tau_{\Gamma} (x) \not\in \Gamma_{\min}$ and $\Gamma = \Gamma_{\min} \cup
[\tau_{\Gamma_{\min}} (x) , \tau_{\Gamma} (x) ]$.
We have $\Gamma_{\min} \cap [ \tau_{\Gamma_{\min}} (x) , \tau_{\Gamma} (x) ]
= \{ \tau_{\Gamma_{\min}} (x) \}$, using $\tau_{\Gamma_{\min}} (\tau_{\Gamma} (x)) = \tau_{\Gamma_{\min}} (x)$ and the latter assertion of Lemma \[lemma:to:be:used\](\[lemma:to:be:used:3\]).
We show the last property $\tau_{\Gamma} (x) \neq \tau_{\Gamma} (y)$. Noting that $\tau_{\Gamma}(x) \;(= \tau_{\Gamma_1}(x)) \not\in \Gamma_0$, we take the connected component $B$ of $X^{{{\operatorname{an}}}}\setminus \Gamma_0$ with $\tau_{\Gamma}(x) \in B$. By Lemma \[lemma:connectedcomponents\], $B$ is the connected component of $X^{{{\operatorname{an}}}}\setminus \{\tau_{\Gamma_0}(x)\}$ with $\tau_{\Gamma}(x) \in B$, and we have $\tau_{\Gamma_0}(B) = \{\tau_{\Gamma_0}(x)\}$ (cf. Lemma \[lemma:retraction:rational\]).
To prove $\tau_{\Gamma} (x) \neq \tau_{\Gamma} (y)$ by contradiction, we assume that $\tau_{\Gamma} (x) = \tau_{\Gamma} (y)$. Then we can show that $y \not\in \Gamma$. Indeed, since $\tau_{\Gamma_0} (y) \neq \tau_{\Gamma_0} (x)$ and $\Gamma_0 \subset \Gamma_1$, we have $\tau_{\Gamma_1} (y) \neq \tau_{\Gamma_1} (x)$. From the obvious equality $\tau_{\Gamma_1} (x) = \tau_{\Gamma_1} (\tau_{\Gamma_1} (x) )$, it follows that $\tau_{\Gamma_1} (y)
\neq
\tau_{\Gamma_1} (\tau_{\Gamma_1} (x))$, and hence $y \neq \tau_{\Gamma_1} (x) = \tau_{\Gamma} (x) = \tau_{\Gamma} (y)$. This shows $y \notin \Gamma$. Thus we take the connected component $A$ of $X^{{{\operatorname{an}}}} \setminus \Gamma$ with $y \in A$. Since we have assumed that $\tau_{\Gamma} (y) = \tau_{\Gamma} (x)$, Lemma \[lemma:connectedcomponents\] tells us that $A$ is the connected component of $X^{{{\operatorname{an}}}} \setminus \{ \tau_{\Gamma} (x)\}
=X^{{{\operatorname{an}}}} \setminus \{ \tau_{\Gamma} (y)
\}
$ with $y \in A$.
We note that $\tau_{\Gamma_0} (x) \notin \overline{A}$. Indeed, since $[\tau_{\Gamma_{\min}} (x) , \tau_{\Gamma} (x) )$ is a connected subspace of $X^{{{\operatorname{an}}}} \setminus \{ \tau_{\Gamma} (x) \}$, $\tau_{\Gamma_{\min}} (x) \in \Gamma_{\min} \subset \Gamma$, and $\Gamma \cap A = \emptyset$, we have $[\tau_{\Gamma_{\min}} (x) , \tau_{\Gamma} (x) ) \cap A = \emptyset$. Since $\overline{A} = A \cup \{ \tau_{\Gamma} (x) \}$ and $\tau_{\Gamma} (x) \neq \tau_{\Gamma_{\min}} (x) $, we have $[\tau_{\Gamma_{\min}} (x) , \tau_{\Gamma} (x) ) \cap
\overline{A} = \emptyset$. On the other hand, since $\tau_{\Gamma_0} (x) \neq \tau_{\Gamma_1}(x)$, Lemma \[lemma:to:be:used\](\[lemma:to:be:used:4\]) tells us that $\tau_{\Gamma_0} (x) \in [\tau_{\Gamma_{\min}}(x), \tau_{\Gamma_1}(x))
= [\tau_{\Gamma_{\min}}(x), \tau_{\Gamma}(x))$. We obtain $\tau_{\Gamma_0} (x) \notin \overline{A}$.
Since $\overline{A}$ is connected and $\tau_{\Gamma_0} (x) \notin \overline{A}$, it follows that $\overline{A}$ is contained in some connected component of $X^{{{\operatorname{an}}}} \setminus \{ \tau_{\Gamma_0} (x) \}$. Since $B$ is a connected component of $X^{{{\operatorname{an}}}} \setminus \{ \tau_{\Gamma_0} (x) \}$ and $ \tau_{\Gamma}(x) \in \overline{A} \cap B$, we have $\overline{A} \subset B$. Then $\tau_{\Gamma_0} ( \overline{A}) \subset
\tau_{\Gamma_0} (B) = \{ \tau_{\Gamma_0} (x) \}$, which implies that $\tau_{\Gamma_0} (y) = \tau_{\Gamma_0} (x)$. However, that contradicts the choice of $\Gamma_0$. Thus $\tau_{\Gamma} (x) \neq \tau_{\Gamma} (y)$.
500
\[rmk:four:types\] By Berkovich’s classification theorem (cf. [@Be1]), one categorizes the points of $X^{{{\operatorname{an}}}}$ into four types. The *type I* points are the classical points of $X^{{{\operatorname{an}}}}$ (cf. §\[subsec:Berko:skeleta\]), i.e., the points in $X(K)$. The [*type II*]{} points are the Shilov points (cf. §\[subsec:Berko:skeleta\]), i.e., the points in $\mathrm{Sh}(X^{{{\operatorname{an}}}}) := \bigcup_{\Gamma} \Gamma_{\Lambda}$, where $\Gamma$ runs through all the compact skeleta of $X^{{{\operatorname{an}}}}$. The [*type III*]{} points are those in $X^{{{\operatorname{an}}}}$ that belong to some compact skeleta but are not Shilov points. Thus the set of type III points is equal to $\bigcup_{\Gamma} \Gamma \setminus \bigcup_{\Gamma} \Gamma_{\Lambda}$, where $\Gamma$ runs through all the compact skeleta of $X^{{{\operatorname{an}}}}$. The [*type IV*]{} points are those in $X^{{{\operatorname{an}}}}$ that are neither of type I, II, or III.
To prove Theorem \[thm:limit:trop:d:intro\], we need to separate all types of points. We have separated points in a skeleton to prove Theorem \[thm:main:unimodular:faithful\], namely, with the terminology in Remark \[rmk:four:types\], we have constructed global sections that separate type II and type III points. To separate two points one of which is type I or type IV, the following is the key proposition. In the proof, we use a base section and a $P$-unimodularity section for a suitable $P \in X(K)$ and with respect to a suitable $B$, which we consider in §\[sec:ft:general:model\].
Let $t (g)$ be as in Theorem \[thm:main:unimodular:faithful\].
\[prop:key:injective\] Let $X$ be a connected smooth projective curve over $K$ of genus $g$, and let $L$ be a line bundle over $X$. Assume that $\deg (L) \geq t (g)$. Then for any $x , y \in X^{{{\operatorname{an}}}}$, there exist nonzero global sections $s_0$ and $ s_1$ of $L$ such that $s_0 (x) \neq 0$ if $x \in X (K)$, $s_0 (y) \neq 0$ if $y \in X (K)$, and $- \log |s_1 / s_0 (x)| \neq - \log |s_1 / s_0 (y)|$.
\[remark:evaluation:globalsection\] In Proposition \[prop:key:injective\], since $s_0$ and $s_1$ are nonzero sections, $s_1 / s_0$ is a nonzero rational function on $X$. It follows that if $x = (x, |\cdot|) \in X^{{\operatorname{an}}}\setminus X (K)$, then $- \log |s_1 / s_0 (x)|$ makes sense as a real number.
[[*Proof.*]{}]{}First, we prove the assertion if $x$ or $y$ is a point in $X(K)$. By symmetry, we assume that $x \in X(K)$. It follows from $\deg (L) \geq t (g)$ and $t(g) \geq 2 g +1$ that $L$ is very ample.
Suppose that $y \in X(K)$. Since $L$ is very ample, there exist global sections $s_0$ and $s_1$ of $L$ such that $s_0 (x) \neq 0$, $s_0 (y) \neq 0$, $s_1 (x) = 0$, and $s_1 (y) \neq 0$. This proves $- \log |s_1 / s_0 (x)| = + \infty \neq - \log |s_1 / s_0 (y)|$.
Suppose that $y = (y, |\cdot|) \not\in X(K)$. Since $L$ is very ample, there exist nonzero global sections $s_0$ and $s_1$ of $L$ such that $s_0 (x) \neq 0$ and $s_1 (x) = 0$. Then Remark \[remark:evaluation:globalsection\] concludes that $- \log |s_1 / s_0 (x)| = + \infty \neq - \log |s_1 / s_0 (y)|$.
Thus in the following we may assume that neither $x$ nor $y$ belongs to $X(K)$. If there exists a skeleton $\Gamma$ such that $x,y \in \Gamma$, then the assertion follows immediately from Theorem \[thm:main:unimodular:faithful\]. Therefore, we may assume that at least one of $x,y$ does not belong to any skeleton. Without loss of generality, we assume that $x$ does not belong to any skeleton.
We fix a minimal skeleton $\Gamma_{\min}$ in $X^{{{\operatorname{an}}}}$. We take a compact skeleton $\Gamma$ that contains $\Gamma_{\min}$ as in Lemma \[lemma:construction:Gamma1\], namely, $\Gamma = \Gamma_{\min} \cup [\tau_{\Gamma_{\min}} (x), \tau_{\Gamma}(x)]$. Since $\tau_{\Gamma_{\min}} (x), \tau_{\Gamma}(x), \tau_{\Gamma}(y)$ are $\Lambda$-rational points (cf. Lemma \[lemma:retraction:rational\]), [@BPR2 Theorem 4.11] gives us a strictly semistable model $\mathscr{X}^0$ of $X$ such that $S ( \mathscr{X}^0) = \Gamma$ and $\tau_{\min} (x)$, $\tau_{\Gamma}(x)$, $\tau_{\Gamma}(y)
\in V ( \mathscr{X}^0)$. To ease notation, we put $w := \tau_{\Gamma} (x)$. Since $w \in V(\mathscr{X}^0) $, there exists $C^0_w \in {\operatorname{\operatorname{Irr}}}( \mathscr{X}^0_s) $ such that $[C^0_w] = w$. Since $\tau_{\Gamma} (x) = [C^0_w] \neq x$, it follows from Lemma \[lemma:retraction:rational\] that ${\operatorname{red}}_{\mathscr{X}^0} (x) \in C^0_{w} (k)
\setminus {\operatorname{Sing}}( \mathscr{X}^0_s)$. We take a section $\sigma^0$ of $\mathscr{X}^0 \to {{\operatorname{Spec}}}(R)$ such that $\sigma^0 (k) \in C^0_w \setminus {\operatorname{Sing}}( \mathscr{X}^0_s )$ and $\sigma^0 (k) \neq {\operatorname{red}}_{\mathscr{X}^0} (x)$. Then $( \mathscr{X}^0 ; \sigma^0)$ is a strictly semistable pair. Set $P := \sigma^0 (K) \in X (K)$. Since $\sigma^0 (k) \in C^0_w \setminus {\operatorname{Sing}}( \mathscr{X}^0_s )$, we have $[w , P)=\Delta ( \sigma^0)$ and $S ( \mathscr{X}^0; \sigma^0) =
\Gamma \cup [w , P)$ (cf. Lemma \[lemma:end=canonialend\](3)).
$$\setlength\unitlength{0.07truecm}
\begin{picture}(150, 90)(0,0)
\qbezier(10, 5)(30, 35)(10, 65)
\multiput(60,45)(2, 0){30}{\line(1,0){1}}
\multiput(60,45)(0, 1){20}{\line(0,1){0.2}}
\put(20, 35){\circle*{1.5}}
\put(20,35){\line(4,1){40}}
\put(60, 45){\circle*{1.5}}
\put(21, 30){$\tau_{\Gamma_{\min}}(x)$}
\put(58, 38){$w = \tau_{\Gamma}(x)$}
\put(40, 55){$\Gamma$}
\put(83, 48){$[w, P)$}
\put(0, 70){$\Gamma_{\min}$}
\put(58, 67){$x$}
\put(120, 43){$P$}
\end{picture}$$ \[figure:for:prop:key:injective\]
Let $A$ be the connected component of $X^{{{\operatorname{an}}}} \setminus \Gamma$ with $x \in A$.
\[claim:disjoint:prop:key:injective\] We have $A \cap [w , P) = \emptyset$.
Indeed, to argue by contradiction, suppose that $A \cap [w , P) \neq \emptyset$. Since $w = \tau_{\Gamma} (x) \in \Gamma$ and $A \cap \Gamma = \emptyset$, we have $A \cap (w , P) \neq \emptyset$. Since $A$ is a connected component of $X^{{{\operatorname{an}}}} \setminus \{ w \}$ by Lemma \[lemma:connectedcomponents\], it follows that $(w , P) \subset A$, and thus $[w , P ] \subset \overline{A} = A \cup \{ w \}$. In particular, $P \in A$. This shows that $$\{ \sigma^0 (k) \} = \{ {\operatorname{red}}_{\mathscr{X}^0} (P) \}
\subset {\operatorname{red}}_{\mathscr{X}^0} (A) \supset \{ {\operatorname{red}}_{\mathscr{X}^0} (x) \}
.$$ By Lemma \[lemma:retraction:rational\], ${\operatorname{red}}_{\mathscr{X}^0} (A)$ is a singleton, so that $\sigma^0 (k) = {\operatorname{red}}_{\mathscr{X}^0} (x)$. This contradicts the choice of $\sigma^0$. Thus Claim \[claim:disjoint:prop:key:injective\] holds.
Now, using Proposition \[prop:main:FT:ends\], we obtain a strictly semistable model $(\mathscr{X} , \mathscr{L})$ of $(X,L)$ and a section $\sigma$ of $\mathscr{X} \to {{\operatorname{Spec}}}(R)$ such that ${{\mathscr{X}}}$ dominates ${{\mathscr{X}}}^0$ and $(\mathscr{X} ; \sigma)$ is a strictly semistable pair with the following properties:
1. $S(\mathscr{X} ) = \Gamma$, $V({{\mathscr{X}}}) \supset V({{\mathscr{X}}}^0)$, and $\sigma (K) = P$;
2. We set $$B :=
\begin{cases}
\{ {\operatorname{red}}_{\mathscr{X}} (x) , {\operatorname{red}}_{\mathscr{X}} (y) \}
&
\text{if ${\operatorname{red}}_{\mathscr{X}} (y) \in \mathscr{X}_s (k) \setminus ({\operatorname{Sing}}( \mathscr{X}_s)
\cup
\{ \sigma (k) \})$,}
\\
\{ {\operatorname{red}}_{\mathscr{X}} (x) \}
&
\text{otherwise}:
\end{cases}$$ Then $\mathscr{L}$ has a base section $\widetilde{s}_0$ with respect to $B$;
3. $\mathscr{L}$ has a $P$-unimodularity section $\widetilde{s}_1$ with respect to $B$.
Here we remark that property $V({{\mathscr{X}}}) \supset V({{\mathscr{X}}}^0)$ in (i) holds, since ${{\mathscr{X}}}$ dominates ${{\mathscr{X}}}^0$. We also remark that ${\operatorname{red}}_{\mathscr{X}} (x) \in
\mathscr{X}_s (k) \setminus ({\operatorname{Sing}}( \mathscr{X}_s)
\cup
\{ \sigma^0 (k) \})$, since ${\operatorname{red}}_{\mathscr{X}^0} (x) \in \mathscr{X}^0_s(k)
\setminus {\operatorname{Sing}}( \mathscr{X}^0_s) $ and ${\operatorname{red}}_{\mathscr{X}^0} (x) \neq \sigma^0 (k)$.
For $i=0,1$, set $s_i := {\left.{\widetilde{s}_i}\right\vert_{{X}}}$. We show that these $s_0, s_1$ have required properties. Since we assume that neither $x$ nor $y$ belongs to $X(K)$, we have $s(x) \neq 0$ and $s(y) \neq 0$ for any nonzero global section $s$ of $L$, whence $s_0(x) \neq 0$ and $s_0(y) \neq 0$. It remains to show that $- \log | s_1 / s_0 (x)|
\neq - \log | s_1 / s_0 (y)|$.
In this step, we consider $- \log | s_1 / s_0 (x)|$: the goal in this step is to show that $$\label{eqn:compari:x:w:1}
- \log | (s_1 / s_0) (x)| = - \log | (s_1 / s_0) (w)|,$$ where we recall that $w := \tau_{\Gamma}(x)$.
We use the notation in Remark \[rmk:four:types\]: $\mathrm{Sh}(X^{{\operatorname{an}}}) $ denotes the set of Shilov points of $X^{{{\operatorname{an}}}}$. To prove (\[eqn:compari:x:w:1\]), we first show that for any $u \in A \cap \mathrm{Sh}(X^{{\operatorname{an}}})$, $$\label{eqn:compari:x:w:2}
- \log | s_1 / s_0 (u)|
= - \log | s_1 / s_0 (w)|.$$ We take any $u \in A \cap \mathrm{Sh}(X^{{\operatorname{an}}})$. By the definition of $\tau_{\Gamma}$, we have $\tau_{\Gamma}(A) = \{w\}$, Then since $\tau_{\Gamma}(u) = w = \tau_{\Gamma}(w)$, let $[u, w]$ denotes the geodesic segment (see §\[subsection:geodesic:line\]). Since $u \in \mathrm{Sh}(X^{{\operatorname{an}}})$, Lemma \[lemma:to:be:used\] shows that $\Gamma \cup [w , u]$ is a compact skeleton and $\Gamma \cap [w , u] = \{ w \}$. Noting [@BPR2 Theorem 4.11], we take a strictly semistable model $\mathscr{X}'$ such that $S ( \mathscr{X}^\prime ) = \Gamma \cup [w , u]$ and $V( \mathscr{X}' ) = V ( \mathscr{X} ) \cup \{ u \}$; we remark that by this equality, the identity on $X$ extends to a morphism $\mu : \mathscr{X}^\prime \to \mathscr{X}$. Let $\sigma^\prime$ be the section of $\mathscr{X}^\prime \to {{\operatorname{Spec}}}(R)$ with $\sigma = \mu \circ \sigma^\prime$. Then $( \mathscr{X}^\prime ; \sigma^\prime
)
$ is a strictly semistable pair, and we have $
S ( \mathscr{X}^\prime ;
\sigma^\prime
)
=
\Gamma \cup \Delta(\sigma^\prime)$. We are going to show that, for $( \mathscr{X}^\prime ; \sigma^\prime
)$, $\mu^{\ast} (\widetilde{s}_0)$ is a base section of $\mu^{\ast} ( \mathscr{L})$ with respect to the empty set $\emptyset$ and $\mu^{\ast} (\widetilde{s}_1)$ is a $P$-unimodularity section of $\mu^{\ast} ( \mathscr{L})$ with respect to $\emptyset$.
By Lemma \[lemma:connectedcomponents\], $A$ is a connected component of $X^{{{\operatorname{an}}}} \setminus \{ w \}$, and we have $u \in A$. Since $(w,u]$ is a connected subspace of $X^{{{\operatorname{an}}}} \setminus \{ w \}$ with $u \in (w,u]$, it follows that $(w,u] \subset A$. Thus ${\operatorname{red}}_{\mathscr{X}} (( w ,u ])
\subset {\operatorname{red}}_{\mathscr{X}} (A) = \{ {\operatorname{red}}_{\mathscr{X}} (x) \}$ (cf. Lemma \[lemma:retraction:rational\]), which is contained in $B$. Here, by the definition of $\mathscr{X}'$, we note that $\mu : \mathscr{X}' \to \mathscr{X}$ is not an isomorphism over ${\operatorname{red}}_{\mathscr{X}} (x)$ but an isomorphism except over ${\operatorname{red}}_{\mathscr{X}} (x)$. Since ${\operatorname{red}}_{\mathscr{X}} (x) \in B$ and $\widetilde{s}_0$ is a base section of $\mathscr{L}$ with respect to $B$, it follows that $\mu^{\ast} ( \widetilde{s}_0 )$ is a base section of $\mu^{\ast}
( \mathscr{L})$ with respect to $\emptyset$.
To see that $\mu^{\ast} ( \widetilde{s}_1 )$ is a $P$-unimodularity section of $\mu^{\ast}
( \mathscr{L})$ with respect to $\emptyset$, let $\mathscr{W}_P$ be the stepwise vertical divisor associated to $P$ defined in (\[align:stepwise:for:ends\]). Since $\widetilde{s}_1$ is a $P$-unimodularity section of $\mathscr{L}$ with respect to $B$, there exists an open neighborhood of $\mathscr{U} \subset \mathscr{X}$ of $B \cup {\operatorname{Sing}}( \mathscr{X}_s ) \cup \{ \sigma (k) \}$ such that ${\operatorname{div}}(\widetilde{s}_1) - \mathscr{W}_P - \sigma (R)$ is trivial on $\mathscr{U}$. The pullback of this Cartier divisor by $\mu$ equals ${\operatorname{div}}(\mu^{\ast} (\widetilde{s}_1)) -
\mu^{\ast} ( \mathscr{W}_P ) - \mu^{\ast} ( \sigma (R) )$, which is trivial on $\mu^{-1} (\mathscr{U})$. We remark that $\mu^{-1} (\mathscr{U})$ is an open neighborhood of ${\operatorname{Sing}}( \mathscr{X}'_s) \cup \{ \sigma' (k) \}$. Let $C_w$ be the irreducible component of $\mathscr{X}_s$ with $[C_w] = w$. Since $\tau_{\Gamma} (x) = w$, Lemma \[lemma:retraction:rational\] tells us that ${\operatorname{red}}_{\mathscr{X}} (x) \in C_{w} (k) \setminus {\operatorname{Sing}}(\mathscr{X}_s)$. We take $\varpi_w \in R$ such that $- \log |\varpi_w|
= {\operatorname{ord}}_{C_w} (\mathscr{W}_P)$. Then on some neighborhood of ${\operatorname{red}}_{\mathscr{X}} (x)$, the Cartier divisor $\mathscr{W}_P$ is defined by $\varpi_w$. Thus on some neighborhood of $\mu^{-1} ( {\operatorname{red}}_{\mathscr{X}} (x) )$, $\mu^{\ast} ( \mathscr{W}_P ) $ is defined by $\varpi_w$. Further, ${\operatorname{red}}_{\mathscr{X}} (x) \neq \sigma (k)$. By the definition of the stepwise vertical divisor associated to $P$, it follows that $\mu^{\ast} ( \mathscr{W}_P ) $ equals the the stepwise vertical divisor $\mathscr{W}_P' $ on $\mathscr{X}'$ associated to $P$. Furthermore, since $\mu$ is isomorphism over $\sigma (R)$, we have $\mu^{\ast} ( \sigma (R) ) = \sigma' (R)$. Thus ${\operatorname{div}}(\mu^{\ast} (\widetilde{s}_1)) -
\mathscr{W}_P' - \sigma' (R)$ is trivial over $\mu^{-1} ( \mathscr{U})$. This proves that $\mu^{\ast} (\widetilde{s}_1)$ is a $P$-unimodularity section of $\mu^{\ast} ( \mathscr{L})$ with respect to $\emptyset$.
Note by Claim \[claim:disjoint:prop:key:injective\] that $[w , u] \cap [w , P) = \{ w \}$. Then by Lemma \[lemma:faithful:ends2\](2), it follows that $- \log | s_1 / s_0 (u)| = - \log | s_1 / s_0 (w)|$. We have shown .
Recall that, since $X^{{{\operatorname{an}}}}$ is a locally connected space, $A$ is open in $X^{{{\operatorname{an}}}}$. Since $\mathrm{Sh} ( X^{{{\operatorname{an}}}})$ is dense in $X^{{{\operatorname{an}}}}$, $A \cap \mathrm{Sh} ( X^{{{\operatorname{an}}}}) $ is dense in $A$. Since $- \log | s_1 / s_0 |$ is a continuous function (with values in ${{\mathbb{T}}}$) on $A$, the equality holds for any $u \in A$ as well as for points in $A \cap \mathrm{Sh} ( X^{{{\operatorname{an}}}})$. Since $x \in A$, this gives .
In this step, we consider $- \log | s_1 / s_0 (y)|$. Noting that $\tau_{\Gamma_{\min}}(x) = \tau_{\Gamma_{\min}}(\tau_{\Gamma}(x))$ and $w:= \tau_{\Gamma}(x)$, we consider the geometric segment $[\tau_{\Gamma_{\min}} (x) , w]$ (see §\[subsection:geodesic:line\]). We first treat the case where $\tau_{\Gamma} (y) \in [\tau_{\Gamma_{\min}} (x) , w]$, and then the case where $\tau_{\Gamma} (y) \not\in [\tau_{\Gamma_{\min}} (x) , w]$.
Suppose that $\tau_{\Gamma} (y) \in [\tau_{\Gamma_{\min}} (x) , w]$. By Lemma \[lemma:faithful:ends\](i), the function $- \log | s_1 / s_0| : [\tau_{\Gamma_{\min}} (x) , w]
\to {{\mathbb{R}}}$ is an affine function with $- \log | s_1 / s_0 ( \tau_{\Gamma_{\min}} (x))| = 0$ and $- \log | s_1 / s_0 ( w)| > 0$. By the definition of $\Gamma$, we have $\tau_\Gamma(y) \neq w$, so that $
- \log | s_1 / s_0 (\tau_{\Gamma_1} (y) )| <
- \log | s_1 / s_0 (w)|
$. Thus by (\[eqn:compari:x:w:1\]), $$\begin{aligned}
\label{eqn:compari:x:y}
- \log | s_1 / s_0 (x)| >
- \log | s_1 / s_0 (\tau_{\Gamma} (y))|.\end{aligned}$$ If $y = \tau_{\Gamma} (y)$, then concludes that $- \log | s_1 / s_0 (x)| \neq
- \log | s_1 / s_0 (y)|$.
Suppose now that $y \neq \tau_{\Gamma} (y)$. Let $A^\prime$ be the connected component of $X^{{{\operatorname{an}}}} \setminus \Gamma$ with $y \in A^\prime$. By Lemma \[lemma:connectedcomponents\], $A^\prime$ is the connected component $X^{{{\operatorname{an}}}} \setminus \{ \tau_{\Gamma} (y) \}$ with $y \in A^\prime$. Then by the same argument as in Step 1, $ - \log |s_1 / s_0 ( y )|
=
- \log |s_1 / s_0 ( \tau_{\Gamma}
(y))|$, where we use ${\operatorname{red}}_{\mathscr{X}} (y) \in B$ if ${\operatorname{red}}_{\mathscr{X}} (y) \in \mathscr{X}_s (k)
\setminus (
{\operatorname{Sing}}( \mathscr{X}_s ) \cup
\{ \sigma^0 (k) \} )$. Together with (\[eqn:compari:x:w:1\]), this concludes that $- \log |s_1 / s_0 (y)| \neq - \log |s_1 / s_0 ( x )|$.
Suppose that $\tau_{\Gamma} (y) \notin [\tau_{\Gamma_{\min}} (x) , w]$. Since $\Gamma = \Gamma_{\min} \cup [\tau_{\Gamma_{\min}} (x), w]$ and $\tau_{\Gamma} (y) \in \Gamma$, $\tau_{\Gamma} (y) \in \Gamma_{\min}$. By Lemma \[lemma:faithful:ends\](iii), we have $- \log |s_1 / s_0 (\tau_{\Gamma} (y))| = 0$. If $y = \tau_{\Gamma} (y)$, then $- \log |s_1 / s_0 (y)| = 0$; since $- \log |s_1 / s_0 (x)|
= - \log |s_1 / s_0 (w)| > 0$ by , this concludes $- \log |s_1 / s_0 (x)| \neq - \log |s_1 / s_0 (y)| $.
Suppose that $y \neq \tau_{\Gamma} (y)$. Let $A^{\prime}$ be the connected component of $X^{{{\operatorname{an}}}} \setminus
\Gamma$ with $y \in A^{\prime}$. Then by the same argument as in Case 1 (or Step 1), one shows that the restriction of $- \log |s_1/s_0|$ to $A^{\prime}$ equals the constant function $0$. Since $y \in A^{\prime}$, this concludes $- \log |s_1 / s_0 (x)| \neq - \log |s_1 / s_0 (y)| $.
500
Proof of Theorem \[thm:limit:trop:d:intro\] {#subsec:proof:thm:limit:trop:d:intro}
-------------------------------------------
Let $Y \subset \mathbb{A}^N = {{\operatorname{Spec}}}(K[z_1, \ldots, z_N])$ be a (closed) connected smooth affine curve. From here on, we regard $\mathbb{A}^N$ as an open subscheme of ${{\mathbb{P}}}^N$ via the embedding $(z_1 , \ldots , z_N ) \mapsto (1 : z_1 : \cdots : z_N)$. Let ${{\mathcal{O}}}_{{{\mathbb{P}}}^N} (1)$ denote the tautological line bundle over ${{\mathbb{P}}}^N$. Since $Y \subset \mathbb{A}^{N} \subset {{\mathbb{P}}}^N$, we regard $Y$ as a subscheme of ${{\mathbb{P}}}^N$. Let $X$ be the closure of $Y$ in ${{\mathbb{P}}}^N$. Recall that the degree $d$ of $Y$ is defined as the the degree of $X$ in ${{\mathbb{P}}}^N$. Assume that $Y$ has smooth compactification in ${{\mathbb{P}}}^N$, i.e., $X$ is smooth. Let $g$ denote the genus of $X$. We also call $g$ the genus of $Y$. For any integer $m$, set ${{\mathcal{O}}}_X (m) := {\left.{{{\mathcal{O}}}_{{{\mathbb{P}}}^N}(m)}\right\vert_{{X}}}$, which is a very ample line bundle over $X$.
We say that $Y$ is *degenerate* if it is contained in some hyperplane of $\mathbb{A}^N$ and that $X$ is *degenerate* if it is contained in some hyperplane of $\mathbb{P}^N$. Then $Y$ is degenerate if and only if $X$ is degenerate.
If $X$ is non-degenerate, then $N \leq d$. Indeed, if $X$ is non-degenerate, then the restriction $H^0
( {{\mathbb{P}}}^N , {{\mathcal{O}}}_{{{\mathbb{P}}}^N}(1) ) \to H^0 ( X , {{\mathcal{O}}}_X (1) )$ is injective, and since $h^{0} (X , {{\mathcal{O}}}_X (1)) \leq \deg ( {{\mathcal{O}}}_X (1) ) + 1 = d+1$ and $h^0 ( {{\mathbb{P}}}^N , {{\mathcal{O}}}_{{{\mathbb{P}}}^N}(1) ) = N+1$, we have $d \geq N$.
\[prop:injective:affine\] Let $Y \subset \mathbb{A}^N = {{\operatorname{Spec}}}(K[z_1, \ldots, z_N])$ be a closed connected smooth affine curve that has smooth compactification. Let $d$ be the degree of $Y$, and let $g$ be the genus of $Y$. Assume that $Y$ is non-degenerate. Let $\ell$ be an integer. Suppose that $$\begin{aligned}
\label{align:forell}
\ell \geq
\begin{cases}
\lceil t (g)/d \rceil
&
\text{if $N = 1, 2$,}
\\
\max
\{ \lceil t (g)/d \rceil, d + 1 - N \}
& \text{if $N \geq 3$.}
\end{cases}\end{aligned}$$ Then for any $x ,y \in Y^{{{\operatorname{an}}}}$ with $x \neq y$, there exists a polynomial $f ( z_1 , \ldots , z_N ) \in
K[z_1 , \ldots , z_N]$ of degree at most $\ell$ such that $- \log |f (x)| \neq - \log |f (y)|$.
[[*Proof.*]{}]{}Let $X$ be the closure of $Y$ in ${{\mathbb{P}}}^N$. Then $X$ is a connected smooth projective curve over $K$ of genus $g$ and degree $d$. We denote by $\jmath: X \to {{\mathbb{P}}}^N$ the natural embedding. Let $Z_0, \ldots, Z_N$ denote the homogeneous coordinate functions of ${{\mathbb{P}}}^N$.
We note that the restriction $
\jmath^*:
H^0 ( {{\mathbb{P}}}^N , {{\mathcal{O}}}_{{{\mathbb{P}}}^N}(\ell) )
\to H^0 (X , {{\mathcal{O}}}_{X} (\ell))$ is surjective. Indeed, if $N = 1$, then $X = {{\mathbb{P}}}^1$ and the surjectivity is obvious. If $N = 2$, then the cokernel of $\jmath^*$ is a subspace of $H^{1} ( {{\mathbb{P}}}^2 , {{\mathcal{O}}}_{{{\mathbb{P}}}^2}(\ell - d) )$. Since $H^{1} ( {{\mathbb{P}}}^2 , {{\mathcal{O}}}_{{{\mathbb{P}}}^2}(\ell - d) ) = 0$, we see that $\jmath^*$ is surjective. If $N \geq 3$, then since $\ell \geq d + 1 - N$, [@GLP Theorem] tells us that $\jmath^*$ is surjective. (Here we use the assumption that $Y$ is non-degenerate.)
We have $\deg ({{\mathcal{O}}}_X(\ell)) = d \ell$. Since $\ell$ is an integer with $\ell \geq t (g)$, Proposition \[prop:key:injective\] gives us nonzero $s_0 , s_1 \in H^0 (X, {{\mathcal{O}}}_X(\ell))$ such that $s_0 (x) \neq 0$, $s_0 (y) \neq 0$, and $- \log |s_1 / s_0 (x)| \neq - \log |s_1 / s_0 (y)|$. Since $\jmath^*: H^0 ( {{\mathbb{P}}}^N , {{\mathcal{O}}}_{{{\mathbb{P}}}^N}(\ell) ) \to H^{0}
(X , {{\mathcal{O}}}_X (\ell))$ is surjective, there exist $F_0 (Z_0, \ldots , Z_N), F_1(Z_0, \ldots , Z_N)
\in H^0 ( {{\mathbb{P}}}^N , {{\mathcal{O}}}_{{{\mathbb{P}}}^N}(\ell))$ such that ${\left.{F_0}\right\vert_{{X}}} = s_0$ and ${\left.{F_1}\right\vert_{{X}}} = s_1$.
For $i = 0, 1$, we set $f_i (z_1 , \ldots , z_N) := F_i (1 , z_1 , \ldots , z_N) \in K[z_1, \ldots, z_N]$. Since $F_i$ is a homogeneous polynomial of degree $\ell$, $f_i \in K[z_1, \ldots, z_N]$ is a polynomial of degree at most $\ell$. Then we have $$- \log | (f_1 / f_0) (x)| = - \log |s_1 / s_0 (x)|
\neq - \log |s_1 / s_0 (y)| = - \log | (f_1 / f_0) (y)|.$$ It follows that $- \log |f_0 (x)| \neq - \log |f_0 (y)|$ or $- \log |f_1 (x)| \neq - \log |f_1 (y)|$. This completes the proof.
500
\[remark:m0e0\] Let $Y$, $d$, and $g$ be as in Proposition \[prop:injective:affine\]. Let $X$ be the closure of $Y$ in ${{\mathbb{P}}}^N$, so that $X$ is a connected smooth projective curve in ${{\mathbb{P}}}^N$ of genus $g$ and degree $d$. Then we have the following sufficient condition for $\ell$ to satisfy (\[align:forell\]).
1. Suppose that $N=1$. Then $Y = {\mathbb{A}}^1$, $d = 1$ and $g = 0$. In this case any integer $\ell \geq 1$ satisfies .
2. Suppose that $N=2$. Since $X$ is smooth, we have $g = (d-1)(d-2)/2$. If $d = 1, 2$, then $g = 0$ and $t(g)= 1$. If $d = 3$, then $g=1$ and $t(g) = 3$. If $d \geq 4$, then $t(g) = 3g-1 = (3 d^2 -9d + 4)/2$. Thus if $\ell$ is an integer with $$\ell
\geq
\begin{cases}
1 & \text{if $d = 1, 2 , 3$},
\\
\left\lceil
\frac{3d^2 - 9 d + 4}{2d}
\right\rceil
& \text{if $d \geq 4$},
\end{cases}$$ then $\ell$ satisfies (\[align:forell\]).
3. Suppose that $N \geq 3$. Suppose also that $Y$ is non-degenerate. Then $d \geq N$. Dividing $d-1$ by $N-1$, we write $$d - 1 = m_0 (N-1) + \epsilon_0,$$ where $m_0$ and $\epsilon_0$ are integers with $m_0 \geq 1$ and $0 \leq \epsilon_0 \leq N-2$. Let $\pi(d, N)$ be Castelnuovo’s number, which is defined by $$\pi(d, N) := \frac{( m_0 + 1) (\epsilon_0 + d - 1)}{2} - (d-1).$$ Then by Castelnuovo’s bound theorem (see [@ACGH Chap. III §2]), we have $$\begin{aligned}
\label{align:AGM}
g \leq
\pi(d, N). \end{aligned}$$ (In [@ACGH Chap. III §2], Castelnuovo’s number $\pi(d, N)$ is defined to be $\frac{m_0(m_0-1)}{2}(N-1) + m_0 \epsilon_0$, and one sees that $\frac{( m_0 + 1) (\epsilon_0 + d - 1)}{2} - (d-1) = \frac{m_0(m_0-1)}{2}(N-1) + m_0 \epsilon_0$.)
Since $d \geq N \geq 3$, we obtain $$t (g)/d
\leq
\begin{cases}
1 & \text{if $g=0,1$}
\\
\frac{3( m_0 + 1) (\epsilon_0 + d - 1)}{2d} - \frac{3d-2}{d}
& \text{otherwise.}
\end{cases}$$ Thus if $\ell$ is an integer with $$\ell
\geq
\max
\left\{
\left\lceil
\frac{3( m_0 + 1) (\epsilon_0 + d - 1)}{2d} - \frac{3d-2}{d}
\right\rceil
,
d+1-N
\right\}
,$$ then $\ell$ satisfies .
We start the proof of Theorem \[thm:limit:trop:d:intro\]. Recall that $D$ is an integer given by . Our goal is to prove that the map $$\mathrm{LTrop}_{\leq D} : Y^{{{\operatorname{an}}}} \to \varprojlim_{\iota \in I_{\leq D}} {\operatorname{Trop}}(Y, \iota)$$ in is an homeomorphism.
### Injectivity {#injectivity .unnumbered}
The injectivity essentially amounts to the following proposition.
\[prop:injective:affine:key\] Let $Y \subset \mathbb{A}^N = {{\operatorname{Spec}}}(K[z_1, \ldots, z_N])$ be a closed connected smooth affine curve that has smooth compactification. Let $d$ be the degree of $Y$, and let $g$ be the genus of $Y$. Further, let $D$ be as in . Then for any $x ,y \in Y^{{{\operatorname{an}}}}$ with $x \neq y$, there exists a polynomial $f ( z_1 , \ldots , z_N ) \in
K[z_1 , \ldots , z_N]$ of degree at most $D$ such that $- \log |f (x)| \neq - \log |f (y)|$.
[[*Proof.*]{}]{}First, we assume that $Y$ is non-degenerate. If $N = 1$, then the assertion immediately follows from Proposition \[prop:injective:affine\] and Remark \[remark:m0e0\](1). If $N = 2$, then Proposition \[prop:injective:affine\] and Remark \[remark:m0e0\](2) shows the assertion. Thus we may and do assume that $N \geq 3$. Recall that $D := \max \{ d-2 , 1\}$. By non-degeneracy assumption of $Y$, we have $d \geq N \geq 3$, and thus $D = d-2$. With the notation in Remark \[remark:m0e0\](3), we are going to show that $$\label{eqn:prop:injective:affine:key}
D \geq
\max
\left\{
\left\lceil
\frac{3( m_0 + 1) (\epsilon_0 + d - 1)}{2d} - \frac{3d-2}{d}
\right\rceil
,
d + 1 - N
\right\}.$$ Indeed, by Proposition \[prop:injective:affine\] and Remark \[remark:m0e0\](3), the assertion follows from this inequality.
Let us prove (\[eqn:prop:injective:affine:key\]). By the inequality of arithmetic and geometric means, we have $$(N - 1) (m_0 + 1) (\epsilon_0 + d -1)
\leq
\left(
\frac{(N-1) (m_0 + 1) + (\epsilon_0 + d - 1)}{2}
\right)^2
.$$ Since $(N-1) (m_0 + 1) + (\epsilon_0 + d - 1)
= 2(d-1) + N-1$ and $3 \leq N \leq d$, we have $$\begin{aligned}
(m_0 + 1) (\epsilon_0 + d -1)
& \leq
\frac{(d-1)^2}{N-1} + (d-1) + \frac{N -1}{4} \\
& \leq
\frac{(d-1)^2}{2} + \frac{5 (d -1)}{4}
= \frac{2d^2+d - 3}{4}.\end{aligned}$$ It follows that $$\frac{3( m_0 + 1) (\epsilon_0 + d - 1)}{2d} - \frac{3d-2}{d}
\leq
\frac{3(2d^2+d - 3)}{8d}
- \frac{3d-2}{d}
=
\frac{6d^2 - 21 d + 7}{8d}
.$$ Further, since $
(d - 2) -
\frac{6d^2 - 21 d + 7}{8d}
=
\frac{2d^2 + 5 d - 7}{8d} \geq 0
$, we have $$d - 2
\geq
\left\lceil
\frac{3( m_0 + 1) (\epsilon_0 + d - 1)}{2d} - \frac{3d-2}{d}
\right\rceil
.$$ Since $d - 2 \geq d + 1 - N$, we obtain . This completes the proof of the proposition when $Y$ is non-degenerate.
It remains to consider the case where $Y$ is degenerate. Let $W$ be the smallest linear subspace of ${{\mathbb{P}}}^N$ such that $X \subset W$. Let $r$ denote the dimension of $W$. Since $Y \subset W$ and $Y = X \cap \mathbb{A}^N$, $W \cap \mathbb{A}^{N}$ is an affine subspace of $\mathbb{A}^N$ of dimension $r$ and $Y$ is a closed subvariety of this affine space. We fix an isomorphism $\phi : W \cap \mathbb{A}^{N} \to \mathbb{A}^r$. Let $w_1 , \ldots , w_r$ denote the affine coordinates of $\mathbb{A}^r$. We regard $\mathbb{A}^r$ as an open subscheme of ${{\mathbb{P}}}^r$ via $(w_1 , \ldots , w_r) \mapsto (1 : w_1 : \cdots : w_r)$. Then $\phi$ extends to a unique isomorphism $W \to {{\mathbb{P}}}^r$. Note that the closure of $\phi (Y)$ in ${{\mathbb{P}}}^r$ is smooth and non-degenerate and has degree $d$. Further, note that the integer $D'$ defined by the formula (\[eqn:thm:limit:trop:d\]) for $\phi (Y) \subset {{\mathbb{P}}}^r$ equals $D$. It follows from what we have shown above that there exists a polynomial $h$ on $w_1 , \ldots , w_r$ of degree at most $D$ such that $- \log |h (\phi (x))| \neq - \log |h (\phi (y))|$. Since $W \cap \mathbb{A}^{N}$ is an affine subspace of $\mathbb{A}^N$, $\phi^{\ast} (w_1) , \ldots , \phi^{\ast} (w_r)$ are the restrictions of some polynomials of degree $1$ on $z_1 , \ldots , z_N$. It follows that there exists a polynomial $f$ on $z_1 , \ldots , z_r$ of degree at most $D$ such that ${\left.{f}\right\vert_{{Y}}} = \phi^{\ast} ( h)$. Then we have $- \log |f (x)| \neq - \log |f (y)|$. Thus the proof is complete.
500
We prove that $\mathrm{LTrop}_{\leq D}$ is injective. We take any $x, y \in Y^{{{\operatorname{an}}}}$ with $x \neq y$. It suffices to show that there exists $\iota \in I_{\leq D}$ such that $\pi_{\iota} (x) \neq \pi_{\iota}(y)$. Take any $\iota_0 \in I_{\leq D}$, and we write $\iota_0 = ( {\left.{f_1}\right\vert_{{Y}}} , \ldots , {\left.{f_n}\right\vert_{{Y}}} )$ for some $f_1 , \ldots , f_n \in K [z_1 , \ldots , z_N ]$ of degree at most $D$. By Proposition \[prop:injective:affine:key\], there exists $f (z_1 , \ldots , z_N )$ such that $- \log | f ( x) | \neq - \log | f ( y) |$. Set $\iota := ({\left.{f}\right\vert_{{Y}}} , {\left.{f_1}\right\vert_{{Y}}} , \ldots , {\left.{f_n}\right\vert_{{Y}}} )$. Then $\iota$ is an affine embedding of $Y$ of degree at most $D$, and $\pi_{\iota} (x) \neq \pi_{\iota} (y) $. This shows that $\mathrm{LTrop}_{\leq D}$ is injective.
### Surjectivity {#surjectivity .unnumbered}
We prove that $\mathrm{LTrop}_{\leq D}$ is surjective. Take any $(b_\iota)_{\iota \in I_{\leq d}} \in \varprojlim_{\iota \in I_{\leq D}} {\operatorname{Trop}}(Y, \iota)$. For each $\iota \in I_{\leq d}$, we set $A_\iota := \pi_\iota^{-1}(b_\iota)$. Since $\pi_\iota\colon Y^{{{\operatorname{an}}}} \to {\operatorname{trop}}(Y, \iota)$ is surjective, we have $A_\iota \neq \emptyset$ for any $\iota \in I_{\leq d}$. We claim that $\{A_{\iota}\}_{\iota \in I_{\leq D}}$ has a finite intersection property, i.e., for any finitely many elements $\iota_1, \ldots, \iota_k \in I_{\leq D}$, we have $
A_{\iota_1}\cap\cdots\cap A_{\iota_k} \neq \emptyset
$. For $\iota_j: Y^{{{\operatorname{an}}}} \hookrightarrow {\mathbb{A}}^{n_j}$, we set $\iota = (\iota_1, \ldots, \iota_k): Y^{{{\operatorname{an}}}} \hookrightarrow {\mathbb{A}}^{n_1+\cdots+n_k}$, and we write ${\operatorname{pr}}_j: {\mathbb{A}}^{n_1+\cdots+n_k} \to {\mathbb{A}}^{n_j}$ for the natural projection. Since $\iota_j = {\operatorname{pr}}_j \circ \iota$, we have $\iota_j \leq \iota$, and the induced map ${\operatorname{Trop}}(Y, \iota) \to {\operatorname{Trop}}(Y, \iota_j)$ is the restriction to ${\operatorname{Trop}}(Y, \iota)$ of the the natural projection ${\operatorname{Trop}}({\operatorname{pr}}_j): {{\mathbb{T}}}^{n_1+\cdots+n_k} \to {{\mathbb{T}}}^{n_j}$. By the definition of the inverse limit, we have ${\operatorname{Trop}}({\operatorname{pr}}_j)(b_{\iota}) = b_{\iota_j}$. It follows that $A_{\iota} \subset A_{\iota_j}$, and we obtain $A_{\iota_1}\cap\cdots\cap A_{\iota_k} \supset A_{\iota} \neq \emptyset$.
Following Bourbaki [@bourbakiGT], we say that a continuous map between topological spaces is *proper* if it is universally closed. We remark that a fiber of a proper map between Hausdorff spaces is compact (see, for example, [@bourbakiGT §10.2, Theorem 1]).
We fix any $\iota_0 \in I_{\leq D}$. Since $\pi_{\iota_0}: Y^{{{\operatorname{an}}}} \to {\operatorname{Trop}}(Y, \iota_0)$ is proper (see [@payne Lemma 2.1, Proposition 2.2]) and since $Y^{{{\operatorname{an}}}}$ and ${\operatorname{Trop}}(Y, \iota_0)$ are Hausdorff spaces, $A_{\iota_0}$ is compact. Consider the family $\{A_{\iota} \cap A_{\iota_0} \}_{\iota \in I_{\leq D}}$ of subspaces of $A_{\iota_0}$. Since $\{A_{\iota} \}_{\iota \in I_{\leq D}}$ has a finite intersection property, so does $\{A_{\iota} \cap A_{\iota_0} \}_{\iota \in I_{\leq D}}$. Since $A_{\iota} \cap A_{\iota_0}$ is compact for any $\iota \in I_{\leq D}$, it follows that $\bigcap_{\iota \in I_{\leq D}} A_{\iota} =
\bigcap_{\iota \in I_{\leq D}} ( A_{\iota} \cap A_{\iota_0} )
\neq
\emptyset$. We take a $y \in \bigcap_{\iota \in I_{\leq D}} A_{\iota}$. Then we have $\pi_\iota(y) = b_\iota$ for any $\iota \in I_{\leq D}$, and thus $\mathrm{LTrop}_{\leq D} (y)
=
\left(\varprojlim_{\iota \in I_{\leq D}}\pi_\iota\right) (y) = (b_\iota)_{\iota \in I_{\leq d}}$. Thus $\mathrm{LTrop}_{\leq D}$ is surjective.
### Homeomorphism {#homeomorphism .unnumbered}
We have shown that the map $\mathrm{LTrop}_{\leq D}$ in is continuous and bijective. To conclude that it is a homeomorphism, it suffices to show that it is a closed map. Since each tropicalization map $\pi_\iota: Y^{{{\operatorname{an}}}} \to {\operatorname{Trop}}(Y, \iota )$ is proper, it follows from [@bourbakiGT §10.2, Corollary 4] that the map $$\prod_{\iota \in I_{\leq D}} \pi_{\iota}: Y^{{{\operatorname{an}}}} \to \prod_{\iota \in I_{\leq D}} {\operatorname{Trop}}(Y, \iota ),
\quad y \mapsto (\pi_\iota(y))_{\iota \in I_{\leq D}}$$ is proper. In particular, $\prod_{\iota \in I_{\leq D}} \pi_{\iota}$ is a closed map. Since the image of the map $\prod_{\iota \in I_{\leq D}} \pi_{\iota}$ is contained in $\varprojlim_{\iota \in I_{\leq D} } {\operatorname{Trop}}(Y, \iota )$ and the topology of $\varprojlim_{\iota \in I_{\leq D} } {\operatorname{Trop}}(Y, \iota )$ is the subspace topology of $\prod_{\iota \in I_{\leq D}} {\operatorname{Trop}}(Y, \iota)$, we see that $\mathrm{LTrop}_{\leq D}$ is a closed map. Thus we complete the proof.
\[remark:morethanD\] It follows from the above proof that $\mathrm{LTrop}_{\leq D'}$ is a surjective closed continuous map for any positive integer $D'$; if $D' \geq D$, then $\mathrm{LTrop}_{\leq D'}$ is injective, in addition, and hence it is a homeomorphism.
We finish this paper by remarking that in the course of the proof of Theorem \[thm:limit:trop:d:intro\], we have shown the following as a corollary of Theorem \[thm:main:unimodular:faithful\]. Note that with the notation below, we have $Y^{{{\operatorname{an}}}} \setminus Y(K) = X^{{{\operatorname{an}}}}\setminus X(K)$ and thus any skeleton $\Gamma$ of $X^{{{\operatorname{an}}}}$ is contained in $Y^{{{\operatorname{an}}}}$.
\[prop:last:remark\] Let $Y \subset \mathbb{A}^N = {{\operatorname{Spec}}}(K[z_1, \ldots, z_N])$ be a connected smooth affine curve that has smooth compactification $X$. Let $d$ be the degree of $Y$. Define $D$ by . Then for any skeleton $\Gamma$ of $X^{{{\operatorname{an}}}}$, there exist nonzero polynomials $f_1, \ldots f_m \in K[z_1, \ldots, z_N]$ of degree at most $D$ such that $$\psi: Y^{{{\operatorname{an}}}} \to {{\mathbb{R}}}^m, \quad
p = (p, |\cdot|) \mapsto (-\log|f_1(p)|, \ldots, -\log|f_m(p)|)$$ gives a faithful tropicalization of $\Gamma$.
[99]{}
O. Amini, [*Reduced divisors and embeddings of tropical curves*]{}, Trans. Amer. Math. Soc. [**365**]{} (2013), no. 9, 4851–4880.
O. Amini, M. Baker, E. Brugallé, J. Rabinoff, [*Lifting harmonic morphisms I metrized complexes and Berkovich skeleta*]{}, Res. Math. Sci. [**2**]{} (2015), Art. 7, 67 pp.
O. Amini and L. Caporaso, [*Riemann–Roch theory for weighted graphs and tropical curves*]{}, Adv. Math. (2013), 1–23.
E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris, [*Geometry of algebraic curves. Vol. I.*]{}, Grundlehren der Mathematischen Wissenschaften, [**267**]{}, Springer-Verlag, 1985.
M. Baker, [*Specialization of linear systems from curves to graphs, with an appendix by Brian Conrad*]{}, Algebra Number Theory [**2**]{} (2008), 613–653.
M. Baker and S. Norine, [*Riemann–Roch and Abel–Jacobi theory on a finite graph*]{}, Adv. Math. (2007), 766–788.
M. Baker, S. Payne, J. Rabinoff, [*On the structure of non-Archimedean analytic curves*]{}, Tropical and non-Archimedean geometry, 93–121, Contemp. Math., [**605**]{}, Amer. Math. Soc., Providence, RI, 2013.
M. Baker, S. Payne and J. Rabinoff, [*Nonarchimedean geometry, tropicalization, and metrics on curves*]{}, Algebr. Geom. [**3**]{} (2016), 63–105.
M. Baker and J. Rabinoff, [*The skeleton of the Jacobian, the Jacobian of the skeleton, and lifting meromorphic functions from tropical to algebraic curves*]{}, Int. Math. Res. Not. IMRN [**2015**]{}, no. 16, 7436–7472.
V. Berkovich, [*Spectral theory and analytic geometry over non-Archimedean fields*]{}, Mathematical Surveys and Monographs, vol. [**33**]{}, American Mathematical Society, Providence, RI, 1990.
V. G. Berkovich, [*Étale cohomology for non-Archimedean analytic spaces,*]{} Inst. Hautes Études Sci. Publ. Math. [**78**]{} (1993), 5–161.
V. G. Berkovich, [*Smooth p-adic analytic spaces are locally contractible*]{}, Invent. Math. [**137**]{} (1999), no. 1, 1–84.
V. G. Berkovich, [*Smooth p-adic analytic spaces are locally contractible. II.*]{}, Geometric aspects of Dwork theory, Walter de Gruyter and Co. KG, Berlin, 2004, pp. 293–370.
S. Bosch, U. Güntzer, R. Remmert, Non-Archimedean Analysis, Grundlehren der mathematischen Wissenschaften 261, Springer-Verlag (1984).
S. Bosch, W. Lütkebohmert, [*Stable reduction and uniformization of abelian varieties I*]{}, Math. Ann. (1985), no. 3, 349–379.
N. Bourbaki, [Elements of mathematics, general topology, Chapters 1–4]{}, Springer.
L. Caporaso, Y. Len, M. Melo, [*Algebraic and combinatorial rank of divisors on finite graphs*]{}, J. Math. Pures Appl. (9) [**104**]{} (2015), no. 2, 227–257.
D. Cartwright, D. Jensen, S. Payne, [*Lifting divisors on a generic chain of loops*]{}, Canad. Math. Bull. [**58**]{} (2015), no. 2, 250–262.
F. Cools, J. Draisma, S. Payne, E. Robeva, [*A tropical proof of the Brill-Noether theorem*]{}, Adv. Math. [**230**]{} (2012), no. 2, 759–776.
M. Cueto and H. Markwig, [*How to repair tropicalizations of plane curves using modifications*]{}, Exp. Math. [**25**]{} (2016), no. 2, 130–164.
P. Deligne and D. Mumford, [*The irreducibility of the space of curves of given genus*]{}, Inst. Hautes Études Sci. Publ. Math. No. [**36**]{} (1969), 75–109.
K. Fujiwara, F. Kato, Foundations of Rigid Geometry I. arXiv:1308.4734, a book to appear from European Mathematical Society Publishing House.
A. Gathmann and M. Kerber, [*A Riemann-Roch theorem in tropical geometry*]{}, Math. Z. [**259**]{} (2008), no. 1, 217–230.
W. Gubler, [*Tropical varieties for non-archimedean analytic spaces*]{}, Invent. Math. [**169**]{} (2007), no. 2, 321–376.
W. Gubler, [*Non-archimedean canonical measures on abelian varieties*]{}, Compositio Math. [**146**]{} (2010), 683–730.
W. Gubler, [*A Guide to Tropicalizations*]{}, Algebraic and Combinatorial Aspects of Tropical Geometry, Contemp. Math., [**589**]{}, Amer. Math. Soc., Providence, RI, 2013, 125–189.
W. Gubler, J. Rabinoff, and A. Werner, [*Skeletons and tropicalizations*]{}, Adv. Math. [**294**]{} (2016), 150–215.
L. Gruson, R. Lazarsfeld, and C. Peskine, [*On a Theorem of Castelnuovo, and the Equations Defining Space Curves*]{}, Invent. math. [**72**]{} (1983), 491–506.
C. Haase, G. Musiker, and J. Yu, [*Linear systems on tropical curves*]{}, Math. Z. [**270**]{} (2012), no. 3-4, 1111–1140.
R. Hartshorne, [*Algebraic geometry*]{}, Graduate Texts in Mathematics, No. [**52**]{}. Springer-Verlag, New York-Heidelberg, 1977.
J. Hladký, D. Král, and S. Norine, [*Rank of divisors on tropical curves*]{}, J. Combin. Theory Ser. A [**120**]{} (2013), no. 7, 1521–1538.
E. Hrushovski, F. Loeser, and B. Poonen, [*Berkovich spaces embed in Euclidean spaces*]{}, Enseign. Math. [**60**]{} (2014), no. 3-4, 273–292.
E. Katz, H. Markwig, and T. Markwig, [*The $j$-invariant of a plane tropical cubic*]{}, J. Algebra [**320**]{} (2012), 853–875.
E. Katz, H. Markwig, and T. Markwig, [*The tropical $j$-invariant*]{}, LMS J. Comput. Math. [**12**]{} (2009), 275–294.
S. Kawaguchi, and K. Yamaki, [*Rank of divisors on hyperelliptic curves and graphs under specialization*]{}, Int. Math. Res. Not. [**2015**]{} (2015), 4121–4176.
S. Kawaguchi, and K. Yamaki, [*Algebraic rank on hyperelliptic graphs and graphs of genus $3$*]{}, Kyoto J. Math. [**56**]{} (2016), no. 1, 177–196.
S. Kawaguchi, and K. Yamaki, [*Effective faithful tropicalizations associated to adjoint line bundles*]{}, preprint 2016, available at arxiv:1612.01099.
M. Kontsevich and Y. Soibelman, [*Affine structures and non-Archimedean analytic spaces*]{}, In The unity of mathematics, 321–385. Progr. Math., [**244**]{}. Birkhäuser, Boston, 2006.
Y. Luo, [*Rank-determining sets of metric graphs*]{}, J. Combin. Theory Ser. A [**118**]{} (2011), no. 6, 1775–1793.
D. Maclagan and B. Sturmfelds, [*Introduction to Tropical Geoemtry*]{}, Graduate Studies in Mathematics, [**161**]{}, Amer. Math. Soc., Providence, RI, 2015.
H. Matsumura, [*Commutative ring theory*]{}, Cambridge Studies in Advanced Mathematics, [**8**]{}, Cambridge University Press, 1986.
G. Mikhalkin and I. Zharkov, [*Tropical curves, their Jacobians and theta functions*]{}, Curves and abelian varieties, 203–230, Contemp. Math., [**465**]{}, Amer. Math. Soc., Providence, RI, 2008.
S. Payne, [*Analytification is the limit of all tropicalizations*]{}, Math. Res. Lett. [**16**]{} (2009), no. 3, 543–556.
M. Raynaud, [*Spécialisation du foncteur de Picard*]{}, Inst. Hautes Études Sci. Publ. Math. (1970), 27–76.
I. Tyomkin, [*Tropical geometry and correspondence theorems via toric stacks*]{}, Math. Ann. [**353**]{} (2012), no. 3, 945–995.
T. Wagner, [*Faithful tropicalization of Mumford curves of genus two*]{}, Beitr. Algebra Geom., to appear.
K. Yamaki, [*Geometric Bogomolov conjecture for abelian varieties and some results for those with some degeneration with an appendix by Walter Gubler The minimal dimension of a canonical measure*]{}, Manuscr. Math. [**142**]{} (2013), 273–306.
K. Yamaki, [*Strict supports of canonical measures and applications to the geometric Bogomolov conjecture*]{}, Compos. Math. [**152**]{} (2016), no. 5, 997–1040.
[^1]: The first named author partially supported by KAKENHI 15K04817 and 25220701, and the second named author partially supported by KAKENHI 26800012.
|
---
abstract: 'We investigate the thermally activated magnetization switching of small ferromagnetic particles driven by an external magnetic field. For low uniaxial anisotropy the spins can be expected to rotate coherently, while for sufficient large anisotropy they should behave Ising-like, i. e. , the switching should then be due to nucleation. We study this crossover from coherent rotation to nucleation for the classical three-dimensional Heisenberg model with a finite anisotropy. The crossover is influenced by the size of the particle, the strength of the driving magnetic field, and the anisotropy. We discuss the relevant energy barriers which have to be overcome during the switching, and find theoretical arguments which yield the energetically favorable reversal mechanisms for given values of the quantities above. The results are confirmed by Monte Carlo simulations of Heisenberg and Ising models.'
address: |
Theoretische Tieftemperaturphysik, Gerhard-Mercator-Universität-Duisburg, 47048 Duisburg/ Germany\
e-mail: uli@thp.uni-duisburg.de
author:
- 'D. Hinzke and U. Nowak'
title: Magnetization switching in a Heisenberg model for small ferromagnetic particles
---
Pacs: 75.10.Hk, 75.40.Mg, 64.60.Qb
Introduction
============
The size of magnetic particles plays a crucial role for the density of information storage in magnetic recording media. Sufficiently small particles become single-domain particles, which improves their quality for magnetic recording. On the other hand, when the particles are too small they become superparamagnetic and no information can be stored (see e. g. [@chantrell] for a review). Hence, much effort has recently been focused on the understanding of small magnetic particles, especially since recent experimental techniques allow for the investigation of isolated single-domain particles [@salling; @lederman; @wernsdorfer; @wernsdorfer2].
In this paper focus is on the reversal of ferromagnetic particles of finite size. We investigate the influence of the size and anisotropy of the particle on the possible reversal mechanisms, two extreme cases of which are coherent rotation and nucleation. The latter mechanism has been a subject of common interest in recent years [@stauffer; @tomita; @rikvold; @acharyya; @garcia; @richards], studied mainly theoretically in Ising models, which can be interpreted as a classical Heisenberg model in the limit of infinite anisotropy. It is the aim of this paper to study the crossover from magnetization reversal due to nucleation for high anisotropy to coherent rotation [@neel; @brown] for lower anisotropy.
Throughout the paper we will consider a finite, spherical three- dimensional system of magnetic moments. These magnetic moments may represent atomic spins or block spins following from a coarse graining of the physical lattice [@nowak]. Our system is defined by a classical Heisenberg Hamiltonian, $${\cal H} = - J \sum_{\langle ij \rangle} {\bf S}_i \cdot {\bf S}_j
-d \sum_i (S^z_i)^2 -{\bf B} \cdot \sum_i {\bf S}_i.
\label{e:ham}$$ where the ${\bf S}_i$ are three-dimensional vectors of unit length. The first sum which represents the exchange of the spins, is over nearest neighbors with the exchange coupling constant $J$. The second sum represents an uniaxial anisotropy which favors the $z$-axis as easy axis of the system (anisotropy constant $d>0$). The last sum is the coupling of the spins to an applied magnetic field, where ${\bf
B}$ is the strength of the field times the absolute value of the magnetic moment of the spin. We neglect dipolar interaction. Although in principle it is possible to consider dipolar interaction in a Monte Carlo simulation [@nowak] this needs much more computational effort due to the long range of the dipolar interaction and, hence, exceeds current computer capacities . Therefore, the validity of our results is restricted to particles which are small enough to be single-domain in the remanent state [@aharony].
In the following, we will investigate the thermally activated reversal of a particle which is destabilized by an magnetic field pointing in a direction antiparallel to the initial magnetization which is parallel to the easy axis of the system. Due to the finite temperature and the magnetic field, after some time the particle will reverse its magnetization, i. e. the $z$-component of the magnetization will change its sign.
In section \[s:theory\], we determine the energy barriers which have to be overcome by thermal fluctuations for the two cases of coherent rotation and nucleation within a classical theory. By a comparison of the energy barriers, we derive where the crossover from one mechanism to the other occurs.
In section \[s:numerics\] we compare our theoretical considerations with numerical results from Monte Carlo simulations of Heisenberg and Ising models, and we relate the lifetime of the metastable state to the theoretical energy barriers for the different reversal mechanisms.
Theory {#s:theory}
======
Coherent rotation
-----------------
We give here a brief summary of the results of the theory of Néel [@neel] and Brown [@brown], since we need this concept for the further progress of our theoretical considerations.
Let us consider a spherical, homogeneously magnetized particle of radius $R$. The simplest theoretical description for the reversal of such a particle is to assume that the reversal mechanism is coherent rotation, i. e. a uniform rotation of all spins of the particle. This reversal process can be described by an angle of rotation $\theta$ between the easy axis of the system – which in our case will be antiparallel to the direction of the magnetic field – and the magnetization of the particle. The increase of the energy during the reversal is then $$\Delta E = -d V \cos^{2}{\theta} - BMV \cos{\theta}.$$ Since this equation should be comparable to Eq. \[e:ham\], the anisotropy constant $d$ is an anisotropy energy per unit cell (spin), and $V= (4/3) \pi R^3$ is the volume of the particle as a number of unit cells. $B$ is - as before - the absolute value of the applied field times the magnetic moment of a unit cell and, hence, $M$ the spontaneous magnetization per magnetic moment. The energy barrier which has to be overcome is due to the anisotropy of the system. It is the maximum of $\Delta E$ with respect to $\theta$: $$\Delta E_{cr} = \frac{4 \pi R^3d}{3}-{\frac {4\pi R^3BM}{3}}
+\frac{\pi R^3B^2M^2}{3d}
\label{e:ecr}$$ The corresponding lifetime of the metastable state is then $$\tau \sim \exp \left( \frac{\Delta E_{cr}}{T} \right)
\label{e:activation}$$ for temperature $T \ll \Delta E_{cr}$. The two equations above are physically relevant only for $$d > M B/2.
\label{e:thermal}$$ Otherwise there is no (positive) energy barrier, and hence the reversal is spontaneous without the need for thermal activation. This is the region of nonthermal reversal.
Nucleation
----------
For a system with a sufficient large anisotropy, it might be energetically favorable to divide into parts with opposite directions of magnetization parallel to the easy axis in order to minimize the anisotropy energy barrier. This kind of reversal mechanism is called nucleation [@becker] (see [@rikvold] for a recent review). The simplest case of a reversal process driven by nucleation for a system of finite size is the growth of one single droplet starting at one point of the boundary of the system (see also [@richards] for a corresponding calculation for a two-dimensional system). Due to the growth of the droplet a domain wall will cross the system, and the energy barrier which has to be overcome is caused by the domain-wall energy. We assume that the domain wall will have a curvature defined by a radius $r$ (see Fig. \[f:droplet\]).
Then the surface of the domain wall is $F = 2 \pi rh$ and the volume of the droplet (the shaded region in Fig. \[f:droplet\]) is $V_d = \pi H^2 (3R-H)/3 + \pi h^2 (3r-h)/3$. The energy increase during the reversal of the particle is $$\Delta E = 2 \sigma F - 2 M B V_d,$$ where $\sigma$ is the energy density of the domain wall. Furthermore, it is $h = r - \sqrt {r^2 -2HR + H^2}$, since the quantities in Fig.\[f:droplet\] are not independent. Hence the energy increase can be expressed in terms of $H$, which is a measure for the penetration depth of the domain wall and its curvature $r$. These two quantities define the geometry of the droplet. Next we determine that curvature $r_{\mbox{min}}$ which minimizes the energy increase by the condition $\frac {\partial {\Delta E}} {\partial{r}} = 0$, yielding the physically relevant solution $$r_{\mbox{min}} = \frac {2\sigma} {MB}
\label{e:rmin}$$ which is also the radius of a critical droplet for classical nucleation in a bulk material.
The energy barrier $\Delta E_w$ which has to be overcome during the reversal is the maximum of the energy increase with respect to $H$. From the condition $\frac {\partial {\Delta E}} {\partial{H}} = 0$, the physically relevant solution $$H_{\mbox{max}} = R \Big(1 - \frac{x}{\sqrt{x^2+4}} \Big)$$ follows with $x = MBR/\sigma$. Inserting the two conditions above into the formula for the energy increase yields the energy barrier for nucleation $$\begin{aligned}
\Delta E_n & = & \frac{4\pi R^2 \sigma}{3x^2\sqrt{x^2+4}} \Big(x^4
+ (4-x^3)\sqrt{x^2+4} +2x^2 -8 \Big) \nonumber \\
&&
\label{e:edw}\end{aligned}$$ This expression has two important limits. The first is the limit of infinite system size, $$\lim_{R \rightarrow \infty} \Delta E_n = \frac{16 \pi \sigma^3}{3M^2B^2},$$ where we obtain half of the energy barrier of the classical nucleation theory for a bulk system. The reduction by a factor of $\frac{1}{2}$ is due to the fact that for open boundary conditions only one-half of a critical droplet has to enter the system from the boundary.
The other interesting limit is that of small magnetic fields, where Eq.\[e:edw\] can be expanded with respect to $x = MBR/\sigma$ resulting in $$\Delta E_n \approx 2 \pi R^2 \sigma - \frac{4 \pi B R^3 M}3
+\frac {3\pi B^2R^4 M^2}{8\sigma} + \ldots$$ This means that for a small field $B$ the energy barrier of a nucleation process is the energy of a flat domain wall in the center of the particle plus corrections which start linearly in $B$. In contrast to the Néel-Brown theory, here, for vanishing magnetic field, the energy barrier is proportional to the cross-sectional area of the particle rather than its volume (see also the work of Braun [@braun]), which consequently reduces the coercivity of the particle.
Comparison of coherent rotation and nucleation
----------------------------------------------
Comparing the two energy barriers for coherent rotation (Eq.\[e:ecr\]) and nucleation (Eq.\[e:edw\]), we can evaluate which reversal mechanism has the lower activation energy for a given set of values of $R$, $B$, $\sigma$, and $d$. A corresponding diagram is shown in Fig. \[f:diagram\] for a system of radius $R=4$ spins, where we set $M=1/$spin and $\sigma = J/\mbox{spin}$.
=4.5cm
For large anisotropy the reversal is dominated by nucleation – the particle behaves like an Ising system. The crossover line $d_c(B,\sigma,R)$ which separates the region of reversal by nucleation from the region of coherent rotation can be determined by the condition that here the energy barriers for a nucleation process and for coherent rotation are equal. This condition results in $$\begin{aligned}
\label{e:dw}
d_c & = & \frac{\sigma}{R} \Big(\frac{2}{x^2}
+\frac{1}{\sqrt{x^2+4}} \big(1 -\frac{4}{x^2} +\frac{x^2}{2}\big) \nonumber\\
&& + \frac{1}{x^2} \sqrt{-3x^2+8 + (2x^2-4)\sqrt{x^2+4}} \Big).\end{aligned}$$ For vanishing magnetic field, the formula above has a finite limit, $$\lim_{B\rightarrow 0} (d_c) = \frac{3\sigma}{2R}.
\label{e:dclb}$$ For large particle size, Eq. \[e:dw\] has the simple asymptotic form $$\lim_{R\rightarrow \infty} (d_c) = \frac{BM}{2}
\label{e:nt}$$ which, interestingly, is also the limit for nonthermal reversal (see Eq. \[e:thermal\]). This means that for increasing particle size the region where a thermally activated reversal by coherent rotation occurs vanishes. For an infinite particle size the reversal is always either nonthermal or it is driven by nucleation – depending only on the ratio of the magnetic field to the anisotropy.
All the considerations above can be expected to be relevant only for sufficiently low temperatures. For higher temperatures the situation is more complicated. That is, for the nucleation regime, a crossover from single-droplet to multi-droplet nucleation – with different energy barriers) – is discussed in the literature (see [@binder; @ray; @sides] and references therein).
Apart from that, in a system with a given finite anisotropy the domain walls may be extended to a certain domain wall width $\xi$. For large anisotropy $\xi$ becomes as small as one lattice constant. This is the Ising case, where the domain-wall energy density is $\sigma \approx
J/$spin. For smaller anisotropy the domain walls become more extended, and $\sigma $ decreases. Hence the crossover from nucleation to coherent rotation may be softened by the occurrence of extended domain walls. In this sense, pure coherent rotation could also be interpreted as a domain-wall-driven reversal, where the width of the domain wall is larger than the particle size. Obviously, our theoretical considerations discuss only two extreme cases. How realistic they are has to be tested numerically.
Monte Carlo Simulation {#s:numerics}
======================
Method
------
Due to the many degrees of freedom of a spin system, numerical methods have to be used for a detailed microscopic description of the system. Since we are especially interested in the thermal properties of the system, we use Monte Carlo methods [@heermann] for the simulation of the magnetic particle. Although a direct mapping of the time scale of a Monte Carlo simulation on experimental time scales is not possible, this method provides information on the dynamical behavior of the system since it solves the master equation for the irreversible behavior of the system [@reif].
We consider spins on a simple cubic lattice of size $L\times L \times
L$, and simulate spherical particles with radius $R = L/2$ and open boundary conditions on this lattice.
One single spin flip of our Monte Carlo procedure consists of three parts. First, a spin is chosen randomly and a trial step is made (the role of which we will discuss below). Second, the change of the energy of the system is computed according Eq. \[e:ham\]. Third, the trial step is accepted with the probability from the heat-bath algorithm. Let us call one sweep through the lattice and performing the procedure explained above once per spin one Monte Carlo step (MCS).
Since we are interested in different reversal mechanisms, we designed a special algorithm which can simulate all of them efficiently. We use three different kinds of trial steps: First, a trial step in any spin direction uniformly distributed in spin space. This step does not depend on the initial direction of the spin. It samples the whole phase space efficiently and guarantees ergodicity. Second, a small step within a limited circular region around the initial spin direction. This step can efficiently simulate the coherent rotation. Third, a reflection of the spin. This step guarantees that, in the limit of large anisotropy, our algorithm crosses over to an efficient simulation of an Ising-like system. For each Monte Carlo step we use one of these different trial steps. Our algorithm then consists of a series of Monte Carlo steps using the different trial steps above. Altogether, our algorithm is ergodic, and it guarantees that all possible reversal mechanisms may occur in the system and can be simulated efficiently. As we tested by comparing simulations with different combinations of trial steps, and as we will demonstrate in the section \[s:results\_heise\], for a two-spins system, this algorithm does not artificially change our results.
Simulations of Heisenberg systems are much more time consuming, than e. g. , those of Ising systems, since the Heisenberg system has many more degrees of freedom. Apart from that, to obtain results which are comparable to our theoretical considerations we have to perform simulations in the limit of low temperatures, $T \ll T_c$, where the critical temperature $T_c$ is 1.44$J$ for anisotropy $d=0$ [@ciftan]. Here, the metastable lifetimes are long – for single runs up to $5\times10^7$MCS in our simulation. Therefore, for the Heisenberg system we had to restrict ourselves to rather small system sizes, $L=4,8,12$. However, we tried to minimize the statistical error by performing an average over many Monte Carlo runs ($100 \ldots 1000$). Since the theoretical considerations which we want to proof are for finite system sizes, and since, hence, the radius of the particle is a variable of the theory we belive that the rather small system sizes of the simulation are no disadvantage. Apart from that, for comparison we also performed Monte Carlo simulations of an Ising model where we also used larger system sizes of up to $L=28$.
The simulations were performed on an IBM-RS6000 workstation cluster and on two Parsytec CC parallel computers (8 and 24 PPC604 nodes, respectively).
Results for the Heisenberg system {#s:results_heise}
---------------------------------
We start our simulation with an initial spin configuration where all spins are pointing up (all spins ${\bf S}_i = (0,0,1)$). The magnetic field ${\bf B} = (0,0,-B)$ destabilizes the system, and after some time the magnetization of the system will reverse. The metastable lifetime $\tau$ is defined by the condition $M_z(\tau) = 0$ where $M_z$ is the $z$-component of the magnetization ${\bf M} = (1/N) \sum_i {\bf S}_i$.
=4.5cm
First, we tested our algorithm by simulating the simplest imaginable system that can show both coherent rotation and nucleation – a two-spin system. Here the energy barrier is $\Delta E_n =
2(J-B)$ for nucleation (i. e. , the spins are antiparallel during the reversal) and $\Delta E_{cr} = 2d(1-B/(2d))^2$ for coherent rotation (i. e. , the spins are always parallel during the reversal). For low temperatures we expect a behavior following thermal activation as in Eq. \[e:activation\] with the energy barriers above. Figure \[f:two\_spins\] demonstrates that in the limit of low temperatures, we actually obtain constant slopes for the $\ln \tau$ vs. $1/T$ data, and the slopes agree perfectly with the theoretical energy barriers above. Hence our simulation is in agreement with the theoretical expectations.
Now we turn to the simulation of larger systems.
=4.cm
Fig. \[f:spin\] shows spin configurations of simulated systems of size $R=6$ spins at the metastable lifetime $\tau$. For simplicity, only one central plane of the three-dimensional system is shown. The $z$-axis of the spin components is pointing up. For sufficient low anisotropy (Fig.\[f:spin\]a) the spins rotate nearly coherently. At the metastable lifetime $\tau$ the magnetization vector of the system points to any given direction in the $x-y$-plane. Therefore, as horizontal component of the spins we show here that component of the $x-y$-plane of the spin space that has the largest
=8.cm
contribution. For larger anisotropy (Fig.\[f:spin\]b) the reversal is driven by nucleation. Since it is $M_z(\tau)=0$, the domain wall at that time is in the center of the system dividing the particle into two oppositely magnetized parts of equal size.
Fig. \[f:mag\_curves\] shows the corresponding time dependence of the $z$-component of the magnetization, its absolute value $M = |{\bf M}|$, and its planar component $M_{\perp} = \sqrt{M_x^2+M_y^2}$ for the same simulation from which the spin configurations of Fig.\[f:spin\] stem. For the case of coherent rotation there is a continuous growth of the planar component of the magnetization during the reversal, while the absolute value of the magnetization remains nearly constant – apart from a small dip at the lifetime $\tau \approx 19000$MCS. For the case of nucleation the planar component of the magnetization is nearly constant zero except of a small hump at the lifetime $\tau \approx
34000$MCS. Here the absolute value of the magnetization breaks down.
These results lead us to the following approach to characterize the reversal mechanisms numerically: we determine the absolute value of the magnetization at the lifetime, $M(\tau)$. In order to obtain reasonable results we have to take an average over many runs, so that we define a quantity $\mu = [M(\tau)]$ where the square brackets denote an average over many Monte Carlo runs (or systems). This quantity should go to zero for a nucleation-driven reversal, and should be finite for coherent rotation. The maximum value of $\mu$ in the limit of low anisotropy should be the spontaneous magnetization.
In order to confirm our theoretical results numerically, we simulated $\mu$ for different values of the anisotropy $d=0.2 \ldots 2J/$spin, the magnetic field $B = 0.7 \ldots 1.2J$, and the system size $R=2 \ldots
6$ spins. We took an average over 100 ($R=6$) to 1000 ($R=2$) runs. Fig. \[f:total\_mag\] shows the results for the anisotropy dependence of $\mu$ for different system sizes and the lowest temperatures that have been simulated ($T = 0.71 \ldots 0.04J$ depending on $d$). The influence of the temperature on the simulations will be discussed later in connection with Fig. \[f:tau\_heise\].
=8.0cm
As expected, for small anisotropy, $\mu$ tends to a finite limit while with increasing anisotropy the curves converge to zero. This effect is stronger the larger the system size is — a behavior that appears to be analogous to the finite size behavior of a system undergoing a phase transition. The corresponding finite-size scaling ansatz [@heermann] is $$\mu = R^{-\beta/\nu} \tilde{\mu}^{\pm}\left(|d_{\infty} - d| R^{1/\nu}\right)$$ where the scaling function $\tilde{\mu}^{\pm} \sim x^{\beta}$ for $x
\rightarrow \infty$ so that
=8.0cm
in the limit of infinite system size it is $\mu \sim (d_{\infty}
-d)^{\beta}$ for $d < d_{\infty}$. In order to test if this scaling form can be applied to the data shown in Fig. \[f:total\_mag\], we present a corresponding scaling plot in Fig. \[f:scaling\]. The data collapse rather well, using $d_{\infty} = B/2 = 0.35J/\mbox{spin}$, $\beta/\nu = 0.5 \pm 0.1$, and $1/\nu = 0.9 \pm 0.1$.
Obviously, in the limit $R \rightarrow \infty$ $\mu(d)$ behaves like an order parameter at a second-order phase transition: it is zero for $d > d_{\infty}$ and finite for $d < d_{\infty}$, following $\mu \sim
(d_{\infty}-d)^{\beta}$.
The fact that for infinite system size the transition occurs at $d_{\infty} = B/2$ is in agreement with our theoretical considerations, since for $R \rightarrow \infty$ the region of thermally activated coherent rotation vanishes and the crossover from nucleation to (nonthermal) coherent rotation occurs at $d_c(R \rightarrow \infty) =
B/2$ (Eq. \[e:nt\]). That is, only for $d<B/2$ will the particle rotate coherently, and $\mu$ must be finite. However, to what degree the crossover from nucleation to coherent rotation for infinite system size may be described as a phase transition must be left for future research.
In the following we will restrict ourselves on systems of finite size. In order to differentiate numerically between the two reversal mechanisms for systems of finite size, we use a criterion that also comes from a study of phase transitions: we define the inflection point of the curves $d(B)$ as that value $d_c(B,R)$ where the crossover from nucleation to coherent rotation occurs. These points are shown in Fig. \[f:diagram\]. For large $B$ they agree very well with the theoretical line. For lower $B$ the numerical values are slightly too small. This systematic deviation might be due to the fact that the theoretical energy barrier for nucleation is overestimated assuming $\sigma = J/$spin: due to occurrence of extended domain walls for lower anisotropy the domain wall energy might be reduced. This also reduces the energy barrier of the nucleation process and consequently, here crossover to nucleation occurs earlier.
Apart from the numerical determination of the cross overline $d_c$ discussed above, we also tried to compute the relevant energy barriers directly. During the simulation, temperature plays a crucial role. The larger $d$ is, the larger is the energy barrier which has to be overcome by thermal activation and, hence, the higher the temperature has to be during the simulation in order to obtain results within a given computing time. On the other hand, we have to simulate as low temperatures as possible to see the behavior that is described by our theoretical considerations. Therefore, varying the anisotropy, we have to adjust the temperature. Fig. \[f:tau\_heise\] shows the temperature dependence of the metastable lifetime for different anisotropies. To save computer time, instead of an average of the lifetimes of the individual Monte Carlo runs, we calculated the median [@stauffer]. In the limit of low temperature we expect a behavior following thermal activation as in Eq. \[e:activation\], with the energy barrier following from the theoretical consideration in section \[s:theory\].
=7.0cm
As Fig. \[f:tau\_heise\] shows, this dependence (i. e. , constant slopes in Fig. \[f:tau\_heise\] for low $T$) can hardly be observed. All curves have a finite curvature even for the lowest simulated temperature except of that for $d = 0.4J$/spin, where the energy barrier is zero. We conclude that we could not reach low enough temperatures within the simulations and, hence, we analyse our data in the following way: We take the local slope of $(\ln \tau)(1/T)$ as temperature-dependent energy barrier. These energy barriers versus $d$ are shown in Fig. \[f:energy\_d\] for three different temperatures, and they are compared with the theoretical results. The magnetic field is $B=1J$. Hence, for $d < 0.5J/$spin – the nonthermal region (Eq. \[e:nt\]) – it is $\Delta E = 0$. In the regime for thermally activated coherent rotation, i. e. , between $d <
0.5J/$spin and the crossover anisotropy $d_c$ (Eq. \[e:dw\]), the energy barrier increases following Eq. \[e:ecr\]. Above $d_c$ which is roughly $0.65J/$spin, here the energy barrier remains constant since in the nucleation regime it does not depend on $d$ (Eq. \[e:edw\]).
=8.0cm
Comparing our numerical results with this theoretical curve, we find agreement as far as the crossover from nonthermal reversal to thermally driven coherent rotation is concerned. Also, we observe that in principal the numerical results seem to converge with decreasing temperature. However, within these simulation we cannot confirm the theoretical curve very accurately, especially the asymptotic behavior of the nucleation regime. We conclude that the main reason for this quantitative deviation is that we do not obtain the asymptotic low-temperature energy barriers, since these depend still on the temperature.
One additional reason for deviations in the coherent rotation regime might be that, in order to compare the results of our simulation with the theoretical results, we had to estimate the domain-wall energy density $\sigma$, which here we simply set to $\sigma = J/$spin. This estimate might be to large for a Heisenberg system which can develop extended domain walls with a lower domain-wall energy. We could try to fit $\sigma$ in such a way that it we obtain a reasonable agreement with the numerical data, but this would not solve the problem mentioned above, namely, that we are not in the asymptotic low-temperature regime.
Results for the Ising system
----------------------------
In order to confirm our theoretical results for the energy barrier of the nucleation regime, it is much more straightforward to simulate an Ising system directly instead of a Heisenberg system with large anisotropy. Therefore, we performed a standard Monte Carlo simulation of an Ising system defined by the Hamiltonian
$${\cal H} = - J \sum_{\langle ij \rangle} S_i S_j - B \sum_i S_i$$
with $S_i = \pm 1$. As before, we simulated spherical particles with radius $r = L/2$ on a simple cubic lattice of size $L\times L \times
L$ with open boundary conditions. For the Ising system, $L$ was varied from $L = 4$ to $L = 32$. We used the same methods as above, performing averages over 50-100 Monte Carlo runs depending on the system size.
Fig. \[f:tau\_ising\] shows the resulting temperature dependence of the metastable lifetime. In our data for the Ising system, a thermal activation corresponding to Eq.\[e:activation\] can be much better extracted than for the Heisenberg system. For low enough temperatures, straight lines can be fitted to the data the slope of which determine the activation barrier $\Delta E_n$.
=7.5cm
For a comparison of our numerically determined activation barriers with Eq.\[e:edw\] for the theoretical energy barriers of the nucleation process, once more we have to estimate the domain-wall energy density $\sigma$. In an Ising system, the domain-wall width is reduced to one lattice constant and, hence, the domain-wall energy cannot be reduced by an extended width of the wall. But, on the other hand, on a cubic lattice the energy of a domain wall per spin depends on the direction of the domain wall with respect to the axis of the lattice. This is $J/$spin for a wall parallel to the axis but, larger for walls in diagonal directions. Thus, we can expect $\sigma$ to be little larger than $J/$spin, and in the following it is set to $\sigma =
1.2J$.
=8.4cm
Fig. \[f:energy\_x\] compares our numerical data with Eq. \[e:edw\]. Since $\Delta E_n/(R^2\sigma)$ depends only on the variable $x = MBR/\sigma$, the data for different system sizes collapse on one single curve, and the agreement of our numerical data with the theoretical curve is satisfactory.
Conclusions
===========
We investigate the magnetization reversal of a classical Heisenberg system for single-domain ferromagnetic particles. Varying the anisotropy one expects different reversal mechanisms, the extreme cases of which are coherent rotation and nucleation, which in the case of a single-droplet-nucleation is a reversal by domain-wall motion. We studied the crossover from switching due to nucleation for high anisotropy, high fields, and large systems to coherent rotation for lower anisotropy, lower fields, and smaller systems. By a comparison of the relevant energy barriers, we derive a formula which estimates where the crossover from the one mechanism to the other occurs.
If we insert the material parameters for CoPt [@nowak], $\sigma =
0.004\mbox{J/m}^2$, and $d = 200$kJ/$\mbox{m}^3$ in Eq. \[e:dclb\], we find that for vanishing magnetic field a reversal by nucleation should occur for particles with a radius larger than 15nm. While at the moment we are not aware of any direct measurement of this crossover, experimental hints of the occurrence of different, particle-size-dependent reversal mechanism were published by Wernsdorfer et al. [@wernsdorfer].
As one important result, we found that in the limit of large particle size the region where a thermally activated coherent rotation occurs vanishes. This means that for large particles the rotation is always either nonthermal or driven by nucleation, depending only on the ratio of the driving field to the anisotropy. Second, in the nucleation regime the energy barrier is reduced, since here – in contrast to the Néel-Brown theory – for vanishing magnetic field the energy barrier is proportional to the cross-sectional area of the particle rather than its volume.
We confirmed the result above by simulations. We also tried to determine the relevant energy barriers numerically. For the case of the Heisenberg model we could hardly reach the low-temperatures limit which one needs in order to see the simplest, lowest-energy reversal mechanism. However, for an Ising model, i. e. , in the limit of infinite anisotropy we could establish our formulas for the energy barrier of a single-droplet nucleation process.
[**Acknowledgments:**]{} We thank K. D. Usadel for helpful suggestions and critically reading of the manuscript. The work was supported by the Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 166 and through the Graduiertenkolleg “Heterogene Systeme”.
R. W. Chantrell and K. O’Grady, in *Applied Magnetism*, edited by R. Gerber, C. D. Wright, and G. Asti (Kluwer Academic Publishers, Dordrecht, 1994), p. 113 C. Salling, R. O’Barr, S. Schultz, I. McFadyen, and M. Ozaki. J. Appl. Phys. [**75**]{}, 7986 (1994) M. Lederman, S. Schultz, and M. Ozaki, Phys. Rev. Lett. [**73**]{}, 1986 (1994) W. Wernsdorfer, K. Hasselbach, D. Mailly, B. Barbara, A. Benoit, L. Thomas, and G. Suran, J. Mag. Mag. Mat. [**140**]{}, 389 (1995) W. Wernsdorfer, E. Bonet Orozco, K. Hasselbach, A. Benoit, B. Barbara, N. Demoncy, A. Loiseau, H. Pascard, and D. Mailly, Phys. Rev. Lett. [**78**]{}, 1791 (1997) D. Stauffer, Int. J. Mod. Phys. C [**3**]{}, No.5, 1059 (1992) H. Tomita and S. Miyashita, Phys. Rev. B [**46**]{}, 8886 (1992). P. A. Rikvold and B. M. Gorman, in *Annual Reviews of Computational Physics I*, edited by D. Staufer (World Scientific, Singapore, 1994), p. 149 M. Acharyya and B. Chakrabarti, Phys. Rev. B [ **52**]{}, 6550 (1995) D. García-Pablos, P. García-Mochales, N. García, and P. A. Serena, J. Appl. Phys. [**79**]{}, 6019 (1996) H. L. Richards, M. Kolesik, P. A. Lindg[å]{}rd, P. A. Rikvold, and M. A. Novotny, Phys. Rev. B [**55**]{}, 11521 (1997) L. Néel, Ann. Geophys. [**5**]{}, 99 (1949) W. F. Brown, Phys. Rev. [**130**]{}, 1677 (1963) U. Nowak, J. Heimel, T. Kleinefeld, and D. Weller, Phys. Rev. B [**56**]{}, 8143 (1997) A. Aharoni, J. Appl. Phys. [**63**]{}, 5879 (1988) R. Becker and W. Döring, Ann. Phys. (Leipzig) [**24**]{}, 719 (1935) H. B. Braun, Phys. Rev. Lett. [**71**]{}, 3557 (1993) K. Binder and H. Müller-Krumbhaar, Phys. Rev. B [ **9**]{}, 2328 (1974) T. S. Ray and J. -S. Wang, Physika A [**167**]{}, 580 (1990) H. L. Richards, S. W, Sides, M. A. Novotny, and P. A. Rikvold, J. Mag. Mag. Mat. [**150**]{}, 37 (1995) K. Binder and D. W. Heermann, *Monte Carlo Simulation in Statistical Physics* (Springer-Verlag, Berlin, 1988), p. 21 F. Reif, *Fundamentals of statistical and thermal physics* (McGraw-Hill Book Company, New York 1965), p. 548 R. G. Brown and M. Ciftan, Phys. Rev. Lett. [**76**]{}, 1352 (1996)
|
---
abstract: 'In the last decade, many medical companies and research groups have tried to convert passive capsule endoscopes as an emerging and minimally invasive diagnostic technology into actively steerable endoscopic capsule robots which will provide more intuitive disease detection, targeted drug delivery and biopsy-like operations in the gastrointestinal(GI) tract. In this study, we introduce a fully unsupervised, real-time odometry and depth learner for monocular endoscopic capsule robots. We establish the supervision by warping view sequences and assigning the re-projection minimization to the loss function, which we adopt in multi-view pose estimation and single-view depth estimation network. Detailed quantitative and qualitative analyses of the proposed framework performed on non-rigidly deformable ex-vivo porcine stomach datasets proves the effectiveness of the method in terms of motion estimation and depth recovery.'
author:
- |
Mehmet Turan$^{1}$, Evin Pinar Ornek$^{2}$, Nail Ibrahimli $^{2}$, Can Giracoglu$^{2}$, Yasin Almalioglu$^{3}$,\
Mehmet Fatih Yanik $^{4}$, and Metin Sitti$^{5}$[^1][^2][^3] [^4][^5]
bibliography:
- 'mybibfile.bib'
title: '**Unsupervised Odometry and Depth Learning for Endoscopic Capsule Robots** '
---
Introduction {#sec:intro}
============
Advancements in various fields of science and technology in the last decade has opened new pathways for non-invasive examination of patient’s body and detailed investigation about diseases. Hospitals are using innovative ways to provide accurate data from inside of the human body. As an emerging example, various diseases such as colorectal cancer and inflamatory bowel disease are diagnosed by the usage of swallowable capsule endoscopes, which are non-invasive, painless, suitable to be used for long duration screening purposes which can access difficult body parts (e.g.,small intestines) better than standard endoscopy. Such benefits make swallowable, non-tethered capsule endoscopes an exciting alternative over standard endoscopy [@sitti2015biomedical; @turan2017endovo].
[0.4]{} ![Unsupervised training approach consists of two separate neural networks, one for depth prediction and another one for multi-view pose estimation. It requires unlabeled image sequences from different temporal points to establish a supervision basis. Models produce pose estimation between two views from different perspectives parameterized as 6-DoF motion, and depth prediction as a disparity map for a given view. []{data-label="fig:networks"}](figures/training.png "fig:"){width="1\linewidth"}
[0.4]{} ![Unsupervised training approach consists of two separate neural networks, one for depth prediction and another one for multi-view pose estimation. It requires unlabeled image sequences from different temporal points to establish a supervision basis. Models produce pose estimation between two views from different perspectives parameterized as 6-DoF motion, and depth prediction as a disparity map for a given view. []{data-label="fig:networks"}](figures/testing.png "fig:"){width="1\linewidth"}
Current capsule endoscope technology employed in GI tract monitoring and disease detection consists of passive devices which are locomated by random peristaltic motions. The doctor would have an easier access to fine-scale body parts and could make more intuitive and correct diagnosis in case of a precise and reliable control over the position of the capsule. Many research groups attempted to build remotely controllable active endoscopic capsule robot systems with additional functionalities such as local drug delivery, biopsy and other medical functions [@goenka2014capsule; @turanendovmfuse2017; @TURAN20181861; @turanendosensor2017; @Turan2017; @Turan2018; @turan2017endovo; @nakamura2008capsule; @munoz2014review; @carpi2011magnetically; @keller2012method; @mahoney2013managing; @yim2014biopsy; @petruska2013omnidirectional; @DBLP:journals/corr/TuranAKS17; @DBLP:journals/corr/TuranPJAKS17; @DBLP:journals/corr/TuranAJAKS17; @DBLP:journals/corr/TuranAAKS17], which are, on the other hand, heavily dependent on a real-time and precise pose estimation capability.
![Demonstration of the active endoscopic capsule robot operation using MASCE (Magnetically actuated soft capsule endoscope) designed for disease detection, drug delivery and biopsy-like operations in the upper GI-tract. MASCE is composed of a RGB camera, a permanent magnet, an empty space for drug chamber and a biopsy tool. Electromagnetic coils based actuation unit below the patient table exerts forces and torques to execute the desired motion. Doctor operates the screening, drug delivery and biopsy processes in real-time using the live video stream onto the medical workstation and the controller joystick to maneuver the endoscopic capsule to the desired position/orientation and to execute desired therapeutic actions such as drug release and biopsy.[]{data-label="fig:goal"}](figures/goal.png){width="\columnwidth"}
In this work, we propose a novel real-time localization and depth estimation approach for endoscopic capsule robots which mimic the remarkable ego-motion estimation and scene reconstruction capabilities of human beings by training an unsupervised deep neural network. The proposed network consists of two simultaneously trained sub networks, the first one assigned for depth estimation via encoder-decoder strategy, the second assigned to regress the camera pose in 6-DoF. The model observes sequences of monocular images and aims to interpret them to estimate executed camera motion in 6-DoF and the depth map of the observed scene as shown in Fig. \[fig:networks\]. Our framework estimates the camera motion and depth information in an end-to-end and unsupervised fashion directly from input pixels. Training is performed using only unlabeled monocular frames in a similar way to prior works such as [@szeliski1999prediction; @zhou2017unsupervised; @flynn2016deepstereo].
We formulate the entire pose estimation and map reconstruction pipeline for endoscopic capsule robots as a consistent and systematic learning concept which can improve its performance every day by collecting streamed data belonging to numerous patients undertaken to endoscopic capsule robot and standard endoscopy investigations in hospitals over the world. This way, we want to mimic and transfer a continuous learning functionality from medical doctors into medical robots domain, where experience and adaptation to unexpected novel situations can be much more critical to real-world scenarios.
![Illustration of the experimental setup. MASCE is a magnetically actuated robotic capsule endoscope prototype which has a ringmagnet on the body. An electromagnetic coil array consisting of nine coils is used for the actuation of the MASCE. The ringmagnet exerts magnetic force and torque on the capsule in response to the external magnetic field provide by the electromagnetic coil array. Magnetic torque and forces are also used to release drug, as well. OptiTrack system consisting of eight infrared cameras is employed for the ground truth pose estimation. An opened and oiled porcine stomach simulator is used to represent human stomach.[]{data-label="fig:opti"}](figures/opti.png){width="\columnwidth"}
To summarize, main contributions of our paper are as follows:
- To best of our knowledge, this is the first unsupervised odometry and depth estimation approach for both the endoscopic capsule robots and hand-held standard endoscopes.
- Since the network learns in a fully unsupervised manner, no ground truth pose and/or depth values are required to train the neural network.
- Neither prior knowledge nor parameter tuning is needed to recover the trajectory and depth, contrary to traditional visual odometry(VO) and deep learning(DL) based supervised odometry approaches.
- We simultaneously train a reliability mask which identifies pixels distorted by camera occlusions, non-rigid organ deformations and/or non-Lambertian surface. Such a mask is very crucial for vision based methods applied on endoscopic type of images since occlusions, non-rigid deformations and specularities violating Lambertian surface properties commonly occur in endoscopic types of images.
Evaluations we made on non-rigidly deformable porcine stomach videos prove the success of our depth estimation and localization approach. As the outline of this paper, the previous work in endoscopic capsule odometry is discussed in Section \[sec:background\]. Section \[sec:method\] introduces the proposed method with its mathematical background in detail and the unsupervised DL architecture. Section \[sec:results\] shows our experimental quantitative and qualitative results achieved for 6-DoF localization and depth recovery. Finally, Section \[sec:conclusion\] mentions some bottlenecks and gives future directions for our project. Our code will be made available at https://github.com/mpi/deep-unsupervised-endovo.
Background {#sec:background}
==========
![Training input consists of sequential images from different perspectives, which are noted by $<I_{t-1}, I_{t}, I_{t+1}>$. After view synthesis creates the supervision baseline, PoseNet is trained to estimate relative motion change between $<I_{t-1}, I_{t}>$ and $<I_{t}, I_{t+1}>$, whereas DispNet learns to predict depth for the target image $<I_{t}>$. []{data-label="fig:seq"}](figures/seq.png){width="\columnwidth"}
In the last decade, several localization methods [@than2012review; @fluckiger2007ultrasound; @rubin2006sonographic; @kim2008noninvasive; @yim20133] were proposed to calculate the 3D position and orientation of the endoscopic capsule robot such as fluoroscopy [@than2012review], ultrasonic imaging [@fluckiger2007ultrasound; @rubin2006sonographic; @kim2008noninvasive; @yim20133], positron emission tomography (PET) [@than2012review; @yim20133], magnetic resonance imaging (MRI) [@than2012review], radio transmitter based techniques and magnetic field based techniques. The common drawback of these localization methods is that they require extra sensors and hardware design. Such extra sensors have their own drawbacks and limitations if it comes to their application in small scale medical devices such as space limitations, cost aspects, design incompatibilities, biocompatibility issues and the interference of the sensors with the activation system of the device.
As a solution of these issues, a trend of VO methods have attracted the attention for endoscopic capsule localization. A classic VO pipeline typically consists of many hand-engineered parts such as camera calibration, feature detection, feature matching, outliers rejection (e.g. RANSAC), motion estimation, scale estimation and global optimization (bundle adjustment). Although some state-of-the-art algorithms based on this traditional pipeline have been developed and proposed for endoscopic VO task in the past decades, their main deficiencies such as tracking failures in low textured areas, sensor occlusion issues, lack of handling non-rigid organ deformation still remain. In last couple of years, DL techniques have been dominating many computer vision related tasks with numerous promising result, e.g. object detection, object recognition, classification problems etc. Contrary to these high-level computer vision tasks, VO is mainly working on motion dynamics and relations across sequence of images, which can be defined as a sequential learning problem.
Our proposed method solves several issues faced by typical VO pipelines, e.g the need to establish a frame-to-frame feature correspondence, vignetting artefacts, motion blur, specularity or low signal-to-noise ratio (SNR). We think that DL based endoscopic VO approach is more suitable for such challenge areas since the operation environment(GI tract) has similar organ tissue patterns among different patients which can be learned by a sophisticated machine learning approach easily. Even the dynamics of common artefacts such as non-rigidness, sensor occlusions, vignetting, motion blur and specularity across frame sequences could be learned and used for a better pose estimation, whereas our unsupervised odometry learning method additionally solves the common problem of missing labels on medical datasets from inner body operations [@turanendovmfuse2017; @turanendosensor2017].
METHOD {#sec:method}
======
{width="\textwidth"}
Different from supervised VO learning [@turan2017endovo; @turanendovmfuse2017; @turanendosensor2017], where camera poses and/or depth ground truths are required to train the neural network, the core idea underlying our unsupervised pose and depth prediction method is to make use of the view synthesis constraint as the supervision metric, which forces the neural network to synthesize target image from multiple source images acquired from different camera poses. This synthesis is performed using estimated depth image, estimated target camera pose values in 6-DoF and nearby color values from source images. In addition, a reliability mask is trained to detect sensor occlusions, non-rigid deformations of the soft organ tissue and lack of textures inside the explored organ.
View synthesis as supervision metric
------------------------------------
To provide a supervision to the neural network, view synthesis is accomplished by training with consecutive images. As input, we take a sequence of 3 consecutive frames, and choose the middle frame as a target frame. Sequences are denoted by $<I_{t-1}, I_{t}, I_{t+1}>$ where $I_{t}$ is the target view and rest of images are source views $I_{s} = <I_{t-1}, I_{t+1}>$, which are used to render the target image (see Fig. \[fig:seq\]). The objective function of the view synthesis is: $$\mathcal{L}_{vs} = \sum_{s}\sum_p | I_t (p) - \hat{I}_s(p) |$$ where $p$ is pixel coordinate, and $\hat{I}_s$ is the source view $I_s$ warped to the target view making use of the estimated depth image $\hat{D}_t$ and $4\times 4$ camera transformation matrix $\hat{T}_{t\rightarrow s}$ [@fehn2004depth]. Let $p_t$ represent the homogeneous pixel coordinates in the target view, and $K$ be the camera intrinsics matrix.
$p_t$ is projected coordinate on the source view and $p_s$ is acquired by: $$p_s \sim K \hat{T}_{t\rightarrow s} \hat{D}_t(p_t) K^{-1} p_t$$
Note that the value of $p_s$ is not discrete. To find the expected intensity value at that position, bilinear interpolation among four discrete neighbors of $p_s$ is used [@zhou2016view]: $$\hat{I}_s(p_t) = I_s(p_s) = \sum_{i\in\{top,bottom\}, j\in\{left,right\}}w^{ij}I_s(p_s^{ij})$$
Let $w^{ij}$ be the proximity value between projected and neighboring pixels summing up to one and $\hat{I}_s$ be the estimated mean intensity for projected pixel $p_s$.
View synthesis approach assumes that camera sensor is not occluded, non-rigid deformations are avoided and explored organ surface obeys Lambertian surface rules enabling photometric error minimization between target and source views. These assumptions are frequently violated in endoscopic type of videos:
1. Sensor occlusions occur often due to peristaltic organ motions.
2. Inner organs have in general a non-rigid structure meaning deformations cannot be completely avoided.
3. Organ fluids cause specularities which violate the Lambertian surface rules.
To overcome these, we trained a soft reliability mask which labels each target-source pixel pair as reliable to be used for view-synthesis or believed to violate assumptions because of being affected by occlusions, non-rigid deformations and/or specularities. Incorporating the soft-reliability mask $\hat{E}_s$, the view synthesis equation is updated as: $$\mathcal{L}_{vs} = \sum_{<I_{t-1}, I_{t+1}> \in \mathcal{S}}\sum_p \hat{E}_s(p) | I_t (p) - \hat{I}_s(p) |~.
\label{eq:exp}$$
Minimizing this energy function without regularizer will force mask to be zero across the whole image domain. To overcome this problem and obtain a reasonable mask, a regularization term is to use which describes the prior knowledge about reliability mask. Hence, let $\mathcal{L}_{reg}(\hat{E}_s^l)$ be the regularization term that minimizes the cross-entropy loss and prevents trivial solutions. Finally, since gradients are derived from differences between four neighbors and corresponding pixel intensities of source and target frames, a smoothness loss $\mathcal{L}^l_{smooth}$ is needed. The multiscale pyramid and smoothness loss for gradients are extracted from larger spatial regions. This leads to the following energy function: $$\mathcal{L}_{final} = \sum_l \mathcal{L}_{vs}^l + \lambda_s \mathcal{L}^l_{smooth} + \lambda_e \sum_{s} \mathcal{L}_{reg}(\hat{E}_s^l)~
\label{eq:final}$$ Here, $s$ indexes source images, $l$ indexes images from different scales, $\lambda_s$ is the regularization weight for depth smoothness, and $\lambda_e$ is the weight for reliability mask.
Network architecture {#sec:network_arch}
--------------------
[0.4]{} {width="\textwidth"}
[0.4]{} {width="\textwidth"}
[0.4]{} {width="\textwidth"}
[0.4]{} {width="\textwidth"}
As mentioned earlier, our problem is estimating odometry in a textureless scene by using only sequenced RGB frames as input. Since classical methods fail to cope with this problem, we use DL methods where we get our motivation from recent works [@vijayanarasimhan17] and [@zhou2017unsupervised] which propose improvements by autoencoder based architectures. Our overall DL model as shown in Fig. \[fig:deeparchitecture\] consists of two end-to-end frameworks.
The first architecture is employed to predict single-view depths by creating disparity map outputs. The encoder-decoder convolutional layers are followed by a prediction layer, whose outputs are constrained by $1/(\alpha * sigmoid(x)+\beta)$ with $\alpha = 10$ and $\beta = 0.1$ to ensure that predictions occur in a desirable interval.
The second network tries to estimate relative pose, parameterized by SE(3) motions between views, and the reliability mask. The encoder part for pose estimation and reliability mask are same, where they share weights in the first five feature extractor convolutional layers and divide into two tracks afterwards. Pose is estimated by encoder’s $6*(N-1)$ channels, as translation and rotation parameters. The decoder part consists of five deconvolutional layers and generates multiscale mask predictions. There are four output channels for each prediction layer, and each two of them predict the reliability for input source-target pairs by softmax normalization.
Both networks are trained and optimized jointly. On the other hand, both networks can be tested and evaluated independently. Testing and training pipelines are illustrated in Fig. \[fig:networks\].
EVALUATION AND RESULTS {#sec:results}
======================
Dataset and Transfer learning
-----------------------------
We used transfer learning to have an initialization for neural network weights since we lack huge amounts of labeled data. For pretraining, DL model proposed by Zhou et al.[@zhou2017unsupervised] is employed. The model is implemented with publicly available Tensorflow framework and pretrained with the KITTI dataset. Batch normalization is used for all of the layers except the outputs. Adam optimization is chosen to increase the convergence rate, with $\beta_1 = 0.9$, $\beta_2 = 0.999$, learning rate of $0.1$ and mini-batch size of $8$. We used the model which was trained with $50K$ images and converged after 150K iterations. The model requires sequential images with size 128 x 416. On top of the model pretrained by a KITTI dataset, we fine-tuned the architecture with our domain data from endoscopic capsule robot by employing a GeForce GTX 1070 model GPU. Our dataset was collected in an experimental setup for an ex-vivo parcine stomach shown in Fig. \[fig:opti\] and it contains $12K$ frames with ground truth odometry obtained by OptiTrack visual tracking system. In this experiment, we fix the length of input image sequences into three frames. We used $10K$ frames for training, $1K$ for cross validation and $1K$ for evaluation and testing.
Pose estimation and Odometry benchmark
--------------------------------------
Our pose estimation network is tested with $1K$ frames. The network outputs the pose predictions as 6-DoF motion (Euclidean coordinates for translation and rotation) between sequences. Ground truth data was established with the OptiTrack mechanism. Some examples from odometry outputs can be seen in Fig. \[fig:trajs\]. Here, we illustrate only short sequences qualitatively. It can be seen that the main trajectory results successfully differentiate the major displacements with a minor amount of drift.
[0.48]{} {width="\textwidth" height="1.5in"}
[0.48]{} {width="\textwidth" height="1.5in"}
![Sample disparity map estimations from ex-vivo porcine stomach dataset. Even though depth estimations lack fine-scale details in low textured areas, major depth differences were successfully caught. []{data-label="fig:res_depth"}](figures/Depth_List2.png){width="\columnwidth"}
We compare our ego-motion estimation method with monocular ORB-SLAM [@mur2015orb], Deep EndoVO [@turan2017endovo], LSD SLAM [@engel2014lsd] using Absolute Trajectory Error (ATE) [@mur2015orb] for the alignment with the ground truth. As shown in Fig. \[fig:trajs\] and error bars in Fig. \[fig:err\_trans\], \[fig:err\_rot\], our method outperforms ORB SLAM and LSD SLAM which are state-of-the art widely used SLAM methods. Because of the geometric and photometric properties of scenes, these methods fail to find and match proper keypoints. Magnetic localization also outperforms ORB-SLAM and LSD-SLAM, because magnetic localization does not depend on textural geometry of the scene. Even though the proposed method is unsupervised, its translational and rotational accuracies are comparable with Deep EndoVO approach which is a supervised odometry learning method.
Depth Estimation
----------------
The neural network model creates depth estimation as a disparity map for a given view. Some estimation results can be seen in Fig. \[fig:res\_depth\]. It is clear that major depth differences are captured by the network. However, since stomach surface is non-Lambertian and the light source is attached to camera, it becomes more challenging to reproduce a robust algorithm. In the disparity map output of the network, it is observable that there are minor errors at some low textured regions or on high gradient parts such as sharp edges. However, our improvement on overall depth estimation with fine-tuning can be seen in Fig. \[fig:finetune\].
CONCLUSIONS {#sec:conclusion}
===========
[0.15]{} ![Disparity map outputs before and after fine-tuning on top of KITTI. (a) shows the estimation without fine-tuning. Since there is no object in front of the camera in KITTI images, the resulting disparity maps have a dark region in the center. Moreover, the disparity map has a poor quality. After transfer learning and training with porcine stomach dataset in addition to KITTI images, the quality of the disparity map drastically increases and the dark hole in the center of the image dissapears (c). []{data-label="fig:finetune"}](figures/BeforeTrain.png "fig:"){width="1\linewidth"}
[0.15]{} ![Disparity map outputs before and after fine-tuning on top of KITTI. (a) shows the estimation without fine-tuning. Since there is no object in front of the camera in KITTI images, the resulting disparity maps have a dark region in the center. Moreover, the disparity map has a poor quality. After transfer learning and training with porcine stomach dataset in addition to KITTI images, the quality of the disparity map drastically increases and the dark hole in the center of the image dissapears (c). []{data-label="fig:finetune"}](figures/Original.png "fig:"){width="1\linewidth"}
[0.15]{} ![Disparity map outputs before and after fine-tuning on top of KITTI. (a) shows the estimation without fine-tuning. Since there is no object in front of the camera in KITTI images, the resulting disparity maps have a dark region in the center. Moreover, the disparity map has a poor quality. After transfer learning and training with porcine stomach dataset in addition to KITTI images, the quality of the disparity map drastically increases and the dark hole in the center of the image dissapears (c). []{data-label="fig:finetune"}](figures/AfterTrain.png "fig:"){width="1\linewidth"}
In this paper we applied unsupervised DL method for estimating VO and depth for endoscopic capsule robot videos. Even though our method performs comparably well to supervised EndoVO method and outperforms existing state of the arts SLAM algorithms ORB and LSD SLAM, some playroom for the improvements of the method still remains:
- Accuracy of the results can be improved by increasing sequence size of inputs. As well, additional training data generated by augmentation techniques could improve the performance of the method for cases where non-rigid deformations, occlusions and heavy specularities exist.
- Since our capsule robot also uses rolling shutter camera, instead using KITTI dataset captured by global shutter camera, we could also incorporate Cityscapes dataset captured by rolling shutter camera.
- The quality of estimated depth maps can be improved by combining the depth output of our method with shading based depth estimation. In that way, a more realistic and therapeutically relevant 3D reconstruction of the explored inner organ could be achieved.
- The dependency of the proposed method on the camera intrinsics matrix makes it rather impractical to be used for random videos streaming from hospitals with unknown calibration matrix.
- It would be interesting to extend our network to perform further tasks such as tissue segmentation and disease detection.
[^1]: $^{1}$Mehmet Turan is with Physical Intelligence Department, Max Planck Institute for Intelligent Systems, Germany [turan@is.mpg.de]{}
[^2]: $^{2}$Evin Pinar Ornek, Nail Ibrahimli, Can Giracoglu is with the Informatics Faculty, Technical University of Muenich, Germany [evin.oernek, nail.ibrahimli, can.giracoglu@tum.de]{}
[^3]: $^{3}$Yasin Almalioglu is with Computer Science Department, University of Oxford, Oxford, UK [yasin.almalioglu@cs.ox.ac.uk]{}
[^4]: $^{4}$M. Fatih Yanik is with the Department of Information Technology and Electrical Engineering, Zurich, Switzerland [yanik@ethz.ch]{}
[^5]: $^{5}$Metin Sitti is with Physical Intelligence Department, Max Planck Institute for Intelligent Systems, Germany [sitti@is.mpg.de]{}
|
---
abstract: 'We report the polarization-dependent electromagnetic response from a series of novel terahertz (THz) metasurfaces where asymmetry is introduced through the displacement of two adjacent metallic arms separated by a distance $\delta$. For all polarization states, the symmetric metasurface exhibits a low quality (Q) factor fundamental dipole mode. By breaking the symmetry, we experimentally observe a secondary dipole-like mode with a Q factor nearly $9\times$ higher than the fundamental resonance. As $\delta$ increases, the fundamental dipole mode $f_{1}$ redshifts and the secondary mode $f_{2}$ blueshifts creating a highly transmissive spectral window. Polarization-dependent measurements reveal a full suppression of $f_{2}$ for all asymmetries at $\theta \geq 60^\circ$. Furthermore, at $\delta \geq 60 \text{ }\mu\text{m}$, we observe a polarization selective electromagnetic induced transparency (EIT) for the fundamental mode. This work paves the way for applications in filtering, sensing and slow-light devices common to other high Q factor THz metasurfaces with EIT-like response.'
address: |
Electro-Optics Department, University of Dayton, 300 College Park Avenue, Dayton, OH, 45469, USA\
Department of Physics $\&$ Astronomy, Howard University, 2355 6th St. NW, Washington, DC, 20059, USA\
Physics Department, University of Dayton, 300 College Park Avenue, Dayton, OH, 45469, USA\
burrowj2@udayton.edu\
thomas.searles@howard.edu
author:
- 'Joshua A. Burrow, Riad Yahiaoui, Andrew Sarangan, Imad Agha, Jay Mathews and Thomas A. Searles'
title: 'Polarization-dependent electromagnetic responses of ultrathin and highly flexible asymmetric terahertz metasurfaces'
---
[99]{}
D. L. Medlin, K. Hattar, J.A. Zimmerman, F. Abdeljawad, and S. M. Foiles, “Defect character at grain boundary facet junctions: Analysis of an asymmetric $\Sigma = 5$ grain boundary in Fe,” In Acta Mater. [**124**]{}, 383–396 (2017).
A. Jost, M. Bendias, J. Böttcher, E. Hankiewicz, C. Brune, H. Buhmann, L. W. Molenkamp, J. C. Maan, U. Zeitler, N. Hussey, and S. Wiedmann, “Electron-hole asymmetry of the topological surface states in strained HgTe,” Proc. Natl. Acad. Sci. U.S.A. [**114**]{}(13), 3381–3386 (2017).
V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett. [**99**]{}(14), 147401 (2007).
R. Singh, I. Al-Naib, M. Koch, and W. Zhang, “Asymmetric planar terahertz metamaterials,” (12), 13044–13050 (2010). R. Singh, I. Al-Naib, M. Koch, and W. Zhang, “Sharp Fano resonances in THz metamaterials,” (7), 6312–6319 (2011). I. Al-Niab, R. Singh, C. Rockstuhl, F. Lederer, S. Delprat, D. Rocheleau, M. Chaker, T. Ozaki, and R. Morandotti, “Excitation of a high-Q subradiant resonance mode in mirrored single-gap asymmetric split ring resonator terahertz metamaterials,” Appl. Phys. Lett. [**101**]{}(7), 071108 (2012). C. Jansen, I. Al-Naib, N. Born, and M. Koch, “Terahertz metasurfaces with high q-factors,” Appl. Phys. Lett., [**98**]{}(5), 051109 (2011). R. Singh, W. Cao, I. Al-Naib, L. Cong, W. Withayachumnankul, and W. Zhang, “Ultrasensative terahertz sensing with high Q-factor Fano resonances in metasurfaces,” Appl. Phys. Lett. [**105**]{}(17), 171101 (2014). N. Born, I. Al-Naib, M. Scheller, C. Jansen, J. V. Moloney, and M. Koch, “Trapped eigen modes in terahertz asymmetric metamolecules,” 2014 39th International Conference on Infrared, Millimeter, and Terahertz waves (IRMMW-THz), Tucson, AZ, 1-2 (2014). M. Manjappa, S. Y. Chiam, L. Cong, A. A. Bettiol, W. Zhang, and R. Singh, “Tailoring the slow light behavior in terahertz metasurfaces,” Appl. Phys. Lett. [**106**]{}(18), 181101 (2015). S. Yang, C. Tang, Z. Liu, B. Wang, C. Wang, J. Li, L. Wang, and C. Gu, “Simultaneous excitation of extremely high-Q-factor trapped and octopolar modes in terahertz metamaterials,” (14), 15938–15946 (2017). X. Yin, T. Feng, S. Yip, Z. Liang, A. Hui, J. C. Ho, and J. Li, “Tailoring electromagnetically induced transparency for terahertz metamaterials: From diatomic to triatomic structural molecules,” Appl. Phys. Lett. [**103**]{}(2), 021115 (2013). Z. Zhao, X. Zheng, W. Peng, H. Zhao, J. Zhang, Z. Luo, and W. Shi, “Localized slow light phenomenon in symmetry broken terahertz metamolecule made of conductively coupled dark resonators,” Opt. Mat. Express. **7**(6), 1950–1961 (2017). L. Cong, M. Manjappa, N. Xu, I. Al-Naib, W. Zhang, and R. Singh, “Fano Resonances in Terahertz Metasurfaces: A Figure of Merit Optimization,” Adv. Optical Mater. [**3**]{}(11), 1537–1543 (2015).
Y. K. Srivastava, M. Manjappa, L. Cong, W. Cao, I. Al-Naib, W. Zhang, and R. Singh, “Ultrahigh-Q Fano Resonances in Terahertz Metasurfaces: Strong Influence of Metallic Conductivity at Extremely Low Asymmetry,” Adv. Optical Mater. [**4**]{}(3). 1–7 (2015).
M. Gupta, V. Savinov, N. Xu, L. Cong, G. Dayal, S. Wang, W. Zhang, N. I. Zheludev, and R. Singh, “Sharp Toroidal Resonances in Planar Terahertz Metasurfaces,” Adv. Mater. [**28**]{}(37), 8206–8211 (2016).
M. Gupta, Y. K. Srivastava, M. Manjappa, and R. Singh, “Sensing with toroidal metamaterial,” Appl. Phys. Lett. [**110**]{}(12), 121108 (2017). L. Cong, Y. K. Srivastava, and R. Singh, “Tailoring the multipoles in THz toroidal metamaterials,” Appl. Phys. Lett. [**111**]{}(8), 081108 (2017).
N. Xu, R. Singh, and W. Zhang, “High-Q lattice mode matched structural resonances in terahertz metasurfaces,” Appl. Phys. Lett. [**109**]{}(2), 021108 (2016). S. Yang, Z. Liu, X. Xia, Y. E, C. Tang, Y. Wang, and C. Gu, “Excitation of ultrasharp trapped-mode resonances in mirror-symmetric metamaterials,” Phys. Rev. B **93**(23), 235407 (2016). N. Zhang, Q. Xu, S. Li, C. Ouyang, X. Zhang, Y. Li, J. Gu, Z. Tian, J. Han, and W. Zhang, “Polarization-dependent electromagnetic responses in an A-shape metasurface,” (17) 20689–20697 (2017). R. Yahiaoui, J. P. Guillet, F. de Miollis, and P. Mounaix, “Ultra-flexible multiband terahertz metamaterial absorber for conformal geometry applications,” **38**(5), 4988–4990 (2013).
R. Yahiaoui, K. Takano, F. Miyamaru, M. Hangyo, and P. Mounaix,“Terahertz metamolecules deposited on thin flexible polymer: design, fabrication and experimental characterization” J. Opt. [**16**]{}(9), 094014 (2014). R. Yahiaoui, S. Tan, L. Cong, R. Singh, F. Yan and W. Zhang, “Multispectral terahertz sensing with highly flexible ultrathin metamaterial absorber,” J. Appl. Phys. **118**(8), 083101 (2015). H. Tao, A. C. Strikwerda, K. Fan, C. M. Bingham, W. J. Padilla, X. Zhang, and R. D. Averitt, “Terahertz metamaterials on free-standing highly flexible polyimide substrates,” J. Phys. D [**41**]{}(23), 232004 (2008). S. Lee, S. Kim, T. Kim, Y. Kim, M. Choi, S. H. Lee, J. Kim, and B. Min, “Reversibly stretchable and tunable terahertz metamaterials with wrinkled layouts," Adv. Mater. [**24**]{}(26), 3491–3497 (2012).
J. Li, C. M. Shah, W. Withayachumnankul, B. Ung, A. Mitchell, S. Sriram, M. Bhaskaran, S. Chang, and D. Abbott, “Mechanically tunable terahertz metamaterials,” Appl. Phys. Lett. [**102**]{}(12), 121101 (2013).
F. Zhang, S. Feng, K. Qiu, Z. Liu, Y. Fan, W. Zhang, Q. Zhao, and J. Zhou, ”Mechanically stretchable and tunable metamaterial absorber," Appl. Phys. Lett. [**106**]{}(9), 091907 (2015).
CST Microwave Studio, (http://www.cst.com).
J. Zhang, Z. Peng, A. Soni, Y. Zhao, B. Peng, J. Wang, M. S. Dresselhaus, Qihua, and Xiong, “Raman spectroscopy of few-quintuple layer topological insulator $\text{Bi}_{2}\text{Se}_{3}$ nanoplatelets,” Nano Lett., [**11**]{}(6), 2407–2414 (2011).
I. Al-Naib, Y. Yang, M. M. Dignam, W. Zhang, and R. Singh, “Ultra-high Q even eigenmode resonances in terahertz metamaterials,” Appl. Phys. Lett. [**106**]{}(1), 011102 (2015). H. Jung, C. In, H. Choi, and H. Lee, “Anisotropy modeling of terahertz metamaterials: polarization dependent resonance manipulation by meta-atom cluster,” Sci. Rep. [**4**]{}, 5217 (2014).
R. Yahiaoui, M. Manjappa, Y. K. Srivastava, and R. Singh “Active control and switching of broadband electromagnetically induced transparency in symmetric metadevices,” Appl. Phys. Lett. [**111**]{}(2), 021101 (2017).
S. Wang, L. Xia, H. Mao, X. Jiang, S. Yan, H. Wang, D. Wei, H.-L. Cui, and C. Du, “Terahertz Biosensing Based on a Polarization-Insensitive Metamaterial,” IEEE Photon. Tech. Lett. **9**(28), 986–989 (2016).
M. Amin and A. D. Khan, “Polarization selective electromagnetic-induced transparency in the disordered plasmonic quasicrystal structure,” J. Phys. Chem. C [**119**]{}(37), 21633-21638 (2015). A. D. Khan and M. Amin, “Polarization Selective Multiple Fano Resonances in Coupled T-Shaped Metasurface,” IEEE Photon. Tech. Lett. [**29**]{}(19), 1611–1614 (2017).
Introduction
============
Symmetry plays an important role in many facets of modern society including art, architecture, fashion and mathematics. Similarly, breaking symmetry can also lead to important implications with respect to the physical properties of materials. Recently, physicists have demonstrated the importance of asymmetry in the defects of crystals [@Medlin17], the transport properties of strained topological insulators [@Jost17] and the electromagnetic response of metamaterials [@Fedotov07; @Singh10; @Jansen11; @Singh11; @Al-Niab12; @Singh14; @Born14; @Manjappa15; @Yang17; @Yin13; @Zhao17]. Specifically for metamaterials, asymmetric unit cells result in Fano-like high Q factor modes desirable for applications in communications and sensing. Several methods have been explored for the achievement of high Q factor resonances with respect to planar metamaterials such as capacitative gap translation[@Cong15; @Singh10; @Manjappa15; @Srivastava15], toroidial designs[@Gupta16; @Gupta17; @Cong17], varying the periodicity [@Xu16; @Yang16], and mirrored asymmetric macro-cells[@Al-Niab12; @Yang16].
For asymmetric THz planar metamaterials, Singh *et al.* showed an increase in the Q factor of a quadropole mode when the incident **E**-field is horizontally oriented with respect to the capacitive gap versus when the incident **E**-field is vertically polarized [@Singh10]. In subsequent studies for asymmetric metasurfaces, the polarization dependence for normally incident parallel and perpendicular **E**-fields have shown transmission and frequency modulation along with excitation of new high Q resonances [@Singh11; @Al-Niab12]. However, to our knowledge, there has not been a study with respect to multiple polarized orientations of the incoming THz wave for asymmetric metasurfaces. Polarization-dependent measurements on A-shaped resonators revealed that transmission windows arose for completely different physical mechanisms from horizontally and perpendicularly polarized incident waves [@Zhang17]. Specifically for the perpendicular polarization, the transmission window stemmed from the plasmon-induced transparency effect whereas for horizontal polarization the window appeared due to the coupling of a localized LC resonance and a nearby higher order mode.
In this article, the polarization dependence of the THz response for ultrathin and highly flexible asymmetric metasurfaces is explored experimentally and presented with supporting numerical simulations. Due to the extreme mechanical flexibility of the substrate, the investigated structure is very appropriate for non-planar applications. Recently, there have been various efforts devoted to demonstrate flexible metamaterials[@Yahiaoui13; @Yahiaoui14; @Yahiaoui15]. The use of ultrathin and highly flexible substrates has provided an unprecedented route to achieve active tunability in the frequency of metamaterials due to modifications in the profiles and the periodicities of the structures when the substrates are stretched [@Tao08; @Lee12; @Li13; @Zhang15]. Here, asymmetry is introduced in a novel way by shifting the metal braces rather than shifting the position of the capacitive gap. From this approach, there is a significant amplitude modulation of a secondary dipole-like resonance as $\delta$ is increased. In addition, this high Q factor mode decreases linearly in Q and increases exponentially in modulation depth as one increases the shift. Furthermore, there is the presence of a polarization selective EIT seen for the fundamental mode at high asymmetries.
Materials and Methods
=====================
![(a) Geometrical description of the CSRR unit cell indicating the asymmetry parameter, and microscope images of CSRR (b) $\delta = 0\text{ }\mu\text{m}$, (c) $\delta = 20\text{ }\mu\text{m}$, (d) $\delta = 40\text{ }\mu\text{m}$, (e) $\delta = 60\text{ }\mu\text{m}$, and (f) $\delta = 80\text{ }\mu\text{m}$.[]{data-label="fig:1"}](1){width="\linewidth"}
The unit cell for the symmetric circle split ring resonator at $\delta = 0\text{ }\mu\text{m}$ (CSRR00) is shown in Fig. \[fig:1\](a). It features four capacitative gaps uniformly distributed along a metallic ring with the following geometric parameters: periodicity $\text{P} = 300\text{ }\mu\text{m}$, outer ring radius $r_{o} = 125\text{ }\mu\text{m}$, inner ring radius $r_{i} = 107.5 \text{ }\mu\text{m}$ and square gap length $t = 35 \text{ }\mu\text{m}$. The unit cell is then repeated periodically to form a metasurface of 100 nm thick Ag structures deposited using electron beam evaporation and patterned using optical lithography on an ultra-flexible polyimide substrate of $50.8\text{ } \mu\text{m}$ thickness. Such periodic structures do not diffract normally incident electromagnetic radiation for frequencies lower than 1 THz. Microscope images of the resulting devices are depicted in Fig. \[fig:1\](b)-(f) for $\delta = 0,\text{ } 20,\text{ } 40,\text{ } 60, \text{ and } 80\text{ }\mu\text{m}$, where the asymmetry parameter $\delta$ indicates the translational offset between two adjacent metallic arms. The total active surface area of each of the fabricated devices was 1 cm $\times$ 1 cm.
The planar metamaterial devices were characterized using a continuous-wave (CW) THz spectrometer (Teraview CW Spectra 400) that emits linearly polarized collimated THz radiation. In this setup, spectroscopy is achieved by incrementally varying the difference frequency of two near-IR diode lasers via a temperature tuning technique that is smooth and mode hop free for a spectral resolution of 100 MHz. The nominal spectral resolution is then governed by the precision of the laser frequency control and not by a mechanical delay stage as found in a conventional THz time-domain spectroscopy (TDS) setup. With a spectral resolution higher than THz TDS, we were able to resolve extremely narrow spectral features. The transmission from each sample was determined as $T(f) = P_{MM}(f)/P_{sub}(f)$, where $P_{MM}(f)$ and $P_{sub}(f)$ are the filtered THz power spectra of the planar metamaterial and polyimide substrate respectively. To investigate the polarization dependence of the symmetric and asymmetric devices, each sample was rotated by an angle $\theta$ to mimic various incident polarization states in which $\theta$ represents the angle from the vertical polarization.
To compliment the experimental results, numerical calculations were carried out using CST Microwave Studio [@CST], a finite difference time domain based technique used to model the governing equations of electrodynamics. In these calculations, the primitive cells of the designed metasurfaces were illuminated by a normally incident plane wave with an appropriate polarization of the electric and magnetic fields, as depicted in the inset of Fig. \[fig:1\](b). Periodic boundary conditions were imposed in the numerical model to mimic a 2D infinite structure. The material parameters used for the lossy Ag metal layer and ultra-flexible dielectric substrate were $\sigma = 6.1\times10^{7} \text{ S/m}$ and $\epsilon = 3.3 + i0.03$, respectively.
Results and Discussions
=======================
Fig. \[fig:2\](a)-(e) compares the measured (solid) and simulated (dashed) transmission spectra under vertical **E**-field polarization for CSRR00, CSRR20, CSRR40, CSRR60, and CSRR80. For the symmetric geometry (CSRR00), a fundamental resonant frequency $f_{1} \sim 0.65\text{ THz }$ was observed with an experimental quality factor of $Q = 4.13$. Due to the four-fold rotational symmetry imposed on the design, the performance of the device was insensitive to the polarization of the incident wave. After introducing asymmetry ($\delta \geq 20 \mu\text{m}$), $f_{1}$ redshifts with increasing asymmetry and a second mode $f_{2}$ is excited near 0.76 THz. For $f_{2}$, there was a blueshift in frequency and the transmission exponentially increased with increasing $\delta$. The latter result infers an enhancement in modulation depth for $f_{2}$ that is not seen for $f_{1}$. In general, the experimental results are well supported by numerical simulations. However, the numerical results indicate the existence of $f_{2}$ in the symmetric case, which is weakly excited and almost imperceptible in simulation and not observed experimentally. Other slight discrepancies in the results of simulations and measurements as seen in Fig. 2 can be attributed to small variations in the unit cells of each metasurface after fabrication and dispersions being unaccounted for during simulations.
![Measured (solid) and simulated (dashed) transmission spectra under $E_{y}$ polarization for (a) CSRR00, (b) CSRR20, (c) CSRR40, (d) CSRR60, and (e) CSRR80. Experimentally measured resonant frequency (f), (g) Q factor and modulation depth of $f_{1}$ and (h) Q factor and modulation depth of $f_{2}$ as a function of $\delta$.[]{data-label="fig:2"}](Figure_2.jpg){width="\linewidth"}
A Fano fit was applied to each resonance with the form: $$F(f) = A\frac{[\Lambda + (f - f_{0})/\Delta f]^{2}}{1 + (f - f_{0})^{2}}$$ where $f_{0}$ is the resonant frequency, $\Lambda$ yields the lineshape asymmetry parameter and $\Delta f$ is the full width half maximum (FWHM) of the mode [@Zhang11]. With the extracted Fano lineshape parameters from Eqn. 1, the Q factor and modulation depth for $f_{1}$ and $f_{2}$ are calculated and compared to one another with respect to the asymmetry parameter $\delta$. The Q factor of each resonance is defined as $Q = f_{i}/\Delta f \text{ for } i \in \{1,2\}$ where $f_{i}$ is the resonant frequency and $\Delta f$ is the FWHM. By tailoring the asymmetry in the structure, the frequency shift, modulation depth and Q factor of both $f_{1}$ and $f_{2}$ are tuned and each attribute is plotted in Fig. \[fig:2\](f), (g) and (h), respectively. Both $f_{1}$ and $f_{2}$ shift linearly with increasing asymmetry at a rate of -1.058 GHz/$\mu$m for $f_{1}$ and 1.236 GHz/$\mu$m for $f_{2}$ as illustrated in Fig. \[fig:2\](f). Furthermore, there is an increasing flattop ultra-broadband transmission window as the asymmetry increases indicated by the shaded area in Fig. \[fig:2\](f). At $\delta = 80 \mu$m, the transmission bandwidth is maximal at $\sim$ 200 GHz. This result shows great promise for these asymmetric metasurfaces for applications as frequency selective devices, or THz bandpass filters.
With respect to Q factor, at $\delta = 20 \mu$m the Q factor for $f_{1}$ is 8.4 and decreases to 5.3 at 80 $\mu$m. These values are on the order of fundamental modes (LC, dipole, etc.) for symmetric metasurfaces [@Singh10; @Xu16]. The Q factor of $f_{2}$ is 35.7 at 20 $\mu$m and decreases exponentially to 15.8 at 80 $\mu$m. As a comparison, the Q factor observed for the asymmetric case at 20 $\mu$m is one of highest experimental Q factors reported and more than 8$\times$ greater than that of $f_{1}$. For THz metasurfaces, high experimental Q factors have been reported for toroidal (Q = 42.5)[@Gupta16] and low asymmetric gap translations (Q = 49)[@Srivastava15] while other high Q factor modes introduced by asymmetry are on the order of 20-30 [@Fedotov07; @Al-Niab12; @Al-Naib15]. To evaluate how the Q factors evolve with increasing asymmetry, a linear fit was applied to the Q factor relation for $f_{1}$ revealing a $-0.049\mu\text{m}^{-1}$ slope. Similarly, for the Q factor of $f_{2}$, an exponential decay was observed and the fit equation was given by $Q = 15.05 + 52.60\exp-0.047\delta$. Decreases in Q factor with increasing degrees of asymmetry have previously been reported [@Fedotov07; @Singh11; @Jansen11] and attributed to increases in radiation loss from an enlarged variation in the surface charge distributions between two excited split-ring resonators and a resultant suppression of the dipole moment in the far field [@Yang17]. To further analyze the results, the modulation depth was defined as $MD = A - \text{min}(T)$ where $A$ is the maximum value of the asymmetric lineshaped transmission dip nearest to the resonant frequency. A proportional relationship between modulation depth and asymmetry was previously reported for trapped and octopolar modes[@Yang17; @Al-Naib15]. For $f_{1}$, there is a subtle exponential decrease from 0.95 at 20 $\mu$m to 0.93 at 80 $\mu$m indicating that the coupling strength of $f_{1}$ is nearly unaffected by the asymmetric modification. However, for $f_{2}$, there was enhancement of the modulation depth up to 40% with increasing $\delta$.
![CSRR00 (a) experimental and simulation transmission spectra and (b) surface current (white arrows) and magnetic field distribution $\sim 0.65$ THz. CSRR80 (c) experimental and simulation transmission spectra, surface currents (white arrows) and magnetic field distributions at (d)$f_{1}\sim0.58$ THz and (e) $f_{2}\sim 0.8$ THz.[]{data-label="fig:3"}](Figure_3.png){width="\linewidth"}
To gain understanding of the physical mechanism for the electromagnetic responses, the surface currents and magnetic field distributions for each metasurface were simulated and results for the CSRR00 and CSRR80 for vertical **E**-field polarization are shown in Fig. \[fig:3\](b), (d), and (e). For the CSRR00 device, the surface current **J** of $f_{1} \sim 0.65$ THz portrays a strong dipolar nature with $\textbf{J}$ converging to the left gap. This is similar to the dipole resonance seen in other two gap CSRRs [@Fedotov07; @Born14]. For the CSRR80 device near $f_{2} \sim 0.8$ THz, the simulated surface current **J** and magnetic field |**H**| amplitude indicate a dipole oscillation in the upper left and lower right arms of the CSRR where the surface current converges to the opposite gap from $f_{1}$. Intuitively, the redshift of the fundamental mode $f_{1}$ could be attributed to the increase in the surface current path of the metasurface, while $f_{2}$ is blueshifted because of the decrease in the current path. These frequency shifts of opposite signs suggest that $f_{2}$ originated from the structural asymmetry of each meta-atom rather than from mimicking its fundamental mode. Indeed, if $f_{2}$ originated from other higher order resonances, $f_{2}$ must be redshifted as much as $f_{1}$ is redshifted because all the higher order frequencies depend on the wavelengths of their fundamental frequencies. However, in our structure, upon increasing the asymmetry ($\delta \geq 20\text{ } \mu\text{m}$) $f_{1}$ is redshifted while $f_{2}$ is blueshifted, thus giving rise to a broadband transparency window \[see Fig. 3(c)\]. Moreover, $f_{2}$ vanished for the CSRR00 when the current paths became perfectly symmetrical. Similar behavior has previously been reported in anisotropic THz metamaterials [@Jung14].
![Evolution of the measured transmission spectra for different values of the shift $\delta$ = 20, 40, 60 $\text{and } 80\text{ }\mu\text{m}$, respectively and under different polar angles $\theta$ varying from $0^{\circ}$ to $90^{\circ}$.[]{data-label="fig:4"}](Figure_4.jpg){width="0.9\linewidth"}
To further understand the nature of the resonances, the effect of polarization was measured by rotating each device to mimic various linear polarization states. Fig. \[fig:4\](a) shows the transmission spectra for $\delta = 20\text{ (purple)}, 40\text{ (blue)}, 60\text{ (green)} \text{ and }80\text{ } \mu$m (red) under a vertically polarized THz wave and the **E** is parallel to the direction of asymmetric shift. Fig. \[fig:4\](b) - (f) depicts the experimental transmittance as the device was rotated counter clockwise by $\theta$ until the $\textbf{E}$ field was orthogonal to shift direction. For the cases when $\delta = 20 \text{ and } 40 \text{ }\mu\text{m}$, the resonant frequency and minimum transmission amplitude of $f_{1}$ oscillates sinusoidally with respect to polar angle $\theta$ and can easily be understood by Malus’ Law. However, for $\delta \geq 60 \text{ }\mu\text{m}$ under the cross polarization ($\theta = 45^{\circ}$) excitation state, a mode splitting of $f_{1}$ was observed. This polarization selective splitting observed for high asymmetries is indicative of electromagnetic induced transparency (EIT) previously seen for A-shaped THz metasurfaces [@Zhang17] and other plasmonic systems [@Yahiaoui17; @Amin15]. The change in coupling strength of the left and right modes for $f_{1}$ continues for $\theta = 60^{\circ}$and then returns to one resonance above $\theta = 75^{\circ}$. Mode $f_{2}$ solely exhibits transmittance amplitude modulation and is fully suppressed when the $\textbf{E}$-field is orthogonal to the direction of translation. The vertical dotted lines indicate the frequency invariance (i.e. the frequency is independent of the polarization) of $f_{2}$ for all asymmetric cases where $\theta \leq 45^{\circ}$. The resonant frequencies of $f_{2}$ are 0.76, 0.78, 0.81, and 0.83 THz for CSRR20, CSRR40, CSRR60, and CSRR80. It is well known that frequency invariance with respect to the polarization insensitivity of high Q factor modes can be advantageous to metasurface sensing applications [@Wang16].
As previously stated, when $\theta = 45^{\circ}$ and upon increasing the asymmetry, the spectral response of the metasurface has a similar signature to the EIT effect, where a transmission peak appears at about 0.657 THz with an amplitude of $57\%$ between two resonance dips at around 0.587 THz and 0.7 THz, respectively \[see Fig. 4(c)\]. Figure 5(a)-(c) shows the simulated (dashed lines) and measured (solid lines) transmission spectra for different values of the asymmetry $\delta = 0, 60$ and $80~\mu\text{m}$, respectively and under a fixed polar angle of $45^{\circ}$. Although there are minor differences in amplitude and bandwidth of the resonances (probably due to microscopic imperfections induced during the fabrication process), the transmission spectra obtained from numerical simulations confirm the trends obtained from measured data. To gain a deeper insight into the physics of the EIT-like effect, we have plotted in Figs. 5(e)-(g) the magnetic field distributions at $f_{l}$, $f_{t}$ and $f_{h}$, respectively. Here, we define the lower resonance frequency, the transmission peak, and the higher resonance frequency as $f_{l}$, $f_{t}$ and $f_{h}$. To better guide the eye, the directions of the THz induced surface currents are indicated by white arrows in Figs. 5(e)-(g). For the lower frequency dip at $f_{l}$, the CSRR is highly radiative. It exhibits parallel surface current distributions on the right and left arms and on the upper and lower arms as well. This surface current distribution possesses a strong field confinement suggesting direct dipole coupling with the external incident radiation wave.
![Measured (solid) and simulated (dashed) transmission spectra excited by cross polarized THz radiation for (a) CSRR00, (b) CSRR60, and (c) CSRR80 devices. (d) Simulated group delays associated with the CSRR80 (red solid line) that exhibit EIT at 0.657 THz and the reference kapton film (gold dashed line), respectively. (e)-(g) Magnetic field distributions at $f_{l}$, $f_{t}$ and $f_{h}$, respectively for the CSRR80. The corresponding induced surface currents are indicated by white arrows.[]{data-label="fig:5"}](Figure_5.png){width="\linewidth"}
Regarding the transmission peak at $f_{t}$, it is found that the surface currents in the right and left arms are strongly suppressed and mainly flow along the upper and lower arms, as shown in Fig. 5(f). The corresponding magnetic field mainly concentrates in the upper and lower arms as well. In this case, the CSRRs couple weakly with the incident THz radiation. Therefore, it can be deduced that the transparency window appears due to the suppression of the dipole response of the right and left arms of the CSRRs. For the higher frequency dip at $f_{h}$, the surface currents flow along the right and left arms, where the corresponding magnetic field is mainly localized \[see Fig. 5(g)\]. Thus, it can be deduced from the current and magnetic field distributions that dipole oscillations are excited by the incident wave for $f_{h}$.
Although the experimental portion of this investigation was carried out using a CW source, it is worth noting the group delay response for pulsed THz slow light applications. The group delay $\tau_{g}$, retrieved from the simulated transmission spectrum of CSRR80, is represented in Fig. 5(d). The group delay of the kapton substrate is also plotted in Fig. 5(d) for comparison. The group delay was calculated using $\tau_{g}=-d\phi(\omega)/d\omega$, where $\phi(\omega)$ is the transmission phase and $\omega=2\pi f$ with $f$ representing the frequency. One can observe that at frequencies away from the resonances, the group delay of THz radiation tends to be equal to that of kapton film. At the resonance frequencies of 0.587 THz and 0.7 THz the metamaterial demonstrates negative group delay. In the vicinity of the transparency peak, large positive group delays are obtained, indicating potential use in slow light applications. For example, at around 0.657 THz where 57% of EIT transmission is achieved, the THz radiation experiences a delay of about 2.6 ps, which is equivalent to the time delay of a 780 $\mu\text{m}$ distance of free space propagation.
Conclusions
===========
In summary, we have experimentally and numerically characterized the polarization-dependent electromagnetic responses of ultra-flexible asymmetric THz metasurfaces and demonstrated a high experimental Q factor ($\sim$35.7) resonance in the transmission spectrum. The modulation depth of the asymmetric mode was increased by displacing adjacent metallic arms and also by modifying the interaction between the metasurface and the incident linear polarization state of incident THz radiation. The induced Fano shaped resonance was investigated through simulations, polarization dependence and asymmetry, revealing the dipole like and frequency invariant nature of the mode. The asymmetric metasurfaces studied here show great promise for low cost devices based platforms that include THz bandpass filters, biological sensors and slow-light devices.
Funding {#funding .unnumbered}
=======
National Science Foundation (NSF) (ECCS-1541959, ECCS-1710273, ECCS-1709200); NASA Ohio Space Grant (NNX15AL50H); Air Force Office of Scientific Research (FA9550-16-1-0346).
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to acknowledge support from Air Force Research Laboratory Minority Leadership Program. This work was performed in part at the Harvard Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Coordinated Infrastructure Network (NNCI). T. A. S. acknowledges support from the CNS Scholars Program.
|
---
abstract: 'For discrete-time linear systems subject to parametric uncertainty described by random variables, we develop a sampling-based Stochastic Model Predictive Control algorithm. Unlike earlier results employing a scenario approximation, we propose an offline sampling approach in the design phase instead of online scenario generation. The paper highlights the structural difference between online and offline sampling and provides rigorous bounds on the number of samples needed to guarantee chance constraint satisfaction. The approach does not only significantly speed up the online computation, but furthermore allows to suitably tighten the constraints to guarantee robust recursive feasibility when bounds on the uncertain variables are provided. Under mild assumptions, asymptotic stability of the origin can be established.'
address:
- 'Institute for Systems Theory and Automatic Control, University of Stuttgart, Germany'
- 'CNR-IEIIT, Politecnico di Torino, Italy'
author:
- Matthias Lorenzen
- Fabrizio Dabbene
- Roberto Tempo
- Frank Allgöwer
bibliography:
- 'IEEEabrv.bib'
- 'GlobalBib.bib'
title: Stochastic MPC with Offline Uncertainty Sampling
---
,
,
,
stochastic model predictive control, receding horizon control, control of constrained systems, stochastic control, data-based control
Introduction
============
In recent years, Stochastic Model Predictive Control (SMPC) has received significant attention for constrained control of systems, where a probabilistic description of disturbances and uncertainties is given [@Kouvaritakis2016_MPCBook]. The online solution of stochastic constrained control remains a challenging problem, which, among other methods, has been addressed using probabilistic approximations, for which significant progress has been made over the last two decades [@Tempo2012_RandAlgForAnalysisAndDesign]. These probabilistic approximations often employ sampling-based methods, which have the advantage of being computationally tractable and are able to handle general types of stochastic uncertainty, including parametric uncertainty that enters nonlinearly in the system dynamics and constraints or has a non-convex support. For stochastic optimization problems that are convex in the optimization variables, in [@Calafiore2006_ScenarioApproachRobContrDesign] and subsequently [@Calafiore2010_RandomConvexPrograms; @Campi2008_ExactFeasibilityOfRandSolOfUncertainConvexProgs; @Campi2011_SampleAndDiscardApprTpCCOpt] the authors studied the so-called “scenario approach”. Tight bounds on the necessary number of samples were derived to provide probabilistic guarantees that the *solution* of the approximate sampling-based program satisfies the original chance constrained problem [@Alamo2009_RandStrategiesforProbSolOfUncertaintFeasAndOptProbs; @Calafiore2010_RandomConvexPrograms; @Campi2008_ExactFeasibilityOfRandSolOfUncertainConvexProgs; @Campi2011_SampleAndDiscardApprTpCCOpt]. These results have been applied to SMPC for linear systems with stochastic uncertainty, e.g. [@Calafiore2013StochasticMPC] , and are shown to be less conservative compared to robust outer approximations [@Deori2014_CompApproachesToRMPC]. While sampling allows for nearly arbitrary uncertainty in the system, two key issues are (i) the large number of constraints that are generated, which increases the computational and memory requirements significantly [@Zhang2014_OnSampleSizeOfRandMPCwAppl], thereby reducing its applicability in SMPC [@Mesbah2014_StochasticNonlinearMPC], and (ii) the lack of guaranteed recursive feasibility [@Deori2014_CompApproachesToRMPC; @Schildbach2014_ScenarioApproachForSMPC].
The first problem has been addressed through exploiting structural properties of the optimization program in SMPC, e.g. [@Schildbach2014_ScenarioApproachForSMPC; @Zhang2014_OnSampleSizeOfRandMPCwAppl]. In [@Schildbach2014_ScenarioApproachForSMPC] only the first step violation probability is considered, which provides bounds on the closed-loop constraint violation, but increases the probability of not being recursively feasible. Furthermore, the lack of rigorous guarantees for asymptotic stability is a critical point. Structural problems of recursive feasibility in SMPC have been highlighted in [@Primbs2009_StochRecedingHorizonContrOfConstrLinSysWithMultNoise] and a solution using a combination of robust and stochastic constraint tightening for linear systems with parametric disturbances has been proposed in [@Fleming2014_StochasticTubeMPCforLPVSysWithSetInclCond]. The main contribution of this paper is to propose a solution that alleviates the two mentioned disadvantages. The SMPC controller design is based on a sampling approach in the *offline* design, as suggested in [@Mayne2014_MPCSurvey]. In contrast to previously mentioned online sampling approaches, scenarios are generated offline and only *necessary* samples are kept for online optimization. Since the tight bounds derived in the scenario approach cannot be directly used, we reformulate the problem to exploit results from statistical learning theory. We provide precise statements on the necessary sample complexity such that, within a user-specified confidence, constraint satisfaction is guaranteed. Following an approach similar to [@Lorenzen2015_improvedConstrTighteningForSMPC; @Lorenzen2015_ConstrTighteningAndStabInSMPC], we extend the idea of an additional first-step constraint to guarantee recursive feasibility. Conditions are stated under which the closed-loop system is asymptotically stable with probability one.
Preliminary results have been presented in the conference paper [@Lorenzen2015_ScenarioBasedStochasticMPC], where an offline sampling approach has been shown to be advantageous with respect to computational requirements and recursive feasibility, but lacking rigorous bounds on the sample complexity. In contrast, here we first illustrate the difference between online and offline uncertainty sampling in receding horizon control and based thereon provide bounds on the number of samples to guarantee chance constraint satisfaction when the uncertainty is sampled offline. A first step constraint to guarantee recursive feasibility has previously been introduced in [@Lorenzen2015_ConstrTighteningAndStabInSMPC]. The paper introduces a non-conservative constraint tightening and discusses the relation to performance and stability in Stochastic MPC for systems with additive disturbance.
The remainder of this paper is organized as follows. Section \[sec:ProbSetup\] introduces the scenario receding horizon problem and briefly reviews the underlying theory. In Section \[sec:offlineSampling\] the difference between offline and online sampling is highlighted and suitable bounds on the sample complexity are derived. Based thereon, the SMPC design is given, starting with a suitable cost and constraint reformulation, followed by the derivation of additional constraints for recursive feasibility. The section ends with the complete SMPC algorithm and closed-loop properties. Finally, Section \[sec:Concl\] provides some conclusions and proposes directions for future work.
#### Notation {#notation .unnumbered}
The notation employed is standard. Uppercase letters are used for matrices and lower case for vectors. $[A]_j$ and $[a]_j$ denote the $j$-th row and entry of the matrix $A$ and vector $a$, respectively. Blackboard boldface letters (e.g. ${\mathbb{W}}$) denote sets, with the exception of ${\mathbb{P}}$ and ${\mathbb{E}}$ for probability measure and expected value. ${\mathbb{P}}^n$ denotes the $n$-fold product probability measure, and $\operatorname{supp}(\cdot)$ the support of a random variable. The set of integers from $i$ to $j$ is denoted by ${\mathbb{N}}_i^{j} := \{i,i+1,\dots,j\}$. We use the double index notation to denote predicted states and inputs, e.g. $x_{l|k}$ for state $x$ predicted $l$ steps ahead from time $k$ with $x_{0|k} = x_k$. Bold letters, e.g. $\mathbf x_k = [x_{0|k}^\top \dots x_{T-1|k}^\top]^\top$, are used to denote the stack vector of $T$ predicted values.
Problem Setup and Preliminary Results {#sec:ProbSetup}
=====================================
The uncertain discrete-time system to be controlled is given by $$x_{k+1} = A(q_k) x_k + B(q_k) u_k \label{eqn:xUncertainSystem}$$ with state $x_k \in {\mathbb{R}}^{n}$, control input $u_k \in {\mathbb{R}}^{m}$ and uncertainty $q_k \in {\mathbb{R}}^{n_q}$. The system matrices $A(q_k)$ and $B(q_k)$, of appropriate dimensions, are subject to stochastic parametric disturbance. The parameter $q_k$ can enter nonlinearly under the following assumption on $A(q_k)$ and $B(q_k)$.
\[ass:uncertaintySetSys\] The parameters $q_k \in {\mathbb{R}}^{n_q}$ for $k \in {\mathbb{N}}$ are realizations of independent and identically distributed (iid) multivariate, real valued random variables $Q_k$. Let ${\mathbb{G}}= \{ (A(q_k),B(q_k))\}_{q_k \in \operatorname{supp}(Q_k)}$, a polytopic outer approximation $\bar {\mathbb{G}}= \operatorname{co}\{(A^j,B^j)_{j\in {\mathbb{N}}_1^{N_c}}\}\supseteq {\mathbb{G}}$ exists and is known.
For the sake of robust recursive feasibility, we assume a known bound on the set ${\mathbb{G}}$. This assumption could be relaxed to e.g. a confidence region for the parameters, which subsequently leads to a notion of probabilistic recursive feasibility. Since the robust outer bound is used only for a one-step prediction, choosing a large outer bound does not lead to as conservative constraint tightening as in Robust MPC.
The system is subject to individual chance constraints on the state and hard constraints on the input
$$\begin{aligned}
{\mathbb{P}}\{ [H_x]_j x_{l|k} &\le 1 ~|~ x_k \} \ge 1-\varepsilon_j, \quad &&\forall l\in {\mathbb{N}}_+ \label{seqn:probConstraints} \\
H_u u_{l|k} &\le 1, \quad &&\forall l\in {\mathbb{N}}\label{seqn:inputConstraints}
\end{aligned}$$
\[eqn:origConstraints\]
with $\varepsilon_j \in (0,1)$ for $j \in {\mathbb{N}}_1^{p}$.
The control objective is to design a stabilizing receding horizon control, which guarantees constraint satisfaction and minimizes $J_\infty$, the expected value of an infinite horizon quadratic cost $$ J_\infty = \sum_{i=0}^\infty {\mathbb{E}}\left\{x_i^\top Qx_i + u_i^\top Ru_i \right\}
\label{eqn:infHorizonCost}$$ with $Q\in {\mathbb{R}}^{n\times n}$, $Q \succ 0$, $R\in {\mathbb{R}}^{m\times m}$, $R \succ 0$.
Online Sampling-based SMPC algorithm
------------------------------------
The SMPC design we propose is closely related to the previously mentioned Scenario MPC [@Calafiore2013StochasticMPC; @Schildbach2014_ScenarioApproachForSMPC], where the uncertainty is sampled *online*. We briefly recall the most relevant results and assumptions of this approach.
The design relies on the standard assumption of the existence of a suitable terminal set ${\mathbb{X}}_T$ and an asymptotically stabilizing control gain for .
There exists a terminal set ${\mathbb{X}}_T \allowbreak=\allowbreak \{x ~|~ {H_T x \le 1}\}$, which is robustly forward invariant for under the control law $u_k=Kx_k$. Given any $x_k \in {\mathbb{X}}_T$, the state and input constraints are satisfied and there exists $P \in {\mathbb{R}}^{n\times n}$ such that $$Q + K^\top R K + {\mathbb{E}}\left[ A_{cl}(Q_k)^\top P A_{cl}(Q_k) \right] - P \preceq 0$$ with $A_{cl}(Q_k) = A(Q_k) + B(Q_k)K$.
A parameterized feedback policy $$u_{l|k} = Kx_{l|k} + v_{l|k}
\label{eqn:prestabContr}$$ is employed, where, for a given $x_{0|k} = x_k$, the correction terms $\{v_{l|k}\}_{l\in{\mathbb{N}}_{0}^{T-1}}$ are determined by the SMPC algorithm as the minimizer of the expected finite horizon cost $$\begin{gathered}
J_T(x_k,\mathbf{v}_{k}) \\
= {\mathbb{E}}\left\{ \sum_{l=0}^{T-1} \left( x_{l|k}^\top Qx_{l|k} + u_{l|k}^\top Ru_{l|k} \right) + x_{T|k}^\top P x_{T|k} ~|~ x_k\right\}.
\label{eqn:finiteHorizonCostFnc}\end{gathered}$$ Note that $J_T(x_k,\mathbf{v}_k)$ is a convex function in $\mathbf v_k$ which can be computed explicitly, as shown in Section \[sec:MainRes\].
Instead of evaluating the chance constraints directly, in [@Calafiore2013StochasticMPC; @Schildbach2014_ScenarioApproachForSMPC] a sampling-based approximation is used. Let $\mathbf Q_k=\{Q_k,Q_{k+1},\dots, Q_{k+T-1}\}$ and let $\mathbf{q}^{(i)}_k=\{q^{(i)}_{0|k} ,q^{(i)}_{1|k},\dots, q^{(i)}_{T-1|k}\}$, $i\in{\mathbb{N}}_1^{N_s}$ denote randomly drawn samples from $\mathbf Q_k$. The *finite horizon scenario program* is
\[eqn:SampleBasedMPCOptimization\] $$\begin{aligned}
\min_{\mathbf v_k} ~& J_T\left( x_k,\mathbf v_k \right) \label{seqn:SampleBasedMPCCost}\\
\text{ s.t. }
& x_{l+1|k}^{(i)} = A(q^{(i)}_{l|k})x_{l|k}^{(i)} + B(q^{(i)}_{l|k})u_{l|k}^{(i)}, \quad x_{0|k}^{(i)}=x_k\nonumber\\
& u_{l|k}^{(i)}=Kx_{l|k}^{(i)}+v_{l|k}\nonumber\\
&\begin{aligned}
& H_x x_{l|k}^{(i)} \le 1 \quad &&\forall l \in {\mathbb{N}}_1^{T-1}\\
& H_u u_{l|k}^{(i)} \le 1 \quad &&\forall l \in {\mathbb{N}}_0^{T-1}\\
& H_T x_{T|k}^{(i)} \le 1 \\
& \text{for all } i \in {\mathbb{N}}_1^{N_s}.
\end{aligned}\label{seqn:SampleBasedMPCConstr}
\end{aligned}$$
As shown in the following proposition, probabilistic guarantees that the solution to satisfies the original chance constraints can be derived, see [@Alamo2015_RandAlgForDesignOfUncertainSys; @Calafiore2010_RandomConvexPrograms; @Campi2008_ExactFeasibilityOfRandSolOfUncertainConvexProgs; @Tempo2012_RandAlgForAnalysisAndDesign] for the underlying theory, and [@Alamo2015_RandAlgForDesignOfUncertainSys] for the particular sample complexity given below.
\[thm:scenario\] Given $f(x,\theta):({\mathbb{R}}^d \times S_\theta) \rightarrow {\mathbb{R}}$, convex in $x$ for any fixed $\theta \in S_\theta$ and assume $\theta \sim {\mathbb{P}}$ with support $S_\theta$. Let $\Theta = \{\theta^{(1)}, \dots,\theta^{(N_s)}\}$ be a multisample of $\theta$ with the sample complexity $$N_s \ge N(d,\varepsilon,\delta) = \frac{1}{\varepsilon}\frac{\e}{\e-1}\left( \ln\frac{1}{\delta} + (d-1) \right)
\label{eqn:numberOfSamples}$$ for $\varepsilon,\delta\in(0,1)$ and $\e$ being the Euler’s number. For $$\begin{aligned}
x^* = \arg \min_x &~c^\top x \\
\text{s.t. } & f(x,\theta^{(i)}) \le 0, \quad i\in{\mathbb{N}}_1^{N_s},
\end{aligned}$$ with confidence $1-\delta$, it holds ${\mathbb{P}}\{f(x^*,\theta)\le0\}\ge 1-\varepsilon$.
This result can be applied to receding horizon control, in particular the finite horizon program , leading to the following corollary, originally derived in [@Calafiore2013StochasticMPC Proposition 3.1 b)].
\[prop:NumberOfSamples\] Let $N_s\ge N(Tm,\varepsilon,\delta)$ with $\varepsilon \le \varepsilon_j$ be chosen according to and $\mathbf v_k^{*}$ be a feasible, optimal solution for . If $\mathbf v_k^{*}$ is applied to the discrete time dynamical system for a finite horizon of length $T$, then, with at least confidence $1-\delta$, the original chance constraints are satisfied for $l=0,\dots,T-1$.
When applying $\mathbf v_k^*$ in a finite horizon control problem, the terminal region constraint and hard input constraints are only met with probability $1-\varepsilon$ and confidence $1-\delta$, as well. When the control law is applied in a receding horizon fashion, then the hard constraints are met as long as remains feasible [@Calafiore2013StochasticMPC]. Note that new scenarios need to be generated at each sampling time.
In [@Schildbach2013_RandomizedSolutions] the results are extended to problems involving multiple chance constraints, which could be applied here to decrease the conservativeness and number of samples.
Offline Uncertainty Sampling for SMPC {#sec:offlineSampling}
=====================================
In the first part of this section, the difference between online and offline uncertainty sampling in SMPC is illustrated, followed by bounds on the number of samples to guarantee chance constraint satisfaction when the uncertainty is sampled offline. In the second part, the [offline]{} sampling based SMPC design is derived and the system theoretic properties, recursive feasibility, constraint satisfaction and stability are proven.
\[ex:circConstr\] Consider the following linear system with parameter $\gamma \in (0,1)$ $$x_{k+1} =
\begin{bmatrix}
1 & \gamma\\ -\gamma & 1
\end{bmatrix}
x_k +
\begin{bmatrix}
1 & 0\\ 0 & 1
\end{bmatrix} u_k$$ and a single chance constraint $${\mathbb{P}}\left\{
\begin{bmatrix}
\cos(\alpha) & \sin(\alpha)
\end{bmatrix} u_k \le 1
\right\} \ge 1-\varepsilon
\label{eqn:exChanceConstr}$$ where $\alpha \sim \mathcal U[0,2\pi]$. For $\varepsilon < 0.5$ the chance constraint set is equal to the constraint $\|u_k\| \le \frac{1}{\cos(\varepsilon\pi)}$. The situation is depicted in Figure \[fig:subsetExample\], where a realization of sampled constraints is given by the blue lines and the area inside the red circle is the feasible region of the chance constraint.
Let the running cost be given by ${\ell}(x_k,u_k) = \|x_k\|^2 + \|u_k\|^2$, then, for $\|x_k\|$ sufficiently large, the optimal input is at a vertex of the polytope given by the sampled constraints. If we draw new samples at each time step, the probability of the chance constraint being satisfied equals the probability of one *random* vertex being inside the red circle. This holds with confidence $1-\delta$ by Theorem \[thm:scenario\], if the sample complexity is chosen according to . In contrast, when the same set of samples is used and $\|x_0\|$ is sufficiently large, then the optimal solution will switch *deterministically* between the vertices, depending on the system dynamics and cost. For some values of $\gamma$ and the data of the sampled constraint, it will be ‘trapped’ at one vertex.
Note that the system and constraint can easily be transformed into the form and .
![Feasible values $u$ for the chance constraint (solid red), the corresponding robust constraint (dotted gray), and a realization of sampled constraints (solid blue) with one vertex violating the chance constraint.[]{data-label="fig:subsetExample"}](RandSubset)
The previous example highlights the difference between offline and online sampling in Stochastic MPC. Let, as defined above, $\mathbb Q_k = \{\mathbf{q}^{(i)}_k\}_{i=1,\ldots,N_s}$ be the samples used to solve the finite horizon scenario program at time $k$ and define $\mathcal F_k=\{q_0, \mathbb Q_0, \dots, q_k, \mathbb Q_k\}$. For $k>0$, the state $x_k$ is a (nonlinear) function not only of all past uncertainty realizations $q_0,\dots,q_{k-1}$, but as well in the sampled scenarios $\mathbb{Q}_0,\dots,\mathbb{Q}_{k-1}$. If the scenarios $\mathbb Q_k$ are sampled independently, then for each realization of $\mathcal F_k$, the SMPC optimization reduces to a scenario approximation of a stochastic optimization program with the random variable $q_k$. Using conditional probabilities, the scenario theory and Theorem \[thm:scenario\] can be applied, as shown e.g. in [@Calafiore2013StochasticMPC]. In contrast, if the sampled scenarios are kept constant $\mathbb Q_k = \mathbb Q$, the argument does not hold true and the assumptions of the scenario approach are not satisfied. In fact, due to the system dynamics, each sampled constraint $H_x x_{l|k}^{(i)} \le 1$ depends on *all* sampled scenarios as $x_{l|k}$ is a function in $\mathbb Q$. Instead of using the scenario results, which guarantee chance constraint satisfaction only for the *optimal solution*, in the following, an approach based on an *inner approximation of the feasible set* is proposed. The number of samples is chosen such that, with confidence $1-\delta$, the sampled constraints are an inner approximation of the chance constraints and hence, with confidence $1-\delta$, each *feasible* solution to the finite horizon scenario program satisfies the original chance constraints.
A Sampling Subset Result {#ssec:subsetResult}
------------------------
Let $a$ be a multivariate random variable with realizations in ${\mathbb{R}}^{1 \times d}$, distribution ${\mathbb{P}}$ and $\varepsilon,\delta \in (0,1)$. Define the set $\mathbb{C}^{P}$ described by a single chance constraint $$\mathbb{C}^{P} = \left\{ x \in {\mathbb{R}}^d ~|~ {\mathbb{P}}\{ a x \le 1 \}\ge1-\varepsilon \right\} \label{eqn:linChanceConstr}$$ as the set of those $x$, where the constraint $a x \le 1$ is satisfied with probability $1-\varepsilon$. Let $a^{(i)}$, $i=1,\dots,N_s$ be $N_s$ iid copies of $a$ and define the second (random) set $$\mathbb{C}^{S} = \left\{ x \in {\mathbb{R}}^d ~|~ a^{(i)}x \le 1, \quad i\in {\mathbb{N}}_1^{N_s} \right\}. \label{eqn:linCCSampledSet}$$ We derive the following proposition on the probability of the set $\mathbb{C}^{S}$, described by sampled constraints, being a subset of the set $\mathbb{C}^{P}$, described by the chance constraint. The result is a corollary from [@Alamo2009_RandStrategiesforProbSolOfUncertaintFeasAndOptProbs Corollary 4, Theorem 8] and is based on statistical learning theory. The proof on the bound of the VC-Dimension of the class of linear half spaces of the form $\{x ~|~ \theta x\le 1\}$ is given in Appendix \[app:VCDim\].
\[prop:subsetProb\] For any $\varepsilon \in (0,1)$, $\delta \in (0,1)$ and $$N_s \ge \tilde N(d,\varepsilon,\delta) = \frac{5}{\varepsilon}\left(\ln \frac{4}{\delta} + d \ln \frac{40}{\varepsilon}\right)
\label{eqn:boundSubsetNumSamples1}$$ it holds $${\mathbb{P}}^{N_s}\left\{ \mathbb{C}^S \subseteq \mathbb{C}^P \right\} \ge 1-\delta.$$ If $\varepsilon \in (0,0.14)$, the result holds true for the lower bound $$N_s \ge \frac{4.1}{\varepsilon}\left( \ln\frac{21.64}{\delta} + 4.39 d \log_2\left( \frac{8 \e}{\varepsilon} \right) \right).
\label{eqn:boundSubsetNumSamples2}$$
Note that, while Theorem \[thm:scenario\] guarantees, with confidence $1-\delta$, satisfaction of the original chance constraint for the optimal solution of the sampled program, Proposition \[prop:subsetProb\] implies that, with confidence $1-\delta$, *all* feasible points of the sampled constraints satisfy the chance constraint in . In the following section, this result is used to derive a Stochastic MPC scheme based on offline uncertainty sampling which guarantees chance constraint satisfaction and robust recursive feasibility. While the necessary number of samples (or ) is larger than , the samples are drawn offline and redundant samples can be removed.
Offline Sampling SMPC Design {#sec:MainRes}
----------------------------
For the following analysis, we explicitly solve equation with prestabilizing input for the predicted states $x_{1|k}, \dots, x_{T|k}$ and predicted inputs $u_{0|k}, \dots, u_{T-1|k}$. With suitable matrices $\Phi^0_{l|k}(\mathbf q_k)$, $\Phi^u_{l|k}(\mathbf q_k)$ and $\Gamma_l$ (given in Appendix \[app:matrices\]) we get $$\begin{aligned}
x_{l|k}(\mathbf q_k) &= \Phi^0_{l|k}(\mathbf q_k)x_k + \Phi^u_{l|k}(\mathbf q_k) \mathbf v_k\\ u_{l|k}(\mathbf q_k) &= K x_{l|k} + v_{l|k} \\
&= K\Phi^0_{l|k}(\mathbf q_k)x_k + (K\Phi^u_{l|k}(\mathbf q_k)+\Gamma_l) \mathbf v_k. \end{aligned}
\label{eqn:xuexplicit}$$
#### Cost Function and Constraint Evaluation {#cost-function-and-constraint-evaluation .unnumbered}
Given the solution , the expected value of the cost can be solved explicitly offline, leading to a quadratic, finite horizon cost function $$J_T(x_{k},\mathbf{v}_{k}) = \begin{bmatrix} x_k^\top &\mathbf{v}_k^\top \end{bmatrix} \tilde Q \begin{bmatrix} x_k \\ \mathbf{v}_k \end{bmatrix} \label{eqn:finiteHorizonCostFncDet}$$ in the deterministic variables $x_{k}$ and $\mathbf v_k$ with appropriate $\tilde Q$ given in Appendix \[app:matrices\]. Using the sampling approach developed in Section \[ssec:subsetResult\], an inner approximation for the chance constraints can be derived in the form of linear constraints on $x_k$ and $\mathbf v_k$. As before let $\delta \in (0,1)$ be a probabilistic level and for each constraint let $\mathbf q^{(i)}$ be independently drawn samples from $\mathbf Q_k$. For $l\in{\mathbb{N}}_1^{T-1}$ and $j\in {\mathbb{N}}_1^p$, the chance constraints describe the sets $${\mathbb{X}}_l^{P,j} = \left\{ x_k, \mathbf v_k ~|~ {\mathbb{P}}\{[H_x]_j x_{l|k}(\mathbf q_k) \le 1\} \ge 1-\varepsilon_j \right\}
\label{eqn:chanceConstrSet}$$ for which, with confidence $1-\delta$, $${\mathbb{X}}_l^{S,j} = \left\{ x_k, \mathbf v_k ~|~ [H_x]_j x_{l|k}(\mathbf q^{(i)}) \le 1, \quad i\in{\mathbb{N}}_1^{N_{l}} \right\}
\label{eqn:sampledChanceConstrSet}$$ is an inner approximation, if $N_l \ge \tilde N(n+lm,\varepsilon_j,\delta)$.
Similarly, the input and terminal constraints can be approximated. As shown in [@Lorenzen2015_improvedConstrTighteningForSMPC; @Lorenzen2015_ConstrTighteningAndStabInSMPC], for stability and constraint satisfaction it suffices to consider stochastic approximations of the hard constraints on the *predicted* input and terminal constraint. However, the hard input constraint is enforced for the *applied* input $u_{0|k} = v_{0|k}$. Choose $\varepsilon_h \in (0,1)$ and define $$\begin{aligned}
{\mathbb{U}}_l^{S,j} &= \left\{ x_k, \mathbf v_k ~|~ [H_u]_j v_{l|k}(\mathbf q^{(i)}) \le 1, \quad i\in{\mathbb{N}}_1^{N_l^u} \right\}, \\
{\mathbb{X}}_T^{S,j} &= \left\{ x_k, \mathbf v_k ~|~ [H_T]_j x_{T|k}(\mathbf q^{(i)}) \le 1, \quad i\in{\mathbb{N}}_1^{N_T} \right\}
\end{aligned}
\label{eqn:sampledInputAndTermConstr}$$ for $l\in{\mathbb{N}}_0^{T-1}$ and $N_T \ge \tilde N(n+Tm,\varepsilon_h,\delta)$, $N_l^u \ge \tilde N(n+lm,\varepsilon_h,\delta)$.
Note that all sets are described by linear constraints, which, due to the sampling procedure, are in general highly redundant. Since the samples are drawn offline, redundant constraints can be easily removed offline with the following algorithm. More sophisticated constraint removal methods already implemented for Matlab can be found e.g. in the MPT toolbox [@MPT3_Toolbox].
Constraint $H x \le h$ $H \in {\mathbb{R}}^{n_c \times n}$, $h\in {\mathbb{R}}^{n_c}$ solve $$\begin{aligned}
h_i^{*} = \max_x~ & [H]_i x\\
\text{ s.t. } & [H]_k x \le [h]_k \quad \forall k\in {\mathbb{N}}_1^{n_c}\setminus i
\end{aligned}$$ $H \gets H\setminus[H]_i$ $h \gets h\setminus[h]_i$ $H,~h$
In the following, we assume that the intersection of the sampled constraint sets and with redundant constraints removed is given by $${\mathbb{D}}= \left\{ x_k, \mathbf v_k ~|~
\begin{bmatrix} \tilde H & H \end{bmatrix}
\begin{bmatrix} x_k \\ \mathbf v_k \end{bmatrix} \le h \right\}.
\label{eqn:nonredundantConstraints}$$
#### Recursive Feasibility {#recursive-feasibility .unnumbered}
Scenario MPC based on repeatedly solving , cannot guarantee recursive feasibility and hence neither constraint satisfaction nor asymptotic stability of the origin [@Deori2014_CompApproachesToRMPC; @Schildbach2014_ScenarioApproachForSMPC]. In contrast, sampling the uncertainty offline allows to suitably augment the constraints in such a way that recursive feasibility is recovered.
Similar to [@Lorenzen2015_ScenarioBasedStochasticMPC; @Lorenzen2015_ConstrTighteningAndStabInSMPC], a first step constraint is added to , as follows. Let $${\mathbb{C}}_T = \left\{ \begin{bmatrix} x_k \\ v_{0|k} \end{bmatrix} \in {\mathbb{R}}^{n+m} ~\left|~
\begin{array}[h]{l}
\exists v_{1|k}, \ldots, v_{T-1|k} \in {\mathbb{R}}^m, \\
\text{s.t. } (x_k, \mathbf v_k) \in {\mathbb{D}}\end{array}\right.
\right\}$$ be the $T$-step set and feasible first input. The set can be computed via a direct projection, which may introduce numerical difficulties, or recursively with the option for suitable approximations to alleviate the computational complexity in each projection step. The set ${\mathbb{C}}_T$ defines the set of feasible states and first inputs of the finite horizon scenario program with given, fixed samples $\mathbf{q}^{(i)}$. Let $${\mathbb{C}}_{T,x}^\infty = \{x ~|~ H_{\infty}x\le h_{\infty}\}$$ be a (maximal) robust control invariant polytope for the system with the constraint $(x,u) \in {\mathbb{C}}_T$. Given the polytopic outer bound $\bar {\mathbb{G}}$, this set can be computed via standard recursions; for algorithms and their finite termination see [@Blanchini2015_SetTheoreticMethodsInControl Section 5.3][@Kerrigan2000_Thesis_RobustConstraintSatisfaction-InvariantSetsAndPredictiveContr] and references therein[^1].
In order for the SMPC optimization to be robustly recursively feasible, the additional constraint set $${\mathbb{D}}_R = \left\{ x_k, \mathbf v_k ~|~
H_{\infty}A_{cl}^{j} x_k + H_{\infty}B^{j}v_{0|k} \le h_\infty, ~ j\in {\mathbb{N}}_1^{N_c}
\right\}
\label{eqn:firstStepConstraint}$$ with $A^j,~B^j$ from Assumption \[ass:uncertaintySetSys\] and $A_{cl}^j = A^j + B^jK$ needs to be intersected with .
#### Sampling-Based SMPC Algorithm {#sampling-based-smpc-algorithm .unnumbered}
The complete sampling-based SMPC algorithm can be divided into two parts: (i) an offline computation of the involved sets and removal of redundant constraints and (ii) the repeated online optimization. In the following, we present the algorithm and state its control theoretic properties.\
*Offline:* Compute the expected value to determine the explicit cost matrix $\tilde Q$ in . Draw a sufficiently large number of samples to determine the sampled constraints and . Remove redundant constraints to get . Determine the first step constraint .\
*Online:* For each time step $k \in {\mathbb{N}}$
1. Measure the current state $x_k$.
2. Determine the minimizer of the quadratic cost subject to the linear constraints and $$\begin{aligned}
\mathbf{v}_k^* ~&= \arg \min_{\mathbf{v}_k}
\begin{bmatrix} x_k^\top &\mathbf{v}_k^\top \end{bmatrix} \tilde Q \begin{bmatrix} x_k \\ \mathbf{v}_k \end{bmatrix}\\
\text{s.t.} ~
&(x_k,\mathbf v_k) \in {\mathbb{D}}\cap {\mathbb{D}}_R.
\end{aligned}
\label{eqn:MPCOptProg}$$
3. Apply $u_k = Kx_k + v^*_{0|k}$.
The redundant constraints obtained combining and can be removed offline as well. They are kept separate here to emphasize the conceptually different constraints.
Properties of the SMPC Algorithm {#sec:MainRes2}
--------------------------------
\[prop:recFeas\] Let ${\mathbb{V}}(x_k) = \left\{ \mathbf v_k \in {\mathbb{R}}^{Tm} ~|~ (x_k,\mathbf v_k) \in {\mathbb{D}}\cap {\mathbb{D}}_R \right\}$. If $\mathbf v_k \in {\mathbb{V}}(x_k)$, then ${\mathbb{V}}(x_{k+1}) \neq \emptyset$ for every realization $q_k$ and $x_{k+1} = A_{cl}(q_k)x_k + B(q_k)v_{0|k}$.
From $(x_k, \mathbf v_k) \in {\mathbb{D}}_R$ it follows $x_{k+1} \in {\mathbb{C}}_{T,x}^\infty$ robustly and by construction ${\mathbb{C}}_{T,x}^\infty \subset \{ x ~|~ {\mathbb{V}}(x)\neq \emptyset\}$.
\[prop:closedLoopConstraintViolation\] If $x_0 \in {\mathbb{C}}_{T,x}^\infty$, the closed-loop system under the proposed SMPC control law satisfies the hard input constraints robustly and, with confidence $1-\delta$, the probabilistic constraint for all $k\ge1$.
Hard input constraint satisfaction follows from robust recursive feasibility (Proposition \[prop:recFeas\]) and the constraint $H_u u_{0|k} \le 1$ which does not rely on sampling.
For all $j=1,\dots,p$ we have ${\mathbb{D}}\subseteq {\mathbb{X}}_1^{S,j}$ and by Proposition \[prop:subsetProb\], with confidence $1-\delta$, it holds ${\mathbb{X}}_1^{S,j} \subseteq {\mathbb{X}}_1^{P,j}$. Hence, for all feasible $(x_k,\mathbf v_k) \in {\mathbb{D}}$ the chance constraint ${\mathbb{P}}\{ [H_x]_j x_{1|k} \le 1 ~|~ x_k \} \ge 1-\varepsilon$ is satisfied, which suffices for .
Note that, due to sampling offline, the confidence $1-\delta$ remains the same for all times $k\ge1$. This determines the probability that chance constraint satisfaction does not hold and should therefore be chosen sufficiently small.
We have shown that the SMPC algorithm remains recursively feasible, i.e. that a solution to can be found. A stronger assumption usually given, namely that the candidate solution $\tilde v_{l|k+1}=v^*_{l+1|k}$ with $\tilde v_{T-1|k+1}=0$ remains feasible at time $k+1$, is in general impossible to guarantee with sampling-based SMPC. Yet, asymptotic stability of the origin can be proved under a further assumption on the probability that the candidate solution remains feasible.
\[ass:upperLowerCostBound\] Let $V_T(x)$ be the optimal value function of and let $P_l, P_u \in {\mathbb{R}}^{n\times n}$, $P_l\succ 0$, $P_u \succ 0$, be such that $x^{\top}P_l x \le V_T(x) \le x^{\top}P_u x$ holds $\forall x \in {\mathbb{C}}_{T,x}^\infty$.
If the candidate solution does not remain feasible, the matrices in Assumption \[ass:upperLowerCostBound\] allow to derive bounds on the cost increase. The matrix $P_l$ is naturally given by the unconstrained infinite horizon cost, while the upper bound can be computed taking into account the vertices of the feasible set ${\mathbb{C}}_{T,x}^\infty$.
\[prop:asStab\] Let $\varepsilon_f$ be an upper bound on the probability that the candidate solution does not remain feasible and let $P_l,~ P_u \in {\mathbb{R}}^{n\times n}$ satisfy Assumption \[ass:upperLowerCostBound\]. If $$\begin{bmatrix}
Q - \frac{\varepsilon_f}{1-\varepsilon_f}(A^\top P_u A - P_l) & -\frac{\varepsilon_f}{1-\varepsilon_f}A^\top P_u B\\
-\frac{\varepsilon_f}{1-\varepsilon_f}B^\top P_u A & R - \frac{\varepsilon_f}{1-\varepsilon_f}(B^\top P_u B)\\
\end{bmatrix} \succ 0
\label{eqn:posdefStabCond}$$ holds for all $A,B \in {\mathbb{G}}$, the origin is asymptotically stable with probability 1 under the proposed SMPC scheme.
It suffices to check for the vertices of the set $\bar {\mathbb{G}}$, since it can be recast as an LMI in $A$ and $B$ using Schur complement. If $\bar {\mathbb{G}}$ is given by interval matrices, the results in [@Alamo2008_NewVertexResultForRobustProbsWithIntervalMatrices] can be applied to further reduce the number of LMIs that need to be checked.
To prove asymptotic stability, the optimal value function of the online optimization program can be used as a stochastic Lyapunov function. Let $V_T(x_k) = J_T(x_k,\mathbf{v}_{k}^*)$ be the optimal value of at time $k$.
We first consider the case that the candidate solution remains feasible at time $k+1$. Let ${\mathbb{E}}\{ V_T(x_{k+1}) | x_k,\allowbreak \mathbf{\tilde v}_{k+1} \text{ feasible}\}$ be the expected optimal value at time $k+1$, conditioning on the state at time $k$ and feasibility of the candidate solution $\tilde v_{l|k+1}=v^*_{l+1|k}$ with $\tilde v_{T-1|k+1}=0$ $$\begin{aligned}
& {\mathbb{E}}\left\{ V_T(x_{k+1}) ~|~ x_k, \mathbf{\tilde v}_{T|k+1}~\text{feasible} \right\} - V_T(x_k) \\
\le&~ {\mathbb{E}}\left\{ J_T(x_{k+1}, \mathbf{\tilde v}_{T|k+1}) ~|~ x_k \right\} - V_T(x_k) \\
=&~ {\mathbb{E}}\left\{\sum_{l=0}^{T-1} \left( \|\tilde x_{l|k+1}\|_Q^2 + \|\tilde u_{l|k+1}\|_R^2 \right) + \|\tilde x_{T|k+1}\|_{P}^2 ~|~ x_k \right\} \\
& - {\mathbb{E}}\left\{ \sum_{l=0}^{T-1} \|x_{l|k}^*\|_Q^2 + \|u_{l|k}^*\|_R^2 + \|x_{T|k}^*\|_P^2 ~|~ x_k \right\} \\
=&~{\mathbb{E}}\left\{ \|x_{T|k}^*\|_{(Q+K^\top R K)}^2 + \|A_{cl}(q_{k+T}) x_{T|k}^*\|_P^2 \right.\\
&\quad \left. - \|x_{0|k}\|_Q^2 - \|u_{0|k}^*\|_R^2 - \|x_{T|k}^*\|_P^2 ~|~ x_k \right\} \\
\le& - \|x_{k}\|_Q^2 - \|u_{k}\|_R^2.
\end{aligned}$$ In case the candidate solution does not remain feasible, Assumption \[ass:upperLowerCostBound\] can be employed to compute an upper bound of the Lyapunov function increase $$\begin{aligned}
& {\mathbb{E}}\left\{ V_T(x_{k+1}) ~|~ x_k, \mathbf{\tilde v}_{T|k+1}~\text{not feasible} \right\} - V_T(x_k) \\
\le & \max_{(A,B)\in{\mathbb{G}}}\|Ax_{k}+Bu_k\|_{P_u}^2 - \|x_{k}\|_{P_l}^2.
\end{aligned}$$ Let $\lambda_{\min}$ be a lower bound of the smallest eigenvalue of the matrix in Proposition \[prop:asStab\] for all $(A,B)\in{\mathbb{G}}$. Then, by the law of total probability, it holds $$\begin{aligned}
& {\mathbb{E}}\left\{ V_T(x_{k+1}) ~|~ x_k \right\} - V_T(x_k) \\
\le & -\left(1-\varepsilon_f)( \|x_{k}\|_Q^2 + \|u_{k}\|_R^2 \right) \\
& \quad
+ \varepsilon_f \left( \max_{(A,B)\in{\mathbb{G}}}\|Ax_{k}+Bu_k\|_{P_u}^2 - \|x_{k}\|_{P_l}^2 \right)\\
\le & -(1-\varepsilon_f)\lambda_{\min}\|x_k\|_2^{2},
\end{aligned}$$ which is a sufficient condition for asymptotic stability with probability 1.
Conclusions and Future Work {#sec:Concl}
===========================
We introduced an offline sampling-based Stochastic MPC scheme for linear systems subject to a parametric disturbance, which can be described by a multivariate random process. Unlike previous contributions that are based on online sampling and scenario approximations of the stochastic optimization program, we proposed an easy-to-implement offline sampling scheme for constraint design. We provided insight into the difference between online and offline sampling in sampling-based SMPC. In particular we have shown that the question of finding a lower bound on the necessary sample complexity can be reduced to the question of a sample approximation being a subset of the original chance constrained set. Results from statistical learning theory have been employed to give explicit bounds, thereby guaranteeing, with a user chosen confidence, constraint satisfaction in closed loop operation. The approach has the advantage to allow further constraint modifications to guarantee robust recursive feasibility and, under further assumptions, asymptotic stability with probability 1 of the origin. The main disadvantage compared to Scenario MPC is the increased complexity and the computational requirements in the design phase as well as the restriction to iid disturbance sequences. Alleviating the latter, e.g. through allowing disturbance sequences that are generated by a dynamical system driven by white noise, and considering unbounded disturbances in combination with probabilistic guarantees of recursive feasibility are future, open research topics.
VC-Dimension {#app:VCDim}
============
Note that half-spaces of the form $\{x \in {\mathbb{R}}^n ~|~ ax \le 1 \}$ always include the origin, which reduces the VC-Dimension from $n+1$ to $n$.
The VC-Dimension of the class of linear half-spaces which include the origin, $\mathbb H_0 = \{x \in {\mathbb{R}}^n ~|~ ax \le 1 \}$, $a\in {\mathbb{R}}^{1 \times n}$ is less or equal than $n$.
We show that a set $\mathbb X = \{x_1,\dots,x_{n+1}\}$ of cardinality $n+1$ cannot be shattered by $\mathbb H_0$. Let $\mathbb X$ be given. By Radon’s Theorem the set $\tilde {\mathbb X} = \{0\} \cup {\mathbb{X}}$ can be partitioned into two set $\mathbb X_1$ and $\mathbb X_2$ such that $\operatorname{co}({\mathbb X_1}) \cap \operatorname{co}(\mathbb X_2) \neq \emptyset$, where $\operatorname{co}({\mathbb{X}})$ denotes the convex hull of the elements of $\mathbb X$. Without loss of generality assume $0 \in \mathbb X_1$. Assume it exists $a \in {\mathbb{R}}^{1 \times n}$ such that $ax_i \le 1$ for all $x_i \in \mathbb X_1$ and $ax_j > 1$ for all $x_j \in \mathbb X_2$, hence $ax \le 1$ for all $x \in \operatorname{co}(\mathbb X_1)$ and $ax > 1$ for all $x \in \operatorname{co}(\mathbb X_2)$ which contradicts $\operatorname{co}({\mathbb X_1}) \cap \operatorname{co}(\mathbb X_2) \neq \emptyset$.
Matrices {#app:matrices}
========
For completeness, the matrices used in the paper are given in the following. We use $I_n$ to denote the $n\times n$ identity matrix and $0_{n \times m}$ to denote a zero matrix in ${\mathbb{R}}^{n \times m}$.
#### Solution matrices {#solution-matrices .unnumbered}
The matrices $\Phi^0_{l|k}$, $\Phi^u_{l|k}$ and $\Gamma_l$ are obtained by solving the dynamics with prestabilizing input explicitly for the predicted state $x_{l|k}$ and input $u_{l|k}$. For $l=0$ we obtain $\Phi^0_{0|k}= I_n$ and $\Phi^u_{0|k}=0_{n\times mT}$. For $l\ge 1$ with the notation $A_{cl}(q_{l|k}) = A(q_{l|k})+B(q_{l|k})K$ and $A^{cl}_{l|k} = A_{cl}(q_{l|k})$ we have $$\begin{aligned}
\Phi^{0}_{l|k} &= A^{cl}_{l-1|k} A^{cl}_{l-2|k} \cdots A^{cl}_{0|k}, \\
\Phi^u_{l|k} &= \begin{bmatrix} A^{cl}_{l-1|k}\cdots A^{cl}_{1|k} B_{0|k} & \dots & B_{l-1|k} & 0_{ln\times (T-l)m} \end{bmatrix}.
\end{aligned}$$ The matrix $\Gamma_l$ selects the $l-th$ entry in the stack vector $\mathbf v_k$$$\Gamma_l = \begin{bmatrix}
0_{m\times lm} & I_m & 0_{m\times (T-l-1)m}
\end{bmatrix}.$$
#### Cost matrix {#cost-matrix .unnumbered}
Let $$\Phi_T(\mathbf q_k) =
\begin{bmatrix}
\Phi^{0}_{0|k} & \Phi^u_{0|k} \\
\vdots & \vdots \\
\Phi^{0}_{T|k} & \Phi^u_{T|k} \\
\end{bmatrix}, \quad
\Gamma =
\begin{bmatrix}
0_{mT \times n} & I_{mT \times mT} \\
\end{bmatrix},$$ $\bar Q = I_T \otimes Q$, $\bar R = I_T \otimes R$ and $\bar K = I_{T} \otimes K$. The explicit cost matrix $\tilde Q$ in is then given by $$\begin{aligned}
\tilde Q =
&{\mathbb{E}}\Bigl\{ \Phi_T(\mathbf q_k)^{\top}
\begin{bmatrix}
\bar Q & 0_{nT \times n}\\ 0_{n \times nT} & P
\end{bmatrix}
\Phi_T(\mathbf q_k) \\
& \qquad +
\begin{bmatrix}
\bar K \Phi_{T-1}(\mathbf q_k) + \Gamma
\end{bmatrix}^{\top}
\bar R
\begin{bmatrix}
\bar K \Phi_{T-1}(\mathbf q_k) + \Gamma
\end{bmatrix}
\Bigr\}
\end{aligned}
\label{eqn:costMatrix}$$ where the expected value can be solved to the desired accuracy using an appropriate numerical integration rule.
[^1]: Matlab implementations of those algorithms as part of a toolbox can be found in [@MPT3_Toolbox; @Kerrigan2000_Thesis_RobustConstraintSatisfaction-InvariantSetsAndPredictiveContr].
|
---
abstract: 'We consider the <span style="font-variant:small-caps;">$\Pi$-free Deletion</span> problem parameterized by the size of a vertex cover, for a range of graph properties $\Pi$. Given an input graph $G$, this problem asks whether there is a subset of at most $k$ vertices whose removal ensures the resulting graph does not contain a graph from $\Pi$ as induced subgraph. Many vertex-deletion problems such as <span style="font-variant:small-caps;">Perfect Deletion</span>, <span style="font-variant:small-caps;">Wheel-free Deletion</span>, and <span style="font-variant:small-caps;">Interval Deletion</span> fit into this framework. We introduce the concept of *characterizing a graph property $\Pi$ by low-rank adjacencies*, and use it as the cornerstone of a general kernelization theorem for <span style="font-variant:small-caps;">$\Pi$-Free Deletion</span> parameterized by the size of a vertex cover. The resulting framework captures problems such as <span style="font-variant:small-caps;">AT-Free Deletion</span>, <span style="font-variant:small-caps;">Wheel-free Deletion</span>, and <span style="font-variant:small-caps;">Interval Deletion</span>. Moreover, our new framework shows that the vertex-deletion problem to perfect graphs has a polynomial kernel when parameterized by vertex cover, thereby resolving an open question by Fomin et al. \[JCSS 2014\]. Our main technical contribution shows how linear-algebraic dependence of suitably defined vectors over $\mathbb{F}_2$ implies graph-theoretic statements about the presence of forbidden induced subgraphs.'
author:
- 'Bart M.P. Jansen'
- 'Jari J.H. de Kroon'
bibliography:
- 'references-utf8.bib'
title: 'Preprocessing Vertex-Deletion Problems: Characterizing Graph Properties by Low-Rank Adjacencies'
---
Introduction {#sec:introduction}
============
Preliminaries {#sec:preliminaries}
=============
Framework based on low-rank adjacencies {#sec:framework}
=======================================
Incidence vectors and characterizations
---------------------------------------
As a first step towards our kernelization framework for <span style="font-variant:small-caps;">$\Pi$-free Deletion</span>, we introduce an incidence vector definition (${\mathrm{inc}}$) that characterizes the neighborhood of a given vertex. Compared to the vector encoding used by Agrawal et al. [@AGRAWAL2019] for <span style="font-variant:small-caps;">Interval Deletion</span>, our vector definition differs because it supports arbitrarily large subsets (they consider subsets of size at most two), and because an entry of a vector simultaneously prescribes which neighbors should be present, and which neighbors should *not* be present.
\[def:inc\] Let $G$ be a graph with vertex cover $X$ and let $c \in \mathbb{N}$. Let $Q',R' \subseteq X$ such that $|Q'| + |R'| \leq c$. We define the *$c$-incidence vector* ${\mathrm{inc}^{c,(Q',R')}_{(G,X)}}(u)$ for a vertex $u \in V(G) \setminus X$ as a vector over $\mathbb{F}_2$ that has an entry for each $(Q,R) \in X \times X$ with $Q \cap R = \emptyset$ such that $|Q|+|R| \leq c$, $Q' \subseteq Q$ and $R' \subseteq R$. It is defined as follows:
$${\mathrm{inc}^{c,(Q',R')}_{(G,X)}}(u)[Q,R] =
\begin{cases}
1 & \text{if $N_G(u) \cap Q = \emptyset$ and $R \subseteq N_G(u)$,}\\
0 & \text{otherwise.}
\end{cases}$$
We drop superscript $(Q',R')$ if both $Q'$ and $R'$ are empty sets. The intuition behind the superscript $(Q',R')$ is that it projects the entries of the full incidence vector ${\mathrm{inc}^{c}_{(G,X)}}$ to those for supersets of $Q', R'$. The $c$-incidence vectors can be naturally summed coordinate-wise. For ease of presentation we do not define an explicit order on the coordinates of the vector, as any arbitrary but fixed ordering suffices.
If the sum of some vectors equals some other vector with respect to a certain graph $G$, then this equality is preserved when decreasing $c$ or taking induced subgraphs of $G$.
\[note:incsubgraph\]\[note:largerrank\] Let $G$ be a graph with vertex cover $X$, let $c \in \mathbb{N}$, and let $D \subseteq V(G)$ be disjoint from $X$. If $v \in V(G) \setminus(D \cup X)$ and ${\mathrm{inc}^{c}_{(G,X)}} (v) = \sum_{u \in D} {\mathrm{inc}^{c}_{(G,X)}} (u)$, then
- ${\mathrm{inc}^{c'}_{(G,X)}} (v) = \sum_{u \in D} {\mathrm{inc}^{c'}_{(G,X)}} (u)$ for any $c' \leq c$, and
- ${\mathrm{inc}^{c}_{(H,X \cap V(H))}} (v) = \sum_{u \in D} {\mathrm{inc}^{c}_{(H,X \cap V(H))}} (u)$ for any induced subgraph $H$ of $G$ that contains $D$ and $v$.
For the first point, observe that for any vertex $v \notin (D \cup X)$, the vector ${\mathrm{inc}^{c'}_{(G,X)}}(v)$ is simply a projection of ${\mathrm{inc}^{c}_{(G,X)}}(v)$ to a subset of its coordinates. Hence if the complete vector of $v$ is equal to the sum of the complete vectors of $u \in D$, then projecting the vector of both $v$ and of the sum to the same set of coordinates, yields identical vectors.
For the second point, observe that since $X$ is a vertex cover of $G$, we have $N_G(v) \subseteq X$ for all $v \in V(G) \setminus X$. Moreover, if $H$ is an induced subgraph of $G$ containing $D$ and $v$, then $X_H := X \cap V(H)$ is a vertex cover of $H$. Hence for any $u \in V(H) \setminus X_H$ the $c$-incidence vector ${\mathrm{inc}^{c}_{(H,X \cap V(H))}}(u)$ is well-defined. If $Q,R$ are disjoint sets for which ${\mathrm{inc}^{c}_{(H,X \cap V(H))}}(u)[Q,R]$ is defined, then $Q,R \subseteq X_H$, so the adjacencies between $u$ and $Q \cup R$ in the induced subgraph $H$ are identical to those in $G$, which implies ${\mathrm{inc}^{c}_{(G,X)}}(u)[Q,R] = {\mathrm{inc}^{c}_{(H,X \cap V(H))}}(u)[Q,R]$. Hence when we replace a $c$-incidence vector with subscript $(G,X)$ by a vector with subscript $(H,X \cap V(H))$, we essentially project the vector to a subset of its coordinates without changing any values. For the same reason as above, this preserves the fact that the vectors of $D$ sum to that of $v$.
We are ready to introduce the main definition, namely characterization of a graph property $\Pi$ by rank-$c$ adjacencies for some $c \in \mathbb{N}$. In our framework, this replaces characterization by $c$ adjacencies in the framework of Fomin et al. [@FOMIN2014468] (\[thm:oldframework\]).
\[def:rankc\] Let $c \in \mathbb{N}$ be a natural number. Graph property $\Pi$ is characterized by rank-$c$ adjacencies if the following holds. For each graph $H$, for each vertex cover $X$ of $H$, for each set $D \subseteq V(H) \setminus X$, for each $v \in V(H) \setminus (D \cup X)$, if
- $H - D \in \Pi$, and
- ${\mathrm{inc}^{c}_{(H,X)}} (v) = \sum_{u \in D} {\mathrm{inc}^{c}_{(H,X)}} (u)$ when evaluated over $\mathbb{F}_2$,
then there exists $D' \subseteq D$ such that $H - v - (D \setminus D') \in \Pi$. If there always exists such set $D'$ of size 1, then we say $\Pi$ is characterized by rank-$c$ adjacencies with *singleton replacements*.
Intuitively, the definition demands that if we have a set $D$ such that $H-D \in \Pi$, and the $c$-incidence vectors of $D$ sum to the vector of some vertex $v$ over $\mathbb{F}_2$, then there exists $D' \subseteq D$ such that removing $v$ from $H-D$ and adding back $D'$ results in a graph that is still contained in $\Pi$. For example, in \[sec:perfect\] we show that the graph property ‘containing an odd hole or odd-anti-hole’ is characterized by rank-4 adjacencies. Using our framework, this leads to a polynomial kernel for <span style="font-variant:small-caps;">Perfect Deletion</span> parameterized by vertex cover. Other examples of graph properties which are characterized by a rank-$c$ adjacencies for some $c \in \mathcal{O}(1)$ include ‘containing a cycle’ and ‘being wheel-free’. On the other hand, we will show in Theorem \[thm:awfnorankc\] that the property ‘containing an induced wheel whose size is 3 or at least 5’ cannot be characterized by rank-$c$ adjacencies for any finite $c$.
A generic kernelization
-----------------------
Our kernelization framework for <span style="font-variant:small-caps;">$\Pi$-free Deletion</span> relies on a single reduction rule presented in \[alg:reduce\]. It assigns an incidence vector to every vertex outside the vertex cover and uses linear algebra to select vertices to store in the kernel. Let us therefore recall the relevant algebraic background. A *basis* of a set $S$ of $d$-dimensional vectors over a field $\mathbb{F}$ is a minimum-size subset $B \subseteq S$ such that all ${\mathbf{v}} \in S$ can be expressed as linear combinations of elements of $B$, i.e., ${\mathbf{v}} = \sum _{{\mathbf{u}} \in B} \alpha_{{\mathbf{u}}} \cdot {\mathbf{u}}$ for a suitable choice of coefficients $\alpha_{{\mathbf{u}}} \in \mathbb{F}$. When working over the field $\mathbb{F}_2$, the only possible coefficients are $0$ and $1$, which gives a basis $B$ of $S$ the stronger property that any vector ${\mathbf{v}} \in S$ can be written as $\sum_{{\mathbf{u}} \in B'} {\mathbf{u}}$, where $B' \subseteq B$ consists of those vectors which get a coefficient of $1$ in the linear combination.
Our reduction algorithm repeatedly computes a basis of the incidence vectors of the remaining set of vertices, and stores the vertices corresponding to the basis in the kernel.
Let $Y_1 := V(G) \setminus X$. Let $V_i = \{{\mathrm{inc}^{c}_{(G,X)}}(y) \mid y \in Y_i\}$ and compute a basis $B_i$ of $V_i$ over $\mathbb{F}_2$. \[alg:reduce:basisline\] For each ${\mathbf{v}} \in B_i$, choose a unique vertex $y_{{\mathbf{v}}} \in Y_i$ such that ${\mathbf{v}} = {\mathrm{inc}^{c}_{(G,X)}}(y_{{\mathbf{v}}})$. Let $A_i := \{y_{{\mathbf{v}}} \mid {\mathbf{v}} \in B_i\}$ and $Y_{i+1} = Y_i \setminus A_i$. $G[X \cup \bigcup_{i=1}^\ell A_i]$
\[prop:reducetimegraphsize\] For a fixed $c \in \mathbb{N}$, \[alg:reduce\] runs in polynomial time in terms of $\ell$ and the size of the graph, and returns a graph on $\mathcal{O}(|X| + \ell \cdot |X|^c)$ vertices.
Observe that for each $i$, the vectors in $V_i$ have at most $2^c \cdot |\binom{X}{\leq c}| = \mathcal{O} (|X|^c)$ entries and therefore the rank of the vector space is $\mathcal{O}(|X|^c)$. Hence each computed basis contains $\mathcal{O}(|X|^c)$ vectors. For constant $c$, this means that each basis can be computed in polynomial time using Gaussian elimination. The remaining operations can be done in polynomial time in terms of $\ell$ and the size of the graph. Since $|A_i| \in \mathcal{O}(|X|^c)$ for each $i \in [\ell]$, the resulting graph has $\mathcal{O}(|X| + \ell \cdot |X|^c)$ vertices.
\[thm:framework\] If $\Pi$ is a graph property such that:
$\Pi$ is characterized by rank-$c$ adjacencies,
\[enum:framework2\] every graph in $\Pi$ contains at least one edge, and
\[enum:framework3\]there is a non-decreasing polynomial $p: \mathbb{N} \rightarrow \mathbb{N}$ such that all graphs $G$ that are vertex-minimal with respect to $\Pi$ satisfy $|V(G)| \leq p({\mathop{\textsc{vc}}}(G))$,
then <span style="font-variant:small-caps;">$\Pi$-free Deletion</span> parameterized by the vertex cover size $x$ admits a polynomial kernel on $\mathcal{O}((x+p(x))\cdot x^c)$ vertices.
Consider an instance $(G,X,k)$ of <span style="font-variant:small-caps;">$\Pi$-free Deletion</span>. Note that if $k \geq |X|$, then we can delete the entire vertex cover to get an edgeless graph, which is $\Pi$-free by , and therefore we may output a constant size <span style="font-variant:small-caps;">yes</span>-instance as the kernel. If $k < |X|$, let $G'$ be the graph obtained by the procedure <span style="font-variant:small-caps;">Reduce</span>($G$,$X$,$\ell := k+1+p(|X|)$,$c$). By \[prop:reducetimegraphsize\] this can be done in polynomial time and the resulting graph contains $\mathcal{O}((|X|+p(|X|))\cdot |X|^c)$ vertices. All that is left to show is that the instance $(G',X,k)$ is equivalent to the original instance. Since $G'$ is an induced subgraph of $G$, it follows that if $(G,X,k)$ is a <span style="font-variant:small-caps;">yes</span>-instance, then so is $(G',X,k)$. In the other direction, suppose that $(G',X,k)$ is a <span style="font-variant:small-caps;">yes</span>-instance with solution $S$. We show that $S$ also is a solution for the original instance.
For the sake of contradiction assume that this is not the case. Then the graph $G-S$ contains an induced subgraph that belongs to $\Pi$. Let $P$ be a minimal set of vertices of $G-S$ for which $G[P] \in \Pi$ and that minimizes $|P \setminus V(G')|$. Since $S$ is a solution for $(G',X,k)$, it follows that there exists a vertex $v \in P \setminus V(G')$. Moreover we have that $v \notin X$, since the graph $G'$ returned by \[alg:reduce\] contains all vertices of $X$. The set $P \cap X$ is a vertex cover for $G[P]$, therefore by property we have that $|P| \leq p({\mathop{\textsc{vc}}}(G[P])) \leq p(|X|)$. Since the vertex sets $A_1, \ldots, A_\ell$ computed in the <span style="font-variant:small-caps;">Reduce</span> operation are disjoint, and since $|S| \leq k$, it follows that there exists an $i \in [k + 1 + p(|X|)]$ such that the set of vertices $A_i$ corresponding to basis $B_i$ is disjoint from both $S$ and $P$.
As $v \notin V(G')$ implies $v \notin \bigcup_{i=1}^\ell A_i$, in each iteration of line \[alg:reduce:basisline\] the vectors of the computed vertex set $A_i$ span the vector of $v$. Hence, since we work over $\mathbb{F}_2$, there exists $D \subseteq A_i \subseteq V(G')$ such that ${\mathrm{inc}^{c}_{(G,X)}}(v) = \sum_{u \in D} {\mathrm{inc}^{c}_{(G,X)}}(u)$. Consider the graph $H := G[P \cup D]$. Since $H$ is an induced subgraph that includes $D$ and $D$ is disjoint from $X$, by \[note:incsubgraph\] it follows that ${\mathrm{inc}^{c}_{(H,X \cap V(H))}}(v) = \sum_{u \in D} {\mathrm{inc}^{c}_{(H,X \cap V(H))}}(u)$. Moreover $H-D \in \Pi$ as $H-D=G[P]$. By the definition of rank-$c$ adjacencies it follows that there exists $D' \subseteq D$ such that $P' = H - v - (D \setminus D') \in \Pi$. But since $|P'\setminus V(G')| < |P \setminus V(G')|$, this contradicts the minimality of $P$. Therefore $S$ must be a solution for the original instance.
Properties of low-rank adjacencies
----------------------------------
In this section we present several technical lemmata dealing with low-rank adjacencies. These will be useful when applying the framework to various graph properties. The next lemma shows that if $\Pi$ is characterized by low-rank adjacencies with singleton replacements, then the edge-complement graphs are as well.
\[lemma:complementgraphprop\] Let $\Pi$ be a graph property that is characterized by rank-$c$ adjacencies with singleton replacements. Let $\overline{\Pi}$ be the graph property such that $G \in \Pi$ if and only if $\overline{G} \in \overline{\Pi}$. Then $\overline{\Pi}$ is characterized by rank-$c$ adjacencies with singleton replacements.
Let $H$ be a graph with vertex cover $X$. Let $D \subseteq V(H) \setminus X$ be a set such that $H-D \in \overline{\Pi}$. Consider some vertex $v \in V(H) \setminus (D \cup X)$ such that ${\mathrm{inc}^{c}_{(H,X)}} (v) = \sum_{u \in D} {\mathrm{inc}^{c}_{(H,X)}} (u)$. Let $X' = V(H) \setminus (D \cup \{v\})$.
\[claim:complement:sums\] We have ${\mathrm{inc}^{c}_{(H,X')}} (v) = \sum_{u \in D} {\mathrm{inc}^{c}_{(H,X')}} (u)$.
Since vertices outside $X$ are independent, neither $v$ nor any vertex in $D$ is adjacent to any vertex in $X' \setminus X$. So for any disjoint $Q,R \subseteq X'$ with $R \cap (X' \setminus X) \neq \emptyset$ we have ${\mathrm{inc}^{c}_{(H,X')}}(v)[Q,R] = {\mathrm{inc}^{c}_{(H,X')}}(u)[Q,R] = 0$ for all $u \in D$ by definition, while for $R \cap (X' \setminus X) = \emptyset$ we have ${\mathrm{inc}^{c}_{(H,X')}}(u)[Q,R] = {\mathrm{inc}^{c}_{(H,X)}}(u)[Q \cap X,R]$ for any $u \in D \cup \{v\}$.
Let $H'$ be obtained from $H$ by (1) taking the edge complement, and then (2) turning $H'[D \cup \{v\}]$ back into an independent set (the complement made it a clique). Note that $X'$ is a vertex cover of $H'$.
We have ${\mathrm{inc}^{c}_{(H',X')}} (v) = \sum_{u \in D} {\mathrm{inc}^{c}_{(H',X')}} (u)$.
Immediate from \[claim:complement:sums\] since ${\mathrm{inc}^{c}_{(H',X')}}(u)[Q,R] = {\mathrm{inc}^{c}_{(H,X')}}(u)[R,Q]$ for all $u \in D \cup \{v\}$.
Observe that $H' - D$ is the edge-complement of $H - D$, so $H'- D \in \Pi$. Together with the previous claim, since $\Pi$ is characterized by rank-$c$ adjacencies with singleton replacements, it follows that there exists $v' \in D$ such that $G' := H'- v - (D \setminus \{v'\}) \in \Pi$. Since $G'$ contains only a single vertex of $\{v\} \cup D$, none of its edges were edited during step (2) above, so that $G := H - v - (D \setminus \{v'\})$ is the edge-complement of $G'$, implying $G \in \overline{\Pi}$. This shows that $\overline{\Pi}$ is characterized by rank-$c$ adjacencies with singleton replacements.
\[prop:graphpropertymerge\] proves closure under taking the union of two characterized properties.
\[prop:graphpropertymerge\] Let $\Pi$ and $\Pi'$ be graph properties characterized by rank-$c_{\Pi}$ and rank-$c_{\Pi'}$ adjacencies (with singleton replacements), respectively. Then the property $\Pi \cup \Pi'$ is characterized by rank-$\max(c_{\Pi},c_{\Pi'})$ adjacencies (with singleton replacements).
Consider a graph $H$ with vertex cover $X$ and set $D\subseteq V(H) \setminus X$ such that $H-D \in \Pi \cup \Pi'$. Let $v \in V(H) \setminus (D \cup X)$ be some vertex such that ${\mathrm{inc}^{\max(c_\Pi,c_{\Pi'})}_{(H,X)}} (v) = \sum_{u \in D} {\mathrm{inc}^{\max(c_\Pi,c_{\Pi'})}_{(H,X)}} (u)$. By \[note:largerrank\], we have ${\mathrm{inc}^{c_\Pi}_{(H,X)}} (v) = \sum_{u \in D} {\mathrm{inc}^{c_\Pi}_{(H,X)}} (u)$. If $H-D \in \Pi$, then there exists $D' \subseteq D$ such that $H-v-(D\setminus D') \in \Pi$ and hence, $H-v-(D\setminus D') \in \Pi \cup \Pi'$ (in case of singleton replacements, $D'$ is replaced by $\{v'\}$ for some $v' \in D$). The case $H - D \in \Pi'$ is symmetric.
While the *intersection* of two graph properties which are characterized by a *finite number* of adjacencies is again characterized by a finite number of adjacencies [@FOMIN2014468 Proposition 4], the same does not hold for low-rank adjacencies; there is no analog of \[prop:graphpropertymerge\] for intersections.
In a graph $G$, we say that vertices $u$ and $v$ *share adjacencies* to a set $S$, if $N_G(u) \cap S = N_G(v) \cap S$. The following lemma states that when we have a set $D$ whose $c$-incidence vectors sum to the vector of $v$, then for any set $S$ of size up to $c$ there exists a nonempty subset $D'\subseteq D$ whose members all share adjacencies with $v$ to $S$.
\[prop:adjacencyshare\] Let $G$ be a graph with vertex cover $X$, let $D \subseteq V(G)$ be disjoint from $X$, and let $c \in \mathbb{N}$. Consider a vertex $v \in V(G) \setminus(D \cup X)$. If ${\mathrm{inc}^{c}_{(G,X)}} (v) = \sum_{u \in D} {\mathrm{inc}^{c}_{(G,X)}} (u)$, then for any set $S \subseteq V(G)$ with $|S|\leq c$ there exists $D' \subseteq D$, such that:
- $|D'| \geq 1$ is odd,
- each vertex $u \in D'$ shares adjacencies with $v$ to $S$, and
- ${\mathrm{inc}^{c,(Q',R')}_{(G,X)}} (v) = \sum_{u \in D'} {\mathrm{inc}^{c,(Q',R')}_{(G,X)}} (u)$, where $Q' = (S \setminus N_G(v)) \cap X$ and $R' = S \cap N_G(v)$.
For any vertex $d \in D$ that does not share adjacencies with $v$ to $S$, the vector ${\mathrm{inc}^{c,(Q',R')}_{(G,X)}} (d)$ is the vector containing only zeros. Let $D' \subseteq D$ be the set of vertices that do share adjacencies with $v$ to $S$. Clearly ${\mathrm{inc}^{c,(Q',R')}_{(G,X)}} (v) = \sum_{u \in D'} {\mathrm{inc}^{c,(Q',R')}_{(G,X)}} (u)$, as removing all-zero vectors does not change the sum. Since ${\mathrm{inc}^{c,(Q',R')}_{(G,X)}} (v)[Q',R'] =1$ and ${\mathrm{inc}^{c,(Q',R')}_{(G,X)}} (u)[Q',R'] =1$ for all $u \in D'$, $|D'| \geq 1$ must be odd.
Our framework adapts \[thm:oldframework\] by replacing characterization by $c$ adjacencies by rank-$c$ adjacencies. From the following statement we can conclude that our framework extends \[thm:oldframework\].
\[lemma:c\_to\_rankc\] A graph property $\Pi$ characterized by $c$ adjacencies is also characterized by rank-$c$ adjacencies with singleton replacements.
Let $\Pi$ be a graph property characterized by $c$ adjacencies. We show that $\Pi$ is characterized by rank-$c$ adjacencies. Let $G$ be a graph with vertex cover $X$ and $D \subseteq V(G) \setminus X$ be a set such that $G-D \in \Pi$. Let $v \in V(G) \setminus (D \cup X)$ be a vertex such that ${\mathrm{inc}^{c}_{(G,X)}}(v) = \sum_{u \in D} {\mathrm{inc}^{c}_{(G,X)}}(u)$.
Since $\Pi$ is characterized by $c$ adjacencies, there exists a set $B$ of size at most $c$ such that all graphs obtained by changing adjacencies between $v$ and $V(G) \setminus B$ are also contained in $\Pi$. By \[prop:adjacencyshare\] there exists $w \in D$ that shares adjacencies with $v$ to $B$. Now consider the graph $G-v-(D\setminus \{w\})$. This graph is isomorphic to $G-D$ where $w$ is matched to $v$ and the adjacencies between $v$ and $V(G) \setminus B$ are changed. But then by the definition of characterization by $c$ adjacencies it follows that $G-v-(D\setminus \{w\}) \in \Pi$.
Using the framework {#sec:using}
===================
In this section we give some results using our framework, which are listed in \[table:results\]. We give polynomial kernels for <span style="font-variant:small-caps;">Perfect Deletion</span>, <span style="font-variant:small-caps;">AT-free Deletion</span>, <span style="font-variant:small-caps;">Interval Deletion</span>, <span style="font-variant:small-caps;">Even-hole-free Deletion</span>, and <span style="font-variant:small-caps;">Wheel-free Deletion</span> parameterized by vertex cover.
Perfect Deletion {#sec:perfect}
----------------
Let $\Pi_P$ be the set of graphs that contain an odd hole or an odd anti-hole. The <span style="font-variant:small-caps;">$\Pi_P$-free Deletion</span> problem is known as the <span style="font-variant:small-caps;">Perfect Deletion</span> problem. It was mentioned as an open question by Fomin et al. [@FOMIN2014468], since one can show that $\Pi_P$ is not characterized by a finite number of adjacencies. In this section we show that $\Pi_P$ is characterized by rank-4 adjacencies with singleton replacements. Following this result, we show that it admits a polynomial kernel using \[thm:framework\]. First, we give a lemma that will be helpful in the proof later on. We say that a vertex *sees an edge* if it is adjacent to both of its endpoints.
\[lemma:odd(anti)hole\_(odd)evenseen\] Let $G$ be a graph, $P = (v_1,...,v_n)$ where $n \geq 4$ is even be an induced path in $G$, and let $y$ be a vertex not on $P$ that is adjacent to both endpoints of $P$ and sees an even number of edges of $P$. Then $G[V(P) \cup \{y\}]$ contains an odd hole as induced subgraph.
We prove the claim by induction on $n$. Consider the case that $n=4$. If $y$ would be adjacent to exactly one of $v_2$ or $v_3$, then $y$ would see a single edge $\{v_1,v_2\}$ or $\{v_3,v_4\}$ respectively. If $y$ would be adjacent to both $v_2$ and $v_3$, then $y$ would see all three edges of $P$. Since $y$ sees an even number of edges of $P$, it follows that $y$ is only adjacent to $v_1$ and $v_4$. Then $G[V(P) \cup \{y\}]$ induces an odd hole.
In the remaining case we assume that the claim holds for $n' < n$, where $n \geq 6$ and both $n'$ and $n$ are even. Suppose that $y$ sees both edges $\{v_1,v_2\}$ and $\{v_{n-1},v_n\}$, then $P' = (v_2,...,v_{n-1})$ is an induced path on an even number of vertices such that $y$ is adjacent to both of its endpoints and $y$ sees an even number edges in $P'$. By the induction hypothesis $G[V(P') \cup \{y\}]$ contains an odd hole, therefore $G[V(P) \cup \{y\}]$ contains an odd hole as well. If $y$ does not see both $\{v_1,v_2\}$ and $\{v_{n-1},v_n\}$, then assume without loss of generality that $y$ does not see the last edge $\{v_{n-1},v_n\}$. Let $v_j$ for $1 \leq j < n-1$ be the largest index before $n$ for which $y$ is adjacent to $v_j$. If $j$ is odd, then $G[\{v_j,...,v_n,y\}]$ induces an odd hole. Otherwise $P' = (v_1,...,v_j)$ is an induced path on an even number of vertices, $y$ is adjacent to both of its endpoints, and $y$ sees an even number of edges in $P'$; hence the induction hypothesis applies. In all cases we get that $G[V(P) \cup \{y\}]$ contains an odd hole.
Before we show the proof that $\Pi_P$ is characterized by rank-4 adjacencies with singleton replacements, we give some intuition for the replacement argument. Suppose we want to replace a vertex $v$ of some odd hole, and we have a set $D$ where each vertex in $D$ is adjacent to both neighbors of $v$ in the hole. Furthermore, the 4-incidence vectors of $D$ sum to the vector of $v$. Then there must exist some vertex in $D$ that sees an even number of edges of the induced path between the neighbors of $v$. This together with \[lemma:odd(anti)hole\_(odd)evenseen\] would result in a graph that contains an odd hole. \[fig:matrix\] adds to this intuition.
![Graph $G$ with vertex cover $X = \{p,q,v_1,\ldots, v_4\}$, containing an odd hole $H = \{v,p,v_1,...,v_4,q\}$, such that $\mathrm{inc}^{4,(\emptyset, \{p,q\})}_{(G,X)}(v) = \sum _{i=1}^7 \mathrm{inc}^{4,(\emptyset, \{p,q\})}_{(G,X)}(y_i)$. All edges $\{p,y_i\}$ and $\{q,y_i\}$ for $i \in [7]$ exist, but not all are drawn. The table shows entries of the vectors $\mathrm{inc}^{4,(\emptyset, \{p,q\})}_{(G,X)}(u \notin X)$. Vertex $y_2$ sees an even number of edges ($\{p,v_1\}$ and $\{v_4,q\}$), and $(v_1,...,v_4,y_2)$ is an odd hole.[]{data-label="fig:matrix"}](figures/matrix.pdf)
Let $\Pi_{OH}$ be the set of graphs that contain an odd hole. We show that $\Pi_{OH}$ is characterized by rank-4 adjacencies with singleton replacements.
\[thm:oddholerankc\] $\Pi_{OH}$ is characterized by rank-$4$ adjacencies with singleton replacements.
Consider some graph $H$ with vertex cover $X$ and let $D \subseteq V(H) \setminus X$ such that $H-D \in \Pi_{OH}$. Let $v$ be an arbitrary vertex in $V(H) \setminus (D \cup X)$ such that ${\mathrm{inc}^{4}_{(H,X)}}(v) = \sum_{u \in D} {\mathrm{inc}^{4}_{(H,X)}}(u)$. We show that $H-v-(D\setminus \{v'\}) \in \Pi_{OH}$ for some $v' \in D$.
Let $C$ be an odd hole in $H-D$. If $v \notin V(C)$, then for every $v' \in D$ we have $H-v-(D\setminus \{v'\}) \in \Pi_{OH}$. So suppose that $v \in C$. Let $C = (v,p,v_1,...,v_{n-3},q)$, where $|V(C)|=n$. Consider the induced path $P = (p,v_1,...,v_{n-3},q)$. We have that $v$ is adjacent to $p$ and $q$. Let $D' \subseteq D$ be a set that shares adjacencies with $v$ to $
\{p,q\}$ such that $|D'| \geq 1$ is odd and ${\mathrm{inc}^{4,(\emptyset,\{p,q\})}_{(H,X)}}(v) = \sum_{u \in D'} {\mathrm{inc}^{4,(\emptyset,\{p,q\})}_{(H,X)}}(u)$. Such set exists by \[prop:adjacencyshare\]. Since $C$ is an odd hole, $|V(P)|$ is even. Hence by Lemma \[lemma:odd(anti)hole\_(odd)evenseen\], $G[V(P) \cup \{u\}]$ contains an odd hole if there exists some $u \in D'$ that sees an even number of edges of $P$. Suppose for the sake of contradiction that every vertex in $D'$ sees an odd number of edges of $P$. Let $E_u$ be the set of edges in $P$ that are seen by $u \in D'$. Then $\sum_{u \in D'} |E_u|$ is odd as it is a sum of an odd number of odd numbers. Let $D'_{\{u,w\}} \subseteq D'$ be the set of vertices that see edge $\{u,w\} \in E(P)$. In order to satisfy $\sum_{u \in D'}{\mathrm{inc}^{(\emptyset,\{p,q\})}}(u)[\emptyset,\{p,q,u,w\}] = {\mathrm{inc}^{(\emptyset,\{p,q\})}}(v)[\emptyset,\{p,q,u,w\}] = 0$ over $\mathbb{F}_2$, for $\{u,w\} \in E(P)$, we require $|D'_{\{u,w\}}|$ to be even. But then $\sum_{e \in E(P)} |D'_e| = \sum_{u \in D'} |E_u|$ would also need to be an even number. This contradicts the fact that $\sum_{u \in D'} |E_u|$ is odd. Therefore there must exist some $u \in D'$ that sees an even number of edges in $P$.
Let $\Pi_{OAH}$ be the set of graphs that contain an odd anti-hole. Then $\Pi_P = \Pi_{OH} \cup \Pi_{OAH}$. From applications of \[lemma:complementgraphprop\] and \[prop:graphpropertymerge\] we get the following.
\[cor:perfect\] Graph properties $\Pi_{OH}$, $\Pi_{OAH}$, and $\Pi_P$ are characterized by rank-4 adjacencies with singleton replacements.
<span style="font-variant:small-caps;">Perfect Deletion</span> parameterized by the size of a vertex cover admits a polynomial kernel on $\mathcal{O}(|X|^5)$ vertices.
By \[cor:perfect\] we have that $\Pi_P$ is characterized by rank-$4$ adjacencies with singleton replacements. Each graph in $\Pi_P$ contains at least one edge. For each odd hole or odd anti-hole $H$, we have $|V(H)| \leq 2\cdot {\mathop{\textsc{vc}}}(H)$. Therefore by \[thm:framework\] it follows that <span style="font-variant:small-caps;">$\Pi_P$-free Deletion</span> and hence <span style="font-variant:small-caps;">Perfect Deletion</span> parameterized by vertex cover admits a polynomial kernel on $\mathcal{O}(|X|^5)$ vertices.
A variation of \[thm:oddholerankc\] presented in \[app:evenhole\] shows that the set $\Pi_{EH}$ of graphs containing an even hole are characterized by rank-3 adjacencies, which leads to a kernel for <span style="font-variant:small-caps;">Even-hole-free Deletion</span> parameterized by the size of a vertex cover of $\mathcal{O}(|X|^4)$ vertices.
AT-free Deletion {#sec:atfree}
----------------
In his dissertation, Köhler [@KOHLER_ATFREE] gives a forbidden subgraph characterization of graphs without asteroidal triples. This forbidden subgraph characterization consists of 15 small graphs on 6 or 7 vertices each, chordless cycles of length at least 6, and three infinite families often called asteroidal witnesses. Let $\Pi_{AT}$ be the set of graphs that contain an asteroidal triple. A technical case analysis leads to the following results.
\[thm:atfreerankc\] $\Pi_{AT}$ is characterized by rank-$8$ adjacencies with singleton replacements.
\[thm:atfreekernel\] <span style="font-variant:small-caps;">AT-free Deletion</span> parameterized by the size of a vertex cover admits a polynomial kernel on $\mathcal{O}(|X|^9)$ vertices.
Every graph in $\Pi_{AT}$ contains at least one edge. By \[thm:atfreerankc\] it follows that $\Pi_{AT}$ is characterized by rank-8 adjacencies with singleton replacements. Each small graph has at most 7 vertices. For each cycle $C$, we have $|V(C)| \leq 2\cdot {\mathop{\textsc{vc}}}(C)$. Finally each asteroidal witness consists of an induced path with 2 or 3 additional vertices, hence there exists $c \in \mathbb{N}$ such that for $G \in \Pi_{AT}$, $|V(G)| \leq c\cdot {\mathop{\textsc{vc}}}(G)$. Hence by \[thm:framework\], <span style="font-variant:small-caps;">AT-free Deletion</span> parameterized by the size of a vertex cover admits a polynomial kernel on $\mathcal{O}(|X|^9)$ vertices.
Interval Deletion {#sec:ivd}
-----------------
<span style="font-variant:small-caps;">Interval Deletion</span> does not fit in the framework of Fomin et al. [@FOMIN2014468], since one can show that its forbidden subgraph characterization is not characterized by a finite number of adjacencies. It was shown to admit a polynomial kernel by Agrawal et al. [@AGRAWAL2019]. We show that our framework captures this result. Consider the graph property $\Pi_{IV} = \Pi_{AT} \cup \Pi_{C_{\geq 4}}$, where $\Pi_{AT}$ is the set of graphs that contain an asteroidal triple as in \[sec:atfree\] and $\Pi_{C_{\geq 4}}$ is the set of graphs that contain an induced cycle of length at least 4. Making a graph $\Pi_{IV}$-free makes it chordal and AT-free, therefore <span style="font-variant:small-caps;">$\Pi_{IV}$-free Deletion</span> corresponds to <span style="font-variant:small-caps;">Interval Deletion</span>.
\[thm:intervalkernel\] <span style="font-variant:small-caps;">Interval Deletion</span> parameterized by the size of a vertex cover admits a polynomial kernel on $\mathcal{O}(|X|^9)$ vertices.
Every graph in $\Pi_{IV}$ contains at least one edge. By \[thm:atfreerankc\], $\Pi_{AT}$ is characterized by rank-8 adjacencies. Furthermore, $\Pi_{C_{\geq 4}}$ is characterized by 3 adjacencies as shown by Fomin et al. [@FOMIN2014468 Proposition 3], and therefore by \[lemma:c\_to\_rankc\] also by rank-3 adjacencies. Therefore by \[prop:graphpropertymerge\], it follows that $\Pi_{IV}$ is also characterized by rank-8 adjacencies. Each vertex minimal graph in $\Pi_{C_{\geq 4}}$ is a cycle $C$, for which we have $|V(C)| \leq 2\cdot {\mathop{\textsc{vc}}}(C)$. Recall that $\Pi_{AT} = \Pi_S \cup \Pi_{C_{\geq 6}} \cup \Pi_{AW}$. Each vertex minimal graph in $\Pi_S$ contains at most 7 vertices. Finally each asteroidal witness consists of an induced path with 2 or 3 additional vertices, hence there exists $c \in \mathbb{N}$ such that for $G \in \Pi_{IV}$, $|V(G)| \leq c\cdot {\mathop{\textsc{vc}}}(G)$. Therefore by \[thm:framework\], <span style="font-variant:small-caps;">Interval Deletion</span> parameterized by the size of a vertex cover admits a polynomial kernel on $\mathcal{O}(|X|^9)$ vertices.
(Almost) Wheel-free Deletion {#sec:wheelfree}
----------------------------
Let $\Pi_{W_{\geq 3}}$ be the set of graphs that contain a wheel of size at least 3 as induced subgraph. Then <span style="font-variant:small-caps;">Wheel-free Deletion</span> corresponds to <span style="font-variant:small-caps;">$\Pi_{W_{\geq 3}}$-free Deletion</span>. Using a similar argument as in \[thm:evenholefreerankc\], we obtain a characterization by rank-4 adjacencies.
\[thm:wheelfreerankc\] $\Pi_{W_{\geq 3}}$ is characterized by rank-4 adjacencies.
Every graph that contains a wheel contains at least one edge. For every wheel $W_n$, we have $|V(W_n)| \leq 2\cdot {\mathop{\textsc{vc}}}(W_n)$. Therefore by \[thm:framework\] we obtain:
<span style="font-variant:small-caps;">Wheel-free Deletion</span> parameterized by the size of a vertex cover admits a polynomial kernel on $\mathcal{O}(|X|^5)$ vertices.
It turns out that this good algorithmic behavior is very fragile. Let $\Pi_{W_{\neq 4}}$ be the set of graphs that contain a wheel of size $3$, or at least $5$. Then <span style="font-variant:small-caps;">$\Pi_{W_{\neq 4}}$-free Deletion</span> corresponds to <span style="font-variant:small-caps;">Almost Wheel-free Deletion</span>. While $\Pi_{W_{\geq 3}}$ can be characterized by rank-4 adjacencies, the following shows that $\Pi_{W_{\neq 4}}$ is not characterized by adjacencies of any finite rank, and therefore does not fall within the scope of our kernelization framework.
\[thm:awfnorankc\] $\Pi_{W_{\neq 4}}$ is not characterized by rank-$c$ adjacencies for any $c \in \mathbb{N}$.
This is not a deficiency of our framework; we prove that the problem does not have any polynomial compression, and therefore no polynomial kernel, unless [NP $\subseteq$ coNP/poly]{}.
\[thm:almostwheelfreenopolykernel\] <span style="font-variant:small-caps;">Almost Wheel-free Deletion</span> parameterized by vertex cover does not admit a polynomial compression unless $\mathrm{coNP} \subseteq \mathrm{NP} / \mathrm{poly}$.
This suggests that the condition of being characterized by low-rank adjacencies is the right way to capture kernelization complexity.
Conclusion {#sec:conclusion}
==========
Appendix
========
Even-hole-free Deletion {#app:evenhole}
-----------------------
An even hole is a cycle $C_n$, where $n \geq 4$ is an even number. Let $\Pi_{EH}$ be the set of graphs that contain an even hole as induced subgraph. Then the <span style="font-variant:small-caps;">Even-hole-free Deletion</span> problem corresponds to the <span style="font-variant:small-caps;">$\Pi_{EH}$-free Deletion</span> problem.
\[lemma:evenhole\_evenseen\] Let $G$ be a graph, $P = (v_1,...,v_n)$ where $n \geq 3$ is odd be an induced path in $G$, and let $y$ be a vertex not on $P$ that is adjacent to both endpoints of $P$ and sees an even number of vertices of $P$. Then $G[V(P) \cup \{y\}]$ contains an even hole as induced subgraph.
We prove the claim by induction on $n$. Consider the case that $n=3$. Since $y$ is adjacent to both $v_1$ and $v_3$, the only way for it to see an even number of vertices of $P$ is for $y$ not to be adjacent to $v_2$. Therefore $G[V(P) \cup \{y\}]$ induces an even hole and the statement holds.
In the remainder, assume that the claim holds for $n' < n$, where $n \geq 5$ and both $n'$ and $n$ are odd. If $y$ is not adjacent to $v_i$ for any $1 < i < n$, then $G[V(P) \cup \{y\}]$ induces an even hole and the statement holds. Otherwise, since $y$ sees an even number of path vertices it sees at least two more. Let $j > 1$ be the smallest index such that $y$ is adjacent to $v_j$ and let $j' < n$ be the largest index such that $y$ is adjacent to $v_{j'}$. If $j$ is odd, then $G[\{v_1,...,v_j,y\}]$ induces an even hole and the statement holds. If $j'$ is odd, then $G[\{v_{j'},...,v_n,y\}]$ induces an odd hole and the statement holds. Otherwise, both $j$ and $j'$ are even. But then $P' = (v_j,...,v_{j'})$ is a path of odd length at least 3, where $y$ is adjacent to both endpoints and sees an even number of vertices. So by the induction hypothesis, the statement holds.
\[thm:evenholefreerankc\] $\Pi_{EH}$ is characterized by rank-3 adjacencies with singleton replacements.
Consider some graph $H$ with vertex cover $X$ and let $D \subseteq V(H) \setminus X$ such that $H-D \in \Pi_{EH}$. Let $v$ be an arbitrary vertex in $V(H) \setminus (D \cup X)$ such that ${\mathrm{inc}^{3}_{(H,X)}}(v) = \sum_{u \in D} {\mathrm{inc}^{3}_{(H,X)}}(u)$. We show that $H-v-(D\setminus \{v'\}) \in \Pi_{EH}$ for some $v' \in D$.
Let $C$ be an even hole in $H-D$. If $v \notin V(C)$, then for every $v' \in D$ we have $H-v-(D\setminus \{v'\}) \in \Pi_{EH}$. So suppose that $v \in C$. Let $C = (v,p,v_1,...,v_{n-3},q)$, where $|V(C)|=n$. Consider the induced path $P = (p,v_1,...,v_{n-3},q)$. We have that $v$ is adjacent to $p$ and $q$. Let $D' \subseteq D$ be a set that shares adjacencies with $v$ to $
\{p,q\}$ such that $|D'| \geq 1$ is odd and ${\mathrm{inc}^{3,(\emptyset,\{p,q\})}_{(H,X)}}(v) = \sum_{u \in D'} {\mathrm{inc}^{3,(\emptyset,\{p,q\})}_{(H,X)}}(u)$. Such set exists by \[prop:adjacencyshare\]. Since $C$ is an even hole, $|V(P)|$ is odd. Hence by Lemma \[lemma:evenhole\_evenseen\], $G[V(P) \cup \{u\}]$ contains an odd hole if there exists some $u \in D'$ that sees an even number of vertices of $P$. Suppose for the sake of contradiction that every vertex in $D'$ sees an odd number of vertices of $V(P)$. Then also every vertex in $D'$ sees an odd number of vertices of $V(P) \setminus \{p,q\}$. Let $V_u$ be the set of vertices in $V(P)\setminus \{p,q\}$ that are seen by $u \in D'$. Then $\sum_{u \in D'} |V_u|$ is odd as it is a sum of an odd number of odd numbers. Let $D'_u \subseteq D'$ be the set of vertices that see vertex $u \in V(P) \setminus \{p,q\}$. In order to satisfy $\sum_{u \in D'}
{\mathrm{inc}^{(\emptyset,\{p,q\})}}(u)[\emptyset,\{p,q,u\}] = {\mathrm{inc}^{(\emptyset,\{p,q\})}}(v)[\emptyset,\{p,q,u\}] = 0$ over $\mathbb{F}_2$ , for $u \in V(P) \setminus \{p,q\}$, we require $|D'_u|$ to be even. But then $\sum_{u \in V(P) \setminus \{p,q\}} |D'_u| = \sum_{u \in D'} |V_u|$ would also need to be an even number. This contradicts the fact that $\sum_{u \in D'} |V_u|$ is odd. Therefore there must exist some $u \in D'$ that sees an even number of vertices of $P$.
Since every graph that contains an even hole has at least one edge and for every hole $H$, we have $|V(H)| \leq 2\cdot {\mathop{\textsc{vc}}}(H)$. By \[thm:framework\] we obtain:
<span style="font-variant:small-caps;">Even-hole-free Deletion</span> parameterized by the size of a vertex cover admits a polynomial kernel on $\mathcal{O}(|X|^4)$ vertices.
Omitted material of Section \[sec:atfree\]
------------------------------------------
Let $\Pi_S$ be the set of graphs that contain at least one of these small graphs as induced subgraph, let $\Pi_{C_{\geq 6}}$ be the set of graphs that contain a chordless cycle of length at least 6, and let $\Pi_{AW}$ be the set of graphs that contain a so called asteroidal witness. Then $\Pi_{AT} = \Pi_S \cup \Pi_{C_{\geq 6}} \cup \Pi_{AW}$.
Fomin et al. [@FOMIN2014468 Proposition 3] show that $\Pi_{C_{\geq 6}}$ is characterized by 5 adjacencies. By \[prop:infiniteset\_finitegraphs\] we have that $\Pi_S$ is characterized by 6 adjacencies. Therefore we focus our attention on the asteroidal witnesses, which are shown in \[fig:aws\].
\[def:aws\] The different types of *asteroidal witnesses* (AWs) are constructed as follows.
- $\dagger_z$-AW: A graph $G$ such that $V(G) = \{t_l,t_r,t,c\} \cup \{b_1,...,b_z\}$, where $t_l = b_0$ and $t_r = b_{z+1}$, $E(G) = \{\{t,c\}\} \cup \{\{c,b_i\} \mid i \in [z]\} \cup \{\{b_{i-1},b_i\} \mid i \in [z+1]\}$, and $z \geq 2$.
- $\ddagger_z$-AW: A graph $G$ such that $V(G) = \{t_l,t_r,t,c_1,c_2\} \cup \{b_1,...,b_z\}$, where $t_l = b_0$ and $t_r = b_{z+1}$, $E(G) = \{\{t,c_1\},\{t,c_2\},\{c_1,c_2\}\} \cup \{\{c_j,b_i\} \mid i \in [z], j\in \{1,2\}\} \cup \{\{b_{i-1},b_i\} \mid i \in [z+1]\}$, and $z \geq 1$.
- $\diamond_z$-AW: A graph $G$ such that $V(G) = \{t_l,t_r,t,c_1,c_2\} \cup \{b_1,...,b_z\}$, where $t_l = b_0$ and $t_r = b_{z+1}$, $E(G) = \{\{t,c_1\},\{t,c_2\}\} \cup \{\{c_j,b_i\} \mid i \in [z], j\in \{1,2\}\} \cup \{\{b_{i-1},b_i\} \mid i \in [z+1]\}$, and $z \geq 1$.
Here $t_l$, $t_r$, and $t$ are called *terminal* vertices, with *top* vertex $t$. These three vertices form an asteroidal triple. Vertices $c$, $c_1$, and $c_2$ are called *center* vertices. Finally $b_i$ for $i \in [z]$ are called *path* vertices.
![Asteroidal witnesses.[]{data-label="fig:aws"}](figures/dAW_ddAW_diamondAW.pdf)
The difficulty with these asteroidal witnesses lies with the top vertex $t$. If $t$ would be adjacent to a single path vertex $b_i$ for some $i \in [z]$, then the resulting graph no longer contains an asteroidal triple. It is possible to show that, because of this difficulty, no finite adjacency characterization exists for asteroidal witnesses. We show that if we have a set of vertices whose vectors sum to the vector of $t$, then we can find a valid replacement in this set.
\[thm:AWtopreplace\] Let $G$ be a graph with vertex cover $X$, containing $O = x_z$-AW for some $x \in \{\dagger,\ddagger,\diamond\}$ according to Definition \[def:aws\] with $t \notin X$. Let $D \subseteq V(G)$ disjoint from $O$ and $X$ such that ${\mathrm{inc}^{8}_{(G,X)}}(t) = \sum_{y \in D} {\mathrm{inc}^{8}_{(G,X)}}(y)$. Then $G[(V(O) \cup \{y\}) \setminus \{t\}]$ contains an asteroidal triple for some $y \in D$.
Let $G' := G[O \cup D]$ and $X' := X \cap V(G')$. By \[note:incsubgraph\] we have that ${\mathrm{inc}^{8}_{(G',X')}}(t) = \sum_{y \in D} {\mathrm{inc}^{8}_{(G',X')}}(y)$. Let $C$ be the center vertices of $O$; then $t$ is adjacent to the center vertices in $C$ and not to the other terminals $t_l$ and $t_r$. Let $D' \subseteq D$ be a set of vertices that share adjacencies with $t$ to $\{t_l,t_r\} \cup C$ such that ${\mathrm{inc}^{8,(T,C)}_{(G',X')}}(t) = \sum_{u \in D'} {\mathrm{inc}^{8,(T,C)}_{(G',X')}}(u)$ where $T = \{t_l,t_r\}\cap X'$. Such $D'$ exists by \[prop:adjacencyshare\]. If there are distinct $w,w' \in D'$ whose vectors are identical, then $\sum_{u \in D'} {\mathrm{inc}^{8,(T,C)}_{(G',X')}}(u) = \sum_{u \in D' \setminus \{w,w'\}} {\mathrm{inc}^{8,(T,C)}_{(G',X')}}(u)$. Therefore we can remove all duplicate vectors, and hence assume that the vectors for vertices in $D'$ are unique. If there exists $y \in D'$ that is not adjacent to any $b_i$ for $i \in [z]$, then $G[(V(O) \cup \{y\}) \setminus \{t\}]$ induces an $x_z$-AW and we are done. Otherwise, let $q_y$ be the smallest index such that $y$ is adjacent to $b_{q_y}$ and let $q_y'$ be the largest index such that $y$ is adjacent to $b_{q_y'}$. Let $y^* \in D'$ be a vertex that maximizes $q'_{y^*} - q_{y^*}$. If $q'_{y^*} = q_{y^*}$, then $y^*$ is adjacent to a single vertex $b_{q_{y^*}}$. However, since $t$ is not adjacent to this vertex, there must exist $y \in D'$ that is adjacent to the same vertex $b_{q_{y^*}}$ in order to satisfy the vector sum. But then $y$ and $y^*$ have the same projected incidence vector, a contradiction. It remains to deal with the case that $q'_{y^*} > q_{y^*}$.
If there exists $y \in D'$ (possibly $y^*$ itself) adjacent to $b_{q_{y^*}}$ and $b_{q_{y^*}'}$, but nonadjacent to some $b_i$ with $q_{y^*} + 1 < i < q_{y^*}'-1$, then $t_l$, $t_r$, and $b_i$ form an asteroidal triple in the graph $G[(V(O) \cup \{y\}) \setminus \{t\}]$, since $y$ provides a path between $t_l$ and $t_r$ avoiding $N(b_i)$. Therefore suppose there is no such $y$. Then any vertex $y \in D'$ adjacent to $b_{q_{y^*}}$ and $b_{q_{y^*}'}$ is also adjacent to all vertices in between, except possibly $b_{q_{y^*}+1}$ and $b_{q_{y^*}'-1}$. Let $Q = (\{t_l,t_r,b_{q_{y^*}+1},b_{q_{y^*}'-1}\} \setminus N_G(y^*)) \cap X$ and $R = C \cup \{b_{q_{y^*}}, b_{q_{y^*}'}\} \cup (\{b_{q_{y^*}+1},b_{q_{y^*}'-1}\} \cap N_G(y^*))$. We have $|Q|+|R| \leq 8$. The entry ${\mathrm{inc}^{8,(T,C)}_{(G',X')}}(y^*)[Q,R] = 1$ since we defined $Q$ and $R$ based on the neighborhood of $y^*$, but ${\mathrm{inc}^{8,(T,C)}_{(G',X')}}(t)[Q,R] = 0$ since $t$ is not adjacent to the path vertices $b_{q_{y^*}}, b_{q'_{y^*}}$. So there must exist another $y'\in D'$ with ${\mathrm{inc}^{8,(T,C)}_{(G',X')}}(y')[Q,R] = 1$. By the maximality of the range seen by $y^*$, we have $q_{y^*} = q_{y'}$ and $q_{y^*}' = q_{y'}'$. But then $y^*$ and $y'$ have the same projected incidence vector, since they agree on the adjacency to $b_{q_{y^*}}, b_{q_{y^*}+1}, b_{q'_{y^*}-1}, b_{q'_{y^*}}$ and are adjacent to all vertices in between. This contradicts the assumption that all vectors for vertices in $D'$ are unique.
### Proof of Theorem \[thm:atfreerankc\]
The graph property $\Pi_{AT}$ is characterized by rank-$8$ adjacencies with singleton replacements.
Consider some graph $H$ with vertex cover $X$ and let $D \subseteq V(H) \setminus X$ such that $H-D \in \Pi_{AT}$. Let $v$ be an arbitrary vertex in $V(H) \setminus (D \cup X)$ such that ${\mathrm{inc}^{8}_{(H,X)}}(v) = \sum_{u \in D} {\mathrm{inc}^{8}_{(H,X)}}(u)$. We show that $H-v-(D\setminus \{v'\}) \in \Pi_{AT}$ for some $v' \in D$. The statement for the cases where $H-D \in \Pi_S$ and $H-D \in C_{\geq 6}$ follow from the fact that they are characterized by 6 and 5 adjacencies respectively. In the remainder we have $H-D \in \Pi_{AW}$.
Suppose that $H-D$ contains $O = \dagger_z$-AW for some $z \geq 2$. We do a case distinction on $v$. If $v \notin V(O)$, then for any $v' \in D$ we have $H-v-(D\setminus \{v'\}) \in \Pi_{AT}$, so suppose that $v \in V(O)$. If $v$ is top terminal $t$, then by \[thm:AWtopreplace\] we have $H-v-(D\setminus \{v'\}) \in \Pi_{AT}$ for some $v' \in D$. If $v$ is a center vertex $c$, then $v$ is adjacent to $t$, $b_1$, and $b_2$ and not to $t_l$ and $t_r$. Let $D' \subseteq D$ be the set of vertices share adjacencies with $v$ to $\{t,b_1,b_2,t_l,t_r\}$. Such set $D'$ exists by \[prop:adjacencyshare\]. Consider any $v' \in D'$, then $\{t,t_l,t_r\}$ still is an asteroidal triple in the graph $H-v-(D\setminus \{v'\})$ and the statement holds. This adjacency argument works for all cases besides the top vertex $t$. We give the set of adjacencies that needs to shared. For $v = t_l$, the adjacencies to share are $\{b_z,c,t,t_r,b_1\}$. The case $v=t_r$ is symmetric. For $v = b_i$ for some $1 < i < z$, share adjacencies to $\{t,t_l,t_r,b_{i-1},b_{i+1},c\}$. For $v=b_1$, share adjacencies to $\{t,t_r,t_l,b_2,c\}$. Again the case for $v=b_z$ is symmetric.
Finally suppose that $H-D$ contains a $O = \ddagger_z$-AW or $O = \diamond_z$-AW for some $z \geq 1$. If $v \notin V(O)$, then for any $v' \in D$ we have $H-v-(D\setminus \{v'\}) \in \Pi_{AT}$, so suppose that $v \in V(O)$. If $v$ is top terminal $t$, then by \[thm:AWtopreplace\] we have $H-v-(D\setminus \{v'\}) \in \Pi_{AT}$ for some $v' \in D$. For the remaining cases of $v$, again the adjacency argument is sufficient to prove the statement. If $v$ is a center vertex $c_1$, then share adjacencies to $\{t_r,t,t_l\}$. The case $v=c_2$ is symmetric. For $v=t_l$, share adjacencies to $\{c_2,t_r,t,c_1,b_1\}$. Again the case $v=t_r$ is symmetric. For $v = b_i$ for some $1 < i < z$, share adjacencies to $\{t,b_{i-1},b_{i+1}\}$. For $v=b_1$, share adjacencies to $\{t,t_l,b_{i+1}\}$. The case $v=b_z$ is again symmetric. Note that for all of these case, the set of adjacencies $S$ that some vertex in $D$ needs to share with $v$ contains at most 8 elements. This completes the proof.
Omitted material of Section \[sec:wheelfree\]
---------------------------------------------
### Proof of Theorem \[thm:wheelfreerankc\]
The graph property $\Pi_{W_{\geq 3}}$ is characterized by rank-4 adjacencies.
Consider some graph $H$ with vertex cover $X$ and set $D \subseteq V(H) \setminus X$ such that $H-D \in \Pi_{W_{\geq 3}}$. Let $v$ be an arbitrary vertex in $V(H) \setminus (D \cup X)$ such that ${\mathrm{inc}^{4}_{(H,X)}}(v) = \sum_{u \in D} {\mathrm{inc}^{4}_{(H,X)}}(u)$. We show that $H-v-(D\setminus D') \in \Pi$ for some $D' \subseteq D$.
Since $H-D \in \Pi_{W_{\geq 3}}$, it contains a wheel $W_n$ for some $n \geq 3$ as induced subgraph. If $v \notin V(W_n)$, then for any $D' \subseteq D$, $H-v-(D\setminus D') \in \Pi_{W_{\geq 3}}$. So suppose $v \in V(W_n)$, then $v$ sees three vertices of $W_n$ that induce a complete graph if $n=3$, or a $P_3$ otherwise. Let $p$, $q$, and $r$ be these vertices. Let $D' \subseteq D$ be the set of vertices that share adjacencies with $v$ to $\{p,q,r\}$ such that ${\mathrm{inc}^{(\emptyset,\{p,q,r\})}}(v) = \sum_{u \in D'} {\mathrm{inc}^{(\emptyset,\{p,q,r\})}}(u)$. Such set $D'$ exists by \[prop:adjacencyshare\]. If $n=3$, then any $d \in D'$ that is adjacent to $p$, $q$, and $r$ is a valid replacement. Suppose that $n > 3$. If $|D'| = 1$, then this vertex must have the same adjacencies as $v$ with respect to the wheel, and hence is a valid replacement. Finally if $|D'| > 1$, then any two vertices in $D'$ together with $p$, $q$, and $r$ induce a $W_4$, since $p$, $q$, and $r$ induce a $P_3$.
Almost Wheel-free Deletion {#app:almostwheelfree}
--------------------------
|
---
abstract: 'We report the results of [*Hubble Space Telescope*]{} Wide Field and Planetary Camera 2 broadband F555W and F702W photometric and F555W polarimetric observations of the “Cloverleaf” QSO H1413+1143. This is a four-component gravitationally-lensed broad absorption line (BAL) QSO. Observations were obtained at two epochs in March 1999 and June 1999 separated by $\approx 100$ days. The observations were photometrically and polarimetrically calibrated using the standard “pipeline” calibration procedures implemented at the Space Telescope Science Institute. The goal of our program was to detect any [*relative*]{} changes among the components and between the two epochs. Over this time baseline we detected an $\approx 0.07$ mag dimming in component D of the lensed image, which we interpret as evidence for microlensing. In March 1999 we find significant evidence for a difference in the relative linear polarization of component D in comparison to the other three components; in June 1999 the combined polarization of the Cloverleaf components was lower. In March 1999 the apparently microlensed component D has a rotated polarization position angle and a somewhat higher degree of polarization than the other three components. We suggest that this difference in polarization is due to microlensing magnification of part of a scattered-light (i.e. polarized) continuum-producing region. The results indicate that in the Cloverleaf the size-scale of the polarized scattered-light region exceeds $\approx 10^{16}$ cm but lies interior to the region producing the broad emission lines ($< 10^{18}$ cm).'
author:
- 'Kyu-Hyun Chae, David A. Turnshek, Regina E. Schulte-Ladbeck, Sandhya M. Rao, and Olivia L. Lupie'
title: 'Hubble Space Telescope Observations of the Gravitationally Lensed Cloverleaf Broad Absorption Line QSO H1413+1143: Imaging Polarimetry and Evidence for Microlensing of a Scattering Region$^1$'
---
Introduction
============
Broad absorption line (BAL) QSOs comprise $\approx$ 10% of the objects in optically selected QSO samples. Their defining characteristics are deep, high-velocity (usually $< 0.1c$) absorption troughs blueward of high-ionization broad emission lines (BELs) in species such as , , , and (e.g. Turnshek 1988 and Weymann et al. 1991). They also have a distribution of linear polarizations which peaks at a significantly higher polarization in comparison to non-BAL radio-quiet QSOs (Schmidt & Hines 1999; Hutsemékers, Lamy & Remy 1998; Goodrich 1997; Turnshek 1988); indeed the origin of much of the polarization in most non-BAL QSOs is probably the Galactic interstellar medium.
The observed polarization properties of BAL QSOs depend on, and therefore contain relevant astrophysical information on, the geometries and physical properties of the inner regions of these QSOs (i.e. the narrow emission-line region, the dusty torus, the BAL region, the BEL region, any scattering regions, the thermal accretion disk, and any region producing non-thermal emission). Recent studies on the polarization properties of BAL QSOs (Schmidt & Hines 1999; Hutsemékers et al. 1998; Ogle 1998; Goodrich 1997) have shown that: (1) their continuum polarizations are, on average, significantly higher than non-BAL radio-quiet QSOs, with the degree of polarization rising mildly toward shorter wavelengths, (2) the BALs are more highly polarized than the continuum, with position angle rotations observed in the BAL troughs, (3) BELs in [*some*]{} BAL QSO spectra are polarized, but the degree of polarization is lower than in the continuum and the polarization position angles are not necessarily similar, and (4) there is some evidence that the degree of polarization is positively correlated with the BAL QSO’s balnicity index (defined by Weymann et al. 1991) and the presence of low-ionization BALs, with objects having higher balnicity indices and/or low-ionization BALs being more polarized on average.
While it is generally agreed that the scattered continuum is, in large part (if not totally), responsible for the observed net polarizations observed in BAL QSOs (e.g. Goodrich & Miller 1995; Ogle 1998; Schmidt & Hines 1999), the size-scales and geometries of the regions containing scattering particles (i.e. electrons and/or dust particles) are not yet well constrained. However, the gravitationally-lensed Cloverleaf BAL QSO H1413+1143 ($z_{em} \approx 2.55$) is at present a unique laboratory for study of the polarization mechanism. Its four components are known to have a combined net continuum polarization which has varied between $1.5 - 3.5$% over a decade (Goodrich & Miller 1995) and one of the four components (component D) shows evidence for microlensing in the form of light variability (see Angonin et al. 1990, Arnould et al. 1993, Remy et al. 1996, Ostensen et al. 1997, and new evidence presented here) and differences in BEL and BAL profile characteristics (Chae & Turnshek 1999 and references therein).
We know that, in principle, microlensing could alter the net polarization of a single gravitationally-lensed component, since this process selectively produces additional magnification of a small region of an Einstein ring radius on the source plane. The resulting polarization properties during microlensing would depend on the detailed geometry of the region where the polarized light originates (see Belle & Lewis 2000 for examples). Consequently, the observed component polarization properties during microlensing can be used to constrain the projected sizes of the scattering regions and their projected distances from any central black hole (i.e. the central region of the accretion disk).
In this [*Letter*]{}, we report the results of [*Hubble Space Telescope (HST)*]{} Wide Field and Planetary Camera 2 (WFPC2) broadband F555W and F702W photometric and F555W polarimetric observations of the Cloverleaf. We find new evidence for microlensing of component D and show that component D has significantly different relative polarization properties than the other three components. In §2 we describe our observations and data analysis; in §3 we present the results; and in §4 we discuss the implications for models of the production of polarized light in BAL QSOs. We note that this paper is the fifth in a series of [*HST*]{} results on the Cloverleaf by members of our group. Earlier results include constraints on the sizes and shapes of absorbing regions producing the BALs (Turnshek 1995), constraints on the component image locations and magnifications (Turnshek et al. 1997), constraints on the properties of the intervening absorbers seen in the component spectra (Monier, Turnshek & Lupie 1998), and implications for gravitational-lens models of the Cloverleaf, including a constraint on BEL region size scales (Chae & Turnshek 1999).
Observations and Data Analysis
==============================
For this study the Cloverleaf QSO was observed at two closely-spaced epochs, 15-16 March 1999 and 23-24 June 1999. Broadband F555W filter observations were made with the Wide Field Camera 2 (WFC2) with either no polarizer or one of four polarizers (POLQ, POLQN18, POLQN33, POLQP15). Also, F702W filter observations were made with the Planetary Camera 2 (PC2) and no polarizer. The observations are summarized in Table 1 and numbered as a referencing convenience. Each F555W observation consisted of taking two sets of four dithered images. The pixel position of component A in the second set was chosen to coincide with that of component D in the first set. In addition to the normal procedures for identifying cosmic ray contamination of WFPC2 images, all pipeline processed images were also individually examined. Cosmic rays present near the Cloverleaf components were removed interactively and the data were replaced using interpolation. After sky subtraction, each set of images was combined via the variable-pixel linear reconstruction algorithm, or the “drizzling” algorithm (Fruchter & Hook 1998; Fruchter et al. 1997). Point spread function (PSF)-fitting photometry showed that the photometric results of the two sets for each observation were consistent with each other. Finally, all eight images in the two sets were drizzled to form one image. The procedure of taking dithered images and combining them via drizzling partially restores the PSF, thereby facilitating more accurate PSF-fitting photometry. Using the final drizzled image for each observation, an empirical PSF was constructed iteratively from the Cloverleaf components themselves, and then the Cloverleaf components were fitted simultaneously using this empirical PSF.[^1] For the above procedures, the Image Reduction and Analysis Facility (IRAF) packages DAOPHOT and STSDAS were used. The reliability of the above PSF-fitting photometry was tested by visually examining the PSF-subtracted region and computing pixel statistics within the subtracted region; the mean pixel data number ([*DN*]{}) value within the region was $|\overline{DN}|
\lesssim 0.1\sigma$ in all cases.
The observations were photometrically and polarimetrically calibrated using the standard “pipeline” calibration procedures implemented at the Space Telescope Science Institute. The goal of our program was not to make absolute measurements, but to detect any relative changes among the components and between the two epochs. Thus, the errors we quote based on our differencing procedures are statistical in nature. Our relative measurements and the resulting interpretation should not be affected by a small systematic error in the pipeline calibration.
The derived F555W photometric results without polarizers (i.e.observation numbers 1, 2, 3 and 11, 12, 13) were found to be highly consistent with one another. This served as an independent test, confirming the reliability of our method of photometry. The F702W photometric observations without polarizers (i.e. observation numbers 4, 5, 6 and 16, 17, 18) allowed us to fill up the remaining time available for exposures in a number of the orbits using a filter that had previously been used to observe the Cloverleaf, but without the goal of deriving photometry using drizzling. We found the statistical errors to be considerably reduced using the drizzling method.
In order to determine the Stokes parameters (I, Q, U), the F555W observations with polarizers were incorporated into the WFPC2 polarization calibration model (Biretta & McMaster 1997). The Stokes parameters are related to the degree of polarization ($p$) and the position angle ($PA$) via the relations $p = (u^2 + q^2)^{1/2}$ and $PA = (1/2)
\tan^{-1}(u/q) + n\pi/2$ where $q \equiv Q/I$, $u \equiv U/I$, and $n =
0,1,2$ for $u \geq 0$ and $q \geq 0$, $u \geq 0$ and $q \leq 0$ (or, $u <
0$ and $q < 0$), and $u \leq 0$ and $q \geq 0$, respectively. For the March epoch, six observations were incorporated simultaneously using a $\chi^2$ fitting technique to determine the Stokes parameters.[^2] The $\chi^2$ is defined by $$\chi^2 = \sum_i \left( \frac{C_i^{\mbox{\scriptsize obs}} -
C_i^{\mbox{\scriptsize mod}}(I,Q,U)}{\sigma_i} \right)^2,$$ where $i = 7,8,9,10,11,12$ are the March observation numbers, $C_i^{\mbox{\scriptsize obs}}$ and $C_i^{\mbox{\scriptsize
mod}}$ are the observed and calibration-model-predicted counts, respectively, and $\sigma_i$ are the statistical errors in the observed counts. The minimum $\chi^2$ value was $\chi_{\mbox{\scriptsize min}}^2 < 7$, with 3 degrees of freedom for all components, indicating that there is a reasonable match between the model and the data. The $1\sigma$ statistical errors for each Stokes parameter were estimated using $\Delta\chi^2
=\chi^2 -\chi_{\mbox{\scriptsize min}}^2 = 1$ (e.g. Press et al.1992), and the statistical errors for $p$ and $PA$ were estimated assuming Gaussian error propagation. For the June epoch ($i = 19, 20, 21$), the Stokes parameters were determined by solving $C_i^{\mbox{\scriptsize obs}}
= C_i^{\mbox{\scriptsize mod}}(I,Q,U)$ for I, Q and U,$^5$ with the statistical errors for the Stokes parameters being estimated using Gaussian error propagation of the photometric errors via these equations.
The results for the June epoch are much less reliable in comparison to the March results. There are two reasons for this. First, the June measurements were made at only three polarization angles, while the March measurements were made at six polarization angles. Second, according to the pipeline calibration, the degree of polarization is evidently smaller in June 1999, which gives rise to a lower signal-to-noise ratio for a fixed photometric error. Thus, the June results should not be given high weight. Since the March data show the polarization of components A, B and C to be similar, for the June epoch we report only the results for the combined measured polarization of components A, B and C, and we report the measured polarization of component D separately.
Results
=======
Below we consider the results of our WFPC2 observations in two separate parts. First, we consider any evidence for photometric variations among the four lensed components of the Cloverleaf between the June 1999 and March 1999 epochs (§3.1, Table 2a,b). Second, we consider any evidence for differences in the linear polarization among the four lensed components (§3.2, Table 3).
Brightness Variation of Component D: Evidence for Microlensing
--------------------------------------------------------------
The photometric observations of the Cloverleaf without polarizers were used to search for and measure any brightness variations over the time baseline ($\approx 100$ days) which separated the two epochs of observation. As noted earlier, the individual F555W photometric results for each epoch were very consistent with one another; the average F555W photometric results for each epoch are reported in Table 2a,b. The average F702W photometric observations are also reported in Table 2a,b, but are less accurate. The photometric results are shown in Figure 1. They clearly indicate that component D decreased its brightness over the $\approx 100$ day time baseline, while the other three components either remained about the same brightness (i.e. components B and C) or slightly increased their brightness (i.e. component A). Assuming an insignificant wavelength dependence to the brightness variation over the F555W and F702W passbands, using variance-weighting we find $\Delta m_D = 0.074\pm 0.004$ mag and $\Delta m_A = -0.023\pm 0.004$ mag. Evidence for brightness variations in components B and C are at the 2$\sigma$ level of significance or less.
Recent detailed work on modeling the Cloverleaf lens and time delays (Chae & Turnshek 1999) suggests that in all reasonable models component C is the leading component and component D is the trailing component, with the maximum time delay being model-dependent and lying in the range $\approx 7-41$ days. However, the predicted time delays between components C and A or components C and B are always a significant fraction (at least $\approx 30$%) of the predicted time delay between components C and D. Given these lens models and observations of brightness variations in other BAL QSOs (Sirola et al. 1997), we believe that these new photometric data on the Cloverleaf are not likely to be consistent with simply an intrinsic variation in the source BAL QSO’s brightness coupled with time delays among the four components. The most likely cause of the light variation of component D is microlensing, which is consistent with the findings of others who have studied brightness variations in the Cloverleaf (Angonin et al.1990; Arnould et al. 1993; Remy et al. 1996; Ostensen et al. 1997).
Polarimetric Results
--------------------
For the March 1999 and June 1999 epochs of observation, Table 3 gives the measured normalized Stokes parameters $q$ and $u$, the degree of linear polarization ($p$), the corrected degree of linear polarization ($p_{\mbox{\scriptsize corr}}$) which takes into account the bias toward measuring higher polarization in low signal-to-noise data (Wardle & Kronberg 1974), and the polarization position angle ($PA$). For the March observations the derived $q$ and $u$ Stokes parameters are shown in Figure 2, and it is seen that the polarizations of components A, B and C differ from the polarization of component D.
The main points of the polarimetric results are: (1) The polarizations of components A, B, and C in March 1999 have no appreciable differences. (2) As is clearly seen in Figure 2, in March 1999 component D has a significantly different relative polarization in comparison to the other three components. The [*relative*]{} normalized Stokes parameters between component D and the combination of the other components in March 1999 is $\Delta q_{\mbox{\scriptsize D,$\overline{\mbox{ABC}}$}}
= -1.26\pm 0.54\%$ and $\Delta u_{\mbox{\scriptsize D,$\overline{\mbox{ABC}}$}}
= -2.32\pm 0.61\%$ (Table 3), which is a difference in polarization at a level of significance of 4.5$\sigma$. (3) The pipeline calibrated data suggest that the net polarization of the Cloverleaf changed between the March 1999 and June 1999 epochs, with the polarization being smaller during the June epoch.
Discussion
==========
These results are the first observational ones which address resolved polarization measurements in a gravitationally-lensed QSO and, owing to our interpretation (§3.1 and below), the first to report evidence for microlensing of a polarized-light region in a QSO.
Before we examine the implications of our interpretation, we should comment on some issues which can affect the interpretation. For example, polarization induced by any dust which is present along the sight-line toward the Cloverleaf is unlikely to be responsible for the added component of polarization that appears to be present in component D.[^3] There are several reasons for this. First, multicolor (F336W, F702W, F814W) [*HST*]{} WFPC2 observations have shown that differential dust reddening is, in fact, present across the Cloverleaf components. Component B is the most reddened and component C is the least reddened, with the F336W$-$F814W color index being $0.56\pm 0.04$ mag between components B and C. The reddening of components A and D are intermediate (see figure 3 in Turnshek et al. 1997). The source of this differential reddening may be dust in the interstellar medium of the lensing galaxy. However, since interstellar polarization is normally proportional to the amount of reddening, we would not expect the change in polarization of component D to be related to the reddening. Second, if the source of the polarization was the lens, it would seem unlikely that the induced polarization would be different only for component D. Third, interstellar polarization would not be expected to be time-variable, but the observations indicate that the polarization changed between the March 1999 and June 1999 epochs. Fourth, Faraday rotation due to Galactic or cosmologically intervening plasma that may be present along the sight-line to component D could not reasonably give rise to any rotation in the polarization position angle. The observed change in position angle for component D, $\Delta \theta = 32 \pm 7$ deg at $\lambda = 5300$ Å (for the F555W filter), would require a medium with a rotation measure of ${\mbox{RM}} = (2.0 \pm 0.4)
\times 10^{12}$ radians m$^{-2}$; however, for example, this is eight orders of magnitude larger than one of the highest values ever measured for an extragalactic radio source (e.g. 3C 295, Perley & Taylor 1991).
Consequently, we have argued that the $\approx 0.07$ mag relative decrease in brightness of component D over the $\approx 100$ day interval between March 1999 and June 1999 (Figure 1) is evidence for microlensing of component D. In this scenario, component D evidently faded due to the motion of a microlens, causing it to be less magnified in June 1999. This is consistent with the earlier conclusions of Chae & Turnshek (1999, also §3.2), who interpreted the lower equivalent widths of the BELs in component D (observed with HST FOS in both June 1993 and December 1994) as evidence for microlens magnification of just the continuum of component D (not the BELs). Now, relying on the new results presented here, we can refine some of the conclusions of Chae & Turnshek (1999), and place better qualitative constraints on the size-scale of the polarized scattered-light region in relation to our understanding of the size-scales of the BEL and continuum-producing regions in QSOs.
We should point out that in all published individual spectra of the Cloverleaf components, the equivalent widths of the BELs in component D are smaller than observed in the other components. The first of four sets of component spectra were obtained in the spring of 1989 (Angonin et al. 1990) and this trend continues up until at least the spring of 2000, when HST-STIS spectra taken by Monier (PI) and several of the authors continued to show component D to have BELs with lower equivalent widths. This suggests that some level of microlens magnification of the continuum-producing region (but not the BELs) of component D is common over a relatively long time baseline.
The Cloverleaf’s macrolens by itself achromatically amplifies all light lying within $\approx 10^{19-20}$ cm of the central source into four point-like image components (see figure 1 of Chae & Turnshek 1999). Observations indicate that the macrolensed region includes the BEL region. This is consistent with expectations since the size of the region producing BELs like and Ly$\alpha$ is estimated to be $\approx 10^{18} L_{46}^{0.5}$ cm from the central photoionizing source (Murray & Chaing 1998; Kaspi et al. 2000), where $L_{46}$ is the lensed QSO luminosity in units of $10^{46}$ ergs s$^{-1}$. The source QSO luminosity in the Cloverleaf is not well-constrained because observations only provide results on the relative component amplifications; however, the lens models suggest that the luminosity is likely to be of order $L_{46}$. Evidently the Cloverleaf is polarized because an asymmetric continuum scattering region also lies within the macrolensed region. The asymmetry is a requirement since the net polarization is non-zero. This region is not static; changes in parts of it must give rise to the variable continuum polarization which is seen. The fact that we have evidence for a variation in the net polarization over a $\approx
100$ day interval in the observed frame ($\approx 30$ day interval in proper time) suggests that the size scales involved which lead to changes in polarization are $< 10^{17}$ cm. However, each part of this asymmetric scattering region by itself would be expected to give rise to highly-polarized continuum light ($p_{scatt,cont}
> 10$%), but when averaged over the entire asymmetric region there would be a much smaller net polarization.
For the purpose of illustration we note that if $\approx
80$% of the flux of component D was composed of the continuum plus BELs (as seen in the nearly identical spectra of components A, B and C), and the remaining $\approx 20$% of the flux of component D was solely composed of microlensed scattered continuum (i.e. no BEL flux), then the polarization of the scattered continuum component would have to be $p_{scatt,cont} \approx 13$% at $PA \approx 117$ deg to match the observations. This degree of polarization is reasonable for scattering.
Owing to the fact that the BEL equivalent widths in component D are observed to be smaller in comparison to the other components, the observed microlensed scattered-light region evidently does not scatter appreciable BEL flux. This suggests that the size-scale of the polarized scattered-light region ($R_{scatt}$) must be less than the size-scale of the region producing BELs, i.e., $R_{scatt} <
10^{18} L_{46}^{0.5}$ cm. At the same time, in order to produce the observed change in the polarization of component D relative to the other three components, the polarized scattered-light region must lie beyond the inner continuum-producing region which it reflects, far enough so that the inner continuum-producing region is not microlensed. This is because, if the entire scattering region were to lie within the microlensed region, there would be a near constant magnification across the region during microlensing, the continuum light and scattered continuum light would be similarly amplified, and the polarization would remain unchanged among the four components during microlensing. Note that microlensing of an unpolarized central continuum-producing region (e.g. the continuum from the thermal accretion disk) is also not a possibility, since this would reduce the polarization in component D, which is not observed. These constraints are new; they were not addressed in the discussion of Chae & Turnshek (1999) because component polarization information was unavailable.
The size of the microlensed region on the source plane will be of order the Einstein ring size. Following Chae & Turnshek (1999), the Einstein ring size on the source plane produced by a microlensing star is given by $\eta_0 \approx 2 \times 10^{16}
(M/M_{\odot})^{0.5} h_{75}^{-0.5}$ cm, where $M$ is the mass of the microlens. This is $\approx 8$ light-days for a solar mass star, which is larger than the expected size of the continuum emitting region from any accretion disk. We would expect the size of the microlensed polarized scattering region to lie beyond this region.
Taken together our results therefore suggest that the size-scale of the polarized scattered-light region in the Cloverleaf is $$2 \times 10^{16} (M/M_{\odot})^{0.5} h_{75}^{-0.5} < R_{scatt} <
10^{18} L_{46}^{0.5} \ \ cm.$$
From these results it is clear that a more rigorous future program dedicated to monitoring (photometric, spectroscopic, and polarimetric) the Cloverleaf would hold promise for providing valuable constraints on models of the inner regions of QSOs (i.e the BAL region, the BEL region, and the scattered-light region producing the polarization).
We are grateful for helpful communications throughout the course of this work from Drs. S. Casertano, J. Krist, P. Massey, E. Monier, and J. Walsh. We also thank Dr. S. Mao for helpful discussions regarding microlensing and comments on the manuscript.
Angonin, M.-C., Remy, M., Surdej, J., & Vanderriest, C. 1990, A&A, 233, L5
Arnould, P., et al. 1993, in [*Proceedings of the 31st Liege International Astrophysical Colloquium, Gravitational Lenses in the Universe*]{}, eds. J. Surdej et al. (Liege: Univ. Liege, Inst. d’Astrophys), p169
Belle, K. E., & Lewis, G. F. 2000, PASP, 112, 320
Biretta, J., & McMaster, M. 1997, in [*Instrument Science Report WFPC2 97-11*]{} (STScI, Baltimore, MD)
Chae, K.-H., & Turnshek, D. A. 1999, , 514, 587
Fruchter, A. S., Hook, R. N., Busko, I. C., & Mutchler, M. 1997, in [*1997 HST Calibration Workshop*]{}, eds. S. Casertano et al. (STScI, Baltimore, MD) p518
Fruchter, A. S., & Hook, R. N. 1998, astro-ph/9808087
Goodrich, R. W., & Miller, J. S. 1995, , 448, L73
Goodrich, R. W. 1997, , 474, 606
Hutsemékers, D., Lamy, H., & Remy, M. 1998, A&A, 340, 371
Kaspi, S., et al. 2000, , 533, 631
Monier, E. M., Turnshek, D. A., & Lupie, O. L. 1998, , 496, 177
Murray, N., & Chiang, J. 1998, , 494, 125
Ogle, P. M. 1998, Ph.D. Thesis, Caltech
Ostensen, R., et al. 1997, A&AS, 126, 393
Perley, R. A., & Taylor, G. B. 1991, , 101, 1623
Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992, [*Numerical Recipes in FORTRAN*]{} (Cambridge University Press), p689
Remy, M., Gosset, E., Hutsémekers, D., Revenaz, B., & Surdej, J. 1996, in [*IAU Symp. 173, Astrophysical Applications of Gravitational Lensing*]{}, eds. C. Kochanek & J. Hewitt (Dordrecht: Kluwer), p261
Schmidt, G. D., & Hines, D. C. 1999, , 512, 125
Turnshek, D. A. 1988, in [*QSO Absorption Lines: Probing the Universe*]{}, eds. J. C. Blades, D. Turnshek, & C. Norman (Cambridge University Press), p17
Turnshek, D. A. 1995, in [*QSO Absorption Lines, ESO Workshop*]{}, ed. G. Meylan (Berlin: Springer), p317
Turnshek, D. A., Lupie, O. L., Rao, S. M., Espey, B. R., & Sirola, C. J. 1997, , 485, 100
Wardle, J. F. C., & Kronberg, P. P. 1974, , 194, 249
Weymann, R. J., Morris, S. L., Foltz, C. B., & Hewett, P. C. 1991, , 373, 23
**Table 1. WFPC2 Photometric and Polarimetric Observations**
=0.5em to
[cccccc]{}
F555W & 15-16 Mar 1999 & $0.000 \pm 0.003$ & $0.193 \pm 0.003$ & $0.300 \pm 0.004$ & $0.292 \pm 0.003$\
& 23-24 Jun 1999 & $-0.025 \pm 0.003$ & $0.188 \pm 0.003$ & $0.281 \pm 0.004$ & $0.368 \pm 0.003$\
F702W & 15-16 Mar 1999 & $0.000 \pm 0.007$ & $0.171 \pm 0.007$ & $0.320 \pm 0.008$ & $0.385 \pm 0.008$\
& 23-24 Jun 1999 & $-0.004 \pm 0.008$ & $0.163 \pm 0.008$ & $0.344 \pm 0.009$ & $0.446 \pm 0.009$\
[ccccc]{}
F555W & $-0.025 \pm 0.004$ & $-0.005 \pm 0.004$ & $-0.019 \pm 0.006$ & $0.076 \pm 0.004$\
F702W & $-0.004 \pm 0.011$ & $-0.008 \pm 0.011$ & $0.024 \pm 0.012$ & $0.061 \pm 0.012$\
F555W+F702W & $-0.023 \pm 0.004$ & $-0.005 \pm 0.004$ & $-0.010 \pm 0.005$ & $0.074 \pm 0.004$\
**Table 3. F555W Relative Polarimetric Results for the Cloverleaf Components$^a$**
=0.7em to
$^a$ Quoted errors are relative statistical errors.\
$^b$ The combined result of components A, B, and C.\
$^c$ We use the formula of Wardle & Kronberg (1974) to correct the bias toward higher polarization degree in low S/N ratio data: $$p_{\mbox{\scriptsize corr}} = p \left[ 1-\left(\frac{\sigma_p}{p}\right)^2
\right]^{1/2}.$$ Our derived errors in $PA$ are in good agreement with their general prescription for estimating the $PA$ errors under such conditions: $\sigma_{PA} \approx 28.65\sigma_p/p$ deg.
[^1]: We could not use the “Tiny TIM” software to generate PSFs since it does not have the capability of incorporating the effects of the polarizing filters. We found that all individual images taken with the POLQN33 polarizer (for both epochs) and the POLQP15 polarizer (for the March epoch) had substantially modified PSFs in terms of FWHM and peak pixel value in comparison to images taken without polarizers.
[^2]: For the March data we did not use the web-based WFPC2 polarization calibration tool or the IRAF $IMPOL$ package, since the former can take only three relative counts at one time while the latter takes the images themselves as input using only aperture photometry, which is less accurate for the “crowded” Cloverleaf field. It was possible to use the web-based WFPC2 polarization calibration tool for the June data.
[^3]: Interstellar polarization within the Milky Way is not significant for the Cloverleaf (e.g. Hutsemékers et al. 1998).
|
---
abstract: 'A [*latin bitrade*]{} $(T^{{\diamond}},\, T^{{\otimes}})$ is a pair of partial latin squares which are disjoint, occupy the same set of non-empty cells, and whose corresponding rows and columns contain the same set of entries. A genus may be associated to a latin bitrade by constructing an embedding of the underlying graph in an oriented surface. We report computational enumeration results on the number of spherical (genus 0) latin bitrades up to size $24$.'
author:
- |
Aleš Drápal[^1]\
Department of Mathematics\
Charles University\
Sokolovská 83\
186 75 Praha 8\
Czech Republic\
\
Carlo Hämäläinen[^2]\
Department of Mathematics\
Charles University\
Sokolovská 83\
186 75 Praha 8\
Czech Republic\
[`c`arlo.hamalainen@gmail.com]{}\
\
Dan Rosendorf\
Department of Mathematics\
University of Wisconsin, Madison, USA
title: |
**An enumeration of spherical\
latin bitrades**
---
Introduction
============
A [*latin bitrade*]{} $(T^{{\diamond}},\, T^{{\otimes}})$ is a pair of partial latin squares which are disjoint, occupy the same set of non-empty cells, and whose corresponding rows and columns contain the same set of entries. One of the earliest studies of latin bitrades appeared in [@DrKe1], where they are referred to as [*exchangeable partial groupoids*]{}. Latin bitrades are prominent in the study of [*critical sets*]{}, which are minimal defining sets of latin squares ([@BaRe2],[@codose],[@Ke2]) and the intersections between latin squares ([@Fu]). A genus may be associated to a latin bitrade, and those of genus zero are known as [*spherical*]{} latin bitrades. In this paper we report on the enumeration of the number of $\tau$-isomorphism classes of spherical latin bitrades up to size $\left| T^{{\diamond}} \right| = 24$.
For completeness we note that separated spherical latin bitrades are equivalent to cubic $3$-connected bipartite planar graphs [@planareulerian]. Our enumeration method does not consider equivalences where rows, columns, and symbols change their roles (this corresponds to changing the $3$-colouring of the related cubic 3-connected bipartite graph), nor does it consider equivalences where $(T^{{\diamond}},\, T^{{\otimes}})$ and $(T^{{\otimes}},\, T^{{\diamond}})$ are equivalent (this corresponds to changing the $2$-colouring of the related cubic 3-connected bipartite graph). Thus our enumeration method counts a larger class of combinatorial objects compared to the method in [@holtonetc].
Latin bitrades
==============
A [*partial latin square*]{} $P$ of order $n > 0$ is an $n \times n$ array where each $e \in \{ 0, 1, \dots, n-1 \}$ appears at most once in each row, and at most once in each column. A [*latin square*]{} $L$ of order $n > 0$ is an $n \times n$ array where each $e \in \{ 0, 1, \dots, n-1 \}$ appears exactly once in each row, and exactly once in each column. It is convenient to use setwise notation to refer to entries of a (partial) latin square, and we write $(i,j,k) \in P$ if and only if symbol $k$ appears in the intersection of row $i$ and column $j$ of $P$. In this manner, $P \subseteq A_1 \times A_2 \times A_3$ for finite sets $A_i$, each of size $n$. It is also convenient to interpret a (partial) latin square as a multiplication table for a (partial) binary operator ${\diamond}$, writing $i {\diamond}j = k$ if and only if $(i,j,k) \in T = T^{{\diamond}}$.
\[defnBitradeA123\] Let $T^{{\diamond}}$, $T^{{\otimes}} \subseteq A_1 \times A_2 \times A_3$ be two partial latin squares. Then $(T^{{\diamond}},\, T^{{\otimes}})$ is a [*bitrade*]{} if the following three conditions are satisfied:
- $T^{{\diamond}} \cap T^{{\otimes}} = \emptyset$;
- for all $(a_1,\, a_2,\, a_3) \in T^{{\diamond}}$ and all $r$, $s \in \{1,\, 2,\, 3\}$, $r \neq s$, there exists a unique $(b_1,\, b_2,\, b_3) \in T^{{\otimes}}$ such that $a_r=b_r$ and $a_s=b_s$;
- for all $(a_1,\, a_2,\, a_3) \in T^{{\otimes}}$ and all $r$, $s \in \{1,\, 2,\, 3\}$, $r \neq s$, there exists a unique $(b_1,\, b_2,\, b_3) \in T^{{\diamond}}$ such that $a_r=b_r$ and $a_s=b_s$.
Conditions [[(R2)]{.nodecor}]{}and [[(R3)]{.nodecor}]{}imply that each row (column) of $T^{{\diamond}}$ contains the same subset of $A_3$ as the corresponding row (column) of $T^{{\otimes}}$. A bitrade $(T^{{\diamond}},\, T^{{\otimes}})$ is [*indecomposable*]{} if whenever $(U^{{\diamond}},\,U^{{\otimes}})$ is a bitrade such that $U^{{\diamond}}\subseteq
T^{{\diamond}}$ and $U^{{\otimes}}\subseteq
T^{{\otimes}}$, then $(T^{{\diamond}},\, T^{{\otimes}})=
(U^{{\diamond}},\, U^{{\otimes}})$. Bijections $A_i \rightarrow A'_i$, for $i = 1$, $2$, $3$, give an [*isotopic*]{} bitrade, and permuting each $A_i$ gives an [*autotopism*]{}. We refer to the bijections $A_1 \rightarrow A'_1$, $A_2 \rightarrow A'_2$, $A_3 \rightarrow A'_3$ as an [*isotopism*]{}.
In [@Dr9; @ales-geometrical] there is a representation of bitrades in terms of three permutations $\tau_i$ acting on a finite set (see also [@hamalainen2007] for another proof). For $r \in \{1,\, 2,\, 3\}$, define the map $\beta_r \colon T^{{\otimes}} \rightarrow
T^{{\diamond}}$ where $(a_1,\, a_2,\, a_3) \beta_r = (b_1,\, b_2,\, b_3)$ if and only if $a_r \neq b_r$ and $a_i = b_i$ for $i \neq r$. By Definition \[defnBitradeA123\] each $\beta_r$ is a bijection. Then $\tau_1,\, \tau_2,\, \tau_3\colon T^{{\diamond}} \rightarrow
T^{{\diamond}}$ are defined by $$\begin{aligned}
\tau_1 &= \beta_2^{-1}\beta_3, \qquad
\tau_2 = \beta_3^{-1}\beta_1, \qquad
\tau_3 = \beta_1^{-1}\beta_2. \label{eqnTau}\end{aligned}$$ We refer to $[\tau_1,\, \tau_2,\, \tau_3]$ as the $\tau_i$ [*representation*]{}. We write ${\textnormal{Mov}}(\pi)$ for the set of points that the (finite) permutation $\pi$ acts on.
Generally we will assume that a bitrade is [*separated*]{}, that is, each row, column, and symbol is in bijection with a cycle of $\tau_1$, $\tau_2$, and $\tau_3$, respectively.
\[defnT1234\] Let $\tau_1$, $\tau_2$, $\tau_3$ be (finite) permutations and let ${\Gamma}= {\textnormal{Mov}}(\tau_1) \cup {\textnormal{Mov}}(\tau_2) \cup {\textnormal{Mov}}(\tau_3)$. Define four properties:
1. $\tau_1 \tau_2 \tau_3 = 1$;
2. if $\rho_i$ is a cycle of $\tau_i$ and $\rho_j$ is a cycle of $\tau_j$ then $\left| {\textnormal{Mov}}(\rho_i) \cap {\textnormal{Mov}}(\rho_j) \right| \leq 1$, for any $1 \leq i < j \leq 3$;
3. each $\tau_i$ is [fixed-point-free]{};
4. the group $\langle \tau_1,\, \tau_2,\, \tau_3 \rangle$ is transitive on ${\Gamma}$.
By letting $A_i$ be the set of cycles of $\tau_i$, we have the following theorem, which relates Definition \[defnBitradeA123\] and \[defnT1234\].
\[theoremDrapalTauStructure\] A separated bitrade $(T^{{\diamond}},\, T^{{\otimes}})$ is equivalent (up to isotopism) to three permutations $\tau_1$, $\tau_2$, $\tau_3$ acting on a set ${\Gamma}$ satisfying [[(T1)]{.nodecor}]{}, [[(T2)]{.nodecor}]{}, and [[(T3)]{.nodecor}]{}. If [[(T4)]{.nodecor}]{}is also satisfied then the bitrade is indecomposable.
To construct the $\tau_i$ representation for a bitrade we simply evaluate Equation . In the reverse direction we have the following construction:
\[constructionTauToBitrade\] Let $\tau_1$, $\tau_2$, $\tau_3$ be permutations satisfying Condition [[(T1)]{.nodecor}]{}, [[(T2)]{.nodecor}]{}, and [[(T3)]{.nodecor}]{}. Let ${\Gamma}= {\textnormal{Mov}}(\tau_1) \cup {\textnormal{Mov}}(\tau_2) \cup {\textnormal{Mov}}(\tau_3)$. Define $A_i = \{ \rho \mid \textnormal{$\rho$ is a cycle of $\tau_i$} \}$ for $i = 1$, $2$, $3$. Now define two arrays $T^{{\diamond}}$, $T^{{\otimes}}$: $$\begin{aligned}
T^{{\diamond}}= \{( {\rho}_1,\, {\rho}_2,\, {\rho}_3 )\mid
&\text{ $\rho_i \in A_i$ and
$\left| {\textnormal{Mov}}(\rho_1) \cap {\textnormal{Mov}}(\rho_2) \cap {\textnormal{Mov}}(\rho_3) \right| \geq
1$} \} \\
T^{{\otimes}} = \{( {\rho}_1,\, {\rho}_2,\, {\rho}_3 ) \mid
&\text{ $\rho_i \in A_i$ and $x$, $x'$, $x''$ are distinct points of
${\Gamma}$ such} \\
&\text{ that $x\rho_1=x'$, $x'\rho_2=x''$, $x''\rho_3=x$} \}. \end{aligned}$$ By Theorem \[theoremDrapalTauStructure\] $(T^{{\diamond}},\, T^{{\otimes}})$ is a bitrade.
\[exampleIntercalateRep\] The smallest bitrade $(T^{{\diamond}},\, T^{{\otimes}})$ is the [*intercalate*]{}, which has four entries. The bitrade is shown below: $$\begin{aligned}
T^{{\diamond}} = \begin{array}{c|cccc}
{\diamond}& 0 & 1 \\
\hline
0 & 0 & 1 \\
1 & 1 & 0
\end{array}
& \quad &
T^{{\otimes}} = \begin{array}{c|cccc}
{\otimes}& 0 & 1 \\
\hline
0 & 1 & 0 \\
1 & 0 & 1
\end{array}
$$ The $\tau_i$ representation is $\tau_1 = (000,011)(101,110)$, $\tau_2 = (000,101)(011,110)$, $\tau_3 = (000,110)(011,101)$, where we have written $ijk$ for $(i,j,k) \in T^{{\diamond}}$ to make the presentation of the $\tau_i$ permutations clearer. By Construction \[constructionTauToBitrade\] with ${\Gamma}= \{ 000, 011, 101, 110 \}$ we can convert the $\tau_i$ representation to a bitrade $(U^{{\diamond}},\, U^{{\otimes}})$: $$\begin{aligned}
U^{{\diamond}} &= \begin{array}{c|cccc}
{\diamond}& (000,101) & (011,110) \\
\hline
(000,011) & (000,110) & (011,101) \\
(101,110) & (011,101) & (000,110)
\end{array} \\
U^{{\otimes}} &= \begin{array}{c|cccc}
{\otimes}& (000,101) & (011,110) \\
\hline
(000,011) & (011,101) & (000,110) \\
(101,110) & (000,110) & (011,101)
\end{array}\end{aligned}$$ In this way we see that row $0$ of $T^{{\diamond}}$ corresponds to row $(000,011)$ of $U^{{\diamond}}$, which is the cycle $( 000,011 )$ of $\tau_1$, and so on for the columns and symbols.
\[exspherical\] The following bitrade is spherical: $$\begin{aligned}
T^{{\diamond}} =
\begin{array}{c|ccccc}
{\diamond}& 0 & 1 & 2 & 3 & 4\\
\hline
0 & 0 & ~ & 2 & ~ & 4\\ 1 & ~ & ~ & ~ & 4 & 2\\ 2 & 1
& 3 & 0 & 2 & ~\\ 3 & 4 & 1 & ~ & 3 & ~\\ 4 & ~ & ~ & ~ & ~ & ~
\end{array}
& \quad &
T^{{\otimes}} =
\begin{array}{c|ccccc}
{\otimes}& 0 & 1 & 2 & 3 & 4\\
\hline 0 & 4 & ~ & 0 & ~ & 2\\ 1 & ~ & ~ & ~ & 2 & 4\\ 2 & 0 & 1 & 2 & 3 & ~\\ 3 & 1 & 3 & ~ & 4 & ~\\ 4 & ~ & ~ & ~ & ~ & ~\\\end{array}
$$ Here, the $\tau_i$ representation is $$\begin{aligned}
\tau_1 &= (000, 022, 044)(134, 142) (201,213,232,220) (304,333,311) \\
\tau_2 &= (000,304,201)(213,311)(022,220)(134,232,333)(044,142) \\
\tau_3 &= (000,220)(201,311)(022,232,142)(213,333)(044,134,304)\end{aligned}$$
If $Z = [\tau_1,\, \tau_2,\, \tau_3]$ then the inverse is denoted by $Z^{-1} = [\tau_1^{-1},\, \tau_2^{-1},\, \tau_2 \tau_1]$. This is equivalent to exchanging $(T^{{\diamond}},\, T^{{\otimes}})$ for $(T^{{\otimes}},\,
T^{{\diamond}})$ and relabelling entries as the following lemma shows:
Let $(T^{{\diamond}},\, T^{{\otimes}})$ be a separated bitrade with representation $Z = [\tau_1,\, \tau_2,\, \tau_3]$. Then the inverse bitrade $(T^{{\otimes}},\, T^{{\diamond}})$ has representation, denoted $Z^{-1}$. This representation is isomorphic to $[\tau_1^{-1},\, \tau_2^{-1},\, \tau_2 \tau_1]$.
Observe that Definition \[defnBitradeA123\] does not specify the order of the two partial latin squares in a bitrade so the bitrade $(T^{{\otimes}},\, T^{{\diamond}})$ is well-defined. For this bitrade we have permutation representation $[\nu_1,\, \nu_2,\, \nu_3]$. Using Equation we find that $\nu_1 = \beta_2 \beta_3^{-1}$ and $\nu_2 = \beta_3 \beta_1^{-1}$. Taking the conjugate of $\nu_r$ by $\beta_3^{-1}$ gives us a representation that acts on the set $T^{{\diamond}}$. In particular, $\beta_3^{-1} \nu_1 \beta_3 = \beta_3^{-1} \beta_2 \beta_3^{-1} \beta_3 =
\beta_3^{-1} \beta_2 =
(\beta_2^{-1} \beta_3)^{-1} = \tau_1^{-1}$ and $\beta_3^{-1} \nu_2 \beta_3 = \beta_3^{-1} \beta_3 \beta_1^{-1} \beta_3 =
\beta_1^{-1} \beta_3 =
(\beta_3 \beta_1^{-1})^{-1} = \tau_2^{-1}$. This shows that the representation $[\nu_1,\, \nu_2,\, \nu_3]$ is isomorphic to $[\tau_1^{-1},\, \tau_2^{-1},\, \tau_2 \tau_1]$.
Let $D$ be the graph with vertices ${\Gamma}$ and directed edges $(x, y)$ where $x \tau_i = y$ for some $i$. A rotation scheme can be imposed on this graph, by ordering the edges as shown in Figure \[figDrapalRotationScheme\]. This turns $D$ into an oriented combinatorial surface.
![Rotation scheme for the combinatorial surface. Dashed lines represent one or more edges, where $\mu_i$ is a cycle of $\tau_i$. The vertex in the centre is $x$.[]{data-label="figDrapalRotationScheme"}](drRotationScheme-2.pdf)
Define $\textnormal{order}(\tau_i) = z(\tau_1) + z(\tau_2) + z(\tau_3)$, the total number of cycles, and $\textnormal{size}(\tau_i) = \left| {\Gamma}\right|$. By some basic counting arguments we find that there are $\text{size}(\tau_i)$ vertices, $3 \cdot \text{size}(\tau_i)$ edges, and $\text{order}(\tau_i) + \text{size}(\tau_i)$ faces. Then Euler’s formula $V-E+F=2-2g$ gives $$\label{eqnDrapalGenus}
\text{order}(\tau_i) = \text{size}(\tau_i) + 2 - 2g$$ where $g$ is the genus of the combinatorial surface. We say that the bitrade $T$ has genus $g$. If $g = 0$ then we say that the bitrade is [*spherical*]{}.
Slide expansion
===============
Let $[\tau_1,\, \tau_2,\, \tau_3]$ be a spherical bitrade on the set ${\Gamma}$. Choose some $x \in {\Gamma}$ and fix a direction $j \in
\{1,2,3\}$. Set $k = j+1$ and $l = k+1$ (calculating modulo $3$). Suppose that (1) the $j$-cycle at $x$ has length at least $3$ and (2) the $k$-cycle through $x$ and the $l$-cycle through $w = x \tau_j$ do not meet in any common point. Then the [*slide expansion*]{} at $x \in {\Gamma}$ in direction $j$ is the following augmentation of the cycles of $\tau_1$, $\tau_2$, and $\tau_3$ by a new point $u \notin {\Gamma}$: $$\begin{aligned}
\tau_j &\colon (a, x, w, b, \dots) \mapsto (a, u, b, \dots) \textnormal{ and } (x, w) \\
\tau_k &\colon (\dots, x, z, \dots) \mapsto (\dots, x, u, z, \dots) \\
\tau_l &\colon (\dots, y, w, \dots) \mapsto (\dots, y, u, w, \dots).\end{aligned}$$ We denote the result of the slide expansion on $X = [\tau_1,\, \tau_2,\, \tau_3]$ at vertex $x$ in direction $j$ by $\textnormal{slide}(X, x, j)$.
A separated bitrade $[\tau_1,\, \tau_2,\, \tau_3]$ is [*bicyclic*]{} if there exists $j$ such that $\tau_j$ consists of only two cycles. These bicyclic bitrades form the root nodes of our enumeration procedure due to the following result:
Let $Z = [\tau_1,\, \tau_2,\, \tau_3]$ be a spherical bitrade. Then there exists a sequence $W_0$, $W_1$, …, $W_{\ell} = Z$ where $W_0$ is a bicyclic bitrade and $W_{i+1}$ is the result of a slide expansion on either $W_{i}$ or $W^{-1}_i$, for $0 \leq i < \ell$.
The opposite of a slide expansion is a [*slide contraction*]{}. For a spherical bitrade $[\tau_1,\, \tau_2,\, \tau_3]$ on the set ${\Gamma}$, choose some $u \in {\Gamma}$ and fix a direction $j \in
\{1,2,3\}$. Set $k = j+1$ and $l = k+1$ (calculating modulo $3$). The slide contraction at $u$ in direction $j$ modifies the cycles of $\tau_1$, $\tau_2$, and $\tau_3$ as follows: $$\begin{aligned}
\tau_j &\colon (a, u, b, \dots) \textnormal{ and } (x, w) \mapsto (a, x, w, b, \dots) \\
\tau_k &\colon (\dots, x, u, z, \dots) \mapsto (\dots, x, z, \dots) \\
\tau_l &\colon (\dots, y, u, w, \dots) \mapsto (\dots, y, w, \dots).\end{aligned}$$ A necessary condition for a slide contraction to be valid is that the $u$-cycle in $\tau_k$ and $\tau_l$ has length at least $3$ (this avoids the creation of a fixed point when $u$ is removed). In general it is necessary to check the bitrade conditions [[(T1)]{.nodecor}]{}, [[(T2)]{.nodecor}]{}, and [[(T3)]{.nodecor}]{}to ensure that a slide contraction is valid.
Enumeration
===========
To enumerate separated spherical bitrades we must fix the notion of $\tau$-isomorphism. Two bitrades $[\tau_1,\, \tau_2,\, \tau_3]$ and $[\nu_1,\, \nu_2,\, \nu_3]$ on the same set ${\Gamma}$ are $\tau$-isomorphic if there is a permutation $\theta \in \textnormal{Sym}({\Gamma})$ such that $\tau_i^{\theta} = \nu_i$ for all $i \in \{1,2,3\}$. A $\tau$-automorphism of a bitrade is defined similarly.
We follow the presentation in [@classificationalgorithms]. In the context of an algorithm for enumerating combinatorial objects, the [*domain*]{} of a search is the finite set ${\Omega}$ that contains all objects considered by the search. The search space is conveniently modelled by a rooted tree, with nodes corresponding to elements of ${\Omega}$, and nodes joined by an edge if they are related by one search step. The root node is the starting node of the search. We write $C(X)$ for the set of all child nodes of $X$, and $p(X)$ for the parent node of $X$.
To reduce the search time we use the method of [*canonical augmentation*]{} [@mckay], following the presentation in [@classificationalgorithms]. In terms of the search tree, the ordered pair $(X, p(X))$ characterises the augmentation used to generate $X$ from the node $p(X)$ during the search. In our case, this augmentation $(X, p(X))$ is the result of a slide expansion on $p(X)$. The goal of the canonical augmentation algorithm is to choose one of the possible parents of $X$ to be the canonical parent. Formally, for any nonroot node $X$, the [*canonical parent*]{} $m(X) \in {\Omega}$ must satisfy:
1. for all nonroot objects $X$, $Y$ it holds that $X {\cong}Y$ implies $(X, m(X)) {\cong}(Y, m(Y))$.
The next property captures the canonical augmentations in the search tree:
1. a node $Z$ occurring in the search tree is generated by canonical augmentation if $(Z, p(Z)) {\cong}(Z, m(Z))$.
Algorithm \[alg1\] gives pseudo-code for the general canonical augmentation search algorithm. Algorithm \[alg2\] is a more explicit version for enumerating separated spherical latin bitrades. The correctness of Algorithm \[alg1\] relies on two properties:
1. for all nodes $X$, $Y$ it holds that if $X {\cong}Y$, then for every $Z \in C(X)$ there exists a $W \in C(Y)$ such that $(Z,X) {\cong}(W,Y)$.
2. for every nonroot node $X$, there exists a node $Y$ such that $X {\cong}Y$ and $(Y,m(Y)) {\cong}(Y,p(Y))$.
Condition [[(I1)]{.nodecor}]{}says that isomorphic nodes have isomorphic children. Condition [[(I2)]{.nodecor}]{}says that every nonroot node is generated by some canonical augmentation.
report $X$ (if applicable) select any $Z \in \mathcal{Z}$ CANAUG-TRAVERSE($Z$)
report $X$ (if applicable) select any $Z \in \mathcal{Z}$ CANAUG-SPHERICAL($Z$) CANAUG-SPHERICAL($Z^{-1}$) CANAUG-SPHERICAL($Z$)
When implemented on a search tree satisfying [[(I1)]{.nodecor}]{}and [[(I2)]{.nodecor}]{}, the algorithm [CANAUG-TRAVERSE]{.nodecor} reports exactly one node from every isomorphism class of nodes.
Canonical form
--------------
The canonical form $\hat{Z}$ is a relabelling of $Z$ such that $\hat{Z} = \hat{Y}$ if and only if $Z {\cong}Y$. We need to be able to efficiently compute the canonical form in order to construct the canonical parent of a bitrade.
To compute the canonical form of a bitrade $Z = [\tau_1,\, \tau_2,\,
\tau_3]$, we fix a starting point $x \in {\Gamma}$. We then perform a breadth-first traversal of the bitrade’s underlying directed graph. Vertices are labelled from the set $\{ 1, \dots, \left| Z \right| \}$ in the order that they are visited. At a vertex $v$ we first visit the $\tau_1$ cycle at $v$, then the $\tau_2$ cycle at $v$, and then the $\tau_3$ cycle at $v$ (only if the cycle has not been visited before). As we traverse a cycle we label unseen vertices and append the cycle (using the new vertex labels) to an array $C$. After traversing a cycle we append a marker (here, $-1$) to denote the end of the cycle. While traversing a $\tau_i$ cycle at the vertex $v$, we check if the neighbouring $\tau_{j}$ and $\tau_{k}$ cycle at $v$ has been visited ($j$, $k \neq i$). If it has not, we push $(j,v)$ and $(k,v)$ onto the queue, making sure that $j < k$. The lexicographically least $C = C(x)$ is the canonical form $\hat{Z}$ of $Z$. Since $\hat{Z}$ encodes just the cycle structure of the $\tau_i$ permutations, it follows that $\hat{Z} = \hat{Z'}$ if and only if $Z {\cong}Z'$.
The canonical parent $m(Z)$ is now computed as follows. First we find the canonical form $\hat{Z}$ of $Z$. In our case, we choose the lexicographically maximal pair $(j, u)$ such that a slide contraction may be performed on $\hat{Z}$ at $u$ in direction $j$. Let $Z_0$ be the result of this contraction. Then $m(Z) = \kappa^{-1}(Z_0)$ where $\kappa \colon Z \mapsto \hat{Z}$ is the canonical relabelling map.
Correctness
-----------
Condition [[(I1)]{.nodecor}]{}holds since a slide expansion $\textnormal{slide}(X, x, j)$ is equivalent to $\textnormal{slide}(Y, y, j)$ where $X {\cong}Y$ and $y$ is the image of $x$ under the isomorphism. In other words, the slide expansion relies only on the cycle structure of the bitrade, not its specific labelling. Condition [[(I2)]{.nodecor}]{}holds because the canonical parent $m(Z)$ is chosen only based on the canonical form of $Z$ and so is independent of isomorphism.
Algorithm \[alg2\] produces bicyclic bitrades at each step where even-sized bitrades are being generated. The following lemma and corollary are necessary to show that no $\tau$-isomorphism class of bicyclic bitrades is produced more than once.
\[bicyclic\_structure\] Let $[\tau_1,\, \tau_2,\, \tau_3]$ be a bitrade such that $\tau_j$, for some $j \in \{1,2,3\}$, consists of exactly two cycles. Let one of the cycles be $(x_0, \dots, x_{n-1})$. Then the other cycle of $\tau_j$ can be expressed as $(x'_{n-1}, \dots, x'_0)$ in such a way that $\tau_{j+1}$ consists of cycles $(x_i, x'_i)$ and $\tau_{j-1}$ consists of cycles $(x_{i+1}, x'_i)$, for $0 \leq i < n$.
\[corbicyclic\] Let $Z$ be a bicyclic bitrade. Then there is no $X$ such that $\textnormal{slide}(X, x, j) = Z$ or $\textnormal{slide}(X, x, j) = Z^{-1}$ for any $x$, $j$.
The slide contraction shortens the cycle at some point $u$ in $\tau_{j+1}$ and $\tau_{j-1}$. By Lemma \[bicyclic\_structure\] at least one of the cycles in $\tau_{j+1}$ or $\tau_{j-1}$ has length $2$, so the slide contraction would introduce a fixed point, contradicting [[(T3)]{.nodecor}]{}.
It can be shown that for any bicyclic bitrade $Z$ we have $Z {\cong}Z^{-1}$. Corollary \[corbicyclic\] shows that lines 11–15 of Algorithm \[alg2\] are correct.
Implementation and relation to other work
=========================================
See [@code] for C++ computer code. The MPI library is used to allow parallel processing of the search tree on a Linux cluster. The size of the automorphism group ${\textnormal{Aut}(Z)}$ is bounded by the size of ${\Gamma}$. Further, the number of homogeneous bitrades is quite small, so usually the automorphism group is even smaller. We just compute the elements of ${\textnormal{Aut}(Z)}$ explicitly.
Cavenagh and Lisonek [@planareulerian] showed that spherical bitrades are equivalent to planar Eulerian triangulations. To verify our results (for small $n$) we used plantri [@plantri] to generate planar Eulerian triangulations. We then applied a simultaneous vertex $3$-colouring and face $2$-colouring, giving a bitrade $(T^{{\diamond}},\,
T^{{\otimes}})$. Then each conjugate of $(T^{{\diamond}},\, T^{{\otimes}})$ was produced, and finally isomorphic copies removed.
The number of $\tau$-isomorphism classes of spherical bitrades with $\left| T^{{\diamond}} \right| = n$ is given below:
$n$ \# iso classes $e_3(n_v)$
----- ---------------- ------------
4 1 1
6 3 1
7 1 1
8 6 2
9 9 2
10 30 8
11 51 8
12 198 32
13 470 57
14 1623 185
15 4830 466
16 16070 1543
17 51948 4583
18 175047 15374
19 588120 50116
20 2015226 171168
21 6933048 582603
22 24123941 2024119
23 84428820 7057472
24 297753519 24873248
The third column is the value $e_3(n_v)$, which is the number of isomorphism classes of eulerian plane triangulations with connectivity at least $3$ (see Table 6 in [@fastgen] for values $n_f = 2n$ in their first column and also [@A000108]). The enumeration resulting in $e_3(n_v)$ considers isomorphic any two graphs that are related by a permutation of the face 2-colouring, changing the orientation of the permutations, and permuting the cyclic ordering. So for large values of $n$ there will be at most $2 \cdot 2 \cdot 3 = 12$ ways to relabel one of their graphs to obtain one of our bitrades. For example, with $n =
24$ we have $e_3(n_v) = 24873248$ and $12 \cdot 24873248 = 298478976 \approx 297753519$. Also, Wanless [@wanlessenumeration] enumerated spherical bitrades up to size $19$, under various various equivalences, but not $\tau$-isomorphism as studied in this paper.
[10]{}
Number of unlabelled trivalent 3-connected bipartite planar graphs with 2n nodes. <http://www.research.att.com/~njas/sequences/A000108>.
J. A. Bate and G. H. J. van Rees. Minimal and near-minimal critical sets in back-circulant [L]{}atin squares. , 27:47–61, 2003.
G Brinkmann and B D McKay. plantri (software). <http://cs.anu.edu.au/~bdm/plantri>.
Gunnar Brinkmann and Brendan D. McKay. Fast generation of planar graphs. , 58(2):323–357, 2007.
Nicholas J. Cavenagh and Petr Lisonek. Planar eulerian triangulations are equivalent to spherical latin bitrades. , 115(1):193–197, 2008.
Joan Cooper, Diane Donovan, and Jennifer Seberry. Latin squares and critical sets of minimal size. , 4:113–120, 1991. Combinatorial mathematics and combinatorial computing (Palmerston North, 1990).
A. Dr[á]{}pal and T. Kepka. Exchangeable partial groupoids. [I]{}. , 24(2):57–72, 1983.
Ale[š]{} Dr[á]{}pal. On elementary moves that generate all spherical latin trades. To appear in Comment Math Univ Car.
Ale[š]{} Dr[á]{}pal. Geometry of latin trades. Manuscript circulated at the conference Loops, Prague, 2003.
Ale[š]{} Dr[á]{}pal. Geometrical structure and construction of latin trades. , 9(3):311–348, 2009.
H-L. Fu. . PhD thesis, Auburn University, 1980.
Carlo Hämäläinen. Spherical bitrade enumeration code. <http://bitbucket.org/carlohamalainen/spherical>.
Carlo Hämäläinen. . in [M]{}athematics, Department of Mathematics, The University of Queensland, 2007. http://carlo-hamalainen.net/phd/hamalainen-20071025.pdf.
Derek A. Holton, Bennet Manvel, and Brendan D. McKay. Hamiltonian cycles in cubic 3-connected bipartite planar graphs. , 38:279–297, 1985.
Petteri Kaski and Patric R. J. [Ö]{}sterg[å]{}rd. . Number 15 in Algorithms and Computation in Mathematics. Springer-Verlag, Berlin Heidelberg, 2006.
A. D. Keedwell. Critical sets in latin squares and related matters: an update. , 65:97–131, 2004.
Brendan D. McKay. Isomorph-free exhaustive generation. , 26(2):306–324, 1998.
Ian M. Wanless. A computer enumeration of small [L]{}atin trades. , 39:247–258, 2007.
[^1]: Supported by grant MSM 0021620839
[^2]: Supported by Eduard Čech center, grant LC505.
|
---
abstract: |
We construct three-qubit entanglement witnesses with relatively simple structures. Despite their simplicity, these witnesses are capable of detecting a number of bound entangled states more effectively. To illustrate this, two families of bound entangled three-qubit states, introduced by Kay \[Phys. Rev. A 83, 020303 (2011)\] and Kye \[J. Phys. A: Math. Theor. 48, 235303 (2015)\], are considered and it is shown that these witnesses are able to detect the entanglement of all states in the families.\
\
[**PACS numbers**]{}: 03.67.Mn, 03.65.Ud, 42.50.Dv\
[**Key Words**]{}: Entanglement witness, Bound entangled state.
---
0 mm 0 mm 0 mm 0 mm 0 mm 160 mm .65 cm
\
\
Introduction
============
Detection of the entanglement in multi-qubit states is a fundamental issue in quantum information science. Multi-qubit entangled states are required in a wide range of quantum information processing tasks such as quantum teleportation [@aa; @ab] and quantum cryptography [@1c]. The problem has been fully solved for two-qubit states [@1]-[@3]. For more than two qubits, however, the situation is more involved [@4]-[@10]. Even for the simplest case of three-qubit system, there exist six inequivalent types of entanglement [@4a].
There exist several different approaches to the detection of quantum entanglement. One of them which is of special interest, uses the notion of an entanglement witness [@20]. The entanglement witness is a Hermitian operator whose expectation values over all separable states are non-negative and there exist some entangled states for which the expectation value is negative. In the latter case, one says that the state is detected by the entanglement witness. The entanglement witness approach has its own advantages: it turns out that any entangled state is detected by some entanglement witnesses [@19] and the entanglement witness enables us to detect quantum states without having enough information about them.
Here, following the method of [@7] and [@21], we construct entanglement witnesses for three-qubit states. The witnesses have relatively simple structures, but they are nevertheless capable to detect a number of bound entangled states more effectively. To illustrate this, we consider the two families of bound entangled three-qubit states introduced by A. Kay [@16] and S. H. Kye [@18] and show that the witnesses are capable to detect the entanglement of all states in the families.
It should be noted that our witnesses are members of a family of optimal linear entanglement witnesses which was introduced by Jafarizadeh et. al. [@22] for constructing nonlinear entanglement witnesses. Here, an alternative demonstration is offered for them and it is shown that they are more suitable for the detection of bound entanglement.
The paper is structured as follows: In Section 2 we give a brief description of the three-qubit states and the two families of bound entangled states introduced by A. Kay and S. H. Kye . In Section 3 we present the entanglement witnesses and apply them to the two families of bound entangled states. Finally, the paper is ended with our discussions and conclusion in section 4.
Three-qubit states
==================
By definition, a three-qubit pure state $|\psi\rangle$ is separable or pure product if it can be written as a tensor product of three single qubit pure states, $|\psi\rangle=|\phi_{1}\rangle\otimes|\phi_{2}\rangle\otimes|\phi_{3}\rangle$. As for the three-qubit mixed state $\rho$, it is said to be separable if it can be expressed as a convex sum of separable pure states. Finally, a three-qubit mixed or pure state is said to be a PPT entangled or bound entangled state if it is entangled and its partial transposition on any qubit is positive.
Here, we consider two families of the the three-qubit bound entangled states. The first family is the one introduced by A. Kay [@16]: $$\label{e2}
\rho_{_{Kay}}(a):=\frac{1}{8+8a}\left(
\begin{array}{cccccccc}
4+a & 0 & 0 & 0 & 0 & 0 & 0 & 2 \\
0 & a & 0 & 0 & 0 & 0 & 2 & 0 \\
0 & 0 & a & 0 & 0 & -2 & 0 & 0 \\
0 & 0 & 0 & a & 2 & 0 & 0 & 0 \\
0 & 0 & 0 & 2 & a & 0 & 0 & 0 \\
0 & 0 & -2 & 0 & 0 & a & 0 & 0 \\
0 & 2 & 0 & 0 & 0 & 0 & a & 0 \\
2 & 0 & 0 & 0 & 0 & 0 & 0 & 4+a \\
\end{array}
\right)$$ Note that for any $a\geq 2$, the matrix $\rho_{_{Kay}}(a)$ is a valid quantum state and has a positive partial transposition for any bipartition. It was proved in [@16] that for $a\geq 2\sqrt{2}$ the state is separable. Furthermore, it was shown numerically that the state is entangled for $2\leq a\leq 2.828$. Finally, it was proved analytically that for $2\leq a< 2\sqrt{2}$ the state is entangled [@17].
The second family is the one introduced by S. H. Kye [@18]: $$\label{e2a}
\rho_{_{Kye}}(b,c):=\frac{1}{6+b+c}\left(
\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\
0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 \\
0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & b & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & c & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\
0 & -1 & 0 & 0 & 0 & 0 & 1 & 0 \\
-1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}
\right)$$ with $b$ and $c$ strictly positive real parameters such that $bc\geq 1$. It is clear that the state is of PPT whenever $bc=1$.
Entanglement witnesses
======================
To construct entanglement witnesses for the bound entangled states $\rho_{_{Kay}}(a)$ and $\rho_{_{Kye}}(b,c)$, let us consider the observables (Hermitian operators) $$\label{e3}
A_{1}=\sigma_{z}\otimes\sigma_{z}\otimes I,\; A_{2}=\sigma_{x}\otimes(\sigma_{x}\otimes\sigma_{x}-\sigma_{y}\otimes\sigma_{y})
,\;A_{3}=\sigma_{y}\otimes(\sigma_{x}\otimes\sigma_{y}-\sigma_{y}\otimes\sigma_{x}),$$ where $I$ is the single qubit identity operator and $\sigma_{i}, i=x,y,z$ are the usual single qubit Pauli operators. Now, if any three-qubit state $\rho$ is characterized by the expectation values of the operators $A_{i}$ over it $(\langle A_{1}\rangle_{\rho}, \langle A_{2}\rangle_{\rho}, \langle A_{3}\rangle_{\rho} )$, where $\langle A_{i}\rangle_{\rho}:=\mathrm{Tr}(\rho A_{i})$, then we have the following observation.
[**[ Observation]{}:**]{} For any separable three-qubit state $\sigma$, the following inequality is fulfilled $$\label{e4}
\langle A_{1}\rangle_{\sigma}+\frac{1}{\sqrt{2}}(\langle A_{2}\rangle_{\sigma}+\langle A_{3}\rangle_{\sigma})\leq1.$$ The inequality may be violated by entangled states.
Obviously the observation remains valid when the observables $A_{i}$ are replaced with the ones that come from all the possible permutations of $\sigma_{x}, \sigma_{y}$ and $\sigma_{z}$ and also when they are replaced with $-A_{i}$.
**Proof:** As the expectation value of an operator is linear in the state, it suffices to prove that the inequality (\[e4\]) is satisfied by pure product states. For the pure product state $|\psi\rangle=|\phi_{1}\rangle\otimes|\phi_{2}\rangle\otimes|\phi_{3}\rangle$, we have $$\label{e5}
\begin{array}{c}
\hspace{-3.7cm}\langle A_{1}\rangle_{|\psi\rangle}+\frac{1}{\sqrt{2}}(\langle A_{2}\rangle_{|\psi\rangle}+\langle A_{3}\rangle_{|\psi\rangle})=
\langle\phi_{1}|\sigma_{z}|\phi_{1}\rangle\langle\phi_{2}|\sigma_{z}|\phi_{2}\rangle\\
+\frac{1}{\sqrt{2}}\langle\phi_{1}|\sigma_{x}|\phi_{1}\rangle(\left\langle\phi_{2}|\sigma_{x}|\phi_{2}\rangle
\langle\phi_{3}|\sigma_{x}|\phi_{3}\rangle-\langle\phi_{2}|\sigma_{y}|\phi_{2}\rangle
\langle\phi_{3}|\sigma_{y}|\phi_{3}\rangle\right)\\
\hspace{.1cm}+\frac{1}{\sqrt{2}}\langle\phi_{1}|\sigma_{y}|\phi_{1}\rangle(\langle\phi_{2}|\sigma_{x}|\phi_{2}\rangle
\langle\phi_{3}|\sigma_{y}|\phi_{3}\rangle-\langle\phi_{2}|\sigma_{y}|\phi_{2}\rangle
\langle\phi_{3}|\sigma_{x}|\phi_{3}\rangle).
\end{array}$$ Now let us introduce the notation $$\label{e6}
x_{i}:=\langle\phi_{i}|\sigma_{x}|\phi_{i}\rangle \quad,\quad y_{i}:=\langle\phi_{i}|\sigma_{y}|\phi_{i}\rangle \quad,\quad z_{i}:=\langle\phi_{i}|\sigma_{z}|\phi_{i}\rangle.$$ Then Eq. (\[e5\]) is rewritten as $$\label{e7}
\langle A_{1}\rangle_{|\psi\rangle}+\frac{1}{\sqrt{2}}(\langle A_{2}\rangle_{|\psi\rangle}+\langle A_{3}\rangle_{|\psi\rangle})=z_{1}z_{2}+\frac{1}{\sqrt{2}}[x_{1}(x_{2}x_{3}-y_{2}y_{3})
+y_{1}(x_{2}y_{3}-y_{2}x_{3})].$$ By the well-known fact that $x_{i}^{2}+y_{i}^{2}+z_{i}^{2}=1$, it is convenient to use the parametrization $$\label{}
x_{i}:=\sin\theta_{i}\cos\varphi_{i} \quad,\quad y_{i}:=\sin\theta_{i}\sin\varphi_{i} \quad,\quad z_{i}:=\cos\theta_{i},$$ where $\theta_{i}\in[0, \pi]$ and $\varphi_{i}\in[0, 2\pi]$ are real parameters. With respect to this parametrization, Eq. (\[e7\]) takes the form $$\label{e8}
\begin{array}{c}
\hspace{-5.5cm}\langle A_{1}\rangle_{|\psi\rangle}+\frac{1}{\sqrt{2}}(\langle A_{2}\rangle_{|\psi\rangle}+\langle A_{3}\rangle_{|\psi\rangle})=\cos\theta_{1}\cos\theta_{2} \\
\hspace{.4cm}+\frac{1}{\sqrt{2}}\sin\theta_{1}\sin\theta_{2}\sin\theta_{3} [\cos\varphi_{1}\cos(\varphi_{2}+\varphi_{3})-\sin\varphi_{1}\sin(\varphi_{2}-\varphi_{3})].
\end{array}$$ Obviously, the right hand side of Eq. (\[e8\]) can not exceed 1 and takes the maximum value 1 for $\varphi_{3}=-\varphi_{2}=\frac{\pi}{4}$, $\varphi_{1}=\frac{\pi}{4}$, $\theta_{1}=\theta_{2}$, and $\theta_{3}=\frac{\pi}{2}$.
The observation provides a necessary criterion for the separability of three-qubit states whose violation is a sufficient criterion for the entanglement. Furthermore, we infer from the inequality (\[e4\]) that the observable $$\label{e9}
W_{1}:=I-A_{1}-\frac{1}{\sqrt{2}}(A_{2}+A_{3})$$ with $I$ the three-qubit identity operator, has a non-negative expectation value over any three-qubit separable state and therefore it is an entanglement witness.
To see the detection power of $W_{1}$, let us consider the state $\rho_{_{Kay}}(a)$ of (\[e2\]). For this state, we have $${\mathrm{Tr}}(W_{1}\rho_{_{Kay}}(a))=1-\frac{2\sqrt{2}+1}{a+1}.$$ When this expectation value is negative it is said that $\rho_{_{Kay}}(a)$ is detected by $W_{1}$. Obviously, the expectation value is negative for $2\leq a< 2\sqrt{2}$ and therefore for these values of $a$ the state $\rho_{_{Kay}}(a)$ is entangled. This result is in excellent agreement with the one of [@17].
As another example, let us consider the state $\rho_{_{Kye}}(b,c)$ of (\[e2a\]). For this state, we use the following witness $$\label{e10}
W_{2}:=I+A_{1}+\frac{1}{\sqrt{2}}(A_{2}+A_{3})$$ which is inferred from the observation when the $A_{i}$’s are replaced with $-A_{i}$’s. For this state, we have $${\mathrm{Tr}}(W_{2}\rho_{_{Kye}}(b,c))=-\frac{8(\sqrt{2}-1)}{6+b+c}.$$ Obviously this expectation value is negative for any allowed values of $b$ and $c$. Therefore, the state $\rho_{_{Kye}}(b,c)$ is detected by $W_{2}$ and therefore it is entangled.
Discussions and conclusion
==========================
We present entanglement witnesses for the entanglement of three-qubit states and find that, despite the simplicity, they are relatively strong in detecting a number of bound entangled states. We show that they detect all the states in the families of bound entangled states introduced by A. Kay and S. H. Kye. It would be interesting to apply the method to other three-qubit bound entangled states and extend it to more than three qubits systems.
[**Acknowledgements:**]{} We acknowledge very valuable discussions with M. A. Jafarizadeh and also thanks M. A. Fasihi for the interesting and useful discussions.
[10]{} F. Verstraete, H. Verschelde, Phys. Rev. Lett. **90**, 097901 (2003). S. Lee, J. Joo, J. Kim, Phys. Rev. A **76**, 012311 (2007). R. Cleve, D. Gottesman, and H.-K. Lo, Phys. Rev. Lett. **83**, 648 (1999). A. Peres, Phys. Rev. Lett. **77**, 1413 (1996). M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A **223**, 8 (1996). W.K. Wootters, Phys. Rev. Lett. **80**, 2245 (1998). W. Dür and I. Cirac, Phys. Rev. A **61**, 042314 (2000). W. Dür, G. Vidal, and J. I. Cirac, Phys. Rev. A **62**, 062314 (2000). A. Acín, D. Bru[ß]{}, M. Lewenstein, and A. Sanpera, Phys. Rev. Lett. **87**, 040401 (2001). G. Tóth, and O. Gühne, Phys. Rev. A **72**, 022340 (2005). M.A. Jafarizadeh, G. Najarbashi, Y. Akbari, and H. Habibian, Eur. Phys. J. D **47**, 233–255 (2008). M. A. Jafarizadeh, Y. Akbari, K. Aghayar, A. Heshmati, and M. Mahdian, Phys. Rev. A **78**, 032313 (2008). M. A. Jafarizadeh, M. Mahdian, A. Heshmati, and K. Aghayar, Eur. Phys. J. D **50**, 107 (2008). O. Gühne, and M. Seevinck, New J. Phys. **12**, 053002 (2010). O. Gühne, Phys. Lett. A **375**, 406 (2011). K. H. Han and S. H. Kye, J. Phys. A: Math. Theor. **50**, 145303 (2017). M. Horodecki, P. Horodecki, R. Horodecki, Phys. Lett. A **223**, 1 (1996). M. Lewenstein, B. Kraus, J.I. Cirac, P. Horodecki, Phys. Rev. A **62**, 052310 (2000). A. Kay, Phys. Rev. A **83**, 020303 (2011). S. H. Kye, J. Phys. A: Math. Theor. **48**, 235303 (2015). O. Gühne, Phys. Lett. A **375**, 406 (2011).
|
---
abstract: 'Understanding heterogeneous multivariate time series data is important in many applications ranging from smart homes to aviation. Learning models of heterogeneous multivariate time series that are also human-interpretable is challenging and not adequately addressed by the existing literature. We propose grammar-based decision trees (GBDTs) and an algorithm for learning them. GBDTs extend decision trees with a grammar framework. Logical expressions derived from a context-free grammar are used for branching in place of simple thresholds on attributes. The added expressivity enables support for a wide range of data types while retaining the interpretability of decision trees. In particular, when a grammar based on temporal logic is used, we show that GBDTs can be used for the interpretable classification of high-dimensional and heterogeneous time series data. Furthermore, we show how GBDTs can also be used for categorization, which is a combination of clustering and generating interpretable explanations for each cluster. We apply GBDTs to analyze the classic Australian Sign Language dataset as well as data on near mid-air collisions (NMACs). The NMAC data comes from aircraft simulations used in the development of the next-generation Airborne Collision Avoidance System (ACAS X).'
author:
- 'Ritchie Lee[^1]'
- 'Mykel J. Kochenderfer[^2]'
- 'Ole J. Mengshoel[^3]'
- 'Joshua Silbermann[^4]'
bibliography:
- 'master.bib'
- 'ritchie.bib'
title: Interpretable Categorization of Heterogeneous Time Series Data
---
Introduction.
=============
Heterogeneous multivariate time series data arises in many applications including driverless cars, smart homes, robotic servants, and aviation. Understanding these datasets is important for designing better systems, validating safety, and analyzing failures. However, knowledge discovery in heterogeneous multivariate time series datasets is very challenging because it typically requires two major data mining problems to be addressed simultaneously. The first problem is how to handle multivariate *heterogeneous* time series data, where the variables are a mix of numeric, Boolean, and categorical types. The second problem is the need for *interpretability*. That is, humans must be able to understand and reason about the information captured by the model. While these problems have been explored separately in the literature, we are not aware of any methods that address both interpretability and heterogeneous multivariate time series datasets together.
Rule-based methods such as decision trees [@Breiman1984] and decision lists [@Rivest1987] are very intuitive because they use symbolic rules for decision boundaries. However, they do not support time series data. Motif discovery methods, such as shapelet [@Ye2011] and subsequence [@Senin2013] discovery, find recurring patterns in sequential data. These methods are less interpretable than rule-based methods as they report prototypes of patterns rather than state relationships. These methods have also not been applied to heterogeneous multivariate time series data. On the other hand, models that support heterogeneous multivariate time series, such as recurrent neural networks [@Gers2000], are not interpretable.
To simultaneously address the problems of interpretability and heterogeneous time series data, we increase the expressiveness of decision trees by allowing decision rules to be any logical expression. Traditional decision trees partition the input space using simple thresholds on attributes, such as $(x_1 < 2)$ [@Breiman1984]. However, these partitioning rules have limited expressiveness and cannot be used to express more complex logical relationships, such as those between heterogeneous attributes or across time. Allowing more expressive rules in the decision tree increases its modeling capability while retaining interpretability. In the past, first order logic rules have been used in decision trees [@Blockeel1998]. We generalize this idea and allow any logical language that can be specified using a context-free grammar. In particular, we show that decision trees combined with temporal logic can be very effective as an interpretable model for heterogeneous multivariate time series data. Temporal logic, combined with other methods, have been applied to heterogeneous time series data in aviation [@Schumann2013].
The key contributions of this paper are:
- a framework that combines decision trees with a grammar;
- a training algorithm for that leverages existing algorithms, such as [@Koza1992] and [@ONeil2003];
- an alternative view of the model as categorization, which is the combination of clustering data into similar groups and providing explanations for them; and
- experiments with and temporal logic on two datasets, showing that learn reasonable explanations while producing competitive classification performance compared to baselines.
Related Work. {#sec:related}
=============
Interpretable Models.
---------------------
A variety of interpretable models for static data have been proposed in the literature. Regression models [@Schielzeth2010], generalized additive models [@Lou2012], and Bayesian case models [@Kim2014] have been recently proposed as models with interpretability. These models aid interpretability by stating decision boundaries in terms of basis functions or representative instances (prototypes). Bayesian networks have also been used for prediction and data understanding [@Basak2016]. Our approach aims to achieve better interpretability by representing decision boundaries symbolically using (temporal) logic rules rather than in terms of basis functions, examples, or statistical weights.
Rule-based models, such as decision trees [@Breiman1984], decision lists [@Rivest1987], and decision sets [@Lakkaraju2016], are easy to understand because their decision boundaries are stated in terms of input attributes and simple logical operators. These methods provide good interpretability, but do not capture the temporal aspect of time series data.
Time Series Models.
-------------------
Time series analysis has focused on dynamic time warping, hidden Markov models, dictionary-based approaches, and recurrent neural networks [@Kadous2002][@Bagnall2017][@Gers2000]. Shapelets [@Ye2011] and subsequence search [@Senin2013] have been proposed for univariate time series classification. These approaches search for simple patterns that are most correlated with the class label. Interpretability comes from identifying a prototype of a recurring subsequence pattern. Implication rules [@Shokoohi2015] and simple logical combinations of shapelets [@Mueen2011] have been proposed to extend the shapelets approach. Learning prototypes or statistical weights are not as interpretable as symbolic rules.
Grammars and Decision Trees.
----------------------------
The combination of decision trees and grammars has been proposed in the past [@Wong2006][@Motsinger2010]. These prior works use a grammar to optimize traditional decision trees as a whole where the splits are simple thresholds over individual attributes. The resulting tree is a traditional decision tree. In contrast, uses a grammar to optimize node splits and the resulting decision tree contains a (temporal) logic rule at each node split. This distinction is important because the added expressivity of the logical expressions is what allows support for multivariate heterogeneous time series datasets.
Preliminaries. {#sec:prelims}
==============
Notation.
---------
A multi-dimensional time series dataset $D$ consists of $m$ record-label pairs $((r_1,l_1),(r_2,l_2),...(r_m,l_m))$, where a record $r$ is a two-dimensional matrix of $n$ attributes by $T$ time steps and a label $l$ is its discrete class label. A trace $\vec{x}_i$ is the $i$’th row of a record and represents the time series of that attribute. Logical and comparison operators are given broadcast semantics where appropriate. For example, the comparison operator in $\vec{x}_i < c$ compares each element of $\vec{x}_i$ to $c$ and returns a vector of the same size as $\vec{x}_i$. Similarly, the logical operator in $\vec{x}_i \wedge \vec{x}_j$ operates elementwise. The temporal operators $\text{F}$ and $\text{G}$ are *eventually* and *globally*, respectively. *Eventually* returns true if any value in the argument vector is true. *Globally* returns true if all values in the argument vector are true.
Context-Free Grammar.
---------------------
A defines a set of rules that govern how to form expressions in a formal language, such as [@Gabbay1980]. The grammar defines the syntax of the language and provides a means to generate valid expressions. A $G$ is defined by a 4-tuple $(\mathscr{N}, \mathscr{T}, \mathscr{P}, \mathscr{S})$, where $\mathscr{N}$ is a set of *non-terminals*; $\mathscr{T}$ is a set of *terminals*; $\mathscr{P}$ is a set of *production rules*, which are rules governing the substitution of non-terminals; and $\mathscr{S}$ is the special *start* symbol that begins a derivation. The derivation is commonly represented as a tree structure called a derivation tree. We assume that the semantics of the symbols in the grammar are defined.
Grammar-Based Expression Search. {#sec:exprsearch}
--------------------------------
Grammar-based expression search (GBES) is the problem of finding an expression $e^*$ from a grammar that minimizes a given fitness function $f(e)$, i.e., $e^*={\operatornamewithlimits{arg\,min}}_e{f(e)}$ [@McKay2010]. The formulation is extremely general due to the flexibility and expressiveness of grammars and the arbitrary choice of fitness function. We describe three existing algorithms as follows. **Monte Carlo.** Monte Carlo generates expressions by repeatedly selecting non-terminals in the partial expression and applying a production rule chosen uniformly at random amongst possible rules. When no non-terminals remain, the fitness of the generated expression is evaluated and the expression with the best fitness is reported. A maximum depth is typically used to ensure that the process terminates.
**Grammatical Evolution.** Grammatical evolution (GE) [@ONeil2003] is an evolutionary algorithm that is based on a sequential representation of the derivation tree. Specifically, defines a transformation from a variable-length integer array to a sequence of rule selections in a . Then it uses a standard to search over integer arrays. We use one-point crossover and uniform mutation [@Poli2008].
**Genetic Programming.** Genetic programming (GP) is an evolutionary algorithm for optimizing trees [@Whigham1995]. Genetic operators are defined specifically for trees and thus do not require any transformations of the derivation tree. Our implementation uses a crossover operator that exchanges compatible subtrees between two individuals and a mutation operator that replaces entire subtrees with randomly-generated ones. We use tournament selection for selecting individuals [@Koza1992].
Grammar-Based Decision Trees. {#sec:gbdt}
=============================
Grammar-based decision trees (GBDTs) extend decision trees with a grammar framework to allow general logical expressions to be used as the branching rules in a decision tree. The domain of the logical expressions is constrained using a . In this paper, we consider grammars based on temporal logic for the classification of heterogeneous multivariate time series data. The grammar can be easily adapted to the characteristics of the data or the application.
Model.
------
is a binary decision tree where each non-leaf node contains a logical expression. Each branch from a non-leaf node is associated with a possible outcome of the logical expression at that node. Leaf nodes are associated with a particular class label for prediction. We use the following notation: A tree $\mathcal{T}$ is either a leaf with class $l$, which we denote by $\mathcal{T}=\textsc{Leaf}(l)$; or it is an internal node with expression $e \in G$, true branch $\mathcal{T}^+$, and false branch $\mathcal{T}^-$, which we denote by $\mathcal{T}=\textsc{INode}(e, \mathcal{T}^+, \mathcal{T}^-)$.
**Prediction.** As in a traditional decision tree, prediction is performed in a top-down fashion starting at the root. At each node, the logical expression is evaluated on the record and the result dictates which branch is followed. The process continues recursively until a leaf node is reached where the class label at the leaf node is used for prediction.
**Categorization.** While decision trees are traditionally used for classification, they can also be used for categorization, which is the combined task of clustering and explaining data. A categorization of the data can be extracted from a by considering each leaf node of the tree to be a separate category. The description of the category is then the conjunction of all branch expressions between the root and the leaf node of interest. The members of the cluster are the records where the cluster’s description holds true. Since partitions are mutually exclusive, the clusters do not overlap.
**Example.** shows an example of a with an accompanying expressed in . The describes a simple temporal logic. It assumes that the data has four attributes $\vec{x}_1,\vec{x}_2,\vec{x}_3,\vec{x}_4$, where the attributes $\vec{x}_1$ and $\vec{x}_2$ are Boolean vectors and $\vec{x}_3$ and $\vec{x}_4$ are vectors of real numbers. The grammar contains two types of rules. *Expression rules*, such as $\textsc{Ev} ::= \text{F}(\textsc{VB})$, contain partially-formed expressions that contain terminal and non-terminal symbols. Non-terminal symbols are substituted further using the appropriate production rules. *Or Rules*, such as $B ::= \textsc{Ev} \mid \textsc{Gl}$, contain the symbol $\mid$, which delineates the different possible substitutions.
Each non-leaf node of the contains a logical expression derived from the grammar. The expression dictates which branch should be followed. Leaf nodes show the predicted class label. In , a data record where $\text{F}(\vec{x}_1 \wedge \vec{x}_2)$ is true and $\text{G}(\vec{x}_3 < 5)$ is false would be predicted to have class label 2. We also label each leaf node with a unique cluster number. For example, the righmost leaf node in is labeled as cluster 4. The overall expression that describes cluster 4 is $\neg \text{F}(\vec{x}_1 \wedge \vec{x}_2) \wedge \neg \text{F}(\vec{x}_1 \wedge (\vec{x}_3 < 2))$.
[0.5]{} ![Grammar-based decision tree example.[]{data-label="fig:gbdtexample"}](gbdtexample.pdf "fig:"){width="1.0\columnwidth"}
[0.4]{} $$\begin{aligned}
\textsc{Start} &::= \textsc{B}\\
\textsc{B} &::= \textsc{Ev} \mid \textsc{Gl}\\
\textsc{Ev} &::= \text{F}(\textsc{VB})\\
\textsc{Gl} &::= \text{G}(\textsc{VB})\\
\textsc{VB} &::= \vec{x}_1 \mid \vec{x}_2 \mid \textsc{And} \mid \textsc{Lt}\\
\textsc{And} &::= \textsc{VB} \wedge \textsc{VB}\\
\textsc{Lt} &::= \textsc{VR} < C\\
\textsc{VR} &::= \vec{x}_3 \mid \vec{x}_4\\
\textsc{C} &::= 1 \mid 2 \mid 3 \mid 4 \mid 5
\end{aligned}$$
Grammars for Heterogeneous Time Series.
---------------------------------------
In the framework, logic expressions are evaluated on a data record and produce a Boolean output. The symbols of the expression can refer to fields of the record, constants, or functions. We adopt a subset of , a formal logic with temporal operators often used in the analysis of time series data [@Gabbay1980]. Temporal logic provides an elegant framework for defining expressions that support heterogeneous multivariate time series data. For example, the logic rule $\text{F}(\vec{x}_1 \wedge (\vec{x}_3 < 2))$ in contains a Boolean vector $\vec{x}_1$ and a vector of real values $\vec{x}_3$. The rule integrates these two different data types elegantly by first evaluating $(\vec{x}_3 < 2)$ using an elementwise comparison to produce a Boolean vector. Then, the result is combined elementwise via conjunction with $\vec{x}_1$ to produce another Boolean vector. Finally, the temporal operator $\text{F}$ collapses the Boolean vector to a single Boolean output. The constants in the grammar can be preselected based on discretizing over the range of each real-valued attribute in the dataset. Alternatively, some algorithms, such as Monte Carlo and genetic programming, also support drawing random values during search [@Koza1992].
For simplicity, the example in included only a small number of operators. The grammar can be readily extended to include other operators including disjunct $\vee$, greater than $>$, equals $=$, where equality is important for categorical attributes, and even arbitrary user-defined functions. Depending on the application, a generic temporal logic grammar can be used (as we have done with the Auslan dataset) or the grammar can be tailored to the application (as we have done with the dataset). This paper considers very simple temporal operators that operate over the entire time series. More sophisticated temporal logics, such as [@Koymans1990], can be used with to discover more intricate patterns in the data.
Natural Language Descriptions.
------------------------------
Logical expressions can sometimes be dense and hard to parse. In many cases, we can improve interpretability by providing natural language descriptions of the expressions. One method to automatically translate expressions into English sentences is to map expression rules and terminal symbols in the to corresponding sentence fragments and then use the structure of the expression’s derivation tree to assemble the fragments. shows an example mapping that could be used with the grammar in .
$$\begin{aligned}
\text{F}(\textsc{VB}) &:= \text{``at some point, $\textsc{VB}$"}\\
\text{G}(\textsc{VB}) &:= \text{``for all time, $\textsc{VB}$"}\\
\textsc{VB} \wedge \textsc{VB} &:= \text{``$\textsc{VB}$ and $\textsc{VB}$}"\\
\textsc{VR} < C &:= \text{``$\textsc{VR}$ is less than $C$"}\\
\vec{x}_1 &:= \text{``advisory is active"}\\
\vec{x}_2 &:= \text{``pilot is responding"}\\
\vec{x}_3 &:= \text{``vertical rate"}\\
\vec{x}_4 &:= \text{``airspeed"}
\end{aligned}$$
Applying the mapping in , the decision rules in become the following natural language descriptions: $\text{F}(\vec{x}_1 \wedge \vec{x}_2)$ is translated to “at some point, \[advisory is active\] and \[pilot is responding\]"; $\text{G}(\vec{x}_3 < 5)$ is translated to “for all time, \[vertical rate\] is less than 5"; and $\text{F}(\vec{x}_1 \wedge (\vec{x}_3 < 2))$ is translated to “at some point, \[advisory is active\] and \[\[vertical rate\] is less than 2\]". We include square brackets to help the reader disambiguate nested sentence components.
Induction of GBDTs. {#sec:gbdtinduction}
===================
Induction of a is performed top-down as in traditional decision tree learning. However, there is a difference in our use of as a subroutine to find the best partition expression. The induction algorithm begins with a single (root) node containing the entire dataset. is then used to search a for the partitioning expression that yields the best fitness. The expression is evaluated on each record and the dataset is partitioned into two child nodes according to the results of the evaluation. The process is applied recursively to each child until all data records at the node are either correctly classified or a maximum tree depth is reached. The mode of the training labels is used for class label prediction at a leaf node. The induction algorithm is shown in .
$R \gets \Call{Split}{G, f, D, d}$ $\Call{Tree}{R}$ $\hat{e}^* \gets \Call{GBES}{G, f, D}$ $(D^+, D^-) \gets \Call{SplitData}{D, \hat{e}^*}$ $child^+ \gets \Call{Split}{G, f, D^+, d - 1}$ $child^- \gets \Call{Split}{G, f, D^-, d - 1}$ $\Call{INode}{\hat{e}^*, child^+, child^-}$
In , <span style="font-variant:small-caps;">GBDT</span> (line 3) is the main entry point to the induction algorithm. It returns a <span style="font-variant:small-caps;">Tree</span> object containing the root node of the induced decision tree. <span style="font-variant:small-caps;">Split</span> (line 6) attempts to partition the data into two parts. It first tests whether the terminal conditions are met and if so returns a <span style="font-variant:small-caps;">Leaf</span> object that predicts the mode of the labels. The partitioning terminates if the maximum depth has been reached or if all class labels are the same, which is tested by the <span style="font-variant:small-caps;">IsHomogeneous</span> function (line 7). The <span style="font-variant:small-caps;">GBES</span> function (line 9) uses to search for the expression that minimizes the fitness function $f(e)$, i.e., $e^*={\operatornamewithlimits{arg\,min}}_e{f(e)}$. <span style="font-variant:small-caps;">SplitData</span> (lines 12-13) evaluates the expression on each record and partitions the data into two parts according to whether the expression holds. Then, <span style="font-variant:small-caps;">Split</span> (lines 12-13) is called recursively on each part. <span style="font-variant:small-caps;">Split</span> returns an <span style="font-variant:small-caps;">INode</span> object containing the decision expression and the children of the node.
Fitness Function. {#sec:fitness}
-----------------
In <span style="font-variant:small-caps;">GBES</span> (line 9), we evaluate the desirability, or *fitness*, of an expression according to two competing objectives. On the one hand, we want expressions that partition the data so that the resulting leaf nodes have the same ground truth class labels. Splits that induce high homogeneity tend to produce shallower trees and thus shorter global expressions at leaf nodes. They also produce classifiers with better predictive accuracy when the maximum tree depth is limited. To quantify homogeneity, we use the Gini impurity metric following the framework [@Breiman1984]. On the other hand, we want to encourage interpretability by minimizing the length and complexity of expressions. Shorter and simpler expressions are generally easier to interpret. We use the number of nodes in the derivation tree as a proxy for the complexity of an expression $e$. The two objectives are combined linearly into a single (minimizing) fitness function given by $$f(e) = w_1 I_G + w_2 N_e$$ where $f$ is the fitness function, $w_1 \in \mathbb{R}$ and $w_2 \in \mathbb{R}$ are weights, and $N_e$ is the number of nodes in the derivation tree of $e$. The total Gini impurity, $I_G$, is the sum of the Gini impurity of each partition that results from splitting the data using expression $e$. It is given by $$I_G = \sum_{L \in \{L^+,L^-\}}\sum_{b \in B}f^b_L (1-f^b_L)$$ where $f^b_L$ is the fraction of labels in $L$ that are equal to $b$; $L^+$ are the labels of the records on which $e$ evaluates to true, i.e., $L^+=[l \mid \textsc{Evaluate}(e,r), (r,l) \in D]$; $L^-$ are the labels of the records on which $e$ evaluates to false, i.e., $L^-=[l \mid \neg\textsc{Evaluate}(e,r), (r,l) \in D]$; and $B = \{$True, False$\}$. We use square brackets with set construction syntax to indicate that $L^+$ and $L^-$ are vectors and can have duplicate elements.
Computational Complexity.
-------------------------
The most computationally expensive part of training is evaluating the fitness of an expression since it involves visiting each record in the dataset and then computing statistics. also requires a large number of expression evaluations to optimize the decision expression at each decision node. The deeper the tree, the more nodes need to be optimized. However, as the tree gets deeper, the nodes operate on increasingly smaller fractions of the dataset. In fact, while the number of nodes grows exponentially with tree depth, the number of records that must be evaluated at each level remains constant (the size of the dataset). Overall, the computational complexity of induction is $O(m \cdot N_{GBES} \cdot d)$, where $m$ is the number of records in the dataset, $N_{GBES}$ is the number of logical expressions evaluated in each , and $d$ is the depth of the decision tree.
Experiments.
============
We evaluate on the Australian Sign Language dataset and data from an aircraft encounter simulator. Both experiments use the same metrics and baselines.
**Metrics.** For classification performance, we report the accuracy, precision, recall, and F1 scores. We use ‘macro’ averaging, which calculates the mean of the binary metrics, giving equal weight to each class. For interpretability, we report the average number of terminals in each rule of the decision tree. For example, $\text{F}(\vec{x}_1 \wedge \vec{x}_2)$ has four terminals: $\text{F}$, $\vec{x}_1$, $\wedge$, and $\vec{x}_2$. Rules containing fewer terminals are considered easier to interpret. A *trial* is a run of an experiment initialized with a unique random seed and train-test split. We perform random trials using stratified randomly-sampled 70/30 train-test splits of the data and report the metric’s mean over the trials on the testing set.
**Baselines.** We compare our algorithm against decision tree, random forest, and Gaussian naive Bayes classifiers from Scikit-Learn [@Scikitlearn]. Random forest and Gaussian naive Bayes models have low interpretability. We preprocess the data before applying the baseline algorithms as follows: z-score normalize each real-valued attribute; convert Boolean attributes to \[-1.0,1.0\]; zero-pad so that all records have the same length in the time dimension; and reshape the time series data (size $m \times n \times T$) into non-time series data (size $m \times nT$) by concatenating time slices. can directly handle heterogeneous multivariate time series data of varying length and thus does not require this preprocessing.
Australian Sign Language. {#sec:auslan}
-------------------------
We analyze the classic Australian Sign Language (“Auslan”) dataset from the repository [@UCI][@Kadous2002]. The data originates from participants wearing instrumented gloves while signing specific words in Australian sign language. The dataset is a multivariate time series of varying episode length containing 22 continuous attributes, 95 words with 27 examples each. We extract eight words (classes) from the dataset: *hello, please, yes, no, right, wrong, same, different*.
**Grammar.** We use a generic temporal logic grammar that includes all attributes and the following operators: $\text{F}$, $\text{G}$, $\implies$, $\neg$, $\vee$, $\wedge$, $=$, $<$, $\leq$, $>$, $\geq$. Since attributes may have different ranges, constant values used in comparison operators must be specialized to each attribute. To address this issue, we consider the range of each attribute in the dataset and uniformly discretize it into 10 points. Attributes are compared with their corresponding set of discretized values. The data itself is not being discretized. Discretization is only used to generate the threshold constants in the grammar.
### Results.
We performed 20 random trials for each algorithm and report the mean results in Table \[tab:auslanmetrics\]. , decision tree, and random forest models were trained to a maximum depth $d=6$ and random forest used 32 trees. -GP performed the best overall both in terms of classification and interpretability metrics. Decision tree performed poorly on this dataset because key characteristics of the attributes may be shifted in time due to variations in the speed of signing. Random forest can partially overcome this issue by using a larger number of trees. The temporal logic rules in can more effectively capture the underlying temporal properties of the data.
**** **GBDT-MC** **GBDT-GE** **GBDT-GP** **Decision Tree** **Random Forest** **Gaussian NB**
---------------- ------------- ------------- ------------- ------------------- ------------------- -----------------
Accuracy 0.9840 0.9847 **0.9868** 0.7585 0.9838 0.6862
F1-Score 0.9840 0.9845 **0.9867** 0.7085 0.9838 0.6674
Precision 0.9857 0.9872 **0.9884** 0.6869 0.9851 0.7025
Recall 0.9840 0.9847 **0.9868** 0.7672 0.9841 0.6867
Avg. Terminals 3.70 3.74 **3.59** — — —
\[tab:auslanmetrics\]
The strength of lies in its ability to categorize and explain data. not only provides interpretable expressions about the data’s distinguishing properties but also a hierarchical organization of the properties. shows the resulting from the best-performing trial. We compare the learned model to the videos of the signed words to intuitively verify correctness[^5]. At the root node, partitions *same* and *different* from the other words identifying the fully bent left pinky as the distinguishing property. Subsequently, *different* is distinguished from *same* by looking at whether the right palm is ever facing upwards. Of the remaining words, *hello* is isolated using the fully straight middle finger and then *wrong* is identified using the straight right pinky. The remaining four words are grouped *right* with *yes* and *please* with *no* using the yaw position of the right hand. The word *yes* is distinguished from *right* by the bent right thumb. Lastly, *no* is distinguished from *please* using the combined property on right forefinger bend, right yaw angle, and $x$-position of the right hand.
Collision Avoidance Application. {#sec:application}
--------------------------------
Airborne collision avoidance systems are mandated worldwide on all large transport and cargo aircraft to help prevent mid-air collisions. They monitor the airspace around an aircraft and issue alerts, called , if a conflict is detected. recommends a safety action to the pilot, for example, instructing the pilot to climb at 1500 feet per minute. To address the growing needs of the national airspace, the is currently developing and testing a new aircraft collision avoidance system, called the next-generation . promises a number of improvements over current systems including a reduction in collision risk while simultaneously reducing the number of unnecessary alerts [@Kochenderfer2012next].
One of the primary safety metrics of airborne collision avoidance systems is the likelihood of , defined as two aircraft coming closer than 500 feet horizontally and 100 feet vertically. Efficient algorithms have been developed to generate large datasets of and non- instances in simulation [@Lee2015]. However, while it is straightforward to observe that has occurred, discovering and categorizing relevant properties of is challenging. We apply to analyze simulated aircraft encounters to discover the most predictive properties of and categorize encounters accordingly. The results of our study are used to help the development team better understand the for validating safety and informing development.
**Dataset.** We analyze a dataset that contains simulated two-aircraft mid-air encounters [@Lee2015]. The dataset contains 10,000 encounters with 863 and 9,137 non-. The class imbalance is due to the rarity of and the difficulty in generating encounters. Each encounter has 77 attributes collected at for 50 time steps. The attributes include numeric, categorical, and Boolean types representing the state of the aircraft, pilot commands, and the state and output of the collision avoidance system for each aircraft. The data was generated during the stress testing of prototype version 0.9.8.
**Grammar.** We craft a custom for the dataset, building on the one presented in . We include temporal logic operators *eventually* and *globally* ; elementwise logical operators *conjunct* $\wedge$, *disjunct* $\vee$, *negation* $\neg$, and *implies* $\implies$; comparison operators *less than* $<$, *less than or equal to* $\leq$, *greater than* $>$, *greater than or equal to* $\geq$, and *equal* $=$; mathematical functions *absolute value* $|x|$, *difference* $-$, and *sign* $\textsc{sign}$; and *count* $\textsc{count}$ (which returns the number of true values in a Boolean vector).
In addition to dividing the attributes by data type, the grammar further subdivides the attributes by their physical representations. This enables comparison of attributes with constant values that have appropriate scale and resolution. For example, even though aircraft heading and vertical rate are both real-valued, aircraft heading should be compared to values between and , whereas vertical rate should be compared to values between and feet per second.
### Results.
We performed 20 random trials for each algorithm and report the mean results in Table \[tab:acasxmetrics\]. , decision tree, and random forest were trained to a maximum depth $d=4$. Random forest used 64 trees. -GP again performed the best overall. The temporal logic rules in are able to more effectively capture the underlying patterns and are robust to temporal variations. Random forest performed very poorly at predicting . The reason is that random forest randomly subsamples attributes, which works poorly when there are many attributes but only a small subset of attributes are predictive.
**** **GBDT-MC** **GBDT-GE** **GBDT-GP** **Decision Tree** **Random Forest** **Gaussian NB**
---------------- ------------- ------------- ------------- ------------------- ------------------- -----------------
Accuracy 0.9577 0.9583 **0.9587** 0.9378 0.9251 0.8705
F1-Score 0.8765 0.8768 **0.8779** 0.8038 0.6130 0.7126
Precision 0.8505 0.8546 0.8553 0.8025 **0.8961** 0.6799
Recall **0.9102** 0.9050 0.9053 0.8083 0.5778 0.8187
Avg. Terminals 5.54 5.56 **5.38** — — —
\[tab:acasxmetrics\]
shows a visual overview of a categorization from one of the resulting trees. The figure shows plots of altitude versus time, which, while cannot fully capture the high-dimensionality of the encounter data, is generally most informative since issues only in the vertical direction. Since we are mainly interested in categorizing , we consider only the six categories out of the 16 total categories produced by the tree. shows the first five encounters for each of the six categories, where each row is a separate category. The categories are labeled in ascending order starting at category 1 at the top row. The first row has only two plots because category 1 only contained two encounters. We describe categories 1, 5, and 6 as follows.
**Category 1.** In this category, the two aircraft maintain altitude separation for most of the encounter, then one aircraft accelerates rapidly toward the other aircraft as they approach. The aggressive maneuvering causes vertical separation to be lost rapidly and results. shows the spike in climb rate at 34 seconds, only 5 seconds before . The large spike in vertical rate near is characteristic of encounters in this category.
Aggressive last-minute maneuvering is known to be problematic. With the pilot’s five-second response time, there is insufficient time remaining for the collision avoidance system to resolve the conflict. It is unlikely that any reasonable collision avoidance system would be able to resolve such . Fortunately, these encounters are extremely rare outside of simulation since pilots do not typically maneuver so aggressively in operations.
**Category 5.** In these encounters, the aircraft cross in altitude early in the encounter without active advisories, then maneuver back toward each other to cause . Since the aircraft are predicted to cross safely and appear to vertically diverge following the crossing, the collision avoidance system witholds to allow the encounter to resolve naturally and reduce the number of unnecessary alerts. However, after crossing, the aircraft change course and maneuver toward each other, which results in . Because the aircraft are already close in altitude, the vertical maneuvering in these encounters is less aggressive than those in previous categories. Due to the late maneuvering and the pilot response delay, pilot 2 does not start to comply with the issued until within five seconds of . shows the vertical profile of an encounter in this category where the aircraft cross in altitude at 19 seconds and then maneuver to at 38 seconds.
**Category 6.** In these encounters, the aircraft receive initial at different times and the aircraft cross in altitude during the pilot response delay of the first . As the pilots start responding to their advisories, the maneuvers actually result in bringing them closer together rather than further apart. A late revision to the advisory is issued. However, the encounter ultimately results in . shows the vertical profile of an encounter in this category. The aircraft cross in altitude during pilot 2’s response delay period at 21 seconds and occurs at 39 seconds.
Issuing advisories can be tricky in cases where the aircraft are approximately co-altitude due to uncertainties in the aircraft’s estimated positions and future intentions. In these encounters, the problem arises as one aircraft maneuvers to cross in altitude immediately after an initial is issued to the other aircraft. Operationally, this is a rare case as the positions of the aircraft and the timing of the maneuver need to coincide.
[0.315]{}
[0.33]{}
[0.33]{}
Conclusions.
============
This paper introduced , a framework that combines decision trees and for interpretable classification and categorization. addresses the need for interpretable models that can support heterogeneous multivariate time series data. We validated our approach on the classic Australian Sign Language dataset from and a dataset from the flight logs of an aircraft encounter simulator. performed well on both classification and interpretability metrics and produced highly-interpretable categorizations of the data. The results of the study were used to inform the development of .
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Neal Suchy at the FAA, Michael Owen and Cindy McLain at MIT Lincoln Laboratory, and the team. This work was supported by the Project under NASA ARMD’s .
[^1]: Carnegie Mellon University Silicon Valley.
[^2]: Stanford University.
[^3]: Carnegie Mellon University Silicon Valley.
[^4]: Johns Hopkins University Applied Physics Laboratory.
[^5]: <https://www.youtube.com/watch?v=_5NbYyUlcHU>
|
---
abstract: 'In this paper, we study the asymptotic behavior of the colored Jones polynomials evaluated at roots of unity for a special class of knots. We show that certain limit is zero as predicted by the volume conjecture.'
author:
- Qihou Liu
date: 'July 25,2008'
title: |
On the colored Jones polynomials\
of certain cable of the torus knots
---
Introduction {#intro}
============
In[@KT], Kashaev obtained an explicit expansion of the colored Jones polynomials evaluated at roots of unity for the torus knots. As a consequence, it is shown that the volume conjecture holds for such knots, —that is: $$\lim_{N\to \infty} \frac{\log\left|\left(\frac{J_N^\sigma
(A,T)}{[N+1]}\right)_{A=\exp\left(\frac{\pi i}{2(N+1)}\right)}\right|}N=0$$
Using a formula from Morton[@MO], one can show that the quantity inside $\log$ function is bounded from above by a polynomial of $N$. To complete the argument, however, one need to bound this quantity from below. In a recent preprint by Roland van der Veen [@VV], the author claims a proof for those knots with volume zero. An upper bound is obtained. We still need to see how the lower bound would be treated. In this paper, we extend Kashaev’s method to give an expansion of colored Jones polynomials for certain cable of the torus knots. Then we show that the above limit is zero based on this estimation. The whole paper is organized as follows. The next two sections contain some definitions and a formula we shall use—without proof. Section 4 states the main theorem and its corollary. In section 5 and 6 we carry out evaluation of the polynomials and some estimations. The proof of the main theorem is finished in section 7.
Basic definitions and notations {#def}
===============================
A cable expansion formula {#expansion}
=========================
Main theorem
============
Integral representation of the polynomial and its value at root of unity
========================================================================
Study of the asymptotic behavior of the double integral {#asymptotics}
=======================================================
Proof of theorem 3 {#proof}
==================
[99]{} Blanchet, C.; Habegger, N.; Masbaum, G.; and Vogel, P., [*Three-manifold invariants derived from the Kauffman bracket*]{}. Topology 1992 [**31**]{}, no. 4, 685–699.
Blanchet, C.; Habegger, N.; Masbaum, G.; and Vogel, P., [*Topological quantum field theories derived from the Kauffman bracket*]{}. Topology 1995 [**34**]{}, no. 4, 883–927.
Kashaev, R.M.; Tirkonnen, O., [*Proof of the Volume Conjecture for Torus Knots*]{}. Journal of Mathematical Sciences 2003 [**115**]{}, no.1, 2033-2036
Lickorish, W. B. R., An introduction to knot theory. Graduate Texts in Mathematics,[**175**]{}. Springer-Verlag, New York, 1997.
Lickorish, W. B. R., [*Quantum invariants of 3-manifolds*]{}. Handbook of geometric topology, 707–734, North-Holland, Amsterdam, 2002.
Lickorish, W. B. R., [*The skein method for three-manifold invariants*]{}. J. Knot Theory Ramifications 1993 [**2**]{} , no. 2, 171-194.
Liu, Q., [*On the colored Jones polynomials of certain links*]{}. Ph.D. thesis, 2008.
Masbaum, G., and Vogel, P., [*3-valent graphs and the Kauffman bracket*]{}. Pacific J. Math 1994 [**164**]{}, 361-381.
Morton, H.R., [*The colored Jones function and Alexander polynomial for the torus knots*]{}. Proc.Cambridge Philos.Soc. 1995 [**117**]{}, no.1, 129-135.
Murakami, H., and Murakami, J.,[*The colored Jones polynomials and the simplicial volume of a knot*]{}. Acta Math. 2001, [**186**]{}, 85-104.
van der Veen, R., [*A cabling formula for the colored Jones polynomial*]{}. preprint, arXiv:math.GT/0807.2679.
|
---
abstract: |
We overview numerous algorithms in computational $D$-module theory together with the theoretical background as well as the implementation in the computer algebra system <span style="font-variant:small-caps;">Singular</span>. We discuss new approaches to the computation of Bernstein operators, of logarithmic annihilator of a polynomial, of annihilators of rational functions as well as complex powers of polynomials. We analyze algorithms for local Bernstein-Sato polynomials and also algorithms, recovering any kind of Bernstein-Sato polynomial from partial knowledge of its roots. We address a novel way to compute the [Bernstein-Sato]{}polynomial for an affine variety algorithmically. All the carefully selected nontrivial examples, which we present, have been computed with our implementation. We address such applications as the computation of a zeta-function for certain integrals and revealing the algebraic dependence between pairwise commuting elements.
**Mathematics Subject Classification (2010).** 13P10, 14F10, 68W30.
**Keywords.** $D$-modules, non-commutative Gröbner bases, annihilator ideal, $b$-function, Bernstein-Sato polynomial, Bernstein-Sato ideal.
author:
- Daniel Andres
- Michael Brickenstein
- Viktor Levandovskyy
- 'Jorge Martín-Morales'
- Hans Schönemann
title: 'Constructive $D$-module Theory with [Singular]{}'
---
Introduction
============
Constructive $D$-module theory has been dynamically developing throughout the last years. There are new approaches, algorithms, implementations and applications. Our work on the implementation of procedures for $D$-modules started in 2003, motivated among other factors by challenging elimination problems in non-commutative algebras, which appear [e. g.]{}in algorithms for computing Bernstein-Sato polynomials. We reported on solving several challenges in [@LM08]. A non-commutative subsystem <span style="font-variant:small-caps;">Singular:Plural</span> [@Plural] of the computer algebra system <span style="font-variant:small-caps;">Singular</span> provides a user with possibilities to compute numerous Gröbner bases-based procedures in a wide class of non-commutative $G$-algebras [@LS03]. It was natural to use this functionality in the context of computational $D$-module theory. Nowadays we present a $D$-module suite in <span style="font-variant:small-caps;">Singular</span> consisting of the libraries `dmod.lib`, `dmodapp.lib`, `dmodvar.lib` and `bfun.lib`. There are many useful and flexible procedures for various aspects of $D$-module theory. These libraries are freely distributed together with <span style="font-variant:small-caps;">Singular</span> [@Singular].
There are several implementations of algorithms for $D$-modules, namely the experimental program <span style="font-variant:small-caps;">kan/sm1</span> by N. Takayama [@KAN], the `bfct` package in <span style="font-variant:small-caps;">Risa/Asir</span> [@Asir] by M. Noro [@Noro02] and the package `Dmodules.m2` in <span style="font-variant:small-caps;">Macaulay2</span> by A. Leykin and H. Tsai [@dmodMac2]. We aim at creating a $D$-module suite, which will combine flexibility and rich functionality with high performance, being able to treat more complicated examples.
In this paper we do not present any comparison between different computer algebra systems in the realm of $D$-modules, referring to [@LM08] and [@ALM09]. However, comparison in the latter articles shows, that our implementation is superior to <span style="font-variant:small-caps;">kan/sm1</span> and <span style="font-variant:small-caps;">Macaulay2</span> and in many cases more powerful than <span style="font-variant:small-caps;">Risa/Asir</span>.
Here is the list of problems we address in this paper:
- $s$-parametric annihilator of $f$ (Section \[sAnn\], see also [@LM08; @ALM09]),
- annihilator of $f^{\alpha}$ for $\alpha \in {{\mathbb C}}$ (Section \[anniA\], see also [@SST00]),
- annihilator of a polynomial function $f$ and of a rational function $f/g$ (Section \[anniA\]),
- $b$-function with respect to weights for an ideal (Section \[bfctIdeal\], see also [@ALM09]),
- global and local Bernstein-Sato polynomials of $f$ (Section \[bspolyA\]),
- partial knowledge of [Bernstein-Sato]{}polynomial (Section \[checkrootA\], see also [@LM08]),
- Bernstein operator of $f$ (Section \[Boperator\]),
- logarithmic annihilator of $f$ (Section \[logannA\]),
- Bernstein-Sato ideals for $f = f_1 \cdot \ldots \cdot f_m$ (Section \[BMI\], see also [@LM08]),
- annihilator and Bernstein-Sato polynomial for a variety (Section \[bsvarA\], see also [@ALM09]).
We describe both theoretical and implementational aspects of the problems above and illustrate them with carefully selected nontrivial examples, computed with our implementation in <span style="font-variant:small-caps;">Singular</span>. In Section \[newBM\], we give yet another alternative proof for the algorithm by [Briançon-Maisonobe]{}for computing $\operatorname{Ann}_{D_n[s]}(f^s)$, presented in [@ALM09]. Notably, this delivers additional structural information. In Section \[Boperator\], we compare several approaches for the computation of Bernstein operators. Using the method of principal intersection, we formalize several methods for computing [Bernstein-Sato]{}polynomials. Following Budur et al. [@BMS06] and [@ALM09], we report on the implementation of two methods for the computation of [Bernstein-Sato]{}polynomials for affine varieties in a framework, which is a natural generalization of our approach to the algorithm by [Briançon-Maisonobe]{}.
*Notations*. Throughout the article ${{\mathbb K}}$ is assumed to be a field of characteristic zero. By $R$ we denote the polynomial ring ${{\mathbb K}}[x_1,\ldots,x_n]$ and by $f\in R$ a non-constant polynomial.
We consider the $n$-th Weyl algebra as the algebra of linear partial differential operators with polynomials coefficients. That is $D_n = D(R) = {{\mathbb K}}\langle x_1,\dots,x_n,\partial_1,\dots,\partial_n \mid \{ \partial_i x_i = x_i \partial_i +1, \partial_i x_j = x_j \partial_i, i \not=j \} \rangle$. We denote by $D_n[s] = D(R) \otimes_{{{\mathbb K}}} {{\mathbb K}}[s_1,\ldots,s_n]$ and drop the index $n$ depending on the context.
The ring $R$ is a natural $D_n(R)$-module with the action $$x_i \bullet f(x_1,\ldots,x_n) = x_i \cdot f(x_1,\ldots,x_n), \quad
{\partial }_i \bullet f(x_1,\ldots,x_n) = \frac{{\partial }f(x_1,\ldots,x_n)}{{\partial }x_i}.$$
Working with monomial orderings in elimination, we use the notation $x \gg y$ for [“$x$ is greater than any power of $y$”]{}.
Given an associative ${{\mathbb K}}$-algebra $A$ and some monomial well-ordering on $A$, we denote by $\operatorname{lm}(f)$ (resp. $\operatorname{lc}(f)$) the leading monomial (resp. the leading coefficient) of $f\in A$. Given a left Gröbner basis $G \subset A$ and $f\in A$, we denote by $\operatorname{NF}(f,G)$ the normal form of $f$ with respect to the left ideal ${}_{A}\langle G \rangle$. We also use the shorthand notation $h {\rightarrow}_{H} f$ (and $h {\rightarrow}f$, if $H$ is clear from the context) for the reduction of $h\in A$ to $f\in A$ with respect to the set $H$. If not specified, under *ideal* we mean *left ideal*. For $a,b \in A $, we use the Lie bracket notation $[a,b]:=ab - ba$ as well as the skew Lie bracket notation $[a,b]_{k}:=ab - k\cdot ba$ for $k\in{{\mathbb K}}^{*}$.
It is convenient to treat the algebras we deal with in a bigger framework of $G$-algebras of Lie type.
\[GalgLie\] Let $A$ be the quotient of the free associative algebra ${{\mathbb K}}\langle x_1,\ldots,x_n\rangle$ by the two-sided ideal $I$, generated by the finite set $\{x_jx_i - x_ix_j-d_{ij} \} \; \forall 1\leq i<j \leq n$, where $d_{ij} \in {{\mathbb K}}[x_1,\ldots,x_n]$. $A$ is called a [*$G$–algebra of Lie type*]{} [@LS03], if\
$\bullet$ $\forall \; 1\leq i < j < k \leq n$ the expression $d_{ij}x_k - x_k d_{ij} + x_j d_{ik} - d_{ik}x_j + d_{jk}x_i - x_i d_{jk}$ reduces to zero modulo $I$ and,\
$\bullet$ there exists a monomial ordering $\prec$ on ${{\mathbb K}}[ x_1,\ldots,x_n]$, such that $\operatorname{lm}(d_{ij}) \prec x_i x_j$, $\forall i < j$.
$G$-algebras are also known as *algebras of solvable type* [@KW; @HLi] and *PBW algebras* [@BGV]. We often use the following.
\[prodCrit\] Let $A$ be a $G$-algebra of Lie type and $f,g\in A$. Suppose $\operatorname{lm}(f)$ and $\operatorname{lm}(g)$ have no common factors, then $\operatorname{spoly}(f,g) {\rightarrow}_{\{f,g\}} [f,g]$.
Challenges for Gröbner bases engines of <span style="font-variant:small-caps;">Singular</span>
===============================================================================================
Since the very beginning of implementation of algorithms for $D$-modules in <span style="font-variant:small-caps;">Singular</span> there have been intensive interaction with the developers of <span style="font-variant:small-caps;">Singular</span>. Numerous challenging examples and open problems from constructive $D$-module theory were approached both on the level of libraries and in the kernel of <span style="font-variant:small-caps;">Singular</span> and <span style="font-variant:small-caps;">Singular:Plural</span>. This resulted in several enhancements in kernel procedures and, among other, motivated M. Brickenstein to develop and implement the generalization of his `slimgb` [@SlimBa06] (slim Gröbner basis) algorithm to non-commutative $G$-algebras. Indeed `slimgb` is a variant of Buchberger’s algorithm. It is designed to keep polynomials *slim*, that is short with small coefficients. The algorithm features parallel reductions and a strategy to minimize the weighted lengths of polynomials. A weighted length function of a polynomial can be seen as measure for the intermediate expression swell and it can consider not only the number of terms in a polynomial, but also their coefficients and degrees. Considering the degrees of the terms inside the polynomials, `slimgb` can often directly (that is, without using Gröbner Walk or similar algorithms) compute Gröbner bases with respect to [e. g.]{}elimination orderings. The procedure `slimgb` demonstrated very good performance on examples from the realm of $D$-modules [@LM08], which require computations with elimination orderings.
As it will be seen in the paper, various computational questions, arising in $D$-module theory, use much more than Gröbner bases only. Among other, a transformation matrix between two bases (called <span style="font-variant:small-caps;">Lift</span> in [@GPS08]), the kernel of a module homomorphism (called <span style="font-variant:small-caps;">Modulo</span> in [@GPS08]) and so on must be applied for complicated examples. On the other hand, the standard `std` routine for Gröbner bases, generalized to non-commutative $G$-algebras, is used together with `slimgb` for a variety of problems. Since the beginning of development of the $D$-module suite in <span style="font-variant:small-caps;">Singular</span>, these functions have been enhanced: they became much faster and more flexible. The effect of the use of the generalized Chain Criterion (cf. [@GPS08]) in Gröbner engines is even bigger in the non-commutative case, due to the discarding of multiplications, which complexity is increased, compared with the commutative case. On the contrary, the generalized Product Criterion (Lemma \[prodCrit\]) plays a minor role in the implementation, since the complete discarding of a pair generalizes to the computation of a Lie bracket of the pair members.
The concept of *ring list*, introduced in <span style="font-variant:small-caps;">Singular</span> in 2004, enormously simplified the process of creation and modification of rings (like changing the monomial module ordering, regrouping of variables, modifying non-commutative relations, working with parameters of the ground field etc.). Especially in the $D$-module setting we modify rings often, create a new one from existing rings and equip a new ring with a new ordering. Thus, with ring lists the development of such procedures became much easier and the corresponding code became much more manageable.
We have to mention, that in the meantime the implementation of non-commutative multiplication in the kernel of <span style="font-variant:small-caps;">Singular:Plural</span> has been improved as well.
$s$-parametric annihilator of $f$ {#sAnn}
=================================
Recall Malgrange’s construction for $f = f_1 \cdots f_p\in {{\mathbb K}}[x_1,\ldots,x_n]$. Consider the algebra $W_{p+n}$, being the $(p+n)$-th Weyl algebra $${{\mathbb K}}\langle \{t_j, {\partial t}_j \mid 1\leq j \leq p\} \mid\{ [{\partial t}_j, t_k]=\delta_{jk} \} \rangle \otimes_{{{\mathbb K}}} {{\mathbb K}}\langle \{x_i, {\partial }_i \mid 1\leq i \leq n\} \mid\{ [{\partial }_i, x_k]=\delta_{ik} \} \rangle.$$ Moreover, consider the left ideal in $W_{p+n}$, called Malgrange ideal $$I_f := \langle \; \{ \; t_j - f_j, \sum^p_{j=1} \frac{\partial f_j}{\partial x_i} {\partial t}_j + \partial_i \;\mid 1\leq j\leq p, 1\leq i\leq n \}\; \rangle.$$
Then for $s=(s_1,\ldots,s_p)$ we denote $f^s := f_1^{s_1} \cdots f_p^{s_p}$. Let us compute $$I_f \cap {{\mathbb K}}[\{t_j {\partial }t_j\}]\langle x_i, {\partial }x_i \mid [{\partial }_i, x_i]=1 \rangle \subset D_n[\{t_j {\partial }t_j \}]$$ and furthermore, replace $t_j {\partial }t_j$ with $-s_j-1$. The result is known ([e. g.]{}[@SST00]) to coincide with the $s$-parametric annihilator of $f^s$, $\operatorname{Ann}_{D_n[s]}(f^s) = \{ Q(x,{\partial }, s) \in D_n[s] \mid Q\bullet f^s = 0\}$.
There exist several methods for the computation of $s$-parametric annihilator of $f^s$.
Oaku and Takayama
-----------------
The algorithm by Oaku and Takayama [@Oaku97; @SST00] was developed in a wider context and uses homogenization. Consider the ${{\mathbb K}}$-algebras $T:={{\mathbb K}}[t_1,\ldots,t_p]$, $D'_p:= D(T)$ and $H := D_n \otimes _{{\mathbb K}}D'_p \otimes _{{\mathbb K}}{{\mathbb K}}[u_1,\ldots,u_p,v_1,\ldots,v_p]$. Moreover, let $I$ below be the $(u,v)$-homogenized Malgrange ideal, that is the left ideal in $H$ $$I = \left\langle \{t_j - u_j f_j, \sum^p_{k=1} \frac{{\partial }f_k}{{\partial }x_i} u_k {{\partial }t}_j + \partial_i, u_j v_j -1 \} \right\rangle.$$ Oaku and Takayama proved, that $\operatorname{Ann}_{D_n[s]}(f^s)$ can be obtained in two steps. At first $\{u_j, v_j\}$ are eliminated from $I$ with the help of Gröbner bases, thus yielding $I' = I \cap (D_n \otimes _{{\mathbb K}}D'_p)$. Then, one calculates $I' \cap (D_n \otimes _{{\mathbb K}}{{\mathbb K}}[\{-t_j {\partial }t_j -1 \}])$ and substitutes every appearance of $t_j {\partial }t_j$ by $-s_j-1$ in the latter. The result is then $\operatorname{Ann}_{D_n[s]}(f^s)$.
Briançon and Maisonobe {#BM}
----------------------
Consider $S_p = {{\mathbb K}}\langle \{ {\partial }t_j, s_j \} \mid {\partial }t_j s_k = s_k {\partial }t_j - \delta_{jk} {\partial }t_j \rangle$ (the $p$-th shift algebra) and $S' = D_n \otimes_{{{\mathbb K}}} S_p$. Moreover, consider the following left ideal in $S'$: $$I = \left\langle \{s_j + f_j {\partial }t_j, \sum^p_{k=1} \frac{\partial f_k}{\partial x_i} {\partial t}_k + \partial_i \} \right\rangle.$$ Briançon and Maisonobe proved [@BM02] that $\operatorname{Ann}_{D_n[s]}(f^s) = I\cap D_n[s_1,\ldots,s_p]$ and hence the latter can be computed via the left Gröbner basis with respect to an elimination ordering for $\{{\partial }t_j\}$.
Another alternative proof of Briançon-Maisonobe’s method {#newBM}
--------------------------------------------------------
Here we give yet another [@ALM09] computer algebraic proof for the method by Briançon and Maisonobe.
Throughout this section, we assume $1\leq i \leq n$ and $1\leq j \leq p$. Define $$E := {{\mathbb K}}\langle \{t_j, {\partial }t_j, x_i, {\partial }_i, s_j \}\mid \{[{\partial }_i, x_i]=1, [{\partial }t_j, t_j]=1,
[t_k, s_j] = \delta_{jk} t_j, [{\partial }t_k, s_j] = -\delta_{jk} {\partial }t_j\} \rangle.$$ Let $B = D_n[s]$ be a subalgebra of $E$, generated by $\{ x_i, {\partial }_i, s_j \}$. Then the [Briançon-Maisonobe]{}method requires to prove [@ALM09], that $$\langle \{t_j - f_j, \sum^p_{j=1} \frac{\partial f_j}{\partial x_i} {\partial t}_j + \partial_i, f_j \partial t_j + s_j \} \rangle \cap D_n[s] = \operatorname{Ann}_{D_n[s]}(f^s).$$
Let us define the following polynomials and sets: $$g_i :=\partial_i + \sum^p_{k=1} \frac{\partial f_k}{\partial x_i} {\partial t}_k, \
G = \{g_i\}, T = \{t_j - f_j\}, S = \{s_j + f_j {\partial }t_j \}.$$ Let $\Lambda$ be a (possibly empty) subset of $\{1,\ldots,p\}$. Define $M_\Lambda:=G \cup \{ t_k - f_k \mid k \in \Lambda\} \cup \{s_j + f_j {\partial }t_j \mid j \in \{1,\ldots,p\} \setminus\Lambda \}$.
1. \[thm.a\] For any $\Lambda$, the elements of $M_\Lambda$ commute pairwise. In particular, so do $G \cup T$ and $G \cup S$.
2. \[thm.b\] Consider an ordering $\prec$, satisfying $\{t_j \} \gg \{ x_i \} $, $\{{\partial }_i, s_j \} \gg \{ x_i, {\partial }t_j \}$. Then any subset of $G \cup T \cup S$ is a left Gröbner basis with respect to $\prec$. In particular, so is the set $M_\Lambda$ for any $\Lambda$.
3. \[thm.c\] The elements of $M_\Lambda$ are algebraically independent.
4. \[thm.d\] For any $\Lambda$, the Krull (and hence the Gel’fand-Kirillov) dimension of ${{\mathbb K}}[M_\Lambda]$ is $n+p$.
5. \[thm.e\] For any $\Lambda$, ${{\mathbb K}}[M_\Lambda]$ is a maximal commutative subalgebra of $E$.
\[thm.a\] Computing commutators between elements, we obtain $$\begin{gathered}
[g_i,g_k] = {{\partial }t}_j \sum_j [{\partial }_i, \frac{{\partial }f_j}{{\partial }x_k} ] + {{\partial }t}_j \sum_j [\frac{{\partial }f_j}{{\partial }x_i}, {\partial }_k]
= {{\partial }t}_j \sum_j ([{\partial }_i, \frac{{\partial }f_j}{{\partial }x_k} ] - [{\partial }_k, \frac{{\partial }f_j}{{\partial }x_i} ]) = 0,\\
[t_k - f_k,g_i] = \sum_j \frac{{\partial }f_j}{{\partial }x_i} [t_k,{\partial }t_j] - [f_k, {\partial }_i]=0, \qquad
[t_i - f_i,t_k - f_k] = 0,\\
[s_i + f_i {\partial }t_i , s_j + f_j {\partial }t_j] = f_j [s_i, {\partial }t_j] - f_i [s_j, {\partial }t_i] = 0, \qquad
[s_j + f_j {\partial }t_j,g_i] = [s_j, {\partial }_i] + \\ + {\partial }t_j [f_j, {\partial }_i] + \sum^p_{k=1} \frac{{\partial }f_k}{{\partial }x_i} [s_j, {{\partial }t}_k] + [f_j {\partial }t_j, \sum^p_{k=1} \frac{{\partial }f_k}{{\partial }x_i} {{\partial }t}_k] = \frac{{\partial }f_j}{{\partial }x_i} {{\partial }t}_j - [{\partial }_i, f_j] {\partial }t_j = 0.\\
\intertext{The only nonzero commutator arises from }
[t_k - f_k, s_j + f_j {\partial }t_j] = [t_k,s_j] + f_j [t_k, {\partial }t_j] - [f_k,s_j] -[f_k, f_j {\partial }t_j] = \delta_{jk} (t_k -f_k).\end{gathered}$$ However, according to the definition, only one of these elements belongs to $M_\Lambda$ for any $\Lambda$.
\[thm.b\] We run Buchberger’s algorithm by hands. Due to the ordering property, for each pair the generalized Product Criterion is applicable. Hence using \[thm.a\] we see, that most $s$-polynomials reduce to commutators, which are zero except $\operatorname{spoly}(t_k - f_k, s_j + f_j {\partial }t_j) = \delta_{jk} (t_k -f_k)$, which reduces to zero modulo the first polynomial. Thus, any subset including $M_\Lambda$ is indeed a Gröbner basis.
\[thm.c\] Using pairwise commutativity, we employ the Commutative Preimage Theorem from [@Lev05]. It states, that the ideal of algebraic dependencies between pairwise commuting elements $\{h_k\mid 1\leq k \leq m\}\subset E$ can be computed as $$E \otimes_{{{\mathbb K}}} {{\mathbb K}}[c_1,\ldots,c_m] \supset \langle \{ h_i - c_i \} \rangle \cap {{\mathbb K}}[c_1,\ldots,c_m],$$ where $c_i$ are new commutative variables, adjoint to $E$. In this elimination problem one requires an ordering on $E \otimes_{{{\mathbb K}}} {{\mathbb K}}[c]$, preferring variables of $E$ to $c_i$’s. For such an ordering, one needs to compute a Gröbner basis. Now, take $\{h_i\}:=M_\Lambda$, $1\leq i \leq p+n$, and run Buchberger’s algorithm with respect to the same ordering as in \[thm.b\]. Thus we are again in the situation, where the Product Criterion applies, hence $[h_i - c_i, h_k - c_k] = 0$ since $[h_i, h_k]=0$ by \[thm.b\] and $c_i$ are central. Hence, $\{ h_i - c_i \}$ is a left Gröbner basis and by the elimination property $\langle \{ h_i - c_i \} \rangle \cap {{\mathbb K}}[c_1,\ldots,c_m] = 0$, that is $\{h_i\}$ are algebraically independent.
\[thm.d\] By \[thm.c\], $M_\Lambda$ generates a commutative ring with no algebraic dependence between its elements, so the Krull dimension is the cardinality of $M_\Lambda$, that is $n+p$. Since $M_\Lambda$ is isomorphic to a commutative polynomial ring by \[thm.c\], its Gel’fand-Kirillov dimension over the field ${{\mathbb K}}$ is $n+p$ as well.
\[thm.e\] With respect to the ordering from \[thm.b\], the leading monomials of the generators are $\{{\partial }_1,\ldots,{\partial }_n\} \cup \{ t_k \mid k\in\Lambda \} \cup \{s_j \mid j\not\in\Lambda \}$. Assume, that there exists an element in $E\setminus {{\mathbb K}}[M_\Lambda]$, which commutes with all elements in $M_\Lambda$. Then its leading monomial must belong to the subalgebra $F$, generated by $\{x_1,\ldots,x_n\} \cup \{ s_k \mid k\in\Lambda \} \cup \{t_j \mid j\not\in\Lambda \} \cup \{{\partial }t_1,\ldots,{\partial }t_p\}$. Since the center of $E$ is ${{\mathbb K}}$, we consider centralizers of elements. Taking $F'= \cap_{k\in\Lambda } C(t_k-f_k) \cap F$, we see that an element from it can have no $\{{\partial }t_k, s_k \mid k\in\Lambda \}$. Considering $F''=\cap_i C(g_i) \cap F' $, we exclude $\{x_1,\ldots,x_n\}$. Thus we are left with the subalgebra, generated by $\tilde{F} = \{{\partial }t_j, s_j \mid j\not\in\Lambda \}$. But no element of it can commute with $\{ s_j + f_j {\partial }t_j \mid j\not\in\Lambda \}$ except constants. Hence the claim.
We want to eliminate both $\{t_j\}$ and $\{{\partial }t_j\}$ from an ideal, generated by $G \cup S \cup T$. By using an elimination ordering for $\{t_j\}$ we proved in \[thm.b\] above, that $G \cup S \cup T$ is a Gröbner basis. Hence, the elimination ideal is generated by $G \cup S$ and we can proceed with eliminating $\{{\partial }t_j\}$ from $G \cup S$, which is exactly the statement of [Briançon-Maisonobe]{}in Section \[BM\].
Implementation
--------------
We use the following acronyms in addressing functions in the implementation: *OT* for Oaku and Takayama, *LOT* for Levandovskyy’s modification of Oaku and Takayama [@LM08] and *BM* for [Briançon-Maisonobe]{}. Moreover, it is possible to specify the desired Gröbner basis engine (`std` or `slimgb`) via an optional argument.
For the classical situation $f = f_1$, the procedure `Sannfs(f)` computing $\operatorname{Ann}_{D_n[s]}(f^s) \subset D_n[s]$ uses a [“minimal user knowledge”]{} principle and chooses one of three mentioned algorithms. Alternatively, one can call the corresponding procedures `SannfsOT, SannfsLOT, SannfsBM` directly.
For the annihilator of $f = f_1 \cdots f_p$, see Section \[BMI\].
\[exSannfs\] We demonstrate, how to compute the $s$-parametric annihilator with `Sannfs`. This procedure takes a polynomial in a commutative ring as its argument and returns back a Weyl algebra of the type `ring` containing an object of the type `ideal` called `LD`. Note, that the latter ideal is a set of generators and not a Gröbner basis in general.
LIB "dmod.lib";
ring r = 0,(x,y),dp; // set up the commutative ring
poly f = x^3 + y^2 + x*y^2; // define the polynomial
def D = Sannfs(f); setring D; // call Sannfs and change to ring D
LD = groebner(LD); LD; // compute and print Groebner basis
==> LD[1]=2*x*y*Dx-3*x^2*Dy-y^2*Dy+2*y*Dx
==> LD[2]=2*x^2*Dx+2*x*y*Dy+2*x*Dx+3*y*Dy-6*x*s-6*s
==> LD[3]=x^2*y*Dy+y^3*Dy-2*x^2*Dx-3*x*y*Dy-2*y^2*s+6*x*s
==> LD[4]=x^3*Dy+x*y^2*Dy+y^2*Dy-2*x*y*s-2*y*s
==> LD[5]=2*y^3*Dx*Dy+3*x^3*Dy^2+x*y^2*Dy^2-4*x^2*Dx^2-8*x*y*Dx*Dy-2*x^2*Dx
-4*y^2*Dx*s+6*x*y*Dy+12*x*Dx*s-10*x*Dx-6*y*Dy+12*s
Annihilators of polynomial and rational functions {#anniA}
=================================================
Annihilator of $f^{\alpha}$ for $\alpha \in {{\mathbb C}}$ {#annfA}
----------------------------------------------------------
It is known ([e. g.]{}[@SST00]) that for any $\alpha \in {{\mathbb C}}$, $D_n/\operatorname{Ann}_{D_n}(f^{\alpha})$ is a holonomic $D$-module. In the procedure `annfspecial` from `dmod.lib` we follow Algorithm 5.3.15 in [@SST00]. Given $f$ and $\alpha$, we compute $\operatorname{Ann}_{D_n[s]}(f^s) \subset D_n[s]$, the [Bernstein-Sato]{}polynomial of $f$ (cf. Section \[globalBS\]) and its minimal integer root $s_0$. Then, if $\alpha- (s_0 + 1) \in {{\mathbb N}}$, according to Algorithm 5.3.15 in [@SST00] we have to compute a certain syzygy module in advance. Otherwise, $\operatorname{Ann}_{D_n}(f^{\alpha}) = \operatorname{Ann}_{D_n[s]}(f^s) \mid_{s=\alpha}$ is obtained via substitution.
\[exAnnSpecial\] In this example we show, how one computes the annihilator of $2xy$.
LIB "dmod.lib"; option(redSB); option(redTail);
ring r = 0,(x,y),dp; poly g = 2*x*y;
def A = Sannfs(g); setring A; // compute Ann(g^s)
LD = groebner(LD); LD; // GB of the ideal Ann(g^s)
==> LD[1]=y*Dy-s
==> LD[2]=x*Dx-s
def B = annfs0(LD,2*x*y); setring B; // compute BS polynomial
BS; // the list of roots and multiplicities of BS polynomial
==> [1]:
==> _[1]=-1
==> [2]:
==> 2
// so, the minimal integer root is -1
setring A; // need to work with Ann(g^s) again
ideal I = annfspecial(LD,2*x*y,-1,1);
// the last argument 1 indicates that we want to compute f^1
print(matrix(I)); // condensed presentation
==> Dy^2, y*Dy-1, Dx^2, x*Dx-1
Alternative for an annihilator of $f^m$
---------------------------------------
Computing a syzygy module in the previous algorithm can be expensive. Therefore we note, that for $\alpha=m \in {{\mathbb N}}$ we better use an easier approach.
\[annPoly\] Let $g\in {{\mathbb K}}[x_1,\ldots,x_n]$. Consider the homomorphism of left $D_n$-modules $\psi: D_n \rightarrow D_n/\langle {\partial }_1,\ldots,{\partial }_n\rangle$, $\psi(1) = g$. Then $\operatorname{Ann}_{D_n}(g) = \ker \psi$.
Note, that ${{\mathbb K}}[x_1,\ldots,x_n] \cong D_n/\langle {\partial }_1,\ldots,{\partial }_n\rangle$ as left $D_n$-modules. Hence we can view $g$ as the image of $1$ under $\psi$. Then $\operatorname{Ann}_{D_n}(g) = \{a\in D_n \mid a\bullet g = 0\} = \{a\in D_n \mid ag \in \langle {\partial }_1,\ldots,{\partial }_n\rangle \} = \ker \psi$.
Hence, given any element $f\in{{\mathbb K}}[x_1,\ldots,x_n]$, $\operatorname{Ann}_{D_n}(f)$ can be computed via the kernel of a module homomorphism (algorithm `Modulo`) which amounts to just one Gröbner basis computation. Moreover, it does not use elimination and hence is clearly more efficient in the special case $g=f^n$ for $f\in R, n\in{{\mathbb N}}$, than the more general method in Section \[annfA\]. Notably this method can be generalized to various other operator algebras, see [@SLZ09] for details. The corresponding procedure in `dmodapp.lib` is called `annPoly`.
\[annPolyCheckRoot\] Yet another improvement can be achieved in the computation of the minimal integer root of the [Bernstein-Sato]{}polynomial with the algorithms from Theorem \[checkRoot\] below. Namely, since we know, that for an integer root, say $\alpha$, of the [Bernstein-Sato]{}polynomial of a polynomial in $n \geq 2$ variables $-n+1 \leq \alpha \leq -1$ holds (by [@Saito94; @Varcenko81]) and $-1$ is always a root, we can run the `checkRoot` procedure (which is just one Gröbner basis computation with an arbitrary ordering, see Section \[checkrootA\]) starting from $\alpha = -n+1$ to $\alpha=-2$. We stop at the first affirmative answer from `checkRoot` or output $-1$ if no positive answer appears. Thus, one executes `checkRoot` at most $n-2$ times.
\[heurAnnFa\]
$f\in{{\mathbb C}}[x_1,\ldots,x_n]$, $\alpha\in{{\mathbb C}}$ $\operatorname{Ann}_{D_n}(f^{\alpha})$ **if** $\alpha \in {{\mathbb C}}\setminus ({{\mathbb Z}}\cap [-n+1,-1])$ **then** $\operatorname{Ann}_{D_n} (f^{\alpha}) =
\begin{cases}
\langle {\partial }_1,\ldots,{\partial }_n\rangle & \text{ if } \alpha=0, \\
\ker(D_n \overset{1\mapsto f^{m}}{\longrightarrow} D_n/\langle {\partial }_1,\ldots,{\partial }_n\rangle) & \text{ if } \alpha=m\in{{\mathbb N}}, \quad \text{(cf. \ \ref{annPoly})},\\
\operatorname{Ann}_{D_n[s]}(f^{s}) \mid_{s=\alpha} & \text{ if } \alpha\in ({{\mathbb C}}\setminus {{\mathbb Z}}) \cup ({{\mathbb Z}}\cap (-\infty,-n]),
\end{cases}$\
**else** (that is $\alpha \in {{\mathbb Z}}\cap [-n+1,-1]$) $\mu := \min\{\beta \in {{\mathbb Z}}_{<0} \mid b_f(\beta)=0\}$ $\operatorname{Ann}_{D_n} (f^{\alpha}) =
\begin{cases}
\text{Procedure \ref{annfA} with \ref{annPolyCheckRoot}} & \text{ if }
\mu +1 \leq \alpha \leq -1, \\
\text{Procedure \ref{annPolyCheckRoot} and} \operatorname{Ann}_{D_n[s]} (f^{s}) \mid_{s=\alpha} &
\text{ if } -n+1 \leq \alpha \leq \mu. \\
\end{cases}$ **end if** $\operatorname{Ann}_{D_n} (f^{\alpha})$
Annihilator of a rational function
----------------------------------
In order to compute the annihilator $I$ of a rational function $\frac{f}{g}$ (it is known that $D_n/I$ is holonomic) we use the following lemma.
\[annRat\] Let $f, g\in {{\mathbb K}}[x_1,\ldots,x_n]\setminus\{0\}$. Consider the homomorphism of left $D_n$-modules $\tau: D_n \rightarrow D_n/\operatorname{Ann}_{D_n}(g^{-1}), q \mapsto qf$. Then $\operatorname{Ann}_{D_n}(\frac{f}{g}) = \ker\tau$.
For $q\in\ker\tau=\{ q\in D_n\mid qf \in \operatorname{Ann}_{D_n}(g^{-1})\}$, $(qf) \bullet g^{-1} = q\bullet (fg^{-1})$, hence $\operatorname{Ann}_{D_n}(\frac{f}{g}) = \ker\tau$.
We compute $\operatorname{Ann}_{D_n}(g^{-1})$ with Algorithm \[heurAnnFa\] above. Although in the case, when $-1$ is not the minimal integer root of the [Bernstein-Sato]{}polynomial of $g$, we have to use expensive algorithms like \[annfA\], we know no other methods to compute the annihilator in Weyl algebras. Also, no general algorithm for computing a complete system of operator equations (with operators including along partial differentiation also partial ($q$-)differences et cetera) with polynomial coefficients, annihilating a rational function, is known to us. In our opinion, the existence of an algorithm for $\operatorname{Ann}_{D_n}(g^{-1})$ shows the intrinsic naturality of $D$-modules compared with other linear operators acting on ${{\mathbb K}}[x]$. The algorithm is implemented in `dmodapp.lib` and the corresponding procedure is called `annRat`.
\[exAnnRat\] In this example we demonstrate the computation of annihilators of a rational function. The procedure `annRat` takes as arguments polynomials in a commutative ring and returns a Weyl algebra (of type `ring`) together with an object of type `ideal` called `LD` (cf. Example \[exSannfs\]). Note, that `LD` is given in a Gröbner basis.
LIB "dmodapp.lib";
ring r = 0,(x,y),dp;
poly g = 2*x*y; poly f = x^2 - y^3; // we will compute Ann(g/f)
option(redSB); option(redTail); // get reduced minimal GB
def B = annRat(g,f); setring B;
LD; // Groebner basis of Ann(g/f)
==> LD[1]=3*x*Dx+2*y*Dy+1
==> LD[2]=y^3*Dy^2-x^2*Dy^2+6*y^2*Dy+6*y
==> LD[3]=9*y^2*Dx^2*Dy-4*y*Dy^3+27*y*Dx^2+2*Dy^2
==> LD[4]=y^4*Dy-x^2*y*Dy+2*y^3+x^2
==> LD[5]=9*y^3*Dx^2-4*y^2*Dy^2+10*y*Dy-10
$b$-function with respect to weights for an ideal {#bfctIdeal}
=================================================
Let $0 \neq w \in {{\mathbb R}}^n_{\geq 0}$ and consider the $V$-filtration $V = \left\{ V_m \mid m \in {{\mathbb Z}}\right\}$ on $D_n$ with respect to $w$, where $V_m$ is spanned by $\left\{ x^{\alpha} {\partial }^{\beta} \mid -w \alpha + w \beta \leq m \right\}$ over ${{\mathbb K}}$. That is, $x_i$ and ${\partial }_i$ get weights $-w_i$ and $w_i$ respectively. Note that then the relation ${\partial }_i x_i = x_i {\partial }_i + 1$ is homogeneous of degree $0$. It is known that the associated graded ring $\bigoplus_{m \in {{\mathbb Z}}} V_m / V_{m-1}$ is isomorphic to $D_n$, which allows us to identify it with the Weyl algebra.
From now on we assume, that $I \subset D_n$ is an ideal such that $D_n/I$ is a holonomic module. Since holonomic $D$-modules are cyclic ([e. g.]{}[@Cou95]), for each holonomic $D$-module $M$ there exists an ideal $I_M$ such that $M \cong D_n/I_M$ as $D$-modules.
Let $0 \neq w \in {{\mathbb R}}^n_{\geq 0}$. For a non-zero polynomial $$p = \sum_{\alpha, \beta \in {{\mathbb N}}_0^n} c_{\alpha \beta} x^{\alpha} {\partial }^{\beta} \in D_n
\quad \text{with all but finitely many } c_{\alpha \beta} = 0$$ we put $m = \max_{\alpha, \beta} \{ -w \alpha + w \beta \mid c_{\alpha \beta} \neq 0\} \in {{\mathbb R}}$ and define the *initial form* of $p$ with respect to the weight $w$ as follows: $${\operatorname{in}_{(-w,w)}}(p) := \sum_{\alpha,\beta \in {{\mathbb N}}_0^n:~ -w \alpha + w \beta = m} c_{\alpha \beta} x^{\alpha} {\partial }^{\beta}.$$ For the zero polynomial, we set ${\operatorname{in}_{(-w,w)}}(0) := 0$. Additionally, the ideal ${\operatorname{in}_{(-w,w)}}(I) := {{\mathbb K}}\cdot \{ {\operatorname{in}_{(-w,w)}}(p) \mid p \in I \}$ is called the *initial ideal* of $I$ with respect to $w$.
Let $0 \neq w \in {{\mathbb R}}^n_{\geq 0}$ and $s := \sum_{i=1}^n w_i x_i {\partial }_i$. Then ${\operatorname{in}_{(-w,w)}}(I) \cap {{\mathbb K}}[s]$ is a principal ideal in ${{\mathbb K}}[s]$. Its monic generator $b_{I,w}(s)$ is called the *global $b$-function* of $I$ with respect to the weight $w$.
\[b(s) is not zero\] The global $b$-function is nonzero.
We will give a proof of this well-known result in Section \[intersection\].
Following its definition, the computation of the global $b$-function of $I$ with respect to $w$ can be done in two steps:
1. Compute the initial ideal $I'$ of $I$ with respect to $w$.
2. Compute the intersection of $I'$ with the subalgebra ${{\mathbb K}}[s]$.
We will discuss both steps separately, starting with the initial ideal. It is important to mention, that although this procedure has been described in [@SST00], this approach was completely treated by Noro in [@Noro02], accompanied with a very impressive implementation in `Risa/Asir`.
Computing the initial ideal {#initial ideal}
---------------------------
In order to compute the initial ideal, the method of weighted homogenization is proposed in [@Noro02], which we will describe below.
Let $u,v \in {{\mathbb R}}^n_{>0}$. The $G$-algebra $D_{(u,v)}^{(h)} := {{\mathbb K}}\langle x_1,\ldots,x_n,{\partial }_1,\ldots,{\partial }_n, h \mid \{x_j x_i = x_i x_j, {\partial }_j {\partial }_i \\= {\partial }_i {\partial }_j, x_i h = h x_i, {\partial }_i h = h {\partial }_i, {\partial }_j x_i = x_i {\partial }_j + \delta_{i,j} h^{u_i + v_j} \} \rangle$ is called the *$n$-th weighted homogenized Weyl algebra* with weights $u,v$, [i. e.]{}$x_i$ and ${\partial }_i$ get weights $u_i$ and $v_i$ respectively.
For $p = \sum_{\alpha, \beta} c_{\alpha \beta} x^{\alpha} \partial^{\beta} \in D_n$ we define the *weighted homogenization* of $p$ as follows: $$H_{(u,v)}(p) = \sum_{\alpha, \beta} c_{\alpha \beta} h^{\deg_{(u,v)}(p) - (u \alpha + v \beta)} x^{\alpha} {\partial }^{\beta}.$$ This definition naturally extends to a set of polynomials. Here, $\deg_{(u,v)}(p)$ denotes the weighted total degree of $p$ with respect to weights $u,v$ for $x,{\partial }$ and weight $1$ for $h$.
For a monomial ordering $\prec$ on $D_n$, which is not necessarily a well-ordering, we define an associated homogenized global ordering $\prec^{(h)}$ in $D_{(u,v)}^{(h)}$ by setting $h \prec^{(h)} x_i, h \prec^{(h)} {\partial }_i$ for all $i$ and, $$\begin{aligned}
{2}
p \prec^{(h)} q \quad
&\text{if} \quad &\deg_{(u,v)}(p) &< \deg_{(u,v)}(q)\\
&\text{or} &\deg_{(u,v)}(p) &= \deg_{(u,v)}(q)
\quad \text{and} \quad
p_{\mid_{h=1}} \prec q_{\mid_{h=1}}.\end{aligned}$$
Note that for $u = v = (1,\ldots,1)$ this is exactly the standard homogenization as in [@SST00] and [@CN97]. Analogue statements of the following two theorems can be found in [@SST00] and [@Noro02] respectively.
\[Theorem: dehomogenization\] Let $F$ be a finite subset of $D_n$ and $\prec$ a global ordering. If $G^{(h)}$ is a Gröbner basis of $\langle H_{(u,v)}(F) \rangle$ with respect to $\prec^{(h)}$, then ${G^{(h)}}_{\mid_{h=1}}$ is a Gröbner basis of $\langle F \rangle$ with respect to $\prec$.
\[Theorem: initial ideal\] Let $\prec$ be a global monomial ordering on $D_n$ and $\prec_{(-w,w)}$ the non-global ordering defined by $$\begin{aligned}
{2}
x^{\alpha} {\partial }^{\beta} \prec_{(-w,w)} x^{\gamma} {\partial }^{\delta} \quad
&\text{if}\quad & -w \alpha + w \beta &< -w \gamma + w \delta\\
&\text{or} &-w \alpha + w \beta &= -w \gamma + w \delta
\quad\text{and}\quad x^{\alpha} {\partial }^{\beta} \prec x^{\gamma} {\partial }^{\delta}.\end{aligned}$$ If $G^{(h)}$ is a Gröbner basis of $H_{(u,v)}(I)$ with respect to $\prec^{(h)}_{(-w,w)}$, then the set $\{ {\operatorname{in}_{(-w,w,0)}}(g) \mid g \in G^{(h)}\}$ is a Gröbner basis of ${\operatorname{in}_{(-w,w,0)}}(H_{(u,v)}(I))$ with respect to $\prec^{(h)}$.
Let $f' \in {\operatorname{in}_{(-w,w,0)}}(H_{(u,v)}(I))$ be $(-w,w,0)$-homogeneous. There exist $f \in H_{(u,v)}(I)$ and $g \in G^{(h)}$ such that $f' = {\operatorname{in}_{(-w,w,0)}}(f)$ and $\operatorname{lm}_{\prec^{(h)}_{(-w,w)}}(g) \mid \operatorname{lm}_{\prec^{(h)}_{(-w,w)}}(f)$. Since $f,g$ are $(u,v)$-homogeneous, we have $$\operatorname{lm}_{\prec^{(h)}_{(-w,w)}}(g) = \operatorname{lm}_{\prec^{(h)}}({\operatorname{in}_{(-w,w,0)}}(g)),\qquad
\operatorname{lm}_{\prec^{(h)}_{(-w,w)}}(f) = \operatorname{lm}_{\prec^{(h)}}({\operatorname{in}_{(-w,w,0)}}(f)),$$ which concludes the proof.
Summarizing the results from this section, we obtain the following algorithm to compute the initial ideal.
\[InitialIdeal\]
$I\subset D_n$ a holonomic ideal, $0 \neq w \in {{\mathbb R}}^n_{\geq 0}$, $\prec$ a global ordering on $D_n$, $u,v \in {{\mathbb R}}^n_{>0}$ A Gröbner basis $G$ of ${\operatorname{in}_{(-w,w)}}(I)$ with respect to $\prec$ $\prec^{(h)}_{(-w,w)} :=$ the homogenized ordering as defined in theorem \[Theorem: initial ideal\] $G^{(h)} :=$ a Gröbner basis of $H_{(u,v)}(I)$ with respect to $\prec^{(h)}_{(-w,w)}$ $G = {\operatorname{in}_{(-w,w)}}({G^{(h)}}_{\mid_{h=1}})$
Intersecting an ideal with a principal subalgebra {#intersection}
-------------------------------------------------
We will now consider a much more general setting than needed to compute the global $b$-function. Let $A$ be an associative ${{\mathbb K}}$-algebra. We are interested in computing the intersection of a left ideal $J \subset A$ with the subalgebra ${{\mathbb K}}[s]$ of $A$ where $s \in A$ is an arbitrary non-constant element. This intersection is always generated by one element since ${{\mathbb K}}[s]$ is a principal ideal domain. In other words, we want to find the monic generator $b \in A$ such that $\langle b \rangle = J \cap {{\mathbb K}}[s]$.
For this section, we will assume that there is an ordering on $A$ such that there exists a finite left Gröbner basis $G$ of $J$.
Then we can distinguish between the following four situations:
1. \[sit 1\] No leading monomials of elements in $G$ divide the leading monomial of any power of $s$.
2. \[sit 2\] There is an element in $G$ whose leading monomial divides the leading monomial of some power of $s$. In this situation, we have the following sub-situations.
1. \[sit 2.1\] $J \cdot s \subset J$ and $\dim_{{{\mathbb K}}}(\mathrm{End}_{A}(A/J)) < \infty$.
2. \[sit 2.2\] One of the two conditions in 2.1. does not hold.
1. \[sit 2.2.1\] The intersection is zero.
2. \[sit 2.2.2\] The intersection is not zero.
\[la:zero intersection\] If there exists no $g \in G$ such that $\operatorname{lm}(g)$ divides $\operatorname{lm}(s^k)$ for some $k \in {{\mathbb N}}_0$, then $J \cap {{\mathbb K}}[s] = \{ 0 \}$.
The lemma covers the first case above. In the second case however, we cannot in general state whether the intersection is trivial or not as the following example illustrates.
The converse of the previous lemma is wrong. For instance, consider ${{\mathbb K}}[x,y]$ and $J = \langle y^2 + x \rangle$. Then $J \cap {{\mathbb K}}[y] = \{ 0 \}$ while $\{ y^2 + x\}$ is a Gröbner basis of $J$ for any ordering.
In situation \[sit 2.1\] though, the intersection is not zero as the following lemma shows, inspired by the sketch of the proof of Theorem \[b(s) is not zero\] in [@SST00].
\[lemma minimal polynomial\] Let $J \cdot s \subset J$ and $\dim_{{{\mathbb K}}}(\mathrm{End}_{A}(A/J)) < \infty$. Then $J \cap {{\mathbb K}}[s] \neq \{ 0 \}$.
Consider the right multiplication with $s$ as a map $A/J {\rightarrow}A/J$ which is a well-defined $A$-module endomorphism of $A/J$ as $a-a' \in J$ implies that $(a-a')s \in J \cdot s \subset J$, which holds by assumption for all $a,a' \in A$. Since $\mathrm{End}_{A}(A/J)$ is finite dimensional, linear algebra guarantees that this endomorphism has a well-defined non-zero minimal polynomial $\mu$. Moreover, $\mu$ is precisely the monic generator of $J \cap {{\mathbb K}}[s]$ as $\mu(s) = [0]$ in $A/J$, hence $\mu(s) \in J \cap {{\mathbb K}}[s]$, and $\deg(\mu)$ is minimal by definition.
\[End(A/J)\] In particular, the lemma holds if $A/J$ itself is a finite dimensional $A$-module. In the case where $A$ is a Weyl algebra and $A/J$ is a holonomic module, we know that $\dim_{{{\mathbb K}}}(\mathrm{End}_{A}(A/J))$ is finite (cf. [@SST00]).
For situation \[sit 2.1\], we have reduced our problem of intersecting an ideal with a subalgebra generated by one element to a problem from linear algebra by the proof of the lemma, namely to the one of finding the minimal polynomial of an endomorphism.
Let $0 \neq w \in {{\mathbb R}}^n_{\geq 0}, J := {\operatorname{in}_{(-w,w)}}(I)$ for a holonomic ideal $I \subset D_n$ and $s := \sum_{i=1}^n w_i x_i {\partial }_i$. Without loss of generality let $0 \neq p = \sum_{\alpha,\beta} c_{\alpha,\beta} x^{\alpha} {\partial }^{\beta} \in J$ be $(-w,w)$-homogeneous. Then we obtain for every monomial in $p$ by using the Leibniz rule $$x^{\alpha} {\partial }^{\beta} x_i {\partial }_i = x^{\alpha+e_i} {\partial }^{\beta+e_i} + \beta_i x^{\alpha} {\partial }^{\beta}
= ({\partial }_i x_i^{\alpha_i+1} - (\alpha_i + 1) x_i^{\alpha_i}) \frac{x^{\alpha}}{x_i^{\alpha_i}} {\partial }^{\beta}
+ \beta_i x^{\alpha} {\partial }^{\beta}$$ $$= ({\partial }_i x_i - (\alpha_i + 1) + \beta_i) x^{\alpha} {\partial }^{\beta}
= (x_i {\partial }_i - \alpha_i + \beta_i) x^{\alpha} {\partial }^{\beta}.$$ Put $m = -w \alpha + w \beta$ for some term $c_{\alpha,\beta} x^{\alpha} {\partial }^{\beta}$ in $p$ where $c_{\alpha,\beta}$ is non-zero. Since $p$ is $(-w,w)$-homogeneous, $m$ does not depend on the choice of this term. Hence, $$\begin{aligned}
p \cdot s
&= p \sum_{i=1}^n w_i x_i {\partial }_i
= \sum_{i=1}^n w_i \sum_{\alpha,\beta} (x_i {\partial }_i - \alpha_i + \beta_i) c_{\alpha,\beta} x^{\alpha} {\partial }^{\beta} \\
&= s \cdot p + \sum_{i=1}^n \sum_{\alpha,\beta} w_i (- \alpha_i + \beta_i) c_{\alpha,\beta} x^{\alpha} {\partial }^{\beta}
= (s + m) \cdot p \in J.\end{aligned}$$ Since $D_n/J$ is holonomic (cf. [@SST00]) and $J \cdot s \subset J$, Remark \[End(A/J)\] and Lemma \[lemma minimal polynomial\] yield the claim.
If one knows in advance that the intersection is not zero, the following algorithm terminates.
\[PrincipalIntersect\]
$s \in A, J \subset A$ a left ideal such that $J \cap {{\mathbb K}}[s] \neq \{ 0 \}$. $b \in {{\mathbb K}}[s]$ monic such that $J \cap {{\mathbb K}}[s] = \langle b \rangle$ $G :=$ a finite left Gröbner basis of $J$ (assume it exists) $i := 1$ $b := s^i + \sum_{j=0}^{i-1} a_j s^j$ $i := i+1$
Note that because $\operatorname{NF}(s^i,G) + \sum_{j=0}^{i-1} a_j \operatorname{NF}(s^j,G) = 0$ is equivalent to $s^i + \sum_{j=0}^{i-1} a_j s^j \in J$, the algorithm searches for a monic polynomial in ${{\mathbb K}}[s]$ that also lies in $J$. This is done by going degree by degree through the powers of $s$ until there is a linear dependency. This approach also ensures the minimality of the degree of the output. The algorithm terminates if and only if $J \cap {{\mathbb K}}[s] \neq \{ 0 \}$. Note that this approach works over any field.
The check whether there is a linear dependency over ${{\mathbb K}}$ between the computed normal forms of the powers of $s$ is done by the procedure `linReduce` in our implementation.
### An enhanced computation of normal forms
When computing normal forms of the form $\operatorname{NF}(s^i,J)$ like in algorithm \[PrincipalIntersect\] we can speed up the reduction process by making use of the previously computed normal forms.
\[NF computing 1\] Let $A$ be a ${{\mathbb K}}$-algebra, $J \subset A$ a left ideal and let $f \in A$. For $i \in {{\mathbb N}}$ put $r_i = \operatorname{NF}(f^i,J)$, $q_i = f^i - r_i \in J$ and $c_i = \frac{\operatorname{lc}(q_i r_1)}{\operatorname{lc}(r_1 q_i)}$ provided $r_1 q_i \not=0$. For $r_1 q_i=0$ we put $c_i=0$. Then we have for all $i \in {{\mathbb N}}$ $$r_{i+1}
= \operatorname{NF}(f r_i,J)
= \operatorname{NF}([f^i-r_i,r_1]_{c_i} + r_i r_1,J).$$
As a consequence, we obtain the following result for some ${{\mathbb K}}$-algebras of special importance.
\[NF computing 2\] If $A$ is a $G$-algebra of Lie type ([e. g.]{}a Weyl algebra), then $$r_{i+1} = \operatorname{NF}(f r_i,J) = \operatorname{NF}([f^i-r_i,r_1] + r_i r_1,J) \text{ holds}.$$ If $A$ is commutative, we have $ r_{i+1} = \operatorname{NF}(r_i r_1,J) = \operatorname{NF}(r_1,J)^{i+1}= \operatorname{NF}(r_1^{i+1},J)$.
Note, that computing Lie bracket $[f,g]$ both in theory and in practice is easier and faster, than to compute $[f,g]$ as $f\cdot g - g \cdot f$, see [e. g.]{}[@LS03].
### Applications
Apart from computing global $b$-functions, there are various other applications of Algorithm \[PrincipalIntersect\].
#### Solving Zero-dimensional Systems.
Recall that an ideal $I \subset {{\mathbb K}}[x_1,\ldots,x_n]$ is called zero-dimensional if one of the following equivalent conditions holds:
- ${{\mathbb K}}[x_1,\ldots,x_n] / I$ is finite dimensional as a ${{\mathbb K}}$-vector space.
- For each $1 \leq i \leq n$ there exist $0 \neq f_i \in I \cap {{\mathbb K}}[x_i]$.
- The cardinality of the zero-set of $I$ is finite.
In order to compute the zero-set of $I$, one can use the classical triangularization algorithms. These algorithms require to compute a Gröbner basis with respect to some elimination ordering (like lexicographic one), which might be very hard.
By Algorithm \[PrincipalIntersect\], a generator of $I \cap {{\mathbb K}}[x_i]$ can be computed without these expensive orderings. Instead, any ordering, hence a better suited one, may be freely chosen.
A similar approach is used in the celebrated FGLM algorithm (cf. [@FGLM93]).
#### Computing Central Characters and Algebraic Dependence.
Let $A$ be an associative ${{\mathbb K}}$-algebra. Intersection of a left ideal with the center of $A$, which is isomorphic to a commutative ring, is important for many algorithms, among other for the computation of central character decomposition of a finitely presented module (cf. [@Lev05] for the theory and [@ALM09] for an example with Principal Intersection). In the situation, where the center of $A$ is generated by one element (which is not seldom), we can apply Algorithm \[PrincipalIntersect\] to compute the intersection (known to be often quite nontrivial) without engaging much more expensive Gröbner basis computation, which use elimination.
Consider the quantum algebra $U'_q(\mathfrak{so}_3)$ (as defined by Fairlie and Odesskii) for $q^2$ being the $n$-th root of unity. It is known, that then, in addition to the single generator $C$ of the center present over any field, three new elements $Z_i$, depending on $n$ will appear. Since $U'_q(\mathfrak{so}_3)$ has Gel’fand-Kirillov dimension $3$, four commuting elements in it obey a single polynomial algebraic dependency (the ideal of dependencies in principal). Computing such a dependency is a very tough challenge for Gröbner bases. But as we see, it is quite natural to apply Principal Intersection.
LIB "ncalg.lib"; LIB "bfun.lib";
def A = makeQso3(5); // below Q^2 is the 5th root of unity
setring A;
// central elements, depending on Q in their classical form:
ideal I = x5+(Q3-Q2+2)*x3+(Q3-Q2+1)*x, y5+(Q3-Q2+2)*y3+(Q3-Q2+1)*y,
z5+(Q3-Q2+2)*z3+(Q3-Q2+1)*z;
I = twostd(I); // two-sided Groebner basis
poly C = 5*xyz+(4Q3-3Q2+2Q-1)*x2+(-Q3+2Q2-3Q+4)*y2+(4Q3-3Q2+2Q-1)*z2;
poly v = vec2poly(pIntersect(C,I),1); // present vector as poly
poly t = subst(v,x,C); t; // t as a polynomial in C of size 42
==> 3125*x5y5z5+(3125Q3-3125Q2+6250)*x5y5z3+(3125Q3-3125Q2+6250)*x5y3z5+ ...
// present matrix of cofactors of t as an element of I:
matrix T = lift(I,t);
poly a = 125*(25*I[1]*I[2]*I[3]+(Q3-7Q2+8Q-4)*(I[1]^2+I[2]^2+I[3]^2));
a-t; // a expresses t in the subalgebra gen. by I[1..3]
==> 0
// define univariate ring over algebraic extension:
ring r = (0,Q),c,dp; minpoly = Q4-Q3+Q2-Q+1;
poly v = fetch(A,v); // map v from A to a univariate poly in c
factorize(v);
The latter factorization delivers the final touch to the answer: the algebraic dependency is described by the equation $C^{2} \cdot (C+4q^{3}-3q^{2}-3q+4)\cdot (C+3q^{3}-q^{2}-q+3)^{2} = 3125\cdot Z_1 Z_2 Z_3+125\cdot(q^3-7q^2+8q-4)\cdot (Z_1^2 + Z_2^2 + Z_3^2)$.
Bernstein-Sato polynomial of $f$ {#bspolyA}
================================
Global Bernstein-Sato polynomial {#globalBS}
--------------------------------
One possibility to define the Bernstein-Sato polynomial of a polynomial $f \in {{\mathbb K}}[x_1,\ldots,x_n]$ is to apply the global $b$-function for specific weights.
Let $b_{I_f,w}(s)$ denote the global $b$-function of the univariate Malgrange ideal $I_f$ of $f$ (cf. Section \[sAnn\]) with respect to the weight vector $w = (1,0,\ldots,0) \in {{\mathbb R}}^{n+1}$, that is the weight of ${\partial }_t$ is $1$. Then $b_f(s) = (-1)^{\deg(b_{I_f,w})} b_{I_f,w}(-s-1)$ is called the *global $b$-function* (*Bernstein-Sato polynomial*) of $f$.
By Theorem \[b(s) is not zero\], $b_f(s) \neq 0$ holds. Moreover, it is known that all roots of $b_f(s)$ are negative rational numbers. Kashiwara proved this result for local Bernstein-Sato polynomials over ${{\mathbb C}}$ [@Kashiwara76/77]. This fact together with Theorem \[localBS\] below and classical flatness properties imply the claim for the global case over an arbitrary field of characteristic $0$.
The following theorem gives us another option to define the Bernstein-Sato polynomial.
\[bfct and ann\] The Bernstein-Sato polynomial $b_f(s)$ of $f$ is the unique monic polynomial of minimal degree in ${{\mathbb K}}[s]$ satisfying the identity $$P \bullet f^{s+1} = b_f(s) \cdot f^s \qquad \text{for some operator } P \in D_n[s].$$
Since $P \cdot f - b_f(s) \in \operatorname{Ann}_{D_n[s]}(f^s)$ holds, $b_f(s)$ is the monic polynomial satisfying $$\label{bernstein2}
\langle b_f(s) \rangle = \operatorname{Ann}_{D_n[s]}(f^s) + \langle f \rangle \cap {{\mathbb K}}[s].$$
Summarizing, there are several choices for computing the Bernstein-Sato polynomial:
1. Compute either\
(a) $J = \operatorname{in}_{(-w,w)}(I_f)$ or\
(b) $J = \operatorname{Ann}_{D_n[s]}(f^s) + \langle f \rangle$.
2. Intersect $J$ with ${{\mathbb K}}[s]$ by\
(a) the classical elimination-driven approach or\
(b) using Algorithm \[PrincipalIntersect\].
It is very interesting to investigate the approach for the computation of [Bernstein-Sato]{}polynomial, which arises as the combination of the two methods:
1. $\operatorname{Ann}_{D_n[s]}(f^s)$ via Briançon-Maisonobe (cf. [@LM08]),
2. $(\operatorname{Ann}(_{D_n[s]}f^s) + \langle f \rangle) \cap {{\mathbb K}}[s]$ via Algorithm \[PrincipalIntersect\].
For an efficient computation of $\operatorname{in}_{(-w,w)}(I_f)$ using the method of weighted homogenization as described in Section \[initial ideal\], Noro proposes [@Noro02] to choose the weights $\hat{u} = (\deg_u(f), u_1, \ldots, u_n )$, $\hat{v} = (1, \deg_u(f) - u_1 + 1, \ldots, \deg_u(f) - u_n + 1)$, such that the weight of $t$ is $\deg_u(f)$ and the weight of ${\partial }_t$ is $1$. Here, $u \in {{\mathbb R}}^n_{>0}$ is an arbitrary vector and $\deg_u(f)$ denotes the weighted total degree of $f$ with respect to $u$. The vector $u$ may be chosen heuristically in accordance to the shape of $f$ or by default, one can set $u = (1,\ldots,1)$.
Implementation
--------------
For the computation of Bernstein-Sato polynomials, we offer the following procedures in the <span style="font-variant:small-caps;">Singular</span> library `bfun.lib`:
`bfct` computes ${\operatorname{in}_{(-w,w)}}(I_f)$ using weighted homogenization with weights $\hat{u},\hat{v}$ for an optional weight vector $u$ (by default $u = (1,\ldots,1))$ as described above, and then uses Algorithm \[PrincipalIntersect\], where the occurring systems of linear equations are solved by the procedure `linReduce`.
`bfctAnn` computes $\operatorname{Ann}_{D_n[s]}(f^s)$ via Briançon-Maisonobe and intersects $\operatorname{Ann}_{D_n[s]}(f^s) + \langle f \rangle$ with ${{\mathbb K}}[s]$ analogously to `bfct`.
`bfctOneGB` computes the initial ideal and the intersection at once using a homogenized elimination ordering, a similar approach has been used in [@HH01].
For the global $b$-function of an ideal $I\subset D_n$, `bfctIdeal` computes ${\operatorname{in}_{(-w,w)}}(I)$ using standard homogenization, [i. e.]{}weighted homogenization where all weights are equal to $1$, and then proceeds the same way as `bfct`. Recall that $D_n/I$ must be holonomic as in [@SST00].
All these procedures work as the following example illustrates for `bfct` and the hyperplane arrangement $xyz(z-y)(y+z)$.
LIB "bfun.lib";
ring r = 0,(x,y,z),dp; // commutative ring
poly f = x*y*z*(z-y)*(y+z);
list L = bfct(f);
print(matrix(L[1])); // the roots of the BS-polynomial
==> -1,-5/4,-3/4,-3/2,-1/2
L[2]; // the multiplicities of the roots above
==> 3,1,1,1,1
Local [Bernstein-Sato]{}Polynomial
----------------------------------
Here we are interested in what kind of information one can obtain from the local $b$-functions for computing the global one and conversely. In order to avoid theoretical problems we will assume in this paragraph that the ground field ${{\mathbb K}}={{\mathbb C}}$.
Several algorithms to obtain the local $b$-function of a hypersurface $f$ have been known without any Gröbner bases computation but under some conditions on $f$. For instance, it was shown by Malgrange [@Malgrange75] that the minimal polynomial of $-{\partial }_t t$ acting on some vector space of finite dimension coincides with the reduced (local) Bernstein polynomial, assuming that the singularity is isolated.
The algorithms of Oaku [@Oaku97] used Gröbner bases for the first time. Recently, Nakayama presented some algorithms, which use the global $b$-function as a bound and obtain a local $b$-function by Mora resp. approximate division [@Nakayama09], see also the work of Nishiyama and Noro [@NN10].
\[localBS\] Let $b_{f,P}(s)$ be the local $b$-function of $f$ at the point $P\in{{\mathbb C}}^n$ and $b_f(s)$ the global one. Then it is verified that $b_f(s) = \operatorname{lcm}_{P\in{{\mathbb C}}^n} b_{f,P}(s) = \operatorname{lcm}_{P\in\Sigma(f)} b_{f,P}(s)$, where $\Sigma(f) = V(\langle f,\frac{{\partial }f}{{\partial }x_1}, \ldots, \frac{{\partial }f}{{\partial }x_n} \rangle)$ denotes the singular locus of $V(f)$.
Assume, that $\Sigma(f)$ consists of finitely many isolated singular points (the dimension of the corresponding defining ideal is $0$). Then the computation of the global $b$-function with Theorem \[localBS\] becomes effective. Moreover, one needs just an algorithm for computing the local $b$-function of a hypersurface, having an isolated singularity at the origin.
The [Singular]{} library [gmssing.lib]{}, developed and implemented by M. Schulze [@Schulze04], contains the procedure [bernstein]{}, which computes the local $b$-function at the origin. It returns the list of roots and corresponding multiplicities.
Let $\mathcal{C}$ be the curve in ${{\mathbb C}}^2$ given by $f = (x^3-y^2)(3x-2y-1)(x+2y)$. This curve has three isolated singular points $p_1 = (0,0)$, $p_2 = (1,1)$ and $p_3 = (1/4,-1/8)$.
LIB "gmssing.lib";
// note that one must use a local ordering for calling 'bernstein'
ring r = 0,(x,y),ds; // ds stands for a local degrevlex ordering
poly f = (x^3-y^2)*(3x-2y-1)*(x+2y);
list L = bernstein(f); // local b-function at the origin p_1
print(matrix(L[1]));
==> -11/8,-9/8,-1,-7/8,-5/8
L[2];
==> 1,1,2,1,1
Moving to the corresponding points we also compute $b_{f,P_2}(s)$ and $b_{f,P_3}(s)$. $$\begin{aligned}
b_{f,P_1}(s) &= (s+1)^2 (s+5/8) (s+7/8) (s+9/8) (s+11/8)\\
b_{f,P_2}(s) &= (s+1)^2 (s+3/4) (s+5/4)\\
b_{f,P_3}(s) &= (s+1)^2 (s+2/3) (s+4/3)\end{aligned}$$ From this information and using Theorem \[localBS\], the global $b$-function is $$b_f(s)=(s+2/3) (s+5/8) (s+3/4) (s+7/8) (s+1)^2 (s+4/3) (s+5/4) (s+9/8) (s+11/8).$$
Moreover, [gmssing.lib]{}, allows one to compute invariants related to the Gauss-Manin system of an isolated hypersurface singularity.
In the non-isolated case the situation is more complicated. For computing the local $b$-function in this case (which is important on its own) we suggest using two methods: Take the global $b$-function as an upper bound and a local version of the [checkRoot]{} algorithm, see below. Another method is to use a local version of `principalIntersect`, which is under development. Despite the existence of many algorithms, the effectiveness of the computation of local $b$-functions is still to be drastically enhanced.
Partial knowledge of [Bernstein-Sato]{}polynomial {#checkrootA}
-------------------------------------------------
As we have mentioned, several algorithms for computing the $b$-function associated with a polynomial have been known. However, in general it is very hard from computational point of view to obtain this polynomial, and in the actual computation a limited number of examples can be treated. For some applications only the integral roots of $b_f(s)$ are needed and that is why we are interested in obtaining just a part of the [Bernstein-Sato]{}polynomial.
Recall the algorithm [checkRoot]{} for checking whether a rational number is a root of the $b$-function of a hypersurface from [@LM08]. Equation was used to prove the following result.
([@LM08])\[checkRoot\] Let $R$ be a ring whose center contains ${{\mathbb K}}[s]$ as a subring. Let us consider $q(s)\in {{\mathbb K}}[s]$ a polynomial in one variable and $I$ a left ideal in $R$ satisfying $I\cap {{\mathbb K}}[s]\neq 0$. Then $
( I + R \langle q(s) \rangle )\cap {{\mathbb K}}[s] = I\cap {{\mathbb K}}[s] + {{\mathbb K}}[s] \langle q(s) \rangle.
$ In particular, using the above equation , we have $$\big(\, \operatorname{Ann}_{D_n[s]}(f^s)+ D_n[s]\cdot \langle f, q(s)\rangle \,\big) \; \cap {{\mathbb K}}[s] = \langle b_f(s),q(s)\rangle.$$ As a consequence, let $m_\alpha$ be the multiplicity of $\alpha$ as a root of $b_f(-s)$ and let us consider the ideals $J_i = \operatorname{Ann}_{D_n[s]}(f^s) + \langle f, (s+\alpha)^{i+1} \rangle \subseteq D_n[s]$, $i=0,\ldots,n$, then $[\, m_\alpha > i \Longleftrightarrow (s+\alpha)^i \notin J_i \,]$.
Once we know a system of generators of the annihilator of $f^s$ in $D_n[s]$, the last theorem provides an algorithm for checking whether a given rational number is a root of the $b$-function of $f$ and for computing its multiplicity, using Gröbner bases for differential operators.
This algorithm is much faster, than the computation of the whole Bernstein polynomial via Gröbner bases, because no elimination ordering is needed for computing a Gröbner basis of $J_i$, once one knows a system of generators of $\operatorname{Ann}_{D_n[s]}(f^s)$. Also, the element $(s+\alpha)^{i+1}$, added as a generator, seems to simplify tremendously such a computation. Actually, when $i=0$ it is possible to eliminate the variable $s$ in advance and we can perform the whole computation in $D_n$. Let us see an example.
Let $A$ be the matrix given by $$A = \left(\begin{array}{cccc}
x_1 & x_2 & x_3 & x_4\\
x_5 & x_6 & x_7 & x_8\\
x_9 & x_{10} & x_{11} & x_{12}
\end{array}\right).$$ Let us denote by $\Delta_i$, $i=1,2,3,4$, the determinant of the minor resulting from deleting the $i$-th column of A, and consider $f= \Delta_1 \Delta_2 \Delta_3 \Delta_4$. The polynomial $f$ defines a non-isolated hypersurface in ${{\mathbb C}}^{12}$. Therefore, from [@Saito94] (see also [@Varcenko81]), the set of all possible integral roots of $b_f(-s)$ is $\{11,10,9,8,7,6,5,4,3,2,1\}$. It is known that $\operatorname{Ann}_{D_n[s]}(f^s) = \operatorname{Ann}^{(1)}_{D_n[s]}(f^s)$ (see Section \[logannA\]) and this fact can be used to simplify the computation of the annihilator.
LIB "dmod.lib";
ring R = 0,(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12),dp;
matrix A[3][4] = x1,x2,x3,x4, x5,x6,x7,x8, x9,x10,x11,x12;
poly Delta1 = det(submat(A,1..3,intvec(2,3,4)));
... // analogous for Delta2 ... Delta4
poly f = Delta1*Delta2*Delta3*Delta4;
def D = Sannfslog(f); setring D; // logarithmic annihilator
poly f = imap(R,f); number alpha = 11;
checkRoot1(LD1,f,alpha);
==> 0
Using the algorithm [checkRoot]{} we have proved that the minimal integral root of $b_f(s)$ is $-1$. This example was suggested by F. Castro-Jiménez and J. M. Ucha for testing the Logarithmic Comparison Theorem. A nice introduction to this topic can be found, for instance, in [@Torrelli07].
Let $g$ be the polynomial resulting from $f$ by substituting $x_1, x_2, x_3, x_4, x_5, x_9$ with $1$. One can show that $b_g(s)$ divides $b_f(s)$ (see [@LM10] for details). Using the [checkRoot]{} algorithm we have found that $
(s+1)^4 (s+1/2) (s+3/2) (s+3/4) (s+5/4)
$ is a factor of $b_g(s)$ and therefore a factor of $b_f(s)$.
Using the notation from Section \[bfctIdeal\], given a holonomic $D$-module $D/I$, it is verified that $(\operatorname{in}_{(-w,w)}(I) + \langle q(s) \rangle )\cap {{\mathbb K}}[s] = \langle b_{I,w}(s),q(s)\rangle$, although Theorem \[checkRoot\] cannot be applied, since $s=\sum w_i x_i {\partial }_i$ does not commute with all operators. For some applications like integration and restriction the maximal and the minimal integral root of the $b$-function of $I$ with respect to some weight vector have to be computed, see [@SST00]. However, the above formula cannot be used to find the set of all integral roots, since no upper/lower bound exists in advance. For instance, as it was suggested by N. Takayama, $I=\langle t{\partial }_t + k\rangle$, $k\in{{\mathbb Z}}$ is $D_1$-holonomic and one has $\operatorname{in}_{(-1,1)}(I)\cap {{\mathbb C}}[s] = \langle s+k\rangle$ with $s=t{\partial }_t$.
We close this section by mentioning that there exist some well-known methods to obtain an upper bound for the Bernstein-Sato polynomial of a hypersurface singularity once we know, for instance, an embedded resolution of such singularity [@Kashiwara76/77]. Therefore using this result by Kashiwara and the [checkRoot]{} algorithm, it is possible to compute the whole Bernstein-Sato polynomial without elimination orderings, see Example 1 in [@LM08]. We investigate different methods in conjunction with the further development of the `checkRoot` family of algorithms in [@LM10].
Bernstein operator of $f$ {#Boperator}
=========================
We define the Bernstein-Sato polynomial $b_f(s)$ to be the monic generator of a principal ideal, hence it is unique. But the so-called $B$-operator $P(s) \in D_n[s]$ from Theorem \[bfct and ann\] is not unique.
Let $G$ be a left Gröbner basis of $\operatorname{Ann}_{D_n[s]}(f^{s+1})$ and define a *Bernstein operator* to be the result of the reduced normal form $\operatorname{NF}(P(s), G)$ of some $B$-operator $P(s)$. Then, for a fixed monomial ordering on $D_n[s]$, the Bernstein operator is uniquely determined.
Suppose that there is another $Q(s) \in D_n[s]$, such that the identities $$P(s)f^{s+1} = b_f(s)f^s \text{ and } Q(s)f^{s+1} = b_f(s)f^s \text{ hold in the module } D_n[s]/\operatorname{Ann}_{D_n[s]}(f^s).$$ Then $(P(s)-Q(s))f^{s+1} = 0$, that is $P(s)-Q(s) \in \operatorname{Ann}_{D_n[s]} (f^{s+1})$. Hence, the set $\{ R(s) \in D_n[s] \mid R(s)f^{s+1} = b_f(s)f^s \}$ can be viewed as an equivalence class and we can take the reduced normal form of any such operator (with respect to $G$) to be the canonical representative of the class. Since the reduced normal form with respect to a fixed monomial ordering on $D_n[s]$ is unique, so is the Bernstein operator $\operatorname{NF}(R(s), G)$.
Note, that we can obtain the left Gröbner basis of $\operatorname{Ann}_{D_n[s]} (f^{s+1})$ via substituting $s$ with $s+1$ in the left Gröbner basis of $\operatorname{Ann}_{D_n[s]} (f^{s})$.
One can compute the Bernstein operator from the knowledge of $\operatorname{Ann}_{D_n[s]} (f^{s})$ and $b_f(s)$ by the following methods.
$B$-operator via lifting
------------------------
The algorithm <span style="font-variant:small-caps;">Lift</span>$(F,G)$ computes the transformation matrix, expressing the set of polynomials $G$ via the set $F$, provided $\langle G \rangle \subseteq \langle F \rangle$. It is a classical application of Gröbner bases.
Suppose, that $\operatorname{Ann}_{D_n[s]} (f^s)$ is generated by $h_1,\ldots,h_m$ and $b_f(s)$ is known. The output of <span style="font-variant:small-caps;">Lift</span>$(\{f,h_1,\ldots,h_m\}, \{b_f(s)\})$ is the $1\times (m+1)$ matrix $(a,b_1,\ldots,b_m)$. Then a $B$-operator is computed as $P(s) = \operatorname{NF}(a, \operatorname{Ann}_{D_n[s]} (f^s))$.
Because of Equation , if $(a,b_1,\ldots,b_m)$ is the output of <span style="font-variant:small-caps;">Lift</span> as in the statement, $$b_f(s) = af + \sum_{i=1}^m b_i h_i \text{ holds},$$ hence the first element of such a matrix is a $B$-operator. Thus, the Bernstein operator is obtained via $\operatorname{NF}(a, \operatorname{Ann}_{D_n[s]} (f^{s+1}))$.
However, we have to mention, that the <span style="font-variant:small-caps;">Lift</span> procedure is quite expensive in general.
Note that another method for the computation of a $B$-operator using lifting techniques is given by applying Algorithm 8 of [@NN10] with $a(x)=1$.
$B$-operator via kernel of module homomorphism
----------------------------------------------
1\. Consider the $D_n[s]$-module homomorphism $$\varphi: D_n[s] {\longrightarrow}D_n[s]/ (\operatorname{Ann}_{D_n[s]} (f^s) + \langle b_f(s) \rangle), \quad 1 \mapsto f,$$ then for $u\in\ker \varphi$, $uf \in \operatorname{Ann}_{D_n[s]} (f^s) + \langle b_f(s) \rangle$. That is, there exist $a, b_i \in D_n[s]$, such that $$uf = a b_f(s) + \sum_{i=1}^m b_i h_i,$$ However, we are interested in such $u$, that $a\in{{\mathbb K}}$. This is possible, but the 2nd method above proposes a more elegant solution. Also one has to say, that in this case we have to compute a Gröbner basis of $\operatorname{Ann}_{D_n[s]} (f^s) + \langle b_f(s) \rangle$ as an intermediate step and also the kernel of a module homomorphism with respect to the latter. This combination is, in general, quite nontrivial to compute. In the Gröbner basis computation a monomial ordering, preferring $x,{\partial }x$ over $s$ seems to be better because of numerous applications of the Product Criterion.
2\. Consider the $D_n[s]$-module homomorphism $$\vartheta: D_n[s]^2 {\longrightarrow}D_n[s]/ \operatorname{Ann}_{D_n[s]} (f^s), \quad \epsilon_1 \mapsto b_f(s), \epsilon_2 \mapsto f.$$ Then $\operatorname{ker}\vartheta = \{ (u,v)^T \in D_n[s]^2 \mid u b_f(s) + v f \in \operatorname{Ann}_{D_n[s]} (f^s) \}$ is a submodule of $D_n[s]^2$. Indeed, $\ker \vartheta$ has many generators. In order to get a vector of the form $(k,u(s))$ for $k\in{{\mathbb K}}$, we perform another Gröbner basis computation for a submodule with respect to a module monomial ordering, giving preference to the first component over the second one. Since in the reduced basis there is a single element of the form $(k,v(s))\subset \operatorname{ker}\vartheta$ with $k\not=0$, it follows that $P(s) = v(s) k^{-1}$.
This algorithm is implemented in `dmod.lib` as `operatorModulo`. The approach via lifting is used in the procedure `operatorBM`, which computes all the Bernstein data. The procedures can be used as follows.
Consider the Reiffen curve $f = x^2 + y^3 + x y^2 \in {{\mathbb K}}[x,y]$. At first we use `operatorBM` and compare the length of an $B$-operator computed via lift to the length of the Bernstein operator.
LIB "dmod.lib";
ring r = 0,(x,y),dp;
poly F = x^2 + y^3 + x*y^2;
def A = operatorBM(F); setring A;
size(PS); // size of B-operator
==> 238
ideal LD2 = subst(LD,s,s+1); // Ann(F^{s+1})
LD2 = groebner(LD2); // LD is not a Groebner basis
poly PS2 = NF(PS,LD2); size(PS2); // size of Bernstein operator
==> 41
So, computing with lifting potentially computes much longer operators. Let us compare with `operatorModulo`.
poly PS3 = operatorModulo(F,LD,bs); size(PS3);
==> 41
The size of the operator, returned by `operatorModulo` need not be minimal ([e. g.]{}by disabling some of the interactive options of <span style="font-variant:small-caps;">Singular</span> one can get a polynomial of length 50 in this example), but it is in general much shorter, than the one, delivered by `operatorBM`. Let us check the main property of the $B$-operator and print its highest terms:
NF(PS3*F - bs, subst(LD2,s,s-1));
==> 0
108*PS2; // i.e. the Bernstein operator
==> 6*x*Dx^2*Dy+9*y*Dx^2*Dy-2*x*Dx*Dy^2-4*y*Dx*Dy^2-y*Dy^3+ ...
In the last line we see the terms of highest degree with respect to ${\partial }x, {\partial }y$.
Gröbner free method
-------------------
As a consequence of Theorem \[bfct and ann\], one obtains that $P \cdot f - b_f(s) \in \operatorname{Ann}_{D_n[s]}(f^s)$ and $P, b_f(s) \notin \operatorname{Ann}_{D_n[s]}(f^s)$. If we fix an ordering such that $b_f(s) = \operatorname{NF}(b_f(s), \operatorname{Ann}_{D_n[s]}(f^s))$ holds, we may rewrite this relation to $b_f(s) = \operatorname{NF}(P \cdot f, \operatorname{Ann}_{D_n[s]}(f^s))$. Hence, we can compute $P$ by searching for a linear combination of monomials $m \in D_n[s]$ that satisfy this equality when multiplied with $f$ from the right side. Using the results from the beginning of this section, one only needs to consider monomials which span $D_n/\operatorname{Ann}_{D_n[s]}(f^{s+1})$ as ${{\mathbb K}}$-vector space. We get the following algorithm.
\[BOperator\]
$f \in {{\mathbb K}}[x_1,\ldots,x_n]$, the Bernstein-Sato polynomial $b_f(s)$ of $f$ $P \in D_n[s]$, the Bernstein operator of $f$ $d := 0$ $M_d := \{ m \in D_n[s] \mid m \text{ monomial}, \deg(m) \leq d, \operatorname{lm}(p) \nmid m \,\forall\, p \in \operatorname{Ann}_{D_n[s]}(f^{s+1}) \}$ $P := \sum\limits_{m \in M_d} a_m m - b_f(s)$ $d := d+1$
The search for the coefficients $a_m$ can be done using `linReduce` (cf. Algorithm \[PrincipalIntersect\]) as one is in fact looking for a linear dependency between the Bernstein-Sato polynomial and the elements $m \cdot f$ in the vector space $D_n[s] / \operatorname{Ann}_{D_n[s]}(f^s)$.
Note, that Algorithm \[BOperator\] can be extended to one searching for both $B$-operator and [Bernstein-Sato]{}polynomial simultaneously. We have to mention, that both algorithms of this kind are well suited for the search of operators and [Bernstein-Sato]{}polynomials in the case, when both of them are of relatively low total degree.
Computing integrals and zeta functions
--------------------------------------
Given a simplex $C\subset{{\mathbb K}}^n$ (for ${{\mathbb K}}={{\mathbb R}},{{\mathbb C}}$) and $f\in {{\mathbb K}}[x_1,\ldots,x_n]$, we can define $\zeta(s) := \int_C f(x)^s dx$. Since $P(s)\bullet f^{s+1} = b_f(s)f^s$, we obtain $$\zeta(s) = \int_C f(x)^s dx = \frac{1}{b_f(s)} \int_C P(s)\bullet f(x)^{s+1} dx$$ expanding the latter with [e. g.]{}the chain rule, we come to an in general inhomogeneous recurrence relation for $\zeta(s)$, which involves coefficients in ${{\mathbb K}}[s]$. Since $P(s)$ is globally defined (and is, of course, independent on $C$), one can obtain a generic formula for all integrals of this type.
Let $f = x^2-x \in {{\mathbb K}}[x]$. Then the Bernstein operator reads as $P(s) = (2x-1){\partial }_x - 4(s+1)$ and $b_f(s)=s+1$. Any simplex in ${{\mathbb K}}^1$ is the interval $[a,b]=:C$. $$\zeta(s) = \int_C f(x)^s dx = \frac{1}{s+1} \int_C ((2x-1){\partial }_x - 4(s+1))\bullet f(x)^{s+1} dx$$ $$= \frac{1}{s+1} \int_C (2x-1){\partial }_x \bullet f(x)^{s+1} dx - 4 \zeta(s+1)$$ By the chain rule, $\int_C (2x-1) ({\partial }_x \bullet f(x)^{s+1}) dx = (2x-1)f(x)^{s+1}\mid_C - 2\int_C f(x)^{s+1}) dx$, hence $$\zeta(s) = \frac{1}{s+1} \cdot (2x-1)f(x)^{s+1}\mid_C - \frac{2}{s+1}\zeta(s+1) - 4 \zeta(s+1),$$ and thus $$(4s+6) \zeta(s+1) + (s+1) \zeta(s) = (2b-1)(b^2-b)^{s+1} - (2a-1)(a^2-a)^{s+1}$$ The right hand side, say $R(s)$, satisfies the homogeneous recurrence $R(s+2) - (a^2-a+b^2-b) R(s+1) + (a^2-a)(b^2-b) R(s) = 0$ of order 2. Substituting the left hand side into it, we obtain a homogeneous recurrence with polynomial coefficients of order 3: $$\begin{gathered}
(a^2-a)(b^2-b)(s+1) \zeta(s)
- ( (s+2)(a^2-a+b^2-b) - (4s+6)(a^2-a)(b^2-b)) \zeta(s+1)\\
- ( (4s+10)(a^2-a+b^2-b) - (s+3) ) \zeta(s+2)
+ (4s+14) \zeta(s+3)
= 0.\end{gathered}$$
To guarantee the uniqueness of a solution to this equation, we need to specify 3 initial values, which can be easily done. However, such recurrences very seldom admit a closed form solution, thus most information about $\zeta(s)$ is contained in the recurrence itself.
Logarithmic annihilator of $f$ {#logannA}
==============================
Given a polynomial $f\in{{\mathbb K}}[x]={{\mathbb K}}[x_1,\ldots,x_n]$, consider the left ideal $\operatorname{Ann}^{(1)}_{D_n[s]}(f^s) \subseteq D_n[s]$ generated by those differential operators $P(s)\in D_n[s]$ of total order (in the partials) less than or equal to one, which annihilate $f^s$. This ideal is clearly contained in $\operatorname{Ann}_{D_n[s]}(f^s)$ and can be generated by elements of the form $$P(s) = a_0(x,s) + a_1(x,s) {\partial }_{x_1} + \cdots + a_n(x,s) {\partial }_{x_n} \in D_n[s],$$ where $(a_0,a_1,\ldots,a_n)\in \operatorname{syz}_{{{\mathbb K}}[x,s]}(f,s\frac{{\partial }f}{{\partial }x_1}, \cdots,s\frac{{\partial }f}{{\partial }x_n})$. Therefore, for each $f\in {{\mathbb K}}[x]$ one can compute, by using Gröbner bases in ${{\mathbb K}}[x,s]$, a system of generators of $\operatorname{Ann}^{(1)}_{D_n[s]}(f^s)$. The corresponding procedure in `dmod.lib` is called `Sannfslog`. Let us see the Reiffen curve $f=x^4+y^5+xy^4$ as an example with [Singular]{}.
LIB "dmod.lib";
ring R = 0,(x,y),dp;
poly f = x^4+y^5+x*y^4;
def A = Sannfslog(f); setring A; LD1;
==> LD1[1]=4*x^2*Dx+5*x*Dx*y+3*x*y*Dy-16*x*s+4*y^2*Dy-20*y*s
==> LD1[2]=16*x*Dx*y^2-125*x*Dx*y-4*x^2*Dy+4*Dx*y^3+5*x*y*Dy+12*y^3*Dy-100*y^2*Dy
-64*y^2*s+500*y*s
// now we compute the whole annihilator with Sannfs and compare
setring R; def B = Sannfs(f); setring B;
map F = A,x,Dx,y,Dy,s;
ideal LD1 = F(LD1);
LD1 = groebner(LD1);
simplify( NF(LD,LD1), 2);
==> _[1]=36*y^3*Dx^2-36*y^3*Dx*Dy+1125/4*x*y*Dx^2-315/4*x*y*Dx*Dy+ ...
And the latter polynomial is not an element of $\operatorname{Ann}^{(1)}_{D_n[s]}(f^s)$ but of $\operatorname{Ann}^{(2)}_{D_n[s]}(f^s) = \operatorname{Ann}_{D_n[s]}(f^s)$.
The annihilator up to degree $k$
--------------------------------
More generally, for a given $k\geq 1$ one can consider the left ideal $\operatorname{Ann}^{(k)}_{D_n[s]}(f^s) \subseteq D_n[s]$ generated by the differential operators $P(s)\in D_n[s]$ of total order less than or equal to $k$, such that $P(s)$ annihilate $f^s$. The tower of ideals $$\operatorname{Ann}^{(1)}_{D_n[s]} (f^s) \subsetneq \cdots
\subsetneq \operatorname{Ann}^{(k_0)}_{D_n[s]} (f^s) = \operatorname{Ann}_{D_n[s]} (f^s)$$ has been recently studied by Narváez in [@Narvaez08]. It is an open problem to find the minimal integer $k_0$ satisfying the above condition without computing the whole annihilator.
Computationally the annihilator up to degree $k$ can be obtained using Gröbner bases in ${{\mathbb K}}[x,s]$ as follows. Consider $P(s) = \sum_{|\beta|\leq k} a_\beta {\partial }^{\beta} \in\operatorname{Ann}^{(k)}_{D_n[s]}(f^s)$ and let $g_\beta(x,s) \in {{\mathbb K}}[x,s]$ be the polynomial defined by the formula ${\partial }^\beta\cdot f^s = g_\beta \cdot f^{s-|\beta|}$. Then $(a_\beta)_{|\beta|\leq k} \in \operatorname{syz}(g_\beta f^{k-|\beta|})_{|\beta|\leq k}$. Eventually, the polynomials $g_\beta(x,s)$ can be computed using the expression given in Lemma \[formulaFS\].
Given $\beta=(\beta_1,\ldots,\beta_n)\in{{\mathbb N}}^n$, as usual $|\beta| = \beta_1+\cdots+\beta_n$, $\beta! = \beta_1! \cdots \beta_n!$ and $\Delta_x^\beta = \frac{1}{\beta!} {\partial }_x^\beta$. A [*partition*]{} of $\beta$ is a way of writing $\beta$ as a sum of integral vectors with non-negative entries. Two sums which only differ in the order of their summands are considered to be the same partition. If $\beta = \sigma_1+\ldots+\sigma_k$ with $\sigma_i\neq 0$, then $\sigma$ is said to be a partition of [*length*]{} $k$. The set of all partitions of $\beta$ (resp. of length $k$) is denoted by $\mathcal{P}(\beta)$ (resp. $\mathcal{P}(\beta; k)$). Obviously $\mathcal{P}(\beta) = \cup_{k=1}^{|\beta|} \mathcal{P}(\beta; k)$. Finally, we write $\ell(\sigma) := (\ell_{\sigma\tau})_{\tau}$, where $\ell_{\sigma\tau}$ is the number of times that $\tau$ appears in $\sigma$.
\[formulaFS\] Using the above notation for all non-zero $\beta\in{{\mathbb N}}^n$ we have, $$\begin{split}
\Delta_x^{\beta} \cdot f^s = &
\sum_{k=1}^{|\beta|} \binom{s}{k} \sum_{\sigma\in\mathcal{P}(\beta; k)}
\frac{1}{\ell(\sigma)!}\ \Delta_x^{\sigma_1}(f)\cdots \Delta_x^{\sigma_k}(f)
\cdot f^{s-k}\,.
\end{split}$$
This formula was suggested by Narváez and can be proved by induction on $|\beta|$. Similar expressions appear in [@MN98 Prop. 5.3.5] and [@Mebkhout97 Prop. 2.3.2].
However, despite this almost closed form, the set of polynomials, between which we have to compute syzygies, is growing fast and the size of polynomials increases. This results in quite hard computations even with the mentioned enhancements.
Bernstein-Sato ideals for $f = f_1 \cdot \ldots \cdot f_m$ {#BMI}
==========================================================
Using the results from [@GHU05], which we confirmed through intensive testing (cf. [@LM08]), it follows, that the method by [Briançon-Maisonobe]{}is the most effective one for the computation of $s$-parametric annihilators where $f = f_1$. Because of the structure of annihilators in the situation $f = f_1 \cdots f_p$, $p>1$, basically the same principles stand behind the corresponding algorithms. Hence, we decided to implement only [Briançon-Maisonobe]{}’s method for the $s$-parametric annihilator $\operatorname{Ann}_{D_n[s]}(f^s) \subset D_n[s]$, where $s = (s_1,\ldots,s_p)$. The corresponding procedure in `dmod.lib` is called `annfsBMI`. It computes both $\operatorname{Ann}_{D_n[s]} f^s \subset D_n[s]$ and the Bernstein-Sato ideal in ${{\mathbb K}}[s]$, which is defined as $$\mathcal{B}(f) =
(\operatorname{Ann}_{D_n[s_1,\ldots,s_p]}(f_1^{s_1} \cdots f_p^{s_p}) + \langle f_1 \cdots f_p \rangle ) \cap {{\mathbb K}}[s_1,\ldots,s_p].$$
In contrary to the case $f=f_1$, in general the ideal $\mathcal{B}(f)$ need not be principal. However, it is an open question to give a criterion for the principality of $\mathcal{B}(f)$. Armed with such a criterion, one can apply a generalization of the method of Principal Intersection \[PrincipalIntersect\] to multivariate subalgebras [@ALM10] and thus replace expensive elimination above by the computation of a minimal polynomial. Otherwise we still can apply the Principal Intersection, which, however, will deliver only one polynomial to us. As in the case $f=f_1$ it is an open question, which strategy and which orderings should one use in the computation of the annihilator and of the [Bernstein-Sato]{}ideal in order to achieve better performance.
We reported in [@LM08] on several challenges, which have been solved with the help of our implementation. Namely, the products $(x^3+y^2)(x^2+y^3)$ and $(x^2+y^2+y^3)(x^3+y^2)$ give rise to principal [Bernstein-Sato]{}ideals.
Let us consider the following example from [@RT10], which is quite challenging to compute indeed.
LIB "dmod.lib"; ring r = 0,(x,y,z),dp;
ideal F = z, x^5 + y^5 + x^2*y^3*z;
def A = annfsBMI(F); setring A;
LD; // prints the annihilator in D[s1,s2]
BS; // prints the Bernstein-Sato ideal
We do not show the output here because of its size. But from the output one can see, the Bernstein-Sato ideal is of dimension 1 in $D_3[s_1,s_2]$ since its Gröbner basis consists of three elements $\{15625 s_1 s_2^8 + 17 \text{ l.o.t.}, 3125 s_1^2 s_2^7 + 24 \text{ l.o.t.}, 625 s_1^5 s_2^6 + 42 \text{ l.o.t.}\}$. Notably, every generator factorizes into linear factors and each factor involves either $s_1$ or $s_2$, which happens quite seldom in general.
In general, quite a little is known about [Bernstein-Sato]{}ideals. Their dimensions, principality, factorization of generators and primary decomposition constitute open problems from the theoretical side.
Bernstein-Sato polynomial for a variety {#bsvarA}
=======================================
Now we proceed to the construction of the Bernstein-Sato polynomial of an affine algebraic variety. We refer to [@BMS06] for the details of the complete construction for arbitrary varieties and to [@ALM09] for the details about annihilator-driven algorithms.
Given two positive integers $n$ and $r$, for the rest of this section we fix the indices $i,j,k,l$ ranging between $1$ and $r$ and an index $m$ ranging between $1$ and $n$.Let $f=(f_1,\ldots,f_r)$ be an $r$-tuple in ${{\mathbb K}}[x]^r$. Here, $s=(s_1,\ldots,s_r)$, $\frac{1}{f} = \frac{1}{f_1\cdots f_r}$ and $f^s = f_1^{s_1}\cdots f_r^{s_r}$. Let us denote by ${{\mathbb K}}\langle S \rangle$ the universal enveloping algebra $U(\mathfrak{gl}_{\,r})$, generated by the set of variables $S=(s_{ij})$, $i,j=1,\ldots,r$, with $s_{ii} = s_i$, subject to relations $[s_{ij}, s_{kl}] = \delta_{jk} s_{il} - \delta_{il} s_{kj}$. We denote by $D_n\langle S \rangle := D_n \otimes_{{{\mathbb K}}} {{\mathbb K}}\langle S \rangle$, which is a $G$-algebra of Lie type by [e. g.]{}[@LS03]. Then the free ${{\mathbb K}}[x,s,\frac{1}{f}]$-module of rank one generated by the formal symbol $f^s$ has a natural structure of a left $D_n \langle S \rangle$-module: $$s_{ij} \bullet (G(s)\cdot f^s) = s_i \cdot G(s+\epsilon_j-\epsilon_i)
\frac{f_j}{f_i} \cdot f^s \in {{\mathbb K}}[x,s,\frac{1}{f}]\cdot f^s,$$ where $G(s) \in {{\mathbb K}}[x,s,\frac{1}{f}]$ and $\epsilon_j$ stands for the $j$-th standard basis vector.
For every $r$-tuple $f=(f_1,\ldots,f_r)\in {{\mathbb K}}[x]^r$ there exists a non-zero polynomial in one variable $b({\sigma})\in {{\mathbb K}}[{\sigma}]$ and $r$ differential operators $P_1(S),\ldots,P_r(S)\in D_n\langle S \rangle$ such that $$\label{BSvariety}
\sum_{k=1}^r P_k(S) f_k \cdot f^s = b(s_1+\cdots+s_r) \cdot f^s \in {{\mathbb K}}[x,s,\frac{1}{f}]\cdot f^s.$$
The [*Bernstein-Sato polynomial*]{} $b_f({\sigma})$ of $f=(f_1,\ldots,f_r)$ is defined to be the monic polynomial of lowest degree in the variable ${\sigma}$ satisfying the equation . It turns out, that every root of the [Bernstein-Sato]{}polynomial is rational, as in the case of a hypersurface. Let $I$ be the ideal generated by $f_1,\ldots, f_r$ and $Z$ the affine algebraic variety associated with $I$ in ${{\mathbb K}}^n$. Then it can be verified, that $b_f({\sigma})$ is independent of the choice of a system of generators of $I$, and moreover that $b_Z({\sigma}) = b_f({\sigma}- \operatorname{codim}Z + 1)$ depends only on $Z$.
Now, let us denote by $\operatorname{Ann}_{D_n\langle S\rangle}(f^s)$ the left ideal of all elements $P(S) \in D_n\langle S \rangle$ such that $P(S)\bullet f^s = 0$. We call this ideal the [*annihilator*]{} of $f^s$ in $D_n \langle S \rangle$. From the definition of the [Bernstein-Sato]{}polynomial it becomes clear that $$(\operatorname{Ann}_{D_n \langle S \rangle} (f^s) + \langle f_1,\ldots,f_r \rangle)\cap
{{\mathbb K}}[s_1+\cdots+s_r] = \langle b_f(s_1+\ldots+s_r)\rangle.$$ Since the final intersection can be computed with Principal Intersection \[PrincipalIntersect\], the above formula provides an algorithm for computing the Bernstein-Sato polynomial of affine algebraic varieties, once we know a Gröbner basis of the annihilator of $f^s$ in $D_n \langle S \rangle$.
\[SannFsVar\] Let $f=(f_1,\ldots,f_r)$ be an $r$-tuple in ${{\mathbb K}}[x]^r$ and $D_n \langle \partial t, S \rangle$ the ${{\mathbb K}}$-algebra generated by $D_n$, $\partial t$ and $S$ with the non-commutative relations of $D_n\langle S \rangle$, described above and additional relations $[s_{ij},{\partial }{t_k}]= \delta_{jk} {\partial }{t_i}$ (${\partial }{t_i}$ commute mutually with the subalgebra $D_n$). Then the annihilator of $f^s$ in $D_n \langle S\rangle$ can be expressed as follows: $$\bigg[ D_n \langle \partial t, S \rangle
\Big( s_{ij} + {\partial }{t_i} f_j \,
,\ \partial_{m} + \sum_{k=1}^r
\frac{\partial f_k}{\partial x_m} \partial {t_k} \left|
\begin{array}{c} 1\leq i,j\leq r \\ 1\leq m\leq n \end{array}\right.
\Big)\bigg] \cap D_n \langle S\rangle.$$
Note, that this result and its proof [@ALM09] can be presented as natural generalization of the algorithm for computing $\operatorname{Ann}_{D_n[s]}(f^s)$ with the method of [Briançon-Maisonobe]{}(cf. Section \[sAnn\]).
As Budur et al. point out [@BMS06 p. 794], the Bernstein-Sato polynomial for varieties coincides, up to shift of variables, with the $b$-function in [@SST00 p. 194], if the weight vector is chosen appropriately, see also [@Shibuta08]. Algorithms for computing the $b$-function have been already discussed in Section \[bfctIdeal\], so the procedure `bfctIdeal` can be immediately applied to this situation. Hence, like for the case of a hypersurface, we have two essentially different ways to compute [Bernstein-Sato]{}polynomials for varieties. The comparison of these two methods is the subject of further research.
In the new <span style="font-variant:small-caps;">Singular</span> library `dmodvar.lib`[^1], we present the implementations of the following algorithms
- `SannfsVar`, which computes $\operatorname{Ann}_{D_n \langle S \rangle} (f^s)$ according to the Theorem \[SannFsVar\],
- `bfctVarIn`, which computes $b_f(s_1+\ldots+s_r)$ using initial ideal approach,
- `bfctVarAnn`, which computes $b_f(s_1+\ldots+s_r)$ using annihilator-driven approach.
Let $TX = V(x_0^2+y_0^3, 2x_0 x_1 + 3 y_0^2 y_1) \subset {{\mathbb C}}^4$ the tangent bundle of $X = V(x^2+y^3)\subset {{\mathbb C}}^2$. Then the Bernstein-Sato polynomial of $TX$ can be computed as follows:
LIB "dmodvar.lib";
ring R = 0,(x0,x1,y0,y1),Dp;
ideal F = x0^2+y0^3, 2*x0*x1+3*y0^2*y1;
bfctVarAnn(F); // annihilator-driven approach
// alternatiely, one can run
bfctVarIn(F); // approach via initial ideal
In both cases we obtain the polynomial $$b_{TX}({\sigma}) = ({\sigma}+1)^2 ({\sigma}+1/3)^2 ({\sigma}+2/3)^2 ({\sigma}+1/2) ({\sigma}+5/6) ({\sigma}+7/6).$$ The annihilator ideal can be computed via executing, in addition to the first 3 lines of the above code, the following code:
def S = SannfsVar(F); // returns a ring
setring S; // in this ring, ideal LD is the annihilator
option(redSB); LD = groebner(LD); // reduced GB of LD
There are 15 generators in the Gröbner basis of $\operatorname{Ann}F^s$:
$3y_{0}^{2}{\partial }x_{1}-2x_{0}{\partial }y_{1},\quad
3y_{0}^{2}{\partial }x_{0}+6y_{0}y_{1}{\partial }x_{1}-2x_{0}{\partial }y_{0}-2x_{1}{\partial }y_{1},\quad
x_{0}y_{0}{\partial }x_{1}{\partial }y_{0}-x_{0}y_{0}{\partial }x_{0}{\partial }y_{1}+x_{1}y_{0}{\partial }x_{1}{\partial }y_{1}-2x_{0}y_{1}{\partial }x_{1}{\partial }y_{1},\quad
3y_{0}y_{1}{\partial }x_{1}^{2}-x_{0}{\partial }x_{1}{\partial }y_{0}+x_{0}{\partial }x_{0}{\partial }y_{1}-x_{1}{\partial }x_{1}{\partial }y_{1},\quad
3x_{0}y_{0}y_{1}{\partial }x_{0}{\partial }x_{1}{\partial }y_{1}+6x_{0}y_{1}^{2}{\partial }x_{1}^{2}{\partial }y_{1}-x_{0}^{2}{\partial }x_{1}{\partial }y_{0}^{2}+x_{0}^{2}{\partial }x_{0}{\partial }y_{0}{\partial }y_{1}-2x_{0}x_{1}{\partial }x_{1}{\partial }y_{0}{\partial }y_{1}+x_{0}x_{1}{\partial }x_{0}{\partial }y_{1}^{2}-x_{1}^{2}{\partial }x_{1}{\partial }y_{1}^{2}+3x_{1}y_{0}{\partial }x_{1}^{2}+3x_{0}y_{1}{\partial }x_{1}^{2}-3y_{0}y_{1}{\partial }x_{1}{\partial }y_{1},\quad
6x_{0}y_{1}^{2}{\partial }x_{1}^{2}{\partial }y_{0}{\partial }y_{1}+3x_{0}y_{0}y_{1}{\partial }x_{0}^{2}{\partial }y_{1}^{2}-3x_{1}y_{0}y_{1}{\partial }x_{0}{\partial }x_{1}{\partial }y_{1}^{2}+6x_{0}y_{1}^{2}{\partial }x_{0}{\partial }x_{1}{\partial }y_{1}^{2}-x_{0}^{2}{\partial }x_{1}{\partial }y_{0}^{3}+x_{0}^{2}{\partial }x_{0}{\partial }y_{0}^{2}{\partial }y_{1}-2x_{0}x_{1}{\partial }x_{1}{\partial }y_{0}^{2}{\partial }y_{1}+x_{0}x_{1}{\partial }x_{0}{\partial }y_{0}{\partial }y_{1}^{2}-x_{1}^{2}{\partial }x_{1}{\partial }y_{0}{\partial }y_{1}^{2}+3x_{1}y_{0}{\partial }x_{1}^{2}{\partial }y_{0}+3x_{0}y_{1}{\partial }x_{1}^{2}{\partial }y_{0}+9x_{0}y_{1}{\partial }x_{0}{\partial }x_{1}{\partial }y_{1}-6y_{0}y_{1}{\partial }x_{1}{\partial }y_{0}{\partial }y_{1}+3y_{0}y_{1}{\partial }x_{0}{\partial }y_{1}^{2}+6y_{1}^{2}{\partial }x_{1}{\partial }y_{1}^{2}+3x_{1}{\partial }x_{1}^{2}+3y_{1}{\partial }x_{1}{\partial }y_{1},\quad
6x_{0}y_{1}^{2}{\partial }x_{1}^{3}{\partial }y_{1}-x_{0}^{2}{\partial }x_{1}^{2}{\partial }y_{0}^{2}+2x_{0}^{2}{\partial }x_{0}{\partial }x_{1}{\partial }y_{0}{\partial }y_{1}-2x_{0}x_{1}{\partial }x_{1}^{2}{\partial }y_{0}{\partial }y_{1}-x_{0}^{2}{\partial }x_{0}^{2}{\partial }y_{1}^{2}+2x_{0}x_{1}{\partial }x_{0}{\partial }x_{1}{\partial }y_{1}^{2}-x_{1}^{2}{\partial }x_{1}^{2}{\partial }y_{1}^{2}-3x_{0}y_{0}{\partial }x_{0}{\partial }x_{1}^{2}+3x_{1}y_{0}{\partial }x_{1}^{3}+3x_{0}y_{1}{\partial }x_{1}^{3}-2x_{0}{\partial }x_{1}{\partial }y_{0}{\partial }y_{1}+x_{0}{\partial }x_{0}{\partial }y_{1}^{2}-3x_{1}{\partial }x_{1}{\partial }y_{1}^{2}+6y_{0}{\partial }x_{1}^{2},\quad
s_{22}-x_{1}{\partial }x_{1}-y_{1}{\partial }y_{1},\quad
6s_{21}-3x_{0}{\partial }x_{1}-2y_{0}{\partial }y_{1},\quad
6s_{11}-3x_{0}{\partial }x_{0}+3x_{1}{\partial }x_{1}-2y_{0}{\partial }y_{0}+4y_{1}{\partial }y_{1},\quad
s_{12}x_{0}+3y_{0}y_{1}^{2}{\partial }x_{1}-x_{0}x_{1}{\partial }x_{0}+x_{1}^{2}{\partial }x_{1}-x_{0}y_{1}{\partial }y_{0},\quad
s_{12}y_{0}{\partial }y_{1}-x_{1}y_{0}{\partial }x_{1}{\partial }y_{0}+2x_{1}y_{1}{\partial }x_{1}{\partial }y_{1}-y_{0}y_{1}{\partial }y_{0}{\partial }y_{1}+2y_{1}^{2}{\partial }y_{1}^{2}-y_{0}{\partial }y_{0}+4y_{1}{\partial }y_{1},\quad
3s_{12} y_0^2+6 x_1 y_0 y_1 {\partial }x_1-3 y_0^2 y_1 {\partial }y_0+6 y_0 y_1^2 {\partial }y_1-2 x_0 x_1 {\partial }y_0,\quad
s_{12}x_{1}{\partial }y_{1}^{2}-3s_{12}y_{0}{\partial }x_{1}+3x_{1}y_{0}y_{1}{\partial }x_{0}{\partial }x_{1}{\partial }y_{1}+6x_{1}y_{1}^{2}{\partial }x_{1}^{2}{\partial }y_{1}-3y_{0}y_{1}^{2}{\partial }x_{1}{\partial }y_{0}{\partial }y_{1}+3y_{0}y_{1}^{2}{\partial }x_{0}{\partial }y_{1}^{2}+6y_{1}^{3}{\partial }x_{1}{\partial }y_{1}^{2}-x_{0}x_{1}{\partial }x_{1}{\partial }y_{0}^{2}+x_{0}x_{1}{\partial }x_{0}{\partial }y_{0}{\partial }y_{1}-2x_{1}^{2}{\partial }x_{1}{\partial }y_{0}{\partial }y_{1}-x_{1}y_{1}{\partial }y_{0}{\partial }y_{1}^{2}+3x_{1}y_{0}{\partial }x_{0}{\partial }x_{1}+3x_{1}y_{1}{\partial }x_{1}^{2}-6y_{0}y_{1}{\partial }x_{1}{\partial }y_{0}+9y_{0}y_{1}{\partial }x_{0}{\partial }y_{1}+18y_{1}^{2}{\partial }x_{1}{\partial }y_{1}-2x_{1}{\partial }y_{0}{\partial }y_{1}+3y_{0}{\partial }x_{0},\quad
3s_{12}y_{0}y_{1}{\partial }x_{1}-s_{12}x_{1}{\partial }y_{1}-3x_{1}y_{0}y_{1}{\partial }x_{0}{\partial }x_{1}-3y_{0}y_{1}^{2}{\partial }x_{0}{\partial }y_{1}+x_{1}^{2}{\partial }x_{1}{\partial }y_{0}+x_{1}y_{1}{\partial }y_{0}{\partial }y_{1}-3y_{0}y_{1}{\partial }x_{0}+x_{1}{\partial }y_{0}$.
This ideal belongs to the ${{\mathbb K}}$-algebra $D_4\langle S \rangle$ in 12 variables, as defined in the beginning of this section. By executing
GKdim(LD);
we obtain, that the Gel’fand-Kirillov dimension of $D_4\langle S \rangle / \operatorname{Ann}F^s$ is 6, the half of the Gel’fand-Kirillov dimension of $D_4\langle S \rangle$. However, $D_4\langle S \rangle / \operatorname{Ann}F^s$ is not a generalized holonomic $D_4\langle S \rangle$-module, since the annihilator of this module contains a central element $s_{12} s_{21}-s_{11} s_{22}-s_{11}$ and hence is not zero.
[10]{}
D. Andres, V. Levandovskyy, and J. Martín-Morales. Principal intersection and [B]{}ernstein-[S]{}ato polynomial of an affine variety. In J. P. May, editor, [*Proc. of the International Symposium on Symbolic and Algebraic Computation (ISSAC’09)*]{}, pages 231–238. ACM Press, 2009.
D. Andres, V. Levandovskyy, and J. Martín-Morales. Effective methods for the computation of [B]{}ernstein-[S]{}ato polynomials for hypersurfaces and affine varieties. <http://arxiv.org/abs/1002.3644>, 2010.
R. Bahloul and T. Oaku. Local [B]{}ernstein-[S]{}ato ideals: algorithm and examples. , 45(1):46–59, 2010.
I. N. Bernstein. Modules over a ring of differential operators. [A]{}n investigation of the fundamental solutions of equations with constant coefficients. , 5(2):89–101, 1971.
J. Briançon and P. Maisonobe. Remarques sur l’idéal de [B]{}ernstein associé à des polynômes. , 2002.
M. Brickenstein. . In [*Rhine Workshop on Computer Algebra*]{}, pages 55–66, 2006. Proceedings of RWCA’06, Basel, March 2006.
N. Budur, M. Musta[ţ]{}[ǎ]{}, and M. Saito. Bernstein-[S]{}ato polynomials of arbitrary varieties. , 142(3):779–797, 2006.
J. Bueso, J. G[ó]{}mez-Torrecillas, and A. Verschoren. . Kluwer Academic Publishers, 2003.
F. Castro-Jim[é]{}nez and L. Narv[á]{}ez-Macarro. Homogenising differential operators. , 1997.
S. Coutinho. Cambridge Univ. Press., 1995.
W. Decker, G.-M. Greuel, G. Pfister, and H. Sch[ö]{}nemann. . <http://www.singular.uni-kl.de>, 2010.
J. C. Faugere, P. Gianni, D. Lazard, and T. Mora. Efficient computation of zero-dimensional [G]{}r[ö]{}bner bases by change of ordering. , 16(4):329–344, 1993.
J. Gago-Vargas, M. Hartillo-Hermoso, and J. Ucha-Enr[í]{}quez. Comparison of theoretical complexities of two methods for computing annihilating ideals of polynomials. , 40(3):1076–1086, 2005.
G.-M. Greuel, V. Levandovskyy, and H. Sch[ö]{}nemann. . <http://www.singular.uni-kl.de>, 2006.
G.-M. Greuel and G. Pfister. . Springer, 2nd edition, 2008. With contributions by O. Bachmann, C. Lossen and H. Sch[ö]{}nemann.
M. I. Hartillo-Hermoso. About an algorithm of [T]{}. [O]{}aku. In [*Ring theory and algebraic geometry ([L]{}eón, 1999)*]{}, volume 221 of [*Lecture Notes in Pure and Appl. Math.*]{}, pages 241–250. Dekker, New York, 2001.
A. Kandri-Rody and V. Weispfenning. Non-commutative [G]{}r[ö]{}bner bases in algebras of solvable type. , 9(1):1–26, 1990.
M. Kashiwara. -functions and holonomic systems. [R]{}ationality of roots of [$B$]{}-functions. , 38(1):33–53, 1976/77.
V. Levandovskyy. On preimages of ideals in certain non–commutative algebras. In G. Pfister, S. Cojocaru, and V. Ufnarovski, editors, [ *Computational Commutative and Non-Commutative Algebraic Geometry*]{}. IOS Press, 2005.
V. Levandovskyy and J. Martín-Morales. Computational [$D$]{}-module theory with [<span style="font-variant:small-caps;">singular</span>]{}, comparison with other systems and two new algorithms. In [*Proc. of the International Symposium on Symbolic and Algebraic Computation (ISSAC’08)*]{}. ACM Press, 2008.
V. Levandovskyy and J. Martín-Morales. Algorithms for checking rational roots of $b$-functions and their applications. <http://arxiv.org/abs/1003.3478>, 2010.
V. Levandovskyy and H. Sch[ö]{}nemann. Plural — a computer algebra system for noncommutative polynomial algebras. In [*Proc. of the International Symposium on Symbolic and Algebraic Computation (ISSAC’03)*]{}, pages 176 – 183. ACM Press, 2003.
H. Li. . Springer, 2002.
B. Malgrange. Le polynôme de [B]{}ernstein d’une singularité isolée. In [*Fourier integral operators and partial differential equations (Colloq. Internat., Univ. Nice, Nice, 1974)*]{}, pages 98–119. Lecture Notes in Math., Vol. 459. Springer, Berlin, 1975.
Z. Mebkhout. Sur le théorème de finitude de la cohomologie [$p$]{}-adique d’une variété affine non singulière. , 119(5):1027–1081, 1997.
Z. Mebkhout and L. Narv[á]{}ez-Macarro. Le théorème de continuité de la division dans les anneaux d’opérateurs différentiels. , 503:193–236, 1998.
H. Nakayama. Algorithm computing the local $b$ function by an approximate division algorithm in [$\widehat{\mathcal{D}}$]{}. , 44(5):449–462, 2009. Spanish National Conference on Computer Algebra.
L. Narváez-Macarro. Linearity conditions on the [J]{}acobian ideal and logarithmic-meromorphic comparison for free divisors. In [*Singularities I, Algebraic and Analytic Aspects*]{}, pages 245–269. Contemporary Mathematics, 474, AMS, 2008.
K. Nishiyama and M. Noro. Stratification associated with local $b$-functions. , 45(4):462 – 480, 2010.
M. Noro. An efficient modular algorithm for computing the global [$b$]{}-function. In [*Mathematical software ([B]{}eijing, 2002)*]{}, pages 147–157. World Sci. Publ., River Edge, NJ, 2002.
M. Noro, T. Shimoyama, and T. Takeshima. <span style="font-variant:small-caps;">Risa/Asir</span>, an open source general computer algebra system. <http://www.math.kobe-u.ac.jp/Asir>, 2006.
T. Oaku. Algorithms for the [$b$]{}-function and [$D$]{}-modules associated with a polynomial. , 117/118:495–518, 1997.
M. Saito. On microlocal [$b$]{}-function. , 122(2):163–184, 1994.
M. Saito, B. Sturmfels, and N. Takayama. , volume 6 of [*Algorithms and Computation in Mathematics*]{}. Springer-Verlag, Berlin, 2000.
K. Schindelar, V. Levandovskyy, and E. Zerz. Exact linear modeling using [O]{}re algebras. , 2010. To appear.
M. Schulze. A normal form algorithm for the [Brieskorn]{} lattice. , 38(4):1207–1225, 2004.
T. Shibuta. An algorithm for computing multiplier ideals. <http://arxiv.org/abs/math/0807.4302>, 2008.
N. Takayama. <span style="font-variant:small-caps;">kan/sm1</span>, a [G]{}r[ö]{}bner engine for the ring of differential and difference operators. <http://www.math.kobe-u.ac.jp/KAN/index.html>, 2003.
T. Torrelli. Logarithmic comparison theorem and [D]{}-modules: an overview. In [*Singularity theory*]{}, pages 995–1009. World Sci. Publ., Hackensack, NJ, 2007.
H. Tsai and A. Leykin. -modules package for <span style="font-variant:small-caps;">Macaulay</span> 2 – algorithms for [D]{}–modules. <http://www.ima.umn.edu/~leykin/Dmodules>, 2006.
A. N. Varchenko. Asymptotic [H]{}odge structure on vanishing cohomology. , 45(3):540–591, 1981.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Francisco Castro-Jiménez, José-María Ucha, Gert-Martin Greuel, Enrique Artal, José-Ignacio Cogolludo and Luis Narváez for fruitful discussions and insightful remarks concerning our work.
Daniel Andres\
Lehrstuhl D für Mathematik, RWTH Aachen, Templergraben 64, 52062 Aachen, Germany\
e-mail: `Daniel.Andres@math.rwth-aachen.de`
Michael Brickenstein\
Mathematisches Forschungsinstitut Oberwolfach, Schwarzwaldstr. 9-11, 77709 Oberwolfach-Walke, Germany\
e-mail: `brickenstein@mfo.de`
Viktor Levandovskyy\
Lehrstuhl D für Mathematik, RWTH Aachen, Templergraben 64, 52062 Aachen, Germany\
e-mail: `Viktor.Levandovskyy@math.rwth-aachen.de`
Jorge Martín-Morales\
Department of Mathematics-I.U.M.A., University of Zaragoza, C/ Pedro Cerbuna, 12 - 50009, Zaragoza, Spain\
e-mail: `jorge@unizar.es`
Hans Schönemann\
Fachbereich Mathematik, TU Kaiserslautern, Erwin-Schrödinger-Str. 48, 67632 Kaiserslautern, Germany\
e-mail: `hannes@mathematik.uni-kl.de`
[^1]: it will be distributed with the next release of [Singular]{}
|
---
author:
- Rong Ge
- 'Sham M. Kakade'
- Rahul Kidambi
- Praneeth Netrapalli
title: 'The Step Decay Schedule: A Near Optimal, Geometrically Decaying Learning Rate Procedure For Least Squares[^1]'
---
Introduction
============
Problem Setup {#sec:setup}
=============
Main results {#sec:comparison}
============
Experimental Results {#sec:exp}
====================
Conclusions and Discussion {#sec:conc}
==========================
#### Acknowledgments:
Rong Ge acknowledges funding from NSF CCF-$1704656$, NSF CCF-$1845171$ (CAREER), Sloan Fellowship and Google Faculty Research Award. Sham Kakade acknowledges funding from the Washington Research Foundation for Innovation in Data-intensive Discovery, NSF Award 1740551, and ONR award N$00014$-$18$-$1$-$2247$. Rahul Kidambi acknowledges funding from NSF Award $1740822$.
[61]{} \[1\][\#1]{} \[1\][`#1`]{} urlstyle \[1\][doi: \#1]{}
A. Agarwal, P. L. Bartlett, P. Ravikumar, and M. J. Wainwright. Information-theoretic lower bounds on the oracle complexity of stochastic convex optimization. *IEEE Transactions on Information Theory*, 2012.
Z. Allen-Zhu. How to make the gradients small stochastically. *CoRR*, abs/1801.02982, 2018.
D. Anbar. *On Optimal Estimation Methods Using Stochastic Approximation Procedures*. University of California, 1971. URL <http://books.google.com/books?id=MmpHJwAACAAJ>.
N. S. Aybat, A. Fallah, M. G[ü]{}rb[ü]{}zbalaban, and A. E. Ozdaglar. A universally optimal multistage accelerated stochastic gradient method. *CoRR*, abs/1901.08022, 2019.
F. R. Bach. Adaptivity of averaged stochastic gradient descent to local strong convexity for logistic regression. *Journal of Machine Learning Research (JMLR)*, volume 15, 2014.
F. R. Bach and E. Moulines. Non-asymptotic analysis of stochastic approximation algorithms for machine learning. In *NIPS 24*, 2011.
F. R. Bach and E. Moulines. Non-strongly-convex smooth stochastic approximation with convergence rate [O]{}(1/n). In *NIPS 26*, 2013.
A. Benveniste, M. Metivier, and P. Priouret. *Adaptive Algorithms and Stochastic Approximations*. Springer texts in Stochastic Modelling and Applied Probability, 1990.
B. Bharath and V. S. Borkar. Stochastic approximation algorithms: overview and recent trends. *Sādhanā*, 1999.
V. Borkar. *Stochastic approximation*. Cambridge Books, 2008.
L. Bottou and O. Bousquet. The tradeoffs of large scale learning. In *NIPS 20*, 2007.
S. Bubeck. Theory of convex optimization for machine learning. *CoRR*, abs/1405.4980, 2014.
D. Davis and D. Drusvyatskiy. Robust stochastic optimization with the proximal point method. *CoRR*, abs/1907.13307, 2019.
D. Davis, D. Drusvyatskiy, and V. Charisopoulos. Stochastic algorithms with geometric step decay converge linearly on sharp functions. *CoRR*, abs/1907.09547, 2019.
A. D[é]{}fossez and F. R. Bach. Averaged least-mean-squares: Bias-variance trade-offs and optimal sampling distributions. In *Artifical Intelligence and Statistics (AISTATS)*, 2015.
O. Dekel, R. Gilad-Bachrach, O. Shamir, and L. Xiao. Optimal distributed online prediction using mini-batches. *Journal of Machine Learning Research (JMLR)*, volume 13, 2012.
A. Dieuleveut and F. R. Bach. Non-parametric stochastic approximation with large step sizes. *The Annals of Statistics*, 2015.
A. Dieuleveut, N. Flammarion, and F. R. Bach. Harder, better, faster, stronger convergence rates for least-squares regression. *CoRR*, abs/1602.05419, 2016.
J. C. Duchi, E. Hazan, and Y. Singer. Adaptive subgradient methods for online learning and stochastic optimization. *Journal of Machine Learning Research*, 12:0 2121–2159, 2011.
R. Frostig, R. Ge, S. M. Kakade, and A. Sidford. Competing with the empirical risk minimizer in a single pass. In *COLT*, 2015.
S. Ghadimi and G. Lan. Optimal stochastic approximation algorithms for strongly convex stochastic composite optimization i: A generic algorithmic framework. *SIAM Journal on Optimization*, 2012.
S. Ghadimi and G. Lan. Optimal stochastic approximation algorithms for strongly convex stochastic composite optimization, ii: Shrinking procedures and optimal algorithms. *SIAM Journal on Optimization*, 230 (4), 2013.
S. Ghadimi and G. Lan. Optimal stochastic approximation algorithms for strongly convex stochastic composite optimization, ii: shrinking procedures and optimal algorithms. *SIAM Journal on Optimization*, 2013.
J. L. Goffin. On the convergence rates of subgradient optimization methods. *Mathematical Programming*, 13:0 329–347, 1977.
N. J. A. Harvey, C. Liaw, Y. Plan, and S. Randhawa. Tight analyses for non-smooth stochastic gradient descent. *CoRR*, 2018. URL <http://arxiv.org/abs/1812.05217>.
E. Hazan and S. Kale. Beyond the regret minimization barrier: optimal algorithms for stochastic strongly-convex optimization. *Journal of Machine Learning Research (JMLR)*, volume 15, 2014.
K. He, X. Zhang, S. Ren, and J. Sun. Identity mappings in deep residual networks. In *ECCV (4)*, Lecture Notes in Computer Science, pages 630–645. Springer, 2016.
K. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image recognition. In *CVPR*, pages 770–778, 2016.
P. Jain, S. M. Kakade, R. Kidambi, P. Netrapalli, and A. Sidford. Parallelizing stochastic approximation through mini-batching and tail-averaging. *arXiv preprint arXiv:1610.03774*, 2016.
P. Jain, S. M. Kakade, R. Kidambi, P. Netrapalli, V. K. Pillutla, and A. Sidford. A markov chain theory approach to characterizing the minimax optimality of stochastic gradient descent (for least squares). *CoRR*, 2017. URL <http://arxiv.org/abs/1710.09430>.
P. Jain, S. M. Kakade, R. Kidambi, P. Netrapalli, and A. Sidford. Accelerating stochastic gradient descent. *arXiv preprint arXiv:1704.08227*, 2017.
P. Jain, D. Nagaraj, and P. Netrapalli. Making the last iterate of sgd information theoretically optimal. *CoRR*, 2019. URL <http://arxiv.org/abs/1904.12443>.
R. Johnson and T. Zhang. Accelerating stochastic gradient descent using predictive variance reduction. In *NIPS 26*, 2013.
A. Kulunchakov and J. Mairal. A generic acceleration framework for stochastic composite optimization. *CoRR*, abs/1906.01164, 2019.
H. J. Kushner and D. S. Clark. *Stochastic Approximation Methods for Constrained and Unconstrained Systems.* Springer-Verlag, 1978.
H. J. Kushner and G. Yin. Stochastic approximation and recursive algorithms and applications. *Springer-Verlag*, 2003.
S. Lacoste-Julien, M. W. Schmidt, and F. R. Bach. A simpler approach to obtaining an o(1/t) convergence rate for the projected stochastic subgradient method. *CoRR*, 2012. URL <http://arxiv.org/abs/1212.2002>.
T. L. Lai. Stochastic approximation: invited paper, 2003.
E. L. Lehmann and G. Casella. *Theory of Point Estimation*. Springer Texts in Statistics. Springer, 1998. ISBN 9780387985022.
L. Li, K. Jamieson, G. DeSalvo, A. Rostamizadeh, and A. Talwalkar. Hyperband: A novel bandit-based approach to hyperparameter optimization. *The Journal of Machine Learning Research*, 180 (1):0 6765–6816, 2017.
L. Ljung, G. Pflug, and H. Walk. *Stochastic Approximation and Optimization of Random Systems*. Birkhauser Verlag, Basel, Switzerland, Switzerland, 1992. ISBN 3-7643-2733-2.
J.-I. Nagumo and A. Noda. A learning method for system identification. *IEEE Transactions on Automatic Control*, 1967.
D. Needell, N. Srebro, and R. Ward. Stochastic gradient descent, weighted sampling, and the randomized kaczmarz algorithm. *Mathematical Programming*, 2016.
A. S. Nemirovsky and D. B. Yudin. *Problem Complexity and Method Efficiency in Optimization*. John Wiley, 1983.
Y. E. Nesterov. A method for unconstrained convex minimization problem with the rate of convergence ${O}(1/k^2)$. *Doklady AN SSSR*, 269, 1983.
G. Neu and L. Rosasco. Iterate averaging as regularization for stochastic gradient descent. *CoRR*, 2018. URL <http://arxiv.org/abs/1802.08009>.
B. T. Polyak and A. B. Juditsky. Acceleration of stochastic approximation by averaging. *SIAM Journal on Control and Optimization*, volume 30, 1992.
J. G. Proakis. Channel identification for high speed digital communications. *IEEE Transactions on Automatic Control*, 1974.
M. Raginsky and A. Rakhlin. Information-based complexity, feedback and dynamics in convex programming. *IEEE Transactions on Information Theory*, 2011.
A. Rakhlin, O. Shamir, and K. Sridharan. Making gradient descent optimal for strongly convex stochastic optimization. In *ICML*, 2012.
H. Robbins and S. Monro. A stochastic approximation method. *The Annals of Mathematical Statistics*, vol. 22, 1951.
S. Roy and J. J. Shynk. Analysis of the momentum lms algorithm. *IEEE Transactions on Acoustics, Speech and Signal Processing*, 1990.
D. Ruppert. Efficient estimations from a slowly convergent robbins-monro process. *Tech. Report, ORIE, Cornell University*, 1988.
O. Shamir. Open problem: Is averaging needed for strongly convex stochastic gradient descent? In *COLT*, 2012.
O. Shamir and T. Zhang. Stochastic gradient descent for non-smooth optimization: Convergence results and optimal averaging schemes. *CoRR*, abs/1212.1824, 2012.
R. Sharma, W. A. Sethares, and J. A. Bucklew. Analysis of momentum adaptive filtering algorithms. *IEEE Transactions on Signal Processing*, 1998.
A. W. Van der Vaart. *Asymptotic statistics*, volume 3. Cambridge university press, 2000.
B. Widrow and M. E. Hoff. *Adaptive switching circuits*. Defense Technical Information Center, 1960.
B. Widrow and S. D. Stearns. *Adaptive Signal Processing*. Englewood Cliffs, NJ: Prentice-Hall, 1985.
Y. Xu, Q. Lin, and T. Yang. Accelerate stochastic subgradient method by leveraging local error bound. *CoRR*, abs/1607.01027, 2016.
T. Yang, Y. Y. 0006, Z. Yuan, and R. Jin. Why does stagewise training accelerate convergence of testing error over sgd? *CoRR*, abs/1812.03934, 2018.
[^1]: This paper appears in the proceedings of the conference in Neural Information Processing Systems (NeurIPS), $2019$, held in Vancouver Canada.
|
---
author:
- Yao Shen
- 'Yu-Zhu Chen'
title: Optical Properties of Synthetic Cannabinoids with Negative Indexes
---
In 1968, Veselago originally introduced the concept of negative index material (NIM) which exhibit unusual optical property[@x1]. For this kind of material, the permittivity and permeability can be negative simultaneously in electromagnetic field, which might bring about the phenomenon of negative refraction. Pendry *et al.* [@x2; @x3; @x4; @x5; @Luo13] developed Veselago’s theory, their research pointed out that the configuration of split-ring resonators (SRRs) [@x3] which had non-trivial symmetry breaking [@Fang15] is an effective way to realize negative refraction. Negative refraction has became more and more popular these decades, new researches and applications come to practice. For instance, the perfect lenses which are made by NIM can focus all Fourier components of image, both their propagating and evanescent waves are amplified, the quality of image and the detection sensitivity are improved[@x6-1; @x6-2; @x6-3; @x6-4; @Ma10]. After the theoretical derivation, Shelby *et al.* [@x6] gave the experimental realization two years later. According to Veselago’s theory, molecules with SRRs configurations such as extended metal atom chains (EMACs), have led a fresh direction of physical and chemical research [@x7; @x71; @x72; @x73; @x8; @x9; @x10; @x10-1; @x10-2; @x10-3; @x10-4; @x10-5; @x10-6; @x10-7; @x10-10; @x10-11; @x10-12; @x10-13; @x10-14; @x10-15; @me-1; @me-2; @me-3; @Zhang14]. Recently many fabulous works in different fields come forth, such as invisible cloaking [@e1; @e2; @e2-1; @e2-2; @e2-3; @e2-4], liquid crystal magnetic control [@e3], anapole moment [@e4], detection of latent fingermarks [@newme; @add1; @add2], etc. Inspired by these ideas, negative refraction plays more and more important role in forensic science field since it is a non-damaged optical method to test materials. More importantly, a small categories of molecules could show negative respondence to eletromagnetic fields [@Dong15]. Utilizing this property, these categories of molecules can be distinguished from others. This is significant to differentiate psychoactive drugs in forensic science. As a matter of fact, according to the theoretical derivation and calculation, we point out that the main ingredients of synthetic cannabinoids, the category of zepam, zolpidem and caffeine show various degree of negative refraction while other psychoactive drugs can not. We discuss the result of synthetic cannabinoids, zolpidem and caffeine quantitatively and the category of zepam qualitatively.
This paper is organized as following: In section II, we previously derive the permittivity and permeability of the main ingredients of synthetic cannabinoids, zolpidem and caffeine theoretically. Subsequently in section III, the numerical results of permittivity and permeability are discussed. Finally, the main results are concluded and future work is predicted in section IV.
Theoretical Derivation {#sec:theory .unnumbered}
======================
Hückel Model and Tight-binding Method {#hückel-model-and-tight-binding-method .unnumbered}
-------------------------------------
According to Pendry’s research, molecules which have the broken ring configurations can realize special optical property which is called negative refraction. We find that the main ingredients of synthetic cannabinoids such as JWH-018 (K2), JWH-073 (K2), AKB48 (K3) and AM2201 (K3) have the architecture of SRRs, therefore, their permittivity and permeability can be negative simultaneously in electromagnetic field. Moreover, two more compounds zolpidem and caffeine can also reveal negative refraction. In this section, we mainly discuss three kinds of model of synthetic cannabinoids, zolpidem and caffeine.
Figure \[fig:two\](a) demonstrates the configuration of AKB48 (apinaca) molecule, and Figure \[fig:two\] (b) is the structure called AM2201. The systematic name of AKB48 is N-(1-adamantyl)-1-pentyl-1H-indazole-3-carboxamide), and that of AM2201 is 1-(5-fluoropentyl)-3-(1-naphthoyl)indole. They are the main ingredients of K3 (synthetic cannabinoids).
![(a)The molecule of AKB48 (Apinaca, $C_{23}H_{31}N_{3}O$). AKB48 is N-(1-adamantyl)-1-pentyl-1H-indazole-3-carboxamide. (b)The molecule of AM2201 ($C_{24}H_{22}FNO$). AM2201 is 1-(5-fluoropentyl)-3-(1-naphthoyl)indole. \[fig:two\]](molecule)
Figure \[fig:two-1\](a) shows the structure of JWH-018 molecule, Figure \[fig:two-1\](b) gives us JWH-073 molecule. JWH-018 is 1-pentyl-3-(1-naphthoyl)indole and JWH-073 is naphthalen-1-yl-(1-butylindol-3-yl)methanone. They are the main ingredients of K2 (synthetic cannabinoids).
![(a)The molecule of JWH-018. JWH-018 is 1-pentyl-3-(1-naphthoyl)indole ($C_{24}H_{23}NO$). (b)The molecule of JWH-073. JWH-073 is naphthalen-1-yl-(1-butylindol-3-yl)methanone ($C_{23}H_{21}NO$). \[fig:two-1\]](molecule2)
Both K2 and K3 are synthetic cannabinoids. The central parts of them are the broken ring of pentagons. These broken pentagons contribute to the negative refraction mostly, because all these phenomena come from the $\pi$ electrons of conjugate ring of these molecules which are called single-nitrogen-substituted heterocyclic annulenes [@x11]. For AKB48, two carbon atoms of the pentagon are substituted with two nitrogen atoms, while for AM2201, JWH-018 and JWH-073, only one carbon atom of the pentagon is replaced by one nitrogen atom. The simplified model are given in following Figure \[three\]. The left figure is the simplified model of AKB48 and the ringt one is the model of AM2201, JWH-018 and JWH-073. All the atoms of the pentagon are in the same plane and be marked as Figure \[three\].
![The Simplified models of synthetic cannabinoids. (a) The model of AKB48 is 1H-pyrazole ($C_{3}H_{4}N_{2}$) on the left. (b) The models of AM2201, JWH-018 and JWH-073 are all pyrrole ($C_{4}H_{5}N$) on the right. The origin is set at the center of the pentagons. \[three\]](5N25NK3)
The simplified model of AKB48 has two nitrogen atoms which are located at site $4$ and $5$. The other model has only one nitrogen atom which is placed at site $5$. These two models are called 1H-pyrazole and pyrrole respectively.
There are two other cases in psychoactive drugs which also have broken conjugate ring. The simplified model of zolpidem and caffeine is given in Figure \[three-1\]. There are also two nitrogen atoms instead of two carbon atoms, but their positions are quite different from 1H-pyrazole. Therefore, these dissimilar simplified models show different degree of negative refraction.
![The Simplified models of zolpidem and caffeine are both 1H-imidazole ($C_{3}H_{4}N_{2}$). 1H-imidazole is different from 1H-pyrazole, althougt they have the same molecular formula. The nitrogen atoms are located at different sites. \[three-1\]](5N2)
The Hückel Hamiltonian of these five $\pi$ electrons conjugate systems are described as[@x12]
$$\mathcal{H}=\sum_{j=1}^{5}\alpha_{j}\left|j\right\rangle \left\langle j\right|+\sum_{j=1}^{5}\beta_{j,j+1}\left(\left|j\right\rangle \left\langle j+1\right|+\left|j+1\right\rangle \left\langle j\right|\right),\label{eq:Hhu}$$
where $j$ denotes the site lable, $\alpha$ is site energy and $\beta$ is the resonant integral. When $j=5$, we have $j+1=1$, because of the ring configuration. We choose $\alpha_{C}=-6.7eV$, $\alpha_{N}=-7.9eV$ [@H0]. The resonant integral given by Harrison expression [@H1; @H2] $$\beta_{ij}=-0.63\frac{\hbar^{2}}{md_{ij}^{2}}\label{eq:beta}$$ where $m$ is the mass of electrons and $d_{ij}$ expresses the bond length. All data come from NIST [@H3].
- 1H-pyrazole (AKB48)
$$\mathcal{H}=\sum_{j=1}^{3}\alpha_{C}\left|j\right\rangle \left\langle j\right|+\alpha_{N}(\left|4\right\rangle \left\langle 4\right|+\left|5\right\rangle \left\langle 5\right|)+\sum_{j=1}^{5}\beta_{j,j+1}\left(\left|j\right\rangle \left\langle j+1\right|+\left|j+1\right\rangle \left\langle j\right|\right),\label{eq:Hhu-1}$$
where $\beta_{1,2}=-2.545eV$, $\beta_{2,3}=-2.393eV$, $\beta_{3,4}=-2.708eV$, $\beta_{4,5}=-2.632eV$, $\beta_{5,1}=-2.598eV$.
- pyrrole (AM2201, JWH-018 and JWH-073)
$$\mathcal{H}=\sum_{j=1}^{4}\alpha_{C}\left|j\right\rangle \left\langle j\right|+\alpha_{N}\left|5\right\rangle \left\langle 5\right|+\sum_{j=1}^{5}\beta_{j,j+1}\left(\left|j\right\rangle \left\langle j+1\right|+\left|j+1\right\rangle \left\langle j\right|\right),\label{eq:Hhu-2}$$
where $\beta_{1,2}=-2.516eV$, $\beta_{2,3}=-2.393eV$, $\beta_{3,4}=-2.516eV$, $\beta_{4,5}=-2.560eV$, $\beta_{5,1}=-2.560eV$.
- 1H-imidazole (zolpidem and caffeine)
$$\mathcal{H}=\sum_{j=1,3,4}\alpha_{C}\left|j\right\rangle \left\langle j\right|+\alpha_{N}(\left|2\right\rangle \left\langle 2\right|+\left|5\right\rangle \left\langle 5\right|)+\sum_{j=1}^{5}\beta_{j,j+1}\left(\left|j\right\rangle \left\langle j+1\right|+\left|j+1\right\rangle \left\langle j\right|\right),\label{eq:Hhu-3}$$
where $\beta_{1,2}=-2.783eV$, $\beta_{2,3}=-2.516eV$, $\beta_{3,4}=-2.582eV$, $\beta_{4,5}=-2.534eV$, $\beta_{5,1}=-2.582eV$.
By diagonalizing the Hückel Hamiltonian , we calculate the energy levels of these models as Figure \[four\].
$$\begin{aligned}
\mathcal{H} & = & \sum_{k=1}^{5}\varepsilon_{k}\left|\psi_{k}\right\rangle \left\langle \psi_{k}\right|,\end{aligned}$$
where $$\left|\psi_{k}\right\rangle =\sum_{j=1}^{5}C_{ki}\left|j\right\rangle$$ $\varepsilon_{k}$ is the eigen value and $\left|\psi_{k}\right\rangle $ is the eign states. Not losing the generality, we shift all energy levels by $\alpha_{C}$ for simplicity.
- 1H-pyrazole (AKB48)
$\varepsilon_{1}=-5.762eV$, $\varepsilon_{2}=-2.109eV$, $\varepsilon_{3}=-1.966eV$, $\varepsilon_{4}=3.465eV$, $\varepsilon_{5}=3.972eV$.
- pyrrole (AM2201, JWH-018 and JWH-073)
$\varepsilon_{1}=-5.330eV$, $\varepsilon_{2}=-1.972eV$, $\varepsilon_{3}=-1.590eV$, $\varepsilon_{4}=3.708eV$, $\varepsilon_{5}=3.983eV$.
- 1H-imidazole (zolpidem and caffeine)
$\varepsilon_{1}=-5.734eV$, $\varepsilon_{2}=-2.472eV$, $\varepsilon_{3}=-1.743eV$, $\varepsilon_{4}=3.694eV$, $\varepsilon_{5}=3.855eV$.
All energy levels are nondegenerated. We have five non-interacting $\pi$ electrons fill in five energy levels. On the basis of Pauli exclusion principle, the energy of ground state is $E_{0}=2\varepsilon_{1}+2\varepsilon_{2}+\varepsilon_{3}$. And the second-quantization form of the ground state can be expressed as
$$\left|\Psi_{0}\right\rangle =a_{1\uparrow}^{\dagger}a_{1\downarrow}^{\dagger}a_{2\uparrow}^{\dagger}a_{2\downarrow}^{\dagger}a_{3\uparrow}^{\dagger}\left|0\right\rangle ,$$
where $\left|0\right\rangle $ is vacuum state and $a_{p\sigma}^{\dagger}$ is the creation operator of the orbital $p$ with spin $\sigma$ ($\sigma=\uparrow,\downarrow$).
![(a) The energy levels of 1,3-cyclopentadiene ($C_{5}H_{6}$). (b) The energy levels of pyrrole ($C_{4}H_{5}N$). (c) The energy levels of 1H-pyrazole ($C_{3}H_{4}N_{2}$). (d) The energy levels of 1H-imidazole ($C_{3}H_{4}N_{2}$). \[four\]](energy)
We only consider the single-excitation with no flip of electronic spin. All these three kinds of systems have fourteen single-excitation states.
The single-excitation states of the system which denote the single electron excitation from $p$ to $q$ can be assumed as $$\left|\Psi_{n}\right\rangle =\left|\Psi_{qp}^{\sigma}\right\rangle =a_{q\sigma}^{\dagger}a_{p\sigma}\left|\Psi_{0}\right\rangle ,$$ where $p=1,2,3$, $q=3,4,5$ and $\sigma=\uparrow,\downarrow$. The coresponding energies are $$E_{n}=E_{0}+\varepsilon_{q}-\varepsilon_{p}.$$
The ground state and single-excitation states span a subspace. In this subspace, the Hamiltonian reads
$$\mathcal{H}=\sum_{n=0}^{14}E_{n}\left|\Psi_{n}\right\rangle \left\langle \Psi_{n}\right|.$$
Perturbation Theory in Electromagnetic Field {#perturbation-theory-in-electromagnetic-field .unnumbered}
--------------------------------------------
When the molecules are placed in a time-dependent electromagnetic field, the total Hamiltonian with the dipole approximation can be written as $$H=H_{0}-\vec{\mu}\cdot\vec{E}_{0}\cos(\vec{k}\cdot\vec{r}-\omega t)-\vec{m}\cdot\vec{B}_{0}\cos(\vec{k}\cdot\vec{r}-\omega t),$$ where $\vec{k}$ is the wave vector, $\vec{\mu}$ is the electric dipole moment, $\vec{m}$ is the magnetic dipole moments and $H_{0}=\mathcal{H}$ as defined above is the Hamiltonian without electromagnetic field. Because the spatial scale of electromagnetic wave length is much longer than that of the molecules, we assume that the spatial part of the electromagnetic field can be neglected.
$$H\simeq H_{0}-\vec{\mu}\cdot\vec{E}_{0}\cos(\omega t)-\vec{m}\cdot\vec{B}_{0}\cos(\omega t).$$
By a unitary transformation, the Hamiltonian can be simplified into a time independent one in Schroüdinger picture. In this picture, the state and operator are $\left|\Psi^{\prime}\right\rangle =U^{\dagger}\left|\Psi\right\rangle ,$ and $A^{\prime}=U^{\dagger}AU$. The Hamiltonian of the system can be rewritten as $$\begin{aligned}
H^{'} & = & U^{\dagger}HU-iU^{\dagger}\dot{U}\nonumber \\
& \simeq & \sum_{n=1}^{14}E_{n}\left|\Psi_{n}\right\rangle \left\langle \Psi_{n}\right|+\left(E_{0}+\omega\right)\left|\Psi_{0}\right\rangle \left\langle \varPsi_{0}\right|+H^{\prime\prime},\end{aligned}$$ where $$U=\exp\left(i\omega\left|\Psi_{0}\right\rangle \left\langle \Psi_{0}\right|t\right),$$
$$\begin{aligned}
H^{\prime\prime} & = & -\frac{1}{2}\sum_{n=1}^{14}\left[\right.\left(\vec{\mu}_{n0}\cdot\vec{E}_{0}\left|\varPsi_{n}\right\rangle \left\langle \varPsi_{0}\right|+\vec{\mu}_{0n}\cdot\vec{E}_{0}\left|\varPsi_{0}\right\rangle \left\langle \varPsi_{n}\right|\right)\nonumber \\
& & +\left(\vec{m}_{n0}\cdot\vec{B}_{0}\left|\Psi_{n}\right\rangle \left\langle \Psi_{0}\right|+\vec{m}_{0n}\cdot\vec{B}_{0}\left|\Psi_{0}\right\rangle \left\langle \Psi_{n}\right|\right)\left.\right],\\
\vec{\mu}_{ns} & = & \left\langle \Psi_{n}\right|\vec{\mu}\left|\Psi_{s}\right\rangle ,\\
\vec{m}_{ns} & = & \left\langle \Psi_{n}\right|\vec{m}\left|\varPsi_{s}\right\rangle .\end{aligned}$$
Moreover, according to the perturbation theory, the molecular ground state in Schroüdinger picture becomes $$\left|\Psi_{0}^{\prime}\right\rangle =U^{\dagger}\left|\Psi_{0}\right\rangle =\left|\Psi_{0}\right\rangle +\sum_{n=1}^{14}\frac{\left\langle \Psi_{n}\right|H^{\prime\prime}\left|\Psi_{0}\right\rangle }{E_{0}+\omega-E_{n}}\left|\Psi_{n}\right\rangle ,$$
### Derivation of Permittivity {#derivation-of-permittivity .unnumbered}
The electric dipole moment in Schroüdinger picture is $$\vec{\mu}^{\prime}=U^{\dagger}\vec{\mu}U=\sum_{n=1}^{14}\left(\vec{\mu}_{n0}e^{i\omega t}\left|\Psi_{n}\right\rangle \left\langle \Psi_{0}\right|+\vec{\mu}_{0n}e^{-i\omega t}\left|\varPsi_{0}\right\rangle \left\langle \Psi_{n}\right|\right).$$ And its expectation value in ground state reads $$\left\langle \Psi_{0}^{'}\right|\vec{\mu}^{\prime}\left|\Psi_{0}^{'}\right\rangle =-\textrm{Re}\sum_{n=1}^{14}\frac{\vec{\mu}_{n0}\cdot\vec{E}_{0}}{E_{0}+\omega-E_{n}}\vec{\mu}_{0n}e^{-i\omega t}.$$ So the electric moment for $N$ identical molecules can be expressed as $$\vec{P}=-\sum_{s=1}^{N}\sum_{n=1}^{14}\frac{\left[\vec{\mu}_{n0}(s)\cdot\vec{E}_{0}\right]\vec{\mu}_{0n}(s)}{E_{0}+\omega-E_{n}}.$$
According to the theory of the electric displacement field in the electromagnetic field, we have $$\vec{D}=\varepsilon\vec{E}_{0}=\varepsilon_{0}\varepsilon_{r}\vec{E}_{0}=\varepsilon_{0}\vec{E}_{0}+\frac{\vec{P}}{V},$$ where $V$ is the volume and $\vec{P}$ is the electric moment. Thus, the relative permittivity can be written as $$\varepsilon_{ij}^{r}\equiv\delta_{ij}-\sum_{s=1}^{N}\sum_{n=1}^{14}\frac{\vec{\mu}_{0n}(s)\cdot\hat{e}_{i}\vec{\mu}_{n0}(s)\cdot\hat{e}_{j}}{\varepsilon_{0}V\left(E_{0}+\omega-E_{n}\right)},\textrm{ for }i,j=x,y,z,\label{eq:ep}$$ where $\hat{e}_{i}$ is the unit vector of the lab coordinate system.
The origin of the simplified model is set at the center of the pentagons (see Figure \[three\], \[three-1\]). As we known, the electric dipole moment reads $$\vec{\mu}=-\sum_{j=1}^{5}e\vec{r}_{j},$$ where$-e$ is the electric charge and $\vec{r}_{j}$ is the location vector. Therefore, in the eign states of molecules, the electric dipole moment operators are
$$\begin{aligned}
\vec{\mu}_{mn}=\left\langle \Psi_{m}\right|\vec{\mu}\left|\Psi_{n}\right\rangle & = & -e\sum_{j=1}^{5}\left\langle \Psi_{m}\right|\vec{r_{j}}\left|\Psi_{n}\right\rangle =-e\sum_{j=1}^{5}\left\langle \psi_{f}\right|\vec{r_{j}}\left|\psi_{g}\right\rangle =-e\sum_{j=1}^{5}C_{fj}^{*}C_{gj}\vec{r}_{j},\end{aligned}$$
where $\psi_{g}$ denotes single electron orbital.
### Derivation of Permeability {#derivation-of-permeability .unnumbered}
As we known, the magnetic dipole moment can be written as $$\vec{m}=\frac{-e}{2m_{e}}\vec{L},$$ where $\vec{L}$ is the angular momentum of the system. According to the the Heisenberg equations, we have $L_{x}=L_{y}=0,$ $$\begin{aligned}
\nonumber \\
\nonumber \\
L_{z} & = & xp^{y}-yp^{x}\nonumber \\
& = & \frac{1}{2}\left(xp^{y}+p^{y}x-yp^{x}-p^{x}y\right)\nonumber \\
& = & im_{e}\left(x\mathcal{\mathcal{H}}y-y\mathrm{\mathit{\mathcal{H}}}x\right).\end{aligned}$$ In this case, the magnetic dipole moment can be rewritten as
$$\begin{aligned}
\vec{m} & = & \frac{-e}{2m_{e}}L_{z}\hat{e}_{z}\nonumber \\
& = & \frac{-ie}{2}\left(x\mathcal{\mathcal{H}}y-y\mathrm{\mathit{\mathcal{H}}}x\right)\hat{e}_{z}\nonumber \\
& = & \frac{-ie}{2}\sum_{p,q}\sum_{n}E_{n}\left(x_{pn}y_{nq}-y_{pn}x_{nq}\right)\left|\Psi_{p}\right\rangle \left\langle \Psi_{q}\right|\hat{e}_{z},\end{aligned}$$
where $x_{pn}=\left\langle \psi_{p}\right|x\left|\psi_{n}\right\rangle $, $y_{nq}=\left\langle \psi_{n}\right|y\left|\psi_{q}\right\rangle $.
Similarly in Schroüdinger picture, the expectation value of magnetic dipole moment in ground state is
$$\left\langle \Psi_{0}^{'}\right|\vec{m}^{\prime}\left|\Psi_{0}^{'}\right\rangle =-\textrm{Re}\sum_{n=1}^{14}\frac{\vec{m}_{n0}\cdot\vec{B}_{0}}{E_{0}+\omega-E_{n}}\vec{m}_{0n}e^{-i\omega t}.$$
So the magnetic dipole moment for $N$ identical molecules can be expressed as $$\vec{M}=-\sum_{s=1}^{N}\sum_{n=1}^{27}\frac{\mu_{0}\vec{m}_{n0}(s)\cdot\vec{B}_{0}}{E_{0}+\omega-E_{n}}\vec{m}_{0n}(s).$$
The magnetic induction in a volume $V$ can be expressed as $$\vec{B}=\mu\vec{H}=\mu_{0}\mu_{r}\vec{H}=\mu_{0}\vec{H}+\mu_{0}\frac{\vec{M}}{V},$$ where $\vec{H}$ is magnetic field intensity.
The relative permeability of system can be expressed as $$\mu_{ij}^{r}\equiv\delta_{ij}-\mu_{0}\sum_{s=1}^{N}\sum_{n=1}^{14}\frac{\vec{m}_{0n}(s)\cdot\hat{e}_{i}\vec{m}_{n0}(s)\cdot\hat{e}_{j}}{V\left(E_{0}+\omega-E_{n}\right)},\textrm{ for }i,j=x,y,z.\label{eq:mu}$$
### Analysis of Analytical Results {#analysis-of-analytical-results .unnumbered}
For negative refraction, the permittivity and permeability of the system should be negative simultaneously. On the basis of the analytical results given above (\[eq:ep\]) and (\[eq:mu\]), the second parts of these two equations should be greater than unity, even far exceed. In this case, we reckon that the denominators are much smaller than the numerator apparently which means $E_{0}+\omega-E_{n}\approx0$ and $\omega\approx E_{n}-E_{0}$. $E_{0}$ is the ground state energy of initial state. When the molecules are placed in electromagnetic field, the electromagnetic field gives them a driving frequency $\omega$ which made them transit to the final state with energy $E_{n}$. Here only single electron excitation is taken into account. The negative refraction, or in another word, simultaneous negative permittivity and permeability occurs while the driving frequency of electromagnetic field $\omega$ approximately equal to the difference of the energy $E_{n}-E_{0}$ .
Numerical Simulation of Permittivity and Permeability\[sec:result\] {#numerical-simulation-of-permittivity-and-permeabilitysecresult .unnumbered}
===================================================================
![The numerical simulation of (a) permittivity and (b) permeability of pyrrole ($C_{4}H_{5}N$, including AM2201, JWH-018 and JWH-073) vs the frequency $\omega$ of the electromagnetic field. The axis $o$ is one main axis in the $xy$ plane while the axis $p$ is another main axis in this plane. (a) The permittivity $\varepsilon_{o}^{r}$ (red solid line) is negative around the resonance frequency, while $\varepsilon_{p}^{r}$ (blue dashed line) is always certain constant. (b) The permeability along $z$ direction $\mu_{z}^{r}$ can be negative around the resonance frequency simultaneously. \[fig:re1\]](5N)
![The numerical simulation of (a) permittivity and (b) permeability of AKB48 (1H-pyrazole ($C_{3}H_{4}N_{2}$)) vs the frequency $\omega$ of the electromagnetic field. The axis $o$ is one main axis in the $xy$ plane while the axis $p$ is another main axis in this plane. (a) The permittivity $\varepsilon_{o}^{r}$ (red solid line) can be negative around the resonance frequency. (b) The permeability along $z$ direction $\mu_{z}^{r}$ can also be negative around the resonance frequency simultaneously.\[fig:re2\]](5NN)
![The numerical simulation of (a) permittivity and (b) permeability of zolpidem and caffeine (1H-imidazole ($C_{3}H_{4}N_{2}$)) vs the frequency $\omega$ of the electromagnetic field. The axis $o$ is one main axis in the $xy$ plane while the axis $p$ is another main axis in this plane. (a) The negative permittivity $\varepsilon_{o}^{r}$ (red solid line) and constant $\varepsilon_{p}^{r}$ (blue dashed line) are shown. (b) The negative permeability along $z$ direction $\mu_{z}^{r}$ is calso given.\[fig:re3\]](5NN2)
The analitical results of section \[sec:theory\] suggest that in certain frequency regime the molecules of certain psychoactive drugs can demonstrate negative refraction in the UV-vis region. These psychoactive drugs mainly include synthetic cannabinoids, zolpidem and caffeine. In this section, we show the numerical results and analyses to corroborate it. These molecules of psychoactive drugs can be simplified into three kinds of two dimensional models (see Figure \[three\] and Figure \[three-1\]). Figure \[fig:re1\] gives the numerical simulation of permittivity and permeability of pyrrole ($C_{4}H_{5}N$) which is the simplified model of AM2201, JWH-018 and JWH-073. Figure \[fig:re2\] is the result of AKB48 (1H-pyrazole ($C_{3}H_{4}N_{2}$)). Figure \[fig:re3\] shows the permittivity and permeability of zolpidem and caffeine (1H-imidazole ($C_{3}H_{4}N_{2}$)). All these permittivities $\varepsilon_{r}$ are in the $xy$ plane, while permeabilities $\mu_{r}$ are in $z$ direction of the systems. We suppose that the life time of the excited-state is $\tau=10$ns [@Tokuji09]. For pyrrole ($C_{4}H_{5}N$) the main contribution comes from the transition to the fifth excited state, e.g. a electron transits from $\varepsilon_{3}$ to $\varepsilon_{5}$ with no flip. And the third transition makes the major contribution to negative indexes of 1H-pyrazole ($C_{3}H_{4}N_{2}$). While for zolpidem and caffeine, which hails from the transition to the forth excited state. All these molecules display negative permittivity and permeability simultaneously, thereby bring negative refraction consequently. For negative magnetic response, the permeability of AKB48 is much more obvious than that of the others (about $10$ times). While the result of permittivity is reversed (about $3$ to $5$ times in $10^6$). The phenomenon of negative refraction of AM2201, JWH-018, JWH-073, zolpidem and caffeine are of significant to AKB48. To achieve negative permittivity and permeability simultaneously, the pickwidths of AKB48 in the figure are wider than others. In other words, the pickwidths reveals that AKB48 is much easier to realize negative refraction at the present stage.
Conclusion and Discussion \[sec:CONCLUSION\] {#conclusion-and-discussion-secconclusion .unnumbered}
============================================
In this paper we put forward a new method to identify certain kinds of psychoactive drugs which have the structures called split-ring resonators. This configuration might induce negative permittivity and permeability simultaneously, consequently negative refraction in the UV-vis region. When light with certain frequency transmit through the transparent media which has psychoactive drugs with negative indexes on it, the refracted light goes on the opposite side comparing with the ordinary case. Namely in common case, when the incident light transmit onto the surface of media , the refracted light lies on the contrary side of normal. But for negative refraction material, the refracted light and incident light lie on the same side of normal. This phenomenon made us capable of distinguishing certain families of psychoactive drugs from others. The optical method has its advantage which is non-damaged and would not cumber the detection of DNA in forensic science. Here we use Hückel model to deal with the systems. As a result, we conclude that the main ingredients of synthetic cannabinoids, zolpidem and caffeine can show various degree of negative refraction, thereby can be distinguished from others. In psychoactive artificial cannabinoid families e.g. AM-xxx, HU-xxx, JWH-xxx, CP xx, other small quantity of ingredients e.g. AM-694, JWH-250 can also give the result of negative refraction but they are unconspicuous. It is a remarkable fact that the category of zepam which has a broken ring of heptagon could also give negative refraction in all psychoactive drugs. But the category of zepam is not a prevalent abused drug as synthetic cannabinoids and others, therefore we only give analytic derivation and numerical simulation of three simplified models.
Acknowledgement {#acknowledgement .unnumbered}
===============
The research was supported by NSF of China under Grant No. 11575125. We thank for the useful discussion of Pro.Wu-Sheng Dai from Department of Physics, Tianjin University.
[References]{} Veselago, V. G. The electrodynamics of substances with simultaneously negative values of $\varepsilon$ and $\mu$. *Sov. Phys. Uspekhi* **10**, 509-514 (1968).
Pendry, J. B. Negative refraction makes a perfect lens. *Phys. Rev. Lett.* **85**, 3966-3969 (2000).
Pendry, J. B., Holden, A. J., Robbins, D. J. & Stewart, W. J. Magnetism from conductors and enhanced nonlinear phenomena. *IEEE Trans. Microwave Theory Tech*. **47**, 2075-2084 (1999).
Pendry, J. B., Holden, A. J., Stewart, W. J. & Youngs, I. Extremely low frequency plasmons in metallic mesostructures. *Phys. Rev. Lett.* **76**, 4773-4776 (1996).
Pendry, J. B., Holden, A. J., Robbins, D. J. & Stewart, W. J. Low frequency plasmons in thin-wire structures. *J. Phys. Condens. Matter* **10**, 4785-4809 (1998).
Luo, Y., Zhao, R. K., Fernandez-Dominguez, A. I., Stefan, A. M. & John, P. B. Harvesting light with transformation optics. *Sci. Chin. Inf. Sci*. **56**, 120401 (2013).
Fang, Y. N., Shen, Y., Ai, Q. & Sun, C. P. Negative refraction induced by Möbius topology. *Preprint arXiv*:1501.05729.
Pendry, J. B., Holden, A. J., Robbins, D. J., Stewart, W. J. Magnetism from conductors and enhanced nonlinear phenomena. *IEEE Trans.* *Microwave Theory Tech.* **47**, 2075, (1999).
Cubukcu, E., Zhang, S., Park,Y.-S., Bartal, G. & Zhang, X. Split ring resonator sensors for infrared detection of single molecular monolayers. *Appl. Phys. Lett.* **95**, 043113 (2009).
Clark, A. W., Glidle, A., Cumming, D. R. S. & Cooper, J. M. Plasmonic split-ring resonators as dichroic nanophotonic {DNA} biosensors. *J. Am. Chem. Soc.* **131**, 176150-17619 (2009).
Pryce, I. M., Kelaita, Y. A., Aydin, K., Briggs, R. M. & Atwater, H. A. Compliant metamaterials for resonantly enhanced infrared absorption spectroscopy and refractive index sensing. *ACS Nano* **5**, 8167-8174 (2011).
Ma, C. B., Aguinaldo, R. & Liu, Z. W. Advances in the hyperlens. *Chin. Sci. Bull*. **55**, 2618-2624 (2010).
Shelby, R. A., Smith, D. R. & Schultz, S. Experimental verification of a negative index of refraction. *Science* **292**, 77-79 (2001).
Ropp, C. *et al*. Positioning and immobilization of individual quantum dots with nanoscale precision. *Nano Lett.* **10**, 4673-4679 (2010).
Cao, J. J. et al. Dielectric optical-controllable magnifying lens by nonlinear negative refraction. Sci. Rep. 5, 11892 (2015).
Bi, K. et al. Negative and near zero refraction metamaterials based on permanent magnetic ferrites. Sci. Rep. 4, 4139 (2014).
Philippe, F. D., Murray, T. W., Prada, C. Focusing on plates: controlling guided waves using negative refraction. Sci. Rep. 5, 11112 (2015).
Smith, D. R., Pendry, J. B. & Wiltshire, M. C. K. Metamaterials and negative refractive index. *Science* **305**, 788-792 (2004).
Decker, M., Linden, S. & Wegener, M. Coupling effects in low-symmetry planar split-ring resonator arrays. *Opt. Lett.* **34**, 1579-1581 (2009).
Pryce, I. M., Aydin, K., Kelaita, Y. A., Briggs, R. M. & Atwater, H. A. Highly strained compliant optical metamaterials with large frequency tunability. *Nano Lett.* **10**, 4222-4227 (2010).
Chen, W. T. *et al*. Optical magnetic response of upright plasmonic molecules in 3D metamaterial. SPIE Newsroom **2011**.
Liu, N., Liu, H., Zhu, S. & Giessen, H. Stereometamaterials. *Nat. Photonics* **3**, 157-162 (2009).
Fleming, G. R. & Wolynes, P. G. Chemical dynamics in solution. *Phys. Today* **43**, 36-43 (1990).
Szabo, A. & Ostlund, N. S. *Modern quantum chemistry: Introduction to advanced electronic structure theory* (Dover, New York, 1996).
Greenwood, H. H. *Computing methods in quantum organic chemistry* (Wiley-Interscience, Germany, 1972).
Pantazis, D. A. & McGrady, J. E. A three-state model for the polymorphism in linear tricobalt compounds. *J. Am. Chem. Soc*. **128**, 4128-4135 (2006).
Pyrka, G. J., El-Mekki, M. & Pinkerton, A. A. Structure of the linear trinuclear copper complex, dichlorotetrakis-(di-2-pyridylamido)tricopper. *J. Chem. Soc., Chem. Commun.* 84-85 (1991).
Peng, S.-M. *et al*. One-dimensional metal string complexes. *J. Magn. Magn. Mater*. **209**, 80-83 (2000).
Tsai, T.-W., Huang, Q.-R., Peng, S.-M. & Jin, B.-Y. Smallest electrical wire based on extended metal-atom chains. *J. Phys. Chem. C* **114**, 3641-3644 (2010).
Chae, D.-H. *et al*. Vibrational excitations in single trimetal-molecule transistors. *Nano Lett*. **6**,165-168 (2006).
Chen, I.-W. P. *et al*. Conductance and stochastic switching of ligand-supported linear chains of metal atoms. *Angew. Chem. Int. Edit.* **45**, 5814-5818 (2006).
Chen, C. C. *et al*. Fabrication of three dimensional split ring resonators by stress-driven assembly method. *Opt. Express* **20**, 9415-9420 (2012).
Ishikawa, A. & Tanaka, T. J. Two-photon fabrication of three-dimensional metallic nanostructures for plasmonic metamaterials. *Laser Micro Nanoeng.* **7**, 11-15 (2012).
Shen, Y. & Jin, B.-Y. Correspondence between Gentile oscillators and N-annulenes. *J. Phys. Chem. A* **117**,12540-12545 (2013).
Shen, Y., Dai, W. S. & Xie, M. Intermediate-statistics quantum bracket, coherent state, oscillator, and representation of angular momentum [\[]{}SU(2)[\]]{} algebra. *Phys. Rev. A* **75**, 042111 (2007).
Shen, Y., Ai, Q. & Long, G. L. The relation between properties of Gentile statistics and fractional statistics of anyon. *Phys. A* **389**,1565-1570 (2010).
Zhang, S. & Zhang, Y. Broadband unidirectional acoustic transmission based on piecewise linear acoustic metamaterials. *Chin. Sci. Bull*. **59**, 3239-3245 (2014).
Pendry, J. B., Schurig, D. & Smith, D. R. Controlling electromagnetic fields. *Science* **312**, 1780-1782 (2006).
Schurig, D. *et al*. Metamaterial electromagnetic cloak at microwave frequencies. *Science* **314**, 977-980 (2006).
Huang, Y. & Gao, L. Equivalent permittivity and permeability and multiple Fano resonances for nonlocal metallic nanowires. *J. Phys. Chem. C* **117**, 19203-19211 (2013).
Droulias, S. & Yannopapas, V. Broad-band giant circular dichroism in metamaterials of twisted chains of metallic nanoparticles. *J. Phys. Chem. C* **117**, 1130-1135 (2013).
Xue, H. J., Wu, R. L. & Yu. Y. Abnormal absorption and energy flow of electromagnetic wave in ultrathin metal films. *J. Phys. Chem. C* **118**, 18257-18262 (2014).
Yannopapas, V. & Psarobas, I. E. Ordered arrays of metal nanostrings as broadband super absorbers. *J. Phys. Chem. C* **116**, 15599-15603 (2012).
Kaelberer, T., Fedotov, V. A., Papasimakis, N., Tsai, D. P. & Zheludev, N. I. Toroidal dipolar response in a metamaterial. *Science* **330**, 1510-1512 (2010).
Zhang, F. *et al*. Microwave Conference, 2008. EuMC 2008. 38th European 2008; Vol. *1*, 801-804.
Shen, Y.; Ai, Q. Optical properties of drug metabolites in latent fingermarks. Sci. Rep. 6, 20336 (2016).
Groeneveld, G., de Puit, M., Bleay, S., Bradshaw, R. & Francese, S. Detection and mapping of illicit drugs and their metabolites in fingermarks by MALDI MS and compatibility with forensic techniques. *Sci. Rep.* **5**, 11716 (2015).
Wei,T. T. *et al*. Metabonomic analysis of potential biomarkers and drug targets involved in diabetic nephropathy mice. *Sci. Rep.* **5**, 11998 (2015).
Dong, W. B., Wu, R. B., Yuan, X. H., Li, C. W. & Tarn, T.-J. The modelling of quantum control systems. *Sci. Bull*. **60**, 1493 (2015).
Shen, Y., Ko, H.-Y., Ai, Q., Peng, S.-M. & Jin, B.-Y. Molecular split-ring resonators based on metal string complexes. *J. Phys. Chem. C* **118**, 3766-3773 (2014).
Salem, L. *The molecular orbital theory of conjugated systems* (W. A. Benjamin, New York, 1966).
Tokuji, S. A. *et al*. Facile formation of a benzopyrane-fused [\[]{}28[\]]{} hexaphyrin that exhibits distinct Möbius aromaticity. *J. Am. Chem. Soc*. **131**, 7240-7241 (2009).
Hawkea, L., Kalosakasa, G. & Simserides, C. Empirical LCAO parameters for pi molecular orbitals in planar organic molecules. Mol. Phys. 107, 17551771 (2009).
Harrison, W. A. Electronic structure and the properties of solids 2nd edn (Dover, New York, 1989).
Harrison, W. A. Elementary electronic structure (World Scientific, New Jersey, 1999).
NIST Chemistry Webbook and references therein. Available at: http://webbook.nist.gov/chemistry/. (Accessed: 4th December 2015).
Author contributions statement {#author-contributions-statement .unnumbered}
==============================
Yao Shen designed the project, wrote the main manuscript text and did the calculations. Yu-Zhu Chen reviewed the manuscript.
Additional information {#additional-information .unnumbered}
======================
Competing financial interests: The authors declare no competing financial interests.
|
---
abstract: 'Statistics of the single-particle levels in a deformed Woods-Saxon potential is analyzed in terms of the Poisson and Wigner nearest-neighbor distributions for several deformations and multipolarities of its surface distortions. We found the significant differences of all the distributions with a fixed value of the angular momentum projection of the particle, more closely to the Wigner distribution, in contrast to the full spectra with Poisson-like behavior. Important shell effects are observed in the nearest neighbor spacing distributions, the larger the smaller deformations of the surface multipolarities.'
author:
- 'J.P. Blocki'
- 'A.G. Magner[^1]'
title: |
CHAOTICITY AND SHELL EFFECTS IN THE\
NEAREST-NEIGHBOR DISTRIBUTIONS
---
PACS: 21.60. Sc, 21.10. -k, 21.10.Pc, 02.50. -r
INTRODUCTION
============
The microscopic many-body interaction of particles of the Fermi systems such as heavy nuclei is rather complicated. Therefore, several theoretical approaches to the description of the Hamiltonian which are based on the statistical properties of its discrete levels are applied for solutions of the realistic problems. For a quantitative measure of the degree of chaoticity of the many-body forces, the statistical distributions of the spacing between the nearest neighboring levels were introduced, first of all in relation to the so called Random Matrix Theory [@porter; @wigner; @brody; @mehta; @aberg]. Integrability (order) of the system was associated usually to the Poisson-like exponentially decreasing dependence on the spacing variable with a maximum at zero while chaoticity was connected more to the Wigner-like behavior with the zero spacing probability at zero but with a maximum at some finite value of this variable.
On the other hand, many dynamical problems, in particular, in nuclear physics can be reduced to the collective motion of independent particles in a mean field with a relatively sharp time-dependent edge called usually as the effective surface within the microscopic-macroscopic approximation [@myersswiat; @strut]. We may begin with the basic ideas of Swiatecki and his collaborators [@myersswiat; @wall; @blsksw1; @blshsw; @blsksw2; @jasw; @maskbl]. In recent years it became apparent that the collective nuclear dynamics is very much related to the nature of the nucleonic motion. This behavior of the nucleonic dynamics is important in physical processes like fission or heavy ion collisions where a great amount of the collective energy is dissipated into a chaotic nucleonic motion. We have to mention here also very intensive studies of the one-body dissipative phenomena described largely through the macroscopic wall formula (w.f.) for the excitation energy [@wall; @blsksw1; @blsksw2; @blshsw; @jasw; @maskbl] and also quantum results [@blsksw1; @blsksw2; @maskbl; @kiev2010]. The analytical w.f. was suggested originally in Ref. [@wall] on the basis of the Thomas-Fermi approach. It was re-derived in many works based on semiclassical and quantum arguments, see Refs [@koonrand; @maggzhfed; @gmf] for instance. However, some problems in the analytical study of a multipolarity dependence of the smooth one-body friction and its oscillating corrections as functions of the particle number should be still clarified. In particular, we would like to emphasize the importance of the transparent classical picture through the Poincare sections and Lyapunov exponents showing the order-chaos transitions [@arvieu; @blshsw; @heiss; @heiss2] and also quantum results for the excitation energy [@blsksw2; @maskbl; @kiev2010]. Then, the peculiarities of the excitation energies for many periods of the oscillations of the classical dynamics were discussed for several Legendre polynomials, see Ref. [@kiev2010]. as the classical measures of chaoticity. The shell correction method [@strut; @FH] was successfully used to describe the shell effects in the nuclear deformation energies as functions of the particle numbers. This is important also for understanding analytically the origin of the isomers in fission within the periodic orbit theory (POT) [@myab]. We should expect also that the deviations of the level density near the Fermi surface, like shell effects, from an averaged constant should influence essentially the nearest neighbor spacing distribution (NNSD). For a further study of the order-chaos properties of the Fermi systems, it might be worth to apply the statistical methods of the description of the single-particle (s.p.) levels of a mean-field Hamiltonian within microscopic-macroscopic approaches (see for instance Refs [@heiss1; @heiss3; @nazm1; @nazm2]).
The statistics of the spacing between the neighboring levels and their relation to the shell effects depending on the specific properties of the s.p. spectra, as well as the multipolarity and deformation of the shape surfaces should be expected. The quantitative measure of the order (or symmetry) can be the number of the single-valued integrals of motion, except for the energy (degree of the degeneracy of the system, see also Refs [@strutmag; @smod; @myab; @brbhad]). If the energy is the only one single-valued integral of motion one has the completely chaotic system [@gutz]. For the case of any such additional integral of motion, say the angular momentum projection for the azimuthal symmetry, one finds the symmetry enhancement that is important for calculations of the level density as the basic s.p. characteristics.
Our purpose now is to look at the order-chaos properties of the s.p. levels in terms of the Poisson and Wigner distributions with focus to their dependence on the multipolarities, equilibrium deformations and shell effects in relation to the integrability of the Hamiltonian through the comparison between the spectra with the fixed angular momenta of particles and full for the Woods-Saxon potential.
SPECTRA AND LEVEL DENSITIES
===========================
We are going now to study the statistical properties of the s.p. spectra of the eigenvalue problem, $$H \phi_i = \varepsilon_i \phi_i ,\qquad\qquad H=T+V,
\label{eigenprob}$$ where $H$ is a static mean-field Hamiltonian with the operator of the kinetic energy $T$ and deformed axially-symmetric Woods-Saxon (WS) potential, $$V\equiv V_{WS}\left({\bf r}\right)=-
\frac{V_0}{1+ \exp\left\{\left[r-R(\theta)\right]/a\right\}}\;,
\label{wspot}$$ $r,\theta,\varphi$ are the spherical coordinates of the vector ${\bf r}$. Following Refs [@blsksw1; @blshsw; @jasw; @maskbl], the shape of the WS-potential surface is defined by the effective radius $R(\theta)$ given by: $$R(\theta)=\frac{R^{}_0}{\lambda}\left[1+\alpha\; \sqrt{\frac{4\pi}{5}}\;
Y^{}_{n0}\left(\theta\right)
+\alpha_1 \sqrt{\frac{4\pi}{3}}\;
Y^{}_{10}\left(\theta\right)\right].
\label{radst}$$ Here, $\lambda$ is a normalization factor ensuring volume conservation, and $\alpha^{}_1$ stands for keeping a position of the center of mass for odd multipolarities and $R^{}_0$ is the radius of the equivalent sphere, $Y^{}_{n0}(\theta)=
\sqrt{(2n+1)/4\pi}\;P^{}_n\left(\cos\theta\right)$ are the spherical functions. $P^{}_n\left(\cos\theta\right)$ are the Legendre polynomials and $\alpha$ is the deformation parameter independent on time. For diagonalization of the Hamiltonian with the WS potential (\[wspot\]), the expansion over a basis of the deformed harmonic oscillator is used as shown in Ref. [@maskbl].
Figs \[fig1\] and \[fig2\] show two examples for the full spectra of the s.p. energies $\varepsilon_i$ and for the fixed angular momentum projection $m=0$ versus the deformation parameter $\alpha$ for the $P_2$ and $P_5$ shapes, respectively. The spectra for $P_3$ and $P_4$ are very similar to the $P_5$ case and therefore, they are not shown. As seen from Fig. \[fig1\], there are clear shell effects in the full spectra at small deformations, approximately at $\alpha \siml 0.1$ for all multipolarities from the $P_2$ shape to the $P_5$ one. With increasing deformation $\alpha$, the shell gaps become less pronounced and slowly changed in the region $\alpha \approx 0.1-0.4$ for all these multipolarities. Much more differences can be found in comparison of Fig. \[fig1\] for full spectra and Fig. \[fig2\] for $m=0$ levels only. The shell effects are seen here too but much less pronounced. The spectrum of levels with $m=0$ becomes more uniform with increasing multipolarity $n$.
The key quantity for calculations of the NNSD $P(S)$ is the level density $g(\varepsilon)$, see Appendix A and Refs [@wigner; @porter; @brody; @aberg; @mehta]. For these calculations one may apply the Strutinsky shell correction method writing $$g^{}_{\Gamma}(\varepsilon)=\tilde{g}(\varepsilon) +
\delta g^{}_\Gamma(\varepsilon).
\label{totlevden}$$ The smooth part $\tilde{g}(\varepsilon)$ is defined by the Strutinsky smoothing procedure [@strut; @FH]. The so called plateau condition (stability of values of the smooth level density $\tilde{g}$ as function of the averaging parameters: Gaussian width $\Gamma$ and the degree of the correction polynomial $M$ takes place at $\Gamma=20-40$ MeV and $M=4-8$). Figs \[fig3\] and \[fig4\] for full spectra and for fixed angular momentum projection $m=0$ show the typical examples of the level densities (smooth component and the total density with the oscillating part) for the same degree of the Legendre polynomials $n=2$ and $5$ at deformations $\alpha=0.1$ and $0.4$, in correspondence with spectra presented in Figs \[fig1\] and \[fig2\], respectively. In Fig. \[fig4\] for the case of the specific $m=0$ levels, one has somewhat larger Gaussian width parameters of the smooth level density $\tilde{g}$ in the total density (Eq. (\[totlevden\])) than those for the full spectra in Fig. \[fig3\]. As seen clearly from Figs \[fig3\] and \[fig4\] the smooth level density for the $m=0$ spectra is more flat (besides of relatively small remaining oscillations because of much less levels with the fixed $m=0$), as compared to the full-spectra results. This more flat behavior for the fixed $m$ value is due to the loss of the symmetry. Therefore one expects the system with the fixed angular momentum $m$ to be more chaotic. The differences between Fig. \[fig3\] and Fig. \[fig4\] are quite remarkable. On the other hand differences between pictures within Fig. \[fig3\] or Fig. \[fig4\] are less notable and as one can see shell effects are still remaining at bigger deformations and multipolarities.
NEAREST-NEIGHBOR SPACING DISTRIBUTIONS
======================================
Following the review paper [@brody], the distribution ${\cal P}(S)$ for the probability of finding the spacing $S$ between the nearest neighboring levels is given by (see also Refs [@wigner; @porter; @aberg; @mehta] and Appendix A) $${\cal P}(S)=g(S)\;
\exp\left(-\int_0^S g(x)\;\mbox{d}x\right)/\aleph.
\label{genwig}$$ The key quantity $g(S)$ can be considered as the density of the s.p. levels counted from a given energy, say, the Fermi energy $E_F$. $D$ is a mean uniform distance between neighboring levels so that $1/D$ is the mean density of levels. $\aleph$ is the normalization factor for large enough maximal value of $S$, $S_{max}$, $$\aleph=\int_0^{S_{max}}\mbox{d}x g(x)
\;exp\left(-\int_0^x g(y)\mbox{d}y\right)/D.
\label{norm}$$ This normalization factor $\aleph$ can be found from the normalization conditions: $$\int \mbox{d}x {\cal P}(x)=
\int \mbox{d}x x {\cal P}(x)= 1\;.
\label{normcond}$$ \[Notice that for convenience we introduced the dimensionless probability ${\cal P}$ in contrast to that of Ref. [@brody] denoted as $P(S)$, see Eq. (1.3) there.\]
The Poisson law follows if we take constant for the level density, $g(S)=1/D$, in Eq. (\[genwig\]), $${\cal P}(S)
=\exp\left(-S/D\right)\;.
\label{pois}$$ Wigner’s law follows from the assumption of the linear level density, proportional to $S$, $${\cal P}(S)
=\left(\pi S/2 D\right)\;\exp\left(-\pi S^2/4 D^2\right)\;.
\label{wig}$$ Both distributions are normalized to one for large enough maximal value of $S$, $S_{max}=\infty$ to satisfy Eq. (\[normcond\]).
The level density in fact is not a constant or $\propto S$. The combination of the Poisson and Wigner distributions was suggested in Ref. [@berrob] by introducing one parameter. For our purpose to keep a link with the properties of the level density, like smooth and shell components [@strut], it is convenient to define the probability ${\cal P}(S)$ (Eq. (\[genwig\])) for a general linear level-density function through two parameters ${\cal A}$ and ${\cal B}$, $$g(S)=\left({\cal A} + {\cal B}S/D\right)/D\;.
\label{denlin}$$ Substituting Eq. (\[denlin\]) into the general formula (Eq. (\[genwig\])) one obtains explicitly the analytical result in terms of the standard error functions, $\mbox{erf}(z)=2\int_0^z \mbox{d}x\;\exp(-x^2_{})/\sqrt{\pi}$, $${\cal P}(S)=\left(1+{\cal B} \xi/{\cal A}\right)\;
\exp\left(-{\cal B} \xi^2_{}/2 - {\cal A} \xi\right)/
\left[\aleph_0 + {\cal B}\;\aleph_1/{\cal A}\right]\,,
\label{pslinPW}$$ $$\begin{aligned}
\aleph_0&=&
\int_0^{\cal C} \mbox{d}\xi \;
\exp\left(-\frac{{\cal B}}{2} \xi^2 - {\cal A}\xi\right)=
\sqrt{\frac{\pi}{2{\cal B}}}\;\exp\left(\frac{{\cal A}^2}{2{\cal B}^2}\right)\;
\mbox{erf}\left(\frac{{\cal A}+{\cal B}{\cal C}}{\sqrt{2{\cal B}}}\right),
\nonumber\\
\aleph_1&=&
\int_0^{\cal C} \mbox{d}\xi \xi\;\exp\left(-\frac{{\cal B}}{2} \xi^2 -
{\cal A} \xi\right)=
-\frac{1}{{\cal B}}\left[\exp\left(-\frac{{\cal B}}{2} {\cal C}^2 -
{\cal A} {\cal C}\right)+ {\cal A}\;\aleph_0\right].
\label{pslin}\end{aligned}$$ where $\xi=S/D$, ${\cal C}=S_{max}/D$ is the maximal value of $\xi$. For large ${\cal C} \rightarrow \infty$ one has simply $\aleph_0 \rightarrow \sqrt{\pi/2{\cal B}}\;
\exp\left({\cal A}^2/2{\cal B}^2\right)$ and $\aleph_1 \rightarrow -{\cal A}\aleph_0/{\cal B}$. Taking the limits ${\cal A} \rightarrow 1$, ${\cal B}\rightarrow 0$ and ${\cal A} \rightarrow 0$, ${\cal B}\rightarrow 1$ in (\[pslinPW\]) one simply finds exactly the standard Poisson (Eq. (\[pois\])) and Wigner (Eq. (\[wig\])) distributions. In this way the constants ${\cal A}$ and ${\cal B}$ are measures of the probability to have Poisson and Wigner distributions (Eq. (\[denlin\])).
NUMERICAL RESULTS
=================
Figs \[fig5\] and \[fig6\] show the corresponding NNSD $P(S)$ (Eq. (\[genwig\])). Again, in accordance with spectra (see Figs \[fig1\], \[fig2\]) and level-density calculations in Figs \[fig3\] and \[fig4\], the dramatic changes are observed between Fig. \[fig6\] for the NNSD $P(S)$ with the $m=0$ and Fig. \[fig5\] for those of the full spectra ones. Results presented by heavy dots in Fig. \[fig5\] look more close to the Poisson distribution and those in Fig. \[fig6\] are more close to the Wigner one.
There are a large difference in numbers ${\cal A}$ and ${\cal B}$ which measure the closeness of the distributions $P(S)$ for the neighboring levels spacing to the standard ones, Poisson (1,0) and Wigner (0,1). However, in Fig. \[fig6\] all distributions are more close to the Wigner in shape having a maximum between zero and large compared to $D$ value $S_{max}$ with respect to $D$ \[ $S_{max}={\cal C}D$, see immediately after Eq. (\[pslin\])\] than monotonous exponential-like decrease similar to the Poisson behavior in Fig. \[fig5\]. Notice that we have more pronounced Wigner-like distribution with increasing multipolarity $n$ and deformation $\alpha$ in Fig. \[fig6\], especially remarkable at $P_5 $ surface distortions and large enough deformation $\alpha=0.4$, see last plot $(d)$ in Fig. \[fig6\]. Including all the angular momentum projections $m$ for all desired multipolarities and deformations one has clearly Poisson-like behavior though they differ essentially in numbers ${\cal A},{\cal B}$ from the standard ones (1,0), see Fig. \[fig6\].
The reason for this can be understood looking at the Poincare sections shown in Fig. \[fig7\] [@arvieu; @blshsw; @kiev2010]. The upper row is related to a small deformation and lower row corresponds to a large deformation. The projection of the angular momentum is $m=0$ in all pictures of Fig. \[fig7\]. Difference is remarkable for the integrable spheroidal cavity and other non-integrable (in the plane of the symmetry axis) shapes. As seen from comparison of upper and lower plot lines, with increasing deformation $\alpha$ and multipolarity $n$ we find more chaotic behavior and we should expect therefore the NNSD closer to the Wigner distribution (\[wig\]). This is in agreement with the NNSD calculations for the fixed $m=0$, see Fig. \[fig6\]. Notice that similar properties of the NNSD for other potentials and constraints were discussed in Refs [@heiss1; @heiss3; @nazm1; @nazm2].
The difference between the NNSD calculations with the realistic level densities by the Strutinsky shell-correction method (see Eq. (\[totlevden\])) for the considered WS potential and those with their idealistic linear behavior (Eq. (\[denlin\])) can be studied in terms of the general formula $P(S)$ (Eq. (\[genwig\])). In particular, the shell effects related to the inhomogeneity of the s.p. levels near the Fermi surface for all desired multipolarities and deformations are found to be significant, also in relation to the fixed quantum number $m$.
Figs \[fig8\] and \[fig9\] show the results of these calculations corresponding to Figs \[fig5\] and \[fig6\]. Notice that in the case of the full spectra, see Fig. \[fig8\], one has Poisson-like distributions corresponding to the smooth density (dashed) with a similar behavior as NNSD shown by heavy dots, in contrast to Fig. \[fig9\] where we find rather big differences between these curves. The shell effects are measured by the differences between the solid curve related to the total level density with the shell components and the dashed one for the smooth level density of Figs \[fig3\] (all $m$) and \[fig4\] (with $m=0$), see correspondingly Figs \[fig8\] and \[fig9\]. With increasing deformations one has slightly decreasing the shell effects, in contrast to the multipolarity dependence for which there is almost no change of the shell effects at the same deformations.
CONCLUSIONS {#concl}
===========
We studied the statistics of the neighboring s.p. levels in the WS potential for several typical multipolarities and deformations of the surface shapes and deformations, as compared with the standard Poisson and Wigner distributions $P(S)$. For the sake of comparison, we derived analytically the combine asymptotic Poisson-Wigner distribution $P(S)$ related to the general linear dependence of the corresponding s.p. level density. We found the significant differences between distributions for a fixed value of the angular momentum projection $m$ of the particle and those accounting all possible values of $m$. For the case of the fixed $m=0$ we obtained distributions $P(S)$ more close to the Wigner shape with the maximum between $S=0$ and a maximal large value of $S$, the more pronounced the larger multipolarity and deformation of the potential surface. We found also that the full spectra distributions $P(S)$ look Poisson-like in a sense that they have maximum at $S=0$ and almost exponential decrease as a function of the energy near the Fermi surface. Our results clarify the widely extended opinion of the relation of the distributions (Poisson or Wigner) to the integrability of the problem (the integrable or chaotic one). All considered potentials are axially-symmetric but they are the same [*non-integrable*]{} ones in the plane of the symmetry axis. However, the degree of the symmetry (classical degeneracy [@myab; @strutmag; @smod] ${\cal K}$, i.e. the number of the single-valued integrals of motion besides of the energy) for the case of the full spectra, Figs \[fig1\], \[fig3\], \[fig5\], ( ${\cal K}=1$, a mixed system) is higher than for the fixed angular momentum $m$ (${\cal K}=0$ like for the completely chaotic system). Notice that integrability is not only one criterium of chaoticity. The measure of the differences of the distributions $P(S)$ between Poisson and Wigner standard ones depends also on the properties of the energy dependence of the level density (from constant to proportional-to-energy dependence). From comparison between the general distribution $P(S)$ related to the smooth level density obtained by the Strutinsky shell correction method and the statistics of the neighboring s.p. levels one finds that all of them are more close to the Poisson-like behavior. This shows that the energy dependence of the smooth level density differs much from the linear functions. We obtained also large shell effects in the distributions $P(S)$ in nice agreement with those in the key quantity in this analysis, - level density dependencies on the energy near the Fermi surface.
As to perspectives, it might be necessary to use the combined microscopic-macroscopic approaches [@myersswiat; @strut] to clear up the results more systematically and analytically. Our quantum results can be interesting for understanding the one-body dissipation at slow and faster collective dynamics with different shapes like the ones met in nuclear fission and heavy-ion collisions.
Acknowledgements
================
We thank S. Aberg, V.A. Plujko and S.V. Radionov for valuable discussions.
**Appendix A: A derivation of the NNSD** \[appA\]
We introduce first the level density, $g(E)$, as the number of the levels ${\mbox d} N$ in the energy interval \[$E,E+{\mbox d} E$\] divided by the energy interval, $g(E)={\mbox d} N/{\mbox d} E$. With the help of this quantity one can derive the NNSD $P(S)$ as the probability density versus the spacing $S$ between the nearest neighboring levels. Specifying $P(S)$ to the problem with the known s.p. spectra of the Hamiltonian, one can split the energy interval $\Delta E$ under the investigation into many small (equivalent for simplicity) parts $\Delta S \ll \Delta E$. Each of $\Delta S$ nevertheless contain many energy levels, $\Delta S \gg D$. Then, we find the number of the levels which occur inside of the small interval $\Delta S$. Normalizing these numbers by the total number of the levels inside the total energy interval $\Delta E$ one obtains the distribution which we shall call as the probability density $P(S)$. Notice that the result of this calculation depends on the energy length of the selected $\Delta S$. In our calculations, we select $\Delta S$ by the condition of a sufficient smoothness of the distribution $P(S)$. Such procedure is often used for the statistical treatment of the experimentally obtained spectrum with the fixed quantum numbers like the angular momentum, parity and so on [@brody].
Following mainly Ref. [@aberg], let us calculate first the intermediate quantity $f(S)$ as the probability that there is no energy level in the energy interval \[$E,E+S$\]. According to the general definition of the level density mentioned above, $g(S) {\mbox d} S$ can be considered as the probability that there is one energy level in $[E+S, E+S+{\mbox d} S]$. Then, $$f(S+{\mbox d} S)=f(S)\left(1-g(S) {\mbox d} S\right)\;,
\label{f}$$ which leads to the differential equation for $f(S)$, $${\mbox d} f= -g(S) {\mbox d} S f(S).
\label{diff}$$ Solving this equation one gets $$f(S)=C \exp\left(-\int_0^S g(x) {\mbox d} x\right)\;.
\label{fsol}$$ Let $P(S) \mbox{d} S$ denote the probability that the next energy level is in $[E+S, E+S+\mbox{d} S]$, $$P(S) {\mbox d} S = f(S) g(S) {\mbox d} S\;.
\label{defps}$$ Then, substituting Eq. (\[fsol\]) into Eq. (\[defps\]) one finally arrives at the general distribution: $$P(S)=C g(S) \exp\left(-\int_0^S g(S') {\mbox d} S'\right).
\label{psol}$$ The boundary conditions in solving the differential equation (\[diff\]) accounts for the meaning of the NNSD $P(S)$ and its argument as the spacing between the nearest neighbor levels as shown in the integration limit in Eq. (\[psol\]). The constant $C$ is determined from the normalization conditions (Eq. (\[norm\])).
[99]{}
E.P. Wigner, Proc. Cambridge Philos. Soc. [**47**]{}, 790 (1951); Canadian Math. Congress Proceedings, University of Toronto Press, Toronto, Canada, 174 (1957); [*ibid.*]{} SIAM Rev. [**9**]{}, 1 (1967).
C.E. Porter, [*Statistical theories of spectra: fluctuations*]{}, Academy Press, New York (1965).
T.A. Brody et al., Rev. Mod., [**53**]{}, 385 (1981).
M.L. Mehta, [*Random Matrices*]{}, Academic Press, San Diego, New York, Boston, London, Sydney, Tokyo, Toronto (1991).
S. Aberg, [*Quantum Chaos*]{}, (Mathematical Physics, Lund, Sweden, 2002).
W.D. Myers and W.J. Swiatecki, Nucl. Phys. [**81**]{}, 1 (1966); [*ibid.*]{} Ann. Phys. (N.Y.) [**55**]{}, 395 (1969).
V.M. Strutinsky, Nucl. Phys. A [**95**]{}, 420 (1967); [*ibid.*]{} [**122**]{}, 1 (1968).
J. Blocki, Y. Boneh, J.R. Nix, J. Randrup, M. Robel, A.J. Sierk, and W.J. Swiatecki, Ann. Phys. (N.Y.) [**113**]{}, 330 (1978).
J. Blocki, J. Skalski, W.J. Swiatecki, Nucl. Phys. B [**594**]{}, 137 (1995).
J. Blocki, J.-J. Shi and W.J. Swiatecki, Nucl. Phys. A [**554**]{}, 387 (1993).
J. Blocki, J. Skalski, W.J. Swiatecki, Nucl. Phys. A [**618**]{}, 1 (1997).
C. Jarzynski and W.J. Swiatecki, Nucl. Phys. A [**552**]{}, 1 (1993).
P. Magierski, J. Skalski and J. Blocki, Phys. Rev. C [**56**]{}, 1011 (1997).
J.P. Blocki, A.G. Magner, and I.S. Yatsyshyn, At. Nucl. and Energy, [**11**]{}, 239 (2010); [*ibid*]{} Int. J. Mod. Phys. E [**20**]{}, 292 (2011).
S.E. Koonin and J. Randrup, Nucl. Phys. A [**289**]{}, 475 (1977).
A.G. Magner, A.M. Gzhebinsky, S.N. Fedotkin, Phys. Atom. Nucl. [**70**]{}, 647 (2007); [*ibid*]{} [**70**]{}, 1859 (2007).
A.M, Gzhebinsky, A.G. Magner, and S.N. Fedotkin, Phys. Rev. C [**76**]{}, 064315 (2007).
R. Arvieu, F. Brut, J. Carbonell, and J. Touchard, Phys. Rev. A [**35**]{}, 2389 (1987); [*ibid.*]{} Nucl. Phys. A [**545**]{}, C497 (1992).
W.D. Heiss, R.G. Nazmitdinov, and S. Radu, Phys. Rev. Lett. [**72**]{}, 2351 (1994).
W.D. Heiss and R.G. Nazmitdinov, Phys. Rev. Lett. [**73**]{}, 1235 (1994).
M. Brack et al., Rev. Mod. Phys., [**44**]{}, 320 (1972).
A.G. Magner, I.S. Yatsyshyn, K. Arita, and M. Brack, Phys. Atom. Nucl., [**74**]{}, 1475 (2011).
W.D. Heiss, R.G. Nazmitdinov, and S. Radu, Phys.Rev. C [**52**]{}, 3032 (1995).
W.D. Heiss and R.G. Nazmitdinov, Physica D, [**118**]{}, 134 (1998).
A. Hamoudi, R.G. Nazmitdinov, E. Shahaliev, and Y. Alhassid, Phys. Rev. C [**65**]{}, 064311 (2002).
R.G. Nazmitdinov, E.I. Shahaliev, M.K. Suleymanov, and S. Tomsovic, Phys. Rev. C [**79**]{}, 054905 (2009).
V.M. Strutinsky and A.G. Magner, Sov. Phys. Part. Nucl., [**7**]{}, 138 (1976).
V.M. Strutinsky, A.G. Magner, S.R. Ofengenden, and T. D[ø]{}ssing, Z. Phys. A [**283**]{}, 269 (1977).
M. Brack and R.K. Bhaduri, [*Semiclassical Physics. Frontiers in Physics*]{}, [**96**]{}, Addison-Wesley, Reading, MA. (1997); 2nd edition, Westview Press, Boulder (2003).
M. Gutzwiller, J. Math. Phys. [**12**]{}, 343 (1971); [*Chaos in Classical and Quantum Mechanics*]{}, Springer-Verlag, N.Y. (1990).
M.V. Berry and M. Robnik, J. Phys. A [**17**]{}, 2413 (1984).
![ The s.p. energy levels $\varepsilon_i$ in the WS potential ($V_0=200 $ MeV, $R_0=6.622$ fm, $a=0.1$ fm) as function of the deformation $\alpha$ for the $P_2$ (left) and $P_5$ (right) shapes (Eq. (\[radst\])).[]{data-label="fig1"}](fig1a.eps "fig:"){width="45.00000%"} ![ The s.p. energy levels $\varepsilon_i$ in the WS potential ($V_0=200 $ MeV, $R_0=6.622$ fm, $a=0.1$ fm) as function of the deformation $\alpha$ for the $P_2$ (left) and $P_5$ (right) shapes (Eq. (\[radst\])).[]{data-label="fig1"}](fig1b.eps "fig:"){width="45.00000%"}
![ The same s.p. spectrum of levels as in Fig. \[fig1\] but with the projection of the angular momentum $m=0$.[]{data-label="fig2"}](fig2a.eps "fig:"){width="45.00000%"} ![ The same s.p. spectrum of levels as in Fig. \[fig1\] but with the projection of the angular momentum $m=0$.[]{data-label="fig2"}](fig2b.eps "fig:"){width="45.00000%"}
![ The level densities $g_\Gamma(S)$ (Eq. (\[totlevden\])) as a function of the energy counted from the Fermi level ($S=0$) for a given particle number $N$ for spectra of the s.p. levels of Fig. \[fig1\] for the $P_2$ $(a,b)$ and $P_5$ $(c,d)$ shapes at the small $\alpha=0.1$ (left, $a,c$) and larger $\alpha=0.4$ (right,$b,d$) deformations; dashed is the smooth density $\tilde{g}(S)$ (Eq. (\[totlevden\])); solid is the total density $g_\Gamma(S)$ (Eq. (\[totlevden\])) ($\Gamma=3$ MeV, $M=0$ for $\delta g_\Gamma(S)$).[]{data-label="fig3"}](fig3a.eps "fig:"){width="45.00000%"} ![ The level densities $g_\Gamma(S)$ (Eq. (\[totlevden\])) as a function of the energy counted from the Fermi level ($S=0$) for a given particle number $N$ for spectra of the s.p. levels of Fig. \[fig1\] for the $P_2$ $(a,b)$ and $P_5$ $(c,d)$ shapes at the small $\alpha=0.1$ (left, $a,c$) and larger $\alpha=0.4$ (right,$b,d$) deformations; dashed is the smooth density $\tilde{g}(S)$ (Eq. (\[totlevden\])); solid is the total density $g_\Gamma(S)$ (Eq. (\[totlevden\])) ($\Gamma=3$ MeV, $M=0$ for $\delta g_\Gamma(S)$).[]{data-label="fig3"}](fig3b.eps "fig:"){width="45.00000%"}
![ The level densities $g_\Gamma(S)$ (Eq. (\[totlevden\])) as a function of the energy counted from the Fermi level ($S=0$) for a given particle number $N$ for spectra of the s.p. levels of Fig. \[fig1\] for the $P_2$ $(a,b)$ and $P_5$ $(c,d)$ shapes at the small $\alpha=0.1$ (left, $a,c$) and larger $\alpha=0.4$ (right,$b,d$) deformations; dashed is the smooth density $\tilde{g}(S)$ (Eq. (\[totlevden\])); solid is the total density $g_\Gamma(S)$ (Eq. (\[totlevden\])) ($\Gamma=3$ MeV, $M=0$ for $\delta g_\Gamma(S)$).[]{data-label="fig3"}](fig3c.eps "fig:"){width="45.00000%"} ![ The level densities $g_\Gamma(S)$ (Eq. (\[totlevden\])) as a function of the energy counted from the Fermi level ($S=0$) for a given particle number $N$ for spectra of the s.p. levels of Fig. \[fig1\] for the $P_2$ $(a,b)$ and $P_5$ $(c,d)$ shapes at the small $\alpha=0.1$ (left, $a,c$) and larger $\alpha=0.4$ (right,$b,d$) deformations; dashed is the smooth density $\tilde{g}(S)$ (Eq. (\[totlevden\])); solid is the total density $g_\Gamma(S)$ (Eq. (\[totlevden\])) ($\Gamma=3$ MeV, $M=0$ for $\delta g_\Gamma(S)$).[]{data-label="fig3"}](fig3d.eps "fig:"){width="45.00000%"}
![ The same as in Fig. \[fig3\] but for levels with the fixed projection of the angular momentum $m=0$.[]{data-label="fig4"}](fig4a.eps "fig:"){width="45.00000%"} ![ The same as in Fig. \[fig3\] but for levels with the fixed projection of the angular momentum $m=0$.[]{data-label="fig4"}](fig4b.eps "fig:"){width="45.00000%"}
![ The same as in Fig. \[fig3\] but for levels with the fixed projection of the angular momentum $m=0$.[]{data-label="fig4"}](fig4c.eps "fig:"){width="45.00000%"} ![ The same as in Fig. \[fig3\] but for levels with the fixed projection of the angular momentum $m=0$.[]{data-label="fig4"}](fig4d.eps "fig:"){width="45.00000%"}
![ The distributions of spacing of the neighboring levels $P(S)$ represented by heavy dots [@brody] vs the energies $S$ for the same spectra as in Figs \[fig1\] and \[fig3\]. Solid curve is a standard Poisson distribution (Eq. (\[pois\])) and a dotted one is a standard Wigner distribution (\[wig\]). Numbers in brackets $({\cal A},{\cal B})$ show ${\cal A}$ and ${\cal B}$ of Eq. (\[pslinPW\]). Dashed curve corresponds to a linear approximation to the level density (Eq. (\[denlin\])); other notations are the same as in Figs \[fig3\] and \[fig4\].[]{data-label="fig5"}](fig5a.eps "fig:"){width="45.00000%"} ![ The distributions of spacing of the neighboring levels $P(S)$ represented by heavy dots [@brody] vs the energies $S$ for the same spectra as in Figs \[fig1\] and \[fig3\]. Solid curve is a standard Poisson distribution (Eq. (\[pois\])) and a dotted one is a standard Wigner distribution (\[wig\]). Numbers in brackets $({\cal A},{\cal B})$ show ${\cal A}$ and ${\cal B}$ of Eq. (\[pslinPW\]). Dashed curve corresponds to a linear approximation to the level density (Eq. (\[denlin\])); other notations are the same as in Figs \[fig3\] and \[fig4\].[]{data-label="fig5"}](fig5b.eps "fig:"){width="45.00000%"}
![ The distributions of spacing of the neighboring levels $P(S)$ represented by heavy dots [@brody] vs the energies $S$ for the same spectra as in Figs \[fig1\] and \[fig3\]. Solid curve is a standard Poisson distribution (Eq. (\[pois\])) and a dotted one is a standard Wigner distribution (\[wig\]). Numbers in brackets $({\cal A},{\cal B})$ show ${\cal A}$ and ${\cal B}$ of Eq. (\[pslinPW\]). Dashed curve corresponds to a linear approximation to the level density (Eq. (\[denlin\])); other notations are the same as in Figs \[fig3\] and \[fig4\].[]{data-label="fig5"}](fig5c.eps "fig:"){width="45.00000%"} ![ The distributions of spacing of the neighboring levels $P(S)$ represented by heavy dots [@brody] vs the energies $S$ for the same spectra as in Figs \[fig1\] and \[fig3\]. Solid curve is a standard Poisson distribution (Eq. (\[pois\])) and a dotted one is a standard Wigner distribution (\[wig\]). Numbers in brackets $({\cal A},{\cal B})$ show ${\cal A}$ and ${\cal B}$ of Eq. (\[pslinPW\]). Dashed curve corresponds to a linear approximation to the level density (Eq. (\[denlin\])); other notations are the same as in Figs \[fig3\] and \[fig4\].[]{data-label="fig5"}](fig5d.eps "fig:"){width="45.00000%"}
![ The same as in Fig. \[fig5\] but for the spectra of Figs \[fig2\], \[fig4\].[]{data-label="fig6"}](fig6a.eps "fig:"){width="45.00000%"} ![ The same as in Fig. \[fig5\] but for the spectra of Figs \[fig2\], \[fig4\].[]{data-label="fig6"}](fig6b.eps "fig:"){width="45.00000%"}
![ The same as in Fig. \[fig5\] but for the spectra of Figs \[fig2\], \[fig4\].[]{data-label="fig6"}](fig6c.eps "fig:"){width="45.00000%"} ![ The same as in Fig. \[fig5\] but for the spectra of Figs \[fig2\], \[fig4\].[]{data-label="fig6"}](fig6d.eps "fig:"){width="45.00000%"}
![ Poincare sections $v_\rho$ vs $\rho$ for spheroid, $P_2$ and $P_5$ shapes at the small deformation $\alpha=0.1$ (upper row) and large deformation $\alpha=0.4$ (lower row) for the projections of the angular momentum on the symmetry axis $m=0$.[]{data-label="fig7"}](fig7.eps){width="90.00000%"}
![ The general distribution $P(S)$ (Eq. (\[genwig\])) vs the energies $S$ for the same spectra as in Figs \[fig1\], \[fig3\], \[fig5\]; dashed are the distributions $P(S)$ related to the Strutinsky smooth density $\tilde{g}(S)$ and solid is the total level density $g_\Gamma(S)$ (Eq. (\[totlevden\])); dots are the same as in Figs \[fig5\] and \[fig6\].[]{data-label="fig8"}](fig8a.eps "fig:"){width="45.00000%"} ![ The general distribution $P(S)$ (Eq. (\[genwig\])) vs the energies $S$ for the same spectra as in Figs \[fig1\], \[fig3\], \[fig5\]; dashed are the distributions $P(S)$ related to the Strutinsky smooth density $\tilde{g}(S)$ and solid is the total level density $g_\Gamma(S)$ (Eq. (\[totlevden\])); dots are the same as in Figs \[fig5\] and \[fig6\].[]{data-label="fig8"}](fig8b.eps "fig:"){width="45.00000%"}
![ The general distribution $P(S)$ (Eq. (\[genwig\])) vs the energies $S$ for the same spectra as in Figs \[fig1\], \[fig3\], \[fig5\]; dashed are the distributions $P(S)$ related to the Strutinsky smooth density $\tilde{g}(S)$ and solid is the total level density $g_\Gamma(S)$ (Eq. (\[totlevden\])); dots are the same as in Figs \[fig5\] and \[fig6\].[]{data-label="fig8"}](fig8c.eps "fig:"){width="45.00000%"} ![ The general distribution $P(S)$ (Eq. (\[genwig\])) vs the energies $S$ for the same spectra as in Figs \[fig1\], \[fig3\], \[fig5\]; dashed are the distributions $P(S)$ related to the Strutinsky smooth density $\tilde{g}(S)$ and solid is the total level density $g_\Gamma(S)$ (Eq. (\[totlevden\])); dots are the same as in Figs \[fig5\] and \[fig6\].[]{data-label="fig8"}](fig8d.eps "fig:"){width="45.00000%"}
![ The same as in Fig. \[fig8\] but for spectra of Figs \[fig2\], \[fig4\] and \[fig6\].[]{data-label="fig9"}](fig9a.eps "fig:"){width="45.00000%"} ![ The same as in Fig. \[fig8\] but for spectra of Figs \[fig2\], \[fig4\] and \[fig6\].[]{data-label="fig9"}](fig9b.eps "fig:"){width="45.00000%"}
![ The same as in Fig. \[fig8\] but for spectra of Figs \[fig2\], \[fig4\] and \[fig6\].[]{data-label="fig9"}](fig9c.eps "fig:"){width="45.00000%"} ![ The same as in Fig. \[fig8\] but for spectra of Figs \[fig2\], \[fig4\] and \[fig6\].[]{data-label="fig9"}](fig9d.eps "fig:"){width="45.00000%"}
[^1]: magner@kinr.kiev.ua
|
---
abstract: |
Lattice data for the QCD equation of state and the magnetic susceptibility computed near the crossover transition at zero magnetic field are used to determine the input parameters of a five dimensional Einstein-Maxwell-Dilaton holographic model. Once the model parameters are fixed at zero magnetic field, one can use this holographic construction to study the effects of a magnetic field on the equilibrium and transport properties of the quark-gluon plasma. In this paper we use this model to study the dependence of the crossover temperature with an external magnetic field. Our results for the pressure of the plasma and the crossover temperature are in quantitative agreement with current lattice data for values of the magnetic field $0 \le eB \lesssim 0.3$ GeV$^2$, which is the relevant range for ultrarelativistic heavy ion collision applications.\
**Keywords:** Holography, thermodynamics, magnetic field, equation of state, magnetic susceptibility, crossover transition.
author:
- Romulo Rougemont
- Renato Critelli
- Jorge Noronha
title: Holographic calculation of the QCD crossover temperature in a magnetic field
---
Introduction
============
Recent relativistic heavy ion collision experiments [@expQGP1; @expQGP2; @expQGP3; @expQGP4] have produced a strongly coupled quark-gluon plasma (QGP) [@QGP] whose physical properties are currently under intense investigation (see [@reviewQGP1; @reviewQGP2] for recent reviews). The study of the equilibrium and transport properties of the QGP as functions of parameters such as the temperature $T$, chemical potential(s), and (electro)magnetic fields are of great relevance for the characterization and understanding of this new state of QCD matter. In particular, very strong magnetic fields up to $\mathcal{O}\left(0.3\,\textrm{GeV}^2\right)$ are expected to be created in the early stages of noncentral relativistic heavy ion collisions [@noncentralB1; @noncentralB2; @noncentralB3; @noncentralB4; @noncentralB5; @noncentralB6][^1] and even much larger magnetic fields of $\mathcal{O}\left(4\,\textrm{GeV}^2\right)$ may have been produced in the early stages of the Universe [@universe1; @universe2] (see also Fig. 10 in [@latticedata0]). Moreover, magnetic fields up to $\mathcal{O}\left(1\,\textrm{MeV}^2\right)$ are present in the interior of very dense neutron stars known as magnetars [@magnetar]. Therefore, the study of the effects of strong magnetic fields on the QGP has sparked a large amount of interest in the community in recent years [@Agasian:2008tb; @Mizher:2010zb; @Evans:2010xs; @Preis:2010cq; @Fukushima:2012xw; @Fukushima:2012kc; @Bali:2012zg; @Blaizot:2012sd; @Callebaut:2013ria; @Bali:2013esa; @Bonati:2014ksa; @Fukushima:2013zga; @Machado:2013rta; @Fraga:2013ova; @Andersen:2013swa; @Bali:2013owa; @Ferreira:2013oda; @Ruggieri:2014bqa; @Ferreira:2014kpa; @Farias:2014eca; @Ayala:2014iba; @Ayala:2014gwa; @Ferrer:2014qka; @Kamikado:2014bua; @Yu:2014xoa; @Braun:2014fua; @Mueller:2015fka; @Endrodi:2015oba] (for extensive reviews and other references, see for instance, [@Fraga:2012rr; @reviewfiniteB1; @reviewfiniteB2; @reviewfiniteB3]).
Since the properties of a strongly coupled QGP cannot be reliably studied using perturbative techniques one has to resort to nonperturbative approaches that are valid at strong coupling. Interestingly enough, contrary to what happens in the case of a nonzero baryon chemical potential where the sign problem of the fermion determinant prevents the application of the Monte Carlo importance sampling method in lattice simulations (for a review see [@fodorreview]), in the case of a nonzero magnetic field (at vanishing baryon chemical potential) standard lattice techniques may be employed to study the equilibrium properties of QCD in the $(T,B)$-plane, see for instance, [@latticedata0; @latticedata2; @latticedata3].
Another nonperturbative method that is suited to study strongly coupled non-Abelian gauge theories is the holographic anti-de Sitter/conformal field theory (AdS/CFT) correspondence (also known as the gauge/gravity duality) [@adscft1; @adscft2; @adscft3]. The correspondence has been employed to obtain useful insights into the properties of the strongly coupled QGP, as recently reviewed in [@solana; @adams]. A very attractive feature of the gauge/gravity duality is that it may be easily employed to compute transport coefficients of strongly coupled non-Abelian gauge theory plasmas (see, for instance, [@GN2; @conductivity; @hydro; @gubser2; @finitemu; @baryondiff; @Finazzo:2015xwa]), which is a challenging task to perform on the lattice [@Meyer:2011gj].
A top-down holographic dual for $\mathcal{N}=4$ super Yang-Mills theory (SYM) in the presence of an external constant magnetic field was proposed in [@DK1; @DK2; @DK3] and calculations for different physical observables in this scenario were carried out, for instance, in [@DK-applications1; @DK-applications2; @DK-applications3; @DK-applications4]. However, the QGP formed in heavy ion collisions [@reviewQGP1; @reviewQGP2] probes the temperature region within which the QCD plasma is highly nonconformal [@latticedata1] (when $T \sim 150-300$ MeV). Therefore, in order to make contact with realistic heavy ion collision applications, one needs to develop holographic models that are able to capture some of the relevant aspects of the physics of the strongly coupled QGP near the QCD crossover [@Aoki:2006we]. One possible way to accomplish this within holography is to deform the boundary quantum field theory by turning on a dynamical scalar field in the bulk whose boundary value sources a relevant operator in the gauge theory. Near the boundary the scalar field approaches zero and conformal invariance is recovered in the ultraviolet. In the infrared, however, the holographic dual gauge theory generated by such deformation behaves very differently than a conformal plasma and may be tuned to display some of the properties of QCD in the strong coupling regime.
In this work we construct a nonconformal anisotropic bottom-up holographic model that is suited for the study of a QCD-like plasma at nonzero magnetic field and vanishing chemical potential(s). Our model is built up on classical nonconformal anisotropic black brane solutions to the Einstein-Maxwell-Dilaton (EMD) model defined with a negative cosmological constant and in the presence of an external constant magnetic field. This constitutes a sequel to the studies of strongly coupled nonconformal plasmas via black brane solutions initiated by [@GN1; @GN2] in the case of finite temperature, zero magnetic field, and vanishing chemical potential[^2], which was later extended in [@gubser1; @gubser2] and also [@finitemu] to take into account the presence of a nonzero baryon chemical potential at zero magnetic field[^3]. This type of nonconformal model has been used in the last years to investigate how different observables of phenomenological relevance to the QGP and the physics of heavy ion collisions vary near the QCD crossover transition. In fact, after the original calculations in [@GN1; @GN2], which included the evaluation of the bulk viscosity at zero baryon chemical potential and zero magnetic field [@GN2], a series of other quantities were computed within this type of holographic model such as the heavy quark free energy [@Noronha:2009ud; @Noronha:2010hb], the energy loss of highly energetic probes [@Ficnar:2010rn; @Ficnar:2011yj; @Ficnar:2012yu], the Debye screening mass [@stefanovacuum], the electric conductivity [@conductivity], a large set of first and second order viscous hydrodynamic transport coefficients [@hydro], the spectrum of quasinormal modes [@Janik:2015waa] and the thermal photon production rate [@yang-muller]. In the context of the holographic models developed in [@gubser1; @gubser2] and [@finitemu] as extensions of the original models [@GN1; @GN2], taking into account the presence of a nonvanishing baryon chemical potential, we mention the calculation of the holographic critical point in the $(T,\mu_B)$-plane and the associated critical exponents [@gubser1], the evaluation of the holographic equation of state, the heavy quark drag force, the Langevin diffusion coefficients, the jet quenching parameter, the energy loss of light quarks and an estimate of the equilibration time in the baryon-rich strongly coupled QGP [@finitemu], the evaluation of the bulk viscosity [@gubser2], as well as the baryon susceptibility, baryon conductivity, thermal conductivity, baryon diffusion [@baryondiff], and the thermal photon and dilepton production rates [@Finazzo:2015xwa] at finite baryon chemical potential and zero magnetic field. Here we add one more entry to this family of nonconformal black hole solutions by taking into account, for the first time, the presence of a magnetic field in the nonconformal, QCD-like gauge theory.
Our model is a bottom-up holographic setup in which the dilaton potential and the Maxwell-Dilaton gauge coupling are dynamically fixed in order to describe lattice data at zero chemical potential(s) and vanishing magnetic field, which should be contrasted with top-down models coming from compactifications of known string theory solutions. Although in bottom-up models the holographic dual is not precisely known, the fact that these models may be constructed using some phenomenological input from QCD makes it possible that at least part of the physics of the boundary gauge field theory resembles, even at the quantitative level, QCD in the strong coupling limit. Thus, one may regard such constructions as holographic effective theories that are engineered to model some specific aspects of QCD phenomenology. Once the model parameters are fixed, these theories can be used to make predictions about observables that are currently beyond the scope of lattice calculations, such as most of the second order hydrodynamic coefficients [@hydro].
This paper is organized as follows. In Section \[sec2\] we describe in detail the construction of our holographic model and how the dilaton potential and the Maxwell-Dilaton gauge coupling can be determined by lattice data for the $(2+1)$-flavor lattice QCD equation of state and magnetic susceptibility at zero magnetic field, respectively. With the holographic model parameters fully specified, we proceed in Section \[sec3\] to obtain the holographic equation of state at nonzero magnetic field and present results for the temperature and magnetic field dependence of the entropy density and the pressure. We find that the deconfinement temperature in our holographic model decreases with an increasing magnetic field, as recently observed on the lattice. Moreover, our model results for the pressure and the crossover temperature are in quantitative agreement with current lattice data up to $eB \lesssim 0.3$ GeV$^2$, which is the relevant range of magnetic fields for heavy ion collisions. We present our conclusions in Section \[conclusion\] where we also point out other applications to be pursued in the near future using the anisotropic nonconformal holographic model developed here.
Throughout this paper we use natural units $c=\hbar=k_B=1$ and a mostly plus metric signature.
The holographic model {#sec2}
=====================
Assuming as usual that charm quarks are not relevant in the crossover transition, in QCD there are three different chemical potentials associated with three independent globally conserved charges. These different chemical potentials are the three lighter quark chemical potentials $\mu_u$, $\mu_d$, $\mu_s$ or, equivalently, the baryon chemical potential $\mu_B$, the electric charge chemical potential $\mu_Q$, and the strangeness chemical potential $\mu_S$. For each nonzero chemical potential in the gauge theory there must be a nonzero temporal component of the associated gauge field in the bulk. It is also clear that an Abelian magnetic field $B$ in the gauge theory should come from a nonzero spatial component of the gauge potential in the electric charge sector.[^4]
In the present work we solely focus on the electric charge sector at $B \neq 0$ with $\mu_Q=\mu_B=\mu_S=0$, which may be described by the following EMD action $$\begin{aligned}
S&=\frac{1}{16\pi G_5}\int_{\mathcal{M}_5}d^5x\sqrt{-g}\left[R-\frac{1}{2}(\partial_\mu\phi)^2-V(\phi) -\frac{f(\phi)}{4}F_{\mu\nu}^2\right] +S_{\textrm{GHY}}+S_{\textrm{CT}},
\label{2.1}\end{aligned}$$ where $S_{\textrm{GHY}}$ is the Gibbons-Hawking-York action [@ghy1; @ghy2] needed to establish a well-posed variational problem with Dirichlet boundary condition for the metric, and $S_{\textrm{CT}}$ is the counterterm action that can be constructed using the holographic renormalization procedure [@ren1; @ren2; @ren3; @ren4; @ren5]. These two boundary terms contribute to the total on-shell action but not to the equations of motion and, since we shall not need to compute the total on-shell action in the present work, we do not need to worry about their explicit form here. Also, as we are going to discuss in detail in Section \[sec2.4\], we shall dynamically fix the gravitational constant $G_5$, the dilaton potential $V(\phi)$, and the Maxwell-Dilaton gauge coupling $f(\phi)$, by solving the equations of motion for the EMD fields with the requirement that the holographic equation of state and magnetic susceptibility at zero magnetic field match the corresponding lattice QCD results.
In , the metric field in the bulk is dual to the stress-energy tensor of the boundary field theory while the dilaton field is introduced in order to dynamically break the conformal symmetry of the gauge theory in the infrared. The Abelian gauge field in the bulk is employed here to introduce an external magnetic field at the boundary, which we take to be constant and uniform in the $\hat{z}$-direction and, as stated before, in the present work we set all the chemical potentials to zero. The constant and uniform magnetic field breaks the $SO(3)$ rotational invariance of the gauge theory down to $SO(2)$ rotations around the $\hat{z}$-axis implying that the Ansatz for the bulk metric must be anisotropic and translationally invariant. Also, at zero temperature this Ansatz must be invariant under boosts in the $(t,z)$-plane though this symmetry is not present at nonzero temperature. Based on these symmetry properties, which are phenomenologically dictated by the corresponding symmetry content present in current lattice QCD calculations defined on the $(T,B)$-plane, we take the following black brane Ansatz for the bulk fields in[^5] : $$\begin{aligned}
ds^2&=e^{2a(r)}\left[-h(r)dt^2+dz^2\right]+e^{2c(r)}(dx^2+dy^2)+\frac{e^{2b(r)}dr^2}{h(r)},\nonumber\\
\phi&=\phi(r),\,\,\,A=A_\mu dx^\mu=\mathcal{B}xdy\Rightarrow F=dA=\mathcal{B}dx\wedge dy,
\label{2.2}\end{aligned}$$ where the radial location of the black brane horizon, $r_H$, is given by the largest root of the equation $h(r_H)=0$ and in our coordinates the boundary of the asymptotically AdS$_5$ spacetime is located at $r\rightarrow\infty$. In we have already fixed a convenient gauge for the Maxwell field, which in the present case is a prescribed non-dynamical field. Also, for simplicity, we shall adopt units where the asymptotic AdS$_5$ radius is equal to one.
Using , the equations of motion obtained from may be expressed as follows $$\begin{aligned}
\phi''+\left(2a'+2c'-b'+\frac{h'}{h}\right)\phi'-\frac{e^{2b}}{h} \left(\frac{\partial V(\phi)}{\partial\phi}+\frac{\mathcal{B}^2e^{-4c}}{2}\frac{\partial f(\phi)}{\partial\phi}\right)&=0,\label{2.3}\\
a''+\left(\frac{14}{3}c'-b'+\frac{4}{3}\frac{h'}{h}\right)a' +\frac{8}{3}a'^2+\frac{2}{3}c'^2+\frac{2}{3}\frac{h'}{h}c'
+\frac{2}{3}\frac{e^{2b}}{h} V(\phi)-\frac{1}{6}\phi'^2&=0,\label{2.4}\\
c''-\left(\frac{10}{3}a'+b'+\frac{1}{3}\frac{h'}{h}\right)c' +\frac{2}{3}c'^2-\frac{4}{3}a'^2-\frac{2}{3}\frac{h'}{h}a'
-\frac{1}{3}\frac{e^{2b}}{h} V(\phi)+\frac{1}{3}\phi'^2&=0,\label{2.5}\\
h''+\left(2a'+2c'-b'\right)h'&=0,\label{2.6}\end{aligned}$$ where the prime denotes a derivative with respect to the radial direction. Using these equations of motions one can also derive a useful constraint $$\begin{aligned}
a'^2+c'^2-\frac{1}{4}\phi'^2+\left(\frac{a'}{2}+c'\right)\frac{h'}{h}+4a'c'
+\frac{e^{2b}}{2h}\left(V(\phi)+\frac{\mathcal{B}^2e^{-4c}}{2}f(\phi)\right)=0.
\label{2.7}\end{aligned}$$ The equation of motion for the Maxwell field is automatically satisfied by the Ansatz . Moreover, $b(r)$ has no equation of motion and, thus, it can be freely chosen to take any value due to reparametrization invariance. In the next Section we specify a subsidiary condition for $b(r)$ that defines a convenient gauge for the metric that will be used in the numerical calculations carried out in the present work.
Ultraviolet expansions {#sec2.1}
----------------------
For the calculation of physical observables in the gauge theory one needs to obtain the near-boundary, far from the horizon expansions for the bulk fields $a(r)$, $c(r)$, $h(r)$, and $\phi(r)$. In the present work, we use the domain-wall gauge defined by the subsidiary condition $b(r)=0$. At the boundary the dilaton field goes to zero in such a way that $V(\phi(r\rightarrow\infty)\rightarrow 0)=-12$ and $f(0)$ is a finite positive constant[^6]. Also, the metric blackening factor, $h(r)$, must go to a constant at the boundary, which we denote by[^7] $h(r\rightarrow\infty)=h_0^{\textrm{far}}$.
Moreover, since we are interested in asymptotically AdS$_5$ solutions to the equations of motion , , , and , at the boundary one finds $a(r\rightarrow\infty)=c(r\rightarrow\infty)$. In the domain-wall gauge $b(r)=0$, the leading order near-boundary expression for $a(r)$ (and also $c(r)$) is linear in $r$ [@gubser1; @gubser2] such that at lowest order in $\phi(r\rightarrow\infty)\rightarrow 0$ we may consider the following leading order far from the horizon ultraviolet asymptotics $$\begin{aligned}
V(\phi)\approx -12,\,\,\, f(\phi)\approx f(0),\,\,\,h(r)\approx h_0^{\textrm{far}},\,\,\,
a(r)\approx a_0^{\textrm{far}}+a_{-1}^{\textrm{far}}r,\,\,\, c(r)\approx c_0^{\textrm{far}}+c_{-1}^{\textrm{far}}r,
\label{2.8}\end{aligned}$$ where $a_{-1}^{\textrm{far}}=c_{-1}^{\textrm{far}}$, as discussed above. Indeed, by substituting into the equations of motion and taking the asymptotic limit of large $r$ (where the ultaviolet expansions hold), one concludes that $$\begin{aligned}
a_{-1}^{\textrm{far}}=c_{-1}^{\textrm{far}}=\frac{1}{\sqrt{h_0^{\textrm{far}}}}.
\label{2.9}\end{aligned}$$ In order to obtain the next to leading order term for $h(r)$ and also the first terms for $\phi(r)$ in the ultraviolet expansions for the bulk fields, we consider the first backreaction of the near-boundary fields expressed in and on the equations of motion[^8]. In fact, we first consider the next to leading order near-boundary expansion for the dilaton potential $$\begin{aligned}
V(\phi)\approx -12+\frac{m^2}{2}\phi^2,\,\,\, m^2=-\nu\Delta,
\label{2.10}\end{aligned}$$ where $\Delta$ is the ultraviolet scaling dimension of the gauge invariant operator dual to the bulk dilaton field and we defined $\nu=d-\Delta$, where $d=4$ is the dimension of the boundary. We shall see in Section \[sec2.4\] that a good description of lattice data can be achieved by taking $\Delta\approx 3$ ($\nu\approx 1$). One can now show that the far from horizon ultraviolet asymptotics for the bulk fields may be written as $$\begin{aligned}
a(r)&\approx\alpha(r)+\cdots,\nonumber\\
c(r)&\approx\alpha(r)+(c_0^{\textrm{far}}-a_0^{\textrm{far}})+\cdots,\nonumber\\
h(r)&\approx h_0^{\textrm{far}}+h_4^{\textrm{far}}e^{-4\alpha(r)}+\cdots,\nonumber\\
\phi(r)&\approx \phi_Ae^{-\nu\alpha(r)}+\phi_Be^{-\Delta\alpha(r)}+\cdots,
\label{2.11}\end{aligned}$$ where we defined $\alpha(r)=a_0^{\textrm{far}}+r/\sqrt{h_0^{\textrm{far}}}$ while $\cdots$ denotes subleading terms. We note that the ultraviolet asymptotics are in agreement with our numerical solutions. By comparing these numerical solutions to one can determine the ultraviolet coefficients $a_0^{\textrm{far}}$, $c_0^{\textrm{far}}$, $h_0^{\textrm{far}}$ and $\phi_A$, which are needed to compute the thermodynamical observables in Sections \[sec2.3\] and \[sec2.4\].
Infrared expansions {#sec2.2}
-------------------
Now we consider the infrared, near-horizon expansions for the bulk fields $a(r)$, $c(r)$, $h(r)$, and $\phi(r)$. Near the horizon all the bulk fields in are assumed to be smooth and we may consider the Taylor expansions $$\begin{aligned}
X(r)=\sum_{n=0}^\infty X_n(r-r_H)^n,
\label{2.12}\end{aligned}$$ where $X=\left\{a,c,h,\phi\right\}$.
In order to numerically solve the equations of motion , , , and we need to specify the boundary conditions $X(r_{\textrm{start}})$ and $X'(r_{\textrm{start}})$, where $r_{\textrm{start}}$ is a value of the radial coordinate that is slightly above the horizon[^9]. In this paper we work with Taylor expansions up to second order, which are sufficient to perform the numerical integrations if $r_{\textrm{start}}$ is close enough to $r_H$. Therefore, we must determine 12 Taylor coefficients in order to specify $X(r_{\textrm{start}})$ and $X'(r_{\textrm{start}})$ at second order. One of these 12 coefficients, namely, $\phi_0$, is one of the two initial conditions of the problem[^10]. Four of these 12 coefficients, namely, $a_0$, $c_0$, $h_0$, and $h_1$ and also the radial location of the black hole horizon, $r_H$, may be fixed by rescaling the bulk coordinates while taking into account also the fact that $h(r)$ vanishes at the horizon. For definiteness, we adopt here numerical coordinates fixed in such a way that $$\begin{aligned}
r_H=0;\,\,\,a_0=c_0=h_0=0,\,\,\,h_1=1.
\label{2.13}\end{aligned}$$ Note that $r_H=0$ may be obtained by rescaling the radial coordinate while $h_0=0$ comes from the fact that $h(r)$ has a simples zero at the horizon. Also, $h_1=1$ may be obtained by rescaling $t$ while $a_0=0$ may be arranged by rescaling $(t,z)$ by a common factor. Similarly, $c_0=0$ may be arranged by rescaling $(x,y)$ by a common factor. After this, the remaining 7 coefficients in the near-horizon Taylor expansions for the bulk fields can be fixed on-shell as functions of the initial conditions $(\phi_0,\mathcal{B})$ by substituting the second order Taylor expansions into the equations of motion and setting to zero each power of $r_{\textrm{start}}$ in the resulting algebraic equations[^11].
With $X(r_{\textrm{start}})$ and $X'(r_{\textrm{start}})$ determined as discussed above, the equations of motion are numerically integrated from $r_{\textrm{start}}$ near the horizon up to some numerical ultraviolet cutoff $r_{\textrm{max}}$ near the boundary. We used $r_{\textrm{start}}=10^{-8}$ and $r_{\textrm{max}}=10$ to numerically solve the equations of motion. It is important to remark, however, that even before reaching $r_{\textrm{conformal}}=2$ the numerical backgrounds we considered in the present work have already reached the ultraviolet fixed point corresponding to the AdS$_5$ geometry. This fact is used in Section \[sec2.3\] to reliably obtain the ultraviolet coefficients in and it will be also employed in Section \[sec2.4\] to properly compute the holographic magnetic susceptibility numerically.
Coordinate transformations and thermodynamical observables {#sec2.3}
----------------------------------------------------------
Let us now introduce the so-called “standard coordinates” of the domain-wall metric gauge, $\tilde{b}(\tilde{r})=0$, where variables with $\sim$ refer to quantities evaluated in these standard coordinates where the background reads $$\begin{aligned}
d\tilde{s}^2&=e^{2\tilde{a}(\tilde{r})}\left[-\tilde{h}(\tilde{r})d\tilde{t}^2+d\tilde{z}^2\right]+ e^{2\tilde{c}(\tilde{r})}(d\tilde{x}^2+d\tilde{y}^2)+\frac{d\tilde{r}^2}{\tilde{h}(\tilde{r})},\nonumber\\
\tilde{\phi}&=\tilde{\phi}(\tilde{r}),\,\,\, \tilde{A}=\tilde{A}_\mu d\tilde{x}^\mu=\hat{B}\tilde{x}d\tilde{y}\Rightarrow \tilde{F}=d\tilde{A}=\hat{B}d\tilde{x}\wedge d\tilde{y},
\label{2.14}\end{aligned}$$ and the boundary is at $\tilde{r}\rightarrow\infty$ while the horizon is at $\tilde{r}=\tilde{r}_H$. The “hat” in $\hat{B}$ accounts for the fact that this is the magnetic field measured in units of the inverse of the AdS radius squared, while $B$ shall be used to denote the boundary magnetic field measured in physical units, as we shall discuss in Section \[sec2.4\]. In the standard coordinates, the ultraviolet asymptotics for the bulk fields are given by [@gubser1; @gubser2] (see also [@finitemu]) $$\begin{aligned}
\tilde{a}(\tilde{r})&\approx\tilde{r}+\cdots,\nonumber\\
\tilde{c}(\tilde{r})&\approx\tilde{r}+\cdots,\nonumber\\
\tilde{h}(\tilde{r})&\approx 1+\cdots,\nonumber\\
\tilde{\phi}(\tilde{r})&\approx e^{-\nu\tilde{r}}+\cdots.
\label{2.15}\end{aligned}$$
The standard coordinates (in which $h(r)$ goes to one at the boundary) are the coordinates where we obtain standard holographic formulas for the gauge theory’s physical observables such as the temperature and the entropy density. However, in order to obtain numerical solutions for the bulk fields one needs to give numerical values for all the infrared near-horizon Taylor expansion coefficients, which in turn requires rescaling these standard coordinates, as discussed in the previous Section. The numerical solutions are obtained in the numerical coordinates described by the Ansatz with the ultraviolet asymptotics , while standard holographic formulas for physical observables are obtained in the standard coordinates described by the background with the ultraviolet asymptotics . One may relate these two sets of coordinates by equating $\tilde{\phi}(\tilde{r})=\phi(r)$, $d\tilde{s}^2=ds^2$ and $\hat{B}d\tilde{x}\wedge d\tilde{y}=\mathcal{B}dx\wedge dy$ and this leads to the following relations[^12] (by comparing the near-boundary asymptotics and for $r\rightarrow\infty$) $$\begin{aligned}
\tilde{r}&=\frac{r}{\sqrt{h_0^{\textrm{far}}}}+a_0^{\textrm{far}}-\ln\left(\phi_A^{1/\nu}\right),\nonumber\\
\tilde{t}&=\phi_A^{1/\nu}\sqrt{h_0^{\textrm{far}}}t,\nonumber\\
\tilde{x}&=\phi_A^{1/\nu}e^{c_0^{\textrm{far}}-a_0^{\textrm{far}}}x,\nonumber\\
\tilde{y}&=\phi_A^{1/\nu}e^{c_0^{\textrm{far}}-a_0^{\textrm{far}}}y,\nonumber\\
\tilde{z}&=\phi_A^{1/\nu}z;\nonumber\\
\tilde{a}(\tilde{r})&=a(r)-\ln\left(\phi_A^{1/\nu}\right),\nonumber\\
\tilde{c}(\tilde{r})&=c(r)-(c_0^{\textrm{far}}-a_0^{\textrm{far}})-\ln\left(\phi_A^{1/\nu}\right),\nonumber\\
\tilde{h}(\tilde{r})&=\frac{h(r)}{h_0^{\textrm{far}}},\nonumber\\
\tilde{\phi}(\tilde{r})&=\phi(r);\nonumber\\
\hat{B}&=\frac{e^{2(a_0^{\textrm{far}}-c_0^{\textrm{far}})}}{\phi_A^{2/\nu}}\mathcal{B}.
\label{2.16}\end{aligned}$$
The temperature of the plasma is given by the black brane horizon’s Hawking temperature $$\begin{aligned}
\hat{T}=\frac{\sqrt{-\tilde{g}'_{\tilde{t}\tilde{t}} \tilde{g}^{\tilde{r}\tilde{r}}\,'}}{4\pi}\biggr|_{\tilde{r}=\tilde{r}_H}= \frac{e^{\tilde{a}(\tilde{r}_H)}}{4\pi}|\tilde{h}'(\tilde{r}_H)|=\frac{1}{4\pi\phi_A^{1/\nu}\sqrt{h_0^{\textrm{far}}}},
\label{2.17}\end{aligned}$$ while the entropy density is obtained via the Bekenstein-Hawking’s relation [@bek1; @bek2] $$\begin{aligned}
\hat{s}=\frac{S}{V}=\frac{A_H/4G_5}{V}=\frac{\int_{\textrm{horizon}}d^3\tilde{x}\sqrt{\tilde{g}(\tilde{r}=\tilde{r}_H, \,\tilde{t}\,\textrm{fixed})}}{4G_5V} = \frac{2\pi}{\kappa^2}e^{\tilde{a}(\tilde{r}_H)+2\tilde{c}(\tilde{r}_H)} = \frac{2\pi e^{2(a_0^{\textrm{far}}-c_0^{\textrm{far}})}}{\kappa^2\phi_A^{3/\nu}},
\label{2.18}\end{aligned}$$ where we defined $\kappa^2=8\pi G_5$ and used , , and .
One can see from , , and that the only ultraviolet coefficients in the numerical coordinates which we need to fix by fitting the numerical solutions with are $a_0^{\textrm{far}}$, $c_0^{\textrm{far}}$, $h_0^{\textrm{far}}$, and $\phi_A$. The numerical solutions for $h(r)$ converge quickly to their asymptotic values at large $r$ and we may reliably set $h_0^{\textrm{far}}=h(r_{\textrm{conformal}})$. With $h_0^{\textrm{far}}$ fixed in this way, we may fix $a_0^{\textrm{far}}$, $c_0^{\textrm{far}}$, and $\phi_A$, respectively, by employing the fitting functions $a(r)=a_0^{\textrm{far}}+r/\sqrt{h_0^{\textrm{far}}}$, $c(r)=c_0^{\textrm{far}}+r/\sqrt{h_0^{\textrm{far}}}$, and $\phi(r)=\phi_Ae^{-\nu a(r)}$ in the interval $r\in[r_{\textrm{conformal}}-1,r_{\textrm{conformal}}]$. We were able to obtain good fits for the near-boundary behavior of the numerical solutions using this fitting scheme.
Also, it is important to remark that there is an upper bound on the initial condition $\mathcal{B}$ for a given value of the initial condition for the dilaton $\phi_0$. In fact, for values of $\mathcal{B}$ above this bound, all the numerical backgrounds we generated failed to be asymptotically AdS$_5$. Such a bound, which we denote by $\mathcal{B}\le\mathcal{B}_{\textrm{max}}(\phi_0)$, may be numerically constructed by interpolating a list with pairs of points $\left\{\left(\phi_0^i,\mathcal{B}_{\textrm{max}}^i\right),\,i=1,2,3,\cdots\right\}$ and the corresponding result is presented in Fig. \[fig1\].
![The curve corresponds to the upper bound for the initial condition $\mathcal{B}$ as a function of the initial condition for the dilaton $\phi_0$, below which the solutions of the equations of motion are asymptotically AdS$_5$. This curve depends on the chosen profiles for the dilaton potential $V(\phi)$ and gauge coupling function $f(\phi)$ to be discussed in the next Section. \[fig1\]](bound_Bmax.pdf)
In the next Section we explain how one can express the thermodynamical quantities $\hat{B}$, $\hat{T}$, and $\hat{s}$ in physical units[^13] using the lattice data for the equation of state and the magnetic susceptibility at zero magnetic field.
Fixing the Maxwell-Dilaton gauge coupling using lattice data for the magnetic susceptibility at zero magnetic field {#sec2.4}
-------------------------------------------------------------------------------------------------------------------
Refs. [@hydro; @finitemu] discussed in detail how to dynamically fix the dilaton potential, $V(\phi)$, and the gravitational constant, $\kappa^2$, using the recent lattice data [@latticedata1] for the QCD equation of state with $(2+1)$-flavors. We refer the reader to those papers for the details about this procedure. The results are $$\begin{aligned}
V(\phi)=-12\cosh(0.606\,\phi)+0.703\,\phi^2-0.1\,\phi^4+0.0034\,\phi^6,\,\,\,\kappa^2 = 8\pi G_5 = 12.5\,.
\label{2.19}\end{aligned}$$ From the dilaton potential specified above one obtains the dilaton mass $m^2\approx-3$, as anticipated in Section \[sec2.1\].
We remark that, although the present EMD construction do not explicit introduce fundamental flavors at the dual boundary quantum field theory, the dilaton potential in Eq. was adjusted in order to quantitatively *mimic* the $(2+1)$-flavor lattice QCD equation of state and its crossover. This *mimicking* procedure was originally introduced in [@GN1] (see also [@Yaresko:2015ysa] for more recent discussions), where it was also discussed how different choices for the dilaton potential may *emulate* not only the QCD crossover, as done in the present work, but also first and second order phase transitions, which may be useful for a large variety of different physical systems.
In the present paper, we employ the same procedure used in [@hydro; @finitemu] to express the holographically determined thermodynamical observables in physical units, i.e., we find the temperature at which our speed of sound squared, $c_s^2$, displays a minimum (at zero magnetic field) and match it to the corresponding lattice QCD result [@latticedata1] $$\begin{aligned}
\lambda=\frac{T_{\textrm{min.}\,c_s^2}^{\textrm{lattice}}}{T_{\textrm{min.}\,c_s^2}^{\textrm{BH}}}\approx \frac{143.8\,\textrm{MeV}}{0.173} \approx 831\,\textrm{MeV}.
\label{2.20}\end{aligned}$$ In what follows, we relate any black hole thermodynamical observable, $\hat{X}$, with its counterpart in physical units, $X$, with mass dimension \[MeV$^p$\], by taking $X = \lambda^p \hat{X}$ \[MeV$^p$\]. This prescription respects the fact that dimensionless ratios, such as $s/T^3$, must give the same result regardless of the units. A comparison between our holographic results for the speed of sound squared, $c_s^2(T,B=0)$, and the (normalized) pressure, $p(T,B=0)/T^4$ (at zero magnetic field) and the corresponding lattice QCD results from [@latticedata1] is shown in Fig. \[fig2\]. One can see that the holographic model provides a good description of the lattice data in the absence of an external magnetic field.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Holographic calculation of the speed of sound squared $c_s^2$ and the (normalized) pressure $p/T^4$. The data points correspond to lattice QCD results from [@latticedata1] computed at zero magnetic field. \[fig2\]](cs2B0.pdf "fig:"){width="48.00000%"} ![Holographic calculation of the speed of sound squared $c_s^2$ and the (normalized) pressure $p/T^4$. The data points correspond to lattice QCD results from [@latticedata1] computed at zero magnetic field. \[fig2\]](pT4B0.pdf "fig:"){width="46.00000%"}
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
In order to fully determine our holographic model and include the effects from a magnetic field we also need to dynamically fix the Maxwell-Dilaton gauge coupling $f(\phi)$. This can be done using the recent lattice data [@latticedata2] for the magnetic susceptibility of QCD with $(2+1)$-flavors evaluated at zero magnetic field. In order to compute the magnetic susceptibility in our holographic model we follow the same general steps discussed in [@donos]: we substitute the Ansatz into the action and calculate the second derivative of the on-shell action with respect to the magnetic field, dividing the result by the entire spacetime volume of the boundary. In order to obtain the bare magnetic susceptibility we plug the on-shell numerical solutions into the expression obtained in the previous step[^14], $$\begin{aligned}
\chi_{\textrm{bare}}(T,B)=-\frac{\partial^2 f_{\textrm{bare}}}{\partial B^2}=-\frac{1}{V_{\textrm{bdy}}}\frac{\partial^2 S_{E,\,\textrm{bare}}^{\textrm{on-shell}}[B]}{\partial B^2} &=
\frac{1}{V_{\textrm{bdy}}}\frac{\partial^2 S_{\textrm{bare}}^{\textrm{on-shell}}[B]}{\partial B^2}\nonumber\\
& = -\frac{1}{2\kappa^2}\int_{\tilde{r}_H}^{\tilde{r}^{\textrm{fixed}}_{\textrm{max}}}d\tilde{r} f(\tilde{\phi}(\tilde{r})) e^{2(\tilde{a}(\tilde{r})-\tilde{c}(\tilde{r}))}\biggr|^{\textrm{on-shell}},
\label{2.21}\end{aligned}$$ where $f_{\textrm{bare}}$ is the bare free energy density and, formally, one should take the limit $\tilde{r}^{\textrm{fixed}}_{\textrm{max}}\rightarrow\infty$. However, in numerical calculations, $\tilde{r}^{\textrm{fixed}}_{\textrm{max}}$ must be a fixed ultraviolet cutoff for all the geometries in order to ensure that the ultraviolet divergence in is independent of the temperature. Since we are interested here in calculating the magnetic susceptibility at zero magnetic field where $a(r)=c(r)$, one obtains from $$\begin{aligned}
\chi_{\textrm{bare}}(T,B=0)= -\frac{1}{2\kappa^2}\int_{\tilde{r}_H}^{\tilde{r}^{\textrm{fixed}}_{\textrm{max}}}d\tilde{r} f(\tilde{\phi}(\tilde{r}))\biggr|^{\textrm{on-shell}}.
\label{2.22}\end{aligned}$$ In order to regularize we follow the same procedure adopted on the lattice [@latticedata2] and subtract from the vacuum contribution at zero temperature. Clearly, this removes the ultraviolet divergences since those are temperature independent. More precisely, we subtract the geometry corresponding to $(T_{\textrm{small}},B)\approx(0.005\,\textrm{MeV},0)$, which is generated by the initial conditions $(\phi_0,\mathcal{B})=(7.8,0)$; this is the asymptotically AdS$_5$ geometry with the lowest temperature and zero magnetic field which we could reach in our numerical computations[^15]. Therefore, we obtain the following holographic formula for the magnetic susceptibility at zero magnetic field (which is valid for any EMD model of the kind considered here) $$\begin{aligned}
\chi(T,B=0)&=\chi_{\textrm{bare}}(T,B=0)-\chi_{\textrm{bare}}(T_{\textrm{small}},B=0)\nonumber\\
&=-\frac{1}{2\kappa^2}\left[\left(\int_{\tilde{r}_H}^{\tilde{r}^{\textrm{fixed}}_{\textrm{max}}}d\tilde{r} f(\tilde{\phi}(\tilde{r}))\right)\biggr|_{T,B=0}-\left(\textrm{same}\right)\biggr|_{T_{\textrm{small}},B=0} \right]^{\textrm{on-shell}}
\nonumber\\
&=-\frac{1}{2\kappa^2}\left[\left(\frac{1}{\sqrt{h_0^{\textrm{far}}}}\int_{r_{\textrm{start}}}^{r^{\textrm{var}}_{\textrm{max}}} dr f(\phi(r))\right)\biggr|_{T,B=0}-\left(\textrm{same}\right)\biggr|_{T_{\textrm{small}},B=0} \right]^{\textrm{on-shell}},
\label{2.23}\end{aligned}$$ where $\tilde{r}^{\textrm{fixed}}_{\textrm{max}}$ must be chosen in such a way that the upper limits of integration in the numerical coordinates satisfy $r_{\textrm{conformal}}\le r^{\textrm{var}}_{\textrm{max}}=\sqrt{h_0^{\textrm{far}}}\left[\tilde{r}^{\textrm{fixed}}_{\textrm{max}}- a_0^{\textrm{far}}+\ln\left(\phi_A^{1/\nu}\right)\right]\le r_{\textrm{max}}$ for all the geometries considered. We found that for $\tilde{r}^{\textrm{fixed}}_{\textrm{max}} \sim 33$ such requirement is met. We also checked that one can vary the value of the ultraviolet cutoff $\tilde{r}^{\textrm{fixed}}_{\textrm{max}}$ and the results for the holographic magnetic susceptibility do not change, which confirms the stability of our numerical procedure.
We can now use many different trial profiles for $f(\phi)$ to evaluate over the zero magnetic field background solutions, trying to holographically fit the recent lattice data from [@latticedata2] for the magnetic susceptibility of $(2+1)$-flavor QCD at zero magnetic field. We found that a good description of the lattice data can be obtained by fixing $$\begin{aligned}
f(\phi)=1.12\,\textrm{sech}(1.05\,\phi-1.45),
\label{2.24}\end{aligned}$$ with the corresponding results displayed in Fig. \[fig3\].
![Holographic calculation of the magnetic susceptibility at zero magnetic field and comparison with lattice data from [@latticedata2] (we consider 10.9 times the data available in table III in [@latticedata2], which corresponds to the magnetic susceptibility in natural units - see footnote 1 in [@latticedata2]). \[fig3\]](chimagB0.pdf)
With the dilaton potential and the Maxwell-Dilaton gauge coupling dynamically fixed by the description of adequate lattice data at zero magnetic field, our holographic model is now fully determined. This setup may be employed to investigate the physics of the dual quantum field theory at finite temperature and nonzero magnetic field with vanishing chemical potential(s).
We finish this Section by mentioning some limitations of the holographic model presented here:
- The model cannot describe phenomena directly related to chiral symmetry and its breaking/restoration (such as $T=0$ magnetic catalysis [@Gusynin:1994re; @Gusynin:1995nb; @Miransky:2002rp]). This could be studied by adding flavor D-branes in the bulk (see, for instance, Ref. [@ihqcd-veneziano]);\
- The model cannot properly describe hadron thermodynamics (which sets in at low temperatures, below $T \sim 150 $ MeV) and the effects of magnetic fields at low temperatures (for a study of the hadron resonance gas in a magnetic field see [@Endrodi:2013cs]). Moreover, in this holographic model asymptotic freedom is replaced by conformal invariance at sufficiently high temperatures. Furthermore, for high enough magnetic fields the nonlinear nature of the DBI action for the D-branes should be taken into account [@vacilao];\
- In Appendix \[apa\], we present a brief discussion on the behavior of electric field response functions in the present EMD model, which indicates that this simple model is not versatile enough to simultaneously cover in a quantitative way both the magnetic and electric sectors of the QGP.
With these limitations in sight, we expect that the present bottom-up holographic model will be mostly useful to understand the effects of magnetic fields on the QGP within the range $T \sim 150-400$ MeV and $eB\lesssim 1$ GeV$^2$.
Holographic QCD thermodynamics at nonzero magnetic field {#sec3}
========================================================
In this Section the results for the holographic equation of state at nonzero magnetic field are presented. The formulas needed to compute the observables shown below were presented in the last Section. Here, we define the pressure as the temperature integral of the entropy density performed while keeping the magnetic field fixed[^16] $$\begin{aligned}
p(T,B)=\int_{T_{\textrm{ref}}}^T dT' s(T',B),
\label{3.1}\end{aligned}$$ where we took a low reference temperature, $T_{\textrm{ref}}=22$ MeV, in agreement with what was done in [@hydro; @finitemu] to obtain the fit for the dilaton potential and the gravitational constant . By doing so, the holographic curves for the pressure in Fig. \[fig4\] (and also Fig. \[fig2\]) actually correspond to differences with respect to reference pressures calculated at $T_{\textrm{ref}}$ for each value of the magnetic field.
In Fig. \[fig4\] we show our holographic results for the normalized entropy density, $s/T^3$, and pressure, $p$, and compare them to recent lattice data [@latticedata3] for $eB = 0$, $0.3$, and 0.6 GeV$^2$. It is important to remark, however, that the above convention to calculate the pressure is not exactly the same used in [@latticedata3] since in the pressure (difference) vanishes at $T=T_{\textrm{ref}}=22$ MeV while in the calculation carried out in [@latticedata3] the pressure goes like $\sim\mathcal{O}\left((eB)^4\right)$ for $T\rightarrow 0$ and, therefore, one should expect that the differences between these two calculations[^17] become more pronounced at low temperatures and large magnetic fields, as seen in Fig. \[fig4\]. However, even for $eB=0.6$ GeV$^2$, we do find a reasonable agreement for the pressure at large temperatures ($T > 200$ MeV).
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![(Color online) Holographic calculation for the normalized entropy density, $s/T^3$, and pressure, $p$, in the presence of an external magnetic field. The solid, dashed, and dot-dashed curves correspond to magnetic fields $eB=0$, $0.3$, and $0.6$ GeV$^2$, respectively. The data points correspond to the lattice calculations for these quantities performed in [@latticedata3].[]{data-label="fig4"}](sT3_finite_B.pdf "fig:"){width="48.00000%"} ![(Color online) Holographic calculation for the normalized entropy density, $s/T^3$, and pressure, $p$, in the presence of an external magnetic field. The solid, dashed, and dot-dashed curves correspond to magnetic fields $eB=0$, $0.3$, and $0.6$ GeV$^2$, respectively. The data points correspond to the lattice calculations for these quantities performed in [@latticedata3].[]{data-label="fig4"}](p_finite_B.pdf "fig:"){width="48.00000%"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
On the other hand, when it comes to the ratio $s/T^3$, the agreement between our holographic results and the lattice is only at the qualitative level. This is in part due to the uncertainties in the holographic description of this observable already at $B=0$: the holographic model parameters were chosen to describe the lattice data for the pressure and the speed of sound squared at $B=0$ and not[^18] $s/T^3$. In any case, one can see that $s/T^3$ increases with an increasing magnetic field, which is the general behavior observed on the lattice [@latticedata3]. Moreover, note that the curve $s/T^3$ becomes steeper near the transition region for increasing values of the magnetic field, which is again in agreement with the general trend observed on the lattice [@Endrodi:2015oba].
$eB$ $[$GeV$^2]$ $T_c(eB)$ $[$MeV$]$
------------------ ---------------------
0 158.2
0.1 157.6
0.2 154.9
0.3 153.2
0.4 151.3
0.5 149.9
: Deconfinement temperature (defined by the inflection point of $s/T^3$) for different values of the magnetic field in the bottom-up holographic model. \[tab1\]
![Deconfinement temperature (defined by the inflection point of $s/T^3$) for different values of the magnetic field in the bottom-up holographic model. The data points correspond to the lattice calculation performed in [@latticedata3].[]{data-label="TcB"}](TcB.pdf)
As discussed in [@latticedata3], the inflection point of $s/T^3$ may be used to characterize the crossover temperature as a function of the magnetic field[^19]. Correspondingly, the peak in $T\partial_T(s/T^3)$ may be used to estimate the crossover temperature as a function of the magnetic field in our holographic model. We used our results for $s/T^3$ to find how the crossover temperature changes with a magnetic field and the results are displayed in table \[tab1\] and in Fig. \[TcB\]. One can see in Fig. \[TcB\] that in our model the crossover temperature decreases with an increasing magnetic field, as found on the lattice [@latticedata0; @latticedata3], but a quantitative agreement with the data from [@latticedata3] occurs only for $eB \lesssim 0.3$ GeV$^2$.
Some general comments regarding the crossover found in our holographic model are in order at this point. Depending on the chosen dilaton potential, the black hole solutions may or may not have a minimum temperature, as detailed discussed, for instance, in Refs. [@GN1; @stefanovacuum; @Yaresko:2015ysa; @Charmousis:2010zz]. In the case there is some minimum temperature below which the black hole solutions do not exist, the system generally features a first order Hawking-Page phase transition [@Hawking:1982dh] to the thermal gas phase at some critical temperature a little bit higher than the minimum temperature for the existence of the black hole solutions. Also, in this case, the black hole solutions are not unique and there is at least one unstable branch of black hole solutions above this minimum temperature. But for some choices of the dilaton potential the temperature of the black hole solutions may monotonically decrease as a function of the radial position of the horizon until going to zero, in which case the black hole solutions are unique and thermodynamically preferred over the thermal gas solution and the system does not feature any phase transition at nonzero temperature (at least at zero magnetic field and vanishing chemical potentials): this is the case realized in our EMD model. Note also this is indeed the adequate situation to mimic the QCD crossover instead of the pure Yang-Mills first order phase transition. In fact, by analyzing our dilaton potential according to the general criteria discussed in [@Charmousis:2010zz], one notes that in the deep infrared our dilaton potential goes like $V(\phi\to\infty)\sim -e^{0.606\phi}$, in which case at each finite value of temperature (at zero magnetic field and vanishing chemical potentials) there exists a unique black hole solution and this corresponds to the true ground state of the system, having a larger pressure than the thermal gas solution. Moreover, since within the region of the $(T,B)$-phase diagram analyzed in our manuscript the pressure of the plasma increases with $B$ (as also seen on the lattice, see Fig. \[fig4\]), within this region the black hole solutions are always thermodynamically preferred and do correspond to the true ground state of the system.
As a technical detail, in order to obtain the curves in Fig. \[fig4\] we used a large grid of initial conditions with 720,000 points taking 900 equally spaced points in the $\phi_0$-direction starting from $\phi_0=0.3$ and going up to $\phi_0=7.8$, and 800 equally spaced points in the $\frac{\mathcal{B}}{\mathcal{B}_{\textrm{max}}(\phi_0)}$-direction starting from $\frac{\mathcal{B}}{\mathcal{B}_{\textrm{max}}(\phi_0)}=0$ and going up to $\frac{\mathcal{B}}{\mathcal{B}_{\textrm{max}}(\phi_0)}=0.99$. A large number of points was required to obtain sufficiently smooth curves for $s/T^3$ that allowed for the extraction of the crossover temperature and its dependence on the magnetic field. However, smooth curves for $p$ could be obtained using much smaller (and faster) grids.
Concluding remarks and perspectives {#conclusion}
===================================
In this paper we developed, for the first time, a bottom-up holographic model that provides a quantitative description of the crossover behavior observed in the equation of state and in the magnetic susceptibility of a QCD plasma with $(2+1)$-flavors at zero magnetic field. We employed this model to study how an Abelian magnetic field $B$ affects the thermodynamic properties of this strongly coupled plasma (at zero chemical potentials). In the presence of the magnetic field the plasma becomes anisotropic and we used the inflection point of the holographically calculated $s/T^3$ curve to determine how the crossover temperature is affected by the external magnetic field. We found that the crossover temperature decreases with an increasing magnetic field, which agrees with the general behavior recently observed on the lattice. Our model calculations display some level of quantitative agreement with the lattice data for values of the magnetic field up to $eB \lesssim 0.3$ GeV$^2$, which is the expected range achieved in ultrarelativistic heavy ion collisions.
We believe that this agreement with the lattice data can be further improved toward larger values of $eB$ if one tries to carefully match the lattice thermodynamic calculations at $B=0$ by simultaneously taking into account different observables such as the pressure and the speed of sound squared, as we have done in the present approach, with the addition of the entropy density and the trace anomaly in a global fit; in this sense, our choice for the holographic model parameters (fixed at $B=0$) may be systematically improved.
An interesting feature of our holographic model that distinguishes it from other constructions (such as [@Ballon-Bayona:2013cta; @Mamo:2015dea]) is that the suppression of the crossover temperature with the external magnetic field found here is directly tied to a *quantitative description* of near crossover lattice QCD thermodynamics at $B=0$. It would be desirable to generalize the present holographic model by taking into account the contribution of the chiral condensate. Moreover, motivated by the recent studies in Refs. [@Endrodi:2015oba; @cohen], one could also investigate if this model indicates the existence of a critical point in the $(T,B)$-plane at higher values of the magnetic field[^20].
The holographic setup constructed here may be employed to obtain estimates for the magnetic field dependence of many other physical observables relevant to the strongly coupled QGP. For instance, one could generalize the calculation of transport coefficients performed in [@DK-applications2] and obtain a quantitative estimate of how the anisotropic shear (and bulk) viscosity coefficients vary with the external magnetic field around the QCD crossover transition.
Recently, the effects of an external magnetic field on the equilibration dynamics of strongly coupled plasmas have been studied using holography [@DK-applications4; @Mamo:2015aia]. In this context, it would be interesting to see how the quasinormal mode spectrum in our nonconformal plasma varies with an external magnetic field. Given that our model can capture the nonconformal behavior of the QGP near the crossover transition, with and without the external magnetic field, a detailed study of the quasinormal modes in this model may shed some light on the thermalization process that takes place in an anisotropic nonconformal strongly magnetized QGP. We hope to report results in this direction in the near future.
We thank I. A. Shovkovy and E. S. Fraga for discussions about the effects of magnetic fields on the QGP and F. Bruckmann and G. Endrodi for comments on the manuscript. We also thank the authors of Ref. [@latticedata3] for making their lattice results avaliable to us. This work was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq). J. N. thanks the Columbia University Physics Department for its hospitality.
Electric susceptibility and conductivity for different coupling functions {#apa}
=========================================================================
In order to check further limitations of the present EMD model (some of which have been discussed at the end of Section \[sec2.4\]), we compare in this Appendix the results for the magnetic susceptibility, and also the electric susceptibility and DC electric conductivity for two different profiles of the Maxwell-Dilaton electric coupling function $f(\phi)$. The first profile is given in Eq. , which was fixed by fitting lattice data [@latticedata2] for the magnetic susceptibility at $B=0$, as discussed before. The second profile was fixed in Ref. [@Finazzo:2015xwa] by fitting lattice data [@Borsanyi:2011sw] for the electric susceptibility also at $B=0$, $$\begin{aligned}
f(\phi)=0.0193\,\textrm{sech}(-100\,\phi)+0.0722\,\textrm{sech}(10^{-7}\,\phi).
\label{A1}\end{aligned}$$
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![[(Color online) EMD magnetic susceptibility (top left), electric susceptibility (top right) and DC electric conductivity (bottom) for two different choices of the Maxwell-Dilaton electric coupling function $f(\phi)$: the full curves were obtained by using $f(\phi)$ given in Eq. , while the dashed curves were obtained by employing $f(\phi)$ given in Eq. . All the lattice data displayed in these plots refer to $(2+1)$-flavor QCD (lattice data for the electric conductivity are taken from [@Aarts:2014nba]).]{} \[figlimitations\]](chimags.pdf "fig:"){width="45.00000%"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![[(Color online) EMD magnetic susceptibility (top left), electric susceptibility (top right) and DC electric conductivity (bottom) for two different choices of the Maxwell-Dilaton electric coupling function $f(\phi)$: the full curves were obtained by using $f(\phi)$ given in Eq. , while the dashed curves were obtained by employing $f(\phi)$ given in Eq. . All the lattice data displayed in these plots refer to $(2+1)$-flavor QCD (lattice data for the electric conductivity are taken from [@Aarts:2014nba]).]{} \[figlimitations\]](chiQs.pdf "fig:"){width="45.00000%"}
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![[(Color online) EMD magnetic susceptibility (top left), electric susceptibility (top right) and DC electric conductivity (bottom) for two different choices of the Maxwell-Dilaton electric coupling function $f(\phi)$: the full curves were obtained by using $f(\phi)$ given in Eq. , while the dashed curves were obtained by employing $f(\phi)$ given in Eq. . All the lattice data displayed in these plots refer to $(2+1)$-flavor QCD (lattice data for the electric conductivity are taken from [@Aarts:2014nba]).]{} \[figlimitations\]](sigmaQs.pdf "fig:"){width="45.00000%"}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
At $B=0$, the holographic formulas for the electric susceptibility and the DC electric conductivity are given respectively by, $$\begin{aligned}
\frac{\chi_2^Q}{T^2}&=\frac{1}{16\pi^2} \frac{s}{T^3} \frac{1}{f(0)\int_{r_H}^\infty dr\, e^{-2a(r)}f^{-1}(\phi(r))},\label{A2}\\
\frac{\sigma_Q}{T}&=\frac{2\pi\sqrt{h_0^{\textrm{far}}}f(\phi_0)}{\kappa^2},\label{A3}\end{aligned}$$ and we refer the reader to consult Ref. [@Finazzo:2015xwa] for a discussion on the derivation of these formulas[^21].
One can see from the results shown in Fig \[figlimitations\] that a simple EMD holographic model cannot give simultaneously good quantitative descriptions of electric and magnetic field response functions: by adjusting the electric coupling $f(\phi)$ in order to fit the magnetic susceptibility at $B=0$, one is able to attain a good description of the QCD thermodynamics at finite $B$, as shown in Section \[sec3\], but response functions to an applied electric field are not well described in a quantitative way within such prescription. On the other hand, if one adjusts the electric coupling $f(\phi)$ in order to match the electric susceptibility, one is not able to obtain a good quantitative agreement with lattice data for the magnetic susceptibility. It would be certainly interesting to think about the construction of some holographic model versatile enough to quantitatively cover the entire electric-magnetic sector of the QGP, which is something that our simple EMD model is not able to do. We must remark, however, that up to now, our EMD model is the only holographic approach available in the literature which is able to match in a quantitative way the behavior of many magnetic field related observables calculated on the lattice.
[99]{}
BRAHMS collaboration, I. Arsene et al., Nucl. Phys. A **757**, 1 (2005), \[arXiv:nucl-ex/0410020\].
PHENIX collaboration, K. Adcox et al., Nucl. Phys. A **757**, 184 (2005), \[arXiv:nucl-ex/0410003\].
PHOBOS collaboration, B. B. Back et al., Nucl. Phys. A **757**, 28 (2005), \[arXiv:nucl-ex/0410022\].
STAR collaboration, J. Adams et al., Nucl. Phys. A **757**, 102 (2005), \[arXiv:nucl-ex/0501009\].
M. Gyulassy and L. McLerran, Nucl. Phys. A **750**, 30 (2005), \[arXiv:nucl-th/0405013\].
U. Heinz and R. Snellings, Ann. Rev. Nucl. Part. Sci. [**63**]{}, 123 (2013), \[arXiv:1301.2826 \[nucl-th\]\]. E. Shuryak, \[arXiv:1412.8393 \[hep-ph\]\].
D. E. Kharzeev, L. D. McLerran and H. J. Warringa, Nucl. Phys. A **803**, 227 (2008), \[arXiv:0711.0950 \[hep-ph\]\].
K. Fukushima, D. E. Kharzeev and H. J. Warringa, Phys. Rev. D **78**, 074033 (2008), \[arXiv:0808.3382 \[hep-ph\]\].
V. Skokov, A. Y. .Illarionov and V. Toneev, Int. J. Mod. Phys. A **24**, 5925 (2009), \[arXiv:0907.1396 \[nucl-th\]\].
K. Tuchin, Adv. High Energy Phys. **2013**, 490495 (2013), \[arXiv:1301.0099 \[hep-ph\]\].
W. -T. Deng and X. -G. Huang, Phys. Rev. C **85**, 044907 (2012), \[arXiv:1201.5108 \[nucl-th\]\].
J. Bloczynski, X. -G. Huang, X. Zhang and J. Liao, Phys. Lett. B **718**, 1529 (2013), \[arXiv:1209.6594 \[nucl-th\]\].
U. Gursoy, D. Kharzeev and K. Rajagopal, Phys. Rev. C [**89**]{}, no. 5, 054905 (2014), \[arXiv:1401.3805 \[hep-ph\]\]. T. Vachaspati, Phys. Lett. B **265**, 258 (1991).
D. Grasso and H. R. Rubinstein, Phys. Rept. **348**, 163 (2001), \[arXiv:astro-ph/0009061\].
G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz, S. Krieg, A. Schafer and K. K. Szabo, JHEP **02** (2012) 044, \[arXiv:1111.4956 \[hep-lat\]\].
R. C. Duncan and C. Thompson, Astrophys. J. **392**, L9 (1992).
N. O. Agasian and S. M. Fedorov, Phys. Lett. B [**663**]{}, 445 (2008), \[arXiv:0803.3156 \[hep-ph\]\]. A. J. Mizher, M. N. Chernodub and E. S. Fraga, Phys. Rev. D [**82**]{}, 105016 (2010), \[arXiv:1004.2712 \[hep-ph\]\]. N. Evans, T. Kalaydzhyan, K. Y. Kim and I. Kirsch, JHEP [**01**]{}, 050 (2011), \[arXiv:1011.2519 \[hep-th\]\]. F. Preis, A. Rebhan and A. Schmitt, JHEP [**03**]{}, 033 (2011), \[arXiv:1012.4785 \[hep-th\]\]. K. Fukushima and J. M. Pawlowski, Phys. Rev. D [**86**]{}, 076013 (2012), \[arXiv:1203.4330 \[hep-ph\]\]. G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz and A. Schafer, Phys. Rev. D [**86**]{}, 071502 (2012), \[arXiv:1206.4205 \[hep-lat\]\]. K. Fukushima and Y. Hidaka, Phys. Rev. Lett. [**110**]{}, no. 3, 031601 (2013), \[arXiv:1209.1319 \[hep-ph\]\]. J. P. Blaizot, E. S. Fraga and L. F. Palhares, Phys. Lett. B [**722**]{}, 167 (2013), \[arXiv:1211.6412 \[hep-ph\]\]. N. Callebaut and D. Dudal, Phys. Rev. D [**87**]{}, no. 10, 106002 (2013), \[arXiv:1303.5674 \[hep-th\]\]. G. S. Bali, F. Bruckmann, G. Endrodi, F. Gruber and A. Schaefer, JHEP [**04**]{}, 130 (2013), \[arXiv:1303.1328 \[hep-lat\]\]. C. Bonati, M. D’Elia, M. Mariti, M. Mesiti, F. Negro and F. Sanfilippo, Phys. Rev. D [**89**]{}, no. 11, 114502 (2014), \[arXiv:1403.6094 \[hep-lat\]\]. K. Fukushima and P. Morales, Phys. Rev. Lett. [**111**]{}, 051601 (2013), \[arXiv:1305.4115 \[hep-ph\]\]. C. S. Machado, F. S. Navarra, E. G. de Oliveira, J. Noronha and M. Strickland, Phys. Rev. D [**88**]{}, 034009 (2013), \[arXiv:1305.3308 \[hep-ph\]\]. E. S. Fraga, B. W. Mintz and J. Schaffner-Bielich, Phys. Lett. B [**731**]{}, 154 (2014), \[arXiv:1311.3964 \[hep-ph\]\]. J. O. Andersen, W. R. Naylor and A. Tranberg, JHEP [**04**]{}, 187 (2014), \[arXiv:1311.2093 \[hep-ph\]\]. G. S. Bali, F. Bruckmann, G. Endrodi and A. Schafer, Phys. Rev. Lett. [**112**]{}, 042301 (2014), \[arXiv:1311.2559 \[hep-lat\]\]. M. Ferreira, P. Costa and C. Providência, Phys. Rev. D [**89**]{}, no. 3, 036006 (2014), \[arXiv:1312.6733 \[hep-ph\]\]. M. Ruggieri, L. Oliva, P. Castorina, R. Gatto and V. Greco, Phys. Lett. B [**734**]{}, 255 (2014), \[arXiv:1402.0737 \[hep-ph\]\]. M. Ferreira, P. Costa, O. Lourenço, T. Frederico and C. Providência, Phys. Rev. D [**89**]{}, no. 11, 116011 (2014), \[arXiv:1404.5577 \[hep-ph\]\]. R. L. S. Farias, K. P. Gomes, G. I. Krein and M. B. Pinto, Phys. Rev. C [**90**]{}, no. 2, 025203 (2014), \[arXiv:1404.3931 \[hep-ph\]\]. A. Ayala, M. Loewe, A. J. Mizher and R. Zamora, Phys. Rev. D [**90**]{}, no. 3, 036001 (2014), \[arXiv:1406.3885 \[hep-ph\]\]. A. Ayala, M. Loewe and R. Zamora, Phys. Rev. D [**91**]{}, no. 1, 016002 (2015), \[arXiv:1406.7408 \[hep-ph\]\]. E. J. Ferrer, V. de la Incera and X. J. Wen, Phys. Rev. D [**91**]{}, no. 5, 054006 (2015), \[arXiv:1407.3503 \[nucl-th\]\]. K. Kamikado and T. Kanazawa, JHEP [**01**]{}, 129 (2015), \[arXiv:1410.6253 \[hep-ph\]\]. L. Yu, J. Van Doorsselaere and M. Huang, Phys. Rev. D [**91**]{}, no. 7, 074011 (2015), \[arXiv:1411.7552 \[hep-ph\]\]. J. Braun, W. A. Mian and S. Rechenberger, \[arXiv:1412.6025 \[hep-ph\]\]. N. Mueller and J. M. Pawlowski, Phys. Rev. D **91**, 116010 (2015), \[arXiv:1502.08011 \[hep-ph\]\]. G. Endrodi, JHEP **07** (2015) 173, \[arXiv:1504.08280 \[hep-lat\]\]. E. S. Fraga, Lect. Notes Phys. [**871**]{}, 121 (2013), \[arXiv:1208.0917 \[hep-ph\]\]. D. Kharzeev, K. Landsteiner, A. Schmitt and H. -U. Yee, Lect. Notes Phys. [**871**]{}, 1 (2013). J. O. Andersen, W. R. Naylor and A. Tranberg, \[arXiv:1411.7176 \[hep-ph\]\].
V. A. Miransky and I. A. Shovkovy, Phys. Rept. [**576**]{}, 1 (2015), \[arXiv:1503.00732 \[hep-ph\]\]. Z. Fodor and S. D. Katz, \[arXiv:0908.3341 \[hep-ph\]\].
C. Bonati, M. D’Elia, M. Mariti, F. Negro and F. Sanfilippo, Phys. Rev. D **89**, 054506 (2014), \[arXiv:1310.8656 \[hep-lat\]\].
G. S. Bali, F. Bruckmann, G. Endrodi, S. D. Katz and A. Schafer, JHEP **08** (2014) 177, \[arXiv:1406.0269 \[hep-lat\]\].
J. Maldacena, Adv. Theor. Math. Phys. **2**, 231 (1998), \[arXiv:hep-th/9711200\].
S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B **428**, 105 (1998), \[arXiv:hep-th/9802109\].
E. Witten, Adv. Theor. Math. Phys. **2**, 253 (1998), \[arXiv:hep-th/9802150\].
J. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopal and U. A. Wiedemann, \[arXiv:1101.0618 \[hep-th\]\].
A. Adams, L. D. Carr, T. Schaefer, P. Steinberg and J. E. Thomas, New J. Phys. **14**, 115009 (2012), \[arXiv:1205.5180 \[hep-th\]\].
S. S. Gubser, A. Nellore, S. S. Pufu and F. D. Rocha, Phys. Rev. Lett. **101**, 131601 (2008), \[arXiv:0804.1950 \[hep-th\]\].
S. I. Finazzo and J. Noronha, Phys. Rev. D **89**, 106008 (2014), \[arXiv:1311.6675 \[hep-th\]\].
S. I. Finazzo, R. Rougemont, H. Marrochio and J. Noronha, JHEP **02** (2015) 051, \[arXiv:1412.2968 \[hep-ph\]\].
O. DeWolfe, S. S. Gubser and C. Rosen, Phys. Rev. D **84**, 126014 (2011), \[arXiv:1108.2029 \[hep-th\]\].
R. Rougemont, A. Ficnar, S. Finazzo and J. Noronha, \[arXiv:1507.06556 \[hep-th\]\].
R. Rougemont, J. Noronha and J. Noronha-Hostler, Phys. Rev. Lett. **115**, 202301 (2015), \[arXiv:1507.06972 \[hep-ph\]\].
S. I. Finazzo and R. Rougemont, \[arXiv:1510.03321 \[hep-ph\]\]. H. B. Meyer, Eur. Phys. J. A [**47**]{}, 86 (2011), \[arXiv:1104.3708 \[hep-lat\]\]. E. D’Hoker and P. Kraus, JHEP [**10**]{}, 088 (2009), \[arXiv:0908.3875 \[hep-th\]\]. E. D’Hoker and P. Kraus, Class. Quant. Grav. [**27**]{}, 215022 (2010), \[arXiv:1006.2573 \[hep-th\]\]. E. D’Hoker and P. Kraus, Lect. Notes Phys. [**871**]{}, 469 (2013), \[arXiv:1208.1925 \[hep-th\]\]. G. Basar, D. E. Kharzeev, Phys. Rev. D **85**, 086012 (2012), \[arXiv:1202.2161 \[hep-th\]\].
R. Critelli, S. I. Finazzo, M. Zaniboni and J. Noronha, Phys. Rev. D **90**, 066006 (2014), \[arXiv:1406.6019 \[hep-th\]\].
R. Rougemont, R. Critelli and J. Noronha, Phys. Rev. D **91**, 066001 (2015), \[arXiv:1409.0556 \[hep-th\]\].
J. F. Fuini III and L. G. Yaffe, JHEP **07** (2015) 116, \[arXiv:1503.07148 \[hep-th\]\].
Sz. Borsanyi, G. Endrodi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti and K. K. Szabo, JHEP **08** (2012) 053, \[arXiv:1204.6710 \[hep-lat\]\].
Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz and K. K. Szabo, Nature [**443**]{}, 675 (2006), \[hep-lat/0611014\]. S. S. Gubser and A. Nellore, Phys. Rev. D **78**, 086007 (2008), \[arXiv:0804.0434 \[hep-th\]\].
S. I. Finazzo and J. Noronha, Phys. Rev. D **90**, 115028 (2014), \[arXiv:1411.4330 \[hep-th\]\].
U. Gursoy, E. Kiritsis and F. Nitti, JHEP **02** (2008) 019, \[arXiv:0707.1349 \[hep-th\]\].
U. Gursoy and E. Kiritsis, JHEP **02** (2008) 032, \[arXiv:0707.1324 \[hep-th\]\].
U. Gursoy, E. Kiritsis, L. Mazzanti and F. Nitti, JHEP **05** (2009) 033, \[arXiv:0812.0792 \[hep-th\]\].
O. DeWolfe, S. S. Gubser and C. Rosen, Phys. Rev. D **83**, 086005 (2011), \[arXiv:1012.1864 \[hep-th\]\].
T. Alho, M. Jarvinen, K. Kajantie, E. Kiritsis, C. Rosen and K. Tuominen, JHEP **04** (2014) 124, \[arXiv:1312.5199 \[hep-ph\]\].
G. Veneziano, Nucl. Phys. B **117**, 519 (1976).
J. Noronha, Phys. Rev. D [**81**]{}, 045011 (2010), \[arXiv:0910.1261 \[hep-th\]\]. J. Noronha, Phys. Rev. D [**82**]{}, 065016 (2010), \[arXiv:1003.0914 \[hep-th\]\]. A. Ficnar, J. Noronha and M. Gyulassy, Nucl. Phys. A [**855**]{}, 372 (2011), \[arXiv:1012.0116 \[hep-ph\]\]. A. Ficnar, J. Noronha and M. Gyulassy, J. Phys. G [**38**]{}, 124176 (2011), \[arXiv:1106.6303 \[hep-ph\]\]. A. Ficnar, J. Noronha and M. Gyulassy, Nucl. Phys. A [**910-911**]{}, 252 (2013), \[arXiv:1208.0305 \[hep-ph\]\]. R. A. Janik, G. Plewa, H. Soltanpanahi and M. Spalinski, Phys. Rev. D **91**, 126013 (2015), \[arXiv:1503.07149 \[hep-th\]\]. D. L. Yang and B. M[ü]{}ller, \[arXiv:1507.04232 \[hep-th\]\].
J. W. York Jr., Phys. Rev. Lett. **28**, 1082 (1972).
G. W. Gibbons and S. W. Hawking, Phys. Rev. D **15**, 2752 (1977).
M. Henningson and K. Skenderis, JHEP **07** (1998) 023, \[arXiv:hep-th/9806087\].
S. de Haro, K. Skenderis and S. N. Solodukhin, Commun. Math. Phys. **217**, 595 (2001), \[arXiv:hep-th/0002230\].
K. Skenderis, Class. Quant. Grav. **19**, 5849 (2002), \[arXiv:hep-th/0209067\].
I. Papadimitriou, JHEP **08** (2011) 119, \[arXiv:1106.4826 \[hep-th\]\].
J. Lindgren, I. Papadimitriou, A. Taliotis and J. Vanhoof, JHEP **07** (2015) 094, \[arXiv:1505.04131 \[hep-th\]\].
J. D. Bekenstein, Phys. Rev. D **7**, 2333 (1973).
S. W. Hawking, Commun. Math. Phys. **43**, 199 (1975); \[Erratum-ibid. **46**, 206 (1976)\].
R. Yaresko, J. Knaute and B. Kämpfer, Eur. Phys. J. C [**75**]{}, no. 6, 295 (2015), \[arXiv:1503.09065 \[hep-ph\]\]. M. Blake, A. Donos and N. Lohitsiri, JHEP **08** (2015) 124, \[arXiv:1502.03789 \[hep-th\]\].
V. P. Gusynin, V. A. Miransky and I. A. Shovkovy, Phys. Rev. Lett. [**73**]{}, 3499 (1994) \[Phys. Rev. Lett. [**76**]{}, 1005 (1996)\], \[hep-ph/9405262\]. V. P. Gusynin, V. A. Miransky and I. A. Shovkovy, Nucl. Phys. B [**462**]{}, 249 (1996), \[hep-ph/9509320\]. V. A. Miransky and I. A. Shovkovy, Phys. Rev. D [**66**]{}, 045006 (2002), \[hep-ph/0205348\]. G. Endrödi, JHEP [**04**]{}, 023 (2013), \[arXiv:1301.1307 \[hep-ph\]\]. M. Jarvinen and E. Kiritsis, JHEP **03** (2012) 002, \[arXiv:1112.1261 \[hep-ph\]\].
C. Charmousis, B. Gouteraux, B. S. Kim, E. Kiritsis and R. Meyer, JHEP [**11**]{}, 151 (2010), \[arXiv:1005.4690 \[hep-th\]\]. S. W. Hawking and D. N. Page, Commun. Math. Phys. [**87**]{}, 577 (1983). A. Ballon-Bayona, JHEP [**11**]{}, 168 (2013), \[arXiv:1307.6498 \[hep-th\]\]. K. A. Mamo, JHEP [**05**]{}, 121 (2015), \[arXiv:1501.03262 \[hep-th\]\]. T. D. Cohen and N. Yamamoto, Phys. Rev. D **89**, 054029 (2014), \[arXiv:1310.2234 \[hep-ph\]\].
K. A. Mamo and H. U. Yee, Phys. Rev. D **92**, 105005 (2015), \[arXiv:1505.01183 \[hep-ph\]\]. S. Borsanyi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti and K. Szabo, JHEP [**1201**]{}, 138 (2012), \[arXiv:1112.4416 \[hep-lat\]\]. G. Aarts, C. Allton, A. Amato, P. Giudice, S. Hands and J. I. Skullerud, JHEP [**1502**]{}, 186 (2015), \[arXiv:1412.6411 \[hep-lat\]\].
[^1]: It is not clear at the moment if the electromagnetic fields present in the early stages of heavy ion collisions remain strong enough to directly affect equilibrium and transport properties of the plasma produced at later stages. However, see [@Gursoy:2014aka] for a recent study on how the QGP’s electric conductivity may actually delay the decay of the magnetic field in the medium.
[^2]: These nonconformal solutions can also be adapted to study the vacuum properties of the gauge theory, as recently discussed in [@stefanovacuum]. This was studied in detail earlier in [@ihqcd-1; @ihqcd-2; @hot-ihqcd] in the case of similar bottom-up models at zero and finite temperature concerning pure glue Yang-Mills theory.
[^3]: See also [@ihqcd-veneziano] for a bottom-up holographic model at finite temperature, nonzero chemical potential, and zero magnetic field in the Veneziano limit [@veneziano].
[^4]: See, for instance, [@DK3] for a construction with finite $B$ and finite electric charge density in the context of Einstein-Maxwell-Chern-Simons theories.
[^5]: As we shall discuss soon, $\mathcal{B}$ is one of the two initial conditions controlling the temperature and the external magnetic field at the boundary quantum field theory. The other initial condition corresponds to the value of the dilaton field evaluated at the black brane horizon, $\phi_0$. The set of initial conditions $(\phi_0,\mathcal{B})$ is nontrivially related to the thermodynamical pair $(T,B)$ in the gauge theory. In Sections \[sec2.3\] and \[sec2.4\] we discuss how one can relate $\mathcal{B}$ to the external magnetic field at the boundary gauge theory, $B$.
[^6]: Note that in the Maxwell-Dilaton gauge coupling $f(\phi)$ plays the role of an inverse effective gauge coupling squared and, therefore, it must correspond to a positive-definite function.
[^7]: This constant is equal to one in the so-called “standard coordinates” of the domain-wall gauge, which we shall discuss soon. Here we are considering general coordinates where this constant may be different than one. We shall also see later how to relate these two sets of coordinates.
[^8]: This procedure may be repeated to obtain all the other subleading terms in the ultraviolet expansions. However, we only need the first few terms in these expansions to compute the thermodynamical observables.
[^9]: The horizon is a singular point of the equations of motion and, thus, we need to initialize the numerical integrations slightly above it.
[^10]: As discussed before, the other initial condition is $\mathcal{B}$.
[^11]: In practice, we set to zero the following 7 terms: $\mathcal{O}(r_{\textrm{start}}^{0})$, $\mathcal{O}(r_{\textrm{start}}^{1})$, and $\mathcal{O}(r_{\textrm{start}}^{2})$ in , $\mathcal{O}(r_{\textrm{start}}^{-1})$ in , $\mathcal{O}(r_{\textrm{start}}^{-1})$ and $\mathcal{O}(r_{\textrm{start}}^{0})$ in , and $\mathcal{O}(r_{\textrm{start}}^{0})$ in .
[^12]: As mentioned in [@gubser1], if $\phi_A<0$ one must replace $\phi_A\mapsto|\phi_A|$ in these relations.
[^13]: Note from , , and that $\hat{B}$, $\hat{T}$, and $\hat{s}$ are proportional to $\phi_A^{-2/\nu}$, $\phi_A^{-1/\nu}$ and $\phi_A^{-3/\nu}$, respectively. Correspondingly, their counterparts in physical units (without the “hat”) are given in MeV$^2$, MeV, and MeV$^3$, respectively. This is related to the fact that the leading mode for the dilaton field, $\phi_A$, corresponds to the insertion of a relevant deformation in the quantum field theory, which is responsible for generating an infrared scale that breaks the conformal invariance of the theory at low energies [@gubser2].
[^14]: As mentioned in footnote 7 of [@donos], the Euclidean action has the opposite sign of the Lorentzian action.
[^15]: Note that $\phi_0=7.8$ corresponds to the local minimum of our dilaton potential . For $\phi_0>7.8$, our dilaton potential becomes non-monotonic and, in practice, we took $\phi_0=7.8$ as the upper bound for the initial condition $\phi_0$ in our numerical calculations to avoid complications with extra singular points in the equations of motion.
[^16]: As discussed in detail in Section 2 of [@latticedata3] this corresponds to the isotropic pressure in the so-called “$B$-scheme” where the magnetic field is kept fixed during compression. Also, this corresponds to the anisotropic pressure in the direction of the magnetic field in the so-called “$\Phi$-scheme” where the magnetic flux is kept fixed during compression.
[^17]: Note that in [@latticedata3] the pressure was obtained from the renormalized free energy density. Here, we could have done the analogous holographic procedure by calculating the free energy density from the holographically renormalized on-shell action for the EMD model. This is, however, a much more laborious calculation than the one we have carried out here where we first calculated the entropy density using the Bekenstein-Hawking’s relation and then we calculated the pressure (difference) using Eq. .
[^18]: Probably a better agreement with $B\neq 0$ lattice data may be obtained by improving the choice of the model parameters through a global fit to $B=0$ lattice data for the pressure, the entropy density, the speed of sound, and the trace anomaly.
[^19]: Since the crossover is not a genuine phase transition, the free energy is analytic in the region where the degrees of freedom change from a hadron gas to a deconfined plasma. Thus, the definition of the crossover temperature $T_c$ depends on the observable one uses to characterize it. Different observables can give in principle different values for $T_c$ and one may use them to obtain a band defining the crossover region [@latticedata0; @latticedata3].
[^20]: Note from Fig. \[fig4\] that for the values of $B$ considered here we only have a smooth analytical crossover, as also seen on the lattice.
[^21]: We use that $a(r_H)=a_0=0$ and $\phi(r_H)=\phi_0$.
|
---
abstract: |
We introduce a non-symmetric operad $\mathcal{N}$, whose dimension in degree $n$ is given by the Catalan number $c_{n-1}$. It arises naturally in the study of coalgebra structures defined on compatible associative algebras. We prove that any free compatible associative algebra admits a compatible infinitesimal bialgebra structure, whose subspace of primitive elements is a $\mathcal{N}$-algebra. The data $({\rm As},{\rm As}^2, \mathcal{N})$ is a good triple of operads, in J.-L. Loday’s sense. Our construction induces another triple of operads $({\rm As},{\rm As}_2,{\rm As})$, where ${\rm As}_2$ is the operad of matching dialgebras.
Motivated by A. Goncharov’s Hopf algebra of paths $P(S)$, we introduce the notion of bi-matching dialgebras and show that the Hopf algebra $P(S)$ is a bi-matching dialgebras.
address: 'Instituto de Matemáticas y Física, Universidad de Talca, Avda. Lircay s/n, Talca, Chile'
author:
- Sebastián Márquez
title:
- COMPATIBLE ASSOCIATIVE BIALGEBRAS
- COMPATIBLE ASSOCIATIVE BIALGEBRAS
---
[^1]
Introduction {#s:int .unnumbered}
============
In the present work, we study the existence of coalgebra structures on compatible associative algebras. A [*compatible associative algebra*]{} over a field $\KK$ is a vector space $A$ equipped with two associative products, $\cdot$ and $\circ$, satisfying that the sum $$(1)\ x \ast y:=x \cdot y + x \circ y,$$ is an associative product.\
In [@Strohmayer-Operads; @of; @compatible; @structures], H. Strohmayer developed the general notion of compatible algebraic structures, and denotes the operad of compatible associative algebras by $\rm{As}^2$. He computed the Koszul dual of ${\rm As}^2$, denoted by $^2{\rm As}$, which is a set theoretical operad, by arising an operadic partition poset (see [@ParttionPosets]). The same author showed, using B. Valettes results, that $^2{\rm As}$ is a Koszul operad, and therefore $\rm{As}^2$ is Koszul operad, too.
Applying H. Strohmayers work, V. Dotsenko obtained in [@Dotsenko-Compatible; @associative; @product] the dimensions of the operad $\rm{As}^2$ and computed the characters of $\rm{As}^2(n)$, both as an $S_n$-module and as an $S_n\times SL_2$-module. The dimension of $\rm{As}^2(n)$ is $n!$ times the Catalan number $c_n$.
The Catalan numbers are sequence of natural numbers, named after the Belgian mathematician Eugène Charles Catalan (1814-1894). There exist many counting problems in combinatorics whose solution is given by them, a large description of the combinatorial objects described by Catalan numbers may be found in [@Stanley]. In particular, the $n^{th}$ Catalan number describes the number of plane rooted trees.
We look for coalgebra structures which are compatible with condition $(1)$, and therefore may be defined on any free compatible associative algebra. In [@LodayRonco-On; @the; @structure; @of; @cofree; @Hopf; @algebras], J.-L. Loday and M. Ronco introduced the notion of unital infinitesimal bialgebra, as an associative unital algebra $(C, *, u)$ equipped with a coassociative coproduct $\Delta: C\longrightarrow C\otimes C$ satisfying that : $$\Delta (x*y)= \sum x_{(1)}\otimes (x_{(2)}*y) + (x*y_{(1)})\otimes y_{(2)}\ -\ x\otimes y,$$ for $x,y\in C$, where $\Delta (x)=\sum x_{(1)}\otimes x_{(2)}$ and $\Delta (y)=\sum y_{(1)}\otimes y_{(2)}$.
A [*compatible infinitesimal bialgebra*]{} is a compatible algebra $(A, \cdot, \circ)$ equipped with a coassociative coproduct $\Delta$, satisfying the unital infinitesimal relation with both associative products.
We give an explicit construction of free objects in the category of $\rm{As}^2$-algebras, easier to work with than the one described in [@Dotsenko-Compatible; @associative; @product]. Using it, we describe a canonical coproduct $\Delta$ on any free $\rm{As}^2$-algebra, which satisfies the unital infinitesimal condition with both products.
Following J.-L. Loday (see [@Loday-GeneralizedBialgebras]) and R. Holtkamp (see [@Holtkamp]), we know that there exists an algebraic operad which describes the subspace of primitive elements of a compatible associative bialgebra, we call this new structure a ${\mathcal N}$-algebra. The operad ${\mathcal N}$ is non-symmetric, and the dimension of the $\KK$-vector space ${\mathcal N}_n$ is the Catalan number $c_{n-1}$. We prove that:
1. there exists an operad homomorphism from ${\mathcal N}$ to the operad of compatible associative algebras,
2. the subspace of primitive elements of any compatible associative bialgebra has a natural structure of ${\mathcal N}$ algebra,
3. the free compatible associative algebra over a vector space $V$ is isomorphic, as a coalgebra, to the cofree conilpotent coalgebra spanned by the free ${\mathcal N}$ algebra over $V$.
As a consequence of the previous results, we prove that the category of conilpotent compatible associative algebras is equivalent to the category of ${\mathcal N}$ algebras, which gives a good triple of operads $({\rm As},{\rm As}^2, \mathcal{N})$ as defined by J.-L. Loday in [@Loday-GeneralizedBialgebras].
A [*matching dialgebra*]{}, previously studied by C. Bai, L. Guo and Y.Zhang in [@Zhang-The; @category; @and; @operad; @of; @matching; @dialgebras], is a compatible associative algebra $(A, \cdot, \circ)$ satisfying that: $$(x \cdot y) \circ z=x\cdot (y \circ z) \text{ and } (x\circ y) \cdot z=x\circ (y \cdot z),$$ for all the elements $x,y,z \in A$.
Motivated by [*the path algebra*]{} introduced by A. Goncharov in [@Goncharov-Galois; @symmetries; @of; @fundamental; @groupoids; @and; @noncommutative; @geometry], we define the notion of [*bi-matching dialgebras*]{} ,and show that the path algebra is a example of a bi-matching dialgebra, which is obtained from [*semi-homomorphism*]{} of algebras. Finally, studying the subcategory of compatible associative bialgebras satisfying that their underlying compatible associative algebra is a matching dialgebra, we obtain another triple of operads $({\rm As},{\rm As}_2,{\rm As})$.
The manuscript is organized as follows: Section 1 we construct the free compatible associative algebra over a vector space $V$.
In Section 2 we develop the notion of compatible infinitesimal bialgebra, while in Section 3 we introduce the notion of ${\mathcal N}$-algebra and prove the structure theorem for compatible infinitesimal bialgebras.
Matching dialgebras are defined in Section 4, where we describe A. Goncharovs path Hopf algebra as the main example of this type of structure.\
Acknowledgment {#acknowledgment .unnumbered}
--------------
I would like to express my thanks to Prof. M. Ronco for motivating me to work on this problem and for her constant contributions to it, and to Prof. A. Labra for many useful comments and for encouraging me to continue my research work. My special thanks to the University of Talca for the support provided during this period.
Notations {#notations .unnumbered}
=========
All vector spaces and algebras considered in the manuscript are over a field $\KK$. Given a set $X$, we denote by $\KK[X]$ the vector space spanned by $X$. For any vector space $V$, we denote by $V^{\otimes n}$ the tensor product of $V\otimes \dots \otimes V$, $n$ times, over $\KK$. In order to simplify notation, we shall denote an element of $V^{\otimes n}$ by $x_1\cdots x_n$.
If $n$ is a positive integer, we denote by $[n]$ the set $\{1,\ldots, n\}$. The symmetric group of permutations of $[n]$ is denoted by $S_n$. Given a permutation $\sigma \in S_n$, we write $\sigma=(\sigma(1),\ldots,\sigma(n))$, identifying $\sigma$ with its image.
The free compatible associative algebra {#freeCompAssAlg}
=======================================
In [@OperadLieComp], V. Dotsenko and A. Khoroshkin computed the dimensions of the components for the operad of the [*compatible Lie algebras*]{} and for the [*bi-Hamiltonian operad*]{}.
A [*compatible Lie algebra*]{} is a $\KK$-vector space $A$, equipped with two Lie brackets $[,]$ and $\{,\}$ , satisfying that their sum is also a Lie bracket.
When the compatible Lie algebra $A$ is equipped with a commutative and associative product $\cdot$ such that the brackets are both derivations for the product $\cdot$, we say that $A$ is a [*bi-Hamiltonian algebra*]{}. In particular, a bi-Hamiltonian algebra is an analogue of a double structure of Poisson algebra, which appears naturally in certain examples of integrable systems.
As in the classical case, there exists a functor from the category of $\rm{As}^2$-algebras over $\KK$ to the category of compatible Lie algebras over the same field. If $(A, \cdot, \circ)$ is a compatible associative algebra, then the Lie brackets given by $$\begin{array}{rll} [x,y]&=x \cdot y -y\cdot x , \\
\{x,y\}&=x \circ y -y\circ x, \\
\end{array}$$ define a structure of compatible Lie algebra on the underlying vector space of $A$.
In this section, we construct the free compatible associative algebra on a vector space $V$ by means of planar rooted trees. We recall the definition of compatible associative algebra, already described in Introduction.
\[def:AssComp\] A *compatible associative algebra* is a vector space $A$ together with two associative products $\pun :A\otimes A\to A$ and $\cir:A\otimes A\to A$, satisfying that their sum $*:=\pun + \cir $ is an associative product, too.
Note that the if the product $*=\pun + \cir $ is associative, then all linear combinations $\lambda\pun\ +\ \mu\ \cir $ are associative products, for any coefficients $\lambda$ and $\mu $ in $\KK$.
\[compatibilidad\] The condition $\pun + \cir $ is an associative product, is equivalent to: $$x\cir(y\pun z)+x\pun(y\cir z)=(x\cir y)\pun z+(x\pun y)\cir z,$$ for all elements $x, y, z \in A$.
V. Dotsenko in [@Dotsenko-Compatible; @associative; @product] constructed associative products on vector spaces spanned by trees using R. Grossman and R.G. Larsons constructions (see [@Grossman-larson_Hopf-algebraic; @structure; @of; @families; @of; @trees]), but the associative products are compatible only in certain cases and quite difficult to deal with.
We give a different construction of the free associative compatible algebra, applying Dotsenkos results on its dimensions, by means of planar rooted trees.
The vector space ${\rm As}^2(V)$
--------------------------------
Let $V$ be a vector space with basis $X=\{a_i\}_{i \in I}$. Denote by $T{}^{X}_{n}$ the set of planar rooted trees with $(n+1)$ vertices, whose vertices different from the root are colored by the elements of $X$. For instance:
{width="12cm"}
Consider the vector space ${\rm{As}}^2(V)=\KK[\bigcup_{n\geq 1}T{}^{X}_{n}]=\bigoplus_{n\geq
1}^{} \KK[T{}^{X}_{n}]$, whose basis is the set $\bigcup_{n\geq 1}T{}^{X}_{n}$ of all planar rooted colored trees.
For any tree $t$ in $T{}^{X}_{n}$, we say that $t$ has degree $n$ and we write $\vert t\vert =n$. We consider the tree $t$ oriented from bottom to top.
Given a vertex $v \in t$, we say that a vertex $v' \in t$ is a child of $v$ if $v'$ is directly connected to the vertex $v$.
Given a tree $t$, the set of vertices of $t$ is denoted by ${\rm Vert}(t)$ and the root of $t$ by ${\rm root}(t)$. The subset ${\rm Vert}(t)\setminus \{{\rm root}(t)\}$ of ${\rm Vert}(t)$ is denoted by ${\rm Vert}^{\ast}(t)$.
We define two associative products in ${\rm As}^2(V)$.
\[ProductoPunto\] Let $t, w$ be trees in ${\rm{As}}^2(V)$. Define $t \pun w$ as the tree obtained by identifying the roots of $t$ and $w$. Extending this binary operation by linearity, we get an associative product $$\cdot :{\rm{As}}^2(V) \otimes {\rm{As}}^2(V) \rightarrow {\rm{As}}^2(V).$$
\[arbolesIrreEn$As^2$\] Note that any tree $t$ in ${\rm{As}}^2(V)$ may be written in a unique way as $t=t^{1}\pun \ldots \pun t^{r}$, where $r \geq 1$ and the root of each $t^{i}$ has only one child, for each $i \in \{1,\ldots, r\}$. Clearly, we have that $|t|=\displaystyle{\sum_{i=1}^{r}|t^i|} $.
When the root of a tree $t \in T^X_n$ has a unique child, we say that $t$ is [*irreducible*]{}. We identify the elements of the basis $X$ with the trees of degree one (which are irreducible).
Denoting by ${\rm Irr}$ the vector space spanned by the set of all irreducible trees in ${\rm{As}}^2(V)$, we have that $({\rm{As}}^2(V),\cdot)$ is free over ${\rm Irr}$ as an associative algebra. The set of irreducible trees of degree $n$ is denoted by ${\rm Irr}_n$.
Second product {#ProductoCirculo}
--------------
Let $t$ and $w$ be trees in ${\rm{As}}^2(V)$, with $t=t^1 \pun \ldots \pun t^r$ as described in Remark \[arbolesIrreEn$As^2$\]. A second product $t\cir w$ is defined proceeding by induction on the degree $n$ of $w$.
If $n=1$, then $w=a$, for some $a \in X$. In this case, the element $t \circ w $ is the tree obtained by replacing the root of $t$ by the vertex, colored with $a$, and adding a new root.\
{width="6cm"}
Assume now that $n > 1$, and that the product $t \circ w$ has been defined for any $|w| < n$.
Let $w=w^{1} \pun \ldots \pun w^{m}$ be the unique decomposition of $w$ as a product of irreducible trees.
If $m=1$, then $w=w^{1}=u \circ a$, where $u$ is a tree such that $|u|=n-1$ and $a$ is an element of the basis $X$. Applying a recursive argument, we may suppose that $t \circ u$ is already defined. The product $t \circ w$ is the element $$t \cir w :=t \cir (u \cir a ) = ( t \cir u ) \cir a .$$
For $m > 1$, the element $t \circ w$ is defined by the following formula:
$$\begin{split}
t \circ w &= \sum _{i=1}^{m}((t \cdot w^1 \cdot \ldots \cdot w^{i-1})\circ w^{i})\cdot \ldots \cdot w^{m}\\
& \quad -\sum_{i=2}^{m} t\cdot ((w^{1} \pun \ldots \pun w^{i-1})\circ w^{i})\cdot w^{i+1}\cdot \ldots \cdot w^{m}\\
\end{split}$$
Note that the recursive hypothesis states that each term of the previous formula is well defined.
Let $a_1, \ldots , a_n$ be elements of the basis $X$, with $n\geq 2$. Consider the tree $w$ given by\
{width="9cm"}
If $n=2$, then we have that $$t \cir w =t \cir (a_1 \pun a_2)= (t
\cir a_1) \pun a_2 - t \pun (a_1 \cir a_2) + (t \pun a_1
)\cir a_2,$$ therefore we get
{width="8cm"}
In general, for $w=a_1\cdot \ldots \cdot a_n$, we get the formula:
{width="10cm"}
The vector space ${\rm As}^2(V)$, equipped with the products $\pun $ and $\cir$ defined in \[ProductoPunto\] and \[ProductoCirculo\], is the free associative compatible algebra on $V$.
Note that the definition of the product $\circ$ implies that the products $\cdot$ and $\circ$ satisfy the compatibility condition.
Let us prove that the product $\circ$ is associative. Let $t_1, t_2, t_3$ be trees in ${\rm As}^2(V)$. To see that $t_1\circ (t_2 \circ t_3)= (t_1\circ t_2) \circ t_3$, we proceed by induction on the degree of the third term $t_3$.
If the degree of $t_3$ is one, the assertion is follows easily from Definition \[ProductoCirculo\].
Suppose that $n=|t_3| > 1$. If $t_3$ is an irreducible tree, then $t_3=w \circ a$, where $w$ is a tree with $|w|=n-1$, and $a$ is an element of the basis $X$. Applying a recursive argument, we get the following identities: $$\begin{array}{rll} t_1 \circ (t_2 \circ t_3)&=t_1 \circ (t_2 \circ (w \circ a))& \\
&=t_1 \circ ((t_2 \circ w) \circ a)\\
&=(t_1 \circ (t_2 \circ w)) \circ a\\
&=((t_1 \circ t_2) \circ w) \circ a\\
&=(t_1 \circ t_2) \circ (w \circ a)\\
&=(t_1 \circ t_2) \circ t_3,\\
\end{array}$$ which imply the result.
Suppose now that $t_3=w \pun z$, where $w$ and $z$ are trees of degree smaller than $|t_3|$.
By the compatibility condition and a recursive argument, we get that: $$\begin{array}{rll}
t_1 \circ (t_2 \circ t_3)&=t_1\circ (t_2 \circ (w\pun z))& \\
&=t_1\circ ((t_2 \circ w)\pun z -t_2\pun(w \circ z)+ (t_2 \pun w)\circ z) \\
&=(t_1\circ(t_2\circ w))\pun z-t_1\pun ((t_2\circ w)\circ z)+(t_1\pun (t_2 \circ w))\circ z \\
&\quad -(t_1\circ t_2 )\pun (w \circ z)+ t_1 \pun (t_2 \circ (w \cir z))-(t_1 \pun t_2 )\circ (w \circ z )\\
&\quad +t_1\circ ((t_2 \pun w)\circ z)\\
&=((t_1 \circ t_2) \circ w)\pun z-(t_1 \circ t_2 )\pun (w \circ z )\\
&\quad +(t_1 \circ (t_2 \pun w )-(t_1 \pun t_2 ) \circ w + t_1 \pun (t_2 \circ w))\circ z.\\
\end{array}$$
As $(t_1\circ t_2)\pun w = t_1 \circ (t_2 \pun w )-(t_1 \pun t_2 ) \circ w + t_1 \pun (t_2 \circ w)$, we conclude that
$$\begin{array}{rll}
t_1 \circ (t_2 \circ t_3)&=((t_1 \circ t_2) \circ w)\pun z-(t_1 \circ t_2 )\pun (w \circ z )+ ((t_1\circ t_2)\pun w)\circ z& \\
&=(t_1 \circ t_2)\circ (w \pun z )\\
&=(t_1 \circ t_2)\circ t_3.\\
\end{array}$$
To end the proof, we need to see that ${\rm As}^2(V)$ is free as associative compatible algebra. Let $A$ be an associative compatible algebra and let $f :V \rightarrow A$ be a linear map. The homomorphism $\widetilde{f} :{\rm As}^2(V) \rightarrow A$ is defined in a recursive way.
Let $t$ be a tree in ${\rm As}^2(V)$. If $|t|=1$ then $t=a$ with $a \in X$ and therefore $\widetilde{f}(t)=f(a)$.
Suppose that $|t|>1$. If $t=t'\circ a$, for some $a \in X$, is irreducible, we define $$\widetilde{f}(t)=\widetilde{f}(t')\circ f(a),$$ which is well defined by a recursive argument.
If $t=t^1 \pun \ldots \pun t^r$ for some $r >1$, then we can assume that $\widetilde{f}(t^i)$ is defined, for $1\leq i \leq r$, and set $\widetilde{f}(t)=\widetilde{f}(t^1) \pun \ldots \pun \widetilde{f}(t^r)$.
To see that $\widetilde{f}$ is unique, consider $g:{\rm As}^2(V) \rightarrow A$, a homomorphism of compatible associative algebras such that $g(a)=f(a)$, for $a \in V$. Let $t$ be a tree in ${\rm As}^2(V)$. If $|t|=1$, then $t=a$, with $a \in X$. So, by definition of $\widetilde{f}$, $g(t)=\widetilde{f}(t)$. Suppose that $|t|>1$. We have that $t=t'\circ a$, for some $a \in X$, or $t=t^1 \pun \ldots \pun t^r$, for some $r >1$. Applying a recursive argument, we have that $g(t)=\widetilde{f}(t)$. This show that $\widetilde{f}$ is unique, which ends the proof.
Compatible infinitesimal bialgebras {#SeccionComInfBial}
===================================
In this section, we introduce [*compatible infinitesimal bialgebras*]{}, which uses the notion of unital infinitesimal bialgebra introduced by J.-L Loday and M. Ronco in [@LodayRonco-On; @the; @structure; @of; @cofree; @Hopf; @algebras]. To work in the more general context, we do not assume the existence of unity. So, an infinitesimal bialgebra is an associative algebra $(H, \cdot)$ equipped with a coassociative coproduct $\Delta :H\longrightarrow H\otimes H$ satisfying $$\Delta(x \cdot y)= x_{(1)}\otimes (x_{(2)} \cdot y) + (x \cdot y_{(1)})\otimes y_{(2)} + x\otimes y,$$ for $x,y\in H$, with $\Delta (x)= x_{(1)}\otimes x_{(2)}$ and $\Delta(y)=y_{(1)}\otimes y_{(2)}$ for $x, y \in H$.
An element $x \in H$ is called [*primitive*]{} when $\Delta(x)=0$.
\[InfComp\][A [*compatible infinitesimal bialgebra*]{} over $\KK$ is an associative compatible algebra $(H, \pun ,\cir )$ equipped with a coassociative coproduct $\Delta :H\longrightarrow H\otimes H$ such that $(H,\pun, \Delta )$ and $(H,\cir ,\Delta )$ are both unital infinitesimal bialgebras.]{}
\[BuenaDefUIAA\] The notion of compatible infinitesimal bialgebra is well-defined.
A direct computation shows that: $$\begin{array}{rl} (1) \quad \Delta((x\cdot y)\circ z) &= \quad x_{(1)}\otimes (x_{(2)}\cdot y)\circ z+x\cdot y_{(1)}\otimes y_{(2)}\circ z+x\otimes (y\circ z) \\
& +(x\cdot y)\circ z_{(1)}\otimes z_{(2)}+(x\cdot y)\otimes z,\\
(2) \quad \Delta((x\circ y)\cdot z) &= \quad x_{(1)}\otimes (x_{(2)}\circ y)\cdot z+x\circ y_{(1)}\otimes y_{(2)}\cdot z+x\otimes (y\cdot z) \\
& +(x\circ y)\cdot z_{(1)}\otimes z_{(2)}+(x\circ y)\otimes z,\\
(3) \quad \Delta(x\cdot (y\circ z)) &= \quad x_{(1)}\otimes x_{(2)}\cdot( y\circ z)+x\cdot y_{(1)}\otimes y_{(2)}\circ z+x\otimes (y\circ z) \\
& +x\cdot (y\circ z_{(1)})\otimes z_{(2)}+(x\cdot y)\otimes z,\\
(4) \quad \Delta(x\circ( y\cdot z)) &= \quad x_{(1)}\otimes x_{(2)}\circ( y\cdot z)+x\circ y_{(1)}\otimes y_{(2)}\cdot z+x\otimes (y\cdot z) \\
& +x\circ (y\cdot z_{(1)})\otimes z_{(2)}+(x\circ y)\otimes z.\\
\end{array}$$ Using the compatibility condition between the products $\cdot$ and $\circ$, we get that:
1. $x_{(1)}\otimes (x_{(2)}\cdot y)\circ z+x_{(1)}\otimes (x_{(2)}\circ y)\cdot z= x_{(1)}\otimes x_{(2)}\cdot( y\circ z)+x_{(1)}\otimes x_{(2)}\circ( y\cdot z),$
2. $(x\cdot y)\circ z_{(1)}\otimes z_{(2)}+(x\circ y)\cdot z_{(1)}\otimes z_{(2)}=x\cdot (y\circ z_{(1)})\otimes z_{(2)}+x\circ (y\cdot z_{(1)})\otimes z_{(2)},$
which implies that $$\Delta((x\cdot y)\circ z+(x\circ y)\cdot z)=\Delta(x\cdot (y\circ z)+x\circ( y\cdot z)).$$
\[$As^2(V)$ ComoCompInfBialgebra\] Let $V$ be a vector space, the free associative compatible algebra $\rm{As}^2(V)$ has a natural structure of compatible infinitesimal bialgebra.
The coproduct $\Delta : {\rm As}^2(V)\rightarrow {\rm As}^2(V) \otimes {\rm As}^2(V)$ is defined by induction on the degree of a tree $t$ in ${\rm As}^2(V)$.
For $t=a \in X$, its image is $\Delta(t)=0$. When $|t|>1$, we consider two cases:
1. for $t=t' \circ a$, with $a \in X$, we define $$\Delta(t)=t'_{(1)}\otimes t'_{(1)}\circ a + t' \otimes a.$$
2. for $t=t' \pun t''$ with $|t'|<|t| \text{ and }|t''|< |t|$, we have that $$\Delta(t)=t'_{(1)}\otimes t'_{(2)}\pun t'' + t' \pun t''_{(1)}\otimes t''_{(2)}+t' \otimes t''.$$
Lemma \[BuenaDefUIAA\] and the inductive hypothesis, state that $\Delta$ is well defined. Note that if $\Delta(t)=t_{(1)} \otimes t_{(2)}$ then $|t_{(1)}|<|t| \text{ and }|t_{(2)}| < |t|$.
To see that $\Delta$ is coassociative, we proceed by induction on degree of $t$. Let $t$ be a tree. For $|t|=1$ the result is immediate.\
For $|t|>1$, we consider two case:
First, if $t$ is an irreducible tree, then $t=t'\circ a$, with $a \in X$. So, we have that: $$\begin{array}{rll}
(\Delta \otimes {\rm Id})\Delta (t)&=(\Delta \otimes {\rm Id})(t'_{(1)}\otimes t'_{(2)} \circ a+t'\otimes a)& \\
&=\Delta(t'_{(1)})\otimes t'_{(2)} \circ a+\Delta(t')\otimes a \\
&=t'_{(1)(1)}\otimes t'_{(1)(2)}\otimes t'_{(2)}\circ a+t'_{(1)}\otimes t'_{(2)}\otimes a. \\
\end{array}$$
Applying the recursive hypothesis to $t'$, we write $$(\Delta \otimes {\rm Id})\Delta (t)=t'_{(1)}\otimes t'_{(2)}\otimes t'_{(3)}\circ a+t'_{(1)}\otimes t'_{(2)}\otimes a.$$
On the other hand, using a similar argument to computer $({\rm Id} \otimes \Delta)\Delta(t)$, we have that $$\begin{array}{rll}
({\rm Id} \otimes \Delta)\Delta (t)&=({\rm Id}\otimes \Delta)(t'_{(1)}\otimes t'_{(2)} \circ a+t'\otimes a& \\
&=t'_{(1)}\otimes \Delta (t'_{(2)} \circ a)+t'\otimes \Delta (a) \\
&=t'_{(1)}\otimes t'_{(2)(1)}\otimes t'_{(2)(2)}\circ a+t'_{(1)}\otimes t'_{(2)}\otimes a \\
&=t'_{(1)}\otimes t'_{(2)}\otimes t'_{(3)}\circ a+t'_{(1)}\otimes t'_{(2)}\otimes a ,\\
\end{array}$$ which gives the expected result.
Second, if $t$ is reducible tree, then $t=t'\pun t''$, with $|t'|<|t| \text{ and }|t''|< |t|$. Applying the recursive hypothesis to $t'$ and $t''$, we have that $$\begin{array}{rl}
(\Delta \otimes {\rm Id})\Delta (t)&=\quad (\Delta\otimes {\rm Id} )(t'_{(1)}\otimes t'_{(2)} \pun t''+t'\pun t''_{(1)}\otimes t''_{(2)}+t'\otimes t'') \\
&=\quad t'_{(1)(1)}\otimes t'_{(1)(2)}\otimes t'_{(2)} \pun t''+t'_{(1)}\otimes t'_{(2)}\pun t''_{(1)}\otimes t''_{(2)} \\
& + t'\pun t''_{(1)(1)}\otimes t''_{(1)(2)}\otimes t''_{(2)}+t'\otimes t''_{(1)}\otimes t''_{(2)}+ t'_{(1)}\otimes t'_{(2)}\otimes t''\\
&=\quad t'_{(1)}\otimes t'_{(2)}\otimes t'_{(3)} \pun t''+t'_{(1)}\otimes t'_{(2)}\pun t''_{(1)}\otimes t''_{(2)}\\
& + t'\pun t''_{(1)}\otimes t''_{(2)}\otimes t''_{(3)}+t'\otimes t''_{(1)}\otimes t''_{(2)}+ t'_{(1)}\otimes t'_{(2)}\otimes t''\\
\end{array}$$ and, $$\begin{array}{rl}
({\rm Id} \otimes \Delta)\Delta (t)&=\quad ({\rm Id}\otimes \Delta)(t'_{(1)}\otimes t'_{(2)} \pun t''+t'\pun t''_{(1)}\otimes t''_{(2)}+t'\otimes t'') \\
&=\quad t'_{(1)}\otimes t'_{(2)(1)}\otimes t'_{(2)(2)} \pun t''+t'_{(1)}\otimes t'_{(2)}\pun t''_{(1)}\otimes t''_{(2)} \\
& + t'_{(1)}\otimes t'_{(2)}\otimes t''+t'\cdot t''_{(1)}\otimes t''_{(2)(1)}\otimes t''_{(2)(2)}+ t'\otimes t''_{(2)}\otimes t''_{(2)}\\
&=\quad t'_{(1)}\otimes t'_{(2)}\otimes t'_{(3)} \pun t''+t'_{(1)}\otimes t'_{(2)}\pun t''_{(1)}\otimes t''_{(2)} \\
& + t'_{(1)}\otimes t'_{(2)}\otimes t''+t'\cdot t''_{(1)}\otimes t''_{(2)}\otimes t''_{(3)}+ t'\otimes t''_{(2)}\otimes t''_{(2)}.\\
\end{array}$$
So, we conclude that $$(\Delta \otimes {\rm Id})\Delta (t)=({\rm Id} \otimes \Delta)\Delta (t),$$ which ends the proof.
Formula for the coproduct $\Delta$
----------------------------------
We want to give an explicit formula for the coproduct $\Delta$, for which we previously describe an order on the vertices of a tree.
Given a tree $t$, we consider the set ${\rm Vert}(t)$ ordered by the [*level order*]{}, that is, the vertices of $t$ are ordered by reading the vertices of $ t $ from left to right and from top to bottom. For instance, if $t$ is the tree
{width="6cm"}
then ${\rm Vert}(t)$ is ordered by $a<b<c<d<e<{\rm root}(t)$.
Given a tree $t$ and a vertex $v \in {\rm Vert}^{\ast}(t)$, we denote by $e_{v}$ the edge of $t$ with final vertex $v$.
\[ContraccionDunArbol\] Let $A=\{v_1,\ldots,v_l\}$ be a ordered subset of ${\rm Vert}^{\ast}(t)$ with $v_1< \cdots <v_l$ respect to the level order.
We define $t_A$ as the tree of the degree $l$ such that ${\rm Vert}^{\ast}(t_A)=A$, which is obtained from $t$ by deletions of the vertices that are not in $A$ and by successive contractions of the edges of which these are the final vertex.
When $t$ is the tree
{width="8cm"}
\[FormulaCoproducto\] Let $t$ be a tree of degree $n$, with ${\rm Vert}^{\ast}(t)$ given by the ordered vertices $a_1<\cdots <a_n$. The coproduct $\Delta(t)$ is given by the formula $$\Delta(t)=\sum_{i=1}^{n-1}t_{\{a_1, \ldots,a_i\}}\otimes t_{\{a_{i+1}, \ldots,a_{n}\}}.$$
Let $t$ be a tree of degree $n$ and $a_1<\cdots <a_n$ its vertices, different of the root, ordered by the level order.
We prove the assertion by induction on $n$. For $n=1$, $t=a \in V$. So, $\Delta(a)=0$ and the assertion is true.
For $n>1$, we consider two cases. First, if $t$ is an irreducible, then $t=t'\circ a_n$, where $t'$ is a tree of degree $(n-1)$, with vertices $a_1,\ldots,a_{n-1}$. In this case, note that $t'=t_{\{a_1,\ldots,a_{n-1}\}}$ and the vertices $a_1,\ldots,a_{n-1}$ preserve the order that they originally had in $t$. Moreover, if $1\leq k \leq l \leq n-1$, then $t'_{\{a_k,a_{k+1},\ldots,a_l\}}=t_{\{a_k,a_{k+1},\ldots,a_l\}}$.
By definition of coproduct $\Delta$ and by a recursive argument, we have that $$\begin{array}{rcl}
\Delta(t) & = & \Delta(t'\circ a_n) \\
& = & \Delta(t')\circ a_n+t'\circ \Delta(a_n)+t'\otimes a_n\\
& = & \displaystyle \sum_{i=1}^{n-2}t'_{\{a_1,\ldots,a_i\}}\otimes t'_{\{a_{i+1},\ldots,a_{n-1}\}}\circ a_n+t'\otimes a_n \\
& = & \displaystyle \sum_{i=1}^{n-2}t_{\{a_1,\ldots,a_i\}}\otimes t_{\{a_{i+1},\ldots,a_{n-1}\}}\circ a_n+t_{\{a_1,\ldots,a_{n-1}\}}\otimes a_n\\
& = & \displaystyle \sum_{i=1}^{n-1}t_{\{a_1, \ldots,a_i\}}\otimes t_{\{a_{i+1}, \ldots,a_{n}\}},\\
\end{array}$$ because $t_{\{a_{i+1},\ldots,a_{n-1}\}}\circ a_n= t_{\{a_{i+1},\ldots,a_{n}\}}$, for $i=1,\ldots,n-2$.
If $t$ is a reducible tree, then we may write $t=t'\cdot t''$, where $t'$ and $t''$ are trees of degree smaller than $n$. If $t'$ is of degree $l$, then ${\rm Ver}(t')=\{a_1,\ldots,a_l\}$ and ${\rm Ver}(t'')=\{a_{l+1},\ldots,a_{l+m}\}$, where $n=l+m$. Note that the vertices of $t'$ and $t''$ preserve the order that they had in $t$. Moreover, $t'=t_{\{a_1,\ldots,a_l\}}$ and $t''=t_{\{a_{l+1},\ldots,a_{l+m}\}}$.
By definition of the coproduct $\Delta$ and by a recursive argument, we obtain that: $$\begin{array}{rcl}
\Delta(t) & = & \Delta(t'\cdot t'') \\
& = & \Delta(t')\cdot t''+t'\cdot \Delta(t'')+t'\otimes t''\\
& = & \displaystyle \sum_{i=1}^{l-1}t'_{\{a_1,\ldots,a_i\}}\otimes t'_{\{a_{i+1},\ldots,a_{l}\}}\cdot t''\\
& &+\displaystyle \sum_{j=1}^{m-1}t'\cdot t''_{\{a_{l+1},\ldots,a_{l+j}\}}\otimes t''_{\{a_{l+j+1},\ldots,a_{l+m}\}}+t'\otimes t'' \\
& = & \displaystyle \sum_{i=1}^{l-1}t_{\{a_1,\ldots,a_i\}}\otimes t'_{\{a_{i+1},\ldots,a_{l}\}}\cdot t''\\
& &+\displaystyle \sum_{j=1}^{m-1}t'\cdot t''_{\{a_{l+1},\ldots,a_{l+j}\}}\otimes t_{\{a_{l+j+1},\ldots,a_{l+m}\}}+t'\otimes t''.
\end{array}$$ As $t'_{\{a_{i+1},\ldots,a_{l}\}}\cdot t''=t_{\{a_{i+1},\ldots,a_n\}}$, for $i=1,\ldots,l-1$, and\
$t'\cdot t''_{\{a_{l+1},\ldots,a_{l+j}\}}= t_{\{a_1,\ldots,a_{l+j}\}}$, for $j=1,\ldots,m-1$, we get $$\Delta(t)=\sum_{i=1}^{n-1}t_{\{a_1, \ldots,a_i\}}\otimes t_{\{a_{i+1}, \ldots,a_{n}\}},$$ which ends the proof.
When $t$ is the tree
{width="8.5cm"}
the coproduct $\Delta(t)$ is given by
{width="11cm"}
\[$As^2(v)$EslibreComoAlgAss\] Consider $H_V={\rm As}^2(V)$ with the coproduct $\Delta$ as in Proposition \[$As^2(V)$ ComoCompInfBialgebra\]. The triples $(H_V,\cdot, \Delta)$ and $(H_V,\circ, \Delta)$ are graded infinitesimal bialgebras, with the natural graduation of $H_V$. In particular, the vector space $V$ is the component of degree one. By definition, $\Delta(v)=0$, for all $v \in V$. So, $(H, \cdot, \Delta)$ and $(H, \circ , \Delta)$ are conilpotent infinitesimal bialgebras.
Applying the result obtained by J.-L. Loday and M. Ronco in [@LodayRonco-On; @the; @structure; @of; @cofree; @Hopf; @algebras], Theorem 2.6 , we obtain that $H$ is isomorphism to $\overline{T}({\rm Prim } H)$.
In particular, the associative algebras $({\rm As}^2(V), \cdot)$ and $({\rm As}^2(V), \circ)$ are free as associative algebras. This result is an alternative proof to that obtained by Dotsenko in [@Dotsenko-Compatible; @associative; @product], using operad theory .
Structure theorem for compatible infinitesimal bialgebras
=========================================================
Our aim is to prove that any conilpotent compatible infinitesimal bialgebra can be reconstructed from the subspace of its primitive elements.
We introduce the notion of $\mathcal{N}$-algebra, which describes the algebraic structure of the subspace of primitive elements of any compatible infinitesimal bialgebra.
\[def:$N$-algebra.\] A [*$\mathcal{N}$-algebra*]{} is a vector space $V$ equipped with $n$-ary operations $N_{n}:V^{\otimes n}\longrightarrow V$, for $n\geq 2$, which satisfy the following conditions: $$\begin{array}{l}
(1)\thinspace N_n(x_1,\ldots,N_2(x_n,x_{n+1}))=\displaystyle \sum_{i=1}^{n-1}N_{i+1}(x_1,\ldots,N_{n-i+1}(x_{i},\ldots,x_n),x_{n+1}), \\
(2)\thinspace \text{ for } n \geq 3\\
\quad N_2(x_1,N_n(x_2,\ldots,x_{n+1}))=N_n(N_2(x_1,x_2),x_3,\ldots,x_{n+1})\\
\hspace{1.4cm} -\displaystyle{ \sum_{i=3}^{n} N_i(x_1,N_{n+2-i}(x_2,\ldots ,x_{n+3-i}),x_{n+3-i+1}, \ldots ,x_{n+1})},\\
(3)\thinspace \text{ for } r, n \geq 3\\
\quad N_n(x,y_1\ldots,y_{n-2},N_r(z,t_1,\ldots,t_{r-2},w)) =\\
\hspace{1.4cm} N_r(N_n(x,y_1,\ldots,y_{n-2},z),t_{1},\ldots,t_{r-2},w)\\
\hspace{1.4cm} + \displaystyle{ \sum_{i=1}^{n-2} N_{r+i}(x,y_1,\ldots,y_{i-1},N_{n-i}(y_{i},\ldots ,y_{n-2},z),t_{1},\ldots,t_{r-2},w)}\\
\hspace{1.4cm} - \displaystyle{ \sum_{i=1}^{r-2} N_{n+r-i-1}(x,y_1,\ldots,y_{n-2},N_{i+1}(z,t_1,\ldots ,t_{i}),t_{i+1},\ldots,t_{r-2},w)},\\
\end{array}$$
For instance, the relations in low degrees give:
1. $N_2$ is an associative product.
2. $N_3(x,y,N_2(z,t))=N_2(N_3(x,y,z),t)+N_3(x,N_2(y,z),t)$.
3. $N_2(x,N_3(y,z,t))=N_3(N_2(x,y),z,t)-N_3(x,N_2(y,z),t)$.
4. $N_3(x,y,N_3(z,t,w))=N_3(N_3(x,y,z),t,w)+N_4(x,N_2(y,z),t,w)-N_4(x,y,N_2(z,t),w)$.
Let $\mathcal{N}$ be the algebraic operad of the $\mathcal{N}$-algebras. It is clear that the operad $\mathcal{N}$ is regular. So, the $S_n$-module $\mathcal{N}(n)$ is of the form $\mathcal{N}(n)=\mathcal{N}_n \otimes \KK[S_n]$ for some vector space $\mathcal{N}_n$, where $\KK[S_n]$ is the regular representation of $S_n$.
\[Dimension $N$\] The dimension of the vector space $\mathcal{N}_n$ is equal to the Catalan number $c_{n-1}$.
Denote by $|\mathcal{N}_n|$ the dimension of $\mathcal{N}_n$, as a $\KK$-vector space. We know that $|\mathcal{N}_n|$ is the dimension of the subspace of homogeneous elements of degree $n$ of the free ${\mathcal N}$ algebra on one generator $x$. From the Definition \[def:$N$-algebra.\], it is clear that the vector space $\mathcal{N}_n$ has a basis formed by all the elements of type: $$N_r(M_1(\ldots),\ldots,M_{r-1}(\ldots),x),$$ where each $M_i$ is an element in the basis of ${\mathcal N}_{m_i}$ and $m_1+\cdots+m_{r-1}=n-1$. So, we get that $$|\mathcal{N}_n|=\sum |\mathcal{N}_{m_1}|\cdot \ldots \cdot |\mathcal{N}_{m_{r-1}}|$$ where the sum is taken over all the families $\{m_i\}_{1\leq i \leq r-1}$ such that $m_1+\cdots+m_{r-1}=n-1$.
In particular, $|\mathcal{N}_1|=1$ and, for $n>1$, we have that: $$\begin{array}{rll}
|\mathcal{N}_n|&=\sum c_{m_1}\cdot \ldots \cdot c_{m_{r-1}}& \\
&=c_{n-1},\\
\end{array}$$ by a recursive hypothesis, which implies that $|\mathcal{N}_1|=c_0$ and that the integers $|\mathcal{N}_n|$ are defined by same equation than the Catalan numbers. We may conclude that $|\mathcal{N}_n|=c_{n-1}$, for $n\geq 1$.
\[definition As N-algebra\] Given a compatible associative algebra $(A ,\pun ,\cir)$ , the $n$-ary operations $N_{n}:A^{\otimes n}\longrightarrow A$ on $A$ are defined as follows: $$N_n(x_1,\ldots , x_n)=(x_1\pun \ldots \pun x_{n-1})\cir x_n -x_1\pun ((x_2\pun \ldots \pun x_{n-1})\cir x_n)$$
\[Observacion$N$en$As^2$\] The operations $N_n$ satisfy the following relations:
1. $N_2(x,y)=x \cir y - x \pun y$.
2. $N_n(x_1,\ldots , x_n)=N_3(x_1,x_2\pun \ldots \pun x_{n-1},x_n)$, for any $n \geq 4$.
\[Prop$N$en$As^2$\] Let $(A ,\pun ,\cir)$ be a compatible associative algebra. For any family of elements $x_1, x_2, x_3 , x_4 , x_5 \in A$ we have that: $$\begin{array}{l}
(1)\enspace N_2(N_2(x_1, x_2),x_3)=N_2(x_1,N_2(x_2,x_3)), \\
(2)\enspace N_3(x_1,x_2,N_2(x_3,x_4))=N_2(N_3(x_1,x_2,x_3),x_4)+N_3(x_1,N_2(x_2,x_3),x_4),\\
(3)\enspace N_3(x_1,x_2,N_3(x_3,x_4,x_5))=N_3(N_3(x_1,x_2,x_3),x_4,x_5)\\
\hspace{3cm} +N_4(x_1,N_2(x_2,x_3),x_4,x_5)-N_4(x_1,x_2,N_2(x_3,x_4),x_5)
\end{array}$$
The first relation states that $N_2$ is an associative product, which is true because $N_2$ is linear combination of the products $\pun$ and $\cir$.
Let us prove the second statement. The proof of the other ones is obtained in an analogous way.
Let us denote the element $N_2(x,y)$ as $x\ast y=x\circ y-x\cdot y$. We have that, $$\begin{array}{rcl} N_3(x_1,x_2,N_2(x_3,x_4) &=&(x_1\cdot x_2)\circ (x_3\ast x_4)-x_1\cdot(x_2\circ(x_3\ast x_4))\\
&=&(x_1\cdot x_2)\circ x_3\circ x_4-(x_1\cdot x_2)\circ (x_3 \cdot x_4)\\
& &-x_1\cdot (x_2\circ x_3\circ x_4)+x_1\cdot (x_2 \circ (x_3 \cdot x_4)).
\end{array}$$
Applying the compatibility condition between $\cdot$ and $\circ$, we get:
1. $(x_1 \cdot x_3)\circ (x_3 \cdot x_4)=((x_1 \cdot x_2)\circ x_3)\cdot x_4 - x_1 \cdot x_2 \cdot (x_3 \circ x_4) +(x_1 \cdot x_2 \cdot x_3)\circ x_4,$
2. $x_1\cdot (x_2\circ (x_3 \cdot x_4))=x_1 \cdot (x_2 \circ x_3 )\cdot x_4 -x_1\cdot x_2 \cdot (x_3 \circ x_4)+x_1 \cdot ((x_2 \cdot x_3 )\circ x_4).$
So, we obtain that $$\begin{array}{rcl} N_3(x_1,x_2,N_2(x_3,x_4) &=&(x_1\cdot x_2)\circ x_3\circ x_4-((x_1 \cdot x_2)\circ x_3)\cdot x_4\\
& &-(x_1 \cdot x_2 \cdot x_3)\circ x_4-x_1\cdot (x_2\circ x_3\circ x_4)\\
& &+x_1 \cdot (x_2 \circ x_3 )\cdot x_4 +x_1 \cdot ((x_2 \cdot x_3 )\circ x_4).
\end{array}$$
Regrouping the terms, we get: $$\begin{array}{rcl} N_3(x_1,x_2,N_2(x_3,x_4) &=&N_3(x_1,x_2,x_3)\circ x_4-N(x_1,x_2,x_2)\cdot x_4\\
& &+(x_1\cdot (x_2 \ast x_3))\circ x_4-x_1 \cdot ((x_2 \ast x_3)\circ x_4\\
&=&N_2(N_3(x_1,x_2,x_3),x_4)+N_3(x_1,N_2(x_2,x_3),x_4),
\end{array}$$ which proves the equality.
\[lema$N$en$As^2$\] Let $(A, \cdot, \circ)$ be an compatible associative algebra and let $\{N_n\}_{n\geq 2}$ be the family of products introduced in Definition \[definition As N-algebra\]. For elements $x, y , z \in A$, we have that:
1. $N_2(x\pun y,z)=N_3(x,y,z)+x \pun N_2(y,z)$.
2. $N_2(x,y \pun z)=N_3(x,y,z)+N_2(x,y) \pun z$.
The formulas are obtained by a straightforward computation, using the definition of the operations $N_n$s.
The following result is immediate to prove.
\[InduccionLemmaNen$As^2$\] Let $A$ be a compatible associative algebra $A$. For any family of elements $x_1, \ldots, x_n \in A$, we have that:
1. $N_2(x_1\pun \ldots \pun x_{n-1},x_n)=N_n(x_1,\ldots,x_n)
+x_1\pun N_{n-1}(x_2,\ldots,x_n)+ \ldots + x_1\pun \ldots \pun x_{n-2}\pun N_2(x_{n-1},x_n)$,
2. $N_2(x_1,x_2\pun \ldots \pun x_n)=N_n(x_1,\ldots,x_n)
+N_{n-1}(x_1,\ldots,x_{n-1})\pun x_n+ \ldots + N_2(x_1,x_2)\pun x_3\pun \ldots \pun x_n $,
Let $(A ,\pun ,\cir)$ be a compatible associative algebra with $n$-ary operations $N_{n}$ introduced in Definition \[definition As N-algebra\]. The data $(A,\{ N_n\})$ is a $\mathcal{N}$-algebra.
We apply Remark \[Observacion$N$en$As^2$\] together with Proposition \[Prop$N$en$As^2$\] and Proposition \[InduccionLemmaNen$As^2$\].
Let us prove the relation $(1)$ of Definition \[def:$N$-algebra.\]. The proofs of the remaining relations follow by similar arguments.
Let $A$ be a compatible associative algebra and consider $x_1, \ldots,x_n,x_{n+1} \in A$, with $n\geq 3$. The equality was proved in Proposition \[Prop$N$en$As^2$\] for $n=3$. Let $n>3$, by Remark \[Observacion$N$en$As^2$\], we have that : $$N_n(x_1,x_2,\ldots,x_{n-1},N_2(x_n,x_{n+1}))=N_3(x_1,x_2\cdot\ldots \cdot x_{n-1},N_2(x_n,x_{n+1})).$$ So, from Proposition \[Prop$N$en$As^2$\], we obtain that
$$\begin{array}{ll} N_3(x_1,x_2\cdot \ldots \cdot x_{n-1},N_2(x_n,x_{n+1})) &=N_2(N_3(x_1,x_2\cdot \ldots \cdot x_{n-1},x_n),x_{n+1})\\
& +N_3(x_1,N_2(x_2\cdot \ldots \cdot x_{n-1},x_n),x_{n+1}).
\end{array}$$
Applying Proposition \[Prop$N$en$As^2$\] and Remark \[Observacion$N$en$As^2$\] to the second term, we get: $$\begin{array}{l} N_3(x_1,N_2(x_2\cdot \ldots \cdot x_{n-1},x_n),x_{n+1})\\
\hspace{3cm}=\displaystyle\sum_{i=2}^{n-1}N_3(x_1,x_2\cdot\ldots \cdot x_{i-1}\cdot N_{n-i+1}(x_i, \ldots, x_n),x_{n+1})\\
\hspace{3cm}=\displaystyle\sum_{i=2}^{n-1}N_{i+1}(x_1,x_2,\ldots, x_{i-1}, N_{n-i+1}(x_i, \ldots, x_n),x_{n+1}).
\end{array}$$
To end the proof it suffices to apply Remark \[Observacion$N$en$As^2$\] to first term, which implies that $$N_n(x_1,\ldots,N_2(x_n,x_{n+1}))=\displaystyle{ \sum_{i=1}^{n-1}N_{i+1}(x_1,\ldots,N_{n-i+1}(x_{i},\ldots,x_n),x_{n+1})}.$$
\[ElementosPrimComInfBialg\] Let $H$ be a compatible infinitesimal bialgebra with coproduct $\Delta$. If the elements $x_1, \ldots , x_n$ in $H$ are primitive, then $N_n(x_1,\ldots , x_n)$ is primitive, too. Therefore, we have that ${\mbox {\it Prim}(H)}$ is a $\mathcal{N}$-subalgebra of $(H, \{N_n\})$.
The cases $n=2$ and $n=3$ are obvious.
Suppose that $n\geq 4$ and that the elements $x_1, \ldots , x_n$ are primitive elements in $H$. Recall that $$\begin{array}{l}(\ast)\enspace N_n(x_1,\ldots , x_n)= N_3(x_1,x_2\pun \ldots \pun x_{n-1},x_n)\\
\hspace{3cm}= (x_1\pun x_2\pun \ldots \pun x_{n-1})\cir x_n -x_1\pun ((x_2 \pun \ldots \pun x_{n-1})\cir x_n).\\
\end{array}$$ Applying a recursive argument on $n\geq 2$, it is immediate to verify that $$\Delta(x_1 \pun \ldots \pun x_n)= \displaystyle{ \sum_{i=1}^{n-1}(x_1\pun \ldots \pun x_i)\otimes (x_{i+1}\pun \ldots \pun x_n}).$$
Applying the formula above to $(\ast)$ we obtain that : $$\begin{array}{rcl} \Delta((x_1\pun \ldots \pun x_{n-1})\cir x_n) &=& \displaystyle{ \sum_{i=1}^{n-2}(x_1\pun \ldots \pun x_i)\otimes ((x_{i+1}\pun \ldots \pun x_{n-1}})\cir x_n) \\
&+& (x_1 \pun \ldots \pun x_{n-1})\otimes x_n,
\end{array}$$ and $$\begin{array}{rcl} \Delta((x_2\pun \ldots \pun x_{r-1}) \cir x_r) &=&\displaystyle{ \sum_{i=1}^{n-2}(x_2\pun \ldots \pun x_i)\otimes ((x_{i+1}\pun \ldots \pun x_{n-1}})\cir x_n) \\
&+& (x_2 \pun \ldots \pun x_{n-1})\otimes x_n,
\end{array}$$ for $n\geq 3$.
Therefore, we may conclude that $$\begin{array}{l} \Delta(x_1\pun ((x_2 \pun\ldots \pun x_{r-1}) \cir x_r)) =\displaystyle{ \sum_{i=2}^{n-2}(x_1\pun \ldots \pun x_i)\otimes ((x_{i+1}\pun \ldots \pun x_{n-1}})\cir x_n) \\
\hspace{3cm}+ (x_1 \pun \ldots \pun x_{n-1})\otimes x_n + x_1\otimes ((x_2\pun \ldots \pun x_{n-1})\cir x_n),
\end{array}$$ and thus $\Delta(N_n(x_1,\ldots, x_n))=0$, which ends the proof.
\[descomposicionEn$As^2(V)$\] Let $V$ be a vector space. Denote by $H_V$ the free compatible associative algebra ${\rm As}^2(V)$ with the compatible infinitesimal bialgebra structure given in Proposition \[$As^2(V)$ ComoCompInfBialgebra\]. By remark \[$As^2(v)$EslibreComoAlgAss\], $(H_V,\Delta, \cdot)$ is isomorphic as bialgebra to $\overline{T}({\rm Prim}(H_V)$. By identifying the product $\cdot$ with the concatenation product in $\overline{T}({\rm Prim}(H_V)$, we have tha any element $x \in H_V$ is written in a unique way as a linear combination of elements of the type $x_1\cdot \ldots \cdot x_r$, where $x_i \in {\rm Prim}(H_V)$, for each $1 \leq i \leq r$.
\[PrimisgeneratedBy$V$\] Let $V$ be a vector space and $H_V$ the compatible infinitesimal algebra as in Remark \[descomposicionEn$As^2(V)$\]. As $\mathcal{N}$-algebra, ${\rm Prim}(H_{V})$ is generated by $V$
Let $N_V$ be the sub-$\mathcal{N}$-algebra of $H_V$ generated by $V$. Since any element $x \in V$ is a primitive element, $N_V \subseteq {\rm Prim}(H_{V})$.
Let us see that any element $x \in H_V$ can written as a linear combination of elements of the type $x_1\cdot \ldots \cdot x_r$, where $x_i \in N_V$, for each $1\leq i \leq r$. It is sufficient to verify the assertion for any tree $t \in H_V$. We prove the statement by induction on degree of $t$.
If $|t|=1$, then $t=a$, for some element $a$ in the basis $X$ of $H_V$. Suppose $|t|=n$, with $n>1$.
If $t$ is a reducible tree, then $t=t_1\cdot \ldots \cdot t_r$, where $r>1$ and $t_i$ is a tree of degree smaller than $n$, for each $1\leq i \leq r$. By a recursive argument, the assertion is true for each $t_i$, which implies the result for $t$.
Now, if $t$ is an irreducible tree, then $t=t'\circ a$, where $t'$ is a tree of degree $(n-1)$ and $a$ is an element in the basis $X$. Applying a recursive argument, $t'$ is a linear combination of elements of the type $x_1\cdot \ldots \cdot x_r$, where $x_1, \ldots, x_r \in N_V$, with $1\leq r \leq n-1$. So, $t=t'\circ a$ is a linear combination of elements of the type $(x_1\cdot \ldots \cdot x_r)\circ a$, where $x_1, \ldots, x_r \in N_V$, with $1\leq r \leq n-1$ and $a$ an element of the basis $X$.
Now, consider the operation $$N_{r+1}(x_1, \ldots ,x_r,a)=(x_1\cdot \ldots \cdot x_r)\circ a-x_1\cdot ((x_2 \cdot \ldots \cdot x_r)\circ a),$$ applied on the elements $x_1, \ldots ,x_r,a$. We have that $$(x_1\cdot \ldots \cdot x_r)\circ a=N_{r+1}(x_1, \ldots ,x_r,a)+x_1\cdot ((x_2 \cdot \ldots \cdot x_r)\circ a).$$
Applying a recursive argument to $x_1$ and $(x_2 \cdot \ldots \cdot x_r)\circ a$, in the right side of the previous equality, we get the result.
By Remark \[descomposicionEn$As^2(V)$\], this implies that ${\rm Prim}(H_V)$ is generated, as $\mathcal{N}$-algebra by $N_V$, which implies that it is generated by $V$. This ends the proof.
\[$N$algebraLibre\] Let $V$ be vector space and let $H_{V}$ be the free associative compatible algebra ${\rm As}^2(V)$, spanned by $V$. The $\mathcal{N}$-algebra ${\rm Prim}(H_{V})$, of primitive elements of $H_{V}$, is the free $\mathcal{N}$-algebra on $V$.
Note that, by Lemma \[PrimisgeneratedBy$V$\], as $\mathcal{N}$-algebra, ${\rm Prim}(H_{V})$ is graded and generated by $V$. Denote by ${\rm Prim}(H_{V})_{n}$ the subspace of homogeneous elements of degree $n$ of ${\rm Prim }(H_V)$.
By Proposition \[Dimension $N$\], to see that ${\rm Prim}(H_{V})$ is the free $N$-algebra on $V$ it suffices to show that the dimension of ${\rm Prim}(H_{V})_{n}$ is equal to $(\mbox{dim}V)^{n}c_{n-1}$.
Let us compute the dimension of ${\rm Prim}(H_V)_{n}$. Recall from [@LodayRonco-On; @the; @structure; @of; @cofree; @Hopf; @algebras] the linear operator $e$. Since $(H_V, \cdot, \Delta)$ is a conilpotent infinitesimal bialgebra, we can define $e: H \rightarrow H$ given by $$e(x)=x-x_{(1)}\cdot x_{(2)}-x_{(2)}\cdot x_{(2)}\cdot x_{(3)}+\cdots ,$$ where $\Delta(x)=x_{(1)}\otimes \cdots \otimes x_{(n)}$, for all $x \in H_V$. Consider the set $$B_n=\{e(t) | t \text{ is an irreducible tree of degree } n \text{ in } H_{V}\}.$$
Let us prove that the set $B_n$ is a basis of ${\rm Prim }(H_V)_{n}$.
From [@LodayRonco-On; @the; @structure; @of; @cofree; @Hopf; @algebras], Proposition 2.5, we have that $B_n \subseteq e(H_{V})={\rm Prim}(H_{V})$ and for any reducible tree $t=t_1 \cdot \ldots \cdot t_r$, $$e(t)=e(t_1\cdot \ldots \cdot t_r)=0.$$
So, $e({\mbox {Irr}}) =e(H_{V})={\rm Prim}(H_{V})$, because all element $x \in H_V$ can be written as a linear combination of elements in $\bigcup _{n\geq 1}{\mbox {Irr}_n}$.
On the other hand, the same result asserts that, if $t$ is a irreducible tree of degree $n$, then $$e(t)=t-t_{(1)}\cdot e(t_{(2)}).$$
So, if $t_1$ and $t_2$ are different irreducible trees in $H_{V}$, then $e(t_1)\neq e(t_2)$. In particular, since the number of irreducible trees of degree $n$ is equal to $(\mbox{dim}V)^{n}c_{n-1}$, we have that $|B_n|=(\mbox{dim}V)^{n}c_{n-1}$.
Let us see that the set $B_n$ is linearly independent. Note that in particular $|B_n|=c_{n-1}$. To simplify the notation, denote $l=|B_n|$ and let $\{t^1, \ldots ,t^l\}$ be the set irreducible trees of degree $n$ in $H_{V}$. We have that $B_n=\{e(t^1), \ldots ,e(t^l)\}$.
Let $\{\alpha_1,\ldots ,\alpha_{l} \}$ be a family of elements in the field $\KK$ and suppose that $$\alpha_1e(t^1) +\ldots+\alpha_le(t^l)=0.$$ Since $e(t^{i})=t^{i}- t^{i}_{(1)}\cdot e(t^{i}_{(2)})$, for any $1\leq i \leq l$, we have that $$\alpha_1t^{1} +\ldots+\alpha_lt^{l}=\sum_{i=1}^{l}\alpha_i ( t^{i}_{(1)}\cdot e(t^{i}_{(2)})).$$
But this is possible only if $\alpha_i=0$, for all $1 \leq i \leq l $, because the right side is linear combination of reducible trees. So, $B_n$ is linearly independent and we may conclude that it is a basis of ${\rm Prim}(H_{V})_{n}$. The dimension of ${\rm Prim}(H_{V})_{n}$ is equal to $(\mbox{dim}V)^{n}c_{n-1}$, which ends the proof.
Proposition \[$N$algebraLibre\] and Proposition \[$As^2(v)$EslibreComoAlgAss\] imply the following structure theorem.
\[TeoremaDstructuraComInfBial\] Let $V$ be vector space and let $\mathcal{N}(V)$ be the free $\mathcal{N}$-algebra generated by $V$. The free associative compatible algebra ${\rm As}^2(V)$ is isomorphic to $T^{c}(\mathcal{N}(V))$.
Let $(A,\cdot,\circ,\Delta)$ be a compatible infinitesimal bialgebra. Consider $x\ast y=\alpha(x\cdot y)+\beta(x \circ y)$ a linear combination of the products $\cdot$ and $\circ$, where $\alpha$ and $\beta$ are elements in the field $\KK$.
A direct compute shows that $$\Delta(x\ast y)=x_{(1)}\otimes x_{(2)} \ast y + x \ast y_{(1)}\otimes y_{(2)} + (\alpha+\beta )x \otimes y.$$
In particular, when $x\ast y=x\cdot y-x\circ y$, $(A, \cdot, \ast, \Delta)$ is a compatible associative algebra with coalgebra structure satisfying:
1. $\Delta(x \cdot y)=x_{(1)}\otimes x_{(2)}\cdot y+x\cdot y_{(1)}\otimes y_{(2)}+x\otimes y,$
2. $\Delta(x \ast y)=x_{(1)}\otimes x_{(2)}\ast y+x\ast y_{(1)}\otimes y_{(2)}$.
So, $\Delta$ is infinitesimal unitary with respect to the product $\cdot$, in the Loday-Ronco’s sense, and infinitesimal with respect to the product $\ast$, in the Joni-Rota’s sense.
This notion of bialgebras is equivalent to the notion of bialgebras that we have given in Definition \[InfComp\].
Matching Dialgebras {#MatchingDialgebras}
===================
In this section, we consider a particular case of compatible associative algebras, the [*matching dialgebras*]{}. In [@Zhang-The; @category; @and; @operad; @of; @matching; @dialgebras], Y. Zhang, Ch. Bai and L. Guo studied the operad of matching dialgebras. They constructed the free matching dialgebras on a vector space $V$ by defining a matching dialgebra structure on the double tensor space $\overline{T}(\overline{T}(V))$.
In the same work, the authors proved that the operad of matching dialgebras is Koszul and compute the complex which gives the homology groups.
The aim of the present section is to study the notion of bialgebras in matching dialgebras.
Motivated by the [*path Hopf algebra*]{} $P(S)$ described by A.B. Goncharov in [@Goncharov-Galois; @symmetries; @of; @fundamental; @groupoids; @and; @noncommutative; @geometry], we introduced [*bi-matching dialgebras*]{}. We show that the Goncharov’s Hopf algebras is part of a family of bi-matching dialgebras, which can be constructed from a bialgebra $(H,\cdot, \Delta)$(in the usual sense) and a [*right semi-homomorphism*]{} $R:H\rightarrow H$, which is a coderivation with respect to the coproduct $\Delta$.
We also develop the notion of compatible infinitesimal bialgebra in a matching dialgebras. In particular, a free matching dialgebra is a compatible infinitesimal bialgebra, which we obtain another example of a Lodays good triple of operads (see [@Loday-GeneralizedBialgebras]).
A [*matching dialgebra*]{} is a vector space $A$ with two associative products $\cdot$ and $\circ $ such that $$(x \cdot y)\circ z=x\cdot(y \circ z) \text{, } (x\circ y) \cdot z = x\circ (y \cdot z)$$ for all $x , y, z \in A$.
We recall from [@Zhang-The; @category; @and; @operad; @of; @matching; @dialgebras] the notion of [*right semi-homomorphism*]{} of algebra.
This type of linear map gives an interesting family of examples of matching dialgebras.
\[semi-homomorfismo\] Let $(A,\cdot)$ be an associative algebra. A $\KK$-linear map $R:A\rightarrow A$ is a [*right semi-homomorphism*]{} if it satisfies the condition $$R(x \cdot y)=R(x)\cdot y, \text{ for all } x, y \in A.$$
\[observacionSemi-homo\] Note that if $(A, \cdot)$ is an associative algebra and $R:A\rightarrow A$ is a right semi-homomorphism, then $(A, \cdot, \circ)$ is a matching dialgebra with the product $\circ: A \otimes A \rightarrow A $ given by $x\circ y:=x \cdot R(y)$ (see [@Zhang-The; @category; @and; @operad; @of; @matching; @dialgebras]).
\[unidadEnUnSemi-Homm\] If $(A, \cdot)$ is an associative algebra and $a$ is an element in $A$, then the map $R:A \rightarrow A$ defined as $R(x):=a \cdot x$, for $x\in A$, is a right semi-homomorphism.
In particular, when $(A, \cdot)$ is a unital associative algebra with unit $e \in A$ and $R:A\rightarrow A$ is a right semi-homomorphism, then the linear map $R$ is completely determined by the action of $R$ on the unit $e$. Indeed, if $x \in A$, then $R(x)=R(e \cdot x)=R(e) \cdot x.$ So, for the case of unital associative algebra $A$, any right semi-homomorphism $R: A \rightarrow A$ is given by $R(x):=a \cdot x$, where $a$ is some element in $A$.
The free matching dialgebra
---------------------------
The free matching dialgebra over a vector space $V$ is a quotient of the free compatible associative algebra ${\rm As}^2(V)$. In particular, we may define an explicit compatible infinitesimal bialgebra structure on the free objects of the category of the matching dialgebras.
Given a vector space $V$, with basis $X$, let $T_{n}^{X}$ be the set of planar rooted trees with $(n+1)$ vertices, whose non-root vertices are colored by the elements of $X$.
In the Subsection \[freeCompAssAlg\], we define a compatible associative algebra structure on the vector space spanned $\displaystyle \bigcup_{n\geq 1}T_{n}^{X}$ of colored planar rooted trees, where $X$ is a basis of $V$, and proved that ${\rm As}^2(V)$ is the free compatible associative algebra over $V$.
The free matching dialgebra over $V$ may be obtained as the quotient ${\rm As}^2(V)$, by the ideal spanned by the elements $(x \cdot y)\circ z=x\cdot(y \circ z)$ and $(x\circ y) \cdot z = x\circ (y \cdot z)$ , for $x,y$ and $z$ in $V$.
We want to find a set of trees which gives a set of representatives of the classes of ${\rm As}^2(V)$ modulo these relations.
\[arbolesGradoBajo$As_2$\] In low degree, we identify the trees:
{width="4.5cm"}
because $a\cdot(b \circ c)=(a \cdot b) \circ c$. In degree four, we have that
{width="7cm"}
In the general case, we have the following result.
\[BaseDialgebra\] Any tree $t \in T_{n}^{X}$[, for $n\geq 1$,]{} is equivalent to a tree of the type $$t=t^1\cdot \ldots \cdot t^r,$$ where each $t^k$ is a tree of the form $t^k=a^k_1 \circ \ldots \circ a^k_{n_k}$, with $1\leq k \leq r$ and $n_1+\ldots +n_r=n$.
For $n=3$, the result was proved in \[arbolesGradoBajo$As_2$\]. For $n>3$, suppose that the assertion is true for any tree of degree strictly less than $n$. If $t$ is an irreducible tree, then $t=t'\circ a$, with $|t'|=n-1$ and $ a $ an element of degree one. Applying a recursive argument to $t'$, we get $$t=(t'_1\cdot \ldots \cdot t'_r)\circ a =t'_1\cdot \ldots \cdot (t'_r\circ a).$$
If $t$ is a reducible tree, then $t=t'\cdot t''$, where $t'$ and $t''$ are trees of degree strictly less than $n$. So, applying the inductive hypothesis to $t'$ and $t''$, we obtain the assertion for $t$, which ends the proof.
We denote by $D_{n}^X$ the set of all trees of degree $n$, described in Proposition \[BaseDialgebra\] and by $D^X$ the set $\bigcup_{n\geq 1}D{}^{X}_{n}$.
For instance, in degree three, we have that:
{width="9.3cm"}
\
The free matching dialgebras is the vector space ${\rm{As}}_2(V)=\bigoplus_{n\geq 1}^{} \KK[D{}^{X}_{n}]$, whose basis is the set $D^X$.
Given two elements $t=t^1 \cdot \ldots \cdot t^r$ and $w=w^1\cdot \ldots \cdot w^s$ in $D^X$ of degree $n$ and $m$, respectively, we have that $t \cdot w$ is the tree in $D_{n+m}^X$ that is obtained by identifying the roots of $t$ and $w$, while that $t\circ w$ is the tree $$t \circ w = t^1 \cdot \ldots \cdot t^{r-1}(t^r \circ w^1)\cdot w^2\ldots \cdot w^s,$$ where $t^r \circ w^1$ is the tree that is obtained by identify the root of $t^r$ with the only leaf of the tree $w^1$. For instance,
{width="8cm"}
In [@Zhang-The; @category; @and; @operad; @of; @matching; @dialgebras], Y. Zhang, C. Bai and L. Guo defined the free matching dialgebra over the vector space $V$ as the double tensor space $\overline{T}(\overline{T}(V))$.
For a vector space $W$, $\overline{T}_{\ast}(W)=\displaystyle \bigoplus_{n\geq 1}W^{\otimes n}$ denotes the non-unitary tensor algebra, where the tensor product is denoted by $\otimes_{\ast}$.
Under this notation, the double tensor space is $\overline{T}_{\ast_1}(\overline{T}_{\ast_2}(V))$. The products $\cdot$ and $\cdot$ are defined as follows:\
For $u=u_1\otimes_{\ast_1}\cdots \otimes_{\ast_1}u_m$ and $v=v_1\otimes_{\ast_1}\cdots \otimes_{\ast_1}v_n$ in $\overline{T}_{\ast_1}(\overline{T}_{\ast_2}(V))$ with $u_i, v_j \in \overline{T}_{\ast_2}(V)$, for $1\leq i \leq m$, $1\leq j \leq n$, define:
1. $u \cdot v= u_1\otimes_{\ast_1}\cdots \otimes_{\ast_1}u_m \otimes_{\ast_1} v_1\otimes_{\ast_1}\cdots \otimes_{\ast_1}v_n$, the tensor product $\otimes_{\ast_1}$.
2. $u \circ v= u_1\otimes_{\ast_1}\cdots \otimes_{\ast_1}(u_m \otimes_{\ast_2} v_1)\otimes_{\ast_1}\cdots \otimes_{\ast_1}v_n.$
In [@Zhang-The; @category; @and; @operad; @of; @matching; @dialgebras], it is showed that $(\overline{T}_{\ast_1}(\overline{T}_{\ast_2}(V)), \cdot, \circ)$ is a matching dialgebra, which is free on the vector space $V$. The identification between both versions of the free matching dialgebras is clear. In our description, the tensors of first type are the trees of the type:
{width="3cm"}
\
where $a_i \in X$, for $1 \leq i \leq n$, are element in the basis $X$. In this context, we will call [*irreducible tree* ]{} to the trees of this type.
For $V=\KK$, we identify the free matching dialgebra ${\rm As}_2(\KK)$ with the partition algebra $$C=\displaystyle\bigoplus_{n\geq 1}C^{n},$$ where $C^n$ is vector space generated by all the ordered partition of a positive integer $n$. We denote by $c_{(n_1, \ldots,n_l)}$ the ordered partition $n=n_1+ \cdots + n_l$ of $n$. The products $\cdot$ and $\circ$ are given respectively by $$c_{(n_1, \ldots,n_l)}\cdot c_{(m_1, \ldots,m_k)}:=c_{(n_1, \ldots,n_l,m_1, \ldots, m_k)}$$ and $$c_{(n_1, \ldots,n_l)}\circ c_{(m_1, \ldots,m_k)}:=c_{(n_1, \ldots,n_l+m_1, \ldots, m_k)}.$$ Since the non-symmetric operad ${\rm As}_2$ is completely described by the free matching algebra on $V=\KK$, we have that $({\rm As}_2)_n=C^n$. In particular, the dimension of $({\rm As}_2)_n$ is $2^n$.
Bi-matching dialgebras
----------------------
We introduce the notion bi-matching dialgebras. For this, we requires the following proposition, which shows that the operad of matching dialgebras is a Hopf operad.
\[morfismo de dialgebra\] If $(A, \cdot , \circ )$ is a matching dialgebras, then $A\otimes A$ is a matching dialgebras with the products defined by:
1. $(a_1\otimes a_2)\cdot (b_1 \otimes b_2)=a_1\cdot b_1 \otimes a_2\cdot b_2$ and
2. $(a_1\otimes a_2)\ast (b_1 \otimes b_2)=a_1\cdot b_1 \otimes a_2\circ b_2+a_1\circ b_1 \otimes a_2\cdot b_2.$
The proof follows by direct computation.
\[observacionsobre morfismos de dialgebras\] Note that in Proposition \[morfismo de dialgebra\], the associativity of the product $\ast$ requires the compatibility condition between the products $\cdot $ and $\circ$.\
The following notion of bialgebra was originally introduced by A.B. Goncharov in [@Goncharov-Galois; @symmetries; @of; @fundamental; @groupoids; @and; @noncommutative; @geometry].
A bi-matching dialgebra is a matching dialgebra $(H,\cdot,\circ)$ equipped with a coassociative coproduct $\Delta: H \rightarrow H\otimes H$ such that $\Delta$ is morphism of matching dialgebras with respect to the matching dialgebra structure of $H\otimes H$ defined in Proposition \[morfismo de dialgebra\].
Let $(H, \cdot, \Delta)$ be a bialgebra and let $R:H \rightarrow H$ be a right semi-homomorphism. If $R$ is a coderivation for the product $\cdot$, then $$\Delta(x \circ y)=\Delta(x)\ast \Delta(y),$$ for any $x,y\in H$, where $x\circ y=x \cdot R(y)$ is the product defined in Remark \[observacionSemi-homo\].
By a straightforward calculation, we get: $$\begin{array}{rll} \Delta(x\circ y)&=\Delta(x \cdot R(y))& \\
&=\Delta(x)\cdot \Delta(R(y))\\
&=x_{(1)}\otimes x_{(2)}\cdot (R(y_{(1)})\otimes y_{(2)}+ y_{(1)}\otimes R(y_{(2)}))\\
&=x_{(1)}\cdot R(y_{(1)}) \otimes x_{(2)}\cdot y_{(2)} + x_{(1)} \cdot y_{(1)}\otimes x_{(2)}\cdot R(y_{(2)})\\
&=x_{(1)}\circ y_{(1)} \otimes x_{(2)}\cdot y_{(2)} + x_{(1)} \cdot y_{(1)}\otimes x_{(2)}\circ y_{(2)}\\
&=\Delta(x)\ast \Delta(y),
\end{array}$$ which proves the formula.
\[SemiHomoEnG-L\] Consider the Grossman-Larson’s Hopf algebra $H=\KK[\mathcal{T}]$ with basis the set of all non-planar rooted trees $\mathcal{T}$ described in [@Grossman-larson_Hopf-algebraic; @structure; @of; @families; @of; @trees]. Recall that the tree $e$ with one vertex is the unit for the product defined in $H$. Consider the linear map $R:H\rightarrow H$ such that, for any rooted tree $t$, $R(t)$ is the sum of trees obtained from $t$ by attaching one more outgoing edge and vertex to each vertex of $t$, which is originally defined on the Connes-Kreimer’s Hopf algebra in [@Connes-Kreimer; @HopfAlgebras].
In [@Panaite-Relating; @G-L; @y; @C-K], Proposition 2.2, F. Panaite showed the linear map $R$ is a right semi-homomorphism for $H$. In fact, $R(x)=R(e)\cdot x$, for all $x \in H$. In his work F. Panaite showed that $R$ is a coderivation for the coproduct $\Delta$ defined in $H$. Indeed, since $R(e)$ is a primitive element, we have that $$\begin{array}{rll} \Delta(R(x))&=\Delta(R(e) \cdot x)& \\
&=\Delta(R(e))\cdot \Delta(x)\\
&=(R(e)\otimes e +e \otimes R(e))\cdot x_{(1)}\otimes x_{(2)}\\
&=R(e)\cdot x_{(1)}\otimes x_{(2)}+x_{(1)}\otimes R(e)\cdot x_{(2)}\\
&=R( x_{(1)})\otimes x_{(2)}+x_{(1)}\otimes R( x_{(2)}),\\
\end{array}$$ and $R$ is a coderivation. So, $(H, \cdot, \circ, \Delta)$ is a bialgebra, where $\circ$ is the associative product induced by $R$ and the compatibility condition between the products $\cdot$ and $\circ$ with the coproduct $\Delta$ is as in Remark \[observacionsobre morfismos de dialgebras\].
\[R(e)Primitivo\] The previous result obtained by F. Panaite may be generalized to any bialgebra $(H, \cdot, \Delta)$ with unit $e \in H$, that is, if $R:H\rightarrow H$ is a right semi-homomorphism and $R(e)$ is a primitive element of $H$, then $R$ is a coderivation. The proof is similar to that given in the Example \[SemiHomoEnG-L\].
Let $H=\KK[X]$ be the $\KK$-algebra of polynomial in one variable, with the usual product and the coproduct given by: $$\Delta (X^n):=\sum _{i=0}^n\binom{n}{i}X^{n-i}\otimes X^{i},$$ with the homomorphism $R$ defined by $R(X^n)=X^{n+1}$. As $R(1)=X$ is a primitive element, we get that $R$ is a coderivation.
The Goncharov’s Hopf algebra
----------------------------
Let us describe the path algebra $P(S)$, introduced by A. B. Goncharov in [@Goncharov-Galois; @symmetries; @of; @fundamental; @groupoids; @and; @noncommutative; @geometry], which motivates our notion of bialgebra, described in Remark \[observacionsobre morfismos de dialgebras\].
Let $S$ be a finite set. Denote by $P(S)$ the $\KK$-vector space with basis $$p_{s_0, \ldots , s_n}, \quad {\rm for}\ n\geq 1,\ {\rm and}\ s_k \in S, \text{ for } k=0, \ldots ,n.$$
The associative product $\cdot: P(S) \otimes P(S) \rightarrow P(S)$ is defined as follows: $$p_{a,X,b} \cdot p_{c, Y, d}=\begin{cases}
p_{a,X,Y,d} & \text{, for}\ b=c,\\
0 & \text{, for}\ b\neq c,\\
\end{cases}$$ where the letters $a,b,c,d$ denote elements, and $X$ and $Y$ denote sequences, possibly empty, of elements of the set $S$. In particular, $p_{a,b}=p_{a,x}\cdot p_{x,b}$, for $x\in S$, and the unit for this product is the element $e=\sum_{i \in S}p_{i,i}$.
The coproduct $\Delta: P(S) \rightarrow P(S) \otimes P(S)$ is given by: $$\Delta(p_{a,x_1, \ldots ,x_n,b})=\sum_{k=0}^{n}\sum_{\sigma \in Sh(k,n-k)}p_{a,x_{\sigma(1)}, \ldots ,x_{\sigma (k)}, b}\otimes p_{a,x_{\sigma(k+1)}, \ldots ,x_{\sigma (n)}, b}.$$
For instance, $\Delta(p_{a,b})=p_{a,b}\otimes p_{a,b}$, for $a,b\in S$, and $$\Delta(e)=\sum_{i\in S}p_{i,i}\otimes p_{i,i}\neq e\otimes e.$$ The linear map $R:P(S)\rightarrow P(S)$, given by: $$R(e)=\sum_{i \in S}p_{i,i,i}$$ is a right semi-homomorphism.
With the definition above, we get that $R(p_{a,X,b})=p_{a,a,X,b}$, for any element $p_{a,X,b}$ of the basis. So, $R$ induces a new associative product $\circ :P(S)\otimes P(S) \rightarrow P(S)$ by setting $x\circ y=x\cdot R(y)$, that is: $$p_{X,b} \circ p_{c,Y}=\begin{cases}
p_{X,b,Y}, & {\rm for}\ b=c,\\
0, & {\rm for}\ b\neq c,\\
\end{cases}$$ where $b, c \in S$, and $X$ and $Y$ are sequences of elements of $S$.
The right semi-homomorphism $R:P(S) \rightarrow P(S)$ is a coderivation.
Note that for any element $i \in S$, we have that $\Delta(p_{i,i,i})=p_{i,i,i}\otimes p_{i,i} + p_{i,i}\otimes p_{i,i,i},$ therefore: $$\Delta(R(e))=\sum_{i\in S}p_{i,i,i}\otimes p_{i,i} + p_{i,i}\otimes p_{i,i,i}.$$
Let $x=p_{a,X,b}$ be an element of the basis of $P(S)$. By definition of the coproduct $\Delta$, we have that the element $\Delta(x)$ is a sum of tensors of type $$p_{a,X',b}\otimes p_{a,X'',b},$$ where $X'$ and $X''$ are (possibly empty) ordered subsequences of $X$.
Using the Sweedler’ notation, we write $$\Delta(x)=x_{(1)}\otimes x_{(2)}= p_{a,X_{(1)},b}\otimes p_{a,X_{(2)},b}.$$
Computing $\Delta(R(x))$, we obtain that: $$\begin{array}{rll} \Delta(R(x))&=\Delta(R(e) \cdot x)& \\
&=(\sum_{i\in S}p_{i,i,i}\otimes p_{i,i} + p_{i,i}\otimes p_{i,i,i})\cdot x_{(1)}\otimes x_{(2)}\\
&=(\sum_{i\in S}p_{i,i,i}\otimes p_{i,i} + p_{i,i}\otimes p_{i,i,i})\cdot p_{a,X_{(1)},b}\otimes p_{a,X_{(2)},b}\\
&=p_{a,a,a}\cdot p_{a,X_{(1)},b}\otimes p_{a,a}\cdot p_{a,X_{(2)},b} + p_{a,a}\cdot p_{a,X_{(1)},b}\otimes p_{a,a,a}\cdot p_{a,X_{(2)},b} \\
&= p_{a,a,X_{(1)},b}\otimes p_{a,X_{(2)},b} + p_{a,X_{(1)},b}\otimes p_{a,a,X_{(2)},b} \\
&=R(x_{(1)})\otimes x_{(2)} + x_{(1)}\otimes R(x_{(2)}), \\
\end{array}$$ which ends the proof.
Notion of compatible infinitesimal bialgebra in matching dialgebras
-------------------------------------------------------------------
We consider the notion of compatible infinitesimal bialgebra in matching dialgebras. A direct compute shows that this notion of bialgebra is well-defined in a matching dialgebra.
Let $(A, \circ, \Delta)$ be an infinitesimal bialgebra. The product $\circ$ and the coproduct $\Delta$ may be extended to $\overline{T}(A)=\bigoplus _{n\geq 1}A^{\otimes n}$ as follows:
1. $(a_1\ldots a_n) \circ (b_1\ldots b_m) = a_1\ldots a_{n-1}(a_n \circ b_1)b_2\ldots b_m$ and
2. $\Delta(a_1\ldots a_n)=\displaystyle \sum_{i=1}^{n-1}a_1\ldots a_{i-1}\Delta(a_i)a_{i+1}\ldots a_n+ \displaystyle\sum_{i=1}^{n-1}a_1\ldots a_{i}\otimes a_{i+1}\ldots a_n$
If we denoted by $\cdot$ the concatenation product in $\overline{T}(A)$, then $(\overline{T}(A), \cdot, \circ)$ is a matching dialgebra, and $\Delta$ is infinitesimal for both products.
In particular, consider the free matching dialgebra ${\rm As}_2(V)=\overline{T}(\overline{T}(V))$. In this case, $A=\overline{T}(V)$ is an infinitesimal bialgebra with the concatenation product and the deconcatenation coproduct. Identifying the tree $a_1\circ \ldots \circ a_n$ with a tensor in $\overline{T}(V)$ and the product $\circ$ with the concatenation product, we get: $$\Delta (a_1\circ \ldots \circ a_n):=\sum _{i=1}^{n-1}(a_1\circ \ldots \circ a_i)\otimes (a_{i+1}\circ \ldots \circ a_n).$$
Thus, extending $\Delta$ to $\overline{T}(\overline{T}(V))$, we have that ${\rm As}_2(V)$ is a compatible infinitesimal bialgebra.
The explicit formula for the coproduct $\Delta$ is given by: $$\Delta(t)=\sum_{i=1}^{n-1}t_{\{a_1,\ldots,a_i\}}\otimes t_{\{a_{i+1},\ldots,a_{n}\}},$$ described in Proposition \[FormulaCoproducto\], which extends the deconcatenation coproduct of $\overline{T}(V)$.
When $t$ is the tree
{width="5.5cm"}
\
the coproduct $\Delta(t)$ is given by:
{width="12cm"}
The primitive part of the compatible infinitesimal bialgebra ${\rm As}_2(V)$ is generated by the associative product $\ast$ given by $x \ast y=x\circ y - x\cdot y$ together with the $\mathcal{N}$-operations of superior degree. By the compatibility conditions defining a matching dialgebras, the $\mathcal{N}$-algebra structure of ${\
rm As}_2(V)$ is reduced only associative product $\ast$. Thus, we get a good triple of operads $( {\rm As}, {\rm As}_2, {\rm As})$, in the Loday’s sense (see [@Loday-GeneralizedBialgebras]).
[X]{} Carinena, J. F., Grabowski, J., & Marmo, G. (2000). Quantum bi-Hamiltonian systems. International Journal of Modern Physics A, 15(30), 4797-4810. Connes, A., & Kreimer, D. (1999). Hopf algebras, renormalization and noncommutative geometry. In Quantum field theory: perspective and prospective (pp. 59-109). Springer Netherlands. Dotsenko, V. (2009). Compatible associative products and trees. Algebra & Number Theory, 3(5), 567-586. Dotsenko, V. V., & Khoroshkin, A. S. (2007). Character formulas for the operad of two compatible brackets and for the bi-Hamiltonian operad. Functional Analysis and Its Applications, 41(1), 1-17. Ginzburg, V., & Kapranov, M. (1994). Koszul duality for operads. Duke mathematical journal, 76(1), 203-272. Goncharov, A. B. (2005). Galois symmetries of fundamental groupoids and noncommutative geometry. Duke Mathematical Journal, 128(2), 209-284. Grossman, R., & Larson, R. G. (1989). Hopf-algebraic structure of families of trees. Journal of Algebra, 126(1), 184-210. R. Holtkamp, [*On Hopf algebra structures over free operads*]{}, Advances in Mathematics 207 (2006), 544-565. Joni, S. A., & Rota, G. C. (1979). Coalgebras and bialgebras in combinatorics. Studies in Applied Mathematics, 61(2), 93-139. Loday, J. L., & Vallette, B. (2012). Algebraic operads (Vol. 346). Springer Science & Business Media. Loday, J. L. (2006). Generalized bialgebras and triples of operads. arXiv preprint math/0611885. Loday, J. L., & Ronco, M. (2006). On the structure of cofree Hopf algebras. Journal fur die reine und angewandte Mathematik (Crelles Journal) 592, 123-155. Milnor, J. W., & Moore, J. C. (1965). On the structure of Hopf algebras. Annals of Mathematics, 211-264. Odesskii, A., & Sokolov, V. (2006). Algebraic structures connected with pairs of compatible associative algebras. International Mathematics Research Notices, 2006, 43734. Panaite, F. (2000). Relating the Connes–Kreimer and Grossman–Larson Hopf algebras built on rooted trees. Letters in Mathematical Physics, 51(3), 211-219. R. Stanley, [*Catalan numbers*]{}, Cambridge University Press (2015). Strohmayer, H. (2008). Operads of compatible structures and weighted partitions. Journal of Pure and Applied Algebra, 212(11), 2522-2534. Vallette, B. (2007). Homology of generalized partition posets. Journal of Pure and Applied Algebra, 208(2), 699-725. Zhang, Y., Bai, C., & Guo, L. (2013). The category and operad of matching dialgebras. Applied Categorical Structures, 21(6), 851-865.
[^1]: The authors work was partially supported by the Project FONDECYT Regular 1130939 and by MathAmSud 17Math-05 LIETS.
|
---
abstract: 'Using *ab initio* density functional theory and quantum transport calculations based on nonequilibrium Green’s function formalism we study structural, electronic, and transport properties of hydrogen-terminated short graphene nanoribbons (graphene flakes) and their functionalization with vanadium atoms. Rectangular graphene flakes are stable, having geometric and electronic structures quite similar to that of extended graphene nanoribbons. We show that a spin-polarized current can be produced by pure, hydrogenated rectangular graphene flakes by exploiting the spatially-separated edge states of the flake using asymmetric, non-magnetic contacts. Functionalization of the graphene flake with magnetic adatoms such as vanadium also leads to spin-polarized currents even with symmetric contacts. We observe and discuss sharp discontinuities in the transmission spectra which arise from Fano resonances of localized states in the flake.'
author:
- 'H. [Ş]{}ahin'
- 'R. T. Senger'
title: 'First principles calculations of spin-dependent conductance of graphene flakes'
---
Introduction
============
In the last two decades various forms of carbon nanostructures, namely buckyballs, carbon nanotubes, and lately graphene have attracted a great deal of interest due to their novel fundamental properties and possible applications in electronics. With the developments in preparation and synthesis techniques carbon-based nanostructures have emerged as one of the most promising materials for non-silicon electronics.
The fabrication of graphene sheets[@novo1] and observation of their unusual properties such as a half-integer quantum-Hall effect have attracted much interest in electronic transport properties of this type of two dimensional graphitic materials. Observed gapless energy spectrum and high mobility electron transport[@novo2; @zhang; @berger] are the most remarkable features of graphene. It was shown by tight-binding calculations considering the $\pi$ bands that in the electronic energy dispersion of graphene, energy is linearly dependent on the wave vector around the Fermi level[@wallace] which makes it a unique material.
In recent experimental studies graphene nanoribbons (GNRs) with narrow widths (10-70 nm) have been realized.[@han; @li] Li *et al* report producing ultra narrow ribbons with widths down to a few nanometers.[@li] In addition to high carrier mobilities that are higher than those in commercial silicon wafers, existence of width-dependent energy band gaps makes the graphene nanoribbons a potentially useful structure for various applications. The width dependence of the band gap and transport properties in quasi-one dimensional narrow GNRs have been studied theoretically.[@son; @barone; @areshkin; @sasaki; @abanin; @lee; @miyamoto; @cresti; @kudin; @ezawa]
Graphene nanoribbons having $r$ zigzag rows ($r$-ZGNRs) are predicted as semiconductors having a narrowing band gap with the increasing width of the ribbon. Armchair-edged ribbons (AGNRs) are also semiconducting with direct band gaps.[@son] Density functional theory (DFT) calculations predict a high density of states around the valence and conduction band edges of ZGNRs, which derive from the states that are localized at the edges of the ribbon, and lead to non-zero magnetic moments on the carbon atoms. The existence of such peculiar edge-states in graphitic materials had been discussed in some earlier works.[@kobayashi; @fujita; @nakada1; @wakabayashi1; @nakada2; @wakabayashi2] It has been established that the ground state of a ZGNR is anti-ferromagnetic with ferromagnetically ordered spin polarization along the edges.[@lee] Due to the fact that the states near the Fermi level are derived from the edge states and their linear combinations, external fields have more significant effects on the edge states. The nature of the interactions between the magnetized edges of ZGNR was studied, concluding that the main contribution to the local moments comes from dangling bonds, and their interaction is determined by tails of the edge-localized $\pi$ states.[@lee] Due to the edge effects, graphene ribbons show different one dimensional transport properties from those of carbon nanotubes.
Since GNRs have long spin-correlation lengths and good ballistic transport characteristics they can be considered as a promising active material of spintronic devices.[@gambi; @dora; @vindi; @delin] In particular ZGNRs, known to have large spin polarizations at the opposite edges of the ribbon, may be utilized to create spin-dependent effects such as spin polarized currents without the need of ferromagnetic electrodes or other magnetic entities.
Hydrogen termination of the edge atoms by forming strong $\sigma$ bonds would be important for the structural and electronic stability of the graphene ribbon. Both first-principles and tight binding calculations showed that the termination of edges with hydrogen atoms removes the electronic states related to the dangling bonds.[@kawai] However, there are no qualitative changes in the electronic structure and the magnetic order of the ZGNRs with hydrogen atom termination,[@kudin] except for a narrowing of the band gap as we calculated.
Modification of electronic structure by impurities, adatoms and external fields is another potential of graphene and graphitic structures for applications in nanoelectronics. Effects on electronic properties and magnetic behavior of graphene by the adsorption of foreign atoms has been considered in some previous works.[@lehti; @nord; @heg; @lee2; @hod1; @gunlycke; @kan; @sevincli] Sizeable gap opening by hydrogen adsorption to Stone-Wales defect sites of graphene has been reported.[@dup] In the case of substitutional boron atom[@endo; @herna] many of the electronic properties have been studied and it is suggested that GNRs may be used as spin filter devices.[@martins] Based on the spin polarized ground state of ZGNRs their possible application as a spin-valve device was proposed by using tight binding,[@rojas; @rycerz] and $\mathbf{k}\cdot\mathbf{p}$ calculations.[@brey] It is also predicted by using first-principles calculations that the electric current flowing on the ribbon can be made completely spin polarized under in-plane homogeneous electric fields.[@cohen; @gunlycke2]
In this study we consider electronic transport properties of short graphene nanoribbons (graphene flakes), and their functionalization with vanadium atoms. Electronic and magnetic properties of various types of nanometer-sized graphene flakes have been reported before.[@wlwang; @shemella; @ezawa2; @fernandez; @jiang; @hod2; @hod3; @rudberg; @booth] We restrict our considerations, however, to rectangular flakes only, which are finite segments of perfect GNRs. In particular we calculate spin-dependent transmission spectra of the flakes when the electrodes make partial contacts along the zigzag rows of the flakes. The geometrical asymmetry of the contacts lead to a polarization in the spin states of transmitted electrons. Depending on the contact geometry and the electrode thickness the spin polarization of the transmission shows various forms around the Fermi level, including perfect (100%) polarization at certain energy ranges. Then we perform first-principles calculations to investigate the optimized geometry and electronic properties of the flakes with adsorbate vanadium atoms. We discuss the binding sites for single and double adsorbate atoms on the graphene flake. By using the relaxed geometry of adsorbate-flake system, we calculate their conductance spectrum by using model metallic electrodes. The adsorbate vanadium atoms introduce additional states around the Fermi level modifying the conductance spectrum of the flake by breaking the spin symmetry of the conduction electrodes even when the electrodes make uniform contacts with the flake. We also find traces of Fano resonances in the conductance spectrum of the graphene flakes arising from the coupling of the extended states of the quantum channel with the localized states of the flake, which acts like a scattering quantum dot. A local density of states (LDOS) analysis show that the discontinuity features with sharp peaks and dips in the conductance spectra are associated with perfectly localized states of the structure in the electrode-flake-electrode geometry.
Calculation Methods
===================
We performed first-principles total energy calculations to obtain electronic structure and equilibrium geometries of rectangular graphene flakes, with and without adatoms, based on the pseudopotential density-functional theory.[@kohn] The spin-dependent exchange-correlation potential is approximated within the generalized gradient approximation[@perdev] (GGA). The software package Atomistix ToolKit[@atk] (ATK), which employs local numerical basis orbitals and nonequilibrium Green’s function formalism to calculate quantum conductance in electrode-device-electrode geometry, has been used in all total energy and transport calculations.
Geometry optimizations of graphene flakes have been done by relaxing all atomic positions in supercell geometries with sufficient vacuum spaces (10 Å minimum) to prevent the interactions with periodic images of the structure. The criteria of convergence for total energy and Hellman-Feynman forces were $10^{-4}$ eV and 0.005 eV/Å, respectively. The electrostatic potentials were determined on a real-space grid with a mesh cutoff energy of 150 Ry. Double-zeta double-polarized basis sets of local numerical orbitals were employed which gives consistent results with previous calculations[@durgun] that used plane-wave basis sets. In cases of extended nanoribbon calculations the Brillouin zone has been sampled with (51,1,1) points within the Monkhorst-Pack k-point sampling scheme.[@monk]
![(Color online) Geometry and spin-dependent charge density of the graphene flake cut from 4-ZGNR. The edges are saturated with hydrogen atoms. Green (dark) and yellow (light) regions denote the local majority spin-type of the charge density. Possible adsorption sites of adatoms are labelled as E1, H1–H4. Electrodes (carbon chain or gold bar) are attached to the flake along the directions A, Z1–Z4.[]{data-label="zgnr4-vanadium"}](1q.eps)
The geometric structure of the graphene flake considered in this study is shown in Fig. \[zgnr4-vanadium\]. It is a finite segment of 4-ZGNR structure having 4 zigzag rows of carbon atoms along its axis. The sides along the cut direction have armchair shape. All edge carbon atoms have been saturated with hydrogen atoms for a better structural and electronic stability of the flake. Possible adsorption sites of adatoms have been denoted in the figure. The magnetic moments of the atoms and the structure as a whole have been calculated by using Mulliken analysis.
The spin-dependent transport properties of the graphene flake were calculated based on non-equilibrium Green’s function formalism as implemented in ATK. Ballistic conductance spectrum of the flake is given within Landauer-Büttiker formalism as $$G(E)=G_0 \sum_\sigma T_\sigma (E)$$ where $G_0=e^2/h$ is the conductance quantum and $T_\sigma$ ($\sigma=\textrm{up, down}$) are the spin-dependent transmission spectra. For each spin state $\sigma$ the transmission probabilities are calculated as $$T_\sigma(E)=\textrm{Tr}[\Gamma_L G^r \Gamma_R G^a]_\sigma,$$ where $G^r$ and $G^a$ are the retarded and advanced Green’s functions, $\Gamma_L$ and $\Gamma_R$ are the contact broadening functions associated with the left and right electrodes, respectively. The broadening functions are anti-Hermitian components of the self-energy terms of the contacts, $\Sigma_L$ and $\Sigma_R$, which depend on the electrode’s surface Green’s function and the contact-molecule (graphene flake) bonding: $\Gamma_{L,R}=i(\Sigma_{L,R}-\Sigma^\dagger_{L,R})/2$. The spatially-separated spin-states of the graphene flake are anticipated to break the spin symmetry in these conductance calculations when contacts to the electrodes are not symmetric at the armchair edges.
In order to exploit spin-dependent characteristics of the graphene flake the structure needs to be partially contacted with thin metallic electrodes. We used linear carbon atomic chains as model electrodes. Carbon atomic chains which are known to be metallic[@tongay] are expected to make reasonably good contact with the flake. The possible contact sites of the electrodes are denoted in Fig. \[zgnr4-vanadium\]. Semi-infinite carbon atomic chains have been rigidly attached to the flake at one or more sites along the A, Z1–Z4 directions after removing the hydrogen atoms at the contact sites. Optimum electrode-flake distances were obtained by relaxation of the supercell consisting of the flake and 4-buffer carbon atoms per each carbon chain attached on both sides of the flakes.
As an alternative to carbon-chain electrodes we also considered gold electrodes to test the robustness of our conductance calculations. The gold electrodes were devised by taking a finite cross-section bar of atoms oriented in the \[001\] direction of fcc crystal. The semi-infinite gold bars have a unit cell of nine atoms in two layers of four and five atoms. We have added single gold atoms to the ends of the electrodes to get atomically sharp tips. Contact regions of the electrodes to the flake were geometrically optimized.
Results and Conclusions
=======================
For the generation of spin polarized currents using a graphene flake, formation of spin-ordered edge-localized states found along the zigzag edges is the key mechanism. Therefore, we first calculate the ground state magnetic ordering in ZGNR structures and compare it to that of the flake. In their ground state, all the carbon atoms of ZGNRs have antiparallel magnetic moments with their nearest neighbors, and their magnitudes rapidly decrease as we go away from the edges. For instance, according to the Mulliken population analysis moments of the atoms in the edge zigzag row of a 4-ZGNR alternate between 1.21$\mu_{\textrm{\tiny{B}}}$ and -0.10$\mu_{\textrm{\tiny{B}}}$ per atom. In the second zigzag row the corresponding values are 0.19$\mu_{\textrm{\tiny{B}}}$ and -0.07$\mu_{\textrm{\tiny{B}}}$. Relatively high magnetic moments are found on the atoms that are under-coordinated. Upon saturation of these carbons with hydrogen atoms the moments on the edge carbon atoms are reduced but the qualitative properties of the magnetic ordering of the ZGNRs persisted. For comparison, in the hydrogen-saturated 4-ZGNR the edge-row carbon atoms have alternating moments of 0.26$\mu_{\textrm{\tiny{B}}}$ and -0.06$\mu_{\textrm{\tiny{B}}}$. In the second row the sequence of moments reduce to 0.07$\mu_{\textrm{\tiny{B}}}$ and -0.04$\mu_{\textrm{\tiny{B}}}$. The hydrogen atoms too attain minute moments of about 0.01$\mu_{\textrm{\tiny{B}}}$, aligned antiparallel to those of the carbon atoms they are attached to. The spin-polarized edge structure of hydrogenated ZGNRs persists for wider ribbons, even with a slight increase in the moments of edge atoms. Hydrogenization also decreases the energy band gap of the $r$-ZGNRs. For instance, we calculated that the band gap values for $r=$3, 4, 5 reduce from 1.25, 1.15, 1.00, to 0.70, 0.65, 0.60 in units of eV, respectively.
Geometric and electronic structures of finite segments of ZGNRs, namely rectangular graphene flakes, are quite similar to those of extended ribbons. Fig. \[zgnr4-vanadium\] shows the particular graphene flake (C$_{52}$H$_{20}$) considered for the calculations in this study. Additional armchair edges of the flakes are also saturated with hydrogen atoms. Spin-unrestricted ground state of the flake corresponds to the antiferromagnetic spin coupling between the zigzag edges of the flake. The total energy of this state is 0.31 eV lower than that of the ground state of a spin-restricted calculation, which is consistent with the calculations on extended ZGNRs.[@cohen] We find that away from the armchair edges of the flake the magnetic moments of the zigzag-edge carbon atoms are close to that of infinite ZGNR structure, but tend to decrease toward the armchair edges. For instance, the moments of the six carbon atoms located at the upmost zigzag edge of the flake shown in Fig. \[zgnr4-vanadium\] are 0.09, 0.21, 0.29, 0.29, 0.21, 0.09 in units of $\mu_{\textrm{\tiny{B}}}$. The magnetic moment profile of the carbon atoms rapidly converge to the ribbon values for longer flakes.
We calculate the spin-dependent ballistic conductance of the graphene flake when contacted with metallic electrodes. Since zero bias conductance and transmission probability spectra are related by a simple multiplicative factor in Eq. 1, we frequently use these terms as equivalent quantities in the following discussions. By exploiting the spatial separation of spin states at the opposite zigzag edges of the flake we show that spin-polarized currents can be achieved without using ferromagnetic electrodes and magnetic entities. By making partial contacts to the armchair sides of the flake the spin symmetry in the coupling of the flake’s electronic states to the electrodes is broken. Consequently the transmission spectrum of the electrons through the flake becomes dependent on their spin state. For a clear demonstration of this effect stripped off complications that may arise from electrode-graphene interactions we chose to use simple, model electrodes. Carbon linear atomic chains are appealing in that respect, being metallic, making naturally good contact with graphene, and having atomically sharp tips.
![(Color online) Spin-dependent transmission spectrum of the graphene flake without (top panels) and with (middle panels) hydrogen saturation of the edge atoms. The carbon linear chain electrodes are connected to the Z1 sites (see Fig. \[zgnr4-vanadium\]). The spectrum when the carbon chain electrodes are replaced by gold leads are shown in the bottom panels. Fermi level is set to zero.[]{data-label="passivation"}](2q.eps)
In Fig. \[passivation\] we plot equilibrium transmission spectra of the bare and hydrogenated graphene flakes with single carbon atomic chain electrodes asymmetrically contacted to one of the edge zigzag rows of the flakes. For comparison, the spectrum of hydrogenated flake with more realistic gold electrodes is also presented. Presence of dangling bonds in the bare flake is reflected in the transmission spectrum as several sharp peaks about 0.7-1.0 eV above the Fermi level. These peaks are removed after hydrogenization of the flake, and are replaced by a single broad peak degenerate for both spin channels. We should note that within the chosen convention of labelling spin states the electrodes are contacted at the flake edge where up-spin states were localized. Therefore, the up-spin states around the Fermi level couple more effectively with the electrodes leading to formation of a broader peak for up-spin channel in the differential conductance spectrum, and consequently the spin polarization of the current for small bias voltages is expected to reflect the flake’s local majority-spin type of the contact side. With the gold electrodes, the basic features in the transmission spectrum remain unchanged except for the transmission peaks being narrower. The narrower peaks are due to weaker coupling of carbon atoms to gold electrodes.
![(Color online) Spin-dependent transmission spectra of the hydrogen-saturated graphene flake when the electrodes make partial contact at the Z1 and Z2 sites (top panels), and when the contacts are uniform (bottom panels). Fermi level is set to zero.[]{data-label="z1z2-z1z2z3z4"}](3q.eps)
The effect of electrode thickness and its contact location on the transmission spectrum is further demonstrated in Fig. \[z1z2-z1z2z3z4\]. When the carbon electrodes make a wider contact with the hydrogenated flake at Z1 and Z2 sites the spin asymmetry in the conductance spectrum is blurred with broader and overlapping peaks, but still the up-spin channel has a larger contribution to the conductance of the flake at energies around the Fermi level. In case of uniformly contacting electrodes (Z1 to Z4) the transmission spectrum is identical for both spin channels as expected. Similarly, conductance of the flake when the electrodes are contacted along the armchair direction (contact site A in Fig. \[zgnr4-vanadium\]) is spin symmetric (not shown).
Another mechanism to break spin symmetry in the conductance of graphene flakes is to introduce magnetic impurities or adatoms. Here we present a systematic study of adsorption of vanadium atoms on graphene flake. Single atoms of several elements are adsorbed on carbon nanotube surfaces with relatively high binding energies and magnetic moments.[@durgun] Similar properties can be expected for their adsorption on graphene. Vanadium is a suitable element to consider due to its high binding energy and spin polarization. The ground state electronic configuration of vanadium is \[Ar\]4s$^2$3d$^3$, and it is a chemically active element with a magnetic moment of 3$\mu_{\textrm{\tiny{B}}}$.
Adsorption site $\mu_{\textrm{\tiny{V}}}$ $\mu_{\textrm{\tiny{tot}}}$ $E_\textrm{\tiny{b}}$ (eV) $d_{\textrm{\tiny{G-V}}}$ (Å)
-------------------------------------- ------------------------------- --------------------------------- -------------------------------- ----------------------------------- --
**E1** ($\mu_{\textrm{\tiny{V}}}<0$) -3.8 -4.9 1.59 1.94
**E1** ($\mu_{\textrm{\tiny{V}}}>0$) 3.8 3.0 1.58 1.94
**H2** ($\mu_{\textrm{\tiny{V}}}<0$) -3.9 -4.3 1.01 1.86
**H2** ($\mu_{\textrm{\tiny{V}}}>0$) 3.9 4.4 1.02 1.87
**H1-H2** (t-t) -2.5, 2.0 0.0 2.09 1.79, 1.71
**H1-H3** (t-t) 3.6, -3.6 0.0 1.42 1.77, 1.77
**H1-H4** (t-t) -1.9, 1.8 0.0 2.41 1.74, 1.74
**H2-H2** (t-b) 3.5, -3.5 0.0 1.12 1.75, 1.75
**E1-E1** (t-b) 3.7, -3.6 0.9 1.28 1.98, 1.96
: Calculated properties of vanadium adatoms on the graphene flake. See Fig. \[zgnr4-vanadium\] for the notation of adsorption sites. The magnetic order in the flake before the adsorption is such that the magnetic moments of the carbon atoms at the top edge are up ($\mu_{\textrm{\tiny{C}}}>0$). In cases of double adsorbate atoms their locations relative to the flake have been denoted by top (t) or bottom (b). Magnetic moments of the vanadium adatoms ($\mu_{\textrm{\tiny{V}}}$) and the total moments of the structures $\mu_{\textrm{\tiny{tot}}}$ are in units of Bohr magneton ($\mu_{\textrm{\tiny{B}}}$). $E_\textrm{\tiny{b}}$ is the binding energy per vanadium adatom, and $d_{\textrm{\tiny{G-V}}}$ is the equilibrium distance of the adatoms to the plane of graphene flake.[]{data-label="tab:polarize"}
We investigated the adsorption of single or double vanadium atoms on the graphene flake. Possible adsorption sites of vanadium atoms on the graphene flake are shown in Fig. \[zgnr4-vanadium\]. When there are two vanadium adatoms their location relative to the flake are denoted by top (t) or bottom (b). Our results are summarized in Table I. Binding energies per adatom are calculated by using the expression $E_\textrm{\tiny{b}}=(E$\[G\]$+nE$\[V\]$-E$\[G+$n$V\])/$n$. Here, $n$ is the number of adsorbed V atoms, $E$\[G\] and $E$\[V\] are the total energies of the graphene flake (C$_{52}$H$_{20}$) and a free vanadium atom, respectively. Total energy of the relaxed structure when foreign atoms are adsorbed on graphene is denoted by $E$\[G+$n$V\]. Like most of the 3$d$ transition metal elements (Sc, Ti, Cr, Mn, Fe, Co) vanadium atoms are adsorbed on hollow sites of two dimensional graphene sheet. Due to the edge effects, however, on the graphene flake vanadium atoms are not adsorbed on the H1, H3, or H4 sites but rather they migrate to the E1 site after relaxation. We found that vanadium atom can be adsorbed also on the H2 site, but the binding energy is less than the one at the E1 site. H2-site adsorption is also off-center; equilibrium position of the vanadium atom tends toward one of the hexagon sides.
Adsorbed vanadium atoms modify the spin-relaxed charge density of the flake, however the planer geometric structure of the graphene flake is quite robust to vanadium adsorption, the displacements of even the nearest neighbor carbon atoms are less than 0.05 Å after geometry relaxation. As shown in Table I depending on the initial spin configurations of the flake and vanadium we identify two distinct magnetic states with close energies for each adsorption site of E1 and H2. In its ground state a vanadium atom adsorbed at the E1 site has a magnetic moment antiparallel with the moments of the edge carbon atoms as depicted in Fig. \[single-van\](a).
![(Color online) Majority spin-type of local charge density of the flake with single vanadium adatoms. The upper zigzag edge of the flake was spin-up polarized before the adsorption. The vanadium atom has a down magnetic moment in (a) and (c) and an up moment in (b) and (d). The lowest energy configuration for a single vanadium atom is shown in (a).[]{data-label="single-van"}](4q.eps)
Binding of two adsorbate vanadium atoms on the same side of the flake is always stronger than that of being on different sides. For both single-sided and double-sided cases, two vanadium adatom configurations with antiparallel magnetic moment have larger binding energies than the cases with parallel magnetic moments.
![(Color online) Spin-dependent transmission spectra of the hydrogen-saturated graphene flake with an adsorbed vanadium atom at the E1 site in its ground state. Top panels: The electrodes make partial contacts at the Z1 sites. Bottom panels: When the contacts are uniform.[]{data-label="up-vanadium"}](5q.eps)
By means of magnetic dipol-dipol interactions stronger bindings occur between the edge carbon atoms and vanadium adatom. Considerable change in the spin charge density of flake with vanadium adsorbate atoms \[see Fig. \[single-van\]\] also modifies the conductance spectrum of the structure. In Fig. \[up-vanadium\] it is seen that the adsorption of a vanadium atom at the E1 site with spin-down magnetic moment enhances the down-spin transmission channel around the Fermi level. Spin polarization of the conductance spectrum is present even when the electrodes make uniform contacts with the flake.
![(Color online) Spin polarization in the transmission spectra of the graphene flake. The type and contact geometries of the electrodes are given as insets. In the lowest panel there is a single vanadium adatom in its ground state.[]{data-label="polarization"}](6q.eps)
In Fig. \[polarization\] we present an alternative view of the spin asymmetry in the conductance spectra by the plots of energy-dependent relative spin-polarization of the transmission probabilities, $(T_{\textrm{\small{up}}}-T_{\textrm{\small{down}}})/(T_{\textrm{\small{up}}}+T_{\textrm{\small{down}}})$, calculated for four different cases. Partial contacts of the electrodes and presence of magnetic adatoms both produce spin polarizations in the spectra. When the flake is contacted from the Z1 sites either with carbon chain or gold electrodes the polarization is almost 100% for an energy window of $\sim$0.8 eV around the Fermi level. We also see that while uniformly contacted flake does not exhibit spin polarization, it can be generated by a single vanadium adatom.
![(Color online) Local density of states (LDOS) isosurfaces calculated for particular energy values of the up-spin transmission spectra of partially contacted graphene flake. The $f$ value next to each plot is the maximum value of LDOS obtained in units of states$/($Å$^3$.eV), and can be taken as a measure of relative magnitudes among the plots. The plotted isosurfaces correspond to LDOS values of $f\times 10^{-2}$. []{data-label="ldos"}](7q.eps)
Before we conclude, we would like to discuss several discontinuity features found in the transmission spectra, a clear example of which can be seen in top panels of Fig. \[z1z2-z1z2z3z4\]. These are in the form of a pulsative variation with very narrow dips and adjacent peaks in the transmission function, reminiscent of a resonant effect. Similar resonances have been recently reported in the transmission spectra of carbon nanotubes with a single iron or vanadium adsorbed atom on them.[@furst] The underlying mechanism of these variations in the transmission function is the Fano resonances of localized states of the structure with extended band states of the electrodes with matching symmetry. The origin of the localized states contributing to the resonances can be either the localized $d$ states of the adatoms as in Ref. and in the vanadium adatom parts of the present study, or more generally they can be the localized states of a quantum dot coupled to the continuous states of a quantum channel.[@xu] For instance, in formation of the resonances found in Fig. \[z1z2-z1z2z3z4\] the graphene flake acts like a scattering quantum dot coupled to the quantum channel of carbon-chain electrodes.
We infer about the character of these resonances by calculating local density of states (LDOS) of the structure at resonance and off-resonance energies of the transmission spectra. In Fig. \[ldos\] we present LDOS isosurface plots for up-spin transmission spectra of three particular contact geometries. LDOS are perfectly confined at the flake region only at resonance energies. At off-resonance energies we always find contribution of extended states of the metallic electrodes.
In summary, we have performed an *ab initio* study of spin-resolved conductance spectra of rectangular graphene flakes. We have shown that spin-polarized currents can be obtained by making the electrodes contact the flake partially along the zigzag-edge directions of the flake, or by introducing transition-metal adatoms such as vanadium to alter the symmetry in spin-dependent scattering rates of the transmitted electrons. In the particular model system considered in this study (a C$_{52}$H$_{20}$ flake cut from 4-ZGNR) a 100% spin-polarization in the conductance spectrum was achieved for both model carbon atomic chain and more realistic gold bar electrodes. Traces of Fano resonances have been found in the transmission functions. We believe that the qualitative features of producing spin-polarized currents in graphene flakes can be verified experimentally with flakes of larger sizes and with realistic electrodes. An understanding and realization of spin-dependent effects in graphitic structures are essential toward the development of graphene-based spintronic devices.
This work was supported by TÜBİTAK under Grant No. 106T597, and through TR-Grid e-Infrastructure Project. RTS acknowledges financial support from Turkish Academy of Sciences (TÜBA) GEBİP program.
[99]{}
K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. Khotkevich, S. V. Morozov, and A. K. Geim, Science **306**, 666 (2004).
K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, A. A. Firsov, Nature (London) **438**, 197 (2005).
Y. Zhang, Y.-W. Tan, H. L. Stormer, and Philip Kim, Nature (London) **438**, 201 (2005).
C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, Science **312**, 1191 (2006).
P. R. Wallace, Phys. Rev. **71**, 622 (1947).
M. Y. Han, B. Özyilmaz, Y. Zhang, and P. Kim, Phys. Rev. Lett. **98**, 206805 (2007).
X. Li, X. Wang, L. Zhang, S. Lee, and H. Dai, Science **319**, 1229 (2008).
Y. Miyamoto, K. Nakada, and M. Fujita, Phys. Rev. B **59**, 9858 (1999).
H. Lee, Y. Son, N. Park, S. Han, and J. Yu, Phys. Rev. B **72**, 174431 (2005).
Y. W. Son, M. L. Cohen, and S. G. Louie, Phys. Rev. Lett. **97**, 216803 (2006).
V. Barone, O. Hod, and G. E. Scuseria, Nano Lett. **6**, 2748 (2006).
K. I. Sasaki, S. Murakami, and R. Saito, J. Phys. Soc. Jpn. **75**, 074713 (2006).
D. A. Abanin, P. A. Lee, and L. S. Levitov, Phys. Rev. Lett. **96**, 176803 (2006).
D. A. Areshkin, D. Gunlycke, and C. T. White, Nano Lett. **7**, 204 (2007).
A. Cresti, G. Grosso, and G. P. Parravicini, Phys. Rev. B **76**, 205433 (2007).
K. N. Kudin, ACS Nano **2**, 516 (2008).
M. Ezawa, Phys. Rev. B **73**, 045432 (2007).
K. Kobayashi, Phys. Rev. B **48**, 1757 (1993). M. Fujita, K. Wakabayashi, K. Nakada, and K. Kusakabe, J. Phys. Soc. Jap. **65**, 1920 (1996). K. Nakada, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B **54**, 17954 (1996). K. Wakabayashi, M. Sigrist, and M. Fujita, J. Phys. Soc. Jap. **67**, 2089 (1998). K. Nakada, M. Igami, and M. Fujita, J. Phys. Soc. Jap. **67**, 2388 (1998). K. Wakabayashi, M. Fujita, H. Ajiki, and M. Sigrist, Phys. Rev. B **59**, 8271 (1999).
P. Gambardella, A. Dallmeyer, K. Maiti, M. C. Malagoli, W. Eberhardt, K. Kern, and C. Carbone., Nature (London) **416**, 301 (2002).
J. Dorantes-Davila and G. M. Pastor, Phys. Rev. Lett. **81**, 208 (1998).
A. Vindigni, A. Rettori, M. G. Pini, C. Carbone, and P. Gambardella, Appl. Phys. A **82**, 385 (2006).
A. Delin, E. Tossati, and R. Weht, Phys. Rev. Lett. **92**, 057201 (2004).
T. Kawai, Y. Miyamoto, O. Sugino and Y. Koga , Phys. Rev. B **62**, R16349 (2000).
P. O. Lehtinen, A. S. Foster, A. Ayuela, A. Krasheninnikov, K. Nordlund, and R. M. Nieminen, Phys. Rev. Lett. **91**, 017202 (2003).
K. Nordlund, J. Keinonen, and T. Mattila, Phys. Rev. Lett. **77**, 699 (1996).
M. Heggie et al., Electrochem. Soc. Proc. **6**, 60 (1998).
Y. H. Lee, S. G. Kim, and D. Tomanek, Phys. Rev. Lett.**78**, 2393 (1997).
O. Hod, V. Barone, J. E. Peralta, and G. E. Scuseria, Nano Lett. **7**, 2295 (2007). D. Gunlycke, J. Li, J. W. Mintmire, and C. T. White, Appl. Phys. Lett. **91**, 112108 (2007). E. J. Kan, Z. Y. Li, J. L. Yang, and J. G. Hou, J. Am. Chem. Soc. **130**, 4224 (2008).
H. Sevincli, M. Topsakal, E. Durgun, and S. Ciraci , Phys. Rev. B **77**, 195434 (2008).
E. J. Duplock, M. Scheffler, and P. J. D. Lindan, Phys. Rev. Lett. **92**, 225502 (2004).
M. Endo, T. Hayashi, S. H. Hong, T. Enoki, and M. S. Dresselhaus, J. Appl. Phys. **90**, 5670 (2001).
E. Hernandez, P. Ordejon, I. Boustani, A. Rubio, and J. A. Alonso, J. Chem. Phys. **113**, 3814 (2000).
T. B. Martins, R. H. Miwa, A. J. R. da Silva, and A. Fazzio, Phys. Rev. Lett. **98**, 196803 (2007).
A. Rycerz, J. Tworzydlo, and C. W.Beenakker, Nature (London) **547**, 172 (2007).
F. M. Rojas, D. Jacob, J. F. Rossier, and J. J. Palacios, Phys. Rev. B **74**, 195417 (2006).
L. Brey and H. A. Fertig, Phys. Rev. B **73**, 235411 (2006).
Y. Son, M. L. Cohen, and S. G. Louie, Nature (London) **444**, 347 (2006).
D. Gunlycke et al., Appl. Phys. Lett **92**, 163109 (2008).
W. L. Wang, S. Meng, and E. Kaxiras, Nano Lett. **8**, 241 (2008).
P. Shemella, Y. Zhang, M. Mailman, P. M. Ajayan, and S. K. Nayak, Appl. Phys. Lett. **91**, 042101 (2007). M. Ezawa, Phys. Rev. B **76**, 245415 (2007). J. Fernandez-Rossier and J. J. Palacios, Phys. Rev. Lett. **99**, 177204 (2007). D. en Jiang, B. G. Sumpter, and S. Dai, J. Chem. Phys. **127**, 124703 (2007). O. Hod, J. E. Peralta, and G. E. Scuseria, Phys. Rev. B **76**, 233401 (2007). O. Hod, V. Barone, and G. E. Scuseria, Phys. Rev. B **77**, 035411 (2008). E. Rudberg, P. Salek, and Y. Luo, Nano Lett. **7**, 2211 (2007). E. Rudberg, P. Salek, and Y. Luo, Nano Lett. **8**, 2442 (2008).
W. Kohn and L. J. Sham, Phys. Rev. **140**, A1133 (1965); P. Hohenberg and W. Kohn, Phys. Rev. **136**, B864 (1964).
J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. **77**, 3865 (1996).
Developed by Atomistix Company, Copenhagen, Denmark. http://www.atomistix.com
E. Durgun, S. Dag, V. M. K. Bagci, O. Gulseren, T. Yildirim, and S. Ciraci, Phys. Rev. B **67**, 201401(R) (2003).
H. J. Monkhorst and J. D. Pack, Phys. Rev. B **13**, 5188 (1976).
S. Tongay, R. T. Senger, S. Dag, and S. Ciraci, Phys. Rev. Lett. **93**, 136404 (2004).
J. A. Fürst, M. Brandbyge, A.-P. Jauho, and K. Stokbro, arXiv:0801.3997v1 \[cond-mat.mtrl-sci\], (unpublished).
H. Xu and W. Sheng, Phys. Rev. B **57**, 11903 (1998).
|
---
bibliography:
- 'bibliography.bib'
---
[**** ]{}\
John M. Aiken^1,3^ Riccardo De Bin^2^ Morten Hjorth-Jensen^1,\ 3,\ 4^ Marcos D. Caballero^1,\ 3,\ 5^
**1** University of Oslo Department of Physics, Centre for Computing in Science Education, Blindern, Oslo, Norway\
**2** University of Oslo Department of Mathematics and Statistics, Blindern, Oslo, Norway\
**3** Michigan State University Department of Physics and Astronomy, East Lansing, Michigan, USA\
**4** Michigan State University, National Superconducting Cyclotron Laboratory and Facility for Rare Ion Beams, East Lansing, Michigan, USA\
**5** CREATE for STEM Institute, Michigan State University, East Lansing, Michigan 48824\
\* j.m.aiken@fys.uio.no
Abstract {#abstract .unnumbered}
========
The time it takes a student to graduate with a university degree is mitigated by a variety of factors such as their background, the academic performance at university, and their integration into the social communities of the university they attend. Different universities have different populations, student services, instruction styles, and degree programs, however, they all collect institutional data. This study presents data for [160,933]{}$\;$ students attending a large American research university. The data includes performance, enrollment, demographics, and preparation features. Discrete time hazard models for the time-to-graduation are presented in the context of Tinto’s Theory of Drop Out. Additionally, a novel machine learning method: gradient boosted trees, is applied and compared to the typical maximum likelihood method. We demonstrate that enrollment factors (such as changing a major) lead to greater increases in model predictive performance of when a student graduates than performance factors (such as grades) or preparation (such as high school GPA).
Introduction
============
University students must meet a number of objectives to obtain degrees and in many cases this can prolong their time at the university [@yue2017rethinking] or they drop out altogether [@braxton2000influence]. During their studies, American students may take on substantial financial obligations when choosing to pursue degrees and extending beyond the “four-year degree” can greatly increase the cost of obtaining that degree [@abel2014recent]. Thus understanding the paths that students take towards degree completion can help faculty and administrators better serve student populations to meet their educational goals. Tinto’s Theory of Drop Out [@tinto1975dropout] has seen large acceptance in its ability to describe the factors that influence a student’s path towards degree completion [@pascarella1983predicting; @nora1990testing; @cabrera1993college; @desjardins1999event; @ishitani2003longitudinal]. Tinto theorized that a student’s college drop out decision is mediated by two conglomerate features: 1) educational goal commitment, and 2) institutional commitment. He further theorized that these commitments are dynamic. Students begin their university studies with initial commitments that are then mediated by how students participate in the academic and social systems of the university. Tinto suggested that using institutional data such as that collected by university registrars, could quantify these relationships and provide predictive models for determining student’s at-risk to drop out or alternatively, graduate.
This paper examines student ultimate success at university. We define this success as obtaining a bachelor’s degree. It uses 20 years of institutional data for [160,933]{}$\;$students attending a large enrollment American research university. This paper examines two specific research questions:
1. In the context of Tinto’s Theory of Drop Out [@tinto1975dropout], what factors (such as grades, participation in a major, student background, etc.) contribute to the time it takes to obtain a degree at a large enrollment research university in the United States? How does the contribution of these factors change for longer durations until graduation?
2. How can recent innovations in statistics and machine learning, such as gradient boosting and xgboost, improve educational model performance?
We focus on a comparison of student participation in an academic system, their involvement in the social system, and their initial conditions due to high school training. We demonstrate that enrollment factors (e.g., changing a major) and academic performance are more important to predicting when a student graduates than pre-college experiences (e.g., high school GPA). Additionally, we compare traditional statistical modeling (maximum likelihood estimation) to new techniques from machine learning (gradient boosting [@chen2016xgboost]) and demonstrate that the machine learning methods are more effective at estimating the function predicting time-to-graduation.
It is important to use an analysis technique that respects the dynamics assumptions of Tinto’s Theory of Drop Out. In this paper we use a Discrete Time Hazard Model framework [@yamaguchi1991event]. Discrete time hazard model is useful when the classic Cox regression model assumption that events happen on a continuous interval no longer hold valid [@yamaguchi1991event]. This is true when data has highly discretized time intervals such as semesters enrolled. Traditionally this modeling is done with logistic regression via maximum likelihood estimation [@yamaguchi1991event]. In this paper we compare two models that fit the logistic model including logistic regression (maximum likelihood estimation) and a gradient boosted tree model (xgboost, [@chen2016xgboost]).
Background
==========
In this section Tinto’s theory[@tinto1975dropout] and the use of that theory will be described, the use of discrete time hazard model [@yamaguchi1991event] to predict when students graduate and/or drop out will be described, and the value of a new machine learning method known as gradient boosting will be described [@chen2016xgboost]. The functional descriptions of the discrete time hazard model and gradient boosting are described in Section \[sec:methods\].
Tinto’s Theory of Drop Out
--------------------------
Tinto’s Theory of Drop Out describes a students intent to drop out based on the interplay of two quantities: 1) a student’s commitment to education, and 2) a student’s commitment to a specific institution. A student’s commitment to education is mediated by the initial state of a student entering the university and the dynamics that occur while the student attends university. These dynamics are dictated by a student’s participation and acceptance into the social and academic communities that are at a university. A student’s commitment to an institution is tempered by many factors such as the educational goals available at an institution (e.g., a technical university degree offerings versus a liberal arts university), family commitment to a university, and social acceptance at the university. While Tinto’s theory explicitly attempts to describe why students drop out from college, it is not uncommon that this theory is used to study student graduation from college given that graduation is an alternative outcome to dropping out [@ishitani2003longitudinal; @scott2005pitfalls; @yue2017rethinking]. In this paper we focus on a student’s commitment to education as the framing for features that predict when a student will graduate if they do so.
It is common to characterize Tinto’s theory of dropout as a discrete time hazard problem [@desjardins1999event; @ishitani2003longitudinal; @scott2005pitfalls; @chen2013stem; @yue2017rethinking]. Tinto’s theory posits that a student’s educational commitment changes over time as they work toward’s a degree and that effects, such as high school GPA, that may be strong in the beginning of a degree program are weak toward’s the end of a degree program. Discrete time hazard modeling provides a systematic method for examining the various effects on the probability to graduate over time [@yamaguchi1991event]. To that end, discrete time hazard modeling has been used to examine the dynamic impact of various effects that Tinto predicts effect a student’s educational commitment. College GPA has been demonstrated to have a profound but diminishing time varying effect on graduation [@desjardins1999event; @chen2012institutional; @yue2017rethinking]. Financial aid and money spent on student services has also demonstrated a time varying (diminishing) effect on the probability to graduate [@chen2012institutional; @yue2017rethinking]. First generation student’s are considered high risk for drop out but this effect diminishes over time indicating that first generation students need specific resources other students may not [@ishitani2003longitudinal]. Non-traditional enrollment factors such as delaying enrollment, working while enrolled, and stopping education for some period of time can all have a negative time varying effect on graduation [@scott2005pitfalls]. In each case these studies showed a statistically significant time varying impact that supports the dynamic claims of Tinto’s theory.
A student’s preparation has long been known to impact a student’s ability to graduate. Preparation is assessed in many ways and can be represented as the experiences a student has in school and also their present innate ability to perform a specific function. Math preparation correlates with both performance in university and graduating [@trusty2003high; @gaertner2014preparing]. The same is true for physics and english preparation [@bamberg1978composition; @sadler2001success; @hazari2007gender]. High school GPA and SAT scores typically account for some but not all of student success at university (GPA, etc.) [@zwick2005predicting; @aiken2019modeling; @zabriskie2019using]. Preparation for university often can be experienced differentially as well. Women in STEM (Science, Technology, Engineering, and Mathematics) degree programs report having less access to laboratory experiences in high school, are encouraged towards science by their father’s differently, and overall have a different preparation than their male counterparts [@seymour2000talking; @hazari2007gender].
A student’s demographics can include the student’s gender, race, the financial support they can expect from their family, and if a student is the first in their family to attend college. Race has long been shown to be a factor in whether a student graduates or not. Black students are less likely to graduate than fellow white students when adjusted for socio-economic status and academic ability and they are more likely to have unwelcoming experiences in STEM programs while at university [@alexander1982social; @seymour2000talking; @yue2017rethinking]. Female students are also likely to have unwelcoming experiences that cause them to switch from STEM programs [@seymour2000talking]. However they have been show to be equally likely to graduate or more likely to graduation in comparison to their male counterparts [@alexander1982social; @seymour2000talking]. The financial support both in terms of loans and scholarships and the socio-economic status of a student’s family have long been known to be a factor in university graduation with students who have more financial support typically being more likely to graduate [@alexander1982social; @sewell1967socioeconomic; @smith2001factors; @rowan2007predictors; @yue2017rethinking]. First generation students are less likely to participate in extra-curricular activities at university [@choy2001students; @pascarella2004first] and are more likely to dropout even when adjusting for race, family income, gender, and preparation [@ishitani2003longitudinal]. Ultimately, American students from different backgrounds often see different success rates at university due to the different experience they have at the university due to their race, gender, or socio-economic status.
Gradient boosting
-----------------
In this paper we use a novel method known as “gradient boosting”. Gradient boosting can solve a large number of statistical modeling problems including logistic problems as found in this paper. Logistic problems are a group of statistical problems that assume that a sigmoidal function made from parameters $\theta$ and data $\boldsymbol{x}$ (i.e., $[1+e^{-\theta^{T} \boldsymbol{x}}]^{-1}$) approximates the probability $P(Y|X)$. In plain language, the model uses input data $X$ to determine how likely the outcome, $Y$, is to be true. In the context of this paper that would mean the input data $X$ is used to predict whether a student will graduate in the following semester. In education research, the maximum likelihood method has commonly been used to find the solutions to logistic problems. Maximum likelihood solves the logistic regression problem by picking parameters $\theta$ that maximize the log likelihood function [@hastie2005elements]. Since there is no closed form solution to this likelihood equation the solution is found iteratively via some optimization algorithm [@hastie2005elements]. Maximum likelihood is the typical default solution for logistic problems in most statistical packages such as statsmodels [@seabold2010statsmodels] in python and glm in R [@glmr].
Gradient boosting produces the solution to the logistic problem in a different way. The log-likelihood is maximized (or, in machine learning terms, the negative log-likelihood is minimized) in small steps: at each step, called boosting iteration, the gradient of the log-likelihood is computed to identify the best direction for the maximization, and it is then fed to a base learner, through which the model is updated. Different choices of base learners (in this paper we will use trees) provide different solutions, guaranteeing maximum flexibility [@buhlmann2007boosting]. An early stop, through a tuning parameter which controls the number of iterations, prevents overfitting and provide a better bias-variance trade-off [@mayr2014extending]. If the base learner is weak enough in comparison to the signal-to-noise, it can be shown that, at least for a continuous response, boosted models outperform their unboosted versions in term of MSE [@buhlmann2003boosting]. Several modifications of the boosting algorithm have been proposed, including stochastic gradient boosting, that uses column and row subsampling to further avoid overfitting and even better deal with the bias-variance trade-off issue [@suen2005combining].
Gradient boosting has been demonstrated to produce better fit models in comparison to traditional methods across a number of domains both in binary classification and in hazard modeling [@lombardo2015binary; @xu2017pdc; @ma2017prioritizing]. Boosted logistic models have been demonstrated to be more effective at fitting data than models that use maximum likelihood estimation [@de2017data]. The structure of educational data can be highly complex often containing many sub groups that have different effects [@bryk1989toward]. It is likely that gradient boosting can provide better parameter estimation for education research questions as well.
In this paper we use the Extreme Gradient Boosting (xgboost) algorithm [@chen2016xgboost]. Xgboost implements stochastic gradient boosting [@friedman2002stochastic] with column and row-wise subsampling, regularization, decision tree base learners with a custom tree split finding algorithm, and an in-model data imputation system for missing data. The specifics of xgboost are explained in Section \[sec:methods\].
Data Set
========
The data in this study comes from registrar information from a large enrollment American research university [@aiken2019modeling]. It includes timestamped course grades, demographics, majors declared, degrees awarded, and preparation information for [160,933]{}$\;$students for the years 1992 to 2012. Student’s are included in the study if they remain enrolled for at least 5 semesters. The 5 semester mark is chosen because a large number of students(18.3%) remain enrolled for only four or less semesters taking core courses [@yue2017rethinking]. These students do not graduate the university and likely transfer elsewhere as their institutional commitment may be low [@tinto1975dropout]. 88.02% students who remain enrolled for at least 5 semesters graduate from this university.
In this study, data is organized by semester into of four categories: demographics, preparation, enrollment, and performance (Fig. \[fig:features\]). Demographics and preparation are considered “static” features. That is, they are not changing over the course of the study. The enrollment and performance features are considered “dynamic”. That is, they are cumulative and values can change as students progress in their studies.
The demographics data includes gender, race, cohort year, and the median family income for the zipcode the high school the student attended. The gender category is binary: male or female. The race category uses the IPEDs definitions [@ipedsdefinitions] which is then reduced to a binary category of white/asian or other. This is due to two reasons: 1) the university has a primarily white/asian population, and 2) in this paper we establish model comparison’s via out-of-sample prediction accuracy. Thus due to the small minority population, out-of-sample, per semester data may not include all racial groups with sufficient numbers ($>$100, [@poldrack2019establishment]). The cohort year is the year the student begins their first courses. The median income comes from the 2011 American Community Survey 5-Year Estimate [@2011census]. If the student has a reported high school GPA, then we also know the zip code of the high school the student attended. This zipcode is matched to the 2011 census data zip codes. The census data includes the median incomes for families living in that zipcode. The log of the median income is then recorded per student. If the zip code data is missing the median income is imputed as described in Section \[sec:methods\]. This method is not an attempt to provide a course grained estimate of the socio-economic status of the student. Instead, it is a method to obtain a measure of the socio-economic status of the high school the student attended. Student learning can be impacted not only by a student’s personal socio-economic status, but also the socio-economic status of the learning community they belong to [@caldas1997effect]. We do not have individual level socio-economic data such as parental income. Nor do we have data on financial aid status.
The preparation data includes the per student reported high school GPA, a math placement score, and if amount of Advanced Placement credits if any. The math placement score comes from a 30 item test that the university requires students to take upon admission. This test has been used for the entirety of the study. A higher score will place the student in a higher math course up to calculus 1. High school GPA is also reported for each student. In both the cases of high school GPA and math placement score the data can be missing for several reasons. High school GPA is not required to be reported by applicants to this university, thus high school GPA is not always recorded for each student. If students have transfer courses for higher math courses (e.g., calculus 1) then the student does not need to take a placement test to determine if they can begin in calculus 1. In the case of missing data the math placement score and the high school GPA are imputed as described in Section \[sec:methods\].
The enrollment features are organized per semester. They include whether a student changed their major in that semester, the number of currently enrolled majors, the ratio of credit hours to a full time load (12 credit hours), the total registered semester credit hours, the total cumulative credit hours accrued, the non-major credits the student has registered for in this semester, the major credits hours enrolled per semester, whether the student was enrolled in the previous semester, and the cumulative number of skipped semesters.
The performance features are organized per semester. They include the current semester GPA, the cumulative GPA, the current semester’s GPA for courses outside of the student’s current major, and the current semester’s GPA for courses within the student’s current major.
![A comparison of a subset of features for students who graduate versus those who do not based on first year enrolled. Since 1992 the student population has become more diverse racially, students have had increasing high school GPAs and math placement scores, and the time it takes students who graduate to graduate has been decreasing. Students who graduate typically take more credit hours than those who don’t, typically are better prepared as measured by high school GPA and math placement score, and are less diverse than the total university population.[]{data-label="fig:rawdata"}](rawdata.pdf)
![Students typically graduate within the window of 8 to 12 semesters. The bars represent the number of students across all cohort years who graduate in the following semester. The green line represents the cumulative fraction of students who have graduated.[]{data-label="fig:gradrates"}](gradrates.pdf)
Methods {#sec:methods}
=======
In this paper we have used a discrete time hazard modeling framework for all models that are presented. We begin by comparing the logistic regression model to a gradient boosted (xgboost) [@chen2016xgboost] model to produce the most predictive model. In this section we will describe the discrete time hazard modeling framework, the logistic regression equation, the xgboost model equation, and the statistics and evaluation methods that we have used to determine the xgboost model is the best predictive model. We will then describe the methods we used to evaluate what features the model uses to produce predictions. The superior xgboost model is then used to describe what factors are most predictive of students graduating in a particular semester.
Discrete Time Hazard Models
---------------------------
Discrete time hazard models are, essentially, logistic regression (or other classification) models calculated per time step for time changing data. They control for time to event predictions when the times to events are simultaneous or explicitly discrete and there is new data being collected over the course of the study period [@yamaguchi1991event]. Traditional hazard analysis such as Cox regression is not capable of doing this type of analysis due to the duration being measured being a common value amongst many students [@yamaguchi1991event]. This is due to the likelihood function of a continuous event duration model assumes independent and unique durations (i.e., the time it takes to graduate). When these durations are not unique, they can lead to overfitting. In this study the time unit used is a semester [@yue2017rethinking]. Thus the models attempt to predict whether a student will graduate in the immediate following semester. If a student drops out from the university, they are not included in following semester data set.
Logistic Regression
-------------------
The logistic regression equation used in this paper is as follows:
$$\log \frac{y(t)}{1-y(t)} = \beta_{0}(t) + \boldsymbol{\beta}_{S}^T \boldsymbol{X}_{S} + \boldsymbol{\beta}_{D}(t)^{T} \boldsymbol{X}_{D}(t)$$
Where $y(t)$ is the likelihood to graduate in the following semester and $\boldsymbol{\beta}_{S}$ and $\boldsymbol{X}_{S}$ are static features indicated by subtext $S$ (Fig. \[fig:features\]). The dynamic terms (indicated by subtext $D$) are calculated per semester $t$ from semester 5 until semester 16 [@yue2017rethinking]. $\epsilon$ is then the irreducible random error. This is effectively an iteratively calculated multinomial logistic regression model [@yue2017rethinking]. The model is fit using the maximum likelihood estimation method. No regularization is used.
The logistic regression model is built using the statsmodels library in python [@seabold2010statsmodels].
Gradient Boosted Trees
----------------------
Xgboost is an implementation of a stochastic gradient boosting machine [@friedman2001greedy; @friedman2002stochastic; @chen2016xgboost] that can also be used to attempt to solve the logistic equation. Gradient boosted machines can be seen as models that are iteratively fit on the current residuals starting from the null model. The additive collection of these models produce the output. Xgboost uses decision trees as its base learners.
The gradient boosted model is thus:
$$\log \frac{y(t)}{1-y(t)} = F_{0} + \sum_{m}^{M}h_{m}(\boldsymbol{X}(t))F_{m}(\boldsymbol{X}(t))$$
Where $m$ is the iteration index, $h_{m}(X)$ is the previous iterations residual model and $F_{m}(X)$ is the current iteration’s model fit to previous iteration’s residuals. The total number of iterations is set by $M$.
A gradient boosted machine can use any learner for the iterative procedure. In this paper we use the decision tree learner as implemented in Xgboost [@chen2016xgboost]. Decision trees are models that use a tree like structure to fit data and produce regressions and classifications. For each “leaf” node in a tree, a specific decision is made that separates data into two diverging paths. Each node represents a single variable. Categorical variables (e.g., gender) are simply split by the category. Continuous variables (e.g., semester GPA) are split by a decision boundary (e.g., GPA $>$ 3.0). This boundary is typically determined through iteratively fitting the model to find the best boundary.
In Xgboost, each decision tree is trained from a randomly sampled set of rows and columns. Each tree is grown up to a maximum depth using a leaf growing algorithm that estimates whether an additional leaf will produce a better or worse tree [@chen2016xgboost]. Thus there is no separation between static and dynamic features as in the logistic regression model for each semester. All xgboost models are built using the xgboost library in python [@chen2016xgboost].
Xgboost has a built in algorithm to deal with missing data [@chen2016xgboost]. When missing, data is imputed for each tree by selecting the most common decision path for a node. Because a feature may appear in many trees in the model, the decision boundary can be very different per tree in comparison to an mean imputation scheme.
Xgboost models have hyperparameters that define how the model is to be constructed prior to the model being trained. These hyperparameters govern two classes of model design: 1) how the boosting functions, and 2) how the trees are grown and structured. In the case of boosting, the most common hyperparameter used is the number of total boosting learners. For the tree structure this includes the maximum depth a tree can grow to and the fraction of variable and row sub-selection. Typically these hyperparameters are determined by grid search when the total combination of hyperparameters is below 1000 combinations [@geron2017hands; @bergstra2012random].
Because the learners in the xgboost model are decision trees and not linear models, the xgboost model does not report typical coefficient values like in a traditional logistic regression. Instead they report feature “importances” known as “gain”. Each time a variable is used in a tree, the tree is built optimally by splitting in the optimum location. The increase in accuracy due to this split is the gain. The feature importance for a specific variable reported then is the average gain across all the instances that the variable is used in in the model. Xgboost uses a custom split finding algorithm that compares the utility of growing a tree to its maximum depth based on increase in accuracy [@chen2016xgboost].
Model Evaluation
----------------
The goal of this study is to produce a predictive model of when student’s graduate. We use a number of techniques to verify the veracity of the model and increase its accuracy (Fig. \[fig:model\_eval\]). These include: splitting the data into a training/testing sets, imputing missing data, and picking custom thresholds for the predicted probability of graduation. We also limit the over-estimation of a variable’s impact on when a student graduates by weighting explanation methods with the $F_{1}$-score [@hastie2005elements]. The following section will explain these techniques in detail.
The data processing and model evaluation follows the procedure shown in the flowchart in Fig. \[fig:model\_eval\]. Data for the discrete time hazard model models are:
1. queried from the pathways database [@aiken2019modeling]
2. Continuous values such as high school GPA and semester GPA are rescaled using the z-score. The means and standard deviations are determined by either the starting year cohort (high school GPA) or the timestamp for the enrolled semester (semester GPA). The median income is scaled using the logarithm.
3. Data is split into training (50%) and testing (50%) sets. Since the discrete time hazard models are evaluated per semester data is split by student and not by semester. Thus a student in training data in semester 5 is also in training data for subsequent semesters.
4. Missing data that is input into the logistic regression model is imputed from the means calculated during the rescaling step. Missing data for the xgboost model is not imputed prior to fitting the model since the xgboost algorithm has an imputation engine built in.
5. Each model is trained and evaluated using the $F_{1}$ score. In-sample $F_{1}$ scores are used to determine the best threshold for splitting the predicted probability distributions for classification (i.e., does the student gradute in the next semester).
6. Models using the selected threshold are then evaluated using out-of-sample test data to compute the $F_{1}$ score.
Several data handling procedures were used to prevent model overfitting, data leakage, and poor predictive performance due to unbalanced class issues. To evaluate the predictive capability of each model, data in this paper is separated into training and test sets for each semester that the discrete time hazard model model is applied. Separating data into partitioned training and testing sets is important to prevent a too optimistic evaluation of the model error[@hastie2005elements]. Without having hold out data to evaluate the model performance, we have no way of knowing how a model will predict the outcome of data it has not seen before [@hastie2005elements; @Hofman486; @poldrack2019establishment]. Using hold-out data is atypical in recommendations within education research for assessing predictive ability [@raudenbush2002hierarchical; @van2019modernizing] and is not used typically in modern papers using discrete time hazard analysis (e.g., [@yue2017rethinking]).
Data leakage could occur between two semesters if, for example, a student in semester 6 is in a training set and the same student is in a test set in semester 7 [@poldrack2019establishment]. This is due to some of the features being cumulative thus they would carry forward information from the previous training period into the next test period. Thus, students are identified prior to model training as belonging to the training or testing data set. Students in the test data set are only given to the model to evaluate model predictive performance. Students do not always have an entry in the database for their high school GPA or their math placement score. This can be due to a variety of reasons such as the high school GPA was not reported, the reported high school GPA was self reported and potentially untrustworthy, and the math placement score was not given to the student because they transferred math credits from another institution. To prevent errors associated with removing data [@rubin1976inference], this data is imputed in one of two ways. For the logistic model data is imputed from the mean for the student’s cohort year. For the xgboost model data is imputed within the model. Xgboost will actively impute missing data within each tree learner based on the most likely branch for each decision node (see Section \[sec:methods\] and [@chen2016xgboost] for more details).
The predictive ability of models in this paper is estimated using the $F_{1}$-score and the recall [@goutte2005probabilistic]. The $F_{1}$-score is the harmonic mean of the precision and recall and is calculated with the following equation:
$$\begin{aligned}
F_{1}\text{-score} = 2 \frac{\text{Pr} \cdot \text{Re}}{\text{Pr} + \text{Re}} \\
Pr = \frac{\text{true positive}}{\text{true positive} + \text{false positive}} \\
Re = \frac{\text{true positive}}{\text{true positive} + \text{false negative}}\end{aligned}$$
The $F_{1}$ score is over the range 0 to 1 with 1 being perfect performance and 0 being random performance. Precision is defined as the ratio of true positives to the sum of true positives and false positives. That is, it is the ratio of true predicted graduations per semester to the sum of the true predicted graduations and those predicted to graduate but actually do not. Precision is over the range 0 to 1 with 1 being perfect performance and 0 being random performance. Recall is defined as the ratio of true positives to the sum of true positives and false negatives. That is, it is the ration of the number of students predicted to graduate to the total number of people who actually graduated per semester. Recall is over the range 0 to 1 with 1 being perfect performance and 0 being random performance.
We use the $F_{1}$-score over other statistics (such as the area under the reciever operating curve (AUC), [@hastie2005elements]) because it balances the trade off between high precision (which includes falsely labeling students as graduated) and high recall (which includes falsely labeling students as not graduating). We use the Recall score in the cases of understanding model performance for sub-groups (such as under-represented minorities) because it gives a more direct interpretation of how accurately the model selects the graduating case.
Students are enrolled for many semesters however they only graduate in a single semester. Thus the majority class in each semester is that a student will *not* graduate in the following semester. This unbalanced class issue can lead to under-fitting of the model specifically such that models simply predict the majority class (in this case not graduating) far too often [@weiss2007cost]. We attempted to use an over-sampling technique [@chawla2002smote] to address this issue however this did not increase model performance. Instead we choose custom thresholds for the model probability distributions for predicting when a student graduated. The custom thresholds are picked using the $F_{1}$ scores calculated from the training data [@guyon2006introduction]. The best threshold is calculated via a grid-search between a threshold of 0 to 1 using increments of 0.01. This still, typically, does not increase $F_{1}$ scores.
Beyond predicting when a student will graduate we are interested in the factors that explain why a student was predicted to graduate (or not graduate). A typical logistic regression model using maximum likelihood produces coefficients that represent the magnitude and direction a particular feature given the other variables. For gradient boosted trees these values are calculated differently. Xgboost uses instead the average gain in prediction accuracy across all instances a feature is used in each tree learner. This is commonly called the “feature importance”. Additionally we can estimate how much a feature, depending on individual values, contributes to the predicted probability of graduating. In this case we use a method called “partial dependence” that was originally developed to use with gradient boosted models [@friedman2001greedy]. Both the xgboost model and the logistic regression model have a different level of confidence in the prediction per semester (Fig. \[fig:f1scores\]). Due to this variable confidence our confidence in feature importances and partial dependences varies as well. In order to account for this confidence we have weighted the feature importances and partial dependences with the out-of-sample $F_{1}$ score. This is explained below.
We estimate the effects of model features on prediction using the using the weighted mean of the feature importances. For all 11 semesters the feature importance is weighted by the overall $F_{1}$-score for the semester following this equation:
$$\boldsymbol{\beta}_{weighted} = \sum_{i=0}^{N=11} F_{1}^{T} \boldsymbol{\beta}_{i}
\label{equation:weightedfeats}$$
This then allows for the cumulative weighted features in Fig. \[fig:feat\_cumweights\] and the weighted features in Fig. \[fig:weighted\_feats\].
We also estimate the effects of model features on prediction using the partial dependence [@friedman2001greedy]. A partial dependence plot represents the average model output for a single variable ($S$) across the entire feature space ($C$) in the context of all other features [@friedman2001greedy]. This means that for a specific feature $S$, the exact values of $S$ are first, fixed for all rows in the data set for each unique value of $S$. Then the model is calculated per value. The average contribution of a specific unique value to the overall predicted probability of graduation is then considered the partial dependence. Partial dependence is then assumed to be a continuous function over the entire feature space of $S$. Partial dependence is estimated using the following equation as:
$$\hat{f}_{x_{S}}(x_{s}) = \frac{1}{n} \sum_{i=1}^{n} \hat{f}(x_{S}, x_{C}^{i})$$
The partial dependence is calculated for each semester and is weighted by the $F_{1}$-score for that semester (Fig. \[fig:cumcredits\_pdp\], \[fig:cumgrades\_pdp\]) in a similar fashion to equation \[equation:weightedfeats\].
Partial dependence allows the researcher to see the direct contribution to the predicted probability of a given variable value. Because variables often have many values, they can give fluctuating contributions to the predicted probability. In linear models, it is assumed that variables produce linear contributions to the partial dependence [@friedman2001greedy; @hastie2005elements] and thus coefficients of a linear model represent unit increases [@hastie2005elements; @hosmer2013applied]. For nonlinear models such as gradient boosted trees, this assumption is relaxed and variables can produce nonlinear contributions to the predicted probability.
Results
=======
In this study we have two broad research questions: 1) exploring the effectiveness of a gradient boosted logistic model in comparison to traditionally solved logistic models, and 2) quantifying the components of Tinto’s Theory of Drop Out in the context of who does or does not graduate at this university during the study period. In this section we will first describe the effectiveness of the gradient boosted model in comparison to the traditional maximum likelihood model. Then we will describe the results that are drawn from the gradient boosted model in the context of Tinto’s theory.
Gradient boosting
-----------------
The models in this study attempt to predict whether a student will graduate or not during the observation period of being enrolled for 5-16 semesters. The xgboost model is generally more effective than the logistic regression model except in the last semesters (15-16) studied as assessed by the out-of-sample $F_{1}$-score (Fig. \[fig:f1scores\]). Xgboost is particularly more effective in highly imbalanced case of students graduating (approximately 5% per semester (Fig. \[fig:gradrates\])) within $\leq$8 semesters (Fig. \[fig:f1scores\]). In the more balanced case (semesters 8-14 have approximately 20% of the students eligible to graduate do so (Fig. \[fig:gradrates\])), Xgboost still performs better than the logistic regression model (Fig. \[fig:f1scores\]). In the final semesters (semesters 15-16) logistic regression outperforms xgboost slightly.
Xgboost is more effective at correctly predicting the graduating semester of female students as assessed by the out-of-sample recall for all semesters except for semester 16 (Fig. \[fig:recalls\]). Xgboost is also more effective at correctly predicting the graduating semester of under represented minority students for all semesters (Fig. \[fig:recalls\]). Additionally, Xgboost is able to handle missing data cases better than the logistic regression model for all semesters (Fig. \[fig:recalls\]). Due to the effectiveness of xgboost over logistic regression, the remaining results will focus on the xgboost model.
Effects on Time-to-Graduation
-----------------------------
Tinto’s Theory of Drop Out posits two overarching quantities as governing student choice to drop out or continue to graduation: 1) educational goal commitment, and 2) institutional commitment. Students initially start university with these commitments. These commitments are then mediated by a student’s participating and integration into the academic system and the social system of a given institution. Generally speaking (Fig. \[fig:rawdata\]), graduating students take more credit hours per semester ($N(grad)=15.1\pm0.01$, $N(no-grad)=13.17\pm0.05$), enroll for more semesters ($N(grad)=10.57\pm0.01$, $N(no-grad)=8.79\pm0.05$), are more likely to be white or asian ($N(grad)=86.7\%$, $N(no-grad)=75.3\%$), and more likely to be female ($N(grad)=54.2\%$, $N(no-grad)=48.9\%$).
In this study, we characterize student’s initial commitments as being mediated via the following quantities: high school GPA, math placement score, the number of AP credit hours the student possesses, the median income of the high school the student attended, gender, and race. Preparation is about half as important in predicting when a student graduates in comparison to enrollment factors (Fig. \[fig:feat\_cumweights\]). For early graduating students ($\leq$8 semesters), both math placement scores and high school GPAs were important in predicting that a student will graduate (Fig. \[fig:weighted\_feats\]). While students are likely to have more AP credits on average if they graduate in 8-10 semesters ($AP_{8-10}=4.06\pm 0.03, AP_{other}=2.26\pm 0.01$), having AP credits was not an important indicator for predicting graduation during any semester in comparison to other features (Fig. \[fig:feat\_cumweights\], \[fig:weighted\_feats\]). While white and asian students and female students are more likely to graduate (Fig., \[fig:rawdata\]), demographics are not major indicators for predicting when a student graduates for any semester (Fig. \[fig:feat\_cumweights\], \[fig:weighted\_feats\]). This is also true for the median incomes of the zipcode the students went to high school.
In this study academic integration is represented by a combination of features including the cumulative credit hours, cumulative mean grade; and the per-semester fraction above or below full-time credit hour, mean grade, non-major GPA. A student’s enrollment and performance are the most predictive features when predicting the semester when a student graduates (Fig. \[fig:feat\_cumweights\], \[fig:weighted\_feats\]). A student’s cumulative credit hours is most important to predicting whether a student graduates “on-time” (after 8 semesters) or not and is overall important for prediction. The cumulative credit hours is less important for prediction the further a student’s credits are away from 120 total credit hours (Fig. \[fig:cumcredits\_pdp\]). A student’s average grade is most important for predicting that students will graduate within 10 semesters and is overall important for predicting when a student will graduate (Fig. \[fig:feat\_cumweights\], \[fig:weighted\_feats\]). Having above average grades is important for predicting students who graduate in 9-10 semesters however this is less important for other semesters (Fig. \[fig:cumgrades\_pdp\]).
In this study we use per semester major GPA, major credit hours, cumulative number of majors, and whether a student changed major in that semester or not to represent their social integration into the departments and learning communities associated with their chosen degree program. Student’s performance within their major was not as an important indicator for predicting graduation in comparison to overall grade and cumulative credit hours (Fig. \[fig:feat\_cumweights\]). Changing a major has increased importance for predicting students who graduate later (Fig. \[fig:weighted\_feats\]).
![Out-of-sample $F_{1}$ scores for all models per semester. Xgboost clearly increases the precision and recall tradeoff for the bulk of semesters. Error bars are the bootstrapped standard deviation of the $F_{1}$ score.[]{data-label="fig:f1scores"}](f1scores.pdf)
![Recall scores for sub-populations within the data. The Xgboost model consistently labels women, under represented minorities, and students with missing data better than logistic regression. In Fig. \[fig:f1scores\], there is a large disparity between the logistic regression model and the xgboost model for semesters 5-7. This may be due to the imputation method used with the logistic regression model and the correlation between graduating and missing data during those semesters. However, the gaps in model performance for women, minorities, and missing data in later semesters with no correlation are likely not due to imputation and demonstrate a strong preference for using the xgboost model over the logistic regression model.[]{data-label="fig:recalls"}](groups_recall.pdf)
![Average feature importance for all graduating semesters for the xgboost model. Feature importances are weighted by the $F_{1}$ scores per semester. Enrollment factors and the cumulative average grade are more likely to predict when a student graduates than other factors.[]{data-label="fig:feat_cumweights"}](average_weighted_feature_importance.pdf)
![Xgboost feature importances for predicting if a student will graduate in the next semester. Feature importances are weighted by the $F_{1}$ score calculated from test data. The row-wise sum of these features would produce Fig. \[fig:feat\_cumweights\]. By far the most important feature for graduating “on time” (within 8 semesters) is having enough credit hours. Student preparation is important for graduating “early” ($<$8 semesters). Students who change majors are more likely to graduate later and thus this feature becomes more important in later graduating semesters.[]{data-label="fig:weighted_feats"}](weighted_featureimportance.pdf)
![The partial dependence per semester for the cumulative credit hours a student has obtained. The stronger the partial dependence, the more contribution that value of cumulative credit hours has on the predicted probability that a student will graduate in the given semester. The partial dependence has been weighted by the per semester $F_{1}$ score. Having a total number of credit hours close to 120 is highly predictive of graduating if students graduate between 8 and 10 semesters of enrollment. Outside of this window the impact of the total number of credit hours diminishes. This is likely due to students having additional credit hours that do not count towards a degree such as when they change their major.[]{data-label="fig:cumcredits_pdp"}](cumcredits_pdp.pdf)
![The partial dependence per semester for the cumulative average GPA a student has obtained. The stronger the partial dependence, the more contribution that the cumulative average GPA has on the predicted probability that a student will graduate in the given semester. The partial dependence has been weighted by the per semester $F_{1}$ score. Having a far above average GPA is a major contribution to graduating if a student graduates between semesters 9 and 10. This is likely due to these students never failing a course and having a high commitment to their chosen major.[]{data-label="fig:cumgrades_pdp"}](cumgrades_pdp.pdf)
Discussion
==========
This paper presents two complimentary results: 1) gradient boosting is a useful tool for predicting when students graduate in comparison to traditional statistical algorithms, 2) students who actively integrate into their academic and social communities is the primary effect that predicts when a student graduates thus following Tinto’s Theory. This section will discuss why the xgboost model outperforms logistic regression, the implications that come with the results of the model, and compare it to other studies that have predicted time-to-graduation using Tinto’s Theory of Drop Out and hazard modeling more broadly.
Why does gradient boosting produce better fitted models than maximum likelihood estimation?
-------------------------------------------------------------------------------------------
In almost every case, the xgboost model predicts graduation better than the logistic regression model. This can be for several reasons. First, the xgboost model fitting procedure is a slower iterative procedure (1000 iterations) than the maximum likelihood estimation (typically 5-6 iterations) that the logistic regression model uses. Thus, in each iteration the xgboost model can focus on the local neighborhood of feature details within the training data that the logistic regression model may miss [@nielsen2016tree].
Second, the xgboost model uses a custom imputation engine [@chen2016xgboost]. Whenever data is missing and an learner is using a feature with missing data, the missing data is imputed to be the most likely choice in the decision tree. Because there are many learners, they can account for different local patterns in the data. The data for the logistic regression model is imputed from the mean of the student’s starting year cohort. This mean imputation likely loses information that the xgboost model is able to attend to. It could be that a more sophisticated imputation model will increase the logistic regression model performance. Typically if data missingness correlates with the outcome variable, then data should be imputed to prevent bias due to the missing data [@sterne2009multiple]. In our case, data missingness for both high school GPA and math placement score correlates with early graduation rates (see supplemental Fig. \[fig:sup\_corr\]). Thus the logistic regression model performance would likely increase for semester’s 5-7 with a more sophisticated imputation model. However this would not explain the substantial improvements xgboost makes over the logistic model for women and under represented minorities in later semesters.
Third, xgboost penalizes leafs within the tree learners that are fit on few examples from the training data [@chen2016xgboost]. Additionally, xgboost weights on class labels as well. This is especially useful in the highly unbalanced case of predicting when a student graduates. Because the weight of training data from students who do not graduate is tuned via a grid search, this prevents the xgboost model from overfitting on the majority class simply due to having more representative samples.
Effects on Time-to-Graduation
-----------------------------
Tinto’s theory suggests that students have an initial level of intent to graduate from an institution upon entry [@tinto1975dropout; @braxton2000influence]. This intent is the combination of a students family background (e.g., socio-economic status), individual attributes (e.g., academic ability, race), and pre-college experiences (e.g., high school GPA). This intent is then tempered by the student’s at-college experience such as social integration [@kezar2014higher], financial support [@ishitani2003longitudinal], and academic performance.
### Effects on the initial conditions of educational commitment
Student’s backgrounds can set up a wide variety of contributions to their initial educational and institutional commitments. In this study we assess these initial conditions using a student’s high school GPA, math placement score, the number of AP credit hours the student possesses, the median income of the high school the student attended, gender, and race. This university uses a math placement test to determine a student’s incoming math ability to place them in the appropriate math course. This test was designed at the university and has been used for the entire study period. This test was not designed with psychometrics in mind. Thus, while the test is representative of some measurement of math ability, it is unclear how much the test is representative of math ability. However, we see that a student’s math placement score is the most important attribute that predicts when a student will graduate that is not a performance or enrollment variable \[fig:weighted\_feats\]). This could be similar to [@chen2013stem] evidence that the starting math course is very important to staying in STEM. In [@chen2013stem], students taking lower level math courses at college had increased likelihood of leaving the STEM major they were enrolled in for a different major. In our case, students who take remedial math courses simply have more courses to take and thus must remain at the university longer. Given that students in our study who come from higher income communities are able to remain at university longer before graduating, this indicates that remedial mathematics should be examined more in terms of the cost for a student. It should also be noted that in this study we are predicting all students time to graduation even those who are not pursuing quantitative degrees. The result that math ability has a dominant effect in the context of other demographic and preparation variables is consistent with the literature which claims that math ability has an sizeable effect on student performance at university [@gaertner2014preparing].
A student’s high school GPA is similarly important to a student’s math ability in predicting when they graduate (Fig. \[fig:weighted\_feats\],\[fig:feat\_cumweights\]). This is consistent with some literature that finds that students with higher high school GPAs are more likely to graduate within 4 years [@ishitani2003longitudinal]. [@desjardins1999event] found in a similar study that high school GPA had little to no effect on predicting when a student drops out from university. In both this paper and [@desjardins1999event] the study uses data from university’s that serve primarily students who come from the state the university resides in. Thus this may indicate that the effect of high school GPA on college success is geographically dependent on the quality of high school preparation for university.
Perhaps counter-intuitively having AP credits is not as important as a student’s high school GPA or math ability (Fig. \[fig:feat\_cumweights\]). AP credits directly count for college credits. Students with more AP credit have fewer credit hours necessary to graduate. In this study, having AP credits is an indicator that a student will graduate in four years (8-9 semesters). However it is not a strong indicator that a student will graduate early in comparison to a student’s math placement score and high school GPA. [@desjardins1999event] found that having transfer credits had no effect on on students dropping out. This was also true in a study of physics students who change their majors [@aiken2019modeling]. It could be that AP credits are too random in whether they count for credit or not. A student may choose to take a course anyways they have AP credit for if its in a sequence (e.g., introductory physics) because they may feel ill prepared for the second semester course. It could also be that some students who take AP courses do so with the intent to take a minor or dual major. In this case, the AP courses “free” up more time at university to be able to graduate in four years.
Financial aid is one of the top reasons students leave the university without a degree [@ishitani2003longitudinal]. In this study we do not know if a student has access to financial aid or not. However, we do include the median income for families that live in the zip code of the high school the student attended according to the 2011 census American Community Survey 5- year estimate [@2011census]. This is a rough estimate of the socio-economic status of the high school that the student attends. A course grained measurement like this does not capture all of the nuances of individual students financial support and in our case, we see that this feature was less important in comparison to performance and enrollment features. In our study, we find that coming from higher income high school’s is an indicator that a student will take longer to graduate. [@caldas1997effect] found that the socio-economic status of a student’s peers had an approximately equal effect as a student’s individual socio-economic status on high school student exit examination scores. While this effect is small, it may indicate that students from higher income regions have access to more resources and thus can spend more time in college before entering the work force. This is a similar result to [@ishitani2003longitudinal] which found that students who come from low income households are likely to graduate within 4 years in comparison to students from higher income households.
### Dynamic contributions to a student’s academic integration
A student’s academic integration is defined by Tinto [@tinto1975dropout] as being a combination of performance in university and their intellectual development. In this study we characterize performance through both cumulative GPA measurements and per semester GPA measurements. Further, we split this into major and non-major GPAs.
Performance as measured by GPA has had a demonstrated primary effect on graduating [@desjardins1999event; @chen2012institutional; @yue2017rethinking]. [@desjardins1999event] found that there was a decaying impact on drop out due to GPA. The longer a student was enrolled, the less their GPA was likely to be a strongly influencing factor on dropping out. In our study we find somewhat different results, namely students with above average cumulative GPAs are more likely to graduate on time (Fig. \[fig:cumgrades\_pdp\]). However this effect diminishes with time. It could be that this peak effect is due to two reasons: 1) high performing students are more likely to graduate on time [@yue2017rethinking], and 2) failing a single course significantly sets back both the time to graduate and a student’s cumulative GPA.
There is an interplay between the cumulative credit hours and the cumulative mean grade. In semester 8, the single most predictive feature is cumulative credit hours (Fig. \[fig:weighted\_feats\]). However by semester 10, cumulative credit hours has exchanged the highest rank with the cumulative mean grade. This has a couple implications. First, the decaying impact of grades as noted by [@desjardins1999event] in this case begins later at semester 10. Second, it may be that to graduate within 8 semesters (4 years), there are very few paths other than a strictly laid out course schedule with no deviations and no failing grades. Whereas graduating within 5 years allows more leeway for students to take additional courses that could be due to, for example, receiving a minor.
Second, there is a transition across graduating semesters of what is important (Fig. \[fig:weighted\_feats\]). Early graduation is predicated more by a student’s initial conditions than anything else. By semester 8, the overwhelming effect on successfully predicting students graduating in this semester is their cumulative credit hours. Past semester 8, there is a strong combination of features that are predictive of graduating. This suggests that while Tinto’s theory indicates that there is a dynamic contribution of on-campus interactions to student educational and institutional commitments, these dynamic contributions may not matter that much for students with very strong backgrounds who wish to graduate early.
### Dynamic contributions to a student’s social integration
A student’s social integration is defined by Tinto [@tinto1975dropout] as being a combination of peer group interactions and faculty interactions. In this study we use a course grained measurement of peer group and faculty interactions by measuring the total per semester credit hours a student registers for in their major. Additionally, we note when a student changes their major. Changing a major is a distinct situation where a student will leave their peer and faculty group for a new group and thus may indicate low social integration.
In this study students who attend the university and take a recommended load of courses and pass each course were likely to graduate in 8-10 semesters. When students altered from this path (e.g., a student changes their major) they increased the amount of time it took to graduate or did not graduate during the study period. [@chen2013stem] provides, at least for STEM students, a complementary explanation as to why a student may graduate later. This university has a very large enrollment (typically $>$50000 students), a strong research program, and a strong greek life. In each case these may contribute to social integrations given there is more opportunity at this university than some others for meeting new people, participating in research, or participating in social events.
While not being the largest effect, students who take higher amounts of major credit hours were more likely to graduate in semesters 8 and 9 (Fig. \[fig:weighted\_feats\]). This is sensible given’s Tinto’s theory since taking more major credit hours both works toward’s graduation and is an indication of a strong integration into a peer and faculty group. Within the literature, STEM students who took fewer STEM courses in the first year, were enrolled in less challenging math courses in the first year, and performed poorly in their STEM courses in comparison to non-STEM courses were highly likely to switch majors. Tinto would describe these students as having a reduced educational commitment due to not integrating into the social community and/or lack in the academic engagement of their chosen degree program. In our study we find similar conclusions, higher performance and the number of major credit hours is a more likely indicator of graduating in four years and this effect diminishes over time after the 4 year (8-9 semesters) mark (Fig. \[fig:weighted\_feats\]). Thus, it may be likely that performance in a major and the frequency of major courses taken may be an indicator of lower social and academic engagement described by Tinto [@tinto1975dropout].
In this study we have highlighted that changing a major can have a profound effect on the time it takes to graduate. Changing a major impacts a student’s time-to-graduation and is predictive of students who graduate later (Fig. \[fig:weighted\_feats\]). A student who changes their major could do so due to low academic integration [@chen2013stem]. However this low academic integration could represent low social integration as well [@seymour2000talking]. Student’s who perform poorly may ask themselves if they “belong” in a major. In many cases STEM students who demonstrate high academic performance change their major due to reasons associated with social and personal interactions at university [@seymour2000talking; @chen2013stem]. This especially affects women and under represented minorities [@seymour2000talking]. In this study changing a major does not show a strong correlation with a student’s race or gender ($\rho_{race}=-0.07, \rho_{gender}=-0.03$). In many cases, the experiences that lead to changing a major are more nuanced than a single variable can contain thus while race and gender are a strong component to major change as reported in qualitative interviews [@seymour2000talking], this may not be captured in binary variables. Ultimately, changing majors has been a poorly investigated and deserves more investigation. Future work will look at using Tinto’s theory as a framework for investigating student’s changing their major.
### Limitations on this Study
Given that a perfect $F_{1}$-score is 1 and in this paper we report $F_{1}$-scores around 0.5 at the maximum, it is likely that some of these effects such as financial aid or social integration have large effects on when a student graduates. It is common in some studies, (e.g., [@yue2017rethinking]) to use individual heterogeneity models or “frailty” models to assess unmeasured contributions to graduation [@vaupel1979impact]. In our case, we attempted initially to use gamma distributed frailty terms [@yamaguchi1991event; @hougaard1995frailty]. These proved to provide no increase in model performance thus were removed from subsequent analysis. Future work will consider what contributes to the unobserved heterogeneity (e.g., direct measurements of peer social engagement such as social network centrality).
Additionally, this paper has focused on a student’s educational commitment (per [@tinto1975dropout]) as opposed to their institutional commitment. Much of this is mediated by the fact that 88% of students in the study do in fact graduate, there may be a subset of students who leave simply due to the fact they become disillusioned with the institution and want to pursue a degree elsewhere. Thus there is likely other factors that are not observed that effects when a student a graduates. Many of these factors, such as life events such as family hardship, are necessarily hard to observe for an entire university population. There are also likely factors that are particular to this university that are not common or not impacting at other institutions. Future work will investigate the effect of student social network belonging and the amount of financial aid available to the students.
Conclusion
==========
This paper has presented a discrete time hazard model predicting when a student will graduate. It uses a novel method to calculate the logistic regression called gradient boosting which has been shown to provide better fits than traditional maximum likelihood. While using more sophisticated imputation with the logistic regression model might have increased performance, the ease of use of xgboosts built-in imputation engine provides a strong advantage in it’s use to researchers. Given the utility of gradient boosting in this setting, especially in providing better predictions for under-served populations, this paper recommends that this method be more prevalent in the education research community. Additionally, other methods should be examined such as artificial neural networks [@gonzalez2002artificial].
Additionally, this paper used the partial dependence method to examine two of the model variables. Partial dependence tells us the contributions to the predicted probability of the model for the entire feature space. This method allows us to examine the entirety of continuous variables such as GPA instead of reducing them to a single value associated with a model coefficient. This method too should see much broader use in the education research community.
This paper follows Tinto’s theory of drop out that predicts social integration and college participation are more likely to impact a student’s commitment to graduation than second order effects such as preparation. Future work will connect student academic performance, participation in course work, with social network metrics and financial aid information. Future work will also examine the specific effects that remedial math have on student retention in a major, their time to graduation, and if participating in remedial math courses lowers the likelihood to graduate.
Acknowledgements
================
This project was supported by the Michigan State University College of Natural Sciences including the STEM Gateway Fellowship and the Lappan-Phillips Foundation, the Association of American Universities, and the Norwegian Agency for Quality Assurance in Education (NOKUT), which supports the Center for Computing in Science Education. This project has also received support from the INTPART project of the Research Council of Norway (Grant No. 288125). Morten Hjorth-Jensen is supported by the U.S. National Science Foundation (Grant No. PHY-1404159).
Supplemental
============
![The fraction of missing data and the Spearman rank correlation between whether data is missing and the outcome variable of graduating the following semester. There is a small correlation in the early semesters (5-7) between whether students are graduating and if they have missing data or not.[]{data-label="fig:sup_corr"}](sup_corr.pdf)
|
---
abstract: 'We introduce the concept of *Conversion/Preference Games*, or CP games for short. CP games generalize the standard notion of *strategic games*. First we exemplify the use of CP games. Second we formally introduce and define the CP-games formalism. Then we sketch two ‘real-life’ applications, namely a connection between CP games and gene regulation networks, and the use of CP games to formalize implied information in Chinese Wall security. We end with a study of a particular fixed-point construction over CP games and of the resulting existence of equilibria in possibly infinite games.'
author:
- Stéphane Le Roux
- |
Pierre Lescanne\
Université de Lyon, CNRS (LIP), ENS de Lyon,\
46 allée d’Italie, 69364 Lyon, France\
- |
René Vestergaard\
School of Information Science, JAIST, 1-1 Asahidai,\
Nomi, Ishikawa 923-1292, Japan
title: 'Conversion/Preference Games'
---
Introduction {#sec:intro}
============
We give a stand-alone account of Conversion/Preference games or CP games, as originally used in [@LeRouxLescanneVestergaard:RGT-Nash]. CP games are built from a set of *players* and a set of (game) *situations*. The ability of the players to change a situation to another is formalised in *conversion* relations. A *preference* relation dictates how the players compare the different situations against each other. The three main aims of this article are to show i) that discrete Nash-style game theory is possible and natural, ii) that the two basic CP concepts of *Conversion* and *Preference* are of wider interest, and iii) that game-theoretic notions are both applicable and relevant in situations where no payoff function need exist, or where the payoff concept would dramatically alter what aspects of the game are being considered.
Basic concepts {#sec:basic}
==============
To start with, let us give the two main notions of games. First, a game involves *players*. Second, a game is characterized by situations. In CP games these situations will be called *situations* or sometimes *synopses* or *game situations*. A player can move from one situation to another, but she[^1] does that under some constraints as she has no total freedom to perform her moves, therefore a relation called *conversion* is defined for each player; it tells what moves a player is allowed to perform. Conversion of player [*Alice*]{} will be written ${"-->"_{{\textit{Alice}}}}$. As such, conversion tells basically the rules of the game. In chess it would say *“a player can move her bishop along a diagonal”*, but it does not tell the game line of the player. In other words it does not tell why the player chooses to move or to “convert” her situation. Another relation called *preference* compares situations in order for a player to choose a « better » move or to perform a « better » conversion. Preference of player [*Beth*]{} will be written ${".>"_{{\textit{Beth}}}}$ and when we write $s{".>"_{{\textit{Beth}}}}s'$ we mean that [*Beth*]{} prefers $s'$ to $s$ or, rather than $s$, she chooses $s'$ or in situation $s$ she is attracted toward situation $s'$. Preference (or choice) is somewhat disconnected from conversion, a player can clearly prefer a situation she cannot move to and vice versa she can move to a situation she does not prefer. Moreover players may share the same conversion relation, but this not a rule and the may share or not the same preference relation or not. Those different situation sill be illustrated by examples throughout the article.
A key concept in games is this of *equilibrium*. As a player can convert a situation, she can convert it into a situation she likes better, in the sense that she prefers the new situation she converted to. A player is *happy* in a situation, if there is no situation she can convert into and she prefers. A situation is an equilibrium if each player is happy with this situation. We will see that this concept of equilibrium captures and generalizes the concept known as *Nash equilibrium* in strategic games, hence the name *abstract Nash equilibrium*.
Some examples {#sec:examples}
=============
Let us present the above concepts of conversion, preference and equilibrium through examples. We will introduce a new concept called *change of mind*.
A simple game on a square {#sec:square-game}
-------------------------
As an introduction, we will look at variations of a simple game on a board.
### A first version
Imagine a simple game where [*Alice*]{} and [*Beth*]{} play using tokens on a square. We number the four positions as $1$, $2$, $3$ and $4$.
$$\xymatrix { &*++[o][F-]{1}\ar@{-}[dl]\ar@{-}[dr]\\ *++[o][F-]{4}
\ar@{-}[dr]&& *++[o][F-]{2}\ar@{-}[dl]\\ &*++[o][F-]{3} } \qquad
\qquad \qquad \xymatrix {
&*++[o][F-]{{{\color{red} `(!)}}}\ar@{-}[dl]\ar@{-}[dr]\\
*++[o][F-]{\phantom{`(!)}} \ar@{-}[dr]&&
*++[o][F-]{{\textcolor{blue}{`[!]}}}\ar@{-}[dl]\\ &*++[o][F-]{\phantom{`(!)}} }$$
Assume that player [*Alice*]{} has a red round token and that player [*Beth*]{} has a blue squared token. The two players place their tokens on vertices and then they move along edges. They can also decide not to move. Assume that [*Alice*]{} and [*Beth*]{} never put their token on a vertex taken by the other player and a position further than this impossible situation is better than a position closer. In other words, a position with Alice on vertex $i$ and Beth on vertex $j$ with $i-j$ even is preferred to a position with $i-j$ odd.
$
\begin{psmatrix}[colsep=.4cm,rowsep=.4cm]
&&&& [name=42]{\mathbf{4|2}}\\
&&& [name=43]{\mathbf{4|3}} && [name=41]{\mathbf{4|1}}\\
&& [name=12]{\mathbf{1|2}} &&&& [name=32]{\mathbf{3|2}} \\
& [name=13]{\mathbf{1|3}} &&&&&& [name=31]{\mathbf{3|1}} \\
&& [name=14]{\mathbf{1|4}} &&&& [name=34]{\mathbf{3|4}} \\
&&& [name=23]{\mathbf{2|3}} && [name=21]{\mathbf{2|1}} \\
&&&& [name=24]{\mathbf{2|4}}
\ncarc[arrows=<->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=-30]{42}{41}
\ncarc[arrows=<->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=30]{42}{43}
\ncarc[arrows=<->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=-30]{12}{13}
\ncarc[arrows=<->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=-30]{13}{14}
\ncarc[arrows=<->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=30]{32}{31}
\ncarc[arrows=<->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=30]{31}{34}
\ncarc[arrows=<->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=30]{23}{24}
\ncarc[arrows=<->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=-30]{21}{24}
\ncarc[arrows=<->,linewidth=.03,linestyle=dashed,linecolor=red,arcangle=-40]{42}{12}
\ncarc[arrows=<->,linewidth=.03,linestyle=dashed,linecolor=red,arcangle=40]{42}{32}
\ncarc[arrows=<->,linewidth=.03,linestyle=dashed,linecolor=red,arcangle=-40]{13}{43}
\ncarc[arrows=<->,linewidth=.03,linestyle=dashed,linecolor=red,arcangle=40]{31}{41}
\ncarc[arrows=<->,linewidth=.03,linestyle=dashed,linecolor=red,arcangle=40]{13}{23}
\ncarc[arrows=<->,linewidth=.03,linestyle=dashed,linecolor=red,arcangle=-40]{31}{21}
\ncarc[arrows=<->,linewidth=.03,linestyle=dashed,linecolor=red,arcangle=-40]{14}{24}
\ncarc[arrows=<->,linewidth=.03,linestyle=dashed,linecolor=red,arcangle=40]{34}{24}
\end{psmatrix}
\quad
\begin{psmatrix}[colsep=.4cm,rowsep=.4cm]
&&&& [name=42]{\mathbf{4|2}}\\
&&& [name=43]{\mathbf{4|3}} && [name=41]{\mathbf{4|1}}\\
&& [name=12]{\mathbf{1|2}} &&&& [name=32]{\mathbf{3|2}} \\
& [name=13]{\mathbf{1|3}} &&&&&& [name=31]{\mathbf{3|1}} \\
&& [name=14]{\mathbf{1|4}} &&&& [name=34]{\mathbf{3|4}} \\
&&& [name=23]{\mathbf{2|3}} && [name=21]{\mathbf{2|1}} \\
&&&& [name=24]{\mathbf{2|4}}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=-30]{12}{13}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=60]{12}{42}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=-30]{12}{31}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=30]{12}{24}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=-30]{43}{13}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=40]{43}{42}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=-30]{43}{31}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=30]{43}{24}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=30]{41}{13}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=-60]{41}{42}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=30]{41}{31}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=-30]{41}{24}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=30]{32}{13}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=-40]{32}{42}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=30]{32}{31}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=-30]{32}{24}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=-30]{34}{13}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=60]{34}{42}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=-30]{34}{31}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=30]{34}{24}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=-30]{21}{13}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=40]{21}{42}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=-30]{21}{31}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=30]{21}{24}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=30]{14}{13}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=-60]{14}{42}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=30]{14}{31}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=-30]{14}{24}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=30]{23}{13}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=-40]{23}{42}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=30]{23}{31}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=-30]{23}{24}
\end{psmatrix}
$
$$\begin{psmatrix}[colsep=.4cm,rowsep=.4cm]
&&&& [name=42]{\mathbf{4|2}}\\
&&& [name=43]{\mathbf{4|3}} && [name=41]{\mathbf{4|1}}\\
&& [name=12]{\mathbf{1|2}} &&&& [name=32]{\mathbf{3|2}} \\
& [name=13]{\mathbf{1|3}} &&&&&& [name=31]{\mathbf{3|1}} \\
&& [name=14]{\mathbf{1|4}} &&&& [name=34]{\mathbf{3|4}} \\
&&& [name=23]{\mathbf{2|3}} && [name=21]{\mathbf{2|1}} \\
&&&& [name=24]{\mathbf{2|4}}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,arcangle=-30]{12}{13}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,arcangle=30]{14}{13}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,arcangle=-30]{21}{24}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,arcangle=30]{23}{24}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,arcangle=30]{32}{31}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,arcangle=-30]{34}{31}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,arcangle=-30]{43}{42}
\ncarc[arrows=->,linewidth=.03,linecolor=red,arcangle=40]{12}{42}
\ncarc[arrows=->,linewidth=.03,linecolor=red,arcangle=-40]{14}{24}
\ncarc[arrows=->,linewidth=.03,linecolor=red,arcangle=40]{21}{31}
\ncarc[arrows=->,linewidth=.03,linecolor=red,arcangle=40]{43}{13}
\ncarc[arrows=->,linewidth=.03,linecolor=red,arcangle=-40]{23}{13}
\ncarc[arrows=->,linewidth=.03,linecolor=red,arcangle=-40]{32}{42}
\ncarc[arrows=->,linewidth=.03,linecolor=red,arcangle=40]{34}{24}
\ncarc[arrows=->,linewidth=.03,linecolor=red,arcangle=-40]{41}{31}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,arcangle=30]{41}{42}
\end{psmatrix}
\qquad
\begin{psmatrix}[colsep=.4cm,rowsep=.4cm]
&&&& [name=42]{\mathbf{4|2}}\\
&&& [name=43]{\mathbf{4|3}} && [name=41]{\mathbf{4|1}}\\
&& [name=12]{\mathbf{1|2}} &&&& [name=32]{\mathbf{3|2}} \\
& [name=13]{\mathbf{1|3}} &&&&&& [name=31]{\mathbf{3|1}} \\
&& [name=14]{\mathbf{1|4}} &&&& [name=34]{\mathbf{3|4}} \\
&&& [name=23]{\mathbf{2|3}} && [name=21]{\mathbf{2|1}} \\
&&&& [name=24]{\mathbf{2|4}}
\ncarc[arrows=->,linewidth=.03,arcangle=-30]{12}{13}
\ncarc[arrows=->,linewidth=.03,arcangle=30]{14}{13}
\ncarc[arrows=->,linewidth=.03,arcangle=-30]{21}{24}
\ncarc[arrows=->,linewidth=.03,arcangle=30]{23}{24}
\ncarc[arrows=->,linewidth=.03,arcangle=30]{32}{31}
\ncarc[arrows=->,linewidth=.03,arcangle=-30]{34}{31}
\ncarc[arrows=->,linewidth=.03,arcangle=-30]{43}{42}
\ncarc[arrows=->,linewidth=.03,arcangle=40]{12}{42}
\ncarc[arrows=->,linewidth=.03,arcangle=-40]{14}{24}
\ncarc[arrows=->,linewidth=.03,arcangle=40]{21}{31}
\ncarc[arrows=->,linewidth=.03,arcangle=40]{43}{13}
\ncarc[arrows=->,linewidth=.03,arcangle=-40]{23}{13}
\ncarc[arrows=->,linewidth=.03,arcangle=-40]{32}{42}
\ncarc[arrows=->,linewidth=.03,arcangle=40]{34}{24}
\ncarc[arrows=->,linewidth=.03,arcangle=-40]{41}{31}
\ncarc[arrows=->,linewidth=.03,arcangle=30]{41}{42}
\end{psmatrix}$$
The game has 12 situations, which we write $i|j$ for ${1\le i,j\le 4}$ and $i\not=j$. The above pictured situation corresponds to $1|2$. The two conversions are described by Figure \[fig:conv\_pref\_square\] left. In this figure, ${{\color{red} \dashrightarrow}}$ is [*Alice*]{}’s conversion and ${\textcolor{blue}{\dashrightarrow}}$ is [*Beth*]{}’s conversion.
In this game, both players share the same preference, namely the following: since a player does not want her token on a position next to the other token, she prefers a situation where her token is on the opposite corner of the other token. This gives the preference given in Figure \[fig:conv\_pref\_square\] right. The arrow from $1|2$ to $1|3$ means players prefer $1|3$ to $1|2$.
From the conversion and the preference we build a relation that we call *change of mind*. [*Alice*]{} can change her mind from a situation $s$ to a new one $s'$, if she can convert $s$ into the new situation $s'$ and rather than $s$ she chooses $s'$. Changes of mind for [*Alice*]{} and [*Beth*]{} are given in Figure \[fig:chg\_of\_mind\_square\] on the left. In this figure, ${{\color{red} \xymatrix{\ar@{->}[r]&}}}$ is [*Alice*]{}’s change of mind and ${\textcolor{blue}{\xymatrix{\ar@{->}[r]&}}}$ is [*Beth*]{}’s conversion. The *(general) change of mind* is the union of the *agent change of mind*, it is given by Figure \[fig:chg\_of\_mind\_square\] on the right. The equilibria are the end points (or “minimal point”) for that relation, namely $1|3$, $4|2$, $3|1$ and $2|4$. This means that no change of mind arrows leave those nodes. In these situations players have their tokens on opposite corners and they do not move. An equilibrium like $1|3$ which is an end point is called an *Abstract Nash Equilibrium*.
$$\begin{psmatrix}[colsep=.4cm,rowsep=.4cm]
&&&& [name=42]{\mathbf{4|2}}\\
&&& [name=43]{\mathbf{4|3}} && [name=41]{\mathbf{4|1}}\\
&& [name=12]{\mathbf{1|2}} &&&& [name=32]{\mathbf{3|2}} \\
& [name=13]{\mathbf{1|3}} &&&&&& [name=31]{\mathbf{3|1}} \\
&& [name=14]{\mathbf{1|4}} &&&& [name=34]{\mathbf{3|4}} \\
&&& [name=23]{\mathbf{2|3}} && [name=21]{\mathbf{2|1}} \\
&&&& [name=24]{\mathbf{2|4}}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=30]{41}{42}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=30]{42}{43}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=-30]{12}{13}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=-30]{13}{14}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=-30]{31}{32}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=-30]{34}{31}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=30]{23}{24}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=30]{24}{21}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=red,arcangle=-40]{42}{12}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=red,arcangle=-40]{32}{42}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=red,arcangle=40]{43}{13}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=red,arcangle=40]{31}{41}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=red,arcangle=40]{13}{23}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=red,arcangle=40]{21}{31}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=red,arcangle=-40]{14}{24}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=red,arcangle=-40]{24}{34}
\end{psmatrix}
\quad
\begin{psmatrix}[colsep=.4cm,rowsep=.4cm]
&&&& [name=42]{\mathbf{4|2}}\\
&&& [name=43]{\mathbf{4|3}} && [name=41]{\mathbf{4|1}}\\
&& [name=12]{\mathbf{1|2}} &&&& [name=32]{\mathbf{3|2}} \\
& [name=13]{\mathbf{1|3}} &&&&&& [name=31]{\mathbf{3|1}} \\
&& [name=14]{\mathbf{1|4}} &&&& [name=34]{\mathbf{3|4}} \\
&&& [name=23]{\mathbf{2|3}} && [name=21]{\mathbf{2|1}} \\
&&&& [name=24]{\mathbf{2|4}}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=30]{41}{42}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=30]{41}{43}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=30]{42}{43}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=-30]{12}{13}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=30]{12}{14}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=-30]{13}{14}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=-30]{31}{32}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=-30]{34}{31}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=30]{34}{32}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=30]{23}{24}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=30]{23}{21}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=30]{24}{21}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=-40]{42}{12}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=-40]{32}{42}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=30]{32}{12}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=40]{43}{13}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=40]{43}{23}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=40]{31}{41}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=40]{13}{23}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=40]{21}{31}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=40]{21}{41}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=-40]{14}{24}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=40]{14}{34}
\ncarc[arrows=->,linewidth=.03,linestyle=dotted,arcangle=-40]{24}{34}
\end{psmatrix}$$ $$\begin{psmatrix}[colsep=.4cm,rowsep=.4cm]
&&&& [name=42]{\mathbf{4|2}}\\
&&& [name=43]{\mathbf{4|3}} && [name=41]{\mathbf{4|1}}\\
&& [name=12]{\mathbf{1|2}} &&&& [name=32]{\mathbf{3|2}} \\
& [name=13]{\mathbf{1|3}} &&&&&& [name=31]{\mathbf{3|1}} \\
&& [name=14]{\mathbf{1|4}} &&&& [name=34]{\mathbf{3|4}} \\
&&& [name=23]{\mathbf{2|3}} && [name=21]{\mathbf{2|1}} \\
&&&& [name=24]{\mathbf{2|4}}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,arcangle=30]{41}{42}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,arcangle=30]{42}{43}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,arcangle=-30]{12}{13}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,arcangle=-30]{13}{14}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,arcangle=-30]{31}{32}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,arcangle=-30]{34}{31}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,arcangle=30]{23}{24}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,arcangle=30]{24}{21}
\ncarc[arrows=->,linewidth=.03,linecolor=red,arcangle=-40]{42}{12}
\ncarc[arrows=->,linewidth=.03,linecolor=red,arcangle=-40]{32}{42}
\ncarc[arrows=->,linewidth=.03,linecolor=red,arcangle=40]{43}{13}
\ncarc[arrows=->,linewidth=.03,linecolor=red,arcangle=40]{31}{41}
\ncarc[arrows=->,linewidth=.03,linecolor=red,arcangle=40]{13}{23}
\ncarc[arrows=->,linewidth=.03,linecolor=red,arcangle=40]{21}{31}
\ncarc[arrows=->,linewidth=.03,linecolor=red,arcangle=-40]{14}{24}
\ncarc[arrows=->,linewidth=.03,linecolor=red,arcangle=-40]{24}{34}
\end{psmatrix}
\quad
\begin{psmatrix}[colsep=.4cm,rowsep=.4cm]
&&&& [name=42]{\mathbf{4|2}}\\
&&& [name=43]{\mathbf{4|3}} && [name=41]{\mathbf{4|1}}\\
&& [name=12]{\mathbf{1|2}} &&&& [name=32]{\mathbf{3|2}} \\
& [name=13]{\mathbf{1|3}} &&&&&& [name=31]{\mathbf{3|1}} \\
&& [name=14]{\mathbf{1|4}} &&&& [name=34]{\mathbf{3|4}} \\
&&& [name=23]{\mathbf{2|3}} && [name=21]{\mathbf{2|1}} \\
&&&& [name=24]{\mathbf{2|4}}
\ncarc[arrows=->,linewidth=.03,arcangle=30]{41}{42}
\ncarc[arrows=->,linewidth=.03,arcangle=30]{42}{43}
\ncarc[arrows=->,linewidth=.03,arcangle=-30]{12}{13}
\ncarc[arrows=->,linewidth=.03,arcangle=-30]{13}{14}
\ncarc[arrows=->,linewidth=.03,arcangle=-30]{31}{32}
\ncarc[arrows=->,linewidth=.03,arcangle=-30]{34}{31}
\ncarc[arrows=->,linewidth=.03,arcangle=30]{23}{24}
\ncarc[arrows=->,linewidth=.03,arcangle=30]{24}{21}
\ncarc[arrows=->,linewidth=.03,arcangle=-40]{42}{12}
\ncarc[arrows=->,linewidth=.03,arcangle=-40]{32}{42}
\ncarc[arrows=->,linewidth=.03,arcangle=40]{43}{13}
\ncarc[arrows=->,linewidth=.03,arcangle=40]{31}{41}
\ncarc[arrows=->,linewidth=.03,arcangle=40]{13}{23}
\ncarc[arrows=->,linewidth=.03,arcangle=40]{21}{31}
\ncarc[arrows=->,linewidth=.03,arcangle=-40]{14}{24}
\ncarc[arrows=->,linewidth=.03,arcangle=-40]{24}{34}
\end{psmatrix}$$
### A second version
We propose a second version of the game, where moves of the token can only be made clockwise. This implies to change the conversion changes, but also the preference, as a player does want not to be threatened by another token placed before hers clockwise and prefers a situation that places this token as far as possible. The conversions, the preferences and the changes of mind are given in Figure \[fig:conv\_pref\_square2\] (page ). If one looks at the equilibrium, one sees that there is no fixed position where players are happy. To be happy the players have to move around for ever, one chasing the other. It is not really a cycle, but a perpetual move. We also call that an equilibrium. It is sometimes called a *dynamic equilibrium* or a *stationary state*.
### A third version
The third version is meant to present an interesting feature of the change of mind. In this version, we use the same rules as the second one, except that we suppose that the game does not start with both token on the board. Actually it starts as follows. [*Alice*]{} has put her token on node $1$ (this game positions is described as $1|`w$). Then [*Beth*]{} chooses a position among $2$, $3$ or $4$. The conversion is given in Figure \[fig:conv\_CoM\_square3\] left. [*Beth*]{} may choose not to play, but in this case she loses, in other words, she prefers any position to $1|`w$. We do not draw the preference relation, as it would make for an entangled picture. The change of mind is given on Figure \[fig:conv\_CoM\_square3\] right (page ). There is again a dynamic equilibrium and one sees that this dynamic equilibrium is not the whole game, indeed one enters the perpetual move after at least one step in the game.
$
\begin{psmatrix}[colsep=.35cm,rowsep=.35cm]
&&&&& [name=42]{\mathbf{4|2}}\\
&&&& [name=43]{\mathbf{4|3}} && [name=41]{\mathbf{4|1}}\\
&&& [name=12]{\mathbf{1|2}} &&&& [name=32]{\mathbf{3|2}} \\
&[name=1w]{\mathbf{1|\omega}}& [name=13]{\mathbf{1|3}} &&&&&& [name=31]{\mathbf{3|1}} \\
&&& [name=14]{\mathbf{1|4}} &&&& [name=34]{\mathbf{3|4}} \\
&&&& [name=23]{\mathbf{2|3}} && [name=21]{\mathbf{2|1}} \\
&&&&& [name=24]{\mathbf{2|4}}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=50]{1w}{12}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=blue]{1w}{13}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=-50]{1w}{14}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=30]{41}{42}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=30]{42}{43}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=-30]{12}{13}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=-30]{13}{14}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=-30]{31}{32}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=-30]{34}{31}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=30]{23}{24}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=blue,arcangle=30]{24}{21}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=red,arcangle=-40]{42}{12}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=red,arcangle=-40]{32}{42}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=red,arcangle=40]{43}{13}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=red,arcangle=40]{31}{41}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=red,arcangle=40]{13}{23}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=red,arcangle=40]{21}{31}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=red,arcangle=-40]{14}{24}
\ncarc[arrows=->,linewidth=.03,linestyle=dashed,linecolor=red,arcangle=-40]{24}{34}
\end{psmatrix}
\begin{psmatrix}[colsep=.35cm,rowsep=.35cm]
&&&&& [name=42]{\mathbf{4|2}}\\
&&&& [name=43]{\mathbf{4|3}} && [name=41]{\mathbf{4|1}}\\
&&& [name=12]{\mathbf{1|2}} &&&& [name=32]{\mathbf{3|2}} \\
&[name=1w]{\mathbf{1|\omega}}& [name=13]{\mathbf{1|3}} &&&&&& [name=31]{\mathbf{3|1}} \\
&&& [name=14]{\mathbf{1|4}} &&&& [name=34]{\mathbf{3|4}} \\
&&&& [name=23]{\mathbf{2|3}} && [name=21]{\mathbf{2|1}} \\
&&&&& [name=24]{\mathbf{2|4}}
\ncarc[arrows=->,linewidth=.03,arcangle=50]{1w}{12}
\ncarc[arrows=->,linewidth=.03]{1w}{13}
\ncarc[arrows=->,linewidth=.03,arcangle=-50]{1w}{14}
\ncarc[arrows=->,linewidth=.03,arcangle=30]{41}{42}
\ncarc[arrows=->,linewidth=.03,arcangle=30]{42}{43}
\ncarc[arrows=->,linewidth=.03,arcangle=-30]{12}{13}
\ncarc[arrows=->,linewidth=.03,arcangle=-30]{13}{14}
\ncarc[arrows=->,linewidth=.03,arcangle=-30]{31}{32}
\ncarc[arrows=->,linewidth=.03,arcangle=-30]{34}{31}
\ncarc[arrows=->,linewidth=.03,arcangle=30]{23}{24}
\ncarc[arrows=->,linewidth=.03,arcangle=30]{24}{21}
\ncarc[arrows=->,linewidth=.03,arcangle=-40]{42}{12}
\ncarc[arrows=->,linewidth=.03,arcangle=-40]{32}{42}
\ncarc[arrows=->,linewidth=.03,arcangle=40]{43}{13}
\ncarc[arrows=->,linewidth=.03,arcangle=40]{31}{41}
\ncarc[arrows=->,linewidth=.03,arcangle=40]{13}{23}
\ncarc[arrows=->,linewidth=.03,arcangle=40]{21}{31}
\ncarc[arrows=->,linewidth=.03,arcangle=-40]{14}{24}
\ncarc[arrows=->,linewidth=.03,arcangle=-40]{24}{34}
\end{psmatrix}
$
Strategic games {#sec:strat_games}
---------------
In this presentation of strategic games we do not use payoff functions, but directly a preference relation (See Section 1.1.2 of [@osborne04a] for a discussion) and we present several games.
### The Prisoner’s Dilemma
The problem is stated usually as follows
> Two suspects, A and B, are arrested by the police. The police have insufficient evidence for a conviction, and, having separated both prisoners, visit each of them to offer the same deal: if one acts as an informer against the other (*finks*) and the other remains *quiet*, the betrayer goes free and the quiet accomplice receives the full sentence. If both stay quiet, the police can sentence both prisoners to a reduced sentence in jail for a minor charge. If each finks, each will receive a similar intermediate sentence. Each prisoner must make the choice of whether to fink or to remain quiet. However, neither prisoner knows for sure what choice the other prisoner will make. So the question this dilemma poses is: What will happen? How will the prisoners act?
Each prisoner can be into two states, either *fink* ($F$) or be *quiet* ($Q$). Each prisoner can go from $Q$ to $F$ and vice-versa, hence the following conversion, where ${{{\color{red} \dashrightarrow}}}$ is prisoner A conversion and ${{\textcolor{blue}{\dashrightarrow}}}$ is prisoner B conversion (Figure \[fig:prisoner\_conv\_pref\] left). Each prisoner prefers to go free over being sentenced and prefers a light sentence to a full sentence. Hence the preference are given in Figure \[fig:prisoner\_conv\_pref\] right, where ${\raisebox{2pt}{\hspace{-30pt}
\begin{psmatrix}
&[name=a]&[name=b]\ncline[arrows=->,linewidth=.04,linecolor=red,linestyle=dotted,arrowscale=.5]{a}{b}
\end{psmatrix}
}}$ is prisoner A preference and ${\raisebox{2pt}{\hspace{-23pt}
\begin{psmatrix}
&[name=a]&[name=b]\ncline[arrows=->,linewidth=.015,linecolor=blue,linestyle=dotted,arrowscale=.5]{a}{b}
\end{psmatrix}
}}$ is prisoner B preference.
$$\begin{psmatrix}[colsep=2cm,rowsep=2cm]
&[name=QQ]{Q,Q} & [name=QF]{Q,F}\\
&[name=FQ]{F,Q} & [name=FF]{F,F} \psset{nodesep=5pt}
\ncline[arrows=<->,linewidth=.03,linecolor=blue,linestyle=dashed]{QQ}{QF}
\ncline[arrows=<->,linewidth=.03,linecolor=blue,linestyle=dashed]{FQ}{FF}
\ncline[arrows=<->,linewidth=.08,linecolor=red,linestyle=dashed]{QQ}{FQ}
\ncline[arrows=<->,linewidth=.08,linecolor=red,linestyle=dashed]{QF}{FF}
\end{psmatrix}
\qquad
\begin{psmatrix}[colsep=2cm,rowsep=2cm]
&[name=QQ]{Q,Q} & [name=QF]{Q,F}\\
&[name=FQ]{F,Q} & [name=FF]{F,F} \psset{nodesep=3.5pt}
\ncline[arrows=->,linewidth=.03,linecolor=blue,linestyle=dotted]{QQ}{QF}
\ncarc[arrows=->,linewidth=.08,linecolor=red,linestyle=dotted,
arcangle=-30]{QF}{QQ}
\ncline[arrows=->,linewidth=.03,linecolor=blue,linestyle=dotted]{FQ}{FF}
\ncarc[arrows=->,linewidth=.08,linecolor=red,linestyle=dotted,
arcangle=30]{FF}{FQ}
\ncline[arrows=->,linewidth=.08,linecolor=red,linestyle=dotted]{QQ}{FQ}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,linestyle=dotted,
arcangle=30]{FQ}{QQ}
\ncline[arrows=->,linewidth=.08,linecolor=red,linestyle=dotted]{QF}{FF}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,linestyle=dotted,
arcangle=-30]{FF}{QF}
\ncarc[arrows=->,linewidth=.08,linecolor=red,linestyle=dotted,
arcangle=-25]{FF}{QQ}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,linestyle=dotted,
arcangle=25]{FF}{QQ}
\ncarc[arrows=->,linewidth=.08,linecolor=red,linestyle=dotted,
arcangle=30]{QF}{FQ}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,linestyle=dotted,
arcangle=30]{FQ}{QF}
\end{psmatrix}
\qquad$$
From this we get the change of mind of Figure \[fig:prisoner\_CoM\]. One sees clearly that the only equilibrium is $F,F$ despite both prefer $Q,Q$ as shown on Figure \[fig:prisoner\_conv\_pref\] right.
$$\begin{psmatrix}[colsep=2cm,rowsep=2cm]
&[name=QQ]{Q,Q} & [name=QF]{Q,F}\\
&[name=FQ]{F,Q} & [name=FF]{F,F}
\psset{nodesep=5pt}
\ncline[arrows=->,linewidth=.03,linecolor=blue]{QQ}{QF}
\ncline[arrows=->,linewidth=.03,linecolor=blue]{FQ}{FF}
\ncline[arrows=->,linewidth=.08,linecolor=red]{QQ}{FQ}
\ncline[arrows=->,linewidth=.08,linecolor=red]{QF}{FF}
\end{psmatrix}
\quad
\begin{psmatrix}[colsep=2cm,rowsep=2cm]
&[name=QQ]{Q,Q} & [name=QF]{Q,F}\\
&[name=FQ]{F,Q} & [name=FF]{\color{green}F,F}
\psset{nodesep=5pt}
\ncline[arrows=->,linewidth=.03]{QQ}{QF}
\ncline[arrows=->,linewidth=.03]{FQ}{FF}
\ncline[arrows=->,linewidth=.03]{QQ}{FQ}
\ncline[arrows=->,linewidth=.03]{QF}{FF}
\end{psmatrix}$$
Such an equilibrium is called a Nash equilibrium in strategic game theory.
The paradox comes from the fact that $F,F$ is an equilibrium despite the fact one has:
$
\begin{psmatrix}[colsep=1.5cm,rowsep=1.5cm] [name=QQ]{Q,Q} &
[name=FF]{F,F}
\psset{nodesep=3.5pt}
\ncarc[arrows=->,linewidth=.06,linecolor=red,linestyle=dotted, arcangle=-15]{FF}{QQ}
\ncarc[arrows=->,linewidth=.025,linecolor=blue,linestyle=dotted, arcangle=15]{FF}{QQ}
\end{psmatrix}
$
in the preference.
### Matching Pennies
This second example is also classic. This is a simple example of strategic game where there is no singleton equilibrium. As an equilibrium can contain more than one situation, we call singleton equilibrium a CP equilibrium which contains only one situation. This boils down to the kind of equilibrium we have introduced so far.
> The game is played between two players, Player A and Player B. Each player has a penny and must secretly turn the penny to heads ($H$) or tails ($T$). The players then reveal their choices simultaneously. If the pennies match (both heads or both tails), Player A wins. If the pennies do not match (one heads and one tails), Player B wins.
The conversion is similar to this of the prisoner’s dilemma (Figure \[fig:match\_pen\_conv\_pref\_CoM\] left) and the preference is given by who wins (Figure \[fig:match\_pen\_conv\_pref\_CoM\] center).
$$\begin{psmatrix}[colsep=2cm,rowsep=2cm]
& [name=HH]{H,H} & [name=HT]{H,T}\\
& [name=TH]{T,H} & [name=TT]{T,T}
\psset{nodesep=5pt}
\ncline[arrows=<->,linewidth=.03,linecolor=blue,linestyle=dashed]{HH}{HT}
\ncline[arrows=<->,linewidth=.03,linecolor=blue,linestyle=dashed]{TH}{TT}
\ncline[arrows=<->,linewidth=.08,linecolor=red,linestyle=dashed]{HH}{TH}
\ncline[arrows=<->,linewidth=.08,linecolor=red,linestyle=dashed]{HT}{TT}
\end{psmatrix}
\qquad
\begin{psmatrix}[colsep=2cm,rowsep=2cm]
& [name=HH]{H,H} & [name=HT]{H,T}\\
& [name=TH]{T,H} & [name=TT]{T,T}
\psset{nodesep=5pt}
\ncline[arrows=->,linewidth=.03,linecolor=blue,linestyle=dotted]{HH}{HT}
\ncline[arrows=->,linewidth=.03,linecolor=blue,linestyle=dotted]{TT}{TH}
\ncline[arrows=->,linewidth=.08,linecolor=red,linestyle=dotted]{TH}{HH}
\ncline[arrows=->,linewidth=.08,linecolor=red,linestyle=dotted]{HT}{TT}
\ncarc[arrows=->,linewidth=.08,linecolor=red,linestyle=dotted,arcangle=30]{HT}{HH}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,linestyle=dotted,arcangle=30]{HH}{TH}
\ncarc[arrows=->,linewidth=.08,linecolor=red,linestyle=dotted,arcangle=30]{TH}{TT}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,linestyle=dotted,arcangle=30]{TT}{HT}
\end{psmatrix}$$
$$\begin{psmatrix}[colsep=2cm,rowsep=2cm]
& [name=HH]{H,H} & [name=HT]{H,T}\\
& [name=TH]{T,H} & [name=TT]{T,T}
\psset{nodesep=5pt}
\ncline[arrows=->,linewidth=.03,linecolor=blue]{HH}{HT}
\ncline[arrows=->,linewidth=.03,linecolor=blue]{TT}{TH}
\ncline[arrows=->,linewidth=.08,linecolor=red]{TH}{HH}
\ncline[arrows=->,linewidth=.08,linecolor=red]{HT}{TT}
\end{psmatrix}
\qquad
\begin{psmatrix}[colsep=2cm,rowsep=2cm]
& [name=HH]{H,H} & [name=HT]{H,T}\\
& [name=TH]{T,H} & [name=TT]{T,T}
\psset{nodesep=5pt}
\ncline[arrows=->]{HH}{HT}
\ncline[arrows=->]{TT}{TH}
\ncline[arrows=->]{TH}{HH}
\ncline[arrows=->]{HT}{TT}
\end{psmatrix}$$
Change of mind for matching pennies is in Figure \[fig:match\_pen\_conv\_pref\_CoM\] right. One notices that there is a cycle. This cycle is the equilibrium. No player has clear mind of what to play and changes her minds each time she loses.
### Scissors, Paper, Stone {#sec:sc_pap_st}
Here we present the famous game known as *scissors, paper, stone*. It involves two players, [*Alice*]{} and [*Beth*]{} who announce either *scissors* ($C$) or *paper* ($P$) or *stone* ($T$) with the rules that *stone beats scissors, scissors beat paper, and paper beats stone*. There are nine situations (see below), one sees that [*Alice*]{} may convert her situation $C,P$ into $P,P$ or $T,P$ and the same for the other situations. The conversion is given below left. Since the rules, it seems clear that [*Alice*]{} prefers $P,P$ to $T,P$ and $C,P$ to $P,P$, hence the preference given below right with ${\raisebox{2pt}{\hspace{-30pt}
\begin{psmatrix}
&[name=a]&[name=b]\ncline[arrows=->,linewidth=.04,linecolor=red,linestyle=dotted,arrowscale=.5]{a}{b}
\end{psmatrix}
}}$ is [*Alice*]{}’s preference and ${\raisebox{2pt}{\hspace{-23pt}
\begin{psmatrix}
&[name=a]&[name=b]\ncline[arrows=->,linewidth=.015,linecolor=blue,linestyle=dotted,arrowscale=.5]{a}{b}
\end{psmatrix}
}}$ is [*Beth*]{}’s preference. To avoid a cumbersome diagram, in the preference we do not put the arrows deduced by transitivity.
$$\begin{psmatrix}[colsep=1.2cm,rowsep=1.2cm]
&[name=CC]{C,C} & [name=CP]{C,P} & [name=CT]{C,T} \\
&[name=PC]{P,C} & [name=PP]{P,P} & [name=PT]{P,T} \\
&[name=TC]{T,C} & [name=TP]{T,P} & [name=TT]{T,T}
\psset{nodesep=5pt}
\ncline[arrows=<->,linewidth=.03,linecolor=blue,linestyle=dashed]{CC}{CP}
\ncline[arrows=<->,linewidth=.03,linecolor=blue,linestyle=dashed]{CP}{CT}
\ncarc[arrows=<->,linewidth=.03,linecolor=blue,linestyle=dashed,arcangle=30]{CC}{CT}
\ncline[arrows=<->,linewidth=.03,linecolor=blue,linestyle=dashed]{PC}{PP}
\ncline[arrows=<->,linewidth=.03,linecolor=blue,linestyle=dashed]{PP}{PT}
\ncarc[arrows=<->,linewidth=.03,linecolor=blue,linestyle=dashed,arcangle=30]{PC}{PT}
\ncline[arrows=<->,linewidth=.03,linecolor=blue,linestyle=dashed]{TC}{TP}
\ncline[arrows=<->,linewidth=.03,linecolor=blue,linestyle=dashed]{TP}{TT}
\ncarc[arrows=<->,linewidth=.03,linecolor=blue,linestyle=dashed,arcangle=-30]{TC}{TT}
\psset{arrowscale=.6}
\ncline[arrows=<->,linewidth=.08,linecolor=red,linestyle=dashed]{CC}{PC}
\ncline[arrows=<->,linewidth=.08,linecolor=red,linestyle=dashed]{PC}{TC}
\ncarc[arrows=<->,linewidth=.08,linecolor=red,linestyle=dashed,arcangle=-30]{CC}{TC}
\ncline[arrows=<->,linewidth=.08,linecolor=red,linestyle=dashed]{CP}{PP}
\ncline[arrows=<->,linewidth=.08,linecolor=red,linestyle=dashed]{PP}{TP}
\ncarc[arrows=<->,linewidth=.08,linecolor=red,linestyle=dashed,arcangle=-30]{CP}{TP}
\ncline[arrows=<->,linewidth=.08,linecolor=red,linestyle=dashed]{CT}{PT}
\ncline[arrows=<->,linewidth=.08,linecolor=red,linestyle=dashed]{PT}{TT}
\ncarc[arrows=<->,linewidth=.08,linecolor=red,linestyle=dashed,arcangle=30]{CT}{TT}
\end{psmatrix}
\qquad
\begin{psmatrix}[colsep=1.2cm,rowsep=1.2cm]
&[name=CC]{C,C} & [name=CP]{C,P} & [name=CT]{C,T} \\
&[name=PC]{P,C} & [name=PP]{P,P} & [name=PT]{P,T} \\
&[name=TC]{T,C} & [name=TP]{T,P} & [name=TT]{T,T}
\psset{nodesep=5pt,arrowinset=.8}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,linestyle=dotted,arcangle=50]{CC}{CT}
\ncarc[arrows=->,linewidth=.08,arrowscale=.6,linecolor=red,linestyle=dotted,arcangle=-30]{CT}{CC}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,linestyle=dotted,arcangle=-20]{CP}{CC}
\ncarc[arrows=->,linewidth=.08,arrowscale=.6,linecolor=red,linestyle=dotted,arcangle=-20]{CC}{CP}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,linestyle=dotted,arcangle=20]{PP}{PC}
\ncarc[arrows=->,linewidth=.08,arrowscale=.6,linecolor=red,linestyle=dotted,arcangle=20]{PC}{PP}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,linestyle=dotted,arcangle=-20]{PT}{PP}
\ncarc[arrows=->,linewidth=.08,arrowscale=.6,linecolor=red,linestyle=dotted,arcangle=-20]{PP}{PT}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,linestyle=dotted,arcangle=-50]{TC}{TT}
\ncarc[arrows=->,linewidth=.08,arrowscale=.6,linecolor=red,linestyle=dotted,arcangle=30]{TT}{TC}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,linestyle=dotted,arcangle=20]{TT}{TP}
\ncarc[arrows=->,linewidth=.08,arrowscale=.6,linecolor=red,linestyle=dotted,arcangle=20]{TP}{TT}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,linestyle=dotted,arcangle=50]{TC}{CC}
\ncarc[arrows=->,linewidth=.08,arrowscale=.6,linecolor=red,linestyle=dotted,arcangle=-30]{CC}{TC}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,linestyle=dotted,arcangle=-20]{CC}{PC}
\ncarc[arrows=->,linewidth=.08,arrowscale=.6,linecolor=red,linestyle=dotted,arcangle=-20]{PC}{CC}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,linestyle=dotted,arcangle=20]{CP}{PP}
\ncarc[arrows=->,linewidth=.08,arrowscale=.6,linecolor=red,linestyle=dotted,arcangle=20]{PP}{CP}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,linestyle=dotted,arcangle=-20]{PP}{TP}
\ncarc[arrows=->,linewidth=.08,arrowscale=.6,linecolor=red,linestyle=dotted,arcangle=-20]{TP}{PP}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,linestyle=dotted,arcangle=-50]{TT}{CT}
\ncarc[arrows=->,linewidth=.08,arrowscale=.6,linecolor=red,linestyle=dotted,arcangle=30]{CT}{TT}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,linestyle=dotted,arcangle=20]{PT}{TT}
\ncarc[arrows=->,linewidth=.08,arrowscale=.6,linecolor=red,linestyle=dotted,arcangle=20]{TT}{PT}
\end{psmatrix}$$
From the above conversion and preference, one gets the following change of mind. $$\begin{psmatrix}[colsep=1.2cm,rowsep=1.2cm]
&[name=CC]{C,C} & [name=CP]{C,P} & [name=CT]{C,T} \\
&[name=PC]{P,C} & [name=PP]{P,P} & [name=PT]{P,T} \\
&[name=TC]{T,C} & [name=TP]{T,P} & [name=TT]{T,T}
\psset{nodesep=5pt,arrowinset=.8}
\end{psmatrix}
\ncline[arrows=->]{CP}{CC}
\ncline[arrows=->]{CP}{CT}
\ncarc[arrows=->,arcangle=30]{CC}{CT}
\ncline[arrows=->]{PT}{PP}
\ncline[arrows=->]{PP}{PC}
\ncarc[arrows=->,arcangle=-30]{PT}{PC}
\ncline[arrows=->]{TC}{TP}
\ncline[arrows=->]{TT}{TP}
\ncarc[arrows=->,arcangle=-30]{TC}{TT}
\ncline[arrows=->]{PC}{CC}
\ncline[arrows=->]{PC}{TC}
\ncarc[arrows=->,arcangle=-30]{CC}{TC}
\ncline[arrows=->]{TP}{PP}
\ncline[arrows=->]{PP}{CP}
\ncarc[arrows=->,arcangle=30]{TP}{CP}
\ncline[arrows=->]{CT}{PT}
\ncline[arrows=->]{TT}{PT}
\ncarc[arrows=->,arcangle=30]{CT}{TT}$$
One sees also perpetual moves as in the matching pennies of which it is a generalization.
### Strategic games as CP games
A strategic game is a specific kind of CP games. To be a strategic game, a CP game has to fulfill the following conditions.
1. Each situation is a $n$-Cartesian product, where $n$ is the number of players. The constituents of the Cartesian product are called *strategies*.
2. Conversion for player $a$, written ${"-->"_{a}}$, is any change along the $a$-th dimension, i.e., $(s_1,...,s_a,...,s_n) {"-->"_{a}}
(s_1,...,s_a',...,s_n)$. Hence in strategic games, conversion is an equivalent relation, namely
- symmetric, ($s {"-->"_{a}} s'$ implies $s' {"-->"_{a}} s$),
- transitive, ($s {"-->"_{a}} s'$ and $s' {"-->"_{a}} s''$ imply $s
{"-->"_{a}} s''$),
- and reflexive ($s {"-->"_{a}} s$).
Blink and you lose {#sec:blink}
------------------
*Blink and you lose* is a game played on a simple graph with two undifferentiated tokens. There are three positions: $$\xymatrix { *++[o][F-]{{{\color{green}`(!)}}{{\color{green}`(!)}}}\ar@{-}[r] &
*++[o][F-]{\phantom{`(!)}\phantom{`(!)}} } \qquad\qquad \xymatrix {
*++[o][F-]{{{\color{green}`(!)}}\phantom{`(!)}}\ar@{-}[r] &
*++[o][F-]{{{\color{green}`(!)}}\phantom{`(!)}} } \qquad\qquad \xymatrix {
*++[o][F-]{\phantom{`(!)}\phantom{`(!)}}\ar@{-}[r] &
*++[o][F-]{{{\color{green}`(!)}}{{\color{green}`(!)}}} }$$ There are two players, *Left* and *Right*. The leftmost position above is the winning position for *Left* and the rightmost position is the winning position for *Right*. In other words, the one who owns both token is the winner. Let us call the positions $L$, $C,$ and $R$ respectively. One plays by taking a token on the opposite node.
### A first tactic: Foresight {#sec:ByL1}
A player realizes that she can win by taking the opponent’s token faster than the opponent can react, i.e., player *Left* can convert $C$ into $L$ by outpacing player *Right*. Player *Right*, in turn, can convert $C$ into $R$. This version of the game has two singleton equilibria: $L$ and $R$. This is described by the following conversion $$\begin{psmatrix}[colsep=1.2cm,rowsep=1.2cm,nodesep=5pt]
& [name=L]{L} & [name=C]{C}& [name=R]{R}
\ncline[arrows=->,linewidth=.03,linecolor=blue,linestyle=dashed]{C}{R}
\psset{arrowscale=.5}\ncline[arrows=->,linewidth=.08,arrowscale=.6,linecolor=red,linestyle=dashed]{C}{L}
\end{psmatrix}$$ preference is $$\begin{psmatrix}[colsep=1.2cm,rowsep=1.2cm,nodesep=5pt]
& [name=L]{L} & [name=C]{C}& [name=R]{R}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,linestyle=dotted,arcangle=30]{C}{R}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,linestyle=dotted,arcangle=30]{L}{C}
\psset{arrowscale=.5}
\ncarc[arrows=->,linewidth=.08,arrowscale=.6,linecolor=red,linestyle=dotted,arcangle=30]{C}{L}
\ncarc[arrows=->,linewidth=.08,arrowscale=.6,linecolor=red,linestyle=dotted,arcangle=30]{R}{C}
\end{psmatrix}$$
where $\begin{psmatrix}[colsep=1cm,arrowscale=.5]
[name=C]& [name=R]
\ncline[arrows=->,linewidth=.08,arrowscale=.6,linecolor=red,linestyle=dotted,arcangle=30]{C}{R}
\end{psmatrix}$ is the preference for *Left* and $\begin{psmatrix}[colsep=1cm]
[name=C]& [name=R]
\ncline[arrows=->,linewidth=.03,linecolor=blue,linestyle=dotted,arcangle=30]{C}{R}
\end{psmatrix}$ is the preference for *Right*. The change of mind is then: $$\begin{psmatrix}[colsep=1.2cm,rowsep=1.2cm,nodesep=5pt]
& [name=L]{L} & [name=C]{C}& [name=R]{R}
\ncline[arrows=->]{C}{L}
\ncline[arrows=->]{C}{R}
\end{psmatrix}$$ and one sees that there are two equilibria: namely $L$ and $R$, which means that players have taken both token and keep them.
### A second tactic: Hindsight {#sec:ByL2}
A player, say *Left*, analyzes what would happen if she does not act. In case *Right* acts, the game would end up in $R$ and *Left* loses. As we all know, people hate to lose so they have an aversion for a losing position. Actually *Left* concludes that she could have prevented the $R$ outcome by acting. In other words, it is within *Left*’s power to convert $R$ into $C$. Similarly for player *Right* from $L$ to $C$. $$\begin{psmatrix}[colsep=1.2cm,rowsep=1.2cm,nodesep=5pt]
& [name=L]{L} & [name=C]{C}& [name=R]{R}
\ncline[arrows=->,linewidth=.03,linecolor=blue,linestyle=dashed]{R}{C}
\psset{arrowscale=.5}\ncline[arrows=->,linewidth=.08,arrowscale=.6,linecolor=red,linestyle=dashed]{L}{C}
\end{psmatrix}$$
We call naturally *aversion* the relation that escapes from positions a player does not want to be, especially a losing position. Aversion deserves its name as it works like conversion, but flies from bad position. We get the following change of mind: $$\begin{psmatrix}[colsep=1.2cm,rowsep=1.2cm,nodesep=5pt]
& [name=L]{L} & [name=C]{C}& [name=R]{R}
\ncline[arrows=->]{L}{C}
\ncline[arrows=->]{R}{C}
\end{psmatrix}$$ where $C$ is singleton equilibrium or an Abstract Nash Equilibrium.
### A third tactic: Omnisight {#sec:ByL3}
The players have both hindsight and foresight, resulting in a CP game $$\begin{psmatrix}[colsep=1.2cm,rowsep=1.2cm,nodesep=5pt]
& [name=L]{L} & [name=C]{C}& [name=R]{R}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,linestyle=dashed,arcangle=30]{C}{R}
\ncarc[arrows=->,linewidth=.03,linecolor=blue,linestyle=dashed,arcangle=30]{L}{C}
\psset{arrowscale=.5}
\ncarc[arrows=->,linewidth=.08,arrowscale=.6,linecolor=red,linestyle=dashed,arcangle=30]{C}{L}
\ncarc[arrows=->,linewidth=.08,arrowscale=.6,linecolor=red,linestyle=dashed,arcangle=30]{R}{C}
\end{psmatrix}$$ with one change-of-mind equilibrium covering all outcomes thus, no singleton equilibrium (or Abstract Nash Equilibrium) exists. $$\begin{psmatrix}[colsep=1.2cm,rowsep=1.2cm,nodesep=5pt]
& [name=L]{L} & [name=C]{C}& [name=R]{R}
\ncarc[arrows=->,arcangle=30]{C}{R}
\ncarc[arrows=->,arcangle=30]{L}{C}
\ncarc[arrows=->,arcangle=30]{C}{L}
\ncarc[arrows=->,arcangle=30]{R}{C}
\end{psmatrix}$$
### A four tactic: Defeatism {#sec:defeat}
One of the player, say *Left*, acknowledges that she will be outperformed by the other, *Right* in this case. She is so terrified by her opponent that she returns the token when she has it. This yields the following conversion: $$\begin{psmatrix}[colsep=1.2cm,rowsep=1.2cm,nodesep=5pt]
& [name=L]{L} & [name=C]{C}& [name=R]{R}
\ncline[arrows=<-,linewidth=.03,linecolor=blue,linestyle=dashed]{R}{C}
\psset{arrowscale=.5}\ncline[arrows=->,linewidth=.08,arrowscale=.6,linecolor=red,linestyle=dashed]{L}{C}
\end{psmatrix}$$ We get the following change of mind: $$\begin{psmatrix}[colsep=1.2cm,rowsep=1.2cm,nodesep=5pt]
& [name=L]{L} & [name=C]{C}& [name=R]{R}
\ncline[arrows=->]{L}{C}
\ncline[arrows=->]{C}{R}
\end{psmatrix}$$ where $R$ is a singleton equilibrium or an Abstract Nash Equilibrium.
### Relation with evolutionary games {#sec:evol}
In [@nowak04:_evolut_dynam_of_biolog_games] (Fig. 2, p. 795), Nowak and Sigmund comment a similar situation in evolutionary games. They call the first tactic, *bistability*, the second tactic, *coexistence*, the third tactic, *neutrality* and the fourth tactic, *dominance* and exhibit the same pictures.
The changes of mind corresponding to the four tactics and their correspondence with evolutionary games with two strategies can be summarized as follows
[||l|l|l||]{} **Blink you lose** & **Change of Mind** & **Evol. games**\
Foresight &
\[name=L\][L]{} & \[name=C\][C]{}& \[name=R\][R]{}
& Bistablity\
Hindsight &
\[name=L\][L]{} & \[name=C\][C]{}& \[name=R\][R]{}
& Coexistence\
Omnisight &
\[name=L\][L]{} & \[name=C\][C]{}& \[name=R\][R]{}
& Neutrality\
Defeatism &
\[name=L\][L]{} & \[name=C\][C]{}& \[name=R\][R]{}
& Dominance\
The $`l$ phage as a CP game {#sec:l_phage}
---------------------------
The $`l$ phage is a game inspired from biology [@ptashne04:_genet_switc; @ptashne01:_genes_and_signal]. The origin of the game will be given in Section \[sec:grn\_as\_cpg\], here we give just *the rules of the game.*
There are three players $cI$, $cro$ and $Env$. The game can be seen as a game with two tokens moving on two graphs where each player may choose to move one of the two tokens[^2]. $Env$ moves one token from the bottom position. The conversion is therefore the same for the three players[^3] and is given by the following rightmost diagram: $$\xymatrix @R 10pt { *++[o][F-]{2}\\ &*++[o][F-]{1}&\\
*++[o][F-]{1}\ar@{-}[uu]\\ &*++[o][F-]{0}\ar@{-}[uu]\\
*++[o][F-]{0}\ar@{-}[uu]\\ cI & cro } \qquad\quad \xymatrix @R 11.5pt
{ *++[o][F-]{\phantom{`(!)}}\\ &*++[o][F-]{\phantom{`(!)}}&\\
*++[o][F-]{{{\color{green}`(!)}}}\ar@{-}[uu]\\
&*++[o][F-]{{{\color{green}`(!)}}}\ar@{-}[uu]\\
*++[o][F-]{\phantom{`(!)}}\ar@{-}[uu]\\ cI & cro } \xymatrix{ {\langle}cI_2,cro_0 {\rangle}\ar@{<-->}[r] \ar@{<-->}[d] & {\langle}cI_2,cro_1
{\rangle}\ar@{<-->}[d] \\ {\langle}cI_1,cro_0 {\rangle}\ar@{<-->}[r] \ar@{<-->}[d] & {\langle}cI_1,cro_1 {\rangle}\ar@{<-->}[d] \\ {\langle}cI_0,cro_0 {\rangle}\ar@{<-->}[r] & {\langle}cI_0,cro_1 {\rangle}}$$ The preference is difficult to describe as an actual game to be played, it comes from the genetics and is specific to each player. The philosophy is as follows: a gene prefers a position if it is “pushed forward” that position. \[preference\] $$\xymatrix{{\langle}cI_2,cro_0 {\rangle}\ar@{.}@/_/[d]|{\rotatebox{270}{$\succ$}}
\ar@{.}@/_2.5pc/[dd]|{\rotatebox{270}{$\succ$}} & {\langle}cI_2,cro_1 {\rangle}\ar@{.}@/_/[d]|{\rotatebox{270}{$\succ$}}
\ar@{.}@/^2.5pc/[dd]|{\rotatebox{270}{$\succ$}}
\ar@{.}[l]|{\rotatebox{180}{$\succ$}}\\ {\langle}cI_1,cro_0 {\rangle}\ar@{.}@/_/[u]|{\rotatebox{90}{$\succ$}} & {\langle}cI_1,cro_1 {\rangle}\ar@{.}@/_/[u]|{\rotatebox{90}{$\succ$}}\ar@{.}[l]|{\rotatebox{180}{$\succ$}}\\
{\langle}cI_0,cro_0 {\rangle}& {\langle}cI_0,cro_1 {\rangle}} \qquad \qquad \qquad \xymatrix{ {\langle}cI_2,cro_0 {\rangle}& {\langle}cI_2,cro_1 {\rangle}\ar@{.}[d]|{\rotatebox{270}{$\supset$}}
\ar@{.}@/^2.5pc/[dd]|{\rotatebox{270}{$\supset$}} \\ {\langle}cI_1,cro_0 {\rangle}& {\langle}cI_1,cro_1 {\rangle}\ar@{.}[d]|{\rotatebox{270}{$\supset$}} \\ {\langle}cI_0,cro_0 {\rangle}& {\langle}cI_0,cro_1 {\rangle}} $$ $$\xymatrix{ {\langle}cI_2,cro_0 {\rangle}& {\langle}cI_2,cro_1 {\rangle}\\ {\langle}cI_1,cro_0 {\rangle}& {\langle}cI_1,cro_1 {\rangle}\\ {\langle}cI_0,cro_0 {\rangle}\ar@{.}[u]|{\rotatebox{270}{$\triangleleft$}}
\ar@{.}[r]|{\rotatebox{180}{$\triangleleft$}}& {\langle}cI_0,cro_1 {\rangle}} $$ \[preference2\] From the conversion and the preferences one deduces three changes of mind. $$\xymatrix{{\langle}cI_2,cro_0 {\rangle}\ar@{-}@/_/[d]|{\rotatebox{270}{$\succ$}} & {\langle}cI_2,cro_1 {\rangle}\ar@{-}@/_/[d]|{\rotatebox{270}{$\succ$}}
\ar@{-}[l]|{\rotatebox{180}{$\succ$}}\\ {\langle}cI_1,cro_0 {\rangle}\ar@{-}@/_/[u]|{\rotatebox{90}{$\succ$}} & {\langle}cI_1,cro_1 {\rangle}\ar@{-}@/_/[u]|{\rotatebox{90}{$\succ$}}\ar@{-}[l]|{\rotatebox{180}{$\succ$}}\\
{\langle}cI_0,cro_0 {\rangle}& {\langle}cI_0,cro_1 {\rangle}} \qquad \qquad \qquad \xymatrix{ {\langle}cI_2,cro_0 {\rangle}& {\langle}cI_2,cro_1 {\rangle}\ar@{-}[d]|{\rotatebox{270}{$\supset$}} \\ {\langle}cI_1,cro_0 {\rangle}&
{\langle}cI_1,cro_1 {\rangle}\ar@{-}[d]|{\rotatebox{270}{$\supset$}} \\
{\langle}cI_0,cro_0 {\rangle}& {\langle}cI_0,cro_1 {\rangle}} $$ $$\xymatrix{ {\langle}cI_2,cro_0 {\rangle}& {\langle}cI_2,cro_1 {\rangle}\\ {\langle}cI_1,cro_0 {\rangle}& {\langle}cI_1,cro_1 {\rangle}\\ {\langle}cI_0,cro_0 {\rangle}\ar@{-}[u]|{\rotatebox{270}{$\triangleleft$}}
\ar@{-}[r]|{\rotatebox{180}{$\triangleleft$}}& {\langle}cI_0,cro_1 {\rangle}} $$ from which we deduce the (general) change of mind of the game: $$\xymatrix{{{\color{green}{\langle}cI_2,cro_0 {\rangle}\ar@{->}@/_/[d]}} & {\langle}cI_2,cro_1 {\rangle}\ar@{->}@/_/[d] \ar@{->}[l]\\ {{\color{green}{\langle}cI_1,cro_0
{\rangle}\ar@{->}@/_/[u]}} & {\langle}cI_1,cro_1 {\rangle}\ar@{->}@/_/[u]\ar@{->}[l]\ar@{->}[d] \\ {\langle}cI_0,cro_0 {\rangle}\ar@{->}[u] \ar@{->}[r] & {{\color{red} {\langle}cI_0,cro_1 {\rangle}}} }$$ One sees one singleton equilibrium namely ${\langle}cI_0,cro_1{\rangle}$ (called the *lyse*) and one dynamic equilibrium namely $\{{\langle}cI_2,cro_0{\rangle}, {\langle}cI_1,cro_0{\rangle}\}$ (called the *lysogen*).
Formal presentation of CP games {#sec:CP_games}
===============================
To define a CP game we have to define four concepts:
- a set [$\mathcal{A}$]{} of *agents*,
- a set § of *situations*,
- for every agent $a$ a relation ${"-->"_{a}}$ on §, called *conversion*,
- for every agent $a$ a relation ${".>"_{a}}$ on §, called *preference*.
From these relations we are going to define a relation called *change of mind*. Before let us introduce formally what a game is.
A game is a 4-uple ${\langle}{\ensuremath{\mathcal{A}}}, {\ensuremath{\mathcal{S}}},({"-->"_{a}})_{a`:{\ensuremath{\mathcal{A}}}}, ({".>"_{a}})_{a`:{\ensuremath{\mathcal{A}}}}{\rangle}$.
For the first version of the square game we have:
- ${\ensuremath{\mathcal{A}}}= \{{\textit{Alice}}, {\textit{Beth}}\}$,
- ${\ensuremath{\mathcal{S}}}= \{1|2, 1|3, 1|4, 2|3, 2|4, 2|1, 3|4, 3|1, 3|2, 4|1, 4|2,
4|3\}$,
- Conversions ${"-->"_{{\textit{Alice}}}}$ and ${"-->"_{{\textit{Beth}}}}$ are given by Figure \[fig:conv\_pref\_square\] left,
- ${".>"_{{\textit{Alice}}}}$ is the same as ${".>"_{{\textit{Beth}}}}$ and this relation is given by Figure \[fig:conv\_pref\_square\] right.
Abstract Nash equilibrium or singleton equilibrium {#sec:sing_seq}
--------------------------------------------------
Let us look at a first kind of equilibria.
A *singleton equilibrium* is a situation $s$ such that: $$`A a`:{\ensuremath{\mathcal{A}}}, s'`:{\ensuremath{\mathcal{S}}}\quad . \quad (s {"-->"_{a}} s') \ \Longrightarrow\
\neg(s {".>"_{a}} s').$$ We write ${{\mathrm{Eq}^{{aN}}_{{{\ensuremath{{\mathrm{G}}_{{}}}}}}}({s})}$ (aN stands for *abstract Nash*).
In the previous paragraphs, we have seen examples of singleton equilibria. If we are at such an equilibrium, this is fine, but if not, we may wonder how to reach an equilibrium. If $s$ is not an equilibrium, this means that $s$ fulfills $$`E s'`:{\ensuremath{\mathcal{S}}}\quad . \quad s {"-->"_{a}} s' \wedge s {".>"_{a}} s'$$ which is the negation of $$`A s'`:{\ensuremath{\mathcal{S}}}\quad . \quad (s {"-->"_{a}} s') \ \Longrightarrow\ \neg(s
{".>"_{a}} s').$$ The relation $s {"-->"_{a}} s' \wedge s {".>"_{a}} s'$ between $s$ and $s'$ is a derived one. Let us call it *change of mind for $a$* and write it ${\xymatrix @C 15pt{\ar@{->}[r]&}_{a}}$. We say that $a$ changes her mind, because she is not happy with $s$ and hopes that following ${\xymatrix @C 15pt{\ar@{->}[r]&}_{a}}$ she will reach not necessary the equilibrium, but at least a better situation. Actually since we want to make everyone happy, we have to progress along all the ${\xymatrix @C 15pt{\ar@{->}[r]&}_{a}}$’s. Thus we consider a more general relation which we call just *change of mind* and which is the union of the ${\xymatrix @C 15pt{\ar@{->}[r]&}_{a}}$’s. We define this new relation as the union of the changes of mind of the agents. $${\xymatrix @C 20pt{\ar@{->}[r]&}}\ {\triangleq}\
\bigcup_{a`:{\ensuremath{\mathcal{A}}}} {\xymatrix @C 15pt{\ar@{->}[r]&}_{a}}.$$ Now suppose that we progress along ${\xymatrix @C 20pt{\ar@{->}[r]&}}$. What happens if we reach an $s$ from which we cannot progress further? This means $$`A a`:{\ensuremath{\mathcal{A}}}, s'`:{\ensuremath{\mathcal{S}}}\quad . \quad \neg (s {"-->"_{a}} s' \wedge s {".>"_{a}}
s')$$ in other words, $s$ is an equilibrium. Hence to reach an equilibrium, we progress along ${\xymatrix @C 20pt{\ar@{->}[r]&}}$ until we are stuck. In graph theory, a vertex from which there is no outgoing arrow is called an *end point* or a *sink*. In relation theory it is called a *minimal element*: $$\xymatrix @C 12pt @R 12pt {\ar@{->>}[dr] && \ar@{->>}[dl]\\ &`(!)
&\\ \ar@{->>}[ur] && \ar@{->>}[ul]}$$ Thus we look for end points in the graph.
Dynamic equilibrium {#sec:dyn_eq}
-------------------
Actually this progression along ${\xymatrix @C 20pt{\ar@{->}[r]&}}$ is not the panacea to reach an equilibrium. Indeed it could be the case that this progression never ends, since we enter a perpetual move (think at the square game 2nd version, Figure \[fig:conv\_CoM\_square3\]). Actually we identify this perpetual move as a second kind of equilibrium.
### Strongly connected components {#sec:SCC}
Here it is relevant to give some concepts of graph theory. A graph[^4] is *strongly connected*, if given two nodes $n_1$ and $n_2$ there is always a path going from $n_1$ to $n_2$ and a path going from $n_2$ to $n_1$. Not all the graphs are strongly connected, but they may contain some maximal subgraphs that are strongly connected; “maximal” means that one cannot add nodes without breaking the strong connectedness. Such a strongly connected subgraph is called a *strongly connected components*, *SCC* in short.
$$\begin{psmatrix}[colsep=.4cm,rowsep=.4cm]
&&&&& [name=42]{\mathbf{4|2}}\\
&&&& [name=43]{\mathbf{4|3}} && [name=41]{\mathbf{4|1}}\\
&&& [name=12]{\mathbf{1|2}} &&&& [name=32]{\mathbf{3|2}} \\
&[name=1w]{\mathbf{{{\color{red} 1|\omega}}}}& [name=13]{\mathbf{1|3}} &&&&&& [name=31]{\mathbf{3|1}} \\
&&& [name=14]{\mathbf{1|4}} &&&& [name=34]{\mathbf{3|4}} \\
&&&& [name=23]{\mathbf{2|3}} && [name=21]{\mathbf{2|1}} \\
&&&&& [name=24]{\mathbf{2|4}}
\ncarc[arrows=->,linewidth=.03,arcangle=50]{1w}{12}
\ncarc[arrows=->,linewidth=.03]{1w}{13}
\ncarc[arrows=->,linewidth=.03,arcangle=-50]{1w}{14}
\ncarc[arrows=->,linewidth=.03,arcangle=30]{41}{42}
\ncarc[arrows=->,linewidth=.03,arcangle=30]{42}{43}
\ncarc[arrows=->,linewidth=.03,arcangle=-30]{12}{13}
\ncarc[arrows=->,linewidth=.03,arcangle=-30]{13}{14}
\ncarc[arrows=->,linewidth=.03,arcangle=-30]{31}{32}
\ncarc[arrows=->,linewidth=.03,arcangle=-30]{34}{31}
\ncarc[arrows=->,linewidth=.03,arcangle=30]{23}{24}
\ncarc[arrows=->,linewidth=.03,arcangle=30]{24}{21}
\ncarc[arrows=->,linewidth=.03,arcangle=-40]{42}{12}
\ncarc[arrows=->,linewidth=.03,arcangle=-40]{32}{42}
\ncarc[arrows=->,linewidth=.03,arcangle=40]{43}{13}
\ncarc[arrows=->,linewidth=.03,arcangle=40]{31}{41}
\ncarc[arrows=->,linewidth=.03,arcangle=40]{13}{23}
\ncarc[arrows=->,linewidth=.03,arcangle=40]{21}{31}
\ncarc[arrows=->,linewidth=.03,arcangle=-40]{14}{24}
\ncarc[arrows=->,linewidth=.03,arcangle=-40]{24}{34}
\end{psmatrix}$$
The graph below has six SCC’s: $$\xymatrix{
&&&{{\color{darkbrown} `(!)}}\ar@{->>}@/^/[dr]&&&&`(!)\ar@{->>}@/^/[dl]\\
{{\color{vertfonce} `(!)}} \ar@{->>}@/^/[dr] &&&&
{{\color{green}`(!)}}\ar@{->>}@/^/@(ur,ul)[rr]\ar@{->>}@/^/[dl]&&
{{\color{green}`(!)}}\ar@{->>}@/^/@(dl,dr)[ll]\ar@{->>}@/^/[dr]\\ &
{{\color{red} `(!)}}\ar@{->>}@/^/[ddrr] &&{{\color{red} `(!)}}\ar@{->>}@/^/[ll]
\ar@{->>}@/^/[dr]&&&& {\textcolor{blue}{`(!)}}\ar@{->>}@/^/[dl]&\\
&&&&{{\color{red} `(!)}}\ar@{->>}@/^/[dl]&& {\textcolor{blue}{`(!)}}\ar@{->>}@/^/[rr]&&
{\textcolor{blue}{`(!)}}\ar@{->>}@/^/[ul]\\
&{{\color{red} `(!)}}\ar@{->>}@/^/[uu]&&{{\color{red} `(!)}}\ar@{->>}@/^/[ll]\ar@{->>}@/^/[uu]&&&&&
}$$ The graph of Figure \[fig:conv\_CoM\_square3\] has two SCC’s (Figure \[fig:2SCC\]).
From a graph, we can deduce a new graph, which we call the *reduced graph* (or *condensation* [@Baase78]), whose nodes are the SCC’s and the arcs are given as follows: there is an arc from an SCC $S_1$ to an SCC $S_2$ (assuming that $S_1$ is different from $S_2$), if there exists a node $n_1$ in $S_1$, a node $s_2$ in $S_2$ and an arc between $n_1$ and $n_2$. By construction the reduced graph has no cycle and its strongly connected components are singletons. The reduced graphs associated with the graphs given above are as follows: $$\xymatrix @C 12pt @R 12pt {
&&&{{\color{darkbrown} `(!)}}\ar@{->>}@/^/[drr]&&&&`(!)\ar@{->>}@/^/[dll]\\
{{\color{vertfonce} `(!)}} \ar@{->>}@/^/[drrr]
&&&&&{{\color{green}`(!)}}\ar@{->>}@/^/[dll]\ar@{->>}@/^/[drr] \\ &&&
{{\color{red} `(!)}}&&&& {\textcolor{blue}{`(!)}} } \qquad\qquad \xymatrix @R 12pt
{&\\{{\color{red} \{1|`w\}}} \ar@{-->>}[r] & \{1|3, 1|2, 1|4,...\}}$$
### Dynamic equilibria as strongly connected components {#sec:dyn_eq_as_SCC}
At the price of extending the notion of equilibrium, we can prove that there is always an equilibrium in finite non degenerated games, i.e., in games with a finite non zero number of game situations. Indeed given a graph, we compute its reduced graph. Then in this reduced graph, we look for end points. There is always such an end point since in a finite acyclic graph (the reduced graph is always acyclic) there exists always at least an end point.
We write ${{\mathrm{Eq}^{{CP}}_{{{\ensuremath{{\mathrm{G}}_{{}}}}}}}({A})}$ to say that the subset $A$ of situations is a CP equilibrium. We can now split equilibria into two categories?
1. *CP Equilibria* (i.e., *Dynamic equilibria*) are equilibria associated with an SCC and may contain many situations.
2. *Abstract Nash Equilibria* (aka *Singleton equilibria*) are equilibria associated with an SCC that contains exactly one situation, i.e., associated with an SCC which is a singleton, hence the name singleton equilibrium. There are specific dynamic equilibria.
Tarjan [@Tarjan] has shown that the reduced graph can be computed in linear time w.r.t. the numbers of nodes and edges of the original graph. Therefore CP equilibria can be computed in linear time in the number of game situations and edges in the change of mind relation, which provides an efficient algorithm to compute CP equilibria.
What are CP good for? {#sec:motivations}
=====================
After the success of strategic games over years, one may wonder why we introduce a new concept, namely CP games. The first nice feature is a theorem that says that *there always exists an equilibrium*. We know that pure strategic games do not enjoy that property and that to obtain such equilibria, Nash had to extend the concept of strategic game to this of probabilistic games. Similarly we have relaxed the notion of equilibrium to this of CP equilibrium.
Beside abstract Nash equilibria that are really like those of strategic games. CP games have other equilibria that biologists called *dynamic equilibria* and that correspond to phenomena they actually consider. Physicists speak about *stationary states* in that case.
Economists know that the concept of payoff is somewhat artificial[^5]. In CP games *the concept of payoff is completely abandoned*, no number are attached to situations and a general relation between situations is proposed instead.
In normal form strategic games, moves from one situation to another are tightly ruled and strong restrictions are imposed, namely right and left moves for one player, back and forth moves for another player and up and down moves for a third, etc., unlike CP games where *very general moves between situations ruled by the conversions are allowed*, like diagonal moves for instance or on the opposite more restricted moves like horizontal or vertical moves to a neighbor situation only (see the $\lambda$ phage). The flexibility of the conversions and the preferences makes possible to formalize many situations, like some that occur in biology.
It is known that games are a good framework to analyze models where the principle of causality fails. CP games allow analyzing a larger class of models.
Gene regulation networks as CP games {#sec:grn_as_cpg}
====================================
In the $`l$ phage, levels $0, 1, 2$, for a gene, correspond to levels of activation or levels of concentration of the corresponding protein. Thus $cI$ has three levels. $0$ corresponds to the gene being inactive (the protein is absent), $1$ corresponds to the gene being moderately active (the protein is present but moderately concentrated), $2$ corresponds to the gene being highly active (the protein is concentrated). On the other hand, $cro$ has two levels of activation, corresponding to the gene being inactive or active. $Env$ has only one level, it is always active. A gene can move from one level at a time, as translated by the conversion relation on page . It has been shown that $cI$ is a repressor for $cro$ and a repressor for itself at level $2$ and an activator for itself at level $1$. This leads to the preference $\xymatrix{\ar@{.}[r]|{\succ}&}$ for $cI$ on the left of diagram on page . On the other hand, it has been shown that $cro$ is a repressor for $cI$, this leads to the preference and an activator for itself at level $1$. This leads to the preference $\xymatrix{\ar@{.}[r]|{\supset}&}$ for $cro$ on the right of diagram on page . Moreover when both genes are inactive, the *environment* may lead to activate either $cI$ or $cro$, this leads to the preference $\xymatrix{\ar@{.}[r]|{\triangleright}&}$ of the diagram on page .
The two equilibria correspond to two well-known states of the $`l$ phage: the *lyse* and the *lysogen*, which the phage always reaches. In particular the lysogen $\{{\langle}cI_2,cro_0{\rangle}, {\langle}cI_1,cro_0{\rangle}\}$ is a relatively stable state, where the phage seems inactive (dormant state). This is due to the fact that the concentration of the protein associated to $cI$ is controlled: if it is too concentrated, a repression process makes the concentration to decrease and vice-versa if the concentration is too low an activation process makes it to increase. These antagonistic actions maintain the concentration at an intermediate level and the associated state is stable. The state ${\langle}cI_0,cro_1{\rangle}$ corresponds to what is called the *lyse* of the $`l$ phage.
What is amazing in the presentation as CP games is that these states are actually computed as CP equilibria. Somewhat connected approaches are [@Thomas73; @ThieffryThomas:DynBRN-II-phage-lambda].
Chinese Wall information security and corporate liability
=========================================================
A main claim of this article is that CP games, simple as they are, is a natural formalism whose conversion/preference distinction is of wider relevance. We shall further justify the claim in this section, with an example chosen because of the succinctness of its CP-game presentation and because the conversion/preference distinction is of stand-alone interest in the context of the example, without any consideration of equilibrium analysis.\
The concept of Chinese Wall information security pertains to the prevention of insider trading, and more generally the insulation of insider knowledge [@BrewerNash:ChineseWall89]. Chinese Wall requirements are codified in laws in many countries, and are interesting to informaticians in part because Chinese Wall security is different from military-style *need to know*. In particular, any information is in principle accessible to the subjects in question, but access is only granted if the subject is not already in possession of information that could create a conflict of interest. Formally, we consider a set of *subjects*, $P$ (for people), a set of *interests classes*, $I$, a set of *companies*, $C$, with *interest classification* function $\mathcal{I}:C\rightarrow I$, and a set of *objects*, $O$, with *ownership* function $\mathcal{C}:O\rightarrow C$. The typical scenario is that the subjects are consultants and the interest classes consists, for example, of `bank`, `oil company`, etc., with the requirement that no consultant handles objects, i.e., information, for more than one, e.g., `bank`. In other words, we are considering a game played by the subjects, ${\ensuremath{\mathcal{A}}}=P$, over complete accounts of what objects each subject has had access to, ${\ensuremath{\mathcal{S}}}=\otimes_{p\in P} 2^O$. Writing $s_p$ for the $p$-projection of an $s\in{\ensuremath{\mathcal{S}}}$, [@BrewerNash:ChineseWall89 Axiom 2] that governs when a subject, $p$, is allowed to gain access to an object amounts to the conversion relation where $s{"-->"_{p}}s'$ iff $$\begin{array}{l}
\forall p' \,.\, p\neq p' \Rightarrow s_{p'} = s'_{p'}\\
\wedge\\
\exists o \,.\, s'_p = s_p\cup\{o\}
\wedge
(\forall o' \in s_p \,.\,
\mathcal{I}(\mathcal{C}(o)) \neq \mathcal{I}(\mathcal{C}(o'))
\vee
\mathcal{C}(o) = \mathcal{C}(o'))
\end{array}$$ In words, subject $p$ may convert $s$ to $s'$ if the situations only differ by some object, $o$, being added to the $p$-projection and, for all other objects in $s_p$, $o$ either belongs to a different interest class or hails from the same company. By [@BrewerNash:ChineseWall89 Axiom 3], we are only interested in situations that can be reached from the situation where no subject has had access to any object, $\overrightarrow{\emptyset}$. With this, [@BrewerNash:ChineseWall89 Theorem 2] says: “A subject can at most have access to one company dataset in each conflict of interest class”. Formally, we have the following.
A state, $s\in {\ensuremath{\mathcal{S}}}$, has *no insider trading* if $$\begin{aligned}
\mathrm{NIT}(s) &\triangleq& \forall p\in P \,.\,
\forall o_1, o_2 \in s_p \,.\,
\mathcal{I}(\mathcal{C}(o_1)) \neq \mathcal{I}(\mathcal{C}(o_2))
\vee
\mathcal{C}(o_1)=\mathcal{C}(o_2)\end{aligned}$$
Given a *Chinese Wall CP game form*, ${\langle}P, \otimes_{p\in P} 2^O, ({"-->"_{p}})_{p\in P}{\rangle}$, no derived *Chinese Wall CP game* can reach a situation with insider trading. $$\forall({".>"_{p}})_{p\in P} \,.\, \forall s \,.\,
\overrightarrow{\emptyset} {\xymatrix @C 15pt{\ar@{->}[r]&}_{}}\!\!^*\ s
\Rightarrow \mathrm{NIT}(s)$$
While logically straightforward, the point of the CP-game version of the theorem is that it is universally quantified over the family of preference relations. In other words, the theorem explicitly states that for a company that implements Chinese Wall regulations for its subjects (the conversion relations), no matter what those employees may be tempted to do (the preference relations), insider trading can only take place if one of the employees breaks the company’s rules. This means that the CP-game version of the result formalizes a notion of corporate liability protection, which is directly relevant to the study of Chinese Wall information security.
Conversion or preference, how to choose? {#sec:choose}
========================================
The attentive reader may have noticed that what counts to compute equilibria is the *change of mind* and that keeping the same set of equilibria there is some freedom on the conversion and the preference provided one keeps the same change of mind. More precisely, we have $$\begin{array}{rcl@{\qquad}l}
{\xymatrix @C 15pt{\ar@{->}[r]&}_{a}} & = & {"-->"_{a}} \cap\ {".>"_{a}}\\
& = & ({"-->"_{a}} \cup \ R)\ \cap {".>"_{a}} & \textrm{when~} R \ \cap
{".>"_{a}} = \emptyset \\
& = & {"-->"_{a}} \cap\ ({".>"_{a}} \cup T) & \textrm{when~} T \ \cap
{"-->"_{a}} = \emptyset
\end{array}$$
On another hand, one notices that in some examples, the preference is independent of the agent whereas, in others, the conversion is independent of the agent. It seems that this is correlated with the domain of application. In particular, we may emit the following hypothesis. In biology, conversion is physics and chemistry, whereas preference is the part that cannot be explained by physics and chemistry, then we may induce that change of mind (combination of physics and true biology) is life. Indeed since physics and chemistry is the same for everyone, it makes sense to say that conversion is the same for everybody, whereas, due to evolution and biological effects, preference, changes with agents.
Fixed point construction, and equilibria in infinite games {#sect:discrete-FP}
==========================================================
For proving the existence of a fixed point for every probabilistic game, Nash [@Nash50] used Kakutani’s fixed point theorem. Since we deal with discrete games, we present in this section a proof of the existence of equilibrium based on a Tarski fixed-point theorem [@Tarski:55]. Recall that Tarski’s theorem uses an *update function*, say $f$, on a lattice and builds a fixed point starting from an element, say $a$, by iteration, $a$, $f(a)$, ..., $f^n(a)$, ... Here the lattice is the powerset $\mathscr{P}({\mathcal{S}})$ of situations ordered by the subset order.
In analogy to Nash’s update function, the function takes a subset of situations and creates a new subset based on how the agents would like to improve upon the old subset.
Given a game [${\mathrm{G}}_{{}}$]{} and a subset ${\mathrm{C}}\subseteq{\mathcal{S}}$ of the set of situations, let ${{\mathscr{U}}({{\mathrm{C}}})}
\;{\mathop{\triangleq}}\;\bigcup{}_{{\mathrm{s}}\in{\mathrm{C}}}
\{{\mathrm{s}}'\mid\,{{{\mathrm{s}}}{{\rightarrow_{}^*}}{{\mathrm{s}}'}}\}$.
With this, we have the following result, covering all CP games.
\[lem:lattice-FP\] Given (any) [${\mathrm{G}}_{{}}$]{}, ${\mathscr{U}}$ has a complete lattice of fixed points.[^6]
Not all fixed points will correspond to equilibria but the equilibria are the least, non-empty fixed points of the update function.
\[lem:least-FP-EqN\] Given [${\mathrm{G}}_{{}}$]{}, with change-of-mind relation ${{\rightarrow_{}^{}}}$. $$\begin{array}{c}
{{\mathrm{Eq}^{{CP}}_{{{\ensuremath{{\mathrm{G}}_{{}}}}}}}({{\mathrm{C}}})}\\
\Updownarrow\\
{{\mathscr{U}}({{\mathrm{C}}})}={\mathrm{C}}\ \wedge\
{\mathrm{C}}\neq\emptyset
\ \wedge\
(\forall {\mathrm{C}}' \,.\,
\emptyset\subsetneq{\mathrm{C}}'\subsetneq{\mathrm{C}}\Longrightarrow
{{\mathscr{U}}({{\mathrm{C}}'})}\not\subseteq{\mathrm{C}}')
\end{array}$$
By two direct arguments. The only interesting step is from bottom to top and showing that, for any two ${{\mathrm{s}}_1,{\mathrm{s}}_2\in{\mathrm{C}}}$, we have ${{{{\mathrm{s}}_1}{{\rightarrow_{}^*}}{{\mathrm{s}}_2}}}$. We first note that ${\mathscr{U}}$ is post-fixpointed: ${{\mathrm{C}}\subseteq{{\mathscr{U}}({{\mathrm{C}}})}}$, idempotent: ${{{\mathscr{U}}({{{\mathscr{U}}({{\mathrm{C}}})}})}={{\mathscr{U}}({{\mathrm{C}}})}}$, and order-preserving: ${{\mathrm{C}}_1\subseteq{\mathrm{C}}_2\,\Longrightarrow\,{{\mathscr{U}}({{\mathrm{C}}_1})}\subseteq{{\mathscr{U}}({{\mathrm{C}}_2})}}$. By order-preservation and ${{{\mathscr{U}}({{\mathrm{C}}})}={\mathrm{C}}}$, we have ${{{\mathscr{U}}({\{{\mathrm{s}}_1\}})}}\subseteq{\mathrm{C}}$. If ${\neg({{{\mathrm{s}}_1}{{\rightarrow_{}^*}}{{\mathrm{s}}_2}})}$, then ${{\mathrm{s}}_2\in{\mathrm{C}}\setminus{{\mathscr{U}}({\{{\mathrm{s}}_1\}})}}$, i.e., ${{{\mathscr{U}}({\{{\mathrm{s}}_1\}})}\subsetneq{\mathrm{C}}}$. By post-fixpointed-ness, ${{{\mathscr{U}}({\{{\mathrm{s}}_1\}})}}$ is non-empty and, by assumption of least-ness, we may therefore conclude ${{{\mathscr{U}}({{{\mathscr{U}}({\{{\mathrm{s}}_1\}})}})}\not\subseteq{{\mathscr{U}}({\{{\mathrm{s}}_1\}})}}$. This contradicts idempotency, and thus ${{{{\mathrm{s}}_1}{{\rightarrow_{}^*}}{{\mathrm{s}}_2}}}$.
CP equilibria are therefore *atomic*, in the sense that neither anything smaller nor anything bigger will have the same defining properties. For finite [${\mathrm{G}}_{{}}$]{}, a counting argument shows that the complete lattice of ${\mathscr{U}}$-fixed point will have least, non-empty elements, thus guaranteeing existence. For the infinite case, e.g., the following unbounded change-of-mind relation will not lead to the existence of least, non-empty elements in the fixed-point lattice because all tails are fixed points.
$$\xymatrix{
\bullet \ar@{->>}@/^.8pc/[r]&~\bullet \ar@{->>}@/^.8pc/[r]&~\bullet \ar@{->>}@/^.8pc/[r]&~~\bullet\ldots
}$$
However there are infinite cases where the existence of a CP equilibrium can be guaranteed, namely when there exists an SCC ${\mathrm{C}}$, which is extremal for the reduced change of mind.
More generally, a sufficient condition for the existence of ${\mathrm{Eq}^{CP}_{{\ensuremath{{\mathrm{G}}_{{}}}}}}$ is that some $s$ can reach only finitely many other elements in ${\ensuremath{\mathcal{S}}}$ using ${{\rightarrow_{}^*}}$. The condition is also necessary if we restrict attention to finite ${\mathrm{Eq}^{CP}_{{\ensuremath{{\mathrm{G}}_{{}}}}}}$. In particular, finite games have ${\mathrm{Eq}^{CP}_{{\ensuremath{{\mathrm{G}}_{{}}}}}}$. Because of the role played by reduced graphs above, also games with finite reduced graphs have ${\mathrm{Eq}^{CP}_{{\ensuremath{{\mathrm{G}}_{{}}}}}}$.
Conclusion {#sec:conclusion}
==========
We have presented conversion preference games as a strict extension of strategic games and we have proved that in a finite CP game an equilibrium always exists and under some conditions in infinite games as well. This theory is infancy and we expect it to generate as many theorems as the classical Nash game theory. See for instance [@LeRouxPhD08].
[10]{}
Sara Baase. . Addison-Wesley Longman Publishing Co., Inc., Boston, MA, USA, 1978.
D. F. C. Brewer and M. J. Nash. The [C]{}hinese [W]{}all security policy. In [*IEEE Symposium on Security and Privacy*]{}, pages 206–214, 1989.
John F. Nash[, Jr.]{} Equilibrium points in n-person games. , 36:48–49, 1950.
M. A. Nowak and K. Sigmund. Evolutionary dynamics of biological games. , 3003:793–798, February 2004.
Martin J. Osborne. . Oxford, 2004.
Mark Ptashne. . Cold Spring Harbor Laboratory Press, April 2004. 3rd edition.
Mark Ptashne and Alexander Gann. . Cold Spring Harbor Laboratory Press, November 2001.
Stéphane Le Roux. . PhD thesis, École normale supérieure de Lyon (France), January 2008.
St[é]{}phane Le Roux, Pierre Lescanne, and Ren[é]{} Vestergaard. A discrete [N]{}ash theorem with quadratic complexity and dynamic equilibria. Research Report IS-RR-2006-006, JAIST, May 2006.
Robert E. Tarjan. Depth first search and linear graph algorithms. , pages 146–160, Januar 1972.
Alfred Tarski. A lattice-theoretical fixpoint theorem and its applications. , 5:285–309, 1955.
Denis Thieffry and Ren[é]{} Thomas. Dynamical behaviour of biological regulatory networks [II]{}: Immunity control in bacteriophage lambda. , 57:277–297, 1995.
Ren[é]{} Thomas. Boolean formalization of genetic control circuits. , 42(3):563–585, 1973.
[^1]: See the preface of [@osborne04a] for the use of personal pronouns
[^2]: In the asynchronous version.
[^3]: Note the difference with the square game where players had different conversions and the same preference. The fact that the conversion is the same for everybody seems to be a feature of biologic game. Moreover notice also that, unlike in strategic games, the conversion is not transitive.
[^4]: In this paper, when we say “graph” , we mean always “oriented graph” or “digraph”.
[^5]: See for instance Osborne’s introduction of his textbook [@osborne04a].
[^6]: We note that complete lattices are non-empty by definition.
|
---
abstract: 'We calculate the branching ratio of the non-resonant $B \to D^- \pi^+\pi^+\pi^- $ decay using a simple model based on the framework of the factorization approach where one can ommit the nonfactorization effects. In naive factorization approach, there are only two tree diagrams for this decay mode. In the first diagram, the matrix element of decay mode is factorized into a $B\to D$ form factor multiplied by a $3\pi$ decay constant and in the second diagram, the matrix element is factorized into a $B\to D\pi$ form factor multiplied by a $2\pi$ decay constant, We assume that in the rest frame of B meson, the $D$ meson remains stationary, so we obtain the value $(3.47\pm0.14)\times 10^{-3}$ for the branching ratio of the $B \to D^- \pi^+\pi^+\pi^- $ decay mode, while the experimental results are $(3.9\pm1.9)\times10^{-3}$.'
author:
- |
Mohammad Rahim Talebtash[^1], Hossein Mehraban[^2]\
Physics Department, Semnan University\
P.O.Box 35195-363, Semnan, Iran
title: ' Non-resonant four-body decay of $B \to D^- \pi^+\pi^+\pi^- $ '
---
Introduction
============
The three-body and four-body decays of $B$ meson are more complicated than two-body case, specifically where both non-resonant and resonant contribution exist. In this types the theoretical momentums of the output particles are not directly calculable. In four-body cases, the momentums and form factors are written in terms of five variables: two invariant-mass of particles variable and three angles.\
In the previous works [@1; @2], for some specific three-body decays assumed that one of the output mesons due to its heavy weigh, remains stationary and the two other output mesons move back to back and the amplitude of this three decay modes can be calculate. In this work we extended this assumption to non-resonant four-body decays and calculate the branching ratio of $B \to D^- \pi^+\pi^+\pi^- $ decay in naive factorization approach where we can ignore the non-factorization effects. We assume that the $D$ output meson, due to its heavy weight, remains stationary and the three pion mesons move in a plane and we have directly resolved their momentums and we calculated the branching ratio of $D^- \pi^+\pi^+\pi^-$ and predicted $(3.47\pm0.14)\times 10^{-3}$ which is compatible with the experimental result $(3.9\pm1.9)\times 10^{-3}$.\
This paper is organized as follows. In section 2 we consider the amplitudes of the $D^- \pi^+\pi^+\pi^-$ decay and we present the transition matrix needed and the kinematics of the four-body decays. In section 3 we give the numerical results, and in the last section we have a summary.
Non-resonant amplitudes of the $B \to D^-\pi^+\pi^+\pi^-$
=========================================================
For the four-body $B \to D^-\pi^+\pi^+\pi^-$, the Feynman diagrams are shown in Fig. 1, where both panel show the $\bar{b}\to \bar{c}u\bar{d}$ tree transitions [@3; @4; @5]. Under the factorization approach, the contribution of the diagram (a) in decay amplitude consists of the $<B\to D^->\times<0\to\pi\pi\pi>$ distinct factorization term, where $<B\to D>$ denotes the form factor of $B$ to $D$ meson transitional, and the contribution of the diagram (b) consists of a $<B\to \pi\pi>\times <0\to \pi \pi>$ distinct factorization term, where $<B\to \pi\pi>$ denotes the two-meson transition matrix element.
![Feynman diagram for the $B \to D^- \pi^+\pi^+\pi^- $ decay.[]{data-label="fig1"}](1.jpg)
So the matrix elements of the $B \to D^-\pi^+\pi^+\pi^-$ decay amplitudes are given by $$\begin{aligned}
|\mathcal{M}|&=& i \frac{G_F}{\sqrt{2}} V_{cb}V_{ud}\Big{\{}a_1<\pi\pi\pi|(\bar{d}u)_-|0><D|(\bar{c}b)_-|B>\\ \nonumber
&&+a_2<D\pi|(\bar{c}b)_-|0><\pi\pi|(\bar{d}u)_-|B>\Big{\}},\end{aligned}$$ Where in factorization, the general expression of the $a_i$ quantities in terms of effective Wilson coefficients is [@ai] $$\begin{aligned}
a_1&=&c_1+\frac{1}{3}c_2,\nonumber \\
a_2&=&c_2+\frac{1}{3}c_1.\end{aligned}$$
Form factors and decay constants
--------------------------------
According to Fig. 1(a), the form factor of $B$ meson to $D$ meson is defined as fallow [@6] $$\begin{aligned}
<D(p_4)|(\bar{c}b)_{(V-A)}|B(P)>&=&\Big{[}(p_B+p_4)_\mu-\frac{m_B^2-m_D^2}{(p_B-p_4)^2}(p_B-p_4)_\mu\Big{]}F_1(q^2) \nonumber \\
&&+\frac{m_B^2-m_D^2}{(p_B-p_4)^2}(p_B-p_4)_\mu F_0(q^2),\end{aligned}$$ and the point like 3-body matrix element is assumed to be [@7]
$$\begin{aligned}
<\pi(p_1)\pi(p_2)\pi(p_3)|(\bar{d}u)_{(V-A)}|0>&=&\frac{2i}{f_\pi}\Big{[}p_{2\mu}-\frac{(p_B-p_4).p_2}{(p_B-p_4)^2-m_\pi^2}(p_B-p_4)_\mu\Big{]}. \nonumber \\\end{aligned}$$
According to Fig. 1(b), for the current-induced process, the two-meson transition matrix element $<\pi\pi|(\bar{d}u)_{(V-A)}|B>$ has the fallowing general expression [@7; @8]
$$\begin{aligned}
<\pi\pi|(\bar{d}u)_{(V-A)}|B>&=& i r (p_B-p_1-p_2)_\mu+i\omega_+(p_2+p_1)_\mu+i\omega_-(p_2-p_1)_\mu, \nonumber \\\end{aligned}$$
where the form factors $r, \omega_+$ and $\omega_-$ for the non-resonant decay are given by $$\begin{aligned}
r &=& \frac{f_B}{2f_\pi^2}-\frac{f_B}{f_\pi^2}.\frac{p_B.(p_2-p_1)}{(p_B-p_1-p_2)^2-m_B^2}+\frac{2g f_{B_S^*}}{f_\pi^2}.\sqrt{\frac{m_B}{m_{B_S^*}}}.\frac{(p_B-p_1).p_1}{(p_B-p_1)^2-m_{B_s^*}^2} \nonumber \\
&&+\frac{4g^2f_B}{f_\pi^2}.\frac{m_Bm_{B_S^*}}{(p_B-p_1-p_2)^2-m_B^2}.\frac{p_1.p_2-p_1.(p_B-p_1)p_2.(p_B-p_1)/m_{B_s^*}^2}{(p_B-P_1)^2-m_{B_S^*}^2}, \nonumber \\\end{aligned}$$ $$\begin{aligned}
\omega_+&=&-\frac{g}{f_\pi^2}.\frac{f_{B_S^*}m_{B_S^*}\sqrt{m_Bm_{B_S^*}}}{(p_B-p_1)^2-m_{B_S^*}}\Bigg{[}1-\frac{(p_B-p_1).p_1}{m_{B_S^*}^2}\Bigg{]}+\frac{f_B}{2f_\pi^2},\end{aligned}$$ $$\begin{aligned}
\omega_-&=&-\frac{g}{f_\pi^2}.\frac{f_{B_S^*}m_{B_S^*}\sqrt{m_Bm_{B_S^*}}}{(p_B-p_1)^2-m_{B_S^*}}\Bigg{[}1+\frac{(p_B-p_1).p_1}{m_{B_S^*}^2}\Bigg{]}.\end{aligned}$$ The tow-body $<D\pi|(\bar{c}b)_{(V-A)}|0>$ matrix element can be related to the $D$ and $\pi$ matrix element of the weak interaction current: $$\begin{aligned}
<D(p_4)\pi(p_3)|(\bar{c}b)_{(V-A)}|0>&=&(p_4-p_3)F^{D\pi}(q^2),\end{aligned}$$ where $F^{D\pi}(q^2)$ is the $D$ to $\pi$ transition form factor and determined from experiment [@9].
The kinematics of four-body decay
---------------------------------
For four-body $B$ decays, we can work in the rest frame of the B meson. By following the definition given in [@10; @11], when B meson decays to final-state mesons, the kinematics of the four-body decays can be described in term of five independent variables where the momenta and masses are defined as $$\begin{aligned}
P(p,M)&\to&P_1(p_1,m_1)P_2(p_2,m_2)P_3(p_3,m_3)P_4(p_4,m_4) ,\end{aligned}$$ with $p^2=M^2$ and $p_i^2=m_i^2$ where $i=1,2,3$ and $4$. We choose the set of independent variables as $\{ s_{12},s_{34},\theta_1,\theta_3,\phi\}$ which have the following geometrical meaning (see Fig. (2))
- $s_{12} = (p_1+ p_2)^2$, is the invariant-mass of particles 1 and 2;
- $s_{34} = (p_3 + p_4)^2$, is the invariant-mass of particles 3 and 4;
- $\theta_1 (\theta_3)$, is the angle between the three-momentum of particle 1 (particle 3) with respect to the direction of $\vec{p}_{12} =\vec p_1 + \vec p_2$ ($ \vec p_{34} = \vec p_3 +\vec p_4)$ defined in the rest frame of the decaying particle (see fig.(2));
- $\phi$ is the angle between the planes defined by particles (1, 2) and (3, 4) also in the rest frame of the decaying particle.
![Kinematics of four-body decays in the rest frame of the decaying particle, $\Sigma_{i=1}^4\vec{P}_i=0$. We have defined $\vec{p_{ij}} = \vec {p_i} + \vec{p_j}$, such that $\vec{p_{12}} + \vec{p_{34}} = 0$.[]{data-label="fig2"}](2.jpg)
With this choice of kinematics, the differential decay rate in the rest frame of the decaying particle can be written as: $$\begin{aligned}
d\Gamma&=&\frac{X\beta_{12}\beta{34}}{4(4\pi)^6m_B^3}\bar{|\mathcal{M}|^2}\frac{1}{n!}ds_{12}ds_{34}d\cos \theta_1d\cos \theta_3d\phi,\end{aligned}$$ where $\beta_{12}(\beta_{34}$) is the velocity of particle 1 (particle 3) in the center of mass frame of particles 1 and 2 (3 and 4) and $X=[(P^2-S_{12}-S_{34})^2-4S_{12}S_{34}]^{1/2}$, the $n!$ factor accounts for identical particles in the phase space and $\bar{|\mathcal{M}|}$ is the amplitude of the decay (Eq.(1)).\
For considering the $B \to D^- \pi^+\pi^+\pi^- $ decay mode, we work in the rest frame of the $B$ meson and assume that the $D^-$ meson due to its heavy weight, has a small momentum and can easily be neglected, so the three pions move in a plane. Hence the momentums of $B$ meson and final-state mesons are given by $$\begin{array}{cc}
p_{1}=(p_{\pi1}^0,\vec{p}_{\pi1}),& p_4=(m_D,\vec{0}),\\
p_{2}=(p_{\pi2}^0,\vec{p}_{\pi2}), &p_B=(m_B,\vec{0}) \\
p_{3}=(p_{\pi3}^0,\vec{p}_{\pi3}),
\end{array}$$ where $\vec{p}_{\pi1}=\vec{p}_{\pi2}=\vec{p}_{\pi3}$ and $p_{\pi1}^0=p_{\pi2}^0=p_{\pi3}^0=(m_B-m_D)/3$. Now we can easily integrate the decay rate with respect to all independent variables and we obtain a simple experiment $$\begin{aligned}
\Gamma&=&\frac{X\beta_{12}\beta_{34}}{4(4\pi)^6m_B^3}\bar{|\mathcal{M}|^2}\frac{8\pi}{3!}s_{12}s_{34}.\end{aligned}$$
Numerical results
=================
The Wilson coefficients $c_i$ have been calculated in different schemes. In this paper we use consistently the naive dimensional regularization (NDR) scheme. The values of $c_i$ at the scales $\mu=m_b$ at the leading order (LO) are [@12]
$$\begin{array}{cc}
c_1=1.117 & c_2=-0.268 .
\end{array}$$
The parameter $g$ in the form factors, determined from the $D^*\to D \pi$ decay is $$\begin{aligned}
g&=&0.3\pm0.1.\end{aligned}$$ For the elements of the CKM matrix, we use the values of the Wolfenstein parameters and obtain [@13] $$\begin{array}{cc}
V_{ud}=0.9745 & V_{cb}=0.0415
\end{array}$$ The meson masses, decay constants and form factors needed in our calculations are taken as (in units of GeV) [@14; @15] $$\begin{array}{cccc}
m_B=5.279 &m_{B^*_s}=5.413 & m_D=1.87 & m_\pi=0.139 \\
f_B=0.176 & f_{B^*_s}=0.220 & f_\pi=0.13 & F^{B\to D}_0=0.52 \\
F^{B\to D}_1=0.52.
\end{array}$$ Using the parameters relevant for the $B \to D^- \pi^+\pi^+\pi^- $ decay, we calculate the branching ratio of this decay which is $$\begin{aligned}
B(B \to D^- \pi^+\pi^+\pi^-)=(3.47\pm0.14)\times 10^{-3}\end{aligned}$$ where the experimental value is [@14] $$\begin{aligned}
B(B \to D^- \pi^+\pi^+\pi^-)_{expt}=(3.9\pm1.9)\times 10^{-3}.\end{aligned}$$
Summary
=======
In this work, we have considered four-body $B\to D^- \pi^+\pi^+\pi^-$ decays under the assumption that the $D$ meson has small and negligible momentum due to its heavy weight. This decay mode have contribution of $a_1$ and $a_2$ coefficients. According to the Feynman diagrams, each contribution factorized into a form factor and a decay constant. Under the assumption of negligible momentum of D meson, the three output pion mesons move in a plane and we have directly resolved their momentums and we calculated the branching ratio of $D^- \pi^+\pi^+\pi^-$ and predicted $(3.47\pm0.14)\times 10^{-3}$ which is compatible with the experimental result $(3.9\pm1.9)\times 10^{-3}$.
[10]{} Behnam Mohammadi and Hossein Mehraban,*“Study of the $B_s^0 \to J/\psi K^+K^-$ decay”*, Phys.Rev. D **89**(9):095026, 2014. Behnam Mohammadi and Hossein Mehraban. *“Three-body decays of $ B^{0(+)} \to K^{*0(+)} \pi^+ \pi^-$”*. Eur.Phys.J. A**50**:122, 2014. Michael Gronau, Oscar F. Hernandez, David London, and Jonathan L. Rosner. *“Broken SU(3) symmetry in two-body B decays”*. Phys.Rev. D**52**:6356[6373]{}, 1995. Mahiko Suzuki. *“Selection rules in three-body B decay from factoriza-tion”*. Phys.Rev. D**62**:091502, 2000. S. Fajfer, R.J. Oakes, and T.N. Pham. *“The Penguin operators in non-resonant $B^- \to M anti-M \pi^- (M=\pi^-, K^-, K^0)$ decays”*. Phys.Rev. D**60**:054029, 1999. K. Terasaki. *“Nonfactorizable long distance contributions in color sup-pressed decays of B mesons”*. Int.J.Mod.Phys. A **13**:4325[4338]{}, 1998. A. Ali, C. Greub, *“An analysis of two-body non-leptonic B decays involving light mesons in the standard model”*. Phys.Rev.D. **57**:2996-3016, 1998. \[arXiv:hep-ph/9707251\] Hai-Yang Cheng and Kwei-Chou Yang. *“Nonresonant three-body decays of D and B mesons”*. Phys.Rev. D **66**:054015, 2002. B. Bajc, S. Fajfer, R.J. Oakes, T.N. Pham, and Sasa Prelovsek. *“The CP violating asymmetry in $B^+ \to M anti-M \pi^+$ decays”*. Phys.Lett. B **447**:313 320, 1999. Tomoko Iwashita. *“Search for a $J/\psi \eta$ resonance in$B^{\pm} \to J/\psi \eta K^{\pm}$ decay at Belle”*. 2012. N. Quintero, G. Lopez Castro, and D. Delepine.*“ Lepton number violation in top quark and neutral B meson decays”*. Phys.Rev. D **84**:096011,2011. Gabriel Lopez Castro and Nestor Quintero. *“Lepton number violating four-body tau lepton decays”*. Phys.Rev. D **85**:076006, 2012. M. Beneke, G. Buchalla, M. Neubert, and Christopher T. Sachrajda. *“QCD factorization in $B \to \pi K, \pi \pi$ decays and extraction of Wolfenstein parameters”*. Nucl.Phys. B **606**:245321, 2001. Claude Amsler et al. Review of Particle Physics. Phys.Lett. B **667**:11340, 2008. K.A. Olive et al. Review of Particle Physics. Chin.Phys. C **38**:090001, 2014. Ying-Ying Fan, Wen-Fei Wang, Shan Cheng, and Zhen-Jun Xiao. *“Semileptonic decays $B \to D^{(*)}l v$ in the perturbative QCD factorization approach”*. Chin.Sci.Bull. **59**:125132, 2014.
[^1]: mtalebtash@students.semnan.ac.ir
[^2]: hmehraban@semnan.ac.ir
|
---
abstract: 'We completely determine the free infinite divisibility for the Boolean stable law which is parametrized by a stability index $\alpha$ and an asymmetry coefficient $\rho$. We prove that the Boolean stable law is freely infinitely divisible if and only if one of the following conditions holds: $0<\alpha\leq\frac{1}{2}$; $\frac{1}{2}<\alpha\leq\frac{2}{3}$ and $2-\frac{1}{\alpha}\leq\rho \leq \frac{1}{\alpha}-1$; $\alpha=1,~\rho=\frac{1}{2}$. Positive Boolean stable laws corresponding to $\rho =1$ and $\alpha \leq \frac{1}{2}$ have completely monotonic densities and they are both freely and classically infinitely divisible. We also show that continuous Boolean convolutions of positive Boolean stable laws with different stability indices are also freely and classically infinitely divisible. Boolean stable laws, free stable laws and continuous Boolean convolutions of positive Boolean stable laws are non-trivial examples whose free divisibility indicators are infinity. We also find that the free multiplicative convolution of Boolean stable laws is again a Boolean stable law.'
author:
- 'Octavio Arizmendi[^1] and Takahiro Hasebe[^2]'
title: Classical and free infinite divisibility for Boolean stable laws
---
Mathematics Subject Classification: 46L54, 60E07
Introduction
============
In probability theory, infinitely divisible distributions play a central role since they arise as the limit distributions of very general limit theorems. In free probability theory [@Voi86; @Voi87] there is an analogous notion of infinite divisibility. Moreover, the so-called Bercovici-Pata bijection [@BePa] maps classically infinitely divisible probability measures to freely infinitely divisible ones. Let $ID(\ast)$ be the class of all classically infinitely divisible distributions on $\mathbb{R}$ and $ID(\boxplus)$ be the class of all freely infinitely divisible distributions on $\mathbb{R}$.
Classically and freely infinitely divisible probability measures are typically quite different. For example, measures in $ID(\boxplus)$ cannot have more than one atom, while there are many examples of purely atomic measures in $ID(\ast)$. Also, there are many measures in $ID(\boxplus)$ with compact support, while every non trivial measure in $ID(\ast)$ has unbounded support. So one might expect that, perhaps with very particular exceptions, some probability measures belong to the “classical world” and some to the “free world”. This paper, however, discovers a family of measures lying on the intersection of $ID(\boxplus)$ and $ID(\ast)$. More interestingly, the examples considered here come from the “Boolean world”, another framework of non-commutative probability.
The Boolean convolution was defined in [@S-W] and Boolean stable laws, denoted by $\mathbf{b}_\alpha^\rho$, were classified. Here $\alpha$ is the stability index and $\rho$ is an asymmetry coefficient (see Section \[prel\] below).
In this paper we consider the classical and free infinite divisibility of Boolean stable laws. On the free probability side, we completely determine which Boolean stable laws are freely infinitely divisible.
\[T1\] The Boolean stable law $\mathbf{b}^\rho_\alpha$ is freely infinitely divisible if and only if:
\(1) $0<\alpha\leq\frac{1}{2}$,
\(2) $\frac{1}{2}<\alpha\leq\frac{2}{3}$, $2-\frac{1}{\alpha}\leq\rho \leq \frac{1}{\alpha}-1$,
\(3) $\alpha=1,~\rho=\frac{1}{2}$.
We use analytical methods to prove Theorem \[T1\] but also show a reproducing property of Boolean stable laws from which one can give a much simpler proof of part *(1)* in Theorem \[T1\], when $\alpha=1/n$, for $n$ an integer greater than $1$.
Boolean stable laws not only provide new examples of measures which are infinitely divisible in the free sense but also in the classical one. More explicitly, we prove the following.
\[T3\] Boolean stable laws on $[0,\infty)$ are $*$-infinitely divisible for $\alpha \leq 1/2$.
Until now, there were only $5$ known examples of measures in $ID(\boxplus)\cap ID(\ast)$: Cauchy distribution, free $1/2$-stable law, Gaussian, chi-square with one degree of freedom, $F$-distribution with 1,2 degrees of freedom [@AHS; @BBLS; @PAS]. Theorems \[T1\] and \[T3\] give a family of probability measures which are infinitely divisible with respect to both classical and free convolutions. Moreover, continuous Boolean convolutions of Boolean stable laws also belong to $ID(\boxplus)\cap ID(\ast)$ as we prove in Theorems \[mix1\] and \[mix2\] below.
Theorem \[T1\] has another consequence regarding the free divisibility indicator of Belinschi and Nica [@BN] (see Section \[prel\] for a brief explanation on this indicator). Recently, we were able to prove in [@AH2] that any Boolean stable law $\mathbf{b}_\alpha^\rho$ has free divisibility indicator $0$ or $\infty$, depending on whether or not $\mathbf{b}_\alpha^\rho$ is in $ID(\boxplus)$. The first non trivial example of a measure with divisibility indicator $\infty$ was found from this observation: free and Boolean stable laws with parameter $1/2$. Theorem \[T1\] broadly generalizes the result of [@AH2].
Apart from this introduction this note is organized as follows. Section \[prel\] presents basic preliminaries needed in this paper. In Section \[S3\] we consider the free infinite divisibility of Boolean stable laws and continuous Boolean convolutions of them. In Section \[S4\], we prove the classical infinite divisibility of some Boolean stable laws. Finally, Section \[S2\] is on a reproducing property of Boolean stable laws with respect to the free multiplicative convolution.
Preliminaries {#prel}
=============
Boolean convolution and Boolean stable laws {#boolean-convolution-and-boolean-stable-laws .unnumbered}
-------------------------------------------
We denote by $\mathcal{P}$ the set of Borel probability measures on $\mathbb{R}$. The upper half-plane and the lower half-plane are respectively denoted by $\mathbb{C}^+$ and $\mathbb{C}^-$. Let $G_\mu(z) = \int_{\mathbb{R}}\frac{\mu(dx)}{z-x}$ $(z \in \mathbb{C}^+)$ be the Cauchy transform of $\mu \in \mathcal{P}$ and $F_\mu(z)$ its reciprocal $\frac{1}{G_\mu(z)}$. The dilation operator $D_b$ $(b>0)$ is defined on $\mathcal{P}$ so that $$\int_{\mathbb{R}}f(x)D_b (\mu)(dx)=\int_{\mathbb{R}}f(bx)\mu(dx)$$ for any bounded, continuous function $f$ on $\mathbb{R}$.
The *Boolean convolution* $\uplus$ was defined in [@S-W] as follows. Let $K_\mu(z)$ be the energy function defined by $$K_\mu(z)=z-F_\mu(z),~~z\in \mathbb{C}^+$$ for $\mu \in \mathcal{P}$. The Boolean convolution $\mu \uplus \nu$ is characterized by $K_{\mu \uplus \nu} (z)=K_\mu(z)+K_\nu(z)$. A Boolean convolution semigroup $\{\mu^{\uplus t}\}_{t \geq 0}$ can always be defined for any probability measure $\mu$, corresponding to the energy transform $tK_\mu$.
Boolean stable laws can be defined in terms of self-similarity. We include only non trivial measures, that is, measures which are different from a point measure $\delta_a$.
1\) A non trivial measure $\mu$ is said to be *Boolean (or $\uplus$-) strictly stable* if $\mu \uplus \mu=D_{2^{1/\alpha}}(\mu)$ for some $\alpha$. We call $\alpha$ the index of stability.
2\) A non trivial measure is said to be *$\uplus$-stable* if $\mu\uplus\delta_a$ is $\uplus$-strictly stable for some $a \in \mathbb{R}$.
Boolean strictly stable distributions were characterized in [@S-W] in terms of the energy function. General $\uplus$-stable distributions can be classified as follows.
Probability measures $\{{\normalfont\textbf{b}}_{\alpha}^\rho: \alpha \in (0,2], \rho \in [0,1]\}$ defined by the following energy functions are $\uplus$-stable:
i\) $K(z)= - e^{i\pi \rho\alpha}z^{-\alpha+1}$ for $\alpha\in(0,1)$ and $\rho \in [0,1]$;
ii\) $K(z)= -2\rho i +\frac{2(2\rho-1)}{\pi}\log z$ for $\alpha =1$ and $\rho \in [0,1]$;
iii\) $K(z)= e^{i(\alpha-2)\rho\pi} z^{-\alpha+1}$ for $\alpha\in(1,2]$ and $\rho \in [0,1]$.
The index of stability of ${\normalfont\textbf{b}}_{\alpha}^\rho$ is $\alpha$. For any $\uplus$-stable law $\mu$, there exist $a \in \mathbb{R}, b >0, \alpha \in(0,2], \rho\in [0,1]$ such that $\mu= \delta_a \uplus D_b({\normalfont\textbf{b}}_{\alpha}^\rho)$.
As shown in [@AH2], the transformation $\mu \mapsto \delta_a \uplus D_b(\mu)$ does not change the free infinite divisibility (free divisibility indicator, more strongly), so that the above cases i)–iii) are enough for our purpose.
The parameter $\rho$ is called an *asymmetry coefficient* of a $\uplus$-stable law. The measure ${\normalfont\textbf{b}}_{\alpha}^\rho$ is supported on $[0,\infty)$ if and only if $\rho =1$, $\alpha \in (0,1]$ and it is symmetric if and only if $\rho=\frac{1}{2}$.
Free additive convolution {#free-additive-convolution .unnumbered}
-------------------------
The *free additive convolution* $\boxplus$ was introduced by Voiculescu in [@Voi86] for compactly supported measures on $\mathbb{R}$ and is defined as follows. If $X_{1},X_{2}$ are free random variables following probability distributions $\mu_1, \mu_2$ respectively, then the probability distribution of $X_1+X_2$ is denoted by $\mu_1 \boxplus \mu_2$ and is called the free additive convolution. The free additive convolution was later extended to all probability measures in [@Be-Vo] and it is characterized as follows.
For any $\beta>0$, there exists $M>0$ such that the reciprocal Cauchy transform $F_\mu$ has a right inverse map $F^{-1}_\mu$ defined in $\Gamma_{\beta, M}:=\{z \in \mathbb{C}^+: \text{Im}\,z >M,~ \beta|\text{Re}\,z| <\text{Im}\,z\}$. Let $\phi_\mu(z)$ be the Voiculescu transform of $\mu$ defined by $$\phi_\mu(z) = F_\mu^{-1}(z)-z,~~z\in\Gamma_{\beta,M}.$$ The free convolution $\mu \boxplus \nu$ of probability measures $\mu$ and $\nu$ is the unique probability measure such that $\phi_{\mu \boxplus \nu}(z) = \phi_\mu (z) + \phi_\nu(z)$ in a common domain.
Free infinite divisibility and divisibility indicator {#S9 .unnumbered}
-----------------------------------------------------
A probability measure $\mu$ is said to be *freely (or $\boxplus$-) infinitely divisible* if for each $n\in\mathbb{N}$ there exist $\mu_n \in \mathcal{P}$ such that $\mu=\mu_n^{\boxplus n}.$ We will denote by $ID(\boxplus)$ the set of $\boxplus$-infinitely divisible measures.
It is known that a probability measure $\mu$ belongs to $ID(\boxplus)$ if and only if $\phi_\mu$ extends analytically to $\mathbb{C}^+$ with values in $\mathbb{C}^- \cup \mathbb{R}$. The free infinite divisibility of $\mu$ is also equivalent to the existence of probability measures $\mu^{\boxplus t}$ for $0 \leq t < \infty$ which satisfy $\phi_{\mu^{\boxplus t}}(z) = t\phi_\mu(z)$ [@Be-Vo], while the partial semigroup $\{\mu^{\boxplus t}\}_{t\geq 1}$ always exists for any $\mu \in \mathcal{P}$ [@NS97].
A particularly important subset of $ID(\boxplus)$ is the class of free regular measures.
\[characterization regular\] A probability measure $\mu $ is *free regular* if $\mu ^{\boxplus t}$ exists as a probability measure on $[0,\infty)$ for all $t>0.$
The importance of this class is based on the following properties: this class coincides with the distributions of free subordinators; this class is closed under free multiplicative convolution [@AHS]. Moreover,
\[closure\] Let $\mu $ be free regular and $\sigma$ be freely infinitely divisible. Then $\mu \boxtimes \sigma $ is freely infinitely divisible.
Recall that the semigroup $\{\mathbb{B}_t \}_{t \geq 0}$, introduced by Belinschi and Nica [@BN], is defined to be $$\mathbb{B}_t(\mu) = \Big(\mu^{\boxplus (1+t)}\Big) ^{\uplus\frac{1}{1+t}},~~\mu\in\mathcal{P}.$$ $\mathbb{B}_1$ coincides with the Bercovici-Pata bijection $\Lambda_B$ from the Boolean convolution to the free one. The reader is referred to [@BePa] for the definition of $\Lambda_B$. Let $\phi(\mu)$ denote the free divisibility indicator defined by $$\phi(\mu):=\sup \{t \geq 0: \mu \in \mathbb{B}_t(\mathcal{P}) \}.$$ A probability measure $\mu$ is $\boxplus$-infinitely divisible if and only if $\phi(\mu) \geq 1$. Moreover $\mu^{\boxplus t}$ exists for $t \geq \max\{1-\phi(\mu),0 \}$.
The following property was proved in [@AH2].
\[prop08\] Let $\mu$ be a probability measure on $\mathbb{R}$. Then $\phi(\mu^{\uplus t}) = \frac{1}{t}\phi(\mu)$ for $t > 0$. In particular, $\phi(\mu) = \sup\{ t >0: \mu^{\uplus t} \in ID(\boxplus)\}$.
As a consequence, any Boolean stable law $\mathbf{b}_\alpha^\rho$ has free divisibility indicator $0$ or $\infty$, depending on whether or not $\mathbf{b}_\alpha^\rho$ is in $ID(\boxplus)$. From this observation the first non-trivial example of measures with $\phi(\mu) =\infty$ was found.
[@AH2] \[T2\] The measure $\mathbf{b}_{1/2}^{\rho}$ is $\boxplus$-infinitely divisible for any $\rho \in [0,1]$. In particular, $\phi(\mathbf{b}_{1/2}^{\rho})=\infty$. Moreover, $\mathbf{b}^1_{1/2}$ is free regular.
We will generalize this proposition in the following sections.
Free infinite divisibility of Boolean stable laws {#S3}
=================================================
In this section we completely solve the problem of the free infinite divisibility of the Boolean stable law, that is, we prove Theorem \[T1\]. It suffices to consider $\textbf{b}_\alpha^\rho$ because the free divisibility indicator and hence free infinite divisibility is invariant with respect to the dilation and Boolean convolution with a point measure (see Proposition 3.7 of [@AH2]).
To prove the free infinite divisibility of some Boolean stable laws, the following subclass of $\boxplus$-infinitely divisible distributions will be useful.
A probability measure $\mu$ is said to be in class $\mathcal{UI}$ if $F_\mu$ is univalent in $\mathbb{C}^+$ and, moreover, $F_\mu^{-1}$ has an analytic continuation from $F_\mu(\mathbb{C}^+)$ to $\mathbb{C}^+$ as a univalent function.
The following property was used in [@BBLS]. See also [@AH1] for applications.
\[lem1\] $\mu \in \mathcal{UI}$ implies that $\mu$ is $\boxplus$-infinitely divisible. Moreover, $\mathcal{UI}$ is closed with respect to the weak convergence.
We now go on to the main results.
\[prop90\] Let $0<\alpha\leq\frac{1}{2}$, then $\mathbf{b}_\alpha^\rho$ belongs to $\mathcal{UI}$ for any $\rho \in [0,1]$.
In the proof, the following fact is useful.
\[prop91\] Let $\gamma \subset \mathbb{C}$ be a Jordan curve and $J_\gamma$ be the bounded open set surrounded by $\gamma$. Let $f: J_\gamma \to \mathbb{C}$ be an analytic map which extends to a continuous map from $\overline{J_\gamma}$ to $\mathbb{C}$. If $f$ is injective on $\gamma$, then $f$ is injective in $\overline{J_\gamma}$ and $f(J_\gamma)$ is a bounded Jordan domain surrounded by the Jordan curve $f(\gamma)$.
For the proof, the reader is referred to [@Buc], pp. 310, where the above fact is proved for $\gamma = \{z\in\mathbb{C}: |z|=1\}$. The general case follows from the Carathéodory theorem (or Osgood-Taylor-Carathéodory theorem), see [@Buc], pp. 309.
We introduce some notations: $l_\theta$ will denote the half line $\{re^{i\theta}:r \geq 0\}$; $\mathbb{C}_{A}$ will denote the region $\{z \in \mathbb{C}\setminus\{0\}: \arg z \in A \}$ for $A \subset \mathbb{R}$, $\sup\{|x-y|: x,y\in A\} \leq 2\pi$.
Assume that $\alpha \in(0,\frac{1}{2})$. The case $\alpha =\frac{1}{2}$ follows from the fact that $\mathcal{UI}$ is closed under the weak convergence. Let $\phi:=\alpha\rho\pi \in[0,\alpha \pi]$ and $F(z):=z+e^{i\phi}z^{1-\alpha}$. A key to the proof is the angles $$\theta_1:= -\frac{\phi}{1-\alpha},~~~~\theta_2:=\frac{\pi-\phi}{1-\alpha}.$$ Since $\phi \in [0,\alpha \pi]$, it holds that $\theta_1 \in (-\pi, 0]$ and $\theta_2 \in [\pi, 2\pi)$. Also we can see $\pi < \theta_2 - \theta_1 < 2\pi$. It holds that $F(l_{\theta_1}\cup l_{\theta_2})\subset\mathbb{C}^-\cup\mathbb{R}$ since $$\text{Im}\, F(re^{i\theta_1}) = r \sin\theta_1 \leq 0,~~~~\text{Im}\, F(re^{i\theta_2}) = r \sin\theta_2 \leq 0,~~~r \geq 0.$$ It is sufficient to prove that $F$ is injective on $l_{\theta_1} \cup l_{\theta_2}$ for the following reason. Since $F(z) =z(1+o(1))$ uniformly as $|z| \to \infty$, $z \in \mathbb{C}_{[\theta_1,\theta_2]}$, $F$ is injective on $\{z \in \mathbb{C}: \arg z \in [\theta_1,\theta_2], |z| >R\}$ for sufficiently large $R>0$. Therefore, if $F$ is injective on $l_{\theta_1} \cup l_{\theta_2}$, it is also injective on the curve $\Gamma_R:= \{re^{i\theta_1}: 0\leq r \leq R \}\cup \{re^{i\theta_2}: 0\leq r \leq R \} \cup \{Re^{i\theta}: \theta \in [\theta_1,\theta_2] \}$. Now Proposition \[prop91\] implies that $F$ is injective in the bounded domain surrounded by $\Gamma_R$ for any large $R>0$. Since $F(l_{\theta_1} \cup l_{\theta_2}) \subset \mathbb{C}^-\cup \mathbb{R}$ and $F(z) = z(1+o(1))$ as $|z|\to \infty$, each point of $\mathbb{C}^+$ is surrounded by $F(\Gamma_R)$ exactly once for large $R>0$. This means that $F$ is injective in $\mathbb{C}_{(\theta_1,\theta_2)}$ and $F(\mathbb{C}_{(\theta_1,\theta_2)}) \supset \mathbb{C}^+$ and hence $F^{-1}$ analytically extends to $\mathbb{C}^+$ as a univalent map.
Now let us prove that $F$ is injective on $l_{\theta_1} \cup l_{\theta_2}$. It holds that $F(re^{i\theta_1}) = re^{i\theta_1} + r^{1-\alpha} \in \mathbb{C}_{[\theta_1,0]}$ and similarly $F(re^{i\theta_2}) \in \mathbb{C}_{[\pi,\theta_2]}$ for any $r >0$. Since $0< \theta_2 -\theta_1 < 2\pi$, the intersection $\mathbb{C}_{[\theta_1,0]}\cap \mathbb{C}_{[\pi,\theta_2]}$ is empty. This means $F(l_{\theta_1}) \cap F(l_{\theta_2}) = \{0\}$.
Therefore, we only have to prove that $F$ is injective on each $l_{\theta_k}$. Suppose $z,w \in l_{\theta_1}$ satisfy both $z\neq w$ and $F(z)=F(w)$. With the notation $z = re^{i\theta_1}$ and $w = Re^{i\theta_1}$, it holds that $$\label{eq90}
e^{i\theta_1}=- \frac{R^{1-\alpha} -r^{1-\alpha}}{R-r} <0,$$ a contradiction. Hence $F$ is injective on $l_{\theta_1}$. Similarly $F$ is injective on $l_{\theta_2}$.
A similar method is applicable to a continuous Boolean convolution of Boolean stable laws defined by $$F_{\textbf{b}(\sigma)}(z) = z + \int_{(0,\frac{1}{2}]} e^{i\alpha \pi}z^{1-\alpha} \sigma(d\alpha),$$ where $\sigma$ is a positive finite measure on $(0,\frac{1}{2}]$. Symbolically one may write $$\textbf{b}(\sigma) = \int^{\uplus}_{(0,\frac{1}{2}]} \textbf{b}_\alpha^{1} \sigma(d\alpha).$$ The function $F_{\textbf{b}(\sigma)}$ maps $\mathbb{C}^+$ into itself and satisfies $\lim_{y \to \infty}\frac{F_{\textbf{b}(\sigma)}(iy)}{iy} =1$, so that $F_{\textbf{b}(\sigma)}$ indeed defines a probability measure $\textbf{b}(\sigma)$.
\[mix1\] The measure ${\normalfont \textbf{b}}(\sigma)$ is in $\mathcal{UI}$. Moreover, $\phi({\normalfont\textbf{b}}(\sigma)) = \infty$ and $\mathbf{b}(\sigma)$ is free regular.
We follow the notation of Proposition \[prop90\]. Let $F(z):=z + \int_{(0,\frac{1}{2}]} e^{i\alpha \pi}z^{1-\alpha} \sigma(d\alpha)$. We may assume that $\sigma$ is supported on $(0, \beta]$, where $\beta \in (0,\frac{1}{2})$; a general $\sigma$ can be approximated by such measures. Let $\theta_\beta:=-\frac{\beta\pi}{1-\beta} \in (-\pi,0)$. Then the proof of Proposition \[prop90\] holds with the following modifications. From direct calculation, we can prove that $F(l_{\theta_\beta}\cup l_{\pi}) \subset \mathbb{C}^- \cup \mathbb{R}$ and $F(l_{\theta_\beta}) \cap F(l_\pi) = \{0\}$. Assume that $z=re^{i\theta_\beta} ,w=Re^{i\theta_\beta}$ satisfy $z \neq w$ and $F(z) =F(w)$, then $$e^{i\theta_\beta} = \int_{(0,\beta]}e^{i((\alpha+1)\pi +(1-\alpha) \theta_\beta)}\frac{R^{1-\alpha}-r^{1-\alpha}}{R-r} \sigma(d\alpha).$$ Since $(\alpha+1)\pi +(1-\alpha) \theta_\beta=\pi-\frac{\beta-\alpha}{1-\beta}\pi \in (\theta_\beta+\pi, \pi]$, the above relation causes a contradiction and this proves the injectivity on $l_{\theta_\beta}$. Similarly, $F$ is injective on $l_\pi$.
The property $\phi(\mathbf{b}(\sigma)) =\infty$ follows from Proposition \[prop08\] in combination with the property $\textbf{b}(\sigma)^{\uplus t} = \textbf{b}(t\sigma) \in ID(\boxplus)$. The free regularity is a consequence of Theorem 13 [@AHS].
Let $\frac{1}{2}<\alpha\leq\frac{2}{3}$ and $2-\frac{1}{\alpha} \leq \rho \leq \frac{1}{\alpha}-1$, then $\mathbf{b}_\alpha^\rho$ is freely infinitely divisible, but $\mathbf{b}_\alpha^\rho \notin\mathcal{UI}$.
Let $\phi:= \alpha \rho\pi$ and $F=F_{\mathbf{b}_\alpha^\rho}$. First we assume that $(2\alpha-1)\pi < \phi < (1-\alpha)\pi$. This inequality implies $\theta_1 \in (-\pi,0)$ and $\theta_2 \in (\pi,2\pi)$. We will prove that $\phi(z)=F^{-1}(z)-z$ has an analytic continuation to $\mathbb{C}^+$ with values in $\mathbb{C}^- \cup \mathbb{R}$.
We follow the notation of Proposition \[prop90\], but for simplicity, let $U_1:= \mathbb{C}_{(\theta_1,\pi)}$ and $U_2:=\mathbb{C}_{(0,\theta_2)}$. In this case, $\theta_2-\theta_1 > 2\pi$, so that we have to consider a Riemannian surface made of $U_1$ and $U_2$, glued on $\mathbb{C}^+$. Let us consider the analytic maps $F_i:=F|_{U_i}$ for $i=1,2$.
Following the arguments of Proposition \[prop90\], we can prove that $F_i$ is univalent in $U_i$ for $i=1,2$. For example, let us consider $i=1$. From simple observations, we find $F_1(re^{i\theta_1}) = re^{i\theta_1} + r^{1-\alpha} \in \mathbb{C}_{(\theta_1,0)}$ and $F_1(-r) =-r + r^{1-\alpha} e^{i(\phi+(1-\alpha)\pi)} \in \mathbb{C}_{(\phi + (1-\alpha)\pi,\pi)}$ for any $r>0$. Hence $F(l_{\theta_1}) \cap F(l_{\pi}) = \{0\}$.
If $z = re^{i\theta_1}, w=Re^{i\theta_1}$ satisfy $z \neq w$, $F_1(z) =F_1(w)$, then (\[eq90\]) holds, which contradicts the property $\theta_1 \in (-\pi,0)$. If $z,w \in l_\pi$, $z \neq w$, then $e^{i(\phi+(1-\alpha)\pi)} = \frac{R-r}{R^{1-\alpha}-r^{1-\alpha}} >0$, a contradiction. The above arguments imply $F_1$ is univalent in $U_1$ and $F_1(U_1) \supset \mathbb{C}_{(0, \phi+(1-\alpha)\pi)}$. Similarly one can show that $F_2$ is univalent in $U_2$ and $F_2(U_2) \supset \mathbb{C}_{(\phi,\pi)}$. Notice that $F_1(U_1) \cup F_2(U_2) \supset \mathbb{C}^+$. These properties still hold for $\phi\in\{(2\alpha-1)\pi, (1-\alpha)\pi\}$ by taking limits.
The map $F^{-1}$ extends analytically to $\mathbb{C}^+$ as follows: $$F^{-1}(z):= \left \{ \begin{array}{cc}
F^{-1}_1(z), & z\in F_1(U_1)\cap \mathbb{C}^+, \\
F^{-1}_2(z), & z\in F_2(U_2)\cap \mathbb{C}^+.
\end{array}\right.$$ This map is well defined for the following reason. Notice that $F_1(U_1\setminus \mathbb{C}^+)\cap F_2(U_2\setminus \mathbb{C}^+) \subset \mathbb{C}^-$. Indeed, $F_1(U_1\setminus \mathbb{C}^+)\subset\mathbb{C}_{(\theta_1, \phi)}$ and $F_2(U_2\setminus \mathbb{C}^+)\subset\mathbb{C}_{(\phi+(1-\alpha)\pi, \theta_2)}$. If $z\in F_1(U_1)\cap F_2(U_2)\cap \mathbb{C}^+$, then there are $z_1\in U_1$ and $z_2\in U_2$ with $z=F_1(z_1)=F_2(z_2)$. Either $z_1$ or $z_2$ shall be in $\mathbb{C}^+$ because $F_1(U_1\setminus \mathbb{C}^+)\cap F_2(U_2\setminus \mathbb{C}^+) \subset \mathbb{C}^-$. If $z_1\in\mathbb{C}^+$(resp. $z_2\in\mathbb{C}^+$), by definition $F_1(z_1)=F_2(z_1)$(resp. $F_1(z_2)=F_2(z_2)$) and hence $z_1=z_2$ from the univalence of $F_2$.
Now, we see that $\phi(z):=F^{-1}(z)-z$ maps $\mathbb{C}^+$ into $\mathbb{C}^- \cup \mathbb{R}$. If $z\in F(\mathbb{C}^+)$, then there is $w\in\mathbb{C}^+$ such that $\phi(z)=w-F(w)$ and then $\text{Im}\, \phi(z)=\text{Im}\, w-\text{Im}\, F(w)\leq 0$ since $\text{Im}\, F(w) > \text{Im}\, w$. If $z\notin F(\mathbb{C}^+)$ either $z\in F_1(U_1)$ or $z\in F_2(U_2)$. Assume without loss of generality that $z\in F_1(U_1)$. Then there is $w\in U_1\setminus \mathbb{C}^+$ such that $\phi(z)=w-F_1(w)=w-z$ and then since $\text{Im}\, z>0$ and $\text{Im}\, w\leq0$ we have $\text{Im}\, \phi(z)\leq 0.$ This proves that $\phi(z)=F^{-1}(z)-z$ maps $\mathbb{C}^+$ into $\mathbb{C}^- \cup \mathbb{R}$ and therefore $\mathbf{b}^\rho_\alpha$ is freely infinitely divisible.
We prove that $\mathbf{b}^\rho_\alpha\notin\mathcal{UI}$. The sheets $U_1$ and $U_2$ have an intersection on $\mathbb{C}^-$. If we take a $w$ from the intersection with sufficiently small $|w|$, then we can prove $F_1(w) \neq F_2(w)$ and $F_1(w), F_2(w) \in \mathbb{C}^+$. This means $F^{-1}$ is not univalent in $\mathbb{C}^+$.
The case $(\alpha,\rho) =(1,\frac{1}{2})$ corresponds to the Cauchy distribution, which is freely infinitely divisible. Now we show that $\textbf{b}_\alpha^\rho$ does not belong to $ID(\boxplus)$ for the remaining parameters. First we consider the case $\alpha \geq 1$.
Let $\mu$ be $\boxplus$-infinitely divisible and $\mu_t:=\mu^{\boxplus t}$. For $s \leq t$, a subordination function $\omega_{s,t}: \mathbb{C}^+\to\mathbb{C}^+$ exists so that it satisfies $F_{\mu_s} \circ \omega_{s,t} = F_{\mu_t}$. This relation is equivalent to $$\label{eq1}
F_{\mu_t}(z)= \frac{t/s}{t/s-1}\omega_{s,t}(z)-\frac{z}{t/s-1}.$$ It is proved in Theorem 4.6 of [@Bel3] that $\omega_{s,t}$ and hence $F_{\mu_t}$ extends to a continuous function from $\mathbb{C}^+\cup \mathbb{R}$ into itself.
Boolean stable laws $\mathbf{b}_\alpha^\rho$ are not $\boxplus$-infinitely divisible in the following cases:
\(1) $\alpha \in (1, 2]$,
\(2) $\alpha = 1$ and $\rho \neq \frac{1}{2}$.
In these cases $F_{\textbf{b}_\alpha^\rho}(0) = \infty$, and hence $F_{\textbf{b}_\alpha^\rho}$ does not extend to a continuous function on $\mathbb{C}^+ \cup \mathbb{R}$.
It only remains to discuss the case $\frac{1}{2}<\alpha<1$.
Let $\frac{1}{2}<\alpha<1$, then $\mathbf{b}^\rho_\alpha$ is not $\boxplus$-infinitely divisible unless $\alpha\leq\frac{2}{3}$ and $2-\frac{1}{\alpha}\leq\rho\leq\frac{1}{\alpha}-1$.
Let $\phi:=\alpha\rho\pi$ and $F(z):=F_{\normalfont \textbf{b}_\alpha^\rho}(z)$. Suppose $\phi<(2\alpha-1)\pi$. A key quantity here is the angle $\theta_3:=\frac{\phi+\pi}{\alpha}$. It holds that $\pi<\theta_3 < 2\pi$ and $$\begin{aligned}
F(re^{i\theta_3})&=&re^{i\theta_3}+e^{i\phi}r^{1-\alpha} e^{i(1-\alpha)\theta_3}\\ \notag
&=& (r-r^{1-\alpha})e^{i\theta_3}.\end{aligned}$$ Hence $F$ is not injective on $l_{\theta_3}$ and $F(l_{\theta_3}) = \{re^{i\theta_3}: r \geq -\alpha(1-\alpha)^{\frac{1-\alpha}{\alpha}}\}$. On the other hand, $F(re^{i\phi/\alpha}) = (r +r^{1-\alpha})e^{i\phi/\alpha}$ and then $F$ is bijective from $l_{\phi/\alpha}$ onto itself. Let us prove that the map $F$ is univalent in $\mathbb{C}_{(\phi/\alpha,\theta_3)}$.
We fix $\theta\in (\phi/\alpha,\theta_3)$. Then $ \theta > \phi+ (1-\alpha)\theta > \theta -\pi$ and hence $F(re^{i\theta}) =re^{i\theta} + r^{1-\alpha}e^{i(\phi+(1-\alpha)\theta)} \in \mathbb{C}_{(\phi+(1-\alpha)\theta, \theta)}$. Since $\phi+(1-\alpha)\theta > \phi/\alpha$, $F(l_\theta)$ does not intersect $l_{\phi/\alpha}$ except at $0$. Similarly to the proof of Proposition \[prop90\], $F$ is injective on $l_{\theta}$, to conclude $F$ is univalent in $\mathbb{C}_{(\phi/\alpha, \theta)}$.
Therefore, $F$ maps $\mathbb{C}_{(\phi/\alpha, \theta_3)}$ onto $\mathbb{C}_{(\phi/\alpha,\theta_3)} \setminus \{rz_0: r \in [0,1] \}$, where $z_0 = -\alpha(1-\alpha)^{\frac{1-\alpha}{\alpha}}e^{i\theta_3}$. We can find $0< r_1<r_2$ such that $F(r_1e^{i\theta_3})=F(r_2e^{i\theta_3})=\frac{z_0}{2} \in \mathbb{C}^+$, so that $$\lim_{\epsilon \rightarrow 0^+}F^{-1}\left(\frac{z_0}{2}+\epsilon\right)=r_1e^{i\theta_3}$$ but $$\lim_{\epsilon \rightarrow 0^-}F^{-1}\left(\frac{z_0}{2}+\epsilon \right)=r_2e^{i\theta_3}.$$ Hence the Voiculescu transform does not extend to $\mathbb{C}^+$ continuously.
The case $\phi>(1-\alpha)\pi$ is analogous by choosing $\theta_4=\frac{\phi-\pi}{\alpha} \in (-\pi,0)$ instead of $\theta_3$.
Classical infinite divisibility of positive Boolean stable laws {#S4}
===============================================================
For the convenience of the reader we start recalling basic facts on classical infinite divisibility and completely monotonic functions. For $\mu,\nu\in\mathcal{P}$, let $\mu\ast\nu$ denote the classical convolution of $\mu$ and $\nu$.
A probability measure $\mu$ is said to be *$\ast$-infinitely divisible* if for each $n\in\mathbb{N}$ there exist $\mu_n$ such that $\mu=\mu_n^{\ast n}.$ We will denote by $ID(\ast)$ the set of $\ast$-infinitely divisible measures.
A function $f: (0,\infty) \to \mathbb{R}$ is said to be *completely monotonic (c.m.)*, if it possesses derivatives $f^{(n)}(x)$ for all $ n = 0, 1,2, 3,\dots$ and if $$(-1)^nf^{(n)}(x)\geq 0, ~~x > 0,~~ n=0,1,2,\dots.$$
The importance of complete monotonicity in relation to classical infinite divisibility is given by the following well known criterion (see e.g. [@Fell]).
Let $\mu$ be a probability distribution on $[0,\infty)$ with absolutely continuous density w.r.t. the Lebesgue measure. If the density is c.m. then the measure is in $ID(\ast)$.
We will use the following well known properties of c.m. functions.
\[cm10\] (1) If $f(x)$ and $g(x)$ are c.m., then $f(x)g(x)$, $f(x)+g(x)$ are also c.m.
\(2) Let $f(x)$ be c.m. and $h(x)$ be a positive differentiable function. If $h'(x)$ is c.m. then $f(h(x))$ is also a c.m. function.
The density of the Boolean stable law on $[0,\infty)$ can be calculated as $$\textbf{b}_\alpha^1 (dx)= \frac{1}{\pi} \frac{\sin(\alpha\pi) x^{\alpha-1}}{x^{2\alpha}+2\cos(\alpha\pi)x^\alpha +1}dx,~~~x>0,$$ for $\alpha \in (0,1)$. This density is c.m. for $\alpha \leq 1/2$ by basic properties of c.m. functions, see Proposition \[cm10\]. Indeed, the function $h(x)=x^{2\alpha}+2 \cos(\alpha \pi)x^\alpha + 1$ has a c.m. derivative $h'(x)$, so that $\frac{1}{h(x)}$ is c.m. The function $\frac{x^{\alpha-1}}{h(x)}$ is the multiplication of two c.m. functions and hence is c.m. Thus we have shown the classical infinite divisibility together with the free infinite divisibility:
The Boolean stable law ${\normalfont \bf b}_\alpha^1$ on $[0,\infty)$ belongs to $ID(\boxplus)\cap ID(\ast)$ for $\alpha \leq 1/2$.
We note that $\textbf{b}_\alpha^{1}$ is a *generalized gamma convolution* (GGC) which is a class of $ID(\ast)$. This fact follows from Theorem 2 of [@Bon].
More generally the complete monotonicity is still true for continuous Boolean convolutions of Boolean stable laws $\textbf{b}(\sigma)$ and hence one concludes the following:
\[mix2\] The continuous Boolean convolution ${\normalfont \bf b}(\sigma) = \int^{\uplus}_{(0,\frac{1}{2}]} {\normalfont \bf b}_\alpha^1 \sigma(d\alpha)$ is in $ID(\boxplus)\cap ID(\ast)$ for any non-negative finite measure $\sigma$ on $(0,\frac{1}{2}]$.
The density of $\textbf{b}(\sigma)$ is given by $$\frac{1}{\pi}\cdot\frac{\int_{(0,\frac{1}{2}]}\sin(\alpha\pi)x^{-\alpha}\sigma(d\alpha)}{\left(x^{1/2}+\int_{(0,\frac{1}{2}]}\cos(\alpha\pi)x^{1/2-\alpha}\sigma(d\alpha)\right)^2 +\left(\int_{(0,\frac{1}{2}]}\sin(\alpha\pi)x^{1/2-\alpha}\sigma(d\alpha)\right)^2},~~~x>0.$$ We first consider the case where $\sigma$ is a finite sum of atoms: $\sigma= \sum_{k=1}^m \lambda_k \delta_{\alpha_k}$, $\alpha_k \in (0,\frac{1}{2}]$, $\lambda_k > 0$. Then the density of $\textbf{b}(\sigma)$ is given by $\frac{f(x)}{\pi g(x)}$, where $$\begin{split}
&f(x) = \sum_{k=1}^m \lambda_k \sin(\alpha_k\pi) x^{-\alpha_k},\\
&g(x) = \left(x^{1/2}+\sum_{k=1}^m \lambda_k \cos(\alpha_k\pi) x^{1/2-\alpha_k}\right)^2 + \left(\sum_{k=1}^m \lambda_k \sin(\alpha_k\pi) x^{1/2-\alpha_k}\right)^2.
\end{split}$$
The function $f(x)$ is c.m. $g(x)$ is a finite sum of $\{x^{\beta}:\beta \in [0,1]\}$ with positive coefficients and hence $g'(x)$ is c.m. Therefore, the density $\frac{f(x)}{\pi g(x)}$ is c.m.
For a general finite Borel measure $\sigma$, consider $\sigma_n:= \sum_{k=0}^{n-1} \sigma\left((\frac{k}{2n}, \frac{k+1}{2n}]\right) \delta_{\frac{2k+1}{4n}}$. Then $\frac{d^p}{dx^p}\int_{(0,\frac{1}{2}]}e^{i\alpha \pi}x^{1/2-\alpha}\sigma_n(d\alpha)$ converges to $\frac{d^p}{dx^p}\int_{(0,\frac{1}{2}]}e^{i\alpha\pi}x^{1/2-\alpha}\sigma(d\alpha)$ locally uniformly on $(0,\infty)$ for any integer $p \geq 0$. This approximation argument shows that the denominator of the density of $\textbf{b}(\sigma)$ has c.m. derivative, and hence $\textbf{b}(\sigma)$ itself has c.m. density.
Relations between Boolean, free and classical stable laws and multiplicative convolutions {#S2}
=========================================================================================
We prove a reproducing property of Boolean stable laws with respect to the free multiplicative convolution. The* free multiplicative convolution* $\boxtimes$ of probability measures with bounded support on $[0,\infty)$ is defined as follows [@Voi87]. Let $X_1,X_2$ be positive free random variables following distributions $\mu_1,\mu_2$, respectively. Then the free multiplicative convolution $\mu_1 \boxtimes \mu_2$ is defined by the distribution of $X^{1/2}YX^{1/2}$. More generally, free multiplicative convolution $\mu_1 \boxtimes \mu_2$ can be defined for any two probability measures $\mu_1$ and $\mu_2$ on $\mathbb{R}$, provided that one of them is supported on $[0,\infty)$; see [@BeVo93].
Let $\textbf{s}_\alpha^\rho$ be the free strictly stable law characterized by $$\phi_{\textbf{s}_\alpha^\rho}(z) = -e^{i\alpha\rho\pi}z^{1-\alpha}, ~~z \in \mathbb{C}^+,$$ $\alpha \in (0,1], \rho\in [0,1]$. Results of [@APA] and [@BePa] show a reproducing property for $\boxplus$-strictly stable laws $\mathbf{s}^\theta_{\alpha}$: $$\label{eq13}
\mathbf{s}^\rho_{1/(1+t)}\boxtimes\mathbf{s}^1_{1/(1+s)}=\mathbf{s}^\rho_{1/(1+s+t)}$$ for $\rho \in \{0,\frac{1}{2}, 1 \}$. This property is still unknown for general $\rho \in [0,1]$.
We show that the same reproducing property holds for $\textbf{b}_\alpha^\rho$, including a new result for general $\rho \in [0,1]$.
\(1) For any $s,t>0$ and $\rho \in \{0,\frac{1}{2},1\}$, $\mathbf{b}^\rho_{1/(1+t)}\boxtimes\mathbf{b}^1_{1/(1+s)} =\mathbf{b}^\rho_{1/(1+s+t)}$.
\(2) For any $s,t >0$ and $\rho \in [0,1]$, $\mathbf{s}^\rho_{1/(1+t)}\boxtimes\mathbf{s}^1_{1/(1+s)}$ is a $\boxplus$-strictly stable law with stability index $\frac{1}{1+s+t}$ and $\mathbf{b}^\rho_{1/(1+t)}\boxtimes\mathbf{b}^1_{1/(1+s)}$ is a $\uplus$-strictly stable law with stability index $\frac{1}{1+s+t}$.
\(1) Since $\mathbb{B}_1$ is the Bercovici-Pata bijection from the Boolean convolution to the free one, $\mathbb{B}_1(\textbf{b}_\alpha^\rho) = \textbf{s}_\alpha^\rho$. Moreover, $\mathbb{B}_1$ is a homomorphism with respect to $\boxtimes$ [@BN], and hence (\[eq13\]) implies $$\mathbb{B}_1(\textbf{b}_{1/(1+t)}^\rho \boxtimes \textbf{b}_{1/(1+s)}^1) = \mathbb{B}_1(\textbf{b}_{1/(1+s+t)}^\rho),$$ and hence $\mathbf{b}^\rho_{1/(1+t)}\boxtimes\mathbf{b}^1_{1/(1+s)} =\mathbf{b}^\rho_{1/(1+s+t)}$ from the injectivity of $\mathbb{B}_1$.
\(2) Suppose $\mu$ or $\nu$ is supported on $[0,\infty)$. Then the relation $$\label{eq23}
(\mu \boxtimes \nu)^{\uplus t} = D_{1/t} (\mu^{\uplus t} \boxtimes \nu^{\uplus t})$$ holds for $t>0$, while $$(\mu \boxtimes \nu)^{\boxplus t} = D_{1/t} (\mu^{\boxplus t} \boxtimes \nu^{\boxplus t})$$ holds for $t\geq 1$; see Proposition 3.5 of [@BN]. Using (\[eq23\]), $$\begin{split}
(\mathbf{b}^\rho_{1/(1+t)}\boxtimes\mathbf{b}^1_{1/(1+s)})^{\uplus 2}
&=D_{1/2}\left((\mathbf{b}^\rho_{1/(1+t)})^{\uplus 2}\boxtimes(\mathbf{b}^1_{1/(1+s)})^{\uplus 2}\right)\\
&=D_{1/2}\left(D_{2^{1+t}}(\mathbf{b}^\rho_{1/(1+t)})\boxtimes D_{2^{1+s}}(\mathbf{b}^1_{1/(1+s)})\right)\\
&=D_{2^{1+t+s}}(\mathbf{b}^\rho_{1/(1+t)}\boxtimes\mathbf{b}^1_{1/(1+s)}).
\end{split}$$ By definition, $\mathbf{b}^\rho_{1/(1+t)}\boxtimes\mathbf{b}^1_{1/(1+s)}$ is $\uplus$-strictly stable with stability index $\frac{1}{1+s+t}$. The proof for the free case is similar.
This reproducing property combined with the fact $\mathbf{b}^\rho_{1/2} \in ID(\boxplus)$ shows that some Boolean stable laws are in $ID(\boxplus)$, giving an alternative look at Theorem \[T1\].
The Boolean stable law $\mathbf{b}_{1/n}^\rho$ is freely infinitely divisible if $n \in \{2,3,4,\cdots \}$ and $\rho \in \{0,\frac{1}{2},1\}$.
Since $\mathbf{b}^\rho_{1/2} \in ID(\boxplus)$ and $\mathbf{b}^1_{1/2}$ is free regular by Proposition \[T2\], $\mathbf{b}^\rho_{1/2}\boxtimes \mathbf{b}^1_{1/2}=\mathbf{b}^\rho_{1/3}$ is also freely infinitely divisible by Theorem \[closure\]. In a similar way, by induction, we conclude that $\textbf{b}^\rho_{1/n}$ is $\boxplus$-infinitely divisible for $n \geq 2$.
We note here an interplay among free, Boolean and classical stable laws. Let $\textbf{s}_\alpha$ be a free strictly stable law on $[0,\infty)$ and $\textbf{n}_\alpha$ be a classical strictly stable law defined by $$\begin{split}
&\phi_{\textbf{s}_\alpha} (z) = -e^{i\alpha \pi}z^{1-\alpha},~~~z \in \mathbb{C}^+, \\
&\int_{0}^\infty e^{-xz} \textbf{n}_\alpha(dx)= e^{-z^\alpha}, ~~z >0.
\end{split}$$ Let $\circledast$ denote the classical multiplicative convolution: $$\int_{[0,\infty)}f(x)(\mu \circledast \nu)(dx) = \int_{[0,\infty)^2}f(xy)\mu(dx)\nu(dy)$$ for any bounded continuous function $f$ on $[0,\infty)$. If $\check{\textbf{s}}_\alpha$ and $\check{\textbf{n}}_\alpha$ respectively denote the push-forwards of $\textbf{s}_\alpha$ and $\textbf{n}_\alpha$ by the map $x \mapsto 1/x$, then $\textbf{s}_\alpha \boxtimes \check{\textbf{s}}_\alpha = \textbf{n}_\alpha \circledast \check{\textbf{n}}_\alpha$ for $\alpha \in (0,1]$ as proved in [@BePa]. Moreover, this coincides with a Boolean stable law: $$\textbf{s}_\alpha \boxtimes \check{\textbf{s}}_\alpha = \textbf{n}_\alpha \circledast \check{\textbf{n}}_\alpha = \textbf{b}_\alpha^1.$$ This fact follows from the calculation of the density shown in Proposition A4.4 of [@BePa].
[100]{} O. Arizmendi and T. Hasebe, On a class of explicit Cauchy-Stieltjes transforms related to monotone stable and free Poisson laws, to appear in Bernoulli. arXiv:1108.3438 O. Arizmendi and T. Hasebe, Semigroups related to additive and multiplicative, Boolean and free convolutions. arXiv:1105.3344 O. Arizmendi, T. Hasebe and N. Sakuma, On the law of free subordinators. arXiv:1201.0311 O. Arizmendi and V. Pérez-Abreu, The S-transform for symmetric probability measures with unbounded supports, Proc. Amer. Math. Soc. **137** (2009), 3057–3066. S.T. Belinschi and H. Bercovici, Partially defined semigroups relative to multiplicative free convolution, Int. Math. Res. Notices, No. 2 (2005), 65–101. S.T. Belinschi, M. Bożejko, F. Lehner and R. Speicher, The normal distribution is $\boxplus$-infinitely divisible, Adv. Math. **226**, No. 4 (2011), 3677–3698.
S.T. Belinschi and A. Nica, On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution, Indiana Univ. Math. J. **57**, No. 4 (2008), 1679–1713. H. Bercovici and V. Pata, Stable laws and domains of attraction in free probability theory (with an appendix by Philippe Biane), Ann. of Math. (2) **149**, No. 3 (1999), 1023–1060. H. Bercovici and D. Voiculescu, Lévy-Hinčin type theorems for multiplicative and additive free convolution, Pacific J. Math. **153**, No. 2 (1992), 217–248. H. Bercovici and D. Voiculescu, Free convolution of measures with unbounded support, Indiana Univ. Math. J. **42**, No. 3 (1993), 733–773. L. Bondesson, A general result on infinite divisibility, Ann. Probab. **7**, No. 6 (1979), 965–979. R.B. Burckel, *An introduction to classical complex analysis*, vol. 1, Academic Press, New York, 1979. W. Feller, An Introduction to Probability Theory and its Applications, vol.II. John Wiley and Sons, 1971. A. Nica and R. Speicher, On the multiplication of free $n$-tuples of noncommutative random variables, Amer. J. Math. **118** (1996), 799–837. V. Pérez-Abreu and N. Sakuma, Free generalized gamma convolutions, Electron. Commun. Probab. **13** (2008) 526–539. R. Speicher and R. Woroudi, Boolean convolution, in Free Probability Theory, Ed. D. Voiculescu, Fields Inst. Commun. **12** (Amer. Math. Soc., 1997), 267–280. D. Voiculescu, Addition of certain non-commutative random variables, J. Funct. Anal. **66** (1986), 323–346. D. Voiculescu, Multiplication of certain noncommuting random variables, J. Operator Theory **18** (1987), 223–235.
[^1]: OA: Universität des Saarlandes, FR $6.1-$Mathematik, 66123 Saarbrücken, Germany, email: arizmendi@math.uni-sb.de. Supported by DFG-Deutsche Forschungsgemeinschaft Project SP419/8-1
[^2]: TH: Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan, email: thasebe@math.kyoto-u.ac.jp. Supported by Global COE program “Fostering top leaders in mathematics—broadening the core and exploring new ground” at Kyoto university.
|
---
abstract: 'We provide a rationale for the Planck scale being the minimum scale in the universe, as also its specific numerical values. In the process we answer the question of why the Planck scale is $10^{20}$ times the Compton scale of elementary particles. These considerations show how the Planck scale provides an underpinning for space time.'
author:
- |
B.G. Sidharth\
International Institute for Applicable Mathematics & Information Sciences\
Hyderabad (India) & Udine (Italy)\
B.M. Birla Science Centre, Adarsh Nagar, Hyderabad - 500 063 (India)
title: THE PLANCK SCALE UNDERPINNING FOR SPACE TIME
---
Introduction
============
In the latest approaches such as Quantum Superstrings (or M-Theory) or Quantum Gravity Theories it is generally accepted that the Planck scale defines a minimum scale for the universe [@ven; @kempf; @garay; @crane; @psu; @psp]. In these schemes there is a maximal mass, the Planck mass $m_P \sim 10^{-5}gms$ which is defined by $$m_P = \left(\frac{\hbar c}{G}\right)^{1/2} \approx 10^{-5}gms\label{e1}$$ Using the value for $m_P$ we can define the Planck length $l_P \sim 10^{-33}cms$ and the Planck time $t_P \sim 10^{-42}secs$, which are the Compton lengths and times for the mass in (\[e1\]). It may be mentioned that these values were postulated by Planck himself. Today the values for the minimum scale as given in (\[e1\]) are taken for granted. We first provide a rationale for the numerical value of the Planck scale.\
The Planck Scale
================
Our starting point is the model for the underpinning at the Planck scale for the universe. This is a collection of $N$ Planck scale oscillators (Cf.refs.[@psu; @psp; @uof; @gip; @ng] for details). We do not need to specfify $N$. We have in this case the following well known relations $$R = \sqrt{N}l, Kl^2 = kT,$$ $$\omega^2_{max} = \frac{K}{m} = \frac{kT}{ml^2}\label{e2}$$ In (\[e2\]), $R$ is of the order of the diameter of the universe, $K$ is the analogue of the spring constant, $T$ is the effective temperature while $l$ is the analogue of the Planck length, $m$ the analogue of the Planck mass and $\omega_{max}$ is the frequency-the reason for the subscript $max$ will be seen below. We do not yet give $l$ and $m$ their usual values as given in (\[e1\]) for example, but rather try to deduce these values.\
We now use the well known result that the individual minimal oscillator particles are black holes or mini universes as shown by Rosen [@rosen]. So using the well known Beckenstein temperature formula for these primordial black holes [@ruffini], that is $$kT = \frac{\hbar c^3}{8\pi Gm}$$ in (\[e2\]) we get, $$Gm^2 \sim \hbar c\label{e3}$$ We can easily verify that (\[e3\]) leads to the value $m \sim 10^{-5}gms$. In deducing (\[e3\]) we have used the typical expressions for the frequency as the inverse of the time - the Compton time in this case and similarly the expression for the Compton length. However it must be reiterated that no specific values for $l$ or $m$ were considered in the deduction of (\[e3\]).\
We now make two interesting comments. Cercignani and co-workers have shown [@cer1; @cer2] that when the gravitational energy becomes of the order of the electromagnetic energy in the case of the Zero Point oscillators, that is $$\frac{G\hbar^2 \omega^3}{c^5} \sim \hbar \omega\label{e4}$$ then this defines a threshold frequency $\omega_{max}$ above in which the oscillations become chaotic.\
Secondly from the parallel but unrelated theory of phonons [@huang; @rief], which are also bosonic oscillators, we deduce a maximal frequency given by $$\omega^2_{max} = \frac{c^2}{l^2}\label{e5}$$ In (\[e5\]) $c$ is, in the particular case of phonons, the velocity of propagation, that is the velocity of sound, whereas in our case this velocity is that of light. Frequencies greater than $\omega_{max}$ in (\[e5\]) are meaningless. We can easily verify that (\[e4\]) and (\[e5\]) give back (\[e3\]).\
Finally we can see from (\[e2\]) that, given the value of $l_P$ and using the value of the radius of the universe, viz., $R \sim 10^{27}$, we can deduce that, $$N' \sim 10^{120}\label{e6}$$
The Gauge Hierarchy Problem
===========================
We next consider a long standing puzzle, the so called gauge hierarchy problem, that is why is there such a wide gap between the mass of a Planck particle, $10^{-5}gms$ and the mass of a typical elementary particle $\sim 10^{-25}gms$. We show that the answer to this problem lies in a particular characterization of gravitation. This moreover also provides a picture of a Planck scale underpinning for the entire universe itself.\
In 1997 the author put forward a model in which particles are created fluctuationally in a phase transition from the background Zero Point Field or Dark Energy. This model lead to dramatic consequences: The Universe would be accelerating and expanding with a small cosmological constant. Besides, several longstanding puzzling relations of the so called large number genre which had no explanation whatsoever, were now deduceable from the theory.\
At that time the accepted Standard Big Bang model predicted a dark matter filled decelerating universe–in other words the exact opposite. In 1998 however the first results were announced by Perlmutter, Kirshner and coworkers, from a study of distant supernovae that the universe was indeed accelerating and expanding with a small cosmological constant, contrary to belief. These conclusions were subsequently reconfirmed several times. This work was the breakthrough of the year 1998 of the American Association for the Advancement of Science [@science] while Dark Energy itself was subsequently confirmed through the WMAP and the Sloan Digital Sky Survey. This in fact was the breakthrough of the year 2003 [@science2].\
To recapitulate the author’s model [@mg8; @ijmpa; @ijtp; @csfnc; @csfnc2; @cu; @uof], we give a simple picture and will return to the nuances later.\
Our starting point is the all permeating Zero Point Field or the Dark Energy, from which the elementary particles are created. As Wheeler put it [@mwt], “From the zero-point fluctuations of a single oscillator to the fluctuations of the electromagnetic field to geometrodynamic fluctuations is a natural order of progression...”\
Let us consider, following Wheeler a harmonic oscillator in its ground state. The probability amplitude is $$\psi (x) = \left(\frac{m\omega}{\pi \hbar}\right)^{1/4}
e^{-(m\omega/2\hbar)x^2}$$ for displacement by the distance $x$ from its position of classical equilibrium. So the oscillator fluctuates over an interval $$\Delta x \sim (\hbar/m\omega)^{1/2}$$ The electromagnetic field is an infinite collection of independent oscillators, with amplitudes $X_1,X_2$ etc. The probability for the various oscillators to have emplitudes $X_1, X_2$ and so on is the product of individual oscillator amplitudes: $$\psi (X_1,X_2,\cdots ) = exp [-(X^2_1 + X^2_2 + \cdots)]$$ wherein there would be a suitable normalization factor. This expression gives the probability amplitude $\psi$ for a configuration $B (x,y,z)$ of the magnetic field that is described by the Fourier coefficients $X_1,X_2,\cdots$ or directly in terms of the magnetic field configuration itself by $$\psi (B(x,y,z)) = P exp \left(-\int \int \frac{\bf{B}(x_1)\cdot
\bf{B}(x_2)}{16\pi^3\hbar cr^2_{12}} d^3x_1 d^3x_2\right).$$ $P$ being a normalization factor. Let us consider a configuration where the magnetic field is everywhere zero except in a region of dimension $l$, where it is of the order of $\sim \Delta B$. The probability amplitude for this configuration would be proportional to $$\exp [-(\Delta B)^2 l^4/\hbar c)$$ So the energy of fluctuation in a region of length $l$ is given by finally [@mwt; @r24; @r25] $$B^2 \sim \frac{\hbar c}{l^4}$$ In the above if $l$ is taken to be the Compton wavelength of a typical elementary particle, then we recover its energy $mc^2$, as can be easily verified.\
It may be mentioned that Einstein himself had believed that the electron was a result of such a condensation from the background electromagnetic field (Cf.[@r26; @cu] for details). We will return to this point again. We also take the pion to represent a typical elementary particle, as in the literature.\
To proceed, as there are $N \sim 10^{80}$ such particles in the universe, we get $$Nm = M\label{e1a}$$ where $M$ is the mass of the universe.\
In the following we will use $N$ as the sole cosmological parameter.\
Equating the gravitational potential energy of the pion in a three dimensional isotropic sphere of pions of radius $R$, the radius of the universe, with the rest energy of the pion, we can deduce the well known relation [@csfnc; @r28; @r29] $$R \approx \frac{GM}{c^2}\label{e2a}$$ where $M$ can be obtained from (\[e1a\]).\
We now use the fact that given $N$ particles, the fluctuation in the particle number is of the order $\sqrt{N}$[@r29; @huang; @ijmpa; @ijtp; @r5; @mg8], while a typical time interval for the fluctuations is $\sim \hbar/mc^2$, the Compton time. We will come back to this point later in the context of the minimum Planck scale: Particles are created and destroyed - but the ultimate result is that $\sqrt{N}$ particles are created. So we have, as we saw briefly earlier, $$\frac{dN}{dt} = \frac{\sqrt{N}}{\tau}\label{ex}$$ whence on integration we get, (remembering that we are almost in the continuum region), $$T = \frac{\hbar}{mc^2} \sqrt{N}\label{e3a}$$ We can easily verify that the equation is indeed satisfied where $T$ is the age of the universe. Next by differentiating (\[e2a\]) with respect to $t$ we get $$\frac{dR}{dt} \approx HR\label{e4a}$$ where $H$ in (\[e4a\]) can be identified with the Hubble Constant, and using (\[e2a\]) is given by, $$H = \frac{Gm^3c}{\hbar^2}\label{e5a}$$ Equation (\[e1a\]), (\[e2a\]) and (\[e3a\]) show that in this formulation, the correct mass, radius, Hubble constant and age of the universe can be deduced given $N$ as the sole cosmological or large scale parameter. Equation (\[e5a\]) can be written as $$m \approx \left(\frac{H\hbar^2}{Gc}\right)^{\frac{1}{3}}\label{e6a}$$ Equation (\[e6a\]) has been empirically known as an “accidental” or “mysterious” relation. As observed by Weinberg[@r10], this is unexplained: it relates a single cosmological parameter $H$ to constants from microphysics. We will touch upon this micro-macro nexus again. In our formulation, equation (\[e6a\]) is no longer a mysterious coincidence but rather a consequence.\
As (\[e5a\]) and (\[e4a\]) are not exact equations but rather, order of magnitude relations, it follows, on differentiating (\[e4a\]) that a small cosmological constant $\wedge$ is allowed such that $$\wedge < 0 (H^2)$$ This is consistent with observation and shows that $\wedge$ is very small – this has been a puzzle, the so called cosmological constant problem alluded to, because in conventional theory, it turns out to be huge [@rj]. Some $10^{70}$ times higher in fact! But it poses no problem in this formulation.\
To proceed we observe that because of the fluctuation of $\sim \sqrt{N}$ (due to the ZPF), there is an excess electrical potential energy of the electron, which infact we have identified as its inertial energy. That is [@ijmpa; @r29], $$\sqrt{N} e^2/R \approx mc^2.$$ On using (\[e2a\]) in the above, we recover the well known Gravitation-electromagnetism ratio viz., $$e^2/Gm^2 \sim \sqrt{N} \approx 10^{40}\label{e7}$$ or without using (\[e2a\]), we get, instead, the well known so called Weyl-Eddington formula, $$R = \sqrt{N}l\label{e8}$$ (It appears that this was first noticed by H. Weyl [@r31]). Infact (\[e8\]) is the spatial counterpart of (\[e3a\]). If we combine (\[e8\]) and (\[e2\]), we get, $$\frac{Gm}{lc^2} = \frac{1}{\sqrt{N}} \propto T^{-1}\label{e9}$$ where in (\[e9\]), we have used (\[e3a\]). Following Dirac (cf.also [@r32]) we treat $G$ as the variable, rather than the quantities $m, l, c \,\mbox{and}\,
\hbar$ (which we will call micro physical constants) because of their central role in atomic (and sub atomic) physics.\
Next if we use $G$ from (\[e9\]) in (\[e5a\]), we can see that $$H = \frac{c}{l} \quad \frac{1}{\sqrt{N}}\label{e10}$$ Thus apart from the fact that $H$ has the same inverse time dependance on $T$ as $G$, (\[e10\]) shows that given the microphysical constants, and $N$, we can deduce the Hubble Constant also, as from (\[e10\]) or (\[e5a\]).\
Using (\[e1a\]) and (\[e2a\]), we can now deduce that $$\rho \approx \frac{m}{l^3} \quad \frac{1}{\sqrt{N}}\label{e11}$$ Next (\[e8\]) and (\[e3a\]) give, $$R = cT\label{e12}$$ (\[e11\]) and (\[e12\]) are consistent with observation.\
With regard to the time variation of $G$, the issue is debatable and model dependent. Measurements on the earth and of the planets, and perhaps most accurate of all, Pulsars indicates a value $\sim 10^{-10}$, though values $10^{-11}$ and $10^{-12}$ have also appeared in some studies [@mel; @uzan].\
We return to the question of why the Planck mass is some $10^{20}$ times the mass of an elementary particle, for example pions or Protons or Electrons (in the large number sense).\
It is well known that the Planck mass is defined by (\[e1\]) [@mwt; @cu] Alternatively the Planck mass defines a black hole having the Schwarzchild radius given by $$\frac{2Gm_P}{c^2} \sim l_P \, \sim 10^{-33}cm\label{e2aa}$$ In (\[e2aa\]) $l_P$ is the Planck length.\
While $m_P$ is $\sim 10^{-5}gms$, a typical elementary particle has a mass $m \sim 10^{-25}gms$. (As mentioned in this order of magnitude sense it does not make much difference, if the elementary particle is an electron or pion or proton (Cf.ref.[@mwt])).\
We now recall that as already shown we have [@ijmpa; @ijtp; @fpl; @uof] $$G = \frac{\hbar c}{m^2\sqrt{N}}\label{e3aa}$$ In (\[e3aa\]) $N \sim 10^{80}$ is the well known number of elementary particles in the universe, which features in the Weyl-Eddington relations as also the Dirac Cosmology.\
What is interesting about (\[e3aa\]) is that it shows gravitation as a distributional effect over all the $N$ particles in the universe [@fpl; @uof].\
Let us rewrite (\[e2aa\]) in the form $$G \approx \frac{\hbar c}{m^2_P}\label{e4aa}$$ remembering that the Planck length is also the Compton length of the Planck mass. (Interestingly an equation like (\[e2aa\]) or (\[e4aa\]) also follows from Sakharov’s treatment of gravitation [@sakharov].) A division of (\[e3\]) and (\[e4aa\]) yields $$m^2_P = \sqrt{N} m^2\label{e5aa}$$ Equation (\[e5aa\]) immediately gives the ratio $\sim 10^{20}$ between the Planck mass and the mass of an elementary particle.\
It is interesting that in (\[e3aa\]) if we take $N \sim 1$, then we recover (\[e4aa\]). So while the Planck mass in the spirit of Rosen’s isolated universe and the Schwarzchild black hole uses the gravitational interaction in isolation, as seen from (\[e3aa\]), elementary particles are involved in the gravitational interaction with all the remaining particles in the universe.\
Finally rememebring that $Gm_P^2 \sim e^2$, as can also be seen from (\[e4aa\]), we get from (\[e3aa\]) $$\frac{e^2}{Gm^2} \sim \frac{1}{\sqrt{\bar{N}}}\label{e6aa}$$ Equation (\[e6aa\]) is the otherwise empirically well known electromagnetism-gravitation coupling constant ratio, but here it is deduced from the theory.\
It may be remarked that one could attempt an explanation of (\[e5aa\]) from the point of view of SuperSymmetry or Brane theory, but these latter have as yet no experimental validation [@gor].
The Universe as Planck Oscillators
==================================
What we have tried to argue is that a typical elementary particle like a pion could be considered to be the result of $n \sim 10^{40}$ evanescent Planck scale particles. The argument was based on random motions and also on the modification to the Uncertainity Principle. We will now consider the problem from a different point of view, which not only reconfirms the above result, but also enables an elegant extension to the case of the entire universe itself. Let us consider an array of $N$ particles, spaced a distance $\Delta x$ apart, which behave like oscillators, that is as if they were connected by springs. We then have as seen $$r = \sqrt{N \Delta x^2}\label{e1d}$$ $$ka^2 \equiv k \Delta x^2 = \frac{1}{2} k_B T\label{e2d}$$ where $k_B$ is the Boltzmann constant, $T$ the temperature, $r$ the extent and $k$ is the spring constant given by $$\omega_0^2 = \frac{k}{m}\label{e3d}$$ $$\omega = \left(\frac{k}{m}a^2\right)^{\frac{1}{2}} \frac{1}{r} = \omega_0
\frac{a}{r}\label{e4d}$$ We now identify the particles with Planck masses, set $\Delta x \equiv a =
l_P$, the Planck length. It may be immediately observed that use of (\[e3d\]) and (\[e2d\]) gives $k_B T \sim m_P c^2$, which ofcourse agrees with the temperature of a black hole of Planck mass. Indeed, as noted, Rosen had shown that a Planck mass particle at the Planck scale can be considered to be a universe in itself. We also use the fact alluded to that a typical elementary particle like the pion can be considered to be the result of $n \sim 10^{40}$ Planck masses. Using this in (\[e1d\]), we get $r \sim l$, the pion Compton wavelength as required. Further, in this latter case, using (48) and the fact that $N = n \sim 10^{40}$, and (\[e2d\]),i.e. $k_BT = kl^2/N$ and (\[e3d\]) and (\[e4d\]), we get for a pion, remembering that $m^2_P/n = m^2,$ $$k_ B T = \frac{m^3 c^4 l^2}{\hbar^2} = mc^2,$$ which of course is the well known formula for the Hagedorn temperature for elementary particles like pions. In other words, this confirms the conclusions in the previous section, that we can treat an elementary particle as a series of some $10^{40}$ Planck mass oscillators. However it must be observed from (\[e2d\]) and (\[e3d\]), that while the Planck mass gives the highest energy state, an elementary particle like the pion is in the lowest energy state. This explains why we encounter elementary particles, rather than Planck mass particles in nature. Infact as already noted [@cu], a Planck mass particle decays via the Bekenstein radiation within a Planck time $\sim 10^{-42}secs$. On the other hand, the lifetime of an elementary particle would be very much higher.\
In any case the efficacy of our above oscillator model can be seen by the fact that we recover correctly the masses and Compton scales in the order of magnitude sense and also get the correct Bekenstein and Hagedorn formulas as seen above, and get the correct estimate of the mass of the universe itself, as will be seen below.\
Using the fact that the universe consists of $N \sim 10^{80}$ elementary particles like the pions, the question is, can we think of the universe as a collection of $n N \, \mbox{or}\, 10^{120}$ Planck mass oscillators? We directly deduced this value a little earlier, in fact. This is what we will now show. Infact if we use equation (\[e1d\]) with $$\bar N \sim 10^{120},$$ we can see that the extent $r \sim 10^{28}cms$ which is of the order of the diameter of the universe itself. Next using (\[e4d\]) we get $$\hbar \omega_0^{(min)} \langle \frac{l_P}{10^{28}} \rangle^{-1} \approx m_P c^2 \times 10^{60} \approx Mc^2\label{e5d}$$ which gives the correct mass $M$, of the universe which in contrast to the earlier pion case, is the highest energy state while the Planck oscillators individually are this time the lowest in this description. In other words the universe itself can be considered to be described in terms of normal modes of Planck scale oscillators.\
The above gives a rationale for the figure $10^{120}$ Planck oscillators which is derived from the observed $10^{80}$ elementary particles in the universe and considerations in Section 2.\
3. In fact it has already been shown that the universe has been considered to be a coherent state of $N'$ Planck oscillators, where $M$ is given by (\[e6\]). In fact if we use (\[e2\]) we can deduce that $$M = \sqrt{N'} m_P\label{ey}$$ Further, using considerations in point 2 we can also deduce that $$m = m_P /\sqrt{n}$$ $$l = l_P \sqrt{n}\label{ez}$$ $$\tau = \tau_P \sqrt{n}$$ where $m, l, \tau$ are the mass, the Compton wavelength and Compton time of a typical elementary particle like the pion. This brings us to the question: While the Planck scale may provide the underpinning, in real life it is the Compton scale of elementary particles that we encounter. How do we make this transition. We consider below some other approaches.
Other Approaches
================
1\. It is well known that the energy of the Zero Point Field is given by $$(\Delta B)^2 \geq \hbar c/L^4\label{g1}$$ If in (\[g1\]) $L$ the extent over which the energy is spread, is taken to be the Compton wavelength, we recover the mass of the elementary particle. Alternatively it is known that the spectral density of the vacuum field is given by $$\rho (\omega) = const. \omega^3\label{g2}$$ where Lorentz invariance requires that the constant is given by $$\frac{\hbar}{2\pi^2 c^3}$$ In conventional theory this leads to an unacceptably high energy density of $10^{15}gms$ per $cm^3$, but at the Compton wavelength, we get back the above result.\
2. In a similar vein, we can argue that when the Zero Point Field is taken into account, at the Compton wavelength scale the momentum coordinate commutators which otherwise vanish in Classical Theory go over to the Quantum Mechanical commutators [@bgsdark]. From this point of view the Zero Point Field, as argued by some protogonists of Stochastic Electrodynamics, gives rise to Quantum behavior.\
3. What we have seen above is that from the background Zero Point Field, Plank scale oscillators condense out. Let us suppose that $n$ such particles are formed. We can then use the well known fact that for a collection of ultra relativistic particles, in this case the Planck oscillators, the various centres of mass form a two dimensional disk of radius $l$ given by $$l \approx \frac{\beta}{m_e c}\label{ex17}$$ where in (\[ex17\]) $m_e (\approx m$ in the large number sense) is the electron mass and $\beta$ is the angular momentum of the system. Further $l$ is such that for distances $r < l$, we encounter negative energies (exactly as for the Compton length). It will at once be apparent that for an electron, for which $\beta = \frac{\hbar}{2}$, (\[ex17\]) gives the Compton wavelength. We can further characterize (\[ex17\]) as follows: By the definition of the angular momentum of the system of Planck particles moving with relativistic speeds, we have $$\frac{\hbar}{2} = m_P c \int^l_0 r^2 drd \Theta \, \sim m_P c \sigma l^3 = m_e cl\label{ex18}$$ In (\[ex18\]) we have used the fact that the disk of mass centres is two dimensional, and $\sigma$ has been inserted to stress the fact that we are dealing with a two dimensional density, so that $\sigma$ while being unity has the dimension $$\left[\frac{1}{L^2}\right]$$ The right side of (\[ex18\]) gives the angular momentum for the electron. From (\[ex18\]) we get $$\sigma l^2 m_P = m_e\label{ex19}$$ which ofcourse is correct.\
Alternatively from (\[ex19\]) we can recover $n \sim 10^{40}$, in the large number sense.\
All this shows how the Compton scale of elementary particles emerges from the Planck scale.
Concluding Remarks
==================
The universe which we encounter is at the Compton scale of elementary particles. The various elementary particles which constitute the universe are in a sense incoherent in that they can be treated as independent particles which are not coupled or linked. Yet they occupy a single space time, remembering that at large distances the various particles interact via the relatively weak force of gravitation. This is expressed by the relation $$M = Nm,$$ where $N$ is the number of particles in the universe and $m$ a typical elementary particle mass, $M$ being the mass of the universe.\
However at a higher energy or smaller scale of observation, viz., the Planck scale, the universe is seen as coherent collection of $N' \sim 10^{120}$ Planck oscillators, which in fact provides the underpinning for all of space time. In this case we have, $$M = \sqrt{N'} m_P$$ where we are now speaking of the Planck mass. As described elsewhere (Cf.refs.[@uof; @gip]), we can describe the above difference in the following manner: The most fundamental scale is the Planck scale, represened by wave functions $\psi_\imath$. However by the Random Phase axiom, the superposition of the $\psi_\imath$s reduces to the simpler superposition of $\phi_\imath$s, these latter representing the wave functions at the Compton scale of elementary particles.
[99]{} Veneziano,G., Quantum geometric origin of all forces in string theory. In: Huggett SA, et al., editors. “The Geometric Universe”, Oxford University Press, Oxford, 1988, p.235-43. Kempf, A., in “From the Planck length to the Hubble radius”, Ed. A. Zichichi, World Scientific, Singapore, 2000, pp.613ff. Garay, L.J., Int.J.Mod.Phys.A., 1995, 10 (2), 145-165. Crane, L., and Smolin, L., Nucl.Phys., B267, 1986, p.714. Sidharth, B.G., Found.Phys.Lett., 17 (5), 2004, 503-506. Sidharth, B.G., Found.Phys.Lett., 15 (6), 2002, 577-583. Sidharth, B.G., “The Universe of Fluctuations”, Springer, Berlin, 2005. Sidharth, B.G., (Ed.) “The Great Ideas of Physics”, Springer, in press. Jack Ng, Y., and Van Dam, H., Mod.Phys.Lett.A., 9 (4), 1994, p.335-340. Rosen, N., International Journal of Theoretical Physics, 32 (8), 1993, p.1435-1440. Ruffini, R., and Fang, L.Z., “Basic Concepts in Relativistic Astrophysics”, World Scientific, Singapore, 1983, p.111ff. Cercignani, C., Found.Phys.Lett., Vol.11, No.2, 1998, 189-199. Cercignani, C., Galgani, L., and Scotti, A., Phys.Lett. 38A, 1972, 403. Huang, K., “Statistical Mechanics”, Wiley Eastern, New Delhi 1975, pp.75ff. Reif, F., “Fundamentals of Statistical and Thermal Physics”, McGraw-Hill Book Co., Singapore, 1965. Science, December, 1998. Science, December, 2003. Sidharth, B.G., Proc. of the Eighth Marcell Grossmann Meeting on General Relativity, Ed. T. Piran, World Scientific, Singapore, 1999, p.476-479. Sidharth, B.G., Int.J. of Mod.Phys.A 13(15), 1998, pp2599ff. Sidharth, B.G., International Journal of Theoretical Physics, Vol.37, No.4, 1998, 1307-1312. Sidharth, B.G., Chaos, Solitons and Fractals, 16 (4), 2003, pp.613-620. Sidharth, B.G., Chaos, Solitons and Fractals, 18 (1), 2003, pp.187-201. Sidharth, B.G., “Chaotic Universe: From the Planck to the Hubble Scale”, Nova Science Publishers, Inc., New York, 2001. C.W. Misner, K.S. Thorne and J.A. Wheeler, “Gravitation”, W.H. Freeman, San Francisco, 1973, p.819ff. Sidharth, B.G., Chaos, Solitons and Fractals, 14, 2002, pp.167-169. Sidharth, B.G., Chaos, Solitons and Fractals, 15 (1), 2003, pp.25-28. Castell, L., in “Quantum Theory and Gravitation”, Ed., Nottale, L., “Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity”, World Scientific, Singapore, 1993, p.312. Hayakawa, S., Suppl of PTP Commemmorative Issue, 1965, 532-541. Sidharth, B.G., in “Frontiers of Quantum Physics”, Eds., Lim, S.C., et al, Springer Verlag, Singapore, 1998. Weinberg, S., “Gravitation and Cosmology”, John Wiley & Sons, New York, 1972, p.62. Weinberg, S., Phys.Rev.Lett., 43, 1979, p.1566. Singh, J., “Great Ideas and Theories of Modern Cosmology”, Dover, New York, 1961. Melnikov, V.N., International Journal of Theoretical Physics, (7), 1994, 1569-1579. Melnikov, V.N., Gravitation and Cosmology, Vol.6 (22), 2000, pp.81-92. Uzan, J.P., Rev.Mod.Phys., 75, April 2003, pp.403-455. B.G. Sidharth, “A Note on the Characterization of Gravitation”, Foundation of Physics Letters, May, 2005. A.D. Sakharov, Soviet Physics - Doklady, Vol.12, No.11, 1968, p.1040-1041. E. Golfain, Chaos, Solitons and Fractals, 22 (3), 2004, pp.513-520. Sidharth, B.G., Found.Phys.Lett., 15 (6), 2002, pp.577-583. Goodstein, D.L., “States of Matter”, Dover Publications Inc., New York, 1985, p.160ff. Sidharth, B.G., “Mysterious Dark Energy”, to appear in “Einstein and Poincare”, Eds., Dvoeglazov, V., Apeiron Press, 2005.
|
---
abstract: |
We review $CP$-violating observables in flavor tagged and untagged $D^0 \rightarrow K^{-}\pi^{+}$ decays and evaluate the $CP$ asymmetry difference between the two decays. We note that this commonly neglected difference is not zero in principle and can be significant in future $B$ factory experiments. We also construct an expression to extract the strong phase difference between $\bar{D}^0 \rightarrow
K^-\pi^+$ and $D^0 \rightarrow K^-\pi^+$ decays, independently of existing experimental methods.
address:
- 'Korea University, Seoul 02841'
- 'Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana'
- 'J. Stefan Institute, 1000 Ljubljana'
author:
- 'E. Won'
- 'B. R. Ko'
- 'B. Golob'
title: 'Study of $CP$ violation in flavor tagged and untagged $D^0 \rightarrow K^-\pi^+$ decays '
---
$CP$ asymmetry ,flavor tagged ,flavor untagged ,strong phase 11.30.Er ,12.15.Ff ,13.25.Ft ,14.40.Lb
The violation of combined charge conjugation and parity ($CP$) symmetry in the quark sector through the weak interaction was predicted by the Cabibbo-Kobayashi-Maskawa (CKM) mechanism [@ref:ckm1; @ref:ckm2] and has been experimentally observed in $K$- and $B$-meson systems [@ref:pdg]. On the other hand, $CP$ violation ($CPV$) in charm meson system has not been observed yet and is expected to be small within the standard model (SM) [@ref:burdman]. Therefore, search for $CPV$ in charm meson system naturally provides a window for new physics beyond the SM.
Recent experimental $CP$ asymmetry measurements [@ref:prl_K0H; @ref:prl_PHIPI; @ref:prl_K0PI; @ref:jhep_K0K+; @ref:BaBar_D0hh; @ref:Belle_D0hh; @ref:prl_K0P0; @ref:cdf_D0hh; @ref:Belle_D0hh2] in charm meson decays adapted $D^0\rightarrow
K^-\pi^+$ [^1] (referred to as “untagged”) and $D^{*+}\rightarrow D^0(\rightarrow
K^-\pi^+)\pi^+_{\rm soft}$ (referred to as “tagged” and $\pi^{+}_{\rm{soft}}$ refers to a relatively low momentum charged pion) decays as control samples to correct for asymmetries due to different reconstruction efficiencies between positively and negatively charged tracks.
According to Ref. [@ref:petrov], untagged $D^0 \rightarrow
K^-\pi^+$ reveals $CP$ asymmetry resulting from the interference between the decays with and without $D^0$-$\bar{D}^0$ mixing even without direct $CPV$, namely $A^{\rm
untag}_{CP}=-\sqrt{R_D}y\sin\delta\sin\phi$, which has been considered in Refs. [@ref:prl_K0H; @ref:prl_PHIPI; @ref:prl_K0PI; @ref:jhep_K0K+].[^2] Other measurements [@ref:BaBar_D0hh; @ref:Belle_D0hh; @ref:prl_K0P0; @ref:cdf_D0hh; @ref:Belle_D0hh2] used both tagged and untagged decays with the common assumption that the difference in $CP$ asymmetries between the two decays is zero.
Also, one of the obstacles in interpreting the experimental $D^0$-$\bar{D}^0$ mixing measurements in the decay $\bar{D}^0\rightarrow K^-\pi^+$ is the appearance of the phase difference between $\bar{D}^0\rightarrow K^-\pi^+$ and $D^0\rightarrow
K^-\pi^+$ decays due to the strong interaction. A direct experimental way to extract this strong phase difference has been utilizing a quantum-coherent production of $D^0\bar{D}^0$ pairs from $\psi(3770)$ and their measurements of the strong phase difference are $\cos\delta
= 1.15^{+0.19+0.00}_{-0.17-0.08}$, $\sin\delta =
0.56^{+0.32+0.21}_{-0.31-0.20}$, $\delta =
(18^{+11}_{-19})^{\circ}$ [@ref:cleo] and $\cos\delta=1.02\pm
0.11\pm 0.06\pm 0.01$ [@ref:besIII]. Further precise measurements are highly desired for interpretation of the recent experimental observations of $D^0$-$\bar{D}^0$ mixing in the decay $\bar{D}^0\rightarrow
K^-\pi^+$ [@ref:dmix_lhcb; @ref:dmix_cdf; @ref:dmix_belle].
In this paper, we review $CPV$ in flavor tagged and untagged $D^0\rightarrow K^-\pi^+$ decays, testing the validity of the systematic consideration of $CPV$ in the untagged decays adopted in Refs. [@ref:prl_K0H; @ref:prl_PHIPI; @ref:prl_K0PI; @ref:jhep_K0K+] and the aforementioned assumption in $CPV$ difference between the two decays adopted in Refs. [@ref:BaBar_D0hh; @ref:Belle_D0hh; @ref:prl_K0P0; @ref:cdf_D0hh; @ref:Belle_D0hh2] within the two $CPV$ scenarios in $D^0$ meson system, “$CPV$-allowed” and “No direct $CPV$ in DCS (doubly Cabibbo-suppressed) decays” [@ref:hfag]. We also propose a method to extract the strong phase difference between $\bar{D}^0 \rightarrow
K^-\pi^+$ and $D^0 \rightarrow K^- \pi^+$ decays.
The time evolution of the $D^0$-$\bar{D}^0$ system can be described by the Schrödinger equation [@ref:pdg] $$i \frac{\partial }{\partial t}
\left(
\begin{array}{c}
D^0(t) \\
\bar{D}^0(t)
\end{array}
\right)
=
\Big( \mathbf{M} - \frac{i}{2} \mathbf{\Gamma}
\Big)
\left(
\begin{array}{c}
D^0(t) \\
\bar{D}^0(t)
\end{array}
\right)
\label{eq:mix}$$ where $\mathbf{M}$ and $\mathbf{\Gamma}$ are Hermitian matrices associating with the transitions, $D^0\to D^0$ and $D^0\to\bar{D}^0$. Our mass ($|D_{1,2}\rangle$) and flavor ($|D^0\rangle, |\bar{D}^0\rangle$) eigenstates of neutral $D$ mesons are expressed as [@ref:hfag] $$|D_{1,2}\rangle = p |D^0 \rangle \mp q |\bar{D}^0 \rangle$$ where $p$ and $q$ are complex numbers with the convention $CP|D^0\rangle=-|\bar{D}^0\rangle$ and $CP|\bar{D}^0\rangle=-|D^0\rangle$ under $CP$ conservation. Note that we adopt the convention used in Ref. [@ref:hfag]. The time evolution of the mass eigenstate is given by $|D_i(t) \rangle
= e^{-im_i t - \frac{1}{2}\Gamma_i t}|D_i\rangle$, ($i$=1,2) where $m_i$ and $\Gamma_i$ are the mass and width of $|D_i\rangle$. From these, one usually defines mixing parameters $
x \equiv (m_1 - m_2)/\Gamma = \Delta m/\Gamma
$ and $
y \equiv (\Gamma_1 - \Gamma_2)/2\Gamma = \Delta \Gamma/2\Gamma
$ where $\Gamma \equiv (\Gamma_1 + \Gamma_2)/2$, in order to describe the time evolution of the $D$ meson system and $CP$ asymmetries conveniently.
For the study of $D^0 \rightarrow f$ decay, one defines decay amplitude of an initially produced $D^0$/$\bar{D}^0$ into the final state $f$/$\bar{f}$ to be $\mathcal{A}_f$/$\bar{\mathcal{A}}_{\bar{f}}$ for Cabibbo-favored (CF) decays and $\mathcal{A}_{\bar{f}}$/$\bar{\mathcal{A}}_f$ for DCS decays, respectively, where $f$/$\bar{f}$ stands for $K^- \pi^+$/$K^+\pi^-$ final state. Then, under direct $CP$ conservation in CF decays, $|\mathcal{A}_f|^2=|\bar{\mathcal{A}}_{\bar{f}}|^2$, we can define $$\begin{aligned}
\label{eq:lambda}
\frac{q}{p}\frac{\bar{\mathcal{A}}_f}{\mathcal{A}_f}
&\equiv&
\sqrt{R_D(1-A_D)} R_M e^{-i(\delta-\phi)},
\nonumber \\
\frac{p}{q}\frac{\mathcal{A}_{\bar{f}}}{\bar{\mathcal{A}}_{\bar{f}}}
&\equiv&
\sqrt{R_D(1+A_D)} R^{-1}_M e^{-i(\delta+\phi)},
\\\nonumber \end{aligned}$$ where $R_D$ is the ratio of DCS to CF decay rates, $R_D=\frac{|\mathcal{A}_{\bar{f}}|^2+|\bar{\mathcal{A}}_{f}|^2}{|\mathcal{A}_{f}|^2+|\bar{\mathcal{A}}_{\bar{f}}|^2}$, $A_D$ is the direct $CPV$ in DCS decays, $A_D=\frac{|\mathcal{A}_{\bar{f}}|^2-|\bar{\mathcal{A}}_{f}|^2}{|\mathcal{A}_{\bar{f}}|^2+|\bar{\mathcal{A}}_{f}|^2}$, and $\delta$ is $CP$ conserving strong phase difference between DCS and CF decay amplitudes. $R_M$ and $\phi$ are the magnitude and argument of $q/p$,[^3] where $R_M\ne 1$ indicates $CPV$ in the mixing and $\phi\ne 0$ (nor $\phi\ne\pi$) implies that in the interference of the mixing and decay. Hence, the relations shown in Eq. (\[eq:lambda\]) satisfy the condition “$CPV$-allowed” in Ref. [@ref:hfag]. With the relations given in Eq. (\[eq:lambda\]), we have the expressions of the decay rates up to the second order in the mixing parameters, assuming the mixing parameters are small ($|x|\ll 1$ and $|y|\ll 1$), to be $$\begin{aligned}
\label{eq:decayrate}
&&\Gamma({D}^0 (t) \rightarrow f) = e^{-\Gamma t} |\mathcal{A}_{f}|^2 [1
\nonumber \\
&+&\Gamma t \sqrt{R_D(1-A_D)} R_M \{y \cos{(\delta - \phi)} + x \sin{(\delta -\phi)}\}
\nonumber \\
&+&\Gamma^2 t^2\frac{1}{4}R_D(1-A_D)R^2_M(x^2+y^2)],
\nonumber \\
&&\Gamma(\bar{D}^0 (t) \rightarrow\bar{f}) = e^{-\Gamma t} |\bar{\mathcal{A}}_{\bar{f}}|^2 [1
\nonumber \\
&+&\Gamma t \sqrt{R_D(1+A_D)} R_M^{-1}\{y \cos{(\delta + \phi)} + x \sin{(\delta +\phi)}\}
\nonumber \\
&+&\Gamma^2 t^2\frac{1}{4}R_D(1+A_D)R^{-2}_M(x^2+y^2)],
\nonumber \\
&&\Gamma({D}^0 (t) \rightarrow\bar{f}) = e^{-\Gamma t} |\bar{\mathcal{A}}_{\bar{f}}|^2 [R_D(1+A_D)
\nonumber \\
&+&\Gamma t \sqrt{R_D(1+A_D)} R_M\{y \cos{(\delta + \phi)} - x \sin{(\delta +\phi)}\}
\nonumber \\
&+&\Gamma^2 t^2\frac{1}{4}R^2_M(x^2+y^2)],
\nonumber \\
&&\Gamma(\bar{D}^0 (t) \rightarrow f) = e^{-\Gamma t} |\mathcal{A}_{f}|^2 [R_D(1-A_D)
\nonumber \\
&+&\Gamma t \sqrt{R_D(1-A_D)} R_M^{-1}\{y \cos{(\delta - \phi)} - x \sin{(\delta -\phi)}\}
\nonumber \\
&+&\Gamma^2 t^2\frac{1}{4}R^{-2}_M(x^2+y^2)],\end{aligned}$$ and they are our fundamental relations in the construction of various $CP$ asymmetries described below.
The final state of the untagged decay is the sum of the CF decay $D^0\rightarrow f$, the DCS decay $\bar{D}^0\rightarrow f$, the DCS decay following $D^0$-$\bar{D}^0$ mixing $D^0\rightarrow\bar{D}^0\rightarrow f$, and the CF decay following $D^0$-$\bar{D}^0$ mixing $\bar{D}^0\rightarrow D^0\rightarrow
f$. Thus, the time-integrated decay rates for the untagged case are $$\begin{aligned}
\Gamma^{\textrm{untag}}_f
&=&
\int^\infty_0 dt~
\bigg(\Gamma(D^0 (t) \rightarrow f) + \Gamma(\bar{D}^0 (t) \rightarrow f)\bigg),
\nonumber \\
\bar{\Gamma}^{\textrm{untag}}_{\bar{f}}
&=&
\int^\infty_0 dt~
\bigg(\Gamma(\bar{D}^0 (t) \rightarrow \bar{f}) + \Gamma({D}^0 (t) \rightarrow \bar{f})\bigg).\end{aligned}$$ The $CP$ asymmetry in this case is defined as $$\begin{aligned}
A^{\textrm{untag}}_{CP} \equiv
\frac{
\Gamma^{\textrm{untag}}_f - \bar{\Gamma}^{\textrm{untag}}_{\bar{f}}
}
{
\Gamma^{\textrm{untag}}_f + \bar{\Gamma}^{\textrm{untag}}_{\bar{f}}.
}
\label{eq:acpun}\end{aligned}$$ Note that our definition of $A_{CP}^{\textrm{untag}}$ has an opposite sign of the one in Ref. [@ref:petrov]. The expression for $A^{\rm
untag}_{CP}$ can be evaluated using all the relations given in Eq. (\[eq:decayrate\]) : $$\begin{aligned}
\label{eq:untag}
A^{\textrm{untag}}_{CP}=
&+&2\sqrt{R_D}y\sin\delta\sin\phi
\nonumber \\
&-& 2\sqrt{R_D}(1-R_M)x\sin\delta\cos\phi
\nonumber \\
&-& R_D A_D
\nonumber \\
&-& \sqrt{R_D}A_D y\cos\delta\cos\phi
\nonumber \\
&+& (1-R_M)(x^2+y^2)
\nonumber \\
&-& R_D(1-R_M)(x^2+y^2)
\nonumber \\
&-& \frac{1}{2}R_D A_D(x^2+y^2), \end{aligned}$$ where the factor 2 of the first term is not present and other terms are neglected in Ref. [@ref:petrov]. Using the world average $CPV$ and mixing parameters from HFAG [@ref:hfag_new], $A^{\rm
untag}_{CP}=(+1.27\pm3.83)\times10^{-5}$, thus the magnitude of $A^{\rm untag}_{CP}$ can be at most $8.79\times10^{-5}$ at 95% confidence level (CL) for the scenario “$CPV$-allowed”. The magnitude of $A^{\rm untag}_{CP}$ can be neglected for the current experimental sensitivities [@ref:prl_K0H; @ref:prl_PHIPI; @ref:prl_K0PI; @ref:jhep_K0K+], for example $A^{D^+\to
K^0_S\pi^+}_{CP}=(-0.363\pm0.094\pm0.063\pm0.014\pm0.016)\times10^{-2}$ [@ref:prl_K0PI], where the first uncertainty is statistical, the second is systematic mostly from the usage of the control samples, the third is systematic due to possible $CPV$ in the control sample based on Ref. [@ref:petrov], and the fourth is irreducible systematic due to $K^0_L$-$K^0_S$ regeneration [@ref:k0mat]. The sensitivity of $A^{D^+\to K^0_S\pi^+}_{CP}$ at the super-$B$ factory currently under construction [@ref:superb], however, is expected to be $(1.3\pm0.9\pm0.9\pm1.6)\times10^{-4}$, where the third is systematic due to possible $CPV$ in the control sample given in this paper and others are the same as above. The $CPV$ in the untagged decays will thus become a significant systematic source in the future $CP$ asymmetry measurements. Assuming direct $CP$ conservation $A_D=0$, the relation $1-R_M=(y/x)\tan\phi$ holds, which not only associates $CPV$ and mixing parameters, but also relates $CPV$ in mixing and interference [@ref:icpv0; @ref:icpv1; @ref:icpv2]. Within the SM and also for the case “No direct $CPV$ in DCS decays”, $\cos\phi\cong 1$ is a good approximation in charm meson system. Hereafter the approximation is to be applied for “No direct $CPV$ in DCS decays”. We note that the first and second terms in Eq. (\[eq:untag\]) cancel out each other under direct $CP$ conservation, then $CP$ asymmetry in the decays is expressed as $$\begin{aligned}
\label{eq:untag2}
A^{\textrm{untag}}_{CP} = \frac{y(x^2+y^2)\sin\phi}{x}~\footnotemark\end{aligned}$$ which is free from the strong interaction. Using the world average $CPV$ and mixing parameters from HFAG [@ref:hfag_new], $A^{\rm{untag}}_{CP}=(-3.98\pm7.95)\times 10^{-7}$, hence the magnitude of $A^{\rm untag}_{CP}$ can be at most $1.96\times10^{-6}$ at 95% CL for the scenario “No direct $CPV$ in DCS decays”. We also find $A^{\textrm{untag}}_{CP}=0$ with limiting the relations given in Eq. (\[eq:decayrate\]) up to the first order in $x$ and $y$, which indicates indirect $CPV$ in charm decays is approximately universal [@ref:YAY].
For the tagged analysis, the decay $D^{*+}\to D^0(\to f)\pi^+_{\rm
soft}$ is the sum of the CF decay $D^0\to f$ and the DCS decay following $D^0$-$\bar{D}^0$ mixing $D^0\rightarrow\bar{D}^0\rightarrow
f$. The time-integrated decay rates for the tagged decays are $$\begin{aligned}
\Gamma^{\textrm{tag}}_f
=
\int^\infty_0 dt~
\Gamma({D}^0 (t) \rightarrow f),
\nonumber \\
\bar{\Gamma}^{\textrm{tag}}_{\bar{f}}
=
\int^\infty_0 dt~
\Gamma(\bar{D}^0 (t) \rightarrow \bar{f}). \end{aligned}$$ The $CP$ asymmetry in the tagged decays is defined as $$\begin{aligned}
A^{\textrm{tag}}_{CP} \equiv
\frac{
\Gamma^{\textrm{tag}}_f - \bar{\Gamma}^{\textrm{tag}}_{\bar{f}}
}
{
\Gamma^{\textrm{tag}}_f + \bar{\Gamma}^{\textrm{tag}}_{\bar{f}}
}
\label{eq:acptrs}\end{aligned}$$ and evaluated using the first and second relations given in Eq. (\[eq:decayrate\]) : $$\begin{aligned}
\label{eq:tag}
A^{\textrm{tag}}_{CP}=
&+&\sqrt{R_D}(y\sin\delta-x\cos\delta)\sin\phi
\nonumber \\
&-&\sqrt{R_D}(1-R_M)(y\cos\delta+x\sin\delta)\cos\phi
\nonumber \\
&-&\frac{1}{2}\sqrt{R_D}A_D(y\cos\delta+x\sin\delta)\cos\phi
\nonumber \\
&-&R_D(1-R_M)(x^2+y^2)
\nonumber \\
&-&\frac{1}{2}R_D A_D(x^2+y^2).\end{aligned}$$ Using the world average $CPV$ and mixing parameters from HFAG [@ref:hfag_new], $A^{\rm
tag}_{CP}=(+0.29\pm2.83)\times10^{-5}$, hence the magnitude of $A^{\rm tag}_{CP}$ can be at most $5.85\times10^{-5}$ at 95% CL for the scenario “$CPV$-allowed”. Under direct $CP$ conservation, the asymmetry in the decays can be expressed as $$\begin{aligned}
\label{eq:tag2}
A^{\textrm{tag}}_{CP} = -\frac{\sqrt{R_D}(x^2 + y^2)\cos\delta\sin\phi}{x},\footnotemark\end{aligned}$$ where this can be found in Ref. [@ref:icpv2]. Using the world average $CPV$ and mixing parameters from HFAG [@ref:hfag_new], $A^{\rm tag}_{CP}=(+3.91\pm7.77)\times 10^{-6}$, thus the magnitude of $A^{\rm tag}_{CP}$ can be at most $1.91\times10^{-5}$ at 95% CL for the scenario “No direct $CPV$ in DCS decays”.
Having compared Eq. (\[eq:untag\]) with Eq. (\[eq:tag\]) and Eq. (\[eq:untag2\]) with (\[eq:tag2\]), in general $A^{\textrm{tag}}_{CP}\neq A^{\textrm{untag}}_{CP}$ and the differences between them $\Delta A_{CP}$ are $(-0.98\pm2.20)\times
10^{-5}$ for the “$CPV$-allowed” and $(+4.31\pm8.57)\times 10^{-6}$ for the “No direct $CPV$ in DCS decays”, respectively. Table \[TABLE:RESULTS\] summarizes $CP$-violating observables and their magnitudes from this work.
----------------------------------------------------------------------------------------
“$CPV$-allowed” “No direct $CPV$\
& & in DCS decays”
---------------------- -------------------------------- --------------------------------
$A^{\rm untag}_{CP}$ $(+1.27\pm3.83)\times 10^{-5}$ $(-3.98\pm7.95)\times 10^{-7}$
$A^{\rm tag}_{CP}$ $(+0.29\pm2.83)\times 10^{-5}$ $(+3.91\pm7.77)\times 10^{-6}$
$\Delta A_{CP}$ $(-0.98\pm2.20)\times 10^{-5}$ $(+4.31\pm8.57)\times 10^{-6}$
----------------------------------------------------------------------------------------
: Summary of $CP$-violating observables and their magnitudes.[]{data-label="TABLE:RESULTS"}
Thus, the magnitudes of $\Delta A_{CP}$ can be at most $\mathcal{O}(10^{-4})$ and $\mathcal{O}(10^{-5})$ at 95% CL for the “$CPV$-allowed” and “No direct $CPV$ in DCS decays”, respectively. The upper limit of $\Delta A_{CP}$, $\mathcal{O}(10^{-4})$ can be neglected for the current experimental sensitivities [@ref:BaBar_D0hh; @ref:Belle_D0hh; @ref:prl_K0P0; @ref:cdf_D0hh; @ref:Belle_D0hh2], for example $A^{D^0\to K^+K^-}_{CP}=(-0.32\pm0.21\pm0.09)\times
10^{-2}$ [@ref:Belle_D0hh2], where the first uncertainty is statistical and the second is systematic. The sensitivity of $A^{D^0\to K^+K^-}_{CP}$ at the super-$B$ factory [@ref:superb], however, is expected to be $3\times 10^{-4}$ (statistical) and $1\times 10^{-4}$ (systematic). The $CPV$ difference between the tagged and untagged decays will thus become a significant systematic source in the future $CP$ asymmetry measurements.
For “No direct $CPV$ in DCS decays”, the strong phase difference between $\bar{D}^0 \rightarrow K^-\pi^+$ and $D^0\rightarrow K^-\pi^+$ decays can be obtained by taking the ratio of $A_{CP}^{\textrm{tag}}$ to $A^{\textrm{untag}}_{CP}$. The relation is $$\cos{\delta}= -\frac{y}{\sqrt{R_D}}
\Bigg(
\frac{A^{\textrm{tag}}_{CP}}{A^{\textrm{untag}}_{CP}}
\Bigg),$$ where the strong phase can be expressed in terms of $y$, $\sqrt{R_D}$, and $CP$ asymmetries only. From this equation, one can extract the strong phase in an independent way by measuring the ratio of $A^{\rm
tag}_{CP}$ to $A^{\rm untag}_{CP}$ experimentally. For the evaluation of the expected sensitivity on $\delta$ ($\sigma_{\delta}$), we assign 0.007 and 0.01 for the uncertainties on $A^{\rm untag}_{CP}$ and $A^{\rm tag}_{CP}$ measurements, respectively, where the former is from the current best measurement [@ref:cleoIII] and the latter from our conservative assumption reflecting a conservative experimental uncertainty. We evaluate $\sigma_{\delta}$ as a function of $|A^{\rm tag}_{CP}/A^{\rm
untag}_{CP}|$ by incorporating errors on $A^{\rm tag}_{CP}$, $A^{\rm
untag}_{CP}$ given above, $y$, and $R_D$ and the correlation between $y$ and $R_D$ from Ref. [@ref:hfag_new]. Figure \[FIG:DERR\] shows our evaluation implying that the sensitivity on $\delta$ using the method introduced in this paper would be better than that of current measurements [@ref:cleo; @ref:besIII] depending on $|A^{\rm
tag}_{CP}/A^{\rm untag}_{CP}|$. Furthermore, our evaluation shows $\sigma_{\delta}$ dominates from current sensitivities of $y$ and $R_D$. Regardless of the sensitivity on $\delta$, it is important to have an independent tool as a cross check on existing methods.
![The solid (red) and dashed (blue) lines show the expected sensitivity on $\delta$ in degrees as a function of $|A^{\rm
tag}_{CP}/A^{\rm untag}_{CP}|$ with and without experimental errors on $A^{\rm tag}_{CP}$ and $A^{\rm untag}_{CP}$ measurements. The dotted (black) line is the current best measurement from Ref. [@ref:besIII].[]{data-label="FIG:DERR"}](ecosdelta.eps){height="50.00000%" width="60.00000%"}
To conclude, we have reviewed $CP$ asymmetries in flavor tagged and untagged $D^0 \rightarrow K^- \pi^+$ decays, $CP$ asymmetry in the untagged decays and $CP$ asymmetry difference between the two decays. Both $CP$-violating observables can be ignored for the current experimental sensitivities [@ref:prl_K0H; @ref:prl_PHIPI; @ref:prl_K0PI; @ref:jhep_K0K+; @ref:BaBar_D0hh; @ref:Belle_D0hh; @ref:prl_K0P0; @ref:cdf_D0hh; @ref:Belle_D0hh2], but cannot be neglected in the future super-$B$ factory experiments. However, improving $CPV$ and mixing parameters at the future experiments helps to reduce the systematic uncertainty in the future $CP$ asymmetry measurements. We also constructed an expression for the strong phase difference in terms of $y$, $R_D$, and $CP$ asymmetries. This provides experimental access to the strong phase from measurements of $CP$ asymmetries in flavor tagged and untagged $D^0\rightarrow K^-\pi^+$ decays, which is independent of existing methods [@ref:cleo; @ref:besIII].
We thank Anze Zupanc for comments on the manuscript and Alan Schwartz for providing us with the HFAG output correlations for $A_D=0$. B. R. Ko is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. NRF-2014R1A2A2A01005286) and E. Won is supported by No. NRF-2010-0021174.
Appendix
========
$CP$ asymmetry in flavor tagged $D^0\rightarrow\bar{f}$ decays (referred to as “wrong sign (WS)” decays) can be expressed as $$\begin{aligned}
%\label{eq:tag}
A^{\textrm{tag,WS}}_{CP}=
&-&\frac{1}{\sqrt{R_D}}(y\sin\delta+x\cos\delta)\sin\phi
\nonumber \\
&-&\frac{1}{\sqrt{R_D}}(1-R_M)(y\cos\delta-x\sin\delta)\cos\phi
\nonumber \\
&+&\frac{1}{2\sqrt{R_D}}A_D(y\cos\delta-x\sin\delta)\cos\phi
\nonumber \\
&-&\frac{1}{R_D}(1-R_M)(x^2+y^2)
\nonumber \\
&+&A_D\end{aligned}$$ using the third and fourth relations in Eq. (\[eq:decayrate\]). Under direct $CP$ conservation, the asymmetry is $$\begin{aligned}
\label{eq:tagws}
A^{\textrm{tag,WS}}_{CP} =
&-&\frac{(x^2 + y^2)\cos\delta\sin\phi}{\sqrt{R_D}x}
\nonumber \\
&-&\frac{y(x^2 + y^2)\sin\phi}{R_D x}.\end{aligned}$$ Then, $A^{\rm untag}_{CP}$ in Eq. (\[eq:untag2\]) can be also obtained from Eqs. (\[eq:tag2\]) and (\[eq:tagws\]) because Eq. (\[eq:acpun\]) can be also expressed as $A^{\rm
untag}_{CP}=A^{\rm tag}_{CP}-R_D A^{\rm tag,WS}_{CP}$. By limiting the relations given in Eq. (\[eq:decayrate\]) up to the first order in the mixing parameters, one can get $A^{\rm
tag}_{CP}=R_D A^{\rm tag,WS}_{CP}$ with Eqs. (\[eq:tag2\]) and (\[eq:tagws\]) [@ref:icpv2], thus $A^{\rm untag}_{CP}=0$ reflecting universality of indirect $CPV$ in charm decays in approximation.
[99]{} N. Cabibbo, Phys. Rev. Lett. [**10**]{}, 531 (1963).
M. Kobayashi and T. Maskawa, Prog. Theor. Phys., [**49**]{}, 652 (1973).
J. Beringer [*et al*]{}. (Particle Data Group), Phys. Rev. D [**86**]{}, 010001 (2012).
G. Burdman and I. Shipsey, Annu. Rev. Nucl. Part. Sci. [**53**]{}, 431 (2003).
B. R. Ko [*et al*]{}. (Belle Collaboration), Phys. Rev. Lett. [**104**]{}, 181602 (2010).
M. Starič [*et al*]{}. (Belle Collaboration), Phys. Rev. Lett. [**108**]{}, 071801 (2012).
B. R. Ko [*et al*]{}. (Belle Collaboration), Phys. Rev. Lett. [ **109**]{}, 021601 (2012), Erratum-ibid. [**109**]{}, 119903 (2012).
B. R. Ko [*et al*]{}. (Belle Collaboration), J. High Energy Phys. 02 ([**2013**]{}) 098.
B. Aubert [*et al*]{}. (BaBar Collaboration), Phys. Rev. Lett. [**100**]{}, 061803 (2008).
M. Starič [*et al*]{}. (Belle Collaboration), Phys. Lett. B [**670**]{}, 190 (2008).
B. R. Ko [*et al*]{}. (Belle Collaboration), Phys. Rev. Lett. [**106**]{}, 211801 (2011).
T. Aaltonen [*et al*]{}. (CDF Collaboration), Phys. Rev. D [**85**]{}, 012009 (2012).
B. R. Ko [*et al*]{}. (Belle Collaboration), <arXiv:1011.0352>.
A. A. Petrov, Phys. Rev. D [**51**]{}, 111901 (2004).
D. M. Asner [*et al.*]{} (CLEO Collaboration), Phys. Rev. D [**86**]{}, 112001 (2012).
M. Ablikim [*et al.*]{} (BES III Collaboration), Phys. Lett. B [**734**]{}, 227 (2014).
R. Aaij [*et al.*]{} (LHCb collaboration), Phys. Rev. Lett. [**110**]{}, 101802 (2013).
T. Aaltonen [*et al.*]{} (CDF collaboration), Phys. Rev. Lett. [**111**]{}, 231802 (2013).
B. R. Ko [*et al.*]{} (Belle collaboration), Phys. Rev. Lett. [**112**]{}, 111801 (2014).
Y. Amhis [*et al.*]{} (Heavy Flavor Averaging Group), [arXiv:1207.1158 \[hep-ex\]](arXiv:1207.1158 [hep-ex]) and online update at <http://www.slac.stanford.edu/xorg/hfag/>.
<http://www.slac.stanford.edu/xorg/hfag/charm/FPCP14/results_mix+cpv.html>.
B. R. Ko, E. Won, B. Golob, and P. Pakhlov, Phys. Rev. D [**84**]{}, 111501(R) (2011).
T. Abe [*et al.*]{}, <arXiv:1011.0352>.
M. Ciuchini [*et al*]{}. Phys. Lett. B [**655**]{}, 162 (2007).
Y. Grossman, Y. Nir, and G. Perez, Phys. Rev. Lett. [**103**]{}, 071602 (2009).
A. L. Kagan and M. D. Sokoloff, Phys. Rev. D [**80**]{}, 076008 (2009).
Y. Grossman, A. L. Kagan, and Y. Nir, Phys. Rev. D [**75**]{}, 036008 (2007).
G. Bonvicini [*et al.*]{}, (CLEO Collaboration), Phys. Rev. D [**89**]{}, 072002 (2014).
[^1]: Throughout this paper, the inclusion of the charge-conjugate decay mode is implied unless otherwise stated.
[^2]: Ref. [@ref:prl_K0PI] used untagged $D^0\rightarrow K^-\pi^+\pi^0$ decays as a control sample , thus the consideration was modified with relevant $R_D$ [@ref:pdg] and $\delta_{K\pi\pi}$ [@ref:hfag].
[^3]: In general, $\phi$ includes the weak phase arising from the corresponding complex elements of the CKM matrix. In charm meson decays these are, however, to a very good approximation real.
|
---
abstract: 'The gravitational interaction between two objects on similar orbits can effect noticeable changes in the orbital evolution even if the ratio of their masses to that of the central body is vanishingly small. @Christou2005 observed an occasional resonant lock in the differential node $\Delta \Omega$ between two members in the Himalia irregular satellite group of Jupiter in the $N$-body simulations (corresponding mass ratio $\sim 10^{-9}$). Using a semianalytical approach, we have reproduced this phenomenon. We also demonstrate the existence of two additional types of resonance, involving angle differences $\Delta\omega$ and $\Delta (\Omega+\varpi)$ between two group members. These resonances cause secular oscillations in eccentricity and/or inclination on timescales $\sim$ 1 Myr. We locate these resonances in $(a,e,i)$ space and analyse their topological structure. In subsequent $N$-body simulations, we confirm these three resonances and find a fourth one involving $\Delta \varpi$. In addition, we study the occurrence rates and the stability of the four resonances from a statistical perspective by integrating 1000 test particles for 100 Myr. We find $\sim 10-30$ librators for each of the resonances. Particularly, the nodal resonance found by @Christou2005 is the most stable: 2 particles are observed to stay in libration for the entire integration.'
author:
- Daohai Li
- 'Apostolos A. Christou'
date: 'Received: date / Accepted: date'
subtitle: A Case Study of the Himalia Prograde Group of Jovian Irregular Satellites
title: 'Secular Resonances Between Bodies on Close Orbits:'
---
Introduction {#sec-intro}
============
All planets in the solar system except Mercury and Venus have satellites [see, e.g., @Murray1999]. Satellites are subject to many types of perturbations, for instance, solar forcing and the oblateness of the host planet.
The Sun’s gravity induces a satellite’s orbit to precess at a rate, [see, for example, @Innanen1997] $$\dot \varpi_\odot \approx{3 k^2 m_\odot \over { 4 n a_\odot^3}}\,,$$ where $\varpi$ is the longitude of pericentre of the satellite, $k^2$ the gravitational constant; $m_\odot$ and $a_\odot$ are the solar mass and semimajor axis of the Sun’s relative motion with respect to the planet; $n$ is the satellite’s mean motion and $n=\sqrt{k^2 m / a^3}$ (where $m$ is the mass of the host planet and $a$ the semimajor axis of the satellite). On the other hand, the oblateness of the host planet also causes the orbit to precess at the rate [e.g., @Roy1978] $$\dot \varpi_\mathrm{J_2} \approx {3 \over 2} n J_2 {R^2 \over a^2}\,,$$ where $J_2$ is a measure of the oblateness and $R$ is the equatorial radius of the planet. Equating the two, we find the critical semimajor axis $$a_\mathrm{crit}=\left({2J_2R^2a_\odot^3 m \over m_\odot}\right)^{1/5}\,.$$ For a satellite with $a$ below this value, its orbital evolution is mainly controlled by the planetary oblateness, while one with $a$ above this value is governed by solar perturbations. Based on this, @Burns1986 defined irregular satellites as those whose orbital evolution is dominated by the Sun, i.e., with $a>a_\mathrm{crit}$ [see also, @Goldreich1966]. Thus inherently, this definition mostly places constraints on the average distance from the satellite to the host planet. Apart from the size, their orbits are often highly eccentric and inclined compared to the regular satellites.
The eccentricity and inclination of the irregulars are generally high and their distribution in $[0,1)$ and $[0^\circ,180^\circ)$ is not uniform [see, e.g., @Jewitt2007]. For example, the inclinations of them are far from $90^\circ$. This feature can be explained by Kozai-Lidov dynamics [@Lidov1962; @Kozai1962] that involves the secular perturbation from the Sun. Under solar forcing, the eccentricity and inclination of the satellite are coupled in such a way that the vertical angular momentum $H\propto\sqrt{1-e^2} \, \cos{i}$ is conserved while the eccentricity $e$ and inclination $i$ of the satellite may experience large-amplitude oscillations, especially when $i$ is high. This mechanism causes the absence of satellites with inclinations around $90^\circ$ [@Carruba2002]. We will discuss this effect in detail in Sect. \[semi-sec-total\].
Most irregular satellites reside on retrograde, rather than prograde, orbits; this relates to the so-called evection phenomenon [@Yokoyama2008; @Frouard2010]. Evection is linked to the mean motion of the Sun. If a satellite is too far from the planet, it will be so perturbed by the Sun that its longitude of pericentre $\varpi$ precesses at a rate comparable to the solar mean motion. In this case, the angle $\psi=\varpi-\lambda_\odot$ (where $\lambda_\odot$ is the mean longitude of Sun) may librate; the angular momentum $G \propto \sqrt{1-e^2}$ may oscillate strongly and the eccentricity may grow from $\sim 0$ to 0.6 [@Frouard2010]. Thus the evection effect places constraints on the stability of irregular satellites and influences their orbital distribution. Furthermore, this effect is not symmetrical with respect to the inclination of $90^\circ$ because for prograde orbits, $\varpi=\Omega+\omega$ (where $\Omega$ is the longitude of the ascending node and $\omega$ the argument of pericentre) while for retrograde orbits, $\varpi$ is the difference of the two angles. As a result, the eccentricity of a prograde orbit is excited more efficiently than a retrograde one, providing a possible mechanism to explain why the retrograde regime harbours more irregular satellites [@Nesvorny2003].
A prominent feature of the irregular satellite population is that they form groups or families [@Nesvorny2003; @Sheppard2003]. Members of the same family have similar orbital elements. Families are thought to be the products of collisional evolution. @Nesvorny2003 used the Gauss equations to infer the degree of dispersion within different groups of irregular satellites. They found that the Himalia family is more widely dispersed than expected for a group of collisional fragments. @Beauge2006 and @Beauge2007 developed a high order analytical method and used it to calculate accurate precession rates of irregular satellites and to study the dynamical structure of the Kozai resonance. Based on this method, they determined the locations of secular resonances and compared them with the positions of known irregular satellites; also, a new family around Pasiphae was identified. @Hinse2010 applied the MEGNO technique to jovian irregular satellites and identified chaotic and quasi-periodic regions; they also found some high-order mean motion resonances. Using similar methods, @Frouard2011 reported a number of resonances related to the Great Inequality and revealed the chaotic diffusion in different irregular satellite groups.
All of the above work has assumed the irregular satellites to be massless. However, in a manner similar to asteroid families [e.g., @Carruba2003] a massive member of a satellite family could exert strong perturbations on the smaller members and scatter their orbits; @Christou2005 found that, gravitational scattering by Himalia operating over the age of the solar system could be responsible for the large velocity dispersion observed by @Nesvorny2003. @Christou2005 used this effect to place constraints on the mass of Himalia. He also observed a transient resonant lock between the nodes of Himalia and Lysithea, another satellite in the family. Thus we suppose a massive member could have significant effects on the orbital evolution of the group and this phenomenon needs further study. Here, as a case study, we focus our attention on the Himalia group of Jupiter and particularly on the resonances identified by @Christou2005.
The paper is organised as follows: in Sect. \[nodal-intro-sec\], we introduce the Himalia group, the phenomenon of “nodal libration” and the model used to study it. Then in Sect. \[semi-sec-total\], we describe our semianalytical approach to model the nodal libration and introduce other resonant phenomena. The results of $N$-body simulations are presented in Sect. \[sec-n-body\]. Finally, Sect. \[sec-discussion\] is devoted to the conclusions and discussions.
The Himalia group, a nodal resonance and our model {#nodal-intro-sec}
==================================================
The Himalia group is a prograde jovian irregular satellite group with five members: Himalia, Elara, Lysithea, Leda and Dia. Himalia is the largest member that carries the group name; it is at least 8 times more massive than any of the other four known members[^1]. @Emelyanov2005 studied the perturbation of Himalia on other satellites and estimated the mass to be $k^2 m=(0.28 \pm 0.04)$ km$^3$/s$^2$, consistent with the estimate of @Christou2005 based on its scattering effect. We adopt this mass estimate throughout the paper. The mean semimajor axis, eccentricity and inclination of the group are 0.078 AU, 0.18 and 28$^\circ$, respectively.
In [@Christou2005], where the main purpose was to study the influence of gravitational scattering within the Himalia group, the effects of the Sun, Saturn and jovian oblateness (corrected for the additional mass of Galilean satellites) were considered; Himalia and Elara were assumed to be massive and other family members were massless. In such a configuration, it was found that the difference between the longitudes of the nodes of Lysithea and Himalia $\Delta \Omega=\Omega_\mathrm{L}-\Omega_\mathrm{H}$ could occasionally librate around $\pi$ on a timescale of 1 Myr with the amplitude of the oscillation in inclination being ${}\sim 0.1 ^\circ$ (see figs. 4 and 5 of that work). @Christou2005 also reported that, on entering and leaving the libration, significant changes in the inclination can occur. This provides a fast channel for the evolution of inclination and may contribute to the large relative velocity dispersion in the Himalia group. We refer to this phenomenon as “nodal resonance” or “nodal libration” hereafter.
Here, our goal is to specifically study this dynamical phenomenon. To isolate the relevant dynamics, we work within a restricted four-body problem model containing Jupiter, the Sun, a massive Himalia and a massless test particle; see Fig \[model-illustration\]. Here, all bodies revolve around Jupiter (in the relative sense). The Sun is assumed to be on a circular orbit.
![Illustration of the system that consists of Jupiter, the Sun, Himalia and a particle[]{data-label="model-illustration"}](Fig1){width="70.00000%"}
In the following sections, we tackle this four-body problem with both semianalytical and fully numerical methods.
Semianalytical approach {#semi-sec-total}
=======================
Our strategy is to split the four-body problem of Jupiter-Sun-Himalia-particle into two restricted three-body problems that can be tackled separately using perturbation analysis. The first includes Jupiter, the Sun and Himalia. Since the mass of Himalia is vanishingly small compared with those of Jupiter and the Sun, it is assumed to be also massless in this part. Thus Himalia and the particle can be studied using the same technique. The second three-body problem is comprised of Jupiter, massive Himalia and the massless particle. We describe in detail the methods dealing with the two restricted three-body problems in the following subsections. Then we explain how we combine the two.
In this paper, we use a normalised unit system in which the Solar mass is 1; jovian semimajor axis is 1; gravitational constant is 1; thus the orbital period of Jupiter is approximately $2 \pi$; in practice, we have used $2 \pi$ in time conversion for the $N$-body simulations described in Sect. \[sec-n-body\].
Kozai dynamics in the Jupiter-Sun-satellite restricted three-body problem {#sec-solar}
-------------------------------------------------------------------------
In the first instance, solar perturbations dominate the evolution of Himalia and the particle. A substantial body of literature has dealt with such “hierarchical” planet-Sun-satellite problems and has discussed the Kozai-Lidov mechanism [@Lidov1962; @Kozai1962]. There, the Hamiltonian of the system is expanded in terms of the ratio of semimajor axis of the satellite to that of the Sun (we denote this ratio as $\alpha=a/a_\odot$ hereafter) and only the leading order terms (of order $\alpha^2$; the “quadrupole” Hamiltonian) are retained. Actually, in our simple model, the Sun has zero eccentricity, causing the octupole Hamiltonian (of order $\alpha^3$) to be zero [see, e.g., @Krymolowski1999; @Naoz2013]. Then the mean anomalies of the Sun and Himalia are eliminated. Owing to a “happy coincidence” [@Lidov1976], the quadrupole secular Hamiltonian is integrable. Specifically, this Hamiltonian is $$\label{quad_initial}
F={k^2 m_\odot a^2\over 16 a_\odot^3} \left[ \left(-15e^2\cos \left(2 \omega\right)+9e^2+6\right)\cos ^2 i+15e^2\cos \left(2 \omega\right)-3e^2-2\right],$$ in which $k^2$ is the gravitational constant and $m_\odot$ the mass of the Sun; $a$ and $a_\odot$ are semimajor axes of the satellite and the Sun; $e$, $i$ and $\omega$ are the satellite’s orbital eccentricity, inclination and argument of pericentre, respectively [@Kozai1962; @Naoz2013]; this Hamiltonian and its simplified forms will be referred to as the “Kozai Hamiltonian” hereafter. Apparently, the Hamiltonian $F$ and semimajor axis $a$ are conserved. As the longitude of the ascending node $\Omega$ does not appear in Eq. , the vertical component of the angular momentum $H\propto \sqrt{1-e^2} \,\cos i$ is constant as well. For a dynamical system governed by this Hamiltonian, $\omega$ may librate around $\pm \pi/2$ while the angular momentum $G\propto \sqrt{1-e^2}$ may experience large amplitude oscillation, giving rise to the Kozai resonance. All four giant planets have such satellites [@Nicholson2008]. The libration in $\omega$ requires the inclination to exceed $\approx 40^\circ$; this is not the situation for the Himalia group. Generally, such evolution under solar perturbation is referred to as a “Kozai cycle”. In Kozai theory, the reference plane is the orbital plane of Jupiter; this frame is used throughout the paper.
With this Hamiltonian, we can obtain the expressions for the nodal and apsidal precession rates of the satellite. The Hamiltonian will be further simplified in Subsect. \[sec-analytical\].
Satellite interaction in the Jupiter-Himalia-particle restricted three-body problem {#sec-coorbital}
-----------------------------------------------------------------------------------
Here, we consider the subsystem comprised of Jupiter, Himalia and a massless test particle.
Since the orbits of the two satellites are close and may cross, the traditional expansion of the disturbing function [e.g., @Murray1999] is not applicable here. To tackle this, we resort to treating the problem under the framework of coorbital theory. The crucial differences between coorbital theory and the conventional treatment lie in the expansion in the disturbing function and in the elimination of the mean anomalies. In coorbital theory, no expansion is made with respect to the ratio of semimajor axes. The mean anomalies of the two orbits are eliminated in a combined manner so that the difference in the mean longitudes $\Delta \lambda$ remains. This enables the study of the variation of the relative semimajor axis $\Delta a$ [see, for example, @Message1966; @Morais1999]. When studying the secular evolution of the system, $\Delta \lambda$ is eliminated.
We follow [@Henon1986] [see also, @Namouni1999] and use the relative quantities, e.g., the relative vector eccentricity $\mathbf{e}_\mathrm{r} =\mathbf{e}_\mathrm{P}-\mathbf{e}_\mathrm{H}=(e_\mathrm{P} \cos \varpi_\mathrm{P}-e_\mathrm{H} \cos \varpi_\mathrm{H}, e_\mathrm{P} \sin \varpi_\mathrm{P}-e_\mathrm{H} \sin \varpi_\mathrm{H})$ (where $\varpi$ is the longitude of pericentre) of the particle relative to Himalia.
We assume that Himalia and the particle are on similar orbits (i.e., with similar semimajor axes, eccentricities and inclinations), that their eccentricities and inclinations are much smaller than unity and that their total mass is far smaller than that of the central body, Jupiter. The semimajor axes, eccentricities and inclinations of the members of the group are restricted to small ranges and the eccentricities are relatively small, satisfying the requirement of coorbital theory. However, the inclination of the Himalia group is about 0.5 rad. As the main purpose of this section is to describe the dynamical structure qualitatively but not to provide a precise quantitative description of the orbital evolution, we divide all inclinations by a factor of 3 in this section wherever coorbital theory is involved. In this way, the eccentricities and inclinations are all around 0.17, better suiting the expansion described below. We will briefly discuss what the situation is if we apply the original inclinations in Sect. \[sec-discussion\]. In this setting, the motion of the two satellites can be expressed in a frame whose origin revolves around Jupiter on a circular orbit of semimajor axis $a_0$ ($a_0\in\left(\min(a_\mathrm{H},a_\mathrm{P}),\max(a_\mathrm{H},a_\mathrm{P})\right)$); the frame also rotates along its $z$-axis at an angular velocity of $n_0=\sqrt{k^2 (m_\mathrm{J}+m_\mathrm{H})/a_0^3}$ (where $m_\mathrm{J}$ and $m_\mathrm{H}$ are masses of Jupiter and Himalia, respectively; the particle is massless). Omitting higher order terms in $e$ and $i$ and assuming that the two satellites are not generally interacting, the Cartesian coordinates of Himalia in this rotating frame can be expressed as the following functions of time [@Henon1986; @Namouni1999] $$\label{co-motion}
\left\{
\begin{aligned}
&x_\mathrm{H}=a_\mathrm{H} (h_\mathrm{H} \cos n_0 t +k_\mathrm{H} \sin n_0 t)+(a_\mathrm{H}-a_0)
\\
&y_\mathrm{H}=-2 a_\mathrm{H} (h_\mathrm{H} \sin n_0 t -k_\mathrm{H} \cos n_0 t) - {3 \over 2} (a_\mathrm{H}-a_0) (n_0 t-{ \lambda_\mathrm{H}})
\\
&z_\mathrm{H}=a_\mathrm{H}(p_\mathrm{H} \cos n_0 t +q_\mathrm{H} \sin n_0 t) \, ,
\end{aligned}
\right.$$ where $a_\mathrm{H}$, $h_\mathrm{H}=e_\mathrm{H} \cos{\varpi_\mathrm{H}}$, $k_\mathrm{H}=e_\mathrm{H} \sin{\varpi_\mathrm{H}}$, $p_\mathrm{H}=i_\mathrm{H} \cos{\Omega_\mathrm{H}}$, $q_\mathrm{H}=i_\mathrm{H} \sin{\Omega_\mathrm{H}}$ and $\lambda_\mathrm{H}$ are the regular orbital elements of Himalia; similar equations apply for the particle. From the linearity of these expressions, it can be seen that analogous expressions exist for the motion of the particle relative to Himalia and the relative $e$ and $i$ are constant.
On account of a massive Himalia, the relative eccentricity and inclination begin to evolve. It has been shown the secular evolution of relative $e$ and $i$ observe the following potential [@Luciani1995; @Namouni1999] $$\label{co-potential}
R={{ k^2 \mu(m_\mathrm{J}+m_\mathrm{H}) }\over{2 \pi a_0} } \log \Delta\, ,$$ where when $e_\mathrm{r} \left| \cos \omega_\mathrm{r}\right| \geq \left| a_\mathrm{r}\right|$, $$\label{co-potential-one}
\Delta=2 e_\mathrm{r} i_\mathrm{r} \left| \cos \omega_\mathrm{r}\right| +i_\mathrm{r}^2+ e_\mathrm{r}^2 \, ,$$ and when $e_\mathrm{r} \left| \cos \omega_\mathrm{r}\right| < \left| a_\mathrm{r}\right|$, $$\begin{aligned}
\Delta&=\sqrt{2} \left| a_\mathrm{r}\right| \sqrt{\sqrt{4 e_\mathrm{r}^2 i_\mathrm{r}^2 \sin ^2\omega_\mathrm{r}+\left(-i_\mathrm{r}^2+e_\mathrm{r}^2-a_\mathrm{r}^2\right)^2}+i_\mathrm{r}^2-e_\mathrm{r}^2+a_\mathrm{r}^2}
\\
&+\sqrt{4 e_\mathrm{r}^2 i_\mathrm{r}^2 \sin ^2\omega_\mathrm{r }+\left(-i_\mathrm{r}^2+e_\mathrm{r}^2-a_\mathrm{r}^2\right)^2}+a_\mathrm{r}^2 \, .
\end{aligned}$$ In the above expressions, $\mu=m_\mathrm{H}/m_\mathrm{J}$, $a_\mathrm{r}=(a_\mathrm{P}-a_\mathrm{H})/a_0$, $e_\mathrm{r}=\sqrt{h_\mathrm{r} ^2+ k_\mathrm{r} ^2}$, $i_\mathrm{r}=\sqrt{p_\mathrm{r} ^2+ q_r ^2}$, $\varpi_\mathrm{r}=\tan^{-1} (k_\mathrm{r}/h_\mathrm{r})$, $\Omega_\mathrm{r}=\tan^{-1} (q_\mathrm{r}/p_\mathrm{r})$, $\omega_\mathrm{r}=\varpi_\mathrm{r}-\Omega_\mathrm{r}$, $h_\mathrm{r}=h_\mathrm{P}-h_\mathrm{H}$, $k_\mathrm{r}=k_\mathrm{P}-k_\mathrm{H}$, $p_\mathrm{r}=p_\mathrm{P}-p_\mathrm{H}$ and $q_\mathrm{r}=q_\mathrm{P}-q_\mathrm{H}$ (the relative elements). The equations of motion are then [@Namouni1999] $$\label{co-euqation-motion}
\begin{aligned}
\dot h_\mathrm{r}&= {1\over n_0}{{\partial R}\over{\partial k_\mathrm{r}}} ,\qquad \dot k_\mathrm{r}=-{1\over n_0}{{\partial R}\over{\partial h_\mathrm{r}}} ,
\\
\dot p_\mathrm{r}&= {1\over n_0}{{\partial R}\over{\partial q_\mathrm{r}}} ,\qquad \dot q_\mathrm{r}= -{1\over n_0}{{\partial R}\over{\partial p_\mathrm{r}}} .
\end{aligned}$$
The coorbital secular potential causes variations in $e_r$ and $i_r$; it also gives rise to precession in $\varpi_r$ and $\Omega_r$ [@Namouni1999]. These effects, in turn, are reflected in the evolution of the particle’s elements $(\varpi_\mathrm{P}, e_\mathrm{P})$ and $(\Omega_\mathrm{P},i_\mathrm{P})$.
Secular dynamics of the Jupiter-Sun-Himalia-particle system {#sec-analytical}
-----------------------------------------------------------
Having isolated the relevant dynamics in the two sub-problems discussed in Subsects. \[sec-solar\] and \[sec-coorbital\], we now want to combine them. This is done so that both Himalia and the test particle experience solar perturbation through the Kozai Hamiltonian ; to this we add the gravitational influence on the particle from Himalia by applying to potential and adding the corresponding variation rates to the particle. Thus the equations of motion are $$\label{both-equation-motion}
\dot \sigma_\mathrm{H}=f^\odot(\sigma_\mathrm{H}) \quad \mathrm{and} \quad \dot \sigma_\mathrm{P}=f^\odot(\sigma_\mathrm{P})+f^\mathrm{coorb}(\sigma_\mathrm{r})\,,$$ where $\sigma$ is an arbitrary orbital element; $f^\odot$ refers to the solar perturbation and is a function of the elements of Himalia or the particle; $f^\mathrm{coorb}$ represents the contribution from the coorbital interaction potential and is a function of the relative elements. We now have an eight-dimensional system of $(\varpi_\mathrm{H}, e_\mathrm{H}; \varpi_\mathrm{P}, e_\mathrm{P}; \Omega_\mathrm{H}, i_\mathrm{H}; \Omega_\mathrm{P}, i_\mathrm{P})$.
The evolution of the particle is generally dominated by solar perturbations even when perturbations by both the Sun and Himalia are considered. We use a simple example to demonstrate this point. According to Eqs. and , and for the nominal orbits of Himalia and Lysithea (as the particle), the timescales of precession in $\varpi_\mathrm{P}$ and $\Omega_\mathrm{P}$ induced by the coorbital potential are $\tau_\varpi ^{\mathrm{coorb}} \sim 2 \pi/\dot \varpi ^{\mathrm{coorb}}\approx 2.7 \times 10^7$ and $\tau_\Omega ^{\mathrm{coorb}} \sim 2 \pi/\dot \Omega ^{\mathrm{coorb}}\approx 2.8 \times 10^7$. For solar perturbations, the corresponding timescales are $\tau_\varpi ^{\mathrm{\odot}} \sim 2 \pi/\dot \varpi ^{\mathrm{\odot}}\approx 1.5 \times 10^2$ and $\tau_\Omega ^{\mathrm{\odot}} \sim 2 \pi/\dot \Omega ^{\mathrm{\odot}}\approx 1.4 \times 10^2$. Noting that, according to @Christou2005, the timescale of nodal libration is $\sim 10^6$, thus the effects of a Kozai cycle should be considered as “fast” and can be eliminated when tackling nodal libration.
Assuming that within a Kozai cycle, changes in eccentricity is small ($\Delta e \ll 1$) and applying some algebraic manipulations (see Appendix \[sec-kozai-derivation\] for details), we have the following “averaged” Kozai Hamiltonian $$\label{constant-kozai}
F_\mathrm{K}=-{k^2 m_\odot a^2\over 16 a_\odot^3}\left(2+3e^2\right)\left(1-3\cos ^2i\right),$$ where $e$ and $i$ are the “averaged” quantities over a Kozai cycle compared to the original ones in Eq. . With this Hamiltonian, the precession rates of $\Omega$ and $\varpi$ are constant. Rates of group members are close with similar $a$, $e$ and $i$.
As shown above, the influence of Himalia is several orders of magnitude smaller than that of the Sun. We suggest that, the libration in $\Delta \Omega$ is only possible where $\mathrm {d}\Delta \Omega / \mathrm{d} t$ is vanishingly small under solar perturbations. With the Hamiltonian , we define a surface in $(a,e,i)$ space where all points on it share the same nodal precession rate with Himalia: $\dot {\Omega} (a,e,i)=\dot {\Omega}|_\mathrm{H}$ (since $\dot {\Omega}$ is a function depending only on $a$, $e$ and $i$); we call this a Surface of Equal Precession Rate (SEPR) in $\Omega$. We show the SEPR in Fig. \[equal-precession-hh\]; it is conceptually similar to those discussed in @Williams1981 in the contact of secular resonances in the main asteroid belt. Note that, until now, we do not assume small eccentricity or inclination for the Kozai Hamiltonian; thus the SEPR applies to the real Himalia group, whose inclination is about 0.5 rad.
![The surface of equal precession rate in $\Omega$ in $(a,e,i)$ space. The location of the surface is calculated using the Kozai Hamiltonian without expansion in eccentricity or inclination. Points on the surface share the same precession rate as Himalia in $ \Omega$. The black square represents the location of Himalia[]{data-label="equal-precession-hh"}](Fig2){width="80.00000%"}
However, as the coorbital potential results from an expansion in eccentricity and inclination, combining it with the full expression for the Kozai potential seems superfluous. Hence, we also expand in $e$ and $i$, retaining terms up to order four to arrive at the following Kozai Hamiltonian $$\label{orig-min-kozai}
{ F}=-{k^2 m_\odot a^2\over 16 a_\odot^3}\left(2i^{4}-9e^2i^2-6i^2+6e^2\right) \,.$$ We then rewrite the Hamiltonian in normalised Poincaré conjugate variables, and again we adopt expansions in $e$ and $i$. The variables are $$\bar g= -\varpi=-(\Omega+\omega), \bar G=e^2/2; \quad \bar h = -\Omega, \bar H=i^2/2.$$ Under the new variable set, the Kozai Hamiltonian becomes $$\label{min-kozai}
{ F}=-{{{k{m_\odot}a^{3/2}}}\over{4\sqrt{m_\mathrm{J}}\,{a_\odot}}^3}\left(2 \bar H^2-9 \bar G \bar H-3 \bar H+3 \bar G\right) \,.$$ Since now the variables are the normalised angular momenta, we introduce a factor of $k\sqrt{m_\mathrm{J} a}$ into the coefficient of to preserve the Hamiltonian property. To better separate “slow” and “fast” angles, we apply the following canonical transformation: $$\label{trans2slow}
\begin{aligned}
\bar g_\mathrm{f}&=&\bar g_\mathrm{P}, \quad \bar G_\mathrm{f}&=& \bar G_\mathrm{P}+\bar G_\mathrm{H};
\\
\bar g_\mathrm{s}&=&\bar g_\mathrm{P}-\bar g_\mathrm{H}, \quad \bar G_\mathrm{s}&=& -\bar G_\mathrm{H};
\\
\bar h_\mathrm{f}&=&\bar h_\mathrm{P}, \quad \bar H_\mathrm{f}&=& \bar H_\mathrm{P}+\bar H_\mathrm{H};
\\
\bar h_\mathrm{s}&=&\bar h_\mathrm{P}-\bar h_\mathrm{H}, \quad \bar H_\mathrm{s}&=& -\bar H_\mathrm{H}.
\end{aligned}$$ On the right-hand side of the above expressions, $(\bar g,\,\bar G)$ and $(\bar h,\,\bar H)$ are conjugate normalised Poincaré variables with subscripts “H” and “P” referring to Himalia and the particle, respectively. On the left-hand side, angles (represented with lower-cased letters) with subscript “f” are faster compared to those with subscript “s”, since the former are differences between, while the latter are identical to, the original angles; the conjugate momenta are represented with capital letters.
Thus the Kozai Hamiltonian of both Himalia and the particle assumes the following form $$\label{slow-kozai}
\begin{aligned}
F_\mathrm{S}=C_\mathrm{P} &\left(
2{\bar H_\mathrm{s}}^2+4{\bar H_\mathrm{f}}{\bar H_\mathrm{s}}-9{\bar G_\mathrm{s}}{\bar H_\mathrm{s}}-9{\bar G_\mathrm{f}}{\bar H_\mathrm{s}}-3{\bar H_\mathrm{s}}+2{\bar H_\mathrm{f}}^2 \right.
\\
& \left. -9{\bar G_\mathrm{s}}{\bar H_\mathrm{f}}-9 {\bar G_\mathrm{f}}{\bar H_\mathrm{f}}-3{\bar H_\mathrm{f}}+3{\bar G_\mathrm{s}}+3{\bar G_\mathrm{f}}
\right)
\\
+C_\mathrm{H}&\left(
2{\bar H_\mathrm{s}}^2-9{\bar G_\mathrm{s}}{\bar H_\mathrm{s}}+3{\bar H_\mathrm{s}}-3{\bar G_\mathrm{s}}
\right) \,,
\end{aligned}$$ with coefficients $C_\mathrm{P}=-{{k { m_\odot} a_\mathrm{P}^{3/2}}/(4{ m_\mathrm{J}^{1/2}}{ a_\odot}^3})$ and $C_\mathrm{H}=-{{k { m_\odot} a_\mathrm{H}^{3/2} }/(4 { m_\mathrm{J}^{1/2}}{ a_\odot}^3})$, where $a_\mathrm{H}$ and $a_\mathrm{P}$ are the semimajor axes of Himalia and the test particle, respectively.
The coorbital potential is a branch function. Note that within the satellite group, the quantity $a_\mathrm{r}=(a_\mathrm{P}-a_\mathrm{H})/a_0$ is always small. Taking the orbital elements of Lysithea for example, $a_\mathrm{r} \approx 0.02$ while $e_\mathrm{r}$ is of order $10^{-1}$. Thus the condition $e_{\mathrm{r}} \left| \cos \omega_{\mathrm{r}}\right| < a_{\mathrm{r}}$ is rarely satisfied and the potential form dominates. We omit the other form and presume that the potential only takes this form.
Again, we emphasise that, what matters is the difference between the nodes of the particle and Himalia. Hence, any effects that operate on timescales comparable to $2 \pi/ \dot\Omega$ are considered “fast” (cf. transformation ) and we need to eliminate them. Since we have assumed constant eccentricity and inclination in the Kozai theory and , there is nothing to remove in the Kozai Hamiltonian. However, the short timescale effects in the coorbital potential need elimination. We proceed as follows. First, we substitute the variables in the coorbital potential to express it in the same variables as the Kozai Hamiltonian . Noting that the fast angles are controlled primarily by the Kozai potential, they precess evenly with time (cf. Eq. ). Hence we perform two finite integrations and arrive at the “slow” coorbital potential[^2]: $$\label{slow-coorbital}
\begin{aligned}
R_\mathrm{S}=&{1\over{4 \pi^2}}\int_0^{2 \pi} \mathrm{d} \bar{g}_\mathrm{f} \int_0^{2 \pi} R(\bar g_\mathrm{f}, \bar G_\mathrm{f}; \bar h_\mathrm{f}, \bar H_\mathrm{f}; \bar g_\mathrm{s}, \bar G_\mathrm{s}; \bar h_\mathrm{s}, \bar H_\mathrm{s}) \, \mathrm{d} \bar{h}_\mathrm{f}
\\
=&{\mu n_0\over \pi^2}\left\{\Im\left[ \mathrm{Li}_2({{\rm i} i_\mathrm{r} \over e_\mathrm{r}}) - \mathrm{Li}_2(-{{\rm i} i_\mathrm{r} \over e_\mathrm{r}}) \right] + \log e_\mathrm{r} \right\}\,.
\end{aligned}$$ Here, the variables have the same meaning as in Eqs. and ; ${\rm i}$ is the unit imaginary number; $\Im$ represents the imaginary part; $\mathrm{Li}$ is the polylogarithm function defined as [e.g., @Weisstein1999] $$\mathrm{Li}_n (z)=\sum_{k=1}^\infty {z^k\over k^n}$$ where $z$ is a complex number and $|z|<1$.
We have now reached a system devoid of “fast” effects, i.e., those operating on timescales similar to a Kozai cycle. Note that the coorbital potential is [*not*]{} a Hamiltonian in these variables (we show this in the Appendix \[sec-non-canonical\]). However, we can still write down the equations of motion of the system as $$\label{equation-motion-slow}
\begin{aligned}
\dot {\bar{g}}_\mathrm{s}&=&f_\mathrm{s,g}^\mathrm{\odot}+f_\mathrm{s,g}^\mathrm{coorb}\,, \quad \dot {\bar{G}}_\mathrm{f}&=&f_\mathrm{f,G}^\mathrm{coorb}\,;
\\
\dot {\bar{h}}_\mathrm{s}&=&f_\mathrm{s,h}^\mathrm{\odot}+f_\mathrm{s,h}^\mathrm{coorb}\,, \quad \dot {\bar{H}}_\mathrm{f}&=&f_\mathrm{f,H}^\mathrm{coorb}\,.
\end{aligned}$$ Here $f_\mathrm{s,g}^\mathrm{\odot}$ is the rate of solar-driven variation in $\bar{g}_\mathrm{s}$ and is derived directly from partial differentiation of the Kozai Hamiltonian . The derivation of the coorbital potential contribution $f_\mathrm{f,G}^\mathrm{coorb}$ and $f_\mathrm{s,g}^\mathrm{coorb}$ is not so straightforward. Using the potential and the equations of motion , we can express the variational rates of the relative variables $h_\mathrm{r}$ and $k_\mathrm{r}$ exerted by Himalia. The rates of the orbital elements of the particle are related to these rates through Eq. . Then the transformation allows us to derive the rates $f_\mathrm{f,G}^\mathrm{coorb}$ and $f_\mathrm{s,g}^\mathrm{coorb}$ in Eq. . Analogous relations exist for $\bar h_\mathrm{f}$ and $\bar H_\mathrm{s}$. In this way, we arrive at the above equations of motion . In essence, what we have is a four-dimensional system of $({\bar{g}}_\mathrm{s};{\bar{G}}_\mathrm{f}; {\bar{h}}_\mathrm{s};{\bar{H}}_\mathrm{f})$.
When describing the dynamics of the system, we still use the original orbital elements $\Delta\Omega=\Omega_\mathrm{P}-\Omega_\mathrm{H}$, $\Delta\varpi=\varpi_\mathrm{P}-\varpi_\mathrm{H}$, $e_\mathrm{P}$ and $i_\mathrm{P}$.
General dynamics
----------------
With the equations of motion above, we are able to integrate the system numerically to investigate its evolution[^3]. We run test integrations and find that, at most locations in phase space, the angles $\Delta \Omega$ and $\Delta \varpi$ circulate with small amplitude oscillations in $e_\mathrm{P}$ and $i_\mathrm{P}$. This is not unexpected and results from the fact that, for positions in the $(a,e,i)$ space far from the SEPR, solar perturbations cause $\Delta \Omega$ and $\Delta \varpi$ to circulate fast, while the coorbital potential contributes minor changes to the precession rates and introduces small amplitude variations in $e_\mathrm{P}$ and $i_\mathrm{P}$.
However, when near the SEPR, Himalia can be as efficient as the Sun in changing the orbital evolution; we show this in the following subsections.
Nodal resonance {#sec-nodal}
---------------
In Fig. \[hh-example-semi\], we show an example of nodal libration in $\Delta \Omega$ in our integrations of Eq. . As observed in @Christou2005, the angle $\Delta \Omega$ librates around $\pi$. While the inclination oscillates in concert with the libration in the nodal difference, the eccentricity undergoes only small amplitude and short time-scale variations. In fact, the eccentricity is a quasi-conserved quantity here and in the two-dimensional system in Subsect. \[sec-phase-space\], it is actually constant. The timescale of nodal libration is of order $10^6$ ($\sim$ 1 Myr) and the changes in inclination are of order $10^{-3}$. To prove the postulate that the librator oscillates around the SEPR when in resonance, we plot the signed distance from the particle to the surface in the same figure. From the plot, it is clear that, when above the SEPR, $\Delta \Omega$ increases and vice versa[^4]. Inherently, when in nodal libration, the particle is restricted to be near the surface and to cross it regularly. This also shows the dominance of the Sun.
![An example of nodal libration from the four-dimensional system . Top panel: $e$ (red) and $i$ (blue); bottom panel: $\Delta \Omega$ (red) and the signed “distance” to the SEPR (blue); $x$-axis: time in our adopted unit. Since what matters is not the absolute “distance” but the relative position of the librating particle with respect to the SEPR, we use the quantity $\Delta i$ to represent this relation. $\Delta i$ is the distance from the particle to the point in the SEPR with the same $a$ and $e$. Thus $\Delta i>0$ means the test particle is above the SEPR and vice versa[]{data-label="hh-example-semi"}](Fig3){width="80.00000%"}
Three additional slow angles {#sec-two-resonance}
----------------------------
Using the same method in Subsect. \[sec-analytical\], we find that the SEPR in terms of the pericentre $\dot {\varpi} (a,e,i)=\dot {\varpi}|_\mathrm{H}$ exists. Furthermore, the SEPRs corresponding to the angle combinations ${\mathrm d (\varpi-\Omega) \over \mathrm d t} (a,e,i)={\mathrm d \omega \over \mathrm d t} (a,e,i)={\mathrm d {\omega} \over \mathrm d t}|_\mathrm{H}$ and ${\mathrm d ({\Omega+\varpi}) \over \mathrm d t} (a,e,i)={\mathrm d ({\Omega+\varpi}) \over \mathrm d t}|_\mathrm{H}$ can be readily computed. We present them together with the SEPR in $\Omega$ in Fig. \[equal-precession-all\]. The four surfaces share the same intersection curve.
![The surfaces of equal precession rate in $ \Omega$, $ \omega= (\varpi-\Omega)$, $ \varpi$ and $ (\Omega+\varpi)$ in $(a,e,i)$ space. They are evaluated in the same way as Fig. \[equal-precession-hh\]; the layout of the three axes and the viewing angle are also the same. Particles on these surfaces share the same precession rates in the angles $ \Omega$ (red), $ \omega= (\varpi-\Omega)$ (blue), $ \varpi$ (cyan) and $ (\Omega+\varpi)$ (green) with Himalia, respectively. The black square marks the position of Himalia. We deliberately choose this viewing angle so that, the normals to the surfaces appear perpendicular to the line of sight[]{data-label="equal-precession-all"}](Fig4){width="80.00000%"}
Using our semianalytical model, we find that two of these three angles, $\Delta \omega= \Delta (\varpi-\Omega)=\omega_\mathrm{P}-\omega_\mathrm{H}$ and $\Delta (\Omega+\varpi)=(\Omega_\mathrm{P}+\varpi_\mathrm{P})-(\Omega_\mathrm{H}+\varpi_\mathrm{H})$, can librate. We show examples of these resonances in Figs. \[wmh-example-semi\] and \[hpw-example-semi\]. Interestingly, we do not find libration in the angle $\Delta \varpi=\varpi_\mathrm{P}-\varpi_\mathrm{H}$; instead there is only slow circulation near its SEPR; see Fig. \[ww-example-semi\].
In the case of the $\Delta \omega$ resonance in Fig. \[wmh-example-semi\], while the angle librates about $\pi$, the variations of eccentricity and inclination are anti-correlated. This behaviour is reminiscent of the Kozai resonance [@Kozai1962]. As we will show in the next subsection, the quasi-conserved quantity is $(e_\mathrm{P}^2+i_\mathrm{P}^2)/2$, the component of the angular momentum along the $z$-axis; we plot this quantity in the same figure.
![An illustration of the resonance in the angle $\Delta \omega= \Delta (\varpi-\Omega)$. Top panel: $e$ (red) and $i$ (blue); bottom panel: $\Delta \omega$ (red) and the quasi-conserved quantity $(e_\mathrm{P}^2+i_\mathrm{P}^2)/2$ (blue); $x$-axis: time. This type of resonance is similar to the Kozai resonance[]{data-label="wmh-example-semi"}](Fig5){width="80.00000%"}
For the resonance of $\Delta (\Omega+\varpi)$ in Fig. \[hpw-example-semi\], the angle librates about $0$ while the eccentricity and inclination are correlated. The quasi-conserved quantity in this case is $(e_\mathrm{P}^2-i_\mathrm{P}^2)/2$.
![A representative of the libration in $\Delta (\Omega+\varpi)$. Top panel: $e$ (red) and $i$ (blue); bottom panel: $\Delta (\Omega+\varpi)$ (red) and the quasi-conserved quantity $(e_\mathrm{P}^2-i_\mathrm{P}^2)/2$ (blue); $x$-axis: time[]{data-label="hpw-example-semi"}](Fig6){width="80.00000%"}
What we have shown here are the examples where the angles librate. When outside libration but still close to the SEPRs, the angles circulate slowly and monotonically increase or decrease (since no crossing of the SEPR takes place now) with the actions evolving on the same timescale and in concert, like the case of the angle $\Delta \varpi$ described below.
For particles near the SEPR in $\varpi$, we detect no libration[^5] thus we show an example of slow circulation in Fig. \[ww-example-semi\]. When $\Delta \varpi$ circulates slowly, the eccentricity varies on the same timescale while the inclination experiences only short-period oscillations. In essence, inclination is the quasi-conserved quantity here. In this situation, the test particle cannot cross the SEPR in $ \varpi$, as is shown in the same figure. However, though the distance from the particle to the SEPR remains positive for most of the time, it does become negative for some time. According to the previous reasoning, the distance should not change its sign when in circulation. We suggest that, because of the coorbital potential, the SEPR from Kozai theory does not coincide with the SEPR of the full problem. As will be shown, the coorbital potential causes a small displacement in the surface. Thus although the distance in terms of the SEPR in Kozai potential changes the sign, the distance with respect to the SEPR controlled by both Kozai and coorbital potential does not. See Fig. \[equilibrium-nodal\] for the influence of coorbital potential on the equilibrium points (i.e., the real SEPR). In Sect. \[sec-discussion\] we discuss the reason why no examples of libration in $\Delta\varpi$ are observed here.
![Typical evolution of the slow circulation of the angle $\Delta \varpi$. Top panel: $e$ (red) and $i$ (blue); bottom panel: $\Delta \varpi$ (red) and the signed “distance” to the SEPR in $\varpi$ (blue); $x$-axis: time[]{data-label="ww-example-semi"}](Fig7){width="80.00000%"}
Phase space properties for the three types of resonance {#sec-phase-space}
-------------------------------------------------------
Having established that the three types of resonance exist in the four-dimensional system , we can reduce the system to two dimensions specifically to study the structure of individual resonances when the frequencies are well separated. We elaborate our approach for the nodal resonance and omit the details for the other two cases. Near the SEPR in $ \Omega$ and far from the common intersection curve of the four SEPRs, we have $\mathrm{d} \Delta \Omega / \mathrm{d}t \ll \mathrm{d} \Delta \varpi / \mathrm{d}t $. At these positions, the effect of the Sun in altering $\Delta \varpi$ far outweighs that of the coorbital interaction. Thus the evolution in $\Delta \varpi , e _\mathrm{P}$ is dominated by the Sun. In addition, under solar forcing , there is only (rapid) precession in $\Delta \varpi$ and no evolution in $e$. Hence, $\Delta \varpi$ is now a “fast” angle (compared to $\Delta \Omega$) and $e_\mathrm{P}$ is constant. To isolate the evolution in $(\Delta \Omega, i_\mathrm{P})$, we can perform the following averaging $$\label{nodal-two-dimensional}
<\dot {\bar{h}}_\mathrm{s}>=f_\mathrm{s,h}^\mathrm{\odot}+{1\over 2\pi}\int_0^{2\pi} f_\mathrm{s,h}^\mathrm{coorb}\,\mathrm{d}\bar{g}_\mathrm{s}\,, \quad <\dot {\bar{H}}_\mathrm{f}>={1\over 2\pi}\int_0^{2\pi} f_\mathrm{f,H}^\mathrm{coorb}\,\mathrm{d}\bar{g}_\mathrm{s}$$ to arrive at a two-dimensional system of $({\bar{h}}_\mathrm{s};{\bar{H}}_\mathrm{f})$ (Cf. Eq. ). Here, all effects operating on timescales shorter than that of nodal libration are removed. We plot the evolution in $(\Delta \Omega , i _\mathrm{P})$ space in Fig. \[phase-nodal\]; we elect to show the inclination relative to an empirically-determined value of 0.22 to highlight the amplitude of the variation. The topological structure of the resonance is clear: the system has two equilibrium points – a stable one with $\Delta\Omega=\pi$ and an unstable one with $\Delta\Omega=0$; libration around the stable equilibrium point is allowed (which is the resonance we see). Hence, we suggest that with the coorbital potential, each point in the SEPR in Kozai potential is split into the two equilibrium points. This can be shown by setting the variation rates in to be zero. In Fig. \[equilibrium-nodal\], we show the positions of the two stationary points as functions of varying $e_\mathrm{P}$. The shift of the equilibrium points from the SEPR indicates the influence of the coorbital potential and is of order $10^{-5}$, reflecting the weakness of the coorbital interaction compared to the Kozai dynamics.
![Phase diagram for the nodal resonance in $\Delta \Omega$. All the orbits in this diagram share the same semimajor axis and the same conserved eccentricity, which are those of Lysithea. We subtract 0.22 from $i_\mathrm{P}$ mainly to highlight the topological structure; the value of 0.22 has been empirically chosen. Cf. Fig. \[hh-example-semi\][]{data-label="phase-nodal"}](Fig8){width="80.00000%"}
![The relative positions of the two equilibrium points of the nodal resonance. Left $y$-axis: the intersection of the SEPR in $\Omega$ with the $a=a_\mathrm{P}$ plane (black solid curve) and the inclination of Himalia (horizontal dashed line); right $y$-axis: distance from the equilibrium point with $\Delta \Omega=\pi$ (the stable one) to the black curve in terms of inclination (red) and distance from the equilibrium point with $\Delta \Omega=0$ (the unstable one, blue); $x$-axis: the varying eccentricity as a parameter[]{data-label="equilibrium-nodal"}](Fig9){width="80.00000%"}
Using similar techniques, we generate analogous plots for the resonances involving the angles $\Delta \omega=\Delta(\varpi-\Omega)$ and $\Delta (\Omega+\varpi)$ in Figs. \[phase-wmh\] and \[phase-hpw\]. In dealing with these two types of resonance, transformations to fully isolate the “slow” evolution are needed. The conserved quantities in the two-dimensional systems that are the angular momenta conjugate to the “fast” angles in the four-dimensional systems are $$H_{\Delta \omega}={ e_\mathrm{P}^2+i_\mathrm{P}^2\over 2}$$ for the resonance in $\Delta \omega$ and $$H_{\Delta (\Omega+\varpi)}={ e_\mathrm{P}^2-i_\mathrm{P}^2\over 2}$$ for the resonance in $\Delta (\Omega+\varpi)$, respectively.
![Phase diagram for the resonance in $\Delta \omega=\Delta(\varpi-\Omega)$. Trajectories in this figure have the same semimajor axes and the same conserved quantity $H_{\Delta \omega}={ (e_\mathrm{P}^2+i_\mathrm{P}^2)/ 2}$. Intuitively, for any give $H_{\Delta \omega}$, we can solve for $e_\mathrm{P}$ and $i_\mathrm{P}$ so that, ${\mathrm{d} \Delta \omega /\mathrm{d}t}=0$ and this is a point on the corresponding SEPR in $ \omega$. However, the solution sometimes does not exist, which is the exact case for the $H_{\Delta \omega}$ prescribed by the elements of Lysithea. Thus we choose the point in the SEPR with the same semimajor axis and eccentricity as Lysithea. Then we use the $H_{\Delta \omega}$ of this point for all the trajectories plotted here. Cf. Fig. \[wmh-example-semi\][]{data-label="phase-wmh"}](Fig10){width="80.00000%"}
![Phase diagram for the resonance in $\Delta (\Omega+\varpi)$. Again, the curves possess the same semimajor axes and the same conserved quantity $H_{\Delta (\Omega+\varpi)}={ (e_\mathrm{P}^2-i_\mathrm{P}^2)/2}$ that is chosen in a similar procedure as in Fig. \[phase-wmh\]. Cf. Fig. \[hpw-example-semi\][]{data-label="phase-hpw"}](Fig11){width="80.00000%"}
Similar to the concept resonance width, we use the amplitude of the oscillation of inclination near the separatrix between libration and circulation as an indicator of the strength of the resonance. It is then observed that, the resonance in $\Delta \Omega$ is the strongest, which may be the reason why it was the first to be identified [@Christou2005]. In Fig. \[phase-nodal\], we observe that the amplitude of the variation of inclination in $\Delta \Omega$ resonance can be as high as $\approx 0.005$. The strengths of the resonances in $\Delta \omega$ and $\Delta (\Omega+\varpi)$ are similar to each other: the amplitudes of the oscillation of inclination are $\lesssim 0.001$.
Since there is apparently no libration in $\Delta \varpi$ in the semianalytical model, the structure of the phase space for the circulation should be trivial. Hence, we omit its phase diagram. Similar with the reasoning above, in the case where the slow angle is $\Delta \varpi$, the conserved quantity is $i_\mathrm{P}$.
$N$-body simulations {#sec-n-body}
====================
Mainly due to the expansion in inclination, the semianalytical approach above is formally not accurate for the real Himalia group although it does serve to illustrate the dynamics. We also need to confirm whether these resonances identified by our semianalytical model survive in the full problem. For these reasons we perform extensive $N$-body simulations[^6] to explore the parameter space of the real satellites.
The most important questions we wish to answer here are (i) how well the properties of these resonances are represented by the semianalytical model and, (ii) the possibility for a fictitious member of the family to be involved in such resonances. To explore these, an ensemble of massless test particles are integrated with Jupiter, the Sun and Himalia.
@Beauge2007 used the metric $$d^2=C_a (\Delta a /a)^2+ C_e (\Delta e)^2 + C_i (\Delta \sin i)^2$$ to represent the distance from a satellite to the centre of a family in the proper element space ($C_a, C_e$ and $C_i$ are constant for a specific irregular satellite family). They proposed that for the Himalia family, a satellite with $ d \lesssim $ 320 m/s could be regarded as a member[^7]. If generating test particles with the above criterion only, we produce an ensemble of particles with the eccentricity in the range $(0.03,0.3)$ and inclination between $(23^\circ,33^\circ)$. These are too large, considering that the corresponding ranges for the real family are about $(0.11,0.22)$ and $(27^\circ,29^\circ)$. To represent better the real family and to enhance the number density of particles in $(a,e,i)$ space, we further require the eccentricity to be less than 0.1 and the inclination less than $2^\circ$ from the family mean values; these two values are approximately twice those of the actual family. We do not apply specific restrictions on the semimajor axes as long as they fulfil the distance criterion. The final distribution of the test particles is: $a\in (0.745 \mathrm{AU}, 0.795 \mathrm{AU})$, $e\in (0.06, 0.26)$ and $i \in (26 ^\circ, 30^\circ)$. We generate 1000 such particles. Strictly speaking, the metric is valid for proper elements. We have applied it to osculating elements for simplicity. The particles, along with Jupiter, Sun and Himalia[^8], are integrated for 100 Myr (about $5.3\times 10^7$ in our time unit). Our integrations take place in a jovi-centric reference system. Following [@Hinse2010], we have changed the longitudes of pericentre by adding a $\pi$ so that, the elements are correct in the jovi-centric frame. The solar eccentricity $e_\odot$ has been set to zero.
Our $N$-body runs confirm the existence of the resonances in $\Delta \Omega$, $\Delta \omega$ and $\Delta (\Omega+\varpi)$ and discover another one involving the angle $\Delta \varpi$. We present examples of them in the following subsection. In Subsect. \[n-statistics-sec\], a statistical study of the resonances to address question (ii) stated in the beginning of this section is presented.
Resonances in $N$-body simulations {#sec-n-resonance}
----------------------------------
In Fig. \[n-hh\], we show a typical example of a test particle in (temporary) nodal libration with Himalia. As in [@Christou2005], the quantities $e$, $i$ are averaged in a $\sim 5\times10^4$ time window to suppress the short-period oscllations. Some of these fluctuations come from the Kozai cycle.
The $N$-body simulations agree well with the semianalytical model in reproducing the general features of the nodal resonance, e.g., the centre and period of libration in $\Delta \Omega=\Omega_\mathrm{P}-\Omega_\mathrm{H}$ ($\pi$ and $10^6$ respectively), the corresponding oscillation in inclination and the failure of the resonance to excite the eccentricity. Inspecting the example in Fig. \[n-hh\] in detail, it is found that, when the signed distance to the SEPR in $(a,e,i)$ space, measured in $\Delta i$ (top panel, the solid curve), is around the peak, $\Delta \Omega$ increases and when $\Delta i$ is near the trough, $\Delta \Omega$ decreases. This is consistent with the semianalytical example (Fig. \[hh-example-semi\]) in the sense that when the particle moves above the SEPR with higher inclination, it has higher nodal precession rate than Himalia (while its absolute value is lower since the rate is negative), and vice versa; as $\Delta \Omega$ approaches zero, the particle is pushed to cross the surface, causing the libration. However, in this example, the distance does not oscillate exactly around zero, but slightly below it. We suggest, this is because of the simplifications we made in Sect. \[sec-analytical\] in constructing the semianalytical model and especially, in calculating the SEPR; in the full problem, such oscillations should be about zero. Alternatively, if we use the fit to the data from the $N$-body simulations (see Subsect. \[n-statistics-sec\] and particularly, Eq. and Fig. \[elements-resonance\]), the result turns out to be better. When in libration, the distance $\Delta i$ (the dashed curve in the top panel) does oscillate around zero. Also apparent is that, before about $1.4\times10^7$, the distance remains smaller than zero and $\Delta \Omega$ keeps decreasing; then from roughly $1.9\times10^7$, the angle, after passage through libration, becomes increasing, with the distance above zero.
![An example of the libration in $\Delta \Omega=\Omega_\mathrm{P}-\Omega_\mathrm{H}$ from the $N$-body simulations. Cf. Figs. \[hh-example-semi\] and \[phase-nodal\]. Top panel: distance $\Delta i$ measured as $\Delta i$ to the SEPR; middle panel: $e$ (red) and $i$ (blue); bottom panel: $\Delta \Omega$; $x$-axis: time in our adopted unit. It is evident that, when in libration, $i$ (but not $e$) is modulated on a timescale of $10^6$. The distance $\Delta i$ has the same meaning as in Fig. \[hh-example-semi\]: the solid curve is the distance to the SEPR from semianalytical model ; the dashed one is related to the fitted surface (Cf. Eq. and Fig. \[elements-resonance\]); the horizontal dotted line marks a value of zero for $\Delta i$[]{data-label="n-hh"}](Fig12){width="80.00000%"}
In Fig. \[n-wmh\] we show an example of the resonance in $\Delta \omega=\Delta(\varpi-\Omega)=\omega_\mathrm{P}-\omega_\mathrm{H}$. As observed by our integrations of the semianalytical model (Fig. \[wmh-example-semi\]), the angle $\Delta \omega$ librates around $\pi$. The variation of the eccentricity and inclination is anti-correlated. This is reminiscent of the Kozai cycle.
![An example of the libration in $\Delta \omega=\omega_\mathrm{P}-\omega_\mathrm{H}$. Cf. Figs. \[wmh-example-semi\] and \[phase-wmh\]. Top panel: $e$ (red) and $i$ (blue); bottom panel: $\Delta \omega$; $x$-axis: time[]{data-label="n-wmh"}](Fig13){width="80.00000%"}
We show an example for the resonance in $\Delta (\Omega+\varpi)=(\Omega_\mathrm{P}+\varpi_\mathrm{P})-(\Omega_\mathrm{H}+\varpi_\mathrm{H})$ as the lower panels in Fig. \[n-hpw\]. This test particle evolves from circulation into libration, leaves libration and enters libration again. Here, the evolution of eccentricity and inclination is correlated: maxima (or minima) of $e$ and $i$ occur at the same time. Interestingly, for this particular example, the rate of circulation of the angle $\Delta \omega$ is not much faster than that of $\Delta (\Omega+\varpi)$. In the upper two panels, we show an enlargement of each of the lower two panels of Fig. \[n-hpw\]. The short-period and anti-correlated evolution of eccentricity and inclination appear superposed on the long-period correlated evolution. We believe that such short-period effects are related to the circulation of $\Delta\omega$. Since the resonances are so close together in $(a,e,i)$ space (see Figs. \[equal-precession-all\] and \[elements-resonance\]), it is not surprising such slow circulation of an angle exists inside the libration region of another angle.
![An example of the libration in $\Delta (\Omega+\varpi)=(\Omega_\mathrm{P}+\varpi_\mathrm{P})-(\Omega_\mathrm{H}+\varpi_\mathrm{H})$ from the $N$-body simulations. Cf. Figs. \[hpw-example-semi\] and \[phase-hpw\]. In the lower two panels: top panel: $e$ (red) and $i$ (blue); bottom panel: $\Delta (\Omega+\varpi)$; $x$-axis: time. The two upper panels represent enlargements of each of the bottom two panels: top panel: $e$ (red) and $i$ (blue); bottom panel: $\Delta (\Omega+\varpi)$ (red) and $\Delta \omega$ (blue); $x$-axis: time[]{data-label="n-hpw"}](Fig14){width="80.00000%"}
Apart from the three types of resonance that appear in the semianalytical model and confirmed by our N-body runs here, we observe another type involving the angle $\Delta \varpi=\varpi_\mathrm{P}-\varpi_\mathrm{H}$. We illustrate it with an example in Fig. \[n-ww\]. The resonance in this angle seems to be the weakest as the resonant angle and action do not behave as regular as those in other three resonances. For the resonance in $\Delta \varpi$, we expect that, while the inclination remains nearly constant on the long term, the eccentricity oscillates periodically in accordance with the libration in $\Delta \varpi$. However, the resonance appears to be highly unstable in the sense that, (i) the libration in $\Delta \varpi$ is intermittent and that (ii) the oscillation of eccentricity does not correlate well with the variation of the angle. In Sect. \[sec-discussion\], we will further explain this. The libration in $\Delta \varpi$ will be referred to as “apsidal libration” or “apsidal resonance” thereafter.
![An example of the libration in $\Delta \varpi=\varpi_\mathrm{P}-\varpi_\mathrm{H}$. Cf. Fig. \[ww-example-semi\]. Top panel: $e$ (red) and $i$ (blue); bottom panel: $\Delta \varpi$; $x$-axis: time[]{data-label="n-ww"}](Fig15){width="80.00000%"}
Statistics of the $N$-body simulations {#n-statistics-sec}
--------------------------------------
Using the high number of libration episodes observed in our simulations, we can draw some statistical conclusions.
Out of the 1000 test particles, 37 are found to be in nodal resonance for longer than about $10^6$ (by visual inspection); among those 37, 32 show libration for time spans $\gtrsim 3 \times 10^6$. A duration of $10^6$ corresponds to approximately two cycles of the nodal libration.
To identify episodes of libration automatically in the time series, we implement two different approaches. The first approach uses the discrete Fast Fourier Transform (FFT) algorithm in the package “Fastest Fourier Transform in the West” [@Frigo2005] to extract the periods and amplitudes of the periodic terms in the time series for $\Delta \Omega$ and $i$; then we identify and compare the dominant periods of the two. In the Fourier spectrum, the distribution of power as a function of period around the peak satisfies the normal distribution $$P(x)= A\, e^{ -\frac{(x-\mu_0)^2}{2\sigma_0 ^2} }\,,$$ where $P(x)$ is the strength at period $x$, $A$ a scaling factor, $\mu_0$ the centre of the normal distribution and $\sigma_0$ its dispersion. Taking the logarithm of both sides, we have $$\log {P(x)}=\log A -\frac{(x-\mu_0)^2}{2\sigma_0 ^2}.$$ We fit this quadratic function to the data to extract the strongest period $\mu_0$ and its strength $P(\mu_0)$. We have tried Gaussian fitting of the defined original strength and it often does not converge. To improve the procedure, we instead weigh our data in an iterative process. The weight is designed so that, points with periods closer to $\mu_0$ and/or with larger strength $P$ are more important; the weight function is $$\label{weight-function}
W_n={1+(P_n/\bar P)^M\over 1+ {|x_n - \mu_0 |/ \Delta x}}\,,$$ in which $(x_n,P_n)$ is a (period, strength) combination from the FFT analysis, $W_n$ its weight, $\bar P$ the mean strength of all periods, $\Delta x$ the total coverage in period and $M$ a free parameter that controls the importance of the periods with different strength $P$. The value of $\mu_0$ is updated after each iteration. This process continues until either $\mu_0$ and $P(\mu_0)$ converge or do not. The analysis of inclination is complicated as it often shows multiple local extrema. Thus for inclination, if $M$ is too large, $\mu_0$ and $P(\mu_0)$ can converge towards a local extremum. For $\Delta \Omega$, the fluctuation is smaller, and a larger $M$ allows us to achieve a better estimate of the strongest period. From experiments, we choose $M=20$ for $\Delta \Omega$ and 4 for $i$; we use $(x_n,P_n)$ only when $x_n\in(3\times10^5,9\times10^5)$ to reduce the influences of irrelevant periods. In addition, we apply the criterion that if the fitted $P(\mu_0)$ is too small, we do not recognise the peak as real. In this way, we can identify all the 32 resonant particles with total time in libration $\ge3 \times 10^6$, without any false positives.
Shown in Fig. \[fit-example\] is the FFT analysis and the fitting results of the same test particle as in Fig. \[n-hh\]. Out of the entire integration time of $5.3\times10^7$, the total duration in libration is roughly $7\times10^6$. This is $(7\times10^6)/(5.3\times10^7)\approx 0.13$ of the entire integration and it generates a strong enough signature in FFT analysis that enables us to detect the libration. The period of nodal libration from the fit is $5.31\times 10^5$ for this particle whereas the average period of the 32 cases is $5.73\times 10^5$. As stated above, the determination of libration period in terms of $\Delta \Omega$ is more reliable owing to the higher value of $M$ in the weight function ; the two periods are from the analysis of $\Delta \Omega$.
![The FFT analysis results and the represented fit in period-strength coordinates for $\Delta \Omega$ and $i$. Left $y$-axis: the FFT algorithm generated strength (point, red) and fitted strength (dashed curve, red) for $\Delta \Omega$; right $y$-axis: the FFT algorithm generated strength (point, blue) and fitted strength (dashed curve, blue) for $i$; $x$-axis: period. This is the analysis of the same particle as in Fig. \[n-hh\][]{data-label="fit-example"}](Fig16){width="80.00000%"}
However, the method described above does not work well for the resonances in $\Delta \omega$, $\Delta (\Omega+\varpi)$ and $\Delta \varpi$. We suggest this issue mainly results from the different behaviour of the resonances. For instance, in Fig. \[n-hpw\] for the resonance involving $\Delta (\Omega+\varpi)$, we see that the behaviour in the time range $(3.1\times 10^7,3.25\times10^7)$ and in $(3.36\times10^7,3.5\times10^7)$ are visually different. Indeed, in the former, the particle is closer to the separatrix with larger amplitude and longer period compared to the latter one where the particle is near the libration centre. In nodal libration, the librator tends to stay near the separatrix (see Fig. \[n-hh\]); this may help to generate strong FFT signals[^9]. Also these resonances are weaker with smaller amplitudes in the oscillations of $e$ and/or $i$ compared to the nodal resonance. These two factors may cause the failure of the above method to identify these episodes of libration for the three resonances.
To provide a universal approach valid for all the resonances, we introduce the following method which detects libration depending on the behaviour of the angles only.
The basic idea is that principally, when an angle is in libration, it cannot cross the point that is $\pi$ away from the libration centre and it must cross the libration centre regularly. For instance, in the case of nodal libration (i.e., when the libration centre is $\pi$), the angle $\Delta \Omega$ should not cross 0 and it is supposed to cross $\pi$ periodically. If this state of affairs persists, we recognise it as a libration event. Additionally, we require the frequency of crossing the libration centre and the rate of the librating angle to be within plausible ranges. Therefore, the addition of these filters reduces significantly the number of false positives. In testing the method for nodal libration, we empirically choose the parameters so that it retrieves all the 32 librators found by the FFT-based method and without false positives. In this respect, the two methods are mutually consistent.
With this method, we find 32, 29, 19 and 11 particles in resonances involving $\Delta \Omega$, $\Delta \omega$, $\Delta (\Omega+\varpi)$ and $\Delta \varpi$, respectively.
In the $N$-body simulations, the evolution of eccentricity and inclination often experiences irregular short-period variations. It is then difficult to estimate the strengths of the resonances in terms of the amplitudes of the actions. Instead, we use the term “stability” to describe the resonances in real $N$-body simulations. We define stability as the longest duration of a single libration event from within the set of all such events. We extract the time duration for each libration event of the four resonances; we plot the distribution of durations of all such events in Fig. \[stability-n\]. The nodal resonance is the most stable, as it can lock a test particle in libration for $\gg10^7$. Actually, we even have two particles remaining in libration for the entire integration of $\approx 5.3\times 10^7$; see Fig. \[most-stable\]. The resonances in $\Delta \omega$ and $\Delta (\Omega+\varpi)$ are similarly stable; both can stay in libration for up to $\approx 10^7$. The resonance in $\Delta \varpi$ is the least stable: the libration cannot survive longer than $ \approx 5.6 \times 10^6$. These results correlate with the strength estimates in Subsect. \[sec-phase-space\]: the resonance with the largest amplitude is the most stable.
![The distribution of time spent in resonance of each libration event. Top-left: resonance involving $\Delta \Omega$; bottom-left: $\Delta \omega$; top-right: $\Delta \varpi$; bottom-right: $\Delta (\Omega+\varpi)$; $y$-axis: the numbers of libration events that spend the corresponding time in libration; $x$-axis: logarithmic time in the power of 10[]{data-label="stability-n"}](Fig17){width="80.00000%"}
![A particle that stays in nodal libration for the entire integration duration of $5.3\times10^7$. Top panel: $i$; bottom panel: $\Delta \Omega$; $x$-axis: time[]{data-label="most-stable"}](Fig18){width="80.00000%"}
We compute the mean orbital elements of a librator when inside resonance and plot them in Fig. \[elements-resonance\]. We only use the averaged elements inside the libration, since as noted by @Christou2005, the elements may experience sudden changes in entering and/or leaving libration (e.g., Figs. \[n-hpw\] and \[n-hh-change\]). In the same figure, we show the following quadratic fit for each type of the resonances: $$i= A_0+A_1 a +A_2 e+A_3 a^2+A_4 a e +A_5 e^2
\label{fit-surface}$$ in which the $A_j$, $j=1,\ldots,5$, are the coefficients to be determined. These surfaces can be regarded as the SEPRs in the full problem since they originate from $N$-body simulations. Comparing them with the surfaces from semianalytical theory in Fig. \[equal-precession-all\], we find that the semianalytical model reproduces the relative positions and the overall slopes and shapes of the surfaces fairly well. The main reason for the differences between the two models for large inclination is that, the oscillation of eccentricity inside a Kozai cycle is not small here, contrary to our assumption of small variation of eccentricity in deriving Eq. . The fitted surface for $ \Omega$ has been used to plot the distance to the SEPR in Fig. \[n-hh\] (top panel, dashed curve). As a reference, the real family members are shown as well. As expected, Lysithea is close to the SEPR for $\Omega$.
![The positions of the librators found in $N$-body simulations and the corresponding fitted SEPRs in $(a,e,i)$ space in comparison with all integrated particles and with the real family members. The big round points with different colours represent the librators of the four types. The surfaces with corresponding colours are the fitted SEPRs. The small grey points are the 1000 test particles. The unfilled square, circle, triangle and diamond represent Dia, Elara, Leda and Lysithea, respectively. The black square is Himalia, again[]{data-label="elements-resonance"}](Fig19){width="80.00000%"}
To better highlight the positions of the resonant particles with respect to the SEPRs, we perform another set of simulations with $2\times 400$ test particles. The generation of initial conditions is, by and large, the same as the previous simulations except that here the initial osculating semimajor axes only assume one of the two values $a_1=0.98 $ and $a_2=1.02$ times the family mean semimajor axis. Then we plot the averaged $e$ and $i$ of the librators found in these $2\times400$ particles in the $(e,i)$ plane and compare them with the positions of the curves of equal precession rate (the intersections of the SEPRs with the $a=a_1$ and $a=a_2$ planes) in Fig. \[fixed-a\]. Again, the curves of equal precession rate from semianalytical model (solid curves) agree with those from the fit (dashed curves). Note that, although the initial osculating semimajor axes are uniquely defined, they can vary significantly over the course of the simulations; thus the mean $a$ could be different from $a_1$ and $a_2$. This explains the excursion of a point from its corresponding fitted curve. For example, in the lower panel, the semimajor axis of the curves is exactly $a_1$ while for the points, the semimajor axes may be slightly different.
![The test particles that pass through the four types of resonance in the simulations with fixed initial $a$s and the corresponding curves of equal precession rate. Top panel: results with $a_\mathrm{P}=a_2$; bottom panel: results with $a_\mathrm{P}=a_1$; $y$-axis: $i$; $x$-axis: $e$. Plotted here is the mean eccentricity and inclination of the librators (of the four resonances and for the time slots in libration) as coloured round points. The solid coloured curves are from Kozai potential and the dashed curves are from the fit . Small grey points are all the $2\times 400$ particles[]{data-label="fixed-a"}](Fig20){width="80.00000%"}
With the 1000 and $2\times 400$ test particles, we have 54, 52, 38 and 28 in resonances of $\Delta \Omega$, $\Delta \omega$, $\Delta (\Omega+\varpi)$ and $\Delta \varpi$. They are all considered in fitting while in Figs. \[stability-n\] and \[elements-resonance\], only the 1000 particle integration results are plotted. We have visually checked these code detected librators and confirm that they are actual librators. However, since our method is insensitive to short duration libration events, more short-lived librators are likely.
As noted in @Christou2005, it was possible that the nodal resonant passage had a net effect in the sense that the inclination, after the libration passage, could change $\sim 0.1^\circ$, mostly increasing. In our simulations, we have confirmed the changes of similar amplitude; we find both directions are possible. Here, we show an example in Fig. \[n-hh-change\]. These resonance-induced changes and the cumulative long-term consequences will be addressed in a forthcoming paper.
![An example of the change in inclination after passing through nodal resonance. This particular particle experiences a decrease of $\sim 0.001$ in inclination after the resonant passage[]{data-label="n-hh-change"}](Fig21){width="80.00000%"}
Conclusion and discussion {#sec-discussion}
=========================
We have investigated the dynamics of the Himalia irregular satellite group in a restricted four-body problem scheme where Jupiter, Sun and the satellite Himalia have finite masses while the test particle, representing a fictitious group member, is massless. We combine the Kozai with coorbital theories to construct a unified semianalytical model; the “fast” effects (operating on timescales comparable to that of a Kozai cycle) are removed. Then we arrive at a “slow” four-dimensional system. In this system, we are able to reproduce the main characteristics of a libration in $\Delta \Omega$ first observed in the numerical simulations in @Christou2005. We also identify two additional types of resonance involving the angles $\Delta \omega$ and $\Delta (\Omega + \varpi)$; their locations in phase space can be approximated by the surfaces of equal precession rate (SEPRs). In reducing the systems to two-dimensional ones, we determine the phase space structure for each of the three resonances. Using numerical integrations of the full Newtonian equations of motion, we confirm the results of the semianalytical model regarding the three types of resonance and find a fourth type involving $\Delta \varpi$. We analyse the occurrence rate and stability of these resonances in the $N$-body simulations. The libration periods of these four resonances are all of order $10^6$; specifically, the FFT-based mean period of nodal resonance is $5.73\times 10^5$ ($\sim$ 1.08 Myr). The main properties of the resonances are listed in Table \[resonances-summary\].
[ \*7[>m[0.105]{}]{} @m[0pt]{}@]{} & & & & & & stability\
$\Delta \Omega$ & $\pi$ & $i_\mathrm{P}$ & $e_\mathrm{P}$ & - & $\sim 10 ^{-3}$ & most stable\
$\Delta \omega$ & $\pi$ & $e_\mathrm{P}, i_\mathrm{P}$ & ${( e_\mathrm{P}^2+i_\mathrm{P}^2)/2}$ & & $\sim 10 ^{-4}$ &\
$\Delta (\Omega+\varpi)$ & 0 & $e_\mathrm{P}, i_\mathrm{P}$ & ${( e_\mathrm{P}^2-i_\mathrm{P}^2)/2}$ & correlated & $\sim 10 ^{-4}$ &\
$\Delta \varpi$ & 0 & - & - & - & - & least stable\
In the semianalytical approach, the “evection” terms [@Yokoyama2008 see also Sect. \[sec-intro\]] that relate the mean motion of the Sun with the apsidal rate of the satellite are omitted. The evection effect can be important for satellites far from the planet ($a \gtrsim r_\mathrm{H}/2$; $r_\mathrm{H}$ is the Hill radius of the planet) where the precession rate is fast. In the planar prograde case, the evection potential to the lowest order in $\alpha$ has the form [@Frouard2010] $$R_\mathrm{evec} \sim {{m_\odot \alpha^3}\over{ a_\odot}} e^2 \cos(2 \varpi -2 \lambda_\odot) ,$$ where $\lambda_\odot$ is the Solar mean longitude. We have implemented the evection corrections as proposed in [@Cuk2004] and found that it does not improve the precision of our model significantly. To be exact, with the corrections, we are still unable to approximate the nodal rate accurately enough to locate the position of the SEPR (in the full $N$-body model). Thus the evection correction is dropped. In addition, though the octupole solar perturbation potential is zero owing to the circular orbit assumed for the Sun [e.g., @Krymolowski1999], even higher order terms may contribute to the precession of the node and shift the location of resonance slightly. Regarding our simple way in removing $\omega$ in Eq. and in reaching Eq. , there are semianalytical methods for this purpose [@Henrard1990]. But that approach requires the introduction of new variables that have no apparent physical meaning, which we believe will complicate the interpretation of the results.
Our semianalytical model does not reproduce the apsidal resonance in $\Delta \varpi$. The original Kozai potential is dependent on $e$ only to order 2 but on $i$ to higher orders (through $\cos i$) while in the coorbital potential (Eqs. and ), $e$ and $i$ are symmetric. Thus this shortcoming of the semianalytical model results from the Kozai potential, specifically the truncated form of the solar forcing. The lack of higher order terms in $e$ causes the absence of $\bar G ^2$ in the final Kozai potential while $\bar H ^2$ does exist. Thus after partial differentiation, the rate $\dot {\bar g}$ does not depend on its conjugate angular momentum $\bar G$ but $\dot {\bar h}$ does depend on $\bar H$. Hence we suggest that, the Kozai potential not only gives rise to the SEPRs, it must also play an important role in modulating $\dot {\bar h}$ (thus ${\mathrm{d} \Delta \Omega / \mathrm{d}t}$) through $\bar H$. Therefore, terms like $\bar G ^2$ in the solar perturbation is needed for the apsidal libration, which is only possible in higher order terms in $\alpha=a/a_\odot$. Indeed, if we artificially add a term $\bar G^2$ in the Kozai potential, the resonance in $\Delta \varpi$ can be retrieved in the semianalytical model. We believe this is why we do not observe apsidal libration in the semianalytical model while it does appear in the $N$-body simulations. This is probably why the apsidal resonance is the most unstable and the weakest (see Figs. \[n-ww\] and \[stability-n\]).
Although our semianalytical model is truncated in $e$ and $i$, we test it with real Himalia group inclinations ($\approx0.5$) and find that, the model does reproduce the three resonances at these inclinations. However, by dividing the real inclinations by a factor of 3, the $e$ and $i$ are similarly small and more consistent with the assumptions made in deriving the coorbital interaction potential [@Henon1986; @Namouni1999].
We have simplified the problem to a restricted four-body problem model. In reality, perturbations such as the oblateness of Jupiter and the effects of other planets exist. When these additional factors are considered, we suggest these resonances may still persist. In our analysis, the solar perturbation mainly arrests relative precession and makes the resonances possible, though terms like $\bar H^2$ do contribute directly to give rise to the resonances. On the timescales we are interested in, it has been simplified in the form containing no angles. When a more elaborate model is constructed, probably similar simplifications can be made and small changes in the resonance locations (in the form of the SEPRs) will likely occur. In @Christou2005, Saturn and the jovian oblateness were taken into account and in such a configuration, he found the resonance in $\Delta \Omega$. Including other planets may introduce additional dynamical effects such as secular resonances [see, e.g., @Beauge2007; @Frouard2011].
Similar to high order mean motion resonance [@Moons1997] and secular resonance [@Bottke2001], resonances involving other combinations of $\Delta \Omega$ and $\Delta \varpi$ may exist. They may be weaker and less stable, compared to the resonances described in this paper.
As shown in Figs. \[equal-precession-all\] and \[elements-resonance\], the SEPRs are close together to each other, especially near the massive satellite. Consequently, the resonances may interact with each other and give rise to chaos; e.g., Fig. \[n-hpw\]. Considering that Himalia may give other group members random kicks during close encounters, it is not strange that a test particle may pass through more than one type of resonance if we extend the $N$-body simulations long enough. Actually, in our 100 Myr integration, we observe an example transiting from libration in $\Delta \varpi$ to libration in $\Delta (\Omega+\varpi)$.
Considering that the retrograde irregular satellite groups are more prevalent in the solar system and that, they often have a much more massive member, the resonances explored in this paper for prograde satellites may have their counterparts in the retrograde case. For example, the Ananke and Carme groups of Jupiter have inclinations around $150^\circ$ and $165^\circ$, respectively. However, the most massive members in the two groups are only 0.004 and 0.02 of the mass of Himalia ($m_\mathrm{H}$) and this is possibly too small to allow resonance phenomena to exist. As an experiment, we have run $N$-body simulations to test if a prograde massive satellite of $0.01 m_\mathrm{H}$ can induce nodal resonance and the results is positive – such low masses may work. As the cases of evection phenomenon (described in Sect. \[sec-intro\]), the definition of $\varpi$ becomes the difference between $\Omega$ and $\omega$ for retrograde orbits; it is possible that the structure of the retrograde resonances is different, for instance, the libration centre may change. For the masses and orbital elements mentioned here, see footnote 1.
Recently, @Novakovic2015 reported a nodal resonance similar to the one discussed here. Their resonance is due to Ceres and could be responsible for the inclination dispersion in the Hoffmeister asteroid family. In their case, the main perturbation comes from Jupiter and the mass ratio between Ceres and Jupiter is of order $10^{-7}$; in our situation, the mass ratio $m_\mathrm{H}/m_\odot$ is $10^{-12}$, arguably a more extreme case. Due to the similar nature of the two problems, we suggest that other types of resonance (e.g., the one involving $\Delta \omega$) may exist in the asteroid belt.
Finally, in Fig. \[elements-resonance\] we see that, Dia, Elara and Leda are near the SEPR in $(\Omega+\varpi)$ while Lysithea is close to the surface in $\Omega$. To check whether the real satellites could be affected by the resonances, we perform an additional integration using the orbits of the real moons for 1000 Myr. We find that, Leda does enter the resonance in $\Delta (\Omega+\varpi)$. Thus some members of the actual family may have been affected by these resonances. This will be addressed in a future paper.
DL thanks Dr. Matjia Ćuk for correspondences on the evection phenomenon. We are grateful for the helpful comments from two anonymous referees, increasing the quality of the paper. The authors wish to acknowledge the SFI/HEA Irish Centre for High-End Computing (ICHEC) for the provision of computational facilities and support. Astronomical research at the Armagh Observatory is funded by the Northern Ireland Department of Culture, Arts and Leisure (DCAL). Fig. \[model-illustration\] is produced using LibreOffice Draw; all the other figures are generated with gnuplot.
Derivation of the “averaged” Kozai Hamiltonian {#sec-kozai-derivation}
==============================================
We derive the Kozai Hamiltonian from its original form here. First, we substitute $e$ and $i$ in Eq. in terms of the angular momenta $G$ and $H$. Then we solve for $G$ so $G=G(F,H,\omega,const.)$, where $const.$ refers to combinations of $k$, $m_\odot$, $a$ and $a_\odot$. Note that in this expression, $F$, $H$ and $const.$ are all constant and only $\omega$ and $G$ are variable. Furthermore, it can be shown with this equation that the variation in eccentricity during a Kozai cycle, $\Delta e^2$ is a small quantity (e.g., for Himalia’s nominal orbit, $\Delta e^2\lesssim 0.03$). Therefore, we assume that the angular momentum $G\propto\sqrt{1-e^2}$ can be expressed as $$G^2=b_0+b_1 \cos 2 \omega + \dots ,$$ where the coefficients $b_1\ll b_0$. We can then express $b_0$ and $b_1$ as functions of $F$, $H$ and $const.$ by comparing the above equation with $G=G(F,H,\omega,const.)$. Now, if we further omit the term $b_1 \cos 2 \omega$ we have $$\label{to-solve-F}
G^2=b_0=b_0(F,H,const.)\propto{1-\langle e\rangle^2}\,.$$ Here, $\langle e\rangle$ is not the $e$ in the original Hamiltonian but the averaged $e$ over a Kozai cycle. Meanwhile, we also have $H^2\propto(1-\langle e\rangle^2)\cos^2\langle i\rangle$ and $\langle i\rangle$ is the averaged $i$. Then we solve Eq. for $F$. So now $F$ is a function of $H$, $b_0=G^2$ and $const.$. After substituting $\langle e\rangle$ and $\langle i\rangle$ into it, we arrive at the “averaged” Kozai Hamiltonian . The new eccentricity and inclination are constant under the solar forcing; for simplicity, we still use the old notations $e$ and $i$.
The non-canonical property of coorbital potential {#sec-non-canonical}
=================================================
As stated in Subsect. \[sec-analytical\], the coorbital potential (and hence ) is not canonical in the variables used in Eq. . To prove this, we presume it is canonical and introduce the following canonical transformations. For the first transformation, we name the variables on the left side as “absolute Cartesian” variables. $$\begin{aligned}
\bar{G}_\mathrm{y,H}&=& \sqrt{2 \bar G_\mathrm{H}} \sin{\bar{g}_\mathrm{H}} ,\quad \bar{G}_\mathrm{x,H}&=& \sqrt{2 \bar G_\mathrm{H}} \cos{\bar{g}_\mathrm{H}} ;
\\
\bar{H}_\mathrm{y,H}&=& \sqrt{2 \bar H_\mathrm{H}} \sin{\bar{h}_\mathrm{H}} ,\quad \bar{H}_\mathrm{x,H}&=& \sqrt{2 \bar H_\mathrm{H}} \cos{\bar{h}_\mathrm{H}} ;
\\
\bar{G}_\mathrm{y,P}&=& \sqrt{2 \bar G_\mathrm{P}} \sin{\bar{g}_\mathrm{P}} ,\quad \bar{G}_\mathrm{x,P}&=& \sqrt{2 \bar G_\mathrm{P}} \cos{\bar{g}_\mathrm{P}} ;
\\
\bar{H}_\mathrm{y,P}&=& \sqrt{2 \bar H_\mathrm{P}} \sin{\bar{h}_\mathrm{P}} ,\quad \bar{H}_\mathrm{x,P}&=& \sqrt{2 \bar H_\mathrm{P}} \cos{\bar{h}_\mathrm{P}} .
\end{aligned}$$ On the right-hand sides of the above expressions, $(\bar g,\,\bar G)$ and $(\bar h,\,\bar H)$ are conjugate Poincaré variables with subscripts “H” and “P” referring to Himalia and a particle, respectively. Then another transformation is introduced. The variables on the left are referred to as “relative Cartesian” variables. $$\label{relative-xy}
\begin{aligned}
\bar{G}_\mathrm{y,t}&=&\bar{G}_\mathrm{y,P}/\sqrt{2}+\bar{G}_\mathrm{y,H}/\sqrt{2} ,\quad \bar{G}_\mathrm{x,t}&=&\bar{G}_\mathrm{x,P}/\sqrt{2}+\bar{G}_\mathrm{x,H}/\sqrt{2} ;
\\
\bar{H}_\mathrm{y,t}&=&\bar{H}_\mathrm{y,P}/\sqrt{2}+\bar{H}_\mathrm{y,H}/\sqrt{2} ,\quad \bar{H}_\mathrm{x,t}&=&\bar{H}_\mathrm{x,P}/\sqrt{2}+\bar{H}_\mathrm{x,H}/\sqrt{2} ;
\\
\bar{G}_\mathrm{y,r}&=&\bar{G}_\mathrm{y,P}/\sqrt{2}-\bar{G}_\mathrm{y,H}/\sqrt{2} ,\quad \bar{G}_\mathrm{x,r}&=&\bar{G}_\mathrm{x,P}/\sqrt{2}-\bar{G}_\mathrm{x,H}/\sqrt{2} ;
\\
\bar{H}_\mathrm{y,r}&=&\bar{H}_\mathrm{y,P}/\sqrt{2}-\bar{H}_\mathrm{y,H}/\sqrt{2} ,\quad \bar{H}_\mathrm{x,r}&=&\bar{H}_\mathrm{x,P}/\sqrt{2}-\bar{H}_\mathrm{x,H}/\sqrt{2} .
\end{aligned}$$ This relates the variables with subscript “r” to the variables used in the coorbital theory as given by @Namouni1999; see the paragraph containing Eqs. -.
Now note that, since we assume the coorbital potential is canonical in the original Poincaré variables, it should preserve the Hamiltonian property in the variables above. However, since the coorbital potential contains only the relative elements, i.e., in the relative Cartesian variables, it only depends on the variables with subscript “r”. Hence, if we consider the Jupiter-Himalia-particle system. The quantities with subscript “t” will be conserved. However, this is not possible. Taking $\bar{G}_\mathrm{y,t}=\bar{G}_\mathrm{y,P}/\sqrt{2}+\bar{G}_\mathrm{y,H}/\sqrt{2}$ for instance. Since the test particle is massless, it should have no influence on Himalia and thus the second term on the right hand side $\bar{G}_\mathrm{y,H}/\sqrt{2}$ is constant. From the above reasoning, we know that, $\bar{G}_\mathrm{y,t}$ should be conserved. This means that, the first term on the right hand side, $\bar{G}_\mathrm{y,P}/\sqrt{2}=\sqrt{\bar G_\mathrm{P}} \sin{\bar{g}_\mathrm{P}}$ will no change. Similarly, $\sqrt{\bar G_\mathrm{P}} \cos{\bar{g}_\mathrm{P}}$, $\sqrt{\bar H_\mathrm{P}} \sin{\bar{h}_\mathrm{P}}$ and $\sqrt{\bar H_\mathrm{P}} \cos{\bar{h}_\mathrm{P}}$ should all be constant. If so, the orbital elements of the particle are not evolving under the perturbation of Himalia. Of course this is not true – the coorbital potential is not a Hamiltonian in the variables used here.
However, the level-curve like structure in the phase diagrams Figs. \[phase-nodal\], \[phase-wmh\] and \[phase-hpw\] of the three types of resonance suggests the existence of a conserved quantity (a Hamiltonian) for these two-dimensional systems. We have tried to construct one of the form of $F_\mathrm{T}=F_\mathrm{S}+A \times R_\mathrm{S}$ (see Eqs. and ), where $A$ is a constant depending on some non-changing parameters. For instance, for the case of nodal resonance, $A$ could be a function of the masses, the semimajor axes and the eccentricities (but not a function of inclinations). However, the solution for $A$ seems to be nonexistent (at least for the nodal resonance). Thus we argue that, the conserved quantity of the two-dimensional systems might be of more complicated forms.
Beaugé, C., Nesvorný, D.: [Proper elements and secular resonances for irregular satellites]{}. [ Astron. J.]{} **133**, 2537–2558 (2007)
Beaugé, C., Nesvorný, D., Dones, L.: [A high-order analytical model for the secular dynamics of irregular satellites]{}. [ Astron. J.]{} **131**, 2299–2313 (2006)
Bottke, W. F., Vokrouhlický, D., Broz, M., Nesvorný, D., Morbidelli, A.: [Dynamical spreading of asteroid families by the Yarkovsky effect.]{} [ Science]{} **294**, 1693–1696 (2001)
Burns, J. A.: [ The evolution of satellite orbits]{}. In: Burns, J. A., Matthews, M. S. (eds.) Satellites, pp. 117–158. The University of Arizona Press, Tucson (1986)
Carruba, V., Burns, J. A., Nicholson, P. D., Gladman, B. J.: [On the inclination distribution of the jovian irregular satellites]{}. Icarus **158**, 434–449 (2002)
Carruba, V., Burns, J. A., Bottke, W., Nesvorný, D.: [Orbital evolution of the Gefion and Adeona asteroid families: close encounters with massive asteroids and the Yarkovsky effect]{}. [ Icarus]{} **162**, 308–327 (2003)
Chambers, J. E.: [A hybrid symplectic integrator that permits close encounters between massive bodies]{}. [ Mon. Not. R. Astron. Soc.]{} **304**, 793–799 (1999)
Christou, A. A.: [Gravitational scattering within the Himalia group of jovian prograde irregular satellites]{}. [ Icarus]{} **174**, 215–229 (2005)
Cuk, M., Burns, J. A.: [On the secular behavior of irregular satellites]{}. [ Astron. J.]{} **128**, 2518–2541 (2004)
Emelyanov, N. V.: [The mass of Himalia from the perturbations on other satellites]{}. [ Astron. Astrophys.]{} **438**, L33–L36 (2005)
Frigo, M. and Johnson, S. G.: [The design and implementation of [FFTW3]{}]{}. [Proc. IEEE.]{} **93** pp. 216–231 (2005)
Frouard, J., Fouchard, M., Vienne, A.: [About the dynamics of the evection resonance]{}. [ Astron. Astrophys.]{} **515**, A54 (2010)
Frouard, J., Vienne, A., Fouchard, M.: [The long-term dynamics of the Jovian irregular satellites]{}. [ Astron. Astrophys.]{} **532**, A44 (2011)
Goldreich P.: [History of the lunar orbit]{}. [Rev. Geophys.]{} **4**, 411–439 (1966)
Hahn, J. M., Malhotra, R.: [Neptune’s migration into a stirred-up Kuiper Belt: a detailed comparison of simulations to observations]{}. [ Astron. J.]{} **130**, 2392–2414 (2005)
Hénon, M., Petit, J. M. [Series expansions for encounter-type solutions of Hill’s problem]{}. [ Celest. Mech.]{} **38**, 67–100 (1986)
Henrard, J.: [A semi-numerical perturbation method for separable hamiltonian systems]{}. [ Celest. Mech. Dyn. Astron.]{} **49**, 43–67 (1990)
Hinse, T. C., Christou, A. A., Alvarellos, J. L. A., Goździewski, K.: [Application of the MEGNO technique to the dynamics of Jovian irregular satellites]{}. [ Mon. Not. R. Astron. Soc.]{} **404**, 837–857 (2010)
Innanen, K. A., Zheng, J. Q., Mikkola, S., Valtonen, M. J.: [The Kozai mechanism and the stability of planetary orbits in binary star systems]{}. [ Astron. J.]{} **113**, 1915–1919 (1997)
Jewitt, D., Haghighipour, N.: [Irregular satellites of the planets: products of capture in the early Solar system]{}. [ Annu. Rev. of Astron. Astro.]{} **45**, 261–295 (2007)
Kozai, Y.: [Secular perturbations of asteroids with high inclination and eccentricity]{}. [ Astron. J.]{} **67**, 591–598 (1962)
Krymolowski, Y., Mazeh, T.: [Studies of multiple stellar systems - II. Second-order averaged Hamiltonian to follow long-term orbital modulations of hierarchical triple systems]{}. [ Mon. Not. R. Astron. Soc.]{} **304**, 720–732 (1999)
Lidov, M.: [The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies]{}. [ Planet. Space Sci.]{} **9**, 719–759 (1962)
Lidov, M. L., Ziglin, S. L.: [Non-restricted double-averaged three body problem in Hill’s case]{}. [ Celest. Mech.]{} **13**, 471–489 (1976)
Luciani, J. F., Namouni, F., Pellat, R.: [The kinetics of systems of planetesimals]{}. [ Astrophys. J.]{} **439**, 800–813 (1995)
Message, P. J.: [The dominant features of the long period librations of the trojan minor planets]{}. In: Rosser, J. B. [eds.]{} [ Space Mathematics, Part 2-Lectures in Applied Mathematics, Vol. 6]{}, pp. 70–78. AMS, Providence, Rhode Island (1966)
Moons, M.: [Review of the dynamics in the Kirkwood gaps]{}. [ Celest. Mech. Dyn. Astron.]{} **65**, 175–204 (1997)
Morais, M. H. M.: [A secular theory for Trojan-type motion]{}. [ Astron. Astrophys.]{} **326**, 318–326 (1999)
Murray, C. D., Dermott, S. F.: [Solar System Dynamics]{}. Cambridge University Press, Cambridge (1999)
Namouni, F.: [Secular interactions of coorbiting objects]{}. [ Icarus]{} **314**, 293–314 (1999)
Naoz, S., Farr, W. M., Lithwick, Y., Rasio, F. A., Teyssandier, J.: [Secular dynamics in hierarchical three-body systems]{}. [ Mon. Not. R. Astron. Soc.]{} **431**, 2155–2171(2013)
Nesvorný, D., Alvarellos, J. L. A., Dones, L., Levison, H. F.: [Orbital and collisional evolution of the irregular satellites]{}. [ Astron. J.]{} **126**, 398–429 (2003)
Nicholson, P. D., Ćuk, M., Sheppard, S. S., Nesvorný, D., Johnson, T. V..: [Irregular satellites of the giant planets]{}. In: Barucci, M. A., Boehnhardt, H., Cruikshank, D. P., Morbidelli, A., Dotson, R. (eds.) [ The Solar System Beyond Neptune]{}, pp. 411–424. University of Arizona Press, Tucson (2008)
Novaković, B., Maurel, C., Tsirvoulis, G., Knežević, Z.: [Asteroid secular dynamics: Ceres? fingerprint identified]{}. [ Astrophys. J.]{} **807**, L5 (2015)
Roy, A. E.: [ [Orbital Motion.]{}]{} Adam Hilger, Ltd., Bristol. (1978)
Sheppard, S. S., Jewitt, D. C.: [An abundant population of small irregular satellites around Jupiter.]{} [ Nature]{} **423**, 261–263 (2003)
Weisstein, E. W.: [ Polylogarithm]{}. MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/Polylogarithm.html (1995). Accessed 24 October 2015
Williams, J. G., Faulkner, J.: [The positions of secular resonance surfaces]{}. [ Icarus]{} **46**, 390–399 (1981)
Yokoyama, T., [Vieira Neto]{}, E., Winter, O. C., Sanchez, D. M., Brasil, P. I. D. O.: [On the evection resonance and its connection to the stability of outer satellites]{}. [ Math. Probl. Eng.]{} **2008**, 1–16 (2008)
[^1]: Dia was discovered in 2000 and then lost. It was recently recovered by S. S. Sheppard; see <http://dtm.carnegiescience.edu/news/long-lost-moon-jupiter-found>. For the mass estimates of the members, see <http://ssd.jpl.nasa.gov/?sat_phys_par>. For orbital elements, see <http://ssd.jpl.nasa.gov/?sat_elem>. Parameters of other satellites from the latter two links will be used in Sect. \[sec-discussion\].
[^2]: Though the form appears asymmetric in $e_\mathrm{r}$ and $i_\mathrm{r}$, exchanging $e_\mathrm{r}$ and $i_\mathrm{r}$ will not alter the value.
[^3]: All integrations of the equations of motion derived from Kozai and/or coorbital theory are carried out with a 7th order Runge-Kutta method; the step-size is adjusted with an 8th order error estimate. The single step error tolerance is $10^{-14}$.
[^4]: In fact, the node precesses with negative rates. Thus when above the SEPR, the test particle has smaller absolute precession rate.
[^5]: We generate 500 randomly distributed particles on the SEPR in $\varpi$ and integrate them for $2\times10^6$. Visual check shows no apsidal resonance.
[^6]: All $N$-body simulations are implemented with the general Bulirsch-Stoer algorithm in the MERCURY package [@Chambers1999; @Hahn2005] with a tolerance $10^{-12}$. Over an integration time of $10^8$ yr, the errors in energy and angular momentum are both of order $10^{-10}$.
[^7]: Note that, when they published the paper, four satellites: Himalia, Elara, Leda and Lysithea were confirmed as group members. The recently recovered fifth member, Dia, has the largest semimajor axis and eccentricity. Our simple experiments show that the criterion has to be updated to $ d \lesssim $ 510 m/s to include Dia. In this paper, we stick to the results of @Beauge2007.
[^8]: In the $N$-body simulations, the initial osculating elements of the real objects are taken from the JPL HORIZONS System <http://ssd.jpl.nasa.gov/?horizons>.
[^9]: The particle in Fig. \[most-stable\] described below is not a typical example: for a librator to be stable for $5.3\times10^7$, it has to lie well within the resonance. Most librators only pass through libration.
|
---
abstract: |
Given a probability distribution over a set of $n$ words to be transmitted, the [*Huffman Coding*]{} problem is to find a minimal-cost prefix free code for transmitting those words. The basic Huffman coding problem can be solved in $O(n \log n)$ time but variations are more difficult. One of the standard techniques for solving these variations utilizes a top-down dynamic programming approach.
In this paper we show that this approach is amenable to dynamic programming speedup techniques, permitting a speedup of an order of magnitude for many algorithms in the literature for such variations as mixed radix, reserved length and one-ended coding. These speedups are immediate implications of a general structural property that permits batching together the calculation of many DP entries.
author:
- |
Mordecai J. GOLIN\
Dept of Computer Science\
& Engineering\
Hong Kong UST\
Hong Kong, China\
[*golin@cs.ust.hk*]{}
- |
Xiaoming XU\
Dept of Computer Science\
& Engineering\
IIPL, Shanghai Key Lab\
Fudan University\
Shanghai, China\
[*xuxiaoming@fudan.edu.cn*]{}
- |
Jiajin YU\
Dept of Computer Science\
& Engineering\
IIPL, Shanghai Key Lab\
Fudan University\
Shanghai, China\
[*jiajinyu@fudan.edu.cn*]{}
bibliography:
- 'Golin\_SODA\_09\_Extended.bib'
title: 'A Generic Top-Down Dynamic-Programming Approach to Prefix-Free Coding[^1]'
---
Introduction {#sec:intro}
============
Optimal prefix-free coding, or *Huffman coding*, is a standard compression technique. Consider an *encoding alphabet* $\Sigma = \{\sigma_1, \ldots,
\sigma_r\}$. A [*code*]{} $W=\{w_1,w_2,\ldots,w_n\}$ is a set of *code words* $w_i \in \Sigma^*$. Code $W$ is *prefix-free* if $\forall w,w' \in
W$ $w$ is not a prefix of $w'.$ As an example, $\{01, 00, 100\}$ is a prefix-free code but $\{01, 00, 001\}$ is not, because $00$ is a prefix of $001$.
For $w \in \Sigma^*$, let $|w|$ devote the [ *length*]{} of $w,$ i.e., the number of characters in $w.$ For example $|0101| =4.$
The input to the problem is a discrete probability distribution $P=\{p_1,p_2,\ldots,p_n\}$, $\sum_i p_i = 1,$ $\forall i,\, p_i \ge 0.$
The output is a a prefix-free code $W = \{w_1,w_2,\ldots,w_n\}$ whose expected encoding length $\sum_{i=1}^n p_i |w_i|$ is minimized over all $n$ word prefix-free codes. Formally set $Cost(W,P) = \sum_{i=1}^n w_i p_i$. Then $$Cost(P) = \min_{\stackrel{W' \subseteq \Sigma^*,\, |W'| = n}{\mbox{\footnotesize $W$ is prefix-free}}} Cost(W',P)$$
In [@Hu52], Huffman gave a classical $O(n \log n)$ time greedy algorithm for solving the binary case ($r=2$) of this problem. Huffman also extended the algorithm to solve the general $r$-ary case with the same time bound. If the $p_i$’s are given in sorted order, Huffman’s algorithm can be improved to $O(rn)$ time [@Va76].
The correctness of the Huffman algorithm, although easy to prove, is very strongly dependent upon properties of optimal prefix-free codes. Almost any extra constraint or generalization added to the problem description will invalidate the algorithm’s correctness. Many such constraints/generalizations appear in the literature ([@Abrahams01] is a nice survey) and all require special purpose algorithms to address them.
Some examples of such prefix-free coding problems are *Length-limited* coding e.g, [@Ka61; @LaHi90; @AgSc94+; @Sc98], *Unequal-cost* coding e.g., [@Blach-54; @GR-98; @BGLR-02; @Dumit06] *Mixed-radix* coding [@ChuGill92], *Reserved-length* coding [@baer-2007], and *One-ended* coding [@BY-90; @CSP-94; @ChanG00],
The major observation is that all of the best algorithms known for these problems use some form of dynamic programming (DP) to build an optimal (min-cost) coding tree that corresponds to an optimal code.
These DPs primarily differ in whether they build the tree from the bottom-up or the top-down. The best algorithms for Length-limited and Unequal-cost coding use what essentially reduces to a bottom-up DP model combined with some DP-speedup techniques, e.g., Monge speedups using the SMAWK algorithm of [@AgKl87+] (see [@BGLR-02] for an example of this technique and [@Sc98] for a more sophisticated but more specialized speedup method); the best algorithms for Reserved-length and One-ended coding use a top-down DP approach. (Mixed-radix coding [@ChuGill92] uses a totally different DP approach described later)
Length-Limited and Unequal-Cost coding [*could*]{} be solved using a top-down approach but the bottom-up solutions are better for two reasons. The first is that the bottom-up solutions use a more compact solution space than corresponding top-down ones would. This is due to the exploitation of some very problem-specific combinatorial structures of their corresponding optimal code trees. The second is that their bottom-up DPs turn out to have special properties, e.g., the Monge property, which enable speeding up the calculation of table entries. The top-down DPs used in the last two problems don’t have such a compact representations and they also, before this paper, didn’t seem to possess any special property that would lead to speedups.
The main result of this paper is a revisiting of the generic top-down DP approach for solving prefix-free coding problems. We will show that, in this setup, many natural coding problems will have an obvious and simple batching speedup. That is, we will be able to partition the DP table entries into smaller batches (groups) and exploit relationships between entries within a batch to fill in all of the entries in each batch in $O(1)$ amortized time per entry. This will enable speeding up known solutions to the last three problems by at least one factor of $n$. The interesting observation is that the [*same*]{} speedup technique works for all of these problems. Table \[tab:Results\] lists the speedups.
\[tab:Results\]
----------------------------------- ------------------------------------ -------------------
Problem Previous Best Result This paper
\[.02in\] Mixed Radix Coding $O(n^4 \log n)$ [@ChuGill92] $O(n^3)$
\[.02in\] Reserved Length Coding:
$g$ specific lengths given $O(gn^3)$ [@baer-2007] $O(gn^2)$
\[.02in\] Reserved Length Coding:
at most $g$ lengths allowed $O(g^3 n^3 \log^g n)$ [@baer-2007] $O(g n^2 \log n)$
\[.02in\] One-ended Coding $O(n^3)$ [@ChanG00] $O(n^2)$
\[.02in\]
----------------------------------- ------------------------------------ -------------------
: A comparison of our new results with previous ones
The problems
------------
We start by quickly recalling the standard correspondence between prefix-free codes and trees. Let $r = |\Sigma|$ be the size of the alphabet and consider an $r$-ary tree $T$ in which the [$i^{\mbox{\footnotesize th}}$]{}edge leaving a node is labelled with character $\sigma_i$. Associate with node $v \in T$ the unique word read off walking down the path from the root to $v.$ The set of words $W$ associated with the leaves of $T$ is prefix-free. Conversly, given a prefix-free code $W$ one can build a tree whose leaves are exactly the nodes associated with the words of $W$. See Figure \[Fig:Tree\_rep\]
Given tree $T$ associated with code $W$, denote its leaves by $v_1,v_2,\ldots,v_n$ where $v_i$ is the leaf associated with word $w_i.$ Let $d(v_i)$ be the [*depth*]{} of $v_i$ in $T.$ By the correspondence, $d(v_i) =
|w_i|$ so $$Cost(T,P) = \sum_i d(v_i) p_i = \sum_i |w_i| p_i = Cost(W,P)$$ where $Cost(T,P)$ can be understood as the weighted external-path-length of $T.$ So the prefix-free coding problem is equivalent to finding a tree with minimal external path length. For this reason, most algorithms for finding prefix-free codes are stated as tree algorithms.
We now quickly discuss the problems mentioned in the previous sections and their tree equivalents and then state our new results for these problems.
\
In Mixed-Radix Coding the size of the encoding alphabet used depends upon the position of the character within the codeword. This corresponds to constructing a tree in which the [*arity*]{} (number of children) of an internal node depends upon the level of the node. That is, as part of the problem definition, we are given a sequence $t_0,t_1,\ldots$ of integers, $t_i \ge 2$ such that the maximum arity of a node on level $i$ is $t_i.$
The coding version of the problem was motivated [@ChuGill92] by coding with side-channel information and the tree version by problems in multi-level data storage. Chu and Gill [@ChuGill92] solved this problem by introducing an alphabetic version of it and then solving a special case of the alphabetic version. Their algorithm runs in $O(n^4 \log n)$ time; we will improve this to $O(n^3)$.
\
Recall that $|w_i|$ is the length of the [$i^{\mbox{\footnotesize th}}$]{}codeword. In reserved-length coding there are specific restrictions as to the permitted values of $|w_i|$. There are two versions of this problem. In the first version, the [*given-lengths case*]{}, $\Lambda =
\{\gamma_1,\gamma_2,\ldots, \gamma_g\}$ is given as part of the input and we must find a minimum-cost code such that $\forall i, |w_i|\in \Lambda$. This corresponds to building a min-cost tree in which all leaves are on levels in $\Lambda.$
In the second version, the [*$g$-lengths case*]{}, $\Lambda$ is not given in advance. The restriction now is to find a minimum-cost code under the restriction that $|\Lambda| \le g$, where $\Lambda$ is the set of codeword lengths used. This corresponds to building a tree in which at most $g$ levels may contain leaves.
Baer [@baer-2007] introduces these problems in the context of fast decoding and used a top-down DP approach to solve the first one in $O(|\Lambda| n^3)$ time and the second one in $O(n^3 g^3 \log^g n)$ time . We will reduce these two cases, respectively, to $O(|\Lambda| n^2)$ and $O(n^2 g \log n)$ time.
\
In One-Ended coding the aim is to find a min-cost binary prefix-free code in which every word must end with a [**1**]{}. This corresponds to finding a min-cost tree in which only right leaves (leaves that are the right children of their parents) are labelled with the $p_i$ and counted in the calculation of the cost.
One-Ended Coding was introduced by Berger and Yeung [@BY-90] in the context of self-synchronizing codes. Their algorithm ran in exponential time. This was later improved by De Santis, Capocelli and Persiano [@CSP-94] to another exponential-time algorithm with a smaller exponential base. Chan and Golin [@ChanG00] showed how to use top-down DP to derive an $O(n^3)$ time algorithm. We will reduce this down to an $O(n^2)$ time one.
In Section \[Sec:topdown\] we introduce a new coding problem called [*Generalized Mixed-Radix Coding*]{}, develop a top-down DP approach for solving it and then speed it up by batching. In Section \[sec:MRCandRLC\] we reduce both Mixed-Radix Coding and Reduced Length Coding to (multiple) applications of GMR and thus take adavantage of the DP speedup. In Section \[sec:One\_Ended\] we reduce the running time of One-Ended coding using an almost identical technique. Since the analysis of One-Ended coding is very similar to that of the GMR problem, we do not provide the details in this extended abstract (but they are available in the appendix).
The Top Down DP for Generalized Mixed-Radix Coding {#Sec:topdown}
==================================================
We start by introducing the Generalized Mixed-Radix (GMR) problem, develop a top-down DP for solving it and then see how to speed it up.
In a *generalized mixed radix* tree, both the *arity* of an internal node $v$ and the *length* of an edge leaving $v$ depends on the [*level*]{} of $v$. Figure \[fig:GMR\] illustrates these and other definitions in this section. More formally,
Given a a sequence of arities $R=(r_1,r_2, \ldots)$ and a sequence of edge length $C=(c_1, c_2 \ldots)$, a generalized mixed radix (GMR) tree $T$ *satisfying* $R$, $C$ is a tree in which internal node $v$ at level $i-1$ has at most $r_i$ children and and the length of an edge from $v$ to any of its children is $c_i$.
We now distinguish between the level $\ell(v)$ of a node v, which is the number of edges from the root to $v$, and its depth, which is the weighted path length from the root to $v$. More formally,
The level of node $v$ in tree $T$ is the number of edges on the unique path from the root to $v$ and will be denoted by ${\ell}(v).$ The level of tree $T$ is $${\ell}(T) = \max\{ {\ell}(v) \,\mid\, \mbox{$v$ is a node in $T$}\}
= \max\{ {\ell}(v) \,\mid\, \mbox{$v$ is a leaf in $T$}\}.$$ The [*depth*]{} of a node on level $i$ of the tree will be the sum of the lengths of the edges on the path from the root to level $i$, i.e., $depth(v) = L(i) = \sum_{j=1}^{i} c_i.$ The depth of tree $T$ will be $depth(T) = L({\ell}(T)).$
There is an obvious definition of cost in such trees.
Given $R,C$ as above
Let $T$ be any generalized mixed radix tree $T$ for $R,C$ with $n$ leaves labeled $v_1, v_2, \cdots, v_n$. Then $$\label{eq:L_cost_def}
Cost(T)=\sum_i depth(v_i) \cdot p_i$$
The problem to be solved is, given $P$ and $R,C$, find a min-cost tree $T^*$ with $n$ leaves, i.e., $Cost(T^*)=min \left\{Cost(T) \mid \textit{T has $n$ leaves}
\right \}$
The Basic Top-Down Dynamic Program {#subsec:Basic TD DP}
----------------------------------
In this section we quickly describe the standard top-down DP formulation. Since variations of this formulation have been extensively used before for various coding problems, e.g. [@GR-98; @DolevKY99; @ChanG00; @baer-2007], we only sketch the method but do not rigorously prove its correctness.
In what follows $P=\{p_1,p_2,\cdots ,p_n\}$ with $p_1 \ge p_2 \ge \cdots \ge p_n\ge 0$ is given and fixed. The $p_i$ can be arbitrary weights and are not required to sum to $1$. The $p_i$ sequence is implicitly padded so that, for $i > n,$ $ p_i=0.$ Finally, for $m \ge 0$, set $W_m = \sum_{i > m} p_i.$
We start with some standard simplifying assumptions about min-cost (optimal) trees $T^*$. We will show that there always exists at least one optimal tree satisfying these assumptions. Since our goal is to find any optimal tree, our search can be restricted to trees satisfying the assumptions.
In what follows, an internal node $v$, ${\ell}(v) = i$, in tree $T$ will be [*full*]{} if and only if it has $r_{i+1}$ children in $T.$ Tree $T$ will be [*full*]{} if all of its internal nodes are full.
[*Assumption 1:*]{} If $i < j$ then, in $T^*$, $depth(v_i) \le depth(v_j)$.\
If this was not true we could label $v_i$ with $p_j$ and $v_j$ with $p_i$. The resulting tree has cost no greater than the original one so it remains optimal. We may therefore always implictly assume that leaf weights in trees are non-increasing in the level of the tree.
[*Assumption 2:*]{} There is a full tree $T^*$ with the same cost as the optimal tree for $n$ leaves. $T^*$ has $n'$ leaves where $$\max(n,r_{\ell(T^*)}) \le n' \le n +r_{\ell(T^*)}-1$$
This will be a consequence of the padding of $P.$
Let $T$ be an optimal tree with [*exactly*]{} $n$ leaves. Suppose ${\ell}(T)= {\ell}(v_n) = \ell$.
First note that all internal nodes $v$ with $\ell(v) < \ell-1$ are full. Otherwise we could add a new leaf child of $v$ at level $\ell(v) + 1 < \ell(v_n)$ and label it with $p_n$, creating a tree with smaller cost than optimal tree $T.$ Thus, the only non-full internal nodes in $T$ are on level $\ell-1.$
Next note that we may assume that [*at most*]{} one internal node at level $(\ell-1)$ is not full; otherwise leaves on level $i$ can be shifted to the left, so that all internal nodes on level-$(i-1)$, except possibly the rightmost one, are full. Make this rightmost node full by adding an appropriate number leaves to it and call this new full tree $T^*$. Note that $\ell(T^*) = \ell(T) = \ell$ and $cost(T) = cost(T^*).$ This follows because $P$ was padded by setting $p_i=0$ for $i >n.$ Let $n'$ be the number of leaves in $T^*$.
By definition $n \le n'.$ Since $T$ contains at least one internal node on level $\ell-1$, $T^*$ contains at least $r_{\ell}$ leaves on level $\ell-1.$ Thus $\max(n,r_{\ell}) \le n'$
Furthermore, $ n' \le n + r_{\ell}-1$ because $T^*$ was created by adding at most $r_{\ell}-1$ leaves to $T$.
[*Assumption 3:*]{} ${\ell}(T^*) \le n$.\
This will follow from the fact that we may assume that $\forall i, r_i \ge 2.$
[*Assumption 4:*]{} The cost of a tree is fully determined by its [*leaf sequence*]{}, i.e., the number of leaves on each level. No other structural properties need to be maintained.
This follows directly from the previous assumptions, i.e., the fullness of optimal trees. Although we will talk about constructing [*trees*]{} we will really be constructing the corresponding [*leaf sequences,*]{} e.g., sequences denoting how many leaves are on on each level. Since the cost of a tree is fully determined by its leaf sequence this does not cause any problems.
Suppose that $T'$ is a tree with $n' >n$ leaves. Create $T''$ by pruning the deepest leaves from $T'$ one-by-one, until exactly $n$ leaves remain. Then, by construction, $T''$ is a tree with exactly $n$ leaves such that $cost(T'') \le cost(T').$ This observation, Assumption 2 and Assumption 3, tell us that we can find the optimal tree for $n$ leaves by first finding – for every $\ell \le n$, and every $n'$ satisfying $\max(n,r_{\ell}) \le n' \le n +r_{\ell}-1$– the cost of the min-cost full tree of level at most $ \ell$ with $n'$ leaves. Then we take the minimum cost tree among all such trees and prune it until it has exactly $n$ leaves. The resulting tree wil be the optimal tree for $n$ leaves.
There are only $O(n^2)$ $(\ell,n')$ pairs that need to be examined; given their costs, finding the optimal pair requires only $O(n^2)$ time. (The subsequent pruning operation can easily be done in $O(n)$ time.) The hard part is, for each given $(\ell, n')$ pair, to find the costs of the appropriate min-cost full tree and then, if necessary, build it.
The intuition behind the solution is to build optimal trees top-down, starting with an initial tree – the root – building successively bigger trees level-by-level by making some nodes on the bottom level internal. Part of the specification of these intermediate [*truncated trees*]{} will be an explicit statement of the number of nodes on their bottom levels that will become internal when they are further expanded. The process ends when a tree whose bottom level contains no internal nodes is constructed. Since we are only interested in constructing full trees and the truncation of a full tree up to any level is full, [*this process may implicitly assume that every intermediate tree built is full*]{}.
To transform this intuition into a dynamic program we will need to somehow encode the space of intermediate trees compactly and introduce an appropriate definition of [*cost*]{} for intermediate trees.
\
\[def:signature\] Tree $T$ is an *$i$-level tree* if all nodes $v\in T$ satisfy ${\ell}(v) \le i$.
If $T$ is an $i$-level tree its *$i$-level signature* is the ordered pair $$sig_i(T)=(m,b)$$ in which $$\begin{aligned}
m &=& |\left\{v\in T\,\mid\, \textit{v is a leaf, } {\ell}(v) \le i\right\}|\\
b &=& |\left\{v\in T\,\mid\, \textit{v is an internal node } {\ell}(v) = i\right\}|
\end{aligned}$$ In the above definition, “internal” means that the leaf at the bottom level is tagged as being made internal if the tree grows to its next level.
Let $T$ be an $i$-level tree with $sig_i(T)=(m,b)$. The $i$-level *partial cost* of $T$ is $$\label{eq:partial-cost}
Cost_i(T)=\sum_{t=1}^m depth(v_t)\cdot p_t+ L(i)\cdot\sum_{t=m+1}^n p_t$$
Figure \[fig:GMR\] illustrates this definition.
Consider the possible signatures that could occur. Suppose that $T$ is an $i$-level truncation of some optimal tree $T^*$ with ${\ell}(T^*)=\ell$ and $sig_i(T)=(m,b)$.
If $T$ is a proper truncation of $T^*$, i.e, $b >0$, then ${\ell}(T) < \ell $. Thus every labelled leaf in $T$ is one of $v_1,\ldots,v_n$ in $T^*$ and every one of the $b$ nodes on the bottom level of $T$ that is labelled as “internal” is the ancestor of one of $v_1,\ldots,v_n$ in $T^*.$ Thus $m+b \le n.$
If $T$ is not a proper truncation of $T^*$, i.e., $T=T^*$, then $b=0$ and, from Assumption 2, $\max(n,r_{\ell}) \le m \le n +r_{\ell}-1.$ This motivates
\[def:valid\_1\] $(m,b)$ is a [*valid $i$-level signature*]{} if
- If $b >0$ then $m+b \le n.$
- If $b=0$ then $\max(n,r_{i}) \le m \le n +r_{i}-1.$
Obviously, the number of valid $i$-level signatures is $O(n^2)$.
We can now introduce the DP table.
\[def:OPT\] Let $(m,b)$ be a valid $i$-level signature. Set $OPT^i[m,b]$ to be the minimum $i$-level partial cost over all $i$-level trees $T$ with signature $(m,b)$. More precisely $$OPT^i[m,b] = min\left\{Cost_i(T)\,\mid\, \exists T, \textit{T is a i-level full tree
with}\ sig_i(T)=(m,b)\right\}$$ If no such tree exists, set $OPT^i[m,b] = \infty.$
If $t > n$ then $p_t=0$. Thus, if $T^*$ is an $i$-level full tree with $n' \ge n$ leaves then, by definition, $Cost_i(T^*) = \sum_{t=1}^n depth(v_t) \cdot p_t = Cost(T^*).$ So, $OPT^i[n',0]$ will be the optimal actual cost of an $i$-level full tree with $n' \ge n$ leaves, which is what we want.
\[def:expand\] Let $T'$ be an $(i-1)$-level tree with $sig_{i-1}(T')=(m',b')$.\
Expand $T'$ to a full $i$-level tree by adding all $r_{i} b'$ nodes on level $i$.
For $b$ satisfying $ 0 \le b \le b' r_{i}$, the [*$b^{\mbox{\footnotesize{th}}}$ expansion*]{} of $T'$ is the $i$-level tree created by denoting $b$ of these $r_{i} b'$ nodes as internal and making the remaining $ b' r_{i} - b$ nodes into leaves. We denote this by $$T = \textit{Expand}_i(T',b).$$ Note that $sig_i(T) = (m,b)$ where $m = m' + b' r_{i} - b$.
We now extend the definition of expansions to signatures
\[def:sig\_exp\] If $(m',b'),$ $(m,b)$ are valid signatures such that $ 0 \le b \le b' r_{i}$ and $m = m' + b' r_{i} - b$ we write $$(m',b') {{\stackrel {i} {\rightarrow}}}(m,b).$$
It is easy to prove the following by construction:
\[lem:sig\_conv\] Let $T'$ be an $(i-1)$-level tree with $sig_i(T') = (m',b')$ and $(m,b)$ such that $(m',b') {{\stackrel {i} {\rightarrow}}}(m,b).$ Then $T=Expand_i(T',b)$ is a well-defined $i$-level tree with $sig_i(T) = (m,b)$.
This implies the following corollary, which is the basis of the correctness of the dynamic program.
\[cor:path\] Let $(m_0,b_0) = (0,1)$ be the unique level-$0$ tree with internal root. Then the lemma implies that every sequence $$\label{eq:sequence}
(m_0,b_0) {{\stackrel {1} {\rightarrow}}}
(m_1,b_2) {{\stackrel {2} {\rightarrow}}}
\cdots {{\stackrel {i-1} {\rightarrow}}}
(m_{i-1},b_{i-1}) {{\stackrel {i} {\rightarrow}}}
(m_i,b_i)$$ corresponds to an $i$-level tree $T$ with $sig_i(T) = (m_i,b_i)$ that can be constructed top-down from the root by following the appropriate expansions given by the sequence. In particular, if $(m_i,b_i) = (n',0)$ then the constructed tree has exactly $n'$ leaves.
\[lem:Sum\] Let $T'$ and $T$ be, respectively, $i-1$ and $i$-level trees with $sig_{i-1}(T') = (m',b')$ and $sig_i(T) = (m,b)$. If $(m',b') {{\stackrel {i} {\rightarrow}}}(m,b)$ then $$Cost_{i}(T) = Cost_{i-1}(T') + c_iW_{m'}.$$
[*Proof:* ]{} From Lemma \[lem:sig\_conv\], $T= Expand(T',b)$ so level $i$ contains the leaves $v_{m'+1},\ldots,v_m$ and $$\nonumber
\begin{split}
Cost_i(T) & = \sum_{t=1}^m depth(v_t)\cdot p_t+ L(i)\cdot\sum_{m<t\le
n} p_t\\
& = \sum_{t=1}^{m'} depth(v_t)\cdot p_t+ (L(i-1) + c_i) \left( \sum_{m'<t \le
m} p_t +\sum_{m<t\le n} p_t\right)\\
& =Cost_{i-1}(T')+ c_i\sum_{m'<t\le n} p_t = Cost_{i-1}(T')+ c_i W_{m'}
\end{split}$$ This tells us that the cost of the $n'$-leaf tree associated with sequence (\[eq:sequence\]) can be calculated level by level to be $\sum_{t=1}^{i} c_t W_{m_t}$ where ${m_i} = n'.$ Combining all of the above, we can now write a simple DP that models building optimal trees from the top-down.
\[lem:DP\_basic\] The optimal cost of an $i$-level tree with signature $(m,b)$ satisfies $$\label{eq:basicDP}
OPT^i[m,b] =
\min_{ {\tiny \{(m',b') \,\mid\, (m',b') {{\stackrel {i} {\rightarrow}}}(m,b)\}} }
\Bigl\{ OPT^{i-1}[m',b']+ c_i W_{m'}\bigr\}.$$ Initial conditions are that $OPT^0[0,1] = 0$ with all other entries being set to $\infty$.
The entries $OPT^i[m,b]$ only depend upon the entries $OPT^{i-1}[m,b]$ so the table can be filled in using the order $i=1,2,\ldots,n.$
For any given level $i$ there are only $O(n^2)$ valid $i$-level signatures. From Definition \[def:sig\_exp\], for every $(m,b)$, there are only $O(n)$ signatures such that $(m',b') {{\stackrel {i} {\rightarrow}}}(m,b)$ So, for fixed $i,$ filling in all of the $OPT^i[m,b]$ requires $O(n^3)$ time, with $O ( n^4)$ total time needed to fill in all of the enties $OPT^i[m,b]$, $i \le n$.
We will now see how, for fixed $i$, to calculate the values $OPT^{i}[m,b]$ in only $O(n^2)$ time. This will, as promised, reduce the total running time for filling in all $n$ levels of the table to to $O(n^3)$.
Batching for Speedup {#subsec:batching}
--------------------
We now see how to fill in the DP entries in a faster way. We first need two more definitions.
\[def:old and new sigs\] For $1< d $, define $${{\cal I}}(d) = \left\{ (m,b) \,\mid\, m+ b=d\right\},
\qquad
{{\cal I}}'_i(d) = \left\{ (m',b') \,\mid\, m'+b'r_{i}=d\right\}.$$
For any fixed $i$ and $(m,b) \in {{\cal I}}_i(d)$, the definition of ${{\stackrel {i} {\rightarrow}}}$ implies that\
“$(m',b') {{\stackrel {i} {\rightarrow}}}(m,b)$” if and only if “$(m',b') \in {{\cal I}}'_i(d)$ with $b \le b' r_i$”.
\
This immediately permits rewriting (\[eq:basicDP\]) as
\[lem:DP\_batched\] If $(m,b) \in {{\cal I}}(d)$ for some $d \le n$ and $$\label{eq:batched1}
OPT^i[m,b] =
\min_{ \stackrel {(m',b') \in {{\cal I}}'_i(d)} {b \le b' r_{i}} }
\Bigl\{ OPT^{i-1}[m',b']+ c_i W_{m'}\Bigr\}.$$
We now claim that, for fixed $d \le n$, the calculation of the values $OPT^i[m,b]$ for [*all*]{} $(m,b) \in {{\cal I}}(d)$ can be batched together in in $O(d)=O(n)$ time, i.e., in amortized $O(1)$ time per entry.
For fixed $d\le n$ suppose $(m',b') \in {{\cal I}}'_i(d)$, i.e., $m' + b' r_{i} = d.$ This implies $b' \le \lfloor d/r_{i}\rfloor$. Set $$\forall 0 \le b' \le \lfloor d/r_{i}\rfloor,\quad
\gamma(b')
= OPT^{i-1}[m',\,b'] + c_i W_{m'}
= OPT^{i-1}[d- b' r_{i},\,b'] + c_i W_{d- r_{i}b'}.$$ These can be precalculated in $O(d)$ time. Then (\[eq:batched1\]) just says that for $(m,b) \in {{\cal I}}(d)$, $$\label{eq:gamma_def}
OPT^i[m,b] = \min \left\{ \gamma(b') \ \Big|\ (b/ {r_{i}}) \le b' \le \left\lfloor d/{r_{i}}\right\rfloor \right\}.$$ $(m,b) \in {{\cal I}}(d)$ implies $m = d -b$. Since $b \le r_{i} \lfloor d/r_{i}\rfloor$, $ m \ge t = d - r_{i} \left \lfloor d/{r_{i}} \right \rfloor = d \bmod r_{i}.$ So $${{\cal I}}(d) = \left\{\, X_m \,|\, m = t, t+1, \ldots ,d-1\right\}
\quad\mbox{where}\quad
X_m = \bigl(m,\, d-m\bigr).$$ Then (\[eq:gamma\_def\]) can be rewritten as $$\label{eq:X_OPT}
OPT(X_m) = \min \left \{ \gamma(b') \ \big|\ ({d-m})/{r_{i}} \le b' \le \left \lfloor d/ {r_{i}}\right \rfloor \right\}$$ This immediately yields $$\begin{aligned}
OPT[X_t] &=& \gamma\left(\left \lfloor d /{r_{i}}\right \rfloor\right) \\[0.05in]
\forall m >t,\ OPT[X_{m}] &=&
\left\{
\begin{array}{ll}
OPT[X_{m-1}] & \mbox{if $r_{i} \not | (d-m)$}\\[0.08in]
\min\left( OPT[X_{m-1}], \gamma\left( ({d-m})/{r_{i}}\right)\right)
& \mbox{if $r_{i} | (d-m)$}
\end{array}
\right.\end{aligned}$$ Thus, we can calculate all of the $OPT[X_m]$ for $X_m \in {{\cal I}}(d)$ in $O(d)$ time by working in the order $m=t,t+1,\ldots,d.$
For fixed $i$, to fill in all valid signatures we start by implicitly setting all entries to $\infty$ and then iterate for $d=1,2,3,\ldots$, for each value of $d$ using $O(d)$ time to calculate all of the entries $OPT^{i}[m,b]$ with $m+b =d$. The question is where to stop the iteration.
From Definition \[def:valid\_1\] we know that if $b > 0$ then $d = m+b \le n$ while if $b=0$ then $\max(n, r_i) \le m \le n + r_i -1.$ There are now two cases.
- Then $m+b < 2n$ so stop the iteration at $d = 2n-1$. All of the valid entries will have been filled in using $\sum_{d < 2n} O(d) = O(n^2)$ time.
- In this case if $b=0$ then $r_i \le m \le r_i + n-1$ and $(m',b') {{\stackrel {i} {\rightarrow}}}(m,0)$ implies that either $(m',b') = (m,b)$ or $(m',b') = (m-r_i,1),$ i.e., $(m,0)$ has only two possible predecessors. We can therefore fill in the full table in two phases. In the first, fill in all valid entries $OPT^{i}[m,b]$ with $m+b =d$ for $d=2,3,\ldots, n$ in $O(n^2)$ time. In the second, fill in the all valid entries of the form $OPT^i[m,0]$ with $r_i \le m \le r_i + n-1$ in $O(n)$ time, by checking the two predecessors of each possible entry.
Mixed-Radix Coding and Reserved-Length Coding {#sec:MRCandRLC}
=============================================
We now see how to solve both Mixed-Radix Coding and Reserved-Length Coding via the GMR approach.
Mixed-Radix Coding
------------------
Chu and Gill’s Mixed-Radix Coding problem [@ChuGill92] is exactly the GMR problem restricted to all of the edge costs being identically 1, i.e., $\forall i, c_i=1.$ The algorithm in the previous section solves this in $O(n^3)$ time, improving upon the $O(n^4 \log n)$ time of [@ChuGill92].
Reserved-Length Coding
----------------------
In the [*reserved-length coding problem*]{}, there are restrictions as to permissible codeword lengths. In the tree version of the problem, these become restrictions on the allowable levels on which leaves can appear. More formally. let $$Level(T) = \{{\ell}(v) \,|\, \mbox{$v$ a leaf in $T$}\}.$$ There are two versions of the problem.
In the first version of the problem, the [*given-lengths*]{} case, a set of integers $\Lambda = {\left\{ \gamma_1, \ldots, \gamma_g\right\}}$ is given (w.l.o.g, $0 < \gamma_1 < \gamma_1 <
\gamma_2 < \cdots < \gamma_g$; we may also add $\gamma_0=0$ since, if $n >1,$ the root will never be internal) and we are asked to find a minimum-cost $r$-ary tree among all trees with $Level(T) \subseteq \Lambda$.
In the second version of the problem, the [*$g$-lengths*]{} case, an integer $g$ is given and we are asked to find a minimum-cost $r$-ary tree among all trees with $|Level(T)| \le g.$
### The Given-Lengths Case {#sec:given-length}
Let $T$ be an optimal $r$-ary tree for given $P$ and $\Lambda = {\left\{ \gamma_1, \ldots, \gamma_g\right\}}$.
All leaves in $T$ are at a level $\gamma_i$ for some $\gamma_i \in \Lambda$. Consider any internal node $v$ at level $\gamma_{i-1}$. It has no leaf descendants at any level $\ell$ with $\gamma_{i-1} < \ell < \gamma_{i}.$ We may therefore assume that all of its $r^{\gamma_{i}-\gamma_{i-1}}$ descendants at level $\gamma_{i}$ are in the tree, i.e., that $v$ is the root of a [*complete*]{} subtree of height $\gamma_{i}-\gamma_{i-1}$.
We may therefore create a new tree $T'$ as follows. The root of $T'$ corresponds to the root of $T.$ Nodes in $T'$ at level $i$ are in $1$-$1$ correspondence with nodes at level $\gamma_i$ in $T$ and there is an edge from node $u$ on level $i-1$ to node $v$ on level $i$ in $T'$ if $u$ is the level $\gamma_{i-1}$ ancestor of $v$ in $T.$ See Figure \[Fig:Reserved\] for an illustration.
By construction, $T'$ is a GMR tree with $r_i = r^{\gamma_{i} - \gamma_{i-1}}$ and $c_i = \gamma_{i} - \gamma_{i-1}.$ Furtheremore, the construction can be reversed, with any generalized mixed arity tree with these parameters being transformable into a restricted length tree with the same cost for the given $\Lambda$.
Since there are at most $g$ levels, our generic GMR algorithm solves this problem in $O(g n^2)$ time, improving upon the $O(g n^3)$ algorithm of [@baer-2007] .
### The $g$-lengths case {#sec:given-num-length}
If the levels on which leaves appeared were known to be $\Lambda = {\left\{ \gamma_1, \ldots, \gamma_g\right\}}$ then this is exactly the given-lengths case, which as seen, is equivalent to building an optimal GMR tree with $(r_i, c_i) = (r^{\gamma_{i} - \gamma_{i-1}},\gamma_{i} - \gamma_{i-1}).$ The added complication here is to [*guess*]{} $\Lambda.$
This is equivalent to the problem of building a slightly generalized version of a GMR tree in which, instead of guessing $\Lambda$, we instead, at each level $i$, guess the pair $(r_i, c_i) = (r^t, t)$ for $t \ge 1.$ Any such pair is allowable but once $t$ is chosen, it applies to all nodes on level $i.$ Furthermore, since the tree only needs $n$ leaves we may assume $r^t \le r n$ and thus may restrict $t \le 1+ \log_r n.$
This motivates slightly modifying the GMR model to allow [*choices*]{} of $(r_i,c_i)$.
Recall that the original definition of GMR specifies arities $R=(r_1, r_2, \ldots)$ and edge lengths $C=(c_1, c_2, \ldots)$. We now replace these with $\overline R=(\overline r_1, \overline r_2, \ldots)$ and edge lengths $\overline C=(\overline c_1, \overline c_2, \ldots)$ where $$\overline r_i = \{r_{i,1},\, r_{i,2},\, \ldots, r_{i,\Delta_i}\},
\quad
\overline c_i = \{c_{i,1},\, c_{i,2},\, \ldots, c_{i,\Delta_i}\},$$ are [*sets*]{} of $\Delta_i$ possibilities for level $i.$ A [*permissible tree*]{} is a GMR tree for some sequence $R=(r_{1,j_1}, r_{2j_2}, \ldots)$ and edge lengths $C=(c_{1,j_1}, c_{2,j_2}, \ldots)$ where $\forall i,\, 1 \le j_i \le \Delta_i$.
Given $P,$ an optimal tree would now be a min-cost permissible tree for the given $\overline R$, $\overline C$.
The discussion above tells us that to solve the $g$-lengths problem, it is only necessary is solve this new generalized version of the GMR problem to construct a minimum-cost $g$-level tree where, for every $i,$ $\overline r_i$ and $\overline c_i$ are the sets defined by $$\Delta_i = 2+\lfloor \log_r n\rfloor,\quad
\mbox{ and } \quad
\forall 1 \le j \le \Delta_i, \quad
r_{i,j} = r^{j-1},\
c_{i,j} = j-1.$$
The modifications to the definitions and algorithms are straightforward. Signatures are defined the same way as before. Definition \[def:valid\_1\] of valid signatures needs to be modified to allow $r_{i,j}.$
\[def:valid\_2\] $(m,b)$ is a [*valid $i$-level signature*]{} if
- If $b >0$ then $m+b \le n.$
- If $b=0$ then $\exists j,$ such that $\max(n,r_{i,j}) \le m \le n +r_{i,j}-1$
Note that the number of valid $i$-level signatures is $O(n^2 + n \Delta_{i}).$\
Definition \[def:sig\_exp\] also needs to be slightly generalized:
\[def:sig\_exp\_2\] If $(m',b'),$ $(m,b)$ are, respectively, valid $(i-1)$ and $i$-level signatures such that $ 0 \le b \le b' r_{i,j}$ and $m = m' + b' r_{i,j} - b$, we write $$(m',b) {{\stackrel {i,j} {\rightarrow}}}(m,b).$$
We now, similarly as before, define $$OPT^i[m,b] = min\left\{Cost_i(T)\,\mid\, \exists T, \textit{T is a i-level tree
with}\ sig_i(T)=(m,b)\right\}$$
The only major difference is in the analogue of Lemma \[lem:DP\_basic\], which gives the DP for calculating $OPT^i[m,b]$. This now needs to be split into two phases; the first calculates, for every $j$, the optimimum value of $OPT^i[m,b]$ assuming that $(r_{i},c_{i}) = (r_{i,j}, c_{i,j}).$ The second takes the minimum of this value over all $j.$ More specifically:
\[lem:DP\_basic2\] $$\mbox{For $1 \le j \le \Delta_{i-1}$, set}
\qquad
OPT^{i,j}[m,b] =
\min_{ {\tiny \{(m',b') \,\mid\, (m',b') {{\stackrel {i,j} {\rightarrow}}}(m,b)\}} }
\Bigl\{ OPT^{i-1}[m',b']+ c_{i,j} W_{m'}\bigr\}.$$
The optimal cost of an $i$-level tree with signature $(m,b)$ then satisfies $$OPT^i[m,b] = \min_{1 \le j \le \Delta_i} OPT^{i,j}[m,b]$$ Initial conditions are that $OPT^{0}(0,1) = 0$ with all other entries being set to $\infty$.
Given the values $OPT^{i-1}[m,b]$, the batching speedup of Subsection \[subsec:batching\] now permits, for any fixed $j$, calculating all of the values $OPT^{i,j}[m,b]$ in $O(n^2)$ time. Thus, all of the values $OPT^{i}[m,b]$ can be calculated in $O(\Delta_i n^2)$ time.
The total amount of work required for calculating $OPT^{g}[m,b]$ from scratch is then $O\left(n^2 \sum_{i=0}^g \Delta_i\right)$.
In the $g$-lengths problem, $\Delta_i = O(\log_r n)$ so the total runnnig time for solving the $g$-lengths problem is $O(g n^2 \log n)$, improving the $O(n^2 g^2 \log^{g-1}n)$ running time of the algorithm in [@baer-2007].
One-Ended Coding {#sec:One_Ended}
================
We now consider the problem of constructing minimum-cost binary prefix-free codes having the property that each codeword ends with a “$\bf 1$”. The original algorithms [@BY-90; @CSP-94] for this problem were exponential. [@ChanG00] presented a top-down DP running in $O(n^3)$ time. Using the batched speedup technique developed in Subsection \[subsec:batching\] we can develop a modified top-down DP that reduces the running time to $O(n^2)$
As in Section \[Sec:topdown\], the algorithm will find a min-cost coding tree. We must first modify the code-tree correspondence to reflect the new $1$-ended requirement. Assume that a left edge is labelled with a ‘$0$’ and a right edge with a ‘$1$’. A node is a $0$-node ($1$-node) if the edge connecting it to its parent is labelled by a $0$ ($1$). We will extend this naturally to $0$-leaves and $1$-leaves, and $0$-internal nodes and $1$-internal nodes.
To reflect the $1$-ended restriction on the codes, only $1$ leaves will be labelled with probabilities from $P=\{p_1,\ldots,p_n\}$. Let $v_i$ be the $1$-leaf in tree $T$ labelled with $p_i$. Then we may still write $Cost(T) = \sum_i d(v_i) p_i$. As before, we pad $P$ so that if $i >n$ then $p_i=0.$
These changes require that we naturally modify the definition of full-trees, signatures and expansions. Doing this yields a DP with a size $O(n^2)$ $OP[m,b]$ table. The naive algorithm for filling in this table would require $O(n^3)$ time but a batching argument very similar to that in Subsection \[subsec:batching\] permits filling in the table in $O(n^2)$ time. The step by step modifications required to change the GMR algroithm into one for $1$-ended coding are given in the Appendix.
The One-Ended Coding Algorithm
==============================
We modify the simplifying assumptions on optimal trees $T^*$ of Subsection \[subsec:Basic TD DP\] to reflect the extra requirement of being $1$-ended. Note that not every optimal tree will satisfy these assumptions but we will show that at least one optimal tree will. These assumptions will therefore permit us to restrict the space of trees in which to search. Figure \[Fig:One\_Ended\] illustrates the concepts and definitions introduced here.
[*Assumption 1:*]{} If $i < j$ then, in $T^*$, $depth(v_i) \le depth(v_j)$.\
This is the same as before
[*Assumption 2:*]{} Let $\ell = \ell(T^*).$\
(a) $T^*$ is full.\
(b) All $1$-internals are on levels $< \ell-1$.\
(c) The only $0$-leaves are on level $\ell.$\
(d) There will be at most $n-1$ $1$-leaves in $T^*$ on levels $<\ell$\
(e) The number of $0$-internals on level $\ell-1$ will be $\le n-1$.
Let $T$ be an optimal $1$-ended tree with exactly $n$ $1$-leaves. Erase all $0$-nodes that do not have a $1$-leaf descendant and then make the tree full by adding appropriate missing edges so that every internal node has two children. Note that after doing this, the $1$-sibling of every $0$-leaf exists in $T$ as either an internal node or one of the $n$ $1$-leaves.
Let $v_n$ be the node labelled with $p_n$. We may assume that $\ell(v_n)=\ell$; otherwise we could erase all nodes on level $\ell$ and get a smaller full tree with the same cost which could replace $T.$
Suppose $v$ is a $1$-internal node that has fewer than two $1$-leaf descendents. Since $T$ is full $v$ must have exactly one $1$-leaf descendent. Erase the subtree rooted at $v$. The resulting tree would still have $n$ $1$-leaves but a smaller cost, contradicting the optimality of $T.$ Thus every $1$-internal has at least two 1-leaf descendents. This immediately implies that all $1$-internals have subtrees of height at least $2$ hanging off of them. so all $1$-internal nodes are on levels $< \ell-1$.
If $u$ is a $0$-leaf then $\ell(u) \ge \ell -1$; otherwise, make $u$ the parent of a $1$-leaf at level $\ell(u)+1 < \ell$ and move $p_n$ to this $1$-leaf. This reduces the cost of the tree, contradicting the optimality of $T.$
Thus, all $0$-leaves in $T'$ are on level $\ell-1$ or $\ell$. Let $u$ be any $0$-leaf on level $\ell-1$. Since $T$ is full, its $1$-sibling $v$ is also in $T$, Since $\ell(v) =\ell-1$, we have already seen that $v$ is a $1$-leaf.
By assumption, since $T$ has exactly $n$ $1$-leaves and one of them is at level $\ell,$ the number of $1$-leaves on levels $<\ell$ is at most $n-1$. In particular, since every $0$-leaf on level $\ell-1$ has a $1$-leaf sibling, there are most $n-1$ such $0$-leaves.
Now create $T^*$ from $T$ by making every $0$-leaf on level $\ell-1$ internal. This adds at most $n-1$ new $1$-leaves to the tree; all of the new $1$-leaves added will be at level $\ell$ and be labelled with the padded $0$s, so this does not change the cost. Since $T$ is optimal for $n$ $1$-leaves, $T^*$ is optimal for the number of $1$-leaves it has.
The $T^*$ thus created satisfies the assumption.
The remaining two assumptions will be the same as in the GMR case.
[*Assumption 3:*]{} ${\ell}(T^*) \le n$.
[*Assumption 4:*]{} The optimality of a tree is fully determined by its [*leaf sequence*]{}, i.e., the number of leaves on each level. No other structural properties need to be maintained.
Our characterization of nodes will be slightly different than in the GMR case. A node in $T^*$ will be a [*good* ]{} if it’s a $1$-leaf that gets labelled with one of $p_1,\ldots,p_n.$ It will be [*bad*]{} if it is a $0$-node or a $1$-node that doesn’t get lablelled with one of $p_1,\ldots,p_n$.
We will build trees $T^*$ from the top-down, level-by-level, At step $i$ our current tree $T$ will be the first $i$ levels of $T^*.$ We will identify in $T$ the the number of nodes that are good leaves and the number of nodes, on $T$’s bottom level, that are bad. Note that if $i < \ell(T^*)$ then, in the next expansion of level, [*all*]{} bad nodes on the current bottom level will become internal, each one contributing one new $1$-node and one new $0$-node on the new bottom level. If $i = \ell(T^*)$ then $T=T^*$ and the bad nodes are extra (those that get labelled with $p_j,$ $j >n$) $1$-leaves and $0$-leaves on level $i.$
This motivates us to change the definition of signatures and cost as follows:
\
\[def:signature\_one\_ended\] If $T$ is an $i$-level tree its *$i$-level signature* is the ordered pair $$sig_i(T)=(m,b)$$ in which $$\begin{aligned}
m &=& |\left\{v\in T\,\mid\, \textit{v is a good $1$-leaf, } {\ell}(v) \le i\right\}|\\
b &=& |\left\{v\in T\,\mid\, \textit{v is a bad node, } {\ell}(v) = i\right\}|
\end{aligned}$$ In the above definition, $b$ is counting leaves in $T$ that, if the tree is expanded one level further, will become internal nodes. Let $T$ be the starting ($0$-level) tree containing only the root. Since the root will always be expanded, it is bad, so $sig_0(T) = (0,1).$ This will later be the starting point of our dynamic program.
Let $T$ be an $i$-level tree with $sig_i(T)=(m,b)$. The $i$-level *partial cost* of $T$ is $$\label{eq:partial-cost_one_ended}
Cost_i(T)=\sum_{t=1}^m depth(v_t)\cdot p_t+ i \cdot\sum_{t=m+1}^n p_t$$
As in the GMR case, we want to build an optimal tree $T^*$ that satisfies the assumptions. We start by noting that, by assumptions 2(b) and 2(c), the only bad leaves are on the bottom level $\ell= \ell(T^*).$
The parents of these bad leaves are on level $\ell-1$ and therefore must be $0$-nodes, since by assumption 2(b), all $1$-nodes on level $\ell-1$ are leaves. From assumption 2(e) there are at most $n-1$ such $0$-internals. So, level $\ell$ contains at most $n-1$ bad 1-leaves and $n-1$ bad $0$-leaves.
Now suppose that $T^*$ is being built top-down level-by-level. Let $T$ be the first $i$ levels of $T^*$ with $sig_i(T)=(m,b)$. By definition $m \le n.$ Consider the bad nodes on the bottom level of $T.$ If $i < \ell$ then every bad node is internal. From the fullness of the tree, every bad node on level $i$ must have a $1$-leaf descendent in $T^*$. There are at most $2n-1$ ($n$ good and $n-1$ bad) $1$-leaves in $T$ and each one can appear at most once in some subtree rooted at level $i$ so $b \le 2n-1.$ If $i=\ell$ then $b$ is the number of bad leaves in $T^*$ which is $\le 2(n-1).$ We may therefore assume that $b \le 2n-1.$
This motivates an analogue of Definition \[def:valid\_1\]:
\[def:valid\_one\_ended\] $(m,b)$ is a [*valid signature*]{} if $$m \le n
\qquad\mbox{and}\qquad
b \le 2n-1$$
We also modify Definition \[def:OPT\]:
\[def:OPT\_one\_ended\] Let $(m,b)$ be a valid signature. Set $OPT[m,b]$ to be the minimum $i$-level partial cost over all $i$ and all $i$-level trees $T$ with signature $(m,b)$. More precisely $$OPT[m,b] = min\left\{Cost_i(T)\,\mid\, \exists i,T, \textit{T is a i-level tree
with}\ sig_i(T)=(m,b)\right\}$$ If no such tree exists, set $OPT[m,b] = \infty.$
Note that if $m=n$, then $OPT[n,b] = \sum_{i=1}^n depth(v_t) \cdot p_t$. Thus, from, our previous discussion $$\min \{ OPT^i[n,b] \,\mid\, 1 \le b \le 2n-2\}$$ is the cost of the solution to the one-ended coding problem.
Now suppose that $T'$ is an $(i-1)$-level tree with $sig_{i-1} = (m',b').$ If $T'$ is expanded one more level then the $b'$ bad nodes on level $i-1$ become internal with each one contributing one $1$-node and one $0$-node to level $i$. Between $0$ and $\min(b',n-m')$ of the $1$-nodes can become good $1$-leaves. The remaining $1$-nodes and all $b'$ $0$-nodes are bad. The newly created $i$-level tree has $sig_i(T) = (m,b)$ where $$\label{eq:one_ended_valid}
m' \le m \le n,
\quad
m = m' + 2b' - b,
\quad
1 \le b' \le b \le 2 b'.$$ We therefore define expansions as follows:
\[def:sig\_exp\_one\_ended\] If $(m',b'),$ $(m,b)$ are valid signatures that satisfy (\[eq:one\_ended\_valid\]) we write $$(m',b) \rightarrow (m,b).$$
The next few steps exactly follow the development of Lemmas \[lem:sig\_conv\], \[lem:Sum\], \[lem:DP\_basic\] and Corollary \[cor:path\] with almost identical proofs, so we do not state them explicitly here. The final result is that
$$\label{eq:one_ended_DP}
OPT[m,b] =
\min_{ {\tiny \{(m',b') \,\mid\, (m',b') \rightarrow (m,b)\}} }
\Bigl\{ OPT^{i-1}[m',b']+ W_{m'}\bigr\}.$$
with initial condition $OPT^0(0,1) = 0$. Note that $(m',b') \rightarrow (m,b)$ implies either $m' < m$ or $m=m$ and $b = 2 b'$. So if $(m',b') \rightarrow (m,b)$ then $(m',b')$ is lexicographically smaller than $(m,b)$. Thus, the table can be filled in in lexicographic order to correctly calculate the final values. After filling in the table, the final solution value will be the minimum value of $OPT[n,b],$ where $1 \le b \le 2n-2$. The actual tree corresponding to this solution can be built by backtracking through the table to find the sequence of expansions $$\label{eq:sequence_one_ended}
(m_0,b_0) \rightarrow
(m_1,b_2) \rightarrow
\cdots \rightarrow
(m_{i-1},b_{i-1}) \rightarrow
(n,b)$$ that corresponds to a min-cost one-ended tree.
The DP table has $O(n^2)$ entries and each entry requires $O(n)$ time to calculate, leading to an $O(n^3)$ algorithm for constructing the optimal tree. An algorithm with this running time (based upon a slightly different DP) was given in [@ChanG00].
We now show that it is possible, using the batching technique earlier introduced for the GMR, to fill in this table in $O(n^2)$ time. Define
\[def:one ended old and new sigs\] For $1< d $, define $$\begin{aligned}
{{\cal I}}(d) &=& \left\{ (m,b) \,\mid\, m+ b=d,\ 0 < b,\ m \le n\right\},\\
{{\cal I}}'(d) &=& \left\{ (m',b') \,\mid\, m'+2b'=d,\ 0 < b,\ m \le n \right\}.\end{aligned}$$
For any fixed $d$ $(m,b) \in {{\cal I}}(d)$, the definition of $\rightarrow$ implies that\
“$(m',b') \rightarrow (m,b)$” if and only if “$(m',b') \in {{\cal I}}'(d)$ with $b/2 \le b' \le b$”.
\
Note that if $(m,b) \in {{\cal I}}'(d)$ then $m'+b' < d$ so $(m',b') \in {{\cal I}}(d')$ with $d'<d$. Also, since $m \le n$ and $b \le 2n-1$, we can bound $d \le 3n-1.$ We may therefore fill in the DP table by filling in all the entries in ${{\cal I}}(d)$ as a batch in the order $d=2,3,4,\ldots 3n-1$.
We now see that for fixed $d$ we can, by batching, fill in all of the entries in ${{\cal I}}(d)$ in $O(d)$ time. This will allow filling in the entire table in $O(n^2)$ time, delivering the promised time reduction.
For fixed $d$ suppose $(m',b') \in {{\cal I}}'(d)$, i.e., $m' + 2 b' = d$. Since $ 0 \le m'$ and $0 < b'$ we have $1 \le b' \le d/2.$ For such $b'$ set $$\gamma(b')
= OPT[m',b'] + W_{m'}
= OPT[d- 2b',b'] + W_{d- 2 b'}.$$ Note that since $m'+b' < d$, the values $OPT[m',b']$ have already been calculated and can therefore be looked up. Therefore, all of the $\gamma(b')$ can be calculated using a total of $O(d)$ time. Then, for $(m,b) \in {{\cal I}}(d)$, $$\label{eq:one_ended_gamma_opt}
OPT[m,b] = \min\left\{ \gamma(b') \,\Big|\, \frac b 2 \le b' \le b \right\}$$
Stated this way, this is just a special case of the Range Minimum Query (RMQ) problem.
Given an array $A$ of size $n$, the RMQ problem is to construct a data-structure that, given two indices $i\le j$, will return the index of a smallest valued item in the subarray $A[i]\ldots A[j].$ There are known algorithms, e.g., [@GabowT83; @BenderFPSS05], for $O(n)$ construction of a data-structure that permits $O(1)$ RMQ queries. In our case, the array is the $b'$ indexed set of $\gamma(b')$ values. The array has size $O(d)$ and its values can be constructed in $O(d)$ time. [@GabowT83; @BenderFPSS05] then permit us, in $O(d)$ time, to construct an RMQ data structure. Once this structure is constructed, each value $OPT[m,b]$ can be found in $O(1)$ time. Thus, as promised, all of the $OPT[m,b]$ for $(m,b) \in {{\cal I}}(d)$ can be batched together and evaluated in $O(d)$ time
[^1]: Work of all of the authors was partially supported by Hong Kong CERG grant 613507. Work of the 2nd and 3rd authors also partially supported by a grant from the National Natural Science Foundation of China (No. 60573025) and by the Shanghai Leading Academic Discipline Project, project number B114.
|
---
abstract: 'We study vortex nucleation for minimizers of a Ginzburg-Landau energy with discontinuous constraint. For applied magnetic fields comparable with the first critical field of vortex nucleation, we determine the limiting vorticities.'
address:
- |
H. AydiUniversit[é]{} de Monastir\
Institut sup[é]{}rieur d’informatique de Mahdia\
Km 4, R[é]{}jiche, 5121 Mahdia, Tunisie
- |
A. KachmarUniversit[é]{} Paris-Sud\
D[é]{}partement de math[é]{}matique\
B[â]{}t. 425\
F-91405 Orsay
author:
- Hassen Aydi
- Ayman Kachmar
title: 'Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint. II'
---
Introduction and main results
=============================
In the framework of the Ginzburg-Landau theory, it is proposed to model the energy of an inhomogeneous superconducting sample by means of the following functional (see [@Chetal; @Ru]) $$\label{V-EGL}
\mathcal G_{\varepsilon,H}(\psi,A)=\int_{\Omega}\left(
|(\nabla-iA)\psi|^2+\frac1{2\varepsilon^2}(p(x)-|\psi|^2)^2+ |{\rm
curl}\,A-H|^2\right)\,{{\rm d}}x.$$ Here, $\Omega\subset\mathbb R^2$ is the 2-D cross section of the superconducting sample, assumed to occupy a cylinder of infinite height. The complex-valued function $\psi\in H^1(\Omega;\mathbb
C)$ is called the ‘order parameter’, whose modulus $|\psi|^2$ measures the density of the superconducting electron Cooper pairs (hence $\psi\equiv0$ corresponds to a normal state), and the real vector field $ A=(A_1,A_2)$ is called the ‘magnetic potential’, such that the induced magnetic field in the sample corresponds to ${\rm curl}\, A$.\
The functional (\[V-EGL\]) depends on many parameters: $\frac1\varepsilon=\kappa$ is a characteristic of the superconducting sample (a temperature independent quantity), $H\geq0$ is the intensity of the applied magnetic field (assumed constant and parallel to the axis of the superconducting sample), $p(x)$ is a positive function modeling the impurities in the sample, whose values are temperature dependent. The positive sign of the function $p$ means that the temperature remains below the critical temperature of the superconducting sample.\
In this paper, the function $p$ is a step function. We take $S_1$ an open smooth set such that $\overline S_1\subset\Omega$, $S_2=\Omega\setminus \overline S_1$, and $$\label{p(x)}
p(x)=\left\{\begin{array}{l} 1\quad{\rm if}~x\in S_1\,,\\
a\quad{\rm if}~x\in S_2\,,
\end{array}\right.$$ where $a\in\mathbb R_+\setminus\{1\} $ is a given constant.\
The above choice of $p$ has two physical interpretations (see [@Kach-v; @Ru]):
- $S_1$ and $S_2$ correspond to two superconducting samples with different critical temperatures;
- The superconducting sample $\Omega$ is subject to two different temperatures in the regions $S_1$ and $S_2$, which may happen by cold or heat working $S_2$.
Minimization of the functional (\[V-EGL\]) will take place in the space $$\mathcal H=H^1(\Omega;\mathbb C)\times
H^1(\Omega;\mathbb R^2)\,.$$ That is we do not assume [*a priori*]{} boundary conditions for admissible configurations, but minimizers satisfy [*natural boundary conditions*]{}. We study nucleation of vortices as the applied magnetic field varies in such a manner that, $$\lim_{\varepsilon\to0}\frac{H}{|\ln\varepsilon|}=\lambda\,,\quad\lambda\in\mathbb R_+\,,$$ and we obtain that their behavior is strongly dependent on the parameter $a$, see Theorems \[thm1\] and \[thm2\] below.\
The singularity of the potential $p$ causes a singularity in the energy as $\varepsilon\to0$ of the order of $\varepsilon^{-1}$, whereas the energy due to the presence of $n$ vortices typically diverges like $n\ln{\varepsilon}$. In order to separate the different singularities of the energy, we let $u_{\varepsilon}$ be the (unique) positive minimizer of (\[V-EGL\]) when $H=0$ (see Theorem \[V-thm-kach3\]), the energy of $u_{\varepsilon}$ being of the order ${\varepsilon}^{-1}$. Then, if $(\psi,A)$ is a minimizing configuration of (\[V-EGL\]), it holds that (see Lemma \[V-lem-psi<u\]), $$\mathcal G_{{\varepsilon},H}(\psi,H)= \mathcal G_{{\varepsilon},0}(u_{\varepsilon},0)+\mathcal
F_{{\varepsilon},H}\left(\frac{\psi}{u_{\varepsilon}},A\right)\,,$$ and the configuration $\left(\displaystyle\frac{\psi}{u_{\varepsilon}},A\right)$ minimizes the functional $\mathcal F_{{\varepsilon},H}$ introduced below, $$\label{reducedfunctional*}
\mathcal
F_{{\varepsilon},H}(\varphi,A)=\int_{\Omega}\left(u_{\varepsilon}^2|(\nabla-iA)\varphi|^2+\frac{u_{\varepsilon}^4}{2{\varepsilon}^2}(1-|\varphi|^2)^2
+|{\rm curl}\,A-H|^2\right)\,{{\rm d}}x\,.$$ The leading order term of the minimizing energy of (\[reducedfunctional\*\]) will be described by means of an auxiliary energy introduced in (\[eq-E-lambda\]) below. Let $\mathcal M(\Omega)$ be the space of bounded Radon measures on $\Omega$, i.e. the topological dual of $C_0^0(\Omega)$. A measure $\mu\in\mathcal M(\Omega)$ can be represented canonically as a difference of two positive measures, $\mu=\mu_+-\mu_-$. The [*total variation*]{} and the [*norm*]{} of $\mu$, denoted respectively by $|\mu|$ and $\|\mu\|$, are by definition $|\mu|=\mu_++\mu_-$ and $\|\mu\|=|\mu|(\Omega)$.\
Given $\lambda>0$, we introduce an energy $E_\lambda$ defined on $\mathcal M(\Omega)\cap H^{-1}(\Omega)$ as follows. For $\mu\in\mathcal M(\Omega)\cap H^{-1}(\Omega)$, let $h_\mu\in
H^1(\Omega)$ be the solution of $$\label{eq-h-mu}
\left\{
\begin{array}{l}
-{\rm div}\left(\frac1{p(x)}\nabla h_\mu\right)+h_\mu=\mu\quad{\rm
in}~
\Omega\,,\\
h_\mu=1\quad{\rm on}~\partial\Omega\,,
\end{array}\right.$$ where $p$ is introduced in (\[p(x)\]). Now $E_\lambda(\mu)$ is by definition $$\label{eq-E-lambda}
E_\lambda(\mu)=\frac1{\lambda}\int_\Omega p(x)|\mu|\,{{\rm d}}x+\int_\Omega \left(\frac1{p(x)}|\nabla
h_\mu|^2+|h_\mu-1|^2\right)\,{{\rm d}}x\,.$$
\[thm1\] Given $\lambda>0$, assume that $$\lim_{\varepsilon\to0}\frac{H}{|\ln\varepsilon|}=\lambda\,,$$ then $$\frac{\mathcal F_{\varepsilon,H}}{H^2}\to E_\lambda$$ in the sense of $\Gamma$-convergence. Here the functional $\mathcal
F_{\varepsilon,H}$ is defined in (\[reducedfunctional\*\]) above.
The convergence in Theorem \[thm1\] is precisely described in Propositions \[prop-upperbound\] and \[prop-lowerbound\] below.\
Minimizers of (\[eq-E-lambda\]) can be characterized by means of minimizers of the following problem, $$\label{eq:dual-en}
\min_{\substack{h\in H^1_0(\Omega)\\-{\rm
div}\left(\frac1{p(x)}\nabla h\right)+h\in\mathcal
M(\Omega)}}\int_\Omega\left(\frac{p(x)}{\lambda}\left|-{\rm
div}\left(\frac1{p(x)}\nabla h\right)+h+1 \right|+\frac{|\nabla
h|^2}{p(x)}+|h|^2\right)\,{{\rm d}}x\,.$$ The above functional being strictly convex and continuous, it admits a unique minimizer, and so the functional $E_\lambda$.\
Therefore, as a corollary of Theorem \[thm1\], we may describe the limiting vorticity measure in terms of the minimizer of the limiting energy $E_\lambda$.
\[thm2\] Under the hypothesis of Theorem \[thm1\], if $(\varphi_\varepsilon, A_\varepsilon)$ is a minimizer of (\[reducedfunctional\*\]), then, denoting by $$h_\varepsilon={\rm curl}\,A_\varepsilon\,,\quad
\mu_\varepsilon=h_\varepsilon+
{\rm
curl}(i\varphi_\varepsilon\,,\,(\nabla-iA_\varepsilon)\varphi
_\varepsilon)\,,$$ the ‘induced magnetic field’ and ‘vorticity measure’ respectively, the following convergence hold, $$\begin{aligned}
&&\frac{\mu_\varepsilon}{H}\to \mu_*\quad{\rm in~}
\left(C^{0,\gamma}(\Omega)\right)^*\quad{\rm
for~all~}\gamma\in(0,1)\,,\\
&&\frac{h_\varepsilon}{H}\to h_{\mu_*} \quad{\rm
weakly~in~}H^1_1(\Omega)~{\rm and~strongly~
in~}W^{1,p}(\Omega)\,,~\forall~p<2\,.\end{aligned}$$ Here $\mu_*$ is the unique minimizer of $E_\lambda$.
Sketch of Proof {#sketch-of-proof .unnumbered}
---------------
\
Let us briefly describe the main points of the proofs of Theorems \[thm1\] and \[thm2\], and thus explain what stands behind their statements.\
The starting point is the analysis of minimizers of (\[V-EGL\]) when $H=0$. In this case, (\[V-EGL\]) has, up to a gauge transformation, a unique minimizer $(u_\varepsilon,0)$ where $u_\varepsilon$ is a positive real-valued function. The asymptotic decay of $u_\varepsilon$ as $\varepsilon\to0$ is obtained in Lemma \[interior\].\
When $H>0$, let $(\psi,A)$ be a minimizer of (\[V-EGL\]). Inspired by Lassoued-Mironescu [@LaMi], we introduce a [*normalized density*]{}[^1] $$\varphi=\frac{\psi}{u_\varepsilon}\,.$$ Then $|\varphi|\leq1$ and we are led to the analysis of the functional $\mathcal F_{{\varepsilon},H}(\varphi,A)$. The main difficulty stems from the boundary layer behavior of the weight $u_{\varepsilon}$, which has rapid oscillations comparable to ${\varepsilon}^{-1}$ near the boundary of $S_1$. The challenge is then to construct a test configuration with vortices localized near $\partial S_1$ and having the right amount of energy. Technically, one does that via a Green’s potential $G_{\varepsilon}$ which we were not able to give a good control of it, see Lemma \[Green\]. To overcome this difficulty, we construct vortices situated at a distance $\frac{\ln|\ln{\varepsilon}|}{|\ln{\varepsilon}|}$ away from $\partial S_1$. Those vortices being expected to lie on a union $\gamma$ of closed curves, a nice $L^1$ bound can be shown to hold for $G_{\varepsilon}(x,x)$ on $\gamma$. Then, using a result of [@Kach-v], recalled in Lemma \[integration\], this $L^1$ bound provides us with a family of well separated points serving as the centers of vortices for the test configurations, see Proposition \[prop-vp\]. We use then suitable methods to estimate from above the energy of the constructed configuration, yielding in Proposition \[prop-upperbound\] the first part of Theorem \[thm1\]. This rough analysis will be carried out through Section \[Section:UB\].\
In Section \[Sec:LB\], we establish a lower bound of the energy, see Proposition \[prop-lowerbound\]. This yields the second part of Theorem \[thm1\].\
The proof of Theorem \[thm2\] is a by-product of the proofs of Propositions \[prop-upperbound\] and \[prop-lowerbound\], see Remark \[rem:conc\]. Finally, we close the paper with a discussion concerning the minimization of the energy (\[eq-E-lambda\]).
#### *Remarks on the notation.*
- The letters $C,\widetilde C, M,$ etc. will denote positive constants independent of $\varepsilon$.
- For $n\in\mathbb N$ and $X\subset\mathbb R^n$, $|X|$ denotes the Lebesgue measure of $X$. $B(x,r)$ denotes the open ball in $\mathbb R^n$ of radius $r$ and center $x$.
- $(\cdot,\cdot)$ denotes the scalar product in $\mathbb C$ when identified with $\mathbb R^2$.
- $\mathcal
F_{{\varepsilon},H}(\varphi,A,U)$ means that the energy density of $(\varphi,A)$ is integrated only on $U\subset\Omega$.
- For $\alpha\in(0,1)$ sufficiently small, $S_1^\alpha=\{x\in S_1~:~{\rm dist}(x,\partial S_1)\geq \alpha\}$ and $S_2^\alpha=\{x\in S_2~:~{\rm dist}(x,\partial
S_1)\geq\alpha\}$.
- For two positive functions $a(\varepsilon)$ and $b(\varepsilon)$, we write $a(\varepsilon)\ll b(\varepsilon)$ as $\varepsilon\to0$ to mean that $\displaystyle\lim_{\varepsilon\to0}\frac{a(\varepsilon)}{b(\varepsilon)}=0$.
Preliminary analysis of minimizers {#V-Sec-EnH=0}
==================================
The case without applied magnetic field
---------------------------------------
This section is devoted to an analysis for minimizers of (\[V-EGL\]) when the applied magnetic field $H=0$. We follow closely similar results obtained in [@kach3].\
We keep the notation introduced in Section 1. Upon taking $A=0$ and $H=0$ in (\[V-EGL\]), one is led to introduce the functional $$\label{V-EnH=0}
\mathcal G_\varepsilon(u):=\int_\Omega\left(|\nabla u|^2
+\frac{1}{2\varepsilon^2}(p(x)-u^2)^2\right)\,{{\rm d}}x\,,$$ defined for functions in $H^1(\Omega;\mathbb R)$.\
We introduce $$\label{V-C0}
C_0(\varepsilon)=\inf_{u\in H^1(\Omega;\mathbb R)} \mathcal
G_\varepsilon(u)\,.$$
The next theorem is an analogue of Theorem 1.1 in [@kach3].
\[V-thm-kach3\] Given $a\in\mathbb R_+\setminus\{1\}$, there exists $\varepsilon_0$ such that for all $\varepsilon\in]0,\varepsilon_0[$, the functional (\[V-EnH=0\]) admits in $H^1(\Omega;\mathbb R)$ a minimizer $u_\varepsilon\in
C^2(\overline{S_1})\cup C^2(\overline{S_2})$ such that $$\min(1,\sqrt{a}\,)<u_\varepsilon<\max(1,\sqrt{a}\,)\quad {\rm in}~\overline{\Omega}.$$ If $H=0$, minimizers of (\[V-EGL\]) are gauge equivalent to the state $(u_\varepsilon,0)$.
We state also some estimates, taken from [@kach3 Proposition 5.1], that describe the decay of $u_\varepsilon$ away from the boundary of $S_1$.
\[interior\] Let $k\in\mathbb N$. There exist positive constants $\varepsilon_0$, $\delta$ and $C$ such that, for all $\varepsilon\in]0,\varepsilon_0]$, $$\label{interiordecay}
\left\|(1-u_\varepsilon)\exp\left(\frac{\delta{\rm dist}(x,\partial
S_1)}\varepsilon\right)\right\|_{H^k(S_1)}+\left\|(\sqrt{a}-u_\varepsilon)\exp\left(\frac{\delta
{\rm dist}(x,\partial
S_1)|}\varepsilon\right)\right\|_{H^k(S_2)}\leq\frac{C}{\varepsilon^k}\,.$$
Finally, we mention without proof that the energy $C_0(\varepsilon)$ (cf. (\[V-C0\])) has the order of $\varepsilon^{-1}$, and we refer to the methods in [@kach3 Section 6] which permit to obtain the leading order asymptotic expansion $$C_0(\varepsilon)=\frac{c_1(a)}\varepsilon+c_2(a)+o(1),\quad(\varepsilon\to0),$$ where $c_1(a)$ and $c_2(a)$ are positive explicit constants.
The case with magnetic field {#V-sec-minimizers}
----------------------------
This section is devoted to a preliminary analysis of the minimizers of (\[V-EGL\]) when $H\not=0$. The main point that we shall show is how to extract the singular term $C_0(\varepsilon)$ (cf. (\[V-C0\])) from the energy of a minimizer.
Notice that the existence of minimizers is standard starting from a minimizing sequence (cf. e.g. [@Gi]). A standard choice of gauge permits one to assume that the magnetic potential satisfies $$\label{V-gauge}
{\rm div}\,A=0\quad {\rm in}~\Omega,\quad n\cdot A=0\quad{\rm
on}~\partial\Omega,$$ where $n$ is the outward unit normal vector of $\partial\Omega$.\
With this choice of gauge, one is able to prove (since the boundaries of $\Omega$ and $S_1$ are smooth) that a minimizer $(\psi,A)$ is in $ C^1(\overline\Omega;\mathbb C)\times
C^1(\overline\Omega;\mathbb R^2)$. One has also the following regularity (cf. [@kach3 Appendix A]), $$\psi \in C^2(\overline S_1;\mathbb C)\cup C^2(\overline S_2;\mathbb
C),\quad A\in C^2(\overline S_1;\mathbb R^2)\cup C^2(\overline
S_2;\mathbb R^2).$$
The next lemma is inspired from the work of Lassoued-Mironescu (cf. [@LaMi]).
\[V-lem-psi<u\] Let $(\psi,A)$ be a minimizer of (\[V-EGL\]). Then $0\leq|\psi|\leq u_\varepsilon$ in $\Omega$, where $u_\varepsilon$ is the positive minimizer of (\[V-EnH=0\]).\
Moreover, putting $\varphi=\frac{\psi}{u_\varepsilon}$, then the energy functional (\[V-EGL\]) splits in the form : $$\label{V-splittingEn}
\mathcal G_{\varepsilon,H}(\psi,A)=C_0(\varepsilon)+ \mathcal
F_{\varepsilon,H}(\varphi,A),$$ where $C_0(\varepsilon)$ has been introduced in (\[V-C0\]) and the new functional $\mathcal F_{\varepsilon,H}$ is defined by : $$\label{V-reducedfunctional}
\mathcal F_{\varepsilon,H}(\varphi,A)=\int_\Omega
\left(u_\varepsilon^2 |(\nabla-iA)\varphi|^2
+\frac1{2\varepsilon^2} u_\varepsilon^4(1-|\varphi|^2)^2+ |{\rm
curl}\,A-H|^2\right){{\rm d}}x.\nonumber$$
Upper bound of the energy {#Section:UB}
=========================
Main result
-----------
The objective of this section is to establish the following upper bound.
\[prop-upperbound\] Assume that $\displaystyle\lim_{\varepsilon\to0}\frac{H}{|\ln\varepsilon|}=\lambda$ with $\lambda>0$. Given $\mu\in\mathcal M(\Omega)\cap
H^{-1}(\Omega)$, there exists a family of configurations $\{(\varphi_\varepsilon,A_\varepsilon)\}_\varepsilon$ in $H^1(\Omega;\mathbb C)\times H^1(\Omega;\mathbb R^2)$ such that $\|\varphi_\varepsilon\|_{L^\infty(\Omega)}\leq 1$, $$\label{cv-mesure}
\frac{\mu(\varphi_\varepsilon,A_\varepsilon)}{H}\rightharpoonup\mu\quad{\rm
in}~\left(C_0^{0,\gamma}(\Omega)\right)^*\quad\forall~\gamma\in(0,1),$$ and $$\limsup_{\varepsilon\to 0} F_{\varepsilon,H}(\varphi_\varepsilon,A_\varepsilon)\leq\frac1\lambda\int_{\Omega}p(x)|\mu|\,{{\rm d}}x+\int_\Omega\left(\frac1{p(x)}|\nabla
h_\mu|^2+|h_\mu-1|^2\right)\,{{\rm d}}x\,.$$
Analysis of a Green’s potential
-------------------------------
This section is devoted to an analysis of a Green’s kernel, i.e. a fundamental solution of the differential operator$-{\rm
div}\,\left(\displaystyle\frac1{u_\varepsilon^2(x)}\nabla\right)+1$. The existence and the properties of this function, taken from [@AfSaSe; @Stamp], are given in the next lemma.
\[Green\] For every $y\in\Omega$ and $\varepsilon\in]0,1]$, there exists a unique symmetric function $\Omega\times\Omega\ni(x,y)\mapsto
G_\varepsilon(x,y)\in\mathbb R_+$ such that : $$\label{equation-Green}
\left\{\begin{array}{rl} -{\rm
div}\,\left(\displaystyle\frac1{u_\varepsilon^2(x)}\nabla_x
G_\varepsilon(x,y)\right)+G_\varepsilon(x,y)=\delta_y(x)&{\rm
in}~\mathcal
D'(\Omega),\\
G_\varepsilon(x,y)\big{|}_{x\in\partial\Omega}=0.
\end{array}\right.$$ Moreover, $G_\varepsilon$ satisfies the following properties:
1. There exists a constant $C>0$ such that $$G_\varepsilon(x,y)\leq C\left(|\,\ln|x-y|\,|+1\right)\,,\quad
\forall~(x,y)\in\overline\Omega\times\overline\Omega\setminus\Delta\,,~
\forall~\varepsilon\in]0,1]\,,$$ where $\Delta$ denotes the diagonal in $\mathbb R^2$.
2. The function $v_\varepsilon:
\Omega\times\Omega\ni (x,y)\mapsto
G_\varepsilon(x,y)+\displaystyle\frac{u_\varepsilon^2(x)}{2\pi}\ln|x-y|$ is in the class $C^1(\Omega\times\Omega\,;\mathbb R)$.
3. Given $q\in[1,2[$, there exists a constant $C>0$ such that $$\|v_\varepsilon(\cdot,y)\|_{W^{1,q}(\Omega)} \leq C,
\quad
\forall~y\in\overline\Omega,\quad\forall~\varepsilon\in]0,1]\,.$$
One feature of the function $G_\varepsilon$ is that we know its homogenized limit $G_0$.
\[conv-G0\] Let $1<q<2$. The function $G_\varepsilon$ converges weakly in $W^{1,q}(\Omega\times\Omega)$ to $G_0$ the solution of $$\label{equation-Green0}
\left\{\begin{array}{rl} -{\rm
div}\,\left(\displaystyle\frac1{p(x)}\nabla_x
G_0(x,y)\right)+G_0(x,y)=\delta_y(x)&{\rm in}~\mathcal
D'(\Omega),\\
G_0(x,y)\big{|}_{x\in\partial\Omega}=0.
\end{array}\right.$$ Moreover, $G_{\varepsilon}$ converges locally uniformly to $G_0$ in $\Omega\times
\Omega\backslash \Delta$ where $\Delta$ is the diagonal of $\mathbb{R}^2$.
Since $G_\varepsilon$ and $G_0$ are symmetric, it is sufficient to establish weak convergence of $G_\varepsilon(\cdot,y)$ for $y\in\Omega$ arbitrarly fixed.\
Lemma \[Green\] assures that $G_\varepsilon(\cdot,y)$ is bounded in $W^{1,q}(\Omega)$ uniformly with respect to $y$ and $\varepsilon$. Hence, $G_\varepsilon(\cdot,y)$ converges weakly to a function $G_0(\cdot,y)$ in $W^{1,q}(\Omega)$. Let us prove that $G_0$ solves (\[equation-Green0\]).\
Let $\varphi\in C_0^\infty(\Omega)$. It is sufficient to prove that as $\varepsilon\to0$, $$\begin{aligned}
&&\int_\Omega\frac{1}{u_\varepsilon^2(x)}\nabla_x
G_\varepsilon(x,y) \cdot\nabla _x\varphi(x)\,{{\rm d}}x\to
\int_\Omega\frac{1}{p(x)}\nabla_x
G_0(x,y)\cdot\nabla _x\varphi(x)\,{{\rm d}}x\\
&&\int_\Omega G_\varepsilon(x,y) \varphi(x)\,{{\rm d}}x\to \int_\Omega
G_0(x,y)\,\varphi(x)\,{{\rm d}}x\,.\end{aligned}$$ The second convergence is actually immediate since $W^{1,q}(\Omega)$ comapctly embedds in $L^q(\Omega)$ and $G_\varepsilon(\cdot,y)$ converges weakly to $G_0(\cdot,y)$ in $W^{1,q}(\Omega)$. In order to prove the first convergence we notice that, $$\begin{aligned}
&&\int_\Omega\frac{1}{u_\varepsilon^2(x)}\nabla_x
G_\varepsilon(x,y) \cdot\nabla _x\varphi(x)\,{{\rm d}}x-
\int_\Omega\frac{1}{p(x)}\nabla_x
G_0(x,y)\cdot\nabla _x\varphi(x)\,{{\rm d}}x\\
&&\hskip0.5cm=\int_\Omega\left(\frac{1}{u_\varepsilon^2(x)}-\frac1{p(x)}\right)
\nabla_x G_\varepsilon(x,y) \cdot\nabla _x\varphi(x)\,{{\rm d}}x\\
&&\hskip1cm+
\int_\Omega\frac{1}{p(x)}\nabla_x \left(\mathcal
G_\varepsilon(x,y)-G_0(x,y)\right)\cdot\nabla _x\varphi(x)\,{{\rm d}}x\,,\end{aligned}$$ so it suffices to prove that $$\int_\Omega\left(\frac{1}{u_\varepsilon^2(x)}-\frac1{p(x)}\right)
\nabla_x G_\varepsilon(x,y) \cdot\nabla _x\varphi(x)\,{{\rm d}}x\to
0\quad{\rm as}~\varepsilon\to0\,.$$ Using the uniform bound on $G_\varepsilon(\cdot,y)$ in $W^{1,q}$, we write, $$\begin{aligned}
&&\int_\Omega\left|\left(\frac{1}{u_\varepsilon^2(x)}-\frac1{p(x)}\right)
\nabla_x G_\varepsilon(x,y)
\cdot\nabla _x\varphi(x)\right|\,{{\rm d}}x\\
&&\leq \left(\int_\Omega
\left(\frac{1}{u_\varepsilon^2(x)}-\frac1{p(x)}\right)^p|\nabla\varphi|^p\,
{{\rm d}}x \right)^{1/p}\left(\int_\Omega|\nabla_x
G_\varepsilon(x,y)|^q\,{{\rm d}}x
\right)^{1/q}\\
&&\leq C\left(\int_\Omega
\left(\frac{1}{u_\varepsilon^2(x)}-\frac1{p(x)}\right)^p|\nabla\varphi|^p\,
{{\rm d}}x \right)^{1/p}\,,\quad\frac1p+\frac1q=1\,.\end{aligned}$$ Now, we use the asymptotic behavior of $u_\varepsilon^2$, which converges pointwise to $p(x)$ and remains uniformly bounded from above and below. Therefore, Lebesgue convergence theorem yields the desired convergence.\
To complete the proof, we refer to [@AfSaSe Lemma 3.1] to find the uniform convergence of $G_{\varepsilon}$ to $G_0$.
Next we cite a result from [@Kach-v Lemma 5.5].
\[integration\] Let $(f_\varepsilon)_{\varepsilon\in]0,1]} \subset C([0,1],\mathbb
R_+)$ be a family of continuous functions. Assume that there exists a constant $C>0$ such that $$\|f_\varepsilon\|_{L^1([0,1])}\leq C\ln|\ln\varepsilon|\,,\quad\forall~\varepsilon\in
]0,1]\,.$$ There exist constants $K>0$ and $c_0\in]0,1[$ such that, given a family $(N(\varepsilon))\subset\mathbb N$ satisfying $N(\varepsilon)\gg1$, there exists a family $(\delta(\varepsilon))\subset ]0,1[$ and a sequence $(t_{m}^\varepsilon)_{m\in\mathbb N}\subset ]0,1[$ and $$\begin{aligned}
&&|f_\varepsilon(t_i^\varepsilon)|\leq K\ln|\ln\varepsilon|\,,
\quad \frac{c_0}{N(\varepsilon)}
\leq\left|t_{i+1}^\varepsilon-t_i^\varepsilon\right|\leq
\delta(\varepsilon)+\frac{c_0}{N(\varepsilon)}\,,\\
&&\forall~i\in\{1,2,\cdots,N(\varepsilon)\}\,,\quad\forall~\varepsilon\in
]0,1]\,.\end{aligned}$$
The next proposition provides us with points enjoying useful properties. These points will serve to be the centers of the vortices of the test configuration that we shall construct in the next section.
\[prop-vp\] Let $\mu\in C_0^0(\Omega)$ be continuous and compactly supported in $\Omega$. Assume that $(n(\varepsilon))_{\varepsilon\in(0,1]}\subset \mathbb N$ is a family of integers such that $c_1|\ln\varepsilon|\leq
n(\varepsilon)\leq c_2\varepsilon^{-2}$ for constants $c_1$ and $c_2$ independent of $\varepsilon\in(0,1]$.\
If the restriction $\tilde\mu$ of $\mu$ to $S_1$ (respectively $S_2$) is different from $0$, then there exist constants $c>0$, $C>0$, a family of integers $N(\varepsilon)\in\mathbb N$, points $a_i^\varepsilon$ in $S_1$ (respectively $S_2$) and degrees $d_i^\varepsilon=\pm1$ such that:
1. $N(\varepsilon)=n(\varepsilon)(1+o(1))$ as $\varepsilon\to0$;
2. $|a_i^\varepsilon-a_j^\varepsilon|\geq
\displaystyle\frac{C}{\sqrt{n(\varepsilon)}}\geq c\varepsilon$ for all $i\not=j$;
3. ${\rm dist}\left(a_i^\varepsilon,\partial
S_1\right)\geq\displaystyle\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|}$ and ${\rm dist}(a_i^\varepsilon,\partial\Omega)>c$ for all $i$;
4. $|v_\varepsilon(a_i^\varepsilon,a_i^\varepsilon)|\ll
|\ln\varepsilon|$ for all $i$;
5. $\displaystyle
\frac1{N(\varepsilon)}\sum_{i=1}^{N(\varepsilon)}
\mu_i^\varepsilon\rightharpoonup
2\pi\frac{\tilde\mu}{\|\tilde\mu\|}$ in the weak sense of measures, for $\mu_i^\varepsilon$ any measure supported in $\overline{B(a_i^\varepsilon,c\varepsilon)}$, of constant sign, and such that $\mu_i^\varepsilon(\Omega)=2\pi d_i^\varepsilon$.
#### **Proof.**
For $i=1,2$, we denote by $S_i^\varepsilon=\{x\in S_i~:~{\rm
dist}(x,\partial S_1)\geq
\frac{\ln|\ln\varepsilon|}{|\ln\varepsilon|}\}$. We only present the construction in $S_1^{\varepsilon}$ as it holds exactly in $S_2^{\varepsilon}$.\
We partition $S_1^\varepsilon$ into squares $K$ of side-length $\ell(\varepsilon)$ where $\ell(\varepsilon)$ is chosen such that, $$\frac1{\sqrt{n(\varepsilon)}}\ll\ell(\varepsilon)\ll1\quad{\rm as}~
\varepsilon\to0\,.$$ We denote by $\mathcal K(\varepsilon)$ the family of such squares that lie entirely inside $S_1^\varepsilon$.\
For $K\in\mathcal K(\varepsilon)$, we set $$\lambda_K=n(\varepsilon)\frac{|\mu(K)|}{M_\varepsilon}\,,$$ where $$M_\varepsilon=\sum_{K\in\mathcal K(\varepsilon)}
|\mu(K)|\,.$$ Since $\mu$ is continuous with compact support in $\Omega$, and $\ell(\varepsilon)$ tends to $0$ as $\varepsilon\to0$, we get that $M_\varepsilon\to \|\tilde\mu\|$ as $\varepsilon\to0$.\
Now $\displaystyle\sum_{K\in\mathcal K(\varepsilon)}\lambda_K=n$, hence Lemma 7.4 of [@SaSe] provides us with nonnegative integers $m_\varepsilon(K)$ such that $$\label{eq-mK}
\sum_{K\in\mathcal
K(\varepsilon)}m_\varepsilon(K)=n(\varepsilon)\,,$$ and $$\label{eq-mK-est}
\left|m_\varepsilon(K)-n\frac{|\mu(K)|}{M_\varepsilon}\right|<1\,.$$ Since $\mu$ is bounded, we may find $M>0$ such that $|\mu|(K)\leq
M\ell(\varepsilon)^2$ for all $K\in\mathcal K(\varepsilon)$. Consequently, we infer from (\[eq-mK-est\]) that $$m_\varepsilon(K)\leq 1+nM\ell^2=\mathcal O(nM\ell^2)\,,$$ (recall that $\frac1{n}\ll\ell\ll1$ as $\varepsilon\to0$).\
Now we decompose $\mathcal K(\varepsilon)=\mathcal
K_1(\varepsilon)\cup \mathcal K_2(\varepsilon)$, where $K\in\mathcal K_2(\varepsilon)$ if and only if${\rm dist}(\partial K,\partial S_1)\geq \frac R2$ where $R>0$ is picked small enough so that each connected component of $S_1$ contains a disc of radius $R$.\
Now assume that $K\in \mathcal K_1(\varepsilon)$. Let $p_\varepsilon(K)$ be the least integer less than $\sqrt{m_\varepsilon(K)}$. Then we may pick $p_\varepsilon(K)$ points $b_i^\varepsilon$ lying on one side $\mathcal L$ of $K$ and such that $$\label{eq-bi}
|b_i^\varepsilon-b_j^\varepsilon|\geq
\frac{C\ell(\varepsilon)}{\sqrt{p_\varepsilon(K)}}\geq
\frac{C}{\sqrt{n(\varepsilon)}}\,.$$ Next, we can construct $p_\varepsilon(K)$ segments $L_i^\varepsilon$ in $K$, each passing by $b_i^\varepsilon$ and perpendicular to the side $\mathcal L$ (hence each $L_i$ has length $\ell(\varepsilon)$).\
Since $K\in \mathcal K_1(\varepsilon)$, we may find closed smooth curves $\gamma_i\supset L_i$ in $\Omega$ such that each $\gamma_i$ is the boundary of a simply connected set $U_i\subset\Omega$ and $2\pi R\leq |\gamma_i|\leq |\partial S_1|$. By this way, referring to [@Kach-v Proof of (5.4)], we may get a constant $C'>0$ independent from $\varepsilon$ and such that, for all $i$, $$\int_{\gamma_i}|v_\varepsilon(x,x)|\,{{\rm d}}x \leq
C'\ln|\ln\varepsilon|\,.$$ In particular, it holds that $$\int_{L_i}|v_\varepsilon(x,x)|\,{{\rm d}}x\leq C'\ln|\ln\varepsilon|\,.$$ Let us parameterize $L_i$ by $(0,\ell(\varepsilon)]\ni s \mapsto
x(s)\in L_i$, and let us define the rescaled continuous function $$(0,1]\ni t\mapsto f_\varepsilon(t)=
v_\varepsilon\left(x(\ell(\varepsilon)t),x(\ell(\varepsilon)t)\right)\,,$$ so that $\|f_\varepsilon\|_{L^1(0,1)}\leq
C''(\ell(\varepsilon))^{-1}\ln|\ln\varepsilon|$. Invoking the result of Lemma \[integration\], we may pick $p_\varepsilon(K)$ points $(b_{i,j}^\varepsilon)\subset L_i$ such that[^2] $$|v_\varepsilon(b_{i,j}^\varepsilon,b_{i,j}^\varepsilon)|\leq
K'\frac{\ln|\ln\varepsilon|}{\ell(\varepsilon)}$$ and $$|b_{i,j}^\varepsilon-b_{i,k}^\varepsilon|\geq \frac{c_0}
{\ell(\varepsilon)\sqrt{n(\varepsilon)}}\geq
\frac{c_0}{\sqrt{n(\varepsilon)}} \,.$$ Denoting by $(a_k^\varepsilon)$ the family of the constructed points in all the segments $L_i$, we have actually constructed $[p_\varepsilon(K)]^2$ points $(a_k^\varepsilon)\subset K$ such that $$|v_\varepsilon(a_k^\varepsilon,a_k^\varepsilon)|\leq
K'\frac{\ln|\ln\varepsilon|}{\ell(\varepsilon)}\,,\quad
|a_k^\varepsilon-a_j^\varepsilon|\geq
\frac{C}{\sqrt{n(\varepsilon)}}\,.$$ Recalling the assumptions on $n(\varepsilon)$ and $\ell(\varepsilon)$, we get actually the two desired properties (2) and (3) stated in Proposition \[prop-vp\] above, with the constant $c>0$ chosen sufficiently small that $c<{\rm dist}({\rm supp}\,\mu\,,\partial\Omega)$.\
It remains now to continue the construction of the points $(a_i^\varepsilon)$ filling the squares $K\in \mathcal
K_2(\varepsilon)$. Notice that Corollary 5.3 of [@Kach-v] (more precisely an adjustment of its proof) provides us with a constant $C>0$ such that, for all $K\in\mathcal K_2(\varepsilon)$ and $\varepsilon\in(0,1)$, we have, $$\|v_\varepsilon(x,x)\| _{L^\infty(K)}\leq C\,.$$ Hence, it is sufficient to construct any well separated $m_\varepsilon(K)$ points in this case. This is exactly the case of [@SaSe p. 144]. Now, the integer $N(\varepsilon)$ is defined as $$N(\varepsilon)=\sum_{K\in\mathcal K_1(\varepsilon)}
\left(p_\varepsilon(K)\right)^2+\sum_{K\in\mathcal
K_2(\varepsilon)} m_\varepsilon(K)\,.$$ That $N(\varepsilon)=n(\varepsilon)(1+o(1))$ is due to the assumption we made on $\ell(\varepsilon)$, $\sqrt{n(\varepsilon)}^{-1}\ll\ell(\varepsilon)
\ll 1$ as $\varepsilon\to0$.\
Now, having constructed the family of points $(a_i^\varepsilon)$, we see that property (3) stated in Proposition \[prop-vp\] is just due to our construction of the points being in $S_\varepsilon^1$, and our choice of the constant $c$ being so small that ${\rm dist}({\rm supp}\,\mu\,,\partial\Omega)>c$.\
Now we define the family $(d_i)$. If $\mu(K)\geq 0$, we assign the degree $d_i=1$ to each $a_i\in K$, otherwise we associate the degree $d_i=-1$. The proof of the last property (5) in Proposition \[prop-vp\] is exactly as that given in [@SaSe p. 145].$\Box$\
Proposition \[prop-vp\] will be used in the following context. Let $\mu\in C_0^0(\Omega)$ be continuous and compactly supported in $\Omega$. Take positive integers $n_1(\varepsilon)$ and $n_2(\varepsilon)$ such that $$c_1|\ln\varepsilon|\leq n_1(\varepsilon)+n_2(\varepsilon)\leq
c_2\varepsilon^{-2}\,,$$ for positive constants $c_1$ and $c_2$ independent from $\varepsilon$.\
If the restrictions $\check\mu$ and $\hat\mu$ of $\mu$ to $S_1$ and $S_2$ respectively are both different from $0$, then we get $N_1(\varepsilon)\sim n_1(\varepsilon)$ points $(a_i^\varepsilon)$ in $S_1$ and $N_2(\varepsilon)\sim n_2(\varepsilon)$ points $(b_i^\varepsilon)$ in $S_2$ satisfying properties (1)-(5) of Proposition \[prop-vp\]. In particular, we set $$x_i^\varepsilon=\left\{
\begin{array}{ll}
a_i^\varepsilon&\forall~i\in\{1,2,\cdots,
n_1(\varepsilon)\}\,,\\
b_i^\varepsilon&\forall ~i
\in\{n_1(\varepsilon)+1,n_2(\varepsilon)+2,\cdots,
n_1(\varepsilon)+n_2(\varepsilon)\}\,,
\end{array}\right.$$ and we define the measures $\mu_i^\varepsilon$ by $$\label{eq:measure1}
\mu_i^\varepsilon(x)= \left\{
\begin{array}{cl}
\displaystyle\frac{2d_i}{c^2\varepsilon^2}&{\rm if}~ x\in
B(x_i^\varepsilon,c\varepsilon)
\\
0&{\rm otherwise}\,.
\end{array}\right.$$ If $\check\mu$ (respectively $\hat\mu$) is zero, we may still choose the points $a_i^\varepsilon$ (respectively $b_i^\varepsilon$) arbitrarily so that properties (1)-(4) of Proposition \[prop-vp\] are valid, and we define the corresponding measures $\mu_i^\varepsilon$ to be zero by pure convention.\
With these notations, we get as an immediate consequence that $$\label{eq:measure}
\mu_{\varepsilon}:=\frac1{n(\varepsilon)}
\sum_{i=1}^{N_1(\varepsilon)+N_2(\varepsilon)} \mu_i^\varepsilon
\rightharpoonup 2\pi\frac{\mu}{\|\mu\|}\quad{\rm
in~}(C_0^{0,\gamma}(\Omega))^*\,,$$ where $$\label{eq:n}
n(\varepsilon)=\left\{
\begin{array}{cl}
N_1(\varepsilon)&{\rm if}\quad{\rm supp}\,\mu\subset\overline S_1\,,\\
N_2(\varepsilon)&{\rm if}\quad{\rm supp}\,\mu\subset\overline
S_2\,,\\
N_1(\varepsilon)+N_2(\varepsilon)&{\rm otherwise.}\end{array}
\right.$$
\[corol-cont-green\] Under the hypotheses and notations above, one has $$\limsup_{\varepsilon\to0}\left(\frac1{n(\varepsilon)}\right)^2\sum_{i\not=j}
\int_{B_i\times B_j} G_\varepsilon(x,y){{\rm d}}\mu_i^\varepsilon(x){{\rm d}}\mu_j^\varepsilon(y)\leq
\frac{4\pi^2}{\|\mu\|^2}\int_{\Omega\times\Omega} G_0(x,y){{\rm d}}\mu(x){{\rm d}}\mu(y)\,. \label{Ge-mu}$$ Here, for all $i$, $B_i$ denotes the ball $B(x_i^\varepsilon,c\,\varepsilon)$.
Given $\alpha>0$, let $\Delta_{\alpha}=\{(x,y),\,\,\,|x-y|<\alpha\}$. We have the following decomposition, $$\label{double-somme1}
\frac1{n^2}\sum_{i\not=j} \int_{B_i\times B_j} G_\varepsilon\,{{\rm d}}\mu_i^\varepsilon{{\rm d}}\mu_j^\varepsilon=\int_{\Omega\times\Omega\backslash\Delta_{\alpha}}
G_{\varepsilon}{{\rm d}}\mu_{\varepsilon}{{\rm d}}\mu_{\varepsilon}+\frac1{n^2}\sum_{i\not=j}
\int_{\Delta_\alpha} G_\varepsilon\,{{\rm d}}\mu_i^\varepsilon{{\rm d}}\mu_j^\varepsilon\,.$$ Knowing from Lemma \[conv-G0\] that $G_{\varepsilon}$ converges uniformly to $G_0$ in $\Omega\times\Omega\backslash\Delta_{\alpha}$, we write using (\[eq:measure\]), $$\label{cv-Ge}
\lim_{{\varepsilon}\rightarrow 0}
\int_{\Omega\times\Omega\backslash\Delta_{\alpha}} G_{\varepsilon}{{\rm d}}\mu_{\varepsilon}{{\rm d}}\mu_{\varepsilon}= \frac{4\pi^2}{\|\mu\|^2}\int_{\Omega\times\Omega\backslash\Delta_{\alpha}} G_0 {{\rm d}}\mu \,{{\rm d}}\mu\,.$$ Let us estimate now the last term on the right hand side of (\[double-somme1\]). Since the supports of $\mu_i^{\varepsilon}$ and $\mu_j^{\varepsilon}$ are disjoint for $i\not=j$, we may write by Lemma \[Green\], $$\left|\frac1{n^2} \sum_{i\not=j} \int_{\Delta_{\alpha}} G_{\varepsilon}{{\rm d}}\mu_i^{\varepsilon}{{\rm d}}\mu_j^{\varepsilon}\right|\leq \frac{C}{n^2} \sum_{i\not=j}
\int_{\Delta_{\alpha}} |\,\ln|x-y|\,|\,|\mu_i^{\varepsilon}|(x)
\,|\mu_j^{\varepsilon}|(y)\,{{\rm d}}x\,{{\rm d}}y\,,$$ where $C>0$ is a constant independent from $\alpha$ and ${\varepsilon}$.\
Proposition \[prop-vp\] provides us that the points $a_i^{\varepsilon}$ are well separated, i.e. $|a_i^{\varepsilon}-a_j^{\varepsilon}|\geq
\displaystyle\frac{C}{\sqrt{n}}$. This actually permits us to write (see [@SaSe p. 147] for details), $$\label{A-S-S}
\left|\frac1{n^2}
\sum_{i\not=j}
\int_{\Delta_{\alpha}} G_{\varepsilon}{{\rm d}}\mu_i^{\varepsilon}{{\rm d}}\mu_j^{\varepsilon}\right|\leq
C\int_{\Delta_\alpha}\left(|\,\ln|x-y|\,|+1\right)
{{\rm d}}x\,{{\rm d}}y\,.$$ Substituting (\[cv-Ge\]) and (\[A-S-S\]) in (\[double-somme1\]), we deduce that, $$\begin{aligned}
\limsup_{{\varepsilon}\to0}
\frac{1}{n^2}\sum_{i\not=j}\int_{B_i\times B_j} G_{\varepsilon}{{\rm d}}\mu_i^{\varepsilon}(x)
{{\rm d}}\mu_j^{\varepsilon}(y)&\leq&
\frac{4\pi^2}{\|\mu\|^2}
\int_{\Omega\times\Omega\backslash\Delta_{\alpha}} G_0 {{\rm d}}\mu \,{{\rm d}}\mu\\
&&+C\int_{\Delta_\alpha}\left(|\,\ln|x-y|\,|+1\right)
{{\rm d}}x\,{{\rm d}}y\,,\end{aligned}$$ with $\alpha$ being arbitrary in the interval $(0,1)$. Making $\alpha\to0$ (recall that $\ln|x-y|$ is in $L^1$), we obtain the desired bound of the lemma.
\[def1\] A family of points $(a_i)\subset\Omega$ satisfying Properties (2)-(4) stated in Proposition \[prop-vp\] is said to be a well-distributed family.
Now, let $h_\varepsilon:\Omega\longrightarrow]0,1[$ be the solution of the equation: $$\label{London*}
-{\rm div}\left(\frac1{u_\varepsilon^2}\nabla
h_\varepsilon\right)+h_\varepsilon=0\quad{\rm in}~\Omega\,,\quad
h_\varepsilon=1\quad{\rm on}~\partial\Omega\,,$$ where $u_\varepsilon$ is introduced in Theorem \[V-thm-kach3\]. We define also, $$\label{eq-J0}
J_0(\varepsilon)=\int_{\Omega}\left(\frac{1}{u_\varepsilon^2}|\nabla
h_{\varepsilon}|^2+|h_\varepsilon-1|^2\right)\,{{\rm d}}x\,.$$
Next, we state a remarkable energy-splitting due to Bethuel-Rivi[è]{}re [@BeRi]. We find it in [@Kach-v Lemma 5.7].
\[Beth-Riv\] Consider $(\varphi,A)\in H^1(\Omega;\mathbb C)\times
H^1(\Omega;\mathbb R^2)$ and define $$A'=A-\frac{H}{u_\varepsilon^2}\nabla^\bot h_\varepsilon.$$ Then we have the decomposition of the energy, $$\mathcal F_{\varepsilon,H}(\varphi,A)=H^2
J_0(\varepsilon)+\mathcal L_{\varepsilon,H}(\varphi,A')+\mathcal
R_0 +2 H\int_\Omega (h_\varepsilon-1) \mu(\varphi,A'),
\label{eq-split}$$ where $$\label{eq-R0}
\mathcal R_0=H^2\int_\Omega \frac{1}{u_{\varepsilon}^2}(|\varphi|^2-1) |\nabla
h_\varepsilon|^2,\quad\quad \mu(\varphi,A')=h'+ {\rm
curl}(i\varphi\,,\,(\nabla-iA')\varphi).$$ Here, the functional $\mathcal L_{\varepsilon,H}$ is $$\label{eq-L}
\mathcal L_{\varepsilon,H}(\varphi,A')=\int_\Omega
u_\varepsilon^2|(\nabla-iA')\varphi|^2+|{\rm
curl}\,A'|^2+\frac1{2\varepsilon^2}
u_\varepsilon^4(1-|\varphi|^2)^2.$$
Given $\varepsilon\in (0,1)$, a well-distributed family[^3] of $n=n_1+n_2$ points $(a_i)\subset\Omega$ ($n_i$ points in $S^i_{\varepsilon}$) together with degrees $(d_i)\subset\{-1,1\}$, there exists a configuration $(\varphi, A)$ such that, $\mu_i^\varepsilon$ being the uniform measure on $\partial B_i=\partial B(a_i, c\varepsilon)$ of mass $2\pi d_i$, and letting $$\mu^{\varepsilon}=\sum_{i=1}^n \mu_i^{\varepsilon},$$ we have for some $\alpha\in [0,1[$, $$\label{autreformule}
\begin{split}
\mathcal{L}_{{\varepsilon}, H}(\varphi,A')& \leq c
{\varepsilon}^{\alpha} n |\ln{\varepsilon}|^2+c' n \,\,o(|\ln{\varepsilon}|)+2\pi |\ln{\varepsilon}|
\sum_{i=1}^{n}p(a_i)\\
&+\sum_{i\not=j} \int\int G_\varepsilon(x,y){{\rm d}}\mu_i^{\varepsilon}(x){{\rm d}}\mu_j^{\varepsilon}(y)+\mathcal{O}(n),
\end{split}$$ $$\label{formule-F}
\begin{split}
\mathcal{F}_{{\varepsilon}, H}(\varphi,A)&=\mathcal{L}_{{\varepsilon},
H}(\varphi,A')+H^2 J_0({\varepsilon})+2H\int_\Omega (h_{\varepsilon}-1) {{\rm d}}\mu(x)\\
&+\mathcal{O}\left(n {\varepsilon}H+n{\varepsilon}^2 H^2+(n^{\frac{1}{2}} {\varepsilon}H+{\varepsilon}H^2)
\sqrt{\mathcal{L}_{{\varepsilon}, H}(\varphi,A')}\right).
\end{split}$$ Moreover, for any $0<\gamma\leq 1$, it holds that, $$\label{controlemesure} \frac{1}{n}
\|\mu(\varphi, A)-\mu\|_{C^{0,\gamma}(\Omega)}\leq C {\varepsilon}^{\gamma}
\left( 1+ \varepsilon H+ \sqrt{\frac{\mathcal{F}_{{\varepsilon},H}(\varphi,
A')}{n}}\right).$$ \[lem-pv\]
We construct a test configuration $(\varphi, A)$. We define a function $h$ in $\Omega$ by $h=h'+H h_{\varepsilon}$ where $h_\varepsilon$ has been introduced in (\[London\*\]) and $h'$ is the solution of $$\label{V-Def-f'}
\left\{
\begin{array}{rl}
-{\rm div}\,\left(\displaystyle\frac1{u_\varepsilon^2}\nabla
h'\right)
+h'=\mu^{\varepsilon}&{\rm in}~\Omega,\\
h'=0&{\rm on}~\partial \Omega.\end{array}\right.$$ Note that $h'(x)=\int_{\Omega} G_{\varepsilon}(x,y) {{\rm d}}\mu^{\varepsilon}(y)$. As a consequence, we have $$\label{formule-h'}
\int_\Omega \left(\frac{|\nabla h'|^2}{u_{\varepsilon}^2}+|h'|^2\right)\,{{\rm d}}x=\int_\Omega\int_\Omega G_{\varepsilon}(x,y) {{\rm d}}\mu(x) {{\rm d}}\mu(y).$$ Now we define an induced magnetic potential $A=A'+\frac{H}{u_\varepsilon^2}\nabla^\bot h_\varepsilon$ by taking simply $${\rm curl}\,A'=h'.$$ This choice is always possible as one can take $A'=\nabla^\bot g$ with $g\in H^2(\Omega)$ such that $\Delta g=h'$. We turn now to define an order parameter $\varphi$ which we take in the form $$\label{V-vsol-psi'}
\varphi=\rho\,e^{i\phi},$$ where $\rho$ is defined by $$\label{V-vsol-rho'}
\rho(x)=\left\{
\begin{array}{cll}
0&{\rm if}&x\in \cup_iB(a_i,c\varepsilon),\\
1&{\rm if}&x\not\in\cup_iB(a_i,2c\varepsilon),\\
\displaystyle\frac{|x-a_i|}\varepsilon-1&{\rm if}&\exists\,i~{\rm
s.t.}~x\in B(a_i,2c\varepsilon) \setminus B(a_i,c\varepsilon).
\end{array}\right.$$ The phase $\phi$ is defined (modulo $2\pi$) by the relation: $$\label{V-vsol-phi'}
\nabla\phi-A'=-\frac{1}{u_\varepsilon^2}\nabla^\bot h'\quad{\rm
in}~ \Omega\setminus \cup_iB(a_i,c\varepsilon),$$ and we emphasize here that we do not need to define $\phi$ in regions where $\rho$ vanishes.\
Having defined $(\varphi,A)$ as above, we estimate $\mathcal{L}_{{\varepsilon}, H}(\varphi,A')$. Recall that $$\label{free energy1}
\mathcal{L}_{{\varepsilon}, H}(\varphi,A')=\int_{\Omega}
\left(u_{\varepsilon}^2|\nabla\rho|^2+\rho^2 u_{\varepsilon}^2
|\nabla\phi-A'|^2+|h'|^2+\frac{u_{\varepsilon}^4}{2{\varepsilon}^2}(1-\rho^2)^2\right){{\rm d}}x.$$ From (\[V-vsol-rho’\]) and using the uniform upper bound of $u_{\varepsilon}$, it follows easily that, $$\label{controle-rho}
\int_{\Omega}\left(u_{\varepsilon}^2
|\nabla\rho|^2+\frac{u_{\varepsilon}^4}{2{\varepsilon}^2}(1-\rho^2)^2\right){{\rm d}}x\leq C n.$$ Thanks to (\[V-vsol-rho’\]), (\[V-vsol-phi’\]) and the definition of $h=h'+Hh_\varepsilon$, we have, $$u_{\varepsilon}^2 \rho^2 |\nabla\phi-A'|^2\leq u_{\varepsilon}^2 |\nabla\varphi
-A'|^2=\frac{|\nabla(h-H h_{\varepsilon})|^2}{u_{\varepsilon}^2}\quad {\rm in}~
\Omega\setminus \cup_iB(a_i,c\varepsilon).\,,$$ Replacing this in (\[free energy1\]) and invoking (\[controle-rho\]), we find, $$\label{free energy2}
\mathcal{L}_{{\varepsilon}, H}(\varphi,A')\leq \int_\Omega
\left(\frac{|\nabla(h-H h_{\varepsilon})|^2}{u_{\varepsilon}^2}+|h-H h_{\varepsilon}|^2\right)\,{{\rm d}}x+\mathcal{O}(n).$$ Thanks to (\[formule-h’\]), we deduce the upper bound, $$\label{free energy3}
\mathcal{L}_{{\varepsilon}, H}(\varphi,A')\leq \int_{\Omega\times\Omega} G_{\varepsilon}d\mu(x) d\mu(y)+\mathcal{O}(n)\,.$$ We now decompose the double integral in two, $$\label{double-somme}
\int_{\Omega\times\Omega} G_{\varepsilon}d\mu^{\varepsilon}(x) d\mu^{\varepsilon}(y)=\sum_{i=1}^n
\int_{B_i\times B_i} G_{\varepsilon}{{\rm d}}\mu_i^{\varepsilon}(x)
{{\rm d}}\mu_i^{\varepsilon}(y)+\sum_{i\not=j}\int_{B_i\times B_j} G_{\varepsilon}{{\rm d}}\mu_i^{\varepsilon}(x) {{\rm d}}\mu_j^{\varepsilon}(y).$$ Let us estimate the first term in the right hand side. Writing $G_{\varepsilon}=v_{\varepsilon}-\frac{u_{\varepsilon}^2}{2\pi} \ln |x-y|$, we have $$\int_{B_i\times B_i} G_{\varepsilon}(x,y) {{\rm d}}\mu_i^{\varepsilon}(x)
{{\rm d}}\mu_i^{\varepsilon}(y)=\int_{B_i\times B_i}
\left(v_{\varepsilon}(x,y)-\frac{u_{\varepsilon}^2(x)}{2\pi} \ln |x-y|\right)
{{\rm d}}\mu_i^{\varepsilon}(x) {{\rm d}}\mu_i^{\varepsilon}(y),$$ Assuming $(x,y)\in {\rm supp}\, \mu_i\times {\rm supp}\, \mu_i$, we get $$\begin{aligned}
&&\hskip-1cm\int_{B_i\times B_i} u_{\varepsilon}^2(x) \ln |x-y| {{\rm d}}\mu_i^{\varepsilon}(x)
{{\rm d}}\mu_i^{\varepsilon}(y)\\
&&\hskip1cm=\int_0^{2\pi}\int_0^{2\pi} u_{\varepsilon}^2(a_i+c{\varepsilon}e^{i\theta_1})
\ln|c{\varepsilon}e^{i\theta_1}-c{\varepsilon}e^{i\theta_2}| {{\rm d}}\theta_1
{{\rm d}}\theta_2\\
&&\hskip1cm=c_{\varepsilon}+2\pi \ln{\varepsilon}\int_0^{2\pi}u_{\varepsilon}^2(a_i+c{\varepsilon}e^{i\theta_1}) d\theta_1\,.\end{aligned}$$ Here and in the sequel, $c_{\varepsilon}$ or $C_{\varepsilon}$ denote constants bounded (uniformly in ${\varepsilon}$) from below and above, and that may change from one line to another.\
The points $a_i$ being away from the boundary of $S_1$, Lemma \[interior\] gives that$u_\varepsilon^2(a_i+c{\varepsilon}e^{i\theta_1})$ is exponentially close to $p(a_i)$. Thus, we may write, upon considering the summation, $$\label{sum}
\sum_{i=1}^n
\int_{B_i\times B_i} \frac{u_{\varepsilon}^2(x)}{2\pi} \ln |x-y|
{{\rm d}}\mu_i^{\varepsilon}(x) {{\rm d}}\mu_i^{\varepsilon}(y)=C_{\varepsilon}n+\ln{\varepsilon}\sum_{i=1}^{n}p(a_i)\,.$$ Let us now estimate $\sum_{i=1}^n \displaystyle \int_{B_i\times B_i}
|v_{\varepsilon}(x,y)| {{\rm d}}\mu_i^{\varepsilon}(x) {{\rm d}}\mu_i^{\varepsilon}(y)$. Referring to Corollary 5.3 in [@Kach-v], we know that for some $\alpha\in(0,1)$ and all $\eta\in(0,1)$, $$\|v_{\varepsilon}(\cdot,y)\|_{C^{0,\alpha}(\{x\in\Omega~:~{\rm dist}(x,\partial S_1)\geq \eta
\}}\leq \frac{C_\alpha}{\eta^2}\,.$$ Consequently, the following estimate holds, $$\begin{split}
\sum_{i=1}^n
\int_{B_i\times B_i} |v_{\varepsilon}(x,y)| {{\rm d}}\mu_i^{\varepsilon}(x)
{{\rm d}}\mu_i^{\varepsilon}(y)&\leq \sum_{i=1}^n
\int_{B_i\times B_i} \left(|v_{\varepsilon}(a_i,a_i)|
+C \frac{{\varepsilon}^{\alpha}}{\eta^2}\right){{\rm d}}\mu_i^{\varepsilon}(x)
{{\rm d}}\mu_i^{\varepsilon}(y)\\
&= 4\pi^2 C n \frac{{\varepsilon}^{\alpha}}{\eta^2}+4\pi^2 n
|v_{\varepsilon}(a_i,a_i)|,
\end{split}$$ with $\eta=\frac{2\ln|\ln\varepsilon|}{|\ln\varepsilon|}$.\
By our hypotheses, we know that $|v_{\varepsilon}(a_i,a_i)|\ll|\ln{\varepsilon}|$. Thus, $$\label{cont-v}
\sum_{i=1}^n
\int_{B_i\times B_i} |v_{\varepsilon}(x,y)| {{\rm d}}\mu_i^{\varepsilon}(x) {{\rm d}}\mu_i^{\varepsilon}(y)\leq c
{\varepsilon}^{\alpha} n |\ln{\varepsilon}|^2+c' n \,\,o(|\ln{\varepsilon}|).$$ Combining (\[sum\]) together with (\[cont-v\]), we get (\[autreformule\]).\
The proof of the properties (\[formule-F\])- (\[controlemesure\]) are exactly as that given in [@SaSe p. 140-142].
Proof of proposition \[prop-upperbound\], completed
---------------------------------------------------
We first assume $\mu\neq 0$ is a continuous and compactly supported function. Let $n_1=[\frac{H}{2\pi} |\mu|(S_1)]$ and $n_2=[\frac{H}{2\pi} |\mu|(S_2)]$ where $[\cdot]$ denotes the integer part. We take $n=n_1+n_2$. Since $S_1$ and $S_2$ are disjoint and cover $\Omega$, we have $\|\mu\|=|\mu|(\Omega)=|\mu|(S_1)+|\mu|(S_2)$ and consequently, $$\label{approx-n}
\frac{n}{H}\backsim \frac{\|\mu\|}{2\pi}.$$ Proposition \[prop-vp\] provides us with a well-distributed family of points that serves as an input in Lemma \[lem-pv\]. Thus, we get configurations $(\varphi_{\varepsilon},A_{\varepsilon})$ and associated measures $\mu_i^{\varepsilon}$ such that $$\mu_{\varepsilon}:=\frac{1}{H}\sum_{i=1}^{n_1+n_2} \mu_i^{{\varepsilon}}\rightharpoonup
\mu\,, \quad{\rm weakly~in~}\left(C_0^{0,\gamma}\right)^*\,,$$ and the estimates (\[Ge-mu\]), (\[autreformule\])-(\[formule-F\]) and (\[controlemesure\]) being all valid.\
Since $H$ has the order of $|\ln{\varepsilon}|$, Lemma \[corol-cont-green\] and estimates (\[autreformule\])-(\[approx-n\]) yield, $$\mathcal{L}_{{\varepsilon}, H}(\varphi,A')\leq 2\pi|\ln{\varepsilon}|
\sum_{i=1}^{n}p(a_i)+4\pi^2\frac{n^2}{\|\mu\|^2}\int_{\Omega}\int_{\Omega}
G_0(x,y){{\rm d}}\mu(x){{\rm d}}\mu(y)+o(H^2),$$ Inserting the particular choices of $n_1$, $n_2$, $n=n_1+n_2$ and using the definition of $p$ being piecewise constant,we get, $$\label{autreformule3}
\mathcal{L}_{{\varepsilon}, H}(\varphi,A')\leq H |\ln{\varepsilon}|\int_{\Omega} p(x)
|\mu|(x) {{\rm d}}x+H^2\int_{\Omega}\int_{\Omega} G_0(x,y){{\rm d}}\mu(x){{\rm d}}\mu(y)+o(H^2).$$ Again the hypothesis on $H=\mathcal{O}(|\ln{\varepsilon}|)$ gives that the remainder terms in (\[formule-F\]) are $o(1)$, leading thus to $$\begin{aligned}
{\mathcal F}_{{\varepsilon},H}(\varphi,A)&\leq& H^2 J_0({\varepsilon})+H
|\ln{\varepsilon}|\int_{\Omega} p(x)
|\mu|(x) {{\rm d}}x\\
&&+2 H^2 \int_\Omega (h_{\varepsilon}-1) {{\rm d}}\mu_{\varepsilon}+H^2\int_{\Omega}\int_{\Omega} G_0(x,y){{\rm d}}\mu(x){{\rm d}}\mu(y)+o(H^2).\nonumber\end{aligned}$$ Using (\[eq-J0\]), we rewrite the preceding formula in the following explicit form, $$\label{autreformule4}
\begin{split}
{\mathcal F}_{{\varepsilon},H}(\varphi,A)& \leq H^2 \int_\Omega
\left(\frac{|\nabla h_{\varepsilon}|^2}{u_{\varepsilon}^2}+|h_{\varepsilon}-1|^2\right)\,{{\rm d}}x +H
|\ln{\varepsilon}|\int_{\Omega} p(x) |\mu|(x) {{\rm d}}x\\
& +2 H^2 \int_\Omega (h_{\varepsilon}-1) {{\rm d}}\mu+H^2\int_{\Omega\times\Omega}
G_{\varepsilon}(x,y){{\rm d}}\mu(x){{\rm d}}\mu(y)\\
&+2 H^2 \int_\Omega (h_{\varepsilon}-1) ({{\rm d}}\mu_{\varepsilon}-{{\rm d}}\mu)+
H^2\int_{\Omega\times\Omega} (G_0-G_{\varepsilon}) {{\rm d}}\mu {{\rm d}}\mu+o(H^2).
\end{split}$$ Let us define the functions $U_{\mu,{\varepsilon}}$ and $h_{\mu,{\varepsilon}}$ by $$\label{Umu-e}
\left\{
\begin{array}{rl}
-{\rm div}\,\left(\displaystyle\frac1{u_{\varepsilon}^2}\nabla
U_{\mu,{\varepsilon}}\right)
+U_{\mu,{\varepsilon}}=\mu&{\rm in}~\Omega,\\
U_{\mu,{\varepsilon}}=0&{\rm on}~\partial
\Omega.\end{array}\right.$$ $$\label{hmu-e}
\left\{
\begin{array}{rl}
-{\rm div}\,\left(\displaystyle\frac1{u_{\varepsilon}^2}\nabla
h_{\mu,{\varepsilon}}\right)
+h_{\mu,{\varepsilon}}=\mu&{\rm in}~\Omega,\\
h_{\mu,{\varepsilon}}=1&{\rm on}~\partial \Omega.\end{array}\right.$$ Remarking that $U_{\mu,{\varepsilon}}(x)=\displaystyle\int_{\Omega} G_{\varepsilon}(x,y)
{{\rm d}}\mu(y)$, we get as a consequence $$\label{formule-hmu-e}
\int_\Omega \left(\frac{|\nabla
U_{\mu,{\varepsilon}}|^2}{u_{\varepsilon}^2}+|U_{\mu,{\varepsilon}}|^2\right)\,{{\rm d}}x=\int_\Omega\int_\Omega G_{\varepsilon}(x,y) {{\rm d}}\mu(x) {{\rm d}}\mu(y)\,.$$ Writing $h_{\mu,{\varepsilon}}-1=U_{\mu,{\varepsilon}}+h_{\varepsilon}-1$ and replacing (\[formule-hmu-e\]) in in (\[autreformule4\]) leads, after some calculations to $$\label{autreformule6}
\begin{split}
\frac{{\mathcal F}_{{\varepsilon},H}(\varphi,A)}{H^2}& \leq \int_\Omega
\left(\frac{|\nabla
h_{\mu,{\varepsilon}}|^2}{u_{\varepsilon}^2}+|h_{\mu,{\varepsilon}}-1|^2\right){{\rm d}}x+
\frac{|\ln{\varepsilon}|}{H} \int_\Omega p(x)\, |\mu|(x) {{\rm d}}x\\
& +2 \int_\Omega (h_{\varepsilon}-1) ({{\rm d}}\mu_{\varepsilon}-{{\rm d}}\mu)+\int_{\Omega\times\Omega} (G_0-G_{\varepsilon}) {{\rm d}}\mu {{\rm d}}\mu+o(1).
\end{split}$$ We define $h_{\mu}$ by $$\label{hmu}
\left\{
\begin{array}{rl}
-{\rm div}\,\left(\displaystyle\frac1{p(x)}\nabla h_{\mu}\right)
+h_{\mu}=\mu&{\rm in}~\Omega,\\
h_{\mu}=1&{\rm on}~\partial \Omega.\end{array}\right.$$ By a standard compactness argument similar to that given to Lemma \[conv-G0\], we check that, $$\label{eq:conv1}
\lim_{{\varepsilon}\longrightarrow 0}\int_\Omega \left(\frac{|\nabla
h_{\mu,{\varepsilon}}|^2}{u_{\varepsilon}^2}+|h_{\mu,{\varepsilon}}-1|^2\right){{\rm d}}x=\int_\Omega
\left(\frac{|\nabla h_{\mu}|^2}{p(x)}+|h_{\mu}-1|^2\right){{\rm d}}x.$$ Using a Cauchy-Schwarz inequality, it follows from Lemma \[conv-G0\] and the boundedness of $\mu$ that $$\label{eq:conv2}
\lim_{{\varepsilon}\longrightarrow 0}\int_{\Omega\times\Omega} (G_0-G_{\varepsilon}) {{\rm d}}\mu {{\rm d}}\mu=0.$$ Next, noticing that $\|h_{\varepsilon}-1\|_{C^0_0(\Omega)}\leq 1$, it follows immediately from the convergence of $\mu_{\varepsilon}$ to $\mu$ in $(C_0^0(\Omega))^*$, $$\lim_{{\varepsilon}\longrightarrow 0}\int_\Omega (h_{\varepsilon}-1) ({{\rm d}}\mu_{\varepsilon}-{{\rm d}}\mu)=0\,. \label{cv-h-mu}$$ Replacing (\[eq:conv1\])-(\[cv-h-mu\]) in (\[autreformule6\]) then using $\displaystyle\lim_{{\varepsilon}\to0}\frac{H}{|\ln{\varepsilon}|}= \lambda$, we get finally, $$\limsup_{{\varepsilon}\longrightarrow 0}\frac{{\mathcal
F}_{{\varepsilon},H}(\varphi,A)}{H^2}\leq \frac{1}{\lambda}\int_\Omega
p(x)|\mu|(x) \,{{\rm d}}x+\int_\Omega\left( \frac{1}{p(x)}|\nabla
h_{\mu}|^2+|h_{\mu}-1|^2\right) {{\rm d}}x .$$ Moreover, since (\[controlemesure\]) holds and ${\mathcal
L}_{{\varepsilon},H}(\varphi,A')\leq C |\ln{\varepsilon}|^2$, we have $$\frac{1}{H}
\|\mu(\varphi,A)-\mu_{\varepsilon}\|_{(C_0^{0,\gamma}(\Omega))^*}\leq C
{\varepsilon}^{\gamma}\left(1+\sqrt{\frac{{\mathcal
L}_{{\varepsilon},H}(\varphi,A')}{n}}\right)\leq o(1).$$ We conclude that (\[cv-mesure\]) holds, which finishes the proof in the case where $\mu$ is a continuous and compactly supported function. The general case where $\mu\in \mathcal{M}(\Omega)\cap
H^{-1}(\Omega)$ follows by a standard approximation argument, see [@SaSe p. 149] for details.
Lower Bound {#Sec:LB}
===========
Main result
-----------
The objective of this section is to prove the lower bound stated in Proposition \[prop-lowerbound\] below.\
Given a family of configurations $\{(\varphi_\varepsilon,A_\varepsilon)\}$, we denote by $$\label{eq:current}
j_\varepsilon=\left(i\varphi_\varepsilon,
(\nabla-iA_\varepsilon)\varphi_\varepsilon\right)\,,\quad
h_\varepsilon={\rm curl}\,A_\varepsilon\,.$$
\[prop-lowerbound\] Assume that $\displaystyle\lim_{\varepsilon\to0}\frac{H}{|\ln\varepsilon|}=\lambda$ with $\lambda>0$. Let $\{(\varphi_{\varepsilon},A_{\varepsilon})\}_n$ be a family of configurations satisfying $\mathcal F_{\varepsilon,H}(\varphi_\varepsilon,A_\varepsilon)\leq
C H^2$ and $\|\varphi_\varepsilon\|_{L^\infty(\Omega)}\leq 1$ for a given constant $C>0$.\
Then, up to the extraction of a subsequence $\varepsilon_n$ converging to $0$, one has, $$\forall~\gamma\in(0,1)\quad
\frac{\mu(\varphi_{\varepsilon_n},A_{\varepsilon_n})}{H}\to\mu \quad{\rm
in~}
\left(C^{0,\gamma}(\Omega)\right)^*\,,$$ $$\frac{j_{\varepsilon_n}}{H}\rightharpoonup j\,,\quad
\frac{h_{\varepsilon_n}}{H}\rightharpoonup h\quad
{\rm weakly~in~}L^2(\Omega)\,.$$ Moreover, $\mu={\rm curl}\,j+h$ and $$\label{eq:lb}
\liminf_{\varepsilon\to0} \frac{\mathcal
F_{\varepsilon,H}(\varphi_\varepsilon,A_{\varepsilon})}{H^2} \geq
E_\lambda(\mu)+\int_\Omega\left( p(x)\left|j+\frac1{p(x)}\nabla^\bot
h_\mu\right|^2 +|h-h_\mu|^2 \right)\,{{\rm d}}x\,.$$ Here, the energy $E_\lambda$ and the function $h_\mu$ are introduced in (\[eq-E-lambda\]) and (\[eq-h-mu\]) respectively.
Vortex-balls
------------
In this section we construct suitable ‘vortex-balls’ providing a lower bound of the energy of minimizers of (\[V-EGL\]). Recall the decomposition of the energy in Lemma \[V-lem-psi<u\], which permits us to work with the ‘reduced energy functional’ $\mathcal
F_{\varepsilon,H}$.\
Notice that, by using $(u_\varepsilon,0)$ as a test configuration for the function (\[V-EGL\]), we deduce an upper bound of the form : $$\label{V-upperbound}
\mathcal F_{\varepsilon,H}(\varphi,A)\leq CH^2\,,$$ where $\varphi=\psi/u_\varepsilon$, $(\psi,A)$ a minimizer of (\[V-EGL\]), and $C>0$ a positive constant.\
We recall the hypothesis that there exists a positive constant $C>0$ such that the applied magnetic field $H$ satisfies $$\label{hypothesis-H}
H\leq C|\ln\varepsilon|\,.$$
The upper bound (\[V-upperbound\]) provides us, as in [@SaSe], with the construction of suitable ‘vortex-balls’.
\[V-lem-vortexballs\] Assume the hypotheses (\[hypothesis-H\]). Given an open set $U\subset \Omega$ and a number $p\in]1,2[$, there exists a constant $C>0$ and a finite family of disjoint balls $\{B((a_i,r_i)\}_{i\in
I}$ such that, $(\varphi,A)$ being a configuration satisfying the bound (\[V-upperbound\]), the following properties hold:
1. $\overline{B(a_i,r_i)}\subset U$ for all $i$;
2. $w=\{x\in U~:~|\varphi(x)|\leq
1-|\ln\varepsilon|^{-4}\}
\subset\displaystyle\displaystyle\bigcup_{i\in I}B(a_i,r_i)$.
3. $\displaystyle\sum_{i\in I}r_i\leq
C\,|\ln\varepsilon|^{-10}$.
4. Letting $d_i$ be the degree of the function $\varphi/|\varphi|$ restricted to $\partial
B(a_i,r_i)$ if $B(a_i,r_i)\subset\Omega$ and $d_i=0$ otherwise, then we have: $$\begin{aligned}
\label{V-Eq-LBest}
&&\hskip-0.5cm \int_{B(a_i,r_i)\setminus\omega}
u_\varepsilon^2|(\nabla-iA)\varphi|^2\,{{\rm d}}x+\int_{B(a_i,r_i)}|{\rm
curl}\,A-H|^2\,{{\rm d}}x\geq\\
&&\hskip3.5cm 2 \pi|d_i|
\left(\min_{B(a_i,r_i)}u_\varepsilon^2\right)
\left(|\ln\varepsilon|-C\ln|\ln\varepsilon|\right). \nonumber\end{aligned}$$
5. $\left\|2\pi\displaystyle\sum_{i\in I} d_i\delta_{a_i}-{\rm
curl}\big{(}A+(i\varphi,\nabla_A\varphi)\big{)}
\right\|_{W^{-1,p}_0(U)}\leq C|\ln\varepsilon|^{-4}.$
We follow the usual terminology and call the balls constructed in Proposition \[V-lem-vortexballs\] ‘vortex-balls’. The proof of Proposition \[V-lem-vortexballs\] is actually a simple consequence of the analysis of [@SaSe].
Proof of Proposition \[prop-lowerbound\]
----------------------------------------
Let us consider smooth and open sets $U_1\subset S_1$ and $U_2\subset
S_2$, and let us denote by $U$ their union.\
Applying Proposition \[V-lem-vortexballs\] in $U_1$ and $U_2$ respectively, we get families of balls $B(a_i,r_i)$ and degrees $d_i$ such that, $$\frac{\mathcal
F_{\varepsilon,H}(\varphi_\varepsilon,A_\varepsilon,V_\varepsilon)}{H^2}
\geq 2\pi \frac{|\ln\varepsilon|}{H}
\sum_i\left(\min_{\overline{B(a_i,r_i)}}u_\varepsilon^2\right)
\frac{|d_i|}{H} +o(1)\,,$$ where $V_\varepsilon=\displaystyle\cup_i B(a_i,r_i)$.\
Recall that $u_\varepsilon^2$ converges uniformly to the function $p$ in $U$. Thus, we may rewrite the above lower bound in the following form, $$\label{eq:lb-V}
\frac{\mathcal
F_{\varepsilon,H}(\varphi_\varepsilon,A_\varepsilon,V_\varepsilon)}{H^2}
\geq 2\pi \frac{|\ln\varepsilon|}{H}
\left(\frac{\sum_{a_i\in U_1} |d_i|}{H}+a\frac{\sum_{a_i\in U_2}|d_i|}H\right)
+o(1)\,.$$ Using the bound $\|\varphi_\varepsilon\|_{L^\infty(\Omega)}\leq1$, we write $|(\nabla-iA_\varepsilon)\varphi_\varepsilon|^2\geq
|\varphi_\varepsilon|^2|(\nabla-iA_\varepsilon)|^2\geq
|j_\varepsilon|^2$. Consequently, we have, $$\label{eq:lb-U}
\frac{\mathcal
F_{\varepsilon,H}(\varphi_\varepsilon,A_\varepsilon,U\setminus
V_\varepsilon)}{H^2} \geq \int_{U\setminus
V_\varepsilon}\left(u_\varepsilon^2\left|\frac{j_\varepsilon}{H}\right|^2+
\left|\frac{h_\varepsilon}{H}-1\right|^2\right)\,{{\rm d}}x\,.$$ With $U=\Omega$, we infer from the bound $\mathcal F_{\varepsilon,H}(\varphi_\varepsilon,A_\varepsilon)\leq
CH^2$ that up to the extraction of a subsequence, $\displaystyle\frac{j_\varepsilon}{H}$ and $\displaystyle\frac{h_\varepsilon}{H}$ respectively converge weakly to $j$ and $h$ in $L^2(\Omega)$.\
Thanks again to the bound (\[V-upperbound\]), we deduce that $\displaystyle\frac{\sum_{a_i\in U_1} |d_i|}{H}$ and $\displaystyle\frac{\sum_{a_i\in U_2} |d_i|}{H}$ are bounded. Hence the measures $\displaystyle\frac{\sum_{a_i\in U_1}
d_i\delta_{a_i}}{H}$ and $\displaystyle\frac{\sum_{a_i\in U_2}
d_i\delta_{a_i}}{H}$ are weakly compact in the sense of measures, and thus, up to extraction of subsequences, they respectively converge to measures $\mu_1$ and $\mu_2$ in $(C_0^0(U_1))^*$ and $(C_0^0(U_2))^*$. Thanks to the last property of Proposition \[V-lem-vortexballs\], we get upon setting $\mu={\rm curl}\,j+h$, $$\mu_1=\mu_{|_{U_1}}\,,\quad\mu_2=\mu_{|_{U_2}}\,.$$ Therefore, combining (\[eq:lb-V\])-(\[eq:lb-U\]) and using the uniform convergence of $u_\varepsilon^2$ to $p$ in $U$, we deduce that, $$\begin{aligned}
\frac{\mathcal
F_{\varepsilon,H}(\varphi_\varepsilon,A_\varepsilon,U)}{H^2} &\geq&
\frac{\mathcal
F_{\varepsilon,H}(\varphi_\varepsilon,A_\varepsilon,V_\varepsilon)}{H^2}
+\frac{\mathcal
F_{\varepsilon,H}(\varphi_\varepsilon,A_\varepsilon,U\setminus
V_\varepsilon)}{H^2}\\
&
\geq&\frac1{\lambda}\left(|\mu|(U_1)+a|\mu|(U_2)\right)+\int_U\left(
p(x)|j|^2+|h-1|^2\right)\,{{\rm d}}x+o(1).\end{aligned}$$ Invoking $\mathcal
F_{\varepsilon,H}(\varphi_\varepsilon,A_\varepsilon,\Omega)\geq\mathcal
F_{\varepsilon,H}(\varphi_\varepsilon,A_\varepsilon,U)$, we deduce that, $$\label{eq:lb-1}
\liminf_{\varepsilon\to0} \frac{\mathcal
F_{\varepsilon,H}(\varphi_\varepsilon,A_\varepsilon,\Omega)}{H^2}
\geq
\frac1{\lambda}\left(|\mu|(U_1)+a|\mu|(U_2)\right)+\int_U\left(p(x)|j|^2+|h-1|^2\right)\,{{\rm d}}x\,.$$ The left hand side of (\[eq:lb-1\]) being independent from $U$, and $U_1$, $U_2$ being arbitrary subsets of $S_1$ and $S_2$, we conclude that $$\label{eq:lb-2}
\liminf_{\varepsilon\to0} \frac{\mathcal
F_{\varepsilon,H}(\varphi_\varepsilon,A_\varepsilon,\Omega)}{H^2}
\geq \frac1{\lambda}\int_\Omega p(x){{\rm d}}|\mu|
+\int_\Omega\left(p(x)|j|^2+|h-1|^2\right)\,{{\rm d}}x\,.$$ Now to conclude, we write $$j=-\frac1{p(x)}\nabla^\bot h_\mu+\left(j+\frac1{p(x)}\nabla^\bot
h_\mu\right)\,,\quad
h=h_\mu+(h-h_\mu)\,.$$ Upon substitution in the right hand side of (\[eq:lb-2\]) and using in particular the remarkable identities $${\rm curl}\left(j+\frac1{p(x)}\nabla^\bot h_\mu\right)
+h-h_\mu=0\quad{\rm in~}\Omega\,,\quad h_\mu-1=0\quad{\rm
on~}\partial\Omega\,,$$ we get the desired conclusion (\[eq:lb\]).
\[rem:conc\] Combining the upper and lower bounds of Propositions \[prop-upperbound\] and Proposition \[prop-lowerbound\], then by uniqueness of the minimizer $\mu_*$ of $E_\lambda$ (see Section \[Sec:Min\] below), it is evident that $\mu_*=\mu$ and $h=h_{\mu_*}$. Here $\mu$ and $h$ are given in Proposition \[prop-lowerbound\] above.
Minimization of the limiting energy {#Sec:Min}
===================================
As we explained in the introduction, by convexity and lower semi-continuity, the limiting energy (\[eq-E-lambda\]) admits a unique minimizer $\mu_*$ which is expressed by means of the unique minimizer $h_*$ of (\[eq:dual-en\]) as follows, $$\label{eq:mu*=h*}
\mu_*=-{\rm div}\left(\frac1{p(x)} \nabla h_*\right)+h_*\,.$$
Proceeding as in [@SaSe3; @SaSe], we may get an equivalent characterization of $h_*$. We write $H^1_1(\Omega)$ for the space of Sobolev functions $u$ such that $u-1\in H^1_0(\Omega)$.
\[prop-min\] The minimizer $u_*$ of $$\min_{\substack{u\in H_0^1(\Omega)\\
-{\rm div}(\frac{1}{p(x)}\nabla
u)+u\in\mathcal{M}(\Omega)}}\int_\Omega\left( \frac{
p(x)}{\lambda}\left|-{\rm div}\left(\frac{1}{p(x)}\nabla
u\right)+u+1\right|+\frac{|\nabla u|^2}{p(x)}+|u|^2\right)\,{{\rm d}}x\,,$$ is also the unique minimizer of the dual problem $$\min_{ \substack{v\in H_0^1(\Omega)\\|v|\leq \frac{ p}{2\lambda}}}
\int_\Omega \left(\frac{|\nabla v|^2}{p(x)}+|v|^2+2 v\right)\,{{\rm d}}x\,.$$ For instance, $h_*=u_*+1$ minimizes the energy, $$\min_{\substack{f\in H_1^1(\Omega)\\(f-1)\geq -\frac{ p}{2\lambda}}}
\left(\int_\Omega \frac{|\nabla f|^2}{p(x)}+|f|^2\right)\,,$$ and satisfies $-{\rm div}\left(\frac{1}{p(x)}\nabla
h_*\right)+h_*\geq 0$.
Let us define the lower semi-continuous and convex functional $$\Phi(u)=
\int_\Omega \frac{p(x)}{2\lambda} \left|-{\rm div}
\left(\frac1{p(x)}\nabla u\right)+u+1\right|\,{{\rm d}}x$$ in the Hilbert space $H=H^1_0(\Omega)$ endowed with the scalar product $\langle f,g\rangle_H=\int_\Omega \frac{1}{p(x)}\nabla f\cdot \nabla
g+ f g$. Let us compute its conjugate $\Phi^*$, i.e. $$\Phi^*(f)=\sup_{~\{g~:~\Phi(g)<\infty\}}\langle f,g\rangle-\Phi(g)\,.$$ Indeed, we have, $$\Phi^*(f) \geq \sup_{\eta\in L^2}\int_\Omega f\eta\,{{\rm d}}x-\frac{1}{2\lambda}\int_\Omega p(x) |\eta|\,{{\rm d}}x-\int_\Omega
f\,{{\rm d}}x\,,$$from which we deduce that $$\Phi^*(f)=\left\{
\begin{array}{ll}
-\displaystyle\int_\Omega f\,{{\rm d}}x&{\rm if}~|f|\leq \frac{ p}{2\lambda}\,,\\
+\infty&{\rm otherwise.}
\end{array}\right.$$ By convex duality (see [@SaSe Lemma 7.2]), $$\min_{u\in H}\left(\|u\|_H^2+2\Phi(u)\right)=-\min_{f\in
H}\left(\|f\|_H^2+2\Phi^*(-f)\right)\,,$$ and minimizers coincide. That the measure $\mu_*=-{\rm
div}\left(\frac{1}{p(x)}\nabla h_*\right) +h_*$ is positive is actually a consequence of the weak maximum principle, see [@Ka p. 131]. One may also follow step by step the proof given in [@SaSe3].
Therefore, the limiting vorticity measure $\mu_*$ is positive, and following [@SaSe3], it can be expressed by means of the coincidence set $w_\lambda=w_\lambda^1\cup w_\lambda^2$, $$\label{eq:w-l}
w_\lambda^1=\{ x\in \overline S_1~:~
1-h_*(x)=\frac{1}{2\lambda}\}\,,\quad w_\lambda^2=\{x\in\overline
S_2~:~\frac{1-h_*(x)}{a}=\frac1{2\lambda}\}\,,$$ as follows, $$\label{eq:mu*}
\mu_*=\left(1-\frac{p(x)}{2\lambda}\right){\bf{1}}_{w_\lambda} {{\rm d}}x\,,$$ where $\mathbf 1_{w_\lambda}{{\rm d}}x$ denotes the Lebesgue measure restricted to $w_\lambda$. Furthermore, $h_*$ (the minimizer of (\[eq:dual-en\])) solves, $$\label{eq:h*}
\left\{
\begin{array}{ll} -{\rm div}\left(\frac1{p(x)}\nabla
h_*\right)+h_*=0&{\rm in}~\Omega\setminus w_\lambda\\
h_*=1-\frac{p}{2\lambda}&{\rm in ~}w_\lambda\\
h_*=1&{\rm on~}\partial \Omega\,,
\end{array}\right.$$ and the regularity of $h_*$ is complicated, as $w_\lambda$ may intersect the interior boundary $\partial S_1$ where $h_*$ has gradient singularities (it is expected to satisfy a transmission condition, see the radial case below). But, we know that in the interior of $S_1$ and $S_2$, $h_*$ is locally $C^{1,\alpha}$ for some exponent $\alpha\in(0,1)$. Remark for instance that the measure $\mu_*$ is no more uniform and may be discontinuous in light of $\mu_*=\left(1-\frac{1}{2\lambda}\right)\mathbf {{\rm d}}x$ in $w_\lambda^1$ and $\mu_*=\left(1-\frac{a}{2\lambda}\right){{\rm d}}x$ in $w_\lambda^2$.\
For the sake of a better understanding of the sets $w_\lambda$, $w_\lambda^1$ and $w_\lambda^2$, we introduce the following critical constants (we emphasize their dependence on the material parameter $a$), $$\label{eq:l0}
\lambda_i(a)=\frac{1}{2\displaystyle \max_{x\in \overline{S_i}}
\left(\frac{1-h_0(x)}{p(x)}\right)},\quad\forall~i\in\{1,2\}\,,\quad
\lambda_0(a)=\min(\lambda_1(a),\lambda_2(a))\,.$$ Here we recall that $h_0$ is the solution of $-{\rm
div}\left(\frac{1}{p(x)}\nabla
h_0\right)+h_{0}=0$ in $\Omega$ and $h_0=1$ on $\partial\Omega$. The maximum principle gives that $0<h_0<1$ in $\Omega$. We denote also by $w_0=w_0^1\cup w_0^2$ the set $w_{\lambda_0(a)}=w_{\lambda_0(a)}^1\cup w_{\lambda_0(a)}^2$ introduced in (\[eq:w-l\]).\
The following proposition follows from a weak maximum principle [@Ka p. 131], see [@SaSe3] for a detailed proof (modulo necessary adjustments).
\[prop:w-l\]
1. $w_\lambda$ is increasing with respect to $\lambda$ and $\cup_{\lambda>0}w_\lambda=\Omega$;
2. $w_\lambda^1$ and $w_\lambda^2$ are disjoint;
3. If $\lambda<\lambda_0(a)$ then $h_*=h_0$, $\mu_*=0$ and $w_\lambda=\emptyset$;
4. If $\lambda=\lambda_0(a)$ then $h_*=h_0$, $\mu_*=0$ and $w_\lambda=w_0$;
5. If $\lambda>\lambda_0(a)$ then $\mu_*\not=0$;
6. If $\lambda<\lambda_i(a)$ for some $i\in\{1,2\}$, then $w_{\lambda}^i=\emptyset$.
Next for the sake of illustrating the above results, we give a detailed analysis of $h_0$ in the radial case. Let us take $\Omega=D(0,1)$ the unit disc in $\mathbb{R}^2$, $S_1=D(0,R)$ and $S_2=D(0,1)\setminus \overline {D(0,R)}$ where $0<R<1$. In this case $h_0$ is radially symmetric, $h_0(x)=h_0(|x|)$, and so it solves the following ODE, $$\label{eq:h0(r)}
\left\{\begin{array}{l}
-h''(r)-r^{-1}h'(r)+h_0(r)=0\quad{\rm if}~0<r<R\,,\\
-h''(r)-r^{-1}h'(r)+a h_0(r)=0\quad{\rm if
}~R<r<1\,,\\
h_0(R_-)=h_0(R_+)\,,\quad h_0'(R_-)=\frac1 a h_0'(R_+)\,,\\
h_0'(0)=0\quad h_0(1)=1\,.
\end{array}\right.$$ We look for a power series solution in the form, $$h_0(r)=\sum_{n=0}^\infty a_n r^n\quad{\rm if}~0<r<R\,,\quad
h_0(r)=\sum_{n=0}^nb_n (r-R)^n\quad{\rm if}~R<r<1\,.$$ The sequences $(a_n)$ and $(b_n)$ depend on $a$ and $R$ and all the terms are expressed as functions of the term $a_0$, the two constants $$\alpha=1+\sum_{k=1}^\infty\frac{R^{2k}}{2^k((k+1)!)^2}\,,\quad\beta=\sum_{k=0}^\infty(2k+2)
\frac{R^{2k+1}}{2^k((k+1)!)^2}\,,$$ and the sequence $(\gamma_n(a))$ defined recursively by, $$\gamma_{-3}(a)=1\,,\quad\gamma_{-2}(a)=-\frac{1}R\,,\quad\gamma_{n+1}(a)=
-\frac1R\gamma_n(a)+a\gamma_{n-1}(a)\,,\quad\forall~n\geq-1\,.$$ Actually, we may verify that, $$a_{2n+1}=0\,,\quad
a_{2n+2}=\frac{a_0}{2^n((n+1)!)^2}\quad\forall~n\in\mathbb N\,,$$ $$b_0=a_0\alpha\,,\quad b_1=a\,a_0\beta\,,\quad
b_{n+2}=\frac1{(n+2)!}\left(\gamma_{n-2}(a)b_1+a\gamma_{n-3}(a)b_0\right)\quad\forall~n\geq2\,,$$ and the term $a_0$ is expressed explicitly by, $$a_0=\left[\alpha+
a\left(\beta(1-R)+\sum_{n=0}^\infty\frac1{(n+2)!}\left(\alpha\gamma_{n-2}(a)+\beta\gamma_{n-3}(a)\right)
(1-R)^n\right)\right]^{-1} \,,$$ provided that the sum in the r.h.s. is finite. For instance, this is the case when $\frac12<R<1$ and $a\to0_+$. Actually, in the limit $a\to0_+$, it holds that, $$a_0\to \alpha^{-1}\,,\quad \frac{1-b_0}{a}\to \alpha^{-1}\left(\beta(1-R)+\frac{\beta-\alpha}{R}
\sum_{n=0}^\infty\frac{(-1)^n}{(n+2)!}\left(\frac1R-1\right)^n\right)=:c_0\,.$$ For $R\in(\frac12,1)$ chosen conveniently (close to $\frac12$), it is easy to verify that $c_0>1-\alpha^{-1}$. Therefore, there exists $a_0\in(0,1)$ sufficiently small (depending on $R\in(\frac12,1)$) such that, for all $a\in(0,a_0)$, we have, $$\lambda_2(a)<\lambda_1(a)\,,$$ where $\lambda_1(a)$ and $\lambda_2(a)$ are the critical constants introduced in (\[eq:l0\]). Coming back to the problem of vortex nucleation for the G-L energy (\[reducedfunctional\*\]), we get in light of Theorem \[thm2\] and Proposition \[prop:w-l\], that for a wide range of applied magnetic fields, $$H=\lambda|\ln\varepsilon|(1+o(1))\quad(\varepsilon\to0)\,,\quad
\lambda\in(\lambda_2(a),\lambda_1(a))\,,$$ vortices exist and are pinned in $S_2$. This result is in accordance with that obtained by the second author in [@Kach-v1; @Kach-v], where nucleation of vortices near the critical magnetic field (in the case of the disc) is studied in details.
Acknowledgments {#acknowledgments .unnumbered}
===============
Part of this work was carried out while the second author visited Lebanese University. He wishes to thank A. Mneimneh and R. Talhouk for their hospitality.
[100]{} Pinning phenomena in the Ginzburg-Landau model of superconductivity. [*J. Math. Pures Appl.*]{} [**80**]{} 339-372 (2001). Vortices for a variational problem related to superconductivity. [*Ann. Inst. H. Poincar[é]{} Anal. Non Lin[é]{}aire*]{} [**12**]{} (3) 243-303 (1995). A Ginzburg Landau type model of superconducting/normal junctions including Josephson junctions. [*European J. Appl. Math.*]{} [**6**]{} (2) 97-114 (1996). Superconductors surrounded by normal materials. [*Proc. Roy. Soc. Edinburgh Sec. A,*]{} [**135**]{} 331-356 (2005). Magnetic Ginzburg-Landau functional with disconinuous constraint. [*C. R. Math. Acad. Sci. Paris*]{} [**346**]{} (5-6) 297-300 (2008). Magnetic vortices for a Ginzburg-Landau type energy with disconinuous constraint. (Submitted). [Preprint arxiv.org]{} On the perfect superconducting solution for a generalized Ginzburg-Landau equation. [*Asymptot. Anal.*]{} [**54**]{} (3-4) 125-164 (2007). Springer. Verlag. (1993). Ginzburg-Landau type energy with discontinuous constraint. [*J. Anal. Math.*]{} [**77**]{} 1-26 (1999). Six lectures in superconductivity, [*Boundaries, Interfaces and Transitions. (Banff, AB, 1995),*]{} 163–184, CRM Proc. Lecture Notes, [**13**]{}. [*Amer. Math. Soc., Providence, RI,*]{} 1998. Progress in Nonlinear Differential Equations and their Applications, 70. Birkh[ä]{}user Boston, 2007. A Rigorous Derivation of a Free-Boundary Problem Arising in Superconductivity. [*Annales Scientifiques de l’ENS,*]{} [**33**]{} (2000), 561-592. quations elliptiques du second ordre [à]{} coefficients discontinus. S[é]{}minaire de Math[é]{}matiques Sup[é]{}rieures, No. 16 ([É]{}t[é]{}, 1965). [*Les Presses de l’Universit[é]{} de Montr[é]{}al, Montreal, Que. 326 pp. 1966).*]{}
[^1]: Notice that $\varphi$ and $\psi$ have the same vortices.
[^2]: We apply the lemma with $N(\varepsilon)=\sqrt{n(\varepsilon)}$ so that we get $\sqrt{n(\varepsilon)}\geq p_\varepsilon(K)$ points.
[^3]: See Definition \[def1\] above.
|
---
abstract: 'Real-time bidding (RTB) has become a new norm in display advertising where a publisher uses auction models to sell online user’s page view to advertisers. In RTB, the ad with the highest bid price will be displayed to the user. This ad displaying process is biased towards the publisher. In fact, the benefits of the advertiser and the user have been rarely discussed. Towards the global optimization, we argue that all stakeholders’ benefits should be considered. To this end, we propose a novel computation framework where multimedia techniques and auction theory are integrated. This doctoral research mainly focus on 1) figuring out the multimedia metrics that affect the effectiveness of online advertising; 2) integrating the discovered metrics into the RTB framework. We have presented some preliminary results and discussed the future directions.'
author:
- Xiang Chen
bibliography:
- 'mybib.bib'
title: 'Towards Global Optimization in Display Advertising by Integrating Multimedia Metrics with Real-Time Bidding'
---
Introduction {#sec:introduction}
============
The explosive growth of multimedia data on Internet has created huge opportunities for online advertising. According to the latest Interactive Advertising Bureau’s annual report [@iab_2016], the second-half revenue totaled \$39.8 billion in 2016, which is an increase of \$7.7 billion from the second-half revenue of 2015. Despite the increasing growth of revenue, the effectiveness of online advertising remains debatable. On the one hand, the user gets annoyed by the improper ads. Marketing data [@hubspot_annoy_ad] shows that: the average click-through rate of display ads across all formats and placement is 0.06%; young adults tend to ignore online banner ads; and ad blocking has grown by 41% globally in the year 2015. The unpleasant ad experience further influences their site visits, and their perception of the displayed ad. On the other hand, the advertiser gets harmed by ineffective ad delivery. In terms of cost-per-impression pricing mechanism, the advertiser has to pay for every impression. Due to user’s low engagement towards the displayed ad, a large amount of budget that the advertiser spends on online advertising is wasted.
The majority of online displayed ads are served through RTB. Compared to the RTB system that focuses on the publisher’s revenue (local maximization), we define the global optimization as the optimal trade-offs among all stakeholders. In this regard, the proper ad is not only the highest revenue from the economic point of view, but also the best fit with the context from the multimedia point of view. Fig. \[fig:schematic\_view\] illustrates how the multimedia metrics are incorporated with RTB. Our proposed framework consists of two stages: 1) the first stage uses the second-price (SP) auction model to deal with economic issues; 2) the second stage uses our proposed optimal re-ranking model to select the proper ad. Note that, the independent process of stage I ensures the equilibrium in ad auctions. The multimedia metrics take effect in stage II. Since they are exogenous, our proposed framework does not affect the advertiser’s bidding behavior.
Due to the complex nature of display advertising, we are facing the following challenges:
- **What metrics to select?** User’s engagement (e.g., view, click) towards the displayed ad varies in different context. Understanding the context and user’s behavior becomes the key factor to improve the effectiveness of online advertising.
- **How to integrate the metrics?** Online advertising is a multidisciplinary topic, which involves economics, multimedia and psychology. The metrics represent the stakeholders’ benefit from different domains. An easy but effective solution is the linear combination model. Thus, determining the optimal weights becomes the key issue.
The development of computer vision, multimedia analysis and machine learning techniques brings opportunities to solve these two challenges. Firstly, we select a set of multimedia metrics according to marketing data and consumer psychology literature. We then conduct a user-study experiment to demonstrate the effectiveness of the selected metrics. Secondly, we simulate a real-world advertising system based on an auction dataset and two multimedia datasets. By adding constrains on the changes of metrics when compared with existing system, we propose an optimization model to obtain the optimal weights. To the best of our knowledge, we are the first towards global optimization in display advertising by combining multimedia techniques and auction theory.
The rest of the paper is organized as follows. Section \[sec:related\_work\] reviews the related work. Section \[sec:approach\] describes the proposed optimization framework. Section \[sec:experiments\] presents our preliminary results. Section \[sec:work\_in\_progress\] lists our works in progress. And section \[sec:conclusion\] concludes the paper.
![Overview of the proposed framework [@chen2017RTBoptimizing].](figs/framework.png){width="0.84\linewidth"}
\[fig:schematic\_view\]
Related work {#sec:related_work}
============
From the economics perspective, researchers mainly focus on investigating the strategy of selling ad impressions. The generalized second-price (GSP) auction [@Edelman_2007_2] and the Vickrey-Clark-Groves (VCG) auction [@Parkes_2007] have been widely used on different platforms. As revenue is always the primary concern, the squashing parameter and reserve price have been discussed to increase the revenue of GSP auctions [@Lahaie_2007; @Thompson_2013]. GSP auction model has been proved to be able to improve social welfare as well [@Lahaie_2011].
From the multimedia perspective, researchers mainly focus on increasing user’s engagement towards displayed ads and meanwhile maintaining user’s online experience. Literature in marketing and consumer psychology has already shown that the contextual relevance between the content of the hosting webpage and the ads makes a large difference in their clickability [@chatterjee2003modeling], and it also has a leading effect on user’s online experience [@mccoy2007effects]. According to the utilized information for contextual similarity matching, we can classify most existing contextual advertising systems into three categories: text-based advertising [@li2010pagesense], visual-based advertising [@sengamedu2007vadeo] and targeted advertising [@wang2009argo]. To achieve better contextual relevance, the multimodal approach has been proposed which considers both textual and visual information for video [@mei2007videosense; @guo2009adon] and image advertising [@mei2012imagesense].
The works discussed so far only focus on the benefit of a specific stakeholder, regardless of the other stakeholders’ benefits. Trade-offs among multiple objectives or stakeholders have been discussed in several works. Likhodedov et al. [@Likhodedov_2003] proposed a framework that linearly combines revenue and social welfare in a single-item auction. Bachrach et al. [@Bachrach_2014] proposed a framework that linearly combines relevant, welfare and clicks in sponsored search. Liao et al. [@liao2008adimage] combined revenue and relevance for video advertising.
Our goal is to foster a vigorous and healthy online advertising ecosystem. Different from most existing works that focus on local maximum value, we focus on the global optimization of the whole advertising ecosystem. Display advertising is an interplay among all stakeholders and each stakeholder’s ad experience is an important factor for advertising effectiveness. In the proposed framework, the publisher’s revenue will decrease in a short term in trade of improvements on the benefits of the advertiser and user. The increased ad experience will increase the demand of the advertiser’s advertising needs and the supply of the user’s webpage visits, which will boost the publisher’s revenue in a long run.
![User’s engagement on displayed ad under different combination of multimedia metrics. []{data-label="fig:user_study"}](figs/recall_and_fixation_with_title){width="1\linewidth"}
---------------------- ----------------------------------- ------------------------------ --------------------------- -------------------------------------------------
Method Input Output
Publisher’s revenue Stage I Bid Payment $x_{1,i,z} = $ normalized $p_{i,z}$
Advertiser’s utility Stage I Bid, payment Utility $x_{2,i,z} = $ normalized $(v_{i,z} - p_{i,z})$
Ad memorability MemNet [@khosla2015understanding] Ad image Memorability score $x_{3,i,z} = $ normalized score
Ad CTR Given by data CTR CTR $x_{4,i,z} = $ normalized CTR
Contextual relevance TakeLab [@vsaric2012takelab] Title, keywords, description Semantic similarity score $x_{5,i,z} = $ normalized score
Ad saliency MBS [@zhang2015minimum] Web page snapshot, ad image Saliency map and ratio $x_{6,i,z} = $ normalized ratio
---------------------- ----------------------------------- ------------------------------ --------------------------- -------------------------------------------------
Proposed framework {#sec:approach}
==================
We propose to incorporate multimedia metrics into RTB to improve the effectiveness of display advertising. Our framework consists of two stages (see Fig. \[fig:schematic\_view\]). The first stage is based on the second-price auction model. And the second stage is based on the optimal re-ranking model to select the most appropriate ad. Since the auction model is simple but effective, we focus on the optimal re-ranking model. When adopting the weighted linear combination model, the key issues are: what metrics should be considered and how to determine the optimal weights.
Select Multimedia Metrics {#sec:multimedia_metrics}
-------------------------
As the user is the potential customer to the advertiser, any ad that can deliver the advertiser’s information to the user can be viewed as an effective ad. It has been recognized that the user’s acceptance of a message is sequence of psychology process: attention, comprehension, yielding, retention and action [@alwitt1985psychological]. Motived by this finding, we suggest the following three multimedia metrics that can help improve the effectiveness of online advertising.
**Contextual relevance** has been widely discussed in computational multimedia advertisement [@mei2010contextual]. Contextual relevance refers to the similarity between the content of the host webpage and the ad description provided by the advertiser. It has been proved to be effective in increasing the ad CTR and meanwhile maintaining user’s online experience.
**Visual saliency** measures whether the ad image can be easily spotted out within the host webpage. Through a series of eye-tracking experiments, recent research found that: users intentionally avoid looking at the banner ads [@sajjacholapunt2014influence], and they tend to ignore ads located on the bottom and right area [@owens2011text]. By introducing visual saliency, we can increase the probability that the user will notice the displayed ad intentionally or unintentionally.
**Image memorability** measures how likely the user will recall the ad correctly. It is an important metric for ad branding conception. It has been shown that image memorability is a stable and intrinsic property of images that is shared across different viewers. By introducing image memorability, the advertiser’s benefit will be improved.
To demonstrate the effectiveness of our introduced multimedia metrics, we proposed a novel multimodal framework for contextual video advertising [@xiang2015salad]. Given an online video and a set of ad candidates, we first roughly identify a set of relevant ads based on textual information matching. We then carefully select a set of candidates based on the visual content matching. In this regards, our selected ads are contextually relevant to the video in terms of both textual information and visual content. We finally select the most salient ad as the most appropriate one. We further extend the proposed framework by considering the image memorability.
In the user-study experiment, we selected 20 popular YouTube videos as our test video set, and we collected the products information from Amazon and Taobao to construct the ad dataset. We invited 80 participants to join our experiment, and we used an eye-tracker device to record their eye gaze information. Since user’s behavior is sensitive to the purpose of the experiment, they did not have prior knowledge about the experiment. We also built a browser interface which is similar to YouTube to simulate the real-world online video viewing environment. The experimental result is shown in Fig. \[fig:user\_study\]. By comparing the saliency-based advertising strategy to the text-based and visual-based strategy, we can find that: the average fixation duration on the displayed ad increases with introducing visual saliency. By comparing the memorability enhanced advertising to the text-based and visual-based strategy, we can find that: the average ad recall rate increases with considering image memorability. By comparing the memorability-enhanced advertising to saliency-based advertising, we observe an interesting finding: higher fixation duration does not ensure higher ad recall rate. This finding indicates that the memorability is an intrinsic property of images, which is consistent with the results in [@khosla2015understanding].
Optimal Re-ranking Model {#sec:re-ranking}
------------------------
Besides the multimedia metrics discussed in the previous section, we further consider three metric variables that is commonly discussed in online advertising: the publisher’s revenue, the advertiser’s utility and the ad CTR. Revenue is always the primary concern in display advertising. If advertiser is selected as the winning bidder, the revenue is her paid price, which equals to the bidding price of the next bidder. The utility shows the advertiser’s cost saving. It is defined as the difference between her value and payment. The ad CTR is the more direct variable to show the effectiveness of online advertising. It can be obtained from the historical data. Table \[tab:second\_stage\_var\_spec\] specifies all the metric variables as the inputs of the second stage.
All advertisers are re-ranked in the second stage. Given a set of advertisers in auction $z$,the rank score of advertiser $i, i \in z$, is defined as follows: $$\begin{aligned}
rs_{i, z} = & \ \sum_{k \in K} \omega_k x_{k, i, z},\end{aligned}$$ where $x_{k, i, z}$ is the input value of metric variable $k$, $\omega_k$ is the corresponding weight.
The optimal weights can be obtained from the historical data. The weights of variables in the rank score represent the publisher’s preferences towards the metrics. As described earlier, the publisher needs to sacrifice a certain amount of revenue in the short term in order to increase the benefits of other stakeholders. Let $\theta_{1}$ be the maximum pre-determined loss rate of revenue and let $\theta_{k},\forall k \in K\backslash\{1\}$, be the minimum increase rate of other variables. Given the maximum loss and the minimum increase targets, we can obtain the optimal weights from the training set $\widetilde{Z}$.
Our algorithm can be expressed as follows: $$\begin{aligned}
\max_{\omega_1, \cdots,\omega_6}
& \ \sum_{z \in \widetilde{Z}} rs_{*,z}, \\[0.02in]
\text{subject to}
& \ 0 \leq \omega_{k} \leq 1, \forall k \in K, \label{eq:weight_nonnegative}\\[0.02in]
& \ \sum_{k \in K} \omega_{k} = 1, \label{eq:weight_unity}\\[0.05in]
& \ |\xi_{1, \widetilde{Z}}| \leq |\theta_{1}|, \theta_1 \leq 0, \label{eq:threshold_revenue} \\[0.02in]
& \ \xi_{k, \widetilde{Z}} \geq \theta_{k}, \theta_k \geq 0, \forall k \in K\backslash\{1\}, \label{eq:threshold_others} \end{aligned}$$ where $rs_{*,z}$ is the rank score of the selected advertiser in auction $z$ by our proposed model, $\xi_{k, \widetilde{Z}}$ is the mean of changes of variable $k$ between our proposed model and the ground truth, defined by $$\xi_{k, \widetilde{Z}} = \frac{\sum_{z\in \widetilde{Z}}(x_{k, *, z}-x_{k,\neg,z})}{\sum_{z \in \widetilde{Z}}x_{k,\neg, z}}, \label{eq:changes_of_variables}$$ where $x_{k,*,z}$ is the input value of metric variable $k$ for the selected advertiser in auction $z$ by our proposed model, $x_{k,\neg,z}$ is the input value of metric variable $k$ for the selected advertiser in auction $z$ in the ground truth.
The optimal weights maximize the sum of rank scores of the select advertisers from all auctions in the training set. Eqs. (\[eq:weight\_nonnegative\])-(\[eq:weight\_unity\]) ensure each variable has an impact in the re-ranking but its impact has an upper bound. Eqs. (\[eq:threshold\_revenue\])-(\[eq:threshold\_others\]) further specify the lower bounds of trade-offs where the maximum decrease of the publisher’s revenue and the minimum increases for other variables.
Experiment {#sec:experiments}
==========
To validate the proposed framework, both bidding information (e.g., bid price, ad CTR) and multimedia information (e.g., text for contextual relevance matching, ad image for visual saliency and image memorability) for each ad are needed. We use two independent datasets to simulate the real-world advertising system: the multimedia dataset (data collected from AOL and YouTube) represents the interaction between the publisher and the user, and the auction dataset shows the interplay between the publisher and the advertiser. We compare the proposed framework with the GSP auction model, which has been widely used in existing RTB system. Since revenue is always the major concern in online advertising, it will be interesting to investigate how the loss of revenue will affect the other metrics. In this experiment, we set $\theta_1$ to a non-positive value and $\theta_k=0,k=2\dots6$. We conduct 10-fold cross validation on the both AOL and YouTube dataset and the results are shown in Fig \[fig:theta\_1\_variables\].
From this figure, we can conclude that there exist trade-offs among stakeholders in the proposed framework, and the trade-offs change with the publisher’s specific needs (the threshold values $\theta_k,k=1\dots6$). We have the following observations: 1) the changes of variables in the test set is similar to that of the training set on both AOL and YouTube multimedia datasets. The consistent results confirm our cross-validation; 2) when $\theta_1$ is close to 0, all variables remain unchanged. This is because our weight determination model (see section \[sec:re-ranking\]) is not able to find a solution to the constrains. In other words, the publisher is too stingy to lose revenue and too greedy to achieve improvement on the other variables. In this scenario, we will use the traditional RTB system to select the ad with highest bid price; 3) when $\theta_1$ further decreases, the revenue follows a monotone decreasing pattern and the other variables increase to various degrees. According to our optimal weights determination model, the solution space will increase with the decrease of $\theta_1$. We will select the optimal weights with the highest accumulated rank score; 4) when $\theta_1$ decreases below a threshold value, the changes of metrics remains stable. Note that, the optimal weights changes with the given threshold values. Through analyzing the historical data and the changes of variables will help the publisher decide a threshold of revenue loss.
{width="0.95\linewidth"}
\[fig:theta\_1\_variables\]
Work in progress {#sec:work_in_progress}
================
The framework described above is only applicable to the situation where each webpage has a single ad slot. We are currently working on a generalized framework where a webpage has multiple ad slots. Recently, we have constructed another multimedia dataset from Yahoo and MSN. We find that: the percentage of single slot webpage is about 18.3% (3,390 out of 18,447) in Yahoo, and 13.8% (1,484 out of 10,717) in MSN, respectively. The multi-slot webpage makes the computational framework more complex. In the current RTB setting, these ad slots are sold separately through individual auctions [@Ben_Zwi_2015]. The following scenarios are likely to happen: 1) competitive ads are displayed on the same webpage (e.g., Apple Iphone 7 and Samsung Galaxy S8 are displayed in the same webpage about mobile techniques). The competitive ads have a negative effect on user’s brand conception [@burke1988competitive]; 2) an ad with a higher bid price will be placed at a less salient slot. The eye-tracking experiment has shown that user’s web surfing eye-gaze follows the ’F’ shape [@jakob2006f]. This finding indicates the importances of different ad slot locations within the webpage. We will propose a computational framework that considers both the webpage-ad context and the ad-ad pair context. The limitations above can be acted as the hard/soft constraints.
Conclusion {#sec:conclusion}
==========
In this paper, we propose a computational framework to improve the effectiveness of display advertising by incorporating multimedia metrics. The metrics are: the contextual relevance to ensure user’s online experience as well as increase the ad CTR, the visual saliency to draw user’s attention and the image memorability to enhance user’s brand conception. We first conduct an eye-tracking experiment to demonstrate the effectiveness of the introduced metrics. After that, we discuss how to integrate the multimedia metrics into RTB. We also propose an optimization model to determine the optimal weights of the metrics. Our experimental results show that the proposed framework is able to increase the benefits of the advertiser and the user with just a slight decrease in the publisher’s revenue. However, better engagements of advertisers and users will increase the demand of advertising and supply of webpage visits, which will boost the publisher’s revenue in the long run.
Acknowledgement
===============
This research is supported by the National Research Foundation, Prime Minister’s Office, Singapore under its International Research Centre in Singapore Funding Initiative.
|
---
abstract: 'We introduce the concept of a quantum background and a functor ${\operatorname{\mathbf{QFT}}}$. In the case that the ${\operatorname{\mathbf{QFT}}}$ moduli space is smooth formal, we construct a flat quantum superconnection on a bundle over ${\operatorname{\mathbf{QFT}}}$ which defines algebraic structures relevant to correlation functions in quantum field theory. We go further and identify chain level generalizations of correlation functions which should be present in all quantum field theories.'
address:
- 'Dept. of Mathematics, Yonsei University, Seoul 120-749'
- 'Queens College, City University of New York, Flushing NY 11367'
- 'NYC College of Technology, City University of New York, Brooklyn NY 11201'
author:
- 'Jae-Suk Park'
- John Terilla
- Thomas Tradler
title: 'Quantum backgrounds and ${\operatorname{\mathbf{QFT}}}$'
---
[^1]
[^2]
Introduction {#Introduction}
============
The work described in this paper grew from an effort to understand quantum field theory mathematically. In particular, we sought to understand the way in which deformations of the action functional of a quantum field theory control the correlation functions of the theory. Our efforts led us to consider what we call *quantum backgrounds*, or just *backgrounds* for short. Quantum backgrounds were invented by the first author and introduced during a sequence of lectures [@js2] in 2004 extending his work on flat families of quantum field theories [@js1]. We believe they provide the proper algebraic setting in which local deformations of quantum field theories should be discussed. A background $B$ is a four tuple $B=(P,m,N,\varphi)$ where $P$ is a graded noncommutative ring, $m$ is a degree one element of $P$ satisfying $m^2=0$, $N$ is a graded left $P$ module, and $\varphi$ is a degree zero element of $N$ satisfying $m\varphi=0.$ Additionally, we require that $P$ and $N$ be free $k[[\hbar]]$ modules for a fixed field $k$, and that $P$ be a $k[[\hbar]]$ algebra with $P/\hbar P$ commutative. The condition $m\varphi=0$ is analogous to having specified an initial action functional that satisfies the quantum master equation. In this paper, we develop our ideas about backgrounds alongside a condensed overview of deformation theory; we do this for two reasons.
The first reason is that physical information (such as correlation functions, our generalized correlation functions, and generalized Ward identities) are extracted from the background data with the aid of a set valued functor that is comparable to the classical deformation functor. Essentially, we are interested in the equation $$\label{muphi}
mU\varphi=0$$ where $U$ is an invertible element of $P$ that depends on parameters. We supply two interpretations: a Schrödinger interpretation where $U\varphi$ is regarded as an evolution of $\varphi$, and a Heisenberg interpretation where $U^{-1}mU$ is regarded as an evolution of $m$. From either perspective, the evolution $U$ is not arbitrary and the way it is constrained is intimately tied to the physics of the theory. If the evolution is unobstructed, the constraints provides enough relations to determine the correlation functions up to finite ambiguity. While one might think of $U\phi$ as a deformation of $\phi$, or $U^{-1}mU$ as a deformation of $m$, we prefer the interpretation as an evolution since that concept, along with the definition of a quantum background itself, is evocative of a sort of *graded* quantum mechanics à la Dirac [@Dirac], but it is a quantum mechanics in the moduli space of action functionals for quantum field theories having the same fields.
In this paper, we make restrictions on the form of $U$ and the type of parameter space over which $\varphi$ evolves. Recall the Maurer-Cartan equation $d\gamma+\frac{1}{2}[\gamma, \gamma]=0$ in a differential graded Lie algebra $L$ leads to a covariant set valued functor of parameter rings ${\operatorname{\mathbf{Def}}}_L$ where ${\operatorname{\mathbf{Def}}}_L(A)$ is the set of solutions to the Maurer-Cartan equation with parameters in the ring $A$, modulo a natural equivalence. In the same way, the quantum master equation $mU\varphi=0$ for a background $B$ leads to a set-value functor of $k[[\hbar]]$ parameter rings ${\operatorname{\mathbf{QFT}}}_B$. Standard deformation theory, enriched to account for $\hbar$, is used to prove (Theorem \[main\_thm\]) that the functor ${\operatorname{\mathbf{QFT}}}_B$ is (pro, homotopy) representable by a differential algebra $R$ and a versal solution $U$ to Equation with parameters in $R$. Recall that a functor ${\mathbf{F}}$ is representable by a ring $R$ if it is naturally equivalent to the functor $\hom(R,\,)$, in which case the identity homomorphism ${\operatorname{Id}}:R\to R$ corresponds to an element $U$, which is a versal element in ${\mathbf{F}}(R)$, modulo the obstructions encoded by the differential in $R$. Here, the evolution of $\varphi\leadsto
U\varphi$ resembles a Schrödinger wave function over the space of theories; or $U\varphi$ might be compared to a versal action functional over the space of action functionals.
The main construction in the paper applies when ${\operatorname{\mathbf{QFT}}}_B$ is smooth formal, which in algebraic terms means that the differential in the representing algebra $R$ vanishes; equivalently, there exist no obstructions to constructing a versal action functional (see Definition \[smoothformal\]). In geometric terms, smooth formal means the initial background data comprises a smooth point in the formal moduli space. In this case, we construct (Section \[Flat\_Connection\]) a quantum flat superconnection on a bundle of over the moduli space $$\nabla:D_\Pi\to D_\Pi \otimes_R \Omega$$ Here, $D_\Pi$, $R$, and are the the tangent bundle, ring of functions, and space of differential forms on the moduli space of solutions to Equation , all are free $k[[\hbar]]$ module equivalents of the usual notions in formal geometry. We call the map $\nabla$ a quantum superconnection because: (i) $\nabla$ depends on $\hbar$ and satisfies the $\hbar$-connection equation $$\label{conn_eq}
\nabla(fe)=\hbar(df)e+ (-1)^{|f|} f\nabla(e) \text{ for $f\in R$ and
a section $e$};$$ and is a Quillen-type superconnection since the target involves all of $\Omega$ and not only $\Omega^1$. We prove (Theorem \[conn\_thm\]) that $\nabla^2=0$; i.e., $\nabla$ is flat. The connection one-form part of $\nabla$, call it $\nabla^{1}:D_\Pi \to D_\Pi \otimes_R
\Omega^1$, defines a flat, torsion free connection, which taken together with $\hbar$ in Equation , is similar to the pencil of flat connections on the tangent bundle of a moduli space of topological conformal field theories [@Dub]. But $\nabla^{1}$ will, in general, depend on $\hbar$. There is some evidence for the existence of special coordinates [@js3; @jt2] in which the $\hbar$ dependence of $\nabla^{1}$ reduces it to a pencil of flat connections, equipping the moduli space with the structure of a representation of the operad $H(\overline{\mathcal{M}}_{0,n})$, but this is not proved in the present paper. One can think of the flat quantum superconnection $\nabla$ as furnishing a sort of Frobenius infinity manifold structure to the moduli space associated to a background.
In the physics language, $\nabla^{1}$ encodes, up to a choice of one point functions, all of the $n$ point correlation functions for $n>1$ of all the theories in the moduli space. We have the compelling metaphor: a theory of graded quantum mechanics over the moduli space of quantum field theories is tantamount to the correlation functions of the theories in the space. This is accomplished without defining a measure for the path integral. From our point of view, the path integral amounts to nothing more than a choice of the one point functions, essentially a choice of basis of observables. The connection $\nabla^{1}$ determines the two point functions for every theory in a neighborhood of the moduli space, and the $n$-point functions for $n>2$ are obtained via products and derivatives in the moduli directions. The rest of the superconnection, $\nabla^k$ for $k>1$, defines a homotopy algebraic structure which is a chain level generalization of correlation functions which, to our knowledge, is new in physics. We offer an analogy with differential graded algebras: compare the correlation functions derived from $\nabla^1$ to the associative product in the homology of a differential graded algebra and compare our chain-level generalized correlation functions derived from $\nabla^k$ for $k>1$ to a minimal $A_\infty$ structure defined in the homology of a dga that is quasi-isomorphic to the original dga. We suspect that the quantum background is determined up to quasi-isomorphism by the flat quantum superconnection $\nabla$, but we do not pursue this idea in this paper because $\nabla$ is defined here only for smooth formal backgrounds and so it’s difficult to determine the extent to which $\nabla$ can be interpretted as a “minimal model” for the background until obstructed backgrounds are better understood. Also, we acknowledge that our definition of “quasi-isomorphism” for quantum backgrounds may need some refinement—in particular, may need to encompass the concept of special coordinates.
The second reason that we develop ${\operatorname{\mathbf{QFT}}}$ alongside ${\operatorname{\mathbf{Def}}}$ is our sense that ${\operatorname{\mathbf{QFT}}}$ generalizes ${\operatorname{\mathbf{Def}}}$. The relationship is epitomized in Section \[Application\], where we discuss an example related to differential BV algebras. The deformation theory of quantum field theories (i.e., the functor of $k[[\hbar]]$ algebras ${\operatorname{\mathbf{QFT}}}_B$ for a background $B$) is richer than classical deformation theory (i.e., the functor of $k$-algebras ${\operatorname{\mathbf{Def}}}_L$ for an $L_\infty$ algebra $L$). The fact that when ${\operatorname{\mathbf{QFT}}}_B$ is a smooth functor, the moduli space admits the quantum flat connection encoding correlation functions and homotopy correlation functions is evidence that ${\operatorname{\mathbf{QFT}}}_B$ is richer since the ${\operatorname{\mathbf{Def}}}_L$-moduli space does not naturally carry such an additional structure. The definition of a background is quite general and might be constructed from a wide variety of mathematical data. Our attitude is that the functor ${\operatorname{\mathbf{QFT}}}_B$ provides a notion of quantum deformation theory that in some examples extends the classical deformation theory in the $\hbar$ direction. When it is possible, one should attribute a background $B$ to some initial mathematical data under investigation, rather than an $L_\infty$ algebra $L$ defined over $k$, and study functors of $k[[\hbar]]$-algebras. Our expectation is that correlations, at least in special coordinates, arising from ${\operatorname{\mathbf{QFT}}}_B$ reveal invariants of the initial data that is more subtle than what can be extracted from ${\operatorname{\mathbf{Def}}}_L$ as a functor of $k$-algebras. However, in order to make a precise statement about ${\operatorname{\mathbf{QFT}}}$ generalizing ${\operatorname{\mathbf{Def}}}$, one should have a theory of special coordinates and perhaps generalize even further the kind of parameter rings on which ${\operatorname{\mathbf{QFT}}}_B$ can be defined. One might think of ${\operatorname{\mathbf{Def}}}_L$ as the functor that controls deformations of objects over (formal, graded, differential) commutative spaces whereas ${\operatorname{\mathbf{QFT}}}_B$ controls deformations of objects over something like “quantized spaces.” We will expand on this point of view in another paper.
We thank Michael Schlessinger, James Stasheff, and Dennis Sullivan for many helpful and inspiring conversations and, in particular to Dennis Sullivan, for generous and supportive working conditions at CUNY. The authors also thank the referee for excellent suggestions: what clarity there is in the description of the flat quantum superconnection is due to the referee’s recommendations, and while we were not able, in the time allotted, to convert ourselves to the “functors from rings to simplicial sets” point of view, we recognize that it is a very good suggestion we intend to pursue it.
Categories, functors, and classical deformation theory {#Primer}
======================================================
This section is essentially a technical primer about deformation theory approached via functors of parameter rings. The reader may wish to skip directly to section \[Quantum\_Backgrounds\] where our new ideas are introduced.
Categories {#Categories}
----------
First, we review some of the tools we employ. In order to describe the moduli space of quantum backgrounds, we use functors of parameter rings that are differential $k[[\hbar]]$-algebras, in addition to being differential graded local Artin rings.
### Parameter rings
We will be working with certain parameter rings, which we now describe. Fix a field $k$. Although we have interest in the case that $k$ has characteristic $p$, let us assume for this paper that the characteristic of $k$ is zero. A graded Artin local algebra is defined to be a graded commutative associative unital algebra over $k$ with one maximal ideal satisfying the ascending chain condition. Such algebras are those local algebras whose maximal ideals are nilpotent finite dimensional graded vector spaces. We use ${\mathfrak{m}}_A$ to denote the maximal ideal of an Artin local algebra $A$. The quotient $A/{\mathfrak{m}}_A$ is a field called the residue field of $A$. A differential on $A$ is defined to be a degree one derivation $d:A \to A$ satisfying $d^2=0$. We denote the homology of $A$ by $H(A):={\operatorname{Ker}}(d)/{\operatorname{Im}}(d)$.
Let ${\mathfrak{C}}$ be the category whose objects are differential graded Artin local algebras with residue field $k$ such that $[1]\neq [0]$ in $H(A)$; morphisms are local homomorphisms, i.e. differential graded algebra maps $A\to B$ such that ${\operatorname{Im}}({\mathfrak{m}}_A)={\mathfrak{m}}_B$ and the induced maps on residue fields is the identity. Let ${\widehat{\mathfrak{C}}}$ denote the category whose objects consist of differential graded complete noetherian local algebras $A$ for which $A/({\mathfrak{m}}_{A})^j \in {\mathfrak{C}}$ for every $j$ and whose morphisms are local. Note objects in ${\widehat{\mathfrak{C}}}$ are colimits of objects in ${\mathfrak{C}}$: $A=\underleftarrow{\lim}\, A/({\mathfrak{m}}_{A})^j $.
Let $\Lambda\in {\operatorname{ob}}({\widehat{\mathfrak{C}}})$, concentrated in degree zero with zero differential, and denote the maximal ideal ${\mathfrak{m}}_{\Lambda}$ by $\mu$. We define the category ${\mathfrak{C}}_{\Lambda}$ to be the subcategory of ${\mathfrak{C}}$ consisting of differential graded Artin local $\Lambda$-algebras; morphisms in ${\mathfrak{C}}_\Lambda$ are local differential $\Lambda$-algebra homomorphisms. Let ${\widehat{\mathfrak{C}}}_\Lambda$ be the category of projective limits of ${\mathfrak{C}}_\Lambda$. We will be concerned primarily with $\Lambda=k[[\hbar]]$ or $\Lambda=k$ itself (and ${\mathfrak{C}}_\Lambda={\mathfrak{C}}$).
We call objects in ${\mathfrak{C}}_\Lambda$ parameter rings.
We have functors (“free”, “zero action”, and “underlying”) $$\label{fzu}
{\mathbf{fr}}:{\widehat{\mathfrak{C}}}\to
{\widehat{\mathfrak{C}}}_\Lambda,\quad {\mathbf{z}}:{\widehat{\mathfrak{C}}}\to {\widehat{\mathfrak{C}}}_\Lambda, \text{ and }{\mathbf{un}}:{\widehat{\mathfrak{C}}}_\Lambda \to
{\widehat{\mathfrak{C}}}$$ defined by ${\mathbf{fr}}(A)=A\hat{\otimes}_k \Lambda$, ${\mathbf{z}}(A)=A$ where the action of $\Lambda$ on $A$ is given by $\lambda a=0$ for $\lambda \in
\mu$ and $a\in A$, and ${\mathbf{un}}(A)=A$, the underlying Artin algebra with the $\Lambda$ action forgotten. Note that ${\mathbf{un}}\circ {\mathbf{z}}={\operatorname{Id}}:{\widehat{\mathfrak{C}}}\to
{\widehat{\mathfrak{C}}}$.
### Tensor and fiber products of objects
For $A,B\in {\operatorname{ob}}({\widehat{\mathfrak{C}}}_\Lambda)$, we define $A\otimes B:=
A\widehat{\otimes}_\Lambda B\in {\operatorname{ob}}({\widehat{\mathfrak{C}}}_\Lambda)$ with grading determined by $\deg(a\otimes b)=\deg(a)+\deg(b)$ and differential $d$ defined as $d(a \otimes b)=d(a)\otimes b + (-1)^{\deg(a)}a \otimes
d(b)$, for homogeneous $a\in A$, $b\in B$.
Let $A,B,C \in {\operatorname{ob}}({\mathfrak{C}}_\Lambda)$ and $\alpha\in \hom(A,C)$ and $\beta
\in \hom(B,C)$. We define the fibered product $A\times_C B \in
{\operatorname{ob}}({\mathfrak{C}}_\Lambda)$ by $A \times_C B=\{(a,b):\alpha(a)=\beta(b)\}$. In ${\widehat{\mathfrak{C}}}_\Lambda$, fibered products do not exist since $A \times_C B$ may fail to be noetherian.
### Homotopy equivalence of morphisms and small extensions
Let us now recall the homotopy model for the interval $I$.
Define $I{\operatorname{ob}}({\widehat{\mathfrak{C}}}_\Lambda)$ by $I=k[[u,v]]$, $\deg(u)=0$, $\deg(v)=1$, and $d(p(u)+q(u)v)=p'(u)v.$
One cannot evaluate an arbitrary element of $I$ at particular values of $u$ and $v$, but for the (non-Artin) subalgebra $I'$ consisting of polynomials in $u$ and $v$ and one does have evaluation maps $ev_i:I'\to k$ and $ev_i:{\mathbf{fr}}(I')\to \Lambda$, where $ev_i$ is evaluation at $(i,0)$ for $i=0,1.$
Two morphisms $\tau_0,\tau_1:A \to B$ in ${\widehat{\mathfrak{C}}}_\Lambda$ are called homotopy equivalent, or homotopic, if there exists an Artin subalgebra $B'\subset B \otimes {\mathbf{fr}}(I')$ and a morphism $\tau(u,v):A\to B'$ in ${\widehat{\mathfrak{C}}}_\Lambda$ satisfying $\tau(0,0):=ev_0\tau(u,v)=\tau_0$ and $\tau(1,0):=ev_1\tau(u,v)=\tau_1.$ As suggested by the terminology, homotopy equivalence is an equivalence relation. We denote the set of homotopy equivalence classes of morphisms $A \to B$ by ${\operatorname{hot}}(A,B).$
For $\mathfrak{A}={\mathfrak{C}}_\Lambda,{\widehat{\mathfrak{C}}}_\Lambda$, we define the category $[\mathfrak{A}]$ to be the homotopy category of $\mathfrak{A}$. That is, the category defined by ${\operatorname{ob}}([\mathfrak{A}])= {\operatorname{ob}}(\mathfrak{A})$ and ${\operatorname{mor}}(A,B)={\operatorname{hot}}(A,B)$.
Let $\rho:A \to B$ be a surjective morphism in ${\mathfrak{C}}_\Lambda$.
We say $\rho$ is a small extension if ${\operatorname{Ker}}(\rho)$ is a nonzero ideal $J$ such that ${\mathfrak{m}}_A J=0$. We say that $\rho$ is an acyclic small extension if, in addition, $H(J)=0$.
Functors of parameter rings {#Functors}
---------------------------
We now recall some basic definitions and properties of set valued functors of parameter rings. For details see [@S; @M2].
### Representability
Any covariant functor ${\mathbf{F}}:{\mathfrak{C}}_\Lambda \to {\mathfrak{Sets}}$, the category of sets, can be extended naturally to a functor ${\mathbf{F}}:{\widehat{\mathfrak{C}}}_\Lambda \to {\mathfrak{Sets}}$ by $F(A)=\underleftarrow{\lim}\, F(A/{\mathfrak{m}}_A^j).$ A functor ${\mathbf{F}}:{\widehat{\mathfrak{C}}}_\Lambda \to {\mathfrak{Sets}}$ satisfying $F(A)=\underleftarrow{\lim}\,F(A/{\mathfrak{m}}_A^j)$ is called continuous. We consider continuous, covariant functors ${\mathbf{F}}:{\widehat{\mathfrak{C}}}_\Lambda \to {\mathfrak{Sets}}$ satisfying ${\mathbf{F}}(k)=F(\Lambda)=\{\text{one point}\}$. Such functors form a cateogry ${\mathfrak{Fun}}_\Lambda$ whose morphisms are natural transformations. In the case that $\Lambda=k$, we will denote ${\mathfrak{Fun}}_\Lambda$ simply by ${\mathfrak{Fun}}$.
Precomposition with ${\mathbf{fr}},{\mathbf{z}},$ and ${\mathbf{un}}$ defined in gives functors (which we denote by the same letters) $$\label{fzu2}
{\mathbf{fr}}:{\mathfrak{Fun}}_\Lambda \to {\mathfrak{Fun}}, \quad {\mathbf{z}}:{\mathfrak{Fun}}_\Lambda
\to {\mathfrak{Fun}}\text{ and } {\mathbf{un}}:{\mathfrak{Fun}}\to {\mathfrak{Fun}}_\Lambda.$$
We define a couple for ${\mathbf{F}}\in {\mathfrak{Fun}}_\Lambda$ to be a pair $(A,\xi)$ where $A\in {\widehat{\mathfrak{C}}}_\Lambda$ and $\xi \in {\mathbf{F}}(A).$
For $R\in {\widehat{\mathfrak{C}}}_\Lambda$, we define $h_R:{\widehat{\mathfrak{C}}}_\Lambda \to {\mathfrak{Sets}}$ by $h_R(A)=\hom_\Lambda (R,A)$, and define $[h_R]:[{\widehat{\mathfrak{C}}}_\Lambda]\to {\mathfrak{Sets}}$ by $[h_R(A)]={\operatorname{hot}}_\Lambda(R,A).$
We say that ${\mathbf{F}}\in {\mathfrak{Fun}}_\Lambda$ is represented by $R$ if ${\mathbf{F}}$ is isomorphic to $h_R$ for some $R\in {\widehat{\mathfrak{C}}}_\Lambda$. We say that ${\mathbf{F}}\in {\mathfrak{Fun}}_\Lambda$ is homotopy represented by $R$ if ${\mathbf{F}}$ induces a functor $[{\widehat{\mathfrak{C}}}_\Lambda]\to {\mathfrak{Sets}}$ that is represented by $[h_R]$.
In the original terminology [@S], Schlessinger called representable functors and couples “pro-representable” and “pro-couples”—we drop the prefix. Homotopy representable functors were treated by Manetti in [@M2] and Schlessinger-Stasheff in [@SS].
If ${\mathbf{F}}$ is represented by $R$, then under the identification $\hom(R,R)\simeq {\mathbf{F}}(R)$, there is an element $\Pi\in
{\mathbf{F}}(R)$ corresponding to ${\operatorname{Id}}\in \hom(R,R).$ The element $\Pi$ is versal in the sense that for all $A\in {\widehat{\mathfrak{C}}}_\Lambda$, and for all $\xi \in {\mathbf{F}}(A)$, there exists a unique morphism $\tau:R \to A$ so that $F(\tau)(\Pi)=\xi.$ One may say that ${\mathbf{F}}$ is represented by the couple $(R,\Pi)$.
Likewise, if ${\mathbf{F}}$ is homotopy represented by $R$, then under the identification ${\operatorname{hot}}(R,R)\simeq {\mathbf{F}}(R)$, there is an element $\Pi\in
{\mathbf{F}}(R)$ corresponding to $[{\operatorname{Id}}]\in {\operatorname{hot}}(R,R)$. The element $\Pi$ is versal in the sense that for all $A\in {\widehat{\mathfrak{C}}}_\Lambda$, and for all $\xi \in {\mathbf{F}}(A)$, there exists a morphism $\tau:R \to A$ unique up to homotopy, so that $F(\tau)(\Pi)=\xi.$ One may say that ${\mathbf{F}}$ is homotopy represented by the couple $(R,\Pi)$.
### Smoothness and quasi-smoothness
We say ${\mathbf{F}}\in {\mathfrak{Fun}}_\Lambda$ is smooth if and only if the map ${\mathbf{F}}(\tau):{\mathbf{F}}(A)\to {\mathbf{F}}(B)$ is surjective for all small extensions $\tau:A \to B$.
We say ${\mathbf{F}}\in {\mathfrak{Fun}}_\Lambda$ is quasi-smooth if and only if the map ${\mathbf{F}}(\tau):{\mathbf{F}}(A)\to {\mathbf{F}}(B)$ is surjective for all acyclic small extensions $\tau:A \to B$.
### The Schlessinger condition
Let $A,B,C \in {\operatorname{ob}}({\mathfrak{C}}_\Lambda)$ and $\alpha\in \hom(A,C)$, $\beta \in
\hom(B,C)$ and consider the fibered product $A\times_C B$. For any functor ${\mathbf{F}}\in {\mathfrak{Fun}}_\Lambda$, the versal property of fibered products (of sets) gives a map $$\label{the_eta_map}
\eta:{\mathbf{F}}(A \times_C B) \to {\mathbf{F}}(A)\times_{{\mathbf{F}}(C)} {\mathbf{F}}(B)$$
We say that ${\mathbf{F}}\in {\mathfrak{Fun}}_\Lambda$ satisfies the Schlessinger condition if the map $\eta$ in equation is a bijection.
Readers familiar with Schlessinger’s work may recognize what we call the Schlessinger condition as a version of his condition (H4) (see theorem 2.11 in [@S]). This condition has also been called the *Mayer-Vietoris property* [@Maz] because of the relationship to abstract homotopy theory [@B1].
### Homotopy equivalence {#simdefined}
Let ${\mathbf{G}}\in {\mathfrak{Fun}}_\Lambda$ and $A\in {\operatorname{ob}}({\mathfrak{C}}_\Lambda)$, we define a relation on the set ${\mathbf{G}}(A)$ by setting $\xi_0 \sim \xi_1$ if and only if there exists an Artin subalgebra $A'\subset A\otimes {\mathbf{fr}}(I')$ and a $\xi(u,v) \in {\mathbf{G}}(A')$ with $\xi(0,0)=\xi_0$ and $\xi(1,0)=\xi_1.$ We’re using the shorthand $\xi(i,0)$ for ${\mathbf{G}}(ev_i)(\xi(u,v))$ where $ev_i:A \otimes {\mathbf{fr}}(I') \to A$ is the evaluation map determined by $(u,v)=(i,0)$ for $i=0,1$. We refer the reader to [@M2] for details.
\[equiv\_thm\] If ${\mathbf{G}}$ is quasi-smooth and satisfies the Schlessinger condition, then $\sim$ is an equivalence relation and the functor ${\mathbf{F}}\in
{\mathfrak{Fun}}_\Lambda$ defined by ${\mathbf{F}}(A)={\mathbf{G}}(A)/{\sim}$ induces a functor on the category $[{\widehat{\mathfrak{C}}}_\Lambda]$.
The nontrivial, but standard, part is proving that the homotopy relation is transitive. (see [@M2] theorem 2.8).
### Tangent space and tangent module
For $i\in {\mathbb{Z}}$, we define the parameter ring $E_i$ by $E_i:=k[\epsilon_i]/\epsilon_i^2$ where $\epsilon_i$ has degree $-i$ and $d(\epsilon_i)=0$.
Then, for any ${\mathbf{F}}\in {\mathfrak{Fun}}_\Lambda$, we define the tangent space to ${\mathbf{F}}$ by $$T_{\mathbf{F}}=\oplus_i T_{{\mathbf{F}}}^i:=\oplus_i {\mathbf{F}}(E_i).$$
If ${\mathbf{F}}$ commutes with certain products of rings with trivial multiplicative structures then $T_{\mathbf{F}}$ naturally has the structure of a graded vector space. Specifically, if $$\label{vs_objects}
{\mathbf{F}}(A \times_k B)\simeq {\mathbf{F}}(A)\times {\mathbf{F}}(B)$$ for all rings $A$ and $B$ with ${\mathfrak{m}}_A^2={\mathfrak{m}}_B^2=0$ and $d(A)=d(B)=0$ (the so-called vector space objects in ${\mathfrak{C}}_\Lambda$) then ${\mathbf{F}}(A)$ will be a vector space. For example, if ${\mathbf{F}}$ satisfies the Schlessinger condition, then $T_{\mathbf{F}}$ is a vector space. Also, and importantly, if ${\mathbf{F}}={\mathbf{G}}/{\sim}$ as in proposition \[equiv\_thm\], then $T_{\mathbf{F}}$ is a vector space. Furthermore, ${\mathbf{T}}$ defines a functor from the subcategory of ${\mathfrak{Fun}}_\Lambda$ consisting of those functors satisfying to ${\mathfrak{Vect}}$, the catgory of graded vector spaces.
One can augment the tangent space slightly and obtain a natural $\Lambda$-module structure. For each $\alpha \in \Lambda$, one uses the morphisms $$\oplus_i {\mathbf{fr}}(E_i)=\oplus_i \Lambda[\epsilon_i]/\epsilon_i^2 \overset{\alpha
\cdot}{\longrightarrow} \oplus_i
\Lambda[\epsilon_i]\epsilon_i^2=\oplus_i {\mathbf{fr}}(E_i)$$ to get maps $\alpha:\oplus_i {\mathbf{F}}({\mathbf{fr}}(E_i)) \to \oplus_i {\mathbf{F}}({\mathbf{fr}}(E_i))$.
For any ${\mathbf{F}}\in {\mathfrak{Fun}}_\Lambda$, we define the tangent $\Lambda$ module to ${\mathbf{F}}$ by $$D_{\mathbf{F}}:=\oplus_i D_{\mathbf{F}}^i \oplus_i {\mathbf{F}}({\mathbf{fr}}(E_i)).$$
Like the tangent space, ${\mathbf{D}}$ defines a functor from the subcategory of ${\mathfrak{Fun}}_\Lambda$ consisting of those functors satisfying to ${\mathfrak{Mod}_\Lambda}$, the catgory of $\Lambda$-modules.
\[rep\_thm\] Suppose that ${\mathbf{G}}\in {\mathfrak{Fun}}_\Lambda$ is quasi-smooth and satisfies the Schlessinger condition. Let ${\mathbf{F}}={\mathbf{G}}/{\sim}$ as in proposition \[equiv\_thm\] and suppose that $T_{\mathbf{F}}$ is finite dimensional with basis $\{\xi_i\}$. Let $\{t_i\}$ be the dual basis for $T_{\mathbf{F}}$. Then, there exists a differential $\delta$ on the algebra $R:=\Lambda[[t_i]]$ with $\delta({\mathfrak{m}}_R)\subseteq ({\mathfrak{m}}_R)^2$ and an element $\Xi\in {\mathbf{F}}(R,\delta)$ so that ${\mathbf{F}}$ is homotopy represented by the couple $((R,\delta),\Xi)$.
See theorem 4.5 and corollary 4.6 in [@M2].
Let us recall one more result from the general theory of functors of parameter rings, which provides a theoretical underpinning for the definition of “quasi-isomorphism” in sections \[Def\_Theory\] and \[Quantum\_Backgrounds\].
\[inv\_fun\_thm\] Suppose ${\mathbf{G}},{\mathbf{G}}' \in {\mathfrak{Fun}}_\Lambda$ are quasi-smooth and satisfy the Schlessinger condition. Let $F={\mathbf{G}}/{\sim}$ and $F'={\mathbf{G}}'/{\sim}$ and suppose $\mathbf{n}:{\mathbf{F}}\to {\mathbf{F}}'$ is a natural transformation. Then $\mathbf{n}$ is an isomorphism of functors if and only if $D_{\mathbf{n}}:D_{\mathbf{F}}\to D_{{\mathbf{F}}'}$ is an isomorphism of graded $\Lambda$ modules.
Classical deformation theory {#Def_Theory}
----------------------------
We now recall the basic elements of classical deformation theory. In this subsection we define the category of $L_\infty$ algebras and define a functor ${\operatorname{\mathbf{Def}}}_L$ for each $L_\infty$ algebra $L$. At the end of the section, we highlight the properties of ${\operatorname{\mathbf{Def}}}_L$ (as described in the section \[Functors\] for general functors) as they pertain to $L$.
Now, we set $\Lambda=k$.
### $L_\infty$ algebras
Let $V=\oplus_{j\in \mathbb{Z}} V^j$ be a graded vector space over $k$. As usual, let $V[n]$ denote the shift $V[n]=\oplus_{j\in\mathbb{Z}} V[n]^j$, where $V[n]^j=V^{j+n}$. Let $S^i V$ denote the $i$-th symmetric product of $V$. Note that $\wedge: S^i V[1] \otimes S^j V[1] \to S^{i+j} V[1]$ makes $${\mathrm{C}(V)}:=\oplus_{i=0}^\infty
\mathrm{S}^i(V[1])$$ into a differential graded commutative associative algebra. It is also a graded cocommutative coassociative coalgebra. The coproduct ${\mathrm{C}(V)}\to {\mathrm{C}(V)}\otimes {\mathrm{C}(V)}$ is characterized by the property that it is map of differential graded algebras, and that $x\mapsto x\otimes 1 + 1\otimes x$ for each $x\in V[1]$. Indeed, $C(V)$ is a construction of the free cocommuative coassociative coalgebra over $V[1]$. The versal property characterizing this free construction is: for any graded coalgebra $C$ and any graded linear map $C\to V[1]$, there exists a unique coalgebra map $C \to C(V)$ extending the linear map. This gives an isomorphism of graded vector spaces: $\hom_\mathrm{linear}(C,V)\simeq
\hom_{\mathrm{coalgebra}}(C,{\mathrm{C}(V)}).$
We mention a second identification. Any homomorphism $\sigma :
S^iV[1] \to V$ extends uniquely to a map $\overline{\sigma}\in
{\operatorname{Coder}}({\mathrm{C}(V)})$, and any coderivation $\sigma \in {\operatorname{Coder}}({\mathrm{C}(V)})$ is determined by its components $\underline{\sigma}_i :S^iV[1] \to V$. Thus, $\hom_{\mathrm{linear}}({\mathrm{C}(V)},V)\simeq {\operatorname{Coder}}({\mathrm{C}(V)}).$
An $L_\infty$ algebra $L$ is defined to be a pair $L=(V,D)$ where $V$ is a graded vector space, and $D\in {\operatorname{Coder}}^1({\mathrm{C}(V)})$ is a coderivation of degree $1$, satisfying $D^2=0$. A morphism $\sigma:L\to L'$ between two $L_\infty$ algebras $L=(V,D)$ and $L'=(V',D')$ consists of a degree zero coalgebra map $\sigma:{\mathrm{C}(V)}\to
\mathrm{C}(V')$ with $\sigma\circ D=D'\circ \sigma$. Let us denote the category of $L_\infty$ algebras by ${\mathfrak{L}}$.
We will use the notation $d_i$ for $\underline{D}_i:S^iV[1]\to V$, the $i$-th component of $D$.
### Maurer-Cartan Functor
Let $L=(V,D)\in {\operatorname{ob}}({\mathfrak{L}})$. We define a functor ${\operatorname{\mathbf{MC}}}_L\in {\mathfrak{Fun}}$ by $${\operatorname{\mathbf{MC}}}_L(A)=\{\text{degree $0$ differential coalgebra
maps }{\mathfrak{m}}_A^* \to {\mathrm{C}(V)}\}.$$ Here, ${\mathfrak{m}}_A^*$ is the differential coalgebra ${\mathfrak{m}}_A^*:=\hom({\mathfrak{m}}_A,k)$.
The functor ${\operatorname{\mathbf{MC}}}_L$ satisfies the Schlessinger condition and is quasi-smooth. Therefore, we can form the quotient ${\operatorname{\mathbf{MC}}}_L/{\sim}$, where $\sim$ is the natural gauge equivalence defined in Section \[simdefined\]. Let us make a list of definitions:
\[MC\_remark\] Any coalgebra map ${\mathfrak{m}}_A^* \to {\mathrm{C}(V)}$ is determined by a linear map ${\mathfrak{m}}_A^* \to V$; equivalently, by and element in $V\otimes {\mathfrak{m}}_A$. Then $$\label{MC_remark_eq}
\gamma \in {\operatorname{\mathbf{MC}}}_L \Leftrightarrow d_1(\gamma)+\frac{1}{2!}d_2(\gamma\wedge
\gamma)+\frac{1}{3!}d_3(\gamma\wedge \gamma\wedge \gamma)+\cdots=0.$$ where the $\gamma$ on the right is viewed as an element in $V\otimes
{\mathfrak{m}}_A$ and $d_i:S^i((V \otimes {\mathfrak{m}}_a) [1] \to V\otimes {\mathfrak{m}}_A$ is extended from the components of the $L_\infty$ structure on ${\mathrm{C}(V)}$ using the differential and multiplication in $A$.
Let $L$ be an $L_\infty$ algebra.
1. Define the quotient ${\operatorname{\mathbf{Def}}}_L:={\operatorname{\mathbf{MC}}}_L/{\sim}$ where $\sim$ is the natural gauge equivalence.
2. The assignment $L\mapsto {\operatorname{\mathbf{Def}}}_L$ defines a functor from ${\operatorname{\mathbf{Def}}}:
{\mathfrak{L}}\to {\mathfrak{Fun}}$.
3. We define $H(L)$ to be the vector space $H(L):=T_{{\operatorname{\mathbf{Def}}}_L}[1]$ and call it the homology of $L$. The composition $H:={\mathbf{T}}\circ
{\operatorname{\mathbf{Def}}}$ defines a functor $H:{\mathfrak{L}}\to {\mathfrak{Vect}}$.
In practice, the classical deformation theory of a mathematical object [@S; @K1; @M2] is controlled by an $L_\infty$ algebra $L$. This means that given a mathematical object, for which there exists the concept of a deformation of that object over a parameter ring $A$ and the concept of equivalence of such deformations, there is a bijection of sets $$\left\{
\begin{array}{c}
\text{(equivalence classes of) deformations}\\\text{of the object over
the ring $A$}
\end{array}\right\}
\leftrightarrow {\operatorname{\mathbf{Def}}}_L(A).$$
\[universal\_mc\_thm\] Suppose $L$ is an $L_\infty$ algebra and $T_{{\operatorname{\mathbf{Def}}}_L}=H(L)[-1]$ is finite dimensional with basis $\{\alpha^1, \ldots, \alpha^k\}$. Let $\{s_i\}$ be the dual basis and let $S:=k[[s_1,\ldots, s_k]].$ There exists a differential $d:S\to S$ with $d({\mathfrak{m}}_S)\subset ({\mathfrak{m}}_S)^2$ and $\Gamma \in {\operatorname{\mathbf{MC}}}_L(S)$ such that ${\operatorname{\mathbf{Def}}}_L$ is homotopy represented by the couple $((S,d),\Gamma)$.
Apply proposition \[rep\_thm\] to ${\operatorname{\mathbf{Def}}}_L$.
Let $L$ be an $L_\infty$ algebra and assume $H(L)$ is finite dimensional.
1. We call $L$ finite.
2. A morphism $L \to L'$ is called a quasi-isomorphism if the map ${\operatorname{\mathbf{Def}}}_{L} \to {\operatorname{\mathbf{Def}}}_{L'}$ is an isomorphism of functors, in which case $L$ and $L'$ are called quasi-isomorphic.
3. We call $\Gamma$ in proposition \[universal\_mc\_thm\] a versal solution to the Maurer-Cartan equation.
4. We call $L$ smooth formal if $d=0$ for $d:S\to S$ as in proposition \[universal\_mc\_thm\].
In light of the proposition \[inv\_fun\_thm\], one can assemble the parts of the definition above to say that $L$ and $L'$ are quasi-isomorphic if and only if there exists a morphism $\sigma:L
\to L'$ inducing an isomorphism $H(L) \to H(L')$, which together with a computation establishing $$T_{{\operatorname{\mathbf{Def}}}_L}\simeq {\operatorname{Ker}}({d_1})/{\operatorname{Im}}({d_1}),$$ may be used to give an alternative definition of quasi-isomorphism as a morphism $L\to L'$ inducing an isomorphism in homology.
Quantum backgrounds {#Quantum_Backgrounds}
===================
Quantum backgrounds were invented to provide an algebro-mathematical formulation of quantum field theory. We begin by defining a quantum background (see Definition \[background\_def\]) which one can think of as the input data, leading naturally to a functor ${\operatorname{\mathbf{QFT}}}$ (see Definition \[QFT\_definition\]). From now on, we set $\Lambda=k[[\hbar]]$.
The category ${\mathfrak{B}}$
-----------------------------
We say that a graded associative, unital $\Lambda$-algebra $P$ has a classical limit provided $P$ is free as a $\Lambda$ module and if $K=P/\hbar P$ is a graded, commutative, associative, unital $k$-algebra.
Note that if $P$ has a classical limit, then $P\simeq K\oplus \hbar K
\oplus \hbar^2 K \oplus \cdots$ and $[P,P]\subseteq \hbar P.$
\[background\_def\] We define a background $B$ to be a four-tuple $B=\left( P,N, m,
\varphi \right )$ where
1. $P=\oplus_{i} P^i$ is a graded, associative, unital $\Lambda$-algebra with a classical limit,
2. $N=\oplus_{i}N^i$ is a graded left $P$ module, which is free as a $\Lambda$ module,
3. $m\in P^1$ satisfies $m^2=0$ ($m$ is called a structure),
4. $\varphi \in N^0$ satisfies $m \cdot \varphi=0$ ($\varphi$ is called a vacuum).
A morphism between two backgrounds $B=\left( P,m,N, \varphi \right
)$ and $B'=\left( P',m', N', \varphi' \right )$ consists of a map $\sigma: P \to P'$ of graded $\Lambda$ algebras and a map $\tau:N\to
N'$ of graded $P$ modules (where $N'$ becomes a $P$ module via $\sigma$) with $\tau(\varphi)=\varphi'$ satisfying the compatability $$\tau(ma\varphi)=m'\sigma(a) \varphi' \text{ for all }a\in P.$$ Denote the category of backgrounds by ${\mathfrak{B}}$.
Quantum master equation {#QME-section}
-----------------------
Let $B=(P,N, m, \varphi)$ be a background, let $(A,d) \in
{\operatorname{ob}}({\mathfrak{C}}_\Lambda)$, where ${\mathfrak{C}}_\Lambda$ is the category of differential graded Artin local $\Lambda=k[[\hbar]]$ algebras, and $\pi \in (P\otimes{\mathfrak{m}}_A)^0$. We call the equation $$\label{QME}
\left( e^{{\pi}/{\hbar}}me^{-{\pi}/{\hbar}}-\hbar
e^{{\pi}/{\hbar}}d \left( e^{-{\pi}/{\hbar}}\right)\right)
\varphi=0$$ the quantum master equation, and denote the set of solutions by ${\operatorname{\mathbf{QM}}}_B(A)$: $${\operatorname{\mathbf{QM}}}_B(A)=\left\{ \pi \in (P\otimes {\mathfrak{m}}_A)^0
\text{ such that $\left(
e^{{\pi}/{\hbar}}me^{-{\pi}/{\hbar}}-\hbar
e^{{\pi}/{\hbar}}d \left( e^{-{\pi}/{\hbar}}\right)
\right) \varphi=0$}\right\}.$$ Notice that ${\mathfrak{m}}_k=0$, so ${\operatorname{\mathbf{QM}}}_B(k)=\{0\}$, and any morphism $\tau:A
\to B\in \hom({\mathfrak{C}}_\Lambda)$ gives rise to a map from ${\operatorname{\mathbf{QM}}}_B(A) \to
{\operatorname{\mathbf{QM}}}_B(B)$ given by $\pi \mapsto (1\otimes \tau) \pi.$ Thus, we have defined a functor ${\operatorname{\mathbf{QM}}}_B\in {\mathfrak{Fun}}_\Lambda.$
Definition \[QME\] relies on the fact that both terms $e^{{\Pi}/{\hbar}} m e^{- {\Pi}/{ \hbar}}$ and $\hbar
e^{{\Pi}/{\hbar}} d\left( e^{-{\Pi}/{ \hbar}}\right)$ are well defined elements of $P\otimes A$, a fact that we clarify in two remarks. An example follows in a third remark.
First, we establish the meaning of $\frac{[\eta,\zeta]}{\hbar}$ in $P\otimes A$. The ring $P$ by assumption is free as a $\Lambda$ module, but $A$, owing to its nilpotency, cannot be and so $\hbar$ is a zero divisor in $P\otimes A$. However, in $P$, one has $[P,P]\subseteq \hbar P$, so one has an operator $$\begin{aligned}
\frac{[\,,\,]}{\hbar}:P \times P & \to P\\
\alpha,\beta \mapsto \gamma
\end{aligned}$$ where $\gamma \in P$ is defined by expressing $[\alpha,\beta]=\hbar
\gamma$. Then, for $\eta=\alpha \otimes a$ and $\zeta=\beta\otimes
b$ in $P\otimes A$, one defines $$\frac{[\eta,\zeta]}{\hbar}=\frac{[\alpha \otimes a,\beta\otimes
b]}{\hbar}:=\gamma \otimes ab\in P\otimes A.$$ Then, $\frac{1}{\hbar^j}[\eta_1,[\cdots [\eta_{j-1},\eta_j
]\cdots]\subseteq P\otimes A$ is understood in the same way.
\[UdU-1\] Expanding $e^{{\pi}/{\hbar}} m e^{-{\pi}/{ \hbar}}$ and $\hbar e^{{\pi}/{\hbar}} d\left( e^{- {\pi}/{
\hbar}}\right)$ in terms of repeated commutators: $$e^{{\pi}/{\hbar}} m e^{-{\pi}/{ \hbar}}=\exp\left({\frac{[-,\pi]}
{\hbar}}\right) (m)=m+{\frac{[m,\pi]} {\hbar }}+ {\frac{[[m,\pi],
\pi]} {2 \hbar^2}}+\cdots$$ displays each term as an element of $P\otimes A$, and the sum terminates since ${\mathfrak{m}}_A$ is nilpotent. The same holds for the $d$ term once expanded as $$\hbar e^{{\pi}/{\hbar}} d \left( e^{- {\pi}/{ \hbar}}
\right)=f \left({\frac{[-,\pi]} {\hbar}}\right) (d \pi), \text{ where
}f(x)= \frac{e^x -1}{x}=\sum_{j\geq 0}\frac{x^j}{(j+1)!}.$$
In certain cases, the quantum background $B$ arises from a differential BV algebra $(V,d,\Delta,\cdot)$ with Lie bracket $(\,,\,).$ In these cases, the quantum master equation defined for the background as defined above reduces to the usual quantum master equation seen in the physics literature: $$d\pi+\hbar \Delta \pi+\frac{1}{2}(\pi,\pi)=0$$ for $\pi\in V\otimes {\mathfrak{m}}_A$. This example is illustrated in Section \[Application\].
\[quasismooth\] The functor ${\operatorname{\mathbf{QM}}}_B$ is quasismooth and satisfies the Schlessinger condition.
Since ${\operatorname{\mathbf{QM}}}_B$ is given as a solution set of an equation, ${\operatorname{\mathbf{QM}}}_B$ satisfies the Schlessinger condition. To show that it is quasi-smooth, let $(A,d) \in {\operatorname{ob}}({\mathfrak{C}}_\Lambda)$ and let $C_A=(P\otimes
A) \varphi$ denote the cyclic submodule of $N\otimes A$ generated by the vacuum $\varphi$ ($=\varphi\otimes 1)$. Note that $C_A$ is a complex with differential $D:C_A^j \to C_A^{j+1}$ defined by $$D((\pi \otimes a) \varphi)= (m\pi\otimes a)\varphi+(-1)^{|a|}\hbar
(\pi \otimes d(a)) \varphi.$$
Let $\tau: A \to A'$ be an acyclic small extension with kernel $I$ and let $a' \in {\operatorname{\mathbf{QM}}}_B(A')$. Since $\tau$ is surjective, we can choose an $a \in P\otimes {\mathfrak{m}}_A$ so that $1 \otimes \tau (a)=a'.$ Let $$i=e^{{a}/{\hbar}}me^{-{a}/{\hbar}}-\hbar
e^{{a}/{\hbar}}\delta \left( e^{-{a}/{\hbar}}\right)\in
P_A.$$ Notice that $(1\otimes \tau)(i\varphi)=0$. So, $i\varphi$ is in the kernel of $1\otimes\tau:C_A \to C_{A'}.$ Since the functor $\otimes$ is right-exact, $i\varphi \in C_{I}$, and we can write $i\varphi=i'\varphi$ for some $i'\in I.$ Now, since $a\in {\mathfrak{m}}_A$, we have $a i'=0$ and $e^{-{a}/{\hbar}}i'=i'.$ Now, we have $$\begin{gathered}
i'\varphi=e^{-{a}/{\hbar}}i'\varphi
=e^{-{a}/{\hbar}}i\varphi\\
=e^{-{a}/{\hbar}}\left(e^{{a}/{\hbar}}me^{-{a}/{\hbar}}-\hbar
e^{{a}/{\hbar}}d \left( e^{-{a}/{\hbar}}\right)\right)
\varphi =\left(me^{-{a}/{\hbar}}-\hbar d\left(
e^{-{a}/{\hbar}}\right)\right) \varphi.
\end{gathered}$$ Now, consider $$D(i'\varphi)=\left(m^2e^{-{a}/{\hbar}}-\hbar md
\left(e^{-{a}/{\hbar}}\right) + d m e^{-{a}/{\hbar}}-\hbar
d^2 \left( e^{-{a}/{\hbar}}\right)\right) \varphi=0.$$ Since $I$ is acyclic, $C_{I}$ is acyclic, which implies that there exists a $j\varphi \in C_{I}^{0}$ with $Dj\varphi =i'\varphi.$ Define $a'=a-\hbar j\in (P\otimes {\mathfrak{m}}_A)^0.$ The claim is that $a' \in
{\operatorname{\mathbf{QM}}}_B(A).$ To show this, we compute. First note that $aj=0$ and $j^2=0$ imply that $e^{{a'}/{\hbar}}=e^{{a}/{\hbar}}+j$ and $e^{{-a'}/{\hbar}}=e^{{-a}/{\hbar}}-j$. So $$\begin{gathered}
\biggr(e^{{a'}/{\hbar}}me^{-{a'}/{\hbar}}-\hbar
e^{{a'}/{\hbar}} d \left(
e^{-{a'}/{\hbar}}\right)\biggr) \varphi\\
=
\left(\left(e^{{a}/{\hbar}}-j\right)m\left(e^{-{a}/{\hbar}}+j\right)-\hbar
\left(e^{{a}/{\hbar}}-j\right)d \left(
e^{-{a}/{\hbar}}+j\right)\right)\varphi \\
=i\varphi-jm\varphi-mj\varphi-jmj\varphi+\hbar j d
e^{-{a}/{\hbar}}\varphi -\hbar e^{{a}/{\hbar}} d j \varphi
=(i'-Dj )\varphi =0.
\end{gathered}$$
As a corollary, we make the following definition:
\[QFT\_definition\]Let $B$ be a background.
1. We define ${\operatorname{\mathbf{QFT}}}_B:={\operatorname{\mathbf{QM}}}_B/{\sim}$ where $\sim$ is the natural gauge equivalence.
2. The assignment $B\mapsto {\operatorname{\mathbf{QFT}}}_B$ defines a functor from ${\operatorname{\mathbf{QFT}}}:
{\mathfrak{B}}\to {\mathfrak{Fun}}_\Lambda$.
3. We define $H(B)$ to be the vector space $H(B):=T_{{\operatorname{\mathbf{QFT}}}_B}$ and call it the homology of $B$. The composition $H:={\mathbf{T}}\circ {\operatorname{\mathbf{QFT}}}$ defines a functor $H:{\mathfrak{B}}\to {\mathfrak{Vect}}$. In addition, we define the Dirac module of $B$ to be the $\Lambda$-module $D(B):=D_{{\operatorname{\mathbf{QFT}}}_B}.$ The composition $D:={\mathbf{D}}\circ {\operatorname{\mathbf{QFT}}}$ defines a functor $D:{\mathfrak{B}}\to
{\mathfrak{Mod}_\Lambda}$.
\[H(B)-and-D(B)\] Let $B$ be a background. There are natural isomorphisms $$\begin{gathered}
D(B)\simeq \left \{a\in P :
\frac{[m,a]}{\hbar}\varphi=0\right\}\biggr/ \sim
\label{dirac_computation} \intertext{and} H(B)\simeq \left \{a\in
P \otimes_\Lambda k :
\frac{[m,a]}{\hbar}\varphi=0\right\}\biggr/ \sim
\label{tangent_computation}
\end{gathered}$$ where $a_1 \sim a_0$ if and only if $(a_1-a_0)\varphi = \frac{[m,b]}{\hbar}
\varphi$ for some $b\in P$ (or $P\otimes_\Lambda k$).
We set up the natural isomorphism in homogeneous pieces by $$\xi \leftrightarrow a.$$ For $\xi \in (P \otimes_\Lambda m_{E_i})^0$, with $\xi=a \epsilon_i$, where $a\in P^{-i}\otimes_\Lambda k $ and $\epsilon_i \in m_{E_i}$, one writes $$e^{{a\epsilon_i}/{\hbar}}me^{-{a\epsilon_i}/{\hbar}} \varphi = \left(
m-\frac{1}{\hbar}[m,a]\epsilon_i+\frac{1}{2
\hbar^2}[[m,a],a]\epsilon_i^2+\cdots\right)\varphi =
-\frac{1}{\hbar}[m,a]\epsilon_i\varphi$$ and sees that $\xi \in
{\operatorname{\mathbf{QM}}}_B(E_i)$ if and only if $\frac{[m,a]}{\hbar}\varphi= 0$.
Now we check equivalence. First suppose $\frac{[m,a_0]}{\hbar}\varphi= 0$ and $\frac{[m,a_1]}{\hbar}\varphi=
0$ and $(a_1-a_0)\varphi = \frac{[m,b]}{\hbar}\varphi$ for some $b\in
P.$ Then, define $\xi(u,v)=(a_1u+a_0(1-u)+bv)\epsilon_i.$ One sees that $\xi(0,0)=\xi_0=a_0\epsilon_i$ and $\xi(1,0)=\xi_1=a_1\epsilon_i$. To see that $\xi(u,v)\in
{\operatorname{\mathbf{QM}}}_B(E_i\otimes {\mathbf{fr}}(k[u,v]/(u^2,uv,v^2))$, compute: $$\begin{gathered}
\left(e^{{\xi(u,v)}/{\hbar}}me^{-{\xi(u,v)}/{\hbar}} -\hbar
e^{{\xi(u,v)}/ {\hbar}}\delta\left(e^{-{\xi(u,v)}/{\hbar}}\right)
\right)\varphi \\ =- \left( \frac{[m,\xi_1]}{\hbar}u
\frac{[m,\xi_0]}{\hbar}(1-u)+\frac{[m,b]}{\hbar}v-(\xi_1-\xi_0)v\right)\varphi
=0.\end{gathered}$$ Conversely, let $$\xi(u,v)=(a(u)+b(u)v)\epsilon_i \in P^{-i}\otimes I'$$ for $a(u), b(u)\in P^{-i}\otimes_\Lambda k[u]$ for some $I' \subseteq
k[u,v]$. Then $\xi(u,v)\in {\operatorname{\mathbf{QM}}}_B(E_i\otimes {\mathbf{fr}}(I'))$ implies that $\frac{m,[a(u)]}{\hbar}\varphi=0$ and $a'(u)\varphi=\frac{[m,b(u)]}{\hbar}\varphi$. Define $b=\int_0^1
b(u)du$. Then, for $\xi(0,0)=a_0\epsilon_i$ and $\xi(1,0)=a_1\epsilon_i$, $a_0,a_1\in P^{-i}\otimes_\Lambda k$, we have $\frac{[m,a_0]}{\hbar}\varphi= 0$ and $\frac{[m,a_1]}{\hbar}\varphi= 0$ and $(a_1-a_0)\varphi =
\frac{[m,b]}{\hbar}\varphi$.
The computation for $D(B)$ is the same, except that the correspondence $[\xi] \leftrightarrow [a]$ is setup for $\xi\in (P\otimes_\Lambda m_{{\mathbf{fr}}(E_i)})^0 =P^{-i} \otimes_k m_{E_i}$ with $\xi=ae_i$ with $a\in P^{-i}$ and $e_i\in m_{E_i}$.
Let $C_\Lambda=P\varphi$ and $C_k=(P\otimes_\Lambda k)\varphi$ be the cyclic submodules of $N$ and $N\otimes k$ generated by the vacuum. Then multiplication on the left by $m$ defines differentials $m:C_\Lambda ^j \to C^{j+1}$ and $m:C_k^j\to C_k^{j+1}$. Then the Dirac space and tangent space are isomorphic to the homology of these complexes $D(B)\simeq H(C_\Lambda ,m)$ and $H(B)\simeq H(C_k,m)$.
\[main\_thm\] Suppose that $B$ is a background and $T_{{\operatorname{\mathbf{QFT}}}_B}=H(B)$ is finite dimensional. with basis $\{\xi^1, \ldots, \xi^r\}$. Let $\{t_1,\ldots, t_r\}$ be the dual basis and let $R=\Lambda[[t_1,\ldots, t_r]]$. There exists a differential $\delta:R \to R$ with $\delta({\mathfrak{m}}_R)\subset ({\mathfrak{m}}_R)^2$ and $\Pi \in
{\operatorname{\mathbf{QM}}}_B(R)$ such that ${\operatorname{\mathbf{QFT}}}_B$ is homotopy represented by the couple $((R,\delta),\Pi).$
Apply proposition \[rep\_thm\] to ${\operatorname{\mathbf{QFT}}}_B$.
\[smoothformal\] Let $B$ be a background and assume $H(B)$ is finite dimensional.
1. We call $B$ finite.
2. A morphism of backgrounds $B \to B'$ is called a quasi-isomorphism if ${\operatorname{\mathbf{QFT}}}_{B} \to {\operatorname{\mathbf{QFT}}}_{B'}$ is an isomorphism of functors, in which case we say $B$ and $B'$ are quasi-isomorphic.
3. We call $\Pi$ in proposition \[main\_thm\] a versal solution to the quantum master equation.
4. We call $B$ smooth formal if $\delta=0$ for $\delta$ in proposition \[main\_thm\].
Flat quantum superconnection {#Flat_Connection}
============================
One is able to supply derived algebro-geometric interpretations of structures associated to smooth formal backgrounds. Some of these structures have been unearthed in specific examples, and our aim in this section to explain this structure from a unifying perspective afforded by backgrounds. For example, our main construction is a flat superconnection $\nabla$ on a bundle over the functor ${\operatorname{\mathbf{QFT}}}_B$. The connection one-form part of $\nabla$, which we call $\nabla^{1}$, defines a flat connection which coincides with the connection defined in [@B2] in the setting of quantum periods. Throughout this section we assume that $B$ is a smooth formal background.
Conceptual description
----------------------
First, to define a bundle over a functor ${\mathbf{F}}$ of parameter rings, one assigns an $A$ module $M_\pi$ to each point $\pi\in {\mathbf{F}}(A)$, functorial with respect to ring maps $A\to A'$. In the case of ${\operatorname{\mathbf{QM}}}_B$ for a background $B$ and a point $\pi\in {\operatorname{\mathbf{QM}}}_B(A)$ where $A$ is a parameter ring with zero differential, we can define a new background $B_\pi$ as $$B_\pi:=(B\otimes A, m^\pi, N\otimes A, \varphi \otimes 1)$$ where $m^\pi:=e^{\pi/ \hbar}me^{-\pi/ \hbar}$. To see that $B_\pi$ defines a background, note that $m^\pi$ squares to zero (trivially) and that $m^\pi (\varphi \otimes 1)=0$ by the quantum master equation. Let $$\begin{aligned}
D_\pi&:=D(B_\pi) = \{x \in P\otimes A: \frac{\left[m^\pi,
x\right]}{\hbar} \phi =0\}/\sim \\
&=H(C_A,m^\pi)\end{aligned}$$ where $C_A=(P\otimes A)\varphi$ is the cyclic submodule of $N\otimes
A$ generated by the vacuum. The Dirac space $D_\pi$ of the background $B_\pi$ is an $A$ module. If $f:A\to A'$ is a map of parameter rings, we obtain a map of background $B_\pi \to B_{{\mathbf{F}}(f)(\pi)}$ and hence a map of Dirac spaces. Furthermore, if $\pi\sim\pi'$, the backgrounds $B_\pi$ and $B_{\pi'}$ are quasi-isomorphic, hence have isomorphic Dirac spaces. Thus, the construction of $D_\pi$ from $\pi \in {\operatorname{\mathbf{QM}}}_B(A)$ defines a bundle over the functor ${\operatorname{\mathbf{QFT}}}_B$ when ${\operatorname{\mathbf{QFT}}}_B$ is smooth formal. Call this bundle the versal Dirac bundle and denote it by $\widetilde{D}$. We haven’t defined a background $B_\pi$ for $\pi \in QM_B(A)$ when $A$ is a parameter ring with nonzero differential, but it’s straightforward (but not necessary for smooth backgrounds) to modify the definition of $D_\pi$ for such a point $\pi.$ Set $D_\pi:=H(C_A,m+d)$ where $d$ is the differential in $A$. The versal Dirac bundle resonates with how one thinks of the quantum master equation: a solution of the quantum master equation over a ring $A$ with no differential defines a new background $B_\pi$ defined over $A$. By taking the Dirac space of $B_\pi$ one obtains an $A$ module which is the fiber of a bundle over the moduli space.
Next, we describe the construction of a flat connection on the versal Dirace bundle $\widetilde{D}$ of a smooth formal background. According to Grothendieck, a flat connection on a bundle over a functor is given by a collection of $A$ module isomorphisms $D_0\otimes A \to D_\pi$ for each point $\pi$ over $A$, identifying the fiber over $\pi$ with the fiber over $0$. It will be convenient to use a representing couple $(R,\Pi)$ where we take $R$ to have zero differential. Using the representing couple, we replace $\widetilde{D}$ with the $R$ module $D_\Pi$ and obtain a flat connection by defining an isomorphism $D_0\otimes R \to D_\Pi.$ This suffices to define a flat connection on the versal Dirac bundle since any other point $\pi$ over $A$ is given by a morphism of parameter rings $R\to A$ inducing $D_\Pi \to D_\pi$ and induces an isomorphism $D_0\otimes A \to D_\pi$.
In terms of a representing couple $(R,\Pi)$, we have the moduli space ${\mathcal{M}}={\operatorname{spec}}(R)$ and $R=\Lambda[[H^*]]=k[[\hbar,H^*]]$ where $H$ is the homology of the background and $H^*$ is the linear dual $H^*=\hom_k(H,k)$. Since $H=\hom_\Lambda (R,k[t]/t^2)$ is isomorphic to the derivations of $R$, the vector space $H$ is naturally isomorphic to the tangent space of the formal space ${\mathcal{M}}$ at its base point. Let $D_0=D(B)$ denote the Dirac space of the original background, which is the fiber of the Dirac bundle at the base point. We will prove that $H\otimes_k R \simeq D_\Pi$, which implies that $D_0\simeq H[[\hbar]]$ and $D_0\otimes R\simeq H \otimes_k R \simeq
D_\Pi$, which defines a flat connection on $\widetilde{D}$. From another point of view, bundles over $M$ correspond to $R$ modules. In particular, sections of the versal Dirac bundle $\widetilde{D}$ are elements in the $R$ module $D_\Pi$, which we will also call the versal Dirac bundle. The connection that we define, $\nabla^{1}$, defines a map $$\begin{aligned}
\nabla^{1}:D_0 \times D_\Pi &\to D_\Pi \\
(X,{\mathcal{Y}})&\mapsto \nabla^{1}_X {\mathcal{Y}}\end{aligned}$$ which is $R$ linear in the first coordinate $\nabla^{1}_{fX}{\mathcal{Y}}= f
\nabla^{1}_X {\mathcal{Y}}$ and satisfies the $\hbar$-connection equation in the second $$\nabla^{1}_X(f{\mathcal{Y}}) = \hbar X(f){\mathcal{Y}}+(-1)^{|f|} f \nabla^{1}_X
{\mathcal{Y}}\text{ for $f\in R$, $X\in D$, ${\mathcal{Y}}\in D_\Pi$}.$$ We prove that $\nabla^{1}$ is flat: $$\nabla^{1}_{X}\nabla^{1}_{Y}\mathcal{Z}
-\nabla^{1}_{Y}\nabla^{1}_{X}\mathcal{Z} -
\nabla^{1}_{[X,Y]}\mathcal{Z} =0.$$ Equivalently, we can use $\Omega:=R[[H^*[1]]]=k[[\hbar, H^*, H^*[1]]]$, the module of Kahler differentials on $R$, to write the connection using one-forms: $$\nabla^{1}:D_\Pi \to D_\Pi \otimes_R \Omega^1.$$ In fact, what we define is a “chain level” flat connection on the bundle of cyclic modules $C_R=(P\otimes_\Lambda R)\varphi$ $$\nabla^{chain}:C_R \to C_R\otimes_R \Omega^1$$ which descends to $\nabla^{1}$ on the Dirac space. However, a deeper analysis of $\nabla^{chain}$ on $C_R$ reveals that when $\nabla^{chain}$ is compressed from the chain level to the Dirac space, it manifests as a superconnection $$\nabla:D_\Pi\to D_\Pi \otimes_R
\Omega$$ which can be decomposed as $\nabla=\nabla^{1}+\nabla^2+\nabla^3+\cdots$ with each $\nabla^i:D_\Pi
\to D_\Pi \otimes \Omega^i.$ In a basis of $D_\Pi$, $\nabla^i=d_{dR}+A^i$ where $d_{dR}$ is the de Rham differential in $\Omega$ and $A^i$ is a matrix of $i$-forms. We prove that the quantum superconnection $\nabla$ is flat. The relationship between $\nabla^{1}$ to $\nabla$ is analogous to a familiar one concerning differential graded algebras. Given a dga $(A,d,\cdot)$, the product descends to an associative structure in the homology $H(A,d)$. However, a more refined structure which always exists is a minimal $A_\infty$ structure in $H(A,d)$ that is quasi-isomorphic to the dga $(A,d,\cdot)$. The first part of such a minimal $A_\infty$ structure on $H(A,d)$ coincides the the associative structure in $H(A,d)$, but the $A_\infty$ structure has more information, it determines the dga $(A,d,\cdot)$ up to homotopy equivalence.
The connection $\nabla^1$ is defined canonically from a representing couple $(R,\Pi)$ but our construction of the superconnection depends on an $R$ module decomposition $C_R\simeq D'\oplus E \oplus F$ where $D'\simeq D_\Pi$ are representatives for the Dirac bundle, and $m^\Pi:F \to E$ is an isomorphism. Such a Hodge decomposition of a chain complex is often used in order to obtain a minimal $A_\infty$ structure in the homology of a dga.
Heisenberg versus Schrödinger
-----------------------------
As mentioned above, if $\pi\in {\operatorname{\mathbf{QM}}}_B(A)$ and the differential in $A$ is zero, then we can define a new background with parameters in $A$. The way it was described above was according to the Heisenberg representation. Explicitly, $$B_\pi^{Heis}:=(P\otimes A, m^\pi, N\otimes A, \varphi\otimes 1)$$ becomes a background: $(m^\Pi)^2=0$ (trivially) and the quantum master equation says that $m^\Pi\varphi=0$ (nontrivially). We think of $B_\pi^{Heis}$ as an evolution of $B$ in which the structure $m$ that evolves: $$m\leadsto
m^\pi=U^{-1}mU \text{ where }U=e^{-\pi/\hbar}.$$ The structure $m^\pi$ depends on a parameter $t$ in a parameter ring $R$. For $t=0$, one has the initial structure $m^\pi(0)=m$ (since $\pi\in P\otimes {\mathfrak{m}}_R$), and we imagine as $t$ varies, $m^\pi(t)$ varies over the space ${\operatorname{spec}}(A)$ of structures path connected to the initial one.
There is also a Schrödinger interpretation of the evolution $B$ to the background $$B_\pi^{Schr}:=(P\otimes A, m, N((\hbar))\otimes A,
e^{-{\pi}/{\hbar}} \varphi).$$ In $\widetilde{B}^{Schr}$, it is the vaccuum $\varphi$, rather than the structure $m$, that evolves: $$\varphi \leadsto
U\varphi.$$ We permit the module $N$ and the evolving vacuum $U\varphi=e^{-\pi/\hbar}\varphi$ to have contain negative powers in $\hbar$, but only finitely many. An expansion in terms of $A$ coordinates $t_i$ $$e^{-{\pi}/{\hbar}}\varphi=\sum_{j= 0}^\infty (-1)^j\frac{ \pi^j}
{\hbar^j j!}\varphi=\sum_{ \substack{ i_1\geq 0,\dots, i_n\geq 0 \\
\, \\ j\geq -(i_1+\dots+i_n)}} t_{i_1}\dots t_{i_n} \hbar^j p_j^{
i_1, \dots, i_n} \varphi,$$ for $p_j^{i_1,\dots, i_n}\in P$ shows that $e^{-{\pi}/{\hbar}}
\varphi$ is a formal Laurent series for each fixed total degree of the $t_i$’s. Now we prove the equivalence of the Heisenberg and Schrödinger perspectives
The backgrounds $B_\pi^{Heis}$ and $B^{Schr}_\pi$ are quasi-isomorphic.
The morphism of quantum backgrounds $$(\sigma, \tau):B_\pi^{Heis} \to
B_\pi^{Schr}$$ defined by $$\sigma(\alpha)=U\alpha U^{-1}
\text{ for }\alpha \in P \otimes A \text{ and
}\tau(\psi)= U \psi \text{ for } \psi\in
N\otimes A.$$ The Heisenberg Dirac module is computed by putting the parameters in the chain complex and twisting the differential: $$D(B_\pi^{Heis})=H(C^{Heis}_\pi,m^{\pi})\text{ where }
C^{Heis}_\pi=(P\otimes A)\varphi.$$ In the Schrodinger picture, we twist the complex and leave the differential unchanged: $$D(B_\pi^{Schr})=H(C^{Schr}_\pi,m) \text{ where }C_\pi^{Schr}=(P\otimes
A)e^{-\pi/\hbar}\varphi.$$ The map $D(B_\pi^{Heis})\to D(B^{Schr}_\pi)$ induced by $(\alpha,\tau)$ is an isomorphism $$\begin{gathered}
U^{-1}mUx\varphi=0\Leftrightarrow
mU xU^{-1} U\varphi=0\Leftrightarrow m\sigma(x)\tau(\varphi)=0\intertext{and}
U^{-1}mUx\varphi=y\varphi\Leftrightarrow
mU xU^{-1} U\varphi=U yU^{-1}U \varphi\Leftrightarrow
m\sigma(x)\tau(\varphi)=\sigma(y)\tau(\varphi).\end{gathered}$$
The isomorphism $H \otimes_k R \to D_\Pi$
-----------------------------------------
We now switch to using the Schrodinger representation. Since we use this picture for the remainder, we do not use the superscript $\emph{Schr}$. Set $U:=e^{-\Pi/\hbar}.$ Then the primary chain complex consists of the $R$ module $$C_\Pi=P\otimes R U\varphi$$ with differential given by multiplication by $m$. The closed elements correspond to elements $\mathcal{X}\in P\otimes R$ satisfying $m \mathcal{X}U\varphi=0$ and the exact elements correspond to elements $\mathcal{Y}\in P\otimes R$ satisfying ${\mathcal{Y}}U\varphi=m\mathcal{X}U\varphi.$ Since $H$ is the $k$-vector space of $k$-linear derivations of $R$, $H$ also acts on $P\otimes R$ as $P$ linear derivations. However, a $P$ linear derivation of $R$ maps $C_\Pi\to
\frac{1}{\hbar}C_\Pi$ since differentiating $U$ may introduce one negative power of $\hbar$. However $\hbar H$ acts on $C_\Lambda$ by $P$ linear derivations.
\[connection\_thm\_1\] If $B$ is a smooth formal background represented by the couple $(R,\Pi)$, then the map $\Phi^0:H\to C_\Pi$ defined by $\Phi^0(x)=\hbar x U\varphi$ induces an isomorphism $H\otimes_k R \simeq D_\Pi.$
Since $\Pi$ is a solution to the master equation, $0=mU\varphi$. Apply the derivation $\hbar x$ to get $\hbar x m U \varphi$. Since $m\in P$ and $x$ is a $P$ linear derivation of $R$, $m$ and $x$ commute and we have $0=m\hbar x
U \varphi$. Now, $\hbar x
(U)\varphi=\hbar x(U) U^{-1}U\varphi$ and since $\hbar x(U)U^{-1}\in
P\otimes R$ contains no negative powers of $\hbar$, $\Phi^0=\hbar
x(U)\varphi$ is a closed element in $C_\Pi$. Taking the homology class of $\Phi^0$ defines a map $H\to D_\Pi$.
To see that the map, when the coefficients are extended to $R$ becomes an isomorphism $H\otimes R \to D_\Pi$, first observe that it is injective. If $\Phi^0(s)=\hbar x U\varphi$ satisfies $\hbar x U\varphi=m{\mathcal{Y}}U\varphi$ for some ${\mathcal{Y}}U\varphi \in
C_\Pi$, then reducing this equation modulo $\hbar$ and $H^*$ implies that $x$ satisfies $x\varphi = m y\varphi$ for $y=\mathcal{Y} \mod
\hbar, H^*$.
To see that $\Phi^0$ is surjective $D_\Pi$, note that if $m\mathcal{X}
U \varphi=0$ then reducing this equation modulo $H^*$ yields $mX_1\varphi=0$ for $X_1=\mathcal{X} \mod \hbar H^* \in
(P\otimes R)/H^*\simeq P$. So $X_1$ defines a class in $H\otimes R/H^*$ and $\Phi^0(X_1)=\mathcal{X}
\mod H^*.$ Then, $\mathcal{X}-\Phi^0(X_1) \mod (H^*)^2$ is an $m$ closed element of $(P\otimes R)/(H^*)^2$ hence defines an element $X_2\in H\otimes R/(H^*)^2$. Then $\mathcal{X}=\Phi^0(X_1+X_2) \mod (H^*)^3$, and so on. This builds inductively an element in $X=X_1+X_2+X_3\in H\otimes
R$ with $\Phi^0(X)=\mathcal{X}\varphi.$
By reducing the isomorphism $H \otimes_k R \simeq D_\Pi$ modulo $H^*$, we see that there is no $\hbar$ torsion in $D$.
\[classical-to-quantum\] If $B$ is a smooth background then $D(B)\simeq
H(B)[[\hbar]].$
In coordinates, say $\{x_i\}$ is a basis for $H$ with dual basis $\{t_i\}$, we have $x_i$ acts by $\hbar \frac{\partial}{\partial t_i}$ on $R=k[[\hbar,t_i]]$ and $P[[\hbar, t_i]].$ A versal solution to the quantum master equation $\Pi$ has the form $$\begin{gathered}
\Pi=\sum \left( x_i t_i+ \hbar x^{(1)}_i t_i + \hbar^2 x^{(2)}_i t_i
+ O(\hbar^3) \right) \\
+ \left(x_{ij} t_it_j + \hbar x_{ij}^{(1)}
t_i t_j \hbar^2 x^{(2)}_i t_j+O(\hbar^3)\right)+\cdots\end{gathered}$$ with $x_{i_1, \ldots, i_n}\in P$. The first terms $\{x_i\}$ are the $k$-basis for $H$ and the linear in $t$ terms $\{X_i:=x_i+ \hbar x^{(1)}_i + \hbar^2 x^{(2)}_i
+ \hbar^3\cdots\}$ are a $k[[\hbar]]$ basis for $D$ and $\left\{\mathcal{X}_i:=\hbar \frac{\partial e^{-\Pi/\hbar}}{\partial
t_i}\right\}$ is an $R=k[[\hbar, H^*]]$-basis for $D_\Pi$. In the case that $\Pi$ commutes with its derivatives in $P$, $\mathcal{X}_i=\frac{\partial
\Pi}{\partial t_i}.$ The isomorphisms $H\otimes \Lambda \to D$ and $H\otimes R \to D_\Pi$ are given by $x_i \mapsto X_i$ and $x_i
\mapsto \mathcal{X}_i.$
Recent developments [@js3; @jt2] suggest that the converse to Corollary \[classical-to-quantum\] is true and the concept of special coordinates in string theory is related to the way in which an isomorphism $D(B)\simeq H(B)[[\hbar]]$ can be extended to produce a “special” versal solution to the quantum master equation.
So, as described in the conecptual description of the connection, Theorem \[connection\_thm\_1\] implies that $D_0\otimes R\simeq
H\otimes_k R \simeq D_\Pi$ and therefore defines a flat connection on $D_\Pi.$ Now we translate from the Grothendieck formal geometry picture to the covariant derivative picture of a flat connection. The translation amounts to observing that $D_0=H[[\hbar]]$ is the $\Lambda=k[[\hbar]]$ module of $\Lambda$ linear derivations of $R=k[[\hbar,H^*]]$, hence act as $P$ linear derivations of $C_\Lambda$ and $D_\Lambda$.
\[chain-connection\] Define $\nabla^{chain}: D_0 \times C_\Pi \to C_\Pi$ by $(X,{\mathcal{Y}})\mapsto \nabla^{chain}_X({\mathcal{Y}})=\hbar X({\mathcal{Y}})U^{-1}.$
First note that $\hbar X (\mathcal{Y}U\varphi)=\hbar
X({\mathcal{Y}}U)U^{-1}U\varphi$ and that $\hbar X({\mathcal{Y}}U)U^{-1}$ has no negative powers of $\hbar$. This proves that $\nabla_X:C_\Pi \to C_\Pi$. Now, since $m\in P$ and $X$ acts as a $P$ linear derivation, $\nabla_X(m{\mathcal{Y}})=X(m{\mathcal{Y}})=mX{\mathcal{Y}}$ so $\nabla_X$ is a chain map, hence defines a map $$\nabla^{1}: D_0 \times D_\Pi \to D_\Pi$$
Superconnection on the versal Dirac module
------------------------------------------
First, refashion $\nabla^{chain}$ in terms of differential forms. Let $\Omega^k=R[[H^*]]\otimes_R S^k(H^*[1])$ be the module of $k$ forms on ${\operatorname{spec}}{R}$ and $\Omega:=\Pi_k\Omega^k$ be the module of Kahler differential forms. Let $d_{dR}:\Omega \to \Omega$ denote the de Rham differential; i.e., the shift functor $H\mapsto H[1]$ extended to a $\Lambda$-linear derivation of $\Omega$. Therefore $\hbar d_{dR}:C_\Pi\otimes_R \Omega \to C_\Pi\otimes_R \Omega $ is a square zero $P$ linear derivation. Since $m\in P$, $[m,\hbar d_{dR}]=0$. Therefore, $d_{dR}$ descends to give a map $$\nabla^1:D_\Pi \to D_\Pi \otimes_R \Omega^1$$ This is precisely the same as the connection defined before, expressed in terms of a connection one forms.
However, one can do better. We now explain how to transfer $d_{dR}$ to a superconnection $\nabla: D_\Pi \to D_\Pi\otimes_R \Omega.$ In order to do so, we use one more piece of data. Consider $\Phi^0:H \to C_\Pi$ as in Theorem \[connection\_thm\_1\] defined by $\Phi^0(x)=\hbar x(U).$ Since $\Phi^0(H)$ is closed in $C_\Pi$ and $R\Phi^0(H)\simeq D_\Pi$, we may choose an acyclic complement $E$ in $C_\Pi$ so that $$\label{directsum}
C_\Pi=\Phi^0(H)\oplus E.$$ Now, the equation $$\label{eq1}
m\Phi^0=0$$ implies $m \hbar d_{dR} \Phi^0=0$. Therefore, $$\label{eq2}
\hbar d_{dR} \Phi^0 = A^1 \Phi^0 + m \Phi^1$$ where $A^1:D_\Pi \to
D_\Pi \Omega^1$ and $\Phi^1:H \to E\otimes \Omega^1$. Since $d_{dR}^2=0$, applying $\hbar d_{dR}$ to Equation yields $$(\hbar d_{dR}A^1) \Phi^0 - A^1 \hbar d_{dR}\Phi^0 - m \hbar d_{dR}\Phi^1=0.$$ Using Equation again get $$\begin{gathered}
0=(\hbar d_{dR}A^1) \Phi^0 - A^1 (A^1 \Phi^0 + m \Phi^1) -m
\hbar d_{dR}\Phi^1\\ =(\hbar d_{dR}A^1-A^1A^1 )\Phi^0 + m(\hbar d_{dR}\Phi^1-A^1 \Phi^1).\end{gathered}$$ This one equation splits into the two equations $$\begin{gathered}
\hbar d_{dR}A^1-A^1A^1=0\\
m(\hbar d_{dR}\Phi^1-A^1 \Phi^1)=0\end{gathered}$$ Now, the second equation defines a cycle in $C_\Pi\otimes_R \Omega$ and so implies that $\hbar d_{dR}\Phi^1-A^1 \Phi^1=A^2
\Phi^0 + m \Phi^2$ for unique $A^2:D_\Pi \to D_\Pi \otimes_R \Omega^2$ and $\Phi^2:H \to E\otimes_R \Omega^2.$ Iterating this procedure proves the following theorem:
\[conn-A\] For every versal solution $\Pi$ to the quantum master equation, and every choice of acyclic complement $E$ of $\Phi^0(H)$ in $C_\Pi$, there exist unique maps $A^n:D_\Pi \to D_\Pi\otimes \Omega^n$ for $n\geq 1$ and $\Phi^n:H\to E\otimes \Omega^n$ for $n\geq 0$ satisfying $$\begin{aligned}
\label{d(A)}
\hbar d_{dR}A^n&=\sum_{j=1}^n (-1)^{n+j} A^{n+1-j}A^{j}, \\
\label{d(phi)}
\hbar d_{dR}\Phi^n_i&=\sum_{j=0}^n A^{n+1-j}\Phi^j+m\Phi^{n+1}.
\end{aligned}$$
\[conn\_def\] Define $\nabla:D_\Pi \to D_\Pi\otimes_R \Omega$ by $$\nabla:= \hbar d_{dR}+A^1+A^2+\cdots$$
\[conn\_thm\] $\nabla$ is flat.
Notice, that $\nabla$ satisfies $$\nabla(f{\mathcal{Y}})=\hbar d_{DR}(f)\wedge
{\mathcal{Y}}+(-1)^{|f|} f\nabla({\mathcal{Y}})$$ for any $f\in R$ and ${\mathcal{Y}}\in D_\Pi$. Equation implies $\nabla^2=0$, which is the flatness condition.
As discussed in the next section, this small connection carries all of the information about the correlation functions (except for a choice of one-point functions). However, $\nabla$ carries much more information, what might be thought of as homotopy correlation functions. For example, in the $B$-model, the homotopy correlation functions should give a chain level generalization of the Frobenius manifold structure already discovered [@B2; @BK].
Path integral, correlation functions, and generalizations
=========================================================
In this section, we relate the quantum superconnection to the correlation functions of the quantum field theory defined by the quantum background $B$. We assume $B$ is finite, but not necessarily smooth formal.
Path integral
-------------
\[path-integral\] Let $B=(P,m,N,\varphi)$ be a background. We define a (chain level) path integral pairing for $B$ to be a $k[[\hbar]]$ module map $\int:N \to k[[\hbar]]$, which we denote by $\psi \mapsto \int
\psi,$ satisfying
- ($P$ Module axiom) ${\displaystyle}\int (\alpha\beta)\psi = \int \alpha
(\beta \psi)$ for all $\alpha, \beta \in P$, and $\psi \in N$,
- (Stokes axiom) ${\displaystyle}\int m\psi =0$ for every $\psi \in N$.
We call the condition ${\displaystyle}\int m\psi =0$ Stokes axiom because, if we use $\int_\psi \alpha$ to denote $\int \alpha \psi$, the condition $\int_\psi m=0$ implies $$\int_{\partial \psi} \alpha = \int_\psi D\alpha ,$$ where $\partial$ and $D$ are differentials in $P$ and $N$ defined by $$\partial(\psi)=m\psi \text{ and }D\alpha = m \alpha.$$
Given a finite background $B$ and a ring $(R,\delta)$ which represents ${\operatorname{\mathbf{QFT}}}_B$, one can extend the module $N$ to $C_\Lambda$ which is a left $P$ module with the action of $m$ extended to be that of $m\otimes 1.$ A path integral pairing $\int$ for $B$ extends $$C_\Lambda \to R((\hbar))$$ and extends with module compatibility property and the Stokes property.
If $\Pi$ is a versal solution to the quantum master equation, then $e^{-{\Pi}/{\hbar}}\varphi\in C_\Lambda \otimes R$ and we define the partition function $\mathcal{Z}\in R((\hbar))$ by $$\mathcal{Z}=\int e^{-{\Pi}/{\hbar}}\varphi.$$
Generalized Ward identities
---------------------------
For a couple $((R,\delta),\Pi)$ representing ${\operatorname{\mathbf{QFT}}}_B$, we have the master equation $$me^{-{\Pi}/{\hbar}}\varphi -\hbar\delta(e^{-{\Pi}/{ \hbar}} )\varphi=0.$$ Integrating (and Stokes axiom) gives $$\int \hbar \delta (e^{-{\Pi}/{ \hbar}})\varphi=\hbar\delta \int e^{-{\Pi}/{\hbar}}\varphi=0.$$ The integral is linear over $R$, giving the following differential equation for the partitition function: $$\label{wardequation}
\hbar\delta \mathcal{Z}=0.$$ This non-trivial equation captures the most general symmetries of the integral, including, in some example, the Ward identities. In the case that the background is smooth, the Ward identities vanish. The conclusion is
> *The obstructions to deforming the background manifest themselves as symmetries of the partition function*.
Correlation functions
---------------------
Now suppose that $B$ is smooth formal. Choose $R$ so that $\delta=0$. The tangent space $H=H(B)$ plays the role of the classical observables and the Dirac space $D=D(B)$ plays the role of physical observables. In the smooth case, there is an isomorphism $H[[\hbar]]\to D$, hence we may think of the two classes of observables interchangeably. We think of the Dirac bundle $D_\Pi$ as consisting of the observables for all theories in a neighborhood of $B$ in the moduli space.
For each $n=0,1,2,\ldots $, we define multilinear maps, called $n$-point correlation functions, $${\langle}\cdots {\rangle}:H^{\otimes n}\to R((\hbar))$$ by $$Z_{i_1}\otimes \cdots \otimes Z_{i_n}\mapsto{\langle}Z_{i_1}, \ldots,
Z_{i_n}{\rangle}:=
\int \hbar^nZ_{i_n}\cdots Z_{i_1} e^{-{\Pi}/{\hbar}}\varphi.$$
Here, the technique of computing correlation functions by differentiating a family of action functionals here becomes a definition since each observable $Z_i\in H$ is a derivation of the representing ring $R$. The $n$ point correlation functions are completely determined by the one point correlation functions and the small connection $\nabla^{1}$; i.e., the one-form part of $\nabla$. To see this note that ${\langle}Z_i{\rangle}={\langle}\Phi^0(Z_i){\rangle}$. Applying $\hbar d_{dR}$ gives a one-form $\hbar d_{dR}(Z_i)$ which when evaluated on the tangent vector $Z_j$ gives $${\langle}\hbar d_{dR}(\Phi^0(Z_i))(Z_j){\rangle}={\langle}Z_j,Z_i{\rangle}.$$ The equations in Theorem \[conn-A\] imply $${\langle}Z_j,Z_i{\rangle}={\langle}A^1\Phi^0(Z_i)+m\Phi^1(Z_i){\rangle}=A^1{\langle}\Phi^0(Z_i){\rangle}.$$ Next, one finds that $${\langle}Z_k,Z_j,Z_i{\rangle}=\hbar d_{dR}A^1(Z_k){\langle}\Phi^0(Z_i){\rangle}+A^1A^1{\langle}\Phi^0(Z_i){\rangle}.$$ By recursively differentiating and eliminating the exact terms, one can compute all correlation functions in terms of the $A^i$, and their derivatives.
Homotopy correlation functions
------------------------------
A chain level path integral pairing gives rise to a linear functional on the versal Dirac bundle and this linear functional on $D_\Pi$ is sufficient to determine the correlation functions. So, one may propose to define a path integral, or a “cohomological path integral” to contrast it with a chain level path integral, as any linear functional $F\in \hom(D_\Pi,R)$. This avenue will lead to the definition of the one-point correlation functions as ${\langle}[Z_i]{\rangle}=F([Z_i])$ and the definition of the $n$-point correlation functions by equations already given. However, there is information, beyond correlation functions, that a chain level path integral pairing can detect that is invisible to any cohomological path integral.
To explain, we first make the correlation function into a function of the moduli by $\int \Phi^0$. Then, the $n$ point correlation functions are obtained as derivatives of the one point correlation functions and are inductively determined by integrating Equation for $n=0$: $$\label{one-form-recursion}
\hbar d_{dR} \int \Phi^0 = A^1\int\Phi^0.$$ Thus, one might as well summarize all of the $n$ point correlation functions into the single fundamental correlation function $$\label{correlation-form}
\int \Phi^0$$ with the understanding that the correlation function satisfies Equation .
Note that the information contained in the boundary term $m\Phi^1$ from equation is lost after integration, but may be retained as a correlation one-form $$\int \Phi^1.$$ Think of the correlation one-form as a homotopy $1$-point correlation function, from which many homotopy $n$-point functions can be derived by differentiation. These homotopy correlation functions that may be derived from $\int \Phi^1$ are summarized inductively by integrating equation : $$\hbar d_{dR}\int \Phi^1=A^1 \int \Phi^1+A^2 \int \Phi^0.$$ This discussion suggests an efficient way to handle these correlation and homotopy correlations by defining the correlation $k$ form by $\int \Phi^k$ which satisfies $d_{dR}\int \Phi^k=A^1 \int \Phi^{k}+A^2
\int \Phi^{k-1}+\cdots+A^k\int \Phi^0.$ Better yet:
We define the primary correlation form to be the function $$\int \Phi:H \to \Omega$$ where $\Phi=\Phi^0+\Phi^1+\Phi^2+\cdots. \in C_\Pi \otimes_R \Omega$
The primary correlation form satisfies ${\displaystyle}\nabla \int
\Phi =0.$
Application—dBV algebras {#Application}
========================
There are many situations which give rise to quantum backgrounds. In each, the resulting quantum flat superconnection provides a penetrating tool for investigating the situation. Here, we give one example which we feel may elucidate the many definitions in this paper. We begin with a differential BV algebra and construct a quantum background from it. Then, we indicate how the various features of the background correspond to the dBV algebra. In particular, the quantum superconnection derived from the ${\operatorname{\mathbf{QFT}}}$ functor deeply probes the homotopy theory of the BV algebra. Related ideas appear in [@js1].
Let $(V,d,\Delta,\wedge)$ be a differential BV algebra. This means that $V$ is a graded vector space and $d, \Delta$ are commuting differentials on $V$ with the properties
- $(V,d,\wedge)$ is a differential graded commutative, associative, algebra, and
- $\left(V[-1],d, (\,,\,)\right)$ is a differential graded Lie algebra, where the bracket is defined by $(v,w):=(-1)^{|v|}\Delta(v\wedge w)-(-1)^{|v|}\Delta(v)\wedge
w-v\wedge \Delta(w)$ for homogeneous vectors $v$ and $w$.
We use parentheses $(\,,\,)$ for the Lie bracket, reserving square brackets $[\,,\,]$ always for the graded commutator. For convenience, we assume that the dBV algebra is fairly simple; assume $V=SU$ for a finite dimensional graded vector space $U$ for which $\wedge$ is the associative symmetric product in $SU$. This assumption allows us to operate easily in coordinates. Let $\{q_1, \ldots, q_n\}$ be a homogeneous basis for $U$. Then elements of $V$ are polynomials in the variables $\{q_i\}$, the wedge product is the ordinary graded commutative product of polynomials, and we may abbreviate $q_i\wedge
q_j$ by $q_iq_j$. The operator $d$, as a derivation of the product, is a first order differential operator, and $\Delta$, as a BV operator, is a second order differential operator. Such operators have expressions in the basis $\{q_i\}$ as $$\begin{gathered}
\label{def_of_d}
d=\sum_{i=1}^n a_i(q) \frac{\partial}{\partial q_i} \intertext{ and
} \Delta=\sum_{i,j=1}^n b_{ij}(q)\frac{\partial^2}{\partial q_j
\partial q_i}\label{def_of_delta}\end{gathered}$$ $a_i(q)\in V$ for each $i$ and $b_{ij}(q)\in V$ with $b_{ij}(q)=-(-1)^{|q_i||q_j|}b_{ji}(q)$ for each $i$ and $j$. From such a dBV algebra, we now define a background $B_{V,d,\Delta,\wedge}=(P,m,N,\varphi)$.
The ring $P$.
-------------
We define the ring $P$ as $P=W(U)$, a graded Weyl algebra on the vector space $U$, defined to be $$\begin{gathered}
W(U):=T(U\oplus U^*)[[\hbar]] \big\slash J \intertext{ where $J$ is
the left ideal generated by} [q,q'],\quad [p,p'],\text{ and }
[q,p]=\hbar p(q).\end{gathered}$$ for any $q, q' \in U$, and $p, p' \in U^*$, and $[\,,\,]$ is the graded commutator. The product in $P$ is induced by the tensor algebra. As a $k[[\hbar]]$ module, $P\simeq (S(U\oplus
U^*))[[\hbar]]$ and as a $k$-algebra $P/\hbar P \simeq S(U\oplus
U^*).$ In coordinates, say $\{q_i\}$ is a basis for $U$ and $\{p^i\}$ is the dual basis of $U^*$, each elments of $P$, which is an equivalence class in $T(U\oplus U^*)[[\hbar]]$ is represented uniquely by a polynomial in the $\{q_i\}$ and $\{p^i\}$ in normal ordering with “all the $p$’s on the right.” Multiplication is carried mechanically out by concatinating the two polynomials and using the commutation relation $p^iq_j=(-1)^{|q_j||p^i|}q_jp^i-\hbar\delta^i_j$ repeatedly until obtaining a polynomial with the $p$’s on the right, thereby obtaining the unique representative for the product in $P$.
The Weyl algebra $P=W(U)$ may be familiar as the “canonical quantization” of the symplectic vector space $U\oplus U^*$. The ring $S(U\oplus U^*)$ can be thought of as the ring of (polynomial) functions on a symplectic space, with the Poisson bracket defined by $\{q_i,p^j\}:=\delta_i^j$. Then $P$, isomorphic as a $k[[\hbar]]$ module to $S(U\oplus U^*)[[\hbar]]$, is a one-parameter deformation of $S(U\oplus U^*)$ over $k[[\hbar]]$ with the infinitesimal deformation being the Poisson bracket. The product we defined in $P$ might be called a “star product.”
While it is easy (and tedious) to perform calculations in $P$ using explicit polynomials in $p$ and $q$, one can understand the elements in $P$ and the product in $P$ in a coordinate free way. Observe that $$P/\hbar P\simeq S(U\oplus U^*) \subseteq V\otimes V^* = \hom_k(V,V),$$ and hence, as a $\Lambda$ module, $$\label{weyl_hom}
W(U)\subseteq
\hom_k(V,V)[[\hbar]].$$ So, without choosing a basis for $U$, one can regard elements of $P$ as power series in $\hbar$ with coefficients that are the finite $k$-linear operators on $V$; i.e. finite sums of operators $S^jU \to S^kU$. The identification in equation is as free $\Lambda=k[[\hbar]]$ modules; to acquaint oneself with the multiplication in $P$ from this point of view, we expand $$\alpha\beta = \alpha\wedge \beta + \hbar
\{\alpha,\beta\}+\mathcal O(\hbar^2)$$ for $\alpha,\beta\in
\hom_k(V,V)$. We interpet some of the $\hbar$ terms on the right for a couple of illustrative cases:
Suppose $\alpha:S^jU\to S^kU$ and let $\beta:S^iU \to S^jU$. Then, $$\alpha\beta = \alpha\wedge \beta+(\hbar \text{ terms})+(\hbar^2 \text{
terms})+\cdots +\hbar^j (\alpha \circ \beta)$$ where $\alpha\wedge \beta: S^{i+j}U \to S^{j+k}U$ is the graded commutative wedge product of linear tranformations and $\alpha \circ
\beta: S^i U\to S^kU$ is composition in $\hom(V,V)$. Hence, $\star$ extends the commutative associative product on $\hom(V,V)$ toward the noncommutative composition of homomorphisms. The $\hbar^r$ terms for $r=2,3, \ldots,j-1$ can be understood in terms of some multiple-gluing operations using a topological model. In this model $\hbar$ is related to genus or Euler characteristic [@DTT].
Let $\alpha:S^iU\to U$ and let $\beta:S^jU \to U$. Then, $$\label{item2} [\alpha,\beta]= \hbar
\underline{\left[\overline{\alpha}, \overline{\beta}\right]_G}$$ where the bracket on the left is the graded commutator in $P$ and the bracket $[\,,\,]_G$ on the right is understood by way of the Gerstenhaber bracket in the ${\operatorname{Coder}}(V)$, the Hochschild complex of the algebra $(V[-1], d, \wedge)$. That is, to compute $[\alpha,\beta]$ in $P$, one lifts $\alpha$ and $\beta$ to coderivations $\overline{ \alpha}, \overline{\beta}: V\to V$, takes the graded commutator of those two coderivations to obtain a coderivation $\left[ \overline{ \alpha}, \overline{\beta}\right]_G:
V \to V$, with the final result of $[\alpha,\beta]$ being $\hbar$ times the component $S^{i+j-1}U\to U$ that determines the coderivation $\left[\overline{\alpha}, \overline{\beta}\right]_G$.
The structure $m$.
------------------
We define $m\in P^1$ $$m=\sum_{i=1}^n
a_i(q)p^i +\sum_{i,j=1}^n b_{ij}(q)p^ip^j$$ where $a_i$ and $b_{ij}$ are as in equations and . The fact that $(V,d,\Delta,\wedge)$ is a BV algebra implies that as an element of $P$, $m^2=0$.
The module $N$.
---------------
We have $U^*\subset P$ and the left ideal $PU^*$. The quotient $P/PU^*$ is naturally a $P$ module and we define $N=P/PU^*$. Use $|\alpha{\rangle}$ to denote the image of $\alpha \in
P$ in $P/PU^*$. There is an isomorphism of $k[[\hbar]]$ modules $V[[\hbar]]\simeq N$. In coordinates, this isomorphism is simple: an element of $V[[\hbar]]$ is a polynomial in $q$ which represents a class of polynomials in $P$, and also a class in $P/PU^*$ since $V$ has no $p$’s. To illustrate, a typical element of $P$ might be $\alpha=2\hbar q_1q_2+q_1p^2p^3+\hbar^2 p^3$ and typical element of $N$ might be $\gamma=|2\hbar q_1-q_1q_3{\rangle}$. Then (say $q_1$ and $q_3$ are even and $q_2$ is odd), $$\begin{aligned}
\alpha \cdot \gamma &=|(2\hbar q_1q_2+q_1p^2p^3+\hbar^2 p^3) (2\hbar
q_1-q_1q_3){\rangle}\\
&=|(4\hbar^2 (q_1)^2q_2-2\hbar (q_1)^2q_2q_3+2\hbar (q_1)^2 p^2
p^3-\hbar (q_1)^2p^2+\hbar^2q_1q_3p^3-\hbar^3q_1{\rangle}\\
&=|(4\hbar^2 (q_1)^2q_2-2\hbar (q_1)^2q_2q_3-\hbar^3q_1{\rangle}.\end{aligned}$$
The vacuum $\varphi$
--------------------
The vaccum $\varphi$ is defined to be $|1{\rangle}$. Note that $m\varphi=m|1{\rangle}=|m{\rangle}=|0{\rangle}$, as required.
Relating the background and the dBV algebra
-------------------------------------------
We begin with a summary of the results. It will be convenient to state the summary in terms of several algebras built from $(V,d,\Delta, \wedge)$: $$\begin{aligned}
C&:=(V,d,\wedge) \text{ is a commutative dga over $k$}\\
L&:=(V[-1],d,(\,,\,) \text{ is a dgLa over $k$}\\
L^\hbar&:=(V[-1][[\hbar]], d+\hbar\Delta, (\,,\,)) \text{ is a dgLa
over
$k[[\hbar]]$}\\
H&:={\operatorname{Ker}}(d:V \to V)/{\operatorname{Im}}(d:V\to V) \text{ is
a vector space over $k$}\\
H^\hbar&:={\operatorname{Ker}}(d-\hbar\Delta:V[[\hbar]] \to
V[[\hbar]])/{\operatorname{Im}}(d-\hbar\Delta :V[[\hbar]]\to V[[\hbar]]) \text{ is a
$k[[\hbar]]$ module.}\end{aligned}$$ Note that $H\simeq H(C)\simeq H(L)[1]$. By the minimal model theory for $L_\infty$ algebras, there is an $L_\infty$ structure on $H[-1]$, unique up to quasi-isomorphism, that is quasi-isormorphic to $L$. Note also, $H^\hbar\simeq H(L^\hbar)[1]$, so there is an $L_\infty$ structure (over $k[[\hbar]]$) on $H^\hbar[-1]$ quasi-isomorphic to $L^\hbar$.
On the quantum background side, we denote $B_{V,d,\Delta,\wedge}$ simply by $B$, and we have $$\begin{aligned}
(R&=k[[H^*,\hbar]],\delta):=\text{ a ring representing ${\operatorname{\mathbf{QFT}}}_B$}\\
\Pi&\in {\operatorname{\mathbf{QM}}}_B(R,\delta):=\text{ versal solution to quantum master
equation}\\
(S&=k[[H^*]],\delta_0):=(R\otimes_\Lambda k,
\delta\otimes_\Lambda k) \text{
is the ``classical'' $\hbar=0$ part of $R,\delta$}\\
\Pi_0&\in {\operatorname{\mathbf{QM}}}_B(S,\delta_0):=\text{ the image of $\Pi$ in $S$ is
the ``classical'' part of $\Pi$}.\end{aligned}$$ The most immediate relationships between the background $B$ and the dBV algebra $(V,d,\Delta, \wedge)$ are summarized in Figure 1.
\[table1\]
$H(B)=$ the tangent space to ${\operatorname{\mathbf{QFT}}}_B$ $H$
---------------------------------------------------------------- ---------------------
$D(B)=$ the Dirac space of $B$ $H^\hbar$
The ring $S$ $k[[ H^*]]$
The ring $R$ $k[[ \hbar, H^* ]]$
$(S,\delta_0)$
$\Pi_0$
$(R,\delta)$
$\Pi$
In the case that $B$ is smooth formal, we also have $L$ and $L^\hbar$ smooth formal and the moduli space for ${\operatorname{\mathbf{QFT}}}_B$ and ${\operatorname{\mathbf{Def}}}_L$ are the identified. On the BV side, we may define the following algebras, which can be thought of as families of algebras fibered over the moduli space. $$\begin{aligned}
C_t&:=(V\otimes S,d+(\Pi_0,\,),\wedge) \text{ is a commutative dga
over $k$}\\
L_t&:=(V[-1]\otimes S,d+(\Pi_0,\,),(\,,\,) \text{ is a dgLa over $k$}\\
L_t^\hbar&:=(V[-1][[\hbar]]\otimes R, d+\hbar\Delta +(\Pi,\,),
(\,,\,)
\text{ is a dgLa over $k[[\hbar]]$}\\
H_t&={\operatorname{Ker}}(d+(\Pi_0,\,):V\otimes S\to V\otimes
S)/{\operatorname{Im}}(d+(\Pi_0,\,))
\text{ is an $S$ module}\\
H^\hbar_t&={\operatorname{Ker}}(d-\hbar\Delta+(\Pi,\,):V[[\hbar]]\otimes R\to
V[[\hbar]]\otimes R)/{\operatorname{Im}}(d+(\Pi_0,\,)) \text{ is an $R$ module}\end{aligned}$$ We note that $H_t\simeq H(C_t)$. So, $H_t$ is a commutative associative algebra. Furthermore, by the minimal model theory, $H_t$ has a $C_\infty$ structure, quasi-isomorphic to $C_t$.
On the quantum background side, we have the quantum superconnection $$\begin{aligned}
\nabla&=d_{dR}+A^1+A^2+\cdots:=\text{ the quantum superconnection}\\
\nabla^{1}&=d_{dR}+A^1:=\text{ the small quantum connection}\\
\nabla_0&=d_{dR}+A_0^1+A_0^2+\cdots:=\text{ the ``classical''
$\hbar=0$
part of $\nabla$ }\\
\nabla_0^{1}&=d_{dR}+A_0^1:=\text{ the ``classical'' $\hbar=0$ part of $\nabla^{1}$}\\\end{aligned}$$ We conjecture that there are special coordinates for any smooth formal background. What that means in the present example of a dBV algebra is that there is a particular solution $\Pi$ to the quantum master equation for which the associated flat quantum connection $\nabla$ has no $\hbar$ dependence. This is proved in [@js3] in a certain semiclassical case. When $\nabla^{1}$ does not depend on $\hbar$, the flatness equation expressed in terms of the conncection one form $A$ $$\hbar d_{dR}A+A^2=0$$ decouples into two equations $d_{dR}A=0$ and $A^2=0$. The equation $A^2=0$ together with the torsion-freeness of $\nabla^{1}$ implies that $A$ defines a family of commutative, associative algebra structures on $H$ parametrized by $H$. The condition that $dA=0$ gives additional constraints on the way theses algebra structures vary with their parameters, implying that $H$ has the structure of a Frobenius manifold without a metric [@Manin]. We cojecture that the superconnection in special coordinates affords $H$ with the structure of a minimal algebra over an appropriate resolution of the Frobenius manifold structure.
More general Weyl-type backgrounds
----------------------------------
We illustrate so many details about a background arising from a dBV algebra because the situation may be familiar, but we emphasize that it is only an example. Notice that $m\in P$ obtained from $d$ and $\Delta$ in a dBV algebra is quite special: it is quadratic in $p$ and has no dependence on $\hbar$. One may more generally consider a background $B=(P,m,N,\varphi)$ where $P=W(U)$ is the Weyl algebra on a graded vector space $U$, $m$ is any element, at least linear in $p$ (in order to annihilate the vacuum), satisfying $m^2=0$, $N=SU[[\hbar]]$, and $\varphi=|1{\rangle}$. There seem to be many interesting examples [@DTT]. Furthermore, for a fixed $m$, one may consider different modules, possibly highlighting different aspects of the same structure.
Prospectus
==========
In conclusion, let us make some very brief remarks about future directions, to which we are now turning.
In this paper, we constructed a flat quantum superconnection over the moduli space if the background is smooth formal. Our attention is now focused on the non smooth formal case. In the general case, we intend to construct a quantum connection $\nabla$ which interacts with the differential $\delta$ from proposition \[main\_thm\]. This seems to be necessary to develop a good minimal model theory for quantum backgrounds.
We imagine the ideas in this paper can be applied in two different ways. The first application is rather direct: apply the framework described here to mathematical situations that fit. One noteworthy example arises in symplectic field theory. One might summarize the output of symplectic field theory as an element of square zero in a particular noncommutative ring. In other words, the output of symplectic field theory is a rather good match to the input data of a background. It would be interesting to construct the ${\operatorname{\mathbf{QFT}}}$ moduli space and quantum connection $\nabla$ in this example and interpret these structures in terms of symplectic topology.
A second application is a kind of quantized deformation theory. By this we mean start with a classical mathematical structure and produce a quantum background $B$ whose classical limit is the $L_\infty$ algebra $L$ controlling the given structure’s classical deformations. By “classical limit” we mean that $B$ and $L$ share a relationship much the same as the relationship between $B$ and $L$ exemplified in Section \[Application\]. Then, the ${\operatorname{\mathbf{QFT}}}_B$ moduli space enriches the classical moduli space in the $\hbar$ direction. One advantage is that the superconnection $\nabla$ when expressed in special coordinates, which are invisible from the classical deformation theory, is expected to encode invariants of the original structure.
[10]{}
S. Barannikov. Quantum periods. I. Semi-infinite variations of Hodge structures. , (23):1243–1264, 2001.
S. Barannikov and M. Kontsevich. Frobenius manifolds and formality of [L]{}ie polyvector fields. , (4):201–215, 1998.
E.H. Brown. Abstract homotopy theory. , 119(1):79–85, 1965.
P. Dirac. . Oxford University Press, 1930.
G. Drummond-Cole, J. Terilla, and T. Tradler. Algebras over $cobar(cofrob)$, arXiv:0807.1241v1, 2008.
B. Dubrovin. Geometry of 2[D]{} topological field theories. In [*Integrable systems and quantum groups*]{}, volume 1620 of [ *Lecture notes in mathematics*]{}, pages 120–348. Springer-Verlag, 1996.
M. Kontsevich. Deformation quantization of [P]{}oisson manifolds, [I]{}. math.QA/9709040, 1997.
M. Manetti. Extended deformation functors. , 14:719–756, 2002.
Y. Manin. . Amer. Math. Soc., 1999.
B. Mazur. Introduction to the deformation theory of [G]{}alois representations. In [*Modular Forms and [F]{}ermat’s last theorem*]{}, pages 243–311. Springer-Verlag, 1997.
J.-S. Park. Lectures at [CUNY]{} [G]{}raduate [C]{}enter, 2004.
J.-S. Park. Semi-classical quantum field theories and [F]{}robenius manifolds. , 81(1):41–59, 2007.
J.-S. Park. Flat family of [QFT]{}s and deformations of d-algebras, hep-th/0308130, 2003.
M. Schlessinger. Functors of [A]{}rtin rings. , 130:208–222, 1968.
M. Schlessinger and J. D. Stasheff. Deformation theory and rational homotopy theory. preprint, 1977.
J. Terilla. Smoothness theorem for differential [BV]{} algebras. , 1:693–702, 2008.
[^1]: The first author supported by KOSEF Interdisciplinary Research Grant No. R01-2006-000-10638-0.
[^2]: The second author supported in part by a grant from The City University of New York PSC Research Award.
|
---
author:
- |
Yali Bian[^1]\
Virginia Tech
- |
Michelle Dowling[^2]\
Virginia Tech
- |
Chris North[^3]\
Virginia Tech
bibliography:
- 'template.bib'
title: Evaluating Semantic Interaction on Word Embeddings via Simulation
---
\[sec:intro\]
Semantic interaction (SI) [@Endert:he; @Endert2015] is an interaction technique for non-experts to interact with the underlying algorithms of visual analytics (*VA*) systems [@cook2005illuminating]. With SI, analysts can focus on reasoning and manipulating data in the 2D spatialization instead of directly interacting with the underlying machine learning (*ML*) models [@amershi2012regroup]. It is, therefore, the system’s responsibility to tune the ML models by capturing users’ interactions and inferring the analyst’s intent [@self2016bridging]. These intents are then translated to updated model parameters via semi-supervised machine learning techniques (ML) [@Endert2015], such as metric learning [@6400486]. Hence, analysts can remain concentrated on their sensemaking activities [@pirolli_2005].
In this paper, we investigate two questions about SI: (1) Can word embeddings help SI learn user’s interactive intents better than traditional features? (2) How can we comparatively evaluate alternative SI models, such as in question (1)?
The human-centred approach [@boukhelifa2018evaluation] is the primary evaluation method for SI systems. The SI system is measured by the utility and effectiveness of the application for end-users through studies of human subjects. For example, ForceSPIRE [@Endert:2012:SIV:2207676.2207741] is a visual text analysis system powered by SI, and its effectiveness was measured by analysts’ performance on an intelligence analysis task. This approach is highly dependent on human feedback, which leads to several challenges when comparing the effectiveness of different SI models because: user interactions are difficult to precisely replicate; SI interactions are incremental and iteratively build; does not scale well to compare many model alternatives as needed in ML; and usage differences can mask subtle model performance differences. To cope with these problems, we design and perform an additional quantitative algorithm-centred analysis that simulates human interactions as a complement to human-centered evaluation. In the evaluation, we use the labelled text datasets as the ground truth, since there is no ground truth for the user’s complex intents and concepts. Then the ground truth is used to simulate user’s interactions, and we calculate and compare the accuracy of trained models.
{width="\textwidth"}
In this paper, we first describe two alternative SI models for visual text analysis using different data features as inputs (embedding vectors, and keywords vectors). We then implement a generalized visual text analysis prototype with SI that can take both embedding vectors and keywords vectors as features. We then demonstrate how the two kinds of evaluation methods together provide a complementary evaluation and comparison of these two SI models.
Semantic Interaction with Word Embeddings {#sec:semantic-interaction}
=========================================
We investigate the opportunity of using word embeddings to better support SI. SI’s ability to infer the analyst’s reasoning process is fundamentally limited by the feature space of the underlying machine learning. Text analytics systems with SI, such as ForceSPIRE [@Endert:2012:SIV:2207676.2207741; @7042492] and Cosmos [@DOWLING201949], typically use keywords (such as text terms and phrases) as data features ($SI_{keyword}$ in Fig. \[fig:SI\_pipeline\]a), known as bag-of-words [@zhang2010understanding]. Recently, word embedding techniques [@mikolov2013distributed; @collobert2008unified; @8416973; @Peters:2018], also known as deep learning representations [@LeCun:2015dt; @athiwaratkun2015feature; @7797053; @Yosinski:wc], have shown significant advantages in numerous tasks in natural language processing and information retrieval [@Kusner:2015:WED:3045118.3045221; @Socher13parsingwith; @Le:2014:DRS:3044805.3045025] over bag-of-words features. Therefore, the combination of SI with word embedding ($SI_{embedding}$ in Fig. \[fig:SI\_pipeline\]) might enable better learning ability than $SI_{keyword}$ to update the underlying machine learning models [@mikolov2013efficient]. The intuition is that word embeddings could represent more abstract concepts that are closer to modeling human cognitive reasoning. To test the hypothesis, we focus on the comparisons of $SI_{embedding}$ with $SI_{keyword}$ through the same SI pipeline but with different document features as input:
- **Keyword features used in $\mathbf{SI_{keyword}}$:** We use TF-IDF values as the keyword features, and word hashing to compress the large number of words from the document collection into 300 dimensions.
- **Embedding features used in $\mathbf{SI_{embedding}}$**: We use the pre-trained GloVe model [@pennington2014glove] with 300 dimensions to extract embedding features, and use the “basic averaged word embeddings” method to compute average word embedding for all words in a document [@Le:2014:DRS:3044805.3045025].
Application Prototyping {#sec:implementation:prototyping}
-----------------------
As shown in Fig. \[fig:SI\_pipeline\], $SI_{embedding}$ has a similar structure with $SI_{keyword}$ as they both use numerical vectors to represent documents (embedding vectors, and TF-IDF vectors). We are able to switch between $SI_{keyword}$ and $SI_{embedding}$ in the same SI prototype. We build the prototype upon the foundation of Andromeda [@self2018observation], a visual analytics tool for exploring high-dimensional data projections. The prototype can use either bag-of-words or embedding vectors as features and update the feature weights to capture analysts’ intents. For bag-of-words, the system will up-weight the shared words in the dragged documents in response to the analyst dragging two or more documents closer together. For embedding vectors, the system will up-weight the dimensions of the embedding vectors with similar patterns in the dragged documents.
As shown in Figure \[fig:case-study\]-1, the visualization is a projection of documents. The distance between documents reflects their relative similarity according to a weighted distance metric over their features. At first, documents scatter in the workspace based on their features and an equally-weighted dimension reduction of the high-dimensional representation. If two documents are positioned close to each other in this initial projection, it implies that these two documents are similar based on all vector features. Semantic interaction enables analysts to directly manipulate the projection by moving the documents to express their own domain knowledge about desired similarities. With this interaction, analysts can express the semantic relationships between documents, which thereby informs the underlying weighted distance model and updates the projection. Through this, the projection can be customized according to the learned intent of the analyst.
{width="\textwidth"}
**Model** **$T_{rec}$** **$T_{religion}$** **$T_{sys}$** **$T_{vis}$**
------------------ ------------------------ ------------------------ ------------------------ -------------------- -- -- --
$SI_{embedding}$ (**0.921**, **0.812**) (**0.829**, **0.773**) (**0.895**, **0.774**) (0.958, **0.809**)
$SI_{keyword}$ (0.497, 0.570) (0.576, 0.581) (0.511, 0.584) (**0.961**, 0.793)
[0.245]{} {width="\textwidth"}
[0.245]{} {width="\textwidth"}
[0.245]{} {width="\textwidth"}
[0.245]{} {width="\textwidth"}
Evaluating SI Embedding
=======================
Traditionally, the human-centred approach is a good method to test if the SI model is an effective application for analysts to perform sensemaking tasks. However, comparing two SI models ($SI_{keyword}$ and $SI_{embedding}$) through this approach is inadequate. Thus, we also design a replicable quantitative study to validate $SI_{embedding}$ in comparison to $SI_{keyword}$ in 4 simulated text analysis tasks of different difficulty levels.
User-Centered Qualitative Analysis {#sec:case-study}
----------------------------------
In this section, we present a practical use case about intelligence analysis as our qualitative study to compare $SI_{keyword}$ against $SI_{embedding}$. For this analysis, we engaged an expert in intelligence analysis to review the visual result generated from both $SI_{embedding}$ and $SI_{keyword}$ to provide qualitative feedback, and provide grounding for the quantitative analysis.
### Crescent Dataset
The crescent dataset [@crescent] has 42 fictional intelligence reports regarding a coordinated terrorist plot in Boston, New York, and Atlanta. Only 24 reports are relevant to these plots. The task for our qualitative analysis is to identify these three terrorist threats by using $SI_{embedding}$ or $SI_{keyword}$.
By moving documents in the visualization according to their perceived similarity, analysts express their reasoning process to the system. The ground truth of this task is as follows, which we use to measure the accuracy of the models: **Boston:** cia7, cia8, cia9, cia10, cia11, fbi1, fbi2, fbi21, se3, se4; **New York:** cia11, fbi1, fbi10, fbi13, fbi16, fbi22, fbi25, se2, se3, se4; **Atlanta:** cia4, cia11, fbi1, fbi7, fbi11, fbi15, fbi17, fbi19, fbi20, fbi24, se4, se5; **Irrelevant:** fbi3, fbi4, fbi5, fbi6, fbi8, fbi9, fbi14, fbi18, fbi23, cia1, cia2, cia3, cia5, cia6, se1.
### Case study with $SI_{embedding}$
In this first case study, the analyst performs the task by using the prototype system with embedding as the document features. As shown in Figure \[fig:case-study\]-1, the analyst updates the layout to reflect the perceived similarities between the documents, grouping three documents about “New York” to the top left region of the projection, indicated by the yellow arrows in Figure \[fig:case-study\]-2, three documents about “Atlanta” to the bottom left region indicated by blue arrows, and three documents about “Boston” to the bottom right part indicated by red arrows.
After the layout updates, some semantic relationships are revealed (Figure \[fig:case-study\]-3.a). The left cluster in red circle contains documents about “Boston,” the top right cluster in yellow circle contains documents about “NYSE,” and the bottom right cluster in blue circles contains documents about “Atlanta.” We found that there are semantic mappings between this updated layout and the ground truth; all relevant reports that belong to single plot are well placed into each cluster. Reports about the coordination between two or more plots are between clusters. For example, the report “se3” contains information regarding both the “Boston” and “NYSE” plots, so the document is located between these two clusters. Thus, $SI_{embedding}$ can capture analysts’ intents and update the model features accordingly. The updated layout then reflects the semantic meanings behind the intents.
### Case Study Using $SI_{keyword}$
Mirroring our case study with $SI_{embedding}$, the analyst performs similar interactions with the prototype. As shown in Figure \[fig:case-study\]-3.b, from the updated layout, there are no clear boundaries between different plots. Even after continued interactions, the model is unable to properly differentiate between the three terrorist plots. For example, documents about “Atlanta” and “NYSE” are mixed together. Furthermore, the documents that are pulled out of the big central cluster in panel 4 are primarily the user interacted documents from panel 2. This might indicate that the model is over-fitting based on some specific unimportant keywords.
### Expert Review
Besides comparing with the ground truth as we mentioned in Section \[sec:case-study\], we also asked an expert familiar with the dataset to evaluate the updated layout generated in both $SI_{embedding}$ and $SI_{keyword}$ (as shown in Figure \[fig:case-study\]-3.a and Figure \[fig:case-study\]-3.b). The expert noted that the $SI_{embedding}$ layout provides more meaningful information about there plots than $SI_{keyword}$ layout because the documents from different plots are grouped in different regions. In contrast, the layout generated by $SI_{keyword}$ makes it difficult to clearly distinguish between these plots. For example, documents in the the $SI_{embedding}$ layout regarding coordination between the plots are placed between the relevant groups. This is exemplified by how the report “se3” is between the “NYSE” group and the “Boston” group since it discusses how terrorists involved in both plots communicated. In the layout produced by $SI_{keyword}$, this document lies at the bottom right of the projection, which does not have such an immediately apparent and relevant semantic meaning.
Additionally, several irrelevant reports were distinguished from the groups in the layout produced by $SI_{embedding}$, such as “cia5”, “fbi23” and “fbi14,” even though they share many keywords with other relevant reports. In contrast, the layout produced by $SI_{keyword}$ only causes one irrelevant report (“cia2”) to be pulled away from other reports.
Algorithm-centered Quantitative Analysis {#sec:quantitative-analysis}
----------------------------------------
In this subsection, we describe our quantitative comparison between $S_{keyword}$ and $SI_{embedding}$. We begin with the experimental design, followed by a description of the datasets we choose for the experiment. Finally, we show the evaluation results and discuss the SI inference abilities of the two models.
### Experiment Setup
We need ground truth to evaluate the underlying algorithms used in $SI_{keyword}$ and $SI_{embedding}$ quantitatively. There is no ground truth about users’ complex intents and interactions, such as a labeled intelligence hypothesis in a 2D projection. However, there are sufficient text datasets labelled for classification, such as 20newsgroup (section \[sec:evaluation:quantitative-analysis:dataset\]). The labels can be used to simulate different positions in the projection: for example, documents with negative label can be located at the top left part of the spatial layout, and documents with positive label can be located at the bottom right part. Therefore, we can use classification datasets mapped in the 2D projection as the ground truth, and we only evaluate the inference ability to capture the intents embedded in classification tasks. Then, subsets of the documents from different positions in the 2D projection will be picked and used as simulated semantic interactions from analysts.
Furthermore, since *incremental formalism* [@Shipman1999] is an important aspect of semantic interaction, we must test the models’ ability to learn incrementally over the course of many interactions. To simulate and evaluate the *incremental inference* ability of these models, the selected interactions will be iteratively passed into the SI model. Through these iterative SIs, the underlying models incrementally update to display better results to the analyst.
For example, in the task $T_{vis}$ (sec. \[sec:evaluation:quantitative-analysis:dataset\]), we simulate that the analyst wants to organize documents by separating the two conferences (InfoVis and VAST) in the 2D projection as ground truth: InfoVis documents placed at top left of the projection, and VAST documents placed at bottom right. Then in each loop, five documents from InfoVis collection, and five documents from VAST collection will be picked and simulated as semantic interactions: moving five documents from InfoVis to $(0, 0)$ position and moving another five documents from VAST to $(1, 1)$ position. In each iteration, the SI model ($SI_{embedding}$ or $SI_{keyword}$) is executed to infer updated model parameters from the simulated semantic interactions. Then the inference ability of each model can be calculated by the quality of the underlying model in classifying InfoVis documents from VAST based on the ground truth.
### Datasets and Tasks {#sec:evaluation:quantitative-analysis:dataset}
To perform this quantitative analysis, we use two datasets: 20News dataset [@Lang95] and Vispubdata dataset [@Isenberg:2017:VMC]. The 20 Newsgroup dataset is a collection of newsgroups posts on 20 topics. Based on this dataset, we create three tasks to simulate users’ intents. Each of the 3 tasks are to separate the documents into 2 groups based on 2 pre-determined topics: **$T_{rec}$**: 594 documents from “rec.autos” and 600 documents “rec.motorcycles”; **$T_{sys}$**: 578 documents from “comp.sys.mac.hardware” and 592 documents from “comp.sys.ibm.pc.hardware”; **$T_{religion}$**: 379 documents from “talk.religion.misc” and 599 documents from “soc.religion.christian”. The Vispubdata dataset contains information on IEEE Visualization (IEEE VIS) publications. We select the abstracts of the academic papers published in two conferences from the dataset: InfoVis (IEEE Information Visualization), and VAST (IEEE Visual Analytics Science and Technology). The documents from different conferences are used to represent different concepts (users’ intents), We create one task based on these abstracts to separate InfoVis papers from VAST papers, defined as $T_{vis}$, including 397 papers from InfoVis and 531 papers from VAST.
### Results
In this experiment, we have run the $SI_{embedding}$ and $SI_{keywrod}$ model multiple times on the four tasks with ground truth. After each interaction (involving five document movements), we calculate the model accuracy using the k-nearest-neighbour (kNN) classifier [@Cover:2006:NNP:2263261.2267456] (as done in Dis-Function [@6400486]): using the cross-validation over the data and set k to 3. The average accuracy of the two models over all iterations for fours tasks is shown in Figure \[fig:result\].
Table \[tab:result\] shows the average accuracy of the last interaction (performed multiple times) and average accuracy of $SI_{embedding}$ and $SI_{keyword}$ across the four tasks. $SI_{embedding}$ achieved the highest scores in the first three tasks, indicating that $SI_{embedding}$ provides more accurate representations of the documents after the interactions, indicating its higher inference ability over $SI_{keyword}$. In $T_{vis}$, $SI_{embedding}$ and $SI_{keyword}$ have similar accurracies, meaning their inference abilities were roughly equal in this task.
As shown in the Figure \[fig:result\], we further analyzed the incremental updates from the two models by evaluating each model’s accuracy after every iteration, thereby simulating the incremental analysis process. In the first three tasks, it is shown that using $SI_{embedding}$ can get better performance than $SI_{keyword}$ over all the iteration loops. The accuracy of $SI_{embedding}$ generally increases with each interaction, showing incremental changes to the model’s representation of the user’s intent over time. In contrast to the first three tasks, however, in $T_{vis}$, both $SI_{emebedding}$ and $SI_{keyword}$ have similar accuracy over the interactions. Finally, the desired accuracy should be as close to 1 as possible. $T_{vis}$ is a relatively “easy” task, as reflected by the final accuracies of both models close to 1.0. These results indicate that $SI_{keyword}$ performs well in relatively easy tasks.
Discussion
----------
Our experimental results substantiate our hypothesis that $SI_{embedding}$ is more effective than $SI_{keyword}$ for modeling user intent, and better supports incremental formalism based on users iterative interactions. The human-centered qualitatively evaluation method shows $SI_{embedding}$ is more effective than $SI_{keyword}$ in real world analysis tasks. This provides a direct evaluation of SI models by the observed utility and effectiveness for end-users. The quantitative analysis offers a complementary approach, and provides replicable and scalable evaluations for SI models. It provides more stable and detailed feedback about the SI model performance in tasks of different difficulty levels. Thus, we confirmed that word embedding can better support SI by better capturing the users’ high-level interactive intents.
Conclusion
==========
In this work, we presented $SI_{embedding}$ as an alternative to the traditional bag-of-words model ($SI_{keyword}$) often used in visual text analytics systems. To make a complete and convincing comparison between $SI_{embedding}$ and $SI_{keyword}$, we performed a quantitative evaluation by simulating analysts’ interactions and calculating the accuracy of the underlying trained ML models, as a complement to the traditional user-centered qualitative evaluation. Results indicate that deep learning distributed representations, such as word embedding, can be exploited to improve interactive visual analytics methods such as semantic interaction.
[^1]: e-mail: yali@vt.edu
[^2]: e-mail:dowlingm@vt.edu
[^3]: e-mail:north@vt.edu
|
---
abstract: 'We will use Thue-Siegel method, based on Padé approximation via hypergeometric functions, to give upper bounds for the number of integral solutions to the equation $|F(x , y)| = 1$ as well as the inequalities $\left|F(x , y)\right| \leq h$, for a certain family of irreducible quartic binary forms.'
address: ' Max-Planck-Institut für Mathematik Vivatsgasse 753111 Bonn Germany '
author:
- Shabnam Akhtari
title: 'The Method Of Thue-Siegel For Binary Quartic Forms'
---
\[section\] \[thm\][Proposition]{} \[thm\][Lemma]{} \[thm\][Corollary]{} \[thm\][Conjecture]{}
Introduction
============
In 1909, Thue [@Thu2] proved that if $F(x , y)$ is an irreducible binary form of degree at least $3$ with integer coefficients, and $h$ a nonzero integer, then the equation $F(x , y) = h$ has only finitely many solutions in integers $x$ and $y$ .
In this paper we will consider irreducible binary quartic forms with integer coefficients, i.e. polynomials of the shape $$F(x , y) = a_{0}x^{4} + a_{1}x^{3}y + a_{2}x^{2}y^{2} + a_{3}xy^{3} + a_{4}y^{4}.$$ The discriminant $D$ of $F$ is given by $$D = D_{F} = a_{0}^{6} (\alpha_{1} - \alpha_{2})^{2} (\alpha_{1} - \alpha_{3})^{2} (\alpha_{1} - \alpha_{4})^{2} (\alpha_{2} - \alpha_{3})^{2} (\alpha_{2} - \alpha_{4})^{2} (\alpha_{3} - \alpha_{4})^{2} ,$$ where $\alpha_{1}$ , $\alpha_{2}$, $\alpha_{3}$ and $\alpha_{4}$ are the roots of $$F(x , 1) = a_{0}x^{4} + a_{1}x^{3} + a_{2}x^{2} + a_{3}x + a_{4} .$$ Here, we will recall some well-known fact about the invariants of quartic forms. We refer the reader to [@Cre2] for more details. The invariants of $F$ form a ring, generated by two invariants of weights $4$ and $6$, namely $$I = I_{F} = a_{2}^{2} - 3a_{1}a_{3} + 12a_{0}a_{4}$$ and $$J = J_{F} = 2a_{2}^{3} - 9a_{1}a_{2}a_{3} + 27 a_{1}^{2}a_{4} - 72 a_{0}a_{2}a_{4} + 27a_{0}a_{3}^{2}.$$ These are algebraically independent and every invariant is a polynomial in $I$ and $J$. For the invariant $D$, we have $$27D = 4I^3 - J^2.$$ In what follows, we will just consider the forms $F$ for which the quantity $J_{F}$ is $0$; i.e. for which we have $$27D = 4I^{3}.$$
Let $h$ be a positive integer. The number of solutions in integers $x$ and $y$ of the equation $$\label{12}
\left|F(x , y)\right| = h.$$ will be the focus of our study in this paper. In Section \[TS2\], we will show that to apply a classical theorem of Thue [@Thu2] from Diophantine approximation to a quartic form $F$, one needs to assume $J_{F} = 0$.
\[main2\] Let $F(x , y)$ be an irreducible binary quartic form with integer coefficients and positive discriminant that splits in $\mathbb{R}$. If $J_{F} = 0$, then the Diophantine equation $\left|F(x , y)\right| = 1$ possesses at most $12$ solutions in integers $x$ and $y$ (with $(x , y)$ and $(-x , -y)$ regarded as the same) .
In Section \[FWSD2\], we will summarize the result of our computations for binary forms with small discriminant. We will give some examples for quartic binary forms $F(x , y)$ satisfying the hypotheses of Theorem \[main2\], where $\left|F(x , y)\right| = 1$ has $4$ or $3$ solutions in inegers $x$ and $y$. The author is not aware of any quartic binary form $F$ for which $\left|F(x , y)\right| = 1$ has more than $4$ solutions.
In [@Akho2] different methods are used to give an upper bound $61$ upon the number of integral solutions to the equation $\left|F(x , y)\right| = 1$, where $F$ is an irreducible binary quartic form with no restriction on the value of $J_{F}$ and with $\left|D_{F}\right|$ large enough. Moreover, it is shown in [@Akho2] that if the irreducible binary quartic form $F$ splits in $\mathbb{R}$ and have large discriminant, the Diophantine equation $\left|F(x , y)\right| = 1$ has at most $36$ solutions in integers $x$ and $y$.
\[main22\] Let $F(x , y)$ be a reduced irreducible binary quartic form with integer coefficients and positive discriminant that splits in $\mathbb{R}$. If $J_{F} = 0$, then the inequality $\left|F(x , y)\right| \leq h$ possesses at most $12$ co-prime solutions $(x , y)$, with $ |y| \geq \frac{h^{3/4}}{ (3I)^{1/8}}$.
The definition of a reduced form is given in Section \[EF2\]. It turns out that each quartic binary form is equivalent to a reduced one (see [@Cre2]).
One reason for us to be interested in these results, despite what are apparently quite serious restriction upon $F$, is that we know important families of quartic forms with these properties. For example a solution to the equation $aX^4 - bY^2 = 1$ gives rise to a solution to the Thue equation $$x^4 + 4tx^3y - 6tx^2 y^2 - 4t^2xy^3 + t^2y^4 = t_{1}^2,$$ where $t_{1} | t$. We have applied the methods of this paper to treat the above Thue equation in [@Akh2].
The method of Thue and Siegel based on Padé approximation to binomial functions applies to broad families of binomial Thue equations, and both so-called “quantitative” results (see the works of Evertse [@Eve2; @Eve3], for example) as well as effective results (via effective irrationality measure from Baker [@Bak2; @Bak12] onwards) can be obtained from it. This method has also been used to study binary cubic forms with positive discriminant, for decades (see [@Ev2], [@Ben2]). In $1939$, Krechmar [@Kre2] showed that when the discriminant of quartic form $F(x , y)$ is sufficiently large ( $D_{F} \gg h^{216/5}$), the equation (\[12\]) has at most $20$ solutions in integers $x$ and $y$, provided that $J_{F} = 0$ and all roots of $F(x , 1)$ are real numbers. We will use a refinement of Thue-Siegel method by Evertse [@Ev2] to obtain our results.
The Method Of Thue-Siegel {#TS2}
=========================
The main purpose of this section is to explain why we need the restriction $J_{F} =0$ in the statements of our Theorems. The answer is hidden in the method we use, the method of Thue-Siegel. The relationship between a system of approximations to an arbitrary cubic irrationality and Padé approximations to $\sqrt[3]{1 - x}$ was first established by Thue [@thu352]. Siegel [@Sie282; @Sie292] identified approximating polynomials in Thue’s papers [@thu352; @thu382] with hypergeometric polynomials and applied this method to bounding the number of solutions to Diophantine equation $f(x , y) = k$, for certain binary forms $f(x , y)$ of degree $r$. He also established bounds for the number of solutions to $$ax^{n} - by^{n} = c ,$$ where $n \geq 3$ [@Sie302].
\[thue2\] Suppose that $P(x)$ is a polynomial of degree $n$ and there is a quadratic polynomial $U(x)$ such that $$\label{ch2}
U(x)P''(x) - (n -1)U'(x) P'(x) + \frac{n (n - 1)}{2} U''(x) P(x) = 0 .$$ Let $$Y(x) = 2 U(x) P'(x) - n U'(x)P(x)$$ and $$h = \frac{n^{2} - 1}{4} \big{(}U'(x)^{2} - 2U(x) U''(x) \big{)}.$$ Consider the recurrences $$P_{r+1}(x) = k_{r}Y(x)P_{r}(x) - P(x)^{2} P_{r-1}(x) ;$$ $$Q_{r+1}(x) = k_{r}Y(x)Q_{r}(x) - P(x)^{2} Q_{r-1}(x) ,$$ with the initial conditions $$P_{0}(x) = Q_{0}(x) = \frac{2}{3} h ,$$ $$P_{1}(x) = U(x) P'(x) - \frac{n -1}{2} U'(x) P(x) ,$$ $$Q_{1}(x) = x P_{1}(x) - U(x) P(x) ,$$ where $$c_{1} = \frac{3}{2} , \ c_{2} = \frac{2(2n - 1)(2n +1)}{3 (n - 1) (n + 1)} h , \ k_{r}c_{r} = \frac{2r + 1}{2}$$ and $$\frac{c_{r+1} - c_{r-1}}{k_{r}} = 2h \frac{n^{2}}{(n -1) (n + 1)} .$$ Then polynomials $P_{r}(x)$, $Q_{r}(x)$ are of degree $rn + 1$ and satisfy equation $$\alpha P_{r}(x) - Q_{r}(x) = (x - \alpha )^{2r + 1} R_{r}(x)$$ for a polynomial $R_{r}(x)$ .
To apply Theorem \[thue2\] to the polynomial $P(x) = a_{0}x^{4} + a_{1}x^{3} + a_{2}x^{2} + a_{3}x + a_{4}$, suppose that for a quadratic polynomial $U(x) = u_{2}x^{2} + u_{1}x + u_{0}$, we have
$$\begin{aligned}
0 &=& U(x)P''(x) - 3U'(x) P'(x) + 6 U''(x) P(x) \\ \nonumber
&=& (12a_{0}u_{0}-3a_{1}u_{1}+2a_{2}u_{2}) x^{2} + (6a_{1}u_{0}-4a_{2}u_{1}+6a_{3}u_{2}) x \\ \nonumber
& & + 2a_{2}u_{0}-3a_{3}u_{1}+12a_{4}u_{2}.
\end{aligned}$$
This implies that $$\left( \begin{array}{ccc}
12a_{0} & -3a_{1} & 2a_{2} \\
3a_{1} & -2a_{2} & 3a_{3} \\
2a_{2} & -3a_{3}& 12a_{4}
\end{array} \right)
\left( \begin{array}{c}
u_{0} \\
u_{1} \\
u_{2}
\end{array} \right)= 0 .$$ Therefore, $$\begin{aligned}
& & \textrm{det} \left( \begin{array}{ccc}
12a_{0} & -3a_{1} & 2a_{2} \\
3a_{1} & -2a_{2} & 3a_{3} \\
2a_{2} & -3a_{3}& 12a_{4}
\end{array} \right)\\
& = & 4(2a_{2}^{3} - 9a_{1}a_{2}a_{3} + 27 a_{1}^{2}a_{4} - 72 a_{0}a_{2}a_{4} + 27a_{0}a_{3}^{2})\\ \nonumber
& = &4J = 0 .
\end{aligned}$$
In this paper, we always suppose that $J = 0$. In Section \[resolvent2\], we will show that if $J_{F} = 0$ then there are linear forms $\xi = \xi(x , y)$ and $\eta = \eta(x , y)$ so that $$F(x , y) = \frac{1}{8\sqrt{3IA_{4}}}\left(\xi^{4} - \eta^{4}\right),$$ where the quantity $A_{4}$ is defined in (\[Ai2\]). We will use Padé approximation via hypergeometric polynomials to approximate ${\eta}/{\xi}$ with rational integers. The main idea here is to replace the construction of a family of dense approximations to ${\eta}/{\xi}$, by a family of rational approximations to the function $(1 - z)^{1/4}$. Consider the system of linear forms $R_{r}(z) = -Q_{r}(z) + (1-z)^{1/4}P_{r}(z)$ that approximate $(1-z)^{1/4}$ at $z = 0$, such that $R_{r}(z) = z^{2r+1}\bar{R}_{r}(z)$, $\bar{R}_{r}(z)$ is regular at $z=0$, and $P_{r}(z)$ and $Q_{r}(z)$ are polynomials of degree $r$. Thue [@thu342; @thu352] explicitly found polynomials $P_{r}(z)$ and $Q_{r}(z)$ and Siegel [@Sie282] identified them in terms of hypergeometric polynomials. Refining the method of Siegel, Evertse [@Ev2] used the theory of hypergeometric functions to give an upper bound for the number of solutions to the equation $f(x , y) = 1$, where $f$ is a cubic binary form with positive discriminant. Here we adjust Lemma $4$ of [@Ev2] for quartic forms.
\[hyp2\] Let $r$, $g$ be integers with $r \geq 1$, $g \in \{ 0 , 1 \}$. Put $$\begin{aligned}
\label{AB2}\nonumber
A_{r, g} (z) & = & \sum_{m =0}^{r}{r - g + \frac{1}{4} \choose m} {2r - g - m \choose r - g} (-z)^{m}, \\
B_{r, g} (z) & = & \sum_{m =0}^{r-g}{r - \frac{1}{4} \choose m} {2r - g - m \choose r } (-z)^{m}.
\end{aligned}$$
- There exists a power series $F_{r,g}(z)$ such that for all complex numbers $z$ with $|z| < 1$ $$\label{ABF2}
A_{r,g}(z) - (1 - z)^{1/4} B_{r, g}(z) = z^{2r+1 -g}F_{r,g}(z)$$ and $$\label{F2}
|F_{r,g}(z)| \leq \frac{{r-g+1/4 \choose r+1-g} {r- 1/4 \choose r}}{{2r + 1 - g \choose r}} (1 - |z|)^{-\frac{1}{2}(2r + 1 - g)}.$$
- For all complex numbers $z$ with $|1 - z| \leq 1$ we have $$\label{A2}
|A_{r,g}(z)| \leq {2r - g \choose r}.$$
- For all complex numbers $z \neq 0$ and for $h \in \{1 , 0\}$ we have $$\label{BA2}
A_{r, 0}(z) B_{r+h , 1 , 1}(z) \neq A_{r+h , 1}(z) B_{r , 0}(z).$$
This lemma has been proven in [@Akh2].
Equivalent Forms {#EF2}
================
We will call forms $F_{1}$ and $F_{2}$ equivalent if they are equivalent under $SL_{2}(\mathbb{Z})$-action (i.e. if there exist integers $b$, $c$, $d$ and $e$ such that $$F_{1}(bx + c y , dx + ey) = F_{2}(x , y)$$ for all $x$ and $y$, where $be - cd = \pm 1$). Denote by $N_{F}$ the number of solutions in integers $x$ and $y$ of the Diophantine equation $$\left|F(x , y)\right| = h.$$ Note that if $F_{1}$ and $F_{2}$ are equivalent, then $N_{F_{1}} = N_{F_{2}}$, $I_{F_{1}} = I_{F_{2}}$ and $J_{F_{1}} = J_{F_{2}}$.
Let us define, for a quartic form $F$, an associated quartic form, the Hessian $H$, by $$H(x , y) = \frac{d^{2}F}{dx^{2}} \frac{d^{2}F}{dy^{2}} - \left(\frac{d^{2}F}{dx dy}\right)^{2}.$$ Then $$H(x , y)= A_{0}x^{4} + A_{1}x^{3}y + A_{2}x^{2}y^{2} + A_{3}xy^{3} + A_{4}y^{4},$$ where $$\begin{aligned}
\label{Ai2}\nonumber
A_{0} & = & 3 (8a_{0}a_{2} - 3a_{1}^{2}) ,\\ \nonumber
A_{1} & = & 12(6a_{0}a_{3} - a_{1}a_{2}), \\
A_{2} & = & 6(3a_{1}a_{3} + 24a_{0}a_{4} -2a_{2}^{2}), \\ \nonumber
A_{3} & = & 12(6a_{1}a_{4} - a_{2}a_{3}),\\ \nonumber
A_{4} & = & 3(8a_{2}a_{4} - 3 a_{3}^{2}).
\end{aligned}$$
We have the following identities (see Proposition 5 of [@Cre2]): $$\label{IHF2}
I_{H} = 12^{2} I_{F}^{2},$$ $$\label{JHF2}
J_{H} = 12^{3}(2I_{F}^{3} - J_{F}^{2})$$ and $$D_{H} = 12^{6}J_{F}^{2}D_{F} ,$$ where $H$ is the Hessian of $F$ and $D_{F}$, $D_{H}$ are the discriminants of $F$ and $H$, respectively . From identities in (\[Ai2\]) and using algebraic manipulation, we have $$\begin{aligned}
& & A_{0}A_{3}^{2} - A_{4}A_{1}^{2} \\ \nonumber
& = & 12^{3}(a_{0}a_{3}^{2} - a_{4}a_{1}^{2}) \left( 2a_{2}^{3} - 9a_{1}a_{2}a_{3} + 27 a_{1}^{2}a_{4} - 72 a_{0}a_{2}a_{4} + 27a_{0}a_{3}^{2}\right)\\ \nonumber
& = & 12^{3}(a_{0}a_{3}^{2} - a_{4}a_{1}^{2})J_{F}
\end{aligned}$$ and similarly, $$A_{3}^{3} + 8A_{1}A_{4}^{2} - 4A_{2}A_{3}A_{4} = 12^{3}(a_{3}^{3} + 8a_{1}a_{4}^{2} - 4 a_{2}a_{3}a_{4})J_{F}.$$ When $J_{F} = 0$, we obtain $$\begin{aligned}
\label{22}
A_{0}A_{3}^{2} &=& A_{4}A_{1}^{2},\\ \nonumber
A_{3}^{3} + 8A_{1}A_{4}^{2} &= &4 A_{2}A_{3}A_{4}.
\end{aligned}$$ Therefore, when $A_{3}A_{4} \neq 0$, $$\begin{aligned}
H(x , y) & = &A_{0}x^{4} + A_{1}x^{3}y + A_{2}x^{2}y^{2} + A_{3}xy^{3} + A_{4}y^{4} \\
& = &
\frac{1}{4A_{3}^{2}A_{4}}(2A_{1}A_{4}x^{2} + A_{3}^{2}xy + 2
A_{4}A_{3}y^{2})^{2} \\ \nonumber
& = & \frac{1}{4A_{3}^{2}A_{4}} W(x , y)^2,
\end{aligned}$$ where we define the quadratic form $W(x , y) = 2A_{1}A_{4}x^{2} + A_{3}^{2}xy + 2
A_{4}A_{3}y^{2}$. So we get $$I_{H}= \left(\frac{A_{3}^4 - 16A_{1}A_{4}^2A_{3}}{4A_{3}^2 A_{4}}\right)^2.$$ From (\[IHF2\]), we obtain $$\label{c22}
\left| A_{3}^4 - 16A_{1}A_{4}^2A_{3}\right| = \left|48A_{3}^2 A_{4}I_{F}\right|.$$
In order to make good use of the above identities, we prove the following lemma:
\[A342\] Let $F(x , y)$ be a quartic form with $J_{F}= 0$. There exists a form equivalent to $F(x , y)$, for which $A_{3} A_{4} \neq 0$.
If $A_{4} = 0$, then by (\[22\]) we have $A_{4} = A_{3} = 0$ and therefore, $$H(x , y) = x^{2}(A_{0}x^{2} + A_{1}xy + A_{2}y^{2}) .$$ Let $$x= mX + lY$$ and $$y = pX + q Y,$$ where $m, l, p$ and $q$ are integers satisfying $mq - lp = \pm 1$. Suppose that $\Phi_{1}(X , Y)$ is equivalent to $F(x , y)$ under this substitution with Hessian $$H_{\Phi_{1}}(X , Y) = A'_{0}X^{4} + A'_{1}X^{3}Y + A'_{2}X^{2}Y^{2} + A'_{3}XY^{3} + A'_{4}Y^{4}.$$ We have, $$A'_{4} = H_{\Phi_{1}} (0 , 1) = H_{F}(l , q) =l^{2}(A_{0}l^{2} + A_{1}lq + A_{2}q^{2}).$$ If $H_{F}$ is identically zero then by (\[IHF2\]) and (\[JHF2\]), we will have $I_{F} = J_{F} = D_{F} =0$. But since we have assumed that $F(x , y)$ is irreducible, $H_{F}(x , y)$ is not identically zero. Therefore, the integers $l$ and $q$ can be chosen so that $$A'_{4} = H_{F}(l , q) \neq 0.$$
Let $t \in \mathbb{Z}$ and put $$M = m + lt,$$ $$P = p + qt.$$ Let $\Phi_{2} (X , Y)$ be the equivalent form to $F(x , y)$ under the substitution $$x = MX + lY$$ and $$y = PX + qY,$$ and $H_{\Phi_{2}}(X , Y) = A''_{0}X^{4} + A''_{1}X^{3}Y + A''_{2}X^{2}Y^{2} + A''_{3}XY^{3} + A''_{4}Y^{4}$. Then substituting $x$ by $ MX + lY$ and $y$ by $PX + qY$ in $H_{F}(x , y)$, we find that $A''_{3}$, the coefficient of the term $XY^3$ in $H_{\Phi_{2}}(X , Y)$, is equal to $$\begin{aligned}
\nonumber
A''_{3} & =& 4Ml^{3}A_{0} + (l^{3}P + 3Ml^{2}q)A_{1} + (2l^{2}Pq + 2Mlq^{2}) A_{2} \\ \nonumber
& & + (q^{3}m + 3Pq^{2}l)A_{3} + 4Pq^{3} A_{4} \\ \nonumber
& =& (m + lt)(4l^{3}A_{0} + 3l^{2}qA_{1} + 2lq^{2}A_{2} + q^{3} A_{3}) \\ \nonumber
& & + (p + qt) (l^{3}A_{1} + 2l^{2}qA_{2} + 3lq^{2}A_{3} +4q^{3}A_{4}) \\ \nonumber
& =& K + 4t (l^4 A_{0} + l^3 q A_{1} + l^2 q^2 A_{2}+ l q^3 A_{3}+ q^4 A_{4})\\ \nonumber
& = & K + 4t A'_{4}.
\end{aligned}$$ Since $A'_{4} \neq 0$, the integer $t$ can be chosen so that $A''_{3} \neq 0$.
In the following, we will show that $F(x , y)$ or one of its equivalences (under $GL_{2}(\mathbb{Z})$-action) satisfies $$|A_{4}| < 4I .$$ From now on, we will suppose that $A_{3}A_{4} \neq 0$. Let $$x = mX + lY$$ and $$y = pX + qY ,$$ where $m$, $l$, $p$ and $q$ are integers satisfying $mq - lp = \pm 1$. Let $\Phi( X , Y)$ be equivalent to $F(x , y)$ under this substitution and $$\Phi(X , Y) = a'_{0}X^{4} + a'_{1}X^{3}Y + a'_{2}X^{2}Y^{2} + a'_{3}X Y^{3} + a'_{4} Y^{4} .$$ We observe that $$A'_{4} = H_{\Phi} (0 , 1) = H_{F}(l , q) ,$$ where $H_{\Phi}( X , Y) = A'_{0}X^{4} + A'_{1}X^{3}Y + A'_{2}X^{2}Y^{2} + A'_{3}XY^{3} + A'_{4}Y^{4} $.
To continue, we will be in need of the following Proposition due to Hermite.
\[her2\] Suppose that $f_{11}x^{2} + 2f_{1 2} xy + f_{2 2}y^{2}$ is a binary form with $D = f_{11}f_{22} - f_{1 2}^{2} \neq 0$. Then there is an integer pair $(u_{1} , u_{2}) \neq (0 , 0)$ for which $$0 < |f_{1 1}u_{1}^{2} + 2f_{1 2} u_{1}u_{2} + f_{2 2}u^{2} | < \sqrt{\frac{4}{3}|D|} .$$
See [@Cas2], page 31.
Proposition \[her2\] implies that we can choose $l$ and $q$, such that $$\begin{aligned}
\label{Rr2}\nonumber
0 <|A'_{4}| &=& \frac{1}{|4A_{3}^{2}A_{4}|} (2A_{1}A_{4}l^{2} + A_{3}^{2}lq + 2A_{4}A_{3}q^{2})^{2} \\ \nonumber
& < & \frac{1}{|4A_{3}^{2}A_{4}|} \left|\frac{1}{3}(A_{3}^{4} - 16A_{1}A_{4}^{2}A_{3}) \right| \\
& = & 4|I|,
\end{aligned}$$ where the last equality comes from (\[c22\]).
We have shown that the Hessian of $F$ satisfies the following formula. $$\begin{aligned}
H(x , y) & =& A_{0}x^{4} + A_{1}x^{3}y + A_{2}x^{2}y^{2} + A_{3}xy^{3} + A_{4}y^{4} \\ \nonumber
& = & \frac{1}{4A_{3}^{2}A_{4}}(2A_{1}A_{4}x^{2} + A_{3}^{2}xy + 2A_{4}A_{3}y^{2})^{2}.\end{aligned}$$ We will need some results due to Cremona [@Cre2]. Since we are using different notations in this paper, we will summarize Propositions $6$ and $8$ of [@Cre2] in Lemmas \[P682\] and \[P882\]. In particular, we note that the quartic polynomial $g_{4}(X)$ in [@Cre2] is equal to $\frac{-1}{3}H(x , 1)$ and its leading coefficient is equal to $-A_{0}/3$.
\[P682\] Suppose $F(x , y)$ is a quartic form with invariants $I$ and $J$ and Hessian $H(x , y)$. Let $\phi$ be a root of $X^3 -3I + J$. Then $$-\frac{1}{9}H(x , y) + \frac{4}{3} \phi F(x , y) = m(x , y)^2,$$ where $m(x , y)$ is a quadratic covariant of $F(x , y)$.
See part (vi) of Proposition $6$ of [@Cre2].
\[P882\] Let F(x , y) be a quartic form with real coefficients and the leading coefficient $a_{0}$. Suppose that $F(x , 1) = 0$ has $4$ real roots. Order the roots $\phi_{i}$ of $X^3 -3I + J$, with $4a_{0} \phi_{1} > 4a_{0} \phi_{2} > 4a_{0} \phi_{3}$. Set $\phi = \phi_{2}$. Then $m(x , y)$ is a positive definite quadratic form with real coefficients, where $m(x , y)$ is the covariant of $F(x , y)$ defined in Lemma $\ref{P682}$.
This is part (ii) of Proposition $8$ of [@Cre2]. Note that the quantity $z$ in that Proposition is equal to $-A_{0}$ and therefore a positive value in our case.
Following Definition $4$ of [@Cre2], we say that the quartic form $F(x , y) = a_{0}x^{4} + a_{1}x^{3}y + a_{2}x^{2}y^{2} + a_{3}x y^{3}+ a_{4}y^{4}$ with positive discriminant is *reduced* if and only if the positive definite quadratic form $m(x ,y)$ is reduced. Here, we remark that the real quadratic form $f(x , y) = ax^{2} + bxy + cy^{2}$ is called *reduced* if $$|b| \leq a \leq c .$$
\[red2\] Let $F$ be the quartic form in Theorem $\ref{main22}$ and $H$ be its Hessian. If $F$ is reduced then for integers $x, y$ we have $$\left|H(x , y)\right| \geq 36 \, I y^4.$$
Suppose that our quartic form $F(x , y)$ is reduced. Taking $\phi = 0$ in Lemma \[P682\], we know that the algebraic covariant $\frac{-1}{9} H(x , y)$ is the square of a quadratic form, say $$\frac{-1}{9} H(x , y) = m^2(x , y).$$ We assume that $y \neq 0$. Put $$m(x , y) = y^2 m(z) = y^2 \left(Az^2 + B z + C \right),$$ where $z = \frac{x}{y}$. Note that $m(z)$ assumes a minimum equal to $ \frac{4AC - B^2}{4A} $ at $ z = \frac{-B}{2A}$. Since $$m^{2}(x , y) = \frac{1}{36A_{3}^2A_{4}}(2A_{1}A_{4}x^{2} + A_{3}^{2}xy + 2A_{4}A_{3}y^{2})^{2},$$ by (\[c22\]), we get $$4AC - B^2 = \frac{16A_{1}A_{3}A_{4}^2 - A_{3}^4}{-36A_{3}^2 A_{4}} = \pm\frac{4}{3}I .$$ Recall that $A_{0} < 0 $ and hence, by (\[22\]), $A_{4} < 0$. Since $I > 0$ and $m(x , y)$ is reduced, we have $4AC - B^2 > 0$ and $$A^2 \leq AC \leq \frac{1}{3} (4AC - B^2) = \frac{4}{9} I.$$ Therefore, $$m(x , y) \geq 2\sqrt{I}y^2,$$
So we can assume that $\left| H(x , y)\right| \geq h^{3} 12\sqrt{3I}$ when looking for pairs of solutions $(x , y)$ with $ |y| \geq \frac{h^{3/4}}{ (3I)^{1/8}}$.
Reduction To A Diagonal Form {#RDF2}
============================
Our goal in this section will be to reduce the problem at hand to consideration of diagonal forms over a suitable imaginary quadratic field. The method of Thue-Siegel is particularly well suited for application to such forms. We will show that
\[rF2\] Let $F$ be the binary form in Theorem $\ref{main2}$. Then $$F(x , y) = \frac{1}{96 A_{3}^2 A_{4}\sqrt{-3I}} \left(\xi^{4}(x , y) - \eta^{4}(x , y) \right),$$ where $\xi$ and $\eta$ are complex conjugate linear forms in $x$ and $y$.
Let $H(x , y) = A_{0}x^{4} + A_{1}x^{3}y + A_{2} x^{2}y^{2} + A_{3} xy^{3} + A_{4}y^{4} $ with $A_{3}A_{4} \neq 0,$ be the Hessian of $F(x , y)$. We can factor $2A_{1}A_{4}x^{2} + A_{3}^{2}xy + 2A_{4}A_{3}y^{2}$ over $\mathbb{C}$ as $$\label{HaHa}
\xi (x , y)\eta(x , y) = 2A_{1}A_{4}x^{2} + A_{3}^{2}xy + 2A_{4}A_{3}y^{2},$$ where $\xi$ and $\eta$ are linear forms. So we may write $$x = m \xi + l \eta$$ $$y = p\xi + q \eta,$$ for some $m$, $l$, $p$, $q \in \mathbb{C}$. Therefore, $$\begin{aligned}
F(x , y) & = &F(m\xi +l\eta , p\xi + q\eta) \\
&= &a'_{0}\xi^{4} + a'_{1}\xi^{3}\eta + a'_{2}\xi^{2}\eta^{2} +a'_{3}\xi \eta^{3} + a'_{4}\eta^{4} \\
& = & \Phi(\xi , \eta).
\end{aligned}$$ The Hessian $H'(\xi , \eta)$ of $\Phi(\xi , \eta)$ satisfies $$\begin{aligned}
H'(\xi , \eta) & = & A'_{0}\xi^{4} + A'_{1}\xi^{3}\eta + A'_{2}\xi^{2}\eta^{2} +A'_{3}\xi \eta^{3} + A'_{4}\eta^{4} \\ \nonumber
&= &\Delta^{2} H(x , y) = \frac{\Delta^{2}}{4A_{3}^{2}A_{4}}\xi^{2} \eta^{2} .
\end{aligned}$$ Hence, $$\label{H'2}
A'_{0} = A'_{1} = A'_{3} = A'_{4} = 0 ; \ \ A'_{2} = \Delta^{2} \frac{1}{4A_{3}^{2}A_{4}} .$$
On the other hand, $$A'_{0} = 3(8a'_{0}a'_{2} - 3 a^{'2}_{1})$$ and $$A'_{1} = 12(6a'_{0}a'_{3} - a'_{1}a'_{2}).$$ Using Maple, it is easy to check that for any form $F(x , y)$, $$-10a_{4}A_{0} + 2a_{3} A_{1} - a_{2}A_{2} + a_{1}A_{3} -2a_{0}A_{4} = 6J .$$ So, for $\Phi(\xi , \eta)$, we obtain $$-10a'_{4}A'_{0} + 2a'_{3} A'_{1} - a'_{2}A'_{2} + a'_{1}A'_{3} -2a'_{0}A'_{4} = 6J_{\Phi} = 6\Delta^{4}J_{F} = 0 ,$$ where $a'_{i}$ are the coefficients of $\Phi$ and $A'_{i}$ are the coefficients of its Hessian. Therefore, by (\[H’2\]), $$a'_{2} = 0$$ and from the expressions for $A'_{0}$ and $A'_{4}$ respectively that result from (\[Ai2\]), $$a'_{1} = a'_{3} = 0 ,$$ whereby, $$F(x , y) = \Phi (\xi , \eta) = a'_{0}\xi^{4} + a'_{4}\eta^{4}.$$ Observe that if $$2A_{1}A_{4}x^{2} + A_{3}^{2}xy + 2A_{4}A_{3}y^{2} = (\alpha x + \beta y) (\gamma x + \delta y)$$ then for any complex number $\lambda$, in (\[HaHa\]) we may take $\xi = \lambda (\alpha x + \beta y)$ and $\eta =\mu (\gamma x + \delta y)$, where $\lambda \mu =1$. Our goal now is to determine the values of $\lambda$ and $\mu = \frac{1}{\lambda}$ in $\xi =\lambda (\alpha x + \beta y) $ and $\eta = \mu (\gamma x + \delta y )$, so that $a'_{4} = - a'_{0}$. We have $$\left( \begin{array}{cc}
\lambda \alpha & \lambda \beta \\
\mu \gamma & \mu \delta
\end{array} \right)
\left( \begin{array}{cc}
m & l \\
p & q
\end{array} \right) = \left( \begin{array}{cc}
1 & 0\\
0 & 1
\end{array}\right).$$ Thus, $$\label{matrr2}
\left( \begin{array}{cc}
m & l \\
p & q
\end{array} \right) = \left( \begin{array}{cc}
\lambda \alpha & \lambda \beta \\
\mu \gamma & \mu \delta
\end{array} \right)^{-1}
=
\frac{1}{\lambda \mu (\alpha \delta - \beta \gamma)} \left( \begin{array}{cc}
\mu \delta & -\lambda \beta \\
-\mu \gamma & \lambda \alpha
\end{array} \right)$$ whereby, $$\frac{p}{q} = \frac{-\mu \gamma}{\lambda \alpha}$$ and we get $$q = -\frac{\lambda p \alpha}{\mu \gamma}.$$ Since $\Phi (\xi , \eta) = a'_{0}\xi^{4} + a'_{4}\eta^{4}$, we have $$a'_{0} = \Phi(1 , 0) \qquad \textrm{and} \qquad a'_{4} = \Phi(0 , 1).$$ When $\eta = \mu (\gamma x + \delta y ) = 0$, we have $$m = \frac{-\delta p}{\gamma}$$ and when $\xi =\lambda (\alpha x + \beta y) = 0 $, we have $$l = \frac{-\beta q}{\alpha}.$$ So we can write $$a'_{0} = F(m , p)= F\left(\frac{-\delta p}{\gamma} , p\right) = \frac{p^{4}}{\gamma^{4}}F(-\delta , \gamma)$$ and $$a'_{4} =F (l , q)= F\left(\frac{-\beta q}{\alpha} , q\right) = \frac{q^{4}}{\alpha^{4}}F(-\beta , \alpha).$$ Therefore, if we choose $\lambda$ and $\mu$ so that $\mu ^{8} = \frac{\mu^{4}}{\lambda^{4}} = \frac{F(-\beta , \alpha)}{F(\delta , -\gamma)}$, then $-a'_{0} = a'_{4}$.
We have shown that $F(x , y)$ can be written as $a'_{0}\left(\xi^{4}(x , y) - \eta^{4}(x , y)\right)$, where $$\label{lin2}
\xi = \lambda (\alpha x + \beta y), \, \, \eta =\mu (\gamma x + \delta y)$$ and $\lambda \mu =1$. It remains to calculate the value of $a'_{0}$. Using (\[H’2\]) and (\[Ai2\]), we get $$A'_{2} = \Delta^{2} \frac{1}{4A_{3}^{2} A_{4}}= 6(3a'_{1}a'_{3} + 24 a'_{0}a'_{4} -2a'_{2}) = 144a'_{0}a'_{4}.$$ Substituting $a'_{4}$ by $ -a'_{0}$, we obtain $$a^{'2}_{0} = -\frac{\Delta^{2}}{24^{2}A_{3}^{2}A_{4}},$$ where $\Delta = mq - lp$ is the determinant of the matrix $\left( \begin{array}{cc}
m & l \\
p & q
\end{array} \right)$. Therefore, from (\[matrr2\]) and he fact that $\lambda \mu = 1$, $$a^{'2}_{0} = -\frac{1}{(\alpha \delta - \beta\gamma)^{2} 24^{2}A_{3}^{2}A_{4}}.$$ To calculate $(\alpha \delta - \beta \gamma)^{2}$, we recall that $$2A_{1}A_{4}x^{2} + A_{3}^{2}xy + 2A_{3}A_{4}y^{2} = (\alpha x + \beta y) (\gamma x + \delta y) ,$$ consequently, computing the discriminant of the above quadratic form and by (\[c22\]), $$\label{IA2}
\left|\alpha \delta - \beta \gamma\right|^{2} = \left| A_{3}^{4} - 16 A_{1}A_{4}^{2}A_{3}\right| = \left|48A_{3}^{2}A_{4}I \right|$$ and therefore, $$a'_{0} = \pm \frac{1}{96 A_{3}^{2}A_{4}\sqrt{-3I}} ,$$ where $I = I_{F}$.
We will assume, without loss of generality, that $$\label{a'2}
a'_{0} = \frac{1}{96 A_{3}^{2}A_{4}\sqrt{-3I}}.$$
Resolvent Forms {#resolvent2}
===============
Suppose that $\xi$ and $\eta$ are linear forms in Lemma \[rF2\]. Let us define $$\xi' = \frac{\xi}{(12A_{3}^{2})^{1/4}|A_{4}|^{1/8} }$$ and $$\eta' = \frac{\eta}{(12A_{3}^{2})^{1/4} |A_{4}|^{1/8}} ,$$ so that $$F(x , y) = \frac{1}{8\sqrt{3IA_{4}}} \left(\xi^{'4}(x , y) - \eta^{'4}(x , y) \right).$$ Lemma \[rF2\] can be restated as follows:
\[nr2\] Let $F$ be the binary form in Theorem $\ref{main2}$. Then $$\label{aa2}
F(x , y) = \frac{1}{8\sqrt{3IA_{4}}} \left(\xi^{4}(x , y) - \eta^{4}(x , y) \right).$$ where $\xi$ and $\eta$ are complex conjugate linear forms in $x$ and $y$, with $$\xi^4, \eta^4 \in \mathbb{Q} \left(\sqrt{A_{0}I/3} \right).$$
For the binary form $F(x , y)$ with Hessian $H(x , y)$, the sextic covariant $Q(x , y)$ is defined by $$Q(x , y) = \frac{\delta F}{\delta x} . \frac{\delta H}{\delta y} - \frac{\delta F}{\delta y}. \frac{\delta H}{\delta x}.$$ Since we have taken $H(x , y) =\frac{1}{4A_{3}^{2}A_{4}}(2A_{1}A_{4}x^{2} + A_{3}^{2}xy + 2A_{4}A_{3}y^{2})^{2} $, we may write $$Q(x , y) = \frac{1}{2A_{3}^{2}A_{4}} W(x , y) \psi(x ,y) ,$$ where $$W(x , y) = 2A_{1}A_{4}x^{2} + A_{3}^{2}xy + 2A_{4}A_{3}y^{2}$$ and $$\psi(x , y) = (A_{3}^{2}x + 4A_{3} A_{4}y) \frac{\delta F} {\delta x} - (4A_{1}A_{4}x + A_{3}^{2}y) \frac{\delta F}{\delta y}.$$ We have (see equation (25) of [@Cre2]) $$16 H^{3} + 9 Q^{2} = 4^{4} \times 3^{3} I H F^{2} .$$ We remark that in [@Cre2], $g_{4} = \frac{-1}{3}H$, $g_{6} =\frac{-1}{36} Q$ and the invariants $I$ and $J$ are the negative of our $I$ and $J$, respectively. Since $H(x , y) = \frac{1}{4A_{3}^{2}A_{4}}W(x , y)^2$ is not identically zero, we can divide both sides of the obove identity by $H(x , y)$ to get $$\label{42}
W^{4}(x , y) + 9 A_{3}^{2}A_{4} \psi ^{2}(x , y) = 4^{4} \times 3^{3} I A_{3}^{4}A_{4}^{2} F^{2}(x , y) .$$ Since $W(x , y) = \xi \eta $ and $F(x , y) = \frac{ \xi^{4} - \eta^{4}}{96 A_{3}^2 A_{4}\sqrt{-3I}} $, (\[42\]) implies that $$\xi^4 \eta^4 + 9 A_{3}^{2}A_{4} \psi ^{2}(x , y) = \frac{1}{4} \left(-\xi^{8} - \eta^{8} + 2 \xi^4 \eta^4 \right)$$ and we obtain $$\label{Q2}
(\xi^{4} + \eta^{4})^{2} = -36 A_{3}^{2}A_{4} \psi^{2}(x , y)$$ and therefore, by (\[22\]) $$\xi^{4} + \eta^{4}=\pm \frac{ 6 A_{3}^{2}}{A_{1}}\sqrt{-A_{0}}\psi(x , y) .$$ Note that if all roots of $F(x , 1)$ are real then $I > 0$ and $A_{0} <0$ ( see [@Cre2], Proposition 7). So we may write $$\xi^{4} + \eta^{4} = b\sqrt{-A_{0}},$$ with $b \in \mathbb{Q}$. We have also seen that $$\xi^{4} - \eta^{4} = ia\sqrt{3I}$$ for some even integer $a$. Therefore, for integers $x$, $y$, the quantities $\xi^{4}(x , y)$ and $\eta^{4}(x ,y)$ are complex conjugates and belong to $\mathbb{Q}\left(\sqrt{-A_{0}} , \sqrt{-3I}\right)$. Moreover, $\sqrt{-A_{0}}\,\xi^{4}(x , y)$ and $\sqrt{-A_{0}}\,\eta^{4}(x ,y)$ are algebraic integers in $\mathbb{Q}\left(\sqrt{A_{0}I/3} \right)$. This is because $$\sqrt{-A_{0}} \left(\xi^{4} + \eta^{4}\right) =\pm \frac{ -6 A_{0} A_{3}^{2}}{A_{1}}\psi(x , y) .$$ and by (\[22\]), $A_{1} | A_{0}A_{3}^2$. We will work in the number field $\mathbb{Q}\left(\sqrt{A_{0}I/3} \right)$. We also have $$\begin{aligned}
\frac{\xi^{4}}{\eta^{4}} & = & \frac{b\sqrt{-A_{0}} + ia\sqrt{3I}} {b\sqrt{-A_{0}} - ia\sqrt{3I}} \\ \nonumber
& = & \frac{-A_{0}b^{2} -3a^{2} I + i6ab\sqrt{-A_{0}I/3}}{-A_{0}b^{2} + 3a^{2}I}
\end{aligned}$$ Therefore, $$\frac{\xi^{4}}{\eta^{4}} \in \mathbb{Q}(\sqrt{A_{0}I/3}) .$$ Note that, in (\[lin2\]), we started with two linear forms and continued with their fourth powers. Let the linear form $\xi = \xi(x , y)$ be a fourth root of $\xi^{4}(x , y)$ and define $$\eta(x , y)= \bar{\xi}(x , y).$$ Indeed, $\eta(x , y)$ is a fourth root of $\eta^4$. Hence, when $F(x , 1)$ splits in $\mathbb{R}$, we can define the complex conjugate linear forms $\xi(x , y)$ and $\eta(x , y)$, so that $$\xi^{4} - \eta^{4} = 96A_{3}^{2} A_{4}\sqrt{-3I} F(x , y)$$ and $$\left| \xi \eta\right| = \left| 2A_{1}A_{4}x^{2} + A_{3}^{2}xy + 2A_{4}A_{3}y^{2}\right| .$$ Now let us define $$\xi' = \frac{\xi}{(12A_{3}^{2})^{1/4}|A_{4}|^{1/8} }$$ and $$\eta' = \frac{\eta}{(12A_{3}^{2})^{1/4} |A_{4}|^{1/8}} ,$$ so that $$F(x , y) = \frac{1}{8\sqrt{3IA_{4}}} \left(\xi^{'4}(x , y) - \eta^{'4}(x , y) \right).$$ From (\[Ai2\]), for every pair of integers $(x , y)$, we have $$3 \mid \frac{1}{4A_{3}^{2}A_{4}} W^{2}(x , y) = H(x , y).$$ This gives $$12 A_{3}^{2} A_{4} \mid W^{2}(x , y).$$ By (\[42\]), for every pair of integers $(x , y)$, we have $$16 A_{3}^{2} A_{4} \mid \psi^{2} (x , y) .$$ Using (\[Q2\]), we conclude that the real part of $\xi^{4}$ has the factor $12A_{3}^{2} A_{4}$ . Since $\xi^4 - \eta^4 = a'_{0}F$, by (\[a’2\]), the imaginary part of $\xi^{4}$ has also the factor $12A_{3}^{2} A_{4}$. So $$\frac{\xi^4}{|12A_{3}^{2}A_{4}|} , \frac{\eta^4}{|12A_{3}^{2}A_{4}|} \in \mathbb{Q} \left(\sqrt{-A_{0}} , \sqrt{-3I}\right).$$ By (\[22\]), $$\frac{\sqrt{-A_{4}}\, \xi^4}{|12A_{3}^{2}A_{4}|} , \frac{\sqrt{-A_{4}}\eta^4}{|12A_{3}^{2}A_{4}|} \in \mathbb{Q} \left(\sqrt{A_{0}I/3} \right).$$
We call a pair of complex conjugates $\xi$ and $\eta$ satisfying the identities in Lemma \[nr2\] a pair of *resolvent forms*, and note that if $(\xi , \eta)$ is one pair, there are precisely three others, given by $(i \xi , -i \eta)$ , $(-\xi , -\eta)$ and $(-i\xi , i\eta)$, where $i = \sqrt{-1}$. We will, however, work with $(\xi , \eta)$, a fixed pair of resolvent forms. For the pair of resolvent form $(\xi , \eta)$, we have $$\label{c62}
|\xi \eta| = \left|\frac{2A_{1}A_{4}x^{2} + A_{3}^{2}xy + 2A_{4}A_{3}y^{2}}{\sqrt{12 A_{3}^2 \sqrt{\left|A_{4}\right|}}}\right| = \frac{\left(H(x , y)^2|A_{4}|\right)^{1/4}}{\sqrt{3}} .$$
[**REMARK**]{}. The fact that for integers $x$ and $y$, $\xi^{4}(x , y)$ and $\eta^4(x , y)$ are complex conjugates and belong to an imaginary quadratic field is very crucial for our proof. To satisfy these conditions, when $J_{F} =0$, we only need $I_{F}A_{0} <0$ (see the proof of Lemma \[nr2\]). Proposition $7$ of [@Cre2], guarantees this property for quartic binary forms that split in $\mathbb{R}$. So we may generalize Theorem \[main2\] to all quartic binary forms with $I_{F}A_{0} <0$.
Gap Principles {#GPe2}
==============
Let $\omega$ be a fourth root of unity (for some $j \in \{ 1 , 2, 3, 4\}$, let $\omega= e^{\frac{2j\pi i}{4}}$). We say that the integer pair $(x , y)$ is *related* to $\omega$ if $$\left|\omega - \frac{\eta(x , y)}{\xi(x , y)}\right| = \min_{0 \leq k \leq 3}\left|e^{2k\pi i/4} - \frac{\eta(x , y)}{\xi(x , y)}\right|.$$ Let us define $z = 1 - \left(\frac{\eta(x , y)}{\xi(x , y)}\right)^{4}$, where $(\xi , \eta)$ is a fixed pair of resolvent forms ( in other words, $\frac{\eta}{\xi}$ is a fourth root of $(1-z)$). We have $$|1 - z| = 1 \ \ , \ \ |z| < 2 .$$ Note that $|z| = 2$ is impossible here. Because it would mean $\eta^4 = -\xi^4$, so $F(x , y) = \frac{1}{4 \sqrt{3IA_{4}}} \xi^4$ and hence $D_{F} = 0$.
\[6.12\] Let $\omega$ be a fourth root of unity and the integral pair $(x , y)$ satisfies $F(x , y) = \frac{1}{8\sqrt{3IA_{4}}}(\xi^{4}(x , y) - \eta^{4}(x , y)) =1$, with $$\left|\omega - \frac{\eta(x , y)}{\xi(x , y)}\right| = \min_{0 \leq k \leq 3}\left|e^{2k\pi i/4} - \frac{\eta(x , y)}{\xi(x , y)}\right|.$$ If $|z| \geq 1$ then $$\label{Gap12}
\left|\omega - \frac{\eta(x,y)}{\xi(x,y)}\right| \leq \frac{\pi}{8} |z|.$$ If $|z| < 1$ then $$\label{Gap22}
\left|\omega - \frac{\eta(x,y)}{\xi(x,y)}\right| < \frac{\pi}{12} |z|.$$
Put $$4\theta = \textrm{arg} \left( \frac{\eta(x , y)^{4}}{\xi(x , y)^{4}}\right).$$ We have $$\sqrt{2 - 2\cos(4\theta)} = |z| .$$ Therefore, when $|z| < 2$ we have $$|\theta| < \frac{\pi}{4}$$ and when $|z| < 1$ we have $$|\theta| < \frac{\pi}{12}.$$ Since $$\left|\omega - \frac{\eta(x,y)}{\xi(x,y)} \right| \leq |\theta|,$$ we obtain $$\left|\omega - \frac{\eta(x,y)}{\xi(x,y)}\right| \leq \frac{1}{4} \frac{|4\theta|}{\sqrt{2 - 2\cos(4\theta)}}
\left|1 - \frac{\eta(x,y)^{4}}{\xi(x,y)^{4}}\right|.$$ By differential calculus $\frac{|4\theta|}{\sqrt{2 - 2\cos(4\theta)}} < \frac{\pi}{2}$ whenever $ 0<|\theta| < \frac{\pi}{4} $. Therefore $$\left|\omega - \frac{\eta(x , y)}{\xi(x , y)}\right| <\frac{\pi}{8} |z| ,$$ and from the fact that $\frac{|4\theta|}{\sqrt{2 - 2\cos(4\theta)}} < \frac{\pi}{3}$ whenever $ 0<|\theta| < \frac{\pi}{12} $ , we conclude $$\left|\omega - \frac{\eta(x , y)}{\xi(x , y)}\right| <\frac{\pi}{12} |z| ,$$ as desired.
Suppose that we have distinct solutions to $\left| F(x , y)\right| \leq h$ indexed by $i$, say $(x_{i}, y_{i})$, related to a fixed fourth root of unity $\omega$ with $|\xi(x_{i+1} , y_{i+1})| \geq |\xi(x_{i} , y_{i})|$. Let $$F(x_{i} , y_{i}) = h_{i}, \, \, F(x_{i+1} , y_{i+1}) = h_{i+1}.$$ For brevity, we will write $\eta_{i} = \eta(x_{i} , y_{i})$ and $\xi_{i} = \xi(x_{i} , y_{i})$. We have $$\left(\begin{array}{cc}
\lambda \alpha & \lambda \beta \\
\mu \gamma & \mu \delta
\end{array}\right)
\left(\begin{array}{cc}
x_{1} & x_{2} \\
y_{1} & y_{2}
\end{array} \right)
=
\sqrt{12 A_{3}^2 \sqrt{\left|A_{4}\right|}} \left(\begin{array}{cc}
\xi_{1} & \xi_{2} \\
\eta_{1} & \eta_{2}
\end{array}\right)$$ (see the definition of linear forms $\xi$ and $\eta$ in Section \[resolvent2\]). Since $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ are distinct co-prime solutions, $x_{1}y _{2}- x_{2}y_{1}$ is a nonzero integer. So by (\[IA2\]) and (\[22\]), we get $$\label{lb2}
|\xi_{1} \eta_{2} - \xi_{2} \eta_{1} | = \frac{|(\alpha \delta - \beta \gamma)(x_{1}y _{2}- x_{2}y_{1}) |}{\sqrt{12 A_{3}^2 \sqrt{\left|A_{4}\right|}} } \geq 2\sqrt{I} \left| A_{4} \right|^{1/4}.$$ On the other hand, by (\[Gap12\]) and (\[Gap22\]), we have $$\begin{aligned}
|\xi_{i} \eta_{i+1} - \xi_{i+1} \eta_{i}| & = &\left|\xi_{i}( \eta_{i+1} - \omega \xi_{i+1}) - \xi_{i+1}( \eta_{i} - \omega \xi_{i})\right| \\ \nonumber
& \leq & \left| \xi_{i}\xi_{i+1} \left(\frac{\eta_{i+1}}{\xi_{i+1}} -\omega \right)\right| + \left| \xi_{i}\xi_{i+1} \left(\frac{\eta_{i}}{\xi_{i}} -\omega \right)\right|\qquad (\textrm{by the triangle inequality})\\ \nonumber
&\leq & \frac{\pi}{8}\left(| \xi_{i}\xi_{i+1}z_{i+1}| + | \xi_{i}\xi_{i+1}z_{i}|\right) \qquad (\textrm{from} (\ref{Gap12}))\\ \nonumber
& = & \frac{\pi}{8}\left(| \xi_{i}\xi_{i+1} \frac{\eta_{i+1}^4 - \xi_{i+1}^4}{\xi_{i+1}^4}| + | \xi_{i}\xi_{i+1}\frac{\eta_{i}^4 - \xi_{i}^4}{\xi_{i}^4}|\right) \\ \nonumber
&\leq& \pi h\sqrt{\left|3I\, A_{4}\right|}\left(\frac{|\xi_{i}|}{|\xi_{i+1}^{3}|} + \frac{|\xi_{i+1}|}{|\xi_{i}^{3}|}\right),\end{aligned}$$ the last inequality holding from expression for $F(x , y)$ in Lemma \[nr2\] and since $\left|F(x , y) \right| < h$. Since we assumed $|\xi_{i}| \leq |\xi_{i+1}|$, we get $$|\xi_{i} \eta_{i+1} - \xi_{i+1} \eta_{i}| \leq 2 \pi h\sqrt{\left|3I\, A_{4}\right|} \left( \frac{|\xi_{i+1}|}{|\xi_{i}^{3}|}\right)$$ Combining this with (\[lb2\]), we conclude $$\label{Gap2}
|\xi_{i+1}| \geq \frac{1}{\pi \sqrt{3}h\, \left|A_{4}\right|^{1/4}}|\xi_{i}|^{3}.$$
Let us now assume that there are $4$ distinct solutions to $\left|F(x , y) \right| \leq h$ related to a fixed choice of $\omega$, corresponding to $\xi_{-1}$, $\xi_{0}$, $\xi_{1}$ and $\xi_{2}$, where $|\xi_{-1}| \leq |\xi_{0}| \leq |\xi_{1}| \leq |\xi_{2}| $and $F(x_{i} , y_{i}) = h_{i} $. We will deduce a contradiction, which shows that at most $3$ such solutions can exist. By (\[Gap2\]) and since $\left|h_{i}\right| \leq h$, $$|z_{i+1}| \leq \frac{ 3 \pi^4 |z_{i}|^{3} h^2}{64 I} ,$$ where $z_{i} = 1 - \frac{\eta_{i}^{4}}{\xi_{i}^{4}} = \frac{8h \sqrt{ \left|3I\, A_{4}\right|}}{\xi_{i}^{4}}$. Since $|z_{-1}|\leq 2$, if $I > 36.6 h^2$ then $|z_{0}|$ , $|z_{1}|$ , $ |z_{2}| < 1$. By (\[Gap12\]), $$\begin{aligned}
|\xi_{-1} \eta_{0} - \xi_{0} \eta_{-1}| &=& |\xi_{-1}(\omega \eta_{0} - \xi_{0}) - \xi_{0}( \omega \eta_{-1} - \xi_{-1})| \\
& \leq & 8 h (1 + \frac{\pi}{12})\sqrt{ \left|3I\, A_{4}\right|} \left( \frac{|\xi_{0}|}{|\xi_{-1}^{3}|}\right). \end{aligned}$$ Combining this with (\[lb2\]), we conclude $$|\xi_{0}| \geq \frac{2\sqrt{3}}{5 \pi h\, \left|A_{4}\right|^{1/4}}|\xi_{-1}|^{3} .$$ Similarly, we get $$\begin{aligned}
& & |\xi_{0} \eta_{1} - \xi_{1} \eta_{0}| = |\xi_{0}(\omega \eta_{1} - \xi_{1}) - \xi_{1}(\omega \eta_{0} - \xi_{0})| \\
& \leq & 8 h \sqrt{\left|3I\, A_{4}\right|}\frac{\pi}{12} \left(\frac{|\xi_{0}|}{|\xi_{1}^{3}|} + \frac{|\xi_{1}|}{|\xi_{0}^{3}|}\right) \leq\frac{4 \pi}{3}h\sqrt{\left|3I\, A_{4}\right|} \left( \frac{|\xi_{1}|}{|\xi_{0}^{3}|}\right),\end{aligned}$$ which leads to $$\label{n2}
|\xi_{1}| \geq \frac{3}{2 \pi h\, \left| A_{4}\right|^{1/4}}|\xi_{0}|^{3} \geq \frac{72\sqrt{3}}{2 \pi h^4 (5 \pi)^3 \, \left| A_{4}\right|}|\xi_{-1}|^{9} .$$ Note that $\left|\frac{8 h \sqrt{ \left|3I\, A_{4}\right|}}{\xi^{4}_{-1}}\right| = |z_{-1}| = \left|1 - \left(\frac{\eta_{-1}}{\xi_{-1}}\right)^{4}\right| < 2$ and therefore, $$\label{nn2}
|\xi_{-1}|^{4} > 4h \sqrt{\left|3I\, A_{4}\right|} .$$ Thus, when $I > 36.6 h^2$ we have $$\label{982}
|\xi_{1}| > I^{\frac{9}{8}} \frac{72 \sqrt{3}\left( 4\sqrt{3}\right)^{9/4} \left| A_{4}\right|^{1/8}}{2 \pi (5 \pi)^3 h^{7/4}} > 0.39 \,
\frac{I^{\frac{9}{8}} \,\left| A_{4}\right|^{1/8} }{h^{7/4}}.$$
Recall that By Lemma \[red2\], we can assume that $\left| H(x , y)\right| \geq h^{3} 12\sqrt{3I}$ when looking for pairs of solutions $(x , y)$ with $ |y| \geq \frac{h^{3/4}}{ (3I)^{1/8}}$. This implies $$\left|H(x_{-1} , y_{-1})\right| \geq 12 \frac{h^3\sqrt{3I}}{\left|A_{3}^{2} A_{4} \right|},$$ So by (\[c62\]), $$\left| \xi_{-1}\right|^4 = \frac{H \sqrt{|A_{4}|}}{3} \geq 4h^3\sqrt{\left|3I A_{4}\right|}.$$ Moreover, one may assume that $h>2$, for the case $h=1$ is being addressed when we are treating the Thue equation. Under these assumptions, we have $$|z_{-1}| = \left| \frac{8h \sqrt{ \left|3I\, A_{4}\right|}}{\xi_{i}^{4}} \right|< 1$$ and by (\[n2\]) and Lemma \[6.12\], $$\label{98h2}
|\xi_{1}| > (4\sqrt{3})^{9/4} I^{\frac{9}{8}}h^{11/4} \, \left| A_{4}\right|^{1/8} \left(\frac{ 3}{2 \pi }\right)^4>
4 h^{11/4} I^{\frac{9}{8}}\, \left| A_{4}\right|^{1/8}.$$ Here the point is that the inequality $ |y| \geq \frac{h^{3/4}}{ (3I)^{1/8}}$ provides us with a good enough lower bound (\[98h2\]) for the size of $\xi_{1}$. Hence, to prove Theorem \[main22\], we do not need the assumption $I > 36.6 h^2$.
Some Algebraic Numbers
======================
Combining the polynomials $A_{r , g}$ and $B_{r , g}$ in Lemma \[hyp2\] with the resolvent forms, we will consider the complex sequences $\Sigma_{r,g} $ given by $$\Sigma_{r,g} = \frac{\eta_{2}}{\xi_{2}}A_{r,g}(z_{1}) - \frac{\eta_{1}}{\xi_{1}} B_{r,g}(z_{1})$$ where $z_{1} = 1 - \eta_{1}^{4}/ \xi_{1}^{4}$ . For any pair of integers $(x , y)$, $ \xi^{4}(x , y)$ and $\eta^{4}(x , y)$ are algebraic integers in $\mathbb{Q}(\sqrt{A_{0}I/3})$(see Lemma \[nr2\]). We have seen that $A_{0} < 0$ and one can assume $A_{3} A_{4} \neq 0$ (see Lemma \[A342\]). Therefore from (\[22\]), we have $A_{1} \neq 0$. Define $$\Lambda_{r,g} = \left(9|A_{4}|\right)^{\frac{1-g}{4}}\xi_{1}^{4r + 1- g} \xi_{2} \Sigma_{r,g}.$$ We will show that $\Lambda_{r,g}$ is either an integer in $\mathbb{Q}(\sqrt{\frac{A_{0}I}{3}})$ or a fourth root of such an integer. If $\Lambda_{r,g} \neq 0$, this provides a lower bound upon $|\Lambda_{r,g}|$.
\[122\] For any pair of integer $(s , t)$, we have $$\frac{\xi( s , t)}{\xi(1 , 0)} , \frac{\eta( s , t)}{\eta(1 , 0)} \in \mathbb{Q}(\sqrt{A_{0}I/3})[s , t].$$
By (\[22\]) and (\[IA2\]), we have $$\alpha\delta - \beta\gamma = \sqrt{A_{3}^{4} - 16 A_{1}A_{4}^{2}A_{3} } = \frac{4A_{3}^{2}}{A_{1}}\sqrt{3IA_{0}} = \frac{12A_{3}^{2}}{A_{1}}\sqrt{\frac{IA_{0}}{3}} .$$ Since $$\begin{aligned}
2A_{1}A_{4}x^{2} + A_{3}^{2}xy + 2A_{3}A_{4}y^{2} & = &(\alpha x + \beta y) (\gamma x + \delta y) \\
& = & \sqrt{12 A_{3}^2 \sqrt{|A_{4}|}}\xi(x , y) \eta(x , y) ,\end{aligned}$$ we conclude that $\alpha \gamma$ , $\beta \delta$ , $\alpha\delta+ \beta\gamma \in \mathbb{Z}$. Thus, for integral pair $(s , t)$, we obtain $$\frac{\xi( s , t)}{\xi(1 , 0)} , \frac{\eta( s , t)}{\eta(1 , 0)} \in \mathbb{Q}(\sqrt{A_{0}I/3})[s , t].$$
\[ai2\] If $(x_{1} , y_{1})$ and $(x_{2}, y_{2})$ are two pairs of rational integers then $$\sqrt{3|A_{4}|^{1/2}}\xi(x_{1} , y_{1}) \eta(x_{2} , y_{2}) ,$$ $$\xi(x_{1} , y_{1})^{3} \xi(x_{2} , y_{2})$$ and $$\eta(x_{1} , y_{1})^{3} \eta(x_{2} , y_{2})$$ are integers in $\mathbb{Q}(\sqrt{A_{0}I/3})$.
For any pair of integers $(x , y)$, Lemma \[122\] implies that $$\frac{\xi(x , y)}{\xi(1 , 0)} \in \mathbb{Q}(\sqrt{A_{0}I/3}).$$ Thus, $$\frac{\xi(x_{1} , y_{1})}{\xi(x_{2} , y_{2})} \in \mathbb{Q}(\sqrt{A_{0}I/3}).$$ Since $$\sqrt{3|A_{4}|^{1/2}}\, \xi(x_{2} , y_{2}) \eta(x_{2} , y_{2}) = \frac{\omega(x , y)}{2|A_{3}|} \in \mathbb{Q},$$ the algebraic integer $\sqrt{3|A_{4}|^{1/2}}\xi(x_{1} , y_{1}) \eta(x_{2} , y_{2})$ belongs to $\mathbb{Q}(\sqrt{A_{0}I/3})$.
Let $\xi(x , y) = \epsilon_{1} x + \epsilon_{2} y$. Clearly, $\epsilon_{1}$ and $\epsilon_{2}$ are algebraic integers and so are $\epsilon_{1}^{4}$, $\epsilon_{1}^{3} \epsilon_{2}$, $\epsilon_{1}^{2} \epsilon_{2}^{2}$, $\epsilon_{1} \epsilon_{2}^{3}$ and $\epsilon_{2}^{4}$. Since $\xi^{4}$ is an integer in $\mathbb{Q}(\sqrt{A_{0}I/3})$, we conclude that $\epsilon_{1}^{4}$, $\epsilon_{1}^{3} \epsilon_{2}$, $\epsilon_{1}^{2} \epsilon_{2}^{2}$, $\epsilon_{1} \epsilon_{2}^{3}$ and $\epsilon_{2}^{4}$ are all algebraic integers in $\mathbb{Q}(\sqrt{A_{0}I/3})$.
$\xi(x_{1} , y_{1})^{3} \xi(x_{2} , y_{2})$ is an integer in $\mathbb{Q}(\sqrt{A_{0}I/3})$ , because it can be written as a linear combination with rational integer coefficients in $\epsilon_{1}^{4}$ , $\epsilon_{1}^{3} \epsilon_{2}$, $\epsilon_{1}^{2} \epsilon_{2}^{2}$, $\epsilon_{1} \epsilon_{2}^{3}$ and $\epsilon_{2}^{4}$.
We can similarly show that that $\eta(x_{1} , y_{1})^{3} \eta(x_{2} , y_{1})$ is also an integer in $\mathbb{Q}(\sqrt{A_{0}I/3})$.
For every polynomial $P(z) = a_{n}z^{n} + a_{n-1}z^{n-1} + \ldots + a_{1}z + a_{0}$, we define $$P^{*}(x , y) = x^{n} P(y/x) = a_{0}x^{n} + a_{1}x^{n-1}y + \ldots + a_{n-1}x y^{n-1} + a_{n}y^{n} .$$ Let $A_{r,g}$ and $B_{r,g}$ be as in (\[AB2\]) and $$C_{r,g}(z) = A_{r,g}(1 - z) , \ D_{r , g}(z) = B_{r,g}(1 - z),$$ where $A_{r , g}$ and $B_{r , g}$ are the polynomials in Lemma \[hyp2\]. For $z \neq 0$, we have $D_{r , 0}(z) = z^{r} C_{r,0}(z^{-1})$, hence
$$\begin{aligned}
\label{star2} \nonumber
A^{*}_{r , 0}(z , z - \bar{z})
& = & z^{r} A_{r , 0} (1 - \frac{\bar{z}}{z}) = z^{r} C_{r,0}(\frac{\bar{z}}{z}) \\
& = & \bar{z}^{r} D_{r,0}(\frac{z}{\bar{z}}) = \bar{z}^{r} B_{r,0}(1 - \frac{z}{\bar{z}}) \\ \nonumber
& = & B^{*}_{r , 0} (\bar{z} , \bar{z} - z) = \bar{B}^{*}_{r,0}(z , z -\bar{z} ) .\end{aligned}$$
\[242\] For any pair of integers $(x , y)$, $$A^{*}_{r,g}(\xi^{4}(x , y), \xi^{4}(x , y) -\eta^{4}(x , y))$$ and $$B^{*}_{r,g}(\xi^{4}(x , y), \xi^{4}(x , y) -\eta^{4}(x , y) )$$ are algebraic integers in $\mathbb{Q}(\sqrt{A_{0}I/3})$.
It is clear that $$A^{*}_{r,g}(\xi^{4}(x , y), \xi^{4}(x , y) -\eta^{4}(x , y))$$ and $$B^{*}_{r,g}(\xi^{4}(x , y), \xi^{4}(x , y) -\eta^{4}(x , y) )$$ belong to $\mathbb{Q}(\sqrt{A_{0}I/3})$. So we need only show that they are algebraic integers. This follows immediately from Lemma 4.1 of [@Chu2] since $$\xi^{4}(x , y) -\eta^{4}(x , y) = 8 h \sqrt{3I A_{4}}F(x , y).$$
We now proceed to show that for any $r \in \mathbb{Z}$, $\Lambda_{r,0}$ and $\Lambda_{r , 1}^{4}$ are integers in $\mathbb{Q}(\sqrt{A_{0}I/3})$. $$\begin{aligned}
\Lambda_{r,g} & = &\left(9|A_{4}|\right)^{\frac{1-g}{4}} \xi_{1}^{4r} \xi_{1}^{1 - g}\xi_{2} \Sigma_{r,g}\\ \nonumber
& = & \left(9|A_{4}|\right)^{\frac{1-g}{4}}\left(\xi_{1}^{1 -g} \eta_{2}A^{*}_{r,g}(\xi_{1}^{4} , \xi_{1}^{4} - \eta_{1}^{4}) - \xi_{1}^{-g}\xi_{2} \eta_{1} B^{*}_{r,g}(\xi_{1}^{4} , \xi_{1}^{4} - \eta_{1}^{4})\right).\end{aligned}$$ For $g=0$, we have $$\Lambda_{r,0} = \left(9|A_{4}|\right)^{\frac{1}{4}}\left(\xi_{1} \eta_{2}A^{*}_{r,0}(\xi_{1}^{4} , \xi_{1}^{4} - \eta_{1}^{4}) - \xi_{2} \eta_{1} B^{*}_{r,0}(\xi_{1}^{4} , \xi_{1}^{4} - \eta_{1}^{4})\right)$$ By Lemma \[ai2\], $\left(9|A_{4}|\right)^{\frac{1}{4}} \left(\xi_{1} \eta_{2}\right)$ and $\left(9|A_{4}|\right)^{\frac{1}{4}} \left(\xi_{2} \eta_{1}\right)$ are integers in $\mathbb{Q}(\sqrt{A_{0}I/3})$. They are also complex conjugates. From (\[star2\]), Lemma \[242\] and the characterization of algebraic integers in quadratic number fields, we conclude that $\Lambda_{r , 0} \in \mathbb{Z} \sqrt{A_{0}I/3}$. By Lemma \[ai2\] and Lemma \[242\], $\Lambda_{r , 1}^{4}$ is an algebraic integer in $\mathbb{Q}(\sqrt{A_{0}I/3})$. Next we will show that $\Lambda^{4}_{r , 1}$ is not an integer when $\Sigma_{r , 1}$ is nonzero.
Suppose $\Lambda_{r,1}^{4} \in \mathbb{Z}$. Then we have for some $\rho \in \{ \pm1 , \pm i \}$, that $\rho \Lambda_{r,1} = \bar{\Lambda}_{r,1}$. Hence by the definition of $\Lambda_{r,1}$ and since $\xi_{i}$ and $\eta_{i}$ are complex conjugates, $$\begin{aligned}
\nonumber
\rho \Sigma_{r , 1} &= &\xi_{1}^{-4r} \xi_{2}^{-1} \bar{\Lambda}_{r,1} \\ \nonumber
& =& \xi_{1}^{-4r} \xi_{2}^{-1} \eta_{1}^{4r}\eta_{2} \left( \frac{\xi_{2}}{\eta_{2}} A_{r,1} \left(1 - \frac{\xi_{1}^{4}}{\eta_{1}^{4}} \right) - \frac{\xi_{1}}{\eta_{1}}B_{r , 1} \left( 1 - \frac{\xi_{1}^{4}}{\eta_{1}^{4}} \right) \right)\\ \nonumber
& =& \xi_{1}^{-4r} \xi_{2}^{-1} \eta_{1}^{4r}\eta_{2} \left( \frac{\xi_{2}}{\eta_{2}} A_{r,1} \left(1 - \frac{\xi_{1}^{4}}{\eta_{1}^{4}} \right) - \frac{\xi_{1}}{\eta_{1}}B_{r , 1} \left( 1 - \frac{\xi_{1}^{4}}{\eta_{1}^{4}} \right) \right)\\ \nonumber
&=& \frac{\eta_{1}^{4r}}{\xi_{1}^{4r}} \left(A_{r,1}\left( 1- \frac{\xi_{1}^{4}}{\eta_{1}^{4}}\right) - \frac{\xi_{1} \eta_{2}}{\xi_{2}\eta_{1}} B_{r ,1} \left( 1- \frac{\xi_{1}^{4}}{\eta_{1}^{4}}\right) \right) .
\end{aligned}$$ This, together with Lemmas \[ai2\] and \[242\], implies that $$\label{ronf}
\rho\Sigma_{r,1} \in \mathbb{Q}(\sqrt{A_{0}I/3}).$$ We have, by definition, $$\Sigma_{r,g} = \frac{\eta_{2}}{\xi_{2}}A_{r,g}(z_{1}) - \frac{\eta_{1}}{\xi_{1}} B_{r,g}(z_{1}) = \frac{\eta}{\xi} \left[\frac{\eta_{2}/\eta}{\xi_{2}/\xi}A_{r,g}(z_{1}) - \frac{\eta_{1}/\eta}{\xi_{1}/\xi} B_{r,g}(z_{1})\right],$$ where $\eta = \eta(1 , 0)$ and $\xi = \xi (1 , 0)$. By Lemmas \[122\] and \[242\], $$\frac{\eta_{2}/\eta}{\xi_{2}/\xi}A_{r,g}(z_{1}) - \frac{\eta_{1}/\eta}{\xi_{1}/\xi} B_{r,g}(z_{1}) \in \mathbb{Q}(\sqrt{A_{0}I/3}).$$ Hence $$\label{fff}
\mathfrak{f} = \mathbb{Q}(\sqrt{A_{0}I/3} , \rho\Sigma_{r,g}) = \mathbb{Q}(\sqrt{A_{0}I/3} , \rho\frac{\xi}{\eta}).$$ If we choose complex number $X$ so that $\xi (X , 1) = \eta(X , 1)$ then by Lemma \[122\], $X \in \mathfrak{f}$. We have $F (X , 1) = \frac{1}{8\sqrt{3IA_{4}}}(\xi^{4}( X , 1) - \eta^{4}(X , 1) )= 0$. Since we have assumed that $F$ is irreducible, $X$ has degree $4$ over $\mathbb{Q}$. But from (\[ronf\]) and the definition of number field $\mathfrak{f}$ in (\[fff\]), $$X \in \mathfrak{f} = \mathbb{Q}(\sqrt{A_{0}I/3}).$$ This contradicts the fact that $X$ has degree $4$ over $\mathbb{Q}$. We conclude that $\Lambda_{r,1}$ can not be a rational integer.
From the well-known characterization of algebraic integers in quadratic fields, we may therefore conclude that, If $\Lambda_{r,g} \neq 0$, then for $g \in \{ 0 , 1\}$ $$\label{ub2}
|\Lambda_{r,g}| \geq 2^{\frac{-g}{4}}(-A_{0}I/3)^{\frac{1}{2} - \frac{3g}{8}}.$$
Approximating Polynomials
=========================
In order to apply (\[lb2\]), we must make sure that $\Lambda_{r , g}$ or equivalently $\Sigma_{r , g}$ does not vanish. First we will show that for small $r$, $\Sigma_{r , 0} \neq 0$.
\[nv12\] Suppose that $(x , y)$ is a pair of solutions to $F(x , y) = \pm 1$ with $I > 135 $ or a pair of solutions to $\left| F(x , y) \right| \leq h$ with $ |y|> \frac{h^{3/4}}{(3I)^{1/8}}$ . For this pair of solutions and $r \in \{1 , 2, 3, 4, 5\}$, we have $$\Sigma_{r,0} \neq 0.$$
Let $r \in \{1 , 2, 3, 4, 5 \}$. Suppose that $\Sigma_{r , 0} = 0$. From (\[ABF2\]), we can find for each $r$, a polynomial $F_{r}(z) \in \mathbb{Q}[z]$, satisfying $$A_{r,0}(z)^{4} - (1 -z)B_{r,0}^{4} = z^{2r+1} F_{r}(z).$$ In fact, using Maple, we have $$A_{1}(z) = 4 A_{1, 0}(z) = 8 - 5z ,$$ $$B_{1}(z) = 4 B_{1 , 0}(z) = 8 - 3z ,$$ $$F_{1}(z) = 320 - 320z + 81z^{2} ,$$ $$A_{2}(z) = \frac{32}{3} A_{2,0}(z) = 64 -72z + 15 z^{2},$$ $$B_{2}(z)= \frac{32}{3} B_{2,0}(z) = 64 -56z + 7 z^{2},$$ $$F_{2}(z) = 86016 - 172032z + 114624z^{2} - 28608z^{3} + 2401z^{4},$$ $$A_{3}(z)= 128A_{3,0}(z)= 2560 - 4160z + 1872z^{2} -195z^{3},$$ $$B_{3}(z)= 128B_{3,0}(z)=2560 - 3520z + 1232z^{2} - 77z^{3},$$ $$\begin{aligned}
F_{3}(z) = & & 14057472000 - 42172416000z \\\nonumber
& & + 48483635200z^{2} - 26679910400z^{3} \\ \nonumber
& &+ 7150266240z^{4}
- 839047040z^{5} \\\nonumber
& & + 35153041z^{6}, \end{aligned}$$ $$\begin{aligned}
A_{4}(z) &= & \frac{2048}{5}A_{4,0}(z) \\
& = & 28672 - 60928z + 42432z^{2} - 10608z^{3} + 663z^{4},
\end{aligned}$$ $$\begin{aligned}
B_{4}(z) & = & \frac{2048}{5}B_{4,0}(z)\\
& =& 28672 - 53760z + 31680z^{2} -6160z^{3} + 231z^{4},
\end{aligned}$$ $$\begin{aligned}
F_{4}(z) = & &13989396348928- 55957585395712z \\\nonumber
& & + 91916125077504z^{2} - 79896826347520z^{3} \\\nonumber
& & +39463764078592z^{4} -11050000539648z^{5}\\\nonumber
& & + 1648475542656z^{6} - 113348764800z^{7} \\\nonumber
& & + 2847396321z^{8},\end{aligned}$$ $$\begin{aligned}
A_{5}(z) & = & \frac{8192}{21}A_{5,0}(z) \\ \nonumber
& = &98304 - 258048z + 243712z^{2} \\\nonumber
& & - 99008z^{3} + 15912z^{4} - 663z^{5},
\end{aligned}$$ $$\begin{aligned}
B_{5}(z) &= &\frac{8192}{21}B_{5,0}(z)\\
& = &98304 - 233472z + 194560z^{2}\\ \nonumber
& & - 66880z^{3} + 8360z^{4} -209z^{5}.
\end{aligned}$$ and $$\begin{aligned}
F_{5}(z) = & &121733331812352 - 608666659061760z \\\nonumber
& & + 1301756554248192z^{2}-1555026262622208z^{3}\\\nonumber
& & +1136607561252864z^{4} -523630732640256z^{5} \\ \nonumber
& & + 151029162176512z^{6} -26204424888320z^{7} \\ \nonumber
& & +2515441608384z^{8} - 113971885760z^{9} \\ \nonumber
& & +1908029761z^{10}.
\end{aligned}$$ We also define $A_{r}^{*}$ and $B_{r}^{*}$ via $$A_{r}^{*}(x , y) = x^{r}A_{r}(y/x) ,$$ and $$B_{r}^{*}(x , y) = x^{r}B_{r}(y/x).$$ Since $\Sigma_{r , 0} $ is assumed to be zero, $$\frac{\eta_{2}^{4}}{\xi_{2}^{4}} = \frac{\eta_{1}^{4} (B_{r}^{*}(\xi_{1}^{4}, \xi_{1}^{4} -\eta_{1}^{4}))^{4}} {\xi_{1}^{4} (A_{r}^{*}(\xi_{1}^{4} , \xi_{1}^{4} - \eta_{1}^{4}))^{4}} .$$ Let $\mathfrak{I}_{r}$ be the integral ideal in $\mathbb{Q}(\sqrt{IA_{0}/3})$ generated by $\xi_{1}^{4} (A^{*}(\xi_{1}^{4} , \xi_{1}^{4} - \eta_{1}^{4}))^{4}$ and $\eta_{1}^{4} (B^{*}(\xi_{1}^{4} , \xi_{1}^{4} - \eta_{1}^{4}))^{4}$ and $N(\mathfrak{I}_{r})$ be the absolute norm of $\mathfrak{I}_{r}$. Since the ideal generated by $\xi_{1}^{4} (A_{r}^{*}(\xi_{1}^{4} , \xi_{1}^{4} - \eta_{1}^{4}))^{4} -
\eta_{1}^{4}( B_{r}^{*}(\xi_{1}^{4} , \xi_{1}^{4} - \eta_{1}^{4}))^{4}$ divides $(\xi_{2}^{4} -\eta_{2}^{4}). \mathfrak{I}_{r}$, we obtain $$\begin{aligned}
& & |\xi_{1}|^{4(4r+1)} | A_{r}^{4}(z_{1}) - (1 - z_{1}) B_{r}^{4}(z_{1})| \\ \nonumber
& = & |\xi_{1}^{4} (A_{r}^{*}(\xi_{1}^{4} , \xi_{1}^{4} - \eta_{1}^{4}))^{4} - \eta_{1}^{4} (B_{r}^{*}(\xi_{1}^{4} , \xi_{1}^{4} - \eta_{1}^{4}))^{4}.\end{aligned}$$ Since $\mathfrak{I}_{r}$ is an imaginary quadratic field, by (\[a’2\]), we get $$|\xi_{1}|^{4(4r+1)} | A_{r}^{4}(z_{1}) - (1 - z_{1}) B_{r}^{4}(z_{1})| \leq N(\mathfrak{I}_{r})^{1/2} |\xi_{2}^{4} - \eta_{2}^{4}|$$ By (\[ABF2\]), $$A_{r}^{4}(z_{1}) - (1 - z_{1}) B_{r}^{4}(z_{1}) = z_{1}^{2r+1} F_{r}(z_{1}) ,$$ and so we conclude $$|z_{1}|^{2r+1} |F_{r}(z_{1})| \leq N(\mathfrak{I}_{r})^{1/2} |\xi_{2}^{4} - \eta_{2}^{4}| |\xi_{1}|^{-4(4r+1)} ;$$ i.e. $$1 \leq \frac{ N(\mathfrak{I}_{r})^{1/2} |\xi_{2}^{4} - \eta_{2}^{4}| |\xi_{1}|^{-4(4r+1)}}{ |z_{1}|^{2r+1} |F_{r}(z_{1})| }.$$ Since $\xi_{1}^{4} = (\xi_{1}^{4} - \eta_{1}^{4}) (1 - \frac{\eta_{1}^{4}}{\xi_{1}^{4}} )^{-1} = (\xi_{1}^{4} - \eta_{1}^{4}) z_{1}^{-1}$we obtain $$1 \leq \frac{ N(\mathfrak{I}_{r})^{1/2} |\xi_{2}^{4} - \eta_{2}^{4}||\xi_{1}^{4} - \eta_{1}^{4}|^{-4r-1} |z_{1}|^{2r}}{|F_{r}(z_{1})| }.$$ Noting that $|z_{1}| = \left |\xi_{1}^{-4}\right|\left|\xi_{1}^{4} - \eta_{1}^{4}\right|$ and $\left|\xi_{i}^{4} - \eta_{i}^{4}\right| = \left|8h\sqrt{3IA_{4}}F(x , y)\right|$, we obtain for $r \in \{ 1 , 2 , 3 , 4 , 5\}$, $$\label{442}
|\xi_{1}|^{8r} \leq \frac{ (N(\mathfrak{I}_{r})^{1/2} |\xi_{1}^{4} - \eta_{1}^{4}|^{-4r-1})|8h\sqrt{3IA_{4}}|^{2r+1}}{|F_{r}(z_{1})|}.$$
To estimate $N(\mathfrak{I}_{r})^{1/2} $, we choose a finite extension $\mathbf{M}$ of $\mathbb{Q}(\sqrt{A_{0}I/3})$ so that the ideal generated by $\xi_{1}^{4}$ and $\xi_{1}^{4} - \eta_{1}^{4}$ in $\mathbf{M}$ is a principal ideal, with generator $p$, say. We denote the extension of $\mathfrak{I}_{r}$ to $\mathbf{M}$, by $\mathfrak{I}'_{r}$. Let $\mathfrak{r}_{r}$ be the ideal in $\mathbf{M}$ generated by $A_{r}^{*}(u , v)$ and $B_{r}^{*}(u, v)$, where $u = \frac{\xi_{1}^{4}}{p}$ and $v = \frac{\xi_{1}^{4} - \eta_{1}^{4}}{p}$. Since $A_{r}^{*}(x , x- y) = B_{r}^{*}(y , y-x)$, $$\begin{aligned}
\label{ss2}
p^{4r+1}\mathfrak{r}_{r}^{4} B_{r}^{*}(0 , 1)^{4} &\subset & p^{4r+1}\mathfrak{r}_{r}^{4} (u , B_{r}^{*}(0 , v)^{4}) ( u - v, B_{r}^{*}(0 , v)^{4}) \\ \nonumber
&\subset & p^{4r+1}\mathfrak{r}_{r}^{4} (u , B_{r}^{*}(0 , v)^{4}) ( u - v, A_{r}^{*}(v , v)^{4}) \\ \nonumber
&\subset & p^{4r+1}\mathfrak{r}_{r}^{4}( u , u - v) (u , B_{r}^{*}(u , v)^{4}) ( u - v, A_{r}^{*}(u , v)^{4}) \\ \nonumber
&\subset & p^{4r+1} (uA^{*}(u , v)^{4} , (u - v)B_{r}^{*}(u , v)^{4}) = \mathfrak{I}'_{r},
\end{aligned}$$ where $(m_{1} , \ldots , m_{n})$ denote the ideal in $\mathbf{M}$ generated by $m_{1} , \ldots , m_{n}$.
We have $$A_{1}^{*}( x , y) - B_{1}^{*}(x , y) = -2y.$$ Therefore, $$2(v) \subset (A_{1}^{*}( u , v) , B_{1}^{*}(u , v)) \subset \mathfrak{r}_{1},$$ where $(v)$ is the ideal generated by $v$ in $\mathbf{M}$. Since $B_{1}^{*}(0 , 1) = -3$, it follows from (\[ss2\]) that $$1296(\xi_{1}^{4} - \eta_{1}^{4})^{5} \subset 1296 p (\xi_{1}^{4} - \eta_{1}^{4})^{4} = p^{5}16v^{4}B_{1}^{*}(0 , 1)^{4} \subset \mathfrak{I}'_{1}.$$ For $r = 2$, we first observe that $$B_{1}^{*}(x , y)A_{2}^{*}( x , y) -A_{1}^{*}(x , y) B_{2}^{*}(x , y) =-10y^{3}$$ and $$(-32x + 7y)A_{2}^{*}( x , y) -(-32x+15y) B_{2}^{*}(x , y) = 80xy^{2}.$$ Therefore, by (\[ss2\]) we have $$80(v)^{2} \subset (-10v^{3} , 80uv^{2}) \subset (A_{2}^{*}( u , v) , B_{2}^{*}(u , v)) \subset \mathfrak{r}_{2} .$$ Since $B_{2}^{*}(0 , 1) =7$, we have $$80^{4}\times 7^{4} (\xi_{1}^{4} - \eta_{1}^{4})^{9} \subset 80^{4}\times 7^{4} p (\xi_{1}^{4} - \eta_{1}^{4})^{8} = 80^{4}p^{9}v^{8} B_{2}^{*}(0 , 1)^{4} \subset \mathfrak{I}'_{2}.$$ When $r = 3$, we have $$B_{2}^{*}(x , y)A_{3}^{*}( x , y) -A_{2}^{*}(x , y) B_{3}^{*}(x , y) =-210y^{5}$$ $$\begin{aligned}
& & (1616x^{2}-1078xy+77y^{2})A_{3}^{*}(x , y) \\ \nonumber
&- &(1616x^{2}-1482xy+195y^{2}) B_{3}^{*}(x,y) \\ \nonumber
& = & -16800x^{2}y^{3}.\end{aligned}$$ Substituting $77$ for $B_{3}^{*}(0 , 1)$, we conclude $$\begin{aligned}
& & 16800^{4}\times 77^{4} (\xi_{1}^{4} - \eta_{1}^{4})^{13} \subset 16800^{4}\times 77^{4} p (\xi_{1}^{4} - \eta_{1}^{4})^{12} \\ \nonumber
& = & 16800^{4}p^{13}v^{12} B_{3}^{*}(0 , 1)^{4} \subset \mathfrak{I}'_{3}.\end{aligned}$$ For $r = 4$, setting $$\begin{aligned}
G_{4}(x , y)=14178304x^{3}-15889280x^{2}y+4071760xy^{2}-162393y^{3}, \\
H_{4}(x , y) = 14178304x^{3}-19433856x^{2}y+6714864xy^{2}- 466089y^{3},
\end{aligned}$$ we may verify that $$B_{3}^{*}(x , y)A_{4}^{*}( x , y) -A_{3}^{*}(x , y) B_{4}^{*}(x , y) =-6006y^{7}$$ and $$G_{4}(x , y)A_{4}^{*}(x , y) - H_{4}(x , y)B_{4}^{*}(x , y) = -150678528y^{4}x^{3}.$$ These two identities imply that $$150678528 ^{4}\times 231^{4} (\xi_{1}^{4} - \eta_{1}^{4})^{17} \subset 150678528^{4}\times 231^{4} p (\xi_{1}^{4} - \eta_{1}^{4})^{16}.$$ Since this latter quantity is equal to $150678528^{4}p^{17}v^{16} B_{4}^{*}(0 , 1)^{4}$, from (\[ss2\]) it follows that $$150678528 ^{4}\times 231^{4} (\xi_{1}^{4} - \eta_{1}^{4})^{17} \subset \mathfrak{I}'_{4}.$$ Finally, for $r = 5$, set $$\begin{aligned}
& & G_{5}(x , y) \\ \nonumber
& = & 43706368x^{4}-69346048x^{3}y+32767856x^{2}y^{2} \\ \nonumber
& & -4764782{x}y^{3}+123519y^{4},\end{aligned}$$ $$\begin{aligned}
& & H_{5}(x , y)\\ \nonumber
& = & 43706368x^{4}-80272640x^{3}y+46006896x^{2}y^{2}\\ \nonumber
& & -8845746xy^{3}+391833y^{4} . \end{aligned}$$ Then we have $$B_{4}^{*}(x , y)A_{5}^{*}( x , y) -A_{4}^{*}(x , y) B_{5}^{*}(x , y) =-14586y^{7}$$ and $$G_{5}(x , y)A_{5}^{*}(x , y) - H_{5}(x , y)B_{5}^{*}(x , y) = - 134424576 y^{5}x^{4}.$$ These two identities imply that $$134424576^{4}\times 209^{4} (\xi_{1}^{4} - \eta_{1}^{4})^{21} \subset 134424576^{4}\times 209^{4} p (\xi_{1}^{4} - \eta_{1}^{4})^{20}.$$ So by (\[ss2\]), $$134424576^{4}\times 209^{4} (\xi_{1}^{4} - \eta_{1}^{4})^{21} \subset 134424576 ^{4}p^{21}v^{20} B_{5}^{*}(0 , 1)^{4} \subset \mathfrak{I}'_{5}.$$ From the preceding arguments, we are thus able to deduce the following series of inequalities : $$N(\mathfrak{I}_{1})^{1/2} |\xi_{1}^{4} - \eta_{1}^{4}|^{-5} \leq 1296,$$ $$N(\mathfrak{I}_{2})^{1/2} |\xi_{1}^{4} - \eta_{1}^{4}|^{-9} \leq 560^{4},$$ $$N(\mathfrak{I}_{3})^{1/2} |\xi_{1}^{4} - \eta_{1}^{4}|^{-13} \leq (77\times16800)^{4},$$ $$N(\mathfrak{I}_{4})^{1/2} |\xi_{1}^{4} - \eta_{1}^{4}|^{-17} \leq (231\times150678528)^{4}$$ and $$N(\mathfrak{I}_{5})^{1/2} |\xi_{1}^{4} - \eta_{1}^{4}|^{-21} \leq (134424576 \times 209)^{4}.$$ Substituting any of these in (\[442\]) provides a contradiction to inequality (\[982\]) when $I > 135$ and a contradition to (\[98h2\]) when $|y| > \frac{h^{3/4}}{(3I)^{1/8}}$. Note that under both assumptions $I > 135$ and $|y| > \frac{h^{3/4}}{(3I)^{1/8}}$, the function $|z| = \left |\xi^{-4}\right| \left|8h\sqrt{3IA_{4}}F(x , y)\right|$ is small. This makes $\left|F_{r}(z)\right|$ large enough for our contradictions.
\[nv2\] If $r \in \mathbb{N}$ and $h \in \{ 0, 1 \}$, then at most one of $\left\{ \Sigma_{r,0}, \Sigma_{r+h,1} \right\}$ can vanish.
Let $r$ be a positive integer and $h \in \{ 0 , 1\}$ . Following an argument of Bennett [@Ben2], we define the matrix $\mathbf{M}$: $$\mathbf{M} =
\left( \begin{array}{ccc}
A_{r,0}(z_{1}) & A_{r+h , 1}(z_{1}) & \frac{\eta_{1}}{\xi_{1}} \\
A_{r,0}(z_{1}) & A_{r+h , 1}(z_{1}) & \frac{\eta_{1}}{\xi_{1}} \\
B_{r,0}(z_{1}) & B_{r+h , 1}(z_{1}) & \frac{\eta_{2}}{\xi_{2}}
\end{array} \right) .$$ The determinant of $\mathbf{M}$ is zero because it has two identical rows. Expanding along the first row, we get $$\begin{aligned}
0 & = &A_{r, 0}(z_{1}) \Sigma_{r+h , 1} - A_{r+h, 1}(z_{1}) \Sigma_{r , 0} \\ \nonumber
& & + \frac{\eta_{2}}{\xi_{2}}( A_{r, 0}(z_{1}) B_{r+h ,1}(z_{1}) - A_{r+h, 1}(z_{1}) B_{r ,0}(z_{1}) ).\end{aligned}$$ If $\Sigma _{r,0} = 0$ and $\Sigma _{r + h,1} = 0$ then $A_{r, 0}(z_{1}) B_{r+h ,1}(z_{1}) - A_{r+h, 1}(z_{1}) B_{r ,0}(z_{1}) = 0$ which contradicts part (iii) of Lemma \[hyp2\].
An Auxiliary Lemma
==================
We now combine the upper bound for $\Lambda_{r,g}$ obtained in (\[ub2\]) with the lower bounds from Lemma \[hyp2\] to prove the following lemma.
\[c2\] If $\Sigma_{r,g} \neq 0$, then $$c_{1}(r,g) |\xi_{1}|^{4r+1-g}|\xi_{2}|^{-3} + c_{2}(r,g) |\xi_{1}|^{-4r-3(1-g)}|\xi_{2}| > 1 ,$$ where we may take $$c_{1}(1 , 0) = 4\pi h\left(\frac{3\left|A_{4}\right|^{3/2}}{\left|A_{0}\right|}\right) ^{1/2}$$ and $$\begin{aligned}
c_{2}(1 , 0) =
27\,h^3 \left(\frac{3\left|A_{4}\right|^{1/2}}{\left|A_{0}\right|}\right) ^{1/2} (9\sqrt{3I
\left| A_{4}\right|})^{2}\frac{5}{128}
\end{aligned}$$ and for $(r , g) \neq (1, 0)$, $$c_{1}(r , g) = 2 \sqrt{\pi} \, h \left(\frac{3\left|A_{4}\right|^{3/2}}{\left|A_{0}\right|}\right) ^{1/2} \left(\frac{3|A_{4}|}{|A_{0}|^{3/2}}\right) ^{-g/4}\frac{4^{r}} {\sqrt{ r}}$$ and $$\begin{aligned}
&& c_{2}(r , g) = \\
&&27\, h^{2r + 1 - g} \left(\frac{3\left|A_{4}\right|^{1/2}}{\left|A_{0}\right|}\right) ^{1/2} \left(\frac{3|A_{4}|}{|A_{0}|^{3/2}}\right)^{-g/4}(9\sqrt{3I \left| A_{4}\right|})^{2r-g}\frac{\sqrt{2}}{\sqrt{r}\pi4^{r}}.
\end{aligned}$$
By the definition of $\Lambda_{r , g}$ and (\[ABF2\]), we can write $$|\Lambda_{r,g}|= (9|A_{4}|)^{(1-g)/4} |\xi_{1}|^{4r+1-g} |\xi_{2}| \left|(\frac{\eta_{2}}{\xi_{2}} - \omega)A_{r,g}(z_{1}) + \omega z_{1}^{2r+1-g}F_{r,g}(z_{1})\right|.$$ Since $|1 - z_{1}| = 1$ , $|z_{1}| \leq 1$ and $|z_{i}| = \frac{8 h \sqrt{3I}}{|\xi^{4}_{i}|}$, by (\[F2\]), (\[A2\]) and inequality (\[Gap22\]), we have $$\label{cc2}
|\Lambda_{r,g}| \leq (9|A_{4}|)^{(1-g)/4} |\xi_{1}|^{4r+1-g} |\xi_{2}| \,\mathfrak{L},$$ where $\mathfrak{L}$ is equal to $$\begin{aligned}
{2r - g \choose r} \frac{2\pi h \sqrt{3I \left| A_{4}\right|}}{3|\xi^{4}_{2}|} + \frac{{r-g+1/4 \choose r+1-g} {r - 1/4 \choose r}}{{2r+1-g \choose r}}
\left( \frac{9 h \sqrt{3I \left| A_{4}\right|}}{|\xi^{4}_{1}|}\right)^{2r+1-g} .
\end{aligned}$$ Comparing this with (\[ub2\]), we obtain $$c_{1}(r,g) |\xi_{1}|^{4r+1-g}|\xi_{2}|^{-3} + c_{2}(r,g) |\xi_{1}|^{-4r-3(1-g)}|\xi_{2}| > 1 ,$$ where we may take $c_{1}$ and $c_{2}$ so that $$\begin{aligned}
& & c_{1}(r,g) \geq \\
& &2 \pi\, h \left(\frac{3 \left| A_{4}\right|^{3/2} }{\left|A_{0}\right|}\right) ^{1/2} \left(\frac{3|A_{4}|}{\left|A_{0}\right|^{3/2}}\right) ^{-g/4} {2r \choose r}
\end{aligned}$$ and $$\begin{aligned}
& & c_{2}(r,g) \geq \\
& & 27\, h^{2r+1-g}\left(\frac{3 \left| A_{4}\right|^{1/2}}{\left|A_{0}\right|}\right) ^{1/2} \left(\frac{3|A_{4}|}{|A_{0}|^{3/2}}\right)^{-g/4}(9\sqrt{3I\left| A_{4}\right|})^{2r-g} \frac{{r-g+1/4 \choose r+1-g} {r - 1/4 \choose r}}{{2r+1-g \choose r}} .\end{aligned}$$ Substituting $r = 1$ and $g = 0$, we get the desired values for $c_{1}(1 , 0)$ and $c_{2} (1 , 0)$. Let us apply the following version of Stirling’s formula (see Theorem (5.44) of [@Str2]): $$\frac{1}{2\sqrt{k}}4^{k} \leq {2k \choose k} < \frac{1}{\sqrt{\pi k}} 4^{k} ,$$ for $k \in \mathbb{N}$. This leads to the stated choice of $c_{1}$ immediately.
To evaluate $c_{2}(r , g)$, we first note that $${2r+1-g \choose r} \geq {2r \choose r} \geq \frac{4^{r}}{2\sqrt{r}}.$$ Next we will show that $$\label{c112}
{r-g+1/4 \choose r+1-g} {r - 1/4 \choose r} < \frac{1}{\sqrt{2}\pi r} ,$$ for $r \in \mathbb{N}$ and $g \in \{ 0 , 1\}$, whence we may conclude that $$\frac{{r-g+1/4 \choose r+1-g} {r - 1/4 \choose r}}{{2r+1-g \choose r}} <\frac{\sqrt{2}}{\sqrt{r}\pi4^{r}}.$$ This leads immediately to the stated choice of $c_{2}$. It remains to show (\[c112\]). Let us set $$X_{r} = {r - 3/4 \choose r} {r - 1/4 \choose r} = \frac{y_{r}}{r},$$ whereby $$X_{r+1} = {r +1/4 \choose r+1} {r + 3/4 \choose r+1} = \left( \frac{r^{2} + r + 2/9}{r^{2 }+ r} \right)\frac{y_{r}}{r+1}.$$ This implies $$y_{1} = 3/16 \ , \ y_{r} = \frac{3}{16} \prod_{k = 1}^{r -1} \frac{k^{2} + k + 3/16}{k^{2} + k}.$$ Since $$\prod_{k = 1}^{\infty} \frac{k^{2} + k + 3/16}{k^{2} + k} = \frac{16}{3 \Gamma(1/4) \Gamma(3/4)} =\frac{16}{3\sqrt{2}\pi},$$ we obtain $$X_{r} < \frac{1}{\sqrt{2}\pi r}.$$ For $r \in \mathbb{N}$, we have $${r - 3/4 \choose r} >{r + 1/4 \choose r+1}.$$ So when $ g \in \{0 , 1\}$, $${r-g+1/4 \choose r+1-g} {r - 1/4 \choose r} \leq X_{r},$$ which completes the proof.
Proof of the Main Theorems
==========================
Let us now assume that there are $4$ distinct solutions $(x_{i} , y_{i})$ to reduced form $$\left|F(x , y)\right| \leq h$$ related to $\omega$ with $|y_{i}| > \frac{h^{3/4}}{ (3I)^{1/8}}$, corresponding to $\xi_{-1}$, $\xi_{0}$, $\xi_{1}$ and $\xi_{2}$, where we have ordered these in nondecreasing modulus. We will deduce a contradiction, implying that at most $3$ such solutions can exist. Then Theorem \[main22\] will be proven, since there are $4$ choices of $\omega$.
We will show that $|\xi_{2}|$ is arbitrarily large in relation to $|\xi_{1}|$. By (\[982\]) and (\[98h2\]), we know that $|\xi_{1}|$ is large and hence $|\xi_{2}|$ is arbitrarily large, a contradiction.
\[fin2\] Let $F(x , y)$ be the quartic form. Suppose that $(x_{1} , y_{1})$ and $(x_{2} , y_{2})$ are $2$ pairs of solutions to $\left|F(x , y)\right| \leq h$, both related to $\omega$, a fixed fourth root of unity. Put $\xi_{j} = \xi(x_{j} , y_{j})$. Assume further that either
- $F(x , y)$ is the quartic form in Theorem $\ref{main22}$ with $$4 \left|A_{4}\right|^{1/8} h^{11/4} I^{9/8} < |\xi_{1}| < |\xi_{2}| ,$$ or
- $F(x , y)$ is the quartic form in Theorem $\ref{main2}$ with $I > 135$ and $$\label{982m}
0.39 \left|A_{4}\right|^{1/8} h^{11/4} I^{9/8} < |\xi_{1}| < |\xi_{2}|.$$
Then, for each positive integer $r$, $$|\xi_{2}| >\frac{4^{r}\sqrt{r}}{27} \frac{|A_{0}|^{1/8}}{\left(3\left|A_{4}\right|^{1/2}\right)^{1/2}h^{2r+1}}(9\sqrt{3I\left|A_{4}\right|})^{-2r} |\xi_{1}|^{4r + 3}.$$
We will use the upper bound (\[98h2\]) for case (i) and the upper bound (\[982m\]) for case (ii). Note that (\[982m\]) is a generalization for the upper bound (\[982\]) obtained to treat the equation $|F(x , y)| = 1$. By (\[n2\]), $|\xi_{2}| \geq \frac{3|\xi_{1}|^{3}}{2 \pi h \left|A_{4}\right|^{1/4}}$. This implies $$c_{1}(1 , 0) |\xi_{1}|^{5} |\xi_{2}|^{-3} \leq 4\, h^4 \pi \times 12^{3/2}\left( \frac{3}{\left|A_{0}\right|}\right)^{1/2} \left|A_{4}\right| \left| \xi_{1} \right|^{-4}$$ Therefore, by (\[98h2\]) or (\[982m\]) and from the fact that $\left| A_{4}\right| < 4 I$, we obtain $$c_{1}(1 , 0) |\xi_{1}|^{5} |\xi_{2}|^{-3} < 0.01$$ Lemma \[nv12\] implies that $\Sigma_{1 , 0} \neq 0$. So we may apply Lemma \[c2\] to get $$c_{2}(1 , 0) I^{3} |\xi_{1}|^{-7}|\xi_{2}| > 0.99 .$$ One may now conclude $$|\xi_{2}| > \frac{0.99}{c_{2}(1 , 0)} |\xi_{1}|^{7} > 0.93\, h^{-3} \left( \frac{3\left|A_{4}\right|^{1/2}}{\left|A_{0}\right|}\right)^{-1/2}(9\sqrt{3I \left|A_{4}\right|})^{-2} |\xi'_{1}|^{7}.$$ This proves the lemma for $r = 1$. Moreover, we may conclude that $$c_{1}(2 ,0)|\xi_{1}|^{9}|\xi_{2}|^{-3} < \frac{18\, h^{10} \sqrt{\pi}\times 16 \times (5 \times 27)^{3} \left|A_{4}\right|}{|A_{0}|^{2}127^{3}\sqrt{2}} \left( 9\sqrt{3I \left|A_{4}\right|} \right)^{6} \left| \xi_{1} \right|^{-12}.$$ Since $\left| A_{4} \right| \leq 4I$ , by (\[98h2\]) or (\[982m\]) we have $$c_{1}(2 ,0)|\xi_{1}|^{9}|\xi_{2}|^{-3} < 0.1.$$ Via Lemmas \[c2\] and \[nv12\], we obtain $$|\xi_{2}| > \frac{0.9}{c_{2}(1 , 0)} |\xi_{1}|^{11}.$$ This leads to the proof of the Lemma for $r = 2$, after substituting the value of $c_{2}(2 , 0)$. To complete the proof, we use induction on $r$. Suppose that for some $r\geq 2$, $$|\xi_{2}| >\frac{4^{r}\sqrt{r}}{27} \frac{|A_{0}|^{1/8}}{\left(3\left|A_{4}\right|^{1/2}\right)^{1/2}h^{2r+1}}(9\sqrt{3I \left|A_{4}\right|})^{-2r} |\xi_{1}|^{4r + 3}.$$ Then $$\begin{aligned}
& & c_{1}(r+1 ,0)|\xi_{1}|^{4r+5}|\xi_{2}|^{-3} < \\
& & \frac{18\, \sqrt{\pi} \times 27^{3} \left|A_{4}\right| h^{6r+4}}{|A_{0}|^{7/8}4^{2r-1}\sqrt{ (r + 1)} r\sqrt{r}} \left( 9\sqrt{3I \left|A_{4}\right|} \right)^{6r} \left| \xi_{1} \right|^{-8r -4}.\end{aligned}$$ By (\[98h2\]) or (\[982m\]), we have $$c_{1}(r+1 ,0) |\xi_{1}|^{4r+5}|\xi_{2}|^{-3} < 0.1.$$ If $\Sigma_{r+1 , 0} \neq 0$, then by Lemma \[c2\], $$c_{2}(r+1 , 0) |\xi_{1}|^{-4(r+1)-3}|\xi_{2}| > 0.9.$$ Hence, $$\begin{aligned}
& & |\xi_{2}| > \frac{0.9}{c_{2}(r +1 , 0)} |\xi_{1}|^{4(r+1) + 3}\\
& > & \frac{4^{r+1}\sqrt{r+1}}{27\,h^{2r+3}}\left(\frac{\left|A_{0}\right|}{3\left|A_{4}\right|^{1/2}}\right)^{1/2}(9\sqrt{3I})^{-2r-2} |\xi_{1}|^{4r + 7}.
\end{aligned}$$ If, however, $\Sigma_{r+1, 0} = 0$, then by Lemma \[nv2\], both $\Sigma_{r+1,1}$ and $\Sigma_{r+2 ,1}$ are both non-zero and by Lemma \[nv12\], we have $r > 5$. Using the induction hypothesis, we get $$c_{1}(r + 1,1) |\xi_{1}|^{4r+4}|\xi_{2}|^{-3} < 0.01$$ and thus by Lemma \[c2\], (\[22\]) and (\[982\]), we conclude $$c_{2}(r+1,1) |\xi_{1}|^{-4r-4}|\xi_{2}| >0.99 .$$ So, we obtain $$|\xi_{2}| > \frac{4^{r+1}\sqrt{r+1}|A_{0}|^{1/8}}{27\, h^{2r+2} \left|3A_{4}\right|^{1/4} }\left( 9 \sqrt{3I \left|A_{4}\right|}\right)^{-2r-1} |\xi_{1}|^{4(r+1)}.$$ Consequently, $c_{1}(r+2 , 1) |\xi_{1}|^{4r+8}|\xi_{2}|^{-3}$ is less than $$\frac{2 \, \sqrt{\pi}\times 27 \left(3 \left|A_{4}\right|\right) \left( 9 \sqrt{3I \left|A_{4}\right|}\right)^{6r+3} h^{6r+7}
}{4^{2r+1}(r+1)\sqrt{(r+1)\,(r+2)\,|A_{0}|} }
|\xi_{1}|^{-8r-4}
< 0.1.$$ A final application of Lemma \[c2\] implies $$c_{2}(r+2,1)|\xi_{1}|^{-4r-8}|\xi_{2}| >0.9$$ or $$|\xi_{2}| > \frac{0.9}{c_{2}(r+2 , 1)} |\xi_{1}|^{4r+8} .$$ It follows that $$|\xi_{2}| >\frac{\sqrt{r+2}\, 4^{r+2}}{27} \frac{|A_{0}|^{1/8}}{ 3^{1/4}h^{2r+4}}\left(9\sqrt{3I \left|A_{4}\right|}\right)^{-2r-3} |\xi_{1}|^{4(r+ 1) + 4}.$$ Since $|\xi_{1}| > 4 I^{9/8}h^{11/4} \left|A_{4}\right|^{1/8}$, we conclude that $$|\xi_{2}| >\frac{4^{r+1}\sqrt{r+1}}{27} \frac{|A_{0}|^{1/8}}{\left(3\left|A_{4}\right|^{1/2}\right)^{1/2}}(9\sqrt{3I \left|A_{4}\right|})^{-2r-2} |\xi_{1}|^{4r + 7}.$$
Forms With Small Discriminant {#FWSD2}
=============================
To finish the proof of Theorem \[main2\], we need to study the quartic forms $F(x , y)= a_{0}x^{4} + a_{1}x^{3}y + a_{2}x^{2}y^{2} + a_{3}xy^{3} + a_{4}y^{4}$ with $0 < I_{F} \leq 135$ and $A_{0}= 3(8a_{0}a_{2} - 3a_{1}^{2}) < 0$.
We followed an algorithm of Cremona, in Section 4.6 of [@Cre2], which gives all inequivalent integer quartics with given invariant $I$ and $J = 0$. Using Magma, we counted the number of solutions to $$\left| F(x , y)\right| =1,$$ for all reduced quartic forms $F$ with $I_{F} \leq 135$ and $J_{F} = 0$. Regarding $(x , y)$ and $(-x , -y)$ as the same, we didn’t find any form $F$ for which there are more than $4$ solutions to $F(x , y) = \pm 1$. Our programming was not efficient in the sense that it solves more than one equation from some equivalent classes. While reading the earlier versions of this paper, the referee has verified these computations in a very efficient way and kindly shared his results with the author. The following table contains all representatives of the complete set of binary forms $F$ with $I_{F} \leq 135$ and $J_{F} = 0$ that split in $\mathbb{R}$.
$F(x , y)$ $I_{F}$
----------------------------------------------- ---------
$x^4 - x^3y - 6x^2y^2 + xy^3 + y^4$ $51$
$x^4 + 2x^3y - 6x^2y^2 - 2xy^3 + y^4$ $60$
$x^4 - 12 x^2 y^2 + 16 x y^3 - 4y^4$ $96$
$x^4 + 8 x^3 y + 6 x^2 y^2 - 4 x y^3 - 2 y^4$ $108$
$x^4 + x^3y - 15 x^2 y^2 + 18 x y^3 - 4 y^4$ $123$
are representatives of the complete set of binary forms $F$ with $I_{F} \leq 135$ and $J_{F} = 0$ that split in $\mathbb{R}$. To solve the Thue equations $F(x , y)= \pm 1$ for forms $F$ in above table, we may also use PARI since all of the binary forms in the table are monic.
If $$F(x , y) = x^4 -x^3y -6x^2y^2 + xy^3+y^4$$ then $I_{F} = 51$ and the solutions are: $$(-1 , 0), (0 , 1), (1 , 2), (-2 , 1).$$ Note that we can write $$F(x , y) = x^4 -x^3y -6x^2y^2 + xy^3+y^4 = \xi^4(x , y) - \eta^{4}(x , y),$$ so that $\frac{\eta(x , y)}{\xi(x , y)} = \frac{x - iy}{x+iy}$ and we have $\frac{\eta(-1, 0)}{\xi(-1 , 0)} = 1$, $\frac{\eta(0 , 1)}{\xi(0 , 1)} = -1$, $\frac{\eta(1 , 2)}{\xi(1 , 2)} = -\frac{3+4i}{5}$ and $\frac{\eta(-2 , 1)}{\xi(-2 , 1)} = \frac{3+4i}{5}$. This means $(-1 , 0)$ is related to $\omega = 1$, $(0 , 1)$ is related to $\omega = - 1$, $(1 , 2)$ is related to $\omega = -i$ and $(-2 , 1)$ is related to $\omega = i$. Therefore, related to each root of unity there is one pair of solution.
If $$F(x , y) = x^4 + 2x^3y - 6x^2y^2 - 2xy^3 + y^4$$ Then $I_{F} = 60$ and the solutions are $(1 , 0)$ and $(0 , 1)$.
If $$F(x , y) = x^4 - 12x^2y^2 + 16xy^3 - 4y^4$$ then $I_{F} = 96$ and $$F(x , y)= 1$$ has $4$ solutions $(5 , 2)$, $(1 , 3)$, $(1 , 1)$, $(1 , 0)$ and the equation $$F(x , y) = -1$$ has no solution.
If $$F(x , y) = x^4 + 8 x^3 y + 6 x^2 y^2 - 4 x y^3 - 2 y^4$$ then $I_{F} = 108$ and the solutions are $(1 , 0)$ and $(-1 , 1)$.
If $$F(x , y) = x^4 + x^3y - 15 x^2 y^2 + 18 x y^3 - 4 y^4$$ then $I_{F} = 123$ and the solutions are $(1 , 1)$ and $(1 , 0)$.
Acknowledgments
===============
The author would like to thank Professor Michael Bennett for his support and insightful comments. The author is indebted to the anonymous referee for his very careful reading and valuable comments on the earlier version of this paper. The referee’s suggestions certainly improved both presentation and mathematical contents of this manuscript
[999]{}
S. Akhtari. The Diophantine equation $aX^{4} - bY^{2} = 1$. (2007), to appear in J. Reine Angew. Math.
S. Akhtari, A. Togbe and G. Walsh. The Diophantine equation $aX^{4} - bY^{2} = 2$. Acta. Arith. $31$ (2008), 145-169.
S. Akhtari, R. Okazaki. The quartic Thue equations. (2008), to appear in J. Number Theory.
A. Baker. Contributions to the theory of Diophantine equations. I. On the representation of integers by binary forms, Philos. Trans. R. Soc. Lond. Ser. A $263$ (1967/1968), 173–191.
Baker. A. The theory of linear forms in logarithms. InTranscendence Theory: Advances and Applications. Baker. A., Masser. D. W. ed. Academic Press 1977.
M.A. Bennett. On the representation of unity by binary cubic forms. Trans. Amer. Math. Soc. $353$ (2001), 1507-1534.
J.W. Cassels. An Introduction to the Geometry of Numbers. Springer-Verlag,1959.
G.V. Chudnovsky. On the method of Thue-Siegel. Ann. of Math. II Ser. $117$ (1983), 325-382.
J.E. Cremona. Reduction of binary cubic and quartic forms. LMS JMC $2$ (1999), 62-92.
J.H. Evertse. On the equation $ax\sp{n}-by\sp{n}=c$. Compositio Math. $47$ (1982), no. 3, 289–315.
J.H. Evertse. On the representation of integers by binary cubic forms of positive discriminant. Invent. Math.$73$(1983), 117-138.
J.H. Evertse. Upper bounds for the numbers of solutions of Diophantine equations. Mathematical Centre Tracts, $168$. Mathematisch Centrum, Amsterdam, 1983.
V. Krechmar. On the superior bound of the number of representation of an integer by binary forms of the fourth degree (Russian). Bull. Acad. Sci. URSS, Ser. Math. (1939), 289-302
W. Ljunggren. On the representation of integers by certain binary cubic and biquadratic forms. Acta. Arith. XVII (1971)
C.L. Siegel. Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss. $1$ (1929).
C.L. Siegel. Einige Erläuterungen zu Thues Untersuchungen über Annäherungswerte algebraischer zahlen und diophantische Gleichungen, Akad. D. Wiss. in Gottingen II, mat.-Phys. 1. 1970. no $8$, Gott. (1970), 169-195.
C.L. Siegel. Die Gleichung $ax^{n} - by^{n} = c$, Math. Ann. $114$ (1937), 57-68.
K. Stromberg. An Introduction to Classical Real Analysis. Wadsworth international mathematical series, 1981.
A. Thue. Bemerkungen Über gewisse Näherungsbrüche algebraischer Zahlen, Kra. Vidensk. Selsk. Skrifter. I. mat. nat. Kl. (1908), no $3$, Kra, 1908.
A. Thue. Über Annäherungenswerte algebraischen Zahlen. J. reine angew. Math. 135 (1909), 284-305.
A. Thue. Über rationale Annaherungswerte der reellen Wüzel der ganzen Funktion dritten Grades $x^{3}-ax-b$. Kra. Vidensk. Selsk. Skrifter. I. Mat. Nat. Kl. (1908), no. $6$, Kra, 1908.
A.Thue. Ein Fundamental Theorem zur Bestimmung vou Annäherungs-Werten aller Würzeln gewisser ganzer Funktionen. J. Reine Angew. Mat. 138 (1910), 96-108.
|
0.6cm
[ **Majorana Neutrinos, CP-Violation, Neutrinoless Double-Beta and** ]{}
[ **Tritium-Beta Decays** ]{}
S. Pascoli$^{(a,b)}$ [^1], S. T. Petcov$^{(a,b)}$ [^2]
Introduction
============
With the accumulation of more and stronger evidences for oscillations of the atmospheric[@SKatm00] and solar[@SKYSuz00SNO] neutrinos, caused by neutrino mixing (see, e.g., [@BiPe87]), the problem of the nature of massive neutrinos emerges as one of the fundamental problems in the studies of neutrino mixing. Massive neutrinos can be Dirac or Majorana particles. In the former case they possess a conserved lepton charge and distinctive antiparticles, while in the latter there is no conserved lepton charge and massive neutrinos are truly neutral particles identical with their antiparticles (see, e.g., [@BiPe87]). Thus, the question of the nature of massive neutrinos is directly related to the question of the basic symmetries of the fundamental particle interactions.
The present and upcoming neutrino oscillation experiments will allow to make a big step forward in understanding the patterns of neutrino mass squared differences and of $\nu$-mixing, but will not be able to determine the absolute values of the neutrino masses and to answer the question regarding the nature of massive neutrinos. The [$\mbox{}^3 {\rm H}$ $\beta$-decay ]{}experiments, studying the electron spectrum, are sensitive to the electron (anti-)neutrino mass $m_{\nu_e}$ and can give information on the absolute value of neutrino masses. The present bounds on $m_{\nu_e}$ from the Troitzk [@MoscowH3] and Mainz [@Mainz] experiments read (at 95% C.L.): $m_{\nu_e} < 2.5
{\mbox{$ \ \mathrm{eV} \ $}}$ [@MoscowH3] and $m_{\nu_e} < 2.9 {\mbox{$ \ \mathrm{eV} \ $}}$ [@Mainz]. There are prospects to increase the sensitivity of the [$\mbox{}^3 {\rm H}$ $\beta$-decay ]{}experiments to $m_{\nu_e} \sim (0.3 - 1.0)$ eV [@KATRIN] (the KATRIN project).
The problem of the nature of massive neutrinos can be addressed in experiments studying processes in which the total lepton charge $L$ is not conserved and changes by two units, $\Delta L = 2$. The process most sensitive to the existence of massive Majorana neutrinos (coupled to the electron) is the neutrinoless double $\beta$ (${\mbox{$(\beta \beta)_{0 \nu} $}}-$) decay of certain even-even nuclei (see, e.g., [@BiPe87]): $(A,Z) \rightarrow (A,Z+2) + e^{-} + e^{-}$. If the ${\mbox{$(\beta \beta)_{0 \nu} $}}-$ decay is generated only by the left-handed charged current weak interaction through the exchange of virtual light massive Majorana neutrinos, the probability amplitude of this process is proportional to the “effective Majorana mass parameter”: $${\mbox{$\left| < \! m \! > \right| \ $}}\equiv \left| | U_{\mathrm{e} 1}|^2~ m_1 +
|U_{\mathrm{e} 2}|^2~e^{i \alpha_{21}} m_2 +
| U_{\mathrm{e} 3}|^2~e^{i \alpha_{31}} m_3 \right|,
\label{meff}$$
where $m_j$ is the mass of the Majorana neutrino $\nu_j$, $U_{\mathrm{e}j}$ is the element of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino (lepton) mixing matrix [@BP57; @MNS62], and $\alpha_{j1}$, $j=2,3$, are two Majorana CP-violating phases [@BHP80]. If CP-parity is conserved we have [@LW81; @BNP84] $\alpha_{j1} =k \pi$, $k=0,1,2...$. Many experiments are searching for ${\mbox{$(\beta \beta)_{0 \nu} $}}-$decay. No indications that this process takes place were found so far. A most stringent constraint on the value of was obtained in the $^{76}$Ge Heidelberg-Moscow experiment [@76Ge00] $ {\mbox{$\left| < \! m \! > \right| \ $}}< 0.35 \ \mathrm{eV}$ ($90\%$ C.L.). The IGEX collaboration has obtained [@IGEX00] ${\mbox{$\left| < \! m \! > \right| \ $}}< (0.33 \div 1.35) \ \mathrm{eV}$ ( $90\%$ C.L.). Higher sensitivity to the value of ${\mbox{$\left| < \! m \! > \right| \ $}}$ is planned to be reached in several ${\mbox{$(\beta \beta)_{0 \nu} $}}-$decay experiments of a new generation [@NEMO3]. The NEMO3 and the planned CUORE experiments aim to achieve a sensitivity to values of ${\mbox{$\left| < \! m \! > \right| \ $}}\cong 0.1~$eV. A sensitivity to ${\mbox{$\left| < \! m \! > \right| \ $}}\cong 10^{-2}~$eV, is planned to be reached in the GENIUS and EXO experiments.
The present article represents a continuation of the studies of the physical implications of the possible future results on and on $m_{\nu_e}$ performed in Refs. [@BPP1; @BPP2; @PPW]. The article is based on the work done in [@PPW]. We generalize the results, derived in [@PPW] by using the best fit values of the input neutrino oscillation parameters, to the case when some of these parameters take values in their 90% C.L. allowed regions. Earlier studies on the subject were performed, e.g., in Refs. [@bbpapers1; @bbpapers2; @bbpapers3]. Recent relevant studies include Ref. [@Czakon01].
Constraining or Determining the Lightest Neutrino Mass $m_1$ and/or the Majorana CP-Violating Phases
====================================================================================================
In this Section, assuming $3-\nu$ mixing, we discuss the information that future -decay and/or $^3$H $\beta-$decay experiments can provide on the lightest neutrino mass $m_1$ and on the CP-violation generated by the Majorana CP-violating phases $\alpha_{21}$ and $\alpha_{31}$. We number the three neutrinos with definite mass $\nu_j$ in such a way that their masses obey $m_1 < m_2 < m_3$. The two cases of neutrino mass spectrum are analyzed: spectrum with normal hierarchy, ${\mbox{$ \Delta m^2_{\odot} \ $}}\equiv {\mbox{$ \ \Delta m^2_{21} \ $}}$, and with inverted hierarchy, ${\mbox{$ \Delta m^2_{\odot} \ $}}\equiv {\mbox{$ \ \Delta m^2_{32} \ $}}$, ${\mbox{$ \Delta m^2_{\odot} \ $}}> 0$ being the neutrino mass squared difference characterizing the solar neutrino oscillations (see, e.g., Refs. [@BPP1; @PPW]). In both cases we use in the analysis which follows fixed values of the mixing parameters $|U_{\mathrm{e}3}|^2$ (or $| U_{\mathrm{e}1}|^2$) and of the angle $\theta_{\odot}$, which are constrained by the CHOOZ and the solar neutrino data [^3], respectively. We allow and , which characterizes the oscillations of atmospheric neutrinos, to vary within their 90% C.L. allowed intervals found in the analyses of the solar and atmospheric neutrino data in Refs. [@Gonza3nu; @ConchaSNO]. We denote the minimal and maximal values in these intervals by ${\mbox{$(\Delta m^2_{\odot})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}$, ${\mbox{$ (\Delta m^2_{\odot})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}$, ${\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}$ and ${\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}$. The results thus obtained are summarized in Fig. 1 (normal neutrino mass hierarchy) and in Fig. 2 (inverted hierarchy).
Normal Mass Hierarchy: ${\mbox{$ \Delta m^2_{\odot} \ $}}\equiv {\mbox{$ \ \Delta m^2_{21} \ $}}$
--------------------------------------------------------------------------------------------------
If ${\mbox{$ \Delta m^2_{\odot} \ $}}= \Delta m^2_{21}$, for any given solution of the solar neutrino problem LMA MSW, LOW-QVO, SMA MSW, can lie anywhere between 0 and the present upper limits, as Fig. 1 indicates. This conclusion does not change even under the most favorable conditions for the determination of , namely, even when , , $\theta_{\odot}$ and $\theta$ are known with negligible uncertainty. The further conclusions in the case of the LMA MSW solution of the solar neutrino problem, which are illustrated in Fig. 1, are now summarized.
[**Case A.**]{} An experimental upper limit on , ${\mbox{$\left| < \! m \! > \right| \ $}}< {\mbox{$\left| < \! m \! > \right| \ $}}_{exp}$, will determine a maximal value of $m_1$, $m_1 < (m_1)_{max}$. The value $(m_1)_{max}$ is fixed by one of the two equalities: $$\begin{aligned}
\label{maxm1nhLMA}
{\mbox{$\left| < \! m \! > \right| \ $}}_{ \! exp} \! &=
\left| \! \left( \! m_1\cos^2 \! \! \theta_\odot - \sqrt{m_1^2 + {\mbox{$ (\Delta m^2_{\odot})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}\! }
\sin^2 \! \theta_\odot \! \right) \! (1 - {\mbox{$|U_{\mathrm{e} 3}|^2$ }}\! \! ) -
\sqrt{m_1^2 + {\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}} {\mbox{$|U_{\mathrm{e} 3}|^2$ }}\! \!
\right|; \\
\label{maxm1nhLMAcos0}
{\mbox{$\left| < \! m \! > \right| \ $}}_{ \! exp} \! &=
\left| \! \left( \! m_1 \cos^2 \! \! \theta_\odot - \sqrt{m_1^2 + {\mbox{$(\Delta m^2_{\odot})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}\! }
\sin^2 \! \theta_\odot \! \right) \! (1 - {\mbox{$|U_{\mathrm{e} 3}|^2$ }}\! \! ) +
\sqrt{m_1^2 + {\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}} {\mbox{$|U_{\mathrm{e} 3}|^2$ }}\! \!
\right|.\end{aligned}$$ Eq. (\[maxm1nhLMA\]) is valid for $\cos 2\theta_\odot > ({\mbox{$ \Delta m^2_{\odot} \ $}}/ m_1^2 ) \sin^4 \theta_\odot$, otherwise eq. (\[maxm1nhLMAcos0\]) should be used.
For the quasi-degenerate mass spectrum one has $m_1 \gg {\mbox{$ \Delta m^2_{\odot} \ $}},{\mbox{$\Delta m^2_{\mathrm{atm}} \ $}}$, $m_1 \cong m_2 \cong m_3 \cong m_{\nu_e}$, and [@PPW; @Czakon01] $$(m_1)_{max} \cong \frac{{\mbox{$\left| < \! m \! > \right| \ $}}_{exp}}
{\left|\cos 2\theta_\odot (1 - {\mbox{$|U_{\mathrm{e} 3}|^2$ }})
- {\mbox{$|U_{\mathrm{e} 3}|^2$ }}\right|}~.
\label{maxm1nhLMAQD}$$ If $ |\cos 2\theta_\odot (1 - {\mbox{$|U_{\mathrm{e} 3}|^2$ }})
- {\mbox{$|U_{\mathrm{e} 3}|^2$ }}|$ is sufficiently small, the upper limit on $m_{\nu_e}$ obtained in $^3$H $\beta-$decay experiments could yield a more stringent upper bound on $m_1$ than the bound from the limit on .
[**Case B.** ]{} A measurement of ${\mbox{$\left| < \! m \! > \right| \ $}}= ({\mbox{$\left| < \! m \! > \right| \ $}})_{exp}
\gtap 0.02$ eV would imply that $m_1 \gtap 0.02$ eV and thus a mass spectrum with partial hierarchy or of quasi-degenerate type [@BPP1]. The lightest neutrino mass will be constrained to lie in the interval, $(m_1)_{min} \leq m_1 \leq (m_1)_{max}$, where $(m_1)_{max}$ and $(m_1)_{min}$ are determined respectively by eq. (\[maxm1nhLMA\]) or (\[maxm1nhLMAcos0\]) and by the equation [@PPW]: $$(m_1\cos^2 \! \theta_\odot + \sqrt{m_1^2 + {\mbox{$ (\Delta m^2_{\odot})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}\! }
\sin^2 \! \theta_\odot) (1 - {\mbox{$|U_{\mathrm{e} 3}|^2$ }}\! \! ) \! + \!
\sqrt{m_1^2 + {\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}\! } {\mbox{$|U_{\mathrm{e} 3}|^2$ }}\! \! \!
= {\mbox{$\left| < \! m \! > \right| \ $}}_{ \! exp}.
\label{minm1nhLMA01}$$ The limiting values of $m_1$ correspond to the case of CP-conservation. If it holds that ${\mbox{$ \Delta m^2_{\odot} \ $}}\ll m_1^2$, (i.e., for ${\mbox{$ \Delta m^2_{\odot} \ $}}\ltap 10^{-4}~{\rm eV^2}$), $(m_1)_{min}$ to a good approximation is independent of $\theta_{\odot}$, and for $\sqrt{{\mbox{$\Delta m^2_{\mathrm{atm}} \ $}}} |U_{\mathrm{e}3}|^2 \ll m_1$, which takes place in the case we consider as $|U_{\mathrm{e}3}|^2 \ltap 0.05$, we have $(m_1)_{min} \cong ({\mbox{$\left| < \! m \! > \right| \ $}})_{exp}$. For $|U_{\mathrm{e}3}|^2 \ll \cos2\theta_{\odot}$, which is realized in Fig. 1 for $|U_{\mathrm{e}3}|^2 \ltap 0.01$, practically all the region between $(m_1)_{min}$ and $(m_1)_{max}$, $(m_1)_{min} < m_1 < (m_1)_{max}$, corresponds to violation of the CP-symmetry. If $|U_{\mathrm{e}3}|^2$ is non-negligible with respect to $\cos2\theta_{\odot}$, e.g., if $|U_{\mathrm{e}3}|^2 \cong (0.02 - 0.05)$ for the values of $\cos2\theta_{\odot}$ used to derive the right panels in Fig. 1, one can have $(m_1)_{min} < m_1 < (m_1)_{max}$ if the CP-symmetry is violated, as well as in two specific cases of CP-conservation [@PPW]. One of these two CP-conserving values of $m_1$, corresponding to $\eta_{21} = - \eta_{31} = -1$, can differ considerably from the two limiting values (see Fig. 1). In general, the knowledge of the value of alone will not allow to distinguish the case of CP-conservation from that of CP-violation.
[**Case C.**]{} It might be possible to determine whether CP-violation due to the Majorana phases takes place in the lepton sector if both and $m_{\nu_e}$ are measured. Since prospective measurements are limited to $(m_{\nu_e})_{exp} \gtap 0.35$ eV, the relevant neutrino mass spectrum is of quasi-degenerate type, $m_1 \cong m_2 \cong m_3 \cong m_{\nu_e}$ (see, e.g., [@BPP1]) and one has $m_1 > 0.35$ eV. If we can neglect $|U_{\mathrm{e}3}|^2$ (i.e., if $\cos 2\theta_\odot \gg |U_{\mathrm{e}3}|^2$), a value of $m_{\nu_e} \cong m_1$, satisfying $(m_1)_{min} < m_{\nu_e} < (m_1)_{max}$, where $(m_1)_{min}$ and $(m_1)_{max}$ are determined by eqs. (\[minm1nhLMA01\]) and (\[maxm1nhLMA\]) or (\[maxm1nhLMAcos0\]), would imply that the CP-symmetry does not hold in the lepton sector. In this case one would obtain correlated constraints on the CP-violating phases $\alpha_{21}$ and $\alpha_{31}$ [@BPP1; @Rodej00]. This appears to be the only possibility for demonstrating CP-violation due to Majorana CP-violating phases in the case of ${\mbox{$ \Delta m^2_{\odot} \ $}}\equiv {\mbox{$ \ \Delta m^2_{21} \ $}}$ under discussion [@PPW]. In order to reach a definite conclusion concerning CP-violation due to the Majorana CP-violating phases, considerable accuracy in the measured values of and $m_{\nu_e}$ is required. For example, if the oscillation experiments give the result $\cos2\theta_{\odot} \leq 0.3$ and ${\mbox{$\left| < \! m \! > \right| \ $}}= 0.3$ eV, a value of $m_{\nu_e}$ between 0.3 eV and 1.0 eV would demonstrate CP-violation. This requires better than 30% accuracy on both measurements. The accuracy requirements become less stringent if the upper limit on $\cos2\theta_{\odot}$ is smaller.
If $\cos2\theta_{\odot} > |U_{\mathrm{e}3}|^2$ but $|U_{\mathrm{e}3}|^2$ cannot be neglected in , there exist two CP-conserving values of $m_{\nu_e}$ in the interval $(m_1)_{min} < m_{\nu_e} < (m_1)_{max}$ [@PPW]. The one that can significantly differ from the extreme values of the interval corresponds to the specific case of CP-conservation with $\eta_{21} = - \eta_{31} = -1$ (Fig. 1).
[**Case D.** ]{} A measured value of $m_{\nu_e}$, $(m_{\nu_e})_{exp} \gtap 0.35$ eV, satisfying $(m_{\nu_e})_{exp} > (m_1)_{max}$, where $(m_1)_{max}$ is determined from the upper limit on , eq. (\[maxm1nhLMA\]) or (\[maxm1nhLMAcos0\]), in the case the -decay is not observed, might imply that the massive neutrinos are Dirac particles. If -decay has been observed and measured, the inequality $(m_{\nu_e})_{exp} > (m_1)_{max}$, would lead to the conclusion that there exist contribution(s) to the -decay rate other than due to the light Majorana neutrino exchange (see, e.g., [@bb0nunmi] and the references quoted therein) that partially cancel the one from the Majorana neutrino exchange.
A measured value of , $( {\mbox{$\left| < \! m \! > \right| \ $}})_{\mathrm{exp}} \gtap 0.01 \ \mathrm{eV}$, and a measured value of $m_{\nu_e}$ or an upper bound on $m_{\nu_e}$ such that $m_{\nu_e} < (m_1)_{min}$, where $(m_1)_{min}$ is determined by eq. (\[minm1nhLMA01\]), would imply that there are contributions to the -decay rate in addition to the ones due to the light Majorana neutrino exchange (see, e.g., [@bb0nunmi]), which enhance the -decay rate and signal the existence of new $\Delta L =2$ processes beyond those induced by the light Majorana neutrino exchange in the case of left-handed charged current weak interaction.
[**Case E.**]{} An actual measurement of ${\mbox{$\left| < \! m \! > \right| \ $}}\ltap 10^{-2}$ eV is unlikely, but it is illustrated in Fig. 1 to show the interpretation of such a result. There always remains an upper limit on $m_1$, $(m_1)_{max}$, determined by eq. (\[maxm1nhLMA\]) or by eq. (\[maxm1nhLMAcos0\]). For ${\mbox{$|U_{\mathrm{e} 3}|$}}^2 \geq {\mbox{ $\widehat{|U_{\mathrm{e} 3}|^2}$}}$, where ${\mbox{ $\widehat{|U_{\mathrm{e} 3}|^2}$}}\equiv (\sqrt{{\mbox{$(\Delta m^2_{\odot})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}} \sin^2 \theta_\odot )
/ \sqrt{{\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}}$, the cancellations between the different terms allow to have $(m_1)_{min}=0$. Both the cases of CP-conservation and CP-violation are allowed. In principle, there exists the possibility to have regions of the parameter space where only the case of CP-violation is allowed. However, in order to establish that and $m_1$ lie in those regions, they must be known with a precision which is far beyond that aimed at in the currently planned future experiments.
If ${\mbox{$|U_{\mathrm{e} 3}|$}}^2 < {\mbox{ $\widehat{|U_{\mathrm{e} 3}|^2}$}}$, and lies in the interval ${\mbox{$\left| < \! m \! > \right| \ $}}_{-}~ \leq {\mbox{$\left| < \! m \! > \right| \ $}}\leq {\mbox{$\left| < \! m \! > \right| \ $}}_{+}$, defined by $$\begin{aligned}
{\mbox{$\left| < \! m \! > \right| \ $}}_{+} &= |\sqrt{{\mbox{$ (\Delta m^2_{\odot})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}}
(\sin^2 \theta_\odot)_{\mbox{}_\mathrm{MAX}} (1 - {\mbox{$|U_{\mathrm{e} 3}|^2$ }}) +
\sqrt{{\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}} {\mbox{$|U_{\mathrm{e} 3}|^2$ }}|, \\
{\mbox{$\left| < \! m \! > \right| \ $}}_{-} &= |\sqrt{{\mbox{$(\Delta m^2_{\odot})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}}
(\sin^2 \theta_\odot)_{\mbox{}_\mathrm{MIN}} (1 - {\mbox{$|U_{\mathrm{e} 3}|^2$ }}) -
\sqrt{{\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}} {\mbox{$|U_{\mathrm{e} 3}|^2$ }}|, \end{aligned}$$ where $(\sin^2 \theta_\odot)_{\mbox{}_\mathrm{MIN}}$ and $(\sin^2 \theta_\odot)_{\mbox{}_\mathrm{MAX}}$ are respectively the minimal and maximal allowed values of $\sin^2 \theta_\odot$ in the LMA solution region, the lower limit on $m_1$ goes to zero. All the CP-parity patterns in the case of CP-conservation as well as the violation of the CP-symmetry are possible. If ${\mbox{$|U_{\mathrm{e} 3}|$}}^2 < {\mbox{ $\widehat{|U_{\mathrm{e} 3}|^2}$}}$ and ${\mbox{$\left| < \! m \! > \right| \ $}}< {\mbox{$\left| < \! m \! > \right| \ $}}_{-}~$, $(m_1)_{min}$ is determined by the following equation: $${\mbox{$\left| < \! m \! > \right| \ $}}_{exp} = \left| (m_1\cos^2 \theta_\odot - \sqrt{m_1^2 + {\mbox{$(\Delta m^2_{\odot})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}}
\sin^2 \theta_\odot) (1 - {\mbox{$|U_{\mathrm{e} 3}|$}}^2) +
\sqrt{m_1^2 + {\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}} {\mbox{$|U_{\mathrm{e} 3}|$}}^2
\right|.
\label{minm1nhLMA}$$ Under the above conditions, the case of CP-conservation corresponding to $\eta_{21} = \pm \eta_{31} = 1$ will be excluded. At the same time both the case of CP-conservation with $\eta_{21} = \pm \eta_{31} = - 1$ and that of CP-violation will be allowed.
It should be noted also that one can have ${\mbox{$\left| < \! m \! > \right| \ $}}= 0$ for $m_1 = 0$ in the case of CP-invariance if $\eta_{21} = - \eta_{31}$ and the relation $\sqrt{{\mbox{$ \Delta m^2_{\odot} \ $}}}
\sin^2 \theta_\odot (1 - |U_{\mathrm{e}3}|^2) =
\sqrt{{\mbox{$\Delta m^2_{\mathrm{atm}} \ $}}} |U_{\mathrm{e}3}|^2$ holds. Finally, there would seem to be no practical possibility to determine the Majorana CP-violating phases.
The analysis of the [*Cases A - E*]{} for the LOW-QVO solution of the solar neutrino problem leads to the same qualitative conclusions as those obtained above for the LMA MSW solution. The conclusions differ, however, in the case of the SMA MSW solution and we will discuss them next briefly [@PPW]. An experimental upper limit on ([*Case A*]{}) in the range ${\mbox{$\left| < \! m \! > \right| \ $}}_{exp} \geq 10^{-2}$ eV, would imply in the case of the SMA MSW solution, $m_1 < {\mbox{$\left| < \! m \! > \right| \ $}}_{exp}(1 - 2 {\mbox{$|U_{\mathrm{e} 3}|^2$ }})^{-1}$. For values of ${\mbox{$\left| < \! m \! > \right| \ $}}\gtap 10^{-2}$ eV, the maximum and minimum values of $m_1$ are extremely close: $(m_1)_{min} \cong {\mbox{$\left| < \! m \! > \right| \ $}}_{exp}$. As a result, a measurement of ([*Case B*]{}) practically determines $m_1$, $m_1 \cong {\mbox{$\left| < \! m \! > \right| \ $}}$. However, no information about CP-violation generated by the Majorana phases can be obtained by the measurement of (or of and $m_{\nu_e}$) [@BPP1]. If both ${\mbox{$\left| < \! m \! > \right| \ $}}\gtap 0.02$ eV and $m_{\nu_e}\gtap 0.35$ eV would be measured ([*Case C*]{}), the relation $m_1 \cong ({\mbox{$\left| < \! m \! > \right| \ $}})_{exp}
\cong (m_{\nu_e})_{exp}$ should hold. The conclusions in the [*Cases D*]{} and [*E*]{} are qualitatively the same as for the LMA MSW solution.
Inverted Mass Hierarchy: ${\mbox{$ \Delta m^2_{\odot} \ $}}\equiv {\mbox{$ \ \Delta m^2_{32} \ $}}$
----------------------------------------------------------------------------------------------------
Consider next the possibility of a neutrino mass spectrum with inverted hierarchy, which is illustrated in Fig. 2. A comparison of Fig. 1 and Fig. 2 reveals two major differences in the predictions for : if ${\mbox{$ \Delta m^2_{\odot} \ $}}\equiv {\mbox{$ \ \Delta m^2_{32} \ $}}$, i) even in the case of $m_1 \ll m_2 \cong m_3$ (i.e., even if $m_1 \ll 0.02$ eV), can exceed $\sim 10^{-2}$ eV and can reach the value of $\sim 0.08$ eV [@BPP1], and ii) a more precise determination of , , $\theta_{\odot}$ and $\sin^2\theta = |U_{e1}|^2$, can lead to a lower limit on the possible values of [@BPP1]. For the LMA and the LOW-QVO solutions, ${\rm min}({\mbox{$\left| < \! m \! > \right| \ $}})$ will depend, in particular, on whether CP-invariance holds or not in the lepton sector, and if it holds - on the relative CP-parities of the massive Majorana neutrinos. All these possibilities are parametrized by the values of the two CP-violating phases, $\alpha_{21}$ and $\alpha_{31}$, entering into the expression for . The existence of a significant lower limit on the possible values of depends crucially in the cases of the LMA and LOW-QVO solutions on the minimal value of $|\cos2\theta_{\odot}|$, $|\cos2\theta_{\odot}|_{\mbox{}_\mathrm{MIN}}$, allowed by the data. Up to corrections $\sim 5\times 10^{-3}$ eV, the minimum value of is for these two solutions (see, e.g., [@BPP1]): $$({\mbox{$\left| < \! m \! > \right| \ $}})_{\mbox{}_\mathrm{MIN}} \simeq
\left| \sqrt{{\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}} | ( \cos 2 \theta_\odot)|_{\mbox{}_\mathrm{MIN}}
(1 - {\mbox{$|U_{\mathrm{e} 1}|^2$ }}) \right|.
\label{minmeffLA}$$ The min() in eq. (\[minmeffLA\]) is reached in the case of CP-invariance and $\eta_{21} = - \eta_{31} = \pm 1$.
We shall discuss next briefly the implications of the results of future -decay and [$\mbox{}^3 {\rm H}$ $\beta$-decay ]{}experiments. We follow the same line of analysis we have used for neutrino mass spectrum with normal hierarchy. Consider the case of the LMA MSW solution of the solar neutrino problem.
[**Case A.**]{} An experimental upper limit on , ${\mbox{$\left| < \! m \! > \right| \ $}}< {\mbox{$\left| < \! m \! > \right| \ $}}_{exp}$, which is larger than the minimal value of , ${\mbox{$\left| < \! m \! > \right| \ $}}_{min}^{ph}$, predicted by taking into account all uncertainties in the values of the relevant input parameters (, , $\theta_{\odot}$, etc.), ${\mbox{$\left| < \! m \! > \right| \ $}}_{exp} \geq {\mbox{$\left| < \! m \! > \right| \ $}}_{min}^{ph}$, will imply an upper limit on $m_1$, $m_1 < (m_1)_{max}$. The latter is determined by the equality: $$\left| \! \left( \! \sqrt{m_1^2 + {\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}\! \! \! \! - {\mbox{$ (\Delta m^2_{\odot})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}\!}
\! \cos^2 \theta_\odot - \sqrt{m_1^2 + {\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}\!}
\! \sin^2 \theta_\odot \! \right) \! (1 - |U_{\mathrm{e}1}|^2 \!)
\pm m_1 |U_{\mathrm{e}1}|^2 \right| \! = \! {\mbox{$\left| < \! m \! > \right| \ $}}_{ \! exp},
\label{maxm1ih}$$ where the $+$ ($-$) corresponds to a negative (positive) value of the expression in the big round brackets [@PPW]. For the quasi-degenerate neutrino mass spectrum ($m_1 \gg {\mbox{$ \Delta m^2_{\odot} \ $}},{\mbox{$\Delta m^2_{\mathrm{atm}} \ $}}$, $m_1 \cong m_2 \cong m_3 \cong m_{\nu_e}$), $(m_1)_{max}$ is given by eq. (\[maxm1nhLMAQD\]) in which $|U_{\mathrm{e}3}|^2$ is replaced by $|U_{\mathrm{e}1}|^2$. Correspondingly, the conclusion that if $|\cos 2\theta_\odot (1 - |U_{\mathrm{e}1}|^2)
- |U_{\mathrm{e}1}|^2|$ is sufficiently small, the upper limit on $m_1 \cong m_{\nu_e}$, obtained in $^3$H $\beta-$decay, can be more stringent than the upper bound on $m_1$, implied by the limit on , remains valid.
An experimental upper limit on , which is smaller than the minimal possible value of , ${\mbox{$\left| < \! m \! > \right| \ $}}_{exp} < {\mbox{$\left| < \! m \! > \right| \ $}}_{min}^{ph}$, would imply that either i) the neutrino mass spectrum is not of the inverted hierarchy type, or ii) that there exist contributions to the -decay rate other than due to the light Majorana neutrino exchange (see, e.g., [@bb0nunmi]) that partially cancel the contribution from the Majorana neutrino exchange. The indicated result might also suggest that the massive neutrinos are Dirac particles.
[**Case B.**]{} A measurement of ${\mbox{$\left| < \! m \! > \right| \ $}}= ({\mbox{$\left| < \! m \! > \right| \ $}})_{exp}
\gtap \sqrt{{\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}}(1 - {\mbox{$|U_{\mathrm{e} 1}|^2$ }})
\cong (0.04 - 0.08)$ eV, where we have used the 90% C.L. allowed regions of and $|U_{\mathrm{e}1}|^2$ from [@Gonza3nu], would imply the existence of a finite interval of possible values of $m_1$, $(m_1)_{min} \leq m_1 \leq (m_1)_{max}$, with $(m_1)_{max}$ and $(m_1)_{min}$ given respectively by eq. (\[maxm1ih\]) and by the equality: $$m_1 {\mbox{$|U_{\mathrm{e} 1}|^2$ }}\! \! \! \! + \!
\left ( \! \sqrt{m_1^2 + {\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}\! \! \! \! \! - {\mbox{$(\Delta m^2_{\odot})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}\! \! }
\cos^2 \theta_\odot
+ \! \sqrt{m_1^2 + {\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}\! \! }
\sin^2 \theta_\odot \right ) \! (1 - {\mbox{$|U_{\mathrm{e} 1}|^2$ }}\! \! )
= {\mbox{$\left| < \! m \! > \right| \ $}}_{ \! exp}.
\label{minm1ih}$$ In this case $m_1 \gtap 0.04$ eV and the neutrino mass spectrum is with partial inverted hierarchy or of quasi-degenerate type [@BPP1]. The limiting values of $m_1$ correspond to CP-conservation. For ${\mbox{$ \Delta m^2_{\odot} \ $}}\ll m_1^2$, i.e., for ${\mbox{$ \Delta m^2_{\odot} \ $}}\ltap 10^{-4}~{\rm eV^2}$, $(m_1)_{min}$ is to a good approximation independent of $\theta_{\odot}$ and we have: $\sqrt{((m_1)_{min})^2 + {\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}}~
(1 - {\mbox{$|U_{\mathrm{e} 1}|^2$ }})\cong ({\mbox{$\left| < \! m \! > \right| \ $}})_{exp}$.
For negligible $|U_{\mathrm{e}1}|^2$ (i.e., $|U_{\mathrm{e}1}|^2 \ltap 0.01$ for the values of $\cos2\theta_{\odot}$ in Fig. 2), essentially all of the interval between $(m_1)_{min}$ and $(m_1)_{max}$, $(m_1)_{min} < m_1 < (m_1)_{max}$, corresponds to violation of the CP-symmetry. If the term $\sim m_1 |U_{\mathrm{e}1}|^2$ cannot be neglected in eqs. (\[maxm1ih\]) and (\[minm1ih\]) (i.e., if $|U_{\mathrm{e}1}|^2 \cong (0.02 - 0.05)$ for the values of $\cos2\theta_{\odot}$ in Fig. 2), there exists for a fixed ${\mbox{$\left| < \! m \! > \right| \ $}}_{exp}$ two CP-conserving values of $m_1$ in the indicated interval [@PPW], one of which differs noticeably from the limiting values $(m_1)_{min}$ and $(m_1)_{max}$ and corresponds to $\eta_{21} = - \eta_{31} = 1$ (Fig. 2).
In general, measuring the value of alone will not allow to distinguish the case of CP-conservation from that of CP-violation. In principle, a measurement of $m_{\nu_e}$, or even an upper limit on $m_{\nu_e}$, smaller than $(m_1)_{max}$, could be a signal of CP-violation, as Fig. 2 (upper panels) shows. However, unless $\cos2\theta_{\odot}$ is very small, the required values of $m_{\nu_e}$ are less than prospective measurements. For example, as is seen in Fig. 2, middle left panel, for $\cos2\theta_{\odot} = 0.1$ and ${\mbox{$\left| < \! m \! > \right| \ $}}= 0.03$ eV, one needs to find $m_{\nu_e} < 0.35$ eV to demonstrate CP-violation.
If the measured value of lies in the interval $({\mbox{$\left| < \! m \! > \right| \ $}}_{-})_{\mbox{}_\mathrm{MAX}} \leq {\mbox{$\left| < \! m \! > \right| \ $}}\leq ({\mbox{$\left| < \! m \! > \right| \ $}}_{+})_{\mbox{}_\mathrm{MIN}}$, where $$\begin{aligned}
\vspace{-0.2cm}
({\mbox{$\left| < \! m \! > \right| \ $}}_{+})_{\mbox{}_\mathrm{MAX}} =
| \sqrt{{\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}- {\mbox{$(\Delta m^2_{\odot})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}}
\cos^2 \theta_\odot + \sqrt{{\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}}
\sin^2 \theta_\odot | (1 - {\mbox{$|U_{\mathrm{e} 1}|^2$ }}), \\
\vspace{-0.2cm}
({\mbox{$\left| < \! m \! > \right| \ $}}_{-})_{\mbox{}_\mathrm{MIN}} =
| \sqrt{{\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}- {\mbox{$ (\Delta m^2_{\odot})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}}
\cos^2 \theta_\odot - \sqrt{{\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}}
\sin^2 \theta_\odot | (1 - {\mbox{$|U_{\mathrm{e} 1}|^2$ }}),
\label{0m1ih}\end{aligned}$$ we would have $(m_1)_{min} = 0$. Furthermore, if one finds that ${\mbox{$\left| < \! m \! > \right| \ $}}< ({\mbox{$\left| < \! m \! > \right| \ $}}_{+})_{\mbox{}_\mathrm{MIN}}$ with $ ({\mbox{$\left| < \! m \! > \right| \ $}}_{+})_{\mbox{}_\mathrm{MIN}} =
| \sqrt{{\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}- {\mbox{$ (\Delta m^2_{\odot})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}}
\cos^2 \theta_\odot + $ $ \sqrt{{\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}}
\sin^2 \theta_\odot | (1 - {\mbox{$|U_{\mathrm{e} 1}|^2$ }})$, the case of CP-conservation corresponding to $ \eta_{21} = \eta_{31} = \pm 1$ will be excluded. If $({\mbox{$\left| < \! m \! > \right| \ $}}_{-})_{\mbox{}_\mathrm{MAX}}
< {\mbox{$\left| < \! m \! > \right| \ $}}\! \! < \! ({\mbox{$\left| < \! m \! > \right| \ $}}_{\! +})_{\mbox{}_\mathrm{MIN}}$, where $({\mbox{$\left| < \! m \! > \right| \ $}}_{\! -})_{\mbox{}_\mathrm{MAX}} =
\left| \! \sqrt{{\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}\! \! \! - {\mbox{$(\Delta m^2_{\odot})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}\! } \!
\cos^2 \! \theta_\odot - \sqrt{{\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}\! \! }
\sin^2 \theta_\odot \right|$ $ (1 - {\mbox{$|U_{\mathrm{e} 1}|^2$ }})$ and $0 < m_1 < (m_1)_{\mathrm{max}}$, one will conclude that the CP-symmetry is violated.
[**Cases C.**]{} As Fig. 2 indicates, the discussions and conclusions are identical to the ones in the same cases for the neutrino mass spectrum with normal hierarchy, except that $(m_1)_{max}$ and $(m_1)_{min}$ are determined by eqs. (\[maxm1ih\]) and (\[minm1ih\]), and $|U_{\mathrm{e}3}|^2$ must be substituted by $|U_{\mathrm{e}1}|^2$ in the relevant parts of the analysis.
[**Case D.**]{} If $m_{\nu_e}$ is measured and $(m_{\nu_e})_{exp} \gtap 0.35 \ \mathrm{eV}$ but the -decay is not observed or is observed and $(m_{\nu_e})_{exp} > (m_1)_{max}$, where $(m_1)_{max}$ is determined by eq. (\[maxm1ih\]), the same considerations and conclusions as in the Case D for the normal hierarchy mass spectrum apply.
A measured value of , $( {\mbox{$\left| < \! m \! > \right| \ $}})_{\mathrm{exp}} \gtap 0.1 \ \mathrm{eV}$, in the case when the measured value of $m_{\nu_e}$ or the upper bound on $m_{\nu_e}$ are such that $m_{\nu_e} < (m_1)_{min}$, where $(m_1)_{min}$ is determined by eq. (\[minm1ih\]), would lead to the same conclusions as in the Case D for the normal hierarchy mass spectrum.
[**Case E.**]{} It is possible to have a measured value of ${\mbox{$\left| < \! m \! > \right| \ $}}\ltap 10^{-2}$ eV in the case of the LMA MSW solution and neutrino mass spectrum with inverted hierarchy under discussion only if $\cos2\theta_{\odot}$ is rather small, $\cos2\theta_{\odot}\ltap 0.2$. A measured value of ${\mbox{$\left| < \! m \! > \right| \ $}}< {\mbox{$\left| < \! m \! > \right| \ $}}_{min}^{ph}$ would imply that either the neutrino mass spectrum is not of the inverted hierarchy type, or that there exist contributions to the -decay rate other than due to the light Majorana-$\nu$ exchange that partially cancel the contribution from the Majorana-$\nu$ exchange.
The above conclusions hold with minor modifications (essentially of the numerical values involved) for the LOW-QVO solution as well. In the case of the SMA MSW solution we have, as is well-known, $\sin^2\theta_{\odot} << 1$ and ${\mbox{$ \Delta m^2_{\odot} \ $}}\ltap 10^{-5}~{\rm eV^2}$ (see, e.g., [@ConchaSNO]). Consequently, the analog of eq. (\[maxm1nhLMAQD\]) in [*Case A*]{} reads $(m_1)_{max} \cong {\mbox{$\left| < \! m \! > \right| \ $}}_{exp}
(1 - 2|U_{\mathrm{e}1}|^2)^{-1}$. The conclusions in the [*Cases B - D*]{} are qualitatively the same as in the case of neutrino mass spectrum with normal hierarchy. In particular, a measured value of ${\mbox{$\left| < \! m \! > \right| \ $}}> {\mbox{$\left| < \! m \! > \right| \ $}}_{+} \cong \sqrt{{\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}}
(1 - {\mbox{$|U_{\mathrm{e} 1}|^2$ }})$, would essentially determine $m_1$, $m_1 \cong ({\mbox{$\left| < \! m \! > \right| \ $}})_{exp}$. No information about CP-violation generated by the Majorana phases can be obtained by the measurement of , or of and $m_{\nu_e}$. If both ${\mbox{$\left| < \! m \! > \right| \ $}}$ and $m_{\nu_e}\gtap 0.35$ eV are measured, the relation $m_1 \cong ({\mbox{$\left| < \! m \! > \right| \ $}})_{exp}
\cong (m_{\nu_e})_{exp}$ should hold. If it is found that ${\mbox{$\left| < \! m \! > \right| \ $}}= \sqrt{{\mbox{$\Delta m^2_{\mathrm{atm}} \ $}}} (1 - {\mbox{$|U_{\mathrm{e} 1}|^2$}})$, one would have $0 \leq m_1 \leq (m_1)_{max}$, where $(m_1)_{max}$ is determined by eq. (\[maxm1ih\]) in which effectively $\sin^2\theta_{\odot} = 0$, $\cos^2\theta_{\odot} = 1$, and ${\mbox{$ \Delta m^2_{\odot} \ $}}= 0$. Finally, a measured value of ${\mbox{$\left| < \! m \! > \right| \ $}}< ({\mbox{$\left| < \! m \! > \right| \ $}}_{-})_{\mathrm{MIN}} \cong ({\mbox{$\left| < \! m \! > \right| \ $}}_{+})_{\mathrm{MIN}}
\cong \sqrt{{\mbox{$(\Delta m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}} (1 - {\mbox{$|U_{\mathrm{e} 1}|^2$ }})$ would either indicate that there exist new additional contributions to the -decay rate, or that the SMA MSW solution is not the correct solution of the solar neutrino problem.
Conclusions
===========
If -decay will be detected by present or upcoming experiments, we will conclude that neutrinos are massive Majorana particles and that the total lepton charge $L$ is not conserved. The observation of the -decay with a rate corresponding to ${\mbox{$\left| < \! m \! > \right| \ $}}\gtap 0.02~$eV, which is in the range of sensitivity of the future -decay experiments, can provide unique information on the type of neutrino mass spectrum and on the absolute values of neutrino masses. With additional information on the value of neutrino masses from [$\mbox{}^3 {\rm H}$ $\beta$-decay ]{}experiments or the type of neutrino mass spectrum, one could obtain also information on the CP-violation in the lepton sector, and - if CP-invariance holds - on the relative CP-parities of the massive Majorana neutrinos. The possibility of establishing CP-nonconservation requires high precision measurements. Given the precision of the future planned -decay and [$\mbox{}^3 {\rm H}$ $\beta$-decay ]{}experiments, it holds for a limited range of the values of the parameters involved.
Acknowledgements
================
S. P. and S. T. P. would like to thank L. Wolfenstein for stimulating discussions which are at the origin of this paper. S. P. is grateful to the organizers of this Workshop, for having being invited to such an interesting meeting which took place in a nice and stimulating atmosphere. The work of S.P. was partly supported by the Marie Curie Fellowship of the European Community program HUMAN POTENTIAL under contract number HPMT-CT-2000-00096.
References {#references .unnumbered}
==========
[99]{} SuperKamiokande Coll., talk by H. Sobel at [*Neutrino 2000*]{}, to appear in the Proceedings.
SuperK Coll., talk by Y. Suzuki at [*Neutrino 2000*]{}, to appear in the Proceedings; SNO Coll., Q.R. Ahmad [*et al.*]{}, nucl-ex/0100615.
S.M. Bilenky and S.T. Petcov, [*Rev. Mod. Phys.*]{} [**59**]{} (1987) 67.
V. Lobashov *et al.*, talk at [*Neutrino 2000*]{}, to appear in the Proceedings.
C. Weinheimer *et al.*, talk at [*Neutrino 2000*]{}, to appear in the Proceedings.
A. Osipowicz et al., (KATRIN Project), hep-ex/0109033.
B. Pontecorvo, Zh. Eksp. Teor. Fiz. [**33**]{}, 549 (1957), and [**34**]{}, 247 (1958).
Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. Phys. 28 (1962) 870.
S. M. Bilenky *et al.*, [*Phys. Lett.*]{} [**B94**]{} (1980) 495.
L. Wolfenstein, [*Phys. Lett.*]{} [**B107**]{} (1981) 77.
S. M. Bilenky, N. P. Nedelcheva and S. T. Petcov, [*Nucl. Phys.*]{} **B247** (1984) 589; B. Kayser, [*Phys. Rev.*]{} [**D30**]{} (1984) 1023;
L. Baudis *et al.*, [*Phys. Rev. Lett.*]{}[**83**]{} (1999) 41; H. V. Klapdor-Kleingrothaus *et al.*, submitted for publ. to [*Phys. Lett.*]{} B and talk given at NOW2000, Otranto, Italy, 09/2000.
C.E. Aalseth, F.T. Avignone III et al., [*Physics of Atomic Nuclei*]{} [**63**]{} (2000) 1225.
C. Marquet *et al.*, (NEMO3 Coll.) to appear in [*Nucl. Phys.*]{} B (Proc. Suppl.) [**87**]{} (2000); E. Fiorini, [*Phys. Rep.*]{} [**307**]{} (1998) 309; H. V. Klapdor-Kleingrothaus *et al.*, [*J. Phys.*]{} G **24** (1998) 483; L. Baudis *et al.*, [*Phys. Rep.*]{} [**307**]{} (1998) 301; M. Danilov *et al.*, [*Phys. Lett.*]{} [**B480**]{} (2000) 12.
S.M. Bilenky, S. Pascoli and S.T. Petcov, [*Phys. Rev.*]{} [**D64**]{} (2001) 053010, (hep-ph/0102265).
S.M. Bilenky, S. Pascoli and S.T. Petcov, [*Phys. Rev.*]{} [**D64**]{} (2001) 113003, (hep-ph/0104218).
S. Pascoli, S.T. Petcov and L. Wolfenstein, hep-ph/0110287.
S. T. Petcov and A. Yu. Smirnov, [*Phys. Lett.*]{} **B322** (1994) 109.
S.M. Bilenky et al., [*Phys. Rev.*]{} **D54** (1996) 4432; S.M. Bilenky et al., [*Phys. Lett.*]{} [**B465**]{} (1999) 193; V. Barger and K. Whisnant, [*Phys. Lett.*]{} **B456** (1999) 194; H. Minakata and O. Yasuda, [*Nucl. Phys.*]{} **B523** (1998) 597; T. Fukuyama *et al.*, [*Phys. Rev.*]{} **D57** (1998) 5844 and hep-ph/9804262;
F. Vissani, [*JHEP*]{} **06** (1999) 022; M. Czakon *et al.*, hep-ph/0003161; H. V. Klapdor-Kleingrothaus, H. Pas and A. Yu. Smirnov, [*Phys. Rev.*]{} **D63** (2001) 073005. M. Czakon *et al.*, hep-ph/0110166.
W. Rodejohann, hep-ph/0008044.
C. Gonzalez-Garcia *et al.*, [*Phys. Rev.*]{} [**D63**]{} (2001) 033005. G.F. Fogli *et al.*, hep-ph/0106247; J.N. Bachall, M.C. Gonzalez-Garcia and C. Peña-Garay, hep-ph/0106258; A. Bandyopadhyay *et al.*, hep-ph/0106264; M.V. Garzelli and C. Giunti, hep-ph/0108191.
K.S. Babu and R.N. Mohapatra, Phys. Rev. Lett. 75 (1995) 2276; H. Pas et al., Phys. Lett. B398 (1977) 311 and [*ibid.*]{} B459 (1999) 450.
[^1]: To be published in the Proceedings of the Conference NANP’01, III International Conference on Non-Accelerator New Physics, Dubna (Russia), June 19-23 2001.
[^2]: Also at: Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria
[^3]: In our further discussion we assume $\cos2\theta_{\odot} \geq 0$, which is favored by the analyses of the solar neutrino data [@ConchaSNO]. The modification of the relevant formulae and of the results in the case $\cos2\theta_{\odot} < 0$ is rather straightforward.
|
---
abstract: 'Partitioning a graph into groups of vertices such that those within each group are more densely connected than vertices assigned to different groups, known as *graph clustering*, is often used to gain insight into the organisation of large scale networks and for visualisation purposes. Whereas a large number of dedicated techniques have been recently proposed for static graphs, the design of *on-line* graph clustering methods tailored for evolving networks is a challenging problem, and much less documented in the literature. Motivated by the broad variety of applications concerned, ranging from the study of biological networks to the analysis of networks of scientific references through the exploration of communications networks such as the World Wide Web, it is the main purpose of this paper to introduce a novel, computationally efficient, approach to graph clustering in the evolutionary context. Namely, the method promoted in this article can be viewed as an incremental eigenvalue solution for the spectral clustering method described by Ng. *et al.* (2001). The incremental eigenvalue solution is a general technique for finding the approximate eigenvectors of a symmetric matrix given a change. As well as outlining the approach in detail, we present a theoretical bound on the quality of the approximate eigenvectors using perturbation theory. We then derive a novel spectral clustering algorithm called *Incremental Approximate Spectral Clustering* (IASC). The IASC algorithm is simple to implement and its efficacy is demonstrated on both synthetic and real datasets modelling the evolution of a HIV epidemic, a citation network and the purchase history graph of an e-commerce website.'
author:
- Charanpal Dhanjal
- Romaric Gaudel
- Stéphan Clémençon
bibliography:
- 'references.bib'
title: 'Efficient Eigen-updating for Spectral Graph Clustering'
---
Introduction
============
*Graph-mining* has recently received increasing attention in the machine-learning literature, motivated by application domains such as the Internet, social networks, epidemics of transmissible infectious diseases, sensor and biological networks. A crucial task in exploratory analysis and data visualization is *graph clustering* [@schaeffer2007graph; @fortunato2010community], which aims to partition the vertices in a graph into groups or clusters, with dense internal connections and few connections between each other. There is a large body of work on graph clustering. A possible approach is to consider a certain measure that quantifies community structure and formulate the clustering issue as an optimisation problem (which is generally NP-hard), for which fairly good solutions can be obtained recursively or by using adequate metaheuristics, see [@newman2004finding; @newman04fastAlg; @flake2004graph; @satuluri2009scalable] for example. A variety of approaches to graph clustering exist, such as those based on modularity maximisation for instance, see [@newman2004finding; @newman2006modularity]. In this paper, focus is on the *spectral clustering* approach [@von2007tutorial], which performs well empirically, is often simple to implement and benefits computationally from the availability of fast linear algebra libraries. The general principle of spectral clustering is to compute the smallest eigenvectors of some particular matrix $\Lm$ (refer to Section \[sec:spectral\_clustering\] for further details) and then cluster the vertices based on their representation in the corresponding eigen-space. The popularity of this approach arises from the fact that the obtained clustering of vertices is closely connected to the spectral relaxation of the minimization of the normalised cut criterion, see [@shi2000normalized].
In many applications such as communications networks (e.g. the Web and Internet), biological networks (of proteins, metabolic reactions, *etc.*), social networks or networks of scientific citations for instance, the graphs of interest slowly change with time. A naive approach to this incremental problem is to cluster each graph in the sequence separately, however this is computationally expensive for spectral clustering as the cost of solving the eigenvalue problem is $\mathcal{O}(n^3)$ at each iteration, where $n$ is the number of vertices. There has been some previous work on the incremental spectral clustering problem, for example [@valgren2007incremental; @ning2010incremental; @ning2007incremental; @kong2011fast] however only [@ning2010incremental; @ning2007incremental; @kong2011fast] update the eigen-system. In this paper we propose an efficient method for clustering a sequence of graphs which leverages the eigen-decomposition and the clustering on the previous graph to find the new clustering. Firstly, a fast approximation of a rank-$k$ eigen-decomposition of $\Lm_{t+1}$ knowing that of $\Lm_t$ is derived from the Singular Value Decomposition (SVD) updating approach used for Latent Semantic Indexing in [@zha1999updating]. Here, the update is efficient to compute when the change (defined in the subsequent analysis) between $\Lm_t$ and $\Lm_{t+1}$ is small. Secondly the clustering of vertices is updated accordingly to the new eigen-space. The efficiency of the complete approach, in terms of clustering accuracy, is demonstrated on synthetic and real data. We point out that the clustering approach was first outlined in [@dhanjal11cluster]. Here we provide a theoretical analysis on the quality of the eigen-approximation, as well as a more extensive empirical study of the algorithm.
The paper is organised as follows. Standard spectral clustering and SVD updating approaches are recalled in Sections \[sec:spectral\_clustering\] and \[sec:svd\_update\]. Then Section \[sec:our\_work\] details the proposed eigen-decomposition update and Section \[sec:quality\] studies the accuracy of the resulting approximate eigenvectors using perturbation theory. In Section \[sec:incrementalCluster\] we show how the eigen-decomposition updates can be applied to spectral clustering. Numerical results are gathered in Section \[sec:exp\], and the paper ends with Section \[sec:conclusion\] discussing results and planned future work.
**Notation**: A bold uppercase letter represents a matrix, e.g. $\Xm$, and a column vector is displayed using a bold lowercase letter e.g. $\xv$. The transpose of a matrix or vector is written $\Xm^T$. The concatenation of the columns of two matrices $\Am$ and $\Bm$ is written $[\Am \quad \Bm]$. $\Am[I, I]$ represent the submatrix of $\Am$ formed using the row and columns indexed by $I$, and $\Am[I, :]$ and $\Am[:, I]$ are submatrices formed using the rows and columns $I$ respectively. The matrix $\Am_k$ is that formed using the largest $k$ eigenvectors and eigenvalues of $\Am$, however $\Imat_p$ is the $p\times p$ identity matrix.
Graph Clustering {#sec:spectral_clustering}
================
Consider an undirected graph $G = (V, E)$, composed of a set of vertices $V=\{v_1, \ldots, v_n\}$ and edges $E \subseteq V \times V$ such that for every edge $(v_i, v_j)$ there is also an edge $(v_j, v_i)$. One way of representing the edges of $G$ is using an *adjacency matrix* $\Am \in \{0, 1\}^{n \times n}$ which has $\Am_{ij} = 1$ if there is an edge from the $i$-th to the $j$-th vertex and $\Am_{ij} = 0$ otherwise. More generally, the *weight matrix* $\Wm \in \mathbb{R}^{n \times n}$ allows one to assign nonzero numerical values on edges, and thus $\Wm_{ij} \neq 0$ when there is an edge from the $i$-th to the $j$-th vertex.
In the perspective of spectral clustering, a useful way of representing $G$ is through its Laplacian matrix [@chung1997spectral]. There are several definitions of the Laplacian matrix, however we are interested in the *normalised Laplacian matrix*, which is symmetric and positive semi-definite. We recall its definition below for clarity.
The *unnormalised Laplacian matrix* of a graph $G$ is defined as $\Lm = \Dm - \Wm$ where $\Dm$ is the degree matrix with supposedly nonzero diagonal entries $\Dm_{ii} = deg(v_i)$, denoting by $deg(v_i)=\sum_{j}\Wm_{ij}$ the degree of the $i$-th vertex, and zeros elsewhere, and $\Wm$ is the weight matrix. The normalised Laplacian matrix of a graph $G$ is then defined as $$\tilde{\Lm} = \Dm^{-\frac{1}{2}}\Lm\Dm^{-\frac{1}{2}}.$$
The normalised Laplacian matrix is used in the spectral clustering approach of Ng et al. [@ng2001spectral]. The algorithm computes the Laplacian and then finds the $k$ smallest eigenvectors which are used for clustering in conjunction with the $k$-means algorithm, see Algorithm \[alg:normCluster\]. The Laplacian matrix is often sparse and one can use power or Krylov subspace methods such as the Lanczos method to find the eigenvectors. There are several variants of Algorithm \[alg:normCluster\], such as that of [@shi2000normalized] which uses the so-called random walk Laplacian and clusters the smallest eigenvectors in a similar way. One of the motivations for these clustering methods is from the spectral relaxation of the minimisation of the normalised cut criterion.
Graph $G$ with weight matrix $\Wm \in \mathbb{R}^{n \times n}$, number of clusters $k$ Find $k$ smallest eigenvectors $\Vm_k = [\vv_1, \ldots, \vv_k]$ of normalised Laplacian $\tilde{\Lm}$ Normalise the rows of $\Vm_k$, i.e. $\Vm_k \leftarrow \diag(\Vm_k\Vm_k^T)^{-\frac{1}{2}}\Vm_k$ Cluster rows of $\Vm_k$ with the $k$-means algorithm Cluster membership vector $\cv \in \{1,\ldots,k\}^n$
\[alg:normCluster\]
A naive approach to spectral clustering on a sequence of graphs has a large update cost due to the computation of the eigen-decomposition of $\tilde{\Lm}_t$ at each iteration $t$. A more efficient approach is to update relevant eigenvectors from one iteration to the next. In [@valgren2007incremental] the authors use the spectral clustering of [@ng2001spectral] however they do not update the eigenvectors incrementally but instead the clustering directly. The iterative clustering approach of *Ning et al.* [@ning2010incremental; @ning2007incremental] uses the spectral clustering method given in [@shi2000normalized] and updates eigenvectors incrementally. The algorithm incrementally updates the solution of the generalised eigenproblem $\Lm \vv = \lambda \Dm \vv$ by finding the derivatives on the eigenvalues/vectors with perturbations in all of the quantities involved. An iterative refinement algorithm is given for the eigenvalues and eigenvectors given a change in the edges or vertices of a graph. One then clusters the resulting $k$ smallest eigenvectors using $k$-means clustering. In order to limit errors which can build up cumulatively the authors recompute the eigenvectors after every $R$-th graph in the sequence. A modification of the algorithm is proposed in [@kong2011fast] which improves efficiency by modelling clusters using a set of representative points (a similar strategy is used in [@yan2009fast] for example to cluster points in $\mathbb{R}^d$).
A disadvantage of the approach of [@ning2010incremental; @ning2007incremental] lies in the fact that, to update an eigenvector, one must invert a small matrix for each weight change in the graph which makes updates costly. The size of this matrix is proportional to the number of neighbours of the vertices incident to the changed edge. To be more precise, the cost of updating an eigenvector in [@ning2010incremental] is $\mathcal{O}(\bar{N}^2 n + \bar{N}^3)$ in which $\bar{N}$ is the average size of the “spatial neighbourhood” of an edge, i.e. the average number of rows/columns incident to the edge. It follows that to update $k$ eigenvectors following $r$ edge changes, the complexity is $\mathcal{O}(rk(\bar{N}^2 n + \bar{N}^3))$. In contrast, the approach presented in our paper has a smaller update cost for a set of vertex or edge weight changes between $\Lm_t$ and $\Lm_{t+1}$, since changes are considered in a batch fashion, as will later become clear. A further problem with the *Ning et al.* approach is that eigenvectors are updated independently of one another and hence one loses the orthogonality $\vv_i^T\Dm\vv_j = \delta(i, j)$, where $\delta$ is the Kronecker delta function (taking the value $1$ if $i=j$ and $0$ otherwise), and vectors can become correlated for example.
Another way of improving the efficiency of spectral clustering is to compute approximate eigen-decompositions at each iteration, for example by using the Nyström approach [@williams01nystrom]. The Nyström method is used for spectral graph clustering in [@fowlkes2004spectral] in conjunction with image segmentation. To estimate the eigenvalues and eigenvectors of $\Am \in \mathbb{R}^{n \times n}$, one first finds a matrix $\Am[I, I]$ in which $I \in \{1, \ldots n\}^m$ is a set of indices selected uniformly at random. If we assume that $\Am[I, I]$ is positive definite, we can take its square root $\Am[I, I]^{1/2}$. One then defines $$\Sm = \Am[I, I] + \Am[I, I]^{-1/2}\Am[I, \bar{I}]\Am[I, \bar{I}]^T\Am[I, I]^{-1/2},$$ where $\bar{I}$ is the complement of $I$, and diagonalises $\Sm$ using the eigen-decomposition $\Sm = \Um\Lambdam\Um^T$. Let $\Vm = \Am[:, I] \Am[I, I]^{-1/2} \Um \Lambdam^{-1/2}$, then it can be shown that the Nyström approximation of $\Am$, given by $\tilde{\Am}_I = \Am[:, I]\Am[I, I]^{-1}\Am[I, :]$, is equivalent to $\Vm \Lambdam\Vm^T$. If $\Am$ is indefinite a more complicated two-step procedure is required, see [@fowlkes2004spectral] for details. The resulting approximation is applied to find the first few eigenvectors of the normalised Laplacian matrix at a total cost of $\mathcal{O}(nm^2 + m^3)$. Several efficiency improvements based on this approach are proposed in [@li2011time] which finds the largest $k$ approximate eigenvectors of $\Dm^{-\frac{1}{2}}\Wm\Dm^{-\frac{1}{2}}$ at a reduced time and space complexity.
The quality of the resulting approximation is determined by the extent the submatrix $\Am[\bar{I}, \bar{I}]$ is spanned by $\Am[I, :]$. One would naturally expect a good approximation for example when $\Am[I, :]$ spans the space of $\Am$. The choice of sampling of $I$ also effects approximation quality, and empirical and theoretical work in [@kumar2009sampling] suggests random sampling without replacement over the non-uniform sampling in [@drineas2005nystrom; @drineas2006fast]. Hence for our later empirical work, we will use uniform random sampling without replacement. In addition to [@kumar2009sampling], error bounds for the Nyström method are presented in [@drineas2005nystrom; @mahdavi2012improved] in terms of the matrix approximation error using the Frobenius or spectral norm (given by $\|\Am\|_F = \sqrt{\tr(\Am^T\Am)}$ and $\|\Am \|_2 = \sqrt{\lambda_{max}(\Am^\Tm\Am)}$ respectively). One of the disadvantages of applying the Nyström based approaches of [@fowlkes2004spectral; @li2011time] on graphs is that by sampling only a subset of columns of the normalised Laplacian one can exclude important edges which help define clusters. These approaches are naturally more effective when a set of points $\{\xv_1, \ldots, \xv_n\} \in \mathbb{R}^d$ is mapped into a weighted graph using for example the Gaussian weighted distance $\Wm_{i, j} = \exp(-\|\xv_i - \xv_j\|^2)/2\sigma^2$, for $\sigma \in \mathbb{R}^+$, which is very informative about the relative positions of the points.
Another class of algorithms which are potentially useful for spectral clustering is the randomised SVD [@halko2011finding]. Notice that the SVD of a symmetric positive semi-definite matrix is identical to its eigendecomposition and hence SVD algorithms can be applied to spectral clustering. We recount an algorithm from [@halko2011finding] which is used in conjunction with kernel Principal Components Analysis (KPCA, [@sch98nonlinear]) in [@yun2011nystrom]. Algorithm \[alg:randomSVD\] provides the associated pseudo-code. The purpose of the first three steps is to find an orthogonal matrix $\Qm$ such that the projection of $\Am$ onto $\Qm$ is a good approximation of $\Ym$, whose columns are random samples from the range of $\Am$, under a rank-$k$ projection. This projection is then used in conjunction with $\Qm$ so that one need only find the SVD of the smaller matrix $\Bm$. The complexity of this approach is $\mathcal{O}((qr+r^2)(m + n))$, which for a square matrix of size $n$ collapses simply to $\mathcal{O}((qr+r^2)n)$
Matrix $\Am \in \mathbb{R}^{m \times n}$, number of projection vectors $r$, exponent $q$ Generate a random Gaussian matrix $\Omegam \in \mathbb{R}^{n \times r}$ Create $\Ym = (\Am\Am^T)^q \Am \Omegam$ by alternative multiplication with $\Am$ and $\Am^T$ \[step:power\] Compute $\Ym = \Qm\Rm$ using the QR-decomposition Form $\Bm = \Qm^T\Am$ and compute SVD $\Bm = \hat{\Um}\Sigmam\Vm^T$ Set $\Um = \Qm\hat{\Um}$ Approximate SVD $\Am \approx \Um\Sigmam\Vm^T$
\[alg:randomSVD\]
SVD Updating {#sec:svd_update}
============
An important aspect of the incremental clustering method described later lies in its ability to efficiently compute the eigenvectors of a matrix from the eigenvectors of a submatrix, and for clarity’s sake, we outline the SVD-updating algorithm of [@zha1999updating]. The SVD of $\Am \in \mathbb{R}^{m \times n}$ is the decomposition $$\Am=\Pm \Sigmam \Qm^T,$$ where $\Pm = [\pv_1, \ldots, \pv_m]$, $\Qm = [\qv_1, \ldots, \qv_n]$ and $\Sigmam = \diag(\sigma_1, \ldots, \sigma_r)$ are respectively the orthogonal matrices of left and right singular vectors and a diagonal matrix of singular values $\sigma_1 \geq \sigma_2 \geq, \ldots, \geq \sigma_r$, with $r = \min(m, n)$.
In [@zha1999updating], the authors use the SVD of $\Am$ to approximate the SVD of $[\Am \quad \Bm]$, with $\Bm \in \mathbb{R}^{m \times p}$, without recomputing the SVD of the new matrix. It is known that the best $k$-rank approximation of $\Am$, using the Frobenius norm error, is given by its SVD, namely $$\Am_k = \Pm_k\Sigmam_k\Qm_k^T,$$ where $\Pm_k$, $\Qm_k$, $\Sigmam_k$ correspond to the $k$ largest singular values. The general idea of the algorithm is to write $$[\Am_k \quad \Bm] = \Sm\Lambdam\Rm^T,$$ in which $\Sm$ and $\Rm$ are matrices with orthonormal columns. One then takes the rank-$k$ SVD $\Lambdam = \Gm_k \Gammam_k \Hm_k^T$ and $(\Sm\Gm_k) \Gammam_k (\Rm\Hm_k)^T$ is the rank-$k$ approximation for $[\Am \quad \Bm]$. The dimensionality of $\Lambdam$ is generally much smaller than that of $[\Am_k \quad \Bm]$ and hence the corresponding SVD approximation is computationally inexpensive.
This approach is more accurate that those presented in [@berry1995using; @o1994information], however it comes at additional computational cost. Furthermore, the authors prove that when the matrix $[\Am \quad \Bm]^T[\Am \quad \Bm]$ has the form $\Xm + \alpha^2 \Imat$ in which $\Xm$ is symmetric positive semi-definite with rank-$k$ then the rank-$k$ approximation of $[\Am \quad \Bm]$ is identical to that of $[\Am_k \quad \Bm]$.
Incremental Eigen-approximation {#sec:our_work}
===============================
In this section we address three types of updating problem upon the largest $k$ eigenvectors of a symmetric matrix. The updates are general operations, however they will be explained in the context of spectral clustering later in Section \[sec:incrementalCluster\]. Assuming that $\Ym_1, \Ym_2 \in \mathbb{R}^{n \times p}$ and one does not have direct access to $\Am \in \mathbb{R}^{m \times n}$ and $\Bm \in \mathbb{R}^{m \times p}$ but only to the matrices $\Cm = \Am^T\Am$, $\Am^T\Bm$ and $\Bm^T\Bm$, the updates are:
1. Addition of a low-rank symmetric matrix $\Cm \rightarrow \Cm + \Um$ where $\Um = \Ym_1\Ym_2^T + \Ym_2\Ym_1^T$
2. Addition of rows and columns $\Am^T\Am \rightarrow [\Am \quad \Bm]^T[\Am \quad \Bm]$
3. Removal of rows and columns $[\Am \quad \Bm]^T[\Am \quad \Bm] \rightarrow \Am^T\Am$
We have written the updates above in terms of a symmetric matrix $\Am^T\Am$ to improve notation. Note that any positive semi-definite symmetric matrix $\Mm$ can be decomposed into the form $\Mm = \Am^T\Am$, where $\Am$ has real entries, for example by using an eigen-decomposition or incomplete Cholesky factorisation.
Addition of a Low-rank Symmetric Matrix {#subsec:eigenAdd}
---------------------------------------
The first type of eigen-approximation we are interested in is the addition of a low-rank symmetric matrix. One computes the eigen-decomposition of $\Cm$ and then approximates the rank-$k$ decomposition of $\Cm_k + \Um$ where $\Um = \Ym_1\Ym_2^T + \Ym_2\Ym_1^T$ and $\Cm_k$ is the approximation of $\Cm$ using the $k$ largest eigenvectors (also known as the best $k$-rank approximation of $\Cm$). A similar but not applicable update is considered for the SVD case in [@zha1999updating] in which the rank-$k$ approximation of $\Am_k + \hat{\Ym}_1\hat{\Ym}_2^T$ is found from $\Am_k$ in which $\hat{\Ym}_1 \in \mathbb{R}^{m \times j}$ and $\hat{\Ym}_2 \in \mathbb{R}^{n \times j}$ for some $j$. The general idea in our case is to find a matrix with orthonormal columns $\tilde{\Qm}$ such that $\Cm_k + \Um = \tilde{\Qm} \Deltam \tilde{\Qm}^T$ for a square matrix $\Deltam$. To this purpose, we first project the columns of $\Ym_1$ into the space orthogonal to the $k$ largest eigenvectors $\Qm_k$ of $\Cm$ (note the deviation from standard notation), a process known as *deflation*. Assuming that eigenvectors have unit norm, the matrix $\Ym_1$ is thus deflated as follows:
$$\bar{\Ym}_1 = (\Imat - \Qm_k\Qm_k^T)\Ym_1,$$ at a cost of $\mathcal{O}(npk)$. Note that $\bar{\Ym}_1\Thetam_1$ for some $\Thetam_1$, is orthogonal to $\Qm_k$ since $\Qm_k^T\bar{\Ym}_1\Thetam_1 = (\Qm_k^T\Ym_1 - \Qm_k^T\Ym_1)\Thetam_1 = \textbf{0}$. If we take the SVD $\bar{\Ym}_1 = \bar{\Pm}_1\bar{\Sigmam}_1 \bar{\Qm}^T_1$ then $\bar{\Pm}_1$ is orthogonal to $\Qm_k$ since $\bar{\Pm}_1 = \bar{\Ym}_1\bar{\Qm}_1\bar{\Sigmam}^{-T}_1$ assuming $\bar{\Sigmam}_1$ has nonzero diagonal entries.
At the next stage we would like to orthogonalise the columns of $\Ym_2$ with respect to both $\Qm_k$ and $\bar{\Pm}_1$. Hence we deflate $\Ym_2$ in the following way:
$$\bar{\Ym}_2 = (\Imat - \bar{\Pm}_1\bar{\Pm}_1^T - \Qm_k\Qm_k^T)\Ym_2,$$ at cost $\mathcal{O}(npk)$, where we have used the fact that $\bar{\Pm}_1$ is orthogonal to $\Qm_k$. Proved in a similar way to the step used earlier, the matrix in the column space of $\bar{\Ym}_2$, $\bar{\Ym}_2\Thetam_2$ for some $\Thetam_2$, is orthogonal to $\Qm_k$ and $\bar{\Pm}_1$. Hence, we compute the SVD $\bar{\Ym}_2 = \bar{\Pm}_2\bar{\Sigmam}_2 \bar{\Qm}^T_2$ and note that the matrices $\bar{\Pm}_1$, $\bar{\Pm}_2$ and $\Qm_k$ are mutually orthogonal and span the space spanned by $\Cm_k + \Um$. This allows one to write
$$\Cm_k + \Um = \tilde{\Qm} \Deltam \tilde{\Qm}^T$$
as required with $\tilde{\Qm} = [\Qm_k \; \bar{\Pm}_1 \; \bar{\Pm}_2] \in \mathbb{R}^{n \times (k+2p)}$ and $\Deltam = \tilde{\Qm}^T (\Cm_k + \Um) \tilde{\Qm}$, or equivalently
$$\Deltam =
\left[ \begin{array}{c c c} \Omegam_k + \Qm_k^T\Um\Qm_k & \Qm_k^T\Um\bar{\Pm}_1& \Qm_k^T\Ym_1\bar{\Qm}_2\bar{\Sigmam}_2 \\
\bar{\Pm}_1^T\Um\Qm_k & \bar{\Pm}_1^T\Um\bar{\Pm}_1 & \bar{\Sigmam}_1\bar{\Qm}_1^T\bar{\Qm}_2\bar{\Sigmam}_2 \\
\bar{\Sigmam_2}\bar{\Qm}_2^T\Ym_1^T\Qm_k & \bar{\Sigmam}_2\bar{\Qm}_2^T\bar{\Qm}_1\bar{\Sigmam}_1 & \textbf{0} \end{array} \right],$$
in which $\Deltam \in \mathbb{R}^{(k+2p) \times (k+2p)}$. We take the rank-$k$ eigen-decomposition $\Deltam_k = \Hm_k\Pim_k\Hm_k^T$ and then the final eigen-approximation is given by $(\tilde{\Qm} \Hm_k) \Pim_k (\tilde{\Qm} \Hm_k)^T$ in which it is easy to verify that the columns of $\tilde{\Qm} \Hm_k$ are orthonormal.
The deflation and SVD of $\bar{\Ym}_1$ and $\bar{\Ym}_2$ cost $\mathcal{O}(npk)$ and $\mathcal{O}(np^2)$ respectively and the eigen-decomposition of $\Deltam$ is $\mathcal{O}((k+2p)^3)$. In order to compute $\Deltam$ one can reuse the computations of $\Qm_k^T\Ym_1$, $\Qm_k^T\Ym_2$, $\bar{\Pm}_1^T\Ym_2$ which are used for deflations and also the matrices used for the SVD decompositions. Thus $\Deltam$ is found in $\mathcal{O}(p^3 + p^2k + pk^2)$, and the overall complexity of this algorithm is $\mathcal{O}((k^2+p^2)(p+k) + np(p + k))$. Of note here is that $n$ scales the complexity in a linear fashion, however costs are cubically related to $k$ and $p$.
Addition of Rows and Columns {#subsec:eigenConcat}
----------------------------
In correspondence with the SVD-updating method given above we consider the case in which one has the eigen-decomposition of $\Cm$ and then wants to find the rank-$k$ approximation of $\Em= [\Am \quad \Bm]^T[\Am \quad \Bm]$. Such a process is useful not just in incremental clustering but also in incrementally solving kernel Principal Components Analysis (KPCA, [@sch98nonlinear]) for example. This update can be written in terms of that described above. This is seen by writing the former update $\Am^T\Am \rightarrow [\Am \quad \Bm]^T[\Am \quad \Bm]$ in terms of the latter:
$$\left[\begin{array}{c c}
\Am^T\Am & \textbf{0} \\
\textbf{0} & \textbf{0} \\
\end{array}\right] \rightarrow
\left[\begin{array}{c c}
\Am^T\Am & \textbf{0} \\
\textbf{0} & \textbf{0} \\
\end{array}\right] +
\left[\begin{array}{c c}
\textbf{0} & \Am^T\Bm \\
\Bm^T\Am & \Bm^T\Bm \\
\end{array}\right].$$
The second term on the right hand side can be written as $\Ym_1\Ym_2^T + \Ym_2\Ym_1^T $ where $\Ym_1 = [\textbf{0} \quad \Imat_p]^T$ and $\Ym_2 = [\Bm^T\Am \quad \frac{1}{2}\Bm^T\Bm]^T$. The eigenvectors of the first matrix on the right-hand side are found from those of $\Am^T\Am$ by simply adding $p$ zero rows to the existing eigenvectors, and the corresponding eigenvalues are identical. Additional eigenvectors are standard unit vectors spanning the $p$ new rows with corresponding eigenvalues as zero. A useful insight is that the deflated matrix $\bar{\Ym}_1 = \Ym_1$ and hence its SVD decomposition can be written directly as $\bar{\Ym}_1 = [\textbf{0} \quad \Imat_p]^T \Imat_p \Imat_p$.
### Alternative Approach {#subsec:alternative_removing}
Here we outline a simpler and more direct approach for the addition of rows and columns to a matrix. First let $\Cm = \Qm \Omegam \Qm^T$ in which $\Qm$ is a matrix of eigenvectors and $\Omegam$ is a diagonal matrix of eigenvalues. Note that $\hat{\Em} = [\Am_k \quad \Bm]^T[\Am_k \quad \Bm]$ can be written as $\tilde{\Qm} \Deltam \tilde{\Qm}^T$ for a square matrix $\Deltam$. In our case we have
$$\begin{aligned}
\tilde{\Qm} = \left[ \begin{array}{c c} \Qm_k & \textbf{0} \\ \textbf{0} & \Imat_p
\end{array} \right] \mbox{ and } \Deltam = \left[\begin{array}{c c} \Omegam_k &
\Qm_k^T \Am_k^T\Bm \\ \Bm^T\Am_k\Qm_k & \Bm^T\Bm \\ \end{array} \right],\end{aligned}$$
noting that $\Am_k\Qm_k\Qm_k^T = \Qm_k\Omegam_k\Qm^T_k\Qm_k\Qm_k^T = \Am_k$ since $\Qm^T_k\Qm_k = \Imat$. Furthermore, note that $\Qm_k^T\Am^T\Bm = \Qm_k^T\Qm\Sigmam\Pm^T\Bm = \Qm_k^T\Qm_k\Sigmam_k\Pm_k^T\Bm = \Qm_k^T\Am_k^T\Bm$ and calculating $\Qm_k^T\Am_k^T\Bm$ is $\mathcal{O}(npk)$. It follows that $\Deltam \in \mathbb{R}^{(k+p) \times (k+p)}$ can be found using $\Qm_k$, $\Am^T\Am$, and $\Am^T\Bm$ and $\Bm^T\Bm$. In the final step, and analogously to the SVD case we take the rank-$k$ eigen-decomposition $\Deltam = \Hm_k\Pim_k\Hm_k^T$ at a cost of $\mathcal{O}((k+p)^3)$ and then the rank-$k$ eigen approximation of $\hat{\Em}$ is given by $(\tilde{\Qm} \Hm_k) \Pim_k (\tilde{\Qm} \Hm_k)^T$.
Notice that this eigen-update almost follows directly from the SVD update in [@zha1999updating] however one need not use the QR decomposition of the deflated $\Bm$. Furthermore, the eigen-approximation above is identical to that at the start of Section \[subsec:eigenConcat\] with $\Ym_1 = [\textbf{0} \quad \Imat_p]^T$ and $\Ym_2 = [\Bm^T\Am_k \quad \frac{1}{2}\Bm^T\Bm]^T$. The difference between the method above and that of Section \[subsec:eigenConcat\] is the former uses $\Am_k$ as opposed to $\Am$ in $\Ym_2$ which results in a greater error.
Removing Rows and Columns
-------------------------
Observe that removing rows and columns is equivalent to zeroing the corresponding rows/columns:
$$\hat{\Cm} = \left[\begin{array}{c c}
\Am^T\Am & \Am^T\Bm \\
\Bm^T\Am & \Bm^T\Bm \\
\end{array}\right]
\rightarrow
\left[\begin{array}{c c}
\Am^T\Am & \Am^T\Bm \\
\Bm^T\Am & \Bm^T\Bm \\
\end{array}\right] -
\left[\begin{array}{c c}
\textbf{0} & \Am^T\Bm \\
\Bm^T\Am & \Bm^T\Bm \\
\end{array}\right],$$
and in this form one can see the connection to Section \[subsec:eigenConcat\]. Again, one can write the second term on the right hand side as $\Ym_1\Ym_2^T + \Ym_2\Ym_1^T $ where $\Ym_1 = [\textbf{0} \quad \Imat_p]^T$ and $\Ym_2 = -[\hat{\Bm}^T\hat{\Am} \quad \frac{1}{2}\hat{\Bm}^T\hat{\Bm}]^T$ where $\hat{\Bm}^T\hat{\Am}$ and $\hat{\Bm}^T\hat{\Bm}$ are found using the rank-$k$ approximation of $\hat{\Cm}$. Since we are updating the rank-$k$ approximation of $\hat{\Cm}$, the final eigen-approximation will have zero elements in the eigenvectors at rows corresponding to those row/columns that are deleted.
Discussion {#subsec:update_interpretation}
----------
A similar eigen-update to that of Section \[subsec:alternative\_removing\] is considered in [@kwok03incrementalEig] and used in conjunction with PCA [@hotelling33pca] and KPCA in [@zhao2006novel]. In PCA, one starts with a set of examples $T = \{\xv_1, \xv_2, \ldots, \xv_n\}$ where $\xv_i \in \mathbb{R}^d$. These examples are centered, and then one finds the $k$ largest eigenvectors $\uv_1, \ldots, \uv_k$ of the covariance matrix $\Cm = 1/n \sum_{i=1}^n \tilde{\xv}_i\tilde{\xv}_i^T$, where $\tilde{\xv}$ is a centered example. KPCA functions similarly, except that one finds the largest eigenvectors of a kernel matrix $\Km \in \mathbb{R}^{n \times n}$ which is computed using the centered examples. In the incremental setting one approximates the eigenvectors of the covariance or kernel matrix on the addition of a new set of examples. In [@zhao2006novel] the authors phrase the problem as a series of SVD updates, and we use a more direct approach in Section \[subsec:alternative\_removing\]. Furthermore, whereas [@zhao2006novel] examines the addition of rows and columns to a positive semi-definite matrix, our work is more general in that we additionally consider removal of rows and columns and addition of a low-rank symmetric matrix. One key novelty of this paper is the update of Section \[subsec:eigenAdd\] which encapsulates all three of these updates and yet does not simply follow from the SVD updating work of [@zha1999updating].
To formalise this notion, the updates outlined above can be written in terms of the addition of a low-rank symmetric matrix $\Um$ to a positive semi-definite matrix $\Cm$. Our approximation method computes $\Cm_k + \Um$ via its expression as $\tilde{\Qm} \Hm_k\Pim_k\Hm_k^T \tilde{\Qm}^T$ where $\tilde{\Qm}$ is a matrix with orthogonal columns spanning $\Cm_k + \Um$ and $\Hm_k$ and $\Pim_k$ represent the largest $k$ eigenvector and eigenvalues of a matrix $\Deltam=\tilde{\Qm}^T \left(\Cm_k + \Um\right) \tilde{\Qm}$. The following lemma shows the consequence of this approach.
Decompose a matrix $\Zm = \Fm\Lambdam\Fm^T$ where $\Fm$ is *any* matrix with orthonormal columns $\Fm^T\Fm = \Imat$. For some $k$ find the best rank-$k$ eigen-approximation $\Lambdam_k = \Hm_k\Pim_k\Hm_k^T$ in which $\Hm_k$ and $\Pim_k$ are the largest $k$ eigenvectors and eigenvalues of $\Lambdam$, and let $\hat{\Zm} = \Fm\Hm_k\Pim_k\Hm_k^T\Fm^T$. Then the best rank-$k$ eigen-decompostion of $\Zm$ is given by: $$\Zm_k = \Um_k\Sm_k\Um_k^T = \hat{\Zm}.$$ \[lem:update\]
Note that $\Lambdam = \Fm^T\Zm\Fm$ due to the orthogonality of $\Fm$. Let $\uv$, $s$ be the eigenvectors and eigenvalues of $\Zm$, and define $\uv = \Fm\vv$ for some $\vv$ then $\Fm^T\Zm\Fm\vv = s\vv$. This implies that the eigenvalues of $\Lambdam$ are the same as those of $\Zm$ and the eigenvectors are related by $\Um = \Fm\Vm$. Hence we have $\Vm = \Hm$ and $\Sm = \Pim$ which implies $\hat{\Zm} = \Fm\Hm_k\Pim_k\Hm_k^T\Fm^T = \Um_k \Sm_k \Um_k^T = \Zm_k$ as required. The only condition on $\Fm$ is that the eigenvectors of $\Zm$ are in the column space of $\Fm$. This must be the case however since $\Zm = \Um\Sm\Um^T = \Fm\Vm\Sm\Vm^T\Fm^T$.
Hence, the update $\Cm_k + \Um$ of Section \[subsec:eigenAdd\] is identical to the best rank-$k$ approximation of $\Cm_k + \Um$.
Eigen-approximation Quality {#sec:quality}
===========================
This section aims to bound the quality of the proposed eigen-approximation approach. As mentioned in Section \[subsec:update\_interpretation\], the proposed approach replaces the expected best rank-$k$ approximation of a matrix $\Am + \Bm$ by the best rank-$k$ approximation of $\Am_k + \Bm$, where $\Am_k$ denotes the best rank-$k$ approximation of $\Am$. Hence, our objective is to control the difference between $(\Am + \Bm)_k$ and $(\Am_k + \Bm)_k$.
The result in Section \[subsec:update\_interpretation\] gives us a first insight into the approximation error of the updates described: the residual matrix is that corresponding to the eigenvectors and eigenvalues after $k$. One can see that for any matrix $\Cm$
$$\| \Cm - \Cm_k \|_F^2 = \|\Cm_{k^\bot} \|_F^2 = \sum_{i=k+1}^n \omega_i^2,$$ where $\omega_i$ is the $i$th eigenvalue of $\Cm$ (unless otherwise stated eigenvalues are always given in descending order) and $\Cm_{k^\bot}$ is the approximation of $\Cm$ using eigenvectors/eigenvalues after $k$. This implies that $\Cm$ is well approximated by the largest $k$ eigenvectors if the sum of the square of the remaining eigenvalues is small.
This certainly gives us insight into when our eigen-updating approach will be accurate, however the kind of matrices we will work with do not have this property in general. We now turn to matrix perturbation theory [@stewart1990matrix] in order to learn more about the approximated eigenvectors of the updated matrix. In a nutshell, it lies in controlling the *angle* between two *invariant subspaces*. Before giving the corresponding theorem, the following subsection introduces the necessary notions of invariant subspaces of a matrix and of the canonical angles between subspaces.
Invariant Subspaces and Canonical Angles
----------------------------------------
We begin by introducing the simple concept of an invariant subspace.
The subspace $\mathcal{X}$ is an *invariant subspace* of $\Am$ if $\Am \mathcal{X} \subset \mathcal{X}$.
It can also be shown that if the columns of $\Xm$ form a basis for $\mathcal{X}$ of $\Am$ then there is a unique matrix $\Lm$ such that $\Am\Xm = \Xm\Lm$. The matrix $\Lm$ is a representation of $\Am$ with respect to the basis $\Xm$, and it has identical eigenvalues to $\Am$. A useful decomposition in perturbation theory is to reduce $\Am$ to a block diagonal form. Let $\Xm_1$ be an orthogonal matrix which spans the invariant subspace of $\Am$, $\mathcal{X}_1$, and assume that we have a matrix $\Ym_2$ such that $[\Xm_1 \Ym_2]$ is unitary and $\Ym_2$ spans the space orthogonal to $\mathcal{X}_1$, then this allows us to write
$$[\Xm_1 \Ym_2]^T \Am [\Xm_1 \Ym_2] =\left[ \begin{array}{c c} \Lm_1 & \Hm \\ \mathbf{0} & \Lm_2 \end{array} \right],
\label{eqn:blockInvSub}$$
in which $\Lm_1 = \Xm_1^T\Am\Xm_1$, $\Lm_2 = \Ym_2^T\Am\Ym_2$ and $\Hm = \Xm_1^T\Am\Ym_2$. The above equation is known as the *reduced form* of $\Am$ with respect to $[\Xm_1 \Ym_2]$. The proof of why the bottom left block of this matrix is zero is straightforward, see [@stewart1990matrix] for details. This gives us the knowledge to define a *simple invariant subspace*.
Let $\mathcal{X}$ be an invariant subspace of $\Am$ and consider the reduced form of Equation , then $\mathcal{X}$ is a *simple invariant subspace* of $\Am$ if there are no common eigenvalues between $\Lm_1$ and $\Lm_2$.
Notice that a simple invariant subspace has a complementary space, defined as follows.
Let the simple invariant subspace $\mathcal{X}_1$ have the reduced form of Equation with respect to the orthogonal matrix $[\Xm_1 \Ym_2]$. Then there exist $\Xm_2$ and $\Ym_1$ such that $[\Xm_1 \Xm_2]^{-1} = [\Ym_1 \Ym_2]^T$ and $$\Am = \Xm_1\Lm_1\Ym_1^T + \Xm_2\Lm_2\Ym_2^T,$$ where $\Lm_i = \Ym_i^T\Am\Xm_i$, $i=1,2$. This form of $\Am$ is known as the *spectral resolution* of $\Am$ along $\mathcal{X}_1$ and $\mathcal{X}_2$.
This allows us to introduce a theorem essential to our main result. However, first we must define the notion of angle between two subspaces.
Let $\Xm_1, \Ym_1 \in \mathbb{R}^{n \times \ell}$ with $\Xm_1^T\Xm_1 = \Imat$ and $\Ym_1^T\Ym_1 = \Imat$. If $2\ell \leq n$, there are unitary matrices $\Qm, \Um_{11}, \Vm_{11}$ such that $$\Qm\Xm_1\Um_{11} = \left( \begin{array}{c} \Imat_\ell \\ \textbf{0} \\ \textbf{0} \end{array} \right)
\text{ and }
\Qm\Ym_1\Vm_{11} = \left( \begin{array}{c} \Gammam \\ \Sigmam \\ \textbf{0} \end{array} \right),$$ in which $\Gammam = \diag(\gamma_1, \ldots, \gamma_\ell)$ with $0 \leq \gamma_1 \leq \ldots \leq \gamma_\ell$, $\Sigmam = \diag(\sigma_1, \ldots, \sigma_\ell)$ with $\sigma_1 \geq \cdots \geq \sigma_\ell \geq 0$, and $\gamma_i^2 + \sigma_i^2 = 1$, $i=1,\ldots,\ell$. If $2\ell > n$ then $\Qm, \Um_{11}, \Vm_{11}$ can be chosen so that $$\Qm\Xm_1\Um_{11} = \left( \begin{array}{c c} \Imat_{n-\ell} & \textbf{0} \\ \textbf{0} & \Imat_{2\ell - n} \\ \textbf{0} & \textbf{0} \end{array} \right)
\text{ and }
\Qm\Ym_1\Vm_{11} = \left( \begin{array}{c c} \Gammam & \textbf{0} \\ \textbf{0} & \Imat_{2\ell - n} \\ \Sigmam & \textbf{0} \end{array} \right),$$ in which $\Gammam = \diag(\gamma_1, \ldots, \gamma_{n-\ell})$ with $0 \leq \gamma_1 \leq \ldots \leq \gamma_{n-\ell}$, $\Sigmam = \diag(\sigma_1, \ldots, \sigma_{n-\ell})$ with $\sigma_1 \geq \cdots \geq \sigma_{n-\ell} \geq 0$, and $\gamma_i^2 + \sigma_i^2 = 1$, $i=1,\ldots,{n-\ell}$. \[thm:ssBases\]
Geometrically, let $\mathcal{X}_1$ and $\mathcal{Y}_1$ be subspaces of dimension $\ell$ and $\mathcal{Q}$ be a unitary transformation. Then the matrices $\Qm\Xm_1\Um_{11}$ and $\Qm\Ym_1\Vm_{11}$, with $\Xm_1 \in \mathcal{X}_1$, $\Xm_2 \in \mathcal{X}_2$, $\Qm \in \mathcal{Q}$ , form bases of $\mathcal{QX}_1$ and $\mathcal{QY}_1$ and $\sigma_i$ and $\gamma_i$ can be regarded as sines and cosines of the angles between the bases. We now define a measure of similarity between subspaces.
Let $\mathcal{X}$ and $\mathcal{Y}$ be subspaces of the same dimension, then the *canonical angles* between the subspaces are the diagonal entries of the matrix $\sin^{-1} \Sigmam=\diag(\sin^{-1}(\sigma_1), \ldots, \sin^{-1}(\sigma_{n-\ell}))$ where $\Sigmam$ is the matrix defined in Theorem \[thm:ssBases\].
It follows that $\Sigmam$ is a measure of how two subspaces differ.
Angle Between Exact and Updated rank-$k$ Approximations
-------------------------------------------------------
We are now ready to state the main theorem required for our result.
\[thm:canAngle\] Let $\Am$ be a Hermitian matrix with spectral resolution given by $[\Xm_1 \Xm_2]^T \Am [\Xm_1 \Xm_2] = \diag(\Lm_1, \Lm_2)$ where $[\Xm_1 \Xm_2]$ is unitary. Let $\Zm \in \mathbb{R}^{n \times k}$ have orthonormal columns, $\Mm$ be any Hermitian matrix of order $k$ and define the residual matrix as $\Rm = \Am\Zm - \Zm\Mm$. Let $\lambda(\Am)$ represent the set of eigenvalues of $\Am$ and suppose that $\lambda(\Mm) \subset [\alpha, \beta]$ and for some $\delta > 0$, $\lambda(\Lm_2) \subset \mathbb{R} \setminus [\alpha - \delta, \beta + \delta]$. Then, for any unitary invariant norm, we have: $$\|\sin \Theta(\mathcal{R}(\Xm_1), \mathcal{R}(\Zm)) \| \leq \frac{\|\Rm\|}{\delta},$$ where $\mathcal{R}(\cdot)$ is the column space of a matrix.
Before we introduce the main result we present a result by Weyl [@weyl1912asymptotische] which characterises the perturbation in the eigenvalues of a matrix.
\[thm:weyl\] Define $\Am \in \mathbb{R}^{n \times n}$ and let $\tilde{\Am} = \Am + \Em$ be its perturbation. The eigenvalues of $\Am$ and $\Em$ are given by $\lambda_i$ and $\epsilon_i$, $i=1,\ldots, n$, respectively. Then the eigenvalues of $\tilde{\Am}$ are, for $i=1, \ldots, n$, $\tilde{\lambda}_i \in [\lambda_i + \epsilon_n, \lambda_i + \epsilon_1]$.
We can now present our main result which is closely related to the Davis-Kahan theorem [@davis1970rotation].
Consider a positive semi-definite matrix $\Am \in \mathbb{R}^{n \times n}$ with eigenvalues $\omega_1, \ldots, \omega_n$ and corresponding eigenvectors $\Qm = [\qv_1, \ldots, \qv_n]$. Let $\Bm \in \mathbb{R}^{n \times n}$ be symmetric with eigenvalues $\epsilon_i$ and $(\Am + \Bm)$ be positive semi-definite with eigen-decomposition $\Um \Gammam \Um^T$ where $\gamma_i$ are eigenvalues, $i=1, \ldots, n$. Fix integer $k$, let $\Am_k + \Bm$ have decomposition $\Vm\Pim\Vm^T$ with eigenvalues $\pi_1, \ldots, \pi_n$ and assume $\gamma_k \neq \gamma_{k+1}$. Then the following bounds hold on the canonical angles between the subspaces defined by $\Um_k$ and $\Vm_k$, assuming $\pi_k > \hat{\gamma}_{k+1}$,
$$\|\sin \Theta(\mathcal{R}(\Um_k), \mathcal{R}(\Vm_k)) \|_F \leq \frac{\sqrt{\tr(\Vm_k^T\Am^2_{k^\bot}\Vm_k)}}{\pi_k - \hat{\gamma}_{k+1}},$$
$$\|\sin \Theta(\mathcal{R}(\Um_k), \mathcal{R}(\Vm_k)) \|_2 \leq \frac{\sqrt{\lambda_{max}(\Vm_k^T\Am^2_{k^\bot}\Vm_k)}}{\pi_k - \hat{\gamma}_{k+1}},$$
where $ \hat{\gamma}_{k+1} = \omega_{k+1} + \pi_{k+1}$. \[thm:pertTheorem\]
We will start by considering the first bound. It is clear that $\Um_k$ is a simple invariant subspace for $(\Am + \Bm)$. The spectral resolution of $\Am + \Bm$ is given by $[\Um_k \Um_{k^\bot}]$ since this matrix is unitary and we have $\Am+\Bm = \Um_k\Um_k^T(\Am +\Bm)\Um_k\Um_k^T + \Um_{k^\bot}\Um_{k^\bot}^T(\Am+\Bm)\Um_{k^\bot}\Um_{k^\bot}^T$. In the reduced form $\Lm_1 = \Gammam_k$ and $\Lm_2 = \Gammam_{k^\bot}$. Furthermore, we set $\Mm = \Pim_k$ and $\Zm = \Vm_k$. The residual matrix is given by $\Rm = (\Am+\Bm)\Vm_k - \Vm_k \Pim_k = \Am_{k^\bot}\Vm_k$ and
$$\begin{aligned}
\|\Rm\|_F &=& \sqrt{\tr(\Vm_k^T\Am^2_{k^\bot}\Vm_k)}. \end{aligned}$$
We know that the eigenvalues of $\Mm$ fall within the range $[\pi_k, \pi_1]$ and those of $\Lm_2$ are bounded using Theorem \[thm:weyl\] in the range $\gamma_i \in [\pi_i + \omega_n, \pi_i+\omega_{k+1}]$ for $i=k+1, \ldots, n$. Considering also the perturbation of eigenvalues of $\Am$ we can write $\gamma_i \leq \hat{\gamma}_i = \min(\omega_{k+1} + \pi_{k+1}, \omega_{k+1} + \epsilon_1) = \omega_{k+1} + \pi_{k+1}$ given $\pi_{k+1} \leq \omega_{k+1} + \epsilon_1$. It follows that $\delta = \pi_k - \hat{\gamma}_{k+1}$ and plugging into Theorem \[thm:canAngle\] gives the required result.
The second bound is proved similarly except in this case we have $$\begin{aligned}
\|\Rm\|_2 &=& \sqrt{\lambda_{max}(\Vm_k^T\Am^2_{k^\bot}\Vm_k)}.\end{aligned}$$
Thus we have a bound on the angle between the subspace of the first $k$ eigenvectors of $\Am_k+ \Bm$ and the corresponding eigenvectors of it perturbation $\Am+\Bm$ without explicitly requiring the eigen-decomposition of $\Am+\Bm$. Provided the eigenvalues of $\Vm_k^T\Am^2_{k^\bot}\Vm_k$ are small and the eigengap $\pi_k - \hat{\gamma}_{k+1}$ is large one can be sure that the two subspaces have small canonical angles. One can see that under small perturbations $\Vm_k$ is close to $\Qm_k$ and hence $\Vm_k^T\Qm_{k^\bot}$ is small, resulting in tight bounds in the angles. In the case that the matrices involved correspond to normalised Laplacians this result corresponds well with similar results outlining a perturbation-based motivation of spectral cluster (see e.g. [@von2007tutorial]) which state that if the value of $\gamma_k - \gamma_{k+1}$ is large then one might reasonably expect a good clustering. The bound becomes loose when this eigengap is small, however in this case the clusters are less distinct even when computing the exact eigenvectors.
Incremental Cluster Membership {#sec:incrementalCluster}
==============================
We now return to the eigenproblem of Algorithm \[alg:normCluster\], $\tilde{\Lm}\vv = \lambda \vv$, in which we are interested in the eigenvectors with the smallest eigenvalues. Define the *shifted Laplacian* as
$$\hat{\Lm} = 2\Imat - \tilde{\Lm} = \Imat + \Dm^{-\frac{1}{2}}\Wm\Dm^{-\frac{1}{2}},$$ which is positive semi-definite since $\tilde{\Lm}$ is positive semi-definite with largest eigenvalue $2$. Note that by negating a matrix one negates the eigenvalues, leaving the eigenvectors the same, and similarly an addition of $\sigma\Imat$ increases the eigenvalues by $\sigma$ leaving the eigenvectors intact. Since we are interested in the smallest eigenvectors of $\tilde{\Lm}$ they correspond exactly to the maximum eigenvectors of $\hat{\Lm}$ and we can use the eigen-update methods described above. Observe that the shifted Laplacian is a normalised version of the *signless Laplacian* [@haemers2004enumeration] defined as $\Lm^{+} = \Dm + \Wm$, and this can be seen from $\hat{\Lm} = \Dm^{-\frac{1}{2}}(\Dm + \Wm)\Dm^{-\frac{1}{2}} = \Dm^{-\frac{1}{2}}\Lm^{+}\Dm^{-\frac{1}{2}}$.
Putting the ingredients together allows us to outline an efficient incremental method for performing graph clustering called Incremental Approximate Spectral Clustering (IASC), see Algorithm \[alg:incNormCluster\]. At a high level the algorithm is quite simple: in the initialisation steps one computes the shifted Laplacian matrix for the first graph $\hat{\Lm}_1$ and then performs $k$-means clustering using the largest $k$ eigenvectors of this matrix. For the $t$-th successive graph, $t > 1$, we use the eigen-update methods above to approximate the largest eigenvectors of $\hat{\Lm}_t$ using the approximate eigenvectors computed at the previous iteration in step \[step:eigenConcat\]. For these updates, it is simple to recover the matrices $\Ym_1$ and $\Ym_2$ given a change in edge weights. Note that we use the first method of Section \[subsec:eigenConcat\] to compute eigenvector upon the addition of row and columns. Furthermore, we store the first $\ell$ eigenvectors of $\hat{\Lm}_1$, with $\ell
\geq k$, and recompute eigenvectors every $T$ iterations in order to reduce cumulative errors introduced in the loop at the expense of increased computation. When $\ell = n$ one recovers the exact eigenvectors at each iteration and Algorithms \[alg:incNormCluster\] and \[alg:normCluster\] become nearly equivalent. In the case that there is a significant eigengap between eigenvalues, one can fix both $\ell$ and $k$ according to Theorem \[thm:pertTheorem\].
Graphs $G_1, \ldots, G_T$ of sizes $n_1, \ldots, n_T$, no. clusters $k$, approximation rank $\ell \geq k$, eigen-decomposition recomputation step $R$ Compute the shifted Laplacian for $G_1$, $\hat{\Lm}_1$ Find $\ell$ largest eigenvectors of $\hat{\Lm}_1$, $\Vm_{\ell}^{(1)}$, let $\Vm_{k}^{(1)}$ be the first $k$ cols of $\Vm_{\ell}^{(1)}$ Normalise the eigenvector rows, $\Vm_{k}^{(1)} \leftarrow \diag(\Vm_{k}^{(1)}(\Vm_{k}^{(1)})^T)^{-\frac{1}{2}}\Vm_{k}^{(1)}$ Use $k$-means on rows of $\Vm_{k}^{(1)}$ and store indicators $\cv_1 \in \{1,\ldots,k\}^{n_1}$ Compute shifted Laplacian for $G_t$, $\hat{\Lm}_t$ \[step:computeLaplacian\] Recompute eigenvectors of $\hat{\Lm}_t$ Use rank-$\ell$ eigen-approximation of Section \[sec:our\_work\] \[step:eigenConcat\] Normalise rows of $\Vm_{k}^{(t)}$, $\Vm_{k}^{(t)} \leftarrow \diag(\Vm_{k}^{(t)}(\Vm_{k}^{(t)})^T)^{-\frac{1}{2}}\Vm_{k}^{(t)}$ Use $k$-means on $\Vm_{k}^{(t)}$ with initial centroids $\cv_{t-1}$, store $\cv_t \in \{1,\ldots,k\}^{n_t}$ Cluster membership $\cv_1 \in \{1,\ldots,k\}^{n_1}, \ldots, \cv_T \in \{1,\ldots,k\}^{n_T}$
\[alg:incNormCluster\]
The complexity of Algorithm \[alg:incNormCluster\] is dictated by the sparsity of the graphs and the extent of the change between successive graphs. We ignore the steps before the for-loop since in general they do not impact the overall complexity. At iteration $t$ and step \[step:computeLaplacian\] one can compute $\hat{\Lm}_t$ from the weight and degree matrix at a cost of $\mathcal{O}(n_t + |E_t|)$. In the following step if there is a change between edges incident to vertices $S = \{v_{I_1}, \ldots ,v_{I_\ell}\}$ then the rows and columns corresponding to the union of the neighbours of $S$, $n(S)$, will change in the corresponding shifted Laplacian. In this case $p = |n(S)|$ in $\Ym_1, \Ym_2 \in \mathbb{R}^{n \times p}$ and if the neighbourhood of the vertices with changed edges $n(S)$ is small, this update can be efficiently computed as outlined in Section \[sec:our\_work\]. The cost of $k$-means is $\mathcal{O}(k^2ns)$ where $s$ is the number of iterations required for convergence.
Computational Results {#sec:exp}
=====================
In this section we study the clustering quality of five incremental strategies: the naive one which computes the exact eigenvectors at each iteration, denoted `Exact`, Ning et al.’s method (`Ning`) since it is a competing incremental strategy, `IASC`, `Nyst`, which uses the Nyström eigen-decomposition approximation of the shifted Laplacian, since it is often used in spectral clustering, and likewise the Randomised SVD method of Algorithm \[alg:randomSVD\] denoted `RSVD`. First, however we observe the quality of the eigenvectors found using our eigen-updating approach.
Quality of Approximate Eigenvectors {#sec:eigen_quality}
-----------------------------------
We compare `Nyström` and `RSVD` approximations with our eigen-updating methods on a synthetic dataset generated in the following way: The initial graph contains 4 clusters of size 250 which are generated using an Erdös-Rényi [@erdos1959random] process with edge probability $p=0.1$. The Erdös-Rényi process creates edges independently randomly with a fixed probability $p$ for all pairs of vertices. For each successive graph we then add 50 random edges to simulate “noise” in the clusters for a total of 100 graphs. We then compute the largest $k=4$ eigenvectors of the shifted Laplacian using the full eigen-decomposition, the `Nyström` method, `RSVD` and the eigen-updating approach of Section \[sec:our\_work\]. For the `Nyström` method we use $m = 900$ randomly selected columns, for `RSVD` we use $r=\{100,~900\}$ random projections, and for the eigen-updating approach we update based on $\ell=\{4,~300\}$ approximate eigenvectors and eigenvalues found for the previous graph. The quality of the approximations is measured by $\|\sin \Theta(\mathcal{R}(\Um_k), \mathcal{R}(\Vm_k))\|_F$ where $\Um_k$ stands for the exact eigenvectors and $\Vm_k$ corresponds to the eigenvectors returned by one of the three considered eigen-approximation strategies. Applying Corollary I.5.4 of [@stewart1990matrix], this norm is equal to $\|\Um_{k^\bot}\Vm_k\|_F$. The experiment is repeated with results averaged over 20 iterations.
Observe that the quality of the approximation decreases with the noise amplitude, see Figure \[fig:nystromEigPlot\]. For the three approximation strategies, the more eigenvectors/columns/random projection they consider, the smaller the canonical angles with the exact decomposition as one might expect. `Nyström` leads to a poor approximation whilst using 90% of the columns. `RSVD` has poor results with 100 random projections, but is close to the exact decomposition with 900 random projections. The eigen-update strategy always does a better job than keeping the initial decomposition, and in this case there does not appear to be a large gain when moving from 300 eigenvectors to just 4. In the case in which we use the complete set of eigenvectors we would expect IASC to coincide with exact eigen-decomposition.
### Perturbation Bound
We now turn to demonstrating the effectiveness of the bound of Theorem \[thm:pertTheorem\] on a synthetic dataset containing 150 vertices. The initial graph contains 3 clusters of size 50 which are generated using an Erdös-Rényi process with edge probability $p=0.3$, resulting in 1092 edges. For each successive graph we add 10 random edges and in total there is a sequence of 80 graphs. We then compute the bound of Theorem \[thm:pertTheorem\] to measure the difference between the canonical angles of the real and approximated eigenvectors of the shifted Laplacian. We compare the results to the bound using the real eigenvalues of the Laplacian, i.e. $\delta = \pi_k - \gamma_{k+1}$. This process is repeated 50 times with different random seeds and the results are averaged.
Figure \[fig:pertBoundPlot\] shows the resulting bounds for this sequence of graphs. The bound diverges slowly from the precise bound for many of the initial graphs, for example $\|\sin \Theta(\mathcal{R}(\Um_k), \mathcal{R}(\Vm_k))\|_F$ is bounded by $0.616\pm0.106$ versus $0.352\pm0.024$ at the 50th graph and the equivalent results for the 2-norm are $0.434\pm0.007$ and $0.248\pm0.02$. Soon after this point, we see a large divergence in the precise and approximate bounds, although the approximate bounds become trivial after the 78th graph for the Frobenius norm and the 73rd for the 2-norm corresponding to the addition of 770 and 730 edges respectively to the initial graph. When we look at the precise bound, one can see that at the last graph $\|\sin \Theta(\mathcal{R}(\Um_k), \mathcal{R}(\Vm_k))\|_2 \leq 0.39$ despite nearly doubling the number of edges which points to the precision of the eigen-approximation in this case.
Clustering on Synthetic Data
----------------------------
To evaluate the clustering approaches, two separate synthetic datasets are considered, both of which are generated by an Erdös-Rényi process. The first dataset, called `3clust`, is based on a graph of 3 clusters of $60$ vertices each. This dataset allows us to compare the clustering quality as the clusters become more/less distinct and also the addition and removal of edges. In the corresponding graph, any possible edge between two points from the same cluster occurs with probability $p_c = 0.3$ and the probability of edges between vertices in different clusters is selected from $p_g \in \{0.1, 0.2\}$. To generate a sequence of graphs we first allow only 20 vertices per cluster and then add 5 vertices to each cluster at a time until each one is of size 60. We then reverse the process so that the 10th graph is the same as the 8th, and the 11th is the same as the 7th etc., which allows us to test the clustering methods upon the removal of vertices.
The second dataset, `discrepancy`, aims to examine the clustering methods on a more complex set of hierarchical clusters. Here $180$ vertices are split into 3 clusters of equal size, and inside each cluster there are 3 sub-clusters. The initial graph is empty, and at each iteration $t\in\{1,\;\ldots,\;T\}$, $T=23$, an edge $\{v_i,v_j\}$ is added with a probability $p_\varepsilon+(p-p_\varepsilon)\ell t/T$, where $\ell$ is set to $1$ if $v_i$ and $v_j$ are in the same subgroup, $0.5$ if $v_i$ and $v_j$ are in the same group, and $0$ otherwise. In our case we set $p_\varepsilon = 0.0005$ and $p = 0.01$.
For both datasets and the clustering approaches, $k$-means is run with $k$ corresponding to the number of clusters (3 and 9 for `3clust` and `discrepancy` respectively). For `IASC` we fix the number of eigenvectors $\ell \in \{3, 6, 12, 24\}$ with the `3clust` dataset and $\ell \in \{9, 72\}$ with `discrepancy` in order to test the approximation quality as this parameter varies. With `Nyström` we sample $m=90$ columns of the Laplacian matrix to find the approximate eigenvectors on `3clust` and select $m \in \{9, 72\}$ for `discrepancy`. `RSVD` uses $r=24$ random projections on `3clust` and $r \in \{9, 72\}$ for `discrepancy`. On `discrepancy`, the approximation methods start with the 3rd graph in the sequence to allow the initial graph to contains enough edges. The experiments are repeated 50 times with different random graphs constructed using the methods described above, and the results are averaged. Clustering accuracy is measured through the *Rand Index* [@Rand71] between the finest true clustering $\cal C$ and the learned one $\widehat{\cal C}$. Rand Index corresponds to the proportion of true answers to the question “*Are vertices $v_i$ and $v_j$ in the same cluster ?*”. More formally, it is given by $$\operatorname{RandIndex}({\cal C},\widehat{\cal C}) =
\frac{\left|\left\{
v_i,v_j\in V :\; v_i \neq v_j,\;
\delta({\cal C}(v_i), {\cal C}(v_j))\neq \delta(\widehat{\cal C}(v_i),
\widehat{\cal C}(v_j)) \right\}\right|}{\left|\left\{v_i,v_j\in V :\; v_i \neq v_j\right\}\right|}
,$$ where ${\cal C}(v_i)$ stands for the cluster index of vertex $v_i$ after clustering ${\cal C}$, $\delta$ is the Kronecker delta function and $| \mathcal{E}|$ denotes the cardinality of any finite set $\mathcal{E}$. The evaluation is completed by the computation of $\|\sin \Theta(\mathcal{R}(\Um_k), \mathcal{R}(\Vm_k))\|_F$, see Section \[sec:eigen\_quality\] for more details. The canonical angles for `Ning` are not given as `Ning` uses the random-walk Laplacian which (i) differs from the normalised Laplacian used with other approaches and (ii) is not symmetric, leading to non-orthogonal eigenvectors.
\
The errors on `3clust` are shown in Figure \[fig:syntheticErrors\]. The first point to note is that the errors decrease as the cluster size increases up until graph 8 and then increase again as vertices are removed. This is explained by the fact that as the cluster size increases there are more edges within each cluster relative to those between clusters, hence they becomes easier to identify. Notice also that the approximation methods perform worse (in terms of Rand Index) than `exact` except for `Ning` and `IASC` $\ell=3$ when vertices are removed. This exception is due to a coincidence: for any iteration $i \in 8,...,16$, clustering results are better when using the eigen-decomposition at time 8 as they are based on more data. On the other hand, `Ning` and `IASC`, $\ell=3$, do not accurately update the eigen-decomposition and hence the eigenvectors are close to those at graph 8. As a consequence, both approaches lead to better results than `exact`. Apart from this exception, `IASC`, $\ell = 24$, and `RSVD` have results close to `exact`, while `Nyström` leads to a bad Rand Index score for $p=0.2$. `Ning` has non-smooth behaviour for any value of $p$. Lastly, the results of `Nyström` when $p=0.1$ demonstrate the limit of canonical angles to measure the clustering quality of an approximation approach: having large canonical angles values does not necessarily imply a Rand Index score far from `exact`.
It is worth noting that since $\ell$ is a fixed value for `IASC`, the approximation of the largest eigenvectors of the shifted Laplacian represents a smaller fraction of the total sum of eigenvalues as the cluster size increases. When using 24 eigenvectors for the approximation on the 2nd largest graph one requires approximately $14.5\%$ of the dimensionality of the Laplacian, yet Rand Index is close to that of `exact`.
The results on `discrepancy` (Figure \[fig:discrepancyErrors\]) show that `Ning` generally does not generate accurate clustering until the 12th graph when eigenvectors are recomputed, however it becomes competitive after this point. This is partly due to the fact that one solves a different eigen-system to the other methods which is not as effective at clustering for the initial graphs. As we increase the number of columns used for the Nyström approach or the random projections used by `RSVD`, the Rand Index values improve for the first few graphs in which the clusters are generally not well defined. In contrast, `IASC` results are relatively accurate with 72 eigenvectors. Surprisingly at the final graph we see that with just 9 eigenvectors `IASC` results in the most accurate clustering. At this point more eigenvectors seem to make the solution worse, with the exact eigen-decomposition being less accurate than `IASC` with 72 eigenvectors. One explanation is that an accurate eigen-decomposition fits “noise” in the Laplacian which is excluded with `IASC` and a low value of $\ell$.
Real-world Graphs
-----------------
We now apply the clustering methodology analysed previously on several real datasets.
### Setup
Here we use the clustering methods on three real datasets. The first one represents individuals in Cuba who are detected as HIV positive between the period 1986 and 2004, see [@Auvert07] for details of the related database. Edges in the graph indicate the occurrence of a sexual contact between two individuals as determined using *contact tracing*, whereby contacts of an infected person are identified and tested. The full sexual contact graph at the end of 2004 consists of 5389 people however it is strongly disconnected and we consider the growth of the largest component of size 2387. We find graphs of detected individuals at 1 month intervals, starting when the graph contains at least 500 vertices. At any particular time point, we consider the component containing first person detected in the largest component at the end of the recorded epidemic.
The next dataset is the high energy physics theory citation network from the Arxiv publications database [@gehrke2003overview] for the period February 1992 to March 2002. The full dataset contains 27,770 papers with 352,807 edges and a graph is generated as follows: if a paper cites another, an edge is made from the former to the latter. Not all of the papers present in the dataset have publication dates and hence we use only those with dates and label cited papers with the date of the oldest citing paper. Taking the largest component, the final resulting graph consists of 15,112 vertices and 193,826 edges. To track the evolution of the connected component we start with the oldest paper, and consider the graphs of connected papers at 1 month intervals, starting with a graph of at least 500 vertices.
The final dataset, called `Bemol`, was first introduced in [@richard10] and corresponds to the purchase history of an e-commerce website over a period of almost two years. The initial dataset is a bipartite graph between users and products composed of more than 700,000 users and 1,200,000 products. In the current experiment we focus on the first 10,000 users, and a graph is constructed between users with edge weights corresponding to the number of commonly purchased products between two users. Taking a maximum of 500 purchases per iteration in the graph sequence we focus on graphs 500 to 600.
To test the clustering methods we run each on the sequences of evolving graphs under a variety of parameters. As the selection of the number of clusters is a complex issue and outside the scope of this paper and we manually choose this value for each dataset. The experiment is run using $k = 25$ clusters for `HIV`, $k=50$ for `Citation` and $k=100$ for `Bemol`. For `HIV` and `IASC`, $\ell \in \{25, 50, 100\}$ and $R = 10$ and for `Ning` we recompute exact eigenvectors after every $10$ iterations. When using `Citation` and `Bemol`, eigenvector recomputations are performed every 20 iterations and $\ell \in \{100, 200, 500\}$. With `Nyström` the number of columns is chosen from $m \in \{1000, 1500\}$ for `HIV` and $m \in \{2000, 5000\}$ for the other datasets. Finally, we apply the randomised SVD method with $q=2$ and $r \in \{1000, 1500\}$ for `HIV` and $r \in \{2000, 5000\}$ for the remaining datasets. Note that in our implementation of $k$-means clustering, $k$ represents an upper bound on the number of clusters found. To evaluate the learned clusters we use measures of *modularity* and *$k$-way normalised cut*. Let $\dv_i = \sum_j \Wm_{ij}$ and $r = \sum_i \dv_i$, then the modularity is defined as $$Q = \frac{1}{2r} \sum_{i,j} \left(\Wm_{ij} - \frac{\dv_i \dv_j}{2r}\right) \delta(\cv_i, \cv_j),$$ where $\cv \in \mathbb{R}^{n}$ is the cluster indicator vector and $\delta$ is the Kronecker delta function. Intuitively modularity is the difference in the sum of edges weights within a cluster and the expected edge weights assuming the same weight distribution $\dv$ for each vertex. The $k$-way normalised cut is $$N = \frac{1}{k} \sum_{\ell=1}^k \frac{\sum_{ij} \Wm_{ij} \delta(\cv_i, \cv_\ell) (1 - \delta(\cv_j, \cv_\ell))}{\sum_{ij} \Wm_{ij}\delta(\cv_i, \cv_\ell)}.$$ A cut between two clusters $A$ and $B$ is the sum of the weights between the clusters and the normalised cut is this sum divided by the sum of the weights of all edges incident to vertices in cluster $A$. Hence the $k$-way normalised cut is the mean normalised cut between each cluster and its complementary vertices. To summarise, the greater the modularity, the better, and the lower the $k$-way normalised cut the better.
All experimental code is written in Python and we use an Intel Core i7-2600K at 3.40GHz with 16GB of RAM to conduct the simulations. The Laplacian matrices are stored in compressed sparse row representation and eigenvectors are found using Implicitly Restarted Lanczos Method in ARPACK [@lehoucq1998arpack] which computes only the required eigenvectors and not the full eigen-decomposition.
### Results
Figure \[fig:RealDataResults\] show the resulting modularities and $k$-way normalised cuts for all datasets however we begin by studying $\texttt{HIV}$. For both `IASC` and `Ning`, since eigenvectors are recomputed every 10 iterations, this can manifest itself as sudden changes in the modularities and $k$-way normalised cuts. These changes are more pronounced with `Ning`. We see a close correspondence of `IASC` and `exact` for both the modularity and normalised cut. As we observed with the toy datasets, a lower value of $\ell$ seems to improve results with the final graph having a cut of $0.09$ with `IASC` $\ell=100$ versus $0.10$ for `exact`. Notice that `IASC` matches or improves results on `exact`, while keeping only 5% or fewer of the final number of eigenvectors. `Ning` fares badly in terms of the modularity of the resulting clustering with a value of 0.73 versus 0.82 for the `exact` approach at the final graph. However, with the cut measure `Ning` provides the best clustering, albeit with a more unstable curve than the other methods. `Nyström` does not provide a good approximation of the largest eigenvectors when the rank of the Laplacian exceeds the number of columns sampled. In contrast, `RSVD` is broadly competitive with `IASC` when using 1500 random projections. One of the reasons for the effectiveness of `RSVD` is Step \[step:power\] of Algorithm \[alg:randomSVD\] which helps to ensure the column space of $\Ym$ is close to that generated using the largest eigenvectors of $\Am$, see [@halko2011finding] for further details.
A similar picture emerges with `Citation` and we see again that `IASC` is close to `exact` in terms of both measures. Note that it was too costly to compute clustering using `Ning` on this dataset and `Bemol`. We ran `Ning` on `Citation` for 337,920s before terminating the experiment: a computational time of at least 7.5 times more than the next most costly approach of `RSVD`, $r = 5000$, which took 44,880s. On `Citation`, when $\ell\in\{200, 500\}$ we obtain a close match to `exact` in general. The results are impressive when we consider the change in the graphs between eigenvector recomputations: the first graph is of size 555, and the 19th is 2855, an increase of 2300. With the 60th graph there are 10063 vertices and 12135 at the 79th. Looking at the `Nyström` curves, we again observe poor clustering performance even when $m=5000$. Furthermore, `RSVD` $r = 5000$ can compete well with the `exact` method particularly when considering the cut measure although the modularity using `RSVD` suffers after approximately the 60th graph.
Finally consider the `Bemol` graphs in which it is difficult to find clear clusters, although they become more distinct over time. This is evident when looking at the `exact` curves for example: the modularity increases slightly from 0.27 to 0.32 whereas the $k$-way normalised falls from 0.74 to 0.65 from beginning to end. In contrast to the other datasets `exact` is improved upon by both `IASC` and `Nyström` (when $m = 5000$) respectively. One of the reasons that the Nyström and `RSVD` methods are effective on this data is because there are many edges and one can sample them out without affecting the clustering significantly. Note however that as we have seen in the other plots, `Nyström` is rather unstable compared to the other methods. Furthermore, the eigenvector updates every 20 iterations are noticeable in the cluster measures with `IASC`.
To conclude the analysis, Figure \[fig:RealDataTimes\] shows the timings of the eigenvector computations of the clustering methods for the datasets. With `HIV` we can make a comparison with `Ning` and one can see that a cost of the method is a computation time which exceeds that of `exact`. `IASC` has a cumulative computation time of 62.0s, `exact` took 256.4s compared to 382.1s for `Ning`. With the `Bemol` and `Citation` datasets the Nyström approach costs the least in term of computation when $m = 2000$ however exceeds the time taken for `exact` when $m=5000$. In this case the time is dominated by the eigen-decomposition of a matrix in $\mathbb{R}^{m \times m}$. Of note also is that `RSVD` with $r=5000$ exceeds the time required for `exact` yet this was the number of random projections required for competitive performance. `IASC` improves over `exact` over the whole sequence of graphs as one does not recompute the eigenvectors at every iteration. Notice that the “staircase” effect in the `IASC` curves correspond the computation of the exact eigenvectors. Observe that on `Bemol`, `IASC` $\ell = 200$ takes 4,956 seconds in total for eigenvector computations versus 31,182 for `exact`, a speedup factor of 6.29 for a similar cluster quality. The equivalent improvement is 2.26 on `Citation`.
To emphasise the conditions in which `IASC` can be effective, we again cluster over the `Citation` data however graphs are recorded at 5 day intervals. The parameters are identical to those used above except we set $R= 50$. Figure \[fig:citation5day\] shows the resulting clustering qualities and eigenvector computation timings for `IASC` and `RSVD`. As one might expect we observe that `RSVD` is competitive to `IASC`, $\ell = 100$, when $r = 5000$, and only competitive with $r=2000$ until approximately the 400th graph. The timings however show that `IASC` is faster than `RSVD` for all values of $\ell$ over all graphs. Furthermore, when $\ell = 100$ `IASC` took 9640 seconds compared to 32,006 and 158,510 seconds with `RSVD` using $r$ values for $2000$ and $5000$ respectively.
Discussion {#sec:conclusion}
==========
We have presented a novel incremental method for spectral graph clustering which updates the eigenvectors of the Laplacian in a computationally efficient way. Such an algorithm is useful for finding clusterings in time evolving graphs such as biological and social networks, and the Internet. A key part of the algorithm is a general way to approximate the first $k$ eigenvectors of a perturbed symmetric matrix given the eigenvectors of the original matrix. The resulting clustering algorithm, IASC, can be easily implemented using a standard linear algebra library.
We analysed IASC in both theoretical and empirical respects. Using perturbation theory, we showed when the canonical angles between the real and approximate subspaces generated by our update algorithm are close. Furthermore, IASC is examined empirically relative to the computation of exact eigenvectors for each graph, the method of Ning et al., randomised SVD, and the Nyström approach. On 2 toy and 3 real datasets we show that IASC can often match the cluster accuracy of the exact approach using a small fraction of the total number of eigenvectors and at a much reduced computational cost.
This work has opened up several perspectives for further study. The first is the analysis of the update of eigenvectors for a modularity matrix and other cluster quality criteria. As we have shown, the quality of the updates would depend on the spectrum of the matrices in question. Another interesting line of research is the issue of how to choose the number of clusters in the time evolving graphs.
|
---
abstract: 'We investigated the spin-dependent transport properties of Fe/MgAl$_2$O$_4$/Fe(001) magnetic tunneling junctions (MTJs) on the basis of first-principles calculations of the electronic structures and the ballistic conductance. The calculated tunneling magnetoresistance (TMR) ratio of a Fe/MgAl$_2$O$_4$/Fe(001) MTJ was about 160%, which was much smaller than that of a Fe/MgO/Fe(001) MTJ (1600%) for the same barrier thickness. However, there was an evanescent state with $\Delta _1$ symmetry in the energy gap around the Fermi level of normal spinel MgAl$_2$O$_4$, indicating the possibility of a large TMR in Fe/MgAl$_2$O$_4$/Fe(001) MTJs. The small TMR ratio of the Fe/MgAl$_2$O$_4$/Fe(001) MTJ was due to new conductive channels in the minority spin states resulting from a band-folding effect in the two-dimensional (2-D) Brillouin zone of the in-plane wave vector ($k_{\parallel}$) of the Fe electrode. Since the in-plane cell size of MgAl$_2$O$_4$ is twice that of the primitive in-plane cell size of bcc Fe, the bands in the boundary edges are folded, and minority-spin states coupled with the $\Delta _1$ evanescent state in the MgAl$_2$O$_4$ barrier appear at $k_{\parallel}$=0, which reduces the TMR ratio of the MTJs significantly.'
author:
- Yoshio Miura
- Shingo Muramoto
- Kazutaka Abe
- Masafumi Shirai
title: 'A first-principles study of tunneling magnetoresistance in Fe/MgAl$_2$O$_4$/Fe(001) magnetic tunnel junctions'
---
Introduction
============
Recent advances in tunneling magnetoresistance (TMR) in magnetic tunnel junctions (MTJs) with a single-crystal MgO barrier and bcc Fe ferromagnetic electrodes make possible the fabrication of ultrahigh-speed and high-density magnetic random access memory(MRAM) devices[@2004Yuasa-NM; @2004Parkin-NM; @2007Ikeda-IEEE]. In Fe/MgO/Fe(001) MTJs, the crystal momentum parallel to the layer is conserved because of the 2-D periodicity of the system, and the tunneling conductance strongly depends on the symmetry of the propagating states in the bcc Fe electrode because of the complex band structures of MgO, which causes a slow decay of the evanescent $\Delta _1$ state in the barrier layer. Furthermore, electrons with an in-plane wave vector $k_{\parallel}=(0,0)$ (normal incidence with respect to the plane) dominate the tunneling, and bcc Fe has a 100% spin polarization in the $\Delta _1$ state at the Fermi level. Thus, Fe/MgO/Fe(001) MTJs act as spin filters for the current, yielding much larger TMR ratios of over 1000% in their ground state[@2001Butler-PRB; @2001Mathon-PRB].
Achieving a high tunneling magnetoresistive (TMR) ratio in MgO-based MTJs requires epitaxial growth of the MgO layer with a correct crystalline orientation on ferromagnetic electrodes (bcc Fe and Co$_x$Fe$_{1-x}$). However, it has been difficult to grow Fe/MgO/Fe(001) MTJs epitaxially owing to the relatively large lattice mismatch (5%) between rock-salt-type MgO and bcc Fe. Recently, an off-stoichiometric, normal spinel MgAl$_2$O$_4$ barrier was grown epitaxially on a single-crystal Co$_2$FeAl$_{0.5}$Si$_{0.5}$ and bcc Fe to explore new materials for the barrier layer of MTJs[@2009Shan-PRL; @2010Sukegawa-APL]. Since normal spinel MgAl$_2$O$_4$ has a lattice constant $a$ of 8.16Å, the lattice mismatch with bcc-type ferromagnetic metals such as bcc Fe and Co-based full Heusler alloys is very small, i.e., less than 1% for a 45$^{\circ}$ (001) in-plane rotation, indicating that MgAl$_2$O$_4$ has the potential to overcome the problem associated with MgO barriers. In fact, a large room-temperature TMR ratio of over 100% was reported in Co$_2$FeAl$_{0.5}$Si$_{0.5}$/MgAl$_2$O$_4$/CoFe(001) and Fe/MgAl$_2$O$_4$/Fe(001) MTJs. Furthermore, a relatively large bias voltage for one-half the zero-bias TMR ratio was obtained at room temperature (RT), which is about twice that reported for MgO-based MTJs.
The band structures of bulk MgAl$_2$O$_4$ were investigated by first-principles density functional calculations[@1991Xu-PRB; @2005Khenata-PLA], and the results were compared with experimental data from vacuum ultraviolet measurements[@1990Bortz-PS]. Normal spinel MgAl$_2$O$_4$ has an indirect band gap of about 7.8eV (6.5 eV in calculations), and the Mg-O bond length (1.919Å) is shorter than that in MgO (2.102Å), implying a stronger bond, and hence a larger band gap. However, the coherent tunneling properties of MTJs are still unknown for MgAl$_2$O$_4$. In particular, it is necessary to elucidate the symmetry-dependent tunneling through the evanescent states of MgAl$_2$O$_4$ and the physical cause of the relatively large TMR ratio obtained in recent experiments involving Fe/MgAl$_2$O$_4$/Fe(001) MTJs[@2010Sukegawa-APL].
This work aims to determine the coherent tunneling properties of MgAl$_2$O$_4$ and the cause of the large TMR ratios in Fe/MgAl$_2$O$_4$/Fe(001) MTJs. To this end, we investigated the spin-dependent transport properties of Fe/MgAl$_2$O$_4$/Fe(001) MTJs on the basis of first-principles density functional calculations of the electronic structures and the ballistic conductance.
Computational details
=====================
We prepared a super cell of a Fe/MgAl$_2$O$_4$/Fe(001) MTJ containing 11 atomic layers of bcc Fe and 9 atomic layers of MgAl$_2$O$_4$. The in-plane lattice parameter of the super cell was fixed at 5.733Å, which corresponds to twice the lattice constant of bcc Fe (2.865Å). Since the lattice constant of spinel-type MgAl$_2$O$_2$O is $a$=8.16Å, the lattice mismatch between bcc Fe and MgAl$_2$O$_4$ for a 45$^{\circ}$ (001) in-plane rotation is less than 1.0%. We performed first-principles calculations of the super cell using density functional theory within the generalized-gradient approximation for exchange-correlation energy[@1996Perdew-PRB]. In order to facilitate structure optimization, which is important for determining the interface structure, we adopted plane-wave basis sets along with the ultrasoft pseudopotential method by using the quantum code ESPRESSO [@espresso]. The number of [**k**]{} points was taken to be 5$\times$5$\times$1 for all cases, and Methfessel-Paxton smearing with a broadening parameter of 0.01Ry was used. The cutoff energies for the wavefunction and charge density were set to 30Ry and 300Ry, respectively. These values are large enough to deal with all the elements considered here within the ultrasoft pseudopotential method.
For the conductance calculations, we considered an open quantum system consisting of a scattering region having a MgAl$_2$O$_4$ barrier and junctions with bcc Fe attached to left and right semi-infinite electrodes corresponding to bulk bcc Fe. The conductance was obtained by solving the scattering equation with infinite boundary conditions in which the wavefunction of the scattering region and its derivative were connected to the Bloch states of each electrode[@1999Choi-PRB]. Since our system is repeated periodically in the $xy$ plane and propagating states can be assigned by an in-plane wave vector $k_{\parallel}=(k_x,k_y)$ index, different $k_{\parallel}$ do not mix and can be treated separately. Furthermore, we neglected the spin-orbit interaction and noncollinear spin configuration. Thus, we solved the scattering equations for some fixed $k_{\parallel}$ and spin index on the basis of Choi and Ihm’s approach [@1999Choi-PRB; @2004Smogunov-PRB].
Results and Discussion
======================
{height="0.18\textheight" width="50.00000%"}
{height="0.25\textheight" width="45.00000%"}
------------------------------ --------------- ----------------- --------------- --------------- ------
TMR ratio \[%\]
Majority spin Minority spin Majority spin Minority spin
Fe/MgAl$_2$O$_4$/Fe(001) MTJ 0.507 0.0245 0.0994 0.105 160
Fe/MgO/Fe(001) MTJ 0.113 0.00300 0.00346 0.00324 1630
------------------------------ --------------- ----------------- --------------- --------------- ------
First, we investigated the stable structure of Fe/MgAl$_2$O$_4$(001) interfaces. Normal spinel MgAl$_2$O$_4$ has two different types of cation sites: tetrahedral Mg sites (A-sites) and octahedral Al sites (B-sites). This results in two types of termination for Fe/MgAl$_2$O$_4$(001) junctions, namely, the A-site (Mg) termination and B-site (Al-O) termination. On the basis of formation energy calculations for optimized surface structures of MgAl$_2$O$_4$(001), we found that B-site termination is thermodynamically more stable than A-site termination. This was also reported for other normal spinel compounds, such as Fe$_3$O$_4$(001)[@2005Fonin-PRB]. Thus, we considered only B-site termination for the Fe/MgAl$_2$O$_4$(001) interface. In B-site termination, there are four possible configurations, with the Fe atoms positioned on top of the Al atoms (Al-top), Al-Al hollow, Al-O hollow and the O atoms (O-top) of MgAl$_2$O$_4$. In order to determine the stable interfacial configuration at Fe/MgAl$_2$O$_4$(001) junctions, we minimized the total energy by relaxing all atomic positions except for those in the electrode region by changing the longitudinal size of the super cell. Figure 1 shows schematics of the Fe/MgAl$_2$O$_4$/MgO(001) interfacial structure with B-site termination. We found that the O-top configuration is the most stable because of the hybridization between the 3$d_{\rm 3z^2-r^2}$ orbital of Fe and the $p_z$ orbital of O. Therefore, in the present study on spin-dependent conductance and the TMR effect, we adopted the O-top configuration for the B-site terminated Fe/MgAl$_2$O$_4$(001) interface.
Figure 2 shows the local density of states (LDOS) of Fe in MgAl$_2$O$_4$/Fe(001), MgO/Fe(001) and the bulk region. The interfacial LDOS of Fe at the interface with MgAl$_2$O$_4$ are modified from those in the bulk region, owing to the appearance of the nonbonding Fe-3$d_{\rm x^2-y^2}$ and Fe-3$d_{\rm xy}$ states around the Fermi level. The spin moment of interfacial Fe is 3.08$\mu _{\rm B}$, which is larger than that in the bulk region (2.49$\mu _{\rm B}$). Similar results have been obtained for MgO/Fe(001) interfaces[@2001Butler-PRB; @2010Saito-PRB].
We investigated the tunneling conductance of the Fe/MgAl$_2$O$_4$($\sim$1nm)/Fe(001) MTJ. Figure 3 plots the in-plane wave vector ($k_{\rm \parallel}$) dependence of the tunneling conductance at the Fermi level for the Fe/MgAl$_2$O$_4$/Fe(001) MTJ in parallel and anti-parallel magnetization configurations. We confirmed in Fig. 3(a) that the majority-spin conductance for the parallel magnetization has a broad peak around the center of the 2-D Brillouin zone. This is the typical behavior of the coherent tunneling conductance of $\Delta _1$ electrons at $k_{\rm \parallel}$=(0,0). Furthermore, we found in Fig. 3(b) that the minority-spin conductance for the parallel magnetization also shows a broad peak at $k_{\rm \parallel}$=(0,0). This result is very different from that for the Fe/MgO/Fe(001) MTJ[@2001Butler-PRB; @2001Mathon-PRB], where the $k_{\rm \parallel}$-dependence of the minority-spin conductance for the parallel magnetization shows hotspot-like peaked structures in the 2-D Brillouin zone with no peak at $k_{\rm \parallel}$=(0,0) because of the absence of the $\Delta _1$ band around the Fermi level. Since the tunneling conductance in the anti-parallel magnetization shows a combination of the features observed in the majority and minority spin channels, we obtained a broad peak at $k_{\rm \parallel}$=(0,0) in the $k_{\rm \parallel}$ dependence of the conductance for the anti-parallel magnetization as shown in Fig. 3(c). In Table 1, we showed the majority- and minority-spin conductance and the TMR ratios for Fe/MgAl$_2$O$_4$/Fe(001) MTJs and Fe/MgO/Fe(001) MTJs. The calculated minority-spin conductance for the Fe/MgAl$_2$O$_4$/Fe(001) MTJ for parallel magnetization was one order of magnitude higher than that for the Fe/MgO/Fe(001) MTJ. This led to TMR ratio of 160% for the Fe/MgAl$_2$O$_4$/Fe(001) MTJ, which was much lower than that for the Fe/MgO/Fe(001) MTJ (1600%) with the same barrier thickness.
{height="0.6\textheight" width="30.00000%"}
Figure 4 shows the imaginary part of the out-of-plane wave vector $k_z$ ($\kappa$) at the Fermi levels of bulk MgAl$_2$O$_4$ and MgO as a function of the in-plane wave vector $k_x$. The positions of the Fermi level in the band gap were determined from the LDOS of oxygen atoms in the center of the barrier layer of Fe/MgAl$_2$O$_4$/Fe(001) and Fe/MgO/Fe(001) MTJs. Since propagating states in the metal electrode couple to evanescent states ($\kappa $) in the barrier layer and decay as $\sim e^{-\kappa z}$, the evanescent state with the smallest decay parameter, $\kappa _{\rm min}$, makes the largest contribution to the tunneling conductance in MTJs. We found that at ($k_x$,$k_y$)=(0,0), MgAl$_2$O$_4$ has the smallest $\kappa$ with $\Delta _1$ symmetry in the gap connecting the top of the $\Delta _1$ valence band to the bottom of the $\Delta _1$ conduction band, which is similar to the case for rock-salt-type MgO. This could be due to the hybridization between O 2$p_z$ and Mg 2$p_z$, generating the energy gap along the $\Delta $ line. Furthermore, the lowest $\kappa$ of MgAl$_2$O$_4$ showed a $k_{\rm x}$ dependence similar to that for MgO. This suggests the possibility that the large TMR ratio stems from the coherent tunneling properties and the half-metallic character of bcc Fe on the $\Delta _1$ state in Fe/MgAl$_2$O$_4$/Fe(001), like in Fe/MgO/Fe(001) MTJs. These results are inconsistent with our calculation results, where the TMR ratio of the Fe/MgAl$_2$O$_4$/Fe(001) MTJ (160%) was one order of magnitude smaller than that of the Fe/MgO/Fe(001) MTJ (1600%).
{height="0.18\textheight" width="35.00000%"}
To further elucidate these results, we show in Fig. 5 the band dispersion of bulk bcc Fe with a tetragonal unit cell ($a$=5.733Å, $c/a$=0.5), corresponding to the unit cell of the electrode region in the Fe/MgAl$_2$O$_4$/Fe(001) MTJ. Since bcc Fe with a primitive lattice constant of 2.867 Åhas minority-spin states with $s$ orbital character at the boundary edge of the 2-D Brillouin zone of $k_{\parallel}$, these minority-spin states appear at $k_{\parallel}$=(0,0) because of the band folding that occurs when the in-plane cell size of bcc Fe is twice that of the primitive cell size. The new conductive channels at $k_{\parallel}$=(0,0) couple with the $\Delta _1$ evanescent states of MgAl$_2$O$_4$, giving the slowest decay in the barrier layer. This induces a relatively large conductance in the anti-parallel magnetization configuration, resulting in the reduction of the TMR ratio. Thus, we concluded that the TMR effect in Fe/MgAl$_2$O$_4$/Fe(001) MTJs is intrinsically smaller than that of Fe/MgO/Fe(001) MTJs because of the band folding effect in the minority-spin states of the Fe electrode.
![(a) Majority- and (b) minority-spin band dispersion of the bcc Fe electrode in Fe/MgAl$_2$O$_4$/Fe MTJs with a tetragonal unit cell ($a$=5.733Å, $c/a$=0.5) along the \[001\] direction at $k_{//}$=(0,0), where the bold lines indicate the band with $\Delta _1$ symmetry in the tetragonal unit cell, which corresponds to the unit cell of MgAl$_2$O$_4$.](figure5.eps){height="0.27\textheight" width="47.00000%"}
We calculated the projected density of states of the tunneling wavefunction (TDOS) on the $\Delta _1$($s$) and $\Delta _5$($p_y$) states for the Fe/MgAl$_2$O$_4$/Fe(001) and Fe/MgO/Fe(001) MTJs. Figure 5 presents the majority-spin TDOS at the Fermi level at $k_{\parallel}$=(0,0) in the barrier regions as a function of the distance from the left junction of the MTJs in parallel magnetization. We found that the TDOS of the Fe/MgAl$_2$O$_4$/Fe(001) MTJ showed a slower decay for the wavefunction of the $\Delta _1$ evanescent state than for the $\Delta _5$ state. The same was observed in the case of the Fe/MgO/Fe(001) MTJ. This is clear evidence of the significant contribution of tunneling electrons with $\Delta _1$ symmetry, as compared to those with $\Delta _5$ symmetry in MgAl$_2$O$_4$. Furthermore, the decay rate of the TDOS of $\Delta _1$ states in the interior of the barrier layer is almost the same for both Fe/MgAl$_2$O$_4$/Fe(001) and Fe/MgO/Fe(001) MTJs. These results indicate that MTJs with a MgAl$_2$O$_4$ barrier can potentially show a large TMR ratio, comparable to that for Fe/MgO/Fe(001) MTJs, if band folding of the Fe electrode can be suppressed.
{height="0.2\textheight" width="50.00000%"}
Finally, we discuss a possibility of the coherent tunneling and the band folding in recent experiments of MTJs with the MgAl$_2$O$_4$ barrier. Sukegawa, [*et al.*]{}, reported TMR ratios of 117% at RT and 165% at 15 K for Fe/MgAl$_2$O$_x$(2nm)/Fe(001) MTJs[@2010Sukegawa-APL]. They showed that the MgAl$_2$O$_x$ barrier layer formed a normal-spinel structure with a (001) orientation, and the Fe/MgAl$_2$O$_x$(2nm)/Fe(001) MTJ had a epitaxial relationship between the bcc-Fe(001)\[110\] and the MgAl$_2$O$_x$(001). This means that the coherent tunneling occurs through the evanescent $\Delta _1$ state in the barrier layer. Thus, the experimentally observed small TMR ratio in the Fe/MgAl$_2$O$_x$(2nm)/Fe(001) MTJ compared to that of the Fe/MgO/Fe(001) MTJ[@2004Yuasa-NM; @2004Parkin-NM; @2007Ikeda-IEEE] can be attributed to appearance of new conductive channels in the anti-parallel magnetization due to the band folding effect. We believe that a modification of the periodic boundary condition along the in-plane direction by choosing ferromagnetic electrodes without the band folding at the junction with MgAl$_2$O$_4$ is necessary to obtain the large TMR ratio in MTJs with the normal-spinel MgAl$_2$O$_4$.
Summary
=======
We investigated the electronic and transport properties of Fe/MgAl$_2$O$_4$/Fe(001) MTJs on the basis of first-principles density functional calculations. Since normal spinel MgAl$_2$O$_4$ has evanescent states with $\Delta _1$ symmetry in the energy gap around the Fermi level, the Fe/MgAl$_2$O$_4$/Fe(001) MTJ, like Fe/MgO/Fe(001), shows coherent tunneling properties. However, we obtained a TMR ratio of 160% for the Fe/MgAl$_2$O$_4$(1nm)/Fe(001) MTJ, which was much smaller than that of the Fe/MgO(1 nm)/Fe(001) MTJ (1600%). We concluded that the appearance of new conductive channels at $k_{\parallel}$=(0,0) by the band folding effect significantly reduced the TMR ratio of the Fe/MgAl$_2$O$_4$(1nm)/Fe(001) MTJ. The new folded channel in the minority-spin state of the Fe electrode coupled with the $\Delta _1$ evanescent state of the MgAl$_2$O$_4$ barrier, which showed a slow decay in the barrier layer and contributed to the minority-spin conductance. These results indicate that the TMR effect in Fe/MgAl$_2$O$_4$/Fe(001) MTJs is intrinsically smaller than that in Fe/MgO/Fe(001) MTJs. However, if the band folding effect can be suppressed by changing the periodic boundary condition along the in-plane direction, the Fe/MgAl$_2$O$_4$/Fe(001) MTJs show a large TMR ratio, comparable to that for Fe/MgO/Fe(001) MTJs.
We are grateful to H. Sukegawa and S. Mitani of National Institute for Materials Science in Japan for valuable discussions about our work. This work was supported by a Grant-in-Aid for Scientific Research (Nos. 22360014 and 22760003) from MEXT, the Japan Science and Technology (JST) through its Strategic International Cooperative Program under the title “Advanced spintronic materials and transport phenomena (ASPIMATT)”. Y. M. and K. A. gratefully acknowledge supports from Mayekawa Houonkai Foundation.
[10]{}
|
---
abstract: 'Recently reported tentative evidence for a gamma-ray line in the Fermi-LAT data is of great potential interest for identifying the nature of dark matter. We compare the implications for decaying and annihilating dark matter taking the constraints from continuum gamma-rays, antiproton flux and morphology of the excess into account. We find that higgsino and wino dark matter are excluded, also for nonthermal production. Generically, the continuum gamma-ray flux severely constrains annihilating dark matter. Consistency of decaying dark matter with the spatial distribution of the Fermi-LAT excess would require an enhancement of the dark matter density near the Galactic center.'
---
DESY 12-112\
June 2012
[**Decaying vs Annihilating Dark Matter\
in Light of a Tentative Gamma-Ray Line\
\[2mm\]**]{}
[Wilfried Buchmüller and Mathias Garny]{}\
[*[ Deutsches Elektronen-Synchrotron DESY, 22607 Hamburg, Germany]{}* ]{}
Introduction
============
Monochromatic gamma-ray lines have been suggested long ago as signature for pair-annihilation of dark matter (DM) particles [@Bergstrom:1988fp]. During the past decades many DM candidates have been discussed, the most popular ones being weakly interacting massive particles (WIMPs) in the context of supersymmetric extensions of the Standard Model [@Bertone:2004pz]. More recently, also decaying dark matter has been studied in detail [@Takayama:2000uz; @Buchmuller:2007ui; @Cirelli:2012ut; @Arvanitaki:2008hq]. An attractive feature of decaying gravitino dark matter is the consistency with thermal leptogenesis, contrary to standard WIMP dark matter [@Buchmuller:2007ui].
During the past years the Large Area Telescope (LAT) [@Atwood:2009ez], on board the Fermi gamma-ray space telescope, has searched with unprecedented sensitivity for photon lines from 30 MeV to 300 GeV. Stringent constraints on decaying and annihilating dark matter have been obtained from searches in the energy range $30-200$ GeV based on 11 months of data [@Abdo:2010nc], and in the range $7-200$ GeV based on 23 months of data [@Ackermann:2012qk]. The search region used in these analyses covers the whole sky except for the Galactic disk ($|b|>10^\circ$) plus a $20^\circ\times 20^\circ$ region around the Galactic center. A similar analysis has been performed independently based on publicly available data corresponding to 27 months in Ref. [@Vertongen:2011mu]. No indications for gamma-ray lines were found.
In a recent analysis, that is based on optimized search regions around the Galactic center, and takes 43 months of Fermi-LAT data into account, a hint for a gamma-ray feature in the energy range $120-140$ GeV is reported [@Bringmann:2012vr; @Weniger:2012tx]. When interpreted in terms of a gamma-ray line [@Weniger:2012tx], the significance obtained from the statistical uncertainties in the search regions close to the Galactic center is $4.6\sigma$. The significance is reduced to $3.3\sigma$ when correcting for the bias introduced by selecting the search regions. While the excess is currently under an active debate [@Profumo:2012tr; @Tempel:2012ey; @Boyarsky:2012ca; @Bergstrom:2012fi; @Ibarra:2012dw; @Dudas:2012pb], the claim has been further strengthened by a recent analysis [@Su:2012ft], which confirms the existence of an excess, as well as its spectral shape, with even higher statistical significance than the one claimed in Ref. [@Weniger:2012tx]. In addition, indications are found that the excess originates from a relatively narrow region of a few degrees around the Galactic center, possibly with a small offset within the Galactic plane. Hopefully, the question about the existence of the spectral feature and its precise properties will be settled in the near future.
Assuming that the feature is real, the question about its origin is of great interest. While an astrophysical explanation might eventually be identified, a spectral feature in this energy range can arise rather generically from the annihilation or decay of dark matter particles. In this paper we compare decaying and annihilating dark matter based on two prototype models: decaying gravitinos and wino/higgsino-like WIMPs which annihilate predominantly into two W-bosons. In both cases we treat the branching ratios into $\gamma\nu$ (gravitino) and $\gamma\gamma$ (wino/higgsino) final states as free parameters to account for some model dependence. Note that we do not demand thermal freeze-out for WIMPs. Higgsinos and winos can be nonthermally produced in gravitino decays, compatible with leptogenesis [@Ibe:2011aa; @Buchmuller:2012bt], or, alternatively, in moduli decays [@Moroi:1999zb; @Kitano:2008tk; @Acharya:2012tw].
In the following we shall compare interpretations of the tentative 130 GeV photon line in terms of decaying and annihilating dark matter in a sequence of increasing assumptions: We first consider constraints from continuum gamma-rays, which are independent of charged cosmic rays and the dark matter distribution (Section 2). This is followed by a discussion of constraints from antiprotons, which depend on the propagation model (Section 3). We then analyze the implications of the spatial distribution of the Fermi-LAT excess (Section 4) and draw our conclusions (Section 5).
Constraints from continuum gamma-rays
=====================================
Since the dark matter particle is required to be electrically neutral, the annihilation or decay into photons is typically suppressed compared to channels involving for example electroweak gauge bosons or the Higgs boson, as well as quarks or leptons. The subsequent decay and fragmentation of these annihilation or decay products gives rise, among others, to an emission of gamma rays with a broad spectrum in the energy range $\simeq 1-100$ GeV below the line. Therefore, one expects that the line flux is accompanied by an associated continuum flux with the same spatial distribution, whose strength depends on the relative size of the annihilation or decay rates into photons and into other Standard Model particles. As compared to the sharp feature resulting from the direct decay into photons, the continuum emission has a less characteristic spectral shape and is therefore much harder to disentangle from the background gamma-ray flux in the Galactic center region [@Boyarsky:2010dr]. On the other hand, one expects a much larger total flux in the continuum compared to the line, and therefore it is important to check up to which level a continuum emission is acceptable. Apart from the center of the Galaxy, the continuum flux of gamma rays can be constrained, [*e.g.*]{}, by the emission seen within the Galactic halo [@Ackermann:2012rg], from Galaxy clusters [@Ackermann:2010rg; @Ripken:2010ja], or from dwarf galaxies [@Ackermann:2011wa]. In general, a comparison of the various constraints from different targets is affected by the uncertainties related to the dark matter density distribution in the different environments.
We will not enter into the details of this discussion here, but instead follow a very conservative approach. Namely, we consider the same data sets for the photon flux that have been used in Ref. [@Weniger:2012tx] to search for a gamma-ray line signal. However, instead of using only a line as the template, we add a contribution from the continuum flux, and test whether the combined spectrum yields a satisfactory fit to the data. This strategy has the advantage that it is not affected by uncertainties related to the actual dark matter distribution, because both the line and the continuum fluxes are generated from the same source. Concretely, we consider a spectral template for the fit given by $$\label{eq:templateDecayingDM}
\frac{dJ}{dE} = \alpha \left( \delta(E-E_\gamma) + \frac{dN_{EG}}{dE} + \frac{1-\operatorname{BR}_{\gamma}}{N_\gamma\operatorname{BR}_{\gamma}} \frac{dN^\gamma_{cont}}{dE} \right) + \beta \left(\frac{E}{E_\gamma}\right)^{-\gamma} \,,$$ where $E_\gamma=m_{DM}(m_{DM}/2)$, $\operatorname{BR}_\gamma=\sigma v_{\gamma\gamma}/\sigma v_{tot}\,(\tau/\tau_{\gamma\nu})$ and $N_\gamma=2(1)$ for dark matter annihilation (decay). The spectrum of secondary photons originating from the decay and fragmentation of quarks, leptons or electroweak gauge bosons produced in annihilation or decay processes is denoted by $dN^\gamma_{cont}/dE$, while $dN_{EG}/dE$ denotes the extragalactic contribution resulting from the superposition of redshifted photons [@Bertone:2007aw]. Both will be discussed in detail below. The fit parameter $\alpha$ determines the strength of the dark matter flux. The background is assumed to follow a power law within the considered energy range (see [@Profumo:2012tr] for possible caveats). Both the normalization $\beta$ and the slope $\gamma$ of the background $(\propto \beta E^{-\gamma})$ are taken as fit parameters. For a given dark matter profile and search region, $\alpha$ can be easily converted into the dark matter annihilation cross section $\sigma v_{\gamma\gamma}$ or partial lifetime $\tau_{\gamma\nu}$ into monochromatic photons. The continuum spectrum is in general given by the sum of two components. The first one arises from redshifted monochromatic photons with extragalactic origin. For decaying dark matter it is given by [@Bertone:2007aw] $$\label{eq:lineEG}
\frac{dN_{EG}}{dE} = \frac{\Omega_{DM}\rho_{c0}}{\sqrt{\Omega_M}(H_0/c)\bar J_\psi} \, \frac{E^{1/2}}{E_\gamma^{3/2}}\left(1+\frac{\Omega_\Lambda}{\Omega_M}\left(\frac{E}{E_\gamma}\right)^3\right)^{-1/2} \Theta(E_\gamma-E)\;.$$ Here $\Omega_\Lambda=1-\Omega_M\simeq 0.73$ are the cosmological density parameters, assuming a spatially flat $\Lambda$CDM model, $\Omega_{DM}h^2=0.112$ is the dark matter density, $H_0=h 100$km/s/Mpc is the Hubble constant with $h=0.704$, $\rho_{c0}=1.05\cdot 10^{-5} h^2$GeV/cm$^3$ is the critical density, $c$ is the speed of light, and the factor $\bar J_\psi=\Delta\Omega^{-1}\int_{\Delta\Omega}d\Omega\int_{l.o.s.}\rho_{dm}(r)$ appears because of the normalization chosen in (\[eq:templateDecayingDM\]). For the search regions 3 and 4 of [@Weniger:2012tx] and the various density profiles, the prefactor in (\[eq:lineEG\]) is of the order $0.3-0.4$.
The second contribution to the continuum gamma spectrum arises from the weighted sum of the spectra produced by annihilation or decays, excluding $\gamma\gamma(\gamma\nu)$, $$\frac{dN^\gamma_{cont}}{dE} \equiv \frac{1}{\sum_{f\not=\gamma} \operatorname{BR}_f} \sum_{f\not=\gamma} \operatorname{BR}_f \frac{ dN^\gamma_f}{dE} \;,$$ where $dN^\gamma_f/dE$ is the number of photons per energy and per annihilation/decay resulting from each mode, and $\operatorname{BR}_f$ is the corresponding branching ratio. For the line, we use the same shape as in [@Weniger:2012tx] based on the instrument response function of the Fermi-LAT detector (FWHM$=0.136$), which has somewhat broader tails compared to a Gaussian, and also smooth the step function appearing in the extragalactic contribution (\[eq:lineEG\]) accordingly. The continuum flux is determined using PYTHIA 8.1 [@Sjostrand:2007gs]. In order to increase the sensitivity to the continuum flux we use the data presented in [@Weniger:2012tx] over the full energy range $20-300$ GeV, taking the more finely binned data where they are available ($80-200$ GeV). We show our results for both the SOURCE and ULTRACLEAN event classes as defined in the Pass 7 Version 6 data release, corresponding to 43 months of data [@Weniger:2012tx]. The former class yields better statistics due to an increased effective area, and the latter a lower contamination with charged cosmic rays. For both, the zenith-angle cut is $\theta<100^\circ$. As discussed above, the constraints arising from the relative strength of monochromatic and continuum gamma rays are rather insensitive to the dark matter profile. For definiteness, we assume an Einasto profile [@Navarro:2003ew] normalized to a local density $\rho_0=0.4{\mbox{GeV}}/{\mbox{cm}}^3$ unless stated otherwise (see Section \[sec:Profile\] for details). Lastly, we require that $\gamma>2.0$ as a conservative lower limit for the background power law index [@Su:2010qj], while its normalization is freely varied. For performing the fit we use the profile likelihood method as detailed in [@Weniger:2012tx], with $95\%$C.L. statistical errors obtained by varying $\alpha$ (with fixed $\operatorname{BR}_\gamma$) and profiling over $\beta$ and $\gamma$ until the $TS$-value has decreased by 3.84. We checked that our statistical analysis reproduces the best-fit values and confidence intervals in [@Weniger:2012tx] using the smaller energy window and a single line up to $10\%$ for regions 2-4, both data samples and all density profiles. When including $\operatorname{BR}_\gamma$ in the fit, we use $\Delta(TS)=5.99$ to obtain the $95\%$C.L. regions, corresponding to the $\chi^2_{k=2}$ distribution that is expected for two parameters of interest[^1]. We will comment on the dependence on the various assumptions later on.
Decaying dark matter
--------------------
For decaying gravitino dark matter, which we consider as representative example, we take the decay channels $Z\nu,W\ell,h\nu$ apart from $\gamma\nu$ into account. In the appendix, the partial decay widths are given in the case of bilinear R-parity breaking as functions of the gravitino mass, parameters of R-parity breaking, gaugino masses and the higgsino mass parameter $\mu$. According to Eq. (\[line\]), the branching ratio into $\gamma\nu$ is enhanced for hierarchical gaugino masses. For example, in the case of wino NLSP with $M_2 \simeq m_{3/2}$ one obtains the maximal branching ratio, $$\label{BRmax}
\operatorname{BR}^{\rm max}_\gamma \simeq \frac{3\pi\alpha}{2\sqrt{2}G_Fm_{3/2}^2} \ ,$$ which yields $\operatorname{BR}^{\rm max}_\gamma \simeq 3\%$ for $m_{3/2} \simeq 260$ GeV. On the contrary, for higgsino NLSP, with $\mu \simeq m_{3/2} \ll M_1,M_2$, the branching ratio into $\gamma\nu$ is negligible.
![\[fig:fluxGravitino\]Sum of monochromatic and continuum photon flux originating from dark matter decaying into $\gamma\nu, Z\nu, h\nu,
W^\pm\ell^\mp$ with branching ratios $\operatorname{BR}_\gamma=0.1$, $0.01$, and $0.0025$, respectively. The lower lines show the contribution from dark matter decay, and the upper lines the sum of signal and fitted power-law background. The dashed blue line shows the flux when taking the gamma-ray line only, plus the redshifted extragalactic contribution, into account. The gamma-ray flux measured by Fermi-LAT corresponding to the SOURCE (upper figure) and ULTRACLEAN (lower figure) data samples is taken from Ref. [@Weniger:2012tx]. The shaded regions correspond to $95\%$C.L. error bands.](figs/reg3SOURCEGravitino "fig:"){width="90.00000%"}\
![\[fig:fluxGravitino\]Sum of monochromatic and continuum photon flux originating from dark matter decaying into $\gamma\nu, Z\nu, h\nu,
W^\pm\ell^\mp$ with branching ratios $\operatorname{BR}_\gamma=0.1$, $0.01$, and $0.0025$, respectively. The lower lines show the contribution from dark matter decay, and the upper lines the sum of signal and fitted power-law background. The dashed blue line shows the flux when taking the gamma-ray line only, plus the redshifted extragalactic contribution, into account. The gamma-ray flux measured by Fermi-LAT corresponding to the SOURCE (upper figure) and ULTRACLEAN (lower figure) data samples is taken from Ref. [@Weniger:2012tx]. The shaded regions correspond to $95\%$C.L. error bands.](figs/reg3ULTRACLEANGravitino "fig:"){width="90.00000%"}
In order to illustrate the possibilities to explain the tentative gamma-ray line with decaying dark matter, we show in Fig. \[fig:fluxGravitino\] the photon spectrum from gravitino decay, fitted to the flux measured by Fermi LAT within the search region 3 of Ref. [@Weniger:2012tx], for three values of the branching ratio into monochromatic photons. The feature in the spectrum near $130$GeV can be explained for branching ratios of $10\%$ and $1\%$. For the relatively small value $0.25\%$, the continuum gamma-ray contribution becomes so strong that it would overshoot the measured flux if one would require the monochromatic decay mode to account for the feature in the spectrum. In this case, the fit to the observed spectrum becomes worse, and requires a somewhat longer total lifetime, such that the continuum flux is in accordance with the observed one. The best fit values for the partial lifetime, as well as the significance with respect to the background-only case obtained from search regions 3 and 4 are shown in Tab. \[tab:continuum\], for various branching ratios. The significance of the fit remains nearly constant for branching ratios larger than $1\%$, and then steeply decreases. For a branching ratio of $0.1\%$ or smaller, the dark matter component cannot yield a significantly better description than the background only.
![\[fig:pbar\]Combined fit of the monochromatic line at $\approx 130$GeV as well as the continuum gamma-ray spectrum from dark matter decay. The yellow and orange shaded areas are $95\%$C.L. regions for the total dark matter lifetime $\tau$ and the branching ratio $\operatorname{BR}(\psi\to\gamma\nu)$ obtained from the Fermi-LAT data within search regions 3 and 4 or Ref. [@Weniger:2012tx], respectively. The regions excluded from antiproton constraints at $95\%$C.L. based on the PAMELA $\bar p/p$ data are shown as blue shaded regions (see Section \[sec:pbar\]), as well as the expected range for gravitino dark matter (red shaded area). Further explanations are given in the text.](figs/gravitinoPbarSOURCE.pdf "fig:"){width="75.00000%"}\
![\[fig:pbar\]Combined fit of the monochromatic line at $\approx 130$GeV as well as the continuum gamma-ray spectrum from dark matter decay. The yellow and orange shaded areas are $95\%$C.L. regions for the total dark matter lifetime $\tau$ and the branching ratio $\operatorname{BR}(\psi\to\gamma\nu)$ obtained from the Fermi-LAT data within search regions 3 and 4 or Ref. [@Weniger:2012tx], respectively. The regions excluded from antiproton constraints at $95\%$C.L. based on the PAMELA $\bar p/p$ data are shown as blue shaded regions (see Section \[sec:pbar\]), as well as the expected range for gravitino dark matter (red shaded area). Further explanations are given in the text.](figs/gravitinoPbarULTRACLEAN.pdf "fig:"){width="75.00000%"}
We have also performed a fit where the branching ratio is varied together with the parameters $\alpha,\beta,\gamma$ in Eq. (\[eq:templateDecayingDM\]). The resulting $95\%$C.L. contours are shown in Fig. \[fig:pbar\], indicating again that branching ratios below $0.25\%$ are disfavored because of the continuum contribution to the photon flux. For comparison we also show the more stringent constraints resulting from fixing the background parameters $(\beta,\gamma)=(\beta,\gamma)|_{BR_{\gamma\nu}=1}$ to the best-fit values obtained from the fit using a gamma-ray line only (grey dot-dashed line), as well as the constraints obtained when using the smaller energy range $80-200$ GeV (grey dotted line) for region 4. We also note that the results are rather insensitive to the precise choice of the relative branching ratios among $Z\nu,W\ell,h\nu$.
Clearly, the limits on the continuum gamma-ray flux obtained in this way are very conservative ones, while stronger constraints could be obtained, [*e.g.*]{}, by taking photons produced via inverse Compton scattering into account ([*e.g.*]{} [@Cirelli:2012ut]) or extending the energy range below $20$ GeV. However, this would also make the conclusions more dependent on details of the background spectrum and therefore on the astrophysical processes close to the Galactic center. In this sense, the limits presented here may be regarded as a rather robust consistency check, based on the assumption that the background can be modeled by a power law in the energy range $20-300$ GeV.
Annihilating dark matter
------------------------
One of the most studied candidates for dark matter is the lightest neutralino within the MSSM, for which the annihilation cross sections into $\gamma\gamma$ and $\gamma Z$, that yield monochromatic gamma rays, are loop suppressed [@Bergstrom:1997fh]. Consequently, the branching ratios into these final states are typically at the permille level, and the continuum flux dominates the gamma spectrum[^2].
In the following we perform first a model independent analysis in order to determine the requirements on the branching ratio into monochromatic photons, and then check whether some well-motivated scenarios are compatible with these constraints.
In particular, we first assume that dark matter annihilates into $\gamma\gamma$ and $W^+W^-$ final states, giving rise to a line at $E_\gamma=m_{DM}$ and a continuum photon flux in the energy range $E<m_{DM}$. The resulting photon flux is shown in Fig. \[fig:fluxWWgamgam\], for several values of the branching ratio $\operatorname{BR}_{\gamma}=\sigma v_{\gamma\gamma}/(\sigma v_{\gamma\gamma}+\sigma v_{WW})$. Similarly to decaying dark matter, we find that the feature in the Fermi spectrum at $\approx 130$GeV can be well fitted for branching ratios $10\%$ or $1\%$, while the continuum spectrum becomes dominant for $0.25\%$. Note that the photon spectrum exhibits a more pronounced dip between the continuum and monochromatic contributions as compared to decaying dark matter, due to the absence of the extragalactic contribution. The statistical significance and best-fit values of the cross section are given in Tab. \[tab:continuumWWgamgam\] for various branching ratios, and exhibit a similar dependence than for decaying dark matter. One might notice that the significance of the fit always decreases when lowering $\operatorname{BR}_\gamma$, while for decaying dark matter it is slightly better for $\operatorname{BR}_\gamma\sim 1\%$. However, these differences are too small to be taken seriously. A common feature is the drastic decrease in the significance for branching ratios smaller than $1\%$. As before, we also performed a fit with variable branching ratio. The resulting $95\%$C.L. regions are shown in Fig. \[fig:pbarWW\]. Depending on the search region and on the data sample used for the analysis, we find that a branching ratio larger than $0.4-0.8\%$ is required. For comparison, the constraints obtained when using a fixed background contribution (grey dot-dashed line) or when using the smaller energy range $80-200$GeV (grey dotted lines) are shown as well. The sensitivity to the continuum spectrum is essentially lost in the latter case for annihilating dark matter, due to the drop of the flux below the line. On the other hand, we checked that removing the lower limit $\gamma>2.0$ on the background slope does not affect the confidence regions.
![\[fig:fluxWWgamgam\]Sum of monochromatic and continuum photon flux originating from dark matter annihilating into $\gamma\gamma$ and $W^+W^-$ with branching ratios $\operatorname{BR}_{\gamma}=0.1$, $0.01$, and $0.0025$, respectively. The lower lines show the contribution from dark matter annihilation, and the upper lines the sum of signal and fitted power-law background. The dashed blue line shows the flux when taking the line only into account. The Fermi data are the same as in Fig. \[fig:fluxGravitino\].](figs/reg3SOURCEWWgamgam "fig:"){width="90.00000%"}\
![\[fig:fluxWWgamgam\]Sum of monochromatic and continuum photon flux originating from dark matter annihilating into $\gamma\gamma$ and $W^+W^-$ with branching ratios $\operatorname{BR}_{\gamma}=0.1$, $0.01$, and $0.0025$, respectively. The lower lines show the contribution from dark matter annihilation, and the upper lines the sum of signal and fitted power-law background. The dashed blue line shows the flux when taking the line only into account. The Fermi data are the same as in Fig. \[fig:fluxGravitino\].](figs/reg3ULTRACLEANWWgamgam "fig:"){width="90.00000%"}
![\[fig:pbarWW\]Combined fit of the monochromatic line at $\approx 130$GeV as well as the continuum gamma-ray spectrum from dark matter annihilation into $\gamma\gamma$ and $W^+W^-$ final states. The shaded areas correspond to $95\%$C.L. regions (yellow/orange: Fermi-LAT Reg 3/4 best-fit region; blue: PAMELA $\bar p/p$ excluded region, see Section \[sec:pbar\]) for the total annihilation cross section $\sigma v$ and the branching ratio $\sigma v_{\gamma\gamma}/\sigma v$. The prediction from higgsino and wino dark matter (red and orange dots) is shown for illustration, as well as the thermal cross-section (straight black line).](figs/WWgamgamPbarSOURCE.pdf "fig:"){width="75.00000%"}\
![\[fig:pbarWW\]Combined fit of the monochromatic line at $\approx 130$GeV as well as the continuum gamma-ray spectrum from dark matter annihilation into $\gamma\gamma$ and $W^+W^-$ final states. The shaded areas correspond to $95\%$C.L. regions (yellow/orange: Fermi-LAT Reg 3/4 best-fit region; blue: PAMELA $\bar p/p$ excluded region, see Section \[sec:pbar\]) for the total annihilation cross section $\sigma v$ and the branching ratio $\sigma v_{\gamma\gamma}/\sigma v$. The prediction from higgsino and wino dark matter (red and orange dots) is shown for illustration, as well as the thermal cross-section (straight black line).](figs/WWgamgamPbarULTRACLEAN.pdf "fig:"){width="75.00000%"}
As a second generic example, we consider dark matter annihilating into $\gamma Z$ and $\gamma\gamma$, giving rise to two lines at $E_{\gamma,1}=m_{DM}$ and $E_{\gamma,2}=m_{DM}-M_Z^2/(4 m_{DM})$. In addition, annihilations into $W^+W^-$ and $ZZ$ yield a continuum spectrum. Motivated by higgsino dark matter, we fix the relative contribution of the two lines according to $\sigma v_{\gamma Z}/(2\sigma v_{\gamma\gamma})=1.68$ as well as $\sigma v_{ZZ}/\sigma v_{WW}=0.66$. However, we freely vary the weighted branching ratio $\operatorname{BR}_\gamma=(\sigma v_{\gamma\gamma}+0.5\sigma v_{\gamma Z})/\sigma v_{tot}$. In this case we obtain the best fit for dark matter masses in the range $132-137$GeV, depending on the search region and the data sample, and with similar significance than for one line. The resulting $95\%$C.L. regions are shown in Fig. \[fig:pbarHiggsino\], which are very similar to the case of a single line discussed before, indicating that the lower limit $\operatorname{BR}_\gamma \gtrsim 0.4-0.8\%$ is rather insensitive to the detailed assumptions about the cross sections of the various annihilation channels.
![\[fig:pbarHiggsino\]As Fig.\[fig:pbarWW\], but for a model with two lines from annihilation into $\gamma\gamma$ and $\gamma Z$, as well as continuum photons from annihilations into $W^+W^-$ and $ZZ$ final states. The axes correspond to the total cross section $\sigma v$ and to the weighted ratio $(\sigma v_{\gamma\gamma}+0.5\sigma v_{\gamma Z})/\sigma v$, respectively. For the fit we fixed the ratios $\sigma v_{\gamma Z}/(2\sigma v_{\gamma\gamma})=1.68$ and $\sigma v_{ZZ}/\sigma v_{WW}=0.66$ motivated by the higgsino dark matter scenario. The actual prediction from higgsino and wino dark matter is also shown (red and orange dots). PAMELA antiproton constraints are discussed in Section \[sec:pbar\].](figs/higgsinoPbarSOURCE.pdf "fig:"){width="75.00000%"}\
![\[fig:pbarHiggsino\]As Fig.\[fig:pbarWW\], but for a model with two lines from annihilation into $\gamma\gamma$ and $\gamma Z$, as well as continuum photons from annihilations into $W^+W^-$ and $ZZ$ final states. The axes correspond to the total cross section $\sigma v$ and to the weighted ratio $(\sigma v_{\gamma\gamma}+0.5\sigma v_{\gamma Z})/\sigma v$, respectively. For the fit we fixed the ratios $\sigma v_{\gamma Z}/(2\sigma v_{\gamma\gamma})=1.68$ and $\sigma v_{ZZ}/\sigma v_{WW}=0.66$ motivated by the higgsino dark matter scenario. The actual prediction from higgsino and wino dark matter is also shown (red and orange dots). PAMELA antiproton constraints are discussed in Section \[sec:pbar\].](figs/higgsinoPbarULTRACLEAN.pdf "fig:"){width="75.00000%"}
Finally, we briefly remark that both higgsino and wino-like dark matter can be ruled out as an explanation for the tentative gamma line based on the large continuum photon flux obtained in both cases. The cross sections for two representative scenarios are given in Tab. \[tab:scenario\], leading to $\operatorname{BR}_\gamma\sim 0.08\%(0.2\%)$ for higgsino(wino)-like dark matter. Even when allowing for a boost factor or a mixed dark matter scenario in order to achieve a line flux of the required size, the continuum flux overshoots the measured flux at energies around $20$GeV, which is illustrated in Fig. \[fig:fluxHiggsino\]. In particular, we remark that this conclusion is independent of the dark matter distribution, and therefore of the uncertainties associated with it. According to the recent analysis in [@Hryczuk:2011vi], electroweak corrections could even further reduce the $\gamma \gamma$ and $\gamma Z$ cross sections for low-mass wino dark matter by a sizeable amount.
$\mu$ $M_2$ $m_{\chi_0^1}$ $m_{\chi_0^2}$ $m_{\chi_\pm^1}$ $\sigma v_{\gamma\gamma}(\sigma v_{\gamma Z})$ $\sigma v_{WW}(\sigma v_{ZZ})$
--- ------- -------- ---------------- ---------------- ------------------ ------------------------------------------------ --------------------------------
H 139 $1000$ 135.89 144.44 139.20 $1.0(3.4)\cdot 10^{-28}$ $2.1(1.4)\cdot 10^{-25}$
W 400 143 139.79 408.08 139.94 $2.0(10.9)\cdot 10^{-27}$ $3.4(0.0)\cdot 10^{-24}$
: Mass spectrum, annihilation cross sections and defining parameters for two scenarios with higgsino (H) and wino-like (W) LSP. Particle masses are given in GeV, and cross sections in $\text{cm}^3/\text{s}$. The cross sections are rather insensitive to the other parameters, which we fixed to $A=0$, $M_{1,3}=\tilde m = 1$TeV and $\tan\beta=7$ for illustration. For the wino case the $\mu$ parameter was chosen such that the neutralino-chargino mass splitting is of the same order than the minimal splitting expected from loop corrections [@Cirelli:2005uq] in order to yield an estimate for the maximal possible gamma cross sections. []{data-label="tab:scenario"}
![\[fig:fluxHiggsino\]Predicted gamma-ray flux for higgsino dark matter (upper graph) and wino dark matter (lower graph, green dashed lines). The monochromatic gamma-ray flux is too small(large) in the higgsino(wino) case for explaining the Fermi excess at $\approx 130$GeV when assuming an Einasto profile. We also show the flux obtained for a boost factor $BF=5.6$ in the higgsino case and when assuming that the wino accounts only for $45\%$ of the dark matter density, respectively (red solid lines). In both cases, the monochromatic contribution to the flux would yield a good fit to the Fermi data (blue dotted lines). However, the continuum contribution to the gamma-ray flux overshoots the measured flux at lower energies, and therefore both scenarios can be ruled out. The Fermi data are taken from [@Weniger:2012tx].](figs/reg4SOURCEHiggsino "fig:"){width="90.00000%"}\
![\[fig:fluxHiggsino\]Predicted gamma-ray flux for higgsino dark matter (upper graph) and wino dark matter (lower graph, green dashed lines). The monochromatic gamma-ray flux is too small(large) in the higgsino(wino) case for explaining the Fermi excess at $\approx 130$GeV when assuming an Einasto profile. We also show the flux obtained for a boost factor $BF=5.6$ in the higgsino case and when assuming that the wino accounts only for $45\%$ of the dark matter density, respectively (red solid lines). In both cases, the monochromatic contribution to the flux would yield a good fit to the Fermi data (blue dotted lines). However, the continuum contribution to the gamma-ray flux overshoots the measured flux at lower energies, and therefore both scenarios can be ruled out. The Fermi data are taken from [@Weniger:2012tx].](figs/reg4SOURCEWino "fig:"){width="90.00000%"}
Constraints from antiprotons {#sec:pbar}
============================
Constraints that are complementary to the continuum gamma-ray spectrum arise from the production of antiprotons in the decay and fragmentation of the weak gauge bosons, Higgs bosons or quarks produced in dark matter annihilation or decay. The resulting primary flux of antiprotons can be potentially observed in the cosmic ray spectrum measured at the Earth. Antiprotons are produced with a rate per unit of kinetic energy and unit volume at a position $r$ with respect to the center of the Galaxy given by $$Q_{\bar{p}}(E,r)=\left\{ \frac{\rho_{dm}(r)}{m_{DM}}\frac{1}{\tau} \atop \frac{\rho_{dm}(r)^2}{2m_{DM}^2}\,\sigma v_{tot} \right\}\times \sum_f \operatorname{BR}_f \frac{dN_f^{\bar{p}}}{dE}\quad \mbox{for} \quad \left\{\rule{0mm}{5mm}\ { \mbox{decay}\rule{12mm}{0mm}} \atop { \mbox{annihilation}\rule{0mm}{5mm} } \right. \;,
\label{eqn:source-term}$$ where $dN_f^{\bar p}/dE$ is the number of antiprotons per kinetic energy and per decay produced in the annihilation/decay channels $f$ with branching ratio given by $\operatorname{BR}_f$, and $\rho_{dm}(r)$ is the dark matter density profile.
The antiproton-to-proton flux ratio observed by the PAMELA satellite experiment [@Adriani:2010rc] agrees well with the expectation for the secondary antiproton spectrum produced by spallation of ordinary cosmic rays on the interstellar medium. Therefore, upper limits on a possible primary component from dark matter annihilation or decay can be derived (see [*e.g.*]{} [@Donato:2008jk; @Ibarra:2008qg; @Buchmuller:2009xv; @Garny:2011cj; @Evoli:2011id; @Garny:2012]). The main source of uncertainty enters due to the propagation of antiprotons in the Galactic magnetic field from the production point to the Earth. In order to obtain lower limits on the dark matter lifetime and upper limits on the annihilation cross section we follow here the approach described in detail in Refs. [@Garny:2012; @Garny:2011cj]. In particular, we describe the propagation by a cylindrical two-zone diffusion model using three representative sets of parameters leading to minimum, medium and maximum antiproton fluxes consistent with the charged cosmic ray abundances [@Bringmann:2006im]. In order to obtain conservative limits we use the minimal secondary antiproton spectrum from Refs. [@Donato:2001ms; @Garny:2011cj], and use a lower cut of $T>1.5$GeV for the kinetic energy.
For the case of dark matter decaying into $Z\nu,h\nu,W\ell$ with the same branching ratios used above, the $95\%$C.L. lower limits on the dark matter lifetime is also shown in Fig. \[fig:pbar\] as blue shaded regions, for the three sets of propagation parameters and the Einasto dark matter profile. For a branching ratio much smaller than one, the $95\%$C.L. lower limit on the lifetime is given by $\tau \gtrsim (0.29,1.34,3.31)\times 10^{27}$s for the minimum, medium and maximum propagation models, respectively. Compared to the uncertainty arising from the propagation model, the dependence of the lower limit on the dark matter profile is rather weak. As can be seen in Fig. \[fig:pbar\], the constraints from antiprotons require the branching ratio to lie above $\sim\mathcal{O}(1\%,5\%,10\%)$, depending on the propagation model, if we require that the partial lifetime into monochromatic photons accounts for the gamma feature at $130$. Even for the minimal propagation model, these lower limits on the branching ratio are more stringent than the ones obtained from the continuum gamma-ray flux. However, they are also more dependent on astrophysical uncertainties related to propagation as well as the secondary antiproton flux [@Evoli:2011id]. For comparison, we also show the lower limits on the lifetime obtained from just requiring that the primary antiproton flux does not overshoot the observed antiproton-to-proton ratio. The resulting limits are somewhat weaker, $\tau\gtrsim (0.64,2.81,6.62)\times 10^{26}$s. For the most conservative case, [*i.e.*]{} for minimal propagation, the resulting lower limit on the branching ratio is comparable to the one obtained from the continuum gamma-ray flux (see Fig. \[fig:pbar\]).
For the case of annihilating dark matter into $W^+W^-$, the antiproton constraints shown in Fig. \[fig:pbarWW\] correspond to upper limits on the annihilation cross section $\sigma v \lesssim (6.1,0.69,0.25)\times
10^{-25}$cm$^3/$s when taking the secondary flux into account. The resulting lower limits on the branching ratio $\sigma v_{\gamma\gamma}/\sigma v$ are more stringent or comparable to the ones from the continuum flux for maximum or medium propagation, respectively, as can be seen in Fig \[fig:pbarWW\]. In the conservative case, when taking only the primary flux into account, one obtains rather weak limits $\sigma v \lesssim (2.8,0.31,0.12)\times
10^{-24}$cm$^3/$s that are comparable to the continuum flux only in the maximum case. The antiproton constraints are very similar for the case when dark matter annihilates into $\gamma\gamma/\gamma Z/WW/ZZ$ final states (see Fig. \[fig:pbarHiggsino\]), except for the antiprotons from $\gamma Z$ that contribute for very large branching ratios.
Altogether, we find that antiproton constraints are typically the leading ones for decaying dark matter, while the continuum gamma-ray flux is more or equally important for annihilating dark matter.
Morphology of the Fermi excess at 130 GeV {#sec:Profile}
=========================================
Apart from constraints arising from the gamma-ray spectrum or from antiprotons, the Fermi-LAT data also provide valuable information about the spatial distribution of the gamma-ray flux observed towards the Galactic center (the angular resolution of the LAT is $\Delta\theta\sim 0.2^\circ$ [@Weniger:2012tx]). The hints for a feature in the spectrum are based on about 69(57) photons collected within 43 months inside search regions 3(4) (solid angle $\Delta\Omega/4\pi=0.025(0.015)$) defined in [@Weniger:2012tx] and the SOURCE sample, and 53(46) for the ULTRACLEAN sample. Consequently, any information about the detailed spatial distribution of the excess within these regions is at present necessarily limited by statistics. Nevertheless, the amount of data allows to test the compatibility with certain spatial distributions. In particular, in the analysis of [@Su:2012ft] it was found that using a Gaussian template the excess appears to originate from a rather small region $\approx 3^\circ$FWHM, possibly with a small offset from the Galactic center of $1.5^\circ$ along the Galactic plane. Additionally, it was argued that an Einasto or contracted NFW profile yields an even better description, when also taking an offset into account.
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
![Expected spatial profile of the gamma-ray emission from sections along the Galactic plane ($\Delta\ell=0.5^\circ, |b|<5^\circ$, upper graph) and perpendicular to the plane ($\Delta b=0.5^\circ, -5^\circ<\ell<-2^\circ$, lower graph) for decaying and annihilating dark matter and two density profiles centered around $(\ell,b)=(-1.5^\circ,0)$. For illustration, we also show the count maps indicating the shape of the Fermi-LAT excess in the energy range $124.7-133.4$ GeV taken from Ref. [@Su:2012ft].\[fig:spatial\]](figs/spatialDistribution_l "fig:"){width="99.00000%"}
![Expected spatial profile of the gamma-ray emission from sections along the Galactic plane ($\Delta\ell=0.5^\circ, |b|<5^\circ$, upper graph) and perpendicular to the plane ($\Delta b=0.5^\circ, -5^\circ<\ell<-2^\circ$, lower graph) for decaying and annihilating dark matter and two density profiles centered around $(\ell,b)=(-1.5^\circ,0)$. For illustration, we also show the count maps indicating the shape of the Fermi-LAT excess in the energy range $124.7-133.4$ GeV taken from Ref. [@Su:2012ft].\[fig:spatial\]](figs/spatialDistribution_b "fig:"){width="99.00000%"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
In the following, we leave aside the question about the origin of a possible offset in connection with a dark matter explanation. Instead, under the tentative hypothesis that dark matter is indeed responsible for the Fermi feature, we discuss the impact of the apparent morphology on decaying versus annihilating dark matter. The flux of photons arriving at the Earth per unit of energy and solid angle from dark matter decay or annihilation is given by the line-of-sight integral over the dark matter density or the density squared, respectively, along the direction of observation (see e.g. [@Garny:2010eg]), $$\frac{dJ_\gamma}{dE d\Omega} = \frac{1}{4\pi}\delta(E-E_\gamma) \left\{
\begin{array}{ll}\displaystyle
\frac{1}{\tau_{\gamma\nu}m_{DM}}\ {\textstyle\int\limits_{l.o.s.}}ds\,\rho_{dm}(r) & \mbox{decay}\\[2.5ex]
\displaystyle
\frac{2\sigma v_{\gamma\gamma}}{ m_{DM}^2}\ {\textstyle\int\limits_{l.o.s.}}ds\,\frac12\,
\rho_{dm}(r)^2 & \mbox{annihilation}
\end{array}\right.\;,$$ where $r=\sqrt{(r_0-s\cos\xi)^2+(s\sin\xi)^2}$, $r_0\simeq 8.5$kpc is the distance of the Earth from the Galactic center, and $\xi$ the angle measured with respect to the direction towards the center of the Galaxy. As a reference, we consider the density profiles given, respectively, by a generalized Navarro-Frenk-White (NFW) [@Navarro:1995iw] or Einasto profile [@Navarro:2003ew], $$\rho_{dm}(r) \propto \frac{1}{(r/r_s)^\alpha(1+r/r_s)^{3-\alpha}},\ \exp\left(-\frac{2}{\alpha_E}(r/r_s)^{\alpha_E}\right)\,,$$ with $\alpha_E=0.17$ (see e.g.[@Navarro:2008kc]), $\alpha=1$ and scale radius $r_s=20$kpc. We also consider two strongly peaked distributions with $\alpha=(1.3,1.7)$, that may arise from adiabatic contraction. All profiles are normalized such that the local density is $\rho_0=0.4{\mbox{GeV}}/{\mbox{cm}}^3$, except for $\alpha=1.7$ where we use $\rho_0=0.28{\mbox{GeV}}/{\mbox{cm}}^3$ corresponding to the maximal allowed $1\sigma$-value when taking observational constraints from microlensing and measurements of the Galactic dynamics into account [@Iocco:2011jz]. Note that for all other profiles the local density $\rho_0=0.4{\mbox{GeV}}/{\mbox{cm}}^3$ lies within the $1\sigma$-allowed region.
In Ref. [@Su:2012ft] it has been argued that the morphology of the Fermi excess at $\approx 130$GeV is consistent with dark matter annihilation assuming the Einasto or NFW profile, possibly with $\alpha$ slightly larger than one. In order to obtain a similar emission profile for decaying dark matter, a much more peaked dark matter distribution would be necessary. In order to illustrate this point, we show in Fig. \[fig:spatial\] the expected spatial distribution for the contracted NFW profiles. For comparison we also show the corresponding shape for annihilating dark matter, and the count maps taken from [@Su:2012ft]. Qualitatively, it seems that even for the strongly contracted profile with $\alpha=1.7$ the expected distribution for decaying dark matter is too flat, especially for the profile along the Galactic plane. One may speculate that, in the perpendicular direction, the excess could be less peaked. However, a definite statement is presently impossible due to statistical limitations.
![\[fig:pbarp\]Best fit values for the partial dark matter lifetime $\tau_{\gamma\nu}$ for a decay into monochromatic photons, obtained for the search regions 2, 3 and 4 defined in Ref. [@Weniger:2012tx], for various dark matter profiles. The error bars show the 95%C.L. regions. Also shown are the lower limits on the partial lifetime from the most recent Fermi LAT analysis (the lower error bars include the estimated systematic uncertainty) [@Ackermann:2012qk], that are based on a wide search region (Halo Limit). The upper and lower figures correspond to the two Fermi data samples used in [@Weniger:2012tx].](figs/bestFitDecaySOURCE "fig:"){width="85.00000%"}\
![\[fig:pbarp\]Best fit values for the partial dark matter lifetime $\tau_{\gamma\nu}$ for a decay into monochromatic photons, obtained for the search regions 2, 3 and 4 defined in Ref. [@Weniger:2012tx], for various dark matter profiles. The error bars show the 95%C.L. regions. Also shown are the lower limits on the partial lifetime from the most recent Fermi LAT analysis (the lower error bars include the estimated systematic uncertainty) [@Ackermann:2012qk], that are based on a wide search region (Halo Limit). The upper and lower figures correspond to the two Fermi data samples used in [@Weniger:2012tx].](figs/bestFitDecayULTRACLEAN "fig:"){width="85.00000%"}
For decaying dark matter, one would expect also a comparably large signal arising from the Galactic halo, where no excess has been observed. Therefore, it is instructive to compare the halo limit with the required lifetime to explain the excess in the central region. The partial dark matter lifetimes $\tau_{\gamma\nu}$ that are required in order to explain the feature in the Fermi gamma-ray spectrum present within various search regions defined in [@Weniger:2012tx], and for the various assumptions of the radial dark matter distribution, are shown in Fig. \[fig:pbarp\]. Although the best-fit values for the considered search regions are compatible with each other, they are in tension with the lower limits on the lifetime obtained by the recent analysis of the Fermi collaboration [@Ackermann:2012qk], also shown in Fig. \[fig:pbarp\]. Note that the Fermi analysis is based on a rather wide search region, and therefore sensitive to the dark matter flux originating in the halo of the Milky Way. Due to the linear scaling of the flux with the dark matter density in the case of decaying dark matter, one typically expects a larger flux from the halo compared to the center than for annihilating dark matter. Thus, for the case of decaying dark matter the gamma-ray line should have been observed also within the search region used by the Fermi collaboration, provided that the radial distribution of dark matter can be well described by the considered NFW or Einasto profiles. The discrepancy is less pronounced for the contracted profiles which are more cuspy towards the Galactic center. On the other hand, for annihilating dark matter, the cross section required to explain the Fermi gamma-ray feature is compatible with earlier upper limits assuming NFW or Einasto profiles [@Weniger:2012tx].
Clearly, this tension disfavors the explanation of the excess in terms of decaying dark matter. However, because of the large uncertainty of the actual dark matter distribution close to the Galactic center as compared to the halo [@Bertone:2004pz; @Bergstrom:2012fi; @Springel:2008cc], and since one cannot exclude large systematic uncertainties at present [@Ackermann:2012qk], we believe that it would be premature to rule out decaying dark matter as an explanation of the Fermi excess based on this tension.
Conclusion
==========
In light of tentative evidence for a gamma-ray line observed by Fermi-LAT near the Galactic center, we have investigated the consequences for several prototype scenarios of decaying and annihilating dark matter, motivated by supersymmetric models with gravitino, higgsino or wino-like LSP. We find that, independently of the actual dark matter distribution, the consistency of continuum and monochromatic contributions to the photon spectrum from dark matter decay or annihilation requires a branching ratio into monochromatic photons larger than $\operatorname{BR}_\gamma \gtrsim 0.5\%$. Both higgsino and wino dark matter can be ruled out because of a too large continuum flux, independently of the production mechanism, while gravitino dark matter with wino NLSP is compatible. We have also investigated constraints arising from the primary antiproton flux, which depend on the adopted propagation model. The resulting limits are comparable to the ones from continuum gamma-rays for annihilating dark matter, but can be more important for decaying dark matter. The morphology of the tentative excess, if confirmed, could yield valuable information on the distribution of dark matter close to the Galactic center. When assuming conventional NFW or Einasto profile functions, the required partial lifetimes for decaying dark matter $\tau_{\gamma\nu} \sim (1-3)\times 10^{28}$s are in tension with lower limits obtained from the Galactic halo. This tension is less pronounced for more cuspy contracted profiles. If decaying gravitino dark matter is indeed responsible for the excess, the dark matter density should be enhanced in the Galactic center region compared to conventional models.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank S. Bobrovskyi, T. Bringmann, L. Covi, J. Hajer, A. Ibarra, A. Morselli, A. Ringwald and C. Weniger for helpful discussions. We thank M. Grefe for useful comments and are grateful to J. Hajer for cross-checking the Appendix. This work has been supported by the German Science Foundation (DFG) within the Collaborative Research Center 676 “Particles, Strings and the Early Universe”.
Appendix {#appendix .unnumbered}
========
In the following we collect the formulae needed to evaluate the various branching ratios of gravitino LSP decays in supersymmetric models with bilinear R-parity breaking.
The couplings of the gravitino field $\psi_\nu$ to matter and gauge fields fields are given by [@Wess:1992cp; @Moroi:1995fs], $$\label{gravcoup}
\mathcal{L} = \frac{i}{\sqrt{2}M}
\left(\overline{\chi} \gamma^\nu \gamma^\mu (D_\mu \phi) \psi_\nu
+ \mathrm{c.c.}\right) -
\frac{1}{4 M}\overline{\lambda} \gamma^\nu\sigma^{\mu\rho}
\psi_\nu F_{\mu\rho}.$$ Here $\chi$ stands for the left-handed lepton and higgsino doublets $l$, $h_u$, $h_d$, and $\phi$ for the corresponding scalar leptons and Higgs fields $\tilde{l}$, $H_u$, $H_d$; $F_{\nu\rho}$ denotes the fields strengths of $U(1)_Y$ and $SU(2)$ gauge interactions, $B_{\nu\rho}$ and $W^I_{\nu\rho}$, and $\lambda$ the corresponding gauginos $b$ and $w^I$.
In the case of bilinear R-parity breaking the transition from weak to mass eigenstates has been worked out in detail in [@Bobrovskyi:2010ps; @Bobrovskyi:2012xy]. Inserting these field transformations in Eq. (\[gravcoup\]) and shifting the Higgs fields around their vacuum expectation values, one finds for the R-parity breaking gravitino couplings $$\begin{aligned}
\label{Rviol}
{\cal L}_3 \supset
\frac{i}{2{M_{\mathrm{P}}}}
\{&\left(\kappa_{hi}\partial_\mu h + i\kappa_{zi} m_Z Z_\mu\right)
\bar{\nu}_i\gamma^\nu\gamma^\mu \psi_{\nu}
+ i \sqrt{2}\kappa_{wi} m_W W^-_\mu \bar{e}_i\gamma^\nu\gamma^\mu \psi_{\nu}
\nonumber\\
& + i \left(\xi_{zi} \partial_\mu Z_\nu
+ \xi_{\gamma i} \partial_\mu A_\nu\right)
\bar{\nu}_i\gamma^\lambda \sigma^{\mu\nu} \psi_{\lambda} \nonumber\\
& + i \sqrt{2} \xi_{wi} \partial_\mu W^-_\nu
\bar{e}_i\gamma^\lambda\sigma^{\mu\nu} \psi_{\lambda} \} + \mathrm{h.c.} \ .\end{aligned}$$ The couplings $\kappa$ and $\xi$ are determined by the R-parity breaking parameters $\zeta_i$, the gaugino masses $M_{1,2}$ and the $\mu$ parameter [@Bobrovskyi:2010ps; @Bobrovskyi:2012xy], $$\begin{aligned}
\label{coefficients}
\kappa_{hi} &= \zeta_i \ ,\\
\kappa_{zi} &= \zeta_i\left(1
+\sin{2\beta}\frac{m_Z^2\left(M_1\cos^2{\theta_W} + M_2\sin^2{\theta_W}\right)}
{M_1 M_2 \mu}\right) \ ,\\
\kappa_{wi} &= \zeta_i\left(1
+\sin{2\beta}\frac{m_W^2}{M_2 \mu}\right) \ ,\\
\xi_{\gamma i} &= \zeta_i\cos{\theta_W}\sin{\theta_W}
\frac{m_Z\left(M_2 - M_1\right)}{M_1 M_2} \ ,\\
\xi_{z i} &= -\zeta_i
\frac{m_Z\left(M_2\sin^2{\theta_W} + M_1\cos^2{\theta_W}\right)}{M_1 M_2} \ ,\\
\xi_{w i} &= -\zeta_i
\frac{m_W}{M_2} \ ,\end{aligned}$$ where $\theta_W$ is the weak angle. For a gravitino LSP one has $M_1,M_2,\mu \gtrsim m_{3/2}$. Note that the gauge boson couplings $\xi$ satisfy the relation $$\cos{\theta_{\rm W}} \xi_{Zi} + \sin{\theta_{\rm W}} \xi_{\gamma i} = \xi_{W i} \ .$$ The gravitino couplings (\[Rviol\]) have previously been obtained in an effective Lagrangian approach [@Buchmuller:2009xv][^3]. The couplings $\xi = \mathcal{O}(m_Z/m_{3/2})$ correspond to dimension-6 operators and the leading terms of the couplings $\kappa$ to a dimension-5 operator, which gives $$\begin{aligned}
\Gamma(\psi_{3/2}\rightarrow h\nu_i) &\simeq
\Gamma(\psi_{3/2}\rightarrow Z\nu_i) \simeq
\frac{1}{2}\Gamma(\psi_{3/2}\rightarrow W^+e^-_i)\ ,\end{aligned}$$ for $m_Z /m_{3/2} \ll 1$. The subleading contributions $\mathcal{O}(m_Z^2/m_{3/2}^2)$ to $\kappa$ correspond to dimension-7 operators which were not considered in [@Buchmuller:2009xv].
Based on the calculations of [@Covi:2008jy; @Grefe:2008zz] one can now obtain the partial two-body gravitino decay widths by matching coefficients. The result reads $$\begin{aligned}
\Gamma(\psi_{3/2}\rightarrow h\nu_i) &=
\frac{m_{3/2}^3}{384\pi{M_{\mathrm{P}}}^2}\kappa_{hi}^2\beta_h^4\ ,\label{bh}\\
\Gamma(\psi_{3/2}\rightarrow \gamma\nu_i) &=
\frac{m_{3/2}^3}{64\pi{M_{\mathrm{P}}}^2}|\xi_{\gamma i}|^2\ ,
\label{bgamma}\\
\Gamma(\psi_{3/2}\rightarrow Z\nu_i) &=
\frac{m_{3/2}^3}{384\pi{M_{\mathrm{P}}}^2}\beta_Z^2
\left(\kappa_{zi}^2 H_Z
+ 16 \frac{m_Z}{m_{3/2}}{\rm Re}(\kappa_{zi}\xi_{z i}) G_Z
+ 6 |\xi_{z i}|^2 F_Z\right)\ , \label{bz}\\
\Gamma(\psi_{3/2}\rightarrow W^+e^-_i) &=
\frac{m_{3/2}^3}{192\pi{M_{\mathrm{P}}}^2}\beta_W^2
\left(\kappa_{wi}^2 H_W
+ 16 \frac{m_W}{m_{3/2}}{\rm Re}(\kappa_{wi}\xi_{w i})G_W
+ 6 |\xi_{W i}|^2 F_W\right)\label{bw}\ .\end{aligned}$$ The functions $\beta_a$, $H_a$, $G_a$ and $F_a$ ($a=h,Z, W$) are given by [@Covi:2008jy] $$\begin{aligned}
\beta_a &= 1 - \frac{M_a^2}{m_{3/2}^2}\ ,\\
H_a &= 1 + 10 \frac{M_a^2}{m_{3/2}^2}
+ \frac{M_a^4}{m_{3/2}^4}\ ,\\
G_a &= 1 + \frac{1}{2} \frac{M_a^2}{m_{3/2}^2}\ ,\\
F_a &= 1 + \frac{2}{3} \frac{M_a^2}{m_{3/2}^2}
+ \frac{1}{3}\frac{M_a^4}{m_{3/2}^4}\ .\end{aligned}$$
The decay width $\Gamma(\psi_{3/2}\rightarrow \gamma\nu)$ is of particular interest since it determines the strength of the gamma line at the end of the continuous spectrum. Neglecting corrections $\mathcal{O}(m_Z^2/m_{3/2}^2)$, one finds for the branching ratio $$\label{line}
\operatorname{BR}_\gamma \simeq \frac{3\pi\alpha}{2\sqrt{2}G_F}
\frac{(M_1-M_2)^2}{M_1^2M_2^2}\ ,$$ where $\alpha$ is the electromagnetic fine-structure constant and $G_F$ is Fermi’s constant.
![Branching ratios of two-body gravitino decays for two representative examples. Wino NLSP: $M_2 = 1.1\ m_{3/2}$, $M_1 = \mu = 10\ m_{3/2}$, and higgsino NLSP: $\mu = 1.1\ m_{3/2}$, $M_1 = 10\ m_{3/2}$, $M_2 = 1.9\ M_1$ ($\tan\beta=10$ and $m_h=125$GeV in both cases).[]{data-label="branching"}](figs/gravitinoBR){width="85.00000%"}
In Figure \[branching\] the different branching ratios are shown as functions of the gravitino mass for the two representative cases of wino and higgsino NLSPs.
[99]{} L. Bergstrom and H. Snellman, Phys. Rev. D [**37**]{} (1988) 3737. For a review and references, see for example,\
G. Bertone, D. Hooper and J. Silk, Phys. Rept. [**405**]{} (2005) 279 \[hep-ph/0404175\]. F. Takayama and M. Yamaguchi, Phys. Lett. B [**485**]{} (2000) 388 \[hep-ph/0005214\]. W. Buchmuller, L. Covi, K. Hamaguchi, A. Ibarra and T. Yanagida, JHEP [**0703**]{} (2007) 037 \[hep-ph/0702184\]. For a recent discussion and references, see for example,\
M. Cirelli, E. Moulin, P. Panci, P. D. Serpico and A. Viana, arXiv:1205.5283 \[astro-ph.CO\]. A. Arvanitaki, S. Dimopoulos, S. Dubovsky, P. W. Graham, R. Harnik and S. Rajendran, Phys. Rev. D [**79**]{} (2009) 105022 \[arXiv:0812.2075 \[hep-ph\]\]. W. B. Atwood [*et al.*]{} \[LAT Collaboration\], Astrophys. J. [**697**]{} (2009) 1071 \[arXiv:0902.1089 \[astro-ph.IM\]\]. A. A. Abdo [*et al.*]{}, Phys. Rev. Lett. [**104**]{} (2010) 091302 \[arXiv:1001.4836 \[astro-ph.HE\]\]. M. Ackermann [*et al.*]{} \[LAT Collaboration\], arXiv:1205.2739 \[astro-ph.HE\]. G. Vertongen and C. Weniger, JCAP [**1105**]{} (2011) 027 \[arXiv:1101.2610 \[hep-ph\]\]. T. Bringmann, X. Huang, A. Ibarra, S. Vogl and C. Weniger, arXiv:1203.1312 \[hep-ph\]. C. Weniger, arXiv:1204.2797 \[hep-ph\]. S. Profumo and T. Linden, arXiv:1204.6047 \[astro-ph.HE\]. E. Tempel, A. Hektor and M. Raidal, arXiv:1205.1045 \[hep-ph\]. A. Boyarsky, D. Malyshev and O. Ruchayskiy, arXiv:1205.4700 \[astro-ph.HE\]. L. Bergstrom, arXiv:1205.4882 \[astro-ph.HE\]. A. Ibarra, S. Lopez Gehler and M. Pato, arXiv:1205.0007 \[hep-ph\]. A. Rajaraman, T. M. P. Tait and D. Whiteson, arXiv:1205.4723 \[hep-ph\]. E. Dudas, Y. Mambrini, S. Pokorski and A. Romagnoni, arXiv:1205.1520 \[hep-ph\]. J. M. Cline, arXiv:1205.2688 \[hep-ph\]. K. -Y. Choi and O. Seto, arXiv:1205.3276 \[hep-ph\]. H. M. Lee, M. Park and W. -I. Park, arXiv:1205.4675 \[hep-ph\]. B. Kyae and J. -C. Park, arXiv:1205.4151 \[hep-ph\]. B. S. Acharya, G. Kane, P. Kumar, R. Lu and B. Zheng, arXiv:1205.5789 \[hep-ph\]. M. R. Buckley and D. Hooper, arXiv:1205.6811 \[hep-ph\]. X. Chu, T. Hambye, T. Scarna and M. H. G. Tytgat, arXiv:1206.2279 \[hep-ph\]. A. Geringer-Sameth and S. M. Koushiappas, arXiv:1206.0796 \[astro-ph.HE\]. D. Das, U. Ellwanger and P. Mitropoulos, arXiv:1206.2639 \[hep-ph\]. Z. Kang, T. Li, J. Li and Y. Liu, arXiv:1206.2863 \[hep-ph\]. N. Weiner and I. Yavin, arXiv:1206.2910 \[hep-ph\]. L. Feng, Q. Yuan and Y. -Z. Fan, arXiv:1206.4758 \[astro-ph.HE\]. M. Su and D. P. Finkbeiner, arXiv:1206.1616 \[astro-ph.HE\]. M. Ibe and T. T. Yanagida, Phys. Lett. B [**709**]{} (2012) 374 \[arXiv:1112.2462 \[hep-ph\]\]; M. Ibe, S. Matsumoto and T. T. Yanagida, Phys. Rev. D [**85**]{} (2012) 095011 \[arXiv:1202.2253 \[hep-ph\]\]. W. Buchmuller, V. Domcke and K. Schmitz, Phys. Lett. B [**713**]{} (2012) 63 \[arXiv:1203.0285 \[hep-ph\]\]. T. Moroi and L. Randall, Nucl. Phys. B [**570**]{} (2000) 455 \[hep-ph/9906527\]. R. Kitano, H. Murayama and M. Ratz, Phys. Lett. B [**669**]{} (2008) 145 \[arXiv:0807.4313 \[hep-ph\]\]. B. S. Acharya, G. Kane and P. Kumar, Int. J. Mod. Phys. A [**27**]{} (2012) 1230012 \[arXiv:1204.2795 \[hep-ph\]\]. A. Boyarsky, D. Malyshev and O. Ruchayskiy, Phys. Lett. B [**705**]{} (2011) 165 \[arXiv:1012.5839 \[hep-ph\]\]. M. Ackermann [*et al.*]{} \[LAT Collaboration\], arXiv:1205.6474 \[astro-ph.CO\]. M. Ackermann, M. Ajello, A. Allafort, L. Baldini, J. Ballet, G. Barbiellini, D. Bastieri and K. Bechtol [*et al.*]{}, JCAP [**1005**]{} (2010) 025 \[arXiv:1002.2239 \[astro-ph.CO\]\]. J. Ripken, J. Conrad and P. Scott, JCAP [**1111**]{} (2011) 004 \[arXiv:1012.3939 \[astro-ph.HE\]\]. M. Ackermann [*et al.*]{} \[Fermi-LAT Collaboration\], Phys. Rev. Lett. [**107**]{} (2011) 241302 \[arXiv:1108.3546 \[astro-ph.HE\]\]. A. Geringer-Sameth and S. M. Koushiappas, Phys. Rev. Lett. [**107**]{} (2011) 241303 \[arXiv:1108.2914 \[astro-ph.CO\]\].
G. Bertone, W. Buchmuller, L. Covi and A. Ibarra, JCAP [**0711**]{} (2007) 003 \[arXiv:0709.2299 \[astro-ph\]\]. T. Sjostrand, S. Mrenna and P. Z. Skands, Comput. Phys. Commun. [**178**]{} (2008) 852 \[arXiv:0710.3820\]. M. Su, T. R. Slatyer and D. P. Finkbeiner, Astrophys. J. [**724**]{} (2010) 1044 \[arXiv:1005.5480 \[astro-ph.HE\]\]. L. Bergstrom and P. Ullio, Nucl. Phys. B [**504**]{} (1997) 27 \[hep-ph/9706232\]; P. Ullio and L. Bergstrom, Phys. Rev. D [**57**]{} (1998) 1962 \[hep-ph/9707333\]. C. B. Jackson, G. Servant, G. Shaughnessy, T. M. P. Tait and M. Taoso, JCAP [**1004**]{} (2010) 004 \[arXiv:0912.0004 \[hep-ph\]\]. For a model of decaying DM with related signatures, see C. Arina, T. Hambye, A. Ibarra and C. Weniger, JCAP [**1003**]{} (2010) 024 \[arXiv:0912.4496 \[hep-ph\]\]. M. Cirelli, N. Fornengo and A. Strumia, Nucl. Phys. B [**753**]{} (2006) 178 \[hep-ph/0512090\]. A. Hryczuk and R. Iengo, JHEP [**1201**]{} (2012) 163 \[arXiv:1111.2916 \[hep-ph\]\]. O. Adriani [*et al.*]{} \[PAMELA Collaboration\], Phys. Rev. Lett. [**105**]{} (2010) 121101 \[arXiv:1007.0821 \[astro-ph.HE\]\]. A. Ibarra and D. Tran, JCAP [**0807**]{} (2008) 002 \[arXiv:0804.4596 \[astro-ph\]\]. F. Donato, D. Maurin, P. Brun, T. Delahaye and P. Salati, Phys. Rev. Lett. [**102**]{} (2009) 071301 \[arXiv:0810.5292 \[astro-ph\]\]. W. Buchmuller, A. Ibarra, T. Shindou, F. Takayama and D. Tran, JCAP [**0909**]{} (2009) 021 \[arXiv:0906.1187 \[hep-ph\]\]. M. Garny, A. Ibarra and S. Vogl, JCAP [**1107**]{} (2011) 028 \[arXiv:1105.5367 \[hep-ph\]\]. C. Evoli, I. Cholis, D. Grasso, L. Maccione and P. Ullio, arXiv:1108.0664 \[astro-ph.HE\]. M. Garny, A. Ibarra, and D. Tran, arXiv:1205.6783 \[hep-ph\]. T. Bringmann and P. Salati, Phys. Rev. D [**75**]{} (2007) 083006 \[astro-ph/0612514\]. F. Donato, D. Maurin, P. Salati, A. Barrau, G. Boudoul and R. Taillet, Astrophys. J. [**563**]{} (2001) 172 \[astro-ph/0103150\]. T. Bringmann, L. Bergstrom and J. Edsjo, JHEP [**0801**]{} (2008) 049 \[arXiv:0710.3169 \[hep-ph\]\]. M. Garny, A. Ibarra, D. Tran and C. Weniger, JCAP [**1101**]{} (2011) 032 \[arXiv:1011.3786 \[hep-ph\]\]. J. F. Navarro, C. S. Frenk, S. D. M. White, Astrophys. J. [**462** ]{} (1996) 563-575. \[astro-ph/9508025\] J. F. Navarro, C. S. Frenk, S. D. M. White, Astrophys. J. [**490** ]{} (1997) 493-508. \[astro-ph/9611107\].
J. F. Navarro, E. Hayashi, C. Power, A. Jenkins, C. S. Frenk, S. D. M. White, V. Springel, J. Stadel [*et al.*]{}, Mon. Not. Roy. Astron. Soc. [**349** ]{} (2004) 1039. \[astro-ph/0311231\]. A. W. Graham, D. Merritt, B. Moore, J. Diemand, B. Terzic, Astron. J. [**132** ]{} (2006) 2701-2710. \[astro-ph/0608613\].
J. F. Navarro, A. Ludlow, V. Springel, J. Wang, M. Vogelsberger, S. D. M. White, A. Jenkins and C. S. Frenk [*et al.*]{}, arXiv:0810.1522 \[astro-ph\]. F. Iocco, M. Pato, G. Bertone and P. Jetzer, JCAP [**1111**]{} (2011) 029 \[arXiv:1107.5810 \[astro-ph.GA\]\]. V. Springel, J. Wang, M. Vogelsberger, A. Ludlow, A. Jenkins, A. Helmi, J. F. Navarro and C. S. Frenk [*et al.*]{}, Mon. Not. Roy. Astron. Soc. [**391**]{} (2008) 1685 \[arXiv:0809.0898 \[astro-ph\]\]. J. Wess and J. Bagger, [*Supersymmetry and supergravity*]{}, Princeton, USA: Univ. Pr. (1992) 259 p.
T. Moroi, [*Effects of the gravitino on the inflationary universe*]{}, hep-ph/9503210. S. Bobrovskyi, W. Buchmuller, J. Hajer and J. Schmidt, JHEP [**1010**]{} (2010) 061 \[arXiv:1007.5007 \[hep-ph\]\]. S. Bobrovskyi, J. Hajer and S. Rydbeck, in preparation. L. Covi, M. Grefe, A. Ibarra and D. Tran, JCAP [**0901**]{} (2009) 029 \[arXiv:0809.5030 \[hep-ph\]\]. M. Grefe, [*Neutrino signals from gravitino dark matter with broken R-parity*]{}, DESY-THESIS-2008-043.
[^1]: By generating a large sample of data that would be expected in the presence of a true signal, we checked that the actual distribution of the profile likelihood ratio is well described by $\chi^2_{k=1}$ when keeping $\operatorname{BR}_\gamma$ fixed, while it is somewhat steeper than $\chi^2_{k=2}$ when including $\operatorname{BR}_\gamma$ in the fit, lying between a $\chi^2_{k=1}$ and $\chi^2_{k=2}$ distribution. Therefore the confidence regions assuming $\chi^2_{k=2}$ may be regarded as conservative.
[^2]: In some extensions of the Standard Model DM annihilation can also produce intense gamma-ray lines. For recent examples and references see [@Weniger:2012tx; @Dudas:2012pb; @Jackson:2009kg].
[^3]: Note the change in notation compared to [@Buchmuller:2009xv]. We also corrected a typo in Eq. (2.7) of [@Buchmuller:2009xv] in the prefactor of our Eq. (\[Rviol\]).
|
---
abstract:
- 'In this review paper, the theory of synaptic transmission (ST) was developed and discussed. We used the hypothesis of isomorphism between: (a) the cooperative behavior of mediators — acetylcholine molecules (ACh) and cholinoreceptors in a synaptic cleft with binding into mediator-receptor (AChR) complexes, (b) the critical phenomena in confined binary liquid mixtures. The systems of two (or three) nonlinear differential equations were proposed to find the change of concentrations of ACh, AChR complexes, and ferment acetylcholinesterase. The main findings of our study: the linear size of the activation zone was evaluated; the process of postsynaptic membrane activation was described as a cooperative process; different approximations of ACh synchronous release were examined; stationary states and types of singular points were studied for the proposed models of ST; the nonlinear kinetic model with three order parameters demonstrated a strange-attractor behavior. cell-to-cell communication, synaptic transmission, phase transition, binary liquid mixture, critical mixing point, finite-size (confined) systems 82.40.Ck, 87.10.Ed'
- '=3000У цьому огляді була розроблена і обговорена теорія синаптичної передачі (СП). Ми застосували гіпотезу ізоморфізму між: (а) кооперативною поведінкою медіаторів — молекул ацетилхоліну (ACh) і холінорецепторів в синаптичній щілині з утворенням медіатор-рецепторних (AChR) комплексів, (б) критичними явищами в просторово обмежених бінарних рідких сумішах. Були запропоновані системи двох (або трьох) нелінійних диференціальних рівнянь для опису зміни концентрацій ACh, комплексів AChR і ферменту ацетилхолінестерази. Основні результати нашого дослідження: оцінено лінійний розмір зони активації; описано процес активації постсинаптичної мембрани як кооперативний процес; розглянуто різні апроксимаційні функції синхронного вивільнення ACh; досліджено стаціонарні стани і типи особливих точок для запропонованих моделей СП; показано, що нелінійна кінетична модель з трьома параметрами порядку демонструє поведінку дивного атрактора. міжклітинна взаємодія, синаптична передача, фазовий перехід, рідка бінарна суміш, критична точка змішування-розшарування, просторово обмежені системи'
author:
- 'A.V. Chalyi[^1], A.N. Vasilev[^2], E.V. Zaitseva'
- 'О.В. Чалий, О.М. Васильєв, О.В. Зайцева'
date: 'Received January 29, 2017, in final form March 12, 2017'
title:
- Synaptic transmission as a cooperative phenomenon in confined systems
- Синаптична передача як кооперативне явище в просторово обмежених системах
---
“Biophysics is a Tower of Babel”
P.G. de Gennes
The Nobel Laureate in Physics (1991)
Introduction
============
The problems of evolution of living (organic) and non-living (inorganic) nature still attract a great interest and agitate inquisitive human mind. Among these problems, synaptic transmission (ST) and cell-to-cell communication play an absolutely outstanding role. Actually, cell-to-cell communication is of the same fundamental importance in animate nature as intermolecular interaction in non-living nature. It makes it possible to understand the basic principles of a decisive problem, i.e., how our brain works and, in particular, how our nonlinear brain system organizes its thinking function [@1; @2; @3; @4].
The understanding of the processes of cell origin (their typical sizes are about $10^{-6}$ m) came from the study of evolution processes in the living nature at distances intermediate relative to the farthest boundary of the Universe, i.e., more than 10 billion light years or $10^{25}$ m, and at the smallest distances inside the atomic nucleus, i.e., less than $10^{-15}$ m. The cells emerged due to the creation of plasmatic membranes, i.e., the process concerned with the so-called self-assembly of amphiphilic molecules of phospholipids in aquatic environment, which was absolutely determined (see appendix \[sec:Apb\]).
Generation and propagation of electric potentials is the most important phenomenon in living cells and tissues based on excitation of cells, regulation of intracellular processes, muscular contraction, and nerve system functioning. The origin of electric membrane potentials of two types (in particular, the *resting potential* and the *action potential*) may be explained by a specific feature of distribution and diffusion of ions considering their permeability through a membrane [@5; @6; @7; @8; @9; @10].
In 1949–1952 Hodgkin and Huxley [@1] experimentally and theoretically studied the generation and propagation of the action potential. In 1963 they were awarded, together with Eccles, a Nobel Prize in medicine or physiology “…for their discoveries concerning the ionic mechanisms involved in excitation and inhibition in the peripheric and central portions of the cell membrane”.
The problems of ST and cell-to-cell communication have actually two important aspects: *electrical* and *chemical*. The electrical intercellular interaction aspect is associated with the process of the action potential propagation along the nerve fibers (axons). The chemical aspect is connected with chemical reactions between neurotransmitters and receptors in synaptic clefts (SC), or synapses separating dendrites of one neuron from other neurons, and in neuro-muscular junctions [@1; @2; @3; @4].
It is known that the body (soma) of a nervous cell (neuron), similarly to any other cells, consists of mitochondria, ribosomes and many other organelles: nucleus with its genetic information; dendrites emitting synaptic information from their own soma and receiving this information from other nervous cells; axons transmitting information to dendrites of other neurons.
Consider some important characteristics and properties of the cell-to-cell communication and the ST [@1; @2; @3; @4; @5; @6; @7; @8; @9; @10; @11]: 1) size of a human nervous cell (neuron) — from 10 to 100 m (the largest size is typical of Betz cells — giant pyramidal cells discovered in 1874 by a Ukrainian anatomist Volodymyr Betz, who worked at the medical faculty of St. Volodymyr Kyiv University [@11]); 2) size of SC (thickness of synapse) — from 10 to 200 nm; 3) diameter of ion channels in the membrane — up to 10 nm; 4) axon length — up to 1 m; width (diameter) of axons — up to $0.5{-}1$ mm for a giant squid axon; 5) the number of neurons in a human brain — about $10^{11}$ (a more precise number is 86 billions); 6) the number of contacts (synapses) between dendrites of one neuron with the other neurons — up to $1.5 \cdot 10^4$; 7) the number of synapses in a human brain — about $10^{14}{-}10^{15}$; 8) time of a signal exchange — about $10^{-3}$ s; 9) the number of channels in the membrane surface — about $10^2{-}10^3$ per m$^2$; 10) the number of impulses generated by a neuron per 1 second — from 1 to $10^3$; 11) the number of neurotransmitters, say, acetylcholine (ACh) molecules contained in 1 vesicle — about $10^4$; 12) the number of ACh molecules released simultaneously from the axonal presynaptic terminal with approximately $10^3$ vesicles, under the action of one nerve impulse (action potential) — about $10^7$.
In this contribution, we propose a new theoretical approach to the problem of ST, based on the nonlinear kinetic models and the hypothesis of isomorphism between (a) the cooperative behavior of a neurotransmitter-receptor system in the synaptic cleft (SC), (b) the critical phenomena in confined binary mixture near its critical mixing point. This hypothesis was originally proposed in 1989 and was published in [@12]. Further, it was developed in the lecture “Critical phenomena in finite-size systems and synaptic transmission”, presented in 1991 at the ARW NATO “Dynamic phenomena at interfaces, surfaces and membranes” and was published in the Les Houches Series in [@13]. The preface to this book was written by the Nobel Laureate P.G. de Gennes (1991), with its first phrase being used here as an epigraph.
As an application of the above-mentioned hypothesis to cholinergic synapses, the systems of two (or three) nonlinear differential equations were proposed to examine the change of order parameters such as concentrations of ACh, acetylcholine-receptor (R) complexes (AChR), and acetylcholinesterase (AChE), being a ferment which removes ACh from the R-location area. The main findings of our study were as follows: the linear size of the activation zone was evaluated as the correlation length (CL) for a mediator-receptor mixture (section \[sec2\]); the process of the postsynaptic membrane activation was described as a cooperative process (section \[sec3\]); the nonlinear kinetic model of ST with three order parameters ACh, AChR and AChE demonstrated a strange-attractor behavior, different approximations for the function of ACh synchronous release were examined, the stationary states and types of singular points were studied for the proposed models of ST (section \[sec4\]).
Model of ST as phase transition in “transmitter-receptor” system {#sec2}
================================================================
The chemical reactions near the bifurcation (critical) points are of a great importance for the ST mechanisms of nerve impulses. The phenomena occurring near the points (lines) of phase transitions or boundaries of stability demonstrate a universal behavior in physical, chemical, biological and other systems [@14; @15; @16; @17; @18; @19; @20; @21; @22; @23; @24; @25; @26; @27; @28; @34; @35; @36; @37]. The reason is related to cooperative (synergetic) nature of such events, defined by interaction of fluctuations of the characteristic order parameters, their correlations at large spatial and temporal intervals.
To understand this reason more deeply, let us consider the analogy between the equations of motion for 1) systems described by the fluctuation Hamiltonians with two interacting order parameters [@14; @15; @16; @17; @18; @19; @20; @21; @22; @23; @26; @27; @28], and 2) nonlinear reaction-diffusion kinetic models [@6; @13; @19; @29; @30; @31; @32]. To study such systems undergoing phase transitions and critical phenomena, one usually uses the fluctuation Hamiltonian [@14; @15; @16; @17; @18; @19] $$H=\sum_{i}H_{\text{LG}}[\varphi_i]+H_{\text{int}}[\varphi_1,\varphi_2].
\label{2.1}$$ Here, $$H_{\text{LG}}[\varphi_i]=\int\left[\frac{1}{2}a_i\varphi_i^2+\frac{1}{4}b_i\varphi_i^4+\frac{1}{2}c_i(\pmb\nabla\varphi_i)^2\right]\rd V
\label{2.2}$$ is the Landau-Ginzburg Hamiltonian, while the second term in (\[2.1\]) corresponds to the Hamiltonian of interaction between two order parameters $\varphi_i^1$ and $\varphi_i^2$. Taking into account Hamiltonians (\[2.1\]) and (\[2.2\]), one can write kinetic equations, or equations of motion for order parameters $$\frac {\partial{\varphi_i}}{\partial {t}} = -\Gamma_i \frac{\delta H}{\delta \varphi_i}=-\Gamma_i\left(a_i\varphi+b_i\varphi_i^3-c_i\Delta\varphi_i+\frac{\delta H_{\text{int}}}{\delta \varphi_i}\right),
\label{2.3}$$ where $\Gamma_i$ are the Onsager coefficients. Actually, such equations (\[2.3\]) resemble the kinetic equations of reaction-diffusion kinetic models containing both the diffusion term, being proportional to $\Delta \varphi_i$ with the diffusion coefficient $D_i=\Gamma_i c_i$, and the non-linearity, connected with the $\varphi_i^3$ and corresponding nonlinear terms in the functional derivative $\delta H_{\text{int}}/\delta \varphi_i$.
It is known that in the vicinity of the critical state, Hamiltonian-like systems demonstrate a universal behavior in the sense that their characteristics do not depend on the particular form of a short-range potential of interparticle interaction. And in the critical state, these systems demonstrate long-range effects that also do not depend on the particular form of interpartical potential. Concerning the ST problem, our principal position, which is based on experimental data, is that the main characteristics of the system do not qualitatively depend on the parameters of the local interaction, and the system demonstrates a cooperative long-range behavior at the regime of its ordinal functioning that could be interpreted as a critical-like state (see additional considerations in the end of section \[sec3\]).
Convenient theories of cell-to-cell communication and ST are commonly based on the ideas regarding the chemical intermediates (transmitter agents) securing interaction between two neurons in the synaptic cleft or between the motor neuron and muscle fibre in the neuro-muscular junction.
Here, we shall consider the cholinergic synapse and the corresponding neurotransmitter — ACh. The sequence of major events in cholinergic synapse are as follows: ACh is synthesized by cholineacetyltransferase and stored in spheroid vesicles in the presynaptic membrane, then ACh molecules are released and, after their diffusion through the synaptic cleft, they react with R — specific ACh receptors. The binding ACh and R$^*$ into ACh–R$^*$ complexes produces conformal changes in the postsynaptic membrane and, therefore, the change in a membrane electric potential. Finally, ACh is either inactivated by AChE or is removed by diffusion.
Such a scheme of chemical reactions corresponds to the processes in the cholinergic synapse: $$\text{ACh} + \text{R} \stackrel{k_1}{\Longleftrightarrow} \text{ACh}-\text{R}^* \stackrel{k_3}{\Longrightarrow} \text{P} + \text{R}^* \stackrel{k_2}{\Longrightarrow} \text{R}\,.
\label{2.4}$$ Here, R and R$^*$ are receptors in non-active and active states, respectively; P is a product of ACh destruction by the action of AChE.
The kinetic equations describing, in accordance with (\[2.4\]), the temporal evolution of the receptor concentration R$^*$ in the active state (variable $x$) and the concentration of ACh–R$^*$ complexes (variable $y$), can be written in the following form [@12]: $$\frac{\rd x}{\rd t} = k_{3}y - k_{2}x,
\label{2.5}$$ $$\frac{\rd y}{\rd t} = k_{1} [\text{ACh}] ([\text{R}]_{0} - y) - (k_{1} + k_{3}) y,
\label{2.6}$$ where \[ACh\] is concentration of ACh molecules, \[R\]$_{0}$ is initial concentration of receptors in non-active state, $k_{i}$ are coefficients of reactions velocities.
It must be stressed that the process of ACh release from vesicles in the presynaptic membrane into the synaptic cleft is a cooperative phenomenon: about $10^{7}$ ACh molecules are simultaneously released under the action of one nerve impulse (spike). Such a synchronous activation of a large zone of receptors by ACh molecules can be treated in detail as the process which is isomorphic to the critical phenomena in finite-size binary liquid mixtures near the critical mixing point [@13; @14; @19; @20].
Here, we begin to study the ST process starting with the simplest model [@12] with two order parameters and two kinetic equations (\[2.5\]), (\[2.6\]) and then we continue to correspondingly examine more complicated models with three order parameters (and kinetic equations) in section \[sec4\]. It is worthy to mention that the 3rd order parameter associated with the concentration of ferment AChE appears at the last stage of ST process when ACh is inactivated by AChE. Actually, the hypothesis of isomorphism corresponds only to the models of ST process in sections \[sec2\], \[sec3\]. This problem was investigated in detail in chapter 5 of [@29] and was explicitly examined in [@30; @31; @32].
***Correlation function and correlation length in ACh–R$^*$ system in SC.*** The expression for the pair correlation function (CF) $G_{2} (r)$ of the Ornstein-Zernike approximation $G_{2} (r) = A \exp (- r / \xi ) / r$, is valid for 3-dimensional systems with zero boundary conditions and system linear dimension $L_{0} \gg \xi$, where $\xi$ is the correlation length of order parameter fluctuations. As it is known, the concentration is an order parameter of binary liquid mixture near the critical mixing point (see, e.g., [@19; @27]). The CF $G_{2} (r) \rightarrow 0$ if $r = |{r_{1}-r_{2}| \rightarrow \infty} $. The anomalous growth of CL $\xi$ and the long-range behavior of the CF $G_{2} (r) \sim r^{-1}$ takes place at the phase transition or critical points only for spatially infinite systems.
To study the correlation properties of ACh–R$^*$ system, a fundamentally important point is the fact that the SC is a finite-size (confined) system. For obvious reasons, it is clear that the CL $\xi$ in the synaptic cleft cannot exceed the linear size $L_{0}$ of a system. Therefore, an actual problem is to find the correlation properties \[CF $G_2 (r)$ and CL $\xi$\] of ACh–R$^*$ system inside the spatially confined synaptic cleft.
Such a problem was first solved for a plane-parallel geometry of the synaptic cleft in [@16] and then in detail in [@28; @29; @30; @31; @32; @33; @34]. The main contribution to the CF is given by the following expression: $$G_{2} (\rho, z) = \frac{1}{2 \pi L_0} K_0 \left[\rho \left(\kappa ^2 + \frac{n^2 \pi ^2}{4 L_0 ^2} \right)^{1/2}\right] \cos\left(\frac{\pi z}{2 L_0}\right),
\label{2.7}$$ where $\rho=(x^2 +y^2)^{1/2}$, $K_0(u)$ is the cylindrical Macdonald function, and $\kappa=\xi^{-1}$ is the reverse CL of spatially infinite systems. Correlation function (\[2.7\]) demonstrates an oscillatory behavior in the $z$ direction of spatial limitation for plane-parallel layers confirming the theoretical results and computer-simulation studies (see e.g. [@35]) for the radial distribution function $g(r)$ in liquids in restricted geometry.
As is seen from (\[2.7\]), the pair correlation function $G_2(\rho, z)$ of ACh–R$^*$ system in confined geometry has a non-exponential shape. Therefore, it is natural to determine CL $\xi$ of the order parameter fluctuations in such bounded liquids according to the following relation: $$\xi=\sqrt{M_2}\,, \qquad M_2=\frac{\int G_2(r)r^2 \rd r}{\int G_2(r) \rd r}\,,$$ where $M_2$ is the second normalized spatial moment of the pair correlation function.
In the SC with a plane-parallel geometry with thickness $H=2L_0$, taking into account the formula (\[2.7\]) for the CF $G_2(r)$, expression for the Macdonald function $K_{\nu}(u)$ and gamma-function $\Gamma(u)$ [@30] $\int_{0}^{\infty} x^{\mu} K_{\nu}(ax)\rd x = 2^{\mu - 1} a^{- \mu - 1} \Gamma \big((1+\mu+\nu)/2\big) \Gamma \big((1+\mu-\nu)/2\big)$, one can get such a formula for the CL $\xi$ of concentration fluctuations in the SC: $$\xi=\left[\left(\kappa ^2 + \frac{\pi ^2}{4 L_0 ^2}\right)^{-1}+\left(1-\frac{8}{\pi ^2}\right)\left( \frac{L_0}{2}\right) ^2\right]^{1/2}.$$ An anisotropic behavior demonstrated by CL $\xi$ is determined by two contributions $\xi = (\xi_{xy}^2 + \xi_{z}^2 )^{1/2}$, where $\xi_{xy} =1/(\kappa^2 + \pi ^2 / 4 L_0 ^2)^{1/2} $ is the CL in the $xy$ plane and $\xi_z = (L_0/2) (1 - 8/ \pi^2)^{1/2}$ is the CL in the $z$ direction, respectively, parallel (perpendicular) to presynaptic (postsynaptic) membrane. The analysis of the expression for $\xi$ shows that the maximum value of the CL in the $xy$ plane equals $(\xi_{xy})_{\text{max}}= 4L_0 /\pi$ in the mixing point where $\kappa=0$. Usually, the thickness of chemical synapses is about several dozens of nanometers, so we may assume for definiteness that the membrane thickness $H = 2L_0 \approx 20$ nm. Thus, in the case of chemical synapses, the maximum value of the CL in the $xy$ plane equals: $(\xi_{xy})_{\text{max}} \approx 6.4$ nm, while the CL in $z$ direction equals: $\xi_{z} \approx 2.2$ nm. In a neuro-muscular junction, the thickness $H$ is higher, reaching 100 nm, i.e., $L_0\approx 50$ nm. In this case, taking the width of the neuro-muscular junction $2L_0\approx 10^{-7}$ m, one has a maximum value of the CL $\xi_{xy}$: $(\xi_{xy})_{\text{max}} \approx 32$ nm, while the CL $\xi_ z \approx 11$ nm.
It is known that at the end plate of the postsynaptic membrane, about several million receptor molecules are placed with a high packing at a density which is equal approximately to $\rho_{\text{surf.\,density}} \approx 10^4$ molecular receptor per 1 m$^2$. It makes it possible to evaluate the number $N_{\text{ACh--}\text{R}^*}$ of correlated ACh–R$^*$ complexes in the postsynaptic membrane. For this purpose, let us find the surface area $S_{\text{act}} =\pi d_{\text{act}}^2 /4$ of the activation zone in the postsynaptic membrane of the chemical synapse with taking into account its maximum linear size — the diameter $d_{\text{act}}\approx (\xi_{xy})_{\text{max}} \approx 6.4$ nm. As a result, one has $S_{\text{act}}\approx \pi (\xi_{xy}^2)_{\text{max}}/4 \approx 32.2~\text{nm}^2 = 32.2\cdot 10^{-6}$ m$^2$. Then, the number $N_{\text{ACh--}\text{R}^*}$ of correlated receptors or ACh–R$^*$ complexes in the postsynaptic membrane of the chemical synapse equals the product of $\rho_{\text{surf.\,density}}$ and $S_{\text{act}}\,$, i.e., one has $N_{\text{ACh--}\text{R}^*} = \rho_{\text{surf.\,density}}\times S_{\text{act}} \approx 10^4$ receptors$/$m$^2 \times 32.2 \cdot 10^{-6}~$m$^2 \approx 32.2 \cdot 10^{-2}$ receptors. This result definitely shows that there are no correlated receptors in a chemical synapse. Thus, receptors of chemical synapses work independently, without any correlations with each other.
Another situation is realized in the neuro-muscular junctions. Really, due to a larger maximum value of the CL $(\xi_{xy} )_{\text{max}}\approx 32$ nm, the surface area $S_{\text{act}}$ of the activation zone in the neuro-muscular junction is 25 times larger and equals $S_{\text{act}}\approx \pi (\xi_{xy}^2)_{\text{max}}/4 \approx 805$ nm$^2 = 8.05\cdot 10^{-4}$ m$^2$. Then, the number $N_{\text{ACh--}\text{R}^*}$ of correlated complexes equals: $N_{\text{ACh--}\text{R}^*}= \rho_{\text{surf.\,density}}\times S_{\text{act}} \approx 10^4$ receptors$/$m$^ 2 \times 8.05\cdot 10^{-4}$ m$^2 \linebreak \approx 8$, i.e., approximately 8 receptors mutually correlate in the neuro-muscular junction.
Model of postsynaptic membrane activation as cooperative process {#sec3}
================================================================
Here, we consider the model which describes the process of the postsynaptic membrane activation as a cooperative process [@37; @38; @39; @40]. Our model accounts for the interaction of ACh with R on the postsynaptic membrane. Due to this interaction, the receptors on the postsynaptic membrane transfer to the activated state which means generating a signal by the postsynaptic membrane neuron. Basing on this model, we calculate the temporal evolution of the amount of the R$^*$ on the postsynaptic membrane. In particular, we analyze how the system responds to the ACh pulse and also how R upon the postsynaptic membrane interacts with the constant amount of ACh. The most notable result of our modelling is that the key factor which determines the activation process is the total amount of ACh injected into the cleft.
***Basic equation***. The key process in passing the signal through a SC is the biochemical reaction that activates the postsynaptic membrane. This provides a pulse for generating a signal at the adjacent neuron. In a simplified form, we can treat the activation process as the binding of ACh with the R on the postsynaptic membrane. Thus, within this approach, we have some R on the postsynaptic membrane that are ready to interact with ACh, and also we have some R$^*$ (it means that they have already interacted with ACh). If we denote $x$ to be the concentration of the R, and $y$ to be the concentration of ACh, then we can write the following equations to determine the dynamics of ACh and R: $$\frac{\rd x}{\rd t} = k_1(x_0 - x) - k_2 xy,$$ $$\frac{\rd y}{\rd t} = f(t) - k_2 xy.$$ Here, the $x_0$ stands for the total concentration of R on the postsynaptic membrane, the coefficients $k_1$ and $k_2$ are phenomenological parameters of the model. The function $f(t)$ determines the intensity of ACh injection into the SC. We also neglect here the diffusion processes within the synapse (but in general case, these effects should also be taken into account). The system must be supplemented with the initial conditions $x(0)=x_0$ and $y(0)=0$ which means that at the beginning, all the receptors are in the non-activated state and there is no ACh in the SC.
After making substitutions $t=k_1 x_0 t$, $x=x_0 u$ and $y=x_0 v$, we get the following system of differential equations: $$\frac{\rd u}{\rd t} = k(1 - u) - uv,
\label{3.3}$$ $$\frac{\rd v}{\rd t} = j(t) - uv,
\label{3.4}$$ where $k=k_2/(k_1 x_0)$, and $j(t)=f(tx_0^{-1}k_1^{-1})/(k_1x_0^2)$ is the dimensionless function that determines the intensity of the ACh injection into the SC. Initial conditions in this case are of the form $u(0)=1$ and $v(0)=0$.
The system of the nonlinear equations (\[3.3\])–(\[3.4\]) can be solved numerically if we know the function $j(t)$ (and the value of the parameter $k$). Next, we will consider some special situations.
***Acetylcholine in synaptic cleft.*** First of all, it is interesting to clear out what happens if ACh appears in the SC. We mean that the function $j(t)$ is zeroth, but the initial concentration $V$ of the ACh is nonzero, that is $v(0)=V$. In this case, the concentration of R decreases at first and then it goes up to the initial unit value, meanwhile the concentration of ACh goes from the initial value down to zero. The concentration of the R$^*$ changes opposite to the concentration of the R: at first, the concentration of the R$^*$ increases and then it goes down to the zeroth value. This behavior is totally expectable since the presence of ACh causes the interaction with the receptors, so the amount of R$^*$ increase. Due to the deactivation process, the amount of the R$^*$ decreases to the initial zeroth value.
=-1 More realistic situation is observed when some portion (i.e., an impulse) of ACh arrives at the postsynaptic membrane. This process can be modulated with the nonzero function $j(t)$ in equation (\[3.4\]). Unfortunately, we do not know the precise expression for the function $j(t)$. Nevertheless, theoretical and experimental data testify that this function (in the case when a single ACh impulse arrives) should be strictly localized in time. Thus, as the probe function we take the expression of the following form: $j(t)=A \exp [-a(t-t_0)^2]$, where the $A$ constant determines the “strength” of the ACh impulse, the $a$ parameter determines the localization of the impulse and the $t_0$ is the time of the impulse arrival. We will consider this case. In other words, we consider the $j(t)$ function and the initial condition for the amount of the ACh concentration being zero, so we take $v(0)=0$. Increasing the parameter $A$, leads to an increase of the R$^*$ amount. This is a natural effect since increasing the parameter $A$ (when the other parameters are fixed) means increasing the total amount of ACh which is injected into the SC. The amount of the R$^*$ also depends on the value of the parameter $a$. As we can see, increasing the parameter $a$ causes a decrease of the amount of the R$^*$ due to a decrease of the total amount of ACh injected into the cleft. Thus, we can expectedly state that varying the parameters $A$ and $a$ causes the change of the amount of the R$^*$. Nevertheless, it is notable that if $A \sim \sqrt{a}$, then changing the parameter $a$ changes the value of the parameter $A$, but the total amount of ACh injected into the cleft is still the same. Numerical calculations show that this synchronic change of the parameters $A$ and $a$ does not significantly affect the profile of the R$^*$ curve.
Figure \[fig:Fig01\] demonstrates the dependence of the concentration of the activated receptors on time. It is obvious that this behavior is universal in the sense that it does not qualitatively depend on the parameter $A = \sqrt{a}$ and, thus, it could be an argument for the initial preposition regarding the isomorphism hypothesis.
![The concentration of the activated receptors as the function of time. It is assumed that $k=1$, $t_0 = 2$, $A = \sqrt{a}$ and the solid line is for the value $a=10$, the dashed line is for the value $a=50$ and the dotted line is for the value $a=75$.[]{data-label="fig:Fig01"}](Fig01)
This feature can be important for analyzing the ST. Indeed, when studying ACh spreading from the presynaptic membrane through the SC to the postsynaptic membrane, it is important to know the peculiarities of the transportation process and the initial distribution of ACh in the SC. Thus, the question of stability is relative to the ACh distribution profile that could play a crucial role for the ST theory.
Models of ST with M- and N-receptors and effects of diffusion {#sec4}
=============================================================
Here, in this section, we shall study the kinetic models of ST with three order parameters and different types of receptors taken into account, specifically, (a) the muscarinergic (M) receptors and (b) the nicotinergic (N) receptors (see appendix \[sec:Apb\]).
***The 1st kinetic model of ST with 3 order parameters and M-receptors*** takes into account the function of ACh release depending on time, being approximated by using function $\cos (\Omega t)$, where parameter $\Omega$ characterizes a slight deviation of the release speed of ACh from the constant value in accordance with the experimental data [@42; @43; @44]. Since the process of release of ACh in the SC is quite fast, in order to take account of this fact and the gradual slowdown supply of ACh molecules in the SC, trigonometric function was expanded in Taylor series preserving the quadratic term in (\[4.1\]). The corresponding system of differential equations is as follows: $$\frac{\rd x}{\rd t}=M\left(1-\frac{\Omega^2 t^2}{2}\right)-k_1 x(a-y)-k_2 xz- k_3 x,
\label{4.1}$$ $$\frac{\rd y}{\rd t}=k_1 x(a-y)-k_4 xyz,
\label{4.2}$$ $$\frac{\rd z}{\rd t}=k_5 z-k_2 xz-k_4 xyz.
\label{4.3}$$ The first term in (\[4.1\]) describes the speed of release of free ACh molecules into SC, the second term describes the rate of binding of ACh with R ($a$ — total number of M-receptors on the postsynaptic membrane), the third shows the rate of ACh molecules decay at the presence of AChE, the fourth shows the withdrawal of ACh from the synaptic cleft by diffusion and reuptake (return of mediator molecules back to the presynaptic membrane). The following processes are present in (\[4.2\]): the first term defines a positive contribution to the rate of change of the concentration of ACh–R$^*$ complexes by binding the ACh molecules with R, and the second term describes the speed of the decay of ACh–R$^*$ complexes at the presence of AChE. In (\[4.3\]), the first term describes the speed of release of free AChE at the decay of ACh–R$^*$ complexes, the second and third terms have the same meaning as similar terms in (\[4.1\]) and (\[4.2\]). The stationary system of equations (\[4.1\])–(\[4.3\]) has the following solutions:
a\) $x_0 = 0$, $y_0 = g$, $z_0 = 0$ ($g <a$). This solution is classified as a strange attractor, because the roots of the characteristic equation are as follows: $\lambda_1 = - \zeta$, $\lambda_2 = 0$, $\lambda_3 = k_5$, where $\zeta=k_1 a+k_2 g+k_3$ and $k_5$ is the rate of release of AChE. The emergence of this solution can be explained by the fact that the release function of ACh in the SC is considered as a short, small perturbation;
b\) $x_0 = 0$, $y_0 = 0$, $z_0 = h$, where $h$ is the concentration of the enzyme AChE. This solution satisfies system (\[4.1\])–(\[4.3\]) provided that $k_5 = 0$. Indeed, for long periods of time, all ACh molecules and ACh–R$^*$ complexes are split and the AChE release rate will be equal to zero. The roots of the corresponding characteristic equation are as follows: $\lambda_1 = - \zeta$, $\lambda_2 = 0$, $\lambda_3 = 0$. In this case, since $\lambda_1 < 0$, $\lambda_2 = 0$, $\lambda_3 > 0$, this singular point is classified as stable torus.
***The 2nd model with 3 order parameters and N-receptors*** was created to study the mechanism of ST, which involved N-cholinergic receptors (see appendix \[sec:Apb\]). Let us present the processes that occur in the synaptic cleft, with the following scheme of biochemical reactions: $$\text{ACh} + \text{R} \stackrel{k_1}{\Longleftrightarrow} \text{AChR},
\label{4.4}$$ $$\text{AChR} + \text{ACh} \stackrel{k_4}{\Longleftrightarrow} \text{ACh}_2 \text{R} \stackrel{k_4}{\Longleftrightarrow} \text{ACh}_2 \text{R}^* \stackrel{k_5}{\Longrightarrow} \text{R}.
\label{4.5}$$ In other words, at secretion of ACh molecule into the SC being bound to the R (\[4.4\]), this complex binds with another ACh molecule, then the mediator-receptor complex enters an excited state, and then it returns to the initial state (\[4.5\]). As in previous models, the reverse processes are not taken into account. Since the activation of N-cholinergic receptors requires two molecules of the ACh, there is an intermediate inactive ACh–R complex with one ACh molecule. To adequately describe such a system, one should increase the number of variables which leads to complications in kinetic equations and increases their numbers. To avoid this, we assume the concentration of the enzyme AChE to be constant. Thus, we will have the following three nonlinear differential equations for the concentration of ACh molecules ($A$), receptors in the unexcited state ($X_1$) and excited mediator-receptor complexes ($X_2$): $$\dot {A}= M\delta (t)- k_1 AX_1 - k_2 AE - k_3 A - k_4 A[X_0 - (X_1 +X_2)] + k_5 X_2+ k_6 [X_0 - (X_1 +X_2)],
\label{4.6}$$ $$\dot {X_1}= k_5 X_2 - k_1 AX_1 + k_6 [X_0 - (X_1 +X_2)],
\label{4.7}$$ $$\dot {X_2}= k_4 A[X_0 - (X_1 +X_2)] - k_5 X_2\,.
\label{4.8}$$ In the first equation, it is accepted that the concentration $E$ of the enzyme AChE is constant, and the release of ACh in the SC is described by Dirac $\delta$-function \[see the first term of (\[4.6\])\]. The second and fifth terms describe the binding of the ACh with free R and inactive neurotransmitter-receptor complex, respectively, the third term describes the hydrolysis of ACh in the presence of AChE, the fourth term shows the removal of ACh from the cleft by diffusion and reuptake, and the sixth and seventh terms show disconnection of ACh molecules from the active and inactive neurotransmitter-receptor complex, respectively. Terms, which are included in (\[4.7\]): the first and the third mean the same as the sixth and the seventh terms in (\[4.6\]), respectively, and a free ACh binding with R is described by the second term. In (\[4.8\]), the first term describes the transition of an inactive neurotransmitter-receptor complex in the excited state, and the second shows the collapse of neurotransmitter-receptor complex.
The stationary system of equations has a solution: $a=A/A_0=0$, $x_1=X_1/X_0=1$, $x_2=X_2/X_0=0$ ($A_0$ is the total concentration of ACh molecules at the time of release, $X_0$ sets the total concentration of cholinergic receptors in the SC), which corresponds to the real situation when there are no molecules of ACh in the SC, and no mediator-receptor complexes, and all receptors are free. This solution is classified as a stable node because numerical calculation of the roots of the characteristic equation with using experimental data [@42; @43; @44] gives the following result: $\lambda_1 \approx -5$ ms$^{-1}$, $\lambda_2 \approx -3082$ ms$^{-1}$, $\lambda_3 \approx -37$ ms$^{-1}$. The corresponding relaxation times are equal to: $\tau_1 \approx 0.2$ ms, $\tau_2 \approx 3 \cdot 10^{-4}$ ms, $\tau_3 \approx 2.7 \cdot 10^{-2}$ ms. Since these relaxation times differ quite significantly ($\tau_1 / \tau_2 \approx 670$, $\tau_1 / \tau_3 \approx 7.5$), the main contribution to the kinetics of the system approach to equilibrium gives the slowest relaxation process with time 0.2 ms.
***Kinetic models of ST with 3 order parameters and diffusion effects.*** Here, we have also considered non-linear kinetic models of synaptic transmission in the synapses with M-cholinergic receptors and N-cholinergic receptors, where the impact of diffusion processes was considered more consistently. For both models, it is believed that the process of synaptic transmission has the following common features:
1\) molecules of ACh are released in the space of SC very quickly, so the release of ACh function is approximated using the Dirac $ \delta$-function;
2\) ACh molecules then diffuse to the postsynaptic membrane, where they bind with the receptor molecules, which causes excitation of nerve impulses in the next cell;
3\) then, the mediator-receptor complexes decompose.
In the ***3rd kinetic model***, the processes occurring in the cleft of the synapse with M-cholinergic receptors are taken into account. It is assumed that one ACh molecule activates the receptor, and AChE participates in the destruction of neurotransmitter-receptor complexes. The corresponding system of nonlinear differential equations for concentrations of molecules of ACh ($x$), for the ACh–R$^*$ complexes ($y$) and for the enzyme AChE ($z$) is as follows: $$\dot {x}=K \delta (t)-k_1 x(a-y)-k_2 xz- k_3 x + D_1 \Delta x,
\label{4.9}$$ $$\dot {y}=k_1 x(a-y)-k_4 xyz,
\label{4.10}$$ $$\dot {z}=k_5 z-k_2 xz-k_4 xyz - D_3 \Delta z.
\label{4.11}$$ The first term on the right-hand side of the equation (\[4.9\]) describes the rate of ACh release in the SC, the latter shows the removal of neurotransmitter molecules from the cleft by diffusion. The first term in the third equation depending on the sign will have a double meaning: for $k_5 \geqslant 0$ — a rate of release of the molecules of the enzyme AChE; for $k_5 < 0$, it is a speed of binding of the enzyme AChE molecules with OPhC that suppresses the speed of release of enzyme molecules. The last term of equation (\[4.11\]) describes the movement of molecules of the enzyme in the region with high concentration of acetylcholine through lateral diffusion. All other terms in the equations (\[4.9\])–(\[4.11\]) have the same meaning as the corresponding equations in (\[4.1\])–(\[4.3\]).
Singular points of the system (\[4.9\])–(\[4.11\]):
\(a) $x_0 = 0$, $y_0 = g$, $z_0 = 0$, where $g$ is some concentration of R$^*$, it is clear that $g \leqslant a$;
\(b) $x_0 = m$, $y_0 = a$, $z_0 = 0$, $m$ is some free concentration of ACh;
\(c) $x_0 = 0$, $y_0 = g$, $z_0 = h$, $h$ is certain concentration of AChE.
Depending on the sign of the parameter $k_5$ that plays the role of a control parameter of the model, points (a), (b) and (c) have the Lyapunov classification (to assess their respective solutions of the characteristic equation, experimental data from the articles [@42; @43; @44] were used):
- for $k_5 \geqslant 0$ point (a) is classified as a strange attractor, point (b) may be a stable node and a saddle, and point (c) is classified as a strange attractor;
- for $k_5 < 0$ point (a) is classified as a limiting cycle, point (b) is classified as a stable node, and point (c) is classified as a limiting cycle.
It should be noted that lateral diffusion of molecules of the enzyme has less impact on the process of synaptic transmission than the diffusion of mediator molecules.
In the last ***4th model***, the influence of the diffusion of mediator molecules on the synaptic transmission was analyzed in the cleft with N-cholinergic receptors. Therefore, a proper system of nonlinear differential equations of the model is substantially similar to the system of equations (\[4.6\])–(\[4.8\]) of the second model, but differs from it by the presence of diffusion term $\Delta A$ in the first equation.
Stationary system of equations of the model has the same solution as the 2nd model: $a=0$, $x_1 = 1$, $x_2 = 0$, which corresponds to the real situation when there are no molecules of acetylcholine in the synaptic cleft, and no mediator-receptor complexes, and the receptors are all free.
An analysis of the solutions of characteristic equation shows that at any values of the diffusion coefficient $D$, a singular point is classified as a stable node (the numerical evaluation of experimental data was taken from [@42; @43; @44]) because the roots of the characteristic equation are equal to: $\lambda_1 \approx -14$ ms$^{-1}$, $\lambda_2 \approx -3717$ ms$^{-1}$, $\lambda_3 \approx -37$ ms$^{-1}$. The diffusion process does not change the type of a singular point, although it affects the relaxation time values: $\tau_1= 0.07$ ms, $\tau_2 = 2.6 \cdot 10^{-4}$ ms, $\tau_3 = 2.7 \cdot 10^{-2}$ ms. Having compared the estimates given in the relevant 2nd model, we see that at a consistent account of the processes of diffusion, the relaxation times $\tau_1$ and $\tau_2$ decreased approximately $2.8$ and $1.2$ times, while $\tau_3$ did not change.
Conclusions
===========
We have created and studied the non-linear kinetic models of ST and intercellular interaction based on actual processes that occur in the synapse of chemical type and isomorphism of coherent aspects of ST and critical phenomena in liquid systems with restricted geometry. The main results and conclusions can be summarized as follows.
1\. The linear size of the activation zone was evaluated as the CL $\xi$ for a binary mediator-receptor mixture. It was found that approximately 8 receptors mutually correlate in the neuro-muscular junction with its thickness of 100 nm.
2\. The process of the postsynaptic membrane activation was described as a cooperative process. The system of nonlinear equations for concentrations of R and ACh was numerically solved to find its temporal evolution in different specific conditions. The question of stability relative to the ACh distribution profile could play a crucial role for the ST theory.
3\. The analysis of ST models with three order parameters and diffusion effects taken into account showed that if the control parameter changed its sign from positive to negative, the type of singular points also changed — they all got stability, i.e., the bifurcation occurred: a strange attractor turned into the limiting cycle.
4\. Based on the experimental data and numerical calculations for kinetic models with N-receptors, the relaxation times of three order parameters were equal to: $\tau_1= 0.2$ ms, $\tau_2 = 3 \cdot 10^{-4}$ ms, $\tau_3 = 2.7 \linebreak\times 10^{-2}$ ms. Thus, the approach of the system to an equilibrium state was characterized by the slowest relaxation process with time $0.2$ ms. Diffusion processes did not change the types of singular points, but affected the relaxation times $\tau_1= 0.07$ ms, $\tau_2 = 2.6 \cdot 10^{-4}$ ms, while $\tau_3$ did not change.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors wish to thank Professor Yu.V. Holovatch, a famous scientist in the field of nanobiophysics, critical phenomena and physics of complex systems, one of the best representatives of the widely-known scientific school by Academician I.R. Yukhnovskii.
Abbreviations {#sec:Apa}
=============
ST — synaptic transmission; ACh — acytineholine; R — non-activated receptors; R$^*$ — activated receptors; SC — synaptic cleft; AChR — mediator-receptor complexes; AChE — acetylcholinesterase; CF — correlation function; CL — correlation length; N — nicotinergic (receptors); M — muscarinergic (receptors); OPhC — organophosphate compounds
Glossary {#sec:Apb}
========
*Amphiphilic* molecule possesses both hydrophilic and hydrophobic properties. This sort of molecules can form bilayers and micelles in water, so that their polar hydrophilic heads turn to the water molecules, while their hydrophobic tails hide inside the formation.
*Phospholipids* are a sort of amphiphilic molecules.
*Mediator* is a biologically active chemical substance transmitting the nerve impulse from one cell to another.
*Receptor* provides transformation of impacts of environment and inner mediuminto nerve impulse. There are a lot of types of receptors in the human body.
*Acetylcholine* is a neuromediator providing nerve-to-muscle transmission of the nerve impulse. Without going into biophysical details of intercellular interactions [@1; @2; @3; @4; @5; @6; @7; @8; @9; @10], let us briefly note that there are two types of ACh receptors: 1) only one mediator molecule is sufficient for activation receptors of the first type (so-called *muscarinergic receptors*, or *M-receptors*), while 2) activation of receptors of the second type (so-called *nicotinergic receptors*, or *N-receptors*) requires two mediator molecules. Here, we shall consider several nonlinear kinetic models with M- and N-receptors and the effects of diffusion [@40; @41].
*Acetylcholinesterase* is a hydrolytic ferment catalizing the hydrolysis of acetylcholine which lets the cell change to the rest state.
*Organophosphor compounds* inhibit the action of *acetylcholinesterase* which leads to the effects of strong poisoning and even to the death of a human body.
[99]{} Hodgkin A.L., Huxley A.F., J. Physiol., 1952, **117**, No. 4, 500–544; . Kostyuk P.G., Kryshtal O.A., Mechanisms of Electrical Excitability of Nervous System, Nauka, Moscow, 1981 (in Russian). Hubel D.H., In: The Brain, Flanagan D. (Ed.), Freeman W.H. and Company, San Francisco, 1979, 2–14. Nolte J., The Human Brain, Mosby, Missouri, 2002. Bolkenshtein M.V., Biophysics, Nauka, Moscow, 1982 (in Russian). Romanovskii Yu.M., Stepanova N.V., Chernavskii D.S., Mathematical Biophysics, Nauka, Moscow, 1984 (in Russian). Chalyi A.V., Non-Equilibrium Processes in Physics and Biology, Naukova Dumka, Kyiv, 1997 (in Russian). Kostyuk P.G., Zyma V.L., Mahura I.S., Miroshnychenko M.S., Shuba M.F., Biophysics, Oberegy, Kyiv, 2001 (in Ukrainian). Medical Physics. Dynamic and Statistical Models, Bulavin L.A. (Ed.), Publishing Center “Kyiv University”, Kyiv, 2011 (in Ukrainian). Medical and Biological Physics, Chalyi A.V. (Ed.), Nova Knyga, Vinnytsia, 2013 (in Ukrainian). Betz W., Centralblatt für die Medizinischen Wissenschaften, 1874, **12**, 595 (in German). Chalyi A.V., Chernenko L.M., In: Fundamental and Applied Aspects of mm-Band Electromagnetic Radiation in Biology and Medicine, Sitko S.P. (Ed.), Scientific Research Center “Vidguk”, Kyiv, 1989 (in Russian). Chalyi A.V., Chernenko L.M., In: Dynamical Phenomena at Interfaces, Surfaces and Membranes, Beysens D., Boccara N., Forgacs G. (Eds.), Nova Science Publishers, New York, 1993, 457–464. Fisher M.E., In: Critical Phenomena, Green M.S. (Ed.), Academic Press, New York, 1971, 1–99. Stanley H.E., Introduction to Phase Transitions and Critical Phenomena, Glarendon Press, Oxford, 1971. Lakoza E.L., Sysoev V.M., Chalyi A.V., Zh. Eksp. Teor. Fiz., 1973, **65**, 605 (in Russian). Patashinskii A.Z., Pokrovskii V.L., The Fluctuation Theory of Phase Transitions, Pergamon Press, Oxford, 1979. Yukhnovskii I.R., Phase Transitions of the Second Order. Method of Collective Variables, Naukova Dumka, Kyiv, 1983 (in Ukrainian). Anisimov M.A., Critical Phenomena in Liquids and Liquid Crystals, Gordon and Breach Science Publishers, Philadelphia, 1991. Binder K., Annu. Rev. Phys. Chem., 1992, **43**, 33; . Chalyi A.V., Sysoev V.M., Chugaev V.I., Ukr. Fiz. Zh., 1989, **9**, 1425 (in Russian). Yukhnovskii I.R., Kozlovskii M.P., Pylyuk I.V., Microscopic Theory of Phase Transitions in 3-Dimensional Systems, Evrosvit, Lviv, 2001 (in Ukrainian). Folk R., Holovatch Yu., Yavorskii T., Usp. Fiz. Nauk, 2003, **173**, 175 (in Russian);\
\[Phys. Usp., 2003, **46**, No. 2, 169;\
\]. Strogatz S.H., Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, 2nd Edn., Westview Press, Boulder, 2014. Solé R.V., Phase Transitions, Princeton University Press, Princeton, 2011. Order, Disorder and Criticality. Advanced Problems of Phase Transition Theory Vol. 4, Holovatch Yu. (Ed.), World Scientific, Singapore, 2015. Chalyi A.V., Lebed A.V., Non-Homogeneous Liquids near the Critical Point and the Boundary of Stability and Theory of Percolation in Ceramics, Harwood Academic Publishers, London, 1993. Chalyi A.V., Synergetic Principles of Education and Science, Vipol, Kyiv, 2000 (in Ukrainian). Chalyi A.V., Vasilev A.N., J. Mol. Liq., 2000, **84**, 203; . Antonchenko V.Y., Davydov A.S., Ilyin V.V., Basic Principles of Water Physics, Naukova Dumka, Kyiv, 1991 (in Ukrainian). Gradshtein I.S., Ryzhyk I.M., Tables of Integrals, Sums, Series and Products, Nauka, Moscow, 1971 (in Russian). Chalyi A.V., Vasilev A.N., Physics of the Alive, 2000, **10**, 32. Chalyi A.V., Tsekhmister Ya.V., Chalyy K.A., Processes of Self-Regulation and Self-Organization in Fluctuational Models of Open Systems, Bogomolets National Medical University, Kyiv, 2001 (in Ukrainian). Chalyi A.V., Tsekhmister Ya.V., Ukr. Fiz. Zh., 1991, **36**, No. 3, 1271 (in Russian). Chalyi A.V., Tsekhmister Ya.V., Ukr. Fiz. Zh., 1991, **37**, No. 3, 457 (in Russian). Chalyi A.V., Tsekhmister Ya.V., Khizhnyak D.A., Ukr. J. Phys., 1996, **41**, No. 5–6, 627 (in Ukrainian). Chalyi A.V., J. Mol. Liq., 1993, **58**, 179; . Vasilev A.N., Chalyi A.V., Biophysics, 2010, **55**, 600; . Vasilev A.N., Chalyi A.V., Ukr. J. Phys., 2009, **54**, 1183. Chalyi A.V., Bulavin L.A., Chalyy K.A., Chernenko L.M., Vasilev A.N., Zaitseva E.V., In: Soft matter under exogenic impacts, Rzoska S.J., Mazur V.A. (Eds.), Springer, Dordrecht, 2007, 399–412; . Nigmatullin N.R., Snetkov V.A., Nikol’skii E.E., Magazanik L.G., Neurophysiology, 1988, **20**, No. 3, 289;\
. Skorinkin A.I., Shaikhutdinova A.R., Biophysics, 2004, **49**, 872 (in Russian). Gubskyi Yu.I., Biological Chemistry, Ukrmedknyga, Kyiv-Ternopil, 2000 (in Ukrainian). Chalyi A.V., Zaitseva E.V., J. Phys. Stud., 2007, **11**, 322.
[^1]: E-mail: avchal@nmu.ua
[^2]: E-mail: vasilev@univ.kiev.ua
|
---
abstract: 'The accelerating expansion of our universe at present could be driven by an unknown energy component (Dark Energy) or a modification of general relativity (Modified Gravity). In this note we revisit the constraints on a phenomenological model which interpolates between the pure $\Lambda$CDM model and the Dvali-Gabadadze-Porrati (DGP) braneworld model with an additional parameter $\alpha$. Combining the cosmic microwave background (CMB), baryon acoustic oscillations (BAO) and type Ia supernovae (SNIa), as well as some high-redshift observations, such as the gamma-ray bursts (GRB) and the measurements of linear growth factors (LGF), we obtain the tight constraint on the parameter $\alpha=0.254\pm0.153$ ($68\%$ C.L.), which implies that the flat DGP model is incompatible with the current observations, while the pure $\Lambda$CDM model still fits the data very well. Finally, we simulate the future measurements with higher precisions and find that the constraint on $\alpha$ can be improved by a factor two, when compared to the present constraints.'
author:
- 'Jun-Qing Xia'
title: Constraining DGP Gravity from Observational Data
---
Introduction
============
Current cosmological observations, such as the CMB measurements of temperature anisotropies and polarization at high redshift $z\sim1090$ [@Komatsu:2008hk] and the redshift-distance measurements of SNIa at $z<2$ [@Kowalski:2008ez], have demonstrated that the universe is now undergoing an accelerated phase of expansion. The simplest explanation is that this behavior is driven by the cosmological constant or the dynamical dark energy models, which suffers from the severe coincidence and fine-tuning problems [@CCproblem]. On the other hand, this observed late-time acceleration of the expansion on the large scales could also be caused by some modifications of general relativity, such as the scalar-tensor [@ST] and $f(R)$ theories [@fR], and gravitational slip [@slip].
One of the well-known examples is the DGP braneworld model [@DGP], in which the gravity leaks off the four dimensional brane into the five dimensional space-time. On small scales gravity is bound to the four dimensional brane and the general relativity is recovered to a good approximation. In the framework of flat DGP model, the Friedmann equation will be modified as [@DGPCosmology]: $$H^2-\frac{H}{r_c}=\frac{8\pi{G}}{3}\rho_m~,$$ where $r_c=(H_0(1-\Omega_m))^{-1}$ is the crossover scale. At early times, $Hr_c\gg{1}$, the Friedmann equation of general relativity is recovered, while in the future, $H\rightarrow{H_{\infty}}=1/r_c$, the expansion is asymptotically de Sitter. Recently there have been a lot of interests in the phenomenological studies relevant to the DGP model in the literature [@Lue; @DGPRev].
In this note we investigate an interesting phenomenological model, first introduced in Ref.[@mDGP], which interpolates between the pure $\Lambda$CDM model and the DGP model with an additional parameter $\alpha$ and presents the tight constraints from the current observations and future measurements. The paper is organized as follows: In Sec. II we describe the general formalism of the modified gravity model. Sec. III contains the current observations we use, and Sec. IV includes our main global fitting results. In Sec. V we present the forecasts from the future measurements, while Sec. VI is dedicated the conclusions.
General Formalism
=================
In this phenomenological model, assuming the flatness of our universe, the Friedmann equation is modified as [@mDGP]: $$H^2-\frac{H^\alpha}{r_c^{2-\alpha}}=\frac{8\pi{G}}{3}\rho_m~,\label{mDGPeq}$$ where $r_c=H^{-1}_0/(1-\Omega_m)^{\alpha-2}$. Thus, we can straightforwardly rewrite the above equation and obtain the expansion rate as following: $$E^2(z)\equiv\frac{H^2}{H^2_0}=\Omega_m(1+z)^3+\frac{\delta{H^2}}{H^2_0}~,\label{ez}$$ where the last term denotes the modification of the Friedmann equation of general relativity: $$\frac{\delta{H^2}}{H^2_0}\equiv(1-\Omega_m)\frac{H^{\alpha}}{H^{\alpha}_0}=(1-\Omega_m)E^\alpha(z)~.\label{deltah}$$ Furthermore, we can obtain the effective equation of state: $$w_{\rm
eff}(z)\equiv-1+\frac{1}{3}\frac{d\ln{\delta{H^2}}}{d\ln(1+z)}
=-1+\frac{\alpha}{3}(1+z)\frac{E'(z)}{E(z)}~,\label{weff}$$ where the prime denotes the derivative with respect to the redshift $z$.
In Fig.\[fig1\] we illustrate the evolutions of the effective energy density $\Omega_\alpha(z)\equiv{1-\Omega_m(z)}$ and $w_{\rm
eff}(z)$ for different values of parameter $\alpha$. During the matter dominated era, $E(z)$ varies as $(1+z)^{3/2}$, which corresponds to the effective equation of state: $w_{\rm
eff}=-1+\alpha/2$. In the future $z\rightarrow0$, with the matter density $\rho_m\propto{(1+z)^3}\rightarrow0$, we have $w_{\rm
eff}(z)\rightarrow-1$ and $\Omega_\alpha(z)\rightarrow1$.
![The evolutions of effective energy density $\Omega_\alpha(z)$ and effective equation of state $w_{\rm eff}(z)$ for different values of $\alpha$. The black solid lines are for $\alpha=0$ ($\Lambda$CDM), the red dashed lines for $\alpha=0.25$ and the blue dotted lines for $\alpha=1$ (DGP). And the vertical line denotes today ($z=0$).\[fig1\]](Fig1.eps)
Besides the modification of the Friedmann equation, the flat DGP model also changes the growth function of density perturbation $\delta(a)$. Under assumptions of a quasi-static regime and sub-horizon scales, the correct evolution of perturbation was found [@Lue; @Koyama1]: $$\ddot{\delta}+2H\dot{\delta}-4\pi{G}\left(1+\frac{1}{3\beta}\right)\rho_m\delta=0~,\label{groweq}$$ where the dot denotes the derivative with respect to the cosmic time $t$, and the $\beta$ factor is given by: $$\beta=1-2r_cH\left(1+\frac{\dot{H}}{3H^2}\right)~.\label{beta1}$$ However, this phenomenological model Eq.(\[mDGPeq\]) is a parametrization, so the situation is more complicated. One of the possible methods was found by Ref.[@Koyama2]. In order to obtain the growth function of density perturbation within a covariant theory, the authors introduced a correction term and assumed the structure of modified theory of gravity to determine this term. Based on those assumptions, it was consequently found that the $\beta$ factor was: $$\beta=1-\frac{2}{\alpha}(Hr_c)^{2-\alpha}\left(1+(2-\alpha)\frac{\dot{H}}{3H^2}\right)~.\label{beta}$$ In the following analysis, we will use Eq.(\[groweq\]) and Eq.(\[beta\]) to calculate the growth of density perturbation.
Defined the normalized growth $g(a)\equiv\delta(a)/a$, the growth function Eq.(\[groweq\]) can be rewritten as: $$\begin{aligned}
\frac{d^2g}{da^2}&+&\left(\frac{7}{2}-\frac{3}{2}\frac{w_{\rm
eff}(a)}{1+X(a)}\right)
\frac{dg}{ada}+\frac{3}{2}\left[1-\frac{w_{\rm eff}(a)}{1+X(a)}\right.\nonumber\\
&-&\left.\frac{X(a)}{1+X(a)}\left(1+\frac{1}{3\beta}\right)\right]\frac{g}{a^2}=0\end{aligned}$$ where the variable $X(a)$ is the ratio of the matter density to the effective energy density $X(a)=\Omega_m(a)/\Omega_\alpha(a)$. In Fig.\[fig2\] we plot the linear growth factor $g(a)$ as function of scale factor $a$ for different values of $\alpha$. One can see that the linear growth factor has been suppressed obviously as long as $\alpha$ is larger than zero. Thus, in the literature the linear growth has been widely used to study the modified gravity models, especially the DGP model [@DGPGrowth].
![The evolutions of linear growth $g(a)\equiv\delta(a)/a$ for different values of $\alpha$. The black solid lines are for $\alpha=0$ ($\Lambda$CDM), the red dashed lines for $\alpha=0.25$ and the blue dotted lines for $\alpha=1$ (DGP).\[fig2\]](Fig2.eps)
Furthermore, the growth factor can be parameterized as [@gamma]: $$f\equiv\frac{d\ln\delta}{d\ln{a}}=\Omega_m^\gamma~,$$ where $\gamma$ is the growth index. And then the growth function becomes: $$\begin{aligned}
\frac{df}{d\ln
a}&+&\left(\frac{1}{2}-\frac{3}{2}\frac{w_{\rm eff}(a)}{1+X(a)}\right)f+f^2\nonumber\\
&-&\frac{3}{2}\frac{X(a)}{1+X(a)}\left(1+\frac{1}
{3\beta}\right)=0~.\end{aligned}$$ For the pure $\Lambda$CDM model, the theoretical value of $\gamma$ is $6/11\approx0.545$, while $\gamma=11/16=0.6875$ in the flat DGP model [@gammavalue].
In the framework of this phenomenological model, we can easily see that the pure $\Lambda$CDM model and flat DGP model can be recovered when $\alpha=0$ and $\alpha=1$, respectively. In order to be consistent with the cosmological observations, the $\alpha$ term should be very small in the early times, such as the Big Bang Nucleosynthesis (BBN) era ($z\sim10^9$). This limit corresponds to the upper bound: $\alpha<2$ [@mDGP]. On the other hand, when $\alpha<0$, the effective equation of state will become smaller than $-1$, which leads to the instability of linear growth of density perturbation due to the appearance of $\beta$ term Eq.(\[beta\]).
![The evolution of the $\beta$ term $(3\beta)^{-1}$ when fixing $\alpha=-2$.\[fig3\]](Fig3.eps)
At early times, such as the matter dominated era, we have $E(z)\propto(1+z)^{3/2}$ and $\dot{H}/H^2\simeq-3/2$. Thus, Eq.(\[beta\]) becomes $$\beta\simeq1-\frac{(\Omega_m(1+z)^3)^{1-\alpha/2}}{1-\Omega_m}\ll0~,\label{beta+}$$ since $(1+z)^3\gg1$ at $2<z<1000$. By contrast, at late times the matter energy density $\rho_m\propto{(1+z)^3}\rightarrow0$ and the expansion is asymptotically de Sitter, $\dot{H}\rightarrow0$. And then we have $$\beta\simeq1-2/\alpha\Rightarrow\cases{\beta<0~,~~{\rm
for}~~0<\alpha<2\cr\beta>0~,~~{\rm for}~~\alpha<0\cr}~.\label{beta-}$$ Based on Eq.(\[beta+\]) and Eq.(\[beta-\]), we can straightforwardly see that as long as $\alpha<0$, during the evolution of universe the value of $\beta$ should change the sign at one pivot redshift $z_t$, which leads to $\beta|{}_{z_t}=0$ and $(3\beta)^{-1}|_{z_t}\rightarrow\infty$. In Fig.\[fig3\] we have shown the evolution of $(3\beta)^{-1}$ when fixing $\alpha=-2$. There is an obvious singularity at $a\sim0.595$. Therefore, based on these discussions above, we use a tophat prior on $\alpha$ as $0\leq\alpha<2$ in our calculations.
Method and Data
===============
In our calculations we assume a flat space and use a uniform prior on the present matter density fraction of the universe: $0.1<\Omega_m<0.5$. Furthermore, we constrain the Hubble parameter to be uniformly in $4\sigma$ Hubble Space Telescope (HST) region: $0.4<h<1.0$. The resulting plots are produced with CosmoloGUI[^1].
In this section we will list the cosmological observations used in our calculations: CMB, BAO and SNIa measurements, as well as some high-redshift observations, such as the GRB and LGF data. We have taken the total likelihood to be the products of the separate likelihoods ($\mathcal{L}_i$) of these cosmological probes. In other words, defining $\chi^2_{L,i}=-2\log{\mathcal{L}_i}$, we get: $$\begin{aligned}
\chi^2_{L,\rm total}&=&\chi^2_{L,\rm CMB}+\chi^2_{L,\rm
BAO}+\chi^2_{L,\rm SNIa}\nonumber\\
&+&\chi^2_{L,\rm GRB}+\chi^2_{L,\rm LGF}~.\end{aligned}$$ If the likelihood function is Gaussian, $\chi^2_{L}$ coincides with the usual definition of $\chi^2$ up to an additive constant corresponding to the logarithm of the normalization factor of $\mathcal{L}$.
CMB Data
--------
CMB measurement is sensitive to the distance to the decoupling epoch via the locations of peaks and troughs of the acoustic oscillations. Here we use the “WMAP distance information" obtained by the WMAP group [@Komatsu:2008hk], which includes the “shift parameter" $R$, the “acoustic scale" $l_A$, and the photon decoupling epoch $z_\ast$[^2]. $R$ and $l_A$ correspond to the ratio of angular diameter distance to the decoupling era over the Hubble horizon and the sound horizon at decoupling, respectively, given by: $$\begin{aligned}
R&=&\frac{\sqrt{\Omega_mH^2_0}}{c}\chi(z_\ast)~,\\
l_A&=&\frac{\pi\chi(z_\ast)}{\chi_s(z_\ast)}~,\end{aligned}$$ where $\chi(z_\ast)$ and $\chi_s(z_\ast)$ denote the comoving distance to $z_\ast$ and the comoving sound horizon at $z_\ast$, respectively. The decoupling epoch $z_\ast$ is given by [@zast]: $$z_\ast=1048[1+0.00124(\Omega_b h^2)^{-0.738}][1+g_1(\Omega_m
h^2)^{g_2}]~,$$ where $$g_1=\frac{0.0783(\Omega_b h^2)^{-0.238}}{1+39.5(\Omega_b
h^2)^{0.763}}~,~g_2=\frac{0.560}{1+21.1(\Omega_b h^2)^{1.81}}~.$$ We calculate the likelihood of the WMAP distance information as follows: $$\chi^2=(x^{\rm th}_i-x^{\rm data}_i)(C^{-1})_{ij}(x^{\rm
th}_j-x^{\rm data}_j)~,$$ where $x=(R,l_A,z_\ast)$ is the parameter vector and $(C^{-1})_{ij}$ is the inverse covariance matrix for the WMAP distance information shown in Table I.
Table I. Inverse covariance matrix for the WMAP distance information $l_A$, $R$ and $z_{\ast}$. The maximum likelihood values are $R=1.710$, $l_A=302.10$ and $z_\ast=1090.04$, respectively.
$l_A(z_{\ast})$ $R(z_{\ast})$ $z_{\ast}$
------------------- ----------------------- -------------------- --------------------
$l_A(z_{\ast})$ $1.800$ $27.968$ $-1.103$
$R(z_{\ast})$ $5667.577$ $-92.263$
$z_{\ast}$ $2.923$
BAO Data
--------
The BAO information has been already detected in the current galaxy redshift survey. The BAO can directly measure not only the angular diameter distance, $D_A(z)$, but also the expansion rate of the universe, $H(z)$. But current BAO data are not accurate enough for extracting the information of $D_A(z)$ and $H(z)$ separately [@Okumura:2007br]. Therefore, one can only determine the following effective distance [@Eisenstein:2005su]: $$D_v(z)\equiv\left[(1+z)^2D_A^2(z)\frac{cz}{H(z)}\right]^{1/3}~.$$ In this note we use the gaussian priors on the distance ratios $r_s(z_d)/D_v(z)$: $$\begin{aligned}
r_s(z_d)/D_v(z=0.20)&=&0.1980\pm0.0058~,\nonumber\\
r_s(z_d)/D_v(z=0.35)&=&0.1094\pm0.0033~,\end{aligned}$$ with a correlation coefficient of $0.39$, extracted from the SDSS and 2dFGRS surveys [@BAO], where $r_s(z_d)$ is the comoving sound horizon size and $z_d$ is the drag epoch at which baryons were released from photons given by [@zd]: $$z_d=\frac{1291(\Omega_m h^2)^{0.251}}{1+0.659(\Omega_m
h^2)^{0.828}}[1+b_1(\Omega_b h^2)^{b_2}]~,$$ where $$\begin{aligned}
b_1&=&0.313(\Omega_m h^2)^{-0.419}[1+0.607(\Omega_m
h^2)^{0.674}]~,\nonumber\\
b_2&=&0.238(\Omega_m h^2)^{0.223}~.\end{aligned}$$
SNIa Data
---------
The SNIa data give the luminosity distance as a function of redshift $$d_L=(1+z)\int^{z}_0\frac{cdz'}{H(z')}~.$$ The supernovae data we use in this paper are the recently released Union compilation (307 sample) from the Supernova Cosmology project [@Kowalski:2008ez], which include the recent samples of SNIa from the SNLS and ESSENCE survey, as well as some older data sets, and span the redshift range $0\lesssim{z}\lesssim1.55$. In the calculation of the likelihood from SNIa we have marginalized over the nuisance parameter, the absolute magnitude $M$, as done in Ref.[@SNMethod]: $$\bar{\chi}^2=A-\frac{B^2}{C}+\ln\left(\frac{C}{2\pi}\right)~,$$ where $$\begin{aligned}
A&=&\sum_i\frac{(\mu^{\rm data}-\mu^{\rm th})^2}{\sigma^2_i}~,\nonumber\\
B&=&\sum_i\frac{\mu^{\rm data}-\mu^{\rm th}}{\sigma^2_i}~,\nonumber\\
C&=&\sum_i\frac{1}{\sigma^2_i}~.\end{aligned}$$
GRB Data
--------
GRBs can potentially be used to measure the luminosity distance out to higher redshift than SNIa. Recently, several empirical correlations between GRB observables were reported, and these findings have triggered intensive studies on the possibility of using GRBs as cosmological “standard” candles. However, due to the lack of low-redshift long GRBs data to calibrate these relations, in a cosmology-independent way, the parameters of the reported correlations are given, assuming an input cosmology, and obviously they depend on the same cosmological parameters that we would like to constrain. Thus, applying such relations to constrain cosmological parameters leads to biased results. In Ref.[@xiagrb] the circular problem is naturally eliminated by marginalizing over the free parameters involved in the correlations; in addition, some results show that these correlations do not change significantly for a wide range of cosmological parameters [@Firmani]. Therefore, in this paper we use the 69 GRBs sample over a redshift range from $z=0.17-6.60$ published in Ref.[@Schaefer:2006pa] but we keep in mind the issues related to the “circular problem" that are more extensively discussed in Ref.[@xiagrb].
LGF Data
--------
As we point out above, the linear growth factor will be suppressed in the modified gravity model. It will be helpful using the measurements of linear growth factor to constrain the modified gravity models. Therefore, in Table II we list linear growth factors data we use in our analysis: the linear growth rate $f\equiv\Omega^\gamma_m$ from galaxy power spectrum at low redshifts [@Hawkins:2002sg; @Tegmark:2006az; @Ross:2006me; @Guzzo; @daAngela:2006mf] and lyman-$\alpha$ growth factor measurement obtained with the lyman-$\alpha$ power spectrum at $z=3$ [@McDonald:2004xn]. It is worth noting that the data points in Table II are obtained with assuming the $\Lambda$CDM model, thus, one should use these data very carefully, especially for the points obtained from Refs.[@Tegmark:2006az; @Ross:2006me; @daAngela:2006mf]. The corresponding $\chi^2$ is simply given by: $$\chi^2=\sum_i\frac{(f^{\rm th}_i-f^{\rm data}_i)^2}{\sigma^2_i}~.$$
TABLE II. The currently available data for linear growth rates $f$ we use in our analysis.
$z$ $f$ $\sigma$ Ref.
------------- ------------- ------------------ --------------------
0.15 0.51 0.11 [@Hawkins:2002sg]
0.35 0.70 0.18 [@Tegmark:2006az]
0.55 0.75 0.18 [@Ross:2006me]
0.77 0.91 0.36 [@Guzzo]
1.40 0.90 0.24 [@daAngela:2006mf]
3.00 1.46 0.29 [@McDonald:2004xn]
Numerical Results
=================
TABLE III. Constraints on the parameters $\alpha$, $\Omega_m$ and $\gamma$. Here we have shown the mean values and errors from the current observations and the standard derivations from the future measurements.
$\alpha$ $\Omega_m$ $\gamma$
--------------- ----------------- ----------------- -----------------
CMB+BAO+SN $0.263\pm0.175$ $0.276\pm0.018$ $-$
All Real Data $0.254\pm0.153$ $0.277\pm0.017$ $0.570\pm0.205$
Future 0.07 0.005 0.050
![One dimensional constraint on the parameter $\alpha$ from the different current data combinations: CMB+BAO+SN (blue dashed lines) and all real data (red solid lines). \[fig4\]](alpha.eps)
In this section we present our main results of constraints on this phenomenological model from the current observational data, as shown in Table III.
In Fig.\[fig4\] we illustrate the posterior distribution of $\alpha$ from the current data. Firstly, we neglect the high redshift probes. The result shows that the current observations yield a strong constraint on the parameter: $$\alpha=0.263\pm0.175~(1\sigma)~.\label{result1}$$ One can see that the pure $\Lambda$CDM model ($\alpha=0$) still fits data very well at $2\sigma$ uncertainty, which is consistent with the current status of global fitting results [@Komatsu:2008hk; @xiaglobal]. And the $95\%$ upper limit is $\alpha<0.686$, which implies that there is a significant tension between the flat DGP model ($\alpha=1$) and the current observations, which is consistent with other works (e.g. Ref.[@ariel; @otherDGP]). However, unlike other works [@ariel], in our analysis we use the “WMAP distance information" which includes the “shift parameter" $R$, the “acoustic scale" $l_A$, and the photon decoupling epoch $z_\ast$, instead of $R$ only, to constrain this phenomenological model. Recently, many results show that the “WMAP distance information" can give the similar constraints, when compared with the results from the full CMB power spectrum [@compare]. By contrast, $R$ could not be an accurate substitute for the full CMB data and may in principle give some misleading results [@shift].
And then, we include some high redshift probes, such as GRB and LGF data sets. From Table III and Fig.\[fig4\], we can find that the constraint on $\alpha$ becomes slightly tighter: $$\alpha=0.254\pm0.153~(1\sigma)~,\label{result2}$$ and $\alpha<0.541$ at $2\sigma$ confidence level. As we have mentioned before, the effective equation of state of this phenomenological model will depart from the cosmological constant boundary at high redshifts. Therefore, these high redshift observations are helpful to improve the constraints on this phenomenological model.
These results (Eqs.(\[result1\]-\[result2\])) are not surprising. From Fig.\[fig1\] we find that the effective equation of state of the flat DGP model, $w_{\rm
eff}\approx-1+\alpha/2=-0.5$, will depart from the cosmological constant $w=-1$ at high-redshift universe significantly. But the current constraint on $w$ is closed to $w=-1$ [@Komatsu:2008hk; @xiaglobal], so we require the small value of $\alpha$ to match the current observations. There is a small difference that our result slightly favors a non-zero value of $\alpha$, but not significantly, which needs more accurate measurements to verify it further.
In Fig.\[fig5\] we plot the two dimensional constraint in the ($\Omega_m$,$\alpha$) panel. $\Omega_m$ and $\alpha$ are strongly anti-correlated. The reason of this degeneracy is that the constraint mainly comes from the luminosity and angular diameter distance information. From Eq.(\[ez\]) and Eq.(\[deltah\]) we can see that when $\alpha$ is increased, the contribution of last $\alpha$ term to the expansion rate will become large, due to the positive $E(z)$. Consequently, $\Omega_m$ must be decreased correspondingly in order to produce the same expansion rate. When combining those current observational data, the matter energy density has been constrained very stringent: $\Omega_m=0.277\pm0.017~(1\sigma)$, which is also consistent with the current status of global fitting results [@Komatsu:2008hk; @xiaglobal]. Naturally, the constraint on $\alpha$ will also be improved, because of the tight constraint on the matter energy density.
![$68\%$ and $95\%$ constraints in the ($\Omega_m$,$\alpha$) plane from the current observations.\[fig5\]](omm-alpha.eps)
Furthermore, we also investigate the limit on the growth index $\gamma$ and obtain $\gamma=0.570\pm0.205$ at $68\%$ confidence level. Obviously, the growth index of the pure $\Lambda$CDM $\gamma=6/11\approx0.545$ is consistent with this result. However, the theoretical value of growth index in the flat DGP model, $\gamma=11/16=0.6875$, is disfavored.
Future Constraints
==================
Since the present data clearly do not give very stringent constraint on the parameter $\alpha$, it is worthwhile to discuss whether future data could determine $\alpha$ conclusively. For that purpose we have performed an analysis and chosen the fiducial model as the mean values of Table III obtained from the current constraints.
The projected satellite SNAP (Supernova / Acceleration Probe) would be a space based telescope with a one square degree field of view with $10^9$ pixels. It aims to increase the discovery rate for SNIa to about $2000$ per year in the redshift range $0.2<z<1.7$. In this paper we simulate about $2000$ SNIa according to the forecast distribution of the SNAP [@Kim:2003mq]. For the error, we follow the Ref.[@Kim:2003mq] which takes the magnitude dispersion $0.15$ and the systematic error $\sigma_{\rm sys}=0.02\times z/1.7$. The whole error for each data is given by: $$\sigma_{\rm mag}(z_i)=\sqrt{\sigma^2_{\rm
sys}(z_i)+\frac{0.15^2}{n_i}}~,\label{snap}$$ where $n_i$ is the number of supernovae of the $i'$th redshift bin. Furthermore, we add as an external data set a mock dataset of 400 GRBs, in the redshift range $0 < z < 6.4$ with an intrinsic dispersion in the distance modulus of $\sigma_{\mu}=0.16$ and with a redshift distribution very similar to that of Figure 1 of Ref.[@Hooper:2005xx].
![$68\%$ and $95\%$ constraints in the ($\Omega_m$,$\alpha$) plane from the future measurements.\[fig6\]](fut.eps)
For the linear growth factors data, we simulate the mock data from the fiducial model with the error bars reduced by a factor of two. This is probably reasonable given the larger amounts of galaxy power spectrum and lyman-$\alpha$ forest power spectrum data that will become available soon as long with a better control of systematic errors in the next generated large scale structure survey. In addition we also assume a Gaussian prior on the matter energy density $\Omega_m$ as $\sigma=0.007$, which is close to future Planck constraints [@Planck].
From Table III it is clear that the future measurements with higher precision could improve the constraints dramatically. The standard derivation of $\alpha$ is reduced by a factor two. Assuming the mean value remains unchanged in the future, the non-zero value of $\alpha$ will be confirmed around $3\sigma$ confidence level by the future measurements. In addition we also illustrate the two dimensional contour of parameters $\Omega_m$ and $\alpha$ in Fig.\[fig6\]. Comparing with the contour in Fig.\[fig5\], the allowed parameter region has been shrunk significantly. The future measurements could have enough ability to distinguish between the modified gravity model and the pure $\Lambda$CDM model.
Conclusions
===========
As an alternative approach to generate the late-time acceleration of the expansion of our universe, models of modifications of gravity have attracted a lot of interests in the phenomenological studies recently. In this note we investigate an interesting phenomenological model which interpolates between the pure $\Lambda$CDM model and the flat DGP braneworld model with an additional parameter $\alpha$.
Firstly, we find that when $\alpha$ is less than zero, the growth function of density perturbation $\delta(a)$ will appear an apparent singularity. This is because the variable $\beta$ will change the sign during the evolution of our universe. And then the $\beta$ term caused by the modified gravity model will be divergent at some redshift $z_t$.
From the current CMB, BAO and SNIa data, we obtain a tight constraint on the parameter $\alpha=0.263\pm0.175~(1\sigma)$, which implies that the flat DGP model ($\alpha=1$) is incompatible with the current observations, while the pure $\Lambda$CDM model still fits the data very well. When adding the high-redshift GRB and LGF data, the constraint is more stringent $\alpha=0.254\pm0.153~(1\sigma)$, which means that these high redshift observations are helpful to improve the constraints on this phenomenological model.
Finally, we simulate the future measurements with higher precisions to limit this phenomenological model. And we find that these accurate probes will be helpful to improve the constraints on the parameters of the model and could distinguish between the modified gravity model and the pure $\Lambda$CDM model.
[nn]{}
E. Komatsu, [*et al.*]{}, arXiv:0803.0547.
M. Kowalski [*et al.*]{}, arXiv:0804.4142.
S. Weinberg, Rev. Mod. Phys. [**61**]{}, 1 (1989); I. Zlatev, L. M. Wang and P. J. Steinhardt, Phys. Rev. Lett. [**82**]{}, 896 (1999).
S. M. Carroll, V. Duvvuri, M. Trodden, and M. S. Turner, Phys.Rev. D [**70**]{}, 043528 (2004); C. Schimd, J. P. Uzan and A. Riazuelo, Phys. Rev. D [**71**]{}, 083512 (2005).
See e.g. V. Acquaviva, C. Baccigalupi and F. Perrotta, Phys. Rev.D [**70**]{}, 023515 (2004); P. Zhang, Phys. Rev. D [**73**]{}, 123504 (2006); Y. S. Song, H. Peiris and W. Hu, Phys. Rev. D [**76**]{}, 063517 (2007); E. Bertschinger and P. Zukin, Phys. Rev. D [**78**]{}, 024015 (2008); and references therein.
S. F. Daniel, R. R. Caldwell, A. Cooray and A. Melchiorri, Phys.Rev. D [**77**]{}, 103513 (2008); S. F. Daniel, R. R. Caldwell, A. Cooray, P. Serra and A. Melchiorri, arXiv:0901.0919.
G. R. Dvali, G. Gabadadze and M. Porrati, Phys. Lett. B [**485**]{}, 208 (2000).
C. Deffayet, Phys. Lett. B [**502**]{}, 199 (2001).
A. Lue, Phys. Rept. [**423**]{}, 1 (2006).
C. Deffayet, Int. J. Mod. Phys. D [**16**]{} (2008) 2023; R. Durrer and R. Maartens, arXiv:0811.4132; and references therein.
G. Dvali and M. S. Turner, arXiv:astro-ph/0301510.
K. Koyama and R. Maartens, JCAP [**0601**]{}, 16 (2006).
K. Koyama, JCAP [**0603**]{}, 17 (2006).
E. V. Linder, Phys. Rev. D [**72**]{}, 043529 (2005); D. Huterer and E. V. Linder, Phys. Rev. D [**75**]{}, 023519 (2007); D. Polarski, arXiv:astro-ph/0605532; S. A. Thomas, F. B. Abdalla and J. Weller, arXiv:0810.4863.
P. J. E. Peebles, Large-Scale Structure of the Universe, Princeton Univ. Press, 1980; P. J. E. Peebles, Astrophys. J. [**284**]{}, 439 (1984); O. Lahav, P. B. Lilje, J. R. Primack and M. J. Rees, Mon.Not. R. Astron. Soc. [**251**]{}, 128 (1991).
E. V. Linder and R. N. Cahn, Astropart. Phys. [**28**]{}, 481 (2007).
W. Hu and N. Sugiyama, Astrophys. J. [**471**]{}, 542 (1996).
T. Okumura, T. Matsubara, D. J. Eisenstein, I. Kayo, C. Hikage, A. S. Szalay and D. P. Schneider, Astrophys. J. [**677**]{}, 889 (2008).
D. J. Eisenstein [*et al.*]{} Astrophys. J. [**633**]{}, 560 (2005).
W. J. Percival, [*et al.*]{}, Mon. Not. Roy. Astron. Soc. [**381**]{}, 1053 (2007).
D. Eisenstein and W. Hu, Astrophys. J. [**496**]{}, 605 (1998).
E. Di Pietro and J. F. Claeskens, Mon. Not. Roy. Astron. Soc., 1299 (2003).
H. Li, J. Q. Xia, J. Liu, G. B. Zhao, Z. H. Fan and X. Zhang, Astrophys. J. [**680**]{}, 92 (2008).
C. Firmani, V. Avila-Reese, G. Ghisellini and G. Ghirlanda, Rev.Mex. Astron. Astrofis. [**43**]{}, 203 (2007).
B. E. Schaefer, Astrophys. J. [**660**]{}, 16 (2007).
E. Hawkins [*et al.*]{}, Mon. Not. Roy. Astron. Soc. [**346**]{}, 78 (2003); L. Verde [*et al.*]{}, Mon. Not. Roy. Astron.Soc. [**335**]{}, 432 (2002).
M. Tegmark [*et al.*]{}, Phys. Rev. D [**74**]{}, 123507 (2006).
N. P. Ross [*et al.*]{}, arXiv:astro-ph/0612400.
L. Guzzo [*et al.*]{}, Nature [**451**]{}, 541 (2008).
J. da Angela [*et al.*]{}, arXiv:astro-ph/0612401.
P. McDonald [*et al.*]{} Astrophys. J. [**635**]{}, 761 (2005).
J. Q. Xia, H. Li, G. B. Zhao and X. Zhang, Phys. Rev. D [**78**]{}, 083524 (2008).
M. Fairbairn and A. Goobar, Phys. Lett. B [**642**]{}, 432 (2006); Z. K. Guo, Z. H. Zhu, J. S. Alcaniz and Y. Z. Zhang, Astrophys. J., 1 (2006); S. Rydbeck, M. Fairbairn and A. Goobar, JCAP [**0705**]{}, 003 (2007); T. M. Davis [*et al.*]{}, Astrophys. J., 716 (2007); M. S. Movahed, M. Farhang and S. Rahvar, Int. J. Theor. Phys. [**48**]{}, 1203 (2009).
Y. S. Song, Phys. Rev. D [**71**]{}, 024026 (2005); Y. S. Song, I. Sawicki and W. Hu, Phys. Rev. D [**75**]{}, 064003 (2007); W. Fang, S. Wang, W. Hu, Z. Haiman, L. Hui and M. May, Phys. Rev.D [**78**]{}, 103509 (2008); Z. H. Zhu and M. Sereno, arXiv:0804.2917; S. A. Thomas, F. B. Abdalla and J. Weller, arXiv:0810.4863; and references therein.
P. S. Corasaniti and A. Melchiorri, Phys. Rev. D [**77**]{}, 103507 (2008); H. Li, J. Q. Xia, G. B. Zhao, Z. H. Fan and X. Zhang, Astrophys. J. [**683**]{}, L1 (2008).
O. Elgaroy and T. Multamaki, Astron. Astrophys. [**471**]{}, 65 (2007).
A. G. Kim, E. V. Linder, R. Miquel and N. Mostek, Mon. Not. Roy.Astron. Soc. [**347**]{}, 909 (2004).
D. Hooper and S. Dodelson, Astropart. Phys. [**27**]{}, 113 (2007).
Planck Collaboration, arXiv:astro-ph/0604069.
[^1]: URL: http://www.sarahbridle.net/cosmologui/.
[^2]: In the revised version of WMAP5 paper [@Komatsu:2008hk], they also extend the baryon density $\Omega_b
h^2$ into the WMAP distance information. But our calculations are not sensitive to $\Omega_b h^2$ and they also claim that this extension does not affect the constraints. Thus, we fix $\Omega_b
h^2=0.022765$ to be the best fit value obtained by the WMAP group.
|
---
abstract: 'Recent progress in realistic simulations of solar convection have given us an unprecedented opportunity to evaluate the robustness of solar interior structures and dynamics obtained by methods of local helioseismology. We present results of testing the time-distance method using realistic simulations. By computing acoustic wave propagation time and distance relations for different depths of the simulated data, we confirm that acoustic waves propagate into the interior and then turn back to the photosphere. This demonstrates that in the numerical simulations properties of acoustic waves ($p$-modes) are similar to the solar conditions, and that these properties can be analyzed by the time-distance technique. For the surface gravity waves ($f$-mode), we calculate perturbations of their travel times, caused by localized downdrafts, and demonstrate that the spatial pattern of these perturbations (representing so-called sensitivity kernels) is similar to the patterns obtained from the real Sun, displaying characteristic hyperbolic structures. We then test the time-distance measurements and inversions by calculating acoustic travel times from a sequence of vertical velocities at the photosphere of the simulated data, and inferring a mean 3D flow fields by performing inversion based on the ray approximation. The inverted horizontal flow fields agree very well with the simulated data in subsurface areas up to 3 Mm deep, but differ in deeper areas. Due to the cross-talk effects between the horizontal divergence and downward flows, the inverted vertical velocities are significantly different from the mean convection velocities of the simulation dataset. These initial tests provide important validation of time-distance helioseismology measurements of supergranular-scale convection, illustrate limitations of this technique, and provide guidance for future improvements.'
author:
- 'Junwei Zhao, Dali Georgobiani, Alexander G. Kosovichev'
- 'David Benson, Robert F. Stein'
- '[Å]{}ke Nordlund'
title: 'Validation of Time-Distance Helioseismology by Use of Realistic Simulations of Solar Convection'
---
Introduction
============
Time-distance helioseismology, along with other helioseismology techniques, is an important tool for investigating the solar interior structure and dynamics. Since it was first introduced by @duv93, this technique has been used to derive the interior structure and flow fields of relatively small scales, such as supergranulation and sunspots [e.g., @kos96; @kos97; @kos00; @giz00; @zha01; @cou04], and also, at the global scale, such as differential rotation and meridional flows [e.g., @gil97; @cho01; @bec02; @zha04]. These studies, together with other local helioseismology techniques [e.g., @kom04; @bas04; @bra00], have opened a new frontier in studies of solar subsurface dynamics. Meanwhile, modeling efforts of time-distance helioseismology have also been carried out to interpret the time-distance helioseismology measurements, and provide sensitivity kernels used in inversions of the solar interior properties [e.g., @giz02; @jen03; @bir04].
However, despite the observational and modeling efforts, it is difficult to evaluate the accuracy or even correctness of the local helioseismological results because the interior of the Sun is unaccessible to direct observations. Still, there are a couple of approaches that help to evaluate the inverted results. One of these is to compare the inverted solar interior structures with models, e.g., comparing the subsurface flow fields below sunspots [@zha01] with results of sunspot models [@hur00]. Another approach is to compare the flow maps obtained by different helioseismological techniques, e.g., comparing $f$-mode time-distance measurements of near-surface flows with measurements of flows obtained by the ring-diagram technique [@hin04]. However, the first approach is only qualitative, and although the second approach is somewhat quantitative, there is a possibility that different helioseismological techniques may provide similar incorrect results because they do not account for turbulence and rapid variability of the subsurface flows.
A convincing way to validate time-distance helioseismology is to perform measurements and inversions on a set of realistic large-scale numerical simulation data, which not only model the turbulent convective motions of various scale in and beneath the solar photosphere, but also carry acoustic oscillation signals generated by the motions. These simulations have the following properties: the spatial resolution is comparable to or better than in typical helioseismological observations, the size of the computational domain is larger than a typical supergranule, the temporal resolution is sufficiently high to capture useful acoustic oscillation signals, and the time duration is long enough to extract acoustic signals with a satisfactory signal to noise ratio. The helioseismology techniques can then be evaluated by comparing the inverted interior results obtained from analyzing surface acoustic oscillations with the interior structures directly from the numerical simulation.
In this paper, we use realistic three-dimensional simulations in solar convections by @ben06, which were based on the work of @ste00. These simulations have enabled us to directly evaluate the validity of time-distance helioseismology measurements of the quiet Sun convection. In a previous paper, @geo06 [hereafter, Paper I] have analyzed the oscillation properties of these simulations, and found that the power spectrum is similar to the power spectrum obtained from real Michelson Doppler Imager (MDI) high resolution Dopplergrams. Their analysis of time-distance diagrams also showed that the simulated data had time-distance relations similar to those of the real Sun. Furthermore, the near-surface $f$-mode analysis using the simulated data gave surface structures similar to both those obtained from local correlation tracking and those actually in the simulation. Thus, this set of realistic simulations of solar convection by @ben06 is quite suitable for detailed $p$-mode time-distance studies, and allows us to evaluate the accuracy of inverted time-distance results.
In this paper, we introduce the simulated data in §2. Then we check the properties of acoustic propagation in the interior regions of the simulated data, and make sure that acoustic signals seen at the surface do carry information from the interior. We present these analyses in §3. In §4, we calculate the surface sensitivity kernel from this dataset. Then, in §5, we carry out $p$-modes time-distance measurements and inversions to infer interior flow fields, and compare our inverted results with the corresponding simulation data. Discussion follows in §6.
Numerical Simulation Data
=========================
The numerical simulation data we use in this paper were results of computation of multi-scale solar convection in the upper solar convection zone and photosphere [@ben06], using a three-dimensional compressible, radiative-hydrodynamic code, which employs LTE, non-gray radiative transfer and a realistic equation of state and opacities [@ste00].
In the space domain, the simulated data span 48 Mm $\times$ 48 Mm horizontally and 20 Mm vertically, with a horizontal spatial resolution of 96 km/pixel, and a varying vertical spatial resolution from 12 to 75 km/pixel. In the time domain, the data were saved every 10 seconds, but in practice, we used only every third snapshot, i.e., every 30 seconds, because the 10-sec temporal resolution provides an acoustic frequency range far beyond the typical solar oscillation frequencies. The whole simulation used in this analysis lasts 511 minutes. The acoustic $k-\omega$ diagram and the time-distance diagram obtained from this simulated dataset in the photosphere can be found in Paper I. In this paper and Paper I, the level of the height of formation of the center of the $\lambda$676.78 nm Ni line observed by SOHO/MDI is taken as the 0 Mm level for the convenience of description in the following text. It is 200 km above continuum optical depth unity.
Figure \[fg1\] presents a snapshot of vertical velocity from the simulation, showing three horizontal slices taken at the depth of 0, 8, and 16 Mm, and a vertical slice, respectively. One can see small scale granular structures in the photosphere, with downdraft lanes at the granular boundaries, and relatively weaker upflows inside granules. Several megameters below the photosphere, small granular structures disappear and are replaced by larger scale structures, but with similar flow patterns, downdrafts at boundaries and upward flows inside the structure.
Propagation Properties of Acoustic Waves
========================================
It is already clear that at the photospheric level, the simulations carry acoustic oscillations similar to those of the real solar observations, as demonstrated in Paper I. Since the goal of this paper is to evaluate time-distance helioseismology in the interior by use of $p$-modes analysis, it would be useful and necessary to check whether the proper acoustic oscillations exist in the interior of the simulated convection, and whether acoustic waves propagate inside properly.
The solar acoustic waves are excited stochastically by multiple random sources in the upper convection zone. In time-distance helioseismology, coherent wave signals are constructed by calculating a cross-correlation function of oscillations observed at locations, $\mathbf{r_1}$ and $\mathbf{r_2}$, separated by distance $\Delta$ and for time lag $\tau$: $$C(\Delta, \tau) = \int_0^T f(\mathbf{r_1}, t) f(\mathbf{r_2}, t +
\tau) \mathrm{d} t,$$ where $f(\mathbf{r},t)$ is the oscillation signal at location $\mathbf{r}$, $\Delta$ is distance between $\mathbf{r_1}$ and $\mathbf{r_2}$, and $T$ is the duration of the whole sequence. The cross-correlation for each distance is obtained by averaging numerous cross-correlations calculated between pairs of pixels, which are a given distance, $\Delta$, apart.
The time-distance diagram, as shown in Figure 2 in Paper I, and many other time-distance diagrams published in literature, are actually a collection of computed cross-correlations with continuous time lags, displayed as a function of acoustic wave travel distance. According to the conjecture of @ric00, the cross-covariance function may be considered as a signal from a point surface source of some particular spectral properties. Nearly all the previously published time-distance diagrams were obtained at the photospheric level, as no observations or simulations below the photosphere were available before.
With the availability of 3D convection simulation, we are now capable of computing the time-distance diagrams at different depths beneath the photosphere by cross-correlating acoustic signals at a desired depth with signals in the photosphere, and following the equation: $$C(\Delta, \tau, d) = \int_0^T f(\mathbf{r_1}, 0, t) f(\mathbf{r_2}, d,
t + \tau) \mathrm{d} t,$$ where $f(\mathbf{r}, d, t)$ is the oscillation signal at horizontal location $\mathbf{r}$, and at the depth of $d$ and time $t$. Because of the great computational burden, in practice such computations are performed in the Fourier domain. The time-distance diagram at a depth, $d$, shows the time it takes the acoustic wave to travel from a point at the surface to a point at the depth $d$ and a horizontal distance $\Delta$ away.
We have computed nearly 100 such time-distance diagrams at every fifth depth of the simulated data, since the original data have 500 vertical mesh points, but some of these mesh points are above the photosphere and not used in computing these diagrams. Several selected depths are presented in Figure \[fg2\]. The diagram at depth of 0 Mm is actually the time-distance diagram for the photosphere. The evolution of acoustic signals with time, and the propagation along horizontal distance are clearly seen from 0 Mm to a depth of 8 Mm, approximately. The time-distance diagrams match very well with the expected time-distance relationships derived from the ray theory, as indicated by the dashed lines in each diagram. Below 8 Mm, the signals are not so clear as above.
The two-dimensional time-distance diagrams for different depths can be represented as a three-dimensional datacube with two spatial dimensions, horizontal distance $\Delta$ and depth $d$, and time $\tau$. It is interesting to display the spatial images as a time sequence, showing acoustic waves propagating from the surface into the interior. Some selected images of this sequence are displayed in Figure \[fg3\]. The left side of each image is symmetrized with the right side. In order to better show the weak signals in the deeper regions, the signals near the surface are intentionally saturated.
As can be seen from the time-distance diagram at the depth of 0 Mm in Figure \[fg2\], the acoustic waves travel like a wave packet, with an oscillation period of 5 minutes or so, and a width of about 17 minutes [also see @duv97]. In Figure \[fg3\], at $\tau = 1$ min, one can see a very small blue feature at the top, and this is considered to be the first negative oscillation in the wave packet. With the evolution of time the blue feature, i.e., the wavefront, expands in size, and propagates into the interior. It is followed by a signal of the opposite sign, a red feature, which is initiated at $\tau=5$ min. We noted that at some moments of the evolution, the wave fronts become open at their horizontal central region. In the upper 10 Mm or so, the general circular wave shapes are often kept well, but below 10 Mm, the wave structure is often irregular and noisy. Also, it looks like some signals, though weak, are reflected back from the bottom boundary.
It is curious that the central part of the waves is open. This may come from the following reason: Because the horizontal span of the simulation domain is only 48 Mm, the simulated data cannot carry any acoustic wave modes that have a first bounce travel distance longer than 48 Mm. Clearly, the modes which travel a longer first bounce distance contribute most of the signal at the central part of the wavefronts. A simple estimate of the turning depth of the acoustic waves corresponding to a first bounce travel distance of 48 Mm is approximately 15 Mm. This is roughly in agreement with the images in Figure \[fg3\], in which no clear signals deeper than 15 Mm can be seen.
Despite the curious open structures at the central part of wavefronts, the noise outside the wave features and some reflected signals from the bottom, it is quite clear the waves are substantial, clear, and in nice agreement with the theoretical expectations. The constructed wave propagation clearly shows that the waves are refracted back to the surface from different turning points as expected from linear helioseismology theories. It is a great success of the time-distance technique that it is capable of retrieving acoustic wave propagation in the deep, very turbulent interior, and it is also a great success for the simulations that they reproduce the basic properties of acoustic waves in the solar interior. Therefore, we conclude that this simulation dataset is quite suitable for testing the $p$-mode time-distance analysis, while keeping in mind that the deepest acoustic wave probe depth in these simulations is shallower than 15 Mm.
Propagation of Surface Gravity Waves: Scattering and Sensitivity Kernels
========================================================================
The sensitivity functions (or kernels) represent perturbations of travel times to small localized features on the Sun. To evaluate the potential for using surface gravity waves (f-modes) in the simulations for testing the $f$-mode diagnostics, we have calculated empirical sensitivity functions following the recent work of @duv06. Their work has offered us not only another method for checking the helioseismic reasonableness of the simulated data, but also a meaningful approach to compute the sensitivity kernels that is potentially useful for time-distance inversions.
@duv06 computed the sensitivity kernels for $f$-modes by measuring travel time variations around some small magnetic elements from real MDI observations. They found elliptic and hyperbolic structures in the kernels, which are similar to the structures modeled by @giz02 and attributed to wave scattering effects. Because of wave scattering, the travel times are sensitive not only to perturbations in the region of wave propagation between two points but also in some areas away the way path, along a hyperbolic-shaped regions. They concluded that their measurements demonstrated that the Born approximation was suitable for deriving time-distance inversion kernels, and that the wave scattering effect is important and has to be included in the derivation of the sensitivity kernels for helioseismic inversions.
Here, we try to check whether the sensitivity kernel with similar hyperbolic structures can be obtained from the numerical simulations, and thus whether the wave scattering is properly modeled in the simulated data.
Since there are no magnetic elements in the current simulation dataset, we select areas with strong downdrafts as the features to calculate the travel time sensitivity kernels. We average the vertical velocity over the whole time sequence of 511 minutes, and select 50 small areas that have the strongest downward flows. We set these 50 areas as the central features to compute the kernels. We then follow the procedure described as “feature method” in @duv06, and compute travel time variations around these selected downdrafts areas. We follow every step described in this paper, except that we use a longer time series in our computations, and that we use the Gabor wavelet fitting to measure the wave travel times. Although we have only one data sequence, and just 50 selected downdraft areas available for averaging (unlike a number of observational datasets and thousands of magnetic elements in the @duv06 paper), we were able to obtain the sensitivity kernels with a reasonable signal-to-noise ratio. Still, we had to do additional spatial averaging to recover the wave scattering structures.
Figure \[fg4\] presents the results of our measurements a) before and b) after boxcar smoothing. There are some small scale black and white patches in the measured original travel time variations, and large scale structures are not quite clear, although such structures do exist. Such small scale structures are not seen in the kernels obtained from real observations, and this may be possibly caused by the following reason: the numerical simulation data have a much higher spatial resolution than the real observations, and thus our measurements could pick up some small-scale signals that are not resolved in the observational data. Additionally, the convective structures we use for such measurements are much less stable than the real magnetic elements, and this may also contribute some noises to the measurements.
To get a better signal-to-noise ratio for large-scale structures, we applied a boxcar smoothing to the original measurements, and obtain the result shown in Figure \[fg4\]b. The hyperbolic dark and light features appear quite clearly, similar to the sensitivity kernel measured from magnetic elements of real observations, although the locations have slight offset as indicated by the dashed lines. However, the details are not comparable, because our sensitivity kernel is measured around large downdrafts areas, but not for magnetic elements as in Figure \[fg4\]c.
It is important that the hyperbolic structures are found in the $f$-modes sensitivity kernels obtained from the simulated data, because this demonstrates that the wave scattering effect is reproduced in the numerical simulation of convection. However, at present, it is difficult to use the empirical sensitivity kernels in time-distance inversions, because the measured kernels such as the one shown in Figure \[fg4\] may depend on various factors at the downdraft locations, for instance, not only the vertical component of flow, but also the converging flows that are often associated with the downdrafts. This issue requires further investigation.
Time-Distance Helioseismology Test for Sub-Surface Flows
========================================================
To test the current time-distance helioseismology procedure, we measured travel times of acoustic waves using only the vertical velocity at the photospheric level from the simulated data, then inferred the three dimensional velocities in the interior by using the inversion procedure based on a ray approximation [@kos97; @kos00; @zha01], and finally compared the inversion results with the averaged interior velocities from the simulations. Following the typical $p$-mode time-distance measurement schemes [e.g., @zha01], we select the following seven annulus radii to perform our measurements: 8.64 – 10.37, 10.56 – 12.29, 12.48 – 14.21, 14.40 – 16.13, 16.32 – 18.05, 18.24 – 19.97, 20.16 – 21.89 Mm. The greatest annulus is a little smaller than half of the horizontal span of the simulated data. In order to evaluate the previous inversion results, the inversions here employ the ray approximation kernels.
To evaluate the inverted time-distance results, we compare our inversion results for flows with the actual flow fields directly from the simulated data. The inverted results are often given as averages of some depth ranges, for instance, 1 – 2 Mm in Figure \[fg6\]. Hence, the simulated data are also averaged arithmetically in the same depth interval over the 511 minutes. For each target depth, the averaging kernels from the inversion are three dimensional, and Figure \[fg5\] only presents a one dimensional curve corresponding to the central point. In addition to the direct comparisons, it is interesting to convolve the three dimensional averaging kernels with the three dimensional simulated velocities, and see how the resultant velocities compare with the inverted velocities. In the following, we present results from the direct comparison, but give the correlation coefficients of both comparisons in Table \[tb1\].
Horizontal Flow Fields
----------------------
Figures \[fg6\], \[fg7\], and \[fg8\] present the inversion results for the horizontal flow components for in three layers: 1-2 Mm deep, 2-3 Mm, and 4-5 Mm. Comparing the horizontal vector flows, as well as the divergence map computed from the horizontal flows, we find that at the depth of 1 – 2 Mm the inversion results agree quite well with the time averaged simulation results. The results reveal areas of strong flow divergence, which have a typical size of solar supergranulations. This is in good agreement with the results obtained by the time-distance analysis of $f$-modes and by a correlation tracking method, which are presented in Paper I. These divergent areas are nearly in one-to-one correspondence between the inverted results and the simulated data, with similar magnitudes as well. It should be pointed out that the flow maps for the simulated data are displayed after applying a low-pass filtering to only keep structures that have a wavenumber smaller than 0.06 Mm$^{-1}$, in order to better match the time-distance computational procedures, which undergo filtering and smoothing during measurements and inversions.
As the inverted area deepens, the correlation between the inverted results and the simulated data gradually worsens. At the depth of 2 – 3 Mm one can still see those divergent flow patterns in both data, but clearly not as clearly as at the depth of 1 – 2 Mm. However, at the depth of 4 – 5 Mm and deeper, the inverted horizontal flows show no clear correlation with the simulated data.
[ccccc]{} depth & $v_x$ & $v_y$ & divergence & $v_z$\
0 – 1 Mm & 0.72 (0.72) & 0.64 (0.64) & 0.56 (0.56) & -0.72\
1 – 2 Mm & 0.85 (0.87) & 0.76 (0.76) & 0.89 (0.91) & -0.72\
2 – 3 Mm & 0.87 (0.92) & 0.74 (0.83) & 0.78 (0.84) & -0.29\
3 – 4 Mm & 0.74 (0.84) & 0.37 (0.53) & 0.36 (0.50) & 0.34\
4 – 5 Mm & 0.18 (0.51) & 0.25 (0.62) & -0.35 (-0.11) & 0.32\
Table \[tb1\] presents the correlation coefficients between the inverted results and the simulated data at different depths in two horizontal velocity components, the divergence that are computed from the horizontal components, and the vertical velocity (see the next section), separately. It is clear that the shallow regions often have better correlations than the deeper regions, except for the vertical velocity. It is curious but not clear why the two horizontal components have correlation coefficients that differ so much, with the north-south direction often worse than the east-west direction. A similar east-west and north-south asymmetry in time-distance results was also found when comparing with local correlation tracking results [@sva06]. And, the correlation of the divergence maps also differ from that of the horizontal components. It is worthwhile pointing out that after the simulated data are convolved with the inversion averaging kernels, the correlation coefficients are generally improved, significantly in the deeper areas.
Vertical Flow Fields
--------------------
It is often the case, in both real observations and numerical simulations, that in the photosphere the vertical velocity is significantly smaller than the horizontal velocity of the same depth, due to the strong density stratification in the upper solar convection zone. And often, near the surface, the downward flows concentrate in very narrow lanes among boundaries of granules or supergranules. These properties make it more difficult to infer vertical velocities by the local helioseismology techniques, which often involves large area smoothing, because the small magnitude vertical velocity in small areas can be easily smeared out.
Figure \[fg9\] presents results of time-distance inverted results of vertical flows, along with the averaged vertical velocities from simulated data at the corresponding depths. At the depth of 1 – 2 Mm the inverted vertical velocities basically have opposite signs to the simulated velocities. For the other two depths, no clear correlation or anti-correlation is seen. As also presented in Table \[tb1\], the correlation coefficients show that, at shallow depths, inverted vertical flows are in anti-correlation with the simulated data, while at deeper depths, the correlations are positive but weak. It seems that the inversions for the vertical flow fields are completely unsuccessful. This discrepancy is not only due to the small magnitude of vertical velocities, but is also caused by the cross-talk effects discussed in the next section.
Discussion
==========
The realistic simulation of solar convection obtained by @ben06 gives us an unprecedented opportunity to evaluate the time-distance helioseismology technique and some other local helioseismology approaches.
By computing the time-distance diagrams at different depths, as shown in Figures \[fg2\] and \[fg3\], we have confirmed that there are acoustic waves propagating in the interior in the simulated data with properties similar to expected from the helioseismology theory. Although there are some unexplained correlated signal noises, bottom reflections, and open structure at the center of wavefront, these do not affect our time-distance analysis in shallow regions below the surface. The longest annulus used in our analysis is about 21 Mm, which probes into a depth of approximately 6.5 Mm according to ray theory, shallow enough not be affected by those unexplained factors in the simulations. Our measurements of the $f$-mode sensitivity kernel near the surface, shown in Figure \[fg3\], have demonstrated that the convection simulation data have wave scattering properties similar to the real solar observations. These analyses demonstrate that the simulated data have the wave properties that are necessary for time-distance helioseismology analysis.
Using very turbulent convection data at the photospheric level and the time-distance technique based on a ray approximation, we were able to derive the internal flow fields, which are in nice correlation with the simulation results in shallow regions. It is not surprising that the time-distance inversions cannot well resolve properties in larger depths, which was already demonstrated by some artificial data tests [e.g., @kos96; @zha01]. It is believed that the reliability of inversion results highly depends on the number of ray-paths passing through that area, whereas deeper areas have fewer ray-paths passing through and less information brought up to the surface. Additionally, it should be recognized again that the horizontal scale size of 48 Mm limits the deepest ray-path penetration at a depth of approximately 15 Mm, and our longest annulus radius once again limits our deepest probe to a depth of 6.5 Mm or so. Therefore, it is quite reasonable that our inversions give acceptable results until a depth of only 4 Mm or so.
It is also not surprising to see the failure of vertical flows in inversions. Certainly, the small magnitudes of vertical velocity may be one reason. However, we believe that the main reason of the failure is due to the cross-talk effects, as already demonstrated by use of some artificial tests [@zha03]. The divergence inside supergranules speeds up outgoing acoustic waves by the same way as downdrafts do, and similarly, the convergence at boundaries of supergranules slows down outgoing waves from this region similar to what upflows do. The time-distance inversions cannot distinguish the divergence (or convergence) from downward (or upward) flows, especially when the vertical flow is small in magnitude. Although it is believed that some additional constraints, e.g., mass conservation, may help the inversion in resolving vertical flows, current inversion technique of time-distance restricts itself using pure helioseismological measurements. This set of simulated data may give us a very good test ground for the future development of vertical velocity inversion codes.
Still, inversions in this study use the ray approximation kernels in order to evaluate the old results published previously by use of such kernels. With the availability of Born approximation kernels [@bir04], it would be very interesting to test such kernels by this kind of analysis on the current simulated dataset, although some previous experiments [@cou04] showed that the inversion results based on two different kernels did not differ much. It is expected that the Born kernel may give better results in deeper areas, but may not be able to solve the vertical velocity problem caused by the cross-talk effects. New time-distance helioseismology schemes are probably required for the solution.
The numerical simulation of solar convection give us an opportunity to evaluate the time-distance technique in quiet solar regions. However, it would be especially interesting if we could test this helioseismology technique in a magnetized region, as the simulations of magnetoconvection [e.g., @ste06; @sch06] are extended in both spatial and temporal scales to meet the helioseismological measurement requirements. For instance, we can evaluate the accuracy of the inferred sunspot structures and flow fields [@kos00; @zha01], and evaluate various magnetic field effects based on such numerical simulations.
We thank Dr. Tom Duvall for providing us data making Figure \[fg4\]c, and Dr. Takashi Sekii for insightful comments on interpretation of interior wave propagations. We also thank an anonymous referee for suggestions to improve the quality of this paper. The numerical simulations used in this work were made under support by NASA grants NNG04GB92G and NAG512450, NSF grants AST-0205500 and AST-0605738, and by grants from the Danish Center for Scientific Computing. The simulations were performed on the Columbia supercomputer of the NASA Advanced Supercomputing Division.
Basu, S., Antia, H. M., & Bogart, R. S. 2004, , 610, 1157
Beck, J. G., Gizon, L., & Duvall, T. L., Jr. 2002, , 575, L47
Benson, D., Stein, R., & Nordlund, Å. 2006, in H. Uitenbroek, J. Leibacher, R. F. Stein (eds.), [*ASP Conf. Ser.: Solar MHD Theory and Observations: a High Spatial Resolution Perspective, 94*]{}
Birch, A. C., Kosovichev, A. G., & Duvall, T. L., Jr. 2004, , 608, 580
Braun, D. C., & Lindsey, C. 2000, , 192, 307
Chou, D.-Y., & Dai, D.-C. 2001 , 559, L175
Couvidat, S., Birch, A. C., Kosovichev, A. G., & Zhao, J. 2004, , 607, 554
Duvall, T. L., Jr., Birch, A. C., & Gizon, L. 2006, , 646, 553
Duvall, T. L., Jr., Jefferies, S. M., Harvey, J. W., & Pomerantz, M. A. 1993, , 362, 235
Duvall, T. L., Jr., et al. 1997, , 170, 63
Giles, P. M., Duvall, T. L., Jr., Scherrer, P. H., & Bogart, R. S. 1997, , 390, 52
Gizon, L., & Birch, A. C. 2002, , 571, 966
Gizon, L., Duvall, T. L., Jr., & Larsen, R. M. 2000, J. Astrophys. Astron., 21, 339
Georgobiani, D., Zhao, J., Kosovichev, A. G., Bensen, D., Stein, R. F., & Nordlund, Å. 2006, , in press, Paper I
Hindman, B. W., Gizon, L., Duvall, T. L., Jr., Haber, D. A., & Toomre, J. 2004, , 613, 1253
Hurlburt, N. E., & Rucklidge, A. M. 2000, , 314, 793
Jensen, J. M., & Pijers, F. P. 2003, , 412, 257
Komm, R., Corbard, T., Durney, B. R., González Hernández, I., Hill, F., Howe, R., & Toner, C. 2004, , 605, 554
Kosovichev, A. G. 1996, , 461, L55
Kosovichev, A. G., & Duvall, T. L., Jr. 1997, in [*Proceedings of SCORe Workshop: Solar Convection and Oscillations and Their Relationship*]{}, ed. J. Christensen-Dalsgaard & F. Pijpers (Dordrecht: Kluwer), 241
Kosovichev, A. G., Duvall, T. L., Jr., & Scherrer, P. H. 2000, , 192, 159
Rickett, J. E., & Claerbout, J. F. 2000, , 192, 203
Schüssler, M., & Vögler, A. 2006, , 641, L73
Stein, R. F., & Nordlund, Å. 2000, , 192, 91
Stein, R. F., & Nordlund, Å. 2006, , 642, 1246
Švanda, M., Zhao, J., & Kosovichev, A. G. 2006, , submitted
Zhao, J., & Kosovichev, A. G. 2003, in [*Proceeding of SOHO 12/ GONG+ 2002 Workshop, Local and Global Helioseismology: the Present and Future*]{}, ed. H. Sawaya-Lacoste (ESA SP-517; Noordwijk: ESA), 417
Zhao, J., & Kosovichev, A. G. 2004, , 603, 776
Zhao, J., Kosovichev, A. G., & Duvall, T. L., Jr. 2001, , 557, 384
|
---
abstract: 'The success of neural networks on a diverse set of NLP tasks has led researchers to question how much do these networks actually know about natural language. Probes are a natural way of assessing this. When probing, a researcher chooses a linguistic task and trains a supervised model to predict annotation in that linguistic task from the network’s learned representations. If the probe does well, the researcher may conclude that the representations encode knowledge related to the task. A commonly held belief is that using simpler models as probes is better; the logic is that such models will *identify linguistic structure*, but not *learn the task itself*. We propose an information-theoretic formalization of probing as estimating mutual information that contradicts this received wisdom: one should always select the highest performing probe one can, even if it is more complex, since it will result in a tighter estimate. The empirical portion of our paper focuses on obtaining tight estimates for how much information BERT knows about parts of speech in a set of five typologically diverse languages that are often underrepresented in parsing research, plus English, totaling six languages. We find BERT accounts for only at most $5\%$ more information than traditional, type-based word embeddings.'
author:
- |
Tiago Pimentel$^{{\normalfont \text{\textipa{D}}}}$ Josef Valvoda$^{{\normalfont \text{\textipa{D}}}}$ Rowan Hall Maudslay$^{{\normalfont \text{\textipa{D}}}}$ Ran Zmigrod$^{{\normalfont \text{\textipa{D}}}}$\
**Adina Williams$^{{\normalfont \text{\textipa{@}}}}$ Ryan Cotterell$^{{\normalfont \text{\textipa{D}}},{\text{\normalfont \textipa{Q}}}}$**\
$^{{\normalfont \text{\textipa{D}}}}$University of Cambridge $^{{\normalfont \text{\textipa{@}}}}$Facebook AI Research $^{{\text{\normalfont \textipa{Q}}}}$ETH Zürich\
`tp472@cam.ac.uk`, `jv406@cam.ac.uk`, `rh635@cam.ac.uk`,\
`rz279@cam.ac.uk`, `adinawilliams@fb.com`, `rdc42@cl.cam.ac.uk`
bibliography:
- 'acl2020.bib'
title: 'Information-Theoretic Probing for Linguistic Structure'
---
Introduction
============
Neural networks are the backbone of modern state-of-the-art Natural Language Processing (NLP) systems. One inherent by-product of training a neural network is the production of real-valued representations. Many speculate that these representations encode a continuous analogue of discrete linguistic properties, e.g., part-of-speech tags, due to the networks’ impressive performance on many NLP tasks [@belinkov-etal-2017-evaluating]. As a result of this speculation, one common thread of research focuses on the construction of **probes**, i.e., supervised models that are trained to extract the linguistic properties directly [@belinkov-etal-2017-evaluating; @conneau-etal-2018-cram; @peters-etal-2018-dissecting; @zhang-bowman-2018-language; @tenney2019; @naik-etal-2018-stress]. A syntactic probe, then, is a model for extracting syntactic properties, such as part-of-speech, from the representations [@hewitt-liang-2019-designing]. In this work, we question what the goal of probing for linguistic properties ought to be. Informally, probing is often described as an attempt to discern how much information representations encode about a specific linguistic property. We make this statement more formal: We assert that the goal of probing ought to be estimating the mutual information [@cover-thomas] between a representation-valued random variable and a linguistic property-valued random variable. This formulation gives probing a clean, information-theoretic foundation, and allows us to consider what “probing” actually means.
Our analysis also provides insight into how to choose a probe family: We show that choosing the highest-performing probe, independent of its complexity, is optimal for achieving the best estimate of mutual information (MI). This contradicts the received wisdom that one should always select simple probes over more complex ones [@alain2016understanding; @liu-etal-2019-linguistic; @hewitt-manning-2019-structural]. In this context, we also discuss the recent work of who propose [**selectivity**]{} as a criterion for choosing families of probes. define selectivity as the performance difference between a probe on the target task and a control task, writing “\[t\]he selectivity of a probe puts linguistic task accuracy in context with the probe’s capacity to memorize from word types.” They further ponder: “when a probe achieves high accuracy on a linguistic task using a representation, can we conclude that the representation encodes linguistic structure, or has the probe just learned the task?” Information-theoretically, there is *no difference* between learning the task and probing for linguistic structure, as we will show; thus, it follows that one should always employ the best possible probe for the task without resorting to artificial constraints. In support of our discussion, we empirically analyze word-level part-of-speech labeling, a common syntactic probing task [@hewitt-liang-2019-designing; @DBLP:journals/corr/abs-1903-09442], within our framework. Working on a typologically diverse set of languages (Basque, Czech, English, Finnish, Tamil, and Turkish), we show that the representations from BERT, a common contextualized embedder, only account for at most $5\%$ more of the part-of-speech tag entropy than a control. These modest improvements suggest that most of the information needed to tag part-of-speech well is encoded at the lexical level, and does not require the sentential context of the word. Put more simply, words are not very ambiguous with respect to part of speech, a result known to practitioners of NLP [@garrette-etal-2013-real]. We interpret this to mean that part-of-speech labeling is not a very informative probing task.
We also remark that formulating probing information-theoretically gives us a simple, but stunning result: contextual word embeddings, e.g., BERT [@devlin-etal-2019-bert] and ELMo [@peters-etal-2018-deep], contain *the same* amount of information about the linguistic property of interest as the original sentence. This follows naturally from the data-processing inequality under a very mild assumption. What this suggests is that, in a certain sense, probing for linguistic properties in representations may not be a well grounded enterprise at all.
Word-Level Syntactic Probes for Contextual Embeddings {#sec:probescontextualembeddings}
=====================================================
Following , we consider probes that examine syntactic knowledge in contextualized embeddings. These probes only consider a single token’s embedding and try to perform the task using only that information. Specifically, in this work, we consider part-of-speech (POS) labeling: determining a word’s part of speech in a given sentence. For example, we wish to determine whether the word [*love*]{} is a [<span style="font-variant:small-caps;">noun</span>]{} or a [<span style="font-variant:small-caps;">verb</span>]{}. This task requires the sentential context for success. As an example, consider the utterance “love is blind” where, only with the context, is it clear that [*love*]{} is a [<span style="font-variant:small-caps;">noun</span>]{}. Thus, to do well on this task, the contextualized embeddings need to encode enough about the surrounding context to correctly guess the POS.
Notation
--------
Let $S$ be a random variable ranging over all possible sequences of words. For the sake of this paper, we assume the vocabulary ${\mathcal{V}}$ is finite and, thus, the values $S$ can take are in ${\mathcal{V}}^*$. We write ${\mathbf{s}}\in S$ as ${\mathbf{s}}= w_1 \cdots w_{|{\mathbf{s}}|}$ for a specific sentence, where each $w_i \in {\mathcal{V}}$ is a specific word in the sentence and the position $i \in \mathbb{N}^{+}$. We also define the random variable $W$ that ranges over the vocabulary ${\mathcal{V}}$. We define both a sentence-level random variable $S$ and a word-level random variable $W$ since each will be useful in different contexts during our exposition. Next, let $T$ be a random variable whose possible values are the analyses $t$ that we want to consider for word $w_i$ in its sentential context, ${\mathbf{s}}= w_1 \cdots w_i \cdots w_{|{\mathbf{s}}|}$. In this work, we will focus on predicting the part-of-speech tag of the $i^\text{th}$ word $w_i$. We denote the set of values $T$ can take as the set ${\mathcal{T}}$. Finally, let $R$ be a representation-valued random variable for the $i^\text{th}$ word $w_i$ in a sentence derived from the entire sentence ${\mathbf{s}}$. We write ${\mathbf{r}}\in {\mathbb{R}}^d$ for a value of $R$. While any given value ${\mathbf{r}}$ is a continuous vector, there are only a countable number of values $R$ can take. To see this, note there are only a countable number of sentences in ${\mathcal{V}}^*$.=-1
Next, we assume there exists a true distribution $p(t, {\mathbf{s}}, i)$ over analyses $t$ (elements of ${\mathcal{T}}$), sentences ${\mathbf{s}}$ (elements of ${\mathcal{V}}^*$), and positions $i$ (elements of $\mathbb{N}^{+}$). Note that the conditional distribution $p(t \mid {\mathbf{s}}, i)$ gives us the true distribution over analyses $t$ for the $i^{{\text{th}}}$ word in the sentence ${\mathbf{s}}$. We will augment this distribution such that $p$ is additionally a distribution over ${\mathbf{r}}$, i.e., $$\begin{aligned}
\label{eq:true}
p({\mathbf{r}}, t, {\mathbf{s}}, i) = \delta({\mathbf{r}}\mid {\mathbf{s}}, i)\,p(t, {\mathbf{s}}, i) \end{aligned}$$ where we define the augmentation as a Dirac’s delta function $$\begin{aligned}
\delta({\mathbf{r}}\mid {\mathbf{s}}, i) &= \mathbbm{1}\{{\mathbf{r}}={\textsc{bert}\xspace}({\mathbf{s}})_i\}\end{aligned}$$ Since contextual embeddings are a deterministic function of a sentence ${\mathbf{s}}$, the augmented distribution in \[eq:true\] has no more randomness than the original—its entropy is the same. We assume the values of the random variables defined above are distributed according to this (unknown) $p$. While we do not have access to $p$, we assume the data in our corpus were drawn according to it. Note that $W$—the random variable over possible words—is distributed according to the marginal distribution $$p(w) = \sum_{{\mathbf{s}}\in {\mathcal{V}}^*}\sum_{i=1}^{|{\mathbf{s}}|} \delta(w \mid {\mathbf{s}}, i)\,p({\mathbf{s}}, i)$$ where we define the deterministic distribution $$\delta(w \mid {\mathbf{s}}, i) = \mathbbm{1}\{{\mathbf{s}}_i = w\}$$
Probing as Mutual Information {#subsec:probeasMI}
-----------------------------
The task of supervised probing is an attempt to ascertain how much information a specific representation ${\mathbf{r}}$ tells us about the value of $t$. This is naturally expressed as the mutual information, a quantity from information theory: $$\begin{aligned}
{\mathrm{I}}(T; R) = {\mathrm{H}}(T) - {\mathrm{H}}(T \mid R)\end{aligned}$$ where we define the entropy, which is constant with respect to the representations, as $$\begin{aligned}
{\mathrm{H}}(T) = &- \sum_{t \in {\mathcal{T}}} p(t) \log p(t)\end{aligned}$$ and where we define the conditional entropy as $$\begin{aligned}
&{\mathrm{H}}(T \mid R) = \int p({\mathbf{r}}) \,{\mathrm{H}}\left(T \mid R = {\mathbf{r}}\right)d{\mathbf{r}}\\
&=\sum_{{\mathbf{s}}\in {\mathcal{V}}^*} \sum_{i = 1}^{|{\mathbf{s}}|} p({\mathbf{s}}, i) \,{\mathrm{H}}\left(T \mid R = {\textsc{bert}\xspace}({\mathbf{s}})_i \right) \nonumber\end{aligned}$$ the point-wise conditional entropy inside the sum is defined as $${\mathrm{H}}(T \mid R = {\mathbf{r}}) = -\sum_{t \in {\mathcal{T}}} p(t \mid {\mathbf{r}}) \log p(t \mid {\mathbf{r}})$$ Again, we will not know any of the distributions required to compute these quantities; the distributions in the formulae are marginals and conditionals of the true distribution discussed in \[eq:true\].
Bounding Mutual Information
---------------------------
The desired conditional entropy, ${\mathrm{H}}(T \mid R)$ is not readily available, but with a model ${q_{{{\boldsymbol \theta}}}}({\mathbf{t}}\mid {\mathbf{r}})$ in hand, we can upper-bound it by measuring their empirical cross entropy $$\begin{aligned}
\label{eq:ent_estimate}
{\mathrm{H}}(T&\,\mid R) := -\underset{{(t, {\mathbf{r}}) \sim p(\cdot, \cdot)}}{\mathbb{E}} \left[ \log p(t \mid {\mathbf{r}}) \right] \\
&= - \underset{{(t, {\mathbf{r}}) \sim p(\cdot, \cdot)}}{\mathbb{E}} \left[ \log \frac{p(t \mid {\mathbf{r}}) {q_{{{\boldsymbol \theta}}}}(t \mid {\mathbf{r}})}{{q_{{{\boldsymbol \theta}}}}(t \mid {\mathbf{r}})} \right] \nonumber \\
&= - \underset{{(t, {\mathbf{r}}) \sim p(\cdot, \cdot)}}{\mathbb{E}} \left[ \log {q_{{{\boldsymbol \theta}}}}(t \mid {\mathbf{r}}) + \log \frac{p(t \mid {\mathbf{r}})}{{q_{{{\boldsymbol \theta}}}}(t \mid {\mathbf{r}})} \right] \nonumber \\
&= \underbrace{{\mathrm{H}}_{{q_{{{\boldsymbol \theta}}}}}(T \mid R)}_{\textit{estimate}} - \underbrace{\underset{{\mathbf{r}}\sim p(\cdot)}{\mathbb{E}} {\mathrm{KL}}(p(\cdot \mid {\mathbf{r}}) \mid \mid {q_{{{\boldsymbol \theta}}}}(\cdot \mid {\mathbf{r}}))}_{\textit{expected estimation error}} \nonumber\end{aligned}$$ where ${\mathrm{H}}_{{q_{{{\boldsymbol \theta}}}}}(T \mid R)$ is the cross-entropy we obtain by using ${q_{{{\boldsymbol \theta}}}}$ to get this estimate. Since the KL divergence is always positive, we may lower-bound the desired mutual information $$\begin{aligned}
{\mathrm{I}}(T; R) &:= {\mathrm{H}}(T) - {\mathrm{H}}(T \mid R) \nonumber \\
&\geq {\mathrm{H}}(T) - {\mathrm{H}}_{{q_{{{\boldsymbol \theta}}}}}(T \mid R)\end{aligned}$$ This bound gets tighter, the more similar (in the sense of the KL divergence) ${q_{{{\boldsymbol \theta}}}}(\cdot \mid {\mathbf{r}})$ is to the true distribution $p(\cdot \mid {\mathbf{r}})$.
#### Bigger Probes are Better.
If we accept mutual information as a natural measure for how much representations encode a target linguistic task (§\[subsec:probeasMI\]), then the best estimate of that mutual information is the one where the probe ${q_{{{\boldsymbol \theta}}}}(t \mid {\mathbf{r}})$ is best at the target task. In other words, we want the best probe ${q_{{{\boldsymbol \theta}}}}(t \mid {\mathbf{r}})$ such that we get the tightest bound to the actual distribution $p(t\mid {\mathbf{r}})$. This paints the question posed by , who write
“when a probe achieves high accuracy on a linguistic task using a representation, can we conclude that the representation encodes linguistic structure, or has the probe just learned the task?”
as a false dichotomy.[^1] From an information-theoretic view, we will always prefer the probe that does better at the target task, since there is *no difference* between learning a task and the representations encoding the linguistic structure.
Control Functions {#sec:control-functions}
=================
To place the performance of a probe in perspective, develop the notion of a control task. Inspired by this, we develop an analogue we term [**control functions**]{}, which are functions of the representation-valued random variable $R$. Similar to ’s control tasks, the goal of a control function ${\mathbf{c}}(\cdot)$ is to place the mutual information ${\mathrm{I}}(T; R)$ in the context of a baseline that the control function encodes. Control functions have their root in the data-processing inequality [@cover-thomas], which states that, for any function ${\mathbf{c}}(\cdot)$, we have $$\label{eq:data-processing}
{\mathrm{I}}(T; R) \geq {\mathrm{I}}(T; {\mathbf{c}}(R))$$ In other words, information can only be lost by processing data. A common adage associated with this inequality is “garbage in, garbage out.”
Type-Level Control Functions
----------------------------
We will focus on type-level control functions in this paper; these functions have the effect of decontextualizing the embeddings. Such functions allow us to inquire how much the contextual aspect of the contextual embeddings help the probe perform the target task. To show that we may map from contextual embeddings to the identity of the word type, we need the following assumption about the embeddings.
\[ass:one\] Every contextualized embedding is unique, i.e., for any pair of sentences ${\mathbf{s}}, {\mathbf{s}}' \in {\mathcal{V}}^*$, we have $({\mathbf{s}}\neq {\mathbf{s}}') \mid\mid (i \neq j) \Rightarrow {\textsc{bert}\xspace}({\mathbf{s}})_i \neq {\textsc{bert}\xspace}({\mathbf{s}}')_j$ for all $i \in \{1, \ldots |{\mathbf{s}}|\}$ and $j \in \{1, \ldots, |{\mathbf{s}}'|\}$.
We note that \[ass:one\] is mild. Contextualized word embeddings map words (in their context) to ${\mathbb{R}}^d$, which is an uncountably infinite space. However, there are only a countable number of sentences, which implies only a countable number of sequences of real vectors in ${\mathbb{R}}^d$ that a contextualized embedder may produce. The event that any two embeddings would be the same across two distinct sentences is infinitesimally small.[^2] \[ass:one\] yields the following corollary.
\[cor:one\] There exists a function $\emph{{\texttt{id}\xspace}} : {\mathbb{R}}^d \rightarrow V$ that maps a contextualized embedding to its word type. The function $\emph{{\texttt{id}\xspace}}$ is not a bijection since multiple embeddings will map to the same type.
Using \[cor:one\], we can show that any *non-contextualized* word embedding will contain *no more* information than a contextualized word embedding. More formally, we do this by constructing a look-up function ${\mathbf{e}}: V \rightarrow {\mathbb{R}}^d$ that maps a word to a word embedding. This embedding may be one-hot, randomly generated ahead of time, or the output of a data-driven embedding method, e.g. fastText [@fasttext]. We can then construct a control function as the composition of the look-up function ${\mathbf{e}}$ and the id function ${\texttt{id}\xspace}$. Using the data-processing inequality, we can prove that in a word-level prediction task, any non-contextual (type level) word-embedding will contain no more information than a contextualized (token level) one, such as BERT and ELMo. Specifically, we have $$\begin{aligned}
{\mathrm{I}}(T ; R) &\geq \\
&{\mathrm{I}}(T ; {\texttt{id}\xspace}(R)) = {\mathrm{I}}(T; W) \geq {\mathrm{I}}(T; {\mathbf{e}}(W)) \nonumber\end{aligned}$$ This result[^3] is intuitive and, perhaps, trivial—context matters information-theoretically. However, it gives us a principled foundation by which to measure the effectiveness of probes as we will show in \[sec:gain\].
How Much Information Did We Gain? {#sec:gain}
---------------------------------
We will now quantify how much a contextualized word embedding knows about a task with respect to a specific control function ${\mathbf{c}}(\cdot)$. We term how much more information the contextualized embeddings have about a task than a control variable the [**gain**]{}, which we define as $$\begin{aligned}
\label{eq:gain}
{\mathcal{G}}(T, &\,R, {\mathbf{c}}) = {\mathrm{I}}(T; R) - {\mathrm{I}}(T; {\mathbf{c}}(R)) \\
&= {\mathrm{H}}(T \mid {\mathbf{c}}(R)) - {\mathrm{H}}(T \mid R) \geq 0 \nonumber\end{aligned}$$ The gain function will be our method for measuring how much more information contextualized representations have over a controlled baseline, encoded as the function ${\mathbf{c}}$. We will empirically estimate this value in \[sec:experiments\].
Interestingly enough, the gain has a straightforward interpretation.
\[prop:interpretation\] The gain function is equal to the following conditional mutual information $${\mathrm{I}}(T; R \mid {\mathbf{c}}(R)) = {\mathcal{G}}(T, R, {\mathbf{c}})$$
$$\begin{aligned}
\!{\mathrm{I}}(T; R \mid {\mathbf{c}}(R)) &:= {\mathrm{I}}(T; R) - {\mathrm{I}}(T; R; {\mathbf{c}}(R)) \\
&= {\mathrm{I}}(T; R) - {\mathrm{I}}(T; {\mathbf{c}}(R)) \\
&= {\mathcal{G}}(T, R, {\mathbf{c}})\end{aligned}$$
The jump from the first to the second equality follows since $R$ encodes all the information about $T$ provided by ${\mathbf{c}}(R)$ by construction.
\[prop:interpretation\] gives us a clear understanding of the quantity we wish to estimate: It is how much information about a task is encoded in the representations, given some control knowledge. If properly designed, this control transformation will remove information from the probed representations.
Approximating the Gain
----------------------
The gain, as defined in \[eq:gain\], is intractable to compute. In this section we derive a pair of variational bounds on ${\mathcal{G}}(T, R, {\mathbf{e}})$—one upper and one lower. To approximate the gain, we will simultaneously minimize an upper and a lower-bound on \[eq:gain\]. We begin by approximating the gain in the following manner $$\begin{aligned}
\label{eq:approx}
{\mathcal{G}}(T, R, {\mathbf{e}}) &\approx \\
&\underbrace{{{\mathrm{H}}_{{q_{{{\boldsymbol \theta}}2}}}}(T ; {\mathbf{c}}(R)) - {{\mathrm{H}}_{{q_{{{\boldsymbol \theta}}1}}}}(T \mid R)}_{\textit{estimated }{\mathcal{G}}_{{q_{{{\boldsymbol \theta}}}}}(T, R, {\mathbf{e}})} \nonumber\end{aligned}$$ these cross-entropies can be empirically estimated. We will assume access to a corpus $\{(t_i, {\mathbf{r}}_i)\}_{i=1}^N$ that is human-annotated for the target linguistic property; we further assume that these are samples $(t_i, {\mathbf{r}}_i) \sim p(\cdot, \cdot)$ from the true distribution. This yields a second approximation that is tractable: $${{\mathrm{H}}_{{q_{{{\boldsymbol \theta}}}}}}(T ; R) \approx \frac{1}{N} \sum\limits_{i=1}^{N} \log {q_{{{\boldsymbol \theta}}}}(t_i \mid {\mathbf{r}}_i)$$ This approximation is exact in the limit $N \rightarrow \infty$ by the law of large numbers.
We note the approximation given in \[eq:approx\] may be either positive or negative and its estimation error follows from \[eq:ent\_estimate\] $$\begin{aligned}
\label{eq:approx_error}
\Delta &\,= \underset{{\mathbf{r}}\sim p(\cdot)}{\mathbb{E}} {\mathrm{KL}}(p(\cdot \mid {\mathbf{r}}) \mid \mid {q_{{{\boldsymbol \theta}}1}}(\cdot \mid {\mathbf{r}})) \\
&- \underset{{\mathbf{r}}\sim p(\cdot)}{\mathbb{E}} {\mathrm{KL}}(p(\cdot \mid {\mathbf{c}}({\mathbf{r}})) \mid \mid {q_{{{\boldsymbol \theta}}2}}(\cdot \mid {\mathbf{c}}({\mathbf{r}}))) \nonumber \\
&\,= {{\mathrm{KL}}_{{q_{{{\boldsymbol \theta}}1}}}}(T, R) - {{\mathrm{KL}}_{{q_{{{\boldsymbol \theta}}2}}}}(T, {\mathbf{c}}(R)) \nonumber\end{aligned}$$ where we abuse the KL notation to simplify the equation. This is an undesired behavior since we know the gain itself is non-negative, by the data-processing inequality, but we have yet to devise a remedy.=-1
We justify the approximation in \[eq:approx\] with a pair of variational bounds. The following two corollaries are a result of \[thm:variational\_bounds\] in \[appendix:a\].
\[cor:upper\] We have the following upper-bound on the gain $$\begin{aligned}
\label{eq:upper}
{\mathcal{G}}(T, R,{\mathbf{e}}) & \\
&\leq {\mathcal{G}}_{{q_{{{\boldsymbol \theta}}}}}(T, R,{\mathbf{e}})\!+\! {{\mathrm{KL}}_{{q_{{{\boldsymbol \theta}}1}}}}(T, R) \nonumber\end{aligned}$$
\[cor:lower\] We have the following lower-bound on the gain $$\begin{aligned}
\label{eq:lower}
{\mathcal{G}}(T, R,{\mathbf{e}}) & \\
&\geq {\mathcal{G}}_{{q_{{{\boldsymbol \theta}}}}}(T, R,{\mathbf{e}}) - {{\mathrm{KL}}_{{q_{{{\boldsymbol \theta}}2}}}}(T, {\mathbf{c}}(R)) \nonumber\end{aligned}$$
The conjunction of \[cor:upper\] and \[cor:lower\] suggest a simple procedure for finding a good approximation: We choose ${q_{{{\boldsymbol \theta}}1}}(\cdot \mid r)$ and ${q_{{{\boldsymbol \theta}}2}}(\cdot \mid r)$ so as to *minimize* \[eq:upper\] and *maximize* \[eq:lower\], respectively. These distributions contain no overlapping parameters, by construction, so these two optimization routines may be performed independently. We will optimize both with a gradient-based procedure, discussed in \[sec:experiments\].
Understanding Probing Information-Theoretically
===============================================
In \[sec:control-functions\] we developed an information-theoretic framework for thinking about probing contextual word embeddings for linguistic structure. However, we now cast doubt on whether probing makes sense as a scientific endeavour. We prove in \[sec:context\] that contextualized word embeddings, by construction, contain no more information about a word-level syntactic task than the original sentence itself. Nevertheless, we do find a meaningful scientific interpretation of control functions. We expound upon this in \[sec:control-functions-meaning\], arguing that control functions are useful, not for understanding representations, but rather for understanding the influence of sentential context on word-level syntactic tasks, e.g., labeling words with their part of speech.
You Know Nothing, BERT {#sec:context}
----------------------
To start, we note the following corollary
It directly follows from \[ass:one\] that ${\textsc{bert}\xspace}$ is a bijection between sentences ${\mathbf{s}}$ and sequences of embeddings $\langle {\mathbf{r}}_1, \ldots, {\mathbf{r}}_{|{\mathbf{s}}|} \rangle$. As ${\textsc{bert}\xspace}$ is a bijection, it has an inverse, which we will denote as ${\textsc{bert}^{-1}\xspace}$.
\[thm:bert\] The function ${\textsc{bert}\xspace}(S)$ cannot provide more information about $T$ than the sentence $S$ itself.
$$\begin{aligned}
{\mathrm{I}}(T ; S) &\geq {\mathrm{I}}(T; {\textsc{bert}\xspace}(S)) \\
&\geq {\mathrm{I}}(T; {\textsc{bert}^{-1}\xspace}({\textsc{bert}\xspace}(S))) \nonumber \\
&= {\mathrm{I}}(T ; S) \nonumber \end{aligned}$$
This implies ${\mathrm{I}}(T ; S) = {\mathrm{I}}(T; {\textsc{bert}\xspace}(S))$. We remark this is not a BERT-specific result—it rests on the fact that the data-processing inequality is tight for bijections.
While \[thm:bert\] is a straightforward application of the data-processing inequality, it has deeper ramifications for probing. It means that if we search for syntax in the contextualized word embeddings of a sentence, we should not expect to find any more syntax than is present in the original sentence. In a sense, \[thm:bert\] is a cynical statement: the endeavour of finding syntax in contextualized embeddings sentences is nonsensical. This is because, under \[ass:one\], we know the answer *a priori*—the contextualized word embeddings of a sentence contain exactly the same amount of information about syntax as does the sentence itself.
What Do Control Functions Mean? {#sec:control-functions-meaning}
-------------------------------
Information-theoretically, the interpretation of control functions is also interesting. As previously noted, our interpretation of control functions in this work does not provide information about the representations themselves. Actually, the same reasoning used in \[cor:one\] could be used to devise a function ${\texttt{id}\xspace}_s({\mathbf{r}})$ which led from a single representation back to the whole sentence. For a type-level control function ${\mathbf{c}}$, by the data-processing inequality, we have that ${\mathrm{I}}(T; W) \geq {\mathrm{I}}(T; {\mathbf{c}}(R))$. Consequently, we can get an upper-bound on how much information we can get out of a decontextualized representation. If we assume we have perfect probes, then we get that the true gain function is ${\mathrm{I}}(T; S) - {\mathrm{I}}(T; W) = {\mathrm{I}}(T; S \mid W)$. This quantity is interpreted as the amount of knowledge we gain about the word-level task $T$ by knowing $S$ (i.e., the sentence) in addition to $W$ (i.e., the word). Therefore, a perfect probe would provide insights about language and not about the actual representations, which are no more than a means to an end.
Discussion: Ease of Extraction {#sec:ease-extract}
------------------------------
We do acknowledge another interpretation of the work of *inter alia*; BERT makes the syntactic information present in an ordered sequence of words more easily extractable. However, ease of extraction is not a trivial notion to formalize, and indeed, we know of no attempt to do so; it is certainly more complex to determine than the number of layers in a multi-layer perceptron (MLP). Indeed, a MLP with a single hidden layer can represent any function over the unit cube, with the caveat that we may need a very large number of hidden units [@cybenko1989approximation].
Although for perfect probes the above results should hold, in practice ${\texttt{id}\xspace}(\cdot)$ and ${\mathbf{c}}(\cdot)$ may be hard to approximate. Furthermore, if these functions were to be learned, they might require an unreasonably large dataset. A random embedding control function, for example, would require an infinitely large dataset to be learned—or at least one that contained all words in the vocabulary $V$. “Better” representations should make their respective probes more easily learnable—and consequently their encoded information more accessible.
We suggest that future work on probing should focus on operationalizing ease of extraction more rigorously—even though we do not attempt this ourselves. The advantage of simple probes is that they may reveal something about the *structure* of the encoded information—i.e., is it structured in such a way that it can be easily taken advantage of by downstream consumers of the contextualized embeddings? We suspect that many researchers who are interested in less complex probes have implicitly had this in mind.
A Critique of Control Tasks
===========================
While this paper builds on the work of , and we agree with them that we should have control tasks when probing for linguistic properties, we disagree with parts of the methodology for the control task construction. We present these disagreements here.
Structure and Randomness
------------------------
introduce control tasks to evaluate the effectiveness of probes. We draw inspiration from this technique as evidenced by our introduction of control functions. However, we take issue with the suggestion that controls should have [**structure**]{} and [**randomness**]{}, to use the terminology from . They define structure as “the output for a word token is a deterministic function of the word type.” This means that they are stripping the language of ambiguity with respect to the target task. In the case of part-of-speech labeling, [*love*]{} would either be a [<span style="font-variant:small-caps;">noun</span>]{} or a [<span style="font-variant:small-caps;">verb</span>]{} in a control task, never both: this is a problem. The second feature of control tasks is randomness, i.e., “the output for each word type is sampled independently at random.[^4]” In conjunction, structure and randomness may yield a relatively trivial task that does not look at all like natural language.
What is more, there is a closed-form solution for an optimal, retrieval-based “probe” that has zero parameters: [^5] If a word type appears in the training set, return the label with which it was annotated there, otherwise return the most frequently occurring label across all words in the training set. This probe will achieve an accuracy that is 1 minus the out-of-vocabulary rate (the number of tokens in the test set that correspond to novel types divided by the number of tokens) times the percentage of tags in the test set that do not correspond to the most frequent tag (the error rate of the guess-the-most-frequent-tag classifier). In short, the best model for a control task is a pure memorizer that guesses the most frequent tag for out-of-vocabulary words.
What’s Wrong with Memorization?
-------------------------------
propose that probes should be optimised to maximise accuracy *and* selectivity. Recall selectivity is given by the distance between the accuracy on the original task and the accuracy on the control task using the same architecture. Given their characterization of control tasks, maximising selectivity leads to a selection of a model that is bad at memorization. But why should we punish memorization? Much of linguistic competence is about generalization, however memorization also plays a key role [@fodor1974; @nooteboom2002storage; @fromkin2018], with word learning [@carey1978] being an obvious example. Indeed, maximizing selectivity as a criterion for creating probes seems to artificially disfavor this property.
\[tab:results-full\]
What Low-Selectivity Means
--------------------------
acknowledge that for the more complex task of dependency edge prediction, a MLP probe is more accurate and, therefore, preferable despite its low selectivity. However, they offer two counter-examples where the less selective neural probe exhibits drawbacks when compared to its more selective, linear counterpart. We believe both examples are a symptom of using a simple probe rather than of selectivity being a useful metric for probe selection. First, point out that, in their experiments, the MLP-1 model frequently mislabels the word with suffix *-s* as [<span style="font-variant:small-caps;">NNPS</span>]{} on the POS labeling task. They present this finding as a possible example of a less selective probe being less faithful in representing what linguistic information has the model learned. Our analysis leads us to believe that, on contrary, this shows that one should be using the best possible probe to minimize the chance of misrepresentation. Since more complex probes achieve higher accuracy on the task, as evidence by the findings of , we believe that the overall trend of misrepresentation is higher for the probes with higher selectivity. The same applies for the second example discussed in section where a less selective probe appears to be less faithful. The authors show that the representations on ELMo’s second layer fail to outperform its word type ones (layer zero) on the POS labeling task when using the MLP-1 probe. While they argue this is evidence for selectivity being a useful metric in choosing appropriate probes, we argue that this demonstrates yet again that one needs to use a more complex probe to minimize the chances of misrepresenting what the model has learned. The fact that the linear probe shows a difference only demonstrates that the information is perhaps more accessible with ELMo, not that it is not present; see \[sec:ease-extract\].
Experiments {#sec:experiments}
===========
We consider the task of POS labeling and use the universal POS tag information [@petrov-etal-2012-universal] from the Universal Dependencies 2.4 [@11234/1-2988]. We probe the multilingual release of BERT[^6] on six typologically diverse languages: Basque, Czech, English, Finnish, Tamil, and Turkish; and we compute the contextual representations of each sentence by feeding it into BERT and averaging the output word piece representations for each word, as tokenized in the treebank.
Control Functions {#control-functions}
-----------------
We will consider three different control functions. Each is defined as the composition ${\mathbf{c}}= {\mathbf{e}}\circ {\texttt{id}\xspace}$ with a different look-up function. These look-up functions are
- ${\mathbf{e}}_\textit{fastText}$ returns a language specific fastText embedding [@fasttext];
- ${\mathbf{e}}_\textit{onehot}$ returns a one-hot embedding;[^7]
- ${\mathbf{e}}_\textit{random}$ returns a fixed random embedding.[^8]
All of these functions are type level in that they remove the influence of the context on the word.
Probe Architecture
------------------
As expounded upon above, our purpose is to achieve the best bound on mutual information we can. To this end, we employ a deep MLP as our probe. We define the probe as $$\begin{aligned}
{q_{{{\boldsymbol \theta}}}}(t &\mid {\mathbf{r}}) =\\
&\text{softmax}\left( W^{(m)} \sigma\left(W^{(m-1)}\cdots \sigma(W^{(1)}\,{\mathbf{r}})\right)\right) \nonumber\end{aligned}$$ an $m$-layer neural network with the non-linearity $\sigma(\cdot) = {\mathrm{ReLU}}(\cdot)$. The initial projection matrix is $W^{(1)} \in {\mathbb{R}}^{r_1 \times d}$ and the final projection matrix is $W^{(m)} \in {\mathbb{R}}^{|{\mathcal{T}}| \times r_{m-1}}$, where $r_i=\frac{r}{2^{i-1}}$. The remaining matrices are $W^{(i)} \in {\mathbb{R}}^{r_i \times r_{i-1}}$, so we half the number of hidden states in each layer. We optimize over the hyperparameters—number of layers, hidden size, one-hot embedding size, and dropout—by using random search. For each estimate, we train 50 models and choose the one with the best validation cross-entropy. The cross-entropy in the test set is then used as our entropy estimate.
Results
-------
We know ${\textsc{bert}\xspace}$ can generate text in many languages, here we assess how much does it actually know about syntax in those languages. And how much more does it know than simple type-level baselines. \[tab:results-full\] presents this results, showing how much information ${\textsc{bert}\xspace}$, fastText and onehot embeddings encode about POS tagging. We see that—in all analysed languages—type level embeddings can already capture most of the uncertainty in POS tagging. We also see that BERT only shares a small amount of extra information with the task, having small (or even negative) gains in all languages.
${\textsc{bert}\xspace}$ presents negative gains in some of the analysed languages. Although this may seem to contradict the information processing inequality, it is actually caused by the difficulty of approximating ${\texttt{id}\xspace}$ and ${\mathbf{c}}(\cdot)$ with a finite training set—causing ${{\mathrm{KL}}_{{q_{{{\boldsymbol \theta}}1}}}}(T \mid R)$ to be larger than ${{\mathrm{KL}}_{{q_{{{\boldsymbol \theta}}2}}}}(T \mid {\mathbf{c}}(R))$. We believe this highlights the need to formalize *ease of extraction*, as discussed in \[sec:ease-extract\].
Finally, when put into perspective, multilingual ${\textsc{bert}\xspace}$’s representations do not seem to encode much more information about syntax than a trivial baseline. ${\textsc{bert}\xspace}$ only improves upon fastText in three of the six analysed languages—and even in those, it encodes at most (in English) $5\%$ additional information.
Conclusion
==========
We proposed an information-theoretic formulation of probing: we define probing as the task of estimating conditional mutual information. We introduce control functions, which allows us to put the amount of information encoded in contextual representations in the context of knowledge judged to be trivial. We further explored this formalization and showed that, given perfect probes, probing can only yield insights into the language itself and tells us nothing about the representations under investigation. Keeping this in mind, we suggested a change of focus—instead of focusing on probe size or information, we should look at *ease of extraction* going forward. On another note, we apply our formalization to evaluate multilingual ${\textsc{bert}\xspace}$’s syntax knowledge on a set of six typologically diverse languages. Although it does encode a large amount of information about syntax (more than $81\%$ in all languages[^9]), it only encodes at most $5\%$ more information than some trivial baseline knowledge (a type-level representation). This indicates that the task of POS labeling (word-level POS tagging) is not an ideal task for contemplating the syntactic understanding of contextual word embeddings.
Variational Bounds {#appendix:a}
==================
\[thm:variational\_bounds\] The estimation error between ${\mathcal{G}}_{{q_{{{\boldsymbol \theta}}}}}(T, R, {\mathbf{e}})$ and the true gain can be upper- and lower-bounded by two distinct Kullback–Leibler divergences.
We first find the error given by our estimate $$\begin{aligned}
{\mathcal{G}}(T, R, {\mathbf{e}}) &:= {\mathrm{H}}(T ; {\mathbf{c}}(R)) - {\mathrm{H}}(T \mid R) \\
&= {{\mathrm{H}}_{{q_{{{\boldsymbol \theta}}2}}}}(T \mid {\mathbf{c}}(R)) - \underset{{\mathbf{r}}\sim p(\cdot)}{\mathbb{E}} {\mathrm{KL}}(p(\cdot \mid {\mathbf{c}}({\mathbf{r}})) \mid \mid {q_{{{\boldsymbol \theta}}2}}(\cdot \mid {\mathbf{c}}({\mathbf{r}}))) \nonumber \\
& \qquad - {{\mathrm{H}}_{{q_{{{\boldsymbol \theta}}1}}}}(T \mid R) + \underset{{\mathbf{r}}\sim p(\cdot)}{\mathbb{E}} {\mathrm{KL}}(p(\cdot \mid {\mathbf{r}}) \mid \mid {q_{{{\boldsymbol \theta}}1}}(\cdot \mid {\mathbf{r}})) \nonumber\\
&= {{\mathrm{H}}_{{q_{{{\boldsymbol \theta}}2}}}}(T \mid {\mathbf{c}}(R)) - {{\mathrm{KL}}_{{q_{{{\boldsymbol \theta}}2}}}}(T, {\mathbf{c}}(R)) - {{\mathrm{H}}_{{q_{{{\boldsymbol \theta}}1}}}}(T \mid R) + {{\mathrm{KL}}_{{q_{{{\boldsymbol \theta}}1}}}}(T \mid R) \nonumber \\
&= {{\mathrm{H}}_{{q_{{{\boldsymbol \theta}}2}}}}(T \mid {\mathbf{c}}(R)) - {{\mathrm{H}}_{{q_{{{\boldsymbol \theta}}1}}}}(T \mid R) + {{\mathrm{KL}}_{{q_{{{\boldsymbol \theta}}1}}}}(T \mid R) - {{\mathrm{KL}}_{{q_{{{\boldsymbol \theta}}2}}}}(T, {\mathbf{c}}(R)) \nonumber \\
&= \underbrace{{\mathcal{G}}_{{q_{{{\boldsymbol \theta}}}}}(T, R, {\mathbf{e}})}_{\textit{estimated gain}} + \underbrace{{{\mathrm{KL}}_{{q_{{{\boldsymbol \theta}}1}}}}(T \mid R) - {{\mathrm{KL}}_{{q_{{{\boldsymbol \theta}}2}}}}(T, {\mathbf{c}}(R))}_{\textit{estimation error}} \nonumber\end{aligned}$$ Making use of this error, we trivially find an upper-bound on the estimation error as $$\begin{aligned}
\Delta &= {{\mathrm{KL}}_{{q_{{{\boldsymbol \theta}}1}}}}(T \mid R) - {{\mathrm{KL}}_{{q_{{{\boldsymbol \theta}}2}}}}(T, {\mathbf{c}}(R)) \\
&\le {{\mathrm{KL}}_{{q_{{{\boldsymbol \theta}}1}}}}(T \mid R) \nonumber\end{aligned}$$ which follows since KL divergences are never negative. Analogously, we find a lower-bound as $$\begin{aligned}
\Delta &= {{\mathrm{KL}}_{{q_{{{\boldsymbol \theta}}1}}}}(T \mid R) - {{\mathrm{KL}}_{{q_{{{\boldsymbol \theta}}2}}}}(T, {\mathbf{c}}(R)) \\
&\ge - {{\mathrm{KL}}_{{q_{{{\boldsymbol \theta}}2}}}}(T, {\mathbf{c}}(R)) \nonumber\end{aligned}$$
Further Results {#appendix:further_results}
===============
In this section, we present accuracies for the models trained using ${\textsc{bert}\xspace}$, fastText and onehot embeddings, and the full results on random embeddings. \[tab:results-extra\] shows that both BERT and fastText present high accuracies in all languages, except Tamil. Onehot and random results are considerably worse, as expected, since they could not do more than take random guesses (e.g. guessing the most frequent label in the training test) in any word which was not seen during training.
-------------------------------- -------------------------- ---------- -------- -------- ---------------------------------------- -------------------------------------
(r)[2-5]{} (r)[6-7]{} Language ${\textsc{bert}\xspace}$ fastText onehot random ${\mathrm{H}}(T \mid {\mathbf{c}}(R))$ ${\mathcal{G}}(T, R, {\mathbf{c}})$
Basque 0.93 0.93 0.81 0.82 0.80 0.49 (15.4%)
Czech 0.98 0.97 0.91 0.87 0.54 0.46 (13.8%)
English 0.96 0.91 0.84 0.84 0.68 0.47 (13.0%)
Finnish 0.95 0.96 0.80 0.80 0.89 0.65 (20.6%)
Tamil 0.87 0.84 0.66 0.66 1.52 0.94 (29.3%)
Turkish 0.93 0.94 0.79 0.80 0.83 0.50 (16.6%)
-------------------------------- -------------------------- ---------- -------- -------- ---------------------------------------- -------------------------------------
: Accuracies of the models trained on ${\textsc{bert}\xspace}$, fastText, onehot and random embeddings for the POS tagging task.[]{data-label="tab:results-extra"}
[^1]: Assuming that the authors intended ‘or’ here as strictly non-inclusive.
[^2]: Indeed, even if we sampled every embedding randomly from a $d$-dimensional Gaussian, the probability that we would ever sample the same real value is zero.
[^3]: Note that although this result holds in theory, in practice the functions ${\texttt{id}\xspace}$ and ${\mathbf{e}}(\cdot)$ might be arbitrarily hard to estimate. This is discussed in length in \[sec:ease-extract\].
[^4]: But not necessarily uniformly.
[^5]: Actually, to be more precise, it will have $|V|+1$ parameters. One for each word in the vocabulary, plus one for the most frequent label.
[^6]: We used the implementation made available by
[^7]: We initialize random embeddings at the type level, and let them train during the model’s optimization.
[^8]: We generate the random embeddings once before the task, at the type level. Results for this control are in the Appendix.
[^9]: This is measured as the relative difference between ${\mathrm{H}}(T)$ and ${\mathrm{H}}(T\mid R)$. On average, this value is $91\%$.
|
epsf
A metal-insulator transition (MIT) observed in Si metal-oxide-semiconductor field-effect transistors [@KRAV1; @POPO] (Si-MOSFET’s) and other systems [@SIGE1; @SIGE2; @GAAS1; @GAAS2] attracts a great deal of attention since it seems to contradict an important result of the scaling theory by Abrahams [*et al.*]{} [@GANG4] that the conductance of a disordered two-dimensional (2D) system at zero magnetic field goes to zero for $ T \rightarrow 0 $. In the metallic phase in Si-MOSFET’s with high peak electron mobilities of $ \mu _{ \rm peak} \gtrsim 2~{ \rm m }^{2} / { \rm V~s} $, the diagonal resistivity $ \rho_{ xx } $ shows a sharp drop with decreasing temperature from about $ 2~{\rm K} $ [@KRAV1]. Recent experiments [@KRAV2; @PUD1] show that magnetic fields applied parallel to the 2D plane suppress the low temperature metallic conduction in Si-MOSFET’s. Since the parallel magnetic field does not couple the orbital motion of electrons, this fact suggests an important role of the spin of electrons. However, the mechanism of the conduction in the anomalous metallic phase is not clear yet.
The 2D systems that show the MIT [@KRAV1; @POPO; @SIGE1; @SIGE2; @GAAS1; @GAAS2] are characterized by strong Coulomb interaction between electrons. The mean Coulomb energy per electron $ U = ( \pi N _ {s} )^{ 1 / 2 } e ^ { 2 } / 4 \pi \varepsilon _ { 0 } \kappa $ is larger than the mean kinetic energy $ K = \pi \hbar ^ { 2 } N _ { s } / m ^ { \ast } $ by an order of the magnitude around the critical point for the MIT. Here, $ N _ {s} $ is the electron concentration, $ \kappa $ is the relative dielectric constant at the interface, and $ m ^ { \ast } $ is the effective mass of electron. It is estimated that $ U = 120~{ \rm K } $, $ K = 14~{ \rm K } $ and the ratio $ r _ { s } = U / K = 8.3 $ for $ \kappa = 7.7 $ and $ m ^ { \ast } = 0.19 m _ { e } $ at $ N _ {s} = 1 \times 10 ^ { 15 } {\rm m } ^ { -2 } $ in Si-MOSFET’s. The ground state of the insulating phase of high mobility Si-MOSFET’s is considered to be a pinned Wigner solid (WS) [@PUD2; @CHUI]. Magnetic field dependence of the thermal activation energy observed for various angles of the magnetic field was essentially explained by a model based on magnetic interactions in the pinned WS [@OKA1; @OKA2]. Although the quantum fluctuations change the 2D system into a liquid at higher-$ N _ {s} $, electron-electron ([*e-e*]{}) interaction is expected to be still important.
In the conduction band of silicon, the spin-orbit interaction is negligible and the spin polarization $ p = ( N _ { \uparrow } - N _ { \downarrow } ) / N _ { s } $ can always be given in the direction to the magnetic field. Here $ N _ { \uparrow } $ and $ N _ { \downarrow } $ are the concentrations of electrons having an up spin and a down spin, respectively $ ( N _ { s } = N _ { \uparrow } + N _ { \downarrow } ) $. In the present work, we investigate the low temperature conduction in a high mobility Si-MOSFET for various values of $ p $.
The Si-MOSFET sample used has a peak electron mobility of $ \mu _{ \rm peak} = 2.4~{ \rm m }^{2} / { \rm V~s} $ at $ N _ { s } = 4 \times 10 ^ { 15 } { \rm m } ^ { -2 } $ and $ T = 0.3~{ \rm K } $. It has a Hall-bar geometry of total length 3 mm and width 0.3 mm. The estimated SiO$_2$ layer thickness is 98 nm. The diagonal resistivity $ \rho _ { xx } $ as well as the Hall resistivity $ \rho _ { xy } $ was obtained from the linear relationship in the [ *I-V* ]{} characteristics using a four-probe dc method. The source-drain current and the potential difference across the two potential probes separated by 1.5 mm were limited below 10 nA and 0.4 mV, respectively. The sample was mounted on a rotatory thermal stage in a dilution refrigerator together with a GaAs Hall generator and a RuO$_2$ resistance thermometer calibrated in magnetic fields [@OKA2].
As shown in Fig. \[FGMIT\], the MIT is clearly observed in a zero magnetic field.
The critical value of $ \rho _ { xx } $ at the MIT is $ 55~{ \rm k } \Omega $. It is close to $ \rho _ { c } \sim 2 h / e ^ { 2 } $ reported by Kravchenko [*et al.*]{} [@KRAV1]. A high magnetic field of 9 T parallel to the 2D plane drastically increases $ \rho _ { xx } $ and eliminates the positive temperature dependence even in the low-$ \rho _ { xx } $ region [@HT]. The disappearance of the MIT due to the parallel magnetic field seems to resemble that induced by increasing disorder reported by Popović [*et al.*]{} [@POPO].
In non-interacting degenerate 2D Fermi gases, the spin polarization $ p $ increases linearly with the total strength $ B _ { \rm tot } $ of the magnetic field for $ p < 1 $ and reaches $ p = 1 $ at the critical magnetic field of $ B_{ c } = 2 \pi \hbar ^ { 2 } N _ { s } / \mu _ { B } g _ { v } g ^ { \ast } m ^ { \ast } $. Here, $ \mu _ { B } $ is the Bohr magneton $ ( = \hbar e / 2 m _ { e } ) $. The valley degeneracy $ g _ { v } $ is 2 on the (001) surface of silicon and the effective $ g $-factor $ g ^ { \ast } $ is 2.0 in the conduction band in silicon [@KODERA]. According to Landau’s Fermi liquid theory, the [*e-e*]{} interaction changes the effective $ g $-factor and mass of electrons (or quasiparticles) into enhanced values denoted by $ g _ { \rm FL } $ and $ m _ { \rm FL } $, and reduces the critical magnetic field.
The product of $ g _ { \rm FL } $ and $ m _ { \rm FL } $ in (001) Si-MOSFET’s can be determined from the Shubnikov-de Haas oscillations in tilted magnetic fields [@FS]. Figure \[FGOS\] shows the diagonal conductivity $ \sigma _ { xx } = \rho _ { xx } / ( \rho _ { xx } {} ^ { 2 } + \rho _ { xy } {} ^ { 2 } ) $ divided by $ e N _ { s } $ for various angles of the magnetic field to the 2D plane.
The normal component $ B _ { \perp } $ of the magnetic field is fixed at $ 0.80~{ \rm T } $. The upper abscissa is given by a half of the Landau level filling factor $ \nu = N _ { s } \phi _ { 0 } / B _ { \perp } $ ($ \phi _ { 0 } = h / e $) taking into account the valley degeneracy $ g _ { v } = 2 $. We ignore the valley splitting energy since no indication is observed at $ \nu = odd $. For $ B _ { \rm tot } = 0.80~{ \rm T } $, $ \sigma _ { xx } / e N _ { s } $ takes minima at even-numbers of $ \nu / 2 $ where the Fermi energy is located at the middle of a gap between Landau levels having different Landau quantum number and spin index. This energy gap $ \Delta E _ { even } = \hbar e B _ { \perp } / m _ { \rm FL } - g _ { \rm FL } \mu _ { B } B _ { \rm tot } $ decreases with $ B _ { \rm tot } $ and the minima in $ \sigma _ { xx } / e N _ { s } $ at even-numbers of $ \nu / 2 $ fade away at higher-$ B _ { \rm tot } $ due to the level broadening. On the other hand, the spin splitting $ \Delta E _ { odd } = g _ { \rm FL } \mu _ { B } B _ { \rm tot } $ increases with $ B _ { \rm tot } $ and the minima in $ \sigma _ { xx } / e N _ { s } $ at odd-numbers of $ \nu / 2 $ become clearer. We have evaluated the critical angle $ \theta _ { c } $ of the magnetic field to the 2D plane where $ \Delta E _ { odd } $ becomes equal to $ \Delta E _ { even } $ from comparing the depth of the minimum in $ \sigma _ { xx } / e N _ { s } $ at an odd-number of $ \nu / 2 $ with the average of those at $ \nu / 2 - 1 $ and $ \nu / 2 + 1 $ for various values of $ B _ { \rm tot } $. We assumed that the level broadening is a smooth function or independent of $ \nu $. Thus we have $ g _ { \rm FL } m _ { \rm FL } = m _ { e } B _ { \perp } / B _ { \rm tot } = m _ { e } \sin \theta _ { c } $.
In Fig. \[FGAL\], we show the ratio of $ g _ { \rm FL } m _ { \rm FL } $ to $ g ^ { \ast } m ^ { \ast } $ in the non-interacting system.
The enhancement factor $ \alpha = g _ { \rm FL } m _ { \rm FL } / 0.38 m_ { e } $ increases monotonically with the strength of the [*e-e*]{} interaction. Results on much lower mobility samples measured in higher magnetic fields by Fang and Stiles [@FS] are also plotted. No $ B _ { \perp } $-dependence is found between our results at $ 0.8~{ \rm T } $ and at $ 1.2~{ \rm T } $. We believe that these values of $ B _ { \perp } $ are small enough and $ g _ { \rm FL } m _ { \rm FL } $ was determined in the limit of $ B _ { \perp } = 0 $. While $ B _ { \perp } $ may cause a small oscillation of the effective $ g $-factor with the maxima $ g _ { \rm FL } ^ { \rm MAX } $ at $ \nu / 2 = odd $ and the minima $ g _ { \rm FL } ^ { \rm MIN } $ at $ \nu / 2 = even $ [@KOBAYASHI], $ g _ { \rm FL } m _ { \rm FL } $ determined from the critical angle for $ \Delta E _ { odd } = \Delta E _ { even } $ is related to the average of $ g _ { \rm FL } ^ { \rm MAX } $ and $ g _ { \rm FL } ^ { \rm MIN } $. The present method cannot be used in the large-$ r _ { s } $ region where $ g _ { \rm FL } m _ { \rm FL } / m _ { e } $ exceeds unity ($ \alpha > 2.63 $). We use a simple extrapolation function $$\begin{aligned}
\alpha - 1 = 0.2212 r _ { s } + 0.003973 r _ { s } {} ^ { 2 } .
\label{Hex}\end{aligned}$$ represented by the solid line in Fig. \[FGAL\]. The critical magnetic field $ B _ { c } = 2 \pi \hbar ^ { 2 } N _ { s } / \mu _ { B } g _ { v } g _ { \rm FL } m _ { \rm FL } $ is calculated using (\[Hex\]) as shown in the inset to Fig. \[FGAL\].
In Fig. \[FGPARA\], $ \rho _ { xx } $ at $ T = 0.21~{ \rm K } $ in the parallel magnetic field ($ B _ {\perp} = 0 $) is shown.
The diagonal resistivity $ \rho _ { xx } $ increases with $ B _ { \rm tot } $ in the low-$ B _ { \rm tot } $ region, but it takes almost constant values in the high-$ B _ { \rm tot } $ region. The saturation of $ \rho _ { xx } $ occurs around $ B _ { \rm tot } = B _ { c } $ at which the spin polarization is expected to be completed. The result indicates that the mixing of the different spin states causes the reduction of $ \rho _ { xx } $ in the low-$ B _ { \rm tot } $ region. The dashed line represents the critical value of $ \rho _ { xx } $ at the MIT. It was tentatively determined from the sign of the change in $ \rho _ { xx } $ from $ T = 0.21~{ \rm K } $ to $ 0.91~{ \rm K } $. The positive temperature dependence of $ \rho _ { xx } $ below $ 1~{ \rm K} $ was not observed for $ B _ { \rm tot } > B _ { c } $. The positive $ T $-dependence of $ \rho _ { xx } $ in the low-$ B _ { \rm tot } $ region may arise from the scattering related to the spin degree of freedom.
The most important finding has been obtained by rotating the sample in the magnetic field. Results at $ N _ { s } = 1.46 \times 10 ^ { 15 } {\rm m } ^ { -2 } $ are shown in Fig. \[FGROT\].
We focus on the $ \rho _ { xx } $ oscillation that appears in the low-$ B _ { \perp } $ region, while the $ \rho _ { xx } ( B _ { \perp } ) $ curves for different $ B _ { \rm tot } $ approach each other in the higher-$ B _ { \perp } $ region (not shown in Fig. \[FGROT\]) and converge on a single curve for $ B _ { \perp } / N _ { s } \phi _ { 0 } > 0.5 $ with two deep minima at $ B _ { \perp } / N _ { s } \phi _ { 0 } = 0.5 $ ($ \nu = 2 $) and $ B _ { \perp } / N _ { s } \phi _ { 0 } = 1.0 $ ($ \nu = 1 $) as reported by Kravchenko [*et al.*]{} [@TILT]. Unlike ordinary Shubnikov-de Haas oscillations that show $ \rho _ { xx } $ minima at fixed points with $ \nu = { \rm integer } $, the $ \rho _ { xx } $ minima indicated by the black arrows and the white arrows in Fig. \[FGROT\] shift toward high-$ B _ { \perp } $ side as $ B _ { \rm tot } $ increases. Such peculiar behavior was also observed for lower-$ N _ { s } $ down to $ 1.02 \times 10 ^ { 15 } {\rm m } ^ { -2 } $.
In Fig. \[FGMIN\], the positions of the $ \rho _ { xx } $ minima for various $ N _ { s } $ are shown.
Both $ B _ { \perp } $ and $ B _ { \rm tot } $ are normalized by $ N _ { s } \phi _ { 0 } $ and $ B _ { c } $, respectively. The closed (open) symbols correspond to the black (white) arrows in Fig. \[FGROT\]. The value of $ B _ { \perp } / N _ { s } \phi _ { 0 } $ increases linearly with $ B _ { \rm tot } / B _ { c } $ for $ B _ { \rm tot } / B _ { c } < 1 $, but saturates for $ B _ { \rm tot } / B _ { c } > 1 $ where the spin polarization is expected to be completed. We link this behavior to the concentration $ N _ { \uparrow } $ of electrons having an up spin. Assuming that a change in $ p $ by $ B _ { \perp } $ is negligible in the low-$ B _ { \perp } $ region, we have $ N _ { \uparrow } / N _ { \rm s } = (1 + B _ { \rm tot } / B _ { c } ) / 2 $ for $ B _ { \rm tot } / B _ { c } < 1 $ and $ N _ { \uparrow } / N _ { \rm s } = 1 $ for $ B _ { \rm tot } / B _ { c } > 1 $. All the data are distributed along the solid line or the dashed line on which the number of the flux quantum per “spin-up” electron takes a constant value of $ B _ { \perp } / N _ { \uparrow } \phi _ { 0 } = 0.265 $ or $ 0.172 $.
One may attribute the observed $ \rho _ { xx } $ minima to the Shubnikov-de Haas oscillation produced by spin-up electrons. Actually, the effective Landau level filling factor of spin-up electrons $ N _ { \uparrow } \phi _ { 0 } / B _ { \perp } $ at the $ \rho _ { xx } $ minima is close to an even number of 4 or 6. Note that we have the valley degeneracy $ g _ { v } = 2 $ in this system. It is expected that $ \rho _ { xx } $ also takes minima at even-numbers of the effective Landau level filling factor of spin-down electrons. However, we observed no feature even for $ N _ { \downarrow } \phi _ { 0 } / B _ { \perp } = 2 $ represented by the dotted line in Fig. \[FGMIN\]. The disappearance of $ \rho _ { xx } $ minima related to spin-down electrons implies that spin-down electrons do not contribute to the conduction owing to localization or large scattering rate. If that is the case, one should accept an unusual negative dependence of the mobility of spin-up electrons on the carrier concentration $ N _ { \uparrow } $ since the value of $ \rho _ { xx } $ steeply increases with the spin polarization. The $ \rho _ { xx } $ minimum at $ B _ { \perp } / N _ { \uparrow } \phi _ { 0 } \approx 0.265 $ appears even in the insulating region with $ \rho _ { xx } \sim 10 ^ { 6 } ~ \Omega $, where the Landau level separation is considered to be entirely smeared out by the level broadening. Further consideration is required to explain the results consistently.
Finally, we also propose a quite different interpretation of the $ \rho _ { xx } $ minima. In the previous work [@OKA1; @OKA2], we have investigated the activation energy in the insulating phase which depends on $ B _ { \perp } $ and $ B _ { \rm tot } $. Oscillatory behavior linked with $ B _ { \perp } / N _ { s } \phi _ { 0 } $ was explained using a model based on the Aharonov-Bohm effect on the exchange of electrons localized in the pinned Wigner solid [@RVD]. This model may be partly applied to the present case if one assumes that spin-up electrons are localized and form an ordered state in the real space like a Wigner solid. Then, the $ B _ { \rm tot } $-dependence of $ \rho _ { xx } $ can be understood as a result of the change in the concentration of spin-down electrons which carry the charge. While we at this stage do not identify the origin of the $ \rho _ { xx } $ minima specifically, we consider that the effect of the magnetic flux on the system of spin-up electrons can cause an oscillatory perturbation in the mobility of spin-down electrons through some interactions.
In summary, we have studied the magnetotransport in a high mobility Si-MOSFET at electron concentrations around the MIT. The relationship between the resistivity and the spin polarization indicates that the mixing of the different spin states is important for the metallic conduction. We also found the $ \rho _ { xx } $ minima in the $ B _ { \perp } $-dependence, which should be related to the concentration of “spin-up” electrons.
This work is supported in part by Grants-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture, Japan, and High-Tech-Research Center in Gakushuin University.
S. V. Kravchenko, G. V. Kravchenko, J. E. Furneaux, V. M. Pudalov, and M. D’Iorio, Phys. Rev. B [**50**]{}, 8039 (1994); S. V. Kravchenko, W. E. Mason, G. E. Bowker, J. E. Furneaux, V. M. Pudalov, and M. D’Iorio, Phys. Rev. B [**51**]{}, 7038 (1995); S. V. Kravchenko, D. Simonian, M. P. Sarachik, W. Mason, and J. E. Furneaux, Phys. Rev. Lett. [**77**]{}, 4938 (1996); D. Simonian, S. V. Kravchenko, and M. P. Sarachik, Phys. Rev. B [**55**]{}, 13421 (1997). D. Popović, A. B. Fowler, and S. Washburn, Phys. Rev. Lett. [**79**]{}, 1543 (1997). J. Lam, M. D’Iorio, D. Brown, and H. Lafontaine, Phys. Rev. B [**56**]{}, 12741 (1997). P. T. Coleridge, R. L. Williams, Y. Feng, and P. Zawadzki, Phys. Rev. B [**56**]{} 12764 (1997). Y. Hanein, U. Meirav, D. Shahar, C. C. Li, D. C. Tsui, and H. Shtrikman, Phys. Rev. Lett. [**80**]{}, 1288 (1998). M. Y. Simmons, A. R. Hamilton, M. Pepper, E. H. Linfield, P. D. Rose, and D. A. Ritchie, Phys. Rev. Lett. [**80**]{}, 1292 (1998). E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. [**42**]{}, 673 (1979). D. Simonian, S. V. Kravchenko, M. P. Sarachik, and V. M. Pudalov, Phys. Rev. Lett. [**79**]{}, 2304 (1997); Phys. Rev. B [**57**]{}, 9420 (1998). V. M. Pudalov, G. Brunthaler, A. Prinz, and G. Bauer, Pis’ma Zh. Éksp. Teor. Fiz. [**65**]{}, 887 (1997) \[JETP Lett. [**65**]{}, 932 (1997)\]. V. M. Pudalov, M. D’Iorio, S. V. Kravchenko, and J. W. Campbell, Phys. Rev. Lett. [**70**]{}, 1866 (1993). S. T. Chui and B. Tanatar, Phys. Rev. Lett. [**74**]{}, 458 (1995); Phys. Rev. B [**55**]{}, 9330 (1997).
T. Okamoto and S. Kawaji, J. Phys. Soc. Jpn. [**65**]{}, 3716 (1996).
T. Okamoto and S. Kawaji, Phys. Rev. B [**57**]{}, 9097 (1998).
At higher temperatures above $ T \sim 2~{\rm K} $, positive temperature dependence appears at $ 9~{\rm T} $ in the low-$ \rho _ { xx } $ region.
H. Kodera, J. Phys. Soc. Jpn. Suppl. [**21**]{}, 578 (1966).
F. F. Fang and P. J. Stiles, Phys. Rev. [**174**]{}, 823 (1968).
M. Kobayashi and K. F. Komatsubara, Jpn. J. Appl. Phys. Suppl. 2, Pt. 2, 343.
S. V. Kravchenko, D. Simonian, M. P. Sarachik, A. D. Kent, and V. M. Pudalov, Phys. Rev. B [ **58**]{}, 3553 (1998). The model used in Ref. should be partly modified because the value of the bare valley splitting energy was overestimated to be $ \Delta E _ { v } \approx 4 ~ { \rm K } $. Since no indication of the valley splitting was observed in the Shubnikov-de Haas oscillation as shown in Fig. \[FGOS\], we at present evaluate the bare valley splitting energy to be below 0.5 K for $ N _ { s } \leq 5 \times 10 ^ { 15 } { \rm m } ^ { -2 } $. On the other hand, the theoretical values of $ \Delta E _ { v } $ have large uncertainties as reviewed by T. Ando, A. B. Fowler, and F. Stern in Rev. Mod. Phys. [**54**]{}, 437 (1982).
|
---
abstract: 'We propose potential geometry for fabrication of the graphite sheets with atomically smooth edges. For such sheets with Bernal stacking, the electron-electron interaction and topology should cause sufficiently high density of states resulting in the high temperature of either spin ordering or superconducting pairing. [@Pisma]'
author:
- 'G.E. Volovik'
- 'V.M. Pudalov'
title: Graphite on graphite
---
The perspective direction in increasing the critical temperature of superconductivity is the search for and synthesis of the electronic systems with flat band in the energy spectrum. The dispersionless energy spectrum has a singular density of states, which provides the transition temperature being proportional to the coupling constant instead of the exponential suppression.[@KhodelShaginyan1990] There are different potential sources of flat band.
One of them is formation of the so-called Khodel-Shaginyan fermion condensate – the flat band caused by electron-electron interaction in metals [@KhodelShaginyan1990; @Volovik1991; @Nozieres1992]. This is the manifestation of the general phenomenon of the energy level merging due to electron-electron interaction. This effect has been recently suggested [@Dolgopolov2014] to be responsible for merging of the discrete energy levels in two-dimensional electron system in quantizing magnetic fields. Another case of merging was found on the verge of the metal-insulator transition or quantized Hall effect to insulator transition, where energy levels for difference subbands merge instead of repulse [@pudalov-Hall_JETPL_1993; @krav_PRL_1955; @pudalov-termination_SS_1994].
According to Ref. [@Yudin2014] the favourable condition for the formation of such flat band is when the van Hove singularity comes close to the Fermi surface [@KhodelShaginyan1990] (see also Ref. [@Volovik1994] for the simple Landau type model of the formation of such flat band). It is also possible that this effect is responsible for the occurrence of superconductivity with high $T$ observed in the pressurized sulfur hydride [@Drozdov2014; @Drozdov2015]. There are some theoretical evidences [@Pickett2015; @Bianconi2015] that the high-$T_c$ superconductivity takes place at such pressure, when the system is near the Lifshitz transition, i.e. when the Fermi surface is close to van Hove singularity and the bands flatten. That is why it is not excluded that the Khodel-Shaginyan flat band is formed in sulfur hydride at pressure 180-200 GPa giving rise to high-T$_c$ superconductivity.
The flat band can be also formed in such semimetals which contain the Dirac lines. Dirac nodal lines are the topologically protected lines in the energy spectrum, which have zero energy [@Ryu2002; @HeikkilaVolovik2011; @HeikkilaKopninVolovik2011; @SchnyderRyu2011; @HeikkilaVolovik2016]. The flat band appears on the surface of such material due to the topological phenomenon of the bulk-edge correspondence. The boundary of the surface flat band is determined by the projection of the Dirac line in the bulk to the plane of the surface. By now, there are many evidences of the existence of Dirac lines in the electronic spectrum of semimetals [@Weng2014; @Kane2015; @Belopolski2015; @Cha2015; @Bian2015; @Zhao2015; @Hirayama2016; @Huang2016; @LiuBalents2016].
The Dirac nodal lines and the corresponding flat bands exist also in the pair-correlated systems: in cuprate superconductors and in the spectrum of fermionic excitations in the recently discovered polar phase of superfluid $^3$He [@Dmitriev2015]. They lead to the singularity in the density of states of the fermions bound to the vortex.[@Volovik1993; @KopninVolovik1996; @Volovik2016].
Graphite is one of the realizations of the approximate nodal line semimetal (for the review, see [@HeikkilaVolovik2016]). In the model, where some small hopping elements are neglected, the Bernal graphite has two vertical Dirac lines running between the [**H**]{} points in the 3D graphite band structure. According to the bulk-edge correspondence, the Dirac line in bulk along the normal to the graphite planes produces the zero energy states on the lateral boundaries of Bernal graphite, which form the surface flat band. This 2D flat band can be considered as extension of the 1D flat band states in graphene with zigzag edges [@HeikkilaVolovik2016]. Note that in the similar model of the rhombohedral graphite the Dirac line forms a spiral, whose projection on the horizontal boundary marks the boundary of the flat band. In the Bernal graphite model the flat band exists only on the lateral boundary.
In the real graphite the flat band is distorted due to two reasons. The first is due to the higher-order hopping elements, which are neglected in the model. These elements break the symmetry, which supports the topological stability of the Dirac line. In real Bernal and rhombohedral graphite the Dirac line transforms to the electron and hole pockets [@McClure1957]. The topology of the Dirac lines, however, is not fully destroyed: the electron and hole pockets form the continuous chain. Nevertheless, the extension of the Dirac line to the chain of Fermi surfaces distorts the surface flat band. Whether such distortion of the flat band considerably reduces the high transition temperature expected for the flat band materials is to be investigated both numerically and experimentally. The flattening of the spectrum for the five layers of graphene with the rhombohedral ABCAB stacking has been observed experimentally [@Pierucci2015] by epitaxial growing on 3C-SiC(111) on 2$^�$off�axis 6H-SiC(0001). Note that the there can be the combined effect of topology and electron-electron interaction for the stabilization of the flat band, i.e. due to the interaction a part of the approximate flat band may become exact.
However, there is another important source of distortion of the edge flat band in Bernal graphite: the lateral walls of graphite are always rough. This is not important for the rhombohedral graphite, where the (approximate) Dirac lines are not vertical and thus have nonzero projection on the smooth horizontal boundary. That is why we need such configuration of the Bernal graphite, where the lateral walls can be made smooth.
In order to fabricate smooth lateral walls, one obvious way is to cut the graphite edge with a nanometer precision by using scanning tunneling lithography [@tapas_Nat.Nano_2008; @biror_2010]; in this way graphene nanoribbons with smooth edges were obtained in Ref. [@Wang; @PRL; @2008]. The graphite flakes may be shaped using Ar- or He-ion beam etching.
Without relying on lithographic techniques, fabrication of nanoribbons with atomically smooth edges has also been reported via the so-called “bottom-up” processes – epitaxial growth on templated nanofacets [@sprinkle_2010] and by chemical vapor deposition on surface features [@ago_2012; @hayashi_2012; @ago_2013]. Recently, graphene nanoribbons with atomically flat edges have been grown on a semiconductor Ge-wafer [@jacobberger_NatCom_2016].
Another way to grow graphene/graphite sheets orthogonal to the atomically flat graphite substrate is shown schematically in Fig. 1. In order to facilitate such growth, an atomically thin and narrow strip of a catalyst (Pt, Ni or TiC) may be deposited at the flat surface of a crystalline substrate (graphite, (001)Ge, h-BN, etc) in the proper direction. Alternatively, growth in the out-of-plane direction may be initiated by surface features such as trenches, steps or ridges. Then, in the CVD reactor the graphite sheets may start growing perpendicular to the substrate plane and directed along the strip.
The growth conditions in the proposed geometry, of cause, require more thorough consideration. We note, however, that it is similar to the “carbon nanowalls” which are graphite nanostructures with edges comprised of stacked planar graphene sheets standing vertically on a substrate. The sheets form a wall structure with thicknesses in the range of a few nanometers to a few tens of nanometers [@hiramatsu_book].
![Schematic view of the graphite-on-graphite design. [*1*]{} – graphite (BN) substrate, [*2*]{} - graphite (or a multilayered graphene) sheet, or a graphite wall, [*3*]{} – a strip of the catalist. ](GonG.pdf){width="0.5\linewidth"}
\[GoG\]
In the case of success, the interface between the vertical graphite and the horizontal graphite layers may be made smooth. With the appropriate mutual orientations of the interface and crystal axes, the flat interface contains the (approximate) flat band, and may have sufficiently large density of states to produce the high temperature of transition to either magnetic or superconducting state. Graphene itself also may serve as substrate.
At the moment there are many evidences of enhanced superconducting transition temperature, which are related to monolayers [@FeSe], interfaces [@Esquinazi2014; @Esquinazi2013a; @Esquinazi2013b; @Esquinazi2013c; @TangFu2014; @Bozovic], alkanes in contact with a graphite surface [@Kawashima2013], polymer composites with embedded graphene flakes [@Ionov2015; @Ionov2016], etc. The interface itself may provide the factor which together with the electron-electron interaction enhances the effect of the flat band singularity in the electronic density of states leading to unexpected effects.
VP benefited from discussions with K. N. Eltsov and E. D. Obraztsova. The authors acknowledge support by RSCF (\# 16-42-01100).
This is the English translation of the Russian version published in: Pis’ma ZhETF [**104**]{}, 888–891 (2016).
V.A. Khodel and V.R. Shaginyan, Superfluidity in system with fermion condensate, JETP Letters [**51**]{}, 553 (1990).
G.E. Volovik, A new class of normal Fermi liquids, JETP Lett. [**53**]{}, 222 (1991).
P. Nozieres, Properties of Fermi liquids with a finite range interaction, J. Phys. (Fr.) [**2**]{}, 443 (1992).
A.A. Shashkin, V.T. Dolgopolov, J.W. Clark, V.R. Shaginyan, M.V. Zverev and V.A. Khodel, Merging of Landau levels in a strongly-interacting two-dimensional electron system in silicon, Phys. Rev. Lett. [**112**]{}, 186402 (2014).
V.M. Pudalov, M.D’Iorio, J.Campbell, Hall resistane and Quantized Hall effect to insulator transitions in a 2D electron system, Pis’ma ZhETF [**57**]{}(9), 592 (1993); JETP Lett. [**57**]{}(9), 608 (1993).
S.V.Kravchenko, W.Mason, J.Furneaux, V.M.Pudalov, Global phase diagram for the Quantum Hall Effect: an experimental picture, Phys. Rev. Lett., [**75**]{}, 910 (1995).
V.M. Pudalov, M.D’Iorio, Termination of the quantum Hall effect by the electron solid, Surface Science, [**305**]{}, 107 (1994).
D. Yudin, D. Hirschmeier, H. Hafermann, O. Eriksson, A.I. Lichtenstein, M.I. Katsnelson, Fermi condensation near van Hove singularities within the Hubbard model on the triangular lattice, Phys. Rev. Lett. [**112**]{}, 070403 (2014).
G.E. Volovik, On Fermi condensate: near the saddle point and within the vortex core, JETP Lett. [**59**]{}, 830 (1994).
A.P. Drozdov, M.I. Eremets, and I.A. Troyan, Conventional superconductivity at 190 K at high pressures, IEEE/CSC & ESAS SUPERCONDUCTIVITY NEWS FORUM (global edition), January 2015, arXiv:1412.0460.
A.P. Drozdov, M.I. Eremets, I.A. Troyan, V. Ksenofontov, S.I. Shylin, Conventional superconductivity at 203 K at high pressures, Nature [**525**]{}, 73 (2015)
Yundi Quan and W.E. Pickett, Impact of van Hove singularities in the strongly coupled high temperature superconductor H$_3$S, Phys.Rev. B [**93**]{} 104526 (2016).
A. Bianconi and T. Jarlborg, Superconductivity above the lowest Earth temperature in pressurized sulfur hydride, EPL [**112**]{}, 37001 (2015); A. Bianconi and T. Jarlborg, Lifshitz transitions and zero point lattice fluctuations in sulfur hydride showing near room temperature superconductivity, Novel Superconducting Materials [**1**]{}, 15 (2015); T. Jarlborg and A. Bianconi, Breakdown of the Migdal approximation at Lifshitz transitions with a giant zero-point motion in H$_3$S superconductor, Sci. Rep. [**6**]{}, 24816 (2016); A. Bussmann-Holder, J. Kohler, M.-H. Whangbo, A.Bianconi, A.Simon, High temperature superconductivity in sulfur hydride under ultrahigh pressure: A complex superconducting phase beyond conventional BCS, Nov. Supercond. Mater. [**2**]{}, 37–42 (2016).
S. Ryu and Y. Hatsugai, Topological origin of zero-energy edge states in particle-hole symmetric systems, Phys. Rev. Lett. [**89**]{}, 077002 (2002).
T.T. Heikkilä and G.E. Volovik, Dimensional crossover in topological matter: Evolution of the multiple Dirac point in the layered system to the flat band on the surface, JETP Lett. [**93**]{}, 59 (2011).
T.T. Heikkilä, N.B. Kopnin and G.E. Volovik, Flat bands in topological media, JETP Lett. [**94**]{}, 233 (2011); arXiv:1012.0905.
A.P. Schnyder and S. Ryu, Topological phases and flat surface bands in superconductors without inversion symmetry, Phys. Rev. B [**84**]{}, 060504(R) (2011).
T.T. Heikkilä and G.E. Volovik, Flat bands as a route to high-temperature superconductivity in graphite, in: [*Basic Physics of Functionalized Graphite*]{}, Springer 2016, pp. 123–143, arXiv:1504.05824.
Hongming Weng, Yunye Liang, Qiunan Xu, Yu Rui, Zhong Fang, Xi Dai, Yoshiyuki Kawazoe, Topological node-line semimetal in three dimensional graphene networks, Phys. Rev. B [**92**]{}, 045108 (2015).
Youngkuk Kim, B. J. Wieder, C. L. Kane, A. M. Rappe, Dirac line nodes in inversion symmetric crystals, Phys. Rev. Lett. [**115**]{}, 036807 (2015).
M. Neupane, I. Belopolski, M.M. Hosen, D.S. Sanchez, R. Sankar, M. Szlawska, S.-Y. Xu, K. Dimitri, N. Dhakal, P. Maldonado, P.M. Oppeneer, D. Kaczorowski, F. Chou, M.Z. Hasan, and T. Durakiewicz, Observation of topological nodal fermion semimetal phase in ZrSiS, Phys. Rev. B [**93**]{}, 201104 (2016).
Y.-H. Chan, C.-K. Chiu, M. Chou, and A. P. Schnyder, Ca$_3$P$_2$ and other topological semimetals with line nodes and drumhead surface state, Phys. Rev. B [**93**]{}, 205132 (2016),
G. Bian, T.-R. Chang, R. Sankar, S.-Y. Xu, H. Zheng, T. Neupert, C.-K. Chiu, S.-M. Huang, G. Chang, I. Belopolski, et al., Topological nodal-line fermions in spin-orbit metal PbTaSe$_2$, Nature communications [**7**]{}, 10556 (2016).
J. Zhao, R. Yu, H. Weng, and Z. Fang, Topological node-line semimetal in compressed black phosphorus, Phys. Rev. B [**94**]{}, 195104 (2016).
M. Hirayama, R. Okugawa, T. Miyake, and S. Murakami, Topological Dirac nodal lines in fcc Calcium, Strontium, and Ytterbium, arXiv:1602.06501.
H. Huang, J. Liu, D. Vanderbilt, and W. Duan, Topological nodal-line semimetals in alkaline-earth stannides, germanides, and silicides, Phys. Rev. B [**93**]{}, 201114 (2016).
Jianpeng Liu and L. Balents, Correlation and transport phenomena in topological nodal-loop semimetals, arXiv:1609.05529.
V.V. Dmitriev, A.A. Senin, A.A. Soldatov, A.N. Yudin, Polar phase of superfluid $^3$He in anisotropic aerogel, Phys. Rev. Lett. [**115**]{}, 165304 (2015).
G.E. Volovik, Superconductivity with lines of gap nodes: Density of states in the vortex, Pisma ZhETF [**58**]{}, 457 (1993); JETP Lett. [**58**]{}, 469 (1993).
N.B. Kopnin and G.E. Volovik, Singularity of the vortex density of states in $d$-wave superconductors, Pisma ZhETF [**64**]{}, 641 (1996); JETP Lett. [**64**]{}, 690 (1996).
G.E. Volovik, Condensation of fermion zero modes in the vortex, Pisma ZhETF [**104**]{}, 201 (2016); JETP Lett. [**104**]{} (2016).
J.W. McClure, Band structure of graphite and de Haas-van Alphen effect, Phys. Rev. [**108**]{}, 612-618 (1957).
D. Pierucci, H. Sediri, M. Hajlaoui, J.-C. Girard, T. Brumme, M. Calandra, E. Velez-Fort, G. Patriarche, M.G. Silly, G. Ferro, V. Souliere, M. Marangolo, F. Sirotti, F. Mauri, and A. Ouerghi, Evidence for flat bands near the Fermi level in epitaxial rhombohedral multilayer graphene, ACS Nano [**9**]{}, 5432 (2015).
L. Tapasztoŕ, G. Dobrik, Ph. Lambin, L.P. Biroŕ, Tailoring the atomic structure of graphene nanoribbons by scanning tunnelling microscope lithography. Nat Nanotechnol. [**3**]{}, 397 (2008).
L. P. Biroŕ, P. Lambin, Nanopatterning of graphene with crystallographic orientation control, Carbon [**48**]{}, 2677 (2010).
Xinran Wang, Yijian Ouyang, Xiaolin Li, Hailiang Wang, Jing Guo, and Hongjie Dai, Room-temperature all-semiconducting sub-10-nm graphene nanoribbon field-effect transistors, Phys. Rev. Lett. [**100**]{}, 206803 (2008).
M. Sprinkle, et al. Scalable templated growth of graphene nanoribbons on SiC. Nat. Nanotechnol. [**5**]{}, 727–731 (2010)
Ago, H., Ito, Y., Tsuji, M., Ikeda, K. Step-templated CVD growth of aligned graphene nanoribbons supported by a single-layer graphene film, Nanoscale [**4**]{}, 5178–5182 (2012).
Hayashi, K., Sato, S., Ikeda, M., Kaneta, C., Yokoyama, N. Selective graphene formation on copper twin crystals, J. Am. Chem. Soc. [**134**]{}, 12492–12498 (2012).
Ago, H. et al. Lattice-oriented catalytic growth of graphene nanoribbons on heteroepitaxial nickel films. ACS Nano [**7**]{}, 10825–10833 (2013).
R. M. Jacobberger, B. Kiraly, M. Fortin-Deschenes, et al., Direct oriented growth of armchair graphene nanoribbons on germanium, Nat. Commun. [**6**]{}, 8006 (2015).
M. Hiramatsu, M. Hori, Carbon Nanowalls, Springer-Verlag/Wien, 2010. X. Shi, Z.-Q. Han, X.-L. Peng, P. Richard, T. Qian, X.-X. Wu, M.-W. Qiu, S.C. Wang, J.P. Hu, Y.-J. Sun, H. Ding, Enhanced superconductivity accompanying a Lifshitz transition in electron-doped FeSe monolayer, arXiv:1606.01470.
P. Esquinazi, T.T. Heikkila, Yu.V. Lysogorskiy, D.A. Tayurskii, G.E. Volovik, On the superconductivity of graphite interfaces, Pis’ma ZhETF [**100**]{}, 374 (2014); JETP Lett. [**100**]{}, 336 (2014).
P. Esquinazi, Graphite and its hidden superconductivity, Papers in Physics [**5**]{}, 050007 (2013).
A. Ballestar, J. Barzola-Quiquia, T. Scheike, and P. Esquinazi, Josephson-coupled superconducting regions embedded at the interfaces of highly oriented pyrolytic graphite, New J. Phys. [**15**]{}, 023024 (2013).
A. Ballestar, T.T. Heikkilä, and P. Esquinazi, Interface size dependence of the Josephson critical behaviour in pyrolytic graphite, Supercond Sci. Technol. [**27**]{}, 115014 (2014)
E. Tang and L. Fu, Strain-induced partially flat band, helical snake states, and interface superconductivity in topological crystalline insulators, Nature Phys. [**10**]{}, 964 (2014).
A. Gozar, G. Logvenov, L. Fitting Kourkoutis, A. T. Bollinger, L. A. Giannuzzi, D. A. Muller, and I. Bozovic, Interface superconductivity between a metal and a Mott insulator, Nature [**455**]{}, 782 (2008).
Yasushi Kawashima, Possible room temperature superconductivity in conductors obtained by bringing alkanes into contact with a graphite surface, AIP Advances [**3**]{}, 052132 (2013).
A.N. Ionov, Pis’ma ZhTF [**41**]{}(13), 79 (2015). Techn. Phys. Lett. [**41**]{}, 651 (2015)
A.N. Ionov, Josephson-like behavior of the current-voltage characteristics of multi-graphene flakes embedded in polysterene, J. Low Temp. Phys. [**185**]{}, 515 (2016).
|
---
abstract: 'We construct geometric maps from the cyclic homology groups of the (compact or wrapped) Fukaya category to the corresponding $S^1$-equivariant (Floer/quantum or symplectic) cohomology groups, which are natural with respect to all Gysin and periodicity exact sequences and are isomorphisms whenever the (non-equivariant) open-closed map is. These [*cyclic open-closed maps*]{} give (a) constructions of geometric smooth and/or proper Calabi-Yau structures on Fukaya categories (which in the proper case implies the Fukaya category has a cyclic $\ainf$ model in characteristic 0) and (b) a purely symplectic proof of the non-commutative Hodge-de Rham degeneration conjecture for smooth and proper subcategories of Fukaya categories of compact symplectic manifolds. Further applications of cyclic open-closed maps, to counting curves in mirror symmetry and to comparing topological field theories, are the subject of joint projects with Perutz-Sheridan [@GPS1:2015; @GPS2:2015] and Cohen [@CohenGanatra:2015].'
author:
- Sheel Ganatra
bibliography:
- 'math\_bib.bib'
title: 'Cyclic homology, $S^1$-equivariant Floer cohomology, and Calabi-Yau structures'
---
[^1]
Introduction
============
This paper concerns the compatibility between chain level $S^1$ actions arising in two different types of Floer theory on a symplectic manifold. The first of these $C_{-*}(S^1)$[^2] actions is induced geometrically on the [*Hamiltonian Floer homology chain complex*]{} $CF^*(M)$, formally a type of Morse complex for an action functional on the free loop space, through rotating free loops. The homological action of $[S^1]$ is known as the *BV operator* $[\Delta]$, and the $C_{-*}(S^1)$ action can be used to define [*$S^1$-equivariant Floer homology theories*]{} — see e.g., [@Seidel:2010fk; @Bourgeois:2012fk][^3]. The second $C_{-*}(S^1)$ action lies on the [*Fukaya category*]{}, and has discrete or combinatorial origins, coming from the hierarchy of compatible cyclic $\Z/k\Z$ actions on cyclically composable chains of morphisms between Lagrangians. A (categorical analogue of a) fundamental observation of Connes’, Tsygan, and Loday-Quillen that such a structure, which exists on any category $\cc$, can be packaged into a $C_{-*}(S^1)$ action on the [*Hochschild homology chain complex*]{} $\r{CH}_*(\cc)$ of the category (see e.g., [@Connes:1985aa; @Tsygan:1983aa; @Loday:1984aa; @McCarthy:1994aa; @Keller:1999aa]) The associated operation of multiplcation by (a cycle representing) $[S^1]$ is frequently called the [*Connes’ $B$ operator*]{} $B$, and the corresponding $S^1$-equivariant homology theories are called [*cyclic homology groups*]{}.
A relationship between the Hochschild homology of the Fukaya category $\mc{F}$ and Floer homology is provided by the so-called [*open-closed string map*]{} [@Abouzaid:2010kx] $$\label{openclosedmap}
\oc: \r{CH}_*(\mathcal{F}) \ra CF^{*+n}(M).$$ Our main result is about the compatibility of $\oc$ with $C_{-*}(S^1)$ actions. Namely, we prove that $\oc$ can be made (coherently homotopically) $C_{-*}(S^1)$-equivariant:
\[thm:mainresult1\] The map $\oc$ admits a geometrically defined ‘[*$S^1$-equivariant enhancement*]{}’, to an $\ainf$ homomorphism of $C_{-*}(S^1)$-modules, $\widetilde{\oc} \in \mathrm{RHom}_{C_{-*}(S^1)}^n(\r{CH}_*(\f),
\r{CF}^*(M))$.
Theorem \[thm:mainresult1\] implies (but is not implied by) the statement (see Theorem \[thm:homology\]) that $[\oc]$ intertwines homological actions of $[S^1]$.
Note that $\oc$ typically does not have the property of strictly intertwining the $C_{-*}(S^1)$ actions. In particular, the homomorphism $\widetilde{\oc}$ involves extra data recording coherently higher homotopies between the two $C_{-*}(S^1)$ actions. This explains our use of the term ‘enhancement’.
It can be shown using usual invariance arguments that this enhancement is canonical: any two such geometrically defined enhancements are homotopic.
To explain the consequences of Theorem \[thm:mainresult1\] to cyclic homology and equivariant Floer homology, recall that there are a variety of [*$S^1$-equivariant homology*]{} chain complexes (and homology groups) that one can associate functorially to an $\ainf$ $C_{-*}(S^1)$ module $P$. For instance, denote by $$\label{orbitfixedpointtate}
P_{hS^1},\ P^{hS^1},\ P^{Tate}$$ the [*homotopy orbit complex*]{}, [*homotopy fixed point complex*]{}, and [*Tate complex*]{} constructions of $P$, described in §\[sec:equivariantgroups\]. When applied to the Hochschild complex $\r{CH}_*(\cc, \cc)$, the constructions by definition recover complexes computing [*(positive) cyclic homology*]{}, [*negative cyclic homology*]{}, and [*periodic cyclic homology*]{} groups of $\cc$ respectively (see §\[subsec:cyclic\]). Similarly the group $H^*(CF^*(M)_{hS^1})$ is the [*$S^1$-equivariant Floer cohomology*]{} studied (for the symplectic homology Floer chain complex); see e.g., [@Viterbo:1999fk; @Seidel:2010fk; @Bourgeois:2012fk]. The groups $H^*(CF^*(M)^{hS^1})$ and $H^*(CF^*(M)^{Tate})$ have also been studied in recent work in Floer theory [@Seidel:2018aa; @Zhao:2019aa; @Albers:2016aa]. Functoriality of the constructions and homotopy invariance properties of $C_{-*}(S^1)$ modules (see Cor. \[S1homotopyinvariance\] and Prop. \[functorialitysequences\]) immediately implies the result announced in the abstract:
\[cor:cyclicOC\] Let $HF^{*, +/-/\infty}_{S^1}(M)$ denote the (cohomology of the) homotopy orbit complex, fixed point complex, and Tate complex construction applied to $CF^*(M)$, and let $\r{HC}^{+/-/\infty}(\cc)$ denote the corresponding positive/negative/periodic cyclic homology groups. Then, $\widetilde{\oc}$ induces *cyclic open-closed maps* $$[\widetilde{\oc}^{+/-/\infty}]: \mathrm{HC}^{+/-/\infty}_*(\mathcal{F}) \ra HF^{*+n,+/-/\infty}_{S^1}(M),$$ naturally compatible with respect to the various periodicity/Gysin exact sequences, which are isomorphisms whenever $\oc$ is.
The map is frequently an isomorphism, allowing one to recover in these cases closed string Floer/quantum homology groups from open string, categorical ones [@BEE1published; @ganatra1_arxiv; @GPS1:2015; @afooo]. In such cases, Theorem \[thm:mainresult1\] and Corollary \[cor:cyclicOC\] allows one to further categorically recover the $C_{-*}(S^1)$ as well as the associated equivariant homology groups (in terms of the cyclic homology groups of the Fukaya category).
There are other $S^1$-equivariant homology functors, to which our results apply tautologically as well. For instance, consider the contravariant functor $P \mapsto (P_{hS^1})^{\vee}$; when applied to $\r{CH}_*(\cc)$ this produces the [*cyclic cohomology*]{} chain complex of $\cc$.
We have been deliberately vague about which Fukaya category and which Hamiltonian Floer homology groups Theorem \[thm:mainresult1\] applies to, as it applies in several geometric settings. The simplest of these are
1. If $M$ is compact, $\mathcal{F}$ is the usual Fukaya category of compact Lagrangians (or a summand thereof). $CF^*(M)$, the Hamiltonian Floer complex of any Hamiltonian, is quasi-isomorphic to the [ *quantum cohomology*]{} ring with its trivial $C_{-*}(S^1)$ action.
2. If $M$ is non-compact and Liouville, one could take $\mathcal{F} = \mathcal{W}$ to be the [*wrapped Fukaya category*]{} and $CF^*(M) = SC^*(M)$ to be the [ *symplectic cohomology ring*]{} with its (typically highly non-trivial) $C_{-*}(S^1)$ action,
3. If $M$ is non-compact and Liouville, one could take $\mathcal{F} \subset \mathcal{W}$ to be the [*Fukaya category of compact Lagrangians*]{}. In this case, the map $\oc$ factors through $CF^*(M) = H^*(M, \partial^{\infty} M)$ the [*relative cohomology group*]{} with its trivial $C_{-*}(S^1)$ action. In fact, $\oc$ further factors through the [*symplectic homology chain complex*]{} $SC_*(M) = (SC^*(M))^{\vee}$.
For example, in case (2) above, when the relevant $[\oc]$ map is an isomorphism, Corollary \[cor:cyclicOC\] computes various $S^1$-equivariant symplectic cohomology groups[^4] in terms of cyclic homology groups of the wrapped Fukaya category.
To keep this paper a manageable length, we implement the map $\widetilde{\oc}$ and prove Theorem \[thm:mainresult1\] in the technically simplest settings: when $M$ is compact, we assume it is aspherical or monotone and we restrict to those Lagrangians that don’t bound discs (‘tautologically unobstructed’) or those that are monotone (if $M$ is), and when $M$ is non-compact, we assume it is Liouville (non-compact, exact, convex at $\infty$) and consider exact Lagrangians therein, e.g., settings (2) and (3) above. However, our methods are entirely orthogonal to the usual analytic difficulties faced in constructing Fukaya categories and open-closed maps in general contexts, and we expect they should extend immediately to other settings. For instance, in the setting of relative Fukaya categories of compact projective Calabi-Yau manifolds, an adapted version of our construction will appear in joint work with Perutz-Sheridan [@GPS2:2015].
There are other settings in which Fukaya categories are now well-studied, for instance Fukaya categories of Lefschetz fibrations (and more general LG models), or more generally partially wrapped Fukaya categories. We do not discuss these situations in our paper, but expect suitable versions of Theorem \[thm:mainresult1\] to hold in such settings too.
\[rmk:thmvariations\] One can consider variations on Theorem \[thm:mainresult1\]. As a notable example, let $M$ denote a (noncompact) Liouville manifold, and $\mathcal{F}$ the Fukaya category of compact Lagrangians in $M$. Then there is a non-trivial refinement of the map $\r{HH}_*(\mathcal{F}) \ra H^*(M, \partial M)$, which can be viewed as a pairing $\r{HH}_*(\mathcal{F}) \times H^*(M) \to \K$, to a pairing $$\r{CH}_*(\f) \otimes SC^*(M) \ra \K.$$ (note symplectic cohomology does not satisfy Poincaré duality, so this is [*not*]{} equivalent to a map to symplectic cohomology). Our methods also imply that this pairing admits an $S^1$-equivariant enhancement, with respect to the diagonal $C_{-*}(S^1)$ action on the left and the trivial action on the right. Passing to adjoints, we obtain cyclic open closed maps from $S^1$-equivariant symplectic cohomology to cyclic [*cohomology*]{} groups of $\f$, and from cyclic homology of $\f$ to equivariant symplectic [ *homology*]{}. See §\[sec:compactopenclosed\] for a more details.
Beyond computing equivariant Floer cohomology groups in terms of cyclic homology theories, we describe in the following subsection two applications of Theorem \[thm:mainresult1\] to the structure of Fukaya categories.
We anticipate additional concrete applications of Theorem \[thm:mainresult1\] and its homological shadow, Theorem \[thm:homology\]. For instance, one can study the compatibility of open-closed maps with *dilations* in the sense of [@Seidel:2010uq], which are elements $B$ in $SH^*(M)$ satisfying $[\Delta] B = 1$; the existence of dilations strongly constrains intersection properties of embedded Lagrangians [@Seidel:2014aa]. Theorem \[thm:homology\], or rather the variant discussed in Remark \[rmk:thmvariations\], implies [*if there exists a dilation, e.g., an element $x \in SH^1(M)$ with $[\Delta] x = 1$, then on the Fukaya category of compact Lagrangians $\mc{F}$, there exists $x' \in (\r{HH}_{n+1}(\mc{F}))^{\vee}$ with $x'
\circ [B] = [tr]$*]{}, where $tr$ is the geometric weak proper Calabi-Yau structure on the Fukaya category (see §\[intro:cystructures\]).
Calabi-Yau structures on the Fukaya category {#intro:cystructures}
--------------------------------------------
Calabi-Yau structures are a type of cyclically symmetric duality structure on a dg or $\ainf$ category $\cc$ generalizing the notion of a nowhere vanishing holomorphic volume form on a complex algebraic variety $X$ in the case $\cc = perf(X)$. As is well understood, there are two (in some sense dual) types of Calabi-Yau structures on $\ainf$ categories:
1. [*proper Calabi-Yau structures*]{} [@Kontsevich:2009ab] can be associated to [ *proper*]{} categories $\cc$ (those which have cohomologically finite-dimensional morphism spaces), abstract and refine the notion of integration against a nowhere vanishing holomorphic volume form. For $\cc = perf(X)$ with $X$ a proper $n$-dimensional variety, the resulting structure in particular induces the Serre duality pairing with trivial canonical sheaf $\r{Ext}^*(\mathcal{E}, \mathcal{F})
\times \r{Ext}^*(\mathcal{F}, \mathcal{E}) \to \K[-n]$, Roughly, a proper Calabi-Yau structure on $\cc$ (of dimension $n$) is a map $[\widetilde{tr}]: \r{HC}^{+}_*(\cc) \to \K[-n]$ satisfying a non-degeneracy condition.
2. [*smooth Calabi-Yau structures*]{} [@Kontsevich:uq; @Kontsevich:CY] can be associated to [*smooth*]{} categories $\cc$ (those with perfect diagonal bimodule), and abstract the notion of the nowhere vanishing holomorphic volume form itself, along with the induced identification (by contraction against the volume form) of polyvectorfields with differential forms. Roughly, a smooth Calabi-Yau structure on $\cc$ (of dimension $n$) is a map $cotr: \K[n] \to \r{HC}^-_*(\cc)$, or equivalently an element $[\widetilde{\sigma}]$ or “$[vol_{\cc}]$” in $\r{HC}^-_{-n}(\cc)$, satisfying a non-degeneracy condition.
Precise definitions are reviewed in §\[section:cystructures\]. When $\cc$ is simultaneously smooth and proper, it is a folk result that the notions are equivalent; see [@GPS1:2015]\*[Prop. 6.10]{}.
In general, Calabi-Yau structures may not exist and when they do, there may be a non-trivial space of choices (see [@Menichi:2009aa] for an example). A Calabi-Yau structure in either form induces non-trivial identifications between Hochschild invariants of the underlying category $\cc$.[^5] Moreover, categories with Calabi-Yau structures (should) carry induced 2-dimensional chain level TQFT operations on their Hochschild homology chain complexes, associated to moduli spaces of Riemann surfaces with marked points [@Costello:2006vn; @Kontsevich:2009ab] (in the smooth case, this has not yet been established, but is ongoing work [@Kontsevich:uq; @Kontsevich:CY]); if the category is proper and non-smooth (respectively smooth non-proper) the resulting TQFT is incomplete in that every operation must have at least one input (respectively output). In the smooth and proper case, Calabi-Yau structures play a central role in the mirror symmetry-motivated question of recovering Gromov-Witten invariants from the Fukaya category and to the related question of categorically recovering Hamiltonian Floer homology with all of its (possibly higher homotopical) operations. See [@Costello:2006vn; @Costello:2009aa; @Kontsevich:2008aa] for work around these questions in the setting of abstract topological field theories and [@GPS1:2015] for applications of Calabi-Yau structures to recovering genus-0 Gromov-Witten invariants from the Fukaya category.
\[rem:cyclic\] A closely related to (1), and well studied, notion is that of a [*cyclic $\ainf$ category*]{}: this is an $\ainf$ category $\cc$ equipped with a chain level perfect pairing $$\langle -, - \rangle : \hom(X,Y) \times \hom(Y,X) \to
\K[-n]$$ such that the induced correlation functions $$\langle \mu^d(-, -, \ldots, -), - \rangle$$ are strictly (graded) cyclically symmetric, for each $d$ see e.g., [@Costello:2006vn; @Fukaya:2010aa; @Cho:2012aa]. Although the property of being a cyclic $\ainf$ structure is not a homotopy invariant notion (i.e., not preserved under $\ainf$ quasi-equivalences), cyclic $\ainf$ categories and proper Calabi-Yau structures turn out to be weakly equivalent [*in characteristic 0*]{}, in the following sense. Any cyclic $\ainf$ category carries a canonical proper Calabi-Yau structure, and Kontsevich-Soibelman proved that a proper Calabi-Yau structure on any $\ainf$ category $\cc$ determines a (canonical up to quasi-equivalence) quasi-isomorphism between $\cc$ and a cyclic $\ainf$ category $\widetilde{\cc}$ [@Kontsevich:2009ab Thm. 10.7]. When $\r{char}(\K)
\neq 0$, the two notions of proper Calabi-Yau and cyclic $\ainf$ differ in general, due to group cohomology obstructions to imposing cyclic symmetry. In such instances, it seems that the notion of a proper Calabi-Yau structure is the “correct” one (as it is a homotopy invariant notion and, by Theorem \[mainthm3\], the compact Fukaya category always has one).
As a first application of Theorem \[thm:mainresult1\], we verify the longstanding expectation that various compact Fukaya categories possess geometrically defined canonical Calabi-Yau structures:
\[mainthm3\] The Fukaya category of compact Lagrangians has a canonical geometrically defined proper Calabi-Yau structure over any ground field $\K$ (over which the Fukaya category and $\widetilde{\oc}$ are defined).
In fact, this proper Calabi-Yau structure is easy to describe in terms of the cyclic open-closed map (c.f., Cor. \[cor:cyclicOC\]): it is the composition of the map $\widetilde{\oc}^+: \r{HC}^+_*(\mc{F}) \ra
H^{*+n}(M, \partial M)( ( u ) ) / u H^{*+n} (M,\partial M)[ [ u ] ]$[^6] with the linear map to $\K$ which sends the top class $PD(pt) \cdot u^0 \in H^{2n}(M, \partial M)$ to $1$, and all other generators $\alpha \cdot u^{-i}$ to 0. See §\[section:cystructures\] for more details.
As a consequence of the discussion in Remark \[rem:cyclic\], specifically [@Kontsevich:2009ab Thm. 10.7], we deduce that
If $\r{char}(\K) = 0$, then the Fukaya category of compact Lagrangians admits a (canonical up to equivalence) cyclic $\ainf$ (minimal) model.
In the case of compact symplectic manifolds and over $\K = $ a Novikov field containing $\R$, Fukaya [@Fukaya:2010aa] constructed a cyclic $\ainf$ model of the Floer cohomology algebra of a single compact Lagrangian, which will be extended to multiple objects by Abouzaid-Fukaya-Oh-Ohta-Ono [@afooo].
In order to construct (chain level) 2d-TFTs on the Hochschild chain complexes of categories, Kontsevich-Soibelman [@Kontsevich:2009ab] partly show (on the closed sector) that a proper Calabi-Yau structure can be used instead of the (weakly equivalent in characteristic 0) cyclic $\ainf$ structures considered in [@Costello:2006vn]. One might similarly hope that, for applications of cyclic $\ainf$ structures to disc counting/open Gromov-Witten invariants developed in in [@Fukaya:2011aa], a proper Calabi-Yau structure was in fact sufficient. See [@Cho:2012aa] for related work.
Turning to smooth Calabi-Yau structures, in §\[subsec:smoothCY\], we will establish the following existence result for smooth Calabi-Yau structures, which applies to wrapped Fukaya categories of non-compact (Liouville) manifolds as well as to Fukaya categories of compact manifolds:
\[thm:smoothCY\] Suppose our symplectic manifold $M$ is [*non-degenerate*]{} in the sense of [@ganatra1_arxiv], meaning that the map $[\oc]: \r{HH}_{*-n}(\f) \ra HF^*(M)$ hits the unit $1 \in HF^*(M)$. Then, its (compact or wrapped) Fukaya category $\f$ possesses a canonical, geometrically defined [*strong smooth Calabi-Yau structure*]{}.
Once more, the cyclic open-closed map gives an efficient description of this structure: it is the unique element $\r{HC}^-_{-n}(\f)$ mapping via $\widetilde{\oc}^-$ to the geometrically canonical lift $\widetilde{1}\in H^*(CF^*(M)^{hS^1})$ of the unit $1 \in CF^*(M)$ described in §\[sec:interior\].[^7]
One can study other Fukaya categories of non-compact Lagrangians, such as [*Fukaya categories of Landau-Ginzburg (LG) models $(X,\pi: X \to \C)$*]{}, and more generally [*partially wrapped Fukaya categories*]{} (such as wrapped Fukaya categories of [*Liouville sectors*]{}). In contrast to compact Fukaya categories or wrapped Fukaya categories of Liouville manifolds, such categories are typically not Calabi-Yau in either sense, even if they are smooth or proper; indeed they typically arise as homological mirrors to perfect/coherent complexes on non-Calabi-Yau varieties. Instead, one might expect such categories to admit [*pre-Calabi-Yau structrues*]{} [@Kontsevich:uq; @Kontsevich:CY] or [*relative Calabi-Yau structures*]{} [@brav_dyckerhoff_2019], structures not discussed further here.
The notion of a smooth Calabi-Yau structure, or sCY structure, will be studied further in forthcoming joint work with R. Cohen [@CohenGanatra:2015], and used to compare the wrapped Fukaya category of a cotangent bundle and string topology category of its zero section as [*categories with sCY structures*]{} (in order to to deduce a comparison of topological field theories on both sides).
Noncommutative Hodge-de-Rham degeneration for smooth and proper Fukaya categories
---------------------------------------------------------------------------------
For a $C_{-*}(S^1)$ module $\mathcal{P}$, there is a canonical Tor spectral sequence converging to $H^*(\mathcal{P}_{hS^1})$ with first page $H^*(\mathcal{P}) \otimes_{\K} H^*(\K_{hS^1}) \cong H^*(\mathcal{P})
\otimes_{\K} H_*(\mathbb{CP}^\infty)$. When applied to the Hochschild complex $\mathcal{P} =
\r{CH}_*(\cc)$ of a (dg/$\ainf$) category $\cc$, the resulting spectral sequence, from (many copies of) $\r{HH}_*(\cc)$ to $\mathrm{HC}^+(\cc)$ is called the [*Hochschild-to-cyclic*]{} or [*noncommutative Hodge-de-Rham (ncHDR) spectral sequence.*]{} The latter name comes from the fact that when $\cc = perf(X)$ is perfect complexes on a complex variety $X$, this spectral sequence is equivalent (via Hochschild-Kostant-Rosenberg (HKR) isomorphisms) to the usual Hodge-to-de-Rham spectral sequence from Hodge cohomology to de Rham cohomology $$H^*(X, \Omega^*_X) \Rightarrow H^*_{dR}(X),$$ which degenerates (as we are in characteristic 0) whenever $X$ is smooth and proper. Motivated by this, Kontsevich formulated the [*noncommutative Hodge-de-Rham (ncHDR) degeneration conjecture*]{} [@Kontsevich:2009ab; @Kontsevich:2008aa]: for any smooth and proper category $\cc$ in characteristic 0, its ncHDR spectral sequence degenerates. A general proof of this fact for $\Z$-graded categories was recently given by Kaledin, following earlier work establishing it in the coconnective case [@Kaledin:2017aa; @Kaledin:2008aa].
Using the cyclic open-closed map, we can give a purely symplectic proof of the nc-HdR degeneration property for those smooth and proper $\cc$ arising as Fukaya categories, including in non-$\Z$-graded cases:
\[thm:hdrfuk\] Let $\mathcal{C} \subset \mathcal{F}(X)$ be a smooth and proper subcategory of the Fukaya category of a compact symplectic manifold over any field $\K$ (over which the Fukaya category and the cyclic open-closed map are defined). Then, the nc Hodge-de-Rham spectral sequence for $\mathcal{C}$ degenerates.
The nc Hodge-de-Rham spectral sequence degenerates at page 1 if and only if $\mathcal{P}$ is isomorphic (in the category of $C_{-*}(S^1)$ modules) to a trivial $C_{-*}(S^1)$-module, e.g., if the $C_{-*}(S^1)$ action is trivializable [@Dotsenko:2015aa]\*[Thm. 2.1]{}. For compact symplectic manifolds $M$, recall that $CF^*(M) \cong H^*(M)$ has a canonically trivial(izable) $C_{-*}(S^1)$ action (see Corollary \[trivialconstantloops\]; this comes from, e.g., the fact that we can choose a $C^2$ small Hamiltonian to compute the complex, all of the orbits of which are constant loops on which geometric rotation acts trivially. Or more directly, we can modify the definition of $\widetilde{\oc}$ to give a map directly to $H^*(M)$ with its trivial $C_{-*}(S^1)$ action, as described in §\[pseudocycles\]).
By earlier work [@GPS1:2015; @Gautogen_arxiv], whenever $\mathcal{A}$ is smoooth, $\oc|_{\mathcal{A}}$ is an isomorphism from $\r{HH}_{*-n}(\mathcal{A})$ onto a non-trivial summand $S$ of $HF^*(M) \cong
\r{QH}^*(M)$; the $C_{-*}(S^1)$ action on this summand is trivial too. Theorem \[thm:mainresult1\] shows that $\widetilde{\oc}|_{\mathcal{A}}$ induces an isomorphism in the category of $C_{-*}(S^1)$ modules between $\r{CH}_{*-n}(\mathcal{A})$ and $S$ with its trivial action, so we are done.
Theorem \[thm:hdrfuk\] holds for a field $\K$ of any characteristic, at least whenever the Fukaya category and relevant structures are defined over $\K$. Moreoever it is true for any grading structure that can be defined on the relevant Fukaya category (for instance, it holds for the $\Z/2$-graded Fukaya category of a monotone symplectic manifold over a field of any characteristic). In contrast, for an arbitrary smooth and proper $\Z/2$-graded dg category in characteristic zero, the noncommutative Hodge-de-Rham degeneration is not yet established (though it is expected). And it is not always true in finite characteristic.
An incomplete explanation for the degeneration property holding for finite characteristic smooth and proper Fukaya categories may be that the Fukaya category over a characteristic $p$ field $\K$ (whenever Lagrangians are monotone or tautologically unobstructed at least) may always admit a lift to second Witt vectors $W_2(\K)$.[^8]
As is described in joint work (partly ongoing) with T. Perutz and N. Sheridan [@GPS1:2015; @GPS2:2015], the cyclic open-closed map $\widetilde{\oc}^-$ can further be shown to be a [*morphism of semi-infinite Hodge structures*]{}, a key step (along with the above degeneration property and construction of Calabi-Yau structure) in recovering Gromov-Witten invariants from the Fukaya category and enumerative mirror predictions from homological mirror theorems.
Outline of Paper
----------------
In §\[section:s1action\], we recall a convenient model for the category of $\ainf$ modules over $C_{-*}(S^1)$ and various equivariant homology functors from this category. In §\[section:open\] we review various Fukaya categories and the $C_{-*}(S^1)$ action on its (and indeed, any cohomologically unital $\ainf$ category’s) [*non-unital Hochschild chain complex*]{}. In §\[section:closed\], we recall the construction of the $\ainf$ $C_{-*}(S^1)$ module structure on the (Hamiltonian) Floer chain complex, following [@Bourgeois:2012fk; @Seidel:2010fk] (note that our technical setup is slightly different, though equivalent). Then we prove our main results in §\[section:openclosed1\]. Some technical and conceptual variations on the construction of $\widetilde{\oc}$ (including Remark \[rmk:thmvariations\]) are discussed at the end of this section, see §\[sec:OCvariants\]. Finally, in §\[section:cystructures\] we apply our results to construct proper and smooth Calabi-Yau structures, proving Theorems \[mainthm3\] and \[thm:smoothCY\].
Conventions {#conventions .unnumbered}
-----------
We work over a ground field $\K$ of arbitrary characteristic (though we note that all of our geometric constructions are valid over an arbitrary ring, e.g., $\Z$). All chain complexes will be graded [*cohomologically*]{}, including singular chains of any space, which hence have negative the homological grading and are denoted by $C_{-*}(X)$. All gradings are either in $\Z$ or $\Z/2$ (in the latter case, degrees of maps are implicitly mod 2).
Acknowledgements {#acknowledgements .unnumbered}
----------------
I’d like to thank Paul Seidel for a very helpful conversation and Nick Sheridan for several helpful discussions about technical aspects of this paper such as signs. Part of this work was revised during a visit at the Institut Mittag-Leffler in 2015, who I’d like to thank for their hospitality.
Complexes with circle action {#section:s1action}
============================
In this section, we review a convenient model for the category of $\ainf$ $C_{-*}(S^1)$ modules, for which the $\ainf$ $C_{-*}(S^1)$ action can be described by a single (hierarchy) of maps satisfying equations. We also describe various equivariant homology complexes in this language in terms of simple formulae. This model appears elsewhere in the literature as [*$\infty$-mixed complexes*]{} or [*$S^1$-complexes*]{} or [*multicomplexes*]{} (we will sometimes adopt the secoond term); see e.g., [@Bourgeois:2012fk; @Zhao:2019aa; @Dotsenko:2015aa] (but note that the first and third references uses homological grading conventions).
Definitions {#circleactionsubsection}
-----------
Let $C_{-*}(S^1)$ denote the dg algebra of chains on the circle with coefficients in $\K$, graded cohomologically, with multiplication induced by the Pontryagin product $S^1 \times S^1 \ra S^1$. This algebra is [*formal*]{}, or quasi-isomorphic to its homology, an exterior algebra on one generator $\Lambda$ of degree -1 with no differential. Henceforth, by abuse of notation we take this exterior algebra as our working model for $C_{-*}(S^1)$ $$C_{-*}(S^1):= \K[\Lambda]/\Lambda^2, \ \ \ \ |\Lambda| = -1,$$ and use the terminology $C_{-*}^{sing}(S^1)$ to refer to usual singular chains on $S^1$.
\[s1strict\] A [*strict $S^1$-complex*]{}, or a [*chain complex with strict/dg $S^1$ action*]{}, is a (unital) differential graded module over $\K[\Lambda]/\Lambda^2$.
Let $(M,d)$ be a strict $S^1$-complex; by definition $(M,d)$ is a co-chain complex (note: all complexes are graded cohomologically by $\Z$ or $\Z/2$; in the latter case all degrees are implicitly modulo 2), and the dg $\K[\Lambda]/\Lambda^2$ module structure is equivalent to specificying the operation of multiplying by $\Lambda$ $$\Delta = \Lambda \cdot -: M_* \ra M_{*-1},$$ which must square to zero and anti-commute with $d$. In other words, $(M,d,\Delta)$ is what is known as a [*mixed complex*]{}, see e.g., [@Burghelea:1986aa; @Kassel:1987aa; @Loday:1992fk].
We will need to work with the weaker notion of an $\ainf$ action, or rather an $\ainf$ module structure over $C_{-*}(S^1) = \K[\Lambda]/\Lambda^2$. Recall that a [*(left) [*$\ainf$ module $M$*]{}*]{} [@Keller:2006ab; @Seidel:2008zr; @Seidel:2008cr; @ganatra1_arxiv] over the associative graded algebra $A = \K[\Lambda]/\Lambda^2$ is a graded $\K$-module $M$ equipped with maps $$\label{ainfmodulemaps}
\mu^{k|1}: A^{\otimes k} \otimes M \ra M,\ \ \ k \geq 0$$ of degree $1-k$, satisfying the $\ainf$ module equations described in [@Seidel:2008cr] or [@ganatra1_arxiv]\*[(2.35)]{}. Since $A = \K[\Lambda]/\Lambda^2$ is unital, we can work with modules that are also [*strictly unital*]{} (see [@Seidel:2008cr]\*[(2.6)]{}); this implies that all multiplications by a sequence with at least one unit elements is completely specified,[^9] and hence the only non-trivial structure maps to define are the operators $$\label{bvinfmaps}
\delta_{k} := \mu^{k|1}_{M}(\underbrace{\Lambda, \ldots, \Lambda}_{k\textrm{ copies}}, -): M \ra M [1-2k],\ k \geq 0.$$ The $\ainf$ module equations are equivalent to the following relations for for each $s \geq 0$, $$\label{inftycomplexequations}
\sum_{i=0}^s \delta_i \delta_{s-i} = 0.$$ We summarize the discussion so far with the following definition:
\[homotopycircleaction\] An [*$S^1$-complex*]{}, or a [*chain complex with a $\ainf$ $S^1$ action*]{}, is a stricly unital (right) $\ainf$ module $M$ over $\K[\Lambda]/\Lambda^2$. Equivalently, it is a graded $\K$-module $M$ equipped with operations $\{\delta_k: M \to M[1-2k]\}_{k \geq 0}$ satisfying, for each $s \geq 0$, the hierarchy of equations .
\[spaceS1actionS1complex\] If $X$ is a topological space with $S^1$ action, then $C_{-*}(X)$ carries a dg $C_{-*}^{sing}(S^1)$ module structure, with module action induced by the action $S^1 \times X \to X$. Under the $\ainf$ equivalence $C_{-*}^{sing}(S^1) \cong \K[\Lambda]/\Lambda^2$, it follows that $C_{-*}(X)$ carries an $\ainf$ (not necessarily dg) $\K[\Lambda]/\Lambda^2$ module structure, which can further be made strictly unital (by [@Lefevre02]\*[Thm. 3.3.1.2]{} or by passing to normalized chains). If one wishes, one can then appeal to abstract strictification results to produce a dg $\K[\Lambda]/\Lambda^2$ module which is quasi-isomorphic as $\ainf$ $\K[\Lambda]/\Lambda^2$ modules to $C_{-*}(X)$. More directly, it turns out one can find an equivalent dg $\K[\Lambda]/\Lambda^2$ module by taking a suitable quotient of the singular chain complex $C_{-*}(X)$ to form [*unordered singular chains*]{} of $X$ (identifying simplices differing by permuting vertices); see e.g., [@CohenGanatra:2015]\*[Appendix]{}.
There are multiple sign conventions for $\ainf$ modules over an $\ainf$ algebra; the most common two conventions appear in e.g., [@Seidel:2008cr]\*[(2.6)]{} and [@Seidel:2008zr]\*[(1j)]{} as well as many other places. These conventions are completely irrelevant for strictly unital $A = \K[\Lambda]/\Lambda^2$ modules, as the the reduced degree of any element in $\bar{A} =
\mathrm{span}_{\K}(\Lambda)$ is zero; hence the (Koszul) signs in various formulae are $+1$ in either convention.
For $s=0$, says simply that the differential $d =
\delta_0$ squares to 0; for $s=1$, implies $\delta:= \delta_1$ anti-commutes with $d$, and for $s=2$, $(\delta)^2 = -
(d\delta_2 + \delta_2 d)$, or that $\delta^2$ is chain-homotopic to zero, but not strictly zero, as measured by the chain homotopy $\delta_2$.
$S^1$-complexes, as strictly unital $\ainf$ modules over the augmented algebra $A = \K[\Lambda]/\Lambda^2$, are the objects of a dg category $dg$ category which we will call $$S^1\mod:= uA\mod$$ (compare [@Seidel:2008cr]\*[p. 90, 94]{} for the definition of $mod(A) = mod(A,\K)$) whose morphisms and compositions we now recall. Denote by $\e: A \ra \K$ the augmentation map, and $\bar{A} = \ker \e =
\mathrm{span}_{\K}(\Lambda)$ the augmentation ideal. Let $M$ and $N$ be two strictly unital $\ainf$ $A$-modules. A [*unital pre-morphism of degree $k$*]{} from $M$ to $N$ of degree $k$ is a collection of maps $ F^{d|1}: \overline{A}^{\otimes d} \otimes M \ra N$, $d \geq 0$, of degree $k-d$, or equivalently since $\dim_{\K}(\overline{A}) = 1$ in degree -1, a collection of operators $$\label{simplifiedpremorphism}
\begin{split}
F &= \{F^d\}_{d \geq 0}\\
F^d &: = F^{d|1}(\underbrace{\Lambda, \ldots, \Lambda}_{d\textrm{ copies}}, -): M \to N[k-2d].
\end{split}$$ The space of pre-morphisms of each degree form the graded space of morphisms in $S^1 \mod$, which we will denote by $\r{Rhom}_{S^1}(-,-)$: $$\label{extcomplex}
\begin{split}
\r{Rhom}_{S^1}(M,N) &:= \bigoplus_{k \in \Z} \r{Rhom}_{S^1}^k(M,N):= \oplus_k \hom_{grVect}(T(\overline{A}[1]) \otimes M, N[k]) \\
&= (\bigoplus_{k \in \Z} \hom_{grVect}( \oplus_{d \geq 0} M[2d], N[k])).
\end{split}$$ There is a differential $\partial$ on described in [@Seidel:2008cr]\*[p. 90]{}; in terms of the simplified form of pre-morphisms , one has $$(\partial F)^s = \sum_{i = 0}^s F^i \circ \delta^M_{s-i} - (-1)^{\deg(F)} \sum_{j=0}^s \delta^N_{s-j}\circ F^j.$$ An [*$\ainf$ $\K[\Lambda]/\Lambda^2$ module homomorphism*]{}, or [*$S^1$-complex homomorphism*]{} is a pre-morphism $F = \{F^d\}$ which is closed, e.g., $\partial F = 0$. In particular, $F$ is an $\ainf$ module homomorphism if the following equations are satisfied, for each $s$: $$\label{s1homomorphismeqns}
\sum_{i = 0}^s F^i \circ \delta^M_{s-i} = (-1)^{\deg(F)} \sum_{j=0}^s \delta^N_{s-j}\circ F^j.$$ Note that the $s=0$ equation reads $F^0 \circ \delta_0^M = (-1)^{\deg(F)}
\delta_0^N \circ F^0$, so (if $\partial F = 0$) $F^0$ induces a cohomology level map $[F^0]: H^*(M) \ra H^{*+\deg(F)}(N)$. A module homomorphism (or closed morphism) $F$ is said to be a [*quasi-isomorphism*]{} if $[F^0]$ is an isomorphism on cohomology. A [*strict*]{} module homomorphism $F$ is one for which $F^k = 0$ for $k > 0$.
There is an enlarged notion of a [*non-unital*]{} pre-morphism (used for modules which aren’t necessarily strictly unital), which is a collection of maps $F^d: A^{\otimes d} \otimes M \to N$ instead of $\overline{A}^{\otimes d} \otimes M \to N$. Any pre-morphism as we’ve defined induces a non-unital pre-morphism by declaring $F^d(\ldots, 1,
\ldots, \mathbf{m}) = 0$. For strictly unital modules, the resulting inclusion is a quasi-isomorphism.
\[derivedhom\] When $M$ and $N$ are $dg$ modules, or strict $S^1$-complexes, $\mathrm{Rhom}_{S^1}(M,N)$ is a [*reduced bar model*]{} of the chain complex of derived $\K[\Lambda]/\Lambda^2$ module homomorphisms, which is one of the reasons we’ve adopted the terminology “$\mathrm{Rhom}$”. In the $\ainf$ setting, we recall that there is no sensible “non-derived” notion of a $\K[\Lambda]/\Lambda^2$ module map (compare [@Seidel:2008cr]).
The composition in the category $S^1 \mod$ $$\r{Rhom}_{S^1}(N,P) \otimes \r{Rhom}_{S^1}(M,N) \ra \r{Rhom}_{S^1}(M,P)$$ is defined by $$(G \circ F)^s = \sum_{j=0}^s G^{s-j} \circ F^j.$$
\[extkk\] If $M$ is any $S^1$-complex, then its endomorphisms $\r{Rhom}_{S^1}(M,M)$ equipped with composition, form a dg algebra. As an example, consider $M = \K$, with trivial module structure (determined by the augmentation $\e: \K[\Lambda]/\Lambda^2 \ra \K$). It is straightforward to compute that, as a dga $$\label{kuidentification}
\r{Rhom}_{S^1}(\K, \K) \cong \K[u], \ |u| = 2.$$ (in terms of the definition of morphism spaces , $u$ corresponds to the unique morphism $G = \{G^d\}_{d \geq 0}$ of degree $+2$ with $G^1 = \mathrm{id}$ and $G^s = 0$ for $s \neq 1$).
In addition to taking the morphism spaces, one can define the (derived) [*tensor product*]{} of $S^1$-complexes $N$ and $M$: using the isomorphism $A \cong
A^{op}$ coming from commutativity of $A = \K[\Lambda]/\Lambda^2$, first view $N$ as a [*right $\ainf$ $A$ module*]{} (see [@Seidel:2008cr]\*[p. 90, 94]{} where the category of right $A$ modules are called $mod(\K, A)$, [@Seidel:2008zr]\*[(1j)]{}, [@ganatra1_arxiv]\*[§2]{}) and then take the usual tensor product of $N$ and $M$ over $A$ (see [@Seidel:2008cr]\*[p. 91]{} or [@ganatra1_arxiv]\*[§2.5]{}). The resulting chain complex has underling graded vector space $$\label{tensorproduct}
\begin{split}
N \otimes_{S^1}^{\mathbb L} M &:= \oplus_{d \geq 0} N \otimes \overline{A}[1]^{\otimes d} \otimes M\\
&= \bigoplus_{d\geq 0} (N \otimes_{\K} M)[2d]
\end{split}$$ (the degree $s$ part is $\bigoplus_{d \geq 0} \bigoplus_{s} N_t \otimes M_{s +
2d-t}$). Let us refer to an element $n \otimes m$ of the $d$th summand of this complex by suggestive notation $n \otimes \underbrace{\Lambda \otimes \cdots
\otimes \Lambda}_{d\textrm{ times}} \otimes m$ as in the first line of . With this notation, the differential on acts as $$\partial(n \otimes\underbrace{\Lambda \otimes \cdots \otimes
\Lambda}_{d} \otimes m) = \sum_{i=0}^d \left( (-1)^{|m|} \delta_i^N n\otimes \underbrace{\Lambda \otimes \cdots \otimes
\Lambda}_{d-i} \otimes m + n\otimes \underbrace{\Lambda \otimes \cdots \otimes
\Lambda}_{d-i} \otimes\delta_i^M m\right).$$ (here our sign convention follows [@ganatra1_arxiv]\*[§2.5]{} rather than [@Seidel:2008cr], though the sign difference is minimal).
Analogously to Remark \[derivedhom\], if $M$ and $N$ are dg $\K[\Lambda]/\Lambda^2$-modules, the chain complex described above computes the ordinary derived tensor product, whose homology is $\mathrm{Tor}_{\K[\Lambda]/\Lambda^2}(M,N)$. While we have therefore opted for the notation $N \otimes_{A}^{\mathbb L} M$ in this more general setting, this (derived) tensor product is typically written in the $\ainf$ literature simply as $N\otimes_{A} M$.
The pairing is suitably functorial with respect to morphisms of the $S^1$-complexes involved, meaning that $-\otimes_{S^1} N$ and $M\otimes_{S^1} -$ both induce dg functors from $S^1\mod$ to chain complexes (compare [@Seidel:2008cr]\*[p. 92]{}). For instance, if $F = \{F^j\}: M_0 \to
M_1$ is a pre-morphism of $S^1$-complexes, then there are induced maps $$\begin{aligned}
F_\sharp: N \otimes_{S^1}^{\mathbb L} M_0 &\ra N \otimes_{S^1}^{\mathbb L} M_1\\
\nonumber n \otimes\underbrace{\Lambda \otimes \cdots \otimes
\Lambda}_{d} \otimes m &\mapsto \sum_{j=0}^d n \otimes\underbrace{\Lambda \otimes \cdots \otimes
\Lambda}_{d-j} \otimes F^j(m);\\
F_\sharp: M_0 \otimes_{S^1}^{\mathbb L} N &\ra M_1 \otimes_{S^1}^{\mathbb L} N\\
\nonumber m \otimes\underbrace{\Lambda \otimes \cdots \otimes
\Lambda}_{d} \otimes n &\mapsto \sum_{j=0}^d (-1)^{\deg(F) \cdot |n|} F^j(m) \otimes\underbrace{\Lambda \otimes \cdots \otimes
\Lambda}_{d-j} \otimes n;\end{aligned}$$ which are chain maps if $\partial(F) = 0$.
Hom and tensor complexes of $S^1$-complexes, as in any category of $\ainf$ modules, satisfy the following strong homotopy invariance properties:
\[homotopyinvariance\] If $F: M \ra M'$ is any quasi-isomorphism of $S^1$-complexes (meaning $\partial(F) = 0$ and $[F^0]: H^*(M) \stackrel{\cong}{\to} H^*(M')$ is an isomorphism), then composition with $F$ induces quasi-isomorphisms of hom and tensor complexes: $$\begin{split}
F \circ \cdot: \r{Rhom}_{S^1}(M', P) &\stackrel{\sim}{\ra} \r{Rhom}_{S^1}(M,P)\\
\cdot \circ F: \r{Rhom}_{S^1}(P, M) &\stackrel{\sim}{\ra} \r{Rhom}_{S^1}(P,M')\\
F_{\sharp}: N \otimes_{S^1}^{\mathbb L} M &\stackrel{\sim}{\ra} N \otimes_{S^1}^{\mathbb L} M'.\\
F_{\sharp}: M \otimes_{S^1}^{\mathbb L} N &\stackrel{\sim}{\ra} M' \otimes_{S^1}^{\mathbb L} N.
\end{split}$$
The proof is a standard argument involving filtration of the above hom and tensor complexes by “length” (with respect to the number of $\overline{A}^{\otimes d}$ factors).
Let $(P, \{\delta_i^P\})$ and $(Q, \{\delta_j^Q\}_j)$ be $S^1$-complexes, and $f: P \to Q$ a chain map of some degree $\deg(f)$ (with respect to the $\delta_0^P$ and $\delta_0^Q$ differentials). An [*$S^1$-equivariant enhancement of $f$*]{} is a degree $\deg(f)$ homomorphism $\mathbf{F} =
\{\mathbf{F}^i\}_{i \geq 0}$ of $S^1$-complexes (e.g., a closed morphism, so $\mathbf{F}$ satisfies ) with $[\mathbf{F}^0] = [f]$.
Note that there are a series of obstructions to the existence of an $S^1$-equivariant enhancement of a given chain map $f$; for instance a necessary condition is the vanishing of $[f] \circ [\delta_1^P] - [\delta_1^Q] \circ
[f] = 0$.
Finally, we note that, just as the product of $S^1$ spaces $X \times Y$ possesses a diagonal action, the (linear) tensor product of $S^1$-complexes is again an $S^1$-complex.
\[diagonalaction\] If $(M,\delta_{eq}^M = \sum_{i=0}^{\infty} \delta_j^M u^j)$ and $(N, \delta_{eq}^N = \sum_{i=0}^{\infty}
\delta_i^N u^i)$ are $S^1$-complexes, then the graded vector space $M \otimes N$ is naturally an $S^1$-complex with $\delta_{eq}^{M \otimes N} = \sum_{i=0}^{\infty} \delta_k^{M
\otimes N} u^k$, where $$\delta_k^{M \otimes N}(\mathbf{m} \otimes \mathbf{n}):=
(-1)^{|\mathbf{n}|}\delta_k^M\mathbf{m} \otimes \mathbf{n} + \mathbf{m} \otimes \delta_k^N \mathbf{n}$$ We call the resulting $S^1$ action on $M \otimes N$ the [*diagonal $S^1$-action*]{}.
We compute $$\delta_j^{M\otimes N} \delta_k^{M \otimes N} (\mathbf{m} \otimes \mathbf{n}) =
\delta_j^M \delta_k^M\mathbf{m} \otimes \mathbf{n} + (-1)^{|\mathbf{n}| + 1}\delta_j^M \mathbf{m} \otimes \delta_k^N \mathbf{n} + (-1)^{|\mathbf{n}|}\delta_k^M\mathbf{m} \otimes \delta_j^N \mathbf{n} + \mathbf{m} \otimes \delta_j^N \delta_k^N \mathbf{n} \\$$ Summing over all $j+k = s$, the middle two terms cancel in pairs and the sums of the leftmost terms (respectively rightmost) terms respectively vanish because $M$ (respectively $N$) is an $S^1$-complex.
\[trivialaction\] Let $M := (M, d)$ be a chain complex over $\K$. The pullback of $M$ along the (augmentation) map $\K[\Lambda]/\Lambda^2 \to \K$ is called the [*trivial $S^1$-complex*]{}, or [*chain complex with trivial $S^1$-action*]{} associated to $M$, and denoted $\underline{M}^{triv}$. Concretely, $\underline{M}^{triv} := (M, \delta_0 = d, \delta_k = 0\textrm{ for }k>0)$.
Equivariant homology groups {#sec:equivariantgroups}
---------------------------
Let $M$ be an $S^1$-complex. Let $\K = \underline{\K}^{triv}$ denote the strict trivial rank-1 $S^1$-complex concentrated in degree 0.
\[defhomotopyorbit\] The [*homotopy orbit complex*]{} of $M$ is the (derived) tensor product of $M$ with $\K$ over $C_{-*}(S^1)$: $$\label{cyclichomologycomplex}
M_{hS^1}:= \K \otimes^{\mathbb L}_{S^1} M.$$
The (strict) morphism of $S^1$-complexes $\epsilon: \K[\Lambda]/\Lambda^2 \to \K$ (here $\K[\Lambda]/\Lambda^2$ comes equipped with structure maps $\delta_k = 0$ for $k \neq 1$ and $\delta_1 = \Lambda \cdot -$) induces by functoriality a chain map from $M$ to $M_{hS^1}$ called the [*projection to homotopy orbits*]{}: $$\label{projectionhomotopyorbits}
pr: M \cong \K[\Lambda]/\Lambda^2 \otimes^{\mathbb L}_{S^1} M \to \K \otimes^{\mathbb L}_{S^1} M = M_{hS^1}$$
\[chainlevelequivarianthomology\] When $M = C_{-*}(X)$, with $S^1$-complex induced by a topological $S^1$ action on $X$ as in Remark \[spaceS1actionS1complex\], the complex computes the Borel equivariant homology of $X$, by the following reasoning: first, the $\ainf$ equivalence between $\K[\Lambda]/\Lambda^2$ and $C_{-*}^{sing}(S^1)$ induces an equivalence $$M_{hS^1} \simeq C_{-*}(X) \otimes^{\mathbb L}_{C_{-*}^{sing}(S^1)} C_{-*}(pt).$$ Next, one observes that $C_{-*}(ES^1) \to C_{-*}(pt)$ is a [*free resolution*]{} of dg $C_{-*}^{sing}(S^1)$ modules (where the $C_{-*}^{sing}(S^1)$ actions are induced by the $S^1$ actions on $ES^1$ and $pt$ respectively), as $S^1$ acts freely on $ES^1$. Hence, the derived tensor product becomes an ordinary tensor product $$C_{-*}(X) \otimes^{\mathbb L}_{C_{-*}^{sing}(S^1)} C_{-*}(pt) = C_{-*}(X) \otimes_{C_{-*}^{sing} S^1} C_{-*}(ES^1).$$ Finally, it is a standard fact in algebraic topology (used in the construction of Eilenberg-Moore type spectral sequences, e.g., [@McCleary:2001fk]\*[Thm. 7.27]{}) that $$C_{-*}(X) \otimes_{C_{-*}^{sing} S^1} C_{-*}(ES^1) \simeq C_{-*}(X \times_{S^1} ES^1) = C_{-*}(X_{hS^1}),$$ which is the usual chain complex computing (Borel) equivariant homology. This gives some justification for the usage of the the ${
}_{hS^1}$ notation in Definition \[defhomotopyorbit\].
The [*homotopy fixed point complex*]{} of $M$ is the chain complex of morphisms from $\K$ to $M$ in the category of $S^1$-complexes: $$\label{negativecyclic}
M^{hS^1}:= \r{Rhom}_{S^1}(\K,M).$$
The morphism of modules $\epsilon: \K[\Lambda]/\Lambda^2 \to \K$ induces a chain map $M^{hS^1} \to M$ called the [*inclusion of homotopy fixed points*]{} $$\label{inclusionhomotopyfixedpoints}
\iota: M^{hS^1} = \r{Rhom}_{S^1}( \K, M) \to \r{Rhom}_{S^1}( \K[\Lambda]/\Lambda^2, M) \cong M.$$
To motivate the usage “homotopy fixed points,” note that in the topological category, the usual fixed points of a $G$ action can be described as $Maps_G(pt, X)$. When $M = C_{-*}(X)$ for $X$ an $S^1$-space, there is a canonical map $C_{-*}(X^{hS^1}) \to (C_{-*}(X))^{hS^1}$. However (in contrast to the case of homotopy orbits discussed in Remark \[chainlevelequivarianthomology\]), this map need not be an equivalence.
Composition in the category $S^1 \mod$ induces a natural action of $$\r{Rhom}_{S^1}(\K,\K) = \K[u]\ \ (|u| = 2) = H^*(BS^1)$$ on the homotopy fixed point complex. There is a third important equivariant homology complex, called the [*periodic cyclic*]{}, or [*Tate*]{} complex of $M$, defined as the localization of $M^{hS^1}$ away from $u = 0$; $$M^{Tate} := M^{hS^1} \otimes_{\K[u]} \K[u,u^{-1}].$$ The Tate construction sits in an exact sequence between the homotopy orbits and fixed points.
\[remark:gysin\] It is straightforward from the viewpoint of $\ainf$ $C_{-*}(S^1)$ modules to explain the appearance of various Gysin and periodicity sequences. Take for instance the [*Gysin exact triangle*]{} $$M \stackrel{pr}{\to} M_{hS1} \to M_{hS1}[2] \stackrel{[1]}{\to}$$ This is a manifestation of a canonical exact triangle of objects in $S^1\mod$: $$\K[\Lambda]/\Lambda^2 \stackrel{\epsilon}{\to} \K \stackrel{u}{\ra} \K[2] \stackrel{[1]}\ra$$ (recall in Remark \[extkk\] it was shown $\r{Rhom}_{S^1}(\K, \K) \cong \K[u]$), pushed forward by the functor $\r{Rhom}_{S^1}(\cdot, M)$. The others exact sequences arise similarly.
As a special case of the general homotopy-invariance properties of $\ainf$ modules stated in Proposition \[homotopyinvariance\], we have:
\[S1homotopyinvariance\] If $F: M \ra N$ is a homomorphism of $S^1$-complexes (meaning a closed morphism), it induces chain maps between equivariant theories $$\begin{aligned}
\label{inducedcyclic}F^{hS^1}: M^{hS^1} &\to N^{hS^1}\\
F_{hS^1}: M_{hS^1} &\to N_{hS^1}\\
\label{inducedtate}F^{Tate}: M^{Tate} &\to N^{Tate}\end{aligned}$$ If $F$ is a quasi-isomorphism of $S^1$-complexes (meaning simply $[F^0]$ is a homology isomorphism), then - are quasi-isomorphisms of chain complexes.
Functoriality further tautologically implies that
\[functorialitysequences\] If $F: M \ra N$ is a homomorphism of $S^1$-complexes, then the various induced maps - intertwine all of the long exact sequences for (equivariant homology groups of) $M$ with those for $N$.
$u$-linear models for $S^1$-complexes {#sec:ulinear}
-------------------------------------
It is convenient to package the data described in the previous two sections into “$u$-linear generating functions”, in the following way: Let $u$ be a formal variable of degree $+2$. Let us use the abuse of notation $$M[ [u ] ] := M \widehat{\otimes}_{\K} \K[u]$$ for the $u$-adically completed tensor product in the category of graded vector spaces; in other words $M [ [ u ] ] := \oplus_k M[ [ u] ]_k$, where $M[ [ u]
]_k = \{\sum_{i=0}^{\infty} m_i u^i\ |\ m_i \in M_{k-2i} \}$. Then, we frequently write an $S^1$-complex $(M, \{\delta_k\}_{k \geq 0})$ as a $\K$-module $M$ equipped with a map $$\delta_{eq}^{(M)} = \sum_{i=0}^{\infty} \delta_i^M u^i: M \to M[ [ u] ]$$ of total degree 1, satisfying $\delta_{eq}^2 = 0$ (where we are implicitly conflating $\delta_{eq}$ with its $u$-linear extension to a map $M[ [ u ] ] \to
M [ [ u ] ]$ in order to $u$-linearly compose and obtain a map $M \to M[ [ u] ]$).
Pre-morphisms from $M$ to $N$ of degree $k$ can similarly be recast as maps $F_{eq} = \sum_{i=0}^{\infty} F_i u^i: M \to N[ [ u] ] $ of pure degree $k$ (so each $F_i$ has degree $k-2i$). The differential on pre-morphisms can be described $u$-linearly as $$\partial(F_{eq}) = F_{eq} \circ \delta_{eq}^M - \delta_{eq}^N \circ F_{eq},$$ and composition is simply the $u$-linear composition $G_{eq} \circ F_{eq}$ (again, one implicitly $u$-linearly extends $G_{eq}$ and then $u$-linearly composes); explicitly $(\sum_{i \geq 0} G_i u^i) \circ (\sum_{j \geq 0} F_j
u^j) = \sum_{k \geq 0} (\sum_{i+j=k} G^i \circ F^j) u^k$.
With respect to this packaging, the formulae for various equivariant homology groups can be given the following, more readable form: $$\begin{aligned}
\label{htopyorbitsU} M_{hS^1} = (M ( ( u )) / u M [ [ u ] ],\ \delta_{eq})\\
M^{hS^1} = (M [ [ u ] ],\ \delta_{eq})\\
M^{Tate} = (M ( (u ) ), \delta_{eq})\end{aligned}$$ where again, we use the abuse of notation $M( ( u) ) = M[ [ u] ] \otimes_{\K
[u]} \K [u, u^{-1}]$ (on the other hand, note that is [*not*]{} completed). As before, any homomorphism (that is, closed morphism) of $S^1$-complexes $F_{eq} = \sum_{i=0}^{\infty} F^i u^i $ induces a $\K[u]$-linear chain map between homotopy-fixed point complexes by $u$-linearly extended composition, and hence, by tensoring over $\K[u]$ with $\K ( ( u )) /
u \K [ [ u ] ]$ or $\K( ( u ))$, chain maps between homotopy orbit and Tate complex constructions. With respect to these explicit complexes, the projection to homotopy orbits and inclusion of fixed points chain maps have simple explicit descriptions as follows: $$\label{projectionhomotopyorbits_explicit}
\begin{split}
pr: M &\to M_{hS^1} \\
\alpha &\mapsto \alpha \cdot u^0;
\end{split}$$
$$\label{inclusionhomotopyfixedpoints_explicit}
\begin{split}
\iota: M^{hS^1} &\to M \\
\sum_{i=0}^\infty \alpha_i u^i &\mapsto \alpha_0.
\end{split}$$
This $u$-linear lossless packaging of the data describing an $S^1$-complex is a manifestation of [*Koszul duality*]{}; in the case of $A =
\K[\Lambda]/\Lambda^2$, it posits that there is a fully faithful embedding, $\r{Rhom}(\K, -) = (-)^{hS^1}$ from $A$-modules into $B:=
\r{Rhom}_{A}(\K,\K) = \K[u]$ modules.
From the $u$-linear point of view, it is easier to observe that the exact triangle of $\K[u]$ modules $\K[[u]] \to \K((u)) \to \K ( ( u ) ) / \K [ [ u]
] = u^{-1} (\K ( ( u ) ) / u \K [ [ u] ])$ induces a (functorial in $M$) exact triangle between equivariant homology chain complexes $M^{hS^1} \to M^{Tate} \to M_{hS^1}[2] \stackrel{[1]}{\to}$.
Circle action on the open sector {#section:open}
================================
The usual and non-unital Hochschild chain complex {#subsec:hochschild}
-------------------------------------------------
Recall that an $\ainf$ category over $\K$, $\cc$ is specified by the following data:
- a collection of objects $\ob \mc{C}$
- for each pair of objects $X,X'$, a graded vector space over $\K$, $\hom_\cc (X,X')$
- for any set of $d+1$ objects $X_0, \ldots, X_d$, higher $\K$-linear composition maps $$\label{compositionmaps}
\mu^d: \hom_\cc (X_{d-1},X_d) \otimes \cdots \otimes \hom_\cc (X_{0},X_1) \ra \hom_\cc (X_0,X_d)$$ of degree $2-d$, satisfying the (quadratic) $\ainf$ relations, for each $k>0$: $$\label{ainfeq} \sum_{i,l} (-1)^{\maltese_i} \mu^{k-l+1}_\cc(x_k, \ldots, x_{i+l+1}, \mu^{l}_\cc(x_{i+l}, \ldots, x_{i+1}), x_i, \ldots, x_1) = 0.$$ with sign $$\label{ainfsign}
\maltese_i := ||x_1|| + \cdots + ||x_i||.$$ where $|x|$ denotes degree and $||x||:= |x| - 1$ denotes reduced degree.
The first two equations of imply that $\mu^1$ is a differential, and the cohomology level maps $[\mu^2]$ are a genuine composition for the (non-unital) category $H^*(\cc)$ with the same objects and morphisms $$\label{cohomologymorphisms}
\mathrm{Hom}_{H^*(\cc)}(X,Y) := H^*(\hom_{\cc}(X,Y), \mu^1).$$ We say that $\cc$ is [*cohomologically unital*]{} if there exist cohomology-level identity morphisms $[e_X] \in \r{Hom}_{H^*(\cc)}(X,X)$ for each object $X$, making $H^*(\cc)$ into a genuine category. We say that $\cc$ is [*strictly unital*]{} if there exist elements ${e}_X^+ \in \hom_\cc(X,X)$, for every object $X$, satisfying $$\label{eq:strictunits}
\begin{split}
\mu^1(e_X^+) &= 0 \\
(-1)^{|y|}\mu^2(e_{X_1}^+,y) &= y = \mu^2(y,e_{X_0}^+)\textrm{ for any }y \in \hom_{\cc}(X_0, X_1) \\
\mu^d(\cdots, e_X^+, \cdots) &= 0 \textrm{ for } d > 2.
\end{split}$$ (we call such elements the chain-level, or strict, identity elements).
The [*Hochschild chain complex*]{}, or [*cyclic bar complex*]{} of an $\ainf$ category $\cc$ is the direct sum of all cyclically composable sequences of morphism spaces in $\cc$: $$\label{cyclichochschild}
\r{CH}_*(\cc):= \bigoplus_{k\geq 0, X_{i_0}, \ldots, X_{i_k} \in \ob
\cc} \hom_\cc(X_{i_k},X_{i_0}) \otimes \hom_\cc(X_{i_k-1},X_{i_k}) \otimes \cdots
\otimes \hom_\cc(X_{i_0},X_{i_1}),$$ The (cyclic bar) differential $b$ acts on Hochschild chains by summing over ways to cyclically collapse elements by any of the $\ainf$ structure maps: $$\label{cyclicbardifferential}
\begin{split}
b (\mathbf{x}_d \otimes &x_{d-1} \otimes \cdots \otimes x_1) =\\
&\sum (-1)^{\#_{k}^d}\mu^{d-i}(x_{k}, \ldots, x_1, \mathbf{x}_d, x_{d-1}, \ldots, x_{k+i+1}) \otimes x_{k+i} \otimes \cdots \otimes x_{k+1} \\
&+ \sum (-1)^{\maltese_{1}^{s}}\mathbf{x}_d \otimes \cdots \otimes \mu^j(x_{s+j+1} \otimes \cdots \otimes x_{s+1}) \otimes x_{s} \otimes \cdots \otimes x_1.
\end{split}$$ with signs $$\begin{aligned}
\maltese_{i}^{k} &:= \sum_{j=i}^k ||x_i||\\
\#_k^d &:= \maltese_1^k \cdot ( 1 + \maltese_{k+1}^d) + \maltese_{k+1}^{d-1} + 1\end{aligned}$$ In this complex, Hochschild chains are (cohomologically) graded as follows: $$\label{hochschildchaingrading}
\deg(\mathbf{x}_d \otimes x_{d-1} \otimes \cdots \otimes x_1) := \deg(\mathbf{x}_d) + \sum_{i=1}^{d-1} \deg(x_i) - d + 1 = |\mathbf{x}_d| + \sum_{i=1}^{d-1} ||x_i||.$$
Frequently the notation $\r{CH}_*(\cc,\cc)$ is used for to emphasize that Hochschild homology is taken here with [*diagonal coefficients*]{}, rather than coefficients in another bimodule.
If $\cc$ is a strictly unital $\ainf$ category, then the chain complex carries a strict $S^1$ action $B: \r{CH}_*(\cc) \to
\r{CH}_{*-1}(\cc)$, involving summing over ways to cyclically permute chains and insert identity morphisms; see Remark \[strictlyunitalB\] below. However, there is a quasi-isomorphic non-unital Hochschild complex of $\cc$, which always carries a strict $S^1$-action (even if $\cc$ is not strictly unital), which we will now describe.
As a graded vector space, the [*non-unital Hochschild complex*]{} consists of two copies of the cyclic bar complex, the second copy shifted down in grading by 1: $$\label{nonunitalcomplex}
\r{CH}_*^{nu}(\cc):= \r{CH}_*(\cc) \oplus \r{CH}_*(\cc)[1]$$ With respect to the decomposition , we sometimes refer to elements as $\sigma := \check{\alpha} + \hat{\beta}$ or $\sigma = (\alpha,
\beta)$, with the notation $\hat{\alpha}$ or $\check{\beta}$ indicating that the element $\alpha$ and $\beta$ belong to the left or right factor respectively. Similarly, we refer to the left and right factors as the [*check factor*]{} and the [*hat factor*]{} respectively.
Let $b'$ denote a version of the differential omitting the “wrap-around terms" (often simply called the [*bar differential*]{}): $$\begin{split}
b' (x_d \otimes &x_{d-1} \otimes \cdots \otimes x_1) =\\
& \sum (-1)^{\maltese_{1}^{s}}x_d \otimes \cdots \cdots \otimes x_{s+j+1} \otimes
\mu^j(x_{s+j} \otimes \cdots \otimes x_{s+1}) \otimes x_{s} \otimes \cdots \otimes x_1\\
& + \sum (-1)^{\maltese_1^{d-j}} \mu^j(x_d, x_{d-1}, \ldots, x_{d-j+1})\otimes x_{d-j} \otimes \cdots \otimes x_1.
\end{split}$$ For an element $\hat{\beta} = x_d \otimes \cdots \otimes x_1$ in the hat (right) factor of , define an element $d_{\wedge \vee}
(\hat{\beta})$ in the check (left) factor of as follows: $$\begin{split}
d_{\wedge \vee} (\hat{\beta}) &:= (-1)^{\maltese_2^d + ||x_1|| \cdot \maltese_2^d + 1} x_1 \otimes x_d \otimes \cdots \otimes x_2 + (-1)^{\maltese_1^{d-1}} x_d \otimes \cdots \otimes x_1.
\end{split}$$ In this language, the differential on the non-unital Hochschild complex can be written as: $$b^{nu}: (\check{\alpha},\hat{\beta}) \mapsto (b(\check{\alpha}) + d_{\wedge \vee} (\hat{\beta}), b'(\hat{\beta}))$$ or equivalently can be expressed via the matrix $$b^{nu} = \left( \begin{array}{cc} b & d_{\wedge \vee} \\ 0 & b' \end{array} \right) .$$ The left factor $\r{CH}_*(\cc) \hookrightarrow \r{CH}_*^{nu}(\cc)$ is by definition a subcomplex. Moreoever, since the quotient complex is the standard $\ainf$ bar complex with differential $b'$, which is acyclic for cohomologically unital $\cc$ (by a standard length filtration spectral sequence argument, compare [@Seidel:2008zr]\*[Lemma 2.12]{} or [@ganatra1_arxiv]\*[Prop. 2.2]{}), it follows that:
\[inclusionquasi\] The inclusion map $\iota: \r{CH}_*(\cc) \hookrightarrow \r{CH}_*^{nu}(\cc)$ is a quasi-isomorphism (when $\cc$ is cohomologically unital).
\[rem:unitinsertion\] The non-unital Hochschild complex of $\cc$ can be conceptually explained in terms of cyclic bar complexes as follows (compare [@Loday:1992fk]\*[§ 1.4.1]{} and [@Sheridan:2015ab]\*[§3.5]{}): first, [*augment*]{} the category $\cc$ by adjoining strict units; meaning consider the $\ainf$ category $\cc^+$ with $\ob\ \cc^+ = \ob
\cc$, $$\hom_{\cc^+}(X,Y) = \begin{cases} \hom_{\cc}(X,Y) & X \neq Y \\
\hom_{\cc}(X,X) \oplus \K \langle e^+_X \rangle & X = Y
\end{cases}$$ whose $\ainf$ structure maps completely determined by the fact that $\cc$ is an $\ainf$ subcategory, and the elements $e^+_X$ elements act as strict units in the sense of . Next, consider the [*normalized (or reduced) Hochschild complex*]{} of the strictly unital category $\cc^+$, $\r{CH}_*^{red}(\cc^+)$, by definition the quotient of $\r{CH}_*(\cc^+)$ by the acyclic subcomplex consisting of $e^+$ terms in any position but the first. Now, take the further quotient of $\r{CH}_*^{red}(\cc^+)$ by the subcomplex of length 1 Hochschild chains of the form $e^{+}_X$ for some $X$. The resulting complex, denoted $\widetilde{\r{CH}_*}(\cc^+)$, is bijective as a chain complexes to $\r{CH}_*^{nu}(\cc)$ via the following chain map $$\label{eq:fbijection}
\begin{split}
f: \widetilde{\r{CH}_*}(\cc^+) &\stackrel{\cong}{\to} \r{CH}_*^{nu}(\cc)\\
f(y_k \otimes \cdots \otimes y_1) &= \begin{cases}
(0, y_{k-1} \otimes \cdots \otimes y_1) & y_k = e^+_X \textrm{ for some }X \\
(y_k \otimes \cdots \otimes y_1, 0) & \textrm{otherwise}
\end{cases}
\end{split}$$ In particular, the differential in $\r{CH}^{nu}(\cc)$ on a Hochschild chain $\hat{\alpha}$ agrees with the (usual cyclic bar) Hochschild differential applied to $e^+_X
\otimes \alpha$ under the correspondence $f$.
Circle action on the Hochschild complex {#subsec:cyclic}
---------------------------------------
The $S^1$ action on the non-unital (or usual) Hochschild complex is built out of several intermediate operations. First, let $t: \r{CH}_*(\cc) \to \r{CH}_*(\cc)$ denote the (signed) cyclic permutation operator on the cyclic bar complex generating the $\Z/k\Z$ cyclic action on the length $k$ expressions $$\label{toperator}
t: x_k \otimes \cdots \otimes x_1 \mapsto (-1)^{||x_1|| \cdot \maltese_2^k + ||x_1|| + ||x_k||} x_1 \otimes x_k \otimes \cdots \otimes x_{2},$$ (this is not a chain map). Let $N$ denote the [*norm*]{} of this operation; that is the sum of all powers of $t$ (this depends on $k$, the length of a given Hochschild chain): $$\label{Noperator}
\begin{split}
N: x_k \otimes \cdots \otimes x_1 = \sigma \mapsto (1 + t + t^2 + \cdots + t^{k-1}) \sigma;
\end{split}$$ Let $s^{nu}: \r{CH}^{nu}_*(\cc) \to \r{CH}^{nu}_{*-1}(\cc)$ be the linear map which sends check chains to the corresponding hat chains, and hat chains to zero: $$\label{snu}
s^{nu}(x_d \otimes \cdots \otimes x_1, y_t \otimes \cdots \otimes y_1) := (-1)^{\maltese_1^d + ||x_d|| + 1}(0, x_d \otimes \cdots \otimes x_1 ).$$ (again, not a chain map), and finally define $$\label{eq:Bnudef}
\begin{split}
B^{nu}: \r{CH}^{nu}_*(\cc) &\to \r{CH}^{nu}_{*-1}(\cc)\\
B^{nu}(x_k \otimes \cdots \otimes x_1,y_l \otimes \cdots \otimes y_1) &:= \sum_{i=1}^k (-1)^{\maltese_1^i \maltese_{i+1}^k + ||x_k|| + \maltese_1^k + 1}(0, x_i \otimes \cdots \otimes x_1 \otimes x_k \otimes \cdots \otimes x_{i+1}))\\
&= s^{nu} (N(x_k \otimes \cdots \otimes x_1), y_l \otimes \cdots \otimes y_1)\\
&= \sum_{i=0}^{k-1} s^{nu} (t^{i}(x_k \otimes \cdots \otimes x_1), y_l \otimes \cdots \otimes y_1)
\end{split}$$
$(B^{nu})^2 = 0$ and $b^{nu} B^{nu} + B^{nu}b^{nu} = 0$. That is, $\r{CH}_*^{nu}(\cc)$ is a strict $S^1$-complex, with the action of $\Lambda = [S^1]$ given by $B^{nu}$.
Let $b_{eq} = b^{nu} + u B^{nu}$ be the strict $S^1$-complex structure on the non-unital Hochschild complex $\r{CH}_*^{nu}(\cc)$, $u$-linearly packaged as in §\[sec:ulinear\]. Using this, we can define cyclic homology groups:
The [*(positive) cyclic*]{} chain complex, the [*negative cyclic*]{} chain complex, and the [*periodic cyclic*]{} chain complexes of $\cc$ are the [*homotopy orbit complex*]{}, [*homotopy fixed point complex*]{}, and [*Tate constructions*]{} of the $S^1$-complex $(\r{CH}_*^{nu}(\cc), b_{eq})$ respectively. That is: $$\begin{aligned}
\r{CC}^+_*(\cc) &:= (\r{CH}_*^{nu}(\cc))_{hS^1} = (\r{CH}_*^{nu}(\cc) \otimes_\K \K ( ( u )) / u \K [ [ u ] ], b_{eq}) \\
\r{CC}^+_*(\cc) &:= (\r{CH}_*^{nu}(\cc))^{hS^1} = (\r{CH}_*^{nu}(\cc) \widehat{\otimes}_\K \K[ [ u ] ], b_{eq}) \\
\r{CC}^\infty_*(\cc) &:= (\r{CH}_*^{nu}(\cc))^{Tate} = (\r{CH}_*^{nu}(\cc) \widehat{\otimes}_\K \K(( u ) ), b_{eq})
\end{aligned}$$ with grading induced by setting $|u| = +2$, and where (as in § \[sec:ulinear\]), $\widehat{\otimes}$ refers to the $u$-adically completed tensor product in the category of graded vector spaces. The cohomologies of these complexes, denoted $\r{HC}^{+/-/\infty}_*(\cc)$, are called the [*(positive), negative,*]{} and [*periodic cyclic homologies*]{} of $\cc$ respectively.
The $C_{-*}(S^1)$ module structure on $\r{CH}_*^{nu}(\cc)$ is suitably functorial in the following sense: Let $\mathbf{F}: \cc \ra \cc'$ be an $\ainf$ functor, There is an induced chain map on non-unital Hochschild complexes $$\label{inducedmapnonunital}
\begin{split}
\mathbf{F}_{\sharp}^{nu}: \r{CH}_*^{nu}(\cc) &\ra \r{CH}_*^{nu}(\cc',\cc')\\
(x,y) &\mapsto (\mathbf{F}_{\sharp}(x), \mathbf{F}_{\sharp}'(y))
\end{split}$$ where $$\begin{aligned}
\mathbf{F}_{\sharp}' (x_k \otimes \cdots \otimes x_0) &:= \sum_{i_1, \ldots, i_s} \mathbf{F}^{i_1}(x_k\cdots) \otimes \cdots \otimes \mathbf{F}^{i_s}(\cdots x_0)\\
\mathbf{F}_{\sharp} (x_k \otimes \cdots \otimes x_0) &:= \sum_{i_1, \ldots, i_s, j} \mathbf{F}^{j + 1 + i_1}(x_j, \ldots, x_0, x_k, \ldots, x_{k-i_1+1}) \otimes \mathbf{F}^{i_2}(\ldots)\otimes \cdots \otimes \mathbf{F}^{i_s}(x_{j+i_s}\ldots, x_{j+1}).\end{aligned}$$ which is an isomorphism on homology if $\mathbf{F}$ is a quasi-isomorphism (indeed, even a Morita equivalence). This functoriality preserves $S^1$ structures:
\[functorialityB\] $\mathbf{F}_{\sharp}^{nu}$ gives a strict morphism of strict $S^1$-complexes, meaning $\mathbf{F}_{\sharp}^{nu} \circ b^{nu} = b^{nu} \circ \mathbf{F}_{\sharp}^{nu}$ and $\mathbf{F}_{\sharp}^{nu} \circ B^{nu} = B^{nu} \circ \mathbf{F}_{\sharp}^{nu}$. In other words, the pre-morphism of $\ainf$ $\K[\Lambda]/\Lambda^2$ modules defined as $$\mathbf{F}_{*}^{d}(\underbrace{\Lambda, \ldots, \Lambda}_d, \sigma) := \begin{cases} \mathbf{F}_{\sharp}^{nu} (\sigma) & d =0 \\
0 & d \geq 1
\end{cases}$$ is closed, e.g., an $\ainf$ module homomorphism.
It is well known that $\mathbf{F}_{\sharp}^{nu}$ is a chain map, so it suffices to verify that $\mathbf{F}_{\sharp}^{nu} \circ B^{nu} = B^{nu}
\circ \mathbf{F}_{\sharp}^{nu}$, or in terms of $$\mathbf{F}_{\sharp}' \circ s^{nu} N = s^{nu} N \circ \mathbf{F}_{\sharp}.$$ We leave this an exercise, noting that applying either side to a Hochschild chain $x_k \otimes \cdots \otimes x_1$, the sums match identically.
\[strictlyunitalB\] If $\cc$ is strictly unital, one can also define an operator $B: \r{CH}_*(\cc)
\to \r{CH}_{*-1}(\cc)$ on the usual cyclic bar complex by $$B = (1-t)s N$$ where up to a sign $s$ denotes the operation of inserting, at the beginning of a chain, the unique strict unit $e_X^+$ preserving cyclic composability: $$\label{degeneracyoperator}
\begin{split}
s: x_k \otimes \cdots \otimes x_1 \mapsto (-1)^{||x_k|| + \maltese_1^k + 1} e_{X_{i_k}}^+ \otimes x_k \otimes \cdots \otimes x_1,\\
\textrm{where }x_k \in \hom_\cc(X_{i_k},X_{i_0}).
\end{split}$$ It can be shown that $B^2 = 0$ and $Bb + bB = 0$, $\r{CH}_*(\cc)$ is a strict $S^1$-complex; moreoever that the quasi-isomorphism $\r{CH}_*(\cc)
\cong \r{CH}_*^{nu}(\cc)$ is one of $S^1$-complexes. In fact $B$ descends to the [*reduced Hochschild complex*]{} $\r{CH}_*^{red}(\cc)$ described in Remark \[rem:unitinsertion\] where it takes the simpler form $$B^{red} = sN$$ (as applying $t s N$ results in a Hochschild chain with a strict unit not in the first position, which becomes zero in $\r{CH}_*^{red}(\cc)$). If $\cc$ was not necessarily strictly unital, following Remark \[rem:unitinsertion\] one can still consider the quotient of the reduced Hochschild complex of the augmented category $\cc^+$, which we called $\widetilde{\r{CH}}_*(\cc^+)$. The discussion here equips this complex with an $S^1$ action $\widetilde{B}^{red}$. Under the bijection $f$ , $\widetilde{B}^{red}$ is sent to $B^{nu}$ and $s$ is sent to $s^{nu}$.
Continuing Remark \[rem:unitinsertion\], suppose we have constructed $\r{CH}^{nu}_*(\cc)$ as $\widetilde{\r{CH}}_*(\cc):= \r{CH}^{red}_*(\cc^+)/ \oplus_X \K \langle e_X^+ \rangle$, the quotient of the reduced Hochschild complex of the augmented category $\cc^+$. Given any $\mb{F}$ as above, extend $\mb{F}$ to an augmented functor $\mb{F}^+$ by mandating that $$\begin{split}
(\mb{F}^+)^1(e^+_X) &= e^+_{\mb{F}X}\\
({\mb{F}^+})^d(\ldots, e^+_X, \ldots ) &= 0.
\end{split}$$ It is easy to see that $(\mb{F}^+)_*: \r{CH}_*(\cc^+) \to \r{CH}_*((\cc')^+)$ descends to a map $\widetilde{\mb{F}}: \widetilde{\r{CH}}_*(\cc^+) \to \widetilde{\r{CH}}_*((\cc')^+)$. Under the bijection , this precisely corresponds to $\mb{F}_{\sharp}^{nu}$ described above. In particular, the fact that strictly unital functors induce strict $S^1$-morphisms between (usual) Hochschild complexes immediately implies Proposition \[functorialityB\].
\[rem:otheroptionsNUcomplex\] There are options besides the non-unital Hochschild complex for seeing the $C_{-*}(S^1)$ option on a Hochschild complex of the Fukaya category. For instance one could:
1. perform a strictly unital replacement (via homological algebra as in [@Seidel:2008zr]\*[§2]{} [@Lefevre02]\*[Thm. 3.2.1.1]{}), and work with the Hochschild complex of the replacement. However, this doesn’t retain a relationship between the $\ainf$ operations and geometric structure, and hence is difficult to use with open-closed maps.
2. Geometrically construct a strictly unital structure on the Fukaya category via constructing [*homotopy units*]{} [@Fukaya:2009qf], which roughly involves building a series of geometric higher homotopies between the operation of $\ainf$ multiplying by a specified geometrically defined cohomological unit, and the operation of $\ainf$ multiplying by a strict unit (which is algebraically defined, but may also be geometrically characterized in terms of forgetful maps) From this one defines a strictly unital $\ainf$ category $\mc{F}^{hu}$ with $\hom_{\mc{F}^{hu}}(X,X) = \hom_{\mc{F}}(X,X) \oplus
\K \langle e^+_X, f_X\rangle$, extending the $\ainf$ structure on $\mc{F}$, such that $e^+_X$ a strict unit and $\mu^1(f_X) = e^+_X - e_X$ for $e_X$ a chosen a cohomological unit.
Remark \[strictlyunitalB\] then equips the usual Hochschild complex $\r{CH}_*(\mc{F}^{hu},\mc{F}^{hu})$ with a strict $S^1$ action. Using this one can construct a cyclic open-closed map with source $\r{CH}_*(\mc{F}^{hu}, \mc{F}^{hu})$, in a manner completely analogous to the construction of $\mc{F}^{hu}$. This option is equivalent to the one we have chosen (and has some benefits), but requires additional technical work (for instance in constructing and working with $\mc{F}^{hu}$).
A construction of homotopy units was introduced in the work of Fukaya-Oh-Ohta-Ono [@Fukaya:2009qf]\*[Ch. 7, §31]{}. See [@ganatra1_arxiv] for an implementation in the (possibly wrapped) exact (or otherwise tautologically unobstructed), multiple Lagrangians setting.
The Fukaya category {#subsec:fukaya}
-------------------
For the purposes of simplifying the discussion of Floer theory, we focus on the technically simplest cases in which (compact or wrapped) Fukaya categories can be defined, namely exact (Liouville) and monotone or aspherical symplectic manifolds. There are nowadays multiple technical models of wrapped Floer homology and wrapped Fukaya categories; we work with the framework of quadratic Hamiltonians as defined in [@Abouzaid:2010kx], where the construction is nearly as simple as (and can be described simultaneously with, with a few minor modifications) that of compact (exact or monotone) Fukaya categories.
Let $(M^{2n},\omega)$ be a symplectic manifold which is [*admissible*]{}, meaning for us that it could be
- Liouville (in particular noncompact), meaning $\omega$ is exact with a fixed choice of primitive $\lambda$, such that flowing out by the Liouville vector field $Z$ (defined by $\iota_Z \omega = \lambda$) induces a diffeomorphism $$\label{cylend}
M \backslash \mathring{\bar{M}} \cong \partial \bar{M} \times [0, \infty)$$ for some codimension-0 manifold-with-boundary $\bar{M}$ (called a Liouville domain whose completion is $M$); or
- compact and either monotone (meaning $[\omega] = 2 \tau c_1(M)$ for some $\tau > 0$) or symplectically aspherical (meaning $\omega(\pi_2(M)) = 0$).
If $M$ is Liouville, we henceforth fix a cylindrical end , and use $r$ to refer to the corresponding $[0,\infty)$ coordinate. If $2c_1(M) = 0$ and we wish to define a $\Z$-graded category, we might optionally equip $M$ with a [*grading structure*]{}, which is a trivialization of the square of the canonical bundle $(\Lambda_{\C}^n T^*M)^{\otimes 2}$. In such an $M$, we will study properly embedded Lagrangians submanifolds $L \subset M$ which are themselves [*admissible*]{} meaning that
- if $M$ is Liouville then $L$ must be exact $\lambda_{L} = df$, equipped with fixed choice of primitive which vanishes outside a compact set (which implies, as in that $L$ is modeled on the cone of a Legendrian near infinity);
- if $M$ is monotone, then we require $L$ to be monotone (meaning $\omega(-)= \rho \mu_L(-): H_2(M,L) \to \R$, where $\omega$ is symplectic area and $\mu_L$ is the Maslov class); and if $M$ is aspherical we require $L$ to be tautologically unobstructed (meaning it bounds no $J$-holomorphic discs for some $J$, which is satisfied if e.g., $\omega(\pi_2(M,L)) = 0$).
More generally we will allow any $M$ and Lagrangians in it which satisfy the Assumptions stated below and in §\[section:closed\]: these amount to requiring either that the possible non-compactness of $M$ and $L$ is of the geometrically tame variety and that $M$/$L$ have no/bound no $J$-holomorphic spheres/discs, or if they do that such spheres/discs can be counted — or shown to not arise in compactifications — using classical methods).
In order to define the Fukaya category we typically equip Lagrangian submanifolds with some extra structure, implicitly depending on the ground field $\K$ and the grading structure we wish to define our category. To this end, we say an [*admissible Lagrangian brane*]{} consists of a properly embedded admissible Lagrangian submanifold $L \subset M$ which is equipped with the following extra two pieces of data (which are only required if one wants to work with $\mathrm{char\ }\K \neq 2$ or with $\Z$ gradings respectively, the latter of which is not relevant to monotone case): $$\begin{aligned}
&\textrm{an orientation and Spin structure; and}\\
&\textrm{a grading in the sense of \cite{Seidel:2000uq} (with respect to a given grading structure on $M$).}
\end{aligned}$$ (these choices of extra data require $L$ to be Spin and satisfy $2c_1(M,L)
= 0$, where $c_1(M,L) \in H^2(M,L)$ is the relative first Chern class respectively).
There are other possible grading structures on $M$ and $L$ one can use to equip the Fukaya category suitable gradings (under geometric hypotheses), for instance $\Z/2k$ gradings, homology class gradings or hybrids thereof (c.f., [@Seidel:2000uq; @Sheridan:2015aa]). We suppress discussion of these, but — seeing as such matters are largely orthogonal to our arguments — we note that our results apply in such contexts as well.
Henceforth, all Lagrangians are admissible Lagrangian branes. Denote by $\ob
\f$ a finite collection of Lagrangians. Choose a (potentially time-dependent) Hamiltonian $H_t: M \to \R$ satisfying, if $M$ is Liouville, the following [*quadratic at $\infty$*]{} condition: $$\label{quadraticatinfinity}
\textrm{$H = r^2$ on the cylindrical end \eqref{cylend} (at least outside a compact subset).}$$ We further assume (for any admissible $M$) that $H$ satisfies the following non-degeneracy condition.
All time-1 chords of $X_{H_t}$ between any pair of Lagrangians in $\ob \f$ are non-degenerate.
It is straightforward to adapt all of our constructions to larger collections of Lagrangians, by for instance, choosing a different Hamiltonian $H_{L_0, L_1}$ for each pair of Lagrangians $L_0, L_1$, and by choosing Floer perturbation data depending on corresponding sequences of objects, see e.g., [@Seidel:2008zr]\*[§9j]{}. We have opted for using a single $H_t$ simply to keep the notation simpler.
For any pair of Lagrangians $L_0, L_1 \in \ob \f$, the set of time 1 Hamiltonian flows of $H$, $\chi(L_0, L_1)$ can be thought of as the critical points of an action functional on the [*path space*]{} from $L_0$ to $L_1$, $\mathcal{P}_{L_0, L_1}$ (this functional is a priori multi-vaued, but it is certainly $\R$-valued in the presence of primitives $\lambda$ for $\omega$ and $f_i$ for $\lambda|_{L_i}$). Given a choice of grading structure on $M$ and grading for each $L_i$ above, elements of $\chi(L_0, L_1)$ can be graded by the [*Maslov index*]{} $$\deg: \chi(L_0, L_1) \ra \Z$$ (in the absence of grading structures this is always well defined mod 2, provided our Lagrangians are oriented, which is automatic if they are Spin). As a graded $\K$-module, the morphism space in the Fukaya category between $L_0$ and $L_1$, also known as the ([*wrapped*]{} if $M$ is Liouville) [*Floer homology cochain complex of $L_0$ and $L_1$ with respect to $H$*]{}, has one (free) generator for each element of $\chi(L_0,L_1)$; concretely $$\hom_{\f}^i(L_0, L_1) = CF^*(L_0, L_1, H_t, J_t):= \bigoplus_{x\in \chi(L_0,L_1), deg(x)=i} |o_x|_{\K},$$ where the [*orientation line*]{} $o_x$ is the real vector space associated to $x$ by index theory (see [@Seidel:2008zr]\*[§11h]{}; a priori $o_x$ depends on a choice of trivialization of $x^*TM$ compatible with the grading structure. However, there is a unique such choice in the presence of a $\Z$-grading, and in the $\Z/2$-graded case any two choices made induce canonically isomorphic orientation lines) and for any one dimensional real vector space $V$ and any ring $\K$, the [*$\K$-normalization*]{} $$\label{normalization}|V|_{\K}$$ is the $\K$-module generated by the two possible orientations on $V$, with the relationship that their sum vanishes (if one does not want to worry about signs, note that $|V|_{\Z/2} \cong \Z/2$ canonically).
The $\ainf$ structure maps arise as counts of parameterized families of (suitably coherently perturbed) solutions to Floer’s equation with source a disc with $d$ inputs and one output. We will quickly summarize the definition and relevant choices required, referring the reader to standard referencers for more details (the basic reference we follow is [@Seidel:2008zr] for Fukaya categories compact exact Lagrangians in Liouville manifolds; see also [@Sheridan:2016] for the mostly straightforward generalization to the monotone case. In the main body of exposition we focus on the (slightly simpler) case of compact (admissible) Lagrangians in compact symplectic manifolds; we detail the additional data and variations required for wrapped Fukaya categories of Liouville manifolds in Remarks \[wrappedFloerdatum\] and \[wrappedainf\]).
For $d \geq 2$, we use the notation $\overline{\mc{R}}^d$ for the (Deligne-Mumford compactified) moduli space of discs with $d+1$ marked points modulo reparametrization, with one point $z_0^-$ marked as negative and the remainder $z_1^+, \ldots, z_d^+$ (labeled counterclockwise from $z_0^-$) marked as positive. Orient the open (interior) locus $\mc{R}^d$ as in [@Seidel:2008zr]\*[§12g]{} and [@Abouzaid:2010kx]. $\overline{\mc{R}}^d$ can be given the structure of a manifold-with-corners, and its higher strata are trees of stable discs with a total of $d$ exterior positive marked points and 1 exterior negative marked point. Denote the positive and negative semi-infinite strips by $$\begin{aligned}
\label{strip1} Z_+ &:= [0,\infty) \times [0,1]\\
\label{strip2} Z_- &:= (-\infty,0] \times [0,1]\end{aligned}$$ One first equips the spaces $\overline{\mc{R}}^d$ for each $d$ with a [*consistent collection of strip-like ends*]{} $\mathfrak{S}$: that is, for each component $S$ of $\overline{\mc{R}}^d$, a collection of maps $\e^{\pm}_k: Z_\pm \ra
S$ all with disjoint image in $S$, chosen so that positive/negative strips map to neighborhoods of positively/negatively-labeled boundary marked points respectively, smoothly varying with respect to the manifold-with-corners structures and compatible with choices made on boundary and corner strata, which are products of lower dimensional copies of $\overline{\mc{R}}^k$’s.
In order to associate transversely cut out moduli spaces of such maps, one studies a parametric family of solutions to Floer’s equation depending on a choice of “Floer (or perturbation) data” over the parameter space. Concretely, a [*Floer datum*]{} for a family of domains (in this case $\overline{\mc{R}}^d$) is a choice of, for each domain $S$ in the parametric family, of
- an $S$-(or domain-)dependent almost complex structure $J_S$ and Hamiltonian $H_S$,
- a one-form $\alpha$ on $S$;
which depends (smoothly) on the particular domain in $\overline{\mc{R}}^d$ (and the position in that domain), and are [*compatible with strip-like ends*]{}, meaning $\alpha$ pulls back to $dt$ and $(H_S, J_S)$ pull back to a fixed choice $(H_t, J_t)$ in coordinates -. One inductively chooses a [*Floer datum for the $\ainf$ structure*]{}, which is a choice of Floer data for the collection of domains $\{\overline{\mc{R}}^d\}_{d \geq 2}$ which is [*consistent*]{}, meaning that the Floer data chosen on a given family of domains $\overline{\mc{R}}^d$ agree smoothly along the boundary and corners (which are products of lower dimensional $\overline{\mc{R}}^k$’s) with previous choices of Floer data made. Such consistent choices exist essentially because spaces of Floer data are contractible).
\[liouvilleFloerdatum\] If $M$ is Liouville and we are studying the Fukaya category of compact exact Lagrangains, there is an additional requirement imposed on any Floer datum one uses; namely one requires that $J_S$ be of [*contact type*]{} in a neighborhood of infinity in the sense of [@Seidel:2008zr]\*[(7.3)]{}, and $H_S$ be either $0$ or of the form $f(r)$ near infinity for some function with non-negative first and second derivatives. The more restrictive types of Floer data chosen for wrapped Fukaya categories in Remark \[wrappedFloerdatum\] of course suffice as well.
\[wrappedFloerdatum\] Following [@Abouzaid:2010kx], we recall the additional data and requirements for Floer data to construction the operations appearing in the wrapped Fukaya category (with quadratic Hamiltonians). If $M$ is a Liouville manifold let $\psi^{\rho}: M \to M$ denote the time $\log(\rho)$ (outward) Liouville flow. One fixes for each $S$ in addition to $(H_S, J_S, \alpha_S)$, a collection of constants $w_k \in \R_{>0}$ for each end called [*weights*]{} (so $w_k$ is the weight associated to the $k$th end), and a map $\rho_S: \partial S \to \R_{>0}$, called the [*time-shifting map*]{}. These should all satisfy:
1. \[wrappedtimeshiftingweight\] $\rho_S$ should be constant and equal to the weight $w_k$ on the $k$th strip-like end;
2. \[wrappedoneform\] The 1-form $\alpha_S$ should be [*sub-closed*]{} meaning $d\alpha_S \leq 0$, equal to $w_k dt$ in the local coordinates on each strip-like end, and 0 when restricted to $\partial S$. By Stokes’ theorem, this condition implies the sum of weights over all negative ends is greater than or equal to the sum of weights over all positive ends, and there should therefore be at least one negative end always (in this case there is one);
3. \[wrappedhamiltonian\] The Hamiltonian term should be quadratic at infinity and pull back to $\frac{H \circ \psi^{w_k}}{w_k^2}$ in coordinates on each end (note that such a Hamiltonian is quadratic if $H$ is by an elementary computation [@Abouzaid:2010kx]\*[Lem. 3.1]{});
4. \[wrappedalmostcomplexstructure\] The almost complex structure term should be of contact-type at infinity and equal to $(\psi^{w_k})^*J_t$ in coordinates on each end.
There is a [*rescaling action*]{} by $(\R_{>0}, \cdot)$ on the space of such surface dependent data, which sends $(\rho_S, \{w_k\}, \alpha_S, H_S, J_S) \mapsto (r \rho_S, \{r w_k\}, r \alpha_S, \frac{H_S \circ \psi^{r}}{r^2},(\psi^{r})^*J_S)$ for $r \in \R_{>0}$. Using this action, one also relaxes the consistency requirement imposed: The Floer datum on $\overline{\mathcal{R}}^d$ must agree smoothly, on a boundary or corner stratum, with [*some rescaling*]{} of the previously made choice ( compare [@Abouzaid:2010kx]\*[Def 4.1]{}).
Given our choices of Floer data, we can define the moduli spaces appearing in the $\ainf$ operations. First for any pair of objects $L_0, L_1$, and any pair of chords $x_0, x_1 \in
\chi(L_0, L_1)$, define $ \widetilde{\mc{R}}^{1}(x_0; x_1)$ to be the moduli space of maps $u: \R_s \times [0,1]_t \to M$ with boundary condition and asymptotics $u(s,0) \in L_0$, $u(s,1) \in L_1$, $\lim_{s \to +\infty} u(s,t) = x_1$ and $\lim_{s \to -\infty} u(s,t) = x_0$ satisfying Floer’s equation for $(H_t, J_t)$: $$\label{coordfreefloer}
(du - X \otimes dt)^{0,1} = 0$$ where $X$ is the Hamiltonian vector field associated to $H_t$ and $(0,1)$ is taken with respect to $J_t$. The translation action on $\R_s$ descends to a map on this moduli space (as the equation satisfied is $s$-independent), and we define the moduli space of (unparametrized) Floer strips to be $$\label{semistablemodulispace}
\mc{R}^1(x_0; x_1) := \widetilde{\mc{R}}^1(x_0; x_1) / \R$$ (with the added convention that whenever we are in a component of $\widetilde{\mc{R}}^1(x_0; x_1)$ with expected dimension 0, this quotient is replaced by the empty set). Now for $d\geq 2$ let $L_0, \ldots, L_d$ be objects of $\f$ and fix any sequence of chords $\vec{x} = \{x_k \in \chi(L_{k-1},L_k)\}$ as well as another chord $x_0 \in
\chi(L_0,L_d)$. We write $\mc{R}^d(x_0; \vec{x})$ for the space of maps $$u: S \ra M$$ with source an arbitrary element $S \in \mc{R}^d$, satisfying moving boundary conditions and asymptotics $$\label{conditionsasymptotic}
\begin{cases}
u(z) \in L_k & \mathrm{if\ }z\in \partial S\textrm{ lies between }z^k\textrm{ and }z^{k+1} \\
\lim_{s\ra \pm \infty} u \circ \e^k(s,\cdot) = x_k
\end{cases}$$ (where the limit above is taken as $s \to +\infty$ if the $k$th end is positive and $-\infty$ if it is negative) and differential equation $$(du - X_S \otimes \alpha_S)^{0,1} = 0$$ where $X_S$ is the Hamiltonian vector field associated to $H_S$ and where $0,1$ is taken with respect to the complex structure $J_S$ (for the choice of consistent Floer datum we have fixed).
The consistency condition imposed on Floer data over the abstract moduli spaces $\overline{\mc{R}}^d$, along with the compatibility with strip-like ends, implies that the (Gromov-type) compactification of the space of maps $\overline{\mc{R}}^d(x_0; \vec{x})$ can be formed by adding the images of the natural inclusions of products of lower-dimensional such moduli spaces: $$\label{boundaryainfinitymoduli}
\overline{\mc{R}}^{d_1}(x_0; \vec{x}_1) \times \overline{\mc{R}}^{d_2}(y; \vec{x}_2) \ra \overline{\mc{R}}^d(x_0; \vec{x})$$ where $y$ agrees with one of the elements of $\vec{x}_1$ and $\vec{x}$ is obtained by removing $y$ from $\vec{x}_1$ and replacing it with the sequence $\vec{x}_2$. Here, we let $d_1$ range from 1 to $d$, with $d_2 = d-d_1 + 1$, with the stipulation that $d_1=1$ or $d_2 = 1$ is the semistable case .
\[wrappedainf\] In the setting of the wrapped Fukaya category (continuing Remark \[wrappedFloerdatum\]), one needs to incorporate the map $\rho_S$ into the Lagrangian boundary conditions and asymptotics specified in Floer’s equation; namely instead of we require $u(z) \in
\psi^{\rho_S(z)} L_k$ if $z\in \partial S$ lies between $z^k$ and $z^{k+1}$; we similarly impose that on the $k$th end, $\lim_{s\ra \pm \infty} u \circ \e^k(s,\cdot) = \psi^{\rho_S(z) := w_k} x_k$. The point is that Liouville flow for time $\log(\rho)$ defines a canonical identification between Floer complexes $$\label{rescaling}
CF^*(L_0,L_1;H,J_t) \simeq
CF^*\left(\psi^\rho L_0, \psi^\rho L_1;
\frac{H}{\rho} \circ \psi^\rho, (\psi^\rho)^* J_t\right).$$ The right hand object is equivalently the (wrapped) Floer complex for $(\psi^\rho L_0,
\psi^\rho L_1)$ for a strip with one-form $\rho dt$ using Hamiltonian $\frac{H}{\rho^2}
\circ \psi^\rho$ and $(\psi^\rho)^* J_t$. Up to Liouville flow, the Floer equation and boundary conditions satisfied on the $k$th strip-like end therefore coincides with the usual Floer equation for $(H_t, J_t)$ between $L_{k-1}$ and $L_k$. In light of this condition and the weakened consistency requirement for Floer data described in Remark \[wrappedFloerdatum\], one can again deduce , that lower-dimensional strata of the Gromov bordification maps can be identified (now possibly using a non-trivial Liouville rescaling) with products of previously defined moduli spaces.
In the graded setting, every connected component of the moduli space $\overline{\mc{R}}^d(x_0;\vec{x})$ has expected (or virtual) dimension $\deg (x_0) + d - 2 - \sum_{1\leq k \leq d} \deg(x_k)$; more generally, this moduli space consists of components of varying expected dimension (a number which can be computed using index theory in terms of the underlying homotopy class of $u$) all of whose mod 2 reductions are $\deg (x_0) + d - 2 - \sum_{1\leq k \leq d} \deg(x_k)$.
\[smoothcompactcor\] For a generically chosen Floer datum for the $\ainf$ structure, the virtual dimension $\leq 1$ components of the moduli spaces $\overline{\mc{R}}^d(x_0;\vec{x})$ are compact manifolds-with-boundary of the given expected dimension. Moreoever, given a fixed $\vec{x}$ this moduli space is empty for all but finitely many $x_0$ (automatic if all of the Lagrangians being considered are compact because there are only finitely many $x_0$ to begin with).
If $M$ is compact (and admissible), these assertions (the last of which is automatic) follow from standard Gromov compactness and transversality methods as in [@Seidel:2008zr]\*[(9k), (11h), Prop. 11.13]{}. In the non-compact case, there is an additional concern that solutions could escape to infinity in the target. To address this, one can e.g., appeal to the [*integrated maximum principle*]{} (compare [@Abouzaid:2010ly]\*[Lemma 7.2]{} or [@Abouzaid:2010kx]\*[§B]{}), which implies that elements of $\mc{R}(x_0; \vec{x})$ have image contained in a compact subset of $M$ dependent on $x_0$ and $\vec{x}$, from where one can again appeal to standard Gromov compactness techniques. (this is strongly dependent on the form of $H$, $J$, and $\alpha$ chosen for our Floer data as in Remark \[wrappedFloerdatum\]). The same result can be used to show that solutions do not exist for $x_0$ of sufficiently negative [*action*]{} compared to $\vec{x}$ (with our conventions, action is bounded above and there are finitely many $x_0$ with action above any fixed level), verifying the last assertion.
Choose a generic Floer datum for the $\ainf$ structure satisfying Lemma \[smoothcompactcor\] and let let $u \in \overline{\mc{R}}^d(x_0; \vec{x})$ be a rigid curve, meaning for us an element of the virtual dimension 0 component (which has dimension 0 in this case). By [@Seidel:2008zr]\*[(11h), (12b),(12d)]{}, given the fixed orientation of $\mc{R}^d$ (in the case $d \geq 2$ that is; for $d=1$ one instead needs to “orient the operation of quotienting by $\R$” as in [@Seidel:2008zr]\*[(12f)]{}), any such element $u
\in \overline{\mc{R}}^d(x_0; \vec{x})$ determines an isomorphism of orientation lines $$\mc{R}^d_{u}: o_{x_d} \otimes \cdots \otimes o_{x_1} \lra o_{x_0}.$$ Defines the $d$th $\ainf$ operation, for $d \geq 1$ $$\mu^d: \hom^*_{\mc{F}}(L_{d-1}, L_d) \otimes \cdots \otimes \hom^*_{\mc{F}}(L_0,L_1) \lra \hom^*_{\mc{F}}(L_0,L_d)$$ as a sum $$\mu^d([x_d], \ldots, [x_1]) :=
\sum_{u \in \overline{\mc{R}}^d(x_0; \vec{x}) \textrm{ rigid}} (-1)^{\bigstar_d}\mc{R}^d_{u} ([x_d], \ldots, [x_1])$$ where the sign is given by $$\bigstar_d = \sum_{i=1}^d i \cdot \deg (x_i)$$ (note that this sum is finite by Corollary \[smoothcompactcor\]). An analysis of the codimension 1 boundary of 1-dimensional moduli spaces along with their induced orientations establishes that the maps $\mu^d$ satisfy the $\ainf$ relations (see [@Seidel:2008zr]\*[Prop. 12.3]{}).
Circle action on the closed sector {#section:closed}
==================================
Floer cohomology and symplectic cohomology {#shsection}
------------------------------------------
Let $M$ be admissible as in §\[subsec:fukaya\]. Given a (potentially time-dependent) Hamiltonian $H: M \ra \R$, *Hamiltonian Floer cohomology* when it is defined is formally the Morse cohomology of the $H$-perturbed action functional on the free loop space of $M$: $\mathcal{A}_H: \mc{L} M \ra \R$. If $\omega$ is exact and comes with a fixed primitive $\lambda$, this functional can be written as: $$x \mapsto -\int_x \lambda + \int_{0}^1 H_t(x(t))dt$$ In general, $\mathcal{A}_H$ may be multi-valued, but $d \mathcal{A}_H$ is always well-defined, leading at least to a Morse-Novikov type theory. We write $\mathcal{O}$ for the set of [*critical points*]{} of $\mathcal{A}_{H_t}$ (when $H_t$ is implicit), which are precisely the time-1 orbits of the associated Hamiltonian vector field $X_H$.
\[assumption:nondegeneratecriticalpoints\] The elements of $\mathcal{O}$ are non-degenerate.
Optionally, given the data of a grading structure on $M$ in the sense of §\[subsec:fukaya\], one can define an absolute $\Z$ grading on orbits by $\deg(y) := n - CZ(y)$, where $CZ$ is the Conley-Zehnder index of $y$ (and such a grading is always well-defined mod 2).
Fix (potentially $S^1$-dependent) almost complex structure $J_t$. In the formal picture, this induces a metric on $\mathcal{L} M$. A [*Floer trajectory*]{} is formally a gradient flowline of $\mathcal{A}_{H_t}$ using the metric induced by $J_t$; concretely it is a map $u: (-\infty, \infty) \times S^1 \ra
M$ satisfying Floer’s equation (which is formally the gradient flow equation for $\mathcal{A}_{H_t}$), and converging exponentially near $\pm \infty$ to a pair of specified orbits $y^{\pm} \in
\mathcal{O}$. In standard coordinates $s,t$ on the cylinder this reads as $$\label{floersequation2} \partial_s u = -J_t (\partial_t u
- X).$$ The following crucial hypothesis is trivial if $M$ is compact (for any $(H_t,J_t)$), and holds only for special classes of $(H_t, J_t)$ if $M$ is non-compact and Liouville:
\[assumption:c0estimate\] For fixed asymptotics $y^+, y^-$, there is a compact set $K \subset M$ depending only on $y^{\pm}$, $H$, and $J$, such that any Floer trajectory $u$ between $y^{\pm}$ has image contained in $K$. Moreoever, from a given $y^+$, there are no Floer trajectories to all but a finite subset of possible $y^-$.
The space of non-constant Floer trajectories between a fixed $y^+$ and $y^-$ modulo the free $\R$ action given by translation in the $s$ direction is denoted $\mathcal{M}(y^-; y^+)$. As in Morse theory, one should compactify this space by allowing *broken trajectories*: $$\overline{\mc{M}}(y^-;y^+) = \coprod \mc{M}(y^-;y^1) \times \mc{M}(y^1;y^2) \times \cdots \times \mc{M}(y^k;y^+).$$ In the graded situation, every component of $\overline{\mc{M}}(y^-; y^+)$ has expected/virtual dimension $\deg(y^-) - \deg(y^+)-1$; in general $\overline{\mc{M}}(y^-; y^+)$ has components of varying virtual dimension (of fixed parity $\deg(y^-) - \deg(y^+) - 1$) depending on the underlying homotopy class of the cylinder. The next Lemma, whose proof is standard for all admissible $M$ (under the Assumptions made), stipulates that aside from the above Morse-type breakings of trajectories, there are no “bad” breakings (such as sphere bubbles), and moreoever that for generic $J_t$ the actual and expected dimensions dimensions of various moduli spaces agree:
\[assumption:transversalitycompactness\] For generic choices of (time-dependent) $J_t$, the virtual dimension $\leq 1$ components of the moduli spaces $\overline{\mathcal{M}}(y^-; y^+)$ are compact manifolds (with boundary) of the given expected dimension.
Putting this all together, the [*Floer co-chain complex*]{} for $(H_t, J_t)$ over $\K$ has generators corresponding to orbits of $H_t$: $$CF^i(M):= CF^i(M;H_t,J_t) := \bigoplus_{y\in \mc{O}, deg(y)=i} |o_y|_{\K},$$ where the [*orientation line*]{} $o_y$ is a real vector space associated to every orbit in $\mc{O}$ via index theory (see e.g., [@Abouzaid:2010kx]\*[§C.6]{}) — as before this index-theoretic definition a priori depends on a choice of trivialization of $y^*TM$ compatible with the grading structure, but any two choices induce isomorphic lines — and $V_{\K}$ is the $\K$-normalization of $V$ as in .
Fixing a $J_t$ as in Lemma \[assumption:transversalitycompactness\], the differential $d:
CF^*(M; H_t,J_t) \lra CF^*(M; H_t,J_t)$ counts rigid elements of the compactified moduli spaces. To fix sign issues, we recall that for a rigid element $u \in \mc{M}(y_0; y_1)$ (meaning $u$ belongs to a component of virtual, hence actual, dimension 0) there is a natural isomorphism between orientation lines induced by index theory (see e.g., [@Seidel:2008zr]\*[(11h), (12b),(12d)]{}, [@Abouzaid:2010kx]\*[Lemma C.4]{}) $$\mu_u: o_{y_1} \lra o_{y_0}.$$ Then, one defines the differential as $$\label{differentialdefinition}
d([y_1]) =
\sum_{u \in \overline{\mc{M}}(y_0;y_1)\textrm{ rigid}} (-1)^{\deg(y_1)} \mu_u ([y_1]).$$ One can show $d^2 = 0$ (under the assumptions made), and we call the resulting cohomology group $HF^*(H_t, J_t)$.
Our (cohomological) grading convention for Floer cohomology follows [@Seidel:2010fk; @Ritter:2013aa; @Abouzaid:2010kx; @ganatra1_arxiv].
### Symplectic cohomology {#subsubsec:sh}
[*Symplectic cohomology*]{} [@Cieliebak:1995fk; @Cieliebak:1996aa; @Floer:1994uq; @Viterbo:1999fk], is Hamiltonian Floer cohomology for a particular class of Hamiltonians on non-compact convex symplectic manifolds. There are several methods for defining this group. We define it here by making the following specific choices of target, Hamiltonian, and almost complex structure:
- $M$ is a Liouville manifold [*equipped with a conical end*]{}, meaning that it comes equipped with a choice of . (this serves primarily as a technical device; the resulting invariants are independent of the specific choice).
- The Hamiltonian term $H_t$ is a sum $H+F_t$ of an [*autonomous Hamiltonian*]{} $H: M \ra \R$ which is [*quadratic at $\infty$*]{}, namely $$\label{eq:quadratic}
H|_{M \backslash \bar{M}}(r,y) = r^2,$$ and a time-dependent perturbation $F_t$ such that on the collar of $M$, $$\label{s1perturbation}
\textrm{for any $r_0 \gg 0$, there exists an $R > r_0$ such that $F(t,r,y)$ vanishes in a neighborhood of $R$}.$$ (for instance, $F_t$ could be supported near non-trivial orbits of $H$, where it is modeled on a Morse function on the circle). We denote by $\mathcal{H}(M)$ the class of Hamiltonians satisfying .
- The almost complex structure should belong to the class $\mc{J}(M)$ of complex structures which are [*(rescaled) contact type*]{} on the cylindrical end , meaning that for some $c > 0$, $$\lambda \circ J = f(r) dr$$ where $f$ is any function with $f(r) > 0$ and $f'(r) \geq 0$.
The following Proposition is well known, and in the formulation described here, appears in e.g., [@Ritter:2013aa; @Abouzaid:2010kx]:
Assumptions \[assumption:nondegeneratecriticalpoints\] and \[assumption:c0estimate\] hold for this $M$ equipped with sufficiently generic choices of Liouville 1-form $\lambda$, $H_t$, and $J_t$.
If $M$, $H_t$, and $J_t$ are specifically as above, we refer to the Floer co-chain complex $CF^*(M, H_t, J_t)$ and $J$ as above as the [*symplectic co-chain complex*]{} $SC^*(M)$ and the resulting cohomology group as [*symplectic cohomology*]{} $SH^*(M)$.
### Relative cohomology {#subsubsec:relh}
Let us instead take $M$ to be any admissible symplectic manifold in the sense of §\[subsec:fukaya\], and fix a (non-degenerate, generic) pair $(H_t, J_t)$ which is arbitrary for compact $M$ and which satisfies the following additional properties if $M$ is Liouville:
- $H$ is linear of very small negative slope near infinity: $$H_t|_{M \backslash \bar{M}}(r, y) = - \lambda r$$ where $r$ is the cylindrical coordinate and $\lambda \ll 1$ is a sufficiently small number (smaller than the length of any Reeb orbit on $\partial \bar{M}$); and
- $J_t$ is (rescaled) contact type near infinity as before.
Then, the following Proposition is well known:
For suitably generic choices of $(H_t, J_t)$ as above, Assumptions \[assumption:nondegeneratecriticalpoints\] and \[assumption:c0estimate\] hold and there is an isomorphism $HF^*(H_t, J_t) \cong H^*(\bar{M}, \partial \bar{M})$ (which equals $H^*(M)$ in case $M$ is compact using the convention then that $\bar{M} = M$ and $\partial \bar{M} = \emptyset$).
The isomorphism can be realized in one of two ways:
- choose $H_t$ as above to be a $C^2$ small (time-independent) Morse function, in which case a well-known argument of Floer [@Floer:1989aa] equates $HF^*(H_t,
J_t)$ with the Morse complex of $H$ by showing that all Floer trajectories must in fact be Morse trajectories of $H|_{\bar{M}}$ (which in turn, as $H$ is inward pointing near $\bar{M}$, compute the relative cohomology)
- Construct a geometric [*PSS morphism*]{} [@Piunikhin:1996aa] $PSS:
H^*(\bar{M}, \partial \bar{M}) \cong H_{2n-*}(M) \to HF^*(H_t, J_t)$.
The cohomological BV operator {#naivesquare}
-----------------------------
The first order BV operator is a Floer analogue of a natural operator that exists on the Morse cohomology of any manifold with a smooth $S^1$ action. Like the case of ordinary Morse theory, this operator exists even when the Hamiltonian and complex structure (c.f. Morse function and metric) are not $S^1$-equivariant.
For $p \in S^1$, consider the following collection of cylindrical ends on $\R
\times S^1$: $$\begin{split}
\e^{+}_p: (s,t) &\mapsto (s+1,t+p),\ \ \ s \geq 0\\
\e^{-}_p: (s,t) &\mapsto (s-1,t),\ \ \ s \leq 0
\end{split}$$
Pick $K: S^1 \times (\R \times S^1) \times M \ra \R$ dependent on $p$, satisfying $$(\e^{\pm}_p)^* K(p, s, \cdot,\cdot) = H(t,m)$$ meaning that $$K_p(s,t,m) = \begin{cases}
H(t+p,m) & s \geq 1\\
H(t,m) & s \leq -1,
\end{cases}$$ so in the range $-1 \leq s \leq 1$, $K_p(s,t,m)$ interpolates between $H_{t+p}(m)$ and $H_t(m)$ (and outside of this interval is independent of $s$).
Similarly, pick a family of almost complex structures $J: S^1 \times (\R \times
S^1) \times M \ra \R$ satisfying $$\begin{aligned}
(\e^{\pm}_p)^* J(p, s, t,m) &= J(t,m)\end{aligned}$$
Now, $x^+, x^- \in \mc{O}$, define $$\mc{M}_{1}(x^+,x^-)$$ to be the following [*parametrized moduli space*]{} of Floer cylinders $$\{p \in S^1, u: S \ra M\ |\ \begin{cases}
\lim_{s \ra \pm \infty} (\e^{\pm}_p)^* u(s,\cdot) = x^{\pm} \\
(du - X_K \otimes dt)^{0,1} = 0.
\end{cases}\}$$
There is a natural bordification by adding broken Floer cylinders to either end $$\label{compactificationBV}
\overline{\mc{M}}_{1}(x^+,x^-) = \coprod \mc{M}(x^+;a_0) \times \cdots \times \mc{M}(a_{k-1};a_k) \times \mc{M}_{1}(a_k,b_1) \times \mc{M}(b_1; b_2) \times \cdots \times \mc{M}(b_{l}; x^-)$$
For non-compact $M$, the following hypothesis is important to verify:
\[assumption:c0bv\] The elements of $\mc{M}_{1}(x^+,x^-)$ satisfy an [*a priori compactness estimate*]{}, meaning for fixed asymptotics $y^+, y^-$, there is a compact set $K \subset M$ depending only on $x^{\pm}$, $H$, and $J$, such that any Floer trajectory $u$ between $x^{\pm}$ has image contained in $K$. Also, from a given $y^+$, there are no Floer trajectories to all but finitely many $y^-$.
In the case of symplectic cohomology Hamiltonians described in § \[subsubsec:sh\], Assumption \[assumption:c0bv\] can be ensured by choosing $K$ carefully as follows. Given that $H_t(M) = H + F_t$ is a sum of an autonomous term and a time-dependent term that is zero at infinitely many levels tending towards infinity, we can ensure that $$\label{quadratichamiltonianlevels}
\textrm{at infinity many levels tending towards infinity, }K_p(s,t,m)\textrm{ is equal to } r^2,$$ and in particular is autonomous. The proof of Assumption \[assumption:c0bv\] follows from [@Abouzaid:2010kx]\*[§B]{}.
As before, $\overline{\mc{M}}_{1}(x^+, x^-)$ contains components of varying expected dimension depending on the underlying homotopy class $\beta$ of a map. Due to the fact that we are studying 1-parametric families of domains and not quotienting by $\R$, the relevant expected dimension is 2 more than the the expected dimension of the components of $\overline{\mc{M}}(x^+, x^-)$ underlying the same homotopy class $\beta$). In particular in the graded case, this expected dimension is $\deg(x_+) - \deg(x^-) + 1$ for every component.
For generic choices of the above data, every component of $\overline{\mc{M}}_{1}(x^+, x^-)$ of virtual dimension $\leq 1$ is a compact manifold-with-boundary of dimension equal to its virtual dimension. (In particular, the boundary of the 1-dimensional components consists of the once-broken trajectories in ).
In the usual fashion, counting rigid elements of this compactified moduli space of maps with the right sign (explained more carefully in the next section) give an operation $\delta_1: CF^*(M) \to CF^{*-1}(M)$, satisfying $$d \delta_1 + \delta_1 d = 0$$ (coming from the fact that the codimension 1 boundary of $\overline{\mc{M}}_{1}(x^+, x_-)$ is $\coprod_y
\overline{\mc{M}}(x^+;y) \times \overline{\mc{M}}_{1}(y;x_-) \cup \overline{\mc{M}}_{1}(x^+;y) \times \overline{\mc{M}}(y;x_-)$). It would be desirable for $\delta_1$ square to zero on the chain level, which would give $(CF^*(M), \delta_0 = d, \delta_1)$ the structure of a [*strict $S^1$-complex*]{}, or [*mixed complex*]{}. However, the $S^1$ dependence of our Hamiltonian and almost complex structure prevent this, in a manner we now explain.
Typically one attempts to prove an geometric/Floer-theoretic operation (such as $\delta_1^2$) is zero by exhibiting that the relevant moduli problem has no zero-dimensional solutions (due to, say, extra symmetries in the equation), or otherwise arises as the boundary of a 1-dimensional moduli space. To that end, we first indicate a moduli space parametrized by $S^1 \times S^1$ which looks like two of the previous parametrized spaces naively superimposed, leading us to call the associated operation we call $\delta_2^{naive}$. The extra symmetry involved in this definition will allow us to easily conclude
\[naivelemma\] $\delta_2^{naive}$ is the zero operation.
For $(p_1,p_2) \in S^1 \times S^1$, consider the following collection of cylindrical ends: $$\begin{split}
\e^{+}_{(p_1,p_2)}: (s,t) &\mapsto (s+1,t+p_1+p_2),\ \ \ s \geq 0\\
\e^{-}_{(p_1,p_2)}: (s,t) &\mapsto (s-1,t),\ \ \ s \leq 0
\end{split}$$
Pick $K: (S^1 \times S^1) \times (\R \times S^1) \times M \ra \R$ dependent on $(p_1,p_2)$, satisfying $$\e^{\pm}_{(p_1,p_2)} K(p_1,p_2, s, \cdot,\cdot) = H(t,m)$$ meaning that $$K_{(p_1,p_2)}(s,t,m) = \begin{cases}
H(t+p_1+p_2,m) & s \geq 1\\
H(t,m) & s \leq -1,
\end{cases}$$ so in the range $-1 \leq s \leq 1$, $K_{p_1+p_2}(s,t,m)$ interpolates between $H_{t+p_1+p_2}(m)$ and $H_t(m)$.
Similarly, pick a family of almost complex structures $J: S^1 \times S^1 \times
(\R \times S^1) \times M \ra \R$ $$\begin{aligned}
\e^{\pm}_{(p_1,p_2)} J(p_1,p_2, s, t,m) &= J(t,m),\end{aligned}$$ such that $$J\textrm{ only depends on the sum }p_1+p_2.$$
Now, for $x^+, x^- \in \mc{O}$, define $$\mc{M}_{2}^{naive}(x^+,x^-)$$ to be the [*parametrized moduli space*]{} of Floer cylinders $$\{(p_1, p_2) \in S^1 \times S^1, u: S \ra M\ |\ \begin{cases}
\lim_{s \ra \pm \infty} (\e^{\pm}_{(p_1,p_2)})^* u(s,\cdot) = x^{\pm} \\
(du - X_K \otimes dt)^{0,1} = 0.
\end{cases}\}$$ For generic choices of $K$ and $J$, this moduli space, suitably compactified by adding broken trajectories, will be (for components of virtual dimension $\leq 1$) a manifold with the correct (expected) dimension; in the graded case this is $\deg(x) - \deg(y)
+ 2$ (the details are similar to the previous section, and will be omitted). Counts of rigid elements in this moduli space will thus, in the usual fashion give a map of degree $-2$, which we call $\delta_2^{naive}$.
Let $(p_1, p_2, u)$ be an element of $\mc{M}_{2}^{naive}(x,y)$. Then, $(p_1-r, p_2 + r, u)$ is an element too, for any $r \in S^1$, as the equation satisfied by the map $u$ only depends on the sum $p_1+p_2$. We conclude that elements of $\mc{M}_{2}^{naive}(x,y)$ are never rigid, and thus that the resulting operations $\delta_2$ is zero.
We would like $\delta_2^{naive}$ to be genuinely equal to $\delta_1^2$, which would imply that $\delta_1^2 = 0$ However, this is only true on the homology level; the lack of $S^1$ invariance of our Hamiltonian and almost complex structure, and the corresponding family of choices of homotopy between $\theta^*H_t$ and $H_t$, over varying $\theta$ , breaks symmetry and ensures that $\delta_1^2 \neq \delta_2^{naive}$ as geometric chain maps. However, there is a geometric chain homotopy, $\delta_2$ between $\delta_1^2$ and $\delta_2^{naive}$ along with a hierarchy of higher homotopies $\delta_k$ forming the $S^1$-complex structure on $CF^*(M)$, which we define in the next section (see in particular the proof of Lemma \[weaks1actionSHlemma\] for the proof of the $S^1$-complex equations, one of which recovers the chain homotopy between $\delta_1^2$ and $\delta_2^{naive}=0$).
The A-infinity circle action {#sec:angledeccylinder}
----------------------------
We turn to a “coordinate-free” definition of the relevant parametrized moduli spaces, which will help us incorporate the construction into open-closed maps.
A [*$r$-point angle-decorated cylinder*]{} consists of a semi-infinite or infinite cylinder $C \subseteq (-\infty,\infty) \times S^1$, along with a collection of auxiliary points $p_1, \ldots, p_r \in C$, satisfying $$\label{heightordering}
(p_1)_s \leq \cdots \leq (p_r)_s,$$ where $(a)_s$ denotes the $s \in (-\infty, \infty)$ coordinate. The [ *heights*]{} associated to this data are the $s$ coordinates $$h_i = (p_i)_s,\ i = 1, \ldots, r$$ and the [*angles*]{} associated to $C$ are the $S^1$ coordinates $$\theta_i := (p_1)_t, \ i \in 1, \ldots, r.$$
The [*cumulative rotation*]{} of an $r$-point angle-decorated cylinder is the first angle: $$\eta := \eta(C, p_1, \ldots, p_r) = \theta_1.$$ The [*$i$th incremental rotation*]{} of an $r$-point angle-decorated cylinder is the difference between the $i$th and $i-1$st angles: $$\kappa^{inc}_i := \theta_i - \theta_{i+1} \textrm{ (where $\theta_{r+1} = 0$)}.$$
\[rpointedmodulispace\] The [*moduli space of $r$-point angle-decorated cylinders*]{} $$\mc{M}_r$$ is the space of $r$-point angle-decorated infinite cylinders, modulo translation.
\[orientationmr\] Note that $C_r$, the space of all $r$-point angle-decorated infinite cylinders (not modulo translation) has a canonical complex orientation. Thus, to orient the quotient space $\mc{M}_r:=C_r/\R$ it is sufficient to give a choice of trivialization of the action of $\R$ on $C_r$. We choose $\partial_s$ to be the vector field inducing said trivialization.
For an element of this moduli space, the angles and relative heights of the auxiliary points continue to be well-defined, so there is a non-canonical isomorphism $$\mc{M}_r \simeq (S^1)^r \times [0,\infty)^{r-1}$$ The moduli space $\mc{M}_r$ thus possesses the structure of an open manifold-with-corners, with boundary and corner strata given by the various loci where heights of the auxiliary points $p_i$ are coincident (we allow the points $p_i$ themselves to coincide; one alternative is to first Deligne-Mumford compactify, and then collapse all sphere bubbles containing multiple $p_i's$. That the result still forms a smooth manifold with corners is a standard local calculation near any such stratum). Given an arbitrary representative $C$ of $\mc{M}_r$ with associated heights $h_1, \ldots, h_r$, we can always find a translation $\tilde{C}$ satisfying $\tilde{h}_r = -\tilde{h}_1$; we call this the [*standard representative*]{} associated to $C$.
Given a representative $C$ of this moduli space, and a fixed constant $\delta$, we fix a positive cylindrical end around $+\infty$ $$\label{posendangles}
\begin{split}
\epsilon^+: [0,\infty) \times S^1 &\ra C\\
(s,t) &\mapsto (s + h_r + \delta, t)\\
\end{split}$$ and a negative cylindrical end around $-\infty$ (note the angular rotation in $t$!): $$\label{negendangles}
\begin{split}
\epsilon^-: (-\infty, 0] \times S^1 & \ra C \\
\nonumber (s,t) &\mapsto ( s - (h_1 - \delta)), t + \theta_1).
\end{split}$$ These ends are disjoint from the $p_i$ and vary smoothly with $C$; via thinking of $C$ as a sphere with two points with asymptotic markers removed, these cylindrical ends correspond to the positive asymptotic marker having angle 0 and the negative asymptotic marker having angle $\theta_1 = \kappa^{inc}_1 +
\kappa^{inc}_2 + \cdots + \kappa^{inc}_r$.
There is a compactification of $\mc{M}_r$ consisting of [*broken $r$-point angle-decorated cylinders*]{} $$\overline{\mc{M}}_r = \coprod_{s} \coprod_{j_1, \ldots, j_s; j_i > 0, \sum j_i = r} \mc{M}_{j_1} \times \cdots \times \mc{M}_{j_s}.$$ The stratum consisting of $s$-fold broken configurations lies in the codimension $s$ boundary, with the manifolds-with-corners structure explicitly defined by local gluing maps using the ends and . Note that the gluing maps, which rotate the bottom cylinder in order to match an end with , induce cylindrical ends on the glued cylinders which agree with the choices of ends made in -.
The compactification $\overline{\mc{M}}_r$ thus has codimension-1 boundary covered by the images of the natural inclusion maps $$\begin{aligned}
\label{anglestratum1}\overline{\mc{M}}_{r-k} \times \overline{\mc{M}}_k &\lra \partial \mc{M}_{r},\ 0 < k < r\\
\label{anglestratum2}\overline{\mc{M}}_r^{i,i+1} &\lra \partial \mc{M}_{r},\ 1 \leq i < r,\end{aligned}$$ where $\overline{\mc{M}}_r^{i,i+1}$ denotes the compactification of the locus where $i$th and $i+1$st heights are coincident $$\mc{M}_r^{i,i+1} := \{C \in \mc{M}_r\ |\ h_i = h_{i+1} \}.$$ With regards to the above stratum, for $r > 1$ there is a projection map which will be relevant, a version of the forgetful map which remembers only the first of the angles with coincident heights: $$\begin{split}
\pi_i: \mc{M}_r^{i,i+1} &\lra \mc{M}_{r-1}\\
(h_1, \ldots, h_i, h_{i+1}=h_i, h_{i+2}, \ldots, h_r) &\longmapsto (h_1, \ldots, h_i, h_{i+2}, \ldots, h_r)\\
(\theta_1, \ldots, \theta_i, \theta_{i+1}, \ldots, \theta_r) &\longmapsto (\theta_1, \ldots, \theta_{i-1}, \theta_{i}, \widehat{\theta_{i+1}}, \theta_{i+2}, \ldots, \theta_r).
\end{split}$$ $\pi_i$ is compatible with the choice of positive and negative ends - and hence $\pi_i$ extends to compactifications $$\label{addmapcompactified}
\pi_i: \overline{\mc{M}}_{r}^{i,i+1} \ra \overline{\mc{M}}_{r-1}.$$ We will equip each $r$-point angle-rotated cylinder $\tilde{C} := (C,p_1,
\ldots, p_r)$ with perturbation data for Floer’s equation or a [*Floer datum*]{} in the sense of the last section, which consists of
- The positive and negative cylindrical ends on $\epsilon^{\pm}: C^{\pm} \ra C$ chosen in -.
- The one-form on $C$ given by $\alpha = dt$.
- A surface-dependent Hamiltonian $H_{\tilde{C}}: C \ra \mc{H}(M)$ compatible with the positive and negative cylindrical ends, meaning that $$(\epsilon^{\pm})^* H_C = H_t,$$ where $H_t$ was the previously chosen Hamiltonian.
- A surface dependent complex structure $J_{\tilde{C}}: C \ra \mc{J}_1(M)$ also compatible with $\epsilon^{\pm}$, meaning that $$(\epsilon^{\pm})^* J_{\tilde{C}} = J_t$$ for our previously fixed choice $J_t$.
A choice of [*Floer data for the $S^1$-action*]{} is an inductive (smoothly varying) choice of Floer data, for each $k$ and each representative $S = (C,p_1, \ldots, p_k)$ of $\overline{\mc{M}}_k$, satisfying the following consistency conditions at boundary strata: $$\begin{aligned}
\label{s1consistency} &\textrm{At a boundary stratum \eqref{anglestratum1}, the datum chosen coincides with the product of Floer data}\\
\nonumber &\textrm{already chosen on lower-dimensional spaces. }\\
\label{s1forgetful} &\textrm{At a boundary stratum \eqref{anglestratum2}, the Floer data coincides with the pull back, via the forgetful} \\
\nonumber &\textrm{map $\pi_i$ defined in \eqref{addmapcompactified}
of the Floer data chosen on $ \overline{\mc{M}}_{k-1}$.}
\end{aligned}$$ Inductively, since the space of choices at each level is non-empty and contractible (and since the consistency conditions are compatible along overlapping strata), universal and consistent choices of Floer data exist. From the gluing map, a representative $S$ sufficiently near the boundary strata inherits cylindrical regions, also known as [*thin parts*]{}, which are the surviving images of the cylindrical ends of lower-dimensional strata. Together with the cylindrical ends of $S$, this determines a collection of cylindrical regions.
Given a choice of Floer data for the $S^1$-action and a pair of asymptotics $(x^+,x^-) \in \mc{O}$ for each $k \geq 1$ there is an associated parametrized moduli space of Floer cylinders with source an arbitrary element of $S \in
\mathcal{M}_k$ (where the Floer equation is with respect to the Hamiltonian $H_S$ and $J_S$, with asymptotics $(x^+,x^-)$: $$\mc{M}_{k}(x^+; x^-):=
\{ S = (C,p_1, \ldots, p_r) \in \mc{M}_k,\ u: C \ra M\ |\
\begin{cases}
\lim_{s \ra \pm \infty} (\e^{\pm})^* u(s,\cdot) &= x^{\pm} \\
(du - X_{H_S} \otimes dt)^{(0,1)_S} &= 0.
\end{cases} \}$$ The consistency condition imposes that the boundary of the Gromov bordification $\overline{\mc{M}}_k(x^+; x^-)$ is covered by the images of the natural inclusions $$\begin{aligned}
\label{goodboundary}\overline{\mc{M}}_r(y; x^-) \times \overline{\mc{M}}_{k-r}(x^+; y) \ra \partial \overline{\mc{M}}_k(x^+; x^-) \\
\label{badboundary}\overline{\mc{M}}_k^{i,i+1} (x^+; x^-) \ra \partial \overline{\mc{M}}_k(x^+; x^-),\end{aligned}$$ along with the usual semi-stable strip breaking boundaries $$\begin{split}
\label{stripbreaking}
\overline{\mc{M}}_k(y; x^-) \times \overline{\mc{M}}(x^+; y) &\ra \partial \overline{\mc{M}}_k(x^+; x^-) \\
\overline{\mc{M}}(y; x^-) \times \overline{\mc{M}}_k(x^+; y) &\ra \partial \overline{\mc{M}}_k(x^+; x^-)
\end{split}$$
For Liouville $M$, as before the following hypothesis is non-trivial:
\[assumption:c0higherbv\] The elements of $\mc{M}_{r}(x^+,x^-)$ satisfy an [*a priori compactness estimate*]{} and are non-empty for only finitely many $x^-$ (given a fixed $x^+$), as described in Assumption \[assumption:c0bv\].
The assumption can be shown to hold using the integrated maximum principle ([@Abouzaid:2010ly]\*[Lemma 7.2]{} or [@Abouzaid:2010kx]\*[§B]{}) provided we impose the further constraint on our Floer data (analogously to ), which we henceforth do in the Liouville case: $$\begin{split}
&\textrm{$H_C$ is equal to $r^2$ or $-\lambda r$ (depending on whether we are in the setting of \S \ref{subsubsec:sh} or \S \ref{subsubsec:relh})} \\
&\textrm{at infinitely many levels of $r$ tending to $\infty$.}
\end{split}$$ (in fact, in the setting of §\[subsubsec:relh\] we can take $H_C$ to be simply equal to $-\lambda r$ for all $r$ outside of a compact set).
In the $\Z$-graded case, the virtual dimension of (every component of) $\overline{\mc{M}}_k(x^+; x^-)$ is $$\deg (x^+) - \deg(x^-) + (2k -1);$$ in the $\Z/2$-graded case every component has virtual dimension of the above parity. Standard methods establish that
For a generic choice of Floer data for the $S^1$-action, the components of virtual dimension $\leq 1$ of the moduli spaces $\overline{\mc{M}}_k(x^+;
x^-)$ are compact manifolds-with-boundary of the correct (expected) dimension.
As usual, signed counts of rigid elements of this moduli space for varying $x^+$ and $x^-$ (using induced maps on orientation lines, twisted as in the differential by $(-1)^{\deg(x_+)}$—see ) give the matrix coefficients for the overall map $$\delta_{k}: CF^*(M) \ra CF^{*-2k + 1}(M).$$ In the degenerate case $k=0$ we define $\delta_0$ to be the (already defined) differential: $$\delta_0: = d: CF^*(M) \to CF^{*+1}(M).$$
\[weaks1actionSHlemma\] For each $k$, $$\label{circleactioneqn}\sum_{i=0}^k \delta_i \delta_{k-i} = 0.$$
The counts of rigid elements associated to the boundary of 1-dimensional components of $\partial \overline{\mc{M}}_k(x^+; x^-)$, along with a description of this codimension 1 boundary - immediately implies that $$(\sum_{i=1}^k \delta_i \delta_{k-i}) + (\sum_i \delta_k^{i,i+1}) + (d \delta_k + \delta_k d) = 0,$$ where $\delta_k^{i,i+1}$ for each $i$ is the operation associated to the moduli space of maps . (Observe that $\delta_2^{1,2}$ is precisely the operation $\delta_2^{naive}$ from §\[naivesquare\]). Note that the consistency condition implies that the Floer datum chosen for any element $S \in \mc{M}_k^{i,i+1}$ only depends on $\pi_i(S)$, where the forgetful map $\pi_i: \mc{M}_k^{i,i+1} \ra
\mc{M}_{k-1}$ has 1-dimensional fibers. Hence given an element $(S,u) \in \overline{\mc{M}}_k^{i,i+1} (x^+; x^-)$, it follows that $(S',u) \in \overline{\mc{M}}_k^{i,i+1}(x^+; x^-)$ for all $S'
\in \pi_i^{-1}\pi_i(S)$. In other words, elements of $\overline{\mc{M}}_k^{i,i+1} (x^+; x^-)$ are never rigid, so the associated operation $\delta_k^{i,i+1}$ is zero.
By definition we conclude:
The pair $(CF^*(M; H_t, J_t), \{\delta_k\}_{k \geq 0})$ as defined above forms an $S^1$-complex, in the sense of Definition \[homotopycircleaction\].
By using continuation maps parametrized by various $(S^1)^k \times
(0,1]^{k}$ (or equivalently, by spaces of angle-decorated cylinders that are not quotiented by overall $\R$-translation), one can prove that
Any continuation map $f: CF^*(M, H_1) \to CF^*(M, H_2)$ enhances to a homomorphism $\mathbf{F}$ of $S^1$-complexes (which is in particular a quasi-isomorphism if $f$ is).
We omit the proof, referring readers to e.g., [@Zhao:2019aa] (though note some notational differences). In particular, the $S^1$-complex defined on the symplectic co-chain complex $SC^*(M)$ or the Hamiltonian Floer complex (with small negative slope if $M$ is non-compact) is an invariant of $M$, up to quasi-isomorphism.
### Relation to earlier definitions in the literature
There is another formulation of the compatibility of Floer data, in terms of these so-called [*rotated cylindrical regions*]{}. Let $top(C)$ to be the maximal $s$ coordinate in $C$ ($+\infty$ if $C$ is positive-infinite) and $bottom(C)$ to be the minimal $s$ coordinate in $C$ ($-\infty$ if $C$ is negative-infinite).
\[cumulativeangle\] Let $(\theta_1, \ldots, \theta_r)$ be a collection of angles. Define the [*$i$th cumulative angle*]{}, for $i$ from $1$ to $r$, via $$\eta_i := \sum_{j = 1}^i \theta_j;$$ or inductively via $$\begin{split}
\eta_1 &:= \theta_1\\
\eta_{i+1} &:= \eta_i + \theta_{i+1}.
\end{split}$$ Also, define $\eta_{-1} = 0$.
The [*($\delta$-spaced) rotated cylindrical regions*]{} for an $r$-point angle-decorated cylinder $(C,p_1, \ldots, p_r)$ consist of the following cylindrical ends and finite cylinders:
- The [*top cylinder*]{} $$\begin{split}
\epsilon^+: [0,\max(top(C) - (h_1 + \delta),0)] \times S^1 &\ra C\\
(s,t) &\mapsto (\min(s + h_1 + \delta,top(C)), t)\\
\end{split}$$
- The [*bottom cylinder*]{} $$\begin{split}
\epsilon^-: [\min(bottom(C) - (h_r - \delta), 0), 0] \times S^1 & \ra C \\
\nonumber (s,t) &\mapsto ( \max(s - (h_r - \delta),bottom(C)), t + \eta_r).
\end{split}$$
- For any $1 \leq i \leq r-1$ satisfying $h_{i+1} - h_i > 3 \delta$, the [*$i$th thin part*]{} $$\begin{split}
\epsilon_i: [h_{i+1} + \delta, h_{i} - \delta] \times S^1 &\ra C\\
(s,t) &\mapsto (s, t + \eta_i)
\end{split}$$
Note that a given $r$-point angle-decorated cylinder may not contain the $i$th thin part, for a given $i \in [1,r-1]$, and indeed may not contain any thin parts. If $\delta$ is implicit, we simply refer to above as the rotated cylinder regions.
Now, fixing an $\delta$, it is easy to give a necessary (but not sufficient!) condition for compatibility with gluing:
We say a Floer datum $(K_C, J_C)$ is is [*$\delta$-adapted*]{} to $(C,p_1, \ldots, p_r, K_t, J_t)$ if for any rotated cylindrical region $\epsilon: C' \ra C$ associated to $(C,p_1, \ldots,
p_r)$ and $\delta$, we have that $$\epsilon^*(K_C, J_C) = (K_t, J_t).$$
(strictly speaking, this condition is only necessary for sufficiently large finite cylinders). If we were to express this condition without using the rotation already built into the cylindrical region maps, we would arrive at the following, which most closely matches [@Bourgeois:2012fk].
Let $(C,p_1, \ldots, p_r)$ be an $r$-point angle-decorated cylinder, and fix a pair $(K_t, J_t)$ of a time-dependent Hamiltonian and almost complex structure. Let $K_C$ and $J_C$ be a $C$ dependent Hamiltonian and almost complex structure. For a positive constant $\delta$, we say that $(K_C,
J_C)$ is [*$\delta$-adapted to $(C,p_1, \ldots, p_r, K_t, J_t)$*]{} if, at a position $z = (s,t)$ $$\begin{split}\label{rotatecondition}
(H_{z}, J_{z}) &= (H_t, J_t) \ \textrm{ for }s > p_1 + \delta;\\
(H_z, J_z) &= \eta_r^*(H_t, J_t) = (H_{t+\eta_r},J_{t+\eta_r}) \ \textrm{ for }s < p_r - \delta;\\
(H_z,J_z) &= \eta_{i}^* (H_z,J_z) \textrm{ if } h_{i+1} - h_{i} > 3 \delta
\textrm{ and } s \in [h_{i+1}+\delta, h_{i} - \delta]
\end{split}$$
The circle action on the interior {#sec:interior}
---------------------------------
From the formal point of view of Floer homology of $M$ as the Morse homology of an action functional on the free loop space $\mc{L} M$, one would espect the contributions to coming from constant loops to be acted on trivially by the $C_{-*}(S^1)$ action (which comes from rotation of free loops). This is indeed the case, as we now review.
Suppose that the Hamiltonian $H_t$ defining $CF^*(M)$ is chosen to be $C^2$-small, time-independent, and Morse in the compact region of $\bar{M}$ (which recall equals $M$ if $M$ is compact). Then, Floer proved that all orbits of $H_t$ inside $\bar{M}$ are (contstant orbits at) Morse critical points of $H$, and all Floer cylinders between such orbits which remain in $\bar{M}$ are in fact Morse trajectories of $H$ [@Floer:1989aa].
Let $C_{Morse}(H)$ denote the Morse complex of $H$. In the setting where $H$ is as in §\[subsubsec:relh\] ($M$ can be Liouville or compact), all contributions to $CF^*(M)$ (both orbits and cylinders) come from $\bar{M}$, so Floer’s argument gives an isomorphisms $$C_{Morse}(H) \cong CF^*(M)$$ In the setting where $H$ is quadratic at infinity (and $M$ is Liouville) as in §\[subsubsec:sh\], one can ensure the collection of orbits coming from $\bar{M}$ is an action-filtered subcomplex (and e.g., the integrated maximum principle will ensure that all cylinders between such orbits lie in $\bar{M}$). Hence, there is an inclusion of subcomplexes $$\label{shsubcomplex}
C_{Morse}(H) \to SC^*(M),$$ which, under smallness constraints on the Floer data for the $S^1$-action gives an [*$S^1$-subcomplex*]{} (meaning the operators $\delta_k$ preserve the subcomplex and in fact the action filtration) [@Zhao:2019aa]\*[Lemma 5.4]{}), hence a morphism of $S^1$-complexes. We will discuss both of the above cases at once: in either case by considering a Hamiltonian which is $C^2$-small on $\bar{M}$ we obtain an inclusion of $S^1$-subcomplexes $$\label{subcomplex}
C_{Morse}(H) \to CF^*(M)$$ with the understanding that in the former case this inclusion is the whole complex.
\[lem:constantloops\] There exists a choice of Floer data for the $S^1$-action so that $C_{Morse}(H)$ becomes a trivial $S^1$-subcomplex; meaning that the various operators $\delta_k$, $k \geq 1$, associated to the $C_{-*}(S^1)$ action strictly vanish on the subcomplex.
By the integrated maximum principle, any Floer trajectory with asymptotics along two generators in $C_{Morse}(H)$ remains in the interior of $\bar{M}$. We can choose the Hamiltonian term of our Floer data on $\mathcal{M}_r$ in this region of $M$ to be autonomous (i.e., $t$ and $s$-independent on the cylinder), $C^2$-small and Morse — in fact equal to $H$; then Floer’s theorem [@Floer:1989aa] again guarantees that any Floer trajectory in $\mathcal{M}_r(x^+; x^-)$ between Morse critical points $x^{\pm}$ is in fact a Morse trajectory of $H$. It follows that for $x^+, x^-$ critical points of $H$, any element $u = (C, \vec{p})$ in the parametrized moduli space of maps $\overline{\mc{M}}_{-k}(x^+,x^-)$ solves an equation that is independent of the choice of parameter $\vec{p}
\in (S^1)^k \times (0,1]^{k-1}$. Namely, $u$ lives in a family of solutions of dimension at least $2k-1$ (given by varying $\vec{p}$), and hence $u$ cannot be rigid. The associated operation $\delta_k$, which counts rigid solutions, is therefore zero.
By invariance of the $S^1$-complex structure on $CF^*(M)$ (up to quasi-isomorphism), we conclude
\[trivialconstantloops\] For $M$ compact and admissible, or Liouville with $(H,J)$ as in §\[subsubsec:relh\], $CF^*(M)$ is quasi-isomorphic to a trivial $S^1$-complex.
For $M$ Liouville with $(H,J)$ as in §\[subsubsec:sh\], the inclusion chain map $$C^*_{Morse}(M) \ra SC^*(M)$$ lifts canonically to a chain map $$(C_{Morse}(H) [ [ u ] ], d_{Morse}) = (C_{Morse}(H))^{hS^1} \to (SC^{*}(M))^{hS^1} = (SC^*(M)[ [ u ] ], \delta_{eq}).$$ inducing a cohomological map $$\label{constantloopsmap}
H^*(M) [ [ u ] ] \ra H^*(SC^{*}(M)^{hS1}).$$
Another possibly more direct way of producing the map $H^*(M) [ [ u ] ] \ra H^*(SC^{*}(M)^{hS1})$ is via an $S^1$-equivariant enhancement $\widetilde{PSS}$ of the PSS morphism $PSS: C^*(M) \to SC^*(M)$. We omit a further description here, and simply note that the resulting map can be shown to coincide cohomologically with the map defined above.
Since the $S^1$-complex structure on $C^*_{Morse}(H)$ is trivial, one can (canonically) split the inclusion of homotopy fixed points map $H^*(C^*_{Morse}(H)^{hS^1}) \to H^*(C^*_{Morse}(H))$ by the map $$H^*(M) \stackrel{[x \mapsto x \cdot 1]}{\ra} H^*(M) [ [ u ] ];$$ the associated composition $$H^*(M) \stackrel{[x \mapsto x \cdot 1]}{\ra} H^*(M) [ [ u ] ] \ra H^*(SC^{*}(M)^{hS1}) \stackrel{[\iota]}{\ra} SH^*(M)$$ coincides with the usual map $H^*(M) \ra SH^*(M)$. In particular, we note that the homotopy fixed point complex of $SC^*(X)$ possesses a canonical (geometrically defined) cohomological element, $$\label{tilde1}
\tilde{1} \in H^*(SC^{*}(M)^{hS^1}),$$ lifting the usual unit $1 \in SH^*(M)$ (under the map $[\iota]$), defined as the image of $1$ under .
Cyclic open-closed maps {#section:openclosed1}
=======================
Open-closed Floer data {#floeropenclosed}
----------------------
Here we review the sort of Floer perturbation data that needs to be be specified on the domains appearing in the open-closed map and their cyclic analogues. The main body of our treatment, following §\[subsec:fukaya\] consists of a (slightly modified) simplification of the setup from [@Abouzaid:2010kx] tailored to the case of Fukaya categories of compact admissible $M$; in Remarks \[liouvilleFloerdatumOC\] and \[wrappedFloerdatumOC\] below we will indicate the modifications we need to make (following [*loc. cit.*]{} and building on Remarks \[liouvilleFloerdatum\], \[wrappedFloerdatum\], and \[wrappedainf\] above) the case of compact Fukaya categories of Liouville manifolds (minor modifications) or wrapped Fukaya categories (slightly more involved modifications). There is one notable deviation from [@Abouzaid:2010kx] in that we allow our interior marked point to have a varying asymptotic marker and choose Floer data depending on this choice (as is done in constructions of BV-type operations in Hamiltonian Floer theory involving such asymptotic markers, see e.g., [@Seidel:2010uq; @Seidel:2014aa]).
Let $S$ be a disc with $d$ boundary punctures $z_1, \ldots, z_d$, marked as positive and an interior marked point $p$ removed, marked as either positive or negative; for the main body of the construction $p$ is negative. We also equip the interior marked point $p$ with an [*asymptotic marker*]{}, that is a half-line $\tau_p \in T_p S$ (or equivalently an element of the unit tangent bundle, defined with respect to some metric). Call any such $S = (S, z_1, \ldots, z_d,
p, \tau_p)$ an [*open-closed framed disc.*]{}
In addition to the notation for semi-infinite strips - , we use the following notation to refer to the positive and negative semi-infinite cylinder: $$\begin{aligned}
A_+ &:= [0,\infty) \times S^1\\
A_- &:= (-\infty,0] \times S^1\end{aligned}$$ A [*Floer datum*]{} on a stable open-closed framed disc $S$ consists of the following choices on each component:
1. A collection of [*strip-like or cylindrical ends*]{} $\mathfrak{S}$ around each boundary or interior marked points respectively of sign matching the sign of the marked point; strip-like ends were defined in §\[subsec:fukaya\] and a (positive resp. negative) cylindrical end is map $$\begin{split}
\delta^{\pm}_j: A_{\pm} &\ra S
\end{split}$$ (so for the main body of the construction we use a negative strip-like end around $p$) All of the strip-like ends around each of the $z_i$ should be positive, and all (strip-like or cylindrical) ends should have disjoint image in $S$. The cylindrical end around $p$ should further should be [ *compatible with the asymptotic marker*]{}, meaning the points with angle zero should asymptotically approach the marker: $$\lim_{s \ra \pm \infty} \delta^{\pm}(s,1) = \tau_p.$$
2. A one-form $\alpha_S$ on $S$, $S$-dependent Hamiltonian function $H_S$ on $M$, and $S$-dependent almost-complex structure $J_S$ on $M$, such that on each strip-like end these data pull back to a given fixed $(dt, H_t, J_t)$ (which we used to define Lagrangian Floer homology chain-complexes) and on the cylindrical end this data pulls back to a given fixed $(dt, H_t^{cyl}, J_t^{cyl})$ which we used to define our Hamiltonian Floer homology chain complex (note: in many cases we could further simplify and choose $(H_t^{cyl}, J_t^{cyl}) = (H_t, J_t)$ given a sufficiently generic choice of $(H_t, J_t)$)
Given a stable open-closed framed disc $S$ equipped with a Floer datum $F_S$, a collection of Lagrangians $\{L_0, \ldots, L_{d-1}\}$, and asymptotics $\{x_1, \ldots, x_d; y\}$ with $x_i$ a chord between $L_{i-1}$ and $L_{i \textrm{ mod }d}$, a map $u: S \ra M$ satisfies [*Floer’s equation for $F_S$ with boundary and asymptotics $\{L_0, \ldots,
L_{d-1}\}, \{x_1, \ldots, x_d; y\}$*]{} if $$\label{floerequationOC}
(du - X_S\otimes \alpha_S)^{0,1} = 0\textrm{ using the Floer data given by $F_S$}$$ (meaning $X_S$ is the Hamiltonian vector field associated to $H_S$, and $0,1$ parts are taken with respect to $J_S$), and $$\label{floerequationocasymptotics}
\begin{cases}
u(z) \in L_i & \textrm{ if $z \in \partial S$ lies counterclockwise from $z_i$ and clockwise from $z_{i+1\ \mathrm{mod}\ d}$ } \\
\lim_{s \ra \pm \infty} u \circ \e^k(s,\cdot) = x_k &\\
\lim_{s \ra -\infty} u \circ \delta(s, \cdot) = y &
\end{cases}.$$ (here $\epsilon^k$ denotes the $k$th strip-like end and $\delta$ denotes the —in this case negative — cylindrical end)
\[liouvilleFloerdatumOC\] If $M$ is Liouville and we are studying the Fukaya category of compact exact Lagrangians, we again impose the additional requirements on Floer data described in Remark \[liouvilleFloerdatum\]. As before the more restrictive types of Floer data chosen for wrapped Fukaya categories in Remark \[wrappedFloerdatumOC\] below would suffice. However, there is additional freedom in the Floer data here which e.g., allows us to define operations in which the interior marked point (and all boundary marked points) are positive, see §\[sec:compactopenclosed\].
\[wrappedFloerdatumOC\] Almost exactly as in Remark \[wrappedFloerdatum\], and following [@Abouzaid:2010kx] in order to associated operations between the wrapped Fukaya category and symplectic cohomology, one needs to make the following modifications to the notion of Floer data. One equips $S$ with strip-like and cylindrical ends as above. Let $\psi^{\rho}$ as before denote the time $\log(\rho)$ Liouville flow on $M$. The modifications to the Floer data are:
- [*Extra choices*]{} of weights and time-shifting maps: Exactly as in Remark \[wrappedFloerdatum\], one associates to a [*weight*]{} $w_k \in \R_{>0}$ to each boundary or interior marked point and a time-shifting map $\rho_S: \partial S \to \R_{>0}$ agreeing with $w_k$ near the $k$th strip-like end.
- [*Modified requirements on one-form*]{}: The 1-form $\alpha_S$ should be subclosed meaning $d \alpha_S \leq 0$, restrict to 0 along $\partial S$, and restrict to $w_k dt$ on each (strip-like or cylindrical) end (as in item \[wrappedoneform\] of Remark \[wrappedFloerdatum\]). It follows by Stokes’ theorem that the weight at the (output) cylindrical end should be greater than the sum of weights over all (input) strip-like ends. In particular, it is not possible for $\alpha_S$ to be subclosed and restrict to 0 along $\partial S$, conditions necessary to appeal to the integrated maximum principle if the interior marked point were also positive. (This is a reflection of the fact that wrapped Fukaya categories do not admit geometric operations with no outputs.)
- [*Modified requirements on Hamiltonians, as in item \[wrappedhamiltonian\] of Remark \[wrappedFloerdatum\]*]{}: The Hamiltonian term should be pull back to $\frac{H \circ \psi^{w_k}}{w_k^2}$ along any strip-like end and to $\frac{H^{cyl} \circ \psi^{w_k}}{w_k^2}$ along the cylindrical end. The Hamiltonian term should also be quadratic at infinitely many levels of tending to infinity (this is a slight weakening of Remark \[wrappedFloerdatum\] coming from the fact that the Hamiltonian used to define $SC^*(X)$ is not quadratic at every level near infinity due to ).
- [*Modified requirements on almost complex structures, as in item \[wrappedalmostcomplexstructure\] of Remark \[wrappedFloerdatum\]*]{} The almost-complex structure should be contact-type at infinity and pull back to $(\psi^{w_k})^*J_t$ along each strip-like end and $(\psi^{w_k})^*J_t^{cyl}$ along the cylindrical end.
Exactly as in Remark \[wrappedFloerdatum\], there is a rescaling action on the space of such Floer data, and we will relax any consistency requirement imposed on Floer data to allow for an arbitrary rescaling when equating different choices of Floer data. Finally, we note the slight modifications to the boundary and asymptotic conditions of Floer’s equation , following Remark \[wrappedainf\]: on the boundary component of $\partial S$ lying counterclockwise from $z_i$ and clockwise from $z_{i+1\textrm{ mod }d}$ we impose $u(z) \in
\psi^{\rho_S(z)} L_i$, and on the $k$th strip-like end, we impose $\lim_{s\ra + \infty} u \circ \e^k(s,\cdot) = \psi^{w_k} x_k$, and on the cylindrical end, we impose $\lim_{s\ra -\infty} u \circ \delta(s,\cdot) = \psi^{w} y$ where $w$ is the weight associated to the interior puncture $p$.
The following “$C^0$-estimate” assumption is again vacuous for $M$ compact and — for non-compact Liouville $M$ — is satisfied (as a consequence of the integrated maximum principle, for instance) by the specific form of Floer data imposed in Remarks \[liouvilleFloerdatumOC\] and \[wrappedFloerdatumOC\]:
\[assumption:c0estimateOC\] For fixed Lagrangians $L_0, \ldots, L_{d-1}$ and asymptotics $x_d, \ldots, x_1, y$, and any smoothly varying family of Floer data over a family of (stable) open-closed framed discs as above, there is a compact set $K \subset M$ depending only on the asymptotics and Floer datum chosen such that such that any Floer solution $u$ from a (stable) open-closed framed discs solving with asymptotics (or the variation for wrapped Fukaya categories described in Remark \[wrappedFloerdatumOC\]) has image contained in $K$. Moreoever, from a given $x_d, \ldots, x_1$ there are no such Floer curves with asymptotics for all but a finite subset of possible $y$.
We will not explicitly refer to this a priori $C^0$ estimate in our construction of the cyclic open-closed map below; however it is a necessary requirement for any appeal to usual Gromov compactness arguments, and is therefore an implicit input in all of our assertions below about moduli spaces being compact.
Non-unital open-closed maps {#sec:nonunital}
---------------------------
We begin by constructing a variant of the open-closed map of [@Abouzaid:2010kx] with source the non-unital Hochschild complex of , which we call the [*non-unital open-closed map*]{} $$\oc^{nu}: \r{CH}^{nu}_{*-n}(\f) \lra CF^*(M).$$ This map actually has a straightforward explanation from the perspective of Remark \[rem:unitinsertion\]: we define the map $\oc^{nu}$ from $\widetilde{\r{CH}}_*(\f))$ by counting discs with an arbitrary number of boundary punctured and one interior puncture asymptotic to an orbit (as in [@Abouzaid:2010kx] with the proviso that we treat the formal elements $e_L^+$ as “fundamental class $[L]$ point constraints (i.e., no constraint)”: we fill back in the relevant boundary puncture and impose no constraints on that marked point. With respect to the decomposition , we define a pair of maps $$\check{\oc} \oplus \hat{\oc}: \r{CH}_*(\f) \oplus \r{CH}_*(\f)[1] \ra CF^*(M)$$ giving the left and right component of the non-unital open-closed map $$\begin{split}
\oc^{nu}: \r{CH}^{nu}_{*-n}(\f) &\lra SC(M),\\
(x,y) &\longmapsto \check{\oc}(x) + \hat{\oc}(y).
\end{split}$$ Since the left (check) factor is a equal to the usual cyclic bar complex for Hochschild homology, $\check{\oc}$ will be defined exactly as the map $\oc$ in [@Abouzaid:2010kx] (this will be briefly recalled below), and the new content is the map $\hat{\oc}$. We will define $\hat{\oc}$ below (and recall the definition of $\check{\oc}$ and prove, extending [@Abouzaid:2010kx] that
\[ocnuchain\] $\oc^{nu}$ is a chain map of degree $n$.
Following [@Abouzaid:2010kx], we refer to the map $\check{\oc}$ viewed as a map from $\r{CH}_*(\f)$ (the left factor of $\r{CH}_*^{nu}(\f)$) as simply $\oc$. It follows from our construction that, assuming Lemma \[ocnuchain\],
\[homologylevelnonunitaloc\] As homology level maps, $[\oc^{nu}] = [\oc]$.
By construction, the chain level map $\oc$ factors as $$\r{CH}_{*-n}(\f) \subset \r{CH}_{*-n}^{nu}(\f) \stackrel{\oc^{nu}}{\lra} CF^*(M).$$ The first inclusion is a quasi-isomorphism by Lemma \[inclusionquasi\], since $\f$ is known to be cohomologically unital.
The moduli space controlling the operation $\check{\oc}$, denoted $$\label{opencloseddm}
\overline{\check{\mc{R}}}_d^{1}$$ is the (Deligne-Mumford compactification of the) abstract moduli space of discs with $d$ boundary positive punctures $z_1,
\ldots, z_d$ labeled in counterclockwise order and 1 interior negative puncture $p_{out}$, with an asymptotic marker $\tau_{out}$ at $p_{out}$ pointing towards $z_d$. The space has a manifold with corners structure, with boundary strata described in [@Abouzaid:2010kx]\*[§C.3]{} (there, the space is called $\overline{\mc{R}}_d^1$)–in short, codimension one strata consist of disc bubbles containing any cyclic subseqeuence of $k$ inputs attached to an element of $\check{\mc{R}}_{d-k+1}^1$ at the relative position of this cyclic subsequence. Orient the top stratum $\mc{R}_d^1$ by trivializing it, sending $[S]$ to the unit disc representative $S$ with $z_d$ and $p_{out}$ fixed at 1 and 0, and taking the orientation induced by the (angular) positions of the remaining marked points: $$\label{openclosedcheckorientation}
-dz_1 \wedge \cdots \wedge dz_{d-1}.$$
![\[fig:oc\_check\]A representative of an element of the moduli space $\check{\mc{R}}^1_{4}$ with special points at 0 (output), $-i$. ](oc_check.pdf)
The moduli space controlling the new map $\hat{\oc}$ is nearly identical to $\check{\mc{R}}_d^1$, but there additional freedom in the direction of the asymptotic marker at the interior puncture $p_{out}$. The top (open) stratum is easiest to define: let $$\label{freemarker}
\mc{R}_d^{1,free}$$ be the moduli space of discs with $d$ positive boundary punctures and one interior negative puncture as in $\check{\mc{R}}_d^1$, but with the asymptotic marker $\tau_{out}$ pointing anywhere between $z_1$ and $z_d$.
\[manifoldcorners\] There is a delicate point in naively compactifying $\mc{R}_d^{1,free}$: on any formerly codimension 1 stratum in which $z_1$ and $z_d$ bubble off, the position of $\tau_{out}$ becomes fixed too, and so the relevant stratum actually should have codimension 2 (and hence does not contribute to the codimension-1 boundary equation for $\hat{\oc}$. Moreoever, there is no nice corner chart near this stratum). For technical convenience, we pass to an alternate, larger (blown-up) model for the compactification in which these strata have codimension 1 but consist of degenerate contributions.
In light of Remark \[manifoldcorners\], we use as motivation and instead define $$\label{hatmodulispace}
\hat{\mc{R}}_{d}^1$$ to be the abstract moduli space of discs with $d+1$ boundary punctures $z_f$, $z_1$, …, $z_d$ and an interior puncture $z_{out}$ with asymptotic marker $\tau_{out}$ pointing towards the boundary point $z_f$, modulo automorphism. We mark $z_f$ as “auxiliary,” but otherwise the space is abstractly isomorphic to $\check{\mc{R}}^1_{d+1}$. Identifying $\hat{\mc{R}}_d^1$ with the space of unit discs with $z_{out}$ and $z_f$ fixed at 1 and 0, the remaining (angular) positions of $z_1, \ldots, z_d$ determine an orientation $$\label{openclosedhatorientation}
-dz_1 \wedge \cdots \wedge dz_d.$$
![A representative of an element of the moduli space $\mc{R}^1_{4,free}$ and the corresponding element of $\hat{\mc{R}}_4^1$.\[fig:oc\_hat\] ](oc_hat.pdf)
The [*forgetful map*]{} $$\pi_f: \hat{\mc{R}}_d^1 \ra \mc{R}_d^{1,free}$$ puts back in the point $z_f$ and forgets it. Since the point $z_f$ is recoverable from the direction of the asymptotic marker at $z_{out}$,
$\pi_f$ is a diffeomorphism.
The perspective of the former space gives us a model for the compactification $$\label{freecompactification}
\overline{\mc{R}}_d^{1,free}$$ as the ordinary Deligne-Mumford compactification $$\label{dmcompactificationocforgotten}
\overline{\hat{\mc{R}}}_{d}^1.$$ As a manifold with corners, is equal to the compactification $\overline{\check{\mc{R}}}_{d+1}^1$, except from the point of view of assigning Floer datum, as we will be forgetting the point $z_f$ instead of fixing asymptotics for it. It is convenient therefore to name components of strata containing $z_f$ differently. At any stratum:
- we treat the main component (containing $z_{out}$ and $k$ boundary marked points) as belonging to $\overline{\hat{\mc{R}}}_{k-1}^1 $ if it contains $z_f$ and $\overline{\check{\mc{R}}}_{k}^1$ otherwise; and
- If the $i$th boundary marked point of any non-main component was $z_f$, we view it as an element of $\mc{R}^{k,f_i}$, the space of discs with 1 output and $k$ input marked points removed from the boundary, with the $i$th point marked as “forgotten,” constructed in Appendix \[discsforgotten\].
- We treat any other non-main component as belonging to $\mc{R}^k$ as usual.
Thus, the codimension-1 boundary of the Deligne-Mumford compactification is covered by the natural inclusions of the following strata $$\begin{aligned}
\label{hatocstrata1} \overline{\mc{R}}^m &\times_i \overline{\hat{\mc{R}}}_{d-m+1}^1\ \ \ 1 \leq i < d-m+1\\
\label{hatocstrata2} \overline{\mc{R}}^{m,f_k}&\times_{d-m+1} \overline{\check{\mc{R}}}^1_{d-m+1} \ \ \ 1 \leq j \leq m,\ 1 \leq k \leq m\end{aligned}$$ where the notation $\times_j$ means that the output of the first component is identified with the $j$th boundary input of the second.
The forgetful map $\pi_f$ extends to a map $\overline{\pi}_{f}$ from the compactification $\overline{\hat{\mc{R}}}_{d}^1$ (to the space of stable framed open-closed discs with $d$ marked points) as follows: we call a component $T$ of a representative $S$ of $\overline{\mc{R}}_{d}^1$ the [*main component*]{} if it contains the interior marked point, and the [*secondary component*]{} if its output is attached to the main component. Then, $\overline{\pi}_f$ puts the auxiliary point $z_f$ back in, eliminates any component which is not main or secondary which has only one non-auxiliary marked point $p$, and labels the positive marked point below this component by $p$. Given a representative $S$ of $\overline{\hat{\mc{R}}}_d^1$, we call $\overline{\pi}_f(S)$ the [*associated reduced surface*]{}. We will study maps from the associated reduced surfaces $\overline{\pi}_f(S)$, parametrized by $S$. To this end, we define a [*Floer datum*]{} on a stable disc $S$ in $\overline{\hat{\mc{R}}}_d^1$ to consist of a Floer datum for the underlying reduced surface $\overline{\pi}_f(S)$ in the sense of §\[floeropenclosed\].
First, in Appendix \[discsforgotten\] we describe an inductive construction of Floer data for (the underlying reduced surfaces of) the compactified moduli space of discs with a forgotten point $\overline{\mc{R}}^{d,f_i}$, for every $d$ and $i$ satisfying: $$\label{forgottencondition}
\begin{split}
&\textrm{For $d>2$, the choice of Floer datum on $\mc{R}^{d,f_i}$
should be pulled back from the forgetful map $\mc{R}^{d,f_i} \ra \mc{R}^{d-1}$. }\\
&\textrm{For $d=2$, the Floer datum on the surface $S$ (with $z_i$ forgotten) should be translation invariant.}
\end{split}$$
Next, we choose a *Floer datum for the non-unital open-closed map* which is an inductive set of choices $(\mathbf{D}_{\check{\oc}}, \mathbf{D}_{\hat{\oc}})$, for each $d \geq 1$ and every representative $S \in \overline{\check{\mc{R}}}_d^1$, $T \in
\overline{\hat{\mc{R}}}_d^1$, of a Floer datum for $S$ and (the associated reduced surface of) $T$ respectively. As usual, these choices should be smoothly varying, and restrict smoothly to previously chosen Floer data on boundary strata. Note that for a given $d$ the boundary strata have components that are either $\overline{\check{\mc{R}}}_{d'}^1$ or $\overline{\hat{\mc{R}}}_{d'}^1$ for $d' < d$, a stratum $\mc{R}^{d'}$ (over which we have chosen a Floer datum for the $\ainf$ structure), or a stratum $\mc{R}^{d,f_i}$ where we have chosen a Floer datum in Appendix \[discsforgotten\] as described above. (As usual for wrapped Fukaya categories we use the notion of Floer data and consistency described in Remark \[wrappedFloerdatumOC\].) Contractibility of the space of choices at every stage (and consistency of the compatibility conditions imposed at corners) ensures as usual that a Floer datum for the non-unital open-closed map exists.
Fixing such a choice, we obtain, for any $d$-tuple of Lagrangians $L_0, \ldots, L_{d-1}$, and asympotic conditions $\vec{x} = (x_{d}, \ldots, x_1),\ x_i \in \chi(L_{i-1},L_{i \textrm{ mod }d})$, $y_{out} \in \mc{O}$ a pair of moduli spaces $$\begin{aligned}
\label{openclosedmaps} &\check{\mc{R}}_d^1(y_{out}; \vec{x})\\
\label{hatopenclosedmaps}&\hat{\mc{R}}_d^1(y_{out}; \vec{x}),\end{aligned}$$ of parametrized families of solutions to Floer’s equation $$\begin{aligned}
\{(S,u)| S \in \mc{R}_d^1: u: S \ra M, (du - X\otimes \alpha)^{0,1} = 0\textrm{ using the Floer datum given by $\mathbf{D}_{\check{\oc}}(S)$} \}\\
\{ (S, u) | S \in \hat{\mc{R}}_d^1, u:\pi_f(S) \ra M | (du - X \otimes \alpha)^{0,1} = 0 \textrm{ using the Floer datum given by $\mathbf{D}_{\hat{\oc}}(S)$} \}\end{aligned}$$ satisfying asymptotic and moving boundary conditions (in either case) as in (with the modification for wrapped Fukaya categories involving Liouville rescalings described in Remark \[wrappedFloerdatumOC\]). The expected dimension of every component of and respectively, is (in the $\Z$-graded case), or agrees mod 2 with (in the $\Z/2$-graded case): $$\begin{aligned}
\deg(y_{out}) - n + d -1 - \sum_{k=1}^{d} \deg(x_k);\\
\deg(y_{out}) - n + d - \sum_{k=1}^{d} \deg(x_k).\end{aligned}$$ As usual there are Gromov-type bordifications $$\begin{aligned}
\label{checkOCmaps}&\overline{\check{\mc{R}}}_d^1(y_{out}; \vec{x});\\
\label{hatOCmaps}&\overline{\hat{\mc{R}}}_d^1(y_{out}; \vec{x}),\end{aligned}$$ which allow semi-stable breakings, as well as maps from strata corresponding to the boundary strata of $\overline{\check{\mc{R}}}_d^1$ and $\overline{\hat{\mc{R}}}_d^1$.
For generic choices of Floer datum for the non-unital open-closed map, the components of and of virtual dimension $\leq 1$ are compact manifolds-with-boundary of dimension agreeing with virtual dimension.
Fix such a Floer datum. At a rigid element $u$ of each of the above moduli spaces, we obtain, using the fixed orientations of moduli spaces of domains - and [@Abouzaid:2010kx]\*[Lemma C.4]{}, isomorphisms of orientation lines $$\begin{aligned}
(\check{\mc{R}}_d^1)_u: o_{x_{d}} \otimes \cdots \otimes o_{x_1} \ra o_{y_{out}}\\
(\hat{\mc{R}}_d^1)_u: o_{x_{d}} \otimes \cdots \otimes o_{x_1} \ra o_{y_{out}}.\end{aligned}$$ These isomorphisms in turn define the $|o_{y_{out}}|_{\K}$ component of the check and hat components of the non-unital open-closed map with $d$ inputs in the lines $|o_{x_d}|_{\K}, \ldots, |o_{x_1}|_{\K}$, up to a sign twist: $$\label{checkocdefinition}
\begin{split}
\check{\oc}_d &([x_d] \otimes \cdot \otimes [x_1]) := \\
&
\sum_{u \in \overline{\check{\mc{R}}}^d_1(y; x_d, \ldots, x_1)\textrm{ rigid}} (-1)^{\check{\star}_d} (\check{\mc{R}}_d^1)_u([x_d], \ldots, [x_1]),\\
\check{\star}_d &:=\deg(x_d) + \sum_{k=1}^d k \deg(x_k)
\end{split}$$
$$\label{hatocdefinition}
\begin{split}
\hat{\oc}_d([x_{d}], \ldots, [x_1]) &:=
\sum_{u \in \overline{\hat{\mc{R}}}_d^1(y_{out}; \vec{x}) \textrm{ rigid}} (-1)^{\hat{\star}_d} (\hat{\mc{R}}_d^1)_u([x_d], \ldots, [x_1]),\\
\hat{\star}_d &:= \sum_{i=1}^d i \cdot \deg(x_i).
\end{split}$$
By analyzing the boundary of one-dimensional components of the moduli spaces $\overline{\check{\mc{R}}}_d^1(y_{out}; \vec{x})$, the consistency condition imposed on Floer data, and a sign analysis, it was proven in [@Abouzaid:2010kx] that
\[occhain\] $\oc:=\check{\oc}$ is a chain map of degree $n$; that is $(-1)^n d_{CF} \circ \check{\oc} = \check{\oc} \circ b$.
Similarly, we prove the following, completing the proof of Lemma \[ocnuchain\]:
\[ochatchainold\] The following equation holds: $$(-1)^n d_{CF} \circ \hat{\oc} = \check{\oc} \circ d_{\wedge \vee} +
\hat{\oc} \circ b'$$
The consistency condition imposed on Floer data implies that the boundary of the 1-dimensional components of $\overline{\check{\mc{R}}}_d^1(y;
\vec{x})$ are covered by the images of the natural inclusions of the rigid (zero-dimensional) components of the moduli spaces of maps coming from the boundary strata , along with (the rigid components of) semi-stable breakings: $$\begin{aligned}
\overline{\hat{\mc{R}}}_d^1(y_1; \vec{x}) \times \overline{\mc{M}}(y_{out}; y_1) &\ra \partial
\overline{\check{\mc{R}}}_d^1(y_{out}; \vec{x}) \label{semistablehat1}\\
\overline{\mc{R}}^1(x_1; x) \times \overline{\hat{\mc{R}}}_d^1(y_{out}; \tilde{\vec{x}}) &\ra \partial
\overline{\check{\mc{R}}}_d^1(y_{out}; \vec{x}),\label{semistablehat2}.\end{aligned}$$ Let $\mu^{d,i}$ be the operation associated to the space of discs with $i$th point marked as forgotten $\mc{R}^{d,f_i}$, which is described in detail in Appendix \[discsforgotten\]. $\mu^{d,i}$ takes a composable sequence of $d-1$ inputs, separated into an $i-1$ tuple and a $d-i$ tuple; in line with Remark \[rem:unitinsertion\] we will use the suggestive notation $$\mu^d(x_d, \ldots, x_{i+1}, e^+, x_{i-1},
\ldots, x_1):= \mu^{d,i}(x_d, \ldots, x_{i+1}; x_{i-1}, \ldots, x_1).$$ [^10] Then, up to sign, by the standard codimension-1 boundary principle for Floer-theoretic operations, we have shown that $$\label{hatocchainequation}
\begin{split}
0 = &d_{CF} \hat{\oc}(x_d, \ldots, x_1) - \sum_{i,j} (-1)^{\maltese_1^i} \hat{\oc} (x_d \otimes \cdots \otimes x_{i+j+1}\otimes \mu^j(x_{i+j}, \ldots, x_{i+1})\otimes x_i\otimes \cdots \otimes x_1)\\
&- \sum_{i,j,k} (-1)^{\sharp_j^k} \check{\oc} ( \mu^{j + k + 1}(x_{j}, \ldots, x_1, e^+, x_d, \ldots, x_{d-k+1}) \otimes x_{d-k} \otimes \cdots \otimes x_{j+1}).
\end{split}$$ with desired signs $$\begin{aligned}
\label{desiredsign1}\maltese_m^n &= \sum_{j=m}^n ||x_i||\\
\label{desiredsign2}\sharp_j^k &= \maltese_1^j \maltese_{j+1}^{d} + \maltese_{j+1}^{d} + 1\end{aligned}$$ However, as shown in Appendix \[discsforgotten\], $$\mu^{j + k + 1}(x_{j}, \ldots, x_1, e^+, x_d, \ldots, x_{d-k+1}) = \begin{cases}
x_1 & j = 1,\ k = 0\\
(-1)^{|x_d|} x_d & j = 0, k = 1\\
0 & \textrm{otherwise}
\end{cases}$$ (in this manner, $e^+$, though a formal element, behaves as a strict unit). So if held, it would follow that $$\begin{split}
\label{lastsumhatoc}
d_{CF} \circ \hat{\oc}(x_d \otimes \cdots \otimes x_1) &=(-1)^{||x_1|| \maltese_2^d + \maltese_2^d + 1} \oc(x_1 \otimes x_d \otimes \cdots \otimes x_2) \\
&+ (-1)^{|x_d| + \maltese_1^d + 1} \check{\oc}(x_d \otimes \cdots \otimes x_1) +\hat{\oc} \circ b' (x_d \otimes \cdots \otimes x_1)\\
&= \check{\oc} ( (-1)^{\maltese_1^d + ||x_d||}(1-t) (x_d \otimes \cdots \otimes x_1)) + \hat{\oc} \circ b'(x_d \otimes \cdots \otimes x_1).\\
&= \left(\check{\oc} \circ d_{\wedge \vee} + \hat{\oc} \circ b'\right) \circ (x_d \otimes \cdots \otimes x_1).
\end{split}$$ So we are done if we establish the signs are exactly -.
Using the notation $$\oc(e^+ \otimes x_d \otimes \cdots \otimes x_1) := \hat{\oc}(x_d \otimes \cdots \otimes x_1),$$ where again $e^+$ is simply a formal symbol referring to the position of the auxiliary (forgotten) input point, we observe that the equation is exactly the equation for $\oc$ being a chain map on inputs of the form $(e^+ \otimes x_d \otimes \cdots \otimes x_1)$ (where we treat an “$e^+$” input as an auxiliary unconstrained point on our domain). The sign verification therefore follows from that of $\check{\oc}$ being a chain map (in [@Abouzaid:2010kx]\*[Lemma 5.4]{}), for we have used identical orientations on the abstract moduli space $\hat{\mc{R}}_d^1$ as on $\check{\mc{R}}_{d+1}^1$, and on $\mc{R}^{d,f_i}$ as on $\mc{R}^d$, and we can even insert a formal degree zero orientation line $o_{e^+}$ into the procedure for orienting moduli spaces of open-closed maps (see [@Abouzaid:2010kx]\*[§C.6]{}), corresponding to the marked point $x_f$. Note that $o_{e^+}$, being of degree zero, commutes with everything, and is just used as a placeholder as if we had an asymptotic condition at $x_f$.
Given that $\check{\oc}$ is already known to be a chain map by [@Abouzaid:2010kx]\*[Lemma 5.4]{}, repeated as Lemma \[occhain\] above, the new part to check is that the $(-1)^n d_{CF} \circ \hat{\oc} = \check{\oc} d_{\wedge \vee} + \hat{\oc} \circ b'$. This is the content of Lemma \[ochatchainold\] above.
An auxiliary operation {#ocs1sec}
----------------------
It will be technically convenient to define an auxiliary operation $$\label{ocS1}
\oc^{S^1}: \r{CH}_{*-n}(\f) \ra CH^{*+1}(M)$$ from the left factor of the non-unital Hochschild complex to Floer co-chains, in which the asymptotic marker $\tau_{out}$ varies freely around the circle. This operation is more easily comparable to the BV operator on Floer cohomology, and moreover, we will show that $\oc^{S^1}$ (and $\hat{\oc}$) can be chosen to satisfy the following crucial identity:
\[ocS1Brelation\] There is an equality of chain level operations: $$\oc^{S^1} = \hat{\oc} \circ B^{nu}.$$
To define , let $$\label{S1modulispace}
\mc{R}^{S^1}_d$$ be the abstract moduli space of discs with $d$ boundary positive punctures $z_1,
\ldots, z_d$ labeled in counterclockwise order and 1 interior negative puncture $p_{out}$, with an asymptotic marker $\tau_{out}$ at $p_{out}$ (or choice of real half line in $T_{p_{out}} D$) which is free to vary. Equivalently, $$\begin{split}
&\textrm{\eqref{S1modulispace} is the space of discs with $z_1, \ldots, z_d$ and
$p_{out}$ as before,
and an extra auxiliary }\\
&\textrm{interior marked point $p_1$ such
that, for a representative with $(p_{out}, z_1)$ fixed at $(0,-i)$, }\\
&\textrm{$|p_1| = \frac{1}{2}$, and the asymptotic marker $\tau_{out}$ points
towards $p_1$.}
\end{split}$$
Abstractly, $$\label{compactifications1}
\mc{R}^{S^1}_d = S^1 \times \mc{R}^1_d,$$ a fact which we use to fix an orientation on the top component of . The Deligne-Mumford type compactification thus has a simple description $$\overline{\mc{R}_d^{S^1}} = \mc{R}^1_d \times S^1.$$ Given an element $S$ of $\mc{R}_d^{S^1}$ and a choice of marked point $z_i$ on the boundary of $S$, we say say that $\tau_{out}$ [*points at $z_i$*]{}, if, when $S$ is reparametrized so that $z_1$ fixed at $-i$ and $p_{out}$ fixed at 0, the vector $\tau_{out}$ is tangent to the straight line from $p_{out}$ to $z_i$. Equivalently, for this representative, $p_{out}$, $p_1$, and $z_i$ are collinear. For each $i$, the locus where $\tau_{out}$ points at $z_i$ forms a codimension 1 submanifold, denoted $$\mc{R}_d^{S^1_i}.$$ The notion compactifies well; if $z_i$ is not on the main component of we say $\tau_{out}$ [*points at $z_i$*]{} if it points at the root of the bubble tree $z_i$ is on. This compactified locus $\overline{\mc{R}_d^{S^1,i}}$ can be identified on the nose with $\overline{\mc{R}}^1_d$ via the map $$\label{pointatimap}
\tau_i: \overline{\mc{R}_d^{S^1,i}} \ra \overline{\mc{R}}^1_d$$ which cyclically permutes the labels of the boundary marked points so that $z_i$ is now labeled $z_0$.
In a similar fashion, we have an invariant notion of what it means for $\tau_{out}$ to point [*between $z_i$ and $z_{i+1}$*]{}; this is a codimension 0 submanifold with corners of , denoted $$\label{pointii1}
\mc{R}_d^{S^1_{i,i+1}}.$$ The compactification has some components that are codimension 1 submanifolds with corners of , when $z_i$ and $z_{i+1}$ both lie on a bubble tree.
Finally, there is a [*free*]{} $\Z_{d}$ action generated by the map $$\label{labelpermute}
\kappa: \overline{\mc{R}_d^{S^1}} \ra \overline{\mc{R}_d^{S^1}}$$ which cyclically permutes the labels of the boundary marked points; for concreteness, $\kappa$ changes the label $z_i$ to $z_{i+1}$ for $i < d$, and $z_d$ to $z_1$. Note that if, on a given $S$, $\tau_{out}$ points between $z_i$ and $z_{i+1}$, then on $\kappa(S)$, $\tau_{out}$ points between $z_{i+1 \rm{\ mod \ }d}$ and $z_{i+2 \rm{\ mod \ }d}$.
\[freediscontinuous\] The action generated by is free and properly discontinuous.
The basic observation arises on the level of uncompactified moduli spaces: since any element of $\mc{R}_d^{S^1}$ has a unit disk representative with $(p_{out},p_1)$ fixed at $(0,\frac{1}{2})$, the positions of the remaining points identify $\mc{R}_d^{S^1}$ with the space of tuples $(z_1, \ldots, z_d)$ of disjoint (cylically ordered) points on $S^1$ (without any further quotienting by automorphism). The action of $\kappa$, which cyclically permutes the labels $z_1, \ldots, z_d$ in this identification, evidently acts freely and properly discontinuously on this locus.
The quotient of the action of $\kappa$ consists of the space of discs with $p_{out}$ and $p_1$ as before, equipped with [*$d$*]{} [*cyclically unordered*]{} or [*unlabeled*]{} boundary marked points. Note that on the open-locus $\mathring{\mc{R}}_d^{S^1}$ where $\tau_{out}$ does not point at a boundary marked point, one can choose a labeling by setting the boundary point immediately clockwise of where $\tau_{out}$ points to be $z_d$. This induces a diffeomorphism $$\mathring{\mc{R}}_d^{S^1}/\kappa \cong \mc{R}_d^{1, free}$$ Similarly, on the complementary locus where $\tau_{out}$ points at a boundary marked point, we can similarly choose a labeling by declaring this boundary marked point to be $z_d$, giving a diffeomorphism (of this locus) with $\check{\mc{R}}_d^1$.
We now choose Floer perturbation data for the family of moduli spaces $\mc{R}_d^{S^1}$; in fact, it will be helpful to re-choose Floer data for the moduli spaces appearing in the non-unital open-closed map to have extra compatibility. To that end, a [*BV compatible Floer datum for the non-unital open-closed map*]{} is an inductive choice $(\mathbf{D}_{\check{\oc}}, \mathbf{D}_{\hat{\oc}},
\mathbf{D}_{S^1})$ of Floer data where $\mathbf{D}_{\check{\oc}}$ and $\mathbf{D}_{\hat{\oc}}$ is a universal and consistent choice of Floer data for the non-unital open-closed map as before, and $\mathbf{D}_{S^1}$ consists of, for each $d
\geq 1$ and every representative $S \in \overline{\mc{R}_d^{S^1}}$, a Floer datum for $S$ varying smoothly over the moduli space. Again, these satisfy the usual consistency condition with respect to previously made choices along lower-dimensional strata. Moreover, there are two additional inductive constraints on the Floer data chosen: $$\begin{aligned}
\label{S1occompatibility}&\textrm{On the codimension-1 loci $\overline{\mc{R}_d^{S^1_i}}$ where
$\tau_{out}$ points at $z_i$, the Floer datum }\\
\nonumber &\textrm{ should agree with the pullback by $\tau_i$ of the existing Floer datum}\\
\nonumber &\textrm{ for the
(check) open-closed map. }\\
\label{S1tauequivariance}&\textrm{The Floer datum should be $\kappa$-equivariant, where $\kappa$ is the map
\eqref{labelpermute}. }
\end{aligned}$$ Also, there is a final a posteriori constraint on the Floer data for the non-unital open-closed map $\mathbf{D}_{\hat{\oc}}$; for $S \in
\overline{\hat{\mc{R}}}_d^1$: $$\label{S1compatibilitycondition}
\begin{split}
&\textrm{the Floer datum on the main component $S_0$ of $\pi_f(S)$ should coincide with the}\\
&\textrm{existing datum chosen on $S_0 \in \mc{R}_d^{1,free} \subset \mc{R}_d^{S^1}$}.
\end{split}$$ By an inductive argument as before, a BV compatible Floer datum for the non-unital open-closed map exists.
To explain the way choices are made (which ensures both existence at every stage and that the requirements above are satisfied): we choose the data for $\mc{R}_d^{S^1}$ prior to choosing that of $\overline{\hat{\mc{R}}}_d^1$ and note that the condition specifies the Floer datum on $\overline{\hat{\mc{R}}}_d^1$ entirely. In particular, the conditions required on the latter Floer datum are compatibile with consistency and the condition . With regards to choosing the data for $\mc{R}_d^{S^1}$, the equivariance constraint , which is compatible with both (which is a $\kappa$-equivariant condition) and with the consistency condition, is also unproblematic in light of Lemma \[freediscontinuous\]: one can pull back a Floer datum from the quotient of $\overline{\hat{\mc{R}}}_d^1$ by $\kappa$.
Fixing a BV compatible Floer datum for the non-unital open-closed map we obtain, for any $d$-tuple of Lagrangians $L_0, \ldots, L_{d-1}$, and asympotics $\vec{x} = (x_{d}, \ldots, x_1)$ ($x_i \in \chi(L_{i},L_{i+1 \mathrm{\ mod\ }d})$), $y_{out} \in \mc{O}$, a moduli space $${\mc{R}}_d^{S^1}(y_{out}; \vec{x}),$$ of parametrized families of solutions to Floer’s equation (with respect to the Floer data chosen) $$\{ (S, u) | S \in \mc{R}_d^{S^1}, u:\pi_f(S) \ra M | (du - X \otimes \alpha)^{0,1} = 0 \}$$ satisfying asymptotic and moving boundary conditions as in (again with modifications of Remark \[wrappedFloerdatumOC\] for wrapped Fukaya categories). Generically the Gromov-Floer compactification $$\overline{\mc{R}}_d^{S^1}(y_{out}; \vec{x})$$ of the components of virtual dimension $\leq 1$ are compact manifolds-with-boundary of the expected dimension; this dimension coincides (mod 2 or exactly depending on whether we are in a $\Z/2$ or $\Z$-graded setting) with $$\deg(y_{out}) - n + d - \sum_{k=1}^{d} \deg(x_k).$$ Each rigid $u \in \overline{{\mc{R}}}_d^{S^1}(y_{out}; \vec{x})$ gives by the orientation from and [@Abouzaid:2010kx]\*[Lemma C.4]{} an isomorphism of orientation lines $$(\mc{R}_d^{S^1})_u: o_{x_{d}} \otimes \cdots \otimes o_{x_1} \ra o_{y_{out}},$$ which gives the $|o_{y_{out}}|_{\K}$ component of the $S^1$ open-closed map with $d$ inputs in the lines $|o_{x_d}|_{\K}, \ldots, |o_{x_1}|_{\K}$, up to a sign twist given below: define $$\begin{split}
\oc^{S^1}([x_{d}], \ldots, [x_1]) &:=
\sum_{u \in \overline{\mc{R}}_d^{S^1}(y_{out}; \vec{x})\textrm{ rigid}} (-1)^{\clubsuit_d} (\mc{R}_d^{S^1})_u([x_d], \ldots, [x_1]),\\
\clubsuit_d &= \sum_{i=1}^d (i+1) \cdot \deg(x_i) + \deg(x_d) + d-1.
\end{split}$$ The proof of Proposition \[ocS1Brelation\], which equates $\oc^{S^1}$ with $\oc \circ B^{nu}$, appears below and is composed of two steps. First, we decompose the moduli space $\mc{R}_d^{S^1}$ into sectors in which $\tau_{out}$ points between a pair of adjacent boundary marked points. It will follow that the sum of the corresponding “sector operations” is exactly $\oc^{S^1}$. The sector operations in turn can be compared to $\hat{\oc}$ via cyclically permuting inputs and an orientation analysis.
We begin by defining the relevant sector operations: For $i\in \Z/(d+1)\Z$, define $$\hat{\mc{R}}^{1}_{d,\tau_i}$$ to be the abstract moduli space of discs with $d+1$ boundary punctures $z_1,
\ldots, z_{i}, z_f, z_{i+1}, \ldots, z_d$ arranged in clockwise order and interior puncture $z_{out}$ with asymptotic marker pointing towards the boundary point $z_f$, which is also marked as “auxiliary.” There is a bijection $$\label{taui}
\tau_i: \hat{\mc{R}}^{1}_{d,\tau_i} \simeq \hat{\mc{R}}^1_d$$ given by cyclically permuting labels, which induces a model for the compactification $\overline{\hat{\mc{R}}^{1}_{d,\tau_i}}$. However, we will use a different orientation than the one induced by pullback: on a slice with fixed position of $z_d$ and $z_{out}$, we take the volume form $$\label{differentorientation}
dz_1 \wedge \cdots \wedge dz_{d-1} \wedge dz_f.$$ By construction, the induced “forgetful map” $$\pi_f^{i}: \hat{\mc{R}}^1_{d,\tau_i} \ra \mc{R}^{S^1_{i,i+1}},$$ is an oriented diffeomorphism that extends to a map between compactifications (note as before that strictly speaking this map does not forget any information, at least on the open locus).
\[inducedorientationslice\] In the case $i=0$, note that this orientation agrees with the previously chosen orientation on $\hat{\mc{R}}^1_d$. To see this, note that we previously defined the orientation on $\hat{\mc{R}}^1_d$ in terms of a different slice of the group action. To compare the forms $dz_1
\wedge \cdots \wedge dz_{d-1} \wedge dz_f$ (coming from the slice with fixed $z_d$ and $z_{out}$) and $-dz_1 \wedge \cdots \wedge dz_d$ (coming from the slice with fixed $z_f$ and $z_{out}$), note that either orientation is induced by the following procedure:
- fix an orientation on the space of discs as above with fixed position of $z_{out}$ (but not $z_f$ or $z_d$): we shall fix the canonical orientation $dz_1 \wedge \cdots \wedge dz_d \wedge dz_f$;
- fix a choice of trivalizing vector field for the remaining $S^1$ action on this space of discs with fixed $z_{out}$: we shall fix $S =
(-\partial_{z_f} -
\partial_{z_1} - \cdots - \partial_{z_d})$; and
- fix a convention for contracting orientation forms along slices of the action: to determine the orientation on a slice of an $S^1$ action, we will contract the orientation on the original space on the right by the trivializing vector field.
Moreoever, this data induces an orientation on the quotient by the $S^1$ action, and also an oriented isomorphism between the induced orientation on any slice and that of the quotient. It follows that on the quotient, the orientation $-dz_1 \wedge \cdots \wedge dz_d$ (from the slice where $z_f$ is fixed), and the orientation $dz_1 \wedge \cdots \wedge dz_{d-1} \wedge dz_f$ (from the slice where $z_d$ is fixed) agree. We conclude these two orientations agree. The author thanks Nick Sheridan for relevant discussions about orientations of moduli spaces.
Choose as a Floer datum for each $\overline{\mc{R}^1_{d,\tau_i}}$ the pulled back Floer datum from $\overline{\hat{\mc{R}}^1_d}$ via ; this system of choices automatically is inductively consistent with choices made on lower strata (inheriting this property from the Floer data on the collection of $\overline{\hat{\mc{R}}^1_d}$). Using this choice, for any $d$-tuple of Lagrangians $L_0, \ldots, L_{d-1}$, and asympotic conditions $ \vec{x} = (x_{d}, \ldots, x_1),\ x_i \in \chi(L_{i},L_{i+1 \rm{\ mod \ }d})$, $y_{out} \in \mc{O}$ we obtain a moduli space $$\mc{R}^1_{d,\tau_i}(y_{out}; \vec{x}) = \hat{\mc{R}}^1_d(y_{out}; (x_{i-1}, \ldots, x_1, x_{d}, \ldots, x_i))$$ of parametrized families of solutions to Floer’s equation $$\{ (S, u) | S \in \hat{\mc{R}}_d^1, u:\pi_f(S) \ra M | (du - X \otimes \alpha)^{0,1} = 0 \textrm{ using the Floer datum for $\pi_f(S)$} \}$$ satisfying asymptotic and moving boundary conditions as in (with the modifications as in Remark \[wrappedFloerdatumOC\] if in the setting of wrapped Fukaya categories), and its Gromov-Floer compactification $$\overline{\mc{R}}^1_{d,\tau_i}(y_{out}; \vec{x}) := \overline{\hat{\mc{R}}^1_d}(y_{out}; (x_{i}, \ldots, x_1, x_{d}, \ldots, x_{i+1})),$$ whose components of virtual dimension $\leq 1$ (at least) are compact manifolds-with-boundary of the correct dimension (which coincides exactly in the graded case and mod 2 in the $\Z/2$ graded case with $\deg(y_{out}) - n + d - \sum_{j=0}^{d} \deg(x_j).$
Each rigid element $u \in \overline{\mc{R}}_{d,\tau_i}^1(y_{out}; \vec{x})$ gives by and [@Abouzaid:2010kx]\*[Lemma C.4]{} an isomorphism of orientation lines $$\label{isomorphismrotated}
(\mc{R}_{d,\tau_i}^1)_u: o_{x_{d}} \otimes \cdots \otimes o_{x_1} \ra o_{y_{out}},$$ which defines the $|o_{y_{out}}|_{\K}$ component of an operation $\hat{\oc}_{d,\tau_i}$ with $d$ inputs in the lines $|o_{x_d}|_{\K}, \ldots, |o_{x_1}|_{\K}$, up to the following sign twist: $$\begin{split}
\hat{\oc}_{d,\tau_i}([x_{d}], \ldots, [x_1]) &:=
\sum_{u \in \overline{\hat{\mc{R}}}_{d,\tau_i}^1(y_{out}; \vec{x})\textrm{ rigid}} (-1)^{\clubsuit_d} (\hat{\mc{R}}_{d,\tau_i}^1)_u([x_d], \ldots, [x_1]),\\
\clubsuit_d &= \sum_{i=1}^d (i+1) \cdot \deg(x_i) + \deg(x_d) + d-1.
\end{split}$$
\[sectordecomposition\] As chain level operations, $$\oc^{S^1} = \sum_i \hat{\oc}_{d,\tau_i}$$
For each $d$, there is an embedding of abstract moduli spaces $$\label{disjointembedding}
\coprod_{i} \hat{\mc{R}}^1_{d,\tau_i} \stackrel{\coprod_i \pi_f^{i}}{\lra} \coprod_i \mc{R}^{S^1_{i,i+1}}_d
\hookrightarrow \mc{R}^{S^1}_d;$$ see Figure \[fig:oc\_s1\_new\].
![ The diffeomorphism between $\hat{\mc{R}}^1_{2,\tau_0} \cup \hat{\mc{R}}^1_{2,\tau_1}$ and the open dense part of $\mc{R}^{S^1}_2$ given by $\mc{R}^{S^1_{0,1}}_2 \cup \mc{R}^{S^1_{1,2}}_2$. The former spaces can in turn be compared to $\hat{\mc{R}}^1_2$ via cyclic permutation of labels. \[fig:oc\_s1\_new\] ](oc_s1_new.pdf)
By construction, this map is compatible with Floer data (this uses the fact that the Floer data on $\mc{R}^{S^1_{i,i+1}}$ agrees with the data on $\hat{\mc{R}}_d^1$ via the reshuffling map $\kappa^{-i}$ by ), and covers all but a codimension 1 locus in the target. Since after perturbation zero-dimensional solutions to Floer’s equation can be chosen to come from the complement of any codimension 1 locus in the source abstract moduli space, we conclude that the two operations in the Proposition, which arise from either side of , are identical up to sign. To fix the signs, note that is in fact an oriented embedding, and all the sign twists defining the operations $\hat{\oc}_{d,\tau_i}$ are chosen to be compatible with the sign twist in the operation $\oc^{S^1}$.
Next, because the Floer data used in the constructions are identical, $\hat{\oc}_{d,\tau_i}(x_{d} \otimes \cdots \otimes x_1)$ agrees with $\hat{\oc}(x_{i} \otimes \cdots \otimes x_1 \otimes x_{d} \otimes \cdots
\otimes x_{i+1})$ up to a sign difference coming from orientations of abstract moduli spaces, cyclically reordering inputs, and sign twists. The following proposition computes the sign difference, and hence completes the proof of Proposition \[ocS1Brelation\]:
\[sectorsignorientation\] There is an equality $$\begin{split}
\hat{\oc}_{d,\tau_i}(x_d \otimes \cdots \otimes x_1)&= \hat{\oc}^d(s^{nu}( t^i (x_d \otimes \cdots x_1)))
\end{split}$$ where $s^{nu}$ is the operation arising from changing a check term to a hat term with a sign twist.
It is evident that $\hat{\oc}_{d,\tau_i}$ agrees with $\hat{\oc}_d \circ s^{nu} \circ
t^i$ up to sign, as the Floer data used in the two constructions are identical. By an inductive argument it suffices to verify the following equalities of signed operations: $$\begin{aligned}
\hat{\oc}_{d,\tau_0} &= \hat{\oc}_d \circ s^{nu}, \label{insertunitsign}\\
\hat{\oc}_{d, \tau_1} &= \hat{\oc}_{d,\tau_0} \circ t \label{cyclicallyrotatesign};
\end{aligned}$$ the remaining sign changes are entirely incremental. For the equality , we simply note that the signs appearing in the operations $\hat{\oc}_{d,\tau_0}([x_d], \ldots, [x_1])$ and $\hat{\oc}_d ([x_d],
\ldots, [x_1])$ differ in the following fashions:
- The abstract orientations on the moduli space of domains agree, as per Remark \[inducedorientationslice\].
- The difference in sign twists is given by $\clubsuit_d -
\hat{\star}_d = \sum_{i=1}^d |x_i| + |x_d| + d-1 = (\sum_{i=1}^d ||x_i||) +
1 + |x_d| = \maltese_1^d + ||x_d||$.
All together, the parity of difference in signs is $\maltese_1^d + ||x_d||$ which accounts for the sign in the algebraic operation $s^{nu}$ (see ); this verifies .
Next, the sign difference between the two operations in the equality is a sum of three contributions:
- The two orientations of abstract moduli spaces from (on the slice where $z_f$ and $z_{out}$ are fixed; see Remark \[inducedorientationslice\]) $-dz_1
\wedge \cdots \wedge dz_d$ to $dz_2 \wedge \cdots \wedge dz_d
\wedge dz_1$ differ by a sign change of parity $$d-1.$$
- For a given collection of inputs, the change in [*sign twisting data*]{} from $\clubsuit_d = \sum_{i=1}^d (i+1) \cdot |x_i| + |x_d| + d-1$ to $\sum_{i=1}^{d-1} (i+1)
|x_{i+1}| + (d+1)|x_1| + |x_1| + d-1 = \sum_{i=2}^d i |x_i| + d
|x_1| + d-1$ ($\clubsuit_d$ for the sequence $(x_2, \ldots, x_d,
x_1)$) induces a sign change of parity $$\sum_{i=2}^d |x_i| + |x_d| + d |x_1| = \sum_{i=1}^d |x_i| + |x_d| + (d-1) |x_1| = \sum_{i=1}^d ||x_i|| + (d-1) || x_1|| + ||x_d|| = \maltese_1^d + (d-1) ||x_1|| + ||x_d||.$$
- Finally, the re-ordering of determinant lines of the inputs induces a sign change of parity $$|x_1| \cdot (\sum_{i=2}^d |x_i|) = ||x_1|| \cdot (\sum_{i=2}^d ||x_i|| ) + \sum_{i=2}^d ||x_i|| + (d-1)||x_1|| + (d-1) = ||x_1|| \maltese_2^d + \maltese_1^d + d||x_1|| + (d-1)$$
The cumulative sign parity is congruent mod 2 to $$||x_1|| \maltese_2^d + ||x_1|| + ||x_d||,$$ which is precisely the sign appearing in $t$ (see ). This verifies .
Combine Lemmas \[sectordecomposition\] and \[sectorsignorientation\] (note the definition of $B^{nu}$ given in ).
Compatibility of homology-level BV operators {#sec:homology}
--------------------------------------------
Before diving into the statement of chain-level equivariance, we prove a homology-level statement. The below Theorem is insufficient for studying, say, equivariant homology groups, but may be of independent interest.
\[thm:homology\] The homology level open-closed map $[\oc^{nu}]$ intertwines the Hochchild and symplectic cohomology BV operators, that is $$[\oc^{nu}] \circ [B^{nu}] = [\delta_1] \circ [\oc^{nu}].$$
Theorem \[thm:homology\] is an immediate consequence of the following chain-level statement:
\[homologylevelchainhomotopy\] The following diagram homotopy commutes: $$\xymatrix{ \r{CH}_{*-n}(\mc{F}, \mc{F}) \ar[d]^{\check{\oc}} \ar@{^{(}->}[r]^{\iota}_{\sim} & \r{CH}_{*-n}^{nu}(\mc{F}, \mc{F}) \ar[r]^{B^{nu}} & \r{CH}_{*-n-1}^{nu}(\mc{F}, \mc{F}) \ar[d]^{\oc^{nu}} \\
CF^*(M) \ar[rr]^{\delta_1} & & CF^{*-1}(M).}$$ where $\iota$ is the inclusion onto the left factor, which is a quasi-isomorphism by Lemma \[inclusionquasi\]. More precisely, there exists an operation $\check{\oc}^1: \r{CH}_{*-n}(\mc{F}, \mc{F}) \ra CF^{*-2}(M)$ satisfying $$(-1)^{n+1} d \check{\oc}^1 + \check{\oc}^1 b = \hat{\oc} B^{nu} \iota - (-1)^n \delta_1 \check{\oc}.$$
Proposition \[homologylevelchainhomotopy\] immediately implies that $[\delta_1] \circ [\check{\oc}] = [\oc^{nu}]\circ [B^{nu}] \circ [\iota]$ where $\iota: \r{CH}_{*-n}(\mc{F}, \mc{F}) \ra \r{CH}_{*-n}^{nu}(\mc{F}, \mc{F})$ is the inclusion of chain complexes. But by Lemma \[inclusionquasi\], $[\iota]$ is an isomorphism and by Corollary \[homologylevelnonunitaloc\] $[\check{\oc}] = [\oc^{nu}]$.
To define $\check{\oc}^1$, consider $$\label{check1equivariant}
{ }_1 \check{\mc{R}}_{d+1}^1,$$ the moduli space of discs with with $d+1$ positive boundary marked points $z_0,
\ldots, z_d$ labeled in counterclockwise order, 1 interior negative puncture $z_{out}$ equipped with an asymptotic marker, and $1$ additional interior marked points $p_1$ (without an asymptotic marker), marked as [*auxiliary*]{}. Also, choosing a representative of an element this moduli space which fixes $z_0$ at $1$ and $z_{out}$ at 0 on the unit disc, $p_1$ should lie [*within a circle of radius $\frac{1}{2}$*]{}: $$\label{ordered1} 0 < |p_1| < \frac{1}{2}.$$ Using the above representative, one can talk about the [*angle*]{}, or [*argument*]{} of $p_1$ $$\theta_i:= \arg(p_i).$$ We require that with respect to the above representative, $$\label{pointing1condition} \textrm{the asymptotic marker on
$z_{out}$ points in the direction $\theta_1$. }$$ For every representative $S \in { }_1 \check{\mc{R}}_{d+1}^1$, $$\label{check1negcylindricalend}
\begin{split} &\textrm{
fix a negative cylindrical end around $z_{out}$ not containing
$p_1$, compatible with the }\\ &\textrm{direction of the
asymptotic marker, or {\it equivalently compatible with the
angle $\theta_1$}.} \end{split}$$ We orient as follows: pick, on a slice of the automorphism action which fixes the position of $z_d$ at 1 and $z_{out}$ at 0, the volume form $$\label{check1orientation} -r_1 dz_1\wedge dz_2 \wedge \cdots
\wedge dz_{d-1} \wedge dr_1 \wedge d\theta_1$$ The compactification of is a real blow-up of the ordinary Deligne-Mumford compactification, in the sense of [@Kimura:1995fk] (see [@Seidel:2010uq] for a first discussion in the context of Floer theory).
The result of this discussion is that the codimension 1 boundary of the compactified check moduli space ${ }_1\overline{ \check{\mc{R}}}_{d+1}^1$ is covered by the images of the natural inclusions of the following strata: $$\begin{aligned}
&\label{check1stratum1} \overline{\mc{R}^s} \times { }_1
\overline{\check{\mc{R}}}_{d-s + 2}^1 \\
&\label{check1stratum2} \overline{ \check{\mc{R}}}_{d}^1 \times \overline{\mc{M}}_{1}\\
&\label{check1stratum3} \overline{\check{\mc{R}}}_{d+1}^{S^1}\end{aligned}$$ The stratum describes the locus which $|p_1| =
\frac{1}{2}$, which is exactly the locus we defined to be the auxiliary operation $\mc{R}_{d+1}^{S^1}$. The strata - have manifold with corners structure given by standard local gluing maps using fixed choices of strip-like ends near the boundary. For this is standard, and for , the local gluing map uses the cylindrical ends and (in other words, one rotates the $1$-pointed angle cylinder by an amount commensurate to the angle of the marked point $z_d$ on the disk before gluing).
We will as usual fix a [*Floer datum for the BV homotopy*]{}, meaning an inductive choice, for every $d \geq
1$, of Floer data for every representative $S_0 \in { }_1
\overline{\check{\mc{R}}}_{d+1}^1$ varying smoothly in $S_0$, which on boundary strata is smoothly equivalent to a product of Floer data inductively chosen on lower dimensional moduli spaces. Such a system of choices exist again by a contractibility argument, and for any such choice, one obtains for any $d$-tuple of Lagrangians $L_0, \ldots,
L_{d-1}$, and asympotic conditions $$\begin{split} \vec{x} &= (x_{d},
\ldots, x_1),\ x_i \in
\chi(L_{i},L_{i+1 \mod d})\\
y_{out} &\in \mc{O} \end{split}$$ a compactified moduli space $$\label{BVcompactifiedspace1}{ }_1 \overline{\check{\mc{R}}}_{d+1}^1(y_{out},
\vec{x})$$ of maps into $M$ with source an arbitrary element $S$ of the moduli space , satisfying Floer’s equation using the Floer datum chosen for the given $S$ as in with asymptotics and boundary conditions as in (with the usual modifications for wrapped Fukaya categories detailed in Remark \[wrappedFloerdatumOC\]). The virtual dimension of every component of ${ }_1
\overline{\check{\mc{R}}}_{d+1}^1(y_{out}, \vec{x}) $ coincides (mod 2 or exactly depending on whether we are in a $\Z/2$ or $\Z$-graded setting) with $$\label{dimBVcompactifiedspace1} \deg(y_{out}) -
n + d +1 - \sum_{i=0}^d \deg(x_i)$$
\[BVhomotopyeqn\] For generic choices of Floer data, the Gromov-Floer compactification of the components of virtual dimension $\leq 1$ of are compact manifolds-with-boundary of expected dimension.
For rigid elements $u$ of the moduli spaces , the orientations , and [@Abouzaid:2010kx]\*[Lemma C.4]{} induce isomorphisms of orientation lines $$\label{BVorientationline1} ({ }_1
\check{\mc{R}}_{d}^1)_u: o_{x_d} \otimes \cdots \otimes o_{x_1}
\ra o_y\\$$ As usual “counting rigid elements $u$,” i.e., summing application of these isomorphisms over all $u$, defines the $|o_{y_{out}}|_\K$ component of an operation $\check{\oc}^1$, up to a sign twist which we specify below: $$\check{\oc}^{1}([x_{d}], \ldots, [x_1]) :=
\sum_{u \in { }_k \overline{\check{\mc{R}}}_d^{1}(y_{out};
\vec{x})\textrm{ rigid}} (-1)^{\check{\star}_d} ({
}_k\check{\mc{R}}_d^{1})_u([x_d], \ldots, [x_1]); \\$$ where the sign is given by $$\check{\star}_d = \deg(x_d) + \sum_{i} i \cdot \deg(x_i).$$ A codimension 1 analysis of the moduli spaces reveals:
\[BVcompactifiedequation1\] The following equation is satisfied: $$(-1)^n\delta_1 \check{\oc} + (-1)^n d \check{\oc}^{1} =
\oc^{S^1} + \check{\oc}^{1} b$$
The boundary of the 1-dimensional components of are covered by the rigid components of the following types of strata:
- spaces of maps with domain lying on the codimension 1 boundary of the moduli space, i.e., in -
- semi-stable breakings, namely those of the form $$\begin{aligned}
&{ }_1\overline{\check{\mc{R}}}_d^1(y_1; \vec{x}) \times
\overline{\mc{M}}(y_{out}; y_1) \\
&\overline{\mc{R}}^1(x_1; x) \times { }_1
\overline{\check{\mc{R}}}_d^1(y_{out}; \tilde{\vec{x}}) \end{aligned}$$
All together, this implies, up to signs, that $$\label{initialcheck1equation} \delta_1 \check{\oc}^1 + d \oc =
\oc^{S^1} + \check{\oc}^{1} b.$$ is of course a shorthand for saying, for a tuple of $d$ cyclically composable morphisms $x_d, \ldots, x_1$, that $$\begin{split}
\sum_{i =0}^1 \delta_i \check{\oc}^{k-i}_d(x_d, \ldots, x_1) &= \oc^{S^1}_d(x_d,
\ldots, x_1) + \sum_{i=1}^{k-1} \check{\oc}^{k,i,i+1}_d (x_d, \ldots,
x_1)\\ &+\sum_{i,s}(-1)^{\maltese_1^s} \check{\oc}^1_{d-i+1} (x_d, \ldots,
x_{s+i+1}, \mu^i(x_{s+i}, \ldots, x_{s+1}), x_s, \ldots, x_1)\\
&+\sum_{i,j} (-1)^{\sharp_j^i} \check{\oc}^1 (\mu^{i + j + 1}(x_i, \ldots,
x_1, x_d, \ldots, x_{d-j}), x_{d-j-1}, \ldots, x_{i+1}). \end{split}$$ Thus, it suffices to verify that the signs coming from the codimension 1 boundary are exactly those appearing in . (in particular, that the terms in, for instance, $\check{\oc}^1 b$ appear with the right sign).
Let us recall broadly how the signs are computed. For any operator $g$ defined above such as $\oc$, $\oc^{S^1}$, $\mu$, $d$, $\delta_1$ etc., we let $g_{\mathrm{ut}}$ denote the [*untwisted*]{} version of the same operator, e.g., the operator whose matrix coefficients come from the induced isomorphism on orientation lines, without any sign twists by the degree of the inputs. So for instance $\mu^d(x_d, \ldots, x_1) =
(-1)^{\sum_{i=1}^d i \deg(x_i)} \mu^d_{\mathrm{ut}}(x_d, \ldots, x_1)$ and so on. The methods described in [@Seidel:2008zr]\*[Prop. 12.3]{} and elaborated upon in [@Abouzaid:2010kx]\*[§C.3, Lemma 5.3]{} and [@ganatra1_arxiv]\*[§B]{}, when applied to the boundary of the 1-dimensional component of the moduli space of maps, $\overline{\check{\mc{R}}}_{d+1}^1(y_{out}, \vec{x}))$, imply the following signed equality: $$\label{untwistedequation}
\begin{split}
0 &= d_{\mathrm{ut}} \check{\oc}^1_{\mathrm{ut}} (x_d, \ldots, x_1) + (\delta_1)_{\mathrm{ut}} \check{\oc}_{\mathrm{ut}}(x_d , \ldots, x_1) \\
& - \oc^{S^1}_{\mathrm{ut}} (x_d, \ldots, x_1) + (-1)^{\mathfrak{f}_d} \check{\oc}^{1} b (x_d, \ldots, x_1)
\end{split}$$ where $$\mathfrak{f}_d := \sum_i (i+1) \deg(x_i) + \deg(x_d) = \check{\star}_d + \maltese_d - d.$$ is an auxiliary sign.
To explain this equation , we note first that the signs appearing in all terms but the last are simply induced by the boundary orientation on the moduli space of domains. The sign appearing in the first term also follows from a standard boundary orientation analysis for Floer cylinders, which we omit (but see e.g., [@Seidel:2008zr]\*[(12.19-12.20)]{} for a version close in spirit). The signs for the first two terms are also exactly as in Lemma \[weaks1actionSHlemma\]. Finally, in the last term, the sign $(-1)^{\mathfrak{f}_d} \check{\oc}^{1} b (x_d \otimes \cdots \otimes x_1)$ (compare [@Seidel:2008zr]\*[(12.25)]{} [@ganatra1_arxiv]\*[(B.59)]{}) appears as a cumulative sum of
- the sign twists which turn the untwisted operations $\check{\oc}^1_{\mathrm{ut}}$ and $\mu^s_{ut}$ into the usual operations $\check{\oc}^1$ and $\mu^s$;
- the Koszul sign appearing in the Hochschild differential $b$; and
- the boundary orientation sign appearing in the relevant (untwisted) term of $\check{\oc}^1 b$, for instance $\check{\oc}^1_{\mathrm{ut}}(x_d, \ldots, x_{n+m+1},
\mu^m_{ut}(x_{n+m}, \ldots, x_{n+1}) x_n, \ldots, x_1)$, which itself is as a sum of two different contributions:
(a) the comparison between the boundary (of the chosen) orientation and the product (of the chosen orientation) on the moduli of [*domains*]{} and
(b) Koszul reordering signs, which measure the signed failure of the method of orienting the moduli of maps (in terms of orientations of the domain and orientation lines of inputs and outputs) to be compatible with passing to boundary strata.
See [@Seidel:2008zr]\*[(12d)]{} for more details in the case of the $\ainf$ structure, and [@Abouzaid:2010kx]\*[§C]{}, [@ganatra1_arxiv]\*[§C]{} for the case of these computations for the open-closed map. We note in particular that the forgetful map $F_1: { }_1 \check{\mc{R}}_d^1
\ra \check{\mc{R}}_d^1$, which forgets the point $p_1$ (and changes the direction of the asymptotic marker to point at $z_d$) has [*complex oriented fibers*]{} (in which just the marked point $p_1$ varies). So the boundary analysis of these “$\check{\oc^1} \circ b$” strata appearing here is identical to the analysis strata appearing in [@Abouzaid:2010kx] and [@ganatra1_arxiv] for the “$\oc\circ b$” strata, which is why we have not repeated it here.
Multiplying all terms of by $(-1)^{\check{\star}_d + \maltese_d -d + 1}$, and noting that, for instance, $\maltese_d -d + 1 + n -2 = \deg(\check{\oc}^1(x_d \otimes \cdots \otimes
x_1))$, so $$\begin{split}
(-1)^{\check{\star}_d + \maltese_d - d + 1} (\delta_1)_{ut}\check{\oc}^1_{\mathrm{ut}}(x_d, \ldots, x_1) &= (-1)^{\deg(\check{\oc}^1(x_d, \ldots, x_1)) - n} (\delta_1)_{ut} (-1)^{\check{\star}_d} \check{\oc}^1_{\mathrm{ut}}(x_d, \ldots, x_1)\\
&= \delta_1 \check{\oc}^1(x_d, \ldots, x_1),
\end{split}$$ (and similarly for the $d \circ \oc^1$ term), it follows that $$\begin{split}
0 &= (-1)^{n}\delta_1 \check{\oc}(x_d, \ldots, x_1) + (-1)^n d \check{\oc}^1 (x_d, \ldots, x_1)\\
& - \check{\oc}^{1} b (x_d, \ldots, x_1) - (-1)^{\check{\star}_d + \maltese_d - d + 1}\oc^{S^1}_{\mathrm{ut}} (x_d, \ldots, x_1),
\end{split}$$ but $\check{\star}_d + \maltese_d - d + 1 = \clubsuit_d$, and hence the last term above is $-\oc^{S^1}(x_d, \ldots, x_1)$ as desired.
The “sector decomposition” performed in Proposition \[ocS1Brelation\] which compares $\oc^{S^1}$ to $\hat{\oc} \circ B^{nu} \circ \iota$, along with Proposition \[BVhomotopyeqn\], immediately implies the result.
The main construction {#sec:equivariance}
---------------------
We now turn to the definition of the (closed) morphism of $S^1$-complexes, and the proof of Theorem \[thm:mainresult1\] and Corollary \[cor:cyclicOC\]. The required data takes the form $$\widetilde{\oc} = \bigoplus_{k\geq 0}
\overline{\K[\Lambda]/\Lambda^2}^{\otimes k}
\otimes \r{CH}_*^{nu}(\f) \ra CF^*(M)[n]$$ which is equivalent, as recalled in §\[circleactionsubsection\] to defining the collection of maps $\widetilde{\oc} = \{\oc^k\}_{k \geq 0}$, or $u$-linearly (see §\[sec:ulinear\]) $\widetilde{\oc} = \sum_{k=0}^\infty \oc^k u^k$, where $$\oc^k = (\check{\oc}^{k} + \hat{\oc}^{k}):= \widetilde{\oc}^{k|1}(\Lambda, \ldots, \Lambda, -):
\r{CH}_*^{nu}(\f) \ra CF^{*+n-2k}(M).$$ (recall from §\[circleactionsubsection\] that $\K[\Lambda]/\Lambda^2$ is our small model for $C_{-*}(S^1)$ and $S^1$-complexes are by definition strictly unital $\ainf$ modules over $\K[\Lambda]/\Lambda^2$). By definition, the case $k=0$ is already covered: $$\begin{split}
\check{\oc}^{0} &:= \check{\oc}\\
\hat{\oc}^{0} &:= \hat{\oc}
\end{split}$$ To handle the general case ($k \geq 0$), we will associate operations to, for each $d$, compactifications of three moduli spaces of domains, in the following order: $$\begin{aligned}
\label{checkequivariant} { }_k \check{\mc{R}}_{d+1}^1\\
\label{S1equivariant} { }_k \mc{R}_{d}^{S^1}\\
\label{hatequivariant} { }_k \hat{\mc{R}}_d^1;\end{aligned}$$ The moduli space will induce an auxiliary operation useful for the proof, whereas and will lead to the desired operations. For $k = 0$, these moduli spaces are simply $\check{\mc{R}}_{d+1}^1$, $\mc{R}_{d}^{S^1}$, and $\hat{\mc{R}}_d^1$ as defined earlier. Inductively, we will construct, and study operations from and simultaneously, and then finally construct . Using these moduli spaces, we will construct the maps $\check{\oc}^{k}$, $\hat{\oc}^k$ and an auxiliary operation $\oc^{S^1, k}$ (which we compare to $\hat{\oc}^{k-1} \circ B^{nu}$ in Proposition \[ocks1Brelation\] below), and then prove that:
\[cyclicOCequations\] The following equations hold, for each $k \geq 0$: $$\begin{aligned}
(-1)^n \sum_{i \geq 0}^k \delta_i \check{\oc}^{k-i} &= \hat{\oc}^{k-1} B^{nu} + \check{\oc}^{k} b\\
(-1)^n \sum_{i \geq 0}^k \delta_i \hat{\oc}^{k-i} &= \hat{\oc}^{k} b' + \check{\oc}^k (1 - t).
\end{aligned}$$ All at once, denoting by $\oc^k =(\check{\oc}^{k} + \hat{\oc}^{k})$ and $\widetilde{\oc} = \sum_{i=0}^\infty \oc^i u^i$, $\delta_{eq} = \sum_{j=0}^\infty \delta_j^{CF}u^j$, and $b_{eq} = b^{nu} + u B^{nu}$ as in § \[sec:ulinear\], we have that $$(-1)^n \delta_{eq} \circ \widetilde{\oc} = \widetilde{\oc} \circ b_{eq}.$$
This will also directly imply our main Theorems, as we will spell out at the bottom of this subsection.
The space is the moduli space of discs with with $d+1$ positive boundary marked points $z_0, \ldots, z_d$ labeled in counterclockwise order, 1 interior negative puncture $z_{out}$ equipped with an asymptotic marker, and $k$ additional interior marked points $p_1, \ldots, p_k$ (without an asymptotic marker), marked as [*auxiliary*]{}. Also, choosing a representative of an element this moduli space which fixes $z_0$ at $1$ and $z_{out}$ at 0 on the unit disc, the $p_i$ should be [*strictly radially ordered*]{}; that is, $$\label{ordered} 0 < |p_1| < \cdots < |p_k| < \frac{1}{2}.$$ Using the above representative, one can talk about the [*angle*]{}, or [*argument*]{} of each auxiliary interior marked point, $$\theta_i:= \arg(p_i).$$ We require that with respect to the above representative, $$\label{pointingcondition} \textrm{the asymptotic marker on
$z_{out}$ points in the direction $\theta_1$ (or towards $z_0$ if
$k=0$).}$$ (equivalently one could define $\theta_{k+1} = 0$, so that $\theta_1$ is always defined). For every representative $S \in { }_k \check{\mc{R}}_{d+1}^1$, $$\label{checknegcylindricalend}
\begin{split}
&\textrm{ fix a negative cylindrical end around $z_{out}$ not containing any $p_i$, compatible with the }\\
&\textrm{direction of the asymptotic marker, or {\it equivalently compatible with the angle $\theta_1$}.}
\end{split}$$ The second moduli space is the moduli space of discs with with $d$ positive boundary marked points $z_1, \ldots, z_{d}$ labeled in counterclockwise order, 1 interior negative puncture $z_{out}$ equipped with an asymptotic marker, and $k+1$ additional interior marked points $p_1, \ldots,
p_k, p_{k+1}$ (without an asymptotic marker), marked as [*auxiliary*]{}. Choosing a representative of an element this moduli space which fixes $z_0$ at $1$ and $z_{out}$ at 0 on the unit disc, the $p_i$ should be [*strictly radially ordered*]{} and [$p_{k+1}$ should lie on the circle of radius $\frac{1}{2}$]{}; that is, $$\label{orderedS1} 0 < |p_1| < \cdots < |p_k| < |p_{k+1}| =
\frac{1}{2}.$$ This asymptotic marker for representative satisfies condition . Abstractly we have that $ {
}_k \mc{R}_{d}^{S^1} \cong S^1 \times { }_k \check{\mc{R}}_{d+1}^1$, where the $S^1$ parameter is given by the position of $p_{k+1}$.
The compactification of is a real blow-up of the ordinary Deligne-Mumford compactification, in the sense of [@Kimura:1995fk] (see [@Seidel:2010uq] for a first discussion in the context of Floer theory).
The result of this discussion is that the codimension 1 boundary of the compactified check moduli space ${ }_k\overline{ \check{\mc{R}}}_{d+1}^1$ is covered by the images of the natural inclusions of the following strata: $$\begin{aligned}
\label{checkstratum1} \overline{\mc{R}^s} \times { }_k
\overline{\check{\mc{R}}}_{d-s + 2}^1 \\
\label{checkstratum2} { }_s\overline{ \check{\mc{R}}}_{d}^1 \times \overline{\mc{M}}_{k-s}\\
\label{checkstratum3}{ }_{k-1} \overline{\check{\mc{R}}}_{d+1}^{S^1}\\
\label{checkstratum4}{ }_k^{i,i+1} \overline{\check{\mc{R}}}_{d+1}^1\end{aligned}$$ The strata -, in which $|p_k| = \frac{1}{2}$ and $|p_i| = |p_{i+1}|$ respectively, describe the boundary loci of the ordering condition and hence come equipped with a natural manifold with corners structure. The strata - have manifold with corners structure given by standard local gluing maps using fixed choices of strip-like ends near the boundary. For this is standard, and for , the local gluing map uses the cylindrical ends and (in other words, one rotates the $r$-pointed angle cylinder by an amount commensurate to the angle of the first marked point $p_{k-s+1}$ on the disk before gluing).
![\[fig:ock\_check\]A representative of an element of the moduli space ${ }_3\check{\mc{R}}^1_{5}$, along with two of the most significant types of degenerations: in the first, the final auxiliary point $|p_3|$ reaches the circle of radius $\frac{1}{2}$, and one obtains an element of ${ }_2\check{\mc{R}}^{S^1}_{5}$. In the second, one of the auxiliary points, $|p_2|$ tends to zero, forcing $p_1$ and $p_2$ into splitting off a copy of $\mathcal{M}_2$. In terms of the cylindrical coordinates near $y_-$, the distance between the height of $p_2$ and that of $p_3$ tends to $\infty$. ](ock_check.pdf)
Associated to the stratum where $p_i$ and $p_{i+1}$ have coincident magnitudes, there is a forgetful map $$\label{picheck} \check{\pi}_i: { }_k^{i,i+1}
\overline{\check{\mc{R}}}_{d+1}^1 \ra { }_{k-1}
\overline{\check{\mc{R}}}_{d+1}^1$$ which simply forgets the point $p_{i+1}$. Since the norm of $p_{i+1}$ and $p_i$ agree on this locus, this amounts to forgetting the argument of $p_{i+1}$ (in particular, the fibers of $\check{\pi}_i$ are one-dimensional).
The $S^1$-moduli space , is abstractly $S^1 \times { }_k
\check{\mc{R}}_{d+1}^1$, and similarly we model its compactification is abstractly by $S^1 \times { }_k\overline{ \check{\mc{R}}}_{d+1}^1$. However, it is preferable to give an explicit description of the boundary strata, which is covered in codimension 1 by the following strata: $$\begin{aligned}
\label{s1stratum1} \overline{\mc{R}^s} \times { }_k \overline{\mc{R}}_{d-s
+ 2}^{S^1} \\ \label{s1stratum2} { }_s\overline{ \mc{R}}_{d}^{S^1} \times
\overline{\mc{M}}_{k-s}\\ \label{s1stratum3}{
}_{k}^{k,k+1}\overline{\mc{R}}_{d+1}^{S^1}\\ \label{s1stratum4}{
}_k^{i,i+1} \overline{\mc{R}}_{d+1}^{S^1}. \end{aligned}$$ Here, and are just versions of the degenerations and , in which a collection of boundary points bubbles off, or a collection of auxiliary points convergest to $z_{out}$ and bubbles off (the fact that the latter occurs in codimension 1 is part of the “real blow-up phenomenon” already discussed). The stratum , is the locus where $|p_i| =
|p_{i+1}|$, for $i < k$, and the stratum , is the locus where $|p_k| = \frac{1}{2} = |p_{k+1}|$.
As in , on strata - where $p_i$ and $p_{i+1}$ have coincident magnitudes, define the map $$\label{piS1} \pi_i^{S^1}: { }_k^{i,i+1}
\overline{\mc{R}}_{d+1}^{S^1} \ra { }_{k-1} \overline{\mc{R}}_{d+1}^{S^1}$$ to be the one forgetting the point $p_{i+1}$ (so as before, this map has one-dimensional fibers), including the case $i=k$ which is .
For an element $S \in { }_{k} \overline{\mc{R}}_{d}^{S^1}$, we say that $p_{k+1}$ [*points at a boundary point $z_i$*]{} if, for any unit disc representative of $S$ with $z_{out}$ at the origin, the ray from $z_{out}$ to $p_{k+1}$ intersects $z_i$. The locus where $p_{k+1}$ points at $z_i$ is denoted $${ }_k\overline{\mc{R}}_{d+1}^{S^1_i},$$ Similarly, we say that $p_{k+1}$ [*points between $z_i$ and $z_{i+1}$*]{} (modulo $d$, so including the case [*$z_d$ and $z_1$*]{}) if for such a representative, the ray from $z_{out}$ to $p_{k+1}$ intersects the portion of $\partial S$ between $z_i$ and $z_{i+1}$. The locus where $p_{k+1}$ points between $z_i$ and $z_{i+1}$ is denoted $${
}_k\overline{\mc{R}}_{d+1}^{S^1_{i,i+1}}.$$
As before in , there is a free and properly discontinuous $\Z_{d}$-action $$\label{labelpermuteequivariant} \kappa: {
}_k\overline{(\mc{R}^1_d)^{S^1}} \ra { }_k\overline{(\mc{R}^1_d)^{S^1}}$$ which cyclically permutes the labels of the boundary marked points; as before, $\kappa$ changes the label $z_i$ to $z_{i+1}$ for $i < d$, and $z_d$ to $z_1$ (compare Lemma \[freediscontinuous\]).
Finally, we come to the third moduli space , the moduli space of discs with with $d+1$ positive boundary marked points $z_f, z_1,
\ldots, z_{d}$ labeled in counterclockwise order, 1 interior negative puncture $z_{out}$ equipped with an asymptotic marker, and $k$ additional interior marked points $p_1, \ldots, p_k$ (without an asymptotic marker), marked as [*auxiliary*]{}, staisfying a [*strict radial ordering*]{} condition as before: for any representative element with $z_f$ fixed at 1 and $z_{out}$ at 0, we require to hold, as well as condition . The boundary marked point $z_f$ is also marked as auxiliary, but abstractly, we see that ${ }_k\hat{\mc{R}}_d^1
\cong { }_k\check{\mc{R}}_{d+1}^1$.
In codimension 1, the compactification $ { }_k \overline{\hat{\mc{R}}}_{d}^1$ has boundary covered by inclusions of the following strata: $$\begin{aligned}
\label{hatstratum1} \overline{\mc{R}^s} \times { }_k
\overline{\hat{\mc{R}}}_{d-s + 2}^1 \\
\label{hatstratum2} \overline{\mc{R}}^{m,f_k}\times_{d-m+1} {
}_k\overline{\check{\mc{R}}}^1_{d-m+1} \ \ \ 1 \leq k \leq m\\
\label{hatstratum3} { }_s\overline{ \hat{\mc{R}}}_{d}^1 \times
\overline{\mc{M}}_{k-s}\\
\label{hatstratum4}{ }_{k-1} \overline{\hat{\mc{R}}}_{d}^{S^1}\\
\label{hatstratum5}{ }_k^{i,i+1} \overline{\hat{\mc{R}}}_{d}^1 \end{aligned}$$ Once more, on strata where $p_i$ and $p_{i+1}$ have coincident magnitudes, define the map $$\label{pihat}
\hat{\pi}_i: { }_k^{i,i+1} \overline{\hat{\mc{R}}}_{d}^1 \ra { }_{k-1}
\overline{\hat{\mc{R}}}_{d}^1.$$ to be the one forgetting the point $p_{i+1}$ (so again, this map has one-dimensional fibers). On the stratum , which is the locus where $|p_{k}| =
\frac{1}{2}$, there is also a map of interest $$\label{hatboundaryforget}
\hat{\pi}_{boundary}: { }_{k-1}\overline{\hat{\mc{R}}}_{d}^{S^1} \ra { }_{k-1} \overline{\mc{R}}_d^{S^1}$$ which forgets the position of the auxiliary boundary point $z_f$.
![\[fig:ock\_hat\]A representative of an element of the moduli space ${ }_4\hat{\mc{R}}^1_{4}$.](ock_hat.pdf)
Denote by ${ }_k{\mc{R}}_d^{1,free}: = { }_k {\mc{R}}_d^{S^1_{d,1}}$ to be the sector of the moduli space ${ }_k\mc{R}^{S^1}_d$ where $p_{k+1}$ points between $z_d$ and $z_1$. The [*auxiliary-rescaling map*]{} $$\label{equivariantforgetful}
\pi_f: { }_k \hat{\mc{R}}_d^1 \ra { }_k{\mc{R}}_d^{1,free},$$ (our replacement of the “forgetful map”) can be described as follows: given a representative $S$ in $ { }_k
\hat{\mc{R}}_d^1$ with $z_{out}$ fixed at the origin, there is a unique point $p$ with $|p|= \frac{1}{2}$ between $z_{out}$ and $z_f$. $\pi_f(S)$ is the element of ${ }_k\mc{R}^{S^1}_d$ with $p_{k+1}$ equal to this point $p$ and with $z_f$ deleted. Of course, $z_f$ is not actually forgotten, because it is determined by the position of $p_{k+1}$. In particular is a diffeomorphism. We extend this map to a map $\overline{\pi}_f$ from the compactification ${ }_k \overline{\hat{\mc{R}}}_d^1$ as in §\[sec:nonunital\], by putting the auxiliary point $z_f$ back in, eliminating any component which is not main or secondary which has only one (non-auxiliary) boundary marked point $q$, and by labeling the positive marked point below this component by $q$.
We orient the moduli spaces - as follows: picking, on a slice of the automorphism action which fixes the position of $z_d$ at 1 and $z_{out}$ at 0, the volume forms $$\begin{aligned}
\label{checkorientation} &-r_1\cdots r_k dz_1\wedge dz_2 \wedge \cdots
\wedge dz_{d-1} \wedge dr_1 \wedge d\theta_1 \wedge \cdots \wedge dr_{k}
\wedge d\theta_k \\
\label{s1orientation} & r_1 \cdots r_k dz_1 \wedge dz_2
\wedge \cdots \wedge dz_{d-1} \wedge d\theta_{k+1} \wedge dr_1 \wedge
d\theta_1 \wedge \cdots \wedge dr_k \wedge d\theta_k \\
\label{hatorientation} &r_1 \cdots r_k dz_1 \wedge dz_2 \wedge \cdots
\wedge dz_{d-1} \wedge dz_f \wedge dr_1 \wedge d\theta_1 \wedge \cdots
\wedge dr_k \wedge d\theta_k. \end{aligned}$$ Above, $(r_i,\theta_i)$ denote the polar coordinate positions of the point $p_i$ (we could equivalently use Cartesian coordinates $(x_i, y_i)$ and substitute $dx_i \wedge dy_i$ for every instance of $r_i dr_i \wedge
d\theta_i$, but polar coordinates are straightforwardly compatible with the boundary stratum where $|p_k| = \frac{1}{2}$).
A [*Floer datum*]{} on a stable disc $S$ in ${ }_k
\overline{\check{\mc{R}}}_{d+1}^1$ or a stable disc $S$ in ${
}_k\overline{\mc{R}}_{d}^{S^1}$ is simply a Floer datum for $S$ in the sense of §\[floeropenclosed\]. A [*Floer datum*]{} on a stable disc $S \in { }_k \overline{\hat{\mc{R}}}_d^1$ is a Floer datum for $\overline{\pi}_f(S)$.
Again we will make a system of choices of Floer data for the above moduli spaces. A [*Floer datum for the cyclic open-closed map*]{} is an inductive sequence of choices, for every $k \geq 0$ and $d \geq 1$, of Floer data for every representative $S_0 \in { }_k
\overline{\check{\mc{R}}}_{d+1}^1$, $S_1 \in {
}_k\overline{\mc{R}}_{d}^{S^1}$, and $S_2 \in { }_k
\overline{\hat{\mc{R}}}_d^1$, varying smoothly in $S_0$, $S_1$ and $S_2$, which satisfies the usual consistency condition: the choice of Floer datum on any boundary stratum should agree with the previously inductively chosen datum along any boundary stratum for which (is possibly a product of moduli spaces for) we have already inductively picked data. Moreover, this choice should satisfy a series of additional requirements: First, for $S_0 \in { }_k \overline{\check{\mc{R}}}_{d+1}^1$, $$\begin{aligned}
\label{checkcompat1} &\textrm{At a boundary stratum of the form
\eqref{checkstratum4}, the Floer datum for $S_0$ is} \\
\nonumber &\textrm{equivalent to the one pulled back from ${
}_{k-1} \overline{\check{\mc{R}}}_{d+1}^1$ via the forgetful map
$\check{\pi}_i$}.
\end{aligned}$$ Next, for $S_1 \in { }_k\overline{\mc{R}}_{d}^{S^1}$, $$\begin{aligned}
\label{s1compat1} &\textrm{On the codimension-1 loci
$\overline{ { }_k(\mc{R}^1_d)^{S^1}_i}$
where $p_{k+1}$ points at $z_i$, the Floer datum }\\
\nonumber &\textrm{should agree with the pullback by $\tau_i$ of the existing Floer datum for the}\\
\nonumber &\textrm{open-closed map. }\\
\label{s1compat2} &\textrm{The Floer datum should be
$\kappa$-equivariant, where $\kappa$ is the map
\eqref{labelpermuteequivariant}. } \\
&\textrm{At a boundary stratum of the form \eqref{s1stratum3} or
\eqref{s1stratum4}, the Floer datum for $S_1$ is} \\
\nonumber &\textrm{conformally equivalent to the one pulled back from
${ }_{k-1} \overline{\mc{R}}_{d+1}^{S^1}$ via the forgetful map
${\pi}_i^{S^1}$ }.
\end{aligned}$$ Finally, for $S_2 \in { }_k \overline{\hat{\mc{R}}}_d^1$, $$\begin{aligned}
\label{hatcompat1} &\textrm{The choice of Floer
datum on strata containing $\mc{R}^{d,f_i}$
components should be}\\
&\nonumber \textrm{constant along fibers of the forgetful map
$\mc{R}^{d,f_i} \ra \mc{R}^{d-1}$. }\\
\label{hatcompat2} &\textrm{The Floer datum on
the main component $(S_2)_0$ of $\pi_f(S_2)$
should coincide with the}\\
\nonumber &\textrm{Floer datum chosen on $(S_2)_0 \in { }_k
\mc{R}_d^{1,free} \subset { }_k \mc{R}_d^{S^1}$}.\\
\label{hatcompat3} &\textrm{At a boundary stratum
of the form \eqref{hatstratum4}, the Floer
datum on the main component} \\
\nonumber &\textrm{of $S_2$ is conformally equivalent to the
one pulled back from ${ }_{k} \overline{\mc{R}}_{d}^{S^1}$ via the forgetful} \\
\nonumber &\textrm{map $\hat{\pi}_{boundary}$ }.\\
\label{hatcompat4} &\textrm{At a boundary stratum of the form
\eqref{hatstratum5}, the Floer datum for $S_2$ is conformally} \\
\nonumber &\textrm{equivalent to the one pulled back from ${
}_{k-1} \overline{\hat{\mc{R}}}_{d+1}^{1}$ via the forgetful map
$\hat{\pi}_i$ }.
\end{aligned}$$ The above system of requirements can be split into three broad categories: the first type concerns the compatibility with forgetful maps of Floer data along the lower strata which were not previously constrained, the second type concerns the equivariance (under a free properly discontinuous action) of the Floer data on ${ }_k\overline{\mc{R}}_{d}^{S^1}$ as well as the relationship between the Floer datum chosen here and the ones chosen on $\overline{\check{\mc{R}}}_{d+1}^1$ and ${ }_k \overline{\hat{\mc{R}}}_d^1$.
A Floer datum for the cyclic open-closed map exists.
Since the choices of Floer data at each stage are contractible, this follows from the straightforward verification that, for a suitably chosen inductive order on strata, the conditions satisfied by the Floer data at various strata do not contradict each other. We use the following inductive order: first, say we’ve chosen a Floer datum for the $\ainf$ structure as in §\[subsec:fukaya\] along with a BV compatible Floer datum for the non-unital open-closed map following §\[ocs1sec\]. In particular, we have chosen Floer data for the the moduli spaces $\overline{\mc{R}}^{d,f_i}$ (per Appendix \[discsforgotten\]), ${ }_0 \check{\overline{\mc{R}}}_d$, the auxiliary moduli space ${ }_0 \overline{\mc{R}}_d^{S^1}$ and (using the conditions above) induced a particular choice of Floer datum on ${ }_0 \hat{\overline{\mc{R}}}_d$. Next, inductively assuming we have made all choices at level $k-1$ $(k>0)$, we first choose Floer data for ${ }_k \check{\overline{\mc{R}}}_d$ for each $d$, then ${
}_k\overline{\mc{R}}_d^{S^1}$ for each $d$ (by pulling back a choice of Floer datum on the quotient by $\kappa$ in order to satisfy the equivariance condition), and finally note that a choice is fixed for ${
}_k \hat{\overline{\mc{R}}}_d$ by the above constraints.
Fixing a Floer datum for the cyclic open-closed map, we obtain, for any $d$-tuple of Lagrangians $L_0, \ldots, L_{d-1}$, and asympotic conditions $$\begin{split} \vec{x} &= (x_{d},
\ldots, x_1),\ x_i \in
\chi(L_{i},L_{i+1 \mod d})\\
y_{out} &\in \mc{O} \end{split}$$ Gromov-Floer compactified moduli spaces $$\begin{aligned}
\label{cycliccompactifiedspace1}{ }_k \overline{\check{\mc{R}}}_{d+1}^1(y_{out},
\vec{x})\\
\label{cycliccompactifiedspace2}{ }_k\overline{\mc{R}}_{d}^{S^1}(y_{out},
\vec{x})\\
\label{cycliccompactifiedspace3}{ }_k
\overline{\hat{\mc{R}}}_d^1(y_{out},
\vec{x}) \end{aligned}$$ of maps into $M$ from an arbitrary element $S$ of the moduli spaces , , and respectively (or rather from $\pi_f(S)$ in the case of ) satisfying Floer’s equation using the Floer datum chosen for the given $S$ as in with asymptotics and Lagrangian boundary conditions as in (again with the modifications as in Remark \[wrappedFloerdatumOC\] for wrapped Fukaya categories, involving Liouville scaling). The virtual dimension of each component of these moduli spaces coincides (mod 2 or exactly, depending on whether we are $\Z/2$ or $\Z$-graded) with $$\begin{aligned}
\label{dimcycliccompactifiedspace1}
\textrm{(for ${ }_k\overline{\check{\mc{R}}}_{d+1}^1(y_{out}, \vec{x})$) }&
\deg(y_{out}) - n + d -1 - \sum_{i=0}^d \deg(x_i) + 2k;\\
\label{dimcycliccompactifiedspace2} \textrm{(for ${ }_k\overline{\mc{R}}_{d}^{S^1}(y_{out}, \vec{x})$) }& \deg(y_{out}) - n
+ d - \sum_{i=0}^d \deg(x_i) + 2k; \\
\label{dimcycliccompactifiedspace3}\textrm{(for ${ }_k
\overline{\hat{\mc{R}}}_d^1(y_{out}, \vec{x})$) }& \deg(y_{out}) - n + d
- \sum_{i=0}^d \deg(x_i) + 2k. \end{aligned}$$
For generic choices of Floer data, the Gromov-Floer compactifications of the components of virtual dimension $\leq 1$ of - are compact manifolds-with-boundary of the expected dimension.
For rigid elements $u$ in the moduli spaces - , (which occurs for asymptotics $(y,
\vec{x})$ satisfying = 0, = 0, or = 0 respectively), the orientations , , and [@Abouzaid:2010kx]\*[Lemma C.4]{} induce isomorphisms of orientation lines $$\begin{aligned}
\label{cyclicorientationline1}
({ }_k \check{\mc{R}}_{d+1}^1)_u: o_{x_d} \otimes \cdots \otimes o_{x_1}
\ra o_y\\
\label{cyclicorientationline2}({ }_k\mc{R}_{d}^{S^1})_u:
o_{x_d} \otimes \cdots \otimes o_{x_1} \ra o_y\\
\label{cyclicorientationline3}({ }_k \hat{\mc{R}}_d^1)_u: o_{x_d}
\otimes \cdots \otimes o_{x_1} \ra o_y. \end{aligned}$$ Summing the application of these isomorphisms over all rigid $u$ (or “counting rigid elements”) defines the $|o_{y_{out}}|_\K$ component of three families of operations $\check{\oc}^k$, ${\oc}^{S^1,k}$, $\hat{\oc}^k$ up to a sign twist specified below: define $$\begin{aligned}
\check{\oc}^{k}([x_{d}], \ldots, [x_1]) &:=
\sum_{u \in { }_k \overline{\check{\mc{R}}}_d^{1}(y_{out};
\vec{x}) \textrm{ rigid}} (-1)^{\check{\star}_d} ({
}_k\check{\mc{R}}_d^{1})_u([x_d], \ldots, [x_1]); \\
\oc^{S^1,k}([x_{d}], \ldots, [x_1]) &:=
\sum_{u \in { }_k \overline{\mc{R}}_d^{S^1}(y_{out}; \vec{x}) \textrm{ rigid}}
(-1)^{\star_d^{S^1}} ({ }_k\mc{R}_d^{S^1})_u([x_d],
\ldots, [x_1]);\\
\hat{\oc}^{k}([x_{d}], \ldots, [x_1]) &:=
\sum_{u \in { }_k \overline{\hat{\mc{R}}}_d^{1}(y_{out}; \vec{x}) \textrm{ rigid}}
(-1)^{\hat{\star}_d} ({ }_k\hat{\mc{R}}_d^{1})_u([x_d], \ldots,
[x_1]). \end{aligned}$$ where the signs are given by $$\begin{aligned}
\check{\star}_d &= \deg(x_d) + \sum_{i} i
\cdot \deg(x_i).\\ \star_d^{S^1} = \clubsuit_d &=
\sum_{i=1}^d (i+1) \cdot \deg(x_i) + \deg(x_d) + d-1 = \check{\star}_d + \maltese_d - 1\\
\hat{\star}_d &= \sum_{i} i \cdot \deg(x_i).\end{aligned}$$ A codimension 1 analysis of the moduli spaces and reveals:
The following equations hold for each $k\geq 0$: $$\begin{aligned}
(-1)^n \sum_{i = 0}^k \delta_i \check{\oc}^{k-i} &= \oc^{S^1,k-1} + \check{\oc}^{k} b\\
(-1)^n \sum_{i = 0}^k \delta_i \hat{\oc}^{k-i} &= \hat{\oc}^{k} b' + \check{\oc}^k (1 - t).\end{aligned}$$
The boundary of the 1-dimensional components of are covered by the (rigid components of) the following types of strata:
- spaces of maps with domain lying on the codimension 1 boundary of the moduli space, i.e., in -
- semi-stable breakings, namely those of the form $$\begin{aligned}
&{ }_k\overline{\check{\mc{R}}}_d^1(y_1; \vec{x}) \times
\overline{\mc{M}}(y_{out}; y_1) \\
&\overline{\mc{R}}^1(x_1; x) \times
{ }_k\overline{\check{\mc{R}}}_d^1(y_{out}; \tilde{\vec{x}})
\end{aligned}$$
All together, this implies, up to sign, that $$\label{initialcheckequation} (-1)^n \sum_{i = 0}^k \delta_i \check{\oc}^{k-i} =
\oc^{S^1,k-1} + \check{\oc}^{k} b + \sum_{i=1}^{k-1}\check{\oc}^{k,i,i+1},$$ where $\check{\oc}^{k,i,i+1}$ is an operation corresponding with some sign twist to . is of course a shorthand for saying, for a tuple of $d$ cyclically composable morphisms $x_d, \ldots, x_1$, that $$\begin{split}
\sum_{i = 0}^k \delta_i \check{\oc}^{k-i}_d(x_d, \ldots, x_1) &=
\oc^{S^1,k-1}_d(x_d, \ldots, x_1) + \sum_{i=1}^{k-1}
\check{\oc}^{k,i,i+1}_d (x_d, \ldots, x_1)\\
&+\sum_{i,s}(-1)^{\maltese_1^s} \check{\oc}^k_{d-i+1} (x_d, \ldots,
x_{s+i+1}, \mu^i(x_{s+i}, \ldots, x_{s+1}), x_s, \ldots, x_1)\\
&+\sum_{i,j} (-1)^{\sharp_j^i} \check{\oc}^k (\mu^{i + j + 1}(x_i,
\ldots, x_1, x_d, \ldots, x_{d-j}), x_{d-j-1}, \ldots, x_{i+1}).
\end{split}$$
We first note that in fact the operation $\check{\oc}^{k,i,i+1} = \sum_d
\check{\oc}_d^{k,i,i+1}$ is zero, because by condition , the Floer datum chosen for elements $S$ in are constant along the one-dimensional fibers of $\check{\pi}_i$. Hence, elements of the moduli space with source in are never rigid (see Lemma \[weaks1actionSHlemma\] for an analogous and more detailed explanation).
Thus, it suffices to verify that the signs coming from the codimension 1 boundary are exactly those appearing in . We can safely ignore studying any signs for the vanishing operations such as $\hat{\oc}^{k,i,i+1}$. The remaining sign analysis is exactly as in Proposition \[BVcompactifiedequation1\]; more precisely note that the forgetful map $\check{F}_k: { }_k \check{\mc{R}}_d^1 \ra { }_1
\check{\mc{R}}_d^1$ which forgets $p_1, \ldots, p_{k-1}$ has complex oriented fibers, and in particular (since the marked points $p_i$ contribute complex domain orientations and do not introduce any new orientation lines) the sign computations sketched in Proposition \[BVcompactifiedequation1\] carry over for any strata whose domain is pulled back from a boundary stratum of ${ }_1
\check{\mc{R}}_d^1$ (in turn, as described in Proposition \[BVcompactifiedequation1\], the sign computations for ${ }_1
\check{\mc{R}}_d^1$ largely reduce to those for ${ }_0 \check{\mc{R}}_d^1$. This verifies .
Similarly, for the hat moduli space, an analysis of the boundary of 1-dimensional moduli spaces of maps tells us, up to sign verification: $$\label{initialhatequation}
(-1)^n\sum_{i = 0}^k \delta_i \hat{\oc}^{k-i} = \hat{\oc}^{k} b' +
\check{\oc}^k (1 - t) + \hat{\oc}^{k,k,k+1} +
\sum_{i=1}^{k-1}\hat{\oc}^{k,i,i+1},$$ where $\hat{\oc}^{k,k,k+1}$ is an operation corresponding with some sign twist to and $\hat{\oc}^{k,i,i+1}$ is an operation corresponding with some sign twist to . The conditions - similarly imply that $\hat{\oc}^{k,k,k+1}$ and $\hat{\oc}^{k,i,i+1}$ are zero, so it is not necessary to even establish what the signs for these terms are.
To verify signs for , we apply the principle discussed in the proof of Lemma \[ochatchainold\], in which by treating the auxiliary boundary marked point $z_f$ as possessing a “formal unit element asymptotic constraint $e_+$,” therefore viewing $\hat{\oc}^k(x_d \otimes \cdots
\otimes x_)$ formally as “$\hat{\oc}^k(e^+ \otimes x_d
\otimes \cdots \otimes x_1)$” the signs for the equations applied to strings of length $d$ $(x_d \otimes
\cdots \otimes x_1)$ follow follow from the sign computations for $\check{\oc}$ applied to strings of length $d+1$ $(e^+ \otimes x_d \cdots \otimes \cdots
x_1)$ (we note that this analysis applies to the term $\hat{\oc}^{k,k,k+1}$ as well, which is the hat version of $\oc^{S^1,k}$; however, the former operation happens to be zero because extra symmetries imply the moduli space controlling this operation is never rigid).
Next, by decomposing the moduli space ${ }_k \mc{R}_d^{S^1}$ into sectors, we can write the auxiliary operation $\oc^{S^1,j}$ in terms of $\hat{\oc}^j$ and the Connes’ $B$ operator:
\[ocks1Brelation\] As chain-level operations, $$\oc^{S^1,k} = \hat{\oc}^k \circ B^{nu}.$$
The proof directly emulates Proposition \[ocS1Brelation\], and as such we will give fewer details. We begin by defining operations $$\hat{\oc}^{k}_{d,\tau_i}$$ associated to various “sectors” of the $k+1$st marked point $p_{k+1}$ of ${ }_k \mc{R}_d^{S^1}$, for $i \in \Z/d\Z$. Once more, to gain better control of the geometry of these sectors in the compactification (when the sector size can shrink to zero), we pass to an alternate model for the compactification: define $${ }_k \mc{R}_{d,\tau_i}^{1}$$ to be the abstract moduli space of discs with $d+1$ boundary punctures, $z_1, \ldots, z_i, z_f, z_{i+1}, \ldots, z_d$ arranged in clockwise order, one interior negative puncture $z_{out}$ with asymptotic marker, and $k$ additional interior auxiliary marked points $p_1, \ldots, p_k$ which are [*strictly ordered*]{}; for a representative fixing $z_0$ at 1 and $z_{out}$ at 0, $$0 < |p_1| < \ldots < |p_k| < \frac{1}{2}.$$ Moreover, as before, $$\textrm{the asymptotic marker on $z_{out}$ points in the direction
$\theta_1$ (or towards $z_f$ if $k=0$).}$$ There is a bijection $$\label{cyclicallypermutesector}
\tau_i: { }_k \mc{R}_{d,\tau_i}^{1} \ra { }_k\hat{\mc{R}}_{d}^1$$ given by cyclically permuting boundary labels, and in particular we also have an [*auxiliary-rescaling map*]{} as in $$\label{equivariantforgetfulsector}
{ }_k \mc{R}_{d,\tau_i}^{1} \ra { }_k \mc{R}_{d}^{S^1_{i,i+1}},$$ which, for a representative with $|z_{out}| = 0$, adds a point $p_{k+1}$ on the line between $z_{out}$ and $z_f$ with $|p_{k+1}|=\frac{1}{2}$ and deletes $z_f$. We choose orientations on ${ }_k \mc{R}_{d,\tau_i}^{1}$ to be compatible with ; more concretely, for a slice fixing the positions of $z_{out}$ and $z_d$, consider the top form $$r_1 \cdots r_k dz_1 \wedge dz_2 \wedge \cdots \wedge dz_{d-1} \wedge dz_d \wedge dz_f \wedge dr_1 \wedge d\theta_1 \wedge \cdots \wedge dr_k \wedge d\theta_k$$ The compactification ${ }_k \overline{\mc{R}}_{d,\tau_i}^{1}$ is inherited from the identification ; the salient point is that we treat bubbled off boundary strata containing the point $z_f$ as coming from $\mc{R}^{d,f_i}$, the moduli space of discs with $i$th marked point forgotten (where the $i$th marked point is $z_f$), constructed in Appendix \[discsforgotten\].
We choose as a Floer datum for ${ }_k
\overline{\mc{R}}_{d,\tau_i}^{1}$ the pulled back Floer datum from , it automatically then exists and is universal and consistent as desired. Moreover we have chosen orientations as in the case $k=0$ so that the auxiliary rescaling map is an oriented diffeomorphism extending to a map between compactifications.
Thus, given a Lagrangian labeling $\{L_0, \ldots, L_{d-1}\}$ and compatible asymptotics $\{x_1, \ldots, x_d; y_{out}\}$ we obtain a moduli space of maps satisfying Floer’s equation with the chosen boundary and asymptotics: $$\label{sectormoduli2}
{ }_k \overline{\mc{R}}_{d,\tau_i}^{1}(y_{out}; \vec{x}) := { }_k \overline{\hat{\mc{R}}}_d^1(y_{out}; x_{i-1}, \ldots, x_1, x_d, \ldots, x_i)$$ a smooth manifold of dimension equal to the dimension of the right hand side, $\deg(y_{out}) - n + d- \sum_{j=1}^{d} \deg(x_j) + 2k$. The isomorphisms of orientation lines $$({ }_k \mc{R}_{d,\tau_i}^{1})_u: o_{x_d} \otimes \cdots \otimes o_{x_1} \ra o_{y_{out}}$$ induced by elements $u$ of the zero-dimensional components of define the $|o_{y_{out}}|_\K$ component of the operation $\hat{\oc}^{k}_{d,\tau_i}$, up to the following sign twist: $$\begin{split}
\hat{\oc}^{k}_{d,\tau_i}([x_{d}], \ldots, [x_1]) &:= \sum_{\deg(y_{out}) = n - d + \sum \deg(x_k) - 2k} \sum_{u \in { }_k\overline{\hat{\mc{R}}}_{d,\tau_i}^{1}(y_{out}; \vec{x})} (-1)^{\hat{\star}_d} ({ }_k {\mc{R}}_{d,\tau_i}^{1})_u([x_d], \ldots, [x_1]),\\
\star_d^{S^1} &= \sum_{i=1}^d (i+1) \cdot \deg(x_i) + \deg(x_d) + d - 1.
\end{split}$$ Now, exactly as in Lemma \[sectordecomposition\], there is a chain-level equality of signed operations $$\label{signedequalityk}
\oc^{S^1,k}_d = \sum_{i=0}^{d-1} \hat{\oc}^{k}_{d,\tau_i}.$$ We recall the geometric statement underlying this; the point is by construction there is an oriented embedding $$\label{disjointembeddingksector}
\coprod_{i} { }_k{\mc{R}}^1_{d,\tau_i} \stackrel{\coprod_i \pi_f^{i}}{\lra} \coprod_i { }_k \mc{R}^{S^1_{i,i+1}}_d \hookrightarrow { }_k\mc{R}^{S^1}_d,$$ compatible with Floer data, covering all but a codimension 1 locus in the target, and moreover all the sign twists defining the operations $\oc^{k}_{d,\tau_i}$ are chosen to be compatible with the sign twist in the operation $\oc^{S^1,k}$ (this uses the fact that the Floer data on ${ }_k\mc{R}^{S^1_{i,i+1}}$ agrees with the data on ${ }_k\hat{\mc{R}}_d^1$ via the cyclic permutation map $\kappa^{-i}$ by ). After perturbation zero-dimensional solutions to Floer’s equation can be chosen to come from the complement of any codimension 1 locus in the source abstract moduli space, implying the equality .
Finally, all that remains is a sign analysis whose conclusion is that $$\hat{\oc}^{k}_{d,\tau_i} = \hat{\oc}^k_d \circ s^{nu} \circ t^i$$ where $s^{nu}$ is the operation arising from changing a check term to a hat term with a sign twist . (some such statement is unsurprising, as the operations are constructed with identical Floer data but potentially different sign twists). The details of this sign comparison are exactly the same as in Proposition \[sectorsignorientation\], including with signs, as when orienting the moduli of maps, the additional marked points $p_1, \ldots
p_{k}$ only contribute complex orientations to the moduli spaces of domains (and no additional orientation line terms).
This is an immediate corollary of the previous two propositions.
We now collect all of this information to finish the proof of our main result.
The pre-morphism $\widetilde{\oc} \in \r{Rhom}_{S^1}^n(\r{CH}_*^{nu}(\f), CF^*(M))$, written $u$-linearly as $\sum_i \oc^k u^k$, where $\oc^k = \check{\oc}^k
\oplus \hat{\oc}^k$ are as constructed above, satisfies $\partial
\widetilde{\oc} = 0$ by Prop. \[cyclicOCequations\], hence $\widetilde{\oc}$ is closed, or an $S^1$-complex homomorphism, also known as an $\ainf$ $C_{-*}(S^1)$ module homomorphism (see § \[circleactionsubsection\]). Note that $[\oc^0] =
[\oc^{nu}] = [\oc]$ where the first equality is by definition and the second is by Corollary \[homologylevelnonunitaloc\], hence $\widetilde{\oc}$ is an enhancement of $\oc$ (as defined in § \[circleactionsubsection\]).
This is an immediate consequence of Theorem \[thm:mainresult1\] and the induced homotopy-invariance properties for equivariant homology groups discussed in §\[section:s1action\], particularly Cor. \[S1homotopyinvariance\] and Prop. \[functorialitysequences\].
Variants of the cyclic open-closed map {#sec:OCvariants}
--------------------------------------
### Using singular (pseudo-)cycles instead of Morse cycles {#pseudocycles}
Let $M$ be Liouville or compact and admissible (in which case by our convention $\bar{M} = M$ and $\partial \bar{M} = \emptyset$,[^11] and let us consider the version of $\widetilde{\oc}$ with target the relative cohomology $H^*(\bar{M}, \partial \bar{M})$ as in §\[subsubsec:relh\]. Instead of using a $C^2$ small Hamiltonian to define the Floer complex computing $H^{*+n}(M, \partial M)$ (which we only did for simultaneous compatibility with the symplectic cohomology case), we can pass to a geometric cycle model for the group, and then build a version of the map $\widetilde{\oc}$ with such a target, which simplifies many of the constructions in the previous section (in the sense that the codimension 1 boundary strata of moduli spaces, and hence the equations satisfied by $\widetilde{\oc}$, are strictly a subset of the terms appearing above). As such, it will be sufficient to fix some notation for the relevant moduli spaces, and state the relevant simplified results.
We denote by $$\begin{aligned}
\label{checkequivariant_cycle} { }_k \check{\mc{P}}_{d+1}^1\\
\label{S1equivariant_cycle} { }_k \mc{P}_{d}^{S^1}\\
\label{hatequivariant_cycle} { }_k \hat{\mc{P}}_d^1;\end{aligned}$$ copies of the abstract moduli spaces - where the interior puncture $z_{out}$ is filled in and replaced by a marked point $\bar{z}_{out}$, [*without any asymptotic marker*]{}. The compactifications of these moduli spaces are exactly as before, except that the auxiliary points $p_1, \ldots, p_k$ are now allowed to coincide with $\bar{z}_{out}$, without breaking off an angle-decorated cylinder / element of $\mc{M}_r$ (in the language of §\[sec:angledeccylinder\]). In other words, the real blow-up of Deligne-Mumford compactifications at $z_{out}$ described in §\[modulispaces\], which was responsible for the boundary strata containing $\mc{M}_r$ factors, [*no longer occurs*]{} (but all other degenerations do occur). Correspondingly the codimension-1 boundaries of compactified moduli spaces have all of the strata as before except for strata containing $\mc{M}_r$’s.
Inductively choose smoothly varying families Floer data as before on these moduli spaces of domains, satisfying all of the requirements and consistency conditions as before (except for any consistency conditions involving $\mc{M}_r$ moduli spaces, which no longer occur on the boundary). For a basis $\beta_1, \ldots, \beta_s$ of smooth (pseudo)-cycles in homology $H_*(M)$ whose Poincaré duals $[\beta_i^\vee]$ generate the cohomology $H^*(\bar{M}, \partial \bar{M})$, one obtains moduli spaces $$\begin{aligned}
\label{checkequivariant_cycle_maps} { }_k \check{\mc{P}}_{d+1}^1(\beta_i; \vec{x})\\
\label{S1equivariant_cycle_maps} { }_k \mc{P}_{d}^{S^1}(\beta_i,\vec{x}) \\
\label{hatequivariant_cycle_maps} { }_k \hat{\mc{P}}_d^1(\beta_i, \vec{x}) ;\end{aligned}$$ of moduli spaces of maps into $M$ with source an arbitrary element of the relevant domain moduli space, satisfying Floer’s equation as before, with Lagrangian boundary and asymptotics $\vec{x}$ as before, [*with the additional point constraint that $z_{out}$ lie on the cycle $\beta_i$*]{}. As before, standard methods ensure that 0 and 1-dimensional moduli spaces are (for generic choices of perturbation data and/or $\beta_i$) transversely cut out manifolds of the “right” dimension and boundary, which is all that we need.
Then, define the coefficient of $[\beta_i^{\vee}] \in H^*(\bar{M}, \partial
\bar{M})$ in $\check{\oc}^k(x_d \otimes \cdots \otimes x_1)$ to be given by signed counts (with the same sign twists as before) of the moduli spaces ; similarly for $\hat{\oc}^k$ and $\oc^{S^1,k}$ using the moduli spaces and . A simplification of the arguments already given (in which the $\delta_k$ operations no longer occur, but every other part of the argument carries through) implies that:
The pre-morphism $\widetilde{\oc} = \sum_{i=0}^{\infty} \oc^k u^k \in
\r{Rhom}_{S^1}^n(\r{CH}_*^{nu}(\mc{F}, \mc{F}), H^*(\bar{M}, \partial
\bar{M}))$ satisfies $$\widetilde{\oc} \circ b_{eq} = 0,$$ where $b_{eq} = b^{nu} + u B^{nu}$. In other words, $\widetilde{\oc}$ is a homomorphism of $S^1$-complexes between $\r{CH}_*^{nu}(\mc{F}, \mc{F})$ with its strict $S^1$ action and $H^*(\bar{M}, \partial \bar{M})$ with its trivial $S^1$ action.
### Compact Lagrangians in non-compact manifolds {#sec:compactopenclosed}
Now let us explicitly restrict to the case of $M$ a Liouville manifold, and denote by $\mc{F} \subset \mc{W}$ the full-subcategory consisting of a finite collection of compact exact Lagrangian branes contained in the compact region $\overline{M}$. By Poincaré duality we may think of the map $\oc$ (and its cyclic analogue, $\widetilde{\oc}$) with target $H^*(\bar{M}, \partial \bar{M})$ as a pairing $\r{CH}_*^{nu}(\mc{F},\mc{F}) \otimes C^*(M) \to \K[n]$. In this case, there is a non-trivial refinement of this pairing to $$\label{ocpairing} \oc_{cpct}: \r{CH}_{*}(\mc{F}) \otimes SC^*(M) \ra \K[-n].$$ where $SC^*(M)$ is the [*symplectic cohomology*]{} cochain complex.
The refinement relies on extra flexibility in Floer theory for compact Lagrangians compared to non-compact Lagrangians (compare Remarks \[liouvilleFloerdatum\] and \[wrappedFloerdatum\]), first alluded to in this form in [@SeidelEquivariant]. This extra flexibility allows us to define operations without outputs — and in particular study a version of the open-closed map where the interior marked point and boundary marked points are all inputs— for instance by Poincaré dually treating some boundary inputs as outputs with “negative weight”.
One way to implement such operations, using the type of Floer data discussed in Remark \[wrappedFloerdatum\], is by allowing the [ *sub-closed one-form $\alpha_S$*]{} used in Floer theoretic perturbations to have complete freedom along boundary conditions corresponding to compact Lagrangians; in contrast along possibly non-compact Lagrangian boundary conditions, $\alpha_S$ is required to vanish in order to appeal to the integrated maximum principle. In particular, if we allow $\alpha_S$ to be non-vanishing along boundary components, Stokes’ theorem no longer implies that $\alpha_S$ being sub-closed implies that the total “output” weights must be greater than the total “input” weights.
The existence of a map $SC^*(M) \to \r{CH}_*(\mc{F})^{\vee}$ is well known. Namely, categories $\cc$ with a [*weak proper Calabi-Yau structure*]{} of dimension $n$ (such as the Fukaya category of compact Lagrangians; see e.g., [@Seidel:2008zr]\*[(12j)]{}) come equipped with isomorphisms between the dual of Hochschild chains and Hochschild co-chains $\r{CH}_*(\cc)^{\vee}[-n] \simeq \r{CH}^*(\cc)$, and the existence of a map $SH^*(M) \to \r{HH}^*(\mc{F})$ was already observed in [@Seidel:2002ys].
The geometric moduli spaces used to establish our main result apply verbatim in this case (with the interior marked point changed to an input, and the ordering of the auxiliary marked points $p_1, \ldots, p_k$ appearing in the cyclic open-closed map reversed). In this case, the operations associated to such moduli spaces imply:
Consider $\r{CH}_*^{nu}(\f) \otimes SC^*(M)$ as an $S^1$-complex with its diagonal $S^1$ action (see Lemma \[diagonalaction\] in § \[circleactionsubsection\]), and $\K = \underline{\K}^{triv} \in S^1\mod$ with its trivial $S^1$-complex structure. Then, the map from $\r{CH}_*(\f)
\otimes SC^*(M)$ to $\K$ can be enhanced to a homomorphism of $S^1$-complexes $$\widetilde{\oc}_{cpct} \in \r{Rhom}_{S^1}^n(\r{CH}_*^{nu}(\f) \otimes SC^*(M), \K)$$ e.g., $\widetilde{\oc}_{cpct}$ satisfies $\partial
\widetilde{\oc}_{cpct} = 0$. In other words, in the notation of §\[sec:ulinear\] there exists a map $\widetilde{\oc}_{cpct, eq} = \sum_{i=0}^\infty \oc_{cpct, i} u^i: \r{CH}_*^{nu}(\f) \otimes SC^*(M) \to \K[ [ u] ])$ of pure degree $n$, with $[\oc_{cpct, 0}] = [\oc_{cpct}]$ such that $$\widetilde{\oc}_{cpct, eq} \circ ( (-1)^{\deg(y)} b_{eq}(\sigma) \otimes y + \sigma \otimes \delta^{SC}_{eq}(y)) = 0.$$
To clarify the relevant moduli spaces used, we define the spaces $$\begin{aligned}
\label{checkequivariant_pairing} { }_k \check{\mc{R}}_{d+1,cpct}^1\\
\label{S1equivariant_pairing} { }_k \mc{R}_{d,cpct}^{S^1}\\
\label{hatequivariant_pairing} { }_k \hat{\mc{R}}_{d,cpct}^1\end{aligned}$$ to be copies of the abstract moduli spaces - where the interior puncture $z_{out}$ is now a [*positive*]{} puncture (still equipped with an asymptotic marker); and all of the other inputs and auxiliary points are as before, except we’ve reversed the order of the labelings $p_1, \ldots, p_k$ (for notational convenience), so the ordering constraints all now read as $0 < |p_k| < \cdots < |p_1| < \frac{1}{2}$. The compactified moduli spaces have boundary strata agreeing with the boundary strata of the compactified -, except now the $\mc{M}_r$ cylinders break “above” the ${ }_{k-r}
\mc{R}_{d+1,cpct}^1$ (equipped with $\check{ }$, $\hat{ }$, or $S^1$ decoration) discs instead of “below.” The reversal of the ordering of auxiliary marked points is designed to be compatible with the ordering of the auxiliary marked points on the $\mc{M}_r$ moduli spaces when it breaks “above” (as in $\mc{M}_r$ the label numbers of the auxiliary marked points increase from top to bottom).
Equipping these moduli spaces with perturbation data satisfying the same consistency conditions and other requirements as before, and counting solutions with sign twists as before defines the terms of the pre-morphism exactly as in the previous subsections (with identical analysis to show that, for instance, the operation corresponding to ${ }_k \mc{R}_{d,cpct}^{S^1}$ is the operation corresponding to ${ }_{k-1} \hat{\mc{R}}_{d,cpct}^1$ composed with the Connes’ B operator, the boundary strata in which $|p_i|$ and $|p_{i+1}|$ are coincident contributes trivially, and so on).
Calabi-Yau structures {#section:cystructures}
=====================
The proper Calabi-Yau structure on the Fukaya category
------------------------------------------------------
Here we review the notion of a [*proper Calabi-Yau structure*]{}, following Kontsevich-Soibelman [@Kontsevich:2009ab], and construct proper Calabi-Yau structures on Fukaya categories of compact Lagrangians (in a compact admissible or Liouville manifold). A proper Calabi-Yau structure induces a chain-level topological field theory operations on the Hochschild chain complex of the given category, controlled by the open moduli space of curves with marked points equipped with asymptotic markers, at least one of which is an input [@Costello:2006vn; @Kontsevich:2009ab]. Note that Costello’s work [@Costello:2006vn] constructing field-theoretic operations has the (a priori stronger) requirement that the underlying $\ainf$ category to be [*cyclic*]{}, but in characteristic zero any proper Calabi-Yau structure determines a unique quasi-isomorphism between the underlying $\ainf$ category and a cyclic $\ainf$ category [@Kontsevich:2009ab Thm. 10.7]; see Remark \[rem:cyclic\] for more discussion.
We say an $\ainf$ category $\mc{A}$ is *proper* (sometimes called *compact*) if its cohomological morphism spaces $H^*(\hom_{\mc{A}}(X,Y))$ have total finite rank over $\K$, for each $X,Y$. Recall that for any object $X \in \mc{A}$, there is an inclusion of chain complexes $\hom(X,X) \ra \r{CH}_*(\mc{A})$ inducing a map $[i]:
H^*(\hom(X,X)) \ra \r{HH}_*(\mc{A})$.
\[weakproperCY\] Let $\mc{A}$ be a proper category. A chain map $tr:
\r{CH}_{*+n}(\mc{A}) \ra \K$ is called a *weak proper Calabi-Yau structure*, or *non-degenerate trace* of dimension $n$ if, for any two objects $X, Y \in \ob \mc{A}$, the composition $$H^*(\hom_{\mc{A}} (X,Y)) \otimes H^{n-*}(\hom_{\mc{A}} (Y,X)) \stackrel{[\mu^2_{\mc{A}}]}{\ra} H^n(\hom_{\mc{A}} (Y,Y)) \stackrel{[i]}{\ra} \r{HH}_n(\mc{A}) \stackrel{[tr]}{\ra \K}$$ is a perfect pairing.
\[ccybimodule\] In the symplectic literature, weak proper Calabi-Yau structures of dimension $n$ are sometimes defined as bimodule quasi-isomorphisms $\mc{A}_{\Delta} \stackrel{\sim}{\ra} \mc{A}^{\vee}[n]$, where $\mc{A}_{\Delta}$ denotes the [*diagonal bimodule*]{} and $\mc{A}^{\vee}$ the [*linear dual diagonal bimodule*]{} — see [@Seidel:2008zr]\*[(12j)]{} and §\[subsec:smoothCY\] for brief conventions on $\ainf$ bimodules (see also [@Tradler:2008fk]). To explain the relationship between this definition and the one above, which has sometimes been called a [*weakly cyclic structure*]{} or [*$\infty$-inner product*]{} [@Tradler:2008fk; @Sheridan:2016], note that for any compact $\ainf$ category $\mc{A}$, there are quasi-isomorphisms (with explicit chain-level models) $$(\r{CH}_*(\mc{A}))^{\vee} = \r{CH}^*(\mc{A}, \mc{A}^{\vee}) \stackrel{\sim}{\leftarrow} \hom_{\mc{A}\!-\!\mc{A}}(\mc{A}_{\Delta}, \mc{A}^{\vee})$$ where $\hom_{\mc{A}\!-\!\mc{A}}$ denotes morphisms in the category of $\ainf$ bimodules (see e.g., [@Seidel:2008cr] or [@ganatra1_arxiv]). Under this correspondence, non-degenerate morphisms from $\r{HH}_*(\mc{A})
\ra \K$ as defined above correspond precisely (cohomologically) to weak Calabi-Yau structures, e.g., those bimodule morphisms from $\mc{A}_{\Delta}$ to $\mc{A}^{\vee}$ which are cohomology isomorphisms.
Remember that the Hochschild chain complex of an $\ainf$ category $\mc{A}$ comes equipped with a natural chain map to the (positive) cyclic homology chain complex, the [*projection to homotopy orbits*]{} $$pr: \r{CH}_*^{nu}(\mc{A}) \ra \r{CC}^+_*(\mc{A})$$ modeled on the chain level by the map that sends $\alpha \mapsto \alpha \cdot
u^0$, for $\alpha \in \r{CH}^{nu}(\mc{A})$ (compare ).
A *(strong) proper Calabi-Yau structure of degree $n$* is a chain map $$\tilde{tr}: \r{CC}^+_*(\mc{A})\ra \K[-n]$$ from the (positive) cyclic homology chain complex of $\mc{A}$ to $\K$ of degree $-n$, such that the induced map $tr = \tilde{tr} \circ pr: \r{CH}_*(\mc{A}) \ra \K[-n]$ is a weak proper Calabi-Yau structure.
Via the model for cyclic chains given as $\r{CC}_*(\mc{A}) :=
(\r{CH}_*^{nu}(\mc{A}) (( u ) ) / u \r{CH}_*^{nu}(\mc{A}) [ [ u ] ], b +
u B^{nu})$, such an element $\tilde{tr}$ takes the form $$\tilde{tr} := \sum_{i=0}^{\infty} tr^k u^k$$ where $$tr^k := (\check{tr}^k \oplus \hat{tr}^k): \r{CH}_*^{nu}(\mc{A}) \ra \K[-n-2k].$$ We now complete the proof of Theorem \[mainthm3\] described and sketched in the Introduction: first, define the putative proper Calabi-Yau structure as the composition: $$\label{trcomposition}
\tilde{tr}: \r{HC}^+_*(\mc{F}) \stackrel{\widetilde{\oc}}{\ra} H^{*+n}(M, \partial M) \otimes_{\K} \K( ( u ) ) / u \K [ [ u ] ] \ra \K$$ where the last map sends $PD(pt) \cdot u^0 \in H^{2n}(\bar{M}, \partial \bar{M})$ to $1$, and other elements to zero. Instead of using a $C^2$ small Hamiltonian to define the Floer complex computing $H^{*+n}(M, \partial M)$ (which we only did for simultaneous compatibility with the symplectic cohomology case), we can pass to a geometric cycle model for the map $\widetilde{\oc}$ (and hence the map $\tilde{tr}$) described in §\[pseudocycles\]. Then the map $\tilde{tr}$ involves counts of the moduli spaces described there where the interior marked point $\bar{z}_{out}$ is [*unconstrainted*]{}, e.g., ${ }_k \check{\mc{P}}_{d+1}^1([M]; \vec{x})$, ${ }_k \hat{\mc{P}}_{d+1}^1([M];
\vec{x})$, and ${ }_k \mc{P}_{d+1}^{S^1}([M]; \vec{x})$; see Figure \[fig:trk\_checkhat\].
The following well-known Lemma verifies the non-degeneracy property of the map $\tilde{tr}$:
\[poincareduality\] The corresponding morphism $[tr]:\r{HH}_{*+n}(\f) \ra \K$ is a non-degenerate trace (or weak proper Calabi-Yau structure).
This is an immediate consequence of Poincaré duality in Lagrangian Floer cohomology, see the references cited above. As a brief sketch, note that $tr \circ \mu^2: \hom(X,Y) \otimes \hom(Y,X)
\ra \K$ is chain homotopic (and hence equal in cohomology) to a chain map which counts holomorphic discs with an interior marked point satisfying an empty constraint, and two (positive) boundary asymptotics on $p$, $q$, with corresponding Lagrangian boundary on $x$ and $y$. Via a further homotopy of Floer data, one can arrange that the generators of $\hom(X,Y)$ and $\hom(Y,X)$ are in bijection (for instance if one is built out of time-1 flowlines of $H$ and one out of time-1 flowlines of $-H$), and the only such rigid discs are constant discs between $p$ and the corresponding $p^{\vee}$.
The above discussion constructs $\tilde{tr}$ and Lemma \[poincareduality\] verifies non-degeneracy.
![\[fig:trk\_checkhat\] An image of representatives of moduli spaces ${ }_3 \check{\mc{P}}_{3}^1([M]; \vec{x})$ and ${ }_2 \hat{\mc{P}}_{4}^1([M]; \vec{x})$, which appear in the map $\tilde{tr}$.](trk_checkhat.pdf)
The smooth Calabi-Yau structure on the Fukaya category {#subsec:smoothCY}
------------------------------------------------------
\#1\#2\#3[\_( [\#2]{}\_[\#3-1]{}, [\#2]{}\_[\#3]{}) \_( [\#2]{}\_0, [\#2]{}\_1)]{} \#1\#2[ ]{} \#1\#2[\^l\_[\#1]{} \_ \^r\_[\#2]{}]{} We give a brief overview of (a categorical version of) the notion of a [*(strong) smooth Calabi-Yau structure*]{}, and construct such smooth Calabi-Yau structures on (wrapped or compact) Fukaya categories under the “non-degeneracy” hypotheses of [@ganatra1_arxiv]. Smooth Calabi-Yau structures were proposed in the work-in-progress [@Kontsevich:uq] (see also [@Kontsevich:CY]; other expositions appear e.g., in [@GPS1:2015; @brav_dyckerhoff_2019] — in the latter work the terminology “left” is used instead of “smooth” and “right” instead of “proper”). A smooth Calabi-Yau structure is expected (analogously to the proper case) by [@Kontsevich:uq] to induce chain-level topological field theory operations on the Hochschild chain complex of the given category, controlled by the open moduli space of curves with marked points equipped with asymptotic markers, at least one of which is an output.[^12]
To state the relevant definitions, we make use of some of the theory of [*$\ainf$ bimodules over a category $\cc$*]{}. We do so without much explanation, instead referring readers to existing references [@Seidel:2008cr; @Tradler:2008fk; @ganatra1_arxiv]. An $\ainf$ bimodule $\eup$ over $\cc$ is a bilinear $\ainf$ functor from $\cc^{op} \times \cc$ to chain complexes, which is roughly the data, of, for every pair of objects $\cc$, a chain complex $(\eup(X,Y), \mu^{0|1|0})$, along with ‘higher multiplication maps’ $\mu^{s|1|t}: \seqc{X}{s} \otimes \eup(X_0, Y_{t}) \otimes
\seqc{Y}{t} \ra \eup(X_s, Y_0)$ satisfying a generalization of the $\ainf$ equations. $\ainf$ bimodules over $\cc$ form a dg category $\bimodcategoryc$ with morphisms denoted $\bimodhomc( \eup, \euq)$ (for dg bimodules over a dg category, this chain complex corresponds to a particular chain model for the ‘derived morphism space’ using the bar resolution). The basic examples of bimodules we require are:
- the [*diagonal bimodule*]{} $\diagc$ which associates to a pair of objects $(K,L)$ the chain complex $\diagc(K,L):= \hom_{\cc}(L,K)$.
- for any pair of objects $A, B$, there is a [*Yoneda bimodule*]{} $\biyon{A}{B}$ which associates to a pair of objects $(K,L)$ the chain complex $\biyon{A}{B}(K,L) := \hom_{\cc}(A, K) \otimes \hom_{\cc}(L,
B)$.
Yoneda bimodules are the analogues of the free bimodule $A \otimes A^{op}$ in the category of bimodules over an associative algebra $A$ (which are the same as $A \otimes A^{op}$ modules). Accordingly, we say a bimodule $\eup$ is [*perfect*]{} if, in the category $\bimodcategoryc$, it is split-generated by (i.e., isomorphic to a retract of a finite complex of) Yoneda bimodules. We say that a category $\cc$ is [*(homologically) smooth*]{} if $\diagc$ is a perfect $\cc$-bimodule.
Recall for what follows that for any bimodule $\eup$ there is a [*cap product action*]{} $$\cap: \r{HH}^*(\cc, \eup) \otimes \r{HH}_*(\cc, \cc) \ra \r{HH}_*(\cc, \eup),
\label{capproducthh}$$ and hence for any class $[\sigma] \in \r{HH}_*(\cc, \cc)$ there is an induced map $$[\cap \sigma]: \r{HH}^*(\cc, \eup) \ra \r{HH}_{*+\deg(\sigma)}(\cc, \eup).
\label{capproducthh}$$ More generally, the cap products acts as $ \r{HH}^*(\cc, \eup) \otimes
\r{HH}_*(\cc, \euq) \ra \r{HH}_*(\cc, \eup \otimes_{\cc} \euq)$; here we are considering $\euq = \diagc$, and then composing with the equivalence $\eup
\otimes_{\cc} \diagc \cong \eup$. See e.g., [@ganatra1_arxiv]\*[§2.10]{} for explicit chain level formulae in the variant case that $\eup = \diagc$, which can be straightforwardly adapted to the general case and then specialized to the case here.
\[weakscy\] Let $\cc$ be a homologically smooth $\ainf$ category. A cycle $\sigma \in
\r{CH}_{-n}(\cc, \cc)$ is said to be a [*weak smooth Calabi-Yau structure*]{}, or a [*non-degenerate co-trace*]{} if, for any objects $K, L$, the operation of capping with $\sigma$ induces a homological isomorphism $$\label{capproduct}
[\cap \sigma]: \r{HH}^*(\cc, \biyon{K}{L}) \stackrel{\cong}{\ra} \r{HH}_{*-n}(\cc, \biyon{K}{L}) \simeq H^*(\hom_{\cc}(K,L)).$$ (this non-degeneracy property only depends on the homology class $[\sigma]$). A cycle in the non-unital Hochschild complex $\sigma \in \r{CH}_{-n}^{nu}(\cc)$ is said to be a weak smooth Calabi-Yau structure if again $[\sigma] \in
H^*(\r{CH}_{-n}^{nu}(\cc)) \cong \r{HH}_{-n}(\cc)$ is non-degenerate in the sense of .
The second isomorphism $\r{HH}_{*-n}(\cc, \biyon{K}{L}) \simeq
H^*(\hom_{\cc}(K,L))$ always holds for cohomologically unital categories (such as the Fukaya category); the content is in the first.
Continuing Remark \[ccybimodule\], there is an alternate perspective on Definition \[weakscy\] using bimodules. Namely, for any bimodule $\eup$, there is a naturally associated [*bimodule dual*]{} $\eup^!$, defined for a pair of objects $(K,L)$ as the chain complex $\eup^!(K,L) :=
\bimodhomc(\eup, \biyon{K}{L})$. The higher bimodule structure is defined in [@ganatra1_arxiv]\*[Def. 2.40]{}; it is an $\ainf$ analogue of defining, for an $A$ bimodule $B$, $B^!:= \mathrm{RHom}_{A \otimes A^{op}}(B, A \otimes
A^{op})$ where $\mathrm{RHom}$ is taken with respect to the outer bimodule structure on $A \otimes A^{op}$ and the bimodule structure on $B^!$ comes from the inner bimodule structure; see e.g., [@Ginzburg:2005aa]\*[§20.5]{}.
We abbreviate $\cc^!:= \cc_{\Delta}^!$, and call $\cc^!$ the [*inverse dualizing bimodule*]{}, following [@Kontsevich:2009ab] (observe $H^*(\cc^!(K,L))
\cong \r{HH}^*(\cc, \biyon{K}{L})$). For a homologically smooth category $\cc$ one notes that there is a quasi-isomorphism $\r{CH}_{-n}(\cc) \simeq \bimodhomc(\cc_{\Delta}^!,\cc_{\Delta})$ (see [@Kontsevich:2009ab]\*[Rmk. 8.11]{} for the case of $\ainf$ algebras), where the equivalence associates to any element, the bimodule morphism whose cohomology level map is the cap product operation . Non-degenerate cotraces correspond precisely then to bimodule quasi-isomorphisms $\cc^!
\stackrel{\sim}{\ra} \cc_{\Delta}$. Further discussion of these structures in the $\ainf$ categorical setting will appear as part of forthcoming work with Cohen [@CohenGanatra:2015].
Let $\iota: \r{CC}^-_*(\cc) \ra \r{CH}_*^{nu}(\cc)$ denote the inclusion of homotopy fixed points chain map from ; concretely as described in this is the chain map sending $\sum_{i=0}^{\infty} \alpha_i u^0
\mapsto \alpha_0$.
Let $\cc$ be a homologically smooth $\ainf$ category. A [*(strong) smooth Calabi-Yau structure*]{} is a cycle $\tilde{\sigma} \in
\r{CC}^-_{-n}(\cc)$ such that the corresponding element $\iota(\tilde{\sigma}) \in \r{CH}_{-n}^{nu}(\cc)$ is a weak smooth Calabi-Yau structure.
Using these definitions and the cyclic open-closed map, we restate and prove Theorem \[thm:smoothCY\]:
Suppose a Liouville manifold is [*non-degenerate*]{} in the sense of [@ganatra1_arxiv], meaning that the map $[\oc]: \r{HH}_{*-n}(\w) \ra SH^*(M)$ hits 1. Then, the wrapped Fukaya category possesses a canonical [ *smooth Calabi-Yau structure*]{}.
In [@ganatra1_arxiv] it was proven that, assuming non-degeneracy of $M$, the map $[\oc]: \r{HH}_{*-n}(\w) \ra SH^*(M)$ is an isomorphism, $\w$ is homologically smooth, and moreover that the pre-image $[\sigma]$ of 1 gives a [*weak smooth Calabi-Yau structure*]{} in the sense described above. Let us briefly recall how the non-degeneracy condition is proven (which is a left slightly implicit in [@ganatra1_arxiv]): First, a geometric morphism of bimodules $\mathcal{CY}: \w_{\Delta} \to \w^![n]$ is constructed and shown in [@ganatra1_arxiv]\*[Thm. 1.3]{} to be a quasi-isomorphism under the given non-degeneracy hypotheses. Then, it is shown that capping with $[\sigma]$ is a one-sided inverse to the homological map $[\mathcal{CY}]$, and thus an isomorphism also, by the following argument: the following commutative (up to an overall sign of $(-1)^{n(n+1)/2}$) diagram is established (which can be thought of as coming from the compatibility of $\oc$ with module structures for Hochschild (co)homology with coefficients in $\biyon{K}{L}$, and which can be extracted from the holomorphic curve theory appearing in [@ganatra1_arxiv]\*[Thm. 13.1]{}): $$\xymatrix{ \r{HH}_{*-n}(\w, \w) \otimes H^*(\hom_{\w}(K,L)) \ar[r]^{(id, [\mathcal{CY}])} \ar[d]^{([\oc], id)}& \r{HH}_{*-n}(\w, \w) \otimes HH^{*+n}(\w, \biyon{K}{L}) \ar[d]^{\cap}\\
SH^*(M) \otimes H^*(\hom_{\w}(K,L)) \ar[r]^{[\mu^2(\co(-), -)]} \ar[r]& H^*(\hom_{\w}(K,L)) = \r{HH}_*(\w, \biyon{K}{L})}.$$ Plugging $[\sigma]$ into the $\r{HH}_*(\w,\w)$ and noting $[\oc]([\sigma]) = 1$ and $\mu^2(1,-)$ is the identity map establishes as desired that $[\sigma \cap (\mathcal{CY}(y))] = [y]$.
To lift the weak smooth Calabi-Yau structure to a (strong) smooth Calabi-Yau structure, first we note that, because $[\oc]$ is an isomorphism, Corollary \[cor:cyclicOC\] implies that there is a commutative diagram of isomorphisms: $$\label{inclusionhomotopyfixedpointsOC}
\xymatrix{
\r{HC}^-_{*-n}(\w) \ar[r]_{[\iota]} \ar[d]_{[\widetilde{\oc}^-]} & \r{HH}_{*-n}(\w) \ar[d]^{[\oc]}\\
H^*(SC^*(M)^{hS^1}) \ar[r]_{[\iota]} & SH^*(M)},$$ where the horizontal maps $\iota$ are the inclusion of homotopy fixed points maps $\iota: P^{hS^1} \ra P$ defined for any $S^1$-complex $P$, sending $\sum_{i=0}^{\infty} \alpha_i u^i \mapsto
\alpha_0$.
In §\[sec:interior\], and specifically , it was shown that there is a canonical geometrically defined element $\tilde{1} \in
H^*(SC^*(M)^{hS^1})$ lifting the unit $1 \in SH^*(M)$— essentially this is because the map $1$ is in the image of the map $H^*(M) \to
SH^*(M)$, which on the chain level (as this map comes from “the inclusion of constant loops into the free loop space” and “constant loops are acted on by $S^1$ trivially”) can be canonically lifted to a map $C^*(M) \to C^*(M)^{hS^1} = C^*(M)[ [ u] ] \to SC^*(M)^{hS^1}$.
Since $[\widetilde{\oc}^-]$ is an isomorphism, it follows that there is a unique (cohomological) element $[\widetilde{\sigma}] \in
\r{HC}^-_{*-n}(\w)$ hitting $\tilde{1}$. By , $\iota(\widetilde{\sigma}) = \sigma$, establishing that (any cycle representing) $\widetilde{\sigma}$ is a smooth Calabi-Yau structure.
Moduli spaces and operations
============================
A real blow-up of Deligne-Mumford space {#modulispaces}
---------------------------------------
We review, in a special case, the compactifications of moduli spaces of surfaces where some interior marked points are equipped with asymptotic markers, which are a real blow-up of Deligne-Mumford moduli space as constructed in [@Kimura:1995fk]. In particular, we show how boundary strata of the abstract compactifications in the sense of [@Kimura:1995fk] can be identified with the specific models of the moduli spaces we use in Section \[section:openclosed1\] (the appearance of the compactifications [@Kimura:1995fk] in Floer theory are not new, see e.g., [@Seidel:2010uq]).
To begin, let $$\mc{M}_{2,0}$$ denote the space of spheres with $2$ marked points $z_1, z_2$ removed and asymptotic markers $\tau_1$, $\tau_2$ around the $z_1$ and $z_2$, modulo automorphism. Fixing the position of $z_1$ and $z_2$ and one of $\tau_1$ or $\tau_2$ gives a diffeomorphism $$\mc{M}_{2,0} \cong S^1.$$ On an arbitrary representative in $\mc{M}_{2,0}$, we can think of the map to $S^1$ as coming from the [*difference in angles*]{} between $\tau_1$ and $\tau_2$ (after, say, parallel transporting one tangent space the other along a geodesic path).
It is convenient to parametrize this difference by a point on the sphere itself, in the following manner (though this will break symmetry between $z_1$ and $z_2$). Let $$\label{m21}
\mc{M}_{2,1}$$ be the space of spheres with $2$ marked points $z_1, z_2$ removed, an extra marked point $y_1$ and asymptotic markers $\tau_1$, $\tau_2$ around the $z_1$ and $z_2$, modulo automorphism such that, for any representative with position of $z_1$, $z_2$, and $y$ fixed, $\tau_2$ is pointing towards $y$. The remaining freedom in $\tau_1$ once more gives a diffeomorphism $\mc{M}_{2,1} \cong S^1$.
We can take a different representative for elements of $\mc{M}_{2,1}$: up to biholomorphism any element of is equal to a cylinder sending $z_1$ to $+\infty$, $z_2$ to $-\infty$) with fixed asymptotic direction around $+\infty$ and an extra marked point $y$ at fixed height freely varying around $S^1$, such that the asymptotic marker at $-\infty$ coincides with the $S^1$ coordinate of $y$. Thus, we obtain an identification $$\label{cylindermodel}
\mc{M}_{2,1} \cong \mc{M}_1$$ where $\mc{M}_1$ is the space in Definition \[rpointedmodulispace\].
Denote by $$\label{equivmoduli}
{ }_k\mc{R}_{d}^1$$ the moduli space of discs $(S, z_1, \ldots, z_d, y, \tau_y, p_1, \ldots, p_k)$ with $d$ boundary marked points $z_1, \ldots, z_d$ arranged in clockwise order, an interior marked point with asymptotic marker $(y,\tau_y)$, and interior marked points with no asymptotic markers $p_1, \ldots, p_k$ satisfying two constraints to be described below, modulo autmorphism. Up to automorphism, every equivalence class of the unconstrained moduli space of such $(S, z_1, \ldots, z_d, y, \tau_y, p_1, \ldots, p_k)$ up to automorphism admits a unique unit disc representative with $z_d$ fixed at $1$ and $y$ at 0; call this the [*$(z_d, y)$ standard representative*]{}. The positions of the asymptotic marker, remaining marked points, and interior marked points identify this unconstrained moduli space with an open subset of $S^1 \times \R^{2k} \times
\R^d$. With respect to this identification, the space consists of those discs satisfying the following (open) “ordering constraint” on the positions of the interior marked points: $$\label{orderingappa}
\textrm{On the standard representative satisfies } 0 < |p_1| < |p_2| < \cdots < |p_k|.$$ along with a (codimension 1) condition on the asymptotic marker: $$\textrm{On the standard representative, $\tau_y$ points at $p_1$}.$$ The condition , which cuts out a manifold with corners of the larger space in which the $p_i$ are unconstrained, is technically convenient, as it reduces the types of bubbles that can occur with $y$). The compactification of interest, denoted $$\label{ksvcompact}
{ }_k\overline{\mc{R}}_{d}^1$$ differs from the Deligne-Mumford compactification in a couple respects: firstly, in the closures of the ordering conditions away from zero, we allow points $p_i$ and $p_{i+1}$ to be coincident without bubbling off (alternatively, we can Deligne-Mumford compactify and collapse the relevant strata).
More interestingly, is a real blow-up of the usual Deligne-Mumford compactification along any strata in which $y$ and $p_i$ points bubble off, in the following precise sense. Let $\Sigma = S_0 \cup_{y^+ = y^-} S_1$ denote a nodal surface, where
- $S_0$ is a sphere containing interior marked points $(y, \tau_y)$, $p_1, \ldots, p_j$, and another marked point $y^+$, and
- $S_1$ is a disc with $d$ boundary marked points $z_1, \ldots, z_d$, and interior marked points $y^-$, $p_{j+1}$, …, $p_{k}$.
To occur as a possible degenerate limit of , the relevant points $p_i$ on $S_0$ and $S_1$ must satisfy an ordering condition: $$\begin{aligned}
\label{orderingappa1}&\textrm{For a representative $S_0'$ of $S_0$ with $y$ and $y^+$ at opposite poles, $0 < |p_1| < \cdots < |p_j| < |y^+|$,}\\
\nonumber &\textrm{where $|p|$ denotes the geodesic distance from $p$ to $y$ for $S_0'$}.\\
\label{orderingappa2}&\textrm{For the $(z_d, y^-)$ standard representative of $S_1$, $0 < |p_{j+1}| < \cdots < |p_k|$}.\end{aligned}$$ Also, $$\textrm{For the same representative $S_0'$, the asymptotic marker $\tau_y$ should point (geodesically) towards $p_1$}.$$ The relevant stratum of consists of all such configurations $S_0 \cup_{y^+ = y^-} S_1$ equipped with an extra piece of information: [*a gluing angle*]{}, which is a real positive line $\tau_{y^+,y^-}$ in $T_{y^+} S_0
\otimes T_{y^-} S_1$, or equivalently, a pair of asymptotic markers around $y^+$ and $y^-$, modulo the diagonal $S^1$ rotation action. Note that the set of gluing angles is $S^1$, giving such strata 1 higher codimension. We note that there is a natural choice of asymptotic marker on $S_1$, coming from the position of $|p_{j+1}|$ (or if $j = s$, the position of $z_d)$ for the standard representative. Fixing such a choice identifies $S_1$ with an element of ${ }_{k-j}\mc{R}^d_1$ and equips $S_0$ with a (freely varying) asymptotic marker $\tau_{y^+}$ at $y^+$. In a manner as in , we can identify $S_0$ up to biholomorphism with an element of $\mc{M}_j$. Hence, we’ve identified the relevant stratum with $$\mc{M}_{j} \times { }_{k-j}\overline{\mc{R}}^{d}_1,$$ which will be useful in defining the relevant pseudoholomorphic curve counts.
Operations with a forgotten marked point {#discsforgotten}
----------------------------------------
We introduce auxiliary degenerate operations that will arise as the codimension 1 boundary of the open-closed map and equivariant structure. This subsection is a very special case of the general discussion in [@ganatra1_arxiv].
Let $d \geq 2$ and $i \in \{1, \ldots, d\}$. The [*moduli space of discs with $d$ marked points with $i$th boundary point forgotten*]{} $$\mc{R}^{d,f_i}$$ is exactly the moduli space of discs $\mc{R}^d$, with $i$th boundary marked point labeled as auxiliary.
The Deligne-Mumford compactification $$\label{dmforgotten}
\overline{\mc{R}}^{d,f_i}$$ is exactly the usual Deligne-Mumford compactification, along with the data of an [*auxiliary label*]{} at the relevant boundary marked point.
For $d > 2$, the [*$i$-forgetful map*]{} $$\label{iforget}
\mc{F}_{d,i}: \mc{R}^{d,f_i} \ra \mc{R}^{d-1}.$$ associates to a surface $S$ the surface obtained by putting the $i$th point back in and forgetting it. This map admits an extension to the Deligne-Mumford compactification $$\overline{\mc{F}}_{d,i}: \overline{\mc{R}}^{d,f_i} \ra \overline{\mc{R}}^{d-1}$$ as follows: eliminate any non-main components with only one non-auxiliary marked point $p$, and label the positive marked point below this component by $p$. We say that any component not eliminated is [*f-stable*]{} and any component eliminated is [*f-semistable*]{}.
The above map is only well-defined for $d > 2$. In the semi-stable case $d =
2$, $\mc{R}^{2,f_i}$ is a point so one can define an ad hoc map $$\mc{F}^{ss}_i: \mc{R}^{2,f_i} \ra pt.$$ which associates to a surface $S$ the (unstable) strip $\Sigma_1 =
(-\infty,\infty) \times [0,1]$ as follows: take the unique representative of $S$ which, after its three marked points are removed, is biholomorphic to the strip $\Sigma_1$ with an additional puncture $(0, 0)$. Then, forget/put back in the point $(0,0)$.
A [*forgotten Floer datum*]{} for a stable disc with $i$th point auxiliary $S \in \overline{\mc{R}}^{d,f_i}$ consists of, for every component $T$ of $S$,
- a Floer datum for $T$, if $T$ does not contain the auxiliary point,
- a Floer datum for $\mc{F}_j(T)$, if $T$ is $f$-stable and contains the auxiliary point as its $j$th input.
- A Floer datum on $\mc{F}^{ss}_i(T)$ which is [*translation invariant*]{}, if $T$ is [ *f-semistable*]{}.
(by [*translation invariant*]{}, we mean the following: note that $\Sigma_1$ has a canonical $\R$-action given by linear translation in the $s$ coordinate. We require $H$, $J$, and the time-shifting map/weights to be invariant under this $\R$ action, and in particular should only depend on $t \in [0, 1]$ at most).
In particular, this Floer datum should only depend on the point $\overline{\mc{F}}_{d,i}(S)$.
Let $i \in \{1,\ldots, d\}$ with $d > 1$. Then the operation associated to $\overline{\mc{R}}^{d,f_i}$ is zero if $d > 2$ and the identity operation $I(\cdot)$ (up to a sign) when $d=2$.
Suppose first that $d>2$, and let $u$ be any solution to Floer’s equation over the space $\mc{R}^{d,f_i}$ with domain $S$. Let Since the Floer data on $S$ only depends on $\mc{F}_{d,i}(S)$, we see that maps from $S'$ with $S' \in
\mc{F}_{d,i}^{-1}(\mc{F}_{d,i}(S))$ also give solutions to Floer’s equation with the same asymptotics. Moreover, the fibers of the map $\mc{F}_{d,i}$ are one-dimensional, implying that $u$ cannot be rigid, and thus the associated operation is zero.
Now suppose that $d = 2$. Then the forgetful map associates to the single point $[S] \in \mc{R}^{2,f_i}$ the unstable strip with its translation invariant Floer datum. Since non-constant solutions can never be rigid (as, by translating, one can obtain other non-constant solutions), it follows that the only solutions are constant ones, and that the resulting operation is therefore the identity.
[^1]: The author was partially supported by the National Science Foundation through a postdoctoral fellowship — grant number DMS-1204393 — and agreement number DMS-1128155. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
[^2]: We use a cohomological grading convention in this paper, so singular chain complexes are [*negatively graded*]{}.
[^3]: Sometimes $S^1$-equivariant Floer theory is instead defined as Morse theory of an action functional on the $S^1$-Borel construction of the loop space. For a comparison between these two definitions, see [@Bourgeois:2012fk].
[^4]: in particular, the usual equivariant symplectic cohomology $SH^*_{S^1}(M) = H^*(SC^*(M)_{hS^1})$, see [@Bourgeois:2012fk] but note differing conventions regarding e.g., homology vs. cohomology
[^5]: In the proper case, there is an induced isomorphism between Hochschild cohomology and the linear dual of Hochschild homology. In the smooth case, there is an isomorphism between Hochschild cohomology and homology without taking duals.
[^6]: Recall that $C^*(M, \partial M)$ has the trivial $C_{-*}(S^1)$ module structure; the homology of the associated homotopy orbit complex is $H^{*+n}(M, \partial M)( ( u ) ) / u H^{*+n} (M,\partial M)[ [ u ] ]$ where $|u| = 2$, as described in §\[section:s1action\].
[^7]: As shown in [@ganatra1_arxiv; @GPS1:2015], if $[\oc]$ hits $1$, then $[\oc]$ is an isomorphism, and hence by Corollary \[cor:cyclicOC\], $[\widetilde{\oc}^-]$ is too. Hence one can speak about the unique element.
[^8]: The author wishes to thank Mohammed Abouzaid for discussions regarding this point.
[^9]: More precisely $\mu^{1|1}(1,\mathbf{m}) =
\mathbf{m}$ and $\mu^{k|1}(\ldots, 1, \ldots, \mathbf{m})
= 0$ for $k > 1$.
[^10]: In fact, when the Fukaya category is equipped with homotopy units, one can ensure that there is a strict unit element $e^+$ in each self-hom space, for which $\mu^k$ with an $e^+$ element admits a geometric description as above. See e.g., [@Fukaya:2009qf] or [@ganatra1_arxiv].
[^11]: Technically we should write $\r{QH}^*(M)$ in latter case, but additively $\r{QH}^*(M) = H^*(M)$ and correspondingly no sphere bubbling occurs in the moduli spaces we define here, so there is no difference for the purposes of this discussion.
[^12]: In contrast, note that in the proper case all operations should have at least one [*input*]{}.
|
---
abstract: |
We analyze a classification of two main families of controllers that are of interest when the feedback loop is subject to switching propagation delays due to routing via a wireless multi-hop communication network. We show that we can cast this problem as a subclass of classical switching systems, which is a non-trivial generalization of classical LTI systems with time-varying delays.
We consider both cases where delay-dependent and delay-independent controllers are used, and show that both can be modeled as switching systems with unconstrained switchings. We provide NP-hardness results for the stability verification problem, and propose a general methodology for approximate stability analysis with arbitrary precision. We finally give evidence that non-trivial design problems arise for which new algorithmic methods are needed.
author:
- 'Raphaël M. Jungers$^1$, Alessandro D’Innocenzo$^2$ and Maria D. Di Benedetto$^2$ [^1][^2][^3]'
title: 'Feedback stabilization of dynamical systems with switched delays$^*$'
---
Introduction {#secIntro}
============
Wireless networked control systems are spatially distributed control systems where the communication between sensors, actuators, and computational units is supported by a wireless multi-hop communication network. The main motivation for studying such systems is the emerging use of wireless technologies in control systems (see e.g. [@akyildiz_wireless_2004], [@SongIECON2010] and references therein) and the recent development of wireless industrial control protocols, such as WirelessHART (`www.hartcomm2.org`) and ISA-100 (`www.isa.org`). The use of wireless Multi-hop Control Networks (MCNs) in industrial automation results in flexible architectures and generally reduces installation, debugging, diagnostic and maintenance costs with respect to wired networks. Although MCNs offer many advantages, their use for control is a challenge when one has to take into account the joint dynamics of the plant and of the communication protocol (e.g. scheduling and routing).
Wide deployment of wireless industrial automation requires substantial progress in wireless transmission, networking and control, in order to provide formal models and analysis/design methodologies for MCNs. Recently, a huge effort has been made in scientific research on Networked Control Systems (see e.g. [@Zhang2001], [@WalshCSM2001], [@SpecialIssueNCS2004], [@Andersson:CDC05], [@MurrayTAC2009], [@Hespanha2007],[@HeemelsTAC10], and references therein for a general overview) and on the interaction between control systems and communication protocols (see e.g. [@Astrom97j1], [@walsh_stability_2002], [@yook_trading_2002], [@TabbaraCDC2007], [@TabbaraTAC2007]). In general, the literature on NCSs addresses non–idealities (e.g. quantization errors, packets dropouts, variable sampling and delay, communication constraints) as aggregated network performance variables, neglecting the dynamics introduced by the communication protocols. In [@Andersson:CDC05], a simulative environment of computer nodes and communication networks interacting with the continuous-time dynamics of the real world is presented. To the best of our knowledge, the first integrated framework for analysis and co-design of network topology, scheduling, routing and control in a MCN has been presented in [@AlurTAC11], where switching systems are used as a unifying formalism for control algorithms and communication protocols. In [@DiBenedettoIFAC11Stab], stabilizability of a MCN has been addressed for SISO LTI plants. In [@PappasCDC2011] a MCN is defined as an autonomous system where the wireless network *itself* acts as a fully decentralized controller.
![State feedback control scheme of a multi-hop control network.[]{data-label="frame"}](./figures/MCNscheme.eps){width="50.00000%"}
In this paper we focus on the effect of routing, and consider a Multi-hop network $\mathcal G$ that provides the interconnection between a state-feedback discrete-time controller $\mathcal C$ and a discrete-time linear plant $\mathcal P$. The network $\mathcal G$ consists of an acyclic graph where the node $v_c$ is directly interconnected to the controller, and the node $v_u$ is directly interconnected to the actuator of the plant. As classically done in multi-hop (wireless) networks to improve robustness of the system with respect to node failures we exploit redundancy of routing paths, namely we assume that the number of paths that interconnect $\mathcal C$ to $\mathcal P$ is greater than one, and that each path is characterized by a delay in forwarding the data. For any actuation data sent from the controller to the plant, a unique routing path of nodes is exploited: since the choice of the routing path usually depends on the internal status of the network, i.e. because of node and/or link failures, we assume that the routing path is non-deterministically chosen. For the above reason, each input signal will be non-deterministically delayed of a finite number of time steps, according to the chosen routing path.
[Our model is strictly related to discrete-time systems with time-varying delay (see e.g. [@LiuCTA2006], [@HetelCDC2007] and [@ShaoTAC2011]). However the existing results cannot be applied to our model because, due to the routing protocol, control commands generated at different time instants can reach the actuator simultaneously, their arrival time can be inverted, and it is even possible that at certain time instant no control commands arrive to the actuator: these aspects generate switching dynamics that cannot be modeled as a time-varying delay. On the other hand, sufficient stability conditions and LMIs based design procedures have been developed for discrete-time switching systems with time-varying delays (see e.g. [@HetelTAC2006] and [@ZhangCTA2008]), which are much more general than our model and do not take into account the specific switching structure induced by the routing protocol. In summary, the specific delayed dynamics introduced by the routing in the network cannot be modeled by a generic time-varying delay system: we prove instead that it can be modeled by a pure switching system, where the switching matrices assume a particular form. In this paper, we exploit the characteristic system structure obtained by our network model to derive tailored results that hold for our switching model. We will address the stability and design problem for two families of controllers.]{}
The first family will be called *delay-dependent controller*, where we assume that the controller is aware of the routing path (namely it can measure the switching signal) and has memory of a finite number of its previous actuation outputs. We prove that the stability analysis problem is NP-hard. Nevertheless, we show that it is possible to compute the worst case growth of the system with arbitrary precision. We prove that for scalar systems the design problem can be solved in a closed form, and provide a counterexample where, even for a 2-dimensional state with 1-dimensional input, there exists a class of switching signals that make the system uncontrollable.
The second family will be called *delay-independent controller*, where we assume that the controller ignores the routing path (namely it ignores the switching signal) and has memory of a finite number of its previous actuation outputs. Differently from the *delay-dependent controller*, in this case it is not trivial to model the closed loop system as a pure switching system. We first prove that this is always achievable, at the cost of augmenting the state space. Thanks to this result, as in the previous case, we show that it is possible to compute the worst case growth of the system with arbitrary precision. We finally provide an example of a scalar system where, depending on the dynamics of the plant, the system can be not stabilizable, stabilizable with memory, and stabilizable without memory. This motivates further studies to develop design methods to guarantee the existence of a stabilizing controller and efficient algorithms to compute it.
From the network point of view, the distinction between the above models depends on the protocol used to route data (see [@YangWiMesh2011] and references therein for an overview on routing protocols for wireless multi-hop networks). If the controller node $v_c$ of $\mathcal G$ is allowed by the protocol to chose a priori the routing path (e.g. source routing protocols), then we can assume that the controller is aware of the routing path and the associated delay, and therefore is also aware of the switching signal. If instead the protocol allows each communication node to choose the next destination node according to the local neighboring network status information (e.g. hop-by-hop routing protocols, such as WirelessHART), then we can NOT assume that the controller is aware of the routing path, and therefore also of the switching signal.
The paper is organized as follows. In Section \[secProbForm\] we provide the problem formulation, and define the two families of controllers introduced above. In Sections \[secDelayDependent\] and \[secDelayIndependent\] we address respectively the *delay-dependent* and the *delay-independent* cases. In Section \[secConclusions\] we provide concluding remarks and open problems for future research.
Problem formulation {#secProbForm}
===================
In this paper we will address the problem of stabilizing a discrete-time LTI system of the form $$x(t+1) = A_P x(t) + B_P u(t),\quad y(t) = x(t),\quad t \geq 0,$$ with $A_P \in \mathbb R^{n \times n}$ and $B_P \in \mathbb R^{n \times m}$, using a *switching linear controller*[^4] $\mathcal C$. We assume that the control signal $v(t)$ generated by $\mathcal C$ is transmitted to the actuator of the plant $\mathcal P$ via a multi-hop control network [@AlurTAC11]. The network $\mathcal G$ consists of an acyclic graph $(V,E)$, where the node $v_c \in V$ is directly interconnected to the controller $\mathcal C$, and the node $v_u$ is directly interconnected to the actuator of the plant $\mathcal P$. In order to transmit each actuation data $v(t)$ to the plant, at each time step $t$ a unique path of nodes that starts from $v_c$ and terminates in $v_u$ is exploited. As classically done in multi-hop (wireless) networks to improve robustness of the system with respect to node failures, we exploit redundancy of routing paths, therefore the number of paths that can be used to reach $v_u$ from $v_c$ is assumed to be greater than one. To each path a different delay can be associated in transmitting data from $v_c$ to $v_u$, depending on the transmission scheduling and on the number of hops to reach the actuator. Since the choice of the routing path usually depends on the internal status of the network (e.g. because of node and/or link failures, bandwidth constraints, security issues, etc.), we assume that the routing path is non-deterministically chosen at each time step $t$. Therefore, the signal $u(t)$ will be non-deterministically delayed of a finite number of time steps, according to the chosen routing path. For the reasons above, we can model the dynamics of a multi-hop control network as follows:
\[def-general-sdsystem\] The dynamics of the interconnected system $N$ can be modeled as follows: $$\label{sdsystem}
x(t+1) = A x(t) + B u(v(t - d_{max}:t),\sigma(t - d_{max}:t)),$$ where $v(t - d_{max}:t), \sigma(t - d_{max}:t))$ represent the latest $d_{max}$ values of $v(\cdot),\sigma(\cdot).$
In the above equation, $\sigma(t)\in D: \,t \geq 0$ is a non-deterministic switching signal, and $A,B$ model the interconnected switching dynamics of the controllability network $\mathcal G_{\mathcal R}$ and of the plant $\mathcal P.$ The set $D=\{d_1,\dots,d_{|D|}\}:\, d_i\in {{\mathbb N}}, d_{max}=\max{(D)}$ is the set of possible delays introduced by all routing paths.
The above model is quite general, and allows representing a wide range of routing communication protocols for (wireless) multi-hop networks [@YangWiMesh2011]. We will prove in the following sections that the model above can be cast as a pure switching system. Therefore, all tools developed for general switching systems can be used for stability analysis and controller design (e.g. [@jungers_lncis; @sun-ge]). However, the particular delay model that we are considering makes our system a special case of general switching systems, endowed with a characteristic matrix structure. In this paper we aim at exploiting such special structure to derive tailored results that hold for our specific switching model. We will show that, already in one of the simplest framework that one could imagine, nontrivial situations and challenging problems occur. As discussed above we think that such results, even if they cannot be applied to general switching systems, have an important impact on analysis and design problems arising in multi-hop control networks.
As discussed in Section \[secIntro\], our switching model is a non-trivial generalization of classical LTI systems where the control signal can be subject to a variable delay before arriving to the plant. As we will see below, there are favorable situations where one can design a controller, just like in the LTI case, which stabilizes the system, no matter what delay actually occurs.\
We emphasize that several variations of this model are possible. For instance, in our setting, it could happen, during the run of a , that at some particular time, no feedback signal comes back to the plant. We assume that in this situation the actuation input to the plant is set to zero. In many practical situations, it is also possible to implement a hold that would keep the previous input signal and resend it to the plant at the particular times for which the switching signal implies that no $v(t)$ from the controller is conveyed towards the plant. We defer the comparison of such variants for further studies.
It remains to clarify what we meant by *switching linear controller* in the control scheme definition. In fact, we have to specify the input that the controller receives in order to compute its output. As we will see, there is not a unique straightforward generalization of classical LTI systems, and we make here an important distinction between the situation where the controller knows at each time step $t$ the switching signal $\sigma(t)$ and the situation where it does not. As illustrated in Section \[secIntro\], this distinction depends on the protocol used to route data in the network. If the controller node selects a priori the whole routing path up to the actuator node, then we assume that the controller is aware of the switching signal $\sigma(t)$ for each time step $t$. If instead each communication node locally selects the next destination node, then we assume that the controller ignores the switching signal.
For classical (i.e. non-switched) feedback systems with fixed delay, it is well known that the system can be neither controllable nor stabilizable if the feedback only depends on $x(t)$ (that is, if the controller does not have a memory of its previous outputs $v(t'),\ t-d_{max}<t'<t-1$). Therefore, we take it for granted here that the controller has a memory of its past $d_{max}$ outputs.
\[def-dep\]\[eq-ut\] A *linear delay-dependent* controller is defined for $\sigma(t)=d$ as: $$\label{eq-vt-dep} v(t)= K(d) \tilde u(t),$$ where $\tilde u(t) = (x(t),u_{1}(t),u_{2}(t),\dots, u_{d_{max}}(t)),$ $$\label{eq-ud} u_{d}(t)=\sum_{t'<t: t'+\sigma(t')=t+d} v(t'),$$ and $K(d)\in {{\mathbb R}}^{m\times (md_{max}+n)}$.
\[def-indep\] A *linear delay-independent* controller is defined as: $$\label{eq-vt} v(t)= K \tilde v(t),$$ where $$\label{eq-def-vtilde}\tilde v(t) = (x(t), v(t-d_{max}),\dots,v(t-1))),$$ and $K\in {{\mathbb R}}^{m\times (md_{max}+n)}$.
The asymmetry between Definitions \[def-dep\] and \[def-indep\] deserves an explanation: in Definition \[def-dep\], it is assumed that the controller knows the previous values of the switching signal, and then can reconstruct $u_d(t)$ by applying equation (\[eq-ud\]). In this situation the controller, in order to compute $v(t)$, can use the values of $u_d(t)$, which is the sum of all past control commands $v(t'), t'<t$ that will reach the actuator after $d$ time steps. In Definition \[def-indep\] however, we suppose that the only available information for the controller is $x(t),$ and this explains why the only variables it can use are $x(t)$ and the $d_{max}$ past control commands $v(t'), t-d_{max} \leq t' \leq t-1$.
We say that a system is *stable* if, for any switching signal $\sigma(t)$ and for any initial condition $\tilde v(0),$ $$\lim_{t\rightarrow \infty}\tilde v(t)=0,$$ where $\tilde v(t)$ is as defined in (\[eq-def-vtilde\]), [and represents an extended state space containing both the state of the plant and the past $d_{max}$ outputs of the controller.]{}
Delay-dependent feedback {#secDelayDependent}
========================
In this section we show that it may be hard or impossible to analyze stability of a . We first show how to model this problem as a switching system stability analysis, and then prove that the problem is NP-hard. We then show examples for which such a controller can be designed, and other for which no controller exists.
Modeling
--------
Since the controller is aware of the delays in the delay-dependent framework, we are allowed to use the variables $u_s(t)$ in the state-space, and it is then easy to write the corresponding closed loop system [from Definitions \[def-general-sdsystem\] and \[def-dep\].]{} We obtain the following equations:
\[prop-delay-dependent-cl\]A and delay-dependent controller can be modeled as a switching system with arbitrary switching signal as follows: $$\label{switched-closed-loop} \tilde u({t+1})= M_{\sigma(t)} \tilde u(t),\quad M_{\sigma(t)} \in \Sigma,$$ where $$\label{eq-sigma}\Sigma=\left\{\begin{pmatrix} A&B&0&\dots & 0\\ 0&0&I&\dots & 0\\0&0&0&\ddots&0\\0&0&0&\dots &I\\0&0&0&\dots &0 \end{pmatrix}+ E(d) K(d):d\in D\right \},$$ and $E(d)$ is the block-column matrix with only zero blocks, except block $(d+1)$ which is equal to the identity if $d\neq0$ and to $B$ if $d=0$. [More precisely, for a same switching signal, the dynamics of and are equal.]{}
We skip the proof, which is obvious for the delay-dependent case. As we will see below, the reduction is less obvious for the delay-independent case, and necessitates an increase of the dimension of the state space.
Analysis
--------
As illustrated above, a can be put in the well studied framework of linear switching systems with arbitrary switching signal. Even though these systems have been at the center of a huge research effort in the last decades [@sun-ge; @LeeD07; @jungers_lncis; @valcher-positive], they are known to be very difficult to handle. Nevertheless, it follows from Proposition \[prop-delay-dependent-cl\] that one can check the stability of a given with arbitrary precision:
\[cor-approx-dep\] For any $\epsilon,$ there exists an algorithm which, given a and delay-dependent feedback, computes in finite time the worst rate of growth of the system, up to an error of $\epsilon.$ More precisely, for any real $r>0$ the algorithm decides in finite time whether
- $\exists K \in {{\mathbb R}}: \forall \sigma, \forall t,\ |\tilde u(t)|\leq K (r+\epsilon)^t$;
- $\exists K \in {{\mathbb R}}, K>0, \exists \sigma:\ |\tilde u(t)|\geq K (r-\epsilon)^t.$
Proposition \[prop-delay-dependent-cl\] reformulates the as a classical switching system, thus it is possible to apply one of the classical stability decision procedures derived in [@protasov-jungers-blondel09 Corollary 3.1] or [@ajprhscc11 Theorem 6.1].
The above corollary provides a tool to approximate the worst rate of growth, by bisection on $r$. Thus, it is possible to decide with an arbitrary precision whether a and delay-dependent feedback is stable. However, no polynomial-time algorithm is known to solve this problem, and hence the proposed solution does not work in polynomial time. This is not surprising in view of our next result: we show that given a system and its controller, it is NP-hard to decide whether the controller asymptotically stabilizes the system.
Unless $P=NP,$ there is no polynomial-time algorithm that, given a linear system $A,$ its control matrices $K_i,$ and a set of delays $D,$ decides whether the corresponding delay-dependent is stable.
Also, the question of whether the system remains bounded is turing-undecidable. This is true even if the matrices have nonnegative rational entries, and the set of delays is $\{0,1\}.$
Our proof works by reduction from the matrix semigroup stability, and the matrix semigroup boundedness, which are well known to be respectively NP-hard and Turing undecidable [@jungers_lncis Theorem 2.4 and Theorem 2.6]. In this problem, one is given a set of two matrices $\Sigma=\{A_1,A_2\}\subset {{\mathbb{Q}}}_+^{n\times n}$ (${{\mathbb{Q}}}_+$ is the set of nonnegative rational numbers) and one is asked whether for any sequence $(i_t)_0^\infty,\ i_t\in [1,2],$ the corresponding product $A_{i_1}A_{i_2}\dots A_{i_T}$ converges to the zero matrix when $T\rightarrow \infty$ (respectively remains bounded when $T\rightarrow \infty$).
Let us consider a particular instance $\Sigma=\{A_1,A_2\}\in {{\mathbb{Q}}}_+^{n\times n}$ of the matrix semigroup stability (resp. boundedness) problem. We build a delay-dependent whose closed loop system can be written as follows: $$\begin{aligned}
\tilde u({t+1})&=&M_i \tilde u(t),\quad M_i\in \Sigma', \end{aligned}$$ where $\Sigma'$ is a set of $2n \times 2n$ matrices. The set $\Sigma'$ is product-bounded (resp. stable) if and only if $\Sigma$ is.
Our construction is as follows: we set $D=\{0,1\}$ as the set of delays, and we build a with a plant of dimension $n$ as follows: the system matrix is given by $$A=0,\, B=I,$$ and the feedback matrix (in block form) by $$K_d= \begin{pmatrix} A_1 &A_2 \end{pmatrix}$$ for $d=0,1.$ Thus, the corresponding closed loop feedback switching system can be expressed from Proposition \[prop-delay-dependent-cl\] as $$\tilde u({t+1})= M_i \tilde u(t)\quad M_i \in \Sigma',$$ where $$\label{eq-np-switching}\Sigma'= \left\{ \begin{pmatrix} A_1 &A_2 \\0&0\end{pmatrix},\begin{pmatrix}0&0\\ A_1 &A_2 \end{pmatrix}\right\}.$$ Writing $\tilde u(t) = (x(t), u_1(t))$ we have that, depending on $\sigma(t)$, either $$\tilde u({t+1})= (A_1x(t)+A_2u_1(t),0)$$ or $$\tilde u({t+1})= (0,A_1x(t)+A_2u_1(t)).$$ It is straightforward to see from Equation (\[eq-np-switching\]) that the set $\Sigma'$ is stable (resp. product bounded) if and only if $\Sigma$ is. Indeed, the blocks in the products of matrices in $\Sigma'$ are arbitrary products of matrices in $\Sigma.$ This concludes the proof.
It is not known (to the best of our knowledge) whether the matrix semigroup stability problem is Turing decidable (say, for matrices with rational entries). Thus, the above proof does not allow us to conclude that the stability problem is undecidable. This is why we only claim that the stability problem is NP-hard, while the boundedness problem is provably Turing undecidable.
Design
------
We complete this section by addressing the *design question:* given a , find the actual value of the delay-dependent controllers $K(d)$ such that the resulting system is stable by considering the particular structure exhibited in (\[eq-sigma\]). As was said above, much less is known in the literature about the design of regular switching systems. We first completely solve the linear delay-dependent controller design for one-dimensional systems ($n=m=1$). Of course, we have to assume that the system is controllable (i.e. $b\neq 0$), since it must already be the case for a fixed delay. For classical LTI systems with fixed delay, in the case where the controller has a memory of its past $d_{max}$ outputs, a solution that drives the trajectory exactly onto the origin in finite time is given by an extension of the Ackermann formula: $$\label{eq-ack}K_{ack,d}=(-a^{d+1}/b,-a^{d},-a^{d-1},\dots,-a).$$ It appears that for a too, there is a solution that reaches exactly the origin in finite time $d_{max}$, where $d_{max}$ is the maximum delay:
Consider System (\[sdsystem\]) with $n=m=1$ and suppose $a,b\neq 0.$ Then, the system is controllable with the following delay-dependent feedback controller: $$K(d)=K_{ack,d_{max}}/(a^{d_{max}-d}),$$ where $K_{ack,d_{max}}$ is the controller as in equation for a system with fixed-delay $d_{max}$. Moreover the system reaches exactly zero at latest at time $t=d_{max}+1$.
Let $\tilde u(t) = (x(t),u_{1}(t),u_{2}(t),\dots, u_{d_{max}}(t))$ be the state of the system. From Equation (\[eq-ack\]) we have, at any time $t,$ $$\begin{aligned}
\label{eq-ackerman-zero} \nonumber(a^{d_{max}+1}/b)x(t) + \sum_{s=1}^{d_{max}} a^{d_{max}-s+1}u_{s}(t)&&\\ +K(\sigma(t))\tilde u(t)(a^{d_{max}-{\sigma(t)}})&=&0. \end{aligned}$$ By Definitions \[def-dep\] and \[sdsystem\] it follows that for $s=1,\dots,d_{max},$ $$\begin{aligned}
\label{eq-us}
u_s(t+1)&=&u_{s+1}(t)\quad \mbox{if }\sigma(t)\neq s\\
\nonumber u_s(t+1)&=&u_{s+1}(t)+v(t)\quad \mbox{if }\sigma(t)= s,\end{aligned}$$ where we fix for conciseness of notations that $u_{d_{max}+1}=0.$ Observe that the last term in the left-hand side of Equation (\[eq-ackerman-zero\]) is equal to $v(t)(a^{d_{max}-{\sigma(t)}}).$ Multiplying that equation by $a,$ and making use of Equation , we obtain: $$\begin{aligned}
0 =& a^{d_{max}+1}/b(ax(t)+bu_1(t)+bz(t))\\
&+\sum_1^{d_{max}}{a^{d_{max}+1-s}u_s(t+1)}\\
=& (a^{d_{max}+1}/b)x({t+1})+\sum_1^{d_{max}}{a^{d_{max}+1-s}u_s(t+1)} \\
=&K_{ack,d_{max}}\tilde u(t+1)\\
=&v({t+1})(a^{d_{max}-{\sigma(t+1)}}),\end{aligned}$$ where, again for conciseness, we introduce the variable $z$ such that $z(t)=v(t)$ if $0 \in D$ and $\sigma(t)=0$, and $z(t)=0$ otherwise. In conclusion, if the controller is applied at time $1,$ the output of the controller at time $2$ is $v_{2}=0.$ Thus, by induction, $$\forall t'>1,\, v({t'})=0.$$ This implies (see Equation (\[eq-ud\])) that $$\forall t''>d_{max},\forall s,\,u_s{(t'')}=0.$$ In turn, since $v_{d_{max+1}}= K_{ack,\sigma({d_{max}+1)}}\tilde u(d_{max}+1)=0,$ this implies that $x_{d_{max}+1}=0.$
We now show that the situation becomes more complex as soon as the dimension of the plant is equal to $2.$ The following is an example of a system with $n=2,m=1$ that is stabilizable with fixed delays, but not with switching delays.
Consider a with the following values: $$A=\begin{pmatrix}0& 2\\2& 0\end{pmatrix},\quad B=\begin{pmatrix}0& 1\end{pmatrix}^T,$$ $$D=\{0,1\},\quad \sigma(t)=t \mbox{ mod } 2.$$
That is, $\sigma= 0$ when $t$ is even, and $1$ when $t$ is odd. Then, if $x(0)=(1, 0),$ the system is not controllable. Indeed, one can show by induction that for any even time $t,$ $x_1(t)=2^t.$
delay-independent feedback {#secDelayIndependent}
==========================
In this section we show that, when the controller does not depend on the delay, the situation is harder because one has to design a single controller that would work for any possible switching signal. Differently from the *delay-dependent controller*, in this case it is not trivial to model the closed loop system as a pure switching system. We first prove that this is always achievable, at the cost of augmenting the state space. This implies, as in the delay-dependent case, that it is possible to compute the worst case growth with arbitrary precision. We finally provide an example of a scalar system where, according to the dynamics of the plant, the system can be not stabilizable, stabilizable with memory, and stabilizable without memory.
Modeling
--------
The delay-independent controller has no access to the variables $u_s(t),$ since one needs the previous values of the switching signal in order to reconstruct these variables. The only variables that the controller can use are its previous outputs $v({t'}):t'<t.$ As done in Proposition \[prop-delay-dependent-cl\], the dynamics of the closed loop system can be written as follows:
$$\label{switched-closed-loop-independent} \tilde v({t+1})= M(\sigma,t) \tilde v(t)\quad M(\sigma,t) \in \Sigma,$$
where $$\begin{aligned}
\label{eq-sigma-cl-independent} \Sigma&=&\left\{\begin{pmatrix} A&0&0&\dots & 0\\ 0&0&I&\dots & 0\\0&0&0&\ddots&0\\K_0&K_1&K_2 &\dots&K_{d_{max}} \end{pmatrix} \right. \\&&\left. +\sum_{t': t'+\sigma(t')=t}E(1) B E^T({d_{max}+1-\sigma(t')})\right \}.\nonumber\end{aligned}$$
However, there are two important differences between the systems defined by Equations (delay-dependent) and (delay-independent). First, the set $\Sigma$ in (\[eq-sigma-cl-independent\]) has a number of matrices that can be exponential in the number of delays $|D|.$ Second, the closed-loop formulation (\[switched-closed-loop-independent\]) is not a switching system with arbitrary switching signal, as the matrix $M(\sigma,t)$ depends not only on $\sigma(t),$ but also on the values $\sigma(t')$ for $t'<t,$ as one can check in (\[eq-sigma-cl-independent\]). Thus, this setting seems harder to represent as a pure switching system because of this correlation in the succession of matrices. Even though recent methods based on LMI criteria have been proposed that offer a natural framework for analyzing switching signals described by a regular language (e.g. [@ajprhscc11; @ajprhscc12; @LeeD07]), it would be convenient to have a formulation of the closed loop switching system without any constraint on the switching signal. Indeed, more methods for analyzing switching systems have been designed in the general framework of unconstrained switching. In the following theorem we show that it is always possible to model system as a pure switching system, at the cost of augmenting the state space.
\[thm-ssreduction-independent\] Any $n$-dimensional and delay-independent control of dimension $m$ and set of delays $D$ can be represented as an switching system with arbitrary switches among $|D|$ matrices, characterized by a $(n+2d_{max}m)$-dimensional state space.
The main idea of the proof is to make use of *both $u_d(t)$ and $v(t)$* in the closed loop state-space representation of the system. Recall that $u_d(t)$ is the sum of the previous outputs of the controller, that are forecast to arrive at the plant at time $t+d,$ and that $v(t)$ is the output of the controller at time $t.$ See Equations (\[eq-ud\]) and (\[eq-vt\]).\
Of course, the controller does not know $u_d(t)$ in this delay-independent setting (we will take that into account in the construction), but this variable is needed in order to reconstruct the feedback signal. On the other hand, $v(t)$ is needed in order to represent the memory of the controller. We now formally describe the state-space, and then the matrices. We set $d_{max}=\max(D)$ as usual.
We define the state-space vector $\tilde w(t)\in {{\mathbb R}}^{n+2d_{max}m}$ as follows: $$\tilde w (t)=(x(t),u_1(t),\dots, u_{d_{max}}(t),v(t-d_{max}),\dots,v(t-1)).$$ Then, for any $d\in D,$ $\tilde w(t+1)$ can be expressed as a linear function of $\tilde w (t).$ Indeed, for any $d\in D,$ the following equations describing the computation of $\tilde w(t+1)$ are linear, and only depend on $\tilde w(t):$ $$\begin{aligned}
\nonumber x(t+1)&=&Ax(t)+Bu_1(t) + B z_{0,d}(t),\\
\nonumber u_s(t+1)&=&u_{s+1}(t) + z_{s,d}(t), \quad 1\leq s\leq d_{max} \\
\nonumber V(t+1)&=& (0,\dots,0,K_0x_{t})^T+\\&&\begin{pmatrix}0&I&\dots&0\\0&0&\ddots&0\\0&\dots&0&I\\K_1 &\dots &&K_{d_{max}}\end{pmatrix}V(t).\end{aligned}$$ In the above equations, $$K=(K_0,K_1,\dots,K_{d_{max}})\in {{\mathbb R}}^{m\times (n+d_{max} m)}$$ is the linear controller, $$V(t)=(v(t-d_{max}),\dots,v(t-1))^T\in{{\mathbb R}}^{d_{max} m}$$ is the memory of the controller, and $z_{s,d}:\, s=0,\dots, d_{max}$ represents the controller output, to be fed back to the plant with a delay $d:$ $$\begin{aligned}
\nonumber z_{s,d}&= &K\begin{pmatrix}x(t)\\V(t) \end{pmatrix} \quad \mbox{ if } s=d,\\\nonumber&=& 0\quad \mbox{otherwise}.\end{aligned}$$
Analysis
--------
Theorem \[thm-ssreduction-independent\] implies that, given a , one can build a corresponding set of matrices, and analyze the stability of the corresponding switching system in order to check for the stability of the given system. In particular the following corollary is equivalent to Corollary \[cor-approx-dep\]:
For any $\epsilon,$ there exists an algorithm which, given a and delay-independent controller, computes in finite time the worst rate of growth of the system, up to an error of $\epsilon.$
Design
------
We end this section by commenting on the design problem, that is, to find suitable values for the entries in $K$ so that the corresponding system is stable. Since it is harder to control such a system than with a delay-dependent controller, this problem might well have no solution, depending on the set $D$ and the actual value of $A$ and $B.$ Below is a simple example, which shows that several situations can be possible, already in the arguably simplest case $n=m=1,$ and $D=\{0,1\}.$
\[ex-indep\] In this example we assume that the controller stores the previous value of $x(t)$ instead of the previous value of $v(t).$ We make this choice for the sake of clarity, in order to have simpler matrices in the equivalent switching system. It is easy to check that in this slightly modified setting, one can still apply the trick of Theorem \[thm-ssreduction-independent\] and obtain the following three-dimensional switching system as a representation of a one-dimensional : $$\label{eq-ssreduction-independent}\begin{pmatrix}x({t})\\ x({t+1})\\ u_1(t+1)\end{pmatrix}= M_{\sigma(t)} \begin{pmatrix}x({t-1})\\ x(t)\\ u_1(t)\end{pmatrix}.$$
where $$M_0=\begin{pmatrix}0& 1 &0\\ bk_1 &a+ bk_2& 1\\0& 0 &0 \end{pmatrix}, \quad M_1=\begin{pmatrix}0 &1& 0\\ 0& a& 1\\ bk_1 &bk_2 &0\end{pmatrix}.$$
The controller stores the value of $x({t-1})$ for one iteration. It then makes use of it and of the current value $x(t)$ for computing its output $v(t).$ If the delay is 1 $v(t)$ is put “in the queue” (third entry of the vector), while if the delay is zero it is directly added in the plant in order to compute $x({t+1}).$ Let us consider $b=1,\ D=\{0,1 \}$. Depending on the other values, we obtain the following cases:
- For $a>3,$ system (\[eq-ssreduction-independent\]) is unstable, whatever controller $K$ is applied. This can be seen by observing that in this case, $trace(M_1)\geq 3,$ hence $M_1$ is unstable. By Theorem \[thm-ssreduction-independent\], the corresponding is uncontrollable.
- For $a<1$ the system is clearly stabilizable without any controller ($k_1=k_2=0$), since it is the case for the autonomous stable dynamical system.
- For a=1.1 the system is controllable without using memory (i.e. $k_1=0$), e.g. by taking $k_2=-0.5,$ and the switching system (\[eq-ssreduction-independent\]) is stable.
- Finally, for $a=2$ the system is still controllable, but in this case one needs $k_1\neq 0$: indeed, if $k_1=0$ one can restrict himself to the $2 \times 2$ lower-right corner of the matrices, and this subsystem is unstable because $trace(M_0)\geq 2$. On the other hand stabilizing controllers exist with $k_1\neq 0,$ as for instance $k_1=0.4,k_2=-1.5.$
In the last two cases, in order to check for the validity of the proposed controller, one can check that the joint spectral radius of the set $\{M_0,M_1\}$ corresponding to the proposed controller is smaller than one, for instance by making use of the JSR toolbox, available on the web [@jsr-toolbox].
Conclusion {#secConclusions}
==========
In this paper, we studied the algorithmic analysis and design of linear discrete-time systems with switched delays. Although many different control strategies are possible, we focused on two simple models and showed that many variate situations can occur. We presented favorable cases, where the problem is algorithmically solvable in polynomial time, and proved that the problem is NP-hard in general. More importantly, we provided an algorithmic procedure to decide stability of a given system with arbitrarily small error (of course, with running time increasing when the error decreases). Our work raises many natural questions. We end this paper by mentioning some of them.\
For the *design questions*, in the one-dimensional case, we showed that the problem is easy in the delay-dependent framework, but the possibility to design a controller in a delay-independent framework seems less straightforward. This leads to our first question:
What are the particular values of $a,b$ and the sets $D$ in the one dimensional case (see Example \[ex-indep\]) for which there is a stabilizing delay-independent linear feedback? Is there an efficient algorithm for deciding the stability/boundedness in this case?
Is the stability/boundedness also NP-hard to decide in the delay-independent framework?
In our framework, the switching signal might cause the feedback signal to be empty at some times. We implemented it as a zero signal, but one might for instance implement a hold, which would in this case repeat the previous feedback signal. What are the situations where the implementation of this hold improves controllability? How to recognize such situations?
How is the situation changed if one is interested in the stability with probability one instead of the worst-case stability?
Suppose that each delay in $D$ appears with a certain probability. Then, one might only require stability with probability one for System (\[sdsystem\]). It is known that the almost sure stability of a classical switching system is ruled by its so-called *Lyapunov Exponent.* Recently, convex optimization techniques have been proposed in order to approximate this quantity [@protasov-jungers-lyap]. So, for the analysis question, these techniques can be applied to the equivalent switching system reformulation that we provided in this paper. We leave the design question for further work.
[1]{}
I. F. Akyildiz and I. H. Kasimoglu Wireless Sensor and Actor Networks: Research Challenges. , 2(4):351–367, 2004.
S. Han, Z. Xiuming, K. M. Aloysius, M. Nixon, T. Blevins and D. Chen Control over WirelessHART network. , pp.2114–2119, 2010.
W. Zhang, M.S. Branicky and S.M. Phillips Stability of Networked Control Systems. , 21(1):84–99, 2001.
G. C. Walsh and H. Ye Scheduling of Networked Control Systems. , 21(1):57–65, 2001.
P. Antsaklis and J. Baillieul Guest Editorial Special Issue on Networked Control Systems. , 49(9):1421–1423, 2004.
M. Andersson, D. Henriksson, A. Cervin and K. Arzen Simulation of Wireless Networked Control Systems. , pp.476–481, 2005.
V. Gupta, A. F. Dana, J. P. Hespanha, R. M. Murray and B. Hassibi Data Transmission Over Networks for Estimation and Control. , 54(8):1807–1819, 2009.
J. P. Hespanha, P. Naghshtabrizi and Y. Xu A Survey of Recent Results in Networked Control Systems. , 95(1):138–162, 2007.
W.P.M.H. Heemels, A.R. Teel, N. van de Wouw and D. Nešić Networked Control Systems With Communication Constraints: Tradeoffs Between Transmission Intervals, Delays and Performance. , 55(8):1781–1796, 2010.
K. Aström and B. Wittenmark Computer-Controlled Systems: Theory and Design. , 1997.
G. C. Walsh, H. Ye and L. G. Bushnell Stability Analysis of Networked Control Systems. , 10(3):438–446, 2002.
J. K. Yook, D. M. Tilbury, N. R. Soparkar, E. Syst and E. S. Raytheon Trading Computation For Bandwidth: Reducing Communication Indistributed Control Systems Using State Estimators. , 10(4):503–518, 2002.
M. Tabbara and D. Nešić Input-to-State & Input-Output Stability of Networked Control Systems. , pp.3321–3326, 2007.
M. Tabbara, D. Nešić and A. R. Teel Stability of Wireless and Wireline Networked Control Systems. , 52(7):1615–1630, 2007.
R. Alur, A. D’Innocenzo, K. H. Johansson, G. J. Pappas and G. Weiss Compositional Modeling and Analysis of Multi-Hop Control Networks. , 56(10):2345–2357, 2011.
M. D. Di Benedetto, A. D’Innocenzo and E. Serra Fault Tolerant Stabilizability of Multi-Hop Control Networks. , pp.5651–5656, 2011.
M. Pajic, S. Sundaram, G. J. Pappas and R. Mangharam Topological Conditions for Wireless Control Networks.
X. G. Liu, R. R. Martin, M. Wu and M. L. Tang Delay-dependent robust stabilisation of discrete-time systems with time-varying delay. , 153(6):689–702, 2006.
W. Jiang, E. Fridman, A. Kruszewski and J.-P. Richard Switching controller for stabilization of linear systems with switched time-varying delays. , Shanghai, P.R. China, December 16-18, pp.7923–7928, 2009.
L. Hetel, J. Daafouz and C. Iung Stability analysis for discrete time switched systems with temporary uncertain switching signal. , New Orleans, LA, USA, December 12-14, pp.5623–5628, 2007.
H. Shao and Q.-L. Han New Stability Criteria for Linear Discrete-Time Systems With Interval-Like Time-Varying Delays. , 56(3):619–625, 2011.
L. Hetel, J. Daafouz and C. Iung Stabilization of Arbitrary Switched Linear Systems With Unknown Time-Varying Delays. , 51(10):1668–1674, 2006.
L. Zhang, P. Shi and M. Basin Robust stability and stabilisation of uncertain switched linear discrete time-delay systems. , 2(7):606–-614, 2008.
Y. Yang, J. Wang and R. Kravets Designing routing metrics for mesh networks. , Santa Clara, CA, 2005.
S. S. Ge and Z. Sun. . Communications and control engineering series. Springer, 2005.
R. M. Jungers. The joint spectral radius, theory and applications. In [*Lecture Notes in Control and Information Sciences*]{}, volume 385. Springer-Verlag, Berlin, 2009.
E. Fornasini and M.-E. Valcher. Stability and stabilizability criteria for discrete-time positive switched systems. , 2012.
J. W. Lee and G. E. Dullerud. Uniformly stabilizing sets of switching sequences for switched linear systems. , 52(05):868–874, 2007.
V. Yu. Protasov, R. M. Jungers, and V. D. Blondel. Joint spectral characteristics of matrices: a conic programming approach. , 31(4):2146–2162, 2010.
P. Parrilo, A. A. Ahmadi, R. M. Jungers and M. Roozbehani. Analysis of the joint spectral radius via lyapunov functions on path-complete graphs. In [*Hybrid Systems: Computation and Control (HSCC’11)*]{}, Chicago, 2011.
P. Parrilo, A. A. Ahmadi, R. M. Jungers and M. Roozbehani. When is a set of [L]{}[M]{}[I]{}s a sufficient condition for stability? , 2012.
V. Yu. Protasov and R. M. Jungers. Convex optimization methods for computing the lyapunov exponent of matrices. , 2011. vailable online at [http://perso.uclouvain.be/raphael.jungers/publis\_dispo/protasov\_jungers%
\_lyapunov.pdf](http://perso.uclouvain.be/raphael.jungers/publis_dispo/protasov_jungers%
_lyapunov.pdf).
G. Vankeerberghen, J. M. Hendrickx, and R. M. Jungers. The [JSR]{} toolbox. 2011. Matlab Central <http://www.mathworks.com/matlabcentral/fileexchange/33202-the-jsr-toolbox>.
J.-P. Richard Time-delay systems: an overview of some recent advances and open problems. , 39:1667–1696, 2003.
[^1]: $^1$ICTEAM Institute, Université catholique de Louvain, Louvain-la-Neuve, Belgium. Email: [`raphael.jungers@uclouvain.be`]{}
[^2]: $^2$Department of Electrical and Information Engineering, Center of Excellence DEWS, University of L’Aquila, Italy. Email: [`alessandro.dinnocenzo@univaq.it`]{}, [`mariadomenica.dibenedetto@univaq.it`]{}
[^3]: $^*$The research leading to these results has received funding from the European Union Seventh Framework Programme \[FP7/2007-2013\] under grant agreement n°257462 HYCON2 Network of excellence. R.J. is supported by the “Communauté francaise de Belgique - Actions de Recherche Concertées”, and by the Belgian Programme on Interuniversity Attraction Poles initiated by the Belgian Federal Science Policy Office. R.J. is a F.R.S.-FNRS fellow.
[^4]: We will clarify in the following what we mean for *switching linear controller.*
|
---
abstract: 'Weak measurements are a new tool for characterizing post-selected quantum systems during their evolution. Weak measurement was originally formulated in terms of von Neumann interactions which are practically available for only the simplest single-particle observables. In the present work, we extend and greatly simplify a recent, experimentally feasible, reformulation of weak measurement for multiparticle observables \[Resch and Steinberg (2004, Phys. Rev. Lett., 92, 130402)\]. We also show that the resulting “joint weak values” take on a particularly elegant form when expressed in terms of annihilation operators.'
address:
- 'Department of Physics, University of Toronto, 60 St. George Street, Toronto ON M5S 1A7, Canada'
- 'Institut für Experimentalphysik, Universität Wien, Boltzmanngasse 5, A-1090 Vienna, Austria'
author:
- 'J.S. Lundeen'
- 'K.J. Resch'
title: Practical measurement of joint weak values and their connection to the annihilation operator
---
,
Weak measurement ,Entanglement ,Post-selection ,Annihilation Operator 03.65.Ta ,02.50.-r ,06.30.-k ,03.67.-a
Weak measurement was originally proposed by Aharanov, Albert and Vaidman (AAV) [@AAV] as an extension to the standard von Neumann (“strong”) model of measurement. A weak measurement can be performed by sufficiently reducing the coupling between the measuring device and the measured system. In this case, the pointer of the measuring device begins in a state with enough position uncertainty that any shift induced by the weak coupling is insufficient to distinguish between the eigenvalues of the observable in a single trial. While at first glance it may seem strange to desire a measurement technique that gives less information than the standard one, recall that the entanglement generated between the quantum system and measurement pointer is responsible for collapse of the wavefunction. Furthermore, if multiple trials are performed on an identically-prepared ensemble of systems, one can measure the average shift of the pointer to any precision – this average shift is called the weak value. A surprising characteristic of weak values is that they need not lie within the eigenvalue spectrum of the observable and can even be complex [@peres; @leggett; @AV]. On the other hand, an advantage of weak measurements is that they do not disturb the measured system nor any other simultaneous weak measurements or subsequent strong measurements, even in the case of non-commuting observables. This makes weak measurements ideal for examining the properties and evolution of systems before post-selection and might enable the study of new types of observables. Weak measurements have been used to simplify the calculation of optical networks in the presence of polarization-mode dispersion [@gisin], applied to slow- and fast-light effects in birefringent photonic crystals [@chiao], and bring a new, unifying perspective to the tunneling-time controversy [@aeweak; @aetree]. Hardy’s Paradox, introduced in Ref. [Hardy]{}, was analyzed in terms of weak values in [@Hardyweak]. In Ref. [@wisemanQED], weak values were used to physically explain the results of the cavity QED experiment described in [@orozco]. The opposing views expressed in Refs. [@scully; @walls] on the role of which-path information and the Heisenberg uncertainty principle in the double-slit experiment are reconciled with the use of weak values in [@wisemanweak]. Weak measurement can be considered the best estimate of an observable in a pre and post-selected system [@estimator].
The von Neumann interaction was originally used to model standard quantum measurement by mathematically describing the coupling between the measured system and the measurement pointer [@vn]. The interaction couples an observable $\hat{A}$ of the quantum system to the momentum $\hat{P}$ of the pointer,
$$\mathcal{H}=g\hat{A}\hat{P}, \label{Hamiltonian}$$
where $g$ is the coupling constant which is assumed to be real to keep $%
\mathcal{H}$ Hermitian. Since $\hat{A}$ and $\hat{P}$ act in different Hilbert spaces we can safely assume they commute. This interaction would be difficult to implement were it not for the fact that typically the measured system itself is used as part of the measurement device. When measuring $\hat{A}$ of a particle an independent degree of freedom of the particle can be used as the pointer. For example, a birefringent crystal can be oriented so that it will displace the position of photon by an amount that depends on the photon’s polarization [@hulet]. Here, $\hat{A}$ is the polarization observable and the pointer is the position of the photon. Another example is the Stern-Gerlach apparatus, where $\hat{A}$ is the spin of the particle and the pointer is the momentum of the particle. If such a measurement strategy were not availabe, one would require a strong controllable interaction between the quantum system and a separate pointer system. This is typically far too technically difficult to implement.
In modern quantum mechanics, we are increasingly interested in a different class of observables than in the above example, in which only a single particle is involved. Often, one would like to measure correlations between observables of distinct particles, like $\hat{S}_{1}\hat{S}_{2},$ the spin of particle one times the spin of particle two. Moreover, any experiment that utilizes or directly measures properties of entanglement is based on such observables and so, much of quantum information and quantum optics deal with these composite or joint observables. The exciting results and complex, rich range of features discovered by studies of entanglement suggests that weak measurement of joint observables should also produce valuable and interesting results. In fact, there already exist a few theoretical ideas for weak measurements that center around joint observables, such as Hardy’s Paradox [@Hardyweak], nonlocality of a single particle [@aetree], and extensions of the Quantum Box Problem [@3boxexp; @aewheeler]. We call the weak value of a joint observable the “joint weak value.” If the composite observable is a product of $N$ single particle observables then the weak value is called the “$N$th-order joint weak value”.
Joint observables are extremely difficult to measure directly with either strong or weak types of measurement. The difficulty lies in the fact that the necessary von Neumann interaction couples two separate observables, and hence particles, to a single pointer. One, therefore, can no longer use the extra degree of freedom on one of the particles as the pointer and so, one requires multiparticle interactions. An approach using multiparticle interactions was outlined in a proposal for a weak measurement experiment with ions but so far there have been no experimental weak measurements of joint observables [@molmer]. On the other hand, experimental strong measurements of joint observables are feasible and even commonplace. This is made possible by employing a different measurement strategy. Instead of measuring the joint observable directly, each single particle observable is measured simultaneously but separately. For example, instead of measuring $%
\hat{S}_{1}\hat{S}_{2}$ directly we can measure $\hat{S}_{1}$ and $\hat{S}%
_{2}$ separately and then multiply the results trial by trial. If one wants to strongly measure the joint observable $\hat{A}_{1}\hat{A}_{2}...%
\hat{A}_{N}=\hat{M}$, instead of using the multiparticle von Neumann Hamiltonian $\mathcal{H}=g\hat{M}\hat{P}$, the general strategy is to simultaneously apply $N$ standard single-particle von Neumann interaction Hamiltonians,
$$\begin{aligned}
\mathcal{H} &=&g_{1}\hat{A}_{1}\hat{P}_{1}+g_{2}\hat{A}_{2}\hat{P}_{2}+... \\
&=&\sum_{j=1}^{N}g_{j}\hat{A}_{j}\hat{P}_{j}. \label{nvn}\end{aligned}$$
Given that we can already perform each of the single-particle Hamiltonians, it is straightforward to implement the total Hamiltonian. This strategy allows one to make projective measurements of $\hat{M}$ which is all that is required to measure the expectation value of $\hat{M}$, $$\left\langle \hat{M}\right\rangle =\left\langle \hat{A}_{1}\hat{A}_{2}...%
\hat{A}_{N}\right\rangle \propto \left\langle \hat{X}_{1}\hat{X}_{2}...\hat{X%
}_{N}\right\rangle , \label{strong}$$where $\hat{X}_{i}$ is the position operator of the pointer and provided all $\hat{A}_{i}$ commute. In other words, the expectation value of $\hat{M}$ is related to the correlation between the positions of all $N$ pointers.
In two earlier works, an analogous strategy was applied to weak measurements [@twoweak; @nweak]. The Hamiltonian in Eq. (\[nvn\]) is utilized in the weak regime to create correlations in the deflections of the $N$ pointers proportional to the weak value. Specifically, the $N$th-order joint weak value was related to two correlations between all $N$ pointer deflections and a complicated combination of lower-order joint weak values. In this work, we show that the $N$th-order joint weak value takes on an elegant and simple form closely related to the strong measurement formula in Eq. (\[strong\]) when expressed entirely in terms of $N$-pointer correlations. This new and simplified form lends itself to a new way of thinking about single and joint weak measurements in terms of expectation values of products of annihilation operators.
We begin by deriving AAV’s formula for the weak value of a single particle observable. AAV based weak measurement on the weak limit of the standard approach to measurement. Specifically, they use the von Neumann interaction in Eq. (\[Hamiltonian\]), which we assume to be constant over some interaction time $t$. The measurement pointer is initially in a Gaussian wavefunction centered at zero, $$\ \left\langle x|\phi \right\rangle =\phi (x)=\left( \frac{1}{\sqrt{2\pi }%
\sigma }\right) ^{\frac{1}{2}}\exp \left( -\frac{x^{2}}{4\sigma ^{2}}\right)
, \label{pointer}$$where $\sigma $ is the rms width of $\left| \phi (x)\right| ^{2}$. In most experiments, quantum mechanical systems are initially prepared in a known initial state $\left| I\right\rangle $. Since this preparation usually involves measuring an ensemble of systems and selecting the subensemble with the correct outcome, this is called pre-selection. For a strong measurement, the von Neumann interaction with a pre-selected system state shifts the mean position of the pointer $\left\langle \hat{X}\right\rangle $ by $gt\left\langle I\right| \hat{A}\left| I\right\rangle $ and leaves $%
\left\langle \hat{P}\right\rangle $ unchanged. AAV considered the case where we further restrict ourselves to the subensemble of system states that are found to be in $\left| F\right\rangle $ after the measurement, a procedure called post-selection. A weak measurement performed between the pre and post-selection can result in very different expectation values than in strong measurements, as we will see.
After the pointer weakly interacts with the initial sytem-pointer state $%
\left| \psi (0)\right\rangle =\left| I\right\rangle \left| \phi
\right\rangle $ the state evolves to
$$\begin{aligned}
\left| \psi (t)\right\rangle &=&\exp \left( \frac{-i\mathcal{H}t}{\hbar }%
\right) \left| I\right\rangle \left| \phi \right\rangle =\left( 1-\frac{i%
\mathcal{H}t}{\hbar }-...\right) \left| I\right\rangle \left| \phi
\right\rangle \label{expansion} \\
&=&\left| I\right\rangle \left| \phi \right\rangle -\frac{igt}{\hbar }\hat{A}%
\left| I\right\rangle \hat{P}\left| \phi \right\rangle -...\end{aligned}$$
We project out the part of the state that is post-selected in state $\left|
F\right\rangle ,$ $$\left\langle F\right| \exp \left( \frac{-i\mathcal{H}t}{\hbar }\right)
\left| I\right\rangle \left| \phi \right\rangle =\left\langle
F|I\right\rangle \left| \phi \right\rangle -\frac{igt}{\hbar }\left\langle
F\right| \hat{A}\left| I\right\rangle \hat{P}\left| \phi \right\rangle -...$$This leaves the state of pointer after the interaction and post-selection. In the limit of an ideal weak measurement, $gt\rightarrow 0,$ $\left|
\left\langle F|I\right\rangle \right| ^{2}=$Prob$_{\text{success}}$ is the probability the post-selection succeeds [@aeweak]. If we renormalize the state and then truncate the amplitude of each term to lowest order in $%
gt $ we get $$\left| \phi _{\text{$fi$}}\right\rangle =\left| \phi \right\rangle -\frac{igt%
}{\hbar }\frac{\left\langle F\right| \hat{A}\left| I\right\rangle }{%
\left\langle F|I\right\rangle }\hat{P}\left| \phi \right\rangle -...,
\label{post select pointer}$$which is just equivalent to dividing by $\left\langle F|I\right\rangle =%
\sqrt{\text{Prob}_{\text{success}}}$. The subscript $fi$, corresponding to final state $\left| F\right\rangle $ and intial state $\left| I\right\rangle
$, labels the final pointer state, with which we can now calculate the expectation value of $\hat{X}$ of the pointer. The terms which contain an expectation value of an odd number of operators go to zero since the pointer is initially an even function about zero. To first order in $gt$, the remaining terms give us $$\begin{aligned}
\left\langle \hat{X}\right\rangle _{\text{$fi$}} &=&\left\langle \phi _{%
\text{$fi$}}\right| \hat{X}\left| \phi _{\text{$fi$}}\right\rangle =\frac{%
-igt}{\hbar }\func{Re}\left( \frac{\left\langle F\right| \hat{A}\left|
I\right\rangle }{\left\langle F|I\right\rangle }\right) \left\langle \phi _{%
\text{$fi$}}\right| \left( \hat{X}\hat{P}-\hat{P}\hat{X}\right) \left| \phi
_{\text{$fi$}}\right\rangle \\
&&+\frac{gt}{\hbar }\func{Im}\left( \frac{\left\langle F\right| \hat{A}%
\left| I\right\rangle }{\left\langle F|I\right\rangle }\right) \left\langle
\phi _{\text{$fi$}}\right| \left( \hat{X}\hat{P}+\hat{P}\hat{X}\right)
\left| \phi _{\text{$fi$}}\right\rangle \\
&=>\func{Re}\left( \frac{\left\langle F\right| \hat{A}\left|
I\right\rangle }{\left\langle F|I\right\rangle }\right) .\end{aligned}$$Here, $\left\langle {}\right\rangle _{\text{$fi$}}$ is used to signify the expectation value of a pointer observable only in the subensemble of measured systems that start in state $\left| I\right\rangle $ and are later post-selected in the state $\left| F\right\rangle .$ Similarly, the momentum expectation value is given by$$\begin{aligned}
\left\langle \hat{P}\right\rangle _{\text{$fi$}} &=&\left\langle \phi _{%
\text{$fi$}}\right| \hat{P}\left| \phi _{\text{$fi$}}\right\rangle =\frac{%
-igt}{\hbar }\func{Re}\left( \frac{\left\langle F\right| \hat{A}\left|
I\right\rangle }{\left\langle F|I\right\rangle }\right) \left\langle \phi _{%
\text{$fi$}}\right| \left( \hat{P}^{2}-\hat{P}^{2}\right) \left| \phi _{%
\text{$fi$}}\right\rangle \\
&&+\frac{gt}{\hbar }\func{Im}\left( \frac{\left\langle F\right| \hat{A}%
\left| I\right\rangle }{\left\langle F|I\right\rangle }\right) \left\langle
\phi _{\text{$fi$}}\right| \left( \hat{P}^{2}+\hat{P}^{2}\right) \left| \phi
_{\text{$fi$}}\right\rangle \\
&=&\frac{\hbar gt}{2\sigma ^{2}}\func{Im}\left( \frac{\left\langle F\right|
\hat{A}\left| I\right\rangle }{\left\langle F|I\right\rangle }\right) .\end{aligned}$$
The shifts from zero in both the $\hat{X}$ and $\hat{P}$ expectation values are proportional to the real and imaginary parts, respectively, of the weak value $\left\langle \hat{A}\right\rangle _{W}$ which is defined as$$\left\langle \hat{A}\right\rangle _{W}\equiv \frac{\left\langle F\right|
\hat{A}\left| I\right\rangle }{\left\langle F|I\right\rangle }.$$In fact, AAV showed that for sufficiently weak coupling $\langle x|\phi
_{fi}\rangle $, the final pointer state, will be $\left( \sqrt{2\pi }\sigma
\right) ^{-\frac{1}{2}}\exp \left( -\left( x-\left\langle \hat{A}%
\right\rangle _{W}\right) ^{2}/4\sigma ^{2}\right) $, unchanged except for a shift by the weak value.
It has been argued that it is the backaction of the measurement on the measured system that leads to a finite $\func{Im}\left\langle \hat{A}%
\right\rangle _{W}$ and thus a nonzero $\left\langle \hat{P}\right\rangle _{%
\text{fi}}$ [@aeweak]. In addition, as the measurement becomes weaker $%
\left\langle \hat{P}\right\rangle _{\text{$fi$}}$ becomes more and more difficult to determine; $\left\langle \hat{P}\right\rangle _{\text{$fi$}}$ decreases with $gt/\sigma ^{2}$ whereas the width $\Delta \hat{P}$ decreases as $1/\sigma $. Some have gone as far as to define the weak value as $\func{%
Re}\left( \frac{\left\langle F\right| \hat{A}\left| I\right\rangle }{%
\left\langle F|I\right\rangle }\right) $ [@wisemanQED]. Nonetheless, we will show that $\left\langle \hat{P}\right\rangle _{\text{$fi$}}$ should not be interpreted as an insignificant artifact of the weak measurement procedure and has an integral role in measuring the $N$th-order joint weak value.
One can express the full weak value in terms of the two expectation values of the pointer,$$\begin{aligned}
\left\langle \hat{A}\right\rangle _{W} &=&\func{Re}\left\langle \hat{A}%
\right\rangle _{W}+i\func{Im}\left\langle \hat{A}\right\rangle _{W} \\
&=&\frac{2\sigma }{gt}\left\langle \frac{1}{2\sigma }\hat{X}+i\frac{\sigma }{%
\hbar }\hat{P}\right\rangle _{\text{$fi$}}. \label{simpleweak}\end{aligned}$$In their derivation of weak values, AAV made the natural choice of a Gaussian for the initial pointer state, as do we. This state also happens to be the ground state $\left| 0\right\rangle $ of a harmonic oscillator with mass $m$ and frequency $\omega $. For illustration, if one reparameterizes the width of the Gaussian in terms of $m\omega $ such that $%
\sigma =\sqrt{\hbar /2m\omega }$ it becomes apparent that the operator in the expectation value in Eq. (\[simpleweak\]) is just the familiar lowering operator, $$\hat{a}=\sqrt{\frac{m\omega }{2\hbar }}\hat{X}+i\sqrt{\frac{1}{2m\omega
\hbar }}\hat{P}.$$The operator in Eq. (\[simpleweak\]) will transform the pointer just as the lowering operator does, even though the pointer is not actually in a harmonic potential. This fact will simplify some of the following calculations. Furthermore, now the weak value can be re-expressed as: $$\left\langle \hat{A}\right\rangle _{W}=\frac{2\sigma }{gt}\left\langle
\hat{a}\right\rangle _{\text{$fi$}}. \label{annihilation}$$
To our knowledge, this is the first time in the literature that this simple but important relationship between the annihilation operator and weak measurement has been described. The reason the annihilation operator is related to the weak value can be understood as follows. When the coupling is sufficiently weak, the expansion in Eq. (\[expansion\]) shows that the largest pointer amplitude is left unchanged in the ground state. The interaction Hamiltonian shifts some of the pointer state into the first excited state by creating a small amplitude, proportional to $gt\hat{A}$,for the $\left| 1\right\rangle $ state. If we restrict ourselves to the post-selected subensemble, as in Eq. (\[post select pointer\]), then this small amplitude changes to be proportional to $gt\left\langle \hat{A}%
\right\rangle _{W}$. The annihilation operator removes the part of the state that is left unchanged by the coupling, leaving only the shifted component. In other words, the annihilation operator isolates only that part of the pointer state that is changed by the interaction.
We now move on to a derivation of $N$th-order joint weak values. In this section, we combine the strategy outlined in the introduction for measuring joint observables with the use of the annihilation operator to extract the weak value. As in previous works, to measure the operator $\hat{M}%
=\prod_{j=1}^{N}\hat{A}_{j}$ we apply $N$ separate von Neumann interactions coupling each $\hat{A}_{j}$ to its own pointer, as in Eq. (\[nvn\]) [twoweak,nweak]{}. To simplify the expressions to come we set all $g_{j}$ to be equal and rewrite the momentum operators $\hat{P}_{j}$ in terms of the respective raising and lowering operators, $\hat{a}_{j}^{\dagger }$ and $%
\hat{a}_{j}$, for each of the pointers, $$\mathcal{H}=i\frac{\hbar g}{2\sigma }\sum_{j=1}^{N}\hat{A}_{j}\left( \hat{a}%
_{j}^{\dagger }-\hat{a}_{j}\right) .$$Now we require $N$ different pointers, all beginning in an initial state defined by Eq. (\[pointer\]). The total initial pointer state can be described by the ground state of $N$ harmonic oscillators: $$\left| \Phi \right\rangle =\prod_{j=1}^{N}\left| \phi _{j}\right\rangle
=\left| 0\right\rangle ^{\otimes N}. \label{N pointer state}$$Continuing, using the number-state notation to describe the pointer, we calculate the state of the combined system after the interaction Hamiltonian is applied,$$\begin{aligned}
\left| \Phi \right\rangle \left| I\right\rangle &\rightarrow &\exp \left(
\frac{-i\mathcal{H}t}{\hbar }\right) \left| 0\right\rangle ^{\otimes
N}\left| I\right\rangle =\left( 1-\frac{i\mathcal{H}t}{\hbar }+...\right)
\left| 0\right\rangle ^{\otimes N}\left| I\right\rangle \label{first order}
\\
&=&\left( 1+\frac{gt}{2\sigma }\sum_{j=1}^{N}\hat{A}_{j}\left( \hat{a}%
_{j}^{\dagger }-\hat{a}_{j}\right) +...\right) \left| 0\right\rangle
^{\otimes N}\left| I\right\rangle \\
&=&\left| 0\right\rangle ^{\otimes N}\left| I\right\rangle +\frac{gt}{%
2\sigma }\sum_{j=1}^{N}\hat{A}_{j}\left| 1_{j}\right\rangle \left|
I\right\rangle +...,\end{aligned}$$where $\left| 1_{j}\right\rangle $ is the state where the jth pointer is in the first-excited state and all the other pointers are in the ground state (e.g. $\left| 0_{1}1_{2}0_{3}...0_{N}\right\rangle $). Here, we have expanded the state in powers of $gt$. Eq. (\[first order\]) shows that to first order, the interaction Hamiltonian coupling the measuring device to the system can displace only one of the $N$ pointers at a time. Simultaneous shifts of multiple pointers come from higher-order terms in the propagator. We are particularly interested in the $N$th term in the expansion, $$\frac{1}{N!}\left( \frac{-i\mathcal{H}t}{\hbar }\right) ^{N}=\frac{1}{N!}%
\left( \frac{gt}{2\sigma }\sum_{j=1}^{N}\hat{A}_{j}\left( \hat{a}%
_{k}^{\dagger }-\hat{a}_{k}\right) \right) ^{N}. \label{Nth term}$$This term is the lowest-order one in the expansion which can simultaneously transfer all $N$ pointers into the first excited state (e.g. $\left|
1_{1}1_{2}1_{3}...1_{N}\right\rangle $). This state, which we label as $%
\left| 1\right\rangle ^{\otimes N},$ is created when each term in the above sum supplies one raising operator. The terms in the sum can contribute the $N$ distinct raising operators in any order and so the portion of Eq. (\[Nth term\]) that creates the $\left| 1\right\rangle ^{\otimes N}$ state is equal to $$\frac{1}{N!}\frac{gt}{2\sigma }\mathcal{\wp }\left\{ \hat{A}_{k}\hat{a}%
_{k}^{\dagger }\right\} _{N},$$where $\mathcal{\wp }\left\{ \hat{L}_{k}\right\} _{N}$ denotes the sum of all $N!$ orderings of the set of $N$ operators $\left\{ \hat{L}%
_{k}\right\} $. Note that these different orderings are only distinct when the operators do not commute. The remaining portions of Eq. (\[Nth term\]) create states where at least one pointer is left in the initial state (e.g. $\left| 2_{1}0_{2}1_{3}...1_{N}\right\rangle $). Projecting onto $%
\left\langle F\right| $ completes the post-selection and leaves us with,$$\begin{aligned}
\left\langle F\right| \exp \left( \frac{-i\mathcal{H}t}{\hbar }\right)
\left| 0\right\rangle ^{\otimes N} &=&\left| 0\right\rangle ^{\otimes
N}\left\langle F|I\right\rangle +\frac{gt}{2\sigma }\sum_{j=1}^{N}\left%
\langle F\right| \hat{A}_{j}\left| I\right\rangle \left| 1_{j}\right\rangle
+... \\
&&+\left( \frac{gt}{2\sigma }\right) ^{N}\frac{1}{N!}\left\langle F\right|
\mathcal{\wp }\left\{ \hat{A}_{k}\right\} _{N}\left| I\right\rangle \left|
1\right\rangle ^{\otimes N}+...\end{aligned}$$ We renormalize the resulting $N$-pointer state $\left| \Phi _{\text{fi}%
}\right\rangle $ and then truncate the amplitude of each term at the lowest nonzero order in $gt,$ $$\left| \Phi _{\text{$fi$}}\right\rangle =\left| 0\right\rangle ^{\otimes N}+%
\frac{gt}{2\sigma }\sum_{j=1}^{N}\frac{\left\langle F\right| \hat{A}%
_{j}\left| I\right\rangle }{\left\langle F|I\right\rangle }\left|
1_{j}\right\rangle +...+\left( \frac{gt}{2\sigma }\right) ^{N}\frac{1}{N!}%
\frac{\left\langle F\right| \mathcal{\wp }\left\{ \hat{A}_{k}\right\}
_{N}\left| I\right\rangle \left| 1\right\rangle ^{\otimes N}}{\left\langle
F|I\right\rangle }+... \label{final N pointer state}$$ This is equivalent to dividing by $\left\langle F|I\right\rangle $, the renormalization constant in the limit of no coupling. In analogy with Eq. (\[annihilation\]), we now wish to take the expectation value of the product of the annihilation operators for all $N$ pointers,
$$\hat{O}\equiv \prod_{j=1}^{N}\hat{a}_{j}.$$In Eq. (\[final N pointer state\]), the $\left| 1\right\rangle ^{\otimes
N} $ state is the lowest order term that does not go to zero when acted on by $\hat{O}$; this term becomes, $$\hat{O}\left| \Phi _{\text{$fi$}}\right\rangle =\left( \frac{gt}{2\sigma }%
\right) ^{N}\frac{1}{N!}\frac{\left\langle F\right| \mathcal{\wp }\left\{
\hat{A}_{j}\right\} _{N}\left| I\right\rangle }{\left\langle
F|I\right\rangle }\left| 0\right\rangle ^{\otimes N}+O\left( \left(
gt\right) ^{N+1}\right) .$$Clearly, to lowest nonzero order the expectation value then becomes,$$\begin{aligned}
\left\langle \hat{O}\right\rangle _{\text{$fi$}} &=&\left\langle \Phi _{%
\text{$fi$}}\right| \hat{O}\left| \Phi _{\text{$fi$}}\right\rangle \\
&=&\left\langle 0\right| \left( \frac{gt}{2\sigma }\right) ^{N}\frac{1}{N!}%
\frac{\left\langle F\right| \mathcal{\wp }\left\{ \hat{A}_{j}\right\}
_{N}\left| I\right\rangle }{\left\langle F|I\right\rangle }\left|
0\right\rangle \\
&=&\left( \frac{gt}{2\sigma }\right) ^{N}\frac{1}{N!}\frac{\left\langle
F\right| \mathcal{\wp }\left\{ \hat{A}_{j}\right\} _{N}\left| I\right\rangle
}{\left\langle F|I\right\rangle }. \label{N expectation}\end{aligned}$$The next lowest order term in the expectation value corresponds to any of the $N$ pointers undergoing an extra pair of transitions (i.e., a pointer is raised to $\left| 2\right\rangle $ and subsequently lowered back to $\left|
1\right\rangle $). Consequently it will be reduced in size by a factor of $%
2\left( \frac{gt}{2\sigma }\right) ^{2}$ compared to the lowest order term. Using Eq. (\[N expectation\]) the $N$th-order joint weak value can now be expressed in the simple formula$$\frac{1}{N!}\left\langle \mathcal{\wp }\left\{ \hat{A}_{j}\right\}
_{N}\right\rangle _{W}=\left\langle \prod_{j=1}^{N}\hat{a}_{j}\right\rangle
_{\text{$fi$}}\left( \frac{2\sigma }{gt}\right) ^{N}.$$
It is often the case that each operator $\hat{A}_{j}$ acts on a different particle, ensuring that all $\hat{A}_{j}$ commute. This allows the further simplification of the $N$th-order joint weak value to$$\left\langle \prod_{j=1}^{N}\hat{A}_{j}\right\rangle _{W}=\left\langle
\prod_{j=1}^{N}\hat{a}_{j}\right\rangle _{\text{$fi$}}\left( \frac{2\sigma }{%
gt}\right) ^{N}. \label{the formula}$$For commuting observables, the magnitude of the simultaneous shift in the $N$ pointers that results from concurrent kicks from all $N$ terms in the Hamiltonian in Eq. (\[nvn\]) is proportional to the shift in one pointer created by a single von Neumann Hamiltonian for measuring operator $\hat{M}$. The role of the annihilation operators is to isolate this simultaneous pointer shift from the total uncorrelated shifts of the $N$ pointers and thus duplicate the action of $\mathcal{H}=g\hat{M}\hat{P}$, without the need for multiparticle interactions.
Since Eq. (\[the formula\]) requires the measurement of the annihilation operator, which is not Hermitian, one might think the expression is, in principle, unmeasurable. In fact, if one expands the annihilation operator in terms of $\hat{X}$ and $\hat{P}$ for each pointer then one is simply left with expectation values of products of $\hat{X}$ or $\hat{P}$ for each pointer. One then measures $\hat{X}$ in one ensemble of pointers and $\hat{%
P}$ in an identically-prepared ensemble.
The expression in Eq. (\[the formula\]) is the central result of this work. As in previous papers, this result shows how one can practically measure a joint weak value even without the multiparticle interactions the AAV method requires [@twoweak; @nweak]. However, this expression is much more elegant and it makes it clear that the annihilation operator plays a key role in joint weak measurements. Specifically, with the use of the annihilation operator, the similarity to the strong measurement expectation value in Eq. (\[strong\]) is apparent. For strong measurement, the equivalent expectation value to the $N$th-order joint weak value is$$\left\langle \prod_{j=1}^{N}\hat{A}_{j}\right\rangle =\left\langle
\prod_{j=1}^{N}\hat{X}_{j}\right\rangle \left( \frac{1}{gt}\right) ^{N}.$$The similarity is striking and makes a good case for the use of the annihilation operator in the understanding of weak values.
Lets compare Eq. (\[the formula\]) to the previous results for the $N$th-order joint weak value [@nweak]. In the previous paper, it was expressed recursively in terms of two $N$th-order correlations between the pointers and to $N$ different joint weak values of order $N-1$. Utilizing this recursive formula, the $N$th-order joint weak value can be expressed purely in terms of the expectation value of position and momentum correlations. This expression includes $2^{N+1}-2$ distinct correlations of various orders, although most will be close to the $N/2$ order as the number of distinct expectation values at each order follows the binomial distribution. In comparison, Eq. (\[the formula\]) relates the $N$th-order joint weak value to $2^{N}$ correlations in the positions and momenta of all $N$ pointers and so requires roughly half the number of expectation values as the final result from the previous paper (but of higher order).
As a specific example of the use of Eq. (\[the formula\]), the weak value of the product of two spin components $S_{1x}S_{2y}$ would be,$$\begin{aligned}
\left\langle S_{1x}S_{2y}\right\rangle _{W} &=&\left( \frac{2\sigma }{gt}%
\right) ^{2}\left\langle \hat{a}_{1}\hat{a}_{2}\right\rangle _{\text{$fi$}}
\\
&=&\left( \frac{2\sigma }{gt}\right) ^{2}\left\langle \left( \frac{1}{%
2\sigma }\hat{X}_{1}+i\frac{\sigma }{\hbar }\hat{P}_{1}\right) \left( \frac{1%
}{2\sigma }\hat{X}_{2}+i\frac{\sigma }{\hbar }\hat{P}_{2}\right)
\right\rangle _{\text{$fi$}}.\end{aligned}$$The real and imaginary parts of the weak value are then$$\begin{aligned}
\func{Re}\left\langle S_{1x}S_{2y}\right\rangle _{W} &=&\left( \frac{1}{gt}%
\right) ^{2}\left( \left\langle \hat{X}_{1}\hat{X}_{2}\right\rangle _{\text{$%
fi$}}-\frac{4\sigma ^{4}}{\hbar ^{2}}\left\langle \hat{P}_{1}\hat{P}%
_{2}\right\rangle _{\text{$fi$}}\right) \\
\func{Im}\left\langle S_{1x}S_{2y}\right\rangle _{W} &=&\frac{2\sigma ^{2}}{%
\hbar }\left( \frac{1}{gt}\right) ^{2}\left( \left\langle \hat{X}_{1}\hat{P}%
_{2}\right\rangle _{\text{$fi$}}+\left\langle \hat{P}_{1}\hat{X}%
_{2}\right\rangle _{\text{$fi$}}\right) .\end{aligned}$$The importance of the pointer momentum shift is demonstrated in the above example. With our measurement technique even the real part of weak value is related to the pointers’ momenta, $\hat{P}_{1}$ and $\hat{P}_{2}$ . In general, the momentum and position observables for each of the $N$ pointers will appear in the expression for the real part of the $N$th-order joint weak value.
Note that like single weak measurements, this method for measuring the $N$th-order joint weak value is not limited to the particular interaction or pointer used in our measurement model [@arbpointer]. For example, one can perform a derivation very similar to the one presented here where a spin, as opposed to position, pointer is used. For a spin pointer, the Hamiltonian would be $\mathcal{H}=-g\hat{A}\hat{S}_{y}=ig\hat{A}(\hat{S}%
_{z}^{+}-\hat{S}_{z}^{-})/2$, where $\hat{S}_{i}^{+}$ and $\hat{S}_{i}^{-}$ are the raising and lowering operators for the $\hat{S}_{i}$ basis. The initial pointer state would be the lowest eigenstate of $\hat{S}_{z},$ with eigenvalue $-\hbar s$. In this case, the expression for the $N$th-order joint weak value in terms of $N$ spin pointers is$$\left\langle \prod_{j=1}^{N}\hat{A}_{j}\right\rangle _{W}=\left\langle
\prod_{j=1}^{N}\hat{S}_{jz}^{-}\right\rangle _{\text{$fi$}}\left( \frac{1}{%
gt\hbar s}\right) ^{N},$$where $\hat{S}_{jz}^{-}$ is the z-basis lowering operator for the jth pointer and all $\hat{A}_{j}$ are assumed to commute. An important advantage of using spin is the absence of unequal coefficients in the expression for the lowering operator. This puts the shifts in the pointer observable and its conjugate on equal footing. Using such a pointer means that the physical shift in the conjugate observable does not become smaller as the measurement becomes weaker. Expectation values are also particularly easy to measure for spins (and polarizations), especially spin 1/2 systems since there are only two basis states which need to be projected onto. For instance, the $N$th-order joint weak value would only require $2^{2N}$ measurements in total if $N$ spin 1/2 pointers were used.
In the present work, we have greatly simplified a recent extension of weak measurement which makes the experimental investigation of composite, or joint observables possible [@twoweak; @nweak]. We have shown that when single and joint weak values are expressed as expectation values of annihilation operators, they take on a surprisingly elegant form very similar to that seen in standard strong measurement. This form is easily generalized to any measurement device in which the initial pointer state is the eigenstate of an appropriate lowering operator. With the extension, the weak measurement of joint observables only requires the same apparatus that one would need to weakly measure each of the component observables separately. Joint observables are central to the detection and utilization of entanglement in multiparticle systems. The weak measurement of these observables should be particularly useful for investigating post-selected systems such as those that have been used produce novel multiparticle entangled states or those that implement quantum logic gates [loqc,exploqc]{}.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by ARC Seibersdorf Research GmbH, the Austrian Science Foundation (FWF), project number SFB 015 P06, NSERC, and the European Commission, contract number IST-2001-38864 (RAMBOQ). We would like to thank Aephraim Steinberg and Morgan Mitchell for helpful discussions.
[99]{} Y. Aharonov, D.Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60 (1988) 1351.
A. Peres, Phys. Rev. Lett. 62 (1989) 2326.
A.J. Leggett, Phys. Rev. Lett. 62 (1989) 2325.
Y. Aharonov, L. Vaidman, Phys. Rev. Lett. 62 (1989) 2327.
N. Brunner, A. Acín, D. Collins, N. Gisin, and V. Scarani, Phys. Rev. Lett. 91 (2003) 180402; N. Brunner, V. Scarani, M. Wegmüller, M. Legré and N. Gisin, quant-ph/0407155.
D. R. Solli, C. F. McCormick, R. Y. Chiao, S. Popescu, and J. M. Hickmann, Phys. Rev. Lett. 92 (2004) 043601.
A.M. Steinberg, Phys. Rev. Lett. 74 (1995) 2405; A.M. Steinberg, Phys. Rev. A 52 (1995) 32.
A.M. Steinberg, Can a falling tree make a noise in two forests at the same time?, in: S. Jeffers, G. Hunter, J.-P. Vigier (Eds.), Causality and Locality in Modern Physics, Kluwer, Dordrecht, 1997, p. 431.
L. Hardy, Phys. Rev. Lett. 68 (1992) 2981.
Y. Aharonov, A. Botero, S. Popescu, B. Reznik, and J. Tollaksen, Phys. Lett. A 301 (2001) 130.
H. M. Wiseman, Phys. Rev. A 65 (2002) 032111.
G.T. Foster, L.A. Orozco, H.M. Castro-Beltran, and H.J. Carmichael, Phys. Rev. Lett. 85 (2000) 3149.
M.O. Scully, B.-G. Englert, and H. Walther, Nature 351 (1991) 111.
E.P. Storey, S.M. Tan, M.J. Collett, and D.F. Walls, Nature 367 (1994) 626.
H.M. Wiseman, Physics Letters A 311 (2003) 285.
L. M. Johansen, Phys. Lett. A 322 (2004) 298.
J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton Univ. Press, Princeton, 1955, pp. 442-445.
N.W.M. Ritchie, J.G. Story, R.G. Hulet, Phys. Rev. Lett. 66 (1991) 1107.
K.J. Resch, J.S. Lundeen, and A.M. Steinberg, Phys. Lett. A 324 (2004) 125.
A.M. Steinberg, quant-ph/0302003, J.D. Barrow, P.C.W. Davies, C.L. Harper Jr. (Eds.), Science and Ultimate Reality: Quantum Theory, Cosmology and Complexity, Cambridge Univ. Press, Cambridge, 2004, in press.
K. Mølmer, Phys. Lett. A 292 (2001) 151.
K. J. Resch and A. M. Steinberg, Phys. Rev. Lett. 92 (2004) 130402.
K.J. Resch, J. Opt. B 6 (2004) 482.
L. M. Johansen, quant-ph/0402050.
E. Knill, R. Laflamme, and G. Milburn, Nature 409 (2001) 46; T. B. Pittman, B. C. Jacobs, and J. D. Franson, Phys. Rev. A 64 (2001) 062311; P. Kok, H. Lee, and J. P. Dowling, Phys. Rev. A 65 (2002) 052104.
K. Sanaka, T. Jennewein, J.W. Pan, K. Resch, and A. Zeilinger, Phys. Rev. Lett. 92 (2004) 017902; J.W. Pan, D. Bouwmeester, H. Weinfurter, A. Zeilinger, Nature 403 (2000) 515; S. Gasparoni, J.W. Pan, P. Walther, T. Rudolph, and A. Zeilinger Phys. Rev. Lett. 93 (2004) 020504; J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning Nature 426 (2003) 264; T. B. Pittman, M. J. Fitch, B. C Jacobs, and J. D. Franson, Phys. Rev. A 68 (2003) 032316; M.W. Mitchell, J.S. Lundeen, and A.M. Steinberg Nature 429 (2004) 16; P. Walther, J.W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, A. Zeilinger, Nature 429 (2004) 158.
|
---
abstract: 'Uncertainty around multimodel ensemble forecasts of changes in future climate reduces the accuracy of those forecasts. For very uncertain forecasts this effect may mean that the forecasts should not be used. We investigate the use of the well-known Bayesian Information Criterion (BIC) to make the decision as to whether a forecast should be used or ignored.'
author:
- 'Stephen Jewson[^1] and Ed Hawkins\'
bibliography:
- 'arxiv.bib'
title: ' Uncertain Climate Forecasts From Multimodel Ensembles: When to Use Them and When to Ignore Them '
---
Introduction
============
The climate predictions produced by numerical models predict *changes* in climate relative to the present day, rather than predicting absolute levels of future climate. The ratio $r$ of the size of the change to the uncertainty around that change is then a measure of the confidence one can have in a forecast. It also determines whether or not it would be better to ignore a forecast, use it, or adjust it in some way. We have studied this question in @jewsonh09c, in which we compared the relative effectiveness of five different ways of using a multimodel numerical model climate forecast, as a function of this ratio $r$. We tested the different methods in terms of the mean squared error of predictions they would produce, using monte carlo simulations. We found that for an ensemble based on 10 truly independent climate models, for small values of this ratio ($r<0.74$) it would be better to ignore the forecast completely. For larger values of this ratio ($0.73<r<2.02$) it is best to adjust the forecast towards zero, and for the largest values of this ratio ($r>2.02$) it is best to use the forecast unadjusted.
These results cannot, however, be used literally as a decision rule. This is because, in practice, we never know the exact value of the ratio $r$ since it is defined in terms of the *actual* mean change in climate and the *actual* uncertainty around that change (which are unknown) rather than the *estimated* change and the *estimated* uncertainty (which are known).
It is tempting to replace the real values of the mean and uncertainty with their estimates, calculate an estimate of $r$, and then use the ranges given above. However, rather than do that it may be better to treat this as a statistical model selection problem, and use standard model selection methods. Perhaps the most standard model selection rule is the Bayesian Information Criterion (BIC) (see many statistics textbooks, such as @wasserman), which gives a way to select between models which have been fitted by maximum likelihood. Two of the five methods used in @jewsonh09c were fitted using maximum likelihood (those that involve either ignoring or using the ensemble mean) with the others using a second stage of fitting for the ‘damping’ parameters. As a first step, therefore, we apply BIC to the two methods from @jewsonh09c that were fitted using maximum likelihood, and postpone to a future study the question of how to apply model selection to the remaining three methods.
In section \[s2\] we present the mathematical setup and the two different methods we will consider for turning multimodel ensemble output into a probabilistic prediction. In section \[s3\] we derive the BIC for each of the methods. In section \[s4\] we compare the derived BIC values, and derive an expression for the critical value of $r$ that determines the threshold between the two methods. In section \[s5\] we apply BIC-based model selection to CMIP derived predictions of UK temperatures, and finally in section \[s6\] we discuss the results.
Two Methods for Interpreting Multimodel Numerical Model Climate Forecasts {#s2}
=========================================================================
Our mathematical setup follows that of @jewsonh09b and @jewsonh09c. We imagine that we have an ensemble of $n$ climate models. Ideally each climate model has been run in a large initial condition ensemble, although in practice that may not matter if the impact of initial condition uncertainty is small relative to model uncertainty.
We use a classical statistical framework in which we consider that the models have the same error statistics and are samples from an infinite population of all possible models with those error statistics. We also assume, perhaps somewhat optimistically, that the mean of this population is equivalent to the true future climate (we call this the ‘perfect ensemble assumption’). We suppose that the variance of this ensemble has been adjusted to compensate for possible dependencies between the models (perhaps using the method of @jewsonh09a) and hence that, because of this adjustment, we can consider the adjusted ensemble members as being independent.
We write the mean of this ensemble as $m$ and the sample variance as $s^2$, where: $$\begin{aligned}
m &=&\frac{1}{n} \sum_{i=1}^{n} x_i\\
s^2&=&\frac{1}{n} \sum_{i=1}^{n} (x_i-m)^2\end{aligned}$$
The uncertainty around the estimate of the ensemble mean can be estimated by $V$, where: $$V=\frac{s^2}{n}$$
We then consider the following two methods for turning this ensemble into a forecast:
Use Ensemble Mean, Use Ensemble Variance {#m1}
----------------------------------------
In this method we make a prediction by fitting a normal distribution to the ensemble using maximum likelihood. The mean and variance ($\mu_1,\sigma_1^2$) of the prediction are then given by:
$$\begin{aligned}
\hat{\mu}_1 &=&m\\
\hat{\sigma}_1^2&=&\frac{1}{n}\sum_{i=1}^{n} (x_i-\mu_1)^2\\
&=&s^2\end{aligned}$$
Note that we deliberately do not use the well known ‘$n-1$’ expression for the variance, since we will need our estimates of the parameters to be maximum likelihood estimates for BIC to apply.
Ignore Ensemble Mean, Use Ensemble Variance {#m2}
-------------------------------------------
In this method we fit a normal distribution to the ensemble using maximum likelihood, but with the assumption that the mean of the distribution is zero. In other words, we ignore the mean of the ensemble, but use the variance.
The mean and variance ($\mu_2,\sigma_2^2$) of the prediction are then given by:
$$\begin{aligned}
\hat{\mu}_2&=&0\\
\hat{\sigma}_1^2
&=&\frac{1}{n} \sum_{i=1}^{n} (x_i-0)^2\\
&=&\frac{1}{n} \sum_{i=1}^{n} (x_i-m+m)^2\\
&=&\frac{1}{n} \left( \sum_{i=1}^{n} (x_i-m)^2 + \sum_{i=1}^{n} m^2 \right)\\
&=&\frac{1}{n} \left( n s^2 + n m^2 \right)\\
&=&s^2+m^2\end{aligned}$$
We do not consider forecasts in which we also ignore the variance of the ensemble, or in which we use the mean but ignore the variance, since doing so would be to assume that the predicted mean change had no uncertainty at all. That would not correspond to our beliefs, and would also give a BIC score of infinity which would be beaten by the other two methods by default.
BIC for each of the models {#s3}
==========================
In this section we now derive BIC values for the two methods defined in section \[s2\].
The definition of BIC is: $$\label{eq1}
B=-2l+k \ln n$$
where $l$ is the likelihood attained at the maximum, $k$ is the number of parameters in the model and $n$ is the number of independent data points used to fit the model (see, for example, @wasserman). When comparing a number of models, the model with the lowest value for the BIC is considered the best model.
As the number of parameters increases models will generally fit the data better and $l$ will increase. Hence the $-2l$ term will decrease while the $k \ln n$ term will increase. Whether the BIC itself increases or decreases depends on whether the extra parameters offer sufficient additional explanatory power to improve predictions, or whether the model becomes overfitted.
Since we have forced independence between the different climate models by inflating the variance the likelihood is given by the density of the multivariate normal distribution for independent samples, which is: $$p(x)=\prod_{i=1}^{n} \frac{1}{\sqrt{2\pi}} \frac{1}{\sigma} \mbox{exp} \left( -\frac{1}{2\sigma^2}(x_i-\mu)^2 \right)$$
The log-likelihood is then: $$\begin{aligned}
l(x)
&=&\ln p(x)\\
&=&\sum_{i=1}^{n} \ln (\frac{1}{\sqrt{2\pi}})+\sum_{i=1}^{n} \ln \frac{1}{\sigma} -\sum_{i=1}^{n} \frac{1}{2\sigma^2}(x_i-\mu)^2\\
&=&-\frac{n}{2}\ln 2\pi-\frac{n}{2} \ln \sigma^2 -\sum_{i=1}^{n} \frac{1}{2\sigma^2}(x_i-\mu)^2\end{aligned}$$
Using this, we now derive expressions for the BIC for each model. The differences in BIC between our two methods arise simply because of the different choices of $\mu$ and $\sigma$ given in sections \[m1\] and \[m2\] above.
BIC for Use Ensemble Mean, Use Ensemble Variance
------------------------------------------------
For this method the log-likelihood is: $$\begin{aligned}
l(x)
&=&-\frac{n}{2}\ln 2\pi-\frac{n}{2} \ln s^2 -\frac{n}{2}\\
&=&-\frac{n}{2} \left [\ln 2\pi+ \ln s^2+n\right]\end{aligned}$$
and there are two estimated parameters $(k=2)$, and so, from equation \[eq1\], the BIC is:
$$\begin{aligned}
B_1
&=& n \left [\ln 2\pi+ \ln s^2+n\right]+2 \ln n\end{aligned}$$
BIC for Ignore Ensemble Mean, Use Ensemble Variance
---------------------------------------------------
For this method the log-likelihood is: $$\begin{aligned}
l(x)
&=&-\frac{n}{2}\ln 2\pi-\frac{n}{2} \ln (s^2+m^2) -\frac{n}{2}\\
&=&-\frac{n}{2} \left [\ln 2\pi+ \ln (s^2+m^2)+n\right]\end{aligned}$$
Note that the only difference from the analogous expression for the previous model is the $m^2$ term, which is due to the bias in the forecast introduced by ignoring the ensemble mean, and which drives the BIC upwards, and tends to make the forecast less favoured.
This time there is only one estimated parameter $(k=1)$, which, on the other hand, tends to makes the forecast *more* favoured, and so the BIC is:
$$\begin{aligned}
B_2
&=& n \left [\ln 2\pi+ \ln (s^2+m^2)+n\right] +\ln n\end{aligned}$$
Comparing BIC values between the two models {#s4}
===========================================
The difference between the BIC scores for our two methods is therefore: $$\begin{aligned}
B_1-B_2
&=& n \left [\ln 2\pi+ \ln s^2+n\right]+2 \ln n-n \left [\ln 2\pi+ \ln (s^2+m^2)+n\right] -\ln n\\
&=& n \left[ \ln s^2 - \ln (s^2+m^2) \right]+\ln n\end{aligned}$$ Setting this equal to zero gives: $$\begin{aligned}
n \left[ \ln s^2 - \ln (s^2+m^2) \right]+\ln n&=&0\\
\ln(s^2+m^2)-ln(s^2)&=&\frac{\ln n}{n}\\
\ln \left[s^2\left(\frac{m^2}{s^2}+1\right)\right]-ln(s^2)&=&\frac{\ln n}{n}\\
\ln \frac{m^2}{s^2}+1&=&\frac{\ln n}{n}\\
\frac{m^2}{s^2}+1&=&\mbox{exp} \left( \frac{\ln n}{n} \right)\\
r_c^2=\frac{m^2}{s^2}&=&\mbox{exp} \left( \frac{\ln n}{n} \right)-1\label{eqrc}\end{aligned}$$
For $r$ below the critical value $r_c$ given by the above expression (equation \[eqrc\]) BIC favours the simpler model (ignore the ensemble mean, use the ensemble variance), and for $r$ above this value it favours the model complex model (use the ensemble mean, use the ensemble variance).
As an example, for $n=10$ ([i.e. an ensemble of size 10]{}) this gives $r_c=\frac{m}{s}=0.509$. Interestingly, this is at a lower point than the crossover between these models in the @jewsonh09c study, which was 0.73. This highlights the difference between model selection and monte carlo simulation methods. In this case using model selection would lead to using the more complex model for smaller changes in predicted climate than using the results of monte-carlo simulations would.
Figure \[fig1\] show the critical values of $r$ versus ensemble size, for a range of different ensemble sizes.
Application to a Prediction of UK Rainfall {#s5}
==========================================
We now apply the ideas developed above to coupled climate model predictions for UK rainfall. These predictions are based on multimodel ensembles generated during the CMIP project [@cmip], but with the ensemble variance inflated using the method described in @jewsonh09a, using assumed correlations between the models of between $0$ and $0.75$.
In figure \[fig2\] we show the value of $r$ for these predictions versus lead time. We also show the critical value of $r$ (which is $0.455$, since we have 14 models in the ensemble) as a horizontal line. We see that, for a correlation of 0.75, the predictions only cross the critical value of $r$ in about 2025. The implication is that before 2025 the spread across the models is so great than the ensemble mean is poorly estimated to the extent that using it to derive the mean of a forecast gives a less good forecast than ignoring it and use a mean change of zero. It is clear, however, that the exact point at which the forecast crosses this threshold would be heavily influenced by how the forecasts are smoothed in time.
Summary {#s6}
=======
We have considered the question of how to make a prediction of future climate using output from a multimodel ensemble of climate models. In particular we have considered what to do when the number of truly independent climate models is small, and the spread between the model results is large, as is often the case for certain variables. Using the standard BIC method for model selection, and making the perfect ensemble assumption, we have compared using the ensemble mean and variance with just using the ensemble variance and setting the mean change in climate to zero. This allows us to derive a simple expression for which of these two predictions to use. We have then applied the expression to CMIP predictions for UK rainfall, and concluded that for those predictions the ensemble mean should be ignored until at least 2025.
It is important to note that model selection is not the same as statistical testing. In statistical testing one typically asks whether the data supports a certain hypothesis (such as whether there are significant changes in UK rainfall), and rejects the hypothesis unless the data is strongly consistent with the hypothesis. In our case we are accepting the hypothesis that there is a change right from the start, rather than testing that hypothesis. We are then trying to decide, at a practical level, whether modelling the change is likely to give a better forecast than ignoring the change. If a predicted change is weak and uncertain, it can be perfectly logical to believe that there is a change (based on independent arguments such as physical reasoning), but that it is impossible to estimate it well enough for it to improve predictions. That appears to be the case for changes in UK rainfall in the near future. As an aside, it is also possible, for slightly stronger and/or less uncertain predictions, that it cannot be proven that a signal is statistically significant, but that because you believe that the signal is real (again, presumably based on independent reasoning) then it is worthwhile to estimate the signal and include it in predictions.
There are a number of directions in which we plan to extend this work. One is to try and include the ‘damped’ predictions described in @jewsonh09c in the model selection process. This cannot be done using BIC, since the damped predictions are not maximum likelihood based. Still, there may be other methods for model selection that one could use instead. The damped prediction methods will hopefully allow model predictions with lower values of $r$ to become usable, and would give more accurate predictions for intermediate values of $r$. Another direction would be to take an objective Bayesian approach to fitting the predictive distribution, in which case the normal becomes a t distribution. Once again BIC can no longer be used, although presumably Bayes Factors, from which BIC is derived, can.
[^1]: *Correspondence email*: `stephen.jewson@rms.com`
|
---
author:
- Aris Karastergiou
- Mark Walker
title: Goonhilly Sparklers
---
Introduction
============
Extreme Scattering Events (ESEs) are a type of transience in which the flux variations are not intrinsic to the source but are caused by radio-wave refraction in ionised gas along the line-of-sight (Fiedler et al 1987; Romani, Blandford and Cordes 1987). To generate these events the ionised gas must have a pressure which is a thousand times higher than the general Interstellar Medium (ISM), in a region of dimensions $\sim1\;$AU. Thus the ESE phenomenon poses a serious challenge to our understanding of the most basic physics of the ISM.
Because of the difficulty in understanding ESEs within a conventional picture of the ISM, we have previously proposed that they are caused by baryonic dark matter (Walker 2007; Walker and Wardle 1998): cold, dense, AU-sized molecular clouds ramming their way through the ISM at high speeds.
ESEs merit further study. Unfortunately there has been no substantial new dataset since the original work of Fiedler et al (1987,1994) using the Green Bank Interferometer. Within the next five years that situation will change drastically as SKA-pathfinder instruments are brought into service. In particular the Variables And Slow Transients (VAST) project, which utilises the Australian Square Kilometre Array Pathfinder (ASKAP: Johnston, Feain and Gupta 2009), will survey a large fraction of the sky on a daily basis. But ASKAP only operates efficiently up to 1.5 GHz, whereas the data we have on ESEs are at 2.7 GHz and above, making it difficult to plan for ESE science with VAST. With Goonhilly we will change that, by discovering a number of new events using 5 GHz data and then studying them at lower frequencies. In the process we will gain some powerful new insights into the physics of ESEs.
-1cm
The Goonhilly facilities
========================
The Goonhilly Earth Station, in Cornwall, UK, was formerly a telecommunications facility. On site are three 30m dishes and some smaller (15m) antennas. The Consortium of Universities for Goonhilly Astronomy plans to instrument two of the 30m dishes for radio astronomy, potentially including various frequency bands within the 1 to 10 GHz range.
In this paper we outline the possibilities for ESE science which may be opened up by operating Goonhilly as an astronomical facility. We consider two hypothetical instruments:${\bf C_1}$: a single 30m dish equipped with a 5 GHz receiver having 1 GHz bandwidth${\bf C_2}$: a pair of 15m dishes operating as an interferometer and receiving across the full 4 to 8 GHz band simultaneously.
ESE science with the ${\bf C_1}$ system
=======================================
To find ESEs we need to monitor the fluxes of a large number of compact radio sources. The combined probability of the two most striking ESEs (those in Q0954+658 and Q1749+096) is $\sim5\times10^{-4}$, so that in a sample of 2,000 extragalactic sources there will typically be one ESE in progress at any given moment. And the event durations are $\sim2\,$months, so the event rate in a sample of 2,000 sources is $\sim6\;{\rm year^{-1}}$.
For daily, year-round monitoring we need to choose sources which are away from the ecliptic. Restricting ourselves to the region with ecliptic latitude greater than $30^\circ$ means that we have $\pi$ steradians available, in principle. But the reality of an alt-az mount is that it can take a long time to slew between sources which are North of the zenith, and those to the South, whereas we need to minimise overheads associated with slewing. The latitude of Goonhilly is approximately $50^\circ$ and we therefore restrict ourselves to the 1.5 sr between declination $50^\circ$ and the North Celestial Pole.
Our program sources must be compact, in order that they can be significantly magnified, so only about one in 5 radio sources will be suitable for our purposes. (Compact sources can be selected on the basis of their radio spectra, which should be inverted or at least flat.) To assemble a sample of 2,000 compact sources we must therefore range down to $S_{min}\simeq35\,$mJy, where the areal density of all radio sources is approximately ${\rm 6,\!000\,sr^{-1}}$ at 5 GHz (Wall 1994).
Assuming that the receiver is a clone of the C-BASS system the flux noise for ${\bf C_1}$ should be approximately $4\,{\rm
mJy\sqrt{s}}$. To detect magnification changes of order 10% with high confidence requires a signal-to-noise ratio of at least 30, which would be achieved in about 12 seconds for a 35 mJy source. (The brightest confusing source in the beam will typically be 0.8 mJy, and this source will normally be steady and will not be detrimental to our study.) Thus the total required on-source time is less than 7 hours per epoch for a sample of 2,000 targets. To this we must add the time required for slewing and settling.
The typical angular separation between targets is less than $2^\circ$. That is not a long slew and it seems reasonable to expect that this can be achieved in under 30 seconds. If so, a sample of 2,000 sources can be monitored on a daily basis using ${\bf
C_1}$. Daily sampling is desirable, even though the events last for many weeks, in order to clearly distinguish ESEs from other forms of variability and, especially, to do so in real time.
Thus with ${\bf C_1}$ operating for one year we can expect to detect 6 ESEs: a significant increase on the current sample of two. But the greatest benefit is not so much the increase in event numbers as the opportunity to identify those events in real time and thus to characterise each one in detail. As well as detailed radio studies (e.g. long baseline imaging), which constrain the ionised gas profiles of the lenses, we aim to test for the presence of underlying neutral gas, e.g. by the associated UV extinction.
ESE science with the ${\bf C_2}$ system
=======================================
The ${\bf C_2}$ system has four times the bandwidth but only half the collecting area of ${\bf C_1}$, so assuming the same system temperature implies that it would take roughly the same integration time to reach the same flux limit (assuming a flat spectrum source).
Confusion noise is potentially larger for ${\bf C_2}$, with the brightest confusing source in the beam being typically about 3.5 mJy. However, if each target is observed at the same hour angle (near transit, say), at every epoch then confusing sources make a constant contribution to the measured visibilities and are not detrimental to our study.
On the basis of experience with C-BASS it is anticipated (Mike Jones, personal communication) that ${\bf C_1}$ should be photometrically stable (i.e. to within the thermal noise) over periods up to about 100 seconds. It will therefore be necessary to intersperse our target sources with some (steep-spectrum, non-compact so non-variable) sources for calibration. Very few such sources would be needed for ${\bf C_2}$, because of the inherent stability of an interferometer, and ${\bf C_2}$ could thus tackle a somewhat larger target sample than ${\bf C_1}$.
But the greatest advantage of ${\bf C_2}$ lies in the broad bandwidth which it covers. Over an octave in radio frequency the refraction angles introduced by the lens change by a factor of four. In turn this means that the monitoring data themselves would yield tight constraints on the electron column-density of the lenses, even in the absence of any detailed real-time follow-up on other telescopes. Thus the ${\bf C_2}$ system is superior to ${\bf C_1}$.
Lastly we note that for a long-term experiment, running for many years and discovering large numbers of events, it will not be possible to use facility-class instruments, like the VLA, to study each event in detail. It is here that the ${\bf C_2}$ system really comes into its own because as a stand-alone experiment it yields much more powerful constraints on the lenses than ${\bf C_1}$.
Fiedler, R. et al 1987 326, 675 Fiedler, R. et al 1994 430, 581 Johnston, S., Feain, I.J., Gupta, N., 2009, ASP Conf Series 407, 446 Romani, R., Blandford, R. D., Cordes, J. M. 1987 328, 324 Walker, M.A. 2007 ASP Conf. Series 365, 299 Walker, M.A., Wardle, M.J. 1998 498, L125 Wall, J.V., 1994, AustJPhys, 47, 625
|
---
abstract: '**Relaxor ferroelectrics, which can exhibit exceptional electromechanical coupling are some of the most important functional materials with applications ranging from ultrasound imaging to actuators and sensors in microelectromechanical devices. Since their discovery nearly 60 years ago, the complexity of nanoscale chemical and structural heterogeneity in these systems has made understanding the origins of their unique electromechanical properties a seemingly intractable problem. A full accounting of the mechanisms that connect local structure and chemistry with nanoscale fluctuations in polarization has, however, remained a need and a challenge. Here, we employ aberration-corrected scanning transmission electron microscopy (STEM) to quantify various types of nanoscale heterogeneity and their connection to local polarization in the prototypical relaxor ferroelectric system Pb(Mg$_{1/3}$Nb$_{2/3}$)O$_{3}$-PbTiO3 (PMN-PT). We identify three main contributions that each depend on Ti content: chemical order, oxygen octahedral tilt, and oxygen octahedral distortion. These heterogeneities are found to be spatially correlated with low angle polar domain walls, indicating their role in disrupting long-range polarization. Specifically, these heterogeneities lead to nanoscale domain formation and the relaxor response. We further locate nanoscale regions of monoclinic distortion that correlate directly with Ti content and the electromechanical performance. Through this approach, the elusive connection between chemical heterogeneity, structural heterogeneity and local polarization is revealed, and the results validate models needed to develop the next generation of relaxor ferroelectric materials.**'
author:
- Abinash Kumar
- 'Jonathon N. Baker'
- 'Preston C. Bowes'
- 'Matthew J. Cabral'
- Shujun Zhang
- Elizabeth Dickey
- 'Douglas L. Irving'
- 'James M. LeBeau'
bibliography:
- 'refs.bib'
title: 'Decoding the complexities of lead-based relaxor ferroelectrics'
---
Relaxor ferroelectrics are distinguished from traditional ferroelectrics by their frequency dependent, diffuse phase transitions that are commonly attributed to the existence of nanoscale order.[@Cohen2006] Among this class of materials, Pb-based systems remain essential as they exhibit the largest known piezoelectric coefficients and outstanding dielectric properties.[@Park1997; @Zhang2012; @Zhang2015] These properties have been commonly attributed to the existence of polar nanoregions (PNRs) in the material that align to the applied field.[@Burns1983; @Burns1983b] This simple model fails, however, to explain relaxor behavior in a variety of materials, including polymer-based systems.[@Yang2014] The recently proposed “polar slush” model overcomes these limitations by considering a multi-domain polar state with low energy, low angle domain walls formed throughout.[@Takenaka2017] Model validation has, however, been hindered by the seemingly endless number of structural characterization studies with differing interpretations, leading these systems to be described as a “hopeless mess” .[@Cohen2006]
Pb(Mg$_{1/3}$Nb$_{2/3}$)O$_{3}$-PbTiO$_3$ (PMN-PT) is the prototypical Pb-based relaxor ferroelectric system and exhibits state-of-the-art properties. For example, piezoelectric constants up to 4100 pC/N have been achieved via Sm doping [@Li2019]. PMN-PT adopts the perovskite structure (formula ABO$_3$) with the A sub-lattice occupied by Pb and the B sub-lattice occupied by the either Mg, Nb, or Ti. Furthermore, as PbTiO$_3$ is added to PMN, the system reaches a morphotropic phase boundary (MPB) near PMN-30PT where the electromechanical properties are maximized.[@Krogstad2018] Deciphering the origins of these enhanced properties is, however, complicated by nanoscale heterogenity of chemistry and structure. While X-ray and neutron scattering are the predominate characterization methods to explain these details, making direct connections between composition, structure, and polarization have proven exceedingly difficult [@Singh2006; @Singh2003; @Thomas1999a; @Kim2012a; @Cowley2011; @Davis2007; @Randall1990]. For example, while nanoscale chemical ordering of the B sub-lattice cations was one of the first signatures revealed [@Randall1990; @C.A.Randall1990; @C.A.Randall21990] and is found to varying degrees in all Pb-based relaxor ferroelectrics[@TakesueN1999; @Goossens2013], its details are still being revealed [@Cabral2018; @Kopecky2016; @Eremenko2019].
Capturing a complete picture of the connection between chemistry, structure, and polarization in relaxor ferroelectrics requires characterization techniques that are able to account for both cation *and* anion sub-lattices. While, distortions of the oxygen sub-lattice previously have been detected with neutron diffraction [@Rosenfeld1995], a combination of X-ray and neutron diffraction recently discovered that correlated oxygen displacements are likely the key to understanding the electromechanical properties [@Krogstad2018]. Diffraction characterization methods, however, lack the spatial resolution required to directly determine the origin of nanoscale polar variation [@Keen2015; @Xu2008]. In contrast, annular dark-field scanning transmission electron microscopy (ADF STEM), Figure \[fig:quiver\]b, can readily image the projected crystal structure at the atomic length scale and is sensitive to chemical ordering [@Cabral2018]. These ADF images are, however, dominated by cation contrast that precludes detailed detection and quantification of light elements such as oxygen. Recently, annular bright field (ABF) [@Findlay2009; @Kim2017] and integrated differential phase contrast (iDPC) [@Lazic2016; @DeGraaf2018] STEM techniques have been introduced to overcome these challenges. As shown in Figure \[fig:quiver\]b, iDPC can visualize the cation and oxygen atom column positions, but lacks direct atomic number interpretability. By simultaneously capturing ADF and iDPC images, however, the resulting datasets are akin to merging results from X-ray and neutron diffraction techniques, and enables the correlation between polarization, chemistry, and structure directly at the atomic scale.
Here, we investigate the structural and chemical origins of relaxor ferroelectric properties in PMN-xPT (x=0, 10 & 30). Through a combination of ADF and iDPC aberration corrected STEM, the projected positions of cation and anion sub-lattices are used to measure the subtle features of nanoscale polarization in these materials. The projected polarization reveals the presence of nanoscale domains that are consistent with a frozen ‘polar slush’ model.[@Takenaka2017] We further quantify the distribution of chemical and structural heterogeneities as a function of Ti content, and a direct correlation is found between the spatial distribution of chemical/structural heterogeneities and polarization domain walls. The heterogeneities are found to inhibit polarization rotation, leading to the formation of low angle domain walls. In combination, these results provide evidence for the underlying mechanisms responsible for yielding relaxor behavior.
Using iDPC STEM, the projected polarization is measured across ten different sample locations using the approach outlined in Supplementary Information Section \[supplsec:projectedPolarization\]. This yields approximately 5,000 analyzed unit cells at each of the three PMN-PT compositions. The high sampling of each composition provides a statistical representation of the underlying structure of the material, aiding in connecting the results presented here to prior diffraction studies. Calculating the center of mass difference between the cations and anions using iDPC STEM, Figure \[fig:quiver\]c shows representative projected polarization maps for PMN, PMN-10PT, and PMN-30PT imaged along [$\left[110\right]$]{}. The projected polarization exhibits regions with similar magnitude and direction, i.e. polar domains, that vary smoothly across each image.
Notably for PMN, the observed polar domains do not decay to a non-polar background, which contrasts with the ’polar nanoregion’ model that assumes PNRs exist in a non-polar matrix. These nanoscale domains vary in size from 2-12 nm and form low angle domain walls between them, which are located using the method described in Supplementary Information Section \[supple:domainwalldistribution\]. This observation is also remarkably similar to reverse Monte Carlo analysis of diffraction data.[@Eremenko2019]. Moreover, a significant majority (72%) of the domain wall angles are in the range of 10-35$^\circ$ , which is in agreement with predictions from the ‘polar slush’ model (see Supplementary Information Section \[supple:domainwalldistribution\]).[@Takenaka2017; @Kim2019] As the fraction of PT increases, the average domain wall angle increases and reflects the onset of ferroelectric behavior. This behavior leads to the mixed relaxor and ferroelectric properties found in PMN-PT materials and expected from the polar slush model.[@Takenaka2017; @Kim2019] The polar slush model does not, however, incorporate the specific structure and chemistry details that drive the formation of the polar domains.
Chemical ordering is revealed in the ADF STEM data, as shown in Figure \[fig:Inhomo\]a,left. Clustered oscillation of B sub-lattice atom column intensity on (111) planes indicates a doubled perovskite lattice where occupancy of Mg and Nb vary systematically. The weaker normalized intensity atom columns (blue) contain more Mg or Ti ($\beta_I$), while those stronger intensity atom columns (red) contain more Nb ($\beta_{II}$). Correlation analysis, shown in Figure \[fig:Inhomo\]a and detailed in Supplementary Information Section \[supplsec:correlationAnalysis\], is used to quantify the relative fraction of these ordered regions. At each composition, the CORs are found throughout and account for 39$\pm$1% of the total projected area in the case of PMN, 37$\pm$2% for 10% of PT, and only 11$\pm$1% for 30% PT. The decreasing chemical order with increasing Ti is in agreement with previous X-ray and neutron scattering results where superlattice reflection intensity also decreases as the composition approaches the MPB.[@Hilton1990]
Recently, Krogstad suggested the presence of anti-ferrodistortive displacements based on diffuse scattering features, i.e. Pb displaces in the opposite direction of its B sub-lattice neighbors [@Krogstad2018]. The chemical and spatial origins of this behavior could not, however, be unambiguously determined. From iDPC STEM data, the Pb atom columns are anti-ferrodistortive relative to the B-sites largely in the CORs, (Supplementary Information Section \[supple:pbDisplacement\]). These anti-ferrodistortive displacements decrease in accordance with the decreasing COR density as PT content increases, and in agreement with Krogstad.
As noted from recent X-ray and neutron scattering experiments, oxygen displacements may hold the key to understanding structure-property relationships in Pb-based relaxors.[@Krogstad2018] To this end, oxygen octahedral distortion (expansion/compression) and tilting (schematically shown in Figure \[fig:Inhomo\]d) are determined from iDPC images, as in Figures \[fig:Inhomo\]b-c. At each composition, distortion and tilting exhibit local ordering, reminiscent of the CORs.
Correlation analysis is used to determine the area fraction of octahedral distortion regions (ODRs), as shown in Figure \[fig:Inhomo\]b. In PMN, 21% of the projected area exhibits distortion ordering, with 63% overlapping the CORs and 27% at the COR boundaries. As PT content increases, the ODRs increase to 28% for PMN-10PT. Furthermore, a similar checkerboard octahedral distortion pattern is predicted for the ordered PMN structure from density functional theory (DFT) (Supplementary information Sections \[supple:dft\] and \[supple:stemImageSims\]).
At PMN-30PT, however, the distortion order changes. Rather than exhibiting the rhombohedral-type, checkerboard distortion pattern (see Figure \[fig:Inhomo\]f), the distortion becomes predominately striped on $\left(110\right)$ planes, which indicates the formation of monoclinic-like unit cells as verified with STEM image simulations (Supplementary Information Figure \[fig:STEM\_crystalSIm\]) and agrees with prior diffraction studies [@Singh2006]. Analyzed across the composition space, monoclinic-type distortion increases with addition of Ti as shown in Figure \[fig:Inhomo\]b. Moreover, this type of local planar distortion should give rise to asymmetric diffuse scattering [@Kreisel2003]. Importantly, Krogstad [@Krogstad2018] found that the oxygen mediated asymmetric diffuse scattering in neutron scattering was the only feature linked to the piezoelectric properties. The results presented here confirm that oxygen displacements are key and they originate in monoclinic-type oxygen octahedral regions.
Beyond expansion and contraction, the oxygen octahedra also tilt with respect to [$\left[110\right]$]{}, as shown in Figure \[fig:Inhomo\]c. Applying correlation analysis, 19% of the total projected area exhibits octahedral tilt ordering in PMN. Furthermore, the B sub-lattice atom column intensities in octahedral tilt regions (OTRs) are normally distributed (see Supplementary Information Section \[supple:intensityDistribution\]). This reveals that the OTRs form in the regions of randomly distributed Mg and Nb instead of the CORs. The anti-phase tilt pattern suggest the presence of local rhombohedral, $R3c$, or orthorhombic, $Pnma$ local symmetry.[@Glazer1972] With increasing PT, the OTR area decreases from 11% to 5% for 10PT and 30PT, respectively. As the fraction of these features decrease with composition, their disappearance also explains another distinct diffuse scattering contribution noted by Krogstad that had both temperature dependent and independent components. While the temperature independent contribution is thought to originate from the CORs, the temperature dependent part then arises from the octahedral tilt ordering that is disrupted by thermal fluctuations. The intensity of this type of diffuse scattering is found to decrease with PT content and follows the same trend as OTRs quantified from STEM.
Combining the ADF and iDPC STEM data, the change in B site chemistry measured with the Mg/Nb/Ti atom column intensity is also found to correlate to the O-O neighbor distance, as shown in Figure \[fig:OODistCor\]. In each case, the B sub-lattice intensity is negatively correlated with the O-O atom column distances [@Sang2015]. This shows that Mg/Ti expands the local oxygen sub-lattice, while Nb leads to contraction. In combination with the response of oxygen at the MPB where the O-O spread increases, the observed correlation indicates the key role of Ti to disrupt the lattice to stabilize the monoclinic phase and increase the piezoelectric coefficient. The Pb/O-Pb/O atom column distances, in comparison, show weak-to-no correlation with variation in B-site chemistry in PMN as shown in Figure \[fig:OODistCor\], likely due to the dominant contribution of Pb on the observed atom column positions (Supplementary Information Section \[supplsec:projectedPolarization\]). In contrast, PMN-10PT exhibits moderate, negative correlation. At the MPB, PMN-30PT, significantly larger distortions are observed that correspond to the increased polarization and piezoelectric coefficient. This correlated structural variation in Pb displacement also explains the diffuse scattering feature found by Krogstad that correlates to the cation size mismatch.
Further analysis of the PMN-10PT $\beta_{I}$ and $\beta_{II}$ normalized intensities (Supplementary Information Figure \[supple:intensityDistribution\]) also reveals that their intensity difference is enhanced with the introduction of Ti. This suggests that Ti initially replaces Nb in the mixed $\beta_I$ sites of the CORs leading to a concomitant decrease in intensity for those atom columns and increased image contrast. Importantly, the preferential segregation of Ti to the CORs has not been previously shown and can be explained by analysis of the B sub-lattice bond lengths from DFT. Any covalent bond has an ideal length or range of lengths it prefers to adopt for any particular coordination environment. For example in oxide perovskites, titanium tends to prefer bond lengths between 1.9 and 2.1 $\text{\AA}$ [@abramov1995chemical; @cole1937lead; @shin2007order; @yoshiasa2016high]. From DFT, Supplementary Figure \[fig:Bsites-within-constraints.png\], the COR structure has more sites close to Ti’s ideal octahedral bonding environment than the disordered structure. These results indicate that Ti can form more, strong bonds in the COR structure than the disordered structure, without changing any other bond lengths or otherwise altering the structures. All else being equal, this strongly suggests that Ti will preferentially incorporate into the CORs, at least on the PMN side of the composition space where CORs form.
The heterogeneity distribution is also found to link directly with the local variation in polarization and domain walls. First, the centers of the CORs, ODRs and OTRs are identified as maxima in the results from correlation analysis as detailed in Supplementary Information Section \[supple:polarCorrelation\]. Qualitatively, Figure \[fig:corPolInhomo\]a-c, the heterogeneity maxima occur at, or very near, the locations of greatest polarization variation, i.e. low angle domain walls.
The observed relationship between ordered heterogeneities and domains walls is validated by quantifying the nearest neighbor distance. As shown in Figure \[fig:corPolInhomo\]d, 95$\%$ of the heterogeneities are within a distance of 1.5-2 unit cells of the nearest domain wall regardless of PT content. Across the composition range investigated, the 1/3 of CORs, 1/3 ODRs, and 1/3 OTRs within this distance of the domain walls. In addition, 1/3 of the domain walls have at least two nearby types of structural or chemical ordering. This indicate that the polar domain walls cannot be explained by a single type of heterogeneity, but requires the consideration of all features identified.
To determine the significance of this observation, consider the null-hypothesis that heterogeneities are randomly distributed with respect to polarization. Taking 10,000 random sets of data equivalent to experiment, but with random heterogeneity locations, the 95$\%$ threshold is not reached until a distance two times larger than experiment, see Figure \[fig:corPolInhomo\]d. Further highlighting the difference, the error bars indicate the maximum and minimum 95$\%$ distances for the 10,000 datasets at each composition. A result similar to experiment is not found any of the random data, indicating that $p$ is significantly less than 0.05, safely discounting the null-hypothesis. Taken together, the results presented here unambiguously reveal that chemical and structural heterogeneities are not randomly distributed with respect to polarization, but instead act as the mechanism to disrupt the formation of long range polarization. Although this role heterogeneity has been postulated, its connection to generating low angle domain walls has not been directly observed until now.[@Randall1990; @Fu2009]
In conclusion, direct quantification of cations and anion atom columns from STEM has revealed key relationships between polarization, structure, and chemistry in the prototypical relaxor ferroelectric system, PMN-PT. The presence of nanoscale domains exhibiting low angle domain walls is in agreement with the recently proposed polar slush model[@Takenaka2017]. The formation of nanoscale polar domain walls arises from the presence of short range chemical and/or structural heterogeneity, which act to disrupt the long range polarization. The introduction of Ti decreases the type and number of heterogeneities, which leads to cooperative alignment of polarization while also increasing the fraction of monoclinic distortion that enhances the piezoelectric coefficient. The combination of nanoscale features explains the origin for relaxor ferroelectric behavior in PMN-PT that lead to the dramatically improved piezoelectric performance. Extending this knowledge, it is anticipated that engineering a combination of short range structural and chemical order are required for advancing the next generation of Pb-free relaxor-ferroelectric materials.
Acknowledgements {#acknowledgements .unnumbered}
================
We gratefully acknowledge support for this work from the National Science Foundation, as part of the Center for Dielectrics and Piezoelectrics under Grant Nos. IIP-1841453 and IIP-1841466. SZ acknowledges support from the Australian Research Council (FT140100698) and the Office of Naval Research Global (N62909-18-12168). PCB was supported by the Department of Defense through the National Defense Science & Engineering Graduate (NDSEG) fellowship program. Computational time and financial support for JNB was provided by AFOSR grant FA9550-17-1-0318. MJC acknowledges support from National Science Foundation as part of the NRT-SEAS under Grant No. DGE-1633587. This work was performed in part at the Analytical Instrumentation Facility (AIF) at North Carolina State University, which is supported by the State of North Carolina and the National Science Foundation (ECCS-1542015). AIF is a member of the North Carolina Research Triangle Nanotechnology Network (RTNN), a site in the National Nanotechnology Coordinated Infrastructure (NNCI). We acknowledge Matthew Hauwiller for useful suggestions while preparing the manuscript.
Author Contributions {#author-contributions .unnumbered}
====================
AK conducted the electron microscopy experiments, data analysis, and image simulations. MC prepared the PMN samples for electron microscopy and collected STEM data. SZ grew the PMN-xPT single crystals. JB and DL performed the DFT calculations and the corresponding analysis. JML and ECD designed the electron microscopy experiments and guided the research. All authors co-wrote and edited the manuscript.
Competing interests {#competing-interests .unnumbered}
===================
The authors declare no competing interests.
Methods {#methods .unnumbered}
=======
Sample Information {#sample-information .unnumbered}
------------------
Pb(Mg$_{1/3}$Nb$_{2/3}$)O$_{3}$-xPbTiO3 (PMN-xPT) single crystals were grown via the high temperature flux method [@Zhang2012]. Samples for electron microscopy were cut from these larger single crystals and oriented along $\left< 011\right>$ and then thinned to electron transparency using mechanical wedge polishing [@Voyles2002a] followed by low temperature, low energy Ar ion milling.
Scanning Transmission Electron Microscopy {#scanning-transmission-electron-microscopy .unnumbered}
-----------------------------------------
STEM imaging was performed with a probe-corrected FEI Titan G2 60-300 kV S/TEM equipped with an X-FEG source at a beam current of 30 pA and probe semi-convergence angle of 19.6 mrad. We used a custom scripting interface to automate the Thermo Fisher Scientific Velox software for simultaneous ADF and iDPC acquisition. ADF images were collected with a semi-collection angle range of 34-205 mrad and iDPC with a with collection semi-angle range of 7-28 mrad. For high image accuracy and precision, the revolving STEM (RevSTEM) method was used.[@Sang2014c; @Dycus2015] Each RevSTEM data set consisted of 20 1024$\times$1024 pixel frames with a 90$^\circ$ rotation between each successive frame. Sample thickness ranged from 6 to 10 nm as determined using position averaged convergent beam electron diffraction [@Lebeau2010]. Supercells were constructed using relaxed structures from DFT, see next section. A repeating unit was cropped and simulated using a custom python based STEM image simulation software. Simulated ADF & iDPC images were convolved with a Gaussian with full-width at half-maximum of 80 pm to approximately account for the finite effective source size [@Lebeau2008].
Density Functional Theory and Image Simulations {#density-functional-theory-and-image-simulations .unnumbered}
-----------------------------------------------
Density functional theory (DFT) calculations of PMN were performed to investigate the atomic structures in three dimensions, and to examine charge localization and bond length distributions. These calculations were performed with the PBE exchange correlation functional in VASP 5.3.3, collinear spin polarization, and a plane wave kinetic energy cutoff of 520 eV. [@Kresse1993; @Kresse1994; @kresse1996efficiency; @Kresse1996a] A single reciprocal space point at $\Gamma$ was used. Projector Augmented Wave pseudopotentials were used with 2, 11, 4, and 6 valence electrons explicitly treated for Mg, Nb, Pb, and O, respectively. Additional details of the calculation are provided in Supplementary Information Section \[supple:dft\].
|
---
author:
- 'V. Matsaev [^1], I. Ostrovskii [^2], and M. Sodin$^{\ast}$'
title: |
Variations on the theme of\
Marcinkiewicz’ inequality
---
Introduction
============
Let $h$ be a smooth function on $\bf R$ with a compact support, and $$g(x) = \hbox{p.v.} \frac{1}{\pi}\, \int_{\bf R} \frac{h(t)}{t-x}\, dt$$ be its Hilbert transform. Set $f=g+ih$ and introduce the function $$u_f(z)=\int_{\bf R} H(zf(t))\, dt\,, \qquad H(z)=\log|1-z|+\hbox{Re}(z)\,,$$ called the [*logarithmic determinant*]{} of genus one. It is subharmonic in $\bf C$, and its Riesz measure is $d\mu_f(\zeta)=d\nu_f(\zeta^{-1})$, where $d\nu_f$ is the distribution measure of $f$: $$\nu_f(E)=\hbox{meas}\left(\{t:\, f(t)\in E\}
\right)\,, \qquad E \ \hbox{is a borelian subset
of} \ \bf C\,,$$ and $\hbox{meas}(\,.\,)$ stands for the Lebesgue measure on $\bf R$. Let $$\mu_f (r)=\mu_f(\{|z|\le r\})=\hbox{meas}\left(\{|f|\ge r^{-1}\} \right)$$ be a (conventional) counting function of $d\mu_f$, and let $$\begin{aligned}
{\mathfrak n}_f(r) &=&
\mu_f \left( \{ |z - ir/2| \le r/2\}\right) +
\mu_f \left( \{ |z + ir/2| \le r/2\}\right) \\
&=& \mu_f\left(\{ |\hbox{Im}(z^{-1})|\ge r^{-1}\}\right)
= \hbox{meas}\left( \{|h|\ge r^{-1}\} \right)\end{aligned}$$ be its [*Levin-Tsuji counting function*]{}, see [@Levin1], [@Tsuji], and [@GO Chapter 1]. Then the classical estimates of the Hilbert transform can be easily rewritten as upper bounds of $\mu_f(r)$ by ${\mathfrak n}_f(r)$.
For example, Marcinkiewicz’ inequality (see [@Koosis Chapter V]) $$m_f(\lambda) \le
C \,\left\{
\frac{1}{\lambda^2}\, \int_0^\lambda sm_h(s) ds
+ \frac{1}{\lambda}\, \int_\lambda^\infty m_h(s) ds
\right\}\,, \qquad 0<\lambda<\infty\,,
\label{0.1}$$ where $$m_f(\lambda) = \hbox{meas}\left(\{|f|\ge \lambda\}\right)
=\nu_f(\{|w|\ge \lambda\}) = \mu_f (\lambda^{-1})\,,$$ and $$m_h(\lambda) = \hbox{meas}\left(\{|h|\ge \lambda\}\right)
=\nu_f(\{|\hbox{Im} w|\ge \lambda\}) = {\mathfrak n}_f (\lambda^{-1})\,,$$ reads: $$\mu_f(r) \le C
\left\{
r \int_0^r \frac{{\mathfrak n}_f(t)}{t^2}\, dt + r^2 \int_r^\infty
\frac{{\mathfrak n}_f(t)}{t^3}\, dt
\right\}\,,
\qquad 0<r<\infty\,.
\label{0.2}$$ From this, one readily obtains $$\mu_f(r) \le Cr \int_0^\infty \frac{{\mathfrak n}_f(t)}{t^2}\, dt\,, \qquad
0<r<\infty\,,
\label{0.3}$$ which is equivalent to Kolmogorov’s weak $L^1$-type inequality: $\lambda m_f(\lambda) \le C||h||_{L^1}$, $0<\lambda<\infty$, and $$\int_0^\infty \frac{\mu_f(t)}{t^{p+1}} dt
\le C(p) \int_0^\infty \frac{{\mathfrak n}_f(t)}{t^{p+1}} dt\,,
\qquad 1<p<2\,,
\label{0.5}$$ which is equivalent to M. Riesz’ inequality: $$||f||_{L^p} \le C(p) ||h||_{L^p}\,.
\label{0.6}$$ The classical proof of inequality (\[0.1\]) is based on the interpolation technique which later served as one of the cornerstones of an abstract theory of interpolation of operators in Banach spaces [@Bennett].
A natural question arises: [*what is special about the subharmonic function $u_f(z)$ which makes inequality (\[0.2\]) valid?*]{} The answer is proposed in [@MS1]: a key fact ensuring this, is the positivity condition: $$u_f(z) \ge 0\,, \qquad z\in {\bf C}\,,
\label{0.7}$$ which can be easily checked with the help of Green’s formula (see [@Essen Lemma 5]) or by applying of the Cauchy residue theorem (see [@MS2]).
This leads to a heuristic principle which suggests that
- [*results about the distribution of the Hilbert transform can be deduced from inequality (\[0.7\]) by using methods of the subharmonic function theory*]{}.
As will be shown, inequality (\[0.7\]) is even too strong, and in many cases it suffices to assume that $u_f(x)\ge 0$ on $\bf R$, or to control the negative part $u_f^-=\max(0,-u_f)$ on $\bf R$.
The principle shows a path to new results on the Hilbert transform. In [@MS2], its application leads to a complete description of the distribution of the Hilbert transform of $L^1$-functions and measures of finite variations. At the same time, putting known estimates of the Hilbert transform into this setting, we arrive at new interesting questions about the argument-distribution of the Riesz measure in certain classes of subharmonic functions. For example, the proofs of the inequalities of Kolmogorov and M. Riesz found in [@MS1] give new bounds for zeros of polynomials. Positivity condition (\[0.7\]) links our work with the theory of uniform algebras and Jensen measures (see [@Gamelin]).
In this work we put forward a new approach to the Marcinkiewicz inequality (\[0.1\]) (or (\[0.2\])). The methods applied in [@MS1], [@MS3] are too rigid for this. Here we use a different technique.
Here and later on, we use the following notations:
$\phi (s) {\lesssim}\psi (s)$ means that there is a positive numerical constant $C$ such that, for each $s>0$, $\phi (s) \le C
\psi (s)$;
$\phi (s) {\lesssim}_{\alpha} \psi (s)$ means the same as above but $C$ may depend on a parameter $\alpha$;
$H(z)=\log|1-z| + \hbox{Re}(z)$ is the canonical kernel of genus one;
$\bf C_\pm$ are the upper and lower half-planes.
Main results
============
Define a subharmonic canonical integral of genus one: $$u(z)=\int_{\bf C} H(z/\zeta)\, d\mu(\zeta)\,,
\label{1.1}$$ where $d\mu$ is a non-negative locally finite measure on $\bf C$ such that $$\int_{\bf C}
\min\left(\frac{1}{|\zeta|},\frac{1}{|\zeta|^2}\right)\,d\mu(\zeta)<\infty\,.
\label{1.2}$$ Let $$\mu (r) = \mu (\{|z|\le r\})$$ be a (conventional) counting function of the measure $d\mu$, and let $${\mathfrak n}(r) = \mu\left( \{ |\mbox{Im}(1/z)|\ge 1/r \} \right)
= \mu\left( \{|z - ir/2| \le r/2 \} \right)
+ \mu\left( \{|z + ir/2| \le r/2 \} \right)$$ be its [*Levin-Tsuji counting function*]{} [@Levin1], [@Tsuji] (see also [@GO Chapter 1]).
Let $M(r,u)=\max_{|z|\le r} u(z)$. Then by the Jensen inequality, $\mu (r) {\lesssim}M(r,u)$. In the opposite direction, a standard estimate of the kernel $$H(z) {\lesssim}\frac{|z|^2}{1+|z|}\,, \qquad z\in{\bf C}\,,$$ yields Borel’s estimate $$M(r,u) {\lesssim}r\int_0^r \frac{\mu(t)}{t^2}\,dt
+ r^2 \int_r^\infty \frac{\mu(t)}{t^3}\,dt \,.
\label{2.borel}$$ In particular, $$M(r,u) = \left\{
\begin{array}{ll}
o(r), & r\to 0\, \\ \nonumber \\
o(r^2), & r\to\infty\,.
\end{array}
\right.
\label{1.M}$$
[Theorem 1]{} Let $u(z)$ be a canonical integral (\[1.1\]) of genus one, then $$M(r,u) {\lesssim}r\int_0^r \frac{{\mathfrak n}(t)}{t^2}\,dt + r^2 \int_r^\infty
\frac{{\mathfrak n}(t)}{t^3}\, dt
+ r^2 \int_r^\infty \frac{\mathfrak m (t,u)}{t^2}\,dt \,,
\label{1.3}$$ where $$\mathfrak m (r,u) =
\frac{1}{2\pi} \int_0^{2\pi} u^-(re^{i\theta}|\sin\theta)|)
\frac{d\theta}{\sin^2\theta}$$is the Tsuji proximity function of $u$.
If the function $u$ is non-negative in $\bf C$, then the proximity function $\mathfrak m(r,u)$ vanishes, and applying Jensen’s inequality we arrive at
[**Corollary 1.** ]{}[*Let $u$ be a canonical integral of genus one which is non-negative in $\bf C$. Then*]{} $$\mu (r) {\lesssim}r\int_0^r \frac{{\mathfrak n}(t)}{t^2}\,dt + r^2 \int_r^\infty
\frac{{\mathfrak n}(t)}{t^3}\, dt\,.$$
As we explained in the introduction, this result immediately yields the classical Marcinkiewicz inequality (\[0.1\]). In this case one can apriori assume that the function $f$ is bounded, so that most of the technicalities needed for the proof of Theorem 1 (see Lemmas 2 and 4 below) are redundant, and our proof of Marcinkiewicz’ inequality, being conceptually new, is comparable in length to the classical one.
There is a curious reformulation of Corollary 1. Let $\mathcal M$ be a measurable space endowed with a locally finite non-negative measure $dm$, let $f$ be a complex-valued measurable function on $\mathcal M$, and let $m_f(\lambda)=m(\{|f|\ge \lambda\})$ be the distribution function of $f$. If $$\int_{\mathcal M} \min(|f|,|f|^2)\, dm <\infty\,,
\label{?}$$ then we define the logarithmic determinant of $f$ of genus one $$u_f(z) = \int_{\mathcal M} H(zf(t))\, dm(t)\,,$$ which is subharmonic in $\bf C$, and moreover is represented by a canonical integral of genus one. Applying Corollary 1, we obtain
[**Corollary 2.** ]{}[*If $f$ satisfies condition (\[?\]), and the logarithmic determinant $u_f$ is non-negative in $\bf C$, then*]{} $$m_f(\lambda) {\lesssim}\left\{
\frac{1}{\lambda^2}\, \int_0^\lambda sm_{{\rm Im}\,f}(s) ds
+ \frac{1}{\lambda}\, \int_\lambda^\infty m_{{\rm Im}\,f}(s) ds
\right\}\,, \qquad 0<\lambda<\infty\,.$$ [*In particular, $$||f||_{L^p(m)} {\lesssim}_p ||{\rm Im}\, f||_{L^p(m)}\,, \qquad 1<p<2\,,$$ and*]{} $$m_f(\lambda) {\lesssim}\frac{||{\rm Im}\, f||_{L^1(m)}}{\lambda}\,, \qquad
0<\lambda<\infty\,.$$
Corollary 1 can also be applied to Jensen measures in $\bf C$. A compactly supported finite measure $\sigma$ in $\bf C$ is called a [*Jensen measure*]{} (with respect to the origin) if for an arbitrary subharmonic function $h$ in $\bf C$ $$h(0) \le \int h d\sigma\,.
\label{10}$$ A simple argument shows that (\[10\]) then holds true for subharmonic functions in a domain $G$ such that $0\in G$ and $\rm{supp}(\sigma)\subset G$. For a harmonic function, the equality sign must occur in (\[10\]). Therefore, $$\sigma ({\bf C}) = \int 1 d\sigma =1\,,$$ that is, $\sigma $ is a probability measure, and $$\int \zeta^k d\sigma (\zeta) = 0\,, \qquad k=1,2,...\,.
\label{20}$$
Define the potential $$V_{\sigma} (z) = \int \log|1-z\zeta| d\sigma (\zeta)\,.
\label{30}$$ Then, due to (\[10\]) and (\[20\]), $$0\le V_\sigma(z) \le \log^+(c|z|)\,, \qquad z\in{\bf C}\,,
\label{40}$$ for some $c>0$. The opposite is also true: if, for some $c>0$, a subharmonic function $V$ satisfies (\[40\]), then it is a potential of a of a Jensen measure [@khabib §4].
Due to condition (\[20\]), every potential $V_\sigma$ of a Jensen measure can be represented by a canonical integral of genus one: $$V_\sigma(z) = \int_{\bf C} H(z/\zeta) d\mu(\zeta)\,, \qquad
d\mu(\zeta) = d\sigma(1/\zeta)\,.$$ Thus, Theorem 1 is applicable to the potential $V_\sigma$, and we obtain
[**Corollary 3.** ]{}[*Let $\sigma$ be a Jensen measure in $\bf C$, $\sigma(\lambda) = \sigma(|z|\ge\lambda)$, $\sigma_I(\lambda) = \sigma(|\rm{Im} z|\ge \lambda)$. Then*]{} $$\sigma (\lambda) {\lesssim}\frac{1}{\lambda^2}
\int_0^\lambda s\sigma_I(s) ds
+ \frac{1}{\lambda} \int_{\lambda}^\infty \sigma_I(s) ds\,.$$
The class of Jensen measures is invariant with respect to the holomorphic mappings. More precisely, let $G$ be a domain which contains the origin and $\rm{supp}(\sigma)$, and let $F$ be an analytic function in $G$, $F(0)=0$. Then the push forward $F_* \sigma$ is defined by $$\int \phi \,dF_* \sigma = \int \phi \circ F\, d\sigma\,,$$ where $\phi $ is an arbitrary continuous function in $\bf C$, and $\phi\circ F$ is a composition of $\phi$ and $F$. By the monotone convergence theorem this equation also holds for semicontinuous functions. The measure $F_*\sigma $ automatically has a compact support since $F$ is bounded on $\rm{supp}(\sigma)$. If $h$ is subharmonic in $\bf
C$, then $h \circ F$ is subharmonic in $G$, and $$\int h\, dF_*\sigma = \int h\circ F\, d\sigma \ge h(F(0)) = h(0)\,.$$ Hence, $F_*\sigma$ is a Jensen measure.
[**Corollary 4.** ]{}[*Let $\sigma$ be a Jensen measure in $\bf C$, and let $f=g+ih$ be an analytic function in a domain $G$ which contains the origin and $\rm{supp}(\sigma)$, and $f(0)=0$. Let $$m_{f,\sigma}(\lambda) = \sigma(|f|\ge \lambda)\,,
\qquad m_{h,\sigma}(\lambda) = \sigma(|h|\ge \lambda)\,.$$ Then*]{} $$m_{f,\sigma}(\lambda) {\lesssim}\frac{1}{\lambda^2}
\int_0^\lambda sm_{h,\sigma}(s) ds
+ \frac{1}{\lambda} \int_{\lambda}^\infty m_{h,\sigma}(s) ds\,.
\label{50}$$
Corollary 4 probably holds true under a weaker (and more natural) assumption $g(0)=0$ rather than $f(0)=0$. In that case, using Theorem 2 (see below) one can get an estimate which is slightly weaker than (\[50\]).
In the next result, we shall not assume that $u(z)$ is non-negative in $\bf C$ and instead introduce the quantity $$\delta (r) = {\mathfrak n}(r) + [u^-(r) + u^-(-r)]$$ which we keep under control.
We assume that the integrals $$\int_0 \frac{\delta (t)}{t^2}\, dt\,, \qquad \mbox{and} \qquad
\int^\infty \frac{\delta (t)}{t^3} (1+\log t)\, dt
\label{1.a}$$ are convergent and define $$\delta^* (r) = r \int_0^r \frac{\delta(t)}{t^2}\, dt +
r^2 \int_r^\infty \frac{\delta (t)}{t^3}
\left(1+\log\frac{t}{r} \right)\,dt
\label{1.*}$$ The function $\delta^*(r)$ does not decrease, $r^{-2}\delta^*(r)$ does not increase, and therefore $$\delta^*(r) \le \delta^*(2r) \le 4\delta^* (r)\,, \qquad 0<r<\infty\,.
\label{1.d}$$
[Theorem 2]{} Let $u(z)$ be an arbitrary subharmonic function in $\bf C$ represented by a canonical integral of genus one. Then $$M(r,u) {\lesssim}r^2
\left[ \int_r^\infty \frac{\sqrt{\delta^*(t)}}{t^2}\,dt \right]^2\,.
\label{1.4}$$
Observe, that the RHSs of (\[1.3\]) and (\[1.4\]) do not depend on the bound for the integral (\[1.2\]). Estimate (\[1.4\]) is slightly weaker than (\[1.3\]); however, it suffices for deriving estimates of M. Riesz and Kolmogorov, as well as of the weak $(p,\infty)$-type estimate (see Corollary 6 below).
Fix an arbitrary $\epsilon>0$. Then by the Cauchy inequality $$\begin{aligned}
\left[ \int_r^\infty \frac{\sqrt{\delta^*(t)}}{t^2}dt \right]^2
&=&
\left[ \int_r^\infty
\frac{\sqrt{\left(1 +
\log^{1+\epsilon}\frac{t}{r}\right)\delta^*(t)}}{t^{3/2}}\,
\frac{dt}{t^{1/2}\sqrt{1+\log^{1+\epsilon}\frac{t}{r}}}
\right]^2 \\ \\
&{\lesssim}_{\epsilon}& \int_r^\infty \frac{\delta^*(t)}{t^3}
\left(1+\log^{1+\epsilon}\frac{t}{r} \right)\,dt \\ \\
&{\lesssim}_{\epsilon}&
\frac{1}{r} \int_0^r \frac{\delta(s)}{s^2}\,ds +
\int_r^\infty \frac{\delta(s)}{s^3} \left(1+\log^{3+\epsilon}\frac{s}{r}
\right)\, ds\,.\end{aligned}$$ Thus we get
[Corollary 5]{} For each $\epsilon>0$, $$M(r,u) {\lesssim}_{\epsilon}
r \int_0^r \frac{\delta(t)}{t^2}\,dt + r^2
\int_r^\infty \frac{\delta(t)}{t^3} \left(1+\log^{3+\epsilon}\frac{t}{r}
\right)\, dt\,.
\label{1.4a}$$
We do not know whether the term $\log^{3+\epsilon}$ is really needed on the RHS of (\[1.4a\]). Apparently, our method does not allow us to omit it. Rewriting (\[1.4a\]) in the form $$M(r,u) {\lesssim}_{\epsilon}
\int_0^1 \frac{\delta (rs)}{s^2}\, ds
+ \int_1^\infty \frac{\delta (rs)}{s^3}
\left( 1+ \log^{3+\epsilon}s \right)\, ds\,,$$ we immediately obtain
[Corollary 6]{} The following inequalities hold for canonical integrals of genus one:
M. Riesz-type estimate: $$\int_0^\infty \frac{\mu(r)}{r^{p+1}}\, dr
{\lesssim}_p \int_0^\infty \frac{M(r,u)}{r^{p+1}}\, dr
{\lesssim}_p\int_0^\infty \frac{\delta(r)}{r^{p+1}}\, dr\,,
\qquad 1<p<2\,,
\label{1.5}$$ weak $(p,\infty)$-type estimate: $$\sup_{r\in (0,\infty)} \frac{\mu(r)}{r^p} {\lesssim}_p
\sup_{r\in (0,\infty)} \frac{M(r,u)}{r^p} {\lesssim}_p
\sup_{r\in (0,\infty)} \frac{\delta (r)}{r^p}\,,
\qquad 1<p<2\,,
\label{1.6}$$ and Kolmogorov-type estimate: $$\sup_{r\in (0,\infty)} \frac{\mu(r)}{r} {\lesssim}\sup_{r\in (0,\infty)} \frac{M(r,u)}{r} {\lesssim}\int_0^\infty \frac{\delta(r)}{r^2}\, dr\,.
\label{1.7}$$
Estimates weaker than (\[1.5\]) and (\[1.7\]) were obtained in [@MS1] and [@MS3] under additional restrictions which now appear to be redundant. Estimate (\[1.6\]) is apparently new.
If we assume that $d\mu$ is supported by $\bf R$, that is, $u(z)$ is harmonic in $\bf C_{\pm}$, then our technique gives a better result:
$0<r<\infty$, $$M(r,u) {\lesssim}r \int_0^r \frac{u^-(t)+u^-(-t)}{t^2}\, dt
+ r^2 \int_r^\infty \frac{u^-(t)+u^-(-t)}{t^3}
\, dt\,.
\label{1.8}$$
Note, that one cannot replace $u^-(x)$ by $u^+(x)$ on the RHS of our estimates. For example, the function $$u_p(z) = r^p \cos p \left(\frac{\pi}{2} - |\theta| \right),
\qquad 1<p<2\,,$$ is subharmonic in $\bf C$, harmonic in the upper and lower half-planes $\bf C_\pm$, represented by the canonical integral of genus one of the measure $d\mu (x) = c_p |x|^{p-1}dx$ ($c_p>0$), and non-positive on $\bf R$.
There is a corollary to Theorem 2 which is parallel to Corollary 2. Let $\mathcal M$ be a measurable space endowed with a locally finite non-negative measure $dm$, and let $f:\, {\mathcal M}\to {\bf R}^{n+1}$, $n\ge 1$, be a measurable function such that $$\int_{\mathcal M} \min(||f||, ||f||^2)\, dt <\infty\,,
\label{1.9a}$$ where $||\,.\,||$ stands for the $n+1$-dimensional Euclidean norm. We start to enumerate the coordinates in ${\bf R}^{n+1}$ with $j=0$, and denote by $e_0$ the vector in ${\bf R}^{n+1}$ with the zeroth coordinate equal one, and other coordinates vanishing. Let $f_j(t)$ be the $j$-th coordinate function of $f(t)$, and ${\hat f}(t) =\left\{\sum_{j=1}^n f_j^2(t) \right\}^{1/2}$. We define the logarithmic determinant $$\begin{aligned}
v_f(x) &=& \int_{\mathcal M}
\big[ \log||e_0-xf(t)||+xf_0(t) \big]\, dm(t) \\ \\
&=& \int_{\mathcal M}
\left[\log\sqrt{1-2xf_0(t)+x^2||f||^2} + xf_0(t)\right]\, dm(t)\,,
\qquad x\in {\bf R}\,,\end{aligned}$$ where the integral converges due to assumption (\[1.9a\]). Then, if the function $v_f(x)$ is non-negative on $\bf R$, we may estimate its distribution function $m_f(\lambda)=m(\{||f||\ge \lambda \})$ by the distribution function $m_{\hat f}=m(\{\hat f \ge \lambda
\})$ of $\hat f$.
For this, observe that $$v_f(x) = \int_{\mathcal M} H(xf_C(t))\, dm(t)\,,
\qquad x\in {\bf R}\,,$$ where $f_C$ is a “complex-valued surrogate” of $f$: $f_C=f_0+i\hat{f}$. That is, $v_f$ has a subharmonic continuation from $\bf R$ to $\bf C$ by a canonical integral of genus one $$u_{f_C}(z) = \int_{\mathcal M} H(zf_C (t))\, dm(t)\,,
\qquad z\in {\bf C}\,.$$ Next, observe that $m_f(\lambda) = m(\{f_0^2+{\hat f}^2\ge \lambda^2\})=m_{f_C}(\lambda)$, and $m_{\hat f}(\lambda) = m_{{\rm Im}\, f_C}(\lambda)$ for $0<\lambda<\infty$. Hence, Theorem 2 is applicable in this situation. For simplicity, we restrict ourselves to the case when $v_f$ is non-negative on the real axis.
[**Corollary 7.** ]{}[*Let $f$ satisfy condition (\[1.9a\]), and let the logarithmic determinant $v_f$ be non-negative on the real axis. Then, for $0<\lambda<\infty$ and $\epsilon>0$,*]{} $$m_f(\lambda) {\lesssim}_{\epsilon}
\frac{1}{\lambda^2} \int_0^\lambda s
\left(1+\log^{3+\epsilon}\frac{\lambda}{s}\right)m_{\hat f}(s)\,ds
+\frac{1}{\lambda} \int_\lambda^\infty m_{\hat f}(s)\,ds\,.$$ [*In particular,*]{} $$||f||_{L^p(m)} {\lesssim}_p ||\hat f||_{L^p(m)}\,, \qquad 1<p<2\,,$$ [*and*]{} $$m_f(\lambda) {\lesssim}\frac{||\hat f||_{L^1(m)}}{\lambda}\,.$$
This corollary may be of some interest in view of the results of Aleksandrov and Kargaev [@AK].
Our third result pertains to a more general class of subharmonic functions represented by a [*generalized*]{} canonical integral of genus one. It gives a Kolmogorov-type estimate which can be applied to a wider class of functions than (\[1.7\]):
[Theorem 3]{} Let $d\mu$ be a non-negative locally finite measure on $\bf C$ such that $$\int_{\{|\zeta| \ge 1\}} \frac{d\mu(\zeta)}{|\zeta|^{2}} < \infty\,,$$ and let there exist a finite principal value integral $$\lim_{\varepsilon\to 0}\,
\int_{\{\varepsilon \le |\zeta|\le 1\}}
\frac{d\mu(\zeta)}{\zeta}\,.$$ Let $$u(z)=\lim\limits_{\varepsilon\to 0} \,\int_{|\zeta|>\varepsilon}
H(z/\zeta)\, d\mu(\zeta)\,,$$ then $$\sup_{0<r<\infty} \frac{M(r,u)}{r} {\lesssim}\int_0^\infty \frac{\delta (t)}{t^2}\, dt +
\limsup\limits_{r\to 0}\, \frac{\mu(r)}{r}\,.
\label{kolmogorov}$$
It is easy to see that if the integral (\[1.2\]) converges at the origin, then the upper limit on the RHS of (\[kolmogorov\]) vanishes, and in this case (\[kolmogorov\]) coincides with (\[1.7\]).
In fact, our proof yields a stronger result $$\int_{-\infty}^\infty \frac{u^+(t)}{t^2}\, dt +
\limsup_{r\to\infty} \frac{M(r,u)}{r}
{\lesssim}\int_0^\infty \frac{\delta (t)}{t^2}\, dt +
\limsup\limits_{r\to 0}\, \frac{\mu(r)}{r}\,,
\label{cart}$$ which gives control over the positive harmonic majorants of $u$ in the upper and lower half-planes. Applying a known technique of functions of Cartwright class [@Levin], [@Koosis1], one can extract from (\[cart\]) information about the asymptotic regularity of $u$ and $\mu$ at infinity and near the origin.
Notice, that one can reformulate Theorem 3 in the spirit of Corollaries 2 and 7. We leave this to the reader.
Auxiliary Lemmas
================
We shall need several known facts about harmonic and subharmonic functions.
[Lemma 1]{} Let $v$ be a subharmonic function in the angle $S=\{z:\, 0<\arg z <
\alpha\}$, $0<\alpha<2\pi$, let $$\limsup_{z\to\zeta, \, z\in S} v^+(z) \le \Phi(|\zeta|)\,, \qquad
\zeta\in \partial S\,;
\label{2.a}$$ and let $$\int_0^\alpha v^+(re^{i\theta}) \sin
\left(\frac{\pi}{\alpha}\theta\right) \, d\theta
= o(r^{\pi/\alpha})\,, \qquad r\to\infty\,.
\label{2.b}$$ Then, for $z=re^{i\theta}\in S$, $$v(re^{i\theta}) \sin\left(\frac{\pi}{\alpha}\theta\right) {\lesssim}_{\alpha}
r^{-\pi/\alpha} \int_0^r \Phi (t) t^{\pi /\alpha -1}\, dt
+ r^{\pi/\alpha} \int_r^\infty \frac{\Phi (t)}{t^{\pi /\alpha +1}}\, dt\,.
\label{2.4}$$ If the majorant $\Phi (t)$ does not decrease, then the factor $\sin(\pi \theta/\alpha)$ on the LHS of (\[2.4\]) can be omitted.
[*Proof:*]{} The general case is easily reduced to the special case when $S=\bf C_+$, so that, without loss of generality, we assume that $\alpha=\pi$. First, we show that $v(z)$ is majorized by the Poisson integral of $\Phi (|t|)$, and then we estimate this integral.
Denote by $h_R(z)$ a harmonic function in the semi-disk $\{\mbox{Im}z>0, |z|<R\}$ with boundary values $h_R(t)=\Phi(|t|)$, $-R<t<R$, and $h_R(Re^{i\theta})=v^+(Re^{i\theta})$, $0<\theta<\pi$. Applying the Poisson-Nevanlinna representation in this semi-disk (see [@GO Chapter 1, Theorem 2.3], [@Levin Section 24.3]), we obtain for $z=re^{i\theta}$, $r<R$, $$v(z)\le h_R(z)
= \int_{-R}^R \Phi (|t|) K_1(z,t)\,dt
+ \int_0^\pi v^+(Re^{i\phi})
K_2(z, Re^{i\phi})\, d\phi\,,
\label{2.c}$$ where $$K_1(z,t) = \frac{r\sin\theta}{\pi}
\left\{ \frac{1}{|z-t|^2} - \frac{R^2}{|R^2-zt|^2}\right\}\,,
\label{*1}$$ $$K_2(z,Re^{i\phi})=\frac{1}{2\pi} \frac{4Rr(R^2-r^2)\sin\phi\sin\theta}
{(R^2+r^2-2Rr\cos(\phi-\theta))(R^2+r^2-2Rr\cos(\phi+\theta))}\,,
\label{*2}$$
By condition (\[2.b\]), the second integral on the RHS of (\[2.c\]) tends to $0$ as $R\to\infty$. Therefore, letting $R\to\infty$ in (\[2.c\]), we obtain $$v(z) \le \frac{r\sin\theta}{\pi}
\int_{-\infty}^{\infty} \frac{\Phi (|t|)}{|z-t|^2}\, dt\,.
\label{2.d}$$ Making use of straightforward estimates of the Poisson kernel, we get $$v(z) \le \frac{2}{\pi r\sin\theta} \int_0^{2r} \Phi (t)\, dt
+ \frac{4r}{\pi} \int_{2r}^\infty \frac{\Phi(t)}{t^2}\, dt\,,$$ and estimate (\[2.4\]) follows.
If the majorant $\Phi (t)$ does not decrease, then we modify the previous argument: $$v(z) \le \frac{4}{r} \int_0^{r/2} \Phi (t)\, dt + \Phi (2r)
+ 4r \int_{2r}^\infty \frac{\Phi (t)}{t^2}\, dt
{\lesssim}\frac{1}{r} \int_0^r \Phi (t)\, dt +
+ r \int_r^\infty \frac{\Phi (t)}{t^2}\, dt\,,$$ completing the proof. $\Box$
The next lemma asserts that under certain conditions the Carleman integral formula [@Levin Lecture 24], [@GO Chapter 1] holds without remainder.
[Lemma 2]{} Let $v(z)$ be a subharmonic function on $D_R=\{z\in {\bf \bar C_+}:\,
|z|\le
R\}$ which satisfies conditions $$\int_0^\pi v^+(re^{i\theta}) \sin\theta\, d\theta = o(r)\,,
\qquad r\to 0\,,
\label{i}$$ and $$\int_0 \frac{\delta (t)}{t^2}\, dt <\infty\,.
\label{ii}$$ Then $$\begin{aligned}
\frac{1}{2\pi} \int_{-R}^R v(t)
\left( \frac{1}{t^2} - \frac{1}{R^2} \right)dt
&+&\frac{1}{\pi R} \int_0^\pi v(Re^{i\phi}) \sin\phi\, d\phi
\nonumber \\ \nonumber \\
&=&
\int_{D_R}
\left( \frac{1}{|\zeta|^2}-\frac{1}{R^2}\right)
{\rm Im}\zeta
\, d\mu(\zeta)\,,
\label{2.6}\end{aligned}$$ where the first integral on the LHS is absolutely convergent.
[*Proof:*]{} We start with the Nevanlinna representation $$\begin{aligned}
\int_{-R}^R v^+(t) K_1(z,t)\,dt
&+& \int_0^\pi v^+(Re^{i\phi}) K_2(z,Re^{i\phi})\, d\phi \\ \\
&=& v(z) +
\int_{-R}^R v^-(t) K_1(z,t)\,dt
+ \int_0^\pi v^-(Re^{i\phi}) K_2(z,Re^{i\phi})\, d\phi \\ \\
&\ & \qquad + \int_{D_R} K_3(z,\zeta)\, d\mu(\zeta)\,,\end{aligned}$$ where the kernels $K_1$ and $K_2$ were defined by (\[\*1\]) and (\[\*2\]), and $$K_3(z,\zeta) = \log\left|
\frac{z-\bar\zeta}{z-\zeta}\,\cdot\,
\frac{R^2-z\bar\zeta}{R^2-z\zeta}\right|\,.
\label{*3}$$
We multiply both the left and right hand sides of the Nevanlinna representation by $r^{-1}\sin\theta$, integrate it with respect to $\theta$ from $0$ to $\pi$ and change the integration order in all terms. We shall use the formulas: $$\begin{aligned}
\frac{1}{r}
\int_0^\pi K_1(re^{i\theta},t) \sin\theta\,d\theta = \frac{1}{2}
\left[\min\left(\frac{1}{t^2}-\frac{1}{r^2} \right)-\frac{1}{R^2}
\right]\,,
\label{*4}\end{aligned}$$ $$\begin{aligned}
\frac{1}{r} \int_0^\pi K_2(re^{i\theta}, Re^{i\phi}) \sin\theta\, d\theta
=\frac{1}{R} \sin\phi\,,
\label{*5}\end{aligned}$$ and $$\begin{aligned}
\frac{1}{r} \int_0^\pi K_3(re^{i\theta},\zeta) \sin\theta\, d\theta
= \pi \mbox{Im}\zeta\,\left[
\min\left(\frac{1}{|\zeta|^2},\frac{1}{r^2}\right)-\frac{1}{R^2}
\right] \,.
\label{*6}\end{aligned}$$
Observe that the RHS of relations (\[\*4\])-(\[\*6\]) are non-decreasing functions of $r^{-1}$. Therefore, making the limit transition $r\to 0$, and using the monotone convergence theorem and condition (\[ii\]) of the lemma, we get $$\begin{aligned}
\frac{1}{2} \int_{-R}^R v^+(t)
\left( \frac{1}{t^2} - \frac{1}{R^2} \right)dt
&+& \frac{1}{R} \int_0^\pi v^+(Re^{i\phi}) \sin\phi d\phi
\nonumber \\ \nonumber \\
&=&\frac{1}{2} \int_{-R}^R v^-(t)
\left( \frac{1}{t^2} - \frac{1}{R^2} \right)dt
+ \frac{1}{R} \int_0^\pi v^-(Re^{i\phi}) \sin\phi\ d\phi \\ \\
&\,& \qquad + \pi \int_{D_R}
\left( \frac{1}{|\zeta|^2}-\frac{1}{R^2}\right)
\mbox{Im}\zeta
\, d\mu(\zeta)\,.\end{aligned}$$ The first and third integrals on the RHS are finite due to condition (\[ii\]). This completes the proof. $\Box$
[Remark]{} Condition (\[i\]) holds true for canonical integrals of genus one defined in (\[1.1\]).
Indeed, if $u(z)$ is such an integral, then due to (\[1.3\]) $$\int_0^{2\pi} u^+(re^{i\theta})\, d\theta = o(r), \qquad r\to 0\,.$$ Since $u(0)=0$, this yields $$\begin{aligned}
\int_0^{2\pi} |u(re^{i\theta})|\, d\theta
&=& 2 \int_0^{2\pi} u^+(re^{i\theta})\, d\theta
- \int_0^{2\pi} u(re^{i\theta})\, d\theta \\ \\
&=&
2 \int_0^{2\pi} u^+(re^{i\theta})\, d\theta
= o(r)\,, \qquad r\to 0\,.\end{aligned}$$
The third lemma was proved in [@MS3] (cf. [@Levin Lecture 26]). Its proof uses the Nevanlinna representation for the semi-disk.
[Lemma 3]{} Let $v(z)$ be a function which is harmonic in $\bf C_+$, subharmonic in $\overline{\bf C}_+$, and satisfies conditions (\[i\]) and (\[ii\]) of Lemma 2. Then, for $z=re^{i\theta}\in {\bf C_+}$, $$v(re^{i\theta}) \sin\theta
{\lesssim}\frac{1}{\pi} \int_0^\pi v^-(2re^{i\varphi}) \sin\varphi \, d\varphi
+ \frac{r}{2\pi} \int_{-2r}^{2r} \frac{v^-(t)}{t^2}\, dt\,.
\label{2.7}$$
The next lemma is a version of the Levin integral formula ([@Levin1], [@GO Chapter 1]) without a remainder.
[Lemma 4]{} Let $v$ be a subharmonic function in $\bf C$ such that $v(z)$ and $v(\bar z)$ satisfy conditions (\[i\]) and (\[ii\]) of Lemma 2. Then $$\frac{1}{2\pi}
\int_0^{2\pi} v(Re^{i\theta}|\sin\theta|) \frac{d\theta}{R\sin^2\theta}
= \int_0^R \frac{{\mathfrak n}(t)}{t^2}\, dt\,,
\label{*7}$$ where ${\mathfrak n}(t)$ is the Levin-Tsuji counting function, and the integral on the LHS is absolutely convergent.
[*Proof:*]{} It suffices to prove that $$\frac{1}{2\pi} \int_0^\pi v(Re^{i\theta}\sin\theta)
\frac{d\theta}{R\sin^2\theta}
=\int_{\left|{\rm Im}\frac{1}{\zeta}\right|>\frac{1}{R}}
\left[\left|{\rm Im} \frac{1}{\zeta}\right|-\frac{1}{R}
\right]\,d\mu(\zeta)\,.
\label{??}$$ Then (\[\*7\]) follows by adding to (\[??\]) a similar formula for the integral from $\pi$ to $2\pi$.
First, we prove that the integral on the LHS of relation (\[\*7\]) is absolutely convergent. Making use of notations introduced in (\[\*1\]), (\[2\]) and (\[\*3\]), observe that the Nevanlinna formula implies that $$|v(z)| \le
\int_{-R}^R |v(t)|K_1(z,t)\, dt + \int_0^\pi |v(Re^{i\phi})|
K_2(z,Re^{i\phi}) \,d\phi + \int_{D_R} K_3(z,\zeta)\,
d\mu(\zeta)\,.$$ We set $z=Re^{i\theta}\sin\theta$, multiply the formula by $(R\sin^2\theta)^{-1}$, integrate it with respect to $\phi$ from $0$ to $\pi$, and change the integration order in all terms. We shall use the following relations: $$\int_0^\pi K_1(Re^{i\theta}\sin\theta, t) \frac{d\theta}{R\sin^2\theta}
= \frac{1}{t^2}-\frac{1}{R^2}\,,$$ $$\int_0^\pi K_2(Re^{i\theta}\sin\theta, Re^{i\phi})
\frac{d\theta}{R\sin^2\theta}
= \frac{2}{R}\sin\phi\,,$$ and $$\int_0^\pi K_3(Re^{i\theta}\sin\theta, \zeta)
\frac{d\theta}{R\sin^2\theta}
= 2\pi \left[ \min
\left(\left|\mbox{Im}\frac{1}{\zeta}\right|,\frac{1}{R}\right)
- \frac{\mbox{Im}\zeta}{R^2}
\right]\,.$$ Using these relations, we verify that $$\begin{aligned}
\int_0^\pi |v(Re^{i\theta}\sin\theta)|\, \frac{d\theta}{R\sin^2\theta}
&\le& \int_{-R}^R |v(t)|
\left(\frac{1}{t^2}-\frac{1}{R^2}\right)\, dt \\ \\
&+& \frac{2}{R} \int_0^\pi |v(Re^{i\phi})|\sin\phi\,d\phi
+2\pi \int_{D_R}\left|\mbox{Im}\frac{1}{\zeta}\right|\, d\mu(\zeta)\,.\end{aligned}$$ The first integral on the RHS is finite due to Lemma 2, and the third is finite due to condition (\[ii\]). That is, the integral on the LHS of (\[\*7\]) is absolutely convergent.
Now, we write the Nevanlinna formula in the form $$v(z)=\int_{-R}^R v(t)K_1(z,t)\, dt + \int_0^\pi
v(Re^{i\phi})K_2(z,Re^{i\phi})\,d\phi
- \int_{D_R} K_3(z,\zeta)\, d\mu(\zeta)\,.$$ Again, we set here $z=Re^{i\theta}\sin\theta$, multiply by $(R\sin^2\theta)^{-1}$, integrate with respect to $\theta$ from $0$ to $\pi$ and change the integration order in all terms. We can do this since we already know that the integrals with $|v|$ instead of $v$ are finite. As a result, we obtain the equation $$\begin{aligned}
\int_0^\pi v(Re^{i\theta}\sin\theta)\frac{d\theta}{R\sin^2\theta}
&=& \int_{-R}^R v(t) \left(\frac{1}{t^2}-\frac{1}{R^2} \right)\, dt
+ \frac{2}{R} \int_0^\pi v(Re^{i\phi})\sin\phi\, d\phi
\nonumber \\ \nonumber \\
&\,& \ - 2\pi \int_{D_R} \left[
\min\left(\left|\mbox{Im}\frac{1}{\zeta}\right|,\frac{1}{R}\right) -
\frac{\mbox{Im}\zeta}{R^2}\right]\, d\mu(\zeta)\,.
\nonumber \\
\label{*8}\end{aligned}$$ Taking into account (\[2.6\]), we get $$\begin{aligned}
\int_0^\pi v(Re^{i\phi}\sin\phi)\frac{d\phi}{R\sin^2\phi}
&=& 2\pi \int_{D_R} \left[ \frac{1}{|\zeta|^2} - \frac{1}{R^2} \right]
\mbox{Im}\zeta \, d\mu(\zeta) \\ \\
&\,&\qquad - 2\pi \int_{D_R} \left[
\min\left(\left|\mbox{Im}\frac{1}{\zeta}\right|,\frac{1}{R}\right) -
\frac{\mbox{Im}\zeta}{R^2}\right]\, d\mu(\zeta)
\\ \\
&=&
2\pi \int_{|{\rm Im}\frac{1}{\zeta}|\ge\frac{1}{R}} \left[
\left|\mbox{Im}\frac{1}{\zeta}\right|
- \frac{1}{R} \right]\, d\mu(\zeta)\,. \end{aligned}$$ Then (\[??\]) follows and the proof is complete. $\Box$
In other words, in the assumptions of Lemma 4, the first fundamental theorem for Tsuji characteristics holds without a remainder term: $${\mathfrak T}(r,u) = \mathfrak m(r,u) + \int_0^r\frac{{\mathfrak n}(t)}{t^2}\,dt\,,
\qquad 0<r<\infty\,,
\label{tsuji}$$ where $${\mathfrak T}(r,u) = \frac{1}{2\pi} \int_0^{2\pi} u^+(re^{i\theta}|\sin\theta)|)
\frac{d\theta}{\sin^2\theta}\,,$$ and $$\mathfrak m (r,u) =
\frac{1}{2\pi} \int_0^{2\pi} u^-(re^{i\theta}|\sin\theta)|)
\frac{d\theta}{\sin^2\theta}\,.$$
The last lemma was proved in a slightly different setting in [@LO] (see also [@GO Lemma 5.2, Chapter 6]):
[Lemma 5]{} Let $u(z)$ be a subharmonic function in $\bf C$, and let $$T(r,u) = \frac{1}{2\pi} \int_0^{2\pi} u^+(re^{i\theta})\,d\theta$$ be its Nevanlinna characteristic function. Then, for $0<R<\infty$, $$\int_R^\infty \frac{T(r,u)}{r^3}\, dr \le \int_R^\infty
\frac{{\mathfrak T}(r,u)}{r^2}\, dr\,.
\label{tsuji1}$$
Proof of Theorem 1
==================
Using monotonicity of $T(r,u)$, Lemma 5, and then Lemma 4, we obtain $$\begin{aligned}
\frac{T(R,u)}{R^2} &\le& 2 \int_R^\infty \frac{T(r,u)}{r^3}\, dr \\ \\
&\stackrel{(\ref{tsuji1})}\le& 2
\int_R^\infty \frac{{\mathfrak T}(r,u)}{r^2}\, dr \\ \\
&\stackrel{(\ref{tsuji})}=& 2 \int_R^\infty \frac{dr}{r^2}
\left( \int_0^r
\frac{{\mathfrak n}(t)}{t^2}\, dt + \mathfrak m(r,u) \right) \\ \\
&=& \frac{2}{R} \int_0^R \frac{{\mathfrak n}(t)}{t^2}\, dt + 2\int_R^\infty
\frac{{\mathfrak n}(t)}{t^3}\, dt
+ 2\int_R^\infty \frac{\mathfrak m(t,u)}{t^2}\,dt\,.\end{aligned}$$ Then the inequality $M(r,u)\le 3T(2r,u)$ completes the proof. $\Box$
Proof of Theorem 2
==================
We split the proof into several parts. Without loss of generality, we assume convergence of the integrals $$\int_0 \frac{\delta(t)}{t^2}\, dt
\qquad \hbox{and} \qquad
\int^\infty \frac{\delta(t)}{t^3} \log t\, dt\,.$$
We define a measure $\mu_1$, ${\rm supp}(\mu_1)\subset {\bf{\bar C}_-}$, by reflecting at the real axis the part of the measure $\mu$ which lies in the upper half-plane. Formally, $$\mu_1(E) = \mu(E\cap {\bf\bar C}_-) + \mu(E^-\cap \bf{\bar C}_-)\,,$$ where $E\subset \bf C$ is a borelian set, and $E^-=\{z:\, \bar z\in E\}$. Then the measure $\mu_1$ also satisfies condition (\[1.1\]) and we denote by $u_1(z)$ its canonical integral of genus one. Observe that $u_1(t)=u(t)$, so that $\delta(t,u_1) = \delta(t,u)$, $t\in {\bf R}$.
Estimate of $u_1^-(iy)$, $y>0$.
-------------------------------
We have $$\begin{aligned}
H(iy/\zeta) &=&
\log \left| 1+ y\, \hbox{Im}\frac{1}{\zeta}
- iy\, \hbox{Re}\frac{1}{\zeta} \right| -
y\, \hbox{Im}\frac{1}{\zeta} \\ \\
&\ge& \log\left| 1 + y\, \hbox{Im}\frac{1}{\zeta}\right|
- y\, \hbox{Im}\frac{1}{\zeta}\,.\end{aligned}$$ Since the RHS is non-positive for $y>0$ and $\zeta\in {\bf {\bar C}_-}$, $$\begin{aligned}
u_1^-(iy) &\le& -\int_{\bf{\bar C}_-}
\left[\log\left| 1 + y\, \hbox{Im}\frac{1}{\zeta}\right|
- y\, \hbox{Im}\frac{1}{\zeta}\right]\,d\mu(\zeta) \nonumber \\ \nonumber \\
&=&-\int_0^\infty \left[ \log\left(
1+\frac{y}{t} \right) - \frac{y}{t}\right] d{\mathfrak n}(t)
\nonumber \\ \nonumber \\
&=& y^2 \int_0^\infty \frac{{\mathfrak n}(t)}{t^2(t+y)}\, dt
\nonumber \\ \nonumber \\
&\le& y\int_0^y \frac{{\mathfrak n}(t)}{t^2}\, dt
+ y^2 \int_y^\infty \frac{{\mathfrak n}(t)}{t^3}\, dt\,.
\label{3.1}\end{aligned}$$
Estimates of $u_1^+(re^{i\theta })$, $0<\theta<\pi$.
----------------------------------------------------
Using harmonicity of the function $u_1$ in the upper half-plane, we transform the lower bound for $u_1$ into the upper bound. We shall show that $$u_1^+(re^{i\theta}) \sin\theta {\lesssim}\delta^* (r)\,
\qquad 0<r<\infty\,, \quad 0<\theta<\pi\,,
\label{3.4}$$ where $\delta^*(r)$ is defined by (\[1.\*\]).
Consider the function $-u_1(z)$ and apply Lemma 1 to the angles $\{0<\arg z < \pi/2\}$ and $\{\pi/2<\arg z <\pi\}$ with $$\Phi (r) = [u_1^-(r) + u_1^-(-r)] + r\int_0^r \frac{{\mathfrak n}(t)}{t^2}\, dt
+ r^2 \int_r^\infty \frac{{\mathfrak n}(t)}{t^3}\, dt\,.$$ Condition (\[2.a\]) holds due to estimate (\[3.1\]), and condition (\[2.b\]) holds due to estimate (\[1.M\]) combined with Jensen’s inequality: $$\int_0^\pi u_1^-(re^{i\theta})\,d\theta
\le \int_0^{2\pi} u_1^+(re^{i\theta})\, d\theta \le
M(r,u_1) = o(r^2)\,, \qquad r\to\infty\,.$$ Therefore, $$\begin{aligned}
-u_1(re^{i\theta}) |\sin 2\theta| &{\lesssim}&
\frac{1}{r^2}\int_0^r \Phi (t) t\, dt
+ r^2 \int_r^\infty \frac{\Phi (t)}{t^3}\, dt
\nonumber \\ \nonumber \\
&{\lesssim}&
\frac{1}{r^2}\int_0^r [u_1^-(t)+u_1^-(-t)] t\, dt
+ r^2 \int_r^\infty \frac{u_1^-(t)+u_1^-(-t)}{t^3}\, dt
\nonumber \\ \nonumber \\
&\ & \qquad +
r \int_0^r \frac{{\mathfrak n}(s)}{s^2}\, ds +r^2 \int_r^\infty \frac{{\mathfrak n}(s)}{s^3}
\left(1+\log\frac{s}{r} \right)\, ds
\nonumber \\ \nonumber \\
&{\lesssim}& \delta^*(r)\,.
\label{4.3.1}\end{aligned}$$
Observe that the factor $|\sin 2\theta|$ on the LHS of (\[4.3.1\]) can be replaced by $\sin\theta$. This follows from inspection of the proof of Lemma 1 (since on the imaginary axis the function $-u(iy)$ has an increasing majorant). Alternatively, one may again apply Lemma 1 to a small angle around the imaginary axis, say in $\{|\theta-\pi/2|<\pi/8\}$. That is, we have $$-u_1(re^{i\theta})\sin\theta
{\lesssim}\delta^*(r)\,.
\label{4.3.2}$$
Using Lemma 3 we obtain $$u_1^+(re^{i\theta}) \sin\theta {\lesssim}\int_0^{\pi} u_1^-(2re^{i\phi}) \sin\phi \,d\phi +
r\int_0^{2r} \frac{u_1^-(t)+u_1^-(-t)}{t^2}\, dt
{\lesssim}\delta^*(r)\,,$$ proving estimate (\[3.4\]).
Estimate of $u^+(re^{i\theta})$, $\theta\ne 0,\pi$.
---------------------------------------------------
Here we prove that, for an arbitrary $\eta >0$, $$\begin{aligned}
u^+(re^{i\theta}) {\lesssim}\frac{\delta^*(r)}{\eta |\sin\theta|} +
\frac{\eta r^2}{\sin^2\theta} \int_r^\infty
\frac{M(t,u)}{t^3}\, dt\,.
\label{5.a}\end{aligned}$$
For this, we shall need several upper bounds for the difference $$D=D(z,\zeta)=H(z/\zeta)-H(z/{\bar \zeta})
=\log\left|\frac{1-z/\zeta}{1-z/\bar\zeta} \right| +
{\rm Re} \left[z\left( \frac{1}{\zeta}
- \frac{1}{\bar \zeta}\right) \right]\,,$$ when $z,\zeta \in {\bf \bar C}_+$.
First, $$D = \log\left|\frac{z-\zeta}{z-\bar\zeta} \right|
+ 2{\rm Im}z\, \left|{\rm Im}\frac{1}{\zeta} \right|
\le 2|z|\, \left|{\rm Im}\frac{1}{\zeta} \right|
\,.
\label{**1}$$ We shall use this estimate when $|z| \left|{\rm Im}\frac{1}{\zeta} \right| \ge 1$.
Next, let $t=|z|/|\zeta|$, $\theta=\arg(z)$, $\phi=\arg(\zeta)$. Then $$\begin{aligned}
D &=& \frac{1}{2} \log
\left[1-\frac{4t\sin\theta\sin\phi}{|1-te^{i(\theta+\phi)}|^2}\right]
+ 2 t\sin\theta\sin\phi \nonumber \\ \nonumber \\
&\le& -\frac{2t\sin\theta\sin\phi}{|1-te^{i(\theta+\phi)}|^2}
+ 2t \sin\theta\sin\phi \nonumber \\ \nonumber \\
&=& 2t\sin\theta\sin\phi\,
\frac{-2t\cos(\theta+\phi)+t^2}
{|1-te^{i(\theta+\phi)}|^2} \nonumber \\ \nonumber \\
&{\lesssim}&
t \sin\theta\sin\phi\, \frac{\max(t,t^2)}{|1-te^{i(\theta+\phi)}|^2}\,.
\label{**2}\end{aligned}$$
If $t\le 1/2$, then $$|1-te^{i(\theta+\phi)}|^2 \gg 1\,,$$ and we obtain $$D {\lesssim}t^2 \sin\theta\sin\phi
{\lesssim}\eta t^2 + \eta^{-1} t^2 \sin^2\phi
= \eta\, \frac{|z|^2}{|\zeta|^2}
+ \frac{|z|^2}{\eta} \left|{\rm Im}\frac{1}{\zeta} \right|^2\,,
\label{**3}$$ with an arbitrary $\eta>0$.
If $t\ge 1/2$, then $$|1-te^{i(\theta+\phi)}|^2 \gg t^2 \sin^2\theta\,,$$ so that (\[\*\*2\]) gives us $$D {\lesssim}t \frac{\sin\phi}{\sin\theta}
{\lesssim}\frac{\eta}{\sin^2\theta} + \frac{t^2}{\eta} \sin^2\phi
=\frac{\eta}{\sin^2\theta} +
\frac{|z|^2}{\eta} \left|{\rm Im}\frac{1}{\zeta} \right|^2\,,
\label{**4}$$ again, with an arbitrary positive $\eta$. We shall use the bounds (\[\*\*3\]) and (\[\*\*4\]) when $|z| \left|{\rm Im}\frac{1}{\zeta} \right| \le 1$.
Now, for $z\in{\bf C}_+$, $r=|z|$, we have $$\begin{aligned}
u(z)-u_1(z) &=& \int_{\bf C_+} D(z,\zeta) \, d\mu(\zeta) \\ \\
&\le&
\left(\int_{\left|{\rm Im}\frac{1}{\zeta} \right|\ge \frac{1}{r}}
+ \int_{\left|{\rm Im}\frac{1}{\zeta} \right|\le \frac{1}{r},
\,|\zeta|\ge 2r} +
\int_{\left|{\rm Im}\frac{1}{\zeta} \right|\le \frac{1}{r},
\, |\zeta|\le 2r}
\right) D(z,\zeta)\, d\mu(\zeta) \\ \\
&{\lesssim}&
r \int_0^{r} \frac{d{\mathfrak n}(t)}{t}
+ \frac{r^2}{\eta} \int_{r}^\infty \frac{d{\mathfrak n}(t)}{t^2}
+ \frac{\eta}{\sin^2\theta} \int_0^{r} d\mu (t)
+ \eta r^2 \int_{r}^\infty \frac{d\mu (t)}{t^2} \\ \\
&{\lesssim}& \frac{\delta(r)}{\eta} +
\frac{\eta r^2}{\sin^2\theta} \int_{r}^\infty
\frac{M(t,u)}{t^3}\, dt\,.\end{aligned}$$
Then, using estimate (\[3.4\]) for $u_1^+(z)$ in the upper half-plane, we obtain estimate (\[5.a\]) for $0<\theta<\pi$. The same argument applies for the lower half-plane, and the proof of (\[5.a\]) is complete.
Integral inequality for $M(r,u)$.
---------------------------------
Here we prove the integral inequality $$M(r,u) {\lesssim}\sqrt{\delta^*(r)\, r^2 \int_r^\infty \frac{M(t,u)}{t^3}dt}\,.
\label{3.12}$$
First, we improve estimate (\[5.a\]) near the real axis. Consider the function $u(z)$ in the angles $\{|\arg z|\le \pi/6\}$ and $\{|\arg z -\pi|\le \pi/6\}$. On the boundary of these angles, $$u(re^{i\theta}) {\lesssim}\Phi (r)\,,
\qquad \theta = \pm\frac{\pi}{6}, \quad \pi \pm \frac{\pi}{6}\,,$$ where $$\Phi (r)
= \eta^{-1} \delta^*(r)
+ \eta r^2 \int_r^\infty \frac{M(t,u)}{t^3}\, dt\,.$$ Applying Lemma 1 to $u(z)$ in these angles, we obtain for $|\theta|\le \pi/8$ and $|\pi -\theta|\le \pi/8$, $$u(re^{i\theta}) {\lesssim}r^{-3} \int_0^r \Phi (t) t^2 dt
+ r^3 \int_r^\infty \frac{\Phi(t)}{t^4}dt
{\lesssim}\Phi (r)\,.$$ The second inequality follows since the function $\Phi (r)$ does not decrease, and the function $r^{-2}\Phi (r)$ does not increase.
Thus, for $0<r<\infty$, $$M(r,u) {\lesssim}\Phi (r)= \eta^{-1} \delta^*(r)
+ \eta r^2 \int_r^\infty \frac{M(t,u)}{t^3}\, dt\,.$$ Choosing $$\eta =
\sqrt{\delta^*(r)}\, : \,
\sqrt{r^2 \int_r^\infty \frac{M(t,u)}{t^3}dt}\,,$$ we obtain inequality (\[3.12\]).
Solution of the integral inequality (\[3.12\]).
-----------------------------------------------
We set $$M_1(r) = \int_r^\infty \frac{M(t,u)}{t^3}\,dt\,.$$ Then $$M(r,u) = -\,r^3 M_1'(r)\,,$$ and inequality (\[3.12\]) takes the form $$-\,M_1'(r)r^2 {\lesssim}\sqrt{\delta^*(r)\,M_1(r)}$$ or $$-\frac{d\sqrt {M_1(r)}}{dr}
{\lesssim}\frac{\sqrt{\delta^*(r)}}{r^2}
\,.$$ Integrating this inequality from $\infty$ to $r$, we obtain $$M_1(r) {\lesssim}\left[
\int_r^\infty \frac{\sqrt{\delta^*(t)}}{t^2}\,dt
\right]^2\,.$$ On the other hand, since $M(r,u)$ does not decrease, $$M_1(r) \ge M(r,u) \int_r^\infty \frac{dt}{t^3}
= \frac{M(r,u)}{2r^2}\,.$$ Therefore, $$M(r,u) {\lesssim}r^2 M_1(r) {\lesssim}r^2 \left[
\int_r^\infty \frac{\sqrt{\delta^*(t)}}{t^2}\,dt
\right]^2\,,$$ completing the proof of Theorem 2. $\Box$
Proof of Theorem 3
==================
We divide the proof into 4 parts. Set $$B:= \limsup_{r\to 0} \frac{\mu(r)}{r}\,,$$ $$C:= \int_0^\infty \frac{\delta (t)}{t^2}\, dt\,.$$ Without loss of generality, we assume that both values $B$ and $C$ are finite.
First, we shall prove the theorem under the additional assumption $${\rm supp}(\mu) \subset \bf{\bar C}_-\,,
\label{**}$$ and till Section 6.4 we assume that the function $u(z)$ is harmonic in ${\bf C}_+$.
The function $u(z)$ has nonnegative harmonic majorants in $\bf C_\pm$.
----------------------------------------------------------------------
Consider the function $$U(z):=-u(z)-\frac{y}{\pi}\int_{-\infty}^\infty
\frac{u^-(t)dt}{(t-x)^2+y^2}\,.$$ This function is harmonic in ${\bf C}_+$ and $U(x)\le 0,\; x\in{\bf R}$. Moreover, for $y>0$, $$\begin{aligned}
U(iy)\le -u(iy)&=&-\lim\limits_{\varepsilon\to
0}\int_{|\zeta|\ge\varepsilon, \,
\zeta\in\bf{\bar C}_-}\left[\log
\left|1-\frac{iy}{\zeta}\right|
+ \mbox{Re}\,\frac{iy}{\zeta}\right]d\mu(\zeta)
\\ \\
&\le& -\lim\limits_{\varepsilon\to 0}\int_{|\zeta|\ge\varepsilon,\,
\zeta\in
\bf{\bar C}_-}\mbox{Re}\,\frac{iy}{\zeta}\,d\mu(\zeta)
\\ \\
&=&y\int_{\bf{\bar C}_-}\mbox{Im}\,
\frac{1}{\zeta}\,d\mu(\zeta)\le Cy\,.\end{aligned}$$
By the Poisson-Nevanlinna representation of harmonic functions in the semi-disk $D_R$ (cf. Section 3), we have $$\begin{aligned}
U(z)&\le&
\frac{1}{2\pi}\int_0^{\pi}U(Re^{i\phi}) K_2(z,Re^{i\phi})\, d\phi
\nonumber \\ \nonumber \\
&\le&\frac{2Rr(R+r)}{\pi(R-r)^3}\int_0^{\pi}u^-(Re^{i\phi})\,d\phi
\nonumber \\ \nonumber \\
&\le&
\frac{4Rr(R+r)}{(R-r)^3}\,T(R,u)\,,
\label{1}\end{aligned}$$ where $T(R,u)$ is the Nevanlinna characteristic of $u$.
Note that, for any $\delta>0$, the function $u$ can be represented in the form $$\begin{aligned}
u(z)&=&\int_{|\zeta|>\delta}\left[\log\left|1-\frac{z}{\zeta}\right|+\mbox{Re}
\frac{z}{\zeta}\right]\,d\mu(\zeta) \nonumber \\
\nonumber \\
&\ & \qquad + \int_{|\zeta|\le \delta}\log\left|1-
\frac{z}{\zeta}\right|\,d\mu(\zeta)+\mbox{Re}\left(z
\int_{|\zeta|<\delta}\frac{d\mu(\zeta)}{\zeta}\right) \nonumber \\
\nonumber \\
&=:&u^{\delta}(z)+v^{\delta}(z)+\mbox{Re}\left(z
\int_{|\zeta|<\delta}\frac{d\mu(\zeta)}{\zeta}\right).
\label{2}\end{aligned}$$ The well-known Borel estimates $$\max_{|z|\le r}u^{\delta}(z)=o(|z|^2), \qquad \max_{|z|\le r}v^{\delta}(z)=o(|z|),\quad
z\to\infty,$$ imply that $T(R,u)=o(R^2),\; R\to\infty$. Therefore, by setting $R=2r$ in (\[1\]), we get $$U^+(z)=o(|z|^2),\qquad z\to\infty,\; \qquad \mbox{Im}z>0.$$
Applying the Phragmén-Lindelöf principle in the angles $\{ 0<\arg z< \pi/2\}$ and $\{ \pi/2<\arg z< \pi \}$, we conclude that $$U(z)\le Cy,\qquad z=x+iy\in {\bf C_+}\,;
\label{3}$$ i.e. $-U(z)+Cy$ is a nonnegative harmonic function in ${\bf C_+}$. Since $u(z) \le -U(z)+Cy$, the function $u(z)$ also has a nonnegative harmonic majorant in ${\bf
C_+}$.
For $z\in{\bf C}_-$, we write $$\begin{aligned}
u(z)-u(\bar{z}) &=&
\lim\limits_{\varepsilon\to 0}\,
\int_{|\zeta|\ge\varepsilon,\zeta\in{\bf C}_-}
\log\left|\frac{1-z/\zeta}{1-z/\bar{\zeta}}\right|+\mbox{Re}\left[z\left(
\frac{1}{\zeta}-\frac{1}{\bar{\zeta}}\right)\right]\,d\mu(\zeta)
\nonumber \\ \nonumber \\
&\le&
\int_{{\bf C}_-}\mbox{Re}\left[z2i\mbox{Im}\frac{1}{\zeta}\right]\,
d\mu(\zeta) \le 2C|y|\,.
\label{4}\end{aligned}$$ Because $u(\bar{z})$ has a nonnegative harmonic majorant in ${\bf C}_-$, we get the desired conclusion.
Estimate of $u(z)$ near the origin.
-----------------------------------
Set $$I(r):=\frac{1}{r}\int_0^{\pi}u(re^{i\theta})\sin\theta\,d\theta.$$ Let us prove that $$\limsup\limits_{r\to 0}I(r){\lesssim}B\,.
\label{5}$$
For any given $\varepsilon>0$, choose a positive $\delta<\varepsilon$ such that $$\mu(r)<(B+\varepsilon)r,\quad\mbox{for}\quad 0<r<\delta.$$ Let us represent $u$ by the formula (\[2\]) with this $\delta$. Since $$u^{\delta}(z)=O(|z|^2),\quad z\to 0,$$ we have $$\limsup\limits_{r\to 0}I(r)\le\limsup\limits_{r\to 0}I^{\delta}(r)+
\left|\int_{|\zeta|<\delta}\frac{d\mu(\zeta)}{\zeta}\right|\,,
\label{6}$$ where $$I^{\delta}(r)=\frac{1}{r}\int_0^{\pi}v^{\delta}(re^{i\theta})\sin\theta\,
d\theta.$$ It suffices to show that $$I^{\delta}(r){\lesssim}B+\varepsilon+
\int_{|\zeta|<\delta}\mbox{Im}\frac{1}{\zeta}\,d\mu(\zeta)\,, \qquad
0<r<\delta.
\label{7}$$ Indeed, if (\[7\]) is valid, then substituting it into (\[6\]), we get $$\limsup\limits_{r\to 0}I(r) {\lesssim}B+\varepsilon+
\int_{|\zeta|<\delta}\mbox{Im}\frac{1}{\zeta}\,d\mu(\zeta)
+\left|\int_{|\zeta|<\delta}\frac{d\mu(\zeta)}{\zeta}\right|.$$ Taking the limit as $\varepsilon\to 0$ (then $\delta\to 0$ as well), we obtain (\[5\]).
To prove (\[7\]), we set for $|z|=r,\; 0<r<\delta$: $$v^{\delta}(z)=\int_{r<|\zeta|<\delta}+\int_{|\zeta|<r}=:v_1^{\delta}(z)
+v_2^{\delta}(z)\,,$$ and $$I_j^{\delta}(r):=\frac{1}{r}\int_0^{\pi}v_j^{\delta}(re^{i\theta})
\sin\theta\,d\theta,\qquad j=1,2\,.$$ Note that $$\begin{aligned}
\int_0^{\pi}\left|\log\left|1-\frac{re^{i\theta}}{\zeta}\right|\right|
\sin\theta\,d\theta
&\le&
2\int_0^{2\pi}\log^+\left|1-\frac{re^{i\theta}}{\zeta}\right|\,d\theta
-\int_0^{2\pi}\log\left|1-\frac{re^{i\theta}}{\zeta}\right|\,d\theta \\ \\
&\le&
4\pi\log\left(1+\frac{\delta}{|\zeta|}\right)+2\pi\log^+\frac{r}{|\zeta|}\,.\end{aligned}$$ This estimate will allow us to change the integration order in the double integrals that arise when estimating $I_j^{\delta}(r),\; j=1,2,$ below.
Write $$I_1^{\delta}(r)=\frac{1}{r}\int_{r<|\zeta|<\delta}d\mu(\zeta)\int_0^{\pi}\log
\left|1-\frac{re^{i\theta}}{\zeta}\right|\sin\theta\,d\theta\,.$$ For $r<|\zeta|$, we have $$\begin{aligned}
\int_0^{\pi}\log\left|1-\frac{re^{i\theta}}{\zeta}\right|\sin\theta\,d\theta
&=&
-\mbox{Re}\sum_{k=1}^{\infty}\frac{r^k}{k\zeta^k}\int_0^{\pi}
e^{ik\theta}\sin\theta\,d\theta \\ \\
&=&
\frac{\pi r}{2}\mbox{Im}\frac{1}{\zeta}+\mbox{Re}\sum_{m=1}^{\infty}
\frac{r^{2m}}{m(4m^2-1)\zeta^{2m}}\,.\end{aligned}$$ Hence $$I_1^{\delta}(r)\le\frac{\pi}{2}\int_{|\zeta|<\delta}\mbox{Im}\frac{1}{\zeta}\,
d\mu(\zeta)+
\sum_{m=1}^{\infty}\frac{r^{2m-1}}{m(4m^2-1)}\int_{r<|\zeta|<\delta}
\frac{d\mu(\zeta)}{|\zeta|^{2m}}\,.$$ Since $$\begin{aligned}
\int_{r<|\zeta|<\delta}\frac{d\mu(\zeta)}{|\zeta|^{2m}}&=&
\int_r^{\delta}\frac{d\mu(t)}{t^{2m}} \\ \\
&\le&
\frac{\mu(\delta)}{\delta^{2m}}
+2m\int_r^{\delta}\frac{\mu(t)}{t^{2m+1}}\,dt \\ \\
&\le& (B+\varepsilon)\delta^{-2m+1} + 2(B+\varepsilon)r^{-2m+1}
< 3(B+\varepsilon)r^{-2m+1}\,,\end{aligned}$$ we get $$I_1^{\delta}(r)\le\frac{\pi}{2}\int_{|\zeta|<\delta}\mbox{Im}
\frac{1}{\zeta}\,d\mu(\zeta) + 3(B+\varepsilon)
\sum_{m=1}^{\infty}\frac{1}{m(4m^2-1)}\,.
\label{8}$$
Further, $$I_2^{\delta}(r)=\frac{1}{r}\int_{|\zeta|<r}d\mu(\zeta)\int_0^{\pi}\log\left|
1-\frac{re^{i\theta}}{\zeta}\right|\sin\theta\,d\theta.$$ For $|\zeta|\le r$, we have $$\begin{aligned}
\int_0^{\pi}\log\left|1-\frac{re^{i\theta}}{\zeta}\right|
\sin\theta\,d\theta
&=&
2\log\frac{r}{|\zeta|}+\int_0^{\pi}
\log\left|1-\frac{\bar{\zeta}e^{i\theta}}{r}\right|\sin\theta\,d\theta
\\ \\
&=&
2\log\frac{r}{|\zeta|}-\mbox{Re}\sum_{k=1}^{\infty}\frac{\bar{\zeta}^k}
{kr^k}\int_0^{\pi}e^{ik\theta}\sin\theta\,d\theta \\ \\
&=& 2\log\frac{r}{|\zeta|}-\frac{\pi}{2r}\mbox{Im}\zeta+\mbox{Re}
\sum_{m=1}^{\infty}\frac{\bar{\zeta}^{2m}}{m(4m^2-1)r^{2m}}\,.\end{aligned}$$ Whence $$\begin{aligned}
I_2^{\delta}(r) &\le&
\frac{1}{r}\int_{|\zeta|<r}\left[2\log\frac{r}{|\zeta|}
+\frac{\pi}{2}+
\sum_{m=1}^{\infty}\frac{1}{m(4m^2-1)}\right]\,d\mu(\zeta)
\nonumber \\ \nonumber \\
&{\lesssim}&
\frac{1}{r}\int_0^r \log\frac{r}{t}d\mu(t)+ \frac{\mu(r)}{r}
\nonumber \\ \nonumber \\
&{\lesssim}& B+\varepsilon
\label{9}\end{aligned}$$ Since $I^{\delta}=I_1^{\delta}+I_2^{\delta}$, the desired inequality (\[7\]) follows from (\[8\]) and (\[9\]).
Estimate of $u^+(z)$ on the real and imaginary axes.
----------------------------------------------------
Let us prove that $$\int_{-\infty}^\infty
\frac{u^+(t)}{t^2}dt+\limsup\limits_{y\to+\infty}\frac
{u^+(iy)}{y}{\lesssim}B+C\,.
\label{10'}$$ Since $u$ has a nonnegative harmonic majorant in ${\bf C}_+$, we have $$\int_{-\infty}^{\infty}\frac{|u(t)|dt}{1+t^2}<\infty\,,$$ and $u$ admits the Poisson representation $$u(re^{i\varphi})=\frac{r\sin\varphi}{\pi}
\int_{-\infty}^{\infty}\frac{u(t)\,dt}
{r^2+t^2-2rt\cos\varphi}+kr\sin\varphi,
\label{11}$$ where $$k=\limsup\limits_{y\to+\infty}\frac{u(iy)}{y}\neq\infty.$$ Note that inequality (\[3\]) implies $$u(iy)\ge-\frac{y}{\pi}\int_{-\infty}^{\infty}\frac{u^-(t)dt}{t^2+y^2}-Cy=
o(y)-Cy, \qquad y\to\infty\,;$$ i.e. $k\ge -C$.
Multiplying (\[11\]) by $\sin\varphi$, integrating against $\varphi$ from $0$ to $\pi$, and taking into account that $$\int_0^{\pi}\frac{\sin^2\varphi\,d\varphi}{r^2+t^2-2rt\cos\varphi}=
\frac{\pi}{2}\min\left(\frac{1}{r^2},\frac{1}{t^2}\right),$$ we get $$\int_0^{\pi}u(re^{i\varphi})\sin\varphi\,d\varphi=
\frac{r}{2}\int_{-\infty}^{\infty}u(t)\min\left(\frac{1}{r^2},\frac{1}{t^2}
\right)\,dt+\frac{k\pi r}{2}.$$ Hence $$\int_{-\infty}^{\infty}u^+(t)\min\left(\frac{1}{r^2},\frac{1}{t^2}\right)
\,dt+k^+\pi
=\int_{-\infty}^{\infty}u^-(t)\min\left(\frac{1}{r^2},
\frac{1}{t^2}\right)\,dt+k^-\pi+2I(r).$$ Letting $r\to 0$, we obtain by the monotone convergence theorem $$\int_{-\infty}^{\infty}\frac{u^+(t)}{t^2}\,dt+k^+\pi=
\int_{-\infty}^{\infty}\frac{u^-(t)}{t^2}\,dt+k^-\pi+2\lim\limits_{r\to 0}
I(r)$$ (it turns out that the last limit exists). Taking into account that $k^-\le C$ and using (\[5\]), we get (\[10’\]).
Concluding steps.
-----------------
From (\[3\]) and (\[10’\]) we obtain $$\limsup\limits_{y\to-\infty}\frac{u^+(iy)}{|y|}\le\limsup\limits_
{y\to+\infty}\frac{u^+(iy)+2Cy}{y}{\lesssim}B+C\,.$$ Since $u$ has nonnegative harmonic majorants in both upper and lower half-planes, the following inequality holds in the whole plane: $$u(z){\lesssim}\frac{|y|}{\pi}\int_{-\infty}^\infty
\frac{u^+(t)\,dt}{(t-x)^2+y^2}
+ (B+C)|y|, \qquad z=x+iy.
\label{12}$$ The assertion of Theorem 3 can be obtained from this inequality and (\[10’\]) by applying a known argument (cf. [@MS1], [@MS3]). First, one applies (\[12\]) and (\[10’\]) to get the upper bound for $u(z)$ in the angles $\{|\arg z \pm \pi/2|\le \pi/4\}$, and then, using the Phragmén-Lindelöf principle, one obtains the upper bound for $u(z)$ in the complementary angles. This gives $u(z){\lesssim}(B+C)|z|$, and completes the proof of estimate (\[kolmogorov\]) for the special case (\[\*\*\]).
Now, let $\mu$ be an arbitrary measure in $\bf C$ satisfying conditions of Theorem 3 and having finite value $C$. As in Section 5, we define the measure $\mu_1$, ${\rm supp}\mu_1\subset {\bf \bar C}_-$, by reflecting with respect to the real axis the part of $\mu$ which charges $\bf C_+$. Since $$\int_{\bf{C}_-} {\rm Im} \frac{1}{\zeta}\, d\mu_1 (\zeta) \le C <\infty\,,$$ the measure $\mu_1$ also satisfies the conditions of Theorem 3, and we can define the corresponding generalized canonical integral $u_1(z)$ of this measure. Then a straightforward estimate (cf. \[4\]) shows that for $z\in \bf{\bar C}_+$ $$u(z) \le u_1(z) + 2|y|\int_{\bf C}
\left| {\rm Im} \frac{1}{\zeta} \right|\, d\mu(\zeta)
{\lesssim}(B+C)|z|\,.$$ The same estimate holds in the lower half-plane, and the general case of Theorem 3 follows. $\Box$
[2]{}
A. Aleksandrov and P. Kargaev,
C. Bennett and R. Sharpley,
M. Essén,
T. W. Gamelin,
A. A. Goldberg and I. V. Ostrovskii,
B. Khabibullin, Math. USSR Izv. [**39**]{} (1992), 1063–1083.
P. Koosis,
P. Koosis,
B. Ya. Levin,
B. Ya. Levin,
B. Ya. Levin and I. V. Ostrovskii, (1960), 427–455. (in Russian). English transl. in Amer. Math. Soc. Transl. (2) [**32**]{} (1963), 323–357.
V. Matsaev and M. Sodin,
V. Matsaev and M. Sodin,
V. Matsaev and M. Sodin,
R. Nevanlinna,
M. Tsuji,
Vladimir Matsaev:
*School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv, 69978, Israel*
matsaev@math.tau.ac.il
Iossif Ostrovskii:
*Department of Mathematics, Bilkent University, 06533 Bilkent, Ankara, Turkey*
iossif@fen.bilkent.edu.tr
and
*Verkin Institute for Low Temperature Physics and Engineering, 310164 Kharkov, Ukraine*
ostrovskii@ilt.kharkov.ua
Mikhail Sodin:
*School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv, 69978, Israel*
sodin@math.tau.ac.il
[^1]: Supported in part by the Israel Science Foundation of the Israel Academy of Sciences and Humanities under Grant No. 93/97-1.
[^2]: Supported in part by the INTAS Project No. 96-0858.
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.