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author:
- |
\
Johann Wolfgang Goethe-Universität Frankfurt am Main\
E-mail:
- |
Francesco Giacosa\
Johann Wolfgang Goethe-Universität Frankfurt am Main\
E-mail:
- |
Dirk H. Rischke\
Johann Wolfgang Goethe-Universität Frankfurt am Main\
E-mail:
title: The chiral partner of the nucleon in the mirror assignment with global symmetry
---
Introduction
============
Effective models which embody chiral symmetry and its spontaneous breakdown at low temperatures and densities are widely used to understand the properties of light hadrons. Viable candidates obey a well-defined set of low-energy theorems [@meissner; @gasioro] but they still differ in some crucial and interesting aspects such as the mass generation of the nucleon and the behavior at non-zero $T$ and $\mu $.
Here we concentrate on a linear sigma model with $U(2)_{R}\times U(2)_{L}$ symmetry and parity-doubled nucleons. The mesonic sector involves scalar, pseudoscalar, vector, and axial-vector mesons. In the baryonic sector, besides the usual nucleon doublet field $N,$ a second baryon doublet $%
N^{\ast }$ with $J^{P}=\frac{1}{2}^{-}$ is included. As first discussed in Ref. [@DeTar:1988kn] and extensively analyzed in Ref. [@jido], in the so-called mirror assignment the nucleon fields $N$ and $N^{\ast }$ have a mass $m_{0}\neq 0$ in the chirally symmetric phase. The chiral condensate $\varphi $ increases the masses and generates a mass splitting of $N$ and $N^{\ast },$ but is no longer solely responsible for generating the masses. Such a theoretical set-up has been used in Ref. [zschiesche]{} to study the properties of cold and dense nuclear matter. The experimental assignment for the chiral partner of the nucleon is still controversial: the well-identified resonances $N^{\ast }(1535)$ and $N^{\ast
}(1650)$ are two candidates with the right quantum numbers listed in the PDG [@PDG], but we shall also investigate the possibility of a very broad and not yet discovered resonance centered at about $1.2$ GeV, which has been proposed in Ref. [@zschiesche].
The aim of the present work is the development of an effective model which embodies the chiral partner of the nucleon in the mirror assignment within the context of global chiral symmetry involving also vector and axial-vector mesons. In this way more terms appear than those originally proposed in Ref. [DeTar:1988kn]{}. Moreover, we restrict our study to operators up to fourth order (thus not including Weinberg-Tomozawa interaction terms). The axial coupling constant of the nucleon can be correctly described. Using recent information about the axial coupling of the partner [@Takahashi] and experimental knowledge about its decay width [@PDG] we can evaluate the mass parameter $m_{0}$ which describes the nucleon mass in the chiral limit. Then, we further evaluate pion-nucleon scattering lengths and we compare them with the experimental values [@schroder].
The model and its implications
==============================
The scalar and pseudoscalar fields are included in the matrix $\Phi =(\sigma
+i\eta )t^{0}+(\overrightarrow{a}_{0}+i\overrightarrow{\pi })\cdot\overrightarrow{%
t}$ and the (axial-)vector fields are represented by the matrices $%
R^{\mu }=(\omega ^{\mu }-f_{1}^{\mu })t^{0}+(\overrightarrow{\rho }^{\mu }-%
\overrightarrow{a_{1}}^{\mu })\cdot\overrightarrow{t}$ and $L^{\mu }=(\omega
^{\mu }+f_{1}^{\mu })t^{0}+(\overrightarrow{\rho }^{\mu }+\overrightarrow{%
a_{1}}^{\mu })\cdot\overrightarrow{t}$ ($\overrightarrow{t}=\frac{1}{2}%
\overrightarrow{\tau },$ where $\overrightarrow{\tau }$ are the Pauli matrices and $t^{0}=\frac{1}{2}1_{2}$). The corresponding Lagrangian describing only mesons reads$$\begin{aligned}
\mathcal{L}_{mes}& =&\mathrm{Tr}\left[ (D_{\mu }\Phi )^{\dagger }(D^{\mu
}\Phi )-m^{2}\Phi ^{\dagger }\Phi -\lambda _{2}\left( \Phi ^{\dagger }\Phi
\right) ^{2}\right] -\lambda _{1}\left( \mathrm{Tr}[\Phi ^{\dagger }\Phi
]\right) ^{2}+c\,(\det \Phi ^{\dagger }+\det \Phi ) \nonumber \\
& +&\mathrm{Tr}[H(\Phi ^{\dagger }+\Phi
)]-\frac{1}{4}\mathrm{Tr}\left[ (L^{\mu \nu })^{2}+(R^{\mu \nu })^{2}\right]
+\frac{m_{1}^{2}}{2}\mathrm{Tr}\left[ (L^{\mu })^{2}+(R^{\mu })^{2}%
\right] \nonumber \\
& +&h_{2}Tr(\Phi ^{\dagger }L_{\mu }L^{\mu }\Phi +\Phi R_{\mu }R^{\mu }\Phi
^{\dagger })+h_{3}Tr(\Phi R_{\mu }\Phi ^{\dagger }L^{\mu })+\mathcal{L}_{three}+\mathcal{L}_{four} \; , \nonumber \\ \label{meslag}\end{aligned}$$ where $D^{\mu }\Phi =\partial ^{\mu }+ig_{1}(\Phi R^{\mu }-L^{\mu }\Phi )$ and $L^{\mu \nu }=\partial ^{\mu }L^{\nu }-\partial ^{\nu }L^{\mu }$, $%
R^{\mu \nu }=\partial ^{\mu }R^{\nu }-\partial ^{\nu }R^{\mu }$ are the field strength tensors of the (axial-)vector fields. $\mathcal{L}_{three}$ and $\mathcal{L}_{four}$ describe 3- and 4-particle interactions of (axial-)vector fields, which are irrelevant for this work, see Ref. [@Denisnew].
The baryon sector involves the baryon doublets $\Psi _{1}$ and $\Psi _{2},$ where $\Psi _{1}$ has positive parity and $\Psi _{2}$ negative parity. In the so-called mirror assignment, $\Psi _{1}$ and $\Psi _{2}$ transform in the opposite way under chiral symmetry, namely: $$\begin{aligned}
\Psi _{1R}& \longrightarrow U_{R}\Psi _{1R},\overline{\Psi }%
_{1R}\longrightarrow \overline{\Psi }_{1R}U_{R}^{\dagger }\;, \;\;\;
\Psi _{2R}& \longrightarrow U_{L}\Psi _{2R},\overline{\Psi }%
_{2R}\longrightarrow \overline{\Psi }_{2R}U_{L}^{\dagger }\;, \label{mirror}\end{aligned}$$and similarly for the left-handed fields. Such field transformations allow to write down a baryonic Lagrangian with a chirally invariant mass term for the fermions, parametrized by $m_{0}$: $$\begin{aligned}
{\cal L}_{bar}&=& \overline{\Psi }_{1L}i\gamma _{\mu }D_{1L}^{\mu }\Psi
_{1L}+\overline{\Psi }_{1R}i\gamma _{\mu }D_{1R}^{\mu }\Psi _{1R}+\overline{%
\Psi }_{2L}i\gamma _{\mu }D_{2R}^{\mu }\Psi _{2L}+\overline{\Psi }%
_{2R}i\gamma _{\mu }D_{2L}^{\mu }\Psi _{2R} \nonumber \\
& -&\widehat{g}_{1}\left( \overline{\Psi }_{1L}\Phi \Psi _{1R}\ +\overline{%
\Psi }_{1R}\Phi ^{\dagger }\Psi _{1L}\right) -\widehat{g}_{2}\left(
\overline{\Psi }_{2L}\Phi ^{\dagger }\Psi _{2R}\ +\overline{\Psi }_{2R}\Phi
\Psi _{2L}\right) \nonumber \\
& -&m_{0}(\overline{\Psi }_{1L}\Psi _{2R}-\overline{\Psi }_{1R}\Psi _{2L}-%
\overline{\Psi }_{2L}\Psi _{1R}+\overline{\Psi }_{2R}\Psi _{1L})\;,
\label{nucl lagra}\end{aligned}$$where $D_{1R}^{\mu }=\partial ^{\mu }-ic_{1}R^{\mu }$, $D_{1L}^{\mu
}=\partial ^{\mu }-ic_{1}L^{\mu }$ and $D_{2R}^{\mu }=\partial ^{\mu
}-ic_{2}R^{\mu }$, $D_{2L}^{\mu }=\partial ^{\mu }-ic_{2}L^{\mu }$ are the covariant derivatives for the nucleonic fields. The coupling constants $%
\widehat{g}_{1}$ and $\widehat{g}_{2}$ parametrize the interaction of the baryonic fields with scalar and pseudoscalar mesons and $\varphi =$ $%
\left\langle 0\left\vert \sigma \right\vert 0\right\rangle =Zf_{\pi }$ is the chiral condensate emerging upon spontaneous chiral symmetry breaking in the mesonic sector. The parameter $f_{\pi }=92.4$ MeV is the pion decay constant and $Z$ is the wavefunction renormalization constant of the pseudoscalar fields [@Strueber].
The term proportional to $m_{0}$ generates also a mixing between the fields $\Psi _{1}$ and $\Psi _{2}.$ The physical fields $N$ and $N^{\ast
},$ referring to the nucleon and to its chiral partner, arise by diagonalizing the baryonic part of the Lagrangian. As a result we have: $$\left(
\begin{array}{c}
N \\
N^{\ast }%
\end{array}%
\right) =\widehat{M}\left(
\begin{array}{c}
\Psi _{1} \\
\Psi _{2}%
\end{array}%
\right) =\frac{1}{\sqrt{2\cosh \delta }}\left(
\begin{array}{cc}
e^{\delta /2} & \gamma _{5}e^{-\delta /2} \\
\gamma _{5}e^{-\delta /2} & -e^{\delta /2}%
\end{array}%
\right) \left(
\begin{array}{c}
\Psi _{1} \\
\Psi _{2}%
\end{array}%
\right) . \label{mixing}$$The masses of the nucleon and its partner are obtained upon diagonalizing the mass matrix $\widehat{M}$:
$$m_{N,N^{\ast }}=\frac{1}{2}\sqrt{4m_{0}^{2}+(\widehat{g}_{1}+\widehat{g}%
_{2})^{2}\varphi ^{2}}\pm \frac{(\widehat{g}_{1}-\widehat{g}_{2})\varphi }{2}%
\;, \label{nuclmasses}$$
i.e., the nucleon mass is not only generated by the chiral condensate $\varphi $ but also by $m_{0}$. The parameter $\delta $ in Eq. (\[mixing\]) is related to the masses and the parameter $m_{0}$ by the expression: $\delta =\mathrm{Arcosh}\left[ \frac{m_{N}+m_{N^{\ast }}}{2m_{0}}%
\right] \;.
$Let us consider two important limiting cases. (i) When $\delta \rightarrow \infty $, corresponding to $m_{0}\rightarrow 0,$ no mixing is present and $N=\Psi _{1},$ $N^{\ast }=\Psi _{2}.$ In this case $%
m_{N}=\widehat{g}_{1}\varphi /2$ and $m_{N^{\ast }}=\widehat{g}_{2}\varphi
/2 $, thus the nucleon mass is generated solely by the chiral condensate as in the linear sigma model. (ii) In the chirally restored phase where $\varphi \rightarrow 0$, one has mass degeneracy $m_{N}=m_{N^{\ast }}=m_{0}.$ When chiral symmetry is broken, $\varphi \neq 0$, a splitting is generated. By choosing $0<$ $\widehat{g}%
_{1}<\widehat{g}_{2}$ the inequality $m_{N}<m_{N^{\ast }}$ is fulfilled.
Note also that, when $g_{1}=c_{1}=c_{2}$ and $h_{1}=h_{2}=h_{3}=0$, the chiral symmetry becomes local [@Ko; @Pisarski]. The corresponding model has been studied in Ref. [@Wilms]. It was not possible to make a clear-cut prediction as to whether the mass of the nucleon is dominantly generated by the chiral condensate or by mixing with its chiral partner. In addition to this, the description of the mesonic decays was not correct in a locally symmetric framework as shown in Ref. [@Parganlija]. Also, the expression for the axial charge reads $g_{A}^{N}=\frac{\tanh \delta }{Z^{2}}$. Since $\left\vert
\tanh \delta \right\vert <1$ for all $\delta $ and $Z>1$, we obtain $%
g_{A}^{N}<1$, at odds with the experimental value $g_{A}^{N}=1.267\pm 0.004$. Thus, in the context of local symmetry one is obliged to introduce terms of higher order such as the Weinberg-Tomozawa one. As a final remark, note that by setting $Z=1$ (which in turn means $g_{1}=0$) the vector mesons and axial-vector mesons drop out and only the (pseudo-)scalar and nucleonic terms survive in the Lagrangian. Then, for $g_{A}^{N^{\ast }}$a value much larger than $0.5$ is predicted, which is in disagreement with lattice data [Takahashi]{}.
In Ref. [@zschiesche] a large value of the parameter $m_{0}$ ($\sim 800$ MeV) is claimed to be needed for a correct description of nuclear matter properties, thus pointing to a small contribution of the chiral condensate to the nucleon mass. Validating this claim through the evaluation of pion-nucleon scattering at zero temperature and density is the subject of the present paper.
The expressions for the axial coupling constants of the nucleon and the partner are given by:$$\begin{aligned}
g_{A}^{N} =\frac{e^{\delta }}{2\cosh \delta }g_{A}^{(1)}+\frac{e^{-\delta }%
}{2\cosh \delta }g_{A}^{(2)}, \; \;
g_{A}^{N^{\ast }} =-\frac{e^{-\delta }}{2\cosh \delta }g_{A}^{(1)}+\frac{%
e^{\delta }}{2\cosh \delta }g_{A}^{(2)} \;, \label{ga}\end{aligned}$$where $$g_{A}^{(1)}=1-\frac{c_{1}}{g_{1}}\left( 1-\frac{1}{Z^{2}}\right) ,\; \;
g_{A}^{(2)}=-1+\frac{c_{2}}{g_{1}}\left( 1-\frac{1}{Z^{2}}\right)$$refer to the axial coupling constants of the bare, unmixed fields $\Psi _{1}$ and $\Psi _{2}.$ Note that when $\delta \rightarrow \infty $ one has $%
g_{A}^{N}=g_{A}^{(1)}$ and $g_{A}^{N^{\ast }}=g_{A}^{(2)}.$ Also, when $%
c_{1}=c_{2}=0$ (or $Z=1$) we obtain the results of Ref. [@DeTar:1988kn]: $g_{A}^{N}=\tanh \delta $ and $g_{A}^{N^{\ast
}}=-\tanh \delta ,$ which in the limit $\delta \rightarrow \infty $ reduces to $g_{A}^{N}=1$ and $g_{A}^{N^{\ast }}=-1.$ However, in our model the interaction with the (axial-)vector mesons generates additional contributions to $g_{A}^{N}$ and $g_{A}^{N^{\ast }}$, which are fixed via the experimental result for $g_{A}^{N}$ and the lattice result for $%
g_{A}^{N^{\ast }},$ see the next section. From the Lagrangian (\[nucl lagra\]) one can compute the decay $N^{\ast }\rightarrow N\pi$ and the scattering amplitudes $a_{0}^{(\pm )}$ [@Gallas].
Results and discussion
=======================
We consider three possible assignments for the partner of the nucleon. (1) The resonance $N^{\ast }(1535)$, with mass $M_{N^{\ast }(1535)}=1535$ MeV and $\Gamma _{N^{\ast }(1535)\rightarrow N\pi }=(67.5\pm 23.6)$ MeV, which – being the lightest baryonic resonance with the correct quantum numbers – surely represents one of the viable and highly discussed candidates for the nucleon partner. (2) The resonance $N^{\ast }(1650),$ with a mass lying just above, $M_{N^{\ast }(1650)}=1650$ MeV and $\Gamma _{N^{\ast
}(1650)\rightarrow N\pi }=(92.5\pm 37.5)$ MeV. (3) A $speculative$ candidate $N^{\ast }(1200)$ with a mass $M_{N^{\ast }(1200)}\sim 1200$ MeV and a very broad width $\Gamma _{N^{\ast }(1650)\rightarrow N\pi }\gtrsim 800$ MeV, such to have avoided experimental detection up to now [@zschiesche].
For all these scenarios, we want to determine the values of the parameters $%
c_{1},$ $c_{2}$, and $m_{0}.$ Beyond the width, which is different in the three cases mentioned above, we also use $g_{A}^{N^{\ast }}=0.2\pm 0.3$, as predicted by lattice QCD [@Takahashi], and $g_{A}^{N}=1.26$. We repeat the evaluation for different values of $Z.$ Remarkably, $m_{0}$ does not depend on $Z$.
Figure \[allem0\_glob\] shows the mass parameter $m_{0}$ as a function of the axial coupling constant of $N^{\ast }$ for different masses of $N^{\ast }$. For the range of $g_{A}^{N^{\ast }}$ given by Ref. [@Takahashi], $%
m_{0}=300-600 $ MeV for $N^{\ast }(1535)$ and $N^{\ast }(1650)$, meaning that half of the nucleonic mass survives in the chirally restored phase. On the contrary, for $N^{\ast }(1200)$ the value for $m_{0}$ lies above $%
1000 $ MeV. This result suggests that the contribution of the chiral condensate to the nucleonic mass should be negative, which is rather unnatural. We can then discard the possibility that a hypothetical, not yet discovered $N^{\ast }(1200)$ is the chiral partner of the nucleon.
According to Ref. [@Glozman], when the fields $N$ and $N^{\ast }$ belong to a parity doublet, $g_{A}^{N}\sim 1$ and $%
g_{A}^{N^{\ast }}\sim -1$. Then, in Ref. [@Glozman] the lattice result of [@Takahashi] is used against the identification of $N^{\ast
}(1535)$ as the partner of the nucleon. However, within our model we can [*still*]{} accommodate $N^{\ast}(1535)$ (or also $N^{\ast }(1650)$) as the partner of the nucleon. The small value of $g_{A}^{N^{\ast }}$ arises because of interactions of the partner with the axial-vector mesons.
In Figure \[Figure\] we plot the isospin-odd and isospin-even scattering lengths as a function of the axial charge $g_{A}^{N^{\ast }}$for $%
Z=1.5$. A comparison of our results to the experimental data on $\pi N$ scattering lengths, as measured in Ref. [@schroder] by precision X-ray experiments on pionic hydrogen and pionic deuterium, yields the following: (a) the isospin-odd scattering length $%
a_{0}^{(-)}$ is close to the experimental range, but we expect an even better result with the introduction of the $\Delta $ resonance. (b) The isospin-even scattering length $a_{0}^{(+)}$ is an order of magnitude smaller than the experimental band $a_{0,\exp }^{(+)}=(-8.85783\pm
7.16)10^{-6}$ MeV. The reason for this is the strong dependence of $%
a_{0}^{(+)}$ on the scalar mesons. Here we use $m_{\sigma }=1370$ MeV. A smaller mass of the sigma meson may be favored, however this result seems to be excluded [@Denisnew].
Summary and outlook
===================
We have computed the pion-nucleon scattering lengths at tree-level in the framework of a globally symmetric linear sigma model with parity-doubled nucleons. Within the mirror assignment the mass of the nucleon originates only partially from the chiral condensate, but also from the mass parameter $%
m_{0}$. Using the lattice results of Ref. [@Takahashi] we find that $m_{0}\simeq300-600$ MeV. Approximately half of the nucleon mass survives in the chirally restored phase. The isospin-odd scattering length lies close to the experimental band, but could be improved in further studies. The isospin-even scattering length is too small: future inclusion of a light tetraquark state gives rise to a large contribution, and thus is expected to improve the results. We also plan to extend our model by including the $\Delta $ resonance, necessary to correctly reproduce the p-wave scattering lengths [@Procura] and to evaluate the radiative $\eta $-photoproduction.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank T. Kunihiro, T. Takahashi and D. Parganlija for useful discussions.
[99]{}
T. T. Takahashi and T. Kunihiro, Phys. Rev. D **78** (2008) 011503 U. G. Meissner, Phys. Rept. **161** (1988) 213. S. Gasiorowicz and D. A. Geffen, Rev. Mod. Phys. **41** (1969) 531. C. DeTar and T. Kunihiro, Phys. Rev. D **39** (1989) 2805. D. Jido, M. Oka, and A. Hosaka, Prog. Theor. Phys. **106** (2001) 873 \[arXiv:hep-ph/0110005\]. D. Zschiesche, L. Tolos, J. Schaffner-Bielich, and R. D. Pisarski, arXiv:nucl-th/0608044. C. Amsler et al. Phys. Lett. B **667** (2008)
H. C. Schroder *et al.*, Eur. Phys. J. C **21** (2001) 473. D. Parganlija, F. Giacosa, and D. Rischke, these proceedings. P. Ko and S. Rudaz, Phys. Rev. D **50** (1994) 6877. S. Strüber and D. H. Rischke, Phys. Rev. D [**77**]{} (2008) 085004 R. D. Pisarski, arXiv:hep-ph/9503330. S. Wilms, F. Giacosa, and D. H. Rischke, *Proceedings of 19th International Conference on Ultra-Relativistic Nucleus-Nucleus Collisions: Quark Matter 2006 (QM2006), Shanghai, China, 14-20 Nov 2006* D. Parganlija, F. Giacosa, and D. H. Rischke, AIP Conf. Proc. **1030** (2008) 160 \[arXiv:0804.3949 \[hep-ph\]\]. S. Gallas, F. Giacosa, and D. H. Rischke, *in preparation* L. Y. Glozman, Phys. Rept. **444** (2007) 1 \[arXiv:hep-ph/0701081\].
M. Procura, B. U. Musch, T. R. Hemmert, and W. Weise, Phys. Rev. D **75** (2007) 014503 \[arXiv:hep-lat/0610105\].
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abstract: |
Let $H$ be a closed, connected subgroup of a connected, simple Lie group $G$ with finite center. The homogeneous space $G/H$ has a [*tessellation*]{} if there is a discrete subgroup $\Gamma$ of $G$, such that $\Gamma$ acts properly discontinuously on $G/H$, and the double-coset space $\Gamma\backslash G/H$ is compact. Note that if either $H$ or $G/H$ is compact, then $G/H$ has a tessellation; these are the obvious examples.
It is not difficult to see that if $G$ has real rank one, then only the obvious homogeneous spaces have tessellations. Thus, the first interesting case is when $G$ has real rank two. In particular, R. Kulkarni and T. Kobayashi constructed examples that are not obvious when $G =
{\operatorname{SO}}(2,2n)^\circ$ or ${\operatorname{SU}}(2,2n)$. H. Oh and D. Witte constructed additional examples in both of these cases, and obtained a complete classification when $G = {\operatorname{SO}}(2,2n)^\circ$. We simplify the work of Oh-Witte, and extend it to obtain a complete classification when $G = {\operatorname{SU}}(2,2n)$. This includes the construction of another family of examples.
The main results are obtained from methods of Y. Benoist and T. Kobayashi: we fix a Cartan decomposition $G = K A^+ K$, and study the intersection $(KHK)
\cap A^+$. Our exposition generally assumes only the standard theory of connected Lie groups, although basic properties of real algebraic groups are sometimes also employed; the specialized techniques that we use are developed from a fairly elementary level.
address:
- 'Department of Mathematics, University of Maryland, College Park, MD 20910 USA'
- 'Department of Mathematics, Oklahoma State University, Stillwater, OK 74078 USA'
author:
- Alessandra Iozzi
- Dave Witte
title: |
Tessellations of homogeneous spaces\
of classical groups of real rank two
---
\#1 latex@info\#1
font@info\#1
Introduction
============
\[PDDefn\] A group $\Gamma$ of homeomorphisms of a topological space $M$ acts *properly discontinuously* on $M$ if, for every compact subset $C$ of $M$, $$\mbox{ $\{\, \gamma \in \Gamma \mid C \cap \gamma C \neq \emptyset \,\}$
is finite.}$$
Classically, a discrete group $\Gamma$ of isometries of a Riemannian manifold $M$ is a crystallographic group if $\Gamma$ acts properly discontinuously on $M$, and the quotient $\Gamma \backslash M$ is compact. The $\Gamma$-translates of any fundamental domain for $\Gamma \backslash M$ form a tessellation of $M$.
These notions generalize to any homogeneous space, even without an invariant metric.
\[TessDefn\] Let
- $G$ be a Lie group and
- $H$ be a closed subgroup of $G$.
A discrete subgroup $\Gamma$ of $G$ is a *crystallographic group* for $G/H$ if
1. $\Gamma$ acts properly discontinuously on $G/H$; and
2. $\Gamma \backslash G/H$ is compact.
We say that $G/H$ has a *tessellation* if there exists a crystallographic group $\Gamma$ for $G/H$.
Crystallographic groups and the corresponding tessellations have been studied for many groups $G$. (A brief recent introduction to the subject is given in [@Kobayashi-survey00].) The classical Bieberbach Theorems [@CharlapBook Chap. 1] deal with the case where $G$ is the group of isometries of Euclidean space ${\mathord{\mathbb{R}}}^n = G/H$. As another example, the Auslander Conjecture [@Abels; @AbelsMargulisSoifer; @FriedGoldman; @Margulis-Auslander; @Tomanov] asserts that if $G$ is the group of all affine transformations of ${\mathord{\mathbb{R}}}^n$, then every crystallographic group has a solvable subgroup of finite index. In addition, the case where $G$ is solvable has been discussed in [@Witte-Solvtess].
In this paper, we focus on the case where $G$ is a simple Lie group, such as ${\operatorname{SL}}(n,{\mathord{\mathbb{R}}})$, ${\operatorname{SO}}(m,n)$, or ${\operatorname{SU}}(m,n)$.
\[standing\] Throughout this paper:
1. \[standing-G\] $G$ is a linear, semisimple Lie group with only finitely many connected components; and
2. \[standing-H\] $H$ is a closed subgroup of $G$ with only finitely many connected components.
\[Hdisconnected\] Because $H/H^\circ$ is finite (hence compact), it is easy to see that $G/H$ has a tessellation if and only if $G/H^\circ$ has a tessellation. Also, if $G/H$ has a tessellation, then $G^\circ / H^\circ$ has a tessellation. Furthermore, the converse holds in many situations. (See §\[disconnected\] for a discussion of this issue.) Thus, there is usually no harm in assuming that both $G$ and $H$ are connected; we will feel free to do so whenever it is convenient. On the other hand, because ${\operatorname{SO}}(m,n)$ is usually not connected (it usually has two components [@HelgasonBook Lem. 10.2.4, p. 451]), it would be somewhat awkward to make this a blanket assumption.
\[classical\] There are two classical cases in which $G/H$ is well known to have a tessellation.
1. If $G/H$ is compact, then we may let $\Gamma = e$ (or any finite subgroup of $G$).
2. \[classical-Borel\] If $H$ is compact, then we may let $\Gamma$ be any cocompact lattice in $G$. (A. Borel [@Borel-CK] proved that every connected, simple Lie group has a cocompact lattice.)
Thus, the existence of a tessellation is an interesting question only when neither $H$ nor $G/H$ is compact. (In this case, any crystallographic group $\Gamma$ must be infinite, and cannot be a lattice in $G$.)
Given $G$ (satisfying [\[standing\][[(]{}\[standing-G\][)]{}]{}]{}), we would like to find all the subgroups $H$ (satisfying [\[standing\][[(]{}\[standing-H\][)]{}]{}]{}), such that $G/H$ has a tessellation. This seems to be a difficult problem in general. (See the surveys [@Kobayashi-survey97] and [@Labourie-survey] for a discussion of the many partial results that have been obtained, mainly under the additional assumption that $H$ is reductive.) However, it can be solved in certain cases of low real rank. In particular, as we will now briefly explain, the problem is very easy if ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G = 0$ or $1$. Most of this paper is devoted to solving the problem for certain cases where ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G =
2$.
If ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G = 0$ (that is, if $G$ is compact), then $G/H$ must be compact (and $H$ must also be compact), so $G/H$ has a tessellation, but this is not interesting. If ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G = 1$, then there are some interesting homogeneous spaces, but it turns out that none of them have tessellations.
\[CalabiMarkusCircle\] $G = {\operatorname{SL}}(2,{\mathord{\mathbb{R}}})$ is transitive on ${\mathord{\mathbb{R}}}^2 - \{0\}$, so ${\mathord{\mathbb{R}}}^2 - \{0\}$ is a homogeneous space for $G$. It does not have a tessellation, for reasons that we now explain.
Let $C$ be the unit circle, so $C$ is a compact subset of ${\mathord{\mathbb{R}}}^2 -
\{0\}$.
We claim that $C \cap gC \neq \emptyset$, for every $g \in G$ (cf. Figure \[circle+ellipse\]). To see this, note that, because $\det g = 1$, the ellipse bounded by $gC$ has the same area as the disk bounded by $C$, so $gC$ cannot be contained in the interior of the disk bounded by $C$, and cannot contain $C$ in its interior. Thus, $gC$ must be partly inside $C$ and partly outside, so $gC$ must cross $C$, as claimed.
Let $\Gamma$ be any discrete subgroup of $G$. The preceding paragraph implies that $C \cap \gamma C \neq \emptyset$, for every $\gamma \in \Gamma$. If $\Gamma$ acts properly discontinuously on ${\mathord{\mathbb{R}}}^2 - \{0\}$, then, because $C$ is compact, this implies that $\Gamma$ is finite. So the quotient $\Gamma \backslash ({\mathord{\mathbb{R}}}^2 - \{0\})$ is not compact. Therefore $\Gamma$ is not a crystallographic group. We have shown that no subgroup of $G$ is a crystallographic group, so we conclude that ${\mathord{\mathbb{R}}}^2 - \{0\}$ does not have a tessellation.
![The Calabi-Markus Phenomenon (Example \[CalabiMarkusCircle\]): $C
\cap gC \neq \emptyset$, for every $g \in {\operatorname{SL}}(2,{\mathord{\mathbb{R}}})$, so no infinite subgroup of ${\operatorname{SL}}(2,{\mathord{\mathbb{R}}})$ acts properly discontinuously.[]{data-label="circle+ellipse"}](circle+ellipse.eps)
This example illustrates the *Calabi-Markus Phenomenon*: if there is a compact subset $C$ of $G/H$, such that $C \cap gC \neq \emptyset$, for every $g \in G$, then no infinite subgroup of $G$ acts properly discontinuously on $G/H$ [[(]{}see \[CalabiMarkus\][)]{}]{}. Thus, $G/H$ does not have a tessellation, unless $G/H$ is compact [[(]{}see \[CDS->notess\][)]{}]{}.
We will see in Section \[CartanSect\] that the following proposition can be proved quite easily from basic properties of the Cartan projection.
[Rrank1-CDS]{}
\[Rrank1-CM\] If ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G = 1$, and $H$ is not compact, then there is a compact subset $C$ of $G/H$, such that $C \cap gC \neq \emptyset$, for every $g \in G$.
\[rank1->notess\] If ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G = 1$, and neither $H$ nor $G/H$ is compact, then $G/H$ does not have a tessellation.
We now consider groups of real rank two. The obvious example is ${\operatorname{SL}}(3,{\mathord{\mathbb{R}}})$, but, in this case, once again, none of the interesting homogeneous spaces have tessellations. Moreover, the same is true when real numbers are replaced by complex numbers or quaternions. The case where $\dim H = 1$ relies on beautiful methods of Y. Benoist and F. Labourie [@BenoistLabourie] or G. A. Margulis [@Margulis-CK], which we describe in Section \[1DSect\].
\[SL3->notess\] If $$\hbox{$G = {\operatorname{SL}}(3,{\mathord{\mathbb{R}}})$, ${\operatorname{SL}}(3,{\mathord{\mathbb{C}}})$, or ${\operatorname{SL}}(3,{\mathord{\mathbb{H}}})$,}$$ and neither $H$ nor $G/H$ is compact, then $G/H$ does not have a tessellation.
It is important to note that some interesting homogeneous spaces do have tessellations.
\[G=LxH\] Suppose $G = L \times H$, and let $\Gamma$ be a cocompact lattice in $L$. Then $\Gamma$ acts properly discontinuously on $L \cong G/H$, and $\Gamma
\backslash G/H \cong \Gamma \backslash L$ is compact. So $G/H$ has a tessellation.
The following easy lemma generalizes this example to the situation where $G$ is a more general product of $L$ and $H$, not necessarily a direct product.
[construct-tess]{}
\[G=LH\] Let $H$ and $L$ be closed subgroups of $G$, such that
- $G = LH$,
- $L \cap H$ is compact; and
- $L$ has a cocompact lattice $\Gamma$.
Then $G/H$ has a tessellation. [(]{}Namely, $\Gamma$ is a crystallographic group for $G/H$.[)]{}
For $G = {\operatorname{SO}}(2,n)$ or ${\operatorname{SU}}(2,n)$, this lemma leads to some interesting examples found by R. Kulkarni [@Kulkarni Thm. 6.1] and T. Kobayashi [@Kobayashi-properaction Prop. 4.9].
\[KulkarniEg\] There are natural embeddings $$\hbox{${\operatorname{SO}}(1,n) \hookrightarrow {\operatorname{SO}}(2,n)$
\qquad and \qquad
${\operatorname{SU}}(1,n) \hookrightarrow {\operatorname{SU}}(2,n)$.}$$ Furthermore, identifying ${\mathord{\mathbb{C}}}^{1+m}$ with ${\mathord{\mathbb{R}}}^{2+2m}$ yields an embedding $${\operatorname{SU}}(1,m) \hookrightarrow {\operatorname{SO}}(2,2m) .$$ Similarly, identifying ${\mathord{\mathbb{H}}}^{1+m}$ with ${\mathord{\mathbb{C}}}^{2+2m}$ yields an embedding $${\operatorname{Sp}}(1,m) \hookrightarrow {\operatorname{SU}}(2,2m) .$$ Thus, we may think of ${\operatorname{SO}}(1,2m)$ and ${\operatorname{SU}}(1,m)$ as subgroups of ${\operatorname{SO}}(2,2m)$; and we may think of ${\operatorname{SU}}(1,2m)$ and ${\operatorname{Sp}}(1,m)$ as subgroups of ${\operatorname{SU}}(2,2m)$.
With the above understanding, we see that ${\operatorname{SO}}(1,2m)$ is the stabilizer of a vector of norm $+1$. Since ${\operatorname{SU}}(1,m)$ is transitive on the set of such vectors, we have $${\operatorname{SO}}(2,2m) = {\operatorname{SO}}(1,2m) \, {\operatorname{SU}}(1,m) .$$ Similarly, $${\operatorname{SU}}(2,2m) = {\operatorname{SU}}(1,2m) \, {\operatorname{Sp}}(1,m) .$$ Then Lemma \[G=LH\] implies that each of the following four homogeneous spaces has a tessellation:
- ${\operatorname{SO}}(2,2m)/{\operatorname{SO}}(1,2m)$,
- ${\operatorname{SO}}(2,2m)/{\operatorname{SU}}(1,m)$,
- ${\operatorname{SU}}(2,2m)/{\operatorname{SU}}(1,2m)$, and
- ${\operatorname{SU}}(2,2m)/{\operatorname{Sp}}(1,m)$.
When discussing ${\operatorname{SO}}(2,n)$ or ${\operatorname{SU}}(2,n)$, we always assume $n > 2$. This causes no harm, because ${\operatorname{SO}}(2,2)$ is locally isomorphic to ${\operatorname{SL}}(2,{\mathord{\mathbb{R}}})
\times {\operatorname{SL}}(2,{\mathord{\mathbb{R}}})$ [@HelgasonBook (x), p. 520], and ${\operatorname{SU}}(2,2)$ is locally isomorphic to ${\operatorname{SO}}(2,4)$ [@HelgasonBook (vi), p. 519].
When $n$ is even, H. Oh and D. Witte [@OhWitte-CK] provided a complete description of all the (closed, connected) subgroups $H$, such that ${\operatorname{SO}}(2,n)/H$ has a tessellation, but their classification is not quite complete when $n$ is odd.
In this paper, we extend the work of Oh and Witte to obtain analogous results for homogeneous spaces of $G = {\operatorname{SU}}(2,n)$. We also give a much shorter proof of the main results of [@OhWitte-CK]. The same techniques should yield significant results for homogeneous spaces of the other simple groups of real rank two, although the calculations seem to be difficult. On the other hand, the groups of higher real rank require different ideas.
Once one knows that a tessellation of $G/H$ exists, it would be interesting to find *all* of the crystallographic groups for $G/H$ and, for each crystallographic group, describe the possible tessellations. These are much more delicate questions, which we do not address at all. (W. Goldman [@Goldman-nonstandard], F. Salein [@Salein], T. Kobayashi [@Kobayashi-deformation], and A. Zeghib [@Zeghib-deSitter] have interesting results in some special cases.)
In the remainder of this introduction, we state the specific results for homogeneous spaces of ${\operatorname{SO}}(2,n)$ and ${\operatorname{SU}}(2,n)$.
\[d(H)-defn\] For any connected Lie group $H$, let $$d(H) = \dim H - \dim K_H ,$$ where $K_H$ is any maximal compact subgroup of $H$. This is well defined, because all the maximal compact subgroups of $H$ are conjugate [@Hochschild-Lie Thm. 15.3.1(iii), pp. 180–181].
\[d(G)\] If $H$ is semisimple, we have the Iwasawa decomposition $H = K_H A_H N_H$ [@HelgasonBook Thm. 6.5.1, pp. 270–271], from which it is obvious that $d(H) = \dim(A_H N_H)$.
This yields the following calculations [[[(]{}see \[d(SU2)\] [and \[d(Sp)\]]{}[)]{}]{}]{}):
- $d \bigl( {\operatorname{SO}}(1,n) \bigr) = n$.
- $d \bigl( {\operatorname{SO}}(2,n) \bigr) = 2n$.
- $d \bigl( {\operatorname{SU}}(1,n) \bigr) = 2n$.
- $d \bigl( {\operatorname{SU}}(2,n) \bigr) = 4n$.
- $d \bigl( {\operatorname{Sp}}(1,n) \bigr) = 4n$.
\[d(H)=dimH\] If $H \subset AN$ (for some Iwasawa decomposition $G = KAN$ of $G$), then $d(H) = \dim H$ (see \[ANsc\] and [\[solvable\][[(]{}\[solvable-nocpct\][)]{}]{}]{}).
\[simdefn\] For subgroups $H_1$ and $H_2$ of $G$, we write $H_1 \sim H_2$ if there is a compact subset $C$ of $G$, such that $H_1 \subset C H_2 C$ and $H_2 \subset
C H_1 C$.
Note that $d$ is not invariant under the equivalence relation $\sim$. For example, the Cartan decomposition $G = KAK$ implies that $G \sim A$, but we have $d(A) = \dim A \neq \dim(AN) = d(G)$.
The following two theorems state a version of the main results for even $n$.
[SUF-known]{}
\[OW-known\] Assume $G = {\operatorname{SO}}(2,2m)$, and let $H$ be a closed, connected, subgroup of $G$, such that neither $H$ nor $G/H$ is compact.
The homogeneous space $G/H$ has a tessellation if and only if
1. $d(H) = 2m$; and
2. either $H \sim {\operatorname{SO}}(1,2m)$ or $H \sim {\operatorname{SU}}(1,m)$.
[SUF-known]{}
\[IW-known\] Assume $G = {\operatorname{SU}}(2,2m)$, and let $H$ be a closed, connected, subgroup of $G$, such that neither $H$ nor $G/H$ is compact.
The homogeneous space $G/H$ has a tessellation if and only if
1. $d(H) = 4m$; and
2. either $H \sim {\operatorname{SU}}(1,2m)$ or $H \sim {\operatorname{Sp}}(1,m)$.
The subgroups $H$ that arise in Theorems \[OW-known\] and \[IW-known\] can also be described more explicitly (cf. \[SOevenTess\] and \[SUevenTess\] below).
T. Kobayashi [@Kobayashi-deformation 1.4] conjectured that if $H$ is reductive and it is impossible to construct a tessellation of $G/H$ by using a generalization of Lemma \[G=LH\] [[(]{}see \[construct-tess\][)]{}]{}, then $G/H$ does not have a tessellation. The following lists three special cases of this general conjecture.
\[notessSU/Sp\] The homogeneous spaces
1. \[notessSU/Sp-SO2/SU1\] ${\operatorname{SO}}(2,2m+1)/{\operatorname{SU}}(1,m)$,
2. \[notessSU/Sp-SU2/Sp1\] ${\operatorname{SU}}(2,2m+1)/{\operatorname{Sp}}(1,m)$, and
3. \[notessSU/Sp-SU2/SU1\] ${\operatorname{SU}}(2,2m+1)/{\operatorname{SU}}(1,2m+1)$
do not have tessellations.
If this conjecture is true, then, for odd $n$, there is no interesting example of a homogeneous space of ${\operatorname{SO}}(2,n)$ or ${\operatorname{SU}}(2,n)$ that has a tessellation.
[SUF->complete]{}
\[SU1m->complete\] Assume $$\hbox{$G = {\operatorname{SO}}(2,2m+1)$ or ${\operatorname{SU}}(2,2m+1)$,}$$ and let $H$ be any closed, connected subgroup of $G$, such that neither $H$ nor $G/H$ is compact.
If Conjecture \[notessSU/Sp\] is true, then $G/H$ does not have a tessellation.
The proof of Theorem \[SU1m->complete\] assumes the following special case proved by R. Kulkarni [@Kulkarni Cor. 2.10]. In short, Kulkarni noted that the Euler characteristic of $\Gamma \backslash G/H$ must both vanish (because the Euler characteristic of $G/H$ vanishes) and not vanish (by the Gauss-Bonnet Theorem). (Other results in the same spirit, obtaining a contradiction from the study of characteristic classes of $\Gamma \backslash
G/H$, appear in [@KobayashiOno].)
\[SO2n/SO1odd-notess\] If $n$ is odd, then ${\operatorname{SO}}(2,n)/{\operatorname{SO}}(1,n)$ does not have a tessellation.
Let us give a more explicit description of the closed, connected subgroups $H$ of ${\operatorname{SO}}(2,2m)$ or ${\operatorname{SU}}(2,2m)$, such that $G/H$ has a tessellation. This shows that if $n$ is even, then the Kulkarni-Kobayashi examples [[(]{}\[KulkarniEg\][)]{}]{} and certain deformations are essentially the only interesting homogeneous spaces of ${\operatorname{SO}}(2,n)$ or ${\operatorname{SU}}(2,n)$ that have tessellations.
\[KANDefn\] Fix an Iwasawa decomposition $G = KAN$. Thus,
- $K$ is a maximal compact subgroup,
- $A$ is the identity component of a maximal split torus, and
- $N$ is a maximal unipotent subgroup.
The following two results are stated only for subgroups of $AN$, because the general case reduces to this [[(]{}see \[HcanbeAN\][)]{}]{}. The reason is basically that $H$ contains a connected, cocompact subgroup that is conjugate to a subgroup of $AN$. (Clearly, if $H'$ is any cocompact subgroup of $H$, then $G/H$ has a tessellation if and only if $G/H'$ has a tessellation.) This is not quite true in general, but the following lemma provides a satisfactory substitute, by showing that it becomes true after enlarging $H$ by a compact amount.
[HcanbeAN]{}
After replacing $H$ by a conjugate subgroup, there is a closed, connected subgroup $H^*$ of $G$, such that $H^*/H$ and $H^*/(AN \cap H^*)^\circ$ are compact, where $(AN \cap H^*)^\circ$ denotes the identity component of $AN
\cap H^*$.
[SUFevenTess]{}
\[SOevenTess\] Assume $G = {\operatorname{SO}}(2,2m)$, and let $H$ be a closed, connected, nontrivial, proper subgroup of $AN$.
The homogeneous space $G/H$ has a tessellation if and only if $H$ is conjugate to a subgroup $H'$, such that either
1. \[SOevenTess-SO\] $H' = {\operatorname{SO}}(1,2m) \cap AN$; or
2. \[SOevenTess-Sp\] $H'$ belongs to a certain family $\{H_B\}$ of deformations of ${\operatorname{SU}}(1,m) \cap
AN$, described explicitly in Theorem \[HBthm\] [(]{}with ${\mathbb{F}}= {\mathord{\mathbb{R}}}$[)]{}.
[SUFevenTess]{}
\[SUevenTess\] Assume $G = {\operatorname{SU}}(2,2m)$, and let $H$ be a closed, connected, nontrivial, proper subgroup of $AN$.
The homogeneous space $G/H$ has a tessellation if and only if $H$ is conjugate to a subgroup $H'$, such that either
1. \[SUevenTess-SU\] $H'$ belongs to a certain family $\{{{H_{[c]}}}\}$ of deformations of ${\operatorname{SU}}(1,2m) \cap AN$, described explicitly in Theorem \[SUegs\]; or
2. \[SUevenTess-Sp\] $H'$ belongs to a certain family $\{H_B\}$ of deformations of ${\operatorname{Sp}}(1,m) \cap
AN$, described explicitly in Theorem \[HBthm\] [(]{}with ${\mathbb{F}}= {\mathord{\mathbb{C}}}$[)]{}.
The proof of Theorem \[SOevenTess\] (or \[OW-known\]) in [@OhWitte-CK] requires a list [@OhWitte-CDS] of all the homogeneous spaces of ${\operatorname{SO}}(2,n)$ that admit a proper action of a noncompact subgroup of ${\operatorname{SU}}(2,n)$. (The list was obtained by very tedious case-by-case analysis. It was extended to homogeneous spaces of ${\operatorname{SU}}(2,n)$ in [@IozziWitte-CDS].) The following proposition [[(]{}\[tessAN->dim>1,2\][)]{}]{} provides an *a priori* lower bound on $\dim H$, and it turns out that the classification of the interesting subgroups of large dimension can be achieved fairly easily (see §\[SUFlargeSect\]). This is the main reason that we are able to give reasonably short complete proofs of Theorems \[OW-known\], \[IW-known\], \[SU1m->complete\], \[SOevenTess\], and \[SUevenTess\].
\[tessAN->dim>1,2\] Suppose $G = {\operatorname{SO}}(2,n)$ or ${\operatorname{SU}}(2,n)$, and let $H$ be a closed, connected, nontrivial subgroup of $AN$. If $G/H$ has a tessellation, then $$\dim H \ge
\begin{cases}
n & \hbox{if $G = {\operatorname{SO}}(2,n)$ and $n$ is even}; \\
n-1 & \hbox{if $G = {\operatorname{SO}}(2,n)$ and $n$ is odd}; \\
2n & \hbox{if $G = {\operatorname{SU}}(2,n)$ and $n$ is even}; \\
2n-2 & \hbox{if $G = {\operatorname{SU}}(2,n)$ and $n$ is odd}.
\end{cases}$$
This research was partially supported by a grant from the National Science Foundation (DMS-9801136). Much of the work was carried out during productive visits to the Isaac Newton Institute for Mathematical Sciences (Cambridge, U.K.); we would like to thank the Newton Institute for the financial support that made the visits possible. A.I. would like to thank the mathematics department of Oklahoma State University for their warm and generous hospitality and Marc Burger both for pointing out a mistake in the original statement and proof of Theorem \[SUFevenTess\] and for many enlightening conversations.
Cartan projection and Cartan-decomposition subgroups {#CartanSect}
====================================================
The main problem in this paper is to determine whether or not a homogeneous space $G/H$ has a tessellation. This requires some method to determine whether or not a given discrete subgroup $\Gamma$ of $G$ acts properly discontinuously on $G/H$. Y. Benoist and T. Kobayashi (independently) demonstrated that the Cartan projection $\mu$ is an effective tool to study this question. It is the foundation of almost all of our work in later sections.
In this section, we introduce the Cartan projection, and describe some of its basic properties. First, however, we recall the notion of a proper action (a generalization of properly discontinuous actions) and of a Cartan-decomposition subgroup. At the end of the section, we use the Cartan projection to briefly discuss the question of when there is a loss of generality in assuming that $G$ is connected.
Proper actions
--------------
A topological group $L$ of homeomorphisms of a topological space $M$ acts *properly* on $M$ if, for every compact subset $C$ of $M$, $$\mbox{ $\{\, g \in L \mid C \cap g C \neq \emptyset \,\}$
is compact.}$$
It is important to note that a *discrete* group of homeomorphisms of $M$ acts properly on $M$ if and only if it acts properly discontinuously on $M$.
For the special case where $M = G/H$ is a homogeneous space, the following lemma restates the definition of a proper action in more group-theoretic terms.
\[proper<>CHC\] A closed subgroup $L$ of $G$ acts properly on $G/H$ if and only if, for every compact subset $C$ of $G$, the intersection $L \cap (CHC)$ is compact.
If $C$ is any compact subset of $G$, then $\overline{C} = CH/H$ is a compact subset of $G/H$; furthermore, any compact subset of $G/H$ is contained in one of the form $\overline{C}$. We have $$\begin{aligned}
\{\, g \in L \mid \overline{C} \cap g \overline{C} \neq
\emptyset \,\}
&= \{\, g \in L \mid (CH) \cap (g CH) \neq \emptyset \,\} \\
&= \{\, g \in L \mid g \in (CH) (CH)^{-1} \,\} \\
&= L \cap (C H C^{-1}) .
\end{aligned}$$
This has the following well-known, easy consequence.
\[CHCproper\] Suppose $H$, $H_1$, $L$, and $L_1$ are closed subgroups of $G$. If
- $L$ acts properly on $G/H$, and
- there is a compact subset $C$ of $G$, such that $H_1 \subset CHC$ and $L_1 \subset CLC$,
then $L_1$ acts properly on $G/H_1$.
Cartan-decomposition subgroups
------------------------------
The following definition describes the subgroups to which the Calabi-Markus Phenomenon applies (cf. Example \[CalabiMarkusCircle\]).
We say that $H$ is a *Cartan-decomposition subgroup* of $G$ if $H \sim
G$ (see Notation \[simdefn\]).
\[AisCDS\] From the Cartan decomposition $G = KAK$, we know that $A$ is a Cartan-decomposition subgroup.
\[CDSconj\] Any conjugate of a Cartan-decomposition subgroup is a Cartan-decomposition subgroup.
\[CalabiMarkus\] If $H$ is a Cartan-decomposition subgroup of $G$, and $\Gamma$ is a discrete subgroup of $G$ that acts properly discontinuously on $G/H$, then $\Gamma$ is finite.
Because $H$ is a Cartan-decomposition subgroup, there is a compact subset $C$ of $G$, such that $CHC = G$. However, from Lemma \[proper<>CHC\], we know that $\Gamma \cap (CHC)$ is finite. Therefore $$\Gamma = \Gamma \cap G = \Gamma \cap (CHC)$$ is finite.
The following well-known, easy fact is a direct consequence of the Calabi-Markus Phenomenon. It is an important first step toward determining which homogeneous spaces have tessellations.
\[CDS->notess\] If $H$ is a Cartan-decomposition subgroup of $G$, such that $G/H$ is not compact, then $G/H$ does not have a tessellation.
The Cartan projection
---------------------
\[A+Defn\]
- If $G$ is connected, let $A^+$ be the (closed) positive Weyl chamber of $A$ in which the roots occurring in the Lie algebra of $N$ are positive [(cf. \[KANDefn\])]{}. Thus, $A^+$ is a fundamental domain for the action of the (real) Weyl group of $G$ on $A$.
- In the general case, let $A^+$ be a closed, convex fundamental domain for the action of the (real) Weyl group of $G$ on $A$, such that $A^+$ is contained in the (closed) positive Weyl chamber of $A$ in which the roots occurring in the Lie algebra of $N$ are positive.
For each element $g$ of $G$, the Cartan decomposition $G = K A^+ K$ implies that there is an element $a$ of $A^+$ with $g \in K a K$. In fact, the element $a$ is unique, so there is a well-defined function $\mu \colon G \to
A^+$ given by $g \in K \, \mu(g) \, K$.
We remark that the function $\mu$ is continuous and proper (that is, the inverse image of any compact set is compact).
The following crucial result of Y. Benoist provides a uniform estimate on the variation of $\mu$ over disks of bounded radius. (A related result was proved, independently and simultaneously, by T. Kobayashi [@Kobayashi-criterion Thm. 3.4].) The proof is both elementary and elegant. However, it requires a bit of notation, so we postpone it to §\[CalcSect-Benoist\] (and, for concreteness, we will assume that $G$ is either ${\operatorname{SO}}(2,n)$ or ${\operatorname{SU}}(2,n)$ in the proof).
\[bddchange\] For any compact subset $C$ of $G$, there is a compact subset $C'$ of $A$, such that $\mu(C g C) \subset \mu(g) C'$, for all $g \in G$.
For subsets $U$ and $V$ of $A^+$, we write $U \approx V$ if there is a compact subset $C$ of $A$, such that $U \subset VC$ and $V \subset UC$.
\[simvsmu\] For any subgroups $H_1$ and $H_2$ of $G$, we have $H_1 \sim H_2$ if and only if $\mu(H_1) \approx \mu(H_2)$.
($\Rightarrow$) Let $C$ be a compact subset of $G$, such that $H_1 \subset
CH_2 C$ and $H_2 \subset CH_1 C$. Choose a corresponding compact subset $C'$ of $A$, as in Proposition \[bddchange\]. Then $$\mu(H_1) \subset \mu(CH_2 C) \subset \mu(H_2) C'$$ and, similarly, $\mu(H_2) \subset \mu(H_1) C'$.
($\Leftarrow$) Let $C$ be a compact subset of $A$, such that $\mu(H_1) \subset
\mu(H_2) C$ and $\mu(H_2) \subset \mu(H_1) C$. Then $$H_1 \subset K \, \mu(H_1) \, K
\subset K \, (\mu(H_2) C) \, K
\subset K \, \bigl( (KH_2 K) C \bigr) \, K$$ and, similarly, $H_2 \subset K H_1 (KCK)$.
The special case where $H_2 = G$ (and $H_1$ is closed and almost connected) can be restated as follows.
\[CDSvsmu\] $H$ is a Cartan-decomposition subgroup of $G$ if and only if $\mu(H) \approx
A^+$.
\[Rrank1-CDS\] Assume that ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G = 1$. The subgroup $H$ is a Cartan-decomposition subgroup of $G$ if and only if $H$ is noncompact.
($\Leftarrow$) We have $\mu(e) = e$, and, because $\mu$ is a proper map, we have $\mu(h) \to \infty$ as $h \to \infty$ in $H$. Because ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G =
1$, we know that $A^+$ is homeomorphic to the half-line $[0,\infty)$ (with the point $e$ in $A^+$ corresponding to the endpoint $0$ of the half-line), so, by continuity, it must be the case that $\mu(H) = A^+$. Then Corollary \[CDSvsmu\] implies that $H$ is a Cartan-decomposition subgroup, but we provide the following direct proof that avoids any appeal to Proposition \[bddchange\].
From the definition of $\mu$, we have $KHK = K \, \mu(H) \, K$. Therefore $$KHK = K \, \mu(H) \, K = K A^+ K = G ,$$ so $H$ is a Cartan-decomposition subgroup (by taking $C = K$ in Definition \[simdefn\]).
By using Lemma \[proper<>CHC\], the proof of Corollary \[simvsmu\] also establishes the following.
\[proper<>mu(L)\] Suppose $H$ and $L$ are closed subgroups of $G$. The subgroup $L$ acts properly on $G/H$ if and only if $\mu(L) \cap \mu(H) C$ is compact, for every compact subset $C$ of $A$.
Disconnected groups {#disconnected}
-------------------
As was mentioned in Remark \[Hdisconnected\], we may assume, without loss of generality, that $H$ is connected. However, it may not be possible to assume that $G$ is connected, because, although there are no known examples, it is possible that the following question has an affirmative answer.
\[assumeGconn?\] Does there exist a homogeneous space $G/H$ (satisfying Assumption \[standing\]), such that $G^\circ/H^\circ$ has a tessellation, but $G/H$ does not have a tessellation?
If $\Gamma$ is a crystallographic group for $G^\circ/H^\circ$, then it is easy to see that $\Gamma \backslash G/H$ is compact. However, the following example shows that $\Gamma$ may not act properly discontinuously on $G/H$.
Let
- $L = H = {\operatorname{SL}}(2,{\mathord{\mathbb{R}}})$,
- $\sigma$ be the automorphism of $L \times H$ that interchanges the two factors (that is, $\sigma(x,y) = (y,x)$),
- $G = (L \times H) \rtimes \langle \sigma \rangle $ (semidirect product), and
- $\Gamma$ be a cocompact lattice in $L$ [[(]{}cf. [\[classical\][[(]{}\[classical-Borel\][)]{}]{}]{}[)]{}]{}.
Then $H = H^\circ$, and $\Gamma$ is a crystallographic group for $G^\circ/H
= (L \times H)/H$ (see Example \[G=LxH\]).
However, $\Gamma \subset L = \sigma^{-1} H \sigma$, so $\Gamma$ does not act properly on $G/H$ [[(]{}see \[proper<>CHC\] [with $C = \{\sigma, \sigma^{-1}\}$]{}[)]{}]{}.
Even so, $G/H$ does have a tessellation, because the diagonal embedding $$\Delta(\Gamma) = \{\, (\gamma,\gamma) \in L \times H \mid \gamma \in
\Gamma \,\}$$ is a crystallographic group for $G/H$. Thus, this example does not provide an answer to Question \[assumeGconn?\].
In this example, $\sigma$ represents an element of the Weyl group of $G$ that does not belong to the Weyl group of $G^\circ$. The following proposition shows that this is a crucial ingredient in the construction.
Let $\Gamma$ be a crystallographic group for $G^\circ/H^\circ$.
If the [(]{}real[)]{} Weyl group of $G$ is same as the [(]{}real[)]{} Weyl group of $G^\circ$, then $\Gamma$ is a crystallographic group for $G/H$.
By assumption, we may choose the same fundamental domain $A^+$ for the Weyl groups of $G$ and $G^\circ$. Let $\mu \colon G \to A^+$ and $\mu^\circ \colon
G^\circ \to A^+$ be the Cartan projections; then $\mu^\circ$ is the restriction of $\mu$ to $G^\circ$. For simplicity, assume, without loss of generality, that $H \subset G^\circ$ (for example, assume $H$ is connected). Then, for any compact subset $C$ of $A$, we have $$\mu(\Gamma) \cap \mu(H) C = \mu^\circ(\Gamma) \cap \mu^\circ(H) C$$ is finite [[(]{}see \[proper<>mu(L)\][)]{}]{}. Thus, $\Gamma$ acts properly discontinuously on $G/H$ [[(]{}see \[proper<>mu(L)\][)]{}]{}, as desired.
A. Borel and J. Tits [@BorelTits-Reductive Cor. 14.6, p. 147] proved that if $G$ is Zariski connected, then every element of the Weyl group of $G$ has a representative in $G^\circ$. Also, any element of the Weyl group must act as an automorphism of the root system. Thus, we have the following corollary.
Let $\Gamma$ be a crystallographic group for $G^\circ/H^\circ$.
If either
- $G$ is Zariski connected, or
- every automorphism of the real root system of $G^\circ$ belongs to the Weyl group of the root system,
then $\Gamma$ is a crystallographic group for $G/H$.
1. If $G = {\operatorname{SO}}(2,n)$, then $G$ is Zariski connected (because ${\operatorname{SO}}(n+2,{\mathord{\mathbb{C}}})$ is connected [@GoodmanWallach Thm. 2.1.9, p. 60]), so $G/H$ has a tessellation if and only if $G^\circ/H^\circ$ has a tessellation.
2. More generally, if $G^\circ = {\operatorname{SO}}(2,n)^\circ$ or ${\operatorname{SU}}(2,n)$ (with $n
\ge 3$), then every automorphism of the real root system of $G^\circ$ belongs to the Weyl group of the root system (cf. Figure \[rootspict\]), so $G/H$ has a tessellation if and only if $G^\circ/H^\circ$ has a tessellation.
If $G = {\operatorname{SL}}(3,{\mathord{\mathbb{R}}}) \rtimes \langle \sigma \rangle$, where $\sigma$ is the Cartan involution of ${\operatorname{SL}}(3,{\mathord{\mathbb{R}}})$, then $\sigma$ represents an element of the Weyl group of $G$ that does not belong to the Weyl group of $G^\circ$, so the proposition does not apply to $G$. However, this does not matter: if neither $H$ nor $G/H$ is compact, then Theorem \[SL3->notess\] implies that $G^\circ/H^\circ$ has no tessellations, so $G/H$ has no tessellations either.
Preliminaries on subgroups of $AN$ {#ANSect}
==================================
This section recalls a technical result that often allows us to assume that $H$ is a subgroup of $AN$. It also recalls some basic topological properties of such subgroups, and also recalls a simple observation relating these subgroups to the root spaces of the Lie algebra ${\mathfrak{\lowercase{G}}}$.
Reduction to subgroups of $AN$
------------------------------
An element $g$ of $G$ is:
- *hyperbolic* if $g$ is conjugate to an element of $A$;
- *unipotent* if $g$ is conjugate to an element of $N$;
- *elliptic* if $g$ is conjugate to an element of $K$.
\[JordanDecomp\] Each $g \in G$ has a unique decomposition in the form $g = auc$, such that
- $a$ is hyperbolic, $u$ is unipotent, and $c$ is elliptic; and
- $a$, $u$, and $c$ all commute with each other.
\[JordanCommute\] If $g = auc$ is the Real Jordan Decomposition of some element $g$ of $G$, then $a$, $u$, and $c$ commute, not only with each other, but also with any element of $G$ that commutes with $g$. This is because the Real Jordan Decomposition of $h^{-1} g h$ is $$h^{-1} g h = (h^{-1} a h) (h^{-1} u h) (h^{-1} c h) :$$ if $h^{-1} g h = g$, then the uniqueness of the Real Jordan Decomposition of $g$ implies $h^{-1} a h = a$, $h^{-1} u h = u$, and $h^{-1} g h = c$.
The following observation is a generalization of the fact that a collection of commuting triangularizable matrices can be simultaneously triangularized.
\[simultaneous\] If $H$ is abelian [(]{}or, more generally, solvable[)]{}, and is generated by hyperbolic and/or unipotent elements, then $H$ is conjugate to a subgroup of $AN$.
Because of the following result, we usually assume $H \subset AN$ (by replacing $H$ with a conjugate of $H'$).
\[HcanbeAN\] If $H$ is connected, then there is a closed, connected subgroup $H'$ of $G$ and a compact, connected subgroup $C$ of $G$, such that
1. \[HcanbeAN-inAN\] $H'$ is conjugate to a subgroup of $AN$;
2. \[HcanbeAN-CH=CH’\] $C H = C H'$ is a subgroup of $G$; and
3. \[HcanbeAN-d(H’)\] $d(H') = d(H)$ [(]{}see Notation \[d(H)-defn\][)]{}.
Moreover, it is easy to see from [[(]{}\[HcanbeAN-CH=CH’\][)]{}]{} that the homogeneous space $G/H$ has a tessellation if and only if $G/H'$ has a tessellation.
First, let us note that every connected subgroup of $AN$ is closed (see [\[solvable\][[(]{}\[solvable-H=Rn\][)]{}]{}]{} and \[ANsc\]), so we do not need to show that $H'$ is closed.
Second, let us note that [[(]{}\[HcanbeAN-d(H’)\][)]{}]{} is a consequence of [[(]{}\[HcanbeAN-inAN\][)]{}]{} and [[(]{}\[HcanbeAN-CH=CH’\][)]{}]{}. To see this, let $K^*$ be a maximal compact subgroup of $CH$ that contains $C$. Then a standard argument shows that $K^* \cap H$ is a maximal compact subgroup of $H$. (Because all maximal compact subgroups of $CH$ are conjugate, there is some $g \in CH$, such that $(g^{-1} K^* g) \cap H$ is a maximal compact subgroup of $H$ that contains $K^* \cap H$. Since $C \subset K^*$, we know that $C$ normalizes $K^*$, so we may assume $g \in H$; thus, $g$ normalizes $H$. Then $g^{-1} (K^* \cap H) g = (g^{-1} K^* g) \cap H$ contains $K^* \cap H$. Because $K^* \cap H$ is compact, this implies that $g$ normalizes $K^* \cap H$. So $K^* \cap H = (g^{-1} K^* g) \cap H$ is a maximal compact subgroup of $H$.) Therefore $$\dim (K^*H/K^*) = \dim \bigl( H/ (K^* \cap H) \bigr)
= \dim H - \dim(K^* \cap H)
= d(H) .$$ Similarly, $ \dim (K^*H'/K) = d(H')$. Since $K^*H = CH = CH' = K^*H'$, we conclude that $d(H') = d(H)$, as desired.
\[HcanbeAN-ss\] Assume $H$ is semisimple.
We have an Iwasawa decomposition $H = K_H A_H N_H$; let $H' = A_H N_H$ and $C = K_H$.
Assume $H = \{h^t\}$ is a one-parameter subgroup.
Let
- $h^t = a^t u^t c^t$ be the Real Jordan Decomposition of $h^t$ [[(]{}see \[JordanDecomp\][)]{}]{};
- $H' = \{ a^t u^t\}$; and
- $C = \overline{\{c^t\}}$ be the closure of $\{c^t\}$.
(Lemma \[simultaneous\] implies that $H'$ is conjugate to a subgroup of $AN$.)
\[HcanbeAN-abel\] Assume $H$ is abelian.
We may write $H$ as a product of one-parameter subgroups: $$H = \{\, h_1^{t_1} h_2^{t_2} \cdots h_r^{t_r} \mid t_1,\ldots,t_r \in
{\mathord{\mathbb{R}}}\,\} .$$ Let $h_j^t = a_j^t u_j^t c_j^t$ be the Real Jordan Decomposition of $h_j^t$ [[(]{}see \[JordanDecomp\][)]{}]{}. Note that $a_j^{t_j}$, $u_j^{t_j}$, and $c_j^{t_j}$ commute, not only with each other, but also with every $a_k^{t_k}$, $u_k^{t_k}$, and $c_k^{t_k}$ [[(]{}see \[JordanCommute\][)]{}]{}. Let $$H' = \{\, (a_1^{t_1} u_1^{t_1}) (a_2^{t_2} u_2^{t_2}) \cdots (a_k^{t_r}
u_k^{t_r}) \mid t_1,\ldots,t_r \in {\mathord{\mathbb{R}}}\,\} ,$$ and let $C = \overline{\{c_1^t\}} \cdots \overline{\{c_1^t\}}$. (Lemma \[simultaneous\] implies that $H'$ is conjugate to a subgroup of $AN$.)
The general case.
From the Levi decomposition [@JacobsonBook p. 91], we know that there is a connected, semisimple subgroup $L$ of $H$ and a connected, solvable, normal subgroup $R$ of $H$, such that $H = LR$ (and $L \cap R$ is finite). Let $U = [H,R]$, so $U$ is a connected, normal subgroup of $H$, and $U$ is conjugate to a subgroup of $N$ (cf. [@JacobsonBook Cor. 2.7.1, p. 51]). By modding out $U$, we (essentially) reduce to the direct product of Cases \[HcanbeAN-ss\] and \[HcanbeAN-abel\].
For $H$ and $H'$ as in Lemma \[HcanbeAN\], Proposition \[AN/H=Rd\] (and \[ANsc\]) implies that if $H' \neq AN$, then $AN/H'$ is not compact; also, Proposition [\[solvable\][[(]{}\[solvable-nocpct\][)]{}]{}]{} (and \[ANsc\]) implies that if $H' \neq e$, then $H'$ is not compact. Therefore:
- $H' = AN$ if and only if $G/H$ is compact; and
- $H' = e$ if and only if $H$ is compact.
Thus, if neither $H$ nor $G/H$ is compact, then $H'$ is a nontrivial, proper subgroup of $AN$.
Topology of solvable groups and their homogeneous spaces
--------------------------------------------------------
Everything is this subsection is well known, though somewhat scattered in the literature. The main results are Propositions \[solvable\] and \[AN/H=Rd\], which, together with Corollary \[ANsc\], show that connected subgroups of $AN$ and their homogeneous spaces are very well behaved topologically. Corollary \[fiberbundle\], on the homology of very simple quotient spaces, is also used in later sections.
We begin with the easy case of abelian groups. This lemma generalizes almost verbatim to solvable groups [[(]{}see \[solvable\][)]{}]{}, but the proof in that generality is not as trivial.
\[abelian\] Let $R$ be a $1$-connected, abelian Lie group.
1. \[abelian-H=Rn\] If $H$ is a connected subgroup of $R$, then $H$ is closed, simply connected, and isomorphic to ${\mathord{\mathbb{R}}}^k$, for some $k$.
2. \[abelian-HcapL\] If $H$ and $L$ are connected subgroups of $R$, then $H \cap L$ is connected.
3. \[abelian-nocpct\] If $C$ is a compact subgroup of $R$, then $C$ is trivial.
Because $R$ is abelian and 1-connected, the exponential map is a Lie group isomorphism from the additive group of the Lie algebra ${\mathfrak{\lowercase{R}}}$ onto $R$.
[[(]{}\[abelian-H=Rn\][)]{}]{} Let $k = \dim H$. Because the exponential map is a Lie group isomorphism (hence a diffeomorphism), and because ${\mathfrak{\lowercase{H}}}$ is a closed $k$-submanifold of ${\mathfrak{\lowercase{R}}}$, we know that $\exp({\mathfrak{\lowercase{H}}})$ is a closed $k$-submanifold of $R$. Of course, $\exp({\mathfrak{\lowercase{H}}})$ is contained in $H$, which is also a $k$-submanifold of $R$. Because the dimensions are the same, we know that $\exp({\mathfrak{\lowercase{H}}})$ is open in $H$. Also, because $\exp({\mathfrak{\lowercase{H}}})$ is closed in $R$, we know that $\exp({\mathfrak{\lowercase{H}}})$ is closed in $H$. Therefore $$\label{abelianPf-H=Rn-exp(H)}
\exp({\mathfrak{\lowercase{H}}}) = H$$ (because $H$ is connected). Finally, we know that $\exp|_{{\mathfrak{\lowercase{H}}}}$ is a diffeomorphism from its domain ${\mathfrak{\lowercase{H}}} {\simeq}{\mathord{\mathbb{R}}}^k$ onto its image $H$.
[[(]{}\[abelian-HcapL\][)]{}]{} From [[(]{}\[abelianPf-H=Rn-exp(H)\][)]{}]{}, we have $\exp({\mathfrak{\lowercase{H}}})
= H$ and, similarly, $\exp({\mathfrak{\lowercase{L}}}) = L$. Also, because $\exp$ is bijective, we have $\exp {\mathfrak{\lowercase{H}}} \cap \exp {\mathfrak{\lowercase{L}}} = \exp( {\mathfrak{\lowercase{H}}} \cap {\mathfrak{\lowercase{L}}} ) $. Therefore $$H \cap L = \exp {\mathfrak{\lowercase{H}}} \cap \exp {\mathfrak{\lowercase{L}}} = \exp( {\mathfrak{\lowercase{H}}} \cap {\mathfrak{\lowercase{L}}} )$$ is connected.
[[(]{}\[abelian-nocpct\][)]{}]{} Because ${\mathord{\mathbb{R}}}^k$ is not compact (for $k > 0$), we know, from [\[abelian\][[(]{}\[abelian-H=Rn\][)]{}]{}]{}, that $C^\circ$ is trivial; so $C$ is finite. Since $R {\cong}({\mathfrak{\lowercase{R}}}, +) {\cong}{\mathord{\mathbb{R}}}^d$ has no elements of finite order, we conclude that $C$ is trivial.
As is usual in the theory of solvable groups, the main results of this section are proved by induction, based on modding out some normal subgroup $L$. To be effective, this method requires an understanding of the quotient space $R/L$. The information we need (even if $L$ is not normal) comes from the following elementary observation, because $R$ is a principal $L$-bundle over $R/L$.
\[trivialbundle\] Let $P$ be a principal $H$-bundle over a manifold $M$.
1. \[trivialbundle-H=Rn\] If $H$ is diffeomorphic to ${\mathord{\mathbb{R}}}^n$, then
1. $P$ is $H$-equivariantly diffeomorphic to $M \times H$, so
2. $P$ is homotopy equivalent to $M$.
2. \[trivialbundle-M=Rn\] If $M$ is diffeomorphic to ${\mathord{\mathbb{R}}}^n$, then
1. $P$ is $H$-equivariantly diffeomorphic to $M \times H$, so
2. $P$ is homotopy equivalent to $H$.
Any principal bundle with a section is trivial [@Husemoller Cor. 4.8.3, p. 48]. If either the fiber or the base is contractible, then there is no obstruction to constructing a section [@Husemoller Thm. 2.7.1(H1), p. 21], so $P$ is trivial: $P {\simeq}M \times H$. (The diffeomorphism can be taken to be $H$-equivariant, with respect to the natural $H$-action on $M \times H$, given by $(m,h)h' = (m,h h')$.) Then the conclusions on homotopy equivalence follow from the fact that ${\mathord{\mathbb{R}}}^n$ is contractible (that is, homotopically trivial).
We recall the long exact sequence of the fibration $H \to R \to R/H$:
\[HtpyExactFibration\] Let $H$ be a closed subgroup of a Lie group $R$. There is a [(]{}natural[)]{} long exact sequence of homotopy groups: $$\cdots \to \pi_1(H) \to \pi_1(R) \to \pi_1(R/H) \to \pi_0(H) \to \pi_0(R)
\to \pi_0(R/H) \to 0 .$$
\[R/Hsc\] Let $H$ be a closed subgroup of a $1$-connected Lie group $R$. The homogeneous space $R/H$ is simply connected if and only if $H$ is connected.
Because $R$ is 1-connected, we have $\pi_1(R) = \pi_0(R) = 0$, so, from [[(]{}\[HtpyExactFibration\][)]{}]{}, we know that the sequence $$0 \to \pi_1(R/H) \to \pi_0(H) \to 0$$ is exact. Thus, $\pi_1(R/H) {\cong}\pi_0(H)$, so the desired conclusion is immediate.
As a step toward Proposition \[solvable\], we prove two special cases that describe the topology of normal subgroups.
\[R=Rd\] If $R$ is a $1$-connected, solvable Lie group, then $R$ is diffeomorphic to ${\mathord{\mathbb{R}}}^d$, for some $d$.
We may assume the group $R$ is nonabelian (otherwise, the desired conclusion is given by Lemma [\[abelian\][[(]{}\[abelian-H=Rn\][)]{}]{}]{}). Then, because $R$ is solvable, there is a nontrivial, connected, proper, closed, normal subgroup $L$ of $R$. Since $R/L$ is simply connected [[(]{}see \[R/Hsc\][)]{}]{}, and $\dim(R/L) < \dim R$, we may assume, by induction on $\dim R$, that $R/L$ is diffeomorphic to some ${\mathord{\mathbb{R}}}^{d_1}$. Therefore
1. \[R=RdPf-R=prod\] $R$ is diffeomorphic to $(R/L) \times L$ and
2. \[R=RdPf-L=R\] $L$ is homotopy equivalent to $R$
[[(]{}see [\[trivialbundle\][[(]{}\[trivialbundle-M=Rn\][)]{}]{}]{}[)]{}]{}. Because $R$ is $1$-connected, [[(]{}\[R=RdPf-L=R\][)]{}]{} implies that $L$ is $1$-connected; hence, $L$ is a 1-connected, solvable Lie group, so we may assume, by induction on $\dim R$, that $L$ is diffeomorphic to some ${\mathord{\mathbb{R}}}^{d_2}$. Thus, [[(]{}\[R=RdPf-R=prod\][)]{}]{} implies that $R$ is diffeomorphic to ${\mathord{\mathbb{R}}}^{d_1} \times
{\mathord{\mathbb{R}}}^{d_2} {\simeq}{\mathord{\mathbb{R}}}^{d_1+d_2}$, as desired.
\[Rnormal\] If $R$ is a $1$-connected, solvable Lie group, then every connected, closed, normal subgroup of $R$ is $1$-connected.
The following proposition is a nearly complete generalization of Lemma \[abelian\] to the class of solvable groups. There are two exceptions:
1. Of course, subgroups of a solvable group may not be abelian, so the conclusion in [\[abelian\][[(]{}\[abelian-H=Rn\][)]{}]{}]{} that $H$ is isomorphic to some ${\mathord{\mathbb{R}}}^k$ must be weakened to the conclusion that $H$ is diffeomorphic to some ${\mathord{\mathbb{R}}}^k$.
2. The intersection of connected subgroups is not always connected [[(]{}see \[HcapLdisconn\][)]{}]{}, so we add the restriction that $L$ is normal to [\[abelian\][[(]{}\[abelian-HcapL\][)]{}]{}]{}. (We remark that no such restriction is necessary if $R \subset AN$, because the exponential map is a diffeomorphism from ${\mathfrak{\lowercase{R}}}$ onto $R$ in this case [@Dixmier-exp; @Saito2].)
\[HcapLdisconn\] Let $$R = {\left\{\,
\begin{pmatrix}
e^{2\pi i t} & x + iy & 0 \\
0 & 1 & 0 \\
0 & 0 & e^t
\end{pmatrix}
\mathrel{\left| \vphantom {\left\{
\begin{pmatrix}
e^{2\pi i t} & x + iy & 0 \\
0 & 1 & 0 \\
0 & 0 & e^t
\end{pmatrix}
\mid
t, x, y \in {\mathord{\mathbb{R}}}\right\} } \right.}
t, x, y \in {\mathord{\mathbb{R}}}\,\right\} } ,
\qquad
g^t =
\begin{pmatrix}
e^{2\pi i t} & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & e^t
\end{pmatrix}
,$$ $$u =
\begin{pmatrix}
1 & 1 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0
\end{pmatrix}
, \qquad
h^t
= u^{-1} g^t u
=
\begin{pmatrix}
e^{2\pi i t} & e^{2\pi i t} - 1 & 0 \\
0 & 1 & 0 \\
0 & 0 & e^t
\end{pmatrix}
.$$ Then $R$, being diffeomorphic to ${\mathord{\mathbb{R}}}^3$, is 1-connected; and $\{g^t\}$ and $\{h^t\}$ are connected subgroups. But $$\{g^t\} \cap \{h^t\} =
{\left\{\,
\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & e^n
\end{pmatrix}
\mathrel{\left| \vphantom {\left\{
\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & e^n
\end{pmatrix}
\mid
n \in {\mathord{\mathbb{Z}}}\right\} } \right.}
n \in {\mathord{\mathbb{Z}}}\,\right\} }$$ is not connected.
\[solvable\] Let $R$ be a $1$-connected, solvable Lie group.
1. \[solvable-H=Rn\] If $H$ is a connected subgroup of $R$, then $H$ is closed, simply connected, and diffeomorphic to some ${\mathord{\mathbb{R}}}^d$.
2. \[solvable-HcapL\] If $H$ and $L$ are connected subgroups of $R$, and $L$ is normal, then $H \cap L$ is connected.
3. \[solvable-nocpct\] If $C$ is a compact subgroup of $R$, then $C$ is trivial.
[[(]{}\[solvable-HcapL\][)]{}]{} We may assume $L$ is nontrivial, so $\dim (R/L) < \dim R$. Thus, by induction on $\dim R$, using [[(]{}\[solvable-H=Rn\][)]{}]{}, we may assume that $HL/L$ is a closed, simply connected subgroup of $R/L$. Then, since $H/(H \cap L)$ is homeomorphic to $HL/L$, we see that $$\label{pi1(H/HcapL)}
\mbox{$H/(H \cap L)$ is simply connected} ,$$ so Lemma \[R/Hsc\] implies that $H \cap L$ is connected.
[[(]{}\[solvable-H=Rn\][)]{}]{} Because $R$ is solvable, there is a connected, closed, proper, normal subgroup $L$ of $R$, such that $R/L$ is abelian. We know that $L$ is 1-connected [[(]{}see \[Rnormal\][)]{}]{}, so, by induction on $\dim R$, we may assume that every connected subgroup of $L$ is closed and simply connected. From [[(]{}\[solvable-HcapL\][)]{}]{}, we know that $H \cap L$ is connected, so we conclude that $H \cap L$ is closed, and $$\label{pi1(HcapL)}
\pi_1(H \cap L) = 0 .$$ From [[(]{}\[HtpyExactFibration\][)]{}]{} (with $H$ in the place of $R$, and $L$ in the place of $H$), together with [[(]{}\[pi1(H/HcapL)\][)]{}]{} and [[(]{}\[pi1(HcapL)\][)]{}]{}, we conclude that $\pi_1(H) = 0$; that is, $H$ is simply connected. So [[(]{}\[R=Rd\][)]{}]{} implies $H$ is diffeomorphic to some ${\mathord{\mathbb{R}}}^d$.
Because both $HL/L$ and $H \cap L$ are closed, it is not difficult to see that $H$ is closed.
[[(]{}\[solvable-nocpct\][)]{}]{} Because $R$ is solvable, there is a connected, closed, proper, normal subgroup $L$ of $R$, such that $R/L$ is abelian. We know that $R/L$ is 1-connected [[(]{}see \[R/Hsc\][)]{}]{}, so $R/L$ has no nontrivial, compact subgroups [[(]{}see [\[abelian\][[(]{}\[abelian-nocpct\][)]{}]{}]{}[)]{}]{}; thus, we must have $C \subset L$. Therefore, $C$ is a compact subgroup of $L$. Then, since $L$ is 1-connected [[(]{}see \[Rnormal\][)]{}]{}, we may conclude, by induction on $\dim R$, that $C$ is trivial.
\[ANsc\] $AN$ is a $1$-connected, solvable Lie group.
Because $G$ is linear, it is a subgroup of some ${\operatorname{GL}}(n,{\mathord{\mathbb{R}}})$. Replacing $G$ by a conjugate, we may assume that $AN$ is contained in the group $B$ of upper triangular matrices with positive diagonal entries [(cf. \[simultaneous\])]{}. The matrix entries provide an obvious diffeomorphism from $B$ onto $({\mathord{\mathbb{R}}}^+)^n \times {\mathord{\mathbb{R}}}^{n(n-1)/2} {\simeq}{\mathord{\mathbb{R}}}^{n(n+1)/2}$, so $B$ is 1-connected. Thus, Proposition [\[solvable\][[(]{}\[solvable-H=Rn\][)]{}]{}]{} implies that $AN$ is simply connected.
The following observation will be used in Sections \[dimHSect\] and \[ExistenceSect\].
\[fiberbundle\] Let $F$ be a connected subgroup of $AN$, and suppose we have a proper, $C^\infty$ action of $F$ on a manifold $M$. Then $M$ and $M/F$ have the same homology.
Because the action is proper, we know that the stabilizer of each point of $M$ is compact. However, $F$ has no nontrivial compact subgroups [[(]{}see [\[solvable\][[(]{}\[solvable-nocpct\][)]{}]{}]{}[)]{}]{}. Thus, the action is free.
Because the action is free, proper, and $C^\infty$, it is easy to see that the manifold $M$ is a principal fiber bundle over the quotient $M/F$ [@Palais-Slice Thm. 1.1.3]. Furthermore, the fiber $F$ of the bundle is contractible [[(]{}see [\[solvable\][[(]{}\[solvable-H=Rn\][)]{}]{}]{}[)]{}]{}, so Lemma [\[trivialbundle\][[(]{}\[trivialbundle-H=Rn\][)]{}]{}]{} implies that $M$ homotopy equivalent to $M/F$. Therefore, the spaces $M$ and $M/F$ have the same homology.
For the special case where $M/F$ is a homogeneous space of a solvable group, the following more detailed result describes the topology of $M/F$, not just its homology.
\[AN/H=Rd\] If $H$ is any connected subgroup of a $1$-connected, solvable Lie group $R$, then $R/H$ is diffeomorphic to the Euclidean space ${\mathord{\mathbb{R}}}^d$, for some $d$.
Because $R$ is solvable, it has a nontrivial, connected, closed, abelian, normal subgroup $L$. Since $L$ is abelian and $H \cap L$ is connected [[(]{}see [\[solvable\][[(]{}\[solvable-HcapL\][)]{}]{}]{}[)]{}]{}, we know that $L/(H \cap L)$ is a 1-connected abelian group [[(]{}see \[R/Hsc\][)]{}]{}, so it is isomorphic to some ${\mathord{\mathbb{R}}}^{d_1}$ [[(]{}see [\[abelian\][[(]{}\[abelian-H=Rn\][)]{}]{}]{}[)]{}]{}.
We know $H$ is closed [[(]{}see [\[solvable\][[(]{}\[solvable-H=Rn\][)]{}]{}]{}[)]{}]{}. Also, since $L$ is nontrivial, we have $\dim (R/L) < \dim R$, so we may assume, by induction on $\dim R$, that $$R/(HL) {\simeq}(R/L) / (HL/L)$$ is diffeomorphic to some ${\mathord{\mathbb{R}}}^{d_2}$.
Now $R$ is a principal $HL$-bundle over $R/(HL)$. Because $R/(HL) {\simeq}{\mathord{\mathbb{R}}}^{d_2}$, this bundle is trivial [[(]{}see [\[trivialbundle\][[(]{}\[trivialbundle-M=Rn\][)]{}]{}]{}[)]{}]{}: $R$ is $HL$-equivariantly diffeomorphic to $R/(HL) \times HL$. Then $$R/H
{\simeq}R/(HL) \times HL/H
{\simeq}R/(HL) \times L/(H \cap L)
{\simeq}{\mathord{\mathbb{R}}}^{d_2} \times {\mathord{\mathbb{R}}}^{d_1}
= {\mathord{\mathbb{R}}}^{d_1+d_2} ,$$ as desired.
$T$-invariant subspaces of ${\mathfrak{\lowercase{A}}} + {\mathfrak{\lowercase{N}}}$
------------------------------------------------------------------------------------
The following well-known observation puts an important restriction on the subspaces of ${\mathfrak{\lowercase{A}}} + {\mathfrak{\lowercase{N}}}$ that are normalized by a torus. It is an ingredient in our case-by-case analysis of all possible subgroups of $AN$ in Sections \[SUFlargeSect\] and \[ProofSect\].
\[rootdecomp\] Let
- $\Phi^+$ be the set of weights of $A$ on ${\mathfrak{\lowercase{N}}}$ [(]{}in other words, the set of all positive real roots of $G$[)]{};
- $T$ be a subgroup of $A$;
- $\omega \in \Phi^+ \cup \{0\}$;
- ${\mathfrak{\lowercase{N}}}^{=\omega} = \bigoplus_{\sigma|_T = \omega|_T} {{\mathfrak{\lowercase{N}}}}_\sigma$, where the sum is over all $\sigma \in \Phi^+ \cup \{0\}$, such that the restriction of $\sigma$ to $T$ is the same as the restriction of $\omega$ to $T$;
- ${\mathfrak{\lowercase{N}}}^{\neq\omega} = \bigoplus_{\sigma|_T \neq \omega|_T}
{{\mathfrak{\lowercase{N}}}}_\sigma$, where the sum is over all $\sigma \in \Phi^+ \cup \{0\}$, such that the restriction of $\sigma$ to $T$ is not the same as the restriction of $\omega$ to $T$.
If ${\mathfrak{\lowercase{U}}}$ is any ${\mathord{\mathbb{R}}}$-subspace of ${\mathfrak{\lowercase{A}}} + {\mathfrak{\lowercase{N}}}$ normalized by $T$, then ${\mathfrak{\lowercase{U}}} = ({\mathfrak{\lowercase{U}}} \cap {\mathfrak{\lowercase{N}}}^{=\omega}) \oplus ({\mathfrak{\lowercase{U}}} \cap {\mathfrak{\lowercase{N}}}^{\neq\omega})$.
Since $T \subset A$, we know that the elements of ${\operatorname{Ad}\nolimits}_G T$ are simultaneously diagonalizable (over ${\mathord{\mathbb{R}}}$), so their restrictions to the invariant subspace ${\mathfrak{\lowercase{U}}}$ are also simultaneously diagonalizable (cf. [@ZariskiSamuel1 Thms. 26 and 27 in §3.12, pp. 167–168]). Thus, ${\mathfrak{\lowercase{U}}}$ is a direct sum of weight spaces: $${\mathfrak{\lowercase{U}}} = \bigoplus_{\psi \in \Psi} {\mathfrak{\lowercase{U}}}_\psi .$$ For each weight $\psi$ of $T$ on ${\mathfrak{\lowercase{U}}}$, we have $${\mathfrak{\lowercase{U}}}_\psi
= {\mathfrak{\lowercase{U}}} \cap {{\mathfrak{\lowercase{N}}}}_\psi
= {\mathfrak{\lowercase{U}}} \cap {{\mathfrak{\lowercase{N}}}}^{= \psi} ,$$ so $${\mathfrak{\lowercase{U}}}_{\omega|_T} = {\mathfrak{\lowercase{U}}} \cap {\mathfrak{\lowercase{N}}}^{=\omega}$$ and $$\bigoplus_{\psi \neq \omega|_T} {\mathfrak{\lowercase{U}}}_\psi
= {\mathfrak{\lowercase{U}}} \cap {\mathfrak{\lowercase{N}}}^{\neq\omega} .$$ The conclusion follows.
Lower bound on the dimension of $H$ {#dimHSect}
===================================
In this section, we prove Corollary \[tess->dim>1,2\], an *a priori* lower bound on $\dim H$. On the way, we recall a result of T. Kobayashi that will also be used several times in later sections, and we establish that crystallographic groups have only one end.
T. Kobayashi’s Dimension Theorem
--------------------------------
The following theorem is essentially due to T. Kobayashi. (Kobayashi assumed that $H$ is reductive, but H. Oh and D. Witte [@OhWitte-CK Thm. 3.4] pointed out that, by using Lemma \[HcanbeAN\], this restriction can be eliminated.) The proof here is based on Kobayashi’s original argument and the modifications of Oh-Witte, but uses less sophisticated topology. Namely, instead of group cohomology and the spectral sequence of a covering space, we use only some basic properties of homology groups of manifolds (including Lemma \[fiberbundle\]). These comments also apply to Theorem \[construct-tess\].
\[noncpctdim\] Let $H$ and $H_1$ be closed, connected subgroups of $G$, and assume there is a crystallographis group $\Gamma$ for $G/H$, such that $\Gamma$ acts properly discontinuously on $G/H_1$. Then:
1. \[noncpctdim-notess\] We have $d(H_1) \le d(H)$.
2. \[noncpctdim-tess\] If $d(H_1) \ge d(H)$, then $\Gamma \backslash G/H_1$ is compact, so $G/H_1$ has a tessellation.
By Lemma \[HcanbeAN\], we may assume $H, H_1 \subset AN$. (So $d(H)
= \dim H$ and $d(H_1) = \dim H_1$ [[(]{}see \[d(H)=dimH\][)]{}]{}.)
From Lemma \[fiberbundle\], we know that $\Gamma \backslash G$ and $\Gamma
\backslash G/H_1$ have the same homology. Therefore $$\max\{\, k \mid {\mathord{\mathcal{H}}}_k(\Gamma \backslash G) \neq 0 \,\}
= \max\{\, k \mid {\mathord{\mathcal{H}}}_k(\Gamma \backslash G/H_1) \neq 0 \,\}
\le \dim G/H_1 ,$$ with equality if and only if $\Gamma \backslash G / H_1$ is compact [@Dold Cor. 8.3.4, p. 260]. Similarly, we have $$\max\{\, k \mid {\mathord{\mathcal{H}}}_k(\Gamma \backslash G) \neq 0 \,\}
= \dim G/H .$$ Combining these two statements, we conclude [[(]{}\[noncpctdim-notess\][)]{}]{} that $\dim G/H \le \dim G/H_1$ and, furthermore, [[(]{}\[noncpctdim-tess\][)]{}]{} that equality holds if and only if $\Gamma \backslash G/H_1$ is compact.
\[noncpct-dim-notess\] Let $H$ and $H_1$ be closed, connected subgroups of $G$, such that $d(H_1) >
d(H)$. If there is a compact subset $C$ of $A$, such that $\mu(H_1) \subset
\mu(H) C$, then $G/H$ does not have a tessellation.
Suppose $\Gamma$ is a crystallographic group for $G/H$. (This will lead to a contradiction.) Because $\Gamma$ acts properly discontinuously on $G/H$, the assumption on $\mu(H_1)$ implies that $\Gamma$ also acts properly discontinuously on $G/H_1$ [(cf. \[proper<>mu(L)\])]{}. So Theorem [\[noncpctdim\][[(]{}\[noncpctdim-notess\][)]{}]{}]{} yields a contradiction.
Crystallographic groups have only one end
-----------------------------------------
It is easy to see that crystallographic groups are finitely generated; we now show that they have only one end [[(]{}see \[tess->1end\][)]{}]{}.
\[1endDefn\] Let $F$ be a finite generating set for an (infinite) group $\Gamma$. We say that $\Gamma$ has *only one end* if, for every partition $\Gamma
= A_1 \cup A_2 \cup C$ of $\Gamma$ into three disjoint sets $A_1$, $A_2$, and $C$, such that $A_1$ and $A_2$ are infinite, but $C$ is finite, there exists $\gamma \in A_1$ and $f \in F \cup F^{-1}$, such that $\gamma f
\in A_2$. (This does not depend on the choice of the generating set $F$.)
The following observation is a straightforward reformulation of Definition \[1endDefn\] (obtained by letting $A_2 = \Gamma \smallsetminus
(A_1 \cup C')$ and $C = C' \smallsetminus A_1$).
\[1endcomplement\] Let $F$ be a finite generating set for an infinite group $\Gamma$. If $A_1$ and $C'$ are subsets of $\Gamma$, such that
- $A_1$ is infinite,
- $C'$ is finite, and
- $A_1 f \in A_1 \cup C'$, for every $f \in F \cup F^{-1}$,
then the complement $\Gamma \smallsetminus A_1$ is finite.
Definition \[1endDefn\] is often stated in the language of Cayley graphs: The *Cayley graph* of $\Gamma$, with respect to the generating set $F$, is the graph $\operatorname{Cay}(\Gamma;F)$ whose vertex set $V$ and edge set $E$ are given by: $$\begin{aligned}
V &= \Gamma; \\
E &= \{\, (\gamma, \gamma f) \mid \gamma \in \Gamma, f \in F \cup F^{-1} \,\}
.
\end{aligned}$$ The group $\Gamma$ has only one end if and only if, for every finite subset $C$ of $\Gamma$, the graph $\operatorname{Cay}(\Gamma; F) \smallsetminus C$ has only one infinite component.
The following lemma is not difficult, but, unfortunately, we do not have a proof that is both short and elementary.
\[HN=AN\] If $HN=AN$, then, for some $x \in N$, the conjugate $x^{-1} H x$ is normalized by $A$.
\[codim>1\] If $d(G) - d(H) \le 1$, and $G/H$ is not compact, then $G/H$ does not have a tessellation.
It suffices to show that $H$ is a Cartan-decomposition subgroup of $G$ [[(]{}see \[CDS->notess\][)]{}]{}.
We may assume, without loss of generality, that $H \subset AN$ [[(]{}see \[HcanbeAN\][)]{}]{}; then $$\dim H + 1 = d(H) + 1 \ge d(G) = \dim(AN)$$ [[[(]{}see \[d(H)=dimH\] [and \[d(G)\]]{}[)]{}]{}]{}. A theorem of B. Kostant [@Kostant Thm. 5.1] implies that $N$ is a Cartan-decomposition subgroup, so we may assume $N \not\subset H$; then $\dim(H \cap N) \le \dim N - 1$. Therefore $$\dim(HN) = \dim H + \dim N - \dim(H \cap N)
\ge \dim H + 1
\ge \dim(AN) .$$ Hence $HN = AN$, so, from Lemma \[HN=AN\], we see that, after replacing $H$ by a conjugate subgroup, we may assume that $H$ is normalized by $A$. Then, letting $\omega = 0$ and $T = A$ in [[(]{}\[rootdecomp\][)]{}]{}, we see that ${\mathfrak{\lowercase{H}}} = ({\mathfrak{\lowercase{H}}} \cap {\mathfrak{\lowercase{A}}}) + ({\mathfrak{\lowercase{H}}} \cap {\mathfrak{\lowercase{N}}})$. Since $HN = AN$, we have ${\mathfrak{\lowercase{H}}} + {\mathfrak{\lowercase{N}}} = {\mathfrak{\lowercase{A}}} + {\mathfrak{\lowercase{N}}}$, so this implies that ${\mathfrak{\lowercase{A}}}
\subset {\mathfrak{\lowercase{H}}}$; therefore $H$ contains $A$. Since $A$ is a Cartan-decomposition subgroup [[(]{}see \[AisCDS\][)]{}]{}, this implies $H$ is a Cartan-decomposition subgroup, as desired.
A topological space $M$ is *connected at $\infty$* if every compact subset $\mathcal{C}$ is contained in a compact subset $\mathcal{C}'$, such that the complement $M \smallsetminus \mathcal{C}'$ is connected.
\[tess->1end\] If $\Gamma$ is a crystallographic group for $G/H$, then $\Gamma$ is finitely generated and has only one end.
Assume, without loss of generality, that $H \subset AN$ [[(]{}see \[HcanbeAN\][)]{}]{}. Then $H$ is torsion free, so $\Gamma$ must act freely on $G/H$; therefore $\Gamma \backslash G/H$ is a compact manifold (rather than an orbifold). Because $\Gamma$ is essentially the fundamental group of $\Gamma \backslash
G/H$ (specifically, $\Gamma {\cong}\pi_1(\Gamma \backslash G/H)/\pi_1(G/H)$), and the fundamental group of any compact manifold is finitely generated [@Raghunathan Thm. 6.16, p. 95], we know that $\Gamma$ is finitely generated.
From the Iwasawa decomposition $G = KAN$, we see that $G/H$ is homeomorphic to $K \times (AN/H)$, and Proposition \[AN/H=Rd\] asserts that $AN/H$ is homeomorphic to ${\mathord{\mathbb{R}}}^d$, for some $d$. Obviously, we must have $d = \dim
(AN) - \dim H$, and we may assume $G/H$ is not compact (otherwise, $\Gamma$ is finite, so the desired conclusion is obvious), so Corollary \[codim>1\] implies that $d > 1$. Thus, we conclude that $G/H$ is connected at $\infty$.
To complete the proof, we use a standard argument (cf. [@Gromov-asymptotic $0.2.C_1$, p. 5]) to show that, because $G/H$ is connected at $\infty$ and $\Gamma \backslash G/H$ is compact, the group $\Gamma$ has only one end. To begin, note that there is a compact subset $\mathcal{C}$ of $G/H$, such that $\Gamma \mathcal{C} = G/H$. Let $$F_0 = \{\, f \in \Gamma \mid \mathcal{C} \cap f \mathcal{C} \neq \emptyset
\,\}$$ (cf. [@PlatonovRapinchuk (ii), p. 195]). Because $\Gamma$ acts properly discontinuously on $G/H$, we know that $F_0$ is finite; let $F$ be a finite generating set for $\Gamma$, such that $F_0
\subset F$.
Suppose $\Gamma = A_1 \cup A_2 \cup C$, with $|A_1| = |A_2| = \infty$ and $|C| < |\infty|$. (We wish to show there exist $\gamma \in A_1$ and $f
\in F$, such that $\gamma f \in A_2$; this establishes that $\Gamma$ has only one end.) Because $G/H$ is connected at $\infty$, there is a compact subset $\mathcal{C}'$ of $G/H$, containing $C \mathcal{C}$, such that $(G/H)
\smallsetminus \mathcal{C}'$ is connected. Because $C \mathcal{C} \subset
\mathcal{C}'$, we have $$(G/H) \smallsetminus \mathcal{C}' = (\Gamma \mathcal{C}) \smallsetminus
\mathcal{C}' \subset A_1 \mathcal{C} \cup A_2 \mathcal{C} .$$ Because $\Gamma$ acts properly discontinuously on $G/H$, we know $A_1
\mathcal{C}$ and $A_2 \mathcal{C}$ are closed (and neither is contained in $\mathcal{C}'$), so connectivity implies that $A_1 \mathcal{C} \cap A_2 \mathcal{C} \neq \emptyset$: there exist $\gamma \in A_1$ and $\gamma' \in A_2$, such that $\gamma \mathcal{C} \cap \gamma' \mathcal{C} \neq \emptyset$. Let $f = \gamma^{-1} \gamma'$; then $\gamma \in A_1$, $\gamma f = \gamma' \in A_2$, and $$\mathcal{C} \cap f \mathcal{C} = \gamma^{-1} (\gamma \mathcal{C} \cap
\gamma' \mathcal{C}) \neq \emptyset ,$$ so $f \in F_0 \subset F$, as desired.
Walls of $A^+$ and a lower bound on $d(H)$
------------------------------------------
\[tess->misswall\] Assume ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G = 2$. Let
- $L_1$ and $L_2$ be the two walls of $A^+$, and
- $\Gamma$ be a crystallographic group for $G/H$.
If $H$ is not compact, then there exists $k \in \{1,2\}$, such that, for every compact subset $C$ of $A$, the intersection $\mu(\Gamma) \cap L_k C$ is finite.
![Proposition \[tess->misswall\]: (a) $\mu(\Gamma)$ cannot be on both sides of $\mu(H)$, because $\Gamma$ has only one end. (b) Therefore, $\mu(\Gamma)$ stays away from $L_k$.[]{data-label="misswallfig"}](two-ends.eps)
Suppose there is a compact subset $C$ of $A$, such that each of $\mu(\Gamma)
\cap L_1 C$ and $\mu(\Gamma) \cap L_2 C$ is infinite. (This will lead to a contradiction.) Let $F$ be a (symmetric) finite generating set for $\Gamma$ [[(]{}see \[tess->1end\][)]{}]{}. We may assume $C$ is so large that $\mu(\gamma F) \subset \mu(\gamma) C$ for every $\gamma \in \Gamma$ [[(]{}see \[bddchange\][)]{}]{}. We may also assume that $C$ is convex and symmetric.
Because $\Gamma$ acts properly on $G/H$, there is a compact subset $\mathcal{C}$ of $A$, such that $\mu(H) C \cap \mu(\Gamma) \subset
\mathcal{C}$ [[(]{}see \[proper<>mu(L)\][)]{}]{}. Furthermore, we may assume that $\mu(L_1)C
\cap \mu(L_2) C \subset \mathcal{C}$.
Let
- $M = \cup_{\gamma \in \Gamma} \mu(\gamma) C \smallsetminus \mathcal{C}$,
- $M_1$ be the union of all the connected components of $M$ that contain a point of $L_1$, and
- $A_1 = \Gamma \cap \mu^{-1}(M_1)$.
Then $A_1$ is infinite (because $\mu(\Gamma) \cap L_1 C$ is infinite). Also, for any $\gamma\in A_1$ and $f \in F$, we have $\mu(\gamma f) \in
\mu(\gamma) C$, so $\gamma f \in A_1 \cup \mu^{-1}(\mathcal{C})$. Since $\Gamma$ has only one end [[(]{}see \[tess->1end\][)]{}]{}, this implies $\Gamma
\smallsetminus A_1$ is finite [[(]{}see \[1endcomplement\][)]{}]{}. Because $\mu(\Gamma) \cap
L_2 C$ is infinite, we conclude that $M_1 \cap L_2 \neq \emptyset$.
Because $\mu(H)$ separates $L_1$ from $L_2$, and every connected component of $M_1$ contains a point of $L_1$, we conclude that $\mu(H) \cap
M_1 \neq \emptyset$. This contradicts the fact that $\mu(H) C \cap \mu(\Gamma)
\subset \mathcal{C}$.
\[tess->missHk\] Assume ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G = 2$. Let
1. $L_1$ and $L_2$ be the two walls of $A^+$; and
2. $H_1$ and $H_2$ be closed, connected, nontrivial subgroups of $G$,
such that $$\mu(H_k) \approx L_k$$ for $k = 1,2$. If $H$ is not compact, then any crystallographic group for $G/H$ acts properly discontinuously on either $G/H_1$ or $G/H_2$.
Suppose $\Gamma$ acts properly discontinuously on **neither** $G/H_1$ **nor** $G/H_2$. (This will lead to a contradiction.) From Proposition \[proper<>mu(L)\], we know there is a compact subset $C$ of $A$, such that each of $\mu(\Gamma) \cap \mu(H_1) C$ and $\mu(\Gamma) \cap
\mu(H_2) C$ is infinite. Then, since $\mu(H_k) \approx L_k$, we may assume (by enlarging $C$) that each of $\mu(\Gamma) \cap L_1 C$ and $\mu(\Gamma) \cap L_2 C$ is infinite. This contradicts the conclusion of Proposition \[tess->misswall\].
\[tess->dim>1,2\] Assume ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G = 2$. Let
1. $L_1$ and $L_2$ be the two walls of $A^+$; and
2. $H_1$ and $H_2$ be closed, connected, nontrivial subgroups of $G$;
such that $$\mu(H_k) \approx L_k$$ for $k = 1,2$. If $G/H$ has a tessellation, and $H$ is not compact, then $$d(H) \ge \min\{ d(H_1), d(H_2) \} .$$
The desired conclusion is obtained by combining Corollary \[tess->missHk\] with Theorem [\[noncpctdim\][[(]{}\[noncpctdim-notess\][)]{}]{}]{}.
For $G = {\operatorname{SL}}(3,{\mathord{\mathbb{R}}})$, there does not exist a connected subgroup $H_k$, such that $\mu(H_k) \approx L_k$ [(cf. \[SL3-B+\])]{}. Thus, Corollary \[tess->dim>1,2\] does not provide a lower bound on $d(H)$ in this case.
One-dimensional subgroups {#1DSect}
=========================
Although the following conjecture does not seem to have been stated previously in the literature, it is perhaps implicit in [@OhWitte-CK].
\[d(H)=1->notess?\] If $d(H) = 1$, then $G/H$ does not have a tessellation.
In this section, we establish that the conjecture is valid in two cases: if either ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G \le 2$ [[(]{}see \[1Drank2\][)]{}]{} or $G$ is almost simple [[(]{}see \[1Dsimple\][)]{}]{}. Each of these illustrates a general theorem: for groups of real rank two, the conjecture follows from a theorem of Y. Benoist and F. Labourie that is based on differential geometry; H. Oh and D. Witte observed that, for simple groups, the conjecture follows from a theorem of G. A. Margulis that is based on unitary representation theory.
The following example is the only case of Conjecture \[d(H)=1->notess?\] that is needed in later sections. (It is used in the proof of Theorem \[SL3->notess\].) Because ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}\bigl( {\operatorname{SL}}(3,{\mathbb{F}}) \bigr) = 2$ and ${\operatorname{SL}}(3,{\mathbb{F}})$ is almost simple, this example is covered both by the theorem of Benoist-Labourie and by the theorem of Margulis, but it would be interesting to have an easy proof.
\[wallinSL3\] Assume $G = {\operatorname{SL}}(3,{\mathbb{F}})$, for ${\mathbb{F}}= {\mathord{\mathbb{R}}}$, ${\mathord{\mathbb{C}}}$, or ${\mathord{\mathbb{H}}}$, and let $$\label{SL3wallSubgrp}
H_1 =
{\left\{\,
\begin{pmatrix}
e^t & 0 & 0 \\
0 & e^t & 0 \\
0 & 0 & e^{-2t}
\end{pmatrix}
\mathrel{\left| \vphantom {\left\{
\begin{pmatrix}
e^t & 0 & 0 \\
0 & e^t & 0 \\
0 & 0 & e^{-2t}
\end{pmatrix}
\mid
t \in {\mathord{\mathbb{R}}}\right\} } \right.}
t \in {\mathord{\mathbb{R}}}\,\right\} }
\subset G
.$$ Then $G/H_1$ does not have a tessellation.
Let us begin with an easy observation.
\[1Drank1\] If $d(H) = 1$ and ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G = 1$, then $G/H$ does not have a tessellation.
We may assume $H \subset AN$ [[(]{}see \[HcanbeAN\][)]{}]{}. From [[(]{}\[Rrank1-CDS\][)]{}]{}, we know that $H$ is a Cartan-decomposition subgroup, so Lemma \[CDS->notess\] implies that $G/H$ must be compact; thus, the trivial group $e$ is a crystallographic group for $G/H$. However, since $$d(G) = \dim A + \dim N \ge 1 + 1 > 1 = d(H)$$ [[(]{}see \[d(G)\][)]{}]{}, and $e$ acts properly discontinuously on $G/G$, this contradicts Theorem [\[noncpctdim\][[(]{}\[noncpctdim-notess\][)]{}]{}]{}.
The proof of Lemma \[1Drank1\] shows that the dimension of every connected, cocompact subgroup of $G$ is at least $d(G)$. This is a result of M. Goto and H.–C. Wang [@GotoWang (1.2), p. 263].
The geometric method of Y. Benoist and F. Labourie
--------------------------------------------------
\[center->notess\] If $H$ is reductive and contains an element of $A$ in its center, then $G/H$ does not have a tessellation.
To illustrate the idea behind Theorem \[center->notess\], we give a direct proof of the following special case (under an additional technical assumption [[(]{}see \[1DA->notessPf-H=Z\][)]{}]{}), which is sufficient for our needs. (Note that Condition [[(]{}\[1DA->notessPf-H=Z\][)]{}]{} is satisfied for the subgroup $H_1$ of Proposition \[wallinSL3\].) Benoist and Labourie prove the general case by using a slightly different 1-form in place of the form $\omega$ that we define in Step \[1DA->notessPf-Omega\].
\[1DA->notess\] If $H$ is a one-dimensional subgroup of $A$, then $G/H$ does not have a tessellation.
Suppose $\Gamma$ is a crystallographic group for $G/H$. (This will lead to a contradiction.) By passing to finite-index subgroups, we may assume that $G$ and $H$ are connected.
We will construct a volume form $\nu$ on $\Gamma \backslash G/H$ that is exact: $\nu = d \xi$. The integral of $\nu$ over $\Gamma \backslash G/H$ is the volume of $\Gamma \backslash G/H$, which is obviously not $0$, but Stokes’ Theorem implies that the integral of any exact form over a closed manifold is $0$. This is a contradiction.
\[1DA->notessPf-Omega\] Construction of a certain $2$-form $\breve{\Omega}$ on $\Gamma \backslash
G/H$.
Let ${\mathfrak{\lowercase{M}}} = {\mathfrak{\lowercase{H}}}^\perp$ be the orthogonal complement to ${\mathfrak{\lowercase{H}}}$, under the Killing form. Then ${\mathfrak{\lowercase{M}}}$ is an $({\operatorname{Ad}\nolimits}_G H)$-invariant subspace of ${\mathfrak{\lowercase{G}}}$, such that ${\mathfrak{\lowercase{G}}} = {\mathfrak{\lowercase{H}}} + {\mathfrak{\lowercase{M}}}$ and ${\mathfrak{\lowercase{H}}} \cap {\mathfrak{\lowercase{M}}} = 0$. Let $\omega_e \colon {\mathfrak{\lowercase{G}}} \to {\mathfrak{\lowercase{H}}}$ be the projection with kernel ${\mathfrak{\lowercase{M}}}$, and let $\omega$ be the corresponding left-invariant ${\mathfrak{\lowercase{H}}}$-valued 1-form on $G$.
The space $G$ is a homogeneous principal $H$-bundle over $G/H$. It is well known [@KobayashiNomizu1 Thm. II.11.1, p. 103] that $\omega$ is the connection form of a $G$-invariant connection on this bundle, and that the curvature form $\Omega$ of this connection is given by $$\label{Omega(M,M)}
\mbox{$\Omega(X, Y) = - \frac{1}{2} \omega \bigl( [X,Y] \bigr)$
\qquad
for $X,Y \in {\mathfrak{\lowercase{M}}}$.}$$ Also, because $H$ is abelian, the Structure Equation [@KobayashiNomizu1 Thm. II.5.2, p. 77] implies $$\label{structeq}
\mbox{$\Omega(X, Y) = d \omega(X,Y)$
\qquad
for $X,Y \in {\mathfrak{\lowercase{G}}}$.}$$ By identifying the $1$-dimensional Lie algebra ${\mathfrak{\lowercase{H}}}$ with ${\mathord{\mathbb{R}}}$, we may think of $\omega$ and $\Omega$ as ordinary (that is, ${\mathord{\mathbb{R}}}$-valued) differential forms.
Because $\omega$ and (hence) $\Omega$ are left-invariant, they determine well-defined forms $\overline{\omega}$ and $\overline{\Omega}$ on $\Gamma
\backslash G$. (Note that $\overline{\omega}$ is a connection form on the principal $H$-bundle $\Gamma \backslash G$ over $\Gamma \backslash G/H$, and the curvature form of this connection is $\overline{\Omega}$.) Because $H$ is abelian, we have $\overline{\Omega}_{gh} = \overline{\Omega}_g$ for all $h
\in H$ (cf. [@KobayashiNomizu1 Prop. II.5.1(c), p. 76]), so the horizontal form $\overline{\Omega}$ determines a well-defined form $\breve{\Omega}$ on the base space $\Gamma \backslash G/H$.
Construction, for a certain $s$, of a certain $s$-form $\breve{\mu}$ on $\Gamma \backslash G/H$.
Identifying ${\mathfrak{\lowercase{H}}}$ with ${\mathord{\mathbb{R}}}$ provides an ordering on the weights of ${\mathfrak{\lowercase{H}}}$. Let
- ${\mathfrak{\lowercase{M}}}_0 = {\mathfrak{\lowercase{C}}}_{{\mathfrak{\lowercase{G}}}}({\mathfrak{\lowercase{H}}}) \cap {\mathfrak{\lowercase{M}}}$ be the $0$ weight space of ${\operatorname{ad}\nolimits}_{{\mathfrak{\lowercase{G}}}} {\mathfrak{\lowercase{H}}}$ on ${\mathfrak{\lowercase{M}}}$;
- ${\mathfrak{\lowercase{M}}}_+$ be the sum of the positive weight spaces of ${\operatorname{ad}\nolimits}_{{\mathfrak{\lowercase{G}}}}
{\mathfrak{\lowercase{H}}}$ on ${\mathfrak{\lowercase{M}}}$;
- ${\mathfrak{\lowercase{M}}}_-$ be the sum of the negative weight spaces of ${\operatorname{ad}\nolimits}_{{\mathfrak{\lowercase{G}}}}
{\mathfrak{\lowercase{H}}}$ on ${\mathfrak{\lowercase{M}}}$;
- $s = \dim {\mathfrak{\lowercase{M}}}_0$; and
- $\mu$ be a nontrivial left-invariant $s$-form on $G$, such that $$\label{mu(H+M)}
\mu( {\mathfrak{\lowercase{H}}} \oplus {\mathfrak{\lowercase{M}}}_+ \oplus {\mathfrak{\lowercase{M}}}_-, *, \ldots, *) = 0 .$$
Because $\mu$ is left-invariant, and $\mu( {\mathfrak{\lowercase{H}}}, *, \ldots, *) = 0$, the form $\mu$ determines a well-defined differential form $\breve{\mu}$ on $\Gamma \backslash G/H$. (We remark that, because ${\mathfrak{\lowercase{G}}} = {\mathfrak{\lowercase{H}}} \oplus {\mathfrak{\lowercase{M}}}_+ \oplus {\mathfrak{\lowercase{M}}}_0
\oplus {\mathfrak{\lowercase{M}}}_-$, Condition [[(]{}\[mu(H+M)\][)]{}]{} implies that the form $\mu$ is unique up to a scalar multiple.)
\[1DA->notessPf-volume\] For a certain $r$, the wedge product $\breve{\mu} \wedge
\breve{\Omega}^{\wedge r}$ is a volume form on $\Gamma \backslash G/H$.
Let $r = \dim {\mathfrak{\lowercase{M}}}_+$. It suffices to show that the restriction of $\Omega$ to ${\mathfrak{\lowercase{M}}}_+ \oplus {\mathfrak{\lowercase{M}}}_-$ is a symplectic form. It is obviously skew, so we need only show that it is nondegenerate. Thus, letting $$\begin{aligned}
{\mathfrak{\lowercase{Z}}}
&= \{\, X \in {\mathfrak{\lowercase{M}}}_+ \oplus {\mathfrak{\lowercase{M}}}_- \mid \Omega (X, {\mathfrak{\lowercase{M}}}_+ \oplus {\mathfrak{\lowercase{M}}}_-) = 0 \bigr) \\
&= \{\, X \in {\mathfrak{\lowercase{M}}}_+ \oplus {\mathfrak{\lowercase{M}}}_- \mid \omega \bigl( [X, {\mathfrak{\lowercase{M}}}_+ \oplus {\mathfrak{\lowercase{M}}}_-] \bigr)
= 0 \,\}
& \mbox{{{\upshape(}\ref{Omega(M,M)}{\upshape)}}} \\
&= \{\, X \in {\mathfrak{\lowercase{M}}}_+ \oplus {\mathfrak{\lowercase{M}}}_- \mid
[X, {\mathfrak{\lowercase{M}}}_+ \oplus {\mathfrak{\lowercase{M}}}_-] \subset {\mathfrak{\lowercase{M}}} \,\}
& \mbox{(definition of~$\omega$)}
\\
&= \{\, X \in {\mathfrak{\lowercase{M}}}_+ \oplus {\mathfrak{\lowercase{M}}}_- \mid
\langle {\mathfrak{\lowercase{H}}} \mid [X, {\mathfrak{\lowercase{M}}}_+ \oplus {\mathfrak{\lowercase{M}}}_-]
\rangle_{\text{Killing}} = 0 \,\}
& \mbox{(${\mathfrak{\lowercase{M}}} = {\mathfrak{\lowercase{H}}}^\perp$)}
,
\end{aligned}$$ we wish to show ${\mathfrak{\lowercase{Z}}} = 0$.
There is no harm in passing to the complexification ${\mathfrak{\lowercase{G}}}^{{\mathord{\mathbb{C}}}}$ of ${\mathfrak{\lowercase{G}}}$. Let ${\mathfrak{\lowercase{T}}}^{{\mathord{\mathbb{C}}}}$ be a Cartan subalgebra of ${\mathfrak{\lowercase{G}}}^{{\mathord{\mathbb{C}}}}$ that contains ${\mathfrak{\lowercase{H}}}$. Because ${\mathfrak{\lowercase{T}}}^{{\mathord{\mathbb{C}}}}$ preserves the Killing form, centralizes ${\mathfrak{\lowercase{H}}}$, and normalizes ${\mathfrak{\lowercase{M}}}_+ \oplus {\mathfrak{\lowercase{M}}}_-$, we know that ${\mathfrak{\lowercase{Z}}}^{{\mathord{\mathbb{C}}}}$ is $({\operatorname{ad}\nolimits}_{{\mathfrak{\lowercase{G}}}^{{\mathord{\mathbb{C}}}}} {\mathfrak{\lowercase{T}}}^{{\mathord{\mathbb{C}}}})$-invariant; thus, ${\mathfrak{\lowercase{Z}}}^{{\mathord{\mathbb{C}}}}$ is a sum of root spaces.
Suppose there exists a nonzero element $X$ of ${\mathfrak{\lowercase{Z}}}^{{\mathord{\mathbb{C}}}}$, such that $X$ belongs to some root space ${\mathfrak{\lowercase{G}}}^{{\mathord{\mathbb{C}}}}_{\alpha}$. (This will lead to a contradiction.) There exists $Y \in {\mathfrak{\lowercase{G}}}^{{\mathord{\mathbb{C}}}}_{-\alpha}$, such that $$\langle t \mid [X,Y] \rangle_{\text{Killing}} = \alpha(t)$$ for all $t \in {\mathfrak{\lowercase{T}}}^{{\mathord{\mathbb{C}}}}$ [@Humphreys-Lie Prop. 8.3(c), p. 37]. Because ${\mathfrak{\lowercase{G}}}^{{\mathord{\mathbb{C}}}}_{\alpha} \subset {\mathfrak{\lowercase{M}}}_+ \oplus {\mathfrak{\lowercase{M}}}_-$, we have $\alpha({\mathfrak{\lowercase{H}}}) \neq 0$. Then $(-\alpha)({\mathfrak{\lowercase{H}}}) \neq 0$, so ${\mathfrak{\lowercase{G}}}^{{\mathord{\mathbb{C}}}}_{-\alpha} \subset {\mathfrak{\lowercase{M}}}_+ \oplus {\mathfrak{\lowercase{M}}}_-$. We now know that $X \in {\mathfrak{\lowercase{Z}}}^{{\mathord{\mathbb{C}}}}$ and $Y \in {\mathfrak{\lowercase{M}}}_+ \oplus {\mathfrak{\lowercase{M}}}_-$, so $\langle {\mathfrak{\lowercase{H}}} \mid [X, Y] \rangle_{\text{Killing}} = 0$. We therefore conclude, from the definition of $Y$, that $\alpha({\mathfrak{\lowercase{H}}}) =
0$. This is a contradiction.
The form $\breve{\Omega}$ is exact: we may write $\breve{\Omega} = d \breve{\phi}$.
Let $\breve{\omega}_0$ be the connection form of a flat connection on $\Gamma
\backslash G$ over $\Gamma \backslash G/H$. (Since the principal bundle is trivial [[(]{}see \[trivialbundle\][)]{}]{}, it is obvious that there is a flat connection.) For any vector field $X$ on $\Gamma \backslash G/H$, let $\widetilde X$ be the lift of $X$ to a vector field on $\Gamma \backslash G$ that is horizontal with respect to the flat connection $\breve{\omega}_0$.
Since $H$ is abelian, there is a well-defined 1-form $\breve{\phi}$ on $\Gamma
\backslash G/H$ given by $$\breve{\phi}(X) = \overline{\omega}(\widetilde X) .$$ Then $$\begin{aligned}
d \breve{\phi}(X,Y)
&= \frac{1}{2} \left(
X \bigl( \breve{\phi}(Y) \bigr) - Y \bigl(
\breve{\phi}(X) \bigr) -
\breve{\phi} \bigl( [X,Y] \bigr) \right)
& \mbox{(definition of~$d$)} \\
&= \frac{1}{2} \left( \widetilde X \bigl( \overline{\omega}(\widetilde Y)
\bigr) - \widetilde Y \bigl( \overline{\omega}(\widetilde X) \bigr) -
\overline{\omega} \bigl( \widetilde{[X,Y]} \bigr) \right)
& \mbox{(definition of~$\breve{\phi}$)} \\
&= \frac{1}{2} \left( \widetilde X \bigl( \overline{\omega}(\widetilde Y)
\bigr) - \widetilde Y \bigl( \overline{\omega}(\widetilde X) \bigr) -
\overline{\omega} \bigl( [\widetilde X,\widetilde Y] \bigr) \right)
& \mbox{($\breve{\omega}_0$ is flat)} \\
&= d \overline{\omega}(\widetilde X, \widetilde Y)
& \mbox{(definition of~$d$)} \\
&= \overline{\Omega}(\widetilde X, \widetilde Y)
& \mbox{{{\upshape(}\ref{structeq}{\upshape)}}} \\
&= \breve{\Omega}(X,Y)
& \mbox{(definition of~$\breve{\Omega}$)} .
\end{aligned}$$
For simplicity, assume that $$\label{1DA->notessPf-H=Z}
\mbox{every hyperbolic element of the center of ${{\mathfrak{\lowercase{C}}}}_{{\mathfrak{\lowercase{G}}}}({\mathfrak{\lowercase{H}}})$
is contained in~${\mathfrak{\lowercase{H}}}$.}$$
\[1DA->notessPf-closed\] We have $d\breve{\mu} \wedge \breve{\Omega}^{\wedge(r-1)} = 0$.
Let
- $X_1,\ldots,X_r$ be a basis of ${\mathfrak{\lowercase{M}}}_+$, and
- $Y_1,\ldots,Y_r$ be the dual basis of ${\mathfrak{\lowercase{M}}}_-$, with respect to the symplectic form $\Omega$ on ${\mathfrak{\lowercase{M}}}_+ \oplus {\mathfrak{\lowercase{M}}}_-$.
Thus, $\Omega_e(X_j,Y_k) = \delta_{j,k}$.
Let $Z_1,\ldots,Z_s$ be a basis of ${\mathfrak{\lowercase{M}}}_0$, write $$[X_j,Y_k] = \sum_\ell a_{j,k}^\ell Z_\ell \pmod {{\mathfrak{\lowercase{H}}} + {\mathfrak{\lowercase{M}}}_+ + {\mathfrak{\lowercase{M}}}_-} ,$$ and define $$W = \sum_j [X_j,Y_j] = \sum_{j,\ell} a_{j,j}^\ell Z_\ell .$$
\[1DA->notessPf-closed-Windep\] $W$ is independent of the choice of the basis $\{X_j\}$ of ${\mathfrak{\lowercase{M}}}_+$ [(]{}with the understanding that $\{Y_j\}$ must be the dual basis of ${\mathfrak{\lowercase{M}}}_-$[)]{}.
Let $$\label{elemop}
X'_j =
\begin{cases}
a X_1 + b X_2 & \mbox{if $j = 1$} \\
X_j & \mbox{if $j \ge 2$}
\end{cases}$$ for some $a,b \in {\mathord{\mathbb{R}}}$ with $a \neq 0$. Then $$Y'_j =
\begin{cases}
(1/a) Y_1 & \mbox{if $j = 1$} \\
Y_2 - (b/a) Y_1 & \mbox{if $j = 2$} \\
Y_j & \mbox{if $j \ge 3$}
,
\end{cases}$$ so $$\begin{aligned}
W'
&= [X'_1,Y'_1] + [X'_2,Y'_2] + \sum_{j \ge 3} [X'_j,Y'_j] \\
&= \bigl( [X_1,Y_1] + (b/a)[X_2,Y_1] \bigr)
+ \bigl( [X_2,Y_2] - (b/a) [X_2,Y_1] \bigr) + \sum_{j \ge 3} [X_j,Y_j] \\
&= \sum_j [X_j,Y_j] \\
&= W .
\end{aligned}$$ Since $X_1,\ldots,X_r$ can be transformed into any other basis of ${\mathfrak{\lowercase{M}}}_+$ by a sequence of elementary operations as in [[(]{}\[elemop\][)]{}]{}, we conclude that $W$ is independent of the choice of basis, as desired.
We have $W \in {\mathfrak{\lowercase{H}}}$.
Substep \[1DA->notessPf-closed-Windep\] implies that $W$ is centralized by $C_G(H)$, so $W$ is in the center of ${\mathfrak{\lowercase{C}}}_{{\mathfrak{\lowercase{G}}}}({\mathfrak{\lowercase{H}}})$.
Let $\sigma$ be a Cartan involution of $G$ with $\sigma(h) = -h$ for $h \in
{\mathfrak{\lowercase{H}}}$. Substep \[1DA->notessPf-closed-Windep\] implies $\sigma(W) = -W$. (From Substep \[1DA->notessPf-closed-Windep\], we see that $W$ depends only on ${\mathfrak{\lowercase{H}}}$ and the chosen identification of ${\mathfrak{\lowercase{H}}}$ with ${\mathord{\mathbb{R}}}$; $\sigma$ reverses the choice of identification.)
Thus, $W$ is a hyperbolic element of the center of ${\mathfrak{\lowercase{C}}}_{{\mathfrak{\lowercase{G}}}}({\mathfrak{\lowercase{H}}})$. By Assumption \[1DA->notessPf-H=Z\], this implies $W \in {\mathfrak{\lowercase{H}}}$, as desired.
Completion of Step \[1DA->notessPf-closed\].
Let $$X_1^*, \ldots, X_r^*, Y_1^*, \ldots, Y_r^*, Z_1^*, \ldots, Z_s^*$$ be the basis of $({\mathfrak{\lowercase{G}}}/{\mathfrak{\lowercase{H}}})^*$ dual to $$X_1, \ldots, X_r, Y_1, \ldots, Y_r, Z_1, \ldots, Z_s .$$ We may assume $\mu = Z_1^* \wedge \cdots \wedge Z_s^*$. Then, because $[{\mathfrak{\lowercase{M}}}_0 + {\mathfrak{\lowercase{M}}}_-, {\mathfrak{\lowercase{M}}}_-] \subset {\mathfrak{\lowercase{M}}}_-$ and $[{\mathfrak{\lowercase{M}}}_0 + {\mathfrak{\lowercase{M}}}_+, {\mathfrak{\lowercase{M}}}_+] \subset {\mathfrak{\lowercase{M}}}_+$, we have $$dZ_\ell^*( {\mathfrak{\lowercase{M}}}_0 + {\mathfrak{\lowercase{M}}}_-, {\mathfrak{\lowercase{M}}}_- ) = 0
= dZ_\ell^*( {\mathfrak{\lowercase{M}}}_0 + {\mathfrak{\lowercase{M}}}_+, {\mathfrak{\lowercase{M}}}_+ ) ,$$ so $$dZ_\ell^* = - \sum_{j,k} a_{j,k}^\ell X_j^* \wedge Y_k^* \pmod{{\mathfrak{\lowercase{M}}}_0^*
\wedge {\mathfrak{\lowercase{M}}}_0^*} .$$ Therefore $$\begin{aligned}
(s+1) \, d \breve{\mu}
&= \sum_{\ell} (-1)^{\ell-1} dZ_\ell^* \wedge Z_1^* \wedge \cdots \wedge
\widehat{Z_\ell^*} \wedge \cdots \wedge Z_s^* \\
&= \sum_{j,k,\ell} (-1)^\ell a_{j,k}^\ell
X_j^* \wedge Y_k^* \wedge Z_1^* \wedge \cdots \wedge \widehat{Z_\ell^*}
\wedge \cdots \wedge Z_s^* .
\end{aligned}$$ From the choice of $Y_1, \ldots, Y_r$, we have $\Omega = 2 \sum_{j=1}^r X_j^* \wedge Y_j^*$, so $$\Omega^{\wedge(r-1)}
= 2^{r-1} (r-1)! \, \sum_{j=1}^r X_1^* \wedge Y_1^* \wedge \cdots \wedge
\widehat{X_j^*} \wedge \widehat{Y_j^*} \wedge \cdots \wedge X_r^* \wedge Y_r^*$$ and $$\Omega^{\wedge r}
= 2^{r} r! \, X_1^* \wedge Y_1^* \wedge \cdots \wedge X_r^* \wedge Y_r^*
.$$ Hence $$\begin{aligned}
(s+1) \, d \breve{\mu} \wedge \breve{\Omega}^{\wedge(r-1)}
&= \frac{1}{2r} \, \sum_{j,\ell}
(-1)^{\ell} a_{j,j}^\ell
Z_1^* \wedge \cdots \wedge \widehat{Z_\ell^*} \wedge \cdots \wedge Z_s^*
\wedge \breve{\Omega}^{\wedge r} \\
&= - \frac{s+2r}{2r} \, \iota_{W} (\breve{\mu} \wedge
\breve{\Omega}^{\wedge r}) .
\end{aligned}$$ Since $W \in {\mathfrak{\lowercase{H}}}$, we have $ \iota_{W} (\mu_e \wedge \Omega_e^{\wedge r}) = 0$, so the desired conclusion follows.
\[1DA->notessPf-volexact\] $\breve{\mu} \wedge \breve \Omega^{\wedge r}$ is exact.
We have $$\begin{aligned}
d \bigl( \breve{\mu} \wedge \breve{\phi} \wedge \breve{\Omega}^{\wedge(r-1)} \bigr)
&= \pm d \breve{\mu} \wedge \breve{\Omega}^{\wedge(r-1)} \wedge \breve{\phi}
\pm \breve{\mu} \wedge d \breve{\phi} \wedge \breve{\Omega}^{\wedge(r-1)}
\pm \breve{\mu} \wedge \breve{\phi} \wedge d \breve{\Omega}^{\wedge(r-1)} \\
&= 0 \wedge \breve{\phi}
\pm \breve{\mu} \wedge \breve{\Omega} \wedge \breve{\Omega}^{\wedge(r-1)}
\pm \breve{\mu} \wedge \breve{\phi} \wedge 0 \\
&= \pm \breve{\mu} \wedge \breve{\Omega}^{\wedge r} .
\end{aligned}$$
A contradiction.
From Step \[1DA->notessPf-volume\], we know that $$\int_{\Gamma \backslash G/H} \breve{\mu} \wedge \breve{\Omega}^{\wedge r}
\neq 0 .$$ On the other hand, Step \[1DA->notessPf-volexact\] implies that this integral is zero. This is a contradiction.
\[oppinv\] Let $\tau$ be the opposition involution in $A^+$; that is, for $a\in A^+$, $\tau(a) = \mu(a^{-1})$ is the unique element of $A^+$ that is conjugate (under an element of the Weyl group) to $a^{-1}$. Thus, for all $h \in G$, we have $$\mu(h^{-1}) = \tau \bigl( \mu(h) \bigr) .$$
See [[(]{}\[SL3-oppinv\][)]{}]{} for an explicit description of the opposition involution in $G = {\operatorname{SL}}(3,{\mathord{\mathbb{R}}})$. For some groups, such as $G = {\operatorname{SO}}(2,n)$, we have $\mu(h^{-1}) = \mu(h)$ for all $h \in G$ [[(]{}see \[mu(h-1)\][)]{}]{}; in such a case, the opposition involution is simply the identity map on $A^+$.
\[1Drank2\] If $d(H) = 1$ and ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G \le 2$, then $G/H$ does not have a tessellation.
Suppose $\Gamma$ is a crystallographic group for $G/H$. (This will lead to a contradiction.)
From [[(]{}\[1Drank1\][)]{}]{}, we know ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G = 2$. Let $L_1$ and $L_2$ be the two walls of $A^+$ and, for $k \in \{1,2\}$, let $H_k = L_k \cup L_k^{-1}$. Because $L_k$ is a ray (that is, a one-parameter semigroup), it is clear that $H_k$ is a subgroup of $A$.
From Proposition \[tess->misswall\], we know that there is some $k \in
\{1,2\}$, such that $$\mbox{$\mu(\Gamma) \cap L_k C$ is finite,}$$ for every compact subset $C$ of $A$. Since $\Gamma = \Gamma^{-1}$, we have $\tau \bigl( \mu(\Gamma) \bigr) = \mu(\Gamma)$, so this implies that $$\mbox{$\mu(\Gamma) \cap \tau(L_k) C$ is finite,}$$ for every compact subset $C$ of $A$. Also, because $L_k \subset A^+$, we have $\mu(L_k) = L_k$, so $$\mu(H_k)
= \mu( L_k \cup L_k^{-1})
= \mu(L_k) \cup \tau \bigl( \mu(L_k) \bigr)
= L_k \cup \tau(L_k) .$$ Therefore $$\mbox{$
\mu(\Gamma) \cap \mu(H_k) C
= \bigl( \mu(\Gamma) \cap L_k C \bigl) \cup \bigl( \mu(\Gamma) \cap
\tau(L_k) C \bigl)
$ is finite,}$$ for every compact subset $C$ of $A$. Hence, Corollary \[proper<>mu(L)\] implies that $\Gamma$ acts properly discontinuously on $G/H_k$. Then, because $d(H) = 1 = d(H_k)$ [[(]{}see \[d(H)=dimH\][)]{}]{}, Theorem [\[noncpctdim\][[(]{}\[noncpctdim-tess\][)]{}]{}]{} implies that $G/H_k$ has a tessellation. This contradicts Corollary \[1DA->notess\].
The representation-theoretic method of G. A. Margulis
-----------------------------------------------------
\[tempered-def\] The subgroup $H$ is *tempered* in $G$ if there exists a (positive) function $f \in L^1(H)$ (with respect to a left-invariant Haar measure on $H$), such that, for every unitary representation $\pi$ of $G$, either
- $|\langle \pi (h) \phi \mid \psi \rangle | \leq f(h) \,
\lVert\phi\rVert \lVert\psi\rVert$ for all $h \in H$ and all $K$-fixed vectors $\phi$ and $\psi$; or
- some nonzero vector is fixed by every element of $\pi(G)$.
For many examples of tempered subgroups of simple Lie groups, see [@Oh-tempered].
\[MargulisTempered\] If $H$ is noncompact and tempered, then $G/H$ does not have a tessellation.
Suppose $\Gamma$ is a crystallographic group for $G/H$. (This will lead to a contradiction.) To simplify the notation somewhat (and because this is the only case we need), let us assume $H = \{h^t\}$ is a one-parameter subgroup of $G$.
Because $G$ and $\Gamma$ are unimodular (recall that $G$ is semisimple [[(]{}see \[standing\][)]{}]{} and $\Gamma$ is discrete), there is a $G$-invariant measure (in fact, a $G$-invariant volume form) on the homogeneous space $\Gamma
\backslash G$ [@Raghunathan Lem 1.4, p. 18]. Thus, the natural representation $\pi$ of $G$ on $L^2(\Gamma \backslash G)$, defined by $$\mbox{
$\bigl( \pi(g) \phi \bigr)(x) = \phi(x g^{-1})$,
\qquad
for $\phi \in L^2(\Gamma \backslash G)$, $g \in G$, $x \in \Gamma
\backslash G$,
}$$ is unitary.
Because $H$ is noncompact, and acts properly on $\Gamma \backslash G$, we know that any compact subset of $\Gamma \backslash G$ has infinitely many pairwise-disjoint translates (all of the same measure), so we see that $$\label{MargulisTemperedPf-notcpct}
\mbox{$\Gamma \backslash G$ is not compact, and has infinite volume} .$$ Therefore, $\pi$ has no nonzero $G$-invariant vectors, so, because $H$ is tempered, we know that there is some $f \in L^1({\mathord{\mathbb{R}}})$, such that $$\label{MargulisTemperedPf-<f}
f(t) \, \lVert\phi\rVert_2 \lVert\psi\rVert_2
\ge \left| \int_{\Gamma \backslash G} \phi( x h^t) \, \psi(x) \, dx \right|$$ for all $t \in {\mathord{\mathbb{R}}}$ and all $K$-invariant $\phi,\psi \in L^2(\Gamma
\backslash G)$.
Because $\Gamma \backslash G/H$ is compact, there is a compact subset $C$ of $G$, such that $\Gamma C H = G$; let $\overline C$ be the image of $C$ in $\Gamma \backslash G$. From the choice of $C$, we know, for each $x \in \Gamma \backslash G$, that there is some $T_x \in {\mathord{\mathbb{R}}}$, such that $$\label{MargulisTemperedPf-Tx}
x h^{T_x} \in \overline{C} .$$ Because $\bigcup_{t=0}^1 \overline{C} h^t K$ is a compact subset of $\Gamma \backslash G$, there is a positive, continuous function $\phi$ on $\Gamma \backslash G$ with compact support, such that $$\label{MargulisTemperedPf-phi(xht)}
\mbox{$\phi(x h^t) \ge 1$ for all $x \in \overline{C}$ and all $t \in
[0,1]$,}$$ and, by averaging over $K$, we may assume that $\phi$ is $K$-invariant.
Fix some large $T \in {\mathord{\mathbb{R}}}^+$. Because $\bigcup_{t = -(T+1)}^{T+1}
\overline{C} h^t K$ is compact, and $\Gamma \backslash G$ has infinite volume [[(]{}see \[MargulisTemperedPf-notcpct\][)]{}]{}, there is some $K$-invariant continuous function $\psi_T$ on $\Gamma \backslash G$, such that $\lVert\psi_T\rVert_2 = 1$, $$\label{MargulisTemperedPf-psi<1}
\mbox{$0 \le \psi_T(x) \le 1$ for all $x \in \Gamma \backslash G$} ,$$ and $$\label{MargulisTemperedPf-Tx>T+1}
\mbox{$|T_x| > T + 1$ for all $x$ in the support of~$\psi_T$} .$$ We have $$\begin{aligned}
\lVert\phi\rVert_2 \int_{|t|>T} f(t) \, dt
&\ge \int_{\Gamma \backslash G} \int_{|t|>T} \phi( x h^t) \, \psi_T(x) \, dt
\, dx
& \mbox{({{\upshape(}\ref{MargulisTemperedPf-<f}{\upshape)}} and $\lVert\psi_T\rVert_2 = 1$)} \\
&\ge \int_{\Gamma \backslash G} \int_0^1 \phi( x h^{T_x + t}) \, \psi_T(x) \,
dt \, dx
& \mbox{{{\upshape(}\ref{MargulisTemperedPf-Tx>T+1}{\upshape)}}} \\
&\ge \int_{\Gamma \backslash G} \psi_T(x) \,
dx
& \mbox{({{\upshape(}\ref{MargulisTemperedPf-Tx}{\upshape)}} and
{{\upshape(}\ref{MargulisTemperedPf-phi(xht)}{\upshape)}})} \\
&\ge 1
& \mbox{({{\upshape(}\ref{MargulisTemperedPf-psi<1}{\upshape)}} and $\lVert\psi_T\rVert_2 = 1$)}
.
\end{aligned}$$ However, because $f \in L^1({\mathord{\mathbb{R}}})$, we know that $\lim_{T \to \infty} \int_{|t|>T} f(t) \, dt = 0$. This is a contradiction.
We state the following well-known result of representation theory without proof. As is explained in [@KatokSpatzier §3, p. 140], it can be obtained by combining work of R. Howe [@Howe Cor. 7.2 and §7] and M. Cowling [@Cowling Thm. 2.4.2]. (The assumption that ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G \ge 2$ can be relaxed: it suffices to assume that $G$ is not locally isomorphic to ${\operatorname{SO}}(1,n)$ or ${\operatorname{SU}}(1,n)$.)
Fix any matrix norm $\lVert \cdot \rVert$ on $G$; for example, we may let $\lVert g \rVert = \max_{j,k} |g_{j,k}|$.
\[expdecay\] If $G$ is almost simple, and ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G \ge 2$, then there are constants $C>0$ and $p>0$ such that, for every unitary representation $\pi$ of $G$, either
1. $ | \langle \pi(g) \phi \mid \psi \rangle|
\leq C \lVert \phi \rVert \lVert \psi \rVert \lVert g\rVert^{-p}$ for all $g \in G$ and all $\pi(K)$-fixed vectors $\phi$ and $\psi$; or
2. some nonzero vector is fixed by every element of $\pi(G)$.
Although we cannot prove Theorem \[expdecay\] here, we present an elementary proof of the following related result, which, unfortunately, is qualitative, rather than quantitative. On the other hand, this simple result applies to all vectors, not only the $K$-fixed vectors, and it applies to all semisimple groups, including ${\operatorname{SO}}(1,n)$ and ${\operatorname{SU}}(1,n)$. It was first proved by R. Howe and C. Moore [@HoweMoore Thm. 5.1] and (independently) R. Zimmer [@Zimmer-orbitspace Thm. 5.2].
\[matcoeffs->0\] If
- $G$ is connected and almost simple;
- $\pi$ is a unitary representation of $G$ on a Hilbert space ${\mathord{\mathcal{H}}}$, such that no nonzero vector is fixed by $\pi(G)$; and
- $\{g_j\}$ is a sequence of elements of $G$, such that $\lVert g_j
\rVert \to \infty$,
then $\langle \pi(g_j) \phi \mid \psi \rangle \to 0$, for every $\phi,\psi
\in {\mathord{\mathcal{H}}}$.
\[matcoeffs->0-A\] Assume $\{g_j\} \subset A$.
By passing to a subsequence, we may assume $\pi(g_j)$ converges weakly, to some operator $E$; that is, $$\langle \pi(g_j) \phi \mid \psi \rangle
\to \langle E \phi \mid \psi \rangle
\mbox{ \ for every $\phi,\psi \in {\mathord{\mathcal{H}}}$} .$$ Let $$\label{horodefn}
U = \{\, v \in G \mid g_j^{-1} v g_j \to e \,\}
\mbox{ \qquad and \qquad}
U^- = \{\, u \in G \mid g_j u g_j^{-1} \to e \,\} .$$ For $u \in U^-$, we have $$\langle E\pi(u) \phi \mid \psi \rangle
= \lim \langle \pi(g_j u) \phi \mid \psi \rangle
= \lim \langle \pi(g_j u g_j^{-1}) \pi(g_j) \phi \mid \psi \rangle
= \lim \langle \pi(g_j) \phi \mid \psi \rangle
= \langle E \phi \mid \psi \rangle ,$$ so $E \pi(u) = E$. Therefore $E \bigl( ({\mathord{\mathcal{H}}}^{U^-})^\perp \bigr) = 0$.
We have $$\langle E^* \phi \mid \psi \rangle
= \langle \phi \mid E \psi \rangle
= \lim \langle \phi \mid \pi(g_j) \psi \rangle
= \lim \langle \pi(g_j^{-1}) \phi \mid \psi \rangle ,$$ so the same argument, with $E^*$ in the place of $E$ and $g_j^{-1}$ in the place of $g_j$, shows that $E^* \bigl( ({\mathord{\mathcal{H}}}^U)^\perp \bigr) = 0$.
Because $\pi$ is unitary, we know that $\pi(g_j)$ is normal (that is, commutes with its adjoint) for every $j$; thus, the limit $E$ is also normal: we have $E^* E = E
E^*$. Therefore $$\lVert E\phi\rVert^2
= \langle E\phi \mid E \phi \rangle
= \langle (E^* E)\phi \mid \phi \rangle
= \langle (E E^*)\phi \mid \phi \rangle
= \langle E^*\phi \mid E^* \phi \rangle
= \lVert E^* \phi\rVert^2 ,$$ so $\ker E = \ker E^* $.
Thus, $$\ker E
= \ker E + \ker E^*
\supset ({\mathord{\mathcal{H}}}^{U^-})^\perp + ({\mathord{\mathcal{H}}}^U)^\perp
= ({\mathord{\mathcal{H}}}^{U^-} \cap {\mathord{\mathcal{H}}}^U)^\perp
= ({\mathord{\mathcal{H}}}^{\langle U, U^- \rangle})^\perp
.$$ By passing to a subsequence of $\{g_j\}$, we may assume $\langle U, U^-
\rangle = G$ [[(]{}see \[horosubseq\][)]{}]{}. Then ${\mathord{\mathcal{H}}}^{\langle U, U^- \rangle} =
{\mathord{\mathcal{H}}}^G = 0$, so $\ker E
\supset 0^\perp
= {\mathord{\mathcal{H}}}$. Hence, for all $\phi,\psi \in {\mathord{\mathcal{H}}}$, we have $$\lim \langle \pi(g_j) \phi \mid \psi \rangle
= \langle E \phi \mid \psi \rangle
= \langle 0 \mid \psi \rangle
= 0 ,$$ as desired.
The general case.
From the Cartan Decomposition $G = KAK$, we may write $g_j = c_j a_j c'_j$, with $c_j,c'_j \in K$ and $a_j \in A$. Because $K$ is compact, we may assume, by passing to a subsequence, that $\{c_j\}$ and $\{c'_j\}$ converge: say, $c_j
\to c$ and $c'_j \to c'$. Then $$\begin{aligned}
\lim \langle \pi(g_j) \phi \mid \psi \rangle
&= \lim \langle \pi(c_j a_j c'_j) \phi \mid \psi \rangle \\
&= \lim \langle \pi(a_j) \pi(c'_j) \phi \mid \pi(c_j)^{-1}
\psi \rangle \\
&= \lim \bigl\langle \pi(a_j) \bigl( \pi(c') \phi \bigr)
\mathrel{\big|} \pi(c)^{-1}
\psi \bigr\rangle \\
&= 0 ,
\end{aligned}$$ by Case \[matcoeffs->0-A\].
The following example illustrates Lemma \[horosubseq\].
Let $G = {\operatorname{SL}}(3,{\mathord{\mathbb{R}}})$, define $H_1$ as in Proposition \[wallinSL3\], and suppose $\{g_j\}$ is some sequence of elements of $H_1$, such that $\lVert g_j \rVert \to \infty$. We may write $$g_j =
\begin{pmatrix}
e^{t_j} & 0 & 0 \\
0 & e^{t_j} & 0 \\
0 & 0 & e^{-2t_j}
\end{pmatrix}
,$$ where $t_j \in {\mathord{\mathbb{R}}}$. By passing to a subsequence, we may assume that either $t_j \to \infty$ or $t_j \to -\infty$. If $t_j \to \infty$, then, in the notation of [[(]{}\[horodefn\][)]{}]{}, we have $$U =
\left\{
\begin{pmatrix}
1 & 0 & * \\
0 & 1 & * \\
0 & 0 & 1
\end{pmatrix}
\right\}
\mbox{ \qquad and \qquad}
U^- =
\left\{
\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
* & * & 1
\end{pmatrix}
\right\}
;$$ if $t_j \to -\infty$, then $U$ and $U^-$ are interchanged. Thus, in either case, ${\mathfrak{\lowercase{U}}}$ is the sum of two root spaces of ${\mathfrak{\lowercase{G}}}$, and ${\mathfrak{\lowercase{U}}}^-$ is the sum of the two opposite root spaces. It is not difficult to see that $[{\mathfrak{\lowercase{U}}}, {\mathfrak{\lowercase{U}}}^-]$ is the sum of ${\mathfrak{\lowercase{A}}}$ and the remaining two root spaces. Therefore, we have $\langle {\mathfrak{\lowercase{U}}}, {\mathfrak{\lowercase{U}}}^- \rangle = {\mathfrak{\lowercase{G}}}$, so $\langle U, U^- \rangle = G$.
\[horosubseq\] If $G$ and $\{g_j\}$ are as in Theorem \[matcoeffs->0\], and $\{g_j\}
\subset A$, then, after replacing $\{g_j\}$ by a subsequence, we have $\langle
U,U^-\rangle = G$, where $U$ and $U^-$ are defined in [[(]{}\[horodefn\][)]{}]{}.
By passing to a subsequence, we may assume $\{g_j\}$ is contained in a single Weyl chamber, which we may take to be $A^+$. Then, by passing to a subsequence yet again, we may assume, for every positive real root $\alpha$, that either $\alpha(g_j) \to \infty$ or $\alpha(g_j)$ is bounded. Let
- $\Phi^+$ be the set of positive real roots;
- $\Delta$ be the set of positive simple real roots;
- $ \Psi = \{\, \alpha \in \Phi^+ \mid
\mbox{$\alpha(g_j)$ is bounded} \,\}$; and
- $T = \cap_{\psi\in \Psi} \ker \psi = \cap_{\psi\in \Psi \cap \Delta}
\ker \psi$.
There is a compact subset $C$ of $A$, such that $\{g_j\} \subset C T$, so, because $\lVert g_j \rVert \to \infty$, we know that $T$ is not trivial.
For each real root $\alpha$, let ${{\mathfrak{\lowercase{N}}}}_\alpha$ be the corresponding root subspace of ${\mathfrak{\lowercase{G}}}$. Then $${\mathfrak{\lowercase{U}}} = \bigoplus_{\alpha \in \Phi^+ \smallsetminus \Psi} {{\mathfrak{\lowercase{N}}}}_\alpha
\mbox{ \qquad and \qquad}
{\mathfrak{\lowercase{U}}}^- = \bigoplus_{\alpha \in \Phi^+ \smallsetminus \Psi} {{\mathfrak{\lowercase{N}}}}_{-\alpha}
.$$ Now, for $\alpha \in \Phi^+$, we have $\alpha \in \Psi$ if and only if $\alpha$ is in the linear span of $\Psi \cap \Delta$. Thus, we see that ${\mathfrak{\lowercase{U}}}$ is precisely the unipotent radical of the standard parabolic subalgebra ${\mathfrak{\lowercase{P}}} = {\mathfrak{\lowercase{C}}}_{{\mathfrak{\lowercase{G}}}}(T) + {\mathfrak{\lowercase{A}}} + {\mathfrak{\lowercase{N}}}$ corresponding to the set $\Psi \cap \Delta$ of simple roots [@BorelTits-Reductive 4.2, pp. 85–86].
Similarly, ${\mathfrak{\lowercase{U}}}^-$ is the unipotent radical of the opposite parabolic algebra ${\mathfrak{\lowercase{P}}}^- = {\mathfrak{\lowercase{C}}}_{{\mathfrak{\lowercase{G}}}}(T) + {\mathfrak{\lowercase{A}}} + {\mathfrak{\lowercase{N}}}^-$. Because $G$ is simple, the unipotent radicals of opposite parabolics generate ${\mathfrak{\lowercase{G}}}$ [@BorelTits-Reductive Prop. 4.11, p. 89], so $\langle U, U^- \rangle =
G$, as desired.
\[1D->tempered\] Assume $G$ is simple, and ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G \ge 2$. If $H$ is a one-parameter subgroup of $AN$, then either
1. $H$ is tempered; or
2. $H \subset N$.
Write $H = \{h^t\}$. From the Real Jordan Decomposition [[(]{}\[JordanDecomp\][)]{}]{}, we may assume, after replacing $H$ by a conjugate subgroup, that $h^t = a^t
u^t$, where $a^t\in A$ is a hyperbolic one-parameter subgroup, and $u^t\in N$ is a unipotent one-parameter subgroup, such that $a^t$ and $u^t$ commute with each other.
We may assume $H \not\subset N$, so $a^t$ is nontrivial. Since the growth of the hyperbolic one-parameter subgroup $a^t$ is exponential, while that of the unipotent one-parameter subgroup $u^t$ is polynomial, there is some $\epsilon
> 0$, such that $$\lVert h^t\rVert = \lVert a^t u^t\rVert > \sqrt{\lVert a^t\rVert} >
e^{\epsilon |t|}$$ for large $t \in {\mathord{\mathbb{R}}}$. Since the function $C/e^{p \epsilon |t|}$ is in $L^1({\mathord{\mathbb{R}}})$, it follows from Theorem \[expdecay\] that $H$ is tempered, as desired.
\[1DinN->notess\] If $d(H) = 1$ and $H \subset N$, then $G/H$ does not have a tessellation.
We have $\dim H = d(H) = 1$ [[(]{}see \[d(H)=dimH\][)]{}]{}, so $H$ is a connected, one-dimensional, unipotent subgroup. Hence, the Jacobson-Morosov Lemma [@HelgasonBook Thm. 9.7.4, p. 432] implies that there exists a connected, closed subgroup $H_1$ of $G$, such that $H_1$ contains $H$, and $H_1$ is locally isomorphic to ${\operatorname{SL}}(2,{\mathord{\mathbb{R}}})$. Then $H$ is a Cartan-decomposition subgroup of $H_1$ [[(]{}see \[Rrank1-CDS\][)]{}]{}, so there is a compact subset $C$ of $A$, such that $\mu(H_1) \subset \mu(H) C$ [[(]{}see \[bddchange\][)]{}]{}. Also, we have $d(H_1)
= 2 > 1 = d(H)$. Therefore, Theorem \[noncpct-dim-notess\] applies.
\[1Dsimple\] If $d(H) = 1$ and $G$ is simple, then $G/H$ does not have a tessellation.
We may assume $H \subset AN$ [[(]{}see \[HcanbeAN\][)]{}]{}, so $\dim H = d(H) = 1$ [[(]{}see \[d(H)=dimH\][)]{}]{}.
- If ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G < 2$, then Lemma \[1Drank1\] applies.
- If $H \subset N$, then Lemma \[1DinN->notess\] applies.
- If ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G \ge 2$ and $H \not\subset N$, then Corollary \[1D->tempered\] implies that $H$ is tempered, so Theorem \[MargulisTempered\] applies.
Homogeneous spaces of ${\operatorname{SL}}(3,{\mathord{\mathbb{R}}})$, ${\operatorname{SL}}(3,{\mathord{\mathbb{C}}})$, and ${\operatorname{SL}}(3,{\mathord{\mathbb{H}}})$ {#SL3notessSect}
===========================================================================================================================================================================
Y. Benoist [@Benoist Cor. 1] and G.A. Margulis (unpublished) proved (independently) that ${\operatorname{SL}}(3,{\mathord{\mathbb{R}}})/{\operatorname{SL}}(2,{\mathord{\mathbb{R}}})$ does not have a tessellation. Using Benoist’s method, H. Oh and D. Witte [@OhWitte-CK Prop. 1.10] generalized this result by replacing ${\operatorname{SL}}(2,{\mathord{\mathbb{R}}})$ with any closed, connected subgroup $H$, such that neither $H$ nor ${\operatorname{SL}}(3,{\mathord{\mathbb{R}}})/H$ is compact. The same argument applies even if ${\mathord{\mathbb{R}}}$ replaced with either ${\mathord{\mathbb{C}}}$ or ${\mathord{\mathbb{H}}}$. However, the proof of Benoist (which applies in a more general context) relies on a somewhat lengthy argument to establish one particular lemma. Here, we adapt Benoist’s method to obtain a short proof of Theorem \[SL3->notess\] that avoids any appeal to the lemma.
\[SL3-oppinv\] Assume $G = {\operatorname{SL}}(3,{\mathbb{F}})$, for ${\mathbb{F}}= {\mathord{\mathbb{R}}}$, ${\mathord{\mathbb{C}}}$, or ${\mathord{\mathbb{H}}}$.
- Let $\tau$ be the opposition involution in $A^+$ [[(]{}see \[oppinv\][)]{}]{};
- Let $B^+=\{\,a\in A^+ \mid \tau(a)=a \,\}$.
More concretely, we have $$\begin{gathered}
A^+ =
{\left\{\,
\begin{pmatrix}
a_1 & 0 & 0 \\
0 & a_2 & 0 \\
0 & 0 & a_3
\end{pmatrix}
\mathrel{\left| \vphantom {\left\{
\begin{pmatrix}
a_1 & 0 & 0 \\
0 & a_2 & 0 \\
0 & 0 & a_3
\end{pmatrix}
\mid
\begin{matrix}
a_1,a_2,a_3 \in {\mathord{\mathbb{R}}}^+ , \\
a_1 a_2 a_3 = 1 , \\
a_1 \ge a_2 \ge a_3
\end{matrix}
\right\} } \right.}
\begin{matrix}
a_1,a_2,a_3 \in {\mathord{\mathbb{R}}}^+ , \\
a_1 a_2 a_3 = 1 , \\
a_1 \ge a_2 \ge a_3
\end{matrix}
\,\right\} }; \\
\tau
\begin{pmatrix}
a_1 & 0 & 0 \\
0 & a_2 & 0 \\
0 & 0 & a_3
\end{pmatrix}
=
\begin{pmatrix}
a_3^{-1} & 0 & 0 \\
0 & a_2^{-1} & 0 \\
0 & 0 & a_1^{-1}
\end{pmatrix}
; \\
B^+ =
{\left\{\,
\begin{pmatrix}
a & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & a^{-1}
\end{pmatrix}
\mathrel{\left| \vphantom {\left\{
\begin{pmatrix}
a & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & a^{-1}
\end{pmatrix}
\mid
\begin{matrix}
\\
a \ge 1 \\
\\
\end{matrix}
\right\} } \right.}
\begin{matrix}
\\
a \ge 1 \\
\\
\end{matrix}
\,\right\} }.
\end{gathered}$$
\[1DinSL3\] If $G = {\operatorname{SL}}(3,{\mathbb{F}})$ and $d(H) = 1$, then $G/H$ does not have a tessellation.
Since ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G = 2$, the desired conclusion follows from Corollary \[1Drank2\]; since $G$ is simple, it also follows from Corollary \[1Dsimple\]. However, we give a proof that requires only the special case described in Proposition \[wallinSL3\], rather than the full strength of [[(]{}\[1Drank2\][)]{}]{} or [[(]{}\[1Dsimple\][)]{}]{}.
Suppose $\Gamma$ is a crystallographic group for $G/H$. (This will lead to a contradiction.) Let $L_1$ and $L_2$ be the two walls of $A^+$. From Proposition \[tess->misswall\], we know that there exists $k \in \{1,2\}$, such that $\mu(\Gamma) C \cap L_k$ is finite, for every compact subset $C$ of $A$.
Because $\Gamma^{-1} = \Gamma$, we have $\tau \bigl( \mu(\Gamma) \bigr) =
\mu(\Gamma)$. On the other hand, $\tau$ interchanges $L_1$ and $L_2$. Thus, the preceding paragraph implies that $\mu(\Gamma) C \cap (L_1 \cup L_2)$ is finite, for every compact subset $C$ of $A$.
For $H_1$ as in [[(]{}\[SL3wallSubgrp\][)]{}]{}, we have $\mu(H_1) = L_1 \cup L_2$, so the conclusion of the preceding paragraph implies that $\Gamma$ acts properly discontinuously on $G/H_1$ [[(]{}see \[proper<>mu(L)\][)]{}]{}. Now Theorem [\[noncpctdim\][[(]{}\[noncpctdim-tess\][)]{}]{}]{} implies $\Gamma \backslash G/H_1$ is compact; thus, $G/H_1$ has a tessellation. This contradicts Proposition \[wallinSL3\].
For completeness, we include the proof of the following simple proposition.
\[SL3-B+\] Assume $G = {\operatorname{SL}}(3,{\mathbb{F}})$. If $H$ is a closed, connected subgroup of $AN$ with $\dim H \ge 2$, then $B^+ \subset \mu(H)$.
Since $H \subset AN$ and $\dim H \ge 2$, it is easy to construct a continuous, proper map $\Phi\colon [0,1] \times {\mathord{\mathbb{R}}}^+ \to H$ such that $\Phi(1,t) = \Phi(0,t)^{-1}$, for all $t \in {\mathord{\mathbb{R}}}^+$ (cf.Figure \[ConstructPhiFig\]). For example, choose two linearly independent elements $u$ and $v$ of ${\mathfrak{\lowercase{H}}}$, and define $$\Phi(s,t) = \exp \bigl( t \cos(\pi s) u + t \sin(\pi s) v \bigr) .$$
![Construction of $\Phi(s,t)$.[]{data-label="ConstructPhiFig"}](htohinv.eps)
If we identify $A$ with its Lie algebra ${\mathfrak{\lowercase{a}}}$, then $A^+$ is a convex cone in ${\mathfrak{\lowercase{a}}}$ and the opposition involution $\tau$ is the reflection in $A^+$ across the ray $B^+$. Thus, for any $a\in A^+$, the points $a$ and $\tau(a)$ are on opposite sides of $B^+$, so any continuous curve in $A^+$ from $a$ to $\tau(a)$ must intersect $B^+$. In particular, for each $t \in {\mathord{\mathbb{R}}}^+$, the curve $$\{\, \mu \bigl( \Phi(s,t) \bigr) \mid 0 \le s \le 1\,\}$$ from $\mu
\bigl( \Phi(0,t) \bigr)$ to $\mu \bigl( \Phi(1,t) \bigr)$ must intersect $B^+$. Thus, we see, from an elementary continuity argument, that $\mu \bigl[ \Phi\bigl( [0,1] \times {\mathord{\mathbb{R}}}^+ \bigr) \bigr]$ contains $B^+$. Therefore, $B^+$ is contained in $\mu(H)$.
![Proposition \[SL3-B+\]: $\mu(H)$ is on both sides of $B^+$, so it must contain $B^+$.[]{data-label="SL3B+Fig"}](B+.eps)
![Proof of Theorem \[SL3->notess\]: $\mu(\Gamma)$ stays away from $B^+$, because $B^+ \subset \mu(H)$. Also, half of $\mu(\Gamma)$ is on each side of $B^+$, because $\tau \bigl( \mu(\Gamma) \bigr) = \mu(\Gamma)$. This contradicts the fact that $\Gamma$ has only one end.[]{data-label="SL3notessFig"}](awayfromB+.eps)
Suppose $\Gamma$ is a crystallographic group for $G/H$. (This will lead to a contradiction.) We may assume $H \subset AN$ [[(]{}see \[HcanbeAN\][)]{}]{}.
Let $F$ be a (symmetric) finite generating set for $\Gamma$, and choose a compact, convex, symmetric subset $C$ of $A$ so large that $$\mu(\gamma F) \subset \mu(\gamma) C$$ for every $\gamma \in \Gamma$ [[(]{}see \[bddchange\][)]{}]{}.
From Lemma \[1DinSL3\], we know that $\dim H \ge 2$, so Proposition \[SL3-B+\] implies that $B^+ \subset \mu(H)$. Then, because $\Gamma$ acts properly on $G/H$, we conclude that $\mu(\Gamma) \cap B^+ C$ is finite [[(]{}see \[proper<>mu(L)\][)]{}]{}. Since $\mu$ is a proper map, this implies that $\Gamma \cap \mu^{-1}(B^+ C)$ is finite.
Let $A_1$ and $A_2$ be the two components of $A^+ \smallsetminus B^+$. Because $\Gamma^{-1} = \Gamma$, we know that $\tau \bigl( \mu(\Gamma) \bigr) =
\mu(\Gamma)$. Then, because $\tau$ interchanges $A_1$ and $A_2$, we conclude that $\tau \bigl( \mu(\Gamma) \cap A_1 \bigr) = \mu(\Gamma) \cap A_2$. Therefore, $\mu(\Gamma) \cap A_1$ and $\mu(\Gamma) \cap A_2$ have the same cardinality, so they must both be infinite. So $$\mbox{each of $\Gamma \cap \mu^{-1}(A_1)$ and $\Gamma \cap \mu^{-1}(A_2)$
is infinite.}$$ Because $\Gamma$ has only one end [[(]{}see \[tess->1end\][)]{}]{}, this implies there exist $$\gamma \in \bigl( \Gamma \cap \mu^{-1}(A_1) \bigr) \smallsetminus \mu^{-1}(B^+
C) ,$$ such that $$\label{SL3->notessPf-gammaf}
\gamma f \in \bigl( \Gamma \cap \mu^{-1}(A_2) \bigr)
\smallsetminus \mu^{-1}(B^+ C) ,$$ for some $f \in F$. Then $\mu(\gamma) \in A_1$, $\mu(\gamma f) \in A_2$, and $$\mu(\gamma f) \in \mu(\gamma F) \subset \mu(\gamma) C .$$ Using the fact that $C$ is symmetric and the fact that $C$ contains the identity element $e$, we conclude that $$\mu(\gamma) \in \bigl( \mu(\gamma f) C \bigr) \cap A_1
\mbox{\qquad and\qquad}
\mu(\gamma f) \in \bigl( \mu(\gamma f) C \bigr) \cap A_2
;$$ therefore $\mu(\gamma f) C$ intersects both $A_1$ and $A_2$. Since $B^+$ separates $A_1$ from $A_2$, and $C$ is connected, this implies that $\mu(\gamma f) C$ intersects $B^+$; hence $\mu(\gamma f) \in B^+ C$. This contradicts the fact that $\gamma f \notin
\mu^{-1}(B^+ C)$ [[(]{}see \[SL3->notessPf-gammaf\][)]{}]{}.
Explicit coordinates on ${\operatorname{{\mathfrak{\lowercase{SO}}}}}(2,n)$ and ${\operatorname{{\mathfrak{\lowercase{SU}}}}}(2,n)$ {#coordsSect}
===================================================================================================================================
From this point on, we focus almost entirely on ${\operatorname{SO}}(2,n)$ and ${\operatorname{SU}}(2,n)$. (The only exception is that some of the examples constructed in Section \[ExistenceSect\] are for other groups.) In this section, we define the group ${\operatorname{SU}}(2,n;{\mathbb{F}})$, which allows us to provide a fairly unified treatment of ${\operatorname{SO}}(2,n)$ and ${\operatorname{SU}}(2,n)$ in later sections.
The group ${\operatorname{SU}}(2,n;{\mathbb{F}})$
-------------------------------------------------
- We use ${\mathbb{F}}$ to denote either ${\mathord{\mathbb{R}}}$ or ${\mathord{\mathbb{C}}}$.
- Let ${q}= {\operatorname{\dim_{{\mathord{\mathbb{R}}}}}}{\mathbb{F}}$, so ${q}\in \{1,2\}$.
- We use ${{\mathbb{F}}_{\text{imag}}}$ to denote the purely imaginary elements of ${\mathbb{F}}$, so $${{\mathbb{F}}_{\text{imag}}}=
\begin{cases}
\hfil 0 & \mbox{if ${\mathbb{F}}= {\mathord{\mathbb{R}}}$} \\
i {\mathord{\mathbb{R}}}& \mbox{if ${\mathbb{F}}= {\mathord{\mathbb{C}}}$} .
\end{cases}$$
- For $\phi \in {\mathbb{F}}$, there exist unique ${\operatorname{Re}}\phi \in {\mathord{\mathbb{R}}}$ and ${\operatorname{Im}}\phi \in {{\mathbb{F}}_{\text{imag}}}$, such that $\phi = {\operatorname{Re}}\phi + {\operatorname{Im}}\phi$. (**Warning:** in our notation, the imaginary part of $a+bi$ is $bi$, *not* $b$.)
- For $\phi \in {\mathbb{F}}$, we use ${\overline{\phi}}$ to denote the conjugate ${\operatorname{Re}}\phi - {\operatorname{Im}}\phi$ of $\phi$. (If ${\mathbb{F}}= {\mathord{\mathbb{R}}}$, then ${\overline{\phi}} = \phi$.)
- For a row vector $x \in {\mathbb{F}}^{n-2}$, or, more generally, for any matrix $x$ with entries in ${\mathbb{F}}$, we use $x^\dagger$ to denote the conjugate-transpose of $x$.
\[SUFDefn\] For $$J =
\begin{pmatrix}
0 & 0 & 0 & \cdots & 0 & 0 & 1 \\
0 & 0 & 0 & \cdots & 0 & 1 & 0 \\
0 & 0 & & & & 0 & 0 \\
\vdots & \vdots & & {\operatorname{Id}}& & \vdots & \vdots \\
0 & 0 & & & & 0 & 0 \\
0 & 1 & 0 & \cdots & 0 & 0 & 0 \\
1 & 0 & 0 & \cdots & 0 & 0 & 0
\end{pmatrix}
\in {\operatorname{SL}}(n+2, {\mathbb{F}}) ,$$ we define $${\operatorname{SU}}(2,n;{\mathbb{F}})
= \{\, g \in {\operatorname{SL}}(n+2,{\mathbb{F}}) \mid g J g^\dagger = J \,\}$$ and $${\operatorname{{\mathfrak{\lowercase{SU}}}}}(2,n;{\mathbb{F}})
= \{\, u \in {\operatorname{{\mathfrak{\lowercase{SL}}}}}(n+2,{\mathbb{F}}) \mid u J + J u^\dagger = 0 \,\}.$$ Then:
- ${\operatorname{SU}}(2,n;{\mathord{\mathbb{R}}})$ is a realization of ${\operatorname{SO}}(2,n)$,
- ${\operatorname{SU}}(2,n;{\mathord{\mathbb{C}}})$ is a realization of ${\operatorname{SU}}(2,n)$, and
- ${\operatorname{{\mathfrak{\lowercase{SU}}}}}(2,n;{\mathbb{F}})$ is the Lie algebra of ${\operatorname{SU}}(2,n;{\mathbb{F}})$.
We choose
- $A$ to consist of the diagonal matrices in ${\operatorname{SU}}(2,n;{\mathbb{F}})$ that have nonnegative real entries,
- $N$ to consist of the upper-triangular matrices in ${\operatorname{SU}}(2,n;{\mathbb{F}})$ with only $1$’s on the diagonal, and
- $K = {\operatorname{SU}}(2,n;{\mathbb{F}}) \cap {\operatorname{SU}}(n+2)$.
A straightforward matrix calculation shows that the Lie algebra of $AN$ is $$\label{SUF-AN}
{\mathfrak{\lowercase{a}}} + {\mathfrak{\lowercase{N}}} =
{\left\{\,
\begin{pmatrix}
t_1 & \phi & x & \eta & {{\mathord{\mathsf{x}}}}\\
0 & t_2 & y & {{\mathord{\mathsf{y}}}}& -{\overline{\eta}} \\
0 & 0 & 0 & -y^{\dagger} & -x^{\dagger} \\
0 & 0 & 0 & -t_2 & -{\overline{\phi}} \\
0 & 0 & 0 & 0 & -t_1 \\
\end{pmatrix}
\mathrel{\left| \vphantom {\left\{
\begin{pmatrix}
t_1 & \phi & x & \eta & {{\mathord{\mathsf{x}}}}\\
0 & t_2 & y & {{\mathord{\mathsf{y}}}}& -{\overline{\eta}} \\
0 & 0 & 0 & -y^{\dagger} & -x^{\dagger} \\
0 & 0 & 0 & -t_2 & -{\overline{\phi}} \\
0 & 0 & 0 & 0 & -t_1 \\
\end{pmatrix}
\mid
\begin{matrix}
t_1,t_2 \in {\mathord{\mathbb{R}}}, \\
\phi,\eta \in {\mathbb{F}}, \\
x,y \in {\mathbb{F}}^{n-2}, \\
{{\mathord{\mathsf{x}}}}, {{\mathord{\mathsf{y}}}}\in {{\mathbb{F}}_{\text{imag}}}\end{matrix} \right\} } \right.}
\begin{matrix}
t_1,t_2 \in {\mathord{\mathbb{R}}}, \\
\phi,\eta \in {\mathbb{F}}, \\
x,y \in {\mathbb{F}}^{n-2}, \\
{{\mathord{\mathsf{x}}}}, {{\mathord{\mathsf{y}}}}\in {{\mathbb{F}}_{\text{imag}}}\end{matrix} \,\right\} }
.$$
\[tworows\] From , we see that the first two rows of any element of ${\mathfrak{\lowercase{a}}} + {\mathfrak{\lowercase{N}}}$ are sufficient to determine the entire matrix. In fact, it is also not necessary to specify the last entry of the second row of the matrix.
\[d(SU2)\] From [[(]{}\[d(G)\][)]{}]{} and [[(]{}\[SUF-AN\][)]{}]{}, we see that $d \bigl( {\operatorname{SU}}(2,n;{\mathbb{F}}) \bigr)
= \dim( {\mathfrak{\lowercase{A}}} +
{\mathfrak{\lowercase{N}}}) = 2 {q}n$.
Because $N$ is simply connected and nilpotent, the exponential map is a diffeomorphism from ${\mathfrak{\lowercase{N}}}$ to $N$ (indeed, its inverse, the logarithm map, is a polynomial [@Hochschild-Algic Thm. 8.1.1, p. 107], so each element of $N$ has a unique representation in the form $\exp u$ with $u
\in {\mathfrak{\lowercase{N}}}$. Thus, each element $h$ of $N$ determines corresponding values of $\phi$, $x$, $y$, $\eta$, ${{\mathord{\mathsf{x}}}}$ and ${{\mathord{\mathsf{y}}}}$ (with $t_1 = t_2 = 0$). We write $$\phi_h, x_h, y_h, \eta_h, {{\mathord{\mathsf{x}}}}_h, {{\mathord{\mathsf{y}}}}_h$$ for these values.
\[simpleroots\] We let $\alpha$ and $\beta$ be the simple real roots of ${\operatorname{SU}}(2,n;{\mathbb{F}})$, defined by $$\mbox{$\alpha(a) = a_{1,1}/a_{2,2}$ and $\beta(a) = a_{2,2}$} ,$$ for a (diagonal) element $a$ of $A$. Thus, the positive real roots (see Figure \[rootspict\]) are $$\begin{cases}
\alpha, \beta, \alpha+\beta, \alpha+2\beta,
& \mbox{if ${\mathbb{F}}= {\mathord{\mathbb{R}}}$} \\
\alpha, \beta, \alpha+\beta, \alpha+2\beta, 2\beta, 2\alpha+2\beta &
\mbox{if ${\mathbb{F}}= {\mathord{\mathbb{C}}}$} . \\
\end{cases}$$ Concretely:
- the root space ${{\mathfrak{\lowercase{N}}}}_\alpha$ is the $\phi$-subspace in ${\mathfrak{\lowercase{N}}}$,
- the root space ${{\mathfrak{\lowercase{N}}}}_\beta$ is the $y$-subspace in ${\mathfrak{\lowercase{N}}}$,
- the root space ${{\mathfrak{\lowercase{N}}}}_{\alpha+\beta}$ is the $x$-subspace in ${\mathfrak{\lowercase{N}}}$,
- the root space ${{\mathfrak{\lowercase{N}}}}_{\alpha+2\beta}$ is the $\eta$-subspace in ${\mathfrak{\lowercase{N}}}$,
- the root space ${{\mathfrak{\lowercase{N}}}}_{2\beta}$ is the ${{\mathord{\mathsf{y}}}}$-subspace in ${\mathfrak{\lowercase{N}}}$ (this is 0 if ${\mathbb{F}}= {\mathord{\mathbb{R}}}$), and
- the root space ${{\mathfrak{\lowercase{N}}}}_{2\alpha+2\beta}$ is the ${{\mathord{\mathsf{x}}}}$-subspace in ${\mathfrak{\lowercase{N}}}$ (this is 0 if ${\mathbb{F}}= {\mathord{\mathbb{R}}}$).
![The real root systems of (a) ${\operatorname{SU}}(2,n;{\mathord{\mathbb{R}}}) = {\operatorname{SO}}(2,n)$ and (b) ${\operatorname{SU}}(2,n;{\mathord{\mathbb{C}}}) = {\operatorname{SU}}(2,n)$.[]{data-label="rootspict"}](BC2-vars.eps)
Let $$\begin{aligned}
{{\mathfrak{\lowercase{D}}}}&= {{\mathfrak{\lowercase{N}}}}_{\alpha+2\beta} + {{\mathfrak{\lowercase{N}}}}_{2\alpha+2\beta} +
{{\mathfrak{\lowercase{N}}}}_{2\beta} \\
&= \{\, z \in {\mathfrak{\lowercase{N}}} \mid \phi_z = 0, x_z = y_z = 0 \,\} \\
&=
{\left\{\,
\begin{pmatrix}
0 & 0 & 0 & \eta & {{\mathord{\mathsf{x}}}}\\
& 0 & 0 & {{\mathord{\mathsf{y}}}}& -{\overline{\eta}} \\
& & \dots
\end{pmatrix}
\mathrel{\left| \vphantom {\left\{
\begin{pmatrix}
0 & 0 & 0 & \eta & {{\mathord{\mathsf{x}}}}\\
& 0 & 0 & {{\mathord{\mathsf{y}}}}& -{\overline{\eta}} \\
& & \dots
\end{pmatrix}
\mid
\begin{matrix}
\eta \in {\mathbb{F}}, \\
{{\mathord{\mathsf{x}}}}, {{\mathord{\mathsf{y}}}}\in {{\mathbb{F}}_{\text{imag}}}\end{matrix}
\right\} } \right.}
\begin{matrix}
\eta \in {\mathbb{F}}, \\
{{\mathord{\mathsf{x}}}}, {{\mathord{\mathsf{y}}}}\in {{\mathbb{F}}_{\text{imag}}}\end{matrix}
\,\right\} }
.
\end{aligned}$$ and, for a given Lie algebra ${\mathfrak{\lowercase{H}}} \subset {\mathfrak{\lowercase{N}}}$, $${{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}= {{\mathfrak{\lowercase{D}}}}\cap {\mathfrak{\lowercase{H}}}.$$ Note that if $\phi_u = 0$ for every $u \in {\mathfrak{\lowercase{H}}}$, then $[{\mathfrak{\lowercase{H}}}, {\mathfrak{\lowercase{H}}}]
\subset {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$ and ${{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$ is contained in the center of ${\mathfrak{\lowercase{H}}}$ [(cf. \[\[u,v\]\])]{}.
By definition [[(]{}see \[A+Defn\][)]{}]{}, we have $$A^+ = \{\, a \in A \mid \alpha(a) \ge 1, \beta(a) \ge 1 \,\} .$$ Therefore, from the definition of $\alpha$ and $\beta$ [[(]{}see \[simpleroots\][)]{}]{}, we see that $$\label{A+}
A^+ = \{\, a \in A \mid a_{1,1} \ge a_{2,2} \ge 1
\,\} .$$
For ${\mathbb{F}}= {\mathord{\mathbb{H}}}$, the division algebra of real quaternions, the group ${\operatorname{SU}}(2,n;{\mathord{\mathbb{H}}})$ is a realization of ${\operatorname{Sp}}(2,n)$. Most of the work in this paper carries over, but the upper bound on $\dim H$ given in Theorem \[maxnolinear\] is not sharp in this case (and it does not seem to be easy to improve this result to obtain a sharp bound). Thus, we have not obtained any interesting conclusions about the nonexistence of tessellations of homogeneous spaces of ${\operatorname{Sp}}(2,n)$.
The subgroups ${\operatorname{SU}}(1,n;{\mathbb{F}})$ and ${\operatorname{Sp}}(1,m;{\mathbb{F}})$
-------------------------------------------------------------------------------------------------
We now describe how the four important families of homogeneous spaces of Example \[KulkarniEg\] are realized in terms of ${\operatorname{SU}}(2,n;{\mathbb{F}})$.
\[SU1nDefn\] Let
- ${\operatorname{SU}}(1,n;{\mathord{\mathbb{R}}}) = {\operatorname{SO}}(1,n)$;
- ${\operatorname{Sp}}(1,n;{\mathord{\mathbb{R}}}) = {\operatorname{SU}}(1,n)$;
- ${\operatorname{SU}}(1,n;{\mathord{\mathbb{C}}}) = {\operatorname{SU}}(1,n)$; and
- ${\operatorname{Sp}}(1,n;{\mathord{\mathbb{C}}}) = {\operatorname{Sp}}(1,n)$.
Then, for an appropriate choice of the embeddings in Example \[KulkarniEg\], we have $$\label{SU1nAN}
{\operatorname{{\mathfrak{\lowercase{SU}}}}}(1,n;{\mathbb{F}}) \cap ({\mathfrak{\lowercase{A}}} + {\mathfrak{\lowercase{N}}}) =
{\left\{\,
\begin{pmatrix}
t & \phi & x & \phi & {{\mathord{\mathsf{x}}}}\\
0 & 0 & 0 & 0 & -{\overline{\phi}} \\
& & \dots
\end{pmatrix}
\mathrel{\left| \vphantom {\left\{
\begin{pmatrix}
t & \phi & x & \phi & {{\mathord{\mathsf{x}}}}\\
0 & 0 & 0 & 0 & -{\overline{\phi}} \\
& & \dots
\end{pmatrix}
\mid
\begin{matrix}
t \in {\mathord{\mathbb{R}}}, \\
\phi \in {\mathbb{F}}, \\
x \in {\mathbb{F}}^{n-2}, \\
{{\mathord{\mathsf{x}}}}\in {{\mathbb{F}}_{\text{imag}}}\end{matrix}
\right\} } \right.}
\begin{matrix}
t \in {\mathord{\mathbb{R}}}, \\
\phi \in {\mathbb{F}}, \\
x \in {\mathbb{F}}^{n-2}, \\
{{\mathord{\mathsf{x}}}}\in {{\mathbb{F}}_{\text{imag}}}\end{matrix}
\,\right\} }$$ and (if $2m \le n$) we have $$\begin{gathered}
\label{Sp1mAN}
{\operatorname{{\mathfrak{\lowercase{Sp}}}}}(1,m;{\mathbb{F}}) \cap ({\mathfrak{\lowercase{A}}} + {\mathfrak{\lowercase{N}}})
= \\
\setcounter{MaxMatrixCols}{15} {\left\{\,
\begin{pmatrix}
t & 0 & x_1 & x_2 & x_3 & x_4 & \dots & x_{2m-3} & x_{2m-2} & 0
& \dots & 0
& \eta & {{\mathord{\mathsf{x}}}}\\
0 & t & -{\overline{x_2}} & {\overline{x_1}} & -{\overline{x_4}} & {\overline{x_3}} & \dots &
-{\overline{x_{2m-2}}} & {\overline{x_{2m-3}}} & 0 & \dots & 0 & -{{\mathord{\mathsf{x}}}}& -{\overline{\eta}} \\
& & & & & & & \dots
\end{pmatrix}
\mathrel{\left| \vphantom {\left\{
\begin{pmatrix}
t & 0 & x_1 & x_2 & x_3 & x_4 & \dots & x_{2m-3} & x_{2m-2} & 0
& \dots & 0
& \eta & {{\mathord{\mathsf{x}}}}\\
0 & t & -{\overline{x_2}} & {\overline{x_1}} & -{\overline{x_4}} & {\overline{x_3}} & \dots &
-{\overline{x_{2m-2}}} & {\overline{x_{2m-3}}} & 0 & \dots & 0 & -{{\mathord{\mathsf{x}}}}& -{\overline{\eta}} \\
& & & & & & & \dots
\end{pmatrix}
\mid
\begin{matrix}
t \in {\mathord{\mathbb{R}}}, \\
x_j \in {\mathbb{F}}, \\
\eta \in {\mathbb{F}}, \\
{{\mathord{\mathsf{x}}}}\in {{\mathbb{F}}_{\text{imag}}}\end{matrix}
\right\} } \right.}
\begin{matrix}
t \in {\mathord{\mathbb{R}}}, \\
x_j \in {\mathbb{F}}, \\
\eta \in {\mathbb{F}}, \\
{{\mathord{\mathsf{x}}}}\in {{\mathbb{F}}_{\text{imag}}}\end{matrix}
\,\right\} }
.
\end{gathered}$$
\[d(Sp)\] From [[(]{}\[d(G)\][)]{}]{}, [[(]{}\[SU1nAN\][)]{}]{} and [[(]{}\[Sp1mAN\][)]{}]{}, we see that
- $ d \bigl( {\operatorname{SU}}(1,n;{\mathbb{F}}) \bigr)
= \dim \bigl( {\operatorname{{\mathfrak{\lowercase{SU}}}}}(1,n;{\mathbb{F}}) \cap ({\mathfrak{\lowercase{A}}} + {\mathfrak{\lowercase{N}}}) \bigr)
= {q}n$ and
- $ d \bigl( {\operatorname{Sp}}(1,m;{\mathbb{F}}) \bigr)
= \dim \bigl( {\operatorname{{\mathfrak{\lowercase{Sp}}}}}(1,m;{\mathbb{F}}) \cap ({\mathfrak{\lowercase{A}}} + {\mathfrak{\lowercase{N}}}) \bigr)
= 2 {q}m$.
Formulas for exponentials and brackets
--------------------------------------
The arguments in later sections often require the calculation of $\exp
u$, for some $u \in {\mathfrak{\lowercase{N}}}$, or of $[u,v]$, for some $u,v \in {\mathfrak{\lowercase{N}}}$. We now provide these calculations for the reader’s convenience.
For $$u =
\begin{pmatrix}
0 & \phi & x & \eta & {{\mathord{\mathsf{x}}}}\\
0 & 0 & y & {{\mathord{\mathsf{y}}}}& -{\overline{\eta}} \\
0 & 0 & 0 & -y^{\dagger} & -x^{\dagger} \\
0 & 0 & 0 & 0 & -{\overline{\phi}} \\
0 & 0 & 0 & 0 & 0 \\
\end{pmatrix}
\in {\mathfrak{\lowercase{N}}}
,$$ we have $$\label{exp}
\exp(u)=
\begin{pmatrix}
{\vphantom{\vrule height 15pt depth 5pt}}1 & \phi & x+\frac{1}{2} \phi y
& \eta -\frac{1}{2} x y^{\dagger} + \frac{1}{2} \phi {{\mathord{\mathsf{y}}}}-
\frac{1}{6} \phi |y|^2
& \vphantom{\vrule height 20pt depth 20pt}
\genfrac{}{}{0pt}{0}
{
- \frac{1}{2} |x|^2
- {\operatorname{Re}}(\phi {\overline{\eta}})
+\frac{1}{24} |\phi|^2 |y|^2
}{
+ \left(
{{\mathord{\mathsf{x}}}}- \frac{1}{6} \phi {{\mathord{\mathsf{y}}}}{\overline{\phi}}
+ \frac{1}{3} {\operatorname{Im}}(x y^{\dagger} {\overline{\phi}} )
\right)
} \\
{\vphantom{\vrule height 15pt depth 5pt}}0 & 1 & y
& {{\mathord{\mathsf{y}}}}-\frac{1}{2} |y|^2
& -{\overline{\eta}} -\frac{1}{2} y x^{\dagger}-\frac{1}{2} {{\mathord{\mathsf{y}}}}{\overline{\phi}}
+\frac{1}{6} |y|^2 {\overline{\phi}} \\
{\vphantom{\vrule height 15pt depth 5pt}}0 & 0 &{\operatorname{Id}}& -y^{\dagger} & -x^{\dagger}+\frac{1}{2}
y^{\dagger} {\overline{\phi}} \\
{\vphantom{\vrule height 15pt depth 5pt}}0 & 0 & 0 & 1 & -\phi^{\dagger} \\
{\vphantom{\vrule height 15pt depth 5pt}}0 & 0 & 0 & 0 & 1 \\
\end{pmatrix}
.$$
When $\phi = 0$, this simplifies to: $$\label{exp(phi=0)}
\exp(u)=
\begin{pmatrix}
{\vphantom{\vrule height 15pt depth 5pt}}1 & 0 & x
& \eta -\frac{1}{2} x y^{\dagger}
& \vphantom{\vrule height 20pt depth 20pt}
{{\mathord{\mathsf{x}}}}- \frac{1}{2} |x|^2 \\
{\vphantom{\vrule height 15pt depth 5pt}}0 & 1 & y
& {{\mathord{\mathsf{y}}}}-\frac{1}{2} |y|^2
& -{\overline{\eta}} -\frac{1}{2} y x^{\dagger} \\
{\vphantom{\vrule height 15pt depth 5pt}}0 & 0 &{\operatorname{Id}}& -y^{\dagger} & -x^{\dagger} \\
{\vphantom{\vrule height 15pt depth 5pt}}0 & 0 & 0 & 1 & 0 \\
{\vphantom{\vrule height 15pt depth 5pt}}0 & 0 & 0 & 0 & 1 \\
\end{pmatrix}
.$$ Similarly, when $y = 0$, we have $$\label{exp(y=0)}
\exp(u)=
\begin{pmatrix}
{\vphantom{\vrule height 15pt depth 5pt}}1 & \phi & x
& \eta + \frac{1}{2} \phi {{\mathord{\mathsf{y}}}}& \vphantom{\vrule height 20pt depth 20pt}
\genfrac{}{}{0pt}{0}
{
- \frac{1}{2} |x|^2
- {\operatorname{Re}}(\phi {\overline{\eta}})
}{
+ \left(
{{\mathord{\mathsf{x}}}}- \frac{1}{6} \phi {{\mathord{\mathsf{y}}}}{\overline{\phi}}
\right)
} \\
{\vphantom{\vrule height 15pt depth 5pt}}0 & 1 & 0
& {{\mathord{\mathsf{y}}}}& -{\overline{\eta}} -\frac{1}{2} {{\mathord{\mathsf{y}}}}{\overline{\phi}} \\
{\vphantom{\vrule height 15pt depth 5pt}}0 & 0 &{\operatorname{Id}}& 0 & -x^{\dagger} \\
{\vphantom{\vrule height 15pt depth 5pt}}0 & 0 & 0 & 1 & -\phi^{\dagger} \\
{\vphantom{\vrule height 15pt depth 5pt}}0 & 0 & 0 & 0 & 1 \\
\end{pmatrix}
.$$
For $$\label{umatrix}
u =
\begin{pmatrix}
0 & \phi & x & \eta & {{\mathord{\mathsf{x}}}}\\
& 0 & y & {{\mathord{\mathsf{y}}}}& -{\overline{\eta}} \\
& & \cdots \\
\end{pmatrix}
\mbox{\qquad and\qquad}
\tilde u =
\begin{pmatrix}
0 & \tilde \phi & \tilde x & \tilde \eta & \tilde {{\mathord{\mathsf{x}}}}\\
& 0 & \tilde y & \tilde {{\mathord{\mathsf{y}}}}& - {\overline{\tilde {\eta}}} \\
& & \cdots \\
\end{pmatrix}
,$$ we have $$\label{[u,v]}
[u, \tilde u] =
\begin{pmatrix}
0 & 0 & \phi \tilde y - \tilde \phi y
& - x \tilde y^{\dagger} + \tilde x y^{\dagger} + \phi \tilde {{\mathord{\mathsf{y}}}}-
\tilde \phi {{\mathord{\mathsf{y}}}}& -2 {\operatorname{Im}}(x \tilde x^{\dagger} + \phi {\overline{\tilde \eta}}
- \tilde \phi {\overline{\eta}}) \\
& 0 & 0 & -2 {\operatorname{Im}}(y \tilde y^{\dagger})
& \tilde y x^{\dagger} - y \tilde x^{\dagger} + \tilde {{\mathord{\mathsf{y}}}}{\overline{\phi}} -
{{\mathord{\mathsf{y}}}}{\overline{\tilde \phi}} \\
& & & \cdots \\
\end{pmatrix}
.$$
\[conjugation\] For $u,v \in {\mathfrak{\lowercase{H}}}$, we have $$\exp(-v) u \exp(v)
= u
+ [u,v]
+ \frac{1}{2} \bigl[ [u,v] , v \bigr]
+ \frac{1}{3!} \Bigl[ \bigl[ [u,v] , v \bigr] , v \Bigr]
+ \cdots
.$$ Combining this with allows us to calculate the effect of conjugating by an element of $N$.
For example, suppose $u \in {\mathfrak{\lowercase{N}}}$, with $\phi_u = 0$ and $y_u = 0$, and suppose $v \in {{\mathfrak{\lowercase{N}}}}_{\alpha+\beta}$. We see, from [[(]{}\[\[u,v\]\][)]{}]{}, that $\phi_{[u,v]} = 0$ and that $x_{[u,v]} = y_{[u,v]} = 0$, so $\bigl[ [u,v] , v \bigr] = 0$ [[(]{}see \[\[u,v\]\][)]{}]{}. Therefore $$\exp(-v) u \exp(v) = u + [u,v] .$$
Calculating the Cartan projection {#CalcSect}
=================================
Y. Benoist [@Benoist Lem. 2.4] showed that calculating values of the Cartan projection $\mu$ is no more difficult than calculating the norm of a matrix [[(]{}see \[mucalc\][)]{}]{}. In this section, we describe this elegant method and some of its consequences, in the special case $G = {\operatorname{SU}}(2,n;{\mathbb{F}})$.
Throughout this section, we assume $G = {\operatorname{SU}}(2,n;{\mathbb{F}})$.
The basic definitions
---------------------
We employ the usual Big Oh and little oh notation: for functions $f_1,f_2$ on a subset $X$ of $G$, we say $$\mbox{\emph{$f_1 = O(f_2)$ for $h \in X$}}$$ if there is a constant $C$, such that, for all $h \in X$ with $\lVert
h\rVert$ large, we have $\lVert f_1(h)\rVert \le C \lVert f_2(h)\rVert$. (The values of each $f_j$ are assumed to belong to some finite-dimensional normed vector space, typically either ${\mathord{\mathbb{C}}}$ or a space of complex matrices. Which particular norm is used does not matter, because all norms are equivalent up to a bounded factor.) We say $$\mbox{\emph{$f_1 = o(f_2)$ for $h \in X$}}$$ if $\lVert f_1(h)\rVert/\lVert f_2(h)\rVert \to 0$ as $h \to \infty$. (We use $h \to \infty$ to mean $\lVert h\rVert \to \infty$.) Also, we write $$f_1 \asymp f_2$$ if $f_1 = O(f_2)$ and $f_2 = O(f_1)$.
We use the following norm on ${\operatorname{SU}}(2,n;{\mathbb{F}})$, because it is easy to calculate. The reader is free to make a different choice, at the expense of changing $=$ to $\asymp$ in a few of the calculations.
For $h \in {\operatorname{SU}}(2,n;{\mathbb{F}})$, we define $\lVert h\rVert$ to be the maximum absolute value among the matrix entries of $h$. That is, $$\lVert h\rVert = \max_{1 \le j,k \le n+2} |h_{j,k}| .$$
\[rhoDefn\] Define $\rho \colon {\operatorname{SU}}(2,n;{\mathbb{F}}) \to {\operatorname{GL}}({\mathbb{F}}^{n+2} \wedge {\mathbb{F}}^{n+2})$ by $\rho(h) = h \wedge h$, so $\rho$ is the second exterior power of the standard representation of ${\operatorname{SU}}(2,n;{\mathbb{F}})$. Thus, we may define $\lVert \rho(h)
\rVert$ to be the maximum absolute value among the determinants of all the $2
\times 2$ submatrices of the matrix $h$. That is, $$\lVert\rho(h)\rVert = \max_{1 \le j,k,\ell,m \le n+2}
\left| \det
\begin{pmatrix}
h_{j,k} & h_{j,\ell} \\
h_{m,k} & h_{m,\ell}
\end{pmatrix}
\right|
.$$
From [[(]{}\[exp\][)]{}]{}, [[(]{}\[exp(phi=0)\][)]{}]{}, and [[(]{}\[exp(y=0)\][)]{}]{}, it is clear that the $2 \times 2$ minor in the top right corner is often larger than the other $2
\times 2$ minors, so we give it a special name.
\[DeltaDefn\] For $h \in {\operatorname{Mat}}_{n+2}({\mathbb{F}})$, define $$\Delta(h) =
\det
\begin{pmatrix}
h_{1,n+1} & h_{1,n+2} \\
h_{2,n+1} & h_{2,n+2}
\end{pmatrix}
.$$
Y. Benoist’s method for using matrix norms to calculate $\mu$ {#CalcSect-Benoist}
-------------------------------------------------------------
\[calcrho\] For $a \in A^+$, we have $\lVert a\rVert = a_{1,1}$ and $\lVert\rho(a)\rVert = a_{1,1} a_{2,2}$.
From [[(]{}\[SUF-AN\][)]{}]{}, we see that $$\label{ajj}
a_{j,j} =
\begin{cases}
1 & \mbox{if $3 \le j \le n$} \\
1/a_{2,2} & \mbox{if $j = n+1$} \\
1/a_{1,1} & \mbox{if $j = n+2$} .
\end{cases}$$ Thus, from [[(]{}\[A+\][)]{}]{}, we see that $$a_{1,1} \ge a_{2,2} \ge a_{j,j}$$ for $j \ge 3$ (and, since $a$ is diagonal, we have $a_{j,k} = 0$ for $j \neq
k$). Therefore, the desired conclusions follow from the definitions of $\lVert
a\rVert$ and $\lVert\rho(a)\rVert$.
\[mu=rho\] We have $$\begin{gathered}
\mu(h) \asymp h, \label{h=mu(h)} \\
\rho \bigl( \mu(h) \bigr) \asymp \rho(h), \label{rho(h)=rho(mu)} \\
\mbox{$\mu(h)_{1,1} \asymp \lVert h\rVert$,
and
$\mu(h)_{2,2} \asymp \lVert \rho(h) \rVert/\lVert h\rVert$,} \label{mucalc}
\end{gathered}$$ for $h \in {\operatorname{SU}}(2,n;{\mathbb{F}})$.
Choose $k_1,k_2 \in K$, such that $\mu(h) = k_1 h k_2$. Because $\lVert xy\rVert = O \bigl( \lVert x\rVert \lVert y\rVert \bigr)$ for $x,y \in {\operatorname{SU}}(2,n;{\mathbb{F}})$, and $\max_{k \in K} \lVert k\rVert < \infty$ (since $K$ is compact), we have $$\lVert \mu(h) \rVert = \lVert k_1 h k_2 \rVert = O \bigl( \lVert h\rVert
\bigr)$$ and $$\lVert h\rVert = \lVert k_1^{-1} \mu(h) k_2^{-1} \rVert = O \bigl(
\lVert\mu(h)\rVert \bigr),$$ so [[(]{}\[h=mu(h)\][)]{}]{} holds. Similarly, we have $$\lVert \rho \bigl( \mu(h) \bigr) \rVert
= \lVert \rho(k_1) \rho(h) \rho(k_2) \rVert
\asymp \lVert\rho(h)\rVert ,$$ so [[(]{}\[rho(h)=rho(mu)\][)]{}]{} holds.
For $a \in A^+$, we know, from [[(]{}\[calcrho\][)]{}]{}, that $a_{1,1} = \lVert a\rVert$ and $ a_{2,2} = \lVert\rho(a)\rVert / a_{1,1}$. Thus, letting $a = \mu(h)$, and using [[(]{}\[h=mu(h)\][)]{}]{} and [[(]{}\[rho(h)=rho(mu)\][)]{}]{}, we see that $$\mu(h)_{1,1} = \lVert\mu(h)\rVert \asymp \lVert h\rVert$$ and $$\mu(h)_{2,2} =
\frac{\lVert \rho \bigl( \mu(h) \bigr) \rVert} {\mu(h)_{1,1}}
\asymp \frac{\lVert\rho(h)\rVert}{\lVert h\rVert} ,$$ as desired.
Proposition \[mu=rho\] generalizes to any reductive group $G$ [@Benoist Lem. 2.3]. However, one may need to use a different representation in the place of $\rho$. In fact, if ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G = r$, then $r$ representations of $G$ are needed; for $G = {\operatorname{SU}}(2,n;{\mathbb{F}})$, we have ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G = 2$, and the two representations we use are $\rho$ and the identity representation $I(h) = h$.
\[gn=hn\] Let $g_n \to \infty$ and $h_n \to \infty$ be two sequences of elements of ${\operatorname{SU}}(2,n;{\mathbb{F}})$. We have $$\mbox{$g_n \asymp h_n$ and $\rho(g_n) \asymp \rho(h_n)$}$$ if and only if $$\mbox{there is a compact subset~$C$ of~$A$, such that, for all $n \in
{\mathord{\mathbb{Z}}}^+$, we have $\mu(g_n) \in \mu(h_n) C$} .$$
($\Rightarrow$) Let $a = \mu(h_n)^{-1} \mu(g_n)$. From [[(]{}\[mucalc\][)]{}]{}, we see that $ \mu(g_n)_{j,j} \asymp \mu(h_n)_{j,j}$ for $j \in \{1,2\}$, so, using [[(]{}\[ajj\][)]{}]{}, we have $$a_{j,j}
= \frac{\mu(g_n)_{j,j}}{\mu(h_n)_{j,j}}
= \begin{cases}
O(1) & \mbox{if $1 \le j \le 2$;} \\
1/1 = 1 & \mbox{if $3 \le j \le n$;} \\
\mu(h_n)_{2,2}/\mu(g_n)_{2,2} = O(1) & \mbox{if $j = n+1$;} \\
\mu(h_n)_{1,1}/\mu(g_n)_{1,1} = O(1) & \mbox{if $j = n+2$} .
\end{cases}$$ Therefore $a = O(1)$, as desired.
($\Leftarrow$) Because $C$ is compact, we have $$\mbox{$\mu(g_n) \asymp \mu(h_n)$ and
$\rho \bigl( \mu(g_n) \bigr) \asymp \rho \bigl( \mu(h_n) \bigr)$}$$ (cf. proof of [[(]{}\[h=mu(h)\][)]{}]{} and [[(]{}\[rho(h)=rho(mu)\][)]{}]{}). Then the desired conclusions follow from [[(]{}\[h=mu(h)\][)]{}]{} and [[(]{}\[rho(h)=rho(mu)\][)]{}]{}.
Because $C$ is compact, we have $g' \asymp g$ and $\rho(g') \asymp \rho(g)$ for any $g' \in CgC$ (cf. proof of [[(]{}\[h=mu(h)\][)]{}]{} and [[(]{}\[rho(h)=rho(mu)\][)]{}]{}). Thus, the desired conclusion follows from Corollary \[gn=hn\].
Because of Proposition \[mu=rho\], we will often need to calculate $\lVert
h\rVert$ and $\lVert\rho(h)\rVert$. The following observation and its corollary sometimes simplifies the work, by allowing us to replace $h$ with $h^{-1}$.
\[mu(h-1)\] We have $\mu(h^{-1}) = \mu(h)$ for $h \in {\operatorname{SU}}(2,n;{\mathbb{F}})$.
Define $J$ as in [[(]{}\[SUFDefn\][)]{}]{}, and choose $k_1,k_2 \in K$, such that $\mu(h) = k_1 h k_2$. For any $a \in A^+$, we see, using [[(]{}\[SUF-AN\][)]{}]{} or [[(]{}\[ajj\][)]{}]{}, that $J a^{-1} J = a$, so $$(J k_2^{-1}) h^{-1} (k_1^{-1} J) = J \, \mu(h)^{-1} \, J = \mu(h) .$$ Note that $\det J = 1$. Also, we have $J^2 = {\operatorname{Id}}$ and $J^\dagger = J$, so it is obvious that $J J J^\dagger = J$ and $J J^\dagger = {\operatorname{Id}}$. Therefore $$J \in {\operatorname{SU}}(2,n;{\mathbb{F}}) \cap {\operatorname{SU}}(n+2) = K .$$ Thus, from the definition of $\mu$, we conclude that $\mu(h^{-1}) = \mu(h)$, as desired.
The following corollary is obtained by combining Lemma \[mu(h-1)\] with Corollary \[gn=hn\].
\[rho(h-1)\] We have $h^{-1} \asymp h$ and $\rho(h^{-1}) \asymp \rho(h)$ for $h \in {\operatorname{SU}}(2,n;{\mathbb{F}})$.
The walls of $A^+$
------------------
For $k \in \{1,2\}$, set $$\label{Lk-defn}
L_k = \{\, a \in A^+ \mid a_{2,2} = a_{1,1}^{k-1} \,\} .$$ From [[(]{}\[A+\][)]{}]{}, we see that $L_1$ and $L_2$ are the two walls of $A^+$. From [[(]{}\[calcrho\][)]{}]{}, we have $$\label{rho(L)}
\mbox{$\rho(a) \asymp \lVert a\rVert^k$ for $a \in L_k$} .$$
We reproduce the proof of the following result, because it is both short and instructive. (Although we have no need for it here, let us point out that the converse of this proposition also holds, and that there is no need to assume $H \subset AN$.) Because of this proposition (and Corollary \[CDS->notess\]), Section \[SUFlargeSect\] will study the existence of curves $h^t$, such that $h^t \asymp \lVert h^t\rVert^k$, for $k
\in \{1,2\}$.
\[CDS<>h\_m\] Let $H$ be a closed, connected subgroup of $AN$ in ${\operatorname{SU}}(2,n;{\mathbb{F}})$. If, for each $k \in \{1,2\}$, there is a continuous curve $h^t$ in $H$, such that $\rho(h^t) \asymp \lVert h^t\rVert^k \to \infty$ as $t \to \infty$, then $H$ is a Cartan-decomposition subgroup.
![Proposition \[CDS<>h\_m\]: if $\mu(H)$ contains a curve near each wall of $A^+$, then it also contains the interior.[]{data-label="mu(H)=wall"}](muhnearwalls.eps)
By hypothesis, there is a continuous, proper map $\Phi\colon \{1,2\}
\times {\mathord{\mathbb{R}}}^+ \to H$, such that $\rho \bigl( \Phi(k,t) \bigr) \asymp
\lVert\Phi(k,t)\rVert^k$. Because $H \subset AN$, we know that $H$ is homeomorphic to some Euclidean space ${\mathord{\mathbb{R}}}^m$ [[(]{}see [\[solvable\][[(]{}\[solvable-H=Rn\][)]{}]{}]{}[)]{}]{}.
Suppose, for the moment, that $\dim H = 1$. (This will lead to a contradiction.) We know that $\rho(h) \asymp h$ for $h \in \Phi(1,{\mathord{\mathbb{R}}}^+)$. Because $h^{-1} \asymp h$ and $\rho(h^{-1}) \asymp \rho(h)$ [[(]{}see \[rho(h-1)\][)]{}]{}, we must also have $\rho(h) \asymp h$ for $h \in \Phi(1,{\mathord{\mathbb{R}}}^+)^{-1}$. There is no harm in assuming $\phi(1,0) = {\operatorname{Id}}$; then $\Phi(1,{\mathord{\mathbb{R}}}^+) \cup \Phi(1,{\mathord{\mathbb{R}}}^+)^{-1} = H$ (because $\dim H = 1$), so we conclude that $\rho(h) \asymp h$ for all $h \in H$. This contradicts the fact that $\rho(h) \asymp \lVert h\rVert^2$ for $h \in \Phi(2,{\mathord{\mathbb{R}}}^+)$.
We may now assume $\dim H \ge 2$. Then, because $H$ is homeomorphic to ${\mathord{\mathbb{R}}}^m$, it is easy to extend $\Phi$ to a continuous and proper map $\Phi'\colon [1,2] \times {\mathord{\mathbb{R}}}^+ \to H$. From and [[(]{}\[gn=hn\][)]{}]{}, we know that the curve $\mu\bigl(\Phi'(k,t) \bigr)$ stays within a bounded distance from the wall $L_k$; say $\operatorname{dist}\bigl[
\bigl(\Phi' (k,t) \bigr), L_k \bigr] < C$ for all $t$. We may assume $C$ is large enough that $\operatorname{dist}\bigl( \Phi'(s,1), e \bigr) < C$ for all $s \in [1,2]$. Then an elementary homotopy argument shows that $\mu \bigl[
\Phi'\bigl( [1,2] \times {\mathord{\mathbb{R}}}^+ \bigr) \bigr]$ contains $$\{\, a \in A^+ \mid \operatorname{dist}(a, L_1 \cup L_2) > C \, \} ,$$ so $\mu \bigl[ \Phi'\bigl( [1,2] \times {\mathord{\mathbb{R}}}^+ \bigr) \bigr] \approx A^+$. Because $\mu(H) \supset \mu \bigl[ \Phi'\bigl( [1,2] \times {\mathord{\mathbb{R}}}^+ \bigr)
\bigr]$, we conclude from Theorem \[CDSvsmu\] that $H$ is a Cartan-decomposition subgroup.
When ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G = 1$, the Weyl chamber $A^+$ has only one point at infinity. Thus, if $H$ is any noncompact subgroup, then the closure of $\mu(H)$ must contain this point at infinity. This is why it is easy to prove that any noncompact subgroup of $G$ is a Cartan-decomposition subgroup [[(]{}see \[Rrank1-CDS\][)]{}]{}.
The idea of Proposition \[CDS<>h\_m\] is that if ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G = 2$, then the points at $\infty$ of the Weyl chamber $A^+$ form a closed interval. If the closure of $\mu(H)$ contains the two endpoints of this interval, then, by continuity, it must also contain all the points in between.
Unfortunately, we have no good substitute for this proposition when ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G
> 2$. The points at $\infty$ of $A^+$ form a closed disk (topologically speaking). It is easy to define a map $f$ from one disk to another, such that the image of $f$ contains the entire boundary sphere, but does not contain the interior of the disk. Thus, it does not suffice to show only that the closure of $\mu(H)$ contains the boundary of the disk at $\infty$; rather, one needs additional homotopical information to guarantee that no interior points are missed.
\[mu(SUorSp)\] Let $G = {\operatorname{SU}}(2,n; {\mathbb{F}})$, and fix some $m \le n/2$. Then $\mu \bigl( {\operatorname{SU}}(1,n;{\mathbb{F}}) \bigr)$ and $\mu \bigl( {\operatorname{Sp}}(1,m;{\mathbb{F}}) \bigr)$ are the two walls of $A^+$.
We have
1. \[mu(SUorSp)-SU\] $\rho(h) \asymp h$ for $h \in {\operatorname{SU}}(1,n;{\mathbb{F}})$; and
2. \[mu(SUorSp)-Sp\] $\rho(h) \asymp \lVert h\rVert^2$ for $h \in {\operatorname{Sp}}(1,m;{\mathbb{F}})$.
Let $H = {\operatorname{SU}}(1,n; {\mathbb{F}})$ or ${\operatorname{Sp}}(1,m;{\mathbb{F}})$. Then $H \cap K$ is a maximal compact subgroup of $H$. From the Cartan decomposition $$H = (K \cap H) (A \cap H) (K \cap H) ,$$ and the definition of $\mu$, we conclude that $\mu(H) = \mu(A \cap H)$. In the notation of , we see (from Definition \[SU1nDefn\]) that $H \cap A = L_k \cup L_k^{-1}$, where $$k =
\begin{cases}
1 & \mbox{if $H = {\operatorname{SU}}(1,n; {\mathbb{F}})$}; \\
2 & \mbox{if $H = {\operatorname{Sp}}(1,m; {\mathbb{F}})$}.
\end{cases}$$ Then, since $\mu(a^{-1}) = \mu(a)$ [[(]{}see \[mu(h-1)\][)]{}]{} and $\mu(a) = a$ for $a \in
A^+$, we conclude that $\mu(H) = L_k$ is a wall of $A^+$. Furthermore, we have $\rho(a) \asymp \lVert a\rVert^k$ for $a \in \mu(H)$ [[(]{}see \[rho(L)\][)]{}]{}, so $\rho(h) \asymp \lVert h\rVert^k$ for $h \in H$ [[(]{}see \[mu=rho\][)]{}]{}.
\[SU1inH\] If there is a continuous curve $h^t \to \infty$ in $H$, such that $\rho(h^t)
\asymp h^t$, then there is a compact subset $C$ of $G$, such that ${\operatorname{SU}}(1,n;{\mathbb{F}})
\subset CHC$.
For any (large) $g \in {\operatorname{SU}}(1,n;{\mathbb{F}})$, we see from continuity (more precisely, from the Intermediate Value Theorem) that there exists $t \in {\mathord{\mathbb{R}}}^+$, such that $$\lVert h^t\rVert = \lVert g\rVert .$$ Then, by assumption and from [\[mu(SUorSp)\][[(]{}\[mu(SUorSp)-SU\][)]{}]{}]{}, we have $$\rho(h^t) \asymp h^t \asymp g \asymp \rho(g) ,$$ so there is a compact subset $C'$ of $A$, such that $$\mu \bigl( {\operatorname{SU}}(1,n;{\mathbb{F}}) \bigr) \subset \{\, \mu(h^t) \mid t \in
{\mathord{\mathbb{R}}}^+\,\} C'
\subset \mu(H) C'$$ [[(]{}see \[gn=hn\][)]{}]{}. Therefore $${\operatorname{SU}}(1,n;{\mathbb{F}}) \subset K \, \mu \bigl( {\operatorname{SU}}(1,n;{\mathbb{F}}) \bigr) \, K
\subset K \, \mu(H) C' \, K
\subset K (KHK) C'K ,$$ as desired.
The following corollary can be proved by a similar argument. (Recall that the equivalence relation $\sim$ is defined in [[(]{}\[simdefn\][)]{}]{}.)
\[HsimSUorSp\] Assume $H$ is not compact.
1. \[HsimSUorSp-SU\] We have $H \sim {\operatorname{SU}}(1,n;{\mathbb{F}})$ if and only if $\rho(h) \asymp h$ for $h \in H$.
2. \[HsimSUorSp-Sp\] We have $H \sim {\operatorname{Sp}}(1,m;{\mathbb{F}})$ if and only if $\rho(h) \asymp \lVert
h\rVert^2$ for $h \in H$.
Because of Proposition \[CDS<>h\_m\], we will often want to show that a curve $h^t$ satisfies $\rho(h^t) \asymp \lVert h\rVert^k$, for some $k \in
\{1,2\}$. The following lemma does half of the work.
\[Owalls\] Let $X$ be a subset of ${\operatorname{SU}}(2,n;{\mathbb{F}})$.
1. \[Owalls-linear\] If $\rho(h) = O(h)$ for $h \in X$, then $\rho(h) \asymp h$ for $h \in X$.
2. \[Owalls-square\] If $\lVert h\rVert^2 = O \bigl( \rho(h) \bigr)$ for $h \in X$, then $\rho(h)
\asymp \lVert h\rVert^2$ for $h \in X$.
From [[(]{}\[calcrho\][)]{}]{} and [[(]{}\[A+\][)]{}]{}, we have $$\lVert a\rVert = a_{1,1} \le a_{1,1} \, a_{2,2} = \lVert \rho(a) \rVert$$ and $$\lVert \rho(a) \rVert = a_{1,1} \, a_{2,2} \le a_{1,1}^2 =
\lVert a\rVert^2$$ for $a \in A^+$. Thus, letting $a = \mu(h)$, and using [[(]{}\[h=mu(h)\][)]{}]{} and [[(]{}\[rho(h)=rho(mu)\][)]{}]{}, we have: $$\lVert h\rVert \asymp \lVert \mu(h) \rVert \le \lVert \rho \bigl( \mu(h)
\bigr) \rVert \asymp \lVert\rho(h)\rVert$$ and $$\lVert \rho(h) \rVert \asymp \lVert \rho \bigl( \mu(h) \bigr) \rVert
\le \lVert \mu(h) \rVert^2 \asymp \lVert h\rVert^2 ,$$ so $h = O \bigl( \rho(h) \bigr)$ and $\rho(h) = O \bigl( \lVert h\rVert^2 \bigr)$. The desired conclusions follow.
For convenience, we record the following simple observation. (For the proof, cf. the proof of [[(]{}\[h=mu(h)\][)]{}]{} and [[(]{}\[rho(h)=rho(mu)\][)]{}]{}.)
\[conjcurve\] Let
- $k \in \{1,2\}$,
- $g \in G$, and
- $h^t \to \infty$ be a continuous curve in $H$.
If $\rho(h^t) \asymp \lVert h^t\rVert^k$, then $\rho(g^{-1} h^t g) \asymp \lVert g^{-1} h^t g \rVert^k$.
Homogeneous functions of the same degree
----------------------------------------
The following well-known, elementary observation is used frequently in the later sections.
\[O(linear)\] Let $V'$ be a subspace of a finite-dimensional real vector space $V$, and let $f_1 \colon V \to W_1$ and $f_2 \colon V \to W_2$ be linear transformations.
1. \[O(linear)-O\] If $f_1^{-1}(0) \cap V' = \{0\}$ [(]{}or, more generally, if $f_1^{-1}(0) \cap V' \subset f_2^{-1}(0)$[)]{}, then there is a linear transformation $f \colon W_1 \to W_2$, such that $f_2(v) = f \bigl( f_1(v)
\bigr)$ for all $v \in V'$. Therefore $f_2 = O(f_1)$ on $V'$.
2. \[O(linear)-=\] If $f_1^{-1}(0) \cap V' = f_2^{-1}(0) \cap V'$, then $f_1 \asymp f_2$ on $V'$.
[[(]{}\[O(linear)-O\][)]{}]{} By passing to a subspace, we may assume $V' = V$. Then, by modding out $f_1^{-1}(0)$, we may assume $f_1$ is an isomorphism onto its image. Define $f' \colon f_1(V) \to W_2$ by $f'(w) = f_2 \bigl( f_1^{-1}(w)
\bigr)$, and let $f \colon W_1 \to W_2$ be any extension of $f'$.
For $v \in V'$, we have $$\lVert f_2(v) \rVert = \lVert f \bigl( f_1(v) \bigr) \rVert
\le \lVert f\rVert \, \lVert f_1(v) \rVert ,$$ so $f_2 = O(f_1)$.
[[(]{}\[O(linear)-=\][)]{}]{} From [[(]{}\[O(linear)-O\][)]{}]{}, we have $f_2 = O(f_1)$ and $f_1 =
O(f_2)$, so $f_1 \asymp f_2$.
Let ${\mathfrak{\lowercase{H}}}$ be a real Lie subalgebra of ${\mathfrak{\lowercase{A}}} + {\mathfrak{\lowercase{N}}}$, and assume there does not exist a nonzero element $u$ of ${\mathfrak{\lowercase{H}}}$, such that $x_u = 0$ and $y_u = 0$. Then there exist ${\mathord{\mathbb{R}}}$-linear transformations $R,S \colon
{\mathbb{F}}^{n-2} \to {\mathbb{F}}$, such that $\eta_u = R(x_u) + S(y_u)$ for all $u \in {\mathfrak{\lowercase{H}}}$. (Similarly, $\phi_u$, ${{\mathord{\mathsf{x}}}}_u$, and ${{\mathord{\mathsf{y}}}}_u$ are also functions of $(x_u,y_u)$.) Furthermore, we have $u \asymp |x_u| + |y_u|$.
The following well-known result is a generalization of the fact that all norms on a finite-dimensional vector space are equivalent up to a bounded factor.
\[QuadrForms\] If $V$ is any finite-dimensional real vector space, and $f_1,f_2 \colon V
\to{\mathord{\mathbb{R}}}$ are two continuous, homogeneous functions of the same degree, such that $f_1^{-1}(0) = f_2^{-1}(0) = \{0\}$, then $f_1 \asymp f_2$.
By continuity, the function $f_1/f_2$ attains a non-zero minimum and a finite maximum on the unit sphere. Because $f_1/f_2$ is homogeneous of degree zero, these values bound $f_1/f_2$ on all of $V \smallsetminus \{0\}$.
Existence of tessellations {#ExistenceSect}
==========================
In this section, we show how to construct several families of homogeneous spaces that have tessellations. All of these examples are based on a method of T. Kobayashi [[(]{}see \[construct-tess\][)]{}]{} that generalizes Example \[KulkarniEg\].
The general Kulkarni-Kobayashi construction
-------------------------------------------
As explained in the comments before Theorem \[noncpctdim\], the following theorem is essentially due to T. Kobayashi.
\[construct-tess\] If
- $H$ and $L$ are closed subgroups of $G$, with only finitely many connected components;
- $L$ acts properly on $G/H$;
- $d(L) + d(H) = d(G)$; and
- there is a cocompact lattice $\Gamma$ in $L$,
then $G/H$ has a tessellation. [(]{}Namely, $\Gamma$ is a crystallographic group for $G/H$.[)]{}
Because $\Gamma$ is a closed subgroup of $L$, we know that it acts properly on $G/H$ [[(]{}see \[CHCproper\][)]{}]{}. Thus, it suffices to show that $\Gamma \backslash
G/H$ is compact.
From Lemma \[HcanbeAN\], we see that there is no harm in assuming $H
\subset AN$, and that there is a closed, connected subgroup $L'$ of $G$, such that
- $L'$ is conjugate to a subgroup of $AN$,
- $d(L') + d(H) = d(G)$, and
- $L'C = LC$, for some compact subset $C$ of $G$.
(Unfortunately, we cannot assume $L \subset AN$: we may not be able to replace $L$ with $L'$, because there may not be a cocompact lattice in $L'$. For example, there is not lattice in $AN$, because any group with a lattice must be unimodular [@Raghunathan Rem. 1.9, p. 21].)
It suffices to show that $L' \backslash G/H$ is compact. (Because $L' \subset
LC$ is compact, and $\Gamma \backslash L$ is compact, this implies that $\Gamma \backslash G/H$ is compact, as desired.)
We know that $L'$ acts properly on $G/H$ [[(]{}see \[CHCproper\][)]{}]{}, so $L' \times H$ acts properly on $G$, with quotient $L' \backslash G/H$. Therefore, Lemma \[fiberbundle\] implies that $L' \backslash G/H$ has the same homology as $G$; in particular, $${\mathord{\mathcal{H}}}_{\dim K}(L' \backslash G/H) {\cong}{\mathord{\mathcal{H}}}_{\dim K}(G) .$$ From the Iwasawa decomposition $G = KAN$, and because $AN$ is homeomorphic to ${\mathord{\mathbb{R}}}^{d(G)}$ [[[(]{}see \[ANsc\] [and \[R=Rd\]]{}[)]{}]{}]{}, we know that $G$ is homeomorphic to $K \times {\mathord{\mathbb{R}}}^{d(G)}$. Since ${\mathord{\mathbb{R}}}^{d(G)}$ is contractible, this implies that $G$ is homotopy equivalent to $K$, so $G$ and $K$ have the same homology; in particular, $${\mathord{\mathcal{H}}}_{\dim K}(G) = {\mathord{\mathcal{H}}}_{\dim K}(K) \neq 0 .$$ Since $$\begin{aligned}
\dim(L' \backslash G/H)
&= \dim G - \dim L' - \dim H \\
&= \dim G - \bigl( d(L') + d(H) \bigr) \\
&= \dim G - d(G) \\
&= \dim(KAN) - \dim(AN) \\
&= \dim K ,
\end{aligned}$$ this implies that the top-dimensional homology of the manifold $L' \backslash
G/H$ is nontrivial. Therefore $L' \backslash G/H$ is compact [@Dold Cor. 8.3.4], as desired.
Our results for $G = {\operatorname{SU}}(2,2m;{\mathbb{F}})$ are based on the following special case of the theorem. The converse of this corollary is proved in Section \[ProofSect\] [[(]{}see \[SUF-known\][)]{}]{}.
Recall the equivalence relation $\sim$, introduced in Notation \[simdefn\].
\[SUevenTessExists\] Let $H$ be a closed, connected subgroup of $G = {\operatorname{SU}}(2,2m;{\mathbb{F}})$. If
- $d(H) = 2{q}m$; and
- either $H \sim {\operatorname{SU}}(1,2m;{\mathbb{F}})$ or $H \sim {\operatorname{Sp}}(1,m;{\mathbb{F}})$,
then $G/H$ has a tessellation.
Let $L_+ = {\operatorname{SU}}(1,2m;{\mathbb{F}})$ and $L_- = {\operatorname{Sp}}(1,m;{\mathbb{F}})$. By assumption, we have $H
\sim L_\varepsilon$, for some $\varepsilon \in \{+,-\}$; let $L =
L_{-\varepsilon}$. Because $\mu(L_+)$ and $\mu(L_-)$ are the two walls of $A^+$ [[(]{}see \[mu(SUorSp)\][)]{}]{}, we know that $L = L_{-\varepsilon}$ acts properly on $G/L_{\varepsilon}$ [[(]{}see \[proper<>mu(L)\][)]{}]{}; since $H \sim L_\varepsilon$, this implies that $L$ acts properly on $G/H$ [[(]{}see \[CHCproper\][)]{}]{}. Also, we have $$d(L) + d(H) = 2{q}m + 2{q}m = d(G) ,$$ [[[(]{}see \[d(Sp)\] [and \[d(G)\]]{}[)]{}]{}]{}, and there is a cocompact lattice in $L$ [[(]{}cf. [\[classical\][[(]{}\[classical-Borel\][)]{}]{}]{}[)]{}]{}. Thus, the desired conclusion follows from Theorem \[construct-tess\].
Deformations of ${\operatorname{SO}}(2,2m)/{\operatorname{SU}}(1,m)$ and ${\operatorname{SU}}(2,2m)/{\operatorname{Sp}}(1,m)$
-----------------------------------------------------------------------------------------------------------------------------
The homogeneous spaces described here were found by H. Oh and D. Witte [@OhWitte-announce Thms. 4.1 and 4.6], [@OhWitte-CK Thm. 1.5].
\[HB-defn\] For any ${\mathord{\mathbb{R}}}$-linear $B \colon {\mathbb{F}}^{n-2} \to {\mathbb{F}}^{n-2}$, we define $${\mathfrak{\lowercase{H}}}_B =
{\left\{\, \begin{pmatrix}
t & 0 & x & \eta & {{\mathord{\mathsf{x}}}}\\
& t & x B & -{{\mathord{\mathsf{x}}}}& -{\overline{\eta}} \\
& & \dots \\
\end{pmatrix}
\mathrel{\left| \vphantom {\left\{ \begin{pmatrix}
t & 0 & x & \eta & {{\mathord{\mathsf{x}}}}\\
& t & x B & -{{\mathord{\mathsf{x}}}}& -{\overline{\eta}} \\
& & \dots \\
\end{pmatrix}
\mid
\begin{matrix}
t \in {\mathord{\mathbb{R}}}, \\
x\in
{\mathbb{F}}^{n-2}, \\
\eta \in {\mathbb{F}}, \\
{{\mathord{\mathsf{x}}}}\in {{\mathbb{F}}_{\text{imag}}}\end{matrix}
\right\} } \right.}
\begin{matrix}
t \in {\mathord{\mathbb{R}}}, \\
x\in
{\mathbb{F}}^{n-2}, \\
\eta \in {\mathbb{F}}, \\
{{\mathord{\mathsf{x}}}}\in {{\mathbb{F}}_{\text{imag}}}\end{matrix}
\,\right\} }
\subset {\mathfrak{\lowercase{A}}} + {\mathfrak{\lowercase{N}}}
.$$ We write $xB$, rather than $Bx$, because $x$ is a row vector.
It is easy to see, using, for instance, the formula for the bracket in , that if $$\label{vBwB}
\mbox{${\operatorname{Im}}\bigl( (vB)(wB)^\dagger \bigr) = -{\operatorname{Im}}(v w^\dagger)$ for every
$v,w \in {\mathbb{F}}^{n-2}$} ,$$ then ${\mathfrak{\lowercase{H}}}_B$ is a real Lie subalgebra of ${\mathfrak{\lowercase{A}}}+{\mathfrak{\lowercase{N}}}$; we let $H_B$ denote the corresponding connected Lie subgroup of $AN$.
From [[(]{}\[d(H)=dimH\][)]{}]{}, we have $$\label{d(HB)}
\begin{split}
d(H_B) &= \dim {\mathfrak{\lowercase{H}}}_B \\
&= \dim {\mathord{\mathbb{R}}}+ \dim {\mathbb{F}}^{n-2} + \dim {\mathbb{F}}+ \dim {{\mathbb{F}}_{\text{imag}}}\\
&= 1 + {q}(n-2) + {q}+ ({q}-1) \\
&= {q}n .
\end{split}$$
\[HB=Sp1m\] Assume $n = 2m$. By comparing [[(]{}\[Sp1mAN\][)]{}]{} with [[(]{}\[HB-defn\][)]{}]{}, we see that there is a ${\mathord{\mathbb{R}}}$-linear map $B_0 \colon {\mathbb{F}}^{2m-2} \to {\mathbb{F}}^{2m-2}$, such that $ {\operatorname{{\mathfrak{\lowercase{Sp}}}}}(1,m;{\mathbb{F}}) \cap ({\mathfrak{\lowercase{A}}} + {\mathfrak{\lowercase{N}}}) = H_{B_0} $ (and $B_0$ satisfies [[(]{}\[vBwB\][)]{}]{}). Thus, in general, $H_B$ is a deformation of ${\operatorname{{\mathfrak{\lowercase{Sp}}}}}(1,m;{\mathbb{F}}) \cap ({\mathfrak{\lowercase{A}}} +
{\mathfrak{\lowercase{N}}})$.
\[HBthm\] Let $B \colon {\mathbb{F}}^{2m-2} \to {\mathbb{F}}^{2m-2}$ be ${\mathord{\mathbb{R}}}$-linear. If
- Condition [[(]{}\[vBwB\][)]{}]{} holds, and
- \[HBthm-xBnotinFx\] $xB \notin {\mathbb{F}}x$, for every nonzero $x \in {\mathbb{F}}^{2m-2}$,
then
1. \[HBthm-mu\] $\rho(h) \asymp \lVert h\rVert^2$ for $h \in H_B$; and
2. \[HBthm-tess\] ${\operatorname{SU}}(2,2m;{\mathbb{F}})/ H_B$ has a tessellation.
[[(]{}\[HBthm-mu\][)]{}]{} Given $h \in H_B$, write $h = au$, with $a \in A$ and $u
\in N$. We may assume that $a_{1,1} \ge 1$ (by replacing $h$ with $h^{-1}$ if necessary [[(]{}see \[rho(h-1)\][)]{}]{}). It suffices to show $\lVert h\rVert^2 = O \bigl(
a_{1,1} a_{2,2} + |\Delta(h)| \bigr)$ (for then $\lVert h\rVert^2 = O \bigl(
\rho(h) \bigr)$, so Lemma [\[Owalls\][[(]{}\[Owalls-square\][)]{}]{}]{} applies).
\[HB-hinN\] Assume $a$ is trivial.
From and [[(]{}\[HB-defn\][)]{}]{}, we see that $$h = O \bigl( |x_h|^2 + |\eta_h| + |{{\mathord{\mathsf{x}}}}_h| \bigr) ,$$ so $$\lVert h\rVert^2 = O \bigl( |x_h|^4 + |\eta_h|^2 + |{{\mathord{\mathsf{x}}}}_h|^2 \bigr) .$$
From [[(]{}\[exp(phi=0)\][)]{}]{} and [[(]{}\[DeltaDefn\][)]{}]{}, we have $$- {\operatorname{Re}}\Delta(h)
= \frac{1}{4} \bigl(|x_h|^2 |y_h|^2 - |x y^\dagger|^2\bigr)
+ \bigl( |\eta_h|^2 + {{\mathord{\mathsf{x}}}}_h {{\mathord{\mathsf{y}}}}_h \bigr) .$$ From [[(]{}\[xBnotinFx\][)]{}]{}, we see that $|x|^2 |xB|^2 - |x (xB)^\dagger|^2 > 0$ for every nonzero $x \in {\mathbb{F}}^{2m-2}$, so Lemma \[QuadrForms\] implies $$|x_h|^4 \asymp |x_h|^2 |y_h|^2 - |x_h y_h^\dagger|^2 .$$ Also, because ${{\mathord{\mathsf{y}}}}_h = - {{\mathord{\mathsf{x}}}}_h$ (and ${{\mathord{\mathsf{x}}}}_h \in {{\mathbb{F}}_{\text{imag}}}$), we have $$|\eta_h|^2 + {{\mathord{\mathsf{x}}}}_h {{\mathord{\mathsf{y}}}}_h = |\eta_h|^2 + |{{\mathord{\mathsf{x}}}}_h|^2 \ge 0 .$$ Thus, $$- {\operatorname{Re}}\Delta(h)
\asymp |x_h|^4 + \bigl( |\eta_h|^2 + |{{\mathord{\mathsf{x}}}}_h|^2 \bigr) ,$$ so $\lVert h\rVert^2 = O \bigl( {\operatorname{Re}}\Delta(h) \bigr) = O \bigl( \Delta(h)
\bigr)$, as desired.
The general case.
From Case \[HB-hinN\], we know $\lVert u\rVert^2 = O \bigl( 1 + |\Delta(u)| \bigr)$. Then, because $\lVert h\rVert \le \lVert a\rVert \lVert u\rVert = a_{1,1} \lVert u\rVert$, we have $$\lVert h\rVert^2
\le a_{1,1}^2 \lVert u\rVert^2
= O \bigl( a_{1,1}^2 ( 1 + |\Delta(u)|) \bigr)
= O \bigl( a_{1,1}^2 + a_{1,1}^2 |\Delta(u)| \bigr).$$ Then, since $a_{1,1} = a_{2,2}$ and $\Delta(h) = a_{1,1} a_{2,2} \Delta(u)$, we conclude that $\lVert h\rVert^2 = O \bigl( a_{1,1}
a_{2,2} + |\Delta(h)| \bigr)$, as desired.
[[(]{}\[HBthm-tess\][)]{}]{} From [[(]{}\[HBthm-mu\][)]{}]{} and [\[HsimSUorSp\][[(]{}\[HsimSUorSp-Sp\][)]{}]{}]{}, we see that $H_B \sim {\operatorname{Sp}}(1,m; {\mathbb{F}})$. Then, because $d(H_B) = {q}(2m)$ [[(]{}see \[d(HB)\][)]{}]{}, Theorem \[SUevenTessExists\] implies that ${\operatorname{SU}}(2,2m;{\mathbb{F}})/H_B$ has a tessellation.
\[Bsymplectic\] Let $B \colon {\mathbb{F}}^{n-2} \to {\mathbb{F}}^{n-2}$ be ${\mathord{\mathbb{R}}}$-linear. Condition [[(]{}\[vBwB\][)]{}]{} holds if and only if either
1. ${\mathbb{F}}= {\mathord{\mathbb{R}}}$; or
2. ${\mathbb{F}}= {\mathord{\mathbb{C}}}$ and $B' \in {\operatorname{Sp}}(2n-4;{\mathord{\mathbb{R}}})$, where $x B' = {\overline{xB}}$ and we use the natural identification of ${\mathord{\mathbb{C}}}^{n-2}$ with ${\mathord{\mathbb{R}}}^{2n-4}$.
Assume ${\mathbb{F}}= {\mathord{\mathbb{R}}}$.
Because ${\operatorname{Im}}z = 0$ for every $z = 0$, it is obvious that [[(]{}\[vBwB\][)]{}]{} holds.
Assume ${\mathbb{F}}= {\mathord{\mathbb{C}}}$.
If [[(]{}\[vBwB\][)]{}]{} holds, then $$\begin{aligned}
{\operatorname{Im}}\bigl( (vB')(wB')^\dagger \bigr)
&= {\operatorname{Im}}\bigl( (\overline{vB})(\overline{wB})^\dagger \bigr) \\
&= {\operatorname{Im}}\overline{\bigl( ({vB})({wB})^\dagger \bigr)} \\
&= -{\operatorname{Im}}\bigl( ({vB})({wB})^\dagger \bigr) \\
&= - \bigl( -{\operatorname{Im}}(vw^\dagger) \bigr) \\
&= {\operatorname{Im}}(vw^\dagger) ,
\end{aligned}$$ so $B'$ is symplectic. The argument is reversible.
\[xBnotinFx\]
- For ${\mathbb{F}}= {\mathord{\mathbb{R}}}$, the assumption that $xB \notin {\mathbb{F}}x$ simply requires that $B$ have no real eigenvalues.
- For ${\mathbb{F}}= {\mathord{\mathbb{C}}}$, we do not know a good description of the linear transformations $B$ that satisfy $xB \notin {\mathbb{F}}x$, although it is easy to see that this is an open set (and not dense). A family of examples was constructed by H. Oh and D. Witte (see \[HBeg\] below).
- If $n$ is odd, then there does not exist $B \colon {\mathbb{F}}^{n-2} \to
{\mathbb{F}}^{n-2}$ satisfying the assumption that $xB \notin {\mathbb{F}}x$. For ${\mathbb{F}}= {\mathord{\mathbb{R}}}$, this is simply the elementary fact that a linear transformation on an odd-dimensional real vector space must have a real eigenvalue. For ${\mathbb{F}}=
{\mathord{\mathbb{C}}}$, see Step \[dim(U)<2npf-U/Z\] of the proof of Proposition \[dim(U)<2n\].
- If $n$ is even, then, by varying $B$, one can obtain uncountably many pairwise non-conjugate subgroups $H_B$, such that ${\operatorname{SU}}(2,n;{\mathbb{F}})/H_B$ has a tessellation. For ${\mathbb{F}}= {\mathord{\mathbb{R}}}$, this is proved in [@OhWitte-CK Thm. 1.5]). For ${\mathbb{F}}= {\mathord{\mathbb{C}}}$, a similar argument can be applied to the examples constructed in [[(]{}\[HBeg\][)]{}]{} below.
\[HBeg\] Assume $n$ is even, let $B' \in {\operatorname{SO}}(n-2; {\mathord{\mathbb{R}}})$, such that $B'$ has no real eigenvalue, and define an ${\mathord{\mathbb{R}}}$-linear map $B \colon {\mathord{\mathbb{C}}}^{n-2} \to
{\mathord{\mathbb{C}}}^{n-2}$ by $xB = {\overline{x}} B'$. Let us verify that $B$ satisfies the conditions of Theorem \[HBthm\] (for ${\mathbb{F}}= {\mathord{\mathbb{C}}}$).
Let $x_1,x_2,y_1,y_2 \in {\mathord{\mathbb{R}}}^{n-2}$. From the definition of $B$, and because $B' \in {\operatorname{SO}}(n-2; {\mathord{\mathbb{R}}})$, we have $$\begin{aligned}
{\operatorname{Im}}\left( \bigl( (x_1 + i x_2)B \bigr) \bigr( (y_1 + i y_2)B \bigr)^\dagger
\right)
&= {\operatorname{Im}}\left( \bigl( (x_1 - i x_2)B' \bigr) \bigr( (y_1 - i y_2) B'
\bigr)^\dagger \right) \\
&= i \left( (x_1 B') (y_2 B')^\dagger - (x_2 B') (y_1 B')^\dagger \right)
\\
&= i ( x_1 y_2^\dagger - x_2 y_1^\dagger ) \\
&= - {\operatorname{Im}}\bigl( (x_1 +i x_2) (y_1 + i y_2)^\dagger \bigr)
.
\end{aligned}$$
Suppose $Bx = \lambda x$, for some $\lambda \in {\mathord{\mathbb{C}}}$. Because $B \in
{\operatorname{SO}}(2n-4;{\mathord{\mathbb{R}}})$, we must have $|\lambda| = 1$. Then $$B'(x + {\overline{\lambda x}})
= B' x + {\overline{\lambda}} B' {\overline{x}}
= {\overline{B' {\overline{x}}}} + {\overline{\lambda}} Bx
= {\overline{Bx}} + {\overline{\lambda}} (\lambda x)
= {\overline{\lambda x}} + x .$$ Because $B'$ has no real eigenvalues, we know that $1$ is not an eigenvalue of $B'$, so we conclude that $x + {\overline{\lambda x}} = 0$. Similarly, because $-1$ is not an eigenvalue of $B'$, we see that $x - {\overline{\lambda x}} =
0$. Therefore $$x = \frac{1}{2} \bigl( (x + {\overline{\lambda x}}) + (x - {\overline{\lambda x}}) \bigr)
= \frac{1}{2} ( 0 + 0 ) = 0 .$$
Deformations of ${\operatorname{SU}}(2,2m)/{\operatorname{SU}}(1,2m)$
---------------------------------------------------------------------
These examples are new for ${\mathbb{F}}= {\mathord{\mathbb{C}}}$, but provide nothing interesting for ${\mathbb{F}}= {\mathord{\mathbb{R}}}$ [[(]{}see [\[Hc\][[(]{}\[Hc-real\][)]{}]{}]{}[)]{}]{}.
\[SUegsDefn\] For $c \in (0,1]$, we define $${{{{\mathfrak{\lowercase{H}}}}_{[c]}}}=
{\left\{\,
\begin{pmatrix}
t & \phi & x & {\operatorname{Re}}\phi + c {\operatorname{Im}}\phi & {{\mathord{\mathsf{x}}}}\\
& 0 & 0 & 0 & * \\
& & \dots \\
\end{pmatrix}
\mathrel{\left| \vphantom {\left\{
\begin{pmatrix}
t & \phi & x & {\operatorname{Re}}\phi + c {\operatorname{Im}}\phi & {{\mathord{\mathsf{x}}}}\\
& 0 & 0 & 0 & * \\
& & \dots \\
\end{pmatrix}
\mid
\begin{matrix}
t \in {\mathord{\mathbb{R}}}, \\
\phi \in {\mathbb{F}}, \\
x \in {\mathbb{F}}^{n-2}, \\
{{\mathord{\mathsf{x}}}}\in{{\mathbb{F}}_{\text{imag}}}\end{matrix}
\right\} } \right.}
\begin{matrix}
t \in {\mathord{\mathbb{R}}}, \\
\phi \in {\mathbb{F}}, \\
x \in {\mathbb{F}}^{n-2}, \\
{{\mathord{\mathsf{x}}}}\in{{\mathbb{F}}_{\text{imag}}}\end{matrix}
\,\right\} }
.$$
It is easy to see, using, for instance, the formula for the bracket in , that ${{{{\mathfrak{\lowercase{H}}}}_{[c]}}}$ is a real Lie subalgebra of ${\mathfrak{\lowercase{A}}}+{\mathfrak{\lowercase{N}}}$ (even without the assumption that $0 < c \le 1$); we let ${{H_{[c]}}}$ be the corresponding connected Lie subgroup of $AN$.
From [[(]{}\[d(H)=dimH\][)]{}]{}, we have $$\label{d(Hc)}
\begin{split}
d({{H_{[c]}}})
&= \dim {{{{\mathfrak{\lowercase{H}}}}_{[c]}}}\\
&= \dim {\mathord{\mathbb{R}}}+ \dim {\mathbb{F}}+ \dim {\mathbb{F}}^{n-2} + \dim {{\mathbb{F}}_{\text{imag}}}\\
&= 1 + {q}+ {q}(n-2) + ({q}-1) \\
&= {q}n .
\end{split}$$
\[Hc\] Let ${\operatorname{{\mathfrak{\lowercase{SU}}}}}(1,n;{\mathbb{F}})$ be embedded into ${\operatorname{{\mathfrak{\lowercase{SU}}}}}(2,n;{\mathbb{F}})$ as in \[SU1nAN\].
1. \[Hc-real\] If ${\mathbb{F}}={\mathord{\mathbb{R}}}$, then $c$ is irrelevant in the definition of ${{{{\mathfrak{\lowercase{H}}}}_{[c]}}}$ (because ${\operatorname{Im}}\phi = 0$); therefore ${{{{\mathfrak{\lowercase{H}}}}_{[c]}}}={\operatorname{{\mathfrak{\lowercase{SU}}}}}(1,n;{\mathord{\mathbb{R}}})\cap ({\mathfrak{\lowercase{A}}} + {\mathfrak{\lowercase{N}}})$.
2. If ${\mathbb{F}}={\mathord{\mathbb{C}}}$, then ${{{\mathfrak{\lowercase{H}}}}_{[1]}} = {\operatorname{{\mathfrak{\lowercase{SU}}}}}(1,n;{\mathord{\mathbb{C}}})\cap ({\mathfrak{\lowercase{A}}} + {\mathfrak{\lowercase{N}}})$.
Thus, in general, ${{{{\mathfrak{\lowercase{H}}}}_{[c]}}}$ is either ${\operatorname{{\mathfrak{\lowercase{SU}}}}}(1,n;{\mathbb{F}})\cap ({\mathfrak{\lowercase{A}}} + {\mathfrak{\lowercase{N}}})$ or a deformation of it.
\[SUegs\] Assume ${\mathbb{F}}= {\mathord{\mathbb{C}}}$, and $n = 2m$ is even. If $c \in (0,1]$, then
1. \[SUegs-linear\] $\rho(h) \asymp h$ for $h \in {{H_{[c]}}}$; and
2. \[SUegs-tess\] ${\operatorname{SU}}(2,2m;{\mathbb{F}})/{{H_{[c]}}}$ has a tessellation.
[[(]{}\[SUegs-linear\][)]{}]{} Given $h \in {{H_{[c]}}}$, it suffices to show that $\rho(h) = O(h)$ [[(]{}see [\[Owalls\][[(]{}\[Owalls-linear\][)]{}]{}]{}[)]{}]{}. Write $h = au$, with $a \in A$ and $u \in N$. We may assume that $a_{1,1} \ge 1$ (by replacing $h$ with $h^{-1}$ if necessary [[(]{}see \[rho(h-1)\][)]{}]{}).
Let $Q \colon {\mathord{\mathbb{C}}}\oplus {\mathord{\mathbb{C}}}^{n-2} \oplus {\mathord{\mathbb{C}}}\to {\mathord{\mathbb{R}}}$ be the real quadratic form $$Q(\phi,x,\eta)=|x|^2+2{\operatorname{Re}}(\phi {\overline{\eta}}) ,$$ and let $V$ be the ${\mathord{\mathbb{R}}}$-subspace of ${\mathord{\mathbb{C}}}\oplus {\mathord{\mathbb{C}}}^{n-2}
\oplus {\mathord{\mathbb{C}}}$ defined by $$V =
{\left\{\,
( \phi,x,\eta)
\mathrel{\left| \vphantom {\left\{
( \phi,x,\eta)
\mid
\begin{matrix}
\phi \in {\mathord{\mathbb{C}}}, \\
x \in {\mathord{\mathbb{C}}}^{n-2}, \\
\eta = {\operatorname{Re}}\phi + c {\operatorname{Im}}\phi
\end{matrix}
\right\} } \right.}
\begin{matrix}
\phi \in {\mathord{\mathbb{C}}}, \\
x \in {\mathord{\mathbb{C}}}^{n-2}, \\
\eta = {\operatorname{Re}}\phi + c {\operatorname{Im}}\phi
\end{matrix}
\,\right\} } .$$
\[SUegspf-Q=x2+phi2\] For $v \in V$, we have $Q(v) \asymp |\phi|^2 + |x|^2$.
For $(\phi,x,\eta) \in V \smallsetminus \{0\}$, we have $$\begin{aligned}
Q(\phi,x,\eta)
&= |x|^2+2{\operatorname{Re}}(\phi {\overline{\eta}}) \\
&= |x|^2 + 2 {\operatorname{Re}}\bigl( \phi ({\overline{{\operatorname{Re}}\phi + c {\operatorname{Im}}\phi}}) \bigr) \\
&= |x|^2 + 2 ({\operatorname{Re}}\phi)^2 - 2 c ({\operatorname{Im}}\phi)^2 \\
&> 0
\end{aligned}$$ (because $c>0$ and ${\operatorname{Im}}\phi$ is purely imaginary). Thus, the restriction of $Q$ to $V$ is positive definite, so the desired conclusion follows from Lemma \[QuadrForms\].
\[SUegspf-u=x2+phi2+xx\] We have $u_{1,n+2} \asymp \bigl( |\phi_u|^2 + |x_u|^2 \bigr) + |{{\mathord{\mathsf{x}}}}_u|$.
From [[(]{}\[exp(y=0)\][)]{}]{} (with ${{\mathord{\mathsf{y}}}}= 0$), we have $${\operatorname{Re}}u_{1,n+2}
= - \left( \frac{1}{2}
|x_u|^2+ {\operatorname{Re}}(\phi_u {\overline{\eta_u}}) \right)
\asymp |x_u|^2+2{\operatorname{Re}}(\phi_u {\overline{\eta_u}})$$ and $${\operatorname{Im}}u_{1,n+2}
= {{\mathord{\mathsf{x}}}}_u .$$ Then, from Step \[SUegspf-Q=x2+phi2\], we see that ${\operatorname{Re}}u_{1,n+2} \asymp
|\phi_u|^2 + |x_u|^2$, so $$u_{1,n+2} \asymp |{\operatorname{Re}}u_{1,n+2}| + |{\operatorname{Im}}u_{1,n+2}|
\asymp \bigl( |\phi_u|^2 + |x_u|^2 \bigr) + |{{\mathord{\mathsf{x}}}}_u| ,$$ as desired.
Completion of the proof.
From Step \[SUegspf-u=x2+phi2+xx\], we have $$h_{1,n+2} = a_{1,1} u_{1,n+2}
\asymp a_{1,1} \bigl( |x_u| + |\phi_u| \bigr)^2 + a_{1,1} |{{\mathord{\mathsf{x}}}}_u|.$$ Also, from [[(]{}\[exp(y=0)\][)]{}]{}, we have $$h_{jk} =
\begin{cases}
O(1) & \mbox{if $j \neq 1$ and $k \neq n+2$} \\
O \bigl( a_1 (|\phi_u| + |x_u|) \bigr) & \mbox{if $j = 1$ and $k \neq n+2$}
\\
O \bigl( |\phi_u| + |x_u| \bigr) & \mbox{if $j \neq 1$ and $k = n+2$} .
\end{cases}$$ Thus, it is easy to see that $$\rho(h) = O \bigl( a_{1,1} |{{\mathord{\mathsf{x}}}}_u| + a_{1,1} (|\phi_u| + |x_u|)^2 \bigr)
= O(h_{1,n+2})
= O(h) ,$$ so the desired conclusion follows from Lemma [\[Owalls\][[(]{}\[Owalls-linear\][)]{}]{}]{}.
[[(]{}\[SUegs-tess\][)]{}]{} From [[(]{}\[SUegs-linear\][)]{}]{} and [\[HsimSUorSp\][[(]{}\[HsimSUorSp-SU\][)]{}]{}]{}, we see that ${{H_{[c]}}}\sim
{\operatorname{SU}}(1,n)$. Then, because $d({{H_{[c]}}}) = 2n$ [[(]{}see \[d(Hc)\][)]{}]{}, Theorem \[SUevenTessExists\] implies that ${\operatorname{SU}}(2,2m;{\mathbb{F}})/{{H_{[c]}}}$ has a tessellation.
Proposition \[HcUncountable\] shows that if ${\mathbb{F}}= {\mathord{\mathbb{C}}}$, then ${{H_{[c]}}}$ is not conjugate to ${H_{[c']}}$ unless $c = c'$ (for $c,c' \in (0,1]$). Thus, Theorem [\[SUegs\][[(]{}\[SUegs-tess\][)]{}]{}]{} implies that, by varying $c$, one obtains uncountably many nonconjugate subgroups ${{H_{[c]}}}$, such that ${\operatorname{SU}}(2,2m)/{{H_{[c]}}}$ has a tessellation.
The product of two rank-one groups
----------------------------------
\[G1xG2-tess\] Let $G = G_1 \times G_2$ be the direct product of two connected, linear, almost simple Lie groups $G_1$ and $G_2$ of real rank one, with finite center, and let $H$ be a nontrivial, closed, connected, proper subgroup of $AN$.
The homogeneous space $G/H$ has a tessellation if and only if, perhaps after interchanging $G_1$ and $G_2$, there is a continuous homomorphism $\sigma \colon
AN \cap G_1 \to AN \cap G_2$, such that $$H = \{\, \bigl(h , \sigma(h) \bigr) \mid h \in AN \cap G_1 \,\} .$$
($\Rightarrow$) We may assume $d(G_1) \ge d(G_2)$ (by interchanging $G_1$ and $G_2$ if necessary).
\[Hcapboth\] Assume $H \cap G_1 \neq e$ and $H \cap G_2 \neq e$.
For $j = 1,2$, we know that $H \cap G_j$ is not compact [[(]{}see [\[solvable\][[(]{}\[solvable-nocpct\][)]{}]{}]{}[)]{}]{}, so Corollary \[Rrank1-CDS\] implies that there is a compact subset $C_j$ of $G_j$, such that $C_j (H \cap G_j)C_j =
G_j$. Then, letting $C = C_1 C_2$, we have $CHC = G$, so Proposition \[CDS->notess\] implies that $G/H$ does not have a tessellation. This is a contradiction.
Assume $H \cap G_1 \neq e$ and $H \cap G_2 = e$.
From Corollaries \[Rrank1-CDS\] and \[CDSvsmu\], we know that there is a compact subset $C$ of $A \cap G_1$, such that $\mu(G_1) \subset \mu(H) C$. Therefore, Corollary \[noncpct-dim-notess\] (with $G_1$ in the place of $H_1$) implies $d(H) \ge d(G_1) = \dim(G_1 \cap AN)$. Then, because $H \cap G_2 = e$ (and $H \subset AN$), we conclude that $H$ is the graph of a homomorphism from $G_1 \cap AN$ to $G_2 \cap AN$, as desired.
Assume $H \cap G_1 = e$.
From Corollary \[tess->dim>1,2\], we know that $\dim H \ge d(G_2)$. Then, since $H \cap G_1 = e$, we conclude that $H$ is the graph of a homomorphism from $G_2 \cap AN$ to $G_1 \cap AN$. Interchanging $G_1$ and $G_2$ yields the desired conclusion.
($\Leftarrow$) We verify the hypotheses of Theorem \[construct-tess\], with $G_2$ in the role of $L$.
Let $\overline{H}$ be the image of $H$ under the natural homomorphism $G \to
G/G_2$. Because $H \subset AN$, we know that $\overline{H}$ is closed [[(]{}see [\[solvable\][[(]{}\[solvable-H=Rn\][)]{}]{}]{}[)]{}]{}. It is well known (and follows easily from [[(]{}\[proper<>CHC\][)]{}]{}) that any closed subgroup acts properly on the ambient group, so this implies that $\overline{H}$ acts properly on $G/G_2$. From the definition of $H$, we have $H \cap G_2 = e$, so we conclude that $H
{\cong}\overline{H}$ acts properly on $G/G_2$; equivalently, $G_2$ acts properly on $G/H$ [(cf. \[proper<>CHC\])]{}.
Because $AN = (AN \cap G_1) \times (AN \cap G_2)$, we have $d(G) = d(G_1) + d(G_2)$. Also, we have $d(H) = \dim H$ [[(]{}see \[d(H)=dimH\][)]{}]{} and, from the definition of $H$, we have $\dim H = \dim(AN \cap G_1) = d(G_1)$. Therefore $$d(H) + d(G_2) = d(G_1) + d(G_2) = d(G) .$$
There is a cocompact lattice in $G_2$ [[(]{}cf. [\[classical\][[(]{}\[classical-Borel\][)]{}]{}]{}[)]{}]{}.
So Theorem \[construct-tess\] implies that $G/H$ has a tessellation.
T. Kobayashi’s examples of higher real rank
-------------------------------------------
T. Kobayashi observed that, besides the examples with $G = {\operatorname{SO}}(2,2n)$ or ${\operatorname{SU}}(2,2n)$ [[(]{}see \[KulkarniEg\][)]{}]{}, Theorem \[construct-tess\] can also be used to construct tessellations of some homogeneous spaces $G/H$ in which $G$ and $H$ are simple Lie groups with ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G > 2$. He found one pair of infinite families, and several isolated examples.
\[Kobayashibigeg\] Each of the following homogeneous spaces has a tessellation:
1. \[Kobayashibigeg-SO4/Sp1\] ${\operatorname{SO}}(4,4n)/{\operatorname{Sp}}(1,n)$;
2. ${\operatorname{SO}}(4,4n)/{\operatorname{SO}}(3,4n)$;
3. ${\operatorname{SO}}(8,8)/{\operatorname{SO}}(8,7)$;
4. ${\operatorname{SO}}(8,8)/\operatorname{Spin}(8,1)$;
5. ${\operatorname{SO}}(4,4)/{\operatorname{SO}}(4,1)$;
6. ${\operatorname{SO}}(4,4)/\operatorname{Spin}(4,3)$;
7. ${\operatorname{SO}}(4,3)/{\operatorname{SO}}(4,1)$;
8. ${\operatorname{SO}}(4,3)/G_{2(2)}$.
It would be very interesting to find other examples of simple Lie groups $G$ with reductive subgroups $H$ and $L$ that satisfy the hypotheses of Theorem \[construct-tess\].
Let $G = {\operatorname{SO}}(4,4n)$ and $H' = {\operatorname{Sp}}(1,n) \cap AN$. From [\[Kobayashibigeg\][[(]{}\[Kobayashibigeg-SO4/Sp1\][)]{}]{}]{}, we know that $G/H$ has a tessellation. H. Oh and D. Witte [@OhWitte-announce Thm. 4.6(2)] pointed out that the deformations $G/H_B$ (where $H_B$ is as in Theorem \[HBthm\], with ${\mathbb{F}}= {\mathord{\mathbb{C}}}$) also have tessellations, but it is not known whether there are other deformations of $G/H'$ that also have tessellations.
It does not seem to be known whether the other examples in Theorem \[Kobayashibigeg\] lead to nontrivial deformations, after intersecting $H$ with $AN$.
Large subgroups of ${\operatorname{SO}}(2,n)$ and ${\operatorname{SU}}(2,n)$ {#SUFlargeSect}
============================================================================
This section presents a short proof of the results we need from [@OhWitte-CDS] and [@IozziWitte-CDS]. Those papers provide an approximate calculation of $\mu(H)$, for every closed, connected subgroup $H$ of ${\operatorname{SO}}(2,n)$ or ${\operatorname{SU}}(2,n)$, respectively, but here we consider only subgroups of large dimension. Also, we do not need a complete description of the entire set $\mu(H)$; we are only interested in whether or not there is a curve $h^t$, such that $\rho(h^t) \asymp \lVert h^t\rVert^k$, for some $k \in
\{1,2\}$. The main results of this section are Theorem \[maxnolinear\] (for $k = 1$) and Theorem \[bestnosquare\] (for $k = 2$). They give a sharp upper bound on $d(H)$, for subgroups $H$ that fail to contain such a curve, and, if $n$ is even, also provide a fairly explicit description of all the subgroups of $AN$ that attain the bound.
Because of the limited scope of this section, the proof here is shorter than the previous work, and we are able to give a fairly unified treatment of the two groups ${\operatorname{SO}}(2,n)$ and ${\operatorname{SU}}(2,n)$. The arguments are elementary, but they involve case-by-case analysis and a lot of details, so they are not pleasant to read.
\[StandingSU2F\] Throughout this section:
1. We use the notation of §\[coordsSect\]. (In particular, ${\mathbb{F}}= {\mathord{\mathbb{R}}}$ or ${\mathord{\mathbb{C}}}$, and ${q}= {\operatorname{\dim_{{\mathord{\mathbb{R}}}}}}{\mathbb{F}}$.)
2. $G = {\operatorname{SU}}(2,n;{\mathbb{F}})$.
3. $n \ge 3$.
4. $H$ is a closed, connected subgroup of $AN$ that is compatible with $A$ [[(]{}see \[compatibleDefn\][)]{}]{}, so $\dim H = d(H)$ [[(]{}see \[d(H)=dimH\][)]{}]{}.
5. $U = H \cap N$. (Note that $U$ is connected [[(]{}see [\[solvable\][[(]{}\[solvable-HcapL\][)]{}]{}]{}[)]{}]{}.)
6. ${\mathfrak{\lowercase{U}}}_{\phi=0} = \{\, u \in {\mathfrak{\lowercase{U}}} \mid \phi_u = 0 \,\}$.
7. We use the notation of §\[CalcSect\]. (In particular, $\lVert\rho(h)\rVert$ is defined in [[(]{}\[rhoDefn\][)]{}]{} and $\Delta(h)$ is defined in [[(]{}\[DeltaDefn\][)]{}]{}.)
8. Except in Subsection \[compatibleSubsect\], $H$ is compatible with $A$ [[(]{}see \[compatibleDefn\][)]{}]{}.
Subgroups compatible with $A$ {#compatibleSubsect}
-----------------------------
Recall that the Real Jordan Decomposition of an element of $G$ is defined in [[(]{}\[JordanDecomp\][)]{}]{}; any element $g$ of $AN$ has a Real Jordan Decomposition $g = au$ (with $c$ trivial). If $a$ is an element of $A$, rather than only conjugate to an element of $A$, we could say that $g$ is “compatible with $A$." We now define a similar, useful notion for subgroups of $AN$. Lemma \[conjtocompatible\] shows there is usually no loss of generality in assuming that $H$ is compatible with $A$, and Lemma \[not-semi\] shows that the compatible subgroups can be described fairly explicitly.
\[compatibleDefn\] Let us say that $H$ is *compatible* with $A$ if $H
\subset T U C_N(T)$, where $T = A \cap (HN)$, $U = H \cap N$, and $C_N(T)$ denotes the centralizer of $T$ in $N$.
In preparation for the proofs of the main results, let us state a lemma that records a few of the nice properties of Jordan components.
\[aucgood\] Let $g = auc$ be the Real Jordan Decomposition of an element $g$ of $G$. Then
1. the Real Jordan Decomposition of ${\operatorname{Ad}\nolimits}g$ is ${\operatorname{Ad}\nolimits}g = ({\operatorname{Ad}\nolimits}a) ({\operatorname{Ad}\nolimits}u) ({\operatorname{Ad}\nolimits}c)$; and
2. $a$, $u$, and $c$ all belong to the Zariski closure of $\langle g
\rangle$.
Therefore:
1. \[aucgood-norm\] $a$, $u$, and $c$ each normalize any connected subgroup of $G$ that is normalized by $g$; and
2. \[aucgood-id\] if $({\operatorname{Ad}\nolimits}g)v \in v + W$, for some $v \in {\mathfrak{\lowercase{G}}}$ and some $({\operatorname{Ad}\nolimits}g)$-invariant subspace $W$ of ${\mathfrak{\lowercase{G}}}$, then $$\mbox{$({\operatorname{Ad}\nolimits}a)v$, $({\operatorname{Ad}\nolimits}u)v$, and $({\operatorname{Ad}\nolimits}c)v$ all belong to $v + W$} .$$
\[conjtocompatible\] $H$ is conjugate, via an element of $N$, to a subgroup that is compatible with $A$.
\[conjtocompatiblePf-algebraic\] Assume, for the Real Jordan Decomposition $h = au$ of each element $h$ of $H$, that $a$ and $u$ belong to $H$.
Let $T$ be a maximal split torus of $H$. (Recall that a split torus is a subgroup consisting entirely of hyperbolic elements.) Then $T$ is contained in some maximal split torus of $G$, that is, in some subgroup of $G$ conjugate to $A$; replacing $H$ by a conjugate, we may assume $T
\subset A$. In other words, we now know that $H \cap A$ is a maximal split torus of $H$.
Given $h \in H$, we have the Real Jordan Decomposition $h = au$. By assumption, $a \in H$; thus, $a$ belongs to some maximal split torus $T'$ of $H$. A fundamental result of the theory of solvable algebraic groups implies that all maximal split tori of $H$ are conjugate via an element of $H \cap N$ [@BorelTits-Reductive Thm. 4.21], so there is some $x \in H \cap N$, such that $x^{-1} a x \in A$. Then $\langle T, x^{-1} a x
\rangle$, being a subgroup of $A$, is a split torus. Thus, the maximality of $T$ implies that $x^{-1} a x \in T$; let $t = x^{-1} a x$. Then $$h = au = x t x^{-1} u = t (t^{-1} x t) x^{-1} u \in T (H \cap N) .$$ Since $h \in H$ is arbitrary, we conclude that $$\label{conjtocompatiblePf-H=TU}
H = T (H \cap N) ,$$ so $H$ is compatible with $A$.
The general case.
Let $$\overline{H}
= \langle\, a, u \mid \mbox{$au = ua \in H$, $a$~hyperbolic, $u$~unipotent}
\,\rangle$$ be the subgroup of $AN$ generated by the Jordan Components of the elements of $H$. (Of course, since every element of $H$ has a Jordan Decomposition, we have $H \subset \overline{H}$.) Then Case \[conjtocompatiblePf-algebraic\] applies to $\overline{H}$, so, replacing $H$ by a conjugate, we may assume $\overline{H} = \overline{T} \,
\overline{U}$, where $\overline{T} = \overline{H} \cap A$ and $\overline{U} =
\overline{H} \cap N$ [[(]{}see \[conjtocompatiblePf-H=TU\][)]{}]{}.
Because $\overline{H}$ normalizes $H$ [[(]{}see [\[aucgood\][[(]{}\[aucgood-norm\][)]{}]{}]{}[)]{}]{}, we know that ${\operatorname{Ad}\nolimits}_G h$ acts as the identity on $\overline{{\mathfrak{\lowercase{H}}}}/{\mathfrak{\lowercase{H}}}$, for all $h \in
H$. Hence, Lemma [\[aucgood\][[(]{}\[aucgood-id\][)]{}]{}]{} implies that ${\operatorname{Ad}\nolimits}_G \overline{h}$ acts as the identity on $\overline{{\mathfrak{\lowercase{H}}}}/{\mathfrak{\lowercase{H}}}$, for all $\overline{h} \in
\overline{H}$; therefore $[\overline{H}, \overline{H}] \subset H$. Also, we have $[\overline{H},\overline{H}] \subset [AN,AN] \subset N$. Thus, letting $U = H \cap N$, we have $$\label{conjtocompatiblePf-comm}
[\overline{H},\overline{H}] \subset H \cap N = U .$$ Because $\overline{T} \subset A$ and $\overline{{\mathfrak{\lowercase{U}}}}$ is $({\operatorname{Ad}\nolimits}_G(T))$-invariant, the adjoint action of $\overline{T}$ on $\overline{{\mathfrak{\lowercase{U}}}}$ is completely reducible, so [[(]{}\[conjtocompatiblePf-comm\][)]{}]{} implies that there is a subspace ${\mathfrak{\lowercase{C}}}$ of $\overline{{\mathfrak{\lowercase{U}}}}$, such that $[\overline{T}, {\mathfrak{\lowercase{C}}}] = 0$ and ${\mathfrak{\lowercase{U}}} +
{\mathfrak{\lowercase{C}}} = \overline{{\mathfrak{\lowercase{U}}}}$. Therefore, $U \, C_{\overline{U}}(\overline{T}) =
\overline{U}$, so $$\label{conjtocompatiblePf-Hbarcompat}
\overline{H}
= \overline{T} \, \overline{U}
= \overline{T} U C_{\overline{U}}(\overline{T})
\subset \overline{T} U C_N(\overline{T}) .$$
Let $\pi \colon AN \to A$ be the projection with kernel $N$, and let $T =
\pi(H)$. Then $$T = \pi(H) \subset \pi(\overline{H}) = \overline{T} ,$$ so $C_N(T) \supset C_N(\overline{T})$. For any $h \in H$, we know, from [[(]{}\[conjtocompatiblePf-Hbarcompat\][)]{}]{}, that there exist $t \in
\overline{T}$, $u \in U$ and $c \in C_N(\overline{T})$, such that $h = tuc$. Because $uc \in N$, we must have $t = \pi(h) \in T$ and, because $C_N(T)
\supset C_N(\overline{T})$, we have $c \in C_N(T)$. Therefore, $h \in T U
C_N(T)$. We conclude that $H \subset T U C_N(T)$, so $H$ is compatible with $A$.
The preceding proposition shows that $H$ is conjugate to a subgroup $H'$ that is compatible with $A$. The subgroup $H'$ is usually not unique, however. The following lemma provides one way to change $H'$, often to an even better subgroup.
\[conjUomega\] Assume that $H$ is compatible with $A$, and let $T = A \cap (HN)$. If $u \in
C_N(T)$, then $u^{-1} H u$ is compatible with $A$.
Let $H' = u^{-1} H u$, $T' = A \cap (H'N)$, and $U' = H' \cap N$. Because $u$ centralizes $T$, we have $$u^{-1} T u = T .$$ Also, because $u \in N$, and $N$ is normal, we have $u^{-1}HuN = HN$, so $$u^{-1} T u = T = A \cap (HN) = A \cap (u^{-1}HuN) = A \cap (H'N) = T' .$$ Since $u \in N$, we have $u^{-1} N u = N$, so $$u^{-1} U u = u^{-1} (H \cap N) u
= (u^{-1} H u) \cap (u^{-1} N u)
= H' \cap N
= U' ,$$ and $$u^{-1} C_N(T) u = C_{u^{-1}Nu}(u^{-1} T u) = C_N(T') .$$ Thus, $$H' = u^{-1} H u
\subset u^{-1} T U C_N(T) u
= \bigl(u^{-1} T u \bigr) \bigl( u^{-1} U u \bigr) \bigl( u^{-1} C_N(T) u
\bigr)
= T' U' C_N(T'),$$ as desired.
\[not-semi\] If $H$ is compatible with $A$, then either
1. \[not-semi-TU\] $H = (H \cap A) \ltimes (H \cap N)$; or
2. \[not-semi-not\] there is a positive root $\omega$, a nontrivial group homomorphism $\psi\colon \ker \omega \to {N}_\omega {N}_{2\omega}$, and a closed, connected subgroup $U$ of $N$, such that
1. \[not-semi-codim1\] $H = \{\, a \, \psi(a) \mid a \in \ker \omega \,\} \, U$;
2. \[not-semi-normal\] $U$ is normalized by both $\ker\omega$ and $\psi( \ker\omega)$; and
3. \[not-semi-disjoint\] $U \cap \psi( \ker\omega) = e$.
Because $H$ is compatible with $A$, we have $H \subset T U C_N(T)$, where $T
= A \cap (HN)$ and $U = H \cap N$. We may assume that $H \neq TU$, for otherwise [[(]{}\[not-semi-TU\][)]{}]{} holds. Therefore $C_N(T) \neq e$. Because ${\mathfrak{\lowercase{n}}}$ is a sum of root spaces, this implies that there is a positive root $\omega$, such that $T \subset \ker\omega$. Because ${\operatorname{\hbox{\upshape${\mathord{\mathbb{R}}}$-rank}}}G = 2$, we have $\dim (\ker\omega) = 1$, so we must have $T =\ker\omega$ (otherwise we would have $T = e$, so $H = U = TU$; hence [[(]{}\[not-semi-TU\][)]{}]{} holds). Therefore, $C_N(T) = U_\omega U_{2\omega}$.
Because $U \subset H \subset T U C_N(T)$, we have $H = U \bigl[H \cap \bigl(T
C_N(T) \bigr) \bigr]$, so there is a nontrivial one-parameter subgroup $\{x^t\}$ in $H \cap \bigl(T C_N(T) \bigr)$ that is not contained in $U$. Because $T$ centralizes $C_N(T)$, we may write $x^t = a^t u^t$ where $\{a^t\}$ is a one-parameter subgroup of $T$ and $\{u^t\}$ is a one-parameter subgroup of $C_N(T)$. Furthermore, this decomposition is unique, because $T
\cap C_N(T) = e$. (In fact, $x^t = a^t u^t$ is the Real Jordan Decomposition of $x^t$.) Define $\psi \colon \ker\omega \to U_\omega U_{2\omega}$ by $\psi(a^t) = u^t$ for all $t \in {\mathord{\mathbb{R}}}$.
[[(]{}\[not-semi-codim1\][)]{}]{} For all $t \in {\mathord{\mathbb{R}}}$, we have $a^t \psi(a^t) = a^t
u^t = x^t \in H$, which establishes one inclusion of [[(]{}\[not-semi-codim1\][)]{}]{}. The other will follow if we show that $\dim H - \dim U = 1$, so suppose $\dim
H - \dim U \ge 2$. Then Lemma [\[dimT\][[(]{}\[dimT-A\][)]{}]{}]{} implies that $A \subset H$, so it follows from Lemma \[rootdecomp\] (with $T = A$ and $\omega = 0$) that $H = A \ltimes (H \cap N)$, contradicting our assumption that $H \neq
TU$.
[[(]{}\[not-semi-normal\][)]{}]{} Because $x^t \in H$, we know that each of $a^t$ and $u^t$ normalizes $H$ [[(]{}see [\[aucgood\][[(]{}\[aucgood-norm\][)]{}]{}]{}[)]{}]{}. Being in $AN$, they also normalize $N$. Therefore, they normalize $H \cap N = U$.
[[(]{}\[not-semi-disjoint\][)]{}]{} Suppose $U \cap \psi(\ker\omega) \neq e$. Because the intersection $U \cap \psi(\ker\omega)$ is connected (see \[not-semi-normal\] and [\[solvable\][[(]{}\[solvable-HcapL\][)]{}]{}]{}), and $\dim(\ker\omega) = 1$, we must have $\psi(\ker\omega) \subset U$. Therefore $a^t = x^t u^{-t} \in HU = H$, so $T \subset H$. This contradicts our assumption that $H \neq TU$.
\[TnormsU\] If $H$ is compatible with $A$, then $A \cap (HN)$ normalizes $H \cap N$.
\[dimT\] If $\dim \bigl( H/(H \cap N) \bigr) \ge \dim A$, then
1. \[dimT-A\] $H$ contains a conjugate of $A$; and
2. \[dimT-CDS\] $H$ is a Cartan-decomposition subgroup.
[[(]{}\[dimT-A\][)]{}]{} Let $\pi \colon AN \to A$ be the projection with kernel $N$, and let $\overline{H}$ be the Zariski closure of $H$. From the structure theory of solvable algebraic groups [@Borel-Algic Thm. 10.6(4), pp. 137–138], we know that $\overline{H} = T \ltimes U$ is the semidirect product of a torus $T$ and and unipotent subgroup $U \subset N$. Replacing $H$ by a conjugate under $N$, we may assume that $T \subset A$. Since $$\dim A \le \dim \bigl( H/(H \cap N) \bigr) = \dim\bigl(\pi(H) \bigr) \le
\dim A ,$$ we must have $\pi(H) = A$, so $$A = \pi(H) \subset \pi(\overline{H}) = \pi(TU) = T \subset
\overline{H}$$ normalizes $H$ [[(]{}see \[Zar-norm\][)]{}]{}. Then, since $\pi(H) = A$, we conclude that $A
\subset H$ [[(]{}see \[rootdecomp\][)]{}]{}.
[[(]{}\[dimT-CDS\][)]{}]{} From [[(]{}\[dimT-A\][)]{}]{}, we see that, by replacing $H$ with a conjugate subgroup, we may assume $A \subset H$. Because $A$ is a Cartan-decomposition subgroup [[(]{}see \[AisCDS\][)]{}]{}, this implies $H$ is a Cartan-decomposition subgroup.
The following basic result was used twice in the above arguments.
\[Zar-norm\] If $H$ is a closed, connected subgroup of $G$, then the Zariski closure of $H$ normalizes $H$.
Subgroups with no nearly linear curve
-------------------------------------
Our goal is to prove Theorem \[maxnolinear\]; we begin with some preliminary results.
First, an observation that simplifies the calculations in some cases, by allowing us to assume that $x_u = 0$.
\[x=0\] Let $u \in {\mathfrak{\lowercase{U}}}$. If ${\operatorname{\dim_{{\mathbb{F}}}}}({\mathbb{F}}x_u + {\mathbb{F}}y_u) \le 1$ and $y_u \neq 0$, then there is some $g \in {N}_\alpha$, such that
1. $x_{g^{-1} u g} = 0$,
2. $\phi_{g^{-1} u g} = \phi_u$, and
3. $y_{g^{-1} u g} = y_u$.
Because ${\operatorname{\dim_{{\mathbb{F}}}}}({\mathbb{F}}x_u + {\mathbb{F}}y_u) \le 1$ and $y_u \neq 0$, there is some $\lambda \in {\mathbb{F}}$, such that $x_u = \lambda y_u$. Let
- $v$ be the element of ${{\mathfrak{\lowercase{N}}}}_\alpha$ with $\phi_\alpha = -\lambda$,
- $g = \exp(v) \in {N}_\alpha$, and
- $w = g^{-1} u g$.
From [[(]{}\[conjugation\][)]{}]{}, we see that
- $\phi_w = \phi_u$,
- $x_w = x_u + \phi_v y_u =
0$, and
- $y_w = y_u$,
as desired.
\[HinN-linear\] If there does not exist a continuous curve $h^t \to \infty$ in $U$, such that $\rho(h^t) \asymp h^t$, then
1. \[HinN-linear-phi=0&eta2=\] for every nonzero element $z$ of ${{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$, we have $|\eta_z|^2 + {{\mathord{\mathsf{x}}}}_z {{\mathord{\mathsf{y}}}}_z
\neq 0$; and
2. \[HinN-linear-phi=0&=0\] for every element $u$ of ${\mathfrak{\lowercase{U}}}_{\phi=0}$, such that ${\operatorname{\dim_{{\mathbb{F}}}}}({\mathbb{F}}x_u + {\mathbb{F}}y_u) = 1$, we have $${{\mathord{\mathsf{x}}}}_u |y_u|^2 + {{\mathord{\mathsf{y}}}}_u |x_u|^2 + 2 {\operatorname{Im}}( x_u y_u^{\dagger}
\eta_u^{\dagger}) \neq 0 .$$
[[(]{}\[HinN-linear-phi=0&eta2=\][)]{}]{} Suppose there is a nonzero element $z$ of ${{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$ with $\Delta(z) = 0$. Let $h^t = \exp(tz) = {\operatorname{Id}}+ tz$ [[(]{}see \[exp(phi=0)\][)]{}]{}. We have $$h_{j,k} =
\begin{cases}
O(t) & \mbox{for all $j,k$}, \\
O(1) & \mbox{if $(j,k) \notin \{1,2\} \times \{n+1,n+2\}$}.
\end{cases}$$ Then, because $\Delta(h^t) = 0$, it is easy to see that $\rho(h) \asymp t$. Also, we have $h^t = {\operatorname{Id}}+ tz \asymp t$, so $\rho(h^t) \asymp t \asymp h^t$, as desired.
[[(]{}\[HinN-linear-phi=0&=0\][)]{}]{} Suppose there is an element $u$ of ${\mathfrak{\lowercase{U}}}_{\phi=0}$, such that ${\operatorname{\dim_{{\mathbb{F}}}}}({\mathbb{F}}x_u + {\mathbb{F}}y_u ) = 1$, and $$\label{xx+yy=0}
{{\mathord{\mathsf{x}}}}_u |y_u|^2 + {{\mathord{\mathsf{y}}}}_u |x_u|^2 + 2 {\operatorname{Im}}( x_u y_u^{\dagger} \eta_u^{\dagger})
= 0 .$$ Let $h = h^t = \exp(tu)$.
\[HinNlinearpf-x=0\] Assume $x_u = 0$.
Because ${\operatorname{\dim_{{\mathbb{F}}}}}({\mathbb{F}}x_u + {\mathbb{F}}y_u ) = 1$, we must have $y_u \neq 0$. Then, from , we know that ${{\mathord{\mathsf{x}}}}_u = 0$. So, from , we see that
- $h_{2,n+1} \asymp |y_u|^2 t^2 \asymp t^2$,
- $h_{j,k} = O (t)$ whenever $(j,k) \neq (2,n+1)$, and
- $h_{j,k} = O(1)$ whenever $j \neq 2$ and $k \neq n+1$.
This implies that $\rho(h) \asymp t^2 \asymp h$.
\[HinNlinearpf-y=0\] Assume $y_u = 0$.
This is similar to Case \[HinNlinearpf-x=0\]. (In fact, this can be obtained as a corollary of Case \[HinNlinearpf-x=0\] by replacing $H$ with its conjugate under the Weyl reflection corresponding to the root $\alpha$.)
Assume $y_u \neq 0$.
Because ${\operatorname{\dim_{{\mathbb{F}}}}}({\mathbb{F}}x_u + {\mathbb{F}}y_u ) = 1$, Lemma \[x=0\] implies there is some $g \in {N}_\alpha$, such that, letting $w = g^{-1} u g$, we have $\phi_w = \phi_u = 0$, $x_w = 0$, and $y_w = y_u \neq 0$. We show below that is satisfied with $w$ in the place of $u$, so, from Case \[HinNlinearpf-x=0\], we conclude that $\rho \bigl( \exp(tw) \bigr)
\asymp \exp(tw)$. Thus, the desired conclusion follows from Lemma \[conjcurve\] (with $k = 1$).
To complete the proof, we now show that is satisfied with $w$ in the place of $u$. (This can be verified by direct calculation, but we give a more conceptual proof.) Because $g^{-1} \in {N}_\alpha$, multiplication by $g^{-1}$ on the left performs a row operation on the first two rows of $h$; likewise, multiplication by $g$ on the right performs a column operation on the last two columns of $h$. These operations do not change the determinant $\Delta(h)$: thus $$\Delta \bigl( \exp(tw) \bigr) = \Delta \bigl( \exp(tu) \bigr) .$$
From and the definition of $\Delta$, we see that $$\Delta \bigl( \exp(tu) \bigr)
= - \frac{1}{4} \bigl( |x_u|^2 |y_u|^2 - |x_u y_u^\dagger|^2 \bigr) t^4
+ \bigl( {{\mathord{\mathsf{x}}}}_u |y_u|^2 + {{\mathord{\mathsf{y}}}}_u |x_u|^2 + 2 {\operatorname{Im}}( x_u y_u^{\dagger}
\eta_u^{\dagger}) \bigr) t^3
+ O(t^2) .$$ Because ${\operatorname{\dim_{{\mathbb{F}}}}}({\mathbb{F}}x_u + {\mathbb{F}}y_u) = 1$, we have $|x_u|^2 |y_u|^2 - |x_u
y_u^\dagger|^2 = 0$, so this simplifies to $$\Delta \bigl( \exp(tu) \bigr)
= \bigl( {{\mathord{\mathsf{x}}}}_u |y_u|^2 + {{\mathord{\mathsf{y}}}}_u |x_u|^2 + 2 {\operatorname{Im}}( x_u y_u^{\dagger}
\eta_u^{\dagger}) \bigr) t^3
+ O(t^2) .$$ Thus, is equivalent to the condition that $\Delta \bigl(
\exp(tu) \bigr) = O(t^2)$. Then, since $$\Delta \bigl( \exp(tw) \bigr) = \Delta \bigl( \exp(tu) \bigr) = O(t^2) ,$$ we conclude that is also valid for $w$.
\[dim(U+Z)<3\] If there does not exist a continuous curve $h^t \to \infty$ in $U$, such that $\rho(h^t) \asymp h^t$, then $$\dim {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}+ \dim {\mathfrak{\lowercase{U}}}/ {\mathfrak{\lowercase{U}}}_{\phi=0} \le 2{q}-1 .$$
Furthermore, if equality holds, and ${\mathbb{F}}= {\mathord{\mathbb{C}}}$, then ${\mathfrak{\lowercase{U}}} = {\mathfrak{\lowercase{U}}}_{\phi=0}$ and $\dim {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}= 3$.
Assume ${\mathbb{F}}= {\mathord{\mathbb{C}}}$.
Because $|\eta_z|^2 + {{\mathord{\mathsf{x}}}}_z {{\mathord{\mathsf{y}}}}_z$ is a quadratic form of signature $(1,3)$ on ${{\mathfrak{\lowercase{D}}}}$, we know, from [\[HinN-linear\][[(]{}\[HinN-linear-phi=0&eta2=\][)]{}]{}]{}, that $\dim {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}\le 3 = 2{q}-1$.
Thus, we may assume ${\mathfrak{\lowercase{U}}} / {\mathfrak{\lowercase{U}}}_{\phi=0} \neq 0$, so there is some $u \in
{\mathfrak{\lowercase{U}}}$, such that $\phi_u \neq 0$.
\[dim(U+Z)<3pf-yy=0\] Assume there exists $z \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$, such that ${{\mathord{\mathsf{y}}}}_z = 0$ and $\eta_z \notin
{\mathord{\mathbb{R}}}\phi_u$.
From [[(]{}\[\[u,v\]\][)]{}]{}, we see that $[u,z] \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$, with ${{\mathord{\mathsf{x}}}}_{[u,z]} = -{\operatorname{Im}}(\phi_u {\overline{\eta_z}}) \neq 0$ and ${{\mathord{\mathsf{y}}}}_{[u,z]} = \eta_{[u,z]} = 0$. This contradicts [\[HinN-linear\][[(]{}\[HinN-linear-phi=0&eta2=\][)]{}]{}]{}.
Assume there exists $z \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$, such that ${{\mathord{\mathsf{y}}}}_z \neq 0$.
From [[(]{}\[\[u,v\]\][)]{}]{}, we see that $[u,z]$ is an element of ${{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$, such that ${{\mathord{\mathsf{y}}}}_{[u,z]} = 0$, and $\eta_{[u,z]} = \phi_u {{\mathord{\mathsf{y}}}}_z$ is a purely imaginary multiple of $\phi_u$. So Subcase \[dim(U+Z)<3pf-yy=0\] applies (with $[u,z]$ in the place of $z$).
Assume ${{\mathord{\mathsf{y}}}}_z = 0$ and $\eta_z \in {\mathord{\mathbb{R}}}\phi_u$, for all $z \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$.
From [\[HinN-linear\][[(]{}\[HinN-linear-phi=0&eta2=\][)]{}]{}]{}, we see that ${{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}\cap
{{\mathfrak{\lowercase{N}}}}_{2\alpha+2\beta} = \{0\}$, so the assumption of this subcase implies $\dim {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}\le 1$. Thus, $$\dim {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}+ \dim {\mathfrak{\lowercase{U}}}/ {\mathfrak{\lowercase{U}}}_{\phi=0} \le 1 + 2 = 3 = 2{q}-1 ,$$ so the desired inequality holds.
If equality holds, then $\dim {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}= 1$ and $\dim {\mathfrak{\lowercase{U}}}/ {\mathfrak{\lowercase{U}}}_{\phi=0} =
2$. Thus, we may choose $z \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$, such that $z \neq 0$, and $u' \in
{\mathfrak{\lowercase{U}}}$, such that ${\mathord{\mathbb{R}}}\phi_u + {\mathord{\mathbb{R}}}\phi_{u'} = {\mathord{\mathbb{C}}}$. From the assumption of this subcase, we know that $\eta_z \in {\mathord{\mathbb{R}}}\phi_u$; thus, $\eta_z \notin {\mathord{\mathbb{R}}}\phi_{u'}$. Therefore, Subcase \[dim(U+Z)<3pf-yy=0\] applies, with $u'$ in the place of $u$.
Assume ${\mathbb{F}}= {\mathord{\mathbb{R}}}$.
Because $\dim {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}\le \dim {{\mathfrak{\lowercase{N}}}}_{\alpha+2\beta} = 1$ and $\dim {\mathfrak{\lowercase{U}}} /
{\mathfrak{\lowercase{U}}}_{\phi=0} \le \dim {{\mathfrak{\lowercase{N}}}}_\alpha = 1$, the desired inequality holds unless ${{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}\neq 0$ and ${\mathfrak{\lowercase{U}}} / {\mathfrak{\lowercase{U}}}_{\phi=0} \neq 0$. Thus, we may assume there is some nonzero $z \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$ and some $u \in {\mathfrak{\lowercase{U}}}$, such that $\phi_u \neq 0$.
Assume $y_u = 0$.
We may assume $|x_u|^2 + \phi_u \eta_u \neq 0$ (by replacing $u$ with $u +
z$, if necessary). Let $h^t = \exp(tu)$. From , we see that $h^t_{1,n+2} \asymp t^{2}$, but $$h^t_{j,k} =
\begin{cases}
O(t) & \hbox{if $(j,k) \neq (1,n+2)$,} \\
O(1) & \hbox{if $j \neq 1$ and $k \neq n+2$} .
\end{cases}$$ Therefore $\rho(h^t) = O(t^2) = O(h^t)$, so Lemma [\[Owalls\][[(]{}\[Owalls-linear\][)]{}]{}]{} implies that $\rho(h^t) \asymp h^t$. This is a contradiction.
Assume $y_u \neq 0$.
Let $v$ be the element of ${{\mathfrak{\lowercase{N}}}}_\beta$ with $y_v = -(1/\phi_u) x_u$, and let $w = \exp(-v) u \exp(v)$. Then $x_w = 0$ [[[(]{}see \[conjugation\] [and \[\[u,v\]\]]{}[)]{}]{}]{}. Thus, by replacing $H$ with the conjugate $\exp(-v) H \exp(v)$ [[(]{}see \[conjcurve\][)]{}]{}, we may assume $x_u = 0$. For any large real number $t$, let $h = h^t$ be the element of $\exp(t u + {{\mathfrak{\lowercase{N}}}}_{\alpha+2\beta})$ that satisfies $\eta_h =
-\phi_h \lVert y_h\rVert^2/{12}$. Then, from , we see that $$h =
\begin{pmatrix}
1 & \phi_h & \frac{1}{2} \phi_h y_h & - \frac{1}{4} \phi_h |y_h|^2
& \frac{1}{8} \phi_h^2 |y_h|^2 \\
& 1 & y_h & - \frac{1}{2} |y_h|^2
& \frac{1}{4} \phi |y_h|^2 \\
& & 1 & -y_h^\dagger
& \frac{1}{2} \phi_h y_h^\dagger \\
& & & 1 & -\phi_h \\
& & & & 1 \\
\end{pmatrix}
.$$ Clearly, we have $h \asymp \phi_h^2 |y_h|^2$.
A calculation shows that $\Delta(h) = 0$, and certain other $2 \times 2$ minors also have cancellation. With this in mind, it is not difficult to verify that $\rho(h) \asymp \phi_h^2 |y_h|^2 \asymp h$ (see [@OhWitte-CDS Case 3 of pf. of 5.12($3 \Rightarrow 2$)] for details). This is a contradiction.
\[dim(U)<2n\] If there does not exist a continuous curve $h^t \to \infty$ in $U_{\phi=0}$, such that $\rho(h^t) \asymp h^t$, then $$\dim {\mathfrak{\lowercase{U}}}_{\phi=0}/ {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}\le
\begin{cases}
{q}(n-2) & \mbox{if $n$ is even} \\
{q}(n-3) & \mbox{if $n$ is odd and $n \ne 3$} \\
{q}-1 & \mbox{if $n = 3$} .
\end{cases}$$ Furthermore,
1. \[dim(U)<2n-eqeven\] if equality holds, and $n$ is even, then ${\operatorname{\dim_{{\mathbb{F}}}}}({\mathbb{F}}x_u + {\mathbb{F}}y_u) = 2$, for every $u \in {\mathfrak{\lowercase{U}}}_{\phi=0} \smallsetminus {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$;
2. \[dim(U)<2n-eq=3\] if equality holds, and $n = 3$, then $\dim {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}\le {q}$.
By passing to a subgroup, we may assume ${\mathfrak{\lowercase{U}}} = {\mathfrak{\lowercase{U}}}_{\phi=0}$. Let $V$ be the projection of ${\mathfrak{\lowercase{U}}}$ to ${{\mathfrak{\lowercase{N}}}}_\beta + {{\mathfrak{\lowercase{N}}}}_{\alpha+\beta}$; then $\dim V = \dim {\mathfrak{\lowercase{U}}}/ \dim {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$.
\[dim(U)<2n-<xy>pf\] Assume ${\operatorname{\dim_{{\mathbb{F}}}}}({\mathbb{F}}x_u + {\mathbb{F}}y_u ) = 2$ for every $u \in {\mathfrak{\lowercase{U}}} \smallsetminus {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$.
Assume $n$ is even.
From Theorem [\[HinN-linear\][[(]{}\[HinN-linear-phi=0&eta2=\][)]{}]{}]{}, we know that $V$ does not intersect ${{\mathfrak{\lowercase{N}}}}_\beta$ (or ${{\mathfrak{\lowercase{N}}}}_{\alpha+\beta}$, either, for that matter), so $$\dim V + \dim {{\mathfrak{\lowercase{N}}}}_\beta
\le \dim ({{\mathfrak{\lowercase{N}}}}_\beta + {{\mathfrak{\lowercase{N}}}}_{\alpha+\beta})
= \dim {{\mathfrak{\lowercase{N}}}}_\beta + \dim {{\mathfrak{\lowercase{N}}}}_{\alpha+\beta} .$$ Therefore $$\dim {\mathfrak{\lowercase{U}}}/ \dim {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}= \dim V \le \dim {{\mathfrak{\lowercase{N}}}}_{\alpha+\beta} = {q}(n-2)
,$$ as desired. (If equality holds, then we have Conclusion [[(]{}\[dim(U)<2n-eqeven\][)]{}]{}.)
\[dim(U)<2npf-<xy>-n=2\] Assume $n$ is odd.
\[dim(U)<2npf-U/Z\] We have $\dim {\mathfrak{\lowercase{U}}} / {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}\le {q}(n-3)$.
Suppose not: then $$\dim V \ge {q}(n-3) + 1 .$$ (This will lead to a contradiction.) Let $X = \{x_v \mid v \in V\}$, so $X$ is a ${\mathord{\mathbb{R}}}$-subspace of ${\mathbb{F}}^{n-2}$. For each $x \in X$, there is some $v =
v(x) \in V$, such that $x_v = x$; define $f(x) = y_{v(x)}$. By the assumption of this case, we know $$V \cap {{\mathfrak{\lowercase{N}}}}_\beta = \{0\} ,$$ so $v(x)$ is uniquely determined by $x$; thus, $f \colon X \to {\mathbb{F}}^{n-2}$ is a well-defined ${\mathord{\mathbb{R}}}$-linear map. Also, again from the assumption of this case, we know that $$\label{TxnotinFx}
\mbox{$f(x) \notin {\mathbb{F}}x$ for every nonzero $x \in X$.}$$ Because $V \cap {{\mathfrak{\lowercase{N}}}}_\beta = 0$, we have $$\dim X = \dim V \ge {q}(n-3) + 1= \dim {\mathbb{F}}^{n-2} - ({q}-1) .$$
If ${\mathbb{F}}= {\mathord{\mathbb{R}}}$ (that is, if ${q}= 1$), this implies $X = {\mathord{\mathbb{R}}}^{n-2}$, so $f$ is defined on all of ${\mathord{\mathbb{R}}}^{n-2}$. Because $n$ is odd, this implies that $f$ has a real eigenvalue, which contradicts .
We may now assume ${\mathbb{F}}= {\mathord{\mathbb{C}}}$. Let
- $E = (X \times {\mathord{\mathbb{C}}}^{n-2})/ {\equiv}$, where $(x,v) \equiv
(-x,-v)$,
- ${\mathord{\mathbb{P}X}}$ be the projective space of the real vector space $X$, and
- $\zeta(x,v) = [x] \in {\mathord{\mathbb{P}X}}$, for $(x,v) \in E$,
so $(E, \zeta)$ is a vector bundle over ${\mathord{\mathbb{P}X}}$.
Define $g \colon X \to {\mathord{\mathbb{C}}}^{n-2}$ by $g(x) = i x$. Any ${\mathord{\mathbb{R}}}$-linear transformation $Q \colon X \to {\mathord{\mathbb{C}}}^{n-2}$ is a continuous function, such that $Q(-x) = - Q(x)$ for all $x \in X$; that is, a section of $(E, \zeta)$. Thus, ${\operatorname{Id}}$, $f$, and $g$ each define a section of $(E, \zeta)$. Furthermore, these three sections are pointwise linearly independent over ${\mathord{\mathbb{R}}}$, because implies that $x$, $f(x)$, and $ix$ are linearly independent over ${\mathord{\mathbb{R}}}$, for every nonzero $x \in X$. On the other hand, the theory of characteristic classes [@MilnorStasheff Prop. 4, p. 39] implies that $(E, \zeta)$ does not have three pointwise ${\mathord{\mathbb{R}}}$-linearly independent sections (see [@IozziWitte-CDS Lem. 8.2] for details). This is a contradiction.
Completion of the proof of Subcase \[dim(U)<2npf-<xy>-n=2\].
From Step \[dim(U)<2npf-U/Z\], we see that the desired inequality holds.
We may now assume $n = 3$ and $\dim {\mathfrak{\lowercase{U}}}/{{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}=
{q}- 1$. Since $\dim {\mathfrak{\lowercase{U}}}/{{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}\le {q}(n-3) = 0$, we must have ${q}= 1$, so ${\mathbb{F}}= {\mathord{\mathbb{R}}}$. Therefore ${{\mathfrak{\lowercase{N}}}}_{2\alpha} = {{\mathfrak{\lowercase{N}}}}_{2\beta} = 0$, so $$\dim {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}\le \dim {{\mathfrak{\lowercase{N}}}}_{\alpha+2\beta} = {q},$$ as desired.
Assume there is some $v \in {\mathfrak{\lowercase{U}}} \smallsetminus {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$, such that ${\operatorname{\dim_{{\mathbb{F}}}}}({\mathbb{F}}x_v + {\mathbb{F}}y_v) = 1$.
\[dimUpf-x=0\] Assume $x_v = 0$.
Since $v \notin {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$, we must have $y_v \neq 0$. Then ${{\mathord{\mathsf{x}}}}_{v+z} \neq 0$ for every $z \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$ (otherwise [\[HinN-linear\][[(]{}\[HinN-linear-phi=0&=0\][)]{}]{}]{} yields a contradiction); this implies $${{\mathord{\mathsf{x}}}}_v \neq 0 ,$$ and $$\mbox{${{\mathord{\mathsf{x}}}}_z = 0$ for every $z \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$.}$$ Because ${{\mathord{\mathsf{x}}}}_v \neq 0$, we know that ${\mathbb{F}}\neq {\mathord{\mathbb{R}}}$; so $${\mathbb{F}}= {\mathord{\mathbb{C}}}.$$ Since ${{\mathord{\mathsf{x}}}}_z = 0$ for every $z \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$, but ${{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}\cap {{\mathfrak{\lowercase{N}}}}_{2\beta} =
0$ [[(]{}see [\[HinN-linear\][[(]{}\[HinN-linear-phi=0&eta2=\][)]{}]{}]{}[)]{}]{}, we must have $\eta_z \neq 0$ for every $z \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$. Therefore $$\dim {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}\le \dim {{\mathfrak{\lowercase{N}}}}_{\alpha+2\beta} = {q}= 2 .$$
Let $p \colon V \to {{\mathfrak{\lowercase{N}}}}_{\alpha+\beta}$ be the natural projection. Note that $$\dim \ker p = 1 .$$ (If $v' \in {\mathfrak{\lowercase{U}}}$, with $x_{v'} = 0$, then there is some $t \in {\mathord{\mathbb{R}}}$, such that ${{\mathord{\mathsf{x}}}}_{v' + tv} = {{\mathord{\mathsf{x}}}}_{v'} + t {{\mathord{\mathsf{x}}}}_v = 0$. We also have $x_{v' + tv}
= 0$, so, from [\[HinN-linear\][[(]{}\[HinN-linear-phi=0&=0\][)]{}]{}]{}, we see that $v' + tv \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$. Thus $v' \in {\mathord{\mathbb{R}}}v + {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$. So $\ker p = ({\mathord{\mathbb{R}}}v + {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}})/{{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$ is 1-dimensional.)
Because ${{\mathord{\mathsf{x}}}}_z = 0$ for every $z \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$, and ${\mathfrak{\lowercase{U}}}$ is a Lie algebra, we see, from [[(]{}\[\[u,v\]\][)]{}]{}, that $p(V)$ must be a totally isotropic subspace for the symplectic form $i {\operatorname{Im}}(x \tilde x^{\dagger})$, so $$\dim p(V) \le \frac{1}{2} \dim {{\mathfrak{\lowercase{N}}}}_{\alpha+\beta} = n-2 .$$ Therefore $$\dim {\mathfrak{\lowercase{U}}}/ {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}= \dim V
= \dim p(V) + \dim \ker p
\le (n-2) + 1
= n-1 .$$ This completes the proof if $n \neq 3$:
- If $n$ is even, then, because $n \ge 4$, we have $n - 1 <
2(n-2) = {q}(n-2)$.
- If $n > 3$ is odd, then $n \ge 5$, so $n-1 \le 2(n-3) = {q}(n-3)$.
Now let $n = 3$, and suppose $\dim V = 2$. (This will lead to a contradiction.) Because equality is attained in the proof above, we must have $\dim p(V) = n-2 = 1$, so there exists $w \in {\mathfrak{\lowercase{U}}}$ with $x_w \neq 0$. For $t \in {\mathord{\mathbb{R}}}$, let $w_t = w + tv$. Then $$\begin{aligned}
{{\mathord{\mathsf{x}}}}_{w_t} |y_{w_t}|^2 + {{\mathord{\mathsf{y}}}}_{w_t}|x_{w_t}|^2 + 2 {\operatorname{Im}}(
x_{w_t}y_{w_t}^{\dagger} \eta_{w_t})
&= t^3 {{\mathord{\mathsf{x}}}}_v |y_v|^2 + O(t^2) \\
&\to
\begin{cases}
+{{\mathord{\mathsf{x}}}}_v \infty & \text{as $t \to \infty$} \\
-{{\mathord{\mathsf{x}}}}_v \infty & \text{as $t \to -\infty$}
.
\end{cases}
\end{aligned}$$ Thus, this expression changes sign, so it must vanish for some $t$. On the other hand, since $n = 3$, we have ${\operatorname{\dim_{{\mathord{\mathbb{C}}}}}}({\mathord{\mathbb{C}}}x + {\mathord{\mathbb{C}}}y) \le 1$ for every $x,y \in {\mathord{\mathbb{C}}}^{n-2} = {\mathord{\mathbb{C}}}$. Thus [\[HinN-linear\][[(]{}\[HinN-linear-phi=0&=0\][)]{}]{}]{} yields a contradiction.
\[dimUpf-y=0\] Assume $y_v = 0$.
This is similar to Subcase \[dimUpf-x=0\]. (In fact, this can be obtained as a corollary of Subcase \[dimUpf-x=0\] by replacing $H$ with its conjugate under the Weyl reflection corresponding to the root $\alpha$.)
Assume $y_v \neq 0$.
Because ${\operatorname{\dim_{{\mathbb{F}}}}}({\mathbb{F}}x_v + {\mathbb{F}}y_v ) = 1$, Lemma \[x=0\] implies there is some $g \in {N}_\alpha$, such that, letting $w = g^{-1} v g$, we have $\phi_w = \phi_u = 0$, $x_w = 0$, and $y_w = y_v \neq 0$. There is no harm in replacing $H$ with $g^{-1} H g$ [[(]{}see \[conjcurve\][)]{}]{}. Then Subcase \[dimUpf-x=0\] applies (with $w$ in the place of $v$).
\[maxnolinear\] Recall that Assumptions \[StandingSU2F\] are in effect.
If there does not exist a continuous curve $h^t \to \infty$ in $H$, such that $\rho(h^t) \asymp h^t$, then $$\label{maxnolinear-<}
\dim H \le
\begin{cases}
{q}n & \hbox{if $n$ is even} \\
{q}(n-1) & \hbox{if $n$ is odd.}
\end{cases}$$ Furthermore, if equality holds, and $n$ is even, then
1. \[maxnolinear-TU\] ${\mathfrak{\lowercase{H}}} = (\ker \alpha) \ltimes {\mathfrak{\lowercase{U}}}$;
2. \[maxnolinear-phi0\] $\phi_u = 0$ for every $u \in {\mathfrak{\lowercase{U}}}$;
3. \[maxnolinear-<xy>\] ${\operatorname{\dim_{{\mathbb{F}}}}}( {\mathbb{F}}x_u + {\mathbb{F}}y_u ) = 2$, for every $u \in {\mathfrak{\lowercase{U}}} \smallsetminus {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$;
4. \[maxnolinear-Z\] $|\eta_z|^2 + {{\mathord{\mathsf{x}}}}_z {{\mathord{\mathsf{y}}}}_z \neq 0$ for every nonzero $z \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$;
5. \[maxnolinear-dimU/Z\] $\dim {\mathfrak{\lowercase{U}}} / {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}= {q}(n-2)$; and
6. \[maxnolinear-dimZ\] $\dim {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}= 2{q}- 1$.
Let $$m =
\begin{cases}
{q}(n-2) & \hbox{if $n$ is even} \\
{q}(n-3) & \hbox{if $n \ge 5$ is odd} \\
{q}-1 & \hbox{if $n = 3$}
.
\end{cases}$$ From Lemmas [\[dimT\][[(]{}\[dimT-A\][)]{}]{}]{} and \[dim(U+Z)<3\], and Proposition \[dim(U)<2n\], we have $$\label{<2d+m}
\begin{split}
\dim H
&\le \dim {\mathfrak{\lowercase{H}}}/{\mathfrak{\lowercase{U}}} + (\dim {\mathfrak{\lowercase{U}}}/{\mathfrak{\lowercase{U}}}_{\phi=0} + \dim {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}})
+ \dim {\mathfrak{\lowercase{U}}}_{\phi=0}/ {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}\\
&\le 1 + (2{q}-1) + m \\
&= 2{q}+m .
\end{split}$$ This implies the desired inequality, unless $n = 3$, ${\mathbb{F}}= {\mathord{\mathbb{C}}}$, and we have equality in both Lemma \[dim(U+Z)<3\] and Proposition \[dim(U)<2n\]. This is impossible, because equality in Lemma \[dim(U+Z)<3\] requires $\dim
{{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}= 3$, but Proposition [\[dim(U)<2n\][[(]{}\[dim(U)<2n-eq=3\][)]{}]{}]{} implies $\dim {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}\le 2$.
In the remainder of the proof, we assume that equality holds in , and that $n$ is even.
Proposition [\[HinN-linear\][[(]{}\[HinN-linear-phi=0&eta2=\][)]{}]{}]{} implies [[(]{}\[maxnolinear-Z\][)]{}]{}.
\[maxnolinearpf-F=C\] Assume ${\mathbb{F}}= {\mathord{\mathbb{C}}}$.
Because equality holds, Lemma \[dim(U+Z)<3\] implies [[(]{}\[maxnolinear-phi0\][)]{}]{} and [[(]{}\[maxnolinear-dimZ\][)]{}]{}. Then Proposition [\[dim(U)<2n\][[(]{}\[dim(U)<2n-eqeven\][)]{}]{}]{} implies [[(]{}\[maxnolinear-<xy>\][)]{}]{} (because ${\mathfrak{\lowercase{U}}} = {\mathfrak{\lowercase{U}}}_{\phi=0}$). Since ${\mathfrak{\lowercase{U}}} = {\mathfrak{\lowercase{U}}}_{\phi=0}$ [[(]{}see \[maxnolinear-phi0\][)]{}]{} and equality holds in , we have $$\label{maxnolinearpf-U/Z}
\dim {\mathfrak{\lowercase{U}}}/ {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}= \dim {\mathfrak{\lowercase{U}}}_{\phi=0}/ {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}= m
= {q}(n-2)$$ and $$\label{maxnolinearpf-dimZ}
\dim {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}= \dim {\mathfrak{\lowercase{U}}}/ {\mathfrak{\lowercase{U}}}_{\phi=0} + \dim {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}= 2{q}- 1 ,$$ so [[(]{}\[maxnolinear-dimU/Z\][)]{}]{} and [[(]{}\[maxnolinear-dimZ\][)]{}]{} hold.
Let $T = A \cap (HN)$. Corollary \[TnormsU\] implies that $T$ normalizes ${\mathfrak{\lowercase{U}}}$, so, from [[(]{}\[maxnolinear-<xy>\][)]{}]{} and Lemma \[rootdecomp\], we see that $T \subset \ker \alpha$. On the other hand, $\dim T = \dim {\mathfrak{\lowercase{H}}} / {\mathfrak{\lowercase{U}}}$, so, from equality in , we conclude that $\dim T = 1$. Therefore $T = \ker \alpha$.
Suppose $\psi \colon \ker \alpha \to {N}_\alpha$ is any continuous group homomorphism, such that $\psi( \ker \alpha)$ normalizes $U$. From [[(]{}\[maxnolinear-<xy>\][)]{}]{} and [[(]{}\[\[u,v\]\][)]{}]{}, we see that $N_{{N}_\alpha}(U) = e$, so $\psi$ must be trivial. This implies that [\[not-semi\][[(]{}\[not-semi-not\][)]{}]{}]{} cannot apply here, so [\[not-semi\][[(]{}\[not-semi-TU\][)]{}]{}]{} yields [[(]{}\[maxnolinear-TU\][)]{}]{}.
Assume ${\mathbb{F}}= {\mathord{\mathbb{R}}}$.
Proposition [\[dim(U)<2n\][[(]{}\[dim(U)<2n-eqeven\][)]{}]{}]{} implies that ${\operatorname{\dim_{{\mathbb{F}}}}}({\mathbb{F}}x_u + {\mathbb{F}}y_u)
= 2$ for every $u \in {\mathfrak{\lowercase{U}}}_{\phi=0} \smallsetminus {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$.
Suppose [[(]{}\[maxnolinear-phi0\][)]{}]{} is false. Then there is some $u \in {\mathfrak{\lowercase{U}}}$, such that $\phi_u \neq 0$. Also, because $\dim {\mathfrak{\lowercase{U}}}_{\phi=0}/{{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}= m >
0$, we may fix some $v \in {\mathfrak{\lowercase{U}}}_{\phi=0} \smallsetminus {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$. Then, letting $w = [u,v]$, we see, from [[(]{}\[\[u,v\]\][)]{}]{}, that $y_w = 0$ and $x_w \neq 0$, so ${\operatorname{\dim_{{\mathbb{F}}}}}({\mathbb{F}}x_w + {\mathbb{F}}y_w) = 1$. This contradicts the conclusion of the preceding paragraph.
Conclusion [[(]{}\[maxnolinear-<xy>\][)]{}]{} follows from [[(]{}\[maxnolinear-phi0\][)]{}]{} and [\[dim(U)<2n\][[(]{}\[dim(U)<2n-eqeven\][)]{}]{}]{}.
Conclusion [[(]{}\[maxnolinear-TU\][)]{}]{} can be established by arguing as in the last two paragraphs of Case \[maxnolinearpf-F=C\].
Equations [[(]{}\[maxnolinearpf-U/Z\][)]{}]{} and [[(]{}\[maxnolinearpf-dimZ\][)]{}]{} establish [[(]{}\[maxnolinear-dimU/Z\][)]{}]{} and [[(]{}\[maxnolinear-dimZ\][)]{}]{}.
Subgroups with no nearly quadratic curve
----------------------------------------
Our goal is to prove Theorem \[bestnosquare\]; we start with two preliminary results.
\[HinN-square\] If there does not exist a continuous curve $h^t \to \infty$ in $U$, such that $\rho(h^t) \asymp \lVert h^t\rVert^2$, then
1. \[HinN-square-indep\] for every element $u$ of ${\mathfrak{\lowercase{U}}}_{\phi=0}$, we have ${\operatorname{\dim_{{\mathbb{F}}}}}( {\mathbb{F}}x_u + {\mathbb{F}}y_u) \le 1$;
2. \[HinN-square-eta2neq\] for every element $z$ of ${{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$, we have $|\eta_z|^2 + {{\mathord{\mathsf{x}}}}_z {{\mathord{\mathsf{y}}}}_z = 0$; and
3. \[HinN-square-y=0+yy=0\] for every element $u$ of ${\mathfrak{\lowercase{U}}}$, such that $\phi_u \neq 0$, $y_u = 0$, and ${{\mathord{\mathsf{y}}}}_u = 0$, we have $|x_u|^2 + 2 {\operatorname{Re}}(\phi_u {\overline{\eta_u}}) \neq 0$.
[[(]{}\[HinN-square-indep\][)]{}]{} Suppose there is an element $u$ of ${\mathfrak{\lowercase{U}}}_{\phi=0}$, such that ${\operatorname{\dim_{{\mathbb{F}}}}}( {\mathbb{F}}x_u + {\mathbb{F}}y_u) = 2$. Let $h^t = \exp(t u)$. Then, from , we see that $h^t = O(t^2)$. Furthermore, $$\Delta(h^t)
= \det
\begin{pmatrix}
\eta_u t - \frac{1}{2} x_u y_u^\dagger t^2
& {{\mathord{\mathsf{x}}}}_u t - \frac{1}{2} |x_u|^2
t^2 \\
{{\mathord{\mathsf{y}}}}_u t - \frac{1}{2} |y_u|^2 t^2 &
-{\overline{\eta_u}} t - \frac{1}{2} y_u x_u^\dagger t^2
\end{pmatrix}
= \frac{1}{4} \bigl| |x_u| |y_u|^2 - |x_u y_u^\dagger|^2 \bigr| t^4
+ O(t^3)
.$$ Because ${\operatorname{\dim_{{\mathbb{F}}}}}( {\mathbb{F}}x_u + {\mathbb{F}}y_u) = 2$, we have $|x_u| |y_u| > |x_u y_u^\dagger|$, so $|x_u|^2 |y_u|^2 - |x_u y_u^\dagger|^2 \neq 0$; therefore $\Delta(h^t) \asymp t^4$, so $$\lVert h^t\rVert^2 = O(t^4) = O \bigl( \Delta(h^t) \bigr) = O\bigl(
\rho(h^t) \bigr),$$ so Lemma [\[Owalls\][[(]{}\[Owalls-square\][)]{}]{}]{} implies that $\rho(h^t) \asymp \lVert h^t\rVert^2$, as desired.
[[(]{}\[HinN-square-eta2neq\][)]{}]{} Suppose there is an element $z$ of ${{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$, such that $|\eta_z|^2 + {{\mathord{\mathsf{x}}}}_z
{{\mathord{\mathsf{y}}}}_z \neq 0$; in other words, we have $\Delta(z) \neq 0$. Let $h^t = \exp(t z) = {\operatorname{Id}}+ tz$ [[(]{}see \[exp(phi=0)\][)]{}]{}. Then $h^t = O(t)$ and $$t^2 \asymp \Delta(z) t^2 = \Delta(h^t) = O \bigl( \rho(h^t) \bigr) ,$$ so $$\lVert h^t\rVert^2 = O(t^2) = O \bigl( \rho(h^t) \bigr) ,$$ so Lemma [\[Owalls\][[(]{}\[Owalls-square\][)]{}]{}]{} implies that $\rho(h^t) \asymp \lVert h^t\rVert^2$, as desired.
[[(]{}\[HinN-square-y=0+yy=0\][)]{}]{} Suppose there is an element $u$ of ${\mathfrak{\lowercase{U}}}$, such that $\phi_u \neq 0$, $y_u = 0$, ${{\mathord{\mathsf{y}}}}_u = 0$, and $|x_u|^2 + 2 {\operatorname{Re}}(\phi_u {\overline{\eta_u}}) = 0$. Let $h^t = \exp(tu)$. From [[(]{}\[exp(y=0)\][)]{}]{}, we see that $h^t = {\operatorname{Id}}+ tu$ (note that, because $|x_u|^2 + 2 {\operatorname{Re}}(\phi_u {\overline{\eta_u}}) = 0$, we have ${\operatorname{Re}}h^2_{1,n+2} = 0$). Then $ h^t = O(t) $ and $$\lVert \rho(h^t) \rVert
\ge \left| \det
\begin{pmatrix}
h^t_{1,2} & h^t_{1,n+2} \\
h^t_{n+1,2} & h^t_{n+1,n+2}
\end{pmatrix}
\right|
= \left| \det
\begin{pmatrix}
t\phi_u & * \\
0 & - t\phi_u^\dagger
\end{pmatrix}
\right|
\asymp t^2 .$$ So $\lVert h^t\rVert^2 = O(t^2) = O \bigl( \rho(h^t) \bigr)$. Thus, Lemma [\[Owalls\][[(]{}\[Owalls-square\][)]{}]{}]{} implies that $\rho(h^t) \asymp \lVert h^t\rVert^2$, as desired.
The following lemma obtains a dimension bound from Condition [\[HinN-square\][[(]{}\[HinN-square-indep\][)]{}]{}]{}.
\[dimV<d(n-2)\] If $V$ is a ${\mathord{\mathbb{R}}}$-subspace of ${\mathbb{F}}^{n-2} \oplus {\mathbb{F}}^{n-2}$, such that ${\operatorname{\dim_{{\mathbb{F}}}}}( {\mathbb{F}}x + {\mathbb{F}}y ) \le 1$ for every $(x,y) \in V$, then either
1. \[dimV<d(n-2)-n>3\] $\dim V \le {q}(n-2)$; or
2. \[dimV<d(n-2)-n=3\] $n = 3$ and $\dim V \le 2{q}$.
Because ${\operatorname{\dim_{{\mathord{\mathbb{R}}}}}}{\mathbb{F}}^{n-2} = {q}(n-2)$, we may assume that there exist nonzero $x_0,y_0 \in {\mathbb{F}}^{n-2}$, such that $(x_0,0) \in V$ and $(0,y_0) \in V$ (otherwise, the projection to one of the factors of ${\mathbb{F}}^{n-2} \oplus {\mathbb{F}}^{n-2}$ is injective when restricted to $V$, so [[(]{}\[dimV<d(n-2)-n>3\][)]{}]{} holds). Then $(x_0,y_0) \in V$, so, by assumption, we have ${\operatorname{\dim_{{\mathbb{F}}}}}({\mathbb{F}}x_0 + {\mathbb{F}}y_0) \le
1$. Because $x_0$ and $y_0$ are nonzero, this implies ${\mathbb{F}}x_0 = {\mathbb{F}}y_0$.
\[dimV<d(n-2)Pf-yinFx\] For all $(x,y) \in V$, we have $y \in {\mathbb{F}}x_0$.
We may assume $y \neq 0$ (otherwise the desired conclusion is obvious). Then, since ${\operatorname{\dim_{{\mathbb{F}}}}}({\mathbb{F}}x + {\mathbb{F}}y ) \le 1$, we conclude that $x \in {\mathbb{F}}y$. Similarly, because $$(x + x_0, y) = (x,y) + (x_0,0) \in V + V = V ,$$ we must have $x+x_0 \in {\mathbb{F}}y$. Therefore $$x_0 = (x+x_0) - x \in {\mathbb{F}}y - {\mathbb{F}}y = {\mathbb{F}}y .$$ Since $x_0 \neq 0$, this implies ${\mathbb{F}}x_0 = {\mathbb{F}}y$, so $y \in {\mathbb{F}}x_0$, as desired.
\[dimV<d(n-2)Pf-V=FxF\] We have $V \subset {\mathbb{F}}y_0 \oplus {\mathbb{F}}x_0$.
Given $(x,y) \in V$, Step \[dimV<d(n-2)Pf-yinFx\] asserts that $y \in {\mathbb{F}}x_0$. By symmetry (interchanging the two factors of ${\mathbb{F}}^{n-2} \oplus
{\mathbb{F}}^{n-2}$), we must also have $x \in {\mathbb{F}}y_0$. So $(x,y) \in {\mathbb{F}}y_0 \oplus {\mathbb{F}}x_0$, as desired.
Completion of the proof.
From Step \[dimV<d(n-2)Pf-V=FxF\], we have $$\dim V \le {\operatorname{\dim_{{\mathord{\mathbb{R}}}}}}({\mathbb{F}}y_0 \oplus {\mathbb{F}}x_0) = 2 {q}.$$ If $n \ge 4$, then [[(]{}\[dimV<d(n-2)-n>3\][)]{}]{} holds; otherwise, [[(]{}\[dimV<d(n-2)-n=3\][)]{}]{} holds.
\[bestnosquare\] Recall that Assumptions \[StandingSU2F\] are in effect.
If there does not exist a continuous curve $h^t \to \infty$ in $H$, such that $\rho(h^t) \asymp \lVert h^t\rVert^2$, then $\dim H \le {q}n$.
Furthermore, if equality holds, then $H$ is of the form $H = T \ltimes U$, where
1. \[bestnosquare-T\] $T = \ker \beta$,
2. \[bestnosquare-U\] ${\mathfrak{\lowercase{U}}} = \bigl( ({{\mathfrak{\lowercase{N}}}}_\alpha + {{\mathfrak{\lowercase{N}}}}_{\alpha+\beta} + {{\mathfrak{\lowercase{N}}}}_{\alpha+2\beta} )
\cap {\mathfrak{\lowercase{U}}} \bigr)
+ {{\mathfrak{\lowercase{N}}}}_{2\alpha+2\beta}$, and
3. \[bestnosquare-x\] $|x_u|^2 + 2{\operatorname{Re}}(\phi_u {\overline{\eta_u}}) \neq 0$ for every $u \in {\mathfrak{\lowercase{U}}}
\smallsetminus {{\mathfrak{\lowercase{N}}}}_{2\alpha+2\beta}$.
Note that $$\dim {\mathfrak{\lowercase{H}}} / {\mathfrak{\lowercase{U}}} \le 1$$ [[(]{}see [\[dimT\][[(]{}\[dimT-A\][)]{}]{}]{}[)]{}]{} and $$\dim {\mathfrak{\lowercase{U}}}/{\mathfrak{\lowercase{U}}}_{\phi=0} \le \dim {{\mathfrak{\lowercase{N}}}}_\alpha = {q}.$$
We have $\dim {\mathfrak{\lowercase{U}}}_{\phi=0} / {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}\le {q}(n-2)$.
Suppose not. Let $V$ be the projection of ${\mathfrak{\lowercase{U}}}_{\phi=0}$ to ${{\mathfrak{\lowercase{N}}}}_{\beta}
+ {{\mathfrak{\lowercase{N}}}}_{\alpha+\beta}$. We have $$\dim V = \dim {\mathfrak{\lowercase{U}}}_{\phi=0} / {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}> {q}(n-2) ,$$ and, for every $u \in {\mathfrak{\lowercase{U}}}_{\phi=0}$ with $x_u \neq 0$, we have ${\operatorname{\dim_{{\mathbb{F}}}}}( {\mathbb{F}}x_u + {\mathbb{F}}y_u ) \le 1$ (see [\[HinN-square\][[(]{}\[HinN-square-indep\][)]{}]{}]{}), so Lemma \[dimV<d(n-2)\] implies that $n = 3$. Therefore $\dim {{\mathfrak{\lowercase{N}}}}_\beta = \dim {{\mathfrak{\lowercase{N}}}}_{\alpha+\beta} = {q}$. Then, because $\dim V > {q}(n-2) = {q}$, we know that $V \cap {{\mathfrak{\lowercase{N}}}}_{\beta} \neq 0$ and $V \cap {{\mathfrak{\lowercase{N}}}}_{\alpha+\beta} \neq 0$; thus, there exist $u,v \in {\mathfrak{\lowercase{U}}}_{\phi=0}$, such that
- $x_u = 0$, $y_u \neq 0$; and
- $x_v \neq 0$, $y_v = 0$.
Therefore $[u,v]$ is a nonzero element of ${{\mathfrak{\lowercase{N}}}}_{\alpha+2\beta}$ [[(]{}see \[\[u,v\]\][)]{}]{}, so $\Delta \bigl( [u,v] \bigr) \neq 0$. This contradicts Lemma [\[HinN-square\][[(]{}\[HinN-square-eta2neq\][)]{}]{}]{}.
We have $\dim {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}\le {q}-1$.
Suppose not: then, because $\dim {{\mathfrak{\lowercase{N}}}}_{2\alpha+2\beta} = {q}-1$, there is some $u \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}\smallsetminus {{\mathfrak{\lowercase{N}}}}_{2\alpha+2\beta}$, and, because $\dim
{{\mathfrak{\lowercase{N}}}}_{2\beta} = {q}-1$, there is some nonzero $v \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$, such that ${{\mathord{\mathsf{y}}}}_v = 0$. We must have $\eta_v = 0$ (otherwise [\[HinN-square\][[(]{}\[HinN-square-eta2neq\][)]{}]{}]{} yields a contradiction); thus $v \in
{{\mathfrak{\lowercase{N}}}}_{2\alpha+2\beta}$. We must have ${{\mathord{\mathsf{y}}}}_u \neq 0$ (otherwise [\[HinN-square\][[(]{}\[HinN-square-eta2neq\][)]{}]{}]{} yields a contradiction). Thus, we see that $$| \eta |^2 + {{\mathord{\mathsf{x}}}}_{u+tv} {{\mathord{\mathsf{y}}}}_{u+tv}
= |\eta_u|^2 + ({{\mathord{\mathsf{x}}}}_u + t {{\mathord{\mathsf{x}}}}_v) (t {{\mathord{\mathsf{y}}}}_u)$$ is nonconstant as a function of $t \in {\mathord{\mathbb{R}}}$, so [\[HinN-square\][[(]{}\[HinN-square-eta2neq\][)]{}]{}]{} yields a contradiction.
The desired inequality.
We have $$\begin{aligned}
\dim {\mathfrak{\lowercase{H}}}
&\le \dim {\mathfrak{\lowercase{H}}}/{\mathfrak{\lowercase{U}}} + \dim {\mathfrak{\lowercase{U}}}/{\mathfrak{\lowercase{U}}}_{\phi=0} + \dim {\mathfrak{\lowercase{U}}}_{\phi=0}/ {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}+ \dim {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}\\
&\le 1 + {q}+ {q}(n-2) + ({q}-1) \\
&= {q}n ,
\end{aligned}$$ as desired.
In the remainder of the proof, we assume that $\dim H = {q}n$.
We must have equality throughout the preceding paragraphs.
\[Vinalpha+beta\] We have $V \subset {{\mathfrak{\lowercase{N}}}}_{\alpha+\beta}$.
Suppose not: then there is some $v \in {\mathfrak{\lowercase{U}}}_{\phi=0}$, such that $y_v \neq
0$. Let $u \in {\mathfrak{\lowercase{U}}} \smallsetminus {\mathfrak{\lowercase{U}}}_{\phi=0}$ and $w = [u,v]$. Then, from , we see that $y_w = 0$ and $x_w \neq 0$, and that $[v,w] \in {{\mathfrak{\lowercase{N}}}}_{\alpha+2\beta} + {{\mathfrak{\lowercase{N}}}}_{2\alpha+2\beta}$. From [\[HinN-square\][[(]{}\[HinN-square-indep\][)]{}]{}]{}, we have $x_v \in {\mathbb{F}}y_v$ and $x_{v+2} \in {\mathbb{F}}y_{v+w} = {\mathbb{F}}y_v$, so $$x_w = x_{v+w} - x_v \in {\mathbb{F}}y_v - {\mathbb{F}}y_v = {\mathbb{F}}y_v .$$ Therefore $x_w y_w ^\dagger \neq 0$, so $\eta_{[v,w]} \neq 0$ [[(]{}see \[\[u,v\]\][)]{}]{}, so [\[HinN-square\][[(]{}\[HinN-square-eta2neq\][)]{}]{}]{} yields a contradiction.
We have ${{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}= {{\mathfrak{\lowercase{N}}}}_{2\alpha+2\beta}$.
From Step \[Vinalpha+beta\], together with the fact that $$\dim V = \dim {\mathfrak{\lowercase{U}}}_{\phi=0} / {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}= {q}(n-2) = \dim
{{\mathfrak{\lowercase{N}}}}_{\alpha+\beta} ,$$ we conclude that $V = {{\mathfrak{\lowercase{N}}}}_{\alpha+\beta}$. Therefore, $${\mathfrak{\lowercase{U}}}_{\phi=0} + {{\mathfrak{\lowercase{D}}}}= V + {{\mathfrak{\lowercase{D}}}}= {{\mathfrak{\lowercase{N}}}}_{\alpha+\beta} + {{\mathfrak{\lowercase{D}}}},$$ so $$\begin{aligned}
{\mathfrak{\lowercase{U}}}
&\supset [{\mathfrak{\lowercase{U}}}_{\phi=0}, {\mathfrak{\lowercase{U}}}_{\phi=0}] \\
&= [{\mathfrak{\lowercase{U}}}_{\phi=0} + {{\mathfrak{\lowercase{D}}}}, {\mathfrak{\lowercase{U}}}_{\phi=0} + {{\mathfrak{\lowercase{D}}}}] \\
&= [{{\mathfrak{\lowercase{N}}}}_{\alpha+\beta} + {{\mathfrak{\lowercase{D}}}}, {{\mathfrak{\lowercase{N}}}}_{\alpha+\beta} + {{\mathfrak{\lowercase{D}}}}] \\
&= [{{\mathfrak{\lowercase{N}}}}_{\alpha+\beta},{{\mathfrak{\lowercase{N}}}}_{\alpha+\beta}] \\
&= {{\mathfrak{\lowercase{N}}}}_{2\alpha+2\beta} .
\end{aligned}$$ Because $\dim {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}= {q}-1 = \dim {{\mathfrak{\lowercase{N}}}}_{2\alpha+2\beta}$, we must have ${{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}= {{\mathfrak{\lowercase{N}}}}_{2\alpha+2\beta}$.
We have ${{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}= {{\mathfrak{\lowercase{N}}}}_{2\alpha+2\beta}$. Let $T = (HN) \cap A$ be the projection of $H$ to $A$. Then there exists $\sigma \in \{\beta, \alpha+\beta, \alpha+2\beta\}$, such that $T = \ker(\alpha- \sigma)$, and, in the notation of Lemma \[rootdecomp\], we have $${\mathfrak{\lowercase{U}}} = ({\mathfrak{\lowercase{U}}} \cap {\mathfrak{\lowercase{N}}}^{=\alpha}) + ({\mathfrak{\lowercase{U}}} \cap {\mathfrak{\lowercase{N}}}^{\neq
\alpha}) .$$
Because $T$ normalizes ${\mathfrak{\lowercase{U}}}$ [[(]{}see \[TnormsU\][)]{}]{}, we know, from Lemma \[rootdecomp\], that ${\mathfrak{\lowercase{U}}} = ({\mathfrak{\lowercase{U}}} \cap {\mathfrak{\lowercase{N}}}^{=\alpha}) + ({\mathfrak{\lowercase{U}}} \cap {\mathfrak{\lowercase{N}}}^{\neq
\alpha})$. Since $\dim {\mathfrak{\lowercase{U}}} / {\mathfrak{\lowercase{U}}}_{\phi=0} = {q}$, we know that ${\mathfrak{\lowercase{U}}} \cap {\mathfrak{\lowercase{N}}}^{=\alpha}$ projects nontrivially (in fact, surjectively) to ${{\mathfrak{\lowercase{N}}}}_\alpha$. On the other hand, we know that ${\mathfrak{\lowercase{U}}} \cap {{\mathfrak{\lowercase{N}}}}_{\alpha} = 0$ (otherwise [\[HinN-square\][[(]{}\[HinN-square-y=0+yy=0\][)]{}]{}]{} yields a contradiction). Therefore ${\mathfrak{\lowercase{N}}}^{=\alpha} \neq {{\mathfrak{\lowercase{N}}}}_\alpha$, so there must be a positive root $\sigma \neq
\alpha$, such that $\sigma|_T = \alpha|_T$. Then $T \subset
\ker(\alpha-\sigma)$; since $\dim T = \dim H/U = 1$, we must have $T =
\ker(\alpha-\sigma)$.
Because ${\mathfrak{\lowercase{U}}} \cap {\mathfrak{\lowercase{N}}}^{\neq\alpha} \subset {\mathfrak{\lowercase{U}}}_{\phi=0}$, we have $$\dim ({\mathfrak{\lowercase{U}}} \cap {\mathfrak{\lowercase{N}}}^{=\alpha})
\ge \dim \frac{{\mathfrak{\lowercase{U}}} \cap {\mathfrak{\lowercase{N}}}^{=\alpha}}{{\mathfrak{\lowercase{U}}}_{\phi=0} \cap {\mathfrak{\lowercase{N}}}^{=\alpha}}
= \dim \frac
{{\mathfrak{\lowercase{U}}} / ({\mathfrak{\lowercase{U}}} \cap {\mathfrak{\lowercase{N}}}^{\neq\alpha})}
{{\mathfrak{\lowercase{U}}}_{\phi=0} / ({\mathfrak{\lowercase{U}}} \cap {\mathfrak{\lowercase{N}}}^{\neq\alpha})}
= \dim \frac{{\mathfrak{\lowercase{U}}}}{{\mathfrak{\lowercase{U}}}_{\phi=0}}
= {q}.$$ Then, since ${\mathfrak{\lowercase{U}}} \cap {{\mathfrak{\lowercase{N}}}}_\alpha = 0$, we must have $$\dim {\mathfrak{\lowercase{N}}}^{=\alpha}
\ge \dim ({\mathfrak{\lowercase{U}}} \cap {\mathfrak{\lowercase{N}}}^{=\alpha}) + \dim {{\mathfrak{\lowercase{N}}}}_\alpha
\ge {q}+ {q}= 2{q}.$$ By inspection, we see that this implies $\sigma \notin \{2\alpha, 2\beta\}$, so we conclude that $\sigma \in \{\beta, \alpha+\beta, \alpha+2\beta\}$, as desired.
\[bestnosquarepf-sigma\] We have $\sigma \in \{ \alpha+\beta, \alpha+2\beta\}$, $T = \ker \beta$, and ${\mathfrak{\lowercase{N}}}^{=\alpha} = {{\mathfrak{\lowercase{N}}}}_\alpha + {{\mathfrak{\lowercase{N}}}}_{\alpha+\beta} +
{{\mathfrak{\lowercase{N}}}}_{\alpha+2\beta}$.
Since $\ker \beta = \ker 2\beta$, it suffices to show $\sigma \neq \beta$. Thus, let us suppose $\sigma = \beta$. (This will lead to a contradiction.) We have ${\mathfrak{\lowercase{N}}}^{=\alpha} = {{\mathfrak{\lowercase{N}}}}_\alpha + {{\mathfrak{\lowercase{N}}}}_\beta$ (and recall that ${\mathfrak{\lowercase{U}}} \cap {{\mathfrak{\lowercase{N}}}}_\alpha = \{0\}$), so there is some $u \in {\mathfrak{\lowercase{U}}}$, such that $\phi_u \neq 0$ and $y_u \neq 0$. Because $V = {{\mathfrak{\lowercase{N}}}}_{\alpha+\beta}$, we have $$\dim \{\, v \in V \mid x_v \in {\mathbb{F}}y_u \,\} = {q}> \dim {{\mathbb{F}}_{\text{imag}}},$$ so there is some $v \in {\mathfrak{\lowercase{U}}}_{\phi=0}$, such that $0 \neq x_v \in {\mathbb{F}}y_u$ and ${{\mathord{\mathsf{y}}}}_v = 0$. Then $[u,v] \in {{\mathfrak{\lowercase{N}}}}_{\alpha+2\beta} + {{\mathfrak{\lowercase{N}}}}_{2\alpha +
2\beta}$, with $\eta_{[u,v]} \neq 0$ [[(]{}see \[\[u,v\]\][)]{}]{}, so [\[HinN-square\][[(]{}\[HinN-square-eta2neq\][)]{}]{}]{} yields a contradiction.
\[bestnosquarepf-TU\] We have $H = (H \cap A) \ltimes ( H \cap N)$.
Suppose not: because $T = \ker \beta$, we conclude that there is some nonzero $w \in {{\mathfrak{\lowercase{N}}}}_\beta + {{\mathfrak{\lowercase{N}}}}_{2\beta}$, such that $w$ normalizes ${\mathfrak{\lowercase{U}}}$ [[(]{}see \[not-semi\][)]{}]{}.
If $y_w \neq 0$, then, because $V = {{\mathfrak{\lowercase{N}}}}_{\alpha+\beta}$, there is some $v
\in {\mathfrak{\lowercase{U}}}_{\phi=0}$, such that $y_w \in {\mathbb{F}}x_v$ and ${{\mathord{\mathsf{y}}}}_v = 0$. Then $[w,v]
\in {{\mathfrak{\lowercase{N}}}}_{\alpha+2\beta} + {{\mathfrak{\lowercase{N}}}}_{2\alpha + 2\beta}$, with $\eta_{[w,v]} \neq
0$ [[(]{}see \[\[u,v\]\][)]{}]{}, so [\[HinN-square\][[(]{}\[HinN-square-eta2neq\][)]{}]{}]{} yields a contradiction.
If $y_w = 0$, then, since $w \neq 0$, we must have ${{\mathord{\mathsf{y}}}}_w \neq 0$. There is some $v \in {\mathfrak{\lowercase{U}}}$ with $\phi_v \neq 0$. Then $[w,v] \in
{{\mathfrak{\lowercase{N}}}}_{\alpha+2\beta} + {{\mathfrak{\lowercase{N}}}}_{2\alpha + 2\beta}$, with $\eta_{[w,v]} \neq 0$ [[(]{}see \[\[u,v\]\][)]{}]{}, so [\[HinN-square\][[(]{}\[HinN-square-eta2neq\][)]{}]{}]{} yields a contradiction.
Completion of the proof.
[[(]{}\[bestnosquare-T\][)]{}]{} From Step \[bestnosquarepf-TU\], We know that $H = T
\ltimes U$, and, from Step \[bestnosquarepf-sigma\], that $T = \ker \beta$.
[[(]{}\[bestnosquare-U\][)]{}]{} Since ${\mathfrak{\lowercase{N}}}^{=\alpha} = {{\mathfrak{\lowercase{N}}}}_\alpha +
{{\mathfrak{\lowercase{N}}}}_{\alpha+\beta} + {{\mathfrak{\lowercase{N}}}}_{\alpha+2\beta}$, it suffices to show ${\mathfrak{\lowercase{U}}} \cap {\mathfrak{\lowercase{N}}}^{\neq \alpha} = {{\mathfrak{\lowercase{N}}}}_{2\alpha+2\beta}$: given $v \in
{\mathfrak{\lowercase{U}}} \cap {\mathfrak{\lowercase{N}}}^{\neq \alpha}$, we wish to show $v \in
{{\mathfrak{\lowercase{N}}}}_{2\alpha+2\beta}$. Because $V = {{\mathfrak{\lowercase{N}}}}_{\alpha+\beta}$, we know that $y_v
= 0$. Thus, all that remains is to show that ${{\mathord{\mathsf{y}}}}_v = 0$. If not, then choosing $u \in {\mathfrak{\lowercase{U}}}$ with $\phi_u \neq 0$, we see that $\eta_{[u,v]} \neq
0$ [[(]{}see \[\[u,v\]\][)]{}]{}. So [\[HinN-square\][[(]{}\[HinN-square-eta2neq\][)]{}]{}]{} yields a contradiction.
[[(]{}\[bestnosquare-x\][)]{}]{} From Lemma [\[HinN-square\][[(]{}\[HinN-square-y=0+yy=0\][)]{}]{}]{}, we know that $|x_u|^2 + 2 {\operatorname{Re}}(\phi_u {\overline{\eta_u}}) \neq 0$ for every $u \in {\mathfrak{\lowercase{U}}}
\smallsetminus {{\mathfrak{\lowercase{N}}}}_{2\alpha+2\beta}$.
Homogeneous spaces of ${\operatorname{SO}}(2,n)$ and ${\operatorname{SU}}(2,n)$ {#ProofSect}
===============================================================================
This section proves two main results. Both assume that $G$ is either ${\operatorname{SO}}(2,n)$ or ${\operatorname{SU}}(2,n)$.
1. Theorem \[SUF->complete\] shows that if $n$ is odd, and one or two specific homogeneous spaces of $G$ do not have tessellations, then no interesting homogeneous space of $G$ has a tessellation.
2. Theorem \[SUFevenTess\] shows that if $n$ is even, then certain deformations of the examples found by R. Kulkarni and T. Kobayashi [[(]{}see \[KulkarniEg\][)]{}]{} are essentially the only interesting homogeneous spaces of $G$ that have tessellations.
The classification results of §\[SUFlargeSect\] (specifically, Theorems \[bestnosquare\] and \[maxnolinear\]) play a crucial role in the proofs.
We use the notation ${\operatorname{SU}}(2,n;{\mathbb{F}})$ of Section \[coordsSect\], to provide a fairly unified treatment of ${\operatorname{SO}}(2,n)$ and ${\operatorname{SU}}(2,n)$.
\[SUF->complete\] Assume $G = {\operatorname{SU}}(2,2m+1; {\mathbb{F}})$ with $m \ge 1$, and let $H$ be any closed, connected subgroup of $G$, such that neither $H$ nor $G/H$ is compact.
If Conjecture \[notessSU/Sp\] is true, then $G/H$ does not have a tessellation.
Assume Conjecture \[notessSU/Sp\] is true, and suppose $\Gamma$ is a crystallographic group for $G/H$. (This will lead to a contradiction.) Let $$\mbox{$H_1 = {\operatorname{SU}}(1,2m+1;{\mathbb{F}})$ and $H_2 = {\operatorname{Sp}}(1,m;{\mathbb{F}})$}$$ [[(]{}see \[SU1nDefn\][)]{}]{}. From [[(]{}\[d(Sp)\][)]{}]{}, we have $d(H_1) = {q}(2m+1)$ and $d(H_2)
= {q}(2m)$, where ${q}= {\operatorname{\dim_{{\mathord{\mathbb{R}}}}}}{\mathbb{F}}$. We may assume that $H \subset AN$ [[(]{}see \[HcanbeAN\][)]{}]{}, and that $H$ is compatible with $A$ [[(]{}see \[conjtocompatible\][)]{}]{}.
Because $H$ is not a Cartan-decomposition subgroup [[(]{}see \[CDS->notess\][)]{}]{}, the contrapositive of Proposition \[CDS<>h\_m\] implies, for some $k \in
\{1,2\}$, that there does **not** exist a continuous curve $h^t \to
\infty$ in $H$, such that $\rho(h^t) \asymp \lVert h^t\rVert^k$. Therefore, either Theorem \[maxnolinear\] (if $k = 1$) or Theorem \[bestnosquare\] (if $k = 2$) implies that $d(H) \le {q}(2m+1) = d(H_1)$.
We consider two cases.
Assume that $\Gamma$ acts properly discontinuously on $G/H_1$.
Theorem [\[noncpctdim\][[(]{}\[noncpctdim-tess\][)]{}]{}]{} (combined with the fact that $d(H) \le
d(H_1)$) implies that $G/H_1$ has a tessellation. This contradicts either Theorem \[SO2n/SO1odd-notess\] (if ${\mathbb{F}}= {\mathord{\mathbb{R}}}$) or Conjecture \[notessSU/Sp\]\[notessSU/Sp-SU2/SU1\] (if ${\mathbb{F}}= {\mathord{\mathbb{C}}}$).
Assume that $\Gamma$ does not act properly discontinuously on $G/H_1$.
From Lemma \[mu(SUorSp)\], we know that $\mu(H_1)$ and $\mu(H_2)$ are the two walls of $A^+$, so Corollary \[tess->missHk\] (combined with the assumption of this case) implies that $\Gamma$ acts properly discontinuously on $G/H_2$. Therefore, since Conjecture \[notessSU/Sp\]\[notessSU/Sp-SO2/SU1\]\[notessSU/Sp-SU2/Sp1\] asserts that $G/H_2$ does not have a tessellation, the contrapositive of Theorem [\[noncpctdim\][[(]{}\[noncpctdim-tess\][)]{}]{}]{} (with $H_2$ in the role of $H_1$) implies that $d(H) > d(H_2) = {q}(2m)$. Hence, the contrapositive of Theorem \[maxnolinear\] implies there is a continuous curve $h^t \to
\infty$ in $H$, such that $\rho(h^t) \asymp h^t$. Thus, there is a compact subset $C$ of $G$, such that $H_1 \subset CHC$ [[(]{}see \[SU1inH\][)]{}]{}. Since $\Gamma$ acts properly discontinuously on $G/H$, this implies that $\Gamma$ acts properly discontinuously on $G/H_1$ [[(]{}see \[CHCproper\][)]{}]{}. This contradicts the assumption of this case.
\[SUFevenTess\] Assume $G = {\operatorname{SU}}(2,2m;{\mathbb{F}})$ with $m \ge 2$, and let $H$ be a closed, connected, nontrivial, proper subgroup of $AN$.
The homogeneous space $G/H$ has a tessellation if and only if either
1. \[SUFevenTess-Sp\] there is an ${\mathord{\mathbb{R}}}$-linear map $B \colon {\mathbb{F}}^{n-2} \to {\mathbb{F}}^{n-2}$, such that
1. \[SUFevenTess-Sp-xB\] ${\operatorname{Im}}\bigl( (vB)(wB)^\dagger \bigr) = -{\operatorname{Im}}(v w^\dagger)$ for every $v,w \in
{\mathbb{F}}^{n-2}$ [[(]{}see \[Bsymplectic\][)]{}]{}, and
2. \[SUFevenTess-Sp-<x,y>\] $xB \notin {\mathbb{F}}x$, for every nonzero $x \in {\mathbb{F}}^{n-2}$ [[(]{}see \[xBnotinFx\][)]{}]{}, and
3. \[SUFevenTess-Sp-HB\] $H$ is conjugate to $H_B$ [(]{}see \[HB-defn\] and \[HB=Sp1m\][)]{}; or
2. \[SUFevenTess-SUR\] ${\mathbb{F}}= {\mathord{\mathbb{R}}}$ and $H$ is conjugate to ${\operatorname{SU}}(1,2m;{\mathord{\mathbb{R}}}) \cap AN$ [[(]{}see \[SU1nDefn\][)]{}]{}; or
3. \[SUFevenTess-SUC\] ${\mathbb{F}}= {\mathord{\mathbb{C}}}$ and there exists $c \in (0,1]$, such that $H$ is conjugate to ${{H_{[c]}}}$ [[(]{}see \[SUegsDefn\][)]{}]{}.
($\Leftarrow$) See [[(]{}\[SUFevenTess-Sp\][)]{}]{} Theorem [\[HBthm\][[(]{}\[HBthm-tess\][)]{}]{}]{}, [[(]{}\[SUFevenTess-SUR\][)]{}]{} Theorem \[SUevenTessExists\] (and \[d(Sp)\]), or [[(]{}\[SUFevenTess-SUC\][)]{}]{} Theorem [\[SUegs\][[(]{}\[SUegs-tess\][)]{}]{}]{}.
($\Rightarrow$) Let $n = 2m$, so $G = {\operatorname{SU}}(2,n;{\mathbb{F}})$. By combining Remark \[d(H)=dimH\], Corollary \[tess->dim>1,2\], Lemma \[mu(SUorSp)\], and Remark \[d(Sp)\], we see that $$\dim H = d(H)
\ge \min \bigl\{\, d \bigl( {\operatorname{SU}}(1,n; {\mathbb{F}}) \bigr), d \bigl( {\operatorname{Sp}}(1,m; {\mathbb{F}}) \bigr)
\, \bigr\}
={q}n .$$
Also, we may assume $H$ is compatible with $A$ [[(]{}see \[conjtocompatible\][)]{}]{}. Because $H$ is not a Cartan-decomposition subgroup [[(]{}see \[CDS->notess\][)]{}]{}, Proposition \[CDS<>h\_m\] implies that one of the following two cases applies.
\[SUFevenTessPf-nolinear\] Assume there does not exist a continuous curve $h^t \to \infty$ in $H$, such that $\rho(h^t) \asymp h^t$.
Since $\dim H \ge {q}n$, Theorem \[maxnolinear\] implies that $\dim H
= {q}n$, and that $H$ is of the form $H=T\ltimes U$ (with $U \subset N$), where
1. \[SUFevenTessPf-TU\] $T = \ker \alpha$;
2. \[SUFevenTessPf-phi0\] $\phi_u = 0$ for every $u \in {\mathfrak{\lowercase{U}}}$;
3. \[SUFevenTessPf-<xy>\] ${\operatorname{\dim_{{\mathbb{F}}}}}( {\mathbb{F}}x_u + {\mathbb{F}}y_u ) = 2$, for every $u \in {\mathfrak{\lowercase{U}}} \smallsetminus {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$;
4. \[SUFevenTessPf-Z\] $|\eta_z|^2 + {{\mathord{\mathsf{x}}}}_z {{\mathord{\mathsf{y}}}}_z \neq 0$ for every nonzero $z \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$;
5. \[SUFevenTessPf-dimU/Z\] $\dim {\mathfrak{\lowercase{U}}} / {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}= {q}(n-2)$; and
6. \[SUFevenTessPf-dimZ\] $\dim {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}= 2{q}- 1$.
\[SUFevenTessSpPf-xx=yy\] We may assume that ${{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}= \{\, z \in {{\mathfrak{\lowercase{D}}}}\mid {{\mathord{\mathsf{x}}}}_z = - {{\mathord{\mathsf{y}}}}_z \,\}$.
Because $\dim {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}= 2{q}- 1$ [[(]{}see \[SUFevenTessPf-dimZ\][)]{}]{}, it suffices to show that ${{\mathord{\mathsf{x}}}}_z = - {{\mathord{\mathsf{y}}}}_z$ for all $z \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$. This is trivially true if ${\mathbb{F}}={\mathord{\mathbb{R}}}$, as ${{\mathord{\mathsf{x}}}}_z,{{\mathord{\mathsf{y}}}}_z \in {{\mathbb{F}}_{\text{imag}}}=\{0\}$ in this case. Thus, we assume ${\mathbb{F}}= {\mathord{\mathbb{C}}}$.
For any $z \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$ with $\eta_z = {{\mathord{\mathsf{y}}}}_z = 0$, we know, from [[(]{}\[SUFevenTessPf-Z\][)]{}]{}, that $z = 0$; therefore, Lemma [\[O(linear)\][[(]{}\[O(linear)-O\][)]{}]{}]{} implies there exist ${\mathord{\mathbb{R}}}$-linear maps $R
\colon {\mathord{\mathbb{C}}}\to i{\mathord{\mathbb{R}}}$ and $S \colon i {\mathord{\mathbb{R}}}\to i {\mathord{\mathbb{R}}}$, such that ${{\mathord{\mathsf{x}}}}_z = R(\eta_z) + S({{\mathord{\mathsf{y}}}}_z)$ for all $z \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$. More concretely, we may say that there exist $\lambda \in {\mathord{\mathbb{C}}}$ and $c \in {\mathord{\mathbb{R}}}$, such that ${{\mathord{\mathsf{x}}}}_z = {\operatorname{Im}}(\lambda \eta_z) + c {{\mathord{\mathsf{y}}}}_z$ for all $z \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$.
Let $v$ be the element of ${{\mathfrak{\lowercase{N}}}}_\alpha$ with $\phi_v = {\overline{\lambda}}/2$, and let $H^* = \exp(-v) H \exp(v)$ be the conjugate of $H$ by $\exp(v)$. Then $H^*$ satisfies the conditions imposed on $H$ (note that $H^*$, like $H$, is compatible with $A$ [[(]{}see \[conjUomega\][)]{}]{}), so there exist $\lambda^* \in
{\mathord{\mathbb{C}}}$ and $c^* \in {\mathord{\mathbb{R}}}$, such that ${{\mathord{\mathsf{x}}}}_{z^*} = {\operatorname{Im}}(\lambda^*
\eta_{z^*}) + c^* {{\mathord{\mathsf{y}}}}_{z^*}$ for all $z^* \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}^*$. Given $z^* \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}^*$ with ${{\mathord{\mathsf{y}}}}_{z^*} = 0$, let $z = \exp(v) z \exp(-v)$. Because ${{\mathord{\mathsf{y}}}}_{z^*} =
0$, we have $\bigl[ [z^*,-v], -v \bigr] = 0$, so, from Remark \[conjugation\] and , we see that $${{\mathord{\mathsf{y}}}}_z = {{\mathord{\mathsf{y}}}}_{z^*} = 0 ,$$ $$\eta_z = \eta_{z^*} - \phi_{-v} {{\mathord{\mathsf{y}}}}_{z^*} = \eta_{z^*} ,$$ and $${{\mathord{\mathsf{x}}}}_z
= {{\mathord{\mathsf{x}}}}_{z^*} + 2 {\operatorname{Im}}(\phi_{-v} {\overline{\eta_{z^*}}})
= {{\mathord{\mathsf{x}}}}_{z^*} + 2{\operatorname{Im}}\bigl( (-{\overline{\lambda}}/2) {\overline{\eta_z}} \bigr)
= {{\mathord{\mathsf{x}}}}_{z^*} + {\operatorname{Im}}( \lambda \eta_z )
= {{\mathord{\mathsf{x}}}}_{z^*} + {{\mathord{\mathsf{x}}}}_z .$$ Therefore $$0 = {{\mathord{\mathsf{x}}}}_{z^*} = {\operatorname{Im}}(\lambda^* \eta_{z^*}) + c^* {{\mathord{\mathsf{y}}}}_{z^*} = {\operatorname{Im}}(\lambda^*
\eta_{z^*}) .$$ Since $\eta_{z^*}$ is arbitrary, this implies $\lambda^* = 0$. Thus, by replacing $H$ with $H^*$, we may assume that $\lambda = 0$. This means that ${{\mathord{\mathsf{y}}}}_z = c {{\mathord{\mathsf{x}}}}_z$ for all $z \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$.
From [[(]{}\[SUFevenTessPf-dimZ\][)]{}]{} (and because ${\mathbb{F}}= {\mathord{\mathbb{C}}}$, so ${q}= 2$), we know that $\dim {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}= 3 > 1$, so there is some nonzero $w \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$, such that ${{\mathord{\mathsf{y}}}}_w = 0$. (So ${{\mathord{\mathsf{x}}}}_w = c {{\mathord{\mathsf{y}}}}_w = 0$.) Then $|\eta_w|^2 + {{\mathord{\mathsf{x}}}}_w {{\mathord{\mathsf{y}}}}_w = |\eta_w|^2 > 0$, so we see, from [[(]{}\[SUFevenTessPf-Z\][)]{}]{}, that $|\eta_z|^2 + {{\mathord{\mathsf{x}}}}_z {{\mathord{\mathsf{y}}}}_z > 0$ for every nonzero $z \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$. Now, since $$\dim {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}= 3 > 2 = \dim {{\mathfrak{\lowercase{N}}}}_{\alpha+2\beta} ,$$ there is some nonzero $z \in {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$, such that $\eta_z = 0$. We have $$0 < |\eta_z|^2 + {{\mathord{\mathsf{x}}}}_z {{\mathord{\mathsf{y}}}}_z = 0 + c {{\mathord{\mathsf{y}}}}_z^2 .$$ Because ${{\mathord{\mathsf{y}}}}_z$ is pure imaginary, we know that ${{\mathord{\mathsf{y}}}}_z^2 < 0$, so this implies that $c < 0$. Thus, replacing $H$ by a conjugate under a diagonal matrix, we may assume $c = -1$, as desired.
\[SUFevenTessPf-U’+Z\] Setting ${\mathfrak{\lowercase{U}}}' = ({{\mathfrak{\lowercase{N}}}}_\alpha + {{\mathfrak{\lowercase{N}}}}_{\alpha+\beta}) \cap {\mathfrak{\lowercase{U}}}$, we have ${\mathfrak{\lowercase{U}}} = {\mathfrak{\lowercase{U}}}' + {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$.
Since $T = \ker \alpha$ [[(]{}see \[SUFevenTessPf-TU\][)]{}]{}, we have
- $\beta|_T = (\alpha+\beta)|_T$,
- $2\beta|_T = (\alpha+2\beta)|_T = (2\alpha+2\beta)|_T$, and
- $\beta|_T \neq 2\beta|_T$.
Thus, in the notation of Lemma \[rootdecomp\], we have ${\mathfrak{\lowercase{N}}}^{=\beta}
\cap {\mathfrak{\lowercase{U}}} = {\mathfrak{\lowercase{U}}}'$ and ${\mathfrak{\lowercase{N}}}^{\neq \beta} \cap {\mathfrak{\lowercase{U}}} = {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$, so ${\mathfrak{\lowercase{U}}} = {\mathfrak{\lowercase{U}}}' \oplus {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$, as desired. (Note that this is a direct sum of vector spaces, not of Lie algebras: we have $[{\mathfrak{\lowercase{U}}}',{\mathfrak{\lowercase{U}}}'] \subset
{{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}$.)
Completion of the proof of Case \[SUFevenTessPf-nolinear\].
For any $u \in {\mathfrak{\lowercase{U}}}'$ with $x_u = 0$, we have $${\operatorname{\dim_{{\mathbb{F}}}}}({\mathbb{F}}x_u + {\mathbb{F}}y_u) = {\operatorname{\dim_{{\mathbb{F}}}}}{\mathbb{F}}y_u \le 1 < 2 ,$$ so $u \in {\mathfrak{\lowercase{U}}}' \cap {{{\mathfrak{\lowercase{D}}}}_{{\mathfrak{\lowercase{H}}}}}= \{0\}$ [[(]{}see \[SUFevenTessPf-<xy>\][)]{}]{}; therefore, Lemma [\[O(linear)\][[(]{}\[O(linear)-O\][)]{}]{}]{} implies there is a ${\mathord{\mathbb{R}}}$-linear map $B \colon
{\mathbb{F}}^{n-2} \to {\mathbb{F}}^{n-2}$, such that $y_u = x_u B$ for all $u \in {\mathfrak{\lowercase{U}}}'$. Then, because $$\dim {\mathfrak{\lowercase{u}}}_{\alpha+\beta} = {\operatorname{\dim_{{\mathord{\mathbb{R}}}}}}{\mathbb{F}}^{n-2} = {q}(n-2) = \dim {\mathfrak{\lowercase{U}}}'$$ [[(]{}see \[SUFevenTessPf-dimU/Z\][)]{}]{}, we must have $${\mathfrak{\lowercase{U}}}' = \{\, u \in {{\mathfrak{\lowercase{N}}}}_\beta + {{\mathfrak{\lowercase{N}}}}_{\alpha+\beta} \mid y_u = x_u B
\,\} .$$ Combining this with [[(]{}\[SUFevenTessPf-TU\][)]{}]{} and the conclusions of Steps \[SUFevenTessSpPf-xx=yy\] and \[SUFevenTessPf-U’+Z\], we see that ${\mathfrak{\lowercase{H}}} = {\mathfrak{\lowercase{H}}}_B$. Therefore $H = H_B$, so Conclusion [[(]{}\[SUFevenTess-Sp-HB\][)]{}]{} holds.
From [[(]{}\[SUFevenTessPf-<xy>\][)]{}]{}, we see that Conclusion [[(]{}\[SUFevenTess-Sp-<x,y>\][)]{}]{} holds.
Letting $z = [u,v]$, for any $u,v \in {\mathfrak{\lowercase{U}}}'$, we see, from , that $${{\mathord{\mathsf{x}}}}_z = -2 {\operatorname{Im}}(x_u x_v^\dagger)$$ and $${{\mathord{\mathsf{y}}}}_z = -2 {\operatorname{Im}}(y_u y_v^\dagger)
= -2 {\operatorname{Im}}\bigl( (x_u B)(x_v B)^\dagger \bigr) .$$ From Step \[SUFevenTessSpPf-xx=yy\], we know that ${{\mathord{\mathsf{y}}}}_z = - {{\mathord{\mathsf{x}}}}_z$, so this implies that Conclusion [[(]{}\[SUFevenTess-Sp-xB\][)]{}]{} holds.
\[SUFevenTessPf-nosquare\] Assume there does not exist a continuous curve $h^t \to \infty$ in $H$, such that $\rho(h^t) \asymp \lVert h^t\rVert^2$.
Since $\dim H \ge {q}n$, Theorem \[bestnosquare\] implies that $\dim H
= {q}n$, and that $H$ is of the form $H=T\ltimes U$, where
1. $T=\ker \beta$,
2. ${\mathfrak{\lowercase{U}}}=(({{\mathfrak{\lowercase{N}}}}_\alpha+{{\mathfrak{\lowercase{N}}}}_{\alpha+\beta}+
{{\mathfrak{\lowercase{N}}}}_{\alpha+2\beta})\cap{\mathfrak{\lowercase{U}}})+{{\mathfrak{\lowercase{N}}}}_{2\alpha+2\beta}$, and
3. \[pdqf\] $|x_u|^2+2{\operatorname{Re}}(\phi_u {\overline{\eta_u}})\neq0$ for every $u\in{\mathfrak{\lowercase{U}}}\smallsetminus{{\mathfrak{\lowercase{N}}}}_{2\alpha+2\beta}$.
Let $${\mathfrak{\lowercase{U}}}' = ({{\mathfrak{\lowercase{N}}}}_\alpha+{{\mathfrak{\lowercase{N}}}}_{\alpha+\beta}+ {{\mathfrak{\lowercase{N}}}}_{\alpha+2\beta})\cap{\mathfrak{\lowercase{U}}}$$ (so ${\mathfrak{\lowercase{U}}} = {\mathfrak{\lowercase{U}}}' \oplus {{\mathfrak{\lowercase{N}}}}_{2\alpha+2\beta}$). Let $Q$ be the sesquilinear form (or bilinear form, if ${\mathbb{F}}= {\mathord{\mathbb{R}}}$) on ${\mathbb{F}}\oplus {\mathbb{F}}^{n-2}\oplus{\mathbb{F}}$ defined by $$Q \bigl( (\phi_1, x_1, \eta_1) , (\phi_2, x_2, \eta_2) \bigr)
= \phi_1 {\overline{\eta_2}} + x_1 x_2^\dagger + \eta_1 {\overline{\phi_2}} .$$ Let $$V_{{\mathfrak{\lowercase{H}}}} = \{\, (\phi_u, x_u, \eta_u) \in {\mathbb{F}}\oplus {\mathbb{F}}^{n-2} \oplus {\mathbb{F}}\mid
u \in {\mathfrak{\lowercase{U}}}' \,\}.$$ From [[(]{}\[pdqf\][)]{}]{}, we see that the restriction of ${\operatorname{Re}}Q$ to $V_{{\mathfrak{\lowercase{H}}}}$ is a (positive-definite) inner product.
Let $V_{{\mathfrak{\lowercase{H}}}}^\perp$ be the $({\operatorname{Re}}Q)$-orthogonal complement to $V_{{\mathfrak{\lowercase{H}}}}$. As a form over ${\mathbb{F}}$, $Q$ has signature $(1,n-1)$. Thus, as a form over ${\mathord{\mathbb{R}}}$, ${\operatorname{Re}}Q$ has signature $\bigl( {q}, {q}(n-1) \bigr)$. Since $$\dim V_{{\mathfrak{\lowercase{H}}}}
= \dim {\mathfrak{\lowercase{H}}} - \dim {\mathfrak{\lowercase{T}}} - \dim {{\mathfrak{\lowercase{N}}}}_{2\alpha+2\beta}
= {q}n - 1 - ({q}-1)
= {q}(n-1) ,$$ we conclude that $V_{{\mathfrak{\lowercase{H}}}}^\perp$ is a ${q}$-dimensional ${\mathord{\mathbb{R}}}$-subspace on which ${\operatorname{Re}}Q$ is negative-definite.
Choose some nonzero $u \in V_{{\mathfrak{\lowercase{H}}}}^\perp$. Multiplying by a real scalar to normalize, we may assume $Q(u,u) = -2$. Because ${\operatorname{SU}}(1,n-1)$ is transitive on the vectors of norm $-1$, there is some $g \in {\operatorname{SU}}({\operatorname{Re}}Q)$, such that $g(u) = (1,0,-1)$. Thus, letting $$\label{ghat}
\hat g =
\begin{pmatrix}
1&0&0\\
0&g&0\\
0&0&1
\end{pmatrix}
\in{\operatorname{SU}}(2,n;{\mathbb{F}}) ,$$ and ${\mathfrak{\lowercase{H}}}^\sharp = {\hat g}^{-1} {\mathfrak{\lowercase{H}}} g$, we have $(1,0,-1) \in V_{{\mathfrak{\lowercase{H}}}^\sharp}$, so, by replacing ${\mathfrak{\lowercase{H}}}$ with the conjugate ${\mathfrak{\lowercase{H}}}^\sharp$, we may assume $u = (1,0,-1)$.
Then $$\label{uperp}
\begin{split}
V_{{\mathfrak{\lowercase{H}}}}
&\subset u^\perp \\
&= (1,0,-1)^\perp \\
&= \{\, (\phi, x, \eta) \mid
{\operatorname{Re}}Q \bigl( (\phi, x, \eta), (1,0,-1) \bigr) = 0 \,\} \\
&= \{\, (\phi, x, \eta) \mid
{\operatorname{Re}}\bigl( \phi (-1) + x (0^\dagger) + \eta (1) \bigr) = 0 \,\} \\
&= \{\, (\phi, x, \eta) \mid {\operatorname{Re}}\eta = {\operatorname{Re}}\phi \,\}
.
\end{split}$$
Assume ${\mathbb{F}}= {\mathord{\mathbb{R}}}$.
By comparing [[(]{}\[uperp\][)]{}]{} and [[(]{}\[SU1nAN\][)]{}]{} (with ${\mathbb{F}}= {\mathord{\mathbb{R}}}$), we conclude that $${\mathfrak{\lowercase{H}}} \subset {\operatorname{SU}}(1,n;{\mathord{\mathbb{R}}}) \cap ({\mathfrak{\lowercase{A}}} + {\mathfrak{\lowercase{N}}}) .$$ By comparing dimensions, we see that equality must hold; this establishes Conclusion [[(]{}\[SUFevenTess-SUR\][)]{}]{}.
Assume ${\mathbb{F}}= {\mathord{\mathbb{C}}}$.
Choose some nonzero $v \in V_{{\mathfrak{\lowercase{H}}}}^\perp$, such that $v$ is $({\operatorname{Re}}Q)$-orthogonal to $u$. Multiplying by a real scalar to normalize, we may assume $Q(v,v) = -2$. By replacing $v$ with $-v$ if necessary, we may assume $\bigl( {\operatorname{Im}}Q(u,v) \bigr)/i \ge 0$.
Because $v$ is $({\operatorname{Re}}Q)$-orthogonal to $u = (1,0,-1)$, we have ${\operatorname{Re}}\eta_v =
{\operatorname{Re}}\phi_v$ [[(]{}see \[uperp\][)]{}]{}. Let $s = ({\operatorname{Im}}\phi_v)/i$ and $t = ({\operatorname{Im}}\eta_v)/i$. Then $$\begin{aligned}
0
&\le \bigl( {\operatorname{Im}}Q(u,v) \bigr)/i \\
&= \Bigl( {\operatorname{Im}}\bigl( (1) {\overline{\eta_v}} + 0 (x_v^\dagger) + (-1) {\overline{\phi_v}}
\bigr) \Bigr)/i \\
&= \bigl( - {\operatorname{Im}}\eta_v + {\operatorname{Im}}\phi_v \bigr)/i \\
&= -t + s,
\end{aligned}$$ so $$\bigl( {\operatorname{Im}}Q(u,v) \bigr)/i = |s-t| .$$ Also, $$\begin{aligned}
-2
&= Q(v,v) \\
& = |x_v|^2 + 2 {\operatorname{Re}}(\phi_v {\overline{\eta_v}}) \\
&= |x_v|^2 + 2 ({\operatorname{Re}}\phi_v)^2 - 2 ({\operatorname{Im}}\phi_v) ({\operatorname{Im}}\eta_v) \\
&\ge - 2 ({\operatorname{Im}}\phi_v) ({\operatorname{Im}}\eta_v) \\
&= 2st
,
\end{aligned}$$ so $st \le -1$. Thus, $s$ and $t$ are of opposite signs so, because $|s| \,
|t| \ge 1$, we have $$\bigl( {\operatorname{Im}}Q(u,v) \bigr) /i = |s-t| = |s| + |t| \ge 2 .$$ Therefore, we may choose $c \in
(0,1]$, such that $${\operatorname{Im}}Q(u,v) = i \left( c + \frac{1}{c} \right) .$$ Let $$w = \left( \frac{i}{c}, 0, - ic \right) .$$ Then $$Q(w,w)
= |x_w|^2 + 2 {\operatorname{Re}}( \phi_w {\overline{\eta_w}})
= 0^2 + 2 \bigl( i / c \bigr) \bigl( i c \bigr)
= -2
= Q(v,v) ,$$ and $$Q(u,w)
= \phi_u {\overline{\eta_w}} + x_u x_w^\dagger + \eta_u {\overline{\phi_w}}
= (1)(i c) + 0 + (-1)(-i / c )
= i \left( c + \frac{1}{c} \right)
= {\operatorname{Im}}Q(u,v) .$$ Hence, there is some $h \in {\operatorname{SU}}(Q)$, such that $h(u) = u$ and $h(v) = w$. Thus, replacing ${\mathfrak{\lowercase{H}}}$ with the conjugate ${\hat h}^{-1} {\mathfrak{\lowercase{H}}} \hat h$ [(cf. \[ghat\])]{}, we may assume $v = w$.
Therefore $$\begin{aligned}
V_{{\mathfrak{\lowercase{H}}}}
&\subset v^\perp \\
&= (cv)^\perp \\
&= (i,0,-ic^2)^\perp \\
&= \{\, (\phi, x, \eta) \mid
{\operatorname{Re}}Q \bigl( (\phi, x, \eta), (i,0,-ic^2) \bigr) = 0 \,\} \\
&= \{\, (\phi, x, \eta) \mid
{\operatorname{Re}}\bigl( \phi (ic^2) + x (0^\dagger) + \eta (-i) \bigr) = 0 \,\} \\
&= \{\, (\phi, x, \eta) \mid {\operatorname{Im}}\eta = c^2 {\operatorname{Im}}\phi \,\}
.
\end{aligned}$$ By combining this with and comparing with [[(]{}\[SUegsDefn\][)]{}]{} (with ${\mathbb{F}}= {\mathord{\mathbb{C}}}$), we conclude that ${\mathfrak{\lowercase{H}}} \subset {{{\mathfrak{\lowercase{H}}}}_{[c^2]}} $. By comparing dimensions, we see that equality must hold; this establishes Conclusion [[(]{}\[SUFevenTess-SUC\][)]{}]{} (because $0 < c^2 \le 1$).
Theorem \[SUevenTess\] can be restated in the following more elementary (but less precise) form.
\[SUF-known\] Let $H$ be a closed, connected subgroup of $G = {\operatorname{SU}}(2,2m; {\mathbb{F}})$ with $m \ge
2$, such that neither $H$ nor $G/H$ is compact, and let ${q}= {\operatorname{\dim_{{\mathord{\mathbb{R}}}}}}{\mathbb{F}}$.
The homogeneous space $G/H$ has a tessellation if and only if
1. \[SUF-known-d(H)\] $d(H) = 2{q}m$; and
2. \[SUF-known-sim\] either $H \sim {\operatorname{SU}}(1,2m;{\mathbb{F}})$ or $H \sim {\operatorname{Sp}}(1,m;{\mathbb{F}})$.
($\Leftarrow$) This is Theorem \[SUevenTessExists\].
($\Rightarrow$) Theorem \[SUFevenTess\]($\Rightarrow$) provides us with three possibilities.
[[(]{}\[SUF-known-d(H)\][)]{}]{} In each case, we have $d(H) = 2 {q}m$ (see \[d(HB)\], \[d(Sp)\], and \[d(Hc)\]).
[[(]{}\[SUF-known-sim\][)]{}]{} In each case, there is some $k \in \{1,2\}$, such that $\rho(h) \asymp \lVert h\rVert^k$ for $h \in H$ (see [\[HBthm\][[(]{}\[HBthm-mu\][)]{}]{}]{}, [\[mu(SUorSp)\][[(]{}\[mu(SUorSp)-SU\][)]{}]{}]{}, and [\[SUegs\][[(]{}\[SUegs-linear\][)]{}]{}]{}). Then Corollary \[HsimSUorSp\] implies either that $H \sim {\operatorname{SU}}(1,2m;{\mathbb{F}})$ (if $k =
1$) or that $H \sim {\operatorname{Sp}}(1,m;{\mathbb{F}})$ (if $k = 2$).
The following proposition shows that no further restriction can be placed on $c$ in the statement of Theorem [\[SUFevenTess\][[(]{}\[SUFevenTess-SUC\][)]{}]{}]{}.
\[HcUncountable\] If ${\mathbb{F}}= {\mathord{\mathbb{C}}}$, then ${{H_{[c]}}}$ is not conjugate to ${H_{[c']}}$, unless $c =
c'$ [(]{}for $c,c' \in (0,1]$[)]{}.
Suppose $g^{-1} {{H_{[c]}}}g = {H_{[c']}}$, for some $g\in G = {\operatorname{SU}}(2,2m)$. Because all maximal split tori in ${H_{[c']}}$ are conjugate, we may assume that $g$ normalizes $\ker\beta$. Since all roots of $\ker\beta$ on both ${{{{\mathfrak{\lowercase{H}}}}_{[c]}}}$ and ${{{\mathfrak{\lowercase{H}}}}_{[c']}}$ are positive, $g$ cannot invert $\ker \beta$, so we conclude that $g$ centralizes $\ker \beta$; that is, $g \in C_G(\ker \beta)$.
In the notation of Case \[SUFevenTessPf-nosquare\] of the proof of Theorem \[SUFevenTess\], define $$S = \{\, \hat h \mid h \in {\operatorname{SU}}(Q) \,\}$$ [(cf. \[ghat\])]{}. Then $C_G( \ker \beta ) = (\ker \beta) S$, so we may assume $g \in S$ (because $\ker \beta$, being a subgroup of ${{H_{[c]}}}$, obviously normalizes ${{H_{[c]}}}$). Write $g = \hat h$. Then, because $g^{-1} {{H_{[c]}}}g = {H_{[c']}}$, we must have $h(V_{{{{{\mathfrak{\lowercase{H}}}}_{[c]}}}}) = V_{{{{\mathfrak{\lowercase{H}}}}_{[c']}}}$; hence $h(V_{{{{{\mathfrak{\lowercase{H}}}}_{[c]}}}}^\perp) =
V_{{{{\mathfrak{\lowercase{H}}}}_{[c']}}}^\perp$.
For any basis $\{u,v\}$ of $V_{{{{{\mathfrak{\lowercase{H}}}}_{[c]}}}}^\perp$ with $Q(u,u) = Q(v,v) = -2$ and ${\operatorname{Re}}Q(u,v) = 0$, we have $${\operatorname{Im}}Q(u,v) = \pm i \bigl( c + (1/c) \bigr) .$$ Similarly for any $({\operatorname{Re}}Q)$-orthonormal basis $\{u',v'\}$ of $V_{{{{\mathfrak{\lowercase{H}}}}_{[c']}}}^\perp$. Because $h \in {\operatorname{SU}}(Q)$, this implies $c + (1/c) = c' + (1/c')$. Because $c,c' \in (0,1]$, we conclude that $c =
c'$, as desired.
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---
abstract: 'We present a decomposition of the generalized binomial coefficients associated with Jack polynomials into two factors: a [*stem*]{}, which is described explicitly in terms of hooks of the indexing partitions, and a [*leaf*]{}, which inherits various recurrence properties from the binomial coefficients and depends exclusively on the skew diagram. We then derive a direct combinatorial formula for the leaf in the special case where the two indexing partitions differ by at most two rows. This formula also exhibits an unexpected symmetry with respect to the lengths of the two rows.'
author:
- Yusra Naqvi
- Siddhartha Sahi
title: A combinatorial formula for certain binomial coefficients for Jack polynomials
---
Introduction {#introduction .unnumbered}
============
This work is motivated by the long-standing quest for direct, combinatorial formulas for binomial coefficients associated with [Jack symmetric polynomials. We recall that Jack polynomials are ]{}multivariate symmetric polynomials $P_{\lambda }=P_{\lambda }^{(\alpha )}(x_{1},\ldots ,x_{M})$, which are indexed by partitions $\lambda $ of length $\leq M$, and whose coefficients belong to the field $\mathbb{F}=\mathbb{Q}\left( \alpha \right)
$ of rational functions in the parameter $\alpha $. In fact the $P_{\lambda }
$ form a homogeneous basis for the algebra of symmetric polynomials $\mathbb{F}\left[ x_{1},\ldots ,x_{M}\right] ^{S_{M}}$, and hold a special place in the algebraic hierarchy of multivariate symmetric polynomials. For $\alpha
=0,\nicefrac{1}{2},1,2,\infty $ they specialize to the symmetric monomials, quaternionic zonal polynomials, Schur functions, real zonal polynomials, and elementary symmetric polynomials, respectively. In turn they, along with the Hall-Littlewood polynomials, were one of the two sources of inspiration for Macdonald’s definition of his two parameter family of symmetric polynomials [@macdonald:95].
For partitions $\lambda $ and $\mu $[, the Jack binomial coefficient]{} $b_{\mu }^{\lambda }=b_{\mu }^{\lambda }\left( \alpha \right) \in \mathbb{F}$ is defined by $$\frac{P_{\lambda }(1+x_{1},\ldots ,1+x_{M})}{P_{\lambda
}(1,\ldots ,1)}=\sum_{\mu }b_{\mu }^{\lambda }\frac{P_{\mu
}(x_{1},\ldots ,x_{M})}{P_{\mu }(1,\ldots ,1)}.$$ In full generality these coefficients were first considered by Lassalle [lassalle:90]{}, although the special cases $\alpha =1$ and $\alpha =2$ occurred in earlier work of Lascoux [@lascoux:82] and Bingham [bingham:74]{}. They were also studied extensively by Okounkov and Olshanski [@okounkov:olshanski:97] who showed that the $b_{\mu }^{\lambda }$ are special values of the interpolation polynomials. These latter polynomials were first defined by one of us in [@sahi:94], and studied in [knop:97, knop:sahi:96, sahi:11]{}. We note that there are anlaogous definition for binomial coefficients for Macdonald symmetric polynomials [@okounkov:97] and for nonsymmetric Jack and Macdonald polynomials [@sahi:98], and we hope to treat these in future work.
The starting point of the present paper is a recursive formula for $b_{\mu
}^{\lambda }$ recently discovered by one of us [@sahi:11]. By way of background, we recall that to a partition $\lambda =\left( \lambda
_{1},\ldots ,\lambda _{M}\right) $ one attaches a Young diagram consisting of a left justified array of boxes, with $\lambda _{i}$ boxes in row $i$. For a box $b$ in $\lambda $ we write arm$_{\lambda }(b)$ and leg$_{\lambda
}(b)$ for the number of boxes to the right of $b$ and below $b$, respectively. Here and in the subsequent discussion, we will find it convenient to set$$r=1/\alpha,$$ and we define the upper and lower $r$-hooks to be $$c_{\lambda }(b)=\text{arm}(b)+r\cdot \text{leg}(b)+r,\quad c_{\lambda }^{\prime
}(b)=\text{arm}(b)+r\cdot \text{leg}(b)+1.$$
We write $\lambda \supset \mu $ if the Young diagram of $\lambda $ contains that of $\mu $, i.e. if $\lambda _{i}\geq \mu _{i}$ for all $i$, and in this case we denote by $\lambda /\mu $ the skew diagram consisting of the boxes of $\lambda $ not in $\mu $ (the shaded boxes in the picture below). $$\ytableausetup{boxsize=1.2em}
\ytableaushort{~~~~~,~~~~,~~~,~}*[*(lightgray)]{0,2+2,1+2}$$ Now a key property of binomial coefficients is that $b_{\mu }^{\lambda }=0$ *unless* $\lambda \supset \mu $. If $\lambda /\mu $ consists of a single box we say that $\lambda ,\mu $ are adjacent, and in this case there is an explicit combinatorial formula due to Kaneko [@kaneko:93] that expresses $b_{\mu }^{\lambda }$ in terms of upper and lower hooks of $\lambda $ and $\mu $. One of the main results of [@sahi:11] asserts that the infinite matrix $\left( b_{\mu }^{\lambda }\right) $ of binomial cofficients is the exponential of the matrix of adjacent binomial coefficients, which leads to the recursive formula for the coefficients alluded to above and recalled in Section \[sec:recur\] below.
However, finding closed-form formulas for $b_{\mu }^{\lambda }$ remains a challenging and active area of ongoing research. For example, as noted in [@sahi:11], after multiplication by a suitable (and precise) factor the coefficients seem to be polynomials in $r$ with positive integer coefficients, but there is currently no proof of this conjecture.
We now briefly describe the main results of this paper. Given partitions $\lambda\supset\mu$ we label the boxes of $\lambda$ as follows: we label the boxes of $\lambda/\mu$ by $S$ and label the remaining boxes of $\lambda$ to indicate whether they share a row $(R)$, column $(C)$, both $(J)$, or neither $(N)$, with the skew boxes: $$\ytableausetup{boxsize=1.2em}
\ytableaushort{NCCCN,RJSS,RSS,N}*[*(lightgray)]{0,2+2,1+2}$$ For convenience we write $C,R,$ etc. for the set of boxes with label $C,R,$ etc. We define the **stem** $K_{\mu }^{\lambda }$ and the **leaf** $L_{\mu }^{\lambda }$ as follows $$K_{\mu }^{\lambda }=\left( \prod_{b\in C}\frac{c_{\lambda }(b)}{c_{\mu }(b)}\right) \left( \prod_{b\in R}\frac{c_{\lambda }^{\prime }(b)}{c_{\mu
}^{\prime }(b)}\right) \left( \prod_{b\in J}\frac{1}{c_{\mu }(b)c_{\mu
}^{\prime }(b)}\right) ,\quad L_{\mu }^{\lambda }=\frac{b_{\mu }^{\lambda }}{K_{\mu }^{\lambda }}.$$
The leaf $L_{\mu}^{\lambda}$ depends only on the skew diagram $\lambda/\mu$. \[prop:leafskew\]
Thus, although the Jack binomial coefficient $b_{\mu }^{\lambda }$ can (and does) vary with $\lambda $ and $\mu $ for fixed $\lambda /\mu$, our result shows that this variation is explicitly described combinatorially by the stem.
The leaf still seems to be a fairly complicated combinatorial object. Our second main result is an explicit formula for leaf in the case where $\lambda /\mu $ consists of (at most) two rows. Our analysis breaks up naturally into two cases — either the two rows of $\lambda /\mu $ share no columns whatsoever, or they have a certain number of overlapping columns. We will refer to these shortly as the “gap” and “overlap” cases. Somewhat remarkably it turns that it is possible to give a uniform formula that covers both cases. To this end we attach a $4-$tuple $(u,d,m,y)$ to a skew shape $\lambda /\mu$ with at most two rows, as follows: $u$ and $d$ are the number of boxes in the upper and lower rows respectively, and $m$ is the number of overlapping columns. In the gap case (when $m$ is 0) we define $y$ to be $$y=y_{\mu }^{\lambda }=\text{arm}_{\mu }(x^{\ast })+r\cdot \text{leg}_{\mu
}(x^{\ast }),$$ and in the overlap case we set $y=0.$ Now, given *any* non-negative integers $u$ and $d\geq m$, let $d^{\prime }=d-m$, and then define: $$L\left( u,d;m,y\right) =\sum_{\ell=0}^{d^{\prime}}\binom{d^{\prime}}{\ell}\prod_{i=0}^{\ell-1}(m+i+1-r)(i+r)\prod_{j=\ell+1}^{d^{\prime}}(y+d^{\prime
}+r-j)(y+u+r+j). \label{eq:Lud}$$
\[thm:main\] Let $\mu \subset \lambda $ be partitions such that $\lambda
/\mu $ consists of at most two rows. Then we have $$L_{\mu }^{\lambda }=L\left( u,d;m,y\right),$$ where $(u,d,m,y)$ is the $4-$tuple associated to ${\lambda}/\mu$ as described above.
We present two examples in Table \[tab:ex\] below.
${\lambda}=(7,3,3,1),\mu=(4,3,1,1)$ ${\lambda}=(6,5,3,1),\mu=(6,2,1,1)$
------------------- ------------------------------------------------------------------- --------------------------------------------------------------------
$\ytableaushort{RJ{x^*}RSSS,NCC,RSS,N}*[*(lightgray)]{4+3,0,1+2}$ $\ytableaushort{NCCCCN,R{x^*}SSS,RSS,N}*[*(lightgray)]{0,2+3,1+2}$
$u$ 3 3
$d$ 2 2
$m$ 0 1
$y$ $1+r$ 0
$K^{\lambda}_\mu$ $\displaystyle \frac{4(7+3r)}{(4+3r)(1+r)^3(2+r)}$ $\displaystyle \frac{3(4+3r)(1+2r)(5+2r)}{r(4+2r)(2+r)(1+r)}$
$L^{\lambda}_\mu$ $60+238r+275r^2+90r^3+9r^4$ $6r$
: Two examples of stems and leaves[]{data-label="tab:ex"}
The numbers $u$ and $d$ of upper and lower boxes in $\lambda /\mu $ play rather different roles in formula (\[eq:Lud\]). However it turns out that there is a remarkable hidden symmetry; to describe this we set$$\tilde{L}\left( u,d\right) :=\frac{L\left( u,d;m,y\right) }{\prod_{i=1}^{d-m}(y+m+i)(y+i-1+2r)}.$$
\[thm:udsym\] If $u,d$ are nonengative integers both $\geq m$, then we have $$\tilde{L}(u,d)=\tilde{L}(d,u).$$
The rest of this paper is arranged as follows. In Section [sec:background]{} we briefly recall some background to partitions, Jack polynomials and their binomial coefficients. Section \[sec:stemleaf\] contains a detailed description of the decomposition of $\lambda $ induced by $\mu $ along with the stem-leaf factorization of $b_{\mu }^{\lambda }$. Here we define the stem combinatorially (Definition \[def:stemleaf\]) and establish that the leaf depends only on $\lambda /\mu $ (Theorem [prop:leafskew]{}). The short Section \[sec:crithook\] serves to define critical boxes and hooks which play an especially important role when $\lambda /\mu $ is a horizontal strip, i.e., when its constituent rows do not overlap along any columns. In Section \[sec:horz\] we study this case in detail, and establish a combinatorial leaf formula which depends on the critical hook (Theorem \[thm:q\]). Section \[sec:overlap\] is devoted to the case where the two skew rows overlap along $m>0$ columns, and in this case $m$ plays an important role in the resulting leaf formula (Theorem [thm:p]{}). In each case, we also establish the symmetry between $u$ and $d$.
Background {#sec:background}
==========
In this short section we collect together some notation and background material for our paper. This has occasioned a slight overlap with the introduction, especially with the concepts that were needed for the formulation of the results. We fix throughout a natural number $n$ and a parameter $\alpha ,$ we write $\mathbb{F}=\mathbb{Q}\left( \alpha
\right) $ and put $r=1/\alpha$.
A *partition* $\lambda $ of length $\leq M$ is a finite sequence of non-negative integers $$\lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{M}\geq 0.$$The $\lambda _{i}$ are referred to as parts of $\lambda $, and we denote their sum by $$\left\vert \lambda \right\vert =\lambda _{1}+\cdots +\lambda _{M}$$We will identify a partition $\lambda $ with its *Young diagram,* which is a left justified array of boxes, with $\lambda _{i}$ boxes in row $i$. For instance, the diagram of $(4,3,2,1)$ (and also of $\left(
4,3,2,1,0\right) $ etc.) is as follows: $$\ytableaushort{~~~~,~~~,~~,~}$$
There are two natural partial orders on the set of partitions. The *dominance* order $\mu \leq \lambda $ is defined by the requirement$$\mu _{1}+\cdots +\mu _{i}\leq \lambda _{1}+\cdots +\lambda _{i}\text{ for
all }i.$$with equality for $i=M,$ i.e. we have $\left\vert \lambda \right\vert
=\left\vert \mu \right\vert .$ The containment order $\mu \subset \lambda $ is defined by$$\mu _{i}\leq \lambda _{i}\text{ for all }i.$$In terms of Young diagrams, dominance means that $\mu $ is obtained from $\lambda $ by moving some boxes to lower rows; while containment means that the diagram of $\lambda $ contains that of $\mu $. For $\mu \subset \lambda $, the *skew diagram* $\lambda /\mu $ consists of boxes of $\lambda $ that are not in $\mu $. If $\lambda /\mu $ consists of exactly one box, we say that $\lambda $ and $\mu $ are adjacent, and we write $\mu \subset
:\lambda $.
For a box $s$ in $\lambda $, we write $\text{arm}_{\lambda }(s)$ for the number of boxes to its right, and $\text{leg}_{\lambda }(s)$ for the number of boxes below it. We define the *lower* and *upper* $r$*-hook* of $s$ to be $$c_{\lambda }(s)=\text{arm}(s)+r\cdot \text{leg}\left( s\right) +r,\quad
c_{\lambda }^{\prime }(s)=\text{arm}(s)+r\cdot \text{leg}\left( s\right) +1.$$
Jack Polynomials
----------------
Jack polynomials form a linear basis for the algebra of symmetric polynomials $\mathbb{F}\left[ x_{1},\ldots ,x_{M}\right] ^{S_{M}}$. They arise as eigenfunctions of the Laplace-Beltrami operator $$D({\alpha}) = \frac{{\alpha}}{2} \sum_i x_i^2 \frac{\partial^2}{\partial x_i^2} + \sum_{i \neq j} \frac{x_i^2}{x_i-x_j} \frac{\partial}{\partial x_i}.$$ It is readily checked that $D$ is upper-triangular with respect to the dominance order on monomial symmetric functions $m_{\lambda }$ in the sense that $$D({\alpha}) m_{\lambda}= \sum_{\mu\leq {\lambda}} c_{{\lambda},\mu} m_\mu.$$ $P_{\lambda }=P_{\lambda }^{(\alpha )}(x_{1},\ldots
,x_{M})$ is the unique eigenfunction of $D$ of the form $$P_{\lambda}=m_{\lambda}+\sum_{\mu\lneq {\lambda}} v_{{\lambda},\mu} m_\mu.$$
We refer the reader to [@jack:69; @knop:sahi:97; @macdonald:95; @stanley:89] for more properties of Jack polynomials, including their role in algebraic combinatorics.
Binomial Coefficients {#sec:recur}
---------------------
For partitions $\mu \subset \lambda $, the *Jack binomial coefficients* $b_{\mu }^{\lambda }$ are defined by the expansion $$\frac{{P}_{\lambda}(1+x_1,\ldots,1+x_M)}{{P}_{\lambda}(1,\ldots,1)}=\sum_\mu b^{\lambda}_\mu \frac{{P}_\mu(x_1,\ldots,x_M)}{{P}_\mu(1,\ldots,1)}.$$
For adjacent partitions $\mu \subset :\lambda $, one knows by [kaneko:93]{} that $b_{\mu }^{\lambda }$ equals the quantity $$\begin{aligned}
\label{a_lamu}
a^\lambda_\mu =\left(\prod_{s \in C} \frac{c_\lambda(s)}{c_\mu(s)} \right) \left( \prod_{s \in R} \frac{c'_\lambda(s)}{c'_\mu(s)} \right),\end{aligned}$$ where $C$ and $R$ are the boxes in $\mu $ that lie in the same column and row, respectively, as the unique box in $\lambda /\mu$.
In [@sahi:11 Theorem 2] one of us established the following recursion for binomial coefficients: $$\begin{aligned}
\left(|{\lambda}|-|\mu|\right) \cdot b^{\lambda}_\mu = \sum_{{\kappa}\subset:{\lambda}} a^{\lambda}_{\kappa}\cdot b^{{\kappa}}_\mu.\end{aligned}$$ This can be reformulated as follows. For $\mu \subset \lambda
$ let $\mathcal{T=T}^{\lambda}_\mu$ be the set of all ascending sequences $T$ of the form $$\mu = \mu_0 \subset: \mu_{1} \subset: \cdots \subset: \mu_n = {\lambda}.$$ For $T$ in $\mathcal{T}$ let $a_{T}$ denote the product of adjacent coefficients $$\begin{aligned}
\label{a_T}
a_T = \prod^{n}_{i=1} a_{\mu_{i-1}}^{\mu_i}.\end{aligned}$$ Then, as noted in the proof of [@sahi:11 Theorem 5], one has $$\begin{aligned}
\label{b_lamu}
n! \cdot b^{\lambda}_\mu = \sum_T a_T.\end{aligned}$$
Stems and Leaves {#sec:stemleaf}
================
We now generalize the sets $C$ and $R$ from (\[a\_lamu\]), as foreshadowed in the introduction.
\[inddecomp\] Given partitions ${\lambda}\supset \mu$, we decompose the Young diagram of ${\lambda}$ relative to $\mu$ into the following sets: $$\begin{aligned}
S{^\lambda_\mu}&={\lambda}/\mu={\left\{(i,j) \in {\lambda}\mid (i,j)\notin \mu\right\}} \mbox{ (\emph{skew} boxes)} \\
R{^\lambda_\mu}&={\left\{(i,j) \in \mu \mid \mu_i < {\lambda}_i \text{ and } \mu'_j = {\lambda}'_j\right\}} \mbox{ (\emph{row} boxes)}, \\
C{^\lambda_\mu}&={\left\{(i,j) \in \mu \mid \mu_i = {\lambda}_i \text{ and } \mu'_j < {\lambda}'_j\right\}} \mbox{ (\emph{column} boxes)}, \\
J{^\lambda_\mu}&={\left\{(i,j) \in \mu \mid \mu_i < {\lambda}_i \text{ and } \mu'_j < {\lambda}'_j\right\}} \mbox{ (\emph{joint} boxes)}, \\
N{^\lambda_\mu}&={\left\{(i,j) \in \mu \mid \mu_i = {\lambda}_i \text{ and } \mu'_j = {\lambda}'_j\right\}} \mbox{ (\emph{neutral} boxes)}.\end{aligned}$$ This collection of sets is called the **decomposition of ${\lambda}$ induced by $\mu$**, and the boxes in the diagram of ${\lambda}$ can be written as the disjoint union $S \cup R \cup C \cup D \cup N$. (We omit the indexing partitions ${\lambda}$ and $\mu$ whenever they are clear from context.)
For ${\lambda}=(8,7,3,3,1)$ and $\mu=(8,4,3,1,1)$, $$\ytableaushort{NCCNCCCN,RJJRSSS,NCC,RSS,N}*[*(lightgray)]{0,4+3,0,1+2}$$
Note that $J$ is determined completely by the skew diagram $S$, since its boxes are precisely those which share both a row and column with some box in $S$. We refer to $S_+ = S\cup J$ as the *completion* of $S$.
\[def:stemleaf\] For $\lambda \supset \mu$ the **stem** is defined as follows: $$K^{\lambda}_\mu= \frac{\displaystyle\left(\prod_{b \in C} \frac{c_\lambda(b)}{c_\mu(b)}\right) \left(\prod_{b \in R} \frac{c'_\lambda(b)}{c'_\mu(b)}\right)}{\displaystyle\left(\prod_{b \in J}c_\mu(b)c'_\mu(b)\right)},$$ and the **leaf** is defined to be the quotient $\displaystyle L^{\lambda}_\mu=\frac{b^{\lambda}_\mu}{K^{\lambda}_\mu}.$
We are now ready to prove our first main theorem.
The leaf $L^{\lambda}_\mu$ depends only on the skew diagram ${\lambda}/\mu$.
Letting $n$ equal $|{\lambda}|-|\mu|$, recall from (\[b\_lamu\]) that $n!\cdot b^\lambda_\mu$ is the sum of multiplicities $a_{T}$ over all ascending sequences $T$ of the form $\mu = \mu_0 \subset: \cdots \subset: \mu_n = {\lambda}$. For each such $T$, we denote by $C^T_i$ and $R^T_i$ those boxes in $\mu_{i-1}$ which lie in the same column and row as the unique box of $\mu_{i}/\mu_{i-1}$. Using (\[a\_T\]) and then (\[a\_lamu\]), we obtain $$\begin{aligned}
a_{T} &= \prod^{n}_{i=1} a_{\mu_{i-1}}^{\mu_i}= \prod^{n}_{i=1} \left(\prod_{s \in C^T_i} \frac{c_{\mu_{i}}(s)}{c_{\mu_{i-1}}(s)} \right) \left( \prod_{s \in R^T_i} \frac{c'_{\mu_{i}}(s)}{c'_{\mu_{i-1}}(s)} \right).\end{aligned}$$ We observe that $a_T$ may also be expressed as a product over all boxes of ${\lambda}$ $$a_{T}=\prod_{s \in {\lambda}} \left( \prod^{n}_{i=1} h^T_i(s) \right),$$ where $$\begin{aligned}
\label{eq:h}
h^T_i(s)=\begin{cases} \frac{c_{\mu_{i}}(s)}{c_{\mu_{i-1}}(s)} &\mbox{ if } s \in C^T_i \\
\frac{c'_{\mu_{i}}(s)}{c'_{\mu_{i-1}}(s)} &\mbox{ if } s \in R^T_i\\
1 &\mbox{ otherwise}.
\end{cases}\end{aligned}$$
Now recall that each box $s$ of ${\lambda}$ lies in precisely one of the five sets from Defintion \[def:stemleaf\]. If $s\in N$, then $s \notin R^T_i \cup C^T_i$ for any sequence $T$ and index $i$, so none of its hook ratios appear as factors in $a_T$. And if $s \in R$, then $s$ is not an element of of $C^T_i$ for any $i$, but the set $I$ of indices $i$ for which $s \in R^T_i$ is non-empty. If $j\notin I$, then $c'_{\mu_j}(s)$ and $c'_{\mu_{j-1}}(s)$ are equal, so $h^T_j(s)=1$. Therefore, in this case we obtain (via telescoping product) $$\prod^{n}_{i=1} h^T_i(s)=\prod_{i=1}^n \frac{c'_{\mu_{i}}(s)}{c'_{\mu_{i-1}}(s)}=\frac{c'_{{\lambda}}(s)}{c'_{\mu}(s)}.$$ By an analogous argument, if $s \in C$, then $$\prod^{n}_{i=1} h^T_i(s)=\prod_{i=1}^n \frac{c_{\mu_{i}}(s)}{c_{\mu_{i-1}}(s)}=\frac{c_{{\lambda}}(s)}{c_{\mu}(s)}.$$ Absorbing the $C$, $R$ and $N$ boxes’ contributions to the multiplicity of $T$ into the stem $K^{\lambda}_\mu$, we have $$a_{T}=K^{\lambda}_\mu \left(\prod_{s\in J} c_\mu(s)c'_\mu(s) \right) \left(\prod_{s \in S_+} \prod^{n}_{i=1} h^T_i(s) \right),$$ whence $$n!L^{\lambda}_\mu= \frac{n!b^{\lambda}_\mu}{K^{\lambda}_\mu}= \left(\prod_{s\in J} c_\mu(s)c'_\mu(s) \right) \sum_{T} \left( \prod_{s \in S_+} \prod^{n}_{i=1} h^T_i(s) \right).$$ Since all indices in the rightmost expression above are sourced from the completion $S_+ = S \cup J$, it follows that the leaf depends only on $S = {\lambda}/\mu.$
In light of Theorem \[prop:leafskew\], we will henceforth denote the leaf by $L_{{\lambda}/\mu}$ rather than $L^{\lambda}_\mu$. Leaves inherit the following recurrence from $b^{\lambda}_\mu.$
Given partitions $\mu \subset {\lambda}$ with $|{\lambda}|-|\mu|=n$, the leaf $L_{{\lambda}/\mu}$ satisfies $$\begin{aligned}
nL_{{\lambda}/\mu}&=\sum_{{\kappa}}L_{{\kappa}/\mu}\prod_{s\in S_+} h^{\lambda}_{\kappa}(s)\ell^{\kappa}_\mu(s), \end{aligned}$$ where the sum is over ${\kappa}$ satisfying $\mu \subset {\kappa}\subset: {\lambda}$, and where $h^{\lambda}_{\kappa}(s)$ and $\ell^{\kappa}_\mu(s)$ are defined by: $$\begin{aligned}
h_\star^\bullet(s)=\begin{cases}
\frac{c_\bullet(s)}{c_\star(s)} &\mbox{ if } s \in C_\star^\bullet \\
\frac{c'_\bullet(s)}{c'_\star(s)} &\mbox{ if } s \in R_\star^\bullet \\
1 &\mbox{ otherwise } \\
\end{cases}
\qquad
\mbox{ and }
\qquad \ell_\star^\bullet(s)=\begin{cases}
c_\bullet(s)c'_\star(s) &\mbox{ if } s \in C_\star^\bullet \\
c'_\bullet(s)c_\star(s) &\mbox{ if } s \in R_\star^\bullet \\
1 &\mbox{ otherwise } \\
\end{cases}\end{aligned}$$
\[prop:leafrec\]
This recurrence follows from (the proof of) Theorem \[prop:leafskew\] by decomposing the set of all ascending sequences $T$ of the form $\mu \subset: \cdots \subset: {\lambda}$ into a sequence $\mu \subset: \cdots \subset: {\kappa}$ of length $(n-1)$ followed by a sequence ${\kappa}\subset: {\lambda}$ of length $1$ .
\[rem:leafrec\] If we decompose $T$ into a sequence $\mu\subset:\nu$ of length $1$ followed by a sequence $\nu\subset: \cdots \subset: {\lambda}$ of length $(n-1)$, we obtain a new dual recurrence $$nrL_{{\lambda}/\mu}=\sum_{\nu}L_{{\lambda}/\nu}\prod_{s\in S_+} h^\nu_\mu(s)\ell^{\lambda}_\nu(s)\prod_{s\in J} \frac{c_\mu(s)c'_\mu(s)}{c_\nu(s)c'_\nu(s)},$$ where the sum is now indexed by all partitions $\nu$ satisfying $\mu \subset: \nu \subset {\lambda}$.
Critical Boxes and Hooks {#sec:crithook}
========================
For the remainder of this paper, we fix partitions $\mu \subset {\lambda}$ with $|{\lambda}|-|\mu|=n$. Letting $N, S, R, C$ and $J$ be the sets from Definition \[inddecomp\], we will henceforth assume that skew diagram $S={\lambda}/\mu$ consists of at most two rows $S_1$ and $S_2$. We call $S$ a [*horizontal strip*]{} if $S_1$ and $S_2$ do not share any columns. Let $u$ be the number of boxes in $S_1$ (the first row) and $d=n-u$ be the number of boxes in $S_2$ (the second row). By convention, if $S$ consists of a single row, then $S_2$ is empty and we have $u=n$ and $d=0$.
Note that $J$ is entirely contained within a single row, and when $S$ is a horizontal strip $J$ contains exactly $d$ boxes. On the other hand, if $S$ is not a horizontal strip, then $S_1$ and $S_2$ overlap in $m \geq 1$ columns and $J$ contains $d-m$ boxes.
The rightmost box of $J$ plays an especially important role in our calculations. We note that its coordinates can be completely determined from the skew diagram $S$ (since it has the same row coordinates as the boxes in $S_1$ and the same column coordinate of the rightmost box in $S_2$).
\[def:crit\] The **critical box** of $S={\lambda}/\mu$ is the rightmost box $x^{\ast}$ of the set $J$. The **critical hook** of $S={\lambda}/\mu$ is the polynomial in $r$ given by $$y = y^{\lambda}_\mu(r) = \text{arm}_\mu(x^*)+r\cdot \text{leg}_\mu(x^*).$$
Note that the critical hook is not itself an upper or lower hook, but rather obtained from these hooks by removing the corner box.
--------------------------------------------------------------------------- -- ---------------------------------------------------------------------------------------
$\ytableaushort{~~~~~~,~{x^{*}}--~,~{|}~,~~}*[*(lightgray)]{0,4+1,0,1+1}$ $\ytableaushort{~~~~~~,~{x^{*}}~~~,~~\uparrow\uparrow,~}*[*(lightgray)]{0,2+3,1+3,0}$
$y = 2+r$ $y =0$
$m=0$ $m=2$
--------------------------------------------------------------------------- -- ---------------------------------------------------------------------------------------
We observe that if $m>0$, then $S_1$ and $S_2$ must be in successive rows. Therefore, $y$ is nonzero only if $m$ is zero, which in turn occurs if and only if $S$ is a horizontal strip. We therefore deal with the cases $m=0$ (the *gap* case) and $m>0$ (the *overlap* case) separately in the next two sections. In each case, our strategy is to find a suitable recurrence relation for the leaf $L_{{\lambda}/\mu}$ and to show that our formula from Theorem \[thm:main\] satisfies this recurrence.
The Gap Case {#sec:horz}
============
Our goal in this section is to prove the following result.
\[thm:q\] Let $\mu \subset {\lambda}$ be partitions with $|\lambda|-|\mu| = n$ such that the skew diagram $\lambda/\mu$ is a horizontal strip consisting of two rows. The leaf $L_{\lambda/\mu}$ is given by $$L_{\lambda/\mu}=\sum^d_{k=0} \binom{d}{k}\prod_{i=0}^{k-1} (i+1-r)(i+r)\prod_{i=k+1}^d(y+d+r-i)(y+u+r+i),$$ where $u$ and $d$ denote the number of boxes in the upper and lower rows of ${\lambda}/\mu$ respectively, while $y$ is the associated critical hook from Definition \[def:crit\].
Under the assumptions of this theorem, the set $J$ contains $d$ boxes $x_1,\ldots,x_d$ in a single row and the critical box $x^*$ is $x_d$. Since the number of overlapping columns $m$ between $S_1$ and $S_2$ is zero, the skew diagram ${\lambda}/\mu$ is determined by $u,d,$ and the critical hook $y = y^{\lambda}_\mu(r)$. It will thus be convenient to denote the leaf $L_{{\lambda}/\mu}$ by ${\mathbf{Q}}^u_d(y)$ throughout this section.
The leaf ${\mathbf{Q}}^u_d(y)$ satisfies the recurrence $$\begin{aligned}
\alpha_0 \cdot {\mathbf{Q}}^u_d(y) = \alpha_1 \cdot {\mathbf{Q}}^{u-1}_{d}(y) + \alpha_2 \cdot {\mathbf{Q}}^{u}_{d-1}(y+1),\end{aligned}$$ where the polynomials $\alpha_\bullet = \alpha_\bullet(y,u,d,r)$ are given by $$\begin{aligned}
\alpha_0(y,u,d,r) & = (u+d)(y+u+r), \\
\alpha_1(y,u,d,r) &= u(y+{u+d}+r), \text{ and} \\
\alpha_2(y,u,d,r) &= d(y+r)(y+u+1)(y+u+2r),\end{aligned}$$ along with the initial condition ${\mathbf{Q}}^0_0(y)=1$. \[prop:leaf1\]
The initial condition follows immediately from observing that when $n=u+d=0$, we have ${\lambda}=\mu$, and the sets $R$, $C$ and $J$ are empty. In this case, the binomial coefficient $b^{\lambda}_\mu$ and its stem $K^{\lambda}_\mu$ both trivially equal $1$, so $L_{{\lambda}/\mu} = Q_0^0(y) = 1$.
Now suppose $n>0$, and let ${\kappa}$ and $\nu$ be the partitions whose Young diagrams are obtained from ${\lambda}$ by removing the rightmost boxes of $S_1$ and $S_2$ respectively. Recall the recursive expression for the leaf from Proposition \[prop:leafrec\]: $$\begin{aligned}
\label{eq:L_lamu_recur}
n L_{{\lambda}/\mu} &= L_{{\kappa}/\mu}\prod_{s\in S_+} h^{\lambda}_{{\kappa}}(s)\ell^{{\kappa}}_\mu(s) + L_{\nu/\mu}\prod_{s\in S_+} h^{\lambda}_\nu(s)\ell^\nu_\mu(s).\end{aligned}$$ Since ${\kappa}/\mu$ has the same critical hook as ${\lambda}/\mu$ but one fewer box in the upper row, we obtain $L_{{\kappa}/\mu}={\mathbf{Q}}^{u-1}_d(y)$. On the other hand, $\nu/\mu$ has one fewer box in the lower row, so its critical hook is $y+1$, and we have $L_{\nu/\mu}={\mathbf{Q}}^{u}_{d-1}(y+1)$. We conclude the argument by establishing $$\frac{\alpha_2(y,u,d,r)}{\alpha_0(y,u,d,r)} = \frac{1}{n}\prod_{s\in S_+} h^{\lambda}_\nu(s)\ell^\nu_\mu(s),$$ and leave the (easier) verification involving $\alpha_1/\alpha_0$ as an exercise. To this end, note that $$S_+ \cap C^{\lambda}_\nu = \{x_d\} \text{ and } S_+ \cap R^{\lambda}_{\nu} = S_2 \cap \nu,$$ so we have an expression for the $h$-product $$\prod_{s\in S_+} h^{{\lambda}}_{\nu}(s) = d\frac{c_{\lambda}(x_d)}{c_{\nu}(x_d)}=\frac{d(y+u+2r)}{(y+u+r)}.$$ And similarly, $$S_+ \cap C^{\nu}_\mu = \varnothing \text{ and } S_+ \cap R^{\nu}_\mu = \{x_d\},$$ which yields an expression for the $\ell$-product $$\prod_{s\in S_+} \ell^{\nu}_{\mu}(s) = c_{\mu}(x_d)c'_\nu(x_d)=(y+u+1)(y+r).$$ The desired result now follows from multiplying the $h$ and $\ell$-products described above, and recalling that $n = u+d$.
Proof of Theorem \[thm:q\]
--------------------------
We will describe a general family of polynomials indexed by $u$ and $d$ and show that they can be suitably modified to satisfy the same recurrence as ${\mathbf{Q}}^u_d(y)$ from Proposition \[prop:leaf1\].
Let ${\mathbf{M}}^u_d$ be the bivariate polynomial given by $${\mathbf{M}}^u_d(z;\theta)=\sum^d_{k=0} \binom{d}{k} \prod^{k-1}_{i=0} (\theta+\rho_i^2)\prod_{i=k+1}^d(z-i)(z+i+u-d),$$ where $\rho_i$ equals $i+\frac{1}{2}.$ \[defn:M\]
We now claim the following.
The polynomials ${\mathbf{M}}^u_d$ satisfy the recurrence $$\alpha'_0\cdot {\mathbf{M}}^u_d = \alpha'_1\cdot {\mathbf{M}}^{u-1}_{d}+ \alpha'_2\cdot {\mathbf{M}}^{u}_{d-1},$$ where the polynomials $\alpha'_\bullet = \alpha'_\bullet(u,d,z,\theta)$ are given by $$\begin{aligned}
\alpha'_0(u,d,z,\theta) &= (u+d)(z+u-d), \\
\alpha'_1(u,d,z,\theta) &= u(z+u),\text{ and} \\
\alpha'_2(u,d,z,\theta) &= d(z-d)(\theta + (z+\rho_{u-d})^2),\end{aligned}$$ along with the initial condition ${\mathbf{M}}^0_0=1$. \[prop:M\]
This recurrence given in Proposition \[prop:M\] can be modified to the recurrence satisfied by ${\mathbf{Q}}^u_d$ in Theorem \[thm:q\] by making the following substitutions: $$\begin{aligned}
z &= y+d+r, \text{ and} \\
\theta &= -\left(r-\frac{1}{2}\right)^2.\end{aligned}$$ Therefore, in order to prove Theorem \[thm:q\], it suffices to prove Proposition \[prop:M\]. To accomplish this, we first establish a different recurrence for ${\mathbf{M}}^u_d$.
The polynomials ${\mathbf{M}}^u_d$ satisfy the recurrence $${\mathbf{M}}^u_d=\left(\theta + (z+\rho_{u-d})^2\right){\mathbf{M}}^u_{d-1}-u(z+u){\mathbf{M}}^{u-1}_{d-1},
\label{eq:M}$$ where ${\mathbf{M}}^u_{0}=1$ for all $u$. \[lem:M\]
For brevity, we define $q_k(\theta) = \prod^{k-1}_{i=0} (\theta+\rho_i^2)$ and $$f^u_d(k) = \binom{d}{k} \prod_{i=k+1}^d(z-i)(z+i+u-d),$$ so ${\mathbf{M}}^u_d=\sum^d_{k=0} f^u_d(k)q_k(\theta)$. Note that the $q_k(\theta)$ form a basis for polynomials in $\theta$ over the field $\mathbb{Q}(z,u,d).$ In fact, we will find it convenient to suppress $\theta$, and write $q_k=q_k(\theta)$. It now suffices to show that the coefficient of $q_k$ on the right side of (\[eq:M\]) also equals $f^u_d(k)$.
Expanding in terms of the $f$’s and $q$’s, the right side of (\[eq:M\]) equals $$\left(\theta + (z+\rho_{u-d})^2\right)\sum_{k=0}^{d-1} f^{u}_{d-1}(k)q_k-u(z+u)\sum_{k=0}^{d-1} f^{u-1}_{d-1}(k)q_k.
\label{eq:rM1}$$ We can absorb the expression $\theta + (z+\rho_{u-d})^2$ outside the first sum into the sum, and then for each $k$, we may rewrite this expression as $$(\theta+\rho_k^2) + \left((z+\rho_{u-d})^2-\rho_k^2\right).$$ With this modification, (\[eq:rM1\]) becomes $$\sum^{d-1}_{k=0} f^{u}_{d-1}(k)q_{k+1} +\sum^{d-1}_{k=0}\left( (z+\rho_{u-d})^2-\rho_{k}^2\right)f^{u}_{d-1}(k)q_k-\sum^{d-1}_{k=0} u(z+u)f^{u-1}_{d-1}(k)q_k.
\label{eq:rM2}$$ Collecting the coefficients of $q_k$ in (\[eq:rM2\]), we have $$f^{u}_{d-1}(k-1)+\left( (z+\rho_{u-d})^2-\rho_{k}^2\right) f^{u}_{d-1}(l)-u(z+u)f^{u-1}_{d-1}(k).
\label{eq:rM3}$$ In order to establish (\[eq:M\]), we must now show that expression (\[eq:rM3\]) equals $f^u_d(k).$
Elementary calculations express each term in (\[eq:rM3\]) as a multiple of $f^u_d(k)$: $$\begin{aligned}
f^{u}_{d-1}(k-1) &= \frac{k(z-k)}{d(z-d)} \cdot f^{u}_d(k), \\
\left((z+\rho_{u-d})^2-\rho_{k}^2\right) \cdot f^{u}_{d-1}(k) &= \frac{(d-k)(z+u-d-k)}{d(z-d)}\cdot f^{u}_d(k), \\
u(z+u) \cdot f^{u-1}_{d-1}(k) &= \frac{u(d-k)}{d(z-d)} \cdot f^{u}_d(k).\end{aligned}$$ (Note that the second calculation requires factoring the expression $(z+\rho_{u-d})^2-\rho_k^2$ as $(z+\rho_{u-d}+\rho_k)(z+\rho_{u-d}-\rho_k)$.) Subtracting the third quantity from the sum of the first two now yields $f^u_d(k)$ on the right side of (\[eq:M\]) and hence concludes the argument.
Although the recurrence of Lemma \[lem:M\] looks quite different from the one in Proposition \[prop:M\] at first glance, it can now be used to yield the desired proof.
We expand the difference $D = \alpha'_0\cdot {\mathbf{M}}^u_d - \alpha'_1\cdot {\mathbf{M}}^{u-1}_d$, using the notation $q_k$ and $f^u_d(k)$ introduced the proof of Lemma \[lem:M\], to get $$D = \sum_{k=0}^{d} \left((u+d)(z+u-d) f^{u}_d(k) q_k- u(z+u)f^{u-1}_d(k) q_k\right)$$ It is readily checked that $(z+u)f^{u-1}_d(k)$ equals $(z+k+u-d)f^u_d(k)$, so we obtain $$\begin{aligned}
D &= \sum_{k=0}^{d} \big( (u+d)(z+u-d)-u(z+k+u-d) \big) f^{u}_d(k) q_k \\
&= d(z-d){\mathbf{M}}^u_d + \sum_{k=0}^{d-1} u(d-k) f^{u}_d(k) q_k.\end{aligned}$$ By the third elementary calculation mentioned at the end of the proof of Lemma \[lem:M\], we know $\frac{u(d-k)}{d(z-d)} \cdot f^{u}_d(k)$ equals $u(z+u) \cdot f^{u-1}_{d-1}(k)$, so we obtain $$\begin{aligned}
D &= d(z-d){\mathbf{M}}^u_d + \sum_{k=0}^{d-1} d(z-d)u(z+u) f^{u-1}_{d-1}(k) q_k \\
&= d(z-d)\left[{\mathbf{M}}^u_d + u(z+u){\mathbf{M}}^{u-1}_{d-1}\right].\end{aligned}$$ Applying Lemma \[lem:M\] yields $D = \alpha'_2\cdot {\mathbf{M}}^u_{d-1}$, as desired.
Symmetry between $u$ and $d$
----------------------------
We now consider some algebraic identities related to the polynomial expression for the leaf $L^\lambda_\mu$ from Theorem \[thm:q\], i.e., $${\mathbf{Q}}^u_d(y) = \sum^d_{k=0} \binom{d}{k}\prod_{i=0}^{k-1} (i+1-r)(i+r)\prod_{i=k+1}^d(y+d+r-i)(y+u+r+i).$$
The parameters $u$ and $d$ appear to be playing remarkably different roles in the recursion of Proposition \[prop:leafrec\] and in our explicit formula from Theorem \[thm:q\]. However, we find a surprising symmetry between $u$ and $d$ involving the polynomials $${\varphi}_k(y)=\displaystyle \prod_{i=1}^k (y+i)(y+i-1+2r).$$ We observe that setting $u=0$ in the recurrence of Proposition \[prop:leaf1\] yields $${\mathbf{Q}}^0_d(y) = (y+1)(y+2r){\mathbf{Q}}^{0}_{d-1}(y+1), \text{ and }{\mathbf{Q}}^0_0(y) = 1.$$ Since ${\varphi}_d$ also satisfies this recurrence, it follows that ${\varphi}_d(y) = {\mathbf{Q}}^0_d(y)$. Moreover, the following ${\varphi}$-identities are easy to verify $$\begin{aligned}
{\varphi}_d(y) &=(y+1)(y+2r) \cdot {\varphi}_{d-1}(y+1), \label{eq:phid1}\\
{\varphi}_d(y) &=(y+d)(y+d+1+2r) \cdot {\varphi}_{d-1}(y), \label{eq:phid2} \text{ and}\\
{\varphi}_d(y+1) &= \frac{(y+d+1)(y+d+2r)}{(y+1)(y+2r)} \cdot {\varphi}_d(y). \label{eq:phid3}\end{aligned}$$
The polynomials ${\mathbf{Q}}^u_d$ and ${\varphi}_d$ satisfy ${\mathbf{Q}}^u_d(y)\cdot {\varphi}_u(y)={\mathbf{Q}}^d_u(y)\cdot {\varphi}_d(y)$. \[thm:symQ\]
We first observe that a new recurrence for ${\mathbf{Q}}^u_d(y)$ may be derived by modifying the proof of Proposition \[prop:leaf1\] as follows. Instead of constructing the intermediate partitions $\kappa$ and $\nu$ by removing a box each from $\lambda$, we produce $\kappa$ by adding the leftmost box of the top row $S_1$ to $\mu$, and produce $\nu$ by adding the rightmost box of $S_2$ to $\mu$. And rather than using the recurrence from Proposition \[prop:leafrec\], we employ the dual recurrence of Remark \[rem:leafrec\]. With these alterations in place, and using precisely the same strategy as before, one obtains $$\bar{\alpha}_0 \cdot {\mathbf{Q}}^u_d(y) = \bar{\alpha}_1 \cdot {\mathbf{Q}}^{u-1}_{d}(y+1) + \bar{\alpha}_2 \cdot {\mathbf{Q}}^{u}_{d-1}(y),$$ with $$\begin{aligned}
\bar{\alpha}_0(y,u,d,r) &= (u+d)(y+d+r),\\
\bar{\alpha}_1(y,u,d,r) &= u(y+r) \text{, and} \\
\bar{\alpha}_2(y,u,d,r) &= d(y+u+d+r)(y+d-1+2r)(y+d),\end{aligned}$$ along with the initial condition ${\mathbf{Q}}^0_0(y)=1$.
We now use this dual recurrence to prove the desired symmetry. We proceed by induction on $n=u+d$, noting that the result holds trivially for $u=d=0$. For $n > 0$, consider our new recurrence with the roles of $u$ and $d$ interchanged: $$\bar{\alpha}_0(y,d,u,r)\cdot {\mathbf{Q}}^d_u(y) = \bar{\alpha}_1(y,d,u,r)\cdot{\mathbf{Q}}^{d-1}_{u}(y+1) + \bar{\alpha}_2(y,d,u,r)\cdot{\mathbf{Q}}^{d}_{u-1}(y).$$ Multiplying throughout by ${\varphi}_d(y)$ and using identity (\[eq:phid1\]) with the first term on the right side, we note that $\bar{\alpha}_0(y,d,u,r)\cdot{\mathbf{Q}}^d_u(y){\varphi}_d(y)$ equals $$\bar{\alpha}_1(y,d,u,r) (y+1)(y+2r)\cdot{\mathbf{Q}}^{d-1}_{u}(y+1){\varphi}_{d-1}(y+1) + \bar{\alpha}_2(y,d,u,r) \cdot {\mathbf{Q}}^{d}_{u-1}(y){\varphi}_d(y).$$ And applying the inductive hypothesis to the expression above yields $$\bar{\alpha}_1(y,d,u,r)(y+1)(y+2r)\cdot{\mathbf{Q}}^{u}_{d-1}(y+1){\varphi}_{u}(y+1) + \bar{\alpha}_2(y,d,u,r)\cdot{\mathbf{Q}}^{u-1}_{d}(y){\varphi}_{u-1}(y).$$ Identities (\[eq:phid2\]) and (\[eq:phid3\]) may now be used to express both summands above in terms of ${\varphi}_u(y)$. Using these identities and the definition of the $\bar{\alpha}_\bullet$’s, our expression for $\bar{\alpha}_0(y,d,u,r) \cdot {\mathbf{Q}}^d_u{\varphi}_d(y)$ simplifies to $$\left[d(y+r)(y+u+1)(y+u+2r)\cdot{\mathbf{Q}}^{u}_{d-1}(y+1) + u(y+u+d+r)\cdot{\mathbf{Q}}^{u-1}_{d}(y) \right]{\varphi}_{u}(y).$$ By Proposition \[prop:leaf1\], the factor within square brackets equals $\alpha_0(y,u,d,r)\cdot {\mathbf{Q}}^u_d(y)$. But since $$\alpha_0(y,u,d,r) = (u+d)(y+u+r) = \bar{\alpha}_0(y,d,u,r),$$ the desired result follows.
The Overlap Case {#sec:overlap}
================
This section is devoted to proving the following result.
\[thm:p\] Let $\mu \subset {\lambda}$ be partitions with $|\lambda|-|\mu| = n$ so that the skew diagram $\lambda/\mu$ consists of two rows which overlap along $m \geq 1$ columns. Then, the leaf $L_{\lambda/\mu}$ is given by $$L_{\lambda/\mu}=\sum^{d-m}_{k=0} \binom{d-m}{k}\prod_{i=0}^{k-1} (m+i+1-r)(i+r)\prod_{j=k+1}^{d-m}(d-m+r-j)(u+r+j),$$ where $u$ and $d$ denote the number of boxes in the upper and lower rows of ${\lambda}/\mu$ respectively.
When the rows $S_1$ and $S_2$ of $S$ overlap in $m$ columns, the set $J$ consists of $d-m$ boxes $x_1,\ldots,x_{d-m}$ (from left to right) in a single row. Recall that in this case the critical hook $y$ equals $0$, so $L^{\lambda}_\mu$ depends only $u$, $d$ and $m$, and since $m$ denotes the number of overlapping columns, we have $m\leq \min\{u,d\}$. It will be convenient to denote $L^{\lambda}_\mu$ by ${\mathbf{R}}^u_d(m)$ in this case. We can then characterize the leaf ${\mathbf{R}}^u_d(m)$ using the following recurrence.
The leaf ${\mathbf{R}}^u_d(m)$ satisfies the recurrence relation $$\beta_0 \cdot {\mathbf{R}}^u_d(m) = \beta_1 \cdot {\mathbf{R}}^{u-1}_{d}(m) + \beta_2 \cdot {\mathbf{R}}^{u}_{d-1}(m-1),$$ where $$\begin{aligned}
\beta_0(m,u,d,r) &= (u+d)(u-m+r), \\
\beta_1(m,u,d,r) &= (u-m)(u+d-m+r), \text{ and}\\
\beta_2(m,u,d,r) &= d(u-m+2r), \end{aligned}$$ with the initial condition ${\mathbf{R}}^u_d(0)={\mathbf{Q}}^u_d(0)$, where ${\mathbf{Q}}^u_d$ denotes the polynomials from Theorem \[thm:q\]. \[prop:leaf3\]
The argument is entirely similar to the one employed in the proof of Proposition \[prop:leaf1\].
Proof of Theorem \[thm:p\]
--------------------------
We provisionally define $$\begin{aligned}
\label{eq:Nshift}
{\mathbf{N}}^u_d(m) = \sum^{d}_{k=0} \binom{d}{k}\prod_{i=0}^{k-1} (m+i+1-r)(i+r)\prod_{j=k+1}^{d}(d-j+r)(u+j+r),\end{aligned}$$ and wish to establish that (up to a simple modification $d \mapsto d-m$) these polynomials satisfy the same recurrence as the one described in Proposition \[prop:leaf3\] for ${\mathbf{R}}^u_d$. For brevity, we will henceforth write $$\begin{aligned}
\label{eq:Nbasis}
{\mathbf{N}}^u_{d}(m) = \sum_{k=0}^{d}g_k(u,d)p_k(m),\end{aligned}$$ where $$\begin{aligned}
p_k(m) &= \prod_{i=0}^{k-1}(m+i+1-r), \text{ and} \label{eq:p}\\
g_k(u,d) &= \binom{d}{k}\prod_{i=0}^{k-1} (r+i)\prod_{j=k+1}^{d}(d-j+r)(u+j+r). \label{eq:f}\end{aligned}$$
We proceed by recalling that the recurrence from Proposition \[prop:leaf3\] expresses ${\mathbf{R}}^u_d(m)$ in terms of ${\mathbf{R}}^{u-1}_d(m)$ and ${\mathbf{R}}^u_{d-1}(m-1)$. The upcoming Propositions \[prop:T1\], \[prop:T2\] and \[prop:T3\] consider corresponding expressions involving ${\mathbf{N}}^u_d(m)$, and show that the first equals the sum of the next two when expanded in terms of the basis $p_k(m)$ of polynomials in $m$ over the field $\mathbb{Q}(r,u,d)$. In each argument it becomes necessary to manipulate the $p_k(m)$ to pass between adjacent indices, and the following lemma allows us to perform such manipulations.
\[lem:xshift\] For any polynomial $h = h(y,u,d,r)$, we have $$\begin{aligned}
(h + m) p_{k}(m) &= \big[(h-k-1+r) p_{k}(m) + p_{k+1}(m) \big], \text{ and} \label{eq:h+}\\
(h - m) p_{k}(m) &= \big[(h+k+1-r) p_{k}(m) - p_{k+1}(m)\big]. \label{eq:h-}\end{aligned}$$
Both statements are derivable from the identity $$\begin{aligned}
p_{k+1}(m) &= (m-r+k+1) \cdot p_{k}(m),\end{aligned}$$ which follows directly from (\[eq:p\]). For instance, (\[eq:h+\]) is verified by noting $$h + m = (h-k-1+r) + (m+ k+1-r).$$
\[prop:T1\] Given ${\mathbf{N}}^u_d(m)$ as in (\[eq:Nbasis\]), we have $$(u+d+m)(u-m+r){\mathbf{N}}^u_d(m) = \sum_{k=0}^d \sum_{i=0}^2 g_k(u,d) \gamma^{1,i}_{k}(u,d)p_{k+i}(m),$$ where the three functions $\gamma^{1,\bullet}_k$ are given by $$\begin{aligned}
\gamma^{1,0}_k(u,d) &= (u+d-k-1+r)(u+k+1) \\
\gamma^{1,1}_k(u,d) &= 2k+3-d-r, \text{ and} \\
\gamma^{1,2}_k(u,d) &= -1.\end{aligned}$$
Let ${\mathbf{X}}$ denote the left side of the desired equality. Applying (\[eq:h-\]) with $h = u+r$ gives $${\mathbf{X}}= (u+d+m)\left[\sum_{k=0}^d g_k(u,d) (u+k+1)p_k(m) - \sum_{k=0}^d g_k(u,d) p_{k+1}(m)\right].$$ By (\[eq:h+\]), with $h = u+d$, the first sum becomes $$\begin{aligned}
{\mathbf{X}}_1 &= (u+d+m)\sum_{k=0}^d g_k(u,d) (u+k+1)p_k(m) \\
&= \sum_{k=0}^d g_k(u,d) \big[(u+d-k-1+r)(u+k+1)p_k(m) + (u+k+1)p_{k+1}(m)\big].\end{aligned}$$ Similarly, the second sum becomes $$\begin{aligned}
{\mathbf{X}}_2 &= -(u+d+m)\sum_{k=0}^d g_k(u,d) p_{k+1}(m)\\
&= - \sum_{k=0}^d g_k(u,d) \big[ (u+d-k-2-r)p_{k+1}(m) + p_{k+2}(m) \big]\end{aligned}$$ Observing that ${\mathbf{X}}= {\mathbf{X}}_1 + {\mathbf{X}}_2$ and collecting the coefficients of the $p_k(m)$ completes the argument.
The proof of our second proposition uses the following elementary consequence of (\[eq:f\]): $$\begin{aligned}
\label{eq:g_ushift}
(u+d+r)\cdot g_k(u-1,d) = (u+k+r)\cdot g_k(u,d).\end{aligned}$$
\[prop:T2\] Given ${\mathbf{N}}^u_d(m)$ as in (\[eq:Nbasis\]), we have $$(u-m)(u+d+r){\mathbf{N}}^{u-1}_d(m) = \sum_{k=0}^d \sum_{i=0}^2 g_k(u,d) \gamma^{2,i}_{k}(u,d)p_{k+i}(m),$$ where the three functions $\gamma^{2,\bullet}_k$ are $$\begin{aligned}
\gamma^{2,0}_k(u,d) &= (u+k+1-r)(u+k+r) \\
\gamma^{2,1}_k(u,d) &= -(u+k+r), \text{ and} \\
\gamma^{2,2}_k(u,d) &= 0.\end{aligned}$$
The left side ${\mathbf{X}}$ is given by $${\mathbf{X}}= (u-m)(u+d+r) \sum_{k=0}^d g_k(u-1,d) p_k(m),$$ and we use (\[eq:g\_ushift\]) to incorporate the $(u+d+r)$ factor within the sum. This gives $${\mathbf{X}}= (u-m)\sum_{k=0}^d (u+k+r) g_k(u,d) p_k(m),$$ and it only remains to incorporate the $(u-m)$ factor. For this purpose, we once again apply (\[eq:h-\]), this time with $h = u$, and obtain the desired result.
Our third proposition requires a new identity which is easily verified via (\[eq:p\]). $$\begin{aligned}
p_k(m-1) &= p_k(m) - k \cdot p_{k-1}(m) \label{eq:p_mshift}\end{aligned}$$
\[prop:T3\] Given ${\mathbf{N}}^u_d(m)$ as in (\[eq:Nbasis\]), we have $$(d+m)(u-m+2r){\mathbf{N}}^{u}_{d}(m-1) = \sum_{k=0}^d \sum_{i=0}^2 g_k(u,d) \gamma^{3,i}_{k}(u,d)p_{k+i}(m),$$ where the three functions $\gamma^{3,\bullet}_k$ are given by $$\begin{aligned}
\gamma^{3,0}_k(u,d) &= (d-k+r)(u+k+r)-(k+1)(u-d+2k+1)-(d-k)(k+r), \\
\gamma^{3,1}_k(u,d) &= (u-d+3k+3), \text{ and} \\
\gamma^{3,2}_k(u,d) &= -1.\end{aligned}$$
Using (\[eq:p\_mshift\]), we can decompose the left side ${\mathbf{X}}$ of the desired equality into two pieces ${\mathbf{X}}_1 - {\mathbf{X}}_2$, where $$\begin{aligned}
{\mathbf{X}}_1 &= (d+m)(u-m+2r)\sum_{k=0}^{d}g_k(u,d)p_k(m), \text{ and} \\
{\mathbf{X}}_2 &= (d+m)(u-m+2r)\sum_{k=0}^{d}g_k(u,d)k p_{k-1}(m).\end{aligned}$$ Applying (\[eq:h-\]) to ${\mathbf{X}}_1$ with $h = u+2r$ yields $${\mathbf{X}}_1 = (d+m)\left[\sum_{k=0}^{d}g_k(u,d)(u+k+1+r)p_k(m)- \sum_{k=0}^d g_k(u,d)p_{k+1}(m)\right].$$ Then by applying (\[eq:h+\]), this time with $h = d$, and collecting $p_\bullet(m)$ terms yields the following equivalent expression for ${\mathbf{X}}_1$: $$\begin{aligned}
\label{eq:LS1}
\sum_{k=0}^{d}g_k(u,d)\big[(d+r-k-1)(u+k+r)p_k(m) + (u-d+2k+2)p_{k+1}(m) - p_{k+2}(m)\big].\end{aligned}$$ Similarly, ${\mathbf{X}}_2$ simplifies to $$\sum_{k=0}^{d}g_k(u,d)k\big[(d-k+r)(u+k+r)p_{k-1}(m) + (u-d+2k+1)p_{k}(m) - p_{k+1}(m)\big].$$ In order to realign the indices of the $p_{k-1}(m)$ terms, we use the identity $$\frac{g_{k-1}(u,d)}{g_k(u,d)} = \frac{k(d-k+r)(u+k-r)}{(d-k+1)(k-1+r)},$$ which follows immediately from (\[eq:f\]). Now, ${\mathbf{X}}_2$ equals $$\sum_{k=0}^{d}g_k(u,d)\Big[\big[(d-k)(k+r)+ (k+1)(u-d+2k-1)\big]p_{k}(m) - (k+1)p_{k+1}(m)\Big].$$ Subtracting this ${\mathbf{X}}_2$ expression from the ${\mathbf{X}}_1$ expression (\[eq:LS1\]) completes the proof.
The polynomials $\gamma^{1,i}_k, \gamma^{2,i}_k$ and $\gamma^{3,i}_k$ from Propositions \[prop:T1\], \[prop:T2\] and \[prop:T3\] (for $i \in \{0,1,2\}$) clearly satisfy $\gamma^{1,i}_k = \gamma^{2,i}_k + \gamma^{3,i}_k$ for all $k$. Relabeling the variable $d$ to $d-m$, it follows that the polynomials ${\mathbf{N}}^u_{d-m}$ satisfy the recurrence from Proposition \[prop:leaf3\]; this observation concludes our proof of Theorem \[thm:p\].
Symmetry between $u$ and $d$
----------------------------
The polynomials ${\mathbf{R}}^u_d$ of Theorem \[thm:p\] given by $${\mathbf{R}}^u_d(m) = \sum^{d-m}_{k=0} \binom{d-m}{k}\prod_{i=0}^{k-1} (m+i+1-r)(i+r)\prod_{j=k+1}^{d-m}(d-m+r-j)(u+r+j)$$ also enjoy a symmetry analogous to the one described in Theorem \[thm:symQ\] for the polynomials ${\mathbf{Q}}^u_d$. In order to describe it, we define the family of functions $$\psi_k(x)= \prod_{i=1}^k (x+i)(i-1+2r).$$
The polynomials ${\mathbf{R}}^u_d$ and $\psi_k$ satisfy ${\mathbf{R}}^u_{d}(m)\psi_{u-m}(m)={\mathbf{R}}^d_{u}(m)\psi_{d-m}(m)$. \[thm:symP\]
Using an argument similar to the one employed in Proposition \[prop:leaf1\], one can show that ${\mathbf{R}}^u_d$ satisfies the recurrence relation $$\beta'_0 \cdot {\mathbf{R}}^u_d(m) = {\mathbf{R}}^{u-1}_{d}(m-1) + \beta'_2\cdot{\mathbf{R}}^{u}_{d-1}(m),$$ where $$\begin{aligned}
\beta'_0(m,u,d,r) &= (u+d)(d-m+r), \text{ and}\\
\beta'_2(m,u,d,r) &= d(d-m)(u+d-m+r)(d-m-1+2r),\end{aligned}$$ along with the initial condition ${\mathbf{R}}^u_d(0)={\mathbf{Q}}^{u}_d(0)$. To derive this recurrence, we construct the intermediate partitions $\kappa$ and $\nu$ by adding the leftmost box of $S_1$ and the rightmost box of $S_2$ to $\mu$ respectively before invoking the dual recurrence of Remark \[rem:leafrec\]. We proceed by induction on $n=u+d$, and note that the result trivially holds when $n=0$. For $n>0$, we exchange the roles of $u$ and $d$ in the recurrence above to obtain $$\beta'_0(m,d,u,r)\cdot{\mathbf{R}}^d_u(m) = {\mathbf{R}}^{d-1}_{u}(m-1) + \beta'_2(m,d,u,r)\cdot{\mathbf{R}}^{d}_{u-1}(m).$$ Multiplying throughout by $\psi_{d-m}(m)$ and using the identity $$\psi_{d-m}(m) = \frac{d}{m}\psi_{d-m}(m-1)$$ with the first term on the right side, we note that $\beta'_0(m,d,u,r)\cdot{\mathbf{R}}^d_u(m)\psi_{d-m}(m)$ is given by $$\frac{d}{m}\cdot{\mathbf{R}}^{d-1}_{u}(m-1)\psi_{d-m}(m-1) + \beta'_2(m,d,u,r){\mathbf{R}}^{d}_{u-1}(m)\psi_{d-m}(m).$$ By the inductive hypothesis, this expression equals $$\frac{d}{m} {\mathbf{R}}^{u}_{d-1}(m-1)\psi_{u-m+1}(m-1)+ \beta'_2(m,d,u,r){\mathbf{R}}^{u-1}_{d}(m)\psi_{u-m-1}(m).$$ Next, we shift the indices of the $\psi$-factors in each term above by using the identities $$\begin{aligned}
\psi_{u-m+1}(m-1) &= m(u-m+2r) \cdot \psi_{u-m}(m), \text{ and } \\
\psi_{u-m}(m) &= u(u-m-1+2r)\cdot \psi_{u-m-1}(m),\end{aligned}$$ and recall the definitions of the $\beta'_\bullet$ to obtain that $\beta'_0(m,d,u,r)\cdot {\mathbf{R}}^d_u(m)\psi_{d-m}(m)$ equals: $$\left[d(u-m+2r)\cdot{\mathbf{R}}^u_{d-1}(m-1)+ (u-m)(u+d-m+r)\cdot{\mathbf{R}}^{u-1}_d(m)\right]\psi_{d-m}(m).$$ An appeal to Proposition \[prop:leaf3\] confirms that the factor within the square brackets above equals $\beta_0(m,u,d,r)\cdot{\mathbf{R}}^u_d(m)$. Finally, we note that $$\beta_0(m,u,d,r) = (u+d)(u-m+r) =\beta'_0(m,d,u,r),$$ and this concludes the proof.
${{\mathbf{R}}}^m_d(m)=\psi_{d-m}(m)$
If $u=m$, then Theorem \[thm:symP\] gives us: ${\mathbf{R}}^m_d(m)\psi_{m-m}(m)={\mathbf{R}}^d_m(m)\psi_{d-m}(m).$ Since $\psi_{0}(m)={\mathbf{R}}^d_m(m)=1$, we have ${{\mathbf{R}}}^m_d(m)=\psi_{d-m}(m)$.
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---
abstract: 'We propose the use of modulated spectra of astronomical sources due to gravitational lensing to probe Ellis wormholes. The modulation factor due to gravitational lensing by the Ellis wormhole is calculated. Within the geometrical optics approximation, the normal point mass lens and the Ellis wormhole are indistinguishable unless we know the source’s unlensed luminosity. This degeneracy is resolved with the significant wave effect in the low frequency domain if we take the deviation from the geometrical optics into account. We can roughly estimate the upper bound for the number density of Ellis wormholes as $n\lesssim 10^{-9}\mbox{AU}^{-3}$ with throat radius $a\sim1{{\rm cm}}$ from the existing femto-lensing analysis for compact objects.'
author:
- 'Chul-Moon Yoo'
- Tomohiro Harada
- Naoki Tsukamoto
title: Wave Effect in Gravitational Lensing by the Ellis Wormhole
---
5.5mm
[ ]{}
introduction {#sec:intro}
============
In a variety of cosmological models based on fundamental theory, exotic astrophysical objects which have not been observed are often predicted. Conversely, an observational evidence for an exotic object would stimulate creative theoretical discussions. Probing these exotic objects and detecting them will give us significant progress of research in fundamental physics. Even if we cannot detect it, giving a constraint on the abundance of the exotic objects is one of powerful means of investigating the nature of our universe.
Generally, the interaction between such unobserved exotic objects and well known matters is very weak or not well established. Thus, only the gravitational interaction would cause reliable observational phenomena. One of the most direct measurements of gravitational effects of an exotic object is gravitational lensing. For instance, massive compact halo objects are probed by using micro-lensing[@Alcock:2000ph; @Tisserand:2006zx; @Wyrzykowski:2009ep]. Cosmic strings are also targets for probing by using gravitational lensing phenomena[@Huterer:2003ze; @Oguri:2005dt; @Mack:2007ae; @2008MNRAS.384..161K; @Christiansen:2010zi; @Tuntsov:2010fu; @Pshirkov:2009vb; @Yamauchi:2011cu; @Yamauchi:2012bc]. In this paper, we propose a way to probe Ellis wormholes[@Ellis:1973yv] by using lensed spectra of astronomical sources.
The Ellis wormhole was first introduced by Ellis as a spherically symmetric solution of Einstein equations with a ghost massless scalar field. The dynamical stability of the Ellis wormhole is discussed in Ref. [@Shinkai:2002gv] and the possible source to support the Ellis geometry was proposed in Ref. [@Das:2005un]. Gravitational lensing by the Ellis wormhole was studied in Refs. [@1984GReGr..16..111C; @1984IJTP...23..335C] and recently revisited by several authors[@Nakajima:2012pu; @Tsukamoto:2012zz]. So far, it has been suggested that Ellis wormholes can be probed by using light curves of gamma-ray bursts [@Torres:1998xd], micro-lensing [@Safonova:2001vz; @Bogdanov:2008zy; @Abe:2010ap] (see also Refs. [@Asada:2011ap; @Kitamura:2012zy]) and imaging observations [@Toki:2011zu; @Tsukamoto:2012xs], while our proposal is the use of spectroscopic observations to probe Ellis wormholes.
In order to fully investigate the lensed spectrum of a point source, the wave effect in gravitational lensing must be taken into account. The wave effect for the point mass lens is discussed in Refs. [@Deguchi:1986zz; @1992grle.book.....S]. Wave effects in gravitational lensing by the rotating massive object[@Baraldo:1999ny], binary system[@Mehrabi:2012dy], singular isothermal sphere[@Takahashi:2003ix] and the cosmic string[@Suyama:2005ez; @Yoo:2012dn] have been considered. In Sec. \[waveform\], we calculate the amplification factor of gravitational lensing by the Ellis wormhole taking the wave effect into account. The geometrical optics limit is analytically presented in Sec. \[GOA\]. The difference in the amplification factor between the point mass lens and the Ellis wormhole lens is discussed in Sec. \[obs\] based on observables. In Sec. \[obsconst\], possible observations to probe Ellis wormholes are listed. Sec. \[summary\] is devoted to a summary.
In this paper, we use the geometrized units in which the speed of light and Newton’s gravitational constant are both unity.
A Derivation of the Lensed Wave Form {#waveform}
====================================
The line element in the Ellis wormhole spacetime can be written by the following isotropic form: $${\mathrm{d}}s^2=-{\mathrm{d}}t^2+\left(1+\frac{a^2}{R^2}\right)^2\left({\mathrm{d}}R^2
+R^2{\mathrm{d}}\Omega^2\right),$$ where $R=a$ corresponds to the throat and we simply call $a$ the throat radius in this paper.[^1]
Assuming the thin lens approximation is valid, we consider the wormhole lens system shown in Fig. \[thinlens\].
![Lens system with thin lens approximation. S, L, and O represent the source, lens, and observer positions, respectively. The path SAB is a ray trajectory which is specified with the vector $\vec \xi$ on the lens plane $\Sigma_{\rm A}$. ${\rm B'}$ is the intersection of the line AO and the plane $\Sigma_{\rm B}$. $\vec \xi'$ is the position vector of the point B on the plane $\Sigma_{\rm B}$. []{data-label="thinlens"}](thinlens.eps)
We use the position vector $\vec X=(X,Y,Z)$ in the flat space. Then, the coordinate $R$ is given by $R=|\vec X-\vec X_{\rm L}|$, where $|\vec X_{\rm L}|$ is the lens position. We set $Z$-axis as the perpendicular direction to the lens plane and the source plane. ${D_{\rm S}}$, ${D_{\rm L}}$, and ${D_{\rm LS}}$ denote the distances from the observer plane to the source plane, from the observer plane to the lens plane, and from the lens plane to the source plane, respectively.
In the geometrical optics limit, we consider light rays emanated from the source. The vector $\vec \xi$ on the lens plane $\Sigma_{\rm A}$ in Fig. \[thinlens\] specifies the light ray which is deflected once at $\vec X=\vec X_{\rm L}+\vec \xi$. Since $\xi:=|\vec \xi|$ can be regarded as the closest approach of the light ray, as is shown in Ref. [@1984GReGr..16..111C], the deflection angle $\alpha$ is given by $$\alpha(\xi)=\pi\left(\frac{a}{\xi}\right)^2+\mathcal O\left(\frac{a}{\xi}\right)^4.$$ As a result, the Einstein radius $\xi_0$ for the Ellis wormhole is given by $$\xi_0=\left(\frac{1}{4}\pi a^2D\right)^{1/3},$$ where $$D=\frac{4{D_{\rm L}}{D_{\rm LS}}}{{D_{\rm S}}}.$$
Since we are interested in the wave effect, that is, the deviation from the geometrical optics limit, we need to treat the wave equation rather than light rays. Neglecting the polarization effect, we consider the scalar wave equation with the frequency $\omega$. The wave equation for the monochromatic wave ${{\rm e}}^{i\omega t}\phi(\vec X)$ is given by $$\omega^2\phi+\left(1+\frac{a^2}{R^2}\right)^{-3}{\partial}_i
\left[\left(1+\frac{a^2}{R^2}\right)\delta^{ij}{\partial}_j\phi\right]
=
-4\pi A ~\delta(\vec X-\vec X_{\rm S}),$$ where $\vec X_{\rm S}$ is the position vector of the point source and $A$ in the source term is a constant which specifies the amplitude. Without the wormhole, we obtain the wave form $\bar \phi_{\rm O}$ at the observer O as follows: $$\bar \phi_{\rm O}=\frac{A}{\sqrt{{D_{\rm S}}^2+\eta^2}}
\exp\left[i\omega \sqrt{{D_{\rm S}}^2+\eta^2}\right]
=
\frac{A}{{D_{\rm S}}}
\exp\left[i\omega {D_{\rm S}}\left(1+\frac{\eta^2}{2{D_{\rm S}}^2}
+\mathcal O\left(\left(\frac{\eta}{{D_{\rm S}}}\right)^4\right)\right)\right],$$ where $\eta=|\vec \eta|$ and $\vec \eta=(X_{\rm S}-X_{\rm L},Y_{\rm S}-Y_{\rm L},0)$ and we consider the case $\eta\ll {D_{\rm S}}$ in this paper.
Our assumptions to calculate the wave form at O are summarized as follows(see Ref. [@1992grle.book.....S]):
- The geometrical optics approximation is valid between the source plane and the plane $\Sigma_{\rm B}$ in Fig. \[thinlens\].
- Thin lens approximation is valid and a ray from the source is deflected once on the lens plane $\Sigma_{\rm A}$.
- Assuming ${D_{\rm S}}\sim{D_{\rm L}}\sim{D_{\rm LS}}\sim D$, we use a non-dimensional parameter $\epsilon$ defined by $\epsilon:=\xi_0/D$, which gives the typical scale of the deflection angle. Then, we assume $1/(\omega D) \ll \epsilon \ll 1$ and $\eta/D= \mathcal O(\epsilon)$.
- On the plane $\Sigma_{\rm B}$, the gravitational potential of the lens object is negligible and $\delta D/D= \mathcal O(\epsilon)$, where $\delta D$ is the distance between the planes $\Sigma_{\rm A}$ and $\Sigma_{\rm B}$.
On the assumptions made above, we calculate the wave form on the plane $\Sigma_{\rm B}$ up to the leading order for the amplitude and next leading terms for the phase part. Applying the Kirchhoff integral theorem between the plane $\Sigma_{\rm B}$ and the observer, we calculate the approximate wave form at O.
The vector $\vec \xi$ specifies the light ray which is deflected once on the plane $\Sigma_{\rm A}$ at $\vec X=\vec X_{\rm L}+\vec \xi$ and reaches the plane $\Sigma_{\rm B}$. The deflection angle is fixed by $\vec \xi$ and the background geometry. The point A in Fig. \[thinlens\] denotes the deflected point. We label the intersection of the deflected light ray and the plane $\Sigma_{\rm B}$ as B, while ${\rm B'}$ in the Fig. \[thinlens\] denotes the intersection of the line AO and the plane $\Sigma_{\rm B}$. As will be mentioned at the end of this section, the dominant contribution to the wave form at O comes from rays which satisfy $\xi\sim \epsilon D$, where $\xi=|\vec \xi|$. Therefore we consider $\vec \xi/D$ as $\mathcal O(\epsilon)$ hereafter.
First, we consider the following ansatz for $\phi$ in the region between the source plane and the plane $\Sigma_{\rm B}$: $$\phi=f(\vec X)~{{\rm e}}^{iS(\vec X)}.$$ On the plane $\Sigma_{\rm B}$, the amplitude $f(\vec X)$ is given by $$\left.f(\vec X)\right|_{\Sigma_{\rm B}}
=\frac{A}{{D_{\rm LS}}}\left(1+\mathcal O(\epsilon)\right).
\label{amplitude1}$$ In the geometrical optics approximation, the phase $S(\vec X)$ satisfies the eikonal equation given by $$\delta^{ij}{\partial}_i S{\partial}_j S =\omega^2\left(1+\frac{a^2}{R^2}\right)^2.$$ At the point B, the phase based on the source position S is given by the following integral $$\left.S\right|_{\rm B}=\int^{\rm B}_{\rm S}
\frac{{\mathrm{d}}x^i}{{\mathrm{d}}l}{\partial}_iSdl,
\label{sourceint1}$$ where we have introduced the optical path length $l$ defined as $$\delta_{ij}\frac{{\mathrm{d}}x^i}{{\mathrm{d}}l}\frac{{\mathrm{d}}x^j}{{\mathrm{d}}l}=1.$$ Since, in the geometrical optics approximation, we find $${\partial}_j S=\omega\left(1+\frac{a^2}{R^2}\right)\delta_{ij}\frac{{\mathrm{d}}x^i}{{\mathrm{d}}l},$$ the integral is given by $$\left.S\right|_{\rm B}=
\omega\int^{\rm B}_{\rm S} dl+\omega a^2\int^{\rm B}_{\rm S}\frac{1}{R^2} dl.
\label{sourceint2}$$ After the calculations explicitly shown in Appendix \[dereq\], we finally obtain the following expression: $$\left.S\right|_B\simeq\omega\left[
{D_{\rm S}}\left(1+\frac{\eta^2}{2{D_{\rm S}}^2}\right)+
\frac{{D_{\rm L}}{D_{\rm S}}}{2{D_{\rm LS}}}\left(\frac{\vec \xi}{{D_{\rm L}}}-\frac{\vec \eta}{{D_{\rm S}}}\right)^2
-r+\frac{\pi a^2}{\xi}\right].
\label{sourceint3}$$ This expression for the phase and Eq. for the amplitude can be used for any value of $\vec \xi$, that is, we have obtained an approximate wave form on the plane $\Sigma_{\rm B}$.
Applying the Kirchhoff integral theorem[@1999prop.book.....B] and neglecting the contribution from the infinity, we express the wave form $\phi_{\rm O}$ at O by the following integral: $$\phi_{\rm O}=-\frac{1}{4\pi}\int_{\Sigma_{\rm B}} {\mathrm{d}}\xi^2 \left\{
\phi_{\rm B}\frac{{\partial}}{{\partial}Z}\left(\frac{{{\rm e}}^{i\omega r}}{r}\right)
-\frac{{{\rm e}}^{i\omega r}}{r}\frac{{\partial}\phi_{\rm B}}{{\partial}Z} \right\},$$ where $\phi_{\rm B}$ is the waveform at B. Since we are interested in only the leading order of the amplitude, we obtain $$\phi_{\rm O}\simeq
-\frac{i\omega A}{2\pi {D_{\rm L}}{D_{\rm LS}}}
\exp\left[i\omega{D_{\rm S}}\left(1+\frac{\eta^2}{2{D_{\rm S}}^2}\right)\right]
\int {\mathrm{d}}\xi^2
\exp\left[i\omega\left\{
+\frac{{D_{\rm L}}{D_{\rm S}}}{2{D_{\rm LS}}}\left(\frac{\vec\xi}{{D_{\rm L}}}-\frac{\vec\eta}{{D_{\rm S}}}\right)^2
+\frac{\pi a^2}{\xi}\right\}\right],$$ where we have used the following approximations: $$\begin{aligned}
&&\frac{{\partial}}{{\partial}Z}\left(\frac{{{\rm e}}^{i\omega r}}{r}\right)\simeq
\frac{i\omega}{r}{{\rm e}}^{i\omega r}\simeq \frac{i\omega}{{D_{\rm L}}}{{\rm e}}^{i\omega r},
\\
&&\frac{{\partial}\phi_{\rm B}}{{\partial}Z}
\simeq \frac{-i\omega A}{{D_{\rm LS}}}{{\rm e}}^{i\left.S\right|_B}. \end{aligned}$$
Defining the amplification factor $F$ by $F:=\phi_{\rm O}/\bar \phi_{\rm O}$, we obtain $$F\simeq \frac{\omega d}{\pi i}\int {\mathrm{d}}x^2
\exp\left[i\omega d\left\{(\vec x-\vec y)^2+\frac{2}{x}\right\}\right],$$ where $$\begin{aligned}
d&=&\frac{\xi_0^2{D_{\rm S}}}{2{D_{\rm LS}}{D_{\rm L}}}=\frac{2\xi_0^2}{D}=2\xi_0\epsilon,
\label{d4wh}\\
\vec x&=&\frac{\vec\xi}{\xi_0}, \\
\vec y&=&\frac{\vec \eta{D_{\rm L}}}{\xi_0 {D_{\rm S}}}. \end{aligned}$$ $d$ gives the optical path difference between the lensed trajectory and the unlensed one in the geometrical optics limit for $\vec\eta=0$.
Introducing a polar coordinate, we rewrite this integral as $$\begin{aligned}
F&\simeq&
\frac{\omega d}{\pi i}{{\rm e}}^{i\omega d y^2}\int^\infty_0{\mathrm{d}}x
x\exp\left[i\omega d\left(x^2+\frac{2}{x}\right)\right]\int^{2\pi}_0{\mathrm{d}}\varphi
\exp\left[-2i\omega d xy \cos\varphi\right]
\label{intxphi}
\\
&=&
-2i\omega d{{\rm e}}^{i\omega dy^2}\int^\infty_0{\mathrm{d}}x
x\exp\left[i\omega d\left(x^2+\frac{2}{x}\right)\right]
J_0(2\omega dxy).
\label{intx}\end{aligned}$$ The integrand is divergent at the infinity on the real axis. This is caused by our approximation associated with $\epsilon$, and not real. If we write down the integrand in a precise form without any approximation, we do not have any divergence. Actually, in the precise form, the contribution from the integral in the region $x\gg 1$ is negligible due to the cancellation of the quasi-periodic integration. For the same reason, the contribution from the integration in the region $x\ll 1$ is also negligible. Since the approximate expression of Eq. is valid in the region $x\ll1/\epsilon$, we can obtain a reliable result by neglecting the contribution from the region $x\gg 1$ in the integral . Practically, to make this integral finite, it is convenient to consider the analytic continuation to the complex plane and take the path of the integral as shown in Fig. \[path\]. Then, this integral can be numerically performed.
![The path of the integral taken in the numerical integration. []{data-label="path"}](path.eps)
Geometrical Optics Approximation {#GOA}
================================
In this section, we derive an approximate form of the amplification factor $F$. In the expression , we apply the stationary phase approximation to the integral with respect to $x$. Then we obtain $$F\simeq\sqrt{\frac{\omega d}{\pi i}}\int^{2\pi}_0
{\mathrm{d}}\varphi \frac{x_0}{\sqrt{1+2/x_0^3}}\exp\left[i\omega d h(x_0)\right],
\label{geointx}$$ where $$h(x)=x^2-2xy\cos\varphi+\frac{2}{x}$$ and $x_{0}=x_{0}(\varphi)>0$ satisfies $$h'(x_0)=0\Leftrightarrow
x_0^3-x_0^2y\cos \varphi-1=0.
\label{x0eq}$$ Note that Eq. has only one positive root as a function of $\varphi$.
In Eq. , we again perform the stationary phase approximation in the integral with respect to $\varphi$, and we obtain $$\begin{aligned}
F\simeq F_{\rm geo}&=&
\Biggl(
\frac{x_+^3}{\sqrt{(x_+^3+2)(x_+^3-1)}}\exp\left[i\omega d\frac{-x_+^3+4}{x_+}\right]\cr
&&+
\frac{x_-^3}{\sqrt{(x_-^3+2)(1-x_-^3)}}\exp\left[i\omega d\frac{-x_-^3+4}{x_-}
-\frac{i\pi}{2}\right]\Biggr), \end{aligned}$$ where $x_\pm$ satisfies $$x_\pm^3\mp x_\pm^2y-1=0.$$ Note that $1<x_+$ and $0<x_-<1$. If we define $\mu_\pm$ and $\theta_\pm$ as $$\begin{aligned}
\mu_\pm&=&\frac{x_\pm^6}{(x_\pm^3+2)(x_\pm^3-1)}, \\
\theta_\pm&=&\omega d\frac{-x_\pm^3+4}{x_\pm}-\frac{\pi}{4}\pm\frac{\pi}{4}, \end{aligned}$$ $F_{\rm geo}$ can be expressed as $$F_{\rm geo}=\sum_\pm\sqrt{|\mu_\pm|}{{\rm e}}^{i\theta_\pm}.$$ $\mu_\pm$ is the magnification factor for each image in the geometrical optics approximation.
As an observable, we focus on $|F|^2$ in this paper. In the geometrical optics approximation, we obtain $$|F_{\rm geo}|^2=|\mu_+|+|\mu_-|+2\sqrt{|\mu_+\mu_-|}\sin(2\omega d \tau(y))
\label{maggeo}$$ with $\tau(y)$ being the following: $$\tau(y):=\frac{\theta_--\theta_++\pi/2}{2\omega d},$$ where note that the definition of $\tau$ can be written in terms of $x_\pm$, which is a function of $y$. $|F|^2$ and $|F_{\rm geo}|^2$ are depicted as functions of $\omega$ for each value of $y$ in Fig. \[waveformwh\].
![$|F|^2$ and $|F_{\rm geo}|^2$ for the wormhole lens, where $T=d\,\tau $. []{data-label="waveformwh"}](why2.eps "fig:") ![$|F|^2$ and $|F_{\rm geo}|^2$ for the wormhole lens, where $T=d\,\tau $. []{data-label="waveformwh"}](why1.eps "fig:")
![$|F|^2$ and $|F_{\rm geo}|^2$ for the wormhole lens, where $T=d\,\tau $. []{data-label="waveformwh"}](why05.eps "fig:") ![$|F|^2$ and $|F_{\rm geo}|^2$ for the wormhole lens, where $T=d\,\tau $. []{data-label="waveformwh"}](why01.eps "fig:")
Comparison with the point mass lens based on observables {#obs}
========================================================
In this paper, we assume the following situations for the observation:
- We can observe the spectrum of a source.
- The unlensed spectrum shape is well known.
Note that, in our analysis, knowledge about the luminosity is not necessary. The amplification factor for the point mass lens is summarized in Appendix \[pml\]. For both point mass and wormhole cases, in the geometrical optics approximation, we obtain the form of Eq. or equivalently Eq. .
In the frequency region where the geometrical optics approximation is valid, there are basically three observables which characterize the form of the amplification factor. The first is the frequency $\omega$, the second is the period of the oscillation of the spectrum as a function of $\omega$, and the third is the ratio $\kappa$ between the amplitude of the oscillation and the mean value. The period of the oscillation of the spectrum makes $T:=\tau(y)d$ an observable. $\kappa$ is given by $$\kappa=\frac{2\sqrt{|\mu_+\mu_-|}}{|\mu_+|+|\mu_-|}$$ from Eq. and plotted as a function of $y$ as is shown in Fig. \[ratio\].
Since $\kappa$ is observable, $y$ and hence $\tau(y)$ can be determined if we have enough accuracy of the observation. As shown in Fig. \[isou\], $\tau(y)$ is a monotonically increasing function of $y$ and close to $2y$ in the region $y<1$.
Then, from another observable $T=\tau(y) d$ we can obtain the value of $d$.
The situation for the point mass lens case is the same as for the Ellis wormhole case. That is, the three observables $\omega$, $T$, and $\kappa$ can be regarded as gravitational lensing by a point mass as well as a wormhole. This fact indicates that we cannot distinguish which is the lens object only by using these three observables in the geometrical optics approximation. This degeneracy is resolved in the small frequency region in which the wave effect becomes significant as is explicitly shown in Fig. \[compare\].
![Comparison between $|F|^2$ and $|F^{\rm p}|^2$. []{data-label="compare"}](compare2.eps "fig:") ![Comparison between $|F|^2$ and $|F^{\rm p}|^2$. []{data-label="compare"}](compare1.eps "fig:")
To simply see this resolution of the degeneracy, we consider the small frequency limit, i.e., $\omega\rightarrow0$. In this limit, we have $F\rightarrow 1$ as explicitly shown in Figs. \[waveformwh\] and \[waveformpo\]. If we can observe the spectrum in any frequency region of interest, the following quantity is an observable: $$\frac{\displaystyle \lim_{\omega\to 0}|F|^2}{\displaystyle \lim_{\omega\to \infty}<|F|^2>}
=1/\mu_{\rm tot}:=1/(|\mu_+|+|\mu_-|),$$ where the bracket $<~>$ denotes the average through several periods. We depict $1/\mu_{\rm tot}$ as a function of $y$ for both the wormhole case and the point mass lens case in Fig. \[magratio\].
![$1/\mu_{\rm tot}$ []{data-label="magratio"}](magratio.eps)
In both cases, $1/\mu_{\rm tot}$ approaches to 0 and 1 in the limits $y\to 0$ and $y\to \infty$, respectively. While $1/\mu_{\rm tot}<1$ is satisfied in all domain of $y$ for the point mass lens, $1/\mu_{\rm tot}$ can exceed 1 for the wormhole case due to the demagnification effect originated from the negative mass density surrounding the Ellis wormhole. The behaviours of $1/\mu_{\rm tot}$ are totally different from each other. This fact shows that they are, in principle, distinguishable.
Observational constraint {#obsconst}
========================
One of the possible sources is gamma-ray bursts, which have been proposed to be used for probing small mass primordial black holes[@1992ApJ...386L...5G; @1993ApJ...413L...7S; @Ulmer:1994ij; @Marani:1998sh] and low tension cosmic strings[@Yoo:2012dn]. Recently, the femto-lensing effects caused by compact objects were searched by using gamma-ray bursts with known redshifts detected by the Fermi Gamma-ray Burst Monitor[@Barnacka:2012bm]. From non-detection of the femto-lensing event, a constraint on the number density of compact objects have been obtained. For a fixed value of the mass $M$ of each compact object, the constraint can be translated into a constraint on $\Omega_{\rm CO}$, where $\Omega_{\rm CO}$ is the average energy density of compact objects in the unit of the critical density $\rho_{\rm cr}$. Then, the abundance of dark compact objects is constrained as $\Omega_{\rm CO}<0.15$ at the 95% confidence level for $M\sim 3\times10^{18}{\rm g}$.
Since the number density $n$ of the compact objects is given by $$n=\frac{\Omega_{\rm CO}\rho_{\rm cr}}{M}
\sim 2\times 10^{-9}{\rm AU}^{-3}
\left(\frac{\Omega_{\rm CO}}{0.15}\right)
\left(\frac{M}{3\times 10^{18}{\rm g}}\right)^{-1}
\left(\frac{\rho_{\rm cr}}{10^{-29}{\rm g/cm^3}}\right),$$ we obtain the constraint for the number dnsity of compact objects with $M\sim 3\times10^{18}{\rm g}$ as $n<2\times 10^{-9}{\rm AU}^{-3}$. Then, we can expect a similar constraint on the number density of Ellis wormholes with the throat radius which gives the same value of $d$ as that for the compact object. The mass of $3\times 10^{18}{\rm g}$ gives $d\sim 5\times10^{-10}{{\rm cm}}$ from Eq. and, from Eq. , the corresponding throat radius $a$ is given by $$a\sim 0.7 {{\rm cm}}\left(\frac{d}{5\times 10^{-10}{{\rm cm}}}\right)^{3/4}
\left(\frac{D}{10^{28}{{\rm cm}}}\right)^{1/4}.$$ Therefore, the number density of Ellis wormholes with $a\sim 1{{\rm cm}}$ must satisfy $n\lesssim 10^{-9}{\rm AU}^{-3}$. Note that this constraint comes from the wave form in the geometrical optics approximation and hence we do not distinguish between the point mass lenses and Ellis wormholes.
Another possible observation to probe Ellis wormholes is the observation of gravitational waves from compact object binaries. The unlensed wave form of the gravitational waves from a compact object binary is well known. From the chirp signal in the inspiral phase, we can obtain the spectrum of the gravitational waves. In order to distinguish the Ellis wormhole from a point mass lens, we need to observe not only the typical interference pattern but also the wave effect in the lensed spectrum. Hence, the spectrum in $d\sim \lambda:=2\pi/\omega$ is necessary to probe the Ellis wormhole. Assuming $d\sim \lambda$, the typical throat radius of Ellis wormholes which can be probed by using gravitational waves is estimated as follows: $$\begin{aligned}
a=d^{3/4}\left(\frac{2D}{\pi^2}\right)^{1/4}
&\sim&\lambda^{3/4}
\left(\frac{2D}{\pi^2}\right)^{1/4}\cr
&\sim&7\times 10^{12} {{\rm cm}}\left(\frac{\lambda}{10^8{{\rm cm}}}\right)^{3/4}
\left(\frac{D}{10^{28}{{\rm cm}}}\right)^{1/4}. \end{aligned}$$
The same estimate is applicable for galactic sources of electro-magnetic waves. We obtain $a\sim10^5{{\rm cm}}$ for galactic radio sources with $\lambda \sim 1{{\rm cm}}$, $a\sim 1$m for galactic optical or infra-red sources and $a\sim 1$cm for galactic X-ray sources. The source must be compact enough to show the clear oscillation behaviour in the spectrum. This fact can be clearly understood by considering the $y$ dependence of the phase in the amplification factor . Since $\tau(y)$ is roughly approximated by $2y$, the period $\delta y$ for one cycle is given by $\delta y\sim \pi/(\omega d)$. The corresponding length scale $\delta \eta$ on the source plane is given by $$\begin{aligned}
\delta \eta
=\delta y \frac{{D_{\rm S}}}{{D_{\rm L}}}\xi_0
&\sim&\frac{\pi}{\omega}\sqrt{\frac{2{D_{\rm LS}}{D_{\rm S}}}{d{D_{\rm L}}}}\cr
&\sim&\sqrt{\frac{\lambda {D_{\rm LS}}{D_{\rm S}}}{2{D_{\rm L}}}}
\sim2\times 10^{11}{{\rm cm}}\left(\frac{\lambda}{1{{\rm cm}}}\right)^{1/2}
\left(\frac{{D_{\rm LS}}{D_{\rm S}}/{D_{\rm L}}}{10 {\rm kpc}}\right)^{1/2}, \end{aligned}$$ where we have assumed $d\sim\lambda$. If the source radius is larger than $\delta \eta$, the interference pattern will be smeared out. Observation of compact galactic sources such as pulsars and white dwarfs might be useful to probe not only dark compact objects but also exotic compact objects such as the Ellis wormhole.
summary
=======
In this paper, we have proposed the probe of Ellis wormholes by using spectroscopic observations. We have assumed that the spectrum of the target source can be measured in enough accuracy and the spectrum shape is well known without lensing, but the luminosity is not necessarily observable. Then, we have discussed the distinguishability of the lensed spectrum from the case of the point mass lens.
We have derived the wave form after the scattering by the Ellis wormhole including the wave effect in the low frequency domain. The geometrical optics limit of the wave form has been also analytically derived. Then, we have found that the Ellis wormhole cannot be distinguished from the point mass lens by using only the high frequency domain in which the geometrical optics approximation is valid. We have also found that this degeneracy is resolved in the low frequency domain in which the wave effect is significant. Possible observational constraints are also discussed and we estimated the upper bound for the number density of Ellis wormholes as $n\lesssim 10^{-9}{\rm AU}^{-3}$ with throat radius $a\sim 1{{\rm cm}}$ from the existing femto-lensing analysis for compact objects.
Finally, we note that our method to probe the Ellis wormhole is complementary to the other methods to probe the Ellis wormhole with micro-lensing [@Safonova:2001vz; @Bogdanov:2008zy; @Abe:2010ap] or the astrometric image centroid displacements[@Toki:2011zu]. These are not feasible for observations on cosmological scales because the time scale of the lens event is too long to detect modulation of the light curve or the displacements. In contrast, the slow relative motion is an advantage for spectroscopic observations. Therefore we may probe the Ellis wormhole on cosmological scales using our method.
Acknowledgements {#acknowledgements .unnumbered}
================
CY is supported by a Grant-in-Aid through the Japan Society for the Promotion of Science (JSPS). The work of NT was supported in part by Rikkyo University Special Fund for Research. TH was supported by the Grant-in-Aid for Young Scientists (B) (No. 21740190) and the Grant-in-Aid for Challenging Exploratory Research (No. 23654082) for Scientific Research Fund of the Ministry of Education, Culture, Sports, Science and Technology, Japan.
Derivation of Eq. {#dereq}
==================
The first term in Eq. can be evaluated as follows. $$\begin{aligned}
\int^{\rm B}_{\rm S} dl=
|\overrightarrow{\rm SA}|+|\overrightarrow{\rm AB}|
&=&|\overrightarrow{\rm SA}|+|\overrightarrow{\rm OA}-\overrightarrow{\rm OB}|\cr
&=&|\overrightarrow{\rm SA}|
+\sqrt{|\overrightarrow{\rm OA}|^2+|\overrightarrow{\rm OB}|^2
-2\overrightarrow{\rm OA}\cdot\overrightarrow{\rm OB}}. \end{aligned}$$ $\overrightarrow{\rm OB}$ and $\overrightarrow{\rm OB'}$ can be written as $$\begin{aligned}
\overrightarrow{\rm OB}&=&
\left[\left(1-\frac{\delta D}{{D_{\rm L}}}\right)
\overrightarrow{\rm OL}
+\vec \eta+\left(\vec \xi-\vec \eta\right)\frac{{D_{\rm LS}}+\delta D}{{D_{\rm LS}}}
-\alpha\delta D\frac{\vec \xi}{\xi}\right](1+\mathcal O(\epsilon^3))\cr
&=&\left[\left(1-\frac{\delta D}{{D_{\rm L}}}\right)
\overrightarrow{\rm OL}
+\vec \xi -\alpha(\xi)\delta D \frac{\vec \xi}{\xi}
+\left(\vec \xi-\vec \eta\right)\frac{\delta D}{{D_{\rm LS}}}\right](1+\mathcal O(\epsilon^3)),\\
\overrightarrow{\rm OB'}&=&\left(1-\frac{\delta D}{{D_{\rm L}}}\right)\overrightarrow{\rm OA}. \end{aligned}$$ From these expressions, we can find $$\begin{aligned}
|\overrightarrow{\rm OB}|&=&
|\overrightarrow{\rm OB'}|\left(1+\mathcal O(\epsilon^3)\right), \\
\overrightarrow{\rm OA} \cdot \overrightarrow{\rm OB}
&=&\overrightarrow{\rm OA} \cdot \overrightarrow{\rm OB'}
\left(1+\mathcal O(\epsilon^3)\right). \end{aligned}$$ Therefore we obtain $$\int^{\rm B}_{\rm S} dl=
\left(|\overrightarrow{\rm SA}|+|\overrightarrow{\rm OA}|
-|\overrightarrow{\rm OB'}|\right)\left(1+\mathcal O(\epsilon^3)\right).$$ Since we find $$\begin{aligned}
|\overrightarrow{\rm SA}|
&=&\sqrt{|\vec \xi-\vec \eta|^2+{D_{\rm LS}}^2}
={D_{\rm LS}}\left(1+\frac{|\vec \xi-\vec \eta|^2}{2{D_{\rm LS}}^2}+\mathcal O(\epsilon^4)\right), \\
|\overrightarrow{\rm AO}|&=&
{D_{\rm L}}\left(1+\frac{\xi^2}{2{D_{\rm L}}^2}+\mathcal O(\epsilon^4)\right), \end{aligned}$$ we obtain the following expression: $$\int^{\rm B}_{\rm S} dl=
\left[
{D_{\rm S}}\left(1+\frac{\eta^2}{2{D_{\rm S}}^2}\right)+
\frac{{D_{\rm L}}{D_{\rm S}}}{2{D_{\rm LS}}}\left(\frac{\vec \xi}{{D_{\rm L}}}-\frac{\vec \eta}{{D_{\rm S}}}\right)^2
-r\right]\left(1+\mathcal O(\epsilon^3)\right),$$ where $r=|\overrightarrow{\rm OB'}|$.
In order to evaluate the second term in Eq. , we first consider the integral between S and A. Letting P be a point on the segment SA, we obtain $$|\overrightarrow{\rm LP}|^2=|\vec \xi+\overrightarrow{\rm AP}|^2
=\left|\vec \xi+\left(1-\frac{l}{l_{\rm SA}}\right) \overrightarrow{\rm AS}\right|^2
=\xi^2+\left(l_{\rm SA}-l\right)^2+2\left(1-\frac{l}{l_{\rm SA}}\right)\vec \xi\cdot
\overrightarrow{\rm AS},$$ where $l_{\rm SA}=|\overrightarrow{\rm SA}|$ and $l=|\overrightarrow{\rm SP}|$. Since $\vec \xi\cdot \overrightarrow{\rm AS}=-\vec \xi\cdot (\vec \xi-\vec \eta)$, we obtain the following expression: $$|\overrightarrow{\rm LP}|^2
=\xi^2+\left(l_{\rm SA}-l\right)^2-2\left(1-\frac{l}{l_{\rm SA}}\right)
\vec \xi \cdot (\vec \xi-\vec \eta).$$ Substituting the above expression of $|\overrightarrow{\rm LP}|^2$ into $R^2$ of the second integral in Eq. with the integral region being from S to A, we obtain $$\begin{aligned}
\int^{\rm A}_{\rm S} \frac{1}{R^2} dl=\int^{l_{\rm SA}}_{0}
\frac{dl}{\xi^2+(l_{\rm SA}-l)^2-2(1-\frac{l}{l_{\rm SA}})
\vec \xi \cdot (\vec \xi-\vec \eta)}.
\label{secondint1}\end{aligned}$$ Then the integral can be performed and evaluated as $$\begin{aligned}
\int^{l_{\rm SA}}_{0}
\frac{dl}{(l_{\rm SA}-l)^2+\xi^2-2(1-\frac{l}{l_{\rm SA}})
\vec \xi \cdot (\vec \xi-\vec \eta)}
&=&\frac{l_{\rm SA}}{\xi {D_{\rm LS}}}\left[\arctan\left(
\frac{-l_{\rm SA}^2+\vec \xi \cdot (\vec \xi-\vec \eta)
+l l_{\rm SA}}{\xi {D_{\rm LS}}}\right)\right]^{l_{\rm SA}}_0\cr
&=&\left(\frac{\pi}{2\xi}-\frac{\vec\xi\cdot\vec\eta}{\xi^2 {D_{\rm LS}}}\right)\left(1+\mathcal O(\epsilon^2)\right). \end{aligned}$$ The contribution from the integral between A and B can be also evaluated by the similar integral. Finally, we obtain the expression .
Point Mass Lens {#pml}
===============
For the point mass case, we have the following expression for the amplification factor [@Deguchi:1986zz; @1992grle.book.....S]: $$\left|F^{\rm po}\right|=\left|{{\rm e}}^{\pi \omega d/2}\Gamma\left(1-i\omega d\right)
{}_1F_1\left(i\omega d,1;i\omega dy\right)\right|,$$ where $\Gamma$ and ${}_1F_1$ are the gamma function and the confluent hyper-geometric function, respectively, and $$d=2M,
\quad
y=\eta\sqrt{\frac{D_{\rm L}}{4MD_{\rm LS}D_{\rm S}}}.
\label{dy4po}$$
In the geometrical optics approximation ($\omega\rightarrow \infty$), we obtain $$\left|F^{\rm po}\right|^2\rightarrow\left|F^{\rm po}_{\rm geo}\right|^2
:=\left|\mu^{\rm po}_+\right|+\left|\mu^{\rm po}_-\right|
+2\sqrt{\left|\mu^{\rm po}_+\mu^{\rm po}_-\right|}\sin(2\omega d \tau_{\rm po}(y)),
\label{eq:pmgeo}$$ where $$\begin{aligned}
\mu^{\rm po}_\pm&=&\pm \frac{1}{4}\left[\frac{y}{\sqrt{y^2+4}}
+\frac{\sqrt{y^2+4}}{y}\pm2\right], \\
\tau_{\rm po}(y)&=&\frac{1}{2}y\sqrt{y^2+4}
+\ln\frac{\sqrt{y^2+4}+y}{\sqrt{y^2+4}-y}. \end{aligned}$$ $|F^{\rm po}|^2$ and $|F^{\rm po}_{\rm geo}|^2$ are depicted as functions of $\omega$ for each value of $y$ in Fig. \[waveformpo\].
![ $|F^{\rm po}|^2$ and $|F^{\rm po}_{\rm geo}|^2$ as functions of $\omega$ for each value of $y$, where $T=d\,\tau_{\rm po}$. []{data-label="waveformpo"}](poy2.eps "fig:") ![ $|F^{\rm po}|^2$ and $|F^{\rm po}_{\rm geo}|^2$ as functions of $\omega$ for each value of $y$, where $T=d\,\tau_{\rm po}$. []{data-label="waveformpo"}](poy1.eps "fig:")
![ $|F^{\rm po}|^2$ and $|F^{\rm po}_{\rm geo}|^2$ as functions of $\omega$ for each value of $y$, where $T=d\,\tau_{\rm po}$. []{data-label="waveformpo"}](poy05.eps "fig:") ![ $|F^{\rm po}|^2$ and $|F^{\rm po}_{\rm geo}|^2$ as functions of $\omega$ for each value of $y$, where $T=d\,\tau_{\rm po}$. []{data-label="waveformpo"}](poy01.eps "fig:")
[10]{}
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[^1]: Since the throat surface area is given by $16\pi a^2$, our definition of the throat radius is half the areal radius of the throat.
|
---
abstract: 'To achieve sparse parametrizations that allows intuitive analysis, we aim to represent deformation with a basis containing *interpretable* elements, and we wish to use elements that have the description capacity to represent the deformation *compactly*. To accomplish this, we introduce in this paper *higher-order momentum distributions* in the LDDMM registration framework. While the zeroth order moments previously used in LDDMM only describe local displacement, the first-order momenta that are proposed here represent a basis that allows local description of affine transformations and subsequent compact description of non-translational movement in a globally non-rigid deformation. The resulting representation contains directly interpretable information from both mathematical and modeling perspectives. We develop the mathematical construction of the registration framework with higher-order momenta, we show the implications for *sparse* image registration and deformation description, and we provide examples of how the parametrization enables registration with a very low number of parameters. The capacity and interpretability of the parametrization using higher-order momenta lead to natural modeling of articulated movement, and the method promises to be useful for quantifying ventricle expansion and progressing atrophy during Alzheimer’s disease.'
author:
- 'Stefan Sommer[^1]'
- 'Mads Nielsen$\,^{*,}$[^2]'
- 'Sune Darkner$\,^*$'
- 'Xavier Pennec[^3]'
bibliography:
- 'bibliography.bib'
title: 'Higher-Order Momentum Distributions and Locally Affine LDDMM Registration'
---
LDDMM, diffeomorphic registration, RHKS, kernels, momentum, computational anatomy
65D18, 65K10, 41A15
Introduction
============
In many image registration applications, we wish to describe the deformation using as few parameters as possible and with a representation that allows intuitive analysis: we search for parametrizations with basis elements that have the *capacity* to describe deformation *sparsely* while being directly *interpretable*. For instance, we wish to use such a representation to compactly describe the progressive atrophy that occurs in the human brain suffering from Alzheimer’s disease and that can be detected by the expansion of the ventricles [@jack_medial_1997; @fox_presymptomatic_1996].
Image registration algorithms often represent translational movement in a dense sampling of the image domain. Such approaches fail to satisfy the above goals: low dimensional deformations such as expansion of the ventricles will not be represented sparsely; the registration algorithm must optimize a large number of parameters; and the expansion cannot easily be interpreted from the registration result.
In this paper, we use *higher-order momentum distributions* in the LDDMM registration framework to obtain a deformation parametrization that increases the *capacity* of sparse approaches with a basis consisting of *interpretable* elements. We show how the higher-order representation model locally affine transformations, and we use the compact deformation description to register points and images using very few parameters. We illustrate how the deformation coded by the higher-order momenta can be directly interpreted and that it represents information directly useful in applications: with low numbers of control points, we can detect the expanding ventricles of the patient shown in Figure \[fig:atrophy1\].
Background
----------
Most of the methods for non-rigid registration in medical imaging model the displacement of each spatial position by either a combination of suitable basis functions for the displacement itself or for the velocity of the voxels. The number of control points vary between one for each voxel [@arsigny_log-euclidean_2006; @hernandez_registration_2009; @christensen_deformable_2002] and fewer with larger basis functions [@rueckert_nonrigid_1999; @bookstein_linear_1999; @durrleman_optimal_2011]. For all methods, the infinite-dimensional space of deformations is approximated by the finite- but high-dimensional subspace spanned by the parametrization of the individual method. The approximation will be good if the underlying deformation is close to this subspace, and the representation will be compact, if few basis functions describe the deformation well. The choice of basis functions play a crucial role, and we will in the rest of the paper denote them *deformation atoms*. Two main observations constitute the motivation for the work presented in this paper:
*Observation 1: Order of the Deformation Model.* In the majority of registration methods, the deformation atoms model the local *translation* of each point. We wish a richer representation which is in particular able to model locally linear components in addition to local translations. The Polyaffine and Log-Euclidean Polyaffine [@arsigny_polyrigid_2005; @arsigny_fast_2009] frameworks pursue this by representing the velocity of a path of deformations locally by matrix logarithms. Ideas from the Polyaffine methods have recently been incorporated in e.g. the Demons algorithm [@vercauteren_diffeomorphic_2009] but, to the best of our knowledge, not in the LDDMM registration framework. We wish to extend the set of deformation atoms used in LDDMM to allow representation of *first-* and *higher-order* structure and hence incorporate the benefits of the Polyaffine methods in the LDDMM framework.
*Observation 2: Order of the Similarity Measure.* When registering DT images, the reorientation is a function of the derivative of the warp; curve normals also contain directional information which is dependent on the warp derivative and airway trees contain directional information in the tree structure which can be used for measuring similarity. These are examples of similarity measures containing *higher-order information*. For the case of image registration, the warp derivative may also enter the equation either directly in the similarity measure [@roche_rigid_2001; @pluim_image_2000] or to allow use of more image information than provided by a sampling of the warp. Consider an image similarity measure on the form $U(\phi)=\int_\Omega F(I_m(\phi^{-1}(x)),I_f(x))dx$. A finite sampling of the domain $\Omega$ can approximate this with $$\tilde{U}^0(\phi)
=
\frac{1}{N}\sum_{k=1}^NF(I_m(\phi^{-1}(x_k)),I_f(x_k))
\ .$$ Letting $\{p_1,\ldots,p_P\}$ be uniformly distributed points around $0$, we can increase the amount of image information used in $\tilde{U}^0(\phi)$ *without* additional sampling of the warp by using a first-order approximation of $\phi^{-1}$: $$\tilde{U}^1(\phi)
=
\frac{1}{NP}\sum_{k=1}^N\sum_{l=1}^NF(I_m(D\phi^{-1}p_l+\phi^{-1}(x_k)),I_f(p_l+x_k))
\ .$$ This can be considered an increase from *zeroth* to *first-order* in the approximation of $U$. Besides including more image information than provided by the initial sampling of the warp, the increase in order allows capture of non-translational information - e.g. rotation and dilation - in the similarity measure. The approach can be seen as a specific case of similarity smoothing; more examples of smoothing in intensity based image registration can be found in [@darkner_generalized_2011]. \
We focus on deformation modeling with the Large Deformation Diffeomorphic Metric Mapping (LDDMM) registration framework which has the benefit of both providing good registrations and drawing strong theoretical links with Lie group theory and evolution equations in physical modeling [@cotter_singular_2006; @younes_shapes_2010]. Most often, high-dimensional voxel-wise representations are used for LDDMM although recent interest in *compact* representations [@durrleman_optimal_2011; @sommer_sparsity_2012] show that the number of parameters can be much reduced. These methods use interpolation of the velocity field by deformation atoms to represent translational movement but deformation by other parts of the affine group cannot be compactly represented.
The deformation atoms are called *kernels* in LDDMM. The kernels are centered at different spatial positions and parameters determine the contribution of each kernel. In this paper, we use the partial derivative reproducing property [@zhou_derivative_2008] to show that partial derivatives of kernels fit naturally in the LDDMM framework and constitute deformation atoms along with the original kernels. In particular, these deformation atoms have a singular *higher-order momentum* and the momentum stays singular when transported by the EPDiff evolution equations. We show how the higher-order momenta allow modeling locally affine deformations, and they hence extend the capacity of sparsely discretized LDDMM methods. In addition, they comprise the natural vehicle for incorporating first-order similarity measures in the framework.
Related Work {#sec:related}
------------
A number of methods for non-rigid registration have been developed during the last decades including non-linear elastic methods [@pennec_riemannian_2005], parametrizations using static velocity fields [@arsigny_log-euclidean_2006; @hernandez_registration_2009], the demons algorithm [@thirion_image_1998; @vercauteren_diffeomorphic_2009], and spline-based methods [@rueckert_nonrigid_1999; @bookstein_linear_1999]. For the particular case of LDDMM, the groundbreaking work appeared with the deformable template model by Grenander [@grenander_general_1994] and the flow approach by Christensen et al. [@christensen_deformable_2002] together with the theoretical contributions of Dupuis et al. and Trouvé [@dupuis_variational_1998; @trouve_infinite_1995]. Algorithms for computing optimal diffeomorphisms have been developed in [@beg_computing_2005], and [@vaillant_statistics_2004] uses the momentum representation for statistics and develops a momentum based algorithm for the landmark matching problem.
Locally affine deformations can be modeled using the Polyaffine and Log-Euclidean Polyaffine [@arsigny_polyrigid_2005; @arsigny_fast_2009] frameworks. The velocity of a path of deformations is here computed using matrix logarithms, and the resulting diffeomorphism flowed forward by integrating the velocity. Ideas from the Polyaffine methods have recently been incorporated in e.g. the Demons algorithm [@vercauteren_diffeomorphic_2009; @seiler_geometry-aware_2011]. In LDDMM, the deformation atoms, the kernels, represent translational movement and the non-translational part of affine transformations cannot directly be represented. We will show how partial derivatives of kernels constitute deformation atoms which allow representing the linear parts of affine transformations. From a mathematical point of view, this is possible due to the partial derivative reproducing property (Zhou [@zhou_derivative_2008]). The partial derivative reproducing property, partial derivatives of kernels, and first-order momenta have previously been used in [@yan_cao_large_2005] to derive variations of flow equations for LDDMM DTI registration, in [@garcin_thechniques_2005] to match landmarks with vector features, and in [@glaunes_transport_2005] to match surfaces with currents. Confer the monograph [@younes_shapes_2010] for information on RKHSs and their role in LDDMM.
In order to reduce the dimensionality of the parametrization used in LDDMM, Durrleman et al. [@durrleman_optimal_2011] introduced a control point formulation of the registration problem by choosing a finite set of control points and constraining the momentum to be concentrated as Dirac measures at the point trajectories. As we will see, higher-order momenta make a finite control point formulation possible which is different in important aspects. Younes [@younes_constrained_2011] in addition considers evolution in constrained subspaces.
Higher-order momenta increase the capacity of the deformation parametrization, a goal which is also treated in sparse multi-scale methods such as the kernel bundle framework [@sommer_sparsity_2012]. This method concerns the size of the kernel in contrast to the order which we deal with here. As we will discuss in the experiments section, the size of the kernel is important when using the higher-order representation as well, and representations using higher-order momenta will likely complement the kernel bundle method if applied together.
Content and Outline
-------------------
We start the paper with an overview of LDDMM registration and the mathematical constructs behind the method. In the following section, we describe registration using higher-order image information and parameterization using higher-order momentum distributions. We then turn to the mathematical background of the method and describe the evolution of the momentum and velocity fields governed by the EPDiff evolution equations in the first-order case. The next sections describes the relation to polyaffine approaches, the effect of varying the initial conditions, and the backwards gradient transport. We then provide examples and illustrate how the deformation represented by first-order atoms can be interpreted when registering human brains with progressing atrophy. The paper ends with concluding remarks and outlook.
LDDMM Registration, Kernels, and Evolution Equations
====================================================
In the LDDMM framework, registration is performed through the action of diffeomorphisms on geometric objects. This approach is very general and allows the framework to be applied to both landmarks, curves, surfaces, images, and tensors. In the case of images, the action of a diffeomorphism $\phi$ on the image $I:\Omega\rightarrow\RR$ takes the form $\phi.I=I\circ\phi^{-1}$, and given a fixed image $I_f$ and moving image $I_m$, the registration amounts to a search for $\phi$ such that $\phi.I_m\sim I_f$. In exact matching, we wish $\phi.I_m$ be exactly equal to $I_f$ but, more frequently, we allow some amount of inexactness to account for noise in the images and allow for smoother diffeomorphisms. This is done by defining a similarity measure $U(\phi)=U(\phi.I_m,I_f)$ on images and a regularization measure $E_1$ to give a combined energy $$E(\phi)=E_1(\phi)+\lambda U(\phi.I_m,I_f)
\label{eq:func-lddmm}
\ .$$ Here $\lambda$ is a positive real representing the trade-off between regularity and goodness of fit. The similarity measure $U$ is in the simplest form the $L^2$-error $\int_\Omega |\phi.I_m(x)-I_f(x)|^2dx$ but more advanced measures can be used (e.g. [@roche_correlation_1998; @wells_multi-modal_1996; @darkner_generalized_2011]).
In order to define the regularization term $E_1$, we introduce some notations in the following: Let the domain $\Omega$ be a subset of $\RR^d$ with $d=2,3$, and let $V$ denote a Hilbert space of vector fields $v: \Omega\to \RR^d$ such that $V$ with associated norm $\|\cdot\|_V$ is included in $L^2(\Omega,\RR^d)$ and admissible [@younes_shapes_2010 Chap. 9], i.e. sufficiently smooth. Given a time-dependent vector field $t\mapsto v_t$ with $$\label{eq:gv_energy}
\int_0^1\|v_t\|^2_V\,dt < \infty$$ the associated differential equation $\partial_t\varphi_{t} = v_t\circ\varphi_{t}$ has with initial condition $\phi_s$ a diffeomorphism $\phi^v_{st}$ as unique solution at time $t$. The set $G_V$ of diffeomorphisms built from $V$ by such differential equations is a Lie group, and $V$ is its tangent space at the identity. Using the group structure, $V$ is isomorphic to the tangent space at each point $\phi\in G_V$. The inner product on $V$ associated to a norm $\|\cdot\|_V$ makes $G_V$ a Riemannian manifold with right-invariant metric. Setting $\phi^v_{00} =
\Id_\Omega$, the map $t\mapsto
\varphi^v_{0t}$ is a path from $\Id_\Omega$ to $\phi$ with energy given by and generated by $v_t$. We will use this notation extensively in the following. A critical path for the energy is a geodesic on $G_V$, and the regularization term $E_1$ is defined using the energy by $$E_1(\phi)
=
\min_{v_t\in V,\phi^v_{01}=\phi}\int_0^1\left\|v_s\right\|_V^2ds
\ ,
\label{eq:reg-lddmm}$$ i.e. it measures the minimal energy of diffeomorphism paths from $\Id_\Omega$ to $\phi$. Since the energy is high for paths with great variation, the term penalizes highly varying paths, and a low value of $E_1(\phi)$ thus implies that $\phi$ is regular.
Kernel and Momentum {#sec:kermom}
-------------------
As a consequence of the assumed admissibility of $V$, the evaluation functionals $\delta_x:v\mapsto v(x)\in \RR^d$ is well-defined and continuous for any $x\in \Omega$. Thus, for any $z\in\RR^d$ the map $z\otimes\delta_x: v\mapsto z^Tv(x)$ belongs to the topological dual $V^*$, i.e. the continuous linear maps on $V$. This in turn implies the existence of spatially dependent matrices $K:\Omega\times\Omega\to
\RR^{d\times d}$, the *kernel*, such that, for any constant vector $z\in\RR^d$, the vector field $K(\cdot,x)z\in V$ represents $z\otimes\delta_x$ and $\ip{K(\cdot,x)z,v}_V=z\otimes \delta_x(v)$ for any $v\in V$, point $x\in\Omega$ and vector $z\in\RR^d$. This latter property is denoted the reproducing property and gives $V$ the structure of a reproducing kernel Hilbert space (RKHS). Tightly connected to the norm and kernels is the notion of *momentum* given by the linear momentum operator $L:V\to V^*\subset L^2(\Omega,\RR^d)$ which satisfies $$\ip{Lv,w}_{L^2(\Omega,\RR^d)}
=
\int_\Omega \big(Lv(x)\big)^Tw(x)dx
=
\ip{v,w}_V$$ for all $v,w\in V$. The momentum operator connects the inner product on $V$ with the inner product in $L^2(\Omega,\RR^d)$, and the image $Lv$ of an element $v\in V$ is denoted the momentum of $v$. The momentum $Lv$ might be singular and in fact $L\big(K(\cdot,y)z\big)(x)$ is the Dirac measure $\delta_y(x)z$. Considering $K$ as the map $z\mapsto
\int_\Omega K(\cdot,x)z(x)dx$, $L$ can be viewed as the inverse of $K$. We will use the symbol $\rho$ for the momentum when considered as a functional in $V^*$ while we switch to the symbol $z$ when the momentum is realized as a vector field on $\Omega$ or for the parameters when the momentum consists of a finite number of singular point measures.
Instead of deriving the kernel from $V$, the opposite approach can be used: build $V$ from a kernel, and hence impose the regularization in the framework from the kernel. With this approach, the kernel is often chosen to ensure rotational and translational invariance [@younes_shapes_2010] and the scalar Gaussian kernel $K(x,y)=\exp(-\frac{\|x-y\|^2}{\sigma^2})\Id_d$ is an often used choice. Confer [@fasshauer_reproducing_2011] for details on the construction of $V$ from Gaussian kernels.
Optimal Paths: The EPDiff Evolution Equations
---------------------------------------------
The relation between norm and momentum leads to convenient equations for minimizers of the energy . In particular, the EPDiff equations for the evolution of the momentum $z_t$ for optimal paths assert that if $\phi_t$ is a path minimizing $E_1(\phi)$ with $\phi_1=\phi$ minimizing $E(\phi)$ and $v_t$ is the derivative of $\phi_t$ then $v_t$ satisfies the system $$\begin{aligned}
&v_t=\int_\Omega K(\cdot,x)z_t(x)dx\ ,
\\&
\frac{d}{dt}z_t=-Dz_tv_t-z_t\nabla\cdot v_t-(Dv_t)^Tz_t\end{aligned}$$ with $Dz_t$ and $Dv_t$ denoting spatial differentiation of the momentum and velocity fields, respectively. The first equation connects the momentum $z_t$ with the velocity $v_t$, and the second equation describes the time evolution of the momentum. In the most general form, the EPDiff equations describe the evolution of the momentum using the adjoint map. Following [@younes_shapes_2010], define the adjoint ${\mathrm{Ad}}_\phi v(x)=(D\phi\,v)\circ \phi^{-1}(x)$ for $v\in V$. The dual of the adjoint is the functional ${\mathrm{Ad}}_\phi^*$ on the dual $V^*$ of $V$ defined by $({\mathrm{Ad}}_\phi^*\rho|v)=(\rho|{\mathrm{Ad}}_\phi(v))$.[^4] Define in addition ${\mathrm{Ad}}_\phi^Tv=K({\mathrm{Ad}}_\phi^*(Lv))$ which then satisfies $\ip{{\mathrm{Ad}}_\phi^Tv,w}=({\mathrm{Ad}}_\phi^*(Lv)|w)$, and let $\nabla_{\phi} U$ denote the gradient of the similarity measure $U$ with respect to the inner product on $V$ so that $\ip{\nabla_{\phi}U,v}_V=\partial_\epsilon U(\psi_{0\epsilon}^v\circ\phi)$ for any variation $v\in V$ and diffeomorphism path $\psi_{0\epsilon}^v$ with derivative $v$. For optimal paths $v_t$, the EPDiff equations assert that $v_t={\mathrm{Ad}}_{\phi_{t1}^v}^Tv_1$ with $v_1=-\frac{1}{2}\nabla_{\phi_{01}^{v}} U$ which leads to the conservation of momentum property for optimal paths. Conversely, the EPDiff equations reduce to simpler forms for certain objects. For landmarks $x_1,\ldots,x_N$, the momentum will be concentrated at point trajectories $x_{t,k}:=\phi_t(x_k)$ as Dirac measures $z_{t,k}\otimes\delta_{x_{t,k}}$ leading to the finite dimensional system of ODE’s $$\begin{split}
& v_t=\sum_{l=1}^NK(\cdot,x_{t,l}))z_{t,l}\ ,
\quad\frac{d}{dt}\phi_t(x_k)=v_t(x_{t,k})\ ,\\
&\frac{d}{dt}z_{t,k}=-\sum_{l=1}^N\nabla_1K(x_{t,l},x_{t,k})z_{t,k}^Tz_{t,l}
\ .
\end{split}
\label{sys:point-epdiff}$$
Registration with Higher-Order Information
==========================================
We here introduce higher-order momentum distributions for registration using higher-order information with the LDDMM framework. We start by motivating the construction by considering the approximation used when computing the similarity measure. We then describe the deformation encoded by higher-order momenta and the evolution equations in the finite case, and we use this to derive a registration algorithm using first-order information. The mathematical background behind the method will be presented in the following sections. \
We will motivate the introduction of higher-order momenta by considering a specific case of image registration: we take on the goal of using a control point formulation [@durrleman_optimal_2011] when solving the registration problem and hence aim for using a relatively sparse sampling of the velocity or momentum field. To achieve this, we will consider the coupling between the transported control points $\{\phi^{-1}(x_1),\ldots,\phi^{-1}(x_N)\}$ and the similarity measure in order to ensure the momentum stays singular and localized at the point trajectories while removing the need for warping the entire image at every iteration of the optimization process.
Considering a similarity measure $U(\phi)=\int_\Omega F(I_m(\phi^{-1}(x)),I_f(x))dx$ as discussed in the introduction, and a finite discretization $\tilde{U}^0(\phi)=1/N\sum_{k=1}^NF(\phi.I_m(x_k),I_f(x_k))$ with a sparse set of control points $\{x_k\}$. While using $\tilde{U}^0(\phi)$ to drive registration of the images will be very efficient in evaluating the warp in few points, it will suffer correspondingly from only using image information present in those points. Apart from not being robust under the presence of noise in the images, the discretization implies that local dilation or rotation around the points $\phi^{-1}(x_k)$ cannot be detected: any variation $v\in V$ of $\phi$ keeping $\phi^{-1}(x_k)$ constant for all $k=1,\ldots,N$ will not change $\tilde{U}^0(\phi)$. Formally, if $\psi_{0\epsilon}$ is a diffeomorphism path that is equal to $\phi$ at $t=0$ and has derivative $v$ at $t=0$, i.e. $\partial_\epsilon\psi_{0\epsilon}=v$ and $\psi_{00}=\phi$, then $$\begin{aligned}
&\partial_\epsilon F( \psi_{0\epsilon}.I_m(x_k),I_f(x_k))
=
\partial_1
F(\phi.I_m(x_k),I_f(x_k))
\cdot
\big(\nabla_{\phi^{-1}(x_k)}I_m\big)^T v(\phi^{-1}(x_k))\end{aligned}$$ which vanishes if $v(\phi^{-1}(x_k))=0$. Here $\partial_1 F$ denotes the derivative of $F:\RR^2\rightarrow\RR$ with respect to the first variable.
A simple way to include more image information in the similarity measure is to convolve with a kernel $K_s$, and thus extend $\tilde{U}^0$ to $$U^1(\phi)
=
\frac{1}{N}\sum_{k=1}^Nc_{K_s}\int_\Omega K_s(p+x_k,x_k)F(\phi.I_m(p+x_k),I_f(p+x_k))dp$$ with $c_{K_s}$ a normalization constant. If $K_s$ is a box kernel, this amounts to a finer sampling of both the image and warp, and hence a finer discretization of the Riemann integral. The kernel $K_s$ should not be confused with the RKHS kernel connected to the norm on $V$ that is used when generating the $V$-gradient. A Gaussian kernel may be used for $K_s$, and more information on using smoothing kernels for intensity based image registration can be found in [@darkner_generalized_2011; @xiahai_zhuang_nonrigid_2011].
The measure $U^1(\phi)$ is problematic since a variation of $\phi$ would affect not only the point $\phi^{-1}(x_k)$ but also $\phi.I_m(p+x_k)$, and $U^1(\phi)$ will therefore be dependent on $\phi.I_m(p+x_k)$ for any $p$ where $K_s(p,x_k)$ is non-zero. In this situation, the momentum is no longer concentrated in Dirac measures located at $\phi_t^{-1}(x_k)$, and it will be necessary to increase the sampling of the warp. However, a first-order expansion of $\phi^{-1}$ yields the approximation $$\tilde{U}^1(\phi)
=
\frac{1}{N}\sum_{k=1}^Nc_{K_s}\int_\Omega
K_s(p+x_k,x_k)F(I_m(D_{x_k}\phi^{-1}p+\phi^{-1}(x_k)),I_f(p+x_k))dp
\ .
\label{eq:tildeU1}$$ The measure $\tilde{U}^1(\phi)$ is now again local depending only on $\phi^{-1}(x_k)$ and the first-order derivatives $D_{x_k}\phi^{-1}$. It offers the stability provided by the convolution with $K_s$, and, importantly, variations $v$ of $\phi$ keeping $\phi^{-1}(x_k)$ constant but changing $D_{x_k}\phi^{-1}$ do indeed affect the similarity measure. This implies that $\tilde{U}^1(\phi)$ is able to catch rotations and dilations and drive the search for optimal $\phi$ accordingly. Please note the differences with the approach of Durrleman et al. [@durrleman_optimal_2011]: when using $\tilde{U}^1(\phi)$ as outlined here, the need for flowing the entire moving image forward is removed and the momentum field will stay singular *directly* thus removing the need for constraining the form of the velocity field.
Evolution and Deformation with Higher-Order Information
-------------------------------------------------------
The dependence on $D\phi$ in the similarity measure $\tilde{U}^1(\phi)$ raises the question of how to represent variations of $D\phi$ in the LDDMM framework. As we will outline here, higher-order momenta appear as the natural choice for such a representation that keeps the benefits of the finite control point formulation. Mathematical details will follow in the next sections.
Recall the reproducing property of the RKHS structure, i.e. $\ip{K(\cdot,x)z,v}_V=z\otimes\delta_x(v)$ for $v\in V$, $x\in\Omega$ and $z\in\RR^d$. Let us define the maps $z\otimes D_x^\alpha:V\rightarrow\RR$ that extend the Diracs $z\otimes\delta_x(v)$ by measuring the *derivative* of $v$ at $x$. These will be denoted *higher-order Diracs*, and we say that the momentum distribution is of higher-order if it is a sum of higher order Diracs. When applying the momentum operator $L$ to the higher-order Diracs, we will get partial derivatives $D_x^\alpha K$ of the RHKS kernel $K$.
In particular, we will see that when using similarity measures such as $\tilde{U}^1(\phi)$, the momentum field will be *a linear combination of higher-order Diracs* and the velocity field will, correspondingly, be *a linear combination of partial derivatives of $K$*. This will imply that the finite dimensional system of ODE’s describing the EPDiff equations in the landmark case will be extended so that the velocity $v_t$ will contain partial derivatives $D_x^\alpha K$. In the first-order case, we will get the velocity $$v(\cdot)=\sum_{l=1}^N\big(K(\cdot,x_l)z_l+\sum_{j=1}^dD^jK(\cdot,x_l)z_l^j\big)$$ where $z_i$ denotes the coefficients of the Dirac measures as in but now the additional vectors $z_i^j$ denote the coefficients of the first-order Diracs $z_i^j\otimes D_{x_i}^j$ for each of the $d$ dimensions $j=1,\ldots,d$. We will later show how these coefficients evolve. Combined with knowledge of how variations of $z_i$ and $z_i^j$ affect the system, we can transport variational information along the optimal paths specified by the EPDiff equations and thus provide the necessary building blocks for a first-order registration algorithm.
Figure \[fig:kernels\] illustrates how the local translation encoded by the kernel is complemented by locally affine deformation when incorporating first-order momenta and corresponding partial derivatives of the kernel. Using the language of deformation atoms, the first-order constructions adds partial derivatives of kernels to the usual set of atoms, and the deformation atoms are thus able to compactly encode expansion, contraction, rotation etc. We can directly *interpret* the coefficients of the first-order momenta as controlling the magnitude of these first-order deformations. In Figure \[fig:shots\] in the experiments section, we give additional illustrations of the deformation encoded by the new atoms.
Algorithm for First-Order Registration
--------------------------------------
In this section, we will derive a registration algorithm for similarity measures incorporating first-order information such as $\tilde{U}^1(\phi)$. Since the algorithm works for general first-order measures, we will again let $U$ denote the similarity measure with $\tilde{U}^1(\phi)$ being just a particular example.
There exists various choices of optimization algorithms for LDDMM registration. Roughly, they can be divided into two groups based on whether they represent the initial momentum/velocity or the entire path $\phi_t$. Here, we take the approach of incorporating first-order momenta with the shooting method of e.g. Vaillant et al. [@vaillant_statistics_2004]. The algorithm will take a guess for the initial momentum, integrate the EPDiff equations forward, compute the similarity measure gradient $\nabla U(\phi)$, and flow the gradient backwards to provide an improved guess.
The registration problem consists of both the similarity measure and the minimal path energy $E_1$. For e.g. landmark based registration, the similarity $U(\phi)$ is most often expressed in terms of $\phi$ directly whether as the similarity measure is usually dependent on the inverse $\phi^{-1}$ for image registration. In the first case, the gradient $\nabla_\phi U$ is known, and, given the initial momentum $z_0$, we can obtain the gradient $\nabla_{z_0}U$ for a gradient descent based optimisation procedure from the backwards transport equations that we derive in Section \[sec:variations\]. For the energy part, it is a fundamental property of critical paths in the LDDMM framework that the energy stays constant along the path. Thus, we can easily compute the gradient from the expressions provided in Section \[sec:hom\]. Given this, the zeroth order matching algorithm in the initial momentum is generalized to zeroth and first-order momenta in Algorithm \[alg:alg1\].
$z_0\gets \mbox{initial guess for initial momentum}$ Solve EPDiff equations forward Compute similarity $U$ Solve backwards the transpose equations Compute the energy gradient $\nabla\|v_0\|^2$ Update $z_0$ from $\nabla\|v_0\|^2+\nabla_{z_0} U$
Traditionally, the similarity measure $U(\phi)$ is in image matching formulated using the inverse of $\phi$, and this approach was taken when formulating the approximation $\tilde{U}^1(\phi)$ in . For this reason, at finite control point formulation is naturally expressed using a sampling $\{x_1,\ldots,x_N\}$ in the *target* image with the algorithm optimizing for the momentum $z_1$ at time $t=1$. The evaluation points $\phi^{-1}(x_k)$ are then generated by flowing *backwards* from $t=1$ to $t=0$, and the gradient of $U(\phi)$ can be computed in $\phi^{-1}(x_k)$ before being flowed *forwards* to update $z_1$. Algorithm \[alg:alg1\] will accommodate this situation by just reversing the integration directions. The control points can be chosen either at e.g. anatomically important locations, at random, or on a regular grid. In the experiments, we will register expanding ventricles using control points placed in the ventricles.
Numerical Integration
---------------------
The integration of the flow equations can be performed with standard Runge-Kutta integrators such as Matlabs `ode45` procedure. With zeroth order momenta only and $N$ points, the forward and backwards system consist of $2dN$ equations. With zeroth and first-order momenta, the forward system is extended to $N(2d+d^2)$ and the backwards system to $2N(d+d^2)$. For $d=3$, this implies an $2.5$ times increase in the size of the forward system and $4$ times increase in the backwards system. As suggested in Figure \[fig:kernels\], the first-order system can be approximated using ensembles of zeroth-order atoms. Such an approximation would for $d=3$ require at least four zeroth-order atoms for each first-order atom making the size of the approximating system equivalent to the first-order system. Due to the non-linearity of the systems, the effect of the approximation introduced with such as an approach is not presently established.
In addition to the increase in the size of the systems, the extra floating point operations necessary for computing the more complicated evolution equations should be considered. The additional computational effort should, however, be viewed against the fact that the finite dimensional system can contain orders of magnitude fewer control points, and the added capacity of deformation description included in the derivative information. In addition and in contrast to previous approaches, we transport the similarity gradient *only* at the control point trajectories, again an order of magnitude reduction of transported information.
Higher-Order Momentum Distributions {#sec:hom}
===================================
We now link partial derivatives of kernels to higher-order momenta using the derivative reproducing property, and we provide details on the EPDiff evolution equations that we outlined in the previous section. We underline that the analytical of structure of LDDMM is not changed when incorporating higher-order momenta, and the evolution equations will thus be particular instances of the general EPDiff equations. These equations in Hamiltonian form constitutes the explicit expressions that allows implementation of the registration algorithm.
We will restrict to scalar kernels when appropriate for simplifying the notation. Scalar kernels are diagonal matrices where all diagonal elements are equal. Thus, we can consider $K(x,y)$ both a matrix and a scalar so that the entries $K^j_i(x,y)$ of the kernel in matrix form equals the scalar $K(x,y)$ if and only if $i=j$ and $0$ otherwise.
Derivative Reproducing Property
-------------------------------
Recall the reproducing property of the RKHS structure, i.e. $\ip{K(\cdot,x)z,v}_V=z\otimes\delta_x(v)$ for $v\in V$, $x\in\Omega$ and $z\in\RR^d$. Zhou [@zhou_derivative_2008] shows that this property holds not only for the kernel but also for its partial derivatives. Letting $D_x^\alpha v$ denote the derivative of $v\in V$ at $x\in\Omega$ with respect to the multi-index $\alpha$, $$D_x^\alpha v
=
\frac{\partial^{|\alpha|}}{\partial_{x^1}^{\alpha_1}\ldots\partial_{x^d}^{\alpha_d}}
v(x)$$ and defining $(D_x^\alpha Kz)(y)=D_x^\alpha (K(\cdot,y)z)$ for $z\in\RR^d$, Zhou proves that $D_x^\alpha Kz\in V$ and that the *partial derivative reproducing property* $$\ip{D_x^\alpha Kz,v}_V=z^TD_x^\alpha(v)
\label{eq:zhou}$$ holds when the maps in $V$ are sufficiently smooth for the derivatives to exist. We denote the maps $z\otimes D_x^\alpha:V\rightarrow\RR$ defined by $z\otimes D_x^\alpha(v):=z^T D_x^\alpha v$ *higher-order Diracs*, and we say that the momentum distribution is of higher order if it is a sum of higher-order Diracs. It follows that $$z\otimes D_x^\alpha
=
\big(
v \mapsto
\ip{D_x^\alpha Kz,v}_V
\big)
\in
V^*
\ .$$ As a consequence, we can connect higher-order momenta and partial derivatives $D_x^\alpha K$ of the kernel. Recall that the momentum map $L:V\rightarrow V^*$ satisfies $\ip{Lv,w}_{L^2}=\ip{v,w}_V$. With the higher-order momenta, $$\ip{L D_x^\alpha Kz,v}_{L_2}
=\ip{D_x^\alpha Kz,v}_V
=z\otimes D_x^\alpha(v)
=\ip{z\otimes D_x^\alpha,v}_{L^2}
\ .$$ Thus $L D_x^\alpha Kz=z\otimes D_x^\alpha$ or, shorter, $L D_x^\alpha K=D_x^\alpha$. That is, partial derivatives of the kernel and higher-order momenta corresponds just as the kernels and Diracs measures in the usual RKHS sense.
Consider a map on diffeomorphisms $U:G_V\rightarrow\RR$ e.g. an image similarity measure dependent on $\phi$. In a finite dimensional setting with $N$ evaluation points $x_k$, $U$ would decompose as $U(\phi)=P\circ Q(\phi)$ with $Q(\phi)=(\phi(x_1),\ldots,\phi(x_N))$ and $P:\RR^{dN}\rightarrow\RR$. Introducing higher-order momenta, we let $Q(\phi)=(D_{x_1}^{\alpha_1}(\phi),\ldots,D_{x_N}^{\alpha_J}(\phi))$ with $J$ multi-indices $\alpha_j$, and decompose $U$ as $U(\phi)=P\circ Q(\phi)$ with $P:\RR^{dNJ}\rightarrow\RR$. We allow $\alpha_j$ to be empty and hence incorporate the standard zeroth order case. The partial derivative reproducing property now lets us compute the $V$-gradient of $U$ as a sum of partial derivatives of the kernel.
Let $\nabla^{kj}P$ denote the gradient with respect to the variable indexed by $D_{x_k}^{\alpha_j}(\phi)$ in the expression for $Q$. Then the gradient $\nabla_\phi U\in V$ of $U$ with respect to the inner product in $V$ is given by $
\nabla_\phi U
=
\sum_{k=1}^N\sum_{j=1}^J
D_{x_k}^{\alpha_j}K\nabla_{Q(\phi)}^{kj}P
$.
\[prop:grad\]
The gradient $\nabla_\phi U$ at $\phi$ is defined by $\ip{\nabla_\phi U, v}=\partial_\epsilon
U(\epsilon v+\phi)$ for all variations $v\in V$. For such $v$, we get using that $$\begin{aligned}
\partial_\epsilon U(\epsilon v+\phi)
&=
\partial_\epsilon P\circ Q(\epsilon v+\phi)
=
\partial_\epsilon P(D_{x_k}^{\alpha_j}(\epsilon v+\phi))
=
\partial_\epsilon P(\epsilon D_{x_k}^{\alpha_j}v+D_{x_k}^{\alpha_j}\phi)
\\
&=
\sum_{k=1}^N\sum_{j=1}^J(\nabla_{Q(\phi)}^{kj} P)^TD_{x_k}^{\alpha_j}v
=
\ip{\sum_{k=1}^N\sum_{j=1}^JD_{x_k}^{\alpha_j}\nabla_{Q(\phi)}^{kj} P,v}_V
\ .
\end{aligned}$$
Momentum and Energy
-------------------
As a result of Proposition \[prop:grad\], the momentum of the gradient of $U$ is $
L\nabla_\phi U
=
\sum_{k=1}^N\sum_{j=1}^J
\nabla_{Q(\phi)}^{kj}P\otimes D_{x_k}^{\alpha_j}
$. In general, if $v\in V$ is represented by a sum of higher-order momenta, the energy $\|v\|_V^2$ can be computed using as a sum of partial derivatives of the kernel evaluated at the points $x_k$. To keep the notation brief, we restrict to sums of zeroth and first-order momenta in the following. If $
v(\cdot)=\sum_{k=1}^N\big(K(x_k,\cdot)z_k+\sum_{j=1}^dD^jK(x_k,\cdot)z_k^j\big)
\ ,
$ we get the energy $$\begin{split}
\|v\|_V^2
&=
\ip{
\sum_{k=1}^N\big(K(x_k,\cdot)z_k+\sum_{j=1}^dD^jK(x_k,\cdot)z_k^j\big)
,
\sum_{k=1}^N\big(K(x_k,\cdot)z_k+\sum_{j=1}^dD^jK(x_k,\cdot)z_k^j\big)
}_V
\\
&=
\sum_{k,l=1}^N
\ip{
K(x_l,\cdot)z_l
,
K(x_k,\cdot)z_k
}_V
+
\sum_{k,l=1}^N\sum_{j,i=1}^d
\ip{
D^jK(x_l,\cdot)z_l^j
,
D^{i}K(x_k,\cdot)z_k^{i}
}_V
\\
&\qquad
+
2\sum_{k,l=1}^N\sum_{j=1}^d
\ip{
D^jK(x_l,\cdot)z_l^j
,
K(x_k,\cdot)z_k
}_V
\\
&=
\sum_{k,l=1}^N
\big(
z_l^TK(x_l,x_k)z_k
+
\sum_{j,i=1}^d
z_k^{i,T}D_2^{i}D_1^jK(x_l,x_k)z_l^j
+
2\sum_{j=1}^d
z_k^TD_1^jK(x_l,x_k)z_l^j
\big)
\end{split}
\label{eq:energy}$$ with $D_q^j K(\cdot,\cdot)$ denoting differentiation with the respect to the $q$th variable, $q=1,2$, and $j$th coordinate, $j=1,\ldots,d$. For scalar symmetric kernels, this expression reduces to $$\begin{aligned}
\|v\|_V^2
&=
\sum_{k,l=1}^N
\big(
z_l^TK(x_l,x_k)z_k
+
\sum_{j,i=1}^d
\big(D_2\nabla_1 K(x_l,x_k)\big)^{i}_jz_k^{i,T}z_l^j
\\
&\quad\
+
2\sum_{j=1}^d
(\nabla_1 K(x_l,x_l))^jz_k^Tz_l^j
\big)
\ .\end{aligned}$$
EPDiff Equations
----------------
It is important to note that higher-order momenta offer a convenient representation for the gradients of maps $U$ incorporating derivative information but since the partial derivatives of kernels are members of $V$ and the higher order momentum in the dual $V^*$, the analytical of structure of LDDMM is not changed. In particular, the adjoint form of the EPDiff equations, i.e. that optimal paths $v_t$ satisfy $v_t={\mathrm{Ad}}_{\phi_{t1}^v}^Tv_1$ with $v_1=-\frac{1}{2}\nabla_{\phi_{01}^{v}} U$, is still valid. The momentum $\rho_1=Lv_1$ is transported to the momentum $\rho_t$ by ${\mathrm{Ad}}_{\phi_{t1}^v}^*p_1$. Because $$\begin{aligned}
(\rho_t|w)
=
(\rho_1|{\mathrm{Ad}}_{\phi_{t1}^v}(w))
=
(\rho_1|(D\phi_{t1}^v\,w)\circ (\phi_{t1}^v)^{-1})
\ ,\end{aligned}$$ if $\rho_1$ is a sum of higher-order Diracs, $\rho_t$ will be sum of higher-order Diracs for all $t$. However, since the time evolution of $(\rho_t|w)$ with the above relation involves derivatives of $D\phi_{t1}^v$, this form is inconvenient for computing $\rho_t$. Instead, we make use of the Hamiltonian form of the EPDiff equations [@younes_shapes_2010 P. 265]. Here, the momentum $\rho_t$ is pulled back to $\rho_0$ but with a coordinate change of the evaluation vector field: the Hamiltonian form $\mu_t$ is defined by $\apply{\mu_t}{w}:=\apply{\rho_0}{(D\phi_{0t}^v)^{-1}(y)w(y)}_y$ where the subscript stresses that $(D\phi_{0t}^v)^{-1}(y)w(y)$ is evaluated as a $y$-dependent vector field. To simplify the notation, we write just $\phi_t$ instead of $\phi_{0t}^v$. Using this notation, the evolution equations become $$\begin{split}
&\partial_t\phi_{t}(y)
=
\sum_{j=1}^d\apply{\mu_t}{K^j(\phi_{t}(x),\phi_{t}(y))}_xe_j
\\
&
\apply{\partial_t\mu_t}{w}
=
-\sum_{j=1}^d\apply{\mu_t}{\apply{\mu_t}{D_2K^j(\phi_{t}(x),\phi_{t}(y))w(y)}_xe_j}_y
\ .
\label{sys:hamilton}
\end{split}$$ The system forms an ordinary differential equation describing the evolution of the path and momentum [@younes_shapes_2010] when $(\rho_0|w)$ does not involve derivatives of $w$, e.g. when $\rho_0$ and hence $\rho_t$ is a vector field $z_t$ and the first equation therefore is an integral $$\partial_t\phi_{t}(y)
=
\int_{\Omega}K(\phi_{t}(y),\phi_{t}(x))z_t(x)dx
\ .$$ For the higher-order case, we will need to incorporate additional information in the system.
Again we restrict to finite sums of zeroth and first-order point measures, and we therefore work with initial momenta on the form $$\rho_0
=
\sum_{k=1}^N
z_{0,k}\otimes\delta_{x_{0,k}}
+
\sum_{k=1}^N
\sum_{j=1}^d
z_{0,k}^j\otimes D^j\delta_{x_{0,k}}
\label{eq:rho0}$$ with $x_{t,r}$ as usual denoting the point positions $\phi_{t}(x_i)$ at time $t$. Then $$\begin{aligned}
\apply{\mu_t}{w}
&=
\apply{\rho_0}{D\phi_{t}(y)^{-1}w(y)}_y
\\
&=
\int_\Omega
\Big(
\sum_{k=1}^N
z_{0,k}\otimes\delta_{x_{0,k}}
+
\sum_{k=1}^N
\sum_{j=1}^d
z_{0,k}^j\otimes D^j\delta_{x_{0,k}}
\Big)
D\phi_{t}(y)^{-1}w(y)
dy
\\
&=
\sum_{k=1}^N
\apply{
\big(
D\phi_{t}(x_{0,k})^{-1,T}
z_{0,k}
+
\sum_{j=1}^d
\big(D^jD\phi_{t}(x_{0,k})^{-1}\big)^T
z_{0,k}^j
\big)\otimes\delta_{x_{0,k}}}{w}
\\
&\quad
+
\sum_{k=1}^N
\sum_{j=1}^d
\apply{D\phi_{t}(x_{0,k})^{-1,T}z_{0,k}^j\otimes D^j\delta_{x_{0,k}}}{w}\end{aligned}$$ showing that $
\mu_t
=
\sum_{k=1}^N
\mu_{t,k}\otimes\delta_{x_{0,k}}
+
\sum_{k=1}^N
\sum_{j=1}^d
\mu_{t,k}^j\otimes D^j\delta_{x_{0,k}}
$ with $$\begin{split}
&\mu_{t,k}
=
D\phi_{t}(x_{0,k})^{-1,T}
z_{0,k}
+
\sum_{j=1}^d
\big(D^jD\phi_{t}(x_{0,k})^{-1}\big)^T
z_{0,k}^j
\\
&\mu_{t,k}^j
=
D\phi_{t}(x_{0,k})^{-1,T}z_{0,k}^j
\ .
\label{eq:mu}
\end{split}$$ The momentum $\rho_t$ can the be recovered as $$\begin{aligned}
\apply{\rho_t}{w}
&
=
\apply{\mu_t}{w\circ\phi_{t}}
=
\big(
\sum_{k=1}^N
\mu_{t,k}\otimes\delta_{x_{0,k}}
+
\sum_{k=1}^N
\sum_{j=1}^d
\mu_{t,k}^j\otimes D^j\delta_{x_{0,k}}
\big)
w\circ\phi_{t}
\\
&=
\sum_{k=1}^N
\mu_{t,k}\otimes\delta_{x_{t,k}}
w
+
\sum_{k=1}^N
\sum_{j=1}^d
\mu_{t,k}^{j,T}
Dw(D^j\phi_{t})(x_{0,k})
\\
&=
\sum_{k=1}^N
\mu_{t,k}\otimes\delta_{x_{t,k}}
w
+
\sum_{k=1}^N
\sum_{j=1}^d
\big(
\sum_{i=1}^d
(D^i\phi_{t})(x_{0,k})^j
\mu_{t,k}^i
\big)
\otimes
D^j\delta_{x_{t,k}}w\end{aligned}$$ and hence the coefficients of the momentum $z_{t,k}$ and $z_{t,k}^j$ (confer ) are given by $
z_{t,k}
=
\mu_{t,k}
$ and $z_{t,k}^j
=
\sum_{i=1}^d
(D^i\phi_{t})(x_{0,k})^j
\mu_{t,k}^i
$. We note that both $z_{t,k}^j$ and $\mu_{t,k}^j$ are coordinate vectors of the first-order parts of the momentum in ordinary and Hamiltonian form respectively. For each point $k$ and time $t$, these coordinate vectors thus represent two $d\times d$ tensors.
Time Evolution
--------------
Even though $\mu_{t,k}$ in depend on the second order derivative of $\phi$, we will show that the complete evolution in the zeroth and first-order case can be determined by solving for the points $\phi_{t}(x_{k,0})$, the matrices $D\phi_{t}(x_{k,0})$, and the vectors $\mu_{t,k}$. This will provide the computational representation we will use when implementing the systems.
Using , $\phi_{t}$ evolves according to $$\begin{aligned}
\partial_t\phi_{t}(y)
&
=
\sum_{i=1}^d
\int_\Omega
\sum_{k=1}^N
\big(
\mu_{t,k}^T\otimes\delta_{x_{0,k}}
+
\sum_{j=1}^d
\mu_{t,k}^j\otimes D^j\delta_{x_{0,k}}
\big)
K^i(\phi_{t}(x),\phi_{t}(y))
dx\,
e_i
\\
&
=
\sum_{i=1}^d
\sum_{k=1}^N
\big(
\mu_{t,k}^T
K^i(\phi_{t}(x_{0,k}),\phi_{t}(y))
+
\sum_{j=1}^d
\mu_{t,k}^{j,T}
D_1K^i(\phi_{t}(x_{0,k}),\phi_{t}(y))
D^j\phi_{t}(x_{0,k})
\big)
e_i
\ .\end{aligned}$$ With scalar kernels, the trajectories $x_{t,k}$ are given by $$\begin{aligned}
\partial_t\phi_{t}(x_{0,k})
&
=
\sum_{l=1}^N
\big(
K(\phi_{t}(x_{0,l}),\phi_{t}(x_{0,k}))
\mu_{t,l}
+
\sum_{j=1}^d
\nabla_1K(\phi_{t}(x_{0,l}),\phi_{t}(x_{0,k}))^T
D^j\phi_{t}(x_{0,l})
\mu_{t,l}^j
\big)
\ .\end{aligned}$$
It is shown in [@younes_shapes_2010] that the evolution of the matrix $D\phi_{t}(x_{k,0})$ is governed by $$\begin{aligned}
&\partial_tD\phi_{t}(y)a
=
\sum_{i=1}^d\apply{\mu_t}{D_2K^i(\phi_{t}(x),\phi_{t}(y))D\phi_{t}(y)a}_xe_i
\ .\end{aligned}$$ Inserting the Hamiltonian form of the higher-order momentum, each component $(r,c)$ (*r*ow/*c*olumn) of the matrix $D\phi_{t}(y)$ thus evolves according to $$\begin{aligned}
&\partial_tD\phi_{t}(y)_r^c
=
\apply{\mu_t}{D_2K^r(\phi_{t}(x),\phi_{t}(y))D\phi_{t}(y)e_c}_x
\\
&\qquad
=
\int_\Omega
\sum_{k=1}^N
\big(
\mu_{t,k}\otimes\delta_{x_{0,k}}
+
\sum_{j=1}^d
\mu_{t,k}^j\otimes D^j\delta_{x_{0,k}}
\big)
D_2K^r(\phi_{t}(x),\phi_{t}(y))D\phi_{t}(y)e_c
dx
\\
&\qquad
=
\sum_{k=1}^N
\mu_{t,k}^T
D_2K^r(\phi_{t}(x_{0,k}),\phi_{t}(y))D\phi_{t}(y)e_c
\\
&\qquad\quad
+
\sum_{k=1}^N
\sum_{j=1}^d
\mu_{t,k}^{j,T}
\big(\sum_{i=1}^d\big(D_1^iD_2K^r(\phi_{t}(x_{0,k}),\phi_{t}(y))\big)\big(D^j\phi_{t}(x_{0,k})\big)^i\big)D\phi_{t}(y)e_c
\ .\end{aligned}$$ With scalar kernels, the evolution at the trajectories is then $$\begin{aligned}
\partial_tD\phi_{t}(x_{0,k})^c
&=
\sum_{l=1}^N
\Big(
\nabla_2K(\phi_{t}(x_{0,l}),\phi_{t}(x_{0,k}))^TD^c\phi_{t}(x_{0,k})
\mu_{t,l}
\\
&\quad
+
\sum_{j=1}^d
\big(D_1\nabla_2K(\phi_{t}(x_{0,l}),\phi_{t}(x_{0,k}))D^j\phi_{t}(x_{0,l})\big)^TD^c\phi_{t}(x_{0,k})
\mu_{t,l}^j
\Big)
\ .\end{aligned}$$
The complete derivation of the evolution of $\mu_t$ is notationally heavy and can be found in the supplementary material for the paper. Combining the evolution of $\mu_t$ with the expressions above, we arrive at the following result:
The EPDiff equations in the scalar case with zeroth and first-order momenta are given in Hamiltonian form by the system $$\begin{split}
&\partial_t\phi_{t}(x_{0,k})
=
\sum_{l=1}^N
\big(
K(x_{t,l},x_{t,k})
\mu_{t,l}
+
\sum_{j=1}^d
\nabla_1K(x_{t,l},x_{t,k})^T
D^j\phi_{t}(x_{0,l})
\mu_{t,l}^j
\big)
\\
&\partial_tD\phi_{t}(x_{0,k})^c
=
\sum_{l=1}^N
\Big(
\nabla_2K(x_{t,l},x_{t,k})^TD^c\phi_{t}(x_{0,k})
\mu_{t,l}
\\
&\qquad\qquad\qquad\quad
+
\sum_{j=1}^d
\big(D_1\nabla_2K(x_{t,l},x_{t,k})D^j\phi_{t}(x_{0,l})\big)^TD^c\phi_{t}(x_{0,k})
\mu_{t,l}^j
\Big)
\\
&\partial_t\mu_{t,k}
=
-
\sum_{l=1}^N
\Big(
\big(
\mu_{t,k}^T
\mu_{t,l}
\big)
\nabla_2K(x_{t,l},x_{t,k})
\\
&\qquad\quad\ \
+\sum_{j=1}^d
\big(
\mu_{t,k}^{j,T}
\mu_{t,l}
\big)
D_2\nabla_2K(x_{t,l},x_{t,k})
D^j\phi_{t}(x_{0,k})
\\
&\qquad\quad\ \
+\sum_{j=1}^d
\big(
\mu_{t,k}^T
\mu_{t,l}^j
\big)
D_1\nabla_2K(x_{t,l},x_{t,k})
D^j\phi_{t}(x_{0,l})
\\
&\qquad\quad\
+\sum_{j,j'=1}^d
\big(
\mu_{t,k}^{j',T}
\mu_{t,l}^j
\big)
D_2
\big(
D_1\nabla_2K(x_{t,l},x_{t,k})
D^j\phi_{t}(x_{0,l})
\big)
D^{j'}\phi_{t}(x_{0,k})
\Big)
\\
&\mu_{t,k}^j
=
D\phi_{t}(x_{0,k})^{-1,T}z_{0,k}^j
\ .
\label{sys:epdiff-hamilton}
\end{split}$$
Note that both $x_{1,k}=\phi_{01}^v(x_{0,k})$ and $D\phi_{01}^v(x_{0,k})$ are provided by the system and hence can be used to evaluate a similarity measure that incorporates first-order information. As in the zeroth order case, the entire evolution can be recovered by the initial conditions for the momentum.
Locally Affine Transformations {#sec:loc-affine}
==============================
The Polyaffine and Log-Euclidean Polyaffine [@arsigny_polyrigid_2005; @arsigny_fast_2009] frameworks model locally affine transformations using matrix logarithms. The higher-order momenta and partial derivatives of kernels can be seen as the LDDMM sibling of the Polyaffine methods, and diffeomorphism paths generated by higher-order momenta, in particular, momenta of zeroth and first-order, can locally approximate all affine transformations with linear component having positive determinant. The approximation will depend only on how fast the kernel approaches zero towards infinity. The manifold structure of $G_V$ provides this result immediately. Indeed, let $\phi(x)=Ax+b$ be an affine transformation with $\det(A)>0$. We define a path $\phi_t$ of finite energy such that $\phi_1\approx \phi$ which shows that $\phi_1\in G_V$ and can be reached in the framework. The matrices of positive determinant is path connected so we can let $\psi_t$ be a path from $\Id_d$ to $A$ and define $\tilde{\psi}_t(x)=\psi_tx+bt$. Then with $\tilde{v}_t(x)=(\partial_t\psi_t)\tilde{\psi}_t^{-1}(x)+b$, we have $\partial_t\tilde{\psi}_t(x)=(\partial_t\psi_t)x+b=\tilde{v}_t\circ\tilde{\psi}_t(x)$ and $$x+\int_0^1\tilde{v}_t\circ\tilde{\psi}_t(x)dt
=
x+\int_0^1(\partial_t\psi_t)x+bdt
=
\phi(x)
\ .$$ Now use that $(\partial_t\psi_t)\tilde{\psi}_t^{-1}(x)=(\partial_t\psi_t)(\psi_t)^{-1}(x-bt)$ and let the $M_t=(m_{1,t}\ldots m_{d,t})$ be the $t$-dependent matrix $(\partial_t\psi_t)(\psi_t)^{-1}$ so that the first term of $\tilde{v}_t(x)$ equals $M_t(x-bt)$. Then choose a radial kernel, e.g. a Gaussian $K_\sigma$, and define the approximation $v_t$ of $\tilde{v}_t$ by $$v_t(x)
=
\sum_{j=1}^dD^j_{\tilde{\psi}_t(0)}K_\sigma(x)m_{j,t} +K_\sigma(\tilde{\psi}_t(0),x)b
\ .
\label{eq:vel-affine}$$ The path $\phi_{01}^v$ generated by $v_t$ then has finite energy, and $$\phi_{01}^v(x)
=
x+\int_0^1v_t\circ\phi_{0t}^v(x)dt
\approx
\phi(x)$$ with the approximation depending only on the kernel scale $\sigma$. Note that the affine transformations with linear components having negative determinant can in a similar way be reached by starting the integration at a diffeomorphism with negative Jacobian determinant.
In the experiments section, we will illustrate the locally affine transformations encoded by zeroth and first-order momenta, and, therefore, it will be useful to introduce a notation for these momenta. We encode the translational part of either the momentum or velocity using the notation $${\mathrm{Tsl}}_{x}(b)=K_{\sigma}(x,\cdot)b$$ and the linear part by $${\mathrm{Lin}}_{x}(M)
=
\sum_{j=1}^d
D^j_{x}K_\sigma(\cdot)m_j$$ with $m_1,m_j$ being the columns of the matrix $M$. Equation can then be written $$v_t(x)
=
{\mathrm{Lin}}_{\tilde{\psi}_t(0)}(M_t) +{\mathrm{Tsl}}_{\tilde{\psi}_t(0)}(b)
\ .$$ We emphasize that though we mainly focus on zeroth and first-order momenta, the mathematical construction allows any order momenta permitted by the smoothness of the kernel at order zero.
Variations of the Initial Conditions {#sec:variations}
====================================
In Algorithm \[alg:alg1\], we used the variation of the EPDiff equations when varying the initial conditions and in particular the backwards gradient transport. We discuss both issues here.
A variation $\delta\rho_0$ of the initial momentum will induce a variation of the system . By differentiating that system, we get the time evolution of the variation. To ease notation, we assume the kernel is scalar on the form $K(x,y)=\gamma(|x-y|^2)$ and write $\gamma_{t,lk}=K(x_{t,l},x_{t,k})$.[^5] Variations of the kernel and kernel derivatives such as the entity $\delta \nabla_1K(x_{t,l},x_{t,k})$ below depend only on the variation of point trajectories $\delta x_{t,l}$ and $\delta x_{t,k}$. The full expressions for these parts are provided in supplementary material for the paper. The variation of the point trajectories in the derived system then takes the form $$\begin{split}
&\partial_t\delta\phi_{t}(x_{0,k})
=
\sum_{l=1}^N
\big(
\delta K(x_{t,l},x_{t,k})
\mu_{t,l}
+
\gamma_{t,lk}
\delta
\mu_{t,l}
\big)
\\
&\qquad
+
\sum_{l=1}^N
\sum_{j=1}^d
\big(
\delta \nabla_1K(x_{t,l},x_{t,k})^T
D^j\phi_{t}(x_{0,l})
\mu_{t,l}^j
+
\nabla_1K(x_{t,l},x_{t,k})^T
\delta D^j\phi_{t}(x_{0,l})
\mu_{t,l}^j
\\
&\qquad\qquad
+
\nabla_1K(x_{t,l},x_{t,k})^T
D^j\phi_{t}(x_{0,l})
\delta
\mu_{t,l}^j
\big)
\end{split}$$ The similar expressions for the evolution of $\delta\mu_{t,k}$ and $\delta
D\phi_{t}(x_{0,k})$ are provided in the supplementary material. The variation of $\mu_{t,k}^j$ is available as $$\begin{aligned}
&\delta\mu_{t,k}^j
=
-\big(
D\phi_{t}(x_{0,k})^{-1}
\delta D\phi_{t}(x_{0,k})
D\phi_{t}(x_{0,k})^{-1}
\big)^Tz_{0,k}^j
+
D\phi_{t}(x_{0,k})^{-1,T}\delta z_{0,k}^j
\ .\end{aligned}$$ However, when computing the backwards transport, we will need to remove the dependency on $\delta z_{0,k}^j$ which is only available for forward integration. Instead, by writing the evolution of $\mu_{t,k}^j$ in the form $$\begin{aligned}
&\partial_t \mu_{t,k}^j
=
\partial_t D\phi_{t}(x_{0,k})^{-1,T}z_{0,k}^j
=
-\big(
D\phi_{t}(x_{0,k})^{-1}
\partial_t D\phi_{t}(x_{0,k})
D\phi_{t}(x_{0,k})^{-1}
\big)^Tz_{0,k}^j
\\
&\qquad
=
-D\phi_{t}(x_{0,k})^{-1,T}
\partial_t D\phi_{t}(x_{0,k})^T
\mu_{t,k}^j
\ ,\end{aligned}$$ we get the variation $$\begin{aligned}
&\partial_t \delta\mu_{t,k}^j
=
-\delta D\phi_{t}(x_{0,k})^{-1,T}
\partial_t D\phi_{t}(x_{0,k})^T
\mu_{t,k}^j
-
D\phi_{t}(x_{0,k})^{-1,T}
\partial_t \delta D\phi_{t}(x_{0,k})^T
\mu_{t,k}^j
\\
&\qquad\qquad
-
D\phi_{t}(x_{0,k})^{-1,T}
\partial_t D\phi_{t}(x_{0,k})^T
\delta \mu_{t,k}^j
\ .\end{aligned}$$
Backwards Transport
-------------------
The correspondence between initial momentum $\rho_0$ and end diffeomorphism $\phi_{01}^v$ asserted by the EPDiff equations allows us to view the similarity measure $U(\phi_{01}^v)$ as a function of $\rho_0$. Let $A$ denote the result of integrating the system for the variation of the initial conditions from $t=0$ to $t=1$ such that $w=A\delta\rho_0\in V$ for a variation $\delta\rho_0$. We then get a corresponding variation $\delta U$ in the similarity measure. To compute the gradient of $U$ as a function of $\rho_0$, we have $$\begin{aligned}
\delta U(\phi_{01}^v)
=
\ip{\nabla_{\phi_{01}^v}U,w}_V
=
\ip{\nabla_{\phi_{01}^v}U,A\delta\rho_0}_V
=
\ip{A^T\nabla_{\phi_{01}^v}U,\delta\rho_0}_{V^*}
\ .\end{aligned}$$ Thus, the $V^*$-gradient of $\nabla_{\rho_0}U$ is given by $A^T\nabla_{\phi_{01}^v}U$. The gradient can equivalently be computed in momentum space at both endpoints of the diffeomorphism path using the map $P$ defined in Proposition \[prop:grad\].
The complete system for the variation of the initial conditions is a linear ODE, and, therefore, there exists a time-dependent matrix $M_t$ such that the ODE $$\begin{aligned}
\partial_t y_t
=
M_t
y_t\end{aligned}$$ has the variation as a solution $y_t$. It is shown in [@younes_shapes_2010] that, in such cases, solving the backwards transpose system $$\partial_t w_t
=
-M_t^T
w_t
\label{sys:backwards}$$ from $t=1$ to $t=0$ provides the value of $A^Tw$. Therefore, we can obtain $\nabla_{\rho_0}U$ by solving the transpose system backwards. The components of $M_t$ can be identified by writing the evolution equations for the variation in matrix form. This provides $M_t^T$ and allows the backwards integration of the system \[sys:backwards\]. The components of the transpose matrix $M_t$ are provided in the supplementary material for the paper.
Experiments {#sec:experiments}
===========
In order to demonstrate the efficiency, compactness, and interpretability of representations using higher-order momenta, we perform four sets of experiments. First, we provide four examples illustrating the type of deformations produced by zeroth and first-order momenta and the relation to the Polyaffine framework. We then use point based matching using first-order information to show how complicated warps that would require many parameters with zeroth order deformation atoms can be generated with very compact representations using higher-order momenta. We underline the point that higher-order momenta allow low-dimensional transformations to be registered using correspondingly low-dimensional representations: we show how synthetic test images generated by a low-dimensional transformation can be registered using only one deformation atom when representing using first-order momenta and using the first-order similarity measure approximation . We further emphasize this point by registering articulated movement using only one deformation atom per rigid part, and thus exemplify a natural representation that reduces the number of deformation atoms and the ambiguity in the placement of the atoms while also reducing the degrees of freedom in the representation. Finally, we illustrate how higher-order momenta in a natural way allow registration of human brains with progressing atrophy. We describe the deformation field throughout the ventricles using few deformation atoms, and we thereby suggest a method for detecting anatomical change using few degrees of freedom. In addition, the volume expansion can be directly interpreted from the parameters of the deformation atoms. We start by briefly describing the similarity measures used throughout the experiments.
For the point examples below, we register moving points $x_1,\ldots,x_N$ against fixed points $y_1,\ldots,y_N$. In addition, we match first-order information by specifying values of $D^j_{x_k}\phi$. This is done compactly by providing matrices $Y_k$ so that we seek $D_{x_k}\phi=Y_k$ for all $k=1,\ldots,N$. The similarity measure is simple sum of squares, i.e. $$U(\phi)
=
\sum_{i=1}^N
\|\phi(x_k)-y_k\|^2
+
\|D_{x_k}\phi-Y_k\|^2$$ using the matrix $2$-norm. This amounts to fitting $\phi$ against a locally affine map with translational components $y_k$ and linear components $Y_k$. For the image cases, we use $L^1$-similarity to build the first-order approximation with the smoothing kernel $K_s$ being Gaussian of the same scale as the LDDMM kernel.
First Order Illustrations
-------------------------
To visually illustrate the deformation generated by higher-order momenta, we show in Figure \[fig:shots\] the generated deformations on an initially square grid with four different first-order initial momenta. The deformation locally model the linear part of affine transformations and the the locality is determined by the Gaussian kernel that in the examples has scale $\sigma=8$ in grid units. Notice for the rotations that the deformation stays diffeomorphic in the presence of conflicting forces. The similarity between the examples and the deformations generated in the Polyaffine framework [@arsigny_fast_2009] underlines the viewpoint that the registration using higher-order momenta constitutes the LDDMM sibling of the Polyaffine framework.
First Order Point Registration
------------------------------
Figure \[fig:matches\] presents simple point based matching results with first-order information. The lower points (red) are matched against the upper points (black) with match against expansion $D_\phi(x_k)=2\Id_2$ and rotation $D_\phi(x_k)=\mathrm{Rot}(v)=\begin{pmatrix}\cos(v),\sin(v)\\-\sin(v),\cos(v)\end{pmatrix}$ for $v=\mp \pi/2$. The optimal diffeomorphisms exhibit the expected expanding and turning effect, respectively. We stress that the deformations are generated using only two deformations atoms with combined 12 parameters. Representing equivalent deformation using zeroth order momenta would require a significantly increased number of atoms and a correspond increase in the number of parameters.
Low Dimensional Image Registration
----------------------------------
We now exemplify how higher-order momenta allow low-dimensional transformations to be registered using correspondingly low-dimensional representations. We generate two test images by applying two linear transformations, an dilation and a rotation, to a binary image of a square, confer the moving images (a) and (e) in Figure \[fig:simple-images\]. By placing one deformation atom in the center of each fixed image and by using the similarity measure approximation , we can successfully register the moving and fixed images. The result and difference plots are shown in Figure \[fig:simple-images\]. The dimensionality of the linear transformations generating the moving images is equal to the number of parameters for the deformation atom. A registration using zeroth order momenta would need more than one deformation atom which would result in a number of parameters larger than the dimensionality. The scale of the Gaussian kernel used for the registration is 50 pixels.
Articulated Motion
------------------
The articulated motion of the finger[^6] in Figure \[fig:finger\] (a) and (b) can be described by three locally linear transformations. With higher-order momenta, we can place deformation atoms at the center of the bones in the moving and fixed images, and use the point positions together with the direction of the bones to drive a registration. This natural and low dimensional representation allows a fairly good match of the images resembling the use of the Polyaffine affine framework for articulated registration [@seiler_geometry-aware_2011]. A similar registration using zeroth order momenta would need two deformation atoms per bone and lacking a natural way to place such atoms, the positions would need to be optimized. With higher-order momenta, the deformation atoms can be placed in a natural and consistent way, and the total number of free parameters is lower than a zeroth order representation using two atoms per bone.
Registering Atrophy
-------------------
Atrophy occurs in the human brain among patients suffering from Alzheimer’s disease, and the progressing atrophy can be detected by the expansion of the ventricles [@jack_medial_1997; @fox_presymptomatic_1996]. Since first-order momenta offer compact description of expansion, this makes a parametrization of the registration based on higher-order momenta suited for describing the expansion of the ventricles, and, in addition, the deformation represented by the momenta will be easily interpretable. In this experiment, we therefore suggest a registration method that using few degrees of freedom describes the expansion of the ventricles, and does so in a way that can be interpreted when doing further analysis of e.g. the volume change.
We use the publicly available Oasis dataset[^7] [@marcus_open_2010], and, in order to illustrate the use of higher-order momenta, we select a small number of patients from which two baseline scans are acquired at the same day together with a later follow up scan. The patients are in various stages of dementia, and, for each patient, we rigidly register the two baseline and one follow up scan [@darkner_generalized_2011].
The expanding ventricles can be registered by placing deformation atoms in the center of the ventricles of the fixed image as shown in Figure \[fig:atrophy1\]. For each patient, we manually place five deformation atoms in the ventricle area of the first baseline 3D volume. It is important to note that though we localize the description of the deformation to the deformation atoms, the atoms control the deformation field throughout the ventricle area. Based on the size of the ventricles, we use 3D Gaussian kernels with a scale of 15 voxels, and we let the regularization weight in be $\lambda=16$. The effect of these choices is discussed below. Each deformation atom consists of a zeroth and first-order momenta. We use $L^1$ similarity to drive the registration [@darkner_generalized_2011][^8] and, for each patient, we perform two registrations: we register the two baseline scans acquired at the same day, and we register one baseline scan against the follow up scan. Thus, the baseline-baseline registration should indicate no ventricle expansion, and we expect the baseline-follow up registration to indicate ventricle expansion. Figure \[fig:atrophy1\] shows for one patient the placement of the control points in the baseline image, the follow up image, the $\log$-Jacobian determinant in the ventricle area of the generated deformation, and the initial vector field driving the registration.
The use of first-order momenta allows us to interpret the result of the registrations and to relate the results to possible expansion of the ventricles. The volume change is indicated by the Jacobian determinant of the generated deformation at the deformation atoms as well as by the divergence of the first-order momenta. The latter is available directly from the registration parameters. We plot in Figure \[fig:atrophy2\] the logarithm of the Jacobian determinant and the divergence for both the same day baseline-baseline registrations and for the baseline-follow up registrations. Patient $1-4$ are classified as demented, patient $5$ and $6$ as non-demented, and all patient have constant clinical dementia rating through the experiment. The time-span between baseline and follow up scan is 1.5-2 years with the exception of 3 years for patient four. As expected, the $\log$-Jacobian is close to zero for the same day baseline-baseline scans but it increases with the baseline-follow up registrations of the demented patients. In addition, the correlation between the $\log$-Jacobian and the divergence shows how the indicated volume change is related directly to the registration parameters; the parameters of the deformation atoms can in this way be directly interpreted as encoding the amount of atrophy.
We chose two important parameters above: the kernel scale and the regularization term. The choice of one scale for all patients works well if the ventricles to be registered are of approximately the same size at the baseline scans. If the ventricles vary in size, the scale can be chosen individually for each patient. Alternatively, a multi-scale approach could do this automatically which suggests combining the method with e.g. the kernel bundle framework [@sommer_multi-scale_2011]. Depending on the image forces, the regularization term in will affect the amount of expansion captured in the registration. Because of the low number of control points, we can in practice set the contribution of the regularization term to zero without experiencing non-diffeomorphic results. It will be interesting in the future to estimate the actual volume expansion directly using the parameters of the deformation atoms with this less biased model.
Conclusion and Outlook {#sec:concl}
======================
We have introduced higher-order momenta in the LDDMM registration framework. The momenta allow *compact* representation of locally affine transformations by increasing the *capacity* of the deformation description. Coupled with similarity measures incorporating first-order information, the higher-order momenta improve the range of deformations reached by sparsely discretized LDDMM methods, and they allow direct capture of first-order information such as expansion and contraction. In addition, the constitute deformation atoms for which the generated deformation is directly interpretable.
We have shown how the partial derivative reproducing property implies singular momentum for the higher-order momenta, and we used this to derive the EPDiff evolution equations. By computing the forward and backward variational equations, we are able to transport gradient information and derive a matching algorithm. We provide examples showing typical deformation coded by first-order momenta and how images can be registered using a very few parameters, and we have applied the method to register human brains with progressing atrophy.
The experiments included here show only a first step in the application of higher-order momenta: the representation may be applied to register entire images; merging the method with multi-scale approaches will increase the description capacity and may lead to further reduction in the dimensionality of the representation. Combined with efficient implementations, higher-order momenta promise to provide a step forward in compact deformation description for image registration.
[^1]: Dept. of Computer Science, Univ. of Copenhagen, Denmark ([sommer@diku.dk]{})
[^2]: BiomedIQ, Copenhagen, Denmark
[^3]: Asclepios Project-Team, INRIA Sophia-Antipolis, France
[^4]: Here and in the following, we will use the notation $(p|v):=p(v)$ for evaluation of the functional $p\in V^*$ on the vector field $v\in V$.
[^5]: The subscript notation is used in accordance with [@younes_shapes_2010]. Please note that $\gamma_{t,lk}$ contains *three* separate indices, i.e. the time $t$ and the point indices $l$ and $k$.
[^6]: X-ray frames from <http://www.archive.org/details/X-raystudiesofthejointmovements-wellcome>
[^7]: <http://www.oasis-brains.org>
[^8]: See also <http://image.diku.dk/darkner/LOI>.
|
---
abstract: 'We study temperature dependence of geometrical (Fiske) and velocity-matching (Eck) resonances in the flux-flow state of small $\mathrm{Bi_2Sr_2CaCu_2O_{8+x}}$ mesa structures. It is shown that the quality factor of resonances is high at low $T$, but rapidly decreases with increasing temperature already at $T > 10$ K. We also study $T$-dependencies of resonant voltages and the speed of electromagnetic waves (the Swihart velocity). Surprisingly it is observed that the Swihart velocity exhibits a flat $T$-dependence at low $T$, following $T-$dependence of the $c$-axis critical current, rather than the expected linear $T$-dependence of the London penetration depth. Our data indicate that self-heating is detrimental for operation of mesas as coherent THz oscillators because it limits the emission power via suppression of the quality factor. On the other hand, significant temperature dependence of the Swihart velocity allows broad-range tunability of the output frequency.'
author:
- 'S. O. Katterwe'
- 'V. M. Krasnov'
title: 'Temperature dependence of geometrical and velocity matching resonances in Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$ intrinsic Josephson junctions'
---
I. Introduction
===============
Single crystals of a cuprate superconductor $\mathrm{Bi_2Sr_2CaCu_2O_{8+x}}$ (Bi-2212) represent natural stacks of atomic scale intrinsic Josephson junctions (IJJs) [@Kleiner92]. Josephson junctions form transmission lines for electromagnetic (EM) waves [@Swihart]. The propagation (Swihart) velocity is $c_0 \simeq c/[L_{\square}
C_{\square}]^{1/2}$, where $c$ is the speed of light in vacuum and $L_{\square}$ and $C_{\square}$ are the inductance and the capacitance per square of the transmission line. $$\begin{aligned}
\label{Lsquare}
L_{\square}= 4\pi \Lambda, \\
\Lambda= t + 2\lambda_S \coth (d/\lambda_S).\end{aligned}$$ Here $t$ and $d$ are thicknesses of the dielectric and superconducting layers, respectively, and $\lambda_S$ is the London penetration depth of the superconductor. In thin-layer junctions $t,d \ll \lambda_S$, $L_{\square} \simeq
8\pi\lambda_S^2/d$ is dominated by a large kinetic inductance of superconducting layers. As a consequence, $c_0$ can be much slower than $c$ - the phenomenon that finds applications in compact superconducting delay lines [@DelayLine]. The Swihart velocity carries a direct information about the London penetration depth. It can be obtained by measuring the propagation (delay) time in a transmission line [@Mason; @Kircher]. However, Josephson junctions provide a much easier way of measuring $c_0$. In Josephson junctions EM waves can be generated in-situ by means of the ac-Josephson effect. At geometrical resonance conditions they form standing waves, leading to appearance of Fiske steps in current-voltage ($I$-$V$) characteristics [@Dmitrenko; @Koshelets; @Cirillo; @FiskeInd; @KatterweFiske]. Fiske step voltages allow simple and direct evaluation of the [*absolute values*]{} of $\lambda_S(T)$ [@Dmitrenko; @Ngai] (unlike surface impedance measurements, which usually provide only relative values [@Lambda; @Lambda2; @Trunin; @Lambda2005]). Such measurements do not require long transmission lines, but can be performed on small $\sim\mu$m-scale junctions. IJJ of $\mu$m sizes, made on high quality Bi-2212 single crystals, are free from crystallographic defects, that can affect $\lambda$ in cuprates [@Pan; @Prozorov]. Therefore, Fiske resonances in small IJJs should provide information about genuine (defect-free) behavior of the penetration depth in cuprates.
Geometrical resonances play also an important role in achieving high power THz EM wave emission from Bi-2212 mesa structures [@Ozyuzer; @Wang; @Nonequilibrium; @Hu; @Klemm; @Breather]. The maximum radiation power from a stack with $N$ junctions is $P_{rad}
\propto N^2 Q^2$ [@TheoryFiske], where $$Q=\omega RC, \label{eq:Q}$$ is the quality factor of the resonance, $\omega$ is the resonant frequency, $R$ the effective damping resistance and $C$ the capacitance of the junctions. The factor $N^2$ is due to constructive interference of $N$ in-phase synchronized junctions [@Note1] and the factor $Q^2$ represents the resonant amplification in each junction by the geometrical resonance. Thus, both the in-phase coherence and the high quality $Q\gg 1$ geometrical resonances are needed for achieving high emission power [@TheoryFiske]. Increment of the emission power is inevitably accompanied by self-heating of the stack. In superconductors this leads to a rapid increment of the quasiparticle (QP) damping, which suppresses $Q$. Self-heating ultimately limits the performance of an oscillator [@Breather]. Clearly, investigation of the quality factor of geometrical resonances and their $T$-dependence has a primary significance for development of high power THz oscillator, based on IJJs.
In this work, we study experimentally $T$-dependencies of geometrical (Fiske) and velocity-matching (Eck) resonances [@Cirillo] in the flux-flow state of small Bi-2212 mesa structures. It is observed that $Q$ of resonances is large at low $T$, but rapidly decreases with increasing temperature already at $T \gtrsim 10$ K $\ll T_c \sim 90$ K, primarily due to enhancement of the quasiparticle damping. Surprisingly, it is observed that resonant voltages, proportional to the Swihart velocity, exhibit a very weak $T$-dependence at low $T$ and do not follow the expected linear $T$-dependence of the effective London penetration depth $\lambda_{ab}(T)$ in Bi-2212 [@Lambda; @Lambda2; @Trunin; @Lambda2005]. We discuss possible origins of such a distinct discrepancy, which to our opinion deserves further experimental and theoretical analysis.
II. Geometrical resonances in stacked Josephson junctions
=========================================================
Stacked Josephson junctions form multilayer transmission lines for electromagnetic waves. The general problem of linear wave propagation in multilayer transmission lines was first considered by Economou [@Economou] and more recently within the inductively coupled junction (ICJ) formalism by Kleiner [@KleinerModes] and Sakai et al., [@SakUstFiske]. In this section we will briefly recollect peculiarities of wave propagation and geometrical resonances in stacked Josephson junctions. In the ICJ model of Sakai, Bodin and Pedersen [@SBP], a layered superconductor is represented by a stack of isotropic superconducting layers with the thickness $d$ and the “intrinsic” penetration depth $\lambda_S$, separated by tunnel barriers with the thickness $t$, the dielectric constant $\epsilon_r$, and the fluctuation-free (maximum) Josephson critical current density $J_{c0}$. The stacking periodicity $s=t+d$ is $\simeq 1.5$ nm for Bi-2212. Properties of inductively coupled stacked Josephson junctions are described by the coupled sine-Gordon equation [@SBP]. The coupling is represented by a tridiagonal coupling matrix $A$ with the of-diagonal terms equal to minus the effective inductive coupling constant between neighbor junctions [@SakUstFiske], $$\label{CouplingConst}
S= \lambda_S \left[ t\sinh\left(
\frac{d}{\lambda_S}\right)+2\lambda_S\cosh\left(
\frac{d}{\lambda_S}\right)\right]^{-1}.$$ For atomic scale IJJs, $S\simeq 0.5-ds/4\lambda_S^2$ is very close to its maximum value 0.5.
A. Eigen-modes in stacked junctions
-----------------------------------
The main difference between single and stacked junctions is the presence of multiple electromagnetic wave modes in the stack. Geometrical resonances in a stack correspond to formation of two-dimensional standing waves [@KleinerModes; @SakUstFiske]. The wave number along the $ab$-planes ($x$-axis) is $k_m=\pi m
/L$, where $L$ is the length of the junctions and $m$ is the number of nodes in the standing wave. In the $c$-axis direction it is given by one of the eigen-modes, $k_n=\pi n/(N+1)s$, $n=1,2,...N$, where $N$ is the number of junctions in the stack. The oscillatory part of the phase difference is: $$\label{GeomModes}
\delta \varphi_{i}(m,n) = a \cos\left(\frac{\pi m x}{L} \right)
\sin\left(\frac{\pi n i}{N+1} \right)e^{j\omega t}.$$ Here $i=1,2,...N$ is the junction index, $a=$const is an amplitude, and $\omega$ is the angular frequency.
Each eigen-mode has a distinct propagation velocity, given by Eq. (3.52) of Ref. [@Economou]. Within the ICJ model they can be written as [@SakUstFiske]: $$\label{c_n}
c_n=c_0\left[1-2S\cos\left( \frac{\pi n}{N+1}\right)
\right]^{-1/2}, ~n=1,2,...N,$$ The Swihart velocity $c_0= \lambda_J \omega_p$ where $$\label{OmegaP}
\omega_p=\left[\frac{8\pi^2 t c J_c}{\Phi_0
\epsilon_r}\right]^{1/2},$$ is the Josephson plasma frequency and $$\label{Lambda_J}
\lambda_J=\left[\frac{\Phi_0c}{8\pi^2 J_{c0}\Lambda}
\right]^{1/2}\simeq\left[\frac{\Phi_0cs}{16\pi^2
J_{c0}\lambda_{ab}^2} \right]^{1/2}$$ is the Josephson penetration depth of a single junction and $$\label{lambda_ab}
\lambda_{ab}\simeq \lambda_S\sqrt{s/d}$$ is the effective London penetration depth for field perpendicular to layers.
Similarly, eigen-modes are characterized by different characteristic lengths [@fluxon] $$\label{LambdaN}
\lambda_n=\lambda_J\left[1-2S\cos\left( \frac{\pi m}{N+1}\right)
\right]^{-1/2}, ~n=1,2,...N.$$ $(\lambda_J/\lambda_n)^{2}$ are eigenvalues of the coupling matrix $A$ [@fluxon]. The shortest, $\lambda_N \simeq \lambda_J/\sqrt{2} \simeq 0.5~
\mu$m for Bi-2212. The longest $\lambda_1$ approaches the effective penetration depth for field parallel to layers $$\label{Lambda_c}
\lambda_c = \left[\frac{\Phi_0c}{8\pi^2 J_{c0} s} \right]^{1/2},$$ for $N\gg \pi\lambda_{ab}/s \simeq 400$. In Bi-2212, $\lambda_c(T=0)\sim 100~ \mu$m $\gg \lambda_{ab}(T=0)\simeq
0.2~\mu$m [@fluxon].
![\[fig1\] (Color online). Spatial distribution of oscillation amplitudes of (a) magnetic field and (b) in-plane currents for the in-phase (squares) and the out-of-phase (circles) modes for a stack with $N=10$ IJJs. Horizontal stripes represent superconducting layers.](Fig1_theory.EPS){width="48.00000%"}
Due to inductive coupling between junctions, the in-plane ($y$-axis) magnetic field is non-local and depends on phase distributions in all junctions: $B_y(i) = (H_0/2)A^{-1}\lambda_J
\partial\varphi_j / \partial x$. Here $H_0=\Phi_0/\pi \lambda_J
\Lambda$ [@fluxon]. Using Eq.(\[GeomModes\]) we obtain for the oscillatory part of magnetic field in the stack: $$\label{By(i)}
B_y(x,z)(m,n) = -\frac{H_0 a \pi m \lambda_n^2}{2L\lambda_J}
\sin\left(k_m x \right) \sin\left(k_n z\right).$$ Here we used the property that $A$ and $A^{-1}$ have the same eigenvectors, and eigenvalues of $A^{-1}$ are $\lambda_n^2/\lambda_J^2$, Eq. (\[LambdaN\]).
The in-plane current density in superconducting layers is obtained from the Maxwell equation $J_x=-(c/4\pi)
\partial B_y/\partial z$: $$\begin{aligned}
\label{Jx(i)}
J_x(x,z)(m,n) = J_{ac}(m,n)\sin\left(k_m x
\right) \cos\left(k_n z\right), \\
J_{ac}(m,n)=\frac{a\Phi_0 c \lambda_n^2 m n}{16 \lambda_{ab}^2
\lambda_J L (N+1)}.\end{aligned}$$ Fig. \[fig1\] shows calculated distributions of the amplitudes of $B_y$ (a) and $J_x$ (b) for modes $n=N$ (open circles) and $n=1$ (squares) for the stack with $N=10$ junctions. Horizontal stripes represent superconducting layers. It is seen that the eigen-modes are characterized by different symmetry along the stacking direction. The slowest $n=N$ mode corresponds to the (almost) out-of-phase state in neighbor junctions $\delta
\varphi_{i} \simeq -\delta \varphi_{i+1}$ . The fastest $n=1$ mode corresponds to the (almost) in-phase state $\delta \varphi_{i}
\simeq \delta \varphi_{i+1}$.
![\[fig2\] (Color online). (a) Calculated slowest and fastest velocities $c_N$ and $c_1$ as a function of the number of junctions in the stack. Calculations are made within the ICJ model for typical Bi-2212 parameters at $T=0$. (b) Calculated temperature dependencies of squares of $c_N$ and $c_1$ for several $N$ along with $\lambda_{ab}^{-2}(T)$, normalized by the corresponding values at $T=0$. ](Fig2_theory.EPS){width="40.00000%"}
Fig. \[fig2\] (a) shows calculated dependence of $c_1$ and $c_N$ on the number of junctions $N$. It is seen that the slowest velocity is almost independent of $N$ [@KatterweFiske] $$\label{cN}
c_N \simeq \frac{c_0}{\sqrt{2}}\simeq c
\left[\frac{ts}{4\varepsilon_r \lambda_{ab}^2 }\right]^{1/2}.$$ To the contrary, the fastest velocity $$\label{c1}
c_1 \simeq c \sqrt{\frac{t}{\varepsilon_r s}}
\left[1+\left(\frac{\pi\lambda_{ab}}{s(N+1)}\right)^2\right]^{-1/2}$$ is growing linearly with $N$ for $N < \pi\lambda_{ab}/s \simeq
400$ [@KatterweFiske]. For $N \gg \pi\lambda_{ab}(T)/s$, it asymptotically approaches the $T$-independent value $c_1(N\rightarrow \infty) =
c[t/\varepsilon_r s]^{1/2}$, close to the speed of light in the dielectric, as shown in Fig. \[fig2\] (a).
Fig. \[fig2\] (b) shows calculated $T$-dependencies of $c_1^2$ and $c_N^2$, Eq. (\[c\_n\]), normalized on the corresponding values at $T=0$, for different $N$. Calculations are made for typical parameters of Bi-2212, using the $[\lambda_{ab}(T)/\lambda_{ab}(0)]^{-2}$ dependence shown by the lowest line obtained from surface impedance measurements [@Lambda; @Lambda2; @Trunin; @Lambda2005; @Prozorov]. As follows from Eq. (\[cN\]), $T$-dependence of the out-of-phase velocity $c_N$ follows $1/\lambda_{ab}(T)$, irrespective of $N$. For IJJs the same is true for all slow modes $n\geqslant 2$.
The speed of the fastest mode, $c_1(T)$, does depends on $N$. For $N < \pi\lambda_{ab}/s \simeq 400$, it maintains the same $T$-dependence $\propto 1/\lambda_{ab}$. The corresponding three curves $[c_N(T)/c_N(0)]^2$, $[\lambda_{ab}(T)/\lambda_{ab}(0)]^{-2}$ and $[c_1(T)/c_1(0)]^2$ for $N=100$ collapse in one in Fig. \[fig2\] (b). For much larger $N$, when $c_1$ approaches $T$-independent speed of light in the dielectrics, see Fig. \[fig2\] (a), $c_1(T)$ becomes flatter at low $T$, as shown in Fig. \[fig2\] (b). However, since $\lambda_{ab}$ diverges at $T\rightarrow T_c$, $c_1$ always vanishes at $T_c$, as seen from the curve with $N=10^4$ in Fig. \[fig2\] (a).
In applied in-plane magnetic field Josephson vortices (fluxons) [@fluxon] enter into the junctions. In strong enough magnetic field fluxons form a regular fluxon lattice in a stack. Usually a triangular lattice is most stable due to fluxon repulsion. However a rectangular lattice can be stabilized via geometrical confinement is small Bi-2212 mesas [@Katterwe]. Motion of fluxons leads to appearance of the flux-flow (FF) branch in the $I$-$V$. Emission of EM waves in the FF state leads to excitation of geometrical resonances [@Koshelets; @Cirillo; @KatterweFiske]. The corresponding Fiske step voltage for the resonant mode $(m,n)$ is $$V_{m,n}(T)=\Phi_0 m c_n(T) /2L. \label{eq:Vmn}$$
The strongest resonance occurs at the velocity matching (VM) condition, when the velocity of fluxons is equal to the velocity of electromagnetic waves [@KatterweFiske]. This leads to appearance of the VM (Eck) step at the end of FF branch [@Cirillo]. The VM voltage is $$\label{VMstep}
V_{VM} \simeq N H s c_n.$$ The $T$-dependencies of both Fiske and VM steps are determined solely by $c_n(T)$, Eq.(\[c\_n\]). Therefore they can be used for accurate detection of the absolute value of $\lambda_{ab}(T)$ (except for the fastest mode at very large $N$, as shown in Fig. \[fig2\] (b)).
B. Connection between the inductively coupled and the Lawrence-Doniach models
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A similar system of coupled sine-Gordon equations was also obtained from the Lawrence-Doniach (LD) model [@LD]. The two main parameters of the LD model are the anisotropy factor $\gamma
=$const$\gg 1$ and the effective London penetration depths $\lambda_{ab}$. The rest of parameters are derived as [@LD]: $\lambda_c=\gamma \lambda_{ab}$, $\lambda_J=\gamma s$, $\omega_p=c/\varepsilon_r^{1/2}\gamma\lambda_{ab}$ and $c_0=cs/\varepsilon_r^{1/2}\lambda_{ab}$.
From comparison with ICJ expressions Eqs. (\[OmegaP\],\[Lambda\_J\],\[lambda\_ab\],\[Lambda\_c\],\[cN\]) it is seen that while the ICJ model contains two $T-$dependent variables $\lambda_{ab}(T)$ and $J_{c0}(T)$, the LD model has only one, $\lambda_{ab}(T)$, which imposes its $T$-dependence on all other variables. Within the range of validity of the LD model, $T_c - T\ll T_c$, the two models are identical because $\lambda_{ab}^{-2}(T)\propto J_{c0}(T) \propto 1-T/T_c$. However, as will be discussed below, $\lambda_{ab}^{-2}(T)$ and $J_{c0}(T)$ have distinctly different $T$-dependencies at low $T$, which does cause a discrepancy between the two models. Essentially it is related to the fact that in cuprates the anisotropy $\gamma(T)=
\lambda_c(T)/\lambda_{ab}(T) \neq$const [@Lambda_abc; @Lambda_c; @Radtke].
![\[fig:1\_4T\] (Color online) $I$-$V$ curves of the mesa-1 at $H=1.4$ T and at different $T=2.0~ \text{K}-15.1$ K. At low $T$, in panels (a) and (b), sequences of hysteretic (high-$Q$) individual Fiske steps are seen at low bias. At higher bias some junctions switch into the QP state, but Fiske steps are still present in the rest of the junctions. The corresponding first four mixed flux-flow-QP branches are marked (QP1-4). At $T=10.1$ K (c) these steps smear out and at $T=15.1$ K (d) individual Fiske steps have vanished, instead a collective, non-hysteretic step is observed.](figure_3a.eps "fig:"){width="23.00000%"} ![\[fig:1\_4T\] (Color online) $I$-$V$ curves of the mesa-1 at $H=1.4$ T and at different $T=2.0~ \text{K}-15.1$ K. At low $T$, in panels (a) and (b), sequences of hysteretic (high-$Q$) individual Fiske steps are seen at low bias. At higher bias some junctions switch into the QP state, but Fiske steps are still present in the rest of the junctions. The corresponding first four mixed flux-flow-QP branches are marked (QP1-4). At $T=10.1$ K (c) these steps smear out and at $T=15.1$ K (d) individual Fiske steps have vanished, instead a collective, non-hysteretic step is observed.](figure_3b.eps "fig:"){width="23.00000%"}\
![\[fig:1\_4T\] (Color online) $I$-$V$ curves of the mesa-1 at $H=1.4$ T and at different $T=2.0~ \text{K}-15.1$ K. At low $T$, in panels (a) and (b), sequences of hysteretic (high-$Q$) individual Fiske steps are seen at low bias. At higher bias some junctions switch into the QP state, but Fiske steps are still present in the rest of the junctions. The corresponding first four mixed flux-flow-QP branches are marked (QP1-4). At $T=10.1$ K (c) these steps smear out and at $T=15.1$ K (d) individual Fiske steps have vanished, instead a collective, non-hysteretic step is observed.](figure_3c.eps "fig:"){width="23.00000%"} ![\[fig:1\_4T\] (Color online) $I$-$V$ curves of the mesa-1 at $H=1.4$ T and at different $T=2.0~ \text{K}-15.1$ K. At low $T$, in panels (a) and (b), sequences of hysteretic (high-$Q$) individual Fiske steps are seen at low bias. At higher bias some junctions switch into the QP state, but Fiske steps are still present in the rest of the junctions. The corresponding first four mixed flux-flow-QP branches are marked (QP1-4). At $T=10.1$ K (c) these steps smear out and at $T=15.1$ K (d) individual Fiske steps have vanished, instead a collective, non-hysteretic step is observed.](figure_3d.eps "fig:"){width="23.00000%"}
III. Experimental
=================
Small mesa structures were fabricated on top of Bi-2212 single crystals with $T_c=82$ K. Twelve mesas with different sizes were fabricated simultaneously on every crystal. All of the studied mesas showed similar behavior. Here we present data for two mesas on the same slightly underdoped Bi-2212 crystal with areas of $2.7\times 1.4~ \mu \text{m}^2$ (mesa-1) and $2.0\times 1.7 ~\mu
up\text{m}^2$ (mesa-2). Both mesas contain $N=12$ IJJs. The results are representative for a large number of mesas made on crystals with different doping and composition (see Table-I in Ref.[@KatterweFiske]). Details of sample fabrication and of the experimental set-up can be found in Ref. [@KatterweFiske].
The magnetic field was applied strictly parallel to the superconducting CuO bilayers, to avoid the intrusion of Abrikosov vortices. Eventual entrance of Abrikosov vortices is immediately obvious in experiment: it causes very strong and irreversible damping of Fiske resonances and of the Fraunhofer modulation of the critical current [@Katterwe]. Essentially, results reported here are observable only in the absence of Abrikosov vortices. Using the rigorous alignment procedure, described in Ref. [@Katterwe], we were able to prevent Abrikosov vortex entrance in fields up to 17 T [@MR; @Polariton]. This is seen from the field-independence of the $c$-axis QP resistance [@MR] and perfect reversibility of all measured characteristics [@Katterwe; @KatterweFiske].
All measurements are made in the 3-probe configuration. To simplify data analysis, a contact or a quasiparticle resistance was subtracted from $I$-$V$ characteristics, as described in Ref. [@Katterwe]. The subtraction is facilitated by the negligible dependence of the QP resistance on the in-plane magnetic fields due to the extremely large anisotropy of Bi-2212 (see. e.g. Fig. 3 (d) in Ref. [@MR]). To do the subtraction, we first carefully measured the corresponding branch of the $I$-$V$ at zero magnetic field. After that we made a high-order polynomial fit of $\ln (I)$ vs. $V$, which is almost linear [@KatterwePRL] and can be fitted with a very high ($\sim \mu$V) accuracy. This fit is then subtracted from the measured $I$-$V$. When studying $T$-dependence, this procedure was repeated at each $T$. Such subtraction simplifies the analysis of Fiske steps, but is not necessary: Fiske steps can be also measured relative to the bias-dependent contact or QP voltages.
IV. Results
===========
Figure \[fig:1\_4T\] shows $I$-$V$ curves (digital oscillograms) for the mesa-1 at $H=1.4$ T and at different $T$ from 2.0 K to 15.1 K. As the current is increased, the $I$-$V$s switch from the zero voltage branch to the flux-flow branch, containing sequences of individual and collective Fiske steps, seen as small sub-branches in Fig. \[fig:1\_4T\] (a), and ending at the velocity-matching step. Detailed discussion of the magnetic field dependence of Fiske and VM steps at low $T$ can be found in Ref. [@KatterweFiske]. Strong hysteresis of Fiske steps indicates high $Q\gg1$ of the geometric resonances. This is facilitated by careful alignment of magnetic field, which prevents penetration of Abrikosov vortices [@MR]. With further increase of current some junctions switch into the QP state, while the rest are remaining in the flux-flow state. This leads to appearance of combined QP-FF families of Fiske steps, four of which are indicated in Fig. \[fig:1\_4T\](a), (QP1-4) with the number corresponding to the number of IJJs in the QP state.
The speed can be obtained directly from resonant voltages using Eqs.(\[eq:Vmn\]) and (\[VMstep\]). The corresponding low-$T$ values for several mesas at different Bi-2212 crystals can be found in Ref. [@KatterweFiske]. Fiske steps in Fig. \[fig:1\_4T\] correspond to slow speed resonances $V_{2,N}=0.27$ mV. At the QP1, QP2 branches another sequence $V_{4,N}=0.54$ mV of individual Fiske steps is seen. As shown in Refs. [@KatterweFiske; @Polariton], the $V_{VM}$ is proportional to the field for 2 T $<H <$ 10 T, consistent with Eq. (\[VMstep\]), before it gets interrupted by phonon-polariton resonances at higher fields [@Polariton]. In this intermediate field range the limiting fluxon velocity is close to the out-of-phase velocity $c_N$.
{width="90.00000%"}
A. Temperature dependence of the quality factor
-----------------------------------------------
As seen from Fig. \[fig:1\_4T\], with increasing temperature, the amplitude of the individual Fiske steps rapidly decreases. At $T=10.1$ K steps are smeared out almost completely and at $T=15.1$ K they vanish. This indicates a substantial reduction of the resonance quality factor. At this temperature only a collective, non-hysteretic Fiske step is visible at $N\times V_{2,N} \simeq
3.2$ mV, see Fig \[fig:1\_4T\] (d).
Figures \[fig:4band6b\] (a) and (b) show $I$-$V$s in a wider $T$-range (a) for the mesa-2 at $H=2.75$ T, and (b) for the mesa-1 at $H=3.85$ T. Collective Fiske steps at $\approx N\times V_{1,N}$ can be seen at low $T$ (indicated by the downward arrows). At higher bias VM steps are observed (indicated by the upward arrows). Both mesas show similar behavior: Sharpness of the collective Fiske and the VM steps rapidly decreases with increasing temperature. This indicates enhancement of damping, also seen from reduction of slopes of $I$-$V$ curves with increasing $T$.
Figure \[fig:4band6b\] (c) shows d$I$/d$V$ curves, numerically calculated from the $I$-$V$ curves from (b). Peaks in conductance correspond to Fiske and VM steps. The decrease of amplitudes of the steps with increasing $T$ is clearly seen, indicating reduction of $Q$ at higher temperatures.
According to the sine-Gordon equation, the initial viscous part of the flux-flow $I$-$V$ should be ohmic with the flux-flow resistance $R_{FF}$ representing the effective damping [@Scott]. Indeed, from Figs. \[fig:4band6b\] (a) and (b) it is seen that the flux-flow $I$-$V$ is nearly ohmic at $10 < V <
20$ mV. This allows accurate evaluation of the bare (non-resonant) $R_{FF}(T)$. It is shown in panel (d) for $V=12$ mV ($\sim $ 1 mV per junction). The $T$-dependence of $R_{FF}$ is almost identical to the low-bias $c$-axis QP resistance $R_{QP}(T, H=0)$ [@KatterwePRL], proving that the $R_{FF}(T)$ dependence is predominantly determined by “freezing out" of quasiparticles. At low $T$ and moderately low $H$ the value of $R_{FF}$ is slightly lower than $R_{QP}$, which may indicate presence of additional damping mechanisms, such as the in-plane QP damping [@KoshelevInplane], or generation of phonons via electrostriction [@Polariton]. At higher $H$, $R_{FF}=R_{QP}$ (see e.g. Fig. 3 (d) from Ref.[@MR]).
Figures \[fig:4band6b\] (e) and (f) represent $T$-dependencies of bare amplitudes of conductance peaks at the collective Fiske step and the VM step, respectively. The peak amplitudes were obtained by subtracting the background flux-flow conductance $R_{FF}^{-1}$. It is seen that resonances in both mesas exhibit similar $T$-dependencies: At low $T$, peaks are high, i.e., quality factors of resonances are large $Q\gg 1$, but they start to rapidly decrease with increasing $T$. Comparison with the effective flux-flow resistance $R_{FF}$, shown in panel (d), indicates that the scale for variation of peak amplitudes is similar to $R_{FF}(T)$. Therefore, both resonances roughly follow Eq. (\[eq:Q\]) with $R \simeq R_{FF}(T)$.
![\[fig3\] (Color online) Normalized temperature dependence of the square of the velocity-matching voltage $V_{VM}\propto c_N$. Solid line represents typical $T$-dependence of $\lambda_{ab}^{-2}$ from Ref. [@Lambda; @Lambda2]. Dashed line represents the $T$-dependence of the fluctuation free Josephson critical current density $J_{c0} \propto \omega_p^2$ from Ref. [@Collapse].](Fig5.EPS){width="40.00000%"}
B. Temperature dependence of the Swihart velocity
-------------------------------------------------
Both Fiske and VM steps in the considered case correspond to propagation of waves, respectively fluxons, with the velocity $\simeq 3.2 \times 10^5$ m/s [@KatterweFiske] is close to the expected value of the slowest out-of-phase velocity $c_N$, Eq. (\[cN\]). It is almost 1000 times slower than $c$, not because of extraordinary large dielectric constant, but because of extraordinary large kinetic inductance of atomically thin superconducting layers in Bi-2212, see Eq. (2). According to Eq. (\[cN\]), $c_N(T)$ should depend solely on $1/\lambda_{ab}(T)$. Thus voltages of Fiske and VM steps should provide a direct information on absolute values of $1/\lambda_{ab}(T)$.
Squares and triangles in Fig. \[fig3\] represent measured $T$-dependencies of $V_{VM}^2$ for both studied mesas. Crosses in Fig. \[fig3\] represent fast geometrical resonance voltages, reported recently by Benseman and co-workers on large Bi-2212 mesas at zero field [@Tim]. Apparently, our data for the slowest resonances coincide with their data for the fast resonance within the measured $T$-range.
Lines in Fig. \[fig3\] represent typical temperature dependencies of $\lambda_{ab}^{-2}$ for cuprates [@Lambda; @Lambda2] and the fluctuation-free $c$-axis critical current density $J_{c0}$ for Bi-2212 IJJs [@Kleiner92; @Collapse]. The latter is similar to $\omega_p^{2}(T)$, measured by the Josephson plasma resonance [@Plasma] and to $\lambda_c^{-2}(T)$ obtained from surface impedance measurements [@Lambda_c], consistent with Eqs.(\[OmegaP\],\[Lambda\_c\]). It is seen that $\lambda_{ab}^{-2}$ and $J_{c0}$ exhibit distinctly different behavior at low $T$: $J_{c0}(T)$ is flat, while $\lambda_{ab}^{-2}(T)$ has a linear $T$-dependence due to the d-wave symmetry of the order parameter [@Lambda; @Lambda2]. Clearly, experimental $V_{VM}^2(T)$ follow $J_{c0}(T)$ rather than the expected $\lambda_{ab}(T)^{-2}$ dependence.
V. Discussion
=============
At low $T$, the obtained speed of EM waves $\simeq 3.2 \times
10^5$ m/s agrees with the expected out-of-phase mode velocity $c_N$, Eq. (\[cN\]) for reasonable parameters $t/\varepsilon_r
=0.1$ nm and $\lambda_{ab}(T=0) \simeq 200$ nm [@Lambda; @Lambda2; @Lambda_abc; @Trunin]. Thus the ICJ model does provide a correct value of the Swihart velocity at low $T$. It also provides correct $T$-dependencies of the Josephson plasma frequency [@Plasma] and $\lambda_c$ [@Lambda_c], $\omega_p(T)\propto \lambda_c^{-1}(T)\propto \sqrt{J_{c0}(T)}$, see Eqs.(\[OmegaP\],\[Lambda\_c\]). Therefore, it is surprising that the $T$-dependence of the effective penetration depth deduced from resonant voltages is different from $\lambda_{ab}(T)$, obtained from surface impedance measurements [@Lambda; @Lambda2; @Lambda_abc; @Trunin]. Below we mention several possible reasons for such a discrepancy.
A. Possible origin of discrepancy with surface impedance measurements
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Derivation of the ICJ model is based on the assumption that field and current distributions within each superconducting layer can be described by the local 2nd London equation [@SBP]. However, this assumption most likely breaks down in atomic scale IJJs (see the condition (4.3) in Ref. [@Economou]).
To understand the reported discrepancy it is, first of all, necessary to understand the difference in local current and field distributions. In surface impedance measurements, the external electromagnetic field is screened at the depth $\lambda_{ab} \sim
200$ nm from the surface of the superconductor. This induces similar (in-phase) screening currents in a fairly large number $N
\sim 130$ of IJJs. To the contrary, at the out-of-phase geometrical resonances the current varies at the atomic scale, as shown in Fig. \[fig1\] (b).
[*i. Non-locality of supercurrent*]{}
The most obvious question is to what extent Cooper pairs are localized in every CuO bi-layer. The very existence of the $c$-axis critical current indicates that the localization is incomplete. This can be particularly significant for the out-of-phase mode, when Cooper pairs are forced to move in opposite directions in neighbor layers, see Fig. \[fig1\] (b). Qualitatively such delocalization will lead to larger effective penetration depth. [*ii. Nonlocal Josephson electrodynamics*]{}
Another type of non-locality in thin layer junctions was considered in Ref. [@Mints]. With decreasing $d$, the effective screening length $\Lambda/2$ Eq.(3) increases and approaches the Pearl length $\lambda_P=\lambda_S^2/d$. To the contrary, the Josephson penetration depth $\lambda_J$ decreases $\propto \Lambda^{-1/2}$ see Eq.(\[Lambda\_J\]). For IJJs $\lambda_J < 1~\mu$m [@Katterwe] is much smaller than $\lambda_P \sim 10~\mu$m even at $T=0$. Such a mismatch changes the dispersion relation of electromagnetic waves [@Mints].
[*iii. Retardation effects*]{}
Retardation effects appear in transmission lines when the time (phase velocity) required to transfer charge within a layer is comparable or faster than that for electromagnetic waves outside the layer [@Economou]. Specific for IJJs is that the out-of-phase electromagnetic wave velocity is so slow $\sim 10^5
~$ m/s, that it becomes comparable to the electronic Fermi velocity. This may affect the dispersion relation.
[*iv. Frequency dependence*]{}
The effective penetration depth in superconductors depends not only on $T$ but also on frequency $\lambda(T,\omega)$. It originates from a significant $(T,\omega)$ dependence of complex conductivity in a superconductor [@Ngai]. The most obvious difference between static and high-frequency $\lambda(T)$ is that the latter does not diverge at $T\rightarrow T_c$, but approaches the finite normal skin-depth. This may flatten-out $T$-dependence of high frequency Fiske steps, compared to static $1/\lambda(T)$ [@Ngai].
Surface impedance measurements are typically performed at $\sim
10$ GHz frequency. In comparison, the studied Fiske and VM step voltages are $\sim 1$ mV per junction, see Fig. \[fig:4band6b\]. According to the ac-Josephson relation, this corresponds to $\sim
500$ GHz. The significant difference in frequencies may lead to a significant difference in the effective $\lambda$.
At even higher THz frequencies, the frequency dependence of the dielectric function $\varepsilon_r(\omega)$ in isolating BiO layers becomes significant. As shown in Ref. [@Polariton], the speed of electromagnetic waves slows down dramatically, when the frequency approaches the transverse optical phonon frequencies.
[*v. Non-linear effects*]{}
Eq.(\[c\_n\]) was derived by linearization of the coupled sine-Gordon equation and is valid for small amplitude EM waves $a
\ll 1$. However, at high quality geometrical resonances the amplitude may be large $a\sim 2\pi$ and non-linearity of the sine-Gordon equation may affect the dispersion relation.
The penetration depth depends on the absolute value of the current density. Close to the depairing current density, $\lambda$ rapidly increases. At geometrical resonances the amplitude of the in-plane current density is given by Eq. (14). It depends on the amplitude $a$, which can be $\sim 1$ for $Q\gg 1$. An estimation for $a=1$ and $L/m = 1 ~\mu$m yields $J_{ac}\sim 10^6$ A/cm$^2$, comparable to the maximum in-plane current density [@Yurgens].
Both types of non-linear effects increase with increasing the quality factor of resonances. Since $Q\gg 1$ only at low $T$, nonlinear corrections can be significant at low $T$, but less so at elevated temperatures.
B. Implications for coherent Josephson oscillators
--------------------------------------------------
As mentioned in the Introduction, stacked IJJs are considered as possible candidates for high power THz oscillators [@Ozyuzer; @Wang; @Nonequilibrium; @Hu; @Klemm; @Breather; @Tim]. A large energy gap in Bi-2212 [@KatterwePRL; @MR] allows generation of electromagnetic radiation with frequencies in excess of $10$ THz. For example, recently polariton generation with frequencies up to $\sim 13$ THz was reported [@Polariton]. Moreover, strong electromagnetic coupling of IJJs facilitates phase-locking of many junctions, which may lead to coherent amplification of the emission power [@Note1].
Realization of a flux-flow oscillator [@Koshelets], based on fluxon motion in the in-plane magnetic field [@Cherenkov; @Bae2007; @FFlowHatano; @FiskeInd], encounter a difficulty, associated with instability of the rectangular fluxon lattice. It can be stabilized by geometrical confinement in small mesas [@Katterwe] or by interaction with infrared optical phonons [@Polariton]. But usually fluxon-fluxon repulsion promotes the triangular fluxon lattice, corresponding to the out-of-phase state, which leads to destructive interference and negligible emission [@TheoryFiske].
High quality geometrical resonances improve operation of a stacked oscillator is several ways: (i) they amplify the emission power $\propto Q^2$ [@TheoryFiske]; (ii) they narrow the radiation linewidth $\propto 1/Q$ [@TheoryFiske]; (iii) they can [*force*]{} phase-locking of junctions. Numerical simulations have demonstrated that large amplitude standing waves, $a\sim 1$, can superimpose their symmetry on the fluxon lattice [@KatterweFiske]. Such a non-linear synchronization requires high $Q$ because $a \propto 1/Q$.
The reported rapid decrease of the quality factor with increasing temperature indicates that self-heating is detrimental for the coherent Josephson oscillator and ultimately limits the emission power from large Bi-2212 mesas [@Breather]. On the other hand, $T$-dependence of the Swihart velocity facilitates fairly broad-range tuning of the resonance frequency, as seen from Fig. \[fig3\]. This may be beneficial for the oscillator [@Tim].
Conclusions
===========
To conclude, we have studied $T$-dependence of geometrical and velocity matching resonances in small Bi-2212 mesa structures. We reported strong $T$-dependence of the quality factors, which is large at low $T$, but rapidly decreases with increasing $T$, already at $T>10$ K. Above $T \sim 60$ K $\sim 0.8 T_c$ resonances are almost fully damped. This observation is consistent with previous observations of strongly underdamped phase dynamics at low $T$ [@Collapse], leading to relatively high macroscopic quantum tunnelling temperature in IJJs [@Lombardi; @MQT; @MQT2], and with the reported collapse into overdamped dynamics at $T/T_c
\sim 0.8$ [@Collapse]. The rapid decrease of $Q(T)$ indicates, that self-heating is detrimental for operation of the coherent THz oscillator and ultimately limits its performance [@Breather]. On the other hand, $T$-dependence of the Swihart velocity facilitates a broad-range tuning of the resonance frequency, which may be beneficial for the oscillator.
Our analysis of $T$-dependence of resonant voltages revealed that the effective penetration depth, that determines the kinetic inductance and the speed of electromagnetic waves in intrinsic Josephson junctions, see Eq. (\[Lsquare\]), is exhibiting a flat $T$-dependence at low, resembling $T$-dependence of the $c$-axis critical current. It is distinctly different from the linear $T$-dependence of $\lambda_{ab}(T)$, obtained from surface impedance measurements [@Lambda; @Lambda2]. We argued that non-trivial physical phenomena, such as break-down of the local London approximation at the atomic scale, are responsible for this distinct discrepancy, which deserves further theoretical consideration.
Acknowledgements: We are grateful to A. Rydh and H. Motzkau for assistance in experiment and to the Swedish Research Council and the SU-Core Facility in Nanotechnology for financial and technical support, respectively.
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|
---
abstract: 'We studied the energy levels of graphene based Andreev billiards consisting of a superconductor region on top of a monolayer graphene sheet. For the case of Andreev retro-reflection we show that the graphene based Andreev billiard can be mapped to the normal metal-superconducting billiards with the same geometry. We also derived a semiclassical quantization rule in graphene based Andreev billiards. The exact and the semiclassically obtained spectrum agree very well both for the case of Andreev retro-reflection and specular Andreev reflection.'
author:
- 'J[ó]{}zsef Cserti [^1]'
- Imre Hagymási
- Andor Kormányos
title: Graphene Andreev Billiards
---
In the well-known Andreev billiards consisting of a normal metal surrounded by a superconductor (NS) the dynamics of the quasiparticles is determined by the so-called [*Andreev retro-reflection*]{} [@Andreev:cikk]. The spectrum of Andreev billiards is described by the Bogoliubov-de Gennes (BdG) equation and has been long studied [@Kosztin:ref; @Lodder-Nazarov:ref] (for review of the topic see, eg, [@Beenakker_LectureNotes]).
The electronic properties of graphene can be described accurately by massless Dirac fermion type excitations using two dimensional relativistic quantum mechanics [@Novoselov_graphene-1; @Zhang_graphene:ref; @Falko_Weak_loc_gr:ref] and also by semiclassical methods [@carmier:245413] (for reviews on the physics of graphene see, e.g., [@review_graphene:cikkek]). In the seminal paper by Beenakker [@beenakker:067007] it has been shown that when monolayer graphene is interfaced with a superconductor then two types of Andreev reflection are possible depending on the ratio of the Fermi energy $E_{\rm F}^{\left(\rm{G}\right)}$ and the electron energy $E$. For $E_{\rm F}^{\left(\rm{G}\right)} > \Delta^{(G)} >E$ the Andreev retro-reflection is dominant as in NS billiards (here $\Delta^{(G)}$ is the superconducting pair potential induced in the graphene). In contrast, when $E_{\rm F}^{\left(\rm{G}\right)} < E < \Delta^{(G)}$, a different type of scattering process takes place at the graphene-superconductor interface, which is named [*specular Andreev reflection* ]{}. The specular Andreev reflection does not exist in NS systems and it is a prominent consequence of the peculiar band structure of the monolayer graphene. Beenakker’s paper has been followed by numerous works [@ref:beenakker-et-al] (for a review on Andreev reflection in graphene see article [@beenakker:1337]). Note that although graphene itself is not superconducting, due to the proximity effect a superconductor can induce non-zero pair-potential in the graphene as well. Indeed, supercurrent has been observed experimentally [@Heersche_supra:ref] between two superconducting electrodes on top of a graphene monolayer. Moreover, experimental results of Ref. [@ref:Miao] attest to the ballistic propagation of quasiparticles in graphene-superconductor hybrid structures.
The most widely studied theoretical model of Andreev billiards is that of a two dimensional electron gas (2DEG) in a quantum dot contacted by a bulk superconductor (see eg. [@Beenakker_LectureNotes]). One of the major obstacles that has thwarted so far the direct comparison of the theoretical predictions and experimental results is the inevitably existing tunnel barrier and mismatch of the Fermi-velocities and effective masses between the 2DEG and the superconductor (often referred to as “non-ideal NS interface” in the literature). This mismatch causes the probability of normal reflection to increase at the NS interface while the probability of the Andreev reflection diminishes significantly. The situation when both normal and Andreev reflection take place at the NS interface is theoretically more difficult to address. In graphene however, when the superconductivity is induced by external superconducting contacts, such mismatch may not exist so that the graphene-superconductor systems may experimentally be ideal to study most of the theoretical predictions made assuming perfect (ie with no mismatch) NS-interfaces.
In this paper we consider [*graphene Andreev billiards*]{} (GABs). In particular, we assume that in a closed region $\cal{D}$ of the graphene sheet the superconducting pair potential is zero and outside this region it takes on a constant value $\Delta^{(G)}$. We demonstrate, in one hand, that when the retro-reflection is the dominant scattering process at the normal graphene-superconductor interface the electronic properties of GABs can indeed be obtained in *semiclassical approximation* from the known results for NS billiards with ideal NS interface. On the other hand, we also calculate the exact spectrum of a GAB for the case when the dominant scattering process is the specular Andreev reflection and we show that it can also be understood using semiclassical considerations.
To see the relation between the energy spectrum of NS billiards and GABs note the following: the dispersion relation of the quasiparticles in the normal ($\Delta^{(N)} = 0$) region of the NS billiards for energies $E<\Delta^{(N)}$ can be linearized around the Fermi energy $E_{\rm F}^{\left(\rm{N}\right)}$ as $E(p)=\pm {v}_{\rm F}^{(N)} ({ p}-p_{\rm F}^{(N)})$, where the sign $+$ ($-$) refers to the electron-like (hole-like) quasiparticles. Here $p$ is the magnitude of the momentum, $p_{\rm F}^{(N)}= \sqrt{2m E_{\rm F}^{\left(\rm{N}\right)}}$ is the Fermi momentum and $v_{\rm F}^{(N)} = p_{\rm F}^{(N)}/m$ is the the Fermi velocity. This linearization is allowed if we are interested in the properties of the bound states ($E<\Delta^{(N)}$) of NS billiards because for typical NS billiards the dimensionless parameter $\Delta^{(N)}/E_F^{(N)}\ll 1$ is much less than unity. The *same linear dispersion* can be found for electron-like (hole-like) quasiparticles in the $\Delta^{(G)}=0$ region for GABs in the retro-reflection regime but with Fermi velocity $v_{\rm F}^{(G)}$ and Fermi momentum $p_{\rm F}^{(G)}$. This simple observation is the core of the intimate relation between the graphene based and normal metal Andreev billiards. As long as the effect of the superconductor in semiclassical approximation can be described by the same way for GABs as for the NS billiards, i.e. by a simple phase shift $-\arccos(E/\Delta^{(N,G)})$, one can expect that when the Andreev retro-reflection is the dominant scattering process the gross features of the energy spectrum of a GAB will closely resemble the spectrum of a NS billiard having the same geometry. This happens because the quasiparticles have linear dispersion in both cases. Moreover, if the Fermi velocities and Fermi momentums are the same i.e., $v_{\rm F} = \hbar k_{\rm F}^{(N)}/m =
E_{\rm F}^{\left(\rm{G}\right)}/(\hbar k_{\rm F}^{(G)})$ and $p_{\rm F}=\sqrt{2m E_{\rm F}^{(N)}}=E_{\rm F}^{(G)}/v_{\rm F}$ the quasiparticles in the $\Delta^{(N,G)}=0$ region will have the same dispersion relation for both NS billiards and GABs. Note that if $p_{\rm F}$ and $v_{\rm F}$ are the same then $E_{\rm F}^{(N)}=E_{\rm F}^{(G)}/2$.
To demonstrate the idea discussed above we consider a simple, circular shape GAB. It consists of normal graphene region of radius $R$ surrounded by superconducting graphene. Owing to the valley degeneracy of the Hamiltonian the full BdG equation for graphene-superconductor systems decouples to two four by four, reduced Hamiltonians that are related to each other by a unitary transformation (see, e.g., [@beenakker:067007]). We now take the one corresponding to the valley ${\bf K}$. Due to the circular symmetry of the setup the reduced Hamiltonian is separable in polar coordinates and therefore the eigenfunctions can be labelled by an integer number $m$ corresponding to the angular momentum quantum number. One can show that the ansatz for the wave functions satisfying the Schr[ö]{}dinger equation for the reduced Hamiltonian in the region where $\Delta^{(G)} =0$, ie, for $r<R$ with energy $E$ are $\Psi_{m}^{\left(\text{N}\right)}(r,\varphi) \! =
\! \left(
c_+^{\left(\text{N}\right)}\,
\chi^{\left(\text{N}\right)}_+(r,\varphi)
+ c_-^{\left(\text{N}\right)} \,
\chi^{\left(\text{N}\right)}_-(r,\varphi)
\right) \!\! \, e^{im\varphi}$, where $\chi^{\left(\text{N}\right)}_+(r,\varphi)
~= {\left[-iJ_m(k_+ r),J_{m+1}(k_+ r)e^{i\varphi},0,0\right]}^T$ and $\chi^{\left(\text{N}\right)}_-(r,\varphi)
~= {\left[0,0, -iJ_m(k_- r),J_{m+1}(k_- r)e^{i\varphi}\right]}^T$ are the two eigenstates, and $k_\pm ~= \left(E_{\rm F}^{\left(\rm{G}\right)} \pm E
\right)/\left(\hbar v_{\rm F} \right)$. In the superconducting region $r>R$ where the pair potential is $\Delta^{(G)}$ the wave function has the form $\Psi_{m}^{\left(\text{S}\right)}(r,\varphi) \!\! =
\hspace{-2mm}\left(
c_+^{\left(\text{S}\right)}\,
\chi^{\left(\text{S}\right)}_+(r,\varphi)
+ c_-^{\left(\text{S}\right)} \,
\chi^{\left(\text{S}\right)}_-(r,\varphi)
\right) \!\! \, e^{im\varphi}$, where $\chi^{\left(\text{S}\right)}_+(r,\varphi) =
{\left[u^{\left(\text{S}\right)}_+,v^{\left(\text{S}\right)}_+\right]}^T$, $u^{\left(\text{S}\right)}_+= \gamma_+ v^{\left(\text{S}\right)}_+$, $v^{\left(\text{S}\right)}_+ = {\left[-i H_m^{(1)}(q_+ r),
H_{m+1}^{(1)}(q_+ r)e^{i\varphi}\right]}^T$. The eigenstate $\chi^{\left(\text{S}\right)}_-(r,\varphi)$ is obtained by the replacement $+ \to -$ and the first kind of Hankel functions to the second one and $q_\pm=\left(E_{\rm F}^{\left(\rm{G}\right)}
\pm i\sqrt{[\Delta^{(G)}]^2-E^2}\right)/(\hbar v_{\rm F})$, while $\gamma_{\pm} = e^{\pm i \arccos \left(E/\Delta^{(G)}\right)} $. Here $J_{m}(x) $ and $H_{m}^{\left(1,2 \right)}(x)$ are the Bessel and the Hankel functions [@Abramowitz:book]. To ensure that the wave function of the bound states is normalizable, the wave function in the superconducting region must go to zero as $r \to \infty$. This condition can be satisfied by choosing the appropriate Hankel function in the eigenstates $\chi^{\left(\text{S}\right)}_\pm(r,\varphi)$ [@Abramowitz:book]. Finally, the unknown coefficients $c_{\pm}^{\left(\text{N}\right)}$ and $c_{\pm}^{\left(\text{S}\right)}$ can be determined from the boundary conditions $\Psi_{m}^{\left(\text{N}\right)}(r=R,\varphi) =
\Psi_{m}^{\left(\text{S}\right)}(r=R,\varphi)$ valid for any $\varphi$. Thus, the condition for non-trivial solutions of the coefficients $c_{\pm}^{\left(\text{N}\right)}$ and $c_{\pm}^{\left(\text{S}\right)}$ can be found from the zeros of a four by four determinant. After some algebra we obtain a quite simple secular equation for the energy levels with fixed angular momentum index $m$:
\[szek0:eq\] $$\begin{aligned}
\label{DGS:eq}
&&\hspace{-10mm} \rm{Im} \left \{\gamma_+ D^{\rm{(+)}}_{\rm{GS}}(m,E) \,
D^{\rm{(-)}}_{\rm{GS}}(m,E) \right \} = 0, \\[1ex]
\label{De-disk_GS:eq}
D^{\rm{(+)}}_{\rm{GS}}(m,E) &=&
\left| \begin{array}{cc}
J_m(k_+ R) & H_m^{(1)}(q_+ R)\\
J_{m+1}(k_+ R) & H_{m+1}^{(1)}(q_+ R)
\end{array} \right|, \end{aligned}$$
and $D^{\rm{(-)}}_{\rm{GS}}(m,E) =
{\left[D^{\rm{(+)}}_{\rm{GS}}(m,-E)\right]}^*$, and ${\rm Im}\{. \}$ and $* $ stand for the imaginary part and the complex conjugation, respectively. Note that $H_m^{(2)}(q_- R) = {\left[H_m^{(1)}(q_+ R)\right]}^*$. The solutions of Eq. (\[szek0:eq\]) for $m=0,\pm 1,\pm 2,\cdots,$ are the exact energy levels of a circular shape GAB. Note that Eq. (\[szek0:eq\]) is valid both in the case of Andreev retro-reflection ($E_{\rm F}^{\left(\rm{G}\right)} > \Delta^{(G)} > E$) and for specular Andreev reflection ($E_{\rm F}^{\left(\rm{G}\right)} < E < \Delta^{(G)}$). One can also notice that the eigenenergies depend only on two dimensionless parameters: $E_{\rm F}^{\left(\rm{G}\right)}/\Delta^{(G)} $ and $\xi_c^{(G)}/R$, where $\xi_c^{(G)}=\hbar v_{\rm F}^{(G)}/\Delta^{(G)}$ is the coherence length in the superconducting graphene.
We now compare the density of states (DOS) $\varrho (E) =
\sum_{nm}\delta(E-E_{nm})$ of a circular shape GAB and of the corresponding NS billiard. For details of the calculation see a similar calculation for NS billiards in Ref. [@sajat_cake:cikk]. It is more convenient to plot the integrated DOS, namely the so-called step function $N(E)= \sum_{nm}\Theta(E-E_{nm})$, where $\Theta(x)$ is the Heaviside function. Our numerical results for $E_{\rm F}^{\left(\rm{N}\right)}/\Delta^{(N)}
= E_{\rm F}^{\left(\rm{G}\right)}/(2\Delta^{(G)})$ are shown in Fig. \[lpecso-retro:fig\]. One can see that the step functions for the considered NS billiard and GAB are indeed very similar.
![\[lpecso-retro:fig\] (color online) The exact step function $N(E)$ for circular shape GAB (red line) and NS (blue line) billiards in the case of Andreev retro-reflection. The parameters for GAB and NS billiard are $E_{\rm F}^{\left(\rm{G}\right)}/\Delta^{(G)} = 10$ and $E_{\rm F}^{\left(\rm{N}\right)}/\Delta^{(N)} = 5$, respectively, and $\xi_c^{(G)}/R = 0.12$ for both cases. The insets show the enlarged parts of the main frame. The trivial factor 2 owing to the valley degeneracy in graphene is not included.](metal_grafen.eps)
It is also clear from Fig. \[lpecso-retro:fig\] that the DOS $\varrho (E) =dN(E)/dE$ shows singularities at certain energies $E_n^{\left(\rm sing \right)}$. Singularities of this kind arise in the case of NS billiards as well (see e.g. Ref. [@sajat_cake:cikk]) and we shall discuss their origin below.
We now demonstrate on the example of circular GABs that at semiclassical level the results for NS billiards and GABs can be mapped into each other by choosing the parameters appropriately. In numerous works [@Melsen; @Lodder-Nazarov:ref; @Heny; @Ihra-Richter1:ref; @ref:adagideli; @sajat_cikkek] it was shown that for NS billiards in semiclassical approximation the step function reads
\[BS\_Stepfn:eq\] $$\begin{aligned}
N_{\rm BS} (E) &=& M \,\sum_{n=0}^\infty \,\left\{1-F[s_n(E)]\right\},
\label{F_s:eq} \\
s_n(E) &=& \frac{n\pi +\arccos \left(E/\Delta^{(N)} \right)}{E/\Delta^{(N)}} \,
\xi_c^{(N)}. \label{s_n:eq} \end{aligned}$$
Here $M$ is the number of open channels in the normal region, $\xi_c^{(N)}=\hbar v_{\rm F}^{(N)}/\Delta^{(N)}$ is the coherence length in the NS system, $F(s) = \int_0^s \, P(s^\prime) \, ds^\prime$ is the integrated path length distribution and $P(s)$ is the classical probability that an electron entering the billiard at the NS interface returns to the interface after a path of length $s$. The path length distribution $P(s)$ is normalized to one, i.e., $\int_0^\infty\, P(s)\, ds =~1$ and one can see that it is a purely geometry-dependent function. In particular, for circular billiards it was found that $P(s) = \frac{1}{{\left( 2R \right)}^2} \,
\frac{s}{\sqrt{1-{\left(s/2R\right)}^2}} \, \Theta(2R-s)$ and $M=2\pi\, k_{\rm F}^{(N)} R$ [@sajat_cake:cikk]. Finally, the quantity $s_n(E)$ in Eq. (\[s\_n:eq\]) depends on the quantization condition for the periodic motion of the electron-hole quasiparicles [@Melsen; @ref:adagideli]. As it has been pointed out in the introduction, in good approximation the quasiparticles have linear dispersion in the non-superconducting region for both GABs and NS billiards. If the effect of the superconductor in GABs can be taken into account by a simple phase shift $-\arccos(E/\Delta^{(G)})$ [@beenakker:1337], expressions of the type of Eq. (\[BS\_Stepfn:eq\]) can be used to calculate the semiclassical approximation of $N(E)$ for GABs as well.
Moreover, employing the same steps as in Ref. [@sajat_cake:cikk], from Eq. (\[szek0:eq\]) one can derive the following semiclassical quantization rule for circular shape GABs:
\[BS\_E\_m:eq\] $$\begin{aligned}
\label{rad_action_quant:eq}
&& \hspace{-7mm} S_+(E) \! - \! \mu_r \, S_-(E)
\! - \! 2\arccos \frac{E}{\Delta^{(G)}}
= 2 \pi \! \left(\!\! n+\frac{1-\mu_r}{4}\!\! \right), \\[2ex]
&& \hspace{-7mm}
S_\pm (E) = 2\sqrt{{\left(\left|k_\pm \right| R\right)}^2-m^2}
-2\left|m \right| \arccos \frac{\left| m \right|}{\left|k_\pm \right| R},
\label{semi_quant:eq} \end{aligned}$$
where $\mu_r = 1, -1$ for Andreev retro-reflection and specular Andreev reflection, respectively, and $n$ is a non-negative integer. Functions $ S_\pm (E)$ are the radial action (in units of $\hbar$) of electrons and holes [@Brack:book] and the term $-2\arccos E/\Delta^{(G)}$ in Eq. (\[rad\_action\_quant:eq\]) accounts for the two Andreev reflections in one period of the orbit, while the second term in the left hand side of Eq. (\[rad\_action\_quant:eq\]) results from the sum and the difference of the Maslov indices $\pi/4$ of the electron-like and hole-like particles for $\mu_r = 1$ and $\mu_r = -1$, respectively.
Formally, in the case of Andreev retro-reflection the quantization condition shown in Eq. (\[BS\_E\_m:eq\]) is the same as for a circular NS billiard [@sajat_cake:cikk], but the meaning of $k_{\pm}$ is different for the two systems (for NS billiards see eg. Ref. [@sajat_cake:cikk]). However, from Eq. (\[BS\_E\_m:eq\]) it is easy to find that if $R^{(N)}/\xi_c^{(N)}=R^{(G)}/\xi_c^{(G)}$ and $E_F^{(N)}/\Delta^{(N)}=E_F^{(G)}/(2 \Delta^{(G)})$ then to first order in $E/\Delta^{(N,G)}$ the quantization condition for circular GABs and NS billiards is the same and the step function $N(E)$ is given by Eq. (\[BS\_Stepfn:eq\]) with coherence length $\xi_c^{(G)}$. The exact and semiclassically calculated $N(E)$ are plotted in Fig. \[lpecso-semi-retro:fig\]. The agreement between the two results is excellent.
![\[lpecso-semi-retro:fig\] (color online) The exact (red line) and the semiclassically (blue line) calculated step function $N(E)$ obtained from Eqs. (\[szek0:eq\]) and (\[BS\_Stepfn:eq\]), respectively for the case of Andreev retro-reflection. The parameters are the same as in Fig. (\[lpecso-retro:fig\]). The insets show the enlarged parts of the main frame. ](grafen_semi.eps)
Moreover, from Eq. (\[BS\_Stepfn:eq\]), we find that the positions of the singularities in the DOS are given by $E_n^{\left(\rm sing \right)}/\Delta^{(G)} = \left(n+1/2 \right)\pi /(1+ 2R/\xi_c^{(G)})$ valid for such integers $n$ that $E_n^{\left(\rm sing \right)} < \Delta^{(G)}$ holds. Note that the position $E_n^{\left(\rm sing \right)}/\Delta^{(G)}$ of the singularities depends only on $R/\xi_c^{(G)}$ but not on $E_{\rm F}^{(G)}/{\Delta^{(G)}}$. Therefore even if $E_{\rm F}^{(N)}/\Delta^{(N)}\neq E_{\rm F}^{(G)}/(2 \Delta^{(G)})$ but $R^{(N)}/\xi_c^{(N)}=R^{(G)}/\xi_c^{(G)}$, the singularities in the DOS for a circular GAB and NS billiard would appear at the same energies.
Next we consider the case of specular Andreev reflection in graphene Andreev billiards. Again, the solutions of Eqs. (\[szek0:eq\]) and (\[BS\_E\_m:eq\]) give the exact and the semiclassically calculated energy levels of circular shape GABs. In Fig. \[lpecso-spec:fig\] the calculated step function $N(E)$ is plotted and as one can see it is completely different from that obtained for the case of Andreev retro-reflection shown in Fig. \[lpecso-retro:fig\].
![\[lpecso-spec:fig\] (color online) The exact (red line) and the semiclassically (blue line) calculated step function $N(E)$ obtained from (\[szek0:eq\]) and (\[BS\_E\_m:eq\]), respectively for specular Andreev reflection. The parameters are $E_{\rm F}^{\left(\rm{G}\right)} = 0$ and $\xi_c^{(G)}/R= 0.03$. The insets show the enlarged parts of the main frame. ](specular_semi.eps)
Moreover, the quantum results in Fig. \[lpecso-spec:fig\] again show very good agreement with the semiclassical ones that can be obtained from Eq. (\[BS\_E\_m:eq\]). This implies that in the case of specular Andreev reflection the DOS depends linearly on the energy for $E\rightarrow 0$. Namely, it can be shown from Eq. (\[BS\_E\_m:eq\]) that in this limit the DOS in semiclassical approximation (without the valley degeneracy) is given by $\rho(E) = 8\frac{E \mathcal{A}}{\pi^3(\hbar v_F)^2} $, where $\mathcal{A}$ is the area of the billiard. It is interesting to note therefore that (apart from the valley degeneracy) $\rho(E)$ is bigger by a factor of $16/\pi^2$ than in the case of neutrino billiards [@ref:berry]. In summary, we calculated the energy levels of graphene based Andreev billiards. We showed that for energy levels corresponding to the case of Andreev retro-reflection the graphene based Andreev billiards in a very good approximation can be mapped to the normal metal-superconducting billiards with the same geometry. We also derived a semiclassical quantization rule in graphene based Andreev billiards and the spectrum obtained from this rule agrees very well with that obtained from the exact quantum calculations for circular shape of GS billiards.
This work is supported by the Hungarian Science Foundation OTKA under the contracts No. T48782 and 75529. A. K. is supported partly by European Commission Contract No. MRTN-CT-2003-504574 and by EPSRC. We gratefully acknowledge fruitful discussions with C. W. J. Beenakker and V. Falko.
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[^1]: e-mail: cserti@complex.elte.hu
|
---
abstract: 'Non-covalent functionalization via physisorption of organic molecules provides a scalable approach for modifying the electronic structure of graphene while preserving its excellent carrier mobilities. Here we investigated the physisorption of long-chain acenes, namely, hexacene and its fluorinated derivative perfluorohexacene, on bilayer graphene for tunable graphene devices using first principles methods. We find that the adsorption of these molecules leads to the formation of localized states in the electronic structure of graphene close to its Fermi level, which could be readily tuned by an external electric field in the range of $\pm$ 3 eV/nm. The electric field not only creates a variable band gap as large as 250 meV in bilayer graphene, but also strongly influences the charge redistribution within the molecule-graphene system. This charge redistribution is found to be weak enough not to induce strong surface doping, but strong enough to help preserve the electronic states near the Dirac point of graphene. Our results further highlight graphene’s potential for selective chemical sensing of physisorbed molecules under the external electric fields.'
author:
- Yuefeng Yin
- Jiri Cervenka
- 'Nikhil V. Medhekar'
bibliography:
- 'ref.bib'
title: Tunable Hybridization Between Electronic States of Graphene and Physisorbed Hexacene
---
Introduction
------------
Graphene—a planar layer of carbon atoms arranged in a hexagonal lattice—exhibits a linear electronic dispersion with the valence and conduction bands touching at the Dirac point[@Novoselov22102004]. As a result of this unique electronic structure, pristine graphene demonstrates an ultrahigh charge carrier mobility in excess of 200,000 cm^2^V^-1^s^-1^, which can be exploited for novel, highly energy efficient electronic devices [@ADMA:ADMA201201587; @ADMA:ADMA201290269; @PhysRevLett.105.266601]. However, the development of graphene-based electronic devices is primarily hindered by the absence of an intrinsic band gap in its electronic structure[@novoselov2012roadmap]. Although various approaches for tailoring the electronic structure of graphene have been pursued in recent years[@C1CS15193B; @doi:10.1021/ar3001487; @C0JM02922J; @zhang2011tailoring], creating a significant band gap while maintaining large charge carrier mobilities in graphene remains a formidable challenge.
One strategy to modify the electronic structure of graphene is to utlilize quantum confinement effects inherent in low dimensional structures such as quasi one-dimensional graphene nanoribbons[@PhysRevLett.98.206805; @wang2008room]. While this strategy can effectively induce the band gap, it also suffers from carrier scattering due to edge imperfections[@doi:10.1021/nl062132h]. Another route is the chemical functionalization of graphene where the addition of covalent bonds to graphene (for example, via hydrogenation and fluorination) changes the hybridization of carbon atoms from $sp^2$ to $sp^3$[@doi:10.1080/00018732.2010.487978]. While such covalent functionalization successfully alters the electronic properties of graphene, it also leads to a severe degradation of its transport properties [@SMLL:SMLL201202196]. Recent demonstrations of heterostructures of graphene with other 2D materials (for example, boron nitride and transition metal dichalcogenides) also provide a possible option, but a consistent production of graphene heterostructure devices is difficult to control on large scale[@doi:10.1021/nn400280c; @doi:10.1021/nl402062j].
Among the various approaches being pursued to modify the electronic structure of graphene, non-covalent functionalization via physisorption of organic molecules offers an interesting pathway[@zhang2011tailoring; @C4NR06470D]. This approach relies on conserving the integrity of the $sp^2$-bonded carbon lattice and thus preserves the linear dispersion of electrons near the Dirac point[@C1CS15193B; @C0JM02922J]. Moreover, the production of devices made of graphene with physisorbed molecules can be readily assisted by molecular self-assembly and can therefore be expected to be scalable.[@Mao2013132; @Colson08042011] Recent studies have suggested that in graphene physisorbed with small molecules such as NO$_\text{2}$ and NH$_\text{3}$, the application of a transverse external electric field can further enhance the tunability of the electronic structure of graphene by affecting the charge redistribution.[@zhang2009direct; @tian2010band; @zhang2011opening; @doi:10.1021/jp212218w; @:/content/aip/journal/jcp/134/4/10.1063/1.3541249; @doi:10.1021/nl2025739] Among the organic compounds that are amenable to physisorption on graphene, aromatic molecules are of particular interest [@PhysRevLett.102.135501; @zhang2011tailoring]. The face-centered parallel stacking of aromatic molecules on graphene surface can lead to a stable hybrid system via van der Waals (vdW) interactions[@Colson08042011], while the enhanced $\pi$-$\pi$ electron interaction is expected to influence the electronic structure of graphene[@zhang2011tailoring]. Moreover, addition of functional groups with high electron or hole affinity to the aromatic molecules has been suggested as an effective approach to induce strong charge doping in graphene.[@chen2007surface; @zhang2011opening; @medina2011tuning; @doi:10.1021/jp1107262; @kozlov2011bandgap]. This can allow a vertical integration of graphene with physisorbed organic molecules with tunable transport characteristics such as charge injection barriers [@wehling2008molecular]. However, recent reports indicate that a strong surface charge doping of graphene by molecules or electric field can cause a significant shift of the Fermi level into the valence or conduction band, and often lead to a severe deformation of the $\pi$ bands of graphene[@tian2010band; @duong2012band]. As a result, the charge carrier mobility of graphene degrades, thus limiting the switching capability of graphene-based semiconducting devices[@duong2012band; @doi:10.1021/jz4010174]. Therefore, identification of suitable organic molecules for non-covalent functionalization of graphene still remains an open challenge for controllable modification of its electronic properties.
Hexacene belongs to the group of acenes, the aromatic compounds formed by linear fusion of benzene rings (C~4n+2~H~2n+4~). Long-chain acenes possess low-lying molecular orbitals that are expected to hybridize with $\pi$ electrons of graphene and thus influence its electronic structure.[@kadantsev2006electronic]. Furthermore, it has been shown that the edges of hexacene can be readily functionalized with chemical groups with widely varying electron or hole affinity.[@doi:10.1021/ja051798v]. Recently, Watanabe [*et al.*]{} reported a successful way to synthesize hexacene that can remain stable up to 300 in dark conditions[@watanabe2012synthesis]. Moreover, organic field effect transistors (OFET) devices made of hexacene have demonstrated a highest charge carrier mobility ever reported for organic semiconductors[@watanabe2012synthesis]. These observations suggest that hexacene can be an attractive candidate for a stable physisorption on graphene. A good fundmental understanding of the electronic interactions between hexacene and graphene is therefore essential.
Here we systematically investigate the effect of physisorbed hexacene and perfluorohexacene (fluorinated hexacene) on the electronic properties of bilayer graphene using first principles density functional theory simulations. We use perfluorohexacene as an effective tool to down-shift the molecular energy levels relative to hexacene, and induce significant $\pi$-$\pi^{\ast}$ interactions and symmetry breaking in bilayer graphene. We examine how the functional groups and adsorption geometry of molecules influence the stability and the electronic structure of the bilayer graphene-molecule system. We show that the adsorption of hexacene and perfluorohexacene on bilayer graphene leads to a significant charge redistribution and the formation of localized states in graphene. By applying an external electric field to bilayer graphene adsorbed with hexacene and perfluorohexacene, we demonstrate that the induced localized states in graphene can be effectively controlled, potentially providing a new strategy for graphene-based sensors for a selective sensing of weakly adsorbed molecules.
Methods {#simset}
-------
![A schematic illustrating (a) single- and (b) dual-molecular adsorption of hexacene and perfluorohexacene on bilayer graphene. The arrow indicates the direction of an applied external electric field. []{data-label="fig1"}](figs/fig1.pdf)
To obtain optimized geometries and the electronic structures of all graphene-molecule systems considered in our study, we employed first principles density functional theory as implemented in Vienna *Ab Initio* Simulation Package [@kresse1996efficiency]. We used the generalized gradient approximation of the Perdew-Burke-Ernzerhof form for the electron exchange-correlation functional[@PhysRevLett.77.3865]. The core and valence electrons were treated using projector augmented wave (PAW) scheme[@PhysRevB.59.1758] with a kinetic energy cut-off of 600 eV for the plane-wave basis set. Since the generalized gradient approximation does not fully account for long-range dispersion interactions[@PhysRevB.81.081408], we used a Grimme’s semi-empirical functional[@JCC:JCC20495] to account for these interactions in the weakly bound graphene-molecule system. To benchmark the accuracy of this functional, we obtained the the equilibrium interlayer distance for pristine bilayer graphene to be 3.23 Å, which is in good agreement with the experimental values[@PhysRev.100.544]. We used a periodic $8\times4$ graphene supercell for investigating the adsorption of hexacene and perfluorohexacene on Bernal-stacked bilayer graphene. A single molecule in this supercell represents a nearly monolayer coverage for hexacene and perfluorohexacene on graphene with a molecular density of 9.846$\times$10^-11^ mol/cm^2^. This magnitude of molecular density is representative of the reported coverage of the aromatic molecules deposited on graphitic or graphene surfaces in various experimental studies[@PhysRevLett.52.2269; @gotzen2010growth; @lee2011surface]. In each case, the periodic images were separated by a 30 Å vacuum, which was found to be large enough to avoid any spurious interactions between the periodic images. All structures were fully relaxed until the ionic forces were smaller than 0.01 eV/Å. Gaussian smearing was used for geometry relaxations, while Blöch tetrahedral smearing was employed for subsequent calculations of electronic structures[@PhysRevB.49.16223]. Finally, for accurate calculations of the electronic structures, we used a fine $6\times12\times1$ $\Gamma$-centred grid for sampling the Brillouin zone.
First, we individually examined the adsorption of hexacene and perfluorohexacene on bilayer graphene using a single-molecular adsorption configuration as shown in Fig. \[fig1\](a). We considered two stacking sequences for this configuration, namely, $ABA$ and $ABC$, using the same notation as in the case of few-layer graphene[@lui2011observation]. We also considered the simultaneous adsorption of hexacene and perfluorohexacene in a dual-molecular configuration shown in Fig. \[fig1\](b). Finally, the influence of external electric fields normal to the plane of graphene was investigated by introducing dipolar sheets at the center of supercells[@neugebauer1992adsorbate].
Results and Discussion {#rd}
----------------------
**Electronic Structure of Hexacene and Perfluorohexacene**
We first obtained the optimized geometries of hexacene and perfluorohexacene molecules and calculated their electronic structure. The relaxed geometries of hexacene and perfluorohexacene are shown in Fig. \[fig2\](a). The average C-H bond length for hexacene and the average C-F bond length for perfluorohexacene were calculated to be 1.09 Å and 1.35 Å respectively. These bond lengths are in close agreement with previous structural calculations reported by Kadantsev et al.[@kadantsev2006electronic]. Figure \[fig2\](b) depicts an energy level diagram showing the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) of hexacene and perfluorohexacene with respect to the position of the Fermi level of graphene. The HOMO-LUMO band gaps of hexacene and perfluorohexacene are narrow, namely, 0.80 eV and 0.59 eV, respectively. The calculated electronic band gap of these molecules is smaller than the experimentally determined band gaps by 0.5 eV – 0.6 eV due to a systematic underestimation of band gap values of semiconducting materials obtained by DFT calculations employing the generalized gradient approximation [@watanabe2012synthesis; @PhysRevLett.51.1884; @PhysRevLett.51.1888].
![(a) Relaxed geometries of hexacene and perfluorohexacene. The C-C, C-H and C-F bond lengths (in Å) are indicated. (b) The energy level diagram of the band alignment between the Fermi level of bilayer graphene, and the HOMO and LUMO of hexacene and perfluorohexacene (in eV). The energy of vacuum is regarded as zero.[]{data-label="fig2"}](figs/fig2.pdf)
[p[4.2cm]{}<p[2.1cm]{}<p[2.7cm]{}<p[5.8cm]{}<]{} & Stacking sequence & Adsorption energy (eV) & Average adsorption distance (Å)\
& ABA &-1.756 & 3.19\
&ABC& -1.751 & 3.19\
& ABA &-2.246 & 3.16\
&ABC&-2.239&3.16\
\[1ex\]
----------------------------
Dual-molecular adosprion
(hexacene+perfluoroxacene)
----------------------------
&\*\[1ex\][ABAB]{} &\*\[1ex\][-4.015]{} &
--------------------------
3.20 (hexacene)
3.17 (perfluorohexacene)
--------------------------
\
&\[1ex\][ABCA]{} &\[1ex\][-4.005]{}&
--------------------------
3.19 (hexacene)
3.16 (perfluorohexacene)
--------------------------
\
\[table4-1\]
**Single- and Dual-Molecular Adsorption on Bilayer Graphene**
Next we obtained optimized ground state configurations of hexacene and pefluorohexacene adsorbed on bilayer graphene in single as well as dual-molecular configurations shown in Fig.\[fig1\]. In each case, the adsorption energy is calculated as $\Delta E=E_\text{{graphene/molecule}}-E_\text{{graphene}}-E_\text{{molecule}},$ where $E_\text{{graphene/molecule}}$ is the total energy of the fully-relaxed graphene-molecule supercell, while $E_\text{{graphene}}$ and $E_\text{{molecule}}$ is the energy of bilayer graphene and isolated molecules in the same supercell, respectively. A negative value of the adsorption energy indicates an exothermic, thermodynamically favorable adsorption. The values for adsorption energy and the average adsorption distance from the nearest graphene layer for each configuration are summarized in Table \[table4-1\]. We find that the adsorption of both hexacene and pefluorohexacene on bilayer graphene is thermodynamically favorable in all configurations considered. The resulting adsorption distance of the molecules from graphene is close to the interlayer distance between graphene layers ($\sim$3.23 Å), indicating that graphene and the molecules are bound by weak van der Waals dispersion interactions. The adsorption distance and the adsorption energies are also nearly unaffected by the change of stacking sequence from ABC to ABA, also suggesting that the molecule-bilayer graphene interactions are largely confined between the molecule and the adjacent graphene layer.
Our results show that pefluorohexacene binds more strongly to bilayer graphene than hexacene (adsorption energy of -2.25 eV vs. -1.76 eV) in a single-molecular adsorption configuration. The stronger binding of pefluorohexacene correlates with a higher electron affinity and chemical reactivity of pefluorohexacene relative to hexacene. Finally, in the case of simultaneous, dual-molecular adsorption of hexacene and perfluorohexacene on bilayer graphene, the adsorption energy of the total hexacene and perfluorohexacene complex is nearly equal to the sum of the adsorption energies of individual molecules in single-molecular adsorption. This observation suggests that the adsorption system can be relatively easily adjusted from a single-molecular configuration to a dual-molecular configuration, or vice versa, at a very low energy cost. The adsorption distances of hexacene and perfluorohexacene in dual- molecular adsorption are found to be slightly larger than for the single-molecular adsorption cases.
**Electronic Structure of Bilayer Graphene upon Adsorption**
![(a) Total density of states of a pristine bilayer graphene. (b-d) Partial densities of states (PDOS) of bilayer graphene upon single-molecular adsorption of hexacene (b), single-molecular adsorption of perfluorohexacene (c), and dual-molecular adsorption of hexacene and perfluorohexacene (d). In all cases, the molecular concentration is 9.85$\times$10^-11^ mol/cm^2^ and represents monolayer coverage. $(\text{Hex})_\text{H}$, $(\text{Hex})_\text{L}$, and $(\text{P-Hex})_\text{H}$, $(\text{P-Hex})_\text{L}$ denote the localized states induced by hybridization with HOMO and LUMO of hexacene, and HOMO and LUMO of perfluorohexacene, respectively. The Fermi level is set to zero in each case. (e-f) Partial charge density plots for the localized states in PDOS of bilayer graphene upon single-molecular adsorption of hexacene (e) and single-molecular adsorption of perfluorohexacene (f). The isosurface level is set to 0.0003 e/Å^3^. []{data-label="fig4"}](figs/fig3.pdf)
Following the structural optimisation, we next investigate the electronic properties of bilayer graphene upon molecular adsorption. Our results show that the molecular stacking sequence did not have any effect on the density of states curves. Consequently, Fig.\[fig4\] presents the partial density of states (PDOS) for bilayer graphene with single-molecular adsorption with ABA stacking and for dual-molecular adsorption with ABCA stacking sequence. We find that the physisorbed hexacene and perflurorohexacene has a negligible effect on the $\pi$ states of bilayer graphene in the vicinity of the Dirac point. No shift in the Fermi level of bilayer graphene was observed upon adsorption of hexacene or perfluorohexacene. The magnitude of the band gap induced in bilayer graphene upon adsorption of these molecules is also negligible (3 meV and 5 meV for hexacene and perflurorohexacene, respectively). These observations point to a weak interaction between graphene and the molecules. Nevertheless, the low lying highest occupied and lowest unoccupied molecular orbitals (HOMO and LUMO) of adsorbed molecules hybridize with $\pi/\pi^*$ states of graphene, giving rise to two localized states near the Fermi level of bilayer graphene (see Fig. \[fig4\](a-d)). These states were found in PDOS of bilayer graphene at -0.18 eV and 0.61 eV after the adsorption of hexacene and at -0.43 eV and 0.18 eV after the adsorption of perfluorohexacene. The position of HOMO and LUMO of hexacene and perfluorohexacene states is slightly altered in comparison to that of isolated molecules (refer Fig.\[fig2\] (b)). To gain further insight into the localized states in graphene, we also plotted the corresponding partial charge densities by integrating the charge density in an energy range $\pm$0.02 eV around the localized peaks. The partial charge density plots (Fig. \[fig4\](e,f)) show the shape of HOMO/LUMO of hexacene/perfluohexacene, and present the signature of hybridization between graphene and the adsorbed molecules. It is evident that the induced localized states in graphene are located on the nearest carbon atoms of the top graphene layer. The presence of these hybridized states near the Dirac point of graphene implies that both electrons and holes can be injected from graphene to molecules at a relative low energy cost. In dual-molecular adsorption configuration (Fig.\[fig4\] (d)), the PDOS can be regarded as a superposition of the energy states from the single-molecular adsorption on graphene. Moreover, we observe that a 8 meV band gap is opened, equal to the sum of the band gap values of individual molecules. This indicates that the interaction between bilayer graphene and acene molecules in dual-molecular configuration is essentially governed by the interaction of a single graphene layer with the adjacent adsorbed molecule.
In order to further assess the influence of molecular adsorption on the electronic properties of graphene, we calculated the charge density difference as defined by $\Delta\rho=\rho_\text{graphene/molecule}-\rho_\text{graphene}-\rho_\text{molecule},$ where $\rho_\text{graphene/molecule}$, $\rho_\text{graphene}$ and $\rho_\text{molecule}$ are the electronic charge densities of the adsorbed system, isolated graphene and the molecule, respectively. With this definition, a positive value of $\Delta\rho$ indicates an accumulation of electronic charge and a negative value indicates a charge depletion. The distribution of the charge density difference for single-molecular adsorption of hexacene and perfluorohexacene is shown in Fig. \[fig5\]. In the case of adsorption of hexacene on bilayer graphene (Fig. \[fig5\] (a)), the charges are depleted from the region 0.6 Å–1.0 Å above the top graphene layer and accumulated close to hexacene in the region 2.6 Å– 2.8 Å above the graphene layer. This charge redistribution primarily arises from the electrostatic interaction between the aromatic rings of hexacene and graphene—the interaction between hydrogen atoms of hexacene and carbon atoms of graphene or hexacene is found to be negligible [@hsun2011electrostatic]. Overall, the interaction between hexacene and bilayer graphene is not strong enough to lead to a significant direct charge transfer between graphene and hexacene, but only to a charge redistribution on carbon atoms of the molecule and the nearest graphene layer. In contrast to hexacene, the charge redistribution is significantly different for perflurorohexacene physisorbed on bilayer graphene as shown in Fig. \[fig5\] (b). Since fluorine atoms are strong electron-attracting groups, the $\pi$ electrons are polarized away from the aromatic rings leading to a relatively electron-deficient aromatic core of perfluorohexacene. Similarly, a strong interaction between fluorine and graphene gives rise to a significant charge depletion from the carbon atoms in graphene close to fluorine atoms in perfluorohexacene, indicating a net charge transfer from graphene to perfluorohexacene. By comparing the charge distribution of hexacene and perfluorohexacene on bilayer graphene, it is evident that interaction between graphene and perfluorohexacene is largely controlled by the presence of fluorine functional groups.
![The distribution of charge density difference for single-molecular adsorption of (a) hexacene and (b) perfluorohexacene on bilayer graphene at monolayer coverage. Red and green isorufaces indicate the accumulation and depletion of electrons at a level of 0.0003 e/Å^3^, respectively. []{data-label="fig5"}](figs/fig4.pdf)
![The planar-averaged ($\Delta\rho_{\text{avg}}(z)$) and integrated planar-averaged ($\overline{\Delta\rho}(z)$) charge density difference curves for single-molecular adsorption of (a) hexacene and (b) perfluorohexacene on bilayer graphene at monolayer coverage. The positions of the top graphene layer (TG) and the molecule are marked by green and yellow dashed lines, respectively. Blue solid lines denote the neutral plane. []{data-label="fig6"}](figs/fig5.pdf)
To quantify the charge transferred from graphene to adosprbed molecules, we plotted the variation of the planar-averaged and the integrated planar-averaged charge density difference as a function of the distance from the basal plane of graphene as shown in Fig. \[fig6\]. The planar-averaged charge density difference $\Delta\rho_{\text{avg}}(z)$ along a plane parallel to the basal plane of graphene is obtained by integrating the charge density difference across the plane, whereas the integrated charge density difference $\overline{\Delta\rho}(z)$ is simply calculated by integrating $\Delta\rho_{\text{avg}}(z)$ curve from the boundary of the periodic box to the position of the plane. The extremum in the integrated charge density difference curve in the region between graphene and the adsorbed molecule (indicated by red lines in Fig. \[fig6\]) denotes the neutral plane and the magnitude of the net charge transfer. According to this analysis, graphene donates 0.035 $e^-$ per molecule to hexacene and a larger fraction 0.050 $e^-$ per molecule to perflurohexacene. The magnitude of net charge transfer obtained here is qualitatively consistent with Bader charge population analysis[@Henkelman2006354], which yields a net charge transfer of 0.02 $e^-$ and 0.18 $e^-$ per molecule from graphene to hexacene and perfluorohexacene, respectively. The differences in the magnitudes of the net charge transfer obtained by these two methods can be attributed to the different ways of calculating the net charges on each ion. Bader charge analysis takes into account both core charges and valence charges, while the charge density difference calculation is more suitable for interpreting the charge redistribution close to the Fermi level [@chan2008first]. Nevertheless, the small magnitude of charge transfer for both molecules is indicative of a weak interaction between the molecules and bilayer graphene.
Finally, few recent experimental studies have reported that as the density of the adsorbed acene molecules is increased beyond the near-monolayer coverage considered here, the molecules tend to tilt rather than maintain a planar orientation on graphene [@lee2011surface; @doi:10.1021/jp3103518]. In order to confirm this observation, we have also studied the adorption of hexacene on bilayer graphene at a nominal molecular concentration of 1.31$\times$10^-10^ mol/cm^2^ using a $8\times3$ graphene supercell. Figure S1 in the Supporting Information shows the adsorption geometry as well as the electronic interaction between hexacene and bilayer graphene at this coverage. We find that hexacene shows a remarkable 11$^\circ$ tilt with respect to the basal plane of graphene, in qualitative agreement with the experimental observations [@lee2011surface; @doi:10.1021/jp3103518]. This tilt arises due to a stronger repulsive intermolecular interaction between neighbouring hexacene molecules at high coverage. The cofacial $\pi$-$\pi$ interactions that contribute to the stability of hexacene adsorption at lower concentrations are disrupted, causing the adsorption at high coverage to be less energetically favorable (adsorption energy -1.534 eV per molecule). Moreover, the electronic interactions between graphene and hexacene also vary spatially, leading to an asymmetric charge redistribution pattern as shown in Fig. S1. Compared to the corresponding patterns at low coverages, a significantly larger charge rearrangement is observed in the region of the molecule where hexacene is closer to the graphene than in the region where hexacene is away from the graphene. This imbalance in the charge redistribution breaks the local symmetry of bilayer graphene, inducing a 54 meV band gap.
[**Effect of the Applied External Electric Field**]{}
The results presented in earlier sections show that in general, the adsorbed aromatic acene molecules interact weakly with bilayer graphene, leading to the formation of localized states and a weak p-type doping of graphene. Since external electric fields can enhance the interactions between adsorbate and substrate[@duong2012band; @doi:10.1021/jp212218w; @:/content/aip/journal/jcp/134/4/10.1063/1.3541249], next we investigate whether the external electric field could be effectively utilized to tune the electronic structure and molecule-specific localized states in bilayer graphene. Specifically, we studied the effect of the electric field in the range of -3 to 3 eV/nm, applied perpendicular to the basal plane of bilayer graphene in the dual-molecular hexacene/bilayer graphene/perfluorohexacene adsorption configuration at a monolayer coverage as shown in Fig. \[fig1\](b). In our notation, a positive electric field is oriented towards hexacene from perfluorohexacene. Figure \[fig7\](a) presents PDOS of bilayer graphene as a function of the strength of the field. The application of the electric field leads to the opening of a considerable band gap in bilayer graphene, as well as to the shift in the energy levels of the localized states arising due to the hybridization with HOMO and LUMO states of the molecules. The external electric field causes an accumulation of electrons in one layer and a depletion of electrons in the other layer of bilayer graphene, thus breaking the interlayer symmetry [@zhang2009direct].. The band gap generated in bilayer graphene is thus a result of the interplay between the field-induced interlayer symmetry breaking and the asymmetric charge transfer between graphene and the adsorbed hexacene and perfluorohexacene molecules. In several cases, the magnitude of band gap is difficult to determine from the PDOS of bilayer graphene alone as the gap region is occupied by the localized states. In order to correctly identify the band gaps, we have calculated the electronic band structure for each case as shown in Fig. \[fig9\]. The localized states can then be readily identified from these band structure diagrams as the flat bands between the $\pi$ and $\pi^{\ast}$ bands of bilayer graphene.
It is evident that both the shape of the PDOS as well as the magnitude of the band gap in bilayer PDOS shows a strong dependence on the strength and the direction of the external electric field. For example, the band gap increases linearly with the electric field till a maximum of $\sim$250 meV when the field is oriented from hexacene towards perfluorohexacene (that is, a negative electric field). For a positive field, the band gap reaches $\sim$200 meV for the field magnitude of 1 eV/nm, then reduces to $\sim$115 meV for the field of 2 eV/nm or greater. This observed contrast in the trend between negative and positive fields can be attributed to the distinct charge transfer behavior of the adsorbed hexacene and perfluorohexacene molecules as shown in Fig. \[fig7\] (b). For instance, for fields $E \geq 2$ eV/nm, the HOMO of perfluorohexacene and the LUMO of hexacene are pinned in the vicinity of the Fermi level. The pinning of these localized states near the Fermi level enhances the driving force for transferring charge from hexacene to perfluorohexacene, which is favored by the application of a positive electric field. Therefore the charge transfer between hexacene and perfluorohexacene is maximized as shown in Fig. \[fig7\] (c). The charge inequivalence between two graphene layers is reduced for positive electric fields greater than 1 eV/nm, leading to a saturation of the band gap. For negative fields on the other hand, the charge transfer trend is reversed. The energy difference between the HOMO of perfluorohexacene and the LUMO of hexacene increases with the strength of the negative electric field. Therefore the charge transfer between the two graphene layers is less affected and the magnitude of band gap increases rapidly with the magnitude of the field.
Figure \[fig9\](a) also shows a comparison between the electronic band structure diagrams of bilayer graphene adsorbed with hexacene and perfluorohexacene to those with the pristine bilayer graphene. Our obtained band structures for pristine graphene are in good agreement with previous works [@doi:10.1021/nl1039499]. It is evident that due to the deformation of $\pi$ and $\pi^{\ast}$ bands, the field-induced band gap in pristine bilayer graphene is no longer located at the K point, but instead along $\Gamma$-X line of the Brillouin zone. However, the $\pi$ and $\pi^{\ast}$ bands of bilayer graphene with dual-molecular adsorption are less deformed close to the Fermi level due to the screening by molecules, indicating that the energy dispersion relationship in the vicinity of the K point is relatively well preserved. This can be clearly seen by comparing the magnitude of band gap at K (E~K~) and the true band gap along $\Gamma$-X line (E~T~) presented in Fig. \[fig9\](b). It can be seen that in general, the range of band gaps that can be induced in bilayer graphene with molecular adsorption (100 meV – 250 meV) is comparable to the pristine bilayer graphene (200 meV – 300 meV). Overall, the band structures follow a similar trend as the DOS plots of dual molecular adsorption on bilayer graphene shown in Fig. \[fig7\] (a), where graphene $\pi$ and $\pi^{\ast}$ states hybridize with the localized HOMO and LUMO bands of the molecules, forming new localized bands with a flat dispersion. These results show that the energy of these localized states can be varied as a function of the strength and polarity of the applied electric field, thereby modifying the electronic structure of bilayer graphene. Finally, for completeness, we briefly compare the electronic structure of bilayer graphene with dual-molecular adsorption under the electric field to that of monolayer graphene. Figure S2 in the Supporting Information shows band structure diagrams for dual-molecular adsorption of hexacene and perfluorohexacene on monolayer graphene as a function of the strength of the electric field. We find that the electronic structure of monolayer graphene with dual-molecular adsorption is significantly different from the bilayer graphene for negative electric fields. The localized states induced in monolayer graphene due to hybidization with the HOMO/LUMO states of the adsorbed molecules are relatively further away from the Fermi level of graphene. Moreover, negative electric fields lead to much lower band gaps in monolayer graphene (less than 50 meV). For positive electric fields, both monolayer and bilayer graphene show a similar magnitude of the band gaps ($\sim$120 meV). When compared with Fig. \[fig9\], these observations highlight the interplay between the charge transfer and the breaking of the symmetry between the top and bottom layer in bilayer graphene. In general, these results suggest that by the application of external electric field to bilayer graphene with dual molecular gating, the electronic structure can be more flexibly controlled, leading to opening of considerable band gaps.
Conclusions
-----------
Here we have used self-consistent density functional theory calculations to study the effect of physisorption of hexacene and its fluorine derivative, perfluorohexacene, on the electronic structure of bilayer graphene. We find that although the overall interaction between graphene and molecules is weak, the adsorption of these molecules results in a significant charge redistribution. This charge redistribution gives rise to the hybridization of HOMO/LUMO energy levels of the molecules with the $\pi$ electrons of graphene, leading to the formation of localized states in bilayer graphene. We have shown that the external electric fields can be used to tune the electronic properties of graphene-molecule system, effectively opening large band gaps of the order of 250 meV in bilayer graphene. Furthermore, external electric fields can also infleunce the energies of the localized states in graphene, an effect that can be utilized in organic field effect transistor (OFET) devices by aligning the electronic states of acene channels with that of graphene electrodes. This effect can also be potentially useful in the sensing of different organic molecules on the surface of graphene transistors. Graphene transistors have proven to be extremely sensitive sensors[@schedin2007detection], but their selectivity remains a major problem for their practical use. In summary, we have shown that hexacene, a stable and high mobility organic electronic material, and its derivatives, are promising candidates for surface electronic structure modification of graphene for potential applications in organic electronics and sensing.
Acknowledgement
---------------
Authors gratefully acknowledge computational support from Monash Sun Grid as well as iVEC and NCI national computing facilities.
Supporting Information
----------------------
Additional figure for the adsorption of hexacene on bilayer graphene in the 8$\times$3 graphene supercell and band structure diagrams for dual-molecular adsorption of hexacene and perfluorohexacene on monolayer graphene.
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abstract: 'Wightman function, the vacuum expectation values of the field square and the energy-momentum tensor are evaluated for a scalar field obeying mixed boundary condition on a spherical brane in $(D+1)$-dimensional Rindler-like spacetime $Ri\times S^{D-1}$, where $Ri$ is a two-dimensional Rindler spacetime. This spacetime approximates the near horizon geometry of $(D+1)$-dimensional black hole in the large mass limit. The vacuum expectation values are presented as the sum of boundary-free and brane-induced parts. Further we extract from the Wightman function for the boundary-free geometry the corresponding function in the bulk $R^{2}\times S^{D-1}$. For the latter geometry the vacuum expectation values of the field square and the energy-momentum tensor do not depend on the spacetime point. For the renormalization of these quantities we use zeta regularization technique. Various limiting cases of the brane-induced vacuum expectation values are investigated.'
author:
- |
A. A. Saharian$^{1}$[^1] and M. R. Setare$^{2}$[^2]\
[*$^1$ Department of Physics, Yerevan State University, Yerevan, Armenia* ]{}\
[*$^2$ Institute for Theoretical Physics and Mathematics, Tehran, Iran*]{}
title: 'Casimir densities for a spherical brane in Rindler-like spacetimes'
---
= 16truecm = 24truecm = -1.3truecm = -2truecm = 1.20cm = 1.60cm
PACS number(s): 03.70.+k, 04.62.+v, 11.10.Kk
Introduction {#sec:Int}
============
Motivated by string/M theory, the AdS/CFT correspondence, and the hierarchy problem of particle physics, braneworld models were studied actively in recent years [@Hora96]-[@Rand99]. In this models, our universe is realized as a boundary of a higher dimensional spacetime. In particular, a well studied example is when the bulk is an AdS space. In the cosmological context, embedding of a four dimensional Friedmann-Robertson-Walker universe was also considered when the bulk is described by AdS or AdS black hole [Nihe99,AdSbhworld]{}. In the latter case, the mass of the black hole was found to effectively act as an energy density on the brane with the same equation of state of radiation. Representing radiation as conformal matter and exploiting AdS/CFT correspondence, the Cardy-Verlinde formula [@Verl00] for the entropy was found for the universe (for the entropy formula in the case of dS black hole see [@Seta02]). Moreover, in the AdS/CFT correspondence, the case of a bulk AdS black hole represents a different phase of the same theory and there is the exciting connection that a transition between an ordinary bulk AdS and a bulk AdS black hole corresponds to the confinement-de confinement transition in the dual CFT [@Witt98]. Therefore it seems interesting to generalize the study of quantum effects due to bulk AdS black holes.
The investigation of quantum effects in braneworld models is of considerable phenomenological interest, both in particle physics and in cosmology. The braneworld corresponds to a manifold with dynamical boundaries and all fields which propagate in the bulk will give Casimir-type contributions to the vacuum energy (for reviews of the Casimir effect see Refs. [@Most97]), and as a result to the vacuum forces acting on the branes. In dependence of the type of a field and boundary conditions imposed, these forces can either stabilize or destabilize the braneworld. In addition, the Casimir energy gives a contribution to both the brane and bulk cosmological constants and, hence, has to be taken into account in the self-consistent formulation of the braneworld dynamics. Motivated by these, the role of quantum effects in braneworld scenarios has received a great deal of attention. For a conformally coupled scalar this effect was initially studied in Ref. [@Fabi00] in the context of M-theory, and subsequently in Refs. [@Noji00a; @Flac01] for a background Randall–Sundrum geometry. The models with dS and AdS branes, and higher dimensional brane models are considered as well [@Noji00b].
In view of these recent developments, it seems interesting to generalize the study of quantum effects to other types of bulk spacetimes. In particular, it is of interest to consider non-Poincaré invariant braneworlds, both to better understand the mechanism of localized gravity and for possible cosmological applications. Bulk geometries generated by higher-dimensional black holes are of special interest. In these models , the tension and the position of the brane are tuned in terms of black hole mass and cosmological constant and brane gravity trapping occurs in just the same way as in the Randall-Sundrum model. Braneworlds in the background of the AdS black hole were studied in [@AdSbhworld]. Like pure AdS space the AdS black hole may be superstring vacuum. It is of interest to note that the phase transitions which may be interpreted as confinement-deconfinement transition in AdS/CFT setup may occur between pure AdS and AdS black hole [@Witt98]. Though, in the generic black hole background the investigation of brane-induced quantum effects is techniqally complicated, the exact analytical results can be obtained in the near horizon and large mass limit when the brane is close to the black hole horizon. In this limit the black hole geometry may be approximated by the Rindler-like manifold (for some investigations of quantum effects on background of Rindler-like spacetimes see [@Byts96] and references therein). In the present paper we investigate the Wightman function, the vacuum expectation values of the field square and the energy-momentum tensor for a scalar field with an arbitrary curvature coupling parameter for the spherical brane on the bulk $%
Ri\times S^{D-1}$, where $Ri$ is a two-dimensional Rindler spacetime. Note that the corresponding quantities induced by a single and two parallel flat branes in the bulk geometry $Ri\times
R^{D-1}$ for both scalar and electromagnetic fields are investigated in [@Cand77]. This problem is also of separate interest as an example with gravitational and boundary-induced polarizations of the vacuum, where all calculations can be performed in a closed form. The paper is organized as follows. In section \[sec:WF\] we consider the positive frequency Wightman function in the region between the brane and Rindler horizon. This function is presented as the sum of boundary-free and boundary-induced parts. The vacuum expectation values for the boundary-free geometry are investigated in section \[sec:bfree\]. The vacuum expectation values induced by a spherical brane are studied in section \[sec:VEVEMT\]. Section \[sec:Conc\] summarizes the main results of the paper.
Wightman function {#sec:WF}
=================
Let us consider a scalar field $\varphi (x)$ propagating on background of $%
(D+1)$-dimensional Rindler-like spacetime $Ri\times S^{D-1}$, where $Ri$ is a two-dimensional Rindler spacetime. The corresponding metric is described by the line element$$ds^{2}=\xi ^{2}d\tau ^{2}-d\xi ^{2}-r_{H}^{2}d\Sigma _{D-1}^{2},
\label{ds22}$$with the Rindler-like $(\tau ,\xi )$ part and $d\Sigma _{D-1}^{2}$ is the line element for the space with positive constant curvature with the Ricci scalar $R=(D-2)(D-1)/r_{H}^{2}$. Line element (\[ds22\]) describes the near horizon geometry of $(D+1)$-dimensional topological black hole with the line element [@Mann97]$$ds^{2}=A_{H}(r)dt^{2}-\frac{dr^{2}}{A_{H}(r)}-r^{2}d\Sigma _{D-1}^{2},
\label{ds21}$$where$$A_{H}(r)=k+\frac{r^{2}}{l^{2}}-\frac{r_{0}^{n+2}}{l^{2}r^{n}},\quad n=D-2,
\label{Ar}$$and the parameter $k$ classifies the horizon topology, with $k=0,-1,1$ corresponding to flat, hyperbolic, and elliptic horizons, respectively. In ([Ar]{}) the parameter $l$ is related to the bulk cosmological constant and the parameter $r_{0}$ depends on the mass of the black hole and on the bulk gravitational constant. In the non extremal case the function $A_{H}(r)$ has a simple zero at $r=r_{H}$. In the near horizon limit, introducing new coordinates $\tau $ and $\rho $ in accordance with$$\tau =\frac{1}{2}A_{H}^{\prime }(r_{H})t,\quad r-r_{H}=\frac{1}{4}%
A_{H}^{\prime }(r_{H})\xi ^{2}, \label{tau}$$the line element is written in the form (\[ds22\]). Note that for a $(D+1)$-dimensional Schwarzschild black hole [@Call88] one has $%
A_{H}(r)=1-(r_{H}/r)^{D-2}$ and, hence, $A_{H}^{\prime }(r_{H})=n/r_{H}$.
The field equation is in the form$$\left( g^{ik}\nabla _{i}\nabla _{k}+m^{2}+\zeta R\right) \varphi (x)=0,
\label{fieldeq1}$$where $\zeta $ is the curvature coupling parameter. Below we will assume that the field satisfies the Robin boundary condition on the hypersurface $%
\xi =a$: $$\left( A+B\frac{\partial }{\partial \xi }\right) \varphi =0,\quad \xi =a,
\label{bound1}$$with constant coefficients $A$ and$\ B$. The Dirichlet and Neumann boundary conditions are obtained as special cases. In accordance with (\[tau\]), the hypersurface $\xi =a$ corresponds to the spherical shell near the black hole horizon with the radius $r_{a}=r_{H}+A_{H}^{\prime }(r_{H})a^{2}/4$. Here we consider the general case of the ratio $A/B$. The application to the braneworld scenario will be given below.
To evaluate the vacuum expectation values of the field square and the energy-momentum tensor we need a complete set of eigenfunctions satisfying the boundary condition (\[bound1\]). Below we shall use the hyperspherical angular coordinates $(\vartheta ,\phi )=(\theta
_{1},\theta _{2},\ldots
,\theta _{n},\phi )$ on $S^{D-1}$ with $0\leq \theta _{k}\leq \pi $, $%
k=1,\ldots ,n$, and $0\leq \phi \leq 2\pi $. In these coordinates the variables are separated and the eigenfunctions can be written in the form$$\varphi _{\alpha }(x)=C_{\alpha }f(\xi )Y(m_{k};\vartheta ,\phi )e^{-i\omega
\tau }, \label{eigfunc1}$$where $m_{k}=(m_{0}\equiv l,m_{1},\ldots m_{n})$, and $m_{1},m_{2},\ldots
m_{n}$ are integers such that$$0\leq m_{n-1}\leq \cdots \leq m_{1}\leq l,\quad -m_{n-1}\leq m_{n}\leq
m_{n-1}, \label{mk}$$$Y(m_{k};\vartheta ,\phi )$ is the surface harmonic of degree $l$ [Erdelyi]{}. Substituting this into Eq. (\[fieldeq1\]) we see that the function $f(\xi )$ satisfies the equation$$\xi \frac{d}{d\xi }\left( \xi \frac{df}{d\xi }\right) +\left( \omega
^{2}-\xi ^{2}\lambda _{l}^{2}\right) f(\xi )=0, \label{feq}$$with the notation$$\quad \lambda _{l}=\frac{1}{r_{H}}\sqrt{l(l+n)+\zeta n(n+1)+m^{2}r_{H}^{2}}.
\label{lambdal}$$The linearly independent solutions to (\[feq\]) are the Bessel modified functions $I_{\pm i\omega }(\lambda _{l}\xi )$ and $K_{i\omega }(\lambda
_{l}\xi )$ with the imaginary order.
In the region $0<\xi <a$ the solution to (\[feq\]) satisfying boundary condition (\[bound1\]) has the form$$f(\xi )=Z_{i\omega }(\lambda _{l}\xi ,\lambda _{l}a)\equiv K_{i\omega
}(\lambda _{l}\xi )-\frac{\bar{K}_{i\omega }(\lambda _{l}a)}{\bar{I}%
_{i\omega }(\lambda _{l}a)}I_{i\omega }(\lambda _{l}\xi ), \label{f2}$$where for a given function $F(z)$ we use the notation$$\bar{F}(z)=AF(z)+bzF^{\prime }(z)=0,\quad b=B/a. \label{fbarnot}$$The coefficient $C_{\alpha }$ in (\[eigfunc1\]) can be found from the normalization condition$$\int \left\vert \varphi _{\alpha }(x)\right\vert ^{2}\sqrt{-g}dV=\frac{1}{%
2\omega }, \label{normcond}$$where the integration goes over the region between the horizon and the sphere. Substituting eigenfunctions (\[eigfunc1\]), using the relation $$\int \left\vert Y(m_{k};\vartheta ,\phi )\right\vert ^{2}d\Omega =N(m_{k})
\label{normsph}$$for spherical harmonics, one finds$$C_{\alpha }=\frac{1}{\pi }\sqrt{\frac{\sinh \omega \pi }{r_{H}^{n+1}N(m_{k})}%
}. \label{Calfa}$$The explicit form for $N(m_{k})$ is given in [@Erdelyi] and will not be necessary for the following considerations in this paper.
First of all we evaluate the positive frequency Wightman function$$G^{+}(x,x^{\prime })=\langle 0\left\vert \varphi (x)\varphi (x^{\prime
})\right\vert 0\rangle , \label{W1}$$where $|0\rangle $ is the amplitude for the corresponding vacuum state. By expanding the field operator over eigenfunctions and using the commutation relations one can see that$$G^{+}(x,x^{\prime })=\sum_{\alpha }\varphi _{\alpha }(x)\varphi _{\alpha
}^{\ast }(x^{\prime }). \label{W2}$$Substituting eigenfunctions (\[eigfunc1\]) with the function (\[f2\]) into this mode sum and by making use the addition theorem$$\sum_{m_{k}}\frac{1}{N(m_{k})}Y(m_{k};\vartheta ,\phi )Y(m_{k};\vartheta
^{\prime },\phi ^{\prime })=\frac{2l+n}{nS_{D}}C_{l}^{n/2}(\cos \theta ),
\label{addtheor}$$for the Wightman function one finds$$\begin{aligned}
G^{+}(x,x^{\prime }) &=&\frac{r_{H}^{1-D}}{\pi ^{2}nS_{D}}\sum_{l=0}^{\infty
}(2l+n)C_{l}^{n/2}(\cos \theta ) \nonumber \\
&&\times \int_{0}^{\infty }d\omega \sinh (\omega \pi )e^{-i\omega (\tau
-\tau ^{\prime })}Z_{i\omega }(\lambda _{l}\xi ,\lambda _{l}a)Z_{i\omega
}^{\ast }(\lambda _{l}\xi ^{\prime },\lambda _{l}a). \label{W3}\end{aligned}$$In (\[addtheor\]), $S_{D}=2\pi ^{D/2}/\Gamma (D/2)$ is the total area of the surface of the unit sphere in $D$-dimensional space, $C_{l}^{n/2}(x)$ is the Gegenbauer or ultraspherical polynomial of degree $l$ and order $n/2$, $%
\theta $ is the angle between directions $(\vartheta ,\phi )$ and $%
(\vartheta ^{\prime },\phi ^{\prime })$, and the sum is taken over the integer values $m_{k},\,k=1,2\ldots $ in accordance with (\[mk\]). To transform the expression on the right of (\[W3\]), we present the product of the functions $Z_{i\omega }$ in the form$$\begin{aligned}
Z_{i\omega }(\lambda \xi ,\lambda a)Z_{i\omega }^{\ast }(\lambda \xi
^{\prime },\lambda a) &=&K_{i\omega }(\lambda \xi )K_{i\omega }(\lambda \xi
^{\prime })+\frac{\pi \bar{K}_{i\omega }(\lambda a)}{2i\sinh \pi \omega }
\nonumber \\
&&\times \sum_{\sigma =-1,1}\frac{I_{i\sigma \omega }(\lambda \xi
)I_{i\sigma \omega }(\lambda \xi ^{\prime })}{\sigma \bar{I}_{i\sigma \omega
}(\lambda a)}. \label{form1}\end{aligned}$$On the base of this formula from (\[W3\]) one finds$$\begin{aligned}
G^{+}(x,x^{\prime }) &=&G_{0}^{+}(x,x^{\prime })+\frac{r_{H}^{1-D}}{2i\pi
nS_{D}}\sum_{l=0}^{\infty }(2l+n)C_{l}^{n/2}(\cos \theta ) \nonumber \\
&&\times \int_{0}^{\infty }d\omega e^{-i\omega (\tau -\tau ^{\prime })}\bar{K%
}_{i\omega }(\lambda _{l}a)\sum_{\sigma =-1,1}\frac{I_{i\sigma \omega
}(\lambda _{l}\xi )I_{i\sigma \omega }(\lambda _{l}\xi ^{\prime })}{\sigma
\bar{I}_{i\sigma \omega }(\lambda _{l}a)}, \label{W4}\end{aligned}$$where the part $$\begin{aligned}
G_{0}^{+}(x,x^{\prime }) &=&\frac{r_{H}^{1-D}}{\pi ^{2}nS_{D}}%
\sum_{l=0}^{\infty }(2l+n)C_{l}^{n/2}(\cos \theta ) \nonumber \\
&&\times \int_{0}^{\infty }d\omega \sinh (\omega \pi )e^{-i\omega (\tau
-\tau ^{\prime })}K_{i\omega }(\lambda _{l}\xi )K_{i\omega }(\lambda _{l}\xi
^{\prime }) \label{W0}\end{aligned}$$does not depend on the parameter $a$ determining the radius of the spherical shell and corresponds to the Wightman function in the situation when the spherical shell is absent. Assuming that the function $\bar{I}_{i\omega
}(\lambda a)$ ($\bar{I}_{-i\omega }(\lambda a)$) has no zeros for $-\pi
/2\leq {\rm arg}\,\omega <0$ ($0<{\rm arg}\,\omega <\pi /2$) we can rotate the integration contour over $\omega $ by angle $-\pi /2$ for the term with $%
\sigma =1$ and by angle $\pi /2$ for the term with $\sigma =-1$. The integrals taken around the arcs of large radius tend to zero under the condition $|\xi \xi ^{\prime }|<a^{2}e^{|\tau -\tau ^{\prime }|}$ (note that, in particular, this is the case in the coincidence limit for the region under consideration). As a result for the Wightman function one obtains$$G^{+}(x,x^{\prime })=G_{0}^{+}(x,x^{\prime })+\langle \varphi (x)\varphi
(x^{\prime })\rangle ^{(b)}, \label{W5}$$where for the sphere-induced part one has$$\begin{aligned}
\langle \varphi (x)\varphi (x^{\prime })\rangle ^{(b)} &=&-\frac{r_{H}^{1-D}%
}{\pi nS_{D}}\sum_{l=0}^{\infty }(2l+n)C_{l}^{n/2}(\cos \theta ) \nonumber
\\
&&\times \int_{0}^{\infty }d\omega \frac{\bar{K}_{\omega }(\lambda _{l}a)}{%
\bar{I}_{\omega }(\lambda _{l}a)}I_{\omega }(\lambda _{l}\xi )I_{\omega
}(\lambda _{l}\xi ^{\prime })\cosh [\omega (\tau -\tau ^{\prime })].
\label{Wb1}\end{aligned}$$For the points away the brane this part is finite in the coincidence limit.
We have investigated the Whightman function in the region between the horizon and the boundary located at $\xi =a$ for an arbitrary ratio of boundary coefficients $A/B$. In the corresponding braneworld scenario the geometry is made up by two slices of the region $0<\xi <a$ glued together at the brane with a orbifold-type symmetry condition analogous to that in the Randall-Sundrum model and the ratio $A/B$ for bulk scalars is related to the brane mass parameter of the field and the extrinsic curvature of the brane. The corresponding formula is obtained by the way similar to that in the case of the Randall-Sundrum braneworld (see, for instance, [@Flac01; @Gher00]). For this we note that in braneworlds the action for a scalar field with general curvature coupling parameter, in addition to the bulk action contains a surface action in the form $\int d^{D-1}x \sqrt{h} (c +\zeta K)\varphi ^2$, where the integration goes over the brane, $h$ is the absolute value of the determinant for the corresponding induced metric, $c$ is the brane mass parameter for the field, and $K$ is the extrinsic curvature scalar for the brane. This action gives $\delta $-type contributions to the field equation located on the brane. Now the eigenfunctions for the quantized bulk scalar field can be written in the form (\[eigfunc1\]), where the function $f(\xi )$ is a solution to the equation which differs from (\[feq\]) by the presence of the term $-(c +\zeta K)\xi ^2 f(\xi )\delta (\xi -a)$ on the right hand side. To obtain the boundary condition for the function $f(\xi )$ we integrate the corresponding equation about $\xi =a$. Assuming that the function $f(\xi )$ is continuous at this point one finds $$\label{discont}
\lim _{\epsilon \to 0} \left. \frac{df}{d\xi }\right| _{\xi =
a-\epsilon }^{\xi =a+\epsilon }=(c +\zeta K)f(a).$$ For an untwisted scalar field we have $f(\xi )=f(2a-\xi )$ and from (\[discont\]) we obtain the boundary condition in the form (\[bound1\]) with $$\label{ABbraneworld}
\frac{A}{B}=\frac{1}{2}\left( c-\frac{\zeta }{a}\right) ,$$ where we have taken into account that for the boundary under consideration $K=-1/a$. For a twisted scalar $f(\xi )=-f(2a-\xi )$ and from (\[discont\]) we obtain the Dirichlet boundary condition. Note that in the braneworld bulk the integration in the normalization integral goes over two copies of the bulk manifold. This leads to the additional coefficient $1/2$ in the expression (\[Calfa\]) for the normalization coefficient $C_{\alpha }$. Hence, the Whightman function in the orbifolded braneworld case is given by formula (\[W5\]) with an additional factor $1/2$ in formulae (\[W0\]), (\[Wb1\]). As it has been mentioned above this function corresponds to the braneworld in the AdS black hole bulk in the limit when the brane is close to the black hole horizon.
Boundary-free geometry {#sec:bfree}
======================
In this section we will consider the vacuum expectation values for the geometry without boundaries. First of all we note that the corresponding Wightman function can be presented in the form$$\begin{aligned}
G_{0}^{+}(x,x^{\prime }) &=&\tilde{G}_{0}^{+}(x,x^{\prime })-\frac{%
r_{H}^{1-D}}{\pi ^{2}nS_{D}}\sum_{l=0}^{\infty }(2l+n)C_{l}^{n/2}(\cos
\theta ) \nonumber \\
&&\times \int_{0}^{\infty }d\omega e^{-\omega \pi }\cos [\omega (\tau -\tau
^{\prime })]K_{i\omega }(\lambda _{l}\xi )K_{i\omega }(\lambda _{l}\xi
^{\prime }), \label{GM1}\end{aligned}$$with the function$$\begin{aligned}
\tilde{G}_{0}^{+}(x,x^{\prime }) &=&\frac{r_{H}^{1-D}}{\pi ^{2}nS_{D}}%
\sum_{l=0}^{\infty }(2l+n)C_{l}^{n/2}(\cos \theta ) \nonumber \\
&&\times \int_{0}^{\infty }d\omega \cosh \{\omega \lbrack \pi -i(\tau -\tau
^{\prime })]\}K_{i\omega }(\lambda _{l}\xi )K_{i\omega }(\lambda _{l}\xi
^{\prime }). \label{GM2}\end{aligned}$$In this formula the $\omega $-integral can be evaluated with the result$$\begin{aligned}
\tilde{G}_{0}^{+}(x,x^{\prime }) &=&\frac{r_{H}^{1-D}}{2\pi nS_{D}}%
\sum_{l=0}^{\infty }(2l+n)C_{l}^{n/2}(\cos \theta ) \nonumber \\
&&\times K_{0}\left( \lambda _{l}\sqrt{\xi ^{2}+\xi ^{\prime 2}-2\xi \xi
^{\prime }\cosh (\tau -\tau ^{\prime })}\right) . \label{GM3}\end{aligned}$$It can be checked that this function is the Wightman function for the bulk geometry $R^{2}\times S^{D-1}$ described by the line element$$ds^{2}=dt^{2}-(dx^{1})^{2}-r_{H}^{2}d\Sigma _{D-1}^{2}, \label{ds31}$$where the coordinates $(t,x^{1})$ are related to the coordinates $(\tau ,\xi
)$ by formulas $t=\xi \sinh \tau $, $x^{1}=\xi \cosh \tau $. To see this we note that the normalized eigenfunctions corresponding to this geometry are given by the formula$$\widetilde{\varphi }_{\alpha }(x)=\frac{Y(m_{k};\vartheta ,\phi
)e^{ik_{1}x^{1}-i\omega _{l}t}}{\sqrt{4\pi \omega _{l}N(m_{k})r_{H}^{n+1}}},
\label{phialfM}$$where $\alpha =(k_{1},m_{k})$ and $\omega _{l}^{2}=k_{1}^{2}+\lambda
_{l}^{2} $, with $\lambda _{l}$ defined by relation (\[lambdal\]). Substituting these functions into the corresponding mode sum and evaluating the $k_{1}$-integral, for the case $|x^{1}-x^{1\prime }|>|t-t^{\prime }|$ one finds$$\begin{aligned}
\tilde{G}_{0}^{+}(x,x^{\prime }) &=&\sum_{\alpha }\widetilde{\varphi }%
_{\alpha }(x)\widetilde{\varphi }_{\alpha }^{\ast }(x^{\prime }) \nonumber
\\
&=&\frac{r_{H}^{1-D}}{2\pi nS_{D}}\sum_{l=0}^{\infty }(2l+n)C_{l}^{n/2}(\cos
\theta )K_{0}\left( \lambda _{l}\sqrt{(x^{1}-x^{1\prime })^{2}-(t-t^{\prime
})^{2}}\right) . \label{GM4}\end{aligned}$$Noting that $\xi ^{2}+\xi ^{\prime 2}-2\xi \xi ^{\prime }\cosh (\tau -\tau
^{\prime })=(x^{1}-x^{1\prime })^{2}-(t-t^{\prime })^{2}$ we see that this formula coincides with (\[GM3\]).
In formula (\[GM1\]), the divergences in the coincidence limit are contained in the term $\tilde{G}_{0}^{+}(x,x^{\prime })$ and, hence, the renormalization is needed for this term only. Now we turn to the evoluation of the vacuum expectation values of the field square and the energy-momentum tensor for the geometry $R^{2}\times S^{D-1}$ described by the line element (\[ds31\]). The amplitude of the corresponding vacuum state we will denote by $|\tilde{0}\rangle $. First of all note that from the problem symmetry it follows that the expectation values $\langle \tilde{0}\vert \varphi
^{2}\vert \tilde{0}\rangle $, $\langle \tilde{0}t\vert T_{i}^{i}\vert \tilde{%
0}\rangle $ do not depend on the point of observation and$$\begin{aligned}
\langle \tilde{0}\vert T_{0}^{0}\vert \tilde{0}\rangle &=&\langle \tilde{0}%
\vert T_{1}^{1}\vert \tilde{0}\rangle , \label{vevEMTM} \\
\langle \tilde{0}\vert T_{2}^{2}\vert \tilde{0}\rangle &=&\cdots =\langle
\tilde{0}\vert T_{D}^{D}\vert \tilde{0}\rangle . \nonumber\end{aligned}$$The component $\langle \tilde{0}\vert T_{2}^{2}\vert \tilde{0}\rangle $ can be expressed through the energy density by using the trace relation$$T_{i}^{i}=D(\zeta -\zeta _{c})\nabla _{i}\nabla ^{i}\varphi
^{2}+m^{2}\varphi ^{2}. \label{TiiM}$$From this relation it follows that$$\langle \tilde{0}\vert T_{2}^{2}\vert \tilde{0}\rangle =\frac{1}{D-1}\left(
m^{2}\langle \tilde{0}\vert \varphi ^{2}\vert \tilde{0}\rangle -2\langle
\tilde{0}\vert T_{0}^{0}\vert \tilde{0}\rangle \right) . \label{vevT22M}$$Hence, it is sufficient to find the renormalized vacuum expectation values of the field square and the energy density. Using the eigenmodes ([phialfM]{}), these quantities are presented as mode sums$$\begin{aligned}
\langle \tilde{0}\vert \varphi ^{2}\vert \tilde{0}\rangle &=&\frac{r_{H}^{-n}%
}{4\pi S_{D}}\int_{-\infty }^{+\infty }dk_{1}\sum_{l}\frac{D_{l}}{\eta
_{l}(r_{H}k_{1})}, \label{modephi2T00} \\
\langle \tilde{0}\vert T_{0}^{0}\vert \tilde{0}\rangle &=&\frac{r_{H}^{-n-2}%
}{4\pi S_{D}}\int_{-\infty }^{+\infty }dk_{1}\sum_{l}D_{l}\eta
_{l}(r_{H}k_{1}), \label{modeT00}\end{aligned}$$with the notation $\eta _{l}(x)=\sqrt{x^{2}+r_{H}^{2}\lambda _{l}^{2}}$. In this formulas $$D_{l}=(2l+D-2)\frac{\Gamma (l+D-2)}{\Gamma (D-1)l!} \label{Dl}$$is the degeneracy of each angular mode with given $l$. Of course, quantities (\[modephi2T00\]), (\[modeT00\]) are divergent and some renormalization procedure is needed. As such a procedure we will use the zeta function technique. Let us define the zeta function$$\begin{aligned}
\zeta (s) &=&\int_{-\infty }^{+\infty }dx\sum_{l=0}^{\infty }D_{l}\eta
_{l}^{-2s}(x) \nonumber \\
&=&\sqrt{\pi }\frac{\Gamma \left( s-\frac{1}{2}\right) }{\Gamma \left(
s\right) }\zeta _{S^{n+1}}\left( s-\frac{1}{2}\right) , \label{zetas}\end{aligned}$$where $$\begin{aligned}
\zeta _{S^{n+1}}(z) &=&\sum_{l=0}^{\infty }D_{l}(r_{H}\lambda _{l})^{-2z}
\nonumber \\
&=&\sum_{l=0}^{\infty }D_{l}\left[ (l+n/2)^{2}+b_{n}\right] ^{-z},
\label{zeta0s}\end{aligned}$$is the zeta function for a scalar field on the spacetime $R\times S^{n+1}$ and $$b_{n}=\zeta n(n+1)-n^{2}/4+m^{2}r_{H}^{2}. \label{bn}$$This function is well investigated in literature (see, for example, [Camp90]{}) and can be presented as a series of incomplete zeta functions. Here we recall that the function $\zeta _{S^{n+1}}(z)$ is a meromorphic function with simple poles at $z=(n+1)/2-j$, where $j=0,1,2,\ldots $ for $n$ even and $0\leq j\leq (n-1)/2$ for $n$ odd. For $n$ even one has $\zeta
_{S^{n+1}}(-j)=0$, $j=1,2,\ldots $. Note that the function $\zeta
_{S^{n+1}}(z)$ can be expressed in terms of the function $$F(z,c,b)=\sum_{l=1}^{\infty }\left[ (l+c)^{2}+b\right] ^{-z}, \label{Fzcb}$$for $n$ even or its derivative for $n$ odd as$$\zeta _{S^{n+1}}(z)=\frac{2^{1-2\eta }}{\Gamma (D-1)}\sum_{j=0}^{[n/2]}\frac{%
b_{j}^{(\eta )}(b_{n})}{(j-z+2\eta )^{2\eta }}\left. \frac{\partial ^{2\eta }%
}{\partial c^{2\eta }}F(z-j-2\eta ,c,b_{n})\right\vert _{c=\eta },
\label{zeta0s1}$$where $\eta =n/2-[n/2]$ and the square brackets mean the integer part of the enclosed expression. In formula (\[zeta0s1\]) the coefficients $%
b_{j}^{(\eta )}(b_{n})$ are defined by the relation$$\prod_{q=1}^{[n/2]}\left[ y-(q+\eta -1)^{2}-b_{n}\right] =%
\sum_{j=0}^{[n/2]}b_{j}^{(\eta )}(b_{n})y^{j}. \label{bneta}$$The formulas for the analytic continuation of the function $F(z,c,b)$ can be found in [@Eliz94; @Eliz90].
Now on the base of formulas (\[modephi2T00\]), (\[modeT00\]) we have $$\langle \tilde{0}\vert \varphi ^{2}\vert \tilde{0}\rangle =\frac{\zeta (1/2)%
}{4\pi S_{D}r_{H}^{D-1}},\quad \langle \tilde{0}\vert T_{0}^{0}\vert \tilde{%
0}\rangle =\frac{\zeta (-1/2)}{4\pi S_{D}r_{H}^{D+1}}. \label{phi2T00}$$By taking into account that the quantities $\zeta _{S^{n+1}}(0)$ and $\zeta
_{S^{n+1}}(-1)$ are finite, from formula (\[zetas\]) we see that at $s=1/2$ and $s=-1/2$ the zeta function $\zeta (s)$ has simple poles with residues $%
\zeta _{S^{n+1}}(0)$ and $\zeta _{S^{n+1}}(-1)/2$, respectively. Hence, in general, the vacuum expectation values of the field square and the energy density contain the pole and finite contributions. The remained pole term is a characteristic feature for the zeta function regularization method. Note that for $n$ even $\zeta _{S^{n+1}}(-1)=0$ and the energy density is finite.
The vacuum expectation value of the field square in the boundary-free geometry $Ri\times S^{D-1}$ is obtained from the Wightman function (\[GM1\]) taking the coincidence limit. Using the relation$$C_{l}^{n/2}(1)=\frac{\Gamma (l+n)}{\Gamma (n)l!}, \label{Cl1}$$for the corresponding quantity one finds$$\langle 0_{0}\vert \varphi ^{2}\vert 0_{0}\rangle =\langle
\tilde{0}\vert
\varphi ^{2}\vert \tilde{0}\rangle -\frac{r_{H}^{1-D}}{\pi ^{2}S_{D}}%
\sum_{l=0}^{\infty }D_{l}\int_{0}^{\infty }d\omega e^{-\omega \pi
}K_{i\omega }^{2}(\lambda _{l}\xi ), \label{phi20}$$where $|0_{0}\rangle $ is the amplitude for the corresponding vacuum state. For large values of $\xi $, by using the asymptotic formulas for the MacDonald function for large values of the argument, we can see that the main contribution into the second term on the right of formula (\[phi20\]) comes from the summand $l=0$ and we obtain$$\langle 0_{0}\vert \varphi ^{2}\vert 0_{0}\rangle =\langle \tilde{0}\vert
\varphi ^{2}\vert \tilde{0}\rangle -\frac{r_{H}^{1-D}e^{-2\lambda _{0}\xi }}{%
\pi ^{2}S_{D}\lambda _{0}\xi }. \label{phi20a}$$In the limit $\xi \rightarrow 0$ the second term on the right of (\[phi20\]) diverges. For small values $\xi $ the main contribution comes from large values $l$ and this term behaves as $(r_{H}/\xi )^{D-1}$. Hence, near the horizon the boundary-free vacuum expectation value of the field square is dominated by the second term on the right of formula (\[phi20\]) and is negative.
Now we turn to the vacuum expectation value of the energy-momentum tensor. The corresponding operator we will take in the form$$T_{ik}=\partial _{i}\varphi \partial _{k}\varphi +\left[ \left( \zeta -\frac{%
1}{4}\right) g_{ik}\nabla _{l}\nabla ^{l}-\zeta \nabla _{i}\nabla _{k}-\zeta
R_{ik}\right] \varphi ^{2}, \label{EMT1}$$with the trace relation (\[TiiM\]). In (\[EMT1\]) $R_{ik}$ is the Ricci tensor for the bulk geometry and for the metric (\[ds22\]) it has components$$\begin{aligned}
R_{ik} &=&0,\quad i,k=0,1;\quad \label{Rik} \\
R_{ik} &=&\frac{n}{r_{H}^{2}}g_{ik},\quad i,k=2,\ldots ,D.\end{aligned}$$On the base of formula (\[EMT1\]) the corresponding vacuum expectation values are presented in the form$$\langle 0_{0}\vert T_{ik}\vert 0_{0}\rangle =\lim_{x^{\prime }\rightarrow
x}\nabla _{i}\nabla _{k}^{\prime }G_{0}^{+}(x,x^{\prime })+ \left[ \left(
\zeta -\frac{1}{4}\right) g_{ik}\nabla _{l}\nabla ^{l}-\zeta \nabla
_{i}\nabla _{k}-\zeta R_{ik}\right] \langle 0_{0}\vert \varphi ^{2}\vert
0_{0}\rangle . \label{EMT2}$$By using decomposition (\[GM1\]), the vacuum energy-momentum tensor is presented in the form$$\langle 0_{0}\vert T_{ik}\vert 0_{0}\rangle =\langle \tilde{0}\vert
T_{ik}\vert \tilde{0}\rangle +\langle T_{ik}\rangle ^{(0)}, \label{Tik0}$$where the second summand on the right is given by formula (no summation over $i$)$$\langle T_{i}^{k}\rangle ^{(0)}=-\frac{\delta
_{i}^{k}r_{H}^{1-D}}{\pi ^{2}S_{D}}\sum_{l=0}^{\infty
}D_{l}\lambda _{l}^{2}\int_{0}^{\infty }d\omega e^{-\omega \pi
}f^{(i)}\left[ K_{i\omega }(\lambda _{l}\xi )\right] .
\label{Tik00}$$In this formula we use the notations$$\begin{aligned}
f^{(0)}[g(z)] &=&\left( \frac{1}{2}-2\zeta \right) \left[ \left( \frac{dg(z)%
}{dz}\right) ^{2}+\left( 1-\frac{\omega ^{2}}{z^{2}}\right) g^{2}(z)\right] +%
\frac{\zeta }{z}\frac{d}{dz}g^{2}(z)+\frac{\omega ^{2}}{z^{2}}g^{2}(z),
\label{f0} \\
f^{(1)}[g(z)] &=&-\frac{1}{2}\left( \frac{dg(z)}{dz}\right) ^{2}-\frac{\zeta
}{z}\frac{d}{dz}g^{2}(z)+\frac{1}{2}\left( 1-\frac{\omega ^{2}}{z^{2}}%
\right) g^{2}(z), \label{f1} \\
f^{(i)}[g(z)] &=&\left( \frac{1}{2}-2\zeta \right) \left[ \left( \frac{dg(z)%
}{dz}\right) ^{2}+\left( 1-\frac{\omega ^{2}}{z^{2}}\right) g^{2}(z)\right] -%
\frac{\lambda _{l}^{2}-m^{2}}{(D-1)\lambda _{l}^{2}}g^{2}(z), \label{fi}\end{aligned}$$with $i=2,3,...,D$ and the indices 0 and 1 correspond to the coordinates $%
\tau $ and $\xi $. Note that for a minimally coupled scalar field the energy density corresponding to (\[Tik00\]) is negative for all values $\xi $. As in the case of the field square, for large values $\xi $ the vacuum expectation values (\[Tik00\]) are exponentially suppressed by the factor $%
e^{-2\lambda _{0}\xi }$. For small values $\xi $ these expectation values behave as $(r_{H}/\xi )^{D+1}$ and diverge on the horizon. This type of horizon divergences is also found in the well-investigated example of a bulk $Ri \times R^{D-1}$ (see references [@Cand77]). Here the situation is similar to that which takes place in the Schwarzshild black hole bulk when the quantum field is prepared in the Boulware vacuum [@Cand80]. In the black hole case quantities characterizing vacuum polarization display this singular behavior because no physical realization is possible of a static system whose size is arbitrarily close to the gravitational radius. The nature of horizon divergences in Rindler-like spacetimes is similar: no physical realization is possible to a coordinate system for which the lines with a fixed $\xi $ are arbitrarily close to the Rindler horizon. In this limit the corresponding particles would move at infinitely large acceleration.
Vacuum expectation values induced by a spherical brane {#sec:VEVEMT}
======================================================
In this section we consider the vacuum expectation values induced by the presence of a spherical brane. On the base of formula (\[W5\]) for the Whightman function, the vacuum expectation value for the field square is presented in the form$$\langle 0|\varphi ^{2}|0\rangle =\langle 0_{0}|\varphi ^{2}|0_{0}\rangle
+\langle \varphi ^{2}\rangle ^{(b)}, \label{phi2b0}$$where the second term on the right is induced by the spherical shell: $$\langle \varphi ^{2}\rangle ^{(b)}=-\frac{r_{H}^{1-D}}{\pi S_{D}}%
\sum_{l=0}^{\infty }D_{l}\int_{0}^{\infty }d\omega \frac{\bar{K}_{\omega
}(\lambda _{l}a)}{\bar{I}_{\omega }(\lambda _{l}a)}I_{\omega }^{2}(\lambda
_{l}\xi ). \label{phi2b}$$This quantity is negative for Dirichlet boundary condition and is positive for Neumann boundary condition. Similar formula can be derived for the vacuum expectation value of the energy-momentum tensor. In accordance with the relation (\[W5\]) we can write$$\langle 0|T_{i}^{k}|0\rangle =\langle 0_{0}|T_{i}^{k}|0_{0}\rangle +\langle
T_{ik}\rangle ^{(b)}, \label{EMT3}$$where $\langle 0_{0}|T_{i}^{k}|0_{0}\rangle $ is the vacuum expectation value for the situation without the spherical shell and $\langle
T_{ik}\rangle ^{(b)}$ is induced by the presence of the sphere. The latter is finite for the points away from the sphere surface and the horizon, and is given by formula (no summation over $i$)$$\langle T_{i}^{k}\rangle ^{(b)}=-\frac{\delta _{i}^{k}r_{H}^{1-D}}{\pi S_{D}}%
\sum_{l=0}^{\infty }D_{l}\lambda _{l}^{2}\int_{0}^{\infty }d\omega \frac{%
\bar{K}_{\omega }(\lambda _{l}a)}{\bar{I}_{\omega }(\lambda _{l}a)}F^{(i)}%
\left[ I_{\omega }(\lambda _{l}\xi )\right] , \label{EMT4}$$where the expressions for the functions $F^{(i)}[g(z)]$ are obtained from the corresponding formulas for the functions $f^{(i)}[g(z)]$ replacing $%
\omega \rightarrow i\omega $. As it has been explained in section \[sec:WF\], the corresponding quantities in the orbifolded braneworld version of the problem are obtained from (\[phi2b\]), (\[EMT4\]) with an additional coefficient $1/2$ and boundary coefficients (\[ABbraneworld\]).
Now let us consider various limiting cases of the general formulas for the brane-induced vacuum expectation values. In the limit $\xi \rightarrow a$ the expectation values for both field square and the energy-momentum tensor diverge. These surface divergences are well known in quantum field theory with boundaries and are investigated for various type of boundary conditions and geometries. For the points near the brane the main contributions come from large values $l$. Using the uniform asymptotic expansions for the Bessel modified function, to the leading order one finds$$\langle \varphi ^{2}\rangle ^{(b)}\approx -\frac{\delta _{B}\Gamma \left(
\frac{D-1}{2}\right) }{(4\pi )^{\frac{D+1}{2}}(a-\xi )^{D-1}},
\label{phi2close}$$for the field square and$$\langle T_{0}^{0}\rangle ^{(b)}\approx \langle T_{2}^{2}\rangle
^{(b)}\approx \frac{D(\zeta -\zeta _{c})\delta _{B}}{2^{D}\pi ^{\frac{D+1}{2}%
}(a-\xi )^{D+1}}\Gamma \left( \frac{D+1}{2}\right) , \label{T00close}$$for the components of the energy-momentum tensor, $\zeta _{c}=(D-1)/4D$ is the curvature coupling parameter for a conformally coupled scalar, and we have introduced the notation $$\delta _{B}=\left\{
\begin{array}{cc}
1, & B=0 \\
-1, & B\neq 0%
\end{array}%
\right. . \label{deltaB}$$These leading terms are the same as those for a flat brane in the Minkowski bulk. They do not depend on the mass and Robin coefficients and have opposite signs for Dirichlet and non-Dirichlet boundary conditions. The leading term in the asymptotic expansion of the component $\langle
T_{1}^{1}\rangle ^{(b)}$ is obtained from (\[T00close\]) by using covariant continuity equation for the tensor $\langle T_{i}^{k}\rangle
^{(b)} $. This term behaves as $(a-\xi )^{-D}$.
For large values of the ratio $a/r_{H}$ the quantity $\lambda _{l}a$ is large and we can replace the Bessel modified functions with this argument by their asymptotics for large values of the argument. This leads to the formulas (no summation over $i$)$$\begin{aligned}
\langle \varphi ^{2}\rangle ^{(b)} &\approx &-\frac{e^{-2\lambda _{0}a}}{%
S_{D}r_{H}^{D-1}}\frac{A-B\lambda _{0}}{A+B\lambda _{0}}\int_{0}^{\infty
}d\omega \,I_{\omega }^{2}(\lambda _{0}\xi ), \label{phi2far} \\
\langle T_{i}^{k}\rangle ^{(b)} &\approx &-\frac{\delta _{i}^{k}\lambda
_{0}^{2}e^{-2\lambda _{0}a}}{S_{D}r_{H}^{D-1}}\frac{A-B\lambda _{0}}{%
A+B\lambda _{0}}\int_{0}^{\infty }d\omega \,F^{(i)}[I_{\omega }(\lambda
_{0}\xi )], \label{Tikfar}\end{aligned}$$with the exponential suppression of the brane-induced vacuum expectation values.
In the near horizon limit, $\xi /r_{H}\ll 1$, with fixed $a/r_{H}$, the main contributions into the $\omega $-integrals come from small values $\omega $. Expanding the functions $I_{\omega }^{2}(\lambda _{l}\xi )$, to the leading order one finds (no summation over $i$)$$\begin{aligned}
\langle \varphi ^{2}\rangle ^{(b)} &\approx &-\frac{r_{H}^{1-D}{\cal I}(a)}{%
2\pi S_{D}\ln (2r_{H}/\xi )},\quad {\cal I}(a)=\sum_{l=0}^{\infty }D_{l}%
\frac{\bar{K}_{0}(\lambda _{l}a)}{\bar{I}_{0}(\lambda _{l}a)}
\label{phi2bnearhor} \\
\langle T_{0}^{0}\rangle ^{(b)} &\approx &-\langle T_{1}^{1}\rangle
^{(b)}\approx -\frac{\zeta r_{H}^{1-D}{\cal I}(a)}{2\pi S_{D}\xi ^{2}\ln
^{2}(2r_{H}/\xi )}, \label{T00nearhor} \\
\langle T_{i}^{i}\rangle ^{(b)} &\approx &\frac{(4\zeta -1)r_{H}^{1-D}{\cal I%
}(a)}{4\pi S_{D}\xi ^{2}\ln ^{3}(2r_{H}/\xi )},\quad i=2,3,\ldots .
\label{Tiinearhor}\end{aligned}$$As we see the brane-induced part in the vacuum expectation value of the field square vanishes at the horizon, whereas the expectation values of the energy-momentum tensor diverge. Recall that near the horizon the boundary free part of the energy-momentum tensor behaves as $\xi ^{-D-1}$ and the vacuum expectation values are dominated by this part. Note that the function ${\cal I}(a)$ is positive for Dirichlet boundary condition and is negative for Neumann boundary condition. In the large mass limit the brane induced vacuum expectation values are exponentially suppressed by the factor $%
e^{-2m(a-\xi )}$.
Conclusion {#sec:Conc}
==========
In this paper, we investigate the quantum vacuum effects produced by a spherical brane in the $(D+1)$-dimensional bulk $Ri\times S^{D-1}$, with $Ri$ being a two-dimensional Rindler spacetime. The corresponding line element (\[ds22\]) describes the near horizon geometry of a non-extremal black hole spacetime defined by the line element (\[ds21\]). The case of a massive scalar field with general curvature coupling parameter and satisfying the Robin boundary condition on the sphere is considered. To derive formulas for the vacuum expectation values of the square of the field operator and the energy-momentum tensor, we first construct the positive frequency Wightman function. This function is also important in considerations of the response of a particle detector at a given state of motion through the vacuum under consideration [@Birr82]. The Wightman function is presented as the sum of the Whightman function for the boundary-free geometry and the term induced by the presence of the spherical brane. For the points away the boundary and horizon the divergences in the coincidence limit are contained in the first term and hence, the renormalization is needed for this term only. In section \[sec:bfree\] we have shown that the the Whightman function for the boundary-free $Ri\times S^{D-1}$ geometry can be presented in the form of a sum of the Whightman function for the boundary-free $%
R^{2}\times S^{D-1}$ geometry plus a term which is finite in the coincidence limit. As a result for the renormalization of the vacuum expectation values of the field square and the energy-momentum tensor it is sufficient to renormalize the corresponding quantities for the geometry $R^{2}\times
S^{D-1}$. The latter are point-independent and as a renormalization procedure we employ the zeta function regularization method. The corresponding zeta function can be expressed in terms of the zeta function $%
\zeta _{S^{n+1}}(z)$ in the geometry $R\times S^{n+1}$, the analytic continuation for which is well investigated in literature. Alternatively the zeta function can be expressed in terms of the function (\[Fzcb\]). The vacuum expectation values of the field square and the energy-momentum tensor are expressed in terms of the zeta function by formulas (\[phi2T00\]). In general, they contain pole and finite contributions. In the minimal subtraction scheme the pole terms are omitted. As a result the vacuum expectation values of the field square and the energy-momentum tensor for the boundary-free $Ri\times S^{D-1}$ geometry are determined by formulas (\[phi20\]), (\[Tik0\]), (\[Tik00\]). On the horizon these expectation values diverge. The leading terms in the near horizon asymptotic expansions behave as $(r_{H}/\xi )^{D-1}$ for the field square and as $(r_{H}/\xi
)^{D+1}$ for the components of the energy-momentum tensor. The vacuum expectation values induced by a spherical brane in the bulk geometry $%
Ri\times S^{D-1}$ are investigated in section \[sec:VEVEMT\]. Near the brane the vacuum expectation values are dominated by the boundary parts and the corresponding components diverge at the brane. For non-conformally coupled scalars the leading terms in the corresponding asymptotic expansions are given by formulas (\[phi2close\]), (\[T00close\]) and are the same as those for an infinite plane boundary in the Minkowski bulk. These terms do not depend on the mass and Robin coefficients and have opposite signs for Dirichlet and non-Dirichlet boundary conditions. For large values of the ratio $a/r_{H}$ the brane-induced vacuum expectation values are exponentially suppressed by the factor $\exp [-2(a/r_{H})\sqrt{\zeta
n(n+1)+m^{2}r_{H}^{2}}]$. For the points near the horizon one has $\xi
/r_{H}\ll 1$ and the brane-induced vacuum expectation value of the field square vanishes as $\ln ^{-1}(2r_{H}/\xi )$. Unlike to the field square, the brane-induced parts in the vacuum expectation values of the energy-momentum tensor diverge on the horizon. In this paper we have considered the vacuum expectation values induced by a spherical brane in the region $0<\xi <a$. By the similar way the corresponding quantities for the region $\xi >a$ may be investigated. It can be seen that in this region the brane-induced quantities can be obtained from those for the region $0<\xi <a$ by the replacements $I_{\omega }\leftrightarrows K_{\omega }$ in formulas ([phi2b]{}) and (\[EMT4\]). In the corresponding braneworld scenario the geometry is made up by two slices of the region $0<\xi <a$ glued together at the brane with a orbifold-type symmetry condition and the ratio $A/B$ for bulk scalars is related to the brane mass parameter of the field by formula (\[ABbraneworld\]). The corresponding formulae for the Wightman function and the vacuum expectation values of the field square and the energy-momentum tensor are obtained from those given above with an additional coefficient $1/2$. They describe the braneworld in the AdS black hole bulk in the limit when the brane is close to the black hole horizon.
Note that in this paper we have considered quantum vacuum effects induced by boundaries in a prescribed background, i.e. the gravitational back-reaction of quantum effects is not taken into account. This back-reaction could have a profound effect on the dynamical evolution of the bulk model. We do not consider this important extension of the theory, but note that the results presented here constitute the starting point for such investigations.
Acknowledgement {#acknowledgement .unnumbered}
===============
The work of AAS was supported by the Armenian National Science and Education Fund (ANSEF) Grant No. 05-PS-hepth-89-70 and in part by the Armenian Ministry of Education and Science Grant No. 0124. AAS is grateful to the Fund Robert Boghossian & Sons.
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[^1]: E-mail: saharyan@server.physdep.r.am
[^2]: E-mail: rezakord@ipm.ir
|
---
abstract: 'This paper presents a closed-form expression for the integral kernels associated with the derivatives of the Ornstein-Uhlenbeck semigroup ${\mathrm{e}}^{tL}$. Our approach is to expand the Mehler kernel into Hermite polynomials and applying the powers $L^N$ of the Ornstein-Uhlenbeck operator to it, where we exploit the fact that the Hermite polynomials are eigenfunctions for $L$. As an application we give an alternative proof of the kernel estimates by [@Portal2014], making all relevant quantities explicit.'
address: |
Delft Institute of Applied Mathematics, Delft University of Technology\
P.O. Box 5031\
2600 GA Delft\
The Netherlands\
j.j.b.teuwen@tudelft.nl
author:
- Jonas Teuwen
bibliography:
- 'library.bib'
title: 'On the integral kernels of derivatives of the Ornstein-Uhlenbeck semigroup'
---
Introduction
============
Much effort [@Harboure2000; @Kemppainen2015; @Maas2010b; @MaasNeervenPortal2011; @MauceriMeda2007; @MauceriMeda2012; @Muckenhaupt; @Pineda2008; @Portal2014; @Sjogren1997; @Teuwen2015] has gone into developing the harmonic analysis of the [*Ornstein-Uhlenbeck operator*]{} $$\label{eq:OU-operator}
L := \frac12 \Delta - \langle x, \nabla \rangle.$$ On the space $L^2({\mathbf{R}}^d, {\mathrm{d}}\gamma)$, where $\gamma$ is the Gaussian measure $$\label{eq:Gaussian-measure}
{\mathrm{d}}\gamma(x) := \pi^{-d/2} {\mathrm{e}}^{-|x|^2} {\mathrm{d}}{x},$$ this operator can be viewed as the Gaussian counterpart of the Laplace operator $\Delta$. Indeed, one has $L = -\nabla^*\nabla$, where $\nabla$ is the the usual gradient and the $\nabla^*$ is its adjoint in $L^2({\mathbf{R}}^d, {\mathrm{d}}\gamma)$. It is a classical fact that the semigroup operators $e^{tL}$, $t>0$, are integral operators of the form $${\mathrm{e}}^{tL} u(\cdot) = \int_{{\mathbf{R}}^d} M_t(\cdot, y) u(y) {\,\mathrm{d}}\gamma(y),$$ where $M_t$ is the so-called [*Mehler kernel*]{} [@Sjogren1997] $$\label{eq:Mehler-kernel-oneformula}
\begin{aligned}
M_t(x, y)
& = \frac{\exp\Bigl(\displaystyle-\frac{|{\mathrm{e}}^{-t}x - y|^2}{1 - {\mathrm{e}}^{-2t}}
\Bigr)}{(1 - {\mathrm{e}}^{-2t})^{d/2}} {\mathrm{e}}^{|y|^2}
\end{aligned}$$ (see [@Teuwen2015] for a representation of $M_t$ which makes the symmetry in $x$ and $y$ explicit).
Developing a Hardy space theory for $L$ is the subject of active current research [@Portal2014; @MauceriMeda2007]. In this theory the derivatives $(d^k/dt^k)e^{t L} = L^N e^{tL}$ play an important role. The aim of the present paper is to derive closed form expressions for the integral kernels of these derivatives, that is, to determine explicitly the kernels $M_t^N$ such that we have the identity $$\label{eq:Mehler}
L^N{\mathrm{e}}^{tL} u(\cdot) = \int_{{\mathbf{R}}^d} M_t^N(\cdot, y) u(y) {\,\mathrm{d}}\gamma(y).$$ Direct application of the derivatives $d^N/dt^N$ to the Mehler kernel yields expressions which become intractible even for small values of $N$. Our approach will be to expand the Mehler kernel in terms of the $L^2$-normalised Hermite polynomials and then to apply $L^N$ to it, thus exploiting the fact that the Hermite polynomials are eigenfunctions for $L$.
As an application of our main result, which is proved in section \[sec:mainresult\] after developing some preliminary material in the sections \[sec:prelim\]-\[sec:Weyl\], we shall give a direct proof for the kernel bounds of [@Portal2014] in section \[sec:application\].
Preliminaries {#sec:prelim}
=============
In this preliminary section we collect some standard properties of Hermite polynomials and their relationship with the Ornstein-Uhlenbeck operator. Most of this material is classical and can be found in [@Sjogren1997; @MR1215939].
Hermite polynomials
-------------------
The [*Hermite polynomials*]{} $H_n$, $n\ge 0$, are defined by Rodrigues’s formula $$\label{eq:Hermite-Rodrigues}
H_n(x) := (-1)^n {\mathrm{e}}^{x^2} \partial_x^n {\mathrm{e}}^{-x^2}.$$ Their $L^2$-normalizations, $$h_n := \frac{H_n}{\sqrt{2^{n} n!}}$$ form an orthonormal basis for $L^2({\mathbf{R}},{\mathrm{d}}\gamma)$. We shall use the fact that the generating function for the Hermite polynomials is given by $$\label{eq:Generating-function-identity} \sum_{n = 0}^\infty \frac{H_n(x)}{n!} t^n ={\mathrm{e}}^{2 tx - t^2}.$$ The relationship with the Ornstein-Uhlenbeck operator is encoded in the eigenvalue identity $L H_n = -n H_n$, from which it follows that for all $t \ge 0$ we have ${\mathrm{e}}^{tL} H_n =
{\mathrm{e}}^{-t n} H_n$. From this one quickly deduces that the Mehler kernel is given by $$\label{eq:Mehler_kernel_non_comp}
M_t(x, y) := \sum_{n = 0}^\infty {\mathrm{e}}^{-t n} h_n(x)h_n(y).$$ We will need two further identities for the Hermite polynomials which can be found, e.g., in [@NIST:DLMF Chapter 18]: the integral representation $$\label{eq:Hermite-integral}
H_n(x) = \frac{(-2i)^n {\mathrm{e}}^{x^2}}{\sqrt\pi} \int_{-\infty}^\infty
\xi^n {\mathrm{e}}^{2ix \xi} {\mathrm{e}}^{-\xi^2} {\,\mathrm{d}}\xi$$ and the “binomial” identity $$\label{eq:Hermite-binomial-type}
H_n(x + y) = \sum_{k = 0}^n \binom{n}{k} (2y)^{n - k} H_k(x) .$$
Hermite polynomials in several variables
----------------------------------------
The Hermite polynomials on ${\mathbf{R}}^d$ are defined, for multiindices $\alpha = (\alpha_1, \dots, \alpha_d)\in{\mathbf{N}}^d$ by the tensor extensions $$\label{eq:Hermite-Rodrigues-Rd}
H_\alpha(x) := \prod_{n = 1}^{d} h_{\alpha_i}(x_i),$$ for $x=(x_1,\dots,x_d)$ in ${\mathbf{R}}^d$. The normalized Hermite polynomials $$\label{eq:Hermite-normalized}
h_\alpha := \frac{H_\alpha}{\sqrt{2^{|\alpha|} \alpha!}}, \ \text{ where } \alpha! = \alpha_1! \cdot \dots \cdot \alpha_d!$$ form an orthonormal basis in ${{L^2({\mathbf{R}}^d, {\mathrm{d}}\gamma)}}$, and we have the eigenvalue identity $$\label{eq:OU-Hermite-action}
LH_{\alpha} = -|\alpha| H_\alpha, \ \text{ where } |\alpha| = \alpha_1 + \dots + \alpha_d.$$ If we consider the action of $L^N {\mathrm{e}}^{tL}$ on a Hermite polynomial $h_\alpha$, through the multinomial theorem applied to $|\alpha|^k$ we get (writing $L_d$ for the operator $L$ in dimension $d$ and $L_1$ for the operator $L$ in dimension $1$) $$\label{eq:reduction-to-d1}
\begin{aligned} L_d^N {\mathrm{e}}^{tL} & h_\alpha(x) = |\alpha|^N {\mathrm{e}}^{-t|\alpha|}
h_{\alpha_1}(x_1) \cdot \hdots \cdot
h_{\alpha_d}(x_d)\\
&= \sum_{|n| = N} \binom{N}{n_1, n_2, \hdots, n_d} \alpha_1^{n_1}
\cdot \hdots \cdot \alpha_d^{n_d} {\mathrm{e}}^{-t\alpha_1}\cdot\hdots\cdot {\mathrm{e}}^{\alpha_d}
h_{\alpha_1}(x_1) \cdot \hdots\cdot
h_{\alpha_d}(x_d)
\\ & = \sum_{|n| = N} \binom{N}{n_1, n_2, \hdots, n_d} L_1^{n_1}
{\mathrm{e}}^{tL_1} h_{\alpha_1}(x_1) \cdot \hdots \cdot L_1^{n_d}
{\mathrm{e}}^{tL_1} h_{\alpha_d}(x_d).
\end{aligned}$$ This implies that we can reduce the question of computing the $d$-dimensional version of the integral kernel to the one-dimension one.
A combinatorial lemma {#sec:setup}
=====================
From now on we concentrate on the Ornstein-Uhlenbeck operator $L$ in one dimension, i.e., in $L^2({\mathbf{R}},{\mathrm{d}}\gamma)$. We are going to follow the approach of [@Sjogren1997]. Recalling the identity $L h_n = -n h_n$, we will apply $L^N$ to the generating function of the Hermite polynomials . A problem which immediately occurs is that $\Delta$ and $\langle x, \nabla \rangle$ do not commute, and because of this we cannot use a standard binomial formula to evaluate $L^N$. Instead, we note that $$L g = -t \partial_t g.$$ In particular this implies that $$L^N g = (-1)^k D_N g,$$ where $$\label{eq:Differential-operator-generated}
D_N := \underbrace{t \partial_t \circ t \partial_t \circ \dots \circ t \partial_t}_{\text{$k$ times}} = (t\partial_t)^k.$$ The following lemma will be very useful.
\[lem:Lk-powers-expanded\] We have $$\label{eq:Expanded-Differentiatial-operator-generated}
D_N = \sum_{n = 0}^N {\genfrac{\{}{\}}{0pt}{}{N}{n}} t^n \partial_t^n, $$ where ${\genfrac{\{}{\}}{0pt}{}{N}{n}}$ are the Stirling numbers of the second kind.
The Stirling numbers of the second kind are quite well-known combinatorial objects. For the sake of completeness we will state their definition and recall some relevant properties below. For more information we refer the reader to [@LINT]. The related Stirling numbers of the first kind will not be needed here.
We begin by recalling the definition of [*falling factorial*]{} $$\label{eq:falling-factorial}
(j)_n := j(j - 1)\dots (j - n + 1) = \frac{j!}{(j - n)!},$$ for non-negative integers $k \geq n$.
\[def:stirling-numbers-second-kind\] For non-negative integers $N \ge n$, the number [*Stirling number of the second kind*]{} ${\genfrac{\{}{\}}{0pt}{}{N}{n}}$ is defined as the number of partitions of an $N$-set into $n$ non-empty subsets.
These numbers satisfy the recursion identity $$\label{eq:Stirling-numbers-second-kind-recursion}
{\genfrac{\{}{\}}{0pt}{}{N}{n}} = N {\genfrac{\{}{\}}{0pt}{}{N - 1}{n}} + {\genfrac{\{}{\}}{0pt}{}{N - 1}{n - 1}}.$$ For all non-negative integers $j$ and $k$ one has the generating function identity $$\label{eq:Stirling-numbers-second-kind-generating-function-1}
j^N = \sum_{n = 0}^N {\genfrac{\{}{\}}{0pt}{}{N}{n}} (j)_n.$$
Weyl Polynomials {#sec:Weyl}
================
Before turning to the proof of lemma \[lem:Lk-powers-expanded\], let us already mention that it only depends on the commutator identity $[t,\partial_t] = -1$. This brings us to the observation that *Weyl polynomials* provide the natural habitat for our expressions. Rougly speaking, a Weyl polynomial is a polynomial in two non-commuting variables $x$ and $y$ which satisfy the commutator identity $[x, y] = -1$. This is made more precise in the following definition.
The [*Weyl algebra*]{} over a field ${\mathbf{F}}$ of characteristic zero is the ring ${\mathbf{F}}\langle x,y\rangle$ of all polynomials of the form $p(x,y) = \sum_{m=0}^M \sum_{n=0}^N c_{mn} x^m y^n$ with coefficients $c_{mn}\in{\mathbf{F}}$ in two noncommuting variables $x$ and $y$ which satisfy the commutator identity $$[x,y]:=xy-yx = -1.$$
We now have the following abstract version of lemma \[lem:Lk-powers-expanded\]:
\[lem:Weyl-expansion\] In the Weyl algebra ${\mathbf{F}}\langle x, y\rangle$ we have the identity $$\label{eq:Weyl-base-thm}
(x y)^m = \sum_{i = 1}^m {\genfrac{\{}{\}}{0pt}{}{m}{i}} x^i y^i,$$ where ${\genfrac{\{}{\}}{0pt}{}{m}{i}}$ are the Stirling numbers of the second kind.
As a preparation for the proof of lemma \[lem:Weyl-expansion\] we make a couple of easy computations. If we set $D := xy$, then $$\begin{aligned}
D x^m &= x^m D + m x^m\label{eq:Weyl-push-1},\\
D y^m &= y^m D - m y^m\label{eq:Weyl-push-2}.\end{aligned}$$ This can be shown by induction on $m$. For instance, note that $$D x^m = x(D + 1)x^{m - 1} = x D x^{m - 1} + x^m.$$ If we take this a bit further and have $p \in {\mathbf{F}}[D]$, then $$\begin{aligned}
\label{eq:Weyl-poly-1}
p(D) x^m &= x^m p(D + m),\\
\label{eq:Weyl-poly-2}
p(D) y^m &= y^m p(D - m).\end{aligned}$$ The $m$-th powers, $m\ge 1$, of $x$ and $y$ satisfy $$\begin{aligned}
\label{eq:Weyl-xmym}
x^m y^m &= \prod_{i = 0}^{m - 1} (D - i),\\
\label{eq:Weyl-ymxm}
y^m x^m &= \prod_{i = 1}^m (D + i).\end{aligned}$$ This can be seen using induction: $$x^{m + 1} y^{m + 1} = x^m D y^m \overset{\eqref{eq:Weyl-poly-2}}{=} x^m y^m (D - m)$$ and $$y^{m + 1} x^{m + 1} = y^m (D + 1) x^m = y^m D x^m + y^m x^m \overset{\eqref{eq:Weyl-poly-1}}{=} y^m x^m (D + (m + 1)).$$
The [*weighted degree*]{} of a monomial $x^m y^n\in {\mathbf{F}}\langle x,y\rangle$ is the integer $m-n$. A polynomial in ${\mathbf{F}}\langle x,y \rangle$ is said to be [*homogeneous of weighted degree $j$*]{} if all its constituting monomials have weighted degree $j$.
Left multiplication by $xy$ is [*homogeneity preserving*]{}, i.e., for all $j\in{\mathbf{Z}}$ it maps the set of homogeneous monomials of weighted degree $j$ into itself. To prove this, first consider a monomial $x^m y^n$ of dweighted egree $j = m-n$. Then, $$(xy) x^m y^n \mathrel{\overset{\eqref{eq:Weyl-push-1}}{=}} (x^m (xy) + m x^m)y^n = x^m(xy)y^n +mx^m y^n = x^{m + 1} y^{n + 1} + m x^m y^n,$$ and we see that weighted degree of homogeneity is indeed preserved. The general case follows immediately. Through we conclude that left multiplication $x^k y^k$ is homogeneity preserving as well, for all non-negative integers $k$. We claim that left multiplication by $x^i y^j$ is homogeneity preserving only if $i = j$. To see this note that $$y x^i y^j = x^i y^{i + j} + i
x^{i - 1} y^j$$ from which we can deduce that $$x^m y^M x^n y^N = x^{n + m} y^{N + M} + \text{ lower order terms}.$$ From which the claim follows.
Finally, a polynomial is homogeneity preserving if and only if all of its constituting monomials have this property. If this were not to be the case we could look at the highest-order non-homogeneous term and note from above $x^m y^M x^n y^N$ would give terms of a lower order in the polynomial expansion which cannot cancel as they have different powers of $x$ or $y$.
It follows from these observations that $$\label{eq:Weyl-G0}
F_0 := \biggl\{\sum_{n=0}^N c_{n} x^n y^n \ \bigg| \ N\in{\mathbf{N}}, \, c_1,\dots,c_N\in{\mathbf{F}}\biggr\}$$ is precisely the set of *homogeneity preserving polynomials* in ${\mathbf{F}}\langle
x, y\rangle$.
Now everything is in place to give the proof of lemma \[lem:Weyl-expansion\].
As $(xy)^k$ is homogeneity preserving, we infer that there are coefficients $a_i^k$ in ${\mathbf{F}}$ such that $$\label{eq:Weyl-lemma-intermediate-step-wanted-form}
(xy)^k = \sum_{i = 0}^k a_i^k x^i y^i.$$ We will apply $x^j$ to the right on both sides of and derive an expression for the $a_i^k$. First note that gives $$(xy)^k x^j = x^j (xy + j)^k,$$ and together with gives $$x^i y^i x^j \overset{\eqref{eq:Weyl-xmym}}{=}
\prod_{\ell = 0}^{i - 1} (xy - \ell) x^j \overset{\eqref{eq:Weyl-poly-1}}{=} x^j \prod_{\ell = 0}^{i - 1} (xy - \ell + j).$$ Hence, to find the coefficients $a_i^k$ it is sufficient to consider $$(xy + j)^k = \sum_{i = 0}^k a_n^k \prod_{\ell = 0}^{i - 1}(xy - \ell + j).$$ Comparing the constant terms on both the left-hand side and right-hand side, we find $$\label{eq:Weyl-result-generating-function}
j^k = \sum_{i = 0}^k a_i^k \prod_{\ell = 0}^{i - 1} (j - \ell) = \sum_{i = 0}^k a_i^k (j)_i,$$ where $(j)_i$ is the falling factorial as in . Comparing with the generating function of the Stirling numbers of the second kind ${\genfrac{\{}{\}}{0pt}{}{k}{i}}$ as given in , we see that $a_i^k = {\genfrac{\{}{\}}{0pt}{}{k}{i}}$. This concludes the proof of lemma \[lem:Weyl-expansion\].
The integral kernel of $L^N {\mathrm{e}}^{t L}$ {#sec:mainresult}
===============================================
As mentioned before, as a first step we would like to apply $D_N$ to the generating function $g(x,t) := {\mathrm{e}}^{-2tx + t^2 } = {\mathrm{e}}^{-(x - t)^2 + x^2}$ for the Hermite polynomials . We first compute the action of $\partial_t^N$ on the generating function.
We have $$\label{eq:derivatives-generating-function-Hermite}
\partial_t^N {\mathrm{e}}^{-(x - t)^2 + x^2} = {\mathrm{e}}^{-(x - t)^2 + x^2} H_N(x - t).$$
We first note that, $$\begin{aligned}
\partial_t {\mathrm{e}}^{-(x - t)^2} &= 2(x - t) {\mathrm{e}}^{-(x - t)^2} = - \partial_x {\mathrm{e}}^{-(x - t)^2}.\end{aligned}$$ Using this we get $$\begin{aligned}
\partial_t^N {\mathrm{e}}^{-(x - t)^2 + x^2} &= {\mathrm{e}}^{x^2} \partial_t^N {\mathrm{e}}^{-(x -
t)^2}\\
&= {\mathrm{e}}^{x^2} \partial_t^{N - 1} \partial_t {\mathrm{e}}^{-(x - t)^2}\\
&= -{\mathrm{e}}^{x^2} \partial_t^{N - 1} \partial_x {\mathrm{e}}^{-(x - t)^2}\\
&= (-)^2{\mathrm{e}}^{x^2} \partial_t^{N - 2} \partial_x^2 {\mathrm{e}}^{-(x - t)^2}\\
&= \dots\\
&= (-1)^N {\mathrm{e}}^{x^2} \partial_x^N {\mathrm{e}}^{-(x - t)^2}.\end{aligned}$$ By a change of variables, $$\partial_t^N {\mathrm{e}}^{-(x - t)^2 + x^2} = (-1)^N {\mathrm{e}}^{-(x - t)^2 + x^2} {\mathrm{e}}^{(x - t)^2} \partial_y^N {\mathrm{e}}^{-y^2} \Bigr|_{y = x - t}.$$ Hence, by , $$\partial_t^N {\mathrm{e}}^{-(x - t)^2 + x^2} = {\mathrm{e}}^{-(x - t)^2 + x^2} H_N(x - t).$$
\[lem:power-Ornstein-Uhlenbeck-generating-Hermite\] For all $x\in{\mathbf{R}}$ and $t>0$ we have $$\label{eq:power-Ornstein-Uhlenbeck-generating-Hermite}
L^N {\mathrm{e}}^{-(x - t)^2 + x^2} = (-1)^N {\mathrm{e}}^{-(x - t)^2 + x^2} \sum_{n = 0}^N {\genfrac{\{}{\}}{0pt}{}{N}{n}} t^n H_n(x - t).$$
This is now easy to prove. Recalling that $L = -t \partial_t$ and using , we get $$\begin{aligned}
L^N {\mathrm{e}}^{-(x - t)^2 + x^2} &= D_N {\mathrm{e}}^{-(x - t)^2 + x^2}\\
&= (-1)^N \sum_{n = 0}^N {\genfrac{\{}{\}}{0pt}{}{N}{n}} t^n \partial_t^n {\mathrm{e}}^{-(x - t)^2 + x^2}\\
&= (-1)^N {\mathrm{e}}^{-(x - t)^2 + x^2} \sum_{n = 0}^N {\genfrac{\{}{\}}{0pt}{}{N}{n}} t^n H_n(x - t).\end{aligned}$$
Our next theorem is the main result of this paper and provides an explicit expression for the integral kernel of $L^N e^{t L}$.
\[th:integral-kernel\] Let $L$ be the Ornstein-Uhlenbeck operator on $L^2({\mathbf{R}}^d,{\mathrm{d}}\gamma)$, let $t>0$, and let $N\ge 0$ be an integer. The integral kernel $M_t^N$ of $L^N {\mathrm{e}}^{tL}$ is given by $$\label{eq:Mehler-kernel-of-powers}
\begin{split}
M_t^N(x, y) &= M_t(x, y) \sum_{|n| = N} \binom{N}{n_1, \dots, n_d} \prod_{i = 1}^d
\sum_{m_i = 0}^{n_i} \sum_{\ell_i = 0}^{m_i} 2^{-m_i}
{\genfrac{\{}{\}}{0pt}{}{n_i}{m_i}} \binom{m_i}{\ell_i}\\
&\quad \times \biggl(-\frac{{\mathrm{e}}^{-t}}{\sqrt{1 - {\mathrm{e}}^{-2t}}} \biggr)^{2m_i - \ell_i}
H_{\ell_i}(x_i) H_{2m_i - \ell_i}\biggl(\frac{x_i {\mathrm{e}}^{-t} - y_i}{\sqrt{1 - {\mathrm{e}}^{-2t}}}\biggr).
\end{split}$$
We first prove the result for $d = 1$. Pulling $L^N$ through the integral expression for $e^{t L}$ involving the Mehler kernel, we must find a suitable expression for the kernel $M_t^N(\cdot, y) = L^N
M_t(\cdot, y)$. Using and the normalization of $H_m$ in we get $$\begin{aligned}
M_t^N(x, y) &= L^N \sum_{m = 0}^\infty \frac{{\mathrm{e}}^{-t m}}{m!}\frac1{2^m} H_m(x) H_m(y)\\
&\overset{\eqref{eq:Hermite-integral}}{=} L^N \sum_{m = 0}^\infty \frac{{\mathrm{e}}^{-t m}}{m!}\frac1{2^m} H_m(x) \frac{(-2i)^m}{\sqrt{\pi}} {\mathrm{e}}^{y^2} \int_{-\infty}^\infty {\mathrm{e}}^{-\xi^2} \xi^m {\mathrm{e}}^{2 i y \xi} {\,\mathrm{d}}\xi\\
&= L^N \frac{{\mathrm{e}}^{y^2}}{\sqrt{\pi}} \int_{-\infty}^\infty {\mathrm{e}}^{-\xi^2} {\mathrm{e}}^{2 i y \xi} \sum_{m = 0}^\infty \frac1{m!} H_m(x) (-i \xi {\mathrm{e}}^{-t})^m{\,\mathrm{d}}\xi\\
&\overset{\eqref{eq:Generating-function-identity}}{=} L^N
\frac{{\mathrm{e}}^{y^2}}{\sqrt{\pi}} \int_{-\infty}^\infty {\mathrm{e}}^{-\xi^2} {\mathrm{e}}^{2
i y \xi} {\mathrm{e}}^{-(x + i \xi {\mathrm{e}}^{-t})^2 + x^2} {\,\mathrm{d}}\xi.\end{aligned}$$ The operator $L^N$ is applied with respect to $x$ here, so by lemma \[lem:power-Ornstein-Uhlenbeck-generating-Hermite\] we get $$\begin{aligned}
M_t^N(x, y) &= \frac{{\mathrm{e}}^{y^2}}{\sqrt{\pi}}
\int_{-\infty}^\infty {\mathrm{e}}^{-\xi^2} {\mathrm{e}}^{2 i y \xi} L^N {\mathrm{e}}^{-(x + i \xi {\mathrm{e}}^{-t})^2 + x^2} {\,\mathrm{d}}\xi\\
&\overset{\eqref{eq:power-Ornstein-Uhlenbeck-generating-Hermite}}{=}(-1)^N
\frac{{\mathrm{e}}^{x^2 + y^2}}{\sqrt{\pi}} \sum_{m = 0}^N {\genfrac{\{}{\}}{0pt}{}{N}{m}}
\int_{-\infty}^\infty {\mathrm{e}}^{2 i y \xi} {\mathrm{e}}^{-(x + i \xi {\mathrm{e}}^{-t})^2} (i
\xi {\mathrm{e}}^{-t})^m H_m(y + i \xi {\mathrm{e}}^{-t}) {\mathrm{e}}^{-\xi^2} {\,\mathrm{d}}\xi,\end{aligned}$$ where in last line we have used the analytic continuation of the algebraic identity . Similarly we can expand $H_m(y + i \xi {\mathrm{e}}^{-t})$ using . This gives $$H_m(y + i \xi {\mathrm{e}}^{-t}) = \sum_{\ell = 0}^m \binom{m}{\ell} H_\ell(y)
(2 i \xi {\mathrm{e}}^{-t})^{m - \ell},$$ so that $M_t^N$ can be written as $$\label{eq:M_k-integral}
(-1)^N \frac{{\mathrm{e}}^{x^2 + y^2}}{\sqrt{\pi}} \sum_{m = 0}^N \sum_{\ell = 0}^m {\genfrac{\{}{\}}{0pt}{}{N}{m}} \binom{m}{\ell} H_\ell(y) 2^{m - \ell} \int_{-\infty}^\infty {\mathrm{e}}^{2 i y \xi - \xi^2} {\mathrm{e}}^{-(x + i \xi {\mathrm{e}}^{-t})^2} (i \xi {\mathrm{e}}^{-t})^{2m - \ell} {\,\mathrm{d}}\xi.$$ If we set $M = 2m - \ell$, this reduces our task to computing the integral $$\begin{aligned}
\notag
\frac{{\mathrm{e}}^{x^2 + y^2}}{\sqrt{\pi}} \int_{-\infty}^\infty &{\mathrm{e}}^{2 i y \xi - \xi^2} {\mathrm{e}}^{-(x + i \xi {\mathrm{e}}^{-t})^2} (i \xi {\mathrm{e}}^{-t})^M {\,\mathrm{d}}{\xi}\\
\label{eq:Hermite-integral-derivation-1}
&= \frac{{\mathrm{e}}^{y^2}}{\sqrt{\pi}} \int_{-\infty}^\infty {\mathrm{e}}^{2 i (x {\mathrm{e}}^{-t} - y) \xi} {\mathrm{e}}^{-(1 - {\mathrm{e}}^{-2 t}) \xi^2} (i \xi {\mathrm{e}}^{-t})^M
{\,\mathrm{d}}{\xi}.\end{aligned}$$ To make the computation less convolved, let us set $$\alpha_t := \sqrt{1 - {\mathrm{e}}^{-2t}}, \text{ and, } \beta_t(x, y) := \frac{x {\mathrm{e}}^{-t} - y}{\sqrt{1 - {\mathrm{e}}^{-2t}}}.$$ This allows us to write the exponential in the integral as $${\mathrm{e}}^{2 i (x{\mathrm{e}}^{-t} - y) \xi} {\mathrm{e}}^{-(1 - {\mathrm{e}}^{-2 t}) \xi^2} = {\mathrm{e}}^{2 i \alpha_t \beta_t(x,y) \xi} {\mathrm{e}}^{-\alpha_t^2 \xi^2}.$$ This reduces the problem, after the substitution $\alpha_t \xi \to \xi$, to computing the integral $$\frac{{\mathrm{e}}^{y^2}}{\sqrt{\pi}} \frac{i^M {\mathrm{e}}^{-M t}}{\alpha_t^{M + 1}}
\int_{-\infty}^\infty {\mathrm{e}}^{2 i \beta_t(x, y) \xi} {\mathrm{e}}^{-\xi^2} \xi^M {\,\mathrm{d}}{\xi}.$$ The final integral is an integral expression for the Hermite polynomials , so $$\begin{aligned}
\ & \frac{{\mathrm{e}}^{y^2}}{\sqrt{\pi}} \frac{i^M {\mathrm{e}}^{-M t}}{\alpha_t^{M + 1}}
\int_{-\infty}^\infty {\mathrm{e}}^{2 i \beta_t(x,y) \xi} {\mathrm{e}}^{-\xi^2} \xi^M {\,\mathrm{d}}{\xi}
\\ & \qquad \overset{\eqref{eq:Hermite-integral}}{=} {\mathrm{e}}^{y^2-\beta_t(x,y)^2}
\frac1{\alpha_t^{M + 1}} \frac{(-1)^M {\mathrm{e}}^{-M t}}{2^M} H_M(\beta_t(x,y)).\end{aligned}$$ Next we note that $\exp(y^2 - \beta_t(x,y)^2) \alpha_t^{-1} = M_t$, the Mehler kernel from . Hence, $$\label{eq:Mehler-kernel-Lk-intermediate-1}
\begin{split}
M_t^N(x, y) &= M_t(x, y) \sum_{m = 0}^N \sum_{\ell = 0}^m
\binom{m}{\ell} {\genfrac{\{}{\}}{0pt}{}{N}{m}}
\biggl(-\frac{{\mathrm{e}}^{-t}}{\sqrt{1 - {\mathrm{e}}^{-2t}}} \biggr)^{2m - \ell}
2^{-m}\\
&\quad \times H_\ell(x) H_{2m - \ell}\biggl(\frac{x {\mathrm{e}}^{-t} - y}{\sqrt{1 - {\mathrm{e}}^{-2t}}}\biggr).
\end{split}$$ Applying we get the result in $d$ dimensions.
An application {#sec:application}
==============
As an application of our main result, in this section we give an alternative proof of the bounds on the kernels $K$ and $\tilde K$ of [@Portal2014] (see the definition below), making the dependence on the parameters more explicit. These kernels play a central role in the study of the Hardy space $H^1({\mathbf{R}}^d,{\mathrm{d}}\gamma)$ in [@Portal2014], where the standard Calderón reproducing formula is replaced by $$u = C \int_0^\infty (t^2 L)^{N + 1} {\mathrm{e}}^{\frac{t^2}{\alpha} L} u \frac{{\,\mathrm{d}}{t}}{t} + \int_{{\mathbf{R}}^d} u {\,\mathrm{d}}\gamma,$$ where $C$ is a suitable constant only depending on $N$ and $\alpha$ (this can be seen by letting $u$ be a Hermite polynomial). The kernels $K$ and $\tilde K$ then occur in several decompositions, and the estimates below allow them to be related to classical results about the Mehler kernel.
We define the kernels $K$ and $\tilde K$ by $$\begin{aligned}
\int_{{\mathbf{R}}^d} K_{t^2, N, \alpha}(x, y) u(y) {\,\mathrm{d}}\gamma(y) &= (t^2 L)^N
{\mathrm{e}}^{\frac{t^2}{\alpha}
L} u(x),\\
\int_{{\mathbf{R}}^d} \tilde{K}_{t^2, N, \alpha, j}(x, y) u(y) {\,\mathrm{d}}\gamma(y) &= (t^2 L)^N
{\mathrm{e}}^{\frac{t^2}{\alpha}
L} t \partial_{x_j}^* u(x).
\end{aligned}$$
Note that the operators on the right-hand sides are indeed given by integral kernels: the first is a scaled version of the operator we have already been studying, and a duality argument implies that the second is given by the integral kernel $$\tilde{K}_{t^2, N, \alpha,j}(x, y) =
t \partial_{x_j} K_{t^2, N, \alpha}(x, y).$$ Thus, both kernels are given as appropriate derivatives of the Mehler kernel.
We begin with a technical lemma which is a rephrased version of [@Portal2014 Lemma 3.4]. One should take note that we define the kernels with respect to the Gaussian measure whereas, [@Portal2014] defines these with respect to the Lebesgue measure.
\[lem:Mehler-alpha-efficient1\] For all $\alpha > 1$ and all $t>0$ and $x, y$ in ${\mathbf{R}}^d$ we have $$\label{eq:Mehler-alpha-efficient2}
\frac{|{\mathrm{e}}^{-\frac{t}{\alpha}}x - y|^2}{1 - e^{-2\frac{t}{\alpha}}}
\geq\frac{\alpha}2 {\mathrm{e}}^{-2t} \frac{|{\mathrm{e}}^{-t}x - y|^2}{1 -
{\mathrm{e}}^{-2t}} - \frac{t^2 \min{(|x|^2, |y|^2)}}{1 - {\mathrm{e}}^{-2\frac{t}{\alpha}}}.$$ Additionally, we have $$\label{eq:Mehler-alpha-efficient3}
\alpha {\mathrm{e}}^{-2t} \leq \frac{1 - {\mathrm{e}}^{-2t}}{1 - {\mathrm{e}}^{-2\frac{t}{\alpha}}} \leq \alpha.$$
Let $N$ be a positive integer, $0 < t < T$. The for all large enough $\alpha>1$ we have
1. If $t |x| \leq C$, then $$\displaystyle |K_{t^2, \alpha, N}(x, y)| \lesssim_{T, N}
\alpha \exp(\frac{\alpha}2C^2)
M_{t^2}(x, y) \exp\biggl(-\frac{\alpha}{8{\mathrm{e}}^{2T}} \frac{|{\mathrm{e}}^{-t^2}x - y|^2}{1 -
{\mathrm{e}}^{-2t^2}} \biggr).$$
2. If $t |x| \leq C$, then $$\displaystyle |\tilde{K}_{t^2, \alpha, N,j}(x, y)|
\lesssim_{T, N} \alpha \exp(\frac{\alpha}2C^2) M_{t^2}(x, y) \exp\biggl(-\frac{\alpha}{8 {\mathrm{e}}^{2T}} \frac{|{\mathrm{e}}^{-t^2}x - y|^2}{1 -
{\mathrm{e}}^{-2t^2}} \biggr).$$
For $K_{t^2, \alpha, N}$, we use Theorem \[th:integral-kernel\] to obtain, after taking absolute values, $$\begin{aligned}
|K_{t^2, \alpha, N}(x, y)| &\leq M_{\frac{t^2}{\alpha}}(x, y) \sum_{|k| = N} \binom{N}{n_1, \dots, n_d}
\prod_{i = 1}^d t^{2k_i }\sum_{\ell_i = 0}^{n_i} \sum_{m_i = 0}^{m_i} 2^{-m_i}
\binom{m_i}{\ell_i} {\genfrac{\{}{\}}{0pt}{}{n_i}{m_i}}\\
&\quad \times \biggl(\frac{{\mathrm{e}}^{-\frac{t^2}{\alpha}}}{\sqrt{1 - {\mathrm{e}}^{-2\frac{t^2}{\alpha}}}} \biggr)^{2m_i - \ell_i}
| H_{\ell_i}(x_i)| \biggl| H_{2m_i - \ell_i}\biggl(\frac{x_i {\mathrm{e}}^{-\frac{t^2}{\alpha}} - y_i}{\sqrt{1 - {\mathrm{e}}^{-2\frac{t^2}{\alpha}}}}\biggr) \biggr |.\end{aligned}$$ Recalling that $\ell_1 + \dots + \ell_d \leq N$, using the assumptions $t \leq T$ and $t |x| \leq
C$ we can bound $t^{2k_i} |H_{\ell_i}(x)|$ by considering the highest order term to obtain $$t^{2k_i} |H_{\ell_i}(x)| \lesssim_{C,N,T} 1.$$ Using we proceed by looking at $$\begin{aligned}
M_{\frac{t^2}{\alpha}}(x, y) &= M_{t^2}(x, y) \bigg(\frac{{1 - {\mathrm{e}}^{-2t^2}}}{{1 - {\mathrm{e}}^{-2\frac{t^2}{\alpha}}}}\bigg)^{1/2}\exp\biggl(\frac{|{\mathrm{e}}^{-t^2} x - y|^2}{1 - {\mathrm{e}}^{-2t^2}}
\biggr) \exp\biggl(-\frac{|{\mathrm{e}}^{-\frac{t^2}{\alpha}} x - y|^2}{1 - {\mathrm{e}}^{-2\frac{t^2}{\alpha}}}
\biggr)\\
&\le \alpha M_{t^2}(x, y) \exp\biggl(\frac{|{\mathrm{e}}^{-t^2} x - y|^2}{1 - {\mathrm{e}}^{-2t^2}}
\biggr) \Bigg[\exp\biggl(-\frac12\frac{|{\mathrm{e}}^{-\frac{t^2}{\alpha}} x - y|^2}{1 - {\mathrm{e}}^{-2\frac{t^2}{\alpha}}}
\biggr) \bigg]^2.
$$ We can now bound the final Hermite polynomial in the expression of the kernel. Setting $M_i = 2m_i - \ell_i$ we get $$\begin{aligned}
\biggl(\frac{{\mathrm{e}}^{-\frac{t^2}{\alpha}}}{\sqrt{1 - {\mathrm{e}}^{-2\frac{t^2}{\alpha}}}} \biggr)^{M_i}
\biggl|H_{M_i}\biggl(\frac{{\mathrm{e}}^{-\frac{t^2}{\alpha}}x_i - y_i}{\sqrt{1 -
{\mathrm{e}}^{-2\frac{t^2}{\alpha}}}}\biggr)\biggr| &\lesssim_N
\biggl(\frac{{\mathrm{e}}^{-\frac{t^2}{\alpha}}}{\sqrt{1
- {\mathrm{e}}^{-2\frac{t^2}{\alpha}}}} \biggr)^{M_i}
\biggl(\frac{|{\mathrm{e}}^{-\frac{t^2}{\alpha}}x_i - y_i|}{\sqrt{1 -
{\mathrm{e}}^{-2\frac{t^2}{\alpha}}}}\biggr)^{M_i}\\
&\le \biggl(\frac{|{\mathrm{e}}^{-\frac{t^2}{\alpha}}x_i - y_i|}{{1 -
{\mathrm{e}}^{-2\frac{t^2}{\alpha}}}}\biggr)^{M_i}.\end{aligned}$$ Also, $$\begin{aligned}
\biggl(\frac{|{\mathrm{e}}^{-\frac{t^2}{\alpha}}x_i - y_i|}{{1 -
{\mathrm{e}}^{-2\frac{t^2}{\alpha}}}}\biggr)^{M_i} \exp\biggl(-\frac1{2}\frac{|{\mathrm{e}}^{-\frac{t^2}{\alpha}} x_i - y_i|^2}{1 - {\mathrm{e}}^{-2\frac{t^2}{\alpha}}}
\biggr) \lesssim 1.\end{aligned}$$ Putting these estimates together, using Lemma \[lem:Mehler-alpha-efficient1\], and taking $\alpha>1$ so large that $$1-\frac{\alpha}{4 {\mathrm{e}}^{2T}} \le -\frac{\alpha}{8 {\mathrm{e}}^{2T}} \ \hbox{ and } \
1 - {\mathrm{e}}^{-2\frac{t^2}{\alpha}} \ge \frac{t^2}{\alpha},$$ we obtain $$\begin{aligned}
|K_{t^2, \alpha, N}(x, y)| & \lesssim_{T, N}\alpha M_{t^2}(x, y) \exp\biggl(\frac{|{\mathrm{e}}^{-t^2} x - y|^2}{1 - {\mathrm{e}}^{-2t^2}}
\biggr) \exp\biggl(-\frac12\frac{|{\mathrm{e}}^{-\frac{t^2}{\alpha}} x - y|^2}{1 - {\mathrm{e}}^{-2\frac{t^2}{\alpha}}}
\biggr)\\
&\leq \alpha M_{t^2}(x, y) \exp\biggl(\frac{|{\mathrm{e}}^{-t^2} x - y|^2}{1 - {\mathrm{e}}^{-2t^2}}
\biggr) \exp\biggl(-\frac{\alpha}{4 {\mathrm{e}}^{2T}} \frac{|{\mathrm{e}}^{-t^2}x - y|^2}{1 -
{\mathrm{e}}^{-2t^2}} \biggr) \exp\biggl(\frac12 \frac{t^4 |x|^2}{1 - {\mathrm{e}}^{-2\frac{t^2}{\alpha}}} \biggr)\\
&= \alpha M_{t^2}(x, y) \exp\biggl(\Big(1-\frac{\alpha}{4 {\mathrm{e}}^{2T}}\Big) \frac{|{\mathrm{e}}^{-t^2}x - y|^2}{1 -
{\mathrm{e}}^{-2t^2}} \biggr) \exp\biggl(\frac12 \frac{t^4 |x|^2}{1 - {\mathrm{e}}^{-2\frac{t^2}{\alpha}}} \biggr)\\
&\leq \alpha M_{t^2}(x, y) \exp\biggl(-\frac{\alpha}{8 {\mathrm{e}}^{2T}} \frac{|{\mathrm{e}}^{-t^2}x - y|^2}{1 -
{\mathrm{e}}^{-2t^2}} \biggr) \exp\biggl(\frac\alpha2 t^2 |x|^2 \biggr)
\\ &\leq \alpha \exp\biggl(\frac\alpha2 C^2 \biggr) M_{t^2}(x, y)
\exp\biggl(-\frac{\alpha}{8 {\mathrm{e}}^{2T}} \frac{|{\mathrm{e}}^{-t^2}x - y|^2}{1 -
{\mathrm{e}}^{-2t^2}} \biggr),\end{aligned}$$ using $t|x| \leq C$ in the last step.
For the bound on $\tilde{K}$ we consider $$\begin{aligned}
t \partial_{x_i} \biggl[H_{\ell_i}(x_i) H_{m_i} \biggl(\frac{x_i {\mathrm{e}}^{-t} - y_i}{\sqrt{1
- {\mathrm{e}}^{-2t}}} \biggr) \biggr] &= t H_{m_i} \biggl(\frac{x_i {\mathrm{e}}^{-t} - y_i}{\sqrt{1
- {\mathrm{e}}^{-2t}}} \biggr) \partial_{x_i} H_\ell(x_i)\\
&\quad + H_{\ell_i}(x_i) t \partial_{x_i} H_{m_i} \biggl(\frac{x_i {\mathrm{e}}^{-t} - y_i}{\sqrt{1
- {\mathrm{e}}^{-2t}}} \biggr).\end{aligned}$$ So, as the first term on the right-hand side just decreases in degree we look at $$\begin{aligned}
t \partial_{x_i} \biggl(\frac{x_i {\mathrm{e}}^{-t} - y_i}{\sqrt{1 - {\mathrm{e}}^{-2t}}}
\biggr)^{m_i} = m_i \biggl(\frac{x_i {\mathrm{e}}^{-t} - y_i}{\sqrt{1 - {\mathrm{e}}^{-2t}}}
\biggr)^{m_i - 1} t \frac{{\mathrm{e}}^{-t}}{\sqrt{1 - {\mathrm{e}}^{-2t}}}\end{aligned}$$ The last term is bounded as $t \downarrow 0$, and the rest of the proof is as before.
Acknowledgments {#acknowledgments .unnumbered}
---------------
This work was partially supported by NWO-VICI grant 639.033.604 of the Netherlands Organisation for Scientific Research (NWO).
The author wishes to thank Alex Amenta and Mikko Kemppainen for inspiring discussions.
|
---
abstract: 'Deep neural networks (DNNs) are notorious for their vulnerability to adversarial attacks, which are small perturbations added to their input images to mislead their prediction. Detection of adversarial examples is, therefore, a fundamental requirement for robust classification frameworks. In this work, we present a method for detecting such adversarial attacks, which is suitable for any pre-trained neural network classifier. We use influence functions to measure the impact of every training sample on the validation set data. From the influence scores, we find the most supportive training samples for any given validation example. A $k$-nearest neighbor ($k$-NN) model fitted on the DNN’s activation layers is employed to search for the ranking of these supporting training samples. We observe that these samples are highly correlated with the nearest neighbors of the normal inputs, while this correlation is much weaker for adversarial inputs. We train an adversarial detector using the $k$-NN ranks and distances and show that it successfully distinguishes adversarial examples, getting state-of-the-art results on four attack methods with three datasets.'
author:
- |
Gilad Cohen\
School of Electrical Engineering\
Tel Aviv University\
Tel Aviv, 69978\
`giladco1@mail.tau.ac.il`\
Guillermo Sapiro\
Electrical and Computer Engineering\
Duke University\
North Carolina, 27708\
`guillermo.sapiro@duke.edu` Raja Giryes\
School of Electrical Engineering\
Tel Aviv University\
Tel Aviv, 69978\
`raja@tauex.tau.ac.il`\
bibliography:
- 'my\_bib.bib'
title: Detecting Adversarial Samples Using Influence Functions and Nearest Neighbors
---
Introduction {#Introduction}
============
Deep Neural Networks (DNNs) are vastly employed in both the academy and industry, achieving state-of-the-art results in many domains such as computer vision [@Krizhevsky2012ImageNetNetworks; @Schroff2015FaceNet:Clustering; @Voulodimos2018DeepReview], natural language processing [@DBLP:journals/corr/BahdanauCB14; @Kim2014ConvolutionalClassification], and speech recognition [@Hinton2012DeepRecognition; @DBLP:journals/corr/ZhangPBZLBC17]. However, studies have shown that DNNs are vulnerable to adversarial examples [@FGSM; @Intriguing], which are specially crafted perturbations on their input. Adversarial attacks generate such examples that fool machine learning models, inducing them to predict erroneously with high confidence, while being imperceptible to humans. Adversarial subspaces of different DNN classifiers tend to overlap, which makes some adversarial examples generated for a surrogate model fool also other different unseen DNNs [@Transferable]. This makes adversarial attacks a real threat to any machine learning model and thus should be kept in mind while deploying a DNN.
The vulnerability of neural networks puts into question their usage in sensitive applications, where an opponent may provide modified inputs to cause misidentifications. For this reason, many methods have been developed to face this challenge. They can be mainly divided into two groups: 1) *proactive* defense methods, which aim at improving the robustness of DNNs to adversarial examples, and 2) *reactive* detection techniques that do not change the DNN but rather try to find whether an attack is associated with a certain input or not.
![The correspondence between the helpful examples based on influence functions and the $k$-nearest neighbours ($k$-NN) in the embedding space of a DNN can help to distinguish adversarial examples from normal ones. We present (using PCA) the embedding space of a DNN for a normal example (black star) with its adversarial version (brown **X**) along with their $k$-NN ($k$=25) and 25 most helpful samples. Note that for the normal example, the helpful samples highly correlate with the $k$-NN in the embedding space. Yet, in the adversarial case, these samples are far from each other. This observation leads us to a technique for detecting adversarial attacks.[]{data-label="images/teaser"}](images/teaser.png){width="\linewidth"}
[**Contribution.**]{} In this work, we focus on the reactive detection problem. We propose a novel strategy for detecting adversarial attacks that can be applied to any pre-trained neural network. The core idea of the algorithm is that there should be a correspondence between the training data and the classification of the network. If this relationship breaks then it is very likely that we are in the case of an adversarial input.
To this end, we use two “metrics” to check the impact of the training data on the network decision. The first is influence functions [@Koh2017UnderstandingBP], which determines how data points in the training set influence the decision of the network for a given test sample. This metric measures how much a small upweighting of a specific training point in the model’s loss function affects the loss of a testing point. Thus, it provides us with a measure of how much a test sample classification is affected by each training sample.
Second, we apply a $k$-nearest neighbor ($k$-NN) classifier at the embedding space of the network. Various recent works [@Deep_kNN_Papernot; @Doring2017_knn_convergence; @to_trust_or_not_a_classifier; @DNN_or_kNN] demonstrate a high correlation between the network softmax output and the decision of a $k$-NN applied at the embedding space of this network (where the neighbors are chosen from the training set). They basically show that the network’s decision relies on the nearest neighbors resemblance in the embedding space. Thus, the distance in that space may serve as a measure for the effect of an example on the network output.
Given the influence function and $k$-NN based measures, we turn to combine them together to generate a novel strategy to detect adversarial examples. The rationale behind our approach is that for a normal input, its $k$-NN training samples (nearest neighbors in the embedding space) and the most helpful training samples (found using the influence function) should correlate. Yet, for adversarial examples this correlation should break and thus, it will serve as an indication that an attack is happening.
Figure \[images/teaser\] illustrates this relationship between the $k$-NN and the most helpful training samples. The black star and brown **X** denote a normal and its corresponding adversarial image from CIFAR-10 validation set; the plot is of the embedding space projected using PCA fitted on the training set. For each sample (normal/adv), we find its 25 nearest neighbors (blue circles/red downward triangles) in the DNN embedding space; in addition, we find its 25 most helpful training examples from the training set (marked as blue squares and red triangles, respectively). Note that the nearest neighbors and the top most helpful training samples of the normal image are very close in the PCA embedding space, whereas the adversarial image does not exhibit the same correspondence between the training samples.
To check the correlation between the two, we pursue the following strategy: For an unseen input sample, we take the most influential examples from the training set chosen by the influence functions. Then, we check their distance ranking in the embedding space of the network (i.e., what value of $k$ will cause $k$-NN to take them into account) and their $L_2$ distance from the input sample’s embedding vector. Finally, we use these $k$-NN features to train a simple Logistic Regression (LR) for detecting whether the input is adversarial or not.
We evaluate our detection strategy on various attack methods and datasets showing its advantage over other leading detection techniques. The results confirm the hypothesis claimed in previous works on the resemblance between $k$-NN applied on the embedding space and the DNN decision, and show how it can be used for detecting adversarial examples.
Related work {#Related work}
============
In this section, we briefly review existing papers on adversarial attacks and defenses, and related theory.
**Theory:** @Towards_DeepLearning_Resistance used the framework of robust optimization and showed results of adversarial training. They found that projected gradient descent (PGD) is an optimal first order adversary, and employing it in the DNN training leads to optimal robustness against any first order attack. @Vulnerability_Input_Dimension demonstrated that DNNs’ vulnerability to adversarial attacks is increased with the gradient of the training loss as a function of the inputs. They also found that this vulnerability does not depend on the DNN model.
@ClassificationRegions studied the geometry and complexity of the functions learned by DNNs and provided empirical analysis of the curvatures of their decision boundaries. They showed that a DNN classifier is most vulnerable where its decision boundary is positively curved and that natural images are usually located in the vicinity of flat decision boundaries. These findings are also supported by @Curvature, who found that positively curved decision boundaries increase the likelihood that a small universal perturbation would fool a DNN classifier.
Some works provided guarantees to certify robustness of the network. @Cross_Lipschitz_regularization formalized a formal upper bound for the noise required to flip a network prediction, while @CertifyingRobustness provided an efficient and fast guarantee of robustness for the worst-case population performance, with high probability.
**Adversarial attacks:** One of the simplest and fastest attack methods is the fast gradient sign method (FGSM) [@FGSM]; in this method the attacker linearly fits the cross entropy loss around the attacked sample and lightly perturbs the image pixels in the direction of the gradient loss. This is a fast one-step attack, which is very easy to deploy on raw input images.
The Jacobian-based saliency map attack (JSMA) [@JSMA] takes a different approach. Instead of mildly changing all image pixels, this attack is crafted on the $L_0$ norm, finding one or two pixels which induce the largest change in the loss and modify only them. This is a strong attack, achieving 97% success rate by modifying only 4.02% of the input features on average. Yet, it is iterative and costly.
Deepfool proposed by @DeepFool is a non-targeted attack[^1] that creates an adversarial example by moving the attacked input sample to its closest decision boundary, assuming an affine classifier. In reality most DNNs are very non linear, however, the authors used an iterative method, linearizing the classifier around the test sample at every iteration. Compared to FGSM and JSMA, Deepfool performs less perturbations to the input. It was also employed in the Universal Perturbations attack by @Dezfooli17Universal, which is an iterative attack that aims at fooling a group of images using the same minimal, universal perturbation applied on all of them.
@CarliniWagner2017Towards proposed a targeted attack[^2] (denoted as CW) to impact the defensive distillation method [@Distillation]. The CW attack is resilient against most adversarial detection methods. In another work @Carlini2017BypassingTen provided an optimization framework, which includes a defense-specific loss as a regularization term . This optimization-based attack is argued to be the most effective to date for a white-box threat model, here the adversary knows everything related to the trained DNN: training data, architecture, hyper-parameters, weights, etc.
**Adversarial defenses:** A wide range of proactive defense approaches have been proposed, including adversarial (re)training [@FGSM; @Kurakin2017AdversarialML; @tramer2018ensemble; @Shaham2018RobustOptimization; @Virtual_Adversarial_Training], distillation networks [@Distillation], gradient masking [@tramer2018ensemble], feature squeezing [@Xu2018FeatureSD], network input regularization [@InputGradients; @Jakubovitz2018ImprovingDR], output regularization [@Cross_Lipschitz_regularization], adjusting weights of correctly predicted labels [@BANG], and Parseval networks [@Parseval].
However, those defenses can be evaded by the optimization-based attack [@Carlini2017BypassingTen], either wholly or partially. Since there are no known intrinsic properties that differentiate adversarial samples from regular images, proactive adversarial defense is extremely challenging. Instead, recent works have focused on reactive adversarial detection methods, which aim at distinguishing adversarial images from natural images, based on features extracted from DNN layers [@Metzen17detecting; @Li2017AdversarialED; @Rouhani2018TowardsSD] or from a learned encoder [@Meng2017MagNetAT]. @Feinman2017DetectingAS proposed a LR detector based on Kernel density and Bayesian uncertainty features.
@LID characterized the dimensional properties of the adversarial subspaces regions and proposed to use a property called Local Intrinsic Dimentionaloty (LID) . LID describes the rate of expansion in the number of data objects as the distance from the reference sample increases. The authors estimated the LID score at every DNN layer using extreme value theory, where the smallest NN distances are considered as extreme events associated with the lower tail of the data samples’ underlying distance distribution. Given a pretrained network and a dataset of normal examples, the authors applied on every sample: 1) Adversarial attack. 2) Addition of Gaussian Noise. The natural and noisy images were considered as negative (non-adversarial) class and the adversarial images were considered as positive class. For each image (natural/noisy/adversarial) they calculated a LID score at every DNN layer. Lastly, a LR model was fitted on the LID features for the adversarial detection task.
@Deep_kNN_Papernot proposed the Deep $k$-Nearest Neighbors (D$k$NN) algorithm to estimate better the prediction, confidence, and credibility for a given test sample. Using a pretrained network, they fitted a $k$-NN model at every layer. Next, they used a left-out calibration set to estimate the nonconformity of every test sample for label j, counting the number of nearest neighbors along the DNN layer which differs from j. They showed that when an adversarial attack is made on a test sample, the real label displays less correspondence with the $k$-NN labels from the DNN activations along the layers.
@Mahalanobis_adv_detection trained generative classifiers using the DNN activations of the training set on every layer to detect adversarial examples by applying a Mahalanobis distance-based confidence score. First, for every class and every layer, they computed the empirical mean and covariance of the activations induced by the training samples. Next, using the above class-conditional Gaussian distributions, they calculated the Mahalanobis distance between a test sample and its nearest class-conditional Gaussian. These distances are used as features to train a LR classifier for the adversarial detection task. The authors claimed that using the Mahalanobis distance is significantly more effective than the Euclidean distance employed by @LID and showed improved detection results.
Method {#Method}
======
We hypothesize that the DNN predictions are influenced by the $k$-NN of the training data in their hidden layers, especially in the embedding layer, which is the penultimate activation layer in the DNN classifier. If so, in order to fool the network, an adversarial attack must move the test sample towards a “bad” subspace in the embedding space, where harmful training data can cause the network to misclassify the correct label. To inspect our hypothesis, we fitted a $k$-NN model on the DNN’s activation layers, and also employed the influence functions as used in [@Koh2017UnderstandingBP].
Influence functions can interpret a DNN by pointing out which of the training samples helped the DNN to make its prediction, and which training samples were harmful, i.e., inhibited the network from its prediction. @Koh2017UnderstandingBP suggested to measure the influence a train sample $z$ has on the loss of a specific test sample $z_{test}$, by the term: $$\label{I_up_loss}
I_{up,loss}(z, z_{test}) = -\nabla_{\theta}L(z_{test}, \theta)^TH_{\theta}^{-1}\nabla_{\theta}L(z, \theta),$$ where $H$ is the Hessian of the machine learning model, $L$ is its loss, and $\theta$ are the model parameters.
For each test example $z_{test}$, we calculate Eq. per each training example $z$ in the training set. Then, we sort all $I_{up,loss}(z, z_{test})$ scores, determining the top $M$ helpful and harmful training examples for a specific $z_{test}$. Next, for each of the 2x$M$ selected training points we find its rank and distance from the testing example by fitting a $k$-NN model on the embedding space using all the training examples’ embedding vectors. We feed the embedding vector of each test sample $z_{test}$ to the $k$-NN model to extract all the nearest neighbors’ ranks (denoted as $\mathcal{R}$) and distances (denoted $\mathcal{D}$) of the examples in the training set. The $\mathcal{R}$ and $\mathcal{D}$ features can also be extracted from any other hidden activation layer within the DNN, and not solely from the embedding vector. $\mathcal{R}^{M\uparrow}$, $\mathcal{D}^{M\uparrow}$ and $\mathcal{R}^{M\downarrow}$, $\mathcal{D}^{M\downarrow}$ are all the ranks and distances of the helpful and harmful training examples, respectively.
We apply an adversarial attack on $z_{test}$ and repeat the aforementioned process on the new, crafted image. Both the normal and adversarial features ($\mathcal{R}^{M\uparrow}$, $\mathcal{D}^{M\uparrow}$, $\mathcal{R}^{M\downarrow}$, $\mathcal{D}^{M\downarrow}$) are used to train a LR classifier for the adversarial detection task. The detector training scheme is described in Algorithm \[alg:NNIF\].
Training set ($X_{train}$, $Y_{train}$) and validation set ($X_{val}$, $Y_{val}$) Pre-trained DNN with L activation layers and parameters $\theta$ $M$: Number of top influence samples to collect Detector($\mathcal{R}^{M\uparrow}$, $\mathcal{D}^{M\uparrow}$, $\mathcal{R}^{M\downarrow}$, $\mathcal{D}^{M\downarrow}$)
$N_{train} = |X_{train}|$, $N_{val} = |X_{val}|$
Initialize: $R_{norm}^+$=\[\], $D_{norm}^+$=\[\], $R_{norm}^-$=\[\], $D_{norm}^-$=\[\] Initialize: $R_{adv}^+$=\[\], $D_{adv}^+$=\[\], $R_{adv}^-$=\[\], $D_{adv}^-$=\[\] ($X_{val}^{adv}, Y_{val}^{adv}$) := adversarial attack on ($X_{val}$, $Y_{val}$)
Fit $k$-NN\[*l*\] model on layer *l*. $k=N_{train}$ $\mathcal{R}^{M\uparrow}$, $\mathcal{D}^{M\uparrow}$, $\mathcal{R}^{M\downarrow}$, $\mathcal{D}^{M\downarrow}$ := <span style="font-variant:small-caps;">NNFeatures</span>($x_i$, $k\text{-NN}$\[*l*\]) $R_{norm}^+$.append($\mathcal{R}^{M\uparrow}$), $D_{norm}^+$.append($\mathcal{D}^{M\uparrow}$), $R_{norm}^-$.append($\mathcal{R}^{M\downarrow}$), $D_{norm}^-$.append($\mathcal{D}^{M\downarrow}$) $\mathcal{R}^{M\uparrow}$, $\mathcal{D}^{M\uparrow}$, $\mathcal{R}^{M\downarrow}$, $\mathcal{D}^{M\downarrow}$ := <span style="font-variant:small-caps;">NNFeatures</span>($x_i$, $k\text{-NN}$\[*l*\]) $R_{adv}^+$.append($\mathcal{R}^{M\uparrow}$), $D_{adv}^+$.append($\mathcal{D}^{M\uparrow}$), $R_{adv}^-$.append($\mathcal{R}^{M\downarrow}$), $D_{adv}^-$.append($\mathcal{D}^{M\downarrow}$)
$NNIF_{pos}$ = ($R_{adv}^+$, $D_{adv}^+$, $R_{adv}^-$, $D_{adv}^-$) $NNIF_{neg}$ = ($R_{norm}^+$, $D_{norm}^+$, $R_{norm}^-$, $D_{norm}^-$)
Detector($\mathcal{R}^{M\uparrow}$, $\mathcal{D}^{M\uparrow}$, $\mathcal{R}^{M\downarrow}$, $\mathcal{D}^{M\downarrow}$) = train a classifier on ($NNIF_{pos}$, $NNIF_{neg}$)
Initialize: $R^+$=\[\], $D^+$=\[\], $R^-$=\[\], $D^-$=\[\]
$\mathcal{R}$, $\mathcal{D}$ := Apply $k\text{-NN}$ on activation layer *l*, get training examples’ ranks and $L_2$ distances out of activations of sample $x_i$ $H_{inds}^+$, $H_{inds}^-$ := $\textsc{InfluenceFunction}((x_i, y_i), (X_{train}, Y_{train}))$ $R^+$.append($\mathcal{R}[j]$) $D^+$.append($\mathcal{D}[j]$) $R^-$.append($\mathcal{R}[j]$) $D^-$.append($\mathcal{D}[j]$) **return** $R^+$, $D^+$, $R^-$, $D^-$
We name our adversarial detection method as Nearest Neighbor Influence Functions (NNIF). We assume that the training, validation, and testing sets are not contaminated with adversarial examples, as in [@Carlini2017BypassingTen]. We start by generating an adversarial validation set from the normal validation set (step 4). The $M$ most helpful and harmful training examples associated with the validation image prediction (either normal or adversarial) are found using the influence function in step 22 (see supp. material for the <span style="font-variant:small-caps;">InfluenceFunction</span> procedure). The NNIF features are then evaluated by the $k$-NN model, extracting the ranks and distances (from $\mathcal{R}$ and $\mathcal{D}$) of the most influential training points found above. This is done for both the normal validation images (step 8) and for the adversarial images (step 12). This scheme can be carried out on the embedding layer alone, or employed for all $L$ activation layers within the DNN.
Finally, a LR classifier is trained using the NNIF features. Images from the test set are classified to either adversarial (positive) or normal (negative) based on the NNIF features extracted from the $M$ most helpful/harmful training examples, ($\mathcal{R}^{M\uparrow}$, $\mathcal{D}^{M\uparrow}$, $\mathcal{R}^{M\downarrow}$, $\mathcal{D}^{M\downarrow}$).
Training our NNIF detector is very time consuming, requiring us to calculate Eq. on the entire training set for every validation image, having a time complexity of $\mathcal{O}(N_{train} \cdot N_{val})$, where $N_{train}$ and $N_{val}$ are the size of the training and validation sets, respectively. For an adversarial detection the complexity time is $\mathcal{O}(N_{train})$, since we need to find the top $M$ helpful/harmful training examples for every new incoming test image.
@Deep_kNN_Papernot focused on improving credibility and robustness in DNN. They used the nearest neighbors in the activation layers for interpretability. As an additional competing strategy, we convert their original D$k$NN algorithm [@Deep_kNN_Papernot] to an adversarial detection method. This is done by collecting the empirical p-values calculated in the D$k$NN strategy and formulating a reactive adversarial detector by training a LR model on these features. While NNIF also use nearest neighbors, instead of inspecting the labels of the nearest neighbors, we examine the correlation between them and the image’s most helpful/harmful training examples using the influence functions.
Results
=======
{width="\linewidth"}
This section shows the power of our NNIF adversarial detector against four adversarial attack strategies: FGSM, JSMA, DeepFool, and CW (using $L_2$ norm), as introduced in Section \[Related work\]. We selected these attacks for our experiments due to their effectiveness, diversity, and popularity. We applied these attacks on three datasets: CIFAR-10, CIFAR-100 [@CIFAR], and SVHN [@SVHN]. NNIF performance is compared to the state-of-the-art LID and Mahalanobis detectors (Section \[Related work\]) and also to the D$k$NN adversarial detector (Section \[Method\]). Before presenting our results, we first describe the experimental setup used in our analysis.
Experimental setup
------------------
**Training and Testing:** Each of the three image datasets was divided into three subsets: *training* set, *validation* set, and *testing* set, containing 49k, 1k, and 10k images respectively. Since our NNIF method is time consuming (especially the procedure <span style="font-variant:small-caps;">InfluenceFunction</span> in Algorithm \[alg:NNIF\]), we randomly selected 49k and 1k *training* and *validation* samples, respectively, from the official SVHN training set and 10k testing samples from the official SVHN testing set. Any *validation* or *testing* image not correctly classified by the DNN was discarded. For every image in the *validation* and *testing* sets, we generated adversarial examples using the four attack methods (FGSM, JSMA, DeepFool, CW), as describe in Step 4 in Algorithm \[alg:NNIF\]. Then, an equal number of normal and adversarial *validation* images were used to train a LR classifier, which was later applied on the remaining *testing* images for calculating the detectors metrics. We used the *cleverhans* library [@cleverhans] to carry out all the adversarial attacks. The image RGB values were scaled to \[0, 1\].
Since the D$k$NN method requires a calibration set, we randomly selected 33% of the *validation* set examples (after discarding the misclassifications) for calibrating it. Note that although @Deep_kNN_Papernot showed that the nearest neighbors can qualitatively detect adversarial attacks (see Fig. 7 in [@Deep_kNN_Papernot]), they did not formalize an adversarial detector. We employ their empirical $p$-values as features for the adversarial detection task.
**Training DNNs:** We trained all DNNs on the *training* set while decaying the learning rate using the *validation* set’s accuracy score. All the DNNs used in our experiments are Resnet-34 [@RESNET] with global average pooling layer prior to the embedding space. The embedding vector was multiplied by a fully-connected layer for the logits calculation. We trained all three datasets for 200 epochs, with $L_2$ weight regularization of 0.0004, using a Stochastic Gradient Decent optimizer with momentum 0.9 and Nesterov updates. For evaluation we used the model checkpoint with the best (highest) validation accuracy on the image classification task. We follow the checklist in [@OnEvaluatingAdvRobustness2019] and report the full DNN validation/test accuracies for the clean models when not under attack and the attacks success rates (see supp. material). These DNNs perform close to the state-of-the-art and thus are sufficient for being used in an adversarial study without fine tuning [@Feinman2017DetectingAS].
**Parameter tuning:** The number of neighbors ($k$) for LID and DkNN, the noise magnitude ($\epsilon$) for the Mahalanobis method, and the number of top influence samples to collect ($M$) for NNIF were chosen using nested cross validation within the *validation* set, based on the AUC values of the detection ROC curve. We tuned $k$ for DkNN using an exhaustive grid search between \[10, ${N}/{\#classes}$\], where N is the dataset size and $\#classes$ is the number of classes. For LID the number of nearest neighbors was tuned using a grid search over the range \[10, 40) while using a minibatch size of 100 (as in [@LID]). For the Mahalanobis method we tuned $\epsilon$ using an exhaustive grid search in log-space between \[$1E^{-5}$, $1E^{-2}$\], and $M$ was tuned using a grid search over \[10, 500\]. The selected parameters are presented in the supp. material.
Running <span style="font-variant:small-caps;">InfluenceFunction</span> in Algorithm \[alg:NNIF\] for an entire *training* set is very slow. Thus, for every *testing* set we randomly selected only 10k out of the 49k samples in the *training* set and calculated $I_{up,loss}$ (Eq. ) just for them. Although this is a coarse approximation of the real nearest neighbors distribution in the *training* set on the DNN embedding space, this approximation is sufficient for achieving new state-of-the-art adversarial detection. We emphasize that this approximation was done only for the *testing* set, and not for the *validation* set.
**Activation layers:** The LID, Mahalanobis, and NNIF detectors can be trained using either features from the embedding space alone or using all the activation layers in the network. The D$k$NN detector portrays very poor results when it is applied on all the DNN’s features (data not shown) and therefore, we present all the D$k$NN results by training features from the embedding space alone.
Detection of adversarial attacks {#Detection of adversarial attacks}
--------------------------------
Figure \[images/all\_defenses\_small\] shows the discrimination power (AUC score) of the four inspected adversarial detectors: D$k$NN (black), LID (blue), Mahalabolis (green) and NNIF (red), on three popular datasets: CIFAR-10, CIFAR-100, and SVHN. We compare between the detection scores calculated for four adversarial attacks: FGSM, JSMA, Deepfool, and CW. The solid bars correspond to detections where only the penultimate activation layer was utilized. In some cases, considering all the layers in the DNN activations boosts the LID/Mahalanobis/NNIF scores; this is portrayed as a complementary hatched patterned bar above the solid bar.
Our method surpasses all other detectors for distinguishing Deepfool and CW attacks, for all the datasets. On FGSM and JSMA our NNIF detector also demonstrates state-of-the-art results, matching the Mahalanobis detector’s performance. Table \[detection\_scores\] summarizes the AUC scores of the detectors using features from all the DNN’s activation layers. The only exception is the D$k$NN method, which is employed only on the embedding space. In the supp. material, we provide a similar table for the obtained AUC scores using only the DNN’s penultimate layer.
Ablation study
--------------
$\mathcal{R}^{M\uparrow}$ $\mathcal{D}^{M\uparrow}$ $\mathcal{R}^{M\downarrow}$ $\mathcal{D}^{M\downarrow}$ AUC(%) acc(%)
--------------------------- --------------------------- ----------------------------- ----------------------------- ----------- -----------
82.11 77.03
66.14 61.47
83.25 78.44
99.79 97.68
**99.82** 97.51
99.79 99.29
99.81 97.34
98.27 96.69
97.73 97.21
98.28 96.73
97.62 97.12
99.79 97.73
99.81 97.78
99.79 97.71
**99.82** **97.86**
: Ablation test for adversarial attack detection: Calculating AUC score and accuracy for selected features. Attacking CIFAR-10 dataset using Deepfool.[]{data-label="ablation_table"}
To quantify the contribution of each one of the features ($\mathcal{R}^{M\uparrow}$, $\mathcal{D}^{M\uparrow}$, $\mathcal{R}^{M\downarrow}$, $\mathcal{D}^{M\downarrow}$) on the NNIF method performance, we conducted an ablation study on CIFAR-10 dataset, detecting FGSM, JSMA, Deepfool, and CW adversarial attacks. Table \[ablation\_table\] shows the AUC and accuracy results for Deepfool attack using features from the DNN’s embedding space only. The complete table with all the attacks is presented in the supp. material.
Our analysis shows that the most influential feature is $\mathcal{D}^{M\uparrow}$, which is the $L_2$ distance from the most helpful training examples on the embedding space. In most cases, our NNIF detector performance using $\mathcal{D}^{M\uparrow}$ is nearly as good as the performance upon utilizing all four features. The least important feature is $\mathcal{R}^{M\downarrow}$, which barely helps the adversarial detection. Intuitively it makes sense because we have noticed that the classes of the most harmful training examples always differ from the normal examples’ class and mostly differ from the adversarial examples’ class, and thus their rankings ($\mathcal{R}^{M\downarrow}$) are expected to be high for both cases (normal/adversarial). On the other hand, the distances from the most harmful training examples ($\mathcal{D}^{M\downarrow}$) are beneficial for the detection. The most helpful ranks ($\mathcal{R}^{M\uparrow}$) is a beneficial feature when used by itself, alas incorporating it with $\mathcal{D}^{M\uparrow}$ did not improve the results. We therefore deduce that the information added by $\mathcal{R}^{M\uparrow}$ can already be inferred from $\mathcal{D}^{M\uparrow}$ in our detector.
We also show that the features $\mathcal{R}^{M\uparrow}$, $\mathcal{D}^{M\uparrow}$, $\mathcal{D}^{M\downarrow}$ affect every attack differently. We calculated the probability density functions for these three features on CIFAR-10, applying the Deepfool and CW attacks (shown in the supp. material). From these histograms it can be easily observed that $\mathcal{R}^{M\uparrow}$ or $\mathcal{D}^{M\uparrow}$ are more useful for detecting Deepfool adversarial attacks than CW ones. On the other hand, the $\mathcal{D}^{M\downarrow}$ feature discriminates CW attacks better than Deepfool ones.
A deployment of any learning based detector on systems is risky since an attacker could potentially have access to the LR classifier’s parameters. Thus, it is helpful to deploy instead a detector which inspects only one feature and applies a simple thresholding. Our results show that this scheme is possible with NNIF using only the $\mathcal{D}^{M\uparrow}$ feature for all attacks.
Generalization to other attacks
-------------------------------
To evaluate how well our detection method can be transferred to unseen attacks, we trained LR classifiers on the features obtained using the FGSM attack, and then evaluated the classifies on JSMA, Deepfool, and CW attacks. The AUC scores are shown in Table \[generalization\_table\]. It can be observed that our NNIF method shows the best generalization on Deepfool and CW, however the Mahalanobis method transferred better to the JSMA attack. Table \[generalization\_table\] results were collected using only the penultimate layer in the DNN (the embedding vector); we provide a similar generalization table with all the DNN layers in the supp. material. Notice that the generalization is weaker for all methods in this case.
Discussion and conclusions {#Discussion and conclusions}
==========================
In this paper, we addressed the task of detecting adversarial attacks. We showed that for normal (untempered) images, there exists a strong correlation between their nearest neighbors in the DNN’s embedding space and their most helpful training examples, found using influence functions. Our empirical results show that the $L_2$ distance from a test image embedding vector to its most helpful training inputs ($\mathcal{D}^{M\uparrow}$) is a strong measure for the detection of adversarial examples. The aforementioned distance combined with the nearest neighbors ranking order of the training inputs were used to achieve a new state-of-the-art adversarial detection performance for sophisticated attacks (Deepfool and CW) on three datasets: CIFAR-10, CIFAR-100, and SVHN.
One possible avenue for future research is to inspect how the nearest neighbors are correlated with the most helpful/harmful training examples using different distance metrics or by employing a transform on the DNN embedding vectors. We emphasize that we used the $L_2$ distance throughout our analysis, thus, we suspect that using another distance metric such as Mahalanobis [@Mahalanobis_adv_detection] could improve our results further.
Another open issue for future research is the long computation time, which is required to calculate the influence functions for the entire training set. It is obvious that in order to deploy our NNIF algorithm, a significant improvement in computation time is needed, especially for real time applications or systems, which mandate fast detection pace. A possible solution to this problem may be a form of hash map from the nearest neighbors to the most influence training examples. Every training example can be encoded with a probability vector for its influence on a specific class; then, instead of employing a simple $k$-NN search in the embedding space, we can average over the probability of each class.
Method {#supp_Method}
======
The main paper proposes a new reactive detection method for adversarial images: the Nearest Neighbors Influence Functions (NNIF). Our detector utilizes a influence functions algorithm as shown in [@Koh2017UnderstandingBP] to measure the contribution of each training sample to a test samples prediction. Their algorithm is summarized in Algorithm \[alg:InfluenceFunction\]. For measuring the influence a train sample $z$ has on the loss of a specific test sample $z_{test}$, [@Koh2017UnderstandingBP] approximate $I_{up,loss}(z, z_{test})$ in Eq. , where $H$ is the Hessian of the machine learning model, $L$ is its loss, and $\theta$ are the model parameters. Eq. is repeated throughout the training set, calculating $I_{up,loss}$ for every training sample. For our NNIF algorithm only the top $M$ helpful training examples ($H_{inds}^+$) and the top $M$ harmful training examples ($H_{inds}^-$) are chosen for further processing.
Experimental setup
==================
The DNNs clean accuracies, when not under attack, are shown in Table \[clean\_acc\]. The FGSM, JSMA, Deepfool, and CW success rates are shown in Table \[attack\_rates\]. Note that the success rates of all attacks are higher for CIFAR-100. This makes sense since CIFAR-100 dataset has 100 classes instead of 10, and it is thus more vulnerable to misclassifications.
The paper explains how we tuned the hyper-parameters for the four inspected algorithms: D$k$NN, LID, Mahalanobis, and our NNIF method.
For the D$k$NN and LID algorithms we tuned the number of neighbors ($k$), for the Mahalanobis algorithm we tuned the noise magnitude ($\epsilon$), and for our NNIF method we set the number of top influence samples to collect ($M$). All parameters were chosen using nested cross entropy validation within the validation set, based on the AUC values of the detection ROC curve. The results are shown in Table \[parameters\].
Test sample $(x_i, y_i$) and a training set ($X_{train}$, $Y_{train}$) $M$: Number of top influence samples to collect $H_{inds}^+$, $H_{inds}^-$ $N_{train} = |X_{train}|$ Initialize $H_{inds}^+$=\[\], $H_{inds}^-$=\[\] Initialize $I_{up,loss}$ = zeros\[$N_{train}$\] $I_{up,loss}[j] = -\nabla_{\theta}L(x_i, \theta)^TH_{\theta}^{-1}\nabla_{\theta}L(x_j, \theta)$ sort($I_{up,loss}[j]$) $j_{m}^+$ = Training example index of $I_{up,loss}[N_{train}-m]$ $H_{inds}^+$.append($j_{m}^+$) $j_{m}^-$ = Training example index of $I_{up,loss}[m]$ $H_{inds}^-$.append($j_{m}^-$) **return** $H_{inds}^+$, $H_{inds}^-$
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Detection of adversarial attacks {#detection-of-adversarial-attacks}
================================
Figure \[images/auc\_cifar10\] presents two ROC curves for classification of Deepfool and CW adversarial attacks on the CIFAR-10 dataset. One can observe that our NNIF method (solid red line) achieves better classification power over the previous state-of-the-art methods.
Table \[detection\_scores\_last\_layer\] presents the AUC scores for the adversarial detection of FGSM, JSMA, Deepfool, and CW attacks on CIFAR-10, CIFAR-100, and SVHN datasets. These results was obtained by using DNN’s features from only the embedding space. A similar table with detectors which were trained on the the entire DNN’s features is in the main paper.
Ablation study
==============
To inspect how the four learned features influence our adversarial detection we conducted an ablation study on CIFAR-10 dataset, for all the attacks: fast gradient sign method (FGSM) [@FGSM], Jacobian-based saliency map attack (JSMA) [@JSMA], Deepfool [@DeepFool], and Carlini Wagner (CW) attack [@CarliniWagner2017Towards]. The results are shown in Table \[Supp\_ablation\_table\]. From these results, one may conclude that the most beneficial feature is $\mathbb{D}^{M\uparrow}$, which is the $L_2$ distance from the most helpful training examples on the deep neural network (DNN) embedding space.
Figure \[images/histograms\_full\] shows the probability density functions for $\mathbb{R}^{M\uparrow}$, $\mathbb{D}^{M\uparrow}$, and $\mathbb{D}^{M\downarrow}$ features on CIFAR-10 for the Deepfool and CW adversarial attacks. From these histograms, it can be easily observed that $\mathbb{R}^{M\uparrow}$ or $\mathbb{D}^{M\uparrow}$ are more useful for detecting Deepfool adversarial attacks than CW attacks. On the other hand, the $\mathbb{D}^{M\downarrow}$ feature discriminates CW attacks better than Deepfool attacks. This is also supported by the results on Table \[Supp\_ablation\_table\]: For $\mathbb{R}^{M\uparrow}$ or $\mathbb{D}^{M\uparrow}$ alone NNIF detects Deepfool better than CW ($98.27\% > 81.91\%$ and $99.79\% > 97.27\%$), however, for $\mathbb{D}^{M\downarrow}$ NNIF is able to detect CW attacks better than Deepfool attacks ($89.97\% > 82.11\%$).
Generalization to other attacks
===============================
The main paper measures the NNIF method transferability from one attack (FGSM) to other, unseen attacks (JSMA, Deepfool, and CW), where all the features are extracted from the penultimate activation layer. Here we provide a similar table where all the DNN’s activation layers are employed for this comparison (Table \[Supp\_generalization\_table\]), except of D$k$NN which only utilizes features from the DNN’s embedding space.
The generalization results in Table \[Supp\_generalization\_table\] does not have a definite winner method. The D$k$NN, Mahalanobis, and our NNIF methods demonstrate the best transferability for various setups. The LID detector does not generalize as good as the others for any setup.
![Probability density functions of the most helpful ranks ($\mathbb{R}^{M\uparrow}$, top row), most helpful distances ($\mathbb{D}^{M\uparrow}$, middle row), and the most harmful distances ($\mathbb{D}^{M\downarrow}$, bottom row), on CIFAR-10 for the Deepfool and CW attacks. The features for the normal (untempered) images that were correctly classified by the network are shown in blue. The features for the adversarial images are shown in orange. The features for the normal images that were misclassified by the network are shown in green (in the middle row).[]{data-label="images/histograms_full"}](images/histograms_full.png){width="\linewidth"}
[^1]: Non-targeted attacks are adversarial attacks which aim to make the prediction incorrect regardless of the spricifc erroneous class.
[^2]: Targeted attacks are adversarial attacks which aim to make the prediction classified to a particular erroneous class.
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abstract: 'Structure-preserving geometric algorithm for the Vlasov-Maxwell (VM) equations is currently an active research topic. We show that spatially-discretized Hamiltonian systems for the VM equations admit a local energy conservation law in space-time. This is accomplished by proving that for a general spatially-discretized system, a global conservation law always implies a discrete local conservation law in space-time when the algorithm is local. This general result demonstrates that Hamiltonian discretizations can preserve local conservation laws, in addition to the symplectic structure, both of which are the intrinsic physical properties of infinite dimensional Hamiltonian systems in physics.'
author:
- Jianyuan Xiao
- Hong Qin
- Jian Liu
- Ruili Zhang
bibliography:
- 'enecons\_pic.bib'
title: 'Local Energy Conservation Law for Spatially-Discretized Hamiltonian Vlasov-Maxwell System'
---
\#1[Fig. \#1]{} \#1[Eq. (\[\#1\])]{} \#1[Sec. \#1]{} \#1[Ref. ]{} \#1[W\_[\_[0]{} I]{}( \#1 )]{} \#1[W\_[\_[1]{} J]{}( \#1 )]{} \#1[W\_[\_[1]{} I]{}( \#1 )]{} \#1[W\_[\_[1]{} J’]{}( \#1 )]{} \#1[W\_[\_[2]{} K]{}( \#1 )]{}
The dynamics of a collection of charged particles and electromagnetic fields is governed by the well-known Vlasov-Maxwell (VM) equations $$\begin{aligned}
{1}
\frac{\partial f_{s}}{\partial t}+\bfv\cdot\nabla f_{s}+\frac{q_{s}}{m_{s}}\left(\bfE+\bfv\times\bfB\right)\cdot\frac{\partial f_{s}}{\partial\bfv}=0\thinspace, & \text{}\label{eq:VM}\\
\frac{\partial\bfB}{\partial t}=-\nabla\times\bfE\thinspace,\\
\frac{\partial\bfE}{\partial t}=\nabla\times\bfB-\sum_{s}\int\rmd\bfv q_{s}f_{s}\left(\bfx,\bfv,t\right)\bfv\thinspace,\label{eq:AL}\end{aligned}$$ where $\bfE$ and $\bfB$ are electromagnetic fields, $f_{s}$, $m_{s}$ and $q_{s}$ are the number density distribution function, mass and charge of the $s$’th particle, respectively. Here the permittivity and permeability are set to 1 for simple notation. This set of equations is also a Hamiltonian Partial Differential Equation (PDE) [@morrison1980maxwell; @weinstein1981comments; @marsden1982hamiltonian], which means that solutions of the equations conserve the symplectic structure [@marsden1998multisymplectic; @marsden2013introduction] and various types of invariants. As one of these invariants, the Hamiltonian itself is conserved, i.e., $$\begin{aligned}
\frac{\partial}{\partial t}\int\rmd\bfx\left(\frac{E^{2}+B^{2}}{2}+\sum_{s}\int\rmd\bfv\left(\frac{1}{2}m_{s}v^{2}f_{s}\left(\bfx,\bfv;t\right)\right)\right)=0~.\end{aligned}$$ For the Vlasov-Maxwell equations and many other Hamiltonian PDE systems, there are also local conservation laws, which can be written in the following form $$\begin{aligned}
\frac{\partial p}{\partial t}+\nabla\cdot\bfu=0~,\end{aligned}$$ where $p$ is a scalar field or a component of a tensor or vector field, and $\bfu$ is the flux corresponding to $p$. As an important example, the local energy conservation law for the Vlasov-Maxwell system reads $$\begin{aligned}
& & \frac{\partial}{\partial t}\left(\frac{E^{2}+B^{2}}{2}+\sum_{s}\int\rmd\bfv\left(\frac{1}{2}m_{s}v^{2}f_{s}\left(\bfx,\bfv;t\right)\right)\right)+\nonumber \\
& & \nabla\cdot\left(\bfE\times\bfB+\sum_{s}\int\rmd\bfv\left(\frac{1}{2}m_{s}v^{2}\bfv f_{s}\left(\bfx,\bfv;t\right)\right)\right)=0~.\label{EqnENEcons}\end{aligned}$$ It can be verified directly by using the Vlasov-Maxwell equations -. The conservation law can also be obtained by using Noether’s theorem and the weak Euler-Lagrangian equations [@qin2014field]. Local conservation laws are more fundamental and practical than global conservation laws, because we often consider systems without global conservations, for example, systems with open boundaries and particle sources. Nowadays, Particle-In-Cell (PIC) simulations [@birdsall1991plasma; @hockney1988computer] are commonly used in the investigation of Vlasov-Maxwell systems [@Cary93; @nieter2004vorpal; @Xiang08; @germaschewski2016plasma; @huang16; @Qiang2016; @Oeftiger16; @Planche16], and advanced structure-preserving geometric algorithms based on variational or Hamiltonian discretization have been developed recently [@Squire4748; @squire2012geometric; @xiao2013variational; @kraus2013variational; @evstatiev2013variational; @Shadwick14; @xiao2015variational; @xiao2015explicit; @crouseilles2015hamiltonian; @Qin15JCP; @he2015hamiltonian; @he2016hamiltonian; @qin2016canonical; @Webb16; @kraus2016gempic; @xiao2016explicit]. Some of these structure-preserving PIC schemes are able to bound the global energy errors for all simulation time-steps and are effective for solving multi-scale problems [@xiao2013variational; @xiao2015explicit; @xiao2016explicit]. For these algorithms, the corresponding spatially-discretized systems conserve global energy as a consequence of time symmetry admitted by the system. However, as an essential physical property, the discrete local energy conservation has not yet been discussed. If a discrete system has a local conservation law, it implies that at every grid point there is a conservation law, which is much stronger than just one global conservation law for the entire system. There have been some investigations on the discrete local energy conservation law associated with Yee’s FDTD schemes for Maxwell’s equations [@chew1994electromagnetic; @de1995poynting]. However, the local conservation law for discrete particle-field systems is still an unexplored topic. In this paper, we prove a discrete local energy conservation law for the spatially-discretized Vlasov-Maxwell system described in Ref. [@xiao2015explicit]. This is accomplished by proving in the Appendix a theorem stating that for a general spatially-discretized system, a global conservation law always implies a discrete local conservation law in space-time when the algorithm is local. Here, an algorithm is called local if the time-advance of a field at a grid point or a particle involving only its neighboring grid points and particles. With this theorem, for any geometric spatial-discretizations with local algorithms, we only need to search for global conservation laws and then the corresponding local conservation laws are automatically satisfied. This general result demonstrates that Hamiltonian discretizations can preserve local conservation laws in space-time, in addition to the symplectic structure, both of which are the intrinsic physical properties of many important infinite dimensional Hamiltonian systems in physics.
The idea of structure-preserving spatial discretization of the Vlasov-Maxwell equations can be traced back to Lewis [@lewis1970energy] who proposed a spatial-discretized Lagrangian for Vlasov plasmas. Today, modern geometric discretizations for constructing PIC schemes are based on Discrete Exterior Calculus (DEC) [@hirani2003discrete; @desbrun2005discrete], interpolation forms [@whitney1957geometric; @desbrun2008discrete; @xiao2015explicit] or Finite Element Exterior Calculus (FEEC) [@arnold2006finite; @arnold2010finite; @monk2003finite] to ensure the conservation of charge, the gauge invariance, and the symplectic structure [@Squire4748; @squire2012geometric; @xiao2015explicit; @he2016hamiltonian; @kraus2016gempic; @xiao2016explicit]. We start from the Lagrangian of the spatially-discretized Vlasov-Maxwell system in Ref. [@xiao2015explicit], $$\begin{gathered}
L_{sd}=\frac{1}{2}\left(\sum_{J}\left(-\dot{\bfA}_{J}-\sum_{I}{\nabla_{\mathrm{d}}}_{JI}\phi_{I}\right)^{2}-\sum_{K}\left(\sum_{J}\CURLD_{KJ}\bfA_{J}\right)^{2}\right)\Delta V+\\
\sum_{s}\left(\frac{1}{2}m_{s}\dot{\bfx}_{s}^{2}+q_{s}\left(\dot{\bfx}_{s}\cdot\sum_{J}W_{\sigma_{1J}}\left(\bfx_{s}\right)\bfA_{J}-\sum_{I}W_{\sigma_{0I}}\left(\bfx_{s}\right)\phi_{I}\right)\right)~.\end{gathered}$$ Here, integers $I$, $J$ and $K$ are indices of grid points in a cubic-mesh, and $\nabla_{\mathrm{d}}$, $\CURLD$ and $\DIVD$ are the discrete gradient, curl and divergence operators defined as follows, $$\begin{aligned}
\left({\nabla_{\mathrm{d}}}\phi\right)_{i,j,k} & = & [\phi_{i+1,j,k}-\phi_{i,j,k},\phi_{i,j+1,k}-\phi_{i,j,k},\phi_{i,j,k+1}-\phi_{i,j,k}]~,\label{EqnDEFGRADD}\\
\left(\CURLD\bfA\right)_{i,j,k} & = & \left[\begin{array}{c}
\left({A_{z}}_{i,j+1,k}-{A_{z}}_{i,j,k}\right)-\left({A_{y}}_{i,j,k+1}-{A_{y}}_{i,j,k}\right)\\
\left({A_{x}}_{i,j,k+1}-{A_{x}}_{i,j,k}\right)-\left({A_{z}}_{i+1,j,k}-{A_{z}}_{i,j,k}\right)\\
\left({A_{y}}_{i+1,j,k}-{A_{y}}_{i,j,k}\right)-\left({A_{x}}_{i,j+1,k}-{A_{x}}_{i,j,k}\right)
\end{array}\right]^{T}~,\label{DEFCURLD}\\
\left({\DIVD}\bfB\right)_{i,j,k} & = & \left({B_{x}}_{i+1,j,k}-{B_{x}}_{i,j,k}\right)+\left({B_{y}}_{i,j+1,k}-{B_{y}}_{i,j,k}\right)+\nonumber \\
& & \left({B_{z}}_{i,j,k+1}-{B_{z}}_{i,j,k}\right)~.\label{EqnDEFDIVD}\end{aligned}$$ These operators are local linear operators on the discrete fields $\phi_{I}$, $\bfA_{J}$ and $\bfB_{K}$. Functions $W_{\sigma_{0J}}$, $W_{\sigma_{1I}}$ and $W_{\sigma_{2K}}$ are interpolation functions (Whitney forms) for 0-forms, e.g., scalar potential, 1-forms, e.g., vector potential, and 2-forms, e.g., magnetic fields, respectively. These discrete operators and interpolation functions satisfy the following properties [@whitney1957geometric; @hirani2003discrete; @desbrun2008discrete; @xiao2015explicit], $$\begin{aligned}
\nabla\sum_{I}W_{\sigma_{0I}}\left(\bfx\right)\phi_{I} & = & \sum_{I,J}W_{\sigma_{1J}}\left(\bfx\right){\nabla_{\mathrm{d}}}_{JI}\phi_{I}~,\label{EqnD0to1FORMAPP}\\
\nabla\times\sum_{J}W_{\sigma_{1J}}\left(\bfx\right)\bfA_{J} & = & \sum_{J,K}W_{\sigma_{2K}}\left(\bfx\right){\CURLD}_{KJ}\bfA_{J}~,\label{EqnD1to2FORMAPP}\\
\nabla\cdot\sum_{K}W_{\sigma_{2K}}\left(\bfx\right)\bfB_{K} & = & \sum_{K,L}W_{\sigma_{3L}}\left(\bfx\right){\DIVD}_{LK}\bfB_{K}~.\label{EqnD2to3FORMAPP}\end{aligned}$$ Periodic boundary in all three directions are adopted to simplify the discussion. Because time $t$ does not explicitly appear in the Lagrangian, the total energy $$\begin{aligned}
H_{sd} & = & \frac{\partial L_{sd}}{\partial\dot{q}}\dot{q}^{T}-L_{sd}~\\
& = & \frac{1}{2}\Delta V\left(\sum_{J}\bfE_{J}^{2}+\sum_{K}\bfB_{K}\right)+\sum_{s}\frac{1}{2}m_{s}\dot{\bfx}_{s}^{2}~\end{aligned}$$ is conserved, i.e., $\dot{H}_{sd}=0$. Here, $q$ is the generalized coordinates, i.e., $q=[\bfA_{J},\phi_{I},\bfx_{s}]$, and $\bfE_{J}$ and $\bfB_{K}$ are discrete electromagnetic fields defied as $$\begin{aligned}
\bfE_{J} & = & -\dot{\bfA}_{J}-\sum_{J}{\nabla_{\rmd}}_{JI}\phi_{I}~,\\
\bfB_{K} & = & \sum_{J}\CURLD_{KJ}\bfA_{J}~.\end{aligned}$$ The corresponding Poisson bracket for this system is [@xiao2015explicit] $$\begin{gathered}
\left\{ F,G\right\} =\frac{1}{\Delta V}\sum_{J}\left(\frac{\partial F}{\partial\bfE_{J}}\cdot\sum_{K}\frac{\partial G}{\partial\bfB_{K}}\CURLD_{KJ}-\sum_{K}\frac{\partial F}{\partial\bfB_{K}}\CURLD_{KJ}\cdot\frac{\partial G}{\partial\bfE_{J}}\right)+\\
\sum_{s}\frac{1}{m_{s}}\left(\frac{\partial F}{\partial\bfx_{s}}\cdot\frac{\partial G}{\partial\dot{\bfx}_{s}}-\frac{\partial F}{\partial\dot{\bfx}_{s}}\cdot\frac{\partial G}{\partial\bfx_{s}}\right)+\\
\sum_{s}\frac{q_{s}}{m_{s}\Delta V}\left(\frac{\partial F}{\partial\dot{\bfx}_{s}}\cdot\sum_{J}W_{\sigma_{1J}}\left(\bfx_{s}\right)\frac{\partial G}{\partial\bfE_{J}}-\frac{\partial G}{\partial\dot{\bfx}_{s}}\cdot\sum_{J}W_{\sigma_{1J}}\left(\bfx_{s}\right)\frac{\partial F}{\partial\bfE_{J}}\right)+\\
-\sum_{s}\frac{q_{s}}{m_{s}^{2}}\frac{\partial F}{\partial\dot{\bfx}_{s}}\cdot\left[\sum_{K}W_{\sigma_{2K}}\left(\bfx_{s}\right)\bfB_{K}\right]\times\frac{\partial G}{\partial\dot{\bfx}_{s}}~.\label{eq:22}\end{gathered}$$ With this Poisson bracket, the time evolution of the system is $$\begin{aligned}
\dot{g}=\{g,H_{sd}\}~,\label{EqnDEM}\end{aligned}$$ where $g=[\bfE_{J},\bfB_{K},\bfx_{s},\dot{\bfx}_{s}].$ Now we introduce the discrete local energy $\epsilon_{I}$ at the $I$’th grid, $$\begin{aligned}
\varepsilon_{I}=\sum_{s}\frac{1}{2}m_{s}\dot{\bfx}_{s}^{2}\WZERO{\bfx_{s}}+\bfE_{I}^{2}+\bfB_{I}^{2}.\end{aligned}$$ The evolution of $\varepsilon_{I}$ is $$\begin{aligned}
\dot{\varepsilon}_{I} & = & \left\{ \varepsilon_{I},H_{sd}\right\} ~,\end{aligned}$$ or more specifically, $$\begin{aligned}
\dot{\varepsilon}_{I} & = & \sum_{s}q_{s}\left(\dot{\bfx}_{s}\cdot\sum_{J'}\bfE_{J'}\WONEJp{\bfx_{s}}\right)\WZERO{\bfx_{s}}+\sum_{s}\frac{1}{2}m_{s}\dot{\bfx}_{s}^{2}\left(\dot{\bfx}_{s}\cdot\nabla\WZERO{\bfx_{s}}\right)+\nonumber \\
& & \bfE_{I}\cdot\sum_{K}\CURLD_{KI}\bfB_{K}-\bfE_{I}\cdot\sum_{s}q_{s}\dot{\bfx}_{s}\WONEA{\bfx_{s}}-\bfB_{I}\cdot\sum_{J}\CURLD_{IJ}\bfE_{J}~.\label{EqnDEPSLDT}\end{aligned}$$ We will see that the right hand side of can be written as a discrete divergence of a discrete vector field, which means that is a discrete energy conservation law.
Let us divide the RHS of into three terms, $$\begin{aligned}
\dot{\varepsilon}_{I} & = & T_{1}+T_{2}+T_{3}~,\end{aligned}$$ where $$\begin{aligned}
T_{1} & = & \bfE_{I}\cdot\sum_{K}\CURLD_{KI}\bfB_{K}-\bfB_{I}\cdot\sum_{J}\CURLD_{IJ}\bfE_{J}~,\\
T_{2} & = & \sum_{s}\frac{1}{2}m_{s}\dot{\bfx}_{s}^{2}\left(\dot{\bfx}_{s}\cdot\nabla\WZERO{\bfx_{s}}\right)~,\\
T_{3} & = & \sum_{s}q_{s}\left(\dot{\bfx}_{s}\cdot\sum_{J'}\bfE_{J'}\WONEJp{\bfx_{s}}\right)\WZERO{\bfx_{s}}-\bfE_{I}\cdot\sum_{s}q_{s}\dot{\bfx}_{s}\WONEA{\bfx_{s}}~.\end{aligned}$$ Firstly, for $T_{1}$, we can check that this term can be written as a discrete divergence $$\begin{aligned}
T_{1}=-\sum_{K}\DIVD_{IK}(\bfE\times*\bfB)_{K}~,\end{aligned}$$ where $\bfE\times*\bfB$ is defined as $$\begin{aligned}
(\bfE\times*\bfB)_{i,j,k} & = & \left[\begin{array}{c}
{E_{y}}_{i,j,k}{B_{z}}_{i-1,j,k}-{E_{z}}_{i,j,k}{B_{y}}_{i-1,j,k}\\
{E_{z}}_{i,j,k}{B_{x}}_{i,j-1,k}-{E_{x}}_{i,j,k}{B_{z}}_{i,j-1,k}\\
{E_{x}}_{i,j,k}{B_{y}}_{i,j,k-1}-{E_{y}}_{i,j,k}{B_{x}}_{i,j,k-1}
\end{array}\right]^{T}.\end{aligned}$$ Next, we investigate $T_{2}$ which represents the energy flow of particles, $$\begin{aligned}
T_{2} & = & \sum_{s}\frac{1}{2}m_{s}\dot{\bfx}_{s}^{2}\left(\dot{\bfx}_{s}\cdot\nabla\WZERO{\bfx_{s}}\right)\nonumber \\
& = & \sum_{s}\frac{1}{2}m_{s}\dot{\bfx}_{s}^{2}\left(\dot{\bfx}_{s}\cdot\sum_{J}\WONE{\bfx_{s}}{\nabla_{d}}_{JI}\right)\nonumber \\
& = & \sum_{J}{\nabla_{d}}_{JI}\sum_{s}\frac{1}{2}m_{s}\dot{\bfx}_{s}^{2}\dot{\bfx}_{s}\cdot\WONE{\bfx_{s}}\nonumber \\
& = & \sum_{J}{\nabla_{d}}_{JI}\mathbf{S}_{J}=-\sum_{K}\DIVD_{IK}\left(*\mathbf{S}\right)_{K}~,\label{eq:T2}\end{aligned}$$ where $$\begin{aligned}
\mathbf{S}_{J} & = & \sum_{s}\frac{1}{2}m_{s}\dot{\bfx}_{s}^{2}\dot{\bfx}_{s}\cdot\WONE{\bfx_{s}}~,\\
\left(*\mathbf{S}\right)_{K} & = & [{S_{x}}_{i-1,j,k},{S_{y}}_{i,j-1,k},{S_{z}}_{i,j,k-1}]~.\end{aligned}$$ Thus this term is also a discrete divergence. Finally, let us look at $T_{3}$, which appears only in the discrete particle-field system. It is $$\begin{aligned}
T_{3} & = & \sum_{s}q_{s}\left(\dot{\bfx}_{s}\cdot\sum_{J}\bfE_{J}\WONE{\bfx_{s}}\right)\WZERO{\bfx_{s}}-\bfE_{I}\cdot\sum_{s}q_{s}\dot{\bfx}_{s}\WONEA{\bfx_{s}}\\
& = & \sum_{s}q_{s}\left(\sum_{J'}\dot{\bfx}_{s}\cdot\bfE_{J}\WONE{\bfx_{s}}\WZERO{\bfx_{s}}-\bfE_{I}\cdot\dot{\bfx}_{s}\WONEA{\bfx_{s}}\right)~.\\
& = & \sum_{s}q_{s}F\left(s\right)\left(\WZERO{\bfx_{s}}-\frac{p\left(s,I\right)}{F\left(s\right)}\right)~,\end{aligned}$$ where $$\begin{aligned}
F\left(s\right) & = & \sum_{I}p\left(s,I\right)\thinspace,\\
p\left(s,I\right) & = & \bfE_{I}\cdot\dot{\bfx}_{s}\WONEA{\bfx_{s}}~.\end{aligned}$$ From the definition of $\WZERO{\bfx}$, we have $\sum_{I}\WZERO{\bfx_{s}}=1~.$ Therefore, $$\begin{aligned}
\sum_{I}\left(\WZERO{\bfx_{s}}-\frac{p\left(s,I\right)}{F\left(s\right)}\right) & = & 0~,\end{aligned}$$ which means that the sum of $T_{3}$ over all spatial grid points vanishes. Now we invoke the two theorems approved in the Appendix, which states that a discrete sum-free field must be a discrete divergence field and that if the sum-free field is local, then the divergence field is local. Therefore, for each particle, $T_{3}$ can be expressed as discrete divergence of a discrete vector field $\mathbf{G}_{s}$, $$\begin{aligned}
T_{3} & = & \sum_{s}q_{s}\sum_{J'}\DIVD_{IJ'}{\mathbf{G}_{s}}_{J'}F\left(s\right)~.\end{aligned}$$ For a particular $s$, $\WZERO{\bfx_{s}}$, $\WONE{\bfx_{s}}$ are local near $\bfx_{s}$, so is $\WZERO{\bfx_{s}}-p\left(s,I\right)/F(s)$ . Thus $\bfG_{s}$ is also local near $\bfx_{s}$.
Finally, we obtain the local energy conservation law for the spatial-discretized Vlasov-Maxwell system. $$\begin{aligned}
& & \frac{\partial}{\partial t}\left(\sum_{s}\frac{1}{2}m_{s}\dot{\bfx}_{s}^{2}\WZERO{\bfx_{s}}+\bfE_{I}^{2}+\bfB_{I}^{2}\right)+\nonumber \\
& & \sum_{K}\DIVD_{IK}\left((\bfE\times*\bfB)_{K}+\left(*\mathbf{S}\right)_{K}-\sum_{s}q_{s}{\mathbf{G}_{s}}_{K}F\left(s\right)\right)=0~.\end{aligned}$$ In conclusion, we started from the Hamiltonian theory of the spatially-discretized Vlasov-Maxwell system, used the property of Whitney forms, and derived a discrete local energy conservation law. In practice the spatially-discretized Vlasov-Maxwell system also need a temporal discretize to become a numerical scheme, after which the total energy will not be an exact invariant. However if we apply a symplectic integrator to perform the temporal discretize, then the total energy error can be bounded within a small value for all simulation time-steps [@Ruth83; @Feng85; @Feng86; @Hairer02]. Investigation on discrete conservation laws for such systems are planned for future work.
a discrete sum-free field is a discrete divergence field
========================================================
In this appendix, we prove the following two theorems.
\[LemR\] If a discrete scalar field $R$ is sum-free, i.e., $$\begin{aligned}
\sum_{I}R_{I} & = & 0~,\end{aligned}$$ then it can be expressed as a discrete divergence of a discrete vector field $\mathbf{G}$, i.e., $$\begin{aligned}
R_{I} & = & \sum_{J}\DIVD_{IJ}\mathbf{G}_{J}~.\label{EqnRIGJ}\end{aligned}$$
The discrete field $R$ in Theorem \[LemR\] can also be a function of the continuous spatial coordinate $\bfx$, such as the Whitney form $W_{\sigma_{0J}}(\bfx).$ A discrete field or a field component $F(\bfx)$ is called local, if for every $\bfx_{0}$, there exists a positive constant $C$ such that $$\begin{aligned}
F_{I}(\bfx_{0})=0\thinspace,\forall I\in\left\{ J|\quad|\bfx_{J}-\bfx_{0}|>C\right\} .\end{aligned}$$
\[LemR2\] If a discrete sum-free scalar field $R$ is local, then there exists a local discrete vector field $\bfG$ such that Eq. holds.
These two theorems also imply that a (local) discrete sum-free vector field or tensor field can be expressed as a discrete divergence of a (local) tensor or high order tensor, because for each component of the vector field or tensor field, Theorems \[LemR\] and \[LemR2\] apply. To prove these two theorems, we first prove the following three lemmas.
The discrete scalar field $R_{x}(i',j',k')$ defied as $$\begin{aligned}
R_{xI}(i',j',k')=\left\{ \begin{array}{lc}
1, & \textrm{if }I=[i'+1,j',k']~,\\
-1, & \textrm{if }I=[i',j',k']~,\\
0, & \textrm{otherwise}~,
\end{array}\right.\end{aligned}$$ can be expressed as a composition of $\DIVD$ and a discrete vector field $\mathbf{G}_{x}(i',j',k')$, $$\begin{aligned}
R_{xI}(i',j',k') & = & \sum_{J}\DIVD_{IJ}\mathbf{G}_{xJ}(i',j',k')~.\end{aligned}$$
Let $\bfG_{x}\left(i',j',k'\right)$ be $$\begin{aligned}
\bfG_{xJ}\left(i',j',k'\right)=\left\{ \begin{array}{lc}
[-1,0,0], & \textrm{if }J=[i'+1,j',k']~,\\
\left[0,0,0\right], & \textrm{otherwise}~.
\end{array}\right.\end{aligned}$$ Then it is straightforward to verify that $$\begin{aligned}
R_{xI}(i',j',k') & = & \sum_{J}\DIVD_{IJ}\mathbf{G}_{xJ}(i',j',k')~.\end{aligned}$$
Using the similar technique, we can construct $\bfG_{y}\left(i',j',k'\right)$ and $\bfG_{z}\left(i',j',k'\right)$ for scalar fields $R_{y}(i',j',k')$ and $R_{z}(i',j',k')$ as well, i.e., $$\begin{aligned}
R_{yI}(i',j',k') & = & \left\{ \begin{array}{lc}
1, & \textrm{if }I=[i',j'+1,k']~,\\
-1, & \textrm{if }I=[i',j',k']~,\\
0, & \textrm{otherwise}~,
\end{array}\right.\\
R_{zI}(i',j',k') & = & \left\{ \begin{array}{lc}
1, & \textrm{if }I=[i',j',k'+1]~,\\
-1, & \textrm{if }I=[i',j',k']~,\\
0, & \textrm{otherwise}~,
\end{array}\right.\\
\bfG_{yJ}\left(i',j',k'\right) & = & \left\{ \begin{array}{lc}
[0,-1,0], & \textrm{if }J=[i',j'+1,k']~,\\
\left[0,0,0\right], & \textrm{otherwise}~,
\end{array}\right.\\
\bfG_{zJ}\left(i',j',k'\right) & = & \left\{ \begin{array}{lc}
[0,0,-1], & \textrm{if }J=[i',j',k'+1]~,\\
\left[0,0,0\right], & \textrm{otherwise}~.
\end{array}\right.\end{aligned}$$
The discrete scalar field $R^{\dagger}\left(I_{1},I_{2}\right)$ defined as $$\begin{aligned}
R_{I}^{\dagger}\left(I_{1},I_{2}\right) & = & \left\{ \begin{array}{lc}
-1, & \textrm{if }I=I_{1}~,\\
1, & \textrm{if }I=I_{2}~,\\
0, & \textrm{otherwise}
\end{array}\right.\end{aligned}$$ can be written in the following form $$\begin{aligned}
R_{I}^{\dagger}\left(I_{1},I_{2}\right) & = & \sum_{I'\in Y_{x}}a_{x}R_{xI}(I')+\sum_{I'\in Y_{y}}a_{y}R_{yI}(I')+\sum_{I'\in Y_{z}}a_{z}R_{zI}(I')~,\label{EqnRDAGGER}\end{aligned}$$ where $Y_{x}$, $Y_{y}$ and $Y_{z}$ are some indices sets, and $a_{x}$, $a_{y}$ and $a_{z}$ are some integers.
Choose $Y_{x}$, $Y_{y}$ and $Y_{z}$ as $$\begin{aligned}
Y_{x} & = & \left\{ [i,j,k]|\min(i_{1},i_{2})\leq i<\max(i_{1},i_{2}),j=j_{2},k=k_{2}\right\} ~,\\
Y_{y} & = & \left\{ [i,j,k]|i=i_{1},\min(j_{1},j_{2})\leq j<\max(j_{1},j_{2}),k=k_{2}\right\} ~,\\
Y_{z} & = & \left\{ [i,j,k]|i=i_{1},j=j_{1},\min(k_{1},k_{2})\leq k<\max(k_{1},k_{2})\right\} ~.\end{aligned}$$ Choose $a_{x}$, $a_{y}$ and $a_{z}$ as $$\begin{aligned}
a_{x} & = & \left\{ \begin{array}{lc}
1, & \textrm{if }i_{1}<i_{2}~,\\
-1, & \textrm{otherwise}~,
\end{array}\right.\\
a_{y} & = & \left\{ \begin{array}{lc}
1, & \textrm{if }j_{1}<j_{2}~,\\
-1, & \textrm{otherwise}~,
\end{array}\right.\\
a_{z} & = & \left\{ \begin{array}{lc}
1, & \textrm{if }k_{1}<k_{2}~,\\
-1, & \textrm{otherwise}~.
\end{array}\right.\end{aligned}$$ It is straightforward to verify that holds.
A sum-free scalar field $R$ can be written as $$\begin{aligned}
R_{I} & = & \sum_{[I',I'']\in Z}b_{[I',I'']}R_{I}^{\dagger}\left(I',I''\right)~,\label{EqnSUMRB}\end{aligned}$$ where $Z$ is some index-pair set.
If $\forall I$ such that $R_{I}=0$, then $Z$ can be chosen as $\varnothing$. Otherwise let $Y_{d}=\left\{ I_{1},I_{2},I_{3},\dots\right\} $ is the index set such that if $I\in Y_{d}$, $R_{I}\neq0$ and $I\notin Y_{d}$, $R_{I}=0$. We can choose the index-pair set $Z$ as $$\begin{aligned}
Z=\left\{ [I',I'']|I'=I_{1},I''\in Y_{d}\textrm{ and }I''\neq I_{1}\right\} ,\end{aligned}$$ where $I_{1}$ is one arbitrarily chosen element in $Y_{d}$, and the corresponding $b_{[I',I'']}$ is $$\begin{aligned}
b_{[I',I'']}=R_{I''}.\end{aligned}$$ Using the fact that
$$R_{I_{1}}+\sum_{I''\in Y_{d}\textrm{ and }I''\neq I_{1}}R_{I''}=\sum_{I}R_{I}=0~,$$
we have
$$R_{I_{1}}=-\sum_{I''\in Y_{d}\textrm{ and }I''\neq I_{1}}R_{I''}~.$$
Then it is straightforward to verify that holds.
Composing these three lemmas, we can see that the Theorems \[LemR\] and \[LemR2\] are proved.
This research is supported by National Magnetic Confinement Fusion Energy Research Project (2015GB111003, 2014GB124005), National Natural Science Foundation of China (NSFC-11575185, 11575186, 11305171), JSPS-NRF-NSFC A3 Foresight Program (NSFC-11261140328), Chinese Scholar Council (201506340103), Key Research Program of Frontier Sciences CAS (QYZDB-SSW-SYS004), and the GeoAlgorithmic Plasma Simulator (GAPS) Project.
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---
abstract: 'For an optically thick metallic film, the transmission for both $s$- and $p$-polarized waves is extremely low. If the metallic film is coated on both sides with a finite dielectric layer, light transmission for $p$-polarized waves can be enhanced considerably. This enhancement is not related to surface plasmon-polaritions. Instead, it is due to the interplay between Fabry-Perot interference in the coated dielectric layer and the existence of the Brewster angle at the dielectric/metallic interface. It is shown that the coated metallic films can act as excellent polarizers at infrared wavelengths.'
author:
- 'Dezhuan Han, Xin Li, Fengqin Wu,'
- Jian Zi
title: Enhanced transmission of optically thick metallic films at infrared wavelengths
---
Metal surfaces are highly reflective over a very wide range of wavelengths. This is the reason why metals are commonly used as mirrors in our daily life and in optical technologies as well. It is known that the transmission of an optically thick metallic film is very low for wavelengths in the visible range or below. However, for $p$-polarized waves transmission could be very high when incident light excites the coupled surface plasmon-polaritons (SPPs) on both sides of the metallic film.[@dra:85] SPPs are a kind of electromagnetic excitations existing at the interface between a metal and a dielectric medium.[@rae:88] For a flat metallic surface, $p$-polarized incident waves cannot directly excite SPPs owing to the wavevector mismatch between incident waves and SPPs. Normally, an attenuated total reflection technique is adopted to generate evanescent waves which can excite SPPs.[rae:88]{} Extraordinary transmission has been also found for a metallic film perforated with subwavelength hole arrays[@ebb:98] or slits.[@por:99] It is believed that the excitation and coupling of SPPs on both surfaces of the metallic film play an important role in such an extraordinary transmission.
In this letter, we report theoretically an enhanced transmission of $p$-polarized waves at infrared wavelengths for an optically thick metallic film coated with a dielectric layer on both sides. It is found that the enhanced transmission is not due to SPPs. Instead, it relies on the interplay between Fabry-Perot interference in the dielectric layer and the existence of a Brewster angle window at the dielectric/metallic interface.
In our numerical simulations, without loss of generality, metallic films are assumed to be Ag. The dielectric constant of Ag is described by the Drude model$$\varepsilon (\omega )=1-\frac{\omega _{p}^{2}}{\omega ^{2}+i\gamma \omega },$$where $\omega _{p}$ is the plasma frequency and $\gamma $ is the parameter related to the energy loss. For Ag, the parameters used are $\omega
_{p}=1.15\times 10^{16}$ rad/s and $\gamma =9.81\times 10^{13}$ rad/s, which are obtained by fitting to the experimental data[@pal:85] at near and mid infrared wavelengths.
![(color online). Reflectance spectra of $p$-polarized waves incident from a dielectric medium (refractive index 1.5) upon a Ag surface at different incident angles. Labels indicate different wavelengths. Inset shows the calculated Brewster angles from Eq. (\[bre\]) and those obtained from the reflectance spectra. []{data-label="fig1"}](fig1.eps){width="7.cm"}
At a dielectric/dielectric interface, there exists a Brewster angle, at which the reflectance for $p$-polarized waves is zero. To show the fact that a metallic surface also possesses a Brewster angle, the reflectance spectra for $p$-polarized waves incident from a dielectric medium with a refractive index of $1.5$ upon a Ag surface at different incident angles are shown in Fig. \[fig1\]. For $s$-polarized waves, the reflection at the Ag surface is very high and increases monotonically with increasing incident angle. For $p$-polarized waves, however, there exist dips in the reflectance spectra for incident angles near 90$^{o}$. The positions of the dips can be viewed as the Brewster angle at the Ag surface. Unlike the dielectric surface, reflectance at the metallic surface is nonzero at the Brewster angle. At short wavelengths (visible or near infrared), dips in the reflectance spectra are rather shallow so that the Brewster angle is not well-defined. Dips become sharper with the increase in wavelength. The Brewster angle is wavelength dependent and shifts upward with the increasing wavelength. It approaches 90$^{o}$ at the long wavelength limit. For a dielectric/dielectric interface, the Brewster angle is given by $\tan \theta
_{B}=n_{2}/n_{1}$, where $n_{1}$ and $n_{2}$ are the refractive indices of two dielectric media. For the dielectric/metallic interface, the Brewster angle cannot be given by a simple formula.[@bed:01] However, for wavelengths considered here it is found that the Brewster angle at the dielectric/metallic interface can be well approximated by the following relation$$\tan \theta _{B}=\left( \frac{\left\vert \varepsilon _{2}\right\vert }{%
\varepsilon _{1}}\right) ^{1/2}\text{,} \label{bre}$$where $\varepsilon _{1}$ and $\varepsilon _{2}$ are the dielectric constants of the dielectric and metallic media, respectively. In Fig. \[fig1\], Brewster angles at different wavelengths obtained from the above relation and from the reflectance spectra are also given for comparison. It can be found that Brewster angles predicted by Eq. (\[bre\]) agree well with those obtained from the reflectance spectra.
The existence of the Brewster angle at a metallic surface can manifest a higher transmission of $p$-polarized waves for a metallic film at the Brewster incident angle with respect to other incident angles. Figure [fig2]{} shows the transmittance spectra of $p$-polarized waves for a 80 nm Ag film immersed in a dielectric medium at different incident angles. For $s$-polarized waves, transmission is extremely low for all incident angles. Thus, the 80 nm Ag film is optically thick enough to block $s$-polarized waves. For $p$-polarized waves, the situation, however, is a bit different. The overall transmission is still very low, especially for incident angles not close to grazing angles. It is interesting to note that there exist some maxima in the transmittance spectra at some grazing incident angles close to 90$^{o}$. For a fixed wavelength transmittance increases monotonically for incident angles varying from normal incidence, reaches a maximum at a certain grazing incident angle, and then decreases monotonically up to the 90$^{o}$ incident angle. It is obvious that the positions of the transmission peaks in Fig. \[fig2\] coincide well with the Brewster angles at the dielectric/metallic interface.
![(color online). Transmittance spectra of $p$-polarized waves as for a 80 nm Ag film immersed in a dielectric medium (refractive index 1.5) at different incident angles. []{data-label="fig2"}](fig2.eps){width="7.cm"}
We now consider a Ag film coated with a dielectric layer symmetrically on both sides. The refractive index of the coated dielectric layer is taken to be 1.5. From the above discussions it is known that the Brewster angle is close to 90$^{o}$. If $p$-polarized waves are incident from air upon the dielectric layer, the refracted angle may be smaller than the Brewster angle at the dielectric/metallic interface since the dielectric constant of air is smaller than that of the dielectric medium. As a result, we may not access the Brewster angle at the dielectric/metallic interface. To exclude this possibility, two prisms with a higher refractive index than the coated dielectric layer are introduced to situate on both coated dielectric layers. Practically, it is not an easy job to get grazing incidence close to 90$^{0}$. The introduction of prisms, however, can overcome this difficulty. This can be easily seen from Snell’s law $$\sin \theta _{1}/\sin \theta _{2}=n_{d}/n_{p},$$where $n_{p}$ and $n_{d}$ are the refractive indices of the prism and the coated dielectric layer, respectively; $\theta _{1}$ is the incident angle in the prism; $\theta _{2}$ is the refracted angle in the dielectric layer, also the incident angle upon the metallic film. Without loss of generality, the refractive index of prisms is chosen to be 3. Thus, the critical angle at the prism/dielectric interface is 30$^{o}$. For $\theta _{1}$ larger than this critical angle, evanescent waves will be generated. This configuration has been commonly used to excite SPPs.[@rae:88]
![(color online). Transmittance spectra of $p$-polarized waves for a coated Ag film with a thickness of 80 nm at different incident angles from the prism. Labels indicate different wavelengths. The thickness of the coated dielectric layer is determined from Eq. (\[d-fp\]). The refractive indices of the coated dielectric and the prism are 1.5 and 3, respectively.[]{data-label="fig3"}](fig3.eps){width="7.cm"}
In the coated layer, Fabry-Perot interference is expected owing to the multiple reflection between the prism/dielectric and dielectric/metallic interfaces. When traversing across the coated dielectric layer, two successively transmitted waves have a phase difference $$\delta =\frac{2\pi }{\lambda _{0}}2n_{d}d\cos \theta _{2}+\phi ,$$where $d$ is the thickness of the coated dielectric layer, $\lambda _{0}$ is the wavelength in vacuum, and $\phi $ is an additional phase shift due to the metallic surface. Transmission into the metallic film can be enhanced if $\delta =(2m+1)\pi $, where the integer $m$ takes the value of $0,1,2,...$. Combining this effect with the Brewster angle window, enhanced transmission should be expected.
In Fig. \[fig3\] transmittance spectra for a coated 80 nm Ag film situated between two prisms at difference incident angles are shown. To get enhanced transmission the choice of the thickness of the coated dielectric layer is crucial. At the Brewster angle, it is found that the additional phase difference due to the metallic surface is $\pi /2$. Thus, the minimal thickness of the coated dielectric layer that renders an enhanced transmission is determined from$$d=\frac{\lambda _{0}}{8n_{d}\cos \theta _{B}}. \label{d-fp}$$It is obvious from Fig. \[fig3\] that transmission is largely enhanced with respect to the same metallic film situated in a dielectric medium. The enhanced factor of the transmission peak for the wavelength of 4 $\mu $m is over 16 and that for the wavelength of 10 $\mu $m is more than 5. It should be noted that transmission is also dependent on the thickness of the metallic film. Higher transmission can be obtained if we reduce the thickness of the metallic film. For smaller thickness of the metallic film, the minimal thickness of the coated dielectric layer that renders enhanced transmission may deviate somewhat from that obtained from Eq. (\[d-fp\]) owing to the coupling of the two dielectric/metallic interfaces.
![(color online). Transmittance spectrum of a $p$-polarized wave of $%
\protect\lambda _{0}=4$ $\protect\mu $m and degree of polarization for a coated Ag film with a thickness of 40 nm as a function of the incident angles from the prism. The thickness of the dielectric layer is 6.45 $%
\protect\mu $m.[]{data-label="fig4"}](fig4.eps){width="7.5cm"}
As shown above, $p$-polarized waves possess a high transmission for a coated metallic film, while $s$-polarized waves have an extremely low transmission. This feature can render infrared polarizers with excellent performance possible. To obtain a good performance, the thickness of the coated dielectric layer and the metallic film should be chosen properly. A good polarizer is characterized by a high transmission and a high degree of polarization, defined by$$P=\frac{T_{p}-T_{s}}{T_{p}+T_{s}},$$where $T_{p}$ and $T_{s}$ are the transmittance of $p$- and $s$-polarized waves, respectively. For a perfect polarizer, the degree of polarization should be $1$. In Fig. \[fig4\], the transmittance spectrum of a $p$-polarized wave with $\lambda _{0}=4$ $\mu $m for a coated Ag film with a thickness of 40 nm and the degree of polarization of the system are shown. A thickness of 6.45 $\mu $m for the dielectric layer is chosen in order to obtain a maximal transmission at certain incident angle. It is obvious that this polarizer has a high transmittance (over 70%) for the $p$-polarized wave around the incident angle of 29.93$^{o}$ from the prism, while its degree of polarization is nearly perfect.
In the visible range, conventional polarizers have excellent performance and are easily attainable. In the infrared regime, on the contrary, the conventionally used metallic wire-type polarizers are costly and their performance is less satisfactory with respect to those in the visible range.[@ben:78] Our results indicate that the coated metallic film could act as an excellent polarizer at infrared wavelengths. The results shown in Fig. \[fig4\] are for $\lambda _{0}=4$ $\mu $m. For other infrared wavelengths, we can also obtain satisfactory performance for a coated metallic film provided that the thicknesses of the coated dielectric layer and the metallic film are properly chosen.
In summary, we studied theoretically the transmission of optically thick metallic films. It was shown that there exists a Brewster angle window at the dielectric/metallic interface. Incorporated with the Fabry-Perot interference, a coated metallic film can have a largely enhanced transmission. This feature can render excellent polarizers at infrared wavelengths possible.
This work was supported by CNKBRSF, NSFC, PCSIRT, and Shanghai Science and Technology Commission, China. We thank Dr. S. Meyer for a critical reading of the manuscript.
[99]{} R. Dragila, B. Luther-Davies, and S. Vukovic, Phys. Rev. Lett. **55**, 1117 (1985).
H. Raether, *Surface Plasmons* (Springer-Verlag, Berlin, 1988).
T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, Nature (London) **391**, 667 (1998).
J. A. Porto, F. J. García-Vidal, and J. B. Pendry, Phys. Rev. Lett. **83**, 2845 (1999).
*Handbook of Optical Constants of Solids,* edited by* *E. D. Palik (Academic Press, New York, 1985).
D. Bedeaux and J. Vlieger, *Optical Properties of Surfaces* (Imperial College Press, London, 2001), p. 64.
J. M. Bennett and H. E. Bennett, *Handbook of Optics,* Ed. by W. Driscoll (McGraw-Hill, New York, 1978).
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abstract: 'Macroscopic traffic models are necessary for simulation and study of traffic’s complex macro-scale dynamics, and are often used by practitioners for road network planning, integrated corridor management, and other applications. These models have two parts: a link model, which describes traffic flow behavior on individual roads, and a node model, which describes behavior at road junctions. As the road networks under study become larger and more complex — nowadays often including arterial networks — the node model becomes more important. Despite their great importance to macroscopic models, however, only recently have node models had similar levels of attention as link models in the literature. This paper focuses on the first order node model and has two main contributions. First, we formalize the multi-commodity flow distribution at a junction as an optimization problem with all the necessary constraints. Most interesting here is the formalization of input flow priorities. Then, we discuss a very common “conservation of turning fractions” or “first-in-first-out” (FIFO) constraint, and how it often produces unrealistic spillback. This spillback occurs when, at a diverge, a queue develops for a movement that only a few lanes service, but FIFO requires that all lanes experience spillback from this queue. As we show, avoiding this unrealistic spillback while retaining FIFO in the node model requires complicated network topologies. Our second contribution is a “partial FIFO” mechanism that avoids this unrealistic spillback, and a (first-order) node model and solution algorithm that incorporates this mechanism. The partial FIFO mechanism is parameterized through intervals that describe how individual movements influence each other, can be intuitively described from physical lane geometry and turning movement rules, and allows tuning to describe a link as having anything between full FIFO and no FIFO. Excepting the FIFO constraint, the present node model also fits within the well-established “general class of first-order node models” for multi-commodity flows. Several illustrative examples are presented.'
author:
- 'Matthew A. Wright'
- Gabriel Gomes
- Roberto Horowitz
- 'Alex A. Kurzhanskiy'
bibliography:
- '../../../traffic.bib'
title: 'On node models for high-dimensional road networks'
---
[**Keywords**]{}: macroscopic first order traffic model, first order node model, multi-commodity traffic, dynamic traffic assignment, dynamic network loading
Introduction {#sec_intro}
============
Common node models and their drawbacks {#sec_node_review}
======================================
A node model for dimensionality management {#sec_node_model}
==========================================
Examples {#sec_examples}
========
Conclusion {#sec_conclusion}
==========
Acknowledgements {#sec_acknowledgement .unnumbered}
================
We would like to express great appreciation to our colleagues Elena Dorogush and Ajith Muralidharan for sharing ideas, Ramtin Pedarsani, Brian Phegley and Pravin Varaiya for their critical reading and their help in clarifying some theoretical issues. This research was funded by the California Department of Transportation.
Notation {#app_notation}
========
MISO node solution algorithm {#app_merge}
============================
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abstract: 'We analyse non-adiabatic non-Abelian holonomic transformations of spin-qubits confined to a linear time dependent harmonic trap with time dependent Rashba interaction. For this system exact results can be derived for spin rotation angle which also enables exact treatment of white gate-noise effects. We concentrate in particular on the reliability of cyclic transformations quantified by fidelity defined by the probability that the qubit after one full cycle remains in the ground-state energy manifold. The formalism allows exact analysis of spin transformations that optimise final fidelity. Various examples of time dependent fidelity probability distributions are presented and discussed.'
address:
- |
J. Stefan Institute, Ljubljana, Slovenia\
lara.ulcakar@ijs.si
- |
Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia\
J. Stefan Institute, Ljubljana, Slovenia\
anton.ramsak@fmf.uni-lj.si
author:
- 'Lara Ul[č]{}akar'
- Anton Ramšak
title: 'Effects of noise on fidelity in spin-orbit qubit transformations '
---
Introduction
============
Spintronics, as a new branch of electronics, is a quantum information technology promising better performance with smaller power consumption.[@Wolf2001; @Zutic2004; @Rashba2007] The spin of electrons plays the central role[@Awschalom2013] and the main challenge is to manipulate the spin of a single electron precisely and locally. Employing magnetic fields, a natural way of spin rotation, usually cannot be applied locally in a small region so other mechanisms should be applied. A possible such solution is to use semiconductor heterostructures[@Winkler2003; @Engel2007] with spin-orbit interaction (SOI) and particularly strong Rashba interaction[@Dresselhaus1955; @Rashba1960] that can be tuned externally using voltage gates.[@Nitta1997; @Schapers1998; @Nitta1999; @Schliemann2003; @Wunderlich2010; @gomez12; @pavlowski16; @pavlowski16b; @fan16]
Recently a simple scheme for the spin-qubit manipulation was proposed in which an electron is driven along a linear quantum wire with time dependent spin-orbit interaction, tuned by external time-dependent potential.[@Cadez2013; @Cadez2014] One limitation of such linear systems is posed by fixed axis of spin rotation, but it can be eliminated in quantum ring structures, exhibiting a rich range of phenomena.[@Buttiker1983; @Fuhrer2001; @Aronov1993; @Qian1994; @Malshukov1999; @Richter2012; @Nagasawa2013; @Saarikoski2015] For quantum ring structures consisting of a narrow ring with superimposed time dependent harmonic trap and controllable time dependent Rashba interaction exact solutions were presented most recently.[@kregar16; @kregar16b]
In linear as well as in ring systems controlled by external gates there are several possible sources of noise which can not be avoided. In particular, noise can be induced due to fluctuating electric fields, caused by the piezoelectric phonons[@sanjose06; @sanjose08; @huang13; @echeveria13] or due to phonon-mediated instabilities in molecular systems with phonon assisted potential barriers, which introduce noise in the confining potentials.[@mravlje06; @mravlje08] For qubits realised as spin of electrons carried by surface acoustic waves the noise can be caused by the electron-electron interaction.[@rejec00; @jefferson06; @giavaras06] Since exact solutions for qubit manipulation scheme considered here are possible, the analysis of environment effects can for some sources of noise be performed analytically.[@ulcakar17]
The paper is organised as follows. After the introduction is in Section 2 presented the model where also a brief overview of the exact solution together with the analysis of effects due to white noise is revealed. Section 3 is devoted to the fidelity of qubit transformations. The derivation of influences of noise on fidelity is presented in detail and explicit examples are given. Results are summarised in Section 4.
Model, exact solution and white noise
=====================================
We consider an electron in a quantum wire confined in a harmonic trap.[@Cadez2013; @Cadez2014] The centre of such one-dimensional quantum dot, $\xi(t)$, can be arbitrarily translated along the wire by means of time dependent external electric fields. Spin-orbit Rashba interaction couples the electron spin with orbital motion, resulting in the Hamiltonian $$\label{H}
H(t)=\frac{p^{2}}{2m^{*}}I+\frac{m^{*}\omega^{2}}{2}[x-\xi(t)]^{2}I+\alpha(t)p\, \mathbf{n}\boldsymbol{\cdot\sigma},$$ where $m^{*}$ is the electron effective mass, $\omega$ is the frequency of the harmonic trap, $\alpha(t)$ is the strength of spin-orbit interaction, possibly time dependent due to appropriate time dependent external electric fields. The spin rotation axis $\mathbf{n}$ is fixed and depends on the crystal structure of the quasi-one-dimensional material used and the direction of the applied electric field.[@nadjperge12] $\boldsymbol{\sigma}$ and $I$ are Pauli spin matrices and unity operator in spin space, respectively, and $p$ is the momentum operator. Exact solution of the time dependent Schr[ö]{}dinger equation corresponding to the Hamiltonian equation is given by[@Cadez2014] $$\begin{aligned}
\label{psi}
|\Psi_{ms}(t)\rangle&=&e^{-i[\theta(t) I+\phi(t) \mathbf{n}\cdot\boldsymbol{\sigma}/2)]}
\mathcal{A}_{\alpha}\mathcal{X}_{\xi}|\psi_{m}(x)\rangle|\chi_{s}\rangle,
\\
\theta(t)&=&{\omega_{m}t+\phi_{\alpha}(t)+{\phi_{\xi}(t)}+m^{*}\dot{a}_{c}(t)a_{c}(t)/\omega^{2}},\\
\mathcal{A}_{\alpha}&=&
e^{-i\dot{a}_{c}(t)p\mathbf{n}\cdot\boldsymbol{\sigma}/\omega^2}
e^{-im^{*}a_{c}(t)x\mathbf{n}\cdot\boldsymbol{\sigma}},\\
\mathcal{X}_{\xi}&=&e^{im^{*}[x-x_{c}(t)]\dot{x}_{c}(t)}e^{-ix_{c}(t)p}I.\end{aligned}$$ Here $\psi_{m}(x)$ represents the $m$-th eigenstate of a harmonic oscillator with eigenenergy $\omega_{m}=(m+1/2)\omega$ and $|\chi_{s}\rangle$ is spinor of the electron in the eigenbasis of operator $\sigma_z$. The phase $\phi_{\xi}(t)=-\int_{0}^{t}L_{\xi}(t')\mathrm{d}t'$ is the coordinate action integral, where $L_{\xi}(t)=m^{*}\dot{x}_{c}^{2}(t)/2-m^{*}\omega^{2}[x_{c}(t)-\xi(t)]^{2}/2$ is the Lagrange function of a driven harmonic oscillator and $x_{c}(t)$ is the solution to the equation of motion of a classical driven oscillator $$\label{eq:xoscillator}
\ddot{x}_c(t)+\omega^{2}x_{c}(t)=\omega^{2}\xi(t).$$ Another phase factor is the SOI action integral phase $\phi_{\alpha}(t)=-\int_{0}^{t}L_{\alpha}(t')\mathrm{d}t'$, with $L_{\alpha}(t)=m^{*}\dot{a}_{c}^{2}(t)/(2\omega^2)-m^{*}[a_{c}(t)-\alpha(t)]^{2}/2+m^{*}\alpha^2(t)/2$ being the Lagrange function of another driven oscillator, satisfying $\ddot{a}_{c}(t)+\omega^{2}a_{c}(t)=\omega^{2}\alpha(t).
$
In this paper we consider particularly interesting cyclic transformations with periodic drivings $\xi(T)=\xi(0)$ and $\alpha(T)=\alpha(0)$ with zero values and time derivatives of responses $x_c$ and $a_c$ at times $t=0$ and $t=T$. The spin-qubit is for such drivings rotated around $\mathbf{n}$ by the angle $\phi=-2m^{*}\int_{0}^{T}\dot{a}_{c}(t')\xi(t')\mathrm{d}t'$.[@Cadez2014]
We assume noise in the driving function $\xi(t)=\xi^{0}(t)+\delta\xi(t)$ consisting of ideal driving part without noise $\xi^{0}(t)$ with superimposed stochastic part with vanishing mean $\langle\delta\xi(t)\rangle=0$. We consider the Ornstein-Uhlenbeck coloured noise[@wang45; @masoliver92] characterized by the autocorrelation function $\langle\delta\xi(t')\delta\xi(t'')\rangle={{\sigma_{\xi}^2} \over {2\tau_\xi} }e^{ |t'-t''|/\tau_\xi}$, with noise intensity $\sigma^2_\xi$ and correlation time $\tau_\xi$. A general solution of equation $x_{c}(t)$ with $x_c(0)=\xi^0(0)$ and ${\dot{x}_c}(0)=0$ is given by $$\label{eq:harmoicsolution}
x_{c}(t)=\xi^0(0)+\omega\int_{0}^{t}\sin[\omega(t-t')]\xi(t'){\rm d}t',$$ which due to the noise term $\delta \xi$ is normally distributed with the variance evaluated as equal-times autocorrelation function, $$\sigma_x^2(t)=\omega^2\lim_{\Delta t \to0} \langle\int_{0}^{t}\sin[\omega(t-t')]\delta\xi(t'){\rm d}t' \int_{0}^{t+\Delta t}\sin[\omega(t-t'')]\delta\xi(t''){\rm d}t'' \rangle.$$ For the Ornstein-Uhlenbeck noise considered here the integrals can be evaluated exactly. Nevertheless, here we consider only the white noise limit where $\tau_\xi \to 0$ and $\langle\delta\xi(t')\delta\xi(t'')\rangle={\sigma_{\xi}^2}\delta(t'-t'')$ leading to the variances $$\label{1d}
\sigma_x^2(t)=\frac{1}{4}\omega\sigma_{\xi}^{2}\left(2\omega t-{{\sin{2\omega t}}}\right)
\quad {\rm and}\quad
\sigma_{\dot{x}}^{2}(t)=\frac{1}{4}\omega^{3}\sigma_{\xi}^{2}\left[2\omega t+\sin(2\omega t)\right],$$ corresponding to ${x}_{c}(t)$ and $\dot{x}_{c}(t)$, respectively.
Additionally to the coordinate noise is also normally distributed noise in SOI driving function $\alpha(t)=\alpha^{0}(t)+\delta\alpha(t)$, where $\alpha^{0}(t)$ is ideal noiseless driving. SOI noise $\delta\alpha(t)$ is similar to the previous case of spatial driving and is again of the Ornstein-Uhlenbeck type of autocorrelation function $\langle\delta\alpha(t')\delta\alpha(t'')\rangle$ with noise intensity $\sigma^2_\alpha$ and correlation time in the white noise limit $\tau_\alpha \to 0$, leading to the time-dependent variances $\sigma_{a}^{2}(t)= { (\sigma_\alpha/ \sigma_{\xi})^{2}} \sigma_{x}^{2}(t)$ and $\sigma_{\dot{a}}^{2}(t)= { ( \sigma_\alpha/ \sigma_{\xi})^{2}} \sigma_{\dot{x}}^{2}(t)$ for $a_{c}(t)$ and $\dot{a}_{c}(t)$, respectively.
Fidelity of noisy qubit transformations
=======================================
As an example of effects of noise to spin-qubit transformations we consider driving corresponding to the class of circular paths in two dimensional coordinate-SOI space ${\cal{C}}_{\rm ad} \sim \alpha^0[\xi]$, $$\xi^0(t) = \xi_0 \cos( \omega t/n) \quad {\rm and} \quad \alpha^0(t) = \alpha_0 \sin( \omega t/n), \label{krog}$$ where $n\ge2$ is integer, and the period of the transformation is $T=2\pi n/\omega$. Periodic responses represent contours ${\cal{C}}\sim a_c^0[\xi]$, where $$a^0_c(t)=\alpha_0\frac{n \left[n \sin \left( \omega t/n\right)-\sin ( \omega t )\right]}{n^2-1},$$ with the phases given by the area in the coordinate-SOI plane, $$\begin{aligned}
\phi_{\rm ad}&=&-2m^{*}\int_{0}^{T}\dot{\alpha}^0(t')\xi(t')\mathrm{d}t'=-2m^* \!\oint_{{\cal C}_{\rm ad}} \!\!\!\!\! \alpha^0[\xi] {\rm d}\xi=-2\pi m^*\xi_0\alpha_0,\label{phiAd}\\
\phi^0 &=&-2 m^* \!\oint_{{\cal C}} a_c^0[\xi] {\rm d}\xi={ n^2\over n^2 - 1} \phi_{\rm ad}.\label{fi}\end{aligned}$$ The adiabatic angle $\phi_\mathrm{ad}$ corresponds to the one when circular driving is of type $n\to\infty$. Transformation angle $\phi=\phi^0+\delta \phi$ is due to the noise distributed normally around the mean $\phi^0$, with the variance after one cycle given by[@ulcakar17] $$\label{sigma0}
\frac{\sigma_{\phi,n}^{2}}{\phi_{\rm ad}^2}= \frac{n(1+n^{2})}{\pi(n^{2}-1)^{2}}\frac{\omega\sigma_{\xi}^2}{ \xi_0^2} +
\frac{2 n^{3}}{\pi(n^{2}-1)^{2}}\frac{\omega\sigma_{\alpha}^2}{ \alpha_0^2}+
n^2\left(\frac{\omega\sigma_{\xi}\sigma_{\alpha}}{\xi_0\alpha_0}\right)^2.$$
In figure \[fig1\](a) are shown spin orbit responses as a function of time and in figure \[fig1\](b) is shown the contour ${\cal C}$ for the case of circular driving equation with $n=6$. In both panels the dashed black lines denote noiseless spin orbit driving $\alpha^0(t)$ and the red line noiseless spin orbit response $a_c^0(t)$. The focus is on the set of $10$ spin orbit responses $a_c(t)$ to $10$ different realisations of white noise in $\alpha(t)$. $\sigma_a^2(t)$ manifests as a spread of these curves around the ideal noiseless red line. Bullets correspond to initial $[a_c^0(0),\xi^0(0)]$ and final noiseless values $[a_c^0(T),\xi^0(T)]$ of noiseless response and show that final values of $a_c(T)$ deviate from the desired ones. The noisy response is not periodic, resulting in open loop in parameter space unlike the case of noiseless $\mathcal{C}$ and noiseless adiabatic driving $\mathcal{C}_\mathrm{ad}$. Consequently the angle of spin rotation cannot be expressed as an area enclosed by the contour as in equation and in figure \[fig1\](b) pink shaded. It should be noted that in general the total angle of spin rotation $\phi$ is less prone to noise because the noisy curves oscillate around the ideal value and so contributions to final error partially cancel out.[@ulcakar17]
This analysis of spin-rotation angle demonstrated that due to gate noise in the driving functions, spin transformations are not completely faithful. For non-adiabatic qubit manipulations the electron state is determined by the time-dependent Hamiltonian during the evolution and is in general a superposition of excited states, ultimately becoming the ground state when the transformation is complete. Therefore in addition to correct transformation of the spin direction, one has also to take care that the electron state has not left the starting energy manifold at the final time. As shown in Refs.[@Cadez2013; @Cadez2014; @kregar16] such motions in parametric space can easily be performed if the driving functions are appropriately chosen. Here an important question is relevant: how well does the final state of the electron relax to the desired final state energy manifold after the transformation if the driving function is not ideal as in the presence of noise?
In order to demonstrate how to answer this question in general we consider the qubit wave function $|\Psi_{0\frac{1}{2}}(t)\rangle$, equation , which is at $t=0$ in the ground state of the harmonic quantum dot (with $m=0$) and spin $\frac{1}{2}$. We observe its relaxation to the ground state manifold that is spanned by two basis states[@Cadez2013] of time dependent Hamiltonian equation at time $t$, $$|\widetilde\Psi_{0s}\rangle=e^{-im^{*}[x-\xi(t)]\alpha(t)\mathbf{n}\boldsymbol{\cdot\sigma}}|\psi_{0}[x-\xi(t)]\rangle|\chi_{s}\rangle.$$ As the appropriate measure of the relaxation accuracy we define fidelity $F=\langle\Psi_{0\frac{1}{2}}(t)|P_0|\Psi_{0\frac{1}{2}}(t)\rangle$, where $P_0=\sum_{s}|\widetilde\Psi_{0s}\rangle\langle\widetilde\Psi_{0s}|$ is the projector onto the ground state manifold. We choose $\mathbf{n}$ perpendicular to the $z$-axis and a straightforward derivation leads to the expression for overlaps of $|\Psi_{0\frac{1}{2}}(t)\rangle$ with the basis states at time $t$, $$\langle\widetilde\Psi_{0\pm \frac{1}{2}}(t)|\Psi_{0\frac{1}{2}}(t)\rangle=\frac{1}{2}[e^{-\frac{1}{2}E_{+}(t)}\pm e^{-\frac{1}{2}E_{-}(t)}],$$ where $$E_{\pm}(t)=\frac{m^{*}}{2\omega}\{[\omega(x_{c}(t)-\xi(t))\pm\dot{a}_{c}(t)/\omega]^{2}+[\dot{x}_{c}(t)\mp(a_{c}(t)-\alpha(t))]^{2}\}$$ resembles classical energy with additional terms for spin-orbit coupling and is equal to the classical energy if the spin-orbit driving is constant.[@Cadez2013] Ideal qubit transformations with spin-fidelities ${\cal F}_s=|\langle\widetilde\Psi_{0s}|\Psi_{0\frac{1}{2}}\rangle|^2=\delta_{s\frac{1}{2}}$ are achieved by applying ideal drivings, where the energies $E_\pm$ vanish at final time $t=T$, [*i.e.*]{}, when $x_c=\xi$, $a_c=\alpha$, $\dot{x}_{c}=0$, and $\dot{a}_{c}=0$.
The fidelity at arbitrary time $t$ is obtained by summation over final spin states, $$F(t)=\sum_s {\cal F}_s(t)=\frac{1}{2}[e^{-E_{+}(t)}+e^{-E_{-}(t)}].$$ The presence of noise in spin-orbit and spatial driving terms makes fidelity a random quantity, $F(t)=F^0(t)+\delta F(t)$, where $F^0(t)$ represents the result of noiseless driving and $\delta F(t)$ is the deviation from this value. Fidelity is therefore characterized by some probability density function $\frac{\mathrm{d}P(F)}{\mathrm{d}F}$. It can be calculated from the probability density for variables $E_{\pm}$ which are functions of independent random variables and normally distributed. The probability density functions for $E_{\pm}$ can at time $t$ be calculated using the formula $$\begin{aligned}
\frac{\mathrm{d}P_\pm(E)}{\mathrm{d}E}{\Bigr\rvert_{t }}&=&\int\!\!\int\!\!\int\!\!\int
\delta[E-E_\pm(x_{c},\dot{x}_{c},a_{c},\dot{a}_{c})]
\\
& &\times
\frac{\mathrm{d}P_x(x_{c})}{\mathrm{d}x_{c}}\frac{\mathrm{d}P_{\dot{x}}(\dot{x}_{c})}{\mathrm{d}\dot{x}_{c}}
\frac{\mathrm{d}P_a(a_{c})}{\mathrm{d}a_{c}}\frac{\mathrm{d}P_{\dot{a}}(\dot{a}_{c})}{\mathrm{d}\dot{a}_{c}}
\mathrm{d}x_{c}\mathrm{d}\dot{x}_{c}\mathrm{d}a_{c}\mathrm{d}\dot{a}_{c}.\nonumber\end{aligned}$$ The result is obtained by first calculating the characteristic functions, $$\begin{aligned}
p_\pm(k)&=&\int_{-\infty}^\infty \frac{\mathrm{d}P_\pm(E)}{\mathrm{d}E} e^{i k E} {\rm d}E={2\sigma_{1}^{-1}\sigma_{2}^{-1}\over\sqrt{(2 \sigma_1^{-2}-ik)(2\sigma_2^{-2}-ik)}},\end{aligned}$$ with $$\begin{aligned}
\sigma_{1}^{2}(t)&=&\left(\frac{2m^{*}}{\omega}\right)
[\omega^{2}\sigma_{x}^{2}(t)+\sigma_{\dot{a}}^{2}(t)/\omega^{2}],\\
\sigma_{2}^{2}(t)&=&\left(\frac{2m^{*}}{\omega}\right)
[\sigma_{\dot{x}}^{2}(t)+\sigma_{a}^{2}(t)].\end{aligned}$$ Note the equality $p_+(k)=p_-(k)$ which after the inverse Fourier transform yields equal functional forms for $E_+$ and $E_-$, $$\frac{\mathrm{d}P_\pm(E_\pm)}{\mathrm{d}E_\pm}{\Bigr\rvert_{t }}=2\sigma_{1}^{-1}\sigma_{2}^{-1}I_{0}[(\sigma_{1}^{-2}-\sigma_{2}^{-2})E_\pm]e^{-(\sigma_{1}^{-2}+\sigma_{2}^{-2})E_\pm},$$ where $I_{0}(z)$ is the modified Bessel function of the first kind.
Since the fidelity is a sum of two dependent random variables, its probability distribution is calculated from the joint probability distribution function for those two variables, which in general cannot be evaluated analytically. However, one can examine $\frac{\mathrm{d}P}{\mathrm{d}F}$ exactly when $\sigma_{x}^{2}(t)=\sigma_{\dot{a}}^{2}(t)/\omega^{4}$ and $\sigma_{\dot{x}}^{2}(t)=\sigma_{a}^{2}(t)$, which is satisfied for $t=T$ if the coordinate and the SOI driving noise intensities are equal, [*i.e.*]{}, $\sigma_\alpha=\omega\sigma_{\xi}$. In this case $E_{+}$ and $E_{-}$ become [*independent*]{} random variables and $\frac{\mathrm{d}P}{\mathrm{d}F}$ can be calculated as the convolution of probability distributions for $e^{-E_{+}}$ and $e^{-E_{-}}$. At $t=T$ the exact result for $F\geq\frac{1}{2}$ is given by $$\label{eq:fidelity2d}
\frac{\mathrm{d}P(F)}{\mathrm{d}F}{\Bigr\rvert_{t =T}}=2\sigma_{F}^{-4}[B(\frac{1}{2F},\sigma_{F}^{-2},\sigma_{F}^{-2})-B(1-\frac{1}{2F},\sigma_{F}^{-2},\sigma_{F}^{-2})](2F)^{2\sigma_{F}^{-2}-1},$$ where $B(x,a,b)$ is the incomplete beta function and $\sigma_{F}^{-2}=\sigma_{1}^{-2}+\sigma_{2}^{-2}$. For $F <\frac{1}{2}$ the probability distribution is $\frac{\mathrm{d}P}{\mathrm{d}F}=2\sigma_{F}^{-4}B(\sigma_{F}^{-2},\sigma_{F}^{-2})(2F)^{2\sigma_{F}^{-2}-1}$, where $B(a,b)$ is the beta function. In practice where noise intensities are small the most relevant regime is $\sigma_{F}\to0$ for which the probability distribution equation (\[eq:fidelity2d\]) simplifies to $\frac{\mathrm{d}P}{\mathrm{d}F} \propto (1-F)F^{2\sigma_{F}^{-2}}$. Due to similar dependence of ${\cal F}_s$ and $F$ on $E_\pm$ it is easy to derive analytical results also for spin-fidelity probability distributions $\frac{\mathrm{d}P_s}{\mathrm{d}{\cal F}_s}$ (not shown here).
In figure \[fig2\](a) different realizations of noisy fidelity (black lines) are compared to the noiseless one (red) for $n=2$. One can observe that noisy fidelity starts to deviate from noiseless one for $t/T\gtrsim0.1$, reaches maximum deviation at $t/T\sim0.5$ and then deviations are again lowered when approaching $t\to T$. The same quantities are presented in figure \[fig2\](c) for circular driving with $n=8$ where noiseless curve is denoted with blue colour. Figure \[fig2\](b) shows noiseless curves $a_c[\xi]$ in parametric space during the transformation with $n=2$ (red), $n=8$ (blue, dashed) and $n\to\infty$ (black, dashed), the latter corresponding to the adiabatic limit. Bullets denote initial and final values of $a_c(t)$ and $\xi(t)$. Note that the motion is periodic with period $T$ and that $a_c(0)=a_c(T)=\alpha(0)=\alpha(T)$ as is manifested also in noiseless fidelity being equal to $1$ at $t=0$ and $t=T$, as a demonstration that the system returns to the ground state manifold with probability $1$. This can be seen from positions of bullets in figures \[fig2\](a) and 2(c). Figure \[fig2\](d) shows the probability density distribution of fidelity at times $t/T=0.5$ (orange), $t/T=0.75$ (green) and $t=T$ (black). It should be mentioned that the distribution for $t=T$ is given also by exact formula, equation (\[eq:fidelity2d\]). The colour code of distributions corresponds to the code of the shading of fidelity spreading around the noiseless value in figure \[fig2\](a). Distributions are centred around noiseless values and their variances are proportional to spreadings observed in figure \[fig2\](a), the distribution at $t/T=0.5$ having the largest variance which is lower at $t/T=0.75$ and even lower at $t=T$.
Summary
=======
We presented an analysis of spin-qubit non-adiabatic manipulation of an electron traped in a moving linear harmonic trap and in the presence of time dependent Rashba interaction. One of the main challenges here is a precise tuning of driving fields since the electron starting from the ground state should after performing one cycle with time-dependent Hamiltonian return to the ground state, although during the cycle the state of the electron is a superposition of excited eigenstates of the moving trap.
The problem is even more subtle because there will always be present some noise in driving functions, which means that spin-qubit transformation will always deviate from the ideal one. Since for the model considered here exact solutions are available for a broad class of drivings, we concentrated also to the exact analysis of the influence of small deviations from ideal qubit manipulation. In particular, we focused to an explicit example and demonstrated how one can analyse the effects of a general noise to the transformation angle and we showed the results for the Ornstein-Uhlenbeck type of noise.
An example, considered in detail, is the case of circular driving in the space of parameters for which exact analytical formulae are given and analysed for white noise. In view of the fact that for non-adiabatic regimes a non-trivial point is the ability of the system to return to the ground state after an arbitrary time-dependent driving, our analysis was focused to the fidelity – the overlap of the actual wave function with the desired ideal. For white noise explicit formulae are derived for symmetric noise intensities in position and spin-orbit driving functions. A detailed derivation and analysis of fidelity is presented. Additionally, analytical results are illustrated by special cases of driving together with numerically generated noisy drivings and the corresponding responses.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors acknowledge support from the Slovenian Research Agency under contract no. P1-0044.
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**In this paper we apply topology optimization to micro-structured superhydrophobic surfaces for the first time. It has been experimentally observed that a droplet suspended on a brush of micrometric posts shows a high static contact angle and low roll-off angle. To keep the fluid from penetrating the space between the posts, we search for an optimal post cross section, that minimizes the vertical displacement of the liquid-air interface at the base of the drop when a pressure difference is applied. Topology optimisation proves effective in this framework, showing that posts with a branching cross-section are optimal, which is consistent with several biologic strategies to achieve superhydrophobicity. Through a filtering technique, we can also control the characteristic length scale of the optimal design, thus obtaining feasible geometries**.
{#section .unnumbered}
Introduction {#introduction .unnumbered}
============
Superhydrophobicity is a remarkable natural phenomenon, recently analysed [@marmur2008hygrophilic; @deGennes2003capillarity; @nosonovsky2007multiscale; @mchale2007cassie; @patankar2004transition; @kusumaatmaja2008collapse] and reproduced artificially [@deGennes2003capillarity; @tuteja2008omniphobic; @bico1999pearl; @bico2002wetting; @extrand1995inclined] by numerous research groups. Superhydrophobic surfaces show very large static contact angles and small roll-off angles for water, and these properties are usually associated with self-cleaning surfaces.
A micro- and/or nano-scale texture is usually at the origin of superhydrophobicity [@bormashenko2007pigeon; @gao2009wetting]. A drop can reach several different equilibrium states on a textured substrate, as sketched in Fig. \[fig:intro\].
The effective minimum energy configuration depends on the chemical and geometrical properties of the liquid-solid interface. We will now focus on superhydrophobicity, which is usually associated to the Cassie-Baxter state [@cassie1944wettability]. In this configuration, the drop is suspended by the protruding features, so that its base is in contact with a heterogeneous solid-air substrate. The apparent static contact angle $\theta_{CB}$ , according to Cassie-Baxter theory, is given by: $$\label{Cassie}
\cos \theta_{CB} = f_{sl} \cos \theta_Y - (1-f_{sl}),$$ a weighted average between the contact angle for the solid substrate ($\theta_Y$) and for air ($\theta_{air}=180^o$), where $f_{sl}$ represents the wetted solid surface per base area of the drop.
If a drop in the Cassie-Baxter state is perturbed, for instance if a pressure difference is applied between the drop and the environment, the liquid-air interface will bulge, and eventually the liquid will begin to flow along the side of the posts when the angle $\theta_{bend}$ (see Fig. \[fig:bend\]) exceeds the contact angle $\theta_Y$ .
This effect is particularly important for inherently hydrophilic materials, for which a heterogeneous wetting state can be achieved through overhanging structures (see Fig.) \[fig:bend\]B), even if the global energy minimum will be a Wenzel state [@tuteja2008omniphobic] (Fig. \[fig:intro\]). Maximising the robustness of the suspended drop configuration upon applied pressure is therefore fundamental for effective superhydrophobic surfaces.
The research about “Cassie mode” superhydrophobicity has so far been characterized by a strong dichotomy. On one hand, complex hierarchical structures have been fabricated and tested experimentally, but their modelling is hard, since the structures are usually rough and non periodic. [@emami2011random] On the other hand, there is an active research for the optimal post shape to achieve a robust Cassie state, which however usually relies on simple shape perturbations to conventional cylindrical or square posts. In this paper, we take a step in bridging this gap, applying the tools of topology optimization.
Topology optimization [@bendsoe2003topology] is a structural optimization method with no intrinsic constraint on the topology of the solution, which has been applied in such different fields as structural mechanics [@bendsoe2003topology], photonic crystal design [@sigmund2008opttopopt] and microfluidic devices [@olesen2006topopt]. We will here apply it to obtain the texture that minimizes the deformation of the liquid-air interface under applied pressure, thus making the suspended state as robust as possible. We will see that this approach generates interesting branching structures, which resemble natural and experimentally tested superhydrophobic structures. However, the symmetry and length scale of the optimal design can be tuned in the numeric optimization procedure, leading to a better understanding and control of such features.
Modelling and numeric setup {#modelling-and-numeric-setup .unnumbered}
===========================
In this work we will restrict our analysis to a unit cell for a square array of posts (Fig.\[fig:sketch\]), neglecting finite size effects at the edge of the drop. We will consider a two dimensional picture, in which the liquid-air interface is flat and suspended on top of the posts (z=0) in the unperturbed configuration, and bulges between the posts to a depth $S(\vec x)$ upon applying a pressure difference $\Delta P$. Such a pressure difference across the liquid-air interface can arise for different reasons, such as the Laplace pressure due to the drop curvature or the pressure upon impact of a drop on the substrate. We also introduce non dimensional unit for length $l$, surface tension $\sigma$ and pressure $P$ as follows:
$$\begin{split}
\sigma &= \sigma_{0}\bar \sigma ,\\
l&= L_0 \bar l , \\ \
P&= \frac{\sigma_{0}}{L_0} \bar P = P_0 \bar P.
\label{nondimensional}
\end{split}$$
Here $L_0$ is the characteristic length of the system, which we will take as the side of the unit cell (typically few $\mu\textrm{m}$), and $\sigma_0$ can be taken as the surface tension of the liquid considered ($\textrm{72.9} \; \textrm{mJ} /\textrm{m}^2$ for water at $\textrm{20}\; {\ensuremath{^\circ}}\textrm{C}$). Moreover, since typically $L_0 << l_c=\sqrt{\frac{\sigma}{\rho g}}$, where $l_c$ is the capillary length for the liquid considered, we can neglect gravity.
Let us first consider a simple geometry, such as a cylindrical post ( cross section is shown in Fig.\[fig:sketch\]B ). The deflection of the liquid-air interface among posts can then be described by the Young-Laplace equation [@zheng2005pressure] $$\label{bareset}
\begin{cases}
{\bm{\nabla}}\cdot \left( \frac{{\bm{\nabla}}S(\vec x)}{|{\bm{\nabla}}S(\vec x)|} \right) = \Delta P& \text{on D} \\
S(\vec x)=0& \text{on} \;\partial \text{D}_1 \\
{\bm{\nabla}}S(\vec x)\cdot \vec n=0& \text{on} \;\partial \text{D}_2. \\
\end{cases}$$
A Dirichlet boundary condition $S(\vec x)=0$ is used at the boundary of the solid structure $\partial \text{D}_1$ to represent that the interface is pinned on the ridge of the post. A Von Neumann condition ${\bm{\nabla}}S(\vec x)\cdot \vec n=0$ is applied on the boundary of the unit cell $\partial \text{D}_2$ to account for the symmetry of the post array (in the following, we will also exploit the symmetry of the cell to work only on one eighth of the domain).
For the optimization procedure, we will now slightly modify this setup. We still consider a “solid” support ($S(\vec x)=0$, red dot in Fig.\[fig:sketch\]C ) in the centre of the domain, but now the post cross section is allowed to change around it, in order to provide an optimal support for the interface. The distribution of material at point $\vec x$ inside the cell is described by the design variable $\gamma(\vec x)$, a scalar field which ranges from 0 (completely solid) to 1 (completely empty) through intermediate values.
The field $\gamma(\vec x)$ will be coupled to Eqs.\[bareset\], leading to the following formulation of the problem: $$\label{topset}
\begin{cases}
{\bm{\nabla}}\cdot \left( K(\gamma) \frac{{\bm{\nabla}}S(\vec x)}{|{\bm{\nabla}}S(\vec x)|} \right) = \Delta P& \text{on D} \\
S(\vec x)=0& \text{on} \;\partial \text{D}_1 \\
{\bm{\nabla}}S(\vec x)\cdot \vec n=0& \text{on} \;\partial \text{D}_2. \\
\end{cases}$$ Where $K(\gamma)$ is defined as: $$\label{kappa}
K(\gamma)= 1+\frac{(K_{max}-1)\cdot q \cdot(1-\gamma)}{(q+\gamma)}$$
Given the form of Eq. \[topset\] and \[kappa\], it is possible to understand the effect of the design variable $\gamma(\vec x)$ on the solution $S(\vec x)$. Where $\gamma(\vec x) = 0$, $K(\gamma) $ is equal to $ K_{max}$, which is fixed to be a large value. The value $\Delta P$ on the right side of Eq.4 becomes then negligible, and the liquid-air interface $S(\vec x)$ does not deform significantly. We therefore recover the “solid” condition $S(\vec x) \simeq 0$. On the other hand, If $\gamma(\vec x) = 1$ (“empty space”), $K(\gamma) =1$, and we recover the Young-Laplace equation out of the support of the post, as in Eqs.\[bareset\]. Intermediate values of $K(\gamma)$ do not have a direct physical interpetation, but are required for a smooth optimization procedure. The interpolation between these two extreme ranges is controlled by the parameter q in Eq. \[kappa\]. By choosing a sufficiently small value (here $ 10^{-4}$), it is possible to drive the optimization procedure to give a well defined “solid-empty” binary design. [@bendsoe2003topology].
This formulation also resembles a 2D optimal heat conduction problem, where $K(\gamma)$ corresponds to the distribution of conducting material [@gersborg2006heat].
We eventually need to define an objective function, i.e. a quantity whose minimization with respect to $\gamma(\vec x)$ will maximise the support to the liquid-air interface. We choose this quantity, called $\Phi \left[ S(\vec x),\gamma(\vec x), \Delta P \right]$, to be the squared integral displacement of the interface (for a given pressure difference $\Delta P$ and material distribution $\gamma(\vec x)$):
$$\label{eq:objective}
\Phi \left[ S(\vec x),\gamma(\vec x), \Delta P \right] = \int_{D} \, S^2(\vec x) \, dA.$$
With this choice, we do not control directly the angle between the interface and the side of the post, which is indeed what would trigger the penetration of the liquid among posts. However, Eq. \[eq:objective\] is easy to evaluate through the optimization procedure, and its minimization naturally constrains the maximum bending angle of the interface [@lobaton2007curvature], although there might be fluctuations along the post ridge.
At every iteration, the topology optimization code changes the value of $\gamma(\vec x)$ over the domain and evaluates $\Phi \left[ S(\vec x),\gamma(\vec x), \Delta P \right]$ and the sensitivity $\frac{\delta\Phi}{\delta\gamma(\vec x)}$. We then use this information as input to find the configuration of $\gamma(\vec x)$ that minimizes the objective function $\Phi$, using the method of moving asymptotes (MMA) [@svanberg1987mma]. Details on the sensitivity analysis and the implementation of the code can be found in the paper by Olesen et al. [@olesen2006topopt]. We will also introduce a constraint on the maximum solid fraction per unit cell as: $$\label{constraint}
\int_{D} \, 1 - \gamma(\vec{x}) \, dA \leq f_{sl}.$$ Remembering Cassie-Baxter relation $\cos \theta_{CB} = f_{sl} \cos \theta_Y - (1-f_{sl})$ , Eq. \[constraint\] can conveniently be interpreted as a constraint on the static contact angle shown by a surface patterned in this way.
The specific coupling $K(\gamma)$ we use in Eqs. \[topset\] will generate a structure connected to the boundary $\partial \text{D}_1$, which “radiates” the support to the $\gamma \simeq 0$ regions [@gersborg2006heat]. This effectively make our analysis a shape optimization with many degrees of freedom, while the general topology optimization routine we use could as easily generate disconnected topologies.
There are a few reasons for the choice of connected design. First, it is well known that dense and thin posts, ideally down to the nanometer scale, offer increasingly better support to drops in the Cassie-Baxter state [@deGennes2003capillarity; @zheng2010small]. However, it is perhaps more interesting to optimize the shape of a *single* texture element, which can then be scaled up or down in size according to fabrication and performance constraints. Second, if we are interested in obtaining a hydrophobic behaviour from hydrophilic materials, overhanging structures are required. In this perspective, the central support in our optimisation can be considered as the stem of the post (see Fig. \[fig:bend\]), while we effectively optimise the cross section of the top plate. Eventually, we argue that connected structures would show higher mechanical robustness than hair-like features, in particular to buckling and shear loads. This latter property is of great relevance for practical fabrication purposes, since most practical application would include significant stresses for the substrates.[@cavalli2012parametric]
A final remark regards the length scales in the optimal design: at every iteration in the optimisation routine we calculate a smoothed version $\tilde{\gamma}(\vec x)$ of the design variable $\gamma(\vec x)$, applying a diffusion step [@lazarov2011filters]:
$$\label{diffusion}
L_{diff}^2\nabla ^2 \tilde{\gamma}(\vec x) = \tilde{\gamma}(\vec x)- \gamma(\vec x).$$
While calculating the sensitivity, $\tilde{\gamma}(\vec x)$ is then used. This process allows to control the minimum size of the features appearing in the optimal design. As we will discuss in the next section, without filtering small length scale features would appear in the optimal design, ideally down to the mesh scale. However, these small solid features surrounded by empty space are transformed by the diffusion step in a homogeneous area with intermediate $\gamma(\vec x)$ value, and thus are penalized by the $K(\gamma)$ function, which favours a binary solid-empty solution. The main advantage of this technique is its formulation in terms of a partial differential equation, which relies on the same numeric tools used for Eqs. \[topset\].
The actual implementation of our optimization routine uses a Matlab code, that relies on the commercial software COMSOL to solve the partial differential equations at every iteration step.
Discussion of optimized designs {#discussion-of-optimized-designs .unnumbered}
===============================
In the following, the pressure difference acting on the interface has been fixed as $\Delta P = P_0$ . In Fig. \[fig:comparison\] We compare the performance of a cylindrical post (A) and an optimized design (C) inside a unit cell. The surface plots displayed on the right (B-D) show the vertical displacement $S(\vec x)$ obtained through Eqs.\[bareset\]. For both structures, the solid fraction is $f_{sl}=0.25$. It is easy to see the enhanced performance of the topology optimized structure, with the mean displacement reduced by a factor 10. It is clear that the branching in the optimal structure increases the length of the contact line, where the surface tension acts on the side of the post to balance the effect of the applied pressure difference. This result in a reduction of the interface deformation. However, we think that just choosing a meandering cross section would not improve dramatically the performance. Lobaton and Salamon [@lobaton2007curvature], for instance, considered a simpler sinusoidal perturbation to a circular cross section. While significantly increasing the contact line length, such a shape modification showed modest improvement in the critical pressure value. The added feature of our optimal designs is the convenient placement of the branches, that adjust to the cell shape (here a square unit cell, however analogous solution have been tested for hexagonal lattices) to reduce the size of the gaps between solid features. We therefore argue that the significant reduction in the surface displacement arises from the interplay of optimal location of the main branches and increased contact line length coming from the secondary branches.
This physical picture makes it easy to understand the effect of the filtering length $L_{diff}$ on the optimal design. The designs shown in the upper row of Fig. \[fig:filter\] were obtained by solving for the domain shown in the bottom row. The yellow dots have a radius equal to $L_{diff}$.
It can be seen that, for any value of $L_{diff}$, the structure branches along the diagonals of the square cell, thus filling the largest gap between two posts. If the resolution is sufficiently fine, further branching appears, with new branches filling the gap among the diagonals. The process continues for even smaller length scales and we get an overall quasi-fractal behaviour.
It is possible to see how the filtering procedure constrains the minimal length scale in the optimal design. This allows to obtain structures suitable for fabrication, i.e. with a feasible amount of branching.
The fractal-like structures resemble several biologic surfaces (such as the lotus leaf), which use analogous (although three dimensional) multi-scale structures to achieve their superhydrophobic properties. A three dimensional optimization would be very intensive in terms of computation. It is however possible to complement the suggestions from topology optimization with general knowledge from superhydrophobic surfaces, to get an even more effective texture. Indeed, most artificial and natural superhydrophobic surfaces are characterized by a micron scale texture with superimposed nanometric roughness. The cross sections shown here should be considered an optimal micron scale pattern, over which nano-grass features can be grown, thus achieving a multi-layer support for the interface (this procedure is currently been considered in collaboration with Nis K. Andersen and Rafael Taboryski, and will be the subject of a future publication).
In Fig. \[fig:solidfraction\], we eventually analyse the dependence of the mean interface displacement $\langle s \rangle = \sqrt{\frac{1}{D} \int_{D} \, S^2(\vec x) \, dA}$ on the solid fraction $f_{sl}$ for a fixed filter length $L_{diff}=0.75\; h_{mesh}$, where $h_{mesh}$ is the characteristic mesh size. An increasing branching for larger solid fraction is clearly seen in the optimal designs, which results in a better support for the interface. In the chart we compare the mean displacement for the optimal design to the displacement for a post of circular cross section and same $f_{sl}$. We can see that the optimised design always performs better than the simple circular cross section, and even more so for large solid fractions, which is again a consequence of the higher degree of branching in the optimised configuration.
Conclusion and outlook {#conclusion-and-outlook .unnumbered}
======================
In this paper we applied topology optimisation to the stability of superhydrophobic surfaces. We found that this technique is very effective for the task. Branching structures are found to be optimal to support hydrostatic pressure for a Cassie-Baxter state, in a two dimensional analogy to natural structures. We also analysed the effect of a solid fraction constraint on the optimal design, as well as the use of a PDE filter to obtain designs suitable for fabrication. Further work will include the fabrication and characterization of such optimised microtextured surfaces. Preliminary fabrication results obtained at DTU Nanotech suggest that the optimal shapes can be reproduced with a high degree of precision using common lithographic techniques. A further step will be to use a cost effective procedure, such as injection moulding, to produce the same designs.
This research is funded by the NanoVation consortium. The authors thank Kristian E. Jensen and Rafael J. Taboryski for useful suggestions and discussions.
|
---
abstract: 'We present and analyze an approach for distributed stochastic optimization which is statistically optimal and achieves near-linear speedups (up to logarithmic factors). Our approach allows a communication-memory tradeoff, with either logarithmic communication but linear memory, or polynomial communication and a corresponding polynomial reduction in required memory. This communication-memory tradeoff is achieved through minibatch-prox iterations (minibatch passive-aggressive updates), where a subproblem on a minibatch is solved at each iteration. We provide a novel analysis for such a minibatch-prox procedure which achieves the statistical optimal rate regardless of minibatch size and smoothness, thus significantly improving on prior work.'
bibliography:
- 'minibatch.bib'
title: 'Memory and Communication Efficient Distributed Stochastic Optimization with Minibatch-Prox'
---
Acknowledgement {#acknowledgement .unnumbered}
===============
Research was partially supported by an Intel ICRI-CI award and NSF awards IIS 1302662 and BIGDATA 1546500. We would like to thank Ohad Shamir for discussions about Distributed SVRG and Tong Zhang for discussions about minibatch-prox.
|
---
abstract: 'The introduction of Dark Matter-neutrino interactions modifies the Cosmic Microwave Background (CMB) angular power spectrum at all scales, thus affecting the reconstruction of the cosmological parameters. Such interactions can lead to a slight increase of the value of $H_0$ and a slight decrease of $\sigma_8$, which can help reduce somewhat the tension between the CMB and lensing or Cepheids datasets. Here we show that it is impossible to solve both tensions simultaneously. While the 2015 Planck temperature and low multipole polarisation data combined with the Cepheids datasets prefer large values of the Hubble rate (up to $H_0 = 72.1^{+1.5}_{-1.7} \rm{km/s/Mpc}$, when $N_{\rm{eff}}$ is free to vary), the $\sigma_8$ parameter remains too large to reduce the $\sigma_8$ tension. Adding high multipole Planck polarization data does not help since this data shows a strong preference for low values of $H_0$, thus worsening current tensions, even though they also prefer smaller value of $\sigma_8$.'
author:
- Eleonora Di Valentino
- Céline Bœhm
- Eric Hivon
- 'François R. Bouchet'
title: ' Reducing the $H_0$ and $\sigma_8$ tensions with Dark Matter-neutrino interactions.'
---
Introduction
============
In the standard cosmological framework, dark matter is assumed to be collisionless. In practice this means that one arbitrarily sets the dark matter interactions to zero when predicting the angular power spectrum of the Cosmic Microwave Background (CMB). However this treatment is at odds with the principle behind dark matter direct and indirect detection, where one explicitly assumes that dark matter (DM) interacts with ordinary matter. This is also in contradiction with the thermal hypothesis which relies on dark matter annihilations to explain the observed dark matter relic density.
A more consistent approach consists in accounting for dark matter interactions and test whether they can be neglected by looking at their effects on cosmological observables. DM interactions in the early Universe damp the primordial dark matter fluctuations through the collisional damping mechanism [@deLaix:1995vi; @Boehm:2000gq; @Boehm:2004th]. They also affect the evolution of the other fluid(s) which the DM is interacting with. The two effects simultaneously impact the distribution of light and matter in the early Universe [@Boehm:2001hm] and eventually affect structure formation in the dark ages [@Boehm:2003xr]. They can also modify how our own cosmic neighborhood should look like [@Schewtschenko:2015rno; @Schewtschenko:2014fca; @Boehm:2014vja; @Vogelsberger:2015gpr; @Cyr-Racine:2015ihg] and change the estimates of the cosmological parameters needed to account for the observed CMB anisotropies.
The so-called “cut-off” scale at which one notices departures from the Lambda$+$Cold DM model (LCDM) predictions in the matter power spectrum is governed by the ratio of the elastic scattering cross section (corresponding to the dark matter scattering off the species $i$, normalised to the Thomson cross section $\sigma_T$) to the dark matter mass. We refer to this ratio as $$u_i = \frac{\sigma_{DM-i}}{\sigma_T} \ \left(\frac{m_{\rm{DM}}}{100 \rm{GeV}}\right)^{-1}.$$ The larger $u_i$, the higher the cut-off scale [@Boehm:2000gq; @Boehm:2001hm; @Boehm:2004th]. Dark matter-radiation interactions is the most interesting case among all interacting DM scenarios. Since radiation dominates the energy in the Universe for a very long time, such interactions erase the dark matter fluctuations on relatively large-scales for $u \ll 1 $ and also change the way the CMB looks like across the sky [@Boehm:2000gq; @Boehm:2001hm; @Chen:2002yh; @Boehm:2004th; @Sigurdson:2004zp; @Mangano:2006mp; @Serra:2009uu; @CyrRacine:2012fz; @Diamanti:2012tg; @Diamanti:2012tg; @Blennow:2012de; @Dolgov:2013una; @Wilkinson:2013kia; @Dvorkin:2013cea]. Dark matter-baryon and dark matter self-interactions can also erase the DM fluctuations but the $u$ ratio needs to be of order 1 to produce the same effects as the one considered here [@Boehm:2000gq; @Boehm:2004th], given that there are many less baryons than radiation in the Universe and baryons are non-relativistic.
In what follows, we focus on Dark Matter-neutrino interactions and study their impact on the cosmological parameters (in particular the Hubble rate $H_0$, the effective number of relativistic degrees of freedom ${N_{\mathrm{eff}}}$ and the linear matter power spectrum value at 8 Mpc, $\sigma_8$). Previous analyses [@Wilkinson:2014ksa] indicated that Dark Matter-neutrino interactions prefer higher values of $H_0$ with respect to LCDM estimates. The higher ${N_{\mathrm{eff}}}$, the higher $H_0$. Therefore we investigate whether DM-$\nu$ interactions could at least partially solve the current tensions arising between the CMB and late-time (i.e. strong lensing [@Bonvin:2016crt] and Cepheids [@R16]) measurements of the $H_0$ value. We also study whether DM-$\nu$ interactions could reduce the tension between the CMB-inferred value of $\sigma_8$ and large-scale-structure surveys, owing to the damping they induce.
In what follows, we consider the Planck 2015 data from the full mission duration, both the recommended Temperature plus low multipole polarisation information, as well as the complete spectral information, thereby including also the high multipole polarisation information which the Planck team considers as preliminary due to the presence of small but detectable low level residual systematics of ${\cal{O}}(1) \ \mu\rm{K}^2$ [@Aghanim:2015xee]. We briefly remind the reader of the expected impact of the DM interactions on the cosmological parameters in Section \[sec:dmnurecap\]. In Section \[sec:method\], we present the method used to analyze the data and give the results in Sections \[sec:temp\], \[sec:pol\] and \[sec:r16\]. We conclude in Section \[sec:conclusion\].
Impact of the DM-$\nu$ interactions on the cosmological parameters \[sec:dmnurecap\]
====================================================================================
The Dark Matter-neutrino interactions have five distinct effects on the temperature and polarisation angular power spectra. These were explained in Ref. [@Wilkinson:2014ksa] and can be seen in Fig. \[fig1\].
Schematically, one can understand the impact of a DM-$\nu$ coupling on the cosmological parameters as follows. On one hand, the DM-$\nu$ interactions induce a damping of the DM fluctuations at small-scales (i.e. at high multipoles). On the other hand, they prevent the neutrino free-streaming, till the neutrinos kinetically decouple from the DM. This last effect enhances the peaks at low multipoles, where the CMB temperature angular power spectrum is best measured. The greater the elastic scattering cross section (or the lighter the dark matter), the more pronounced are these two effects. Hence the fit to the data imposes an upper limit on the strength of these interactions.
The enhancement of the first few peaks is less pronounced in a younger Universe. Hence scenarios with DM-$\nu$ interactions are compatible with the data, when the value of $H_0$ is larger than the value estimated using the LCDM model. One should also observe a damping of the DM primordial fluctuations at small-scales because of the impact of neutrinos on the DM fluid. This effect translates into a damped oscillating matter power spectrum [@Boehm:2001hm] and thus leads to a smaller value of the $\sigma_8$ parameter than that in the LCDM scenario.
![The temperature and polarization CMB angular power spectra in the presence of Dark Matter-neutrino interactions.[]{data-label="fig1"}](test_u.pdf "fig:"){width="9cm"} ![The temperature and polarization CMB angular power spectra in the presence of Dark Matter-neutrino interactions.[]{data-label="fig1"}](test_uEE.pdf "fig:"){width="9cm"} ![The temperature and polarization CMB angular power spectra in the presence of Dark Matter-neutrino interactions.[]{data-label="fig1"}](test_uTE.pdf "fig:"){width="9cm"}
Finally, we note that the difference in TE spectra between $u=10^{-3}$ and $u=10^{-4}$ is of the order of the same order of magnitude as Planck sensitivity (e.g. ${\cal{O}}(1) \ \mu\rm{K}^2$). Therefore Planck’s angular power spectra alone are not sufficient to establish a preference for lower values of the $u$ ratio. However the suppression of power that such values ($u=10^{-3}$ and $u=10^{-4}$) induce in the matter power spectrum are very different. Using the $\sigma_8$ value together with the angular power spectra, we can rule out $u=10^{-3}$.
Method \[sec:method\]
=====================
The Boltzmann equations in presence of Dark Matter-neutrino interactions were given in e.g. Ref. [@Mangano:2006mp; @Serra:2009uu]. To ensure the full treatment of the Boltzmann hierarchy, we use a modified version of the Boltzmann code [class]{}[^1] [@Lesgourgues:2011re; @Blas:2011rf], that incorporates the Dark Matter-neutrino interactions [@Wilkinson:2014ksa]. We perform our analysis in three main steps.
In our first analysis, we use the six cosmological parameters of the Standard Model (namely the baryonic density $\Omega_bh^2$, the dark matter density $\Omega_ch^2$, the ratio between the sound horizon and the angular diameter distance at decoupling $\Theta_{s}$, the reionization optical depth $\tau$, the spectral index of the scalar perturbations $n_\mathrm{S}$, the amplitude of the primordial power spectrum $A_\mathrm{S}$) plus the ratio $u \equiv u_{\nu}= \sigma_{DM-\nu}/m_{\rm{DM}}$.
In a second step, we consider eight free parameters, i.e the seven parameters mentioned above $+$ either the effective number of relativistic degrees of freedom ${N_{\mathrm{eff}}}$ or the total neutrino mass ${{\Sigma}m_{\nu}}$. Our rationale for doing this is that adding a dark radiation component (${N_{\mathrm{eff}}}> 3.046$ [@std_neff]) as in Ref. [@Wilkinson:2014ksa; @darkradiation; @DiValentino:2015sam] or allowing the sum of the neutrino masses ${{\Sigma}m_{\nu}}$ to depart from the benchmark value taken by the Planck collaboration (${{\Sigma}m_{\nu}}= 0.06$ eV) could reduce current tensions on the age of the Universe.
Our last analysis uses nine free parameters, namely the seven mentioned above $+$ ${N_{\mathrm{eff}}}$ $+$ ${{\Sigma}m_{\nu}}$. Note that we use a logarithmic prior to constrain the $u$ parameter and flat priors for the other parameters (i.e. $\Omega_bh^2$, $\Omega_ch^2$, $\Theta_{s}$, $\tau$, $n_\mathrm{S}$, $A_\mathrm{S}$, ${N_{\mathrm{eff}}}$ and ${{\Sigma}m_{\nu}}$).
To understand the impact of the polarisation data, we start by analysing the full range of the 2015 temperature power spectrum ($2\leq\ell\leq2500$) plus the low multipoles polarization data ($2\leq\ell\leq29$) [@Aghanim:2015xee]. We will refer to this analysis as the “Planck TT + lowTEB” datasets. We then perform a second analysis, which we will refer to as “Planck TTTEEE + lowTEB”, where we include the Planck high multipole polarization data [@Aghanim:2015xee]. Finally, we perform a third analysis where we include the 2015 Planck measurements of the CMB lensing potential power spectrum $C^{\phi\phi}_\ell$ [@Ade:2015zua]. This last analysis will be referred to as the “lensing” dataset.
The scenarios for which the $H_0$ tension between the model-dependent Planck value and that inferred from the observations of Cepheids variables [@R16] appears to be less than $2\sigma$ are analysed again. This time, we assume a Gaussian prior on $H_0$ (i.e. $H_0=73.24\pm1.75 \ \rm{km \ s^{-1} \ Mpc^{-1}}$) and refer to this set of analysis as “R16”.
Results based on the “Planck TT + lowTEB” datasets only {#sec:temp}
=======================================================
We now present the results of our analyses using the Planck low multipole polarisation data. The $68 \% $ confidence level (c.l.) limits on the cosmological parameters for the DM-$\nu$ scenario are shown in Table \[tab:temp\]. For comparison, we also display the $68 \% $ c.l. constraints obtained by the Planck collaboration [@planck2015] for collisionless LCDM[^2] in Table \[tab:temp\_nou\].
”Weak” interactions are expected to erase primordial scales which have not been observed yet. Hence our analysis is bound to exclude only the strongest DM-$\nu$ interactions. This translates into an upper bound on the $u$ parameter of $u < 10^{-4.1}$ (or $u<10^{-4.0}$, using the lensing datasets), corresponding to a DM-$\nu$ elastic cross section of $\sigma \simeq 3-6 \ 10^{-31} \ \left(m_{\rm{dm}}/\rm{GeV}\right) \rm{cm^2}$. This result is similar to the limit derived in Ref. [@Escudero:2015yka], using the 2013 Planck temperature data. Furthermore, we find that the Planck data prefer low values of the Hubble constant $H_0$, even in the presence of DM interactions (see Table. \[tab:temp\]). This is at odds with the conclusions from Ref. [@Wilkinson:2013kia] but a possible explanation is that the 2013 Planck data relied on the (low $l$) WMAP polarisation data, while the 2015 Planck data rely on Planck’s polarisation data.
We observe in addition that the introduction of the DM-$\nu$ interactions breaks the well-known degeneracy between $H_0$ and the clustering parameter $\sigma_8$. The Hubble constant slightly increases while the clustering parameter $\sigma_8$ slightly decreases in presence of such interactions. For example, we find $\sigma_8=0.825\, _{-0.015}^{+0.014}$ (see the first column of the Table \[tab:temp\]) while the Planck collaboration found $\sigma_8=0.829\,\pm 0.014$ using the same dataset combination (see the first column of the Table \[tab:temp\_nou\]) for collisionless LCDM.
When we allow ${N_{\mathrm{eff}}}$ to vary, we obtain ${N_{\mathrm{eff}}}=3.14\,_{-0.35}^{+0.32}$ for ${{\Sigma}m_{\nu}}=0.06$ eV (see the third column of the Table \[tab:temp\] and Fig. \[fig2\]). This result is a bit higher than the Standard Model value (${N_{\mathrm{eff}}}= 3.046$) but it does remain compatible with it nonetheless. The Hubble rate then shifts from $H_0 = 68.0\,^{+2.6}_{-3.0} \ \rm{km \ s^{-1} \ Mpc^{-1} }$ to $H_0 = 68.3^{+2.6}_{-3.2} \ \rm{km \ s^{-1} \ Mpc^{-1}}$ (see the third columns of Table \[tab:temp\_nou\] and Table \[tab:temp\] respectively). In this case, the tension between the local measurements of $H_0$ ($H_0=73.24 \pm 1.75 \ \rm{km \ s^{-1} \ Mpc^{-1}}$) [@R16], and the Planck $\Lambda$CDM value [@planck2015] is somewhat reduced. Therefore we can reasonably combine the Planck datasets with the R16 datasets and perform a new analysis. The results are given in Table \[tab:R16\].
The introduction of DM-$\nu$ interactions is also compatible with heavier neutrinos. This is an important point since it was noted in Ref. [@Giusarma:2014zza; @sigma8; @DiValentino:2015sam] that massive neutrinos could alleviate the tension between Planck and the weak lensing measurements from the CFHTLenS survey [@Heymans:2012gg; @Erben:2012zw] and KiDS-450 [@Hildebrandt:2016iqg]. Assuming DM-$\nu$ interactions and the Planck TT + lowTEB + lensing dataset, we obtain ${{\Sigma}m_{\nu}}< 1.6 \ \rm{eV}$ at $95\%$ c.l. (see the sixth column of Table \[tab:temp\]) instead of ${{\Sigma}m_{\nu}}<0.675 \ \rm{eV}$ for LCDM (see the sixth column of Table \[tab:temp\_nou\]).
For that same combination of datasets (Planck TT + lowTEB + lensing), both the Hubble constant $H_0$ and the clustering parameter $\sigma_8$ increase with respect to the Standard Model ($\Lambda$CDM + ${{\Sigma}m_{\nu}}$) value. However, the value of $\sigma_8$ thus obtained remains small enough to partially reduce the tension with the weak lensing measurements. We obtain $\sigma_8 = 0.787\,^{+0.036}_{-0.030}$ which is much lower than the Planck value $\sigma_8= 0.8149\,\pm0.0093$, which was reported by the collaboration for LCDM only (i.e LCDM $+$ fixed values of ${N_{\mathrm{eff}}}$ and ${{\Sigma}m_{\nu}}$) using the Planck TT + lowTEB + lensing dataset. Using the Planck TT+lowTEB dataset only and the definition $S_8 \equiv\sigma_8 \sqrt{\Omega_m/0.3}$, we find $S_8 =0.826\, _{-0.028}^{+0.033}$. Hence adding the DM-$\nu$ interactions does reduce the tension with the KiDS-450 measurements ($S_8=0.745\pm0.039$ [@Hildebrandt:2016iqg]), to about $1.7\sigma$.
Finally, varying ${N_{\mathrm{eff}}}$ and ${{\Sigma}m_{\nu}}$ simultaneously allows to reduce the $H_0$ and $\sigma_8$ tensions. Using the Planck TT + lowTEB datasets, we find that $H_0=66.2\, _{-3.7}^{+4.0}$ and $\sigma_8= 0.792\,_{-0.040}^{+0.060}$, as shown in the seventh column of the Table \[tab:temp\]. The tension with other $H_0$ measurements is then about $1.6\sigma$. The new value for $S_8$ (namely $S_8=0.826\, _{-0.027}^{+0.033}$) also reduces the tension with KiDS-450 [@Hildebrandt:2016iqg] to about $1.7\sigma$.
Results with the polarization data {#sec:pol}
==================================
In Table \[tab:pol\], we report the $68 \% $ c.l limits on the DM-$\nu$ scenario obtained using the polarisation data. For comparison, we also give the $68 \% $ c.l. limits[^3] obtained by the Planck collaboration [@planck2015] for the LCDM scenario in Table \[tab:pol\_nou\].
Assuming fixed values of ${N_{\mathrm{eff}}}$ and ${{\Sigma}m_{\nu}}$, we find that the use of the Planck polarization data generally slightly improves the constraints of the strength of the Dark Matter-neutrino interactions (see Fig. \[fig3\]). For example, instead of $u<10^{-4.0}$, we now find $u< 10^{-4.3}$ using the Planck TTTEEE + lowTEB + lensing datasets and the scenario with nine parameters (see last column of Table \[tab:pol\]). The rest of the parameters remain compatible with $\Lambda$CDM values.
Similarly to the analysis performed in Section \[sec:temp\], we also vary the effective number of relativistic degrees of freedom ${N_{\mathrm{eff}}}$. However it remains consistent with the standard model value, and so does the Hubble constant $H_0$ in this case (see, for example, the third column of the Table \[tab:pol\] and \[tab:pol\_nou\]).
Adding the polarization data however helps to relax the bounds on massive neutrinos. The latter shifts from ${{\Sigma}m_{\nu}}<0.492 \ \rm{eV}$ for the Planck TTTEEE + lowTEB datasets without interactions to ${{\Sigma}m_{\nu}}<1.9 \ \rm{eV}$ for the same combination of datasets in presence of interactions (see the fifth column of Table \[tab:pol\_nou\] and Table \[tab:pol\] respectively). Furthermore, $\sigma_8$ decreases a bit in presence of interactions. We find $\sigma_8= 0.797\,_{-0.023}^{+0.049}$ in presence of interactions ($\sigma_8= 0.789\,_{-0.020}^{+0.036}$ if we add the lensing dataset) versus $\sigma_8= 0.812\,^{+0.039}_{-0.017}$ (or $\sigma_8= 0.783\,_{-0.020}^{+0.040}$ if we add the lensing dataset) in LCDM, as shown in the fifth and six columns of Tables \[tab:pol\] and \[tab:pol\_nou\]. Here again, we find that the tension with the weak lensing measurements is reduced. We obtain $S_8=0.832\, _{-0.022}^{+0.029}$ in presence of interactions (and letting ${{\Sigma}m_{\nu}}$ free to vary) for the Planck TTTEEE+lowTEB datasets, which decreases the tension with KiDS-450 to $1.8\sigma$.
Finally, we observe a small shift in both values of $H_0$ and $\sigma_8$ with respect to LCDM when we either vary ${N_{\mathrm{eff}}}$, ${{\Sigma}m_{\nu}}$ or both simultaneously. The upper bound on ${{\Sigma}m_{\nu}}$ is also relaxed with respect to the collisionless $\Lambda$CDM. Adding the lensing dataset, we obtain $H_0=65.4\, ^{+2.2}_{-2.0}$ and $\sigma_8= 0.784\,_{-0.024}^{+0.035}$ ($S_8=0.820_{-0.015}^{+0.019}$), as shown in the eight column of the Table \[tab:pol\]. Whilst the new value of $H_0$ does not remove completely the tensions between the different observation datasets, it does reduce the $S_8$ tension to $1.7\sigma$.
We note that when the Dark Matter-neutrino interactions are introduced, the scalar spectral index ($n_S$) gets very slightly shifted towards smaller values, for all the dataset combinations and parameters considered in this paper. This shift is due to the fact that the interactions change all the acoustic peaks (see Fig. \[fig1\] and discussion in Section II). In fact, they increase the low multipoles due to the suppression of neutrino free-streaming and decrease the high multipoles due to the collisional damping. Therefore, in order to reconcile the prediction of this model with the observed angular power spectra, the increase in the Hubble constant needs to be compensated by a change in the spectrum tilt.
Results with R16 {#sec:r16}
================
In this Section, we analyse again the models for which the tension between the 2015 Planck and Riess et al. 2016 [@R16] value of $H_0$ is less than $2\sigma$. These correspond to the scenarios where ${N_{\mathrm{eff}}}$ is free to vary, when we ignored the high multipole polarisation data. Applying a Gaussian prior on the value of $H_0$, we obtain new constraints on the cosmological parameters ($68 \% $ C.L.) for the interacting DM scenario, as shown in Table \[tab:R16\].
We find that all the cosmological parameters are shifted towards higher values, as can be seen by comparing the results from Table \[tab:R16\] with Table \[tab:temp\]. Moreover, owing to the very well-known degeneracy between $H_0$ and ${N_{\mathrm{eff}}}$ (see Fig. \[fig2\]), we find an indication for a dark radiation at about $2\sigma$ by imposing the R16 prior. In particular, we find ${N_{\mathrm{eff}}}=3.54\,\pm 0.20$ for the $\Lambda$CDM + $u$ + ${N_{\mathrm{eff}}}$ scenario and ${N_{\mathrm{eff}}}=3.56\,_{-0.26}^{+0.19}$ for the $\Lambda$CDM + $u$ + ${N_{\mathrm{eff}}}$ + ${{\Sigma}m_{\nu}}$ model. A dark radiation component can be explained by the existence of some extra relic component, such as a sterile neutrino or a thermal axion [@darkradiation; @DiValentino:2015wba; @DiValentino:2016ikp; @Giusarma:2014zza; @Archidiacono:2011gq]. However, in these models, an increase in the value of ${N_{\mathrm{eff}}}$ may not be related to the presence of an additional species. It could be related to dark matter annihilations into neutrinos as they would reheat the neutrino fluid and mimic an increase in the value of ${N_{\mathrm{eff}}}$ [@Boehm:2012gr; @Boehm:2013jpa].
Conclusion \[sec:conclusion\]
=============================
In the $\Lambda$CDM model, dark matter is assumed to be collisionless. This means that one arbitrarily sets the dark matter interactions to zero to interpret the CMB temperature and polarisation angular power spectra and determine the cosmological parameters. Here we relaxed the collisionless assumption and studied the impact of DM-$\nu$ interactions on the cosmological parameters.
We performed a similar analysis to [@Wilkinson:2014ksa]. However this time, we used the full 2015 Planck data [@planck2015] as they include both the high and low multipoles polarization spectra and are more precise than the 2013 data. In general, we observe that the introduction of dark matter-neutrino interactions can break the existing correlation between $H_0$ and $\sigma_8$. They can increase the value of $H_0$ and simultaneously decrease the value of $\sigma_8$, thus potentially reducing the current tensions between the Planck data and other measurements. However our main conclusions are three-fold.
- The high multipole polarisation data prefer LCDM-like models, though they do also predict a smaller value for $\sigma_8$ than LCDM.
- The DM-$\nu$ interactions do help to reduce the tension between the CMB and weak lensing estimates [@Heymans:2012gg; @Erben:2012zw; @Hildebrandt:2016iqg] of the $\sigma_8$ value, whatever the CMB dataset under consideration. This is particularly true when ${N_{\mathrm{eff}}}$ and/or ${{\Sigma}m_{\nu}}$ are kept as free parameters. However ${N_{\mathrm{eff}}}$ remains compatible with the Standard Model value, unless one also adds the Cepheids measurements.
- DM-$\nu$ interactions can also help to reduce the tensions between the CMB and Cepheid measurements of the Hubble constant, if one disregards the high multipole polarisation data. The combination of the CMB $+$ Cepheid datasets leads to a Hubble rate value of about $72.1^{+1.5}_{-1.7} \ \rm{km} \, \rm{s}^{-1} \, \rm{Mpc}^{-1}$ when ${N_{\mathrm{eff}}}$ is free to vary (and $71.9^{+1.6}_{-1.8} \ \rm{km} \, \rm{s}^{-1} \, \rm{Mpc}^{-1}$ when both ${N_{\mathrm{eff}}}$ and ${{\Sigma}m_{\nu}}$ are free). Under these conditions, ${N_{\mathrm{eff}}}$ can become as large as ${N_{\mathrm{eff}}}=3.54 \, {\pm 0.20}$ or ${N_{\mathrm{eff}}}=3.56\,_{-0.26}^{+0.19}$ if ${{\Sigma}m_{\nu}}$ can vary. In the latter case, we find that the sum of neutrino masses could reach up to 0.87 eV but the $\sigma_8$ parameter remains too high to reduce both the $H_0$ and $\sigma_8$ tensions simultaneously.
Finally we note that whatever the datasets used and hypothesis that we made, the DM-$\nu$ elastic scattering cross section cannot exceed $\sigma_{\rm{DM}} \lesssim 3 \ 10^{-31} - 6 \ 10^{-31} \ (m_{\rm{DM}/\rm{GeV}}) \ \rm{cm^2}$.
To conclude, DM-$\nu$ interactions do not enable to solve both the $H_0$ and $\sigma_8$ tensions simultaneously, but they can reduce them slightly nonetheless, if we ignore the high multipole polarisation data. Furthermore the combination of the low multipole and Cepheid data [@R16] show that such interactions have the potential to solve the $H_0$ tension, if we ignore the $\sigma_8$ tension. Should there be a good reason to ignore the high multipole polarisation data, one could potentially establish a link between the DM abundance and the neutrino masses [@Boehm:2006mi; @Ma:2006km; @Farzan:2009ji; @Farzan:2010mr; @Arhrib:2015dez]. The DESI [@Levi:2013gra] and Euclid[^4] surveys should be able to determine whether such relatively large interactions were present in the early Universe [@Escudero:2015yka]. Such high values of the $u$ ratio would question our understanding of structure formation, as it is expected that there would be little satellite companions left in the Milky Way [@Boehm:2014vja].
{width="1.5\columnwidth"}
{width="1.5\columnwidth"}
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors would like to thank J.L. Bernal, M. Escudero, M. Gerbino, A. Riess, L. Verde and R. Wilkinson for very useful discussions. EDV acknowledges support from the European Research Council in the form of a Consolidator Grant with number 681431. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation.
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|
---
abstract: 'Within the framework QCD sum rules, we use the least square fitting method to investigate the first radial excitations of the nucleon and light mesons such as $\rho$, $K^{*}$, $\pi$ , $\varphi$. The extracted masses of these radial excitations are consistent with the experimental data. Especially we find that the decay constant of $\pi(1300)$, which is the the first radial excitation of $\pi$, is tiny and strongly suppressed as a consequence of chiral symmetry.'
author:
- 'Jin-Feng Jiang'
- 'Shi-Lin Zhu'
bibliography:
- 'refs.bib'
title: Radial excitations of mesons and nucleons from QCD sum rules
---
INTRODUCTION {#sec2}
============
The method of QCD sum rules has been widely used to extract the resonance information in hadron physics [@Shifman1979]. This formalism is usually applied to study the ground state in a specific channel due to the limitation of theoretical accuracy and difficulty of numerical analysis. The excitations of mesons have been studied within finite energy sum rules in the literature [@Krasnikov1982; @Kataev1983; @Gorishnii1984]. Recently, there are some attempts to study the excitations of the heavy-light mesons using the method of QCD sum rules [@Gelhausen2014].
The radial excitations have the same spin-parity as the ground state. Experimentally many radial excitations of mesons and baryons have been established [@Olive2014]. Sometimes it is quite difficult to identify the radial excitations of hadrons. For example, the situation of the radial excitations of the vector charmonium above 4 GeV becomes quite unclear after so many charmonium-like XYZ states have been reported experimentally in the past decade. Theoretical investigations of the radial excitations are also very challenging.
In this work, we shall study the first radial excitations of the light mesons and nucleon within the framework of the QCD sum rule formalism. We explicitly keep two poles in the usual spectrum representation. Then we employ the least square method in the numerical analysis to extract the resonance information of the first radial excited state. The extracted masses of the radial excitations of the light mesons and nucleon agree with the experimental data quite well.
The paper is organized as follows. In Section \[sec2\], we introduce the QCD sum rule formalism and our least-square method. The numerical results are presented in Sections \[sec3rho\]-\[sec3nucleon\]. The last section is a short summary.
Formalism {#sec2}
=========
Within the framework of the QCD sum rule approach, we study the correlation function at the quark level $$\Pi\left(q\right)={\rm
i}\int\mathrm{d}^{4}x\,\mathrm{e}^{\mathrm{i}qx}\left\langle
0\left|T\left\{ j\left(x\right)j^{\dagger}\left(0\right)\right\}
\right|0\right\rangle$$ where $j\left(x\right)$ is the interpolating current with the same quantum numbers as the hadrons. The above correlation function satisfies the dispersion relation $$\Pi(q^{2})=\frac{1}{\pi}\int_{s_{min}}ds\frac{\mathrm{Im}\Pi(s)}{s-q^{2}-\mathrm{i}\epsilon},$$ At the quark gluon level, the correlation function can be calculated with the operator product expansion. The gluon and quark condensates appear as higher dimensional operators in this expansion. At the hadron level, the spectral density of the correlation function can be expressed in terms of the hadron masses and couplings. Due to the quark hadron duality, we get an equation called the QCD sum rule which relates the correlation function at the quark gluon level to the physical states. After making Borel transformation to the sum rule in the momentum space, one gets $$\Pi^{\prime}\left(M^{2}\right)=\frac{1}{\pi}\int\mathrm{e}^{-\nicefrac{s}{M^{2}}}\mathrm{Im}\Pi\left(s\right){\rm
ds}$$ where $M$ is the Borel parameter.
The spectral density usually takes the one-pole approximation $$\rho(s)\equiv\frac{1}{\pi}\mathrm{Im}\Pi\left(s\right)=f\delta\left(s-m^{2}\right)+\rho_{continuum}\theta\left(s-s_{0}\right),\label{eq:usual
spectrum}$$ where $m$ is the mass of the ground state and $s_0$ is the threshold parameter. Above $s_0$, the spectral density at the hadron level is replaced by the spectral density derived at the quark-gluon level. Now the sum rule reads $$f\mathrm{e}^{-\nicefrac{m^{2}}{M^{2}}}=\Pi^{\prime}\left(M^{2}\right)-\int_{s_{0}}^{\infty}\mathrm{e}^{-\nicefrac{s}{M^{2}}}\rho^{\text{OPE}}\left(s\right){\rm
ds}.\label{eq:sum rule}$$ The usual numerical method in QCD sum rule analysis is to differentiate Eq. (\[eq:sum rule\]) with respect to $\nicefrac{1}{M^{2}}$ and divide the resulting equation by Eq. (\[eq:sum rule\]) $$m^{2}=\frac{\int_{0}^{s_{0}}\mathrm{e}^{-\nicefrac{s}{M^{2}}}s\rho^{\text{OPE}}\left(s\right){\rm
ds}}{\int_{0}^{s_{0}}\mathrm{e}^{-\nicefrac{s}{M^{2}}}\rho^{\text{OPE}}\left(s\right){\rm
ds}}.$$ One usually plots the variation of the mass versus $M^2$ and $s_0$ to find a working window.
However, the method described above can only be applied to the ground states. In order to extract the resonance information of the first radial excitation, we modify the above spectral density and explicitly keep the pole of the first radial excitation in the spectrum. Now the modified spectral density reads $$\rho(s)\equiv\frac{1}{\pi}\mathrm{Im}\Pi\left(s\right)=f_{1}\delta\left(s-m^{2}\right)+f_{2}\delta\left(s-m^{\prime2}\right)+\rho_{continuum}\theta\left(s-s_{0}^{\prime}\right).\label{eq:
spectrum}$$ To simply the numerical analysis, we use the zero width approximation for both the ground state and first radial excitation. The parameters $f_{1}$ and $f_{2}$ are related to the coupling parameters while $m$ and $m'$ are the masses of the ground state and the first radial excitation respectively. Now the sum rules read $$\begin{aligned}
& \int\mathrm{e}^{-\nicefrac{s}{M^{2}}}\rho_{ground}\left(s\right)\mathrm{d}s+\int\mathrm{e}^{-\nicefrac{s}{M^{2}}}\rho_{\text{excitation}}\left(s\right)\mathrm{d}s+\int_{s_{0}}^{\infty}\mathrm{e}^{-\nicefrac{s}{M^{2}}}\rho_{\text{continuum}}\left(s\right)\mathrm{d}s\\
& =\Pi^{\prime}\left(M^{2}\right)=\Pi^{\prime\text{perturbation}}\left(M^{2}\right)+\Pi^{\prime\text{condensates}}\left(M^{2}\right).\end{aligned}$$ The usual numerical method cannot be applied here because the modified spectrum has two mass parameters. We use the least square method [@Narison1984] to fit these masses and decay parameters. The detail of the method are described below.
As usual in the sum rule analysis, one has to find an optimal working interval of the Borel parameter $M^{2}$. The lower boundary of $M^2$ is chosen to ensure the convergence of the operator product expansion while the upper boundary is chosen to make the continuum contribution remain subleading.
To get an optimal interval of the Borel parameter $M^{2}$, we set $$\left|\frac{\int_{s_{0}}^{\infty}\mathrm{e}^{-\nicefrac{s}{M^{2}}}\rho_{\text{continuum}}\left(s\right)\mathrm{d}s}{\Pi^{\prime}\left(M^{2}\right)}\right|\leq\alpha_{1}$$ which ensures that the continuum contribution remains subleading and determines the upper boundary and $$\left|\frac{\Pi^{\prime\text{condensates}}\left(M^{2}\right)}{\Pi^{\prime}\left(M^{2}\right)}\right|\leq\alpha_{2}$$ which ensures that the OPE is reliable and determines the lower boundary. The two boundaries determine the optimal interval of $M^{2}$ for our numerical analysis.
The number $\alpha_{1}$ and $\alpha_{2}$ is chosen to ensure a rational contribution of continuum and higher order OPE terms. For the meson case, we set $\alpha_{1}=\alpha_{2}=\alpha$ to get a reasonable interval of $M^{2}$. We use different values for $\alpha_{1}$ and $\alpha_{2}$ in the nucleon case. Note that we always try a smaller $\alpha$ in the excitation case since the continuum contribution decreases as the threshold parameter $s_0$ increases. If no reasonable interval of $M^{2}$ can be got in any way, the sum rule may not be appropriate in our numerical method.
We rewrite the sum rule as $$\begin{aligned}
& \int\mathrm{e}^{-\nicefrac{s}{M^{2}}}\rho_{ground}\left(s\right)\mathrm{d}s+\int\mathrm{e}^{-\nicefrac{s}{M^{2}}}\rho_{excitation}\left(s\right)\mathrm{d}s\\
& =g\left(M^{2},s_{0}\right)=\Pi^{\prime}\left(M^{2}\right)-\int_{s_{0}}^{\infty}\mathrm{e}^{-\nicefrac{s}{M^{2}}}\rho_{continuum}\left(s\right)\mathrm{d}s\end{aligned}$$ which separates the part of expression with physical parameters from the part with just Borel parameter $M^{2}$ and threshold $s_{0}$.
With the above expression of $g\left(M^{2},s_{0}\right)$, we can generate a series of points $\left\{\left(M_{i}^{2},
g\left(M_{i}^{2},s_{0}\right)\right)\right\} $ by choosing a set $\left\{
M_{i}^{2}\right\} $ within the optimal interval of $M^{2}$. We uniformly choose $N$ points in the optimal interval of $M^{2}$. The number $N$ is chosen to be $20$ or even larger.
With the sets $\left\{\left(M_{i}^{2},
g\left(M_{i}^{2},s_{0}\right)\right)\right\} $, we use the least-square method which minimizes the sum of the squares of the difference between the two sides of the sum rules $$\sum_{i=1}^{N}\frac{\left|f_{1}\mathrm{e}^{-\frac{m^{2}}{M_{i}^{2}}}+f_{2}\mathrm{e}^{-\frac{m^{\prime2}}{M_{i}^{2}}}-g\left(M_{i}^{2},s_{0}\right)\right|^{2}}{N}=min\label{eq:least
square}$$ to get the best fit of the resonance parameters of the ground state and first radial excitation.
The masses of the ground states of the light mesons and nucleon are measured precisely experimentally. The extracted masses from the traditional QCD sum rule formalism with the one-pole approximation agree with the experimental data very well. In our analysis we first use the least square method to reproduce the resonance parameters of the ground states. As expected, the resulting masses are consistent with experimental data and those extracted from the traditional QCD sum rule analysis.
Then we use the extracted masses of the ground states as inputs to extract the resonance parameters of the radial excited states since the less parameters in the fitting will cost less computing resource and lead to relatively more stable results. Moreover, we do not fix the masses of the ground states in Eq. (\[eq:least square\]) in our numerical analysis. Instead we allow them to vary around the experimental central value within $\pm5\%$. In this way, we extract the resonance parameters of the first radial excited states numerically.
We analyze several light mesons and nucleon in the following section. The sum rules of the light mesons can be found in the pioneer paper [@Shifman1979]. The nucleon sum rule with the radiative corrections can be found in Ref. [@Drukarev2009]. We collect these sum rules in the appendix.
In our analysis we use the following values for the various condensates and parameters [@Olive2014; @Ioffe2003; @Shifman1979]: $\left\langle
\bar{q}q\right\rangle
\left(2\text{GeV}\right)=-\left(277_{-10}^{+12}\text{MeV}\right)^{3}$, $\left\langle
0\right|m_{u}\bar{u}u+m_{d}\bar{d}d\left|0\right\rangle
=-\frac{1}{2}f_{\pi}^{2}m_{\pi}^{2}=-1.7\times10^{-4}{\rm GeV}^{4}
$, $m_{s}\left(2\text{GeV}\right)=(95\pm5)\text{MeV}$, $
\nicefrac{\left\langle \bar{s}s\right\rangle }{\left\langle
\bar{q}q\right\rangle }=0.8\pm0.3$, $ \left\langle
0\right|\frac{\alpha_{s}}{\pi}G_{\mu\nu}^{a}G_{\mu\nu}^{a}\left|0\right\rangle
=0.012_{-0.012}^{+0.006}{\rm GeV}^{4}$, $ \left\langle
0\right|\alpha_{s}\left(\bar{u}\gamma_{\alpha}\gamma_{5}t^{a}u-\bar{d}\gamma_{\alpha}\gamma_{5}t^{a}d\right)^{2}\left|0\right\rangle
=\frac{32}{9}\alpha_{s}\left\langle
0\right|\bar{q}q\left|0\right\rangle ^{2}\simeq6.5\times10^{-4}{\rm
GeV}^{4}$, $ \left\langle
0\right|\alpha_{s}\left(\bar{u}\gamma_{\alpha}\gamma_{5}t^{a}u-\bar{d}\gamma_{\alpha}\gamma_{5}t^{a}d\right)
\sum_{q=u,d,s}\bar{q}\gamma_{\alpha}t^{a}q\left|0\right\rangle
\simeq-\frac{32}{9}\alpha_{s}\left\langle
0\right|\bar{q}q\left|0\right\rangle ^{2}\simeq-6.5\times10^{-4}{\rm
GeV}^{4}$, $
\alpha_{s}\left(Q^{2}\right)=\nicefrac{4\pi}{\left(b\ln\left(\nicefrac{Q^{2}}{\Lambda^{2}}\right)\right)}
$, $\Lambda=0.1\text{GeV}$, $
\alpha_{s}\left(m_{Z}\right)=0.1184\pm0.0007 $, $\alpha_{s}\left(1.5\text{GeV}\right)=0.353\pm0.006$.
The $\rho$ meson {#sec3rho}
================
The interpolating current for the $\rho$ meson is $$j_{\mu}^{\left(\rho\right)}=\frac{1}{2}\left(\bar{u}\gamma_{\mu}u-\bar{d}\gamma_{\mu}d\right),$$ and the resulting sum rule can be found in the appendix. The usual single-pole spectral density reads $$\rho^{\left(\rho\right)}\left(s\right)=6\pi^{2}f_{\rho}^{2}\delta\left(s-m_{\rho}^{2}\right)+\frac{3}{2}\left(1+\frac{\alpha_{s}\left(s\right)}{\pi}\right)\theta\left(s-s_{0}\right)$$ We also need the double-pole spectral density $$\rho^{\left(\rho\right)}\left(s\right)=6\pi^{2}f_{\rho}^{2}\delta\left(s-m_{\rho}^{2}\right)+6\pi^{2}f_{\rho^{\prime}}^{2}\delta\left(s-m_{\rho^{\prime}}^{2}\right)+\frac{3}{2}\left(1+\frac{\alpha_{s}\left(s\right)}{\pi}\right)\theta\left(s-s_{0}\right),$$ where $f_{\rho}$ and $f_{\rho^{\prime}}$ are defined as $$\left\langle 0\left|\bar{q}\gamma_{\mu}q\right|\rho\right\rangle
=m_{\rho}f_{\rho}\epsilon_{\mu},\left\langle
0\left|\bar{q}\gamma_{\mu}q\right|\rho^{\prime}\right\rangle
=m_{\rho^{\prime}}f_{\rho^{\prime}}\epsilon_{\mu}^{\prime}$$ where $q=u,d$.
We first use the least square method and the traditional one-pole spectrum representation with $\alpha=0.2$ and $N=40$ to extract the mass and decay constant of the $\rho$ meson. The results are listed in Table \[rho ground state\]. The parameter $f_{1}$ is related to the decay constant in Eq. (\[eq:usual spectrum\]). The values of “min” are the sum of the squares of the differences in Eq. (\[eq:least square\]). Only when the value of “min” is much smaller than the parameters $f^2_{1}$, $f^2_{2}$ etc, the fit and the extracted decay constants are reliable.
$s_{0}$[\[]{}$\text{GeV}^{2}$[\]]{} 1.2 1.3 1.4 1.5 1.6
------------------------------------------- ----------- ----------- ----------- ----------- -----------
$M_{min}^{2}$[\[]{}$\text{GeV}^{2}$[\]]{} 0.43 0.43 0.43 0.43 0.43
$M_{max}^{2}$[\[]{}$\text{GeV}^{2}$[\]]{} 0.74 0.82 0.88 0.94 1.00
$m$[\[]{}GeV[\]]{} 0.74 0.75 0.75 0.76 0.77
$f_{\rho}$[\[]{}MeV[\]]{} 187 190 193 197 201
$f_{1}$[\[]{}$\text{GeV}^{2}$[\]]{} 2.06 2.13 2.22 2.30 2.39
min[\[]{}$\text{GeV}^{4}$[\]]{} $10^{-5}$ $10^{-5}$ $10^{-5}$ $10^{-5}$ $10^{-5}$
: The mass and decay constant of the $\rho$ ground state with $\alpha=0.2$ and $N=40$. \[rho ground state\]
We collect the fitting results with the double-pole spectrum in Table \[rho excitation\]. Note that the parameter $m$ in Table \[rho excitation\] is the input to extract the information of the excited state. We use $\alpha=0.1$ in this case. The threshold $s_{0}$ plays the role of including the first radial excitation in the spectrum while excluding the contribution from the higher excitations. To check the consistency of our fitting and dependence of our results on $s_0$, we vary $s_{0}$ in a range. A reliable fitting requires that the mass $m^{\prime}$ and decay constant $f_{\rho^{\prime}}$ of the first radial excitation should not vary too much with $s_0$.
$s_{0}$[\[]{}$\text{GeV}^{2}$[\]]{} 2.3 2.4 2.5 2.6 2.7
------------------------------------------- ----------- ----------- ----------- ----------- -----------
$M_{min}^{2}$[\[]{}$\text{GeV}^{2}$[\]]{} 0.50 0.50 0.50 0.50 0.50
$M_{max}^{2}$[\[]{}$\text{GeV}^{2}$[\]]{} 1.00 1.04 1.10 1.14 1.18
$m$[\[]{}GeV[\]]{} 0.76 0.76 0.76 0.76 0.76
$m^{\prime}$[\[]{}GeV[\]]{} 1.24 1.29 1.35 1.38 1.40
$f_{\rho}$[\[]{}MeV[\]]{} 196 197 198 198 198
$f_{\rho^{\prime}}$[\[]{}MeV[\]]{} 130 141 152 161 170
$f_{1}$[\[]{}$\text{GeV}^{2}$[\]]{} 2.3 2.3 2.3 2.3 2.3
$f_{2}[\mbox{GeV}^{2}]$ 1.0 1.2 1.4 1.5 1.7
min[\[]{}$\text{GeV}^{4}$[\]]{} $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-6}$ $10^{-6}$
: Masses and decay constants of the $\rho$ ground state and first radial excitation with $\alpha=0.1$ and $N=40$. \[rho excitation\]
From Table \[rho ground state\] we have $$m=\left(0.76\pm0.01\right)\text{GeV},f_{\rho}=\left(194\pm6\right)\text{MeV}$$ which agrees with the $\rho$ meson mass from PDG $m=0.77$GeV [@Olive2014] and the experimental measurement of the $\rho$ meson decay constant [@Becirevic2003] $$f_{\rho}^{\text{exp}}\simeq216\left(5\right)\text{MeV}.$$ In order to reduce the dependence on the threshold parameter $s_0$, the extracted values of $m$ and $f_{\rho}$ are the average values of the numerical values in Table \[rho ground state\]. From Table \[rho excitation\] we have $$m^{\prime}=\left(1.33\pm0.07\right)\text{GeV},f_{\rho}=\left(197\pm1\right)\text{MeV},f_{\rho^{\prime}}=\left(151\pm16\right)\text{MeV}.$$ From PDG, the mass of the first radial excitation is $m^{\prime}=1.47$ GeV and its width is $\Gamma=0.40$ GeV. Our extracted $\rho^{\prime}$ mass is consistent with the experimental data. At present, the decay constant of $\rho^{\prime}$ has not been measured yet.
The $\pi$ and $A_1$ mesons {#sec3pi}
==========================
We adopt the axial current for the pion and $A_1$ mesons $$j_{5\mu}^{A_1}=\bar{u}\gamma_{\mu}\gamma_{5}d,$$ and the resulting sum rule can be found in the appendix. Besides the $a_1$ pole, the pion also contributes to this sum rule due to the partial conservation of the axial vector current. As a Goldstone boson, the pion mass is tiny. Especially in the sum rule analysis, $m_\pi^2$ is much much less than the Borel parameter $M^2$. We can safely ignore the pion mass and let it be zero in the numerical analysis.
The usual spectrum representation is $$\rho\left(s\right)=\pi f_{\pi}^{2}\delta\left(s\right)+\pi
f_{A_{1}}^{2}\delta\left(s-m_{A_{1}}^{2}\right)+\frac{1}{4\pi}\left(1+\frac{\alpha_{s}\left(s\right)}{\pi}\right)\theta\left(s-s_{0}\right).$$ Our modified spectrum representation reads $$\rho^{\left(\pi\right)}\left(s\right)=\pi
f_{\pi}^{2}\delta\left(s\right)+\pi
f_{\pi^{\prime}}^{2}\delta\left(s-m_{\pi^{\prime}}^{2}\right)+\pi
f_{A_{1}}^{2}\delta\left(s-m_{A_{1}}^{2}\right)+\frac{1}{4\pi}\left(1+\frac{\alpha_{s}\left(s\right)}{\pi}\right)\theta\left(s-s_{0}\right),$$ where $f_{\pi}$, $f_{\pi^{\prime}}$ $f_{A_{1}}$ are defined as $$\left\langle 0\left|j_{\mu}^{\pi}\right|\pi\right\rangle
=if_{\pi}p_{\mu},\left\langle
0\left|j_{\mu}^{\pi}\right|\pi^{\prime}\right\rangle
=if_{\pi^{\prime}}p_{\mu}^{\prime},\left\langle
0\left|j_{\mu}^{\pi}\right|A_{1}\right\rangle
=m_{A_{1}}f_{A_{1}}\epsilon_{\mu}^{\prime}.$$
In the fitting, we use the least square method and the traditional spectrum representation with $\alpha=0.3$ and $N=80$ to extract the $A_1$ mass and decay constant. The results are listed in Table \[pion ground state\].
$s_{0}$[\[]{}$\text{GeV}^{2}$[\]]{} 1.3 1.40 1.50 1.60 1.70
------------------------------------------- ----------- ----------- ----------- ----------- -----------
$M_{min}^{2}$[\[]{}$\text{GeV}^{2}$[\]]{} 0.52 0.52 0.52 0.52 0.52
$M_{max}^{2}$[\[]{}$\text{GeV}^{2}$[\]]{} 1.16 1.20 1.28 1.36 1.44
$m_{A_{1}}$[\[]{}GeV[\]]{} 1.14 1.18 1.22 1.26 1.28
$f_{\pi}$[\[]{}MeV[\]]{} 134 135 136 137 137
$f_{A_{1}}$[\[]{}MeV[\]]{} 124 139 153 166 175
$f_{1}$[\[]{}$\text{GeV}^{2}$[\]]{} 0057 0.057 0.058 0.059 0.059
$f_{2}[\mbox{GeV}^{2}]$ 0.048 0.060 0.074 0.087 0.096
min[\[]{}$\text{GeV}^{4}$[\]]{} $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-8}$
: The mass and decay constant of the $A_1$ meson. We use the least square method and the traditional spectrum representation with $\alpha=0.3$ and $N=80$. \[pion ground state\]
In order to extract the resonance parameters of the first excitation of the pion meson, we employ the modified spectrum and allow $f_{A_{1}}$ and $m_{A_{1}}$ to vary around the experimental data within $\pm5\%$. The numerical results are listed in Table \[pion excitation\].
$s_{0}$[\[]{}$\text{GeV}^{2}$[\]]{} 2.0 2.1 2.2 2.3 2.4 2.5 2.6
------------------------------------------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
$M_{min}^{2}$[\[]{}$\text{GeV}^{2}$[\]]{} 0.63 0.63 0.63 0.63 0.63 0.63 0.63
$M_{max}^{2}$[\[]{}$\text{GeV}^{2}$[\]]{} 1.28 1.34 1.38 1.44 1.50 1.56 1.62
$m_{A_{1}}$[\[]{}GeV[\]]{} 1.29 1.29 1.29 1.29 1.29 1.29 1.29
$m_{\pi}^{\prime}$[\[]{}GeV[\]]{} 1.34 1.36 1.31 1.34 1.41 1.43 1.46
$f_{\pi}$[\[]{}MeV[\]]{} 121 122 123 123 124 125 126
$f_{A_{1}}$[\[]{}MeV[\]]{} 248 248 248 248 248 248 248
$f_{\pi^{\prime}}$[\[]{}MeV[\]]{} 0.2 0.3 0.1 2.3 0.2 0.7 0.1
$f_{1}$[\[]{}$\text{GeV}^{2}$[\]]{} 0.05 0.05 0.05 0.05 0.05 0.05 0.05
$f_{2}[\mbox{GeV}^{2}]$ 0.19 0.19 0.19 0.19 0.19 0.19 0.19
$f_{3}[\mbox{GeV}^{2}]$ $10^{-7}$ $10^{-7}$ $10^{-8}$ $10^{-5}$ $10^{-7}$ $10^{-6}$ $10^{-8}$
min[\[]{}$\text{GeV}^{4}$[\]]{} $10^{-5}$ $10^{-5}$ $10^{-5}$ $10^{-5}$ $10^{-5}$ $10^{-5}$ $10^{-5}$
: Masses and decay constants of the $A_1$ ground state and the first radial excitation of the pion with $\alpha=0.2$ and $N=80$. \[pion excitation\]
From Table \[pion ground state\] we have $$m_{A_{1}}=\left(1.22\pm0.06\right)\text{GeV},f_{\pi}=\left(135\pm1\right)\text{MeV},f_{A_{1}}=\left(151\pm20\right)\text{MeV}.$$ From PDG we have $m_{A_{1}}=1.23$ GeV and $\Gamma_{A_{1}}=0.40$ GeV. We note that the $A_1$ mass from the fitting is in rough agreement with the experimental data. The extracted pion decay constant agrees with the experimental data [@Olive2014] $$f_{\pi}^{\text{exp}}=130\text{MeV}.$$ However, the extracted $A_1$ decay constant is only half of the experimental data [@Wingate1995] $$f_{A_{1}}^{\text{exp}}=254\left(20\right)\text{MeV}.$$
To extract the first radial excitation of the pion meson, we use the experimental data of the $A_1$ decay constant as input in the numerical analysis. The results are collected in Table \[pion excitation\]. We have $$m_{\pi^{\prime}}=\left(1.38\pm0.06\right)\text{GeV},f_{\pi}=\left(123\pm1\right)\text{MeV},f_{\pi^{\prime}}=\left(0.6\pm0.8\right)\text{MeV}.$$ The resulting mass of the pion radial excitation agrees with the PDG value very nicely: $m_{\pi^{\prime}}=1.30$ GeV and $\Gamma_{\pi^{\prime}}=0.40$ GeV [@Olive2014]. Note that the extracted numerical value of $f_{\pi^{\prime}}$ is not reliable since the parameter $f^2_{3}$ is even smaller than the “min”. In this case, we may get a upper bound $$\left|f_{3}\right|<\sqrt{min}\sim0.0032\mbox{GeV}^{2}$$ Accordingly, we get the upper bound for $f_{\pi^{\prime}}$ $$f_{\pi^{\prime}}<0.032\mbox{GeV}$$ If the value of $f_{\pi^{\prime}} $ is larger than 0.032 GeV, we should be able to extract its value through the least square fitting method.
In other words, our numerical analysis demonstrates that the decay constant of the pion radial excitation $\pi^{\prime}$ is much smaller than the pion decay constant around 130 MeV. This interesting fact was also noticed by previous theoretical work including lattice simulations [@Andrianov1998; @Elias1997; @Maltman2002; @McNeile2006; @Volkov1997; @Holl2004; @Holl2005; @Qin2012; @Narison2014; @Kataev1983; @Gorishnii1984]. In fact, the suppression of the $\pi^{\prime}$ decay constant is a consequence of the chiral symmetry breaking. In the chiral limit, the decay constants of the pion and its radial excitations satisfy the following relation [@Dominguez1977] $$f_{\pi_{n}}m_{\pi_{n}}^{2}=0,$$ where $m_{\pi_{n}}$ ($n\geq 1$) is the mass of the pion radial excitation. The pion ground state is massless in the chiral limit as a Goldstone boson, hence its decay constant can be large and nonzero. For the pion radial excitation, its mass is large and nonzero. Therefore its decay constant has to vanish, i.e., $f_{\pi_{1}}=0$.
The $K^{*}$ meson {#sec3kaon}
=================
The interpolating current for the $K^{*}$ meson is $$j_{\mu}^{\left(K^{*}\right)}=\bar{u}\gamma_{\mu}s$$ and the resulting sum rule can be found in the appendix. The usual single-pole spectral density reads $$\mathrm{\rho}\left(s\right)=\pi
f_{K^{*}}^{2}\delta\left(s-m_{K^{*}}^{2}\right)+\frac{1}{4\pi}\left(1+\frac{\alpha_{s}\left(s\right)}{\pi}\right)\theta\left(s-s_{0}\right).$$ Our modified spectrum representation reads $$\rho^{\left(K^{*}\right)}\left(s\right)=\pi
f_{K^{*}}^{2}\delta\left(s-m_{K^{*}}^{2}\right)+\pi
f_{K^{*\prime}}^{2}\delta\left(s-m_{K^{*\prime}}^{2}\right)+\frac{1}{4\pi}\left(1+\frac{\alpha_{s}\left(s\right)}{\pi}\right)\theta\left(s-s_{0}\right),$$ where $f_{K^{*}}$ and $f_{K^{*\prime}}$ are defined as $$\left\langle
0\left|j_{\mu}^{\left(K^{*}\right)}\right|K^{*}\right\rangle
=m_{K^{*}}f_{K^{*}}\epsilon_{\mu},\left\langle
0\left|j_{\mu}^{\left(K^{*}\right)}\right|K^{*\prime}\right\rangle
=m_{K^{*\prime}}f_{K^{*\prime}}\epsilon_{\mu}^{\prime}.$$
$s_{0}$[\[]{}$\text{GeV}^{2}$[\]]{} 1.4 1.5 1.6 1.6 1.8
------------------------------------------- ----------- ----------- ----------- ----------- -----------
$M_{min}^{2}$[\[]{}$\text{GeV}^{2}$[\]]{} 0.63 0.63 0.63 0.63 0.63
$M_{max}^{2}$[\[]{}$\text{GeV}^{2}$[\]]{} 1.10 1.18 1.28 1.36 1.44
$m$[\[]{}GeV[\]]{} 0.88 0.89 0.90 0.90 0.91
$f_{K^{*}}$[\[]{}MeV[\]]{} 202 206 210 215 219
$f_{1}$[\[]{}$\text{GeV}^{2}$[\]]{} 0.13 0.13 0.14 0.14 0.15
min[\[]{}$\text{GeV}^{4}$[\]]{} $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-7}$ $10^{-7}$
: The mass and decay constant of the $K^{*}$ ground state. We use the least square method and the traditional spectrum representation with $\alpha=0.3$ and $N=20$. \[K star ground state\]
$s_{0}$[\[]{}$\text{GeV}^{2}$[\]]{} 2.3 2.4 2.5 2.6 2.7 2.8
------------------------------------------- ----------- ----------- ----------- ----------- ----------- -----------
$M_{min}^{2}$[\[]{}$\text{GeV}^{2}$[\]]{} 0.63 0.63 0.63 0.63 0.63 0.63
$M_{max}^{2}$[\[]{}$\text{GeV}^{2}$[\]]{} 1.40 1.46 1.52 1.58 1.64 1.70
$m$[\[]{}GeV[\]]{} 0.89 0.89 0.89 0.89 0.89 0.89
$m^{\prime}$[\[]{}GeV[\]]{} 1.22 1.25 1.27 1.33 1.29 1.37
$f_{K^{*}}$[\[]{}MeV[\]]{} 200 201 202 200 207 207
$f_{K^{*\prime}}$[\[]{}MeV[\]]{} 139 146 153 162 159 172
$f_{1}$[\[]{}$\text{GeV}^{2}$[\]]{} 0.13 0.13 0.13 0.12 013 0.13
$f_{2}[\mbox{GeV}^{2}]$ 0.06 0.07 0.08 0.08 0.08 0.09
min[\[]{}$\text{GeV}^{4}$[\]]{} $10^{-8}$ $10^{-8}$ $10^{-8}$ $10^{-8}$ $10^{-8}$ $10^{-8}$
: Masses and decay constants of the $K^{*}$ ground state and the first radial excitation with $\alpha=0.2$ and $N=20$. \[K star excitation\]
The results from the first spectrum representation are listed in Table \[K star ground state\] and those from the modified spectrum are listed in Table \[K star excitation\]. From Table \[K star ground state\] we have $$m=\left(0.89\pm0.01\right)\text{GeV},f_{K^{*}}=\left(210\pm7\right)\text{MeV}.$$ From Table \[K star excitation\] we have $$m^{\prime}=\left(1.28\pm0.06\right)\text{GeV},f_{K^{*}}=\left(203\pm3\right)\text{MeV},f_{K^{*\prime}}=\left(155\pm11\right)\text{MeV},$$ where m is an input parameter in Table \[K star excitation\]. The decay constant of the $K^{*}$ was measured to be [@Becirevic2003] $$f_{K^{*}}^{\text{exp}}\simeq217\text{MeV}.$$ From PDG, the mass and width of the $K^{*\prime}$ are $m^{\prime}=1.41$ GeV, $\Gamma=0.232$ GeV respectively. Clearly our extracted $f_{K^{*}}$ from both fitting agrees with the data. The extracted $m^{\prime}$ is also consistent with data.
The $\varphi$ meson {#sec3phi}
===================
The interpolating current for the $\varphi$ meson is $$j_{\mu}^{\left(\varphi\right)}=-\frac{1}{3}\bar{s}\gamma_{\mu}s,$$ and the resulting sum rule can be found in the appendix. The usual spectrum representation is $$\rho\left(s\right)=\frac{1}{9}\pi
f_{\varphi}^{2}\delta\left(s-m_{\varphi}^{2}\right)+\frac{1}{36\pi}\left(1+\frac{\alpha_{s}\left(s\right)}{\pi}\right)\theta\left(s-s_{0}\right).$$ We also use the modified spectrum representation $$\rho^{\left(\varphi\right)}\left(s\right)=\frac{1}{9}\pi
f_{\varphi}^{2}\delta\left(s-m_{\varphi}^{2}\right)+\frac{1}{9}\pi
f_{\varphi^{\prime}}^{2}\delta\left(s-m_{\varphi^{\prime}}^{2}\right)+\frac{1}{36\pi}\left(1+\frac{\alpha_{s}\left(s\right)}{\pi}\right)\theta\left(s-s_{0}\right)$$ where $f_{\varphi}$ and $f_{\varphi^{\prime}}$ are defined as $$\left\langle 0\left|\bar{s}\gamma^{\mu}s\right|\varphi\right\rangle
=m_{\varphi}f_{\varphi}\epsilon_{\mu},\left\langle
0\left|\bar{s}\gamma^{\mu}s\right|\varphi^{\prime}\right\rangle
=m_{\varphi^{\prime}}f_{\varphi^{\prime}}\epsilon_{\mu}^{\prime}.$$
We use the least square method and the traditional spectrum representation with $N=20$. Note that there does not exist a working interval of $M^{2}$ for $\alpha=0.2$. So we use $\alpha=0.3$ here. The results from the first spectrum representation are listed in Table \[phi ground state\] and those from the modified spectrum are listed in Table \[phi excitation\], where m is the input parameter in Table \[phi excitation\].
$s_{0}$[\[]{}$\text{GeV}^{2}$[\]]{} 1.7 1.8 1.9 2.0 2.1 2.2
------------------------------------------- ----------- ----------- ----------- ----------- ----------- -----------
$M_{min}^{2}$[\[]{}$\text{GeV}^{2}$[\]]{} 0.87 0.87 0.87 0.87 0.87 0.87
$M_{max}^{2}$[\[]{}$\text{GeV}^{2}$[\]]{} 1.66 1.78 1.90 2.00 2.12 2.24
$m$[\[]{}GeV[\]]{} 1.02 1.03 1.03 1.04 1.05 1.06
$f_{\varphi}$[\[]{}Mev[\]]{} 217 221 226 231 236 240
$f_{1}$[\[]{}$\text{GeV}^{2}$[\]]{} 0.016 0.017 0.018 0.019 0.019 0.020
min[\[]{}$\text{GeV}^{4}$[\]]{} $10^{-9}$ $10^{-9}$ $10^{-9}$ $10^{-9}$ $10^{-9}$ $10^{-9}$
: The mass and decay constant of the $\varphi$ ground state. We use the least square method and the traditional spectrum representation with $N=20$ and $\alpha=0.3$ here. \[phi ground state\]
$s_{0}$[\[]{}$\text{GeV}^{2}$[\]]{} 3.4 3.5 3.6 3.7 3.8 3.9 4.0
------------------------------------------- ------------ ------------ ------------ ------------ ------------ ------------ -----------
$M_{min}^{2}$[\[]{}$\text{GeV}^{2}$[\]]{} 1.15 1.15 1.15 1.15 1.15 1.15 1.15
$M_{max}^{2}$[\[]{}$\text{GeV}^{2}$[\]]{} 2.02 2.08 2.16 2.22 2.28 2.36 2.42
$m$[\[]{}GeV[\]]{} 1.00 1.00 1.00 1.00 1.00 1.00 1.00
$m^{\prime}$[\[]{}GeV[\]]{} 1.45 1.64 1.52 1.55 1.62 1.50 1.51
$f_{\varphi}$[\[]{}MeV[\]]{} 203 221 210 213 218 203 202
$f_{\varphi^{\prime}}$[\[]{}MeV[\]]{} 215 215 222 226 230 240 246
$f_{1}$[\[]{}$\text{GeV}^{2}$[\]]{} 0.014 0.017 0.015 0.016 0.017 0.014 0.014
$f_{2}[\mbox{GeV}^{2}]$ 0.016 0.016 0.017 0.018 0.018 0.020 0.021
min[\[]{}$\text{GeV}^{4}$[\]]{} $10^{-10}$ $10^{-10}$ $10^{-10}$ $10^{-10}$ $10^{-10}$ $10^{-10}$ $10^{-9}$
: Masses and decay constants of the $\varphi$ ground state and the first radial excitation with $\alpha=0.2$ and $N=20$. \[phi excitation\]
From PDG, the mass and width of the $\varphi$ ground state are $m=1.020$ GeV and $\Gamma=0.004$ GeV while $m^{\prime}=1.68$ GeV, $\Gamma=0.20$ GeV for first radial excitation. The decay constant of ground state was measured to be [@Becirevic2003] $$f_{\varphi}^{\text{exp}}=233\text{MeV}.$$ From Table \[phi ground state\] we have $$m=\left(1.04\pm0.02\right)\text{GeV},f_{\varphi}=\left(229\pm9\right)\text{MeV}.$$ From Table \[phi excitation\] we have $$m^{\prime}=\left(1.54\pm0.07\right)\text{GeV},f_{\varphi}=\left(210\pm8\right)\text{MeV},f_{\varphi^{\prime}}=\left(228\pm11\right)\text{MeV}.$$ The decay constant of the $\varphi$ meson from both fittings agrees with the data very well while the extracted mass of the first radial excitation is in rough agreement with the data.
The nucleon {#sec3nucleon}
===========
The interpolating current for the nucleon is $$\eta=\epsilon^{abc}\left[u^{aT}Cd^{b}\right]\gamma^{5}u^{c}-\epsilon^{abc}\left[u^{aT}C\gamma^{5}d^{b}\right]u^{c}$$ and the resulting sum rule [@Drukarev2009] can be found in the appendix. The usual spectrum representation for the nucleon is $$\rho^{\left(N\right)}\left(s\right)=\beta_{N}^{2}\delta\left(s-m^{2}\right)+\rho_{continuum}\left(s\right)\theta\left(s-s_{0}\right)$$ where $$\begin{aligned}
\rho_{\text{continuum}}\left(s\right) & = & \frac{1}{\pi}\mathrm{Im}\Pi\left(s\right)\nonumber \\
& = & \frac{s^{2}}{4\left(2\pi\right)^{4}}\left(1+\frac{71}{12}\frac{\alpha_{s}}{\pi}-\frac{\alpha_{s}}{\pi}\mathrm{ln}\frac{s}{\mu^{2}}\right)+\frac{1}{\left(2\pi\right)^{2}}\frac{1}{8}\left\langle \frac{\alpha_{s}}{\pi}G^{2}\right\rangle -\frac{2\left\langle \bar{q}q\right\rangle ^{2}}{9}\frac{\alpha_{s}}{\pi}\frac{1}{s}\end{aligned}$$
We also use the modified spectrum representation $$\rho\left(s\right)=\beta_{N}^{2}\delta\left(s-m^{2}\right)+\beta_{N^{\prime}}^{2}\delta\left(s-m^{\prime2}\right)+\rho_{continuum}\left(s\right)\theta\left(s-s_{0}\right),$$ where $\beta_{N}^{2}=32\pi^{4}\lambda_{N}^{2}$, $\beta_{N^{\prime}}^{2}=32\pi^{4}\lambda_{N^{\prime}}^{2}$ and $\lambda_{N}$ is the overlapping amplitude of the interpolating current with the nucleon state.
$s_{0}$[\[]{}$\text{GeV}^{2}$[\]]{} 1.80 1.85 1.90 1.95 2.0
------------------------------------------- ----------- ----------- ----------- ----------- -----------
$M_{min}^{2}$[\[]{}$\text{GeV}^{2}$[\]]{} 0.7 0.7 0.7 0.7 0.7
$M_{max}^{2}$[\[]{}$\text{GeV}^{2}$[\]]{} 1.52 1.54 1.58 1.62 1.64
$m$[\[]{}GeV[\]]{} 0.89 0.91 0.93 0.94 0.96
$\beta_{N}^{2}$ 1.9 2.0 2.1 2.2 2.4
min $10^{-4}$ $10^{-4}$ $10^{-4}$ $10^{-4}$ $10^{-4}$
: The mass of the nucleon ground state with $\alpha_{1}=0.8$, $\alpha_{2}=0.4$ and $N=20$. \[nucleon ground state\]
$s_{0}$[\[]{}$\text{GeV}^{2}$[\]]{} 2.1 2.15 2.20 2.25 2.30 2.35 2.40
------------------------------------------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
$M_{min}^{2}$[\[]{}$\text{GeV}^{2}$[\]]{} 0.83 0.83 0.83 0.83 0.83 0.83 0.83
$M_{max}^{2}$[\[]{}$\text{GeV}^{2}$[\]]{} 1.36 1.38 1.40 1.42 1.44 1.48 1.50
$m$[\[]{}GeV[\]]{} 0.929 0.929 0.929 0.929 0.929 0.929 0.929
$m^{\prime}$[\[]{}GeV[\]]{} 1.45 1.47 1.48 1.50 1.52 1.53 1.55
$\beta_{N}^{2}$ input 2.1 2.1 2.1 2.2 2.2 2.2 2.2
$\beta_{N^{\prime}}^{2}$ 0.67 0.86 1.06 1.28 1.50 1.76 2.00
min $10^{-5}$ $10^{-5}$ $10^{-5}$ $10^{-5}$ $10^{-5}$ $10^{-5}$ $10^{-5}$
: The masses of the nucleon ground state and the first radial excitation with $\alpha_{1}=0.7$, $\alpha_{2}=0.3$ and $N=20$.\[nucleon excitation\]
The results from the first spectrum representation are listed in Table \[nucleon ground state\] and those from the modified spectrum are listed in Table \[nucleon excitation\]. To get stable results, we have used the nucleon mass and $\beta_{N}^{2}=2.1$ from Table \[nucleon ground state\] as input in the numerical analysis of the first radial excitation. From PDG, the nucleon mass is $m=0.938$ GeV while the mass and width of its first radial excitation are $m^{\prime}=1.44$ GeV and $\Gamma^{\prime}=0.300$ GeV. From Table \[nucleon ground state\] we have $$m=\left(0.93\pm0.03\right)\text{GeV}.$$ From Table \[nucleon excitation\] we have $$m^{\prime}=\left(1.50\pm0.04\right)\text{GeV}$$ which is in rough agreement with the data.
Summary {#sec4}
=======
In short summary, we have attempted to extract the masses of the first radial excited states of the light mesons and nucleon. In our modified hadronic spectral density, we explicitly keep the pole of the first radial excited states together with the ground state. Requiring that the operator product expansion converge and the continuum contribution be subleading leads to the optimal working interval of the Borel parameter $M^{2}$. Then a series of “data” points (or pseudo-data points) were produced within this working interval of $M^{2}$. Using the usual one-pole spectral density, we can extract the mass of the ground state with the least square fitting method, which agrees with the experimental data. Then we use these “data” points and the mass of the ground state as input parameters to extract the mass and the decay constant of the first radial excited state by the least square method, which are in good agreement with the available data.
The QCD sum rule method has its inherent accuracy limit due to the various approximations adopted within this framework, such as the truncation of the the OPE series of the correlation function, the assumption of the quark-hadron duality, the omission of the decay width in the spectral density, the factorization of the four quark condensates and the uncertainties of the values of the various condensates etc. In our analysis we only include the uncertainty from the fitting using the least square method itself. The least square method with the modified spectrum representation allows us to extract useful information of the first radial excitations, which depends on the accuracy of the sum rules. It will be very interesting to explore whether such a formalism can be applied to the other hadrons.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
This project is supported by the National Natural Science Foundation of China under Grant No. 11261130311.
QCD sum rules of the light mesons and nucleon {#qcd-sum-rules-of-the-light-mesons-and-nucleon .unnumbered}
=============================================
For the $\rho$ meson $$\begin{aligned}
& \int\mathrm{d}s\,\mathrm{e}^{-\nicefrac{s}{M^{2}}}\rho\left(s\right)=\frac{3}{2}M^{2}\biggl[1+\frac{\alpha_{s}\left(M\right)}{\pi}+\frac{4\pi^{2}\left\langle 0\left|m_{u}\bar{u}u+m_{d}\bar{d}d\right|0\right\rangle }{M^{4}}\nonumber \\
& +\frac{1}{3}\pi^{2}\frac{\left\langle 0\left|\frac{\alpha_{s}}{\pi}G_{\mu\nu}^{a}G_{\mu\nu}^{a}\right|0\right\rangle }{M^{4}}-2\pi^{3}\frac{\left\langle 0\left|\alpha_{s}\left(\bar{u}\gamma_{\alpha}\gamma_{5}t^{a}u-\bar{d}\gamma_{\alpha}\gamma_{5}t^{a}d\right)^{2}\right|0\right\rangle }{M^{6}}\\
& -\frac{4}{9}\pi^{3}\frac{\left\langle 0\left|\alpha_{s}\left(\bar{u}\gamma_{\alpha}t^{a}u+\bar{d}\gamma_{\alpha}t^{a}d\underset{q=u,d,s}{\sum}\bar{q}\gamma_{\alpha}t^{a}q\right)\right|0\right\rangle }{M^{6}}\biggr]\nonumber\end{aligned}$$ For the $\pi$ meson $$\begin{aligned}
& \int\mathrm{e}^{-\nicefrac{s}{M^{2}}}\rho\left(s\right)\mathrm{d}s=\frac{M^{2}}{4\pi}\biggl[1+\frac{\alpha_{s}\left(M\right)}{\pi}+\frac{1}{3}\pi^{2}\frac{\left\langle 0\left|\frac{\alpha_{s}}{\pi}G_{\mu\nu}^{a}G_{\mu\nu}^{a}\right|0\right\rangle }{M^{4}}\nonumber \\
& +\frac{4\pi^{3}\alpha_{s}\left\langle 0\left|\bar{u}\gamma_{\alpha}\gamma_{5}t^{a}d\bar{d}\gamma_{\alpha}\gamma_{5}t^{a}u\right|0\right\rangle }{M^{6}}\\
& -\frac{4}{9}\pi^{3}\alpha_{s}\frac{\left\langle 0\left|\left(\bar{u}\gamma_{\alpha}t^{a}u+\bar{d}\gamma_{\alpha}t^{a}d\underset{q=u,d,s}{\sum}\bar{q}\gamma_{\alpha}t^{a}q\right)\right|0\right\rangle }{M^{6}}\biggr]\nonumber\end{aligned}$$ For the $K^{*}$ meson $$\begin{aligned}
& \int\mathrm{d}s\,\mathrm{e}^{-\nicefrac{s}{M^{2}}}\rho^{\left(K^{*}\right)}\left(s\right)=\frac{M^{2}}{4\pi}\biggl[1+\frac{\alpha_{s}\left(M\right)}{\pi}+\frac{14}{3}\frac{\pi^{2}\left\langle 0\left|m_{u}\bar{u}u+m_{s}\bar{s}s\right|0\right\rangle }{M^{4}}\nonumber \\
& +\frac{1}{3}\pi^{2}\frac{\left\langle 0\left|\frac{\alpha_{s}}{\pi}G_{\mu\nu}^{a}G_{\mu\nu}^{a}\right|0\right\rangle }{M^{4}}-2\pi^{3}\frac{\left\langle 0\left|\alpha_{s}\left(\bar{u}\gamma_{\alpha}\gamma_{5}t^{a}u-\bar{s}\gamma_{\alpha}\gamma_{5}t^{a}s\right)^{2}\right|0\right\rangle }{M^{6}}\\
& -\frac{4}{9}\pi^{3}\frac{\left\langle 0\left|\alpha_{s}\left(\bar{u}\gamma_{\alpha}t^{a}u+\bar{s}\gamma_{\alpha}t^{a}s\underset{q=u,d,s}{\sum}\bar{q}\gamma_{\alpha}t^{a}q\right)\right|0\right\rangle }{M^{6}}\biggr]\nonumber\end{aligned}$$ For the $\varphi$ meson $$\begin{aligned}
& \int\mathrm{e}^{-\nicefrac{s}{M^{2}}}\rho^{\left(\varphi\right)}\left(s\right)\mathrm{d}s=\frac{M^{2}}{36\pi}\biggl[1+\frac{\alpha_{s}\left(M\right)}{\pi}-\frac{6m_{s}^{2}\left(M\right)}{M^{2}}+\frac{8\pi^{2}\left\langle 0\left|m_{s}\bar{s}s\right|0\right\rangle }{M^{4}}\nonumber \\
& +\frac{1}{3}\pi^{2}\frac{\left\langle 0\left|\frac{\alpha_{s}}{\pi}G_{\mu\nu}^{a}G_{\mu\nu}^{a}\right|0\right\rangle }{M^{4}}-\frac{448}{81}\pi^{3}\alpha_{s}\left(\mu\right)\frac{\left\langle 0\left|\bar{q}q\right|0\right\rangle ^{2}}{M^{6}}\biggr]\end{aligned}$$ For the nucleon [@Drukarev2009] $$\begin{aligned}
\tilde{A}_{0}+\tilde{A}_{4}+\tilde{A}_{6}+\tilde{A}_{8} & = &
\beta_{N}^{2}\mathrm{e}^{-\nicefrac{m^{2}}{M^{2}}}+\beta_{N^{\prime}}^{2}\mathrm{e}^{-\nicefrac{m^{\prime2}}{M^{2}}}\end{aligned}$$ where $$\begin{aligned}
\tilde{A}_{0}\left(M^{2},W^{2}\right) & = & M^{6}E_{2}\left[1+\frac{\alpha_{s}}{\pi}\left(\frac{53}{13}-\mathrm{ln}\frac{W^{2}}{\mu^{2}}\right)\right]\\
& & -\frac{\alpha_{s}}{\pi}\left[M^{4}W^{2}\left(1+\frac{3W^{2}}{4M^{2}}\right)\mathrm{e}^{-\frac{W^{2}}{M^{2}}}+M^{6}\varepsilon\left(-\frac{W^{2}}{M^{2}}\right)\right]\\
\tilde{A}_{4}\left(M^{2},W^{2}\right) & = & \frac{bM^{2}E_{0}}{4L}\\
\tilde{A}_{6}\left(M^{2},W^{2}\right) & = &
\frac{4}{3}a^{2}\left[1-\frac{\alpha_{s}}{\pi}\left(\frac{5}{6}+\frac{1}{3}\left(\mathrm{ln}\frac{W^{2}}{\mu^{2}}+\varepsilon\left(-\frac{W^{2}}{M^{2}}\right)\right)\right)\right]\end{aligned}$$
$$a=-\left(2\pi\right)^{2}\left\langle \bar{q}q\right\rangle
,b=\left(2\pi\right)^{2}\left\langle
\frac{\alpha_{s}}{\pi}G^{2}\right\rangle
,\beta_{N}=\left(2\pi\right)^{4}\lambda_{N}^{2},\alpha_{s}\left(1\text{GeV}\right)\approx0.37$$
$$E_{0}=1-\mathrm{e}^{-x},E_{2}=1-\left(1+x+\frac{1}{2}x^{2}\right)\mathrm{e}^{-x}$$ with $x=\nicefrac{W^{2}}{M^{2}}$, $\varepsilon\left(x\right)=\underset{n}{\sum}\frac{x^{n}}{n\cdot
n!}$,
$$L=\frac{\ln\left(\nicefrac{M^{2}}{\Lambda^{2}}\right)}{\ln\left(\nicefrac{\mu^{2}}{\Lambda^{2}}\right)}$$
|
---
abstract: 'The Australian Space Eye is a proposed astronomical telescope based on a CubeSat platform. The Space Eye will exploit the low level of systematic errors achievable with a small space based telescope to enable high accuracy measurements of the optical extragalactic background light and low surface brightness emission around nearby galaxies. This project is also a demonstrator for several technologies with general applicability to astronomical observations from nanosatellites. Space Eye is based around a 90 mm aperture clear aperture all refractive telescope for broadband wide field imaging in the $i''$ and $z''$ bands.'
author:
- Anthony Horton
- Lee Spitler
- Naomi Mathers
- Michael Petkovic
- Douglas Griffin
- Simon Barraclough
- Craig Benson
- Igor Dimitrijevic
- Andrew Lambert
- Anthony Previte
- John Bowen
- Solomon Westerman
- 'Jordi Puig-Suari'
- Sam Reisenfeld
- Jon Lawrence
- Ross Zhelem
- Matthew Colless
- Russell Boyce
bibliography:
- '/home/ajh/Documents/Papers/library.bib'
title: 'The Australian Space Eye: studying the history of galaxy formation with a CubeSat'
---
INTRODUCTION {#sec:intro}
============
In December 2014 the Advanced Instrumentation Technology Centre at the Australian National University hosted the AstroSats 2014 workshop. The workshop was held to explore concepts for astronomical nanosatellite missions, with the goal of identifying viable missions that could be funded within existing Australian grant schemes. Concepts were required to be justifiable by the expected scientific results alone, i.e. without reference to the development of technology or accumulation of expertise, valuable as those outcomes would be. Additionally proposed spacecraft were to be no larger than a CubeSat. The Australian Space Eye was conceived in response to this call for proposals.
Meeting the combination of scientific value, cost and size constraints is very challenging, essentially it means identifying a compelling science programme that is within the capabilities of a very small instrument in low earth orbit but which could not be done with (much larger) ground based instruments for comparable cost. The two most well known benefits of situating an astronomical telescope above the Earth’s atmosphere are accessing wavelengths that are absorbed by the Earth’s atmosphere and avoiding the degradation of spatial resolution caused by atmospheric turbulence. Neither of these are of great relevance to an astronomical CubeSat however, at least one conceived within the AstroSats constraints. The gamma ray, X-ray, far ultraviolet and infrared wavelength regions all require exotic optics and/or image sensors which are not compatible with the cost, volume, thermal and power limitations of the platform. For this reason we limited our consideration to the near ultravoilet to very near infrared wavelength region ($\sim200$–) that is accessible with silicon based image sensors and conventional optics. Within this wavelength range there is no spatial resolution advantage to being above the Earth’s atmosphere either, the fundamental diffraction limit of a CubeSat sized telescope aperture ($\lesssim\SI{90}{\milli\metre}$ diameter) is greater than atmospheric seeing at the same wavelengths.
There are however two other effects of the atmosphere which are relevant to a CubeSat based optical telescope: atmospheric scattering and emission. Atmospheric scattering spreads the light from astronomical objects in a similar way to scattering from the instrument’s optics however the impact is compounded by the fact that the distribution of aerosols in the atmosphere is spatially and temporally variable, the amount of scattered light surrounding a given source can vary by several percent of the source brightness even in good ‘photometric’ conditions, on timescales of minutes to months. The variability in the scattering makes it difficult to accurately characterise and subtract [@McGraw2010], thereby introducing problematic systematic errors. Similar issues arise from atmospheric emission, particularly at longer wavelengths ($>\SI{700}{\nano\metre}$). Here the atmosphere glows increasingly brightly, primarily due to line emission from OH$^*$ molecules in the mesosphere, reducing the sensitivity of ground based telescopes. This emission is sensitive to dynamic processes in the upper atmosphere (e.g. gravity waves) and consequenty it is also spatially and temporally variable on timescales of minutes and longer [@Moreels2008] which makes accurate subtraction difficult. As a result of these effects locating a telescope in space not only helpfully reduces both scattered light and sky background levels but crucially makes both far more stable. Above the atmosphere the only sources of scattered light are associated with the instrument itself and the dominant source of sky background is the zodiacal light [@Leinert1998] which, while it does exhibit large scale spatial and seasonal variations, is much less variable, more uniform and more predictable than atmospheric emission.
The scientific competitiveness of small telescope for certain types of observations is well proven. For example while large astronomical telescopes are able to detect extremely faint compact or point-like objects their sensitivity to diffuse emission is limited by systematic errors from a number of sources. A significant contribution to these systematic errors is contamination with light from brighter objects within/near the instrument’s field of view, caused by a combination of diffraction, scattered light and internal reflections. The difficulties caused by these effects have been discussed by, for example, Sandin [@Sandin2014; @Sandin2015] and by Duc et al [@Duc2014], who noted that the simpler optics of small telescopes suffer less from internal reflections. In addition small telescopes can more readily be constructed with unobscured apertures to minimise diffraction, and simpler optics also make them less prone to internal scattering of light. The resulting competitiveness of optimised small telescopes in the context of ‘low surface brightness’ (LSB) imaging has been demonstrated by the Dragonfly Telephoto Array, an astronomical imaging system based on an array of telephoto camera lenses [@Abraham2014]. This instrument exhibits less diffraction/scattering/internal reflections than other telescopes for which comparable data are available [@Abraham2014; @Sandin2014] and the system has produced impressive results, including the discovery of a new class of ‘ultra-diffuse galaxies’[@VanDokkum2015]. Two of this paper’s authors (AJH & LS) are currently assembling a ground based instrument based on the same principles, the Huntsman Telephoto Array[@Horton2016].
We conclude that a CubeSat based astronomical telescope may be scientifically competitive, especially an optimised telescope performing measurements that are typically limited by systematic errors. We identify the 700– wavelength region as particularly promising as it is accessible to instrumentation compatible with the constraints of a CubeSat mission while ground based instruments would be hampered by bright atmospheric emission, in terms of both reduced sensitivity and increased systematic errors. We have based the Australian Space Eye proposal on two science goals which would be well served by a telescope operating in this regime, the measurement of extragalactic background light and imaging of low surface brightness structures around nearby galaxies. These are discussed in more detail in the next section.
SCIENCE CASE {#sec:aims}
============
Extragalactic Background Light
------------------------------
A fundamental measurement of the universe is its total luminosity, which contains the entire history of all sources of light, from signatures of the Big Bang in the cosmic microwave background and, at optical wavelengths, all past radiative processes including light from stars and even black hole accretion disks. Although the microwave background has been securely measured the cosmic optical background (the total luminosity at optical wavelengths) has not due to our non-ideal vantage point - we are embedded in the dust cloud of the solar system. This dust scatters sunlight into the observer’s telescope and so we must separate the background light we want to measure from the light scattered by this foreground dust, the so-called Zodiacal light. This has proved very challenging (see Cooray 2016[@Cooray2016]), though there have been a few attempts (e.g. Bernstein 2007[@Bernstein2007], Aharonian et al. 2006[@Aharonian2006] and Tsumura et al. 2010[@Tsumura2010]). What is needed is an instrument that can measure and separate the Zodiacal and extragalatic background light.
One approach for disentangling the Zodiacal and extragalactic light exploits spectral features of the former. As Zodiacal light is scattered sunlight its spectrum is similar to that of the Sun, subject to reddening due to the wavelength dependence of the scattering strength[@Giavalisco2002; @Aldering2001; @Leinert1998]. This includes the strong Calcium triplet absorption lines near , which will be present in the Zodiacal light spectrum with the same equivalent width as in the solar spectrum but will be absent from the spectrum of the extragalactic background due to the effects of cosmological redshifts. Consequently the observed equivalent width of absorption lines in the total background light (Zodiacal plus extragalactic) provided a measure of the relative contributions of the two components. This was the goal of one of the instruments of the CIBER sounding rocket experiment[@Zemcov2013], a narrowband imaging spectrometer based on a tilted objective filter[@Korngut2013], however the measurement proved extremely difficult to make with the limited total exposure time of a sounding rocket campaign. An orbital space telescope, able to accumulate a year or more of observing time, promises secure measurements of both the absolute total sky brightness and the relative contributions of the Zodiacal light and extragalactic background.
Low Surface Brightness Galaxies
-------------------------------
A new ground-based imaging system called the Dragonfly Telephoto Array[@Abraham2014] uses commercial-off-the-shelf camera lenses to reach $16\times$ (3 magnitudes) fainter surface brightness levels than existing telescopes at optical wavelengths (0.4–, $g'$ and $r'$ bands). The key innovation is the use of unobscured refracting optics, which significantly reduce scattered and diffracted light compared to conventional large reflecting telescopes and bring the limiting systematic uncertainties down to (32 AB mag./arcsecond$^2$ in $g'$)[@Sandin2014]. The Dragonfly observing system has led to a number of important results, including finding an entirely new class of galaxy (e.g. van Dokkum, Abraham, Merritt 2014[@VanDokkum2014], van Dokkum et al. 2015[@VanDokkum2015]).
By adapting the Dragonfly concept to observing galaxies at longer wavelengths of light from space we can constrain the stellar population properties of these galaxies. The Australian Space Eye will target several nearby galaxies in order to obtain images at the lowest possible surface brightness levels and detect extremely faint structures in their outskirts. When the Space Eye $i'$ and $z'$ band imaging is combined with optical $g'$ and $r'$ band imaging from the Dragonfly and Huntsman[@Horton2016] ground-based observing facilities we will have valuable information about the stellar age and chemical content of these faint structures.
SPACE EYE CONCEPT
=================
The Australian Space Eye is a nano-satellite based on the CubeSat form factor standard[@Hevner2011] housing an optical imaging telescope. The specifications of the system are driven by the aims discussed in Sec. \[sec:aims\], subject to the constraints discussed in Sec. \[sec:intro\]. An artist’s impression of Space Eye is shown in Fig. \[fig:render\].
![\[fig:render\]Artist’s impression of the Australian Space Eye in orbit over the Tasman Sea.](figures/ASE-2016-a.png){width="\textwidth"}
Optical Payload
---------------
### Requirements {#sec:payloadreqs}
The scientific goals described in Sec. \[sec:aims\] call for a wide field imaging instrument with moderate spatial resolution, an exceptionally ‘clean’ and stable point spread function (PSF), and the highest low light sensitivity achievable within the other constraints of the platform. The imager must be capable of both broadband imaging in the $i'$ and $z'$ bands (approximately 700– and 850– respectively) as well as measurement of the strength of the Calcium absorption lines in the sky background spectrum.
The optical system is expected to occupy approximately 50% of the spacecraft internal volume. The optical axis will be aligned with the long axis of the spacecraft body and positioned on the centreline to assist with centre of mass positioning. The bulk of the optical payload is therefore confined to an approximately $300 \times 100 \times \SI{100}{\milli\metre}$ volume, however 1– of the remaining of internal volume will be available for payload electronics. The largest practical optical aperture within these constraints is $\sim\SI{90}{\milli\metre}$ in diameter.
Considerations of cosmic variance, sky background gradients and the angular extents of galaxy groups/clusters result in a desire for a field of view at least across, preferably close to .
We have selected a image scale of , this is close to Nyquist sampling of the diffraction limited PSF at the Space Eye aperture size and operating wavelengths ($1.22 \lambda / D = \SI{2.0}{\arcsecond}$–). Coarser spatial sampling could increase the signal to noise ratio for diffuse sources however we are wary of significantly undersampling the PSF due to the potential impact on the accuracy of PSF fitting and subsequent point source subtraction. We will show that Space Eye can still be sky background noise limited at this pixel scale. is also a good match to the pixel scale of ground based low surface brightness imaging facilities such as the Dragonfly Telephoto Array[@Abraham2014] and Huntsman Telephoto Array[@Horton2016].
[|l|c|]{}
------------------------------------------------------------------------
Field of view & $>\SI{1}{\degree}$ (goal $\sim\SI{2}{\degree}$)\
------------------------------------------------------------------------
Pixel scale &\
------------------------------------------------------------------------
Wavelength range & 700–\
------------------------------------------------------------------------
Aperture diameter &\
------------------------------------------------------------------------
Optical system dimensions & $300 \times 100 \times \SI{100}{\milli\metre}$ max\
------------------------------------------------------------------------
Payload electronics volume & $<\SI{2}{U}$\
To accomplish the required measurements of both broadband $i'$ and $z'$ surface brightness and Calcium absorption we proposed to use a set of 6 slightly modified $i'$ and $z'$ band filters with band edges positioned around the strong absoprtion line at . More details are discussed in Sec. \[sec:mosaic\].
For Space Eye the requirement for a clean and stable PSF translates to minimising all sources of stray light, including surface and bulk scattering, internal reflections, stray sunlight/Earthshine/moonlight and diffraction.
The highest possible sensitivity will be achieved when the optical aperture is as large as it can be within the space constraints of the CubeSat, the optical throughput and quantum efficiency (QE) of the image sensor are close to 100%, and the total instrumental noise is below the Poisson noise from the sky background light.
### Image sensor
![\[fig:cis115\]CIS-115 CMOS image sensor, photo credit e2v.](figures/cis115.png){width="50.00000%"}
The choice of image sensor is key, not only is it the biggest variable in determining the overall sensitivity of the system but the specifications for many of the other subsystems (optics, data handling, downlink capacity, thermal control, power, etc.) depend on the specifications of the sensor.
Astronomical instruments operating at these wavelengths typically use charge-coupled device (CCD) image sensors however for Space Eye we have selected the CIS115 CMOS image sensor from e2v[@Jorden2014]. The CIS115 is a back-side illuminated image sensor with $2000 \times 1504$ pixels on a pitch. The main specifications, taken from the draft e2v datasheet, are given in Tab. \[tab:cis115\].
[|l|c|]{}
------------------------------------------------------------------------
Number of pixels & $2000 \times 1504$\
------------------------------------------------------------------------
Pixel size & square\
------------------------------------------------------------------------
Quantum efficiency at &\
------------------------------------------------------------------------
Dark current & at\
------------------------------------------------------------------------
Read noise & at per channel\
------------------------------------------------------------------------
Well depth & (linear), (saturation)\
------------------------------------------------------------------------
Non-linearity & $\pm\SI{4}{\percent}$\
------------------------------------------------------------------------
Operating temperature & –+\
------------------------------------------------------------------------
Power consumption & $\sim\SI{40}{\milli\watt}$\
The CIS115 is a new image sensor developed with space based scientific imaging in mind, in particular the ESA JUICE mission. In general its performance is close to that of back side illuminated CCD image sensors operated in inverted mode, the main difference being a 2–$3\times$ higher read noise when compared to the best CCDs. CMOS image sensors offer a number of advantages for a CubeSat space telescope however, including lower power consumption and greater resistance to radiation induced damage than CCDs. Of particular importance to Space Eye is the relatively small pixel size, versus the 12– of available scientific CCDs. This factor of 2 reduction in pixel size enables a corresponding reduction in the focal length of the optics required for a given on-sky pixel scale which has a significant impact on the design of the optics as discussed in the next section. An additional advantage is the ability to operate regions of the sensor independently of others, opening up the possibility of using small regions for high frame rate autoguiding/star tracking while simultaneously taking a long exposure with the rest of the sensor (‘on-chip guiding’).
![\[fig:qe\]Predicted CIS115 quantum efficiency as a function of wavelength at both and , data from Soman et al.[@Soman2014]](figures/CIS115QE.pdf){width="80.00000%"}
Predicted quantum efficiency data for the CIS115 are plotted in Fig. \[fig:qe\]. These data are reproduced from Soman et al.[@Soman2014] and we regard them as conservative estimates, Wang et al.[@Wang2014] report slightly higher peak QE from both their own and e2v’s measurements of a prototype device (CIS107). The decline in QE for wavelengths greater than is due to the increasing transparency of silicon at these wavelengths and is common to all silicon based image sensors. In this wavelength range only deep depletion or high rho CCDs offer significantly higher QE, due to the greater effective thickness of their photosensitive layer. We consider these devices unsuitable for Space Eye however as those currently available have large pixels and, more importantly, require deep cooling ($<\SI{-100}{\celsius}$) to control dark current due to their non-inverted operation. Deep depletion/high rho CMOS image sensors show promise however at the time of writing no devices suitable for Space Eye were known to the authors.
![\[fig:dc\]CIS115 dark current as a function of temperature, based on the model from Wang et al.[@Wang2014]](figures/CIS115DC.pdf){width="80.00000%"}
The dark current of the CIS107 prototype was measured by Wang et al. between + and and they found the expected exponential temperature dependence with a halving of dark current for every drop. Their best fit model is shown in Fig. \[fig:dc\]. The dark current of cooled production CIS115 image sensors may be somewhat lower, e2v’s draft datasheet suggests a typical dark current that is $\sim\SI{50}{\percent}$ lower at that also falls more rapidly with cooling, halving with each 5.5– temperature drop. Based on these data we confidently expect a dark current of $\lesssim\SI{0.04}{\el\per\pix\per\second}$ at . Wang et al. also note a helpful reduction in read noise on cooling, to approximately .
### Telescope optics
[|l|c|]{}
------------------------------------------------------------------------
Field of view & $\SI{1.67}{\degree} \times \SI{1.25}{\degree}$\
------------------------------------------------------------------------
Effective focal length &\
------------------------------------------------------------------------
Aperture diameter &\
------------------------------------------------------------------------
Focal ratio & $f/5.34$\
------------------------------------------------------------------------
Wavelength range & 700–\
------------------------------------------------------------------------
Overall length &\
![\[fig:optics\]Cross section optical layout diagram of the baseline design for the Australian Space Eye.](figures/spie-layout.png){width="80.00000%"}
![\[fig:ensquared\]Ensquared energy as a function of half-width for the baseline optical design.](figures/ensquared.png){width="80.00000%"}
The combination of the requirements from Tab. \[tab:payload\] and the image sensor specifications from Tab. \[tab:cis115\] enable us to determine the remaining requirements for the telescope optics. The resulting specifications are summarised in Tab. \[tab:optics\]. The biggest consideration for the design of the Space Eye optics is minimising the wings of the PSF due to internal scattering, reflections and diffraction. For this reason we have decided to use an all refractive design. Abraham and van Dokkum[@Abraham2014] have argued that refractive designs have a fundamental advantage over reflecting or catadioptric telescopes in this respect, and comparisons of telescope PSF measurements by Sandin[@Sandin2014] appear to support this view. Note that the relatively small pixels of the CIS115 image sensor are essential to allow a refractive design to be used. With and pixels the required effective focal length is about twice the length of the space available for the optics, necessitating a moderately telephoto design but not presenting any insuperable problems. With the larger pixels typical of back side illuminated CCDs the focal length would have to be approximately four times the length of the available space, in which case folded light path reflective or catadioptric designs would be the only practical options.
We have produced a baseline optical design for Space Eye which is illustrated in figure \[fig:optics\]. The design consists of 6 elements in 4 groups with two CaF$_2$ elements (L2 and L3) and two aspheric surfaces (L3-S1 and L4-S1). Six elements were required to achieve the desired image quality within the overall length constraints however we are able to limit the number of vacuum-glass interfaces to 7 which will assist with minimising surface scattering and internal reflections/ghosting. The image quality is essentially diffraction limited across the full field of view, as confirmed by the ensquared energy plots in Fig. \[fig:ensquared\]. The calculation was performed using the Huygens-Fresnel method with the Zemax OpticStudio software. The ensquared energies are polychromatic (700–) and have been calculated for 5 field points at a range of positions from the centre to the corner of the field of view.
Bulk scattering will be minimised through the use of high purity, high homogeneity lens blanks. Super polishing and high performance anti-reflection coatings will be used to minimise surface scattering and internal reflections. We are pursuing the possibility of using nano-structured anti-reflection coatings (as employed in some high end DSLR lenses) for this purpose, these are capable of very low reflectivities for a wide range of wavelengths and angles of incidence. Internal knife-edge baffles will be used, along with an external deployable baffle designed to prevent illumination of the optics by sunlight, Earthshine or moonlight during observations. A 700– bandpass filter will be applied to the 1st lens surface. A full stray light analysis will be performed as part of a general review of the optical design when the project is confirmed however we are confident that we will be able to acheive the required clean and stable PSF.
The telescope will include a bistable optical shutter between the L5 and L6 elements to enable on orbit image sensor dark current measurements.
### Mosaic filter {#sec:mosaic}
In order to provide the necessary data to allow both broadband $i'$ and $z'$ band imagery and seperate measurements of the Zodiacal light and extragalactic components of the sky background we propose the use of 6 broadband filters, 3 variants on the $i'$ filter and 3 variants on the $z'$ filter. Nominal transmission profiles for these filters are shown in Fig. \[fig:filters\], together with a model of the Zodiacal light photon spectral flux density. The filters differ in their red cutoff wavelength in the case of the $i'$ filters and their blue cutoff wavelength in the case of the $z'$ filters. The cutoffs are chosen to bracket the strongest of the Calcium triplet absorption lines () as well as an adjacent region of continuum. The pairwise surface brightness differences between filters provide information on the relative strength of the Calcium absorption that can used together with sky background models to separate the Zodiacal light and extragalactic components. Meanwhile the sum of the 3 $i'$ or $z'$ band filters gives deep broadband data with effective filter response close to the standard $i'$ or $z'$ filter bands.
Multiband astronomical imaging is typically accomplished by taking full frame images through each individual filter sequentially, i.e. temporal multiplexing. This approach requires a set of full frame filters and some sort of filter exchange mechanism, e.g. a filter wheel. For space based instruments, and especially those intended for nanosatellites, there are strong incentives to avoid introducing mechanisms if at all possible due to complexity, cost and potential for failure. Consequently we favour a spatial multiplexing based approach in which the field of view is divided up amongst the 6 different filters by a fixed mosaic filter at the focal plane, an example layout of which can be seen in Fig. \[fig:mosaic\]. By taking sequences of 6 images with pointing offsets equal to the dimensions of the filter regions contiguous images can be built up for each filter.
![\[fig:mosaic\]Example mosaic filter layout. Each filter covers an area of $333 \times \SI{376}{\pix}$, $\SI{16.7}{\arcminute} \times \SI{18.8}{\arcminute}$ on sky.](figures/mosaic.pdf){width="40.00000%"}
![\[fig:filters\]Nominal filter transmission profiles shown together with a model of the Zodiacal light photon spectral flux density.](figures/filters.pdf){width="80.00000%"}
### Performance modelling {#sec:perf}
In order to determine appropriate operational parameters and predict the approximate sensitivity of Space Eye we have developed performance models for the optical payload. These are not full end-to-end simulations but are instead parametric models, suitable for efficiently exploring parameter space ahead of the more detailed analysis to follow.
As noted in Sec. \[sec:payloadreqs\] obtaining the maximum sensitivty for a fixed telescope aperture size requires both maximising the end-to-end efficiency (optical throughput and image sensor QE) and ensuring the instrumental noise sources do not add significantly to the fundamental Poisson noise of the light received from the sky, which at these wavelengths is predominantly Zodiacal light. For the purpose of these calculations we use a model for the Zodiacal light based on that used by the Hubble Space Telescope Exposure Time Calculator[@Giavalisco2002]. The starting point is a solar spectrum from Colina, Bohlin and Castelli[@Colina1996], to which we apply a normalisation, reddening and spatial dependency following the prescription of Leinert et al.[@Leinert1998] with the revised parameters from Aldering[@Aldering2001]. Using the aperture size, estimated optical throughput, filter transmission profiles, pixel scale and image sensor quantum efficiency we can predict the observed Zodiacal light signal (in photo-electrons per second per pixel) and then, with the estimated image sensor dark current and read noise values, we can predict the signal to noise ratio (SNR). By comparing the predicted SNR with the value it would have in the absence of dark current and read noise we can quantify the degree to which instrumental noise is effecting sensitivity. The key results are summarised in Fig. \[fig:relsnr\], which shows the SNR relative to the SNR without instrumental noise for the $i2$ and $z2$ filters for a range of image sensor temperatures and sub-exposure time.
![\[fig:relsnr\]Predicted SNR relative to the SNR without instrumental noise for the $i2$ and $z2$ filters as a function sub-exposure time for a range of image sensor temperatures. The calculation uses the ecliptic pole Zodical light surface brightness values.](figures/relsnr.pdf){width="80.00000%"}
The calculation uses the Zodiacal light model surface brightness values for the eclipitc poles, which are close to the minimum possible value and therefore place the most stringent demands on instrumental noise. Due primarily to the falling image sensor QE the $z'$ band is more sensitive to instrumental noise and drives the selection of operating parameters. Based on these results we have selected a nominal exposure time for individual science exposures of and an image sensor operating temperature of .
Using these parameters we then calculate the predicted sensitivity limit for Space Eye, specifically the source surface brightness spectral flux density that would correspond to a signal to noise of 1 per pixel, for each filter. This is plotted as a function of total exposure time in Fig. \[fig:sens\], with a horizontal scale that runs from (i.e. a single exposure) to . The latter value corresponds to the approximate total exposure time per filter per celestial hemisphere for a 2 year duration mission, assuming the duty cycle discussed in Sec. \[sec:obsconst\]. Of more relevance to the broadband imaging science goals, we have also calculated the predicted sensitivity limit from summing the 3 $i'$ and $z'$ filters. At the end of the 2 year mission we predict ultimate $1\sigma$ per pixel sensitivity limits of and for $i'$ and $z'$ bands respectively, equivalent to 30.5 and 29.6 in AB magnitude surface brightness units.
![\[fig:sens\]Predicted surface brightness spectral flux density corresponding to a signal to noise ratio of 1 per pixel as a function of total exposure time per filter. The calculcation is for the ecliptic pole Zodiacal light surface brightness, an image sensor temperature of and sub-exposures of .](figures/sens.pdf){width="80.00000%"}
The Zodiacal light model includes spatial (and seasonal) variations, allowing us to calculate how the predicted sensitivity varies with position on the sky. For extragalactic sources, such as other galaxies and the extragalactic background light, we must also take into account extinction by Galactic dust. For this we use the all sky dust reddening (E(B-V)) maps from the Planck Legacy Archive[@Abergel2014] and convert to extinctions at our filter wavlengths using the prescription of Fitzpatrick[@Fitzpatrick1999], with $R=3.1$. Fig \[fig:sensmap\] shows the resulting $i'$ band relative sensitivity maps for regions near the ecliptic poles at the times of the equinoxes and solstices. Due to the seasonal variation in Zodiacal light regions close to the north ecliptic pole are preferred from February to July, and close to the south ecliptic pole from August to January. The tightest constraints come the Galactic dust, in order to minimise its effects target should be chosen from within regions between 15.5 and 16.5 hours Right Ascension and +55 and +60 degrees declination in the north, and between 4 and 5 hours Right Ascension and -50 and -60 degrees declination in the south.
![\[fig:sensmap\]Predicted $i'$ band sensitivity relative to the ecliptic pole, zero extinction case. The maps show $\SI{45}{\degree}\times\SI{45}{\degree}$ gnomonic projections centred on the north and south ecliptic poles in March, June, September and December, with a colourmap running from halved sensitivity (black) to full sensitivity (white). Equatorial coordinate grids are also shown, with 1 hour spacing in RA and in dec, as well as the outlines of the designated target regions.](figures/i_rel_sens.pdf){width="53.50000%"}
Spacecraft Bus
--------------
The Space Eye spacecraft bus will be based on the Tyvak Endeavour platform. This highly integrated, high performance platform incorporates almost all the systems required for a functional 3– Cubesat, including Command and Data Handling (C & DH), Electrical Power System (EPS) and thermal management, Attitude Determination, Control and Navigation System (ADCNS), and structural and mechanical parts. In a typical configuration the avionics package, including battery modules, occupies approximately of volume. As noted in Sec. \[sec:payloadreqs\] the Space Eye telescope will occupy a volume leaving a final $\sim\SI{2}{U}$ of volume for other mission specific hardware, e.g. the image sensor control electronics, image sensor thermal control system (Sec. \[sec:thermal\]), image stabilisation system/ADCS 2nd stage (Sec. \[sec:adcs\]) and communication equipment.
An initial power budget analysis has indicated that Space Eye will require body mounted solar panels on the largest face of the spacecraft body plus wide deployable panels along both long edges, as shown schematically in figure \[fig:render\]. Based on a preliminary analysis we believe that standard UHF telemetry/command and S-band downlink systems will be sufficient given the expected data rates (see section \[sec:conops\])[@Reisenfeld2015].
### Attitude Determination & Control {#sec:adcs}
![\[fig:adcs\]Main components of the standard Tyvak Endeavour attitude determination and control system (ADCS). Space Eye’s ADCS will be based on an updated Endeavour system.](figures/adcs.png){width="70.00000%"}
The main technical challenge for astronomical imaging from CubeSats is instrument pointing stability. Long exposures are required to prevent image sensor noise overwhelming the faint signals from the sky (see Sec. \[sec:perf\]) and the instrument must be kept stable to within less than 1 pixel for the duration of the exposures to avoid blurring. For Space Eye the required exposure times are and corresponds to , these requirements are well beyond the capabilities of any current commercially available CubeSat ADCS system. Improvements are required in both the attitude determination and attitude control aspects.
The standard Tyvak Endeavour main attitude determination system is based on a set of two orthogonal Tyvak-developed star trackers and a three-axis MEMS gyro. These sensors are paired with a set of three Tyvak reaction wheels as primary attitude control actuators. The system also incorporates three magnetic torque coils for wheel desaturation and a set of sun sensors and magnetometers to provide coarse attitude determination. Attitude determination and control computations are performed on a dedicated processor on the Endeavor main board. Pointing stability of the Endeavour platform is primarily driven by rate noise from the gyroscope in the attitude control loop. Disturbances from the reaction wheels also contribute jitter to a lesser extent; however, reaction wheel induced jitter can be mitigated with placement and isolation. Modest software and firmware updates to the current Endeavour platform achieve () stability in simulation.
Tyvak has investigated several paths to achieve arcsecond level pointing stability with the Endeavour platform. The highest technology readiness level path for arcsecond level pointing of an imaging payload is a two-stage control system with piezoelectric actuators driving lateral translation of the focal plane assembly (FPA) in a high bandwidth second stage. The two-stage approach is well suited to CubeSats since it is orbit agnostic and CubeSat rideshares generally cannot select their orbit.
The piezoelectric second stage consists of the FPA (image sensor, interface board, mosaic filter and field flatenner lens), piezoelectric actuators driving lateral translation of the FPA, a dedicated processor for calculating guide star centroids and performing control calculations, and the electrical and power system for the piezoelectric actuators.
Space Eye will use fine star tracking (autoguiding) in the main telescope focal plane to provide the precision pointing information required. Dedicated sensors adjacent to the main image sensor could be used however the CIS115 image sensor and its control electronics are capable of continuously reading out sub-regions of the image sensor for fine star tracking while the remainder of the sensor is simultaneously doing a long science exposure. A fraction of the science image is lost in this way however avoiding a dedicated set of fine star tracking sensors and associated control electronics reduces complexity considerably.
Preliminary design for the dual-stage controller indicates the control bandwidth of the piezoelectric second stage needs to be greater than with centroiding noise of less than . Control inputs (centroids from guide stars) need to be computed at roughly and with delay of no more than roughly .
Specific piezoelectric actuators have not been analyzed however the required actuator throw is within COTS hardware capability. Significant work remains to refine guide star sensor selection, actuator selection, packaging, and the algorithms to drive the second stage control system.
### Thermal control {#sec:thermal}
[m[0.45]{} m[0.45]{}]{} ![\[fig:thermalpics\]Configuration of the spacecraft showing the deployed thermal IR radiator baffle (left), and spacecraft attitude and solar illumination during (northern) summer solstice in a 14:00 LTAN orbit without Earth IR avoidance manoeuvres outside of the observation windows (right).](figures/radiator.png "fig:"){width="45.00000%"} & ![\[fig:thermalpics\]Configuration of the spacecraft showing the deployed thermal IR radiator baffle (left), and spacecraft attitude and solar illumination during (northern) summer solstice in a 14:00 LTAN orbit without Earth IR avoidance manoeuvres outside of the observation windows (right).](figures/orbit.png "fig:"){width="45.00000%"}
![\[fig:thermalplots\] Predicted image sensor and thermal radiator temperature variations during several orbits with a thermal IR baffle, both with and without Earth IR avoidance manoeuvres outside of the observation window. When Earth IR avoidance manoeuvres are used the image sensor temperature remains below the nominal operating temperature throughout the observation windows.](figures/thermal.pdf){width="80.00000%"}
Cooling the image sensor below the temperature where the dark current is a negligible contributor to the system radiometric noise budget (see Sec. \[sec:perf\]) is both critical to the scientific performance of the mission as well as a very difficult engineering challenge within the resource constraints of a CubeSat spacecraft. In order to address the feasibility of achieving this requirement, a number of configurations of the spacecraft, attitude control strategies and Concept of Operations need to be assessed.
In order to cool the detector over an extended period of time a radiator needs to be integrated onto an external face of the spacecraft to reject heat to deep space. Through the orbit the radiator will in general be in radiative exchange with deep space, the Earth, the solar heat load from the direct view of the radiator to the Sun as well as the solar energy reflected from the Earth to the radiator (i.e. the Albedo heat load). The overall efficacy of a particular radiator design will then depend on the area, thermal conductivity and thermo-optical properties of the radiator as well as the relative geometry of the radiator surface to the Earth, Sun and any other appendages of the spacecraft. Ultimately, these geometric factors depend on the accommodation of the subsystems within the spacecraft, the attitude of the spacecraft, the season and the orbital parameters of the spacecraft and the location of the spacecraft in its orbit.
The most favorable orbit, from the point of view of thermal control of the detector radiator, is dawn-dusk Sun synchronous. In this orbit the detector radiator can be placed on the opposite side of the spacecraft to the main solar array. has a very good view factor towards deep space and is protected from direct solar illumination. There are several drawbacks of baselining this orbit which makes it unattractive from the mission level perspective. Firstly, only a relatively small minority of CubeSat launches are dawn-dusk and therefore adopting it involves a risk of incurring a significant programmatic delay to wait for a suitable launch opportunity. Secondly, due to the fact that the sequencing of the deployments of the primary and secondary payloads is planned to optimize the orbital parameters of the primary payload, there is generally an injection error of the CubeSat with respect to a fully sun synchronous orbit. Finally, without a propulsion system, as the orbit decays it will drift further and further away from the ideal orbit and compromise the thermal performance. In the light of these considerations it was decided to abandon the dawn-dusk sun-synchronous orbit approach and consider Sun-synchronous orbits with a Local time of the Ascending Node (LTAN) closer to local noon.
A Thermal Mathematical Model (TMM) of the spacecraft and orbit was built using ESATAN TMS. For the purpose of the analysis we assumed a Sun-synchronous circular orbit with a 96.7 minute period and LTAN of 14:00. The initial analysis showed that a $\SI{100}{\milli\metre} \times \SI{100}{\milli\metre}$ radiator on the anti-Sun facing surface of the spacecraft would not provide sufficient cooling to the image sensor due to the thermal IR load from the Earth (and to a lesser extent the solar albedo load). In order to improve the efficiency of the radiator a deployable Thermal IR baffle was modelled (see Fig. \[fig:thermalpics\]). The inside faces of this baffle are to be coated with a low-emissivity coating (for example, vacuum deposited aluminum) to improve the effective view factor of the radiator to deep space while blocking views to Earth. In order to avoid interference with the spacecraft Attitude Determination and Control System (ADCS) the stiffness and first eigenmode of the deployed baffle needs to be outside of the control bandwidth of the fine pointing system.
The TMM indicated that although the thermal IR baffle improves the performance of the radiator, there are periods of the orbit where the radiator is exposed to the Earth and does not provide enough cooling to the image sensor. Fortunately, these periods occur outside the observation windows and therefore the spacecraft is able to slew the radiator away from the Earth without significantly compromising the power generating capacity of the solar arrays. This can be seen in Fig. \[fig:thermalplots\], where the image sensor temperatures are plotted during several orbits with and without the Earth IR avoidance manoeuvres. The design and analysis carried out on the image sensor cooling system shows the fundamental feasibility for this aspect of the mission.
Concept of Operations {#sec:conops}
=====================
The concept of operations for Space Eye is very much a work in progress at the time of writing, however some aspects have been worked out. The science aims of the project require the telescope to obtain repeated long exposure images of a small number of target fields for as long as possible.
Target fields
-------------
The exact positions of the target fields will be chosen based on a number of factors, including the positions of suitable guide stars, avoidance of very bright stars and the locations of galaxies and galaxy groups. It is possible to constrain the general location of the target fields based on sensitivity and systematic error considerations, however. Sensitivity models for the baseline design which include the effects of Zodiacal light and Galactic dust show that the preferred regions of sky are those close to the North ecliptic pole from February to July and close to the South ecliptic pole from August to January, and that the tightest constraints come from Galactic dust (see Sec. \[sec:perf\]. We can use relative sensitivity maps (Fig. \[fig:sensmap\]) to narrow down the optimum target field positions to between 15.5 and 16.5 hours Right Ascension and +55 and +60 degrees declination in the north and between 4 and 5 hours Right Ascension and -50 and -60 degrees declination in the south. Within each of these regions we expect to select 1 or 2 target fields, and each region will be observed for approximately 6 months before switching to the other region.
Operational requirements
------------------------
In order to observe both northern and southern target fields under favourable Zodiacal light conditions we require an operational lifetime of at least 1 year, with a goal of 2 years or more. Based on this lifetime requirement and the results our initial power, thermal and communications budget analyses we have selected a nominal circular Sun-synchronous orbit at altitude, period and LTAN of 14:00.
Observing constraints {#sec:obsconst}
---------------------
Useful scientific data can only be acquired when certain constraints are met. These include image sensor temperature () and minimum angular separations of the target field from the Earth’s disc, the Sun and the Moon (, and respectively). Preliminary analyses indicate that for the nominal orbit and approximate target field positions approximately 3–4 science exposures will be possible during each orbit.
Observing sequence
------------------
The nominal sequence of operations for science observations is as follows:
1. [Slew spacecraft to target field, arcminute level pointing accuracy is sufficient. Roll angle is chosen to orientate the main solar array towards the Sun]{}
2. [Take a short exposure with the main image sensor to locate positions of pre-selected guide stars (2-3) on the image sensor, define regions-of-interest (RoIs) around them.]{}
3. [Begin continuous imaging of the guide star RoIs and start closed loop image stabilisation using guide star centroids (ADCS 2nd stage).]{}
4. [Take a long exposure using the remainder of the main image sensor. Usually this would be a 600s exposure but some shorter exposures will be taken to obtain unsaturated images of the brightest stars.]{}
5. [Repeat 4 until orbital motion causes observing constraints to be no longer satisfied, likely after 3–4 exposures]{}
6. [Perform Earth IR avoidance manoeuvres until observering constraints met again.]{}
7. [Repeat from 1 with pointing offset to the next dither position]{}
Over the course of 6 orbits Space Eye will obtain images at each of 6 dither positions in order to construct contiguous images for each of the 6 filters in the focal plane mosaic. Over the course of a 6 month period spent observing either the northern or southern targets the spacecraft will observe at roll angles spanning a range of as the spacecraft tracks the Sun with its main solar array. This ensures that each position on the sky is observed at a range of field positions, which helps in suppressing residual calibration errors and internal reflections.
Calibration sequences
---------------------
Space Eye will be extensively characterised before launch however these data will be supplemented with on orbit calibrations, including dark current measurements and flat fielding using a combination of Earth streak images, stellar photometry and median sky frames. The on-orbit measurements will account for changes caused by radiation damage, etc. In addition science observations will be interspersed with exposures of bright stars (‘PSF standards’) to regularly characterise the faint, outer parts of the PSF wings in a strategy similar to that reported by Tujillo and Fliri[@Trujillo2015].
Data handling
-------------
Each main image sensor image will be captured at 2000 x 1504 pixel resolution and 16 bits/pixel depth, i.e. each comprises of raw data. Image data will be stored as losslessly compressed Flexible Image Transport System (FITS) files, which typically have compression factors of around 2 for astronomical data. On board data processing will be limited to insertion of relevant spacecraft and instrument status information into FITS file headers, all raw images will be downlinked for processing and analysis on the ground. Based on the assumption of 3–4 science images per orbit the average (compressed) data rate would be $\sim135$– per day. Downlinking this volume of data from a CubeSat using standard S-band communications and 1–2 ground stations would be a challenge, but is achievable[@Reisenfeld2015].
STATUS AND PLANS
================
At the time of writing funding to complete the design, construction, testing, launch, and commissioning of the Australian Space Eye is being sought via the Australian Research Council’s Linkage Infrastructure, Equipment & Facilities (LIEF) national competitive grant scheme.
The LIEF grant scheme allows consortia lead by an Australian higher education institution to seek part funding (typically $\sim50\%$) for a research infrastructure project from the government, with the remainder to come from the institutions of the consortium. The Space Eye LIEF consortium is lead by PI Lee Spitler of Macquarie University and includes astronomers, instrument scientists and engineers from 7 Australian universities (Macquarie University, Australian National University, UNSW, University of Sydney, University of Queensland, Western Sydney University and Swinburne), the Australian Astronomical Observatory, and both Tyvak and California Polytechnic State University, San Luis Obispo from the USA.
The outcome of our grant application is expected in October/November 2016 and, if successful, will be followed by a 3 year construction phase with launch and commissioning planned for H2 2019.
This research made use of Astropy, a community-developed core Python package for astronomy[@Robitaille2013], and the affiliated package ccdproc[@Craig2015]. Some of the results in this paper have been derived using the HEALPix[^1][@Gorski2004] package. This work also used the NumPy[@VanderWalt2011], Scipy[@Jones2001], Matplotlib[@Hunter2007] and IPython[@Perez2007] Python packages.
[^1]: <http://healpix.sf.net/>
|
---
abstract: |
Given two positive integers $n,r$, we define the Gaudin function of level $r$ to be quotient of the numerator of $$\det\Bigl(\, ((x_i-y_j)(x_i-ty_j) \cdots (x_i-t^r y_j))^{-1}
\Bigr)_{i,j=1\ldots n}$$ by the two Vandermonde in $x$ and $y$. We show that it can be characterized by specializing the $x_i$ into the $y_j$ variables, multiplied by powers of $t$. This allows us to obtain the Gaudin function of level $1$ (due to Korepin and Izergin) as the image of a resultant under the the Euler-Poincaré characteristics of the flag manifold. As a corollary, we recover a result of Warnaar about the generating function of Macdonald polynomials.
---
**Gaudin functions, and Euler-Poincaré characteristics**
*Alain Lascoux*
Gaudin functions of arbitrary level
===================================
Let $\x=\{ x_1,\ldots, x_n\}$, $\y=\{ y_1,\ldots, y_n\}$ be two sets of indeterminates of the same cardinality $n$. The Cauchy determinant $\det\bigl( (x-y)^{-1} \bigr)_{x\in \x, y\in\y}$ plays a central rôle in the theory of symmetric functions [@Macdonald].
Generalizations of this determinant appear in the calculation of correlation functions of different physical models. Gaudin [@Gaudin Ch.IV] obtained the determinant $$\det\bigl( (x-y)^{-1} (x-y+\gamma)^{-1} \bigr)_{x\in \x, y\in\y} \, ,$$ $\gamma$ a parameter, for a bose gas in one dimension. Izergin and Korepin [@Izergin] solved the Heisenberg XXZ-antiferromagnetic model with the help of $$\det\bigl( (x-y)^{-1} (x- t y)^{-1} \bigr)_{x\in \x, y\in\y} \, ,$$ and Kirillov and Smirnov [@Kirillov Th.1] wrote more general determinants. These different determinants have led to an abundant literature, in connection with different statistical models and combinatorial enumerations, for example the enumeration of alternating sign matrices displaying some symmetries[@Okada].
In this text, we shall be interested in a purely algebraic generalization, having no physical interpretation to offer. With $\x$ and $\y$ as above, let $t$ be an extra indeterminate and $r$ be a positive integer.
We propose to study the determinant $$\det \left(\,\frac{1}{ (x-y)(x-ty)\cdots (x-t^r y)} \,
\right)_{x\in \x, y\in\y} \ .$$ This is a rational function, which can be written $$\frac{\Delta(\x) \Delta(\y)}{R(\x, \y(1+\cdots+t^r))}\,
F^r_n(\x,\y) \, ,$$ where $F^r_n(\x,\y)$ is a polynomial symmetrical in both $x_1,\ldots, x_n$ and $y_1,\ldots, y_n$, $\Delta(\x)$ being the Vandermonde $\prod_{i<j} (x_j-x_i)$, the resultant $R(\x,\y)$, for two finite sets of indeterminates $\x,\y$, being the product of differences\
$ \prod_{x\in\x}\prod_{y\in\y} (x-y)$, and $\y(1+\cdots+t^r)$ being $\{ t^i y_j :\, 1\leq j\leq n,\, 0\leq i\leq r\}$.
We shall call the function $F_n^r(\x,\y)$ the *Gaudin function of level $r$*, in recognition of the pioneering work of Gaudin so well illustrated in his thesis [@GaudinThese] and his book [@Gaudin].
Let us introduce $n$ sets of indeterminates $$\y^j :=\{ y_j^0=y_j, y_j^1,\ldots, y_j^r \}\ ,\ j=1,\ldots, n \, .$$
Recall that, for any pair $(a,b)$ of indeterminates, the divided difference relative to $a,b$ is the operator (denoted on the right) $$f(a,b) \to f(a,b)\d_{a,b} = \bigl(f(a,b)-f(b,a) \bigr)\, (a-b)^{-1} \, .$$
The determinant $M_n= \bigl| R(x_i,\y^j)^{-1} \bigr|$ is the image of the Cauchy determinant under the product of divided differences $$\prod_{j=1}^n \d_0^j \cdots \d_{r-1}^j \, ,$$ where $\d_i^j$ is relative to the pair $ y_j^i, y_j^{i+1}$.
Indeed, for any $j$, any $x$, one has $$(x-y^0_j)^{-1} \stackrel{\d_0^j}{\longrightarrow}
(x-y^0_j)^{-1}(x-y^1_j)^{-1} \stackrel{\d_1^j}{\longrightarrow}
(x-y^0_j)^{-1}(x-y^1_j)^{-1}(x-y^2_j)^{-1} \cdots \, ,$$ so that the divided differences relative to the alphabet $\y^j$ transform all the terms $(x_i-y^0_j)^{-1}$, inside the Cauchy determinant, into $(x_i-y^0_j)^{-1}\cdots (x_i-y^r_j)^{-1}$.
On the other hand, it is classical, and immediate, that the Cauchy determinant be equal to $\Delta(\x) \Delta(\y)/R(\x,\y)$. It can also be expressed as $$\Delta(\x)\, \left|\frac{1}{R(\x,y)},\, \frac{y}{R(\x,y)},
\ldots,\, \frac{y^{n-1}}{R(\x,y)}, \right|_{y=y_1,\ldots,y_n} \, .$$
Its product by $\prod_i R(\x, \y^i)$ can therefore be written as $$\label{Cauchy2}
\Delta(\x)\, \Bigl| (y_i^0)^j\, R(\x, \y^i-y_i^0) \Bigr|_{i=1,\ldots,n,\,
j=0,\ldots,n-1}
\, .$$
Since each factor $R(\x,\y^i)$ commutes with any divided difference $\d_k^j$, $k=0,\ldots, r\moins 1$, $j=1,\ldots,n$, the image of (\[Cauchy2\]) under $\prod_i \d_0^i \cdots \d_{r-1}^i $ is equal to $M_n\, \prod_i R(\x, \y^i)$: $$\label{Cauchy3}
\frac{M_n}{\Delta(\x) } \, \prod_i R(\x, \y^i) =
\Bigl| (y_i^0)^j\, R(\x, \y^i-y_i) \d_0^i \cdots \d_{r-1}^i
\Bigr| \, .$$
We shall now have recourse to symmetric functions, referring to the last section for more details, as well as [@Cbms; @Macdonald]. For any non negative integer $j$, one has $$(y_i^0)^j\, R(\x, \y^i-y_i^0) \d_0^i \cdots \d_{r-1}^i
= S_{j;\square}( \y^i ; \y^i -\x) \, ,$$ where $\square = [\underbrace{n\moins 1,\ldots,n\moins 1}_r ] = (n\moins 1)^r$, and $S_{j;\square}$ is a multi-Schur function.
Writing $\y^i = (\y^i -\x) +\x$, and expanding by linearity, then $$S_{j;\square}( \y^i ; \y^i -\x) =
\sum_{k=0}^{j} S_k(\x) S_{j-k,\, \square}(\y^i -\x) \, .$$ This identity allows us to transform (\[Cauchy3\]) into $$\label{Cauchy4}
\frac{M_n}{\Delta(\x) } \, \prod_i R(\x, \y^i) =
\Bigl| S_{j,\, \square}( \y^i -\x) \Bigr| \, ,$$ where now the entries of row $i$ are Schur functions in the difference $\y^i -\x$.
Going back to the original variables, that is, specializing each $\y^i$ into $y_i+ty_i+\cdots +t^r y_i$, we obtain the following expression of the Gaudin function of level $r$.
\[GaudinMatLevel\] The function $$F^r_n(\x, \y) :=
\frac{ R(\x, \y(1\plus \cdots \plus t^r))}{\Delta(\x)\Delta(\y)} \,
\det \left(\,\frac{1}{ (x_i-y_j)(x_i-ty_j)\cdots (x_i-t^r y_j)} \, \right)$$ is equal to $$\frac{1}{\Delta(\y)}\,
\Bigl| S_{j,\, \square}(y_i(1\plus \cdots \plus t^r) -\x)
\Bigr|_{i=1,\ldots,n,\, j=0,\ldots,n-1} \, .$$
The original case of Izergin, Korepin is for level $1$, and reads $$\begin{gathered}
F^1_n(\x, \y) = \frac{1}{\Delta(\y)}\,
\Bigl| S_{n-1}(y_i(1\plus t) -\x),\, \\
S_{1,n-1}(y_i(1\plus t) -\x),\, \ldots,
S_{n-1,n-1}(y_i(1\plus t) -\x) \,
\Bigr|_{i=1,\ldots,n} \, .\end{gathered}$$ For example, for $r=1$, $n=3$, the function is $$F^1(\x,\y) \Delta(\y) = \begin{vmatrix}
S_{022}({ y_1\plus ty_1}-\x) & S_{122}({ y_1\plus ty_1}-\x)
&S_{222}({ y_1\plus ty_1}-\x) \\
S_{022}({ y_2\plus ty_2}-\x) & S_{122}({ y_2\plus ty_2}-\x)
&S_{222}({ y_2\plus ty_2}-\x) \\
S_{022}({y_3\plus ty_3}-\x) & S_{122}({y_3\plus ty_3}-\x)
&S_{222}({y_3\plus ty_3}-\x) \\
\end{vmatrix} \, .$$ We have given another expression in [@SLC], separating the variables $\x$ and $\y$.
The determinant $\det\left(
(tx_i-y_j\frac{1}{t})^{-1} (t^2x_i-y_j\frac{1}{t^2})^{-1}
\right)$ specializes into\
$\det\left( (x_i-y_j) (x_i^3-y_j^3) ^{-1} \right)$ when $t=\exp(2\pi\sqrt{\moins 1}/3)$. In that case, $F_n^1(\x,\y)$ becomes the Schur function in the union of $\x$ and $\y$ of index $[0,0,1,1,2,2,\ldots, n\moins 1,n\moins 1]$ (cf. [@Stroganov02; @Stroganov04]). More generally, the Gaudin function of level $r$, when $r$ is odd, displays such a global symmetry. In that case, $(tx-yt^{-1})^{-1}\cdots (t^{r+1}x -yt^{-r-1})^{-1}
= (x-y) (x^{r+2}-y^{r+2})^{-1}$, and the determinant $\det\left( (x_i-y_j)(x_i^{r+2}-y_j^{r+2})^{-1}
\right)$ is equal to $$\frac{\Delta(\x) \Delta(\y)}{\prod x_i^{r+2}-y_j^{r+2}}\,
S_{0,0,\beta,\beta,\ldots, (n\moins 1)\beta, (n\moins 1)\beta}(\x+\y) \, .$$ More general determinants displaying a symmetry in $x_1,\ldots, x_n,y_1,\ldots, y_n$ are given in [@Pfaff Lemma 13] and [@Okada].
We shall now characterize the Gaudin function by specialization. Expanding the determinant expressing $F^r_n(\x, \y)$ by linearity in $\x$, one sees that $F^r_n(\x, \y)$ is a linear combination of terms $ \frac{1}{\Delta(\y)}
\Bigl| y_i^{v_1},\, \ldots,\ldots y_i^{v_n}\Bigr|$,\
$0\leq v_1,\ldots,v_n\leq (n\moins 1)(r\plus 1)$, i.e. is a linear combination of Schur functions of $\y$ indexed by partitions contained in $\boxplus = [\, \underbrace{(n\moins 1)r,\ldots, (n\moins 1)r}_n \, ]$. By symmetry $\x \leftrightarrow \y$, $F^r_n(\x, \y)$ is a linear combination of products of Schur function of $\x$ and of Schur functions of $\y$ indexed by partitions contained in $\boxplus$.
Given any infinite set of indeterminates $\z$, any linear combination of Schur functions in $\x$ with coefficients in $\z$, indexed by partitions $\subseteq \boxplus$, is also a linear combination of Grassmannian Schubert polynomials $Y_v(\x,\z)$, $v \subseteq \boxplus$. As such, it is determined by the $\binom{nr+n-r}{n}$ specializations $\x \subset \{z_1,\ldots, z_{nr+n-r}\}$. In the next theorem, we choose $\z=\{ y_1,\ldots, y_n, ty_1,\ldots, ty_n, t^2 y_1,\ldots, t^2 y_n,
\ldots \}$ to get simple specializations.
\[TheoremSpec\] $F^r_n(\x,\y)$ is the only linear combination of Schur functions in $\x$, with coefficients in $\y$, indexed by partitions contained in $\boxplus$, which has the same specializations $$\x \subset \{ y_1,\ldots, y_n,\ldots, t^r y_1,\ldots, t^r y_n\}$$ than the function $$G_n^r(\x,\y) := \frac{\Delta(\x)}{\Delta(\y)} \prod_i
S_{\square} (y_i(1\plus \cdots\plus t^r) -\x) \, ,$$ where $\square$ is, as before, equal to $(n\moins 1)^r$.
If the specialization of $\x$ contains several occurrences of the same $y_i$ (ignoring the powers of $t$), then all the functions $S_{j,\, \square}(y_i(1\plus \cdots\plus t^r)-\x)$ vanish. Thus, $F_n^r(\x,\y)$ as well as $G_n^r(\x,\y) $ vanish in that case.
Let now $\x'=\{ x'_1=y_1 t^{\epsilon_1}, \ldots,
x'_n=y_n t^{\epsilon_n} \}$, with $0\leq \epsilon_1,\ldots, \epsilon_n \leq r$. In that case, each $y_i(1\plus \cdots\plus t^r)- \x'$ is equal to the difference of two sets of respective cardinalities $r,n-1$, and, according to (\[Factorise\]), $$S_{j,\, \square}(y_i(1\plus \cdots\plus t^r)-\x')
= S_j( x'_i-\x')\, S_{\square}(y_i(1\plus \cdots\plus t^r)-\x') \, .$$ This factorization allows to extract from the determinant expressing $F_n^r(\x',\y)$ the factor $\prod_i S_{\square} (y_i(1\plus \cdots\plus t^r) -\x')$. There remains $\det\bigl(S_j( x'_i-\x') \bigr)$, which is equal to the Vandermonde $\Delta(\x')$. Therefore, the two functions $F_n^r(\x,\y)$, $G_n^r(\x,\y)$ have the same specializations in $\{ y_1,\ldots, t^r y_n\} $. To characterize $F_n^r(\x,\y)$, we need only specialize $\x$ to a subset of the first $nr+n-r$ elements of $\{ y_1,\ldots, t^r y_n\} $, and this finishes the proof of the theorem. QED
Notice that all the specializations occurring in Theorem \[TheoremSpec\] are either $0$, or products of factors $(y_it^k -x_j)$.
When $r=1$, Theorem \[TheoremSpec\] claims that $F_n^1(\x,\y)$ has the same specializations $\x\subset \{ y_1,\ldots,y_n,ty_1,\ldots, ty_n\}$ as $ \Delta(\x) \Delta(\y)^{-1}
\prod_i S_{n-1}(y_i+ty_i-\x)$. We have moreover remarked that one can suppress one letter from $\{ y_1,\ldots,ty_n\}$ to characterize the Gaudin function.
For $n=2$, $r=2$, the expression in Theorem \[GaudinMatLevel\] specializes, for $x_1=y_1,x_2=t^2y_2$, into $$\begin{gathered}
\frac{1}{y_2-y_1}
\begin{vmatrix}
S_{011}(ty_1+t^2y_1-t^2y_2) & S_{111}(ty_1+t^2y_1-t^2y_2) \\
S_{011}(y_2+ty_2-y_1) & S_{111}(y_2+ty_2-y_1)
\end{vmatrix} \\
= t^3 (y_2-y_1)(y_1-ty_2)^2
\begin{vmatrix} 1 & -t^2y_2 \\ 1 & -y_1
\end{vmatrix} = t^3 (y_2-y_1)(y_1-ty_2)^2 (t^2y_2-y_1) \, .\end{gathered}$$ Theorem \[TheoremSpec\] gives on the other hand $$\begin{gathered}
\frac{t^2y_2-y_1}{y_2-y_1}\, S_{011}( ty_1+t^2y_1-t^2y_2)\,
S_{011}(y_2+ty_2-y_1) \\
= \frac{t^2y_2-y_1}{y_2-y_1} (ty_1-t^2y_2)(t^2y_1-t^2y_2) (y_2-y_1)(ty_2-y_1)
\, , \end{gathered}$$ which is, indeed, equal.
Euler-Poincaré characteristics
==============================
We go back to the original Gaudin-Izergin-Korepin determinant [@Gaudin; @Izergin], that is, from now on, we take level $r=1$.
The *Euler-Poincaré characteristics* for a flag manifold under $Gl(\C^n)$, conveniently generalized by Hirzebruch [@Hirzebruch; @Chiy], can be combinatorially interpreted as a summation over the symmetric group $\mfS_n$ : $$\C[t][x_1,\ldots,x_n] \ni f \longrightarrow
f \carre_\omega := \sum_{w\in\mfS_n} \left(
f \frac{\prod_{i<j} (tx_i -x_j)}{x_i-x_j} \right)^w \in \Sym(\x) \, .$$
This morphism is characterized by the fact that the images of dominant monomials $$x^\l := x_1^{\l_1} \cdots x_n^{\l_n}\, ,\ \l_1\geq \cdots \geq \l_n\geq 0$$ are the Hall-Littlewood polynomials [@Macdonald] $c_\l P_\l(\x,t)$, the normalization constants $c_\l$, writing $\l= 0^{m_0} 1^{m_1}\cdots n^{m_n}$, being $$c_\l = \prod_{i=0}^n \prod_{j=1}^{m_i} (1-t^j)(1-t)^{-1} \, .$$
The elementary operators (case $n=2$) are $$\carre_i := \carre_{s_i} :
f \longrightarrow f\, \carre_i = f (tx_i-x_{i+1})\, \d_i$$ and generate the Hecke algebra of the symmetric group (as an algebra of operators on polynomials. This is the description of the affine Hecke algebra that I used with M.P. Schützenberger in [@SymmetryFlag]). The usual generators $ T_i := \carre_i -1 $ satisfy the braid relations, and the Hecke relation $(T_i-t)(T_i+1)=0$, while $\carre_i^2 = (1+t)\carre_i$.
We shall also need an affine operation $\theta$, which is the incrementation of indices on the $x$-variables : $$x_i\, \theta =x_{i+1}\, , \quad \text{periodicity}\ x_{i+n} = x_i t^{-1} \, .$$ Notice that to define Macdonald’s polynomials (see next section), one uses the periodicity $x_{i+n} = q x_i$ with $q$ independent of $t$.
Let $\x,\y$ be two alphabets of cardinality $n$, $f$ be a function of a single variable. Then $$\begin{gathered}
\label{TheoremTheta}
f(x_1)\, R(\x-x_1\, ,\, \y) (1-t\theta)\cdots (1-t^{n-1}\theta)
\carre_\omega \\
= \left(f(x_1) x_2\cdots x_n \d_1\cdots \d_{n-1}\,\right)
F_n^1(\x,\y)\, [n]! \, .\end{gathered}$$
The LHS, as a function of $\y$, belongs to the space generated by the Schur functions of index contained in the partition $(n\moins 1)^n$. It therefore can be determined by computing all the specializations $$\y \subset \{ x_1,\ldots, x_n, x_2 t^{-1},\ldots x_n t^{-1} \}$$ (we do not need to take $x_1t^{-1}$).
To lighten notations, let us take $n=4$. The function $$f(x_1)\, R(x_2\plus x_3\plus x_4 ,\y) (1-t\theta)
(1-t^2\theta)(1-t^3\theta)$$ is equal to $$\begin{gathered}
f(x_1)\, R(x_2\plus x_3\plus x_4 ,\y)
-t {3 \brack 1} f(x_2)\, R(x_3\plus x_4\plus x_1/t ,\y) \\
+t^3 {3 \brack 2} f(x_3)\, R(x_4\plus x_1/t\plus x_2/t ,\y)
-t^6 f(x_4)\, R(x_1/t\plus x_2/t\plus x_3/t ,\y)\end{gathered}$$
The sum under the symmetric group can be written $$\begin{gathered}
\sum_w \left(f(x_1)R(x_2\plus x_3\plus x_4 ,\y)
\frac{\Delta_t(1234)}{\Delta(1234)} \right)^w \\
-t {3 \brack 1} \sum_w \left(f(x_1)R(x_3\plus x_4\plus x_2t^{-1} ,\y)
\frac{\Delta_t(2134)}{\Delta(2134)} \right)^w \\
+t^3 {3 \brack 2} \sum_w \left(f(x_1)R(x_4\plus x_2 t^{-1}\plus x_3t^{-1} ,\y)
\frac{\Delta_t(3214)}{\Delta(3214)} \right)^w \\
-t^6 \sum_w \left(f(x_1)R( x_2 t^{-1}\plus x_3t^{-1}\plus x_4t^{-1} ,\y)
\frac{\Delta_t(4231)}{\Delta(4231)} \right)^w\end{gathered}$$
We shall identify the coefficients of $f(x_1)$ in both members of \[TheoremTheta\]. Consider all the specializations $\y \subset \{ x_1,\ldots, x_4, x_2/t,\ldots, x_4/t \} $ of the LHS. Up to symmetry, the only non-zero specializations are
- $\y\to \{x_1,x_2,x_3,x_4\}$: $-t^6 R(\frac{x_2+x_3+x_4}{t},\y)\sum_{w:\, w_1=1}
\left( \frac{\Delta_t(1234)}{\Delta(1234)} \right)^w $
- $\y\to \{x_1,x_2,x_3,\frac{x_4}{t} \}$: $ t^3 {3 \brack 2} R(\frac{x_2+x_3}{t}+x_4,\y) \sum_{w:\, w_1=1,w_4=4}
\left( \frac{\Delta_t(3214)}{\Delta(3214)} \right)^w $
- $\y\to \{x_1,x_2,\frac{x_3}{t},\frac{x_4}{t} \}$: $-t {3 \brack 1} R(\frac{x_2}{t}+x_3\plus x_4,\y) \sum_{w:\, w_1=1,w_2=2}
\left( \frac{\Delta_t(2134)}{\Delta(2134)} \right)^w $
- $\y\to \{x_1,\frac{x_2}{t},\frac{x_2}{t},\frac{x_4}{t} \}$: $ R(x_2\plus x_3\plus x_4,\y) \sum_{w:\, w_1=1}
\left( \frac{\Delta_t(1234)}{\Delta(1234)} \right)^w $ .
Up to the global factor $$\frac{x_2x_3x_4(1-t)(1-t^2)(1-t^3)}{\Delta_t(1234) \Delta_t(4321)
R(x_1, x_2\plus x_3\plus x_4) }$$ these specializations are respectively equal to $$1,\, \frac{R(x_1\plus x_2\plus x_3, x_4)}{R(x_1\plus x_2\plus x_3, tx_4)},\,
\frac{R(x_1\plus x_2,x_3\plus x_4)}{R(x_1\plus x_2,tx_3\plus tx_4)},\,
\frac{R(x_1, x_2\plus x_3\plus x_4)}{R(x_1, tx_2\plus tx_3\plus tx_4)}\, .$$ They coincide with the specializations of the RHS of (\[TheoremTheta\]), thanks to Theorem \[TheoremSpec\], writing $f(x_1)x_2x_3x_4 \d_1\d_2\d_3$ as $$\frac{f(x_1) x_2x_3x_4}{R(x_1, x_2\plus x_3\plus x_4)}
+ \frac{f(x_2) x_1x_3x_4}{R(x_2, x_1\plus x_3\plus x_4)}
+ \frac{f(x_3) x_1x_2x_4}{R(x_3, x_1\plus x_2\plus x_4)}
+ \frac{f(x_4) x_1x_2x_3}{R(x_4, x_1\plus x_2\plus x_3)} \, .$$
In final, we have checked enough specializations to prove (\[TheoremTheta\]).
Generating functions of Macdonald polynomials
=============================================
The symmetric Macdonald polynomials $P_\l(\x;q,t)$ satisfy a Cauchy formula : $$\label{CauchyMacdo}
\sigma_1\left(\x\y\frac{1-t}{1-q} \right) :=
\prod_{x\in\x, y\in\y} \prod_{i\geq 0}
\frac{1-tq^i xy}{1-q^i xy} =
\sum_\l b_\l P_\l(\x;q,t) P_\l(\y;q,t) \, ,$$ sum over all partitions of length $\ell(\l)\leq n$, the constants $b_\l $ being defined in [@Macdonald VI.4.11].
Let $\tau_q$ be the following incrementation of indices on the $x$-variables : $$x_i\, \tau_q =x_{i+1}\, , \quad \text{periodicity}\ x_{i+n} = qx_i \, .$$
We want to compute $$\sigma_1\left(\x\y\frac{1-t}{1-q} \right)\,
(1-t\tau_q)\cdots (1-t^n\tau_q)\, \carre_\omega \, .$$ Since $$\x \frac{1-t}{1-q} \tau_q = \x \frac{1-t}{1-q} + x_1(t-1) \, ,$$ one has $$\begin{gathered}
\label{fgMacdo}
\sigma_1\left(\x\y\frac{1-t}{1-q} \right)\,
(1-t\tau_q)\cdots (1-t^n\tau_q)\, \carre_\omega \\
= \sigma_1\left(\x\y(1\moins t)\right) (1-t\tau_0)\cdots (1-t^n\tau_0)
\carre_\omega \, \sigma_1\left(\x\y q\frac{1-t}{1-q} \right)\end{gathered}$$
The parameter $q$ has been eliminated from the operation, and we are thus reduced to the case of Hall-Littlewood polynomials, which is treated in the next theorem.
\[ThHL\] The image of the generating function of Hall-Littlewood polynomials $P_\l(\x,t)= P_\l(\x;0,t)$ under $(1-t\tau_0)\cdots (1-t^n\tau_0) \carre_\omega$ is $$\label{fgHL}
\sigma_1\left(\x\y(1\moins t)\right) (1-t\tau_0)\cdots (1-t^n\tau_0)
\carre_\omega
= \sigma_1(\x\y)\, \tF_n^1(\x,\y)\, [n]! \, ,$$ where $\tF_n^1(\x,\y)$ is the Gaudin function $(x_1\cdots x_n)^{n-1}\, F_n^1(\x^\vee, \y)$,\
$\x^\vee=\{ x_1^{-1},\ldots, x_n^{-1}\}$, and $[n]!=(1-t)\cdots (1-t^n)$.
One rewrites $ \sigma_1\left(\x\y(1\moins t)\right)
= R(t\x, \y^\vee) R(\x, \y^\vee)^{-1}$. Notice that $$\begin{gathered}
\frac{ R(tx_i+\cdots+tx_n, \y^\vee)}{ R(x_i+\cdots+x_n, \y^\vee)}
(1-t^n\tau_0) =
\frac{ R(tx_i+\cdots+tx_n, \y^\vee)}{ R(x_i+\cdots+x_n, \y^\vee)}
-\frac{ R(tx_{i+1}+\cdots+tx_n, \y^\vee)}{ R(x_{i+1}+\cdots+x_n, \y^\vee)}\\
= \bigl( R(tx_i,\y^\vee)-t^n R(x_i,\y^\vee) \bigr)
\frac{ R(tx_{i+1}+\cdots+tx_n, \y^\vee)}{ R(x_i+\cdots+x_n, \y^\vee)}
= \frac{f(x_i)}{R(x_i+\cdots+x_n, \y^\vee)} \, \end{gathered}$$ with $f(x_i)$ a polynomial in $x_i$ of degree $n-1$. Multiplying the LHS of (\[fgHL\]) by the function $R(\x,\y^\vee)$, one transforms it into $$\begin{gathered}
\Bigl( f(x_1) R(tx_2+\cdots +tx_n,\y^\vee) -e_1 f(x_2) R(tx_3
+\cdots +tx_n+x_1,\y^\vee) \\
+\cdots +(\moins 1)^{n-1} e_{n-1}
f(x_n) R(x_1+\cdots +x_{n-1}, \y^\vee) \Bigr)\, \carre_\omega\, ,\end{gathered}$$ where $e_1,\ldots, e_{n-1}$ are the elementary symmetric functions of $t,\ldots, t^{n-1}$. One recognizes in this last expression $$f(x_1) R(tx_2+\cdots +tx_n,\y^\vee) (1-t\theta)\cdots
(1-t^{n-1}\theta) \, .$$
We now invoke Theorem \[TheoremTheta\]. Since the function $f(x_1) x_2\cdots x_n \d_1\cdots \d_{n-1}$ is a constant (for degree reasons), the LHS of (\[fgHL\]) is proportional to $F_n^1(\x,\y^\vee)$, that is, is proportional to $\tF_n^1(\x,\y)$. In fact, $f(x) = \prod_j( tx -y_j^{-1})\\ -t^n \prod_j( x -y_j^{-1})$, so that $$f(x_1) x_2\cdots x_n \d_1\cdots \d_{n-1}
= (1-t^n)(y_1\cdots y_n)^{-1} \, .$$
Correcting by the right powers of $(x_1\cdots x_n)$ and $(y_1\cdots y_n)$, one finishes the proof of the theorem.
One can now go back to the case of Macdonald polynomials, and recover a result of Warnaar [@Warnaar Th.3.1].
There holds $$\label{Warnaar}
\sum_\l b_\l P_\l(\x;q,t) P_\l(\y;q,t)
\prod_{i=1}^n (1-q^{\l_i} t^{n-i+1}) =
\sigma_1\left(\x\y\frac{1-t}{1-q} \right)\,
\sigma_1(t\x\y) \tF_n^1(\x,\y) \, .$$
The non symmetric Macdonald polynomials are eigenfunctions of certain commuting Dunkl-type operators $\xi_1,\ldots, \xi_n$, first introduced in [@BGHP] and extensively used by Cherednik [@Cherednik]. The eigenvalues are $q^{\l_1} t^{n-1},\ldots, q^{\l_n}t^0$ for the polynomial $M_\l$ indexed by $\l: \l_1\geq \cdots \geq \l_n\geq0$. Up to normalization, the image of $M_\l$ under $\carre_\omega$ is equal to $P_\l(\x,q,t)$, and for any symmetric function $g$ in $n$ variables, then $$P_\l(\x;q,t)\, g(\xi_1,\ldots, \xi_n)
= P_\l(\x;q,t)\, g( q^{\l_1} t^{n-1},\ldots, q^{\l_n}t^0) \, .$$
Using [@YangRota] that $\carre_\omega \, \xi_i\, \carre_\omega =
\carre_\omega \, t^{n-i}\tau_q\, \carre_\omega$, one sees that $$P_\l(\x;q,t) \prod_{i=1}^n (1-q^{\l_i} t^{n-i+1}) =
P_\l(\x;q,t) (1-t\tau_q)\cdots (1-t^n\tau_q)\, \carre_\omega \, .$$ Therefore, the LHS of (\[Warnaar\]) can be identified with $$\sigma_1\left(\x\y \frac{1\moins t}{1\moins q} \right)
(1-t\tau_q)\cdots (1-t^n\tau_q)\, \carre_\omega \, .$$ Thanks to (\[fgMacdo\]) and (\[fgHL\]), this can be written $$\sigma_1(\x\y) \sigma_1\left(\x\y q\frac{1\moins t}{1\moins q}\right)
\tF_n^1(\x,\y)
= \sigma_1\left(\x\y \frac{1\moins t}{1\moins q} \right)
\sigma_1(t\x\y) \tF_n^1(\x,\y) \, ,$$ which is Warnaar’s formula.
Note: Symmetric functions and Schubert polynomials
==================================================
We use $\lambda$-ring conventions to describe symmetric functions. Given three sets $A,B,C$ of indeterminates (“alphabets”), the generating function of complete functions $S_n(AB-C)$ is $$\sigma_z:= \frac{\prod_{c\in \C} 1-zc}{\prod_{a\in A,b\in B} 1-z ab }
= \sum z^n S_n(AB-C)\, .$$ We write alphabets as sums of the letters composing them. For example, $A(1+t+\cdots+t^r)$ is the alphabet $\{ at^i :\, a\in A,
0\leq i\leq r \}$.
Schur functions $S_v(A-C)$, $v\in \N^n$, are determinants of complete functions: $$S_v(A-C) = \det\bigl( S_{v_j+j-i}(A-C)\bigr)_{i,j=1\ldots n} \, .$$
One generalizes Schur functions to *multi-Schur functions* by taking different alphabets in blocks of columns of the preceding determinant. For example, $S_{v_1; v_2,\ldots, v_n}(A_1-C_1; A-C)$ is the determinant with first column $S_{v_1}(A_1-C_1),\ldots, S_{v_1-n+1}(A_1-C_1)$, and entries $S_{v_j+j-i}(A-C)$ elsewhere.
Multi-Schur functions satisfy some factorization properties [@Cbms Prop. 1.4.3]. We need only the following case, which was much used in classical elimination theory in the 19th century.
Given two finite alphabets $A,B$ of respective cardinalities $\alpha,\beta$, given $j:\, 0\leq j\leq \beta$, then $$\label{Factorise}
S_{j,\beta^{\alpha}}(A-B)= S_j(-B) \prod_{a\in A, b\in B} (a-b)=
(\moins 1)^j e_j(B) \prod_{a\in A, b\in B} (a-b) \, ,$$ where $e_j(B)$ is the elementary symmetric function of degree $j$ in $B$. In particular, $S_{j,\beta^{\alpha}}(A-B)=0$ if $A\cap B\neq \emptyset$.
There are several families of non-symmetric polynomials extending the basis of Schur functions. Of special interest are the *Schubert polynomials* $Y_v(\x,\y)$, $v\in \N^n$, which constitute a linear basis of the ring of polynomials in $x=\{x_1,\ldots, x_n\}$, with coefficients in $y_1,y_2,\ldots, y_\infty$. They can be characterized by vanishing properties related to the Bruhat order.
The subfamily of Schubert polynomials indexed by (increasing) partitions form a basis of the ring of symmetric polynomials [@Cbms]. It satisfies the following property. Given $u\in \N^n:\, 0\leq u_1 \leq \cdots \leq u_n$, let $\y^{<u>} :=\{ y_{1+u_1},\ldots, y_{n+u_n}\}$.
Let $v= [0\leq v_1\leq \cdots \leq v_n]$ be a partition. Then $Y_v(\x,\y)$ is the only symmetric function in $\x$ of degree $|v|= v_1+\cdots+v_n$ such that $ Y_v(\y^{<u>},\y)=0$ for all $u:\, |u|\leq |v|$, $u\neq v$, and $Y_v(\x, \{0,0,\ldots\}) = S_v(\x)$.
More generally, $ Y_v(\y^{<u>},\y) \neq 0$ iff the diagram of $u$ contains the diagram of $v$.
In the case where $\y=\{0,1,2,\ldots\}$ (resp. $\y=\{q^0,q^1,q^2,\ldots\}$), the polynomials $Y_v(\x,\y)$ are called *factorial Schur functions* (resp. *$q$-factorial Schur functions*, and the above vanishing properties are extensively used in [@OkounkovOlshanski].
Acknowledgements {#acknowledgements .unnumbered}
----------------
The author benefits from the ANR project BLAN06-2\_134516.
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[Alain Lascoux]{}\
CNRS, IGM, Université Paris-Est\
77454 Marne-la-Vallée Cedex, France\
Email: Alain.Lascoux@univ-mlv.fr\
|
---
abstract: 'We consider the general scalar field Horndeski Lagrangian coupled to matter. Within this class of models, we present two results **** that are independent of the particular form of the model. **** First, we show that in a Friedmann-Robertson-Walker metric the Horndeski Lagrangian coincides with the pressure of the scalar field. Second, we employ the previous result to identify **** the most general form of the Lagrangian **** that allows for cosmological scaling solutions, i.e. solutions where the ratio of matter to field density and the equation of state remain constant. Scaling solutions of this kind may help solving the coincidence problem since in this case the presently observed ratio of matter to dark energy does not depend on initial conditions, but rather on the theoretical parameters.'
author:
- 'Adalto R. Gomes'
- Luca Amendola
title: The general form of the coupled Horndeski Lagrangian that allows cosmological scaling solutions
---
Introduction
============
The discovery of cosmic acceleration [@Perlmutter_etal_1999; @Riess_etal_1998] has generated many attempts at finding suitable explanations beyond the classical cosmological constant. Many of these are based on scalar fields because they provide a relatively simple way of adding degrees of freedom into the matter Lagrangian with appealing properties: weak clustering, tunable equation of state, isotropy. In fact, scalar fields are also the simplest and most investigated models for the other period of acceleration – inflation.
The simple quintessence scalar field Lagrangian [@Wetterich_1988; @Ratra_Peebles_1988; @Copeland:1997et; @Ferreira_Joyce_1998; @1998PhRvL..80.1582C] has been progressively expanded by including terms coupled to gravity [@Wetterich_1995; @amendola2000; @Baccigalupi_Matarrese_Perrotta_2000; @2006JCAP...12..020A] and terms that are general functions of the kinetic energy [[@kessence]]{}. Building on pioneering results, some authors realized recently that the most general scalar field Lagrangian that produces second order equations of motion is the so-called Horndeski Lagrangian [@Horndeski:1974; @Deffayet:2011gz; @Kobayashi:2011nu], a model that includes four arbitrary functions of the scalar field and its kinetic energy. Models that expand beyond Horndeski have also been proposed [@Gleyzes:2013ooa; @2014PhRvD..89f4046Z]. This Lagrangian is a form of modified gravity since, in general, it changes the Poisson equation, the lensing equation, and the gravitational wave equation.
The number of free functions in the Horndeski Lagrangian makes an exhaustive study very complicate. Several classes of models that exhibit special properties have been already investigated and shown to produce succesful models of dark energy (see e.g. [@2012PhRvD..85j4040C]). In a previous paper [@2014JCAP...03..041G] we posed the question of whether it was possible to find some general property of the Horndeski Lagrangian without solving the equations of motion. In particular, we proposed to identify which classes of Lagrangian contain the so-called scaling solutions, defined by the property that the energy density of matter and scalar field scale in the same way with time, so that their ratio remains constant. A second condition that has also been employed to simplify the treatment is that the field equation of state remains constant during the scaling trajectory. Scaling solutions are particularly interesting because one can hope to employ them to avoid the problem of the coincidence between the present matter and dark energy densities, i.e. the fact that today the two density fractions $\Omega_{m},\Omega_{\phi}$ are very similar. In fact, while this coincidence occurs only today for a cosmological constant model and for all the models in which matter and dark energy scale with time in a different way, and therefore depends in a critical way on the initial conditions, in scaling solutions the “coincidence” depends only the choice of parameters and, if the solution is stable, this can remain valid forever. It is important to remark that we are not demanding that the theory contains *only* scaling solutions, but that scaling solutions are allowed: in general they will contain other non-scaling solutions. Whether these models can successfully explain the entire cosmological sequence of radiation, matter and accelerated eras is still to be seen [@Kscal].
The first example of scaling model is a simple uncoupled scalar field with an exponential potential [@Copeland:1997et; @Ferreira_Joyce_1998]. However, in this case no****acceleration is possible during the scaling regime. Next, one can couple the scalar field and the matter component (or equivalently couple field and gravity) [@Wetterich_1995; @amendola2000]. Several interesting properties of this kind of scaling solutions have been studied in the past, as for instance a similar coupling to neutrinos [@Amendola_Baldi_Wetterich_2008] and the behavior of perturbations [@Amendola:2001rc], and more recently, with multiple dark matter models [@Baldi_2012a; @2015arXiv150205922V].
A generalization of scaling models has been realized in Ref. [@pt] (see also [@ts]). They found in fact that the most general Lagrangian without gravity coupling that contains scaling solutions must have the form $$S=\int d^{4}x\sqrt{-g}\biggl[\frac{1}{2}R+K(\phi,X)\biggr]+S_{m}(\phi,\psi_{i},g_{\mu\nu})\label{action-1}$$ with $$K(\phi,X)=Xg(Xe^{\lambda\phi}),$$ where $X=-\frac{1}{2}\nabla_{\mu}\phi\nabla^{\mu}\phi$, $g$ an arbitrary function and $\lambda$ a constant. $S_{m}$ is the action for the matter fields $\psi_{i}$, which also depends generally on the scalar field $\phi$. The same form applies if the field has a constant coupling to gravity. In Ref. [@Amendola:2006qi] this result has been extended to variable couplings.
In Ref. [@2014JCAP...03..041G] we performed another step towards extending this result to the entire Horndeski Lagrangian. We studied in fact a Lagrangian of type [@Pujolas:2011he; @Deffayet:2010qz; @Kobayashi:2011nu] $$S=\int d^{4}x\sqrt{-g}\biggl[\frac{1}{2}R+K(\phi,X)-G_{3}(\phi,X)\nabla_{\mu}\nabla^{\mu}\phi\biggr]+S_{m}(\phi,\psi_{i},g_{\mu\nu})\label{action-2}$$ denoted KGB model in [@Deffayet:2010qz]. The new term containing $G_{3}$ produces new second order terms in the equation of motion. The addition of the term linear in $\Box\phi\equiv\nabla_{\mu}\nabla^{\mu}\phi$ introduced several new features and enlarged considerably the class of models that allow for accelerated scaling solutions. This paper is devoted to completing our programme by extending the search for scaling Lagrangians to the entire Horndeski class. Unfortunately the mathematics required to achieve this extension is exceedingly tedious. The conclusion is however quite simple to state: we obtain **** the most general form of Horndeski Lagrangian that contains scaling solutions **** among their Friedmann-Lemaitre-Robertson-Walker (FLRW) solutions.
To achieve this generalization we need an intermediate result that is interesting on its own, namely, the equivalence of the Lagrangian with the pressure of the scalar field. This is well known to occur with the term $K(\phi,X)$ [@2000PhRvD..62b3511C; @ArmendarizPicon:2000dh] and also when including $G_{3}$ [@Deffayet:2010qz]. Here we prove it valid for the entire Lagrangian, at least when the metric is restricted to a flat FLRW.
Horndeski Lagrangian and equations of motion
============================================
We consider the Horndeski action $$S=\int d^{4}x\sqrt{-g}\mathcal{L}_{H}+S_{m}(\phi,\psi_{i},g_{\mu\nu})\label{action}$$ where $\mathcal{L}_{H}$ is the Horndeski Lagrangian, which is the most general Lagrangian which has second order equations of motion, defined as $\mathcal{L}_{H}=\mathcal{L}_{2}+\mathcal{L}_{3}+\mathcal{L}_{4}+\mathcal{L}_{5}$, with $$\begin{aligned}
\mathcal{L}_{2} & = & K(\phi,X),\\
\mathcal{L}_{3} & = & -G_{3}(\phi,X)\Box\phi,\\
\mathcal{L}_{4} & = & G_{4}(\phi,X)R+G_{4,X}[(\Box\phi)^{2}-(\nabla_{\mu}\nabla_{\nu}\phi)(\nabla^{\mu}\nabla^{\nu}\phi)]\\
\mathcal{L}_{5} & = & G_{5}(\phi,X)G_{\mu\nu}(\nabla^{\mu}\nabla^{\nu}\phi)-\frac{1}{6}G_{5,X}[(\Box\phi)^{3}-3(\Box\phi)(\nabla_{\mu}\nabla_{\nu}\phi)(\nabla^{\mu}\nabla^{\nu}\phi)\nonumber \\
& & +2(\nabla^{\mu}\nabla_{\alpha}\phi)(\nabla^{\alpha}\nabla_{\beta}\phi)(\nabla^{\beta}\nabla_{\mu}\phi)]\end{aligned}$$ and $\phi$ is a scalar field, $X=-\frac{1}{2}\nabla_{\mu}\phi\nabla^{\mu}\phi$ and $R$ is the Ricci scalar. We consider that there is only one type of pressureless matter of energy density $\rho_{m}=-T_{0}^{0}$, in the Einstein frame, where the energy-momentum tensor is defined by $$T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta S_{m}}{\delta g^{\mu\nu}}.$$ In this frame matter is directly coupled to the scalar field through the function $Q(\phi)$, where $$Q=-\frac{1}{\rho_{m}\sqrt{-g}}\frac{\delta S_{m}}{\delta\phi}.\label{Qdef}$$
We consider a FLRW flat metric with $ds^{2}=-dt^{2}+\mathcal{A}^{2}(t)d\textbf{x}^{2}$, where $\mathcal{A}(t)$ is the scale factor and the scalar field depends only on $t$. In this case we have $X=\dot{\phi}^{2}/2$, $\dot{X}=\dot{\phi}\ddot{\phi}$, $\nabla_{\mu}X\nabla^{\mu}\phi=-2X\ddot{\phi}$, where dot means derivative with respect to the cosmic time $t$. Varying the action with respect to $g_{\mu\nu}$, and defining $H=\dot{\mathcal{A}}/{\mathcal{A}}$, one gets [@DeFelice:2011bh] $$\begin{aligned}
\sum_{i=2}^{5}\mathcal{E}_{i} & = & -\rho_{m},\label{eq_gmunu_a}\\
\sum_{i=2}^{5}\mathcal{P}_{i} & = & 0,\label{eq_gmunu_b}\end{aligned}$$ where $$\begin{aligned}
\mathcal{E}_{2} & = & 2XK_{,X}-K,\\
\mathcal{E}_{3} & = & 6X\dot{\phi}HG_{3,X}-2XG_{3,\phi},\\
\mathcal{E}_{4} & = & -6H^{2}G_{4}+24H^{2}X(G_{4,X}+XG_{4,XX})-12HX\dot{\phi}G_{4,\phi X}-6H\dot{\phi}G_{4,\phi},\\
\mathcal{E}_{5} & = & 2H^{3}X\dot{\phi}(5G_{5,X}+2XG_{5,XX})-6H^{2}X(3G_{5,\phi}+2XG_{5,\phi,X}).\end{aligned}$$ and $$\begin{aligned}
\mathcal{P}_{2} & = & K,\\
\mathcal{P}_{3} & = & -2X(G_{3,\phi}+\ddot{\phi}G_{3,X}),\\
\mathcal{P}_{4} & = & 2(3H^{2}+2\dot{H})G_{4}-12H^{2}XG_{4,X}-4H\dot{X}G_{4,X}-8\dot{H}XG_{4,X}-8HX\dot{X}G_{4,XX}\nonumber \\
& & +2(\ddot{\phi}+2H\dot{\phi})G_{4,\phi}+4XG_{4,\phi\phi}+4X(\ddot{\phi}-2H\dot{\phi})G_{4,\phi X},\\
\mathcal{P}_{5} & = & -2X(2H^{3}\dot{\phi}+2H\dot{H}\dot{\phi}+3H^{2}\ddot{\phi})G_{5,X}-4H^{2}X^{2}\ddot{\phi}G_{5,XX}\nonumber \\
& & +4HX(\dot{X}-HX)G_{5,\phi X}+2[2(\dot{H}X+H\dot{X})+3H^{2}X]G_{5,\phi}+4HX\dot{\phi}G_{5,\phi\phi}.\end{aligned}$$
Varying the action with respect to $\phi$, one gets $$\frac{1}{\mathcal{A}^{3}}\frac{d}{dt}(a^{3}J)=\mathcal{P}-\rho_{m}Q,$$ where $$\begin{aligned}
J & \equiv & \dot{\phi}K_{,X}+6HXG_{3,X}-2\dot{\phi}G_{3,\phi}+6H^{2}\dot{\phi}(G_{4,X}+2XG_{4,XX})-12HXG_{4,\phi X}\nonumber \\
& & +2H^{3}X(3G_{5,X}+2XG_{5,XX})-6H^{2}\dot{\phi}(G_{5,\phi}+XG_{5,\phi X})\\
\mathcal{P} & \equiv & K_{,\phi}-2X(G_{3,\phi\phi}+\ddot{\phi}G_{3,\phi X})+6(2H^{2}+\dot{H})G_{4,\phi}+6H(\dot{X}+2HX)G_{4,\phi X}\nonumber \\
& & -6H^{2}XG_{5,\phi\phi}+2H^{3}X\dot{\phi}G_{5,\phi X}\end{aligned}$$
Now, in order to confront these models with SNIa observations, we isolate from the complete action a term corresponding to the Einstein-Hilbert one. Then the action is rewritten as $$S=\int d^{4}x\sqrt{-g}\biggl(\frac{1}{2}R+\mathcal{L}\biggr)+S_{m}(\phi,\psi_{i},g_{\mu\nu})\label{action_R}$$ with $$\begin{aligned}
\mathcal{L} & = & K(\phi,X)-G_{3}(\phi,X)\Box\phi+\biggl(G_{4}(\phi,X)-\frac{1}{2}\biggr)R+G_{4,X}[(\Box\phi)^{2}-(\nabla_{\mu}\nabla_{\nu}\phi)(\nabla^{\mu}\nabla^{\nu}\phi)]\nonumber \\
& & +G_{5}(\phi,X)G_{\mu\nu}(\nabla^{\mu}\nabla^{\nu}\phi)-\frac{1}{6}G_{5,X}[(\Box\phi)^{3}-3(\Box\phi)(\nabla_{\mu}\nabla_{\nu}\phi)(\nabla^{\mu}\nabla^{\nu}\phi)\nonumber \\
& & +2(\nabla^{\mu}\nabla_{\alpha}\phi)(\nabla^{\alpha}\nabla_{\beta}\phi)(\nabla^{\beta}\nabla_{\mu}\phi)\end{aligned}$$ and we write the Einstein equations as [@tsujikawa_DMDE] $$H^{2}=\frac{1}{3}(\rho_{\phi}+\rho_{m})\label{feq}$$ and $$-2\dot{H}=\rho_{m}+\rho_{\phi}+p.\label{feq2}$$
Comparing Eqs. (\[feq\]), (\[feq2\]) with Eqs. (\[eq\_gmunu\_a\]), (\[eq\_gmunu\_b\]), we arrive to useful definitions for the energy density ($\rho_{DE}$) and pressure ($p$) of dark energy: $$\begin{aligned}
\rho_{\phi} & \equiv & \sum_{i=2}^{5}\mathcal{E}_{i}+3H^{2},\\
p & \equiv & \sum_{i=2}^{5}\mathcal{P}_{i}-(3H^{2}+2\dot{H}),\end{aligned}$$ or also $$\begin{aligned}
\rho_{\phi} & \equiv & 2XK_{,X}-K+6X\dot{\phi}HG_{3,X}-2XG_{3,\phi}+3H^{2}(1-2G_{4})\nonumber \\
& & +24H^{2}X(G_{4,X}+XG_{4,XX})-12HX\dot{\phi}G_{4,\phi X}-6H\dot{\phi}G_{4,\phi}\nonumber \\
& & +2H^{3}X\dot{\phi}(5G_{5,X}+2XG_{5,XX})-6H^{2}X(3G_{5,\phi}+2XG_{5,\phi,X}),\\
p & \equiv & K-2X(G_{3,\phi}+\ddot{\phi}G_{3,X})-(3H^{2}+2\dot{H})(1-2G_{4})-12H^{2}XG_{4,X}\nonumber \\
& & -4H\dot{X}G_{4,X}-8\dot{H}XG_{4,X}-8HX\dot{X}G_{4,XX}+2(\ddot{\phi}+2H\dot{\phi})G_{4,\phi}\nonumber \\
& & +4XG_{4,\phi\phi}+4X(\ddot{\phi}-2H\dot{\phi})G_{4,\phi X}\nonumber \\
& & -2X(2H^{3}\dot{\phi}+2H\dot{H}\dot{\phi}+3H^{2}\ddot{\phi})G_{5,X}-4H^{2}X^{2}\ddot{\phi}G_{5,XX}\nonumber \\
& & +4HX(\dot{X}-HX)G_{5,\phi X}+2[2(\dot{H}X+H\dot{X})+3H^{2}X]G_{5,\phi}+4HX\dot{\phi}G_{5,\phi\phi}.\end{aligned}$$ Defining $$\Omega_{\phi}=\frac{\rho_{\phi}}{3H^{2}},\,\,\,\Omega_{m}=\frac{\rho_{m}}{3H^{2}}\label{Omegaphi}$$ we can rewrite the Friedman equation (Eq. (\[feq\])) as $$\Omega_{\phi}+\Omega_{m}=1.\label{sumOmega}$$ Now we introduce the $e$-folding time $N=\log a$, so that $d/dt=Hd/dN$. Then the equation of motion for the scalar field $\phi$ and matter can be written as (see also [@DeFelice:2011bh]) $$\begin{aligned}
\frac{d\rho_{\phi}}{dN}+3(1+w_{\phi})\rho_{\phi} & =-\rho_{m}Q\frac{d\phi}{dN}\label{rhophi2}\\
\frac{d\rho_{m}}{dN}+3\rho_{m} & =\rho_{m}Q\frac{d\phi}{dN},\label{rhom2}\end{aligned}$$ where $w_{\phi}=p/\rho_{\phi}$.
Scaling Solutions
=================
The condition $\Omega_{\phi}/\Omega_{m}$ constant defines scaling solutions. This is equivalent to $\rho_{\phi}/\rho_{m}$ constant, or to $$\frac{d\log\rho_{\phi}}{dN}=\frac{d\log\rho_{m}}{dN}\label{scaling}$$ Also, from Eq. (\[sumOmega\]) we get that $\Omega_{\phi}$ is a constant. We also assume that on scaling solutions the equation of state parameter $w_{\phi}$ is a constant [@ts]. Subtracting both Eqs. (\[rhophi2\]) and (\[rhom2\]) and using Eq. (\[scaling\]) we get $$\frac{d\phi}{dN}=-\frac{3\Omega_{\phi}w_{\phi}}{Q}\propto\frac{1}{Q}.\label{dphidN}$$ Back to Eqs. (\[rhophi2\]) and (\[rhom2\]) we get $$\frac{d\log\rho_{\phi}}{dN}=\frac{d\log\rho_{m}}{dN}=-3(1+w_{eff}),$$ where $w_{eff}=w_{\phi}\Omega_{\phi}$. Now, from $w_{\phi}$ constant, we have $$\frac{d\log p}{dN}=-3(1+w_{eff})\label{dlogp_dn}$$ At this point we need a crucial statement, namely that the Lagrangian $\mathcal{L}$ is equivalent to the pressure of the scalar field, up to boundary terms. The demonstration that indeed this is true for the entire Horndeski Lagrangian is rather long and tedious and we moved it to the Appendix C. For our purposes, it is enough to show this to be true for a flat FLRW metric, since we are looking for scaling solutions only on such a metric. We conjecture that $\mathcal{L}=p$ for any metric; this indeed can easily be shown to be the case for the $K$ and the $G_{3}$ terms and we will present the general proof elsewhere.
From the equality $\mathcal{L}=p$ we have then $$\begin{aligned}
p & = & K-G_{3}\Box\phi+\biggl(-\frac{1}{2}+G_{4}\biggr)R+G_{4,X}[(\Box\phi)^{2}-\boxtimes\phi]\nonumber \\
& & +G_{5}(\phi,X)(\boxbar\phi)-\frac{1}{6}G_{5,X}[(\Box\phi)^{3}-3(\Box\phi)\boxtimes\phi+2(\boxdot\phi)\end{aligned}$$ with $$\begin{aligned}
\boxtimes\phi & \equiv & (\nabla_{\mu}\nabla_{\nu}\phi)(\nabla^{\mu}\nabla^{\nu}\phi)\\
\boxbar\phi & \equiv & G_{\mu\nu}(\nabla^{\mu}\nabla^{\nu}\phi)\\
\boxdot\phi & \equiv & (\nabla^{\mu}\nabla_{\alpha}\phi)(\nabla^{\alpha}\nabla_{\beta}\phi)(\nabla^{\beta}\nabla^{\mu}\phi)\end{aligned}$$ For FLRW metric, we have $R=6\dot{H}+12H^{2}$. Then, using Einstein equations, we get $$R=\biggl(\frac{1}{w_{\phi}\Omega_{\phi}}-3\biggr)p.$$ That is, for scaling solutions we have $R=\tilde{c}p$, with $\tilde{c}$ a constant.
Now defining $$f(\phi,X)\equiv\biggl[1+\biggl(\frac{1}{2}-G_{4}(\phi,X)\biggr)\tilde{c}\biggr],\label{f_def}$$ we can write the pressure of dark energy as $$\begin{aligned}
p & = & \frac{1}{f(\phi,X)}\biggl[K(\phi,X)-G_{3}(\phi,X)\Box\phi+G_{4,X}[(\Box\phi)^{2}-\boxtimes\phi]\nonumber \\
& & +G_{5}(\phi,X)(\boxbar\phi)-\frac{1}{6}G_{5,X}[(\Box\phi)^{3}-3(\Box\phi)\boxtimes\phi+2(\boxdot\phi)\biggr]\label{eq:p_finv}\end{aligned}$$ That is, $p=p(X,\Box{\phi},\boxtimes\phi,\boxbar\phi,\boxdot\phi,\phi)$
Now we want to find a generalized “master equation” for $p=p(X,\Box{\phi},\boxtimes\phi,\boxbar\phi,\boxdot\phi,\phi)$. Eq. (\[dlogp\_dn\]) gives $$\begin{aligned}
& & \frac{\partial\log p}{\partial\phi}\frac{d\phi}{dN}+\frac{\partial\log p}{\partial\log X}\frac{d\log X}{dN}+\frac{\partial\log p}{\partial\log\Box\phi}\frac{d\log\Box\phi}{dN}+\frac{\partial\log p}{\partial\log\boxtimes\phi}\frac{d\log\boxtimes\phi}{dN}\nonumber \\
& & +\frac{\partial\log p}{\partial\log\boxbar\phi}\frac{d\log\boxbar\phi}{dN}+\frac{\partial\log p}{\partial\log\boxdot\phi}\frac{d\log\boxdot\phi}{dN}=-3(1+w_{eff}).\label{master}\end{aligned}$$ Now we know that
$$\begin{aligned}
\frac{d\phi}{dN} & = & -\frac{3\Omega_{\phi}w_{\phi}}{Q}=3(1+w_{eff})\frac{1}{\lambda Q},\end{aligned}$$
with $$\lambda=-\frac{1+w_{eff}}{w_{eff}}.\label{lambda}$$ We need the partial derivatives ${d\log X}/{dN}$, ${d\log\Box\phi}/{dN}$, ${d\log\boxtimes\phi}/{dN}$, ${d\log\boxbar\phi}/{dN}$ and ${d\log\boxdot\phi}/{dN}$ that are obtained as follows (for details, see Appendix C):
${d\log X}/{dN}$
----------------
From the definition of $X$ and Eq. (\[dphidN\]) we have $$X=\frac{1}{2}\dot{\phi}^{2}=\frac{H^{2}}{2}\biggl(\frac{d\phi}{dN}\biggr)^{2}\propto\frac{H^{2}}{Q^{2}}=\frac{\rho_{\phi}}{3\Omega_{\phi}}\frac{1}{Q^{2}}\propto\frac{p}{Q^{2}},$$ and then $$\begin{aligned}
\frac{d\log X}{dN} & = & \frac{d\log p}{dN}-2\frac{d\log Q}{dN}\nonumber \\
& = & -3(1+w_{\mathrm{eff}})-\frac{2}{Q}\frac{dQ}{d\phi}\frac{d\phi}{dN}\\
& = & -3(1+w_{\mathrm{eff}})-\frac{2}{Q}\frac{dQ}{d\phi}\left(-\frac{3\Omega_{\phi}w_{\phi}}{Q}\right)\\
& = & -3(1+w_{\mathrm{eff}})\left(1+\frac{2}{\lambda Q^{2}}\frac{dQ}{d\phi}\right)\end{aligned}$$
${d\log\Box\phi}/{dN}$
----------------------
We start with (for details here and forthe next terms, see Appendices A and B) $$\Box\phi=-3H\dot{\phi}-\ddot{\phi}\label{Boxphi0}$$ Now, from Eq. (\[dphidN\]) this can be rewritten as $$\Box\phi=\frac{3}{2}(1-w_{eff})\frac{p}{Q}\left(1-\frac{2}{\lambda}\frac{1+w_{\mathrm{eff}}}{1-w_{\mathrm{eff}}}\frac{1}{Q^{2}}\frac{dQ}{d\phi}\right),\label{Boxphi}$$ So far we put no restrictions on the coupling function $Q$. However we find that the analysis is very simplified if we assume $$\frac{1}{Q^{2}}\frac{dQ}{d\phi}=c=const.\label{eqQ}$$ This restricts the coupling to be $$Q(\phi)=\frac{1}{-c\phi+c_{2}},\label{Qsol}$$ with $c_{2}$ constant. From now we we will consider this restriction in $Q(\phi)$. From Eq. (\[Boxphi\]) we have then $$\log\Box\phi=\log p-\log Q+const.,$$ which gives $$\frac{\partial\log\Box\phi}{\partial N}=-3(1+w_{eff})\biggl(1+\frac{1}{\lambda Q^{2}}\frac{dQ}{d\phi}\biggr).$$
${d\log\boxtimes\phi}/{dN}$
----------------------------
We start with $$\boxtimes\phi=(\ddot{\phi})^{2}+3H^{2}(\dot{\phi})^{2}.$$ This results in $$\frac{\partial\log\boxtimes\phi}{\partial N}=-3(1+w_{eff})\biggl(2+\frac{2}{\lambda Q^{2}}\frac{dQ}{d\phi}\biggr).$$
${d\log\boxbar\phi}/{dN}$
-------------------------
We start with $$\boxbar\phi=-3H^{2}\ddot{\phi}-6H\dot{H}\dot{\phi}-9H^{3}\dot{\phi}.$$ This results in $$\frac{\partial\log\boxbar\phi}{\partial N}=-3(1+w_{eff})\biggl(2+\frac{1}{\lambda Q^{2}}\frac{dQ}{d\phi}\biggr).$$
${d\log\boxdot\phi}/{dN}$
-------------------------
We start with $$\boxdot\phi=-\ddot{\phi}^{3}-3H^{3}\dot{\phi}^{3}.$$ This results in $$\frac{\partial\log\boxdot\phi}{\partial N}=-3(1+w_{eff})\biggl(3+\frac{3}{\lambda Q^{2}}\frac{dQ}{d\phi}\biggr).$$ Finally, Eq. (\[master\]) becomes $$\begin{aligned}
& & \biggl(1+\frac{2}{\lambda Q^{2}}\frac{dQ}{d\phi}\biggr)\frac{\partial\log p}{\partial\log X}+\biggl(1+\frac{1}{\lambda Q^{2}}\frac{dQ}{d\phi}\biggr)\frac{\partial\log p}{\partial\log\Box\phi}+\biggl(2+\frac{2}{\lambda Q^{2}}\frac{dQ}{d\phi}\biggr)\frac{\partial\log p}{\partial\log\boxtimes\phi}\nonumber \\
& & +\biggl(2+\frac{1}{\lambda Q^{2}}\frac{dQ}{d\phi}\biggr)\frac{\partial\log p}{\partial\log\boxbar\phi}+\biggl(3+\frac{3}{\lambda Q^{2}}\frac{dQ}{d\phi}\biggr)\frac{\partial\log p}{\partial\log\boxdot\phi}-\frac{1}{\lambda Q}\frac{\partial\log p}{\partial\phi}=1.\label{master2}\end{aligned}$$ As expected, the master equation reduces to the one obtained in Ref. [@Amendola:2006qi] when $G_{3}(\phi,X)=G_{5}(\phi,X)=0$ and $G_{4}(\phi,X)=1/2$.
Solutions for the master equation
=================================
Here, after a convenient *Ansatz*, we derive the general solution for the master equation Eq. (\[master2\]). Remember, however, that there are restrictions in the form of $Q(\phi)$, given by Eq. (\[Qsol\]), that will be taken into account in due course. We start with Eq. (\[master2\]) rewritten as $$\begin{aligned}
& & \biggl(1+\frac{2}{Q}\frac{dQ}{d\psi}\biggr)\frac{\partial\log p}{\partial\log X}+\biggl(1+\frac{1}{Q}\frac{dQ}{d\psi}\biggr)\frac{\partial\log p}{\partial\log\Box\phi}+\biggl(2+\frac{2}{Q}\frac{dQ}{d\psi}\biggr)\frac{\partial\log p}{\partial\log\boxtimes\phi}\nonumber \\
& & +\biggl(2+\frac{1}{Q}\frac{dQ}{d\psi}\biggr)\frac{\partial\log p}{\partial\log\boxbar\phi}+\biggl(3+\frac{3}{Q}\frac{dQ}{d\psi}\biggr)\frac{\partial\log p}{\partial\log\boxdot\phi}-\frac{\partial\log p}{\partial\psi}=1,\label{master3}\end{aligned}$$ where $$\psi=\int_{\phi}du[\lambda Q(u)].\label{psi-def}$$
Now set $$p=XQ^{2}(\phi)\tilde{G}(X,\Box\phi,\boxtimes\phi,\boxbar\phi,\boxdot\phi,\phi).\label{p-tildeg-Q}$$ where $\tilde{G}$ is an arbitray function of its argument. Then for $\tilde{G}\neq0$ we obtain $$\begin{aligned}
& & \biggl(1+\frac{2}{Q}\frac{dQ}{d\psi}\biggr)X\frac{\partial\tilde{G}}{\partial X}+\biggl(1+\frac{1}{Q}\frac{dQ}{d\psi}\biggr)\Box\phi\frac{\partial\tilde{G}}{\partial\Box\phi}+\biggl(2+\frac{2}{Q}\frac{dQ}{d\psi}\biggr)(\boxtimes\phi)\frac{\partial\tilde{G}}{\partial(\boxtimes\phi)}\nonumber \\
& & +\biggl(2+\frac{1}{Q}\frac{dQ}{d\psi}\biggr)(\boxbar\phi)\frac{\partial\tilde{G}}{\partial(\boxbar\phi)}+\biggl(3+\frac{3}{Q}\frac{dQ}{d\psi}\biggr)(\boxdot\phi)\frac{\partial\tilde{G}}{\partial(\boxdot\phi)}-\frac{\partial\tilde{G}}{\partial\psi}=0.\label{masterQ}\end{aligned}$$ where by (\[eqQ\]) the term $\frac{2}{Q}\frac{dQ}{d\psi}$ is a constant. This partial differential equation is linear in $\tilde{G}$. Then the method of separation of variables is justifiable. Here, however, inspired by the expression given by Eq. (\[eq:p\_finv\]) for $p$, we look for solutions of $\tilde{G}$ of the form $$\tilde{G}(X,\Box\phi,\boxtimes\phi,\boxbar\phi,\boxdot\phi,\phi)=\frac{1}{f(\phi,X)}{\tilde{g}(X,\Box\phi,\boxtimes\phi,\boxbar\phi,\boxdot\phi,\phi)}.$$ Then, Eq. (\[masterQ\]) turns into $$\begin{aligned}
& & \frac{1}{f}\biggl\{\biggl[\biggl(1+\frac{2}{Q}\frac{dQ}{d\psi}\biggr)X\frac{\partial f}{\partial X}-\frac{\partial f}{\partial\psi}\biggr]\frac{\tilde{g}}{f}\nonumber \\
& & -\biggl[\biggl(1+\frac{2}{Q}\frac{dQ}{d\psi}\biggr)X\frac{\partial\tilde{g}}{\partial X}+\biggl(1+\frac{1}{Q}\frac{dQ}{d\psi}\biggr)\Box\phi\frac{\partial\tilde{g}}{\partial\Box\phi}+\biggl(2+\frac{2}{Q}\frac{dQ}{d\psi}\biggr)(\boxtimes\phi)\frac{\partial\tilde{g}}{\partial(\boxtimes\phi)}\nonumber \\
& & +\biggl(2+\frac{1}{Q}\frac{dQ}{d\psi}\biggr)(\boxbar\phi)\frac{\partial\tilde{g}}{\partial(\boxbar\phi)}+\biggl(3+\frac{3}{Q}\frac{dQ}{d\psi}\biggr)(\boxdot\phi)\frac{\partial\tilde{g}}{\partial(\boxdot\phi)}-\frac{\partial\tilde{g}}{\partial\psi}\biggr]\biggr\}=0.\end{aligned}$$ For $f\neq0$ this is equivalent to $(\widehat{O}_{1}f)\tilde{G}-\widehat{O}_{2}\tilde{g}=0,$where the operators $\widehat{O}_{1}$ and $\widehat{O}_{2}$ are evident in the equation above. For $G_{4}=1/2$ and $G_{5}=0$ we have $f=1$ and the linear differential equation $\widehat{O}_{2}\tilde{g}=0$ is reduced to a form already solved in our previous paper[@2014JCAP...03..041G] . Now we want to generalize our former result for general $G_{4}$ and $G_{5}$. The simplest choice is to impose that $\tilde{g}$ and $f$ satisfy separately the linear differential equations: $$\widehat{O}_{1}f=0,$$ that is $$\biggl(1+\frac{2}{Q}\frac{dQ}{d\psi}\biggr)X\frac{\partial f}{\partial X}-\frac{\partial f}{\partial\psi}=0$$ with solution (known for the case where $G_{2}\neq0$, $G_{4}=1/2$ and $G_{3}=G_{5}=0)$. $$f(\phi,X)=g_{2}(XQ^{2}e^{\psi}),$$ where $g_{2}$ is a general function. Now we turn to the equation for $\tilde{g}$: $$\widehat{O}_{2}g=0,$$ that is $$\begin{aligned}
& & \biggl(1+\frac{2}{Q}\frac{dQ}{d\psi}\biggr)X\frac{\partial\tilde{g}}{\partial X}+\biggl(1+\frac{1}{Q}\frac{dQ}{d\psi}\biggr)\Box\phi\frac{\partial\tilde{g}}{\partial\Box\phi}+\biggl(2+\frac{2}{Q}\frac{dQ}{d\psi}\biggr)(\boxtimes\phi)\frac{\partial\tilde{g}}{\partial(\boxtimes\phi)}\nonumber \\
& & +\biggl(2+\frac{1}{Q}\frac{dQ}{d\psi}\biggr)(\boxbar\phi)\frac{\partial\tilde{g}}{\partial(\boxbar\phi)}+\biggl(3+\frac{3}{Q}\frac{dQ}{d\psi}\biggr)(\boxdot\phi)\frac{\partial\tilde{g}}{\partial(\boxdot\phi)}-\frac{\partial\tilde{g}}{\partial\psi}=0\end{aligned}$$
We do not want the general solution of this linear differential equation. Indeed, guided by the form of $p$, we look for a solution of the form (all other possibilites lead to trivial constant solutions or are not compatible with the form of the Lagrangian). $$\begin{aligned}
\tilde{g} & = & g_{b}(h_{b})+g_{c}(h_{c})+g_{d}(h_{d})+g_{\ell}(h_{\ell})+g_{e}(h_{e})+g_{j}(h_{j})+g_{m}(h_{m})+g_{n}(h_{n})+g_{p}(h_{p})\nonumber \\
& & +g_{q}(h_{q})+g_{r}(h_{r})+g_{s}(h_{s})+g_{t}(h_{t})+g_{u}(h_{u})+g_{z}(h_{v})+g_{z}(h_{z}),\label{ga-gd-Q}\end{aligned}$$ where $g_{b},g_{c},g_{d},...$ are arbitrary functions and $$\begin{aligned}
h_{b}(X,\psi) & = & f_{1b}(X)f_{3b}(\psi),\label{hbQ}\\
h_{c}(X,\Box\phi) & = & f_{1c}(X)f_{2c}(\Box\phi)\label{hcQ}\\
h_{d}(\Box\phi,\psi) & = & f_{2d}(\Box\phi)f_{3d}(\psi).\label{hdQ}\\
h_{\ell}(X,\Box\phi,\psi) & = & f_{1\ell}(X)f_{2\ell}(\Box\phi)f_{3\ell}(\psi)\label{hlQ}\\
h_{e}(X,\boxtimes\phi) & = & f_{1e}(X)f_{4e}(\boxtimes\phi)\label{heQ}\\
h_{j}(\boxtimes\phi,\psi) & = & f_{3j}(\psi)f_{4j}(\boxtimes\phi)\label{hjQ}\\
h_{m}(X,\boxtimes\phi,\psi) & = & f_{1m}(X)f_{3m}(\psi)f_{4m}(\boxtimes\phi)\label{hmQ}\\
h_{n}(\psi,\boxbar\phi) & = & f_{2n}(\psi)f_{5n}(\boxbar\phi)\label{hnQ}\\
h_{p}(X,\boxbar\phi) & = & f_{1p}(X)f_{5p}(\boxbar\phi)\label{hpQ}\\
h_{q}(X,\boxbar\phi,\psi) & = & f_{1q}(X)f_{3q}(\psi)f_{5q}(\boxbar\phi)\label{hqQ}\\
h_{r}(X,\boxdot\phi) & = & f_{1r}(X)f_{6r}(\boxdot\phi)\label{hrQ}\\
h_{s}(\psi,\boxdot\phi) & = & f_{2s}(\psi)f_{6s}(\boxdot\phi)\label{hsQ}\\
h_{t}(X,\boxdot\phi,\psi) & = & f_{1t}(X)f_{3t}(\psi)f_{6t}(\boxdot\phi)\label{htQ}\\
h_{u}(X,\Box\phi,\boxtimes\phi) & = & f_{1u}(X)f_{2u}(\Box\phi)f_{4u}(\boxtimes\phi)\label{huQ}\\
h_{v}(\psi,\Box\phi,\boxtimes\phi) & = & f_{3v}(\psi)f_{2v}(\Box\phi)f_{4v}(\boxtimes\phi)\label{hvQ}\\
h_{z}(\psi,X,\Box\phi,\boxtimes\phi) & = & f_{1z}(X)f_{3z}(\psi)f_{2z}(\Box\phi)f_{4z}(\boxtimes\phi)\label{hzQ}\end{aligned}$$ In the following we will consider separately each possibility. We will be guided by the general form given by Eq. (\[eq:p\_finv\]) for $p$, which fixes the maximum order of the factors $\Box\phi,\boxtimes\phi,\boxbar\phi$ and $(\boxdot\phi)$.
$g_{b}(h_{b}(X,\psi))$
----------------------
Eq. (\[masterQ\]) gives $$\frac{dg}{dh_{b}}\biggl[\biggl(1+\frac{2}{Q}\frac{dQ}{d\psi}\biggr)\frac{d\log f_{1b}}{d\log X}-\frac{1}{f_{3b}}\frac{df_{3b}}{d\psi}\biggr]=0,$$ which gives $f_{1b}=X^{\alpha}$ and $f_{3b}=e^{\alpha\psi}Q^{2\alpha}$. Then Eq. (\[hbQ\]) gives $$h_{b}=(XQ^{2}(\phi)e^{\psi})^{\alpha}$$ and therefore $$g(h_{b})=g(XQ^{2}(\phi)e^{\psi}),$$ where $g$ is an arbitrary function.
$g_{c}(h_{c}(X,\Box\phi))$
--------------------------
This gives $g_{c}$ constant.
$g_{d}(h_{d}(\Box\phi,\psi))$
-----------------------------
Eq. (\[masterQ\]) gives $$\frac{dg_{d}}{dh_{d}}\biggl[\biggl(1+\frac{1}{Q}\frac{dQ}{d\psi}\biggr)\frac{d\log f_{2d}}{d\log\Box\phi}-\frac{1}{f_{3d}}\frac{df_{3d}}{d\psi}\biggr]=0.\label{master-gd-Q}$$ which gives $f_{2d}=(\Box\phi)^{\alpha}$ and $f_{3d}=Q^{\alpha}e^{\alpha\psi}$. Then Eq. (\[hdQ\]) gives $$h_{d}=\biggl((\Box\phi)e^{\psi}Q\biggr)^{\alpha}$$ and therefore $$g_{d}(h_{d})=g_{d}\biggl((\Box\phi)Q(\phi)e^{\psi}\biggr).$$ Now comparing with Eq. (\[eq:p\_finv\]) for $p$, we see that $g_{d}(h_{d})$ must be at most third order in $\Box\phi$. Then we have $$g_{d}(h_{d})=d_{1}\biggl((\Box\phi)Q(\phi)e^{\psi}\biggr)+d_{2}\biggl((\Box\phi)Q(\phi)e^{\psi}\biggr)^{2}+d_{3}\biggl((\Box\phi)Q(\phi)e^{\psi}\biggr)^{3},$$ with $d_{1},d_{2},d_{3}$ constants.
$g_{\ell}(h_{\ell}(X,\Box\phi,\psi))$
-------------------------------------
Eq. (\[masterQ\]) gives $$\frac{dg_{\ell}}{dh_{\ell}}\biggl[\biggl(1+\frac{2}{Q}\frac{dQ}{d\psi}\biggr)\frac{1}{f_{1\ell}}\frac{df_{1\ell}}{d\log X}+\biggl(1+\frac{1}{Q}\frac{dQ}{d\psi}\biggr)\frac{1}{f_{2\ell}}\frac{df_{2\ell}}{d\log\Box\phi}-\frac{1}{f_{3\ell}}\frac{df_{3\ell}}{d\psi}\biggr]=0$$ which gives $f_{1\ell}=X^{\alpha}$, $f_{2\ell}=(\Box\phi)^{\beta}$ and $f_{3\ell}=e^{(\alpha+\beta)\psi}Q^{2\alpha+\beta}$. Then Eq. (\[hlQ\]) gives $$h_{\ell}=\biggl[X(\Box\phi)^{\beta/\alpha}e^{(1+\beta/\alpha)\psi}Q^{2+\beta/\alpha}\biggr]^{\alpha}$$ and $$g_{\ell}(h_{\ell})=g_{\ell}\biggl(X(\Box\phi)^{n}e^{(1+n)\psi}Q^{2+n}\biggr),$$ where from Eq. (\[eq:p\_finv\]) for $p$, the exponent $n$ can be $1$, $2$ or $3$ for terms linear, quadratic or cubic in $(\Box\phi)$. This leads to $$g_{\ell}(h_{\ell})={\ell_{1}}\biggl(X(\Box\phi)e^{2\psi}Q^{3}(\phi)\biggr)+{\ell_{2}}\biggl(X(\Box\phi)^{2}e^{3\psi}Q^{4}(\phi)\biggr)+{\ell_{3}}\biggl(X(\Box\phi)^{3}e^{4\psi}Q^{5}(\phi)\biggr),$$ with $\ell_{1},\ell_{2},\ell_{3}$ constants
$g_{e}(h_{e}(X,\boxtimes\phi))$
-------------------------------
This gives $g_{e}$ constant.
$g_{j}(h_{j}(\boxtimes\phi,\psi))$
----------------------------------
Eq. (\[masterQ\]) gives $$\frac{dg_{j}}{dh_{j}}\biggl[\biggl(2+\frac{2}{Q}\frac{dQ}{d\psi}\biggr)\frac{d\log f_{4j}}{d\log\boxtimes\phi}-\frac{1}{f_{3j}}\frac{df_{3j}}{d\psi}\biggr]=0.\label{master-gj-Q}$$ which gives $f_{4j}=(\boxtimes\phi)^{\alpha}$ and $f_{3j}=Q^{2\alpha}e^{2\alpha\psi}$. Then Eq. (\[hjQ\]) gives $$h_{j}=\biggl((\boxtimes\phi)e^{2\psi}Q^{2}\biggr)^{\alpha}.$$ Comparing with Eq. (\[eq:p\_finv\]) for $p$, we see that $g_{j}(h_{j})$ must be at most linear in $\Box\phi$. Then we get $$g_{j}(h_{j})={j}\biggl((\boxtimes\phi)Q^{2}e^{2\psi}\biggr),$$ with $j$ a constant.
$g_{m}(h_{m}(X,\boxtimes\phi,\psi))$
------------------------------------
Eq. (\[masterQ\]) gives $$\frac{dg_{m}}{dh_{m}}\biggl[\biggl(1+\frac{2}{Q}\frac{dQ}{d\psi}\biggr)\frac{1}{f_{1m}}\frac{df_{1m}}{d\log X}+\biggl(2+\frac{2}{Q}\frac{dQ}{d\psi}\biggr)\frac{1}{f_{2m}}\frac{d\log f_{2m}}{d\log\boxtimes\phi}-\frac{1}{f_{3m}}\frac{df_{3m}}{d\psi}\biggr]=0$$ which gives $f_{1m}=X^{\alpha}$, $f_{4m}=(\boxtimes\phi)^{\beta}$ and $f_{3m}=e^{(\alpha+2\beta)\psi}Q^{2\alpha+2\beta}$. Then Eq. (\[hmQ\]) gives $$h_{m}=\biggl[X(\boxtimes\phi)^{\beta/\alpha}e^{(1+2\beta/\alpha)\psi}Q^{2+2\beta/\alpha}\biggr]^{\alpha}.$$ Then comparing with Eq. (\[eq:p\_finv\]) we get $$g_{m}(h_{m})={m}\biggl(X(\boxtimes\phi)e^{3\psi}Q^{4}\biggr),$$ with $m$ a constant.
$g_{p}(h_{p}(X,\boxbar\phi))$
-----------------------------
This gives $g_{p}$ constant.
$g_{n}(h_{n}(\boxbar\phi,\psi))$
--------------------------------
Eq. (\[masterQ\]) gives $$\frac{dg_{n}}{dh_{n}}\biggl[\biggl(2+\frac{1}{Q}\frac{dQ}{d\psi}\biggr)\frac{d\log f_{5n}}{d\log\boxbar\phi}-\frac{1}{f_{3n}}\frac{df_{3n}}{d\psi}\biggr]=0.$$ which gives $f_{5n}=(\boxbar\phi)^{\alpha}$ and $f_{3n}=Q^{\alpha}e^{2\alpha\psi}$. Then Eq. (\[hjQ\]) gives $$h_{n}=\biggl((\boxbar\phi)e^{2\psi}Q\biggr)^{\alpha}.$$ Comparing with Eq. (\[eq:p\_finv\]), we see that $g_{n}(h_{n})$ must be at most linear in $(\boxbar\phi)$.Then we get $$g_{n}(h_{n})={n}\biggl((\boxbar\phi)Qe^{2\psi}\biggr),$$ with $n$ a constant.
$g_{q}(h_{q}(X,\boxbar\phi,\psi))$
----------------------------------
Eq. (\[masterQ\]) gives $$\frac{dg_{q}}{dh_{q}}\biggl[\biggl(1+\frac{2}{Q}\frac{dQ}{d\psi}\biggr)\frac{1}{f_{1q}}\frac{df_{1q}}{d\log X}+\biggl(2+\frac{1}{Q}\frac{dQ}{d\psi}\biggr)\frac{1}{f_{5q}}\frac{d\log f_{5q}}{d\log\boxbar\phi}-\frac{1}{f_{3q}}\frac{df_{3q}}{d\psi}\biggr]=0$$ which gives $f_{1q}=X^{\alpha}$, $f_{5q}=(\boxbar\phi)^{\beta}$ and $f_{3q}=e^{(\alpha+2\beta)\psi}Q^{2\alpha+\beta}$. Then Eq. (\[hmQ\]) gives $$h_{q}=\biggl[X(\boxbar\phi)^{\beta/\alpha}e^{(1+2\beta/\alpha)\psi}Q^{2+\beta/\alpha}\biggr]^{\alpha}.$$ Then comparing with Eq. (\[eq:p\_finv\]), we get $$g_{q}(h_{q})={q}\biggl(X(\boxbar\phi)e^{3\psi}Q^{3}\biggr),$$ with $m$ a constant.
$g_{r}(h_{r}(X,\boxdot\phi))$
-----------------------------
This gives $g_{r}$ constant.
$g_{s}(h_{s}(\boxdot\phi,\psi))$
--------------------------------
Eq. (\[masterQ\]) gives $$\frac{dg_{s}}{dh_{s}}\biggl[\biggl(3+\frac{3}{Q}\frac{dQ}{d\psi}\biggr)\frac{d\log f_{6s}}{d\log\boxdot\phi}-\frac{1}{f_{3s}}\frac{df_{3s}}{d\psi}\biggr]=0.$$ which gives $f_{6s}=(\boxdot\phi)^{\alpha}$ and $f_{3n}=Q^{3\alpha}e^{3\alpha\psi}$. Then Eq. (\[hjQ\]) gives $$h_{s}=\biggl((\boxbar\phi)e^{3\psi}Q^{3}\biggr)^{\alpha}.$$ Then comparing with Eq. (\[eq:p\_finv\]), we get $$g_{s}(h_{n})={s}\biggl((\boxdot\phi)Q^{3}e^{3\psi}\biggr),$$ with $s$ a constant.
$g_{t}(h_{t}(X,\boxdot\phi,\psi))$
----------------------------------
Eq. (\[masterQ\]) gives $$\frac{dg_{t}}{dh_{t}}\biggl[\biggl(1+\frac{2}{Q}\frac{dQ}{d\psi}\biggr)\frac{1}{f_{1t}}\frac{df_{1t}}{d\log X}+\biggl(3+\frac{3}{Q}\frac{dQ}{d\psi}\biggr)\frac{1}{f_{6t}}\frac{df_{6t}}{d\log\boxdot\phi}-\frac{1}{f_{3t}}\frac{df_{3t}}{d\psi}\biggr]=0$$ which gives $f_{1t}=X^{\alpha}$, $f_{6t}=(\boxdot\phi)^{\beta}$ and $f_{3t}=e^{(\alpha+3\beta)\psi}Q^{2\alpha+3\beta}$. Then Eq. (\[htQ\]) gives $$h_{t}=\biggl[X(\boxdot\phi)^{\beta/\alpha}e^{(1+3\beta/\alpha)\psi}Q^{2+3\beta/\alpha}\biggr]^{\alpha}.$$ Comparing with Eq. (\[eq:p\_finv\]) for $p$, we see that $g_{t}(h_{t})$ must be at most linear in $(\boxbar\phi)$. Then we get $$g_{t}(h_{t})={t}\biggl(X(\boxdot\phi)e^{4\psi}Q^{5}\biggr),$$ with $t$ a constant.
$g_{u}(h_{u}(X,\Box\phi,\boxtimes\phi))$
----------------------------------------
This gives $g_{u}$ constant.
$g_{v}(h_{v}(\Box\phi,\boxtimes\phi,\psi))$
-------------------------------------------
Eq. (\[masterQ\]) gives $$\frac{dg_{v}}{dh_{v}}\biggl[\biggl(1+\frac{1}{Q}\frac{dQ}{d\psi}\biggr)\frac{d\log f_{1v}}{d\log X}+\biggl(2+\frac{2}{Q}\frac{dQ}{d\psi}\biggr)\frac{d\log f_{4v}}{d\log\boxtimes\phi}-\frac{1}{f_{3s}}\frac{df_{3v}}{d\psi}\biggr]=0.$$ which gives $$g_{v}(h_{v})={v}\biggl((\Box\phi)(\boxtimes\phi)Q^{3}e^{3\psi}\biggr),$$ with $v$ a constant.
$g_{z}(h_{z}(X,\Box\phi,\boxtimes\phi,\psi))$
---------------------------------------------
Eq. (\[masterQ\]) gives $$\begin{aligned}
& & \frac{dg_{z}}{dh_{z}}\biggl[\biggl(1+\frac{2}{Q}\frac{dQ}{d\psi}\biggr)\frac{1}{f_{1z}}\frac{df_{1z}}{d\log X}+\biggl(1+\frac{1}{Q}\frac{dQ}{d\psi}\biggr)\frac{1}{f_{2z}}\frac{df_{2z}}{d\log\Box\phi}\nonumber \\
& & +\biggl(2+\frac{2}{Q}\frac{dQ}{d\psi}\biggr)\frac{1}{f_{4z}}\frac{df_{4z}}{d\log\boxtimes\phi}-\frac{1}{f_{3z}}\frac{df_{3z}}{d\psi}\biggr]=0\end{aligned}$$ which gives $f_{1z}=X^{\alpha}$, $f_{2z}=(\Box\phi)^{\beta}$, $f_{4z}=(\boxtimes\phi)^{\beta}$, and $f_{3t}=e^{(\alpha+3\beta)\psi}Q^{2\alpha+3\beta}$. Then Eq. (\[htQ\]) gives $$h_{z}=\biggl[X(\Box\phi\boxtimes\phi)^{\beta/\alpha}e^{(1+3\beta/\alpha)\psi}Q^{2+3\beta/\alpha}\biggr]^{\alpha}.$$ Then comparing with Eq. (\[eq:p\_finv\]), we get $$g_{t}(h_{z})={z}\biggl(X(\Box\phi\boxtimes\phi)e^{4\psi}Q^{5}\biggr),$$ with $z$ a constant.
General form for Lagrangian with scaling solution
-------------------------------------------------
From the former results and Eq. (\[ga-gd-Q\]) we obtain
$$\begin{aligned}
\tilde{g}(X,\Box\phi,\boxtimes\phi,\boxbar\phi,\boxdot\phi,\phi) & = & g(XQ^{2}(\phi)e^{\psi})+d_{1}\biggl((\Box\phi)Q(\phi)e^{\psi}\biggr)+d_{2}\biggl((\Box\phi)Q(\phi)e^{\psi}\biggr)^{2}\nonumber \\
& & +d_{3}\biggl((\Box\phi)Q(\phi)e^{\psi}\biggr)^{3}+{\ell_{1}}\biggl(X(\Box\phi)e^{2\psi}Q^{3}(\phi)\biggr)+{\ell_{2}}\biggl(X(\Box\phi)^{2}e^{3\psi}Q^{4}(\phi)\biggr)\nonumber \\
& & +{\ell_{3}}\biggl(X(\Box\phi)^{3}e^{4\psi}Q^{5}(\phi)\biggr)+j(Q^{2}(\phi)e^{2\psi}(\boxtimes\phi))+{m}(XQ^{4}(\phi)e^{3\psi}(\boxtimes\phi))\nonumber \\
& & +{n}\biggl((\boxbar\phi)Q(\phi)e^{2\psi}\biggr)+{q}\biggl(X(\boxbar\phi)e^{3\psi}Q^{3}\biggr)+{s}\biggl((\boxdot\phi)Q(\phi)^{3}e^{3\psi}\biggr)\nonumber \\
& & +{t}\biggl(X(\boxdot\phi)e^{4\psi}Q^{5}\biggr)+{v}\biggl((\Box\phi)(\boxtimes\phi)Q^{3}e^{3\psi}\biggr)\nonumber \\
& & +{z}\biggl(X(\Box\phi\boxtimes\phi)e^{4\psi}Q^{5}\biggr).\end{aligned}$$
We have also obtained $$f(\phi,X)=g_{2}(XQ^{2}e^{\psi}),$$ Now we need to compare our results and Eq. (\[p-tildeg-Q\]) , rewritten as $$p=\frac{1}{f(\phi,X)}[XQ^{2}(\phi)\tilde{g}(X,\Box\phi,\boxtimes\phi,\boxbar\phi,\boxdot\phi,\phi)]$$ with the expression of $p$ given by Eq. (\[eq:p\_finv\])**,** rewritten as: $$\begin{aligned}
p & = & \frac{1}{\biggl[1+\biggl(\frac{1}{2}-G_{4}(\phi,X)\biggr)\tilde{c}\biggr]}\biggl[K(\phi,X)-G_{3}(\phi,X)\Box\phi+G_{4,X}[(\Box\phi)^{2}-\boxtimes\phi]\nonumber \\
& & +G_{5}(\phi,X)(\boxbar\phi)-\frac{1}{6}G_{5,X}[(\Box\phi)^{3}-3(\Box\phi)\boxtimes\phi+2(\boxdot\phi)\biggr].\end{aligned}$$ This results in $$\begin{aligned}
1+\biggl(\frac{1}{2}-G_{4}(\phi,X)\biggr)\tilde{c} & = & g_{2}(XQ^{2}e^{\psi})\\
K(\phi,X) & = & XQ^{2}g(XQ^{2}(\phi)e^{\psi})\\
-G_{3}(\phi,X)\Box\phi & = & XQ^{2}\biggl[d_{1}\biggl((\Box\phi)Q(\phi)e^{\psi}\biggr)\nonumber \\
& & +{\ell_{1}}\biggl(X(\Box\phi)e^{2\psi}Q^{3}(\phi)\biggr)\biggr]\\
G_{4,X}[(\Box\phi)^{2}-\boxtimes\phi] & = & XQ^{2}\biggl[d_{2}\biggl((\Box\phi)Q(\phi)e^{\psi}\biggr)^{2}\nonumber \\
& & +{\ell_{2}}\biggl(X(\Box\phi)^{2}e^{3\psi}Q^{4}(\phi)\biggr)\nonumber \\
& & +j(Q^{2}(\phi)e^{2\psi}\boxtimes\phi)\nonumber \\
& & +{m}(XQ^{4}(\phi)e^{3\psi}\boxtimes\phi)\biggr]\\
G_{5}(\phi,X)(\boxbar\phi) & = & XQ^{2}\biggl[{n}\biggl((\boxbar\phi)Q(\phi)e^{2\psi}\biggr)\nonumber \\
& & +{q}\biggl(X(\boxbar\phi)e^{3\psi}Q^{3}\biggr)\biggr]\\
-\frac{1}{6}G_{5,X}[(\Box\phi)^{3}-3(\Box\phi)\boxtimes\phi+2(\boxdot\phi)\biggr] & = & XQ^{2}\biggl[d_{3}\biggl((\Box\phi)Q(\phi)e^{\psi}\biggr)^{3}\nonumber \\
& & +{\ell_{3}}\biggl(X(\Box\phi)^{3}e^{4\psi}Q^{5}(\phi)\biggr)\nonumber \\
& & +{v}\biggl((\Box\phi)(\boxtimes\phi)Q^{3}e^{3\psi}\biggr)\nonumber \\
& & {z}\biggl(X(\Box\phi\boxtimes\phi)e^{4\psi}Q^{5}\biggr)\nonumber \\
& & +{s}\biggl((\boxdot\phi)Q(\phi)^{3}e^{3\psi}\biggr)\nonumber \\
& & +{t}\biggl(X(\boxdot\phi)e^{4\psi}Q^{5}\biggr)\biggr]\end{aligned}$$ Now, in each of the former equations from above we must impose relations between the general constants in order to get a compatible solution for the general functions.
- For $G_{4}$: $$d_{2}=-j,$$ $$\ell_{2}=-m,$$
- For $G_{5}$: $$\begin{aligned}
n & = & 0,\\
\ell_{3} & = & 0,\\
t & = & 0,\\
z & = & 0\\
v & = & -3d_{3},\\
s & = & 2d_{3},\end{aligned}$$
Additionally, consistency between expressions for $G_{5}$ and $G_{5,X}$ gives
$$\begin{aligned}
d_{3} & = & -\frac{1}{3}q.\end{aligned}$$
The above conditions and consistency between expressions for $G_{4}$ and $G_{4,X}$ result in the following expressions for the functions that compose the Horndeski Lagrangian: $$\begin{aligned}
K(\phi,X) & = & XQ^{2}g(XQ^{2}e^{\psi})\\
G_{3}(\phi,X) & = & -d_{1}XQ^{3}e^{\psi}-{\ell_{1}}X^{2}Q^{5}e^{2\psi}\\
G_{4}(\phi,X) & = & h(\phi)+\frac{1}{2}d_{2}X^{2}Q^{4}e^{2\psi}+\frac{1}{3}\ell_{2}Xe^{3\psi}Q^{6}\\
G_{5}(\phi,X) & = & qX^{2}e^{3\psi}Q^{5}.\end{aligned}$$ where $h(\phi)$ is a general smooth function of $\phi$. Then the general scaling Horndeski Lagrangian is $$\begin{aligned}
\mathcal{L}_{H} & = & XQ^{2}g(XQ^{2}e^{\psi})-[d_{1}XQ^{3}e^{\psi}+{\ell_{1}}X^{2}Q^{5}e^{2\psi}]\Box\phi\nonumber \\
& & +\biggl(h(\phi)+\frac{1}{2}d_{2}X^{2}Q^{4}e^{2\psi}+\frac{1}{3}{\ell_{2}}X^{3}e^{3\psi}Q^{6}\biggr)R\nonumber \\
& & +\biggl(d_{2}XQ^{4}e^{2\psi}+\ell_{2}X^{2}e^{3\psi}Q^{6}\biggr)[(\Box\phi)^{2}-(\nabla_{\mu}\nabla_{\nu}\phi)(\nabla^{\mu}\nabla^{\nu}\phi)]\nonumber \\
& & +qX^{2}e^{3\psi}Q^{5}G_{\mu\nu}\nabla^{\mu}\nabla^{\nu}\phi\nonumber \\
& & -\biggl(\frac{1}{6}\biggr)2qXe^{3\psi}Q^{5}[(\Box\phi)^{3}-3(\Box\phi)(\nabla_{\mu}\nabla_{\nu}\phi)(\nabla^{\mu}\nabla^{\nu}\phi)\nonumber \\
& & +2(\nabla^{\mu}\nabla_{\alpha}\phi)(\nabla^{\alpha}\nabla_{\beta}\phi)(\nabla^{\beta}\nabla_{\mu}\phi)].\label{L_scal}\end{aligned}$$ Note that case $d_{2}=\ell_{2}=q=0$ and $h(\phi)=\frac{1}{2}$ recover the solution we have found in Ref. [@2014JCAP...03..041G] for scaling cosmological solutions in the KGB model.
Now, in order to ease the comparison with the literature, let us make the following field redefinitions. First of all take $\psi\to\lambda\psi$. Then
$$\psi(\phi)=\int_{\phi}duQ(u).\label{psi-def2}$$
Now consider $\phi\to\psi(\phi)$, with $\psi(\phi)$ given by Eq. (\[psi-def2\]). This implies
$$\begin{aligned}
X & \to & X_{\psi}=XQ^{2}(\phi)\\
Q\Box\phi & \to & \Box\psi+2\frac{d\log Q}{d\psi}X_{\psi}\\
Q\nabla^{\mu}\nabla^{\nu}\phi & \to & \nabla^{\mu}\nabla^{\nu}\psi-\frac{d\log Q}{d\psi}\partial^{\mu}\psi\partial^{\nu}\psi\\
Q^{2}\boxtimes\phi & \to & \boxtimes\psi-2\frac{d\log Q}{d\psi}\partial_{\mu}\psi\partial_{\nu}\psi\nabla^{\mu}\nabla^{\nu}\psi+4\biggl(\frac{d\log Q}{d\psi}\biggr)^{2}X_{\psi}^{2}\\
Q^{3}\boxdot\phi & \to & \boxdot\psi-\frac{d\log Q}{d\psi}\biggl(\partial^{\mu}\psi\partial_{\alpha}\psi\nabla^{\alpha}\nabla_{\beta}\psi\nabla^{\beta}\nabla_{\mu}\psi+\nabla^{\mu}\psi\nabla_{\alpha}\psi\partial^{\alpha}\partial_{\beta}\psi\nabla^{\beta}\nabla_{\mu}\psi\nonumber \\
& & +\nabla^{\mu}\psi\nabla_{\alpha}\psi\nabla^{\alpha}\nabla_{\beta}\psi\partial^{\beta}\partial_{\mu}\psi\biggr)\nonumber \\
& & -\left(\frac{d\log Q}{d\psi}\right)^{2}\biggl(\nabla^{\mu}\nabla_{\alpha}\psi\partial^{\alpha}\psi\partial_{\beta}\psi\partial^{\beta}\psi\partial_{\mu}\psi+\partial^{\mu}\psi\partial_{\alpha}\psi\nabla^{\alpha}\nabla_{\beta}\psi\partial^{\beta}\psi\partial_{\mu}\psi\nonumber \\
& & +\partial^{\mu}\psi\partial_{\alpha}\psi\partial^{\alpha}\psi\partial_{\beta}\psi\nabla^{\beta}\nabla_{\mu}\psi\biggr)-\left(\frac{d\log Q}{d\psi}\right)^{3}(8X_{\psi}^{2})\\
\frac{1}{Q^{2}}\frac{dQ}{d\phi} & \to & \frac{d\log Q}{d\psi}\end{aligned}$$
Then Eq. (\[L\_scal\]) turns into $$\begin{aligned}
\mathcal{L}_{H} & = & X_{\psi}g(X_{\psi}e^{\lambda\psi})-[d_{1}X_{\psi}e^{\lambda\psi}+{\ell_{1}}X_{\psi}^{2}e^{2\lambda\psi}]\left(\Box\psi+2\frac{d\ln Q}{d\psi}X_{\psi}\right)\nonumber \\
& & +\biggl(h(\psi)+\frac{1}{2}d_{2}X_{\psi}^{2}e^{2\lambda\psi}+\frac{1}{3}{\ell_{2}}X_{\psi}^{3}e^{3\lambda\psi}\biggr)R\nonumber \\
& & +\biggl(d_{2}X_{\psi}e^{2\lambda\psi}+\ell_{2}X_{\psi}^{2}e^{3\lambda\psi}\biggr)\left[(\Box\psi)^{2}-\boxtimes\phi+4\frac{d\ln Q}{d\psi}\left(X_{\psi}\Box\psi+\frac{1}{2}\partial^{\mu}\psi\partial_{\nu}\psi\nabla_{\mu}\nabla^{\nu}\psi\right)\right]\nonumber \\
& & +qX_{\psi}^{2}e^{3\lambda\psi}G_{\mu\nu}\left(\nabla^{\mu}\nabla^{\nu}\psi-\frac{d\ln Q}{d\psi}\partial^{\mu}\psi\partial^{\nu}\psi\right)\nonumber \\
& & -\biggl(\frac{1}{6}\biggr)2qX_{\psi}e^{3\lambda\psi}\biggl\{(\Box\psi)^{3}-3(\Box\psi)\boxtimes\psi+2\boxdot\psi\nonumber \\
& & +\frac{d\ln Q}{d\psi}\biggl[6(\Box\psi)^{2}X_{\psi}-6X_{\psi}\boxtimes\psi+6(\Box\psi)\partial_{\mu}\psi\partial_{\nu}\psi\nabla^{\mu}\nabla^{\nu}\psi\nonumber \\
& & -2\nabla^{\mu}\nabla_{\alpha}\psi\nabla^{\alpha}\nabla_{\beta}\psi\partial^{\beta}\psi\partial_{\mu}\psi-2\partial^{\mu}\psi\partial_{\alpha}\psi\nabla^{\alpha}\nabla_{\beta}\psi\nabla^{\beta}\nabla{}_{\mu}\psi\nonumber \\
& & -2\nabla^{\mu}\nabla_{\alpha}\psi\partial^{\alpha}\psi\partial_{\beta}\psi\nabla^{\beta}\nabla_{\mu}\psi\biggr)\biggr]\nonumber \\
& & +\left(\frac{d\log Q}{d\psi}\right)^{2}\biggl[12X_{\psi}\partial_{\mu}\psi\partial_{\nu}\psi\nabla^{\mu}\nabla^{\nu}\psi-2\nabla^{\mu}\nabla_{\alpha}\psi\partial^{\alpha}\psi\partial_{\beta}\psi\partial^{\beta}\psi\partial_{\mu}\psi\nonumber \\
& & -2\partial^{\mu}\psi\partial_{\alpha}\psi\nabla^{\alpha}\nabla_{\beta}\psi\partial^{\beta}\psi\partial_{\mu}\psi-2\partial^{\mu}\psi\partial_{\alpha}\psi\partial^{\alpha}\psi\partial_{\beta}\psi\nabla^{\beta}\nabla_{\mu}\psi\biggr]\nonumber \\
& & +\left(\frac{d\log Q}{d\psi}\right)^{3}(-8X_{\psi}^{3})\biggr\}.\end{aligned}$$ It could seem that the Lagrangian has an explicit dependence on the coupling $Q(\phi)$. However, remember that in this paper we are considering couplings satisfying Eq. (\[eqQ\]), that is
$$\begin{aligned}
\frac{1}{Q^{2}}\frac{dQ}{d\phi} & = & c.\end{aligned}$$
This is equivalent to
$$\begin{aligned}
\frac{d\log Q}{d\psi} & = & c.\end{aligned}$$
This means the Horndeski Lagrangian can be written as $$\begin{aligned}
\mathcal{L}_{H} & = & X_{\psi}g(X_{\psi}e^{\lambda\psi})-[d_{1}X_{\psi}e^{\lambda\psi}+{\ell_{1}}X_{\psi}^{2}e^{2\lambda\psi}]\left(\Box\psi+2cX_{\psi}\right)\nonumber \\
& & +\biggl(h(\psi)+\frac{1}{2}d_{2}X_{\psi}^{2}e^{2\lambda\psi}+\frac{1}{3}{\ell_{2}}X_{\psi}^{3}e^{3\lambda\psi}\biggr)R\nonumber \\
& & +\biggl(d_{2}X_{\psi}e^{2\lambda\psi}+\ell_{2}X_{\psi}^{2}e^{3\lambda\psi}\biggr)\left[(\Box\psi)^{2}-\boxtimes\phi+4c\left(X_{\psi}\Box\psi+\frac{1}{2}\partial^{\mu}\psi\partial_{\nu}\psi\nabla_{\mu}\nabla^{\nu}\psi\right)\right]\nonumber \\
& & +qX_{\psi}^{2}e^{3\lambda\psi}G_{\mu\nu}\left(\nabla^{\mu}\nabla^{\nu}\psi-c\partial^{\mu}\psi\partial^{\nu}\psi\right)\nonumber \\
& & -\biggl(\frac{1}{6}\biggr)2qX_{\psi}e^{3\lambda\psi}\biggl\{(\Box\psi)^{3}-3(\Box\psi)\boxtimes\psi+2\boxdot\psi\nonumber \\
& & +c\biggl[6(\Box\psi)^{2}X_{\psi}-6X_{\psi}\boxtimes\psi+6(\Box\psi)\partial_{\mu}\psi\partial_{\nu}\psi\nabla^{\mu}\nabla^{\nu}\psi\nonumber \\
& & -2\nabla^{\mu}\nabla_{\alpha}\psi\nabla^{\alpha}\nabla_{\beta}\psi\partial^{\beta}\psi\partial_{\mu}\psi-2\partial^{\mu}\psi\partial_{\alpha}\psi\nabla^{\alpha}\nabla_{\beta}\psi\nabla^{\beta}\nabla{}_{\mu}\psi\nonumber \\
& & -2\nabla^{\mu}\nabla_{\alpha}\psi\partial^{\alpha}\psi\partial_{\beta}\psi\nabla^{\beta}\nabla_{\mu}\psi\biggr)\biggr]\nonumber \\
& & +c^{2}\biggl[12X_{\psi}\partial_{\mu}\psi\partial_{\nu}\psi\nabla^{\mu}\nabla^{\nu}\psi-2\nabla^{\mu}\nabla_{\alpha}\psi\partial^{\alpha}\psi\partial_{\beta}\psi\partial^{\beta}\psi\partial_{\mu}\psi\nonumber \\
& & -2\partial^{\mu}\psi\partial_{\alpha}\psi\nabla^{\alpha}\nabla_{\beta}\psi\partial^{\beta}\psi\partial_{\mu}\psi-2\partial^{\mu}\psi\partial_{\alpha}\psi\partial^{\alpha}\psi\partial_{\beta}\psi\nabla^{\beta}\nabla_{\mu}\psi\biggr]\nonumber \\
& & +c^{3}(-8X_{\psi}^{2})\biggr\}.\label{eq:Lpsi}\end{aligned}$$ That is, when expressed in terms of $\psi$, there is no dependence at all on the Lagrangian in the coupling (at least considering Eq. (\[eqQ\]) to be valid). This can be understood since the definition of the coupling $Q$, given by Eq. (\[Qdef\]), when expressed in terms of $\psi$, assumes the form of a constant coupling:
$$\begin{aligned}
1 & = & -\frac{1}{\rho_{m}\sqrt{-g}}\frac{\delta S_{m}}{\delta\psi}.\end{aligned}$$
This shows that, for the full Horndeski Lagrangian, the Lagrangian with scaling solutions written in terms of $\phi$ and its derivatives, with a general coupling satisfying $\frac{d\log Q}{d\phi}$ constant, is equivalent to the above Lagrangian written in terms of $\psi$ and its derivatives, for the coupling $Q=1$. In other words, if one is interested in scaling solutions with couplings $\frac{d\log Q}{d\phi}$ constant, it is sufficient to work with a Lagrangian given by Eq. (\[eq:Lpsi\]), independently of the particular coupling considered. In that case one can indeed affirm, as done in Ref.[@Amendola:2006qi], that the case of constant coupling $Q=1$ is the most general.
Now we make the redefinition $\bar{\phi}=\psi/\bar{Q},$ with $\bar{Q}$ a constant. This helps to compare with the literature, e.g. [@pt; @ts; @Amendola:2006qi; @2014JCAP...03..041G]. In the exponents that appear in the Lagrangian this is equivalent to a redefinition of $\lambda$. Then, after dropping the bars we can rewrite $\mathcal{L}_{H}$ as
$$\begin{aligned}
\mathcal{L}_{H} & = & Q^{2}Xg(Q^{2}Y)-[d_{1}Q^{2}Y+{\ell_{1}}Q^{4}Y^{2}]\left(Q\Box\phi+2(cQ)QX\right)+\biggl(h(Q\phi)+\frac{1}{2}d_{2}Q^{4}Y^{2}+\frac{1}{3}{\ell_{2}}Q^{6}Y^{3}\biggr)R\nonumber \\
\nonumber \\
& & +\biggl(d_{2}Q^{4}\frac{Y^{2}}{X}+\ell_{2}Q^{4}\frac{Y^{3}}{X}\biggr)\left[Q^{2}(\Box\phi)^{2}-Q^{2}\boxtimes\phi+4(cQ)\left(Q^{2}X\Box\phi+Q^{2}\frac{1}{2}\partial^{\mu}\phi\partial_{\nu}\phi\nabla_{\mu}\nabla^{\nu}\phi\right)\right]\nonumber \\
& & +qQ^{4}\frac{Y^{3}}{X}G_{\mu\nu}\left(Q\nabla^{\mu}\nabla^{\nu}\phi-(cQ)Q\partial^{\mu}\phi\partial^{\nu}\phi\right)\nonumber \\
& & -\biggl(\frac{1}{6}\biggr)2qQ^{2}\frac{Y^{3}}{X^{2}}\biggl\{ Q^{3}(\Box\phi)^{3}-3Q^{3}(\Box\phi)\boxtimes\phi+2Q^{3}\boxdot\phi\nonumber \\
& & +(cQ)\biggl[6Q^{3}(\Box\phi)^{2}X-6Q^{3}X\boxtimes\phi+6Q^{3}(\Box\phi)\partial_{\mu}\phi\partial_{\nu}\phi\nabla^{\mu}\nabla^{\nu}\phi\nonumber \\
& & -2Q^{3}\nabla^{\mu}\nabla_{\alpha}\phi\nabla^{\alpha}\nabla_{\beta}\phi\partial^{\beta}\phi\partial_{\mu}\phi-2Q^{3}\partial^{\mu}\phi\partial_{\alpha}\phi\nabla^{\alpha}\nabla_{\beta}\phi\nabla^{\beta}\nabla{}_{\mu}\phi\nonumber \\
& & -2Q^{3}\nabla^{\mu}\nabla_{\alpha}\phi\partial^{\alpha}\phi\partial_{\beta}\phi\nabla^{\beta}\nabla_{\mu}\phi\biggr)\biggr]\nonumber \\
& & +(cQ)^{2}\biggl[12Q^{3}X\partial_{\mu}\phi\partial_{\nu}\phi\nabla^{\mu}\nabla^{\nu}\phi-2Q^{3}\nabla^{\mu}\nabla_{\alpha}\phi\partial^{\alpha}\phi\partial_{\beta}\phi\partial^{\beta}\phi\partial_{\mu}\phi\nonumber \\
& & -2Q^{3}\partial^{\mu}\phi\partial_{\alpha}\phi\nabla^{\alpha}\nabla_{\beta}\phi\partial^{\beta}\phi\partial_{\mu}\phi-2Q^{3}\partial^{\mu}\phi\partial_{\alpha}\phi\partial^{\alpha}\phi\partial_{\beta}\phi\nabla^{\beta}\nabla_{\mu}\phi\biggr]\nonumber \\
& & +(cQ)^{3}(-8Q^{3}X^{2})\biggr\}.\label{eq:Lpsi-1}\end{aligned}$$
with
$$\begin{aligned}
\lambda & = & -Q\left(\frac{1+w_{\mathrm{eff}}}{w_{\mathrm{eff}}}\right)\\
Y & = & Xe^{\lambda\psi}.\end{aligned}$$
Since the system has symmetry under a simultaneous change of sign of $\lambda$ and $Q$, we will be considering from now on $\lambda>0$. Next, redefining the general constants and functions we get
$$\begin{aligned}
\mathcal{L}_{H} & = & Xg(Y)-[d_{1}Y+{\ell_{1}}Y^{2}]\left(\Box\phi+2cX\right)+\biggl(h(\phi)+\frac{1}{2}d_{2}Y^{2}+\frac{1}{3}{\ell_{2}}Y^{3}\biggr)R\nonumber \\
\nonumber \\
& & +\biggl(d_{2}\frac{Y^{2}}{X}+\ell_{2}\frac{Y^{3}}{X}\biggr)\left[(\Box\phi)^{2}-\boxtimes\phi+4c\left(X\Box\phi+\frac{1}{2}\partial^{\mu}\phi\partial_{\nu}\phi\nabla_{\mu}\nabla^{\nu}\phi\right)\right]\nonumber \\
& & +q\frac{Y^{3}}{X}G_{\mu\nu}\left(\nabla^{\mu}\nabla^{\nu}\phi-c\partial^{\mu}\phi\partial^{\nu}\phi\right)-\frac{q}{3}\frac{Y^{3}}{X^{2}}\biggl\{(\Box\phi)^{3}-3(\Box\phi)\boxtimes\phi+2\boxdot\phi\nonumber \\
& & +c\biggl[6(\Box\phi)^{2}X-6X\boxtimes\phi+6(\Box\phi)\partial_{\mu}\phi\partial_{\nu}\phi\nabla^{\mu}\nabla^{\nu}\phi-2\nabla^{\mu}\nabla_{\alpha}\phi\nabla^{\alpha}\nabla_{\beta}\phi\partial^{\beta}\phi\partial_{\mu}\phi-2\partial^{\mu}\phi\partial_{\alpha}\phi\nabla^{\alpha}\nabla_{\beta}\phi\nabla^{\beta}\nabla{}_{\mu}\phi\nonumber \\
& & -2\nabla^{\mu}\nabla_{\alpha}\phi\partial^{\alpha}\phi\partial_{\beta}\phi\nabla^{\beta}\nabla_{\mu}\phi\biggr)\biggr]+c{}^{2}\biggl[12X\partial_{\mu}\phi\partial_{\nu}\phi\nabla^{\mu}\nabla^{\nu}\phi-2\nabla^{\mu}\nabla_{\alpha}\phi\partial^{\alpha}\phi\partial_{\beta}\phi\partial^{\beta}\phi\partial_{\mu}\phi\nonumber \\
& & -2\partial^{\mu}\phi\partial_{\alpha}\phi\nabla^{\alpha}\nabla_{\beta}\phi\partial^{\beta}\phi\partial_{\mu}\phi-2\partial^{\mu}\phi\partial_{\alpha}\phi\partial^{\alpha}\phi\partial_{\beta}\phi\nabla^{\beta}\nabla_{\mu}\phi\biggr]-8c{}^{3}X^{2}\biggr\}.\label{eq:Lscal2}\end{aligned}$$
Conclusions
===========
This paper offers two insights into the vast realm of the Hornesdki Lagrangians. First, we show that the entire Lagrangian is equivalent to the scalar field pressure (at least in a FLRW metric), extending earlier results valid for the $K,G_{3}$ subclass. This result is then employed to identify the form of the Horndeski Lagrangian that contains scaling solutions in which the ratio of matter to field density and the equation of state are constant. This also generalizes previous results, in particular Ref. [@2014JCAP...03..041G].
The existence of this particular class of solution is interesting since it could represent a solution of the coincidence problem. If the ratio $\Omega_{m}/\Omega_{\phi}$ depends on the fundamental constant of the theory, instead of on initial conditions, then the fact that it is close to unity would no longer be a surprising coincidence. On the other hand, nothing guarantees that such a scaling solution is stable or viable, in the sense of providing a valid cosmology when compared to observations. Indeed we found in Ref. [@2014JCAP...03..041G] that this was not the case for the subclass $K,G_{3}$ because one has either a stable scaling accelerated regime or a well-behaved matter era, but not both. We conjecture that this negative result extends to the full scaling Horndeski Lagrangian but due to its very complicate form a full proof is still to be elaborated.
acknowledgements
================
A. G. thanks FAPEMA and CNPq for financial support and ITP for the hospitality. L.A. acknowledges support from DFG under the TransRegio33 project “The Dark Universe”. L.A. acknowledges interesting discussions with Ignacy Sawicki.
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Appendix A - Some useful relations for flrw metric {#appendix-a---some-useful-relations-for-flrw-metric .unnumbered}
==================================================
Metric determinant
$$\begin{aligned}
\frac{d}{dt}\sqrt{-g} & = & \sqrt{-g}(3H).\end{aligned}$$
Nonvanishing elements of the affine connection:
$$\begin{aligned}
\Gamma_{ij}^{0} & = & A\dot{A}\eta_{ij,}\\
\Gamma_{0j}^{i} & =\Gamma_{j0}^{i}= & H\delta_{j}^{i},\\
\Gamma_{22}^{1} & = & -r,\\
\Gamma_{33}^{1} & = & -r\sin^{2}\theta,\\
\Gamma_{12}^{2} & = & \Gamma_{21}^{2}=\Gamma_{13}^{3}=\Gamma_{31}^{3}=\frac{1}{r},\\
\Gamma_{33}^{2} & = & -\sin\theta\cos\theta,\\
\Gamma_{23}^{3} & = & \Gamma_{32}^{3}=\cot\theta.\end{aligned}$$
Nonvanishing components of Ricci tensor and Ricci scalar:
$$\begin{aligned}
R_{00} & = & -3\frac{\ddot{A}}{A},\\
R_{ij} & = & (A\ddot{A}+2\dot{A}^{2})\eta_{ij},\\
R & = & 6\dot{H}+12H^{2}.\end{aligned}$$
Relation between $R$ and $p$, after using Einstein equations in the above expression of $R$:
$$\begin{aligned}
R & = & p\left(\frac{1}{w_{\phi}\Omega_{\phi}}-3\right).\end{aligned}$$
Components of Einstein tensor:
$$\begin{aligned}
G_{\mu}^{\mu} & = & -(6\dot{H}+12H^{2}),\\
G_{0}^{0} & = & -3H^{2},\\
G_{j}^{i} & = & -(2\dot{H}+3H^{2})\delta_{j}^{i}.\end{aligned}$$
We consider now the operators acting on $\phi$:
$$\begin{aligned}
\Box\phi & = & \nabla_{\mu}\nabla^{\mu}\phi=\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}\nabla^{\mu}\phi)=-3H\dot{\phi}-\ddot{\phi},\\
\boxtimes\phi & = & \nabla_{\mu}\nabla_{\nu}\phi\nabla^{\mu}\nabla^{\nu}\phi=(\ddot{\phi})^{2}+3H^{2}(\dot{\phi})^{2},\\
\boxbar\phi & = & G_{\mu\nu}\nabla^{\mu}\nabla^{\nu}\phi=3H^{2}\ddot{\phi}+6H\dot{H}\dot{\phi}+9H^{3}\dot{\phi},\\
\boxdot\phi & = & (\nabla^{\mu}\nabla_{\alpha}\phi)(\nabla^{\alpha}\nabla_{\beta}\phi)(\nabla^{\beta}\nabla_{\mu}\phi)=-(\ddot{\phi})^{3}-3H^{3}(\dot{\phi})^{3}.\end{aligned}$$
Appendix B - Some useful results for the master equation {#appendix-b---some-useful-results-for-the-master-equation .unnumbered}
========================================================
From Eq. (\[dphidN\]) we have $$\begin{aligned}
\dot{\phi} & = & -\frac{3H}{Q}w_{\phi}\Omega_{\phi},\\
\ddot{\phi} & = & -\frac{3\dot{H}}{Q}w_{\phi}\Omega_{\phi}-\frac{9H^{2}}{Q^{3}}\frac{dQ}{d\phi}(w_{\phi}\Omega_{\phi})^{2}.\end{aligned}$$ Now, with $$\begin{aligned}
\dot{H} & = & -\frac{3}{2}H^{2}(1+w_{eff}),\\
H^{2} & = & \frac{1}{3}\frac{p}{w_{eff}},\end{aligned}$$ we get
$$\begin{aligned}
\ddot{\phi} & = & \frac{3}{2}\frac{p}{Q}(1+w_{eff})\left(1+\frac{2}{\lambda}\frac{1}{Q^{2}}\frac{dQ}{d\phi}\right),\\
\nonumber \\
\nonumber \end{aligned}$$
with
$$\begin{aligned}
\lambda & = & -\frac{1+w_{eff}}{w_{\phi}\Omega_{\phi}}.\end{aligned}$$
Then the last expressions of Appendix A can be rewritten as
$$\begin{aligned}
\Box\phi & = & \frac{3}{2}(1-w_{eff})\frac{p}{Q}\left(1-\frac{2}{\lambda}\frac{1+w_{eff}}{1-w_{eff}}\frac{1}{Q^{2}}\frac{dQ}{d\phi}\right),\\
\boxtimes\phi & = & \frac{p^{2}}{Q^{2}}\left\{ \left[\frac{3}{2}(1+w_{eff})\left(1+\frac{2}{\lambda}\frac{1}{Q^{2}}\frac{dQ}{d\phi}\right)\right]^{2}+3\right\} ,\\
\boxbar\phi & = & \frac{p^{2}}{Q}\left\{ \frac{1}{w_{eff}}\frac{3}{2}(1+w_{eff})\left(3+\frac{2}{\lambda}\frac{1}{Q^{2}}\frac{dQ}{d\phi}\right)-\frac{3}{w_{eff}}\right\} ,\\
\boxdot\phi & = & -\frac{p^{3}}{Q^{3}}\left\{ \left[\frac{3}{2}(1+w_{eff})\left(1+\frac{2}{\lambda}\frac{1}{Q^{2}}\frac{dQ}{d\phi}\right)\right]^{3}-3\right\} .\end{aligned}$$
Now for $\frac{1}{Q^{2}}\frac{dQ}{d\phi}$ constant,
$$\begin{aligned}
\ln\Box\phi & = & \ln p-\ln Q+constant,\\
\ln\boxtimes\phi & = & 2\ln p-2\ln Q+constant,\\
\ln\boxbar\phi & = & 2\ln p-\ln Q+constant,\\
\ln\boxdot\phi & = & 3\ln p-3\ln Q+constant.\end{aligned}$$
Then the corresponding partial derivatives with respect to $N$ are
$$\begin{aligned}
\frac{\partial\ln\Box\phi}{\partial N} & = & \frac{\partial\ln p}{\partial N}-\frac{\partial\ln Q}{\partial N}=-3(1+w_{eff})\left(1+\frac{1}{\lambda Q^{2}}\frac{dQ}{d\phi}\right),\\
\frac{\partial\ln\boxtimes\phi}{\partial N} & = & 2\frac{\partial\ln p}{\partial N}-2\frac{\partial\ln Q}{\partial N}=-3(1+w_{eff})\left(2+\frac{2}{\lambda Q^{2}}\frac{dQ}{d\phi}\right),\\
\frac{\partial\ln\boxbar\phi}{\partial N} & = & 2\frac{\partial\ln p}{\partial N}-\frac{\partial\ln Q}{\partial N}=-3(1+w_{eff})\left(2+\frac{1}{\lambda Q^{2}}\frac{dQ}{d\phi}\right),\\
\frac{\partial\ln\boxdot\phi}{\partial N} & = & 3\frac{\partial\ln p}{\partial N}-3\frac{\partial\ln Q}{\partial N}=-3(1+w_{eff})\left(3+\frac{3}{\lambda Q^{2}}\frac{dQ}{d\phi}\right).\end{aligned}$$
Appendix C - Equality between lagrangian and pressure {#appendix-c---equality-between-lagrangian-and-pressure .unnumbered}
=====================================================
Here we prove in detail the equivalence of the Lagrangian and the scalar field pressure in a FLRW metric. For completeness we rederive the already known results for $K$ and $G_{3}$. The extension to $G_{4},G_{5}$ is new.
Term $K(\phi,X)$
----------------
We start with
$$\begin{aligned}
S_{total} & = & \int d^{4}x\frac{1}{2}R-\int d^{4}x\sqrt{-g}K(\phi,X)\\
& = & S_{EH}+S,\end{aligned}$$
That is, the Lagrangian in the action S is
$$L=K(\phi,X).$$
It is evident then that
$$p=L=K.$$
Term $G_{3}$:
-------------
We start with
$$\begin{aligned}
S_{total} & = & \int d^{4}x\frac{1}{2}R-\int d^{4}x\sqrt{-g}G_{3}(\Box\phi)=\\
& = & S_{EH}+S,\end{aligned}$$
with
$$\begin{aligned}
S & = & -\int d^{4}x\sqrt{-g}G_{3}(\Box\phi).\\
\nonumber \end{aligned}$$
That is, the Lagrangian in the action S is
$$L=-G_{3}(\Box\phi)=-G_{3}(-3H\dot{\phi}-\ddot{\phi}).$$
We have
$$p=-2X(G_{3,\phi}+\ddot{\phi}G_{3,X}).$$
Now we compare with the Lagrangian
$$\begin{aligned}
L & = & -G_{3}\Box\phi\nonumber \\
& = & -\nabla_{\mu}(G_{3}\nabla^{\mu}\phi)+(\nabla_{\mu}G_{3})(\nabla^{\mu}\phi)\nonumber \\
& = & -\nabla_{\mu}(G_{3}\nabla^{\mu}\phi)+p.\end{aligned}$$
We note that the $L=p$, up to a covariant divergence, which integrated in the action, by Gauss’s law results to be zero.
Term $G_{4}:$
-------------
We start with
$$\begin{aligned}
S_{total} & = & \int d^{4}x\frac{1}{2}R+\int d^{4}x\sqrt{-g}\left[\left(G_{4}-\frac{1}{2}\right)R+G_{4,X}[(\Box\phi)^{2}-(\boxtimes\phi)\right]\\
& = & S_{EH}+S=I+III,\end{aligned}$$
The Lagrangian in the action $S$ is
$$L=\left(G_{4}-\frac{1}{2}\right)R+G_{4,X}[(\Box\phi)^{2}-(\boxtimes\phi)].$$
We have
$$\begin{aligned}
p & = & -(3H^{2}+2\dot{H})(1-2G_{4})-12H^{2}XG_{4,X}\nonumber \\
& & -4H\dot{X}G_{4,X}-8\dot{H}XG_{4,X}-8HX\dot{X}G_{4,XX}+2(\ddot{\phi}+2H\dot{\phi})G_{4,\phi}\nonumber \\
& & +4XG_{4,\phi\phi}+4X(\ddot{\phi}-2H\dot{\phi})G_{4,\phi X}.\end{aligned}$$
Now we want to compare $p$ with the Lagrangian. We start with
$$\begin{aligned}
I & = & \int d^{4}x\sqrt{-g}\left(G_{4}-\frac{1}{2}\right)R\\
& = & \int d^{4}x\sqrt{-g}\left(1-2G_{4}\right)(-3H^{2}-2\dot{H})+\int d^{4}x\sqrt{-g}\left(1-2G_{4}\right)(-3H^{2}-\dot{H})\nonumber \\
& = & \int d^{4}x\sqrt{-g}\left(1-2G_{4}\right)(-3H^{2}-2\dot{H})+II+\int d^{4}x\sqrt{-g}\left(1-2G_{4}\right)(-\dot{H}),\end{aligned}$$
where
$$\begin{aligned}
II & = & \int d^{4}x\sqrt{-g}\left(1-2G_{4}\right)(-3H^{2})\\
& = & \int d^{4}x\left(\frac{d}{dt}\sqrt{-g}\right)\left(1-2G_{4}\right)(-H)\nonumber \\
& = & -\int d^{4}x\sqrt{-g}\frac{d}{dt}\left[\left(1-2G_{4}\right)(-H)\right]\nonumber \\
& = & -\int d^{4}x\sqrt{-g}\left[\left(-2G_{4,X}\dot{X}-2G_{4,\phi}\dot{\phi}\right)(-H)+\left(1-2G_{4}\right)(-\dot{H})\right].\end{aligned}$$
Back to $I$: $$\begin{aligned}
I & = & \int d^{4}x\sqrt{-g}\left(1-2G_{4}\right)(-3H^{2}-2\dot{H})\nonumber \\
& & +\int d^{4}x\sqrt{-g}\left[\left(2G_{4,X}\dot{X}+2G_{4,\phi}\dot{\phi}\right)(-H)-\left(1-2G_{4}\right)(-\dot{H})\right]+\int d^{4}x\sqrt{-g}\left(1-2G_{4}\right)(-\dot{H})\nonumber \\
& = & \int d^{4}x\sqrt{-g}\left(1-2G_{4}\right)(-3H^{2}-2\dot{H})\nonumber \\
& & +\int d^{4}x\sqrt{-g}\left(-2H\dot{X}G_{4,X}-2H\dot{\phi}G_{4,\phi}\right).\end{aligned}$$ Now consider the second part of the action $S$:
$$\begin{aligned}
III & = & \int d^{4}x\sqrt{-g}G_{4,X}[(\Box\phi)^{2}-(\boxtimes\phi)]\\
& = & \int d^{4}x\sqrt{-g}G_{4,X}[(-3H\dot{\phi}-\ddot{\phi})^{2}-((\ddot{\phi})^{2}+3H^{2}(\dot{\phi})^{2})]\nonumber \\
& = & \int d^{4}x\sqrt{-g}G_{4,X}[6H^{2}(\dot{\phi})^{2}+6H\dot{\phi}\ddot{\phi}]\nonumber \\
& = & \int d^{4}x\sqrt{-g}G_{4,X}[6H^{2}(\dot{\phi})^{2}]+IV,\end{aligned}$$
where
$$\begin{aligned}
IV & = & \int d^{4}x\sqrt{-g}G_{4,X}[6H\dot{\phi}\ddot{\phi}]\\
& = & 6\int d^{4}x\sqrt{-g}G_{4,X}H\dot{\phi}\left(\frac{d}{dt}\dot{\phi}\right)\nonumber \\
& = & -6\int d^{4}x\frac{d}{dt}\left(\sqrt{-g}G_{4,X}H\dot{\phi}\right)\dot{\phi}\nonumber \\
& = & -6\int d^{4}x\sqrt{-g}\left(3HG_{4,X}H\dot{\phi}+(G_{4,XX}\dot{X}+G_{4,X\phi}\dot{\phi})H\dot{\phi}+G_{4,X}\dot{H}\dot{\phi}+G_{4,X}H\ddot{\phi}\right)\dot{\phi}\nonumber \\
& = & \int d^{4}x\sqrt{-g}\left[\left(-36H^{2}X-12\dot{H}X-6H\dot{X}\right)G_{4,X}-12X\dot{X}HG_{4,XX}-12G_{4,X\phi}HX\dot{\phi}\right].\end{aligned}$$
Then back to $III$:
$$\begin{aligned}
III & = & \int d^{4}x\sqrt{-g}G_{4,X}[(\Box\phi)^{2}-(\boxtimes\phi)]\nonumber \\
& = & \int d^{4}x\sqrt{-g}G_{4,X}[6H^{2}(\dot{\phi})^{2}]+IV\nonumber \\
& = & \int d^{4}x\sqrt{-g}\left[\left(-24H^{2}X-6H\dot{X}-12\dot{H}X\right)G_{4,X}-12X\dot{X}HG_{4,XX}-12HX\dot{\phi}G_{4,X\phi}\right].\end{aligned}$$
Now from $I$ and $III$ we recover the action:
$$\begin{aligned}
S & = & \int d^{4}x\sqrt{-g}\left[\left(G_{4}-\frac{1}{2}\right)R+G_{4,X}[(\Box\phi)^{2}-(\boxtimes\phi)\right]=I+III\\
& = & \int d^{4}x\sqrt{-g}\left(1-2G_{4}\right)(-3H^{2}-2\dot{H})+\int d^{4}x\sqrt{-g}\left(-2H\dot{X}G_{4,X}-2H\dot{\phi}G_{4,\phi}\right)\nonumber \\
& & +\int d^{4}x\sqrt{-g}\left[\left(-24H^{2}X-6H\dot{X}-12\dot{H}X\right)G_{4,X}-12X\dot{X}HG_{4,XX}-12HX\dot{\phi}G_{4,X\phi}\right]\nonumber \\
& = & \int d^{4}x\sqrt{-g}p+\int d^{4}x\sqrt{-g}\{[(3H)(-4H^{2}X)+\frac{d}{dt}(-4HX)]G_{4,X}\nonumber \\
& & +(-4XH)(\dot{X}G_{4,XX}+\dot{\phi}G_{4,X\phi})\}\nonumber \\
& & +\int d^{4}x\sqrt{-g}\{(-2\ddot{\phi}-6H\dot{\phi})G_{4,\phi}-4XG_{4,\phi\phi}+(-4X\ddot{\phi})G_{4,X\phi})\}\nonumber \\
& = & \int d^{4}x\sqrt{-g}p+\int d^{4}x\frac{d}{dt}\left(\sqrt{-g}(-4H^{2}X)G_{4,X}\right)\nonumber \\
& & +\int d^{4}x\sqrt{-g}\{(-2\ddot{\phi}-6H\dot{\phi})G_{4,\phi}-2\dot{\phi}\dot{\phi}G_{4,\phi\phi}-2\dot{\phi}\dot{X})G_{4,X\phi})\}\nonumber \\
& = & \int d^{4}x\sqrt{-g}p+\int d^{4}x\frac{d}{dt}\left(\sqrt{-g}(-4H^{2}X)G_{4,X}\right)\nonumber \\
& & +\int d^{4}x\{-2\sqrt{-g}\left(\frac{d}{dt}(\dot{\phi})\right)G_{4,\phi}-2\left(\frac{d}{dt}(\sqrt{-g})\right)(\dot{\phi}G_{4,\phi})-2\sqrt{-g}\dot{\phi}\frac{d}{dt}\left(G_{4,\phi}\right)\}\nonumber \\
& = & \int d^{4}x\sqrt{-g}p+\int d^{4}x\frac{d}{dt}\left(\sqrt{-g}(-4H^{2}X)G_{4,X}\right)\nonumber \\
& & +\int d^{4}x\frac{d}{dt}\left(-2\sqrt{-g}(\dot{\phi})G_{4,\phi}\right)\nonumber \\
& = & \int d^{4}x\sqrt{-g}p.\end{aligned}$$
That is again $L=p$.
Term $G_{5}$:
-------------
We start now with
$$\begin{aligned}
S_{total} & = & \int d^{4}x\frac{1}{2}R+\int d^{4}x\sqrt{-g}\{G_{5}(\boxbar\phi)-\frac{1}{6}G_{5,X}[(\Box\phi)^{3}-3(\Box\phi)(\boxtimes\phi)+2(\boxdot\phi)]\}\\
& = & S_{EH}+S\end{aligned}$$
with
$$\begin{aligned}
S & = & \int d^{4}x\sqrt{-g}\{G_{5}(\boxbar\phi)-\frac{1}{6}G_{5,X}[(\Box\phi)^{3}-3(\Box\phi)(\boxtimes\phi)+2(\boxdot\phi)]\}.\end{aligned}$$
That is, the Lagrangian in the action $S$ is
$$L=G_{5}(\boxbar\phi)-\frac{1}{6}G_{5,X}[(\Box\phi)^{3}-3(\Box\phi)(\boxtimes\phi)+2(\boxdot\phi)].$$
We have
$$\begin{aligned}
p & = & -2X(2H^{3}\dot{\phi}+2H\dot{H}\dot{\phi}+3H^{2}\ddot{\phi})G_{5,X}-4H^{2}X^{2}\ddot{\phi}G_{5,XX}\nonumber \\
& & +4HX(\dot{X}-HX)G_{5,\phi X}+2[2(\dot{H}X+H\dot{X})+3H^{2}X]G_{5,\phi}+4HX\dot{\phi}G_{5,\phi\phi}.\end{aligned}$$
Now we want to compare $p$ with the Lagrangian:
$$\begin{aligned}
L & = & G_{5}(3H^{2}\ddot{\phi}+6H\dot{H}\dot{\phi}+9H^{3}\dot{\phi})\nonumber \\
& & -\frac{1}{6}G_{5,X}[(-3H\dot{\phi}-\ddot{\phi})^{3}-3(-3H\dot{\phi}-\ddot{\phi})((\ddot{\phi})^{2}+3H^{2}(\dot{\phi})^{2})+2(-(\ddot{\phi})^{3}-3H^{3}(\dot{\phi})^{3})]\\
& = & G_{5}(3H^{2}\ddot{\phi}+6H\dot{H}\dot{\phi}+9H^{3}\dot{\phi})-\frac{1}{6}G_{5,X}\{[-27H^{3}(\dot{\phi})^{3}-(\ddot{\phi})^{3}+3(-3H\dot{\phi})^{2}(-\ddot{\phi})+3(-3H\dot{\phi})(-\ddot{\phi})^{2}]\nonumber \\
& & +[3(3H\dot{\phi}+\ddot{\phi})((\ddot{\phi})^{2}+6H^{2}X)]-2[(\ddot{\phi})^{3}+6H^{3}X(\dot{\phi})]\}\nonumber \\
& = & G_{5}(3H^{2}\ddot{\phi}+6H\dot{H}\dot{\phi}+9H^{3}\dot{\phi})+G_{5,X}(6H^{2}X\ddot{\phi}+2H^{3}X\dot{\phi}).\\
\nonumber \end{aligned}$$
Now
$$\begin{aligned}
V & = & \int d^{4}x\sqrt{-g}G_{5,X}6H^{2}X\ddot{\phi}\\
& = & -\int d^{4}x\frac{d}{dt}\left(\sqrt{-g}G_{5,X}6H^{2}X\right)\dot{\phi}\nonumber \\
& = & -\int d^{4}x\biggl[\frac{d}{dt}\left(\sqrt{-g}\right)G_{5,X}6H^{2}X+\sqrt{-g}\frac{d}{dt}\left(G_{5,X}\right)6H^{2}X\nonumber \\
& & +\sqrt{-g}G_{5,X}\frac{d}{dt}\left(6H^{2}\right)X+\sqrt{-g}G_{5,X}6H^{2}\frac{d}{dt}\left(X\right)\biggr]\dot{\phi}\\
& = & -\int d^{4}x\sqrt{-g}\left[3HG_{5,X}6H^{2}X+(G_{5,XX}\dot{X}+G_{5,X\phi}\dot{\phi})6H^{2}X+G_{5,X}\left(12H\dot{H}\right)X+G_{5,X}6H^{2}\dot{X}\right]\dot{\phi}\nonumber \\
& = & \int d^{4}x\sqrt{-g}\left[-18H^{3}X\dot{\phi}G_{5,X}-12H^{2}X^{2}\ddot{\phi}G_{5,XX}-12H^{2}X^{2}G_{5,X\phi}-12H\dot{H}X\dot{\phi}G_{5,X}-12H^{2}X\ddot{\phi}G_{5,X}\right].\end{aligned}$$
Then, the terms of the Lagrangian depending explicitly on $G_{5,X}$ can be written as
$$\begin{aligned}
& & \int d^{4}x\sqrt{-g}G_{5,X}(6H^{2}X\ddot{\phi}+2H^{3}X\dot{\phi})\\
& = & \int d^{4}x\sqrt{-g}\biggl[G_{5,X}(-18H^{3}X\dot{\phi}-12H\dot{H}X\dot{\phi}-12H^{2}X\ddot{\phi}+2H^{3}X\dot{\phi})+G_{5,XX}(-12H^{2}X^{2}\ddot{\phi})\biggr]\nonumber \\
& & +G_{5,X\phi}(-12H^{2}X^{2})\\
& = & \int d^{4}x\sqrt{-g}\left[G_{5,X}(-16H^{3}X\dot{\phi}-12H\dot{H}X\dot{\phi}-12H^{2}X\ddot{\phi})+G_{5,XX}(-12H^{2}X^{2}\ddot{\phi})+G_{5,X\phi}(-12H^{2}X^{2})\right].\end{aligned}$$
Now
$$\begin{aligned}
VI & = & \int d^{4}x\sqrt{-g}G_{5}3H^{2}\ddot{\phi}\\
& = & 3\int d^{4}x\frac{d}{dt}\left(\sqrt{-g}G_{5}H^{2}\dot{\phi}\right)-3\int d^{4}x\frac{d}{dt}\left(\sqrt{-g}G_{5}H^{2}\right)\dot{\phi}\nonumber \\
& = & -3\int d^{4}x\frac{d}{dt}\left(\sqrt{-g}G_{5}H^{2}\right)\dot{\phi}\nonumber \\
& = & -3\int d^{4}x\left[\frac{d}{dt}\left(\sqrt{-g}\right)G_{5}H^{2}\dot{\phi}+\sqrt{-g}\frac{d}{dt}\left(G_{5}\right)H^{2}\dot{\phi}+\sqrt{-g}G_{5}\frac{d}{dt}\left(H^{2}\right)\dot{\phi}\right]\nonumber \\
& = & \int d^{4}x\sqrt{-g}\left[-9H^{3}G_{5}\dot{\phi}-3G_{5,X}\dot{X}H^{2}\dot{\phi}-3G_{5,\phi}\dot{\phi}H^{2}\dot{\phi}-6G_{5}H\dot{H}\dot{\phi}\right].\end{aligned}$$
Then, from the terms of the Lagrangian depending on $G_{5}$:
$$\begin{aligned}
& & \int d^{4}x\sqrt{-g}[(3H^{2}\ddot{\phi}+6H\dot{H}\dot{\phi}+9H^{3}\dot{\phi})G_{5}]\\
& = & \int d^{4}x\sqrt{-g}\left[-9H^{3}G_{5}\dot{\phi}-3G_{5,X}\dot{X}H^{2}\dot{\phi}-3G_{5,\phi}\dot{\phi}H^{2}\dot{\phi}-6G_{5}H\dot{H}\dot{\phi}+(6H\dot{H}\dot{\phi}+9H^{3}\dot{\phi})G_{5}\right]\nonumber \\
& = & \int d^{4}x\sqrt{-g}\left[-6G_{5,X}XH^{2}\ddot{\phi}-6G_{5,\phi}XH^{2}\right].\end{aligned}$$
Then the action can be written as
$$\begin{aligned}
S & = & \int d^{4}x\sqrt{-g}L\nonumber \\
& = & \int d^{4}x\sqrt{-g}[G_{5}(3H^{2}\ddot{\phi}+6H\dot{H}\dot{\phi}+9H^{3}\dot{\phi})+G_{5,X}(6H^{2}X\ddot{\phi}+2H^{3}X\dot{\phi})]\nonumber \\
& = & \int d^{4}x\sqrt{-g}\{[-6G_{5,X}XH^{2}\ddot{\phi}-6G_{5,\phi}XH^{2}]\nonumber \\
& & +[G_{5,X}(-16H^{3}X\dot{\phi}-12H\dot{H}X\dot{\phi}-12H^{2}X\ddot{\phi})+G_{5,XX}(-12H^{2}X^{2}\ddot{\phi})+G_{5,X\phi}(-12H^{2}X^{2})]\}\nonumber \\
& = & \int d^{4}x\sqrt{-g}\{G_{5,X}(-16H^{3}X\dot{\phi}-12H\dot{H}X\dot{\phi}-18H^{2}X\ddot{\phi})\nonumber \\
& & +G_{5,XX}(-12H^{2}X^{2}\ddot{\phi})+G_{5,X\phi}(-12H^{2}X^{2})+(6XH^{2})G_{5,\phi}-(12XH^{2})G_{5,\phi}\}.\end{aligned}$$
Now we transform the term of the action depending on $G_{5,\phi}:$
$$\begin{aligned}
-\int d^{4}x\sqrt{-g}(12XH^{2})G_{5,\phi} & = & -\int d^{4}x(3H\sqrt{-g})(4HX)G_{5,\phi}\nonumber \\
& = & -\int d^{4}x\frac{d}{dt}\left[(\sqrt{-g})(4HXG_{5,\phi})\right]+\int d^{4}x\sqrt{-g}\frac{d}{dt}\left[4HXG_{5,\phi}\right]\nonumber \\
& = & \int d^{4}x\sqrt{-g}\frac{d}{dt}\left[4HXG_{5,\phi}\right]\nonumber \\
& = & \int d^{4}x\sqrt{-g}\left[(4\dot{H}X+4H\dot{X})G_{5,\phi}+(4HX\dot{X})G_{5,\phi X}+(4HX\dot{\phi})G_{5,\phi\phi}\right].\end{aligned}$$
Then the action is
$$\begin{aligned}
S & = & \int d^{4}x\sqrt{-g}\{G_{5,X}(-16H^{3}X\dot{\phi}-12H\dot{H}X\dot{\phi}-18H^{2}X\ddot{\phi})\nonumber \\
& & +G_{5,XX}(-12H^{2}X^{2}\ddot{\phi})+G_{5,X\phi}(-12H^{2}X^{2})+(6XH^{2})G_{5,\phi}-(12XH^{2})G_{5,\phi}\}\nonumber \\
& = & \int d^{4}x\sqrt{-g}\{G_{5,X}(-4H^{3}X\dot{\phi}-4H\dot{H}X\dot{\phi}-6H^{2}X\ddot{\phi})+(6XH^{2}+4\dot{H}X+4H\dot{X})G_{5,\phi}\nonumber \\
& & +(4HX\dot{\phi})G_{5,\phi\phi}+G_{5,XX}(-4H^{2}X^{2}\ddot{\phi})+G_{5,X\phi}(4HX\dot{\phi}-4H^{2}X^{2})\}\nonumber \\
& & +\int d^{4}x\sqrt{-g}\{G_{5,X}(-12H^{3}X\dot{\phi}-8H\dot{H}X\dot{\phi}-12H^{2}X\ddot{\phi})+G_{5,XX}(-8H^{2}X^{2}\ddot{\phi})+G_{5,X\phi}(-8H^{2}X^{2})\}\\
& = & \int d^{4}x\sqrt{-g}p+S',\end{aligned}$$
where
$$\begin{aligned}
S' & = & \int d^{4}x\sqrt{-g}\{G_{5,X}(-12H^{3}X\dot{\phi}-8H\dot{H}X\dot{\phi}-12H^{2}X\ddot{\phi})+G_{5,XX}(-8H^{2}X^{2}\ddot{\phi})+G_{5,X\phi}(-8H^{2}X^{2})\}\\
& = & \int d^{4}x\sqrt{-g}\{G_{5,X}(-12H^{3}X\dot{\phi}-8H\dot{H}X\dot{\phi}-12H^{2}X\ddot{\phi})+\left(\frac{d}{dt}G_{5,X}\right)(-4H^{2}X\dot{\phi})\}\nonumber \\
& = & \int d^{4}x\{G_{5,X}\sqrt{-g}[-12H^{3}X\dot{\phi}-8H\dot{H}X\dot{\phi}+(-4H^{2}\dot{\phi}\ddot{\phi}\dot{\phi}-4H^{2}X\dot{\phi})]\nonumber \\
& & +\sqrt{-g}\left(\frac{d}{dt}G_{5,X}\right)(-4H^{2}X\dot{\phi})\}\nonumber \\
& = & \int d^{4}x\{G_{5,X}\frac{d}{dt}\left[\sqrt{-g}(-4H^{2}X\dot{\phi)}\right]+\sqrt{-g}\left(\frac{d}{dt}G_{5,X}\right)(-4H^{2}X\dot{\phi})\}\nonumber \\
& = & \int d^{4}x\frac{d}{dt}\left[\sqrt{-g}G_{5,X}(-4H^{2}X\dot{\phi)}\right]=0.\end{aligned}$$
That is, again $L=p$. This completes our proof.
|
---
abstract: 'In this paper we introduce geometric tools to study the families of rational vector fields of a given degree over $\CP^1$. To a generic vector field of such a parametric family we associate several geometric objects: a periodgon, a star domain and a translation surface. These objects generalize objects with the same name introduced in previous works on polynomial vector fields. They are used to describe the bifurcations inside the families. We specialize to the case of rational vector fields of degree $4$.'
address: 'Christiane Rousseau, Département de mathématiques et de statistique, Université de Montréal, C.P. 6128, Succursale Centre-ville, Montréal (Qc), H3C 3J7, Canada.'
author:
- Martin Klimeš
- Christiane Rousseau
title: 'Remarks on rational vector fields on $\CP^1$ '
---
Introduction
============
Polynomial and rational vector fields are important for many questions in holomorphic dynamics. For instance, the formal normal form of a parabolic point of a germ of holomorphic diffeomorphism of codimension $k$ (i.e. a multiple fixed point of multiplicity $k+1$) is given by the time-one map of a rational vector field $\dot z = \frac{z^{k+1}}{1+az^k}$. This is also true for the unfoldings: the formal normal form of a generic $k$-parameter parabolic point of a germ of holomorphic diffeomorphism of codimension $k$ is given by the time-one map of a family of rational vector fields $$\dot z = \frac{P_\eps(z)}{1+a(\eps)z^k}= \frac{z^{k+1}+ \eps_{k-1} z^{k-1} + \dots +\eps_0}{1+a(\eps)z^k},\label{model_family}$$ and generalizations exist for non generic families and/or generic families depending on $m\neq k$ parameters. Similar theorems exist for resonant fixed points of a holomorphic diffeomorphism (when the multiplier at the fixed point is a root of unity of order $q$): there the normal form is the composition of a rotation of order $q$ with the time-one map of a rational vector field.
For many of these problems, the fixed points are all in a small neighborhood of the origin while the poles are far from the origin. Hence the (real) phase portrait of the vector field (i.e. with real time) is close to the one of the polynomial vector field $\dot z = P_\eps(z)$. But there are new problems where this is not the case. For instance, when studying the unfolding of a resonant irregular singular point of a linear system, one of us had to deal with the system with $k=1$ and $a$ that could be as large as $\frac1{\eps}$ [@Kl].
Douady, Estrada and Sentenac [@DS] initiated geometric methods to study the real dynamics of complex polynomial vector fields, by introducing an invariant of analytic classification for generic polynomial vector fields. This invariant, later generalized for all polynomial vector fields (see for instance [@BD]), is composed of two parts: a topological (or combinatorial) part, and an analytic part. The set of structurally stable monic polynomial vector fields is divided in $C_k= \frac{\binom{2k}{k}}{k+1}$ open sets separated by bifurcation surfaces where the Douady-Estrada-Sentenac invariant is discontinuous. Some attempts have been made to generalize this invariant to rational vector fields under the generic condition that they do not exhibit an annulus of periodic orbits [@Tom1]. But in general, the global dynamics of rational vector fields is not yet fully understood.
In the paper [@CR], a new tool was introduced to study polynomial vector fields, namely the *periodgon*, or polygon of the periods. First introduced for the $1$-parameter family of vector fields $\dot z = z^{k+1}-\eps$, the tool was generalized in [@KR] to all polynomial vector fields, and it was analyzed in detail in the particular case of the $2$-parameter family of vector fields $\dot z = z^{k+1}+\eps_1z+\eps_0$. When all singular points of a polynomial vector field are simple the periodgon is constructed as follows: First we associate to each simple singular point its *periodic domain*, defined as the basin of the center at the singular point in the properly rotated vector field. The *periodgon* is defined as the image in the time variable of the complement of the union of the periodic domains of all singular points. It takes the form of a “polygon” on the translation surface of the time coordinate. The periodgon represents the “core” part of the dynamics from which the complete information can be read. Its construction has also its degenerate equivalent in case of vector fields with multiple singular points. In a parametric setting the periodgon bifurcates when the boundary of some periodic domain is the union of several homoclinic loops. In all the examples studied in [@CR], [@KR] and [@R19], the bifurcation diagram of the periodgon produces less open strata than that of the Douady-Estrada-Sentenac invariant.
In this paper we generalize the construction of the periodgon to rational vector fields and we explore its role in describing the structure of the family of rational vector fields of degree $n$. We also introduce a notion of *translation model* of the vector field, as the surface $\CP^1{\smallsetminus}\{\text{singularities of the vector field}\}$ equipped with a translation structure given by the rectifying coordinate for the vector field. This is a natural object that has been previously studied by Muciño-Raymundo and Valero-Valdés [@Mucino; @Mucino-Valero; @Mucino-Valero2], and it appears implicitly also in the work of Douady, Estrada and Sentenac [@DS], and in that of Branner and Dias [@BD]. The translation model is also obtained by regluing the periodgon and attaching semi-infinite cylinders to its sides in place of the periodic domains.
There are immediate differences between polynomial and rational vector fields. A first one is that rational vector fields have in general several poles and, generically, the periodic domain of each singular point is attached to one of them. Hence we have configurations of periodic domains attached to the different poles, and we study these configurations. A second difference is that rational vector fields can have annuli of periodic solutions not included in the basin of a center. Also, while the periodgon of a rational vector field that we define below will be generically unique, the chosen cuts are less natural than in the polynomial case since they are determined not only by the planar organization of the periodic domains, but also by their length. However, the associated translation model is unique and canonical, and it is the important invariant.
After introducing the definitions of periodgon and translation model, and stating some general theorems regarding rational vector fields of degree $d$, we study the case $d=4$.
This leads to us ask the following general questions.
[**Question 1.**]{} What are the possible configurations of periodic domains attached to the poles in a rational vector field of degree $d$, i.e. the partition of periodic domains between the different poles? What are the maximal and the minimal number of periodic domains attached to a pole depending on its order? As a particular case, can we have configurations with no periodic domain attached to a pole?
[**Question 2.**]{} How many open sets in parameter space are necessary to cover the set of all rational vector fields of degree $d$ having a generic periodgon? For instance, the bifurcation creating an annulus of periodic solutions does not need a change of open set in parameter space to describe the periodgon.
[**Question 3.**]{} How many geometric types of periodgon can we have for generic rational vector fields of degree $d$? In this paper we will show that if $d=4$, this number is $3$ (Theorem \[thm:types\]).
[**Question 4.**]{} How many annuli of periodic solutions not surrounding a center are possible for a rational vector field of degree $d$, and what are their relative positions if several are possible?
General theory of rational vector fields on $\CP^1$
===================================================
We consider a rational vector field of the form $$\dot z = \frac{P(z)}{Q(z)}$$ on $\CP^1$. Since the Euler characteristic of $\CP^1$ is equal to $2$, it follows from the Poincaré-Hopf index theorem that the number of zeros minus the number of poles when counted with multiplicity is equal to $2$. Using a Moebius transformation we can suppose that $\infty$ is a regular point, in which case we can suppose that $\deg(P)-\deg(Q)=2$ and we say that the *degree of the vector field* is the degree of $P$. When $\deg(P)=2$, the vector field is polynomial and can be brought by an affine change of coordinate to $\dot z = z^2-\eps$. When $\deg(P)=3$, it is better to move the pole at infinity, thus transforming the vector field in a polynomial vector field. The study of cubic polynomial vector fields was done in [@R]. Hence the first truly rational case occurs when $\deg(P)= 4$. It will be studied below in Section \[sec:deg4\].
Singular points and poles
-------------------------
We consider a rational vector field $\dot z =V(z)= \frac{P(z)}{Q(z)}$ with $P$ and $Q$ relatively prime and $\deg(P)-\deg(Q)=2$. The following propositions are well known: see for instance [@BriTho; @DS; @GGJ1; @Tom2].\
1. The vector field is analytically linearizable in the neighborhood of any simple singular point $z_j$, which can only be:
- a focus if $V'(z_j)\notin \R\cup i\R$;
- a center if $V'(z_j)\in i\R\setminus\{0\}$;
- a radial node if $V'(z_j)\in \R\setminus\{0\}$;
2. Multiple singular points are parabolic points. The local analytic type of a parabolic singular point $z_j$ of order $k+1$ (codimension $k$) is completely determined by its order and the its period (or dynamical residue) $\nu_j=2\pi i\,\res_{z=z_j}\frac{dz}{V(z)}$; in particular, it is locally biholomorphically equivalent to $\dot z= \frac{(z-z_j)^{k+1}}{1+\frac{\nu_j}{2\pi i}(z-z_j)^k}$. It has $2k$ sepal zones, separated alternatingly by $k$ *attractive* and $k$ *repulsive directions* given by $\Im\left((z-z_j)^k\right)=0$ (see Figure \[parabolic-pole\](a)).
3. The poles are given by the zeros of $Q(z)$. In the neighborhood of a pole $p_j$ of order $k-1$, the vector field is analytically conjugate to $\dot z = \frac{1}{(z-p_j)^{k-1}}= \frac{(\bar z-p_j)^{k-1}}{|z-p_j|^{2k-2}}$.
Global organization of trajectories
-----------------------------------
The following proposition is well known (for instance [@Ben],[@Ha66 Theorem 2], [@Stre Theorem 9.4]).\
1. Every periodic trajectory of a rational vector field on $\CP^1$ belongs to a maximal open domain consisting of periodic trajectories (all of the same period) and possibly of a center equilibrium point, the boundary of which is formed by a union of homoclinic or heteroclinic separatrices.
2. All non-periodic trajectories start and terminate at either a singular point or a pole.
\[configurations\] [@Tom2]
1. All separatrices of poles either land at a singular point or merge with another separatrix in a homoclinic or heteroclinic connection.
2. At least one separatrix of some pole lands at each singular point of focus or node type, and at least one separatrix of some pole lands at a parabolic point tangent to each attractive/repulsive direction.
The following lemma (see for exemple [@Ha66 Theorem 1]) is a simple consequence of Poincaré-Hopf index theorem.
\[lemma:configurations\] Each periodic orbit of a rational vector field surrounds $n+1$ zeros and $n$ poles for some $n\in \Z_{\geq0}$ (multiplicities taken into account).
Simple zeros of a rational vector field have index $+1$ and simple poles have index $-1$, since the trajectories are organized as for a saddle point (the line field is the same as that of the vector field $\dot z = C\bar z$). It is known that the sum of the indices of the singularities inside a closed trajectory is equal to $+1$.
A periodic trajectory of a polynomial vector field is inside the basin of a center.
If a periodic trajectory or a polynomial vector field of degree $d$ surrounds more than one singular point, then it must surround a pole. But the only pole is at infinity and has degree $d-2$. Hence the periodic trajectory must surround $d-1$ singular points. Since it divides $\CP^1$ in two components, it is in the basis of the $d$-th singular point which is a center.
The closure in $\CP^1$ of the union of all separatrices of all poles is called the *separatrix graph*. The connected components of its complement are called *zones*.
The topological organization of the real phase portrait of the vector field is completely determined by the separatrix graph. There are four kinds of zones that can occur in a rational vector field on $\CP^1$:
1. *$\alpha\omega$-zone:* all trajectories have the same $\alpha$-limit and the same $\omega$-limit which are two different equilibria (simple or multiple)
2. *odd/even sepal zone:* all trajectories have the same $\alpha$-limit and $\omega$-limit which is a parabolic equilibrium,
3. *odd/even center zone* consisting of periodic counter-clockwise/clockwise periodic orbits of the same period $\nu_j>0$ around a center equilibrium point $z_j$ (which is included in the zone),
4. *periodic annulus* consisting of periodic orbits of the same period $\nu>0$ and surrounding on each side (on $\CP^1$) at least one pole.
The translation model of a rational vector field
------------------------------------------------
In this section we will move to the rectifying coordinate $t$ defined by $$t = t(z) = \int\frac{Q(z)}{P(z)}\;dz.\label{eq:t}$$ The Riemann surface of $t(z)$ is a translation surface with the coordinate $t$ unique up to a translation. The periodgon and the star domain that we define below lie on this surface. The quotient of this surface by the deck transformations of the projection $t(z)\to z$, equipped with the form $dt$, will be called the *translation model* for the vector field.
We define the *translation model* for the vector field $\dot z = \frac{P(z)}{Q(z)}$ as a triple $({\mathcal}S,\Sigma, dt)$ where
1. ${\mathcal}S$ is the smooth manifold $\CP^1{\smallsetminus}\{z:P(z)=0\}$,
2. $\Sigma\subset{\mathcal}S$ is the finite set of poles $\{z:Q(z)=0\}$,
3. ${\mathcal}S{\smallsetminus}\Sigma$ is a translation surface, with local translation atlas given by the restrictions of the map to simply connected domains,
4. the neighborhood of each point of $\Sigma$ corresponding to a pole of order $m-1$ is a cone of angle $2m\pi$,
5. $dt$ is a translation invariant abelian form on ${\mathcal}S{\smallsetminus}\Sigma$.
In another words, the translation model is obtained by cutting $\CP^1{\smallsetminus}\{z:P(z)=0\}$ into simply connected pieces, taking their image by $t(z)$ and gluing them back together to create an abstract translation surface on which the real dynamics of the vector field is represented by the horizontal flow of $\dot t=1$.
The translation model is a very natural object that was previously considered by other authors; in particular it appears in [@Mucino-Valero; @Mucino-Valero2; @Mucino].
Two rational vector fields are globally conjugated (by a Moebius transformation) if and only if their translation models are isomorphic, i.e. if there exists a biholomorphism of the translation surfaces sending each conic singularity to a conic singularity with the same angle and preserving the form $dt$.
There is a bijection $h_l$ between $\CP^1{\smallsetminus}\{z:P_l(z)\}$ and the domain of the translation model of $\dot z = \frac{P_l(z)}{Q_l(z)}$, which is biholomorphic outside of the poles and sends the poles to the conic singularities. Let $H$ be an isomorphism between the translation surfaces. Note that $H$ sends the conic singularities to conic singularities of the same type. Also it preserves the vector field $\dot t =1$. Then $h_2^{-1}\circ H\circ h_1$ is a global conjugacy of the two vector fields on $\CP^1{\smallsetminus}\{z:P_l(z)=0\}$ that extends to the punctures (because it is bounded there), and that is analytic at the poles; hence it is a Moebius transformation.
Periodic, parabolic and annular domains
---------------------------------------
To each equilibrium point $z_j$ one associates its *period* (also called *dynamical residue*) as the “travel time” along a simple loop around the point: $$\nu_j=2\pi i \;{\rm Res}_{z_j} \frac{Q(z)}{P(z)}.$$ By the residue theorem $\nu_1+\dots + \nu_d=0.$
If $z_j$ is a simple equilibrium point, then its period is $\nu_j= 2\pi i\,\frac{Q(z_j)}{P'(z_j)}$, and $z_j$ is a center of the rotated vector field $$\dot z = e^{i\arg \nu_j}\frac{P(z)}{Q(z)}. \label{rotated_vf}$$
1. For a simple equilibrium point $z_j$, the *periodic domain* of $z_j$ is the periodic basin of the center (also called center zone) at $z_j$ of . The boundary of the periodic domain of $z_j$ consists of one or several homoclinic or heteroclinic connections of . Generically it is a single homoclinic loop through one pole, which we then call the *homoclinic loop* of $z_j$.
2. For a multiple equilibrium point $z_j$, the *parabolic domain* of $z_j$ is the union of all the sepal zones of $z_j$ in for all $\beta\in\R$ (see Figure \[parabolic\_domains\]).
3. An *annular domain* is a periodic annulus of for some $\beta\in\R$.
4. An *end* of a periodic/parabolic/annular domain refers to a “corner” of the domain at an adjacent pole. The domain may have several ends at the same pole.
\
Given a rational vector field, $\dot z =\frac{P(z)}{Q(z)}$, we consider the associated family of rotated vector fields $$\label{rotated_beta}
\dot z =e^{i\beta}\frac{P(z)}{Q(z)},\qquad \beta\in \R.$$ Any homoclinic or heteroclinic connection appearing in the rotating family for some $\beta\in\R$ is called a *saddle connection* of the rational vector field. Note that it is oriented.
We will consider simultaneously saddle connections corresponding to different angles of rotation $\beta$.
The parabolic domain of a multiple equilibrium $z_j$ covers a full neighborhood of $z_j$. Its boundary consists of a finite number of saddle connections.
Locally, the sepal zones of the equilibrium $z_j$ rotate with $\beta$, hence their union covers a full neighborhood of $z_j$. For each $\beta$ the sepal zones are bounded by separatrices. A separatrix landing at the equilibrium will be completely contained in a sepal zone for any $\tilde\beta$ either on the left side or on the right side of $\beta$, and sufficiently close to $\beta$. On the other hand, a homoclinic separatrix bordering a sepal zone of $z_j$ for some $\beta$ will be also bordering the parabolic domain $z_j$, as follows from Proposition \[prop:periodicdomains\] below. There is only a finite number of ends to the parabolic domain (there is only a finite number of poles, each can host only finitely many ends), hence the boundary consists of only finitely many saddle connections.
\[prop:periodicdomains\]
1. The periodic and parabolic domains of all the equilibria act as trapping regions for the whole rotating family of vector fields : no trajectory of for any $\beta$ can both enter and leave a periodic or parabolic domain. In particular, no saddle connection can intersect the periodic or parabolic domain of an equilibrium.
2. The periodic, parabolic and annular domains are all disjoint.
3. If the degree is $>2$ and if all singular points are simple and the boundary of each periodic domain consists of exactly one homoclinic loop, then these homoclinic loops are disjoint too.
The proof is the same as that of [@KR] in the polynomial case and uses well-known facts about families of rotated vector fields (see for instance [@Duf] or [@P]), namely that trajectories of the family of rotated vector fields for different values of $\beta\in \R$ can have at most one intersection point. Also, separatrices of a pole for different $\beta$’s can only intersect at the pole.
The following proposition is immediate.
\[prop:translationmodel\] Let us now consider the images of the periodic, parabolic and annular domains by the map .
1. The image of an annular domain is a cylinder of finite area in the translation model, the boundary of which consists of two connected components formed by saddle connections with the same angle $\beta$. In particular, each of the two boundary components contains at least one pole (element of $\Sigma$).
2. The image of a periodic domain minus the singular point is a semi-infinite cylinder (a “hose” of flat pants) in the translation model, the boundary of which has a single component formed by saddle connections with the same angle $\beta=\arg\nu_j$.
3. The image of the parabolic domain of an equilibrium of multiplicity $k+1$ minus the singular point itself is a union of $2k$ half-planes, corresponding to the sepal zones for any fixed $\beta$, and of a finite number of half-strips, corresponding to inter-sepal regions each attached to a single component of the boundary of the domain (see Figure \[fig:par\_domains\]). In total, it covers a punctured cone of angle $2k\pi$ at infinity (in a projective coordinate).
\
1. Each annular domain contains infinitely many saddle connections. More precisely, for each pair of ends on opposite boundaries of the domain, there are $\Z$-many different saddle connections (see Example \[example1\] and Figure \[fig:example1\]): if the $0$-th such connection is selected as (one of) the shortest, then the $n$-th such connection is the one for which the closed curve obtained by composing it with the reverse of the $0$-th one has a turning number $n$. As $n\to\pm\infty$ the angle $\beta_n$ in of the $n$-the saddle connection tends to the angle $\beta$ of the annular domain.
2. There is only a finite number of saddle connections that are not completely contained in one of the annular domains.
\[example1\] We consider the system $$\dot z = i\frac{z^4+1}{1-z^2}.\label{vf_example}$$ It has four centers at $z_j= e^{i\frac{\pi(1+2j)}4}$, $j=0, \dots, 3$, with periods $\pm\frac{\pi}{ \sqrt{2}}$, and two poles at $z=\pm1$. Moreover, the imaginary axis is invariant, and hence a periodic orbit on $\CP^1$. It belongs to a family of periodic orbits. The phase portrait appears in Figure \[fig:example1\]. There is a bi-infinite sequence of $\beta_n$ such that the corresponding rotated vector fields has a saddle connection.
Chains of saddle connections
----------------------------
The following proposition is a form of Poincaré-Hopf lemma for regions bounded by chains of saddle connections.
\[prop:index2\] Let $\Gamma\subset\CP^1$ be a (union of) positively oriented closed curve(s) consisting each of a finite number of a saddle connections forming the boundary of an open set ${\operatorname{int}}(\Gamma)\subset\CP^1$. Then $$\label{formula}
\#_{sc} \Gamma-\tfrac{1}{\pi}\sum_{\Gamma\cap\Sigma}\sphericalangle=2\chi({\operatorname{int}}(\Gamma))-2\sum_{z\in{\operatorname{int}}(\Gamma)}{\operatorname{Ind}}_zX,$$ where
- $\#_{sc} \Gamma$ is total the number of the oriented saddle connections appearing in all components of $\Gamma$ (the same curve may appear twice as two saddle connections with opposite orientations),
- $\sum_{\Gamma\cap\Sigma}\sphericalangle$ is the sum of the angles of ${\operatorname{int}}(\Gamma)$ at its ends (at the points of $\Gamma\cap\Sigma$) measured in the translation model (i.e. each angle is the angle between the saddle connections at the pole on $\CP^1$multiplied by $m+1$, where $m$ is the multiplicity of the pole),
- $\chi$ is the Euler characteristic,
- ${\operatorname{Ind}}_zX$ is the Poincaré-Hopf index of the vector field $X=\frac{P(z)}{Q(z)}\frac{\partial}{\partial z}$ at a point $z$, that is $m$, if $z$ is a singularity of multiplicity $m$, and $-m$, if $z$ is a pole of multiplicity $m$.
We shall show that the formula is stable by the operations
1. adding new points to $\Sigma$ as poles of multiplicity 0;
2. cutting the set ${\operatorname{int}}(\Gamma)$ by additional saddle connections (which then appears twice in the new $\Gamma$ with two opposite orientations).
By this procedure, the domain can be cut into a union of nonsingular triangles and of periodic and parabolic domains. Indeed, with every singularity in ${\operatorname{int}}(\Gamma)$, then ${\operatorname{int}}(\Gamma)$ contains also its periodic/parabolic domain relative to $\Sigma$, since no saddle connection can cut this domain by Proposition \[prop:periodicdomains\]. Therefore it will be enough to show that the formula is valid for these domains. We do this in (3).
1. Adding a new point to $\Sigma$:
1. a point of ${\operatorname{int}}(\Gamma)$: it increases the Poincaré-Hopf index by 0;
2. a point of $\Gamma$: it divides a saddle connection into two pieces and adds an interior angle $\pi$.
2. Cutting ${\operatorname{int}}(\Gamma)$ along a new saddle connection increases the number of saddle connections by 2 while
1. if both endpoints are on $\Gamma$: either it increases the number of connected components of ${\operatorname{int}}(\Gamma)$ by 1 or it decreases the number of holes in ${\operatorname{int}}(\Gamma)$, in both cases increasing the Euler characteristic by 1;
2. if one endpoint is on $\Gamma$ and the other in ${\operatorname{int}}(\Gamma)$: an interior pole of order $m-1$ stops being counted by the double of its Poincaré-Hopf index $m-1$ and becomes counted by an angle $2m\pi$;
3. if both endpoints are in ${\operatorname{int}}(\Gamma)$: this creates a new hole, thus decreasing the Euler characteristic by 1, while also changing how the endpoints are counted.
3. The formula is clear for simply connected domains with no singularities, as well as for periodic domains. Let us show it for parabolic domains. By Proposition \[prop:translationmodel\] for a generic angle $\beta$ the parabolic domain decomposes into a union of $2m$ single-ended sepal zones (corresponding to half-planes with a single point of $\Sigma$ on the boundary line), each contributing by an angle $\pi$, and of a number of inter-sepal regions (corresponding to half-strips with two corner points of $\Sigma$), one for each saddle connection on the boundary of the domain, where the total angle $\pi$ of the half-strip cancels the contribution of the saddle connection.
One can always add any regular point to the set of poles $\Sigma$ as a pole of order zero, thus enlarging the set of saddle connections. In particular, Lemma \[lemma:configurations\] becomes a special case of Proposition \[prop:index2\] if one adds to $\Sigma$ a point on the periodic trajectory.
\[prop:fundamentalgroupoid\] The fundamental groupoid $$\Pi_1({\mathcal}S,\Sigma)=\Pi_1(\CP^1{\smallsetminus}\{\text{singularities}\},\ \{\text{poles}\})$$ is generated by the set of saddle connections. Different saddle connections are not end-point-homotopic.
If two saddle connections are end-point-homotopic, then by Proposition \[prop:index2\] the sum of the angles between them at their end-points is null, meaning that they are equal.
The boundary of each periodic/parabolic domain is a chain of saddle connections forming a simple loop around the singularity with a base-point at a pole. All we need to show is that each two poles are connected by a chain of saddle connections.
To each pole we associate the open domain swept by its separatrices in the rotating family , i.e. all the points on the translation model connected to the pole by a straight segment. If non-empty, the boundary of this domain in ${\mathcal}S$ must contain another pole connected to the original pole by a saddle connection. The union of the domains of all the poles that are connected to the original one by a chain of saddle connections has therefore an empty boundary, hence it is all of ${\mathcal}S$.
Proposition \[prop:fundamentalgroupoid\] makes good sense only if $\Sigma\neq\emptyset$, which can always be assumed potentially after adding to it a pole of order 0 in the case of degree $2$ vector fields.
The *relative homology* $H_1({\mathcal}S,\Sigma;\Z)$, which is the abelianization of the fundamental groupoid $\pi_1({\mathcal}S,\Sigma)$, is therefore also generated by the saddle connections.
The *period map* $\nu: H_1({\mathcal}S,\Sigma;\Z)\to\C$ is given by integration of the time form $dt=\tfrac{Q(z)}{P(z)}dz$ along cycles. The *period* of a saddle connection $\gamma$ is $$\nu_\gamma= \int_\gamma\frac{Q(z)}{P(z)}\;dz\neq 0.$$
Even for generic vector fields with all zeros simple, the knowledge of the multiplicities of its poles in $\Sigma$ and of the period map $\nu: H_1({\mathcal}S,\Sigma;\Z)\to\C$ alone does not determine the translation model. Another combinatorial information expressing the organization of the saddle connections is necessary. In case of polynomial vector fields, the cyclic order in which the periodic domains are attached to the poles provides such information.
The periodgon and star domain
-----------------------------
In order to understand the form of the translation model we shall cut it and unwrap it into a flat simply connected domain on the surface of $t(z)$: the star domain, and its core part: the periodgon, defined below.\
\[def:periodgon\] Consider a rational vector field of degree at least 3, and let $\Sigma$ be the set of poles of positive multiplicity only.
1. *Cuts*:
- For each equilibrium $z_j$, choose an end of the periodic/parabolic domain (several choices may be possible) and cut along a separatrix of inside the domain connecting $z_j$ to the end. If $\nu_j\neq 0$, e.g. if $z_j$ is simple, let $\beta=\arg\nu_j+\frac{\pi}{2}$.
- For each annular domain cut along (one of) the shortest saddle connection(s) inside the domain (not on the boundary).
- For each component of the complement of all the periodic/parabolic/annular domains with at least two poles, keep adding cuts each time along (one of) the shortest heteroclinic saddle connection(s) not intersecting the previous cuts, until the cuts form a tree connecting all the poles.
2. The complement in $\CP^1$ of these cuts is a simply connected domain, the closure of whose image by (a branch of) $t(z)$ is called the *star domain*.
3. The *periodgon* is the part of the star domain that corresponds to the complement of all the periodic and parabolic domains.
4. If in the definition of the star domain and the periodgon we choose instead any set of non-intersecting cuts simply connecting all the poles then the corresponding objects will be called a *generalized star domain* and a *generalized periodgon*.
1. If a periodic/parabolic domain has several ends, or if several shortest saddle connections appear in the construction of the cuts, then several choices are possible, leading to several different periodgons describing the same dynamics. But this is a nongeneric case.
2. Heteroclinic connections between the poles of a rotated vector field always exist for some $\beta$ because of the monotonous movement of the separatrices of a family of rotated vector fields when $\beta$ varies. See Proposition \[prop:fundamentalgroupoid\].
3. The (generalized) periodgon is a compact polygonal domain, possibly degenerate, on the translation surface of $t(z)$ with vertices at the conical points. The projection of the (generalized) periodgon on $\C$-space may have self-intersections. This is because the translation surface of $t(z)$ is ramified at the images of the poles.
4. In the case when equilibria are simple, the boundary of the (generalized) periodgon is formed by $d$ segments corresponding to the period vectors $\nu_1,\dots, \nu_d$ of the $d$ equilibrium points $z_1,\ldots,z_d$ and $(n-1)$ pairs of parallel equal vectors of opposite directions corresponding to the travel times for the cuts between the $n$ poles. And the (generalized) star domain is the union of the periodgon and $d$ infinite branches of respective width $\nu_1, \dots, \nu_d$, which are orthogonal to the respective sides $\nu_1, \dots, \nu_d$.
5. Note that the (generalized) periodgon has no limit in a parametric family when approaching a parabolic point. Indeed, the sides of the periodgon corresponding to the merging points become infinite and their arguments turn with the parameter. For example, in the family of the form $\dot z = z^2 -\eps z+O(z^3)$ (resp. $\dot z = z^2-\eps +O(z^3)$) the merging sides are $\sim \pm\frac{2\pi i}{\epsilon}$ (resp. $\sim \pm\frac{2\pi i}{\sqrt\epsilon}$), while their sum has a finite limit given by the period of the limit parabolic point.
[**Coming back to Example \[example1\].**]{} We consider the system . In Figure \[multiple\_cuts\]. There are infinitely many saddle connections between the two poles, of which only the two arcs in $\R\cup\infty$ are of the shortest length and can each serve as a cut (see Figure \[multiple\_cuts\]), which yields to two different periodgons. Indeed, the left and right sides of the periodgon are glued together, forming a cylinder, on which we have saddle connections that make an arbitrary number of turns around the cylinder.
We consider the system $$\dot z = i\frac{z^4+1.2}{1-z^2},\label{vf_example2}$$ which is a deformation of , and whose periodgon is a deformation of the periodgon of Figure \[multiple\_cuts\](b) corresponding to the red cut in Figure \[multiple\_cuts\](a). The four centers have now become attracting or repelling foci and we still have the two poles at $z=\pm1$. Moreover, the imaginary axis is still invariant and still belongs to a family of periodic orbits. The phase portrait appears in Figure \[fig:example2\](a), and the periodgon and star domain in Figure \[fig:example2\](b).
Bifurcations of the periodgon
-----------------------------
An important question is to understand how the periodgon and star domain depend on the coefficients of $P$ and $Q$ in a rational vector field of degree $d$ and what are their bifurcations. We address this question when $d=4$ in the next section.
The case of a rational vector field of degree $4$ {#sec:deg4}
=================================================
Using an affine change of coordinate we can suppose that the vector field has the form $$\dot z = \frac{z^4+\eta_3z^3+\eta_2z^2+\eta_1z+\eta_0}{1-az^2}.\label{vf_4}$$ There is still one degree of freedom coming from Moebius transformations that can be used to simplify the system: see Proposition \[prop:normal\_form\] below. The system is truly rational when $a\neq0$. The periodgon was described in [@KR] in the particular case $a=\eta_2=\eta_3=0$, and in [@R19] in the case $a=\eta_0=\eta_3=0$. We will start by investigating the case $a\neq0$. As a second step we will consider what happens when we let $a=0$.
Structure theorem
-----------------
\[prop:configuration\_4\] We consider the vector field with $a\neq0$.
1. The only generic configurations of periodic domains are $(3,1)$ and $(2,2)$, namely $3$ (resp. $2$) periodic domains attached to one pole and $1$ (resp. $2$) attached to the other pole. Moreover, these configurations do occur.
2. A periodic orbit either belongs to the basin of a center, or surrounds on each side two singular points and one pole and belongs to an annulus of periodic orbits. The latter case can only occur in the case of a configuration $(2,2)$.
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1. An end of a periodic domain at a simple pole has an opening of $\frac{\pi}2$ in the $z$-coordinate. Let us show that it is impossible to have four periodic domains attached to one pole. Indeed, for this to occur, the four ends of the four periodic domains would need to be all tangent at the pole, and hence the four domains would need to appear simultaneously for the same angle of rotation. But the homoclinic loops bounding four periodic domains cannot occur simultaneously, since a simple pole has only four separatrices.
2. This follows from Lemma \[lemma:configurations\].
We will show that the different types of generic configurations of Proposition \[prop:configuration\_4\] indeed exist in the family .
\[thm:types\] In the case of distinct zeros and poles we have generically three types of periodgon, or equivalently, three geometric types of translation model (see Figure \[pgon\_types\]):
- A periodgon corresponding to configuration $(3,1)$;
- A periodgon corresponding to configuration $(2,2)$ with no annular domain, noted $(2,2)_{NO}$;
- A periodgon corresponding to configuration $(2,2)$ with an annular domain, noted $(2,2)_{AD}$.
In all cases the periodgon is planar, i.e. its identical projection from the translation surface of $t$ to $\C$ has no self-intersections. It contains two parallel sides corresponding to a cut between the two poles.
These three types of periodgon can all bifurcate from the polynomial case $\dot z = P(z)$ when the two poles merge at infinity. In the merging the parallel sides corresponding to the cuts shrink to a point.
The existence of the three configurations within the general family of degree 4 rational vector fields with simple poles will follow from the Realization Theorem (Theorem \[thm:realization\]) and Figure \[pgon\_types\]. The fact that the (generalized) periodgon has no self-intersection is because of the small number of sides which, when negatively oriented, are of the form $\nu_{\sigma(1)}, \nu_{\sigma(2)}, \tau, \nu_{\sigma(3)}, \nu_{\sigma(4)}, -\tau$ in a cyclic order for the configuration $(2,2)$, resp. $\nu_{\sigma(1)}, \nu_{\sigma(2)}, \nu_{\sigma(3)}, \tau,\nu_{\sigma(4)}, -\tau$ for the configuration $(3,1)$, where $\sigma$ is some permutation on the indices, and the fact that the periodgon is bounded.
Projective reduction of the family
-----------------------------------
Simpler forms than can be found when using projective transformations to simplify the system.
\[prop:normal\_form\]
1. Any system with $a\neq0$ can be brought by a Moebius transformation depending analytically on the coefficients $(a, \eta_0, \dots, \eta_3)$ to the form $$\dot z =C(\eps) \frac{z^4+\eps_3z^3+\eps_2z^2+\eps_1z+\eps_0}{1-z^2},\label{vf_4bis}$$ for some $C(\eps)\neq0$.
2. Let be a family with $a\neq0$ depending on the multi-parameter $\eta=( \eta_0, \dots, \eta_3)$ varying in the neighborhood of the origin in $\C^4$.Then there exists a family of Moebius transformations depending analytically on $\eta$ and transforming the family to the form $$\dot z =C(\eps) \frac{z^4+\eps_2z^2+\eps_1z+\eps_0}{1-z^2},\label{vf_4quat}$$ for some $C(\eps)=1+O(\eps)$.
3. Let be a family depending on the multi-parameter $\eta=(a, \eta_0, \dots, \eta_3)$ varying in the neighborhood of the origin in $\C^5$, and let $C\in \C^*$. Then there exists a family of Moebius transformations depending analytically on $\eta$ and transforming the family to the form $$\dot z =C\frac{z^4+\eps_2z^2+\eps_1z+\eps_0}{1-\delta z^2},\label{vf_4ter}$$ where $\delta$ and the $\eps_j$ depend analytically on $\eta$ and vanish for $\eta=0$.
In (1) and (2), using a real (complex) scaling of time depending real-analytically (resp. analytically) on $\eta$ we take make $C= \exp(i\beta(\eps))$ (resp. $C=1$ or any desired value of $C$).
We consider a change of coordinate $Z=d\frac{z+b}{1+cz}$ with $d(1-bc)\neq0$. The transformed system has poles of the form $\pm Z_0$ if and only if $c=ab$.
1. Moreover $Z_0=1$, if and only if $a-d^2=0$. It suffices to take $d=\pm \sqrt{a}\neq0$.
2. By (1) we can start with a system of the form , and consider a change $Z=\frac{z+b}{1+bz}$. The coefficient of $Z^3$ has the form $h(\eta)(b+O(\eta))$, where $h(0)\neq0$. Hence, by the implicit function theorem it will vanish for some analytic function $b(\eta)$.
3. We consider a change $Z=d\frac{z+b}{1+abz}$. The coefficient of $Z^4$ is of the form $\frac{1+O(\eta)}{d^3}$, and hence can be taken equal to $C$ for some nonzero analytic function $d(\eta)$. Then the coefficient of $Z^3$ vanishes for $-4bd+ O(\eta)=0$, which can be solved by the implicit function theorem.
An example
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We consider the following subfamily: $$\dot z =i \frac{(z^2-\eta_1)(z^2+\eta_2)}{1-z^2},\label{centre_organisateur}$$ with $\eta_1,\eta_2\in \R^+$ and $\eta_1>1$ (see Figure \[centre\_org\](a)). The system is symmetric with respect to $i\R$ and reversible with respect to $\R$. The singular points $A =\sqrt{\eta_1}$ and $C =-\sqrt{\eta_1}$ are centers and their periodic domains are each attached to one pole, namely $S_1 = 1$ or $S_2 = -1$. The periodic domains of $B= i \sqrt{-\eta_2}$ and $D=-i\sqrt{-\eta_2}$ are symmetric with respect to $i\R$. Hence they both are bounded by a heteroclinic cycle through the two poles, one side of the heteroclinic cycle being $[-1,1]\subset \R$ (see Figure \[centre\_org\](b)).
There are four ways to draw the periodgon depending on the four ways we take the cuts from each of $B,D$ to one pole $S_j$ (see Figure \[four\_ways\]), to which correspond four star domains and periodgons (see Figure \[four\_periodgons\]).
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A perturbation breaking the symmetry with respect to $i\R$, while keeping the reversibility with respect to $\R$, for instance $$\dot z =i \frac{(z^2+\eta_3z -\eta_1)(z^2+\eta_2)}{1-z^2},$$ with $\eta_3$ real small, will then create the configurations $(3,1)_1$ (with the periodic domains of $B,D$ attached to $S_1$) or $(3,1)_2$ (with the periodic domains of $B,D$ attached to $S_2$) depending on the sign of $\eta_3$. A second perturbation by moving $\eta_2$ in outside $\R$ will keep the symmetry with respect to the origin but will break both the symmetry with respect to $i\R$ and the reversibility with respect to $\R$. It will then create the two configurations $(2,2)$, where either the periodic domain of $B$ is attached to $S_1$ and that of $D$ to $S_2$, or the converse. All these configurations appear in Figure \[perturbed\_pd\]. The corresponding four types of periodgons (possibly with additional deformations) are illustrated in Figure \[deformed\_periodgon\].
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We see the general form of the periodgon: it has four segments corresponding to homoclinic loops around the four singular points and two segments corresponding to a cut between the poles.
Bifurcation between the two configurations $(3,1)$ and $(2,2)$
--------------------------------------------------------------
In the family and Figure \[centre\_org\] we have seen a bifurcation between the two configurations $(3,1)$ and $(2,2)$. But this was a codimension $2$ phenomenon where two singular points had their periodic domains bounded by heteroclinic loops through the two poles. The phenomenon however is of ocodimension $1$ and a bifurcation occurs as soon as one singular point has a periodic domain bounded by a heteroclinic loop through the two poles (see Figure \[bif\_conf\]). The bifurcation is easily described through the change of the periodgon (see Figure \[bif3-1\_2-2\]).
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Bifurcation creating an annular domain
--------------------------------------
When considering a rotated vector field $\dot z = e^{i\alpha} \frac{P(z)}{Q(z)}$ of $\dot z = \frac{P(z)}{Q(z)}$, its corresponding periodgon is obtained by rotating the periodgon of $\dot z = \frac{P(z)}{Q(z)}$ of an angle $-\alpha$. A rotated vector field $\dot z = e^{i\alpha} \frac{P(z)}{Q(z)}$ has an annulus of periodic solutions if there exists inside the periodgon a continuum of parallel segments joining the two parts of the boundary corresponding to the cut between the poles. Such a continuum does not exist in any of the periodgons of Figure \[deformed\_periodgon\]. When moving further from , in the case of the configuration $(2,2)$, the periodgon will be deformed. Is it possible to have a continuous deformation leading smoothly to the formation of an annular domain (for instance as in Figures \[multiple\_cuts\] and \[fig:example2\])? This is the phenomenon that we now study.
We consider a vector field with simple singular points and poles, and such that each periodic domain is attached to only one pole. Let $\nu_1, \nu_2,\nu_3,\nu_4$ be the four period vectors of the singular points of and let $\pm\tau$ be the period of the chosen saddle connection between the two poles along which one cuts. Note that $\nu_1+\nu_2+\nu_3+\nu_4=0$. There exists a permutation $\sigma$ of $\{1, 2, 3, 4\}$ such that the periodgon has, for configuration $(2,2)$, sides given by the vectors $$\nu_{\sigma(1)}, \nu_{\sigma(2)}, \tau, \nu_{\sigma(3)}, \nu_{\sigma(4)}, -\tau,$$ in this circular order. Note that the $\tau$ vector can never be aligned with two $\nu_{\sigma(j)}$ located between the two $\tau$. Indeed, a simple pole has four separatrices. Hence if for some $\beta$ the vector field has a heteroclinic connection, it can have simultaneously at most one homoclinic connection.
We want to characterize those realizable periodgons for which the realized vector field has an annular domain. This comes to characterize geometrically those periodgons for which there exists, inside the periodgon, a continuum of parallel segments joining the two segments $\tau, -\tau$ of the boundary. Each part of the boundary of an annulus of periodic solutions not surrounding a center can be of two kinds:
- a single homoclinic connection;
- a pair of homoclinic connections forming a figure eight (as in Figure \[fig:example1\]).
\[prop:ann\_domain\] The vectors $
v_{12}=\nu_{\sigma(1)}+ \nu_{\sigma(2)}, \tau, v_{23}=\nu_{\sigma(3)}+ \nu_{\sigma(4)}, -\tau$ can be considered as the oriented boundary of a parallelogram, yielding an orientation to the parallelogram. A necessary condition for the existence of an annular domain is that a parallel strip between the two sides $\pm \tau$ be included inside the periodgon. For this to occur, it is necessary that the parallelogram be negatively oriented. Hence a bifurcation creating an annular domain is one that reverses the orientation of the parallelogram. This occurs precisely when the four vectors $v_{12}, \tau, v_{23}, -\tau$ become aligned: see Figure \[fig:bif\_periodic\]. The corresponding bifurcation creating an annular domain is a heteroclinic loop with no return map in a rotated vector field. It appears in Figure \[fig:bif\_per\_bis\].
Note that the parallelogram is positively oriented in all the periodgons of Figure \[deformed\_periodgon\] and that none of the corresponding vector fields has an annular domain.
![A heteroclinic loop with no return map. []{data-label="fig:bif_per_bis"}](Double_HL){width="3.8cm"}
The bifurcation described in Proposition \[prop:ann\_domain\] explicitly occurs inside rational vector fields of degree $4$. Indeed, consider the family $$\dot z = i \frac{\left((z-a)^2+\frac12\right)\left((z^2-2)^2+\frac12\right)}{1-z^2},\label{family_heart}$$ depending on real $a$. The bifurcation of heteroclinic loop occurs for $a\in(-0.8,-0.6)$, see Figure \[fig:heart\]. This special case was called the *heart* by Ilyashenko in [@I] in planar vector fields. The bifurcation diagram in the generic case was studied by Dukov [@D]. The poles are transformed into saddles if one multiplies the vector field by $(1-z^2)(1-\bar z^2)$. One specific feature of the bifurcation diagram is the existence of two semi-infinite sequences of bifurcation curves where heteroclinic connections occur. For each $n\in \N$, there is one heteroclinic connection starting from the first (resp. second) pole to the second (resp. first pole) and making exactly $n$-turns and a fraction around two anti-saddles. The phenomenon has been called *sparkling separatrices* by Ilyashenko [@I].
A heteroclinic loop is a real codimension 2 bifurcation. Hence its bifurcation diagram requires two parameters. This bifurcation diagram indeed occurs in rational vector fields of degree $4$ inside the family $$\dot z = i e^{i\theta} \frac{\left((z-a)^2+\frac12\right)\left((z^2-2)^2+\frac12\right)}{1-z^2}$$ depending on the two parameters $a$ and $\theta$: see Figure \[fig:bif\_diag\]. The bifurcation diagram can also be read from the periodgon: see Figure \[fig:bif\_diag2\].
![The topological $2$-parameter bifurcation diagram of the heteroclinic loop of Figure \[fig:bif\_per\_bis\]: an annulus of period orbits occurs along a half-curve in parameter space. There are two infinite sequences of heteroclinic connections occuring in the upper and lower left quadrants. []{data-label="fig:bif_diag"}](Bif_diag_heart){width="13cm"}
The real codimension $2$ bifurcation of heteroclinic loop of Figure \[fig:bif\_per\_bis\] does not split the parameter space. But since we are only interested in the bifurcation of the shape of the periodgon, regardless of its rotations, we quotient the parameter space by the group of rotations , which in the family are induced by the action $$\begin{cases}z\mapsto e^{-i\frac{\beta}{3}}z,\\
(a,\eta_0,\eta_1,\eta_2,\eta_3)\mapsto(e^{i\frac{2\beta}{3}}a,e^{-i\frac{4\beta}{3}}\eta_0,e^{-i\beta}\eta_1,e^{-i\frac{2\beta}{3}}\eta_2,e^{-i\frac{\beta}{3}}\eta_3).\end{cases}$$ Then this becomes a real codimension $1$ bifurcation, which splits the quotient parameter space.
Realization of a generalized periodgon and star domain
------------------------------------------------------
\[thm:realization\] Let $\nu_1, \nu_2,\nu_3,\nu_4\in \C^*$ such that $\nu_1+\nu_2+\nu_3+\nu_4=0$ and let $\tau\in\C^*$ be given. Let us consider the two (circular) sequences below.
1. $\nu_{\sigma(1)}, \nu_{\sigma(2)}, \tau, \nu_{\sigma(3)}, \nu_{\sigma(4)}, -\tau$ (for configuration $(2,2)$);
2. $\nu_{\sigma(1)}, \nu_{\sigma(2)}, \nu_{\sigma(3)}, \tau,\nu_{\sigma(4)}, -\tau$ (for configuration $(3,1)$).
We consider each sequence as the ordered sequence of sides of a polygon over a translation surface with ramification points at the vertices. As a first condition we ask that the closed polygonal curve with sides given by the sequence is the oriented boundary (in the positive direction, i.e. the domain is to the left of the boundary) of an unbounded open domain in $\C$, the complement of which is a (possibly degenerate) polygon. In particular no sides can intersect transversally in their interior, but some sides can coincide with opposite direction. Let us now attach numbers $1$ or $2$ (representing the two poles) to the vertices of the polygon with the rule that the same number is attached to the two ends of a side $\nu_{\sigma(j)}$ and different numbers are attached to the two ends of $\pm \tau$. We ask that the sum of the inner angles of the periodgon attached to each of the two numbers is equal to:
1. $2\pi$ at each pole for configuration $(2,2)$;
2. $\pi$ (resp. $3\pi$) for the pole attached to $3$ (resp. $1$) sides (resp. side) $\nu_j$ for configuration $(3,1)$.
Then the polygon can be realized as a generalized periodgon of a rational vector field of degree $4$.
Let us build the associated star domain by adding orthogonal semi-infinite strips of width $\nu_{j}$ on the corresponding sides of the polygon, which we endow of the constant vector field $\dot t=1$. We glue together the two sides $\tau$ and $-\tau$, and the sides of the semi-infinite strips (yielding semi-infinite cylinders), thus obtaining a translation surface with conical singularities at the points $1$ and $2$. Removing these two points, the translation surface is conformally equivalent to $\CP^1$ minus six points. Because the total angle is $4\pi$ at each of the points $1$ and $2$, a local uniformizing coordinate is given by $Z= \sqrt{t-t_0}$ with transforms $\dot t=1$ into $\dot Z= \frac1{2Z}$, i.e. the point is a pole. A uniformizing coordinate at the end of the semi-infinite cylinders of width $\nu_j$ is $Z = \exp\left(\frac{ 2\pi it}{\nu_j}\right)$, transforming $\dot t=1$ into $\dot Z= \frac{2\pi i}{\nu_j}Z$, i.e. $z_j$ is a simple singular point. Hence we have constructed a rational vector field of degree $4$ on $\CP^1$.
Note that all conditions enumerated in the theorem are necessary conditions. Hence the theorem is a characterization of realizable generalized periodgons. The realisation map is however not bijective: one needs to identify those generalized periodgons that give rise to the same translation model and therefore the same vector field (up to a Moebius transformation). An open question is to find necessary and sufficient condition for realizable periodgons (not generalized).
Polynomial vector fields of degree 4
------------------------------------
As a consequence we can see what kind of periodgons can appear for a polynomial system of degree $4$.
The periodgon of a polynomial system of degree $4$ is always a planar polygon (its projection from the translation surface of $t(z)$ to $\C$ has no self-intersection). It can be flat (four sides aligned) with three sides in one direction and one in the other as in $\dot z = i z(z^3-1)$, or with two sides in each direction as in $\dot z = (z^2+1)(z^2+2)$.
The periodgon bifurcates generically when two adjacent sides become aligned in opposite directions, as in $\dot z = (z^2-1) (z-i)(z-2i) $. In that case, one of the periodic zone is surrounded by two homoclinic loops (see Figures \[two\_concentric\_homo\] and \[bifur\_periodgon\]). Indeed, we can remark that as soon as one center is surrounded by two homoclinic loops, it separates the other singular points in two groups of, respectively, one and two points. The singular point in the group of $1$ is then surrounded by a homoclinic loop and hence, necessarily a center.
![The phase portrait of $\dot z = (z^2-1) (z-i)(z-2i) $ with two homoclinic loops surrounding $z=2i$.[]{data-label="two_concentric_homo"}](Two_concentric_homo){width="5cm"}
\
Acknowledgements {#acknowledgements .unnumbered}
================
The authors are grateful to Arnaud Chéritat for stimulating discussions.
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abstract: 'Considering fractional fast diffusion equations on bounded open polyhedral domains in $\mathbb{R}^N$, we give a fully Galerkin approximation of the solutions by $C^0$-piecewise linear finite elements in space and backward Euler discretization in time, a priori estimates and the rates of convergence for the approximate solutions are proved, which extends the results of *Carsten Ebmeyer and Wen Bin Liu, SIAM J. Numer. Anal., 46(2008), pp. 2393–2410*. We also generalize the a priori estimates and the rates of convergence to a parabolic integral equation under the framework of *Qiang Du, Max Gunzburger, Richaed B. Lehoucq and Kun Zhou, SIAM Rev., 54 (2012), no. 4, pp. 667–696.*'
address:
- 'School of Mathematics, Tianjin University, Tianjin 300072, P. R. China'
- 'School of Mathematics, Tianjin University, Tianjin 300072, P. R. China'
author:
- Dongxue Li
- Youquan Zheng
title: On the finite element approximation for fractional fast diffusion equations
---
Introduction
============
Suppose $\Omega\subset\mathbb{R}^N$ is a bounded open polyhedral domain and $T \in (0, +\infty)$, we consider the finite element approximation of the following nonlocal nonlinear diffusion problem, $$\label{E:main}
\begin{cases}
u_t = - (-\Delta)^s(|u|^{m-1}u) &\quad \text{ in }\,\, \Omega\times (0,T],\\
u(x, t) = 0 &\quad \text{ on }\,\, (\mathbb{R}^N \setminus \Omega)\times (0,T],\\
u(x, 0) = u_0(x) &\quad \text{ in }\,\, \Omega,
\end{cases}$$ for a function $u(x, t):\mathbb{R}^N \times[0,T]\to \mathbb{R}$, here $u_0(x) = 0$ on $\mathbb{R}^N\setminus\Omega$ and $s\in (0, 1)$.
The fractional Laplacian appeared in (\[E:main\]) is defined as follows $$(-\Delta)^sg(x) = c_{N,s}\text{p.v.}\int_{\mathbb{R}^N}\frac{g(x) - g(z)}{|x - z|^{N+2s}}\, dz,$$ where $c_{N,s} > 0$ is a normalization constant, which can be viewed as an infinitesimal generator of the stable and radially symmetric Lévy processes [@Bertoin1996]. $(-\Delta)^s$ and its generalizations not only play important roles in probability, but also have been widely used to model nonlocal Dirichlet forms [@Applebaum2004], phase transitions [@BatesChmaj1999], nonlocal heat conduction [@Bobaru2010], anomalous diffusions in physics [@MetzlerKlafter2000], and among others in recent years. We refer the interested readers to the well written paper [@DuqiangSIAMReview] for further discussions on more applications as well as related mathematical work.
When $s = 1$, $u_t=\Delta(|u|^{m-1}u)$ is the classical fast diffusion equation for $0<m<1$ and the porous medium equation for $1<m<\infty$, which are degenerate or singular equations, many standard finite element techniques do not work well. The a priori error bounds are often suboptimal and the orders usually converge to zero when parameters approximate zero or infinity. The first error estimates of fully discrete schemes for porous medium equations were obtained in [@Rose2010]. Semi-discretizations in time for the porous medium and fast diffusion equations were discussed in [@EdenMichauxRakotoson], [@Roux1989] and [@RouxMainge]. Fully discrete schemes for the fast diffusion equations were given in [@LeftonWei] and [@RullaWalkington]. In [@Ebmeyer-Liu-2008], the authors used the $C^0$-piecewise linear finite elements in space and the backward Euler time discretization, a priori error estimates in quasi-norms were derived, furthermore, they estimated the rate of convergence. In [@NogHuang2017], an adaptive moving mesh finite element method was studied for the porous medium equations.
In this paper, we consider the fractional case $s\in (0, 1)$, which also includes the fast diffusion equations $(0 < m < 1)$ and the porous medium equations $(1 < m < \infty)$, respectively. The case $m = 1$ has attracted much attention in recent years, we limit ourselves to cite only, for example, [@DuqiangSIAMReview], [@DuGuZhou2013], [@DuHuangLehoucq2014], [@DuYang2016], [@DuYangZhou2017], [@DuZhou2011]. Reference [@DuZhou2011] established a nonlocal functional analytical framework for a linear peridynamic model of a spring network system, Galerkin finite element approximation was applied for numerical approximation. In [@DuGuZhou2013], the nonlocal analogs of first and second Green’s identities, local and nonlocal balance laws and second-order elliptic boundary-value problems, to the classical calculus were established. By exploiting the nonlocal vector calculus given in [@DuGuZhou2013], the authors in [@DuqiangSIAMReview] provided a variational analysis for nonlocal diffusion problems, which were described by a class of parabolic linear integral equation. Finite dimensional approximations using continuous and discontinuous Galerkin methods, condition number and error estimates were also derived. In [@DuHuangLehoucq2014], by introducing a nonlocal operator with the indicator function of a domain, the authors gave a description of initial and initial volume-constrained problems associated with a linear nonlocal convection-diffusion equation. Finite difference schemes as well as Monte Carlo simulations were applied to solve such nonlocal problems. In [@DuYang2016], the Fourier spectral approximations of a nonlocal Allen-Cahn equation were investigated, the authors also provided various error estimates and studied the steady states of the models with different kernels via both numerical simulations and theoretical analysis. References [@DuYangZhou2017] studied a nonlocal-in-time parabolic equation, a semi-discrete finite element approximation was proposed and error estimates were obtained.
For the case $s\in (0, 1)$ and $m\neq 1$, in the well written papers [@deltesoJakobsen2019] and [@deltesoJakobsen2018], the authors dealt with fully discrete numerical methods and provided in [@deltesoJakobsen2019] rigorous analysis of such numerical schemes which covers local and nonlocal, linear and nonlinear, non-degenerate and degenerate, and smooth and non-smooth problems. In [@deltesoJakobsen2018], the authors gave concrete discretizations and verify numerically some important theoretical properties of the methods given in [@deltesoJakobsen2019]. Regularity of solutions for (\[E:main\]) were studied, for example, in [@BonFigalliRoOton] and [@VPFR2017].
In this paper we use the methods developed in [@Ebmeyer-Liu-2008] to study problem (\[E:main\]) for $m\in (0, 1)$, in Section 2, we give a Galerkin approximation of the solutions for (\[E:main\]) by $C^0$-piecewise linear finite elements in space and backward Euler discretization in time, in Section 3, we prove a priori estimates for the approximate solutions and in Section 4, the rates of convergence is showed, which extends the results of [@Ebmeyer-Liu-2008] to the fractional case (\[E:main\]). In Section 5, we show that the results in Section 3 and Section 4 can be generalized to a more general problem then (\[E:main\]) described by parabolic integral equations in the framework of [@DuqiangSIAMReview].
Discretization of problem (\[E:main\]) {#Section2}
======================================
To get a finite element approximation of problem (\[E:main\]), we use the following assumptions:
1. $\Omega \subset \mathbb{R}^N$, $N \ge 1$, is a bounded, convex polyhedral domain.
2. $\partial\Omega = \bigcup_{1\leq k \leq M} \overline{\Gamma}_k$, each $\Gamma_k$ is a $(N - 1)$-dimensional polyhedron and $\Gamma_i\cap\Gamma_k=\phi$ for $i\neq k$.
3. $\partial\Gamma_{k1}\cap \dots \cap \partial\Gamma_{kj} = \phi$ if $j > N$ and $k_1 < \dots <k_j$.
4. $u_0 \in L^{\infty}(\Omega)$ and $u_0 = 0$ on $\mathbb{R}^N\setminus\Omega$.
$u(x,t)$ is called a weak solution of if $$\label{Eq3.1}
\begin{aligned}
& - \int_0^T\int_{\mathbb{R}^N}u\varphi_t\, dx\, dt \\
&\qquad + \int_0^T\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{(|u|^{m-1}u)(x) - (|u|^{m-1}u)(z)}{|x - z|^{N+2s}}(\varphi(x) - \varphi(z))\, dz\, dx\, dt\\
& = \int_{\mathbb{R}^N} u_0\varphi_0\, dx
\end{aligned}$$ holds, where $\varphi(\cdot, T) \equiv 0$ on $\Omega$, $\varphi_0 = \varphi(\cdot, 0)$. Indeed, multiplying by $\varphi$ and integrating it over $\mathbb{R}^N$, then $$\int_0^T\int_{\mathbb{R}^N}u_t\varphi\, dx\, dt = - \int_0^T\int_{\mathbb{R}^N}(-\Delta)^s(|u|^{m-1}u)\varphi\, dx\, dt.$$ First, integrate by parts, we have $$\int_0^T\int_{\mathbb{R}^N}u_t\varphi\, dx\, dt = - \int_{\mathbb{R}^N} u_0\varphi_0\, dx - \int_0^T \int_{\mathbb{R}^N}u\varphi_t\, dx\, dt.$$ Second, there holds $$\begin{aligned}
I &:= \int_0^T\int_{\mathbb{R}^N}(-\Delta)^s(|u|^{m-1}u)\varphi(x)\, dx\, dt\\
&= 2\int_0^T\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{(|u|^{m-1}u)(x) -(|u|^{m-1}u)(z)}{|x - z|^{N+2s}}\, dz\varphi(x)\ dx\, dt\\
&= 2\int_0^T\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{(|u|^{m-1}u)(x) - (|u|^{m-1}u)(z)}{|x - z|^{N+2s}}\varphi(x)\, dz\, dx\, dt\\
&= 2\int_0^T\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{(|u|^{m-1}u)(x) - (|u|^{m-1}u)(z)}{|x - z|^{N+2s}}\big(\varphi(x) - \varphi(z)\big)\, dz\, dx\, dt\\
&\quad + 2\int_0^T\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{(|u|^{m-1}u)(x) - (|u|^{m-1}u)(z)}{|x - z|^{N+2s}}\, dz\varphi(z)\, dz\, dx\, dt.
\end{aligned}$$ Since $$\begin{aligned}
& 2\int_0^T\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{(|u|^{m-1}u)(x) - (|u|^{m-1}u)(z)}{|x - z|^{N+2s}}\, dz\varphi(z)\, dz\, dx\, dt\\
& = -2\int_0^T\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{(|u|^{m-1}u)(z) - (|u|^{m-1}u)(x)}{|x - z|^{N+2s}}\varphi(z)\, dx\, dz\, dt\\
& = -I,
\end{aligned}$$ we get $$I = \int_0^T\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{(|u|^{m-1}u)(x) - (|u|^{m-1}u)(z)}{|x - z|^{N+2s}}\big(\varphi(x) - \varphi(z)\big)\, dz\, dx\, dt.$$
The discrete problem. {#Subsection2.2}
---------------------
In the following, denote the time step as $\tau > 0$. For $n = 0, \cdots, M$, set $t_n = n\tau$, $T = t_M$, $I_n = (t_{n-1},t_n)$, $v_n(x) = v(x,t_n)$, $\bar{v}^0(x) = v_0(x)$ and $$\bar{v}^n(x) = \tau^{-1}\int_{I_n}v(x,s)\, ds\text{ for } n \ge 1.$$ Let $T_h$ be a family of decompositions of $\Omega$ into closed $N-$simplices and $h$ is the mesh-size, moreover, assume $T_h$ is a regular triangulation, i.e., intersection of two non-disjoint nonidentical elements in $T_h$ is a common vertex or edge or surface, and there exists a constant $c > 0$ such that $$|K| \ge c(\text{diam} \,\,\, K)^N \text{ for all simplices }K \in T_h.$$ Define
$S_h(\Omega) =$ {$\phi_h \in C^0(\overline{\Omega}):\phi_h$ is piecewise linear w.r.t. $T_h,\phi_h = 0$ on $\mathbb{R}^N \setminus \Omega$}
and $\Pi_hv \in S_h$ denote the $C^0$-piecewise linear interpolation of $v$. For $s \in (0,1)$, the standard fractional-order Sobolev space is defined as $$H^s(\mathbb{R}^N) := \{u \in L^2(\mathbb{R}^N) : \Vert u\Vert_{L^2(\mathbb{R}^N)} + |u|_{H^s(\mathbb{R}^N)} < \infty\},$$ where $$|u|_{H^s(\mathbb{R}^N)}^2 := \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{(u(y) - u(x))^2}{|y - x|^{N+2s}}\, dy\, dx.$$ Let $P_h : H^s(\mathbb{R}^N) \to S_h$ be the $H^s$-projection onto $S_h$ defined as $$\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{(v - P_hv)(x) - (v - P_hv)(z)}{|x -z |^{N+2s}}\big(\chi(x) - \chi(z)\big)\, dz\, dx = 0 \quad \forall\chi \in S_h.$$ Let $w := |u|^{m-1}u$, then $\partial_tu = - (- \Delta)^sw$ and $$\label{Eq4.1}
\partial_t\psi(w) = - (- \Delta)^s w,$$ where $\psi(y) := |y|^{\frac{1-m}{m}}y$. Let $W_n \in S_h$, $n = 1,2,\ldots$ be the solutions of the following system $$\label{Eq4.2}
\begin{aligned}
& \left(\frac{\psi(W_n) - \psi(W_{n-1})}{\tau},\chi_n \right)\\
& \quad + \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{W_n(x) - W_n(z)}{|x - z|^{N+2s}}\big(\chi_n(x) - \chi_n(z)\big)\, dz\, dx = 0 \quad \forall\chi_n \in S_h,
\end{aligned}$$ where $W_0 := \psi^{-1}(\Pi_h\psi(w_0))$. The finite element approximation $W(x,t)$ of $w(x,t)$ is defined as follows $$\label{Eq4.3}
W(x,t) =
\begin{cases}
W_0(x) &\quad \mathrm{if} \,\, t = 0,\\
W_n(x,t)&\quad \mathrm{if} \,\, t \in (t_{n-1},t_n],\,\, 1 \leq n \leq M.
\end{cases}$$ Then we have
There exist unique functions $W_1,\ldots,W_M \in S_h$ solving .
The proof of is similar to that of [@Ebmeyer-Liu-2008 Lamme 3.1], so we omit it here .
There exists a positive constant $c$ such that $$\begin{aligned}
& \displaystyle\sup_{1\leq n\leq M}\Vert W_n\Vert_{L^{\frac{m+1}{m}}(\mathbb{R}^N)}^{\frac{m+1}{m}} + \tau\displaystyle\sum_{n=1}^M\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{(W_n(x) - W_n(z))^2}{|x - z|^{N+2s}}\, dz\, dx \leq c.
\end{aligned}$$
Choose $\chi_n = W_n$ in and take summation, we obtain $$\begin{aligned}
&\displaystyle\sum_{n=1}^M(\psi(W_n),W_n) + \tau\displaystyle\sum_{n=1}^M\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{(W_n(x) - W_n(z))^2}{|x - z|^{N+2s}}\, dy\, dx = \displaystyle\sum_{n=1}^M(\psi(W_{n-1}),W_n).
\end{aligned}$$ From Young’s inequality, $|\psi(W_{n-1})W_n| \leq \frac{1}{m+1}|W_{n-1}|^{\frac{m+1}{m}} + \frac{m}{m+1}| W_n|^{\frac{m+1}{m}}$. Since $\psi(W_n)W_n = |W_n|^{\frac{m+1}{m}}$, $$\begin{aligned}
&\frac{1}{m+1}\Vert W_M\Vert_{\frac{m+1}{m}}^{\frac{m+1}{m}} + \tau\displaystyle\sum_{n=1}^M\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{(W_n(x) - W_n(z))^2}{|x - z|^{N+2s}}\, dy\, dx \leq \frac{1}{m+1}\Vert W_0\Vert_{\frac{m+1}{m}}^{\frac{m+1}{m}}.
\end{aligned}$$ Clearly, $\Vert W_0\Vert_{\frac{m+1}{m}} \leq c$ for some positive constant $c$.
The quasi-norm. {#section4}
---------------
For $v_1,v_2 \in L^p(\Omega)$ and $p > 1$, define the quasi-norm as $$\Vert v_2\Vert_{(v_1,p)}^2 := \int_{\Omega}(|v_1| + |v_2|)^{p-2}|v_2|^2.$$ From [@Ebmeyer-Liu-2008], we know that $$c_1\Vert v_2\Vert_{L^p(\Omega)}^p \leq \Vert v_2\Vert_{(v_1,p)} \leq \Vert v_2\Vert_{L^p(\Omega)}^{p/2}\qquad \text{for}\,\,\, 1 < p < 2$$ and $$\Vert v_2\Vert_{L^p(\Omega)}^{p/2} \leq \Vert v_2\Vert_{(v_1,p)} \leq c_2\Vert v_2\Vert_{L^p(\Omega)}^p\qquad \text{for}\,\,\, 2 < p < \infty,$$ where $c_1,c_2 > 0$ are constants depending on $\Vert v_1\Vert_{L^p(\Omega)}$ and $\Vert v_2\Vert_{L^p(\Omega)}$. Also, $$\label{Eq5.1}
\begin{aligned}
& \int_0^T\Vert w - v\Vert_{(w,\frac{m+1}{m})}^2 \cong \int_0^T(\psi(w) - \psi(v),w - v)\\
& \cong \int_0^T\Vert \psi(w) - \psi(v)\Vert_{(\psi(w),m+1)}^2 = \int_0^T\Vert u - \psi(v)\Vert_{(u,m+1)}^2
\end{aligned}$$ for all $v \in L^{\frac{m+1}{m}}([0,T] \times \Omega)$, where $u = \psi(w)$.
Now consider the following time independent problem $$\psi(v) +(-\Delta)^sv = f \quad \mathrm{on}\,\, \Omega, \qquad v = 0 \quad\mathrm{on}\,\, \mathbb{R}^N\setminus\Omega$$ for a smooth function $f$. It is easy to see that there exists a unique weak solution satisfying $$\label{Eq5.2}
(\psi(v),\varphi) + \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{v(x) - v(z)}{|x - z|^{N+2s}}(\varphi(x) - \varphi(z))\, dz\, dx = (f,\varphi)$$ for $\forall\varphi \in H_0^s(\Omega) := \{u \in H^s(\mathbb{R}^N) : u \equiv 0 \quad \text{on} \,\,\, \mathbb{R}^N\setminus\Omega\}$. Let $V \in S_h$ be the finite element approximation satisfying $$\label{Eq5.3}
(\psi(V),\chi) + \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{V(x) - V(z)}{|x - z|^{N+2s}}(\chi(x) - \chi(z))\, dz\, dx = (f,\chi)\text{ for }\forall\chi \in S_h.$$ Then there exists a constant $c > 0$ independent of $h$ satisfying $$\label{Eq5.4}
\begin{aligned}
& \Vert v - V\Vert_{(v,\frac{m+1}{m})}^2 + \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{(v(x) - v(z) - (V(x) - V(z)))^2}{|x - z|^{N+2s}}\, dz\, dx\\
& \leq c\displaystyle\inf_{v_h \in S_h}\Big(\Vert v - v_h\Vert_{(v,\frac{m+1}{m})}^2 + \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{(v(x) - v(z) - (v_h(x) - v_h(z)))^2}{|x - z|^{N+2s}}\, dz\, dx\Big).
\end{aligned}$$ Indeed, choose $\varphi = \chi$ in and take difference between and , we obtain $$\label{Eq5.5}
\begin{aligned}
& (\psi(v) - \psi(V),\chi) + \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{v(x) - v(z) - (V(x) - V(z))}{|x - z|^{N+2s}}(\chi(x) - \chi(z))\, dz\, dx = 0.
\end{aligned}$$ For $v_h \in S_h$, from we have $$\begin{aligned}
& (\psi(v) - \psi(V),v - V) + \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{(v(x) - v(z) - (V(x) - V(z)))^2}{|x - z|^{N+2s}}\, dz\, dx\\
& = (\psi(v) - \psi(V),v - v_h) + \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{v(x) - v(z) - (V(x) - V(z))}{|x - z|^{N+2s}}\\
& \qquad \times (v(x) - v(z) - (v_h(x) - v_h(z)))\, dz\, dx.
\end{aligned}$$ From [@Ebmeyer-Liu-2008 Lemma 4.4], $$\begin{aligned}
(\psi(v) - \psi(V),v - v_h) \leq \delta(\psi(v) - \psi(V),v - V) + c_{\delta}(\psi(v) - \psi(v_h),v - v_h).
\end{aligned}$$ Then Young’s inequality implies that $$\begin{aligned}
& \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{v(x) - v(z) - (V(x) - V(z))}{|x - z|^{N+2s}}(v(x) - v(z) - (v_h(x) - v_h(z)))\, dz\, dx\\
& \leq \delta\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{(v(x) - v(z) - (V(x) - V(z)))^2}{|x - z|^{N+2s}}\, dz\, dx \\
& \qquad + c_{\delta}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{(v(x) - v(z) - (v_h(x) - v_h(z)))^2}{|x - z|^{N+2s}}\, dz\, dx.
\end{aligned}$$ Hence is valid.
A priori error estimates. {#section5}
=========================
In this section we prove an a priori error estimate for $w - W$.
\[thm:main5.1\] For any $m > 0$ there exists a constant $c > 0$ independent of $h$ and $\tau$ such that $$\begin{aligned}
& \int_0^T\Vert w - W\Vert_{(w,\frac{m+1}{m})}^2 + \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|(\bar{w}(x) - \bar{w}(z)) - (\overline{W}(x) - \overline{W}(z))|^2}{|x - z|^{N+2s}}\, dz\, dx\\
& \leq c\Bigg(\displaystyle\sum_{n=1}^M\int_{I_n}\Vert w_n - w\Vert_{(w,\frac{m+1}{m})}^2 + \int_0^T\Vert w - P_hw\Vert_{(w,\frac{m+1}{m})}^2 + \Vert \psi(w_0) - \Pi_h\psi(w_0)\Vert_2^2\\
&\quad + \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{\left|\tau\displaystyle\sum_{n=1}^M\left[(\bar{w}^n(x) - \bar{w}^n(z)) - (P_h\bar{w}^n(x) - P_h\bar{w}^n(z))\right]\right|^2}{|x - z|^{N+2s}}\, dz\, dx \Bigg),
\end{aligned}$$ where $\bar{w}(x) = \int_0^Tw(x,t)\, dt$ and $\overline{W}(x) = \int_0^TW(x,t)\, dt$.
Integrating over $I_n$, we find $$\label{Eq6.1}
\tau^{-1}(\psi(w_n) - \psi(w_{n-1})) + \int_{\mathbb{R}^N}\frac{\bar{w}^n(x) - \bar{w}^n(z)}{|x - z|^{N+2s}}\, dz = 0.$$ Let $\chi_n \in S_h$, multiply and by $\tau\chi_n$, take difference and sum over $n$, we get $$\begin{aligned}
& \displaystyle\sum_{n=1}^M(\psi(w_n) - \psi(w_{n-1}) - (\psi(W_n) - \psi(W_{n-1})),\chi_n)\\
& \quad + \displaystyle\sum_{n=1}^M\tau\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{\bar{w}^n(x) - \bar{w}^n(z) - (W_n(x) - W_n(z))}{|x - z|^{N+2s}}(\chi_n(x) - \chi_n(z))\, dz\, dx = 0,
\end{aligned}$$ From the identity $\displaystyle\sum_{n=1}^M(a_n - a_{n-1})b_n = a_Mb_M - a_0b_0 + \displaystyle\sum_{n=0}^{M-1}a_n(b_n - b_{n+1})$, we obtain $$\begin{aligned}
& (\psi(w_M) - \psi(W_M),\chi_M) + \displaystyle\sum_{n=0}^{M-1}(\psi(w_n) - \psi(W_n),\chi_n - \chi_{n+1}) \\
& \quad +\displaystyle\sum_{n=1}^M\tau\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{\bar{w}^n(x) - \bar{w}^n(z) - (W_n(x) - W_n(z))}{|x - z|^{N+2s}}(\chi_n(x) - \chi_n(z))\, dz\, dx \\
& = (\psi(w_0) - \psi(W_0),\chi_0).
\end{aligned}$$ Set $\chi_n = \tau\displaystyle\sum_{k=n}^M(P_h\bar{w}^k - \Pi_hW_k)$ for $0 \leq n \leq M$, since $\Pi_hW_k = W_k$ for all $k \ge 1$ and $\chi_n - \chi_{n+1} = \tau(P_h\bar{w}^n - \Pi_hW_n)$, we have $$\label{Eq6.2}
\begin{aligned}
& \tau\displaystyle\sum_{n=0}^M(\psi(w_n) - \psi(W_n),P_h\bar{w}^n - \Pi_hW_n) \\
& \quad + \tau\displaystyle\sum_{n=1}^M\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{\bar{w}^n(x) - \bar{w}^n(z) - (W_n(x) - W_n(z))}{|x - z|^{N+2s}}\\
& \qquad \times \tau\displaystyle\sum_{k=n}^M(P_h\bar{w}^k(x) - P_h\bar{w}^k(z) - (W_k(x) - W_k(z)))\, dz\, dx \\
& = \left(\psi(w_0) - \psi(W_0), \tau\displaystyle\sum_{n=0}^M(P_h\bar{w}^n - \Pi_hW_n)\right).
\end{aligned}$$ Subtracting $\tau(\psi(w_0) - \psi(W_0),P_h\bar{w}^0 -\Pi_hW_0)$ on both side of , from the identity $$\displaystyle\sum_{n=1}^M\left(a_n\displaystyle\sum_{k=n}^Ma_k \right) = \frac{1}{2}\left(\displaystyle\sum_{n=1}^Ma_n \right)^2 + \frac{1}{2}\displaystyle\sum_{n=1}^M(a_n)^2$$ and the fact that $P_h$ is $H^s-$projection, we obtain $$\begin{aligned}
& \tau\displaystyle\sum_{n=1}^M\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{\bar{w}^n(x) - \bar{w}^n(z) - (W_n(x) - W_n(z))}{|x - z|^{N+2s}}\\
& \quad \times \tau\displaystyle\sum_{k=n}^M(P_h\bar{w}^k(x) - P_h\bar{w}^k(z) - (W_k(x) - W_k(z)))\, dz\, dx \\
\end{aligned}$$ $$\begin{aligned}
& = \tau\displaystyle\sum_{n=1}^M\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{P_h\bar{w}^n(x) - P_h\bar{w}^n(z) - (W_n(x) - W_n(z))}{|x - z|^{N+2s}}\\
& \quad \times \tau\displaystyle\sum_{k=n}^M(P_h\bar{w}^k(x) - P_h\bar{w}^k(z) - (W_k(x) - W_k(z)))\, dz\, dx\\
& = \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\tau\displaystyle\sum_{n=1}^M\Bigg((P_h\bar{w}^n(x) - P_h\bar{w}^n(z) - (W_n(x) - W_n(z))\\
& \quad \times \tau\displaystyle\sum_{k=n}^M(P_h\bar{w}^k(x) - P_h\bar{w}^k(z) - (W_k(x) - W_k(z))\Bigg)\frac{1}{|x - z|^{N+2s}}\, dz\, dx\\
& \ge \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{1}{2}\frac{\left|\tau\displaystyle\sum_{n=1}^M(P_h\bar{w}^n(x) - P_h\bar{w}^n(z) - (W_n(x) - W_n(z))\right|^2}{|x - z|^{N+2s}}\, dz\, dx.
\end{aligned}$$ Hence, from we get $$\label{Eq6.3}
\begin{aligned}
J_1 + J_2 &:= \tau\displaystyle\sum_{n=1}^M(\psi(w_n) - \psi(W_n),w_n - W_n)\\
& \quad + \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{1}{2}\frac{\left|\tau\displaystyle\sum_{n=1}^M(P_h\bar{w}^n(x) - P_h\bar{w}^n(z) - (W_n(x) - W_n(z)))\right|^2}{|x - z|^{N+2s}}\, dz\, dx\\
& \leq \tau\displaystyle\sum_{n=1}^M(\psi(w_n) - \psi(W_n),w_n - P_h\bar{w}^n)\\
& \quad + \left(\psi(w_0) - \psi(W_0), \tau\displaystyle\sum_{n=1}^M(P_h\bar{w}^n - W_n)\right)\\
& =: J_3 + J_4.
\end{aligned}$$ By [@Ebmeyer-Liu-2008 Lemma 4.4], we have $$\begin{aligned}
J_3 &= \displaystyle\sum_{n=1}^M\int_{I_n}(\psi(w_n) - \psi(W_n),w_n - P_hw)\\
& \leq \delta\displaystyle\sum_{n=1}^M\int_{I_n}(\psi(w_n) - \psi(W_n),w_n - W_n)\\
&\quad + c_{\delta}\displaystyle\sum_{n=1}^M\int_{I_n}(\psi(w_n) - \psi(P_hw),w_n - P_hw)\\
& =: J_{31} + J_{32}.
\end{aligned}$$ For $\delta > 0$ sufficiently small, the term $J_{31}$ can be absorbed into the left-hand side of . From [@Ebmeyer-Liu-2008 Lemma 4.3], there holds $$J_{32} \leq c\left[\displaystyle\sum_{n=1}^M\int_{I_n}(\psi(w_n) - \psi(w),w_n - w) + \displaystyle\sum_{n=1}^M\int_{I_n}(\psi(w) - \psi(P_hw),w - P_hw)\right].$$ Hölder’s and Young’s inequalities yield that $$J_4 \leq c_{\delta}\Vert\psi(w_0) - \psi(W_0)\Vert_{L^2(\mathbb{R}^N)}^2 + \delta\left\Vert\tau\displaystyle\sum_{n=1}^M(P_h\bar{w}^n - W_n)\right\Vert_{L^2(\mathbb{R}^N)}^2.$$ By [@DuqiangSIAMReview Lemma 4.2, Lemma 4.3], the term $\delta\Vert\tau\displaystyle\sum_{n=1}^M(P_h\bar{w}^n - W_n)\Vert_2^2$ can be absorbed. Moreover, since $\psi(W_0) = \Pi_h\psi(w_0)$, $\Vert\psi(w_0) - \psi(W_0)\Vert_2^2 = \Vert\psi(w_0) - \Pi_h\psi(w_0)\Vert_2^2$.
Next, from [@Ebmeyer-Liu-2008 Lemma 4.3], we obtain $$(\psi(w) - \psi(w_n),w - W_n) \leq c[(\psi(w) - \psi(w_n),w - w_n) + (\psi(w_n) - \psi(W_n),w_n - W_n)],$$ hence, $$J_1 \ge c\left(\displaystyle\sum_{n=1}^M\int_{I_n}(\psi(w) - \psi(W_n),w - W_n) - \displaystyle\sum_{n=1}^M\int_{I_n}(\psi(w) - \psi(w_n),w - w_n)\right).$$ Furthermore, $$\begin{aligned}
J_2 &= \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{1}{2}\frac{\left|\tau\displaystyle\sum_{n=1}^M(P_h\bar{w}^n(x) - P_h\bar{w}^n(z) - (W_n(x) - W_n(z)))\right|^2}{|x - z|^{N+2s}}\, dz\, dx\\
& \ge \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{1}{4}\frac{\left|\tau\displaystyle\sum_{n=1}^M(\bar{w}^n(x) - \bar{w}^n(z) - (W_n(x) - W_n(z)))\right|^2}{|x - z|^{N+2s}}\, dz\, dx\\
& \quad - \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{1}{2}\frac{\left|\tau\displaystyle\sum_{n=1}^M(\bar{w}^n(x) - \bar{w}^n(z) - (P_h\bar{w}^n(x) - P_h\bar{w}^n(z)))\right|^2}{|x - z|^{N+2s}}\, dz\, dx
\end{aligned}$$ and $$\begin{aligned}
&\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{1}{4}\frac{\left|\tau\displaystyle\sum_{n=1}^M(\bar{w}^n(x) - \bar{w}^n(z) - (W_n(x) - W_n(z)))\right|^2}{|x - z|^{N+2s}}\, dz\, dx \\
& = \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{1}{4}\frac{|(\bar{w}(x) - \bar{w}(z)) - (\overline{W}(x) - \overline{W}(z))|^2}{|x - z|^{N+2s}}\, dz\, dx
\end{aligned}$$ hold. Combine all the estimates together, we finally obtain $$\begin{aligned}
& \int_0^T(\psi(w) - \psi(W),w - W) \\
& \qquad + \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|(\bar{w}(x) - \bar{w}(z)) - (\overline{W}(x) - \overline{W}(z))|^2}{|x - z|^{N+2s}}\, dz\, dx\\
& \leq c\Bigg(\displaystyle\sum_{n=1}^M\int_{I_n}(\psi(w_n) - \psi(w),w_n - w) + \displaystyle\sum_{n=1}^M\int_{I_n}(\psi(w) - \psi(P_hw),w - P_hw)\\
& \qquad + \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{\left|\tau\displaystyle\sum_{n=1}^M(\bar{w}^n(x) - \bar{w}^n(z) - (P_h\bar{w}^n(x) - P_h\bar{w}^n(z)))\right|^2}{|x - z|^{N+2s}}\, dz\, dx\\
& \qquad + \Vert\psi(w_0) - \Pi_h\psi(w_0)\Vert_{L^2(\mathbb{R}^N)}^2\Bigg).
\end{aligned}$$ By , we get the desired estimate.
The proof of Theorem \[thm:main5.1\] also implies the following result.
There is a constant $c>0$ independent of $h$ and $\tau$ such that $$\begin{aligned}
\int_0^T\Vert w - W\Vert_{(w,\frac{m+1}{m})}^2 & \leq c\Bigg(\displaystyle\sum_{n=1}^M\int_{I_n}\Vert w_n - w\Vert_{(w,\frac{m+1}{m})}^2 + \int_0^T\Vert w - P_hw\Vert_{(w,\frac{m+1}{m})}^2 \\
& \qquad + \Vert \psi(w_0) - \Pi_h\psi(w_0)\Vert_2^2\Bigg).
\end{aligned}$$
Indeed, the corollary follows directly from the following estimate $$\begin{aligned}
& \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{\left|\tau\displaystyle\sum_{n=1}^M(P_h\bar{w}^n(x) - P_h\bar{w}^n(z) - (W_n(x) - W_n(z)))\right|^2}{|x - z|^{N+2s}}\, dz\, dx\\
& \qquad + \int_0^T(\psi(w) - \psi(W),w - W) \\
& \leq c\Bigg(\displaystyle\sum_{n=1}^M\int_{I_n}(\psi(w_n) - \psi(w),w_n - w) + \displaystyle\sum_{n=1}^M\int_{I_n}(\psi(w) - \psi(P_hw),w - P_hw)\\
& \qquad + \Vert\psi(w_0) - \Pi_h\psi(w_0)\Vert_{L^2(\mathbb{R}^N)}^2\Bigg).
\end{aligned}$$
The convergence rate. {#section6}
=====================
In the section we discuss the rates of convergence.
\[thm:main6.1\] Let $0 < m < 1$ and $w_0 \in L^{\infty}(\mathbb{R}^N)\cap H^s_0(\Omega).$ Then there is a positive constant $c$ independent of $h$ and $\tau$ such that $$\label{Eq7.1}
\begin{aligned}
\left(\int_0^T\Vert w - W\Vert_{(w,\frac{m+1}{m})}^2 + \displaystyle\sup_{1\leq n\leq N}\left\Vert\int_0^{t_n}(w - W)\right\Vert_{H^s(\mathbb{R}^N)}^2\right)^{1/2} \leq c(\tau + h^s).
\end{aligned}$$
Set $v := |u|^{\frac{m-1}{2}}u$, by [@Ebmeyer-Liu-2008 Lemma 4.1, Lemma 4.2], we have $$\int_{I_n}\Vert w_n - w\Vert_{(w,\frac{m+1}{m})}^2 \leq c\int_{I_n}(\psi(w_n) - \psi(w),w_n - w) \leq c\int_{I_n}\int_{\mathbb{R}^N}|v_n - v|^2.$$ Since $$\int_{I_n}|v_n - v|^2 = \int_{I_n}\left |\int_t^{t_n}v_t\right|^2 \leq \int_{I_n}\int_t^{t_n}|v_t|^2|I_n| \leq \tau^2\int_{I_n}|v_t|^2$$ and $|v_t|^2 = |\partial_t(|u|^{\frac{m-1}{2}}u)|^2$, by the regularity results for nonlinear fractional diffusion equations from [@VPFR2017], we obtain $$\label{Eq7.2}
\displaystyle\sum_{n=1}^M\int_{I_n}\Vert w_n - w\Vert_{(w,\frac{m+1}{m})}^2 \leq c\tau^2\left\Vert \partial_tu^{\frac{m+1}{2}}\right\Vert_{L^2(0,T;L^2(\mathbb{R}^N))} \leq c\tau^2.$$ The same arguments as Section 4.4 in [@Brenner-Scott] and the nonlocal Poincar$\acute{e}$ inequality $\mathbf{I}$ (see [@DuqiangSIAMReview Lemma 4.3]) imply that $$\label{Eq7.3}
\begin{aligned}
& \int_0^T\Vert w - P_hw\Vert_{(w,\frac{m+1}{m})}^2 = \int_0^T\int_{\mathbb{R}^N}(|w| + |w - P_hw|)^{\frac{1-m}{m}}|w - P_hw|^2\\
& \leq c\Bigg(\Vert w\Vert_{L^{\infty(0,T;L^{\infty}(\mathbb{R}^N))}}^{\frac{1-m}{m}}\Vert w - P_hw\Vert_{L^2(0,T;L^2(\mathbb{R}^N))}^2 + \Vert w - P_hw\Vert_{L^{\frac{m+1}{m}}(0,T;L^{\frac{m+1}{m}}(\mathbb{R}^N))}^{\frac{m+1}{m}}\Bigg)\\
& \leq ch^{2s}\Bigg(\int_0^T\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{(w(x) - w(z))^2}{|x - z|^{N+2s}}\, dz\, dx\, dt + \Vert w \Vert_{L^{\frac{m+1}{m}}(0,T;W^{\frac{2ms}{m+1},\frac{m+1}{m}}(\mathbb{R}^N))}^{\frac{m+1}{m}}\Bigg)\\
& \leq ch^{2s}.
\end{aligned}$$ From the regularity results of [@VPFR2017], we have $$\label{Eq7.4}
\begin{aligned}
& \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{\left|\tau\displaystyle\sum_{n=1}^M[(\bar{w}^n(x) - \bar{w}^n(z)) - (P_h\bar{w}^n(x) - P_h\bar{w}^n(z))]\right|^2}{|x - z|^{N+2s}}\, dz\, dx\\
& = \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{\left|(\bar{w}(x) - \bar{w}(z)) - (P_h\bar{w}(x) - P_h\bar{w}(z))\right|^2}{|x - z|^{N+2s}}\, dz\, dx\\
& \leq ch^{2s}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{\left|\bar{w}(x) - \bar{w}(z)\right|^2}{|x - z|^{N+2s}}\, dz\, dx\\
& \leq ch^{2s}.
\end{aligned}$$ Furthermore, since $w_0 \in L^{\infty}(\mathbb{R}^N)\cap H^s_0(\Omega)$, $\psi(w_0) \in H^s_0(\Omega)$ and $$\label{Eq7.5}
\begin{aligned}
& \Vert \psi(w_0) - \Pi_h\psi(w_0)\Vert_{L^2(\mathbb{R}^N)}^2 \leq ch^{2s}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{(\psi(w_0)(x) - \psi(w_0)(z))^2}{|x - z|^{N+2s}}\, dz\, dx \leq ch^{2s}.
\end{aligned}$$ From Theorem \[thm:main5.1\] and -, there holds $$\label{Eq7.6}
\begin{aligned}
& \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|(\bar{w}(x) - \bar{w}(z)) - (\overline{W}(x) - \overline{W}(z))|^2}{|x - z|^{N+2s}}\, dz\, dx + \int_0^T\Vert w - W\Vert_{(w,\frac{m+1}{m})}^2\\
& \leq c(\tau^2 + h^{2s}),
\end{aligned}$$ for $\bar{w}(x) = \int_0^Tw(x,t)\, dt$, $\overline{W}(x) = \int_0^TW(x,t)\, dt$ and $T = t_M$. By the nonlocal Poincar$\acute{e}$ inequality $\mathbf{I}$ (see [@DuqiangSIAMReview Lemma 4.3]) and [@DuqiangSIAMReview Lemma 4.2] again, there exists a positive constant $c$ such that $$\begin{aligned}
& \left\Vert\int_0^{t_n}(w - W)\right\Vert_{L^2(\mathbb{R}^N)}^2 \leq c\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{\left(\int_0^{t_n}(w - W)(x) - \int_0^{t_n}(w - W)(z)\right)^2}{|x - z|^{N+2s}}\, dz\, dx
\end{aligned}$$ and the assertion follows.
Under the assumptions of Theorem \[thm:main6.1\], there exists a positive constant $c$ independent of $h$ and $\tau$ such that $$\Vert u - U\Vert_{L^{m+1}(0,T;L^{m+1}(\mathbb{R}^N))} \leq c(\tau + h^s),$$ where $U = \psi(W)$.
The proof of the above corollary is similar to that of [@Ebmeyer-Liu-2008 Corollary 6.2], hence we omit it.
Under the assumptions of Theorem \[thm:main6.1\], then $$\begin{aligned}
\left(\int_0^T\int_{\mathbb{R}^N}|u - U||w - W| \right)^{\frac{1}{2}} + \left\Vert\int_0^t(w - W)\right\Vert_{L^{\infty}(0,T;H^s(\mathbb{R}^N))} \leq c(\tau + h^s).
\end{aligned}$$
Assume $T_0 \in (t_{n-1},t_n)$, note that $W$ is constant with respect to $t$ in $(t_{n-1},t_n]$, the $M$th equation of can be written as $$\begin{aligned}
& \left(\frac{\psi(W(T_0)) - \psi(W(t_{n-1}))}{\tau},\chi_M \right)\\
&+ \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{W(T_0)(x) - W(T_0)(z)}{|x - z|^{N+2s}}\big(\chi_M(x) - \chi_M(z)\big)\, dz\, dx = 0 \quad \forall\chi_M \in S_h
\end{aligned}$$ Hence, replacing $t_M$ by $T_0$ and $W_M$ by $W(T_0)$ in the proofs of Theorem \[thm:main5.1\] and Theorem \[thm:main6.1\], we obtain $$\begin{aligned}
\left\Vert\int_0^{T_0}(w - W)\right\Vert_{H^s(\mathbb{R}^N)} \leq c(\tau + h^s).
\end{aligned}$$
Some extensions {#section7}
===============
In this section, we indicate how to generalize the results in Section 3 and Section 4 to a more general parabolic integral equation. Let $\Omega_s\subset\mathbb{R}^n$ and $\Omega_c\subset\mathbb{R}^n$ be bounded and open polyhedral domains and $T\in (0, +\infty)$, $\Omega_s$ and $\Omega_c$ have a nonempty common boundary, we consider the following nonlinear diffusion problem $$\label{Eq8.1}
\begin{cases}
u_t = \mathcal{L}(|u|^{m-1}u) &\quad \mathrm{on} \,\, \Omega_s , t > 0,\\
\mathcal{V}u = 0 &\quad \mathrm{on} \,\, \Omega_c, t > 0,\\
u(x,0) = u_0(x) &\quad \mathrm{on} \,\, \Omega_s\cup \Omega_c,
\end{cases}$$ where $u(x,t):(\Omega_s\cup\Omega_c) \times [0,T] \to \mathbb{R}$ and $\mathcal{V}$ denotes a linear operator of constraints acting on a volume $\Omega_c$ which is disjoint from $\Omega_s$.
Given an open bounded subset $\Omega_s \subset \mathbb{R}^n$, $\Omega_c$ is the corresponding [*interaction domain*]{}. For $u(x) : \Omega \to \mathbb{R}$, the action of the linear operator $\mathcal{L}$ on the function $u(x)$ is defined as $$\label{Eq8.2}
\mathcal{L}u(x):= 2\int_{\Omega_s\cup \Omega_c}\big(u(y) - u(x)\big)\gamma (x,y)\, dy\qquad \forall x\in \Omega_s \subseteq \mathbb{R}^n,$$ where the volume of $\Omega_s$ is nonzero and the kernel $\gamma(x,y)$ is a nonnegative symmetric mapping, i.e., $\gamma (x,y) = \gamma (y,x) \ge 0$. We refer the interested readers to [@DuqiangSIAMReview] for more details.
As in [@DuqiangSIAMReview], given positive constants $\gamma_0$ and $\varepsilon$, assume that $\gamma$ satisfies $$\label{Eq8.3}
\begin{cases}
\gamma(x,y) \ge 0 & \quad \forall y \in B_{\varepsilon}(x)\,\, \mathrm{and}\,\, \gamma(x,y) \ge \gamma_0 > 0 \quad \mathrm{when}\,\, y \in B_{\varepsilon/2}(x),\\ \gamma(x,y) = 0 & \quad \forall y \in (\Omega_s\cup\Omega_c)\setminus B_{\varepsilon}(x),
\end{cases}$$ where $B_{\varepsilon}(x) := \{y \in \Omega_s\cup\Omega_c : |y - x| \leq \varepsilon\}$, for all $x \in \Omega_s\cup\Omega_c$. Furthermore, assume that there exist $s \in (0,1)$ and positive constants $\gamma_*$ and $\gamma^*$ such that, for all $x \in \Omega_s$, $$\label{case1}
\frac{\gamma_*}{|y - x|^{n+2s}} \leq \gamma(x,y) \leq \frac{\gamma^*}{|y - x|^{n+2s}} \qquad \text{for}\,\, y \in B_{\varepsilon}(x).$$
For $s \in (0,1)$, define the fractional-order Sobolev space as $$H^s(\Omega_s\cup\Omega_c) := \{u \in L^2(\Omega_s\cup\Omega_c) : \Vert u\Vert_{L^2(\Omega_s\cup\Omega_c)} + |u|_{H^s(\Omega_s\cup\Omega_c)} < \infty\},$$ where $|u|_{H^s(\Omega_s\cup\Omega_c)}^2 := \int_{\Omega_s\cup\Omega_c}\int_{\Omega_s\cup\Omega_c}(u(y) - u(x))^2\gamma(x,y)\, dy\, dx$. Assuming $u_0 \in L^{\infty}(\Omega_s\cup\Omega_c)$, $u(x,t)$ a weak solution of if $$\label{Eq8.5}
\begin{aligned}
& - \int_0^T\int_{\Omega_s\cup \Omega_c}u\varphi_t\, dx\, dt\\
& \quad + \int_0^T\int_{\Omega_s\cup \Omega_c}\int_{\Omega_s\cup \Omega_c}\big((|u|^{m-1}u)(y) - (|u|^{m-1}u)(x)\big)\gamma(x,y)\big(\varphi(y) - \varphi(x)\big)\, dy\, dx\, dt\\
& = \int_{\Omega_s\cup \Omega_c} u_0\varphi_0\, dx,
\end{aligned}$$ where $\varphi(\cdot,T) \equiv 0$ on $\Omega_s$, $\varphi_0 = \varphi(\cdot,0)$. Let $T_h$ be a family of decompositions of $\Omega_s$ into closed $N-$simplices and $h$ is the mesh-size, assume $T_h$ is a regular triangulation and there exists a constant $c > 0$ such that $$|K| \ge c(\text{diam} \,\,\, K)^N \text{ for all simplices }K \in T_h.$$ Introduce the space
$S_h(\Omega_s\cup\Omega_c) =$ {$\phi_h \in C^0(\overline{\Omega}):\phi_h$ is piecewise linear w.r.t. $T_h$, $\mathcal{V}\phi_h = 0$ on $\Omega_c$}.
Let $\Pi_hv \in S_h$ denote the $C^0-$piecewise linear interpolant of the function $v$ and $P_h : H^s(\Omega_s\cup\Omega_c) \to S_h$ be the $H^s-$projection onto $S_h$ defined by $$\int_{\Omega_s\cup \Omega_c}\int_{\Omega_s\cup \Omega_c}\big((v - P_hv)(y) - (v - P_hv)(x)\big)\gamma(x,y)\big(\chi(y) - \chi(x)\big)\, dy\, dx = 0 \quad \forall\chi \in S_h.$$ As in Section 2, set $w := |u|^{m-1}u$, then $\partial_tu = \mathcal{L}w$ and $$\label{Eq8.6}
\partial_t\psi(w) = \mathcal{L}w,$$ where $\psi(s) := |s|^{\frac{1-m}{m}}s$. Let $W_n \in S_h, n = 1,2,\ldots,$ be the solutions of the system $$\label{Eq8.7}
\begin{aligned}
& \Big(\frac{\psi(W_n) - \psi(W_{n-1})}{\tau},\chi_n \Big)\\
&+ \int_{\Omega_s\cup \Omega_c}\int_{\Omega_s\cup \Omega_c}\big(W_n(y) - W_n(x)\big)\gamma(x,y)\big(\chi_n(y) - \chi_n(x)\big)\, dy\, dx = 0 \quad \forall\chi_n \in S_h,
\end{aligned}$$ where $W_0 := \psi^{-1}(\Pi_h\psi(w_0))$. Then the finite element approximation $W(x,t)$ of $w(x,t)$ is defined as $$\label{Eq8.8}
W(x,t) =
\begin{cases}
W_0(x) &\quad \mathrm{if} \,\, t = 0,\\
W_n(x,t)&\quad \mathrm{if} \,\, t \in (t_{n-1},t_n], 1 \leq n \leq N.
\end{cases}$$ The same arguments of Section 3 and Section 4 imply the following results.
\[thm:main7.6.1\] For any $m > 0$ there is a positive constant $c$ independent of $h$ and $\tau$ such that $$\begin{aligned}
& \int_0^T\Vert w - W\Vert_{(w,\frac{m+1}{m})}^2 + \int_{\Omega_s\cup\Omega_c}\int_{\Omega_s\cup\Omega_c}|(\bar{w}(y) - \bar{w}(x)) - (\overline{W}(y) - \overline{W}(x))|^2\gamma(x,y)\, dy\, dx\\
& \leq c\Big(\displaystyle\sum_{n=1}^N\int_{I_n}\Vert w_n - w\Vert_{(w,\frac{m+1}{m})}^2 + \int_0^T\Vert w - P_hw\Vert_{(w,\frac{m+1}{m})}^2 + \Vert \psi(w_0) - \Pi_h\psi(w_0)\Vert_2^2\\
&\quad + \int_{\Omega_s\cup\Omega_c}\int_{\Omega_s\cup\Omega_c}\left|\tau\displaystyle\sum_{n=1}^N[(\bar{w}^n(y) - \bar{w}^n(x)) - (P_h\bar{w}^n(y) - P_h\bar{w}^n(x))]\right|^2\gamma(x,y)\, dy\, dx \Big),
\end{aligned}$$ where $\bar{w}(x) = \int_0^Tw(x,t)\, dt$ and $\overline{W}(x) = \int_0^TW(x,t)\, dt$.
\[thm:main7.6.2\] Let $0 < m < 1$, $w_0 \in L^{\infty}(\Omega_s\cup\Omega_c)\cap H^s(\Omega_s\cup\Omega_c)$ and $\mathcal{V}w_0 = 0$ on $\Omega_c$. Then there is a positive constant $c$ independent of $h$ and $\tau$ such that $$\label{Eq8.15}
\begin{aligned}
\left(\int_0^T\Vert w - W\Vert_{(w,\frac{m+1}{m})}^2 + \displaystyle\sup_{1\leq n\leq N}\left\Vert\int_0^{t_n}(w - W)\right\Vert_{H^s(\Omega_s\cup\Omega_c)}^2\right)^{1/2} \leq c(\tau + h^s).
\end{aligned}$$
\[thm:main7.6.3\] Under the assumptions in the Theorem \[thm:main7.6.1\], there exists a positive constant $c$ independent of $h$ and $\tau$ such that $$\Vert u - U\Vert_{L^{m+1}(0,T;L^{m+1}(\Omega_s\cup\Omega_c))} \leq c(\tau + h^s),$$ where $U = \psi(W)$.
\[thm:main7.6.4\] Under the assumptions in the Theorem \[thm:main7.6.1\], then $$\begin{aligned}
\left(\int_0^T\int_{\Omega_s\cup\Omega_c}|u - U||w - W| \right)^{\frac{1}{2}} + \left\Vert\int_0^t(w - W)\right\Vert_{L^{\infty}(0,T;H^s(\Omega_s\cup\Omega_c))} \leq c(\tau + h^s).
\end{aligned}$$
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---
abstract: 'Circular flush Jets In Cross-Flow were experimentally studied in a water tunnel using Volumetric Particle Tracking Velocimetry, for a range of jet to cross-flow velocity ratios, $r$, from $0.5$ to $3$, jet exit diameters $d$ from $0.8$ $cm$ to $1$ $cm$ and cross-flow boundary layer thickness $\delta$ from $1$ to $2.5$ $cm$. The analysis of the 3D mean velocity fields allows for the definition, computation and study of Counter-rotating Vortex Pair trajectories. The influences of $r$, $d$ and $\delta$ were investigated. A new scaling based on momentum ratio $r_m$ taking into account jet and cross-flow momentum distributions is introduced based on the analysis of jet trajectories published in the literature. Using a rigorous scaling quality factor $Q$ to quantify how well a given scaling successfully collapses trajectories, we show that the proposed scaling also improves the collapse of CVP trajectories, leading to a final scaling law for these trajectories.'
address: 'PMMH, 10, rue Vauquelin 75006 Paris, France'
author:
- 'T. Cambonie, N. Gautier, J.-L. Aider'
bibliography:
- 'LowVelocityRatioScaling.bib'
title: 'Experimental study of Counter-Rotating Vortex Pair Trajectories induced by a Round Jet in Cross-Flow at Low Velocity Ratios'
---
**Key words**: 3D velocimetry, Jet in cross-flow, Low velocity ratio, Trajectory scaling.
Introduction {#intro}
============
Jets In Cross-Flows (JICF) are complex three-dimensional flows which can be found in many engineering applications such as film cooling of turbines and combustors or the control of separated flows over airfoils and ground vehicles ([@Margarson1993], [@Stanislas2006], [@Joseph2011]). The control and understanding of JICF’s is of great industrial interest. Its complexity also makes it a great challenge for academic research. Thus, it has been the subject of many experimental, numerical and theoretical studies over the past fifty years which are well summarized in the recent review by [@AnnR2010].
When studying a JICF, many parameters can be considered, such as the Reynolds numbers of both jet and cross-flow, the diameter of the jet or the velocity ratio. The latter is considered as the key parameter and is defined as $r=\sqrt{\rho_j \overline{V_j}^2/\rho_{\infty} U_{\infty}^2}$ where $\rho_j$,$V_j$ are the jet density and mean exit velocity and $\rho_{\infty}$, $U_{\infty}$ are the free stream density and velocity. When jet and free stream fluid densities are equal, the momentum ratio becomes $r=\overline{V_j}/U_{\infty}$.
The main feature of the mean flow observed in previous studies is the counter-rotating vortex pair (CVP), sketched on Fig. \[fig:SketchCVP\]. CVP are, to our knowledge, always present in time-averaged velocity fields. Moreover, the CVP is the only structure remaining far from the injection site, sometimes persisting as far as a thousand jet diameters as shown by [@Keffer1963]. The CVP has been investigated in detail by [@Chassaing1974], [@Blanchard1999], [@Cortelezzi2001] and [@Marzouk2007]. Characterization of its location through the study of its trajectory is therefore of great interest.
![ Sketch of jet in cross-flow: the CVP and the Horseshoe vortex are the main swirling structures observed in the mean velocity field.[]{data-label="fig:SketchCVP"}](Fig1){width="100.00000%"}
We consider low velocity ratios ($r<3$). Most previous studies focused on higher velocity ratios ($r>2-3$). Low velocity ratios JICF’s were investigated by [@Camussi2002] and [@Gopalan2004]. A significant difference between high and low velocity ratios is the interaction with the boundary layer: at low $r$ the jet interacts with the boundary layer leading to a profound modification of the flow structure. Transition between globally unstable and convectively unstable flow has been shown to exist at $r=3$ by [@Megerian2007]. A transition at very low velocity ratios ($r=0.3$) has been observed by [@Cambonie2012]. It is a transition from a *blown* jet topology to a *classical* jet topology. These transitions could impact the CVP. Our range of velocity ratios is $0.5<r<3$, above the transition from blown jet to classical jet.\
\
To our knowledge there are no parametric studies focusing on CVP trajectory, although they are mentioned as vortex curves and studied by [@Weston1974] and [@Karagozian1986]. The objective of this paper is to define the CVP trajectories in such a way that it can be computed for any velocity ratio and to propose a scaling for these trajectories which takes into account jet and boundary layer momentum distributions, cross-flow boundary layer thickness and jet diameter for low velocity ratios.
Experimental setup {#sec:1}
==================
Water tunnel, jet supply system and geometries {#sec:2}
----------------------------------------------
Experiments were conducted in a hydrodynamic channel in which the flow is driven by gravity. The walls are made of Altuglas for easy optical access from any direction. Upstream of the test section the flow is stabilized by divergent and convergent sections separated by honeycombs. The test section is $80$ $cm$ long with a rectangular cross section $15$ $cm$ wide and $20$ $cm$ high as described in Fig. \[fig:plate\].\
The mean free stream velocity $U_{\infty}$ ranges between $0.9$ to $8.37$ $cm.s^{-1}$ corresponding to $Re_{\infty}=\frac{U_{\infty}d}{\nu}$ ranging between 220 and 660. The quality of the main stream can be quantified in terms of flow uniformity and turbulence intensity. The spatial $\sigma_s$ and temporal $\sigma_t$ standard deviations are computed using a sample of 600 velocity fields. The values are, for the highest free stream velocity featured in our data, $\sigma_s=0.038$ $cm.s^{-1}$ and $\sigma_t=0.059$ $cm.s^{-1}$ which corresponds to turbulence levels $\frac{\sigma_s}{U_{\infty}}=0.15$ $\%$ and $\frac{\sigma_t}{U_{\infty}}=0.23$ $\%$, respectively.\
A custom made plate with a specific leading-edge profile is used to start the cross-flow boundary layer. The boundary layer over the plate is laminar and stationary according to $Re_x=\frac{U_{\infty}x}{\nu}<2100$, where x is the distance to the leading edge of the plate, for the highest free stream velocity case, which is considerably less than the critical value for this profile. The boundary layer characteristics were investigated using 600 instantaneous 3D velocity fields without a jet present for all cross-flow velocities. The average field allows us to compute the boundary layer velocity profiles. The boundary layer thickness $\delta$ varies from $2.5$ $cm$ to $1$ $cm$ for increasing cross-flow velocity.\
These unperturbed fields were used to compute cross flow velocity by averaging longitudinal velocity in the volume field, excluding the boundary layer.
![Definition of the experimental test section. The flow goes from left to right and develops over a raised plate with NACA leading edge. The measurement volume is lit through the upper plate. The three cameras of the V3V system are tracking particles through the side-wall of the channel. The jet nozzle is located $42 cm$ downstream of the leading edge.[]{data-label="fig:plate"}](Fig2){width="95.00000%"}
![2D sketch of the injection site with definitions of the main geometric and physical parameters.[]{data-label="fig:nozzle"}](Fig3){width="50.00000%"}
The jet supply system was custom made. Water enters a plenum and goes through a volume of glass beads designed to homogenize the incoming flow. The flow then goes through a cylindrical nozzle which exits flush into the cross-flow. In the following, we focus on nozzles with different diameters $d$ and different injection lengths $l_n$ (Fig. \[fig:nozzle\], and table \[tab:configs\]). The jet axis is normal to the flow. The mean vertical jet velocity $\overline{V_{j}}$ ranges between $1.9$ and $8$ $cm.s^{-1}$, leading to velocity ratios $r = \overline{V_{j}} / U_{\infty}$ ranging between $0.5$ to $3$. The dimensions of the jet nozzle and flow characteristics for the 22 configurations presented in this study are summarized in table \[tab:configs\].
3D Particle Tracking Velocimetry measurements {#sec:3}
---------------------------------------------
To analyze the mean-flow characteristics of the JICF, we use volumetric particle tracking velocimetry (3DPTV). The method was pioneered by [@WILLERT1992] and further developed by [@Pereira2002]. The set-up was designed and the physical parameters were chosen to optimize the quality of the instantaneous velocity fields, using the methodology of [@Cambonie2013]. We used $50\mu m$ polyamide particles (PSP) for seeding, with a concentration of $5.10^{-2}$ particles per pixel. The flow is illuminated through the upper wall and the particles are tracked using three cameras facing the side wall (Fig. \[fig:plate\]). The three double-frame cameras are $4$ MP with a 12 bit output. Volumetric illumination is generated using a 200 mJ pulsed YaG laser and two perpendicular cylindrical lenses. Synchronization is ensured by a TSI synchronizer. The measurement volume $(l_x,l_y,l_z)$ is $14 \times 6 \times 3$ $cm^3$. The spatial resolution is one velocity vector per millimeter for both the instantaneous and mean three-components velocity field . This resolution might not always allow for the detection of the smallest structures in the flow, especially for higher velocity ratios. Nevertheless the jet diameter has been chosen to ensure a good spatial resolution of the main vortices ($8$ $mm<d<10$ $mm$). The characteristic width of a vortex is the an order of magnitude higher than the spatial resolution allowing us to clearly detect the CVP. The acquisition frequency is 7.5$Hz$. 1000 instantaneous velocity fields are recorded for each configuration to ensure statistical convergence of the mean velocity field.
Trajectory computation {#sec:5}
======================
Visualization of the CVP
-------------------------
To analyze the complex three-dimensional flow, we use the swirling strength criterion $\lambda_{ci}$. It was first introduced by [@Chong1990] who analyzed the velocity gradient tensor and proposed that the vortex core be defined as a region where $\nabla u$ has complex eigenvalues. It was later improved and used for the identification of vortices in three-dimensional flows by [@Zhou1999]. This criterion allows for an effective detection of vortices even in the presence of shear. It is calculated for the entire 3D velocity fields.
![Mean iso-surface of $\lambda_{ci}$ colored by longitudinal vorticity for configuration 7 (velocity ratio $r=1.07$, together with a contour of vertical velocity at $X=10d$. The computed CVP trajectory is shown as a thick black line.[]{data-label="fig:3DTraj"}](Fig4){width="95.00000%"}
Fig. \[fig:3DTraj\] shows a typical example of the main vortical structures present in the mean velocity field using isosurfaces of $1.5\cdot \sigma(\lambda _{Ci}) $ (where $\sigma$ is the spatial standard deviation) colored by the longitudinal vorticity. One can clearly see the two counter-rotating vortices growing downstream of the injection site. The vertical velocity field is also visualized in the $X/d=10$ cross-section showing the strong outflow region induced by the CVP. The CVP creates a well-defined outflow region in its center. Thus a practical way of computing the CVP trajectory is to look for the locus of maximal vertical velocity.
Jet and CVP trajectory {#sec:6}
-----------------------
![Jet centerline trajectories (higher curves), CVP trajectories (lower curves) computed in the numerical simulations of [@Salewski2007]. To the different trajectories correspond different number of cells for the numerical simulations: 3.8 million cells (-), 3.2 million cells ($+$) and 2.4 million cells ($\times$).[]{data-label="fig:CVPunder"}](Fig5){width="75.00000%"}
It is important to stress that CVP trajectory and jet trajectory are distinct entities. [@Muppidi2007], [@Salewski2007], as well as [@Mungal2001] show that the CVP trajectory lies under the jet trajectory.\
There are several ways of defining the jet trajectory: the jet centerline (for circular jets it is the streamline starting at the center of the injection nozzle), the locus of maximum velocity or the locus of maximum concentration. [@Yuan1998] compare these methods and show that although the computed trajectories vary, they show the same behavior.\
Fig. \[fig:CVPunder\] ([@Salewski2007]) features numerical data showing the jet centerline trajectory and the location of the CVP. The CVP does not start at origin ($x=0, y=0$) and is clearly lower than the jet centerline. However both trajectories are parallel for $z/d>8$. This is because the CVP is a structure of the mean flow field, a time average of transient structures in the instantaneous flow as shown by [@FRIC1994].
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![ a) Jet (- -) and CVP(-) trajectories for configuration 10 ($r=1.51$). b) Swirling strength of the strongest vortex for configuration 10. Maximum is indicated by a cross.[]{data-label="fig:JetvsCVPtraj"}](Fig6a "fig:"){width="55.00000%"} ![ a) Jet (- -) and CVP(-) trajectories for configuration 10 ($r=1.51$). b) Swirling strength of the strongest vortex for configuration 10. Maximum is indicated by a cross.[]{data-label="fig:JetvsCVPtraj"}](Fig6b "fig:"){width="55.00000%"}
a) b)
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\[sec:7\]
When the velocity ratio $r$ is high enough, the difference between jet and CVP trajectories can be observed in our data, as shown by Fig. \[fig:JetvsCVPtraj\] a. To compute these trajectories we locate the two vertical velocity maxima in every cross sections. This gives the (y-z)-coordinates of the CVP and jet trajectory for the given abscissa. This computation method is straightforward, easy to implement and applicable at any velocity ratio. It allows us to distinguish vertical velocity created by the CVP and vertical velocity from the jet itself.
Computing CVP trajectories {#sec:8}
---------------------------
The method for CVP trajectory computation featured above is not self-sufficient as it does not yield the start of the trajectory. To determine where to start the trajectory we track the vortex pair, by computing the two maxima of swirling strength $\lambda_{ci}$ in every constant cross section. This allows us to compute the intensity of the vortex pair $I_{CVP}$ along the trajectory of its cores. Fig. \[fig:JetvsCVPtraj\] b shows the intensity of the strongest core for configuration 10. We define the start of the CVP trajectory as the abscissa of the maximum of swirling strength of the strongest vortex core which corresponds to lateral shear on the side on the jet.\
It might seem unduly complicated to track the outflow instead of the vortex cores themselves. Indeed another way of defining the CVP trajectory is by computing the mean trajectory of both streamwise vortex cores, however it is not as practical. Fig. \[fig:VsLci\] a and \[fig:VsLci\] b show CVP trajectories for configuration 10 computed by both methods. Fig. \[fig:VsLci\] a shows that CVP trajectories computed using vertical velocity and $\lambda_{ci}$ are in good agreement.This demonstrates the relevance of detecting the CVP using the outflow. Fig. \[fig:VsLci\] b shows the (more common) case where difference in the strength of the vortex cores induces large fluctuations in computed trajectory using swirling strength. Similarly, tracking only one vortex core is much less reliable. Trajectories extracted with $\lambda_{ci}$ are less reliable, specifically when the intensity of the vortices differs. For our experimental data we obtain considerably better results when considering the locus of vertical velocity maxima than for the locus of $\lambda_{Ci}$ maxima.
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![a) CVP trajectories computed using vertical velocity (-) and $\lambda_{ci}$ (- -) for configuration 3. b) CVP trajectories computed using vertical velocity (-) and $\lambda_{ci}$ (- -) for configuration 2.[]{data-label="fig:VsLci"}](Fig7a "fig:"){width="50.00000%"} ![a) CVP trajectories computed using vertical velocity (-) and $\lambda_{ci}$ (- -) for configuration 3. b) CVP trajectories computed using vertical velocity (-) and $\lambda_{ci}$ (- -) for configuration 2.[]{data-label="fig:VsLci"}](Fig7b "fig:"){width="50.00000%"}
a) b)
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Trajectories were computed in a volume, but they are very close to the symmetry plane. Therefore only the y-component of the trajectory will be analyzed hereafter. We show on Fig. \[fig:alltraj\] a and b all 22 computed trajectories using non-dimensional coordinates ($y/\delta$, $x/d$). Trajectories are widely distributed inside and outside the boundary layer (between 0.5 to 3.5 $\delta$).
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![ a ) Trajectories for configurations 1 to 11. b) Trajectories for configurations 12 to 22. Markers are detailed in table \[tab:configs\].[]{data-label="fig:alltraj"}](Fig8a "fig:"){width="50.00000%"} ![ a ) Trajectories for configurations 1 to 11. b) Trajectories for configurations 12 to 22. Markers are detailed in table \[tab:configs\].[]{data-label="fig:alltraj"}](Fig8b "fig:"){width="50.00000%"}
a) b)
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Definition and relevance of the momentum ratio $r_m$ {#sec:9}
====================================================
In most of the previous studies of JICF, the velocity ratio $r$ is considered as the key parameter despite its limitations: it does not take into account some important features such as the boundary layers of the jet and the cross-flow. Indeed, [@Muppidi2005] have shown that the classic $rd$ scaling was not sufficient to collapse all jet trajectories published in the literature onto a single curve. They suggest that the jet exit velocity profile as well as the cross-flow boundary layer thickness influence the jet. This is supported by the analysis of the influence of jet exit velocity profile on jet trajectories conducted by [@New2006]. To account for momentum distribution in the jet and boundary layer we introduce a momentum ratio $r_m$ integrating the momentum distribution of the jet and cross-flow boundary layer (also mentioned in [@Muppidi2005]), equation \[eq:rm1\]:
$$r_m^2=\frac{\frac{1}{S} \int_S V_j^2 dS} { \frac{1}{\delta} \int_0^{\delta} U_{cf}^2 dy}
\label{eq:rm1}$$
where $U_{cf}(y)$ is the cross-flow velocity at $y$ and $S$ is the jet nozzle exit section. To highlight the difference with the velocity ratio $r$, $r_m$ can be decomposed in three parts:
$$r_m=(\sqrt{r_{m,jet}}\cdot \frac{1}{\sqrt{r_{m,cf}}}) \cdot r
\label{eq:rm2}$$
with
$$\begin{aligned}
r_{m,jet}=\frac{S \int_S V_j^2 dS}{(\int_S V_j dS)^2}= \frac{\overline{V_j^2}}{\overline{V_j}^2}, &&
r_{m,cf}=\frac{\int_0^{\delta} U_{cf}^2 dy}{\delta U_{\infty}^2}= \int_0^{1} (\frac{U_{cf}}{U_{\infty}})^2(\frac{a}{\delta}) da\end{aligned}$$
This decomposition involves two non-dimensional shape factors: $r_{m,cf}$ and $r_{m,jet}$. $r_{m,cf}$ accounts for the momentum distribution in the cross-flow boundary layer ($0<r_{m,cf}<1$, by definition), while $r_{m,jet}$ accounts for the momentum distribution in the jet.\
To quantify the influence of the velocity profiles on these two new shape factors, we use the boundary layer velocity profiles shown on Fig. \[fig:rm\] a for the cross-flow and the velocity profiles shown on Fig. \[fig:rm\] b for the jet. Typical values obtained for $r_{m,cf}$ with the Blasius ($r_{m,cf} = 0.52$) or experimental ($r_{m,cf} = 0.57$) boundary layer profiles are shown on table \[tab:profiles\]. $r_{m,cf} \approx 1$ corresponds to a plug profile. This is coherent with the fact a boundary layer with much momentum near the wall leads to a lower trajectory. In the following, the value of $r_{m,cf}$ is computed using experimental velocity data.
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![(a) : Experimental boundary layer velocity profile for configuration 13, : theoretical Blasius boundary layer profile. b) Theoretical jet velocity profiles as a function of injection length. $l_n=0.5cm, r_{m,jet}=1.04 $ ( $-$ ); $l_n=1cm, r_{m,jet}=1.05$ ( $+$ ); $l_n=2cm, r_{m,jet}=1.07$ ( ); $l_n=3cm, r_{m,jet}=1.09$ ( $-+$ ) []{data-label="fig:rm"}](Fig9a "fig:"){width="50.00000%"} ![(a) : Experimental boundary layer velocity profile for configuration 13, : theoretical Blasius boundary layer profile. b) Theoretical jet velocity profiles as a function of injection length. $l_n=0.5cm, r_{m,jet}=1.04 $ ( $-$ ); $l_n=1cm, r_{m,jet}=1.05$ ( $+$ ); $l_n=2cm, r_{m,jet}=1.07$ ( ); $l_n=3cm, r_{m,jet}=1.09$ ( $-+$ ) []{data-label="fig:rm"}](Fig9b "fig:"){width="50.00000%"}
a) b)
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[|c | c c |c| c | c c |]{} profiles & Blasius& Experimental & &profiles & Plug / Tophat & Parabolic\
$r_{m,cf}$ & $0.52$& $0.57$ && $r_{m,jet} $ & $1$ & $1.33 $\
Our measurement method does not allow for a sufficient resolution of the velocity profiles at the exit of the jet nozzle to satisfactorily compute the value of $r_{m,jet}$ with experimental data. Consequently $r_{m,jet}$ is estimated using the expression for boundary layer thickness in a smooth pipe proposed by [@Mohanty1978]. Knowing the jet velocity and the nozzle injection length we compute the analytical jet exit velocity profiles shown in Fig. \[fig:rm\] b, before computing the associated values for $r_{m,jet}$. Values for $r_{m,jet}$ vary between $1$ (for a top-hat profile) and $1.33$ (for a parabolic profile). [@Muppidi2005] show that a parabolic JICF achieves higher penetration than a top-hat JICF. This is corroborated by experimental work by [@New2006]. Their interpretation is that the thicker shear layers associated with parabolic JICF delay the formation of leading-edge and lee-side vortices. Therefore possible values taken by $r_{m,jet}$ are coherent with the effect of jet velocity profile on jet trajectory. Consider two jet trajectories with identical velocity ratios, boundary layer profiles and jet exit diameter but with different exit velocity profiles: one with a parabolic profile, one with a top-hat velocity profile. The parabolic jet penetrates deeper resulting in a higher overall trajectory. For both cases, values of $r$ are identical. Values of $r_m$ are different, making $r_m$ the more relevant parameter.\
As shown in table \[tab:configs\], we obtain $0.75<r_{m}<4.10$ corresponding to $0.55<r_{m,cf}<0.67$ and $1.05<r_{m,jet}<1.13$ for our configurations.
Influence of experimental parameters on CVP trajectories {#sec:10}
=========================================================
Influence of velocity ratio and boundary layer thickness {#sec:11}
--------------------------------------------------------
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![ a) Influence of the velocity ratio on the CVP trajectories with constant jet exit velocity for $r=0.54$ (-), $r=0.74$ ($-\times$), $r=1.14$ ($-\bullet$), $r=1.62$ ($-+$). b) Influence of the velocity ratio on the CVP trajectories with constant boundary layer thickness and profile for $r=0.51$ (-), $r=0.83$ (-), $r=1.07$ ($- -$), $r=1.24$ ($\cdot \cdot \cdot$).[]{data-label="fig:trajsr"}](Fig10a "fig:"){width="50.00000%"} ![ a) Influence of the velocity ratio on the CVP trajectories with constant jet exit velocity for $r=0.54$ (-), $r=0.74$ ($-\times$), $r=1.14$ ($-\bullet$), $r=1.62$ ($-+$). b) Influence of the velocity ratio on the CVP trajectories with constant boundary layer thickness and profile for $r=0.51$ (-), $r=0.83$ (-), $r=1.07$ ($- -$), $r=1.24$ ($\cdot \cdot \cdot$).[]{data-label="fig:trajsr"}](Fig10b "fig:"){width="50.00000%"}
a) b)
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![ Influence of boundary layer thickness on CVP trajectories: $\delta=1.59d$ ($- \cdot$), $\delta=1.87d$ ($-$), $\delta=2.25d$ (-), $\delta=2.59d$ (-), $\delta=2.79d$ ($-\times$).[]{data-label="fig:trajsdelta"}](Fig11){width="50.00000%"}
Fig. \[fig:trajsr\] a and b show the influence of velocity ratio. The x and y coordinates are scaled by $d$.\
In Fig. \[fig:trajsr\] a the velocity ratio ranges from $r=0.54$ to $r=1.62$, while jet exit velocity and profile are kept constant. Cross-flow velocity changes and therefore boundary layer thickness changes also.\
In Fig. \[fig:trajsr\] b velocity ratio ranges from $r=0.51$ to $r=0.84$, while cross-flow velocity, boundary layer thickness and profile are kept constant. Jet exit velocity and profile change. It should be noted that with a constant injection length it is experimentally impossible to vary jet velocity ratio while keeping jet exit velocity profile and boundary layer thickness constant. In all cases the trajectory of the CVP rises with an increase in velocity ratio.\
\
Fig. \[fig:trajsdelta\] compares CVP trajectories for different values of the boundary layer thickness. All other parameters being equal CVP trajectories penetrate deeper when the cross-flow boundary layer is thicker. The same result has been obtained numerically for jet trajectories by [@Muppidi2005] and observed by [@Cortelezzi2001]. This is explained by the fact a thinner boundary layer has more momentum close to the jet exit. Jet trajectories bend earlier and the resulting CVP is created closer to the wall, thus resulting in an overall lower CVP trajectory.
Influence of jet exit velocity profile through variation of injection length
----------------------------------------------------------------------------
For a constant jet flowrate, changing the injection length modifies the jet exit velocity profile. Fig. \[fig:trajInjectionLength\] shows CVP trajectories for different nozzle injection lengths, while cross-flow velocity and mean jet velocity are kept constant for two different velocity ratios. An increase in injection length leads to more parabolic jet exit velocity profiles as illustrated in Fig. \[fig:rm\] b. Fig. \[fig:trajInjectionLength\] shows that the more parabolic the velocity profile, the higher the CVP trajectory. Although nozzle lengths do not come close to what one would need to ensure a full parabolic profile ($l_n>60d$) the effect on CVP trajectory is significant. This is an important result: even a small modification of the exit velocity profile can change the height of the CVP trajectories significantly. This sensitivity could be due to the low velocity ratios featured for this data. This issue is investigated in section \[sec:16\]. Apart from the discussion on trajectory scaling, this data clearly illustrates how it is possible to obtain higher trajectories without spending more energy, only by modifying the design of the injection.
\[sec:12\]
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![a) Influence of injection length $l_n$ on CVP trajectories for $r=0.96$ and constant boundary layer thickness and profile: $l_n=0.5cm$ ($-\bullet$), $l_n=2cm$ (-), $l_n=3cm$ ($\cdot \cdot \cdot$). b) Influence of injection length $l_n$ on CVP trajectories for $r=1.9 $ and constant boundary layer thickness and profile: $l_n=0.5cm$ ($-+$), $l_n=2cm$ ($- -$), $l_n=3cm$ ($- \cdot -$).[]{data-label="fig:trajInjectionLength"}](Fig12a "fig:"){width="50.00000%"} ![a) Influence of injection length $l_n$ on CVP trajectories for $r=0.96$ and constant boundary layer thickness and profile: $l_n=0.5cm$ ($-\bullet$), $l_n=2cm$ (-), $l_n=3cm$ ($\cdot \cdot \cdot$). b) Influence of injection length $l_n$ on CVP trajectories for $r=1.9 $ and constant boundary layer thickness and profile: $l_n=0.5cm$ ($-+$), $l_n=2cm$ ($- -$), $l_n=3cm$ ($- \cdot -$).[]{data-label="fig:trajInjectionLength"}](Fig12b "fig:"){width="50.00000%"}
a) b)
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Trajectory Scaling {#sec:13}
==================
Scaling quality factor {#sec:14}
----------------------
In order to quantitatively compare how well different scalings collapse trajectories we define a non-dimensional scaling quality factor $Q$. A perfect scaling would collapse all trajectories onto a single curve, in other words the scattering would be null. This can be characterized by a quantitative criterium.\
For a given abscissa $\tilde{x}$ we define $Y(\tilde{x})$ the set of values taken by the trajectories at this abscissa. We define $\left[\tilde{x}_{start}, \tilde{x}_{end}\right]$ the range where trajectories exist. $\tilde{x}_{start}$ is then the first abscissa where an outflow region can be identified, i.e where the first trajectory starts, while $\tilde{x}_{end}$ corresponds to where the longest trajectory ends. This range may change depending on how the abscissa is scaled.\
For our trajectories, the case arises where not all of them are defined for a given $\tilde{x}$. In order to take this into account we introduce $N(\tilde{x})$ and $N_{curves}$, respectively the number of curves defined at abscissa $\tilde{x}$ and the total number of curves considered for scaling. The scaling quality factor is defined as the integral of trajectory scatter relative to the mean over the range where these trajectories exist.
$$Q=\int_{\tilde{x}_{start}}^{\tilde{x}_{end}} \frac{\sigma(Y)}{\overline{Y}}(\tilde{x}). \frac{N(\tilde{x})}{N_{curves}} d \tilde{x}
\label{eq:Q}$$
where $\sigma{(Y)}$ is the standard deviation of Y and $\overline{Y}$ is the mean of Y for a given abscissa $\tilde{x}$. $Q=0$ corresponds to a perfect scaling.\
To take into account the fact that trajectories are not defined over the same spatial range, this relative scatter is weighted by the ratio, $\frac{N(\tilde{x})}{N_{curves}}$. This is done to give more meaning to the collapse of many trajectories than to the collapse of a few. For a set of trajectories defined over the same domain the weight is one, and the definition for $Q$ can be simplified to the expression shown in equation \[eq:Qsimple\]: $$Q=\int_{\tilde{x}_{start}}^{\tilde{x}_{end}} \frac{\sigma(Y)}{\overline{Y}}(\tilde{x}). d \tilde{x}
\label{eq:Qsimple}$$ Normalizing by the mean is necessary to ensure that multiplication of all trajectories by any constant does not change the value of $Q$. This method is applicable to any collection of 2D curves, for any scaling of the x-coordinate. Particularly $Q$ can be used to gauge the efficacy of a given scaling of CVP or jet trajectories.\
For clarity, $Q$ is normalized by its value $Q_0$ taken when the data is not scaled, both in $x$ and in $y$.
Reflexions on previously published jet trajectories {#sec:15}
---------------------------------------------------
To the best of the authors knowledge there are no CVP trajectory data for which jet exit velocity profile, boundary layer thickness and profile are available. However since CVP trajectories follow the same trends as jet trajectories (e.g. deeper penetration with increase in momentum ratio) we will begin our discussion using jet trajectory data published in [@Muppidi2005]. These results were chosen because the varying parameters were boundary layer thickness and jet exit velocity profile for two different velocity ratios. Table \[tab:Muppidiparam\] summarizes the different parameters used by [@Muppidi2005] for their study. The corresponding values of $r_{m,jet},r_{m,cf}$ and $r_m$ were computed using the data presented in their paper.\
The objective is to derive an approach to the scaling of these jet trajectories which can be applied to CVP trajectories. [@Muppidi2005] present a scaling that successfully collapses their trajectories. This scaling uses a parameter $h$ extracted from the data as the y-coordinate at a distance $x=0.05d$. Because CVP trajectories do not start at $x=0$, $h$ is not defined and cannot be used for scaling purposes. Moreover our objective was to validate a more general scaling based on experimental parameters. Thus an alternate scaling was sought.\
Fig. \[fig:MuppidiR\] shows the influence of jet velocity profile and boundary layer thickness for different velocity ratios on jet trajectories. Tophat and parabolic jet exit velocity profiles are used. As shown in table \[tab:Muppidiparam\] values of $r_m$ are higher for the parabolic profile. For the CVP, jet penetration is higher for parabolic velocity profiles and for thicker boundary layers.
case I II III IV V VI VII VIII IX
-------------------- ------- ----------- ------- ----------- ------- ------- ------- ------- ------
Velocity ratio $r$ 1.52 1.52 1.52 1.52 5.7 5.7 5.7 5.7 5.7
$\delta_{80 \%}$ 1.32d 1.32d 0.44d 0.44d 1.32d 1.32d 0.44d 0.44d 6.4d
$r_{m,jet} $ 1.33 1.185 1.33 1.185 1.185 1.33 1.185 1.33 1.33
$r_{m,cf} $ 0.52 0.52 0.52 0.52 0.52 0.52 0.52 0.52 0.52
$r_m$ 2.44 2.29 2.44 2.29 9.16 8.60 9.16 8.60 9.16
Markers - $-\times$ $-*$ $-\cdot $ - $-|$ $--$ $-$
: Parameters for jet trajectories from [@Muppidi2005], and corresponding values for $r_{m,jet},r_{m,cf}$ and $r_m$ obtained using their parameters.[]{data-label="tab:Muppidiparam"}
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![a) Jet trajectories for constant jet exit velocity profile and velocity ratio but varying boundary layer thickness$-\bullet$, $\delta=0.44d$, $-\times$, $\delta=1.32d$. b) Trajectories for constant velocity ratio, boundary layer thickness. Jet exit velocity profile varies between tophat and parabolic. Parabolic : -,$r_m=8.6$. Top-hat : -,$r_m=9.16$ . From [@Muppidi2005] data.[]{data-label="fig:MuppidiR"}](Fig13a "fig:"){width="50.00000%"} ![a) Jet trajectories for constant jet exit velocity profile and velocity ratio but varying boundary layer thickness$-\bullet$, $\delta=0.44d$, $-\times$, $\delta=1.32d$. b) Trajectories for constant velocity ratio, boundary layer thickness. Jet exit velocity profile varies between tophat and parabolic. Parabolic : -,$r_m=8.6$. Top-hat : -,$r_m=9.16$ . From [@Muppidi2005] data.[]{data-label="fig:MuppidiR"}](Fig13b "fig:"){width="50.00000%"}
a) b)
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Scaling of jet trajectory {#sec:16}
-------------------------
Trajectory scaling of a circular jet in cross-flow has been the subject of much research ([@Pratte1967], [@Smith1998], [@Yuan1998], [@Mungal2001], [@Muppidi2005], [@Gutmark2008]), however no scaling is fully satisfactory. Among the most successful scalings, the $rd$ scaling by [@Pratte1967] has proven to collapse most experimental trajectories. For $5 < r <35$, they show the collapse of the centerline trajectory with the $rd$ length scale defined as follows: $$\frac{y}{rd}=A(\frac{x}{rd})^b
\label{eq:rdscaling}$$
where $A=2.05$ and $b=0.28$. However more recent works by [@Muppidi2005] and [@New2006] show that this scaling is not satisfactory for flows where boundary layer thickness and jet exit velocity profile vary. Several attempts were made to scale jet trajectories while accounting for these factors ([@Muppidi2005], [@Gutmark2008]).\
A scaling using $r^{\alpha}$ was introduced by [@Karagozian1986] for high velocity ratios. Similarly we choose to consider a scaling using $r_m^{\alpha}$ to account for jet exit velocity profile, where $\alpha$ quantifies the influence of momentum ratio $r_m$ and is unknown *a priori*. To account for the influence of the boundary layer thickness we introduce, in a manner analogous to [@Muppidi2005] who use $(\frac{h}{d})^C$ , the non-dimensional parameter $(\frac{\delta}{d})^\beta$, where $\beta$ quantifies the influence of $\delta$ on trajectory. This leads to the new scaling described in equation \[eq:Newscl\]:
$$\frac{y}{{r_m}^{\alpha} d (\frac{\delta}{d})^\beta}
\label{eq:Newscl}$$
Reasoning on the physics of the flow and empirical data, it is possible to define upper and lower bounds for $\beta$ and $\alpha$.\
To $\alpha=1,\ \beta=0$ corresponds the scaling $\frac{y}{r_md}$. Having $\beta<0$ would mean the jet penetrates deeper with a decreasing boundary layer thickness, therefore $\beta>0$. Moreover for high velocity ratios where the jet exit profile is usually a plug profile with a fixed boundary layer profile which gives $r_m\propto r$ thus making this scaling equivalent to the $rd$ scaling.\
To $\alpha=1,\ \beta=1$ corresponds the scaling $\frac{y}{r_m \delta}$. Having $\beta>1$ would mean deeper jet penetration with decrease in jet diameter, therefore $\beta>1$. Similarly the data shows how trajectories rise with $d$, therefore $\beta<1$.\
Using the same reasoning we obtain $\alpha>0$, since trajectories rise with $r_m$. There is however no upper bound on $\alpha$.\
\
Fig. \[fig:sclMup\] [c, d]{} shows scaled trajectories using equation \[eq:Newscl\] compared to the classic $rd$ scaling (Fig. \[fig:sclMup\] [a,b]{}). For a given set of jet trajectories we search for $\alpha,\beta$ to obtain the best possible collapse. This is equivalent to minimizing the quality factor $Q$, here used in its simplified form defined in equation \[eq:Qsimple\].\
Note that jet exit velocity profile and boundary layer thickness do not affect trajectories in the same way for different velocity ratios.\
For $r=5.7$ (Fig. \[fig:sclMup\] [a,c]{}), we have $y/(r_m^{1.5} d (\frac{\delta}{d})^{0.05})$ whereas for $r=1.5$ (Fig. \[fig:sclMup\] [b,d]{}) we obtain $y/(r_m^{2.3} d (\frac{\delta}{d})^{0.16})$. Indeed two different sets of exponents are found depending on $r$, i.e $\alpha(r)$ and $\beta(r)$. The exponents for $r_m$ and $(\frac{\delta}{d})$ give us insight into how jet trajectory is influenced by jet exit velocity profile and boundary layer thickness. These results indicate that for low velocity ratios, jet trajectory will be more sensitive to variations of the incoming cross flow boundary layer thickness. While for high velocity ratios boundary layer thickness is less of an issue and the trajectory is mainly influenced by the momentum ratio. The proposed scalings achieve significant collapse as shown in Fig. \[fig:sclMup\]. It also shows how the scaling differs whether high or low velocity ratios are considered.\
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![ a) Jet trajectories without scaling in y for case I (-), case II ($-\times$), case III ($-*$), case IV ($-\bullet$). b) Jet trajectories without scaling for case V (-), case VI (-), case VII ( $- |$ ), case VIII ($--$), case IX ($-$). c) Jet trajectories from a) with scaling for $r_m^{2.23}$ and $(\frac{\delta}{d})^{0.16}$ leading to $Q=20\%$. d) Jet trajectories from b) with scaling for $r_m^{1.55}$ and $(\frac{\delta}{d})^{0.06}$ leading to $Q=26\%$. Each set of trajectories is normalized to make the original and scaled trajectory set comparable.[]{data-label="fig:sclMup"}](Fig14a "fig:"){width="50.00000%"} ![ a) Jet trajectories without scaling in y for case I (-), case II ($-\times$), case III ($-*$), case IV ($-\bullet$). b) Jet trajectories without scaling for case V (-), case VI (-), case VII ( $- |$ ), case VIII ($--$), case IX ($-$). c) Jet trajectories from a) with scaling for $r_m^{2.23}$ and $(\frac{\delta}{d})^{0.16}$ leading to $Q=20\%$. d) Jet trajectories from b) with scaling for $r_m^{1.55}$ and $(\frac{\delta}{d})^{0.06}$ leading to $Q=26\%$. Each set of trajectories is normalized to make the original and scaled trajectory set comparable.[]{data-label="fig:sclMup"}](Fig14b "fig:"){width="50.00000%"}
a) b)
![ a) Jet trajectories without scaling in y for case I (-), case II ($-\times$), case III ($-*$), case IV ($-\bullet$). b) Jet trajectories without scaling for case V (-), case VI (-), case VII ( $- |$ ), case VIII ($--$), case IX ($-$). c) Jet trajectories from a) with scaling for $r_m^{2.23}$ and $(\frac{\delta}{d})^{0.16}$ leading to $Q=20\%$. d) Jet trajectories from b) with scaling for $r_m^{1.55}$ and $(\frac{\delta}{d})^{0.06}$ leading to $Q=26\%$. Each set of trajectories is normalized to make the original and scaled trajectory set comparable.[]{data-label="fig:sclMup"}](Fig14c "fig:"){width="50.00000%"} ![ a) Jet trajectories without scaling in y for case I (-), case II ($-\times$), case III ($-*$), case IV ($-\bullet$). b) Jet trajectories without scaling for case V (-), case VI (-), case VII ( $- |$ ), case VIII ($--$), case IX ($-$). c) Jet trajectories from a) with scaling for $r_m^{2.23}$ and $(\frac{\delta}{d})^{0.16}$ leading to $Q=20\%$. d) Jet trajectories from b) with scaling for $r_m^{1.55}$ and $(\frac{\delta}{d})^{0.06}$ leading to $Q=26\%$. Each set of trajectories is normalized to make the original and scaled trajectory set comparable.[]{data-label="fig:sclMup"}](Fig14d "fig:"){width="50.00000%"}
c) d)
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[|>p[2.5cm]{}| c |p[4cm]{} <|p[2.8cm]{} <| ]{} Scalings & No scaling & y/$rd$ & y/$r_m d$\
cases I to IV ($r=1.5$) & $Q=67.8\% $ & $Q=67.8\%$ & $Q=60.8\%$\
cases V to IX ($r=5.7$)& $Q=46.0\%$ & $Q=46.0\%$ & $Q=36.6\%$\
cases I to IX & $Q=100.0\%$ & $Q=78.4\%$ & $Q=73.3\%$\
\
Scalings & Muppidi y/($rd (\frac{h}{d})^C$) & Gutmark $ y/rd /(r_b\cdot(r^2 \frac{d}{\delta})^{0.45})$$\Leftrightarrow y/(rd \cdot(r_b(r^2 \frac{d}{\delta})^{0.45}))$ & y/$r_m^{\alpha} d (\frac{\delta}{d})^{\beta}$\
cases I to IV ($r=1.5$) & $Q=41.6\%$ &$Q=220.0\%$ & $Q=13.5\%$,$(\alpha=2.23,\beta=0.16)$\
cases V to IX ($r=5.7$) & $Q=32.3\%$ &$Q=393.3\%$ & $Q=12.0\%$,$(\alpha=1.55,\beta=0.05)$\
cases I to IX & $Q=36.3\% $& $Q=336.9 \%$ & $Q=28.0\%$,$(\alpha=1.14,\beta=0.08)$\
Table \[tab:Muppidiscalings\] summarizes the different scalings and how successfully they collapse the data. The proposed scaling achieves similar or better collapse than the scaling proposed by [@Muppidi2005]. It requires the determination of two parameters, whereas the $rd (\frac{h}{d})^C$ scaling requires only one. On the other hand $h$ has to be extracted from the data independently for each trajectory, whereas $r_m$ and $\delta$ are experimental parameters known *a priori*. For this data, the scaling suggested in [@Gutmark2008] does not result in trajectory collapse, on the contrary it increases the dispersion of the curves. It is most likely due to a typographic mistake in the printed scaling formula. For instance, the $\frac{d}{\delta}$ factor has to be inverted to make physical sense.
Scaling of CVP trajectories {#sec:17}
---------------------------
Experimental CVP trajectory data analyzed in section \[sec:9\] and jet trajectory data discussed in section \[sec:16\] show that both types of trajectories behave in the same manner when parameters vary. This is to be expected since the CVP is a structure created by the jet and it seems CVP trajectory follows jet centerline trajectory.\
Nevertheless there are differences between CVP and jet trajectories. CVP trajectories do not start at the jet exit ($x=0,y=0$) and are lower than jet trajectories. Moreover, since jet trajectories ([@Salewski2007]) and CVP trajectories are parallel in the far field, it is impossible for both types of trajectories to assume a power law and retain that parallelism. Nevertheless a power law will be used to scale CVP trajectories, keeping in mind that the starting point abscissa for CVP trajectories vary ($x_{start} \simeq 1.5d$ for most trajectories).\
Since the trajectories of the CVP are influenced by momentum ratio $r_m$, diameter $d$ and boundary layer thickness $\delta$, the scaling described in equation \[eq:Newscl\] is applied to our data.
Determination of the optimal scaling for CVP trajectories {#sec:18}
---------------------------------------------------------
To determine the influence of boundary layer thickness on CVP trajectories we consider configurations 9 to 13. In these cases, boundary layer thickness varies from $1.36d$ to $2.26d$ while $r_m=2.2 \pm 5\%$. Best collapse is obtained for $\beta=0.91$, thus for these cases boundary layer thickness has a significant relative influence on CVP trajectory.\
Fig. \[fig:11to15delta\] shows the highest and lowest of the trajectories before and after scaling by $d (\frac{\delta}{d})^{\beta}$. Collapse is significant ($Q=28.3\%$). $\beta=0.91$ is much higher than the value found in section \[sec:11\] although the velocity ratios are comparable ($r=1.5$ and $r=1.6$). CVP trajectories being lower, interaction with boundary layer would be stronger for this velocity ratio, resulting in a higher value for $\beta$.
![ Highest and lowest trajectories for configurations 9 to 13, before (grey, dotted line) and after (black, solide line) scaling by $d(\delta/d)^{\beta}$ ($Q=28.3\%$). $y$ axis is normalized to help comparison.[]{data-label="fig:11to15delta"}](Fig15){width="65.00000%"}
To go further, a simplifying assumption is made: $\beta$ is assumed to be constant. In other words, the way the boundary layer thickness affects trajectory is considered independent of other parameters such as $r$, $r_{m,jet}$ or $r_{m,cf}$. Of course, this is not strictly true as shown in section \[sec:16\]. Furthermore since some of these trajectories are close or even inside the boundary layer (see Fig. \[fig:trajsr\]) it stands to reason $\beta$ would change with momentum ratio. However data are insufficient for a more thorough analysis of this issue, another extensive parametric study would be required. Nevertheless, based on the data from [@Muppidi2005] analyzed in the previous section, we can expect $\beta$ to be a decreasing function of $r_m$.
To determine the influence of $r_m$ we consider configurations 1 to 8 and 14 to 22. All these configurations feature variations in $r_m$. However these variations are brought about in different ways: variations in $r$ $(0.51<r<3.05)$ by changing jet velocity and cross flow velocity and variations in $r_{m,jet}$ by changing jet velocity profile. We find $\alpha=1.23$ ($Q=13.1\%$). Fig. \[fig:1to1016to24R\] shows the highest and lowest trajectories before and after scaling by $r_m^{\alpha} d (\frac{\delta}{d})^{\beta}$.
![ Lowest and highest CVP trajectories for configurations 1 to 8 and 14 to 22. No scaling (- -) and after scaling (-) leading to $Q=13.1\%$. $y$ axis is normalized to allow visual comparison.[]{data-label="fig:1to1016to24R"}](Fig16){width="65.00000%"}
For all 22 configurations the scaling mentioned gives $Q=13.14\%$. This relatively high value is most likely due to the fact that $\alpha$ and $\beta$ are considered constant when they have been shown to depend on $r$. Furthermore the underlying assumption of a power law scaling, i.e that there exists a scaling such that trajectories can be expressed as $y=A x^b$ where $A$ and $b$ are constant, is erroneous as shown by [@New2006]. However the proposed scaling does allow for significant collapse with a $43\%$ improvement over the $rd$ scaling.
Using the computed values for $\alpha$ and $\beta$, the scaled data are well fitted with a power law as described in equation \[eq:trjct\]: $$\frac{y}{{r_m}^{\alpha} d (\frac{\delta}{d})^{\beta}}=A(\frac{x}{r_m d})^b
\label{eq:trjct}$$
with $A=0.48$, $b=0.42$. Commonly $rd$ scalings of the jet trajectory yield $1.2<A<2.6$ and $0.28<b<0.34$. For CVP trajectories $A$ is lower because the trajectory lies under the jet centerline. Possible uses of this equation are many-fold. For example when devising an experiment involving jets in cross-flow it could be helpful to choose the proper geometrical and physical parameters for a given objective.
Finally, we summarize on Fig. \[fig:allenvelopes\] how the main scalings discussed previously collapse all CVP trajectories. The improvements in collapse brought about by each scalings are clear and quantified by the decrease of the quality factor which is minimum for the scaling based on momentum ratio proposed in this study (Fig. \[fig:allenvelopes\]d). For the range of parameters considered here, equation \[eq:trjct\] allows for a decent approximation of the CVP’s position in space as illustrated on Fig. \[fig:allenvelopes\] d.\
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Whole set of trajectories scaled with different scalings. $y$ axis is normalized to help comparison. a) Trajectories without scaling, b) Trajectories with $rd$ scaling, c) Trajectories with $(\delta/d)^{\beta}$ scaling, d) final scaling together with the trajectory described in equation \[eq:trjct\] (red dotted line).[]{data-label="fig:allenvelopes"}](Fig17a "fig:"){width="50.00000%"} ![Whole set of trajectories scaled with different scalings. $y$ axis is normalized to help comparison. a) Trajectories without scaling, b) Trajectories with $rd$ scaling, c) Trajectories with $(\delta/d)^{\beta}$ scaling, d) final scaling together with the trajectory described in equation \[eq:trjct\] (red dotted line).[]{data-label="fig:allenvelopes"}](Fig17b "fig:"){width="50.00000%"}
a) b)
![Whole set of trajectories scaled with different scalings. $y$ axis is normalized to help comparison. a) Trajectories without scaling, b) Trajectories with $rd$ scaling, c) Trajectories with $(\delta/d)^{\beta}$ scaling, d) final scaling together with the trajectory described in equation \[eq:trjct\] (red dotted line).[]{data-label="fig:allenvelopes"}](Fig17c "fig:"){width="50.00000%"} ![Whole set of trajectories scaled with different scalings. $y$ axis is normalized to help comparison. a) Trajectories without scaling, b) Trajectories with $rd$ scaling, c) Trajectories with $(\delta/d)^{\beta}$ scaling, d) final scaling together with the trajectory described in equation \[eq:trjct\] (red dotted line).[]{data-label="fig:allenvelopes"}](Fig17d "fig:"){width="50.00000%"}
c) d)
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Conclusions {#sec:19}
===========
An experimental study of the CVP trajectories created by a round JICF has been carried out in an hydrodynamic tunnel. 3D3C velocity fields were used to identify the CVP’s and their corresponding outflow regions. The outflow region is used to define and compute CVP trajectories for 22 JICF configurations, including those with a low velocity ratio $r$. The influence of jet velocity and profile as well as cross flow velocity and boundary layer thickness on CVP trajectories is investigated. Parallels are drawn between the behavior of jet and CVP trajectories.\
A more general momentum ratio $r_m$ is introduced as an improvement of the velocity ratio $r$ to take into account the boundary layer and jet exit momentum distributions. The relevance of $r_m$ for jet and CVP trajectories is demonstrated for numerical and experimental data.\
Experimental CVP trajectories and jet trajectories from the literature are scaled and analyzed. The quality of a given scaling is defined and allows for the determination of the relative significance of each parameter (momentum ratio, boundary layer thickness) on trajectories. A new scaling taking into account jet exit momentum distributions, velocity ratio and boundary layer thickness is proposed.\
Finally, a unique trajectory taking into account all relevant parameters is suggested for CVP trajectories.
Acknowledgments
===============
The authors gratefully acknowledge the ADEME (Agence De l’Environnement et la Maitrise de l’Energie) for its financial support, as well the reviewers for their helpful comments.
|
---
abstract: 'We provide an abstract categorical framework that relates the Cuntz semigroups of the C$^*$-algebras $A$ and $A\otimes \mathcal{K}$. This is done through a certain completion of ordered monoids by adding suprema of countable ascending sequences. Our construction is rather explicit and we show it is functorial and unique up to isomorphism. This approach is used in some applications to compute the stabilized Cuntz semigroup of certain C$^*$-algebras.'
address: |
Departament de Matemàtiques\
Universitat Autònoma de Barcelona\
08193 Bellaterra\
Barcelona, Spain.
author:
- Ramon Antoine
- Joan Bosa
- Francesc Perera
title: '**Completions of monoids with applications to the Cuntz semigroup**'
---
Introduction {#introduction .unnumbered}
============
Given a C$^*$-algebra $A$, the structure of the Cuntz semigroup ${\mathrm{W}}(A)$, introduced by J. Cuntz in 1978 ([@cu]), has been intensively studied in recent years notably in relation to the classification program of nuclear C$^*$-algebras. On the one hand because it provides a serious obstruction to the original Elliott Conjecture (see Toms [@tomsann]) but also, on the other hand, because its structure contains large amounts of information coming from the Elliott Invariant, and can, in many cases, be recovered from it (see for example [@bt], [@bpt] and [@dad.toms]). For more details concerning the Cuntz semigroup we refer the reader to [@blac; @kiroramer; @rorfunct].
Coward, Elliott and Ivanescu propose in [@CEI] a modified version of the Cuntz semigroup, $\textrm{Cu}(A)$. They use suitable equivalence classes of countably generated Hilbert modules (which, in the case of stable rank one, amount to isomorphism) to obtain a semigroup strongly related to the classical Cuntz semigroup; it is in fact isomorphic to ${\mathrm{W}}(A\otimes\mathcal{K})$. The advantage of their approach is that they further provide a category Cu for this new semigroup, consisting of positively ordered abelian semigroups with some additional properties of a topological nature. Mainly, $\textrm{Cu}(A)$ is closed under suprema of increasing sequences.
In this paper we enlarge this abstract setting to embrace both ${\mathrm{W}}(A)$ and $\textrm{Cu}(A)$. In Section \[sec:precu\] we build a category of ordered abelian monoids, PreCu, to which the original Cuntz semigroup belongs for a large class of C$^*$-algebras. This category contains Cu as a full subcategory, and differs from it in that monoids need not be closed under suprema of ascending sequences. We subsequently define, in Section \[sec:complecio\], the completion of a monoid in PreCu, in terms of universal properties. Such an object is proved to exist in Section \[sec:existence\] by providing an explicit construction. The completion obtained is thus unique and gives us a functor from the category PreCu to Cu which is a left adjoint of the identity. This, in the particular case of the Cuntz semigroup, allows us to describe $\textrm{Cu}(A)$ as a completion of ${\mathrm{W}}(A)$. This approach is proved to be useful in computing certain Cuntz semigroups as we see in Section \[sec:applications\] where we recover results by Brown and Toms [@bt] in which the stabilized Cuntz semigroup of certain classes of simple, unital, exact and separable C$^*$-algebras is described in terms of K-theory and traces. Another categorical property proved in [@CEI] for Cu is that it admits countable inductive limits in such a way that the functor Cu$(-)$ from the category of C$^*$-algebras (where arbitrary inductive limits always exist) to the category Cu is (countably) continuous. This is not true for the Cuntz semigroup in its classical form. The most basic counterexample is the inductive sequence of C$^*$-algebras defining the compact operators ${\mathcal K}=\lim_{\rightarrow} M_{n}({\mathbb{C}})$. At the level of semigroups, this induces the sequence ${\mathbb{N}}\stackrel{\mathrm{id}}{\to}{\mathbb{N}}\stackrel{\mathrm{id}}{\to}{\mathbb{N}}\to\cdots$, whose limit is not ${\mathrm{W}}(\mathcal{K})={\mathbb{N}}\cup\{\infty\}$. It is natural, however, to expect continuity of the functor ${\mathrm{W}}(-)$ *after* completion, in the case PreCu admits inductive limits. As we will see in Section \[sec:limits\], those do not always exist in PreCu, but this situation can be remedied by defining a smaller category $\mathcal{C}$ sitting between Cu and PreCu, to which ${\mathrm{W}}(A)$ still belongs for a large class of C$^*$-algebras. For this category, we prove that inductive limits always exist and also that we do have continuity of ${\mathrm{W}}(-)$ after completion. This, together with the preceding results, can be applied to compute the stabilized Cuntz semigroup of some direct limits of C$^*$-algebras.
Notation and preliminaries
==========================
Throughout, $M$ will denote a commutative monoid, written additively, with neutral element $0$. We shall also assume that $M$ is equipped with a partial order $\leq$ (compatible with addition) such that $x\geq 0$ for any $x$ in $M$. In particular $\leq$ will extend the algebraic order, that is, if $x+z=y$, then $x\leq y$.
All maps between monoids will be additive ordered maps that preserve $0$. Recall that a monoid map $\varphi\colon M\to N$ is an *order-embedding* if $a\leq b$ whenever $\varphi(a)\leq\varphi(b)$.
Given an increasing sequence $(y_n)$ in $M$, an element $y$ is a *supremum* of $(y_n)$ provided it is a least upper bound. If they exist, suprema of increasing sequences are unique. We shall denote, as is customary, $\sup y_n$ the supremum of the increasing sequence $(y_n)$. Since our considerations might involve different monoids, we will when necessary use $\sup_M$ to mean that the supremum is computed in the monoid $M$.
Let $x$, $y$ be elements in $M$. We write $x\ll y$ if, whenever $(y_n)$ is an increasing sequence in $M$ whose supremum exists in $M$ and $y\leq\sup(y_{n})$, then $x\leq y_{k}$ for some $k$. If $x\ll y$, we shall say that $x$ is *way below* $y$. A sequence $(x_n)$ in $M$ such that $x_n\ll x_{n+1}$ for all $n$ will be called *rapidly increasing*, and an element $x\in M$ such that $x\ll x$ will be called *compact*.
Observe that if $x\leq y$ and $y\ll z$, then $x\ll z$. Likewise, if $x\ll y$ and $y\leq z$, then $x\ll z$.
The previous definition was given by D. Scott in [@scott] for general posets and was first used in connection with the Cuntz subequivalence of positive elements in C$^*$-algebras in [@CEI]. We briefly recall the definitions.
[cf. [@cu]]{} Let $A$ be a C$^*$-algebra, and let $a$, $b\in A_+$. We say that $a$ is Cuntz subequivalent to $b$, in symbols $a\precsim b$, if there is a sequence $(v_n)$ in $A$ such that $a=\lim_nv_nbv_n^*$. We say that $a$ is Cuntz equivalent to $b$ if both $a\precsim b$ and $b\precsim a$ occur. Upon extending this relation to $M_{\infty}(A)_+$, one obtains an abelian semigroup ${\mathrm{W}}(A)=M_{\infty}(A)_+/\!\sim$. Denoting classes by $\langle a\rangle$, the operation and order are given by $$\langle a\rangle+\langle b\rangle=\langle \left(\begin{smallmatrix} a & 0 \\ 0 & b\end{smallmatrix}\right)\rangle=\langle a\oplus b\rangle\,,\,\,
\langle a\rangle\leq \langle b\rangle \text{ if } a\precsim b\,.$$ The semigroup ${\mathrm{W}}(A)$ is referred to as the *Cuntz semigroup*.
For a compact convex set $K$, we shall use $\mathrm{LAff}(K)^{++}$ to refer to those affine, lower semicontinuous functions defined on $K$, with values on $\mathbb{R}^{++}\cup\{\infty\}$. (Here, $\mathbb{R}^{++}$ stands for the strictly positive real numbers.) Note that this is a partially ordered semigroup with the usual pointwise addition and order. We shall denote by $\mathrm{LAff}_b(K)^{++}$ the subsemigroup of $\mathrm{LAff}(K)^{++}$ consisting of those functions that are *bounded*. We shall also use $\mathrm{Aff}(K)^{++}$ to refer to the subsemigroup of $\mathrm{LAff}_b(K)^{++}$, whose elements are those affine, continuous and strictly positive functions defined on $K$.
The category PreCu {#sec:precu}
==================
We start by definining a category of monoids that will center our attention. It is modelled after the category Cu, introduced by Coward, Elliott and Ivanescu in [@CEI], as an abstract setting where the Cuntz semigroup of a stable C$^*$-algebra naturally belongs to. The difference between our definition and theirs is that we do not require all increasing sequences in our monoids to have suprema.
\[def:precu\] Let PreCu be the category defined as follows. Objects of PreCu will be partially ordered abelian monoids $M$ satisfying the properties below:
1. Every element in $M$ is a supremum of a rapidly increasing sequence.
2. The relation $\ll$ and suprema are compatible with addition.
Maps of PreCu are monoid maps preserving
1. suprema of increasing sequences (when they exist), and
2. the relation $\ll$.
In view of Definition \[def:precu\], we may define the category Cu as follows:
\[def:cu\] [(see [@CEI])]{} Let Cu be the full subcategory of PreCu whose objects are those partially ordered abelian monoids (in PreCu) for which every increasing sequence has a supremum.
As proved in [@CEI], for any C$^*$-algebra $A$, the Cuntz semigroup ${\mathrm{W}}(A\otimes\mathcal{K})$ is an object of Cu. It is a natural question to ask whether the Cuntz semigroup ${\mathrm{W}}(A)$ of a C$^*$-algebra $A$ always belongs to the category PreCu.
For a C$^*$-algebra $A$, we shall denote by $\iota\colon A\to A\otimes \mathcal{K}$ the natural inclusion defined by $\iota(a)=a\otimes e_{11}$. This map induces a map at the level of Cuntz semigroups, that we shall also denote by $\iota$, which is an order-embedding. To see this, we identify $A$ with its image inside $A\otimes \mathcal{K}$ and suppose that $\iota(\langle a\rangle)\leq\iota(\langle b\rangle)$. Then $a\precsim b$ in $(A\otimes \mathcal{K})_{+}$, and so $a\precsim b^3$, i.e. $a= \lim v_{n}b^3v^{*}_{n}$ for a sequence $(v_{n})$ in $A\otimes\mathcal{K}$. Right and left multiplication by $a$ implies that $a^3= \lim (av_{n}b)b(bv^{*}_{n}a)$ where now $av_{n}b$ belongs to $A$ since the latter is a hereditary C$^*$-subalgebra of $A\otimes\mathcal{K}$. Therefore $\langle a^{3}\rangle
\leq\langle b\rangle $ in ${\mathrm{W}}(A)$, and so $\langle a\rangle \leq
\langle b\rangle$ in ${\mathrm{W}}(A)$.
We have that any $a\in M_{\infty}(A)_+$ is the limit, in norm, of the increasing sequence $(a-1/n)_+$, so that indeed $\langle a\rangle =\sup\langle (a-1/n)_+\rangle$. But, while $\langle (a-\epsilon)_+\rangle\ll \langle a\rangle$ in ${\mathrm{W}}(A\otimes\mathcal K)$, it is not obvious this is the case anymore in ${\mathrm{W}}(A)$. We have the following
\[lem:waybelowvssuprema\] Let $A$ be a C$^*$-algebra. The following conditions are equivalent:
1. $\langle(a-\epsilon)_+ \rangle\ll\langle a\rangle$ for any $\epsilon>0$ and any $a\in M_{\infty}(A)_+$.
2. $\sup_{{\mathrm{W}}(A)}x_n=\sup_{{\mathrm{W}}(A\otimes\mathcal K)}x_n$ whenever $(x_n)$ is an increasing sequence in ${\mathrm{W}}(A)$ with supremum in ${\mathrm{W}}(A)$.
\(i) $\Rightarrow$ (ii): Let $(x_n)$ be an increasing sequence in ${\mathrm{W}}(A)$ with $x=\sup_{{\mathrm{W}}(A)}x_n$ and $y=\sup_{{\mathrm{W}}(A\otimes \mathcal K)}x_n$. Clearly $y\leq x$. If we write now $x=\langle a\rangle$ for some $a\in M_{\infty}(A)_+$, our assumption implies that, for any $n$, there exists $m$ with $\langle (a-1/n)_+\rangle\leq x_m$. Thus $\langle (a-1/n)_+\rangle\leq y$ for every $n$, whence $x=\langle a\rangle\leq y$.
\(ii) $\Rightarrow$ (i): Let $a\in M_{\infty}(A)_+$, and suppose that $x_n$ is an increasing sequence in ${\mathrm{W}}(A)$ with supremum in ${\mathrm{W}}(A)$. The assumption implies that this agrees with the supremum in ${\mathrm{W}}(A\otimes\mathcal K)$. If $\langle a\rangle\leq \sup x_n$, then as $\langle(a-\epsilon)_+ \rangle\ll\langle a\rangle$ in ${\mathrm{W}}(A\otimes\mathcal K)$ (by the results in [@CEI]), we have $\langle(a-\epsilon)_+ \rangle\leq x_n$ for some $n$ in ${\mathrm{W}}(A\otimes\mathcal K)$, hence also in ${\mathrm{W}}(A)$.
\[dfn:hereditary\] Let $M$ and $N$ be partially ordered monoids. An order-embedding $f\colon M\to N$ will be called *hereditary* if, whenever $x\in N$ and $y\in f(M)$ satisfy $x\leq y$, it follows that $x\in f(M)$.
\[lem:hereditary\] Let $N$ be an object of Cu and $M$ be a partially ordered monoid. Let $f\colon M\to N$ be a hereditary map. Then, for any increasing sequence $(x_{n})$ in $M$ with supremum $x$ in $M$, we have $f(x)=\sup(f(x_{n}))$
Since $x_{n}\leq x$ for all $n$, it follows that $f(x_{n})\leq f(x)$ for all $n$. Therefore $\sup(f(x_{n}))\leq f(x)$. Our assumption on $f$ now implies that $\sup(f(x_{n}))=f(y)$ for some $y\in M$. Using that $f$ is an order-embedding we obtain that $x_n\leq y\leq x$ for all $n$, so $y=x$.
Let $A$ be a C$^*$-algebra. We will say that ${\mathrm{W}}(A)$ is *hereditary* if the map $\iota\colon {\mathrm{W}}(A)\to {\mathrm{W}}(A\otimes\mathcal{K})$ is hereditary.
\[lem:hervssup\] Let $A$ be a C$^*$-algebra such that ${\mathrm{W}}(A)$ is hereditary. Then, given $a\in M_{\infty}(A)_+$ and $\epsilon>0$, we have $\langle (a-\epsilon)_+\rangle\ll\langle a\rangle$ in ${\mathrm{W}}(A)$.
Since ${\mathrm{W}}(A\otimes\mathcal K)$ is an object of Cu, and ${\mathrm{W}}(A)$ is hereditary, we may apply Lemma \[lem:hereditary\] to conclude that suprema of increasing sequences in ${\mathrm{W}}(A)$ agree, when they exist, with suprema in ${\mathrm{W}}(A\otimes\mathcal K)$. The conclusion now follows from Lemma \[lem:waybelowvssuprema\].
\[prop:wainprecu\] Let $A$ be a C$^*$-algebra such that ${\mathrm{W}}(A)$ is hereditary. Then ${\mathrm{W}}(A)$ is an object of PreCu and the map $\iota\colon{\mathrm{W}}(A)\to{\mathrm{W}}(A\otimes\mathcal{K})$ is an order-embedding in PreCu.
By Lemma \[lem:hereditary\], coupled with Lemma \[lem:waybelowvssuprema\], we have that every element $\langle a\rangle$ in ${\mathrm{W}}(A)$ is the supremum of the rapidly increasing sequence $\langle (a-1/n)_+\rangle$.
Assume now that $x\ll y$ and $z\ll t$ in ${\mathrm{W}}(A)$, and write $y=\langle a\rangle$ and $t=\langle b\rangle$. It follows that there is $n$ with $x\leq \langle (a-1/n)_+\rangle$ and $z\leq \langle (b-1/n)_+\rangle$. Therefore, since $$(a-1/n)_+\oplus (b-1/n)_+\sim (a\oplus b-1/n)_+\,,$$ we get $$x+z\leq \langle (a\oplus b-1/n)_+\rangle\ll \langle a\rangle+\langle b\rangle\,.$$
Finally, suppose that $(x_n)$ and $(y_n)$ are increasing sequences in ${\mathrm{W}}(A)$ with suprema $x$ and $y$ respectively. We are to show that $x+y$ is the supremum of $(x_n+y_n)$. Using that ${\mathrm{W}}(A)$ is hereditary, one sees that $(x_n+y_n)$ also has a supremum in ${\mathrm{W}}(A)$. And using that suprema and addition are compatible in ${\mathrm{W}}(A\otimes\mathcal{K})$, we see, using Lemma \[lem:waybelowvssuprema\], that $$\sup_{{\mathrm{W}}(A)}x_n+\sup_{{\mathrm{W}}(A)}y_n=\sup_{{\mathrm{W}}(A\otimes\mathcal{K})}x_n+\sup_{{\mathrm{W}}(A\otimes\mathcal{K})}y_n=\sup_{{\mathrm{W}}(A\otimes\mathcal{K})}(x_n+y_n)=\sup_{{\mathrm{W}}(A)}(x_n+y_n)\,.$$ We have already observed that $\iota$ is an order-embedding. Combining again Lemma \[lem:hereditary\] and Lemma \[lem:waybelowvssuprema\], we see that it preserves suprema of increasing sequences.
Suppose now that $\langle a\rangle\ll\langle b\rangle$. Since $\langle b\rangle =\sup(\langle
(b-\varepsilon)_{+}\rangle)$ in ${\mathrm{W}}(A)$, there exists $\varepsilon >0$ such that $\langle a\rangle \leq \langle (b-\varepsilon)_{+}\rangle$. Applying $\iota$, $$\iota(\langle a\rangle)\leq\iota(\langle (b-\varepsilon)_{+}\rangle)=\langle( b-\varepsilon)_{+})\rangle\ll\langle b\rangle=\iota(\langle b\rangle)\text{ whence }\iota(\langle a\rangle)\ll\iota(\langle b\rangle)\,,$$ so that $\iota$ is a map in PreCu.
\[rem:Anohereditaria\] There are examples of C$^*$-algebras $A$ for which ${\mathrm{W}}(A)$ is not hereditary ([@betal]). The Cuntz semigroup of these algebras contains bounded ascending sequences with no least upper bound. This is needed for ${\mathrm{W}}(A)$ to be hereditary (see Proposition \[prop:heriffC\]), but not for ${\mathrm{W}}(A)$ to be in PreCu.
\[lem:sr1her\] Let $A$ be a C$^*$-algebra with $\mathrm{sr}(A)=1$. Then ${\mathrm{W}}(A)$ is hereditary.
Let $a\in A\otimes\mathcal{K}_+$ and $b\in M_{\infty}(A)_+$, and assume that $a\lesssim b$. We are to show that there exists $c\in M_{\infty}(A)_+$ such that $c\sim a$.
Using that $A\otimes\mathcal{K}$ is the completion of $M_{\infty}(A)$, we obtain that if $a\in (A\otimes\mathcal{K})_{+}$, there exists a sequence $(a_{n})$ belonging to $M_{\infty}(A)_{+}$ such that $a=\lim (a_{n})$, and we may assume that $\|a-a_{n}\|\leq 1/n$. By Lemma 2.2 in [@kiror], there are contractions $d_n$ in $A\otimes\mathcal{K}$ such that $(a-1/n)_{+}=d_{n}a_{n}d_{n}^{*}$. Thus $$(a-1/n)_{+}=d_{n}a_{n}d_{n}^{*}\sim a^{1/2}_{n}d^{*}_{n}d_{n}a^{1/2}_{n}\,,$$ and $b_n:=a_{n}^{1/2}d^{*}_{n}d_{n}a_{n}^{1/2}\in M_{\infty}(A)_{+}$. This implies that $\langle a\rangle=\sup \langle b_n\rangle$ in ${\mathrm{W}}(A\otimes\mathcal{K})$ and that the sequence $(\langle b_n\rangle)$ is rapidly increasing in $W(A\otimes\mathcal{K})$.
Notice that the sequence $(\langle b_n\rangle)$ is bounded above in ${\mathrm{W}}(A)$ by $\langle b\rangle$. Therefore, it also has a supremum $\langle c\rangle$ in ${\mathrm{W}}(A)$, by [@bpt Lemma 4.3]. The arguments in [@bpt] show that for each $n$ there exists $m$ and $\delta_n>0$ with $\delta_n\to 0$ such that $(c-1/n)_{+}\precsim (b_{m}-\delta_{n})_{+}$. This implies then that $$(c-1/n)_{+}\precsim (b_{m}-\delta_{n})_{+}\precsim b_{m}\precsim a$$ in $A\otimes\mathcal{K}$, whence $c\precsim a$.
On the other hand, since clearly $b_{n}\precsim c$ for all $n$, and $\langle a\rangle$ is the supremum in ${\mathrm{W}}(A\otimes\mathcal{K})$ of $\langle b_n\rangle$, we see that $a\precsim c$. Thus $c\sim a$.
For a unital C$^*$-algebra $A$, we denote as usual by ${\mathrm{T}}(A)$ the simplex of normalized traces. Given a trace $\tau\in{\mathrm{T}}(A)$ and $a\in M_{\infty}(A)_+$, we may construct $d_{\tau}(a)=\lim_n\tau(a^{1/n})$. It turns out that $$d_{\tau}\colon M_{\infty}(A)_+\to\mathbb{R}^+\,,\, a\mapsto d_{\tau}(a)$$ is lower semicontinuous and does not depend on the Cuntz class of $a$. Thus, it defines a state on ${\mathrm{W}}(A)$, termed a *lower semicontinuous dimension function*. Let us denote by $LDF(A)$ the set of all lower semicontinuous dimension functions on $A$. Observe also that the function $\hat{a}\colon {\mathrm{T}}(A)\to\mathbb{R^+}$ defined by $\hat{a}=(\tau)=d_{\tau}(a)$ is an element of $\mathrm{LAff}({\mathrm{T}}(A))^+$.
In the case that $A$ is moreover simple, we say that $A$ has *strict comparison* if the order in ${\mathrm{W}}(A)$ is determined by lower semicontinuous dimension functions. Namely, if $d_{\tau}(a)<d_{\tau}(b)$ for all $\tau\in {\mathrm{T}}(A)$, it follows that $a\precsim b$.
\[def:rc\] [([@tomsjfa; @tomsplms])]{} A unital C$^*$-algebra $A$ has *$r$-comparison* if whenever $a$, $b\in M_{\infty}(A)_+$ satisfy $$s(\langle a\rangle)+r<s(\langle b\rangle)\,,\text{ for every $s\in\mathrm{LDF}(A)$}\,,$$ then $\langle a\rangle\leq\langle b\rangle$. The *radius of comparison* of $A$ is $$\mathrm{rc}(A)=\inf\{r\geq 0\mid A \text{ has $r$-comparison}\}\,,$$ which is understood to be $\infty$ if the infimum does not exist.
\[thm:rcher\] [([@betal])]{} Let $A$ be a unital C$^*$-algebra with finite radius of comparison. Then ${\mathrm{W}}(A)$ is hereditary.
Abstract completions {#sec:complecio}
====================
The purpose of this section is to define the notion of completion for an object of PreCu and establish the universal property it satisfies, from which uniqueness of this object will follow. Existence will be proved by constructing a concrete completion and will be carried out in the next section. We also show that the completion process induces a left adjoint functor of the identity.
\[dfn:completion\] Let $M$ be an object of PreCu. We say that a pair $(N,\iota)$ is a *completion* of $M$ if
1. $N$ is an object of Cu,
2. $\iota\colon M\to N$ is an order-embedding in PreCu, and
3. for any $x\in N$, there is a rapidly increasing sequence $(x_{n})$ in $M$ such that $x=\sup\iota(x_{n})$.
\[orderembedding\] Let $M$, $N$ be objects of PreCu and let $\alpha\colon M\to N$ be a map in PreCu. The following conditions are equivalent:
1. $\alpha$ is an order-embedding.
2. $\alpha (x)\ll\alpha(y)$ if and only if $x\ll y$.
\(i) $\Rightarrow$ (ii): If $x\ll y$, then as $\alpha$ is a map in PreCu, it follows that $\alpha (x)\ll\alpha(y)$. Conversely, suppose that $\alpha(x)\ll\alpha(y)$. If $y\leq \sup(z_{n})$, then $\alpha(y)\leq \alpha(\sup(z_n))=\sup\alpha(z_n)$, so that there is $n$ with $\alpha(x)\leq \alpha(z_{n})$. Since $\alpha$ is order-embedding, this implies $x\leq z_{n}$ .
\(ii) $\Rightarrow$ (i): Suppose now that $\alpha(x)\leq\alpha(y)$. Write $x=\sup(x_{n})$, where $x_{n}\ll x_{n+1}$. Since $\alpha$ is a map in PreCu, $\alpha(x)=\sup(\alpha(x_{n}))$ and $\alpha(x_{n})\ll\alpha(x_{n+1})$. Therefore, $\alpha(x_{n})\ll\alpha(x_{n+1})\leq\alpha(y)$. Using the hyphotesis we obtain that $x_{n}\ll y$ for all $n$, and then $x\leq y$.
\[universality\] Let $M$ be an object of PreCu, $P$ an object of $Cu$ and $\alpha\colon M\to P$ a map in PreCu. Then, if $(N,\iota)$ is a completion of $M$, there exists a unique map $\beta\colon N\to P$ in Cu such that $\beta\circ\iota=\alpha$. Moreover, if $\alpha$ is an order-embedding then so is $\beta$.
Since we can write any $x\in N$ as $x=\sup\iota (x_{n})$, where $(x_{n})$ is a rapidly increasing sequence in $M$, define $\beta\colon N\to P$ by $\beta(x)=\sup(\alpha(x_{n}))$. We need to check that $\beta$ is well defined.
Suppose that $x=\sup\iota(x_{n})=\sup \iota(y_{m})$ where $(x_{n})$ and $(y_{m})$ are both rapidly increasing sequences in $M$. Then, for every $n$, there exist $m$ and $k$ such that $$x_{n}\ll y_{m}\ll x_{k}\,$$ Since $\alpha$ is a map in PreCu, we obtain that $$\alpha(x_{n})\ll \alpha(y_{m})\ll \alpha(x_{k})\,.$$ Therefore $\sup \alpha(x_{n})=\sup \alpha(y_{n})$, whence $\beta$ is well-defined.
That $\beta$ is additive and preserves the identity element follows easily from the fact that $\alpha$ belongs to PreCu.
Next, let $x\leq y$ in $N$. Write $x=\sup(\iota(x_{n}))$ and $y=\sup(\iota(y_{n}))$, where $(x_n)$ and $(y_n)$ are rapidly increasing. As before we obtain that for every $n$, there is $m$ with $x_{n}\ll y_{m}$, so then $$\alpha(x_{n})\ll \alpha(y_{m})\leq \sup\alpha(y_k)=\beta(y)\,.$$ Therefore $\beta(x)=\sup(\alpha(x_{n}))\leq\beta(y)$, hence $\beta$ is order-preserving.
If now $x=\sup (x_n)\in M$, with $(x_{n})$ rapidly increasing, apply $\iota$ followed by $\beta$ so that $$\beta(\iota(x))=\beta(\iota(\sup(x_{n})))=\beta(\sup(\iota(x_{n})))=\sup(\alpha(x_{n}))=\alpha(x)\,,$$ which shows $\beta\circ\iota=\alpha$.
Suppose that $x\ll y$, for elements $x$, $y$ in $N$. Write $y=\sup\iota(y_n)$, where $(y_n)$ is a rapidly increasing sequence in $M$. There exists then $n$ such that $x\leq \iota (y_n)$, and since $y_n\ll y_{n+1}$, we may apply $\alpha$ to obtain $\alpha(y_n)\ll \alpha(y_{n+1})$. Now $$\beta(x)\leq\beta(\iota(y_n))=\alpha(y_n)\ll\alpha(y_{n+1})=\beta(\iota(y_{n+1}))\leq\beta(y)\,,$$ hence $\beta (x)\ll\beta (y)$.
It remains to be shown that $\beta$ preserves suprema. Let $\{x_{n}\}$ be an increasing sequence in $N$ and let $x=\sup(x_{n})$. As $\beta$ is order-preserving we readily get $\beta(x_{n})\leq \beta(x)$ for all $n$. Therefore $\sup(\beta(x_{n}))\leq \beta(x)$. Since we can also write $x=\sup\iota(y_{n})$ (for a rapidly increasing sequence $(y_n)$) it follows that, given $n$, there is $m$ with $\iota(y_{n})\leq x_{m}$. Apply $\beta$ to get $\alpha(y_n)=(\beta\circ\iota)(y_{n})\leq \beta(x_{m})\leq \sup(\beta(x_{k}))$, whence $$\beta(x)=\sup(\alpha(y_{n}))\leq \beta(x_{m})\leq \sup(\beta(x_{k}))\,.$$
We now prove that $\beta$ is unique. To this end, assume $\beta'\colon N\to P$ is another map in $Cu$ such that $\beta'\circ\iota=\alpha=\beta\circ\iota$. Let $x=\sup(\iota(x_{n}))$ be an element in $N$. Then $$\beta'(x)=\beta'(\sup\iota(x_{n}))=\sup((\beta'\circ\iota)(x_{n}))=\sup((\beta\circ\iota)(x_{n}))=
\beta(\sup\iota(x_{n}))=\beta(x)\,.$$ Assume finally that $\alpha$ is an order-embedding and suppose that $\beta(x)\leq\beta(y)$. With $x=\sup\iota(x_{n})$, and $y=\sup\iota(y_{n})$ for rapidly increasing sequences $(x_n)$ and $(y_n)$, this implies that, for any $n$, $$\alpha(x_n)\ll\alpha(x_{n+1})\leq\sup\alpha(x_n)\leq\sup\alpha(y_n)\,.$$ There is then $m$ depending on $n$ with $\alpha(x_n)\leq\alpha(y_m)$, that is, $x_n\leq y_m$ (as $\alpha$ is an order-embedding). Thus $\iota(x_{n})\leq \iota(y_{m})\leq y$, and so $x\leq y$. This shows that $\beta$ is an order-embedding.
A standard argument yields the following.
Let $M$ be an object in PreCu. If there is a completion of $M$, then it is unique (up to order-isomorphism).
In view of the corollary above, given an object $M$ of PreCu, we shall write $(\overline{M},\iota)$ to refer to its (unique) completion, in case it exists.
If every object of PreCu has a completion $(\overline{M},\iota)$, then the map $\iota\colon M\to\overline{M}$ induces a covariant functor $C\colon PreCu\to Cu$, which is a left adjoint of the identity, i.e., if $i\colon Cu\to PreCu$ is the inclusion functor, then $C\circ i\sim id_{Cu}$ (where $\sim$ denotes natural equivalence).
Put $$\begin{tabular}{c c c c}
$C : $ & PreCu &$\longrightarrow$ & Cu \\
& $M$& $\mapsto$ & $\overline{M}$ \\
\end{tabular}\\\,.$$ Let $\sigma\colon M\to N$ be a map in PreCu, and let $(\overline{M},\iota)$ and $(\overline{N},\iota_{1})$ be the completions of $M$ and $N$ respectively.
Now use that the map $\iota_{1}\circ\sigma:M\to\overline{N}$ belongs to PreCu and Theorem \[universality\], so that there is a unique morphism $\overline{\sigma}\colon\overline{M}\to\overline{N}$ in Cu such that $\overline{\sigma}\circ\iota=\iota_{1}\circ\sigma$. In fact, if $x=\sup(\iota(x_{n}))$ is an element in $\overline{M}$ (with $(x_n)$ rapidly increasing in $M$), then $$\overline{\sigma}(x)=\sup((\iota_{1}\circ\sigma)(x_{n}))\,.$$
It is a simple matter to check that $\overline\sigma$ satisfies $\overline{\mathrm{id}_M}=\mathrm{id}_{\overline{M}}$ and that $\overline{\sigma\circ\tau}=\overline{\sigma}\circ\overline{\tau}$.
Finally, notice that if $M$ is already an object of Cu, then $(M,\mathrm{id})$ is a completion, hence unique, whence $C\circ i(M)\cong M$.
It follows from the result above that, if every object in PreCu admits a completion and $M$ is in PreCu, then $\overline{\overline M}\cong \overline M$.
Existence of completions in PreCu {#sec:existence}
=================================
In order to construct the completion of an object of PreCu, we basically need to add suprema of every ascending sequence of the given object. This is best captured by using intervals – we recall the definitions below.
Let $M$ be a partially ordered abelian monoid. An *interval* in $M$ is a nonempty subset $I\subseteq M$ which is upward directed and order-hereditary (i.e. if $x\leq y$ and $y\in I$, then $x\in I$). Denote $\Lambda(M)$ the set of all intervals in $M$. Note that $\Lambda(M)$ is equipped with a natural abelian monoid structure, namely if $I,J\in\Lambda(M)$, then $$I+J=\{z\in M\mid z\leq x+y \text{ for some } x\in I, y\in J\}\,.$$
We shall be exclusively concerned with *countably generated* intervals. Those are elements $I$ in $\Lambda (M)$ that have a countable cofinal subset, that is, a countable subset $X$ – that may always be assumed to be upwards directed – such that $I=\{x\in M\mid x\leq y \text{ for some }y\in X\}$. In that case, there is an increasing sequence $(x_n)$ in $M$ with $I=\{x\in M\mid x\leq x_n\text{ for some }n \}$. We will denote by $\Lambda_{\sigma}(M)$ the monoid of countably generated intervals over M. We shall sometimes refer to a rapidly increasing cofinal subset $X$ for an interval $I$, meaning that for any $x$, $y\in X$, there is $z\in X$ with $x$, $y\ll z$. Note that an interval $I$ with rapidly increasing countable cofinal subset may be then written as $I=\{x\in M\mid x\leq x_n\text{ for some }n\}$ where now $(x_n)$ is a rapidly increasing sequence.
For intervals $I$ and $J$ in $\Lambda_{\sigma}(M)$, write $I\precsim J$ if given $x$ in $I$ and $z\ll x$, there exists $y\in J$ such that $z\ll y$. It is clear that this is equivalent to taking $x$ and $y$ above in countable cofinal subsets $X$ and $Y$ for $I$ and $J$ respectively. The relation $\precsim$ is easily seen to be reflexive and transitive, and induces a congruence on $\Lambda_{\sigma}(M)$ by defining $I\sim J$ if $I\precsim J$ and $J\precsim I$.
Put $\overline{M}=\Lambda_{\sigma}(M)/\sim$, and denote the elements of $\overline{M}$ by $[I]$.
\[propo\] Let $M$ be an object in PreCu. Then $\overline{M}$ is a partially ordered monoid.
Define $[I]\leq [J]$ if $I\precsim J$ in $\Lambda_{\sigma}(M)$, which is clearly well-defined, so that $\overline{M}$ becomes a partially ordered set.
Now, let $I$, $J$, $K$, $L\in\Lambda_{\sigma}(M)$ and assume that $[I]\leq [J]$ and $[K]\leq[L]$ in $\overline{M}$. Choose increasing countable cofinal sets $\{x_n\}$, $\{y_n\}$, $\{z_n\}$ and $\{t_n\}$ for $I$, $J$, $K$ and $L$ respectively. Write each term in the above sequences as a supremum of rapidly increasing sequences, namely: $x_n=\sup_m x_n^m$, $y_n=\sup_m y_n^m$, $z_n=\sup_m z_n^m$ and $t_n=\sup_m t_n^m$.
If $x\ll x_{n}+z_{n}$, then there exists $m$ such that $x\leq x^{m}_{n}+z^{m}_{n}\ll x^{m+1}_{n}+z^{m+1}_{n}$. Since $x^{m+1}_{n}\ll x_{n}$ and $z^{m+1}_{n}\ll z_{n}$, we see that $x^{m+1}_{n}\ll y_{p}$ and $z^{m+1}_{n}\ll t_{q}$ for some $p$, $q$. As addition and $\ll$ are compatible, it follows that $x \leq x^{m}_{n}+z^{m}_{n}\ll y_{t}+t_{k}$ where $k=\max\{p,q\}$. Therefore $[I+K]\leq [J+L]$, and we may define $[I]+[J]=[I+J]$. This operation is clearly compatible with the order.
Let $M$ be an object of PreCu. For any $x\in M$, put $I(x)=[0,x]=\{ y\in M\mid y\leq x\}$, which is clearly a countably generated interval. Next, define a map $\iota$ as follows: $$\begin{tabular}{ c c c }
$\iota\colon M$ & $\longrightarrow$ & $\overline{M}$\\
$\,\,\, x$ & $\mapsto$ & $[I(x)]$\,. \\
\end{tabular}$$ Observe that $\iota$ is an order-embedding. Indeed, $\iota$ is additive and preserves the order. If now $[I(x)]\leq [I(y)]$ in $\overline{M}$, then $I(x)\precsim I(y)$ in $\Lambda_{\sigma}(M)$. Write $x=\sup(x_{n})$, where $(x_{n})$ is rapidly increasing. Since $x_{n}\ll x$ for all $n$ we have $x_{n}\leq y$ for all $n$, whence $x\leq y$
\[lem:rapincr\] Let $M$ be an object of PreCu. Then every element in $\Lambda_{\sigma}(M)$ is equivalent to an interval with a rapidly increasing countable cofinal subset.
Let $(x_n)$ be an increasing countable cofinal subset for $I\in \Lambda_{\sigma}(M)$. For each $n$ choose a rapidly increasing sequence $(x_{n}^m)$ with $x_{n}=\sup(x^{m}_{n})$, and then consider the rapidly increasing cofinal subset $(x_n^m)$ now varying $n$ and $m$. Let $J$ be the interval generated by $(x_n^m)$. It is clear that $J\subseteq I$ and therefore $J\precsim I$. Consider $x\in M$ such that $x\ll x_n$. Since $x_n=\sup x_n^m$, there exists $k$ such that $x\ll x_n^k$. Therefore $I\precsim J$.
\[suprem\] Let $M$ be an object in PreCu. Then, every increasing sequence in $\overline{M}$ has a supremum in $\overline{M}$. More concretely, if $([I_n])$ is an increasing sequence in $\overline{M}$, and $X_n$ is a rapidly increasing countable cofinal subset for each $I_n$, then $X=\cup_n X_n$ is a countable cofinal subset for $\sup([I_n])$.
Let $([I_{n}])$ be an increasing sequence in $\overline{M}$, and let $X_n$ be rapidly increasing countable cofinal subsets for each $I_n$ (this is no loss of generality, in view of Lemma \[lem:rapincr\], to assume that $X_n$ is rapidly increasing for every $n$). Let $X=\cup_n X_n$ and put $I=\{x\in M\mid x\leq y\text{ for some }y\in X\}$. We are to show that $I\in\Lambda_{\sigma}(M)$ and that $[I]=\sup([I_n])$.
Let $x_1$, $x_2\in I$. Then there are $n$ and $m$ and $y_1\in X_n$ and $y_2\in X_m$ with $x_i\leq y_i$. We may assume that $n\leq m$. Find $y$ in $X_n$ such that $y_1\ll y$, and since $I_n\precsim I_m$, there is $z$ in $X_m$ with $y_1\ll z$. Since $y_2\in X_m$, there exists $w$ in $X_m$ with $z$, $y_2\ll w$. Then $x_i\leq y_i\ll w$ and $w\in X$. This shows that $I$ is upwards directed. As $I$ is clearly hereditary and $X$ is countable (and nonempty), we conclude that $I\in\Lambda_{\sigma}(M)$.
That $I_n\precsim I$ for all $n$ is clear. Suppose that $J\in \Lambda_{\sigma}(M)$ satisfies $I_n\precsim J$. Choose a countable cofinal subset $Y$ for $J$, and let $x\in X$ and $z\ll x$. There is $n$ with $x\in X_n$, so there is $y\in Y$ with $z\ll y$, and this implies $I\precsim J$. Therefore $[I]=\sup [I_n]$.
\[lem:natural\] Let $M$ be an object of PreCu. If $x\ll y$ in $M$, then $[I(x)]\ll [I(y)]$ in $\overline M$.
Let $[I_n]$ be increasing in $\overline{M}$. Choose a rapidly increasing countable cofinal set $X_n$ for $I_n$ and put $X=\cup_n X_n$. If we let $I=\{x\in M \mid x\leq y\text{ for some }y\in X\}$, we know that $[I]=\sup[I_n]$.
Suppose that $[I(y)]\leq [I]$. Since $x\ll y$, there is $z$ in $X$ with $x\ll z$, and $z\in X_n$ for some $n$. Therefore, there is $n$ such that $w\ll z$ whenever $w\ll x$, which shows that $[I(x)]\leq [I_n]$.
\[rapidament\] Let $M$ be an object in PreCu. Then, every element in $\overline{M}$ is the supremum of a rapidly increasing sequence coming from $M$.
Let $[I]\in\overline{M}$. We may assume that there is a rapidly increasing sequence $(x_n)$ such that $I=\{x\in M\mid x\leq x_n\text{ for some }n\}$. Consider the sequence $([I^n])$ in $\overline{M}$, where $$I^{n}=\{y\in M\mid y\leq x_{n}\}\,.$$ It follows from Lemma \[lem:natural\] that $[I^n]\ll [I^{n+1}]$. Thus, to prove the desired result, it suffices to show that $[I]=\sup([I^{n}])$. It is clear from the definition that $[I^{n}]\leq [I]$ for all $n$. Suppose that $J\in \Lambda_{\sigma}(M)$ has an increasing countable cofinal subset $(y_{m})$ and that $[I^n]\leq [J]$ for all $n$. Given $n$ and $x\ll x_{n}$, we can find $m$ such that $x\ll y_{m}$. Therefore $[I]\leq [J]$, obtaining that $[I]$ is the smallest upper bound of the sequence.
\[rem:rapidlyincreasing\] [The proof in Proposition \[rapidament\] shows that, given $I$ and $J\in\Lambda_{\sigma}(M)$ with respective rapidly increasing countable cofinal subsets $(x_n)$ and $(y_n)$, then, letting $I^n=[0,x_n]$ and $J^n=[0,y_n]$, we have $[I^n+J^n]$ is rapidly increasing and $\sup ([I^n]+[J^n])=I+J$.]{}
Let $M$ be an object of PreCu. Then suprema and $\ll$ are compatible with addition in $\overline{M}$.
Let $([I_{n}])$, $([J_{n}])$ be two increasing sequences in $\overline{M}$, and let $$[I]=\sup([I_{n}])\text{ and }[J]=\sup([J_{n}])\,.$$ Choose rapidly increasing sequences $([I^n])$ and $([J^n])$ such that $[I+J]=\sup ([I^n+J^n])$, and $[I]=\sup [I^n]$, $[J]=\sup [J^n]$ (see Remark \[rem:rapidlyincreasing\]). We can then find $m$ with $[I^n]\leq [I_m]$ and $[J^n]\leq [J_m]$. Thus: $$[I^{n}+J^{n}]=[I^{n}]+[J^{n}]\leq [I_{m}]+[J_{m}]\leq [I]+[J]\,,$$ whence $$[I]+[J]=\sup ([I^n]+[J^n])\leq\sup([I_{n}+J_{n}])\leq [I]+[J]\,.$$
To prove that $\ll$ and addition are compatible we have to check that if $[I]\ll [J]$ and $[K]\ll [L]$ then $[I+K]\ll [J+L]$. If we write $[J]=\sup([J^{n}])$ and $[L]=\sup([L^{n}])$, where $J^n$ and $L^n$ are constructed as in Proposition \[rapidament\] (see also Remark \[rem:rapidlyincreasing\]), so that the corresponding rapidly increasing sequence for $[J+L]$ will be $[J^n+L^n]$. Using that $[I]\ll [J]$ and $[K]\ll [L]$, we find $m$ with $[I]\leq [J^{m}]$ and $[K]\leq[L^{m}]$. Therefore $$[I]+[K]\leq [J^{m}+L^{m}]\ll [J^{m+1}+L^{m+1}]\leq [J]+[L]\,,$$ as desired.
Collecting the results above we, we obtain:
\[thm:completion\] Let $M$ be an object of PreCu. Then $\overline{M}$ is an object of Cu, $$\begin{tabular}{ c c c }
$\iota\colon M$ & $\longrightarrow$ & $\overline{M}$\\
$\,\,\, x$ & $\mapsto$ & $[I(x)]$, \\
\end{tabular}$$ (where $I(x)=[0,x]$) is an order-embedding in PreCu and $(\overline{M},\iota)$ is the completion of $M$.
It is clear by the preceding results that $\overline M$ is an object of Cu, and we already know that $\iota$ is an order-embedding that preserves $\ll$ (see Lemma \[lem:natural\]).
Let $(x_n)$ be an increasing sequence in $M$ with $x=\sup(x_{n})$ in $M$, and let $I=\{x\in M\mid x\leq x_n\text{ for some }n\}$. We first show that $\iota(x)=[I]$. It is clear that $I\precsim I(x)$. Write $x=\sup z_n$, where $(z_n)$ is rapidly increasing in $M$. Thus, if $y\ll x$, then there is $n$ such that $y\ll z_n$, so that $y\leq x_m$ for some $m$. This shows that $I(x)\precsim I$.
Now, let $J\in\Lambda_{\sigma}(M)$ and choose an increasing countable cofinal subset $(y_n)$ for $J$. If $[0,x_n]\precsim J$ for all $n$, we get that for any $n$ and $x\ll x_n$, there is $m$ with $x\ll y_m$, so that $I\precsim J$. It follows that $\iota(x)=\sup\iota(x_n)$.
This, together with Proposition \[rapidament\], shows that $(\overline{M},\iota)$ is the completion of $M$.
\[cor:completion\] Let $M$ be an object of PreCu such that every element is compact. Then $\overline{M}\cong\Lambda_{\sigma}(M)$.
This follows directly from Theorem \[thm:completion\] and the fact that, if every element in $M$ is compact, the relation $\precsim$ defined in $\Lambda_{\sigma}(M)$ reduces to inclusion.
Some applications {#sec:applications}
=================
Our first application of Theorem \[thm:completion\] relates the Cuntz semigroup of $A$ with that of $A\otimes\mathcal K$ in the case that ${\mathrm{W}}(A)$ is hereditary.
\[thm:wacompletion\] Let $A$ be a C$^*$-algebra with ${\mathrm{W}}(A)$ hereditary. Then the pair ${\mathrm{W}}(A)$ is in PreCu and $({\mathrm{W}}(A\otimes\mathcal{K}),\iota)$ is the completion of ${\mathrm{W}}(A)$.
We need to verify that ${\mathrm{W}}(A)$ and $({\mathrm{W}}(A\otimes\mathcal{K}),\iota)$ satisfy the conditions of Definition \[dfn:completion\], and then invoke Theorem \[thm:completion\].
We have already proved in Proposition \[prop:wainprecu\] that, under our assumptions, ${\mathrm{W}}(A)$ is an object of PreCu and that $\iota$ is an order-embedding in PreCu.
Also, as in the proof of Lemma \[lem:sr1her\], if $a\in (A\otimes\mathcal{K})_{+}$, we obtain a sequence $\langle b_n\rangle$ with $b_n \in M_{\infty}(A)$ which is rapidly increasing (both in ${\mathrm{W}}(A)$ and ${\mathrm{W}}(A\otimes\mathcal{K})$ since ${\mathrm{W}}(A)$ is hereditary) and such that $\langle a\rangle=\sup \langle b_n\rangle$ in ${\mathrm{W}}(A\otimes\mathcal{K})$.
It is well known that if a simple unital C$^*$-algebra $A$ has strict comparison if and only if ${\mathrm{W}}(A)$ is *almost unperforated*, that is, whenever $(n+1)x\leq ny$ ($x$, $y\in {\mathrm{W}}(A)$, $n\in \mathbb{N}$), one has $x\leq y$ (see [@rorfunct]). A related property that a partially ordered monoid $M$ might satisfy is that of being *almost divisible*. This means that, for any $x$ in $M$ and any $n\in\mathbb{N}$, there is $y$ in $M$ such that $ny\leq x\leq (n+1)y$.
Recall that a C$^*$-algebra $A$ is $\mathcal{Z}$-stable if it absorbs $\mathcal{Z}$ tensorially, that is, $A\otimes\mathcal{Z}\cong A$, where $\mathcal{Z}$ is the Jiang-Su algebra ([@jiangsu]). Any $\mathcal{Z}$-stable algebra $A$ has the properties that ${\mathrm{W}}(A)$ is almost unperforated and almost divisible (see [@rorijm], [@bpt], [@kngper]). This is also the case for every simple, unital AH-algebra with slow dimension growth ([@bpt], [@tomsplms]).
For the class of simple, separable, finite, exact, unital C$^*$-algebras with strict comparison and such that ${\mathrm{W}}(A)$ is almost divisible, a precise description of ${\mathrm{W}}(A)$ was given in [@bpt] (see also [@aptsantander]), by means of data already contained in the Elliott invariat (i.e. $\mathrm{K}$-Theory and traces). Moreover, the recovery of one from the other has a functorial nature (see [@pt]).
More specifically, consider the set ${\mathrm{V}}(A)\sqcup \mathrm{LAff}({\mathrm{T}}(A))^{++}$. Equip it with an abelian monoid structure and a partial order, as follows. Addition extends the natural operations in both ${\mathrm{V}}(A)$ and $\mathrm{LAff}({\mathrm{T}}(A))^{++}$, and is defined on mixed terms as $[p]+f=\hat{p}+f$ where $\hat{p}(\tau)=\tau(p)$. The order is given by:
1. $\leq$ is compatible with the natural order defined in ${\mathrm{V}}(A)$;
2. if $f,g\in\mathrm{LAff}({\mathrm{T}}(A))^{++}$ we will say that $f\leq g$ if $f(\tau)\leq g(\tau)$ for all $\tau\in {\mathrm{T}}(A)$;
3. If $f\in\mathrm{LAff}({\mathrm{T}}(A))^{++}$ and $[p]\in {\mathrm{V}}(A)$ we will say that $f\leq [p]$ if $f(\tau)\leq \tau(p)$ for all $\tau\in {\mathrm{T}}(A)$;
4. If $f\in\mathrm{LAff}({\mathrm{T}}(A))^{++}$ and $[p]\in {\mathrm{V}}(A)$ we will say $[p]\leq f$ if $\tau(p)<f(\tau)$ for all $\tau\in {\mathrm{T}}(A)$.
The set ${\mathrm{V}}(A)\sqcup\mathrm{LAff}_b({\mathrm{T}}(A))^{++}$ naturally inherits the structure and order just defined, so that the natural inclusion $\iota\colon {\mathrm{V}}(A)\sqcup\mathrm{LAff}_b({\mathrm{T}}(A))^{++}\to {\mathrm{V}}(A)\sqcup\mathrm{LAff}({\mathrm{T}}(A))^{++}$ is an order-embedding.
\[thm:bpt\] [(cf. [@bpt], [@aptsantander])]{} Let $A$ be a simple, unital, separable, exact C$^*$-algebra. Assume that $A$ is finite, has strict comparison and ${\mathrm{W}}(A)$ is almost divisible. Then, there is an ordered monoid isomorphism $${\mathrm{W}}(A)\cong {\mathrm{V}}(A)\sqcup \mathrm{LAff}_{b}({\mathrm{T}}(A))^{++}\,,$$ such that $\langle p\rangle\mapsto [p]$, for a projection $p$, and $\langle a\rangle\mapsto\hat{a}$, for $a$ not equivalent to a projection.
The following lemma is known. Its proof uses Edwards’ separation Theorem (see, e.g. [@poag Theorem 11.12]).
\[LAff2\] Let $K$ be a metrizable Choquet simplex and $g\in\mathrm{LAff}(K)^{++}$. Then $g$ is the pointwise supremum of a strictly increasing sequence $(f_{n})$, where $f_{n}\in \mathrm{Aff}(K)^{++}$ for all $n$.
Let $f$ and $g$ be affine functions on a convex set $K$. We write $f<g$ to mean $f(x)<g(x)$ for every $x$ in $K$.
\[LAff3\] Let $K$ be a compact convex set, and let $f$, $g\in\mathrm{LAff}(K)^{++}$.
1. If $f\ll g$, then $f< g$.
2. If $f<g$ and $f$ is continuous, then $f\ll g$.
(i). Since $g\in\mathrm{LAff}(K)^{++}$ and $K$ is compact, we know that $g$ is bounded away from zero. Therefore, there exists $n_{0}$ such that $g-1/n\in\mathrm{LAff}(K)^{++}$ for all $n\geq n_{0}$, and we may take $n_{0}=1$. Since $\sup_{n}(g-1/n)=g$, there exists $n$ such that $f\leq g-1/n$, whence $f<g$.
(ii). Suppose that $g\leq\sup(g_{n})$ where $(g_{n})$ is an increasing sequence in $\mathrm{LAff}(K)^{++}$. For each $n$, put $\mathcal{U}_{n}:=\{x\in K\mid f(x)< g_{n}(x)\}$, which is open as $f$ is continuous and $g_{n}$ is lower semicontinuous (and so is $g_{n}-f$). Since $f<g$, we see that $\bigcup_{n\geq 1}\mathcal{U}_{n}=K$. Using now that $K$ is compact and that $(g_n)$ is increasing, we find $m\geq 1$ with $K=\mathcal{U}_{m}$. This implies that $f< g_{m}$.
\[lem:whatsthat\] Let $A$ be a simple, separable and unital C$^*$-algebra. Then, the monoid ${\mathrm{V}}(A)\sqcup \mathrm{LAff}({\mathrm{T}}(A))^{++}$ is an object of Cu.
Let $M={\mathrm{V}}(A)\sqcup \mathrm{LAff}({\mathrm{T}}(A))^{++}$. Let us first prove that every increasing sequence $(x_n)$ in $M$ has a supremum. If infinitely many terms of the sequence belong to $\mathrm{LAff}({\mathrm{T}}(A))^{++}$, then the supremum equals the (pointwise) supremum $g$ of the functions that appear in $(x_n)$. For, if $[p]\in{\mathrm{V}}(A)$ is such that $x_n\leq [p]$ for all $n$, then by definition $x_n\leq\hat{p}$ for every $n$ such that $x_n\in\mathrm{LAff}({\mathrm{T}}(A))^{++}$, whence $g\leq \hat{p}$, that is, $g\leq [p]$.
Otherwise, all but finitely many terms in $(x_n)$ belong to ${\mathrm{V}}(A)$. For those, write $x_n=[p_n]$, where $p_n$ are projections in matrices. Then, either the sequence is eventually constant, in which case the supremum belongs to ${\mathrm{V}}(A)$, or else $[p_k]<[p_{k+1}]$ for infinitely many $k$s. In that case $\sup_n x_n=\sup_n \hat{p}_n$. We only need to verify that $[p_k]\leq \sup_n\hat{p}_n$. Indeed, given $k$, there is $l>k$ with $[p_k]<[p_l]$. Simplicity of $A$ ensures that $\hat{p}_k<\hat{p}_l$, whence $\hat{p}_k<\sup_n\hat{p}_n$. Thus $[p_k]\leq\sup_n\hat{p}_n$.
From our observations above, it follows that the (only) compact elements in $M$ are the ones in ${\mathrm{V}}(A)$. Indeed, if $[p]\leq\sup f_n$ for functions $f_n\in \mathrm{LAff}({\mathrm{T}}(A))^{++}$, then $\hat{p}<\sup f_n$, and by compactness we may choose $\epsilon>0$ such that $\hat{p}+\epsilon<\sup f_n$. Since $\hat{p}$ is continuous, Lemma \[LAff3\] implies $\hat{p}<f_k$ for some $k$, that is, $[p]\leq f_k$.
As $A$ is separable, we know that ${\mathrm{T}}(A)$ is metrizable, and it follows from Lemma \[LAff2\] that each element in $\mathrm{LAff}({\mathrm{T}}(A))^{++}$ is the pointwise supremum of a strictly increasing sequence of elements from $\mathrm{Aff}({\mathrm{T}}(A))^{++}$, which again, in view of Lemma \[LAff3\], is a rapidly increasing sequence.
It is easy to verify that suprema and addition in $M$ are compatible. Assume now that $x\ll y$, and $z\ll t$. The only case of interest arises when one of $y$, $t$ belongs to ${\mathrm{V}}(A)$ and the other does not. Assume then that, for example, $y\in{\mathrm{V}}(A)$, and write $t=\sup_n f_n$ for a strictly increasing sequence of affine, continuous functions. Then, $z\leq f_n$ for some $n$, whence $$x+z\leq y+t_n=\hat{y}+f_n\ll\hat{y}+f_{n+1}=y+f_{n+1}\leq y+t\,,$$ showing that $\ll$ and addition are compatible.
Assembling the results above, we obtain the following result that recovers Theorem 2.6 in [@bt].
Let $A$ be a unital, simple, separable, exact C$^*$-algebra. Assume that $A$ is finite, has strict comparison, and ${\mathrm{W}}(A)$ is almost divisible. Then, there is an ordered monoid isomorphism $${\mathrm{W}}(A\otimes\mathcal{K})\cong {\mathrm{V}}(A)\sqcup\mathrm{LAff}({\mathrm{T}}(A))^{++}\,.$$
Let $\iota\colon{\mathrm{V}}(A)\sqcup \mathrm{LAff}_{b}({\mathrm{T}}(A))^{++}\to {\mathrm{V}}(A)\sqcup \mathrm{LAff}({\mathrm{T}}(A))^{++}$ be the natural inclusion. We only need show that the pair $({\mathrm{V}}(A)\sqcup\mathrm{LAff}({\mathrm{T}}(A))^{++},\iota)$ is the completion of ${\mathrm{V}}(A)\sqcup \mathrm{LAff}_{b}({\mathrm{T}}(A))^{++}$ and then invoke Theorem \[thm:bpt\] and Theorem \[thm:wacompletion\].
We know that ${\mathrm{V}}(A)\sqcup \mathrm{LAff}_{b}({\mathrm{T}}(A))^{++}$ is an object of PreCu (by Theorem \[thm:bpt\], Proposition \[prop:wainprecu\] and Lemma \[lem:sr1her\]), and that ${\mathrm{V}}(A)\sqcup \mathrm{LAff}({\mathrm{T}}(A))^{++}$ is an object of Cu (by Lemma \[lem:whatsthat\]). Clearly, the map $\iota$ is an order-embedding in PreCu. Now, any $f\in \mathrm{LAff}(T(A))^{++}$ may be written as $f=\sup(\iota(f_{n}))$ where $f_{n}\in\mathrm{Aff}(T(A))^{++}$. Thus $({\mathrm{V}}(A)\sqcup \mathrm{LAff}({\mathrm{T}}(A))^{++},\iota)$ satisfies the requirements of Definition \[dfn:completion\].
We now turn to the real rank zero situation, and show that, for this class, the Cuntz semigroup (of the stabilisation) is order isomorphic to the monoid of intervals in the projection monoid. This was shown to be the case if moreover the algebra has stable rank one by the third author in [@perijm], but this turns out not to be necessary.
Let $A$ be a $\sigma$-unital C$^*$-algebra with real rank zero. Then $${\mathrm{W}}(A\otimes\mathcal{K})\cong\Lambda_{\sigma}({\mathrm{V}}(A))$$ as ordered monoids.
We may identify ${\mathrm{V}}(A\otimes\mathcal{K})$ with ${\mathrm{V}}(A)$. For each $a\in A\otimes\mathcal{K}_+$ put $I(a)=\{[p]\in{\mathrm{V}}(A)\mid p\precsim a\}=\{[p]\in{\mathrm{V}}(A)\mid p\in\overline{aM_{\infty}(A)a}\}$, which is an element in $\Lambda_{\sigma}({\mathrm{V}}(A))$.
Define $\varphi\colon{\mathrm{W}}(A\otimes\mathcal{K})\to\Lambda_{\sigma}({\mathrm{V}}(A))$ by $\varphi(\langle a\rangle)= I(a)$. The proof of Theorem 2.8 in [@perijm] shows that $\varphi$ is a well defined order embedding.
Now, let $I\in\Lambda_{\sigma}({\mathrm{V}}(A))$ and let $\{[p_n]\}$ be an increasing countable cofinal subset. We of course have that $(\langle p_n\rangle)$ is an increasing sequence in ${\mathrm{W}}(A\otimes\mathcal{K})$. Let $\langle a\rangle=\sup(\langle p_{n}\rangle)$, and let us show that $I(a)=I$.
Let $p\precsim a$ be a projection. Then $\langle
p\rangle\leq\langle a\rangle$. Since $\langle
p\rangle\ll\langle p\rangle\leq\langle a\rangle =\sup(\langle
p_{n}\rangle)$, we get $\langle p\rangle\leq\langle p_{n}\rangle$ for some $n$, that is, $[p]\leq [p_{n}]$ in ${\mathrm{V}}(A)$. Conversely, if $[q]\in I$, then $[q]\leq [p_{n}]$ for some $n$, and therefore $\langle q\rangle \leq
\langle p_{n}\rangle\leq \langle a\rangle$. Thus $[q]\in I(a)$.
Countable inductive limits of monoids {#sec:limits}
=====================================
In this section we answer, in the negative, the question of whether the category PreCu is closed under coutable inductive limits. We subsequently repair this defect by constructing a smaller category that sits between Cu and PreCu, and to which the Cuntz semigroup belongs in interesting cases.
\[PreCunolim\]The category PreCu does not have inductive limits.
We consider for all $i\geq 0$ the following monoids $$S_i=\frac{1}{2^i}{\mathbb{N}}=\left\{\frac{n}{2^i}\ \mid n\in {\mathbb{N}}\right\},$$ with the natural order and addition. It is clear that these discrete monoids are in PreCu (in fact in ${\textrm{Cu}}$ if we add an infinite element) and are all isomorphic to ${\mathbb{N}}$. Consider now the following inductive sequence, where $f_i$ are the natural inclusions:
$$S_1 \stackrel{f_1}{\to} S_2 \stackrel{f_2}{\to} S_3 \stackrel{f_3}{\to} \dots$$
Let $S=\bigcup_{i\geq 1} S_i\subseteq {\mathbb{Q}}^+$ be the standard algebraic inductive limit of the sequence (i.e. the inductive limit in the category of ordered abelian monoids), with the inclusions $\varphi_i\!:\!S_i\to S$ as compatible maps ($\varphi_{i+1}f_i=\varphi_i$). Observe that this monoid is no longer discrete, and, as we will see, it can not be the limit in PreCu (neither in ${\textrm{Cu}}$ by just adding infinity), since properties as $1\ll 1$ in any of the $S_i$ are not preserved in $S$.
Arguing by contradiction, suppose that the sequence $(S_i, f_i)$ has an inductive limit in PreCu, $$T=\displaystyle{\lim_{\rightarrow\text{\tiny PreCu}}}\left(S_i,f_i\right) \text{ with } \psi_i\!:\!S_i\to T \text{ such that }\psi_{i+1}f_i=\psi_i.$$
We will construct two monoids $T_1$, $T_2$ in PreCu with compatible maps $S_i\to T_j$ ($j=1,2$), and use the universal property from $T$ to derive a contradiction.
First, let $S'=S\setminus\{0\}$. Denote its elements by $s'$. Define $T_1:=S\sqcup S'$ with addition extending the addition in $S$ and $S'$ by $(a+b')=(a+b)'$ (primes denoting elements in $S'$), and the order also extending the order in $S$ and $S'$ with $$a'\leq b \text{ if } a\leq b,\ \text{ and } a\leq b' \text{ if } a<b.$$ This last condition is in fact a consequence of extending algebraically the order in $S$ and $S'$. We also define $T_2:=S\sqcup S'$, the same as $T_1$ as sets and even as monoids, but with a different order, again extending the order from $S$ and $S'$ but now $$a'\leq b \text{ if } a< b, \ \text{ and } a\leq b' \text{ if } a<b.$$ Observe that, by doing this, we are just preventing $a$ and $a'$ to be comparable for any $a\in S\setminus\{0\}$.
*$T_1$ and $T_2$ are objects of PreCu.*
It is easy to see, in both situations, that the addition is order preserving and hence that $T_1,T_2$ are ordered abelian monoids. The other properties in PreCu are also easily derived once we know how suprema are constructed, and, with that, when we have $x\ll y$.
Suprema of stationary sequences always exist, so let us consider $x_1\leq x_2\leq \dots \in T_j$ a non stationary sequence with a supremum in $T_j$. Let $\gamma_j\!:\!T_j\to S$ be the order preserving monoid map identifying both copies of $S$, and consider the sequence $(\gamma_j(x_i))_i$ which is also non stationary since we only have two copies of each element. Observe that if $\gamma_j(x_i)<r$ in $S$, then $x_i<r,r'$ both in $T_1$ and $T_2$. Hence, if a supremum exists in $T_j$, it must exist in $S$, say $r=\sup_{S}(\gamma_j(x_i))_i$, and the supremum in $T_j$ must be either $r$ or $r'$. In $T_1$ we have $r'<r$ therefore $\sup_{T_1}(x_n)_n=r'$. However, $r$ and $r'$ are not comparable in $T_2$, and hence we can not have a supremum.
Hence, in $T_1$, non stationary sequences with suprema are the ones which have a supremum by identifying both copies of $S$, and in this case, the supremum is the copy coming from $S'$. In $T_2$ the only sequences with supremum are the stationary ones.
Using this one sees that $T_1$ is a linearly ordered monoid, with two copies of each element in $S\setminus \{0\}$, ordered as $a'<a$, and with $x\ll y$ if $x<y$ or if $x=y\in S$.
Suprema in $T_2$ come only from stationary sequences, whence this is like the discrete situation, so we have $x\ll y$ if $x\leq y$.
With this in mind it is now easy to prove that $T_1$ and $T_2$ are objects in PreCu. Furthermore, considering the natural inclusions in the first copy of $S$, $i_j\colon S\to T_j$ for $j=1,2$, we also obtain compatible maps in PreCu, $i_j\varphi_i\colon S_i\to T_j$ (that is, $i_j\varphi_i=i_j\varphi_{i+1}f_i$).
*$T$ contains a copy of $S$ as an ordered abelian submonoid.*
Consider the following diagram of ordered monoid maps, observing that the only maps that belong to PreCu are the $f_i$s and the $i_1\varphi_j$s: $$\xymatrix@=7ex{
& & & S\ar@{-->}[d]^{\exists !\psi}\ar@/^2pc/[dd]^{\exists ! i_1} \\
S_1\ar@{->}[r]^(.7){f_1} \ar@{->}[urrr]^{\varphi_1} \ar@{->}[drrr]_{i_1\varphi_1} & S_2\ar@{->}[r]^(.6){f_2} \ar@{->}[urr]_{\varphi_2}\ar@{->}[drr]^{i_1\varphi_2} & S_3\ar@{..>}[r]^{\psi_i} \ar@{->}[ur]_{\varphi_3} \ar@{->}[dr]^{i_1\varphi_3} & T \ar@{-->}[d]^{\exists !\varphi} \\
& & & T_1
}$$
Since $T$ has compatible maps $\psi_i$ with the $f_i$, by the universal property of $S$ we obtain a unique ordered monoid homomorphism $\psi\!:\!S\to T$ such that $\psi\varphi_i=\psi_i$. Again, since $T_1$ also has compatible maps $i_1\varphi_i$, now in the category PreCu, by the universal property of $T$ we obtain a unique map $\varphi\!:\!T\to T_1$ such that $\varphi\psi_i=i_1\varphi_i$, and, at the level of monoids, we obtain a unique ordered map $\tilde\varphi\!:\!S\to T_1$ such that $\tilde\varphi\varphi_i=i_1\varphi_i$. By uniqueness this last map should be the inclusion, $i_1=\tilde\varphi$, and since $\varphi\psi\varphi_i=\varphi\psi_i=i_1\varphi_i$ we have, again by uniqueness, $i_1=\varphi\psi$.
Therefore, since the inclusion factors through $T$, $\psi$ is injective and we have an isomorphic copy of $S$ in $T$. We will thus write $T=\psi(S)\sqcup T'$.
*$T'\neq\emptyset$.*
Observe that since the monoids $S_i$ are discrete and all bounded sequences are eventually stationary, all elements are compact (that is, $a\ll a$ for all $a\in S_i$). Now for any $i\geq 1$, $\psi(1)=\psi(\varphi_i(1))=\psi_i(1)$, and since $1\ll 1$ in $S_i$, and $\psi_i$ are PreCu morphisms, we have that $\psi(1)\ll \psi(1)$ also in $T$. Clearly this can be extended to prove that all elements in $\psi(S)$ are compact.
In $T$ we have that $\psi(1)\geq \psi(\frac{2^n-1}{2^n})=\psi(1-\frac{1}{2^n})$ for all $n\geq 1$. Since $\psi$ is injective and compact elements can not be written as suprema of non stationary sequences, we have that $\psi(1)\neq \sup_T(\psi(1-\frac{1}{2^n}))_n$ (this last element might not even exist). But $\psi(1)$ is an upper bound for the sequence and therefore there exists $t'\in T$ such that $\psi(1-\frac{1}{2^{n}})\leq t'$, for all $n\geq 1$, and either $t'<\psi(1)$ or else $t'$ is not comparable with $\psi(1)$. Since $\psi$ is an ordered morphism, and $\psi(S)$ is completely ordered, it is clear that $t'\not\in \psi(S)$, and hence $t'\in T'\neq \emptyset$.
*$T$ cannot be the inductive limit in PreCu of $(S_i,f_i)$.*
Recall that in $T_2$ the only sequences with suprema are the stationary sequences. With this in mind, and with $\psi$ as in Claim 2, it is not difficult to see that the following is a morphism in PreCu : $$\begin{array}{rcl} \gamma: T_2& \longrightarrow &T \\
a & \longmapsto & \psi(a) \text{ if } a\in S \\
a' & \longmapsto & \psi(a) \text{ if } a'\in S'
\end{array}$$ Now consider the following diagram of maps in PreCu, $$\xymatrix@=7ex{
& & T \ar@{-->}[dd]_{\exists ! \varphi} \\
S_i\ar[r]^{f_i}\ar[urr]^{\psi_i}\ar[drr]_{i_2\varphi_i} & S_{i+1}\ar[ur]_{\psi_{i+1}}\ar[dr]^{i_2\varphi_{i+1}} & \\
& & T_2\ar@/_1.5pc/[uu]_\gamma}$$ By the universal property of $T$ there exists a unique map $\varphi\!:\!T\to T_2$ such that $\varphi\psi_i=i_2\varphi_i$. Now observe that $\psi_i(s)=\psi(\varphi_i(s))=\gamma(i_2 \varphi_i(s))$ and therefore, $\gamma\varphi\psi_i=\gamma i_2\varphi_i=\psi_i$. But $\gamma\varphi\neq \text{Id}_T$ since $t'\not\in \gamma(T_2)$. This contradicts the universal property for $T$ leading to two different compatible maps from $T$ to $T$.
Observe that the example of inductive chain used in the proof of the previous Theorem, can be obtained as an inductive chain induced by the Cuntz functor ${\mathrm{W}}(-)$ applied to an inductive chain of C$^*$-algebras.
Consider the $2^\infty$ UHF-Algebra, $A=\lim_{\rightarrow} (M_{2^i}({\mathbb{C}}),g_i)$ with $g_i(x)=\left(\begin{smallmatrix}x & 0 \\ 0 & x\end{smallmatrix}\right)$. The Cuntz semigroup of each matrix algebra ${\mathrm{W}}(M_n({\mathbb{C}}))$ is isomorphic to a cyclic free semigroup with $0$, given by the rank function, $\langle a\rangle \mapsto \text{rank}(a)$ for all $a\in A_+$. Hence, if we make identifications ${\mathrm{W}}(M_{2^i}({\mathbb{C}}))=\frac{1}{2^i}{\mathbb{N}}=S_i$ – by using the weighted rank funtions $\langle a\rangle\mapsto \frac{1}{2^i}\text{rank}(a)$ – the induced maps ${\mathrm{W}}(g_i)$ are the natural inclusions $f_i\colon S_i\to S_{i+1}$. Therefore, we obtain the previous sequence of monoids in PreCu.
Let us thus define a new category, which will be suitable for the Cuntz semigroup for a large class of C$^*$-algebras, and in which inductive limits can be constructed.
Let $\mathcal{C}$ be the full subcategory of PreCu whose objects are monoids $M$ closed by suprema of bounded increasing sequences.
As illustrating examples, observe that $\mathbb{Q}^+$ is an object of PreCu but not of ${\mathcal C}$, $\mathbb{R}^+$ is an object of ${\mathcal C}$ but not of Cu, and finally $\mathbb{R}^+\cup \{\infty\}$ is an object of ${\textrm{Cu}}$.
We have seen in Lemma \[lem:sr1her\] that for stable rank one C$^*$-algebras $A$, ${\mathrm{W}}(A)$ is hereditary and therefore in PreCu. Also, by [@bpt Lemma 4.3] we see that ${\mathrm{W}}(A)$ has suprema of bounded sequences (this was in fact used to prove Lemma \[lem:sr1her\]) and therefore ${\mathrm{W}}(A)$ is an object of $\mathcal C$.
Furthermore, we will see that the category $\mathcal C$ coincides with the category of monoids $M$ for which the inclusion $\iota\colon M\to \overline{M}$ is hereditary.
.
\[prop:heriffC\] Let $M$ be in PreCu. Then the inclusion $\iota\colon M\to\overline{M}$ is hereditary if and only if $M$ is an object of $\mathcal C$, that is, all bounded increasing sequences in $M$ have a supremum.
Suppose the inclusion $\iota\colon M\to \overline{M}$ is hereditary and consider $(x_n)$ a bounded ascending sequence in $M$, say $x_n\leq y\in M$ for all $n\geq 1$. Then, since $\iota$ is a map in PreCu, we obtain $\iota(x_n)\leq \iota (y)$ for all $n\geq 1$. Now, since $\overline{M}$ is in Cu, we have $z=\sup_{\overline{M}}(\iota(x_n))\leq \iota(y)$. Using that $\iota$ is hereditary there exists an element $x\in M$ such that $\iota(x) = z$. Suppose there exists $x'\in M$ such that $x_n\leq x'$ for all $n$. Then $\iota(x_n)\leq \iota(x')$ which implies $\iota(x)=z\leq \iota(x')$. But since $\iota$ is an order-embedding we have $x\leq x'$ and therefore $x=\sup_M (x_n)$.
Now suppose $M$ is a monoid in $\mathcal C$ and consider the map $\iota \colon M\to \overline{M}$. Since $(\overline{M},\iota)$ is a completion of $M$, any $x\in \overline{M}$ can be written as $x=\sup_{\overline{M}}(\iota(x_n))$. Suppose further that $x\leq \iota(y)$ for some $y\in M$. Therefore $\iota(x_n)\leq \iota(y)$ for all $n\geq 1$, and since $\iota$ is an order embedding, we obtain $x_n\leq y$ for all $n\geq 1$. Now since $M$ is in $\mathcal C$ and the sequence is bounded by $y$, there exists $z=\sup_{M}(x_n)$ and since $\iota$ preserves suprema we obtain $\iota(z)=\iota(\sup_M(x_n))=\sup_{\overline{M}}(\iota(x_n)) = x$, obtaining the desired result.
Note that for a C$^*$-algebra $A$, if the embedding ${\mathrm{W}}(A)\to {\mathrm{W}}(A\otimes\mathcal{K})$ is hereditary, by Theorem \[thm:wacompletion\] ${\mathrm{W}}(A)$ is an object of PreCu and ${\mathrm{W}}(A\otimes\mathcal{K})$ is order-isomorphic to $\overline{W(A)}$. Therefore the embedding ${\mathrm{W}}(A)\to \overline{{\mathrm{W}}(A)}$ is also hereditary. On the other hand, it is not clear that if ${\mathrm{W}}(A)$ is an object of PreCu and the embedding ${\mathrm{W}}(A)\to\overline{{\mathrm{W}}(A)}$ is hereditary then ${\mathrm{W}}(A)\to{\mathrm{W}}(A\otimes\mathcal{K})$ is also hereditary
By Proposition \[prop:heriffC\] and the preceding remark, we obtain:
If $A$ is a C$^*$-algebra such that ${\mathrm{W}}(A)$ is hereditary, then ${\mathrm{W}}(A)$ is an object of $\mathcal C$.
The category $\mathcal C$ has countable inductive limits.
Let $(S_i,f_i)_{i\geq 0}$ be an inductive sequence of monoids in $\mathcal C$ and let $S^{\text{alg}}$ be the algebraic inductive limit with compatible maps $\varphi_i:S_i\to S^{\text{alg}}$. Also, we define for any $m>i\geq 0$ the maps $f_{m,i}=f_{m-1}\dots f_i$.
To construct the inductive limit in ${\textrm{Cu}}$, Coward, Elliott and Ivanescu considered the set of ascending sequences (through the morphisms $f_i\!:\!S_i\to S_{i+1}$) with an intertwining relation between the compactly contained elements of the sequence. We will use the same construction but, since we are only interested in obtaining suprema in $S$ for bounded sequences, we should only be interested in sequences which are bounded in some of the $S_i$, and its successive homomorphic images, that is, in $S^{\text{alg}}$. But, in order to maintain the rapidly increasing structure, we will consider ascending sequences with a bound in $S^{\text{alg}}$ for its compactly contained elements.
Let us call a sequence $s=(s_1,s_2,\dots )$ with $s_i\in S_i$ a *bounded ascending sequence* in $(S_i,f_i)$ if $f_i(s_i)\leq s_{i+1}$ and there exists $M_s\in S^{\text{alg}}$ such that, for all $i\geq 0$ and $x\ll s_i$, $\varphi_i(x)\leq M_s$. We will say that $M_s$ is the *bound in* $S^{\text{alg}}$ for $s$, or that $x$ is bounded in $S^{\text{alg}}$. We now define $$S^{(0)}:=\{ s=(s_1,s_2,\dots ) \mid s \text{ is bounded}\}.$$ This set becomes a pre-ordered abelian monoid with componentwise addition and pre-order relation given by $(s_1,s_2,\dots)\precsim (t_1,t_2,\dots)$ if, for any $i$ and $s\in S_i$ with $s\ll s_i$, there exists an $m>i$ such that $f_{m,i}(s)\ll t_m$. Antisymmetrizing the relation $\precsim$ ($(s_i)\sim (t_i)$ if $(s_i)\precsim (t_i)$ and $(t_i)\precsim (s_i)$), we obtain an ordered abelian monoid $S=S^{(0)}/\sim$ which, together with the morphisms $\varphi_i\!:\!S_i\to S$, $\varphi_i(s)=[(0,\dots,0,s,f_i(s),\dots)]$, is the inductive limit of $(S_i,f_i)$ in $\mathcal C$.
This construction is based on the construction in [@CEI] with the difference that we are considering a wider range of monoids (possibly with unbounded sequences), but a smaller subset $S^{(0)}$ (subset of the cartesian product $\prod_{i}S_i$). The proof follows the lines of the one in [@CEI] but with a number of non trivial modifications. For the sake of brevity we will point out where extra care in the construction needs to be taken. First, we need to check that $S^{(0)}$ is closed under addition.
Let $(s_1,s_2,\dots), (t_1,t_2,\dots)\in S^{(0)}$ with bounds in $S^{alg}$ $M_s,M_t$ respectively. For any $i\geq 0$ and $x\ll s_i+t_i$, let us write $s_i,t_i$ as suprema of rapidly increasing sequences in $S_i$, $s_i=\sup_{S_i}(s_i^n)_n$, and $t_i=\sup_{S_i}(t_i^n)_n$. Then $x\ll s_i+t_i=\sup_{S_i}(s_i^n+t_i^n)_n$. Hence, for some $m$ we have that $x\ll s_i^m+t_i^m$. But $s_i^m\ll s_i$ and $t_i^m\ll t_i$ implies that $\varphi_i(s_i^m)\leq M_s$ and $\varphi_i(t_i^m)\leq M_t$. Therefore $\varphi_i(x)\leq \varphi_i(s_i^m+t_i^m)\leq M_s+M_t\in S^{\text{alg}}$ which is a bound in $S^{\text{alg}}$ for $(s_i)+(t_i)$ .
One other important fact is that each element $[(s_i)]\in S$ has a representative $[(\tilde s_i)]$ whose elements are rapidly increasing in the sense that $f_i(\tilde s_i)\ll \tilde s_{i+1}$. This can still be done, since this representative is a Cantor diagonal sequence obtained from rapidly increasing sequences $s_i^n$ with the original elements $s_i$ as suprema. Since those elements are always way below the original ones, the resulting representative is also bounded in $S^{\text{alg}}$.
Also, we should take care in how suprema is constructed.
First we see that if $(s_i)\precsim (t_i)$ and $M_t\in S^{\text{alg}}$ is a bound for $(t_i)$, then $M_t$ is also a bound for $(s_i)$. If $x\ll s_i$, by the $\precsim$ relation, there exists $m$ such that $f_{m,i}(x)\ll t_m$. Now, by the bound for the compactly contained in $t_m$, $\varphi_m(f_{m,i}(x))\leq M_t$. But since the $f_j$ are compatible maps, we obtain $\varphi_i(x)\leq M_t$.
Therefore, a bound in $S$ for an ascending sequence $s^i$ gives us a bound in $S^{alg}$ for all the elements in the sequence, and thus for the computed supremum: Let $[(s_i^1)]\leq [(s_i^2)]\leq\dots \leq [(t_i)]$ be a bounded ascending sequence in $S$. By the above argument the compactly contained elements of all the $s_i^j$ (even for any other representative), are bounded in $S^{\text{alg}}$ by the bound of the $(t_i)$, say $M_t$. But the supremum (as constructed in [@CEI]) is obtained from the components of the $[(s_i^n)]$ (rapidly increasing representatives), which will be bounded in $S^{\text{alg}}$ by $M_t$, and therefore will also led to an element in $S^{(0)}$.
[Let us recall a property of the inductive limit in ${\textrm{Cu}}$, which is still valid in $\mathcal C$ and which should be used later. As in the proof of the previous Theorem, given $s\in S$, choosing a rapidly increasing representative, $s=[(s_i)]\in S$ with $f_i(s_i)\ll s_{i+1}$, we have that $s=\sup_{S}(\varphi_i(s_i))_i$, that is, all elements in $S$ can be written as the supremum of a sequence of elements coming from the $S_i$.]{}
Observe that the inductive sequence in Theorem \[PreCunolim\] is in fact a sequence in $\mathcal C$. To compute the inductive limit $\lim_{\rightarrow\mathcal C}(S_i,f_i)$, just observe that $T=S\sqcup \mathbb{R}^{++}$ with order and addition as $T_1$ in the proof of Theorem \[PreCunolim\], has the desired properties.
If, instead of $(S_i,f_i)$ we had considered the sequence $(S_i\cup\{\infty\},\bar f_i)$ (now objects in ${\textrm{Cu}}$), we would have obtained $\lim_{\rightarrow\mathcal C}(S_i,f_i)=S\sqcup {\mathbb{R} }^{++}\cup\{\infty\}$ which is the inductive limit in ${\textrm{Cu}}$.
One could ask whether or not, the limit in $\mathcal C$ applied to monoids already in ${\textrm{Cu}}$ leads to the inductive limit in ${\textrm{Cu}}$, that is, if $S_i$ are monoids in ${\textrm{Cu}}$, $$\lim_{\rightarrow \mathcal C}S_i\stackrel{?}{=}\lim_{\rightarrow {\textrm{Cu}}}S_i.$$ This is not the case as we see in the following example:
For all $n\geq 0$ let $T_n=\{a_0,a_1,\dots,a_n\}$ with addition $a_i+a_j=a_{\text{max\{i,j\}}}$ and the induced algebraic order. It is not difficult to check that $T_n\in{\textrm{Cu}}$, and now, if we consider the inductive sequence $(T_n,g_n)$ with the natural inclusions as maps, it can be proven, through its defining universal properties, that $$\lim_{\rightarrow \mathcal C}S_i=\{a_0,a_1,a_2,\dots\},\text{ and } \lim_{\rightarrow {\textrm{Cu}}}S_i=\{a_0,a_1,a_2,\dots,a_{\infty}\}.$$ equipped with the same order and addition as before.
\[thm:limitsac\] Let $(S_i,f_i)$ be an inductive sequence of maps in $\mathcal C$. Then $$\overline{ \lim_{\rightarrow \mathcal C} S_i}= \lim_{\rightarrow{\mathcal C}u} \overline{S_i}.$$
Let $(S_i,f_i)$ be an inductive sequence of maps in $\mathcal C$, and $S_{\mathcal C}$ the inductive limit in $\mathcal C$ with compatible maps $\varphi_i\!:\!S_i\to S_{\mathcal C}$. Similarly let $(\overline{S_i}, \overline{f_i})$ be the induced sequence in ${\textrm{Cu}}$, and $S_{{\textrm{Cu}}}$ the inductive limit in ${\textrm{Cu}}$ with compatible maps $\psi_i\!:\overline S_i\to S_{{\textrm{Cu}}}$. Let $\gamma_i\!:\! S_i\to \overline S_i$ be the corresponding inclusions. Now, since $\gamma_{i+1}f_i=\overline{f_i}\gamma_i$ (by definition as in Theorem \[universality\]) consider the following diagram of compatible maps in $\mathcal C$: $$\xymatrix{
& & S_{\mathcal C}\ar@{-->}[ddd]^{\exists!\gamma} \\
S_i\ar[r]^{f_i}\ar@/^/[urr]^{\varphi_i}\ar[d]_{\gamma_i} & S_{i+1}\ar[ur]_{\varphi_{i+1}}\ar[d]^{\gamma_{i+1}} & \\
\overline{S_i}\ar[r]^{\overline{f_i}}\ar@/_/[drr]_{\psi_i} & \overline{S_{i+1}}\ar[dr]^{\psi_{i+1}} & \\
& & S_{{\textrm{Cu}}}
}$$ By the universal property of $S_{\mathcal C}$ there exists a unique map $\gamma\!:\!S_{\mathcal C}\to S_{{\textrm{Cu}}}$ such that $\gamma\varphi_i=\psi_i\gamma_i$.
If $s\in S_{{\textrm{Cu}}}$, then $s$ can be written as a supremum of a rapidly increasing sequence of elements coming from the $\overline S_i$ (see [@CEI]), $s=\sup_{S_{{\textrm{Cu}}}}(\psi_i(s_i))_i$. In turn, since $\overline{S_i}$ is the completion of $S_i$, each of the $s_i$ can be written as a supremum of a rapidly increasing sequence of elements coming from $S_i$, $s_i=\sup_{\overline S_i}(\gamma_i(s_i^j))$. Therefore, since $\psi_i$ are morphisms in ${\textrm{Cu}}$ preserving suprema, $$\begin{gathered}
s=\sup_{S_{{\textrm{Cu}}}}(\psi_i(s_i))_i=\sup_{S_{{\textrm{Cu}}}}(\psi_i(\sup_{\overline S_i}(\gamma_i(s_i^j))_j))_i=
\sup_{S_{{\textrm{Cu}}}}(\sup_{S_{{\textrm{Cu}}}}(\psi_i(\gamma_i(s_i^j)))_j)_i=\sup_{S_{{\textrm{Cu}}}}(\gamma(\varphi_i(s_i^j)))_{i,j},\end{gathered}$$ and we see that each element in $S_{{\textrm{Cu}}}$ can be written as the supremum of elements in $\gamma(S_{\mathcal C})$.
Now let us prove that $\gamma$ is an order embedding.
Suppose $\gamma(s)\leq \gamma(t)$ in $S_{\mathcal C}$ and let $s=[(s_i)], t=[(t_i)]$ with rapidly increasing representatives as in the construction of $S_{\mathcal C}$. Then, recall that in this situation $s=\sup_{S_{\mathcal C}}(\varphi_i(s_i))_i$ and $t=\sup_{S_{\mathcal C}}(\varphi_i(t_i))_i$. Since $\gamma$ is a $\mathcal C$ map and $\gamma\varphi_i=\psi_i\gamma_i$, we obtain $$\sup_{S_{{\textrm{Cu}}}}(\psi_i\gamma_i(s_i))_i\leq \sup_{S_{{\textrm{Cu}}}}(\psi_i\gamma_i(t_i))_i.$$ Similarly as before, in $S_{{\textrm{Cu}}}$ as constructed in [@CEI], this is $[(\gamma_i(s_i))]\leq [(\gamma_i(t_i))]$.
Hence, given $x\ll s_i\in S_i$ we have $\gamma_i(x)\ll \gamma_i(s_i)$. By the order relation in $S_{{\textrm{Cu}}}$ there exists $m\geq i$ such that $\overline{f_{m,i}}\gamma_i(x)\ll \gamma_m(t_m)$ and therefore $\gamma_m(f_{m,i}(x))\ll \gamma_m(t_m)$. But by definition $\gamma_m$ is an order embedding in PreCu, which implies $f_{m,i}(x)\ll t_m$. But this, by the order relation in $S_{\mathcal C}$ implies $[(s_i)]\leq [(t_i)]$ or $s\leq t$. Hence $\gamma$ is an order embedding, $(S_{{\textrm{Cu}}},\gamma)$ is a completion of $S_{\mathcal C}$, and by Theorem \[universality\] we have $S_{{\textrm{Cu}}}\cong \overline S_{\mathcal C}$.
As a consequence, we can now compute the stabilized Cuntz semigroup for some countable inductive limits of $C^*$-algebras in terms of the Classical Cuntz semigroup.
Let $A$ be a C$^*$-algebra such that $A=\lim_{\rightarrow}(A_i,f_i)$ where ${\mathrm{W}}(A_i)$ are hereditary. Then $${\textrm{Cu}}(A)=\overline{\lim_{\rightarrow\mathcal C}({\mathrm{W}}(A_i),{\mathrm{W}}(f_i))}.$$
Using [@CEI Theorem, 2], Theorem \[universality\] and Theorem \[thm:limitsac\] we obtain
$${\textrm{Cu}}(A)=\lim_{\rightarrow{\textrm{Cu}}}({\textrm{Cu}}(A_i),{\textrm{Cu}}(f_i))=\lim_{\rightarrow{\textrm{Cu}}}\left(\overline{{\mathrm{W}}(A_i)},\overline{{\mathrm{W}}(f_i)}\right)=
\overline{\lim_{\rightarrow\mathcal C}({\mathrm{W}}(A_i),{\mathrm{W}}(f_i))}.$$
[99]{} P. Ara, F. Perera, and A. S. Toms, *K-Theory for operator algebras. Classification of C$^*$-algebras*, arXiv preprint: 0902.3381. B. Blackadar, *Comparison theory for simple C$^*$-algebras*. London Math. Soc. Lecture Note Ser. , Cambridge Univ. Press (1989), pp. 21–54. B. Blackadar, L. Robert, A. Tikuisis, A. S. Toms, and W. Winter, in preparation. N. P. Brown and A. S. Toms, *Three applications of the Cuntz semigroup*, Int. Math. Res. Notices. (2007), Vol. 2007, article ID rnm068, 14 pages. N. P. Brown, F. Perera, and A. S. Toms, *The Cuntz semigroup, the Elliott conjecture, and dimension functions on [C]{}$^*$-algebras*, J. reine angew. Math. **621** (2008), pp. 191–211. K.T. Coward, G.A. Elliott and C. Ivanescu. *The Cuntz Semigroup as an invariant for C$^*$-algebras*. J. die reine angew. Math. **623** (2008), pp. 161–193. J. Cuntz, *Dimension functions on simple [C]{}$^*$-algebras*, Math. Ann. **233** (1978), 145–153. M. Dadarlat and A. S. Toms, *Ranks of operators in simple C$^*$-algebras*, arXiv preprint: 0912.0675. K. R. Goodearl, *Partially Ordered abelian Groups with Interpolation*, Math. Surveys and Monographs **20**, Amer. Math. Soc., Providence, 1986. X. Jiang, H. Su, *On a simple unital projectionless [C]{}$^{*}$-algebra*, Amer. J. Math. [**121**]{} (1999), pp. 359–413. E. Kirchberg and M. Rørdam, *Non-simple purely infinite C\*-algebras*. Amer. J. Math. **122** (2000), pp. 637–666 E. Kirchberg, M. Rørdam, *Infinite non-simple [C]{}$^*$-algebras: Absorbing the Cuntz algebra $\mathcal{O}_{\infty}$*, Adv. Math. **167** (2002), pp. 195–264. D. Kucerovsky, P. W. Ng, and F. Perera, *Purely infinite corona algebras of simple C$^*$-algebras*, Math. Ann., **346** (2010), pp. 23–40. F. Perera, *The structure of positive elements for C$^*$-algebras with real rank zero*, International J. Math., **8** (1997), pp. 383–405. F. Perera and A. S. Toms, *Recasting the Elliott Conjecture*, Math. Ann., **338** (2007), pp. 669–702. M. Rørdam, *On the structure of simple [C]{}$^*$-algebras tensored with a UHF-algebra. II*, J. Funct. Anal., **107** (1992), pp. 255–269. M. Rørdam, *The stable and the real rank of $\mathrm{Z}$-absorbing [C]{}$^*$-algebras*, International J. Math. **15** (2004), pp. 1065–1084. D.S. Scott, *Continuous Lattices*, Lecture notes in math. **274** (1972), Springer-Verlag, pp. 97–136. A.S. Toms, *Flat dimension growth for C$^*$-algebras*, J. Funct. Anal. **238** (2006), pp. 678–708. A.S. Toms, *On the classification problem for nuclear C$^*$-algebras*. Ann. Math. (2), **167** (2008), no. 3, pp. 1029–1044. A.S. Toms, *Stability in the Cuntz semigroup of a commutative C$^*$-algebra*, Proc. Lond. Math. Soc., **96** (2008), pp. 1–25.
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abstract:
- |
The hydrodynamic equations for indirect excitons in the double quantum wells are studied taking into account 1) a possibility of an exciton condensed phase formation, 2) the presence of pumping, 3) finite value of the exciton lifetime, 4) exciton scattering by defects. The threshold pumping emergence of the periodical exciton density distribution is found. The role of localized and free exciton states is analyzed in the formation of emission spectra.
self-organization, quantum wells, excitons, phase transition
- '=3000[Проведено аналіз рівнянь гідродинаміки екситонів у квантовій ямі. Рівняння враховують 1) можливість фазового переходу в системі, 2) присутність зовнішньої накачки, 3) скінчений час життя екситонів, 4) розсіяння екситонів на дефектах. Визначно порогову накачку утворення періодичного розподілу екситонної густини. Досліджується вплив локалізованих і вільних екситонів на формування спектрів випромінювання.]{} самоорганізація, квантові ями, екситони, фазовий перехід'
address:
- ' Institute for Nuclear Research, 47 Nauky ave., 03680 Kyiv, Ukraine '
- 'Інститут ядерних досліджень, просп. Науки, 47, 03680 Київ, Україна'
author:
- 'V.I. Sugakov'
- 'В.Й. Сугаков'
date: 'Received April 29, 2014'
title: 'Конденсація екситонів в квантових ямах. Гідродинаміка екситонів. Вплив локалізованих станів дефектів'
---
Introduction
============
Phase transitions in systems of unstable particles are specific examples of non-equilibrium phase transitions and processes of self-organization [@Hak83]. In such a system, particles are created by external sources and disappear due to different reasons. If there is an attractive interaction between the particles, they may create a condensed phase during their lifetime. A steady state may arise in a system if the number of the created particles in the unit time is equal to the number of the disappeared particles. This state is stationary, but it is not equilibrium. The following examples of such systems with unstable particles may be presented: 1) dielectric exciton liquid in crystals; 2) electron-hole liquid in semiconductors; 3) highly excited gas with many excited molecules; 4) vacancies and interstitials in a cryatal created by nuclear irradiation; 5) quark glyuon plasma and others. The finite value of the particles determine some peculiarities of phase transitions in such systems. The main peculiarities are as follows: a) a phase transition in a system of unstable particles may occur if the lifetime is larger than some critical value; b) in the presence of parameters at which two phases coexistent, the sizes of the regions of condensed phases of unstable particles are restricted: c) there is strong spatial correlation between different regions of condensed phases, that is why periodical structures may arise.
The present paper is devoted to an investigation of self-organization processes of the exciton system in semiconductor quantum wells. The appearance of periodical dissipative structures in exciton systems at the irradiations greater than some critical value was shown in the work [@Sug86; @Sug98]. Experimental observation of a periodical distribution of the exciton density was obtained in [@But02; @Gor06] in a system of indirect excitons in semiconductor double quantum wells. An indirect exciton consists of an electron and hole separated over two wells by an electric field. Due to the damping of the electron-hole recombination, indirect excitons have a large lifetime which makes it possible to create a high concentration of excitons at small pumping and to study the manifestation of the effects of exciton-exciton interaction. The authors of the paper [@But02] observed the emission from a double quantum well on the basis of AlGaAs system in the form of periodically situated islands along the ring around the laser spot. In the paper [@Gor06], in which the excitation of a quantum well was carried out through a window in a metallic electrode, a periodical structure of the islands situated along the ring under the perimeter of the window was found in the luminescence spectrum. The islands were observed at a frequency that corresponds to the narrow line arising at a threshold value of pumping [@Lar00]. Afterwards, spatial structures of exciton density distributions were observed in a single wide quantum well [@Tim06], in different types of electrodes that create a periodical potential [@Rem09] or have windows in the shape of a rectangle, two circles and others [@Dem11; @Gor12].
The phenomenon of a symmetry loss and the creation of structures in the emission spectra of indirect excitons urged a series of theoretical investigations [@Lev05; @Par07; @Sap08; @Liu06; @Muk10; @Wil12; @And13; @Bab13]. The main efforts were directed towards the verification of a fundamental possibility of the appearance of the periodicity of the exciton density distribution. A specific explanation of the experiment is presented in two works [@Lev05; @Wil12] with respect to only one experiment [@But02].The authors of the work [@Lev05] considered the instability that arises under the kinetics of level occupations by the particles with the Bose-Einstein statistics. Namely, the growth of the occupation of the level with zero moment should stimulate the transitions of excitons to this level. However, the density of excitons was found greater, and the temperature was found lower than these values observed in the experiments. In the paper [@Wil12] the authors did not take into account the screening between the charges at macroscopic distances.
Our model is based on the appearance of self-organization processes in non-equilibrium systems for excitons with attractive interaction shown in [@Sug86; @Sug98]. Investigations performed in this model [@Sug05; @Sug06; @Cher06; @Sug07; @Cher07; @Cher09; @Cher12] gave the explanations of spatial structures and their temperature and pumping dependencies obtained in different experiments [@But02; @Gor06; @Rem09; @Dem11; @Gor12]. Theoretical approaches of the works [@Sug05; @Sug06; @Cher06; @Sug07; @Cher07; @Cher09; @Cher12] are based on the following assumptions.
1. There is an exciton condensed phase caused by the attractive interaction between excitons. The existence of attractive interaction between excitons is confirmed by the calculations of biexcitons [@Tan05; @Schin08; @Mey08; @Lee09], and by investigations of a many-exciton system [@Loz96]. Nevertheless, there is an experimental work [@Yan07], where the authors explain their experimental results by the existence of a repulsion interaction between excitons. These results come into conflict with our suggestion regarding the attractive interaction. We shall remove this contradiction in section 3.
2. The finite value of the exciton lifetime plays an important role in the formation of a spatial distribution of exciton condensed phases. As usual, the exciton lifetime significantly exceeds the duration of the establishment of a local equilibrium. However, it is necessary to take into account the finiteness of the exciton lifetime in the study of spatial distribution phases in two-phase systems, because the exciton lifetime is less than the time of the establishment of equilibrium between phases. The latter is determined by slow diffusion processes and is long.
Two approaches of the theory of phase transitions were used while developing the theory: the model of nucleation (Lifshits-Slyozov) and the model of spinodal decomposition (Cahn-Hillart). These models were generalized to the particles with the finite lifetime, which is important for interpretation of the experimental results. The involvement of Bose-Einstein condensation for excitons was not required in order to explain the experiments.
In the present paper, the hydrodynamic equation for excitons is investigated for the case of excitons being in a condensed phase. The appearance of an instability of the uniform distribution of the exciton density and the development of nonhomogeneous structures are studied. The effect of defects on spectral positions of the emission spectra of both gas and condensed phases is analysed as well.
Analysis of hydrodynamic equations of exciton condensed phase
=============================================================
Hydrodynamic equations of excitons were obtained and analysed in the work [@Lin92]. Hydrodynamic equations of excitons generalizing the Navier-Stokes equations that take into account the finite exciton lifetime, the pumping of exciton, the existence of an exciton condensed phase and the presence of defects were developed in [@SugUJP]. In the paper, we make some analysis of these equations.
The system is described by the exciton density $n \equiv n(\vec {r},t)$ and by the velocity of the exciton liquid $\vec
{u} \equiv \vec {u}(\vec {r},t)$. The equations for conservation of the exciton density and for the movement of exciton density are basic for the exciton hydrodynamic equations. $$\begin{aligned}
\label{eq8}
&&\frac{\partial n}{\partial t} + \textrm{div}(n\vec {u}) = G -
\frac{n}{\tau _\textrm{ex}}\,,\\
%\end{equation}
%\begin{equation}
\label{eq9}
&&\frac{\partial mnu_i }{\partial t} = - \frac{\partial
\Pi _{ik} }{\partial x_k } - \frac{mnu_i }{\tau _\textrm{sc} }\,,\end{aligned}$$ where $G$ is the pumping (the number of excitons created for unit time in unit area of the quantum well), $\tau_\textrm{ex}$ is the exciton lifetime,$m$ is the exciton mass, $\Pi _{ik} $ is the tensor of density of the exciton flux $$\label{eq10} \Pi _{ik} = P_{ik} + mnu_i u_k - {\sigma }'_{ik} \,,$$ where $P_{ik}$ is the pressure tensor, ${\sigma }'_{ik}$ is the viscosity tensor of tension. In the equation (\[eq9\]), we neglected the small momentum change caused by the creation and the annihilation of excitons.
Introducing coefficients of viscosity and using (\[eq8\]), equation (\[eq9\]) may be rewritten in the form $$\begin{aligned}
\label{eq11} \rho \left[ {\frac{\partial u_i }{\partial t} +
\left( {u_k \frac{\partial }{\partial x_k }} \right)u_i } \right]
= - \frac{\partial P_{ik} }{\partial x_k } + \eta \Delta u_i
%\nonumber\\
+ (\varsigma + \eta / 3)\left( {\frac{\partial
}{\partial x_i }} \right)\textrm{div}\vec {u} - \frac{\rho u_i }{\tau _\textrm{sc}
}\,,\end{aligned}$$ where $\rho=mn$ is the mass of excitons in the unit volume.
We assume that the state of the local equilibrium is realized and the state of the system may be described by free energy, which depends on a spatial coordinate. Let us present the functional of the free energy in the form $$\label{eq12} F = \int {\rd\vec {r}\left[ {\frac{K}{2}\left(\vec{\nabla} n\right)^2 +
f(n)} \right]} .$$
At the given presentation of free energy, the pressure tensor is determined by the formula [@Swi96] $$\label{eq13} P_{\alpha \beta } = \left[ {p - \frac{K}{2}\left(\vec
{\nabla }n\right)^2 - Kn\Delta n} \right]\delta _{\alpha \beta } +
K\frac{\partial n}{\partial x_\alpha }\frac{\partial n}{\partial
x_\beta }\,,$$ where $p = n{f}'(n) - f(n)$ is the equation of the state, $p$ is the isotropic pressure.
Taking into account (\[eq13\]) and introducing coefficients of viscosity, we finally rewrite the equation (\[eq9\]) in the form $$\begin{aligned}
\label{eq14} \frac{\partial u_i }{\partial t} + u_k \frac{\partial
u_i }{\partial x_k } + \frac{1}{m}\frac{\partial }{\partial x_i
}\left( { - K\Delta n + \frac{\partial f}{\partial n}} \right) +
\nu \Delta u_i
%\nonumber\\
+ (\varsigma / m + \nu / 3)\left(
{\frac{\partial }{\partial x_i }} \right)\textrm{div}\vec {u} + \frac{u_i
}{\tau _\textrm{sc} } = 0.\end{aligned}$$ Equations (\[eq8\]), (\[eq14\]) are the equations of the hydrodynamics for an exciton system. It follows from the estimations, made in the work [@Lin92], that the terms with the viscosity coefficients are small and we shall neglect them.
In the case of a steady state irradiation, the equations (\[eq8\]) and (\[eq14\]) have the solution $n=G\tau$, $u=0$. To study the stability of the uniform solution we consider, that the behavior of a small fluctuation of the exciton density and the velocity from these values are as follows: $n\rightarrow n+\delta n\exp[\ri\vec{k}\cdot\vec{r}+\lambda(\vec{k})
t]$, $u=\delta u\exp[\ri\vec{k}\cdot\vec{r}+\lambda(\vec{k})
t]$. Having substituted these expressions in equations (\[eq8\]), (\[eq14\]), we obtain, in the linear approximation with respect to fluctuations, the following expression $$\begin{aligned}
\label{eq14aa} \lambda_{\pm}(\vec{k})=\frac{1}{2}\left[-\left(\frac{1}{\tau_\textrm{sc}}+\frac{1}{\tau_\textrm{ex}}\right)
%\nonumber\\
\pm\sqrt{\left(\frac{1}{\tau_\textrm{sc}}-\frac{1}{\tau_\textrm{ex}}\right)^2-
\frac{4k^2n}{m}\left(k^2K+\frac{\partial^2 f}{\partial n^2}\right)} \right].\end{aligned}$$
It follows from (\[eq14aa\]) that both parameters $\lambda_{\pm}(\vec{k})$ have a negative real part at small and large values of vector $\vec{k}$ and, therefore, the uniform solution of the hydrodynamic equation is stable. The instability with respect to a formation of nonhomogeneous structures arises at some threshold value of exciton density and at some critical value of the wave vector, when ${\partial^2 f}/{\partial n^2}$ becomes negative. The analysis of the equation (\[eq14aa\]) gives the following expression for the critical values of the wave vector $k_\textrm{c}$ and the exciton density $n_\textrm{c}$ $$\begin{aligned}
\label{eq14ab}
&&k_\textrm{c}^4=\frac{m}{Kn_\textrm{c}\tau_\textrm{sc}\tau_\textrm{ex}}\,,\\
%\end{equation}
%\begin{equation}
\label{eq14ac}
&&\frac{k_\textrm{c}^2n_\textrm{c}}{m}\left(k_\textrm{c}^2K+\frac{\partial^2 f(n_\textrm{c})}{\partial
n_\textrm{c}^2}\right)+\frac{1}{\tau_\textrm{sc}\tau_\textrm{ex}}=0.\end{aligned}$$
For stable particles ($\tau_\textrm{ex}\rightarrow\infty$), the equations (\[eq14ab\]), (\[eq14ac\]) give the condition ${\partial^2
f}/{\partial n^2}=0$, which is the condition for spinodal decomposition for a system in the equilibrium case.
Depending on parameters, the equations (\[eq8\]), (\[eq14\]) describe the ballistic and diffusion movement of the exciton system. The relaxation time $\tau _\textrm{sc}$ plays an important role in the formation of the exciton movement. Due to the appearance of nonhomogeneous structures, there exist exciton currents in a system ($\vec{j}=n\vec{u}\neq 0$) even under the uniform steady-state pumping. Excitons are moving from the regions having a small exciton density to the regions having a high density. In the present paper, we shall consider the spatial distribution of exciton density and exciton current in the double quantum well under steady-state pumping. In this case, the exciton carrent is small and we assume the existence of the following conditions $$\begin{aligned}
\label{eq14a}
\frac{\partial u_i }{\partial t}
&\ll u_i/\tau _\textrm{sc}\,,\\
%\end{equation}
% \begin{equation}
\label{eq14b}
u_k \frac{\partial u_i }{\partial x_k } &\ll u_i /\tau
_\textrm{sc}\,.\end{aligned}$$ Particularly, the equation (\[eq14a\]) holds in the study of the steady-state exciton distribution. The fulfilment of equation (\[eq14b\]) will be shown later following some numerical calculations.
Using the conditions (\[eq14a\]) and (\[eq14b\]), we obtain from equation (\[eq14\]) the value of the velocity $\vec {u}$ $$\label{eq15} \vec {u} = - \frac{\tau _\textrm{sc}}{m}\vec {\nabla }\left(
{ - K\Delta n + \frac{\partial f}{\partial n}} \right).$$ As a result, the equation for the exciton density current may be presented in the form $$\label{eq16} \vec {j} = n\vec {u}=- M\nabla \mu,$$ where $\mu = \delta F/\delta n$ is the chemical potential of the system, $M = nD/\kappa T$ is the mobility, $D =
\kappa T\tau _\textrm{sc} /m$ is the diffusion coefficient of excitons.
Therefore, the equation for the exciton density (\[eq8\]) equals $$\begin{aligned}
\label{eq17} \frac{\partial n}{\partial t} = \frac{D}{\kappa T}
\left( -
Kn\Delta ^2n - K\vec {\nabla }n \cdot \vec {\nabla }\Delta n\right)
%\nonumber\\
+\frac{D}{\kappa T}\vec {\nabla }\cdot \left(
n\frac{\partial ^2f}{\partial n^2}\vec {\nabla }n \right) + G -
\frac{n}{\tau _\textrm{ex} }\,.\end{aligned}$$
Just in the form of (\[eq17\]), we investigated a spatial distribution of the exciton density at exciton condensation using the spinodal decomposition approximation by choosing different dependencies $f$ on $n$ [@Cher06; @Cher07; @Cher12]. Thus, our previous consideration of the problem corresponds to the diffusion movement of hydrodynamic equations (\[eq8\]), (\[eq14\]). For the system under study, a condensed phase appears if the function $f(n)$ describes a phase transition. In the papers mentioned above, the examples of such dependencies were given. Here, we analyse another dependence $f(n)$, which is also often used in the theory of phase transitions $$\label{eq18} f = \kappa Tn\left[\ln (n / n_a ) - 1\right] + a\frac{n^2}{2} +
b\frac{n^4}{4} + c\frac{n^6}{6}\,,$$ where $a$, $b$, $c$ are constant values. Three last terms in the formula (\[eq18\]) are the main terms. They arise due to an exciton-exciton interaction and describe the phase transition. The first term was introduced in order to describe the system in a space, where the exciton concentration is small (this is important if such a region exists in a system). At an increase of the exciton density, the term $a{n^2}/{2}$ manifests itself firstly. It contributes the $an$ value to the chemical potential. In our system, the origin of this term is connected with the dipole-dipole interaction, which should become apparent at the beginning with the growth of the density due to its long-range nature. To estimate $a$ for the dipole-dipole exciton interaction in double quantum well we may use the plate capacitor formula $an=4\pi e^2 dn/\epsilon$, where $d$ is the distance between the wells, $\epsilon$ is the dielectric constant. This formula is usually used to determine the exciton density from the experimental meaning of the blue shift of the frequency of the exciton emission with the rise of the density. It follows from the formula that $a=4\pi e^2 d/\epsilon$. When the exciton density grows, the last two terms in (\[eq16\]) begin to play a role. The existence of a condensed phase requires that the value $b$ should be negative ($b<0$). For stability of a system, at large $n$, the parameter $c$ should be positive ($c>0$). It is assumed in the model that the condensed phase arises due to the exchange and Van der Waals interactions. The calculations show that in some region of distances between the wells, these interactions exceed the dipole-dipole repulsion.
Let us introduce dimensionless parameters: $\tilde {n} = n / n_0 ,$ where $n_0 = \left( {a / c} \right)^{1 /
4}$, $\tilde {b} = b / (ac)^{1 / 2}$, $\tilde {\vec {r}} = \vec{r}
/ \xi $, where $\xi = \left( {K / a} \right)^{1 / 2}$ is the coherence length, $\tilde {t} = t / t_0 $, where $t_0 =
{\kappa TK}/({Dn_0 a^2})$, $D_1 = {\kappa T}/({an_0 })$, $\tilde {G} = Gt_0/n_0 $, $\tilde {\tau }_\textrm{ex} = \tau / t_0 $. As a result, the equation (\[eq17\]) is reduced to the form (hereinafter the symbol $\sim $ will be omitted in the equation) $$\begin{aligned}
\label{eq19} \frac{\partial n}{\partial t} &=& D_1 \Delta n -
n\Delta ^2n + n\Delta n\left(1 + 3bn^2 + 5n^4\right)
\nonumber \\
&&- \vec
{\nabla }n\cdot\vec {\nabla }\Delta n+\left(\vec {\nabla }n\right)^2\left(1 +
9bn^2 + 25n^4\right) + G - \frac{n}{\tau _\textrm{ex}}\,.\end{aligned}$$
The solutions of the equation (\[eq19\]) are presented in figure \[fig1\] for the one-dimensional case \[$n(\vec {r},t) \equiv n(z,t)$\] for three values of the steady-state uniform pumping.
![ The spatial dependence of the exciton density at a different value of the pumping: for a continues line $G=0.0055$, for a periodical line $G=0.008$, for a dashed line $G=0.0092$. $D_1=0.03$, $b=-1.9$.[]{data-label="fig1"}](f1_){width="8.6cm"}
The solutions are obtained at the initial conditions $n(z,0) =
0$ and the boundary conditions $n'(0,t) = n'(L,t) = n''(0,t) =
n''(L,t) = 0$, where $L$ is the size of a system. The periodical solution exists in some interval of the pumping $G_{\textrm{c}1} < G < G_{\textrm{c}2} $. At specified parameters, the periodical solution exists at $0.0055 < G < 0.0092$. Outside this region, the solution describes a uniform system: the gas phase at a low pumping and the condensed phase at a large pumping. The upper part of the periodical distribution corresponds to a condensed phase, the lower part corresponds to the gas phase. The size of the condensed phase increases with the change of the pumping from $G_{\textrm{c}1}$ to $G_{\textrm{c}2}$. Figure \[fig2\] shows the spatial dependence of the exciton current calculated by the formula (\[eq16\]). The current equals zero in the centers of the condensed and gas phases and it has a maximum in the region of a transition from the condensed phase to the gas phase.
![ The spatial dependence of the exciton current at $G=0.008$, $D_1=0.03$, $b=-1.9$. []{data-label="fig2"}](f2_){width="8.6cm"}
Let us do some estimations. The results for the currents in figure \[fig2\] are presented in dimensionless units: $\tilde{j}=j/j_0$, where $j_0=n_0u_0$, $u_0=(\tau_\textrm{sc}n_0a)/(m\xi)$ is the unit of the velocity. The exciton density is presented in figure \[fig1\] in dimensionless units ($\tilde{n}=n/n_0$). It is seen in figure \[fig1\] that $\tilde{n}\sim 1$, and the magnitude of $n$ is of an order of $n_0$. Thus, for estimations we may assume that $n_0a$ corresponds to the shift of the luminescence line with an increase of the exciton density, the magnitude of $\xi$ is of the order of the size of the condensed phase. For the following magnitudes of parameters $\tau_\textrm{sc}=10^{-11}$ [s]{}, $n_0a=2\cdot10^{-3}$ [eV]{}, $m=2\cdot10^{-28}$ [g]{}, $\xi=2\cdot 10^{-4}$ [cm]{}, we obtain $u_0\sim 10^6$ [cm/s]{}. According to calculations (see figure \[fig2\]), the magnitudes of the current and the velocity are two orders of magnitude less than their units $j_0$ and $u_0$, so the condition $u\sim10^{4}$ [cm/s]{} takes place. In order to verify the fulfilment of the condition (\[eq14b\]), let us suppose that ($\partial u_i)/(\partial x_k\sim u/l$), where $l$ is the period of a structure. It follows from experiments [@But02; @Gor06] that $l\sim (5\div10)$ m. Using these data we see that the condition ($\ref{eq14b}$) is very well satisfied. This condition is violated at $\tau_\textrm{sc}\geqslant 10^{-9}$ [s]{}. Therefore, the formation of nonuniform exciton dissipative structures in a double quantum well occurs due to the diffusion movement of excitons. To prove the main hydrodynamic equation (\[eq17\]), the last term in equation (\[eq9\])is of importance. It describes the loss of the momentum due to the scattering of excitons by defects and phonons. It is this term that describes the processes that cause a decay of the exciton flux. From the viewpoint of a possibility of the appearance of superfluidity, the situation for excitons is more complicated than that for the liquid helium and for the atoms of alkali metals at ultralow temperatures. In the latter systems, the phonons (movement of particles) are an intrinsic compound part of the system spectrum, the interaction between phonons (particles) is the interaction between the atoms of a system and does not cause the change of the complete momentum of a system and its movement as a whole. Phonons and defects for excitons are external subsystems that brake the exciton movement. Therefore, to create the exciton superfluidity, it is needed that the value of $\tau _\textrm{sc}$ should grow significantly. This is possible for exciton polaritons that weakly interact with phonons; moreover, there is a certain experimental evidence on an observation of the polariton condensation [@Kas06]. For indirect excitons, the critical temperature of a superfluid transition is strongly lowered by inhomogeneities [@Ber06; @Bez11]. Thus, the question regarding the possibility of the superfluidity existence for indirect excitons on the basis of AlGaAs system is open.
Thus, the peculiarities observed at large densities of indirect excitons may be explained by phase transitions in a system of particles having attractive interactions and by the finite value of the lifetime without an involvement of the Bose-Einstein condensation.
Distribution of excitons between localized and delocalized states
=================================================================
According to the experimental results [@Yan07], the frequency of the emission from the islands of a condensed phase, where the exciton density is large, is higher than the frequency of emission from the region between islands, in which the density is less. The authors made the conclusion [@Yan07] that the interaction between excitons is repulsive, and, therefore, the formation of a condensed phase by attractive interaction is impossible. This contradicts the main assumption of our works [@Sug05; @Sug06; @Cher06; @Sug07; @Cher07; @Cher09; @Cher12], though these works explain many experiments. Now, we remove this contradiction taking into account the presence of localized excitons.
The localized states arise due to the presence of residual donors, acceptors, defects, and inhomogeneous thickness of the wells. Their existence is confirmed by the presence of an emission at the frequencies less than the frequency of the exciton band emission and by broadening of exciton lines. At a low temperature and at a small pumping, the main part of the emission band consists of the emission from defect centers, while the part of the exciton emission grows with an increased pumping. Now, we consider the relation between the contribution to the emission band intensity from free excitons and from the excitons (pairs of electrons and holes) localized on the defects. We assume that the localized states are saturable, namely, every center may capture a restricted number of electron-hole pairs. In our calculations we assume that only a single excitation may be localized on a defect. There are no other localized excitations or they have a very low binding energy and are unstable. The dependence of the density of localized states on the energy was chosen in the exponential form, namely $\rho (E) = \alpha N_l \exp( \alpha E)$, where $N_l $ is the density of the defect centers, $E$ is the depth of the trap level. The exciton states (free and localized) are distributed onto levels after the creation of electrons and holes due to an external irradiation and their subsequent recombination and relaxation. Since the time of relaxation is much less than the exciton lifetime, the distribution of excitation between free and localized states corresponds to the thermodynamical equilibrium state. In the considered model, we should obtain a distribution of electron-hole pairs, whose population on a single level may be changed from zero to infinity for $E>0$ (for free exciton states) and from zero to one for $E<0$ (for localized states). Formally, in the considered system, free excitons have Bose-Einstein statistics while localized excitations obey the Fermi-Dirac statistics. At a small exciton density, Bose-Einstein and Boltsmann statistics give similar results for free excitons, but the application of Fermi-Dirac statistics for localized states on a single level for one trap is important. The equation for energy distribution may be found from the minima of a large canonical distribution $$\label{eqa} w(n_k,n_i)=\exp\left(\frac{\Omega+N\mu-E}{\kappa
T}\right),$$ where $N=\Sigma_in_i+\Sigma_kn_k$, $E=\Sigma_in_iE_i+\Sigma_{k,l}n_kE_k$, $n_i=0,1$, $n_k=0,1,\ldots,\infty$, $k$ is the wave vector of the exciton, $l$ designates the singular levels. Parameter $\mu$ is the exciton chemical potential.
![ The dependence of the density of free (thick line) and trapped (thin line) excitons on the pumping. The parameters of the system: $T=2K, N_l=0.001/a_\textrm{ex}^2,
\alpha=300 (\textmd{eV})^{-1}$.[]{data-label="fig3"}](f3_){width="8.6cm"}
The distribution of excitons over free and localized levels is determined from the minimum of the functional (\[eqa\]). As a result, we obtain the following conditions for the mean values of the free exciton density $n$ and the density of localized states $n_L$ $$\begin{aligned}
\label{eqb} n_\textrm{ex}&=\frac{g \nu}{4\pi
E_\textrm{ex}a_\textrm{ex}^2}\int_0^\infty\frac{\rd E}{\exp\left(\frac{E-\mu}{\kappa
T}\right)-1} \,,\\
%\end{equation}
%\begin{equation}
\label{eqc} n_L&=\alpha N_l
\int_{-\infty}^0\frac{\exp\left(\alpha E\right) \rd E}{\exp\left(\frac{E-\mu}{\kappa
T}\right)+1} \,,\end{aligned}$$ where $a_\textrm{ex}=(\hbar^2\varepsilon)/(\mu_\textrm{ex} e^2)$ and $E_\textrm{ex}=(\mu_\textrm{ex} e^4)/(2\varepsilon^2 \hbar^2)$ are the radius and the energy of the exciton in the ground state in the bulk material, $g=4$, $\mu_\textrm{ex}$ is the reduced mass of the exciton, $\nu$ is the ratio of the reduced and the total mass of the exciton. The chemical potential $\mu$ is determined from the condition $$\label{eq6}
n_L + n = G\tau_\textrm{ex}\,,$$ where $G\tau_\textrm{ex}$ is the whole number of excitation (free and localized) per unit surface.
The dependence of distribution of free and localized excitons on the pumping is presented in figure \[fig3\] as a function of the whole number of excitation presented in units of $1/a_\textrm{ex}^2$. Let the exciton radius be equal to 10 [nm]{}. Then, the concentration of the traps and the width of the distribution of trap levels, chosen under calculations of figure \[fig3\], are of the order of $10^9$ [cm]{}$^{-2}$ and 0.003 [eV]{}, correspondingly.
![ (Color online) The distribution of excitations in the traps and in the states of the exciton band. The thick line in figure \[fig4\] (a) corresponds to the energy per a single exciton in the condensed phase. On the right \[figure \[fig4\] (b)\], the upper line describes the whole emission from the island (the emission of both the condensed phase and trapped excitons), the low line describes the emission of the trapped excitons. []{data-label="fig4"}](f4_){width="8.9cm"}
As it is seen in figure \[fig3\], the number of localized excitations at small pumping exceeds the number of free excitons and the emission band is determined by the emission from the traps. With an increase of pumping, the occupation of the trap levels becomes saturated. For the chosen parameters, the concentration of excitations under saturation is of the order of $10^9$ [cm]{}$^{-2}$. The exciton density grows simultaneously with the saturation of the localized levels. As a result, the shortwave part of the emission band should increase with an increased pumping. When the exciton density becomes larger, the collective exciton effects begin to manifest themselves. The equations (\[eqb\]), (\[eqc\]) do not take into account the interactions between excitations, and special models and theories are needed to describe the collective effects. The appearance of a narrow line was observed in [@Lar00] with an increased pumping on the shortwave part of the exciton emission band. Simultaneously, patterns arise in the emission spectra. The narrow line appeared after the localized states become occupied. According to [@Lar00], this line is explained by the exciton Bose-Einstain condensation. According to our model [@Sug05; @Sug07], the appearance of the islands corresponds to the condensed phase caused by the attractive interaction between excitons. The energy per a single exciton in the condensed phase is less than the energy of free excitons (the thick line in figure \[fig4\]), but the gain of the energy under condensation of indirect excitons in AlGaAs system is less than the whole bandwidth, which are formed by the localized and delocalized states. Thus, the energy of photons emitted from the islands of a condensed phase is higher than the energy of photons emitted by traps (see figure \[fig4\]). The excitons cannot leave the condensed phase (the islands) and move to the traps (to the states of lower energy) since the levels of the traps are already occupied. This may be the reason of the results obtained in [@Yan07], where the maximum of the frequency of emission from the islands is higher than the maximum frequency from the regions between the islands *in spite of the attractive interaction between the excitons.*
The qualitative results coincide with the results obtained in [@SugUJP] using another method from the solution of kinetic equations for level distributions at some simple approximation for the probability transition between the levels. Similar behavior of distribution of free and trapped excitons is observed for another energy dependence of the density of localized states.
The results may be used to explain the intensity and temperature dependencies of the exciton emission of dipolar excitons in InGaAs coupled double quantum wells [@Schin13]. The authors observed a growth of the shortwave side of the band with an increased pumping.
Conclusion
==========
Hydrodynamic equations are analyzed for excitons in a double quantum well. The equations take into account the presence of pumping, the finite value of the exciton lifetime and the possibility of a condensed phase formation in a phenomenological model. The equations describe the diffusion and the ballistic movement of an exciton system. It is shown that the spatial nonuniform structures, observed experimentally in double wells on the basis of AlGaAs crystal, may be explained by hydrodynamic equations in the diffusion approximation. The effect of saturable localized states on spectral distribution of the emission from condensed and gas phases is obtained. The theory explains the features of experimental dependencies of the emission spectra from the condensed and gas phases.
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---
abstract: 'We consider a new class of open covers and classes of spaces defined from them, called $\iota $-spaces (“iota spaces"). We explore their relationship with $\epsilon $-spaces (that is, spaces having all finite powers Lindelöf) and countable network weight. An example of a hereditarily $\epsilon $-space whose square is not hereditarily Lindel" of is provided.'
address:
- 'Department of Mathematics, Faculty of Science and Engineering, York University, Toronto, ON, M3J 1P3 Canada'
- 'Department of Mathematics, Faculty of Science and Engineering, York University, Toronto, ON, M3J 1P3 Canada'
- 'Department of Mathematics, Faculty of Science and Engineering, York University, Toronto, ON, M3J 1P3 Canada'
author:
- Natasha May
- Santi Spadaro
- Paul Szeptycki
title: A new class of spaces with all finite powers Lindelöf
---
Introduction
============
A topological space in which each finite power is Lindel" of is called an *$\epsilon $-space*. Equivalently, $X$ is an $\epsilon $-space if every open $\omega $-cover of $X$ has a countable $\omega $-subcover, where a cover of a space $X$ is an *$\omega $-cover* if each finite subset of $X$ is contained in an element of the cover. A natural generalization of an $\omega $-cover can be defined by requiring that disjoint finite sets be separated by a member of the cover. We call a cover ${\mathcal U}$ of a space $X$ an *$\iota $-cover* if for every pair of disjoint finite sets $F, G\subseteq X$, there is a member $U\in {\mathcal U}$ such that $F\subseteq U$ and $G\cap U =\emptyset $. Notice that every space with a countable network is an $\epsilon $-space. Furthermore, we will show that every $T_2$ space with a countable network has the property that every open $\iota $-cover has a countable refinement that is also an $\iota $-cover. Hence, we call such spaces with this property $\iota $-spaces. The motivation for these definitions of $\iota$-cover and $\iota $-space arose when the third named author was trying to make the example in [@SS2] zero dimensional to solve the D-space problem.
We will explore the relationship between $\iota $-spaces and countable network weight, providing a ZFC example of a regular $\iota $-space with no countable network. We also investigate the relationship between $\epsilon $-spaces and $\iota $-spaces, determining an additional property that makes them equivalent. We use the notion of an $\iota $-cover to construct a hereditarily $\epsilon$-space whose square is not hereditarily Lindel" of. Finally, we give an example of a non D-space that has a countable open $\iota $-cover.
Preliminaries
=============
A family of sets $\mathcal{U}$ is an $\omega$-cover of $X$ if for every $F \in [X]^{<\omega}$ there is $U \in \mathcal{U}$ such that $F \subset U$.
A family of sets $\mathcal{U}$ is an *$\iota$-cover* (*$n$-ota cover*) of $X$ if for every $F, G \in [X]^{<\omega}$ ($F, G \in [X]^n$) such that $F \cap G=\emptyset$ there is a member $U \in \mathcal{U}$ such that $F \subset U$ and $G \cap U=\emptyset$.
In the following proposition we collect a few trivial facts about $\iota$-covers and their relationship with $\omega$-covers.
[ ]{}
1. Every $\iota$-cover is an $\omega$-cover.
2. Any open $\omega$-cover of a $T_1$ topological space has an open refinement that is an $\iota$-cover.
3. Any fattening of an $\iota$-cover is an $\omega$-cover.
We call a space $X$ an $\epsilon$-space if every open $\omega$-cover of $X$ has a countable $\omega$-subcover.
\[defiotaspace\] We call a space $X$ an *$\iota$-space* (*$n$-ota space*) if every open $\iota$-cover ($n$-ota cover) of $X$ has a countable refinement which is an $\iota$-cover ($n$-ota cover).
In Definition $\ref{defiotaspace}$ we used *refinement* rather than *subcover* because the class of spaces where every $\iota$-cover has a countable $\iota$-subcover coincides with the class of countable spaces. Indeed if $X$ is uncountable and $T_1$ then $\{X \setminus F : F \in [X]^{<\omega} \}$ is an $\iota$-cover without a countable $\iota$-subcover.
While every space $X$ has a countable open $\omega$-cover (simply consider $\{X\}$), not all spaces have countable $\iota$-covers.
\[noiotacover\] A compact $T_2$ space of size $\omega_1$ without a countable $\iota$-cover.
Let $X=D \cup \{p\}$ be the one-point compactification of a discrete set of size $\aleph_1$, where $p$ is the unique non-isolated point. Suppose by contradiction that $X$ has a countable $\iota$-cover $\mathcal{U}$ and let $\mathcal{U}_p=\{U \in \mathcal{U}: p \in U \}$. The set $\mathcal{U}_p$ is countable and every element of $\mathcal{U}_p$ is a cofinite set. Therefore, the set $\bigcap \mathcal{U}_p$ is uncountable and hence we can fix distinct points $x, y \in \bigcap \mathcal{U}_p$. But then $\mathcal{U}$ has no element containing $\{p,x\}$ and missing $\{y\}$. Therefore $\mathcal{U}$ is not an $\iota$-cover.
In view of Example $\ref{noiotacover}$ it makes sense to consider the following class of spaces.
We call a space $X$ an $\iota_w$-space if it has a countable open $\iota$-cover.
Every $\iota$-space is certainly an $\iota_w$-space, but the converse is far from being true.
\[bound\] Let $X$ be an $\iota_w$-space. Then $|X| \leq \mathfrak{c}$.
Let $\mathcal{U}$ be a countable open $\iota$-cover for $X$. Define a map $f: X \to [\mathcal{U}]^\omega$ as follows: $f(x)=\{U \in \mathcal{U}: x\in U \}$. Since $\mathcal{U}$ is an $\iota$-cover, $f$ is a one-to-one map. Therefore $|X| \leq |[\mathcal{U}]^\omega|=\mathfrak{c}$.
The discrete space of size $\kappa$ is an $\iota_w$-space if and only if $\kappa \leq \mathfrak{c}$.
If $\kappa \leq \mathfrak{c}$ fix a separable metric topology $\tau$ on $\kappa$. Any $\iota$-refinement of $\tau$ provides an open $\iota$-cover of $\kappa$ with the discrete topology. The converse follows from Proposition $\ref{bound}$.
There is, however, a natural relationship between $\iota$-spaces and $\iota_w$-spaces.
\[epsilon+weakiota\] A space $X$ is an $\iota$-space if and only if it is an $\epsilon$-space and an $\iota_w$-space.
The direct implication is trivial. To prove the converse implication, fix a countable $\iota$-cover $\mathcal{C}$ for $X$ and let $\mathcal{U}$ be any open $\iota$-cover. Then $\mathcal{U}$ is also an $\omega$-cover, and since $X$ is an $\epsilon$-space we can find a countable $\omega$-subcover $\mathcal{V}$ of $\mathcal{U}$. The set $\{U \cap V: U \in \mathcal{C}, V \in \mathcal{V} \}$ is then a countable $\iota$-refinement of $\mathcal{U}$.
There are hereditarily Lindelöf spaces which are $\iota_w$-spaces, but not $\iota$-spaces. One such example is the Sorgenfrey line. Indeed, since its topology is a refinement of the topology of the real line, it has a countable $\iota$-cover, but its square is not Lindelöf and hence it’s not an $\iota$-space. Note that by Theorem $\ref{epsilon+weakiota}$ if $X$ is a subspace of the Sorgenfrey Line, then $X$ is an $\epsilon$-space if and only if $X$ is an $\iota$-space.
Let $X$ be a Tychonoff space such that $C_p(X)$ is separable and has countable tightness. Then $X$ is an $\iota$-space.
From [@A], $C_p(X)$ has countable tightness if and only if $X$ is an $\epsilon$-space and $C_p(X)$ is separable if and only if $X$ has a one-to-one continuous map onto a separable metrizable space. It’s easy to see that this last condition is equivalent to $X$ having a coarser second-countable topology. But this easily implies that $X$ has a countable $\iota$-cover, that is, $X$ is an $\iota_w$-space.
Let $X$ be a Tychonoff space. Suppose $C_p(X)$ is hereditarily separable. Then $X$ is an $\iota$-space.
[ ]{} \[subspaceprop\]
1. \[noniotaclosed\] Let $(X, \tau)$ be an $\iota$-space, then every closed subspace is an $\iota$-space.
2. \[noniotaany\] Let $(X, \tau)$ be an $\iota_w$-space, then every subspace is an $\iota_w$-space.
To prove ($\ref{noniotaclosed}$) suppose $Y$ is a closed subspace of the $\iota$-space $X$. Fix an open cover $\mathcal{U}_Y$ of $Y$. Let $\mathcal{U}=\{U \in \tau: U \cap Y \in \mathcal{U}_Y \}$ and $\mathcal{U}^Y=\{U \in \tau: U \cap Y=\emptyset \}$.
Let $$\mathcal{V}=\{(U \setminus F) \cup V: U \in \mathcal{U}, F \in [Y]^{<\omega} , V \in \mathcal{U}^Y \}$$
Then $\mathcal{V}$ is an $\iota$-cover for the whole space $X$ and the trace of any countable $\iota$-refinement of $\mathcal{V}$ on $Y$ is a countable $\iota$-refinement of $\mathcal{U}_Y$.
The proof of ($\ref{noniotaany}$) is similar and even easier.
Let $\{X_i: i \in I\}$ be a family of spaces, where $|X_i| \geq 2$ and $|I| \geq \aleph_1$. Then $\prod_{i \in I} X_i$ is not an $\iota_w$-space.
Simply note that $\prod_{i \in I} X_i$ contains a copy of $2^{|I|}$, which in turn contains a copy of the one-point compactification of a discrete space of size $|I|$ and that this space is not an $\iota_w$-space.
\[productiw\] Let $\{X_i: i < \omega \}$ be a countable family of $\iota_w$-spaces. Then $X:=\prod_{i <\omega} X_i$ is an $\iota_w$-space.
Let $\mathcal{U}_i$ be a countable open $\iota$-cover for $X_i$. Consider two disjoint finite subsets $F$ and $G$ of $X$. For every $(x_n)_{n<\omega} \in F$ let $U_{x_i} \in \mathcal{U}_i$ be an open set containing $x_i$ and missing $\{z \in \pi_{X_i}(F \cup G): z \neq x_i \}$. Let now $U= \bigcup \{\prod_{i <\omega} U_{x_i}: (x_i)_{i<\omega} \in F \}$. Then $U$ contains $F$ and misses $G$. Indeed if $G \cap U$ were non-empty then there would be $(x_n)_{n<\omega} \in F$ such that $(\prod_{n<\omega} U_{x_n}) \cap G \neq \emptyset$, but this contradicts the definition of $U_{x_n}$ for $n<\omega$.
It follows that $\{\bigcup \{\prod_{i<\omega} U_i: U_i \in \mathcal{F}_i\}: \mathcal{F}_i \in [\mathcal{U}_i]^{<\omega} \}$ is a countable open $\iota$-cover for $\prod_{i<\omega} X_i$.
Let $\{X_i: i \in I\}$ be a family of $\iota_w$-spaces. Then $\prod_{i \in I} X_i$ is an $\iota_w$-space if and only if $|I| \leq \omega$.
Countable Network Weight
========================
It is known that $T_2$ spaces with a countable network are $\epsilon$-spaces, so using Theorem \[epsilon+weakiota\] we have the following.
Every $T_2$ space with a countable network is an $\iota$-space.
The converse is not true. We are going to present three counterexamples. The first one has the advantage of being simpler, the second one has the advantage of being regular, and the third one is only consistent, but we present it anyway, because the techniques used in verifying its properties might have independent interest.
There is a $T_2$ $\iota$-space without a countable network.
Let $\mathbb{R}_c$ be the real line with the topology generated by sets of the form $U \setminus C$, where $U$ is a Euclidean open set and $C$ is a countable set of reals.
Suppose by contradiction that $\{N_n: n < \omega \}$ is a countable network for $\mathbb{R}_c$. Without loss of generality we can assume that $N_n$ is infinite for every $n$ and use this to inductively pick $x_n \in N_n \setminus \{x_i: i < n \}$. then $\mathbb{R}_c \setminus \{x_i: i < \omega \}$ is an open set not containing any element of $\{N_n: n < \omega \}$. It follows that $\mathbb{R}_c$ does not have a countable network.
Now $\mathbb{R}_c$ is a refinement of the Euclidean topology on $\mathbb{R}$ and hence it is both an $\epsilon $-space and an $\iota _w$-space. Therefore, by Theorem \[epsilon+weakiota\], $\mathbb{R}_c$ is an $\iota$-space.
There is a regular $\iota$-space without a countable network within the usual axioms of ZFC.
Let $X \subset \mathbb{R}$ be a subset of the reals. By Michael-type space $L(X)$ we mean the refinement of the usual topology on $\mathbb{R}$ obtained by isolating every point of $\mathbb{R} \setminus X$. By Theorem $\ref{epsilon+weakiota}$ every Michael-type space which is an $\epsilon$-space is also an $\iota$-space. It’s easy to see, that if $X$ is a Bernstein set (that is, a set which hits every uncountable closed set of the real line along with its complement), then $L(X)$ is Lindelöf, and Lawrence [@L] proved that there is in ZFC a Bernstein set $X \subset \mathbb{R}$ such that $L(X)$ is an $\epsilon$-space. The techniques used to construct the Bernstein set originated in [@P] and Burke gives the details of the construction in [@B].
The next construction preceded Theorem \[epsilon+weakiota\], but we include it because it may be of independent interest. It gives a recursive construction of an $\iota $-space.
\[Michael\] There is a Michael space, $M_X$, that is an $\iota $-space.
For convenience, call ${\mathcal U}$ an open finite union (ofu)-$\iota $-cover of ${\mathbb Q}$ if
1. $\forall U\in {\mathcal U}$, $U=\bigcup _{i<n}I_i$ where $n\in \omega $, $I_i=(p_i,q_i)$, $p_i,q_i\in {\mathbb Q}$.
2. ${\mathbb Q}\subseteq \bigcup {\mathcal U}$
3. $\forall F,G\in [{\mathbb Q}]^{<\omega }$ such that $F\cap G=\emptyset $, $\exists U=\bigcup _{i<n}I_i\in {\mathcal U}$ such that $F\subseteq U$, $\bar{I}_i\cap G=\emptyset $, $\forall i<n$.
Let $\{{\mathcal U}_\alpha :\alpha <\omega _1\}$ enumerate all (ofu)-$\iota $-covers of ${\mathbb Q}$. Define by recursion $X=\{x_\alpha :\alpha <\omega _1\}$ so that $\forall \alpha <\omega _1$, IH($\alpha )$ holds, where\
\
IH($\alpha $): $\forall \beta <\alpha $, ${\mathcal U}_\beta $ is an $\iota $-cover of ${\mathbb Q}\cup \{x_\xi :\beta <\xi <\alpha \}$.\
\
Let $x_0\in {\mathbb R}\setminus {\mathbb Q}$.\
Fix $\alpha <\omega _1$ and suppose $\{x_\xi :\xi <\alpha \}$ have been defined.\
We must choose $x_\alpha $ so that IH($\alpha +1$) is satisfied. That is, ${\mathcal U}_\beta $ is an $\iota $-cover of ${\mathbb Q}\cup \{x_\xi :\beta <\xi \leq \alpha \}$, $\forall \beta \leq \alpha $.
For ${\mathcal U}\in \{{\mathcal U}_\beta :\beta \leq \alpha \}$, let ${\mathbb Q}_{{\mathcal U}}={\mathbb Q}\cup \{x_\xi :\beta <\xi <\alpha \}$ where ${\mathcal U}={\mathcal U}_\beta $.
Let ${\mathcal T}=\{({\mathcal U},F,G)\in \{{\mathcal U}_\beta :\beta \leq \alpha \}\times [{\mathbb Q}\cup \{x_\xi :\xi <\alpha \}]^{<\omega }\times [{\mathbb Q}\cup \{x_\xi :\xi <\alpha \}]^{<\omega }: F,G\in [{\mathbb Q}_{\mathcal U}]^{<\omega }, F\cap G=\emptyset \}$. Enumerate ${\mathcal T}=\{({\mathcal U}_{\alpha n},F_n,G_n):n\in \omega \}$. For $n\in \omega $, let $\beta _n \leq \alpha $ such that ${\mathcal U}_{\alpha n}={\mathcal U}_{\beta _n}$. Then, $F_n, G_n \in [{\mathbb Q}\cup \{x_\xi :\beta _n<\xi <\alpha \}]^{<\omega }$ and by IH($\alpha )$, ${\mathcal U}_{\alpha n}$ is an $\iota $-cover of ${\mathbb Q}\cup \{x_\xi :\beta _n<\xi <\alpha \}$.\
\
Build sequences $\{q_n:n\in \omega \}\subseteq {\mathbb Q}$, $\{U_n^i:n\in \omega, i<2\}$, $\{I_n:n\in \omega \}$ and $\{V_n:n\in \omega \}$ such that
1. $U_n^i\in {\mathcal U}_{\alpha n} \forall i<2$, $n\in \omega $.
2. $I_n$, $V_n$ open intervals.
3. $q_0\in {\mathbb Q}\setminus (F_0\dot{\cup }G_0)$ and $q_n\in V_{n-1}\cap {\mathbb Q}\setminus (F_n\dot{\cup }G_n)$, $\forall n\geq 1$.
4. $F_n\cup \{q_n\}\subseteq U_n^0$, $U_n^0\cap G_n=\emptyset $ and $F_n\subseteq U_n^1$, $U_n^1\cap (G_n\cup \{q_n\})=\emptyset $.
5. $q_0\in I_0\subseteq U_n^0$ and $q_n\in I_n\subseteq V_{n-1}\cap U_n^0$, $\forall n\geq 1$.
6. $\overline{V}_n\subseteq I_n\setminus (U_n^1\cup \{q_n\})$ such that diam$(V_n)<\frac{1}{n}$ ($\forall n\geq 1)$
Let $q_0\in {\mathbb Q}\setminus (F_0\dot{\cup }G_0)$. Then $F_0\cup \{q_0\}, G_0\in [{\mathbb Q}\cup \{x_\xi :\beta _0<\xi <\alpha \}]^{<\omega }$ so let $U_0^0=\bigcup _{i<k_0}I_i \in {\mathcal U}_{\alpha 0}$ such that $F_0\cup \{q_0\}\subseteq U_0^0$, $G_0\cap U_0^0=\emptyset $. Let $I_0\in \{I_i:i<k_0\}$ such that $q_0\in I_0$. Also, $F_0,G_0\cup \{q_0\}\in [{\mathbb Q}\cup \{x_\xi :\beta _0<\xi <\alpha \}]^{<\omega }$ so let $U_0^1=\bigcup _{i<m_0}I_i$ such that $F_0\subseteq U_0^1$, $(G_0\cup \{q_0\})\cap U_0^1=\emptyset $. Let $V_0$ be an open interval such that $\overline{V}_0\subseteq I_0\setminus (U_0^1\cup \{q_0\})$.\
Fix $n\in \omega $ and suppose $\{q_m:m<n\}$, $\{U_m^i:i<2,m<n\}$, $\{I_m:m<n\}$ and $\{V_m:m<n\}$ have been defined.\
Let $q_n\in V_{n-1}\cap {\mathbb Q}\setminus (F_n\dot{\cup }G_n)$. Then $F_n\cup \{q_n\}, G_n\in [{\mathbb Q}\cup \{x_\xi :\beta _n<\xi <\alpha \}]^{<\omega }$ so let $U_n^0=\bigcup _{i<k_n}I_i \in {\mathcal U}_{\alpha n}$ such that $F_n\cup \{q_n\}\subseteq U_n^0$, $G_n\cap U_n^0=\emptyset $. Let $I_n\subseteq V_{n-1}\cap U_n^0$ be an open interval such that $q_n\in I_n$. Also, $F_n,G_n\cup \{q_n\}\in [{\mathbb Q}\cup \{x_\xi :\beta _n<\xi <\alpha \}]^{<\omega }$ so let $U_n^1=\bigcup _{i<m_n}I_i$ such that $F_n\subseteq U_n^1$, $(G_n\cup \{q_n\})\cap U_n^1=\emptyset $. Let $V_n$ be an open interval of diameter $<\frac{1}{n}$ such that $\overline{V}_n\subseteq I_n\setminus (U_n^1\cup \{q_n\})$.\
Let $x_\alpha \in {\mathbb R}\setminus {\mathbb Q}$ such that $\bigcap _{n\in \omega }\overline{V}_n=\{x_\alpha \}$.\
To see IH$(\alpha +1)$ is satisfied, let $\beta \leq \alpha $ and notice ${\mathcal U}_\beta $ is an $\iota $-cover of ${\mathbb Q}\cup \{x_\xi :\beta <\xi \leq \alpha \}$: Let $F,G\in [{\mathbb Q}\cup \{x_\xi :\beta <\xi \leq \alpha \}]^{<\omega }$ such that $F\cap G=\emptyset $. If $x_\alpha \in F$, let $F^\shortmid =F\setminus \{x_\alpha \}$ and $m\in \omega $ such that $({\mathcal U}_\beta , F^\shortmid , G)=({\mathcal U}_{\alpha m},F_m,G_m)$. Then, $U_m^0\in {\mathcal U}_{\alpha m} $ such that $F_m\subseteq U_m^0, G_m\cap U_m^0=\emptyset $ and $x_\alpha \in V_m\subseteq U_m^0$. Therefore, $U_m^0\in {\mathcal U}_{\beta} $ such that $F\subseteq U_m^0$ and $G\cap U_m^0=\emptyset $. If $x_\alpha \in G$, let $G^\shortmid =G\setminus \{x_\alpha \}$ and $k\in \omega $ such that $({\mathcal U}_\beta , F, G^\shortmid )=({\mathcal U}_{\alpha k},F_k,G_k)$. Then, $U_k^1\in {\mathcal U}_{\alpha k} $ such that $F_k\subseteq U_k^1, G_k\cap U_k^1=\emptyset $ and $x_\alpha \in V_k\subseteq I_k\setminus (U_k^1\cup \{q_k\})$. Therefore, $U_k^1\in {\mathcal U}_{\beta} $ such that $F\subseteq U_k^1$ and $G\cap U_k^1=\emptyset $.
Therefore, by construction, ${\mathcal U}_\beta $ is an $\iota $-cover of ${\mathbb Q}\cup \{x_\xi :\xi >\beta \}$, $\forall \beta <\omega _1$. So, ${\mathcal U}_\beta $ is an $\iota $-cover of a tail of ${\mathbb Q}\cup X$.
Let $M_X={\mathbb Q}\cup X$ with the Michael topology (usual basic open neighbourhoods for ${\mathbb Q}$ and isolate points of $X$). Let ${\mathcal U}$ be an open $\iota $-cover of $M_X$.
For $F,G\in [X]^{<\omega }$ such that $F\cap G=\emptyset $, let ${\mathcal U}_{FG}=\{U\in {\mathcal U}:F\subseteq U, U\cap G=\emptyset \}$.
For any $F,G \in [X]^{<\omega }$ such that $F\cap G =\emptyset $, $\exists \alpha _{FG}<\omega _1$ such that ${\mathcal U}_{\alpha _{FG}}\prec {\mathcal U}_{FG}$.
Note that, ${\mathcal U}_{FG}$ is an $\iota $-cover of ${\mathbb Q}\cup X\setminus (F\cup G)$. So, for each $F^\shortmid , G^\shortmid \in [{\mathbb Q}]^{<\omega }$ such that $F^\shortmid \cap G^\shortmid =\emptyset $, let $U(F^\shortmid ,G^\shortmid )\in {\mathcal U}_{FG}$ such that $F^\shortmid \subseteq U(F^\shortmid ,G^\shortmid )$ and $U(F^\shortmid ,G^\shortmid )\cap G^\shortmid =\emptyset $. For $x\in F^\shortmid $, let $p_{x(F^\shortmid G^\shortmid )}, q_{x(F^\shortmid G^\shortmid )} \in {\mathbb Q}$ such that $x\in I_{x(F^\shortmid G^\shortmid )}=(p_{x(F^\shortmid G^\shortmid )}, q_{x(F^\shortmid G^\shortmid )})\subseteq U(F^\shortmid ,G^\shortmid )$ but $\bar{I}_{x(F^\shortmid G^\shortmid )}\cap G^\shortmid =\emptyset $. Let $V_{F^\shortmid G^\shortmid }=\bigcup _{x\in F^\shortmid }I_{x(F^\shortmid G^\shortmid )}$. Then ${\mathcal V}=\{V_{F^\shortmid G^\shortmid }:F^\shortmid , G^\shortmid \in [{\mathbb Q}]^{<\omega }, F^\shortmid \cap G^\shortmid =\emptyset \}$ is an (ofu)-$\iota $-cover of ${\mathbb Q}$ that refines ${\mathcal U}_{FG}$. So, let $\alpha _{FG}<\omega _1$ such that ${\mathcal V}={\mathcal U}_{\alpha _{FG}}$.\
\
By a closing off argument, let $\bar{\alpha }<\omega _1$ such that ${\mathcal U}_{\bar{\alpha }}\prec {\mathcal U}_{FG}$, $\forall F,G\in [\{x_\xi :\xi \leq \bar{\alpha }\}]^{<\omega }$ such that $F\cap G=\emptyset $.
${\mathcal U}^\shortmid =\{U\cup F:U\in {\mathcal U}_{\bar{\alpha }}, F\in [\{x_\xi :\xi \leq \bar{\alpha }\}]^{<\omega }\}$ is a countable open refinement of ${\mathcal U}$ that is an $\iota $-cover.
To see ${\mathcal U}^\shortmid \prec {\mathcal U}$, let $U\cup F\in {\mathcal U}^\shortmid $. Then $U\in {\mathcal U}_{\bar{\alpha }}$, $F\in [\{x_\xi :\xi \leq \bar{\alpha }\}]^{<\omega }$ so let $G\in [\{x_\xi :\xi \leq \bar{\alpha }\}\setminus F]^{<\omega }$ and since ${\mathcal U}_{\bar{\alpha }}\prec {\mathcal U}_{FG}$, let $U^\shortmid \in {\mathcal U}_{FG}$ such that $U\subseteq U^\shortmid $. Then, $U^\shortmid \in {\mathcal U}$ such that $U\cup F\subseteq U^\shortmid $.
To see ${\mathcal U}^\shortmid $ is an $\iota $-cover, let $F^\shortmid ,G^\shortmid \in [M_X]^{<\omega }$ such that $F^\shortmid \cap G^\shortmid =\emptyset $. Let $F_1^\shortmid =F^\shortmid \cap \{x_\xi :\xi \leq \bar{\alpha }\}$, $F_2^\shortmid =F^\shortmid \cap ({\mathbb Q}\cup \{x_\xi :\xi >\bar{\alpha }\})$, $G_1^\shortmid =G^\shortmid \cap \{x_\xi :\xi \leq \bar{\alpha }\}$, $G_2^\shortmid =G^\shortmid \cap ({\mathbb Q}\cup \{x_\xi :\xi >\bar{\alpha }\})$. Then, $F_2^\shortmid ,G_2^\shortmid \in [{\mathbb Q}\cup \{x_\xi :\xi >\bar{\alpha }\}]^{<\omega }$ such that $F_2^\shortmid \cap G_2^\shortmid =\emptyset $. So let $U\in {\mathcal U}_{\bar{\alpha }}$ such that $F_2^\shortmid \subseteq U$ and $U\cap G_2^\shortmid =\emptyset $. Then, $U\cup F_1^\shortmid \in {\mathcal U}^\shortmid $ such that $F^\shortmid \subseteq U\cup F_1^\shortmid $ and $(U\cup F_1^\shortmid )\cap G^\shortmid =\emptyset $.
We constructed $M_X$ so that any open $\iota $-cover of $M_X$ has a countable open refinement that $\iota $-covers a tail of $M_X$, which is enough to show that $M_X$ is an $\iota $-space since the countable open refinement ${\mathcal U}^\shortmid $ of ${\mathcal U}$ that $\iota $-covers $M_X$ is defined from an $\iota $-cover of a tail or an almost $\iota $-cover. This leads us to our next definition and some useful facts.
\[almostiota\] A space $X$ is [*almost*]{}-$\iota $ if for every open $\iota $-cover ${\mathcal U}$ of $X$, there is a countable open refinement ${\mathcal V}$ of ${\mathcal U}$ and $A\in [X]^\omega $ such that ${\mathcal V}$ is an $\iota $-cover of $X\setminus A$.
Almost-$\iota $ is closed hereditary.
\[almostiotalem\] If $X$ is almost-$\iota $ and has points regular $G_\delta $ then $X$ is an $\iota $-space.
Before proving Lemma \[almostiotalem\], we need the following:
If $X$ is almost-$\iota $ and has points regular $G_\delta $ then $X\setminus F$ is almost-$\iota, $ $\forall F\in [X]^{<\omega }$.
Fix $F\in [X]^{<\omega }$ and let ${\mathcal U}$ be any open $\iota $-cover of $X\setminus F$. Since $X$ has points regular $G_\delta $, let $U_n\subseteq X$ be open such that $F\subseteq U_n$, $\forall n\in \omega $ and $F=\bigcap\overline{U_n}$. For $n\in \omega $, let ${\mathcal U}_n=\{U\cap X\setminus U_n:U\in {\mathcal U}\}$, which is an open $\iota $-cover of $X\setminus U_n$. Thus, since $X\setminus U_n$ is almost-$\iota $ (being a closed subspace of an almost-$\iota $ space) let ${\mathcal U}_n^\shortmid $ be a countable open refinement of ${\mathcal U}_n$ and $A_n\in [X\setminus U_n]^{\omega }$ such that ${\mathcal U}_n^\shortmid $ is an $\iota $-cover of $(X\setminus U_n)\setminus A_n$. Finally, $\forall n\in \omega $, let ${\mathcal V}_n=\{V\setminus \overline{U_n}:V\in {\mathcal U}_n^\shortmid \}$. The following claim finishes the proof:
$\bigcup _{n\in \omega }{\mathcal V}_n$ is a countable open refinement of ${\mathcal U}$ that is an $\iota $-cover of $(X\setminus F)\setminus (\bigcup _{n\in \omega }A_n)$.
Let $F^\shortmid ,G^\shortmid \in [(X\setminus F)\setminus (\bigcup _{n\in \omega }A_n)]^{<\omega }$ such that $F^\shortmid \cap G^\shortmid =\emptyset $. Then, $(F^\shortmid \cup G^\shortmid )\cap F=\emptyset $ and $(F^\shortmid \cup G^\shortmid )\cap \bigcup _{n\in \omega }A_n=\emptyset $. So, $\exists k\in \omega $ such that $(F^\shortmid \cup G^\shortmid )\cap \overline{U_k}=\emptyset $ and $(F^\shortmid \cup G^\shortmid )\cap A_k=\emptyset $. Thus, $F^\shortmid ,G^\shortmid \in [(X\setminus U_k)\setminus A_k]^{<\omega }$ such that $F^\shortmid \cap G^\shortmid =\emptyset $. So, let $U \in {\mathcal U_k}^\shortmid $ such that $F^\shortmid \subseteq U$ and $U\cap G^\shortmid =\emptyset $. Then $U\setminus \overline{U_k}\in {\mathcal V}_k$ such that $F^\shortmid \subseteq U\setminus \overline{U_k}$ and $(U\setminus \overline{U_k})\cap G^\shortmid =\emptyset $.
Let ${\mathcal U}$ be any open $\iota $-cover of $X$. Let ${\mathcal M}$ be a countable elementary submodel of $H_{\theta }$ (for $\theta $ large enough) such that ${\mathcal U}, (X,\tau )\in {\mathcal M}$.
${\mathcal V}=\{V\in {\mathcal M}\cap \tau :V\subseteq U$ for some $U\in {\mathcal U}\}$ is a countable open refinement of ${\mathcal U}$ that is an $\iota $-cover of $X$.
Let $F^\shortmid ,G^\shortmid \in [X]^{<\omega }$ such that $F^\shortmid \cap G^\shortmid =\emptyset $. Let $F=F^\shortmid \cap {\mathcal M}$ and $G=G^\shortmid \cap {\mathcal M}$. By elementarity, ${\mathcal M}\models (X\setminus E$ is almost-$\iota $, $\forall E\in [X]^{<\omega })$. Thus, since $F,G\in {\mathcal M}$, ${\mathcal M}\models X\setminus (F\cup G)$ is almost-$\iota $. Notice ${\mathcal U}_{FG}=\{U\in {\mathcal U}:F\subseteq U, U\cap G=\emptyset \}\in {\mathcal M}$ is an open $\iota $-cover of $X\setminus (F\cup G)$. So, let ${\mathcal V}_{FG}\in {\mathcal M}$ be a countable open refinement of ${\mathcal U}_{FG}$ and $A_{FG}\in [X]^{\omega }\cap {\mathcal M}$ such that ${\mathcal V}_{FG}$ is an $\iota $-cover of $(X\setminus (F\cup G))\setminus A_{FG}$. Then, ${\mathcal V}_{FG},A_{FG}\subseteq {\mathcal M}$. Thus $F^\shortmid \setminus F, G^\shortmid \setminus G\in [(X\setminus (F\cup G))\setminus A_{FG}]^{<\omega }$ such that $F^\shortmid \setminus F \cap G^\shortmid \setminus G =\emptyset $. So let $V\in {\mathcal V}_{FG}$ such that $F^\shortmid \setminus F\subseteq V $ and $V\cap G^\shortmid \setminus G=\emptyset $. Since ${\mathcal V}_{FG}$ refines ${\mathcal U}_{FG}$, $\exists U\in {\mathcal U}_{FG}(V\subseteq U)$. So, by elementarity, let $U\in {\mathcal U}_{FG}\cap {\mathcal M}$ such that $V\subseteq U$. Also, ${\mathcal M}\models (x$ is regular $G_\delta $, $\forall x\in X$), so in particular, since $F\in {\mathcal M}$ is finite, let $U_n\in {\mathcal M}$ such that $F\subseteq U_n$, $\forall n\in \omega $ and $F=\bigcap _{n\in \omega }\overline{U_n}$. Then, since $G^\shortmid \cap F=\emptyset$, $\exists n\in \omega $ such that $G^\shortmid \cap \overline{U_n}=\emptyset $ and $V\cup (U_n\cap U)\in {\mathcal V}$ such that $F^\shortmid \subseteq V\cup (U_n\cap U)$ and $G^\shortmid \cap (V\cup (U_n\cap U))=\emptyset $.
Returning again to the relationship with countable network weight, we have seen some (consistent) counterexamples, but restricting ourselves to the hereditary property raises the natural question.
Is every hereditarily $\iota$-space a space with a countable network?
$\epsilon $-spaces
==================
Theorem \[epsilon+weakiota\] provides us with an instance when $\epsilon $-spaces and $\iota $-spaces are equivalent. We investigate what additional characteristics can be placed on an $\epsilon $-space to ensure it is an $\iota $-space.
Let $\mathcal{U}$ be a cover of a space $X$. We say that $\mathcal{U}$ is a *regular 1-ota cover* if for every $x\neq y \in X$ there is $U \in \mathcal{U}$ such that $x \in U$ and $y \notin \overline{U}$.
\[lemregular\] Let $\mathcal{U}$ be an $\iota$-cover of the regular space $X$. Then $\mathcal{U}$ has a regular 1-ota refinement.
For every $x\neq y \in X$ choose $U(x,y) \in \mathcal{U}$ such that $x \in U(x,y)$ and $y \notin U(x,y)$. Now let $V(x,y)$ be an open set such that $x \in V(x,y) \subset \overline{V(x,y)} \subset U(x,y)$. Then $\mathcal{V}=\{V(x,y): x\neq y \in X \}$ is a regular 1-ota refinement of $\mathcal{U}$.
\[tool\] Let $X$ be a regular space such that $X^2 \setminus \Delta$ is Lindelöf. Then $X$ is 1-ota.
Let $\mathcal{U}$ be a 1-ota cover for $X$ without a countable 1-ota refinement. Let $\mathcal{V}$ be a regular 1-ota refinement of $\mathcal{U}$ having minimal size $\kappa \geq \omega_1$.
Fix an enumeration $\{V_\alpha: \alpha < \kappa \}$ of $\mathcal{V}$ and let $\mathcal{V}_\alpha:=\{V_\beta: \beta \leq \alpha \}$ and:
$$A(\mathcal{V}_\alpha):=\{ (x,y) \in X^2 \setminus \Delta: (\forall U \in \mathcal{V_\alpha}) ((x \in U \wedge y \in \overline{U}) \vee (x \in \overline{U} \wedge y \in U) \vee (\{x,y\} \cap U=\emptyset ) \}.$$
[**Claim.**]{} $A(\mathcal{V}_\alpha) \neq \emptyset$ and $A(\mathcal{V}_\alpha)$ is closed in $X^2 \setminus \Delta$ for every $\alpha < \omega_1$.
The fact that $A(\mathcal{V}_\alpha) \neq \emptyset$ follows from the assumptions about $\mathcal{U}$. To prove that $A(\mathcal{V}_\alpha)$ is closed, let $(x,y) \notin A(\mathcal{V}_\alpha) \cup \Delta$. Then we can find $U_x, U_y \in \mathcal{V}_\alpha$ such that $x \in U_x$, $y \notin \overline{U_x}$, $y \in U_y$ and $x \notin \overline{U_y}$. Now $(U_x \setminus \overline{U_y}) \times (U_y \setminus \overline{U_x})$ is an open neighbourhood of $(x,y)$ which misses $A(\mathcal{V}_\alpha)$.
So $\{A(\mathcal{V}_\alpha): \alpha < \kappa \}$ is an uncountable decreasing sequence of non-empty closed subsets of the Lindelöf space $X^2 \setminus \Delta$ and thus $\bigcap \{A(\mathcal{V}_\alpha): \alpha < \omega_1 \} \neq \emptyset$ which contradicts regularity of $\mathcal{V}$.
Note that the fact that $X^2 \setminus \Delta$ is Lindelöf implies that $X$ has a $G_\delta$ diagonal, whenever $X$ is regular. Indeed, for every $x \in X^2 \setminus \Delta$, let $U_x$ be an open neighbourhood of $x$ such that $\overline{U_x} \cap \Delta = \emptyset$. The family $\{U_x: x \in X^2 \setminus \Delta \}$ covers $X^2 \setminus \Delta$, and hence there is a countable set $C \subset X^2 \setminus \Delta$ such that $X^2 \setminus \Delta = \bigcup \{\overline{U_x}: x \in C \}$ and hence $\Delta = \bigcap_{x \in C} X^2 \setminus \overline{U_x}$, which proves that $\Delta$ is a $G_\delta$ subset of $X$.
The following lemma is not new. For example, the proof of a more general statement can be found in [@BG]. We nevertheless include a quick direct proof of it for the reader’s convenience.
Every countably compact 1-ota space $X$ is metrizable.
Let $\mathcal{U}$ be an $\iota$-cover of $X$. By Lemma $\ref{lemregular}$ we can assume that $\mathcal{U}$ is a regular $\iota$-cover. Let $\mathcal{V}$ be a countable $\iota$-refinement of $\mathcal{U}$ and let $\mathcal{B}=\{ X \setminus \overline{\bigcup \mathcal{F}}: \mathcal{F} \in [\mathcal{V}]^{<\omega}\}$. The set $\mathcal{B}$ is countable. We claim that $\mathcal{B}$ is a base of $X$, proving that $X$ is metrizable.
To see that let $U$ be an open set and $x \in U$. For every $y \in X \setminus U$ choose an open set $U_y \in \mathcal{V}$ such that $y \in U_y$ and $x \notin \overline{U_y}$. The countable set $\{U_y: y \in X \setminus U \}$ covers $X \setminus U$ so we can choose a finite set $F \subset X \setminus U$ such that $X \setminus U \subset \bigcup_{y \in F} U_y$. Then $x \in \bigcap_{y \in F} X \setminus \overline{U_y} \subset U$ and hence $\mathcal{B}$ is a base.
There are no countably compact strong $L$-spaces.
If $X^2$ is hereditarily Lindelöf then $X$ is 1-ota and every countably compact 1-ota space is metrizable and thus separable.
Every compact space with a $G_\delta$ diagonal is metrizable.
If $X$ is a compact space with a $G_\delta$ diagonal then $X^2 \setminus \Delta$ is $\sigma$-compact. Thus $X$ is a compact 1-ota space and hence it’s metrizable.
Generalizing Lemma \[tool\] provides us with a characterization that we are looking for.
Let $\Delta_n=\{(x_1, \dots, x_n) \in X^n: |\{x_1, x_2, \dots, x_n \}|<n\}$. Clearly, $\Delta_n$ is a closed subset of $X^n$.
\[powersthm\] Assume $X$ is a regular space. If $X^{2n} \setminus \Delta_{2n}$ is Lindelöf for every $n\in \omega $ then $X$ is an $\iota$-space.
Suppose that $X^{2n} \setminus \Delta_{2n}$ is Lindelöf. We prove that $X$ is $n$-ota. Indeed, let $\mathcal{U}$ be a $n$-ota cover without a countable $n$-ota refinement. Let $\mathcal{V}$ be a regular $n$-ota refinement of $\mathcal{U}$ having minimal size $\kappa \geq \omega_1$. Enumerate $\mathcal{V}$ as $\{V_\alpha: \alpha < \aleph_1 \}$ and let $\mathcal{V}_\alpha=\{V_\beta: \beta \leq \alpha \}$ and $$A(\mathcal{V_\alpha})=\{(x_1, \dots x_{2n}) \in X^{2n} \setminus \Delta_{2n}: (\forall U \in \mathcal{V}_\alpha)(\{x_1, \dots x_n \} \subseteq U \wedge \{x_{n+1}, \dots , x_{2n}\} \cap \overline{U}\neq \emptyset)$$ $$\vee(\{x_1, \dots, x_n\} \cap \overline{U} \neq \emptyset \wedge \{x_{n+1}, \dots x_{2n}\} \subseteq U) \vee (\{x_1, \dots, x_n \} \nsubseteq U \wedge \{x_{n+1}, \dots x_{2n}\} \nsubseteq U) \}$$ [**Claim.**]{} $A(\mathcal{V}_\alpha)$ is closed.
Let $(x_1, \dots, x_{2n}) \notin A(\mathcal{V}_\alpha) \cup \Delta_{2n}$. Then we can find sets $U_1$ and $U_2$ which are open in $x$ and such that $\{x_1, \dots, x_n\} \subset U_1$, $\{x_{n+1}, \dots x_{2n}\} \cap \overline{U_1}=\emptyset$, $\{x_{n+1}, \dots, x_{2n}\} \subset U_2$ and $\{x_1, \dots, x_n \} \cap \overline{U_2}=\emptyset$. Then $(U_1^n \setminus \overline{U_2^n}) \times (U_2^n \setminus \overline{U_1^n})$ is an open neighbourhood of $(x_1, \dots x_n, x_{n+1}, \dots, x_{2n})$ which misses $A(\mathcal{V}_\alpha)$.
So $\{A(\mathcal{V}_\alpha): \alpha < \kappa \}$ is an uncountable decreasing chain of closed sets in $X^{2n} \setminus \Delta_{2n}$, and hence it has non-empty intersection. This contradicts that $\mathcal{V}$ is a regular $n$-ota cover. So if $X^i \setminus \Delta_i$ is Lindelöf for every $i<\omega$ then $X$ is $i$-ota for every $i<\omega$ and hence an $\iota$-space.
Every $\epsilon$-space with a $G_\delta$ diagonal is an $\iota$-space.
Suppose $X$ has a $G_\delta$ diagonal. Then $\Delta_n$ is a finite union of $G_\delta$ sets, and thus $G_\delta$. It follows that $X^n \setminus \Delta_n$ is a countable union of Lindelöf spaces, and thus Lindelöf. So $X$ is an $\iota$-space by Theorem \[powersthm\].
If $X^{2n} \setminus \Delta_{2n}$ is Lindelöf for some $n$, then $X$ is Lindelöf.
By the proof of Theorem $\ref{powersthm}$, $X$ is an $n$-ota space. But every $n$-ota space is Lindelöf. Indeed, let $\mathcal{U}$ be an open cover for $X$. Let $\mathcal{V}$ be the set of all $n$-sized unions from $\mathcal{U}$. Let $\mathcal{G}$ be an $n$-ota refinement of $\mathcal{V}$ and $\mathcal{F}$ be a countable refinement of $\mathcal{G}$. Then $\mathcal{F}$ naturally induces a countable refinement of the original cover $\mathcal{U}$.
Recall that a space $X$ is a *Lindelöf $\Sigma$-space* if it has a cover $\mathcal{C}$ by compact sets, and a countable family $\mathcal{N}$ of closed subsets of $X$ which is a *network modulo $\mathcal{C}$*, that is, for every $C \in \mathcal{C}$ and every open set $U$ such that $C \subset U$ there is $N \in \mathcal{N}$ such that $C \subset N \subset U$. We will use this notion to provide an instance of when being an $\iota$-space and having a countable network are equivalent. The proof of the following theorem is similar to the proof that every Lindelöf $\Sigma$-space is stable (see, for example, [@Tk]).
Let $X$ be a regular Lindelöf $\Sigma$-space. If $X$ is an $\iota$-space then $X$ has a countable network.
Let $\mathcal{C}$ be a cover of $X$ consisting of compact sets and $\mathcal{N}$ be a countable family which is a countable network for $X$ modulo $\mathcal{C}$. Since $X$ is regular, we can use Lemma $\ref{lemregular}$ to fix a countable regular $\iota$-cover $\mathcal{U}$ for $X$. Let $\mathcal{G}=\{\overline{U}: U \in \mathcal{U} \}$. We claim that the following family is a countable network for $X$:
$$\mathcal{B}=\left \{\bigcap \mathcal{F}: \mathcal{F} \in [\mathcal{G} \cup \mathcal{U}]^{<\omega} \right \}.$$
To see that, let $x \in X$ and $U$ be an open neighbourhood of $x$. Since $\mathcal{C}$ covers $X$, there is a $C \in \mathcal{C}$ such that $x \in C$. If $C \subset U$ then we can find $N \in \mathcal{N}$ such that $x \in C \subset N \subset U$ and we are done. Otherwise, the set $K=C \setminus U$ is compact non-empty. For every $y \in K$ choose a set $G_y \in \mathcal{G}$ such that $y \in G$ but $x \notin G$. Then $\{G_y \cap K: y \in K \}$ has empty intersection, and hence, by compactness of $K$ there is a finite set $S \subset K$ such that $\bigcap_{y \in S} G_y \cap K = \emptyset$. Therefore, $\bigcap_{y \in S} G_y \cap C \setminus U = \emptyset$, and hence $C \subset X \setminus (\bigcap_{y \in S} G_y \setminus U)$. But then there is an $N \in \mathcal{N}$ such that $C \subset N \subset X \setminus (\bigcap_{y \in S} G_y \setminus U)$. Therefore $N \cap \bigcap_{y \in S} G_y \setminus U=\emptyset$ and hence $x \in N \cap \bigcap_{y \in S} G_y \subset U$, which is what we wanted, since $N \cap \bigcap_{y \in S} G_y \in \mathcal{B}$.
Is there a Lindelöf Hausdorff $\Sigma$-space without a countable network which is an $\iota$-space?
Note that a Lindelöf $\Sigma$-space which is an $\iota_w$-space is also an $\iota$-space, since countable products of Lindelöf $\Sigma$-spaces are Lindelöf $\Sigma$.
L-spaces
========
Since every $\iota $-space is an $\epsilon $-space and $\epsilon $-spaces are characterized by having all finite powers Lindel" of, L-spaces are interesting spaces for us to consider. It is conjectured that even in ZFC there is an L-space that is not even an $\epsilon $-space. That is, of course, it is conjectured that Justin Moore’s $L$-space has a finite power which is not Lindelöf in ZFC. Certainly this is consistently known.
\[Lspace\] Consistently, every hereditarily Lindelöf $\iota$-space is separable.
Under $MA_{\omega_1}$, every $L$-space has a finite power which is not Lindelöf. Now, every $\iota$-space is an $\epsilon$-space.
To investigate the consistency of the negation of this statement, we focus on more general classes of spaces that yield L-spaces and provide many counterexamples in topology. These spaces are subspaces of products of the form $2^I$, where $I$ is a set of ordinals. So, the following notation is useful.
- $Fn(I,2)$ is the set of finite partial functions from $I$ into $2$.
- For $\varepsilon \in Fn(I,2)$, $[\varepsilon ]=\{f\in 2^I:\varepsilon \subseteq f\}$ denotes the basic clopen set determined by $\varepsilon $.
- If $b\in [I]^{<\omega }$ such that $b=\{\beta _i:i\in n=\vert b\vert \}$ and $\varepsilon \in 2^n$ then $\varepsilon *b$ denotes the element of $Fn(I,2)$ which has $b$ as its domain and satisfies $\varepsilon *b(\beta _i)=\varepsilon (i)$, $\forall i\in n$.
- For any cardinal $\mu $ and $r\in \omega $ we denote by ${\mathcal D}_\mu ^r(I)$ the collection of all sets $B\in [[I]^r]^\mu $ such that the members of $B$ are pairwise disjoint. We write ${\mathcal D}_\mu (I)=\bigcup \{{\mathcal D}_\mu ^r(I):r\in \omega \}$ and if $B\in {\mathcal D}_\mu (I)$ then $n(B)=\vert b\vert $ for any $b\in B$.
If $B\in {\mathcal D}_\mu (I)$ and $\varepsilon \in 2^{n(B)}$ then $[\varepsilon ,B]=\bigcup \{[\varepsilon *b]:b\in B\}$ is called a ${\mathcal D}_\mu $-set in $2^I$.
$X\subseteq 2^\lambda $ with $\vert X\vert >\omega $ is an HFC space if for every $B\in {\mathcal D}_\omega (\lambda )$ and $\varepsilon \in 2^{n(B)}$, $\vert X\setminus [\varepsilon ,B]\vert \leq \omega $. That is, every ${\mathcal D}_\omega $-set in $2^\lambda $ finally covers $X$.
For any $k\in \omega $, a map $F:\kappa \times \lambda \rightarrow 2$ with $\kappa \geq \omega _1$ and $\lambda \geq \omega $ $(\lambda \geq \omega _1)$ is called an HFC$^k$ (HFC$_w^k$) matrix if for every $A\in {\mathcal D}_{\omega _1}^k(\kappa )$ and $B\in {\mathcal D}_{\omega }(\lambda )$ $(B\in {\mathcal D}_{\omega _1}(\lambda ))$ and for any $\varepsilon _0,\dots ,\varepsilon _{k-1}\in 2^{n(B)}$ there exists $b\in B$ such that $\vert \{a\in A: \forall i\in k(f_{\alpha _i}\supseteq \varepsilon _i*b)\}\vert =\omega _1$, where $\{\alpha _i:i\in k\}$ is the increasing enumeration of the elements of $a$.
$F$ is a strong HFC (HFC$_w$) matrix if it is HFC$^k$ (HFC$_w^k$) for all $k\in \omega $.
$X\subseteq 2^\lambda $ is a strong HFC (HFC$_w$) space if it is represented by a strong HFC (HFC$_w$) matrix, $F$. That is $X=\{f_\alpha :\alpha <\kappa \}$ where $f_\alpha (\gamma )=F(\alpha ,\gamma )$, $\forall \gamma <\lambda $.
\[strongHFChL\][@J] If $X$ is a strong HFC$_w$ space (hence strong HFC) then $X^k$ is hereditarily Lindel" of, $\forall k\in \omega $.
\[strongHFCiota\] Every strong HFC is an $\iota $-space. In fact, if $X^n$ is hereditarily Lindel" of, $\forall n\in \omega $ then $X$ is an $\iota $-space.
Follows from Theorem \[powersthm\].
In contrast
\[notiotaHFC\] There is an HFC with no countable open $\iota $-cover.
By CH, enumerate the collection of ${\mathcal D}_\omega $-sets in $2^{\omega _1}$ by $\{u_\alpha :\alpha <\omega _1\}$ so that for $n<\omega $, $\{\sigma _{ni}:i\in \omega \}\subseteq Fn(\omega _1,2)$ such that {dom$(\sigma _{ni}):i\in \omega \}$ is a pairwise disjoint collection of finite subsets of $\omega $ and $u_n=\bigcup _{i\in \omega }[\sigma _{ni}]$. Moreover, for $\alpha \geq \omega $, $\{\sigma _{\alpha i}:i\in \omega \}\subseteq Fn(\omega _1,2)$ such that {dom$(\sigma _{\alpha i}):i\in \omega \}$ is pairwise disjoint and $u_\alpha =\bigcup _{i\in \omega }[\sigma _{\alpha i}]$. For $\alpha \geq \omega $, let ${\mathcal U}_\alpha =\{u_\beta :\beta <\alpha $, dom$(\sigma _{\beta i})\subseteq \alpha $, $\forall i\in \omega \}$. Enumerate ${\mathcal U}_\alpha =\{v_{\alpha i}:i\in \omega \}$ where each $u\in {\mathcal U}_\alpha $ appears as infinitely many $v_{\alpha i}$’s. Construct HFCs $X=\{x_\alpha :\omega \leq \alpha <\omega _1\}$ and $Y=\{y_\alpha :\omega \leq \alpha <\omega _1\}$ by induction, defining $x\restriction \alpha $, $y\restriction \alpha $ at stage $\alpha $ and letting $x_\alpha (\gamma )=0$, $\forall \gamma \geq \alpha $, $y_\alpha (\gamma )=1$, $\forall \gamma \geq \alpha $.
For $\omega \leq \alpha < \omega _1$, define $\{\sigma ^\alpha _i:i\in \omega \}$ such that
(i)
: if $v_{\alpha i}=u_\beta $ then $\sigma ^\alpha _i=\sigma _{\beta j}$ for some $j\in \omega $.
(ii)
: {dom$(\sigma ^\alpha _i):i\in \omega \}$ is pairwise disjoint.
$v_{\alpha 0}\in {\mathcal U}_\alpha \Rightarrow v_{\alpha 0}=u_\beta $ for some $\beta <\alpha $ so let $\sigma^\alpha _0=\sigma _{\beta 0}$.
Fix $n>0$ and suppose $\{\sigma ^\alpha _i:i<n\}$ have been defined. Again, since $v_{\alpha n}\in {\mathcal U}_\alpha $, let $\gamma <\alpha $ such that $v_{\alpha n}=u_\gamma $, where $u_\gamma =\bigcup \{[\sigma _{\gamma i}]:i\in \omega \}$ with {dom$(\sigma _{\gamma i}):i\in \omega \}$ pairwise disjoint. Thus, let $j_n\in \omega $ such that dom$(\sigma _{\gamma j_n})\cap $dom$(\sigma^\alpha _i)=\emptyset $, $\forall i<n$ and $\sigma ^\alpha _n=\sigma _{\gamma j_n}$. For $\omega \leq \alpha < \omega _1$, define $x_\alpha $, $y_\alpha \in 2^{\omega _1}$ as follows: $x_\alpha (\gamma )=y_\alpha (\gamma )=\sigma ^\alpha _i(\gamma )$, $\forall \gamma \in \bigcup _{i\in \omega }$dom$(\sigma ^\alpha _i)$, $x_\alpha (\gamma )=y_\alpha (\gamma )=0$, $\forall \gamma \in \alpha \setminus \bigcup _{i\in \omega }$dom$(\sigma ^\alpha _i)$ and as above, $x_\alpha (\gamma )=0$, $\forall \gamma \geq \alpha $, $y_\alpha (\gamma )=1$, $\forall \gamma \geq \alpha $.
$X\cup Y$ is an HFC with no countable open $\iota $-cover.
To see $X\cup Y$ is an HFC, fix $\beta <\omega _1$ and show $u_\beta $ is a final cover of $X\cup Y$. Note that $\forall \beta <\omega _1$, $\exists \delta <\omega _1$ such that $u_\beta \in {\mathcal U}_\delta $. So, let $\delta _\beta =\min \{\delta <\omega _1: u_\beta \in {\mathcal U}_\delta \}$. Then $X\cup Y \setminus (\{x_\gamma :\gamma <\delta _\beta \}\cup \{y_\gamma :\gamma <\delta _\beta \})\subseteq u_\beta $. Let ${\mathcal U}=\{U_n:n\in \omega \}$ be any countable open cover of $X\cup Y$. Since $X\cup Y$ is hereditarily Lindel" of (being HFC), let $\sigma _n(i)\in Fn(\omega _1,2)$ such that $U_n=\bigcup _{i\in \omega } [\sigma _n(i)]\cap (X\cup Y)$, $\forall n\in \omega $. Let $\alpha _n=\sup (\bigcup _{i\in \omega}$dom$(\sigma _n(i)))<\omega _1$ and $\alpha =\sup \{\alpha _n:n\in \omega \}<\omega _1$. We claim that $\forall \beta >\alpha $, $x_\beta \in U_n \Leftrightarrow y_\beta \in U_n$, $\forall n\in \omega $ and hence ${\mathcal U}$ is not an $\iota $-cover. Fix $\beta >\alpha $, $n\in \omega $. $$\begin{aligned}
x_\beta \in U_n &\Leftrightarrow (\exists i\in \omega )x_\beta \in [\sigma _n(i)]\\
&\Leftrightarrow (\exists i\in \omega )\sigma _n(i)\subseteq x_\beta \\
&\Leftrightarrow (\exists i\in \omega )x_\beta (\gamma )=\sigma _n(i)(\gamma ) \forall \gamma \in dom(\sigma _n(i)) \\
&\Leftrightarrow (\exists i\in \omega )y_\beta (\gamma )=\sigma _n(i)(\gamma ) \forall \gamma \in dom(\sigma _n(i)) \\%since y_\beta \restriction \beta =x_\beta \restriction \beta and dom(\sigma _n(i))\subseteq \alpha _n\subseteq \alpha \subseteq \beta
&\Leftrightarrow (\exists i\in \omega )\sigma _n(i)\subseteq y_\beta \\
&\Leftrightarrow (\exists i\in \omega )y_\beta \in [\sigma _n(i)]\\
&\Leftrightarrow y_\beta \in U_n\\\end{aligned}$$
This gives us another example of an L-space that is not an $\iota $-space, in fact, not even an $\iota _w$-space. Although we already know consistently (under MA$_{\omega _1}$) that this space is not even an $\epsilon $-space, the argument used to show the space has no countable open $\iota $-cover will be used to show what we really want: there is a hereditarily $\epsilon $-space that is not an $\iota _w $-space. Naively we tried to extend this argument to a strong HFC space (a hereditarily $\epsilon $-space), but along the way we discovered the missing ingredient. Thus Example \[notiotaHFC\] also provides an example of a certainly already known result.
There is a pair of strong HFCs whose union is not a strong HFC.
Let $X=\{x_\alpha :\alpha <\omega _1\}$, $Y=\{y_\alpha :\alpha <\omega _1\}$ be the HFCs from Example \[notiotaHFC\]. Let $f_X:X\rightarrow \omega _1$ such that $f_X(x_\alpha )=\alpha $ and $f_Y:Y\rightarrow \omega _1$ such that $f_Y(y_\alpha )=\alpha $.
$\exists A\in [\omega _1]^{\omega _1}$ such that $\{x_\alpha :\alpha \in A\}$, $\{y_\alpha :\alpha \in A\}$ are strong HFCs.
Let $\vec{N}=\left\langle N_\alpha :\alpha <\omega _1 \right\rangle $ be an $\omega _1$-chain of countable elementary submodels of some $H_\theta $ such that $X,Y,f_X,f_Y \in N_0$ and $\beta <\alpha <\omega _1 \Rightarrow N_\beta \subsetneq N_\alpha $. Define by recursion $Z=\{z_\alpha :\alpha <\omega _1\}\subseteq X$, separated by $\vec{N}$:\
Let $z_0\in X\cap N_0$\
Fix $\alpha >0$ and suppose $\{z_\beta :\beta <\alpha \}$ have been defined such that $z_\beta \in X\cap N_\beta \setminus \bigcup _{\gamma <\beta }N_\gamma $. Since $X$ is uncountable and $\bigcup _{\beta <\alpha }N_\beta $ is countable, $X\setminus \bigcup _{\beta <\alpha }N_\beta \neq \emptyset $. So, by elementarity, let $z_\alpha \in X\cap N_\alpha \setminus \bigcup _{\beta <\alpha }N_\beta \neq \emptyset $.\
To see $Z$ is separated by $\vec{N}$, let $\{z_\alpha ,z_\beta \}\in [Z]^2$. Without loss of generality, suppose $\alpha <\beta $. Then, by construction, $N_\alpha \cap \{z_\alpha ,z_\beta \}=\{z_\alpha \}$ and hence $\exists \alpha <\omega _1$ such that $\vert N_\alpha \cap \{z_\alpha ,z_\beta \}\vert = 1$.\
Then, by Theorem 2.1 of [@S], $Z$ is a strong HFC. Since $Z\in [X]^{\omega _1}$, let $A\in [\omega _1]^{\omega _1}$ such that $Z=\{x_\alpha :\alpha \in A\}$. We claim that $\{y_\alpha :\alpha \in A\}$ is separated by $\vec{N}$ and hence is a strong HFC (again by Theorem 2.1 of [@S]).
$\forall \alpha ,\gamma <\omega _1$, $x_\alpha \in N_\gamma \Leftrightarrow y_\alpha \in N_\gamma $.
Suppose $x_\alpha \in N_\gamma $. Since $f_X\in N_\gamma $, $f_X(x_\alpha )=\alpha \in N_\gamma $ and hence $x_\alpha \restriction \alpha \in N_\gamma $. Recall that $y_\alpha $ is definable from $x_\alpha \restriction \alpha $, $\alpha \in N_\gamma $ since $y_\alpha \restriction \alpha =x_\alpha \restriction \alpha $ and $y_\alpha (\gamma )=1$, $\forall \gamma \geq \alpha $. Hence $y_\alpha \in N_\gamma $. Similarly, $y_\alpha \in N_\gamma \Rightarrow x_\alpha \in N_\gamma $.
Then it is clear $\{y_\alpha :\alpha \in A\}$ is separated by $\vec{N}$. If $\{y_\alpha ,y_\beta \}\in [\{y_\alpha :\alpha \in A\}]^2$ then $\{x_\alpha ,x_\beta \}\in [Z]^2$ so $\exists \gamma <\omega _1$ such that $\vert N_\gamma \cap \{x_\alpha ,x_\beta \}\vert =1\Leftrightarrow \vert N_\gamma \cap \{y_\alpha ,y_\beta \}\vert =1$ (by the note).
Therefore, $\{x_\alpha :\alpha \in A\}, \{y_\alpha :\alpha \in A\}$ are strong HFCs and as in Example \[notiotaHFC\], $\{x_\alpha :\alpha \in A\}\cup \{y_\alpha :\alpha \in A\}$ has no countable $\iota $-cover. Thus, $\{x_\alpha :\alpha \in A\}\cup \{y_\alpha :\alpha \in A\}$ is not an $\iota $-space and hence is not a strong HFC by Corollary \[strongHFCiota\].
Fortunately, considering strong HFC$_w$ spaces and working a little harder provides us with the desired example. In [@J], Juh' asz constructs a strong HFC$_w$ space in a generic extension obtained by adding a Cohen or random real (in fact a generic extension with a slightly more general property). Using this same construction, we obtain two strong HFC$_w$ spaces whose union is a hereditarily $\epsilon $-space but has no countable open $\iota $-cover, hence not $\iota _w$. This gives an example of a space in which every subspace has any finite power Lindel" of, but there are two subspaces whose product is not Lindel" of. In particular, all squares of subspaces are Lindel" of, but there is a rectangle that is not Lindel" of; a hereditarily Lindel" of space whose square is not hereditarily Lindel" of.
In comparison to Definition \[almostiota\],
\[almostepsilon\] A space $X$ is almost-$\epsilon $ if for every open $\omega $-cover ${\mathcal U}$ of $X$, there is a countable ${\mathcal V}\subseteq {\mathcal U}$ and $A\in [X]^\omega $ such that ${\mathcal V}$ is an $\omega $-cover of $X\setminus A$.
\[almostepsilonlem\] If $X$ is almost-$\epsilon $ then $X$ is an $\epsilon $-space.
Let ${\mathcal U}$ be any open $\omega $-cover of $X$ and ${\mathcal M}$ be a countable elementary submodel of some $H_\theta $ ($\theta $ sufficiently large) such that ${\mathcal U}, (X,\tau )\in {\mathcal M}$.
${\mathcal U}\cap {\mathcal M}$ is a countable $\omega $-subcover of ${\mathcal U}$.
Let $F\in [X]^{<\omega }$ and consider ${\mathcal U}_{F\cap {\mathcal M}}=\{U\in {\mathcal U}:F\cap {\mathcal M}\subseteq U\}\in {\mathcal M}$ (since $F\cap {\mathcal M}\subseteq {\mathcal M}$ is finite). Notice that ${\mathcal U}_{F\cap {\mathcal M}}$ is an open $\omega $-cover of $X$ and since, by elementarity, ${\mathcal M}\models X$ is almost-$\epsilon $, let ${\mathcal V}\in {\mathcal M}$ be countable and $A\in [X]^\omega \cap {\mathcal M}$ such that ${\mathcal V}\subseteq {\mathcal U}_{F\cap {\mathcal M}}$ is an $\omega $-cover of $X\setminus A$. Since $A,{\mathcal V}\in {\mathcal M}$ are countable, $A,{\mathcal V}\subseteq {\mathcal M}$. In particular, ${\mathcal V}\subseteq {\mathcal U}\cap {\mathcal M}$. Also, since $A\subseteq {\mathcal M}$, $F\setminus {\mathcal M}\in[X\setminus A]^{<\omega }$ so let $V\in {\mathcal V}\subseteq {\mathcal U}_{F\cap {\mathcal M}}$ such that $F\setminus {\mathcal M}\subseteq V$. Then $V\in {\mathcal U}\cap {\mathcal M}$ such that $F\subseteq V$.
The following alternate characterization of an HFC$_w^k$ space is an adaptation of the characterization of an HFC$_w$ space from [@J].
For any $k\in \omega $, if $X\subseteq 2^\lambda $ with $\vert X\vert >\omega $ and $\lambda >\omega $ is HFC$_w^k$, then $$\begin{aligned}
& \forall B\in {\mathcal D}_{\omega _1}(\lambda ), \tag{$*_k$} \forall \varepsilon _0,\dots ,\varepsilon _{k-1}\in 2^{n(B)} , \exists C\in [B]^\omega , \exists \alpha \in \kappa \\
& (\forall a=\{\alpha _i:i<k\}\in[\kappa \setminus \alpha ]^k )
(\exists b\in C) f_{\alpha _i}\supseteq \varepsilon _i*b, \forall i\in k.\end{aligned}$$
Suppose, by way of contradiction, that there is $B\in {\mathcal D}_{\omega _1}(\lambda )$ and $\varepsilon _0,\dots ,\varepsilon _{k-1}\in 2^{n(B)}$ such that $\forall C\in [B]^\omega $ and $\forall \alpha <\kappa $, $\exists a=\{\alpha _i:i<k\}\in [\kappa \setminus \alpha ]^k$ and $\exists j\in k$ such that $f_{\alpha _j}\nsupseteq \varepsilon _j*b$, $\forall b\in C$. Enumerate $B=\{b_\gamma :\gamma <\omega _1\}$ and let $C_\mu =\{b_\gamma :\gamma <\mu \}$, $\forall \mu <\omega _1$. Then $C_\mu \in [B]^\omega $, $\forall \mu <\omega _1$ so define by recursion $\{\alpha _\mu :\mu <\omega _1\}\subseteq \kappa $ so that $A=\{a_\mu :\mu <\omega _1\}\in {\mathcal D}_{\omega _1}^k(\kappa )$ where, by assumption, $a_\mu =\{\alpha _i^\mu :i<k\}\in [\kappa \setminus \alpha _\mu ]^k$ such that $f_{\alpha _j^\mu}\nsupseteq \varepsilon _j*b$, $\forall b\in C_\mu $, for some $j<k$. Then, since $X$ is HFC$_w^k$, $A\in {\mathcal D}_{\omega _1}^k(\kappa )$, $B\in {\mathcal D}_{\omega _1}(\lambda )$ and $\varepsilon _0,\dots ,\varepsilon _{k-1}\in 2^{n(B)}$, let $b\in B$ such that $\vert \{a\in A:\forall i\in k(f_{\alpha _i}\supseteq \varepsilon _i*b)\}\vert=\omega _1$. But then $b=b_\mu $ for some $\mu <\omega _1$ and $\{a\in A:\forall i\in k(f_{\alpha _i}\supseteq \varepsilon _i*b)\}\subseteq \{a_\gamma :\gamma \leq \mu \}$ (since $b=b_\mu \in C_\gamma $, $\forall \gamma > \mu $), which is countable and hence we have a contradiction.
\[heredepsilonnotiota\] Con(ZFC) $\rightarrow $ Con(ZFC + $ \exists $ hereditarily $\epsilon $-space with no countable open $\iota $-cover).
Construct two HFC$_w$ spaces $X=\{x_\alpha :\alpha <\omega _1\}$ and $Y=\{y_\alpha :\alpha <\omega _1\}$, as in (4.2) of [@J], so that $x_\alpha \restriction \alpha =y_\alpha \restriction \alpha =f_\alpha \restriction \alpha $, where $f_\alpha =F(\alpha ,-)=r\circ h_\alpha $ and $\forall \gamma \geq \alpha $, $x_\alpha (\gamma )=0, y_\alpha (\gamma )=1$.
$X\cup Y$ is hereditarily-$\epsilon $ but has no countable open $\iota $-cover.
Let $Z\subseteq X\cup Y$ and ${\mathcal U}$ be any open $\omega $-cover of $Z$. Without loss of generality, ${\mathcal U}$ consists of finite unions of basic open sets in $2^{\omega _1}$. That is, $\forall U\in {\mathcal U}$, $U=\bigcup _{i<n_U}[\sigma _i^U]$ with $\sigma _i^U\in Fn(\omega _1,2)$. Let ${\mathcal M}$ be a countable elementary submodel of $H_\theta $ (for some large enough $\theta $) such that ${\mathcal U}\in {\mathcal M}$. We claim that ${\mathcal U}\cap {\mathcal M}$ is a countable $\omega $-cover of $Z\setminus (Z\cap {\mathcal M})$ showing that $Z$ is almost-$\epsilon $ and hence an $\epsilon $-space by Lemma \[almostepsilonlem\], as required. To see ${\mathcal U}\cap {\mathcal M}$ is an $\omega $-cover, let $F\in [Z\setminus (Z\cap {\mathcal M})]^{<\omega }$. Enumerate $F=\{f_{\alpha _0},\dots ,f_{\alpha _{n-1}}\}$ for some $n\in \omega $, where $f_{\alpha _i}=x_{\alpha _i}\in Z\cap X$ or $f_{\alpha _i}=y_{\alpha _i}\in Z\cap Y$ so that if $\exists \beta <\omega _1$ such that $x_\beta ,y_\beta \in Z$, $f_\beta =x_\beta $. Notice that this enumeration is not a problem since if $F$ is the original set and there is $U\in {\mathcal U}\cap {\mathcal M}$ such that $\{f_{\alpha _0},\dots ,f_{\alpha _{n-1}}\}\subseteq U$, then, as above, since ${\mathcal U}\cap {\mathcal M}$ is countable, enumerate ${\mathcal U}\cap {\mathcal M}=\{U_k:k\in \omega \}$ where $U_k=\bigcup _{i<n_k}[\sigma _i^k]$ and $\gamma _k=\sup (\bigcup _{i<n_k}$dom$(\sigma _i^k))<\omega _1$. Let $\gamma =\sup \{\gamma _k:k\in \omega \}$. Then, as above, $\forall \beta >\gamma $, $x_\beta \in U_k \Leftrightarrow y_\beta \in U_k$, $\forall k\in \omega $. In particular, $x_\beta \in U \Leftrightarrow y_\beta \in U$ $\forall \beta >\gamma $. Note that $\gamma $ is definable in ${\mathcal M}$ since $U_k\in {\mathcal M}$, $\forall k\in \omega $. Thus, since $F\notin {\mathcal M}$, $\alpha _i>\gamma $, $\forall i<n$ and hence $F\subseteq U$. Since $F\in [Z]^{<\omega }$ and ${\mathcal U}$ is an $\omega $-cover of $Z$, let $U\in {\mathcal U}$ such that $F\subseteq U$. If $U\in {\mathcal M}$ we are done, so suppose $U\notin {\mathcal M}$. Since $F\subseteq U=\bigcup _{i<n_U}[\sigma _i^U]$, we can refine $U$ so that $F\subseteq \bigcup_{i<n}[\tau _i]\subseteq U$ with $f_{\alpha _i}\in [\tau _i]$ $\forall i<n$. We need to define $\tau _0,\dots ,\tau _{n-1}\in Fn(\omega _1,2)$ such that $f_{\alpha _i}\in [\tau _i]\subseteq U$, $\forall i<n$:\
Since $f_{\alpha _0}\in [\sigma _{i_0}^U]$ for some $i_0<n_U$, let $\tau _0=\sigma _{i_0}^U$.\
Fix $m>0$ and suppose $\{\tau _j:j<m\}$ have been defined. Since $f_{\alpha _m}\in [\sigma _{i_m}^U]$, if $\tau _j\neq \sigma _{i_m}^U$, $\forall j<m$, then $\tau _m=\sigma _{i_m}^U$. Otherwise, let $j<m$ such that $\tau _j=\sigma _{i_m}^U$. Let $N=\{i<m: f_{\alpha _m}\in [\tau _i]\}$ and $\gamma _m=\max \{\bigcup dom(\tau _i):i<m\}<\omega _1$. Define $dom(\tau _m)=\bigcup _{i\in N}dom(\tau _i)\cup \{\gamma _m+1\}$ and $\tau _m(\alpha )=\tau _i(\alpha)=f_{\alpha _m}(\alpha )$, $\forall \alpha \in dom(\tau _i), i\in N$, $\tau _m(\gamma _m+1)=f_{\alpha _m}(\gamma _m+1)$.\
Then $F\subseteq \bigcup _{i<n}[\tau _i]\subseteq U$ and we will further refine $\bigcup _{i<n}[\tau _i]$ so that $F\subseteq \bigcup _{i<n}[\varepsilon _i*b]\subseteq \bigcup _{i<n}[\tau _i]\subseteq U$ for $b\in [\omega _1]^k$ $(k\in \omega )$ and $\varepsilon _0,\dots \varepsilon _{n-1}\in 2^k$. Let $b=\bigcup _{i<n}dom(\tau _i)\in [\omega _1]^{<\omega }$, $k=\vert b\vert $ and enumerate $b=\{\beta _i:i<k\}$. For $i<n$, let $\varepsilon _i\in 2^k$ such that $\varepsilon _i(j)=f_{\alpha _i}(\beta _j)$, $\forall j<k$. Then $\varepsilon _i*b(\beta _j)=\varepsilon _i(j)=f_{\alpha _i}(\beta _j) \Rightarrow \varepsilon _i*b\subseteq f_{\alpha _i}$, $\forall i<n$. By absoluteness, $\varepsilon _0,\dots ,\varepsilon _{n-1}\in {\mathcal M}$ and if $b\in {\mathcal M}$ then by elementarity, $\exists U\in {\mathcal U}\cap {\mathcal M}$ such that $F\subseteq U$, so suppose $b\notin {\mathcal M}$. Let $r=b\cap {\mathcal M}$. Then $b\setminus r\neq \emptyset $. Let $m=k- \vert r\vert $ and ${\mathcal D}=\{d\in [\omega _1]^m:\bigcup [\varepsilon _i*(r\cup d)]\subseteq V$, for some $V\in {\mathcal U}\}$. Notice that $b\setminus r\in {\mathcal D}$ and since $b\setminus r\notin {\mathcal M}$, ${\mathcal D}$ is uncountable. Then ${\mathcal D}^\shortmid =\{r\cup d: d\in {\mathcal D}\}$ is an uncountable family of finite subsets of $\omega _1$ so let ${\mathcal B}^\shortmid \subseteq {\mathcal D}^\shortmid $ be an uncountable $\Delta $-system with root $r=b\cap {\mathcal M}$ and let ${\mathcal B}=\{d\setminus r:d\in {\mathcal B}^\shortmid \}\subseteq {\mathcal D}$. Then ${\mathcal B}\in {\mathcal D}_{\omega _1}(\omega _1)$ such that $\forall b^\shortmid \in {\mathcal B}$, $\vert b^\shortmid \vert =k-\vert r\vert =m$. Recall $b\setminus r=\{\beta _j: \vert r\vert \leq j<k\}$. So, reenumerate $b\setminus r=\{\gamma _j:j<m\}$ where $\gamma _j=\beta _{j+\vert r\vert }$, $\forall j<m$. For $i<n$, let $\varepsilon _i^\shortmid \in 2^m$ such that $\varepsilon _i^\shortmid (j)=\varepsilon _i^\shortmid *b\setminus r(\gamma _j)=\varepsilon _i*b\setminus r (\beta _{j+\vert r\vert })=\varepsilon _i(j+\vert r\vert )$, $\forall j<m$. Then, since ${\mathcal B}\in {\mathcal D}_{\omega _1}(\omega _1)$, $\varepsilon _0^\shortmid ,\dots ,\varepsilon _{n-1}^\shortmid \in 2^m$ and by elementarity ${\mathcal M}\models (*_k)$, let ${\mathcal C}\in [\mathcal B]^\omega \cap {\mathcal M}$ and $\alpha \in \omega _1\cap {\mathcal M}$ such that $\forall a=\{\beta _i:i<n\}\in [\omega _1\setminus \alpha ]^n$, $\exists c\in {\mathcal C}$ such that $f_{\beta _i}\in [\varepsilon _i^\shortmid *c]$, $\forall i<n$. In particular, since $F\notin {\mathcal M}$, $\alpha _0,\dots ,\alpha _{n-1}\notin {\mathcal M}$ and since $\alpha \in {\mathcal M}$ we have that $\alpha _i>\alpha $, $\forall i<n$. Thus, $\{\alpha _i:i<n\}\in [\omega _1\setminus \alpha ]^n$ so let $c\in {\mathcal C}$ such that $f_{\alpha _i}\in [\varepsilon _i^\shortmid *c]$, $\forall i<n$. But, since ${\mathcal C}\in {\mathcal M}$ is countable, ${\mathcal C}\subseteq {\mathcal M}$ and hence $c\in {\mathcal C}\cap {\mathcal M}$ such that $\varepsilon _i^\shortmid *c\subseteq f_{\alpha _i}$, $\forall i<n$. Also, since $\varepsilon _i*b\subseteq f_{\alpha _i}$, $\forall i<n$, $f_{\alpha _i}\in [\varepsilon _i\restriction \vert r\vert *r]\cap [\varepsilon _i^\shortmid *c]$, $\forall i<n$. We claim that $[\varepsilon _i\restriction \vert r\vert *r]\cap [\varepsilon _i^\shortmid *c]=[\varepsilon _i*(r\cup c)]$ and hence $f_{\alpha _i}\in [\varepsilon _i*(r\cup c)]$, $\forall i<n \Rightarrow F\subseteq \bigcup_{i<n}[\varepsilon _i*(r\cup c)]$.
‘$\subseteq $’
: Let $g\in [\varepsilon _i\restriction \vert r\vert *r]\cap [\varepsilon _i^\shortmid *c]$. Then $\varepsilon _i\restriction \vert r\vert *r\subseteq g$ and $\varepsilon _i^\shortmid *c\subseteq g$. Recall $r=\{\beta _j: j<\vert r\vert \}$ and enumerate $c=\{\gamma _j:j<m\}$. Then, $g(\beta _j)-\varepsilon _i\restriction \vert r\vert *r(\beta _j)=\varepsilon _i(j)$, $\forall j<\vert r\vert $ and $g(\gamma _j)=\varepsilon _i^\shortmid *c(\gamma _j)=\varepsilon _i^\shortmid (j)=\varepsilon _i(j+\vert r\vert )$, $\forall j<m$. Now, since $r\cup c=\{\beta _j:j<\vert r\vert \}\cup \{\gamma _j:j<m\}$, reenumerate $r\cup c=\{\alpha _j:j<k\}$ so that $\alpha _j=\beta _j$ for $j<\vert r\vert $ and $\alpha _j =\gamma _{j-\vert r\vert }$ for $\vert r \vert \leq j<k$. Then, for $j<\vert r\vert $, $g(\alpha _j)=g(\beta _j)=\varepsilon _i(j)=\varepsilon _i*(r\cup c)(\alpha _j)$ and for $j\geq \vert r\vert $, $g(\alpha _j)=g(\gamma _{j-\vert r\vert })=\varepsilon _i((j-\vert r\vert )+\vert r\vert )=\varepsilon _i(j)=\varepsilon _i*(r\cup c)(\alpha _j)$.
‘$\supseteq $’
: Let $g\in [\varepsilon _i*(r\cup c)]$. Then $g(\alpha _j)=\varepsilon _i*(r\cup c)(\alpha _j)=\varepsilon _i(j)$, $\forall j<k$, where $\alpha _j $ is defined as above, for $j<k$. Then, $g(\beta _j)=g(\alpha _j)=\varepsilon _i(j)=\varepsilon _i*(r\cup c)(\beta _j)=\varepsilon \restriction \vert r\vert *r(\beta _j)$, $\forall j<\vert r\vert $ and hence $g\in [\varepsilon _i\restriction \vert r\vert *r]$. Moreover, $g(\gamma _j)=g(\alpha _{j+\vert r\vert })=\varepsilon _i(j+\vert r\vert )=\varepsilon _i^\shortmid ((j+\vert r\vert )-\vert r\vert )=\varepsilon _i^\shortmid (j)=\varepsilon _i^\shortmid *c(\gamma _j)$, $\forall j<m$ and hence $g\in [\varepsilon _i^\shortmid *c]$.
Since $c\in {\mathcal C}\subseteq {\mathcal D}$, $\bigcup _{i<n}[\varepsilon _i*(r\cup c)]\subseteq V$ for some $V\in {\mathcal U}$. But $\bigcup _{i<n}[\varepsilon _i*(r\cup c)]\in {\mathcal M}$ (since $r,c, \varepsilon _0, \dots ,\varepsilon _{n-1}\in {\mathcal M}$) so by elementarity, let $V\in {\mathcal U}\cap {\mathcal M}$ such that $\bigcup _{i<n}[\varepsilon _i*(r\cup c)]\subseteq V$. Then $\exists V\in {\mathcal U}\cap {\mathcal M}$ such that $F\subseteq \bigcup _{i<n}[\varepsilon _i*(r\cup c)]\subseteq V$.
D-spaces
========
The third named author first considered $\iota $-covers when trying to make the $T_2$ hereditarily Lindel" of non D-space of [@SS2] regular. Since this $T_2$ example is an $\epsilon $-space, he asked in [@SS2] whether every regular (hereditarily) $\epsilon $-space is a D-space. We could ask the same about (hereditarily) $\iota $-spaces. In fact, it remains unclear whether $\iota $-covers could play a role in constructing such a regular, hereditarily Lindel" of non D-space.
There is an $\iota_w$-space which is not a $D$-space.
The example is taken from [@AJW], but we nevertheless present the details of its construction for the reader’s convenience. Erik van Douwen showed in [@vD] that one can put, on every subset of the real line, a locally compact locally countable topology with countable extent which is finer than the topology it inherits from the Euclidean one. Let $B \subset \mathbb{R}$ be a Bernstein set, that is, a set meeting every uncountable closed set along with its complement. Let $X=\mathbb{R}$, where points of $B$ have neighbourhoods as in the van Douwen topology and points of $X \setminus B$ have their usual Euclidean neighbourhoods.
[**Claim 1.**]{} $X$ is Lindelöf and a $D$-space.
To prove that $X$ is Lindelöf, let $\mathcal{U}$ be an open cover of $X$. Let $V=\bigcup \{ U \in \mathcal{U}: U \cap (X \setminus B) \neq \emptyset \}$. Then $V$ covers all but countably many points of $X$. Indeed if $X \setminus V$ were uncountable, then $(X \setminus V) \cap (X \setminus B) \neq \emptyset$ which is a contradiction. But since the topology of $X \setminus B$ is Lindelöf, $\mathcal{V}$ has a countable subcover, and hence $X$ is a Lindelöf space.
To prove that $X$ is a $D$-space, we need the following lemma:
Every countable space $Y$ is a $D$-space.
Let $N: Y \to \tau$ be a neighbourhood assignment and fix an enumeration $\{y_n: n < \omega \}$ of $Y$. Suppose you have picked points $\{y_{k_i}: i \leq n \}$. If $N(\{y_{k_i}: i \leq n \})=Y$ then we are done, otherwise let $y_{k_{n+1}}$ be the least indexed point such that $y_{k_{n+1}} \notin N(\{y_{k_i}: i \leq n \})$. We claim that $N(\{y_{k_i}: i < \omega \})=Y$ and $\{y_{k_i}: i<\omega \}$ is closed discrete. For the first claim, suppose by contradiction that there is a point $y \notin N(\{y_{k_i}: i < \omega\}$. Then there is a $j<\omega$ such that $y$ is the least indexed point (in the original enumeration of $Y$) such that $y \notin \{y_{k_i}: i \leq j \}$. But then $y=y_{k_{j+1}}$, which is a contradiction. The fact that $\{y_{k_i}: i < \omega \}$ is closed discrete follows from the first claim and the fact that $N(y_{k_n}) \cap \{y_{k_i}: i < \omega \}$ is a finite set.
Let $N: X \to \tau$ be an open neighbourhood assignment. Since $X \setminus B$ is a $D$-space we can find a closed discrete set $D_1 \subset X \setminus B$ such that $N(D_1) \supset X \setminus B$. Now $X \setminus N(D_1)$ is a countable closed set. So $X \setminus N(D_1)$ is a $D$-space and hence we can find $D_2 \subset X \setminus N(D_1)$ such that $N(D_2) \supset X \setminus N(D_1)$. Therefore $D=D_1 \cup D_2$ is a closed discrete set such that $N(D)=X$, so $X$ is a $D$-space.
Now let $B_e$ be the Bernstein set $B$ with its usual (Euclidean) topology.
[**Claim 2**]{}. $X \times B_e$ is an $\iota_w$-space but not a $D$-space.
[*Proof of Claim 2*]{}. Since the topology on $X$ refines the topology of the real line, and the real line has a countable open $\iota$-cover, also $X$ has a countable open $\iota$-cover. So, it follows from Theorem $\ref{productiw}$ that $X \times B_e$ is an $\iota_w$-space. To prove that $X \times B_e$ is not a $D$-space, note that it contains the closed copy $\{(x,x): x \in B \}$ of the space $B$ and that $B$ is not a $D$-space, because it has countable extent, but it is uncountable and locally countable and thus not Lindelöf.
Is every (hereditarily) $\iota$-space a $D$-space?
[10]{} O. Alas, L. Junqueira and R. Wilson, *Dually discrete spaces*, Topology Appl. **155** (2008), 1420–1425. A. V. Arkhangel’skiǐ, *Topological function spaces*. Translated from the Russian by R. A. M. Hoksbergen. Mathematics and its Applications (Soviet Series), **78**. Kluwer Academic Publishers Group, Dordrecht, 1992. Z. Balogh and G. Gruenhage, *Base multiplicity in compact and generalized compact spaces* Topology Appl. **115** (2001), 139–151. D. K. Burke, *Covering properties*. Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 347–422, I. Juh' asz, *HFD and HFC type spaces, with applications*, Topology Appl. **126** (2002), 217–262. L. Lawrence, *Lindelöf spaces concentrated on Bernstein subsets of the real line*, Proc. Amer. Math. Soc. **114** (1992), 211–215. T. C. Przymusi' nski, *Normality and paracompactness in finite and countable Cartesian products*, Fund. Math. 105 (1979-80), no. 2, 87–104. D. Soukup and P. J. Szeptycki, *A counterexample in the theory of D-spaces*, Topology Appl. **159** (June 2012), no. 10-11, 2669–2678. L. Soukup, *Certain L-spaces under CH*, Topology Appl. **47** (1992), 1–7. V. Tkachuk, *Lindelöf $\Sigma$-spaces, an omnipresent class*, Revista de la Real Academia de Ciencias Exactas, F’sicas y Naturales. Serie A: Matem' aticas, **104** (2010), 221-244. E. van Douwen, *A technique for constructing honest locally compact submetrizable examples*, Topology Appl. **47** (1992), 179–201.
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abstract: 'For a general vector field we exhibit two Hilbert spaces, namely the space of so called “closed functions” and the space of “exact functions” and we calculate the codimension of the space of “exact functions” inside the larger space of “closed functions”. In particular we provide a new approach for the known cases: the Glauber field and the second-order Ginzburg-Landau field and for the case of the fourth-order Ginzburg-Landau field.'
address: 'Department of Mathematics and Statistics, Queen’s University, Kingston, ON, K7L 3N6, Canada'
author:
- BY Anamaria Savu
title: 'Closed and exact functions in the context of Ginzburg-Landau models'
---
Introduction {#intro}
============
Statistical physics has developed a whole variety of interacting particle systems that capture some aspects of the movement of particles on the microscale. An interacting particle system is usually a complex Markov process with a finite or infinite state space. By taking an appropriate scaling limit of an interacting particle system we expect to derive the evolution of the system on the macroscale, in general a nonlinear partial differential equation. It is fairly well understood the transition from the microscopic scale to macroscopic scale, at least for some systems, and in this notes we take for granted this step.
The most interesting microscopic models constructed so far, lack the so called gradient condition. This condition corresponds to the Fick’s law of fluid dynamics according to which the instantaneous current $w$ of particles over a bond is the gradient $\tau h - h$ of some local function. Since the work of Varadhan [@Var1], Quastel [@Qua1] and Varadhan and Yau [@Var2] on nongradient systems, new ideas have been imposed in the field. The main idea is that a nongradient system has a generalized Fick’s law, also called the fluctuation-dissipation equation, of the form $$w \approx \hat{a}(m)(\tau h -h) + L g,$$ where $\hat{a}$ is the transport coefficient depending on the particle density $m$ in a microscopic cube, $h$ is some local function, and $L$ is the generator of the microscopic dynamics. The $Lg$ part of the approximate equation above is negligible on the macroscale, and is called the fluctuation part of the equation.
One of the main difficulty in finding the scaling limit of a nongradient system is to make rigorous sense of the fluctuation-dissipation equation. As it has been shown in [@Var1], [@Qua1], [@Var2] the current w, the gradient $\tau h -h$ and the fluctuations $Lg$ are elements of the Hilbert space of “closed functions” and the fluctuation-dissipation equation is a consequence of a direct-sum decomposition of this Hilbert space. The gradient part $\tau h-h$ of the current $w$ that survives after taking the scaling limit of the model is just the projection of $w$ onto a one dimensional subspace of the Hilbert space of “closed functions”. The remaining negligible fluctuations $Lg$ are vectors of the Hilbert subspace of “exact function”.
The purpose of the present paper is not to show how the Hilbert space of “closed functions” and “exact functions” arises in the context of interacting particle systems, but rather to motivate the direct-sum decomposition of the Hilbert space of “closed functions” and to find the codimension of the space of “exact functions” inside the space of “closed functions”. We calculate this codimension for an arbitrarily chosen vector field. The three continuum models known as the Glauber system, the second-order Ginzburg-Landau system and the continuum solid-on-solid model, also called the fourth-order Ginzburg-Landau system are covered by our general result. Our approach to establish the direct-sum decomposition of the Hilbert space of “closed functions” is new and differs from the approach used before to study the first two models (see Varadhan [@Var1]). We have followed a different path based on Fourier analysis that has allowed us to handle a general vector field.
The decomposition theorem {#decomp}
=========================
In this section we introduce some terminology and state the main result.
The Hermite polynomials provide an orthogonal basis for the Hilbert space of functions defined on the real axis, that are square integrable with respect to the Gaussian probability measure $\frac{1}{\sqrt{2\pi}}\exp(-\frac{x^2}{2}) dx$. The $i$th Hermite polynomial is defined through $$H_i(x) = \frac{(-1)^i}{i!} \exp\bigg(\frac{x^2}{2}\bigg)
\bigg(\frac{d^i}{dx^i}\exp\bigg(-\frac{x^2}{2}\bigg) \bigg),
\quad i \in \mathbb{N}.$$ We stress that $H_i$ is not normalized to have $L^2$ norm 1 with respect to the probability measure $\frac{1}{\sqrt{2\pi}}\exp(-\frac{x^2}{2}))dx$, but rather $\frac{1}{\sqrt{i!}}.$
There is an extension of Hermite polynomials to more variables. A multi-index is a double-sided infinite sequence $I = \{ i_n \}_{n \in {\mathbb{Z}}}$ of positive integers, with at most finitely many non-zero entries. The degree of a multi-index is $|I|= \sum_{n \in {\mathbb{Z}}} i_n$. Call $\mathcal{I}$ the set of multi-indices and $\mathcal{I}_N$ the set of multi-indices of fixed degree $N$. The multidimensional Hermite polynomials are $$H _I(x) = \Pi_{n\in \mathbb{Z}} H_{i_n}(x_n), \quad I \in { {\mathcal I}}.$$ We make the convention that if a multi-index $I$ has some strictly negative entries then $H_I =0$. Together the multidimensional Hermite polynomials, $\{ H_I \}_{I \in \mathcal{I}}$ form an orthogonal basis for the Hilbert space of functions defined on ${\mathbb{R}^{\mathbb{Z}}}$, that are square integrable with respect to the probability measure $$d\nu_{0}^{gc} =\bigotimes_{i \in {\mathbb{Z}}} \frac{1}{\sqrt{2\pi}}\exp\bigg(-\frac{x_i^2}{2}\bigg)dx_i.$$ It is interesting to note that this Hilbert space is a model for the symmetric Fock space over the space of square summable, double-sided sequences $l^2({\mathbb{Z}})$, and decomposes as a direct sum of the degree $N$ subspaces $${\mathcal H}_N = \{ H_I \; | \; |I|=N \}^c.$$ The superscript on the line above, means that we take the closed linear span of the set.
The shift $\tau$ acts on configurations as $(\tau (x ))_{n} =x_{n+1}$ and on functions as $(\tau f)(x) = f(\tau x)$. $\tau^n$ stands for the $n$-fold composition $\tau \circ \dots \circ \tau$. If a multi-index $I=(i_n)_{n\in \mathbb{Z}}$ has $i_n=0$ for all $n <0$ we shall say that the multi-index is supported on the set of positive integers. We shall use the notation $\delta_n$ for the multi-index that corresponds to the configuration with a single particle at the site $n$. Two multi-indices can be added and the addition is point-wise.
The action of the annihilation, creation, and shift operators on the multidimensional Hermite polynomial $H_I$ is very simple: $$\partial_{n} H_I(x) = H_{I-\delta_n} (x), \quad
(x_n - \partial_{n} ) H_I(x) = H_{I+\delta_n} (x) , \quad
\tau H_I = H_ {\tau^{-1}I } .$$ Above $\partial_n$ stands for the partial derivative with respect to the $n$th coordinate.
Given a double-sided sequence of real numbers $(a_k)_{k \in {\mathbb{Z}}}$, that are all but finitely many zero we introduce the vector field $D_0=\sum_{k \in {\mathbb{Z}}} a_k \partial_k$ with constant coefficients. Translating $a$’s to the left or to the right produces a new sequence that defines the vector field $D_n=\sum_{k \in {\mathbb{Z}}} a_k \partial_{k+n}$, $n \in {\mathbb{Z}}$. Now we have the setup needed to introduce the closed and exact functions.
We shall say that a function $\xi \in L^2({\mathbb{R}^{\mathbb{Z}}}, d\nu_{0}^{gc})$ is closed (or more precisely, $D_0$-closed) if it satisfies in the weak sense $$\label{closedt}
D_n (\tau^m \xi) = D_m (\tau^n \xi)$$ for all integers $m$ and $n$. Let ${\mathcal C}_D$ denote the space of all $D_0$-closed functions.
We shall say that a function $\xi^g \in L^2({\mathbb{R}^{\mathbb{Z}}}, d\nu_{0}^{gc})$ is exact (or more precisely $D_0$-exact) if there is a local function $g$, a function that depends on finitely many co-ordinates, such that $$\label{closed}
\xi^g = D_0 \bigg( \sum_{k \in \mathbb{Z}} \tau^k g \bigg) =
\sum_{k \in \mathbb{Z}} D_0(\tau^k g).$$ Let ${\mathcal E}_D$ denote the closed linear span of the set of $D_0$-exact functions.
Although the infinite sum $\sum_{k \in \mathbb{Z}} \tau^k g$ does not make sense, after applying the differential operator $D_0$ we get a meaningful expression. Since $g$ is a local function, the vector field $D_0$ kills all but finitely many terms of the infinite formal sum.
The terminologies of exact and closed functions are not arbitrarily chosen. We can define formally the form $w = \sum_{n \in {\mathbb{Z}}} \tau^n \xi \; dx_n$ and the boundary operator $ df = \sum_{n \in {\mathbb{Z}}} D_n (f)\; dx_n.$ It is not hard to see, with these new definitions, that the form $w$ is closed ($dw=0$), in the vector calculus sense, if and only if $D_n (\tau^m \xi) = D_m (\tau^n \xi)$, i.e., if and only if $\xi$ is a closed function.
Knowing that any exact function is closed a natural question to ask is about the codimension of the space of exact functions inside the space of closed functions. In this paper we provide the answer for this question.
\[the1\] Let $D_0=\sum_{k \in {\mathbb{Z}}}a_k \partial_k$ be a vector field with constant real coefficients. All but finitely many numbers in the sequence $(a_k)_{k \in {\mathbb{Z}}}$ are zero. The following decomposition results hold:
- If the sum of the coefficients of the vector field $D_0$ is not equal to zero then $${\mathcal C}_D = {\mathcal E}_D.$$
- If the sum of the coefficients of the vector field $D_0$ is equal to zero then $${\mathcal C}_D = \mathbb{R}\bf{1}\oplus {\mathcal E}_D.$$
[**Idea of the proof for the decomposition theorem \[the1\].**]{} We outline the main ideas used to prove the decomposition theorem. We shall show later that a function $\xi$ is $D_0$-closed if and only if the projections $\mathrm{Proj}_{\mathcal{H}_N}\xi$, $N \geq 0$ are $D_0$-closed. Degree $0$ subspace is easy to analyze since it is one dimensional. Any constant function is always $D_0$-closed, but is exact if and only if the sum of the coefficients of $D_0$ is not equal to zero. If the sum of the coefficients of the $D_0$ is equal to zero, then any $D_0$-closed function is orthogonal on the degree $0$ subspace. Therefore the result of the theorem holds if we can prove that a given $D_0$-closed function $\xi$ in $\mathcal{H}_N$, $N \geq 1$, the function $\xi$ can be approximated with $D_0$-exact functions.
We shall investigate the properties of the Fourier coefficients of closed and exact functions, and we shall rather establish that the Fourier coefficients of a closed functions can be approximated in the appropriate sense with Fourier coefficients of exact functions. The ideas will be elaborated in the following sections.
[*Note.*]{} In two cases relevant for statistical physics questions, namely the second-order Ginzburg-Landau vector field $Y_0=\partial_1-\partial_0$ and the fourth-order Ginzburg-Landau vector field $X_0=\partial_1-2\partial_0+\partial_{-1}$, the decomposition result of Theorem \[the1\] is equivalent with the fluctuation-dissipation equation mentioned in the introduction section of the paper.
[*Note.*]{} To get a flavor of the result stated in Theorem \[the1\] we give some examples of exact and closed functions in the case of the fourth-order Ginzburg-Landau field, $X_0=\partial_1-2\partial_0+\partial_{-1}$: $x_n + x_{-n}-2x_0$ are $X_0$-exact, ${\bf 1}$, $x_0$, $x_n+x_{-n}$ are examples of $X_0$-closed but not $X_0$-exact functions. A strange phenomena appears for: besides the function [**[1]{}**]{} there exists another function that is $X_0$-closed and not $X_0$-exact, namely $x_0$. Therefore one may expect that the codimension of the space of exact functions is two. This is not the case and $x_0$ can be approximated with exact functions.
The set of multi-indices {#multiind}
========================
A multi-index $I = \{i_n\}_{n \in {\mathbb{Z}}}$ can be thought of as a configuration of particles sitting on the sites of the lattice ${\mathbb{Z}}$. On top of the site $n$ sit $i_n$ particles. Rather than saying how many particles are at each site, we give the positions of the particles. This way we obtain a vector $$\label{coding}
z_I = ( \underbrace{ n_1 \dots n_1}_{i_{n_1}}, \dots , \underbrace{ n_k \dots n_k}_{i_{n_k}}).$$ that lists, in increasing order, all occupied sites of $I$ repeated according to the number of particles that occupy the site. We assume the only non-zero entries of the multi-index $I$ are $i_{n_1}, \dots i_{n_k}$. Note that the dimension of the vector $z_I$ is the degree of the multi-index $I$. If the multi-index has zero degree then $z_I$ is just a point. We say that $z_I$ is a new coding of the multi-index $I$. This correspondence shows that the set ${ {\mathcal I}}_N$ is bijective with the set of vectors of ${\mathbb{Z}}^N$ with entries in increasing order or is in bijection with the quotient space ${\mathbb{Z}}^N/S_N$, where $S_N$ is the group of permutations of $N$ letters.
For the results that follow we need to say more about the set of multi-indices. We partition the set of multi-indices into orbits with the help of the group action $$\label{action}
{\mathbb{Z}}\times {\mathbb{Z}}^{{\mathbb{Z}}} \longrightarrow {\mathbb{Z}}^{{\mathbb{Z}}} \quad
(n, I) \longmapsto n \cdot I := \tau^n(I -\delta_n + \delta_0).$$ When restricted to ${\mathbb{Z}}\times { {\mathcal I}}$ the map (\[action\]) is not an action any more since the multi-indices that enumerate the basis of the $L^2$ space are constrained to have positive entries.
The orbits of the action (\[action\]) provide a partition of the set of multi-indices ${\mathbb{Z}}^{\mathbb{Z}}$. For each multi-index $I \in \mathcal{I} $ we define ${o(I)}$ to be the shadow of the orbit of $I$ on the set ${ {\mathcal I}}$, i.e., ${o(I)} = \{ J | J= n \cdot I \quad n \in {\mathbb{Z}}\} \cap { {\mathcal I}}= \{ J | J= n \cdot I\quad n\in s(I)\}.$ Here, $s(I)= \{ n \in {\mathbb{Z}}\; | \; i_n \neq 0 \}$ is the finite set of occupied positions of $I$. From now on we will refer to ${o(I)}$ as the orbit of $I$, although this is just a part of the actual orbit of the action. It has the advantage of being finite since the multi-index $I$ has all but finitely many entries zero and there are just finitely many $n$’s that after acting on $I$ give rise to a multi-index with positive entries. All the multi-indices in the same orbit have the same degree. The orbits partition ${ {\mathcal I}}$ and ${ {\mathcal I}}_N$. Call $\mathcal{O}$ the set of orbits, and $\mathcal{O}_N$ the set of orbits containing multi-indices of degree $N$.
It is worth mentioning that inside each orbit ${o(I)}$ there exists a unique representative supported on the positive integers. Denote this multi-index by $R({o(I)})$. To see that this is true let us assume that $I$ has some particles in some negative position, i.e., there exists some $n < 0$ such that $i_n \geq 1$. If $k$ is the leftmost occupied position of $I$ and $k<0$, then $k \cdot I \in {o(I)}$ is supported on the positive integers. Let us call $\mathcal{R}$ the set of all representatives, and $\mathcal{R}_N$ the set of degree-$N$ representatives.
So far the orbit space $\{{o(I)}\}_{I\in { {\mathcal I}}}$ is an abstract object. Fortunately we are able to give a concrete description of the orbit space. For this purpose it is very useful to know that each orbit, $o$ has a unique representative $R(o)$ supported on the positive semi-axis. The vector $z_{R(o)}$ is a point in the positive cone $\mathcal{C}_N^+=\{ z \in {\mathbb{Z}}^N | z=(z_1, \dots, z_N), \; 0 \leq z_1 \leq \dots \leq z_N \}$. Therefore the set of representatives, and in particular the set of orbits $\mathcal{O}_N$, are bijective with the cone $\mathcal{C}_N^+$. Since there is only one multi-index with zero degree, ${\bf 0}=(0)_{n \in {\mathbb{Z}}}$, the sets ${ {\mathcal I}}_0$, ${\mathcal O}_0$ and ${\mathcal R}_0$ contain just a single element. By convention, ${\mathcal C}_0^+$ is just one-point set.
We can say even more about this picture. The cone $\mathcal{C}_N^{+}$, itself is an orbit space, which we shall describe below.
Let us define the transformations that acts on the lattice ${\mathbb{Z}}^N$, for any $1\leq i,j \leq N$, $$\sigma_{i,j}:{\mathbb{Z}}^N \rightarrow {\mathbb{Z}}^N, \quad
\sigma_{i,j}(z_1,\dots ,z_i, \dots, z_j \dots ,z_N) = (z_1,\dots ,z_j, \dots, z_i, \dots ,z_N)$$ $$\mathrm{and} \quad \gamma_1: {\mathbb{Z}}^N \rightarrow {\mathbb{Z}}^N, \quad
\gamma_1 (z_1,z_2,\dots ,z_N) = (-z_1, z_2-z_1\dots ,z_N-z_1).$$ The smallest group generated by $\sigma_{i,j}$, $1 \leq i,j \leq N$ and $\gamma_1$ will be denoted by $\widetilde{S}_N$. To see that ${\widetilde{S}}_N$ is isomorphic with the group of permutations of $N$ letters, we write down the basic relations among the generating transformations: $(\gamma_1 \sigma_{1,2})^3 = \bf{id}$ and $\gamma_1 \sigma_{i,i+1}=\sigma_{i+1,i}\gamma_1$, $1 \leq i \leq N-1$. The group ${\widetilde{S}}_N$ has $S_N$, the group of permutations of $N$ letters as subgroup, and ${\widetilde{S}}_N$ decomposes into left cosets with respect to $S_N$, as ${\widetilde{S}}_N = S_N \cup
\gamma_1 S_N \dots \cup \gamma_N S_N $, where the transformations $\gamma_i$ are $$\gamma_i: {\mathbb{Z}}^N \rightarrow {\mathbb{Z}}^N, \quad \gamma_i (z_1,z_2,\dots ,z_N) = (-z_i, z_2-z_i\dots ,z_N-z_i).$$
It is interesting to note that ${\mathbb{Z}}^N/{\widetilde{S}}_N$ is bijective with the cone $\mathcal{C}_N^+$, as the next argument proves. Any orbit of ${\mathbb{Z}}^N/{\widetilde{S}}_N$ contains at least one vector, let say $z$, with components in increasing order. If this vector does not have positive co-ordinates, it means $z_1 < 0$. But $(-z_1, z_2-z_1, \dots , z_N-z_1)$ is still a point in the orbit of $z$ under the action of ${\widetilde{S}}_N$. We can rearrange the coordinates of the new vector to be in increasing order and hence the orbit of $z$ under the action of ${\widetilde{S}}_N$ contains at least one vector of the cone $\mathcal{C}_N^+$. To see that the orbit of $z$ does not contain more than one vector of $\mathcal{C}_N^+$ we use the coset decomposition of ${\widetilde{S}}_N$. If $z$ is in $\mathcal{C}_N^+$, then rearranging the co-ordinates of $z$ we obtain either the vector $z$ or some vector outside the cone $\mathcal{C}_N^+$. If we act on $z$ or some other vector obtained from $z$ by changing the places of the co-ordinates, with either of the transformations $\gamma_1, \dots \gamma_N$ we get a vector that has at least one negative co-ordinate, so does not belong to $\mathcal{C}_N^+$.
Now we can say that the set of orbits $\mathcal{O}_N$ is bijective with the cone $\mathcal{C}_N^+$, and hence with the quotient space ${\mathbb{Z}}^N/{\widetilde{S}}_N$. The bijection is $ o \in \mathcal{O}_N \mapsto z_{R(o)} \in \mathcal{C}_N^+$.
In addition, if $I$ and $J$ are two multi-indices in the same orbit of the action (\[action\]) then $z_I$ and $z_J$ are in the same orbit of the action of ${\widetilde{S}}_N$ on ${\mathbb{Z}}^N$. Assume that $J=n_j \cdot I$ with $I= \sum_{i=1}^k a_{i} \delta_{n_i}$, where $a_{i} \neq 0$ and $n_1 \leq \dots \leq n_k$. Then $J = \sum_{i=1,...,k, i\neq j} a_{i} \delta_{n_i -n_j}+
(a_{j}-1)\delta_0 + \delta_{-n_j}$, and so $$z_I = ( \underbrace{ n_1, \dots ,n_1}_{a_1}, \dots , \underbrace{ n_k, \dots ,n_k}_{a_k})$$ $$z_J = ( -n_j,\underbrace{ n_1-n_j, \dots ,n_1-n_j}_{a_1}, \dots , \underbrace{ 0, \dots ,0}_{a_j-1},
\dots, \underbrace{ n_k -n_j, \dots ,n_k-n_j}_{a_k}) .$$ It follows that $z_J$ is the image of $z_I$ under some element of ${\widetilde{S}}_N$.
We shall denote by $z \stackrel{S_N}{\sim} z'$ and $z \stackrel{{\widetilde{S}}_N}{\sim} z'$ two lattice points $z$ and $z'$ that have the same image in the quotient space ${\mathbb{Z}}^N/S_N$ and ${\mathbb{Z}}^N/{\widetilde{S}}_N$, respectively.
Before we leave this section it is important to notice the following crucial facts. Let $N \geq 1$. Since $\mathcal{I}_N$ is identified with ${\mathbb{Z}}^N/S_N$ we can think of any function $\hat{\xi}:\mathcal{I}_N \to {\mathbb{R}}$ as being a $S_N$-invariant function $\hat{\xi}:{\mathbb{Z}}^N \to {\mathbb{R}}$, where $ \hat{\xi}(z)=\hat{\xi}(I)$ if $z \stackrel{S_N}{\sim} z_I$. Similarly, since $\mathcal{O}_N$ is identified with ${\mathbb{Z}}^N/{\widetilde{S}}_N$ we can think of any function $c:\mathcal{O}_N \to {\mathbb{R}}$ as being a ${\widetilde{S}}_N$-invariant function ${\widetilde{c}}:{\mathbb{Z}}^N \to {\mathbb{R}}$, where ${\widetilde{c}}(z) = c(o)$ if there exists a multi-index $I \in o$ such that $z \stackrel{S_N}{\sim} z_I$.
Properties of closed functions and of exact functions {#clos-exac}
=====================================================
This section contains a detailed study of closed and exact functions.
**Closed functions.** We start with a very simple but important property of closed functions.
\[lema1\] Assume $D_0$ is a vector field with constant coefficients, $D_0=\sum_{k \in {\mathbb{Z}}} a_k \partial_k$. All but finitely many coefficients of the vector field $D_0$ are zero. A function $\xi \in L^2({\mathbb{R}}^{{\mathbb{Z}}}, d\nu_0^{gc})$ is $D_0$-closed if and only if the projection $Proj_{{\mathcal H}_N} \xi$ onto the degree $N$ subspace $\mathcal{H}_N$ is $D_0$-closed, for any $N \geq 0$.
[**Proof.**]{} We denote by $\partial_j$ the differential operator with respect to the $j^{\mathrm{th}}$ coordinate, and by $\partial_j^*= -\partial_j +x_j$ the adjoint operator of $\partial_j$. The adjoint is taken with respect to the inner product $<,>$ of $L^2({\mathbb{R}}^{{\mathbb{Z}}}, d\nu_0^{gc})$. The operators $\partial_j$ and $\partial_j^*$ are bounded operators when restricted to a degree subspace, although they are unbounded on the whole $L^2({\mathbb{R}}^{{\mathbb{Z}}}, d\nu_0^{gc})$ space.
If $\xi \in \mathcal{H}_N$, with Fourier series $\xi = \sum_{I \in \mathcal{I}_N}
\hat{\xi}_I H_I$, then the image of $\xi$ under the operator $\partial_j$ is $ \partial_j(\xi) = \sum_{I \in \mathcal{I}_N} \hat{\xi}_I H_{I-\delta_j},$ with the convention that if the multi-index $I-\delta_j$ has some negative entries then $H_{I-\delta_j}=0$. For a function $f \in L^2({\mathbb{R}}^{{\mathbb{Z}}}, d\nu_0^{gc})$ denote by $||f||=\sqrt{<f,f>}$ the $L^2$ norm of $f$.
The operators $\partial_j$ and $\partial_j^*$ act on the degree $N$ subspaces as follows: $$\partial_j ({\mathcal H}_N) \subseteq {\mathcal H}_{N-1}, \quad N \geq 1, \quad
\partial_j^* ({\mathcal H}_N) \subseteq {\mathcal H}_{N+1}, \quad N \geq 0.$$
The boundedness of these operators follows from the observation that $$\frac{1}{(N!)^N} \leq \inf_{I \in \mathcal{I}_N} ||H_I||^2 \leq
\sup_{I \in \mathcal{I}_N} ||H_I||^2 \leq 1,$$ and from the existence of two strictly positive constants, $C_1^N$, $C_2^N$, that depend just on $N$ such that $$\label{ineq1}
C_1^N \sum_{I \in \mathcal{I}_N} \hat{\xi}_I^2 \leq ||\xi||^2 \leq
C_2^N \sum_{I \in \mathcal{I}_N} \hat{\xi}_I^2, \quad \quad
C_1^N \sum_{I \in \mathcal{I}_N} \hat{\xi}_I^2 \leq ||\partial_j (\xi)||^2 \leq
C_2^N \sum_{I \in \mathcal{I}_N} \hat{\xi}_I^2 .$$ Indeed $$||\partial_j \xi ||^2 = \sum_{I \in \mathcal{I}_N} \hat{\xi}^2_I ||H_{I -\delta_j}||^2 \leq
\sum_{I \in \mathcal{I}_N} \hat{\xi}_I^2 \leq (N!)^N \sum_{I \in \mathcal{I}_N}\hat{\xi}_I^2 ||H_I||^2 \leq (N!)^N ||\xi||^2.$$ and hence the norm of the operator $\partial_j :\mathcal{H}_N \to \mathcal{H}_{N-1}$ is bounded above by $(N!)^N$.
The vector field $D_0$ with constant coefficients has similar properties: $$D_0({\mathcal H}_N) \subseteq {\mathcal H}_{N-1}, \quad N \geq 1, \quad D_0^*
({\mathcal H}_N) \subseteq {\mathcal H}_{N+1}, \quad N \geq 0.$$
For any function $\xi \in L^2({\mathbb{R}}^{{\mathbb{Z}}}, d\nu_0^{gc})$ and any test function $\phi \in {\mathcal H}_{N-1}$ we have $$<D_n(\tau^m \xi ), \phi > = <\xi,\tau^{-m} (D_n^* \phi)> =
<\mathrm{Proj}_{{\mathcal H}_N}\xi, \tau^{-m} (D_n^* \phi)> =$$ $$\label{qwe111}
= <D_n(\tau^m \mathrm{Proj}_{{\mathcal H}_N}\xi), \phi >,$$ $$<D_m(\tau^n \xi ), \phi > = <\xi, \tau^{-n} (D_m^* \phi)> =
<\mathrm{Proj}_{{\mathcal H}_N}\xi, \tau^{-n} (D_m^* \phi)> =$$ $$\label{qwe1}
= <D_m(\tau^n \mathrm{Proj}_{{\mathcal H}_N}\xi), \phi >.$$ It follows that $D_n(\tau^m \xi ) =D_m(\tau^n \xi )$ in the weak sense if and only if $D_n(\tau^m \mathrm{Proj}_{{\mathcal H}_N}\xi)=D_m(\tau^n \mathrm{Proj}_{{\mathcal H}_N}\xi)$ in the strong sense for all $N \geq 0$.
We recall that a function $\xi$ is closed if and only if $D_n(\tau^m \xi ) =D_m(\tau^n \xi )$ for all $m,n \in {\mathbb{Z}}$, which, by the previous equalities (\[qwe111\]) and (\[qwe1\]), is equivalent to $$D_n(\tau^m \mathrm{Proj}_{{\mathcal H}_N}\xi)=D_m(\tau^n \mathrm{Proj}_{{\mathcal H}_N}\xi)
\quad m,n \in {\mathbb{Z}}\quad N \geq 0.$$ Therefore, a function $\xi$ is closed if and only if $\mathrm{Proj}_{{\mathcal H}_N}\xi$ is closed for all $N \geq 0$.
[*Note.*]{} If $\xi =\sum_{I \in \mathcal{I}_N} \hat{\xi}_I H_I$ is a function inside the space $\mathcal{H}_N$, two norms can be defined for $\xi$: the $L^2$ norm $||\xi||$ and the sum of squared Fourier coefficients $\sum_{I \in \mathcal{I}_N} \hat{\xi}_N^2$. It is important to note the inequality (\[ineq1\]) implies that these two norms define the same topology on the space $\mathcal{H}_N$.
[*Note.*]{} Assume $\xi \in \mathcal{H}_N$ is a $D_0$-closed function, with Fourier series expansion $\xi = \sum_{I \in \mathcal{I}} \hat{\xi}_I H_I$. We calculate, $$D_n \xi = \sum_{I \in { {\mathcal I}}} \bigg[ \sum_{k \in {\mathbb{Z}}} a_k \hat{\xi}_{I+\delta_{(n+k)}}\bigg] H_I, \quad
D_0(\tau^n \xi) = \sum_{I \in { {\mathcal I}}} \bigg[ \sum_{k \in {\mathbb{Z}}} a_k \hat{\xi}_{\tau^n(I+\delta_{k})} \bigg] H_I.$$ Therefore a function is closed if and only if its Fourier coefficients satisfy the relations: $$\label{feq4}
\sum_{k \in {\mathbb{Z}}} a_k \hat{\xi}_{I+\delta_{(n+k)}}=\sum_{k \in {\mathbb{Z}}} a_k \hat{\xi}_{\tau^n(I+\delta_{k})}
\quad n \in {\mathbb{Z}}, \quad I \in {\mathcal I}.$$
[**Construction of exact functions.**]{} It is important to have some examples of functions that are exact. The functions that will be constructed next will be used in the proof of the decomposition theorem \[the1\], to approximate closed functions with exact ones.
\[lem2\] Let $c$ be a function defined on the set of orbits with finite support (i.e., c(o)=0 except for finitely many orbits $o$). The function $$\xi = \sum_{o \in \mathcal{O}} c(o ) D_0 \bigg[ \sum_{n \in {\mathbb{Z}}} \tau^n
H_{R(o)+\delta_0} \bigg]$$ is $D_0$-exact and the Fourier coefficients of $\xi$ are $$\label{feq85}
\hat{\xi}_I= \sum_{k \in {\mathbb{Z}}} a_k c({o(\tau^{-k} I)}).$$
[**Proof.**]{} The function $\xi$, that has been introduced is well-defined since the sum is over a finite set, and is exact as a sum of exact functions. To conclude the lemma we need to calculate the Fourier coefficients of $\xi$. We have, $$\begin{aligned}
\label{feq2}
\xi & = & \sum_{o \in \mathcal{O}} c(o) D_0\bigg[ \sum_{n \in {\mathbb{Z}}}
H_{ \tau^{-n}(R(o)+\delta_0)} \bigg]
= \sum_{o \in \mathcal{O} , n \in {\mathbb{Z}}} c(o) \sum_{k \in {\mathbb{Z}}}
a_kH_{\tau^{-n}(R(o)+\delta_0- \delta_{-n+k})} = \nonumber \\
& & = \sum_{o \in \mathcal{O} , n \in {\mathbb{Z}}} c(o)
\sum_{k \in {\mathbb{Z}}} a_kH_{\tau^{-k}[(-n+k)\cdot R(o)]}
= \sum_{I \in \mathcal{I}} \sum_{k \in {\mathbb{Z}}} a_kc({o(\tau^{k}I)})H_I.
$$
To justify the integration by parts in the calculation above (\[feq2\]) we make the following observation. For any multi-index $I \in \mathcal{I}$ there exists a unique orbit $o \in \mathcal{O}$ and a unique integer $n\in {\mathbb{Z}}$ such that $I=\tau^{-k}[(-n+k) \cdot R(o)]$. This is a consequence of the freeness of the action (\[action\]). Moreover, the orbit $o$ is the same as ${o(\tau^k I)}$. We stress again that the sums in (\[feq2\]) are over finite sets as $c$ has finite support. Actually all computations that we carried out to prove this lemma are valid because $c$ is a function with finite support and the sums are finite, although this wasn’t emphasized each time we used it. Also we have made use of the convention that $H_I=0$ if $I$ is a multi-index with negative entries.
Let $N \geq 1$ be a natural number and $e=(1, \dots , 1) \in {\mathbb{Z}}^N$. In addition if ${\widetilde{c}}$ is a real-valued function defined on ${\mathbb{Z}}^N$, with finite support and ${\widetilde{S}}_N$-invariant then the function $$\label{exact}
\xi_{{\widetilde{c}}} = \sum_{I \in \mathcal{I}_N} \bigg(\sum_{k \in {\mathbb{Z}}}a_k {\widetilde{c}}(z_I-ke)\bigg) H_I$$ is a well-defined $D_0$-exact function in the degree $N$ subspace $\mathcal{H}_N$.
**Proof.** This lemma follows from lemma \[lem2\]. Since ${\widetilde{c}}:{\mathbb{Z}}^N \to {\mathbb{R}}$ is ${\widetilde{S}}_N$-invariant, it makes sense to introduce $c:\mathcal{O} \to {\mathbb{R}}$, where $c(o)= {\widetilde{c}}(z_I)$ if $I$ is a multi-index in the orbit $o$ of degree $N$, and $c(o)=0$ otherwise. We should note that if $I$ is a multi-index in the orbit $o$ then $z_I+ke = z_{\tau^{-k}I}$ and ${\widetilde{c}}(z_I-ke)=c( o(\tau^{k}I))$. Hence the Fourier coefficients of the function $\xi_{{\widetilde{c}}}$ are of the form $\sum_{k \in {\mathbb{Z}}}a_k c(o(\tau^{k}I))$, and the function $\xi_{{\widetilde{c}}}$ is $D_0$-exact.
In the previous lemma an operator has come out in a natural way in our construction of exact functions. Below we provide the exact definition of this operator.
Let $D_0=\sum_{k \in {\mathbb{Z}}} a_k \partial_k$ be a vector field with constant coefficients, all the coefficients being zero except finitely many. The vector field $D_0$ defines an operator $T_{D_0}$ that acts on functions $c:{\mathbb{Z}}^N \to {\mathbb{R}}$ and produces a function $T_{D_0}c:{\mathbb{Z}}^N \to {\mathbb{R}}$, where $$(T_{D_0}c)(z) = \sum_{k \in {\mathbb{Z}}} a_k c(z-ke), \quad z \in {\mathbb{Z}}^N.$$ Above $e$ is the vector $(1,\dots ,1)$ of the lattice ${\mathbb{Z}}^N$.
Proof of the decomposition theorem \[the1\]
===========================================
We start by listing two important properties of the operator $T_{D_0}$ introduced at the end of the previous section.
\[compact\] Let $c$ be a real-valued function defined on the lattice ${\mathbb{Z}}^N$, $N\geq 1$. We assume that the function $c$ is square-summable and $\widetilde{S}_N$- invariant. Then, there exists a sequence $(c_n)_{n \geq 1}$ of real-valued, finitely supported, $\widetilde{S}_N$-invariant functions such that $T_{D_0}c_n \to T_{D_0}c$ as $n \to \infty$ and the convergence is in the Hilbert space topology of $L^2({\mathbb{Z}}^N)$.
[**Proof.**]{} We define a sequence of $\widetilde{S}_N$-invariant regions of the lattice ${\mathbb{Z}}^N$, namely $$\label{hexa}
P_i = \cup_{\gamma \in \widetilde{S}_N} \gamma\{z=(z_1, \dots , z_N) \in {\mathbb{Z}}^N | 0 \leq z_1 \leq \dots \leq z_N \leq i-1 \}, \quad i\geq 1.$$ For the reader convenience we add two pictures of the region $P_i$ in dimension $N=1$, respectively $N=2$. In dimension $N=1$ the region $P_i$ contains the lattice points inside the segment $[-i+1,i-1]$, whereas in dimension $N=2$ the region $P_i$ contains the lattice points inside the hexagon shown below.
(400,50)(-50,0) (70,30) (90,30) (110,30) (130,30) (150,30) (170,30) (190,30) (210,30) (230,30) (60,10)[-i+1]{} (82,10)[-i+2]{} (120,10)[$\dots$]{} (150,10)[0]{} (175,10)[$\dots$]{} (205,10)[i-2]{} (225,10)[i-1]{}
Figure 1. The region $P_i$ in dimension $N=1$.
(400,250)(-50,0) (70,30) (90,30) (110,30) (130,30) (150,30) (70,50) (70,70) (70,90) (70,110) (150,190) (170,190) (190,190) (210,190) (230,190) (230,170) (230,150) (230,130) (230,110) (90,130) (110,150) (130,170) (170,50) (190,70) (210,90) (150,170)(170,170)(190,170)(210,170) (130,150)(150,150)(170,150)(190,150) (210,150) (110,130)(130,130) (150,130)(170,130) (190,130)(210,130) (90,110)(110,110) (130,110)(150,110) (170,110)(190,110) (210,110) (90,90)(110,90) (130,90)(150,90) (170,90)(190,90) (90,70)(110,70) (130,70)(150,70) (170,70)(90,50)(110,50)(130,50) (150,50) (50,15)[(-i+1,-i+1)]{} (50,115)[(-i+1,0)]{} (140,200)[(0,-i+1)]{} (220,200)[(i-1,i-1)]{} (220,115)[(i-1,0)]{} (140,15)[(0,-i+1)]{} (140,115)[(0,0)]{}
Figure 2. The region $P_i$ in dimension $N=2$.
Beside being ${\widetilde{S}}_N$-invariant, the sequence of regions $(P_i)_{i \geq 1}$ defined above, grows to cover the entire lattice ${\mathbb{Z}}^N$ as $i \to \infty$. Define $c_n$ to be $c {\bf 1}_{P_n}$, for $n \geq 1$. Since ${\bf 1}_{P_n}$ is the characteristic function of the region $P_n$ we have immediately that $c_n$ is a finitely supported, ${\widetilde{S}}_N$-invariant function. Square-summability of $c$ implies that $c_n \to c$ as $n \to \infty$ in the topology of $L^2({\mathbb{Z}}^N)$ (the norm $||c-c_n||^2=\sum_{z \notin P_n} c^2(z)$ involves only the values of $c$ outside the region $P_n$, and these values decay to zero as $n \to \infty$ since $c$ is square-summable). Then, obviously, $c_n \to c$ and $Tc_n \to Tc$ as $n \to \infty$ in the topology of $L^2({\mathbb{Z}}^N)$.
Below we discuss certain facts about the Fourier transform of functions defined on the lattice ${\mathbb{Z}}^N$. The Fourier transform of a function $c:{\mathbb{Z}}^N \to \mathbb{C}$ is formally defined to be $$\mathcal{F} c :[-\pi , \pi )^N \to \mathbb{C}, \quad \mathcal{F}c (\alpha ) =
\frac{1}{\sqrt{2\pi }}\sum_{z \in {\mathbb{Z}}^N} c(z) e^{i z \alpha }.$$ In the exponent above $z\alpha$ stands for the dot product $z_1\alpha_1+\dots +z_N \alpha_N$. The reader may consult Rudin [@Rud] for an extended treatment of Fourier transform of functions defined on lattice. We remind the reader that $\mathcal{F}$ is an isometry between the spaces $L^2( {\mathbb{Z}}^N )$ and $L^2 ([-\pi, \pi )^N)$. The space $L^2([-\pi,\pi)^N)$ is considered with respect to the Lebesgue measure on $[-\pi, \pi)^N$. Also if $c$ is invariant under a certain group of transformations then $\mathcal{F}c$ is invariant, as well. Though the symmetry group of $\mathcal{F}c$ might not coincide with the symmetry group of $c$. Indeed if $c$ is symmetric, or $S_N$-invariant then $\mathcal{F}c$ is symmetric. Now suppose that $c$ is ${\widetilde{S}}_N$-invariant then $\mathcal{F}c$ is invariant under the action of the group ${\widetilde{\Sigma}}_N$, generated by the transformations: $$s_{ii+1}:[-\pi,\pi)^N \to [-\pi,\pi)^N, \quad 1 \leq i \leq N-1$$ $$s_{ii+1}(\alpha_1, \dots , \alpha_N) =(\alpha_1, \dots , \alpha_{i+1}, \alpha_{i}, \dots , \alpha_N),$$ and $$g:[-\pi,\pi)^N \to [-\pi,\pi)^N,
g(\alpha_1, \dots , \alpha_N) =(\mathrm{mod}_{2\pi}(-\alpha_1- \dots -\alpha_N) , \alpha_{2}, \dots , \alpha_N).$$ On the line above we used the notation $\mathrm{mod}_{2\pi}(t)$. Any real number $t$ can be written uniquely as $2\pi a+b$, where $a$ is an integer number and $b$ is a real number in the interval $[-\pi, \pi)$. By $\mathrm{mod}_{2\pi}(t)$ we denote the remainder $b$. It is also true that if the Fourier transform $\mathcal{F}c$ is ${\widetilde{\Sigma}}_N$-invariant then $c$ is ${\widetilde{S}}_N$-invariant.
In section \[clos-exac\] we have established that a function $\xi=\sum_{I \in \mathcal{I}_N} \hat{\xi}_I H_I$ is $D_0$-closed if and only if the following holds: $$\label{feq10}
\sum_{k \in {\mathbb{Z}}} a_k \hat{\xi}_{I+\delta_{(n+k)}}=\sum_{k \in {\mathbb{Z}}} a_k \hat{\xi}_{\tau^n(I+\delta_{k})}
\quad n \in {\mathbb{Z}}, \quad I \in {\mathcal I}.$$ Obviously we can use the Fourier coefficients of $\xi$ to construct a $S_N$-invariant function $\hat{\xi}: {\mathbb{Z}}^N \to {\mathbb{R}}$, $\hat{\xi}(z)=\hat{\xi}_I$ if $z \stackrel{S_N}{\sim} z_I$. The relations (\[feq10\]) force our function $\xi$ to satisfy $$\label{feq11}
\sum_{k \in {\mathbb{Z}}} a_k \hat{\xi}(z+ke_1) = \sum_{k \in {\mathbb{Z}}} a_k \hat{\xi}(z-z_1e - (z_1+k)e_1),
\quad z=(z_1, \dots , z_N) \in {\mathbb{Z}}^N.$$ The vectors $e$ and $e_1$ of the lattice ${\mathbb{Z}}^N$ are $(1, \dots , 1)$ and $(1, 0, \dots , 0)$, respectively. After applying the Fourier transform in both sides of the equation (\[feq11\]) we find that $\hat{\xi}$ satisfies $$\label{feq12}
p(e^{-i\alpha_1}) (\mathcal{F}\hat{\xi})(\alpha) = p(e^{i(\alpha_1+\dots + \alpha_N)})(\mathcal{F}\hat{\xi})(g( \alpha)) , \quad
\alpha=(\alpha_1, \dots , \alpha_N) \in [-\pi, \pi)^N,$$ where $p$ is the rational function $p(x)=\sum_{k \in {\mathbb{Z}}} a_k x^k$. To wrap up our argument we can say that the $D_0$-closedness condition implies property (\[feq12\]).
Next we shall establish a crucial fact about functions that satisfy relation (\[feq12\]).
Let $\hat{\xi}$ be a real-valued, $S_N$-invariant function defined on the lattice ${\mathbb{Z}}^N$, $N \geq 1$ that satisfies (\[feq12\]). Then, there exists a sequence $(c_n)_{n \geq 1}$ of real-valued, square-summable, ${\widetilde{S}}_N$-invariant functions defined on the lattice ${\mathbb{Z}}^N$ such that $T_{D_0}c_n \to \hat{\xi}$ as $n \to \infty$ in the topology of $L^2({\mathbb{Z}}^N)$.
[**Proof.**]{} Examples of functions $\hat{\xi}$ satisfying the properties listed in the hypothesis of this lemma, are functions constructed, as explained before in this section, from the Fourier coefficients of closed functions.
The first step towards establishing our result is to solve, at least on a formal level, the equation $T_{D_0}c =\hat{\xi}$. The Fourier transform for functions defined on the lattice will help us to make our guess.
After applying the Fourier transform in each side of the equation $T_{D_0}c =\hat{\xi}$, we get $$p(e^{i(\alpha_1+\dots +\alpha_N)})(\mathcal{F}c)(\alpha) =(\mathcal{F}\hat{\xi})(\alpha), \quad
\alpha=(\alpha_1, \dots , \alpha_N) \in {\mathbb{Z}}^N .$$ where $p$ is the rational function $\sum_{j \in {\mathbb{Z}}} a_j x^j$ canonically associated to $T_{D_0}$.
After multiplying with a high enough power of $x$ the equation $p(x)=0$, we see that any solution $x$ of $p(x)=0$ has to be the root of a certain polynomial. Since the number of roots of any polynomial is finite, the number of solutions of $p(x)=0$ is finite, as well. If the equation $p(x)=0$ has no solutions on the unit circle then the equation $T_{D_0}c=\hat{\xi}$ can be solved in the space $L^2({\mathbb{Z}}^N)$. Indeed the unique, square-summable solution of $T_{D_0}c=\hat{\xi}$ is $$c=\mathcal{F}^{-1}\bigg( \frac{1}{p(e^{i(\alpha_1+\dots +\alpha_N)})}( \mathcal{F}\hat{\xi}) \bigg).$$ That $\hat{\xi}$ has been built out of the Fourier coefficients of a closed function implies that $c$ is ${\widetilde{S}}_N$-invariant. Hence the lemma is proved in this case. We can choose $c_n=c$, for any $n \geq 1$.
A more involved case is when the equation $p(x)=0$ has solutions on the unit circle. If the sum of the coefficients of $p$ is equal to $0$ then the number $1$ is a solution of $p(x)=0$. The cases arising from interacting particle models are of this kind. The difficulty in this case arises because the equation $Tc=\hat{\xi}$ can not be solved in $L^2({\mathbb{Z}}^N)$. To get around this problem we shall consider a slightly modified equation $\mathcal{F}(T_{D_0}c)=(\mathcal{F}\hat{\xi})1_{A_n}$. The function $\mathcal{F}\hat{\xi}$ is multiplied with the characteristic function of a set $A_n$ that in ${\widetilde{\Sigma}}_N$-invariant and carefully chosen to avoid the unit roots of $p$. More precisely, $$A_n= \cap_{\gamma \in {\widetilde{\Sigma}}_N} \bigg\{ \gamma(\alpha)\; \bigg|\; \alpha =(\alpha_1, \dots , \alpha_N) \in [-\pi,\pi]^N,$$ $$| \; \mathrm{mod}_{2\pi}(\alpha_1+\dots +\alpha_N)-r_k|> \frac{1}{n}, \;\; e^{ir_k}
\mathrm{unit}\; \mathrm{root} \; \mathrm{of}\; p \bigg\}.$$
The next table contains the roots of the rational function $p(x)$ in four particular cases.\
Vector field $D_0$ Rational function $p(x)$ Solutions of $p(x)=0$
---------------------------------------- -------------------------- -------------------------------------------------------
$\partial_0$ $1$ none
$\partial_1 - \partial_0$ $x-1$ $1$
$\partial_1-2\partial_0+\partial_{-1}$ $x-2+x^{-1}$ $1$, $1$
$\partial_3-\partial_0$ $x^3-1$ $1$, $\frac{1+\sqrt{-3}}{2}$, $\frac{1-\sqrt{-3}}{2}$
If the vector field $D_0$ is $\partial_1 -\partial_0$ or $\partial_1-2 \partial_0+\partial_{-1}$ then the region $A_n$ in dimension $N=1$ is just $A_n= \{ \alpha \in [-\pi,\pi) \; | \; |\alpha| > \frac{1}{n} \}$,\
Suppose we are given a $Y_0$-closed function $\xi$ with Fourier expansion $\xi= \sum_{I \in \mathcal{I}}\hat{\xi}_I H_I$. We can actually turn our graph into a weighted graph by assigning to each directed edge $(o(I),o(\tau I))$ the weight $\hat{\xi}_I$.
(400,50)(-50,0) (50,27)[\[]{} (130,27)[)]{} (170,27)[(]{} (250,27)[\]]{} (50,30)[(1,0)[80]{}]{} (170,30)[(1,0)[80]{}]{} (50,10)[-$\pi$]{} (150,10)[0]{} (120,10)[$-\frac{1}{n}$]{} (170,10)[$\frac{1}{n}$]{} (250,10)[$\pi$]{}
Figure 3. The region $A_n$ in dimension $N=1$.
(400,250)(-50,0) (70,30)[(1,0)[60]{}]{} (130,30)[(0,1)[80]{}]{} (50,50)[(0,1)[60]{}]{} (50,110)[(1,0)[80]{}]{} (70,30)[(-1,1)[20]{}]{}
(170,30)[(1,0)[60]{}]{} (170,30)[(0,1)[60]{}]{} (230,30)[(-1,1)[60]{}]{}
(250,50)[(-1,1)[60]{}]{} (250,50)[(0,1)[60]{}]{} (250,110)[(-1,0)[60]{}]{}
(250,210)[(0,-1)[60]{}]{} (230,230)[(-1,0)[60]{}]{} (250,210)[(-1,1)[20]{}]{} (250,150)[(-1,0)[80]{}]{} (170,150)[(0,1)[80]{}]{}
(50,150)[(1,0)[60]{}]{} (50,150)[(0,1)[60]{}]{} (110,150)[(-1,1)[60]{}]{}
(130,170)[(-1,1)[60]{}]{} (130,230)[(0,-1)[60]{}]{} (130,230)[(-1,0)[60]{}]{}
Figure 4. The region $A_n$ in dimension $N=2$, as a subset of the square $[-\pi,\pi]^2$.
Let $c_n$ be the unique $L^2({\mathbb{Z}}^N)$ solution of the equation $\mathcal{F}(T_{D_0}c_n)=(\mathcal{F}\hat{\xi})1_{A_n}$, $n \geq 1$. The solution $c_n$ is defined through $$c_n = \mathcal{F}^{-1}\bigg( \frac{1}{p(e^{i(\alpha_1+\dots +\alpha_N)})}
(\mathcal{F}\hat{\xi})(\alpha)1_{A_n}(\alpha) \bigg).$$ The ${\widetilde{\Sigma}}_N$-invariance of the set $A_n$ and the fact that $\hat{\xi}$ is constructed from the Fourier coefficients of a closed function implies that $c_n$ is ${\widetilde{S}}_N$-invariant, $n\geq 1$.
Obviously we have the convergence $\mathcal{F}(T_{D_0}c_n) \to \mathcal{F}\hat{\xi}$ as $n \to \infty$ in the topology of $L^2([-\pi,\pi]^N)$, hence $T_{D_0}c_n \to \hat{\xi}$ as $n \to \infty$ in the topology of $L^2({\mathbb{Z}}^N)$.
**Proof of the decomposition theorem \[the1\]**. Let $\xi \in L^2({\mathbb{R}}^{{\mathbb{Z}}}, d\nu_0^{gc})$ be a $D_0$-closed function. From lemma 4.1 we know that $\mathrm{Proj}_{\mathcal{H}_N}\xi$ is $D_0$-closed, for any $N \geq 0$. $\mathrm{Proj}_{\mathcal{H}_0}\xi$ is a constant function. If the sum of the coefficients of $D_0$ is not equal to zero then $\mathrm{Proj}_{\mathcal{H}_0}\xi$ is $D_0$-exact, otherwise $\mathrm{Proj}_{\mathcal{H}_0}\xi$ is orthogonal on any $D_0$-exact function. Therefore, the decomposition theorem follows as long as we establish that any $D_0$-closed function $\xi \in \mathcal{H}_N$ can be approximated by $D_0$-exact functions, for any $N \geq 1$.
Assume $\xi =\sum_{I \in \mathcal{I}_N} \hat{\xi}_I H_I \in \mathcal{H}_N$, $N \geq 1$. Define the $S_N$-invariant function $\hat{\xi} :{\mathbb{Z}}^N \to {\mathbb{R}}$ through $\hat{\xi}(z) = \hat{\xi}_I$ if $z \stackrel{S_N}{\sim} z_I$, see (3). We use lemmas 5.1 and 5.2 to find a sequence of finitely supported, ${\widetilde{S}}_N$-invariant functions $(c_n)_{n \geq 1}$, such that $T_{D_0} c_n \to \hat{\xi}$ as $n \to \infty$ in the topology of $L^2({\mathbb{Z}}^N)$. But lemmas 4.2 and 4.3 tell us that each of $T_{D_0}c_n$, $n \geq 1$ defines a $D_0$-exact function $\xi_{c_n}$, see (\[exact\]). At the end of lemma 4.1 we noticed that the topology of $\mathcal{H}_N$ and $L^2({\mathbb{Z}}^N)$ are equivalent, hence we can claim that $\xi_{c_n} \to \xi$ as $n \to \infty$ in the Hilbert space topology of $\mathcal{H}_N$, or $L^2({\mathbb{R}}^{{\mathbb{Z}}}, d\nu_0^{gc})$.
Second-order Ginzburg-Landau field and algebraic topology
=========================================================
We conclude with some remarks about the second-order Ginzburg-Landau field $Y_0=\partial_1-\partial_0$ which has been studied in the work of S. R. S. Varadhan, [@Var1]. Our approach places Varadhan’s result in a new light by depicting an algebraic topologic aspect, to be explained below.
In section \[multiind\] we presented an extensive study of the set of multi-indices $\mathcal{I}$. There we partitioned the set of multi-indices $\mathcal{I}$ into disjoint orbits, and we denoted by $\mathcal{O}$ the space of orbits. Below we exhibit a procedure to construct a directed graph that has as vertices the orbits of the set of multi-indices.
A directed graph is a pair $(V, E)$ of two sets, where $V$ is the set of vertices of the graph and $E$ is the set of directed edges. A directed edge is a pair of two vertices $(v_1, v_2)$ where the first vertex indicates the starting point of the edge and the second vertex indicates the tip of the edge. For us we choose $V$ to be the set of orbits $\mathcal{O}$. Also we say that we have a directed edge $(o_1, o_2)$ if there exist a multi-index $I \in o_1$ such that $\tau I \in o_2$. Notice that if there exists an edge between two orbits then the orbits contain multi-indices with identical degrees. Hence our graph will have at least one connected component for each degree $N \geq 0$. We shall show that there exists precisely one connected component for each degree $N \geq 0$.
We would like to have a concrete or geometric presentation of the graph. For this purpose we use the identification of the set of orbits $\mathcal{O}_N$ containing the multi-indices of degree $N$, with the cone $\mathcal{C}_N^+$ of the lattice ${\mathbb{Z}}^N$, $N \geq 0$.
Assume $N=0$, then $\mathcal{O}_N$ contains a single orbit, the orbit of the multi-index $0$ and this orbit contains a single multi-index. Since the multi-index $0$ has the property that $0=\tau 0$, we will have a directed edge going out of and returning to $0$; in other words we have a loop at $0$.
Assume that $N \geq 1$. It can be shown that a directed edge links $z \in \mathcal{C}_N^+$ to $z' \in \mathcal{C}_N^+$ if and only if either $z'=z-e_1-\dots -e_N$ or $z'=z+e_i$ for some $1 \leq i \leq N$. Here $e_i$ is lattice vector $(0,\dots , 1, \dots, 0)$ with the $i$th coordinate $1$. We shall indicate below how this presentation of the graph can be obtained.
Let $o \in \mathcal{O}_N$ be some orbit and $R(o) =\sum_{i=1}^k a_i \delta_{n_i}$, with $a_i \geq 1$ and $0 \leq n_1 < n_2< \dots < n_k$ be the representative of the orbit $o$. Given our rule, the orbit $o$ is connected by an edge going out of $o$ to each of the orbits $ o(\tau R(o))$, $o(\tau(n_1 \cdot R(o)))$, $\dots$, $o(\tau(n_k \cdot R(o)))$. For each of the orbits in the list before we can calculate the representatives and the corresponding point in the cone $\mathcal{C}_N^+$.
For example: the representative of the orbit $o$ is $R(o) =\sum_{i=1}^k a_i \delta_{n_i}$ and the cone point is $$z_{R(o)}=(\underbrace{n_1, \dots , n_1}_{a_1}, \dots , \underbrace{n_k, \dots , n_k}_{a_k}).$$ Assume $n_1 \geq 1$. The representative of the orbit of $\tau R(o)$ is $ \tau R(o) $ and the corresponding cone point is $$z_{\tau R(o)}=(\underbrace{n_1-1, \dots , n_1-1}_{a_1}, \dots , \underbrace{n_k-1, \dots , n_k-1}_{a_k}).$$ We notice that $z_{\tau R(o)} = z_{R(o)}-e_1-\dots - e_N$. Also if $n_1=0$ the representative of the orbit of $\tau R(o)$ is $ (-1) \cdot \tau R(o) $ and the corresponding cone point is $$z_{(-1) \cdot \tau R(o)}=(\underbrace{0, \dots , 0}_{a_1-1}, 1, \dots , \underbrace{n_k-1, \dots , n_k-1}_{a_k}),$$ and $z_{(-1) \cdot \tau R(o)} = z_{R(o)}+e_{a_1}$. Similarly, we can analyze the other orbits connected with $o$.
In particular our discussion proves that for any two given orbits $o_1$ and $o_2$ if there exists a multi-index $I \in o_1$ and $\tau I \in o_2$ then this multi-index is unique. As we will see later that this observation allows us to assign in a unique way a multi-index to any directed edge of our graph.
We include three pictures of the connected components of the directed graph for $N=0$, $N=1$ and $N=2$.
(400,50)(-50,0) (150,60)[(1,0)[5]{}]{} (150,40) (150,20) (149,8)[0]{}
Figure 5. The connected component of the graph for $N=0$.
(400,50)(-50,0) (170,30)[(1,0)[40]{}]{} (50,30)[(1,0)[40]{}]{} (90,30)[(1,0)[40]{}]{} (130,30)[(1,0)[40]{}]{} (50,30)(70,80)(90,30) (90,30)(110,80)(130,30) (130,30)(150,80)(170,30) (170,30)(190,80)(210,30) (50,30) (90,30) (130,30) (170,30) (210,30)
(220,30)[…]{} (50,10)[0]{} (90,10)[1]{} (130,10)[2]{} (170,10)[3]{} (210,10)[4]{}
(70,55)[(-1,0)[5]{}]{} (110,55)[(-1,0)[5]{}]{} (150,55)[(-1,0)[5]{}]{} (190,55)[(-1,0)[5]{}]{}
Figure 6. The connected component of the graph for $N=1$.
(400,200)(-50,0) (170,30)[(1,0)[40]{}]{} (210,30)[(0,1)[40]{}]{} (210,70)[(-1,-1)[40]{}]{} (210,70)[(0,1)[40]{}]{} (210,110)[(-1,-1)[40]{}]{} (210,110)[(0,1)[40]{}]{} (210,150)[(-1,-1)[40]{}]{} (210,150)[(0,1)[40]{}]{} (210,190)[(-1,-1)[40]{}]{} (50,30)[(1,0)[40]{}]{} (90,30)[(0,1)[40]{}]{} (90,70)[(-1,-1)[40]{}]{} (90,30)[(1,0)[40]{}]{} (130,30)[(0,1)[40]{}]{} (130,70)[(-1,-1)[40]{}]{} (130,70)[(0,1)[40]{}]{} (130,110)[(-1,-1)[40]{}]{} (130,30)[(1,0)[40]{}]{} (170,30)[(0,1)[40]{}]{} (170,70)[(-1,-1)[40]{}]{} (170,70)[(0,1)[40]{}]{} (170,110)[(-1,-1)[40]{}]{} (170,110)[(0,1)[40]{}]{} (170,150)[(-1,-1)[40]{}]{} (90,70)[(1,0)[40]{}]{} (130,70)[(1,0)[40]{}]{} (170,70)[(1,0)[40]{}]{} (130,110)[(1,0)[40]{}]{} (170,110)[(1,0)[40]{}]{} (170,150)[(1,0)[40]{}]{} (50,30) (90,30) (90,70) (130,30) (130,70) (130,110) (170,30) (170,70) (170,110) (170,150) (210,30) (210,70) (210,110) (210,150) (210,190)
(220,30)[…]{} (40,10)[(0,0)]{} (80,10)[(1,0)]{} (120,10)[(2,0)]{} (160,10)[(3,0)]{} (200,10)[(4,0)]{}
Figure 7. The connected component of the graph for $N=2$.
Suppose we are given a function $\xi \in L^2({\mathbb{R}}^{{\mathbb{Z}}}, d\nu_0^{gc})$ with Fourier expansion $\xi =\sum_{I \in \mathcal{I}} \hat{\xi}_I H_I$. We can actually turn our directed graph into a weighted graph by assigning to each directed edge $(o_1,o_2)$ the Fourier coefficient $\hat{\xi}_I$ corresponding to the unique multi-index $I$ such that $I \in o_1$ and $\tau I \in o_2$. Note that each Fourier coefficient will be assigned to one and only one edge and each edge will have assigned one and only one Fourier coefficient, since there is a one-to-one correspondence between the edges of our graph and the set $\mathcal{I}$ of multi-indices. For example the edge $(o(I), o(\tau I))$ will have attached the weight $\hat{\xi}_I$.
(400,50)(-50,0) (125,40)[$\hat{\xi}_I$]{} (50,30) (200,30) (50,30)[(1,0)[150]{}]{} (45,10)[$o(I)$]{} (195,10)[$o(\tau I)$]{}
Figure 8. A directed edge and its attached weight.
It is interesting to note that if $\xi$ is $Y_0$-closed then the weights of the graph discussed before sum up to zero along any directed cycle of the graph except the loop of the connected component corresponding to $N=0$. Indeed the closedness condition of $\xi$ imposes no restriction on the coefficient $\hat{\xi}_0$. Also note that the $Y_0$-closedness condition (\[feq4\]) $$\hat{\xi}_{I+\delta_{(n+1)}}-\hat{\xi}_{I+\delta_n} = \hat{\xi}_{\tau^n(I+\delta_{1})}-
\hat{\xi}_{\tau^n(I+\delta_{0})} \quad n \in {\mathbb{Z}}, \quad I \in \mathcal{I}$$ plus the square-integrability of $\xi$ are equivalent to the property that the weights of the graph associated to $\xi$ sum up to zero around any directed cycle of any connected component corresponding to $N \geq 1$. However if $\xi$ is $Y_0$-exact then $\hat{\xi}_0=0$ and hence the weights of the graph sum up to zero around any directed cycle of the graph. If $\xi$ is $Y_0$-exact and is constructed as in lemma \[lem2\] then we can say that in any connected component of the graph all but finitely many weights are zero.
The above can be explained from an algebraic topologic point of view. We can turn our directed graph into a $2$-dimensional $\Delta$-complex (see A. Hatcher’s book on Algebraic Topology, [@Hat]) by attaching enough discs to cycles of the graph such that each of the connected components $N \geq 1$ can be retracted to a single point. We do not attach a disc onto the loop of the connected component $N=0$. After the attaching process the $2$-dimensional $\Delta$-complex can be retracted to the disjoint union of a circle with a countable number of points. The Fourier coefficients of a $Y_0$-exact function form a coboundary of our $2$-dimensional $\Delta$-complex and the Fourier coefficients of a $Y_0$-closed function form a cocycle for our $2$-dimensional $\Delta$-complex. Since the cohomology group $H^1(C, \mathbb{R})$ of a circle $C$ is one-dimensional we expect the space of $Y_0$-exact functions to have codimension one inside the space of $Y_0$-closed functions.
Acknowledgments {#acknowledgments .unnumbered}
===============
I would like to thank Professor George A. Elliott for his suggestion to use a Fourier analysis approach to solve the problem discussed in this paper.
[8]{} C. Kipnis, C. Landim, [*Scaling limits of interacting particle systems*]{}, Springer-Verlag, Berlin, 1999. A. Hatcher, [*Algebraic Topology*]{}, Cambridge University Press, Cambridge, 2002. http://www.math.cornell.edu/ hatcher/AT/ATpage.html J. Quastel, [*Diffusion of color in the simple exclusion process*]{}, Comm. Pure Appl. Math. [**45**]{} (1992), no.6, 623-679. W. Rudin, [*Fourier analysis on groups*]{}, John Wiley and Sons, Inc., New York, 1962. A. Savu, [*Hydrodynamic scaling limit of continuum solid-on-solid model*]{}, preprint, http://front.math.ucdavis.edu/math.PR/0506001. H. Spohn, [*Large scale dynamics of interacting particle systems*]{}, Springer-Verlag, Heidelberg and Berlin, 1991. S. R. S. Varadhan, [*Nonlinear diffusion limit for a system with nearest neighbor interactions II*]{}, Asymptotic problems in probability theory: Stochastic models and diffusion on fractals (Sanda/Kyoto, 1990), 75-128, Pitman Research Notes in Mathematics, 75-128. [**283**]{} Longman Sci. Tech. Harlow, 1993. S. R. S. Varadhan, H. T. Yau, [*Diffusive limit of lattice gas with mixing conditions*]{}, Asian J. Math. [**1**]{} (1997), no.4, 623-678.
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abstract: 'The equation of state (EOS) of nuclear matter, i.e., the thermodynamic relationship between the binding energy per nucleon, temperature, density, as well as the isospin asymmetry, has been a hot topic in nuclear physics and astrophysics for a long time. The knowledge of the nuclear EOS is essential for studying the properties of nuclei, the structure of neutron stars, the dynamics of heavy ion collision (HIC), as well as neutron star mergers. HIC offers a unique way to create nuclear matter with high density and isospin asymmetry in terrestrial laboratory, but the formed dense nuclear matter exists only for a very short period, one cannot measure the nuclear EOS directly in experiments. Practically, transport models which often incorporate phenomenological potentials as an input are utilized to deduce the EOS from the comparison with the observables measured in laboratory. The ultrarelativistic quantum molecular dynamics (UrQMD) model has been widely employed for investigating HIC from the Fermi energy (40 MeV per nucleon) up to the CERN Large Hadron Collider energies (TeV). With further improvement in the nuclear mean-field potential term, the collision term, and the cluster recognition term of the UrQMD model, the newly measured collective flow and nuclear stopping data of light charged particles by the FOPI Collaboration can be reproduced. In this article we highlight our recent results on the studies of the nuclear EOS and the nuclear symmetry energy with the UrQMD model. New opportunities and challenges in the extraction of the nuclear EOS from transport models and HIC experiments are discussed.'
author:
- Yongjia Wang
- 'Qingfeng Li[^1]'
title: Application of microscopic transport model in the study of nuclear equation of state from heavy ion collisions at intermediate energies
---
Introduction
============
Matter with extremely conditions, such as high density, temperature, and isospin asymmetry, is hardly observed on earth, but can be found at various astrophysical objects. Study of the properties of dense matter may provide deep insight into the structure and evolution of astrophysical objects. The nuclear equation of state (EOS) which characterizes the thermodynamic relationship between the binding energy $E$, temperature $T$, density $\rho$, as well as the isospin asymmetry $\delta=(\rho_{n}-\rho_{p})/(\rho_{n}+\rho_{p})$ in nuclear matter (an uniform and infinite system with neutrons and protons) has attracted considerable attention from both nuclear physics and astrophysics communities since long time ago[@BALi08; @Tsang:2012se; @Baldo:2016jhp; @Oertel:2016bki; @Li:2018lpy; @Roca-Maza:2018ujj; @Burrello:2019wyi; @Giuliani:2013ppnp; @Ma:2018wtw; @Ono:2019jxm; @Xu:2019hqg; @Chen:2013uua; @Xu:2009vi; @Gao:2019vby]. At zero temperature, both phenomenological and microscopic model calculations have indicated that the binding energy per nucleon in isospin asymmetric nuclear matter can be well approximated by $E(\rho,\delta)=E(\rho,\delta=0) + E_{sym}(\rho)\delta^{2} + \mathcal{O}(\delta^{4})$. The first term $E(\rho,\delta=0)$ is the binding energy per nucleon in the isospin symmetric nuclear matter, $E_{sym}(\rho)$ is the density-dependent nuclear symmetry energy. Odd-order $\delta$ terms are vanished because of the charge symmetry of nuclear forces. Higher-order terms for $\delta$ are usually negligible for most investigations as the typical value of $\delta$ is about 0.2 in nuclei and nuclear collisions. It is worth noting that higher-order terms in $\delta$ may play an important role in the studying of astrophysical processes[@Cai:2011zn; @Steiner:2006bx; @Pu:2017kjx; @Liu:2018far]. The parabolic approximation would be expected to be valid only at small $\delta$, however based on many theoretical calculations, it turns out that it is fairly satisfied even at $\delta$=1 with moderate density[@BALi08].
It is of great interest to investigate how $E(\rho,\delta=0)$ and $E_{sym}(\rho)$ vary as density, because the information of both $E(\rho,\delta=0)$ and $E_{sym}(\rho)$ are essential for studying the structures and the properties of nuclei and neutron stars, the dynamics of heavy-ion collision, supernovae explosions, as well as neutron star mergers. It is also one of the fundamental goals of the current and future nuclear facilities (e.g., the CSR and HIAF in China, the FRIB in the United States, the RIBF in Japan, the SPIRAL2 in France, the FAIR in Germany) around the world. Practically, the nuclear incompressibility $K_0=9\rho^2\left(\frac{\partial^2E(\rho,\delta=0)}{\partial\rho^2}\right)|_{\rho=\rho_{0}}$, the symmetry energy coefficient $S_0=E_{\rm sym}(\rho_{0})$, and its slope parameter $L=3\rho\left(\frac{\partial{E_{\rm sym}(\rho)}}{\partial\rho}\right)|_{\rho=\rho_{0}}$ and its curvature parameter $K_{sym}$=$9\rho^2\left(\frac{\partial^2{E_{sym}(\rho)}}{\partial\rho^2}\right)|_{\rho=\rho_{0}}$ which characterize how $E(\rho,\delta=0)$ and $E_{sym}(\rho)$ change as density, have attracted considerable attention. Although great endeavors have been made to constrain these parameters, a precise picture of the EOS has still not emerged, especially at high densities, which remains an open challenge for further research.
In the present work, we review and highlight our recent results on the studies of the nuclear EOS of isospin symmetric matter, the medium effects on the nucleon-nucleon cross section, and the density-dependent nuclear symmetry energy based on the ultrarelativistic quantum molecular dynamics (UrQMD) model. This article is organized as follows. In next section, the UrQMD model and its recent updates, as well as observables in HIC are briefly introduced. In Section \[set1\], the influence of the in-medium nucleon-nucleon cross section on observables in HIC at intermediate energies (with beam energy of several hundreds MeV per nucleon) is discussed. Section \[set2\] gives the result of studying the nuclear EOS of isospin symmetric matter from the rapidity-dependent elliptic flow. Constraints on the density-dependent nuclear symmetry energy with the UrQMD model are reviewed and discussed in Section \[set3\]. Finally, a summary and outlook is given in Section\[set5\].
Model description and observables {#set0}
=================================
The UrQMD model has been widely used to study nuclear reactions within a large range of beam energies, from the Fermi energy (tens of MeV per nucleon) up to the highest energy (TeV) presently available at the Large Hadron Collider [@Bass98; @Bleicher:1999xi; @Li:2011zzp; @Li:2012ta]. In the UrQMD model, each hadron can be represented by a Gaussian wave packet [@Bass98]. Usually, the width parameter of 2 fm$^2$ is chosen for simulating collisions with Au. Mean field potential and collision terms are two of the most important ingredients of the UrQMD model. In this section, we briefly discuss the recent updates on these two terms.
Mean field potential
--------------------
After carefully choosing nuclei with a proper binding energy and radius in the initialization, the coordinate $\textbf{r}_i$ and momentum $\textbf{p}_i$ of nucleon $i$ are propagated according to $$\begin{aligned}
\dot{\textbf{r}}_{i}=\frac{\partial \langle H \rangle}{\partial\textbf{ p}_{i}},
\dot{\textbf{p}}_{i}=-\frac{\partial \langle H \rangle}{\partial \textbf{r}_{i}}.\end{aligned}$$ Here, [*$\langle H \rangle$*]{} is the total Hamiltonian function, it consists of the kinetic energy $T$ and the effective interaction potential energy $V$. For studying HICs at intermediate energies, the following density and momentum dependent potential has been widely employed in QMD-like models [@Aichelin:1991xy; @Hartnack:1997ez; @Li:2005gfa; @Zhang:2018rle], $$\label{eq2}
V=\alpha\left(\frac{\rho}{\rho_0}\right)+\beta\left(\frac{\rho}{\rho_0}\right)^{\eta} + t_{md} \ln^2[1+a_{md}(\textbf{p}_{i}-\textbf{p}_{j})^2]\frac{\rho}{\rho_0}.$$
Here $t_{md}$=1.57 MeV and $a_{md}$=500 $c^{2}$/GeV$^{2}$. In present version, $\alpha$, $\beta$, and $\eta$ are calculated using Skyrme parameters via $\frac{\alpha}{2}=\frac{3}{8}t_{0}\rho_{0}$, $\frac{\beta}{\eta+1}=\frac{1}{16}t_{3}\rho_{0}^{\eta}$, and $\eta=\sigma+1$. The parameters $t_{0}$, $t_{1}$, $t_{2}$, $t_{3}$ and $x_{0}$, $x_{1}$, $x_{2}$, $x_{3}$, $\sigma$ are the well-known parameters of the Skyrme force[@Zhang:2006vb; @Zhang:2007gd]. Following recent progress in the study of the density-dependent nuclear symmetry energy and to better describe the recent experimental data at intermediate energies, the surface, the surface asymmetry term, the symmetry energy term obtained from the Skyrme potential energy density functional have been introduced to the present version [@Wang:2013wca; @Wang:2014rva]. It reads as $$\label{urho}
\begin{aligned}
u_{Skyrme}=&u_{sur}+u_{sur,iso}+u_{sym}\\
&=\frac{g_{\text{sur}}}{2\rho_{0}}(\nabla\rho)^{2}+\frac{g_{\text{sur,iso}}}{2\rho_{0}}[\nabla(\rho_{n}-\rho_{p})]^{2}\\
&+\left(A_{\text{sym}}\frac{\rho^{2}}{\rho_{0}}+B_{\text{sym}}\frac{\rho^{\eta+1}}{\rho_{0}^{\eta}}+C_{\text{sym}}\frac{\rho^{8/3}}{\rho_{0}^{5/3}}\right)\delta^2.
\end{aligned}$$ And, the parameters $g_{\text{sur}}$, $g_{\text{sur,iso}}$, $A_{sym}$, $B_{sym}$, and $C_{sym}$ are related to the Skyrme parameters via $$\begin{aligned}
% \nonumber to remove numbering (before each equation)
\frac{g_{sur}}{2} &=& \frac{1}{64}(9t_{1}-5t_{2}-4x_{2}t_{2})\rho_{0}, \\
\frac{g_{sur,iso}}{2} &=& -\frac{1}{64}[3t_{1}(2x_{1}+1)+t_{2}(2x_{2}+1)]\rho_{0},\\
A_{sym} &=& -\frac{t_{0}}{4}(x_{0}+1/2)\rho_{0}, \\
B_{sym} &=& -\frac{t_{3}}{24}(x_{3}+1/2)\rho_{0}^{\eta}, \\
C_{sym} &=& \frac{1}{24}\left(\frac{3\pi^{2}}{2}\right)^{2/3}\rho_{0}^{5/3}\Theta_{sym},\end{aligned}$$ where $\Theta_{sym}=3t_{1}x_{1}-t_{2}(4+5x_{2})$. With the introduction of the Skyrme potential energy density functional, one can easily choose different Skyrme interactions to study properties of dense nuclear matter formed in HICs with the UrQMD model. E.g., to investigate the incompressibility $K_0$ of isospin symmetric nuclear matter, one can select Skyrme interactions which yield similar values of the nuclear symmetry energy but different values of $K_0$. While Skyrme interactions which give similar value of $K_0$ but very different density-dependent nuclear symmetry energy can be selected to study the effect of the nuclear symmetry energy on various observables.
The potentials for produced mesons, i.e., pion and kaon, also can be incorporated into the UrQMD model, it is found that, with considering the kaon potential (including both the scalar and vector aspects) and pion potential, the collective flow of pion and kaon can be reproduced as well, details can be found in Refs.[@Liu:2018xvd; @Du:2018ruo].
The in-medium nucleon-nucleon cross section
-------------------------------------------
Besides the mean field potential, the nucleon-nucleon cross section (NNCS) is one of the most essential ingredients of the transport model as well. In free space, the information of NNCS has been well measured by experiments, but in the nuclear medium, how the NNCS varies with the nuclear density and momentum is still an open question. It is known from many theoretical studies that the NNCS in the in-medium is smaller than that in the free space, however, the degree of this reduction is still far from being entirely pinned down[@lgq14; @Sammarruca:2005tk; @HJS; @CF; @HFZ07; @WGL; @Alm:1995chb; @Mao:1994zza; @Li:2003vd]. One of possible way to obtain the detailed information of the in-medium NNCS is to compare the transport model simulations with the corresponding experimental data. Several different forms of the in-medium NNCS have been used in transport models, such as $\sigma_{NN}^{\text{in-medium}}=(1-\eta\rho/\rho_{0})\sigma_{NN}^{\text{free}}$ with $\eta=0.2$[@gd10; @Zhang:2007gd], $\sigma_{NN}^{\text{in-medium}}=0.85\rho^{-2/3}/\tanh(\frac{\sigma^{\text{free}}}{0.85\rho^{-2/3}})$[@dds], $\mathcal{F}=\sigma_{NN}^{\text{in-medium}}/\sigma_{NN}^{\text{free}}=(\mu_{NN}^{*}/\mu_{NN})^2$, where $\mu_{NN}^{*}$ and $\mu_{NN}$ are the $k$-masses of the colliding nucleons in the medium and in free space [@Li:2005iba; @Feng:2011eu; @Guo:2013fka].
In the present UrQMD model, the in-medium elastic NNCS are treated as the product of a medium correction factor $F$ and the cross sections in free space for which the experimental data are available. The total nucleon-nucleon binary scattering cross sections can thus be expressed as $$\sigma_{tot}^{*}=\sigma_{in}+\sigma_{el}^{*}=\sigma_{in}+F(\rho,p)
\sigma_{el} \label{ecsf}$$ with $$F(\rho,p)=\left\{
\begin{array}{l}
f_0 \hspace{3.45cm} p_{NN}>1 {\rm GeV}/c \\
\frac{F_{\rho} -f_0}{1+(p_{NN}/p_0)^\kappa}+f_0 \hspace{1cm}
p_{NN} \leq 1 {\rm GeV}/c
\end{array}
\right.
\label{fdpup}$$ where $p_{NN}$ denotes the momentum in the two-nucleon center-of-mass (c.o.m.) frame. Here $\sigma_{el}$ and $\sigma_{in}$ are the NN elastic and inelastic cross sections in free space, respectively. We note here that the experimental data of the inelastic cross sections in free space are still be used. Because the probability for a nucleon to undergo inelastic scattering and to become a $\Delta$ is small in HICs around 1 GeV$/$nucleon regime[@Bass:1995pj]. Thus the influence of inelastic channels on nucleonic observables, which are mainly focused on in this work, can be neglected. The density-dependent factor $F_\rho$ is parameterized as $$F_\rho=\lambda+(1-\lambda)\exp[-\frac{\rho}{\zeta\rho_0}]. \label{fr}$$ In this work, $\zeta$=1/3 and $\lambda$=1/6 are used which correspond to FU3 in Ref. [@Li:2011zzp]. To systematically investigate the effect of the in-medium NNCS on various observables in HICs at intermediate energies, four different parametrization sets for $f_{0}$, $p_{0}$ and $\kappa$ in Eq. \[fdpup\] are chosen to obtain different momentum dependences of $F(\rho,p)$. The reduced factors obtained with these parametrization sets are displayed in Fig.\[NNCS\]. Specifically, the parametrization set FU3FP1 was usually applied to investigate HICs around the Fermi energy region where the mean-field potential and the Pauli blocking effects are much more important. It has been found that with FU3FP1 parameter set, the experimental data of both the collective flow and the nuclear stopping power at the Fermi energy domain can be reproduced[@Li:2011zzp; @Guo:2012aa; @Wang:2012sy; @lipc; @Li:2018bus]. While, at higher energies, e.g., FU3FP2 has been used to extract the density-dependent symmetry energy with the elliptic flow data, as calculations with this parametrization set are found to be much more close to the experimental data of collective flows at 400 MeV$/$nucleon[@Russotto:2011hq]. We further introduce the FP4 and FP5 sets which lie roughly between FP1 and FP2. This permits more detailed studies of the momentum dependence of the in-medium NNCS by taking advantage of the large number of new FOPI data.
----------------------------------------------- -- -- --
**Set & $f_0$ & $p_0$ \[GeV/$c$\] & $\kappa$\
FP1 & 1 & 0.425 & 5 \
FP2 & 1 & 0.225 & 3\
FP4 & 1 & 0.3 & 8\
FP5 & 1 & 0.34 & 12\
**
----------------------------------------------- -- -- --
: The parameter sets FP1, FP2, FP4 and FP5 used for describing the momentum dependence of $F(\rho,p)$. []{data-label="tabfp"}
 The medium correction factor $F(\rho,p)$ obtained with the parameterization on the momentum dependence with the four options FP1, FP2, FP4, and FP5 given in Table \[tabfp\] at $\rho$=0.5$\rho_0$, $\rho_0$, and 2$\rho_0$.](fig1-NNCS.eps){width="40.00000%"}
Usually, the UrQMD transport program stops at 150 fm$/c$ and an isospin-dependent minimum spanning tree (iso-MST) method which was introduced by Zhang [*et al.*]{} [@Zhang:2012qm] is used to construct clusters. In this method, if the relative distances and momenta of two nucleons are smaller than $R_{0}$ and $P_{0}$, respectively, they are considered to belong to the same fragment. It is found that with a proper set of $R_{0}$ and $P_{0}$, the fragment mass distribution in HICs at intermediate energies can be reproduced well [@Zbiri:2006ts; @Russotto:2011hq; @Li:2016mqd]. The parameters adopted in this paper are $R_{0}^{pp}$=2.8 fm, $R_{0}^{nn}$=$R_{0}^{np}$=3.8 fm, and $P_{0}$=0.25 Gev/c. We would like to note here that the collective flow and the nuclear stopping power, which will be focused in this work, are insensitive to $R_{0}$ and $P_{0}$, as well as the stopping time, when they are selected in their reasonable ranges [@Wang:2013wca].
Observables
-----------
The directed $v_{1}$ and elliptic $v_{2}$ flows are the two of most widely studied observables in HICs at energy from intermediate energies to the relativistic energies, which can be obtained from the Fourier expansion of the azimuthal distribution of detected particles [@Reisdorf:1997fx; @FOPI:2011aa; @Heinz:2013th], $$\label{v1}
v_{1}\equiv \langle cos(\phi)\rangle=\left\langle\frac{p_{x}}{p_{t}}\right\rangle,$$ $$\label{v2}
v_{2}\equiv \langle cos(2\phi)\rangle=\left\langle\frac{p_{x}^{2}-p_{y}^{2}}{p_{t}^{2}}\right\rangle,$$ in which $p_{x}$ and $p_{y}$ are the two components of the transverse momentum $p_{t}=\sqrt{p_{x}^{2}+p_{y}^{2}}$. And the angle brackets in Eq.\[v1\] and Eq.\[v2\] indicate an average over all considered particles from all events. The directed flow $v_1$ characterizes particle motion (bounce-off or rotational-like) in the reaction plane (defined by the impact parameter $b$ in the $x$-axis and the beam direction $z$-axis), while the elliptic flow $v_2$ describes the emission (squeeze-out) perpendicular to the reaction plane. Both $v_1$ and $v_2$ have complex multi-dimensional structure. For a certain species of particles produced in a nuclear reaction with fixed colliding system, beam energy, and impact parameter, they depend both on the rapidity $y_z$ and the transverse momentum $p_t$. The scaled units $y_0\equiv y/y_{pro}$ and $u_{t0}\equiv u_t/u_{pro}$ (with $u_t=\beta_t\gamma$ the transverse component of the four-velocity and $u_{pro}$ is the velocity of the incident projectile in the c.o.m system of two nuclei) are used instead of $y_z$ and $p_t$ throughout, in the same way as done in the experimental report[@FOPI:2011aa], in order to scale with whole incident energies. The subscript $pro$ denotes the incident projectile in the c.o.m system.
Usually, the slope of $v_1$ and the value of $v_2$ at mid-rapidity ($y_0$$\sim$0) are calculated and compared to the experimental data to extract the nuclear EOS and the in-medium NNCS[@Danie02; @Ollitrault:1997vz]. Roughly speaking, at low beam energies ($\le$ 100 MeV$/$nucleon), the slope of $v_1$ is negative while the $v_2$ is positive, nucleons are more likely to be emitted in the reaction plane and undergo a rotation-like motion[@Andronic:2006ra]. With increasing beam energy, the slope of $v_1$ is increasing to a maximal (positive) value while the $v_2$ is decreasing to a minimal (negative) value at beam energies about 400-600 MeV$/$nucleon[@Andronic:2004cp; @Le]. Further increasing beam energy, the slope of $v_1$ decreases while the $v_2$ increases with beam energy. In general, both $v_{1}$ and $v_{2}$, as well as nuclear stopping power in HICs around 1 GeV$/$nucleon are strongly related to the detailed ingredients of the nuclear EOS and the in-medium NNCS[@Zheng:1999gt; @Persram:2001dg; @Andronic:2004cp; @Gaitanos:2004ic; @Li:2005jy; @Zhang:2006vb; @BALi08; @Li:2011zzp; @Kaur:2016eaf; @Barker:2016hqv; @Basrak:2016cbo].
The nuclear stopping power which measures the efficiency of converting the beam energy in the longitudinal direction into the transverse direction is also one of the most important observables. Serval different definitions$/$quantities of nuclear stopping power have been used and reported in literature, such as the quadrupole momentum tensor $Q_{zz}$=$\sum_{i}2p_z^2(i)-p_x^2(i)-p_y^2(i)$, and the ratio of transverse to parallel energy $R_E$ [@Lehaut:2010zz], the ratio of the variances of the transverse rapidity distribution over that of the longitudinal rapidity distributions $varxz$ [@Reisdorf:2004wg]. In present work, we mainly focus on $varxz$, which reads
$$varxz=\frac{<y_{x}^2>}{<y_{z}^2>} . \label{eqvartl}$$
Here $$<y_{x,z}^2>=\frac{\sum(y^2_{x,z}N_{y_{x,z}})}{\sum
N_{y_{x,z}}}, \label{eqgm}$$ where $<y_{x}^2>$ and $<y_{z}^2>$ are the variances of the rapidity distributions of nucleons in the $x$ and $z$ directions, respectively. $N_{y_{x}}$ and $N_{y_{z}}$ denote the numbers of nucleons in each of the $y_x$ and $y_z$ rapidity bins. Apparently, one expects that for full stopping, the value of $varxz$ will be unity, while it will be zero for full transparency. The excitation function of the stopping power from the Fermi energy to several GeV has shown that $varxz$ first increases to a maximal value (close to but smaller than unity) at beam energy around 800 MeV$/$nucleon then decreases afterwards [@Andronic:2006ra; @Reisdorf:2004wg; @FOPI:2010aa]. Besides the collective flow and the nuclear stopping power, other observables such as particle yield, fragment multiplicity distribution, rapidity distribution, kinetic energy and transverse momentum spectra are also applied widely to deduce the properties of the formed dense nuclear matter.
Influence of the in-medium nucleon-nucleon cross section on observables {#set1}
=======================================================================
 (a) The slope of the directed flow and the elliptic flow (b) at mid-rapidity ($y_0$=0) for light particles up to mass number $A=3$ ($^3H$ and $3^He$) calculated with FU3FP1, FU3FP2, FU3FP4 and FU3FP5 (lines with symbols) parametrizations. The $^{197}$Au+$^{197}$Au collision at $E_{\rm lab}$=250 MeV$/$nucleon with $0.25<b_0<0.45$ is considered as an example. The FOPI experimental data (stars) are from Ref. [@FOPI:2011aa]. ](fp1-fp2-fp4-fp5-flow.eps){width="48.00000%"}
 The nuclear stopping power $varxz$ of free protons, deuterons, tritons, as well as hydrogen isotopes ($Z=1$) produced in central $^{197}$Au+$^{197}$Au collisions at the beam energies 150, 250, and 400 MeV$/$nucleon. Calculations with the FU3FP4 and FU3FP5 sets are compared with the FOPI experimental data (stars) [@FOPI:2010aa]. ](varxz-150-250-400.eps){width="48.00000%"}
To show the effect of the in-medium NNCS on varies observables, $^{197}$Au+$^{197}$Au collisions at beam energies 150, 250, and 400 MeV$/$nucleon for centrality $0<b_0<0.45$ are calculated. The reduced impact parameter $b_0$ is defined as $b_0=b/b_{max}$ with $b_{max} = 1.15 (A_{P}^{1/3} + A_{T}^{1/3})$ fm. The slope of the $v_1$ and $v_2$ at mid-rapidity for light particles are displayed in Fig.\[flow\]. One sees clearly that calculations with FU3FP4 (blue line) and FU3FP5 (red line) are well separated. It implies that the directed and elliptic flows are very sensitive to the momentum dependence of the in-medium NNCS within a narrow region of $p_{NN}=0.2-0.4$ GeV/$c$, as the largest difference between these two parametrizations exist in that narrow region (shown in Fig.\[NNCS\]). Both the slope of $v_{1}$ calculated with FU3FP2 and FU3FP4 and the $v_{2}$ calculated with FU3FP1 and FU3FP5 track each other closely. Large difference between FU3FP2 and FU3FP4 at the low momenta and between FU3FP1 and FU3FP5 at high momenta can be observed in Fig.\[NNCS\]. Thus one may conclude that the slope of the directed flow is not sensitive to the low momentum part while the elliptic flow is not sensitive to the high momentum part of the in-medium NNCS. However, the sensitivity of the collective flow to the FU3FP4 and FU3FP5 sets will be reduced at higher beam energies since they almost overlap at higher relative momentum. Further, it can be seen that both the $v_1$ slope and the $v_2$ of free protons at mid-rapidity can be quite well reproduced with FU3FP5, while that of deuterons and A=3 clusters calculated with FU3FP4 are found to be more close to the experimental data than that with FU3FP5. Consequently, the FU3FP4 and FU3FP5 parametrization sets offer the greatest possible degree of the momentum-dependent in-medium NNCS. Besides the in-medium NNCS, other ingredients in transport models, such as, the initialization, the nuclear EOS, as well as the Pauli blocking effects, may also affect the collective flows and the nuclear stopping power to some extensive, details can be found in our previous publications[@Li:2011zzp; @Wang:2013wca; @lipcjpg].
Figure.\[varxz\] displays the nuclear stopping power $varxz$ of free protons, deuterons, tritons, as well as hydrogen isotopes ($Z=1$) in $^{197}$Au+$^{197}$Au collisions at the beam energies 150, 250, and 400 MeV$/$nucleon. Once again, it is found that the $varxz$ of free protons can be well reproduced both with FU3FP4 and FU3FP5, while the results of other light clusters calculated with FU3FP4 are found to be more close to the experimental data. The $varxz$ obtained with FU3FP5 is smaller than that with FU3FP4. Because FU3FP5 denotes a larger reduction on the NNCS at lower relative momenta than FU3FP4 does, the more violent collision prevailing in FU3FP4 parametrization enhances the nuclear stopping power.
Besides the total in-medium NNCS, the differential cross section, i.e., the angular distribution, in transport model also plays an important role. As discussed in our previous publication[@Wang:2016yti], by comparing the results of the collective flows and stopping power calculated with different angular distributions within the UrQMD model, it is found that both the collective flows and the nuclear stopping power obtained by using the forward-backward peaked differential NNCS are smaller than that with the isotropic one, while the elliptic flow difference between neutrons and hydrogen isotopes can hardly be influenced by the angular distributions. Details can be found in Ref. [@Wang:2016yti].
Determination of the nuclear incompressibility {#set2}
==============================================
 The incompressibility of isospin symmetric nuclear matter from 37 analyses using nuclei structure observations collected by J. R. Stone [*et al.*]{} in Ref.[@Stone:2014wza]. The red circle denotes the result deduced by J. R. Stone [*et al.*]{} in Ref.[@Stone:2014wza]. The dashed line represents the averaged value.](K0-year.eps){width="48.00000%"}
$K_0$ (MeV) $S_0$ (MeV) $L $ (MeV)
-------- ------------- -- ------------- -- ------------ -- --
Skxs15 201 31.9 34.8
MSK1 234 30.0 33.9
SKX 271 31.1 33.2
: \[tab:table1\] Saturation properties of nuclear matter as obtained with the three Skyrme interactions used in studying the incompressibility $K_0$.
The EOS of isospin symmetric nuclear matter can be expanded as $\frac{E}{A}(\rho)=E_0+\frac{K_0}{18}(\frac{\rho-\rho_0}{\rho_0})^2+...$, therefore, a more accurate value of $K_0$ means a better understanding of the nuclear EOS around the normal density. Constraints on $K_{0}$ through comparing experimental data on nuclear structure properties and theoretical model calculations have been summarized in Ref. [@Stone:2014wza], and the results are displayed in Fig.\[k0\]. As can be seen in Fig.\[k0\], most of these constraints indicate that $K_0$ should be in the range 200-300 MeV. While in Ref.[@Stone:2014wza], the authors showed that $250<K_0<315$ MeV can be obtained, based on the up-to-date data on the giant monopole resonance energies. In Ref.[@Khan:2013mga], the authors studied the giant monopole resonance energies of $^{208}$Pb and $^{120}$Sn, based on the constrained Hartree-Fock-Bogoliubov approach, $190<K_0<270$ MeV is found out. Although the incompressibility $K_0$ has been extensively investigated, different models offer a wide range of results for $K_0$, see, e.g., Refs. [@Stone:2014wza; @Khan:2013mga; @Giuliani:2013ppnp] and references therein.
Extraction of the incompressibility $K_0$ with HIC also has a long history, to our best knowledge, the very first studies can be found in 1980s[@Molitoris:1986pp; @Molitoris:1985gs; @Kruse:1985hy; @Aichelin:1986ss; @Stoecker:1986ci; @Cassing:1990dr]. The collective flow and particle (e.g., $\pi$ and kaon) productions are two of the main observables used to extract $K_0$. Using the microscopic Vlasov-Uehling-Uhlenbeck (VUU) model, evidence for a stiff ($K_0 \sim 380$ MeV) nuclear EOS was presented from a comparison with experimental data on pion production and collective sidewards flow by Joseph Molitoris and Horst Stöcker et al. in 1985 [@Molitoris:1986pp; @Molitoris:1985gs; @Kruse:1985hy]. By comparing pBUU model calculations to the directed and elliptic flows in Au+Au at the beam energies from 0.15 to 10.0 GeV$/$nucleon, the most extreme cases (for $K_{0}$ larger than 380 MeV or less than 167 MeV) have been ruled out by Danielewicz *et al*. [@Danie02]. In Refs [@Hartnack:2005tr; @Feng:2011dp], it was found that calculations with the soft EOS ($K_{0}$=200 MeV) are close to the kaon yields and yield ratios.
 The pressure and the binding energy per nucleon in symmetric nuclear matter as a function of density. The lines represent calculations for the Skxs15, MSK1, and SKX interactions. The results obtained by Danielewicz et al.[@Danie02] and Arnaud Le Fèvre et al.[@Fevre:2015fza] are represented by shaded regions.](pressure.eps){width="48.00000%"}
Recently, $v_{2n}$ which relates to the elliptic flow ($v_2$) in a broader rapidity range has been found to be very sensitive to the incompressibility $K_0$[@Fevre:2015fza]. By comparing the FOPI data with the calculations using the isospin quantum molecular dynamics (IQMD) model, a incompressibility $K_0=190 \pm 30$ MeV was extracted[@Fevre:2015fza]. In view of the fact that the collective flow also can be influenced by the in-medium NNCS and the findings from the comparison of the transport models, i.e., results from different transport models are diversified even the same physical inputs are required [@Xu:2016lue], more studies on $v_{2n}$ seems quite necessary.
To constrain the incompressibility $K_0$ using $v_{2n}$, the Skxs15, MSK1, and SKX interactions which give quite similar values of nuclear symmetry energy but the incompressibilities $K_0$ varies from 201 MeV to 271 MeV (see Table \[tab:table1\])[@Dutra:2012mb] are considered. The binding energy per nucleon and the pressure as a function of the density are illustrated in Fig.\[press\]. For comparison, constraints obtained by Danielewicz et al.[@Danie02] and by Fèvre et al.[@Fevre:2015fza] are also displayed with shaded regions.
 The elliptic flow of free protons and deuterons in Au+Au collisions at $E_{\rm lab}=0.4$ GeV$/$nucleon with centrality $0.25<b_0<0.45$ and the scaled transverse velocity $u_{t0}>0.4$. Results calculated with MSK1 together with the FU3FP4 (blue) and FU3FP5 (red) parametrizations of the in-medium NNCS are compared with the FOPI experimental data[@FOPI:2011aa]. Lines are fits to the calculated results assuming $v_2(y_0)=v_{20} + v_{22}\cdot y_0^2 $.](y0-v2-e400.eps){width="48.00000%"}
A good agreement between model calculations and the measured data of the elliptic flow are illustrated in Fig.\[y-v2\] and figures in Ref. [@Wang:2018hsw]. Fig. \[y-v2\] compares the elliptic flow of free protons and deuterons calculated with FU3FP4 and FU3FP5 to the FOPI experimental data. $v_2(y_0)=v_{20} + v_{22}\cdot y_0^2 $ is used to fit the calculated results, the same as in the FOPI analysis. Both the elliptic flow of free protons and deuterons in the whole inspected rapidity range can be reproduced by calculations with FU3FP4 and FU3FP5. As it has been discussed in Refs. [@Fevre:2015fza] and [@Wang:2018hsw], the sensitivity to the incompressibility $K_0$ is enhanced by using the observable $v_{2n}=|v_{20}| + |v_{22}|$. Because a smaller value of $K_0$ leads to smaller values of both $|v_{20}|$ and $|v_{22}|$, as can be found in figures in Refs. [@Fevre:2015fza; @Wang:2018hsw].
 The $v_{2n}$ of free protons produced from $^{197}$Au+$^{197}$Au collisions at $E_{\rm lab}=0.4$, $0.6$, $0.8$, and $1.0$ GeV$/$nucleon are shown as a function of the incompressibility $K_0$. The results obtained from the IQMD model are represented by full triangles [@Fevre:2015fza]. The shaded bands indicate the FOPI experimental data. Three full squares (open squares) denote respectively the results calculated using Skxs15, MSK1, and SKX together with the FU3FP4 (FU3FP5) parametrizations for the in-medium NNCS. The lines are the linear fits to the calculations. Reproduced from Ref. [@Wang:2018hsw].](v2n-p.eps){width="48.00000%"}
 The same as Fig.\[v2n-p\] but for the $v_{2n}$ of deuterons. Reproduced from Ref. [@Wang:2018hsw].](v2n-d.eps){width="48.00000%"}
The $v_{2n}$ of free protons and deuterons are shown as a function of the incompressibility $K_0$ in Figs.\[v2n-p\] and \[v2n-d\]. Calculations with the FU3FP4 and FU3FP5 sets are compared to the FOPI experimental data, as well as to the results calculated with the IQMD model, taken from Ref.[@Fevre:2015fza]. The $v_{2n}$ increases strongly with increasing $K_0$ in both model calculations, it implies that the $v_{2n}$ is very sensitive to the incompressibility $K_0$, though the slope is not exactly the same. Generally, the values of $v_{2n}$ calculated with the UrQMD model are smaller than that with the IQMD model, and the difference become smaller at higher beam energies. Consequently, the extracted $K_0$ with the IQMD model is smaller than that with the UrQMD model. As we have discussed in Ref. [@Wang:2018hsw], the difference may comes from the different collision term in the two models, i.e., the free NNCS is used in the IQMD model, while the UrQMD model incorporates the in-medium NNCS (density- and momentum- dependent). At higher energies, the difference between the two models become smaller, it is because that the in-medium and free cross sections at the higher relative momenta are almost the same, as shown in Fig.\[NNCS\]. Besides, different values of the width of the Gaussian wave packet and different treatments in the Pauli blocking in the two models may also contribute the observed difference in the extraction of $K_0$. Influences of these treatments and/or parameters on the $v_{2n}$ deserve further studies. On average, the central value of the incompressibility $K_0$ is obtained to be 240 (275) MeV for calculations with the FU3FP4 (FU3FP5) parametrization. With a stronger reduction of the in-medium nucleon-nucleon cross section, i.e., FU3FP5, a larger $K_0$ is extracted. It may also explain the reason why the $K_0$ obtained with the UrQMD model is larger than that with the IQMD model. $K_0 = 240 \pm 20$ MeV ($K_0 = 275 \pm 25$ MeV) for the FU3FP4 (FU3FP5) parametrization of the in-medium NNCS, which best describes the $v_{2n}$ of free protons, is extracted. In addition, within both models, it is found that $K_0$ extracted from the $v_{2n}$ of deuterons is smaller than that from $v_{2n}$ of free protons. Furthermore, the extracted $K_0$ from the $v_{2n}$ of deuterons is not sensitive to the beam energy, which is unlikely to that observed for the $v_{2n}$ of free protons. $K_0 = 190 \pm 10$ MeV ($K_0 = 225 \pm 20$ MeV) for the FU3FP4 (FU3FP5) parametrization is obtained from the $v_{2n}$ of deuterons. By combining the error intervals of the results obtained from the $v_{2n}$ of free protons and deuterons, an averaged $K_0 = 220 \pm 40$ MeV is obtained for the FU3FP4 parametrization.
Constraints on the density-dependent nuclear symmetry energy {#set3}
============================================================
 The nuclear symmetry energy for various Skyrme interactions are displayed as a function of density. Symmetry energies used in Ref. [@Russotto:2011hq] with $\gamma$=0.5 and 1.5, and favored in LQMD model [@Feng:2009am] and in IBUU04 model [@Xiao:2009zza] are also shown for comparison. Stars are constraints at 2$\rho_0$ obtained from astrophysical observations by Zhang and Li [@Zhang:2018bwq], Xie and Li [@Xie:2019sqb], and Zhou and Chen [@zhouchen]. The nuclear symmetry energy for Skz4, Skz2, SLy4, MSL0, SkO’, SV-sym34, Ska35s25, Gs, and SkI1 at lower densities are displayed in the inset. The shaded region exhibits the result obtained by Danielewicz *et al.* [@Danielewicz:2013upa]. Five different scattered symbols represent recent constraints obtained by Roca-Maza *et al.* [@RocaMaza:2012mh], Brown[@brown], Zhang *et al.* [@Zhang:2013wna], Wang *et al.*[@Wang:2015kof], and Fan *et al.* [@Fan:2014rha], respectively. ](Esym.eps){width="48.00000%"}
To probe the density-dependent nuclear symmetry energy with HICs, microscopic transport models which provides a bridge between experimental observables and the nuclear symmetry energy are necessary. Many observables have been predicted as sensitive probes for the nuclear symmetry energy, e.g., the yield ratio and the collective flow difference (ratio) between different isospin partners (e.g., proton and neutron, $^3$H$/$$^3$He, $\pi^{-}$/$\pi^{+}$, $K^0/K^+$, and $\Sigma^{-}$/$\Sigma^{+}$), as well as the balance energy of directed flow [@Li:2002qx; @DiToro:2010ku; @Tsang:2008fd; @Lopez:2007; @Xiao:2009zza; @Feng:2009am; @Russotto:2011hq; @Cozma:2011nr; @Xie:2013np; @Cozma:2013sja; @Li:2005zza; @LI:2005zi; @Kumar:2011td; @Gautam:2010da; @Lu:2016htm; @Russotto:2013fza; @Wang:2012sy; @Guo:2012aa; @Wang:2014aba; @Tsang:2016foy]. In spite of the progress made, a precise constraint on the density-dependent nuclear symmetry energy with HICs is still very difficult to achieve due to a) the difficulties in precision experimental measurements, and b) strong model- and observable-dependent results, see, e.g., Refs. [@Tsang:2012se; @Li:2012mw; @Chen:2012pk; @Wolter; @Li:2013ola] for review.
In this section, we review in detail two of our recent studies on the density-dependent nuclear symmetry energy by using the $^3$H$/$$^3$He yield ratio and the elliptic flow ratio between neutrons and hydrogen isotopes. As these two observables probe the nuclear symmetry energy at different density region, we incorporate two groups of Skyrme interactions into the UrQMD model. Group I includes 13 Skyrme interactions for which give quite similar values of the incompressibility $K_0$ but different values of $L$ [@Dutra:2012mb], the saturation properties of these interactions are shown in Table \[GroupI\]. In addition, the slope parameter $L$ at $\rho$=0.08, 0.055, and 0.03 $fm^{-3}$ are also shown in Table \[GroupI\]. Group II includes 21 Skyrme interactions. The saturation properties of these Skyrme interactions are shown in Table \[GroupII\]. Moreover, the SkA and SkI5 which give larger values of the incompressibility $K_0$ are also considered to examine the influence of $K_0$ on the elliptic flow ratio and difference. The density-dependent nuclear symmetry energy from various Skyrme interactions are displayed in Fig.\[esym\]. It can be seen that the selected Skyrme interactions cover the different forms of symmetry energies currently discussed by different theoretical groups. In addition, some recent constraints extracted from nuclear structure properties, e.g., binding energy, neutron skin thickness, and isovector giant quadrupole resonance [@RocaMaza:2012mh; @Zhang:2013wna; @brown; @Fan:2014rha; @Wang:2015kof], and from astrophysical observations [@Zhang:2018bwq; @Xie:2019sqb; @zhouchen] are also displayed for comparison (scatter markers and shaded band).
Result from $^3$H/$^3$He yield ratio
------------------------------------
---------- ------------ -- ------- -- ------- -- --------------- -- ------------- -- ------------- -- --
$\rho=\rho_0$ $\rho=0.01$ $\rho=0.08$
$\rho_{0}$ $K_0$ $S_0$ $L $ $L(\rho)$ $L(\rho)$
Skz4 0.16 230 32.0 5.8 16.5 34.5
Skz2 0.16 230 32.0 16.8 14.2 35.7
SV-mas08 0.16 233 30.0 40.2 10.6 32.8
SLy4 0.16 230 32.0 45.9 12.1 33.2
MSL0 0.16 230 30.0 60.0 8.7 31.6
SkO’ 0.16 222 32.0 68.9 8.9 33.2
SV-sym34 0.159 234 34.0 81.0 8.4 35.7
Rs 0.158 237 30.8 86.4 6.7 31.4
Gs 0.158 237 31.1 93.3 8.6 38.1
Ska35s25 0.158 241 37.0 98.9 6.3 31.7
SkI2 0.158 241 33.4 104.3 7.0 32.4
SkI5 0.156 256 36.6 129.3 6.9 34.5
SkI1 0.16 243 37.5 161.1 3.6 33.4
---------- ------------ -- ------- -- ------- -- --------------- -- ------------- -- ------------- -- --
: \[GroupI\] Group I: Saturation properties of nuclear matter as obtained with the selected 13 Skyrme interactions used to study the $^3$H$/$$^3$He yield ratio. All entries are in MeV, except for density in $fm^{-3}$.
\[skyrme\]
 The transverse component of the four-velocity distributions of $^3$H and $^3$He as well as the corresponding ratio at reaction times t=30 fm/c and 150 fm/c. The results calculated with two extreme cases (Skz4 and SkI1) are displayed.](ut0-dN-log.eps){width="48.00000%"}
 $^3$H/$^3$He ratio as a function of the slope of $E_{sym}(\rho)$ at densities of $\rho$=0.01 $fm^{-3}$ and 0.08$fm^{-3}$, as well as the saturation density $\rho_0$. The lines represent linear fits to calculations. Correspondingly, the Adj. $R^2$ values are also given. The shaded region indicates the FOPI data of $^3$H/$^3$He ratio [@FOPI:2010aa]. ](3h-3he-e400.eps){width="48.00000%"}
Calculations with both the quantum molecular dynamics (QMD) type and the Boltzmann-Uehling-Uhlenbeck (BUU) type transport models have been shown that the yield ratio of $^3$H and $^3$He emitted from HICs can be used to constrain the nuclear symmetry energy[@Chen:2003qj; @Chen:2004kj; @Li:2005kqa; @Zhang:2005sm], but some puzzling inconsistency still exists. For example, the yield of $^3$H calculated with a soft symmetry energy is larger than that with a stiff one based on calculations with two different QMD type models [@Li:2005kqa; @Zhang:2005sm], while the opposite trend is found in Ref. [@Chen:2003qj] with the isospin-dependent BUU (IBUU) model. Recently, a large amount of yield data for protons, $^2$H, $^3$H, $^3$He, and $^4$He produced in HICs at intermediate energies has been measured by the FOPI collaboration[@FOPI:2010aa; @FOPI:2011aa]. This data set offers new opportunities for studying the nuclear symmetry energy by using the $^3$H$/$$^3$He ratio over wide ranges of both beam energy and system size.
The transverse component of the four-velocity $u_{t0}$ distributions of $^3$H and $^3$He as well as the corresponding ratios at reaction times $t$=30 fm$/$c and 150 fm$/$c are shown in Fig.\[ut0\]. It can be seen that, at $t$=30 fm$/$c (the early stage of expansion phase), $^3$H and $^3$He with high $u_{t0}$ are more abundant than that with low $u_{t0}$, these clusters mainly reflect the behavior of symmetry energy at high densities. At early stage, $^3$H and $^3$He consist of protons and neutrons which emitted mainly from the high density region. A stiff (i.e, SkI1) symmetry energy will repel more neutrons and less protons than a soft one (i.e., Skz4). Thus more $^3$H (neutron-rich) can be formed, then higher values of the $^3$H/$^3$He and neutron/proton ratios are obtained. As the reaction proceeds, i.e., at $t$=150 fm$/$c, more and more $^3$H and $^3$He clusters with low-$u_{t0}$ are emitted from low density environment, and finally the ratio reflects the behavior of symmetry energy at sub-saturation densities. Very similar results can be found in calculations with other QMD-type models [@Li:2005kqa; @Kumar:2011td]. Furthermore, with increasing $u_{t0}$, the ratio calculated with Skz4 approach that of SkI1, their order may even be reversed if the residual symmetry potential is large enough. We have checked that the reversed order on the neutron$/$proton ratio is more obvious than that on the $^3$H$/$$^3$He ratio, as one excepted.
Figure \[3h3he-e400\] shows the $^3$H/$^3$He ratios calculated with the 13 selected Skyrme interactions as a function of the slope of $E_{sym}(\rho)$ at three different densities. The line in each bunch represents a linear fit to the calculations, the respective value of the adjusted coefficient of determination (Adj. $R^2$) is also shown. A very strong linearity between the $^3$H/$^3$He ratio and the slope of $E_{sym}(\rho)$ at $\rho$=0.01 $fm^{-3}$ can be observed, which indicates again a strong correlation between them at low densities. The results obtained with Skz4 and Skz2 fall outside the band, while it obtained with MSL0, SkO’, SV-sym34, and Ska35s25 are centered in the experimental band. The symmetry energy obtained with these four interactions also lie quite close to the constraints obtained from other methods, as shown in Fig.\[esym\]. Obviously, the large uncertainty of the experimental data prevents us from getting a tighter constraint on the density-dependent symmetry energy. However, the comparison to experimental $^3$H/$^3$He data as functions of beam energy and system size is possible, supplying a more systematic and thus more consistent information on the symmetry energy. As shown and discussed in our previous publication [@Wang:2014aba], the $^3$H/$^3$He data from different collision systems (i.e., $^{40}$Ca+$^{40}$Ca, $^{96}$Ru+$^{96}$Ru, $^{96}$Zr+$^{96}$Zr) and different beam energies (from 0.09 to 1.5 GeV$/$nucleon) also can be well reproduced by the calculations with MSL0, SkO’, SV-sym34, and Ska35s25. Although a tighter constraint on the density-dependent nuclear symmetry energy is still not obtained from the data of $^3$H/$^3$He yield ratio, partly due to the large uncertainties in the experimental data, a very satisfactory consistency among the presented comparisons is achieved. Furthermore, the results obtained from $^3$H/$^3$He yield ratio is also in agreement with our previous results obtained from the elliptic flow ratio [@Wang:2014rva; @Russotto:2011hq; @Cozma:2013sja], although one should keep in mind that the nuclear symmetry energies at different density regions are extracted from these two observables.
Result from the elliptic flow ratio between neutrons and hydrogen isotopes
--------------------------------------------------------------------------
Using the neutron-proton differential transverse and elliptic flows to probe the isospin-dependent EOS has been proposed almost twenty year ago[@Li:2000bj; @Li:2002qx; @Greco:2002sp], while the first constraint on the $E_{sym}(\rho)$ by using elliptic flows ratio between neutrons and hydrogen isotopes was achieved ten years ago[@Russotto:2011hq; @Trautmann:2009kq]. By comparing the simulations of the UrQMD model and the FOPI/LAND data for Au+Au collisions at 400 MeV$/$nucleon, a moderately soft symmetry energy with a slope of $L$=83$\pm$26 MeV was obtained [@Russotto:2011hq]. Afterwards, new FOPI/LAND experimental data of the elliptic flow ratio between neutrons and all charged particles became available [@Russotto:2016ucm], the UrQMD model also has been updated [@Wang:2013wca]. In this section, we review the results from the UrQMD model in which the Skyrme energy density functional is introduced to obtain parameters in the mean-field potential term.
$\rho_{0}$ $K_0$ $S_0$ $L $ $K_{\rm sym}$
---------- ------------ -- ------- -- ------- -- ------- -- --------------- -- --
Skz4 0.160 230 32.0 5.8 -240.9
BSk8 0.159 230 28.0 14.9 -220.9
Skz2 0.160 230 32.0 16.8 -259.7
BSk5 0.157 237 28.7 21.4 -240.3
SkT6 0.161 236 30.0 30.9 -211.5
SV-kap00 0.16 233 30.0 39.4 -161.8
SV-mas08 0.160 233 30.0 40.2 -172.4
SLy230a 0.16 230 32.0 44.3 -98.2
SLy5 0.16 230 32.0 48.2 -112.8
SV-mas07 0.16 234 30.0 52.2 -98.8
SV-sym32 0.159 234 32.0 57.1 -148.8
MSL0 0.160 230 30.0 60.0 -99.3
SkO’ 0.16 222 32.0 68.9 -78.8
Sefm081 0.161 237 30.8 79.4 -39.5
SV-sym34 0.159 234 34.0 81.0 -79.1
Rs 0.158 237 30.8 86.4 -9.2
Sefm074 0.16 240 33.4 88.7 -33.1
Ska35s25 0.158 241 37.0 98.9 -23.6
SkI1 0.160 243 37.5 161.1 234.7
SkA 0.155 263 32.9 74.6 -78.5
SkI5 0.156 256 36.6 129.3 159.6
: \[GroupII\] Group II: Saturation properties of nuclear matter as obtained with the 21 Skyrme parameterizations used to study the elliptic flow ratio between neutrons and hydrogen isotopes. All entries are in MeV, except for density in $fm^{-3}$.
\[skyrme\]
 Elliptic flow $v_2$ of neutrons and hydrogen isotopes for Au + Au with b $\le$ 7.5 fm and $E_{\rm lab}=0.4$ GeV$/$nucleon as a function of the transverse momentum $p_t$. The rapidity window $|y_0|<0.5$ is chosen the same as the experimental data. The results calculated with two extreme cases (i.e., Skz4 and SkI1) are compared to the FOPI/LAND data reported in Ref. [@Russotto:2011hq].](pt-v2.eps){width="48.00000%"}
A good agreement between the UrQMD calculations and the experimental data can be observed again in Fig.\[pt-v2\] where the $v_2$ of neutrons and hydrogen isotopes as a function of the transverse momentum $p_t$ is displayed. The $v_2$ of neutrons obtained with SkI1 is more negative than that obtained with Skz4, while the opposite trend is observed for hydrogen isotopes. This finding has been reported and discussed widely in Refs.[@Russotto:2011hq; @Cozma:2013sja; @Wang:2014rva; @Cozma:2017bre; @Trautmann:2009kq; @Trautmann:2010at]. It is due to the fact that the nuclear symmetry potential tends to attract protons and expel neutrons in a neutron-rich environment, and the repulsion for neutrons (attraction for protons) are much stronger for the stiff symmetry energy (i.e., SkI1) at densities above $\rho_0$ than that for the soft one (i.e., Skz4). The stronger repulsion interaction results in the more negative elliptic flow at the beam energy studied here.
 (a) Elliptic flow ratio between neutrons and hydrogen isotopes $v_{2}^{n}$/$v_{2}^{H}$ as a function of the transverse momentum $p_t$. Calculations with the indicated 9 Skyrme interactions are compared to the FOPI/LAND data (shaded area) reported in Ref. [@Russotto:2011hq]. (b) The total $\chi^2$ which demonstrates the quality of the fitting procedure is plotted as a function of the slope parameter $L$. The smooth curve is a quadratic fit to the total $\chi^2$, and the horizontal dashed line is used to determine the error of $L$ within a 2-$\sigma$ uncertainty. Reproduced from Ref.[@Wang:2014rva]. ](pt-v2-ratio.eps){width="48.00000%"}
 Elliptic flow ratio at mid-rapidity between neutrons and protons $v_{2}^{n}$/$v_{2}^{p}$ as a function of the slope of the nuclear symmetry energy $L$. Au+Au collisions at $E_{\rm lab}=0.4$ and 1.0 GeV$/$nucleon are displayed. Calculations with the 11 Skyrme interactions listed in Ref.[@wyj-plb2] are shown by solid symbols. The solid lines represent linear fits to the calculations, shaded bands are 95% confidence intervals around the fitted lines. ](v2np-e400-e1000.eps){width="48.00000%"}
Figure \[pt-v2-ratio\] (a) shows the comparison of the measured and the calculated ratios $v_{2}^{n}$/$v_{2}^{H}$ as a function of the transverse momentum $p_t$ ($p_t = u_{t0} \cdot 0.431$ GeV/c at $E_{\rm lab}=400$ MeV/nucleon for nucleons). SV-mas08&FU3FP2 denotes the ratio calculated with SV-mas08 interaction and the FU3FP2 parameterization of the in-medium NNCS, while others are calculated with the FU3FP4 parameterization on the in-medium NNCS. It can be seen that the $v_{2}^{n}$/$v_{2}^{H}$ ratio increases with increasing $L$, and the difference among calculations steadily grows when moving to the low transverse momentum region. The results calculated with SV-sym34 and SkA (give similar value of $L$) are almost overlapped even though the difference in $K_0$ is as large as almost 30 MeV. It illustrates that the elliptic flow ratio is not sensitive to the incompressibility $K_0$. We note here that the elliptic flows of both neutrons and hydrogen isotopes obtained with SkA are larger than that obtained with other interactions, however by taking the ratio, the impact of the incompressibility $K_0$ can be largely canceled out, similar results also can be found in Ref. [@Cozma:2017bre] by using Tübingen QMD model. Furthermore, the ratio obtained with SV-mas08&FU3FP2 lies close to that obtained with SV-mas08 in which the FU3FP4 parameterization on the in-medium NNCS is used, indicating that the influence of the in-medium NNCS on the elliptic flow ratio is quite small, similar result also has been observed in Ref. [@Russotto:2011hq]. Thus, one can conclude that the systematically increasing of $v_{2}^{n}$/$v_{2}^{H}$ as displayed in Fig.\[pt-v2-ratio\] (a) is mainly caused by the increase of the stiffness of the nuclear symmetry energy and not caused by other changes of the isoscalar components of the mean-field potential. Fig. \[pt-v2-ratio\] (b) shows the quality of the fitting to the FOPI/LAND data. The total $\chi^2$ as calculated with the 21 Skyrme interactions are displayed as a function of the slope parameter $L$. It can be seen that the variation of $\chi^2$ with $L$ can be well described with a quadratic fit. The slope parameter is extracted to be $L=89\pm45$ MeV within a 2-$\sigma$ uncertainty. In Ref.[@Wang:2014rva], the four observables $v_2^{n}-v_2^{p}$, $v_2^{n}-v_2^{H}$, $v_2^{n}/v_2^{p}$, and $v_2^{n}/v_2^{H}$ (the $p_t$-integrated results) are displayed as a function of the slope parameter $L$ of the 21 Skyrme interactions. Fairly good linearities between these observables and the slope parameter are observed. Together with the FOPI/LAND data, constraints on the slope parameter $L$ can be achieved. The intervals of $L$=61-137, 44-103, 62-132, and 54-106 MeV are obtained from $v_2^{n}-v_2^{p}$, $v_2^{n}-v_2^{H}$, $v_2^{n}/v_2^{p}$, and $v_2^{n}/v_2^{H}$, respectively. Although these results are largely overlapped with each other, the largest difference among their central values is about 25 MeV (from $v_2^{n}-v_2^{p}$ and $v_2^{n}-v_2^{H}$). The uncertainties of $L$ obtained from $v_2^{n}/v_2^{p}$ and $v_2^{n}/v_2^{H}$ are smaller than that from $v^n_2$-$v^H_2$ and $v^n_2$-$v^p_2$, one of the possible reason is that by taking the ratio the impact from uncertainties in the determination of the reaction plane and in the isoscalar components of the nuclear potential can be largely cancelled out. To our knowledge, the uncertainty of the extracted $L$ using the elliptic flow ratio (difference) is large for two main reasons. (a) The large uncertainty in neutron flow measurements. (b) Contribution of $K_{sym}$ to the elliptic flow ratio has not been disentangled. As $K_{sym}$ becomes more and more important for studying the high-density behavior of nuclear symmetry energy, both $L$ and $K_{sym}$ are expected to affect the elliptic flow ratio. Thus, correlation analyses and more systematic and accurate experimental data of the elliptic flow difference (ratio) are required before achieving a tighter constraint on the density-dependent nuclear symmetry energy. Fig.\[l-v2np\] shows the elliptic flow ratio $v_{2}^{n}$/$v_{2}^{p}$ as a function of the slope parameter $L$ at beam energies of 0.4 and 1.0 GeV$/$nucleon. With increasing beam energy, the sensitivity of $v_{2}^{n}$/$v_{2}^{p}$ to $L$ is reduced, because of the weakened mean field potential effects at higher energies. Moreover, the inelastic collisions (e.g., $n$+$n$ $\rightarrow$ $p$ + $\Delta^-$ $\rightarrow$ $p$ + $n$ +$\pi^-$) may also further reduce the effects of the symmetry potential on the flow of nucleons, as neutrons (protons) can be converted to protons (neutrons).
 Constraints on the slope of $E_{sym}(\rho)$ by using the elliptic flow ratio (difference) between neutrons and protons (hydrogen isotopes, all charged particles). The results obtained by Russotto *et al.* in Refs.[@Russotto:2011hq; @Russotto:2016ucm] and by Cozma *et al.* in Ref. [@Cozma:2013sja] are compared to the results obtained in present work.](L-com.eps){width="48.00000%"}
Constraints on the slope of $E_{sym}(\rho)$ by using the elliptic flow ratio (difference) between neutrons and protons (hydrogen isotopes, all charged particles) are summarized in Fig.\[L-com\]. It is interesting to observe that constraints presented with the updated UrQMD model, and obtained by Cozma *et al.* with the Tübingen QMD model [@Cozma:2013sja], as well as obtained by Russotto *et al.* with previous version of the UrQMD model [@Russotto:2011hq; @Russotto:2016ucm], are well overlapped within the range of 60-85 MeV. Moreover, in a recent study[@Cozma:2017bre], by considering much more theoretical uncertainties in the Tübingen QMD model, e.g., the compressibility $K_0$, the in-medium nucleon-nucleon cross section, and the nucleon effective mass splitting, the slope of $E_{sym}(\rho)$ is extracted to be $L$=84$\pm$30 (exp) $\pm$ 19 (theor) MeV from the elliptic flow ratio between neutrons and protons (hydrogen isotopes). Again, this result also overlaps with $L$=60-85 MeV. Furthermore, it is noticed that the extracted central values of $L$ from $v_2^{n}-v_2^{H}$ ($v_2^{n}/v_2^{H}$) are smaller than that from $v_2^{n}-v_2^{p}$ ($v_2^{n}/v_2^{p}$). The result obtained with from the elliptic flow ratio between neutrons and all charged particles ($v_2^{n}/v_2^{Ch.}$) is also smaller than that with $v_2^{n}-v_2^{H}$ and $v_2^{n}/v_2^{H}$. It is known from the analysis in Ref.[@Russotto:2016ucm] that, the sensitivity densities obtained from $v_2^{n}/v_2^{H}$ and $v_2^{n}/v_2^{Ch.}$ are smaller than that with $v_2^{n}/v_2^{p}$. The difference in the extracted $L$ may also stem from the fact that different range of densities are probed by these different observables. Very recently, a new observable related to the rapidity at which $v_2$ of protons changes sign from negative to positive, is found to be sensitive to the density-dependent nuclear symmetry energy, by comparing the FOPI data and the UrQMD calculations, the slope parameter $L=43 \pm 20$ MeV is extracted [@wyj-plb2]. This result is about 30 MeV smaller than the result summarized in Fig.\[L-com\], the main reason is that this new observable which involves $v_2$ in a broader rapidity range probes the nuclear symmetry energy at a lower density region. Detailed discussions can be found in Ref. [@wyj-plb2].
Summary and outlook {#set5}
===================
We review our recently studies on the nuclear equation-of-state and the in-medium NNCS by using the ultrarelativistic quantum molecular dynamics (UrQMD) model. With incorporating the Skyrme potential energy density functional to obtain parameters in the mean-field potential part of the UrQMD model, three Skyrme interactions which give quite similar values of the nuclear symmetry energy but different values of the incompressibilities $K_0$ are adopted. It is found that the nuclear incompressibility $K_0$ is quite sensitive to the $v_{2n}$. By comparing the FOPI data of the $v_{2n}$ of free protons and deuterons with the UrQMD model calculations, an averaged $K_0 = 220 \pm 40$ MeV is extracted with the FU3FP4 parametrization on the in-medium NNCS. However, remaining systematic uncertainties, partly related to the choice of in-medium NNCS, are of the same magnitude ($\pm 40$ MeV). Overall, the rapidity dependent elliptic flow supports a soft nuclear equation-of-state.
With considering different forms of the density- and momentum-dependent in-medium NNCS in the UrQMD model, their influence on the collective flow and nuclear stopping power is studied as well. It is found that both the collective flow and the nuclear stopping power of free protons can be reproduced with the calculations using the FU3FP5 parametrization on the in-medium NNCS, while the results of light clusters are found to be reproduced well with the FU3FP4 parametrization. The FU3FP4 and FU3FP5 parametrization sets offer the greatest possible degree of the momentum-dependent in-medium NNCS.
We further review the extraction of the density-dependent nuclear symmetry energy by using the experimental data of $^3$H/$^3$He yield ratio and the elliptic flow ratio between neutrons and hydrogen isotopes. It is found that $^3$H/$^3$He yield ratio is sensitive to the nuclear symmetry energy at sub-normal densities, while the elliptic flow ratio between neutrons and hydrogen isotopes is sensitive to the high-density behavior of the nuclear symmetry energy. By comparing the UrQMD calculations with 21 Skyrme interactions to the transverse-momentum dependent elliptic flow ratio $v_{2}^{n}/v_{2}^{H}$, the slope parameter of the density-dependent symmetry energy is extracted to be $L = 89 \pm 45$ MeV within a 2-$\sigma$ confidence limit. The large uncertainty is partly due to the large error bars in the experimental data and to the fact that the effect of $K_{sym}$ on the elliptic flow ratio has not been disentangled. In the near future, we will concentrate on the investigation of $K_{sym}$ with heavy-ion collisions. The $p_t$-integrated elliptic flow ratio and difference $v_2^{n}-v_2^{p}$, $v_2^{n}-v_2^{H}$, $v_2^{n}/v_2^{p}$, and $v_2^{n}/v_2^{H}$, also can be used to obtain constraints on the slope parameter $L$. Overall, $L$=60-85 MeV is found to be overlapped with the constraints obtained with the UrQMD model and the Tübingen QMD model by using the data of the elliptic flow ratio (difference) between neutrons and protons (hydrogen isotopes, all charged particles).
Finally, we would like to point out that to achieve a better understanding about the nuclear equation-of-state and the in-medium nucleon-nucleon cross section by using heavy-ion collisions, a detailed and systematical investigation on how the sensitive observables varies with beam energy and collision system is quite necessary. On one hand, the current and future rare isotope beam facilities (e.g., the CSR and the HIAF in China, the FRIB in the United States, the RIBF in Japan, the SPIRAL2 in France, the FAIR in Germany) around the world, will provide more and more experimental data in the next decades, offering new opportunities for theoretical investigation. On the other hand, uncertainties in transport models, e.g., model-dependent results observed in the comparison of different transport models [@Xu:2016lue; @Zhang:2017esm; @Ono:2019ndq], need to be understood and solved. Endeavors of both experimentalists and theorists are mandatory to achieve a tight constraint on the nuclear equation-of-state.
To probe the high density behavior of the nuclear symmetry energy with heavy-ion collisions, the $\pi^{-}$/$\pi^{+}$ yield ratio and the elliptic flow ratio between neutrons and protons are two of the most popular observables so far, as the corresponding experimental data are available. As the contribution of the curvature parameter $K_{sym}$ becomes more and more important when studying the high-density behavior of the nuclear symmetry energy, constraint on the $K_{sym}$ is quite necessary[@Guo:2018flw; @Guo:2019onu]. The effects of the pion and $\Delta$ potentials[@Li:2015hfa; @Cozma:2016qej; @Zhang:2017mps; @Liu:2018xvd; @Guo:2014fba; @Guo:2015tra; @Yong:2017cdl], the in-medium threshold effects on $\Delta$ resonance production and decay[@Song:2015hua; @Li:2016xix; @Zhang:2017mps; @Li:2017pis; @Cui:2018bkw; @Cui:2018qrg; @Cui2019], and other issues[@Guo:2014usa; @Gao:2018nnp; @Cheng:2016pso] on the $\pi^{-}$/$\pi^{+}$ yield ratio need to be understood before a more reliable constraint on the nuclear symmetry energy can be achieved. For the elliptic flow ratio between neutrons and protons, one of a great challenge for experimental techniques is to measure the flow of neutrons with high precision, while from a theoretical point of view, the influence of the neutron-proton effective mass splitting ought to be isolated in advance[@Li:2015pma; @Li:2018lpy; @Xie:2013bsa; @Xie:2015xma; @Feng:2011pu; @Feng:2018emx; @feng2012; @Gior; @Tong]. In addition, the effects of nucleon-nucleon short-range correlations as well as the associated nucleon momentum distributions in heavy-ion collisions also need to be studied[@Li:2018lpy; @Hen:2014yfa; @Li:2014vua; @Liu:2014tqa; @Yong:2015gma; @Yong:2017zgg; @Yang:2018xtl; @Yong:2018eeq; @Yang:2019jwo].
Besides using heavy-ion collisions, astrophysical observations such as the mass-radius relation and tidal deformability of neutron stars and gravitational waves also can be used to constrain the density-dependent nuclear symmetry energy [@Li:2019xxz; @Zhang:2018bwq; @Xie:2019sqb; @Tsang:2019mlz; @Baiotti:2019sew]. Together with constraints on the nuclear equation of state from observables in both nuclear physics (with nuclear structure properties and heavy-ion collisions) and astrophysics (e.g., neutron stars and their mergers), a more precise picture of the nuclear equation of state in a wider density range will be achieved.
We are very grateful to Zhuxia Li, Hongfei Zhang, Chenchen Guo, Arnaud Le Fèvre, Yvonne Leifels, Wolfgang Trautmann for collaborating with us on some of the topics discussed in this review. The authors acknowledge support by the computing server C3S2 in Huzhou University. The work is supported in part by the National Natural Science Foundation of China (Nos. 11875125, 11947410, 11505057), and the Zhejiang Provincial Natural Science Foundation of China under Grants No. LY18A050002 and No. LY19A050001, and the “Ten Thousand Talent Program" of Zhejiang province.
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[^1]: Corresponding author: liqf@zjhu.edu.cn
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author:
- 'Luis Núñez-Betancourt and Felipe Pérez'
bibliography:
- 'References.bib'
title: '$F$-jumping and $F$-Jacobian ideals for hypersurfaces'
---
Acknowledgments {#acknowledgments .unnumbered}
===============
|
---
abstract: 'The straylight contamination due to the Galactic emission (GSC, Galaxy Straylight Contamination) entering at large angles from the antenna centre direction may be one of the most critical sources of systematic effects in observations of the cosmic microwave background (CMB) anisotropies by future satellite missions like [Planck]{} and MAP. For the Low Frequency Instrument (LFI), this effect is expected to be particularly crucial at the lowest frequency channels. We describe here different methods to evaluate the impact of this effect and compare it with other systematics of instrumental and astrophysical origin. The results are presented in terms of simulated data streams and maps, Fourier series decomposition and angular power spectrum. The contributions within few degrees from the beam centre dominate the GSC near the Galaxy plane. The antenna sidelobes at intermediate and large angles from the beam centre dominate the GSC at medium and high Galactic latitudes. We find a GSC peak at $\sim 15\mu$K and a GSC angular power spectrum above that of the white noise for multipoles $\ell \lsim 10$ albeit smaller than that of CMB anisotropies by a factor larger than $\sim 10$. At large multipoles, the GSC affects the determination of CMB angular power spectrum significantly less than other kinds of instrumental systematics, like main beam distortions and $1/f$ noise. Although the GSC is largest at low Galactic latitudes, the contamination produced by far pattern features at medium and high Galactic latitudes, peaking at $\sim 4\mu$K, has to be carefully investigated, because the combination of low amplitude of Galaxy emission in those regions with the extremely good nominal [Planck]{} sensitivity naturally makes high Galactic latitude areas the targets for unprecedentedly precise estimation of cosmological CMB anisotropy. This paper is based on [Planck]{} LFI activities.'
author:
- '[C. Burigana]{}'
- '[D. Maino]{}'
- '[K.M. Górski]{}'
- '[N. Mandolesi]{}'
- '[M. Bersanelli]{}'
- '[F. Villa]{}'
- '[L. Valenziano]{}'
- '[B.D. Wandelt]{}'
- '[M. Maltoni]{}'
- '[E. Hivon]{}'
date: 'Submitted on A&A, 28.9.2000'
title: '[Planck]{} LFI: Comparison Between Galaxy Straylight Contamination and other Systematic Effects'
---
\#1 \#2 \#3 \#4
=2em =1 by 0em \#1, [*\#2*]{}, [**\#3**]{}, \#4.
\#1
=2em =1 by 0em \#1.
\#1
Introduction
============
After the great success of -DMR (Smoot 1992, Bennet 1996a, Górski 1996) which probed the gravitational instability scenario for structure formation through the detection of CMB anisotropies at few degree scales, and the recent balloon-borne experiments at high sensitivity and resolution on limited sky regions (De Bernardis 2000, Hanany 2000), supporting a universe model with $\Omega_{tot} \sim 1$ (Lange 2000, Balbi 2000, Jaffe 2000), ultimately, future progresses of the CMB anisotropy cosmology will be based on two space missions, MAP (Microwave Anisotropy Probe) (see Bennet 1996b) by NASA and [Planck]{} by ESA, planned to be launched respectively in the years 2001 and 2007.
In particular, the Low Frequency Instrument (LFI, Mandolesi 1998) and the High Frequency Instrument (HFI, Puget 1998) on-board [Planck]{} will cover together a wide frequency range (30$\div$900 GHz) which should significantly improve the accuracy of the subtraction of foreground contamination from the primordial CMB anisotropy, providing at the same time a gold mine of cosmological as well as astrophysical information (e.g. De Zotti 1999a and references therein). To fully reach these scientific goals, great attention has to be devoted to properly reduce and/or subtract all the possible systematic effects. Detailed simulation codes have been developed and are continuously implemented to analyse the impact of several classes of instrumental effects related to the behaviour of the optics, instruments and environment for a wide set of possible scanning strategies (e.g. Burigana 1998, Delabrouille 1998, Maino 1999, Mandolesi 2000a). Ultimately this effort leads to the optimization of the mission design and the production of realistic data streams and simulated maps for data analysis tools as well as a realistic definition of [Planck]{} scientific performance.
In particular, the behaviour of the [Planck]{} antenna patterns both at intermediate and large angles from the directions of beam centres have to be carefully considered. The requirement on the rejection of unwanted radiation coming from directions far from the optical axis (straylight) is stringent for [Planck]{} and does not pertain only the telescope itself, but the entire optical system, including also solar panels, shielding, thermal stability and focal assembly components. The primary sources of error for the LFI are those due to imperfect off-axis rejection by the optical system of radiation from the Sun, Earth, Moon, planets, Galaxy and the spacecraft itself (de Maagt 1998). The variations of the spurious straylight signal during the mission introduce contaminations in the anisotropy measurements.
The antenna response features at large angular scales from the beam centre (far sidelobes) are determined largely by diffraction and scattering from the edges of the mirrors and from nearby supporting structures. Therefore they can be reduced by decreasing the illumination at the edge of the primary, i.e. increasing the edge taper, defined as the ratio of the power per unit area incident on the centre of the mirror to that incident on the edge. Of course, the higher is edge taper, the lower is the sidelobe level and the straylight contamination. On the other hand, increasing the edge taper has a negative impact on the angular resolution for the fixed size of the primary mirror (e.g. Mandolesi 2000a). A trade off between angular resolution and straylight contamination has to be found.
The main aim of this work is to evaluate the impact of the Galactic emission as a source of straylight for [Planck]{} LFI. We will then compare it with the effects generated by other kinds of systematics, the main beam distortion introduced by optical aberrations and the $1/f$ noise related to gain fluctuations in LFI radiometers, and with the astrophysical contamination from the Galaxy and the extragalactic sources in the main beam. At LFI frequencies, the Galaxy straylight contamination (GSC) is expected to be particularly crucial at the lowest frequencies, due to the increase of synchrotron absolute emission and anisotropies with the wavelength. For simplicity, we limit our analysis to the case of the 30 GHz channel, but the methods presented here can be extended to higher frequencies.
In Sect. \[simul\] we briefly describe the basic recipes of our simulation code, discussing the geometrical aspects relevant for the full sky convolution, the format of the optical simulation output performed by the ESA (de Maagt et al. 1998) that are adopted in the present work, the conversion from data streams to maps and the computation of the Fourier modes and of the angular power spectra, two different estimators of the GSC impact. We estimated the expected GSC on the basis of the antenna integrated response from angular regions at different angles from the beam centre, and on the level of Galaxy emission. In Sect. \[acc\_test\] we focus on the integration accuracy of our computations and test the consistency of the code by assuming simplified input maps and antenna patterns. The main results concerning the evaluation of the GSC are presented in Sect. \[results\]. In Sect. \[comparison\] they are compared with the effects introduced by other kinds of instrumental effects and several sources of astrophysical contamination. Finally, we draw our main conclusions in Sect. \[conclusions\].
Simulation of [Planck]{} observations {#simul}
=====================================
The selected orbit for [Planck]{} is a Lissajous orbit around the Lagrangian point L2 of the Sun-Earth system (e.g. Mandolesi 1998). The spacecraft spins at 1 r.p.m. and the field of view of the two instruments is $5^\circ$ around the telescope optical axis at a given angle $\alpha$ from the spin-axis direction, given by a unit vector, $\vec s$, choosen outward the Sun direction. In this work we consider values of $\alpha \sim 80^{\circ}-90^{\circ}$ [^1]. The spin axis will be kept parallel to the Sun–spacecraft direction and repointed by $\simeq 2.5'$ every $\simeq 1$ hour. In addition, a precession of the spin-axis with a period, $P$, of $\simeq 6$ months at a given angle $\beta \sim 10^{\circ}$ about an axis, $\vec f$, parallel to the Sun–spacecraft direction (and outward the Sun) and shifted of $\simeq 2.5'$ every $\simeq 1$ hour, may be included in the effective scanning strategy. These kinds of scanning strategies do not modify the angle between the spin axis and the Sun–spacecraft direction, avoiding possible thermal fluctuations induced by modulations of the Sun illumination, and allow to achieve nearly full or full sky coverage. Hence [Planck]{} will trace large circles in the sky. The detailed distribution in the sky of the number of observations per pixel depends on the adopted scanning strategy, the telescope design and the arrangement of the feed array on the telescope focal surface. The scanning strategy and spacecraft geometry have to be carefully addressed in order to minimize systematic effects before and after the data analysis and for ensuring that the sky coverage be as complete and spatially smooth as possible.
The code we have implemented for simulating [Planck]{} observations for a wide set of scanning strategies is described in detail in Burigana (1997, 1998) and in Maino (1999). Here we consider simple scanning strategies, namely with the spin axis kept always in the ecliptic plane. Under this assumption, the geometrical input parameters relevant for the scanning strategy are the angle $\alpha$ between the spin axis and the telescope line of sight and the beam location in the telescope field of view.
In this study we neglect the small effects introduced on the GSC evaluation by the [Planck]{} orbit by simply assuming [Planck]{} located in L2 and consider spin-axis shifts of 2every two days and 180 samplings per scan circle. These simplifications allow to speed up the simulation without significantly affecting our understanding of the main effects introduced by the GSC, because of the decreasing of the Galaxy fluctuation level at small scales and because the effects of pattern features we want to study here occur at $\sim$ degree or larger scales. In fact, a recent study of synchrotron emission (Baccigalupi et al. 2000a) based on recent high resolution surveys at low at medium latitudes indicates a steeper power law of the total intensity angular power spectrum where diffuse emission dominates.
Coordinate transformations and input antenna pattern {#coord}
----------------------------------------------------
For the simple scanning strategies considered here, the vector $\vec s$, which gives the spin axis direction, and a unit vector $\vec p$, parallel to the telescope line of sight $z$, are simple functions of the time and of three scanning strategy parameters: the spinning frequency $f_s$, the angle $\alpha$ and the orientation of the [Planck]{} telescope line of sight at the beginning of each scan circle that we arbitrarily fix close as much as possible to the positive $z$ ecliptic axis. In additon, we set the rotation of $\vec p$ clockwise with respect to the positive direction of $\vec s$.
On the plane tangent to the celestial sphere in the central direction of the field of view, i.e. on the field of view plane of the [Planck]{} telescope, we choose two coordinates $x$ and $y$, with unit vector $\vec u$ and $\vec v$ respectively, according to the convention that the unit vector $\vec u$ points always toward $\vec s$ and that $x,y,z$ is a standard cartesian frame, referred here as “telescope frame” [^2].
In general, the beam centre will be identified by its unit vector $\vec b$ in the frame $x,y,z$ or equivalently by the coordinates, $x_0,y_0$, of its projection on the plane $x,y$ or, more usually, by its corresponding standard polar coordinates, the colatitude $\theta_B$ and the longitude $\phi_B$. The HFI feedhorns will be located in a circular area at the centre of the focal plane, and LFI feedhorns in a ring around HFI. Therefore the corresponding positions of LFI beams on the sky field of view are significantly off-axis. For a telescope with $\simeq$ 1.5 m aperture the typical 100 GHz LFI beam is located at $\simeq 2.8^{\circ}$ from the optical axis, whereas the 30 GHz beams are at about $\simeq 5^{\circ}$ from it.
The shape of the main beam computed by ESA (de Maagt 1998) is provided in a regular equispaced grid on $x,y$ about the beam centre. We can then perform the convolution of the main beam (within a choosen angle from the beam centre) with the sky signal directly in this frame.
According to the standard output of the “GRASP8” code for optical simulations as performed de Maagt et al. (1998), we describe the antenna pattern response, $J$, at large angles from the beam centre by using two standard polar coordinates $\theta_{bf}$ (between $0^{\circ}$ and $180^{\circ}$) and $\phi_{bf}$ (between $0^{\circ}$ and $360^{\circ}$) referred to the “beam frame”. This corresponds to the standard cartesian “beam frame” $x_{bf},y_{bf},z_{bf}$ which is obtained by the “telescope frame” $x,y,z$ when the unit vector of the axis $z$ is rotated by an angle $\theta_B$ on the plane defined by the unit vector of the axis $z$ and the unit vector $\vec b$ up to reach $\vec b$ [^3]. We use here the polar coordinates $\theta_{bf},\phi_{bf}$ for the convolution of the antenna pattern with the sky signal at large angles from the beam centre. For the antenna pattern at “intermediate” (namely up to few degrees from the beam centre) and “far” (namely for the entire solid angle) angular distances from the beam centre different equispaced grids in $\theta_{bf}$ are available: more refined for the former, because the response variations are stronger close to the beam centre, and less refined for the latter, where the relevant response variations occur on degree or larger angular scales.
The orientation of these frames as the satellite moves is implemented in the code. For each integration time, we determine the orientations in the sky of the telescope frame and of the beam frame and compute the pattern response in each considered sky direction, thus performing a direct convolution with the sky signal.
Optical results {#optic}
---------------
The telescope design of the Carrier Configuration for [Planck]{} – the current baseline in which [Planck]{} and FIRST are lauched together and separate in orbit – is based on the Phase A study. The configuration is a off-axis Gregorian Dragone-Mizuguchi telescope with the main reflector oversized at 1.5 meter projected aperture for reducing the spillover at the primary mirror. The subreflector axis is tilted at $14^\circ$ with respect to the main reflector axis. The off-axis design of [Planck]{} telescope (see Fig. \[sketch\_tel\]) introduces particular features in the full sky antenna response. In Fig.\[sketch\_tel\] a sketch on the symmetry plane of the telescope is reported. For simplicity, a feedhorn is located at the centre of the focal plane. Its pattern on the sky is along the direction of the optical axis of the telescope (see bottom panel of Fig. \[sketch\_tel\]). The unwanted ray directions are shown in the upper panel of Fig.\[sketch\_tel\]. The rays labelled with 1 are coming from the sky directly into the feed. The rays labelled with 2 are those scattered into the feed by the subreflector only. These identified regions are the spillover past the subreflector for rays 1 and the spillover past the main reflector for rays 2. Considering the [Planck]{} Carrier Configuration (de Maagt 1998), the rays 1b and 2b are blocked by the spacecraft shields, which redirect the rays in an angular region close to the main beam. In Fig. \[full\_patt\_fig\] we show a map of the considered full pattern; see the connection beetween the regions marked in Fig. \[sketch\_tel\] and Fig. \[full\_patt\_fig\]. In Fig. \[full\_patt\_cuts\] we show also several cuts at constant azimuths $\phi_{bf}$ from 0to 360 (from the bottom to the top).
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We have conservatively considered the worst case (de Maagt 1998) for what concerns the straylight effect: we use the antenna pattern computed at 30 GHz, the channel with the highest spillover and with the highest Galaxy signal. We also included the shields for the Carrier Configuration.
The main feature is the spillover (2a) at about $90^\circ$ from the main beam (see also Fig. \[full\_patt\_fig\], where the main beam is located at North pole) which shows a response of $\approx -60$ dB with respect to the maximum extending for few tens of degrees in $\theta_{bf}$ and in $\phi_{bf}$ (around $\phi_{bf}=0^\circ$, i.e. always quite close to the direction of the axis $x$ in the “telescope frame”). Another relevant feature is the subreflector spillover (1a), with similar response level and an angular extension close to the main beam ($\theta_{bf}\sim
30^\circ$) as we can see also in Fig.\[sketch\_tel\]. Other features are located on the northern semisphere also, due to the shields which block rays coming from the southern part of the sphere (see de Maagt 1998 for a more detailed discussion of their connection with the optical configuration).
The pattern has been calculated by using the Physical Theory of Diffraction (PTD). Since pattern responses at levels smaller than about $-60$ dB are hard to measure, this is the most accurate method for predicting the side lobe response on the antenna. The validity of the simulations at very small levels of the sidelobes will be tested by measuring the antenna response of a fully representative copy of the [Planck]{} telescope both in compact range and in outdoor far field test range facilities.
Simple estimates {#estimates}
----------------
Taking the level of Galactic emission and the antenna integrated response from angular regions at different angles from the beam centre, we can provide first order estimates of the expected GSC.
The region with $\theta_{bf}$ between $\simeq 1.2\deg$ and $\simeq 2\deg$ contains about 0.5% of the integrated response; the region between $\simeq 1.2\deg$ and $\simeq 5\deg$ contains about 0.6% of the integrated response and all the rest of the far pattern ($\theta_{bf} \gsim 5\deg$) contains about 1% of the integrated response. Of course, the remaining main integrated response falls in the “main” beam (up to $\simeq 1.2\deg$) (see also Sect. \[acc\] for a discussion on the choice of these characteristic angles). In addition, in the main spillover (2a) enters $\approx 0.1 \div 0.2 \%$ of the integrated response.
The sky signal at 30 GHz is known with pixel size of about $2.5\deg$ by -DMR. For the present study at 30 GHz the relevant astrophysical source is the Galaxy emission. We have implemented “small” angular extrapolations (see e.g. Burigana 2000a for further details) for generating Galaxy maps with resolution of about $1.2\deg$, corresponding to Quad-Cube resolution 7.
For simple estimates, we note that in the adopted 30 GHz Galaxy map there are $\sim 13 \sq$ with a signal (in terms of antenna temperature $T_a$) larger the 2 mK, $\sim 73 \sq$ with $T_a > 1.5$ mK and $\sim 230 \sq$ with $T_a > 1$ mK, while the minimum signal is $\sim 0.05$ mK and about the 50% of the sky shows a signal $\sim 0.1$ mK.
By combining these numbers with the percentages of integrated responses falling within the above different angles from the beam centre, we expect to find a contamination peaking at about 10$~\mu$K from the pattern regions between $\simeq 1.2\deg$ and $\simeq 5\deg$ and at few $~\mu$K from the pattern regions outside $\simeq 5\deg$. In particular, in the main spillover (2a) we expect a signal peaking at $\sim 2~\mu$K when it looks at high signal Galactic regions. Similar contributions are expected from the pattern features at few tens of degrees from the beam centre. Of course, smaller contaminations ($\sim 0.5 \mu$K) are expected when the relevant pattern features look at regions with low signal Galactic.
Numerical calculations, like those presented in the next sections, are required for more accurate estimates.
From data streams to sky maps {#streamtomap}
-----------------------------
The input map is converted from its original Quad-Cube pixelisation to equal area, hierarchic HEALPix pixelisation scheme (Górski et al. 1998), adopted in the present work (see also section \[acc\] for details about the nominal resolution of this map). The final output of the simulation code relevant here are 2 matrices with a number of rows equal to the considered number of spin-axis positions $n_s$ for one year of mission ($n_s=180$ here) and a number of columns equal to the number of considered samplings along one scan circle ($n_p=180$ here). In the first matrix, ${\bf N}$, we store the pixel numbers corresponding to the main beam central directions for the considered 180$\times$180 integrations; they are stored in HEALPix pixelisation scheme at $n_{side}=32$ (the number of pixels $n_{pix}$ in a full sky map is related to $n_{side}$ by $n_{pix}=12 n_{side}^2$, Górski 1998). In the second matrix, ${\bf G}$, we store the antenna temperatures “observed” by the considered portion of the antenna pattern for the above pointing positions. We neglect here the receiver noise and all the other systematics. These data streams are the first output of our simulations; they give immediately the impact of GSC and are useful to understand how this effect changes during the mission.
From these data streams it is quite simple to obtain observed simulated maps, that can be visualized for example in mollweide projection: we make use of ${\bf N}$ and ${\bf G}$ to simply coadd the temperatures of those pixels observed several times during the mission. In this way we attribute to each pixel the average of the signals observed when the antenna pattern, due to the scanning strategy, is differently oriented in the sky and Galactic regions with very different signal intensities enter in the intermediate/far sidelobes; of course, by coadding different samples of the same location into pixels the systematic error per pixel is smaller than the systematic error in the most contaminated sample. This is because for different samples of the same location on the sky the sidelobes are pointing towards different regions of the sky, some brighter some fainter.
Power spectra {#streamtomap}
-------------
We can analyse both data streams and maps in terms of power at different scale-lengths or multipoles (or modes).
In order to analyse separately each scan circle of simulated data streams we follow the approach suggested by Puget & Delabrouille (1999) and decompose the time series from the scan circle in Fourier series:
$$g(\xi)=\sum_{m=0}^{m=n_p/2-1} a_m \sqrt{2} {\rm cos}(m\xi+\xi_{0,m}) \, ,$$
where $g(\xi)$ is the signal at the position identified by an angle $\xi$ (between 0 and $2 \pi$) on the considered scan circle. The $2\times n_p$ coefficients $a_m$ and $\xi_{0,m}$ can be easily computed from the time series by solving a linear system. The amplitude, $a_m$, of each mode $m$, analogous to the multipole $\ell$ of a usual spherical harmonic expansion, has to be compared with that of the wanted signal and of other sources of noise. This approach is particularly interesting for the data analysis during the mission, as [Planck]{} will continuously scan different circles in the sky.
For analysing the maps of coadded signals we use the standard approach of computing their angular power spectrum. We produce maps in the HEALPix pixelisation scheme (Górski 1998) which takes advantage from the isolatitude of the pixels for a quick generation of a map from the coefficients $a_{\ell m}$ of the spherical harmonic expansion and vice versa (Muciaccia 1997). \[We will show the angular power spectra in terms of $\delta T_\ell (\nu) = \sqrt{\ell (2 \ell +1)C_\ell(\nu)/4\pi}$\]. This is a very significant test for evaluating the GSC impact on [Planck]{} science, as the estimation of angular power spectrum of CMB fluctuations from the sky maps is one of the main objectives of [Planck]{} mission. It offers also the possibility of directly comparing the GSC with astrophysical contaminations and other sources of instrumental noise. Of course, it produces a “global” estimate of GSC effect, useful in the analysis of the whole sky maps.
Accuracy and tests {#acc_test}
==================
The antenna pattern is theoretically known from optical simulation codes at the desired accuracy and resolution compatible with the available computing time. The adopted grids have a resolution much better than those of currently available sky maps at [Planck]{} frequencies, although extrapolations both in frequency and in angular resolution of existing maps allow to produce more refined simulated maps (e.g. Burigana 1998, 2000a). On the other hand, the knowledge of Galactic emission at high resolution is not particularly relevant here and only small angular extrapolations up to a resolution of about $1\deg$ has been implemented. Given the currently available input map resolution and the adopted simplified scanning strategies, we are able to derive the power of GSC only up to $\ell \sim m \lsim 80$, a mode/multipole range satisfactory for the study of the smooth features in the intermediate and far pattern response. A very good agreement is found with the high resolution computations based on faster Fourier expansion methods (Wandelt & Górski 2000) which will be practically necessary for whole sky convolutions at angular resolutions higher than those adopted in this work. On the other hand, at larger multipoles the GSC effects are dominated by other systematics, as discussed in Sect. 5.
Sky, pattern grids and numerical accuracy {#acc}
-----------------------------------------
We are interested in producing signal data streams as they would be observed separately by different angular regions of the antenna pattern in order to understand the effect of the different pattern features as they project in the sky during the [Planck]{} observations. We have considered here three regions: $0\deg \le \theta_{bf} \le \theta_{bf,1}$ (main pattern); $\theta_{bf,1} \le \theta_{bf} \le \theta_{bf,2}$ (intermediate pattern); $\theta_{bf,2} \le \theta_{bf} \le 180\deg$ (far pattern). We have choosen $\theta_{bf,1} = 1.2\deg$ and $\theta_{bf,2} = 5\deg$. Of course, the choice of $\theta_{bf,1}$ and $\theta_{bf,2}$ has to be appropriate to the considered antenna pattern: for a given telescope design it depends mainly on the considered frequency and only weakly on the exact feed location on the focal surface. For the 30 GHz channel, the main beam can be accurately measured in flight through planets (Mandolesi 1998, Burigana 2000b) up to response levels of $\approx -30$ dB with respect to the peak response; $\theta_{bf} \gsim 1.2\deg$ corresponds to antenna responses lower than $-40$ dB, where the beam response probably becames highly difficult to measure in flight; $\theta_{bf,2} \sim 5\deg$ roughly divides pattern regions where significant response variations occur on angular scales less than $1\deg$ from those where they occur on $\sim$ degree or much larger scales.
The observed antenna temperature is given by
$$T_{a,obs}= {\int J(\theta_{bf},\phi_{bf}) T_a(\theta_{bf},\phi_{bf}) d\Omega
\over \int J(\theta_{bf},\phi_{bf}) d\Omega } \, ,$$
where $J$ and $T_a$ are the antenna response and the sky antenna temperature in the direction given by $\theta_{bf},\phi_{bf}$.
The convolution of antenna response with the sky and the integration of the antenna pattern is simply computed by adding the contributions from all the pixels within the considered solid angle, at resolutions corresponding to $n_{side}=1024, 512, 64$ respectively for the main, intermediate and far pattern in order to take accurately into account the pattern response variations. The main pattern is given in equispaced cartesian coordinates with $\Delta x = \Delta y = 0.0005$. The intermediate pattern and the far pattern are provided in equispaced “GRASP8” polar grids with $\Delta \phi'_{bf} = 10\deg$ and $\Delta \theta'_{bf} = 0.1\deg$ and $0.5\deg$, respectively. When we extract all the pixels in the sky that contribute to the convolution within the considered solid angle, the exact central position of each pixel typically does not coincide with a grid point where the pattern is known. Simple standard bilinear interpolation (e.g. Press 1992) on the pattern grid has been implemented: this is fast, robust and accurate enough for the present purposes.
An estimate of the error introduced by the above discretizations and interpolation/computation methods can be provided by comparing the convolutions obtained with different values of $n_{side}$, for example by increasing $n_{side}$ to 1024 for the intermediate pattern convolution. The numerical error is negligible ($\lsim 1 \mu$K, $\lsim 0.5 \mu$K or $\lsim 0.2 \mu$K for the convolutions with the main, intermediate and far pattern, respectively).
Tests with schematic skies and patterns {#test}
---------------------------------------
Checking the consistency of the part of the code that computes the signal entering in the main beam and in the intermediate pattern is quite direct: we simply expect that the maps extracted from the corresponding data streams are respectively very similar or roughly proportional (according to the fractional signal entering at $1.2\deg \le \theta_{bf} \le 5\deg$) to Galactic emission pattern, except for the beam smoothing.
Testing the validity of the computation of the signal entering the far pattern is not immediate, because it does not reflect in a simple way the Galactic emission pattern. We have verified the consistency of our simulation code by exploring simple cases for which we can easily predict the large scale symmetries of the maps derived from the data streams observed by the far pattern. We have assumed a simple antenna pattern, centred on the optical axis and perfectly symmetric in $\phi_{bf}$, given by the sum of two gaussian shapes, one for the main beam and one for the main spillover located at $90\deg$ from the main beam centre, plus a constant low response level. We have performed tests with the following different very simple input skies: $i)$ a spot at North Galactic pole: it produces a map with a well defined slab on the Galactic plane; $ii)$ a slab on the ecliptic plane: the corresponding map shows a signal maximum at the ecliptic poles and decreasing toward the ecliptic plane; $iii)$ a slab on the Galactic plane: the corresponding map shows a signal maximum at the Galactic poles and decreasing toward the Galactic plane, where it exhibits small longitudinal modulations related to those of the solid angle subtended by the main spillover (this is due to the scanning symmetry with respect to ecliptic coordinates and not with respect to Galactic ones). We have also verified that the angular power spectrum of the these maps presents a main peak at the multipole $\ell = 2$ and secondary peaks at its harmonic frequencies, as expected from the $90\deg$ symmetry of the adopted far pattern.
Simulations results {#results}
===================
We have considered two options for the [Planck]{} Carrier configuration. The first is exactly that considered by de Maagt (1998) with $\alpha = 80\deg$ and including shields. In the second case we have used the same optical results but with $\alpha = 90\deg$: the corresponding results are then only indicative, being not perfectly consistent because the spacecraft design would be slightly different for this configuration; on the other hand, this case is instructive because it allows to start addressing the question of the dependence of GSC on a basic parameter of the scanning strategy. We have considered the antenna pattern at 30 GHz with the beam centre located at $\theta_B=5.62\deg$ and $\phi_B=126.03\deg$.
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Analysis of the scan circle data streams
----------------------------------------
Figures \[scans3\_80\] and \[scans3\_90\] show the absolute signals entering the main, intermediate and far pattern and their ratios for the data streams of three representative scan circles for $\alpha = 80\deg$ and $90\deg$, respectively.
Note that the signal entering the intermediate pattern is roughly proportional to that in the main beam: two main relative maxima typically appear, related to the two crossings of the Galactic plane. The signal from the far sidelobes exhibits a clearly different and shifted angular behaviour, although two main relative maxima are again typically present. These are mainly due to the contributions from the pattern features (1a) in the cases of the left and central panels, and to the main spillover (2a) in the case of the right panels, as they cross the Galactic plane. Note in fact the displacement between the maximum signal from the main and far pattern, of few tens of degrees (several tens of degrees) for the left and central panel (right panel).
We have applied the Fourier series decomposition (see Fig. \[fourier\]) described by eq. (1) to the sum of intermediate and far pattern (i.e. for $\theta_{bf} \ge 1.2\deg$) data streams from the scan circles shown in Figures \[scans3\_80\] and \[scans3\_90\].
The same decomposition has been applied to white noise data streams, computed according to the LFI sensitivity at 30 GHz (e.g. Maino 1999) averaged over a number of scan circles that spans an ecliptic longitude length arc equal to the FWHM $ = 33'$, i.e. essentially the sensitivity corresponding to half year mission. The white noise power is above that of the signal entering at $\theta_{bf} \ge 1.2\deg$, for practically all the modes $m \gsim 3$, becaming $\approx 10$ times larger at $m \approx 10$; this is essentially due to the strong decreasing of Galaxy fluctuations at small angular scales.
No significant differences are found by varying $\alpha$ from $80^{\circ}$ to $90^{\circ}$; for this reason and for sake of simplicity, we will show in what follows only the results for $\alpha = 80^{\circ}$, the angle for which the optical simulations have been appropriately performed.
All simulated data streams for a 1 yr mission are shown in Fig. \[scansall\_80\] for the case $\alpha = 80\deg$ (similar patterns are obtained in the case $\alpha = 90\deg$). In the right panel, note the vertical high signal line at $\lambda \simeq 270^{\circ}$, corresponding to the main spillover (2a), and the two high signal features, close to this line, at about $(\lambda , \beta ) \sim (180^{\circ},0^{\circ})$ and $(\lambda , \beta ) \sim (360^{\circ},-160^{\circ})$, corresponding to the pattern features at few tens of degrees from the beam centre. Note also how the azimuthally asymmetric far pattern reflects in the large difference between the two halfs (along $\lambda$ axis) of the right panel of Fig. \[scansall\_80\] corresponding to the first and second six months of observation. This redundancy can be exploited for an efficient subtraction of GSC in the data analysis.
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Straylight contamination maps and angular power spectrum
--------------------------------------------------------
By coadding the data streams as described in Sect. \[streamtomap\] we can obtain the corresponding maps. This is shown in Fig. 8 for the case $\alpha = 80\deg$, by coadding the simulated data from the whole year.
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As apparent in Fig. 8, the map from intermediate pattern is roughly proportional to that derived from the main pattern; the relative intensities are roughly scaled by the fraction of integrated response entering the two portions of the antenna pattern.
On the contrary, the sky “observed” by the far pattern is very different. The signal is higher close to the Galactic plane, because of the features in the antenna pattern within $10^{\circ} - 20^{\circ}$ from the main beam, and at about $90^{\circ}$ from the Galactic plane, because of the signal entering the main spillover (2a).
We have computed the angular power spectra of these GSC maps (see Fig. \[comp\_gsin\_noise\]) and compared them with the theoretical angular power spectrum of the white noise for a single 30 GHz receiver and for four receivers and with a typical CMB anisotropy angular power spectrum (a tilted - $n_p=0.9$ - power spectrum with standard CDM cosmological parameters and approximately COBE normalized) and with that of Galaxy fluctuations as seen by the main pattern. The most important contamination in terms of angular power spectrum derives from the signal entering in the intermediate pattern when all the sky is considered; on the contrary, if we consider only the regions at $\vert b \vert \ge 30\deg$, more crucial for [Planck]{} main science, the largest contributions to the GSC power spectrum derive from the far sidelobes. In general, the GSC power spectrum is larger than the white noise one at low multipoles ($\ell \lsim 5$) but their ratio becames less than $\simeq 1/10$ at $\ell \simeq 50$ and decreases further at larger multipoles, due to their different dependence on $\ell$ [^4].
1truecm
Comparison with other sources of noise {#comparison}
======================================
Many other sources of contamination, both instrumental and astrophysical in origin, may affect [Planck]{} observations. We approach here a first comparison among the effects introduced by some of these systematics.
Other kinds of instrumental noise {#other_instr}
---------------------------------
The impact of main beam distortions introduced by optical aberrations on [Planck]{} measurements has been carefully studied in several works (e.g. Burigana 1998, 2000a, Mandolesi 1997, 2000a). Burigana (1998) discussed the impact of the main beam distortions for the representative case of an elliptical main beam shape. In general, the absolute rms additional noise, in the range of few $\mu$K, increases with the beam ellipticity. The combined effect of main beam distortions and of Galaxy emission fluctuations increases the additional error at $\sim 30$ GHz by a factor $\simeq 3$ with respect to the case of the essentially pure CMB fluctuation sky at high Galactic latitudes, whereas it produces only a small additional effect at higher LFI frequencies. In addition, the combined effect of main beam distortions and extragalactic source fluctuations is found to be very small at all LFI frequencies (Burigana et al. 2000a) compared to the noise induced by beam distortions in the case of a pure CMB sky. Then, we focus here further on the impact of the main beam distortions on the determination of angular power spectrum of CMB fluctuations by considering the idealized case of the above pure CMB fluctuation sky. The kind and the magnitude of optical distortions depend on the details of the optical design; for aplanatic configurations currently under study (e.g. Villa 1998 and Mandolesi 2000b) the typical main beam shape is roughly elliptical owing to the strong reduction of the coma distortion. We computed a full year simplified simulation both for a pure symmetric gaussian beam with FWHM $= \sigma \sqrt{8{\rm ln}2} =
30'$ and for an elliptical gaussian beam with axial ratio $r = 1.3$ and with the same effective resolution ($\sqrt{\sigma_x
\sigma_y} = \sigma$) of the symmetric beam ($r=1$). We shift the spin axis at steps of 5$'$ and consider a step of 5$'$ between two samplings on the same scan circle. We computed the difference between the maps obtained from the elliptical and the symmetric beam by coadding the corresponding data streams and calculate the angular power spectrum of this difference map in order to understand which range of multipoles is mainly affected. As expected (see Fig. 10), this effect is particularly relevant at quite large multipoles, close to the CMB peak, where GSC power significantly decreases: the magnitude, of course, is related to the value of $r$.
From optical simulations we know that $r$ typically increases with the distance from the beam centre. From the present simulations we infer that a value of $r$ quite smaller than 1.3, say less than $\simeq 1.1$ up to $-3$ dB from the centre and less than $\simeq 1.2$ up to $-20$ dB from the centre, is good enough for avoiding significant contaminations in the data, in agreement with the indications inferred on the basis of the approximations of Burigana (1998) for the rms noise added by a main beam elliptical distortion.
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The $1/f$ noise due to amplifier noise temperature fluctuations induced by gain fluctuations in [Planck]{} LFI receiver and its dependence on the relevant instrumental parameters has been studied by Seiffert (1997). It introduces additional noise in [Planck]{} observations which shows as stripes in final maps (Janssen 1996) owing to the particular [Planck]{} scanning strategy. We have recently carried out detailed studies (Maino 1999 and reference therein) on its effect on [Planck]{} LFI measurements and on the efficiency of destriping algorithms based on the use of the crossings between different scan circles (Bersanelli 1996, Delabrouille 1998) for a wide set of [Planck]{} scanning strategies. We extend here their simulations by relaxing the hypothesis of symmetric beam to study the impact of main beam distortion into the destriping algorithm. In Fig. \[comp\_instr\_noise\] we show the angular power spectrum of the receiver noise before and after applying the destriping algorithm when we include also the above elliptical distortion, for a simple scanning strategy with $\alpha =90\deg$ and a beam location at $\theta_B = 2.8\deg$, $\phi_B=45\deg$, a “mean” choice regarding to the destriping efficiency (Maino et al. 1999). We find that the destriping efficiency is not significantly affected by the additional uncertainty introduced by the “systematic” differences among the observed temperatures resulting from different orientations of the main beam at the crossing points of different scan circles.
1truecm
As evident by comparing Figs. $9\div11$, there is a crucial difference between the angular power spectra of GSC, main beam distortion induced noise and $1/f$ noise. The GSC affects particularly the determination of CMB angular power spectrum at low multipoles, whereas main beam distortions are critical at large multipoles. The $1/f$ noise affect both high and low multipoles, but destriping algorithms are particularly efficient in removing high multipole features in the power spectrum. It is clear that all these effects have to be reduced both via hardware and software. The $1/f$ noise can be reduced independently of the other two, its magnitude being related essentially to the instrument stability and to the scanning strategy (Maino et al. 1999). On the contrary, a compromise has to be reached between GSC and main beam distortion noise, being both related mainly to the optical design. As discussed in Mandolesi (2000a), for a given telescope design, their relative weight is controlled by the edgetaper. The optical design has to be optimized to find a trade off for reducing the combined impact of these two effects.
Astrophysical contamination {#astro}
---------------------------
The impact of foregrounds on the primary cosmological goal of [Planck]{} mission has been extensively studied in literature for what concern both Galactic and extragalactic contaminations, of discrete and diffuse origin; [Planck]{} itself will be a good opportunity for studying cluster physics, many classes of extragalactic and Galactic sources and the diffuse emission from the Galaxy (e.g. De Zotti 1999a and references therein). Many approaches have been studied to separate the different components of the microwave sky and for deriving their angular power spectra (e.g. Tegmark & Esftathiou 1996, Hobson 1998, Baccigalupi 2000b, and references therein).
We consider here the foreground impact on CMB science and their comparison with the effect of instrumental systematics. In Figure \[comp\_astro\_noise\] we report several estimates of the angular power spectra of different astrophysical components.
1truecm
The Galaxy angular power spectrum is known to decrease with multipole order $\ell$: we show here the power spectrum derived from the map observed by the adopted main pattern by cutting (lower thin dotted-dashed line, “CG’) or not (upper thin dotted-dashed line, “WG”) the region at $\vert b \vert \le 30\deg$ and the power spectra proposed by Tegmark & Esftathiou (1996) for free-free (thick dotted line, “FF”), synchrotron (upper thick dotted-dashed line, “S”) and dust (upper thick dashed line, “D”) emission at relevant Galactic latitudes. We show also for comparison the power spectra for synchrotron (lower thick dotted-dashed line, “s”) and dust (lower thick dashed line, “d”) as derived by Prunet (1998) and Bouchet (1998) for a sky patch at medium latitudes. Of course, Galaxy contamination strongly depends on the considered region.
In terms of $\delta T_{\ell}$, Poisson fluctuations from extragalactic unresolved discrete sources increase approximately proportionally to the multipole $\ell$. We show here, separately for radiosource (thin solid lines, “R”) and the far infrared galaxy (thin dashed lines, “IR”), the Poisson fluctuation power spectra predicted by Toffolatti (1998) as recently revised by Toffolatti (1999), De Zotti (1999b) and references therein, on the basis of current source counts and assuming evolution models and spectra in agreement with current data, when sources above 1 Jy (upper curves) or 100 mJy (lower curves) are detected and subtracted. Of course radiosources dominate at low frequencies. We have taken into account here a gaussian (FWHM=$33'$) beam smoothing in all cosmological and astrophysical angular power spectra and consequently neglected it in the receiver noise angular power spectrum.
At low multipoles, Galaxy contamination is larger then instrumental effects, dominated by the GSC (and possible residual $1/f$ noise); on the other hand, if the Galaxy emission and anisotropy can be modelled at few percent accuracy and at high Galactic latitudes, instrumental effects can became comparable with the residual Galaxy contamination.
If not appropriately taken into account in the data analysis, main beam distortions may introduce at high multipoles an additional contamination comparable with that introduced from radiosource fluctuations, after their subtraction at few hundreds mJy level.
Conclusions
===========
We have studied the impact of GSC on [Planck]{} observations at 30 GHz, by considering different and complementary evaluation approaches: absolute and relative quantification of the impact on scan circle data streams, Fourier decomposition of scan circle signals, computation of maps of GSC and evaluation of their angular power spectra. These different methods allow us to focus on different aspects of GSC.
No relevant differences are found by varying the angle $\alpha$ between the spin axis and the telescope line of sight from $80^{\circ}$ to $90^{\circ}$.
Our simulations show that the GSC peaks at values of about $15 \mu$K (a value comparable with the sensitivity per pixel), mainly due to the signal entering at few degrees from the beam centre. Such values are found in the regions quite close to the Galactic plane, where in any case the “direct” (i.e. observed by the main beam) contamination from the Galaxy prevents an accurate knowledge of CMB fluctuations, as it is next to impossible to remove the Galactic signals to accuracy better than $\sim$ %. These large contamination values, although critical for CMB anisotropy measurements near the Galactic plane, are not crucial for the determination of Galaxy emission, which is several order of magnitude larger.
By considering all the pixels in the sky, the typical values of GSC are less than the 50% of the white noise sensitivity per pixel.
The most crucial contamination derives from the signal entering in the far pattern, in spite of its peak values, of about 4$\mu$K, smaller than those obtained for the intermediate pattern regions. In fact, although this effect does not seem to be very large in amplitude (indeed, being nominally subdominant to power spectrum of expected receiver noise) it does dominate the GSC at medium and high Galactic latitudes, which are critical regions for the extraction of the best quality results on CMB anisotropy. This could be also critical for the [Planck]{} polarization measurements which will take advantage from the two patches close to the ecliptic poles where the sensitivity will be several times better than the average, according to the scanning strategy and the feed array arrangement.
As expected on the basis of the behaviour of Galaxy emission angular power spectrum, the GSC affects the determination of the CMB angular power spectrum mainly in the low multipole region and much less at large multipoles, particularly when compared with the other instrumental effects considered here, the main beam distortion and the $1/f$ noise. The additional noise introduced by the main beam distortion can be in principle subtracted in the data analysis, provided that the beam pattern is accurately reconstructed.
Of course, a substantial improvement in the data analysis is necessary to jointly treat all the systematics, of instrumental and astrophysical origin. From the telescope design point of view, the best optimization of the edge taper requires a trade off between the main beam distortion and the GSC effects.
We acknowledge stimulating and helpful discussion with J. Delabrouille and J.L. Puget; we gratefully thank P. de Maagt and J. Tauber for having promptly provided us with their optical simulation results.
[**References**]{}
[^1]: Most recently it was recommended to choose $\alpha = 85^{\circ}$.
[^2]: We note that the “telescope frame” defined above may be or not equivalent to the “primary mirror frame” depending on the absence or presence of a tilt angle between the primary mirror axis and the central direction of the sky field of view. For example, the current optical configuration forseen for [Planck]{} exibits a tilt angle of $\simeq - 3.75^{\circ}$.
[^3]: Properly, the standard of “GRASP8” code for optical simulations is to provide the far pattern in terms of a “colatitude” angle $\theta'_{bf}$ (between $-180^{\circ}$ and $180^{\circ}$) from the “polar” axis parallel $z_{bf}$ and an “azimuthal” angle $\phi'_{bf}$ (between $0^{\circ}$ and $180^{\circ}$) related to $\theta_{bf}$ and $\phi_{bf}$ by: $\phi'_{bf}=\phi_{bf}$ if $\phi_{bf} \le 180^{\circ}$ and $\phi'_{bf}=\phi_{bf}-180^{\circ}$ if $\phi_{bf} > 180^{\circ}$; $\theta'_{bf}=\theta_{bf}$ if $\phi_{bf} \le 180^{\circ}$ and $\theta'_{bf}=-\theta_{bf}$ if $\phi_{bf} > 180^{\circ}$.
[^4]: The increase of the power of GSC from far sidelobes at $\ell \gsim 50$, evident in the plot, is not an intrinsic effect of the GSC but is generated by the “stripe” corresponding to the sky region observed a single time (see also Fig. 8) in the current 1 yr simulation with $\alpha = 80^{\circ}$ and off-axis beam.
|
---
abstract: 'Using redshift space distortion data, we perform model-independent reconstructions of the growth history of matter inhomogeneity in the expanding Universe using two methods: crossing statistics and Gaussian processes. We then reconstruct the corresponding history of the Universe background expansion and fit it to type Ia supernovae data, putting constraints on $({\ensuremath{\Omega_\text{m,0}}},{\ensuremath{\sigma_{8,0}}})$. The results obtained are consistent with the concordance flat-[$\Lambda$CDM]{} model and General Relativity as the gravity theory given the current quality of the inhomogeneity growth data.'
author:
- |
Benjamin L’Huillier,$^1$ Arman Shafieloo,$^{1,2}$ David Polarski,$^3$ Alexei A. Starobinsky$^{4,5}$\
$^{1}$Korea Astronomy and Space Science Institute, Yuseong-gu, Daedeok-daero 776, Daejeon 34055, Korea\
$^{2}$University of Science and Technology, Yuseong-gu 217 Gajeong-ro, Daejeon 34113, Korea\
$^{3}$Laboratoire Charles Coulomb, Université de Montpellier & CNRS UMR 5221,F-34095 Montpellier, France\
$^{4}$L. D. Landau Institute for Theoretical Physics RAS, Moscow 119334, Russia\
$^{5}$National Research University Higher School of Economics, Moscow 101000, Russia
bibliography:
- 'biblio.bib'
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: 'Defying the laws of Gravity I: Model-independent reconstruction of the Universe expansion from growth data'
---
\[firstpage\]
cosmological parameters – large-scale structure of Universe – cosmology: observations – cosmology: theory – gravitation
Introduction
============
The discovery at the end of last century of the late-time accelerated expansion rate of the Universe raised the question of its physical cause. There are two great avenues towards solving this problem: either that it is due to an unknown new physical component dubbed (physical) dark energy, or its origin lies in a modification of the laws of gravity [e.g., @Sahni:1999gb; @2012IJMPD..2130002Y; @2012PhR...513....1C]. However, they represent particular cases of a more general situation like in scalar-tensor gravity, when both a new physical field is introduced for dark energy description and gravity is modified, too (see e.g. @Boisseau:2000pr [@Copeland:2006wr; @Sahni:2006pa]).
In the concordance model the role of dark energy is played by a cosmological constant $\Lambda$ while gravity is described by Einstein’s theory of General Relativity (GR). While GR has been remarkably successful to explain observations in the Solar system (see e.g. @2006JCAP...09..016G), its successful extrapolation to much larger cosmic scales remains unclear. Modified gravity models with modifications of gravity on cosmic scales are not excluded and could well be the solution to the recent acceleration of the Universe. The nature of dark energy and therefore also the correct model of gravity are burning issues of cosmology and theoretical physics in general.
The large-scale structures of the Universe are an ideal laboratory to test gravity, and to distinguish between physical dark energy and modified gravity (which may be also called geometrical dark energy as in @Sahni:2006pa). In particular, redshift-space distortion (RSD) due to galaxy peculiar velocities can be used to estimate the growth factor $f$, which is a key to understanding gravity. For a flat-FLRW Universe with dark energy as a perfect fluid with equation of state $w(z)$, the expansion history $h(z) = H(z)/H_0$ is described by $$\begin{aligned}
\label{eq:expans}
h^2(z) & = {\ensuremath{\Omega_\text{m,0}}}(1+z)^3 + (1-{\ensuremath{\Omega_\text{m,0}}})\exp\left(3\int_0^z \frac{1+w(u)}{1+u}{\ensuremath{\mathrm{d}}}u\right).\end{aligned}$$ In GR, the evolution of the matter overdensity $\delta({\ensuremath{\mathbfit{x}}},z) = (\rho({\ensuremath{\mathbfit{x}}},z)-\bar\rho(z))/\bar\rho(z)$ are governed in the Newtonian approximation by $$\begin{aligned}
\label{eq:growth}
\ddot\delta + 2H\dot\delta & = \frac 3 2 H^2 {\ensuremath{\Omega_\text{m}}}\delta,\end{aligned}$$ where dot stands for a derivative with respect to cosmic time $t$, and ${\ensuremath{\Omega_\text{m}}}(z) = {\ensuremath{\Omega_\text{m,0}}}(1+z)^3/h^2(z)$ is the matter density normalized by the critical density. From eqs. and , it is clear that changing the expansion will also affect the growth evolution. In fact, @1998JETPL..68..757S showed that the Universe expansion history $H(z)$ can be also reconstructed from $\delta(z)$ unambiguously in this case (the situation becomes more complicated in scalar-tensor gravity @Boisseau:2000pr).
In this paper, we aim first to reconstruct the growth history from data, and then to use it in order to deduce the expansion history (assuming GR) and to compare it with the supernovae data. We should note here that there are two different reconstructions involved here: a statistical reconstruction of the growth factor $f(z)$ from observational data on one hand, and on the other hand, the theoretical reconstruction of the background expansion $H(z)$ from $f(z)$. A similar approach was recently applied in @2018arXiv180800377Y. The theoretical reconstruction is described in § \[sec:method\], and the statistical reconstructions together with the results are presented in § \[sec:real\] Our conclusions are drawn in § \[sec:ccl\]. We validate the method on mock data in § \[sec:mock\],
Method {#sec:method}
======
Theoretical Framework
---------------------
From eq. , using that for any function $x(t)$, $$\begin{aligned}
\dot x &= H {\ensuremath{\frac{{\ensuremath{\mathrm{d}}}{x}}{{\ensuremath{\mathrm{d}}}{\ln a}}}}=-H{\ensuremath{\frac{{\ensuremath{\mathrm{d}}}{x}}{{\ensuremath{\mathrm{d}}}{\ln (1+z)}}}},
\intertext{one obtains}
{\ensuremath{\frac{{\ensuremath{\mathrm{d}}}{^2\delta}}{{\ensuremath{\mathrm{d}}}{\ln a^2}}}} & + \left(2+\frac{1}{h(z)}{\ensuremath{\frac{{\ensuremath{\mathrm{d}}}{h}}{{\ensuremath{\mathrm{d}}}{\ln a}}}}\right) {\ensuremath{\frac{{\ensuremath{\mathrm{d}}}{\delta}}{{\ensuremath{\mathrm{d}}}{\ln a}}}} = \frac 3 2 {\ensuremath{\Omega_\text{m}}}(z) \delta.
\intertext{It is convenient to introduce the growth factor}
f & = {\ensuremath{\frac{{\ensuremath{\mathrm{d}}}{\ln \delta}}{{\ensuremath{\mathrm{d}}}{\ln a}}}} = {\ensuremath{\Omega_\text{m}}}^\gamma(a), \label{eq:gamma}\end{aligned}$$ where the last equality defines the growth index $\gamma$. In general, $f=f({\ensuremath{\mathbfit{k}}},z)$ and therefore $\gamma = \gamma({\ensuremath{\mathbfit{k}}},z)$ [e.g., @2009JCAP...02..034G]. However, in GR and for dust-like matter, $\gamma$ is ${\ensuremath{\mathbfit{k}}}$-independent and has weak dependence on $z$, so that $\gamma(z)\simeq 0.55$, with a slight dependence on the equation of state of dark energy $w$ and the matter density parameter [$\Omega_\text{m,0}$]{}. Note, however, that $\gamma$ may not be exactly $z$-independent for quintessence (a scalar field with a potential minimally coupled to gravity) models of dark energy as was shown in @2016JCAP...12..037P.
Observational data on RSD provide us with the product [$f\sigma_8$]{}, where $$\begin{aligned}
{\ensuremath{f\sigma_8}}& = {\ensuremath{\frac{{\ensuremath{\mathrm{d}}}{\sigma_8}}{{\ensuremath{\mathrm{d}}}{\ln a}}}}\\
\intertext{where}
\sigma^2_R(z) & = \frac 1 {2\pi^2}\int_0^\infty P(k,z) W^2_R(k) k^2{\ensuremath{\mathrm{d}}}k \propto\delta^2(z)\end{aligned}$$ is the rms of the density fluctuations smoothed over a radius $R$, usually taken to be [$8 \, h^{-1} \mathrm{Mpc}$]{}.
In @2018PhRvD..98h3526S [see also @2018MNRAS.476.3263L], we used Pantheon and a compilation of growth data to put model-independent constraints on $({\ensuremath{\Omega_\text{m,0}}},{\ensuremath{\sigma_{8,0}}},\gamma)$, where subscript 0 stands for the current value. However, in that paper, we treated $\gamma$ as a constant, effectively performing a consistency test of GR. In fact, one can solve the problem without assuming $\gamma = $ constant. Assuming $\delta$ and $\delta'$ are known, the expansion history $h$ can be uniquely determined via [@1998JETPL..68..757S] $$\begin{aligned}
\label{eq:delta2h2}
h^2(z) & = \left(\frac{1+z}{\delta'(z)}\right)^2 \left(
\delta'^2_0-3{\ensuremath{\Omega_\text{m,0}}}\int_0^z\delta(u){\ensuremath{\left\lvert {\delta'(u)}\right\rvert}}\frac{{\ensuremath{\mathrm{d}}}u}{1+u}\right),
$$ where $'$ denotes a derivative with respect to $z$ (not $\ln a$).
Thus, one can obtain from RSD measurements $$\begin{aligned}
\frac{\delta'(z)}{\delta_0} &= -(1+z)\frac {{\ensuremath{f\sigma_8}}(z)}{{\ensuremath{\sigma_{8,0}}}}, \mbox{and}\\
\delta(z) & = \delta_0\left(1-\frac 1{{\ensuremath{\sigma_{8,0}}}}\int_0^z {\ensuremath{f\sigma_8}}(u)\frac{{\ensuremath{\mathrm{d}}}u}{1+u}\right). \end{aligned}$$
Therefore, for a given reconstruction ${\ensuremath{\widehat{{\ensuremath{f\sigma_8}}}}}(z)$ together with a given $({\ensuremath{\Omega_\text{m,0}}},{\ensuremath{\sigma_{8,0}}})$, the expansion history ${\ensuremath{\hat{h}}}(z)$ is uniquely determined by $$\begin{gathered}
{\ensuremath{\hat{h}}}^2(z|{\ensuremath{\widehat{{\ensuremath{f\sigma_8}}}}},{\ensuremath{\Omega_\text{m,0}}},{\ensuremath{\sigma_{8,0}}}) = \frac{(1+z)^4}{{\ensuremath{\widehat{{\ensuremath{f\sigma_8}}}}}^2(z)}
\left({{\ensuremath{\widehat{{\ensuremath{f\sigma_8}}}}}^2_0}\phantom{\int}\right.\\
\left.
-3{{\ensuremath{\Omega_\text{m,0}}}}
\int_0^z\left({\ensuremath{\sigma_{8,0}}}-
\int_0^u {\ensuremath{\widehat{{\ensuremath{f\sigma_8}}}}}(v)\frac{{\ensuremath{\mathrm{d}}}v}{1+v}\right)
{{\ensuremath{\widehat{{\ensuremath{f\sigma_8}}}}}(u)}
\frac{{\ensuremath{\mathrm{d}}}u}{(1+u)^2}
\right).
\label{eq:h2}\end{gathered}$$
Therefore, reconstructing ${\ensuremath{\widehat{{\ensuremath{f\sigma_8}}}}}(z)$, and exploring the $({\ensuremath{\Omega_\text{m,0}}},{\ensuremath{\sigma_{8,0}}})$ parameter space, we can reconstruct $h(z)$ and the luminosity distance. In the following, we assume a flat-FLRW universe, which is consistent with current model-dependent [@2013PhLB..723....1F; @2018arXiv180706209P] and model-independent[@2014JCAP...03..035R; @2015PhRvL.115j1301R; @2017JCAP...01..015L; @2018JCAP...03..041D; @2018PhRvD..98h3526S] constraints. The luminosity distance is thus $$\begin{aligned}
{\ensuremath{d_\text{L}}}(z) & = \frac c {H_0} (1+z) \int_0^z\frac{{\ensuremath{\mathrm{d}}}x}{h(x)},\\
\intertext{and the corresponding distance modulus}
\mu(z) & = 5 \log_{10}({\ensuremath{d_\text{L}}}/ \SI{1}{Mpc}) + 25,\end{aligned}$$ we can then fit the reconstructed $\mu$ to SNIa data, obtain a [$\chi^2$]{}, and hence put model-independent constraints on $({\ensuremath{\Omega_\text{m,0}}},{\ensuremath{\sigma_{8,0}}})$.
Fit to the data
---------------
After obtaining ${\ensuremath{f\sigma_8}}$ from GP, we then obtain the expansion history $h$ via eq. , for a given choice of $({\ensuremath{\Omega_\text{m,0}}},{\ensuremath{\sigma_{8,0}}})$. We note that, a priori, for a given $({\ensuremath{f\sigma_8}},{\ensuremath{\Omega_\text{m,0}}},{\ensuremath{\sigma_{8,0}}})$, there is no guarantee for $h^2(z)$ to be larger than the matter term ${\ensuremath{\Omega_\text{m,0}}}(1+z)^3$. Therefore, hereafter, we impose the positive dark energy condition that $$\begin{aligned}
\label{eq:pde}
{\ensuremath{\Omega_\text{de}}}(z) & = h^2(z) -{\ensuremath{\Omega_\text{m,0}}}(1+z)^3 \geq 0 \qquad \forall z.\end{aligned}$$
Finally, calculate the [$\chi^2$]{} for all the data: $$\begin{aligned}
{\ensuremath{\chi^2}}_\text{tot} & = {\ensuremath{\chi^2}}_{{\ensuremath{f\sigma_8}}}+{\ensuremath{\chi^2}}_\text{SNIa}.\end{aligned}$$
Results {#sec:real}
=======
{width="\textwidth"}
We used the same data sets as in @2018PhRvD..98h3526S: the Pantheon compilation [@2018ApJ...859..101S], and the compilation of RSD data including: 2dFGRS [@2009JCAP...10..004S], WiggleZ [@2011MNRAS.415.2876B], 6dFGRS [@2012MNRAS.423.3430B], VIPERS [@2013MNRAS.435..743D], the SDSS Main galaxy sample [@2015MNRAS.449..848H], 2MTF [@2017MNRAS.471.3135H], BOSS DR12 [@2017MNRAS.465.1757G], FastSound [@2016PASJ...68...38O], and eBOSS DR14Q [@2019MNRAS.482.3497Z].
Crossing statistics
-------------------
{width="\textwidth"}
In this section, we study the effects of distorting the mean function on the final fit. In practice, this is equivalent to applying the Bayesian interpretation of the crossing statistics formalism to the reconstructed growth history [@2011JCAP...08..017S; @2012JCAP...05..024S; @2012JCAP...08..002S]. In this formalism, the prediction from the theory to be tested, [$\Lambda$CDM]{}+GR in the present case, is multiplied by some hyperfunction $T_N(x|C_0,\dots,C_N)$: $$\begin{aligned}
{\ensuremath{\widehat{{\ensuremath{f\sigma_8}}}}}(x) & = ({\ensuremath{f\sigma_8}})^{\mathrm{{\ensuremath{\Lambda}CDM}}}(x) \times T_N(x|C_0,\dots,C_N),
\intertext{where}
T_N(x|C_0,\dots,C_N) &= \sum_i^N C_i P_i(x), \\
x & = 2\left(\frac{z}{z_\mathrm{max}}-\frac 1 2\right),\end{aligned}$$ $P_i(x)$ is the $i$th order Chebyshev polynomial of the first kind, and $C_i$ are free parameters. The Chebyshev polynomials constitute an orthogonal basis for $x\in[-1,1]$, and as such, can represent any function. The zeroth order controls the absolute scaling, the first order the tilt, and higher order introduce a curvature and inflexion points. In the Bayesian interpretation of the crossing statistics, we are interested in the confidence intervals around the hyperparameters $C_i$. If the $C_i$ are consistent with $C_0=1, C_i=0 (i\geq 1)$, the data have no preference for any departure from the mean function, meaning the model is consistent with the data. In case of significant deviation from $C_0=1,C_i=0 (i\geq 1)$, the data suggest a preferred deformation of the mean function.
Distorting the starting ${\ensuremath{f\sigma_8}}$, we obtain $\hat\mu$ and fit both $\hat\mu$ and [$\widehat{{\ensuremath{f\sigma_8}}}$]{} to the data. As noted by @2014PhRvD..89d3004H, going towards too high orders, one might miss the effects of the lower orders. Therefore, we start by limiting to the first order, i.e., tilting the mean function. We used the `emcee` Monte-Carlo Markov Chain (MCMC) package to explore the parameter space $({\ensuremath{\Omega_\text{m,0}}}, {\ensuremath{\sigma_{8,0}}},C_0,C_1)$, and show the posteriors in Fig. \[fig:mcmc\_xing\]. They are consistent with $C_0=1,C_1=0$, i.e., the data suggest no modifications, and are perfectly consistent with the best-fit [$\Lambda$CDM]{}+GR model. It is interesting to notice that the preferred ${\ensuremath{\Omega_\text{m,0}}}$ is rather high with respect to the Planck value, while the preferred ${\ensuremath{\sigma_{8,0}}}$ is low. We checked that when going to higher orders in the crossing functions, i.e., including $C_2$ and $C_3$, the contours and do not suggest further modifications (i.e., is still consistent with $C_0=1, C_{i>0} = 0$).
Fig. \[fig:crossing\] shows a random selection of 16 crossing functions and their effects on the reconstructed $\mu$ and on $({\ensuremath{\Omega_\text{m,0}}},{\ensuremath{\sigma_{8,0}}})$. Their corresponding crossing parameters $C_i$ are shown as coloured points in Fig. \[fig:mcmc\_xing\]. This gives the reader some intuition on how distorting the growth affects the reconstructed expansion. Small distortions from the best-fit [$\Lambda$CDM]{}+GR case can lead to significantly different $\hat\mu$.
Gaussian Process regression {#res:GP}
---------------------------
We used Gaussian Process regression [GP, @2006gpml.book.....R] to reconstruct ${\ensuremath{f\sigma_8}}(z)$ from the RSD measurements. GP have been increasingly used in cosmology [@2010PhRvD..82j3502H; @2010PhRvL.105x1302H; @2011PhRvD..84h3501H; @2012PhRvD..85l3530S; @2013PhRvD..87b3520S; @2018PhRvD..97l3501J; @2019MNRAS.485.2783L] and other fields of astronomy [e.g. @2019arXiv190102877I]. A Gaussian process is effectively a random sampling on a function space, generalizing random numbers. [GP can be used to reconstruct a smooth function ${\ensuremath{\mathbfit{f}}}_*$ at the test points ${\ensuremath{\mathbfit{x}}}_*$ given a discrete set of observations $(x_i,y_i)$ and a data covariance matrix ${\ensuremath{\mathbfss{C}}}$.]{} For a given kernel, the covariance between pairs of random variables [$\mathbfit{u}$]{} and [$\mathbfit{v}$]{} is thus given by [$\mathbfss{K}$]{}(f([$\mathbfit{u}$]{}),f([$\mathbfit{v}$]{})) = k([$\mathbfit{u}$]{},[$\mathbfit{v}$]{}), where k([$\mathbfit{u}$]{},[$\mathbfit{v}$]{}) is the covariance kernel. The joint-distribution of the training (observed) outputs [$\mathbfit{y}$]{} and the test (reconstructed) output ${\ensuremath{\mathbfit{f}}}_*$ is a Gaussian joint distribution given by $$\begin{aligned}
\begin{bmatrix}
{\ensuremath{\mathbfit{y}}} \\
{\ensuremath{\mathbfit{f}}}_*
\end{bmatrix} &
\sim \mathcal{N}\left(
{\ensuremath{\mathbfit{0}}},
\begin{bmatrix}
{\ensuremath{\mathbfss{K}}}(X,X) + {\ensuremath{\mathbfss{C}}} & {\ensuremath{\mathbfss{K}}}(X,X_*)\\
{\ensuremath{\mathbfss{K}}}(X_*,X) & {\ensuremath{\mathbfss{K}}}(X_*,X_*)
\end{bmatrix}
\right)\end{aligned}$$ where [$\mathbfss{C}$]{} is the covariance of the data.
We use the squared exponential kernel defined as $$\begin{aligned}
k_{\sigma_f,\ell}({\ensuremath{\mathbfit{x}}},{\ensuremath{\mathbfit{y}}}) & = \sigma_f^2\exp{\left(-\frac{{\ensuremath{\left\lvert {{\ensuremath{\mathbfit{x}}}-{\ensuremath{\mathbfit{y}}}}\right\rvert}}^2}{2\ell^2}\right)},\end{aligned}$$ where $(\sigma_f,\ell)$ are two hyperparameters controlling the amplitude and the correlation scale, and thus the deviation from the mean function. For a given $(\sigma_f^2,\ell)$, we can thus generate a number of samples of ${\ensuremath{f\sigma_8}}$ at any redshift $z$.
In practice, we start from the best-fit [$\Lambda$CDM]{} as a mean function, use GP as a sampling of possible growth histories, and then apply the formalism from § \[sec:method\].
In order to prevent fitting the noise, we impose a hard prior on $\ell\in [0.2,1]$.
A notable difference with the work of @2018arXiv180800377Y is in the GP regression itself. While they obtain the mean and one-sigma contours, we follow each individual random sampling of the function space, therefore obtaining a set of plausible (and self-consistent) couples of expansion and growth histories.
{width="\textwidth"}
![Model-independent constraints on $({\ensuremath{\Omega_\text{m,0}}},{\ensuremath{\sigma_{8,0}}})$, that is, allowed contours in the $({\ensuremath{\sigma_{8,0}}},{\ensuremath{\Omega_\text{m,0}}})$ for which we can find at least one ${\ensuremath{\widehat{{\ensuremath{f\sigma_8}}}}}$ and its corresponding [$\hat{h}$]{} that fit the growth and SNIa better than [$\Lambda$CDM]{}. []{data-label="fig:Oms8_current"}](Oms8_current.png){width="\columnwidth"}
The left-hand panel of Fig. \[fig:res\_current\] shows the reconstructed [$\widehat{{\ensuremath{f\sigma_8}}}$]{}, as well as the best fit in orange. The shadowed area shows the envelope of the reconstructed [$\widehat{{\ensuremath{f\sigma_8}}}$]{}. A vast majority of these reconstructions do not fit the data, therefore, we show in red those reconstructions of [$\widehat{{\ensuremath{f\sigma_8}}}$]{} that, together with some appropriate $({\ensuremath{\Omega_\text{m,0}}},{\ensuremath{\sigma_{8,0}}})$, yields a $\chi^2_\text{tot} < {\ensuremath{\chi^2}}_\text{ref}+1$. The middle and right-hand panels show the reconstructed $h^2(z)$ and $\mu-\mu_\text{ref}$, with the same convention.
Due to the oscillations in the reconstructed ${\ensuremath{\widehat{{\ensuremath{f\sigma_8}}}}}$ from GP, the reconstructed shapes of $h$ and $\mu$ have more flexibility than [$\Lambda$CDM]{}, yielding possible better fit to the data, therefore representing a non-exhaustive set of plausible expansion and growth histories. It is worth mentioning here that, since the process of reconstructing $h^2$ from ${\ensuremath{\widehat{{\ensuremath{f\sigma_8}}}}}$ via eq. involves two integrals, it is very sensitive to variations in the growth history and in the choice of $({\ensuremath{\Omega_\text{m,0}}},{\ensuremath{\sigma_{8,0}}})$. In practice, most reconstructed ${\ensuremath{\widehat{{\ensuremath{f\sigma_8}}}}}$ cannot yield any ${\ensuremath{\hat{h}}}$ that fits the SNIa data. This can be seen by the large grey envelope in the three panels of Fig. \[fig:res\_current\] compared to the thinner band of allowed reconstructions. Therefore, it is important to explore the $(\sigma_f^2, \ell)$ parameter space and generate a large number of random realizations.
Fig. \[fig:Oms8\_current\] shows the area of the $({\ensuremath{\Omega_\text{m,0}}},{\ensuremath{\sigma_{8,0}}})$ parameter space that, for at least one reconstructed ${\ensuremath{\widehat{{\ensuremath{f\sigma_8}}}}}$, yields ${\ensuremath{\chi^2}}<{\ensuremath{\chi^2}}_\mathrm{ref}+1$, i.e., which is within the $1\sigma$ region of the best-fit [$\Lambda$CDM]{} model. In Appendix \[sec:mock\], we demonstrate the validity of the algorithm on a simulated data set, successfully recovering the input cosmology.
It is worth noting that, since we need to assume a value for ${\ensuremath{\Omega_\text{m,0}}}$ in order to reconstruct $h$, we can then obtain the (uniquely defined) equation of state $w(z)$ and growth rate $\gamma(z)$, as $$\begin{aligned}
w(z) & = \frac{\tfrac{2}{3}(1+z)\tfrac{h'}{h}-{\ensuremath{\Omega_\text{m}}}(z)}{1-{\ensuremath{\Omega_\text{m}}}(z)}-1,\qquad \mbox{and}\\
\gamma (z) & = \frac{\ln f(z)}{\ln{\ensuremath{\Omega_\text{m}}}(z)}.\end{aligned}$$ The positive dark energy condition ensures that ${\ensuremath{\Omega_\text{m}}}(z)\geq 0$. Due to the low quality of the current growth data, the constraints are very poor. We note that, since two consecutive integrations are needed to obtain $h$, then in theory only one is needed to obtain $h'$, making this method potentially less sensitive to numerical noise, provided that the quality of the $f\sigma_8$ data improve substantially. Equivalently, we can reconstruct $$\begin{aligned}
\mathcal{F}_\text{de}(z) & = \frac{{\ensuremath{\Omega_\text{de}}}(z)}{{\ensuremath{\Omega_\text{de}}}(0)} = \exp\left(3\int_0^z \frac{1+w(x)}{1+x}{\ensuremath{\mathrm{d}}}x\right),\end{aligned}$$ which does not involve derivatives of $h$. Fig. \[fig:wgamma\_current\] shows our obtained reconstructions of $\mathcal{F}_\text{de}(z)$ and $\gamma(z)$. The quality of current data does not constrain these function beyond $z \geq 0.1$.
![Model-independent reconstructions of $\mathcal{F}_\mathrm{de}(z)$ (left) and $\gamma(z)$ (right). []{data-label="fig:wgamma_current"}](wgamma_current_deltamax1.png){width="\columnwidth"}
It is interesting to compare these obtained model-independent constraints on $({\ensuremath{\Omega_\text{m,0}}},{\ensuremath{\sigma_{8,0}}})$ (Fig. \[fig:Oms8\_current\] with those obtained in @2018PhRvD..98h3526S, where the approach was different. In that paper, the authors reconstructed $h(z)$ from the Pantheon SNIa compilation via iterative smoothing, and then obtained ${\ensuremath{\widehat{{\ensuremath{f\sigma_8}}}}}(z)$ by assuming a constant $\gamma = 0.55$ (or varying it as a free, but constant, parameter), and obtain $f={\ensuremath{\Omega_\text{m}}}(z)^\gamma$.
Summary & Conclusion {#sec:ccl}
====================
Using the latest compilation of RSD measurements, we reconstruct the growth history using two model-independent approaches, namely, crossing statistics and Gaussian processes, only assuming a flat-FLRW Universe and general relativity. We then used the method introduced by @1998JETPL..68..757S to reconstruct the expansion history, and fit the corresponding distance moduli to the Pantheon SNIa compilation, and finally obtained model-independent constraints on $({\ensuremath{\Omega_\text{m,0}}},{\ensuremath{\sigma_{8,0}}})$. In addition, it is possible to reconstruct the dark energy equation of state $w(z)$ and the growth rate $\gamma(z)$. Applying the crossing statistics formalism, i.e., multiplying the best-fit [$\Lambda$CDM]{}+GR growth by some hyperfunction, and obtained constraints on ${\ensuremath{\Omega_\text{m,0}}},{\ensuremath{\sigma_{8,0}}}, C_i$, where the $C_i$ are hyperparameters of the crossing hyperfunctions, we find consistency with $C_0=1, C_i=0, i\geq 1$, i.e., the data do not call for any modification to the best-fit. However, the preferred values for ${\ensuremath{\Omega_\text{m,0}}}=0.381^{+0.049}_{-0.113}$ and ${\ensuremath{\sigma_{8,0}}}= 0.68^{+0.15}_{-0.12}$ are respectively higher and lower than the Planck best-fit, although consistent with them. Using Gaussian processes gives similar results. Both approaches suggest no departure from [$\Lambda$CDM]{}+GR. However, future surveys such as the Dark Energy Spectroscopic Instrument are expected to provide more accurate measurements of the growth, and thus, to further constrain the gravity model and its parameters. This approach can be thought of as the reciprocal approach of @2018PhRvD..98h3526S, which uses direct reconstructions of $h$ from the supernovae data to fit the RSD data. Both approaches can be thought of as a mutual consistency test of the data and theory: in the former paper, the reconstructed expansion histories are tested against the growth data, while here, the reconstructed growth is compared to the SNIa data. Our analyses point towards the consistency of the reconstructed [$\Lambda$CDM]{} background evolution (via the mean function) with the growth history inside GR. While in GR, $\gamma = 0.55$ is a very good approximation as long as ${\ensuremath{\Omega_\text{m,0}}}$ is not too small (@2016JCAP...12..037P), it is not valid anymore beyond GR. On the contrary, the present approach, can be applied to non-GR models provided that ${\ensuremath{G_\text{eff}}}$ is known, or can even be used to reconstruct the effective Newton constant [$G_\text{eff}$]{} (L’Huillier et al. in prep), and therefore to constraining modified gravity.
Acknowledgements {#acknowledgements .unnumbered}
================
This work benefited from the Supercomputing Center/ Korea Institute of Science and Technology Information with supercomputing resources including technical support (KSC-2017-C2-0021) and the high performance computing clusters Polaris and Seondeok at the Korea Astronomy and Space Science Institute. A.A.S. was partly supported by the program KP19-270 “Questions of the origin and evolution of the Universe” of the Presidium of the Russian Academy of Sciences (the project number 0033-2019-0005 of the Ministry of Science and Higher Education).
Validation on mock data {#sec:mock}
=======================
{width="\textwidth"}
![Model-independent 1$\sigma$ contours of $({\ensuremath{\Omega_\text{m,0}}},{\ensuremath{\sigma_{8,0}}})$. []{data-label="fig:Oms8_sim"}](Oms8_sim_lcdm.png){width="\columnwidth"}
In order to validate the method, we applied it to a controlled simulated realization of the data, where the input cosmology is known. We generated mock data, following the redshift distribution and errors following the data in @2018PhRvD..98h3526S, assuming a known cosmology of $({\ensuremath{\Omega_\text{m,0}}},{\ensuremath{\sigma_{8,0}}})=(0.3,0.8)$. We fit a flat-[$\Lambda$CDM]{} universe to the total (RSD+SNIa) data, and use this best-fit [$\Lambda$CDM]{} model as a mean function for the GP, and use its [$\chi^2$]{} as a reference. Hereafter, we use subscript $_\text{ref}$ to denote the best-fit model. We applied our pipeline, reconstruct [$f\sigma_8$]{} and $h$, and calculate the $\chi^2$ to the data (mock growth and Pantheon-like SNIa). Fig. \[fig:res\_sim\] is the same as Fig. \[fig:res\_current\] but with our simulated data. In addition to the best-fit in red, the true cosmology is shown in green. Fig. \[fig:Oms8\_sim\] shows the allowed contours for $({\ensuremath{\Omega_\text{m,0}}},{\ensuremath{\sigma_{8,0}}})$, and the true cosmology in green is inside the contours, showing the validity of the method.
\[lastpage\]
|
---
abstract: 'We present a detailed analysis of new ALMA observations of the disk around the T-Tauri star HD143006, which at 46mas (7.6au) resolution reveal new substructures in the 1.25mm continuum emission. The disk resolves into a series of concentric rings and gaps together with a bright arc exterior to the rings that resembles hydrodynamics simulations of a vortex, and a bridge-like feature connecting the two innermost rings. Although our $^{12}$CO observations at similar spatial resolution do not show obvious substructure, they reveal an inner disk depleted of CO emission. From the continuum emission and the CO velocity field we find that the innermost ring has a higher inclination than the outermost rings and the arc. This is evidence for either a small ($\sim8\degree$) or moderate ($\sim41\degree$) misalignment between the inner and outer disk, depending on the specific orientation of the near/far sides of the inner/outer disk. We compare the observed substructures in the ALMA observations with recent scattered light data from VLT/SPHERE of this object. In particular, the location of narrow shadow lanes in the SPHERE image combined with pressure scale height estimates, favor a large misalignment of about $41\degree$. We discuss our findings in the context of a dust-trapping vortex, planet-carved gaps, and a misaligned inner disk due to the presence of an inclined companion to HD143006.'
author:
- 'Laura M. Pérez'
- Myriam Benisty
- 'Sean M. Andrews'
- Andrea Isella
- 'Cornelis P. Dullemond'
- Jane Huang
- 'Nicolás T. Kurtovic'
- 'Viviana V. Guzmán'
- Zhaohuan Zhu
- Tilman Birnstiel
- Shangjia Zhang
- 'John M. Carpenter'
- 'David J. Wilner'
- Luca Ricci
- 'Xue-Ning Bai'
- Erik Weaver
- 'Karin I. Öberg'
title: |
The Disk Substructures at High Angular Resolution Project (DSHARP):\
X. Multiple rings, a misaligned inner disk, and a bright arc in the disk around the TTauri star HD143006
---
Introduction {#sec:intro}
============
High resolution observations of protoplanetary disks have shown that most, if not all, disks host substructures in the form of spiral arms, rings, or azimuthally asymmetric features. These features are imprints of the processes of disk evolution and planet formation, and can be observed from optical [in scattered light, e.g. @avenhaus2018] to millimeter wavelengths [e.g. @rings; @spirals]. In the dust continuum, the most common substructures are multiple bright rings and dark gaps, possibly tracing over-densities and depletion of dust grains, respectively. Rings and gaps are a natural outcome of dynamical interactions with planets [e.g., @Paardekooper2004; @crida2007; @zhu2011], but could also be tracing ice lines [e.g., @zhang2015; @okuzumi2016], or be a result of certain magneto-hydrodynamical processes [e.g., @johansen2009; @dittrich2013; @bai2014; @simon2014; @bethune2017; @suriano2018].
These substructures could play a very important role in the process of planet formation, and in particular, in the first steps towards the growth of planetesimals. Alternatively, substructures may exhibit what is the outcome of planet-disk interaction, and serve as an important diagnostics of ongoing planet formation. Since dust particles experience aerodynamic drag from the gas, in a smooth disk they would rapidly lose angular momentum and drift towards the star before they can grow to the planetesimal size [@brauer2007; @birnstiel2010]. However, dust grains can be maintained in localized particle traps (pressure maxima) which allow them to efficiently grow [@pinilla2012b]. Observationally, these traps appear as a suite of bright and dark rings in the dust continuum [@pinilla2012a], reminiscent of continuum observations of multiple ringed systems [e.g., @ALMA]. In addition to concentric rings, azimuthal asymmetries have also been observed, in particular in protoplanetary disks with dust-depleted large cavities [transition disks; e.g., @casassus2013; @isella2013; @dong2018b]. They are interpreted as azimuthal dust trapping [@birnstiel2013; @Lyra], possibly in a vortex due to the Rossby wave instability [e.g., @li2000; @meheut2012; @zhu2014]. Such an instability can be generated at the edge of a gap created by a massive planet [e.g., @regaly2012; @ataiee2013; @zhu2016], or at the edge of a dead zone [e.g., @kretke2007; @flock2017].
Planet-disk interactions can also leave imprints on the disk gas kinematics [@seba2015] and observations of CO isotopologues have revealed non Keplerian velocities in various disks. Recently, @pinte2018 found a local deformation of the CO disk velocity field of the intermediate-mass young star HD163296, consistent with the spiral wave induced by a 2M$_{\rm{Jup}}$ planet located at 260au. @teague2018 measured, in the same disk, local pressure gradients consistent with gaps carved by a 1M$_{\rm{Jup}}$ planet at 83au and 137au. Other observations of non-Keplerian velocities were explained by the presence of radial flows [@rosenfeld2014], of a warp [@rosenfeld2012; @casassus2015; @walsh2017] or of free-falling gas connecting a strongly misaligned inner disk to the outer disk [@loomis2017]. A misalignment between the inner and outer disk could be induced by a massive companion on an inclined orbit with respect to the plane of the disk [e.g. @bitsch2013; @owen2017], or alternatively, by a misaligned stellar magnetic field [@bouvier2007] in which case the star should be periodically obscured [periodic dippers, see e.g., @cody2014]. In the presence of a strongly misaligned inner disk, the outer disk illumination is drastically affected, as evidenced by shadows observed in scattered light images that trace the surface of the disk [e.g. @marino2015; @benisty2017; @benisty2018; @casassus2018; @pinilla2018].
The focus of this paper is the T Tauri star HD143006, a G-type star [$1.8^{+0.2}_{-0.3}$ M$_{\odot}$, 4-12 Myr old, @preibisch2002; @pecaut2012; @garufi2018; @survey], located in Upper Sco at a distance of $165\pm5$pc [@Gaia2018]. Sub-millimeter continuum observations of the disk at moderate spatial resolution revealed an azimuthal asymmetry and a marginally-resolved cavity [@barenfeld2016], although it is not classified as a transition disk from its spectral energy distribution. A strong near-infrared excess and typical mass accretion rate [$\sim2\times10^{-8}\,M_{\odot}$yr$^{-1}$; @rigliaco2015] indicate that the innermost regions retain dust and gas. Recent scattered light observations of HD143006 show multiple brightness asymmetries: a shadow over half of the disk (the West side) and two additional narrow shadow lanes, both suggestive of a moderately-misaligned inner disk [@benisty2018 and §\[sec:SPHERE\]].
In the following, we present new observations of HD143006 obtained with ALMA as part of DSHARP in §\[sec:obs\], we then present a characterization of its substructures in §\[sec:results\], that we further discuss in §\[sec:discussion\], and the conclusions of our work in §\[sec:conclusions\].
{width="1.\textwidth"}
Observations {#sec:obs}
============
Dust continuum emission at 1.25mm and $^{12}$CO emission in the $J=2-1$ transition were observed with long and short baselines in the ALMA Large Program 2016.1.00484.L. Additional short baselines observations (from project 2015.1.00964.S, P.I. Öberg) in the same dust and gas tracers were combined to increase uv-coverage on the short-spacings. Details on the calibration of visibilities can be found in [@survey]. Observations were centered to R.A.(J2000) = 15h58m36.90s, Dec(J2000) = $-22$d57m15.603s, which are the coordinates of the phase center of the last execution of the long baseline observations. The continuum images presented here used the same imaging parameters as listed in Table 4 of @survey, while the CO images were obtained using a robust parameter of 0.8 with a uv-tapering of $20\;$mas, which was applied to the CO visibilities to enhance extended emission from the disk. The RMS noise in the continuum image is 14.3$\mu$Jybeam$^{-1}$ for a $46\times 45$mas beam with position angle (PA) of $52.1\degree$. In the CO spectral cube we measure an RMS noise of 1.04mJybeam$^{-1}$kms$^{-1}$, for a beam size of $66\times49$mas, PA of $83.6\degree$, and channel width of 0.32kms$^{-1}$. The continuum map is presented in Figure \[fig:cont\], and the full set of CO channel maps in Figure 5.9 of @survey.
{width="1.\textwidth"}
Results {#sec:results}
=======
Dust Continuum Emission {#sec:dust}
-----------------------
The continuum emission from HD143006, presented in Figure \[fig:cont\], is observed close to face-on: the disk resolves into an inner depleted region, three bright rings at roughly $0.05''$ (8au), $0.24''$ (40au), $0.39''$ (64au) from the disk center, two dark annuli at roughly $0.13''$ (22au) and $0.31''$ (51au), and a bright south-east asymmetric feature just outside of the outermost ring at $0.45''$ (74au) from the disk center. This prominent arc is located at a position angle (PA, defined from North toward East) of $\sim140\degree$ and has an azimuthal extent of $\sim30\degree$. From now on, we follow the convention of @rings and label the dark/bright annular features with a letter “D”/“B” followed by their radial location in au. Then, the innermost bright ring is labeled B8, the second bright ring as B40 and the outermost bright ring as B64. The dark annuli between B8 and B40, and between B40 and B64 are labeled D22 and D51, respectively.
The radial profile of the continuum emission presented in @rings indicates that B8 and B40 have a contrast of $\sim42$ and $\sim24$ with respect to the D22; these are one of the highest contrast ring-features of the DSHARP sample. However, these rings are not completely smooth: B8 has two peaks along a PA of $\sim165\degree$ (see panel (a) on Fig.\[fig:cont\]), but this is due to its inclination w.r.t. our line of sight (see §3.4). The outermost rings, B40 and B64, both peak close to a PA of $180\degree$ (better seen in the residual maps of §\[sec:model\]). The bright asymmetric feature outside of B64 (see panel (b) on Fig.\[fig:cont\]) is also uneven in the azimuthal direction: the arc resolves into three peaks along its azimuthal extent. The brightest peak at the center of the arc has a signal-to-noise ratio (SNR) of 48, while the flanking peaks have a SNR of 44 and are separated by $\sim10$au (60mas) from the brightest peak of the arc. Finally, the dark annuli are not completely empty, in particular, inside D22 there is a bridge-like emission feature, with a SNR of 4.9$\sigma$, connecting the B8 and B40 rings (see emission between the rings at a PA of $\sim 300\degree$ in panel (a) of Figure \[fig:cont\]).
Gas emission traced by CO {#sec:gas}
-------------------------
The overall emission from the gaseous component of the HD 143006 disk, as traced by the CO line, is presented in the middle and right panels of Figure \[fig:contCO\] and extends out to $\sim1''$ in radius, twice larger in radial extent than the dust continuum emission. CO emission is detected above 3$\sigma$ from $-0.12$ km s$^{-1}$ to $15.2$ km s$^{-1}$ in the LSRK reference frame, with a systemic velocity of $\sim7.7$ km s$^{-1}$. The integrated intensity of CO, also known as moment 0 map (left panel, Figure \[fig:contCO\]) is not centrally peaked. Since the continuum emission appears to be optically thin [$\tau<0.2$, @rings], the lack of CO emission in the inner disk cannot be fully explained by continuum subtraction of optically-thick dust emission, and rather indicates that some gas depletion occurs in the inner disk.
The intensity-weighted velocity of CO, also known as moment 1 map (right panel, Figure \[fig:contCO\]), shows a clear rotation pattern for a nearly face-on disk with a PA of $\sim165\degree$ (note that the moment0 map has two maxima along a similar PA). However, the disk rotation is not perfectly described by a single geometry, as can be seen on the inset of Figure \[fig:contCO\], right panel, where the velocity field in the inner disk appears different from the outer disk. This difference will be quantified in the next sections.
Given the low inclination of this system and the low SNR of the spectral cube (peak SNR of $\sim 11$), the front and back sides of the CO emitting layers are not clearly separated, a feature observed in other disks with higher line-of-sight inclinations [e.g., @rosenfeld2013; @isellaLP]. Nevertheless, of the two possibilities for the absolute disk geometry, we suggest that the West side of the disk is the one closest to us: at every channel, the West side of the disk appears shorter when projected onto the rotation axis of the disk than the East side [see Figure 5.9 in @survey]
Finally, a close inspection of the channel maps revealed that redshifted emission close to the systemic velocity has a deviation or “kink” from the Keplerian iso-velocity curves that is not present in the blue-shifted channels. This kink is marked by an arrow on the bottom panels (redshifted channels) of Figure \[fig:kink\] (see Appendix) and appears at a PA of between $\sim80\degree-120\degree$.
{width="\textwidth"}
Modeling CO kinematics {#sec:COmodel}
----------------------
We model the Keplerian velocity field in HD143006 using an analytical model of a razor-thin disk in Keplerian motion around a 1.8M$_{\odot}$ star, in order to constrain the *global* geometry of the disk. The model parameters are the inclination ($i$), position angle (PA), systemic velocity ($v_{sys}$), as well as the right ascension and declination offsets ($x_0,y_0$) for the center of rotation. We fix the inner and outer radius of the disk to 0.3 and 250au, respectively. For a given set of parameters we generate a velocity map that is convolved with the beam of our CO observations, which we then compare to the intensity-weighted velocity of CO (right panel of Figure \[fig:contCO\]) pixel by pixel. We perform our fit using an affine invariant MCMC sampler [$emcee$, @emcee], and sample the parameters with uniform priors in the following range: $i$ between \[$0\degree, 90\degree$\], PA between \[$0\degree, 360\degree$\], $v_{sys}$ between \[5 km s$^{-1}$, 10 km s$^{-1}$\], and $x_0,y_0$ between \[-0.2, 0.2\]. The parameter space is explored with 80 walkers and for 4000 steps. From the last 2000 steps we compute the best-fit values, which correspond to the 50$^{th}$ percentile of the samples in the marginalized distributions, while the error bars correspond to the 16$^{th}$ and 84$^{th}$ percentiles. The best-fit parameters can be found on Table\[tab:COmodel\], while Figure \[fig:mom1\] presents the velocity field of the observations, best-fit model, and residual (computed by subtracting the model from the observations). Although the best-fit model reproduces the velocity field of the outer disk, inwards of $\sim0.15''$ there are significant residuals that suggest gas moving at higher speeds than those in the model.
Note that if we separately fit the inner and outer disk kinematics, by masking the velocity field in Figure \[fig:mom1\] inside and outside of $0.15''$ (which corresponds to the mip-point separation between B8 and B40), we find two different disk geometries. First, a fit for the geometry of the outer disk while excluding the inner disk (i.e. masking those pixels inside of $0.15''$), results in a PA consistent with the global fit in Table \[tab:COmodel\] but a slightly more face-on outer disk with an inclination of $14.59\degree\pm0.02\degree$. On the other hand, when we only fit for the disk kinematics inside of $0.15''$ (i.e. masking those pixels outside of $0.15''$), we find $i = 22.84\degree \pm 0.05\degree$ and PA $= 165.6\degree\pm0.1\degree$ for the inner disk. Thus, we constrain a difference between the inner and outer disk geometry, in particular, the inclination of the disk changes from the inner to the outer disk (see Figure \[fig:restrictedmom1\] on the Appendix, for the best-fit model and residuals when the inner or outer disk are masked).
----------- ------------- -- ----------------------------
$x_0$ (mas) $-6 \pm 3$
$y_0$ (mas) $29\pm 2$
$i$ ($^\circ$) $16.2\degree\pm0.3\degree$
PA ($^\circ$) $167\degree\pm1\degree$
$v_{sys}$ km s$^{-1}$ $7.71\pm0.02$
----------- ------------- -- ----------------------------
: Best-fit model to the intensity-weighted velocity field around HD143006[]{data-label="tab:COmodel"}
[ l c r r r r r r ]{} & & & &\
$i$ & ($^\circ$)& & $24.1\pm1.0$ & &\
$PA$ & ($^\circ$)& & $164.3\pm2.4$ & &\
$x_0$ & (mas) & & $-4.4\pm0.2$ & &\
$y_0$ & (mas) & & $23.1\pm0.2$ & &\
& & & & & &\
$I_R$ or $I_A$ & ($\mu$Jy) & & $1.54\pm0.02$ & & $0.64\pm0.01$ & $0.410\pm0.001$ & $1.09\pm0.01$\
$r_R$ or $r_A$ & (au) & & $7.67\pm0.04$ & & $39.95\pm0.03$ & $63.6\pm0.1$ & $74.2\pm0.1$\
$\sigma_R$ or $\sigma_{r,A}$& (au) & & $2.54\pm0.04$ & & $4.2\pm0.1$ & $9.4\pm0.1$ & $4.6\pm0.1$\
$\sigma_{\theta,A}$ & (au) & & - & & - & - & $21.1\pm0.1$\
$\theta_A$ &($^\circ$) & & - & & - & - & $141.4\pm0.1$\
Modeling of continuum emission {#sec:model}
------------------------------
To constrain the observed substructure in this disk, we generate simple morphological models that describe the emission for the observed rings and the south-east arc. These models are constructed by combining the emission of three radially-Gaussian rings (that describe the B8, B40 and B64 features) and a two-dimensional Gaussian, in both the radial and azimuthal direction (that describes the outermost arc feature).
For every ring, its intensity is given by: $$I(r) = I_R \; e^{-(r-r_R)^2/2\sigma_R^2},$$ where $I_R$ is the peak intensity at radius $r_R$, $\sigma_R$ is the ring width, and the ring is assumed to be symmetric along the azimuthal direction. For the asymmetric arc feature, its intensity is given by: $$I(r,\theta) = I_A \; e^{-(r-r_A)^2/2\sigma_{r,A}^2} \; e^{-(\theta-\theta_A)^2/2\sigma_{\theta,A}^2},$$ where $I_A$ is the peak intensity at radius $r_A$ and azimuthal location $\theta_A$, and $\sigma_{r,A}$, $\sigma_{\theta,A}$ are the width of the 2D-Gaussian in the radial and azimuthal direction, respectively. The model adopted for the arc corresponds to the distribution of material in a steady-state vortex [@Lyra].
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Additional parameters of importance are any offset ($x_{0},y_{0}$) that the rings may have from the phase center of the observations defined in §2, and the inclination ($i$) of each feature along a particular position angle (PA) on the sky. Given the evidence for a misalignment between the inner disk and outer disk presented in @benisty2018, as well as the evidence from modeling its velocity field (§3.3), we assume that the features found in the outer disk (i.e. B40, B64, and the arc) share the same center, inclination, and PA, while the innermost ring (B8) may have a different geometry due to the misalignment (i.e. different center, inclination, and PA). This results in a total of 22 free parameters that are constrained with over $34$million visibilities. For a given set of parameters we produce a model image of the disk that is Fourier-transformed and sampled at the same locations in the uv-plane as the observed visibilities; for this we use the publicly available code Galario [@Galario]. To sample the PDF of our model parameters, we use the MCMC sampler $emcee$ [@emcee] with 120 walkers that we ran for 25000 steps. Convergence was checked by measuring the autocorrelation time, which was under 3000 iterations. For each parameter we compute the posterior PDF by marginalizing over all but the parameter of interest over the last 15000 steps.
The best-fit values of the continuum emission model, chosen as the 50$^{th}$ percentile of the PDF, as well as the 1-$\sigma$ uncertainty of each parameter from the 16$^{th}$ and 84$^{th}$ percentile of the PDF, are shown in Table \[tab:dustmodel\]. From the best-fit we construct model visibilities (sampled at the same locations in uv-space as the observed visibilities) and we compute residual visibilities by subtracting the model visibilities from the observations. Images of the best-fit model and residual, obtained with the same imaging parameters as the observations (see §2), are presented in Figure \[fig:modelCont\].
The morphological models employed here can reproduce the observed emission reasonably well, as can be seen in the residual map of Figure \[fig:modelCont\] where leftover emission is minimal and the largest residual is at $\pm9\sigma$. @rings find the rings locations directly on the image, resulting in differences of only 2-3% for the outermost, well-resolved, rings and a 30% difference in the radial location of B8, which due to its small angular size will have much less independent measurements on the image. The radial width of the rings increases as a function of distance from the star: our model constrains radial extents of 6, 10, and 22au (in FWHM) for B8, B40, and B64. When fitting width of each ring directly on the image, @dullemond2018 find a similar but narrower ring widths for B40 and B64, 7% and 30% narrower, respectively.
As the rings are not completely symmetric in the azimuthal direction, for both B40 and B64 the radially-Gaussian rings cannot reproduce the excess of emission seen in the South of each ring. For the south-east arc, our model constrains a narrow extent in the radial direction with a factor of $\sim4$ wider extent in the azimuthal direction, similar to that observed in other disks with arcs observed at lower angular resolution [e.g. @Perez2014]. There are significant residuals at the arc location, in particular, the 2D-Gaussian prescription cannot properly describe the multiple peaks observed along the arc (Fig.\[fig:cont\], panel (b)), which appear also in the residual map at the $\sim6-9\sigma$ level.
Discussion {#sec:discussion}
==========
Substructures in the ALMA images
--------------------------------
The distribution of larger solids, as traced by the ALMA dust continuum observations, reveals a wealth of substructure in the HD143006 disk (§3.4), while the distribution of CO emission exhibits little substructure (only in the innermost disk regions, §3.3). This difference may arise from the fact that the 1.25mm dust emission is optically thin throughout the disk [@rings], while the observed CO emission is optically thick and traces the surface layers of the disk.\
**Bright rings and dark annuli.** For the dark annuli D22 and D51, there is not a corresponding decrement of CO emission in either the channel maps [Figure 5.9 @survey] or the integrated CO emission map (Figure \[fig:contCO\], right panel). Assuming that the dust depletion observed in the outermost dark annulus originates from dynamical clearing by objects embedded in the disk, @Shangjia2018 infer masses of $\sim10-20M_{Jup}$ and $\sim0.2-0.3M_{Jup}$, for planets inside D22 and D51, respectively.
On the other hand, there is a decrement of both CO and dust emission inside B8. In particular, CO emission does not appear to be centrally-peaked, either on the integrated emission map (moment 0 map, left panel of Figure \[fig:contCO\]) or in the peak emission map (moment 8 map, right panel of Figure \[fig:SPHERE\]). Although both these maps are susceptible to beam dilution effects [@weaver2018], which may cause an “false” inner cavity in the gas, the lack of CO emission inside B8 is also observed in the channel maps away from the systemic velocity [see appendix Figure \[fig:kink\], and Figure 5.9 in @survey], meaning that there is some depletion of the CO column density inside of $\sim$15au ($\sim90$mas). We note that the lack of gas emission inside B8 cannot be explained by absorption from optically thick dust in the inner disk, since we also observe a dust emission deficit inside B8 and the emission from dust is optically thin throughout [@rings]. Most likely, the depletion of CO inside B8 is quite large, for example, in transition disk cavities with centrally-peaked CO emission, the depletion has been measured to be more than an order of magnitude [@Marel2015], and in the case of HD143006 we do not observe a centrally peaked CO.
A possible origin for the depleted inner disk is photoevaporation, however, the star is accreting at a moderate rate [$2\times10^{-8} M_{\sun}$ yr$^{-1}$, @rigliaco2015] and the near infrared excess points to a dust-rich inner disk, making this possibility less likely. A perturber inside B8 could deplete both gas and dust, and even misalign the inner disk, a possibility that will be discussed in §4.3.
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**Bright arc in the south-east.** Vortices are regions of higher gas pressure that can very efficiently trap solids [e.g., @barge1995; @klahr1997; @baruteau2016]. We model the arc emission with a prescription for the distribution of dust in a vortex that has reached steady-state [@Lyra], and which should be smooth in the radial and azimuthal direction. However, the prominent arc in the 1.25mm continuum images has further substructure: three separate peaks that are unresolved in the radial direction (see residual map in Figure \[fig:modelCont\]), with a separation of $\sim10$au from the central peak. During their lifetime, vortices may never reach a stable equilibrium. For example, simulations have shown that when the back-reaction of the dust onto the gas is taken into account, vortices may trap particles not in a single but in multiple structures, and it is this dust feedback that would eventually lead to the destruction of the vortex [see, e.g., @fu2014]. Thus, it is not surprising that the observed arc has internal substructure.
The radial and azimuthal extent of the arc constrained by our model is about 5 by 21au. From the midplane temperature profile ($T_{mid}$) assumed in @rings, we estimate that at the vortex location ($\sim74$au), the pressure scale height corresponds to $H_p \sim 5$au (assuming a local sound speed of $c_s = (k_B T_{mid} / \mu m_p)^{0.5}$, and $\mu=2.3$). Thus, the vortex is radially as wide as $H_P$ and it extends azimuthally over a few pressure scale heights, still consistent with the vortex scenario. In particular, the $\sim21$au azimuthal extent is consistent with that of vortices formed at the edge of a gap/cavity, as it has a much smaller width than the $\pi$-wide vortices expected due to instabilities at the dead zone [@regaly2017]. However, the arc is outside of the millimeter dust disk, rather than at the edge of a dark annulus, and it is radially separated from B64 by $\sim 10$au. Such a configuration is similar to that of the disk in HD 135344B, which resolves into a narrow ring with an asymmetric arc outside of the ring [@cazzoletti2018]. Simulations by @lobo2015 show that after a vortex forms at the edge of a planet-induced gap, the surface density can be enhanced further out in the disk than at the initial vortex location. Such density enhancement triggers again the Rossby wave instability and a second generation vortex may form beyond the primary, leaving only the outermost vortex once the first one is damped. Thus, a second-generation vortex may explain the location of the observed arc in the outer disk of HD143006.
We note that the vortex-like structure is not associated to spiral arms in scattered light [@benisty2018], unlike the well-studied cases of MWC758 [@dong2018b], HD142527 [@avenhaus2014] and HD 135344B [@stolker2016], which in addition are classified as transition disks from their spectral energy distribution and whose central stars are Herbig ABe objects.
Substructures seen in scattered light {#sec:SPHERE}
-------------------------------------
Figure \[fig:SPHERE\] presents our continuum image (left), the polarized scattered light coronagraphic image of HD143006 obtained with VLT/SPHERE[^1] (middle), and the peak intensity of the CO line at each velocity, also known as moment 8 map (right). The scattered light image is a good tracer of small micron-sized dust grains located in tenuous disk surface layers, while the ALMA continuum image traces the millimeter-sized dust grains at the midplane. The peak intensity of the (optically thick) gas emission from CO (moment 8 image) also traces the upper disk layers.
Each pixel of the scattered-light image (middle panel, Figure\[fig:SPHERE\]) was scaled by $r^{2}$, where $r$ is the distance to the central star, to compensate for the drop off in stellar illumination and allow a better detection of faint outer disk features. In this image, a broad shadow in the West, covering half of the disk, is present. In addition, the SPHERE image shows from inside out: a gap/cavity beyond the coronagraph radius ($\sim$80mas, 13au), a non-symmetric ring with two narrow shadow lanes aligned along the North-South direction, a gap not completely devoid of emission, and a non-symmetric outer disk. Both the outer disk and the inner ring present an over-brightness along a small range of position angles (PA$\sim$100-170$^\circ$, see contours in Figure \[fig:SPHERE\]).
### Comparison with CO emission
As the CO line is optically thick, its intensity probes the temperature of the emitting gas, which, in turn, depends on the amount of starlight received at the disk surface. In scattered light, the outer part of the disk is not detected in the West, indicating a drop of irradiation. To first approximation, the temperature of the CO emitting layer should scale as the received luminosity to the $1/4$ power, thus, we should also expect an East-West brightness asymmetry in the peak intensity CO map (left panel, Figure \[fig:SPHERE\]). We measure a East-West contrast in the CO peak intensity map of roughly $\sim1.2$. The difference between the level of asymmetry in the scattered-light image and in the moment 8 map is likely related to the different depths of the $\tau\sim$1 layers in the two tracers. We note however that no substructure in the CO channel maps or moment maps is found to coincide with the narrow shadow lanes in the inner ring of the scattered light image.
The absence of an over-brightness in the CO peak intensity map at the location of the millimeter arc suggests that the over-brightness observed in scattered light does not only originate from an effect of a stronger irradiation. And since the continuum emission in the arc is not optically thick ($\tau \sim 0.4$ at the peak) this is unlikely an issue of continuum subtraction of optically thick emission on the CO line. As the inner disk is not probed by the SPHERE data, we cannot further compare the observed depletion of the CO column density inside of $\sim$15au ($\sim90$mas) discussed in §4.1. Interestingly, the “kink” seen in the channel maps (§3.2, also see Figure \[fig:kink\] in the appendix) roughly coincides in radius and azimuthal extent with the over-brightness seen in scattered light along the inner ring.
### Comparison with dust continuum
The comparison between the ALMA continuum and the SPHERE scattered light images indicates striking differences. Apart from the bright prominent arc in the outer disk, none of the features appear co-radial and none of the azimuthal asymmetries seen in scattered light have counterparts in the continuum image. Such differences are expected, as the two images trace distinct layers of the disk and different dust particles. The scattered light image shows the regions of the disk that are directly lit by the star and its appearance strongly depends on the shape of the disk surface, while the 1.25mm continuum image shows the midplane features, whose brightness depend on the dust density, temperature, and opacity. An schematic of the features observed in the HD143006 disk is presented in Figure\[fig:cartoon\] and will be discussed here.\
**Radial distribution.** While B8 is masked by the SPHERE coronagraph, the inner ring in scattered light appears located inside B40 (see middle panel of Figure \[fig:SPHERE\]), indicating that micron-sized grains are extending further in than the mm-sized grains. Such a spatial segregation by particle size is a natural outcome of dust trapping by a massive planet [@rice2006; @zhu2012; @dejuanovelar2013; @pinilla2015] and has already been observed in transition disks [e.g., @garufi2013]. If the inner scattered-light ring traces the edge of the gap, it would be directly illuminated by the star and “puff-up”. We propose that B40 lies in its shadow, which is supported by a darker region seen just beyond the scattered-light inner ring (on the East side of the disk), and that the outer disk re-emerges from the shadow at a larger distance from the star [e.g., @dullemond2004; @isella2018].
We note that a projected radial offset between the features seen in scattered light and in the millimeter could be expected due to the disk inclination and position angle, the opening angle of the scattering surface, and the vertical structure of the ring [@dong2018b]. However, such an effect is considerable only for high disk inclinations. In that case, on the near side of the disk the scattered-light ring should appear inside B40, while on the far side, it should appear outside of it. That is not what we observe: the inner scattered-light ring is inside B40 at all position angles at which it is detected.
It is possible that a companion between B8 and B40 is shaping the disk, leading to the observed radial segregation and to a dust-depleted gap in both tracers of small and large dust grains. In this paper series, @Shangjia2018 constrain a planet mass of $\sim 10-20 M_{Jup}$ based solely on the deep gap observed in ALMA images between B8 and B40. Such a high mass companion is consistent with the hydrodynamic simulations coupled with dust evolution from @dejuanovelar2013, which require a planet more massive than $9M_{Jup}$ to be responsible for the observed radial segregation by particle size. Although we find a strong depletion in small grains that are well coupled to the gas, we do not detect clear depletion in CO emission between B8 and B40, suggesting that dust grains are filtered and that gas can still flow through the gap, similar to what is often seen in transition disks [@pinilla2015b].
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**Azimuthal asymmetries.** The large East/West shadow (and the additional two narrow shadow lanes) observed in the SPHERE image can be explained by a moderate misalignment of $\sim$20-30$^\circ$ between the inner and outer disk [@benisty2018 see also schematics in Figure \[fig:cartoon\]]. As shown in the residual map of Figure \[fig:modelCont\], B40 and B64 are symmetric, and we find no evidence for a counterpart of the East/West shadow in the continuum observations that would indicate inhomogeneities in the dust temperature. This supports the finding of @rings, where the rings in HD143006 are optically thin. Indeed, a configuration in which the inner and outer disks are moderately misaligned (see Figure \[fig:cartoon\]) would lead to e.g. the West side of the disk in the shadow while the East side is irradiated, for the front side of the disk that is facing us. On the backward-facing side of the disk (not seen by the observer) the opposite happens: the East side is in the shadow while the West is irradiated. If the disk is vertically optically thin, and the continuum emission traces the full vertical extent of the disk, the asymmetry will cancel out between the two sides, and the continuum emission would appear symmetric. We note however the presence of residuals (up to 7$\sigma$) in the south (PA$\sim$180$^\circ$) at the location where the East/West shadow starts. As in the CO data, there is no evidence for counterparts to the narrow shadow lanes in the continuum observations, unlike what is observed in two transition disks so far [DoAr44, J1604; @casassus2018; @pinilla2018].
The bright arc in the ALMA image coincides with the over-brightness seen in the outer disk of the SPHERE image (see contours in Figure\[fig:SPHERE\] and diagram in Figure \[fig:cartoon\]). If the arc traces an over-density in large grains, we also expect an enhancement of small particles due to fragmentation, and if turbulent mixing over the vertical height of the disk allows it, a small-particle enhancement should be seen from the surface layers as well. Additionally, the small grains seem to extend further, and over a wider range of position angles, than the larger particles, which is one of the predictions of dust trapping in a vortex [@baruteau2016]. Future observations at shorter/longer ALMA wavelengths can test this idea by measuring the spread/concentration of particles over the arc location.
The inner ring from the scattered-light image shows a strong over-brightness that has no counterpart in the ALMA continuum image (see contours inside B40 in Figure\[fig:SPHERE\]), and that is likely due to an irradiation effect and local change in scale height. Interestingly, it lies over a range of position angles similar to the one over which the arc extends in the ALMA continuum image. If the line-up is not coincidental, it must be related to a radiative effect as any physical structure at such different radii would shear away the alignment quickly due to different angular rotation velocities. However, it is possible that the over-brightness along the first scattered-light ring further extends to the West (in the shadowed region), and that it covers a much broader range of position angles. In that case, the line-up could be coincidental.
In general, such differences likely mean that the small grains in the surface layers are only marginally affected by dynamical processes (while large grains in the midplane are), but are instead very much affected by irradiation processes.
A misalignment between inner and outer disk
-------------------------------------------
Our continuum emission modeling indicates that the inner and outer disk do not share the same geometry, which is expected based on the shadows observed in the SPHERE image [@benisty2018]. The inner disk appears inclined by 24$^\circ$ with a PA of 164$^\circ$, while the outer disk has a lower inclination of 17$^\circ$ with a PA of 176$^\circ$, this difference in geometry is of high statistical significance (see Table \[tab:dustmodel\]). Based on these values, we can estimate the misalignment angle between the inner and outer disks, defined as the angle between the normal vectors to the disks. Assuming the values above, we find a small misalignment of 8$^\circ$. However, this assumes that the inner and outer disks share the same near side of the disk (the side closer to us), which cannot be determined with the continuum data alone. If instead, the near sides do not coincide (e.g., the near side of the outer disk is in the East, while the near side of the inner disk is in the West), the misalignment would be much larger, of 41$^\circ$ (computed using $i=-24^\circ$ for the inner disk, the rest of the parameters being the same as above). In this case, once projected onto the plane of the sky, the inner disk rotation would appear as counter-rotating with respect to the outer disk. This is not something seen in the CO data (at an angular resolution of 66$\times$49mas, $\sim9$au), in particular at the highest velocities. Nevertheless, the fit of the intensity-weighted velocity map with a Keplerian disk model shows residuals inside $0.15''$, supporting a different geometry than the one of the outer disk.
The location of the shadows seen in scattered light depend on the inner and outer disk geometry, and on the height of the scattering surface ($z_{\rm{scat}}$) of the disk where the shadows are seen. Using the equations developed in @min2017, we find a $z_{\rm{scat}}/R$ of $\sim$0.03 and $\sim$0.16 at $\sim$18au (the location of the inner scattered light ring), for misalignment angles of $7\degree$ and $41\degree$, respectively. Since the pressure scale height of the disk, $H_p$, should be smaller than the scattering surface (by a factor of $\sim$2-4), we expect $H_p/R<0.015$ and $<$0.08 at $\sim$18au, for misalignment angles of $7\degree$ and $41\degree$, respectively. From the standard temperature profile assumed in @rings, we estimate that at 18au $H_p/R\sim0.04$. Given the small $H_p/R$ expected for a $7\degree$ misalignment and the larger value of the pressure scale height as estimated above from standard assumptions, we favor the larger misalignment value of $41\degree$.
The overall morphology of the shadows in the scattered light image were reproduced by a circumbinary disk that is broken and misaligned by $\sim$30$^\circ$ [@benisty2018], due to an inclined equal-mass binary [see the hydrodynamical simulations by @facchini2018]. We note however, that the value of the misalignment needed to reproduce the scattered light image is model-dependent, and depends on the exact geometry of the inner disk as well as on the shape of the outer disk rim considered in the model.
The presence of an equal mass binary companion, as well as of any companion with a mass ratio larger than q=0.2 (corresponding to 0.3M$_{\odot}$), can be ruled out as it would have been detected by imaging and interferometric surveys [@kraus2008; @benisty2018]. However, an inclined low-mass stellar companion, as the one detected in the wide gap of HD142527 (with a mass $\sim$0.13M$_{\odot}$), could be responsible for the misalignment [@price2018], and secular precession resonances can result in large misalignments for companions with mass ratio of 0.01-0.1 [@owen2017]. A massive planet could in principle also lead to a misaligned inner disk as long as its angular momentum is larger than that of the inner disk [e.g. @xg2013; @bitsch2013; @matsakos2017], i.e., in cases where the inner disk is depleted.
In any case, it is not clear whether the putative companion responsible for the misalignment in HD143006 should be located in the gap between B8 and B40, or between B8 and the dust sublimation edge. The spectral energy distribution indicates the presence of hot dust close to the sublimation radius, which was spatially resolved by near-infrared interferometry [@lazareff2017], suggestive of the presence of (at least) another ring of small dust in the innermost au where mm-sized grains are depleted.
Conclusions {#sec:conclusions}
===========
From the DSHARP ALMA observations of the HD143006 protoplanetary disk that reach $\sim7$au in spatial resolution, and its comparison with existing scattered-light observations at similar spatial resolution, we conclude the following:
- In terms of substructure, the dust continuum emission from HD143006 reveals three bright rings, two dark gaps, and an arc at the edge of the dusty disk. The CO observations at similar angular resolution exhibit a depletion of gas emission in the inner disk with no significant features, except for a deviation or “kink” from the Keplerian rotation pattern over a few red-shifted channels close to the systemic velocity.
- From different tracers of the disk structure we find further evidence for a misalignment between the inner and outer disk: a fit to the disk Keplerian velocities with a global/single disk geometry does not account well for the inner disk kinematics, while modeling of dust continuum emission results in a more inclined inner disk (as traced by B8) than the outer disk (as traced by B40, B64, and the south-east arc). These findings are in agreement with existing VLT/SPHERE images that suggest a disk misalignment from the presence of shadows.
- The prominent south-east arc in the ALMA 1.25mm image resolves into three peaks along its azimuthal extent. The counterpart to this arc in the scattered-light image shows a broader radial and azimuthal extent, indicative of segregation by particle size as would be expected for a dust-trapping vortex at this location. Future observations at longer wavelengths should be able to test this scenario.
- The bright rings have increasingly larger radial widths with increasing distance from the star. We find evidence for radial segregation by particle size at the outer edge of the gap between B8 and B40, and a strong depletion of small and large grains in the gap. These are consistent with a companion carving the gap. However, no dark annulus is observed in CO emission at this radius, suggesting that dust grains are filtered but gas can still flow through the gap.
Future observations at longer millimeter wavelengths will allow us to determine if there is efficient trapping of dust at the substructures location, if these are due to localized pressure maxima, and in particular to understand if the arc traces a vortex. Observing gas tracers at high spectral resolution will also be fundamental to elucidate the absolute misalignment of the inner disk and to determine the velocity structure at the arc location.
We are thankful to S. Facchini, A. Juhász and R. Teague for insightful discussions. This paper makes use of ALMA data and ). ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. L.P. acknowledges support from CONICYT project Basal AFB-170002 and from FCFM/U. de Chile Fondo de Instalación Académica. M.B. acknowledges funding from ANR of France under contract number ANR-16-CE31-0013 (Planet Forming disks). S. A. and J. H. acknowledge funding support from NASA Exoplanets Research Program grant 80NSSC18K0438. T.B. acknowledges funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 714769. C.P.D. acknowledges support by the German Science Foundation (DFG) Research Unit FOR 2634, grants DU 414/22-1 and DU 414/23-1. V.V.G. and J.C acknowledge support from the National Aeronautics and Space Administration under grant No. 15XRP15\_20140 issued through the Exoplanets Research Program. J.H. acknowledges support from the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1144152. A.I. acknowledges support from the National Aeronautics and Space Administration under grant No. NNX15AB06G issued through the Origins of Solar Systems program, and from the National Science Foundation under grant No. AST-1715719. L. R. acknowledges support from the ngVLA Community Studies program, coordinated by the National Radio Astronomy Observatory, which is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. Z. Z. and S. Z.acknowledges support from the National Aeronautics and Space Administration through the Astrophysics Theory Program with Grant No. NNX17AK40G and Sloan Research Fellowship. Simulations are carried out with the support from the Texas Advanced Computing Center (TACC) at The University of Texas at Austin through XSEDE grant TG-AST130002.
Additional Figures
==================
Here we present additional supporting figures. First, a zoom-in to the channel maps near the systemic velocity that show a “kink” on the red-shifted channels (bottom panels of Figure \[fig:kink\]) that is not present on the blue-shifted channels (top panels of Figure \[fig:kink\]). Second, Figure \[fig:restrictedmom1\] shows how the Keplerian velocity fit differs when the intensity-weighted velocity field map has been masked inside 0.15$''$ (top panels, Figure \[fig:restrictedmom1\]) and masked outside of $0.15''$ (bottom panels, Figure \[fig:restrictedmom1\]).
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[^1]: We note that the SPHERE infrared (1.2$\mu$m) image has a resolution of $37\times$37mas, comparable to the spatial resolution of the ALMA observations [@benisty2018].
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---
abstract: 'We generalize the textbook Kronig-Penney model to realistic conditions for a quantum-particle moving in the quasi-one-dimensional (quasi-1D) waveguide, where motion in the transverse direction is confined by a harmonic trapping potential. Along the waveguide, the particle scatters on an infinite array of regularized delta potentials. Our starting point is the Lippmann-Schwinger equation, which for quasi-1D geometry can be solved exactly, based on the analytical formula for the quasi-1D Green’s function. We study the properties of eigen-energies as a function of particle quasi-momentum, which form band structure, as in standard Kronig-Penney model. We test our model by comparing it to the numerical calculations for an atom scattering on an infinite chain of ions in quasi-1D geometry. The agreement is fairly good and can be further improved by introducing energy-dependent scattering length in the regularized delta potential. The energy spectrum exhibits the presence of multiple overlapping bands resulting from excitations in the transverse direction. At large lattice constants, our model reduces to standard Kronig-Penney result with one-dimensional coupling constant for quasi-1D scattering, exhibiting confinement-induced resonances. In the opposite limit, when lattice constant becomes comparable to harmonic oscillator length of the transverse potential, we calculate the correction to the quasi-1D coupling constant due to the quantum interference between scatterers. Finally, we calculate the effective mass for the lowest band and show that it becomes negative for large and positive scattering lengths.'
author:
- Marta Sroczyńska
- Tomasz Wasak
- Zbigniew Idziaszek
bibliography:
- 'bibliografia.bib'
title: 'Analytically solvable quasi-one-dimensional Kronig-Penney model'
---
Introduction
============
The standard textbook Kronig-Penney (KP) model [@Kronig1931], introduced in courses on solid states physics [@Kittel2005], is defined in one dimension (1D) and provides probably the most simple picture of a crystalline solid. In this approach electron-ion interactions are replaced by rectangular wells or Dirac $\delta$-potentials, allowing for analytical solution of the problem, and derivation of the energy spectrum exhibiting band structure behavior. Despite its educational values, there are numbers of laboratory systems for which KP model can be applied, at least for qualitative description.
In solid states, periodic one-dimensional structures of atoms can be created using scanning tunneling microscopy technique [@Nilius2002; @Ortega2007; @Oncel2008]. In experiments with ultracold atoms, periodic one-dimensional structures of atoms can be realized in optical lattices in the Mott insulator phase [@Bloch2005]. Using two distinct species of atoms and applying the technique of [*species-selective dipole potential*]{}, i.e. an optical potential experienced exclusively by one species [@LeBlanc2007; @Catani2009], in principle one could realize situation resembling conditions of KP model. In such a realization one species of atoms is tightly confined in the optical lattice potential, creating an effective periodic lattice for the atoms of the second species, which do not feel the optical lattice potential. Hybrid systems of ultracold atoms and ions [@Harter2014; @Tomza2019] provide another realization of KP model, where 1D crystal of ions can be combined with ultracold atomic cloud in quasi-1D geometry. Such a system offers a powerful platform for quantum simulations of a solid state including inherently electron-phonon coupling [@Bissbort2013]. Here, the atom-ion interaction results in even and odd scattered waves in 1D geometry [@Idziaszek2007controlled], which leads to a generalized KP model including two separate coupling constants for even and odd partial waves [@Negretti2014].
While the optical lattices potentials are typically far from the assumption of KP model since the characteristic size of barriers is limited by the wavelength of light, in recent experiment subwavelength optical structures were obtained [@Lacki2016; @Wang2018], making it possible to directly trap the atoms in optical lattices of nearly $\delta$-potentials. The common assumption for all these realizations is that motion of atoms in the transverse direction can be neglected due to the strong confinement, making these systems effectively 1D. Nevertheless, at small distances when atoms approach the scattering center (atom of a second species, ion, or a subwavelength optical barrier) the scattering process takes place in three dimensions, and as such requires inclusion of the transverse motion, which is neglected by definition in the standard KP model.
In our work, we generalize KP model to include the motion in the transverse direction, which allows for a realistic description of the scattering process between the particle and scattering centers (impurities) at the lattice nodes. We consider the motion of an atom in quasi-1D geometry in the presence of periodically spaced impurities, either atoms or ions. The impurities are distributed at equidistant points along $z$ at the center of the trap. They are interacting with the moving atom, thus, forming an external periodic potential acting on that particle. In this work, we assume that the mobility of impurities is negligible. The atom can move freely in the $z$ direction, whereas the motion in the perpendicular directions is confined by the harmonic trapping potential, $V_\perp({{\bf r}_\perp}) = \frac{1}{2}m\omega^2 {{\bf r}_\perp}^2$, where by ${{\bf r}_\perp}=(x,y)$ we denote the coordinates perpendicular to the $z$-axis. In the case of atomic impurities, at ultracold temperatures the atom-atom scattering can be accurately modeled by Fermi zero-range pseudopotential [@Fermi1936; @Huang1987] parametrized by the $s$-wave scattering length. In the case of ionic lattice, we employ directly in the calculations the long-range atom-ion polarization potential characterized by the characteristic range of the potential $R^\ast$ [@Idziaszek2009; @Tomza2019].
Solving the Schrödinger equation either with the Green’s function method or numerically using the finite element method, we determine the wavefunctions and spectrum of stationary states. Inclusion of the transverse degrees of freedom permits us to study the interplay between the transverse confinement length and remaining parameters: the scattering length, the characteristic range of the potential, and a lattice constant, which is beyond the scope of the standard KP model.
In the limit of large impurity separations, one can expect that atom-impurity scattering can be described as a single separate scattering process. In such a case the scattering exhibits confinement-induced resonances and the system properties can be obtained from the usual KP model with 1D coupling constant calculated for quasi-1D geometry derived by Olshanii [@Olshanii1998; @Bergeman2003]. In the opposite limit, when lattice spacing is comparable or smaller to the transverse confinement length one expects that scattering processes on neighboring impurities will interfere, and in such a case quasi-1D KP model cannot be reduced to purely 1D KP model.
The paper is structured as follows. In Section II we discuss physical systems where quasi-1D KP model can be realized and we specify the interaction potentials between atoms and impurities belonging to the lattice. In Section III we demonstrate how to derive the ordinary KP model using the Green’s function method. The same technique is later applied to solve analytically the quasi-1D KP model, obtaining equations determining eigenergies as a function of quasi-momentum. Section V presents details of our numerical calculations performed for the model system containing an atom moving in quasi-1D trap and interacting with the infinite lattice of ions. The analytical results derived in Section IV for regularized delta potentials are compared in Section VI to the numerical calculations for the model hybrid atom-ion system in quasi-1D geometry. We analyze the properties of the energy spectrum, the wave functions, and effective mass on the atom-ion scattering length. Section VII presents conclusions, while four appendices present some technical details on analytical calculations for quasi-1D KP model.
Physical system
===============
In this work we consider motion of an atom in quasi-1D geometry in the presence of a periodic lattice of impurities. The Schrödinger equation for the atom of mass $m$ moving in the presence of external trap and lattice of impurities reads $$\label{schro}
-\frac{\hbar^2 \nabla^2}{2m} \psi({{\bf r}}) + V_\perp({{\bf r}_\perp})\psi({{\bf r}}) + V({{\bf r}}) \psi(x) = E \psi(x),$$ where $V({{\bf r}})$ is the periodic potential, with period $L$ in $z$-direction (taken along unit vector $\mathbf{e}_z$), $$\label{latt}
V({{\bf r}})=V({{\bf r}}+ L \mathbf{e}_z) = \sum_{n=-\infty}^{\infty} V_{\mathrm{ai}}({{\bf r}}-{\bf d}_n),$$ given in terms of a sum of individual atom-impurity potentials $V_{ai}({{\bf r}})$. In this notation, the vector $\mathbf{d}_n=(0,0,nL)$ is the position of the $n$-th impurity on the $z$-axis. The transverse confinement potential $V_\perp({{\bf r}_\perp})$ is simply $$V_\perp({{\bf r}_\perp}) = \frac{1}{2}m\omega^2 {{\bf r}_\perp}^2$$ with ${{\bf r}_\perp}=(x,y)$.
Potential for atom-atom scattering
----------------------------------
In the case of atomic impurities in the ultracold regime, the scattering takes place dominantly in $s$-wave, and the atom-impurity collisions can be accurately modeled by Fermi zero-range pseudopotential [@Fermi1936; @Huang1987], V\_[ai]{}(r) = g ([[**r**]{}]{}) r, \[eqn:Fermi\] where the coupling strength $g = 2\pi \hbar^2 a / m$ is related to the $s$-wave scattering length $a$ and the mass of the atom $m$. Note that the mass of the impurity does not enter the considerations, because of the assumption of the immobility of the impurity; formally it is equivalent to taking the limit of infinite impurity’s mass in which case the reduced mass is equal to $m$.
The accuracy of Fermi pseudopotential can be further improved by introducing an energy-dependent pseudopotential [@Blume2002fermi; @Julienne2002effective] V\_[ai]{}(r;E) = ([[**r**]{}]{}) r, which is defined in terms of an energy-dependent scattering length a(E) = - , \[aeff\] where $\delta_0(k)$ is the $s$-wave phase shift and wave-vector $k$ is related to the kinetic energy $E = \hbar^2 k^2/(2m)$. Such a description extends the validity of $\delta$-pseudopotential to finite kinetic energies and to the finite range of the interaction potential, where characteristic range becomes comparable to the external confinement.
Potential for atom-ion scattering
---------------------------------
The long-range part of the atom-ion interaction potential reads V\_[ai]{}(r) -, \[Vai\] where the constant $C_4$ depends on the electric dipole polarizability $\alpha$ of an atom in the electronic ground state $C_4 = \alpha e^2/(8\pi\epsilon_0)$. Such a potential is valid provided that an atom is in its electronic ground state. Otherwise, if it is in an excited state, it has nonzero quadrupole moment, and the charge-quadrupole dominates the long range part of the atom-ion interaction potential [@Mies1973].
The introduced interaction potential defines the length scale $R^*~=~\sqrt{2mC_4/\hbar^2}$ and energy scale $E^*~=~\hbar^2/(2m(R^*)^2)$. Note that these units are defined in a slightly different way than units usually defined in the context of atom-ion collisions [@Idziaszek2009; @Tomza2019], where a reduced mass of the atom-ion system usually enters in the problem, instead of $m$.
At short distances realistic ion-atom interaction appears to be strongly repulsive and the singular potential has to be regularized. One of the possible ways is to introduce a cut-off radius $b$, so that the potential becomes V\_(r) = -, \[eqn:potRegB\] and the relation between $b$ and scattering length is given by [@szmytkowski] a(b)=R\^\*(). One value of the scattering length can be reproduced by many different values of $b$. The choice of $b$ in a given range determines how many bound states it will give. The potential supports $n$ bound states for $b\in (b_{n-1}, b_{n})$, and $b_n = 1/\sqrt{4n^2-1}$.
In our calculations we will also employ another approach based on the quantum-defect method [@Seaton1983; @Greene1982; @Mies1984], where one introduces additional quantum-defect parameter, which is the phase $\varphi$, characterizing the short-range behavior of the wave function for $R_0 \ll r \ll R^*$ [@Idziaszek2007controlled; @Idziaszek2011; @Gao2010]: (r)\~(+). \[QuantDef\]
1D Kronig-Penney model
======================
Before we start to solve Eq. with Green’s function method, we first demonstrate how this method works in the case of a purely 1D system. We expect to reproduce the well-known solution of the strictly 1D KP model. The 1D situation is simpler to handle mathematically because no regularization of the pseudopotential is required. The Schödinger equation is then $$\label{schro1D}
-\frac{\hbar^2}{2m}\frac{\partial}{\partial z} \psi(z) - E \psi(z) = - V_{1D}(z) \psi(z),$$ where $V_{1D}(z) = \sum_n g_{1D}\delta(z-nL)$ and the summation extends over all integer values. The coupling constant $g_{1D}$ is the given parameter of the problem. This equation can be formally solved by the one-dimensional Green’s function $G_{1D}(z,z|E)$ defined by Eq. . The formal solution is given by $$\label{1dsol}
\psi(z) = \mathcal{C}\psi_0(z) + \int\!\!dz'\, G_{1D}(z,z'|E)V_{1D}(z') \psi(z'),$$ where $\psi_0(z)$ is the solution of Eq. with $V_{1D}\equiv0$. Inserting here the potential, we obtain $$\label{eq1D}
\psi(z) = \mathcal{C}\psi_0(z) + \sum_n g_{1D} G_{1D}(z,d_n|E) \psi(d_n),$$ where $d_n = n L$ is the position of the impurity.
Before we go proceed further, we invoke the Bloch theorem, which states that the wavefunction of the hamiltonian with discrete translational symmetry has to be also an eigenfunction of the translation symmetry operator. It means that $\psi(z) = e^{i q z }u_q(z)$ with periodic $u_q(z+L)=u_q(z)$. Consequently, $\psi(z+L) = e^{i\theta}\psi(z)$ and $\psi(z+n L) = e^{in\theta}\psi(z)$ with the phase $\theta = qL$. We use this property to solve Eq. . To this end, we first set $z=d_{n'}$. Next, we have the freedom to choose the Green’s function; we take $G_{1D}^+$ which describes the outgoing scattered wave. It is straightforward to show, that any linear combination $G_p=p G_{1D}^++(1-p) G_{1D}^-$ leads to the same result provided $0\leqslant p \leqslant 1$ (this condition is necessary for $G_p$ to be a Green’s function).
In the considered case, the Eq. leads to $$\label{1dfin}
e^{i\theta n'} = \mathcal{C}\psi_0(d_{n'}) + \sum_n g_{1D} G_{1D}^+(d_{n'},d_n|E) e^{i\theta n},$$ in which, without loss of generality, we set $\psi(d_n) = e^{i \theta n}$, and the angle $\theta = q L$ satisfies, according to Bloch’s theorem, $-\pi\leqslant \theta \leqslant \pi$. This condition for $\theta$ ensures that $q$ is taken from first Brillouin zone.
Assuming now $E=\hbar^2k^2/2m\geqslant0$ and inserting into Eq. an explicit formula for the Green’s function (Eq. in appendix \[Sec:AppG1D\]), we obtain $$1- \frac{m g_{1D}}{\hbar^2}\frac{1}{i k} \sum_n e^{ik L |n|} e^{i \theta n} = \mathcal{C}\psi_0(d_{n'})e^{-i\theta n'}.$$ The right hand side is a function of $n'$, whereas the left-hand side is independent of $n'$. Consequently, we set $\mathcal{C}=0$ and arrive at $$\label{KP1}
\cos{\theta}= \cos{k L} + \frac{m g_{1D}}{\hbar^2} \frac{\sin{ k L}}{k},$$ which is the well-known relation describing the dispersion relation $E(k)=\hbar^2 k^2/2m$ as a function of quasi-momentum $q=\theta L^{-1}$ in the Kronig-Penney model. An analogous calculation for $E= - \hbar^2 \kappa^2/2m<0$ leads to Eq. with $k$ interchanged by $i\kappa$, which is related to analytic continuation of the wavefunction from $E>0$ to $E<0$ passing by $E=0$ on the physical sheet, i.e., in the upper plane of the complex energy $E$.
Quasi-1D Kronig-Penney model
============================
Having established the technique of the Green’s function to solve one-dimensional KP model, we now proceed to solve the quasi-1D case. In analogy to [Eq. ]{}, the formal solution of [Eq. ]{} with the potential [Eq. ]{} is given by the 3D Green’s function: $$\label{3dsol}
\psi({{\bf r}}) = \int\!\! d^3r'\, G_{3D}({{\bf r}},{{\bf r}}'|E)V({{\bf r}}') \psi({{\bf r}}'),$$ where we dropped the homogenous term. The explicit form of the function $G_{3D}$ is presented in Appendix \[G3Ddef\]. Inserting here the form of the potential we are led to $$\label{3dsolg}
\psi({{\bf r}}) = \sum_{n=-\infty}^{+\infty} G_{3D}({{\bf r}},{\mathbf{d}}_n |E) g \gamma_n ,$$ with $$\label{defgamma}
\gamma_n = \frac{\partial}{\partial r_n}\bigg(r_n \psi({{\bf r}}') \bigg)\bigg|_{{{\bf r}}\to {\mathbf{d}}_n}.$$ This equation for the wavefunction resembles the one given by [Eq. ]{}. However, they are different in one important aspect. Due to the 3D interaction with the impurity, the 3D wavefunction $\psi({{\bf r}})$ when ${{\bf r}}$ is in the neighbourhood of the $n$-th impurity behaves as $\propto 1 - a/|{{\bf r}}- {\mathbf{d}}_n|$ in the ultracold regime where the $s$-wave scattering dominates. Consequently, in our contact potential approximation, the wavefunction has a pole at ${{\bf r}}={\mathbf{d}}_n$ and the value of the waverfunction cannot appear in [Eq. ]{} in the same manner as in [Eq. ]{}. Nevertheless, the derivative in the form of $\gamma_n \propto \partial( r\psi)/\partial r$ is finite at ${{\bf r}}={\mathbf{d}}_n$ and is related to the atom-impurity scattering length $a$.
To go further, we invoke the Bloch theorem. To this end, we note that $\gamma_n$ and $\gamma_{n+1}$ are related to the derivatives of $\psi({{\bf r}})$ calculated around ${\mathbf{d}}_n$ and ${\mathbf{d}}_{n+1}$. Since these points are separated by a lattice constant $L$, they differ by a phase $e^{i \theta}$ with $\theta = q L$. Without loss of generality, we may take $\gamma_n = e^{ i n \theta }$. To find the relation between $\theta $ and $E$ we insert [Eq. ]{} into [Eq. ]{}. The resulting expression includes the terms $G_{3D}({\mathbf{d}}_n,{\mathbf{d}}_{n'}|E)$ with $n\neq n'$, because the regularizing operator acts only when ${{\bf r}}\approx {\mathbf{d}}_{n'} $ in $G_{3D}({{\bf r}},{\mathbf{d}}_{n'}|E)$. Consequently, the derivative in [Eq. ]{} modifies only diagonal terms of the Green’s function leaving the off-diagonal unaffected. The resulting expression takes the following form: $$\label{eq3Da}
\gamma_n = g\bigg( \gamma_n \beta(E) + \sum_{n'\neq n} \gamma_{n'} G_{3D}({\mathbf{d}}_{n'}, {\mathbf{d}}_n |E) \bigg),$$ where $$\label{betadef1}
\beta(E) = \frac{\partial }{\partial r}\bigg( r G_{3D}({{\bf r}},0|E) \bigg)\bigg|_{{{\bf r}}=0}.$$ Notice, that due to the translation symmetry along $z$-axis, the Green’s function $G(x,y,z;y',y',z'|E)$ depends only on the distance $|z-z'|$. Moreover, since all the impurities are residing only on the $z$-axis, we may take ${{\bf r}_\perp}={{\bf r}_\perp}'=0$. Therefore, hereafter by writing $G_{3D}(z-z'|E)$ we refer to $G_{3D}(0,0,z;0,0,z'|E)$. Using this new notation, and exploiting the result $\gamma_n = e^{i n \theta }$, we arrive at $$\label{eq3Db}
1 = g\bigg( \beta(E) + \sum_{n'>0} 2 G_{3D}( n' L |E)\, \cos{n'\theta} \bigg),$$ where now $$\label{betadef2}
\beta(E) = \frac{\partial}{\partial z}\bigg(z G_{3D}(z|E)\bigg)\bigg|_{z=0^+}.$$
To evaluate all the necessary quantities we take the “Feynman Green’s function” $G_{3D}^F = (G_{3D}^+ + G_{3D}^-)/2$. The detailed calculations of $\beta(E)$ and the sum with $n'>0$ are given in Appendices \[Appbeta\] and \[AppLambda\]. The resulting expression for $\beta(E)$ is given by $$\label{beta_final}
\beta(E) = - \frac{1}{\pi a_\perp^2} \frac{2m}{\hbar^2} \frac{a_\perp}{4} \zeta_H\bigg( \frac12, 1 - \frac{E-E_{2n^*}}{2\hbar \omega} \bigg),$$ where we introduced the energy-dependent nearest threshold energy $E_{2n^*} = \hbar\omega(2 n^*(E) + 1)$, and $\zeta_H$ is the Hurwitz zeta function. The energy $E_{2n^*}$ plays in the problem an important role. It is the energy corresponding to the $n^*$-th 2D harmonic oscillator state with $m=0$. Note that $E_{2n^*} $ depends on $E$ and that $E_{2n^*}$ is always below $E$, but increasing $n^*$ by one takes $E$ below this new value, i.e., $E_{2n^*} \leqslant E < E_{2(n^*+1)}$. Alternatively, we may write $n^*(E) = {\lfloor (E-\hbar\omega)/2\hbar\omega \rfloor}$, where ${\lfloor x \rfloor}$ is the floor function.
The sum over $n'$ in [Eq. ]{} is conveniently split into two terms, because they differ in asymptotics if $L \gg a_\perp$. Therefore, we define
\[defLambda\] $$\begin{aligned}
\Lambda(E) &\equiv& \sum_{n>0} 2 G_{3D}( n L |E)\, \cos{n\theta} \\
&=& \frac{2m}{\hbar^2}\frac{L}{\pi a_\perp^2}\bigg(\Lambda_p(E) + \Lambda_e(E)\bigg),
\end{aligned}$$
with $$\label{eqn:LambdaP}
\Lambda_p(E) = \sum_{0 \leqslant n \leqslant n^*} \frac{1}{2k_n L} \frac{\sin{k_n L}}{\cos\theta - \cos{k_n L}},$$ with $k_n a_\perp = 2\sqrt{ (E-\hbar\omega)/2\hbar\omega - n}$ and $$\Lambda_e(E) = \sum_{n \geqslant 1} \frac{1}{k_n L} \mathrm{Re}\bigg[\frac{1}{1 - e^{\tilde{k}_n L + i \theta}} \bigg],
\label{eqn:LambdaE}$$ with $\tilde k_n a_\perp = 2\sqrt{n - (E-E_{2n^*})/2\hbar\omega}$. The equation determining energy as a function of $\theta$ is given by $$\label{final_eq}
1 = \frac{a}{a_\perp} C(E) + \frac{4 a L}{a_\perp^2}(\Lambda_p(E) + \Lambda_e(E)),$$ where $C(E) = - \zeta_H(1/2, 1 - {(E-E_{2n^*})}/{2\hbar \omega})$. This can be further transformed to the following form $$\label{final_eq1}
a_{1D}(E) = - 2 L (\Lambda_p(E) + \Lambda_e(E)),$$ where $a_{1D}(E)$ is the one-dimensional energy-dependent scattering length in quasi-1D system $$a_{1D}(E) = - \frac{a_\perp^2}{2 a}\bigg( 1 - C(E) \frac{a}{a_\perp} \bigg).
\label{eqn:a1DOlshaniiE}$$ It is a generalization of the 1D scattering length to the case of finite energies [@Bergeman2003; @Naidon2007]. In the low-energy limit, $E \to \hbar \omega$, it reduces to the well-known result, $$a_{1D} = - \frac{a_\perp^2}{2 a}\bigg( 1 - C \frac{a}{a_\perp} \bigg).\label{eqn:a1DOlshanii}$$ with $C = C(\hbar\omega)= 1.46035\ldots$. which was first derived in the seminal paper by Olshanii [@Olshanii1998].
The quantities above are so defined that $k_n$ and $\tilde k_n$ are always real and positive. $\Lambda_p(E)$ results from all the physically open channels of the 2D radial harmonic oscillator. For fixed energy $E$ only radial excitations with energies up to $E_{2n^*} \leqslant E$ contribute to this term. If energy $E$ is slightly above $\hbar\omega$ only a single channel is open, and all the others are closed. This is why only periodic functions appear in $\Lambda_p$ (and so the subscript $p$). On the other hand, the term $\Lambda_e(E)$ contains exponential function which is related to the fact that it captures the contribution from the closed channels with radial excitations with energies larger than $E$.
Let us now see how, in the limiting case, we reproduce 1D KP model. To this end, we assume that the energy $E= \hbar \omega + \hbar^2 k^2 /2m$ is slightly above the first threshold $E_0 = \hbar\omega$ but far from the second threshold $E_2 = (2 \times 1 + 1)\hbar \omega = 3\hbar\omega$. This means that the kinetic energy $\hbar^2 k^2 /2m \ll \hbar \omega$, $n^*=0$ and $C(E)$ can be approximated by its low-energy limit $C(E) \approx C$. Next, we notice that $k_0 = k$ and $\Lambda_p(E)$ boils down to a single term only: $$\Lambda_p(E) \approx \frac{1}{2 k L} \frac{\sin k L}{\cos\theta - \cos k L}.$$ Now, the function $\Lambda_e$ in the denominator has exponents of $\tilde k_n L = (2L / a_\perp)\sqrt{n - (k L/2)^2}$. If the momentum $k$ is far from the first threshold, then in the limit $L \gg a_\perp$ the exponent is large. Consequently, $\Lambda_e$ is exponentially small, and can be neglected. In such a case, Eq. reduces to determining the energy spectrum in the 1D Kronig-Penney model with the one-dimensional coupling constant $g_{1D} = -\hbar^2/m a_{1D}$ given in terms of the one-dimensional scattering length $a_{1D}$ . This situation corresponds physically to the scattering on the well-separated impurities, where each of the scattering events is described by the quasi-1D result of Olshanii [@Olshanii1998].
In the limit, when distance between impurities is comparable to the transverse scattering length, or smaller: $L \lesssim a_{\perp}$, the contribution from the term $\Lambda_e$ has to be included. Assuming again that the energy is slightly above the first threshold: $\hbar^2 k^2 /2m \ll \hbar \omega$ and $n^*=0$, we have $\tilde k_n a_\perp \approx 2\sqrt{n}$, which after substituting into gives $$\Lambda_e(\theta) \approx \sum_{n \geqslant 1} \frac{a_\perp}{2 \sqrt{n} L} \mathrm{Re}\bigg[\frac{1}{1 - e^{2 \sqrt{n}L/a_\perp + i \theta}} \bigg],
\label{eqn:LambdaEappr}$$ This can be further simplified to \_e ()-(qL)H(2 ),\[eqn:LambdaEapprWithH\] where H(x) = \_[n=1]{}\^ n\^[-1/2]{}(- x ). \[h\] Within this regime Eq. determining energy levels can be expressed in the following form: $$a_{\mathrm{1D}}^{\mathrm{eff}} =- \frac{1}{k} \frac{\sin k L}{\cos\theta - \cos k L},$$ that is equivalent to Eq. of 1D K-P model, with an effective one-dimensional scattering length $$a_{\mathrm{1D}}^{\mathrm{eff}}(\theta)= a_{\mathrm{1D}}+2 L\Lambda_e (\theta).
\label{eqn:a1Deff}$$ This quantity already incorporates the interference effects from the scattering on closely located impurities. For well-separated impurities ($L \to \infty$), $L\Lambda_e (\theta) \to 0$, and $a_{\mathrm{1D}}^{\mathrm{eff}}(\theta)$ reduces to one-dimensional scattering length $a_{\mathrm{1D}}$ derived for a single impurity in quasi-1D geometry. We note that this quantity depends on the quasi-momentum $q$, while we neglected the dependence of $a_{1D}$ on $k$.
Atom moving in ionic lattice
============================
In this section we consider an atom confined in quasi-1D geometry and moving in the potential of a periodic infinite chain of ions. Exploiting the fact that the system is axially symmetric and periodic, according to the Bloch theorem, we can write the solution to the Schrödinger Eq. in cylindrical coordinates $\rho$ and $z$ (**r**) = \[eqn:PsiFk\]. Substituting into the Schrödinger equation with the periodic potential leads to the following equation for $u_q$
-(q\^2 + 2iq + + ) u\_q(, z)+\
+ (m\^2\^2 - )u\_q(, z)+\
-\_[n=-]{}\^V\_(r\_n)u\_q(, z) = E u\_q(, z). \[eqn:SchroedU\]
Numerical calculations
----------------------
In order to find $u_q$ for a given $q$, we solve numerically Eq. on a rectangle of size $L$ along $z$ coordinate and $\rho_{\mathrm{max}}$ along $\rho$ coordinate (see Fig. \[fig:mesh\]). We assume periodic boundary conditions in $z$: $u_q(\rho, -L/2) = u_q(\rho, L/2)$, Dirichlet boundary condition in $\rho_{\mathrm{max}}$: $u_q(\rho_{\mathrm{max}}, z) = 0$ and von Neumann boundary condition at $\rho=0$: $\frac{\partial}{\partial\rho} \frac{u_q(\rho, z)}{\sqrt{\rho}} =
0$. The ion is placed at $z=0$ and $\rho=0$, however, the interaction potential $V({{\bf r}})$ contains contributions from the whole ion chain. In the numerical calculations we apply the regularized form of the atom-ion interaction given by . The value of cut-off radius $b$ was adjusted to reproduce a given value of the $s$-wave scattering length. In the numerical calculations we assumed $\rho_{\mathrm{max}}/a_{\perp} = 5$. We also chose the value of the parameter $b$ such that it supports only one bound state as it gives relatively shallow potential, which is more convenient for numerical calculations.
The calculations are done using the finite element method routines built in Wolfram Mathematica program [@Mathematica]. Due to the fact that potential becomes relatively deep in the vicinity of the ion’s position, and the corresponding wave function is quickly oscillating in that region, in our calculations we have used variable grid size. The mesh refinement function was related to the local de Broglie wavelength $\lambda({{\bf r}},E)=2\pi/\sqrt{2m|E -
V_{\mathrm{ai}}(\rho, z)|/\hbar^2}$), by assuming that area of a single cell in the grid fulfils $\Delta \leq \lambda({{\bf r}},E)^2/N^2$. We have tested several values of $N$ parameter, observing that numerical calculations converge for $N \gtrsim 20$, and for the final results we assumed $N = 22$. An example grid is shown in Fig. \[fig:mesh\].
![An example grid used for the finite element method. The grid size is determined by the local de Broglie wavelength and it becomes very dense close to the location of ion at $z=0$ and $\rho=0$[]{data-label="fig:mesh"}](plots/mesh_v2){width="48.00000%"}
Quasi-1D KP model with energy-dependent scattering length
---------------------------------------------------------
The regularized delta pseudopotential is applicable, provided that the size of external trapping potential is much larger than characteristic range: $a_\perp \gg
R^\ast$. However, the validity of the Fermi pseudopotential can be extended to the regime $a_\perp \sim R^\ast$ by replacing scattering length $a$ by the energy-dependent scattering length $a(E)$ defined in Eq. [@Julienne2002effective; @Blume2002fermi]. Consequently, the energy spectrum for the atom moving in quasi-1D geometry and interacting with the infinite chain of ions can be obtained from Eq. with 1D energy-dependent scattering length $$a_{1D}(E) = - \frac{a_\perp^2}{2 a(E)}\bigg( 1 - C(E) \frac{a(E)}{a_\perp} \bigg).
\label{eqn:a1DOlshaniiE1}$$ Depending on the method of regularization at short distances, the energy-dependent scattering length $a(E)$ for the atom-ion interaction can be evaluated either numerically or analytically. For the regularization based on the cut-off radius, see Eq. , the energy dependent scattering length was evaluated numerically by solving corresponding radial Schrödinger equation with Numerov method. In the framework of the quantum-defect theory, the atom-ion interaction is regularized by introducing some short-range quantum-defect parameter, eg. short-range phase, see Eq. , and the solution of the radial Schrödinger equation can be expressed in terms of Mathieu functions of the imaginary argument [@Vogt1954; @Spector1964; @Levy1963; @Idziaszek2006]. In this way, one can find analytical expression for both $a(E)$ and its low-energy expansion [@Idziaszek2011] a(E)a(0) + (R\^\*)\^2 k, \[eqn:aE\_approx\] where $a(0) = \lim_{k\rightarrow0} a(k)$.
Fig. \[fig:aE\] shows the energy-dependent scattering length for $a(0) = \pm R^\ast$. It compares numerical calculations for regularized potential , with the quantum-defect result \[Eq. (26) in Ref. [@Idziaszek2011]\] parametrized by $\varphi$ and the low-energy expansion . In general, $a(E)$ calculated from the regularized potential agrees well with quantum-defect regularization, and the agreement improves with the number of bound states supported by the regularized potential (smaller values of $b$). In that sense, quantum-defect approach represents the limit of $b \to 0$ of regularization . On the other hand, low-energy expansion is valid only for very small energies $E \ll E^\ast$, and it completely fails to predict low-energy behaviour for $a(0) = R^\ast$, where the energy-dependent scattering length exhibits a resonance at $E \approx 2.53E^*$ (for the quantum-defect method). Actually, position of resonances for the regularized potential slightly differ, and occur at $E=3.76E^*$ (in the case of potential supporting one bound state) and at $E=2.63E^*$ (smaller $b$, potential supporting three bound states). Therefore, in the rest of the paper, the energy-dependent scattering length is evaluated using the quantum-defect method.
![Energy-dependent $s$-wave scattering length calculated from Eq. (black dotted line corresponds to the value of $b$ supporting only one bound state, black dashed line corresponds to $b$ supporting three bound states) compared with the quantum-defect regularization parametrized by $\varphi$ (red dotted line) and low-energy expansion Eq. (blue dotted line). The values of $a(0)$, the corresponding parameter $b$ and short-range phase $\varphi$ are: (a) $a(0)/R^*=-1$, $b/R^*=0.299$ (black dotted), $b/R^*=0.135$ (black dashed), $\varphi = \pi/4$, (b) $a(0)/R^*=1$, $b/R^*=0.431$ (black dotted), $b/R^*=0.156$ (black dashed), $\varphi = -\pi/4$. []{data-label="fig:aE"}](plots/aEa "fig:"){width="48.00000%"} ![Energy-dependent $s$-wave scattering length calculated from Eq. (black dotted line corresponds to the value of $b$ supporting only one bound state, black dashed line corresponds to $b$ supporting three bound states) compared with the quantum-defect regularization parametrized by $\varphi$ (red dotted line) and low-energy expansion Eq. (blue dotted line). The values of $a(0)$, the corresponding parameter $b$ and short-range phase $\varphi$ are: (a) $a(0)/R^*=-1$, $b/R^*=0.299$ (black dotted), $b/R^*=0.135$ (black dashed), $\varphi = \pi/4$, (b) $a(0)/R^*=1$, $b/R^*=0.431$ (black dotted), $b/R^*=0.156$ (black dashed), $\varphi = -\pi/4$. []{data-label="fig:aE"}](plots/aEb "fig:"){width="48.00000%"}
Results {#sec:numerical_results}
=======
Energy spectrum {#subsec:spectrum}
---------------
In the standard 1D KP model, energy spectrum exhibits a band structure, consisting of bands of allowed energies and energy gaps, where motion of a quantum particle is forbidden. The size of energy gaps depends on the amplitude of delta interaction potential, and increases for stronger interactions. In this section we analyse how these basic properties change, when one includes the transverse degrees of freedom, and when the particle-lattice node interaction has a finite range. By solving Eq. , we obtain energy spectrum for an atom in quasi-1D geometry moving along infinite lattice of ions. This is compared to the analytical results of a model assuming regularized contact pseudopotentials from Eq. for both constant and energy-dependent scattering length.
Fig. \[fig:spectra\] shows the band structure calculated for different ratios of the distance $L$ between the ions, characteristic distance $R^*$, to the transverse confinement length $a_{\perp}$. The atom-ion scattering length is expressed in units of $R^\ast$ as this quantity represents solely the properties of atom-ion interaction potential, and it should not be scaled with size of external confinement. The range of parameters corresponds to current experiments on ultracold atom-ion systems, where in order to minimize the effects of micromotion light atoms with heavy ions are preferably combined [@CetinaPRL12]. For instance, for $^6$Li atom interacting with the linear chain of equally spaced $^{174}$Yb ions [@Joger2017] with $L=1.1\mu m$ confined in the transverse direction by harmonic trap with $\omega
~=~2~\pi~\times~100$ kHz, one obtains $L/{a_\perp}~=~15$ and $R^*/a_{\perp}~=~0.5$. For the same $L$ and $\omega$ parameters, in the case of Rb-$^{174}$Yb$^+$ system [@Zipkes2010], one gets $L/{a_\perp}~=~5$, $R^*/a_{\perp}~=~5$.
Analysing Fig. \[fig:spectra\] we observe that for energies $E < 3 \hbar \omega$ the energy spectrum resembles the ordinary 1D KP model. This corresponds to the situation with no excitations in the transverse direction, and the only effect of the transverse confinement is the renormalization of the 1D effective scattering length . Above every threshold energy $E_{2n^*} = \hbar\omega(2 n^*(E) + 1)$, there is a new set of eigenstates appearing in the spectrum, with 2D harmonic oscillator state in $n^\ast$-th excited state and $m = 0$ in the transverse direction. Each such a new set generates a new band structure, with the lowest band starting at $E_{2n^*}$ and creating avoided crossings with band structures with smaller $n^\ast$. Fig. \[fig:spectrumL15b\] presents in more details such avoided crosiings in the range of energies close to the second threshold at $3 \hbar \omega$. Another interesting feature is the size of the energy gaps for larger ion separations: $L=5 R^\ast$ and $L=15^\ast$, which for $a=R^\ast$ is relatively large, as in the case of large coupling constants $g_\mathrm{1D}$ in 1D KP model. In the other case $a=-R^\ast$, the energy gaps are relatively small and the spectrum is weakly affected in comparison to the dispersion relation of a free quantum particle in quasi-1D geometry. This behaviour can be attributed to the renormalization of 1D scattering legth in quasi-1D confinement according to Eq. , which for $a=R^\ast$ and $R^\ast = 0.5 a_\perp$ leads to much larger $g_\mathrm{1D}$ than for $a=-R^\ast$ and $R^\ast = 0.5 a_\perp$. In general, the numerically calculated spectrum for relatively large ion separation $L$ and small rations $R^*/a_{\perp}$ are well approximated by models assuming pseudopotential interaction, in particular when on includes the energy-dependent scattering length. The models assuming zero-range delta interactions break down when range of the potential $R^\ast$ becomes comparable to the transverse confinement $a_\perp$, and for small ion separations $L$.
{width="32.00000%"} {width="32.00000%"} {width="32.00000%"}
![Zoom in of the spectrum for $L/{a_\perp}=15$ with different values of $R^*/{a_\perp}$ and scattering length: (a) $R^*/{a_\perp}=0.1$, $a/R^*=1$, (b) $R^*/{a_\perp}=0.1$, $a/R^*=15$.[]{data-label="fig:spectrumL15b"}](plots/spectrum_zoom_together_v2){width="50.00000%"}
Fig. \[fig:bandsA\] shows the dependence of the band structure (calculated for pseudopotential with energy-independent scattering length) on the scattering length. The black lines represent the boundaries of the spectrum: $qL/\pi=0$ (dotted) and $qL/\pi=1$ (dashed). The allowable values of energies for $qL/\pi\in (0,1)$ lie between these two lines and are marked with colors.
In the case of large separation of the ions, bands are well separated and relatively narrow (Fig. \[fig:bandsA\]a). When the distance between the ions decreases, the bands become more wide and start to overlap, which is visible especially in the case of $L/a_\perp=1$ (Fig. \[fig:bandsA\]c). It is worth noting that there are certain points where the bottom and upper boundaries of the band intersect, such as e.g. the second and third band in Fig. \[fig:bandsA\]b for positive scattering length. Another interesting feature is that the bound states ($E<\hbar\omega$) exist not only for positive scattering lengths, but also for $a<0$, similarly to bound states of two atoms confined in the harmonic traps [@Busch1998; @Idziaszek2005].
![Energy bands as a function of the scattering length with (a) $L/a_{\perp}=15$, (b) $L/a_{\perp}=5$, (c) $L/a_{\perp}=1$. The black lines correspond to $qL/\pi=0$ (dotted) and $qL/\pi=1$ (dashed). The bands are colored in two different colors (pink and blue) for better readability.[]{data-label="fig:bandsA"}](plots/bands_together_v2){width="50.00000%"}
Wavefunctions
-------------
Fig. \[fig:wf\] presents plots of the wavefunctions $|u_q(\rho, z)/\sqrt{\rho}|^2$ at $q=0$ and $q=-\pi/L$ for the three lowest states of the spectrum, evaluated numerically for regularized atom-ion potential , for $L/{a_\perp}=15$, $R^*/{a_\perp}=0.1$ and $a(0)=R^*$. Fig. \[fig:wfpoints\] shows the points on the energy spectrum corresponding to the wavefunctions plotted in Fig. \[fig:wf\]. It is worth noting that at the points (a), (c) and (e) the models assuming pseudopotential with constant and energy-dependent scattering length predict identical values. This is due to fact that at $z=L/2$ the wavefunction vanishes (see Fig. \[fig:wf\]a, c and e) exactly at the position, where the impurity is placed. Therefore, the presence of the impurity at this point does not affect the energy of the system, and the wavefunction are given by noninteracting particle states at this particular values of quasi-momenta.
{width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"} {width="32.00000%"}
![Three lowest states of the spectrum for $L/{a_\perp}=15$, $R^*/{a_\perp}=0.1$ and $a/R^*=1$. Black points refer to the parameters of Fig. \[fig:wf\], where we plot the wavefunctions corresponding to each point on the spectrum.[]{data-label="fig:wfpoints"}](plots/wfpoints){width="48.00000%"}
Effective mass
--------------
Effective mass is calculated by fitting the following dependence to the lowest band of the spectrum E\_0(q)=\_b + |q|\^2 + A|q|\^4+B|q|\^6, \[eqn:meff\_fit\] where $\epsilon_b$ corresponds to the bottom of the energy band, and $A$ and $B$ are coefficients introduced in order to improve the quality of the fitting procedure at larger energies. Eq. is expressed in dimensionless units of harmonic oscillator, and $m_{\mathrm{eff}}$ denotes the effective mass. Fig. \[fig:meff\] shows how does this ratio change for different values of the scattering length. Both in the case of contact pseudopotential and regularized atom-ion potential, for large scattering length the effective mass becomes negative. In the case of the regularized atom-ion potential, the energy-dependent scattering length (horizontal axis) is calculated numerically from Eq. at the energy of the lowest band for $q=0$ ($\epsilon_b$ from Eq. ). The results for contact pseudopotential are calculated with constant scattering length.
{width="\textwidth"}
The effective mass calculated within both models agrees best in case of small $R^*/a_{\perp}$ ratio, when the approximation of the atom-ion potential by contact interaction is applicable. The separation between the ions seems not crucial, however, for $L/a_{\perp}=1$ the results differ slightly in the region $|a|/a_{\perp} \sim 1$. Increasing the value of $R^*/a_{\perp}$ results in larger differences between both approaches, especially for positive scattering length $a$. At $a=0$ and in the limit $R^*/a \to 0$ the effective mass by definition is the same as physical mass $m_{\mathrm{eff}}=m$. At finite $R^*$, $m_{\mathrm{eff}}$ can be different than the physical mass even at $a=0$, which can be observed for the case of $R^*/a_{\perp}=0.5$. This is due to the finite range effects of the atom-ion interaction potential.
Effective scattering length for quasi-1D crystal
------------------------------------------------
Fig. \[fig:a1Deff\] presents effective scattering length $a_{\mathrm{1D}}^{\mathrm{eff}}$ for a particle moving in quasi-1D crystal, determined from eq. with $\Lambda_e$ given by Eq. and Eq. , at constant value of the scattering length. Calculation were performed for $a/a_{\perp}=1$ for three different values of $L/a_{\perp}$. As expected, at large impurity separations $a_{\mathrm{1D}}^{\mathrm{eff}}$ does not depend on quasi-momentum, and its value corresponds to 1D scattering length of Olshanii [@Olshanii1998] for a single scattering centre. At smaller impurity separations $L=1.5 a_\perp$, $a_{\mathrm{1D}}^{\mathrm{eff}}$ starts to depend on the quasi-momentum. Finally at $L=a_\perp$, $a_{\mathrm{1D}}^{\mathrm{eff}}$ varies strongly with quasi-momentum, which indicates that scattering events on separate impurities interferes with each other, and one cannot describe the scattering in quasi-1D geometry using result derived for single scattering centre [@Olshanii1998]. We note that for $L/a_{\perp}=1.5$ and $L/a_{\perp}=3$ approximation works relatively well, but for $L/a_{\perp}=1$ one has to sum up the series in order to determine $\Lambda_e(E)$.
![Effective one-dimensional scattering length $a_{1D}^{\mathrm{eff}}$ as defined in Eq. with $\Lambda_e$ given by Eq. (dotted lines) or Eq. (dashed lines) for $a/a_\perp=1$. Colors correspond to different values of $L/a_\perp$: $L/a_\perp=1$ (black lines), $L/a_\perp = 1.5$ (blue lines) and $L/a_\perp = 3$ (red lines). []{data-label="fig:a1Deff"}](plots/a1DeffAll_v4){width="50.00000%"}
Conclusions
===========
In our work we have generalized the Kronig-Penney model to quasi-1D geometry with harmonic confinement in the transverse direction. Such a system can be realized in experiments either by using tightly confined atoms of another species as affective lattice or using ionic crystal in hybrid atom-ion systems. Assuming regularized delta potential for the interaction, we have derived analytical equation determining energy spectrum as a function of quasi-momentum. In case of ionic crystal we have performed numerical calculations, that in the expected range of parameters agree well with the model assuming pseudopotential interaction.
The main difference with respect to standard 1D KP model is the appearance of multiple overlapping bands, due to additional transverse degrees of freedom. In addition, due to the transverse confinement the effective 1D scattering length exhibits resonances, allowing to reach the strong-coupling regime in KP model, where the energy bands are separated by large gaps, already for atom-impurity scattering length comparable to the transverse confinement size. We have shown that is some regimes of parameters, the effective mass determined at the bottom of the lowest energy band, becomes negative.
In the future studies, we plan to extend our model in order to incorporate the possibility of small oscillation of ions around equilibrium positions. In this approach we plan to include phonon excitations of the ionic crystal, and derive an effective model coupling atomic degrees of freedom with phonon excitations of the lattice. Such a setup can serve as a quantum simulator of the solid-state physics [@Bissbort2013], and using presented results for delta pseudopotential interaction, it should be possible to determine analytically all the parameters of the effective Hamiltonian.
Acknowledgement
===============
This work was supported by the Polish National Science Centre Grant No. 2015/17/B/ST2/00592. The authors would like to thank Krzysztof Jachymski for fruitful discussions.
1D Green’s function $G_{1D}(z,z'|E)$ {#Sec:AppG1D}
====================================
The 1D Green’s functions reads $$\label{G1D}
G_{1D}^\pm(z,z'|E) = \frac{2m}{\hbar^2} \frac{1}{\pm 2 ik }e^{\pm i k |z-z'|},$$ valid for $E>0$ with $k = \sqrt{2 m E/\hbar^2}$. For negative energy the two functions coincide and are exponentially decaying with the distance: $$G_{1D}^\pm(z,z'|E) = \frac{2m}{\hbar^2} \frac{1}{- 2 \kappa}e^{-\kappa |z-z'|},$$ with $\kappa= \sqrt{-2mE/\hbar^2}$ for $E<0$. Notice that it depends only on the distance between the points $z$ and $z'$. It satisfies the following equation $$\label{green_def}
\bigg(-\frac{\hbar^2}{2m} \frac{\partial^2}{\partial z^2} - E \bigg)G_{1D}(z,z'|E) = - \delta(z-z').$$ Formally, the Green’s operator $\hat G(\zeta)$ is defined as the inverse of the operator $(\zeta-\frac{\hbar^2}{2m} \partial^2/\partial z^2)$ for arbitrary $\zeta$ for which the inverse exists. For $E<0$, the Green’s function is defined as $G_{1D}^{+}(z,z'|E) = G_{1D}^{-}(z,z'|E) \equiv {\langle z|} \hat G(E){| z'\rangle}$, whereas $G_{1D}^{\pm}(z,z'|E) = \lim_{\epsilon\to0^+}{\langle z|} \hat G(E\pm i \epsilon){| z'\rangle}$.
3D Green’s function $G_{3D}^{\pm}({{\bf r}},{{\bf r}}'|E)$ {#G3Ddef}
==========================================================
The two 3D Green’s functions $G_{3D}^+$ and $G_{3D}^-$, which describe outgoing and incoming waves at large distances, respectively, for the quasi-1D geometry are given by: $$G_{3D}^{\pm}({{\bf r}},{{\bf r}}'|E) = \sum_{n,m} \phi_{n,m}({{\bf r}_\perp})\phi_{n,m}^*({{\bf r}_\perp}')G_{1D}^{\pm}(z,z'|E-E_n),$$ where the perpendicular variables are denoted by ${{\bf r}_\perp}=(x,y)$, and the summation is taken of the radial quantum number of the 2D harmonic oscillator, $n=0,1,2,\ldots$, and the azimuthal quantum number that, for fixed $n$, takes values $m=-n, -n+2, \ldots, n-2, n$. The eigenenergies of the 2D harmonic oscillator do not depend on the azimuthal quantum number and are given by $E_n = \hbar \omega (n+1)$.
The 2D harmonic oscillator functions are given by $$\begin{aligned}
\phi_{n,m}({{\bf r}_\perp}) &=& \bigg[ \frac{2\alpha^2 p!}{(p+|m|)!} \bigg]^{1/2} e^{-\alpha^2 \rho^2/2} \times\\
&& \times(\alpha \rho)^{|m|} L_{p}^{|m|}(\alpha^2\rho^2)\frac{e^{im\varphi}}{\sqrt{2\pi}},\end{aligned}$$ with harmonic oscillator length ${a_\perp}= \sqrt{\hbar/m\omega}$, $\alpha = 1/{a_\perp}$ and $p = (n-|m|)/2$. Notice that at ${{\bf r}}=0$ the probability density is independent of the radial quantum number, i.e., $|\phi_{n,0}(0)|^2 = 1 / \pi {a_\perp}^2$.
Evaluation of $\beta(E)$ {#Appbeta}
========================
In this appendix, we demonstrate the necessary steps to evaluate the function $\beta(E)$ from Eq. . Using the explicit form of the 3D Green’s function, we can rewrite $$\label{app-beta}
\beta(E) = \frac{\partial}{\partial z}\bigg[ z \sum_{n,m} |\phi_{2n,m}({{\bf r}}_\perp = 0)|^2 G_{1D}^F(z| E - E_{2n})\bigg]\bigg|_{z=0^+},$$ where the sum is over $n=0,1,2,\ldots$ and $m=0$ since $\phi_{\tilde n,m}(0)$ is zero for $|m|>0$ or odd $\tilde n$. For completeness, we note that the Feynman Green’s function is $$\label{GF1D}
G_{1D}^F(z|E) =
\left\{ \begin{array}{ll}
\frac{2m}{\hbar^2} \frac{\sin k|z|}{2k} & \textrm{if $E\geqslant0$,}\\
\frac{2m}{\hbar^2} \frac{e^{- \kappa|z|}}{-2\kappa} & \textrm{if $E<0$,}\\
\end{array} \right.$$ where $k = \sqrt{2m E/\hbar^2}$ for $E\geqslant0$ and $\kappa = \sqrt{2m (-E)/\hbar^2}$ for $E<0$. To proceed, for a fixed $E$, we set $n^*(E)$, so that the following conditions are satisfied: $$\label{nstar}
E_{2n^*} \leqslant E < E_{2n^*+2}.$$ The energy $E$ can thus be written as $E = E_p + E_{2n^*}$, where $E_{2n^*}$ sets the threshold and $E_p$ is the available kinetic energy for the atom, $$E_p = \frac{\hbar^2 p^2}{2m},$$ which satisfies $0 \leqslant E_p < 2\hbar \omega$. Next, in Eq. , we split the sum over $n$ into two contributions: for $n \leqslant n^*$ and $n>n^*$. In the former, the argument of the Green’s function $E-E_{2n}$ is non-negative, and so we use the upper formula from Eq. . In the latter, the sum $E-E_{2n}$ is negative, and so we apply the lower formula from Eq. . We denote the sum with $n \leqslant n^*$ by $\beta_<$ and with $n>n^*$ by $\beta_>$; then $\beta =
\beta_< + \beta_>$. Explicit evaluation gives $$\label{betaM}
\beta_<(E) = \frac{1}{\pi a_\perp^2} \frac{2m}{\hbar^2} \frac{\partial}{\partial z}\bigg[ z \sum_{0 \leqslant n \leqslant n^*} \frac{\sin{p_n z}}{2 p_n} \bigg]\bigg|_{z=0^+},$$ where $z>0$ and $p_n = \sqrt{2m |E- E_{2n}|/\hbar^2}$. The remaining part is given by $$\label{betaP}
\beta_>(E) = \frac{1}{\pi a_\perp^2} \frac{2m}{\hbar^2} \frac{\partial}{\partial z}\bigg[ z \sum_{n > n^*} \frac{e^{-p_n z}}{-2 p_n} \bigg]\bigg|_{z=0^+}.$$ There is always a finite sum in $\beta_<$, and so the derivative can be evaluated explicitly term by term. Due to the condition $z=0^+$, we have $\beta<(E) = 0$, and, therefore, $\beta(E) = \beta_>(E)$.
Now, we rewrite Eq. in the following form: $$\beta(E) = -\frac{a_\perp}{2} \frac{1}{\pi a_\perp^2} \frac{2m}{\hbar^2} \frac{\partial}{\partial x}\bigg[ x \sum_{n > n^*} \frac{e^{ -\alpha_n x}}{\alpha_n}
\bigg]\bigg|_{x=0^+},$$ where we introduced $x = z/a_\perp$ and $\alpha_n = \sqrt{4n - 2 (E-\hbar\omega)/\hbar\omega}$. We can further simplify the sum by shifting the index $n \to n + n^*$. Then, we obtain $$\beta(E) = -\frac{a_\perp}{2} \frac{1}{\pi a_\perp^2} \frac{2m}{\hbar^2} \frac{\partial}{\partial x}\bigg[ x \sum_{n=1}^\infty \frac{e^{ -\tilde\alpha_n x}}{\tilde\alpha_n}
\bigg]\bigg|_{x=0^+},$$ where $\alpha_n = \sqrt{4n - 2 \epsilon}$ and $\epsilon = (E - E_{2n^*})/\hbar\omega$ with $0 \leqslant \epsilon <2$. In this form, the sum can be evaluated explicitly and reads: $$\frac{\partial}{\partial x}\bigg[ x \sum_{n=1}^\infty \frac{e^{ -\tilde\alpha_n x}}{\tilde\alpha_n}
\bigg]\bigg|_{x=0^+} = \frac{1}{2} \zeta_H \bigg(\frac{1}{2}, 1 - \frac{\epsilon}{2}\bigg),$$ where $\zeta_H$ is the Hurwitz zeta function. Finally, we obtain $\beta(E)$ given in Eq. .
Evaluation of $\Lambda(E) = \Lambda_p(E) + \Lambda_e(E)$ {#AppLambda}
========================================================
In this appendix, we demonstrate the evaluation of $\Lambda(E)$ from Eq. , i.e., $$\label{app-lambda}
\Lambda(E) = \sum_{M>0} 2 G_{3D}( M L |E)\, \cos{M\theta},$$ where the sum is over integer positive $M$. The explicit form of the 3D Green’s function reads: $$\begin{aligned}
G_{3D}(ML|E) &=& \frac{1}{\pi a_\perp^2} \bigg[ \sum_{0 \leqslant n\leqslant n^*} G_{1D}^F( M L |E-E_{2n}) + \\
&& + \sum_{n>n^*} G_{1D}^F( M L |E-E_{2n}) \bigg],\end{aligned}$$ where we used the definition of $n^*(E)$, see Eq. . Now, in the first term, $E-E_{2n}$ is non-negative, and so we apply the upper formula from Eq. . In the second term, $E-E_{2n}$ is negative, and here we apply the lower formula from Eq. . In this way, we arrive at $$\begin{aligned}
G_{3D}(ML|E) &=& \frac{1}{\pi a_\perp^2} \frac{2m}{\hbar^2} \bigg[ \sum_{ 0 \leqslant n\leqslant n^*} \frac{\sin( M p_n L )}{2 p_n} + \nonumber\\
&& + \sum_{n>n^*} \frac{e^{-M p_n L }}{-2 p_n} \bigg],
\label{app-g3d}\end{aligned}$$ where $p_n = \sqrt{2m |E-E_{2n}|/\hbar^2}$. Now, we insert the form of the Green’s function from Eq. into Eq. , and write: $$\Lambda(E) = \frac{L}{\pi a_\perp^2} \frac{2m}{\hbar^2}[ \Lambda_p(E) + \Lambda_e(E) ],$$ where
$$\begin{aligned}
\Lambda_p(E) &=& \sum_{M>0 }\sum_{ n=0}^{n^*} \frac{2\cos(M\theta)\sin( M p_n L )}{2 L p_n} \\
\Lambda_e(E) &=& \sum_{M>0} \sum_{n = n^*+1}^\infty \frac{2\cos(M\theta) e^{-M p_n L }}{-2 L p_n}.
\end{aligned}$$
In this expressions, the sums over $M$ can be explicitly evaluated. As a result, we obtain for $\Lambda_p(E)$ and $\Lambda_e(E)$ Eqs. and , respectively.
|
---
author:
- 'Raphaël Lachièze-Rey[^1] and Sergio Vega [^2]'
bibliography:
- 'selfsimilar.bib'
title: Boundary density and Voronoi set estimation for irregular sets
---
**Abstract**
In this paper, we study the inner and outer boundary densities of some sets with self-similar boundary having Minkowski dimension $s>d-1$ in $\mathbb{R}^{d}$. These quantities turn out to be crucial in some problems of set estimation, as we show here for the Voronoi approximation of the set with a random input constituted by $n$ iid points in some larger bounded domain. We prove that some classes of such sets have positive inner and outer boundary density, and therefore satisfy Berry-Esseen bounds in $n^{-s/2d}$ for Kolmogorov distance. The Von Koch flake serves as an example, and a set with Cantor boundary as a counter-example. We also give the almost sure rate of convergence of Hausdorff distance between the set and its approximation.
#### Keywords
Voronoi approximation; Set estimation; Minkowski dimension; Berry-Esseen bounds; self-similar sets
#### MSC 2010 Classification
Primary 60D05, 60F05, 28A80, Secondary 28A78, 49Q15
#### Notations
We designate by $d(.,.)$ the Euclidean distance between points or subsets of $\mathbb{R}^{d}$. The closure, the interior, the topological boundary and the diameter of a set $A\subset\mathbb{R}^{d}$ are designated by ${\textnormal{cl}(A)}$, ${\textnormal{int}(A)}$, $\partial A$, $\text{diam}(A)$ respectively. The open Euclidean ball with center $x$ and radius $r$ in $\mathbb{R}^{d}$ is noted $B(x,r)$.
Given two sets $A,B$, we write $A+B$ for $\{c\in \mathbb{R}^{d}\mid c=a+b, a\in A, b\in B \}$. The Hausdorff distance between $A$ and $B$ is designated by $d_{H}(A,B)$, that is $$d_{H}(A,B)=\inf\{r>0:\,A\subset B+B(0,r),\,B\subset A+B(0,r)\}.$$ ${\textnormal{Vol}}$ is the $d$-dimensional Lebesgue measure and $\kappa_{d}$ is the volume of the Euclidean unit ball. For $s>0,$ $\mathcal{H}^{s}$ is the $s$-dimensional Hausdorff measure on $\mathbb{R}^{d}$.
Throughout the paper, $K\subset \mathbb{R}^{d}$ is a non-empty compact set with positive volume. The letters $c,C$ are reserved to indicate positive constants that depend only on fixed parameters like $K$ or $d$, and which value may change from line to line.
Background {#background .unnumbered}
==========
Set estimation theory is a topic of nonparametric statistics where an unknown set $K$ is estimated, based on partial random information. The random input generally consists in a finite sample $\chi $ of points, either IID variables [@CFR; @Pen07] or a Poisson point process [@HevRei; @JimYuk; @ReiSpoZap]. Based on the information of which of those points belong or not to $K$, one can reconstruct a random approximation $K_{\chi }$ of $K$ and study the asymptotic quality of the approximation. See the recent survey [@KenMol Chap. 11] about related works in nonparametric statistics.
The results generally require the set to be smooth in some sense. In the literature, the set under study is assumed to be convex [@ReiSpoZap; @Sch12], $r$-convex [@CueRod04; @Rod07], to have volume polynomial expansion [@BCCF], positive reach, or a $(d-1)$-rectifiable boundary [@JimYuk]. Another class of regularity assumptions usually needed is that of *sliding ball* or *rolling ball* conditions ([@CFP; @Walth97; @Walth99]). The most common form of this condition is that in every point $x$ of the boundary, there must be a ball touching $x$ and contained either in $K$, in $K^{c}$, or both.
In those works, the random approximation model $K_{\chi }$ can be the union of balls centred in the points of $\chi $ with well tuned radius going to $0$, a level set of the sum of appropriately scaled kernels centred on the random points, or else. Recently, a different model has been used in stochastic geometry, based on the Voronoi tessellation associated with $\chi $. One defines $K_{\chi }$ as the union of all Voronoi cells which centers lie in $K$, assuming that points of $\chi $ fall indifferently inside and outside $K $, as $K$ is unknown. This is equivalent to defining $K_{\chi }$ as the set of points that are closer to $\chi \cap K$ than to $\chi \cap K^{c}$.
This elegant model presents practical advantages in set estimation. For volume estimation the bias and standard deviation rates of the Voronoi approximation seem to be best among all estimators of which the authors are aware of, and hold under almost no assumption on $K$. Regarding shape estimation, Voronoi approximation also consistently estimates $K$ and $\partial K$ in the sense of the Hausdorff distance (Proposition \[consistency-hausdorff\]), and here again convergence rates and necessary assumptions compare favourably to those of other estimators (see Theorem \[theorem-haus\] and the following Remarks).
An heuristic explanation of these features is that the estimator naturally fills in regions inside $K$ where the sample $\chi$ is sparse, without need for convexity-like assumptions on $K$ [@Rod07] or parameter tuning [@BiauCadre09; @CFR; @DevWise].
The reader will find a more formal presentation of Voronoi approximation along with a summary of existing results [@CalChe13; @HevRei; @JimYuk; @ReiSpoZap; @Sch12] in Section \[sec:voronoi\].
Approach and main results {#approach-and-main-results .unnumbered}
=========================
This work was inspired and is closely related to [@LacPec15], in which a central limit theorem and variance asymptotics for ${\textnormal{Vol}}(K_{\chi})$ were obtained for binomial input under very weak assumptions on $K$. Here we slighlty enhance their central limit theorem by showing that ${\textnormal{Vol}}(K_{\chi})$ can be recentered by ${\textnormal{Vol}}(K)$ instead of ${\mathbf{E}}({\textnormal{Vol}}(K_{\chi}))$. Explicitly, for suitable $K$, we have for each $\varepsilon >0$ a constant $C_{\varepsilon >0}$ such that $$\label{eq:berry-esseen-2-intro}
\sup_{t\in \mathbb{R}}\left| {\mathbf{P}}\left( \frac{{\textnormal{Vol}}(K_{\chi _{n}})-{\textnormal{Vol}}(K)}{\sqrt{ {\textnormal{\textbf{Var}}}({\textnormal{Vol}}(K_{\chi _{n}}))}}\geqslant t \right)-{\mathbf{P}}(N\geqslant t) \right| \leqslant C_{\varepsilon }n^{-s/2d}\log(n)^{4-s/d+\varepsilon }$$ where $s$ is the Minkowski dimension of $\partial K$ (see Section \[sec:MinkoContents\]).
We also show that with Poisson input we have the almost sure convergence rates for the Hausdorff distance $$\begin{aligned}
& c \leqslant \liminf\limits_{n\rightarrow +\infty} \frac{d_{H}(K,K_{\chi_{n}})}{(n^{-1} \ln(n))^{1/d}} \leqslant \limsup\limits_{n\rightarrow +\infty} \frac{d_{H}(K,K_{\chi_{n}})}{(n^{-1} \ln(n))^{1/d}} \leqslant C , \label{eq:haus-intro} \\
& c \leqslant \liminf\limits_{n\rightarrow +\infty} \frac{d_{H}(\partial K, \partial K_{\chi_{n}})}{(n^{-1} \ln(n))^{1/d}} \leqslant \limsup\limits_{n\rightarrow +\infty} \frac{d_{H}(\partial K,\partial K_{\chi_{n}})}{(n^{-1} \ln(n))^{1/d}} \leqslant C, \label{eq:haus-intro-2}\end{aligned}$$ thus answering a query raised in [@HevRei] and extending the results obtained in [@CalChe13].
####
The assumptions on $K$ necessary for , and to hold are worth of interest on their own. They are broad enough to allow for irregular $K$, a feature which few estimators possess and is useful in some applications (see [@CFR; @JimYuk], and references therein). Also, they are not specific to Voronoi approximation, and might be crucial for other estimators. They are mainly concerned with the densities of $K$ at radius $r$ in $x$, defined by $$\begin{aligned}
f^{K}_{r}(x)=& \enspace \frac{{\textnormal{Vol}}(K\cap B(x,r))}{{\textnormal{Vol}}(B(x,r))}, \\
f^{K^{c}}_{r}(x)=& \enspace \frac{{\textnormal{Vol}}(K^{c}\cap B(x,r))}{{\textnormal{Vol}}(B(x,r))}.\end{aligned}$$ For ease of notation, we shall simply write $f_{r}$ for $f^{K}_{r}$ and $g_{r}$ for $f_{r}^{K^{c}}$, $K$ being implicit in all of the paper. Boundary densities have already appeared in set estimation theory [@Cue90; @CalChe13; @CFR], where a set $K$ is said to be *standard* whenever $f_{r}\geqslant \varepsilon$ on $K$ for some fixed $\varepsilon > 0$ and all small enough $r$. Here we shall prefer to specify *inner standard* since we are also interested in cases where the inequality $g_{r}>\varepsilon$ holds. In the latter case, $K$ is said to be *outer standard*, and if $K$ is both inner and outer standard $K$ will be said to be *bi-standard*. The condition on $K$ for to hold is essentially bi-standardness, which is a usual assumption in set estimation [@CFR Theorem 1].
The requirement for to hold seems to be new in set estimation theory. [It consists in a positive $\liminf$ [as $r\rightarrow 0$]{} of the quantity $$\frac{1}{{\textnormal{Vol}}(\partial K_{r})}\int_{K^{c}} f_{r} =\frac{1}{{\textnormal{Vol}}(\partial K_{r})} \int_{K}g_{r},$$ where $\partial K_{r}$ is the set of points within distance $r$ from $\partial K$]{}. See Assumption \[roll\] and Proposition \[roll-bis\] for precise statements [and equivalent assertions]{}. The above quantity measures the interpenetration of $K$ and $K^{c}$ along their common boundary, since the greater it is, the more homogenously $K$ and $K^{c}$ are distributed along $\partial K$. This lead us to name the condition of Assumption \[roll\] the *boundary permeability condition*.
####
Study of densities on the boundary is also related with works in geometric measure theory. Points for which $\lim f_{r}(x)$ is $0$ or $1$ are considered resp. as the measure-theoretic exterior and interior of $K$, while other points constitute $\partial ^{*}K$ the *essential boundary* of $K$. Federer [@AFP Th.3.61] proved that if $K$ is a measurable set with finite measure-theoretic perimeter then $f_{r}\rightarrow 1/2$ on most of the essential boundary.
We address here the question of whether comparable results hold if $\partial K$ is an irregular set, with self-similar features. In general, such boundaries have a Hausdorff dimension $s>d-1$ and don’t have finite perimeter. But, because of self-similarity, the densities $f_{r},g_{r}$ should nevertheless have continuous and somehow periodical fluctuations in $r$, and therefore a positive infimum. This is confirmed by Theorem \[theorem1\], which gives, for $K$ with self-similar boundary, a set of conditions under which $f_{r}>\varepsilon $ on the boundary uniformly in $r>0$. It is even proved that a ball with radius $cr$ for some $c>0$ can be rolled inside or outside the boundary, staying within a distance $r$ from the boundary, but not touching it (otherwise self-similar boundaries would be excluded). Theorem \[theorem1\] applies for instance to the Von Koch flake in dimension $2$, which is therefore well-behaved under Voronoi approximation and satisfies , and .
Some sets with self-similar boundary do not fall under the scope of this result, and we also give example of a self-similar set $K_{\text{cantor}}$ with Cantor-like self-similar boundary not satisfying the boundary permeability condition. Simulations we ran suggest that this irregularity of $K_{\text{cantor}}$’s boundary indeed reflects on the behaviour of its Voronoi approximation and prevents the variance of the estimator from satisfying an asymptotic power law like in . This suggests that the boundary permeability condition is indeed significant in set estimation and not merely a contingent constraint due to the methods used to obtain .
Plan {#plan .unnumbered}
====
The plan of the paper is as follows. In Section \[sec:self-similar\], we recall basic facts and definitions about self-similar sets, especially regarding upper and lower Minkowski contents. We then give conditions under which sets with self-similar boundaries are standard. Voronoi approximation is formally introduced in Section \[sec:voronoi\]. We then derive the volume normal approximation for sets with well-behaved boundaries, [as well as]{} Hausdorff distance results. We also develop the counter example $K_{\text{cantor}}$ that satisfies neither the hypotheses of Theorem \[theorem1\] nor the volume approximation variance asymptotics .
Self-similar sets {#sec:self-similar}
=================
Self-similar set theory
-----------------------
This subsection contains a review of some classic results of self-similar set theory. A more precise treatment of the subject and most of the results stated here can be found in [@Fal]. Broadly speaking, a set is self-similar when arbitrarily small copies of the set can be found in the neighbourhood of any of its points. This suggests that a self-similar set should be associated with a family of similitudes.
####
Let $\{\phi_{i}, i \in I\}$ be a finite set of contracting similitudes. Such a set is called an *iterated function system*. Define the following set transformation $$\begin{aligned}
\psi: \mathcal{P}(\mathbb{R}^{d})&\longrightarrow \enspace \mathcal{P}(\mathbb{R}^{d}) \\
E \quad &\longmapsto \enspace \bigcup_{i} \phi_{i}(E).\end{aligned}$$ It is easily seen that $\psi$ is contracting for the Hausdorff metric, which happens to be complete on $\mathcal{K}^{d}$, the class of non-empty compact sets of $\mathbb{R}^{d}$. By a fixed point theorem, there is an unique set $E \in \mathcal{K}^{d}$ satisfying $\psi(E)=E$, which is by definition the self-similar set associated with the $\phi_{i}$.
####
If there is a bounded open set $U$ such as $\psi(U)=\bigcup\phi_{i}(U)\subset U$ with the union disjoint, then necessarily $E \subset {\textnormal{cl}(U)}$ and the $\phi_{i}$ are said to satisfy the *open set condition*. Schief proved in [@Sch94] that we can pick $U$ so that $U \cap E$ is not empty. [This stronger assumption is referred to as the *strong open set condition* in the literature]{}.
####
The similarity dimension of $E$ is the unique $s$ satisfying $$\sum \lambda_{i}^{s}=1$$ where $\lambda_{i}$ is the stretching factor of $\phi_{i}$. When the open set condition holds, this similarity dimension is also the Hausdorff dimension and the Minkowski dimension of $E$. Furthermore, $E$’s upper and lower $s$-dimensional Minkowski contents ([see Subsection \[sec:MinkoContents\]]{}) are finite and positive. This is an easy and probably known result, but since we have not found it explicitly stated and separately proven in the literature, we will do so here in Proposition \[lemma1\]. We will need the following classical lemmae, that we prove for completeness.
\[lemma1\] Let $(U_{i})$ be a collection of disjoint open sets in $\mathbb{R}^{d}$ such that each $U_{i}$ contains a ball of radius $c_{1}r$ and is contained in a ball of radius $c_{2}r$. Then any ball of radius $r$ intersects at most $(1+2c_{2})^{d}c_{1}^{-d}$ of the sets ${\textnormal{cl}(U_{i})}$.
Let $B$ be a ball of center $x$ and radius $r$. If some ${\textnormal{cl}(U_{i})}$ intersects $B$ then ${\textnormal{cl}(U_{i})}$ is contained in the ball $B'$ of center $x$ and radius $r(1+2c_{2})$. If $q$ different ${\textnormal{cl}(U_{i})}$ intersect $B$ then there are $q$ disjoint balls of radius $c_{1}r$ inside $B'$, and by comparing volumes $q\leqslant (1+2c_{2})^{d}c_{1}^{-d}$.
\[lemma2\] Suppose that $E$ and the $\phi_{i}$ satisfy the open set condition with $U$. Then for every $r < 1$ we can find a finite set $\mathcal{A}$ of similarities $\Phi_{k}$ with ratios $\Lambda_{k}$ such that
1. The $\Phi_{k}$ are composites of the $\phi_{i}$.
2. The $\Phi_{k}(E)$ cover $E$.
3. The $\Phi_{k}(U)$ are disjoint.
4. $\displaystyle\sum \Lambda_{k}^{s} =1$ where $s$ is the similarity dimension of $E$.
5. $\min_{i}(\lambda_{i})r \leqslant \Lambda_{k} < r$ for all $k$.
We give an algorithmic proof. Initialise at step $0$ with $\mathcal{A}=\{Id\}$. At step $n$ replace every $\Phi \in \mathcal{A}$ with ratio greater than $r$ by the similarities $\Phi\circ\phi_{i}, i \in I$. Stop when the process becomes stationary, which will happen no later than step $\lceil\ln(r)/\ln({\max(\lambda_{i})})\rceil$.
####
Obviously, point 1 is satisfied. We will prove the next three points by induction. At step $0$, all of $E$ is covered by the $\Phi_{k}(E)$, the $\Phi_{k}(U)$ are disjoint, and the $\Lambda_{k}^{s}$ sum up to $1$. The first property is preserved when $\Phi$ is replaced by the $\Phi\circ\phi_{i}$, since $\Phi(E)=\Phi(\psi(E))=\bigcup \Phi\circ\phi_{i}(E)$. Likewise, the $\Phi\circ\phi_{i}(U)$ are disjoint one from each other because $\Phi$ is one-to-one, and disjoint from the other $\Phi_{k}(U)$ because $\bigcup \Phi\circ\phi_{i}(U)= \Phi(\psi(U)) \subset \Phi(U)$, which yields point 3. For point 4 note that if $\Phi$ has ratio $\Lambda$, then the $\Phi\circ\phi_{i}$ have ratios $\Lambda\lambda_{i}$ and $\Lambda^{s}=\Lambda^{s}\sum\lambda_{i}^{s}=\sum(\Lambda\lambda_{i})^{s}$ so the sum of the $\Lambda_{k}^{s}$ remains unchanged by the substitution. Finally, since $r < 1$, every final set of the process has an ancestor with ratio greater than $r$. This gives the lower bound for point 5; the upper bound comes from the fact that the process ends.
The process in the proof of Lemma \[lemma2\] is often resumed by the formula $$\mathcal{A}=\{\phi_{i_{1}}\circ\phi_{i_{2}}\ldots \circ\phi_{i_{n}} \mid \prod_{k=1}^{n} \lambda_{i_{k}} < r \leqslant \prod_{k=1}^{n-1} \lambda_{i_{k}}\}.$$
Minkowski contents of self-similar sets {#sec:MinkoContents}
---------------------------------------
Recall that the $s$-dimensional lower Minkowski content of a non-empty bounded set $E\subset \mathbb{R}^{d}$ can be defined as $$\liminf_{r>0}\frac{{\textnormal{Vol}}(E+B(0,r))}{r^{d-s}}.$$ Similarly, the $s$-dimensional upper Minkowski content of $E$ is $$\limsup_{r>0}\frac{{\textnormal{Vol}}(E+B(0,r))}{r^{d-s}}.$$ In this paper, when both contents are finite and positive, we will simply say that $E$ has upper and lower Minkowski contents. This leaves no ambiguity on the choice of $s$, since in that case $s$ is necessarily the Minkowski dimension of $E$, i.e $$s= d-\lim\limits_{r\rightarrow 0}\frac{\ln({\textnormal{Vol}}(E+B(0,r)))}{\ln(r)}.$$ We show below that self-similar sets always have upper and lower Minkowski contents. One can find an alternative proof for the lower content in [@Gat00 Paragraph 2.4], it can also be considered a consequence of $\mathcal{H}^{s}(E)>0$, like suggested in [@Mattila].
\[prop1\] Let $E$ be a self-similar set satisfying the open set condition with similarity dimension $s$. Then $E$ has finite positive $s$-dimensional upper and lower Minkowski contents.
As before, let $\phi_{i}$ be the generating similarities of $E$, $\lambda_{i}$ their ratios, $\psi:A\mapsto \bigcup \phi_{i}(A)$ the associated set transformation, and $U$ the open set of the open set condition. Choose any $0 < r < 1$ and define the $\Phi_{k},\Lambda_{k}$ as in Lemma \[lemma2\]. Finally, write $E_{k}=\Phi_{k}(E), U_{k}=\Phi_{k}(U)$.
####
We approximate $E+B(0,r)$ by the sets $E_{k}+B(0,r)$, who are similar to the $E+\Phi_{k}^{-1}(B(0,r))$. By construction $\Phi_{k}^{-1}(B(0,r))$ is a ball with a radius belonging to $[1,(\min_{i}\lambda_{i})^{-1}]$, so that$${\textnormal{Vol}}(B(0,1))\leqslant {\textnormal{Vol}}(E+\Phi_{k}^{-1}(B(0,r))) \leqslant {\textnormal{Vol}}(B(0,\text{diam}(E)+(\min_{i}\lambda_{i})^{-1})),$$ because $E$ is not empty. Applying $\Phi _{k}$ gives $$c\Lambda_{k}^{d}\leqslant {\textnormal{Vol}}(E_{k}+B(0,r)) \leqslant C\Lambda_{k}^{d}.$$
Since $E+B(0,r) \subset \bigcup_{k} E_{k}+B(0,r)$ and $\sum\Lambda _{k}^{s}=1$ we immediately get the upper bound $$\begin{aligned}
{\textnormal{Vol}}(E+B(0,r)) \leqslant &\enspace \sum {\textnormal{Vol}}(E_{k}+B(0,r)) \\
\leqslant &\enspace \sum C\Lambda_{k}^{d} \\
\leqslant &\enspace C\sum \Lambda_{k}^{s}r^{d-s} \\
\leqslant &\enspace Cr^{d-s}.\end{aligned}$$
####
For the lower bound we apply Lemma \[lemma1\] to the disjoint $U_{k}$. Since $U$ is open we can put some ball of radius $c_{1}$ in $U$, and conversely we can put $U$ in some ball of radius $c_{2}$, since $U$ is bounded. This means that each of the $U_{k}$ contains a ball of radius $ r\min_{i}(\lambda_{i})c_{1}\leqslant \Lambda_{k}c_{1}$ and is contained in a ball of radius $rc_{2}\geqslant\Lambda_{k}c_{2}$. So for any $x\in E+B(0,r)$, $B(x,r)$ intersects at most $q$ of the $E_{k}$ (since $E_{k}\subset{\textnormal{cl}(U_{k})}$) with $q$ a positive integer independent of $r$ and $x$. This can be rewritten $\displaystyle\mathbf{1}_{E+B(0,r)}\geqslant \frac{1}{q}\sum\mathbf{1}_{E_{k}+B(0,r)}$. Integrating we get $\displaystyle {\textnormal{Vol}}(E+B(0,r))\geqslant \frac{1}{q} \sum {\textnormal{Vol}}(E_{k}+B(0,r))$ so that
$$\begin{aligned}
{\textnormal{Vol}}(E+B(0,r)) \geqslant &\enspace\frac{1}{q} \sum {\textnormal{Vol}}(E_{k}+B(0,r)) \\
\geqslant &\enspace\frac{c}{q} \sum \Lambda_{k}^d \\
\geqslant &\enspace\frac{c}{q}(\min_{i}\lambda_{i})^{d-s} \sum \Lambda_{k}^{s}r^{d-s} \\
\geqslant &\enspace cr^{d-s}. \\\end{aligned}$$
Boundary regularity {#sec:boundary-density}
-------------------
In order to formulate our result, we introduce the notion of *proper* and *improper* points. A point $x\in \mathbb{R}^{d}$ is proper to $K$ if ${\textnormal{Vol}}(O\cap K) > 0$ for every neighbourhood $O$ of $x$, it is improper to $K$ otherwise. In other words, the set $K^{\text{prop}}$ of proper points of $K$ is the support of the measure ${\textnormal{Vol}}(K\cap \cdot)$. Further use of proper points will be made in Section \[essential2\]. We can already note that $K$ must have no improper points if we want a positive lower bound for the $f_{r}$ on $K$.
Our result holds for self-similar subsets $E$ of $\partial K$ satisfying the following assumption:
\[ass:1\] A [self-similar]{} subset $E$ of $\partial K$ satisfies the strong open set condition with some set $U$ such that $U\cap \partial K\subset E$ and $U \setminus \partial K$ has finitely many connected components.
This assumption can be justified heuristically: if $E$ cuts its neighbourhood into infinitely many connected components, then because of self-similarity it also does so locally, and $K$ and $K^{c}$ are too disconnected to contain the balls mentioned in Theorem \[theorem1\]. Example \[cantor-example\] will show that these concerns are legitimate.
\[theorem1\] Let $K$ be a non-empty compact set with no improper points and ${\textnormal{Vol}}(\partial K)=0$. Let $E$ be a self-similar subset of $\partial K$ for which Assumption \[ass:1\] holds. Then there are constants $\delta, \varepsilon >0$ such that, for all $r< \delta, x\in E$, both $B(x,r)\cap K^{c}$ and $B(x,r)\cap K$ contain a ball of radius $\varepsilon r$.
Let $\phi_{i}$ be the generating similarities of $E$, $\lambda_{i}$ their ratios, $\psi:A\mapsto \bigcup \phi_{i}(A)$ the associated set transformation. [Denote by]{} $V_{j}$ the connected components of $U\setminus E$. Since there are finitely many of them, we can suppose they all contain a ball of radius $\tau>0$. Fix any $0 < r < 1$ and $x \in E$.
####
Lemma \[lemma2\] shows that there is a similarity $\Phi$ with ratio $\Lambda$ such that $\min(\lambda_{i})r \leqslant \Lambda < r$ and $x \in \Phi(E)$. It follows that $\Phi(U) \subset B(x,r)$. We also have $\Phi(U)\cap \partial K = \Phi(U)\cap E = \Phi(U\cap E)$. Indeed, for any point $x'$ of $E$ outside $\Phi(E)$ there is another similarity $\Phi'$ of Lemma \[lemma2\] such that $x'\in\Phi'(E)$ and $\Phi'(U)\cap\Phi(U)=\emptyset$, which implies ${\textnormal{cl}(\Phi'(U))}\cap \Phi(U)=\Phi'({\textnormal{cl}(U)})\cap\Phi(U)=\Phi'(E)\cap \Phi(U)=\emptyset$ so that $x' \notin \Phi(U)$.
####
Consequently, for all $j$, $\Phi(V_{j})$ has no intersection with $\partial K$. So $\Phi(V_{j})\cap {\textnormal{int}(K)}$ and $\Phi(V_{j})\cap K^{c}$ are two disjoint open set sets who cover $\Phi(V_{j})$, and we must have either $\Phi(V_{j}) \subset K$ or $\Phi(V_{j}) \subset K^{c}$.
####
Since there is a point $y$ in $\Phi(U)\cap E$ and $K$ has no improper points, we must have ${\textnormal{Vol}}(K\cap U), {\textnormal{Vol}}(K^{c}\cap U) > 0$. Because ${\textnormal{Vol}}(E)=0$, this can only happen if one of the $\Phi(V_{j})$ is included in $K^{c}$ and another in $K$. Hence $B(x,r)\cap K$, $B(x,r)\cap K^{c}$ each contain a ball of radius $\Lambda\tau$. Since $\Lambda\geqslant \min(\lambda_{i})r$, the conclusion of the theorem holds with $\varepsilon=\min(\lambda_{i})\tau$.
The theorem implies that $K,K^{c}$ have lower density bounds on $E$. More precisely, for appropriate $\delta, \varepsilon > 0$ $$\begin{aligned}
\label{eq:positive-density-boundary}
\forall x\in E, r < \delta, \quad f_{r}(x), g_{r}(x) \geqslant \varepsilon.\end{aligned}$$ This weaker statement is enough for our purposes regarding Voronoi approximation.
####
We show below that the Von Koch flake provides a concrete example of an irregular set satisfying the hypotheses of Theorem \[theorem1\].
\[flake\]
Let $E$ be the self-similar set associated with the direct similarities $\phi_{i}:\mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ sending $S=A_{0}A_{4}$ to $a_{i}=A_{i-1}A_{i}$, for $i=1,2,3,4$, in the configuration of Figure \[VKcurve\]. Such sets $E$ are called Von Koch curves. Looking at the iterates $\psi^{(n)}(S)$ in Figure \[VKiterates\] gives an idea of the general form of the Von Koch curve and of why it is said to be self-similar.
![\[VKcurve\]The generating similitudes of the Von Koch curve. $Z_{2}$ is the center of the similarity $\phi_{2}$.](VKcurve.png)
![\[VKiterates\]The sets $\psi^{(1)}(S),\psi^{(2)}(S),\psi^{(3)}(S)$.](VKiterates.png)
####
Note that the $\psi^{(n)}(S)$ are curves, i.e the images of continuous mappings $\gamma_{n}:[0,1]\rightarrow \mathbb{R}^{2}$. The $\gamma_{n}$ can be chosen to be a Cauchy sequence for the uniform distance between curves in $\mathbb{R}^{2}$. Hence their limit $\gamma$ is also a continuous mapping, $\gamma([0,1])$ is compact and has distance $0$ with $E$ in the Hausdorff metric, so $\gamma([0,1])=E$ which proves that the Van Koch curve is, indeed, a curve. It can also be shown to be a non-intersecting curve (the image of an injective continuous mapping from $[0,1]$ into $\mathbb{R}^{2}$).
####
With a similar reasoning, if we stick three Von Koch curves of same size as in Figure \[VKflake\], we get a closed non-intersecting curve $\mathcal{C}$. Jordan’s curve theorem says $\mathbb{R}^{2}\setminus \mathcal{C}$ has exactly two connected components who both have $\mathcal{C}$ as topological boundary. The closure $K$ of the bounded component is a compact set with no improper points satisfying $\partial K =\mathcal{C}$. $K$ is called a Von Koch flake.
![\[VKflake\] The boundary of the Von Koch flake $K$.](VKflake.png)
####
Now, construct kites $C_{1},C_{2},C_{3}$ on each of the Von Koch curves $E_{1},E_{2},E_{3}$ making $\partial K$ as in Figure \[VKkites\]. It is easy to see that as long as the two equal angles of the lower triangle are flat enough, $C_{i}\cap\partial K=C_{i}\cap E_{i}$. Furthermore, applying Jordan’s curve theorem to the $E_{i}$ and the two upper (resp. lower) edges of the corresponding $C_{i}$ shows that the $C_{i}\setminus E_{i}$ have exactly two connected components. The strong open set condition is also satisfied, so Theorem \[theorem1\] can be applied three times to obtain lower bounds for $f_{r}$ and $g_{r}$ on $\partial K$.
![\[VKkites\] Assumption \[ass:1\] is satisfied with the kite $C$.](VKkites)
Theorem \[theorem1\] is only concerned with the behaviour of $f_{r}$ and $g_{r}$ on $\partial K$, whereas standardness assumption require lower bounds on all of $K$ and $K^{c}$ respectively. The following lemma takes care of this issue.
\[lem:inside\] If for all $r<\delta$ we have $f_{r} \geqslant \varepsilon$ on $\partial K$, then for all $r<\delta$ we have $f_{r} \geqslant 2^{-d}\varepsilon$ on $K$. The same result holds if $f_{r}$ is replaced by $g_{r}$ and $K$ by $K^{c}$.
If $x$ is in $\partial K_{r/2}$ then $B(x,r)$ contains a ball of radius $r/2$ centered on $x'\in\partial K$, so $f_{r}(x)={\textnormal{Vol}}(K\cap B(x,r))\kappa_{d}^{-1}r^{-d} \geqslant {\textnormal{Vol}}(K\cap B(x',r/2))\kappa_{d}^{-1}r^{-d} \geqslant \varepsilon 2^{-d}$. If $x$ is in $K\setminus \partial K_{r/2}$ then the ball $B(x,r/2)$ is contained in $K$ so that $f_{r}(x) \geqslant 2^{-d} \geqslant \varepsilon 2^{-d}$. So in all cases, if $x\in K$ then $f_{r}(x)\geqslant \varepsilon 2^{-d}$. Replacing $K$ by $K^{c}$ gives the result regarding $g_{r}$.
Voronoi approximation {#sec:voronoi}
=====================
In this section, $\chi$ is a locally finite point process [, and $n\geqslant 1$]{}. If $\chi=\chi_{n} =\{X_{1},X_{2},\ldots,X_{n}\}$, where the $X_{i}$ are iid random points uniformly distributed over $[0,1]^{d}$, we speak of binomial input; if $\chi=\chi'_{\lambda }$ is a homogenous Poisson point process of intensity $\lambda >0$ we speak of Poisson input.
Define the Voronoi cell $\upsilon_{\chi}(x)$ of nucleus $x$ with respect to $\chi$ as the closed set of points closer to $x$ than to $\chi$ $$\upsilon_{\chi}(x)=\{y \in \mathbb{R}^{d} : \forall x' \in \chi, d(x,y)\leqslant d(x',y)\}.$$
####
The Voronoi approximation $K_{\chi}$ of $K$ is the closed set of all points which are closer to $K\cap \chi$ than to $K^{c} \cap \chi$. Its name comes from the relation $$K_{\chi}=\bigcup_{x \in \chi \cap K} \upsilon_{\chi}(x).$$
The volume $\varphi (\chi )={\textnormal{Vol}}(K_{\chi })$ first arised in [@KhmTor] as discriminating statistics in the two-sample problem. These authors proved a strong law of large numbers in dimension $1$ for the volume approximation. Explicit rates of convergence in higher dimensions were obtained by Reitzner and Heveling [@HevRei], who proved that if $K$ is convex and compact and ${\chi}={\chi}'_{\lambda}$ then $$\begin{aligned}
{\mathbf{E}}\varphi({\chi})= & \enspace {\textnormal{Vol}}(K), \\
{\textnormal{\textbf{Var}}}(\varphi({\chi}))\leqslant & \enspace C\lambda ^{-1-1/d}S(K), \\
$$ where $S(K)$ is the surface area of $K$, all constants can be made explicit and depend only on $d$. They also studied the quantity $ \varphi _{\text{Per}}(\chi)={\textnormal{Vol}}(K\Delta K_{\chi })$ to estimate the perimeter, after suitable renormalisation. Reitzner, Spodarev and Zaporozhets [@ReiSpoZap] extended these results to sets with finite variational perimeter, and also gave upper bounds for ${\mathbf{E}}|\varphi({\chi}'_{\lambda})^{q}-{\textnormal{Vol}}(K)^{q} | $ for $q\geq 1$. Schulte [@Sch12] obtained a matching lower bound for the variance with convex $K$, i.e. $cS(K)\lambda ^{-1-1/d}\leq {\textnormal{\textbf{Var}}}(\varphi({\chi}))$, and derived the corresponding CLT $$\begin{aligned}
\frac{\varphi({\chi})-{\mathbf{E}}\varphi({\chi})}{\sqrt{{\textnormal{\textbf{Var}}}(\varphi ({\chi})})} \overset{(d)}{\longrightarrow} N.\end{aligned}$$ Very recently, Yukich [@Yuk15] gave quantitative Berry-Esseen bounds for this CLT similar to the ones that are stated here for binomial input.
When dealing with binomial input, which has been less studied than Poisson input, it is necessary to assume that $K\subset (0,1)^{d}$ and redefine $K_{\chi}$ as $$K_{\chi}=\bigcup_{x \in \chi \cap K} \upsilon_{\chi}(x) \cap [0,1]^{d},$$ in order to avoid trivial complications due to possibly infinite cells. Penrose [@Pen07] proved the remarkable fact that for $\chi={\chi}_{n}$ $$\begin{aligned}
{\mathbf{E}}\varphi(\chi) \to & \enspace {\textnormal{Vol}}(K), \\
{\mathbf{E}}(\varphi_{\text{Per}}(\chi)) \to & \enspace 0,\end{aligned}$$ almost surely, with no need for assumptions on $K$’s shape.
####
To further assess the quality of the approximation with binomial input, we must quantify the previous convergence. The unbiasedness of the Poisson case does not occur with binomial input, mainly because of edge effects. Nevertheless those effects seem to decrease exponentially with the distance, like is customary for Voronoi cells. The following result shows that the bias of the estimator $\varphi (\chi _{n})$ decreases geometrically with $n$, therefore it is negligible with respect to the standard deviation, as shown in the following sections. Also, it still holds when $(0,1)^{d}$ is replaced by an arbitrary set $U$ containing $K$ in its interior.
\[th:voronoi-bias\] Assume that $K$ is a compact set with positive volume and let $U$ be an open set containing $K$. Let $\chi _{n}=\{X_{i},1\leqslant i\leqslant n\}$ be iid uniform variables on $U$. Then there is a constant $0<c<1$ depending only on $K$ and $d$ such that for $n\geqslant 1,$ $$\begin{aligned}
\left| {\mathbf{E}}{\textnormal{Vol}}(K_{\chi _{n}})-{\textnormal{Vol}}(K) \right|\leqslant c^{n} .\end{aligned}$$
Let $\chi_{k}=\{X_{i},i\leqslant k\}$. By homogeneity of the problem we can suppose ${\textnormal{Vol}}(U)=1$. The Voronoi approximation $K_{\chi_{n}}$ of $K$ satisfies $$\begin{aligned}
\notag
{\mathbf{E}}({\textnormal{Vol}}(K_{\chi_{n}})) = & \enspace \sum_{i=1}^{n} {\mathbf{E}}(1_{X_{i}\in K}{\textnormal{Vol}}(v_{\chi_{n}}(X_{i})\cap U))\\
\notag= & \enspace n {\mathbf{E}}(1_{X_{n}\in K}{\textnormal{Vol}}(v_{\chi_{n-1}}(X_{n})\cap U)) \\
\label{eq:exp-vol-K-chi-n}= & \enspace n \int_{K} {\mathbf{E}}{\textnormal{Vol}}(v_{\chi_{n-1}}(x)\cap U) dx.\end{aligned}$$ Take $0<r<\frac{1}{2}d(K,U^{c})$. We have for all $x\in K$ $$\begin{aligned}
{\mathbf{E}}({\textnormal{Vol}}(v_{\chi_{n-1}}(x)\cap U))= & \enspace {\mathbf{E}}(\int_{U}1_{y\in v_{\chi_{n-1}}(x)})dy \\
= & \enspace {\mathbf{E}}(\int_{U} 1_{B(y,\|y-x\|)\cap \chi_{n-1}=\emptyset})dy \\
=& \enspace \int_{U} {\mathbf{P}}({B(y,\|y-x\|)\cap \chi_{n-1}=\emptyset})dy \\
=& \enspace \int_{U} (1-{\textnormal{Vol}}(B(y,\|y-x\|)\cap U))^{n-1}dy \\
= & \enspace \int_{B(x,r)} (1-\kappa_{d}\|y-x\|^{d})^{n-1} dy+c_{n}\end{aligned}$$ where $$\begin{aligned}
c_{n}=\int_{U\setminus B(x,r)} (1-{\textnormal{Vol}}(B(y,\|y-x\|)\cap U))^{n-1} dy.\end{aligned}$$ For $y\in U\setminus B(x,r)$, let $B_{y}$ be the ball interiorly tangent to $B(y,\|y-x\|)$ with center on $[x,y]$ [and radius $r$. We have $B_{y}\subset B(y,\|y-x\|)$ by construction]{} and $B_{y}\subset U$ because $B_{y}\subset B(x,2r)$. It follows that $$\begin{aligned}
c_{n}\leqslant \int_{U\setminus B(x,r)}(1-{\textnormal{Vol}}(B_{y}))^{n-1}dy=\int_{U\setminus B(x,r)}(1-\kappa_{d}r^{d})^{n-1}dy\leqslant c_{0}^{n}\end{aligned}$$for some $0<c_{0}<1$, noticing that $\kappa _{d}r^{d} <{\textnormal{Vol}}(U)\leqslant 1$ because $B(x,r)\subset U$. From there, a polar change of coordinates yields $$\begin{aligned}
{\mathbf{E}}({\textnormal{Vol}}(v_{\chi_{n-1}}(x)\cap U)) =&\int_{0}^{r}d\kappa_{d}t^{d-1}(1-\kappa_{d}t^{d})^{n-1} dt + c_{n}\quad \text{(because a $d$-sphere has surface $d\kappa_{d}$)} \\
=& \enspace \left[ -\frac{(1-\kappa_{d} t^{d})^{n}}{n}\right]_{0}^{r} + c_{n} \\
=& \enspace \frac{1}{n} +O(c^{n})\end{aligned}$$ for some $c\in (0,1)$. Reporting in yields the result.
Recalling that the estimator is unbiased if the underlying sample is Poisson in $\mathbb{R}^{d}$, this pleads in favor of Voronoi approximation against other estimators [@DevWise; @Rod07] where the bias is not known and does not seem to be negligeable.
Asymptotic normality {#sec:volume-approx}
---------------------
This subsection is concerned with the results of [@LacPec15], where it is shown that with binomial input, the volume approximation $K_{\chi}$ is asymptotically normal when the number of points tends to $\infty $. Variance asymptotics and upper bounds on the speed of convergence for the Kolmogorov distance are also given.
We begin by stating the boundary regularity condition necessary for these results to hold, which is related to the boundary densities studied in the previous section. As explained in the introduction, it can be seen as a weakened form of the standardness assumption. Define, for all $r > 0$, the boundary neighbourhoods $$\begin{aligned}
\partial K_{r}= & \enspace \partial K + B(0,r), \\
\partial K_{r}^{-}= & \enspace \partial K_{r} \cap K, \\
\partial K_{r}^{+}= & \enspace \partial K_{r} \cap K^{c}.\end{aligned}$$
\[roll\] A set $K$ with no improper points satisfies the boundary permeability condition whenever $$\label{roll1}
\underset{r > 0}{\liminf} \enspace \frac{1}{{\textnormal{Vol}}(\partial K_{r})}\left(\int_{\partial K_{r}^{+}} f_{r}^{2}(x) \mathop{dx} + \int_{\partial K_{r}^{-}} g_{r}^{2}(x)\mathop{dx}\right) > 0.$$
The following proposition gives a more meaningful equivalent for Assumption \[roll\].
\[roll-bis\] Assumption \[roll\] holds if and only if $$\label{roll2}
\liminf\limits_{r > 0} \frac{1}{{\textnormal{Vol}}(\partial K_{r})\kappa_{d}r^{d}}\int_{K\times K^{c}} 1_{||x-y|| \leqslant r} \mathop{dx} \mathop{dy} > 0.$$
Let us begin by establishing the relation between the expression of (\[roll2\]) and $K$’s boundary densities. By Fubini’s theorem $$\begin{aligned}
\int_{K\times K^{c}} \frac{ \mathbf{1}_{||x-y|| \leqslant r}}{\kappa_{d}r^{d}} \mathop{dx} \mathop{dy} = & \enspace \int_{K} \frac{{\textnormal{Vol}}(B(x,r)\cap K^{c})}{{\textnormal{Vol}}(B(x,r))} \mathop{dx} \\
= & \enspace \int_{K^{c}} \frac{{\textnormal{Vol}}(B(x,r)\cap K)}{{\textnormal{Vol}}(B(x,r))} \mathop{dx},\end{aligned}$$ which rewrites simply as $$\int_{K^{c}} f_{r}= \frac{1}{\kappa_{d}r^{d}}\int_{K\times K^{c}} \mathbf{1}_{||x-y|| \leqslant r} \mathop{dx} \mathop{dy} = \int_{K} g_{r},$$ by definition of boundary densities.
####
Consider the function $h_{r}=\mathbf{1}_{K}g_{r}+\mathbf{1}_{K^{c}}f_{r}$. We have $0 \leqslant h_{r} \leqslant 1$ and $h_{r}=0$ outside of $\partial K_{r}$. Applying the Cauchy-Schwarz inequality gives $$\int h_{r}^{2}\leqslant \int h_{r} \leqslant \sqrt{{\textnormal{Vol}}(\partial K_{r})} \sqrt{\int h_{r}^{2}}$$ which rewrites as $$\begin{aligned}
\enspace \frac{1}{{\textnormal{Vol}}(\partial K_{r})}\int_{\partial K^{+}_{r}} f_{r}^{2} + \int_{\partial K^{-}_{r}}g_{r}^{2} \leqslant \enspace \frac{1}{{\textnormal{Vol}}(\partial K_{r})}&\left(\frac{2}{\kappa_{d}r^{d}}\int_{K\times K^{c}} \mathbf{1}_{||x-y|| \leqslant r} \mathop{dx} \mathop{dy}\right) \\
\leqslant &\enspace \sqrt{\frac{1}{{\textnormal{Vol}}(\partial K_{r})}\left( \int_{\partial K_{r}^{+}} f_{r}^{2} + \int_{\partial K_{r}^{-}}g_{r}^{2} \right)},\end{aligned}$$ so that clearly and are equivalent.
If $K$ is bi-standard with constant $\varepsilon$ then (\[roll1\]) holds as well with the left hand being greater than $\varepsilon^{2}$. Hence bi-standarness implies the boundary permeability condition.
\[comparable-sides-boundary\] Note that $${\textnormal{Vol}}(\partial K_{r}^{+}) \geqslant \frac{1}{\kappa_{d}r^{d}}\int_{K\times K^{c}} 1_{||x-y|| \leqslant r} \mathop{dx} \mathop{dy},$$ so that ${\textnormal{Vol}}(\partial K_{r}^{+})\ll {\textnormal{Vol}}(\partial K_{r})$ prevents from being satisfied. Of course, the same reasoning holds with $\partial K_{r}^{-}$ instead. In other words, it is necessary for the boundary permeability condition to be fulfilled that both sides of the boundary have comparable volumes.
####
We reproduce below the result derived in [@LacPec15 Th. 6.1] for Voronoi approximation, modified to measure the distance to the normal of the variable ${\textnormal{Vol}}(K_{\chi _{n}})-{\textnormal{Vol}}(K)$, instead of ${\textnormal{Vol}}(K_{\chi _{n}})-{\mathbf{E}}{\textnormal{Vol}}(K_{\chi _{n}})$ like in the original result. This subtlety is important for dealing with practical applications and obtaining confidence intervals for ${\textnormal{Vol}}(K)$. We deal with Kolmogorov distance, also adapted to confidence intervals, and defined by
$$\begin{aligned}
{d_{\mathcal{K}}}(U,V):=\sup_{t\in \mathbb{R}}\left| {\mathbf{P}}(U\leqslant t)-{\mathbf{P}}(V\leqslant t) \right|,\end{aligned}$$
for any random variables $U,V$.
\[LRP\] Let $K$ be a compact subset of $(0,1)^{d}$. Assume that for some $s < d$ $$\begin{aligned}
\label{eq:minkowski}
0<\liminf_{r>0}r^{s-d} {\textnormal{Vol}}(\partial K_{r})\leqslant \limsup_{r>0}r^{s-d } {\textnormal{Vol}}(\partial K_{r})<\infty,\end{aligned}$$and that $K$ satisfies the boundary permeability condition (Assumption \[roll\]). Then $$\begin{aligned}
\label{eq:var}
0<\liminf_{r>0} \frac{{\textnormal{\textbf{Var}}}({\textnormal{Vol}}(K_{\chi _{n}}))}{n^{-2+s/d}}\leqslant \limsup_{r>0} \frac{{\textnormal{\textbf{Var}}}({\textnormal{Vol}}(K_{\chi _{n}}))}{n^{-2+s/d}}< \infty,\end{aligned}$$ and for all $\varepsilon > 0$ there is $C_{\varepsilon }>0$ such that for all $n\geqslant 1$ $$\begin{aligned}
\label{eq:tcl}
{d_{\mathcal{K}}}\left( \frac{{\textnormal{Vol}}(K_{\chi _{n}})- {\textnormal{Vol}}(K)}{\sqrt{{\textnormal{\textbf{Var}}}\left( {\textnormal{Vol}}(K_{\chi _{n}})\right) }},N\right) \leqslant C_{\varepsilon}n^{-s/2d}\log(n)^{4-s/d+\varepsilon},\end{aligned}$$where $N$ is a standard Gaussian variable.
This result is almost exactly [@LacPec15 Th. 6.1], except that there it is proved that $$\begin{aligned}
\label{eq:tcl-2}
{d_{\mathcal{K}}}\left( \frac{{\textnormal{Vol}}(K_{\chi _{n}})- {\mathbf{E}}{\textnormal{Vol}}(K_{\chi _{n}})}{\sqrt{{\textnormal{\textbf{Var}}}\left( {\textnormal{Vol}}(K_{\chi _{n}})\right) }},N\right) \leqslant C_{\varepsilon} n^{-s/2d}\log(n)^{4-s/d+\varepsilon}.\end{aligned}$$ To have a similar bound involving ${\textnormal{Vol}}(K)$ instead of ${\mathbf{E}}\varphi ({\textnormal{Vol}}(K_{\chi _{n}}))$, let us first remark that for $\delta \in \mathbb{R}$, a random variable $U$, and $V=U+\delta $, $${d_{\mathcal{K}}}(V,N)\leqslant {d_{\mathcal{K}}}(V,N+\delta )+{d_{\mathcal{K}}}(N+\delta ,N)
\leqslant {d_{\mathcal{K}}}(U,N)+(2\pi )^{-1/2} | \delta |,$$ since ${d_{\mathcal{K}}}(V,N+\delta)={d_{\mathcal{K}}}(U,N)$. It follows that $$\begin{aligned}
{d_{\mathcal{K}}}\left( \frac{{\textnormal{Vol}}(K_{\chi _{n}})- {\textnormal{Vol}}(K)}{\sqrt{{\textnormal{\textbf{Var}}}\left( {\textnormal{Vol}}(K_{\chi _{n}}) \right)}},N \right) &\leqslant {d_{\mathcal{K}}}\left( \frac{{\textnormal{Vol}}(K_{\chi _{n}})- {\mathbf{E}}{\textnormal{Vol}}(K_{\chi _{n}})}{\sqrt{{\textnormal{\textbf{Var}}}\left( {\textnormal{Vol}}(K_{\chi _{n}}) \right)}},N \right) \\
&\hspace{3cm}+(2\pi )^{-1/2}\left| \frac{{\mathbf{E}}{\textnormal{Vol}}(K_{\chi _{n}})-{\textnormal{Vol}}(K)}{\sqrt{{\textnormal{\textbf{Var}}}({\textnormal{Vol}}(K_{\chi _{n}}))}} \right| \\
&\leqslant {d_{\mathcal{K}}}\left( \frac{{\textnormal{Vol}}(K_{\chi _{n}})- {\mathbf{E}}{\textnormal{Vol}}(K_{\chi _{n}})}{\sqrt{{\textnormal{\textbf{Var}}}\left( {\textnormal{Vol}}(K_{\chi _{n}}) \right)}},N \right) +\frac{O(c^{n})}{n^{-1+s/2d }}\end{aligned}$$for some $c\in (0,1)$ by Theorem \[th:voronoi-bias\]. Reporting the bounds of yields .
The fact that $[0,1]^{d}$ is the support of the random sampling variables does not seem to have a great importance. Uniformity over $[0,1]^{d}$ eases certain estimates in the proof of [@LacPec15 Th. 6.1] related to stationarity, but is not essential. If the variables are only assumed to have a positive continuous density $\kappa (x)>0$ on an open neighborhood of $\partial K$, it should be enough for similar results to hold. See Theorem \[th:voronoi-bias\], or [@Pen07], for rigourous results in this direction.
If $K$ satisfies all the hypotheses of Theorem \[LRP\] except the boundary permeability condition, then we have $$\begin{aligned}
\sup_{t\in \mathbb{R}}&\left | {\mathbf{P}}\left( \frac{{\textnormal{Vol}}(K_{\chi _{n}})-{\mathbf{E}}{\textnormal{Vol}}(K_{\chi _{n}})}{\sqrt{{\textnormal{\textbf{Var}}}\left( {\textnormal{Vol}}(K_{\chi _{n}}) \right)}} \leqslant t \right)-{\mathbf{P}}(N\leqslant t)\right |\notag\\
\label{eq:tcl-variance}&\hspace{4cm} \leqslant C_{\varepsilon}n^{\varepsilon }(\sigma ^{-2}n^{-2+s/2d}+\sigma ^{-3}n^{-3+s/d}+\sigma ^{-4}n^{-4+s/d})\end{aligned}$$where $\sigma ^{2}$ is the variance of ${\textnormal{Vol}}(K_{\chi _{n}})$. See [@LacPec15 Th. 6.2] for more details.
Set-estimation literature is also concerned with perimeter approximation [@KenMol Sec. 11.2.1]. In the context of Voronoi approximation, the study of the functional ${\textnormal{Vol}}(K_{\chi _{n}}\Delta K)$ has been initiated in [@HevRei; @ReiSpoZap]. Although the result is not formally stated, a bound of the form for this functional is available using the exact same method. One has to work separately to obtain a variance lower bound. Such a result with Poisson input has been derived very recently in the paper of Yukich [@Yuk15].
[Results regarding the volume of the symmetric difference between the set and its approximation can be used to compare Voronoi approximation with another estimator. Indeed, the bound in $n^{-1/d}$ given in [@HevRei] for ${\mathbf{E}}{\textnormal{Vol}}(K_{\chi _{n}}\Delta K)$ is better than the bound in $(nr_{n}^{d})^{-1/2}$ of [@BCP08], who use the Devroye-Wise estimator with a smoothing parameter $r_{n} \ll n^{-1/(d+1)}$.]{}
The consequences of Theorems \[theorem1\] and \[LRP\] for sets $K$ with self-similar boundary are immediate, condition (\[eq:minkowski\]) automatically holds by Proposition \[prop1\].
\[corollary1\] Let $K$ be a compact set such that $\partial K$ is a finite union of self-similar sets satisfying Assumption \[ass:1\]. Then (\[eq:var\]) and (\[eq:tcl\]) hold.
This corollary applies to the Von Koch flake with $s=\ln(4)/\ln(3)$ (Example \[flake\]). The conclusions of Theorem \[LRP\] also apply for instance to the Von Koch anti flake, where three Von Koch curves are sticked together like for building the flake, but here the curves are pointing inwards, and not outwards (Figure \[VKantiflake\]). Assumption \[ass:1\] is not satisfied on the whole boundary, but it is within an open ball of $\mathbb{R}^{d}$ intersecting one and only one of the three curves, and having (\[eq:positive-density-boundary\]) on a self-similar $E$ with same Minkowski dimension as $\partial K$ is actually enough for the boundary permeability condition to hold.
![\[VKantiflake\] The Von Koch antiflake](VKantiflake.png)
In Section \[sec:counter-example\] we exhibit an example of a set $K$ such that $\partial K$ is self-similar and $K$ does not satisfy Assumption \[ass:1\]. We run simulations suggesting that (\[eq:var\]) is also false. Our theorem gives a set of sufficient conditions, but other versions should be valid. For instance, the question of whether a compact set $K\subset \mathbb{R}^{2}$ whose boundary is a locally self-similar Jordan curve satisfies the conclusions of the theorem above seems to be of interest.
Convergence for the Hausdorff distance {#sec:hausdorff}
--------------------------------------
In this subsection we will make use of $r$-coverings and $r$-packings. Consider a collection $\mathcal{B}$ of balls having radius $r$ *and centers belonging to some set* $E\subset \mathbb{R}^{d}$. $\mathcal{B}$ is said to be an $r$-packing of $E$ if the balls are disjoint. It is an $r$-covering if the balls cover $E$.
The size of minimal coverings and maximal packings is closely related to the Minkowski dimension of $E$. A necessary and sufficient condition for $E$ to have upper and lower Minkowski contents is that, for all small enough $r$, we can find an $r$-covering of $E$ with less than $Cr^{-s}$ balls, and an $r$-packing of the same set with more than $cr^{-s}$ balls. More related results can be found in [@Mattila].
####
To estimate with precision $r$ the shape [of]{} a set $E$ by a point process $\chi$ it is often necessary to request that every point of $E$ is at distance less than $r$ of $\chi$. In the context of Voronoi approximation, this is made precise by the following lemma. Note that we only require $\chi$ to be dense enough near $\partial A$. This is, as suggested in the introduction, because Voronoi approximation fills in the interior regions of $K$ where points of $\chi$ are scarce.
\[lemma3\] Let $\chi \subset \mathbb{R}^{d}$ be a locally finite non-empty set.
1. If every point $x$ of ${\textnormal{cl}(\partial K_{r}^{+})}$ satisfies $d(x,\chi)< r$ then $K_{\chi} \subset K + B(0,r)$.
2. If every point $x$ of ${\textnormal{cl}(\partial K_{r}^{+})}$ satisfies $d(x,\chi)< r$ and every point $x$ of ${\textnormal{cl}(\partial K_{r}^{-})}$ satisfies $d(x,\chi\cap K)< r$ then $d_{H}(K,K_{\chi})\leqslant r$.
3. If every point $x$ of ${\textnormal{cl}(\partial K_{r}^{+})}$ satisfies $d(x,\chi\cap K^{c})< r$ and every point $x$ of ${\textnormal{cl}(\partial K_{r}^{-})}$ satisfies $d(x,\chi\cap K)< r$ then $d_{H}(\partial K, \partial K_{\chi}) \leqslant r$.
4. If some point $x\in \partial K$ satisfies $d(x,\chi\cap K)\geqslant 3r$ and $d(x,\chi\cap K^{c})\leqslant r$ then $d_{H}(K,K_{\chi})\geqslant r$ and $d_{H}(\partial K,\partial K_{\chi})\geqslant r$ .
We begin with the first point. Suppose $x\in K_{\chi}$ satisfies $d(x,K)\geqslant r$. Then there is a point $c_{x}\in \chi\cap K$ such that $x\in\upsilon_{\chi}(c_{x})$. The segment joining $c_{x}$ and $x$ contains points from $\partial K$ so we can consider $x_{0}$ the point of ${\textnormal{cl}(\partial K_{r}^{+})}$ closest to $x$ on that segment. We have $d(x_{0},\partial K)=r$ and $x_{0} \in K^{c}$ since otherwise there would be another point of ${\textnormal{cl}(\partial K_{r}^{+})}$ closer to $x$. As a consequence $d(x_{0},c_{x}) \geqslant r$. But then by assumption there is a point $y$ of $\chi$ such that $d(x_{0},y)< r$ and $c_{x}$ isn’t the point of $\chi$ closest to $x$, which is a contradiction. Hence $x\in K_{\chi}$ implies $d(x,K)< r$ and $K_{\chi}\subset K+B(0,r)$.
####
Note that in the setting of points 2 and 3 we can apply the previous argument to $K^{c}$ instead of $K$, [the compacity of $K$ not playing any role in the proof. Along with $(K_{\chi})^{c}=(K^{c})_{\chi}$]{} this yields $K^{c}_{\chi } \subset K^{c}+B(0,r)$, which reformulates as $K \setminus {\textnormal{cl}(\partial K_{r}^{-})} \subset K_{\chi}$ by taking complements. Hence in both cases we have the inclusions $$K_{\chi}\subset K+ B(0,r), \enspace K_{\chi}^{c}\subset K^{c}+B(0,r),$$ and their reformulations $$K\setminus {\textnormal{cl}(\partial K_{r}^{-})} \subset K_{\chi}, \enspace K^{c}\setminus {\textnormal{cl}(\partial K_{r}^{+})} \subset K^{c}_{\chi}.$$
####
To prove the second point it is enough to show that $K \subset K_{\chi} + B(0,r)$. Let $x$ be a point of $K$. If $x\in K\setminus {\textnormal{cl}(\partial K_{r}^{-})}$ then $x$ belongs to $K_{\chi}$. And if $x$ is in ${\textnormal{cl}(\partial K_{r}^{-})}$ then there is a point $y$ of $\chi \cap K$ such that $d(x,y)<r$. In all cases $x\in K_{\chi} + B(0,r)$.
####
We move on to point 3. The two inclusions $K^{c}\setminus {\textnormal{cl}(\partial K_{r})} \subset K_{\chi}^{c}$ and $K\setminus {\textnormal{cl}(\partial K_{r})} \subset K_{\chi}$ also show that if $x$ satisfies $d(x,\partial K) > r$, $x$ is interior to either $K_{\chi}$ or $K_{\chi}^{c}$. Hence $\partial K_{\chi} \subset \partial K+{\textnormal{cl}(B(0,r))}$. Conversely, for every point $x$ of $\partial K$ there are points of both $\chi \cap K$ and $\chi \cap K^{c}$ inside $B(x,r)$, so $B(x,r)$ contains a point of $\partial K_{\chi}$. Hence $\partial K \subset \partial K_{\chi}+B(0,r)$ and $d_{H}(\partial K, \partial K_{\chi})\leqslant r$.
####
Lastly, suppose the requirements of point 4 are met. Let $y$ be a point of $\chi \cap B(x,r) \cap K^{c} $. Then all of the points in $B(x,r)$ are closer to $y$ than to the points outside of $B(x,3r)$. Consequently all points $B(x,r)$ must lie in Voronoi cells centered in $K^{c}$, and $x\notin K_{\chi}+B(0,r)$ so that $d_{H}(K,K_{\chi})\geqslant r$. The fact that $B(x,r)\subset K^{c}_{\chi}$ also implies $d(x,\partial K_{\chi}) \geqslant r$ and $d_{H}(\partial K,\partial K_{\chi})\geqslant r$.
#### {#essential2}
Now we apply this lemma to show almost sure convergence of $K_{\chi}$ in the sense of the Hausdorff distance. To formulate such a result, the concept of proper points (beginning of Section \[sec:boundary-density\]) proves to be useful. Improper points are invisible to the Voronoi approximation $K_{\chi}$ of $K$. Though this has no incidence when measuring volumes, it becomes a nuisance when measuring Hausdorff distances.
The set $K^{\text{prop}}$ of points proper to $K$ can be thought of as the complement of the biggest open set $O$ such that ${\textnormal{Vol}}(O\cap K)=0$, from which it follows that $K^{\text{prop}}$ is compact and that $K_{\chi}=K^{\text{prop}}_{\chi}$ a.s.
\[consistency-hausdorff\] $K_{\chi_{n}}\underset{n\rightarrow +\infty}{\longrightarrow}K^{\text{prop}}$ and $\partial K_{\chi_{n}}\underset{n\rightarrow +\infty}{\longrightarrow}\partial K^{\text{prop}}$ almost surely in the sense of the Hausdorff metric for both Poisson and binomial input.
Since $K_{\chi}=(K^{\text{prop}})_{\chi}$ almost surely and $K^{\text{prop}}$ has no improper points, this is equivalent to the fact that $K_{\chi_{n}}\rightarrow K$ and $\partial K_{\chi_{n}}\rightarrow \partial K$ almost surely when $K$ has no improper points. By the Borel-Cantelli lemma it is enough to show that both series $$\sum_{n\geqslant 1}{\mathbf{P}}(d_{H}(K_{\chi_{n}},K) > r), \enspace \sum_{n\geqslant 1}{\mathbf{P}}(d_{H}(\partial K_{\chi_{n}},\partial K) > r)$$ are convergent for any positive $r$.
Consider $r/2$-coverings $\mathcal{B}^{+},\mathcal{B}^{-}$ of ${\textnormal{cl}(\partial K_{r}^{+})}, {\textnormal{cl}(\partial K_{r}^{-})}$ respectively. Since both sets are compact, these coverings can be made with finitely many balls. Set $\mathcal{B}=\mathcal{B}^{+}\cup \mathcal{B}^{-}$ and $$V=\min\left(\min_{B\in\mathcal{B^{-}}}{\textnormal{Vol}}(B\cap K),\min_{B\in\mathcal{B^{+}}}{\textnormal{Vol}}(B\cap K^{c})\right).$$ Because $K$ and $K^{c}$ have no improper points, $V>0$. If every ball of $\mathcal{B}^{+}$ contains a point of $\chi\cap K^{c}$ and every ball of $\mathcal{B}^{-}$ a point of $\chi\cap K$, then the requirements of points 2 and 3 in Lemma \[lemma3\] are met. The probability of this not happening is bounded by $|\mathcal{B}|(1-V)^{n}$ for binomial input and $|\mathcal{B}|e^{-nV}$ for Poisson input. In all cases the series associated with ${\mathbf{P}}(d_{H}(K_{\chi_{n}},K) > r)$ and ${\mathbf{P}}(d_{H}(\partial K_{\chi_{n}},\partial K) > r)$ converge, as required.
A refinement of the method above gives an order of magnitude for $d_{H}(K,K_{\chi})$ with Poisson input, under assumptions on $\partial K, f_{r}$ and $g_{r}$ resembling those of Theorem \[LRP\]. This requires better estimations of the probability of the points of Lemma \[covering-lemma\] being met, which is the purpose of the following lemma.
\[covering-lemma\] Let $A,\chi$ be non-empty sets, and $\mathcal{B}$ a collection of balls centered on $A$ with radii $r$. Write $\mathcal{B}_{\tau}$ [for]{} the collection of balls having same centers as those of $\mathcal{B}$ but radius $\tau r$, and choose $\tau_{1},\tau_{2}>0$ such that $\tau_{1}+\tau_{2}=1$. If $\mathcal{B}_{\tau_{1}}$ is a $\tau_{1} r$-covering of $A$ and every ball of $\mathcal{B}_{\tau_{2}}$ contains a point of $\chi$, then $A \subset \chi + B(0,r)$.
Let $x$ be a point of $A$. By hypothesis, there is a ball of $\mathcal{B}$ with center $c$ such that $d(x,c)< \tau_{1} r$, and also a point $y$ of $\chi$ such that $d(y,c)< \tau_{2}r$. Hence $d(x,y)< r (\tau_{1}+\tau_{2})$ and $d(x,\chi)< r$. So indeed every point of $A$ is at distance less than $r$ of $\chi$.
This handy lemma is meant to give probability estimations of events of the type $A\subset \chi + B(0,r)$, which are useful outside the context of Voronoi approximation. Typically, $\chi$ is chosen to be a random point process, and the covering $\mathcal{B}_{\tau_{1}}$ is chosen deterministically with as few balls as possible, often $C\tau_{1}^{-d}r^{-d}$. Bounding the probability that a ball of $\mathcal{B}_{\tau_{2}}$ does not intersect $\chi$ then gives an upper bound of the form $${\mathbf{P}}(A \nsubseteq \chi + B(0,r))\leqslant |\mathcal{B}|\max_{B\in \mathcal{B}_{\tau_{2}}} {\mathbf{P}}(B\cap \chi = \emptyset).$$ The estimations obtained in such applications are less sensible to the number of balls in $\mathcal{B}$ than to their size. Hence optimal results are obtained when $\tau_{1}$ is small.
For example, the reader may use Lemma \[covering-lemma\] to derive [@CueRod04 Th. 1] and its counterpart for Poisson input, which are concerned with the order of magnitude of $d_{H}(K,K\cap\chi)$ with $\chi$ an homogenous point process. Note that use of Minkowski contents and boundary densities give slighlty better bounds, which turn out to be optimal, see Remark \[rem:points-vs-voronoi\].
\[theorem-haus\] Suppose that $\partial K$ has Minkowski dimension $s > 0$ with upper and lower contents, and that for all $r$ small enough, $$\begin{aligned}
f_{r} \geqslant & \enspace \varepsilon \enspace \textnormal{on $K$}, \\
g_{r}\geqslant & \enspace \varepsilon \enspace \textnormal{on $K^{c}$}. \end{aligned}$$ Then we have $$\begin{aligned}
&{\mathbf{P}}\left( \alpha \leqslant \frac{d_{H}(K,K_{\chi'_{\lambda}})}{(\lambda^{-1} \ln(\lambda))^{1/d}} \leqslant \beta \right) \underset{\lambda \rightarrow \infty}{\longrightarrow} 1 \\
&{\mathbf{P}}\left( \alpha \leqslant \frac{d_{H}(\partial K, \partial K_{\chi'_{\lambda}})}{(\lambda^{-1} \ln(\lambda))^{1/d}} \leqslant \beta \right) \underset{\lambda \rightarrow \infty}{\longrightarrow} 1\end{aligned}$$ where $\chi'_{\lambda}$ is a Poisson point process of intensity $\lambda$ and $\alpha,\beta$ satisfy $\alpha < \alpha_{K}, \beta > \beta_{K}$ with $$\begin{aligned}
\alpha_{K} = & \enspace\frac{1}{3}\left(\frac{s}{d\kappa_{d}(1-\varepsilon)}\right)^{1/d}, \\
\beta_{K} = & \enspace \left(\frac{s}{d\kappa_{d}\varepsilon}\right)^{1/d}.\end{aligned}$$
The approach of the proof is to tune $r$ in Lemma \[lemma3\] in order to have the events of points 3 happen with high probability. We shall only show the assertions regarding $d_{H}(\partial K,\partial K_{\chi})$, since the exact same arguments hold with $d_{H}(K, K_{\chi})$ as well.
We start with the upper bound. For all $\lambda$ let $\Omega_{\lambda}$ be the event where all the requirements from point 3 of Lemma \[lemma3\] are met with $\chi=\chi'_{\lambda}$, $r=r_{\lambda}=\beta(\lambda^{-1}\ln(\lambda))^{1/d}$. Hence $\{d_{H}(\partial K, \partial K_{\chi})> r \} \subset \Omega_{\lambda}^{c}$. We shall show that ${\mathbf{P}}(\Omega_{\lambda}^{c})\rightarrow 0$.
Choose $\tau_{1},\tau_{2}<1$ so that $\tau_{1}+\tau_{2}=1$ and $\tau_{2}\beta > \beta_{K}$. Let $\mathcal{B}^{+}$ be a collection of balls with radius $r$ and centers on ${\textnormal{cl}(\partial K^{+}_{r})}$. As in Lemma \[covering-lemma\], call $\mathcal{B}^{+}_{\tau}$ the collection of balls with same centers as those of $\mathcal{B}^{+}$, but radius $\tau r$. Define $\mathcal{B}^{-},\mathcal{B}^{-}_{\tau}$ similarily and set $\mathcal{B}=\mathcal{B}^{+}\cup \mathcal{B}^{-}$. Note that $\mathcal{B}$ depends on $\lambda$, but $\tau_{1},\tau_{2}$ do not.
We can and do choose $\mathcal{B}^{+},\mathcal{B}^{-}$ so that $\mathcal{B}_{\tau_{1}}^{+},\mathcal{B}_{\tau_{1}}^{-}$ are coverings of ${\textnormal{cl}(\partial K_{r}^{+})}$ and ${\textnormal{cl}(\partial K_{r}^{-})}$ respectively, and $|\mathcal{B}|$ has less than $C\tau_{1}^{-d}r^{-s}=C\tau_{1}^{-d}(\lambda/\ln{\lambda})^{-s/d}$ balls. Indeed, consider $\tau_{1}r/2$-packings of $\partial K_{r}^{+}$ and $\partial K_{r}^{-}$, both optimal in the sense that no ball can be added without losing the packing property. Because of volume issues, the packings have less than $C\tau_{1}^{-d}r^{-s}$ balls, and because of the optimality assumption doubling the radii of the balls gives the desired $\tau_{1}r$-coverings.
The intersection of $K$ with a ball $B \in \mathcal{B}_{\tau_{2}}^{-}$ of center $x$ has volume exactly $\kappa_{d}(\tau_{2}r)^{d}f_{\tau_{2}r}(x)$. Because $f_{r}\geqslant \varepsilon$ for large enough $\lambda$ and $\tau_{2}\beta > \beta_{K}$ it follows that $${\mathbf{P}}(B \cap \chi \cap K = \emptyset) \leqslant
\exp(-\lambda\tau _{2}^{d}\varepsilon\kappa_{d} r^{d})=\lambda^{-s/d-\delta}$$ for some $\delta>0$. The same bound is valid for ${\mathbf{P}}(B\cap\chi\cap K^{c}=\emptyset), B\in \mathcal{B}_{\tau_{2}}^{+}$.
Applying Lemma \[covering-lemma\] twice with $A=\partial K_{r}^{+},\partial K_{r}^{-}$ successively gives $${\mathbf{P}}(\Omega_{\lambda}^{c}) \leqslant |\mathcal{B}|\lambda^{-s/d-\delta} \leqslant C \tau_{1}^{-d}\ln(\lambda)^{s/d}\lambda^{-\delta}$$ so that, since $\tau_{1}$ is fixed, ${\mathbf{P}}(\Omega_{\lambda}^{c})\rightarrow 0$ as desired.
#### {#section-24}
The proof for the lower bound is quite similar. Fix $\delta > 0$, and redefine $\Omega_{\lambda}$ to be the event where the requirements described in point 4 of Lemma \[lemma3\] are met for $\chi=\chi'_{\lambda}$, $r=r_{\lambda}=\alpha(\ln(\lambda)\lambda^{-1})^{1/d}$ with $\alpha < \alpha_{K}$. Again, we shall show ${\mathbf{P}}(\Omega_{\lambda}^{c})\rightarrow 0$. Let $\mathcal{B}=\mathcal{B}_{\lambda}$ be a $3r$-packing of $\partial K$. We can assume $|\mathcal{B}|\geqslant cr^{-s}$.
The probability of there being no points of $K\cap\chi_{\lambda}$ in a ball $B(x,3r)$ of $\mathcal{B}$ and at least one point of $K^{c}\cap\chi$ in $B(x,r)$ for a point $x$ in the boundary is exactly $$\exp(-\lambda \kappa_{d}(1-g_{3r}(x))3^{d}r^{d})\left(1-\exp(-\lambda \kappa _{d} g_{r}(x) r^{d})\right)$$ because $B(x,3r_{\lambda})\cap K^{c}$ and $B(x,r)\cap K$ are disjoint. So we have the following upper bound, for $\lambda$ big enough $${\mathbf{P}}(\Omega_{\lambda}^{c})\leqslant (1-e^{-\lambda \kappa_{d}(1-\varepsilon)3^{d}r^{d}}(1-e^{-\lambda\kappa_{d} r^{d} \varepsilon}))^{|\mathcal{B}|} .$$ We would like the right hand to go to $0$ with $\lambda$. Taking logarithms this is equivalent to $$|\mathcal{B}|\exp(-\lambda \kappa_{d}(1-\varepsilon)3^{d}r^{d})(1-\exp(-\lambda \kappa_{d} r^{d} \varepsilon)) \underset{\lambda\rightarrow+\infty}{\longrightarrow} +\infty.$$ Because $\exp(-\lambda \kappa_{d}(1-\varepsilon)3^{d}r^{d})=\lambda^{\delta-s/d}$ with $\delta > 0$, $\exp(-\lambda \kappa_{d}r^{d}\varepsilon)\rightarrow 0$ and $|\mathcal{B}|\geqslant c(\lambda/\ln(\lambda))^{s/d}$, it is indeed the case.
The proof and the result call for some comments. Most of them are minor variants on the result which were not included in the proof for clarity’s sake.
\[rem:no-need-minko\] It is possible to dispose of the hypothesis that $\partial K$ has Minkowski upper and lower contents, by using instead the so-called upper and lower Minkowski dimension, which always exist, see [@Mattila]. In particular, we can always do the coverings in the proof with $Cr^{-d}$ balls, so the upper bound still holds after replacing $s$ by $d$ in the expression of $\beta_{K}$. This compares with the result given by Calka and Chenavier in [@CalChe13 Corollary 2]. One can also show, using the fact that $K$ is bounded and has positive volume, that $\mathcal{H}^{d-1}(\partial K)> 0$ so that $s$ can be replaced by $d-1$ in the expression of $\alpha_{K}$. Hence a lower bound also holds with no assumption on $\partial K$’s geometry when $d\geqslant 2$.
[For the results concerned with $d_{H}(\partial K, \partial K_{\chi})$, this is a remarkable feature that to our knowledge no other estimators possess. For instance, in [@CueRod04] a so-called expandability condition is required to obtain similar rates with the Devroye-Wise estimator.]{}
If $s = 0$ and $\partial K$ has Minkowski contents then actually $d=1$, $\partial K$ has a finite number of points, and $d_{H}(K,K_{\chi})$ has order $\lambda^{-1}$ in the sense that for $\lambda$ large enough $${\mathbf{P}}(d_{H}(K,K_{\chi'_{\lambda}})\lambda > t)\leqslant 2|\partial K|\exp(-2\varepsilon t),$$ which is enough to guarantee the existence of moments of all orders for $d_{H}(K,K_{\chi})\lambda$. This is not true of other shape estimators, and is due to the fact that Voronoi approximation only requires $\chi$ to be dense near $\partial K$ and not on all of $K$. If we don’t have Minkowski contents the situation might be more delicate.
Better estimations of the ${\mathbf{P}}(\Omega_{\lambda}^{c})$ in the proof along with an application of the Borel-Cantelli lemma yield the almost sure convergence rates advertised in the introduction. Explicitly $$\alpha_{K} \leqslant \liminf\limits_{n\rightarrow +\infty} \frac{d_{H}(K,K_{\chi'_{n}})}{(n^{-1} \ln(n))^{1/d}} \leqslant \limsup\limits_{n\rightarrow +\infty} \frac{d_{H}(K,K_{\chi'_{n}})}{(n^{-1} \ln(n))^{1/d}} \leqslant \beta'_{K}$$ and similarily for $d_{H}(\partial K, \partial K_{\chi})$, with $\alpha_{K},\beta_{K}$ as in Theorem \[theorem-haus\] and $\beta'_{K}=(\beta_{K}^{d}+(1/\kappa_{d}\varepsilon))^{1/d}$.
For binomial input, some minor changes in the proof give the same upper bound. It can’t be done for the lower bound since we use the fact that $\chi \cap A, \chi\cap B$ are independent when $A$ and $B$ are disjoint and $\chi$ is a Poisson point process.
\[rem:points-vs-voronoi\] Using similar techniques as in the proof above it is possible to show that $$\frac{d_{H}(K,K\cap\chi'_{\lambda})}{(\lambda^{-1} \ln(\lambda))^{1/d}} \overset{{\mathbf{P}}}{\longrightarrow} \left(\frac{2(d-1)}{d\kappa_{d}}\right)^{1/d}$$ if $K$ has no improper points and $\partial K$ is a $\mathcal{C}^{2}$ manifold. Theorem \[theorem-haus\] shows that, under the same assumptions, the above limit can be used as an upper bound for $d_{H}(K,K_{\chi_{\lambda}})(\lambda/\ln(\lambda))^{-1/d}$. Hence, as a shape estimator, $K_{\chi}$ is not worse than $\chi \cap K$. It would be interesting to know if it is better in some sense, a question related to the optimality of the bounds in Theorem \[theorem-haus\].
Applying point 2 of Lemma \[lemma3\] instead of point 3 in the proof of the theorem yields a better result for $d_{H}(K,K_{\chi})$. Specifically if $f_{r}\geqslant \varepsilon_{f}$ on $K$ then $${\mathbf{P}}\left(\frac{d_{H}(K,K_{\chi'_{\lambda}})}{(\lambda^{-1} \ln(\lambda))^{1/d}} \leqslant \beta \right) \underset{\lambda \rightarrow \infty}{\longrightarrow} 1$$ whenever $$\beta > \left(\frac{s}{d\kappa_{d}\varepsilon_{f}}\right)^{1/d}.$$ [Together with Remark \[rem:no-need-minko\] this shows that inner standardness is a sufficient assumption to have convergence rates for $d_{H}(K,K_{\chi})$.]{}
A counter-example {#sec:counter-example}
-----------------
Here we construct a set $K_{\text{cantor}}$ with self-similar boundary not satisfying the boundary permeability condition. This example shows that Theorem \[theorem1\] cannot be generalised by dropping Assumption \[ass:1\], even if the conclusion is weakened.
The example $K$ below is uni-dimensional, but a counter-example in dimension $2$ can be obtained by considering $K\times [0,1]$.
\[cantor-example\] Let $E\subset \mathbb{R}$ the self-similar set generated by the similarities $\phi_{1}:x\mapsto x/3$, $\phi_{2}:x\mapsto (2+x)/3$ who satisfy the open set condition with $U=(0,1)$. $E$ is in fact the Cantor set, and can be characterized as the set of points having a ternary expansion with no ones.
#### {#section-25}
$K_{\text{cantor}}$ will be defined as the closure of open intervals of $[0,1]\setminus E$. The trick is to choose few intervals with quickly decreasing length, so that $f_{r}$ is small on most of $K_{\text{cantor}}$’s boundary, but to distribute them well so that $\partial K_{\text{cantor}} =E$.
#### {#section-26}
To every positive integer $n$ associate the sequence $s'^{n}$ of its digits in base $2$ in reverse order and double the terms to get $s^{n}$. For example, since $6$ is $110$ in base $2$, $s^{6}=(0,2,2)$. This defines a bijection between $\mathbb{N}$ and the set of finite sequences of zeroes and twos ending in 2, with the additional property that $s^{n}$ always has length $l_{n}\leqslant n$. Now for all $n$ define $$\begin{aligned}
a_{n}= \enspace &\frac{1}{3^{n+1}}+ \sum_{k\geqslant 1}\frac{s^{n}_{k}}{3^{k}}\\
b_{n}= \enspace &\frac{2}{3^{n+1}}+ \sum_{k\geqslant 1}\frac{s^{n}_{k}}{3^{k}}\\
A_{n}= \enspace &(a_{n},b_{n})\end{aligned}$$ We have the following ternary expansions $$\begin{aligned}
a_{n} &=\enspace 0.s^{n}_{1}s^{n}_{2}...s^{n}_{l_{n}}000...01 \\
&=\enspace 0.s^{n}_{1}s^{n}_{2}...s^{n}_{l_{n}}000...0022222... \\
b_{n} &=\enspace 0.s^{n}_{1}s^{n}_{2}...s^{n}_{l_{n}}000...02 \\\end{aligned}$$ Now, set $K={\textnormal{cl}(\bigcup A_{n})}$. We claim that $K$ has no improper points, $\partial K = E$ and that $K$ does not satisfy the regularity condition of Theorem \[LRP\].
The first assertion is easy to prove. Being segments, the $A_{n}$ have no improper points to themselves, so $\bigcup A_{n} \subset K^{\text{prop}}$ and $K\subset K^{\text{prop}}$ by taking closures.
#### {#section-27}
For the second assertion we need to show that $\partial K = K\setminus \bigcup A_{n}={\textnormal{cl}(\bigcup \{a_{n},b_{n}\})}$. We already have the obvious $\partial K\subset K\setminus \bigcup A_{n}$. Define $$\begin{aligned}
a'_{n}= \enspace &\frac{1}{3^{n+1}} -\frac{2}{3^{l_{n}}} + \sum_{k\geqslant 1}\frac{s^{n}_{k}}{3^{k}}\\
b'_{n}= \enspace &\frac{2}{3^{n+1}} -\frac{2}{3^{l_{n}}} + \sum_{k\geqslant 1}\frac{s^{n}_{k}}{3^{k}}\\
A'_{n}= \enspace &(a'_{n},b'_{n})\end{aligned}$$ Since for all $n, s_{l_{n}}^{n}=2$, the corresponding ternary expansions are $$\begin{aligned}
a'_{n} &=\enspace 0.s^{n}_{1}s^{n}_{2}...s^{n}_{l_{n}-1}000...01 \\
&=\enspace 0.s^{n}_{1}s^{n}_{2}...s^{n}_{l_{n}-1}000...0022222... \\
b'_{n} &=\enspace 0.s^{n}_{1}s^{n}_{2}...s^{n}_{l_{n}-1}000...02 \\\end{aligned}$$ If $x \in A_{i}\cap A'_{j}$ then every ternary expansion of $x$ has the same digits as the finite ternary expansions of $a_{i},a'_{j}$ up to the first 1, which is impossible. So $\bigcup A'_{n}$ is an open set disjoint from $\bigcup A_{n}$ and hence from $K$. Furthermore, $\bigcup A'_{n}$ is dense near the $a_{n}$, because for all $k,N \in \mathbb{N}^{*}$, we can find an $a'_{k'}$ whose ternary expansion has the same $N$ first digits as the non-terminating expansion of $a_{k}$, so that $d(a_{k},a'_{k'}) \leqslant 1/3^{N}$. A similar argument works for the $b_{n}$, so that the $a_{n},b_{n}$ belong to $\partial K$ and, since the latter is closed, ${\textnormal{cl}(\bigcup \{a_{n},b_{n}\})} \subset \partial K$.
Finally, consider a point $x\in K\setminus \bigcup A_{n}$. For all $r > 0$, $B(x,r)$ contains a point from an $A_{k}$, and since $x \notin A_{k}$, one of the two points $a_{k},b_{k}$ must also be in $B(x,r)$. Consequently, $x$ is also an accumulation point of $\bigcup \{a_{n},b_{n}\}$. We just proved that $K\setminus \bigcup A_{n} \subset {\textnormal{cl}(\bigcup \{a_{n},b_{n}\})}$. Putting this together with the previous two inclusions we get the desired equality.
Since for all $x \in E, N\in \mathbb{N}^{*}$ we can find an $a_{k}$ with the same first $N$ digits as $x$ in base 3, the $a_{n}$ are dense in $E$ and $E \subset \partial K$. Conversely, $\partial K \subset E$, since the $a_{n}, b_{n}$ belong to $E$, who is closed.
#### {#section-28}
For the last assertion, pick any $r >0$ and set $N=2\lceil -\log_{3}(r)\rceil$. Let $X$ be the union of the balls of radius $r$ centered on the endpoints of the $N$ first $A_{n}$. $X$ has area at most $-4r\log_{3}(r)$ and for any $x \in \partial K_{r} \setminus X$, $B(x,r)$ does not intersect the $A_{k}, k\leqslant N$. Since ${\textnormal{Vol}}(\partial K)=0$ $${\textnormal{Vol}}(K\setminus (A_{1}\cup A_{2}\ldots \cup A_{N})) = {\textnormal{Vol}}(\bigcup_{n> N}A_{n}) = \frac{1}{2.3^{N+1}} \leqslant r^{2}.$$ [But ${\textnormal{Vol}}(\partial K_{r})$ has order $r^{1-\ln(2)/\ln(3)}$ and $${\textnormal{Vol}}(\partial K_{r}^{-}) \leqslant {\textnormal{Vol}}(X)+ {\textnormal{Vol}}(K\setminus (A_{1}\cup A_{2}\ldots \cup A_{N})) \leqslant -4r\log_{3}(r) + r^{2}$$ so that ${\textnormal{Vol}}(\partial K_{r}^{-}) \ll {\textnormal{Vol}}(\partial K_{r})$. According to Remark \[comparable-sides-boundary\], this prevents from holding.]{}
#### {#section-29}
Simulations were made for the quality of the Voronoi volume approximation with this set $K$. The magnitude order of the empirical variance of ${\textnormal{Vol}}(K_{\chi_{n}})$ seems to be $n^{\tau}$ with $\tau\approx -1.8$, as shown in Figure \[regression\]. Looking at Theorem \[LRP\], the approximation behaves as if the set had a “nice” fractal boundary of dimension $\approx 0.2$, whereas its real fractal dimension is $1-\ln(2)/\ln(3)\approx 0.37$.
![\[regression\]In blue $\ln({\textnormal{\textbf{Var}}}(K_{\chi_{n}}))$ as a function of $\ln(n)$, in red the associated linear regression. For each $n$, the variance was estimated with 1000 realisations of ${\textnormal{Vol}}(K_{\chi_{n}})$.](regression.png)
Simulations also suggest that a central limit theorem still holds. Such a fact indicates that though the results of Lachieze-Rey and Peccati [@LacPec15] seem to be generalisable, the variance of ${\textnormal{Vol}}(K_{\chi_{n}})$ is indeed related to the behaviour of $f_{r}$ and $g_{r}$ near $\partial K$.
It is possible to construct other sets not satisfying the regularity condition of Assumption \[roll\]. If we don’t require $\partial K$ to be a self-similar set, a much simpler example is given by $$K={\textnormal{cl}(\bigcup_{n\in \mathbb{N}^{*}} \left(\frac{1}{n}-\frac{1}{3^{n}},\frac{1}{n}\right))}.$$ Intentionally, $\partial K$ looks like the set $\{n^{-1}, n\in\mathbb{N}^{*}\}$, who is often given as an example of a countable set with positive Minkowski dimension. $K$ has no improper points, its boundary has Minkowski dimension $1/2$ with upper and lower contents, but $K$ does not satisfy (\[roll1\]) or (\[roll2\]). This can be proved using the same methods as in Example \[cantor-example\]. Again, simulations tend to show that the variance of ${\textnormal{Vol}}(K_{\chi_{n}})$ is about $n^{\tau }$ with $\tau \approx -1,8$ and that a central limit theorem still holds.
[^1]: raphael.lachieze-rey@parisdescartes.fr, Laboratoire MAP5 (UMR CNRS 8145), Université Paris Descartes, Sorbonne Paris Cité
[^2]: Laboratoire MAP5 (UMR CNRS 8145), Université Paris Descartes, Sorbonne Paris Cité
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**Ghost spinors, shadow electrons and the Deutsch Multiverse**
Elena V. Palesheva
Department of Mathematics, Omsk State University\
644077 Omsk-77 RUSSIA\
E-mail: m82palesheva@math.omsu.omskreg.ru\
July 28, 2001\
ABSTRACT
In this article a new solution of the Einstein-Dirac’s equations is presented. There are ghost spinors, i.e. the stress-energy tensor is equal to zero and the current of these fields is non-zero vector. Last the ghost neutrino was found. These ghost spinors and shadow particles of Deutsch are identified. And in result the ghost spinors have a physical interpretation and solutions of the field equations for shadow electrons as another shadow particles are found.
Introduction {#introduction .unnumbered}
============
If in General theory of relativity the right parts of the Einstein’s equations without cosmological constant are equal to zero then one speaks about empty space, or the space is emty if matter is absence. But appears a question: if a gravitational field is generated by the matter whence it appears when matter disappears? This answer is simple: this is because required to distinguish such two at first thought interchangeables each other concepts as substance and matter. Matter generates given structure of World. Herewith the matter is not obliged to have any energy that it occurs when the stress-energy tensor equal to zero. Presence of substance is on the contrary characterized by the non-zero stress-energy tensor. The actual example of matter, which in our world shows the object with zero energy and momentum, i.e. in introduced terminology it is not a substance, serves the neutrino’s ghost. About their existance we can be to speak in light of received our results and results of other authors [@2; @3; @4; @5].
The world which surround us holds the ensemble of riddles, and we all time want to understand the nature of space, in which we are living. So and David Deutsch [@1] makes an attempt logically to explain the phenomenon of an interference of quantum particles and comes to a conclusion about existence of the parallel worlds, in all set representing Multiverse [@1]. More precisely speaking, the assumption of presence in [*our*]{} spacetime of the shadow photons, which identified by him with photons of [*other*]{} universe, one results him in the completed ground of observed interferentional picture. But generally in this case the shadow particles have property of objects with zero stress-energy tensor – it is directly follows from the put experiences – and consequently their existence should be physically is proved. The results which are discribed in this article, namelly the parallel between quantum particles ghost and corresponding shadow particles, give such ground. Some times ago the corresponding to the ghost neutrinos solutions of the Einstein-Dirac’s equations in cases plane-symmetric spacetimes [@2; @5], cyllindrically-symmetric spacetimes [@3], and so in case of wave gravitational field [@4] were found. Herewith spacetime is curved. In [@4] additionally it is showed the existence of the ghost neutrinos in a flat spacetime. The spacetime in this article also is flat.
The description of spasetime geometry and corresponding spinor fields
=====================================================================
We consider the Einstein-Dirac’s equations $$%\label{1}
\left\{\begin{array}{l}
%
{\displaystyle R_{ik}-\frac{1}{2} g_{ik}R=\kappa T_{ik}}
\\
{\displaystyle i\hbar {\gamma}^k\hspace*{-1mm}\left(
\frac{\partial\psi}{\partial x^{\scriptscriptstyle k}}-{\Gamma}_k\psi
\right)-mc\psi =0},
%
\end{array}\right.$$ where $$T_{ik}=\frac{i\hbar c}{4}\left\{{\psi}^*{\gamma}^{(0)}{\gamma}_i
\left(\frac{\partial\psi}{\partial
x^{\scriptscriptstyle k}}-{\Gamma}_k\psi\right)-\left(
\frac{\partial {\psi}^*}{\partial x^{
\scriptscriptstyle
k}}{\gamma}^{(0)}+{\psi}^*{\gamma}^{(0)}{\Gamma}_k
\right){\gamma}_i\psi+\right.$$ $$\label{4}
\left.+{\psi}
^*{\gamma}^{(0)}{\gamma}_k\left(\frac{\partial\psi}
{\partial x^{\scriptscriptstyle i}}-{\Gamma}_i
\psi\right)-\left(\frac{\partial {\psi}^*}
{\partial x^{\scriptscriptstyle i}}{\gamma}^{(0)}+{
\psi}^*{\gamma}^{(0)}{\Gamma}_i\right){\gamma}_k\psi\right\}.$$ Here $\psi$ is a bispinor, simbol ${}^*$ means the Hermite conjugation.
Spacetime geometry are discribed by flat metric $$\label{1}
ds^2={dx^{\scriptscriptstyle 0}}^2+2e^{x^0}dx^{\scriptscriptstyle 0}
dx^{\scriptscriptstyle
3}-{dx^{\scriptscriptstyle 1}}^2-{dx^{\scriptscriptstyle 2}}^2.$$ So Riemann tensor $R^i_{\, klm}$ is zero and the left part of the Einstein’s equations also equal to zero. Owing to the above we receive zero spinor matter stress-energy tensor $T_{ik}$.
In our formulas $${\Gamma}_k=\frac{1}{4}g_{ml}\left(\frac{\partial{\lambda}^{(s)}_r}
{\partial x^{\scriptscriptstyle
k}}\,{\lambda}^l_{(s)}-{\Gamma}^l_{rk}\right)s^{mr},$$ $$s^{mr}=\frac{1}{2}\left({\gamma}^m{\gamma}^r-{\gamma}^r
{\gamma}^m\right),\quad
{\gamma}^k\equiv{\lambda}^k_{(i)}{\gamma}^{(i)},$$ where ${\lambda}^k_{(i)}$ – i-vector of tetrade, ${\gamma}^{(i)}$ – matrixes of Dirac, for which we have the next presentation with matrixes of Pauly $${\gamma}^{(0)}=\left[\begin{array}{cc}I&0\\ 0&-I\end{array}
\right],\quad{\gamma}^{(\alpha)}=
\left[\begin{array}{cc}0&{\sigma}_{\alpha}\\
-{\sigma}_{\alpha}&0\end{array}\right],$$ $${\sigma}_1=\left[\begin{array}{cc}0&1\\ 1&0\end{array}\right],
{\sigma}_2=\left[\begin{array}{cc}0&-i\\ i&0\end{array}\right],
{\sigma}_3=\left[\begin{array}{cc}1&0\\ 0&-1\end{array}\right],
I=\left[\begin{array}{cc}1&0\\ 0&1\end{array}\right].$$
Metric tensor of spacetime can be expressed by vector’s tetrade in the following form [@6 c.373]: $$ds^2=\eta_{ab}\left({\lambda}^{(a)}_idx^{\scriptscriptstyle i}
\right)\left({\lambda}^{(b)}_kdx^{
\scriptscriptstyle k}\right).$$ In this case we have $${\eta}_{ab}=\left[\begin{array}{cccc}1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\
0&0&0&-1\end{array}\right].$$ Then for gravitational field (\[1\]) $${\lambda}^{(0)}_i=(1,0,0,e^{x^0}),\hspace*{0.3cm}
{\lambda}^{i}_{(0)}=(1,0,0,0),$$ $${\lambda}^{(1)}_i=(0,1,0,0),\hspace*{0.6cm}{\lambda}^{i}_{(1)}=(0,1,0,0),$$ $${\lambda}^{(2)}_i=(0,0,1,0),\hspace*{0.6cm} {\lambda}^{i}_{(2)}=(0,0,1,0),$$ $$\hspace*{1.0cm}{\lambda}^{(3)}_i=(0,0,0,e^{x^0}),\hspace*{0.3cm}
{\lambda}^{i}_{(3)}=(-1,0,0,e^{-x^0}).$$ Then ${\Gamma}_1={\Gamma}_2={\Gamma}_3=0$ and $${\Gamma}_0=\frac{1}{2}\left[\begin{array}{cc}0&{\sigma}_3\\
{\sigma}_3&0\end{array}\right].$$
The Ghostes
===========
The ghost neutrinos
-------------------
In this section we will consider a neutrino, i.e. in the Dirac equation (1) we must take $m=0$. Herewith let us expect, that $$\frac{\partial\psi}{\partial x^{\scriptscriptstyle 1}}=
\frac{\partial\psi}{\partial x^{
\scriptscriptstyle 2}}=\frac{\partial\psi}{\partial x^{\scriptscriptstyle
3}}=0,\quad \frac{\partial\psi}{\partial x^{\scriptscriptstyle 0}}=
\alpha\psi,$$ where $\alpha$ is a real constant. For example, the bispinor $$\label{5}
\psi=\left[\begin{array}{c}u_0\\ u_1\\ u_2\\ u_3\end{array}
\right]e^{\alpha x^0+\beta}$$ satisfies this conditions, where $u_i,\beta$ are complex constants. Then the Dirac equation (1) in our case equivalent to $$\left[\begin{array}{cc}I&-{\sigma}_3\\
{\sigma}_3&-I\end{array}\right]\psi=0.$$ Hereinafter we have $$\label{6}
u_0=u_2,\quad u_1=-u_3.$$ For stress-energy tensor (\[4\]) by considering the restrictions on spinor field $\psi$ we get $$\begin{array}{l}
T_{00}={\displaystyle -\frac{i\hbar c}{\mathstrut 2}
\alpha{\psi}^*{\gamma}^{(0)}\left\{{\gamma}_0{\Gamma}_0+
{\Gamma}_0{\gamma}_0\right\}\psi }\\
T_{01}={\displaystyle -\frac{\mathstrut i\hbar c}{\mathstrut 4}
\alpha{\psi}^*{\gamma}^{(0)}\left\{{\gamma}_1{\Gamma}_0+{\Gamma}_0{
\gamma}_1\right\}\psi }\\
T_{02}={\displaystyle -\frac{\mathstrut i\hbar c}{4\mathstrut }
\alpha{\psi}^*{\gamma}^{(0)}\left\{{\gamma}_2{\Gamma}_0+{\Gamma}_0{
\gamma}_2\right\}\psi }\\
T_{03}={\displaystyle -\frac{i\hbar c\mathstrut }{\mathstrut 4}
\alpha{\psi}^*{\gamma}^{(0)}\left\{{\gamma}_3{\Gamma}_0+{\Gamma}_0{
\gamma}_3\right\}\psi }\\
T_{11}=T_{12}=T_{13}=T_{22}=T_{23}=T_{33}=0.
\end{array}$$ And after some transformations $$\label{7}
\begin{array}{l}
T_{00}=T_{03}=0\\
T_{01}={\displaystyle -\frac{i\hbar c}{4}\alpha}\left(\bar{u_0},
\bar{u_1},-\bar{u_2},-\bar{u_3}\right)\left[
\begin{array}{cc}-i{\sigma}_2&0\\ 0&i{\sigma}_2\end{array}\right]\psi \\
\mathstrut \\
T_{02}={\displaystyle -\frac{i\hbar c}{4}\alpha}\left(\bar{u_0},
\bar{u_1},-\bar{u_2},-\bar{u_3}\right)\left[
\begin{array}{cc}-i{\sigma}_1&0\\ 0&i{\sigma}_1\end{array}\right]\psi.
\end{array}$$
In result by using (\[6\]) we insert (\[5\]) in (\[7\]) and then $T_{01}=T_{02}=0$.
And finally we receive that $T_{ik}\equiv0$, i.e. we find a solution of the Einstein-Dirac’s equation corresponding to ghost neutrinos as the current which as known calculated by formula: $$\label{8}
j^{(k)}={\lambda}^{(k)}_i{\psi}^*{\gamma}^{(0)}{\gamma}^i\psi,$$ is non-zero: $$j^{(k)}=\left(2({u_0}^2+{u_1}^2)e^{2\alpha x^0+2\beta},0,0,2({u_0}^2+{u_1}^2)
e^{2\alpha x^0+2\beta}\right).$$
The ghost spinors
-----------------
As ${\Gamma}_1={\Gamma}_2={\Gamma}_3=0$ then Dirac’s equation takes the following form $$\label{9}
i\hbar \left({\gamma}^k\frac{\partial\psi}
{\partial x^{\scriptscriptstyle k}}-{\gamma}^0
{\Gamma}_0\psi\right)-mc\psi =0,$$ After some transformation we get $$\left[\begin{array}{cc}I&-{\sigma}_3\\
{\sigma}_3&-I\end{array}\right]\frac{\partial\psi}
{\partial x^{\scriptscriptstyle
0}}+\left[\begin{array}{cc}0&{\sigma}_1\\ -{\sigma}_1&0\end{array}
\right]\frac{\partial\psi}
{\partial x^{\scriptscriptstyle 1}}+\left[\begin{array}
{cc}0&{\sigma}_2\\ -{\sigma}_2&0
\end{array}\right]\frac{\partial\psi}
{\partial x^{\scriptscriptstyle 2}}+e^{-x^0}\left[
\begin{array}{cc}0&{\sigma}_3\\ -{\sigma}_3&0\end{array}\right]
\frac{\partial\psi}{\partial
x^{\scriptscriptstyle 3}}-$$ $$-\frac{1}{2}\left[\begin{array}{cc}-I&{\sigma}_3\\
-{\sigma}_3&I\end{array}\right]\psi=-i\frac{mc}{\hbar}\psi.$$ Then after non-difficult calculations we have $$\left\{\begin{array}{l}
{\displaystyle u_{3,1}+u_{1,1}-i(u_{3,2}+u_{1,2})+e^{-x^0}
(u_{2,3}+u_{1,3})=-i\frac{mc}{\mathstrut
\hbar}(u_0-u_2)}\\
{\displaystyle u_{0,0}-u_{2,0}+iu_{1,2}-u_{1,1}-e^{-x^0}u_{1,3}+
\frac12u_0-\frac12u_2=-i\frac{
\mathstrut mc}{\mathstrut \hbar}u_2}\\
{\displaystyle u_{2,1}-u_{0,1}+i(u_{2,2}-u_{0,2})-e^{-x^0}
(u_{3,3}-u_{0,3})=-i\frac{
\mathstrut mc}{\mathstrut \hbar}(u_1+u_3)}\\
{\displaystyle -u_{1,0}-u_{3,0}-iu_{0,2}-u_{0,1}+e^{-x^0}u_{0,3}
-\frac12u_1-\frac12u_3=-i\frac{
\mathstrut mc}{\hbar}u_3}\\
\end{array}\right.$$ where $u_{i,k}$ is a partial derivation with respect to $x^k$. Now let us assume that $u_0=u_2$, $u_1=-u_3$. Then we have only following restrictions $$\left\{\begin{array}{l}
{(u_2+u_1)}_{,3}=0\\
{\displaystyle -u_{1,1}+iu_{1,2}-e^{-x^0}u_{1,3}=-i\frac{mc}{\hbar}u_2}\\
{(u_3-u_0)}_{,3}=0\\
{\displaystyle -u_{0,1}-iu_{0,2}+e^{-x^0}u_{0,3}=-i\frac{mc}{\hbar}u_3}
\end{array}\right.$$ Besides this let us take the following condition: $u_0=-u_1=u_2=u_3=u$. And we find $$\left\{\begin{array}{l}
{\displaystyle \frac{\partial u}{\partial
\mathstrut x^{\scriptscriptstyle 1}}=0}\\
{\displaystyle -i\frac{\mathstrut \partial u}
{\partial x^{\scriptscriptstyle 2}}+e^{-x^0}\frac{\partial u}{\partial x^{
\scriptscriptstyle 3}}=-i\frac{mc}{\hbar}u}
\end{array}\right.$$ Let $\partial u/\partial x^{\scriptscriptstyle 3}=0$. Then $${\displaystyle \frac{\partial u}{\partial x^{\scriptscriptstyle
2}}=\frac{mc}{\hbar}u},$$ and $\partial u/\partial x^{\scriptscriptstyle 0}$ is free. Herewith $$u=\exp\left({\frac{mc}{\hbar}x^{\scriptscriptstyle 2}+
\alpha (x^{\scriptscriptstyle
0})}\right),$$ or in more general form $$\label{10}
\psi=\left[\begin{array}{r}1\\ -1\\ 1\\ 1
\end{array}\right]e^{\frac{mc}{\hbar}x^
{\scriptscriptstyle 2}+\alpha (x^{\scriptscriptstyle 0})}.$$
The (\[10\]) is ghost iff $T_{ik}\equiv 0$. Take $Im\ [\alpha
(x^{\scriptscriptstyle 0})]=0$. Then $T_{ik}\equiv 0$. Hence we have a ghost spinor. If $m=0$ then it is ghost neutrino, which is addition to ghost in previous section.
By using (\[8\]) we get that density of current is non-zero: $$j^{(k)}=\left(4e^{2\frac{mc}{\hbar}x^{\scriptscriptstyle 2}+2\alpha (x^{\scriptscriptstyle 0})},
0,0,4e^{2\frac{mc}{\hbar}x^{\scriptscriptstyle 2}+2\alpha
(x^{\scriptscriptstyle 0})}\right),$$ i.e. in spacetime (\[1\]) there exist the flowes with zero energy and momentum.
Parallel Universes
==================
In [@1] the some experiments with light interferention ares discribed. The main point of the Deutsch explanation of these experiments is suggestion about existance of shadow photons, whence Deutsch come to the conclusion about partition of the multiverse on set of parallel worlds. The logically building sequence of discourses and conclusions about presence of set of parallel universes is clearly brought. Though there are some problems with interaction of worlds between itself. In particular very difficult to agree with conclusion about interaction of particles with their own shadow particles only. Actually this is only desired suggestion, which from nowhere does not follow. Except this there is one more minus in offered explanation: if shadow photon acts upon real moreover thereby that given influence is reflected on results of experiment – exactly on interferentional picture moreover direct image – then must be the equations, describing this interaction. Furthermore and suggestion about existance of such photons can be incorrect. Really, the fact of that no sensors could not fix presence of shadow photon as well as a straight line dependency from it received interferentional picture, results in conclusion about a zero energy of its and, as a result, zero stress-energy tensor. All this seamingly speaks about impossibility like to situations. But for full understanding of the presenteded position let us postpone aside photons. After all in begining the problem stood in understanding of interferentional natures in general quantum particles. Simply Deutsch considered the case with photons. Since the interference of quantum particles runs equally then conclusion from [@1] about parallel universes can generalise, for instance, on the spinor fields. But again for this we need the equations, describing a shadow spinor fields, which, as was spoken above, in view of zero energy have and zero stress-energy tensor. But this now does not difficult problem! Because there are so-called the ghost spinors! Thereby, we get at first, a physical interpretation of the ghost spinors – this is a corresponding spinor fields in the parallel universes, and at secondally, a physical motivation of shadow particles – the equations its describing are found. Moreover, we now have got the possibility to do wholly legal transition to statement about existance of parallel universes. Really, since all bodies are consisted by atoms, but atoms are consisted by electrons, neutrons and protons, which are described by the Dirac’s equation, then a presence in space of ghosts for these particles draws a presence of ghosts and for each body, i.e. we get the set of parallel worlds.
Conclusion. {#conclusion. .unnumbered}
===========
In this article the ghost spinors were found, and also its phisical interpretation which takes for basis a parallel worlds is done. But moreover we have to say that hereafter the matter and substence are not a synonyms. So-called ghosts as once are a matter and are not a substence, i.e. there are a flows of particles which not have a characters of the last.
Come back to the question about photons we consider that shadow photons in change from real photons are can not discribed by the Makswell’s equations. This is because from the zero stress-energy tensor following a absence of electric-magnetic field in our spacetime. But here contradictions are absented! After all the real photon is nor than other as a carrier of a certain energy, but energy a shadow photon is a zero. As once in this fact it consists the difference between photons and particles with half-whole spin. May be the problem is solved by the finding for photons the field equations with greater degrees of freedom than the Makswell’s equations.
[99]{}
Deutsch, D. [*The Fabric of Reality.*]{} Allen Lane. The Pengguin Press, 1999.
Davis, T.M., Ray, J.R. [*Ghost neutrinos in plane-symmetric spacetimes.*]{} // J. Math. Phys. 1975. V.16. No.1. P.75-79. Davis, T.M., Ray, J.R. [*Neutrinos in cyllindrically-symmetric spacetimes.*]{} // J. Math. Phys. 1975. V.16. No.1. P.80-81. Guts, A.K. [*A new solution of the Einstein-Dirac equations.*]{} // Izvestia vuzov. Fizika. 1979. No.8. P.91-95 (Russian).
Pechenick, K.R., Cohen, J.M. [*New exact solution to the Einstein-Dirac equations*]{} // Phys. Rev., 1979. D 19, No.6. P.1635-1640.
Landau, L.D., Lifshits, E.M.. [*Theory of field.*]{} Moscow.: Nauka, 1973.
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abstract: 'The paper presents a new algorithmic construction of a finite generating set of rational invariants for the rational action of an algebraic group on the affine space. The construction provides an algebraic counterpart of the moving frame method in differential geometry. The generating set of rational invariants appear as the coefficients of a Gröbner basis, reduction with respect to which allows to express a rational invariant in terms of the generators. The replacement invariants, introduced in the paper, are tuples of algebraic functions of rational invariants. Any invariant, whether rational, algebraic or local, can be rewritten in terms of a replacement invariant by a simple substitution.'
author:
-
nocite: '[@algeom4]'
title: 'Rational, Replacement and Local Invariants of a Group Action'
---
[**Key words:**]{} rational and algebraic invariants, algebraic and Lie group actions, cross-section, Gröbner basis, moving frame method, smooth and differential invariatns.
Introduction
============
We present algebraic constructions for invariants of a rational group action on an affine space, and relate them to their counterparts in differential geometry. The constructions are algorithmic and can easily be implemented in general purpose computer algebra systems or software specialized in Gröbner basis computations. This is illustrated by the <span style="font-variant:small-caps;">maple</span> worksheet available at <http://www.inria.fr/cafe/Evelyne.Hubert/Publi/rrl_invariants.html> where the examples of the paper are treated.
The first construction is for the computation of a generating set of rational invariants. This generating set is endowed with a simple algorithm to express any rational invariant in terms of them. The construction comes into two variants. In the first one we consider the ideal of the graph of the action as did Rosenlicht [@rosenlicht56], Vinberg & Popov [@vinberg89][^1] , and M[ü]{}ller-Quade & Beth [@beth99][^2]. We point out the connections with these previous works in the text. Our proofs are independent and provide an original approach. We show that the coefficients of a reduced Gröbner basis of the ideal of the graph of the action are invariant. We prove that these coefficients generate the field of rational invariants by exhibiting an algorithm for rewriting any rational invariant in terms of them. The second variant provides a purely algebraic formulation of the geometric construction of a *fundamental set of local invariants* on a smooth manifold proposed by Fels and Olver [@olver99], as a generalization of Cartan’s moving frame method. It is also computationally more effective as we reduce to zero the dimension of the polynomial ideal for which a reduced Gröbner basis is computed. This is achieved by adding the ideal of a cross-section to the ideal of the graph.
That latter construction allows to introduce *replacement invariants*, the algebraic counterpart of *normalized invariants* appearing in the geometric construction. A replacement invariant is a tuple of algebraic of functions of rational invariants. Any invariant can be trivially rewritten in their terms by substituting the coordinate functions by the corresponding invariants from this tuple. An *invariantization* map, a computable isomorphism from the set of algebraic functions on the cross-section to the set of algebraic invariants, is defined in terms of replacement invariants.
We use invariantization process to make explicit the connection between the present algebraic construction and the geometric construction of Fels and Olver [@olver99]. We introduce an alternative definition of smooth invariantization which, on one hand, generalizes the one given in [@olver99] and, on the other hand, matches the algebraic construction. We thus provide a bridge between the theory of rational and algebraic invariants [@vinberg89; @derksen02] and the theory of smooth local invariants in differential geometry.
Diverse fields of application of algebraic invariant theory are presented in [@derksen02 Chapter 5]. Some of the applications can be addressed with rational invariants. Their present construction together with the simple rewriting algorithm can bring computational benefits. An application of the moving frame method to classical invariant theory [@HilbertEng; @Gur64; @Sturmfels93] was proposed in [@O99; @Kthesis; @BO00; @KM02]. In these works, however, the geometric formulation of the method is used without adapting it to the algebraic nature of the problem. A purely algebraic formulation of the moving frame method opens new possibilities of its application in classical invariant theory.
The present algebraic formulation provides a new tool for the investigation of the differential invariants of Lie group actions and their applications to differential systems in the line of [@olver99; @hubert05; @ko03; @mansfield01]. This larger project motivates our choice to consider rational actions. Even if we start with an affine or even linear action on the zeroth order jet space, the prolongation of the action to the higher order jet spaces is usually rational.
The paper is structured as follows. In [Section \[groupaction\]]{} we introduce the action of an algebraic group on the affine space and the graph of the action. This leads to a first construction of a set of generating rational invariants. A second version of the construction is given after the introduction of the cross-section to the orbits in [Section \[crossection\]]{}. This second construction gives rise to the replacement invariants in [Section \[invariantization\]]{}, which are used to define a computable invariantization map. In [Section \[felsolver\]]{} we present a geometric construction of local smooth invariants that generalizes the construction of [@olver99] and explicitly relates it to the algebraic construction of the previous sections. [Section \[examples\]]{} provides additional examples.
<span style="font-variant:small-caps;">Acknowledgments:</span> [We would like to thank Liz Mansfield, Peter Olver and Agnes Szanto for discussing the ideas of the paper during the workshop “Differential Algebra and Symbolic Computation” in Raleigh, April 2004, sponsored in part by NSF grants CCR-0306406 and CCF-0347506. We are grateful to Michael Singer for continuing discussion of the project and a number of valuable suggestions.]{}
Graph of a group action and rational invariants {#groupaction}
===============================================
We give a definition of a rational action of an algebraic group over a field ${\ensuremath{\mathbb{K} } }$ on an affine space, and formulate two additional hypotheses necessary to our construction. We recall the definition for the graph of the action. It plays a central role in our constructions. The first variant of the algorithm for constructing a generating set of rational invariants, together with an algorithm for expressing any rational invariant in terms of them, is presented in this section.
For exposition convenience we assume that the field ${\ensuremath{\mathbb{K} } }$ is algebraically closed. The construction proposed in this section relies only on Gröbner basis computations and thus can be performed in the field of definition of the data (usually ${{\ensuremath{\mathbb{Q}}}}$ or ${\ensuremath{\mathbb{F} } }_p$). Outside of [Section \[felsolver\]]{} the terms *open*, *close* and *closure* refer to the Zariski topology.
Rational action of an algebraic group {#agroupaction:def}
-------------------------------------
We consider an algebraic group that is defined as an algebraic variety ${\ensuremath{\mathcal{G}}}$ in the affine space ${\ensuremath{\mathbb{K} } }^l$. The group operation and the inverse are given by polynomial maps. The neutral element is denoted by $e$. We shall consider an action of ${\ensuremath{\mathcal{G}}}$ on an affine space ${\ensuremath{\mathcal{Z}}}={\ensuremath{\mathbb{K} } }^{n}$.
Throughout the paper $\lambda=(\lambda_1, \ldots,\lambda_{l})$ and $z=(z_1,\ldots, z_{{n}})$ denote indeterminates while ${\ensuremath{\bar{\lambda}} }=(\bar{\lambda}_1, \ldots, \bar{\lambda}_{l})$ and ${\ensuremath{\bar{z}} }=(\bar{z}_1,\ldots, \bar{z}_{{n}})$ denote points in ${\ensuremath{\mathcal{G}}}\subset {\ensuremath{\mathbb{K} } }^{l}$ and ${\ensuremath{\mathcal{Z}}}={\ensuremath{\mathbb{K} } }^{{n}}$ respectively. The coordinate ring of ${\ensuremath{\mathcal{Z}}}$ and ${\ensuremath{\mathcal{G}}}$ are respectively ${\ensuremath{\mathbb{K} } }[z_1, \ldots, z_{n}]$ and ${\ensuremath{\mathbb{K} } }[\lambda_1, \ldots,\lambda_{l}]/G$ where $G$ is a radical unmixed dimensional ideal. By ${\ensuremath{\bar{\lambda}} }\cdot{\ensuremath{\bar{\mu}} }$ we denote the image of $({\ensuremath{\bar{\lambda}} },{\ensuremath{\bar{\mu}} })$ under the group operation while ${\ensuremath{\bar{\lambda}} }^{-1}$ denotes the image of ${\ensuremath{\bar{\lambda}} }$ under the inversion map.
\[action:def\] A rational action of an algebraic group ${\ensuremath{\mathcal{G}}}$ on the affine space ${\ensuremath{\mathcal{Z}}}$ is a rational map $g\colon{\ensuremath{\mathcal{G}}}\times {\ensuremath{\mathcal{Z}}}\rightarrow {\ensuremath{\mathcal{Z}}}$ that satisfies the following two properties
1. $g(e,{\ensuremath{\bar{z}} })={\ensuremath{\bar{z}} }$, $ \forall {\ensuremath{\bar{z}} }\in {\ensuremath{\mathcal{Z}}}$
2. $g({\ensuremath{\bar{\mu}} }, g({\ensuremath{\bar{\lambda}} }, z))=g({\ensuremath{\bar{\mu}} }\cdot{\ensuremath{\bar{\lambda}} }, z)$, whenever both $({\ensuremath{\bar{\lambda}} },{\ensuremath{\bar{z}} })$ and $({\ensuremath{\bar{\mu}} }\cdot{\ensuremath{\bar{\lambda}} },{\ensuremath{\bar{z}} })$ are in the domain of definition of $g$.
A rational action is thus uniquely determined by a ${n}$-tuple of rational functions of ${\ensuremath{\mathbb{K} } }(\lambda,z)$ whose domain of definition is a dense open set of ${\ensuremath{\mathcal{G}}}\times{\ensuremath{\mathcal{Z}}}$. We can bring these rational functions to their least common denominator $h\in {\ensuremath{\mathbb{K} } }[\lambda,z]$ without affecting the domain of definition. In the rest of the paper the action is thus given by $$\label{action} g({\ensuremath{\bar{\lambda}} },{\ensuremath{\bar{z}} })=\left(
{g_1(\bar{\lambda}, \bar{z})}, \ldots, {g_{{n}}(\bar{\lambda},
\bar{z})} \right)\mbox{ for } g_1, \ldots, g_{{n}} \in
h^{-1}{\ensuremath{\mathbb{K} } }[\lambda_1, \ldots,\lambda_{l},z_1,\ldots,z_{{n}}]$$
\[groupaction:hyp\] We make the additional assumptions
1. for all ${\ensuremath{\bar{z}} }\in {\ensuremath{\mathcal{Z}}}$, $h(\lambda,{\ensuremath{\bar{z}} })\in {\ensuremath{\mathbb{K} } }[\lambda]$ is not a zero-divisor of $G$. This says that the domain of definition of $g_{{\ensuremath{\bar{z}} }} \colon {\ensuremath{\bar{\lambda}} }\mapsto g({\ensuremath{\bar{\lambda}} },{\ensuremath{\bar{z}} })$ contains a non-empty open set of each component of ${\ensuremath{\mathcal{G}}}$.
2. for all ${\ensuremath{\bar{\lambda}} }\in {\ensuremath{\mathcal{Z}}}$, $h({\ensuremath{\bar{\lambda}} },z)\in {\ensuremath{\mathbb{K} } }[z]$ is different from zero. In other words, for every element ${\ensuremath{\bar{\lambda}} }\in{\ensuremath{\mathcal{G}}}$ there exists $ {\ensuremath{\bar{z}} }\in{\ensuremath{\mathcal{Z}}}$, such that $({\ensuremath{\bar{\lambda}} },{\ensuremath{\bar{z}} })$ is in the domain of definition $g$.
The following three examples serve as illustration throughout the text.
\[scaling:def\] <span style="font-variant:small-caps;">Scaling</span>. Consider the multiplicative group given by ${\ensuremath{G}}= (1-\lambda_1\lambda_2)\subset {\ensuremath{\mathbb{K} } }[\lambda_1, \lambda_2]$. The neutral element is $(1,1)$ and $({\ensuremath{\bar{\mu}} }_1,{\ensuremath{\bar{\mu}} }_2)\cdot ({\ensuremath{\bar{\lambda}} }_1,{\ensuremath{\bar{\lambda}} }_2)^{-1}=({\ensuremath{\bar{\mu}} }_1{\ensuremath{\bar{\lambda}} }_2,{\ensuremath{\bar{\mu}} }_2{\ensuremath{\bar{\lambda}} }_1)$. We consider the scaling action of this group on ${\ensuremath{\mathbb{K} } }^2$. It is given by the following polynomials of ${\ensuremath{\mathbb{K} } }[\lambda_1,\lambda_2,z_1,z_2]$: $g_1 = \lambda_1 z_1, \quad g_2=\lambda_1 z_2.$
\[translation:def\] <span style="font-variant:small-caps;">translation+reflection.</span> Consider the group that is the cross product of the additive group and the group of two elements $\{1,-1\}$, its defining ideal in ${\ensuremath{\mathbb{K} } }[\lambda_1,\lambda_2]$ being ${\ensuremath{G}}=(\lambda_2^2-1)$. The neutral element is $(0,1)$ while $({\ensuremath{\bar{\mu}} }_1,{\ensuremath{\bar{\mu}} }_2)\cdot ({\ensuremath{\bar{\lambda}} }_1,{\ensuremath{\bar{\lambda}} }_2)^{-1}=({\ensuremath{\bar{\mu}} }_1-{\ensuremath{\bar{\lambda}} }_1,{\ensuremath{\bar{\mu}} }_2{\ensuremath{\bar{\lambda}} }_2).$ We consider its action on ${\ensuremath{\mathbb{K} } }^2$ as translation parallel to the first coordinate axis and reflection w.r.t. this axis. It is defined by the following polynomials of ${\ensuremath{\mathbb{K} } }[\lambda_1,\lambda_2,z_1,z_2]$: $ g_1 = z_1+ \lambda_1 , \quad g_2=\lambda_2 z_2.$
\[rotation:def\] <span style="font-variant:small-caps;">rotation.</span> Consider the special orthogonal group given by ${\ensuremath{G}}= (\lambda_1^2+\lambda_2^2-1)\subset {\ensuremath{\mathbb{K} } }[\lambda_1, \lambda_2]$ with $e=(1,0)$ and $({\ensuremath{\bar{\mu}} }_1,{\ensuremath{\bar{\mu}} }_2)\cdot ({\ensuremath{\bar{\lambda}} }_1,{\ensuremath{\bar{\lambda}} }_2)^{-1}=
({\ensuremath{\bar{\mu}} }_1{\ensuremath{\bar{\lambda}} }_1+{\ensuremath{\bar{\mu}} }_2{\ensuremath{\bar{\lambda}} }_2,{\ensuremath{\bar{\mu}} }_2{\ensuremath{\bar{\lambda}} }_1-{\ensuremath{\bar{\mu}} }_1{\ensuremath{\bar{\lambda}} }_2).$ Its linear action on ${\ensuremath{\mathbb{K} } }^2$ is given by the following polynomials of ${\ensuremath{\mathbb{K} } }[\lambda_1,\lambda_2,z_1,z_2]$: $$g_1 = \lambda_1 z_1-\lambda_2 z_2, \quad g_2=\lambda_2
z_1+\lambda_1z_2.$$ An element of the group acts as a rotation around the origin.
Graph of the action and orbits {#graph::orbits}
------------------------------
The *graph of the action* is the image ${\ensuremath{\mathcal{O}}}\subset{\ensuremath{\mathcal{Z}}}\times{\ensuremath{\mathcal{Z}}}$ of the map $({\ensuremath{\bar{\lambda}} },{\ensuremath{\bar{z}} })\mapsto({\ensuremath{\bar{z}} },g({\ensuremath{\bar{\lambda}} },{\ensuremath{\bar{z}} }))$ that is defined on a dense open set of ${\ensuremath{\mathcal{G}}}\times{\ensuremath{\mathcal{Z}}}$. We have ${\ensuremath{\mathcal{O}}}= \{({\ensuremath{\bar{z}} },{\ensuremath{\bar{z}} }') \;|\; \exists {\ensuremath{\bar{\lambda}} }\in {\ensuremath{\mathcal{G}}}\, \mbox{s.t.} \, {\ensuremath{\bar{z}} }'=g({\ensuremath{\bar{\lambda}} }, {\ensuremath{\bar{z}} })\}
\subset {\ensuremath{\mathcal{Z}}}\times{\ensuremath{\mathcal{Z}}}$.
We introduce a new set of variables $Z=(Z_1, \ldots, Z_{{n}})$ and the ideal $ J={\ensuremath{G}}+ (Z-g(\lambda,z)) \subset h^{-1}{\ensuremath{\mathbb{K} } }[\lambda,z,Z]$, where $(Z-g(\lambda,z))$ stands for $\left(Z_1-g_1(\lambda, z), \ldots\right.$, $\left. Z_{{n}}-g_{{n}}(\lambda,
z)\right)$. The set ${\ensuremath{\mathcal{O}}}$ is dense in its closure $\overline{{\ensuremath{\mathcal{O}}}}$, and $\overline{{\ensuremath{\mathcal{O}}}}$ is the algebraic variety of the ideal: $${\ensuremath{O}}= J\cap {\ensuremath{\mathbb{K} } }[z,Z]
=\left({\ensuremath{G}}+ (\,Z-g(\lambda, z)\,) \,\right)\cap {\ensuremath{\mathbb{K} } }[z,Z] .$$
Since $G$ is radical and unmixed dimensional so is $J$ because of the linearity in $Z$. If ${\ensuremath{G}}=\bigcap_{i=0}^{\kappa} {\ensuremath{G}}^{(i)}$ is the prime decomposition of ${\ensuremath{G}}$ then we have the following prime decomposition of $J$: $$\left({\ensuremath{G}}+ (\,Z-g(\lambda, z)\,) \,\right)= \bigcap_{i=0}^{\kappa} \left({\ensuremath{G}}^{(i)} +
(\,Z-g(\lambda, z)\,) \,\right).$$ The prime ideal ${\ensuremath{O}}^{(i)} = \left({\ensuremath{G}}^{(i)} + (\,Z-g(\lambda, z)\,) \,\right) \cap {\ensuremath{\mathbb{K} } }[z,Z]$ is therefore a component of $O$. The ideals ${\ensuremath{O}}^{(i)}$, however, need not be all distinct.
The set ${\ensuremath{\mathcal{O}}}$ is symmetric: if $({\ensuremath{\bar{z}} },{\ensuremath{\bar{z}} }')\in {\ensuremath{\mathcal{O}}}$ then $({\ensuremath{\bar{z}} }',{\ensuremath{\bar{z}} })\in{\ensuremath{\mathcal{O}}}$. By the NullStellensatz the ideal ${\ensuremath{O}}$ is also symmetric: $p(Z,z)\in {\ensuremath{O}}$ if $p(z,Z)\in {\ensuremath{O}}$. Since $J\cap{\ensuremath{\mathbb{K} } }[z]=(0)$, $O\cap{\ensuremath{\mathbb{K} } }[z]=(0)$ and therefore ${\ensuremath{O}}\cap {\ensuremath{\mathbb{K} } }[Z]=(0)$ also.
A set of generators, and more precisely a Gröbner basis [@weisp], for ${\ensuremath{O}}\subset {\ensuremath{\mathbb{K} } }[z,Z]$ can be computed.
\[gbforo\] Let $g'$ be the ${n}$-tuple of numerators of $g$, that is $g'=hg=(h g_1, \ldots, h g_n) \in \left({\ensuremath{\mathbb{K} } }[\lambda,z]\right)^n$. Consider a term order s.t. $z \cup Z \ll \lambda \cup \{y\}$ where $y$ is a new indeterminate. If $Q$ is a Gröbner basis for ${\ensuremath{G}}+ (h\,Z - g') + (y h -1)$ according to this term order then $Q \cap {\ensuremath{\mathbb{K} } }[z,Z]$ is a Gröbner basis of $O$ according the induced term order on $z \cup Z$.
Take $J' = ({\ensuremath{G}}+ (Z-g))\cap {\ensuremath{\mathbb{K} } }[\lambda,z,Z]$ and note that $J' = ({\ensuremath{G}}+ (h\,Z - g')){\ensuremath{\! : \!}}h^\infty$ where $g'$ is the numerator of $g$. Given a basis $\Lambda$ of ${\ensuremath{G}}$ and $g$ explicitly, a Gröbner basis of ${\ensuremath{J}}$ is obtained thanks to [@weisp Proposition 6.37, Algorithm 6.6]. We recognize that $O$ is an elimination ideal of $J'$, namely $O=J'\cap {\ensuremath{\mathbb{K} } }[z,Z]$. A Gröner basis for $O$ is thus obtained by [@weisp Proposition 6.15, Algorithm 6.1].
We mainly use the extension ${\ensuremath{O}}^e$ of $O$ in ${\ensuremath{\mathbb{K} } }(z)[Z]$. If $Q$ is a Gröbner basis of $O$ w.r.t. a term order $z\ll Z$ then $Q$ is also a Gröbner basis for ${\ensuremath{O}}^e$, for the term order induced on $Z$ [@weisp Lemma 8.93]. It is nonetheless often preferable to compute a Gröbner basis of ${\ensuremath{O}}^e$ over ${\ensuremath{\mathbb{K} } }(z)$ directly (see [Example \[derksen\]]{}).
The *orbit* of ${\ensuremath{\bar{z}} }\in{\ensuremath{\mathcal{Z}}}$ is the image ${\ensuremath{\mathcal{O}}}_{{\ensuremath{\bar{z}} }}$ of the rational map $g_{{\ensuremath{\bar{z}} }}\colon{\ensuremath{\mathcal{G}}}\mapsto{\ensuremath{\mathcal{Z}}}$ defined by $g_{{\ensuremath{\bar{z}} }}({\ensuremath{\bar{\lambda}} })=g({\ensuremath{\bar{\lambda}} },{\ensuremath{\bar{z}} })$. We then have the following specialization property (see for instance [@cox Exercise 7]).
\[specialize\] Let $Q$ be a Gröbner basis for ${\ensuremath{O}}^e$ for a given term order on $Z$. There is a closed proper subset $\mathcal{W}$ of ${\ensuremath{\mathcal{Z}}}$ s.t. for ${\ensuremath{\bar{z}} }\in {\ensuremath{\mathcal{Z}}}\setminus\mathcal{W}$ the image of $Q$ under the specialization $z \mapsto {\ensuremath{\bar{z}} }$ is a Gröbner basis for the ideal whose variety is the closure of the orbit of ${\ensuremath{\bar{z}} }$.
Therefore, for ${\ensuremath{\bar{z}} }\in {\ensuremath{\mathcal{Z}}}\setminus\mathcal{W}$, the dimension of the orbits of ${\ensuremath{\bar{z}} }$ is equal to the dimension of ${\ensuremath{O}}^e \subset {\ensuremath{\mathbb{K} } }(z)[Z]$ [@cox Section 9.3, Theorem 8]. In the rest of the paper this dimension is denoted by ${s}$.
\[scaling:graph\] <span style="font-variant:small-caps;">Scaling</span>. Consider the group action of [Example \[scaling:def\]]{}. The set of orbits consists of 1-dimensional punctured straight lines through the origin and a single zero-dimensional orbit, the origin. By elimination on the ideal ${\ensuremath{J}}=( 1-\lambda_1\lambda_2, Z_1- \lambda_1 z_1, Z_2-\lambda_1 z_2)$ we obtain ${\ensuremath{O}}=(z_1Z_2-z_2Z_1)$. Take $\mathcal{W}$ to consist solely of the origin. For ${\ensuremath{\bar{z}} }\in {\ensuremath{\mathcal{Z}}}\setminus \mathcal{W}$ the closure of the orbit of ${\ensuremath{\bar{z}} }$ is the algebraic variety of $({\ensuremath{\bar{z}} }_1Z_2-{\ensuremath{\bar{z}} }_2Z_1)$
\[translation:graph\] <span style="font-variant:small-caps;">translation+reflection.</span> Consider the group action of [Example \[translation:def\]]{}. By elimination on the ideal ${\ensuremath{J}}= (\lambda_2^2-1, Z_1-z_1-\lambda_1, Z_2-\lambda_2 z_2)$ we obtain ${\ensuremath{O}}=(Z_2^2-z_2^2)$. The orbit of a point ${\ensuremath{\bar{z}} }=({\ensuremath{\bar{z}} }_1,{\ensuremath{\bar{z}} }_2)$ with ${\ensuremath{\bar{z}} }_2\neq 0$ consists of two lines parallel to the first coordinate axis, while the latter is the orbit of all points with ${\ensuremath{\bar{z}} }_2= 0$
\[rotation:graph\] <span style="font-variant:small-caps;">rotation.</span> Consider the group action of [Example \[rotation:def\]]{}. The orbits consist of the origin and the circles with the origin as center. By elimination on the ideal ${\ensuremath{J}}= (\lambda_1^2+\lambda_2^2-1, Z_1-\lambda_1 z_1+\lambda_2 z_2,
Z_2-\lambda_2 z_1-\lambda_1z_2)$ we obtain ${\ensuremath{O}}=(Z_1^2+Z_2^2-z_1^2-z_2^2)$.
Rational invariants {#rational}
-------------------
We construct a finite set of generators for the field of rational invariants. Our construction brings out a simple algorithm to rewrite any rational invariant in terms of them. The required operations are restricted to computing a Gröbner basis and normal forms. Those are implemented in most computer algebra systems. We provide a comparison with related results in [@beth99; @rosenlicht56; @vinberg89].
A rational function $r\in {\ensuremath{\mathbb{K} } }(z)$ is a *rational invariant* if $r(g(\lambda,z)) = r(z) \mod G.$
The set of rational invariants forms a field[^3] ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$. We show that the coefficients of the Gröbner basis for ${\ensuremath{O}}^e$ are invariant and generate ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$. The basis is computed using [Proposition \[gbforo\]]{}.
\[invariance\] If $q(z,Z)$ belongs to ${\ensuremath{O}}$ then $q(g({{\ensuremath{\bar{\lambda}} }},z),Z)$ belongs to ${\ensuremath{O}}^e$ for all ${\ensuremath{\bar{\lambda}} }\in {\ensuremath{\mathcal{G}}}$.
A point $({\ensuremath{\bar{z}} },{\ensuremath{\bar{z}} }')\in {\ensuremath{\mathcal{O}}}$ if there exists ${\ensuremath{\bar{\mu}} }\in{\ensuremath{\mathcal{G}}}$ s.t. ${\ensuremath{\bar{z}} }'=g({\ensuremath{\bar{\mu}} },{\ensuremath{\bar{z}} })$. Then for a generic ${\ensuremath{\bar{\lambda}} }\in {\ensuremath{\mathcal{G}}}$, ${\ensuremath{\bar{z}} }'=
g({\ensuremath{\bar{\mu}} }\cdot {\ensuremath{\bar{\lambda}} }^{-1}, g({\ensuremath{\bar{\lambda}} },{\ensuremath{\bar{z}} }))$. Therefore $(g({\ensuremath{\bar{\lambda}} },{\ensuremath{\bar{z}} }),{\ensuremath{\bar{z}} }')\in{\ensuremath{\mathcal{O}}}$. Thus if $q(z,Z)\in {\ensuremath{O}}$ then $q(g({{\ensuremath{\bar{\lambda}} }},{\ensuremath{\bar{z}} }),{\ensuremath{\bar{z}} }')=0$ for all $(\bar{z},{\ensuremath{\bar{z}} }')$ in ${\ensuremath{\mathcal{O}}}$. By the Hilbert NullStellensatz the numerator of $q(g({{\ensuremath{\bar{\lambda}} }},z),Z)$ belongs to ${\ensuremath{O}}$ and therefore $q(g({{\ensuremath{\bar{\lambda}} }},z),Z)\in {\ensuremath{O}}^e$.
Following [@weisp Definition 5.29] a set of polynomials is reduced, for a given term order, if the leading coefficients of the elements are equal to $1$ and each element is in normal form with respect to the others. Given a term order on $Z$, a polynomial ideal in ${\ensuremath{\mathbb{K} } }(z)[Z]$ has a unique reduced Gröbner basis [@weisp Theorem 5.3].
\[invgb\] The reduced Gröbner basis of ${\ensuremath{O}}^e$ with respect to any term order on $Z$ consists of polynomials in ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[Z]$.
Let $Q=\{q_1, \ldots, q_\kappa\}$ be the reduced Gröbner basis of ${\ensuremath{O}}^e$ for a given term order on $Z$. By [Lemma \[invariance\]]{} $q_i(g({\ensuremath{\bar{\lambda}} },z), Z)$ belongs to ${\ensuremath{O}}^e$. It has the same support[^4] as $q_i$. As $q_i(g({\ensuremath{\bar{\lambda}} },z), Z)$ and $q_i(z, Z)$ have the same leading monomial, $q_i(g({\ensuremath{\bar{\lambda}} },z), Z) - q_i(z,Z)$ is in normal form with respect to $Q$. As this difference belongs to ${{\ensuremath{O}}}^e$, it must be $0$. The coefficients of $q_i$ are therefore invariant.
Let us note the construction of a generating set of rational invariants proposed by Rosenlicht [@rosenlicht56]. In the paragraph before Theorem 2, Rosenlicht points out that the coefficients of the Chow form of ${\ensuremath{O}}^e$ over ${\ensuremath{\mathbb{K} } }(z)$ form a set of separating rational invariants. By [@rosenlicht56 Theorem 2] or [@vinberg89 Lemma 2.1] this set is generating for ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$.
Vinberg and Popov showed the existence of a subset of ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[Z]$ that generates ${\ensuremath{O}}^e$ [@vinberg89 Lemma 2.4]. We propose the construction of such a set. They showed furthermore that the set of the coefficients of such a family of generators *separates generic orbits* [@vinberg89 Theorem 2.3] and therefore generates ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$ [@rosenlicht56 Theorem 2],[@vinberg89 Lemma 2.1]. From those results we deduce that the set of coefficients of a reduced Gröbner basis of ${\ensuremath{O}}^e$ generates ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$. The next theorem provides an alternative proof of this result, providing additionally a rewriting algorithm. To prove generation we indeed exhibit an algorithm that allows to rewrite any rational invariant in terms of the coefficients of a reduced Gröbner basis.
In the case of linear actions M[ü]{}ller-Quade and Beth [@beth99] showed that the coefficient of the Gröbner basis of ${\ensuremath{O}}^e$ generate the field of rational invariants. Their proof is based on more general results about the characterization of subfields of ${\ensuremath{\mathbb{K} } }(z)$ obtained in [@quade99]. Our approach is quite different and more direct. The rewriting algorithm we propose, although it was obtained independently, is nonetheless reminiscent of [@quade99 Algorithm 1.10].
\[lolita\] Let $\frac{p}{q}$ be a rational invariant, $p,q\in {\ensuremath{\mathbb{K} } }[z]$. Then $ p(Z)\,q(z)- q(Z)\, p(z) \in {\ensuremath{O}}.$
Since $\frac{p}{q}$ is an invariant $\frac{p({\ensuremath{\bar{z}} })}{q({\ensuremath{\bar{z}} })}=\frac{p( g({\ensuremath{\bar{\lambda}} },{\ensuremath{\bar{z}} }))}{q(g({\ensuremath{\bar{\lambda}} },{\ensuremath{\bar{z}} }))}$ for all $({\ensuremath{\bar{\lambda}} },{\ensuremath{\bar{z}} })$ where this expression is defined. Thus $a({\ensuremath{\bar{z}} }',{\ensuremath{\bar{z}} })=p({\ensuremath{\bar{z}} }')\,q({\ensuremath{\bar{z}} })- q({\ensuremath{\bar{z}} }')\, p({\ensuremath{\bar{z}} })=0$ for all $({\ensuremath{\bar{z}} },{\ensuremath{\bar{z}} }')$ in ${\ensuremath{\mathcal{O}}}=\left\{({\ensuremath{\bar{z}} },{{\ensuremath{\bar{z}} }}')\,|\,
\exists{\ensuremath{\bar{\lambda}} }\in {\ensuremath{\mathcal{G}}}\mbox{ s.~t. }{{\ensuremath{\bar{z}} }}'=g({\ensuremath{\bar{\lambda}} },{\ensuremath{\bar{z}} })\right\}\subset{\ensuremath{\mathcal{Z}}}\times{\ensuremath{\mathcal{Z}}}$. In other words the polynomial $a(Z,z)= p(Z)\,q(z)- q(Z)\, p(z)\in {\ensuremath{\mathbb{K} } }[Z,z] $ is zero at each point of ${\ensuremath{\mathcal{O}}}$. Since the algebraic variety of ${\ensuremath{O}}$ is the closure $\bar{\ensuremath{\mathcal{O}}}$ of ${\ensuremath{\mathcal{O}}}$ and that ${\ensuremath{\mathcal{O}}}$ is dense in $\bar{{\ensuremath{\mathcal{O}}}}$ we can conclude that $a(Z,z)\in {\ensuremath{O}}$ by Hilbert Nullstellensatz.
Assume a polynomial ring over a field is endowed with a given term order. A polynomial $p$ is in *normal form* w.r.t. a set $Q$ of polynomials if $p$ involves no term that is a multiple of a leading term of an element in $Q$. A *reduction* w.r.t. $Q$ is an algorithm that given $p$ returns a polynomial $p'$ in normal form w.r.t. $Q$ s.t. $p = p' + \sum_{q\in Q} a_q \, q$ and no leading term of any $a_q \, q$ is larger than the leading term of $p$. Such an algorithm is detailed in [@weisp Algorithm 5.1]. It consists in rewriting the terms that are multiple of the leading terms of the elements of $Q$ by polynomials involving only terms that are lower. Note that if the leading coefficients of $Q$ are $1$ then no division occurs. When $Q$ is a Gröbner basis w.r.t. the given term order, the reduction of a polynomial $p$ is unique in the sense that $p'$ is then the only polynomial in normal form w.r.t. $Q$ in the equivalence class $p+(Q)$.
\[rewrite\] Consider $\{r_1,\ldots, r_\kappa\} \in {\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$ the coefficients of a reduced Gröbner basis $Q$ of ${\ensuremath{O}}^e$. Then ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }={\ensuremath{\mathbb{K} } }(r_1,\ldots, r_\kappa)$ and we can rewrite any rational invariant $\frac{p}{q}$, with $p,q\in {\ensuremath{\mathbb{K} } }[z]$, in terms of those as follows.
Take a new set of indeterminates $y_1, \ldots, y_\kappa$ and consider the set $Q_y \subset {\ensuremath{\mathbb{K} } }[y,Z]$ obtained from $Q$ by substituting $r_i$ by $y_i$. Let $a(y, Z)=\sum_{\alpha\in{{\ensuremath{\mathbb{N}}}}^{{n}}}a_{\alpha}(y)Z^\alpha$ and $b(y,Z)=\sum_{\alpha\in{{\ensuremath{\mathbb{N}}}}^n}a_{\alpha}(y)Z^\alpha$ in ${\ensuremath{\mathbb{K} } }[y,Z]$ be the reductions[^5] of $p(Z)$ and $q(Z)$ w.r.t. $Q_y$. There exists $\alpha\in {{\ensuremath{\mathbb{N}}}}^{{n}}$ s.t. $b_{\alpha}(r)\neq 0$ and for any such $\alpha$ we have $\frac{p(z)}{q(z)}=\frac{a_\alpha(r)}{b_{\alpha}(r)}$.
It is sufficient to prove the second part of the statement. The Gröbner basis $Q$ is reduced and therefore monic. The sets of leading monomials of $Q$ and of $Q_y$ are equal. If $a(y,Z)$ is the reduction of $p(Z)$ w.r.t. $Q_y$ then $a(r,Z)$, obtained by substituting back $y_i$ by $r_i$, is the normal form of $p(Z)$ w.r.t. $Q$. Similarly for $b(y,Z)$ and $q(Z)$.
As ${\ensuremath{O}}^e\cap{\ensuremath{\mathbb{K} } }[Z]=(0)$, neither $p(Z)$ nor $q(Z)$ belong to ${\ensuremath{O}}^e$ and therefore both $a(r,Z)$ and $b(r,Z)$ are different from $0$. By [Lemma \[lolita\]]{} $q(z)p(Z) \equiv p(z)q(Z) \mod {\ensuremath{O}}^e$ and thus the normal forms of the two polynomials modulo ${\ensuremath{O}}^e$ are equal: $q(z)\,a(r,Z)=p(z)\,b(r,Z)$. Thus $a(r,Z)$ and $b(r,Z)$ have the same support and this latter is non empty since $a,b\neq 0$. For each $\alpha$ in this common support, we have $q(z) a_{\alpha}(r) = p(z) b_{\alpha}(r)$ and therefore $\frac{p(z)}{q(z)} = \frac{a_{\alpha}(r)}{b_{\alpha}(r)}$.
\[scaling:rational\] <span style="font-variant:small-caps;">Scaling</span>. We consider the group action given in [Example \[scaling:def\]]{}. A reduced Gröbner basis of ${\ensuremath{O}}^e$ is $Q=\{ Z_2 -\frac{z_2}{z_1}Z_1\}$. By [Theorem \[invgb\]]{}, ${\ensuremath{\mathbb{K} } }(z_1,z_2)^G={\ensuremath{\mathbb{K} } }(\frac{z_2}{z_1})$.
Let $p = z_1^2+4z_1z_2+z_2^2$ and $q=z_1^2-3z_2^2$. We can check that $p/q$ is a rational invariant and we set up to write $p/q$ as a rational function of $r=z_2/z_1$. To this purpose consider $P=Z_1^2+4Z_1Z_2+Z_2^2$ and $Q=Z_1^2-3Z_2^2$ and compute their normal forms $a$ and $b$ w.r.t. $\{Z_2 - y\,Z_1\}$ according to a term order where $Z_1<Z_2$. We have $a=(1+4y+y^2)Z_1^2$ and $b=(1-3y^2)Z_1^2$. Thus $$\frac{z_1^2+4z_1z_2+z_2^2}{z_1^2-3z_2^2} = \frac{1+4r+r^2}{1-3r^2}
\hbox{ where } r= \frac{z_2}{z_1}$$
\[translation:rational\] <span style="font-variant:small-caps;">translation+reflection</span>. We consider the group action given in [Example \[translation:def\]]{}. A reduced Gröbner basis of ${\ensuremath{O}}^e$ is $Q=\{ Z_2^2 -z_2^2\}$. By [Theorem \[invgb\]]{}, ${\ensuremath{\mathbb{K} } }(z_1,z_2)^G={\ensuremath{\mathbb{K} } }(z_2^2)$.
\[rotation:rational\] <span style="font-variant:small-caps;">Rotation</span>. We consider the group action given in [Example \[rotation:def\]]{}. A reduced Gröbner basis of ${\ensuremath{O}}^e$ is $Q=\{ Z_1^2+Z_2^2 -(z_1^2+z_2^2)\}$. By [Theorem \[invgb\]]{}, ${\ensuremath{\mathbb{K} } }(z_1,z_2)^G={\ensuremath{\mathbb{K} } }(z_1^2+z_2^2)$.
Cross-section and rational invariants {#crossection}
=====================================
Given a cross-section we construct a generating set of rational invariants endowed with a rewriting algorithm. The method is the same as the one presented in previous section but applies to only a section of the graph. In previous section we considered an ideal of the dimension of the generic orbits.Here we consider a zero dimensional ideal. This is computationally advantageous when Gröbner bases are needed.
We use Noether normalization to prove the existence of a cross-section. The construction thus relies on selecting elements of in an open subset of a certain affine space. Therefore the construction does not entail a deterministic algorithm for the computation of rational invariants. Yet the freedom of choice is extremely fruitful in applicative examples.
Though the presentation is done with an algebraically closed field ${\ensuremath{\mathbb{K} } }$ that is therefore infinite, the construction is meant to be realized in characteristic zero (i.e. over ${{\ensuremath{\mathbb{Q}}}}$) or over a sufficiently large field.
Cross-section {#acs}
-------------
Geometrically speaking a *cross-section of degree $d$* is a variety that intersects generic orbits in $d$ simple points. We give a definition in terms of ideals for it is closer to the actual computations. We give its geometric content in a proposition afterward.
\[dcs\] Let ${\ensuremath{P_{}}}$ be a prime ideal of ${\ensuremath{\mathbb{K} } }[Z]$ of complementary dimension to the generic orbits, i.e. if $O^e$ is of dimension ${{s}}$ then ${\ensuremath{P_{}}}$ is of codimension ${s}$. ${\ensuremath{P_{}}}$ defines a *cross-section* to the orbits of the rational action $g:{\ensuremath{\mathcal{G}}}\times{\ensuremath{\mathcal{Z}}}\rightarrow {\ensuremath{\mathcal{Z}}}$ if the ideal ${\ensuremath{I}}^e = {\ensuremath{O}}^e + {\ensuremath{P_{}}}$ of ${\ensuremath{\mathbb{K} } }(z)[Z]$ is radical and zero dimensional. If $d$ is the dimension of ${\ensuremath{\mathbb{K} } }(z)[Z]/{\ensuremath{I}}^e$ as a ${\ensuremath{\mathbb{K} } }(z)$-vector space, we say that ${\ensuremath{P_{}}}$ defines a *cross-section of degree $d$*.
Indeed the algebra ${\ensuremath{\mathbb{K} } }(z)[Z]/{\ensuremath{I}}^e$ is a finite ${\ensuremath{\mathbb{K} } }(z)$-vector space since $I^e$ is zero dimensional [@weisp Theorem 6.54]. A basis for it is provided by the terms in $Z$ that are not multiple of the leading terms of a Gröbner basis of ${\ensuremath{I}}^e$ [@weisp Proposition 6.52]. Let us note here that an ideal of ${\ensuremath{\mathbb{K} } }(z)[Z]$ is zero dimensional iff any Gröbner basis of it has an element whose leading term is $Z_i^{d_i}$, for all $1\leq i\leq n$ [@weisp Theorem 6.54].
The cross-section is thus the variety ${\ensuremath{\mathcal{P}}}$ of ${\ensuremath{P_{}}}$. The geometric properties of this variety are explained by the following proposition.
\[transverse\] Let ${\ensuremath{P_{}}}$ define a cross-section ${\ensuremath{\mathcal{P}}}$ of degree $d$. There is a closed set $\mathcal{S} \subset{\ensuremath{\mathcal{Z}}}$ s.t. the closure of the orbit of any ${\ensuremath{\bar{z}} }\in{\ensuremath{\mathcal{Z}}}\setminus\mathcal{S}$ intersects ${\ensuremath{\mathcal{P}}}$ in $d$ simple points.
Let $Q$ be a reduced Gröbner basis for ${\ensuremath{I}}^e={\ensuremath{O}}^e+{\ensuremath{P_{}}}$. Similarly to [Proposition \[specialize\]]{}, the image $Q_{{\ensuremath{\bar{z}} }}$ of $Q$ under the specialization $z\mapsto{\ensuremath{\bar{z}} }$ is a Gröbner basis for ${\ensuremath{O}}_{{\ensuremath{\bar{z}} }}+{\ensuremath{P_{}}}$ in ${\ensuremath{\mathbb{K} } }[Z]$ for all ${\ensuremath{\bar{z}} }$ in ${\ensuremath{\mathcal{Z}}}$ outside of a closed set $\mathcal{W}$. Thus $I_{{\ensuremath{\bar{z}} }}={\ensuremath{O}}_{{\ensuremath{\bar{z}} }}+{\ensuremath{P_{}}}$ is zero dimensional and the dimension of ${\ensuremath{\mathbb{K} } }[Z]/I_{{\ensuremath{\bar{z}} }}$ as a vector space over ${\ensuremath{\mathbb{K} } }$ is $d$.
By the Jacobian criterion for regularity and the prime avoidance theorem [@eisenbud Corollary 16.20 and Lemma3.3] there is a $n\times n$ minor $f$ of the Jacobian matrix of $Q$ that is not included in any prime divisor of ${\ensuremath{I}}^e$. Therefore $f$ is not a zero divisor in ${\ensuremath{\mathbb{K} } }(z)[Z]/{\ensuremath{I}}^e$ which is a product of fields. There exists thus $f'\in{\ensuremath{\mathbb{K} } }(z)[Z]$ s.t. $f\,f'\equiv 1 \mod {\ensuremath{I}}^e$.
Provided that ${\ensuremath{\bar{z}} }$ is furthermore chosen so that the denominators of $f$ and $f'$ do not vanish, $f$ specializes into a $n\times n$ minor ${f}_{{\ensuremath{\bar{z}} }}$ of the Jacobian matrix of $Q_{{\ensuremath{\bar{z}} }}$ and we have $f_{{\ensuremath{\bar{z}} }}\,f_{{\ensuremath{\bar{z}} }}'\equiv 1 \mod {\ensuremath{I}}_{{\ensuremath{\bar{z}} }}$ for the specialization $f_{{\ensuremath{\bar{z}} }}'$ of $f'$. So $f_{{\ensuremath{\bar{z}} }}$ belongs to no prime divisors of ${\ensuremath{I}}_{{\ensuremath{\bar{z}} }}$ and thus ${\ensuremath{I}}_{{\ensuremath{\bar{z}} }}$ is radical [@eisenbud Corollary 16.20]. We take $\mathcal{S}$ to be the union of $\mathcal{W}$ with the algebraic set associated to the product of the denominators of $f$ and $f'$. That the number of points of intersection is $d$ is shown by [@eisenbud Proposition 2.15].
That property shows that the cross-sections of degree $d=1$ and $d>1$ are respectively the sections and the quasi-sections defined in [@vinberg89 Paragraph 2.5]. The existence of quasi-section is insured by [@vinberg89 Proposition 2.7], while a criterion for the existence of a section is described in [@vinberg89 Paragraph 2.5 and 2.6] Our terminology elaborates on the one used in [@rosenlicht56] and [@olver99].
The discussion of [@vinberg89 Section 2.5] shows that ${\ensuremath{\mathbb{K} } }({\ensuremath{\mathcal{P}}})$ is isomorphic to ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$ when ${\ensuremath{\mathcal{P}}}$ is a cross-section of degree 1. If ${\ensuremath{\mathcal{P}}}$ is a cross-section of degree $d>1$ then ${\ensuremath{\mathbb{K} } }({\ensuremath{\mathcal{P}}})$ is an algebraic extension of ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$ of degree $d$. In [Section \[invariantization\]]{} we shall come back to those points with a constructive angle that relies on the choice of a cross-section. The viewpoint adopted here is indeed the geometric intuition of the moving frame construction in [@olver99]: almost any algebraic variety of complementary dimension provides a cross-section (of some degree).
The existence of a cross-section is proved by Noether normalization theorem and is linked to an alternative definition of the dimension of an ideal [@shafarevich Section 6.2].
\[cross:existence\] A linear cross-section to the orbit is associated to each point of an open set of ${\ensuremath{\mathbb{K} } }^{{s}(n+1)}$, where ${s}$ is the dimension of the generic orbits and ${n}$ the dimension of ${\ensuremath{\mathcal{Z}}}$.
Assume that a Gröbner basis $Q$ of ${\ensuremath{O}}^e$ w.r.t. a term order $Z_1,\ldots, Z_{{s}} \ll Z_{{s}+1},\ldots, Z_{{n}}$ is s.t. an element of $Q$ has leading term $Z_i^{d_i}$ for some $d_i\in {{\ensuremath{\mathbb{N}}}}\setminus\{0\}$ for all ${s}+1\leq i\leq {n}$ and there is no element of $Q$ independent of $\{ Z_{{s}+1},\ldots, Z_{{n}}\}$. Then $Q$ is a Gröbner basis for the extension of ${\ensuremath{O}}^e$ to ${\ensuremath{\mathbb{K} } }(z)(Z_1,\ldots,Z_{{s}})[Z_{{s}+1}, \ldots, Z_{{n}}]$ [@weisp Lemma 8.93]. For $(a_1,\ldots, a_{{s}})$ in an open set of ${\ensuremath{\mathbb{K} } }^{s}$ the specialization $Q_a \subset {\ensuremath{\mathbb{K} } }[Z_{{s}+1}, \ldots, Z_{{n}}]$ of $Q$ under $Z_i\mapsto a_i$ is a Gröbner basis [@cox Exercise 7]. Therefore $Q_a \cup \{Z_1-a_1, \ldots, Z_{{s}}-a_{{s}}\}$ is a Gröbner basis by Buchberger’s criteria [@weisp Theorem 5.48 and 5.66]. It is a Gröbner basis of a zero dimensional ideal [@weisp Theorem 6.54]. We can thus take ${\ensuremath{P_{}}}$ to be generated by $\{Z_1-a_1, \ldots, Z_{{s}}-a_{{s}}\}$.
We can always retrieve the situation assumed above by a change of variables thanks to Noether normalization theorem [@greuel Theorem 3.4.1]. Inspecting the proof we observe that we can choose a change of variables given by a matrix $(m_{ij})_{1\leq i,j\leq {n}}$ with the vector of entries $m_{ij}$ in ${\ensuremath{\mathbb{K} } }^{{{n}}^2}$ outside of some algebraically closed set. The set $\{ a_i - \sum_{1\leq j \leq {n}} m_{ij} Z_j\;|\; 1\leq i\leq {s}\}$ thus defines a cross-section.
The choice of a cross section introduces a non deterministic aspect to the algebraic construction proposed in next section. An analysis of the probability of success in characteristic $0$ would be based on the measure of a correct test sequence [@giusti93a Theorem 3.5 and 3.7.2], [@giusti93b Section 3.2], [@krick01 Section 4.1].
We can computationally test if ${\ensuremath{\mathcal{P}}}$ is a cross-section by checking the properties of ${\ensuremath{I}}^e= \left(G+ P+ (Z-g(\lambda,z))\right)\cap{\ensuremath{\mathbb{K} } }(z)[Z]$, starting with the computation of its Gröbner basis. It is nonetheless worth performing the preliminary necessary test of transversality detailed in [Section \[smooth::section\]]{}. It relies on computing the rank of a matrix.
\[mfidcomp\] Assume that ${\ensuremath{P_{}}}\subset {\ensuremath{\mathbb{K} } }[Z]$ defines a cross-section and that ${\ensuremath{O}}= \bigcap_{i=0}^{\tau} {\ensuremath{O}}^{(i)}$ is the prime decomposition of $O$. Then $${\ensuremath{O}}+ {\ensuremath{P_{}}}= \bigcap_{i=0}^{\tau}({\ensuremath{O}}^{(i)}+{\ensuremath{P_{}}}) \quad \hbox{
and } \quad
({\ensuremath{O}}^{(i)}+{\ensuremath{P_{}}}) \cap {\ensuremath{\mathbb{K} } }[Z] = {\ensuremath{P_{}}}.$$
We can easily check that $ \bigcap_{i=0}^{\tau}({\ensuremath{O}}^{(i)}+{\ensuremath{P_{}}}) \subset {\ensuremath{O}}+ {\ensuremath{P_{}}}$ because ${\ensuremath{O}}+{\ensuremath{P_{}}}$ is radical. The converse inclusion is trivial.
For the second equality, note first that ${\ensuremath{P_{}}}\subset
({\ensuremath{O}}^{(i)}+{\ensuremath{P_{}}})\cap{\ensuremath{\mathbb{K} } }[z,Z]$. The projection of the variety of ${\ensuremath{O}}^{(i)} \subset {\ensuremath{\mathcal{Z}}}\times{\ensuremath{\mathcal{Z}}}$ is thus contained in ${\ensuremath{\mathcal{P}}}$. We show that the projection is exactly ${\ensuremath{\mathcal{P}}}$. We can assume that the numbering is such that ${\ensuremath{O}}^{(i)}= (\left({\ensuremath{G}}^{(i)} + (\,z-g(\lambda, Z)\,) \,\right)
\cap {\ensuremath{\mathbb{K} } }[z,Z]$ where ${\ensuremath{G}}^{(i)}$ is a minimal prime of ${\ensuremath{G}}$ (see [Section \[groupaction\]]{}). By [Asumption \[groupaction:hyp\]]{}, for any ${\ensuremath{\bar{z}} }$ in ${\ensuremath{\mathcal{Z}}}$ and therefore in ${\ensuremath{\mathcal{P}}}$, there exists ${\ensuremath{\bar{\lambda}} }$ in the variety of ${\ensuremath{G}}^{(i)}$ s.t. $g({\ensuremath{\bar{\lambda}} }, {\ensuremath{\bar{z}} })$ is defined. Above each point of ${\ensuremath{\mathcal{P}}}$ there is a point in the variety of ${\ensuremath{O}}^{(i)}$.
Rational invariants revisited {#mango}
-----------------------------
The following theorems provide a construction of a generating set of rational invariants together with an algorithm to rewrite any rational invariant in terms of generators. The method is the same as in [Section \[rational\]]{} but applied to the ideal $I^e$ rather than to ${\ensuremath{O}}^e$. The computational advantage comes from the fact that ${\ensuremath{I}}^e$ is zero dimensional.
If $G$ is a prime ideal we can actually choose a coordinate cross-section that is ${\ensuremath{P_{}}}$ can be taken as the ideal generated by a set of the following form: $\{Z_{j_1}-\alpha_1, \ldots, Z_{j_{{s}}}-\alpha_{{s}}\}$ for $(\alpha_1, \ldots, \alpha_{{s}})$ in ${\ensuremath{\mathbb{K} } }^{{s}}$. In this case we can remove $r$ variables for the computation.
\[cocorico\] The reduced Gröbner basis of ${\ensuremath{I}}^e$ with respect to any term ordering on $Z$ consists of polynomials in ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[Z]$.
The union of a reduced Gröbner basis of ${\ensuremath{O}}^e$ and ${\ensuremath{P_{}}}$ forms a generating set for ${\ensuremath{I}}^e={\ensuremath{O}}^e + {\ensuremath{P_{}}}$. The coefficients of a basis for ${\ensuremath{P_{}}}$ are in ${\ensuremath{\mathbb{K} } }$, while the coefficients of a reduced basis for ${\ensuremath{O}}^e$ belong to ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$ due to [Theorem \[invgb\]]{}. Since the coefficients of a generating set for ${\ensuremath{I}}^e$ belong to ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$, so do the coefficients of the reduced Gröbner basis with respect to any term ordering.
\[rewrite2\] Consider $\{r_1,\ldots, r_\kappa\} \in {\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$ the coefficients of a reduced Gröbner basis $Q$ of ${\ensuremath{I}}^e$. Then ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }={\ensuremath{\mathbb{K} } }(r_1,\ldots, r_\kappa)$ and we can rewrite any rational invariant $\frac{p}{q}$, with $p,q\in {\ensuremath{\mathbb{K} } }[z]$ relatively prime, in terms of those as follows.
Take a new set of indeterminates $y_1, \ldots, y_\kappa$ and consider the set $Q_y \subset {\ensuremath{\mathbb{K} } }[y,Z]$ obtained from $Q$ by substituting $r_i$ by $y_i$. Let $a(y, Z)=\sum_{\alpha\in{{\ensuremath{\mathbb{N}}}}^n}a_{\alpha}(y)Z^\alpha$ and $b(y,Z)=\sum_{\alpha\in{{\ensuremath{\mathbb{N}}}}^n}a_{\alpha}(y)Z^\alpha$ in ${\ensuremath{\mathbb{K} } }[y,Z]$ be the reductions of $p(Z)$ and $q(Z)$ w.r.t. $Q_y$. There exists $\alpha\in {{\ensuremath{\mathbb{N}}}}^m$ s.t. $b_{\alpha}(r)\neq 0$ and for any such $\alpha$ we have $\frac{p(z)}{q(z)}=\frac{a_\alpha(r)}{b_{\alpha}(r)}$.
We can proceed just as in the proof of [Theorem \[rewrite\]]{}. We only need to argue additionally that if $r=\frac{p}{q}\in{\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$, $p$ and $q$ being relatively prime, then $p(Z),q(Z)\notin{\ensuremath{I}}^e$. We prove the result for $p$, the case of $q$ being similar.
By hypothesis $p(z)\, q(g(\lambda,z)) \equiv q(z)\, p(g(\lambda,z)) \mod {\ensuremath{G}}$. Since $p$ and $q$ are relatively prime, $p(z)$ divides $p(g(\lambda,z))$ modulo ${\ensuremath{G}}$, that is there exists $\alpha \in h^{-1}{\ensuremath{\mathbb{K} } }[z,\lambda]$ s.t. $p(g(\lambda,z))\equiv \alpha(\lambda,z) \, p(z)\mod {\ensuremath{G}}$. Therefore if $p$ vanishes at ${\ensuremath{\bar{z}} }\in{\ensuremath{\mathcal{Z}}}$, then it vanishes on ${\ensuremath{\mathcal{O}}}_{{\ensuremath{\bar{z}} }}$. Thus if $p\in{\ensuremath{P_{}}}$, or equivalently if $p$ vanishes on ${\ensuremath{\mathcal{P}}}$, it vanishes on an open subset of ${\ensuremath{\mathcal{Z}}}$ ([Proposition \[transverse\]]{}). So $p$ must be zero. This is not the case and thus $p\notin{\ensuremath{P_{}}}$. Since ${\ensuremath{I}}^e\cap {\ensuremath{\mathbb{K} } }[Z]={\ensuremath{P_{}}}$, it is the case that $p(Z)\notin I^e$
When ${\ensuremath{P_{}}}$ defines a cross-section of degree $1$, the rewriting trivializes into a *replacement*. Indeed, if the dimension of ${\ensuremath{\mathbb{K} } }(z)[Z]/{\ensuremath{I}}^e$ as a ${\ensuremath{\mathbb{K} } }(z)$ vector space is $1$ then, independently of the chosen term order, the reduced Gröbner basis $Q$ for $I^e$ is given by $\{ Z_i -r_i(z) \,|\, 1\leq i\leq n\}$ where the $r_i\in {\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$. In view of [Theorem \[rewrite2\]]{} ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }={\ensuremath{\mathbb{K} } }(r_1, \ldots, r_{n})$ and any rational invariant $r(z)\in {\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$ can be rewritten in terms of $r_i$ by replacing $z_i$ by $r_i$: $$r(z_1,\ldots,z_{n})= r(\, r_1(z), \ldots, r_{{n}}(z)\,),
\quad \forall r\in {\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }.$$ In the next section we generalize this replacement to cross-section of any degree by introducing some special *algebraic invariants*.
\[scaling:cross\] <span style="font-variant:small-caps;">scaling</span>. We carry on with the action considered in [Example \[scaling:def\]]{} and \[scaling:rational\].
Choose ${\ensuremath{P_{}}}=(Z_1-1)$. A reduced Gröbner basis of ${\ensuremath{I}}^e$ is given by $\{Z_1-1, Z_2 -\frac{z_2}{z_1}\}$. We can see that [Theorem \[cocorico\]]{} is verified and that ${\ensuremath{P_{}}}$ defines a cross-section of degree 1. By [Theorem \[rewrite2\]]{} we know that $r=z_2/z_1$ generates the field of rational invariants ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$. In this situation, the cross section is of degree $1$ and we see that the rewriting algorithm of [Theorem \[rewrite2\]]{} is a simple replacement. For all $p\in {\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$ we have $p(z_1,z_2)=p(1,r)$.
\[translation:cross\] <span style="font-variant:small-caps;">translation+reflection</span>. We carry on with the action considered in [Example \[translation:def\]]{} and \[translation:rational\].
Choose ${\ensuremath{P_{}}}=(Z_1-Z_2)$ to define the cross-section. A reduced Gröbner basis of ${\ensuremath{I}}^e$ is given by $\{Z_1-Z_2, Z_2^2 -z_2^2\}$. The cross-section is thus of degree 2.
\[rotation:cross\] <span style="font-variant:small-caps;">rotation</span>. We carry on with the action considered in [Example \[rotation:def\]]{} and \[rotation:rational\].
Choose ${\ensuremath{P_{}}}=(Z_2)$. The reduced Gröbner basis of ${\ensuremath{I}}^e$ w.r.t. any term order is $\{Z_2, Z_1^2-(z_1^2+z_2^2)\}$. We can see that [Theorem \[invgb\]]{} is verified and that ${\ensuremath{P_{}}}$ defines a cross-section of degree 2. By [Theorem \[rewrite2\]]{} we know that $r=z_1^2+z_2^2$ generates the field of rational invariants ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$. In this situation, the rewriting algorithm of [Theorem \[rewrite2\]]{} consists in substituting $z_2$ by $0$ and $z_1^2$ by $r$.
Replacement invariants and invariantization {#invariantization}
===========================================
Given a cross-section ${\ensuremath{\mathcal{P}}}$ of degree $d$ we introduce $d$ distinct ${n}$-tuples of elements that are algebraic over the field of rational invariants. Each $n$-tuple has an important *replacement* property: any rational invariant can be rewritten in terms of its components by a simple substitution of the variables by the corresponding elements from the tuple.
The replacement invariants are used to define a process of *invariantization*, that is a projection from the algebraic functions onto the field of algebraic invariants. This projection can be explicitly computed by algebraic elimination. It gives a constructive approach to the isomorphism $\overline{{\ensuremath{\mathbb{K} } }({\ensuremath{\mathcal{P}}})}\cong{\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }$.
Replacement invariants {#repl}
----------------------
Let ${\ensuremath{\mathcal{P}}}$ be a cross-section of degree $d$ defined by a prime ideal ${\ensuremath{P_{}}}$ of ${\ensuremath{\mathbb{K} } }[Z]$. The field of rational functions on ${\ensuremath{\mathcal{P}}}$ is denoted by ${\ensuremath{\mathbb{K} } }({\ensuremath{\mathcal{P}}})$. It is the fraction field of the integral domain ${\ensuremath{\mathbb{K} } }[Z]/{\ensuremath{P_{}}}={\ensuremath{\mathbb{K} } }[{\ensuremath{\mathcal{P}}}]$. We introduce $d$ replacement invariants associated to ${\ensuremath{\mathcal{P}}}$. We use them to show that ${\ensuremath{\mathbb{K} } }({\ensuremath{\mathcal{P}}})$ is an algebraic extension of degree $d$ of the field of rational invariants ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$.
An *algebraic invariant* is an element of the algebraic closure ${\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }$ of ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$.
A reduced Gröbner basis $Q$ of ${\ensuremath{I}}^e={\ensuremath{O}}^e+{\ensuremath{P_{}}}$ is contained in ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[Z]$ ([Theorem \[cocorico\]]{}) and therefore is a reduced Gröbner basis of ${\ensuremath{I}}^G={\ensuremath{I}}^e\cap{\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[Z]$. The dimension of ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[Z]/{\ensuremath{I}}^G$ as a ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$-vector space is therefore equal to the dimension $d$ of ${\ensuremath{\mathbb{K} } }(z)[Z]/{\ensuremath{I}}^e$ as a ${\ensuremath{\mathbb{K} } }(z)$-vector space. Consequently the ideal ${\ensuremath{I}}^G$ has $d$ zeros ${\ensuremath{\xi} }=({\ensuremath{\xi} }_1, \ldots, {\ensuremath{\xi} }_n)$ with $\xi_i\in {\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }$ [@eisenbud Proposition 2.15]. We call such a tuple $(\xi_1, \ldots, \xi_n)$ a ${\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }$-zero of ${\ensuremath{I}}^G$. A ${\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }$-zero of ${\ensuremath{I}}^G$ is a ${\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }$-zero of ${\ensuremath{I}}^e$ and conversely.
A replacement invariant is a ${\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }$-zero of ${\ensuremath{I}}^G={\ensuremath{I}}^e\cap{\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[Z]$, i.e. a ${n}$-tuple ${\ensuremath{\xi} }=({\ensuremath{\xi} }_1,\ldots, {\ensuremath{\xi} }_n)$ of algebraic invariants that forms a zero of ${\ensuremath{I}}^e$.
Thus $d$ replacement invariants ${\ensuremath{\xi} }^{(1)}, \ldots,{\ensuremath{\xi} }^{(d)}$ are associated to a cross-section of degree $d$. The name owes to next theorem which can be compared with Thomas replacement theorem discussed in [@olver99 page 38].
\[replacement\] Let ${\ensuremath{\xi} }=({\ensuremath{\xi} }_1,\ldots,{\ensuremath{\xi} }_{n})$ be a replacement invariant. If $r \in {\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$ then $r(z_1,\ldots, z_{n})=r({\ensuremath{\xi} }_1,\ldots, {\ensuremath{\xi} }_{n})$ in ${\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }$.
Write $r = \frac{p}{q}$ with $p,q$ relatively prime. By [Lemma \[lolita\]]{}, $p(z)\,q(Z)- q(z)\,p(Z) \in {\ensuremath{O}}^e \subset {\ensuremath{I}}^e$ and therefore $p(Z)- \frac{p(z)}{q(z)}\, q(Z)=p(Z)-r(z) \, q(Z)
\in{\ensuremath{I}}^e$. Since ${\ensuremath{\xi} }$ is a zero of ${\ensuremath{I}}^e$, we have $p({\ensuremath{\xi} })-r(z) \, q({\ensuremath{\xi} })=0$. In the proof of [Theorem \[cocorico\]]{} we saw that $p(Z),q(Z)$ can not belong to $P$ and therefore cannot be zero divisors modulo ${\ensuremath{I}}^e$. Thus $q({\ensuremath{\xi} })\neq 0$ and the conclusion follows.
The field ${\ensuremath{\mathbb{K} } }({\ensuremath{\xi} })$, for any replacement invariant ${\ensuremath{\xi} }$, is an algebraic extension of ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$. Indeed ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }\subset{\ensuremath{\mathbb{K} } }({\ensuremath{\xi} })$ and ${\ensuremath{\xi} }$ is algebraic over ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$. This leads to the following results.
\[IGprime\]$I^G={\ensuremath{I}}^e\cap {\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[Z]$ is a prime ideal of ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[Z]$.
Let $I^{(1)}$ and $I^{(2)}$ be prime divisors of ${\ensuremath{I}}^G$ in ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[Z]$ and consider replacement invariants ${\ensuremath{\xi} }^{(1)}$ and ${\ensuremath{\xi} }^{(2)}$ that are ${\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }$-zeros of $I^{(1)}$ and $I^{(2)}$ respectively. Due to [Theorem \[replacement\]]{} ${\ensuremath{\mathbb{K} } }({\ensuremath{\xi} }^{(i)})={\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }({\ensuremath{\xi} }^{(i)})$. There is therefore a ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$-isomorphism ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[Z]/I^{(i)}\cong{\ensuremath{\mathbb{K} } }({\ensuremath{\xi} }^{(i)})$ for $i=1$ or $2$. On the other hand we have ${\ensuremath{\mathbb{K} } }({\ensuremath{\xi} }^{(i)})\cong{\ensuremath{\mathbb{K} } }({\ensuremath{\mathcal{P}}})$ since $P$ is the ideal of all relationships on the components of ${\ensuremath{\xi} }^{(i)}$ over ${\ensuremath{\mathbb{K} } }$ ([Proposition \[mfidcomp\]]{}). Thus $${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[Z]/I^{(1)}\cong{\ensuremath{\mathbb{K} } }({\ensuremath{\xi} }^{(1)})\cong {\ensuremath{\mathbb{K} } }({\ensuremath{\mathcal{P}}})\cong{\ensuremath{\mathbb{K} } }({\ensuremath{\xi} }^{(2)})\cong
{\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[Z]/I^{(2)}.$$ We have an isomorphism between ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[Z]/I^{(1)}$ and ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[Z]/I^{(2)}$ that leaves ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$ fixed and maps the class of $Z$ modulo $ I^{(1)}$ to the class of $Z$ modulo $I^{(2)}$. Therefore $I^{(1)}=I^{(2)}$ so that ${\ensuremath{I}}^G$ is prime.
\[d-ext\] The field ${\ensuremath{\mathbb{K} } }({\ensuremath{\mathcal{P}}})$ is an algebraic extension of ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$ of degree $d$, the degree of the cross-section ${\ensuremath{\mathcal{P}}}$.
For any replacement invariant ${\ensuremath{\xi} }$ we have ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[Z]/{\ensuremath{I}}^G\cong{\ensuremath{\mathbb{K} } }({\ensuremath{\xi} })\cong {\ensuremath{\mathbb{K} } }({\ensuremath{\mathcal{P}}})$. Since the dimension of ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[Z]/{\ensuremath{I}}^G$ as ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$-vector space is $d$, the field ${\ensuremath{\mathbb{K} } }({\ensuremath{\mathcal{P}}})$ is an algebraic extension of ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$ of degree $d$.
In particular if ${\ensuremath{\mathcal{P}}}$ is a cross-section of degree one we have ${\ensuremath{\mathbb{K} } }({\ensuremath{\mathcal{P}}})\cong {\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$. In all cases we have the isomorphism $\overline{{\ensuremath{\mathbb{K} } }({\ensuremath{\mathcal{P}}})}\cong{\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }$ obtained in [@vinberg89 Section 2.5] by different means.
\[scaling:algebraic\] <span style="font-variant:small-caps;">scaling</span>. Consider the multiplicative group from [Example \[scaling:def\]]{}, \[scaling:graph\], \[scaling:rational\]. We considered the cross-section of degree 1 defined by ${\ensuremath{P_{}}}=(Z_1-1)$. There is single replacement invariant ${\ensuremath{\xi} }=(1,\frac{z_2}{z_1})$ with rational components, which can be read off the reduced Gröbner basis of ${\ensuremath{I}}^e =(Z_1-1, Z_2 -\frac{z_2}{z_1})$. One can check that $r(z_1,z_2)=r(1,\frac{z_2}{z_1})$ for any $r\in {\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }={\ensuremath{\mathbb{K} } }\left(\frac{z_2}{z_1}\right)$.
\[translation:algebraic\] <span style="font-variant:small-caps;">translation+reflection</span>. Consider the group action from [Example \[translation:def\]]{}, \[translation:graph\], \[translation:rational\], \[translation:cross\]. We chose the cross-section defined by ${\ensuremath{P_{}}}=(Z_1-Z_2)$ and found that ${\ensuremath{\mathbb{K} } }(z_2^2)$ was the field of rational invariants. Generic orbits have two components and the cross-section is of degree 2. Since ${\ensuremath{I}}^e=(Z_1-Z_2, Z_2^2-z_2^2)$, the two replacement invariants are $\xi^{(1)}=( z_2,z_2)$ and $\xi^{(2)}=(-z_2,-z_2)$. Though rational functions, their components are not rational invariants but only algebraic invariants. Also $I^e=(Z_1-z_2,Z_2-z_2)\cap(Z_1+z_2,Z_2+z_2)$ is a reducible ideal of ${\ensuremath{\mathbb{K} } }(z)[Z]$, while $I^G$ is an irreducible ideal of ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[Z]$.
\[rotation:algebraic\] <span style="font-variant:small-caps;">rotation</span>. Consider the group action from [Example \[rotation:def\]]{}, \[rotation:graph\], \[rotation:rational\], \[rotation:cross\]. We chose the cross-section defined by ${\ensuremath{P_{}}}=(Z_2)$. Here the cross-section is again of degree 2 but the generic orbits have a single component. Since ${\ensuremath{I}}^e=(Z_2, Z_1^2-z_1^2-z_2^2)$ the two replacement invariants associated to ${\ensuremath{\mathcal{P}}}$ are $\xi^{(\pm)}=(0,\pm\sqrt{z_1^2+z_2^2})$.
Invariantization {#inv}
----------------
In this section we introduce invariantization as a projection from the ring of univariate polynomials over ${\ensuremath{\mathbb{K} } }[z]$ to the ring of univariate polynomials over ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$. It depends on the choice of a cross-section and is computable by algebraic elimination. As this projection extends to univariate polynomials over ${\ensuremath{\mathbb{K} } }({\ensuremath{\mathcal{P}}})$ it can be understood as the computable counterpart to the isomorphism $ \overline{{\ensuremath{\mathbb{K} } }({\ensuremath{\mathcal{P}}})}\cong{\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }$ that follows from [Proposition \[d-ext\]]{}.
We assume throughout this section that the field ${\ensuremath{\mathbb{K} } }$ is of characteristic zero. The ideal of the cross-section ${\ensuremath{\mathcal{P}}}$ is taken alternatively in ${\ensuremath{\mathbb{K} } }[z]$ and in ${\ensuremath{\mathbb{K} } }[Z]$. To avoid confusion we shall use in this section ${\ensuremath{P_{z}}}$ and ${\ensuremath{P_{Z}}}$ to distinguish the two cases. The localization of ${\ensuremath{\mathbb{K} } }[z]$ at ${\ensuremath{P_{z}}}$ is denoted by ${\ensuremath{{\ensuremath{\mathbb{K} } }[z]_{{\ensuremath{\mathcal{P}}}}}}$. In the proof of [Theorem \[rewrite2\]]{} we have shown that ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }\subset{\ensuremath{{\ensuremath{\mathbb{K} } }[z]_{{\ensuremath{\mathcal{P}}}}}}$.
The first approach for invariantization that draws directly on [@olver99] is to consider a replacement invariant ${\ensuremath{\xi} }$ associated to ${\ensuremath{\mathcal{P}}}$ and the following chain of homomorphisms: $$\label{icv}
\begin{array}{ccccc}
{\ensuremath{{\ensuremath{\mathbb{K} } }[z]_{{\ensuremath{\mathcal{P}}}}}}&\stackrel{{\pi}} \longrightarrow & {\ensuremath{\mathbb{K} } }({\ensuremath{\mathcal{P}}}) &\stackrel{{{\ensuremath{ \phi_{{\ensuremath{\xi} }}}}}} \longrightarrow & {\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }\\
r(z) & \longmapsto & r(z)+{\ensuremath{P_{z}}} & \longmapsto & r({\ensuremath{\xi} })
\end{array}$$ The restriction of ${\ensuremath{\iota}}_{\ensuremath{\xi} }={{\ensuremath{ \phi_{{\ensuremath{\xi} }}}}}\circ\pi\colon {\ensuremath{{\ensuremath{\mathbb{K} } }[z]_{{\ensuremath{\mathcal{P}}}}}}\to{\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }$ to ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$ is the identity map by [Theorem \[replacement\]]{}. We call the image of a rational function $r(z)\in{\ensuremath{{\ensuremath{\mathbb{K} } }[z]_{{\ensuremath{\mathcal{P}}}}}}$ under ${\ensuremath{\iota}}_{\ensuremath{\xi} }$ its *[$\xi$ ]{}-invariantization*.
If ${\ensuremath{\mathcal{P}}}$ is a cross-section of degree $d$ there are $d$ distinct associated replacement invariants ${\ensuremath{\xi} }^{(1)}, \ldots, {\ensuremath{\xi} }^{(d)}$. The image ${\ensuremath{\iota}}_{\ensuremath{\xi} }(r(z))=r({\ensuremath{\xi} })$ depends on the chosen replacement invariant ${\ensuremath{\xi} }$. Such is not the case of the minimal polynomial of $r({\ensuremath{\xi} })$ over ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$ which depends only on ${\ensuremath{\mathcal{P}}}$ as we shall see. We therefore define the ${\ensuremath{\mathcal{P}}}$-invariantization as a map taking a univariate polynomial over ${\ensuremath{{\ensuremath{\mathbb{K} } }[z]_{{\ensuremath{\mathcal{P}}}}}}$ to a univariate polynomial over ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$. The connection to the smooth invariantization of [@olver99] is developed in [Section \[felsolver\]]{}.
\[inv:def\] The [$\mathcal{P}$]{}-invariantization ${\ensuremath{\iota}}\alpha$ of a monic univariate polynomial $\alpha\in {\ensuremath{{\ensuremath{\mathbb{K} } }[z]_{{\ensuremath{\mathcal{P}}}}}}[{\zeta}]$ is the squarefree part of $\prod_{i=1}^d\alpha({\ensuremath{\xi} }^{(i)},{\zeta})$, where ${\ensuremath{\xi} }^{(1)}, \ldots, {\ensuremath{\xi} }^{(d)}$ are the $d$ replacement invariants associated to the cross-section ${\ensuremath{\mathcal{P}}}$.
Readers familiar with computer algebra techniques can see that ${\ensuremath{\iota}}\alpha$ belongs to ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[{\zeta}]$ with the following line of arguments. The replacement invariants ${\ensuremath{\xi} }^{(1)}, \ldots, {\ensuremath{\xi} }^{(d)}$ are the $d$ distinct zeros of the zero dimensional prime ideal ${\ensuremath{I}}^G$ of ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[Z]$. By a transcription of the primitive element theorem, see for instance [@greuel Proposition 4.2.2-3], they are thus the images by a polynomial map $\psi: \theta \mapsto (\psi_1(\theta), \ldots,\psi_n(\theta))$ over ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$ of the roots ${\theta}^{(1)}, \ldots, {\theta}^{(d)} \in {\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }$ of an irreducible univariate polynomial of degree $d$ with coefficients in ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$. The coefficients of the polynomial $$\prod_{i=1}^d\alpha({\ensuremath{\xi} }^{(i)},{\zeta}) = \prod_{i=1}^d\alpha(\psi({\theta}^{(i)}),{\zeta})$$ are elements of the field extension ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }(\theta^{(1)}, \ldots,\theta^{(d)})$ of ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$ that are invariant under all permutations of the $\theta^{(i)}$. By [@waerden71 Section 8.1] or [@geddes92 Theorem 8.15], that polynomial belongs to ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[{\zeta}]$ and thus so does its squarefree part ${\ensuremath{\iota}}\alpha$ [@waerden71 Section 8.1].
For a Galois theory oriented reader the details are given below. By definition ${\ensuremath{\iota}}\alpha$ belongs to the extension ${\ensuremath{\mathbb{K} } }({\ensuremath{\xi} }^{(1)},
\ldots, {\ensuremath{\xi} }^{(d)})$, which we denote by ${\ensuremath{\mathbb{K} } }_\xi$. Due to [Theorem \[replacement\]]{} ${\ensuremath{\mathbb{K} } }_\xi={\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }({\ensuremath{\xi} }^{(1)}, \ldots,
{\ensuremath{\xi} }^{(d)})$. In order to prove that ${\ensuremath{\iota}}\alpha \in {\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[{\zeta}]$ we will show that this polynomial is preserved by the Galois group of the extension ${\ensuremath{\mathbb{K} } }_\xi\supset {\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$. We need the following proposition.
Let $\{{\ensuremath{\xi} }^{(1)},\ldots, {\ensuremath{\xi} }^{(d)}\}$ be the set of replacement invariants corresponding to a cross-section ${\ensuremath{\mathcal{P}}}$ of degree $d$. Then the field ${\ensuremath{\mathbb{K} } }_{\ensuremath{\xi} }={\ensuremath{\mathbb{K} } }( {\ensuremath{\xi} }^{(1)},\ldots, {\ensuremath{\xi} }^{(d)})$ is a splitting field of a univariate polynomial $\beta(z,{\zeta})\in
{\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[{\zeta}]$ of degree $d$. The Galois group of the extension ${\ensuremath{\mathbb{K} } }_{\ensuremath{\xi} }\supset {\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$ permutes the $n$-tuples ${\ensuremath{\xi} }^{(1)},\ldots, {\ensuremath{\xi} }^{(d)}$.
Due to the replacement Theorem \[replacement\] one has the equality ${\ensuremath{\mathbb{K} } }({\ensuremath{\xi} }^{(1)})={\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }({\ensuremath{\xi} }^{(1)})$. From Corollary \[d-ext\] it follows that ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }({\ensuremath{\xi} }^{(1)})$ is an extension of degree $d$ of ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$ for $i=1..d$. Since ${\ensuremath{\mathbb{K} } }$ assumed to be of characteristic zero, the components ${\ensuremath{\xi} }^{(1)}_1,\dots,{\ensuremath{\xi} }^{(1)}_n$ of n-tuple ${\ensuremath{\xi} }^{(1)}$ are separable over ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$. Hence there exists a primitive element ${\ensuremath{\theta} }_1\in{\ensuremath{\mathbb{K} } }({\ensuremath{\xi} }^{(1)})$, such that ${\ensuremath{\mathbb{K} } }({\ensuremath{\xi} }^{(1)})={\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }({\ensuremath{\xi} }^{(1)})={\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }({\ensuremath{\theta} }_1)$, where ${\ensuremath{\theta} }_1$ is a root of an irreducible univariate polynomial $\beta(z,{\zeta})\in{\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[{\zeta}]$ of degree $d$ [@cox04 Theorem 5.4.1].
Let $\sigma_{ji}\colon {\ensuremath{\mathbb{K} } }({\ensuremath{\xi} }^{(i)})\to{\ensuremath{\mathbb{K} } }({\ensuremath{\xi} }^{(j)})$ be the ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$-isomorphism induced by exchanging ${\ensuremath{\xi} }^{(i)}$ and ${\ensuremath{\xi} }^{(j)}$. Then ${\ensuremath{\theta} }_j=\sigma_{j1}({\ensuremath{\theta} }_1)$ is a primitive element of the extension ${\ensuremath{\mathbb{K} } }({\ensuremath{\xi} }^{(j)})\supset{\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$. Indeed, since ${\ensuremath{\theta} }_1$ is the primitive element of ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }({\ensuremath{\xi} }^{(1)})$, for each $i=1..n$, there exists polynomial $\psi_i$ over ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$ such that ${\ensuremath{\xi} }^{(1)}_i=\psi_i({\ensuremath{\theta} }_1)$. Since $\sigma_{j1}$ is a ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$-isomorphism, it follows that ${\ensuremath{\xi} }^{(j)}_i=\sigma_{j1}({\ensuremath{\xi} }^{(1)}_i)=\sigma_{j1}(\psi_i({\ensuremath{\theta} }_1))=\psi_i(\sigma_{j1}({\ensuremath{\theta} }_1))=\psi_i({\ensuremath{\theta} }_j)$ for $i=1..n$. Thus ${\ensuremath{\theta} }_j$ is a primitive element of ${\ensuremath{\mathbb{K} } }({\ensuremath{\xi} }^{(j)})\supset{\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$, and so ${\ensuremath{\mathbb{K} } }_{\ensuremath{\xi} }={\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }( {\ensuremath{\theta} }_1,\dots,{\ensuremath{\theta} }_d)$
In addition, we proved that $n$-tuples ${\ensuremath{\xi} }^{(1)},\ldots,
{\ensuremath{\xi} }^{(d)}$ are images of ${\ensuremath{\theta} }_1,\dots,{\ensuremath{\theta} }_d$ under the polynomial map $\psi=(\psi_1,\dots\psi_n):{\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }\to\left[{\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }\right]^n$, where the coefficients of the univariate polynomials $\psi_1,\dots\psi_n$ are in ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$. Since ${\ensuremath{\xi} }^{(1)},\ldots, {\ensuremath{\xi} }^{(d)}$ are distinct tuples, then ${\ensuremath{\theta} }_{1},\ldots, {\ensuremath{\theta} }_{d}$ are distinct elements of ${\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }$. We will now show that ${\ensuremath{\theta} }_1,\dots,{\ensuremath{\theta} }_d$ are roots of the minimal polynomial $\beta\in{\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[{\zeta}]$ that defines ${\ensuremath{\theta} }_1$.
Indeed, since the field ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$ is fixed under $\sigma_{j1}$, for $j=1..d$, then so is the polynomial $\beta$. Thus ${\ensuremath{\theta} }_j=\sigma_{j1}({\ensuremath{\theta} }_1)$ are roots of the polynomial $\beta$. It follows that ${\ensuremath{\mathbb{K} } }_{\ensuremath{\xi} }={\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }({\ensuremath{\theta} }_1,\dots,{\ensuremath{\theta} }_d)$ is the splitting field of an irreducible univariate polynomial $\beta\in{\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[{\zeta}]$ of degree $d$.
The elements of the $Gal({\ensuremath{\mathbb{K} } }_\xi/{\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} })$ permute the roots ${\ensuremath{\theta} }_1,\dots,{\ensuremath{\theta} }_d$ of the polynomial $\beta$, and therefore it permutes the tuples ${\ensuremath{\xi} }^{(j)}=\psi(\theta_j)$ for all $j=1..d$.
Let $\alpha(z,{\zeta}) \in{\ensuremath{{\ensuremath{\mathbb{K} } }[z]_{{\ensuremath{\mathcal{P}}}}}}$ be a univariate polynomial over ${\ensuremath{{\ensuremath{\mathbb{K} } }[z]_{{\ensuremath{\mathcal{P}}}}}}$. Then its ${\ensuremath{\mathcal{P}}}$-invariantization ${\ensuremath{\iota}}\alpha$ is a polynomial over ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$.
The Galois group of the extension ${\ensuremath{\mathbb{K} } }_{\ensuremath{\xi} }\supset
{\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$ induces permutations of the n-tuples ${\ensuremath{\xi} }^{(1)},\ldots,
{\ensuremath{\xi} }^{(d)}$. Thus the polynomial $p({\zeta})=\prod_{i=1}^d\alpha({\ensuremath{\xi} }^{(i)},{\zeta})\in{\ensuremath{\mathbb{K} } }_{\ensuremath{\xi} }[{\zeta}]$ is fixed under $Gal ({\ensuremath{\mathbb{K} } }_{\ensuremath{\xi} }/{\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} })$. Hence its coefficients belong to ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$. By definition ${\ensuremath{\iota}}\alpha$ is the square-free part of $p({\zeta})$, and hence it is also fixed under the Galois group, since it has the same roots in ${\ensuremath{\mathbb{K} } }_{\ensuremath{\xi} }$ as $p({\zeta})$ itself [@cox04 Proposition 5.3.8], and the Galois group permutes these roots. Thus its coefficients of ${\ensuremath{\iota}}\alpha$ are in ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$.
The following properties follow directly from the definition of the map ${\ensuremath{\iota}}$:
1. A ${\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }$-zero of ${\ensuremath{\iota}}\beta$ is a ${\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }$-zero of a $\beta(\xi^{(i)}, {\zeta})$ and conversely.
2. If $\beta \in {\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[{\zeta}]$ then ${\ensuremath{\iota}}\beta=\beta$ since $\beta({\ensuremath{\xi} }^{(i)},{\zeta})=\beta(z,{\zeta})$ by [Theorem \[replacement\]]{}.
3. If $\alpha\equiv\beta\mod{\ensuremath{P_{z}}}$ then ${\ensuremath{\iota}}\alpha={\ensuremath{\iota}}\beta$ since the elements of ${\ensuremath{P_{z}}}$ vanish on all ${\ensuremath{\xi} }^{(i)}$.
The last property shows that ${\ensuremath{\iota}}$ induces a map $\phi$ from the set of monic polynomials of ${\ensuremath{\mathbb{K} } }({\ensuremath{\mathcal{P}}})[{\zeta}]$ to the set monic polynomials of ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[{\zeta}]$ s.t. ${\ensuremath{\iota}}= \phi\circ\pi $.
From the first property it follows that $\beta({\ensuremath{\xi} }^{(i)},{\zeta})$ divides ${\ensuremath{\iota}}\beta(z,{\zeta})$ in ${\ensuremath{\mathbb{K} } }({\ensuremath{\xi} }^{(i)})[{\zeta}] \supset {\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[{\zeta}]$ when $\beta({\ensuremath{\xi} }^{(i)},{\zeta})$ is squarefree. Since ${\ensuremath{\mathbb{K} } }({\ensuremath{\mathcal{P}}})\cong{\ensuremath{\mathbb{K} } }({\ensuremath{\xi} }^{(i)})$ this amounts to the following proposition that is used in [Section \[felsolver\]]{}.
\[factor\] Let $\beta$ be a monic polynomial of ${\ensuremath{{\ensuremath{\mathbb{K} } }[z]_{{\ensuremath{\mathcal{P}}}}}}[{\zeta}]$. If $\beta$ is squarefree when considered in ${\ensuremath{\mathbb{K} } }({\ensuremath{\mathcal{P}}})[{\zeta}]$ then it divides ${\ensuremath{\iota}}\beta(z,{\zeta})$ in ${\ensuremath{\mathbb{K} } }({\ensuremath{\mathcal{P}}})[{\zeta}]$, that is there exists $q(z,{\zeta}) \in{\ensuremath{{\ensuremath{\mathbb{K} } }[z]_{{\ensuremath{\mathcal{P}}}}}}[{\zeta}]$ s.t. ${\ensuremath{\iota}}\beta(z,{\zeta})\equiv q(z,{\zeta})\beta(z,{\zeta}) \mod {\ensuremath{P_{z}}}$.
Also we recognize in the definition of the invariantization map the norm of a polynomial in a algebraic extension [@geddes92 Section 8.8]. We reformulate the results extending those of that text namely:
- ${\ensuremath{\iota}}\beta$ can be computed by algebraic elimination.
- if $\beta({\ensuremath{\xi} }^{(i)},{\zeta})$ is the minimal polynomial over ${\ensuremath{\mathbb{K} } }({\ensuremath{\xi} }^{(i)})\subset{\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }$ of an element in ${\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }$, then ${\ensuremath{\iota}}\beta$ is the minimal polynomial of this element over ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$
The algebraic elimination to compute ${\ensuremath{\iota}}\beta$ can be performed by several techniques. For a strict generalization of [@geddes92 Section 8.8] one could introduce a resultant formula, as developed in [@andrea05]. We propose here a formulation in terms of elimination ideals.
\[inv:comp\] Let $\beta \in {\ensuremath{{\ensuremath{\mathbb{K} } }[z]_{{\ensuremath{\mathcal{P}}}}}}[{\zeta}]$ be a monic polynomial. Then its ${\ensuremath{\mathcal{P}}}$-invariantization ${\ensuremath{\iota}}\beta$ is the squarefree part of the monic generator of $({\ensuremath{I}}^G+\alpha(Z,{\zeta}))\cap {\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[{\zeta}]$ where $\alpha(z,{\zeta})\in {\ensuremath{\mathbb{K} } }[z][{\zeta}]$ is the numerator of $\beta$.
The leading coefficient of $\alpha(Z,{\zeta})\in{\ensuremath{\mathbb{K} } }[Z][{\zeta}]$ does not belong to ${\ensuremath{P_{Z}}}$, and therefore it does not belong to ${\ensuremath{I}}^G$. It follows that $({\ensuremath{I}}^G+\alpha(Z,{\zeta}))\cap {\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[{\zeta}] \neq (0)$ since ${\ensuremath{I}}^G$ is zero-dimensional.
Let $\gamma(z,{\zeta})$ be the monic generator of $({\ensuremath{I}}^G+\alpha(Z,{\zeta}))\cap{\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[{\zeta}]$. We first prove that ${\ensuremath{\iota}}\beta$ divides the squarefree part of $\gamma(z,{\zeta})$. The fact that $\gamma(z,{\zeta})$ belongs to ${\ensuremath{I}}^G+\alpha(Z,{\zeta})$ can be written as $\gamma(z,{\zeta})\equiv q(z,Z,{\zeta})\alpha(Z,{\zeta}) \mod {\ensuremath{I}}^G$ where $q(z,Z,{\zeta})\in {\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[Z,{\zeta}]$. Substituting ${\ensuremath{\xi} }^{(i)}$ for $Z$ we have $\gamma(z,{\zeta})=q'(z,{\ensuremath{\xi} }^{(i)},{\zeta})\beta({\ensuremath{\xi} }^{(i)},{\zeta})$ where $q(z,{\ensuremath{\xi} }^{(i)},{\zeta})$ and $q'(z,{\ensuremath{\xi} }^{(i)},{\zeta})$ differ by the factor in ${\ensuremath{\mathbb{K} } }[{\ensuremath{\xi} }^{(i)}]$ that distinguishes $\alpha({\ensuremath{\xi} }^{(i)},{\zeta})$ from $\beta({\ensuremath{\xi} }^{(i)},{\zeta})$. Therefore all the factors $\beta({\ensuremath{\xi} }^{(i)},{\zeta})$ of ${\ensuremath{\iota}}\beta$ divide $\gamma(z,{\zeta})$. Since ${\ensuremath{\iota}}\beta$ is the squarefree product of $\beta({\ensuremath{\xi} }^{(i)},{\zeta})$ it divides the squarefree part of $\gamma(z,{\zeta})$.
Conversely, we prove that the squarefree part of $\gamma(z,{\zeta})$ divides ${\ensuremath{\iota}}\beta$. The ${\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }$-zeros of $ \alpha (Z,{\zeta})+{\ensuremath{I}}^G$ are the $(n+1)$-tuples $({\ensuremath{\xi} }^{(i)},{\ensuremath{{f}} }_{i,j})$, where ${\ensuremath{{f}} }_{i,j}$, $1\leq j\leq\deg\beta$, are the roots of $\beta({\ensuremath{\xi} }^{(i)},{\zeta})$. Since $\gamma(z,{\zeta})$ belongs to $
\alpha (Z,{\zeta})+{\ensuremath{I}}^G$ its set of ${\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }$-roots includes all the ${\ensuremath{{f}} }_{i,j}$. Thus $\gamma$ and ${\ensuremath{\iota}}\beta$ have the same set of roots. Therefore the squarefree part of $\gamma$ divides ${\ensuremath{\iota}}\beta$
Note that the monic generator of $({\ensuremath{I}}^G+\alpha(Z,{\zeta}))\cap
{\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[{\zeta}]$ is the monic generator of $({\ensuremath{I}}^e+\alpha(Z,{\zeta}))\cap{\ensuremath{\mathbb{K} } }(z)[{\zeta}]$. This latter is an element of the reduced Gröbner basis of $\left( \alpha (Z,{\zeta})+{\ensuremath{I}}^e \right)$ w.r.t a term order that eliminates $Z$. It follows from [Proposition \[cocorico\]]{} that it belongs to ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[{\zeta}]$. Therefore computations over ${\ensuremath{\mathbb{K} } }(z)$ lead to the correct reasult over ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$.
The last proposition provides the computable counterpart of the isomorphism $ \overline{{\ensuremath{\mathbb{K} } }({\ensuremath{\mathcal{P}}})}\cong{\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }$, elements of $ \overline{{\ensuremath{\mathbb{K} } }({\ensuremath{\mathcal{P}}})}$ or ${\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }$ being represented by irreducible monic polynomials over ${\ensuremath{\mathbb{K} } }({\ensuremath{\mathcal{P}}})$ or ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$ respectively.
\[irr\] Let $\alpha$ be a monic polynomial of ${\ensuremath{{\ensuremath{\mathbb{K} } }[z]_{{\ensuremath{\mathcal{P}}}}}}[{\zeta}]$. The polynomial ${\ensuremath{\iota}}\alpha\in {\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }[{\zeta}]$ is irreducible if and only if $\alpha$ is a power of an irreducible polynomial when considered in ${\ensuremath{\mathbb{K} } }({\ensuremath{\mathcal{P}}})[{\zeta}]$.
Note that ${\ensuremath{\iota}}(\beta\,\gamma)$, for $\beta,\,\gamma \in {\ensuremath{{\ensuremath{\mathbb{K} } }[z]_{{\ensuremath{\mathcal{P}}}}}}[{\zeta}]$, is the squarefree part of the product ${\ensuremath{\iota}}\beta\,{\ensuremath{\iota}}\gamma$. So if $\alpha$ considered in ${\ensuremath{\mathbb{K} } }({\ensuremath{\mathcal{P}}})[{\zeta}]$ is the product of two relatively prime factors then ${\ensuremath{\iota}}\alpha$ cannot be irreducible.
We can replace $\alpha$ by its squarefree part when considered in ${\ensuremath{\mathbb{K} } }({\ensuremath{\mathcal{P}}})[{\zeta}]$ without loss of generality and thus assume for the converse implication that $\alpha(z,{\zeta})$ is irreducible there. Let $\bar{\alpha} \in{\ensuremath{\mathbb{K} } }[z][{\zeta}]$ be obtained from $\alpha$ by cleaning up the denominators. Then $\bar{\alpha}(Z, {\zeta})$ is irreducible modulo ${\ensuremath{I}}^G$ so that $\left(\bar{\alpha}(Z,{\zeta})+{\ensuremath{I}}^G\right)$ is prime. The monic generator ${\ensuremath{\iota}}\alpha$ of $\left(\alpha(Z,{\zeta})+{\ensuremath{I}}^G\right)\cap{\ensuremath{\mathbb{K} } }(z)[{\zeta}]$ is thus irreducible.
The following example illustrates various properties of the ${\ensuremath{\mathcal{P}}}$-invariantization map ${\ensuremath{\iota}}$.
<span style="font-variant:small-caps;">scaling</span>. We consider the scaling action defined in [Example \[scaling:def\]]{} and the cross-section defined by the ideal ${\ensuremath{P_{Z}}}=(Z_1^2+Z_2^2-1)$. It is a cross-section of degree $2$. We have $I^e=( Z_1^2-\frac{z_1^2}{z_1^2+z_2^2}, Z_2-\frac{z_2}{z_1}Z_1)$ and therefore the two replacement invariants are $${\ensuremath{\xi} }^{(\pm)} =
\left(\frac{\pm z_1}{\sqrt{z_1^2+z_2^2}},
\frac{\pm z_2}{\sqrt{z_1^2+z_2^2}}\right).$$
The invariantization of $\alpha = {\zeta}-z_{1}$ is ${\ensuremath{\iota}}\alpha= {\zeta}^2-\frac{z_{1}^2}{z_{1}^2+z_{2}^2}$. We have $ {\ensuremath{\iota}}\alpha = ({\zeta}+z_1) \alpha +
\frac{z_1^2}{z_1^2+z_2^2}(z_1^2+z_2^2-1)\equiv ({\zeta}+z_1) \alpha \mod {\ensuremath{P_{z}}}$. We obtained ${\ensuremath{\iota}}\alpha$ by computing the reduced Gröbner basis of the ideal $({\zeta}-Z_{1}, Z_1^2-\frac{z_1^2}{z_1^2+z_2^2},
Z_2-\frac{z_2}{z_1}Z_1 )$ with a term order that eliminates $Z_{1}$ and $Z_2$. Note that, although $\alpha$ defines a polynomial function, its invariantization defines two algebraic invariants $\pm\frac{z_1}{\sqrt{z_1^2+z_2^2}}$.
The invariantization of $\beta= {\zeta}^3+ {\zeta}^2+z_2{\zeta}+1$ is ${\ensuremath{\iota}}\beta= {\zeta}^6+2\,{\zeta}^5+{\zeta}^4+2\,{\zeta}^3+\frac{z_2^2+2z_1^2}{z_1^2+z_2^2}\,{\zeta}^2+1$. We have ${\ensuremath{\iota}}\beta\equiv({\zeta}^3+{\zeta}^2-z_2 {\zeta}+1)\beta \mod {\ensuremath{P_{z}}}$.
In the next two instances the monic polynomial is equal modulo ${\ensuremath{P_{z}}}$ to a polynomial in ${\ensuremath{\overline{{\ensuremath{\mathbb{K} } }(z)}^G} }[{\zeta}]$. As a consequence, the invariantization equals to the original polynomial modulo ${\ensuremath{P_{z}}}$
The polynomial $\gamma={\zeta}-z_1^{2}$ is equal to its [$\mathcal{P}$]{}-invariantization ${\ensuremath{\iota}}\gamma = {\zeta}-\frac{z_{1}^2}{z_{1}^2+z_{2}^2} \equiv \gamma \mod {\ensuremath{P_{z}}}$.
The irreducible polynomial $\delta= {\zeta}^2-\frac{z_1^2+z_2^2-1}{z_2^2}{\zeta}-\frac{z_1^2}{z_2^2}$ becomes a reducible modulo ${\ensuremath{P_{z}}}$: $\delta \equiv {\zeta}^2-\frac{z_1^2}{z_2^2}\mod {\ensuremath{P_{z}}}$. Its invariantization is thus reducible: ${\ensuremath{\iota}}\delta =({\zeta}-\frac{z_1}{z_2})({\zeta}+\frac{z_1}{z_2})
\equiv \delta \mod {\ensuremath{P_{z}}}$.
Local invariants and the moving frame construction {#felsolver}
==================================================
In this section we connect the algebraic algorithms presented in this paper with their original source of inspiration, the Fels-Olver moving frame construction [@olver99]. It is shown in [@olver99] that in the case of a *locally free smooth action* of a Lie group ${\ensuremath{\mathcal{G}}}$ on a manifold ${\ensuremath{\mathcal{Z}}}$, a choice of local cross-section corresponds to a local ${\ensuremath{\mathcal{G}}}$-equivariant map $\rho\colon {\ensuremath{\mathcal{Z}}}\to {\ensuremath{\mathcal{G}}}$. This map provides a generalization of the classical geometrical moving frames[^6] [@cartan]. A moving frame map gives rise to an *invariantization* process, a projection from the set of smooth functions to the set of local invariants.
We introduce an alternative definition of the smooth invariantization process which, on one hand, generalizes the definition given in [@olver99] to non-free, semi-regular actions and, on the other hand, can be effectively reformulated in the algebraic context. We make explicit comparisons with both the moving frame and the algebraic constructions in Section \[smf\] and Section \[smooth::alg\] respectively.
In this section we consider real smooth manifolds. All statements and constructions from this section are applicable to complex manifolds. In the latter case all maps and functions are assumed to be meromorphic.
Local action of a Lie group on a smooth manifold {#LGaction}
------------------------------------------------
We consider a Lie group ${\ensuremath{\mathcal{G}}}$, with identity denoted $e$, and a smooth manifold ${\ensuremath{\mathcal{Z}}}$ of dimension ${n}$. We first review the necessary facts and terminology from the theory of Lie group actions on smooth manifolds. Our presentations is based on [@GOV93; @olver:yellow].
A local action of a Lie group ${\ensuremath{\mathcal{G}}}$ on a smooth manifold ${\ensuremath{\mathcal{Z}}}$ is a smooth map $g\colon\Omega\to{\ensuremath{\mathcal{Z}}}$, where $\Omega\supset\{e\}\times{\ensuremath{\mathcal{Z}}}$ is an open subset of ${\ensuremath{\mathcal{G}}}\times{\ensuremath{\mathcal{Z}}}$, and the map $g$ satisfies the following two properties:
1. $g(e,{\ensuremath{\bar{z}} })={\ensuremath{\bar{z}} }$, $ \forall {\ensuremath{\bar{z}} }\in {\ensuremath{\mathcal{Z}}}$.
2. $g({\ensuremath{\bar{\mu}} }, g({\ensuremath{\bar{\lambda}} }, z))=g({\ensuremath{\bar{\mu}} }\cdot{\ensuremath{\bar{\lambda}} }, z)$, for all ${\ensuremath{\bar{z}} }\in{\ensuremath{\mathcal{Z}}}$ and ${\ensuremath{\bar{\lambda}} },{\ensuremath{\bar{\mu}} }\in{\ensuremath{\mathcal{G}}}$ s. t. $({\ensuremath{\bar{\lambda}} },{\ensuremath{\bar{z}} })$ and $({\ensuremath{\bar{\mu}} }\cdot{\ensuremath{\bar{\lambda}} },{\ensuremath{\bar{z}} })$ are in $\Omega$.
The *orbit* of ${\ensuremath{\bar{z}} }\in{\ensuremath{\mathcal{Z}}}$ is the image ${\ensuremath{\mathcal{O}}}_{{\ensuremath{\bar{z}} }}$ of the smooth map $g_{{\ensuremath{\bar{z}} }}\colon{\ensuremath{\mathcal{G}}}\mapsto{\ensuremath{\mathcal{Z}}}$ defined by $g_{{\ensuremath{\bar{z}} }}({\ensuremath{\bar{\lambda}} })=g({\ensuremath{\bar{\lambda}} },{\ensuremath{\bar{z}} })$. The domain of $g_{{\ensuremath{\bar{z}} }}$ is an open subset of ${\ensuremath{\mathcal{G}}}$ containing $e$.
For every point ${\ensuremath{\bar{z}} }\in{\ensuremath{\mathcal{Z}}}$ the differential $dg_{{\ensuremath{\bar{z}} }}:T{\ensuremath{\mathcal{G}}}|_e
\rightarrow T{\ensuremath{\mathcal{Z}}}|_{{\ensuremath{\bar{z}} }}$ maps the tangent space of ${\ensuremath{\mathcal{G}}}$ at $e$ to the tangent space of ${\ensuremath{\mathcal{Z}}}$ at the point ${\ensuremath{\bar{z}} }$. The tangent space $T{\ensuremath{\mathcal{G}}}|_e$ can be identified with the Lie algebra ${\frak g}$ of ${\ensuremath{\mathcal{G}}}$. Let ${\hat v} \in {\frak g}$ then ${v}({\ensuremath{\bar{z}} })= dg_{{\ensuremath{\bar{z}} }}({\hat v})$ is a smooth vector field on ${\ensuremath{\mathcal{Z}}}$, called the [ *infinitesimal generator*]{} of the ${\ensuremath{\mathcal{G}}}$-action corresponding to ${\hat v}$. The set of all infinitesimal generators for a ${\ensuremath{\mathcal{G}}}$-action form a Lie algebra, such that the map $\hat v\to v$ is a Lie algebra homomorphism.
By $\exp(\epsilon v,{\ensuremath{\bar{z}} })\colon {\ensuremath{\mathbb{R}}}\times{\ensuremath{\mathcal{Z}}}\to{\ensuremath{\mathcal{Z}}}$ we denote the [flow]{} of $v$. The flow is defined as an integral curve of the vector field $v$ with the initial condition ${\ensuremath{\bar{z}} }$. One can prove that every point of the connected component of the orbit ${\ensuremath{\mathcal{O}}}^0_{{\ensuremath{\bar{z}} }}\ni
{\ensuremath{\bar{z}} }$ can be reached from ${\ensuremath{\bar{z}} }$ by a composition of flows of a finite number of infinitesimal generators.
Let $\hat v_1,\dots,\hat v_\kappa$, where $\kappa\geq{s}$ is the dimension of the group, be a basis of the Lie algebra of ${\ensuremath{\mathcal{G}}}$. Then the infinitesimal generators $v_1,\dots,v_\kappa$ span the tangent space to the orbits at each point of ${\ensuremath{\mathcal{Z}}}$.
An action of a Lie group ${\ensuremath{\mathcal{G}}}$ on a smooth manifold ${\ensuremath{\mathcal{Z}}}$ is semi-regular if all orbits have the same dimension.
Throughout this section the action is assumed to be semi-regular. The dimension of the orbits is denoted by ${s}$.
Local invariants
----------------
We give definitions of local invariants and fundamental sets of those. We prove that the existence of a fundamental set of local invariants follows from the existence of a flat coordinate system. The proof is based on standard arguments from differential geometry.
\[linv\] A smooth function $f$, defined on an open subset ${\ensuremath{\mathcal{U}}}\subset{\ensuremath{\mathcal{Z}}}$, is a [ local invariant]{} if $v(f)=0$ for any infinitesimal generator $v$ of the ${\ensuremath{\mathcal{G}}}$-action on ${\ensuremath{\mathcal{U}}}$.
Equivalently $f(\exp (\varepsilon v, {\ensuremath{\bar{z}} }))=f({\ensuremath{\bar{z}} })$ for all ${\ensuremath{\bar{z}} }\in
{\ensuremath{\mathcal{U}}}$, all infinitesimal generator $v$, and all real $\varepsilon$ sufficiently close to zero. If the group ${\ensuremath{\mathcal{G}}}$ is connected, the function $f$ is continuous on ${\ensuremath{\mathcal{Z}}}$, and the condition of Definition \[linv\] is satisfied at every point of ${\ensuremath{\mathcal{Z}}}$ then $f$ is a global invariant on ${\ensuremath{\mathcal{Z}}}$ due to [@olver:yellow Proposition 2.6]. In what follows we neither assume $f$ to be continuous outside of ${\ensuremath{\mathcal{U}}}$, nor ${\ensuremath{\mathcal{G}}}$ to be connected.
A collection of smooth functions $f_1,\dots,f_l$ are functionally *dependent* on a manifold ${\ensuremath{\mathcal{Z}}}$ if for each point ${\ensuremath{\bar{z}} }\in{\ensuremath{\mathcal{U}}}$ there exists on open neighborhood an ${\ensuremath{\mathcal{U}}}$ and a non-zero differentiable function $F$ in $l$ variables such that $F(f_1,\dots,f_l)=0$ on ${\ensuremath{\mathcal{U}}}$. From the implicit function theorem it follows that $f_1,\dots,f_l$ are functionally [ dependent]{} on ${\ensuremath{\mathcal{U}}}$ if and only if the rank of the corresponding Jacobian matrix is less than $l$ at each point of ${\ensuremath{\mathcal{Z}}}$. We say that functions $f_1,\dots,f_l$ are *independent* on ${\ensuremath{\mathcal{Z}}}$ if they are not dependent when restricted to any open subset of ${\ensuremath{\mathcal{Z}}}$. As it is commented in [@olver:yellow p85] functional dependence and functional independence on ${\ensuremath{\mathcal{Z}}}$ do not exhaust the range of possibilities, except for analytic functions. Throughout the section the term [*independent functions*]{} means [*functionally independent functions*]{}. Finally we say that $f_1,\dots,f_l$ are independent at a point ${\ensuremath{\bar{z}} }\in{\ensuremath{\mathcal{Z}}}$ if the rank of the corresponding Jacobian matrix is maximal at ${\ensuremath{\bar{z}} }$. Independence at ${\ensuremath{\bar{z}} }$ implies independence on some open neighborhood of this point. If ${\ensuremath{\mathcal{U}}}$ is an open subset of ${\ensuremath{\mathcal{Z}}}$ and $f_1,\dots,f_n$ are independent at each point of ${\ensuremath{\mathcal{Z}}}$, then these functions provide a coordinate system on ${\ensuremath{\mathcal{U}}}$.
A collection of local invariants on ${\ensuremath{\mathcal{U}}}$ forms a *fundamental set* if they are functionally independent, and any local invariant on ${\ensuremath{\mathcal{U}}}$ can be expressed as a smooth function of the invariants from this set.
The Lie algebra of infinitesimal generators provides an integrable distribution[^7] of smooth vector-fields on ${\ensuremath{\mathcal{Z}}}$, whose integral manifolds are orbits. For a semi-regular action this distribution is of constant rank ${s}$, the dimension of the orbits. It follows from Frobenius theorem that in an open neighborhood ${\ensuremath{\mathcal{U}}}$ of each point there exists a coordinate system $x_1,\dots, x_{s},y_1,\dots,y_{n-{s}} $ such that the connected components of the orbits on ${\ensuremath{\mathcal{U}}}$ are level sets of the last $n-{s}$ coordinates [@spivak70 p. 262] and [@olver:yellow Theorem 1.43]. Such coordinate system is called [*flat, or straightening*]{}. The proof of the following theorem establishes that $y_1,\dots,y_{n-{s}}$ form a fundamental set of local invariants.
\[fund\] Let ${\ensuremath{\mathcal{G}}}$ be a Lie group acting semi-regularly on an $n$-dimensional manifold ${\ensuremath{\mathcal{Z}}}$. Let ${s}$ be the dimension of the orbits. In the neighborhood of each point ${\ensuremath{\bar{z}} }\in {\ensuremath{\mathcal{Z}}}$ there exists a fundamental set of $n-{s}$ local invariants.
By Frobenius theorem there exists a flat coordinate system $x_1,\dots, x_{s}$, $y_1,\dots,y_{n-{s}} $ in a neighborhood ${\ensuremath{\mathcal{U}}}\ni {\ensuremath{\bar{z}} }$. The connected components of the orbits on ${\ensuremath{\mathcal{U}}}$ coincide with the level sets of the last $n-{s}$ coordinate functions. Thus $y_1,\dots,y_{n-{s}}$ are constant on the connected components of the orbits, and therefore they are local invariants, being smooth and functionally independent by definition of a coordinate system. It remains to show that any other invariant is locally expressible in terms of them. Let $v$ be an infinitesimal generator of the group action. Since $v(y_i)=0$ for $i=1..(n-{s})$ then $v=\sum_{i=1}^{{s}}v(x_i){\frac\partial{\partial x_{i}}} $ is a linear combination of the first ${s}$ basis vector fields. Let $v_1=\sum_{i=1}^{{s}}a_{1i}{\frac\partial{\partial x_{i}}} ,\dots,
v_\kappa=\sum_{i=1}^{{s}}a_{\kappa i}{\frac\partial{\partial x_{i}}} $ be a basis of infinitesimal generators of the group action. Without loss of generality we may assume that the first ${s}$ generators $v_1,\dots, v_{s}$ are linearly independent at each point of ${\ensuremath{\mathcal{U}}}$. Let $f( x_1\dots, x_{s},y_1,\dots y_{n-{s}})$ be a local invariant, then $v_j(f)=\sum_{i=1}^r a_{ji}\frac{\partial f}{\partial x_i } =0$ for $j=1..{s}$. This is a homogeneous system of ${s}$ linear equation with ${s}$ unknowns $ \frac{\partial f}{\partial
x_1}, \dots, \frac{\partial f}{\partial x_{s}}$. Since $v_1,\dots,
v_{s}$ are linearly independent at each point, the rank of the system is maximal. Thus $ \left(\frac{\partial f}{\partial x_1}=0,
\dots, \frac{\partial f}{\partial x_n}=0\right)$ is the only solution. Hence $f$ is a function of invariants $y_1,\dots, y_{n-{s}}$.
The existence of a fundamental set of local invariants, therefore, follows from the existence of a flat coordinate system. The proof is not constructive however. The invariantization process, introduced in next section, leads to a different characterization of a fundamental set of invariants. Invariantization, and therefore fundamental invariants, can be effectively computed either by the algorithms of [Section \[invariantization\]]{}, in the case of a rational action of an algebraic group (see [Section \[smooth::alg\]]{}), or by the moving frame method of [@olver99], in the case of a locally free action of a Lie group (see [Section \[smf\]]{}).
Local cross-section and smooth invariantization {#smooth::section}
-----------------------------------------------
We define local cross-sections to the orbits and show that a local cross-section passing through any given point can easily be constructed. A local cross-section gives rise to an equivalence relationship on the ring of smooth functions such that any class has a single representative that is a local invariant. This leads to an *invariantization map*, a projection from the ring of smooth functions to the ring of local invariants. It generalizes the invariantization process defined in [@olver99] to semi-regular actions. Although a possibility of such generalization is indicated in the remarks of [@olver99 Section 4], the precise definitions and theorems, appearing in this section, are new.
\[lsection\] An embedded submanifold ${\ensuremath{\mathcal{P}}}$ of ${\ensuremath{\mathcal{Z}}}$ is a *local cross-section* to the orbits if there is an open set ${\ensuremath{\mathcal{U}}}$ of ${\ensuremath{\mathcal{Z}}}$ such that
- ${\ensuremath{\mathcal{P}}}$ intersects ${\ensuremath{\mathcal{O}}}_{{\ensuremath{\bar{z}} }}^0\cap{\ensuremath{\mathcal{U}}}$ at a unique point $\forall{\ensuremath{\bar{z}} }\in{\ensuremath{\mathcal{U}}}$, where ${\ensuremath{\mathcal{O}}}_{{\ensuremath{\bar{z}} }}^0$ is the connected component of ${\ensuremath{\mathcal{O}}}_{{\ensuremath{\bar{z}} }}\cap
{\ensuremath{\mathcal{U}}}$, containing ${\ensuremath{\bar{z}} }$.
- for all ${\ensuremath{\bar{z}} }\in {\ensuremath{\mathcal{P}}}\cap {\ensuremath{\mathcal{U}}}$, ${\ensuremath{\mathcal{O}}}_{{\ensuremath{\bar{z}} }}^0$ and ${\ensuremath{\mathcal{P}}}$ are transversal and of complementary dimensions.
The second condition in the above definition is equivalent to the following condition on tangent spaces: $T_{{\ensuremath{\bar{z}} }}{\ensuremath{\mathcal{Z}}}=T_{{\ensuremath{\bar{z}} }}{\ensuremath{\mathcal{P}}}\oplus T_{{\ensuremath{\bar{z}} }}{\ensuremath{\mathcal{O}}}_{{\ensuremath{\bar{z}} }}$, $\forall {\ensuremath{\bar{z}} }\in
{\ensuremath{\mathcal{P}}}\cap {\ensuremath{\mathcal{U}}}$.
An embedded submanifold of codimension ${s}$ is locally given as the zero set of ${s}$ independent functions. Assume that $h_1(z),\dots, h_{s}(z)$ define ${\ensuremath{\mathcal{P}}}$ on ${\ensuremath{\mathcal{U}}}$. The tangent space at a point of ${\ensuremath{\mathcal{P}}}$ is the kernel of the Jacobian matrix $J_h$ at this point. A basis of infinitesimal generators $v_1,\dots,v_\kappa$, where $\kappa\geq{s}$ is the dimension of the group, span the tangent space to the orbits at each point of ${\ensuremath{\mathcal{P}}}$. Therefore the submanifold ${\ensuremath{\mathcal{P}}}$ is a local cross-section if and only if the span of the infinitesimal generators $v_1,\dots,v_\kappa$ has a trivial intersection with the kernel of $J_h$ on ${\ensuremath{\mathcal{P}}}$. Equivalently: $$\label{tr} \mbox{ the rank of the } {s}\times\kappa
\mbox{ matrix }
\left(v_j(h_i)\right)_{i=1..{s}}^{j=1..\kappa}=J_h \cdot V
\mbox{ equals to } {s}\mbox{ on } {\ensuremath{\mathcal{P}}},$$ where $V$ is the $n\times \kappa$ matrix, whose $i$-th column consists of the coefficients of the infinitesimal generator $v_i$ in a local coordinate system. In the next theorem we prove the existence of a local cross-section through every point. The first paragraph of the proof provides a [simple practical algorithm to construct a coordinate local cross-section through a point]{}. An algebraic counterpart of this statements is given by Theorem \[cross:existence\].
\[banana\] Let ${\ensuremath{\mathcal{G}}}$ act semi-regularly on ${\ensuremath{\mathcal{Z}}}$. Through every point ${\ensuremath{\bar{z}} }\in{\ensuremath{\mathcal{Z}}}$ there is a local cross-section that is defined as the level set of ${s}$ coordinate functions.
Let $V$ be the $n\times \kappa$ matrix of the coefficients of the infinitesimal generators $v_1,\dots,v_\kappa$ relative to a coordinate system $z_1,\dots,z_k$. The rank of $V$ equals to the dimension of the orbits ${s}$. Thus there exist ${s}$ rows of $V$ that form an ${s}\times\kappa$ submatrix $\hat V$ of rank ${s}$ at the point ${\ensuremath{\bar{z}} }$, and therefore it has rank ${s}$ on an open neighborhood ${\ensuremath{\mathcal{U}}}_1\ni {\ensuremath{\bar{z}} }$. Assume that these rows correspond to coordinate $z_{i_1},\dots,z_{i_{s}}$. Let $(c_1,\dots,c_n)$ be coordinates of the point ${\ensuremath{\bar{z}} }$, then functions $h_1=z_{i_1}-c_{i_1},\dots,h_{s}=z_{i_{s}}-c_{i_{s}}$ satisfy condition [(\[tr\])]{}. The common zero set ${\ensuremath{\mathcal{P}}}$ of these functions contains ${\ensuremath{\bar{z}} }$.
It remains to prove that there exists a neighborhood ${\ensuremath{\mathcal{U}}}\ni{\ensuremath{\bar{z}} }$ such that ${\ensuremath{\mathcal{P}}}$ intersects each connected component of the orbits on ${\ensuremath{\mathcal{U}}}$ at a unique point. Let $x_1,\dots,x_s,y_1,\dots,y_{n-{s}}$ be a flat coordinate system in an open neighborhood ${\ensuremath{\mathcal{U}}}_2\ni {\ensuremath{\bar{z}} }$. Due to [Theorem \[fund\]]{} $y_1,\dots,y_{n-{s}}$ are independent local invariants. We will show that functions $z_{i_1},\dots,z_{i_{s}},y_1,\dots,y_{n-{s}}$ provide a coordinate system an open set ${\ensuremath{\mathcal{U}}}={\ensuremath{\mathcal{U}}}_1\cap{\ensuremath{\mathcal{U}}}_2$ containing ${\ensuremath{\bar{z}} }$. Without loss of generality we may assume that $\{z_{i_1},\dots,z_{i_{s}}\}=
\{z_{1},\dots,z_{{s}}\}$ are the first ${s}$ coordinates. In terms of flat coordinates $z_i=F_i(x,y),\,i=1..{s}$, where $F_i$ are smooth functions on ${\ensuremath{\mathcal{U}}}_2$. Since $v_i(y_j)=0$ for $i=1..\kappa,\,j=1.. n-{s}$, then $$\label{au1}\left(v_j(z_i)\right)^{j=1..\kappa}_{i=1..{s}}=\left(\frac{\partial F_i}{\partial x_r}\right)^{r=1..{s}}_{i=1..{s}}\cdot\left(v_j(x_r)\right)^{j=1..\kappa}_{r=1..{s}}.$$ We note that $\left(v_i(z_j)\right)_{j=1..{s}}^{i=1..\kappa}=\hat V$ is ${s}\times\kappa$ matrix of rank ${s}$ at each point of ${\ensuremath{\mathcal{U}}}$. Matrix $\left(v_j(x_r)\right)^{j=1..\kappa}_{r=1..{s}}$ also has maximal rank ${s}$ on ${\ensuremath{\mathcal{U}}}$. Therefore the matrix $\left(\frac{\partial F_i}{\partial x_r}\right)^{r=1..{s}}_{i=1..{s}}$ is invertible on ${\ensuremath{\mathcal{U}}}$. By looking at the rank of the corresponding Jacobian matrix in flat coordinates, we conclude that functions $z_1,\dots,z_{s},y_1,\dots,y_{n-{s}}$ are independent at each point of ${\ensuremath{\mathcal{U}}}$, and therefore define a coordinate system on ${\ensuremath{\mathcal{U}}}$.
By construction all points on ${\ensuremath{\mathcal{P}}}$ have the same $z$-coordinates. Thus two distinct points of ${\ensuremath{\mathcal{P}}}$ must differ by at least one of the $y$-coordinates. Since $y$ coordinates are constant on the connected components of the orbits on ${\ensuremath{\mathcal{U}}}$, distinct points of ${\ensuremath{\mathcal{P}}}$ belong to distinct connected components of the orbits.
Given a cross-section on ${\ensuremath{\mathcal{U}}}$ one can define a projection from the set of smooth functions on ${\ensuremath{\mathcal{U}}}$ to the set of local invariants.
\[iota\] Let ${\ensuremath{\mathcal{P}}}$ be a local cross-section to the orbits on an open set ${\ensuremath{\mathcal{U}}}$. Let $f$ be a smooth function on ${\ensuremath{\mathcal{U}}}$. The *invariantization* ${\ensuremath{\bar{\iota}}}f$ of $f$ is the function on ${\ensuremath{\mathcal{U}}}$ that is defined, for ${\ensuremath{\bar{z}} }\in {\ensuremath{\mathcal{U}}}$, by ${\ensuremath{\bar{\iota}}}f({\ensuremath{\bar{z}} })=f ({\ensuremath{\bar{z}} }_0),$ where ${\ensuremath{\bar{z}} }_0={\ensuremath{\mathcal{O}}}_{{\ensuremath{\bar{z}} }}^0\cap{\ensuremath{\mathcal{P}}}$.
In other words, the invariantization of a function $f$ is obtained by spreading the values of $f$ on ${\ensuremath{\mathcal{P}}}$ along the orbits. The next theorem shows that ${\ensuremath{\bar{\iota}}}f$ is the unique local invariant with the same values on ${\ensuremath{\mathcal{P}}}$ as $f$.
\[invariantizationII\] Let a Lie group ${\ensuremath{\mathcal{G}}}$ act semi-regularly on a manifold ${\ensuremath{\mathcal{Z}}}$, and let ${\ensuremath{\mathcal{P}}}$ be a local cross-section. Then ${\ensuremath{\bar{\iota}}}f$ is the unique local invariant defined on ${\ensuremath{\mathcal{U}}}$ whose restriction to ${\ensuremath{\mathcal{P}}}$ is equal to the restriction of $f$ to ${\ensuremath{\mathcal{P}}}$. In other words ${\ensuremath{\bar{\iota}}}f|_{\ensuremath{\mathcal{P}}}=f|_{\ensuremath{\mathcal{P}}}$.
For any ${\ensuremath{\bar{z}} }\in{\ensuremath{\mathcal{U}}}$ and small enough $\varepsilon$ the point $\exp(\varepsilon v, {\ensuremath{\bar{z}} })$ belongs to the same connected component ${\ensuremath{\mathcal{O}}}_{{\ensuremath{\bar{z}} }}^0$. Let ${\ensuremath{\bar{z}} }_0={\ensuremath{\mathcal{O}}}_{{\ensuremath{\bar{z}} }}^0\cap{\ensuremath{\mathcal{P}}}$. Then ${\ensuremath{\bar{\iota}}}f\left(\exp(\varepsilon v,
{\ensuremath{\bar{z}} })\right)=f({\ensuremath{\bar{z}} }_0)={\ensuremath{\bar{\iota}}}f({\ensuremath{\bar{z}} })$, and thus ${\ensuremath{\bar{\iota}}}f$ is a local invariant. By definition ${\ensuremath{\bar{\iota}}}f({\ensuremath{\bar{z}} }_0)=f({\ensuremath{\bar{z}} }_0)$ for all ${\ensuremath{\bar{z}} }_0\in
{\ensuremath{\mathcal{P}}}$.
In order to show its smoothness we write ${\ensuremath{\bar{\iota}}}f$ in terms of flat coordinates $ x_1,\dots, x_{s}$, $y_1,\dots,y_{n-{s}}$. By probably shrinking ${\ensuremath{\mathcal{U}}}$, we may assume that ${\ensuremath{\mathcal{P}}}$ is given by the zero-set of smooth independent functions $h_1(x_1,\dots,
x_{s},y_1,\dots,y_{n-{s}}),\dots,$ $ h_{s}(x_1,\dots,
x_{s},y_1,\dots,y_{n-{s}})$. From the transversality condition [(\[tr\])]{} and local invariance of $y$’s, it follows that the first ${s}$ columns of the Jacobian matrix $J_h$ form a submatrix of rank ${s}$. Thus the cross-section ${\ensuremath{\mathcal{P}}}$ can be described as a graph
$x_1=p_1(y_1,\dots,y_{n-{s}}), \dots,
x_{s}=p_{s}(y_1,\dots,y_{n-{s}})$, where $p_1,\dots,p_{s}$ are smooth functions. Then the function
$${\ensuremath{\bar{\iota}}}f(x_1,\dots, x_{s},y_1,\dots,y_{n-{s}} )=f\left(
p_1(y_1,\dots,y_{n-{s}}), \dots,
p_{s}(y_1,\dots,y_{n-{s}}\right),y_1,\dots,y_{n-{s}})$$
is smooth, as a composition of smooth functions.
To prove the uniqueness, assume that an invariant function $q$ has the same values on ${\ensuremath{\mathcal{P}}}$ as $f$, then the invariant function $h={\ensuremath{\bar{\iota}}}f-q$ has zero value on ${\ensuremath{\mathcal{P}}}$. A point ${\ensuremath{\bar{z}} }\in{\ensuremath{\mathcal{U}}}$ can be reached from ${\ensuremath{\bar{z}} }_0={{\ensuremath{\mathcal{P}}}}\cap {\ensuremath{\mathcal{O}}}_{{\ensuremath{\bar{z}} }}^0$ by a composition of flows defined by infinitesimal generators. Without loss of generality, we may assume that it can be reached by a single flow ${\ensuremath{\bar{z}} }=\exp (\epsilon v, {\ensuremath{\bar{z}} }_0)$, where $\exp (\varepsilon v, {\ensuremath{\bar{z}} }_0)\subset {\ensuremath{\mathcal{O}}}_{{\ensuremath{\bar{z}} }}^0$ for all $0\leq
\varepsilon \leq \epsilon$. From the invariance of $h$ it follows that $h\left(\exp (\epsilon v,
{\ensuremath{\bar{z}} }_0)\right)=h({\ensuremath{\bar{z}} }_0)=0$. Thus $q(z)={\ensuremath{\bar{\iota}}}f(z)$ on ${\ensuremath{\mathcal{U}}}$.
Theorem \[invariantizationII\] allows us to view the invariantization process as a projection from the set of smooth functions on ${\ensuremath{\mathcal{U}}}$ to the equivalence classes of functions with the same value on ${\ensuremath{\mathcal{P}}}$. Each equivalence class contains a unique local invariant. The algebraic counterpart of this point of view is described in Section \[inv\].
The invariantization of differential forms can be defined in a similar implicit manner. It has been shown in [@olver99; @ko03] that the essential information about the differential ring of invariants and the structure of differential forms can be computed from the infinitesimal generators of the action and the equations that define the cross-section, without explicit formulas for invariants.
Normalized and fundamental invariants {#normfund}
-------------------------------------
The *normalized invariants* introduced in [@olver99] are the invariantizations of the coordinate functions. They have the replacement property. In the algebraic context they correspond to *replacement* invariants defined in Section \[invariantization\]. This correspondence is made precise by Proposition \[smooth::replacement\]. We show that a set of normalized invariants contains a fundamental set of local invariants.
All results of this subsection are stated under the following assumptions. A manifold ${\ensuremath{\mathcal{P}}}$ is a local cross-section to the ${s}$-dimensional orbits of a semi-regular ${\ensuremath{\mathcal{G}}}$-action on an open ${\ensuremath{\mathcal{U}}}\subset{\ensuremath{\mathcal{Z}}}$, and ${\ensuremath{\bar{\iota}}}$ is the corresponding invariantization map. The set ${\ensuremath{\mathcal{U}}}$ is a single coordinate chart on ${\ensuremath{\mathcal{Z}}}$ with coordinate functions $z_1,\dots,z_n$. By possibly shrinking ${\ensuremath{\mathcal{U}}}$ we may assume that ${\ensuremath{\mathcal{P}}}$ is the zero set of ${s}$ independent smooth functions.
Since our definition of invariantization differs from [@olver99] we restate and prove the replacement theorem.
\[thomas\] If $f(z_1,\dots,z_n)$ is a local invariant on ${\ensuremath{\mathcal{U}}}$ then $f({\ensuremath{\bar{\iota}}}z_1,\dots,{\ensuremath{\bar{\iota}}}z_n)=f( z_1,\dots,z_n)$.
Since ${\ensuremath{\bar{\iota}}}z_1|_{\ensuremath{\mathcal{P}}}=z_1|_{\ensuremath{\mathcal{P}}},\dots,{\ensuremath{\bar{\iota}}}z_n|_{\ensuremath{\mathcal{P}}}=z_n|_{\ensuremath{\mathcal{P}}}$, then $f({\ensuremath{\bar{\iota}}}z_1,\dots,{\ensuremath{\bar{\iota}}}z_n)|_{\ensuremath{\mathcal{P}}}=f( z_1,\dots,z_n)|_{\ensuremath{\mathcal{P}}}$. Thus functions $f({\ensuremath{\bar{\iota}}}z_1,\dots,{\ensuremath{\bar{\iota}}}z_n)$ and $f( z_1,\dots,z_n)$ are both local invariants and have the same value on ${\ensuremath{\mathcal{P}}}$. By Theorem \[invariantizationII\] they coincide.
\[syzygy0\] Let ${\ensuremath{\mathcal{P}}}$ be a local cross-section on ${\ensuremath{\mathcal{U}}}$, given as the zero set of $s$ independent functions $h_1,\ldots,h_s$. Then $h_1({\ensuremath{\bar{\iota}}}z_1, \ldots, {\ensuremath{\bar{\iota}}}z_n)=0, \ldots, h_s({\ensuremath{\bar{\iota}}}z_1, \ldots, {\ensuremath{\bar{\iota}}}z_n)=0$ on ${\ensuremath{\mathcal{U}}}$. If for a differentiable $n$-variable function $f$ we have $f({\ensuremath{\bar{\iota}}}z_1, \ldots, {\ensuremath{\bar{\iota}}}z_n)\equiv 0$ on an open subset of ${\ensuremath{\mathcal{U}}}$, then there exits open ${\cal W}\subset {\ensuremath{\mathcal{U}}}$ such that ${\cal W}\cap{\ensuremath{\mathcal{P}}}\neq\emptyset$ and at each point of ${\cal W}\cap{{\ensuremath{\mathcal{P}}}}$ functions $f$, $h_1, \ldots, h_s$ are not independent.
Since $h({\ensuremath{\bar{\iota}}}z)|_{\ensuremath{\mathcal{P}}}={\ensuremath{\bar{\iota}}}h(z)|_{\ensuremath{\mathcal{P}}}$ and both functions are invariants, one has $h({\ensuremath{\bar{\iota}}}z)={\ensuremath{\bar{\iota}}}h(z)$ by [Theorem \[invariantizationII\]]{}. The latter is zero since $h|_{{\ensuremath{\mathcal{P}}}}=0$. Assume now that there exits a differentiable function $f$ and an open subset of $\cal V\subset {\ensuremath{\mathcal{U}}}$ such that $f({\ensuremath{\bar{\iota}}}z_1, \ldots, {\ensuremath{\bar{\iota}}}z_n)\equiv 0$ on $\cal V$. Since $f({\ensuremath{\bar{\iota}}}z)={\ensuremath{\bar{\iota}}}f( z)$ is invariant, there exists an open ${\cal W}\supset {\cal V}$ such that $f({\ensuremath{\bar{\iota}}}z_1, \ldots, {\ensuremath{\bar{\iota}}}z_n)\equiv 0$ on ${\cal W}$ and ${\cal W}\cap{\ensuremath{\mathcal{P}}}\neq\emptyset$. We conclude that $f(z_1,\dots,z_n)\equiv 0$ on ${\ensuremath{\mathcal{P}}}\cap {\cal W}$. In this case $f$ cannot be independent of $h_1,\ldots,h_s$ at any point of ${\ensuremath{\mathcal{P}}}\cap W$ since otherwise this would imply that ${\ensuremath{\mathcal{P}}}$ is of dimension less then $n-s$.
Let ${\ensuremath{\mathcal{P}}}$ be a local cross-section on ${\ensuremath{\mathcal{U}}}$, given as the zero set of $s$ independent functions. The set $\{{\ensuremath{\bar{\iota}}}z_1,\dots,{\ensuremath{\bar{\iota}}}z_n\}$ of the invariantizations of the coordinate functions $z_1,\dots,z_n$ contains a fundamental set of $n-{s}$ local invariants on ${\ensuremath{\mathcal{U}}}$.
Due to the implicit function theorem, after a possible shrinking ${\ensuremath{\mathcal{U}}}$ and renumbering of the coordinate functions, we may assume that ${\ensuremath{\mathcal{P}}}$ is the zero set of the functions $h_1(z)=z_1-p_1(z_{{s}+1},\dots, z_{n}), \ldots,
h_{s}(z)=z_{s}- p_{s}(z_{{s}+1},\dots,z_{n})$. Therefore ${\ensuremath{\bar{\iota}}}z_1 = p_1({\ensuremath{\bar{\iota}}}z_{{s}+1},\dots, {\ensuremath{\bar{\iota}}}z_{n}),\ldots,
{\ensuremath{\bar{\iota}}}z_{s}= p_k({\ensuremath{\bar{\iota}}}z_{{s}+1},\dots, {\ensuremath{\bar{\iota}}}z_{n})$ by [Theorem \[invariantizationII\]]{}. From [Theorem \[thomas\]]{} we can conclude that any local invariant can be written in terms of ${\ensuremath{\bar{\iota}}}z_{{s}+1},\dots, {\ensuremath{\bar{\iota}}}z_{n}$. Since for every differentiable non-zero $n-s$-variable function $f$, functions $f(z_{{s}+1},\dots, z_{n}), h_1(z),\dots,h_{s}(z)$ are independent at every point of ${\ensuremath{\mathcal{U}}}$, then by [Lemma \[syzygy0\]]{}, ${\ensuremath{\bar{\iota}}}z_{{s}+1},\dots, {\ensuremath{\bar{\iota}}}z_{n}$ are functionally independent on ${\ensuremath{\mathcal{U}}}$.
Relation between the algebraic and the smooth constructions {#smooth::alg}
-----------------------------------------------------------
We establish a connection between the smooth and the algebraic constructions. We show that the normalized invariants ([Section \[normfund\]]{}) can be viewed as smooth representatives of the replacement invariants ([Section \[repl\]]{}), and that algebraic invariantization ([Section \[inv\]]{}) provides a constructive approach to smooth invariantization ([Section \[smooth::section\]]{}).
To be at the intersection of the hypotheses of the smooth and the algebraic settings we consider a real algebraic group, that is the set of real points of an algebraic group defined[^8] over ${{\ensuremath{\mathbb{R}}}}$. It is a real Lie group [@springer89 the Proposition in Chapter 3, Section 2.1.2]. Lie groups appearing in applications often satisfy this property. We also assume that the local action is given by a rational map [(\[action\])]{}, in [Section \[agroupaction:def\]]{}, that satisfies [Asumption \[groupaction:hyp\]]{}. This guarantees semi-regularity of the action on an open set ${\ensuremath{\mathcal{Z}}}$ of ${{\ensuremath{\mathbb{R}}}}^n$ as the orbits of non-maximal dimension are contained in an algebraic set defined by minors of the matrix $V$ of [(\[tr\])]{}, in [Section \[smooth::section\]]{}.
In [Section \[groupaction\]]{} to \[invariantization\] we assumed for convenience of writing that the field of coefficients ${\ensuremath{\mathbb{K} } }$ was algebraically closed. Yet the algebraic constructions of those sections require no extension of the field of definition of the group or the action. With the initial data described above, [Theorem \[invgb\]]{} produces a set of rational invariants in ${\ensuremath{{{{\ensuremath{\mathbb{R}}}}(z)^G}} }$ that generate ${\ensuremath{{{{\ensuremath{\mathbb{R}}}}(z)^G}} }$ by [Theorem \[rewrite\]]{}.
Rational invariants are obviously local invariants. We show that so are smooth representatives of algebraic invariants. The following definition formalizes the notion of a smooth representative of an algebraic function.
A smooth map $F:{\ensuremath{\mathcal{U}}}\rightarrow{{\ensuremath{\mathbb{R}}}}^k$ is a smooth zero of $\{p_1, \ldots, p_\kappa\} \subset {{\ensuremath{\mathbb{R}}}}(z)[\zeta_1,\ldots, \zeta_k]$ if the coefficients of the $p_i$ are well defined on ${\ensuremath{\mathcal{U}}}$ and $p_i({\ensuremath{\bar{z}} },F({\ensuremath{\bar{z}} }))=0$ for all ${\ensuremath{\bar{z}} }\in {\ensuremath{\mathcal{U}}}$. In this case we also say that $F$ is a smooth zero of the ideal $(p_1, \ldots, p_\kappa)$.
\[smz\] Assume $F:{\ensuremath{\mathcal{U}}}\rightarrow{{\ensuremath{\mathbb{R}}}}^k$ is a smooth zero of $\{p_1, \ldots, p_\kappa\} \subset {\ensuremath{{{{\ensuremath{\mathbb{R}}}}(z)^G}} }[\zeta_1,\ldots, \zeta_k]$. If $(p_1, \ldots, p_\kappa)$ is a zero dimensional ideal then the components of $F$ are local invariants.
Let $p\in{\ensuremath{{{{\ensuremath{\mathbb{R}}}}(z)^G}} }[\zeta]$, that is $p(z,\zeta)=\sum_{\alpha\in{{\ensuremath{\mathbb{N}}}}^n}a_{\alpha}(z)\zeta^{\alpha}$, where $a_{\alpha}(z)\in {\ensuremath{{{{\ensuremath{\mathbb{R}}}}(z)^G}} }$. Assume that $p({\ensuremath{\bar{z}} },F({\ensuremath{\bar{z}} }))=0$ for all ${\ensuremath{\bar{z}} }\in {\ensuremath{\mathcal{U}}}$. For any ${\ensuremath{\bar{z}} }\in{\ensuremath{\mathcal{U}}}$ and an infinitesimal generator $v$ there exits $\epsilon>0$, such that ${\exp(\varepsilon v, {{\ensuremath{\bar{z}} }})}\in{\ensuremath{\mathcal{U}}}$ whenever $|\varepsilon|<\epsilon$. Then $p({\exp(\varepsilon v, {{\ensuremath{\bar{z}} }})},{F}({\exp(\varepsilon v, {{\ensuremath{\bar{z}} }})}))=\sum_{\alpha\in{{\ensuremath{\mathbb{N}}}}^n}a_\alpha({\exp(\varepsilon v, {{\ensuremath{\bar{z}} }})}){F}({\exp(\varepsilon v, {{\ensuremath{\bar{z}} }})})^\alpha=0$. Since the coefficients $a_{\alpha}$ are invariant $\sum_{\alpha\in{{\ensuremath{\mathbb{N}}}}^n}a_{\alpha}({\ensuremath{\bar{z}} }){F}({\exp(\varepsilon v, {{\ensuremath{\bar{z}} }})})^\alpha= 0$ for all ${\ensuremath{\bar{z}} }\in{\ensuremath{\mathcal{U}}}$ and small enough $\varepsilon$. Thus for a fixed point ${\ensuremath{\bar{z}} }$ all the values ${F}({\exp(\varepsilon v, {{\ensuremath{\bar{z}} }})})$ for all sufficiently small $\varepsilon$ are the common roots of the set of polynomials $\{p_1, \ldots, p_\kappa\}$. Since by the assumption the number of roots is finite, we conclude that ${F}({\exp(\varepsilon v, {{\ensuremath{\bar{z}} }})})={F}\left(\exp(0 v, {{\ensuremath{\bar{z}} }})\right)={F}( {{\ensuremath{\bar{z}} }})$ and thus the components of ${F}(z)$ are local invariants.
It follows from [Theorem \[banana\]]{} that, through every point of ${\ensuremath{\mathcal{Z}}}$, there exists a local cross-sections defined by linear equations over ${{\ensuremath{\mathbb{R}}}}$. Conversely, we can consider a cross-section ${\ensuremath{\mathcal{P}}}$, defined over ${{\ensuremath{\mathbb{R}}}}$, that has non singular real points, meaning that the real part has the same dimension as the complex part. For any point ${\ensuremath{\bar{z}} }\in{\ensuremath{\mathcal{Z}}}\cap{\ensuremath{\mathcal{P}}}$ where the rank of the matrix [(\[tr\])]{} does not drop, there is a neighborhood ${\ensuremath{\mathcal{U}}}$ on which ${\ensuremath{\mathcal{P}}}$ defines a local cross-section, and such points are dense in ${\ensuremath{\mathcal{P}}}$.
The ${\ensuremath{{{{\ensuremath{\mathbb{R}}}}(z)^G}} }$-zero of the zero dimensional ideal ${\ensuremath{I}}^G= ( G+P+(z-g(\lambda,z)) )\cap {\ensuremath{{{{\ensuremath{\mathbb{R}}}}(z)^G}} }[Z]$ are precisely the replacement invariants. According to the previous proposition the smooth zeros of this ideal are local invariants. We characterize the tuple of normalized invariants as one of them.
\[smooth::replacement\] Let ${\ensuremath{\mathcal{P}}}$ be an algebraic cross-section which, when restricted to an open set ${\ensuremath{\mathcal{U}}}$, defines a smooth cross-section. The tuple of normalized invariants ${\ensuremath{\bar{\iota}}}z=( {\ensuremath{\bar{\iota}}}z_1,\dots,{\ensuremath{\bar{\iota}}}z_n)$ is the smooth zero of the ideal ${\ensuremath{I}}^G$ whose components agree with the coordinate functions on ${\ensuremath{\mathcal{P}}}\cap {\ensuremath{\mathcal{U}}}$.
Let ${\ensuremath{\bar{z}} }\in {\ensuremath{\mathcal{U}}}$ be an arbitrary point, and let ${\ensuremath{\bar{z}} }_0$ be the point of intersection of ${\ensuremath{\mathcal{P}}}$ with the connected component of ${\ensuremath{\mathcal{O}}}_{{\ensuremath{\bar{z}} }}\cap{\ensuremath{\mathcal{U}}}$, containing ${\ensuremath{\bar{z}} }$. Then there exists ${\ensuremath{\bar{\lambda}} }$ in the connected component of the identity of ${\ensuremath{\mathcal{G}}}$, such that ${\ensuremath{\bar{z}} }_0={\ensuremath{\bar{\lambda}} }{\ensuremath{\bar{z}} }$ so that $({\ensuremath{\bar{z}} },{\ensuremath{\bar{z}} }_0)$ is a zero of the ideal $I=O+P$. By definition ${\ensuremath{\bar{\iota}}}z({\ensuremath{\bar{z}} })={\ensuremath{\bar{z}} }_0$ and therefore $({\ensuremath{\bar{z}} },{\ensuremath{\bar{\iota}}}z({\ensuremath{\bar{z}} }))$ is a zero of the ideal $I$ for all ${\ensuremath{\bar{z}} }\in{\ensuremath{\mathcal{U}}}$. Equivalently ${\ensuremath{\bar{\iota}}}z$ is a smooth zero of ${\ensuremath{I}}^G$. By [Theorem \[invariantizationII\]]{} it is the unique tuple of local invariants that agree with the coordinate functions on ${\ensuremath{\mathcal{P}}}\cap{\ensuremath{\mathcal{U}}}$.
Therefore a replacement invariant not only generates algebraic invariants but their smooth representatives also generate local invariants.
\[scaling:smooth\] <span style="font-variant:small-caps;">scaling</span>. The action defined in [Example \[scaling:def\]]{} corresponds to the following action of the multiplicative group ${{\ensuremath{\mathbb{R}}}}^{*}$: $$g : \begin{array}[t]{ccl}
{{\ensuremath{\mathbb{R}}}}^{*} \times {{\ensuremath{\mathbb{R}}}}^2 & \rightarrow &{{\ensuremath{\mathbb{R}}}}^2 \\
(\lambda, z_1,z_2) & \mapsto & (\lambda z_1, \lambda z_2).
\end{array}$$ The action is semi-regular on ${{\ensuremath{\mathbb{R}}}}^2\setminus\{(0,0)\}$.
In [Example \[scaling:cross\]]{} we chose the cross-section ${\ensuremath{\mathcal{P}}}$ defined by $z_1=1$. The cross-section being of degree 1 there is a single associated replacement invariant that corresponds to the tuple $(1,\frac{z_2}{z_1})$ of rational invariants.
Let ${\ensuremath{\mathcal{U}}}=\{(z_1,z_2) \in {{\ensuremath{\mathbb{R}}}}^2 \; |\;z_1\neq 0\}$. The components of the smooth map $F : {\ensuremath{\mathcal{U}}}\rightarrow {{\ensuremath{\mathbb{R}}}}^2$ s.t. $F(z_1,z_2)=(1,\frac{z_2}{z_1})$ are the normalized invariants for the local cross-section ${\ensuremath{\mathcal{P}}}\cap {\ensuremath{\mathcal{U}}}$.
\[translation:smooth\] <span style="font-variant:small-caps;">translation+reflection</span>. The action defined in [Example \[translation:def\]]{} corresponds to the following action of the Lie group ${{\ensuremath{\mathbb{R}}}}\times \{1,1\}$ given by $$g : \begin{array}[t]{ccl}
{{\ensuremath{\mathbb{R}}}}\times \{1,1\} \times {{\ensuremath{\mathbb{R}}}}^2 & \rightarrow &{{\ensuremath{\mathbb{R}}}}^2 \\
(\lambda_1,\lambda_2, z_1,z_2) & \mapsto &
\left( z_1 +\lambda_1, \lambda_2\,z_2 \right) .
\end{array}$$ The action is semi-regular on ${{\ensuremath{\mathbb{R}}}}^2$.
In [Example \[translation:cross\]]{} we chose the cross-section ${\ensuremath{\mathcal{P}}}$ defined by $z_2=z_1$. There are two replacement invariants associated to ${\ensuremath{\mathcal{P}}}$: $\xi^{(\pm)}=( \pm z_2,\pm z_2)$. They both correspond to smooth maps $F^{(\pm)}:{{\ensuremath{\mathbb{R}}}}^2 \rightarrow {{\ensuremath{\mathbb{R}}}}^2$ the components of which are local invariants.
Only $(z_2,z_2)$ coincides with the coordinate functions on ${\ensuremath{\mathcal{P}}}$, that defines a local cross-section on ${\ensuremath{\mathcal{U}}}={{\ensuremath{\mathbb{R}}}}^2$. The normalized invariants are thus $(z_2,z_2)$.
\[rotation:smooth\] <span style="font-variant:small-caps;">rotation</span> The action defined in [Example \[rotation:def\]]{} corresponds to the following action of the additive group ${{\ensuremath{\mathbb{R}}}}$ given by $$g : \begin{array}[t]{ccl}
{{\ensuremath{\mathbb{R}}}}\times {{\ensuremath{\mathbb{R}}}}^2 & \rightarrow &{{\ensuremath{\mathbb{R}}}}^2 \\
(t, z_1,z_2) & \mapsto &
\left(\frac{1-t^2}{1+t^2} z_1 -\frac{2t}{1+t^2} z_2 ,
\frac{2t}{1+t^2} z_1+\frac{1-t^2}{1+t^2} z_2\right) .
\end{array}$$ The action is semi-regular on ${{\ensuremath{\mathbb{R}}}}^2\setminus\{(0,0)\}$.
In [Example \[rotation:cross\]]{} we chose the cross-section ${\ensuremath{\mathcal{P}}}$ defined by $z_2=0$. The replacement invariants associated to the cross-section ${\ensuremath{\mathcal{P}}}$ are the $\overline{{{\ensuremath{\mathbb{R}}}}(z)}^G$-zeros of the ideal ${\ensuremath{I}}^G =(Z_2,Z_1^2-(z_1^2+z_2^2))$.
The smooth maps $F^{(\pm)} : {{\ensuremath{\mathbb{R}}}}^2\setminus\{(0,0)\} \rightarrow {{\ensuremath{\mathbb{R}}}}^2$ s.t. $F^{(\pm)}(z_1,z_2)=(0,\pm \sqrt{z_1^2+z_2^2})$ are smooth zeros of $I^G$. Their components are thus local invariants.
The cross-section ${\ensuremath{\mathcal{P}}}$ defines a local cross-section for instance on ${\ensuremath{\mathcal{U}}}={{\ensuremath{\mathbb{R}}}}^2\setminus \{(z_1,z_2) \; |\;z_1=0, z_2\leq0\}$. As $F^{(+)}|_{{\ensuremath{\mathcal{P}}}\cap{\ensuremath{\mathcal{U}}}} = z_1$, the tuple of normalized invariants are $(0, \sqrt{z_1^2+z_2^2})$ on ${\ensuremath{\mathcal{U}}}$.
We conclude this section by linking the smooth invariantization and the algebraic invariantization introduced in [Section \[inv\]]{}. Recall that the algebraic invariantization was a map that associated a univariate polynomial over ${\ensuremath{{{{\ensuremath{\mathbb{R}}}}(z)^G}} }$ to univariate polynomials over ${\ensuremath{\mathbb{K} } }[z]_{\ensuremath{\mathcal{P}}}$ (Definition \[inv:def\]).
Let ${\ensuremath{\mathcal{P}}}$ be an algebraic cross-section which, when restricted to an open set ${\ensuremath{\mathcal{U}}}$, defines a local cross-section. Let $f:{\ensuremath{\mathcal{U}}}\rightarrow {{\ensuremath{\mathbb{R}}}}$ be a smooth zero of a univariate polynomial $\beta\in{\ensuremath{\mathbb{K} } }(z)[\zeta]$. The smooth invariantization ${\ensuremath{\bar{\iota}}}f$ of $f$ is a smooth zero of the algebraic ${\ensuremath{\mathcal{P}}}$-invariantization ${\ensuremath{\iota}}\beta\in{\ensuremath{{{{\ensuremath{\mathbb{R}}}}(z)^G}} }[\zeta]$ of $\beta$.
The polynomial ${\ensuremath{\iota}}\beta(z,\zeta)=\sum_{i=1}^{k}b_{i}(z)\zeta^{i}$, where $b_{i}\in{\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$. Any point ${\ensuremath{\bar{z}} }\in{\ensuremath{\mathcal{U}}}$ can obtained from the point ${\ensuremath{\bar{z}} }_{0}\in{\ensuremath{\mathcal{P}}}$ by a composition of flows along infinitesimal generators of the group action. The argument will not change if we assume that $ {\ensuremath{\bar{z}} }=\exp(\varepsilon v,{\ensuremath{\bar{z}} }_{0})$ is obtained by the flow along a single vector field. Then from the invariance of $b_{i}(z)$ and local invariance of ${\ensuremath{\bar{\iota}}}{f}(z)$ it follows that $\forall{\ensuremath{\bar{z}} }\in{\ensuremath{\mathcal{U}}}$: $$\begin{aligned}
{\ensuremath{\iota}}\beta({\ensuremath{\bar{z}} },{\ensuremath{\bar{\iota}}}{f}({\ensuremath{\bar{z}} }))
&=&\sum_{i=1}^{k}b_i\left(\exp(\varepsilon v,{\ensuremath{\bar{z}} }_{0})\right)f\left(\exp(\varepsilon
v,{\ensuremath{\bar{z}} }_{0})\right)^i\\
&=&\sum_{i=1}^{k}b_{i}({\ensuremath{\bar{z}} }_0){\ensuremath{\bar{\iota}}}{f}({\ensuremath{\bar{z}} }_0)^i
={\ensuremath{\iota}}\beta\left({\ensuremath{\bar{z}} }_0,{\ensuremath{\bar{\iota}}}{f}(z_{0})\right), \,
\mbox{ where } {\ensuremath{\bar{z}} }_0\in{\ensuremath{\mathcal{P}}}\cap{\ensuremath{\mathcal{U}}}.\end{aligned}$$ From Proposition \[factor\] it follows that ${\ensuremath{\iota}}\beta$ is divisible by $\beta$ when restricted to ${\ensuremath{\mathcal{P}}}$. Thus ${\ensuremath{\iota}}\beta({{\ensuremath{\bar{z}} }}_0,{f}({{\ensuremath{\bar{z}} }}_0))=0,\quad \forall
{{\ensuremath{\bar{z}} }}_0\in{\ensuremath{\mathcal{P}}}\cap{\ensuremath{\mathcal{U}}}$, since $\beta({\ensuremath{\bar{z}} },{f}({\ensuremath{\bar{z}} }))\equiv 0 $ on ${\ensuremath{\mathcal{U}}}$. It follows that ${\ensuremath{\bar{\iota}}}{f}(z)$ is a smooth zero of a polynomial ${\ensuremath{\bar{\iota}}}\beta(z,\zeta)\in {\ensuremath{\mathbb{K} } }(z)^G[\zeta]$.
In particular if $r(z)$ is a rational function that is well defined on ${\ensuremath{\mathcal{U}}}$, then its smooth invariantization ${\ensuremath{\bar{\iota}}}r(z)$ is a smooth zero of the ${\ensuremath{\mathcal{P}}}$-invariantization ${\ensuremath{\iota}}(\zeta - r(z))$ of the polynomial $\zeta-r(z)$. To discriminate the right one we only need to check that its value coincide with the one of $r(z)$ on ${\ensuremath{\mathcal{P}}}\cap{\ensuremath{\mathcal{U}}}$.
Moving frame map {#smf}
----------------
We show that the invariantization map described in Section \[inv\] generalizes the invariantization process described in [@olver99]. The latter is restricted to locally-free actions, and is based on the existence of a local ${\ensuremath{\mathcal{G}}}$-equivariant map $\rho\colon {\ensuremath{\mathcal{U}}}\to {\ensuremath{\mathcal{G}}}$. Although local freeness of the action guarantees the existence of $\rho$, due to the implicit function theorem, it might not be explicitly computable. We review the Fels-Olver construction, and prove that in the case of locally free actions it is equivalent to the one presented in Section \[smooth::section\].
\[lfree\] An action of a Lie group ${\ensuremath{\mathcal{G}}}$ on a manifold ${\ensuremath{\mathcal{Z}}}$ is *locally free* if for every point ${\ensuremath{\bar{z}} }\in {\ensuremath{\mathcal{Z}}}$ its isotropy group ${\ensuremath{\mathcal{G}}}_{{\ensuremath{\bar{z}} }}=\{{\ensuremath{\bar{\lambda}} }\in {\ensuremath{\mathcal{G}}}|{\ensuremath{\bar{\lambda}} }\cdot{\ensuremath{\bar{z}} }={\ensuremath{\bar{z}} }\}$ is discrete.
Local freeness implies semi-regularity of the action, the dimension of each orbit being equal to the dimension of the group. Theorem 4.4 from [@olver99], can be restated as follows in the case of locally free actions.
\[lmf\] A Lie group ${\ensuremath{\mathcal{G}}}$ acts locally freely on ${\ensuremath{\mathcal{Z}}}$ if and only if every point of ${\ensuremath{\mathcal{Z}}}$ has an open neighborhood ${\ensuremath{\mathcal{U}}}$ such that there exists a map $\rho\colon {\ensuremath{\mathcal{U}}}\to {\ensuremath{\mathcal{G}}}$ that makes the following diagram commute. Here the map ${\ensuremath{\bar{\mu}} }\mapsto
{\ensuremath{\bar{\mu}} }\cdot{\ensuremath{\bar{\lambda}} }^{-1}$ is chosen for the action of ${\ensuremath{\mathcal{G}}}$ on itself, and ${\ensuremath{\bar{\lambda}} }$ is taken in a suitable neighborhood (depending on the point of ${\ensuremath{\mathcal{U}}}$) of the identity in ${\ensuremath{\mathcal{G}}}$. $$\xymatrix{
{\ensuremath{\mathcal{U}}}\ar[d]_{\rho} \ar[r]^{{\ensuremath{\bar{\lambda}} }} & {\ensuremath{\mathcal{U}}}\ar[d]^{\rho}\\
{\ensuremath{\mathcal{G}}}\ar[r]_{{{\ensuremath{\bar{\lambda}} }}} & {\ensuremath{\mathcal{G}}}}$$
The map $\rho$ is locally ${\ensuremath{\mathcal{G}}}$-equivariant, $\rho({\ensuremath{\bar{\lambda}} }\cdot {\ensuremath{\bar{z}} })= \rho \cdot
{\ensuremath{\bar{\lambda}} }^{-1}$ for ${\ensuremath{\bar{\lambda}} }$ sufficiently close to the identity, and is called *a moving frame map*. If ${\ensuremath{\mathcal{P}}}$ is a cross-section, then the equation $$\label{mfdef}
\rho({\ensuremath{\bar{z}} })\cdot{\ensuremath{\bar{z}} }\in {\ensuremath{\mathcal{P}}},$$ uniquely defines $\rho({\ensuremath{\bar{z}} })$ in a sufficiently small neighborhood of the identity. In particular, $\rho({\ensuremath{\bar{z}} }_0)=e$ for all ${\ensuremath{\bar{z}} }_0\in {\ensuremath{\mathcal{P}}}$. Reciprocally, a moving frame map defines a local cross-section to the orbits: ${\ensuremath{\mathcal{P}}}=\{\rho({\ensuremath{\bar{z}} })\cdot {\ensuremath{\bar{z}} }\;|\; {\ensuremath{\bar{z}} }\in {\ensuremath{\mathcal{U}}}\}\subset {\ensuremath{\mathcal{U}}}$.
In local coordinates, Condition [(\[mfdef\])]{} gives rise to [ *implicit equations*]{} for expressing the group parameters in terms of the coordinate functions on the manifold. When the group acts locally freely, the local existence of smooth solutions is guaranteed by the transversality condition and the implicit function theorem. Since the implicit function theorem is not constructive, we might nonetheless not be able to obtain explicit formulas for the solution.
In [@olver99 Definition 4.6] the invariantization of a function $f$ on ${\ensuremath{\mathcal{U}}}$ is defined as the function whose value at a point ${\ensuremath{\bar{z}} }\in{\ensuremath{\mathcal{U}}}$ is equal to $f(\rho({\ensuremath{\bar{z}} })\cdot {\ensuremath{\bar{z}} })$. Next proposition shows that this moving frame based definition of invariantization is equivalent to Definition \[iota\] given in terms of cross-section. The advantage of the latter definition is that it is not restricted to locally free actions.
\[foinv\] Let $\rho$ be a moving frame map on ${\ensuremath{\mathcal{U}}}$. Then ${\ensuremath{\bar{\iota}}}f({\ensuremath{\bar{z}} })=f(\rho({\ensuremath{\bar{z}} })\cdot{\ensuremath{\bar{z}} }).$
Local invariance of $f(\rho(z)\cdot
z)$ follows from the local equivariance of $\rho$, i. e. for ${\ensuremath{\bar{\lambda}} }$ sufficiently close to the identity: $$f\left(\rho({\ensuremath{\bar{\lambda}} }\cdot {\ensuremath{\bar{z}} })\cdot({\ensuremath{\bar{\lambda}} }\cdot
{\ensuremath{\bar{z}} })\right)=f\left(\rho( {\ensuremath{\bar{z}} })\cdot{\ensuremath{\bar{\lambda}} }^{-1}\cdot({\ensuremath{\bar{\lambda}} }\cdot
{\ensuremath{\bar{z}} })\right)=f(\rho({\ensuremath{\bar{z}} })\cdot
{\ensuremath{\bar{z}} }.$$ Since $\rho(z_0)=e$ then $ f(\rho({\ensuremath{\bar{z}} }_0)\cdot{\ensuremath{\bar{z}} }_0)=f({\ensuremath{\bar{z}} }_0)$ for all ${\ensuremath{\bar{z}} }_0\in {\ensuremath{\mathcal{P}}}$. Thus $f(\rho(z)\cdot
z)$ is locally invariant and equals to $f$, when restricted to ${\ensuremath{\mathcal{P}}}$. The conclusion follows from Theorem \[invariantizationII\].
Thus the moving frame map offers an approach to invariantization that is constructive up to the resolution of the implicit equations given by [(\[mfdef\])]{}. In the algebraic case the moving frame map is defined by the ideal $${\ensuremath{M}}^e=
\left(\,{\ensuremath{G}}+{\ensuremath{P_{}}}+(Z-g(\lambda,z))\,\right) \;\cap\;{{\ensuremath{\mathbb{R}}}}(z)[\lambda].$$ Indeed, if $({\ensuremath{\bar{z}} },{\ensuremath{\bar{\lambda}} })$ is a zero of $M=M^e\cap{{\ensuremath{\mathbb{R}}}}[z,\lambda]$, in an appropriate open set of ${\ensuremath{\mathcal{Z}}}\times{\ensuremath{\mathcal{G}}}$, then ${\ensuremath{\bar{\lambda}} }\cdot{{\ensuremath{\bar{z}} }}\in {\ensuremath{\mathcal{P}}}$. The action is locally free if and only if ${\ensuremath{M}}^e$ is zero dimensional.
In this case, the smooth zero $F:{\ensuremath{\mathcal{U}}}\rightarrow{\ensuremath{\mathcal{G}}}$ of $M^e$, that is the identity of the group when restricted to ${\ensuremath{\mathcal{P}}}$, provides a moving frame map $\rho$ on ${\ensuremath{\mathcal{U}}}$.
If one can obtain the map $\rho$ explicitly, the invariantization map can be computed using [Proposition \[foinv\]]{}. Even in this favorable case, the expression for $\rho$ often involves algebraic functions which can prove difficult to manipulate symbolically. The purely algebraic approach proposed in [Section \[invariantization\]]{} is more suitable for symbolic computation.
Additional examples {#examples}
===================
We first consider a linear action of $S\!L_2$ on ${\ensuremath{\mathbb{K} } }^7$ taken from [@derksen99]. That latter paper presents an algorithm to compute a set of generators of the algebra of polynomial invariants for the linear action of a reductive group. The ideal ${\ensuremath{O}}= ({\ensuremath{G}}+ (Z-g(\lambda,z))) \cap {\ensuremath{\mathbb{K} } }[z,Z]$, where now $g$ is a polynomial map that is linear in $z$, is also central in the construction as a set of generators of ${\ensuremath{\mathbb{K} } }[z]^G$ is obtained by applying the Reynolds operator, which is a projection from ${\ensuremath{\mathbb{K} } }[z]$ to ${\ensuremath{\mathbb{K} } }[z]^G$, to generators of $O+(Z_1,\ldots,Z_{{n}})$, the ideal of the null cone.
The fraction field of ${\ensuremath{\mathbb{K} } }[z]^G$ is included in ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$ but does not need to be equal. Conversely there is no known algorithm to compute ${\ensuremath{\mathbb{K} } }[z]^G={\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }\cap {\ensuremath{\mathbb{K} } }[z]$ from the knowledge of a set of generators of ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }$.
\[derksen\] We consider the linear action of $S\!L_2$ on ${\ensuremath{\mathbb{K} } }^7$ given by the following polynomials of ${\ensuremath{\mathbb{K} } }[\lambda_1,\ldots,\lambda_4,z_1,\ldots, z_7]$: $$\begin{array}{c} g_1 = \lambda_1 z_1+\lambda_2 z_2,\quad
g_2=\lambda_3z_1+\lambda_4z_2\; \\ g_3 = \lambda_1 z_3+\lambda_2
z_4,\quad g_4=\lambda_3z_3+\lambda_4z_4\; \\ g_5 =
\lambda_1^2z_5+2\lambda_1\lambda_2z_6+\lambda_2^2z_7, \\ g_6 =
\lambda_3\lambda_1 z_5+\lambda_1\lambda_4
+\lambda_2\lambda_3z_6+\lambda_2\lambda_4z_7,\; \\ g_7 =
\lambda_3^2z_5+2\lambda_3\lambda_4z_6+\lambda_4^2
\end{array}$$ the group being defined by ${\ensuremath{G}}=(\lambda_1\lambda_4-\lambda_2\lambda_3-1) \subset
{\ensuremath{\mathbb{K} } }[\lambda_1,\lambda_2,\lambda_3,\lambda_4]$.
The cross-section defined by ${\ensuremath{P_{}}}=(Z_1+1,Z_2,Z_3)$ is of degree one. The reduced Gröbner basis (for any term order) of the ideal ${\ensuremath{I}}^e \subset {\ensuremath{\mathbb{K} } }(z)[Z]$ is indeed given by $ \{Z_1 + 1, Z_2, Z_3, Z_4-r_2, Z_5-{r_3},
Z_6-{r_4}, Z_7 - r_1\} $ where $$\begin{array}{c}
{r_1}={z}_{7}\,{{z}_{1}}^{2}-2\,{z}_{2}\,{z}_{6}\,{z}_{1}+{{
z_2}}^{2}{z}_{5},
\quad
{r}_{2}={z}_{3}\,{z}_{2}-{z}_{1}\,{z}_{4},
\\
{\displaystyle}{r_3}={\frac {{{z}_{3}}^{2}{z}_{7}-2\,{z}_{6}\,{z}_{4}\,{z}_{3}
+{z}_{5}\,{{z}_{4}}^{2}}{ \left({z}_{1}\,{z_4} -{z}_{3}\,{z}_{2} \right) ^{2}}},
\quad
{r}_{4}={\frac {{z}_{1}\,
{z_6}\,{z}_{4}-{z}_{1}\,{z}_{3}\,{z}_{7}+
{z_3}\,{z}_{2}\,{z}_{6}-{z}_{2}\,{z}_{5}\,{z}_{4}}{{z}_{1}\,{z}_{4}-{z}_{3}\,{z}_{2} }}
\end{array}$$
By [Theorem \[rewrite2\]]{}, ${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }={\ensuremath{\mathbb{K} } }(r_1,r_2,r_3,r_4)$. In this case the rewriting of any rational invariant in terms of $r_1,r_2,r_3,r_4$ consists simply of the substitution of $(z_1,z_2,z_3,z_4,z_5,z_6,z_7)$ by $(-1,0,0,r_2,r_3,r_4,r_1)$. We illustrate this by rewriting the five generating polynomial invariants computed in [@derksen99] in terms of $r_1,r_2,r_3,r_4$: $$\begin{array}{c}
{z_2}^{2}z_5-2\,z_2\,z_6\,z_1+z_7\,{z_1}^{2}=r_1,\quad
z_3\,z_2-z_1\,z_4=r_2,
\\
{z_3}^{2}z_7-2\,z_6\,z_4\,z_3+z_5\,{z_4}^{2}=r_3{r_2}^{2}, \,\,
z_1\,z_3\,z_7-z_3\,z_2\,z_6+z_2\,z_5\,z_4-z_1\,z_6\,z_4=r_4\,r_2,
\\
{z_6}^{2}-z_7\,z_5={r_4}^{2}-r_1\,r_3,\quad
\end{array}$$
The reduced Gröbner basis of ${\ensuremath{O}}^e$, relative to the total degree order with ties broken by reverse lexicographical order, has 9 elements:
$$\begin{array}{c}
{{ Z_6}}^{2}
-{ Z_7}\,{ Z_5}+{ r_1}\,{ r_3}-{ r_4}^{2},
\quad
{ Z_6}\,{ Z_4}
+{ r_3}\,{ r_2}\,{ Z_2}-{ r_4}\,{ Z_4}-{ Z_3}\,{ Z_7},
\\
{ Z_5}\,{ Z_4}
-{ Z_3}\,{ Z_6}+{ r_3}\,{ r_2}\,{ Z_1}-{ r_4}\,{ Z_3},
\quad
{ Z_3}\,{ Z_2}
-{ Z_1}\,{ Z_4}-{ r_2},
\\
{ Z_2}\,{ Z_6}
-{ Z_1}\,{ Z_7}+{ r_4}\,{ Z_2}-{\frac{r_1}{ r_2}}\, Z_4,
\quad
{ Z_2}\,{ Z_5}
+{ Z_1}\,{ r_4}-{ Z_6}\,{ Z_1}-{\frac {r_1}{ r_2}}\,{ Z_3} ,
\\
{{ Z_2}}^{2}
+{\frac {{ r_1}}{{ r_3}\,{{ r_2}}^{2}}}\,{{ Z_4}}^{2}
-{\frac {{ Z_7}}{{ r_3}}}
-2\,{\frac {{ r_4}}{{ r_3}\,{ r_2}}}\,{ Z_4}\,{ Z_2},
\quad
{{ Z_1}}^{2}
-{\frac {{ Z_5}}{{ r_3}}}-
2\,{\frac {{ r_4}}{{ r_3}\,{ r_2}}}\,{ Z_3}\,{ Z_1}
+{\frac {{ r_1}}{{ r_3}\,{{ r_2}}^{2}}}\,{{ Z_3}}^{2}
\\
{ Z_2}\,{ Z_1}
-{\frac {{ r_4}}{{ r_3}}}-{\frac {{ Z_6}}{{ r_3}}}
+{\frac {{ r_1}}{{ r_3}\,{{ r_2}}^{2}}}\,{ Z_4}\,{ Z_3}
-2\,{\frac {{ r_4}}{{ r_3}\,{ r_2}}}\,{ Z_4}\,{ Z_1},
\end{array}$$ Though this Gröbner basis is obtained without much difficulty, the example illustrates the advantage obtained by considering the construction with a cross-section: ${\ensuremath{I}}^e$ has a much simpler reduced Gröbner basis than ${\ensuremath{O}}^e$.
We finally take a classical example in differential geometry: the Euclidean action on the second order jets of curves. The variables $x,y_0,y_1,y_2$ stand for the independent variable, the dependent variable, the first and the second derivatives respectively. We shall recognize the curvature as the non constant component of a replacement invariant.
We consider the group defined by ${\ensuremath{G}}=(\alpha^2+\beta^2-1, \epsilon^2-1)
\subset{\ensuremath{\mathbb{K} } }[\alpha,\beta,a,b,\epsilon]$. The neutral element is $(1,0,0,0,1)$, the group operation is\
$ (\alpha',\beta',a', b', \epsilon')\cdot
(\alpha,\beta,a, b,\epsilon) =
(\alpha\alpha'-\beta\beta', \beta\alpha'+\alpha\beta',
a +\alpha a'-\beta b',
b+\alpha a'+\alpha b', \epsilon\,\epsilon')$ and the inverse map $(\alpha,\beta,a, b)^{-1}=
(\alpha,-\beta, -\alpha \, a- b\beta, \beta \,a-\alpha b, \epsilon)$. The rational action on ${\ensuremath{\mathbb{K} } }^4$ we consider is given by the rational functions: $$\begin{array}{c} {\displaystyle}g_1 = \alpha x-\beta y_0 +a,
\quad g_2 = \epsilon \beta x+\epsilon \alpha y_0 + b,\\
{\displaystyle}g_3 = \frac{\beta+\alpha y_1}{\alpha-\beta y_0} ,
\quad g_4 =\frac{y_2}{(\alpha-\beta y_0)^3}.
\end{array}$$
We have $${\ensuremath{O}}= \left(\left( 1+ y_1^2 \right)^3 Y_2^2-\left(1+Y_1^2\right)^3 y_2^2\right)$$ and if we consider the the cross section defined by ${\ensuremath{P_{}}}=(X,Y_0,Y_1)$ the reduced Gröbner basis of ${\ensuremath{I}}^e = {\ensuremath{O}}^e+ {\ensuremath{P_{}}}$ is $$\left\{X,Y_0,Y_1, Y_2^2-\frac{y_2^2}{(1+y_1^2)^3} \right\}.$$ According to [Theorem \[rewrite\]]{} or [Theorem \[rewrite2\]]{} $${\ensuremath{{{\ensuremath{\mathbb{K} } }(z)^G}} }={\ensuremath{\mathbb{K} } }\left(\frac{y_2^2}{(1+y_1^2)^{3}}\right).$$
The two replacement invariants ${\ensuremath{\xi} }=({\ensuremath{\xi} }_1, {\ensuremath{\xi} }_2, {\ensuremath{\xi} }_3, {\ensuremath{\xi} }_4)$ associated to the cross-sections are given by $${\ensuremath{\xi} }_1=0,{\ensuremath{\xi} }_2=0,{\ensuremath{\xi} }_3=0,
{\ensuremath{\xi} }_4=\pm \sqrt{\frac{y_2^2}{(1+y_1^2)^3}}.$$
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[^1]: We are indebted to a referee of the MEGA conference for pointing out this reference that motivated us to push further some of the results.
[^2]: We would like to thank H. Derksen for suggesting comparison with this reference after we made public a first preprint.
[^3]: Though we do not use this fact but rather retrieve it otherwise, it is worth noting that, as a subfield of ${\ensuremath{\mathbb{K} } }(z)$, the field of rational invariants is always finitely generated [@waerden71].
[^4]: The support here is the set of terms in $Z$ with non zero coefficients.
[^5]: For those reductions in ${\ensuremath{\mathbb{K} } }[y,Z]$ the term order on $Z$ is extended to a block order $y \ll Z$ so that the set of leading term of $Q_y$ is equal to the set of leading terms of $Q$.
[^6]: For this reason the map $\rho$ is called *moving frame* in [@olver99]. We adopt the term *a moving frame map.*
[^7]: An integrable distribution is a collection of smooth vector fields, whose span over the ring of smooth functions is closed with respect to Lie bracket.
[^8]: This implicitly means that we know the ideal $G$ ([Section \[agroupaction:def\]]{}) from a set of generators with coefficients in ${{\ensuremath{\mathbb{R}}}}$.
|
---
abstract: 'This work addresses the problem of learning sparse representations of tensor data using structured dictionary learning. It proposes learning a mixture of separable dictionaries to better capture the structure of tensor data by generalizing the separable dictionary learning model. Two different approaches for learning mixture of separable dictionaries are explored and sufficient conditions for local identifiability of the underlying dictionary are derived in each case. Moreover, computational algorithms are developed to solve the problem of learning mixture of separable dictionaries in both batch and online settings. Numerical experiments are used to show the usefulness of the proposed model and the efficacy of the developed algorithms.'
author:
- 'Mohsen Ghassemi, Zahra Shakeri, Anand D. Sarwate, and Waheed U. Bajwa, [^1][^2]'
title: 'Learning Mixtures of Separable Dictionaries for Tensor Data: Analysis and Algorithms'
---
[Ghassemi : Learning Mixtures of Separable Dictionaries]{}
Dictionary learning, Kronecker structure, sample complexity, separation rank, tensor rearrangement.
Introduction
============
Preliminaries and Problem Statement {#sec:problem_statement}
===================================
Identifiability in the Rank-constrained LSR-DL Problem {#sec:NP_identif}
======================================================
Identifiability in the Tractable LSR-DL Problems {#sec:tractable_identif}
================================================
In Section \[sec:problem\_statement\], we introduced two tractable relaxations to the rank-constrained LSR-DL problem: a regularized problem with a convex regularization term and a factorization-based problem in which the dictionary is written in terms of its subdictionaries. Based on our results in Section \[sec:NP\_identif\] for the rank-constrained problem, we now provide results on the local identifiability of the true dictionary $\D^0$ in these problems, i.e., we find conditions under which at least one local minimizer of these problems is located near the true dictionary $\D^0$. Such local identifiability result implies that any DL algorithm that converges to a local minimum of these problems can recover $\D^0$ up to a small error if it is initialized close enough to $\D^0$.
Regularization-based LSR Dictionary Learning {#regularization_section}
--------------------------------------------
Factorization-based LSR Dictionary Learning {#factorization_section}
-------------------------------------------
Computational Algorithms {#sec:algorithms}
========================
In Section \[sec:tractable\_identif\], we showed that the tractable LSR-DL Problems and each have at least one local minimum close to the true dictionary. In this section we develop algorithms to find these local minima. Solving Problems and require minimization with respect to (w.r.t.) $\X\triangleq [\x_1^T,\cdots,\x_L^T]$. Therefore, similar to conventional DL algorithms, we introduce alternating minimization-type algorithms that at every iteration, first perform minimization of the objective function w.r.t. $\X$ (sparse coding stage) and then minimize the objective w.r.t. the dictionary (dictionary update stage).
The sparse coding stage is a simple Lasso problem and remains the same in our algorithms. However, the algorithms differ in their dictionary update stages, which we discuss next.
*Remark.* We leave the formal convergence results of our algorithms to future work. However, we provide a discussion on challenges and possible approaches to establish convergence of our algorithms in Appendix \[app:convergence\].
: A Regularization-based LSR-DL Algorithm {#subsec:stark}
-----------------------------------------
: A Factorization-based LSR-DL Algorithm {#subsec:tefdil}
----------------------------------------
O: An Online LSR-DL Algorithm
-----------------------------
Numerical Experiments {#sec:experiments}
=====================
Conclusion
==========
[10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{}
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[^1]: The authors are with the Department of Electrical and Computer Engineering, Rutgers, The State University of New Jersey, Piscataway, NJ, 08854 USA (Emails: {m.ghassemi, zahra.shakeri, anand.sarwate, waheed.bajwa}@rutgers.edu).
[^2]: Some of the results reported here were presented at IEEE Int. Workshop on Comput. Advances in Multi-Sensor Adaptive Process., 2017 [@ghassemi2017stark]. This work is supported in part by the National Science Foundation under award CCF-1453073 and by the Army Research Office under award W911NF-17-1-0546.
|
---
abstract: 'We present a new method to add long wavelength power to an evolved $N$-body simulation, making use of the Zel’dovich (1970) approximation to change positions and velocities of particles. We describe the theoretical framework of our technique and apply it to a P$^3$M cosmological simulation performed on a cube of $100$ Mpc on a side, obtaining a new “simulation" of $800$ Mpc on a side. We study the effect of the power added by long waves by mean of several statistics of the density and velocity field, and suggest possible applications of our method to the study of the large-scale structure of the universe.'
author:
- Giuseppe Tormen
- Edmund Bertschinger
title: 'Adding Long Wavelength Modes to an $N$-Body Simulation'
---
=0.45truecm
[*Accepted for publication in the Astrophysical Journal*]{}
=0.1truecm
Introduction {#intro}
============
Computer simulations of large scale structure play a fundamental role in cosmology by providing a better understanding of the many issues related to structure formation. The usual setup of an $N$-body simulation can be summarized as follows. One generates some initial conditions for the simulation by placing $N^3$ particles in a cubic box of side $L$. The recipe for assigning an initial position and velocity to each particle is usually the Zel’dovich (1970) approximation. This approximation lets one distribute the particles in the box so that they trace some initial fluctuating density field with power spectrum $P(k)$. If the density fluctuations form a Gaussian random field, as is usually assumed, then $P(k)$ together with the evolution of the expansion scale factor uniquely specifies the cosmological model of the simulation. One then evolves this self gravitating system by numerically integrating the trajectories of all the particles under their mutual gravitational attraction.
Periodic boundary conditions are commonly imposed on the box. Since a periodic function has a discrete Fourier transform, the periodic boundary conditions on the box imply a discrete sampling of $P(k)$: the only density fluctuations present in the evolution are those with wavelength $\lambda = 2\pi/k$ satisfying the usual periodicity requirement: $k^2 = (2\pi n_x / L)^2 + (2\pi n_y / L)^2 + (2\pi n_z / L)^2$ where $n_x$,$n_y$ and $n_z$ are positive integers.
The two parameters $L$ and $N$ determine the dynamical resolution of the simulation. The box size $L$ fixes the force resolution at large scales, since fluctuations on scales $\lambda >L$ are not included in the simulation and so are missed. It also determines the sampling resolution of the power spectrum $P(k)$: a larger $L$ means a denser sampling of fluctuations at all scales, as $\Delta k \propto L^{-1}$. The particle number $N^3$ instead determines the force resolution at small scale since, for a given $L$, it fixes the minimum wavelength of the fluctuations present in the initial conditions, via the Nyquist relation: $k_{max} = \pi N/L$, with $k = k_x$, $k_y$, $k_z$. Larger $N$’s also mean more particles per object of interest, therefore a better resolution in mass.
The ideal configuration is of course a large box size $L$ and a large number of particles $N^3$. Unfortunately, even with the current supercomputer power, computer memory and cpu time impose severe limitations to the values of $N^3$ one can reasonably take. Given the maximum number $N^3$ of particles one can afford, the choice generally made is therefore to specialize the size $L$ of the box to a specific purpose. For example, for a study of the general properties of a cosmological model, where the identity and structure of single objects is of secondary importance compared to the overall large scale structure and motions produced in the model, one will take a large box, i.e. with size $L \approx$ few $\times 100$ Mpc, to the detriment of the small scale resolution in force and in mass. On the other hand, when the interest is focussed on the internal structure of the class of objects under investigation (e.g. galaxies or galaxy clusters) the box size $L$ is taken much smaller, of the order of $\approx$ few $\times 10$ Mpc. In this case there are two kind of disadvantages. First, the missed fluctuations on scales larger than the box are still very important for the formation of structure. Second, since there are only $3$ independent Fourier modes associated with density fluctuations of scale equal to the box size $L$, statistical fluctuations in the density and velocity fields are not negligible.
The missing power on large scales will cause a sort of [*cosmic bias*]{} (in the statistical sense of the term), because the number of high density regions, the strength of the clustering and the amplitude of the peculiar velocities will be systematically [*lower*]{} than in the ideal case of infinite $N$ and $L$. The statistical fluctuations in different realizations of the same initial $P(k)$ on scale $\lambda \simeq L$ will instead introduce a [*cosmic variance*]{} in the simulation, since the measures performed on the density and velocity field will fluctuate around their statistical mean value. Both effects can be dramatic if the volume $L^3$ is too small for the statistic one is considering, and are particularly evident if one is interested in the peculiar velocity field, which receives important contributions form linear density fluctuations of very large scale. For example, in a Standard Cold Dark Matter universe with a dimensionless Hubble parameter $h = 0.5$, and a linear $P(k)$ normalized to an mass density fluctuation (in spheres of radius $8 h^{-1}$ Mpc) $\sigma_8=1$, the linear bulk velocity of a cube of side $100$ Mpc is still well over $500$ km s$^{-1}$. However, a simulation of 100 Mpc on a side will have zero bulk flow on the same scale, since by definition the box is at rest. The missing power on scale $\lambda > 100$ Mpc is responsible for this cosmic bias. A good example of cosmic variance concerns the measure of the Hubble constant itself. This can assume quite different values locally, since different patches of the universe are expected to expand at different [*local*]{} rates. Turner (1992) have shown that, for a Cold Dark Matter universe with [*true*]{} Hubble constant $h=0.8$, the local Hubble constant measured out to $30 h^{-1}$ Mpc in regions comparable to the North Galactic Cap has an estimated value $h = 0.5 - 1.28$ at the 95% confidence level.
The problems listed so far are well known in the literature of cosmological simulations of large-scale structure. However, attempts to solve them by inserting in a small scale simulation the missing large scale power have until now been limited to corrections applied to the mean values of the statistics (like $P(k)$ or the bulk velocity) ([@cc92], [@st93]), or to the velocity field only ([@st95]), but not individually to the velocity [*and*]{} density field. In this paper we propose a new method to cure these large-scale limitations. This method is applicable if the cosmic bias caused by the missing large scale power is produced by density fluctuations which are still in the linear regime. The idea is then to use standard linear theory and the Zel’dovich (1970) approximation to add to each individual particle of an [*evolved*]{} simulation of size $L$ a random realization of the power coming from wavelengths larger than the original box size $L$. From this idea we named our method by the acronym (Mode Adding Procedure). The corresponds to embedding the simulated cube in a much larger (and possibly infinite) one, therefore increasing the volume sampled and decreasing the cosmic bias and variance associated with it. Although there is virtually no upper limit to the scale of the added fluctuations, let us call $L_{big}$ the scale of the largest fluctuations one reasonably wants to introduce.
A first straightforward application of this new method is the construction of different realizations of a very large scale simulation (e.g. 3000 Mpc on a side) from just one evolved medium scale (e.g. 200 Mpc) simulation. From this one could extract artificial redshift surveys of size comparable to the real surveys completed or in program, e.g. the Las Campanas Redshift Survey ([@lan96]), the ESO Slice Project ([@ve94]), the Sloan Digital Sky Survey ([@gw94]) and others. The resulting simulations would have at the same time a scale sufficient to address the issue, and enough resolution on small scale to properly identify galactic halos. The artificial surveys can be used to calibrate the different sources of uncertainties present in the real data (sparse sampling, redshift errors etc.), to estimate the scientific impact expected from the new data, and to compare them with the predictions of different cosmological models, via a number of statistics. Other possible applications include the dynamical study of the role played in the structure formation process by linear density fluctuations on very large scale. Do they couple with nonlinear modes during time evolution? How do they trigger the formation of cosmic structure on all scales? How important are they when studying the velocity field or superclustering phenomena?
The paper is organized as follows. Section 2 is a description of our method. In Section 2.1 we discuss the Fourier space manipulation required to apply our method to a simulation. Section 2.2 describes the corresponding steps performed in position space: a Mode Removing step and a Mode Adding step. In Section 2.3 Lagrangian and Eulerian ways of implementing the technique are considered and discussed. In Section 3 we apply the method to an $N$-body simulation and compare some statistics of the density and velocity field obtained from the simulation with the same statistics applied to a real $N$-body simulation performed on large scale. Section 4 gives a summary of the results and presents some conclusions.
Method {#met}
======
Fourier Space Manipulation {#fsm}
--------------------------
The Fourier space of a periodic simulation can be thought as a cubic lattice. The sampling of the initial $P_\delta(k)$ is made on a regular cubic grid centered on $\vec k =\vec 0$, ($|\vec k |\equiv 2\pi/\lambda$), with inter-grid size $\Delta k=k_{min}=(2\pi /L)$ and extension in every direction determined by the number $N^3$ of particles used, via the Nyquist relation: $k_{max}=\pi N/L$.
The diagram in Fig. 1a shows the central region of this sampling. The dots correspond to the positions where $P_\delta(k)$ is evaluated, and the regular grid divides the Fourier space in cubes of equal side $\Delta k$. Each cube is associated with a discrete Fourier component $\hat\delta_{\vec k}$ of the density fluctuation field $\delta(\vec x)$: the intensity of density fluctuations of wavenumber $\vec k$ is the mean power per mode $\langle |\hat\delta_{\vec k}|^2\rangle \approx P_\delta(k) (\Delta k)^3$. Including in the simulation density fluctuations on scales $\lambda > L$ corresponds to improving the sampling of $P_\delta(k)$ around $\vec k=\vec 0$. Our scheme is thus a kind of mesh-refinement algorithm implemented in Fourier space.
Our approach is the following. We first remove, around $\vec k = \vec 0$, the power associated with the Fourier modes of the original sampling. This means deleting the power of a number of cubes each of side $\Delta k = (2\pi/L)$ in the central region of Fourier space. We then add back new power by filling the same region with a larger number of smaller cubes, each of side $\Delta k^\prime = (2\pi/mL)$ (with $m$ a positive integer). The power per mode of these new cubes is assigned with a random realization of the (Rayleigh distributed) linear power spectrum $P_\delta(k)$ at the corresponding positions; this ensures that the correct amount of power is added to the simulation. The procedure is sketched in Fig. 1b for the case $m=4$: a grid four times finer is substituted for the original one in Fourier space, out to an extension $r_k \equiv k/k_{min}=1$ in each direction. Subtracting a cubic region of extension $r_k$ corresponds to removing $(2r_k+1)^3$ cubes of side $2\pi/L$ from the Fourier space.
Our method adds long-wavelength power to a simulation at the end of time evolution. To ensure that the result is dynamically consistent we must remove and add only power associated with fluctuations that are still in the linear regime. The linearity constraint may be interpreted in different ways. The most general requirement is that the root mean square density fluctuation associated with linear waves must be smaller than unity: $\sigma_\delta < 1$. Another straightforward characteristic of linear waves is that they evolve in agreement with the equations of linear theory. A third requirement is that linear waves should not dynamically couple with any other wave. This issue has been explored by Jain and Bertschinger (1994), who find that for a cold dark matter spectrum of density perturbations mode coupling can transfer significant power to shorter waves from long wavelengths still nearly in the linear regime. We can monitor this by checking that the longest mode in our original box has linearly growing amplitude, but even this is not a rigorous limit because longer waves absent from the box might have caused the amplitude already to depart from linear growth. For now we adopt the practical viewpoint of trying the method and later testing for effects of nonlinear coupling. We will discuss this issue further in the summary at the end of this paper.
Whichever linearity requirement we choose, this will put a limit on the region of Fourier space where we can perform the power substitution. In the example of Fig. 1 we manipulate the power in a cube of extension $r_k=1$; depending on the simulation, we may extend the substituted region up to higher $\vec k$.
The idea: Mode Removing and Mode Adding {#idea}
---------------------------------------
To remove and add the power as described in Section 2.1 we decided to use the displacement field and to perform the power manipulation by mean of the Zel’dovich (1970) approximation. In this approximation each fluid element moves along a straight line with a velocity linearly extrapolated from its initial velocity.
We will work in co-moving coordinates with $\vec q_i$ and $\vec x_i$ being respectively the initial (Lagrangian) and final (Eulerian) position of the $i$-th particle of the original simulation, $i=1,2,\dots, N$. We will denote by $\vec x^\prime$ the final particle positions after the mode removing step and $\vec x^{\prime\prime}$ the final positions after the mode adding step.
Before describing how we perform the mode substitution in practice, we will briefly review some relations between the density, velocity and displacement fields that will be used later. The relation between the Eulerian and Lagrangian position of a fluid element at $\vec x$ is given by the displacement field $\vec \psi$: $\vec x(\vec q,t)=\vec q + \vec \psi(\vec q,t)$. By applying mass conservation and assuming that $\vec x(\vec q\,)$ is one-to-one (no orbit mixing) we can write the exact relation $$\delta(\vec x) =
\left\| {\partial\vec x \over \partial \vec q}\right\|^{-1} - 1 =
-\vec \nabla_q \cdot \vec\psi(\vec q\,) + {\cal O}(\psi^2).
\label{lin1}$$ Note that $\vec \psi$ is the full, nonlinear displacement. This relation is still mixing Lagrangian and Eulerian coordinates; however, in the linear approximation we may consider the displacement to be a function of $\vec x$ and write $$\delta^{\,(1)}(\vec x) = -\vec \nabla_x \cdot \vec\psi^{\,(1)}(\vec x)
\label{lin2}$$ where now both $\delta^{\,(1)}(\vec x)$ and $\vec \psi^{\,(1)}(\vec x)$ are first order quantities. We see how the time dependence of the linear displacement field is that of $\delta^{\,(1)}(\vec x,t)$, that is, considering only the growing mode $D_+(t)$: $\vec \psi^{\,(1)}(t)\propto D_+(t)$. Using this result and taking the time derivative of the mapping $\vec q \longrightarrow \vec x(\vec q\,)$ we obtain to first order the linear relation between the co-moving peculiar velocity $\vec v \equiv d\vec x/dt$ and the displacement field at time $t$: $$\vec v^{\,(1)}(\vec x) = H_0 f(\Omega) \vec \psi^{\,(1)}(\vec x)
\label{lin4}$$ where $f(\Omega)=d\log D_+/d\log a\approx \Omega^{0.6}$. In particular, if $\Omega=1$ the linear displacement field and the linear peculiar velocity field coincide if units of km s$^{-1}$ are used. The Zel’dovich (1970) approximation is then written as: $$\vec x(\vec q\,)= \vec q + D_+(t)\vec \psi^{\,(1)}(\vec q\,)=
\vec q + {D_+(t) \over H_0 f(\Omega)} \vec v^{\,(1)}(\vec q\,)\ .
\label{zel}$$
The steps we follow in practice are the following (all equations are meant at a given time $t$):
[**1.**]{} Starting with a simulation performed with $N$ particles on a cubic volume $L^3$ we compute the co-moving displacements as $\vec \psi_i= \vec x_i - \vec q_i$, $\ \ \ i=1,\dots,N$. We use these displacements to define a displacement field $\vec \psi$ on a regular grid in position space. This can be done in different ways, as we will see in the next subsection.
[**2. Mode Removing**]{}: We decompose the displacement field into the contributions $\vec \psi^{long}$, due to the modes that we are going to subtract from the simulation, and $\vec \psi^{short}$ due to all the other modes: $\vec \psi=\vec \psi^{long} +\vec \psi^{short}$. We subtract the long wavelength power from the simulation by interpolating $\vec \psi^{long}$ to the [*Eulerian*]{} position of each particle and subtracting it, changing in this way each particle’s position. The velocities are changed in the same way, by taking advantage of the linear relation between them and the displacements Eq.(\[lin4\]): $$\vec x^\prime_i=\vec x_i - \vec \psi^{long}(\vec x_i); \ \ \
\vec v(\vec x^\prime_i)=\vec v(\vec x_i)- H_0 f(\Omega)
\vec \psi^{long}(\vec x_i); \ \ \
i=1,2,\dots, N.
\label{erem}$$
Another way to subtract the large scale contribution to the velocities would be to directly decompose the velocity field as we did for the displacement, instead of using the linear relation between the two. We expect the two procedures to give the same result, as long as the basic assumption of removing only linear modes holds. Note that we did not subtract the displacements $\vec\psi^{long}$ at the $\vec q$ (Lagrangian) positions of the particles. If we did that, we would disrupt the nonlinear structures which formed during the evolution of the simulation, and smear out the power on small scales. So far the change in positions does not make use of the Zel’dovich approximation, since the subtracted displacement is the actual one and not the initial one times the growth factor $D_+$. We are, however, assuming linear theory in the relation between the velocity and displacement fields, which is only approximate even for long waves.
[**3. Mode Adding**]{}: We generate a new set of initial conditions consisting of random long-wavelength displacements, $\vec \psi_1^+, \dots, \vec \psi_N^+$ in a cube of side $mL$, by randomly sampling the power spectrum $P_\delta(k)$ only at those positions around $\vec k=\vec 0$ corresponding to the long wavelength modes we are going to add. These displacements are then linearly evolved up to the present time, again interpolated to the [*Eulerian*]{} position of each particle, and added to the positions and velocities as prescribed by the Zel’dovich approximation: $$\vec x^{\prime\prime}_i=\vec x^\prime_i + D_+(t)\vec \psi^+(\vec x^\prime_i); \ \ \
\vec v(\vec x^{\prime\prime}_i)=\vec v(\vec x^\prime_i)+ D_+(t) H_0 f(\Omega)
\vec \psi^+(\vec x^\prime_i); \ \ \
i=1,2,\dots, N
\label{eadd}$$ The final result is a set of particle positions and velocities which now include the effect of density fluctuation waves as long as $mL$.
We stress the importance of adding the long wave power by interpolating $\vec\psi^+$ to the new positions $\vec x'$, not at the original positions $\vec x$. In fact, in the view the $\vec x$ are just some [*wrong*]{} Eulerian positions where the particles stand due to the missing large scale power. Considering only the waves we subtract and add, we can identify $\vec x^\prime_i$ with the Lagrangian position $\vec q$. The correct way to apply the Zel’dovich (1970) approximation is $\vec x=\vec q + D_+\vec \psi^+(\vec q)$, which indeed corresponds to Equation (\[eadd\]). As a check we also tried the alternative formulation: $\vec x^{\prime\prime}_i=\vec x^\prime_i + D_+(t)\vec \psi^+(\vec x_i)$. As expected, the results do not satisfactorily reproduce the linear long wave power one is introducing.
Lagrangian vs Eulerian approach {#laeu}
-------------------------------
As sketched in Section 2.2, we need to define a displacement field on a regular grid. This can be performed in two ways. Starting from a set of $N$ displacements $\vec \psi_1,\dots,\vec \psi_N$ we can assign each displacement $\vec \psi_i$ to its initial position $\vec q_i$: $\vec \psi_i \equiv \vec \psi(\vec q_i)$ and write: $$\vec \psi_L(\vec q\,)={\sum_{i=1}^N \vec \psi_i \delta_D(\vec q-\vec q_i)
\over \sum_{i=1}^N \delta_D(\vec q-\vec q_i)}.
\label{ldisp}$$ where $\delta_D$ is a Dirac delta function. The subscript $L$ stands for [*Lagrangian*]{} because the resulting displacement field is defined on a regular grid of initial positions.
Alternatively, we can assign every displacement $\vec \psi_i$ to the corresponding final position $\vec x_i$: $\vec \psi_i \equiv \vec \psi(\vec x_i)$ and interpolate such displacements onto a regular grid of final positions $\vec x_g$ by mean of a suitable window function $W$; the resulting displacement field will be called [*Eulerian*]{} and indicated by a subscript $E$: $$\vec \psi_E(\vec x_g)= {\sum_{i=1}^N \vec \psi_i
W(\vec x_g, \vec x_i) \over \sum_{i=1}^N W(\vec x_g, \vec x_i)}.
\label{edisp}$$
In subtracting the longest waves from the parent simulation we used the Eulerian displacement field $\vec\psi_E(\vec x)$, with a window function $W$ corresponding to a Triangular Shaped Cloud (TSC) interpolation on a regular cubic grid with $32^3$ mesh points. The mode adding part was performed instead using the Lagrangian field $\vec\psi_L(\vec x)$, since we assigned the displacements generated by the long waves to a grid of initial positions $\vec q$ as is usually done when generating standard initial conditions for a simulation. However, we do not expect the choice between $\vec\psi_L$ and $\vec\psi_E$ to be fundamental to the final result. In fact, the displacement fields we are considering are due only to long, linear waves; $(\vec \psi_L -\vec \psi_E)$ at a given position is a second-order quantity.
In practice, the displacements employed by Equations (\[erem\]) and (\[eadd\]) are computed by interpolating the displacement field from the grid points to the Eulerian particle position: $$\vec \psi(\vec x)= {\sum_{\vec x_g} \vec \psi(\vec x_g)
W(\vec x, \vec x_g) \over \sum_{\vec x_g} W(\vec x, \vec x_g)}
\label{edis2}$$ where $\vec \psi$ may be either $\vec \psi_L$ or $\vec \psi_E$ and $\vec x$ is the particle position. The sum is extended over all the grid points $\vec x_g$ in the simulation. If $\vec \psi_L$ is used, the grid points $\vec x_g$ stand of course for the initial positions $\vec q_i$, $i=1, \dots, N$.
Application to a simulation: MAP8x144 {#sim}
=====================================
As a first application of our technique we will take a medium range, high resolution $N$-body simulation, originally evolved in a periodic cube of side $L$, replicate it $m^3$ times in a larger cube and add to it the missing power from the long wavelengths not sampled in the original cube, up to $\lambda=mL$. We use a P$^3$M $N$-body cosmological simulation of an Einstein-De Sitter cold dark matter universe, with a dimensionless Hubble constant $h=0.5$, evolved in a cube of side $L=100$ Mpc ([@gb94]). The simulation was run with $144^3$ collisionless particles, each with mass $2.3 \times 10^{10} M_\odot$ and a Plummer softening radius of $65$ kpc. We chose the output of the simulation corresponding to a linear normalization $\sigma_8=0.7$. We will refer to this simulation as P3M144. This simulation has high mass and force resolution, and is particularly suited for studying the dynamics of cold dark matter halo formation and the small to medium scale structure. On the other hand, its size is too small to allow a study of the velocity field on large scales through statistics like the bulk flow or the velocity correlation tensor. We will compare statistics of the density and velocity field for P3M144, both before and after applying the , and for a reference $N$-body simulation evolved on a much larger scale. The latter, which we call P3M256, was run with identical cosmological parameters as P3M144, but has $256^3$ particles in a cube of side $L=640$ Mpc and a Plummer softening radius of $160$ kpc.
One way to verify the assumption of large-scale linearity for the fields in P3M144 is to measure the power associated with the long waves that are removed from and added back to it. The density fluctuation and displacement are defined directly in Fourier space respectively as $$\sigma_\delta=\sqrt{\sum P_\delta(k) (\Delta k)^3} \ \ \ \ \hbox{and}
\ \ \ \ \psi_{rms}=\sqrt{\sum P_\delta(k) {(\Delta k)^3 \over k^2}}
\label{pow2}$$ where the sums are extended over the modes under investigation. Referring to the picture of Fourier space in Fig. 1, we tried the mode substitution on P3M144 in a cubic region of extension $r_k = 1$. This corresponds to removing the power associated with the $27$ central cubes of side $\Delta k= (2\pi/100)$ Mpc$^{-1}$ each. The displacement field associated with the removed region of Fourier space is what we called $\vec \psi^{long}$ in Equation (\[erem\]). Its root mean square value computed from P3M144 as in Eq.(\[pow2\]) results $\psi^{long}_{rms}= 7.1$ Mpc. The displacement due to all the wavelengths present in the simulation is $\psi_{rms}=11.2$ Mpc, showing how most of the displacement is due to the long waves present in the simulation. This is in agreement with what we said earlier about the peculiar velocities: both the velocity and displacement fields receive the biggest contribution from large-scale density fluctuations. This fact however does not invalidate our linear theory approximation, because the long wavelength displacement and velocities correspond to nearly uniform (bulk) motions for the particles, with no creation of nonlinear structures such as pancakes. This is also confirmed by the value of the density fluctuation due to the removed modes: $\sigma^{long}_\delta=0.55$ is less than one, as we would expect if the linear approximation applies. We will see if this is small enough when we compare with a larger simulation below.
We computed $P_\delta(k)$ from the simulation at different time-steps and plot in Fig. 2a the growth rates of the lowest modes, normalized to the value of $P_\delta(k)$ at $a(t)=0.1$. Fig. 2b shows the analogous plot for P3M256, for comparison. Note that in this case the plot is normalized to a scale factor $a(t)=0.2$, and the most evolved output of the simulation corresponds to $a(t)=0.7$. From Fig. 2 we deduce that all the modes of P3M144 up to $r_k\sqrt{3}$ are approximately in the linear regime, since their growth rate departs from the linear prediction by less than 20% even at the latest times. On the basis of these two tests we conclude that $r_k=1$ is a good choice for the region of Fourier space of P3M144 where we will carry out our mode substitution.
We recall that the new sampling of the power spectrum around $\vec k=\vec 0$ will have a resolution $m$ times the initial one: $\Delta k^\prime=\Delta k/m$, as shown in Section \[fsm\]. We estimated that $m=8$, twice the resolution of the example in Fig. 1b, is a reasonable choice. In terms of wavelength, this roughly corresponds to removing linear displacements generated by density fluctuations of scale $\lambda=100$ Mpc and to adding back displacements associated with scales $\lambda \in [67,800]$ Mpc. However, since the geometry of Fourier space forced us to remove and add power in cubic regions, there are some shorter wavelengths removed and added as well, corresponding to the edges of the cube. Specifically, we remove modes up to a minimum $\lambda_{min}=58$ Mpc and add modes up to a minimum $\lambda_{min}=38$ Mpc. Considering a maximum fluctuation scale of $800$ Mpc guarantees that we are including most of the power driving the velocity field for a standard cold dark matter model. In fact, a cube of side $800$ Mpc has a bulk flow of roughly $150$ km s$^{-1}$ for $\sigma_8=1$, much lower than the bulk flow of the original box. Taking $8^3$ replicas of the original simulation blows up the total number of particles to more than $1.5 \times 10^9$, too many for us to retain. We chose to keep a different random subsample of $32,000$ particles from each of the 512 replicas of the original simulation, for a total of $16,384,000$ particles over the $(800$ Mpc)$^3$ volume. We will refer to this simulation as MAP8x144.
The displacement and density fluctuation due to the added modes and computed as in Eq. (\[pow2\]) are $\psi_{rms}^{+}=10.1$ Mpc and $\sigma_\delta^+=0.63$ (we recall that the linear normalization for P3M144 is $\sigma_8=0.7$). We find as expected that the added power is larger than the power we subtracted from the simulation: $\psi_{rms}^{+}>
\psi_{rms}^{long}$ and $\sigma_\delta^+ > \sigma_\delta^{long}$. However, the extra power has been added in a way consistent with the power spectrum underlying the simulation and satisfying the requirements of the linear approximation.
The figures quoted for the density fluctuations associated with the removed and added modes might look high when compared to the normalization of the original simulation. In particular, one might wonder how can the long waves introduce an density contrast as big as $\sigma_\delta^+=0.63$, if $\sigma_8=0.7$. The answer to this question is found in the different ways which were used to compute the density contrast. While $\sigma_8=0.7$ refers to a spherical top-hat filter, the figures associated with the long waves were computed with Eq.(\[pow2\]), which gives direct summations of the power of each mode with no filter function to smooth it. Equivalently, the filter function for the spherical top-hat is sufficiently different from the filter function corresponding to the discrete mode distribution we are dealing with that a direct comparison of the two is not possible.
Throughout this section we will compare the results and statistics from MAP8x144 with those from P3M144 and P3M256. We divide the analysis in two parts. The first part is dedicated to the study of the density field: we will show the effect of adding long wavelengths on both the morphology and the statistics. In the second part we will study the velocity field.
The density field
-----------------
To give a first visual impression of the effect of long waves on a distribution of matter, we applied the to a two-dimensional sheet of particles regularly spaced. The result is shown in Figure 3: the modulation produced by the long waves is evident. The displacement field has moved the particles from their grid positions, creating density fluctuations on the required scales; the shortest fluctuations are of the order of 40 Mpc. In the most crowded regions the particle trajectories are relatively close to shell crossing, but have not yet reached it, as can be clearly seen in the right panel, which shows a blow up of the part of slice enclosed by the square imprinted on top of the left panel. The magnified region, $200$ Mpc on a side, is subdivided into four equal parts, each of which has the same size as P3M144. One can see how the effect of long waves on different replicas of the original simulation changes the global distribution of matter, modulating the pre-existing structure in different ways on different copies.
Next we want to make sure that the small scale, nonlinear clustering present in P3M144 is not changed or disrupted when we add the extra long wavelength power. We tested this expectation by applying the to a two dimensional network of filaments, shown in Figure 4. After the action of the the network is still connected, i.e. the topology of the structure has not been modified by long-wave fluctuations. However, the filaments have been stretched here and compressed there, along all three dimensions, with different strength and effect in different places. We can measure how much “stretching” or “compression” the long waves produced on the structure at different positions by looking at the derivative of the displacement field along the filaments. We randomly choose one of the 16 filaments shown in Fig. 4 and computed the three components of the displacement field along it. These are shown in the first panel of Fig. 5.
The displacements plotted in Figure 5 are continuous functions as expected, meaning that there is no “stripping” of two nearby particles due to the long waves. The total three dimensional displacement of a particle is shown in the second panel. Its derivative tells us how much two initially close particles can be taken apart by the long waves. The steepest part of the function is blown up in the third panel. The derivative on the slope between $x=500$ Mpc and $x=520$ Mpc is approximately constant and has a value $ds/dx_0 \approx 0.5$. If we take this as a figure representative of the whole displacement field, it tells us that particles originally in a structure of size $R$ will be taken apart at most by an amount equal to $R/2$, that is half the size of the structure. Hence the identity of e.g. dark matter halos is preserved by the if the mean inter-halo separation is bigger than the size of each individual halo. As this is usually the case, we do not expect the small structures of the halo to be appreciably disturbed by the action of the long linear waves of the , although the final word on this issue can only be given by direct analysis of the halos through some group finding algorithm, which we have not yet done.
In Figure 6 we compare MAP8x144 and P3M256 by showing slices of $640 \times 640 \times 10$ Mpc$^3$. The linear power spectrum normalization is $\sigma_8=0.7$ for all slices. In this Figure we sampled the particles of the three simulations so that the number of particles shown in each slice is approximately equal. Fig. 6a shows a slice from the original P3M144, replicated over such a volume, prior to any mode substitution; the cut shows regions with structure as big as the whole original simulation P3M144, and the periodicity over $100$ Mpc is evident. Fig. 6b shows the same Eulerian slice after we performed the mode substitution. In Fig. 6b most of the periodicity has been disrupted: the long waves have stretched here and compressed there the original pattern of clustering and voids, resulting in a more varied structure. New patterns have developed with a characteristic scale much larger than the original box size. Fig. 6c shows a slice from the P3M256 $N$-body simulation. It is relatively easy for the eye to see the richer range in patterns of this true simulation when compared to Fig. 6b. Such a comparison brings to evidence the residual periodicity of MAP8x144, in the form of cell-like structure of about the size of the original simulation, P3M144. Cells of similar size also appear in Fig. 6c, but they are less evident due to their more irregular distribution. Although the figures give an idea of the performance of the on large scales, they are also slightly misleading in that they do not show how much better MAP8x144 actually is on small scales owing to its better resolution when compared to P3M256. This can only be shown by specific tests, like the ones we are going to present in the next Sections. On the whole, the largest structures which can be identified in Fig. 6b and Fig. 6c have roughly the same size, of the order of about $150$ Mpc. The relatively emptier regions seen in Fig. 6b, in comparison with Fig. 6c, are due to the sparse sampling we had to apply to P3M144 (roughly one particle in 90) in order to reduce the total number of particles in the (800 Mpc)$^3$ box to 16 million.
In Fig. 7 we plot the logarithm of the one point density fluctuation distribution function $f(\delta)$ obtained by computing the density field on a regular lattice with $5$ Mpc spacing using the TSC interpolation scheme. We also computed the density fluctuation for our various simulations; the values obtained are listed on the first row of Table 1. We see that MAP8x144 has a larger number of grid points with no particles and a longer tail of high over-densities. The abundance of empty grid points is due to the sampling problem noticed already in Fig. 6b. In order to produce MAP8x144 we took only $32,000$ particles from each copy of P3M144. This corresponds to an average of $32$ particles contributing to the density value of each grid point; such number is evidently too low to allow a good sampling of very under-dense regions, which turn out completely empty. This does not happen for P3M144, where the average number of particles contributing to a grid point is about $3000$. As for P3M256, the Figure shows that $64$ particles per grid point seem enough for a good sampling, but this is probably due to the larger particle mass of this simulation.
[Density and Velocity Moments]{} \[tab\]
P3M144 MAP8x144 P3M256
-------------------------------- ----------------- ----------------- -----------------
$\langle\delta^2\rangle^{1/2}$ 1.96 1.94 2.00
$\langle|\vec v|\rangle$ 655 km s$^{-1}$ 722 km s$^{-1}$ 699 km s$^{-1}$
$\langle v^2\rangle^{1/2}$ 768 km s$^{-1}$ 827 km s$^{-1}$ 804 km s$^{-1}$
Fig. 8 shows $P_\delta(k)$ for P3M144, both before and after removing the long waves, as well as $P_\delta(k)$ for MAP8x144 and P3M256. The spectra have been computed using a $512^3$ regular grid ($128^3$ for P3M144 after the mode removing); they include both a deconvolution from the interpolating scheme and a shot noise subtraction. The different starting and ending point for the curves are a consequence of the different scales of the simulations.
If our assumption of linearity is correct we should see in $P_\delta(k)$ the same change in power that we performed on the displacement field. In fact we do see that the amplitude of the power spectrum after mode removing has decreased by a factor of about $40$ in the first bin, corresponding to modes with $\Delta k/k_{min} \in [0.5,1.5]$. We note however that the power spectrum after mode removing seems to be slightly lower than the original one: defining $b^2(k)\equiv P_\delta(k)/P_{\delta}^-(k)$, where $P_{\delta}^-(k)$ is the power spectrum of P3M144 after the mode removing step, we found that $b(k) \approx 1.15$ for high $k$. This may be due to some mode coupling between small and large $k$, caused by the fact that some of the subtracted waves are not evolving in a sufficiently linear way. If this is the case, then our tolerance of $\sim 20\%$ departures from linearity shown in Figure 2 would not be enough to ensure accuracy to better than 15%. To test this possibility one could apply the mode removing to a larger simulation, e.g. twice the linear size of P3M144, and see if the effect is still there. On the other hand, the fact that P3M256 also has a power spectrum amplitude which is little lower than that of P3M144 on small scales suggests that the explanation of these differences could be more complicated.
As a check of our method we also tried to subtract from P3M144 a larger number of modes, corresponding to cubes of extension $r_k=2$, $3$ and $4$ in Fourier space. We found that removing shorter and shorter waves from the displacement field does not correspond to removing the equivalent power from the density fluctuations because the linear relation between the displacement and density fluctuation field breaks down for small scales.
The spikes shown by the power spectrum of the simulation at $\log\ k \in [-1,-0.6]$ are an artifact due to the uneven sampling of power (as shown in Fig.1b). This was not taken into account in the way $P_\delta(k)$ is numerically evaluated, since the power summation is made in spherical shells with constant width $\Delta k= (2\pi/800)$ Mpc$^{-1}$. The effect shows up at values of $k$ where the old and new power sampling mix together, and disappears at higher wavenumbers due to the higher number of modes present in each shell.
The global agreement of the power spectra of P3M144, MAP8x144 and P3M256 should imply an equally good agreement of the corresponding mass autocorrelation functions. We indeed found that this statistic differs by less than 35% (or 0.13 in logarithm) between the three simulations over range of pair separations not influenced by small scale force softening or by border effects.
The velocity field
------------------
We would like to test the performance on the velocity field, in the same way as we tested the density field. Unlike the density, the velocity has the advantage of being defined (for single particles) without any smoothing, enabling us to study directly the distribution function of the raw particle velocities. In Fig. 9 we plot the distribution function $f(v)$ of the velocity modulus for the three simulations. The gain in peculiar velocities due to the power associated with long waves is evident. Table 1 lists the first and second moment of the distributions for easy comparison. Besides increasing the peculiar velocities, long waves add coherence to the velocity field, so that the average velocity of a region of size 100 Mpc is roughly zero for the P3M144 simulation, but is of a few hundred km s$^{-1}$ for the simulation.
We would like to measure the velocity power spectrum $P_v(k)$ for the simulations. Unfortunately, this is not a very well defined quantity. In fact, in order to define a velocity field on a regular grid one needs to take the ratio between the momentum and the density fields. If there are no particles in the neighborhood of a grid point, the density and the momentum will be zero there, and the velocity will be undetermined. Therefore, in order to define a velocity field at all grid points one has to smooth the fields on a larger scale, so that some particles contribute to the density and momentum field of every grid point. However, in doing so information is lost on the velocity field at all scales smaller than the smoothing scale. Unlike the case of the density field, where we could subtract the effect of the smoothing scheme by de-convolving $P_\delta(k)$ in Fourier space, here we deal with a ratio of convolved fields, which does not correspond to a simple multiplication in Fourier space. Hence the deconvolution from the interpolating scheme is not possible for the velocity field. This sets a limit on the resolution of the velocity power spectrum at small scales.
In our case, to define the velocity fields of our simulations we evaluated the density and momentum density fields using TSC interpolation onto a 5 Mpc grid, followed by Gaussian smoothing of each with a kernel of size $7$ Mpc before the ratio of fields is taken. Since in linear theory only the longitudinal component of the velocity field $\vec v_\parallel$ (defined by the irrotationality condition: $\vec \nabla \times \vec v_\parallel =0$) is related to the density fluctuation field $\delta(\vec x)$, we considered only the power spectrum of $\vec v_\parallel$. Fig. 10 compares $P_v(k)$ for P3M144, P3M256 and MAP8x144, superimposed on the linear prediction. The difference in amplitude between the latter and the power spectra of the three simulations at high wavenumbers is an effect of the filtering of the velocity field of the former. We can see from the Figure how $P_v(k)$ for the simulation shows some amplitude difference over the spectra of both P3M144 and P3M256 on scales smaller than about $25$ Mpc. This may again be related to the sampling problem discussed before, or to the invalidity of the linear approximation, but our current understanding of the effects of smoothing on the velocity field is too limited to allow a definite interpretation.
From the simulations we also evaluated the pairwise velocity dispersion $\sigma_{v,12}$ as a function of pair separation. This is defined as the second central moment of the velocity field: given pairs of particles with velocities $\vec v_1$ and $\vec v_2$, separated by a distance $r_{12}$, the parallel component of the pairwise velocity dispersion is $$\sigma_{v,12 \parallel} = \left\langle\left[(\vec v_2 -\vec v_1)\cdot
\hat r_{12}\right]^2\right\rangle^{1/2}.$$ where $\hat r_{12} \equiv \vec r_{12}/|\vec r_{12}|$. Since the value of $\sigma^2_{v,12 \parallel}(r)$ is determined by the power associated with density fluctuations on scales $\lambda
\mincir 1/r$, the pairwise velocity dispersion is a suitable statistic to estimate the small scale velocity power of a simulation. Fig. 11 plots $\sigma_{v,12 \parallel}$ for our three simulations; the three curves agree with each other to better than 10%. From this figure our conclusion is that the has not changed the velocity field significantly on small scales. Small differences between the curves are found also for P3M256 and may just reflect statistical fluctuations. The global agreement of this statistic contrasts somewhat with the different amplitudes of the velocity power spectrum between P3M144 and MAP8x144. This difference is not fully explained, but it could again be due to nonlinear effects in the : the original simulation (L = 100 Mpc) might still be too small to guarantee a sufficient linear evolution for its fundamental modes, with $\lambda = L$. Mode coupling would then propagate to small scales any change in the large scale power. Put another way, the pairwise velocity seems more robust than the velocity power spectrum in measuring the small scale power.
Summary and Conclusions
=======================
We have proposed a new method to add to an $N$-body simulation the large scale power associated with scales larger than the volume in which the simulation is performed. We made use of the Zel’dovich approximation (Zel’dovich, 1970) to change each particle’s position and velocity according to the extra power introduced. We tested the method using a simulation of standard cold dark matter, which we called P3M144. It had been evolved in a cube of $100$ Mpc on a side by means of a P$^3$M code within $144^3$ collisionless particles. We replicated the simulation to fill a cube of side $800$ Mpc and added to it the power associated with fluctuations up to scales $\lambda=800$ Mpc. We compared this enlarged simulation, named MAP8x144, with the original P3M144 simulation and with a larger simulation in a volume of $640$ Mpc on a side called P3M256.
We showed both visually and by means of several statistics how the density and velocity field are modified by the addition of long waves: velocities are increased and structures are created with characteristic scales larger than the original box size. The rms velocity of a particle is $v_{rms}=768$ km s$^{-1}$ in P3M144, $v_{rms}=827$ km s$^{-1}$ for MAP8x144, and $v_{rms}=804$ km s$^{-1}$ for P3M256. The equivalent figures for the density field show that the slightly enhances the preexisting clustering.
Our analysis of several statistics shows the effects of long waves in nonlinear simulations. The procedure assumes that long and short waves evolve independently and that the former are describable by the Zel’dovich approximation. However, our results suggest that there may be some transfer of power between long and short wavelengths in our simulations, for example in the power spectra. If so, for accurate results our method may require stronger conditions than those met in our simulations. While a detailed study of this problem is beyond the scope of this paper, we can try to shed more light on this point by examining Fig. 2b, which refers to the simulation P3M256, performed on a cube of $640$ Mpc on a side.
We can identify three intervals of wavenumbers corresponding to different behaviors of the growth rates. The first interval corresponds to fluctuations with $\log (k$ Mpc$) \geq -0.8$ (i.e. $\lambda \leq 40$ Mpc); these grow faster than linear in all plotted outputs, and their growth becomes faster at later times, defining what is usually called the nonlinear regime. A second intermediate interval is approximately $-1.5 \leq \log (k$ Mpc$) \leq -0.8$ (corresponding to scales between $200$ Mpc and $40$ Mpc); fluctuations in this range grow slightly more slowly than the linear theory prediction, the effect becoming most visible at the latest time $a=0.7$. Finally, fluctuations with wavenumber $\log (k$ Mpc$) \leq -1.5$ ($\lambda \magcir 200$ Mpc) maintain a strict linear growth (to within 1%) at all times. The analogous plot for P3M144 (shown in Fig. 2a) may suggest a similar behavior at least for a=0.7, but unfortunately the growth rates are much more unstable, due to the smaller size of the simulation, so that a definite interpretation is not possible.
The existence of these three regimes suggests that some coupling exists between the modes in the intermediate interval of wavenumbers and the modes in the nonlinear regime, with a transfer of power from the former to the latter. Transfer of power between long and short wavelength modes is consistent with the results of Jain and Bertschinger (1994): using second order perturbative calculations for a CDM-like spectrum, they found that mode coupling cause a slight suppression of $P(k)$ at small $k$, and a significant enhancement at high $k$ compared to the linear prediction, with the transition region occurring where the spectral slope is $n\mincir-1$ (that is $k \magcir 0.1$ Mpc) respectively. If this interpretation is correct, then the size of P3M144 ($100$ Mpc on a side) is slightly too small to perform the mode substitution, because the longest waves in the simulation are still weakly coupled with shorter wavelengths. Therefore by subtracting the longest waves from P3M144 using our displacement field technique we have also subtracted some power in the density field from small, nonlinear scales. The small scale power however is not given back with the addition of long waves up to $L_{big}$. In fact, the new large scale power was not present during the simulation, but is added randomly at the end of time evolution, and so has no chance to dynamically enhance or suppress small-scale waves. Fortunately, even if some mode coupling affects the present example, it does not represent a limit of the method but just of the simulation we used to apply the , so the conclusions we drew on the method are still valid. One obvious way to verify this hypothesis is to run the starting with a simulation originally performed on a larger volume, for example $200$ Mpc on a side. In this case we would expect to see no significant power transfer.
On a completely different issue, we would like to stress the point that our end product is not equivalent to a real simulation evolved from initial conditions on a comparable scale. In fact, outside the substituted region of Fourier space the MAP8x144 simulation samples high wavenumbers exactly like P3M144. That is, the density of Fourier modes there is not as high as in an $N$-body simulation actually performed on $800$ Mpc on a side. Moreover, some small scale periodicity may still be present in the final result, even if modulated by the large scale waves.
The can also be applied to any catalog of dark matter halos, or to any class of objects that can be defined in a simulation. In that case, however, one cannot trace back the Lagrangian position of the objects, since they contain different particles, and are defined during – or after – time evolution. What one does instead is to apply in Equations (\[erem\]) and (\[eadd\]) the displacement field obtained by the original particle distribution which is parent to the halo catalog. That is, the displacement field of Equations (\[ldisp\]) and (\[edisp\]) is obtained from the particles.
Finally, the application of the described in this paper concerns the study of very large scale density and velocity fields. However, other applications are possible, with focus on different aspects. In fact, the procedure that we have described here does not require at all the use of a box as big as the longest added wavelengths. Once the mode removing step is performed, one can interpolate the added displacement field to an arbitrary volume of the simulation. For example, in a study that does not require using a very large simulated volume, one can introduce the large-scale power in the original simulation without taking any replica. In such a case all the particles of the original simulation may be used, to preserve the initial high definition and resolution.
We would like to thank Jim Frederic for providing the P3M256 simulation, and Sabino Matarrese and Simon White for useful comments and suggestions on an earlier version of the manuscript. We wish to thank the referee, Michael Strauss, for a helpful referee’s report. Thanks are also given to the National Center for Super-computing Applications which provided our computer time. Financial support was provided by NSF grant AST-9318185 and for G.T. also by the Italian MURST and by an EC-HCM fellowship.
=0.1truecm
Couchman, M. M. P., and Carlberg, R. G. 1992, , 389, 453
Gelb, J., and Bertschinger, E. 1994, , 436, 467
Gunn, J.E., and Weinberg, D.H. 1994, [in]{} Wide–Field Spectroscopy and the Distant Universe, ed. Maddox, S.J. & Aragon–Salamanca, A. (Singapore: World Scientific), p. 3
Jain B., and Bertschinger, E. 1994, , 431, 495
Landy, S.D., Schectman, S.A., Lin, H., Kirshner, R.P., Oemler, A.A. and Tucker, D. 1996, , 456, 1L
Strauss, M. A., Cen, R., and Ostriker, J. P. 1993, , 408, 389
Strauss, M. A., Cen, R., Ostriker, J. P., Lauer, T. R. and Postman, M. 1995, , 444, 507.
Turner, E. L., Cen, R., and Ostriker, J. P. 1992, , 103, 1427
Vettolani, G., et al. 1994, [in]{} Wide–Field Spectroscopy and the Distant Universe, ed. Maddox, S.J. & Aragon–Salamanca, A. (Singapore: World Scientific), p. 115
Zel’dovich, Ya. B. 1970, , 5, 84
**Figure Captions**
|
---
abstract: 'A new class of micromechanically motivated chain network models for soft biological tissues is presented. On the microlevel, it is based on the statistics of long chain molecules. A wormlike chain model is applied to capture the behavior of the collagen microfibrils. On the macrolevel, the network of collagen chains is represented by a transversely isotropic eight chain unit cell introducing one characteristic material axis. Biomechanically induced remodeling is captured by allowing for a continuous reorientation of the predominant unit cell axis driven by a biomechanical stimulus. To this end, we adopt the gradual alignment of the unit cell axis with the direction of maximum principal strain. The evolution of the unit cell axis’ orientation is governed by a first-order rate equation. For the temporal discretization of the remodeling rate equation, we suggest an exponential update scheme of Euler-Rodrigues type. For the spatial discretization, a finite element strategy is applied which introduces the current individual cell orientation as an internal variable on the integration point level. Selected model problems are analyzed to illustrate the basic features of the new model. Finally, the presented approach is applied to the biomechanically relevant boundary value problem of an in vitro engineered functional tendon construct.'
author:
- |
Ellen Kuhl\
Chair for Applied Mechanics\
University of Kaiserslautern, D-67653 Kaiserslautern, Germany\
[ekuhl@rhrk.uni-kl.de]{}\
Krishna Garikipati\
Department of Mechanical Engineering,\
University of Michigan, Ann Arbor, MI 48109, USA\
[krishna@engin.umich.edu]{}\
Ellen M. Arruda\
Department of Mechanical Engineering\
Macromolecular Science and Engineering Program\
University of Michigan, Ann Arbor, MI 48109, USA\
[arruda@umich.edu]{}\
Karl Grosh\
Department of Mechanical Engineering,\
University of Michigan, Ann Arbor, MI 48109, USA\
[grosh@engin.umich.edu]{}
bibliography:
- 'litra.bib'
title: 'Remodeling of biological tissue: Mechanically induced reorientation of a transversely isotropic chain network'
---
\#1 \#1 \#1 \#1 \#1 \#1 \#1
|
---
abstract: 'Recent work suggests that many short-period extrasolar planets may have spin obliquities that are significantly tilted with respect to their orbital planes. These large obliquities are a natural outcome of “secular spin-orbit resonance”, a configuration in which the planetary spin precession frequency matches the frequency of orbit nodal regression, or a Fourier component thereof. While exoplanet spin obliquities have not yet been measured directly, they may be detectable indirectly through their signatures in various observations, such as photometric measurements across the full phase of a planet’s orbit. In this work, we employ a thermal radiative model to explore how large polar tilts affect full-phase light curves, and we discuss the range of unique signatures that are expected to result. We show that the well-studied short-period planets [HD 149026 b]{}, [WASP-12 b]{}, and [CoRoT-2 b]{} all exhibit phase curve features that may arise from being in high-obliquity states. We also constrain the parameters and assess the detectability of hypothetical perturbing planets that could maintain the planets in these states. Among the three planets considered, [CoRoT-2 b]{} has the tightest constraints on its proposed obliquity ($45.8^{\circ} \pm 1.4^{\circ}$) and axial orientation. For [HD 149026 b]{}, we find no significant evidence for a non-zero obliquity, and the phase curve of [WASP-12 b]{} is too complicated by strong tidal distortions for a conclusive assessment.'
author:
- 'Arthur D. Adams'
- Sarah Millholland
- 'Gregory P. Laughlin'
bibliography:
- 'apj-jour.bib'
- 'paper.bib'
title: Signatures of Obliquity in Thermal Phase Curves of Hot Jupiters
---
Introduction {#sec:introduction}
============
Analysis of the full-phase photometry of transiting extrasolar planets has generated a number of insights. By tracking the emission and reflection of light from planets as they trace through their orbits, one may derive important clues regarding the planets’ atmospheric compositions, surface flow patterns, atmospheric thermal responses, cloud coverage, day-to-night heat redistribution efficiency, and more. High signal-to-noise detections of full-phase photometry are now routinely performed in both the optical and in the near-infrared, and the technique has been employed for both detection and characterization [see, e.g. @heng15; @shp17; @dem17].
The climate properties probed by full-phase light curves are highly sensitive to the planet’s orbital and spin geometry. For instance, many short-period ($P\lesssim 5\,{\rm d}$) planets are subject to strong tidal evolution that has produced complete or near-complete orbital circularization [@Rod2010]. Moreover, because time scales for planetary tidal spin evolution are generally shorter than for orbital evolution, it is usually assumed that short-period planets have zero obliquity, and are spinning synchronously for orbits with $e=0$ [see e.g. @Gladman1996] or pseudo-synchronously for orbits with finite eccentricity [see e.g. @hut81; @iva07].
Time-resolved photometry of assumed-synchronous planets on circular orbits suggests that the peak infrared emission from the planet usually lies eastward of the sub-stellar point [e.g. @knu07b; @knu09a; @cow11a; @knu12; @Cowan2012; @zel14; @sch17; @zha18]. By contrast, phase curves in the optical tend to suggest that peak reflectivity occurs westward of the sub-stellar point and closer to the morning terminator [@shp17]. These observations are generally interpreted to imply eastward circumplanetary flow and cloud decks that burn off when advected into the direct beam of instellation.
This composite picture can be tested when the phase curves of eccentric orbits are tracked [e.g. @lau09; @Adams2019a]. A planet with a non-circular orbit cannot be fully tidally de-spun; the surface flows are thus dynamically responsive. Bulk properties of the atmosphere such as the radiative response timescale can be directly inferred [e.g. @Cowan2012; @dew16; @ada18b].
Just as the eccentricity produces observable signatures in the full-phase photometry, so too will the presence of a significant obliquity. It is thus of interest to examine possible obliquity-induced signatures in the context of the data sets that are currently available.
Several authors have developed a mathematical formalism to predict optical observational effects of the relationship between a planet’s spin, orbit, and viewing geometry. @Kawahara2010 constructed a geometric framework for mapping planets (including oblique ones) in reflected/scattered light. This framework was later extended to account for effects such as cloud cover [@Kawahara2011] and generally inclined orbits [@Fujii2012]. Signatures of obliquity have been predicted in Fourier analyses of photometry for directly-imaged planets [@Kawahara2016]. @Schwartz2016 demonstrated the ability to infer spin axis orientation for general albedo maps, and outlined a feasible minimum observing baseline for making robust inferences of the spin axis orientation. @Farr2018 introduced `exocartographer`[^1], a software package that generates reflected light photometry for an arbitrary albedo map and spin geometry and fits a variety of time-sampled data. Another software package, `STARRY`[^2] [@lug19], provides computationally efficient determinations of full-phase light curves, occultation signatures, and transit signals by using a global planetary surface pattern expressed as a sum of spherical harmonics.
Here we consider a similar geometric framework in the near-infrared, where the thermal emission from the planet — rather than reflected light — should be the primary source of the observed flux. @Cowan2012 considered the effects of eccentricity and obliquity for the phase photometry of an Earth-like planet and demonstrated that one can infer the characteristic thermal time scales. They also concluded that a combination of optical and infrared observations would be necessary to accurately measure bulk atmospheric conditions. Most recently, @Ohno2019a [@Ohno2019b] developed a comprehensive shallow water model for describing the atmospheric dynamics and resulting thermal phase variations of planets with arbitrary spin period, spin obliquity, and orbital eccentricity. In short, because these three properties all determine the instantaneous sub-stellar location, the resulting thermal phase variations will be shaped by their combined influence.
While one can model thermal variations for any choice of orbit and spin orientation, we will first restrict ourselves to scenarios that are physically feasible for systems with close-in giant planets. In addition to representing a dynamically plausible configuration, accurate phase curves for short-period giant planets already exist. Moreover, if we infer a particular spin-orbit geometry for a planet from its observed photometry, we may place constraints on the system architecture and make potentially observable dynamical predictions.
In this work, we focus on three short-period giant planets: [HD 149026 b]{}, [WASP-12 b]{}, and [CoRoT-2 b]{}. These all have full-phase thermal light curves obtained with *Spitzer*, and they have features [discussed in detail for [HD 149026 b]{} and [WASP-12 b]{} in @ada18b] that suggest they are potential high obliquity candidates. [HD 149026 b]{}, [WASP-12 b]{}, and [CoRoT-2 b]{} do not currently have any known planetary companions. They would therefore require special dynamical states to have anything other than circular orbits, spin-orbit synchronization, and zero obliquities.
One such dynamical state that may maintain a large obliquity is a “Cassini state”, an equilibrium configuration where the planet’s spin vector stays fixed in the reference frame of its precessing orbit [@Colombo1966; @Peale1969; @Ward1975a]. As an instance of a Cassini state, a secular spin-orbit resonance involves an average commensurability between the frequency of the planet’s spin axis precession and the frequency of its orbit nodal recession (or a Fourier component thereof). The orbital recession may be provided by a number of sources, such as another planet in the system, the stellar quadrupolar gravitational potential [@Fabrycky2007; @MillhollandLaughlin2019], or, early on in the system’s lifetime, the protoplanetary disk [@MillhollandBatygin2019]. The result is a stable state in which a planet may maintain a non-synchronous spin and non-zero obliquity, even in the presence of strong tides.
Cassini states have been invoked to explain, for example, the co-precession of the lunar spin vector and the lunar orbit normal [@DeLaunay1860], the obliquities of Saturn [@Ward2004; @Hamilton2004] and Jupiter [@Ward2006], and the spin precession state of Mercury [@Peale2006]. They have also been considered as a mechanism to inflate the radii of hot Jupiters [@Winn2005], albeit with disputed feasibility [@Levrard2007; @Fabrycky2007]. Recently, [@MillhollandLaughlin2019] showed that high-obliquity Cassini states might be common for planets in short-period, compact systems. In this paper, we examine whether [HD 149026 b]{}, [WASP-12 b]{}, and [CoRoT-2 b]{} may have large obliquities due to their participation in secular spin-orbit resonances, and we examine the potential signatures of such states in their thermal full-phase light curves.
This paper is organized as follows. In §\[sec: Three Unusual Planets\], we introduce three unusual planets – [HD 149026 b]{}, [WASP-12 b]{}, and [CoRoT-2 b]{} – and discuss why they are viable candidates for obliquity investigations. In §\[sec:photometry\] we review these planets’ near-infrared photometric observations, including the analyses of their light curve morphologies. §\[sec:model\] describes the thermal model, which accounts for a general rotation rate and spin axis orientation. This framework is then employed in §\[sec:results\] to re-analyze the *Spitzer* phase photometry. In order to build a physical framework where we could plausibly observe a single-transiting, close-in, oblique planet, we make a feasibility assessment in §\[sec:REBOUND\] wherein a compact, nearly co-planar multi-planet system evolves on short timescales to states of high mutual inclination, with only one planet capable of transiting. Finally, §\[sec:Cassini state\] provides a dynamical analysis of the three case study systems and explores the possibility that a Cassini state may exist between the known planets and an as-yet undiscovered, non-transiting companion.
Three Unusual Planets {#sec: Three Unusual Planets}
=====================
[cccc]{}
$P$ (days) & $2.8758911 \pm 2.5\times10^{-6}$ & $1.09142119 \pm 2.1\times10^{-7}$ & $1.7429935 \pm 1.0\times10^{-6}$\
$e$ & $0$ & $0.0447\pm0.0043$ & $0.0143^{+0.0077}_{-0.0076}$\
$\varpi\left(^\circ\right)$ & N/A & $272.7^{+2.4}_{-1.3}$ & $102^{+17}_{-5}$\
$M_\mathrm{P}$ $\left(M_\mathrm{J}\right)$ & $0.368^{+0.013}_{-0.014}$ & $1.43 \pm 0.14$ & $3.47 \pm 0.22$\
$R_\mathrm{P}$ $\left(R_\mathrm{J}\right)$ & $0.813^{+0.027}_{-0.025}$ & $1.825 \pm 0.094$ & $1.466^{+0.042}_{-0.044}$\
$M_\star$ $\left(M_\odot\right)$ & $1.345 \pm 0.020$ & $1.280 \pm 0.05$ & $0.96 \pm 0.08$\
$R_\star$ $\left(R_\odot\right)$ & $1.541^{+0.046}_{-0.042}$ & $1.630 \pm 0.08$ & $0.906^{+0.026}_{-0.027}$\
$T_{\textrm{eff}}$ (K) & $6160 \pm 50$ & $6300^{+200}_{-100}$ & $5625 \pm 120$\
Ref. & (1) & (2)–(4) & (5)\
\[table:planet\_properties\]
[HD 149026 b]{} orbits its subgiant host in 2.88 days; the planet is slightly more massive (0.38 M$_\mathrm{J}$) but smaller in size (0.74 R$_\mathrm{J}$) than Saturn [@Stassun2017]. Its formation is still a matter of ongoing study, since the bulk density is quite high compared with other short-period giant planets. @Sato2005, who first measured the planet’s radius during transit, proposed that a high core mass of $\sim\!67$ M$_\oplus$ could explain the measurements. Subsequent assessments have inferred similarly high core masses in the range of $\sim\!50$–110 M$_\oplus$ [@Fortney2006; @Ikoma2006; @Broeg2007; @Burrows2007] using combinations of atmospheric and interior modeling. @Ikoma2006 proposed that [HD 149026 b]{}’s high metallicity and modest H/He envelope might either be explained via planetesimal capture and a limited gas supply, if its current state was obtained prior to disk dissipation, or a combination of envelope photoevaporation, Roche lobe overflow, and major collisions after disk dissipation. @zha18 find that the best-fit phase offsets from the *Spitzer* 3.6 $\mu$m and 4.5 $\mu$m full phase photometry are both significantly different from zero and in disagreement with each other. While @zha18 suggested uncharacterized instrumental systematics as a potential cause of the disparity, it is also worth investigating whether unusual global atmospheric dynamics may be at play.
[WASP-12 b]{} is another short-period ($1.09$ days) giant planet, which is tidally distorted due to its density and proximity to its host star [@lis10; @lai10]. There is spectroscopic evidence that the planet is overflowing its Roche lobe [@Fossati2010; @Haswell2012; @Fossati2013; @Jackson2017], and hydrodynamic simulations [@Debrecht2018] suggest that it is undergoing significant atmospheric mass loss. @VonEssen2019 have recently analyzed time variability in the measured optical eclipse depths; with neither cloud albedo nor high temperatures tenable to explain the variability without invoking extreme physical magnitudes, they point cautiously to additional occultation from the atmospheric mass loss as a possible mechanism. Extensive transit observations have also been made (detailed further in §\[sec:photometry\]), whose rapidly advancing ephemerides suggest either apsidal precession of an eccentric orbit or orbital decay [@mac2016; @Patra2017]. Recent observations [@Maciejewski2018] and theoretical investigations [@Bailey2019] are in favor of the orbital in-spiral scenario. [@MillhollandLaughlin2018] proposed that this rapid orbital decay may be due to tidal dissipation in the planet that is strongly enhanced by a high obliquity state.
[CoRoT-2 b]{} is a $3.3 \ M_{\mathrm{J}}$ hot Jupiter with a 1.74-day orbital period and $1.47 \ R_{\mathrm{J}}$ radius that has consistently been measured as anomalously inflated[^3] [@Alonso2008; @Gillon2010; @sou11]. Previous proposed explanations include that the system is very young ($\sim\!30$–40 Myr) and the planet has not yet fully contracted gravitationally, or it is a bit older ($\sim\!130$–500 Myr) and the radius inflation is a transient response from a recent collision [@Guillot2011]. Alternatively, the extreme radius inflation may be the result of obliquity tides, a possibility we discuss later in §\[sec:discussion\]. In addition, @Dang2018 reported *Spitzer* thermal phase observations that robustly indicate that the day-side hotspot is westward of the sub-stellar point, in contrast to the eastward offset that planets typically exhibit. Magnetohydrodynamic effects may be one mechanism for generating an unusual westward offset, but recently [@Hindle2019] inferred that such effects are very unlikely to be a viable mechanism for the specific westward offsets seen at 4.5 $\mu$m in [CoRoT-2 b]{}. @Dang2018 considered that it may arise from non-synchronous rotation, which, as discussed in §\[sec:Cassini state\], would be a consequence of a high-obliquity state.
Photometry {#sec:photometry}
==========
[HD 149026 b]{} has been observed in transit in Strömgren $b$ and $y$ [@Sato2005; @Winn2008], $g$ and $r$ [@Char2006], NICMOS (1.1–2.0 $\mu$m) on the Hubble Space Telescope [@Carter2009], and the 8.0 $\mu$m channel of the Infrared Array Camera (IRAC) on the *Spitzer* Space Telescope [@Nutzman2009]. Secondary eclipses have also been observed in each of the 4 IRAC channels (3.6–8.0 $\mu$m) and the Infrared Spectrograph (IRS) at 16 $\mu$m [@ste12]. @knu09b presented the first phase photometry, which spanned just over half the orbit in 8.0 $\mu$m. Most recently @zha18 published two full-phase observations in the Warm *Spitzer* bands (3.6 and 4.5 $\mu$m). We draw attention to §4.1 of @zha18, where the authors point out that inconsistencies between the bands warrant a fair degree of skepticism. In particular, the positive phase offset, or late minimum, of the 3.6 $\mu$m time series, is difficult to explain with modeling that assumes spin-orbit synchronization.
[WASP-12 b]{} was first discovered in transit via the SuperWASP camera [@heb09], and subsequent transits have been observed in the $V$ band [@cha12], $R$ [@mac11], $J$, $H$, and $K_s$ [@cro11], and at 3.6–8.0 $\mu$m from *Spitzer* [@cam11; @cow12; @ste14]. @man13 also provided transit spectroscopy from WFC3 (1.1–1.7 $\mu$m) on the Hubble Space Telescope. Our work focuses on the full-orbit phase curves available from the warm *Spitzer* (3.6 and 4.5 $\mu$m) channels, originally published in @cow12.
[CoRoT-2 b]{} has been observed in transit [@Alonso2008] and eclipse via photometry in the *Spitzer* IRAC 3.6 [@Deming2011], 4.5, and 8.0 $\mu$m [@Gillon2010] channels. Transit spectra have also been measured [@Bouchy2008; @Czesla2012; @bal15]. Refinement of the stellar parameters for CoRoT-2 led to revised transit depths [@sou11; @sou12; @bal15]. Even with the revised radius, [CoRoT-2 b]{} is estimated to have a density comparable with Jupiter’s, hinting at a possible radius inflation that might be due to uncharacterized tidal processes. Recently, @Dang2018 published a full-phase light curve at 4.5 $\mu$m, which showed an unusual and substantial $23^{\circ}$ westward phase offset. The authors proposed possible astrophysical sources of the offset including westward winds, magnetic effects, or partial cloud cover. Inhomogeneous cloud cover has been studied as a possible explanation of the westward offsets seen in optical light curves [@shp17], and optical follow-up for [CoRoT-2 b]{} may support this hypothesis [@Barstow2018]. Here we offer the alternative hypothesis of a high obliquity state. We show that this could not only produce the observed westward offset, but it would also provide new information about the formation and dynamical evolution mechanisms of close-in giants.
Components of the Oblique Thermal Model {#sec:model}
=======================================
In order to develop a model of thermal phase variations for a planet with non-zero spin obliquity, we build upon the model framework developed for the analysis in @ada18b. Each planet is divided into a grid with cells of dimension $5^\circ\times 5^\circ$ by latitude and longitude. We start with the known system properties of each modeled planet (Table \[table:planet\_properties\]). There are 6 tunable parameters which jointly govern the resulting thermal emission. The first three — the albedo $A$, equilibrium radiative timescale $\tau_{\mathrm{rad}}$, and minimum temperature $T_0$ — most directly control the thermal properties of each cell. They are put into their formal context in §\[sec:model:thermal\]. In short, each cell absorbs a fraction $1-A$ of radiation from the host star and re-radiates as a blackbody with a corresponding characteristic timescale. Its brightness temperature is set both by the time-dependent instellation and any non-stellar heating (e.g. tidal heat emanating from the planet interior), the latter of which is captured with the minimum temperature. The remaining three parameters define the components of the rotation vector. Movement of the cells in the model is set entirely by the rotation; the infrared photospheric layer of the planet rotates with some angular frequency $\omega_{\mathrm{rot}}$ ($\equiv 2\pi/P_{\mathrm{rot}}$ for the rotation period $P_{\mathrm{rot}}$) with some orientation of its axis relative to the orbital plane. This may be written in terms of two spherical angles: ${\epsilon}$, which is the angle between the planet’s axis and the orbit normal, and ${\zeta}$, which is the projected angle of the axis in the orbital plane relative to a reference direction. We set this reference direction along the planet-star line during periastron, such that ${\zeta}=0$ implies the northern hemisphere[^4] summer solstice occurs during periastron [equivalent to $f_\mathrm{sol}$ in @Ohno2019a]. This corresponds to an angle of the axial projection from the line of sight of ${\zeta}_\mathrm{obs} = {\zeta}-\nu_\mathrm{tra}$, for true anomaly at transit $\nu_\mathrm{tra} \equiv \pi/2 - \varpi$; we assume $\varpi \rightarrow \pi/2$ for circular orbits so that ${\zeta}={\zeta}_\mathrm{obs}$. The time evolution of grid cell temperatures is convolved with both the viewing geometry from Earth, which is approximated as edge-on, and the relevant instrumental band profiles, to predict full-orbit light curves for a given set of parameter values. Our fitting routine is detailed in §\[sec:results\].
Thermal Evolution of the Cells {#sec:model:thermal}
------------------------------
The planetary model is initialized at apastron with a uniform surface temperature $T_0$; for planets on circular orbits, we set an arbitrary argument of periastron $\varpi=\pi/2$ such that apastron occurs during secondary eclipse. To calculate the incoming stellar radiation over the orbit, we start with the star-planet separation $$\label{eq:orbit_distance}
r\!\left(t\right) = a \left( \frac{1-e^2}{1+e\cos\nu} \right)$$ where $a$ is the orbital semi-major axis, $e$ the orbital eccentricity, and $\nu = \nu\!\left(t\right)$ the true anomaly. To solve for the true anomaly from the time in orbit we first calculate the mean anomaly $M$, which is directly proportional to time: $M\!\left(t\right) = \omega_{\mathrm{rot}}\left(t - t_{\mathrm{peri}}\right)$ for rotation rate $\omega_{\mathrm{rot}}$ and periastron passage time $t_{\mathrm{peri}}$. The eccentric anomaly $E=E\!\left(t\right)$ is then given by Kepler’s equation $$\label{eq:Kepler}
M\!\left(t\right) = E - e \sin E.$$ There is a direct relation between regular time intervals and regular intervals in mean anomaly, but not for the eccentric or true anomalies; we must calculate the latter numerically. The sine and cosine of the true anomaly, $\nu$ are given by $$\begin{aligned}
\label{eq:true_anomaly}
\begin{split}
\cos\nu &= \frac{\cos E - e}{1-e\cos E}\\
\sin\nu &= \frac{\sqrt{1-e^2}\sin E}{1-e\cos E}.
\end{split}\end{aligned}$$
At each time $t$ in the orbit and longitude/latitude $\left(\phi, \theta\right)$ on the planetary surface, the equilibrium temperature is calculated via $$\label{eq:Teq}
T_{\textrm{eq}}^4\!\left( \phi, \theta, t \right) = \left(1 - A\right) \left( \frac{L_\star}{4\pi \sigma r^2} \right) \cos\alpha_\star + T_0^4$$ where $A$ is the planetary albedo, $L_\star$ the stellar luminosity, $\sigma$ the Stefan-Boltzmann constant, $r=r\!\left(t\right)$ the star-planet separation, $\alpha_\star=\alpha_\star\!\left( \phi, \theta, t \right)$ the local stellar altitude, and $T_0$ the minimum temperature parameter. The stellar altitude $\alpha_\star$ is, relative to the sub-stellar point pointed to by $\hat{r}_\star = \phi_\star\hat{\phi} + \theta_\star\hat{\theta}$ and the unit normal $\hat{n}$ at the position, $$\label{eq:stellar_altitude}
\cos\alpha_\star =
\begin{cases}
\hat{n}\cdot\hat{r}_\star, &\hat{n}\cdot\hat{r}_\star \geq 0 \\
0, &\hat{n}\cdot\hat{r}_\star < 0
\end{cases}$$ where $\hat{n}\cdot\hat{r}_\star \rightarrow \hat{r}\cdot\hat{r}_\star = \cos\theta\cos\theta_\star \left[\cos\left(\phi-\phi_\star\right)-1\right] + \cos\left(\theta-\theta_\star\right)$ for spherical planets, and the sub-stellar point is given by[^5] $$\begin{aligned}
\label{eq:substellar_point}
\begin{split}
\phi_\star =\ &\phi_\star\!\left(t_0\right) - \Big\{\omega_\mathrm{rot}\left(t-t_0\right) + \\ &\arctan\left\{\cos{\epsilon}\tan\left[\left(\nu-{\zeta}\right)-\nu\!\left(t_0\right)\right]\right\} \Big\} \\
\theta_\star =\ &\sin^{-1}\left[\sin{\epsilon}\cos\left(\nu-{\zeta}\right)\right].
\end{split}\end{aligned}$$ For tidally distorted planets, the unit normal $\hat{n}$ no longer matches the position unit vector $\hat{r}$; §\[sec:model:tidal\] covers this case.
Finally, the change in temperature of each cell in time is calculated as $$\label{eq:T}
\dot{T}\!\left( \phi, \theta, t \right) = \frac{T_{\textrm{eq}}}{4 \tau_{\textrm{rad}}} \left\{ 1 - \left[\frac{T\!\left( \phi, \theta, t \right)}{T_{\textrm{eq}}}\right]^4 \right\}\,,$$ which is a differential equation we can evaluate numerically for sufficiently small timesteps. We choose to divide each orbit into 200 timesteps.
Generating Observables {#sec:model:observables}
----------------------
Given a temperature map $T\!\left(\phi,\theta,t\right)$, we solve for the corresponding planet-star flux contrast via $$\label{eq:contrast}
\bar{F}\!\left(t\right) = \frac{1}{\pi}\left(\frac{R_\mathrm{p}}{R_\star}\right)^2 \frac{\iiint w\,B_\lambda\!\left(T\right) V \,d\lambda\,d\theta\,d\phi}{\int w\,B_\lambda\!\left(T_\star\right) \,d\lambda}$$ where $B_\lambda\!\left(T\right)$ is the specific blackbody intensity at a wavelength $\lambda$ and temperature $T$, $w=w\!\left(
\lambda\right)$ the weighted response of the instrumental bandpass at $\lambda$, and $V=V\!\left(\phi,\theta,t\right)$ is the component of the normal vectors of the cells along the line of sight, given by $$\label{eq:visibility}
V =
\begin{cases}
\hat{n}\cdot\hat{r}_\mathrm{obs}, &\hat{n}\cdot\hat{r}_\mathrm{obs} \geq 0 \\
0, &\hat{n}\cdot\hat{r}_\mathrm{obs} < 0
\end{cases}$$ where the sub-observer point is given by $$\begin{aligned}
\label{eq:subobserver_point}
\begin{split}
\phi_\mathrm{obs} &= \phi_\star\!\left(t_\mathrm{ecl}\right) - \omega_\mathrm{rot}\left(t-t_\mathrm{ecl}\right) \\
\theta_\mathrm{obs} &= \sin^{-1}\left[\sin{\epsilon}\cos\!\left(\nu_\mathrm{ecl}-{\zeta}\right)\right]
\end{split}\end{aligned}$$ and $\nu_\mathrm{ecl}=3\pi/2-\varpi$ is the true anomaly during secondary eclipse (i.e. at time $t_\mathrm{ecl}$), and for spherical planets, $\hat{n}\cdot\hat{r}_\mathrm{obs} = \cos\theta\cos\theta_\mathrm{obs} \left[\cos\left(\phi-\phi_\mathrm{obs}\right)-1\right] + \cos\left(\theta-\theta_\mathrm{obs}\right)$. However, as with the stellar altitude, for non-spherical planets the calculation is more involved, as we discuss in the following section.
A Simple Model of Tidal Distortion {#sec:model:tidal}
----------------------------------
A tidally distorted planet can have a significantly aspherical shape. We first outlined a model of tidal asphericity in Appendix A of @ada18b. Here we adopt that work’s primary assumption to model the distorted shape as a prolate spheroid, with the long axis displaced clockwise in the orbital plane from the star-planet line by some lag angle $\lambda \equiv \cos^{-1}\!\left(\hat{r}_\star \cdot \hat{r}_\ell\right)$. We will assume this lag angle is zero for our analysis, but include it for completeness. The lengths of the long and short axes are dictated by the gravitational potential of the star-planet system. Consider a coordinate system where $\hat{z}$ points from the planet center along the long axis, $\hat{y}$ points along the orbit normal, and $\hat{x}$ points along the short axis according to a right-handed coordinate system. Then the potential becomes $$\begin{aligned}
\label{eq:potential}
\begin{split}
\Phi \!\left(\vec{r}\right) = -\frac{GM_\star}{a} \Bigg\{&\left[\left(\frac{z}{a}+\frac{\xi}{1+\xi}\right)^2 + \left(\frac{x^2+y^2}{a^2}\right)\right]^{-1/2} \\ + &\left[\left(\frac{z}{a}+\frac{1}{1+\xi}\right)^2 + \left(\frac{x^2+y^2}{a^2}\right)\right]^{-1/2} \\ + &\frac{1+\xi}{2} \left(\frac{x^2+z^2}{a^2}\right)\Bigg\}
\end{split}\end{aligned}$$ where $a$ is the orbital semi-major axis and $\xi$ is the planet-star mass ratio $M_\mathrm{p} / M_\star$. We then fit the cross-sectional area along $\hat{x}$ and $\hat{y}$ to the observed transit depth to get the long and short axis lengths[^6].
For an ellipsoid the area of each cell is position-dependent, and will therefore affect the surface area over which it radiates. Once we have the planetary semi-major and semi-minor extents (defined as one-half of the long and short axes, respectively), denoted $A_\mathrm{p}$ and $B_\mathrm{p}$, we can calculate the areas of individual cells. We adapt the result from Equation A7 of @ada18b for the area of a cell spanning longitudes $\phi_i$–$\phi_j$ and latitudes $\theta_i$–$\theta_j$, now with a more complicated relationship between the Cartesian coordinates for the planet: $$\begin{aligned}
\label{eq:cell_area}
\begin{split}
S_{ij} &= \, B^2_\mathrm{p} \int_{\phi_i}^{\phi_j} \!\int_{\theta_i}^{\theta_j}\! \cos\theta \, \Big\{ 1 \\ &+ \left(\chi^2-1\right) \left[\frac{1}{4}f\!\left(\phi,\theta\right)+2\left(\hat{r}\cdot\hat{r}_\ell\right)\right] \\ &+ \left(\chi^2-1\right)^2 \left(\hat{r}\cdot\hat{r}_\ell\right)^2 \Big\}^{1/2} \,d\theta\,d\phi,
\end{split}\end{aligned}$$ where $\chi \equiv A_\mathrm{p}/B_\mathrm{p}$ is the axis ratio and $$\begin{aligned}
\label{eq:area_term}
\begin{split}
f\!\left(\phi,\theta\right) &= \left\{\sin\theta\cos\theta_\ell\left[\cos\left(\phi-\phi_\ell\right)-1\right] - \sin\left(\theta-\theta_\ell\right)\right\}^2 \\ &+ \cos^2\theta_\ell \left[\sin\left(\phi-\phi_\ell\right)+1\right].
\end{split}\end{aligned}$$
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**[HD 149026 b]{}** **[WASP-12 b]{}** **[CoRoT-2 b]{}**
{height="5.5cm"} {height="5.5cm"} {height="5.5cm"}
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A second effect of the non-spherical shape is a change in the stellar altitude as a function longitude and latitude. To quantify the change we first write the function representing the shape of the prolate ellipsoid: $$\label{eq:ellipsoid_function}
f\!\left(x,y,z\right) = \frac{x^2+y^2}{B^2_\mathrm{p}} + \frac{z^2}{A^2_\mathrm{p}}.$$ We need a way of expressing these ellipsoidal coordinates in the oblique coordinates (i.e. with respect to the latitude/longitude coordinates defined by the rotation). To do this we note that we can express the oblique positions of both the sub-stellar point, given by equation \[eq:substellar\_point\], and the extreme point of the planet along the long axis, given by $$\begin{aligned}
\label{eq:ellipsoidal_point}
\begin{split}
\phi_\ell &= \phi_\star\!\left(\nu\rightarrow\nu-\lambda\right) \\
\theta_\ell &= \theta_\star\!\left(\nu\rightarrow\nu-\lambda\right). \\
\end{split}\end{aligned}$$ If the lag angle $\lambda>0$, then we can construct our ellipsoidal Cartesian unit vectors entirely with respect to the unit position vectors for these two points: $$\begin{aligned}
\label{eq:ellipsoidal_units}
\begin{split}
\hat{x} &= \frac{\hat{r}_\star - \cos\lambda\hat{r}_\ell}{\sin\lambda} \\
\hat{y} &= \frac{\hat{r}_\ell\times\hat{r}_\star}{\sin\lambda} \\
\hat{z} &= \hat{r}_\ell.
\end{split}\end{aligned}$$ From this the normal unit vector at a given point on the surface is given by $$\begin{aligned}
\label{eq:ellipsoid_normal}
\begin{split}
\hat{n} &= \vec{\nabla}\!f / \left\lVert \vec{\nabla}\!f \right\rVert \\
&= \left\{1 + g^{-1}\!\left[\chi,\left(\hat{r}\cdot\hat{r}_\ell\right)\right]\right\}^{-1/2} \left[\left(\hat{r}\cdot\hat{x}\right)\hat{x} + \left(\hat{r}\cdot\hat{y}\right)\hat{y}\right] \\
&+ \left\{1 + g\!\left[\chi,\left(\hat{r}\cdot\hat{r}_\ell\right)\right]\right\}^{-1/2} \hat{z}
\end{split}\end{aligned}$$ where $$g\!\left[\chi,\left(\hat{r}\cdot\hat{r}_\ell\right)\right] \equiv \chi^{2} \left[\frac{1-\left(\hat{r}\cdot\hat{r}_\ell\right)^2}{\left(\hat{r}\cdot\hat{r}_\ell\right)^2}\right]$$ Then the cosines of the stellar altitude and the visibility are given by the component of $\hat{n}$ along the instellation and observer lines, as in Equations \[eq:stellar\_altitude\] and \[eq:visibility\]. $$\begin{aligned}
\label{eq:ellipsoid_cosalpha}
\begin{split}
\cos\alpha_\ell &= \hat{n} \cdot \hat{r}_\star \\
&= \sin\lambda \left(1 + g^{-1}\right)^{-1/2} \left[\hat{r}\cdot\hat{r}_\star-\cos\lambda\left(\hat{r}\cdot\hat{r}_\ell\right)\right] \\
&+ \cos\lambda \left(1 + g\right)^{-1/2} \left(\hat{r}\cdot\hat{r}_\ell\right)
\end{split}\end{aligned}$$ and the component along the observer line is $$\begin{aligned}
\label{eq:ellipsoid_cosobs}
\begin{split}
V &= \hat{n}\cdot\hat{r}_\mathrm{obs} \\
&= \frac{\cos\left(\nu-\nu_\mathrm{ecl}\right) - \cos\left[\left(\nu-\lambda\right)-\nu_\mathrm{ecl}\right]}{\sin\lambda} \left(1 + g^{-1}\right)^{-1/2} \\
&\times \left[\hat{r}\cdot\hat{r}_\star-\cos\lambda\left(\hat{r}\cdot\hat{r}_\ell\right)\right] \\
&+ \cos\left[\left(\nu-\lambda\right)-\nu_\mathrm{ecl}\right] \left(1 + g\right)^{-1/2} \left(\hat{r}\cdot\hat{r}_\ell\right).
\end{split}\end{aligned}$$
Fit Methods and Results {#sec:results}
=======================
[cccccc]{} $\tau_{\mathrm{rad}}$ (hr) & $7.7^{+10.8}_{-4.6}$ & $2.4^{+24.0}_{-1.3}$ & $0.01^{+0.01}_{-0.01}$ & $65.5^{+24.3}_{-0.3}$ & $23.9^{+2.5}_{-3.1}$\
$T_0$ (K) & $877^{+388}_{-631}$ & $1388^{+155}_{-158}$ & $427^{+166}_{-N/A}$ & $2458^{+84}_{-141}$ & $92^{+209}_{-52}$\
$A$ & $<0.12$ & $0.64^{+0.09}_{-0.05}$ & $<0.06$ & $0.36^{+0.09}_{-0.35}$ & $<0.03$\
${\epsilon}$ ($^\circ$) & $93.9^{+45.1}_{-32.5}$ & $4.2^{+47.7}_{-1.9}$ & $87.9^{+0.6}_{-8.5}$ & $91.2^{+70.1}_{-3.1}$ & $45.8\pm1.4$\
${\zeta}$ ($^\circ$) & $10.6^{+143.7}_{-170.42}$ & $-29.0^{+113.0}_{-149.2}$ & $-39.4^{+4.4}_{-0.9}$ & $-73.0^{+21.2}_{-58.6}$ & $-82.5^{+10.8}_{-7.1}$\
$P_{\mathrm{rot}}/P_{\mathrm{orb}}$ & $1.09\pm0.06$ & $0.39^{+0.08}_{-0.21}$ & $0.77^{+0.14}_{-0.04}$ & $0.94\pm0.01$ & $1.13^{+0.04}_{-0.02}$\
$\tau_{\mathrm{rad}}$ (hr) & $97.2^{+5.1}_{-0.2}$ & $3.7^{+0.4}_{-2.2}$ & $12.7^{+32.2}_{-2.7}$ & $40.0^{+8.1}_{-3.4}$ & $16.9^{+4.7}_{-4.4}$\
$T_0$ (K) & $1172^{+334}_{-111}$ & $1223^{+165}_{-894}$ & $1567^{+134}_{-63}$ & $2204^{+40}_{-14}$ & $102^{+219}_{-75}$\
$A$ & $0.01^{+0.39}_{-0.01}$ & $0.44^{+0.05}_{-0.34}$ & $0.09^{+0.02}_{-0.08}$ & $<0.03$ & $<0.03$\
${\epsilon}$ ($^\circ$) & $4.2^{+57.2}_{-2.7}$ & $36.4^{+28.3}_{-3.4}$ & $0.8^{+6.0}_{-0.4}$ & $21.1^{+8.5}_{-10.1}$ & $2.0^{+8.1}_{-1.2}$\
${\zeta}$ ($^\circ$) & $-4.7^{+7.3}_{-30.2}$ & $69.8^{+3.0}_{-29.5}$ & $-67.0^{+123.4}_{-N/A}$ & $46.6^{+58.3}_{-12.1}$ & $-32.3^{+47.1}_{-20.8}$\
\[table:fits\]
**[HD 149026 b]{}**
-------------------------------------------- -- --
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-------------------------------------------- -- --
**[WASP-12 b]{}**
------------------------------------------- -- --
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------------------------------------------- -- --
**[CoRoT-2 b]{}**
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[ccccrrr]{}\[htb!\] Fixed zero & $0^\circ$ & Synchronous & 1 & 2.62 & 0 & 1.37\
Fixed zero & $0^\circ$ & Free & $1.11\pm0.05$, $0.51^{+0.28}_{-0.03}$ & 0 & 1.26 & 0\
$\lambda$-consistent & $89.3^{+30.2}_{-31.8}$ & Sub-synchronous & 43.41 ($>1.20$) & 1.80 & 4.11 & 0.51\
$\lambda$-consistent & $119.0^{+41.5}_{-49.1}$ & Free & $0.37^{+2.25}_{-0.04}$, $1.21^{+0.40}_{-0.68}$ & 3.51 & 6.32 & 4.39\
Free & $93.9^{+45.1}_{-32.5}$, $4.2^{+47.7}_{-1.9}$ & Sub-synchronous & $7.31$ ($>1.04$), $1.00^{+0.12}_{-0.00}$ & 1.78 & 4.12 & 4.63\
Free & $4.2^{+57.2}_{-2.7}$, $36.4^{+28.3}_{-3.4}$ & Free & $1.09\pm0.06$, $0.39^{+0.08}_{-0.21}$ & 4.09 & 5.35 & 8.13\
Fixed zero & $0^\circ$ & Synchronous & 1 & 593.24 & 91.18 & 680.13\
Fixed zero & $0^\circ$ & Free & $0.91^{+0.01}_{-0.00}$, $0.95\pm0.01$ & 0 & 6.16 & 1.87\
$\lambda$-consistent & $19.4^{+1.7}_{-1.6}$ & Sub-synchronous & $1.002\pm0.001$ & 592.90 & 81.67 & 666.17\
$\lambda$-consistent & $11.1^{+4.5}_{-7.9}$ & Free & $0.910^{+0.009}_{-0.001}$, $0.945^{+0.010}_{-0.005}$ & 68.37 & 6.63 & 66.57\
Free & $93.9^{+45.1}_{-32.5}$, $4.2^{+47.7}_{-1.9}$ & Sub-synchronous & $1.0015^{+0.0005}_{-0.0010}$, $1.0039^{+0.0011}_{-0.0009}$ & 586.68 & 76.60 & 658.98\
Free & $4.4^{+3.3}_{-2.2}$, $27.4^{+6.2}_{-10.8}$ & Free & $0.91\pm0.004$, $0.95\pm0.01$ & 4.31 & 0 & 0\
Fixed zero & $0^\circ$ & Synchronous & 1 & & 259.41 &\
Fixed zero & $0^\circ$ & Free & $1.10^{+0.02}_{-0.03}$ & & 0 &\
Free & $45.8\pm1.4$ & Sub-synchronous & $1.07\pm0.01$ & & 7.56 &\
Free & $2.0^{+8.1}_{-1.2}$ & Free & $1.13^{+0.04}_{-0.02}$ & & 4.06 &\
\[table:AIC\]
For each planet we run two cases: one with 6 free parameters where the rotation rate is unconstrained, and another with 5 parameters where the rotation rate is fixed to the obliquity-dependent equilibrium rotation rate $$\omega_{\mathrm{eq}} = n\frac{2\cos\epsilon}{1+\cos^2\epsilon}
\label{omega eq}$$ where $n \equiv 2\pi/P_\mathrm{orb}$ is the mean motion (see §\[Cassini state constraints\] for more details). Additionally, for [HD 149026 b]{} and [WASP-12 b]{}, which have data in two bands, we run the above two cases where we constrain the axial orientation to be consistent between the bands, for a total of 4 oblique model cases.
We use a Markov-Chain Monte Carlo process to broadly evaluate the likelihood landscape over the relevant parameters and converge on the sets of parameter values for each case with the most favorable likelihoods. Specifically, we employ a Metropolis-Hastings algorithm with simulated annealing [@Kirkpatrick1983]. Annealing introduces a temperature parameter that corresponds to acceptance probability, and serves to both broadly explore the likelihood space at initially high values, and to converge on optimal solutions as it is gradually reduced. After convergence, we continue the MCMC chain without annealing, to estimate uncertainties on the parameter values. The quoted 1-$\sigma$ uncertainties are determined by the 68% ranges on either side of the values. The best-fit values and uncertainties for the models with band-distinct obliquities are listed in Table \[table:fits\]. The resulting model light curves are plotted with these 1-$\sigma$ uncertainties, as well as the 2-$\sigma$ uncertainties, determined by the 95% ranges, in the leftmost columns of Figures \[fig:HD149026b\_curves\]–\[fig:CoRoT-2b\_curves\].
To evaluate the relative fit quality of these oblique models, we calculate the Akaike Information Criteria [@Akaike1973; @Akaike1974] with a second-order correction for small sample sizes [@Sugiura1978; @Hurvich1989; @Hurvich1995]. This is given by $$\mathrm{AICc} \equiv 2 \left[k - \ln\mathcal{L} + \frac{k \left(k+1\right)}{n-k-1}\right],$$ where $k$ is the number of free parameters, $\ln\mathcal{L}$ the log likelihood (and, by extension, the full second term $-2\ln\mathcal{L}$ being equal to the chi-squared statistic), and $n$ the sample size. We express the AIC values relative to the smallest value (i.e. the value of the most favorable model) in each band ($\Delta\mathrm{AICc} \equiv \mathrm{AICc} - \mathrm{AICc}_\mathrm{min}$), in Table \[table:AIC\]. Additionally, to more appropriately compare the models with band-distinct obliquities with the models co-varying in obliquity, we calculate the cumulative AICc values for the combined model across both bands. Following the interpretation of @Burnham2004, $\Delta\mathrm{AICc}\leq2$ indicates a substantial level of evidence for a model (as compared with the most favorable); $4\leq\Delta\mathrm{AICc}\leq7$ implies low evidence, and $\Delta\mathrm{AICc}>10$ effectively implies no evidence. We discuss the results of each planet in the following sub-sections.
[HD 149026 b]{} {#sec:results:HD149026b}
---------------
Following the prescription of the AIC, we find that the preferred model at 3.6 $\mu$m is the model with zero obliquity and unconstrained rotation. This implies that any fit improvement from adding axial orientation parameters are not statistically warranted. Indeed, the least-constrained model, which has a distinct obliquity and unconstrained rotation, returns a nearly upright axis, effectively approximating the non-oblique case. The $\Delta\mathrm{AIC}\sim 4$ is consistent with nearly identical likelihoods, since we are increasing the number of parameters by 2 by including obliquity.
In contrast, the most preferred model at 4.5 $\mu$m is the simplest — zero obliquity and synchronous rotation. This implies that any discernible phase offset in the data is not strong enough to warrant a model that can capture it, either by a sub-synchronous rotation or high obliquity. The non-oblique, unconstrained rotation case is next in line, with a best-fit spin-orbit ratio of nearly two. Moving into the oblique cases, we see that there is a moderate drop-off in support for the models at 4.5 $\mu$m; this is not surprising, since the light curves at 3.6 and 4.5 $\mu$m disagree with the direction of offset.
Interestingly, when we compare the combined model criteria, the combined non-oblique model with unconstrained rotation is on top, but the model with consistent obliquity and sub-synchronous rotation is not far behind. The most likely interpretation is that the 3.6 $\mu$m data are modestly better fit by an axis nearly perpendicular to the orbit normal, rather than simply a slow effective rotation rate. Given the very slow returned rotation for this case, it also suggests that the rotation has a weak effect on the offset at this high of obliquity, since the transverse motion is taken almost completely out of the plane of the sky from our perspective.
[WASP-12 b]{} {#sec:results:WASP-12b}
-------------
Due to its strong tidal distortion, [WASP-12 b]{} is the most complicated of the planets, and we expect that it will be difficult to fully capture its shape only accounting for rotation and obliquity. In sharp contrast with the moderate differences in $\Delta\mathrm{AIC}$ between models in the bands of [HD 149026 b]{}, which imply a dominant effect of the number of parameters, for [WASP-12 b]{} both the synchronous and sub-synchronous models are vastly worse than those with unconstrained rotation. This makes sense given that the light curves in both bands show strong variations between occultations, at least one of which can be approximated using an eastward phase offset. Eastward offsets are more amenable to a super-rotating atmospheric layer than high obliquity, and indeed the models with the most free parameters return small values for obliquity.
AIC however cannot provide a quantification of the absolute fit quality. While our model light curves generally lie within the range of observed fluxes at any given time, the maximum observed amplitude of semi-annual variations exceed what can be captured with a combination of spin-orbit geometry and thermal modeling. Ellipsoidal variations due to the planet’s tidal distortion should only change the observed surface area by $\sim\!10\%$ [@lis10; @lai10], and the complementary effects on the star’s shape should be a further order of magnitude smaller. These correspond to variations on the orders of $\sim\!4\times10^{-4}$ and $\sim\!4\times10^{-5}$, according to @cow12. Indeed, the incorporation of an ellipsoidal distortion to the non-oblique thermal model [e.g. in @ada18b] does not provide enough additional variation to approach the full amplitudes of the data.
[CoRoT-2 b]{} {#sec:results:CoRoT-2b}
-------------
Our analysis of the light curves is more limited for [CoRoT-2 b]{} since it only has one band, but the AIC values do indicate that the non-synchronous models are much better fits to the data. As with the 3.6 $\mu$m fits for [HD 149026 b]{}, the differences in selection criteria between the non-oblique, non-synchronous and oblique models appear to be largely influenced by the AIC’s penalty for additional parameters by introducing axial orientation. From this we infer that the model fits from either a westward wind interpretation (slower than synchronous rotation with little to no obliquity) or a sub-synchronous rotation at moderately high obliquity are of similarly effective quality.
Considering these three cases, we can draw some conclusions. For planets whose phase variations are already amenable to fitting by a non-oblique thermal model, but whose observed westward phase offsets require a slower-than-synchronous effective rotation rate (most readily interpreted as westward winds), our fit quality is at least as good with an oblique model where we assume an obliquity-dependent sub-synchronous rotation (Equation \[omega eq\]). However, particularly for [WASP-12 b]{}, we have no cases where the addition of obliquity improves the quality of fit of a thermal model which has difficulty reproducing all major features of the data.
Dynamical Evolution of a Tightly Packed System {#sec:REBOUND}
==============================================
Having established the feasibility of obliquity to fit the observed phase variations, we now place this interpretation in a dynamical framework in the following sections. @Batygin2016 set a physical framework where a system comprising multiple close-in planets (i.e. $P \lesssim 100$ days) with masses in the super-Earth regime ($1 < M/M_\oplus < 30$) can evolve to have high mutual orbital inclinations. They first determine that, above an initial mass of $\gtrsim 15 M_\oplus$, the innermost planet can undergo runaway accretion under a range of stellar nebular densities, reaching $M_p \sim M_{\mathrm{J}}$ within a timescale $\sim\!10^6$ years, thereby predicting in-situ formation of close-in giant planets. Their model couples this with the evolution of the host star onto the main sequence and determines that outer planets can evolve to high mutual inclinations via nodal regression commensurability. These results constitute a prediction that systems with hot Jupiters and mutually inclined outer planets should be common.
In order to evaluate the feasibility of evolving a dynamically full multi-planet system into a configuration consistent with the existence of secular spin-orbit resonance and associated high obliquities, we performed dynamical simulations using the `REBOUND`[^7] software package with the `IAS15` integrator [@rei12; @rei15]. For our fiducial precursor system, we adopted a known multi-planet system, Kepler-107 [@Rowe2014; @VanEylen2015]. We then statistically explored the dynamical reaction of the configuration to orbital instability triggered by rapid increase in the mass of the innermost planet. We assumed an instantaneous increase from the estimated innermost planet mass of 3.7 $M_\oplus$, to that of Jupiter. We also integrated the system without any mass increase. For both cases we performed 100 separate simulations with a duration of at least $\sim\!10^5$ Earth years ($\sim\!10^7$ orbits of the innermost planet as initialized).
[cccccccc]{} b & $3.179997\pm1.1\times10^{-5}$ & 0.044 & $0.020^{+0.200}_{-0.020}$ & 0.01167 & $0.139\pm0.005$ & $0.34\pm0.24$ & $66.49904\pm0.00200$\
c & $4.901425\pm1.6\times10^{-5}$ & 0.059 & $0.020^{+0.260}_{-0.020}$ & 0.0133 & $0.161\pm0.016$ & $0.78\pm0.29$ & $71.60742\pm0.00174$\
d & $7.958203\pm1.04\times10^{-4}$ & 0.082 & $0.14^{+0.25}_{-0.14}$ & 0.00371 & $0.095\pm0.005$ & $0.27\pm0.24$ & $70.79968\pm0.00612$\
e & $14.749049\pm3.4\times10^{-5}$ & 0.123 & $0.020^{+0.180}_{-0.020}$ & 0.0360 & $0.308\pm0.022$ & $0.90\pm0.28$ & $71.77998\pm0.00134$\
\[table:Kepler-107\]
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The orbital geometry of the Kepler-107 system is not entirely constrained by the transit parameters reported by @Rowe2014. We therefore adopted initial values for the semi-major axes from @Rowe2014 and initial eccentricities from @VanEylen2015. For the angular orbital elements, we used the known transit parameters in conjunction with the following process to randomly generate orbital elements consistent with the observational constraints.
1. Set the line of sight to the positive $x$-axis, the default reference direction in `REBOUND`[^8].
2. Choose a longitude of periastron, $\varpi \sim U\!\left(0,2\pi\right)$.
3. Choose a longitude of ascending node $\Omega$ from a uniform distribution over the set of angles satisfying $$\lvert\sin\Omega\rvert \geq b \left[\frac{R_\star+R_\mathrm{P}}{r_\mathrm{tr}\!\left(e, \varpi\right)}\right]$$ where $b$ is the impact parameter, $R_\star$ and $R_\mathrm{P}$ the stellar and planetary radii, respectively, and $$r_\mathrm{tr} \equiv a\left(\frac{1-e^2}{1+e\sin\varpi}\right)$$ the star-planet separation at the transit mid-point.
4. Calculate the inclination in the range $\left[0, \pi/2\right]$ that solves the transit condition $$\sin i \, \lvert\sin\Omega\rvert = b \left[\frac{R_\star+R_\mathrm{P}}{r_\mathrm{tr}\!\left(e, \varpi\right)}\right].$$
After integrating systems constructed in this manner for $\sim\!10^5$ Earth years, we see a significant evolution of the systems away from their initially nearly co-planar states. We calculate the inclinations (relative to the net system angular momentum vector) for the final orbits of the remaining planets and evaluate the transit conditions along the initial line of sight (Figure \[fig:inclination\_dists\]). In a significant fraction of simulations, some combination of the outer planets ($c$–$e$) no longer transit due to large inclinations which can exceed 20$^\circ$. Out of the 100 simulations with the nominal Kepler-107 masses, only 3 saw an ejection (defined in our simulation is defined as reaching a separation from the star an order of magnitude larger than the initial semi-major axis). The fraction of ejections increases greatly in the inner-Jupiter model, with a significant fraction of trials ejecting $c$ and/or $d$ within $10^5$ years. Half of ejections occurred within the first $\sim\!2.5\times10^4$ years. These results as a whole suggest that, under a significant perturbation such the runaway accretion of the inner planet, it is plausible that a compact multi-transiting planet system such as Kepler-107 could evolve to high mutual inclinations and/or ejections for the outer planets, leaving a majority of cases where only the inner planet could be observed to transit.
The simulations presented here have only accounted for the radii of the star and orbiting planets in terms of collision detection, whose prescribed resolution was a merger. No mergers occurred in any trial. Otherwise, the bodies are effectively treated as point masses. If these bodies are endowed with structure, as formulated for example by @mar02, then secular inclination resonances can act to increase the degree of dynamical instability (leading to agglomerating collisions) and mutual inclinations among remaining planets, thereby increasing the fractions of non-transiting planets [@Batygin2016].\
Cassini State Driven by an Inclined, External Perturber {#sec:Cassini state}
=======================================================
While we have examined the possibility that close-in giant planets have non-zero obliquities, we have not yet discussed in detail how such states may arise. Tidal torques are strongest for these close-in, large-radius planets, and they act to dampen planetary obliquities to zero. Large tilts can be maintained, however, if a planet is locked in a secular spin-orbit resonance involving synchronous precession of its spin vector and orbital angular momentum vector [@Fabrycky2007]. Of particular interest is Cassini State 2 [@Peale1969], where the planet maintains a large obliquity as the spin and orbital axes precess at the same rate on opposite sides of the axis normal to the invariable plane. In the absence of dissipation, these three axes are coplanar, but in the presence of tides, the spin axis is slightly shifted out of the plane.
In this section, we evaluate the possibility that the planets [HD 149026 b]{}, [WASP-12 b]{}, and [CoRoT-2 b]{} have their spin axes trapped in Cassini states due to spin-orbit resonances driven by exterior perturbing planets. The case of a resonance for [WASP-12 b]{} was already discussed in detail by [@MillhollandLaughlin2018] as a theory for the planet’s rapid orbital decay [@mac2016; @Patra2017; @Maciejewski2018; @Bailey2019]. We include elements of [@MillhollandLaughlin2018]’s analysis here for consistency and comparison with the other two systems.
We start by calculating the frequencies of spin and orbital precession. We then examine the constraints on the hypothetical perturbing planets in order for the spin-orbit resonant configurations to be plausible. Table \[system parameter table\] shows the system parameters we adopt in these calculations.
Spin-Orbit Resonant Frequencies
-------------------------------
The torque from the host star on a rotationally-flattened planet will cause the planet’s spin-axis to precess about the orbit normal at a period, $T_{\alpha} = 2\pi/(\alpha\cos\epsilon)$. Here $\epsilon$ is the obliquity and $\alpha$ is the precession constant [@Ward2004], which is given by
$$\alpha = \frac{1}{2}\frac{M_{\star}}{M_{\mathrm{p}}}\left(\frac{R_{\mathrm{p}}}{a}\right)^3\frac{k_2}{C}\omega.
\label{alpha}$$
We have assumed there are no satellites and have defined $M_{\mathrm{p}}$, the planet mass, $C$, the moment of inertia normalized by $M_{\mathrm{p}} {R_{\mathrm{p}}}^2$, and $\omega$, the spin frequency. This expression also assumes that the coefficient of the quadrupole moment of the planet’s gravitational field, $J_2$, takes the form [@Ragozzine2009], $$J_2 = \frac{\omega^2 {R_{\mathrm{p}}}^3}{3 G {M_{\mathrm{p}}}}k_2.$$
The secular spin-orbit resonance requires a commensurability between the planet’s spin-axis precession frequency and its orbit nodal regression frequency, $g = \dot{\Omega}$. Planet-planet interactions are one source of nodal regression. In a two-planet system, the nodes of both planets regress uniformly due to secular perturbations. The frequency is given by Laplace-Lagrange theory to be $$\begin{aligned}
\label{LL g}
\begin{split}
g_{_\mathrm{LL}} = &-\frac{1}{4} b_{3/2}^{(1)}(\alpha_{12})\alpha_{12} \times \\
&\left(n_1\frac{M_{p2}}{M_{\star} + M_{p1}}\alpha_{12} + n_2\frac{M_{p1}}{M_{\star} + M_{p2}}\right),
\end{split}\end{aligned}$$ if the planets are not near mean-motion resonance [@1999ssd..book.....M]. Here, $\alpha_{12} = a_1/a_2$ and $n_i$ is the mean-motion of planet $i$, $n_i^2 = G M_{\star}/a_i^3$. The constant, $b_{3/2}^{(1)}(\alpha_{12})$ is a Laplace coefficient, defined by $$b_{3/2}^{(1)}(\alpha_{12}) = \frac{1}{\pi}\int_{0}^{2\pi}\frac{\cos\psi}{(1-2\alpha\cos\psi + \alpha^2)^{3/2}}d\psi.$$
In addition to planet-planet interactions, the stellar quadrupole gravitational moment also induces orbit nodal recession about the stellar spin vector. In the absence of secular planet interactions, this occurs at the frequency [@Spalding2017] $$g_{\star} = n \frac{k_{2\star}}{2}\left(\frac{\omega_{\star}}{n}\right)^2\left(\frac{R_{\star}}{a}\right)^5.
\label{star g}$$ Accordingly, in a multiple-planet system, the planets’ orbit normal vectors evolve in response to perturbation components at different frequencies. One frequency, $g_{p-p}$, is due to the planet-planet perturbations and the other, $g_{\star}$, is associated with the stellar quadrupole moment. These are close but not exactly equal to the equations \[LL g\] and \[star g\] above, since the analytical expressions only account for one driver of nodal recession at a time. When the stellar equatorial plane is coincident with the plane perpendicular to the total orbital angular momentum vector, these two frequencies add linearly such that $g= g_{p-p} + g_{\star}$, and the nodal recession is uniform. However, if these planes are not coincident (i.e. if there is an angle between the stellar spin vector and the total orbital angular momentum vector), then the frequencies do not add linearly, but rather the nodal recession is a non-uniform superposition of these modes.
The spin-orbit resonance can be encountered and captured when the spin precession is commensurable with either one of these orbital frequency components, $g_{p-p}$ or $g_{\star}$ [@MillhollandLaughlin2019]. In the case of short-period planets, however, the spin axis precession is so fast that spin-orbit resonances induced by planet-planet interactions are much more likely. Although $g_{\star}$ is fast early in the system’s lifetime because the star is rapidly rotating and has not finished contracting [@Batygin2013; @Spalding2017], $g_{\star}$ is significantly smaller for a main-sequence star, and it is generally not fast enough for spin-orbit resonance. To illustrate this explicitly, we calculate $g_{\star}$ for our three case studies. The results are shown in Table \[system parameter table\]. In all three systems, $g_{\star}$ is much smaller than the spin axis precession constant. This clearly indicates that nodal recession must be provided by planet-planet perturbations if these systems are in spin-orbit resonances. We will therefore assume in the remainder of this work that planet-planet resonances are the relevant ones.\
Constraints on the masses and semi-major axes of potential inclined companions
------------------------------------------------------------------------------
Suppose that, in each of the three systems, planet $b$ is in a high-obliquity spin-orbit resonance with its orbital precession induced by secular interactions with an exterior, inclined, and as-yet undetected companion. Existence of this configuration places significant constraints on the masses, semi-major axes, and inclinations of the perturbing companions. These constraints stem from
upholding the Cassini state,
preserving total angular momentum conservation, and
maintaining consistency with existing radial velocity (RV) data.
In the subsections that follow, we review and apply each of these constraints in detail.
### Constraints from Maintaining the Cassini State {#Cassini state constraints}
[cccc]{} $M_{\star} \ [M_{\odot}]$ & $1.345^{(1)}$ & $1.36^{(2)}$ & $0.97^{(3)}$\
$R_{\star} \ [R_{\odot}]$ & $1.541^{(1)}$ & $1.63^{(2)}$ & $0.90^{(3)}$\
$P_{\star} \ [\mathrm{days}]$ & $\sim 13^{(4^*)}$ & $36^{(5)}$ & $4.5^{(3)}$\
$k_{2\star}$ & 0.01$^{(6)}$ & 0.01$^{(6)}$ & 0.01$^{(6)}$\
$a \ [\mathrm{AU}]$ & $0.042^{(4)}$ & $0.02299^{(7)}$ & $0.0281^{(3)}$\
$M_{\rm p} \ [M_{\rm {Jup}}]$ & $0.36^{(4)}$ & $1.41^{(7)}$ & $3.3^{(3)}$\
$R_{\rm p} \ [R_{\rm {Jup}}]$ & $0.725^{(4)}$ & $1.89^{(2)}$ & $1.466^{(3)}$\
$k_2$ & 0.1 & 0.1 & 0.1\
$C$ & 0.2 & 0.2 & 0.2\
$2\pi/{\alpha_{\mathrm{syn}}} \ \mathrm{[yr]}$ & 14.3 & 0.2 & 4.0\
$2\pi/{g_{\star}} \ \mathrm{[yr]}$ & $2.4\times10^5$ & $1.7\times10^5$ & $8.7\times10^4$\
\[system parameter table\]
If a planet is captured in a Cassini state, there will be a resonant commensurability that can be stated as [@Ward2004] $$\lvert g \rvert \approx \alpha\cos\epsilon.
\label{resonance condition}$$ This holds if the planetary obliquity, $\epsilon$, is large compared to the mutual inclination between the orbits. We will take that as an assumption and use this condition to calculate the range of values of each perturbing planet’s mass, $M_{p2}$, and semi-major axis, $a_2$, that allow for resonant commensurability. Table \[system parameter table\] shows the estimates of the spin axis precession periods used in these calculations.
In Figure \[Obliquity heatmap\], we show heatmaps in $M_{p2}$ and $a_2$ space that represent the obliquity of planet $b$ necessary for the resonance to hold in each system. We assume that the nodal recession is driven by interactions with the secondary planet, such that $g = g_{p-p} \approx g_{_\mathrm{LL}}$. We also assume $e_b = 0$ and that the spin rate of planet $b$ is at equilibrium, at which $d\omega/dt = 0$. The equilibrium rate is given by equation \[omega eq\] in the traditional viscous approach to equilibrium tide theory [@Levrard2007]. The quantity $n\equiv2\pi/P_\mathrm{orb}$ is the mean-motion. Combining equations \[alpha\], \[resonance condition\], and \[omega eq\], the resonance condition becomes $$\lvert g \rvert = \alpha_{\mathrm{syn}}\frac{2\cos^2\epsilon}{1+\cos^2\epsilon},$$ where $\alpha_{\mathrm{syn}}$ is the value of $\alpha$ in the case of synchronous rotation, $\omega = n$. The solution for $\epsilon$ is then $$\cos\epsilon = \left(\frac{1}{2\alpha_{\mathrm{syn}}/{\lvert g \rvert} - 1}\right)^{1/2}.$$ This expression can be used to calculate $\epsilon$ for a range of values of $M_{p2}$ and $a_2$.
Figure \[Obliquity heatmap\] shows that if $a_2$ is too small, $\lvert g \vert$ is too large and no resonance is possible (white regions). Alternatively, if $a_2$ is too large, $\lvert g \vert$ is small and $\epsilon \sim 90^{\circ}$, which is unstable in the long-term. Figure \[Obliquity heatmap\] thus allows us to define approximate constraints on the semi-major axes of the hypothetical companion planets. The constraints on their masses are not strong due to the weak dependence of $g_{_\mathrm{LL}}$ on $M_{p2}$ when $M_{p1} \gg M_{p2}$.
### Constraints from Angular Momentum Conservation {#angular momentum conservation}
For the equilibrium tidal theory that we are considering, a planet maintained in an oblique state spins sub-synchronously (equation \[omega eq\]), and the rate at which tides convert orbital energy into heat energy is orders of magnitude larger than it would be in the case of zero obliquity [@Levrard2007; @Wisdom2008]. An oblique tidally dissipating planet migrates inwards towards the star, decreasing $\lvert g \rvert/\alpha$ in the process and further increasing its obliquity. The decrease in orbital angular momentum associated with inward migration must be counteracted such that total angular momentum of the system, $\mathbf{J}$, is conserved. This can be accomplished through gradual alignment of the planetary orbital angular momenta, $\mathbf{L_1}$ and $\mathbf{L_2}$, with one another and with the stellar spin angular momentum, $\mathbf{S_{\star}}$. If we assume for a conservative argument that alignment with $\mathbf{S_{\star}}$ does not play a role, then the second planet must have enough angular momentum to preserve the total conservation. This allows conservative limits to be placed on the orbits of the perturbing planets [@Fabrycky2007].
We begin with an expression for $\mathbf{J}$. We define $i$ to be the angle between $\mathbf{L_1}$ and $\mathbf{L_2}$. In addition, we define $\phi$ to be the angle between $\mathbf{S_{\star}}$ and $\mathbf{L_1}+\mathbf{L_2}$. Then, assuming the planets’ own spin angular momenta are negligible, the magnitude of $\mathbf{J}$ is given by $$\begin{split}
\label{Jsq}
J^2 &= {S_{\star}}^2 + {L_1}^2 + {L_2}^2 + 2{L_1}{L_2}\cos i \\
&+ 2{S_{\star}}({L_1}^2 + {L_2}^2 + 2{L_1}{L_2}\cos i)^{1/2}\cos\phi.
\end{split}$$
Secular interactions between the planets do not change $a_2$ or $e_2$ to first order, so $L_2$ remains fixed. Therefore, as $L_1$ decreases, conservation of $J$ must be upheld by an increase in $S_{\star}$ via tidal spin-up of the star [@Brown2011] or a decrease in $i$ and $\phi$ through reorientation of the orbits. As stated above, we will conservatively assume that stellar spin-up and reorientation between the orbital and stellar spin angular momenta do not play a role. Under this assumption, the maximum possible value of planet $b$’s initial semi-major axis, $\max({a_{1i}})$, may be expressed in terms of $a_2$ and $M_{p2}$, $$\max({a_{1i}}) = a_1\left[1 + 2\left(\frac{M_{p2}}{M_{p1}}\right)\left(\frac{a_2}{a_1}\right)^{1/2}\right],
\label{max a_1i}$$ where $a_1$ corresponds to the present-day value of the semi-major axis. This expression is obtained by assuming that $i$ was initially near $90^{\circ}$ and is currently near $0^{\circ}$. We also assumed $e_2 \approx 0$.
This expression does not hold enough information to constrain $a_2$ and $M_{p2}$ in and of itself, since there is no clear limit on $\max(a_1)$. To develop a constraint, we apply the additional requirement that the initial obliquity of planet $b$ must be near $0^{\circ}$. Though this is not strictly necessary, if it is true, it makes the initial resonant capture scenario easy to explain. (Recall that $\epsilon$ increases as the planet tidally migrates inwards.) For a given $a_2$ and $M_{p2}$, the semi-major axis, $a_{1i}$, at which the initial obliquity is zero satisfies the expression $$g(a_{1i},a_2) = \alpha_1(a_{1i}).$$ There is only a plausible solution if $a_{1i} <= \max({a_{1i}})$. Accordingly, this is the additional constraint that we use in conjunction with equation \[max a\_1i\].
The green lines in Figure \[Obliquity heatmap\] delineate the region that simultaneously upholds the conservative angular momentum limits and maintains the possibility of a small initial obliquity for planet $b$. Significant regions of phase space are ruled out for both the WASP-12 and CoRoT-2 systems. While this is somewhat problematic for the theory, it is important to keep in mind that these tight constraints can be strongly alleviated and/or removed by assuming a non-zero initial obliquity or by allowing realignment with $\mathbf{S_{\star}}$ to account for some degree of the system’s angular momentum conservation.
### Constraints from Radial Velocity Data
An entirely unrelated constraint arises from the set of RV measurements of the systems. The hypothetical companion planets must have RV semi-amplitudes smaller than the current detection limits; otherwise they would have already been discovered. Table \[RV jitter table\] shows current estimates of the RV semi-amplitudes, $K_1$, of the known planets and the RV jitters of the fits, $\sigma_{\mathrm{jit}}$. The hypothetical outer planets must have $K_2 \lesssim \sigma_{\mathrm{jit}}$. Figure \[Obliquity heatmap\] shows that the parameter space in the HD 149026 and WASP-12 systems are somewhat constrained by the RV limits. CoRoT-2’s jitter, however, is so large that hardly any phase space area is ruled out.
[cccc]{} [HD 149026 b]{} & $38.5 \pm 1.2$ & $5.8 \pm 0.6$ & (1)\
[WASP-12 b]{} & $219.9^{+2.2}_{-2.1}$ & $9.1^{+1.8}_{-1.3}$ & (2)\
[CoRoT-2 b]{} & $568^{+23}_{-22}$ & $40^{+14}_{-10}$ & (2)\
\[RV jitter table\]
Given its small $\sigma_{\mathrm{jit}}$, it is worthwhile to examine the RVs of HD 149026 to search for any hints of a second signal. The Lick-Carnegie Exoplanet Survey Team (LCES) collected 70 measurements of Keck/HIRES Doppler velocities [@Butler2017] over 8.5 years. The top panel of Figure \[RV fit, residuals, and periodogram\] shows the Keck RVs phase-folded at the period of planet $b$, $P = 2.8759$ days. The second panel shows the residuals obtained after fitting a circular orbit. Finally, the bottom panel displays a Lomb-Scargle periodogram of the RV residuals. There is a noticeable peak at 12.68 days. This peak may be related to stellar activity, since it is close to the $\sim\!13$ day stellar rotation period suggested by the $v \sin i$ measurement from [@Sato2005]. Alternatively, it may be the signature of an additional planet, and a closer examination of both the rotational signature in the existing spectra, as well as additional Doppler measurements of the star may be warranted.
Summary of Constraints
----------------------
In summary, the combination of the requirements of a secular spin-orbit resonance for planet b, total angular momentum conservation, and RV detection limits place strong constraints on the parameters of hypothetical perturbing planets. The limits from angular momentum conservation are particularly restrictive, though we used very conservative assumptions in those calculations. All three systems could therefore host additional, as-yet-undetectable planets with parameters appropriate for generating high-obliquity Cassini states for their companion giant planets.
Discussion {#sec:discussion}
==========
While most thermal full-phase light curves of close-in giant planets show that the hottest region on the planet is eastward of the sub-stellar point, which is consistent with super-rotating winds, there are now at least two planets ([HD 149026 b]{} and [CoRoT-2 b]{}) with significant westward hotspot offsets in at least one *Spitzer* band [@zha18; @Dang2018]. One way of reconciling these westward offsets is to consider variations in the spin axis orientation, which fundamentally changes the relationship between the instellation variations due to the rotation and those due to the orbit. Planets on very short-period orbits occupy an interesting regime where we expect the rotation and orbital rates to be comparable; changes in the spin geometry can therefore have major effects on the observed phases. In this work, we developed a thermal radiative model to investigate how non-zero planetary obliquities may produce observable signatures in full-orbit phase curves.
It is important to note that we have not considered the degeneracies inherent to a modeling approach that considers only the output 1-D photometry. Many advances have been made recently that focus on first constructing physical 2- or 3-dimensional maps of the planetary flux, then calculating photometric variations [e.g. @Rauscher2018]. These will likely be the most rigorous frameworks for future photometric modeling. Despite these limitations, and acknowledging that there are multiple potential explanations for westward offsets, we showed that, to first order, highly-oblique hot Jupiters should have different phase variation morphologies from those with little to no obliquity.
We focused on a case study of three planets — [HD 149026 b]{}, [WASP-12 b]{}, and [CoRoT-2 b]{} — all of which have previous light-curve observations that exhibit anomalous features [@cow12; @zha18; @Dang2018; @Barstow2018] that we propose may arise from non-zero obliquities. We reanalyzed the existing *Spitzer* photometry using a thermal model that allows the spin orientation to vary freely.
A combination of sufficiently long thermal timescales and spin axes nearly perpendicular to the orbit normal can reproduce the strong westward offset in the 3.6 $\mu$m data of [HD 149026 b]{}. However, these oblique models are not preferred over a non-oblique model with fewer parameters, where the westward offset comes from an effective rotation slower than synchronous. The results are quite similar for [CoRoT-2 b]{}, where offset-capable models are strongly preferred, and oblique models can effectively reproduce the offset with a very tight constraint on the axial orientation, but due to the slight increase in the number of parameters are statistically not preferred over the simpler non-oblique model. For [WASP-12 b]{}, variations near quadrature cannot be effectively reproduced by a simple thermal model even with free spin geometry, though they fare better than models with constrained rotation. This suggests that more complex models with tidal distortion or refined instrumental characterization need to be considered to help explain the phase curves of [WASP-12 b]{}.
Among the three planets we studied, [CoRoT-2 b]{} has the tightest constraints on its proposed obliquity, at $45.8^{\circ} \pm 1.4^{\circ}$. In addition to the westward hotspot offset [@Dang2018], a high obliquity state could also account for the planet’s extremely inflated radius, which is more anomalous than typical hot Jupiters [@Guillot2011]. The enhanced tidal dissipation that accompanies a high obliquity state is more than enough to provide the heat. To second order in eccentricity, the rate of tidal dissipation in a state of equilibrium rotation and according to equilibrium tide theory is [@Levrard2007] $$\frac{\dot{E}_{\mathrm{tide}}(e,\epsilon)}{K} = \frac{2}{1+\cos^2\epsilon}[\sin^2\epsilon + e^2(7+16\sin^2\epsilon)].
\label{dissipation rate}$$ Here $K$ is given by $$K = \frac{3n}{2}\frac{k_2}{Q_n}\left(\frac{G {M_{\star}}^2}{R_{\rm p}}\right)\left(\frac{R_{\rm p}}{a}\right)^6.
\label{tidalK}$$ Using $k_2 = 0.1$ and $C = 0.2$ as in section \[sec:Cassini state\], $Q = 5\times10^5$, $e=0$, and $\epsilon = 45.8^{\circ}$ as suggested by the sub-synchronous fit, the tidal dissipation rate is $\dot{E}_{\mathrm{tide}} = 1.2 \times 10^{28} \ \mathrm{erg \ s^{-1}}$. This rate is a substantial fraction of the $2.7 \times 10^{29} \ \mathrm{erg \ s^{-1}}$ of energy from incident stellar radiation, which indicates that obliquity tides may provide a clean solution to the anomalous radius inflation. Although the thermal phase curve modeling of [WASP-12 b]{} was inconclusive due to tidal distortion complications, this planet also has a highly inflated radius, which might similarly be a product of obliquity tides [@MillhollandLaughlin2018]. Variations in the measured optical eclipse depths for [WASP-12 b]{} point to a possible additional occultation from escaping material [@VonEssen2019], which could correspondingly cause unexpected thermal variability.
If any of these planets do indeed have non-zero obliquities, these configurations could be induced by secular spin-orbit resonances with as-yet undetected exterior planets. We outlined dynamical arguments for each system to identify the regions of parameter space where such obliquity-producing perturbing planets could exist. There is allowable parameter space in each system, particularly in the HD 149026 system. Such results leave open the possibility that these close-in planets may be locked in a stable, high-obliquity spin states.
S.M. is supported by the National Science Foundation Graduate Research Fellowship Program under Grant Number DGE-1122492. This material is based upon work supported by the National Aeronautics and Space Administration through the NASA Astrobiology Institute under Cooperative Agreement Notice NNH13ZDA017C issued through the Science Mission Directorate. We acknowledge support from the NASA Astrobiology Institute through a cooperative agreement between NASA Ames Research Center and Yale University.
This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.
[^1]: <https://github.com/bfarr/exocartographer>
[^2]: <https://rodluger.github.io/starry/>
[^3]: It is important to note that the host star is quite active [@Lanza2009; @Huber2009], and this activity can bias measurements of the planetary radius from transit observations [@Czesla2009; @Huber2010; @Bruno2016].
[^4]: The northern hemisphere is defined according to the axis around which the planet rotates counter-clockwise. Retrograde motion is therefore obtained by setting ${\epsilon}>90^\circ$.
[^5]: For the equivalent derivation in inertial Cartesian coordinates, see §3 in @Dobrovolskis2013.
[^6]: In general the solution yields unequal extents along $\hat{x}$ and $\hat{y}$, while the prolate spheroid assumption implies they are equal. For [WASP-12 b]{}, the differences are minor enough that the corresponding differences in calculated ellipticity are $\lesssim$ 0.01.
[^7]: <https://github.com/hannorein/rebound>
[^8]: Note that in this setup, an inclination of zero will imply the orbit is viewed edge-on, rather than the standard $i=90^\circ$. We adopt this non-standard definition purely for convenience.
|
---
author:
- 'J. R. Brucato, G.A. Baratta,'
- 'G. Strazzulla'
bibliography:
- 'referenc.bib'
date: 'Received 27 February 2006 / Accepted 10 April 2006'
title: An IR study of pure and ion irradiated frozen formamide
---
[The chemical evolution of formamide (HCONH$_2$), a molecule of astrobiological interest that has been tentatively identified in interstellar ices and in cometary coma, has been studied in laboratory under simulated astrophysical conditions such as ion irradiation at low temperature.]{} [To evaluate the abundances of formamide observed in space or in laboratory, the integrated absorbances for all the principal IR features of frozen amorphous pure formamide deposited at 20 K were measured. Further evidence that energetic processing of ices occurring in space is extremely relevant both to astrochemistry and to astrobiology has been found, showing that new molecular species are synthesized by ion irradiation at a low temperature.]{} [Pure formamide were deposited at 20 K and IR transmission spectra measured for different ice thicknesses. The ice thickness was derived by looking at the interference pattern (intensity versus time) of a He-Ne laser beam reflected at an angle of 45 deg by the vacuum-film and film-substrate interfaces. Samples of formamide ice were irradiated with 200 keV H$^+$ ions and IR spectra recorded at different ion fluences.]{} [New molecules were synthesized among which are CO, CO$_2$, N$_2$O, isocyanic acid (HNCO), and ammonium cyanate (NH$_4$$^+$OCN$^-$). Some of these species remain stable after warming up to room temperature.]{}
Introduction
============
Dust particles are mostly formed in circumstellar regions of evolved stars and ejected in the interstellar medium. In dense molecular clouds, most of the gas–phase species are rapidly condensed on refractory cores, forming icy mantles. The frozen mantles are rich in water, the dominant ice in the Universe, which was identified as a component of dust grains (Willner et al. 1982). Space–based infrared observations performed by observatories such as ISO or Spitzer show that new icy species are continuously detected.
Formamide (HCONH$_2$) was observed in the interstellar medium (for a review of molecules observed in interstellar and circumstellar media, see Millar 2004), in the long period comet C/1995 O1 Hale–Bopp (Bockeleé-Morvan 2000), and tentatively in young stellar objects W33A (Schutte et al. 1999) and NGC 7538 IRS9 (Raunier et al 2004). Formamide is very interesting for its active role in prebiotic chemistry. The chemical reactions of simple compounds containing H, C, N, and O, such as formamide, are considered a plausible pathway for synthesis on the Earth of biomolecules under prebiotic conditions (Oparin 1938, Miller 1953, Eschenmoser & Loewenthal 1992, Chyba & McDonald 1995, Saladino et al. 2004). Among the different theories of prebiotic chemistry active in the early Earth (Miller–Urey and Fischer–Tropsh synthesis), the extraterrestrial synthesis of organic compounds and delivery on the planetary surface is an interesting one. A considerable amount of extraterrestrial material was and is continuously delivered on the Earth. It was estimated that between 10$^7$ and 10$^9$ kg yr$^{-1}$ of carbon contained in organic compounds arrived here in the first billion years as interplanetary dust particles (IDPs) (Chyba & Sagan 1992). Moreover, different strategies for synthesizing organic molecules occurring in space consider simple nitrogen–bearing compounds – such as hydrogen cyanide (HCN), isocyanic acid (HNCO), formamide, ammonium cyanide (NH$_4$CN), or mixtures of H$_2$O, CO$_2$, CO, NH$_3$,CH$_4$, CH$_2$O, CH$_3$OH, etc. – as potential astrobiological precursors (Grim & Greenberg 1987, Demyk et al. 1998, Moore & Hudson 2003, Hudson et al. 2001, Palumbo et al 2000, 2004). In this work we focused on pure formamide molecule under simulated astrophysical conditions.
The HCONH$_2$ molecule is formed at room temperature by the hydrolysis of HCN, and it is the most abundant product of the pyrolysis of HCN–polymer. It is a reactive compound both at the carboxyl moiety and at the amino group. The role of formamide as prebiotic precursor of the synthesis of nucleic acid bases has been shown under a variety of conditions. In particular the presence of different inorganic compounds, such as oxides, minerals or cosmic dust analogues, is able to catalyze the formamide condensation that originate, among many compounds, purine and pyrimidine bases, namely, the products of the first primordial steps that might have leaded to the appearance of life as we know it (Yamada et al. 1972, 1975, Saladino et al. 2003, 2004, 2005).
Energetic processing of ices occurring in space is extremely relevant both to astrochemistry and to astrobiology. Processing by UV photons of interstellar/cometary ice analogue mixtures of H$_2$O, CH$_3$OH, CO, and NH$_3$ has shown that during warming up to 200 K formamide, acetamide, ethanol, and nitriles compounds are formed (Bernstein et al. 1995). Moreover, it was suggested that formamide is one of the products of the ion irradiation and phoyolysis of H$_2$O + HCN (Gerakines et al. 2004) or of NH$_3$ + CO (Demyk et al 1998), even at a low temperature (18 K). Unfortunately, a quantitative measurement of the amount synthesized has not been made due to lacking IR band absorbances. Experimental investigations of chemical evolution of pure formamide in simulated interstellar/cometary conditions are therefore deemed necessary.
In this paper we present an IR transmittance spectroscopy study in the 4000–1200 [cm$^{-1}$]{}range of pure formamide deposited at a low temperature (20 K). We measured the absorbances of the major IR bands and irradiated samples of condensed formamide with 200 keV H$^+$ ions at different fluences. New molecules were synthesized and identified in the spectra. The chemical evolution of the irradiated samples were then monitored at increasing temperatures.
[l@\*[4]{}[l]{}]{} & & Integrated absorbance\
[cm$^{-1}$]{}& &[$\mu$m]{}& Assignment & ($\times$10$^{-17}$cm mol$^{-1}$)\
3368 && 2.97 & [$\nu$]{}[$_{1}$]{} asym. NH$_2$ stretch & 13.49 ([$\nu$]{}[$_{1}$]{}+[$\nu$]{}[$_{2}$]{})\
3181 && 3.14 & [$\nu$]{}[$_{2}$]{} sym. NH$_2$ stretch\
2881 && 3.47 & [$\nu$]{}[$_{3}$]{} CH stretch & 0.47\
1708 && 5.85 & [$\nu$]{}[$_{4}$]{} CO stretch & 6.54 ([$\nu$]{}[$_{4}$]{}+[$\nu$]{}[$_{5}$]{})\
1631 && 6.13 & [$\nu$]{}[$_{5}$]{} in plane NH$_2$ scissoring\
1388 && 7.20 & [$\nu$]{}[$_{6}$]{} in plane CH scissoring & 0.68\
1328 && 7.53 & [$\nu$]{}[$_{7}$]{} CN stretch & 0.85\
Experimental procedures
=======================
Gaseous samples of formamide were prepared in a pre–chamber (P $<$ 10$^{-6}$ mbar) and admitted by a gas inlet through a needle valve into the scattering chamber, where they accreted onto a cold (20 K) silicon substrate; the thickness of the deposited films were then measured (see below for details). The infrared transmittance spectra were obtained in the high–vacuum scattering chamber (P $<$ 10$^{-7}$ mbar) interfaced with a FTIR spectrophotometer (Bruker Equinox 55) through IR–transparent windows. The silicon substrate was in thermal contact with a closed–cycle helium cryostat whose temperature can be varied between 10 and 300 K. The vacuum chamber is interfaced with an ion implanter (200 kV; Danfysik 1080–200) that generates ions with energies up to 200 keV (400 keV for double ionization). The ion beam produces a spot that is larger than the area probed by the infrared beam (for more details on the experimental set up see Strazzulla et al. 2001). In this work we used 200 keV H$^+$ ions. In order to avoid macroscopic heating of the target, the current density was maintained in the range of 100 $\mathrm{nA\, cm^{-2}}$ to a few $\mathrm{\mu A\, cm^{-2}}$. The ion fluence in $\mathrm{ions\, cm^{-2}}$ was measured by a charge integrator from the ion current monitored during irradiation. The substrate plane is placed at an angle of 45 degrees with respect to the IR beam and the ion beam so that spectra can be taken in situ, even during irradiation, without tilting the sample. Spectra were taken at selected temperatures in the range of 20–300 K. All the spectra shown below were taken with a resolution of 1 cm$^{-1}$ using a DTGS detector.
The 200 keV H$^+$ ions penetrate about 2 [$\mu$m]{} in formamide calculated using the TRIM program (Transport of Ions in Matter; e.g., Ziegler 1977; Ziegler et al. 1996). The maximum thickness of deposited layers was about 0.49 [$\mu$m]{} (see below) i.e. thinner than the penetration depth of incoming ions. With the same software, it is also possible to calculate the stopping power (i.e. the amount of energy deposited per unit path length) of a given ion in a given target. For 200 keV H$^+$ in formamide we obtained about 50$\times$10$^{-15}$ eV cm$^{2}$/molecula. By multiplying this number times the number of bombarding ions per square centimetre we obtained the amount of energy released to the sample (dose) in eV per molecula. Here we express the dose in eV/16amu, because it is a convenient way to characterize chemical changes and to compare with other experiments with different samples.
![IR transmittance spectrum in optical depth scale of formamide as deposited at 20 K on a silicon substrate (dot line). The band–peak positions and assignment are given in Table 1. The spectrum obtained after irradiation at a dose of 12 eV/16amu with 200 keV H$^+$ ions is also shown (full line).[]{data-label="fig1"}](5095fig1.eps){width="10cm"}
Results
=======
Figure \[fig1\] shows the two spectra of (HCONH$_2$) before and after ion irradiation (12 eV/16 amu deposited by 200 keV H$^+$ ions) at 20 K in the 3800–1200 [cm$^{-1}$]{}range. The peak positions of fundamental vibrational modes of formamide and their assignments are reported in Table \[table1\]. Such assignments were made on the basis of work on gaseous or liquid samples (McNaughton et al. 1999, Rubalcava 1956). The (HCONH$_2$) spectrum is characterized by a couple of intense and broad bands at 3368 and 3181 [cm$^{-1}$]{}, which correspond to asymmetric and symmetric NH$_2$ stretching modes, respectively. The strongest band is peaked at 1708 [cm$^{-1}$]{}and related to the CO–stretching mode of the carbonyl group.
Absorbances
-----------
The experimental set–up allows the thickness to be monitored during accretion by looking at the interference pattern (intensity versus time) given by a He–Ne laser beam reflected at an angle of 45 deg by the vacuum–film and film–substrate interfaces. After the reflection from the substrate, the laser beam follows the same path of the infrared beam coming from the IR source and can be detected by using an external detector placed in the source compartment of the IR spectrometer.
In general the interference curve versus thickness given by the laser beam reflected by the film+substrate assembly is an oscillating function. For absorbing materials, the laser light transmitted into the film and reflected back by the interface film–substrate is attenuated into the film; thus the amplitude of the oscillation in the interference curve exponentially decays with the thickness, and the reflectance approaches its bulk value at a large thickness. In molecular ices at visible wavelengths, the absorption is so low that it can be neglected for a thickness of a few microns, so that the interference curve can be considered a periodic function whose period (distance between two maxima or minima) is given by the relation: $$\Delta d=\frac{\lambda _{0}}{2n_{f}\sqrt{1-sin^2\theta_i/n^2_f}}
\label{eq:inter}$$ where $n_{f}$ is the refractive index of the ice at the laser wavelength $\lambda_{0}$, and $\theta_{i}$ is the incidence angle measured from the normal. The previous relation can be used to measure the thickness if $n_{f}$ is known. The amplitude of the interference curve depends on the refractive index of the ice, the refractive index of the substrate, the incidence angle, and on the polarization of the laser. Hence $n_{f}$ can be derived from the amplitude of the interference curve (intensity ratio between maxima and minima); details of this method can be found in Westley et al (1998) and Baratta and Palumbo (1998).
The absolute accuracy of the thickness measured in this way is about 5$\%$ and is limited mainly by the uncertainties in the knowledge of the refractive index of the substrate at low temperature (silicon in our case) and by the error in measuring the incidence angle of the laser. By following this method we found a refractive index $n_{f}$=1.361 for formamide ice at the laser wavelength of 0.543 [$\mu$m]{}. For an incidence angle of 45 deg, this yields a film thickness of 0.234 [$\mu$m]{} between interference maxima. Once the thickness is measured, the integrated bands absorbance, [*A*]{} (cm mol$^{-1}$), can be computed from the infrared spectrum if the density $\rho$ of the film is known through the formula: $$A=\frac{\int \tau(\nu)d\nu}{N}$$ where $\tau(\nu)$ is the optical depth and $N$ the molecular column density in units of molecules cm$^{-2}$.
In this work we derived the density of formamide ice by using the Lorentz–Lorenz relation: $$L \rho=\frac{n_{f}^{2}-1}{n_{f}^{2}+2}.
\label{eq:inter}$$ For a given material, the quantity $L$ (Lorentz–Lorenz coefficent) is nearly constant for a given wavelength regardless of the material phase and temperature (Wood & Roux 1982). Formamide at 25 C (liquid) has a density $\rho^{liq}$=1.129 gcm$^{-3}$ and a refractive index of $n^{liq}_{f}$=1.446 for the sodium line (0.589 [$\mu$m]{}) (Cases et al. 2001). The corresponding Lorentz–Lorenz coefficient is $L$= 0.2362 cm$^{3}$ g$^{-1}$. By substituting in Eq (\[eq:inter\]), the L coefficient derived for the liquid phase and the refractive index measured by interference, we obtain a density $\rho^{ice}$=0.937 g cm$^{-3}$ for formamide ice. The corresponding integrated band absorbances are given in Table \[table1\], together with band peak positions and assignments. The absorbances were derived by considering two different thickness increased by a factor (from the Snell’s low) of $1/cos\theta_{r}=1/\sqrt{1-sin^{2}\theta_{i}/n^{2}_{f}}$, where $\theta_{r}$ is the refractive angle. The correction takes into account, in a approximate way, the increased path length of the IR beam at an oblique incidence of $\theta_{i}$=45 deg. The corrective factor was derived by assuming a constant value of the refractive index with the wavelength. This approximation neglects the variation of the refractive index in the infrared region due to the vibrations and the contribution of the electronic transitions to the dispersion from the visible (0.543 [$\mu$m]{} ) to the infrared. We estimate a corresponding uncertainty of $\simeq$ 10 $\%$ in the band absorbances.
Ion–induced synthesis of molecules
----------------------------------
Figure \[fig1\] also shows the IR spectrum of formamide after 200 keV proton irradiation at 12 eV/16amu dose. The peak positions and identification of each new molecular species is reported in Table \[table2\].
In the 2430–1850 [cm$^{-1}$]{}spectral region shown better in Fig. \[fig2\], both CO and CO$_2$ absorption bands are observed at 2140 and 2342 [cm$^{-1}$]{}, respectively. The band observed at 2260 [cm$^{-1}$]{}is assigned to the NCO stretching mode of isocyanic acid (HNCO). The band peaked at wavenumbers larger than that observed by Raunier et al. (2004) (2252 [cm$^{-1}$]{}). This may be due to the presence of a shoulder at 2238 [cm$^{-1}$]{}attributed to the synthesis of NO$_2$. The weak band at 2083 [cm$^{-1}$]{}is due to the cyanate anion CN$^-$ stretching mode. A shoulder at about 2100 [cm$^{-1}$]{}is present and tentatively assigned to hydrogen cyanide (HCN). The presence of the SiH stretching band observed at 1997 [cm$^{-1}$]{}stems from the fraction of protons that entirely cross the ice sample and is implanted into the silicon substrate.
Another intense band located at 2165 [cm$^{-1}$]{}is commonly assigned to cyanate anion OCN$^-$ (Grim and Greenberg 1987, Hudson et al. 2001, Broekhuizen et al. 2004), even if other carriers have been proposed by Pendleton et al. (1999). Evidence that the ammonium cyanate complex NH$_4^+$OCN$^-$ is formed is given by the wide band at 1478 [cm$^{-1}$]{}, which is ascribed to NH$_4^+$ (Raunier et al. 2003). Further evidence of ammonium ion synthesis is formed in the smooth and weak peaks at 3074 and 3206 [cm$^{-1}$]{} observed over the very broad feature extending from about 3600 [cm$^{-1}$]{}to 2400 [cm$^{-1}$]{}. This feature is due to the overlapping of a number of NH and CH stretches of newly formed compounds and of a residual amount of formamide still present after ion irradiation. A tentative identification of NH$_3$ molecules is made by the band present at 3376 [cm$^{-1}$]{}due to NH vibration stretch and by a noisy feature at about 1110 [cm$^{-1}$]{}that is not reported in Fig. \[fig1\].
[l@\*[4]{}[l]{}]{} & &\
[[cm$^{-1}$]{}]{}& [[$\mu$m]{}]{} & Vibration & Assignment\
3376& 2.96 & N–H stretch & NH$_3$\
3206& 3.12 & N–H stretch & NH$_4^+$\
3074 & 3.25 & N–H stretch & NH$_4^+$\
2342 & 4.27 & C=O stretch & CO$_2$\
2260 & 4.42 & N=C=O$_{asym.}$ stretch & HNCO\
2238 & 4.47 & N=N stretch & N$_2$O\
2165 & 4.62 & N=C=O$_{asym.}$ stretch & OCN$^-$\
2140 & 4.67 & C$\equiv$O stretch & CO\
2083 & 4.80 & C$\equiv$N stretch & CN$^-$\
1997 & 5.01 & Si–H stretch & SiH\
1478 & 6.77 & N–H$_{sym.}$ bending & NH$_4^+$\
![IR transmittance spectrum in optical depth scale of formamide (20 K) after irradiation at a dose of 12 eV/16amu with 200 keV H$^+$ ions (full line). The spectrum obtained before irradiation is also shown (dotted line).[]{data-label="fig2"}](5095fig2.eps){width="10cm"}
Temperature effects
-------------------
After irradiation, the samples were warmed up (about 1 K/min) to study the evolution of the IR spectrum with temperature, a circumstance that induces a differential sublimation of volatiles and an increase in the synthesized yields likely to be affected by higher mobility. In Fig. \[fig3\] the IR spectra in the range 3700–1200 [[cm$^{-1}$]{}]{} of formamide after irradiation at a dose of 24 eV/16amu at 20 K and after warming up at 125 K, 220 K, and 300 K are shown. Spectra were arbitrarily shifted and, moreover, that at room temperature magnified five times for sake of clarity.
The peak position of OCN$^-$ shifts during warm–up from 2167 [[cm$^{-1}$]{}]{} at 20 K to 2164 [[cm$^{-1}$]{}]{} at 220 K. At room temperature, the OCN$^-$ peak position is about 2167 [[cm$^{-1}$]{}]{}. The peak position dependence on temperature of OCN$^-$ synthesized by ion irradiation of C,N,O–bearing ice mixtures is a commonly observed phenomena. Nevertheless the amount of shift and the trend observed with temperature depend strongly on the specific ice mixture (Palumbo et al. 2000, Palumbo et al. 2004). In particular the peak position of the OCN$^-$ feature observed in the organic residues at room temperature can vary from 2150 to 2168 [[cm$^{-1}$]{}]{} depending on the particular ion–irradiated ice mixture considered (Palumbo et al. 2004). A shift at lower wave number is observed for the N–H$_{sym.}$ bending band peaking at 1437 [[cm$^{-1}$]{}]{} at 300 K.
It is evident that the volatile species are desorbed and the bands of ammonium cyanate are left over. The formation of NH$_4^+$OCN$^-$ agrees with that measured by the reaction at 10 K of co–deposited NH$_3$ and HNCO (Raunier et al. 2004). The band strength of ammonium cyanate increases with temperature, which could be due to additional formation by reaction between NH$_3$ and HNCO synthesized at low T. Further features observed in Fig. \[fig3\] at 220 K correspond to crystalline formamide still present in the sample after irradiation, which has had a structural transition at about 180 K.
![Comparison of the spectra of (from the bottom to the top): formamide after irradiation with a dose of 24 eV/16amu at 20 K, and after warming up at 125 K, 220 K and room temperature (this spectrum was magnified 5 times). The arrows show the bands corresponding to the presence of NH$_4^+$OCN$^-$. Spectra have been arbitrarily shifted for sake of clarity.[]{data-label="fig3"}](5095fig3.eps){width="9cm"}
Discussion
==========
Formamide has been tentatively identified in the ISO–SWS infrared spectra of different astronomical environments. Although it has been synthesized by UV photolysis and proton irradiation of HCN containing ices (Gerakines et al. 2004) or by UV irradiation of HNCO ice (Raunier et al. 2004) and H$_2$O, CO, NH$_3$ ice mixtures (Demyk et al. 1998, Broekhuizen et al. 2004), it has not been possible to precisely determine the amount of synthesized (or observed) formamide because of the lack of measured integrated absorbances. This work covers such a need by measuring the integrated absorbances for all the principal features of frozen amorphous pure formamide deposited at 20 K.
With the hypothesis that solid formamide is present in protostellar sources as might be the case for NGC 7538 IRS9 (Raunier et al. 2004), the experimental results reported here on the ion processing of formamide ice have to be considered when astronomical and laboratory spectra are compared. In fact, the evolution of solids (ices, silicates, and carbonaceous phases) in the circumstellar and interstellar media is governed by a number of processes, such as surface reactions, UV photolysis, particle irradiation, and thermal annealing.
Irradiation in the circumstellar environments of young stars or in the dense interstellar medium was simulated by irradiating pure formamide layers at 20 K. After irradiation, warming the target up causes the sublimation of volatile molecules. This simulates the thermal processing suffered by the dust near a forming star or the passage of the dust from a dense to a diffuse medium. It is important to note that in this case, calculations indicate that in the diffuse interstellar medium the temperature of the dust is still of the order of 10 K, as it is in the dense medium (Greenberg 1971; Mathis et al. 1983); however, the gas density is such that icy mantles cannot be maintained or formed.
The flux of low–energy cosmic rays irradiating grains in the interstellar medium is not well known, although a reasonable estimate was given by Moore (1999). She calculated that in cold dense clouds, ions deposit about 30 eV/molecule (in 10$^8$ years), a factor of 10 less than in the diffuse medium. The dominant contribution of cosmic rays comes mainly from low–energy protons in the MeV range, which lose energy through ionizations and excitations of the target atoms as 200 keV H$^+$ ions do. From the present experimental results we estimated that 64 and 78% of the formamide molecules are destroyed at 12 and 24 eV/16 amu, respectively. This result indicates that the destruction of formamide is a first–order process. Extrapolating to the irradiation dose expected in the dense medium, it is possible to infer that, in first approximation, about 20% of the frozen formamide molecules are able to survive in dense medium on a time scale of 10$^8$ years.
A very important result is that the species produced after irradiation of formamide are mostly the same as those produced in a large number of irradiation experiments conducted by different groups on different icy mixtures containing simple H,O, C, and N bearing molecules (Grim et al., 1987, Demyk et al. 1998, Moore & Hudson 2003, Hudson et al. 2001, Palumbo et al 2000, 2004). Moreover, most of the bands shown in Fig. \[fig2\] have been observed in astronomical spectra (Sandford et al. 1990, Elsila et al. 1997, Novozamsky et al. 2001,Pendleton et al. 1999, Tegler et al. 1993, 1995, van Broekhiuzen 2005). Of particular interest is the synthesis of the ammonium cyanate molecule. In fact the CN stretching in OCN$^-$ is considered responsible for the 2165 [cm$^{-1}$]{}band observed in almost all the observations of molecular clouds and in young stellar objects. Ion and UV processing of ice samples containing N–bearing molecules such as N$_2$, NH$_3$, HCN, and HNCO mixed with the most abundant molecules observed in space – H$_2$O, CO, CO$_2$, and CH$_3$OH – share the yielding of NH$_4$OCN$^-$ with the results of this work.
Ammonium cyanate is theoretically very important since the first synthetic production of an organic from inorganic compounds (Wohler 1828). Once prepared by the gaseous reaction of ammonia and cyanic acid, thermal annealing of ammonium cyanate aqueous solution formed urea (NH$_2$)$_2$CO through dissociation into ammonia and isocyanic acid (Warner and Stitt 1933). Urea was also synthesized at 10 K by UV irradiation of isocyanic acid and tentatively detected in a protostellar object (Raunier et al. 2004).
These results suggest that the chemistry of HCN (in the presence of H$_2$O) has to be considered a preferential route for the prebiotic chemistry even in space.
The plausible pathways for the synthesis of biomolecules in space at low temperature have to be deal with when studying the organic compounds that might have favoured the emergence of life when delivered on the early Earth by comets, meteorites and IDPs. Ion and photon processing of ices containing H, C, N, and O shows a specificity in driving the chemistry in space so that it requires deep investigation, in particular for its important consequences on astrobiological research.
This research has been supported by the Italian Space Agency (ASI), and the Italian Ministero dell’Istruzione, Universit\` a e Ricerca (MIUR).
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abstract: 'The person re-identification task requires to robustly estimate visual similarities between person images. However, existing person re-identification models mostly estimate the similarities of different image pairs of probe and gallery images independently while ignores the relationship information between different probe-gallery pairs. As a result, the similarity estimation of some hard samples might not be accurate. In this paper, we propose a novel deep learning framework, named Similarity-Guided Graph Neural Network (SGGNN) to overcome such limitations. Given a probe image and several gallery images, SGGNN creates a graph to represent the pairwise relationships between probe-gallery pairs (nodes) and utilizes such relationships to update the probe-gallery relation features in an end-to-end manner. Accurate similarity estimation can be achieved by using such updated probe-gallery relation features for prediction. The input features for nodes on the graph are the relation features of different probe-gallery image pairs. The probe-gallery relation feature updating is then performed by the messages passing in SGGNN, which takes other nodes’ information into account for similarity estimation. Different from conventional GNN approaches, SGGNN learns the edge weights with rich labels of gallery instance pairs directly, which provides relation fusion more precise information. The effectiveness of our proposed method is validated on three public person re-identification datasets.'
author:
- Yantao Shen
- 'Hongsheng Li[^1]'
- Shuai Yi
- |
\
Dapeng Chen
- Xiaogang Wang
bibliography:
- 'egbib.bib'
title: |
Person Re-identification with\
Deep Similarity-Guided Graph Neural Network
---
-- -----------------------------
\(a) Conventional Approach.
\(b) Our proposed SGGNN.
-- -----------------------------
Introduction
============
Person re-identification is a challenging problem, which aims at finding the person images of interest in a set of images across different cameras. It plays a significant role in the intelligent surveillance systems.
To enhance the re-identification performance, most existing approaches attempt to learn discriminative features or design various metric distances for better measuring the similarities between person image pairs. In recent years, witness the success of deep learning based approaches for various tasks of computer vision [@krizhevsky2012imagenet; @he2016deep; @ren2015faster; @wu2016model; @srivastava2014dropout; @chu2017multi; @Liuxihui_2017_ICCV; @wu20173d; @yang2017learning; @li2017vip; @kang2017object], a large number of deep learning methods were proposed for person re-identification [@liu2016multi; @Zhou_2017_CVPR; @xiao2016learning; @liu2017quality]. Most of these deep learning based approaches utilized Convolutional Neural Network (CNN) to learn robust and discriminative features. In the mean time, metric learning methods were also proposed [@Bak_2017_CVPR; @bkak2016person; @Yu_2017_ICCV] to generate relatively small feature distances between images of same identity and large feature distances between those of different identities.
However, most of these approaches only consider the pairwise similarity while ignore the internal similarities among the images of the whole set. For instance, when we attempt to estimate the similarity score between a probe image and a gallery image, most feature learning and metric learning approaches only consider the pairwise relationship between this single probe-gallery image pair in both training and testing stages. Other relations among different pairs of images are ignored. As a result, some hard positive or hard negative pairs are difficult to obtain proper similarity scores since only limited relationship information among samples is utilized for similarity estimation.
To overcome such limitation, we need to discover the valuable internal similarities among the image set, especially for the similarities among the gallery set. One possible solution is utilizing manifold learning [@bai2017scalable; @loy2013person], which considers the similarities of each pair of images in the set. It maps images into a manifold with more smooth local geometry. Beyond the manifold learning methods, re-ranking approaches [@Zhong_2017_CVPR; @garcia2015person; @ye2016person] were also utilized for refining the ranking result by integrating similarities between top-ranked gallery images. However, both manifold learning and re-ranking approaches have two major limitations: (1) most manifold learning and re-ranking approaches are unsupervised, which could not fully exploit the provided training data label into the learning process. (2) These two kinds of approaches could not benefit feature learning since they are not involved in training process.
Recently, *Graph Neural Network* (GNN) [@bruna2013spectral; @henaff2015deep; @kipf2016semi; @niepert2016learning] draws increasing attention due to its ability of generalizing neural networks for data with graph structures. The GNN propagates messages on a graph structure. After message traversal on the graph, node’s final representations are obtained from its own as well as other node’s information, and are then utilized for node classification. GNN has achieved huge success in many research fields, such as text classification [@defferrard2016convolutional], image classification [@bruna2013spectral; @oord2016pixel], and human action recognition [@yan2018spatial]. Compared with manifold learning and re-ranking, GNN incorporates graph computation into the neural networks learning, which makes the training end-to-end and benefits learning the feature representation.
In this paper, we propose a novel deep learning framework for person re-identification, named Similarity-Guided Graph Neural Network (SGGNN). SGGNN incorporates graph computation in both training and testing stages of deep networks for obtaining robust similarity estimations and discriminative feature representations. Given a mini-batch consisting of several probe images and gallery images, SGGNN will first learn initial visual features for each image ([*e*.*g*.]{}, global average pooled features from ResNet-50 [@he2016deep].) with the pairwise relation supervisions. After that, each pair of probe-gallery images will be treated as a node on the graph, which is responsible for generating similarity score of this pair. To fully utilize pairwise relations between other pairs (nodes) of images, deeply learned messages are propagated among nodes to update and refine the pairwise relation features associated with each node. Unlike most previous GNNs’ designs, in SGGNN, the weights for feature fusion are determined by similarity scores by gallery image pairs, which are directly supervised by training labels. With these similarity guided feature fusion weights, SGGNN will fully exploit the valuable label information to generate discriminative person image features and obtain robust similarity estimations for probe-gallery image pairs.
The main contribution of this paper is two-fold. (1) We propose a novel Similarity Guided Graph Neural Network (SGGNN) for person re-identification, which could be trained end-to-end. Unlike most existing methods, which utilize inter-gallery-image relations between samples in the post-processing stage, SGGNN incorporates the inter-gallery-image relations in the training stage to enhance feature learning process. As a result, more discriminative and accurate person image feature representations could be learned. (2) Different from most Graph Neural Network (GNN) approaches, SGGNN exploits the training label supervision for learning more accurate feature fusion weights for updating the nodes’ features. This similarity guided manner ensures the feature fusion weights to be more precise and conduct more reasonable feature fusion. The effectiveness of our proposed method is verified by extensive experiments on three large person re-identification datasets.
Related Work
============
Person Re-identification
------------------------
Person re-identification is an active research topic, which gains increasing attention from both academia and industry in recent years. The mainstream approaches for person re-identification either try to obtain discriminative and robust feature [@yi2014deep; @li2014deepreid; @ahmed2015improved; @Su_2017_ICCV; @schumann2017person; @cheng2016person; @Lin_2017_CVPR; @Sun_2017_ICCV; @shen2017learning; @shen2018deep; @chen2018group; @chen2018video; @song2017region; @karaman2014leveraging] for representing person image or design a proper metric distance for measuring similarity between person images [@paisitkriangkrai2015learning; @bkak2016person; @Bak_2017_CVPR; @Liu_2017_ICCV; @Yu_2017_ICCV]. For feature learning, Yi [*et al*.]{} [@yi2014deep] introduced a Siamese-CNN for person re-identification. Li [*et al*.]{} [@li2014deepreid] proposed a novel filter pairing neural network, which could jointly handle feature learning, misalignment, and classification in an end-to-end manner. Ahmed [*et al*.]{} [@ahmed2015improved] introduced a model called Cross-Input Neighbourhood Difference CNN model, which compares image features in each patch of one input image to the other image’s patch. Su [*et al*.]{} [@Su_2017_ICCV] incorporated pose information into person re-identification. The pose estimation algorithm are utilized for part extraction. Then the original global image and the transformed part images are fed into a CNN simultaneously for prediction. Shen [*et al*.]{} [@shen2018end] utilized kronecker-product matching for person feature maps alignment. For metric learning, Paisitkriangkrai [*et al*.]{} [@paisitkriangkrai2015learning] introduced an approach aims at learning the weights of different metric distance functions by optimizing the relative distance among triplet samples and maximizing the averaged rank-k accuracies. Bak [*et al*.]{} [@bkak2016person] proposed to learn metrics for 2D patches of person image. Yu [*et al*.]{} [@Yu_2017_ICCV] introduced an unsupervised person re-ID model, which aims at learning an asymmetric metric on cross-view person images.
Besides feature learning and metric learning, manifold learning [@bai2017scalable; @loy2013person] and re-rank approaches [@Zhong_2017_CVPR; @ye2015ranking; @ye2016person; @garcia2015person] are also utilized for enhancing the performance of person re-identification model, Bai [*et al*.]{} [@bai2017scalable] introduced Supervised Smoothed Manifold, which aims to estimating the context of other pairs of person image thus the learned relationships with between samples are smooth on the manifold. Loy [*et al*.]{} [@loy2013person] introduced manifold ranking for revealing manifold structure by plenty of gallery images. Zhong [*et al*.]{} [@Zhong_2017_CVPR] utilized k-reciprocal encoding to refine the ranking list result by exploiting relationships between top rank gallery instances for a probe sample. Kodirov [*et al*.]{} [@kodirov2016person] introduced graph regularised dictionary learning for person re-identification. Most of these approaches are conducted in the post-process stage and the visual features of person images could not be benefited from these post-processing approaches.
Graph for Machine Learning {#subsec:graph}
--------------------------
In several machine learning research areas, input data could be naturally represented as graph structure, such as natural language processing [@mills2014graph; @liu2018show], human pose estimation [@chu2016crf; @yan2018spatial; @yang2016end], visual relationship detection [@li2017scene], and image classification [@quek2011structural; @pavlidis2013structural]. In [@scarselli2009graph], Scarselli [*et al*.]{} divided machine learning models into two classes due to different application objectives on graph data structure, named *node-focused* and *graph-focused* application. For *graph-focused* application, the mapping function takes the whole graph data $G$ as the input. One simple example for *graph-focused* application is to classify the image [@pavlidis2013structural], where the image is represented by a region adjacency graph. For *node-focused* application, the inputs of mapping function are the nodes on the graph. Each node on the graph will represent a sample in the dataset and the edge weights will be determined by the relationships between samples. After the message propagation among different nodes (samples), the mapping function will output the classification or regression results of each node. One typical example for *node-focused* application is graph based image segmentation [@zheng2015conditional; @liu2015crf], which takes pixels of image as nodes and try to minimize the total energy function for segmentation prediction of each pixel. Another example for *node-focused* application is object detection [@bianchini2005recursive], the input nodes are features of the proposals in a input image.
Graph Neural Network
--------------------
Scarselli [*et al*.]{} [@scarselli2009graph] introduced Graph Neural Network (GNN), which is an extension for recursive neural networks and random walk models for graph structure data. It could be applied for both graph-focused or node-focused data without any pre or post-processing steps, which means that it can be trained end-to-end. In recent years, extending CNN to graph data structure received increased attention [@bruna2013spectral; @henaff2015deep; @kipf2016semi; @niepert2016learning; @yan2018spatial; @defferrard2016convolutional; @liang2016semantic], Bruna [*et al*.]{} [@bruna2013spectral] proposed two constructions of deep convolutional networks on graphs (GCN), one is based on the spectrum of graph Laplacian, which is called spectral construction. Another is spatial construction, which extends properties of convolutional filters to general graphs. Yan [*et al*.]{} [@yan2018spatial] exploited spatial construction GCN for human action recognition. Different from most existing GNN approaches, our proposed approach exploits the training data label supervision for generating more accurate feature fusion weights in the graph message passing.
Method {#sec:method}
======
To evaluate the algorithms for person re-identification, the test dataset is usually divided into two parts: a probe set and a gallery set. Given an image pair of a probe and a gallery images, the person re-identification models aims at robustly determining visual similarities between probe-gallery image pairs. In the previous common settings, among a mini-batch, different image pairs of probe and gallery images are evaluated individually, [*i*.*e*.]{}, the estimated similarity between a pair of images will not be influenced by other pairs. However, the similarities between different gallery images are valuable for refining similarity estimation between the probe and gallery. Our proposed approach is proposed to better utilize such information to improve feature learning and is illustrated in Figure \[fig:main\]. It takes a probe and several gallery images as inputs to create a graph with each node modeling a probe-gallery image pair. It outputs the similarity score of each probe-gallery image pair. Deeply learned messages will be propagated among nodes to update the relation features associated with each node for more accurate similarity score estimation in the end-to-end training process.
In this section, the problem formulation and node features will be discussed in Section \[sec:pro\_for\]. The Similarity Guided GNN (SGGNN) and deep messages propagation for person re-identification will be presented in Section \[sec:SGGNN\]. Finally, we will discuss the advantage of similarity guided edge weight over the conventional GNN approaches in Section \[sec:advan\]. The implementation details will be introduced in \[sec:imp\]
Graph Formulation and Node Features {#sec:pro_for}
-----------------------------------
In our framework, we formulate person re-identification as a *node-focused* graph application introduced in Section \[subsec:graph\]. Given a probe image and $N$ gallery images, we construct an undirected complete graph $G(V,E)$, where $V = \{v_1, v_2, ..., v_N\}$ denotes the set of nodes. Each node represents a pair of probe-gallery images. Our goal is to estimate the similarity score for each probe-gallery image pair and therefore treat the re-identification problem as a node classification problem. Generally, the input features for any node encodes the complex relations between its corresponding probe-gallery image pair.
In this work, we adopt a simple approach for obtaining input relation features to the graph nodes, which is shown in Figure \[fig:embed\](a). Given a probe image and $N$ gallery images, each input probe-gallery image pair will be fed into a Siamese-CNN for pairwise relation feature encoding. The Siamese-CNN’s structure is based on the ResNet-50 [@he2016deep]. To obtain the pairwise relation features, the last global average pooled features of two images from ResNet-50 are element-wise subtracted. The pairwise feature is processed by element-wise square operation and a Batch Normalization layer [@ioffe2015batch]. The processed difference features $d_i$ ($i=1,2,...,N$) encode the deep visual relations between the probe and the $i$-th gallery image, and are used as the input features of the $i$-th node on the graph. Since our task is node-wise classification, [*i*.*e*.]{}, estimating the similarity score of each probe-gallery pair, a naive approach would be simply feeding each node’s input feature into a linear classifier to output the similarity score without considering the pairwise relationship between different nodes. For each probe-gallery image pair in the training mini-batch, a binary cross-entropy loss function could be utilized, $$\label{eq:loss}
L =-\sum_{i=1}^{N}y_i \log(f(d_i))+(1-y_i)\log(1-f(d_i)) ,$$ where $f()$ denotes a linear classifier followed by a sigmoid function. $y_i$ denotes the ground-truth label of $i$-th probe-gallery image pair, with 1 representing the probe and the $i$-th gallery images belonging to the same identity while 0 for not.
Similarity-Guided Graph Neural Network {#sec:SGGNN}
--------------------------------------
Obviously, the naive node classification model (Eq.( \[eq:loss\])) ignores the valuable information among different probe-gallery pairs. For exploiting such vital information, we need to establish edges $E$ on the graph $G$. In our formulation, $G$ is fully-connected and $E$ represents the set of relationships between different probe-gallery pairs, where $W_{ij}$ is a scalar edge weight. It represents the relation importance between node $i$ and node $j$ and can be calculated as, $$\label{eq:egde_weight}
W_{ij} =
\begin{cases}
\frac{\text{exp}(S(g_i,g_j))}{\sum_{j}\text{exp}(S(g_i,g_j))}, \quad i \neq j \\
0, \quad i = j\\
\end{cases},$$ where $g_i$ and $g_j$ are the $i$-th and $j$-th gallery images. $S()$ is a pairwise similarity estimation function, that estimates the similarity score between $g_i$ and $g_j$ and can be modeled in the same way as the naive node (probe-gallery image pair) classification model discussed above. Note that in SGGNN, the similarity score $S(g_i,g_j)$ of gallery-gallery pair is also learned in a supervised way with person identity labels. The purpose of setting $W_{ii}$ to 0 is to avoid self-enhancing.
[m[0cm]{}@m[6cm]{}m[7cm]{}]{} & &\
& (a) Node input feature generating. & (b) Deep message passing of SGGNN.
To enhance the initial pairwise relation features of a node with other nodes’ information, we propose to propagate deeply learned messages between all connecting nodes. The node features are then updated as a weighted addition fusion of all input messages and the node’s original features. The proposed relation feature fusion and updating is intuitive: using gallery-gallery similarity scores to guide the refinement of the probe-gallery relation features will make the relation features more discriminative and accurate, since the rich relation information among different pairs are involved. For instance, given one probe sample $p$ and two gallery samples $g_i$, $g_j$. Suppose that $(p, g_i)$ is a hard positive pair (node) while both $(p, g_j)$ and $(g_i, g_j)$ are relative easy positive pairs. Without any message passing among the nodes $(p, g_i)$ and $(p, g_j)$, the similarity score of $(p, g_i)$ is unlikely to be high. However, if we utilize the similarity of pair $(g_i, g_j)$ to guide the refinement of the relation features of the hard positive pair $(p, g_i)$, the refined features of $(p, g_i)$ will lead to a more proper similarity score. This relation feature fusion could be deduced as a message passing and feature fusion scheme.
Before message passing begins, each node first encodes a deep message for sending to other nodes that are connected to it. The nodes’ input relation features $d_i$ are fed into a message network with 2 fully-connected layers with BN and ReLU to generate deep message $t_i$, which is illustrated in Figure \[fig:embed\](b). This process learns more suitable messages for node relation feature updating,
$$\label{eq:transform}
t_i = F(d_i) \quad \text{for }i=1,2,...,N,$$
where $F$ denotes the 2 FC-layer subnetwork for learning deep messages for propagation.
After obtaining the edge weights $W_{ij}$ and deep message $t_i$ from each node, the updating scheme of node relation feature $d_i$ could be formulated as $$\label{eq:refine_walk}
d_{i}^{(1)} = (1 -\alpha) d_{i}^{(0)} + \alpha \sum_{j = 1}^{N} W_{ij} t_{j}^{(0)} \quad \text{for} \ i=1,2,...,N,$$ where $ d_{i}^{(1)}$ denotes the $i$-th refined relation feature, $d_{i}^{(0)}$ denotes the $i$-th input relation feature and $t_{j}^{(0)}$ denotes the deep message from node $j$. $\alpha$ represents the weighting parameter that balances fusion feature and original feature.
Noted that such relation feature weighted fusion could be performed iteratively as follows, $$\label{eq:refine_walk_iter}
d_{i}^{(t)} =(1 - \alpha) d_{i}^{(t-1)} + \alpha \sum_{j = 1}^{N} W_{ij} t_{j}^{(t-1)} \quad \text{for} \ i=1,2,...,N,$$ where $t$ is the iteration number. The refined relation feature $d_i^{(t)}$ could substitute then relation feature $d_i$ in Eq. (\[eq:loss\]) for loss computation and training the SGGNN. For training, Eq. (\[eq:refine\_walk\_iter\]) can be unrolled via back propagation through structure.
In practice, we found that the performance gap between iterative feature updating of multiple iterations and updating for one iteration is negligible. So we adopt Eq. (\[eq:refine\_walk\]) as our relation feature fusion in both training and testing stages. After relation feature updating, we feed the relation features of probe-gallery image pairs to a linear classifier with sigmoid function for obtaining the similarity score and trained with the same binary cross-entropy loss (Eq. (\[eq:loss\])).
Relations to Conventional GNN {#sec:advan}
-----------------------------
In our proposed SGGNN model, the similarities among gallery images are served as fusion weights on the graph for nodes’ feature fusion and updating. These similarities are vital for refining the probe-gallery relation features. In conventional GNN [@yan2018spatial; @niepert2016learning] models, the feature fusion weights are usually modeled as a nonlinear function $h(d_i, d_j)$ that measures compatibility between two nodes $d_i$ and $d_j$. The feature updating will be $$\label{eq:gnn}
d_{i}^{(t)} =(1 - \alpha) d_{i}^{(t-1)} + \alpha \sum_{j = 1}^{N} h(d_i, d_j) t_{j}^{(t-1)} \quad \text{for} \ i=1,2,...,N.$$ They lack directly label supervision and are only indirectly learned via back-propagation errors. However, in our case, such a strategy does not fully utilize the similarity ground-truth between gallery images. To overcome such limitation, we propose to use similarity scores $S(g_i, g_j)$ between gallery images $g_i$ and $g_j$ with directly training label supervision to serve as the node feature fusion weights in Eq. (\[eq:refine\_walk\]). Compared with conventional setting of GNN Eq. (\[eq:gnn\]), these direct and rich supervisions of gallery-gallery similarity could provide feature fusion with more accurate information.
Implementation Details {#sec:imp}
----------------------
Our proposed SGGNN is based on ResNet-50 [@he2016deep] pretrained on ImageNet [@deng2009imagenet]. The input images are all resized to $256 \times 128$. Random flipping and random erasing [@zhong2017random] are utilized for data augmentation. We will first pretrain the base Siamese CNN model, we adopt an initial learning rate of 0.01 on all three datasets and reduce the learning rate by 10 times after 50 epochs. The learning rate is then fixed for another 50 training epochs. The weights of linear classifier for obtaining the gallery-gallery similarities is initialized with the weights of linear classifier we trained in the base model pretraining stage. To construct each mini-batch as a combination of a probe set and a gallery set, we randomly sample images according to their identities. First we randomly choose $M$ identities in each mini-batch. For each identity, we randomly choose $K$ images belonging to this identity. Among these $K$ images of one person, we randomly choose one of them as the probe image and leave the rest of them as gallery images. As a result, a $K \times M$ sized mini-batch consists of a size $K$ probe set and a size $K \times (M-1)$ gallery set. In the training stage, $K$ is set to 4 and $M$ is set to 48, which results in a mini-batch size of 192. In the testing stage, for each probe image, we first utilize $l2$ distance between probe image feature and gallery image features by the trained ResNet-50 in our SGGNN to obtain the top-100 gallery images, then we use SGGNN for obtaining the final similarity scores. We will go though all the identities in each training epoch and Adam algorithm [@kingma2014adam] is utilized for optimization.
We then finetune the overall SGGNN model end-to-end, the input node features for overall model are the subtracted features of base model. Note that for gallery-gallery similarity estimation $S(g_i, g_j)$, the rich labels of gallery images are also used as training supervision. we train the overall network with a learning rate of $10^{-4}$ for another 50 epochs and the balancing weight $\alpha$ is set to 0.9.
Experiments
===========
Datasets and Evaluation Metrics
-------------------------------
To validate the effectiveness of our proposed approach for person re-identification. The experiments and ablation study are conducted on three large public datasets.
**CUHK03** [@li2014deepreid] is a person re-identification dataset, which contains 14,097 images of 1,467 person captured by two cameras from the campus. We utilize its manually annotated images in this work.
**Market-1501** [@zheng2015scalable] is a large-scale dataset, which contains multi-view person images for each identity. It consists of 12,936 images for training and 19,732 images for testing. The test set is divided into a gallery set that contains 16,483 images and a probe set that contains 3,249 images. There are totally 1501 identities in this dataset and all the person images are obtained by DPM detector [@felzenszwalb2010object].
**DukeMTMC** [@ristani2016MTMC] is collected from campus with 8 cameras, it originally contains more than 2,000,000 manually annotated frames. There are some extensions for DukeMTMC dataset for person re-identification task. In this paper, we follow the setting of [@zheng2017unlabeled]. It utilizes 1404 identities, which appear in more than two cameras. The training set consists of 16,522 images with 702 identities and test set contains 19,989 images with 702 identities.
We adopt mean average precision (mAP) and CMC top-1, top-5, and top-10 accuracies as evaluation metrics. For each dataset, we just adopt the original evaluation protocol that the dataset provides. In the experiments, the query type is single query.
------------------------------------- ------------ ---------- ---------- ---------- ----------
mAP top-1 top-5 top-10
Quadruplet Loss [@Chen_2017_CVPR] CVPR 2017 - 75.5 95.2 99.2
OIM Loss [@xiao2017joint] CVPR 2017 72.5 77.5 - -
SpindleNet [@zhao2017spindle] CVPR 2017 - 88.5 97.8 98.6
MSCAN [@Li_2017_CVPR] CVPR 2017 - 74.2 94.3 97.5
SSM [@bai2017scalable] CVPR 2017 - 76.6 94.6 98.0
k-reciprocal [@Zhong_2017_CVPR] CVPR 2017 67.6 61.6 - -
VI+LSRO [@zheng2017unlabeled] ICCV 2017 87.4 84.6 97.6 98.9
SVDNet [@Sun_2017_ICCV] ICCV 2017 84.8 81.8 95.2 97.2
OL-MANS [@Zhou_2017_ICCV] ICCV 2017 - 61.7 88.4 95.2
Pose Driven [@Su_2017_ICCV] ICCV 2017 - 88.7 98.6 **99.6**
Part Aligned [@zhao2017deeply] ICCV 2017 - 85.4 97.6 99.4
HydraPlus-Net [@Liuxihui_2017_ICCV] ICCV 2017 - 91.8 98.4 99.1
MuDeep [@qian2017multi] ICCV 2017 - 76.3 96.0 98.4
JLML [@gong2017person] IJCAI 2017 - 83.2 98.0 99.4
MC-PPMN [@mao2018multi] AAAI 2018 - 86.4 98.5 **99.6**
Proposed SGGNN **94.3** **95.3** **99.1** **99.6**
------------------------------------- ------------ ---------- ---------- ---------- ----------
: mAP, top-1, top-5, and top-10 accuracies by compared methods on the CUHK03 dataset [@li2014deepreid].[]{data-label="tab:cuhk"}
------------------------------------- ------------ ---------- ---------- ---------- ----------
mAP top-1 top-5 top-10
OIM Loss [@xiao2017joint] CVPR 2017 60.9 82.1 - -
SpindleNet [@zhao2017spindle] CVPR 2017 - 76.9 91.5 94.6
MSCAN [@Li_2017_CVPR] CVPR 2017 53.1 76.3 - -
SSM [@bai2017scalable] CVPR 2017 68.8 82.2 - -
k-reciprocal [@Zhong_2017_CVPR] CVPR 2017 63.6 77.1 - -
Point 2 Set [@Zhou_2017_CVPR] CVPR 2017 44.3 70.7 - -
CADL [@Lin_2017_CVPR] CVPR 2017 47.1 73.8 - -
VI+LSRO [@zheng2017unlabeled] ICCV 2017 66.1 84.0 - -
SVDNet [@Sun_2017_ICCV] ICCV 2017 62.1 82.3 92.3 95.2
OL-MANS [@Zhou_2017_ICCV] ICCV 2017 - 60.7 - -
Pose Driven [@Su_2017_ICCV] ICCV 2017 63.4 84.1 92.7 94.9
Part Aligned [@zhao2017deeply] ICCV 2017 63.4 81.0 92.0 94.7
HydraPlus-Net [@Liuxihui_2017_ICCV] ICCV 2017 - 76.9 91.3 94.5
JLML [@gong2017person] IJCAI 2017 65.5 85.1 - -
HA-CNN [@li2018harmonious] CVPR 2018 75.7 91.2 - -
Proposed SGGNN **82.8** **92.3** **96.1** **97.4**
------------------------------------- ------------ ---------- ---------- ---------- ----------
: mAP, top-1, top-5, and top-10 accuracies of compared methods on the Market-1501 dataset [@zheng2015scalable].[]{data-label="tab:market"}
Comparison with State-of-the-art Methods
----------------------------------------
### Results on CUHK03 dataset.
The results of our proposed method and other state-of-the-art methods are represented in Table \[tab:cuhk\]. The mAP and top-1 accuracy of our proposed method are 94.3% and 95.3%, respectively. Our proposed method outperforms all the compared methods.
Quadruplet Loss [@Chen_2017_CVPR] is modified based on triplet loss. It aims at obtaining correct orders for input pairs and pushing away negative pairs from positive pairs. Our proposed method outperforms quadruplet loss 19.8% in terms of top-1 accuracy. OIM Loss [@xiao2017joint] maintains a look-up table. It compares distances between mini-batch samples and all the entries in the table. to learn features of person image. Our approach improves OIM Loss by 21.8% and 17.8% in terms of mAP and CMC top-1 accuracy. SpindleNet [@zhao2017spindle] considers body structure information for person re-identification. It incorporates body region features and features from different semantic levels for person re-identification. Compared with SpindleNet, our proposed method increases 6.8% for top-1 accuracy. MSCAN [@li2017learning] stands for Multi-Scale ContextAware Network. It adopts multiple convolution kernels with different receptive fields to obtain multiple feature maps. The dilated convolution is utilized for decreasing the correlations among convolution kernels. Our proposed method gains 21.1% in terms of top-1 accuracy. SSM stands for Smoothed Supervised Manifold [@bai2017scalable]. This approach tries to obtain the underlying manifold structure by estimating the similarity between two images in the context of other pairs of images in the post-processing stage, while the proposed SGGNN utilizes instance relation information in both training and testing stages. SGGNN outperforms SSM approach by 18.7% in terms of top-1 accuracy. k-reciprocal [@Zhong_2017_CVPR] utilized gallery-gallery similarities in the testing stage and uses a smoothed Jaccard distance for refining the ranking results. In contrast, SGGNN exploits the gallery-gallery information in the training stage for feature learning. As a result, SGGNN gains 26.7% and 33.7% increase in terms of mAP and top-1 accuracy.
### Results on Market-1501 dataset.
On Market-1501 dataset, our proposed methods outperforms significantly state-of-the-art methods. SGGNN achieves mAP of 82.8% and top-1 accuracy of 92.3% on Market-1501 dataset. The results are shown in Table \[tab:market\].
HydraPlus-Net [@Liuxihui_2017_ICCV] is proposed for better exploiting the global and local contents with multi-level feature fusion of a person image. Our proposed method outperforms HydraPlus-Net by 15.4 for top-1 accuracy. JLML [@gong2017person] stands for Joint Learning of Multi-Loss. JLML learns both global and local discriminative features in different context and exploits complementary advantages jointly. Compared with JLML, our proposed method gains 17.3 and 7.2 in terms of mAP and top-1 accuracy. HA-CNN [@li2018harmonious] attempts to learn hard region-level and soft pixel-level attention simultaneously with arbitrary person bounding boxes and person image features. The proposed SGGNN outperforms HA-CNN by 7.1% and 1.1% with respect to mAP and top-1 accuracy.
### Results on DukeMTMC dataset.
In Table \[tab:duke\], we illustrate the performance of our proposed SGGNN and other state-of-the-art methods on DukeMTMC [@ristani2016MTMC]. Our method outperforms all compared approaches. Besides approaches such as OIM Loss and SVDNet, which have been introduced previously, our method also outperforms Basel+LSRO, which integrates GAN generated data and ACRN that incorporates person of attributes for person re-identification significantly. These results illustrate the effectiveness of our proposed approach.
----------------------------------- ------------ ---------- ---------- ---------- ----------
mAP top-1 top-5 top-10
BoW+KISSME [@zheng2015scalable] ICCV 2015 12.2 25.1 - -
LOMO+XQDA [@liao2015person] CVPR 2015 17.0 30.8 - -
ACRN [@schumann2017person] CVPRW 2017 52.0 72.6 84.8 88.9
OIM Loss [@xiao2017joint] CVPR 2017 47.4 68.1 - -
Basel.+LSRO [@zheng2017unlabeled] ICCV 2017 47.1 67.7 - -
SVDNet [@Sun_2017_ICCV] ICCV 2017 56.8 76.7 86.4 89.9
Proposed SGGNN **68.2** **81.1** **88.4** **91.2**
----------------------------------- ------------ ---------- ---------- ---------- ----------
: mAP, top-1, top-5, and top-10 accuracies by compared methods on the DukeMTMC dataset [@ristani2016MTMC].[]{data-label="tab:duke"}
Ablation Study
--------------
To further investigate the validity of SGGNN, we also conduct a series of ablation studies on all three datasets. Results are shown in Table \[tab:abl\].
We treat the siamese CNN model that directly estimates pairwise similarities from initial node features introduced in Section \[sec:pro\_for\] as the base model. We utilize the same base model and compare with other approaches that also take inter-gallery image relations in the testing stage for comparison. We conduct k-reciprocal re-ranking [@Zhong_2017_CVPR] with the image visual features learned by our base model. Compared with SGGNN approach, The mAP of k-reciprocal approach drops by 4.3%, 4.4%, 3.5% for Market-1501, CUHK03, and DukeMTMC datasets. The top-1 accuracy also drops by 0.8%, 3.1%, 1.2% respectively. Except for the visual features, base model could also provides us raw similarity scores of probe-gallery pairs and gallery-gallery pairs. A random walk [@bai2017scalable] operation could be conducted to refine the probe-gallery similarity scores with gallery-gallery similarity scores with a closed-form equation. Compared with our method, The performance of random walk drops by 3.6%, 4.1%, and 2.2% in terms of mAP, 0.8%, 3.0%, and 0.8% in terms of top-1 accuracy. Such results illustrate the effectiveness of end-to-end training with deeply learned message passing within SGGNN. We also validate the importance of learning visual feature fusion weight with gallery-gallery similarities guidance. In Section \[sec:advan\], we have introduced that in the conventional GNN, the compatibility between two nodes $d_i$ and $d_j$, $h(d_i, d_j)$ is calculated by a non-linear function, inner product function without direct gallery-gallery supervision. We therefore remove the directly gallery-gallery supervisions and train the model with weight fusion approach in Eq. (\[eq:gnn\]) , denoted by *Base Model + SGGNN w/o SG*. The performance drops by 1.6%, 1.6%, and 0.9% in terms of mAP. The top-1 accuracies drops 1.7%, 2.6%, and 0.6% compared with our SGGNN approach, which illustrates the importance of involving rich gallery-gallery labels in the training stage.
To demonstrate that our proposed model SGGNN also learns better visual features by considering all probe-gallery relations, we evaluate the re-identification performance by directly calculating the $l_2$ distance between different images’ visual feature vectors outputted by our trained ResNet-50 model on three datasets. The results by visual features learned with base model and the conventional GNN approach are illustrated in Table \[tab:feature\]. Visual features by our proposed SGGNN outperforms the compared base model and conventional GNN setting significantly, which demonstrates that SGGNN also learns more discriminative and robust features.
Sensitivity Analysis
--------------------
We tried training our SGGNN with different $K$ and also testing with different top-$K$ choices (Table \[table:sense\], rows 2-5). Results show that higher top-$K$ slightly increases accuracy but also increases computational cost.
----------------------------------------------- ---------- ---------- ---------- ---------- ---------- ---------- --
mAP top-1 mAP top-1 mAP top-1
Base Model 76.4 91.2 88.9 91.1 61.8 78.8
Base Model + k-reciprocal [@Zhong_2017_CVPR] 78.5 91.5 89.9 92.2 64.7 79.9
Base Model + random walk [@bai2017scalable] 79.2 91.5 90.2 92.3 66.0 80.3
Base Model + SGGNN w/o SG 81.2 90.6 92.7 93.6 67.3 80.5
Base Model + SGGNN **82.8** **92.3** **94.3** **95.3** **68.2** **81.1**
----------------------------------------------- ---------- ---------- ---------- ---------- ---------- ---------- --
: Ablation studies on the Market-1501 [@zheng2015scalable], CUHK03 [@li2014deepreid] and DukeMTMC [@ristani2016MTMC] datasets.[]{data-label="tab:abl"}
--------------------------- ---------- ---------- ---------- ---------- ---------- ---------- --
mAP top-1 mAP top-1 mAP top-1
Base Model 74.6 90.4 87.6 91.0 60.3 77.6
Base Model + SGGNN w/o SG 75.4 90.4 87.7 91.5 61.7 78.1
Base Model + SGGNN **76.7** **91.5** **88.1** **93.6** **64.6** **79.1**
--------------------------- ---------- ---------- ---------- ---------- ---------- ---------- --
: Performances of estimating probe-gallery similarities by $l_2$ feature distance on the Market-1501 [@zheng2015scalable], CUHK03 [@li2014deepreid] and DukeMTMC [@ristani2016MTMC] datasets.[]{data-label="tab:feature"}
----------- ----- ---------- ----- ------ ------- ------ ------- ------ -------
Top-$K$ $K$ $\alpha$ $t$ mAP top-1 mAP top-1 mAP top-1
Top-$100$ $4$ 0.9 1 82.8 92.3 94.3 95.3 68.2 81.1
Top-100 $3$ 0.9 1 82.0 91.7 94.1 95.2 68.2 80.8
Top-100 $5$ 0.9 1 82.1 91.8 94.2 95.2 68.0 80.6
Top-50 $4$ 0.9 1 80.7 91.3 93.7 95.1 66.6 79.8
Top-150 $4$ 0.9 1 83.6 92.0 94.5 95.3 71.8 83.5
Top-100 $4$ 0.9 2 82.9 91.3 95.1 96.1 68.9 81.7
Top-100 $4$ 0.9 3 81.3 89.3 95.4 96.0 69.0 81.9
Top-100 $4$ 0.5 1 79.8 91.4 92.4 94.2 66.6 81.0
Top-100 $4$ 0.95 1 82.8 92.8 94.3 95.4 68.3 81.6
----------- ----- ---------- ----- ------ ------- ------ ------- ------ -------
: Performances of different $K$ and top-$K$ choices.[]{data-label="table:sense"}
Conclusion
==========
In this paper, we propose Similarity-Guided Graph Neural Neural to incorporate the rich gallery-gallery similarity information into training process of person re-identification. Compared with our method, most previous attempts conduct the updating of probe-gallery similarity in the post-process stage, which could not benefit the learning of visual features. For conventional Graph Neural Network setting, the rich gallery-gallery similarity labels are ignored while our approach utilized all valuable labels to ensure the weighted deep message fusion is more effective. The overall performance of our approach and ablation study illustrate the effectiveness of our proposed method.
Acknowledgements
================
This work is supported by SenseTime Group Limited, the General Research Fund sponsored by the Research Grants Council of Hong Kong (Nos. CUHK14213616, CUHK14206114, CUHK14205615, CUHK14203015, CUHK14239816, CUHK419412, CUHK14207814, CUHK14208417, CUHK14202217), the Hong Kong Innovation and Technology Support Program (No. ITS/121/15FX).
[^1]: Hongsheng Li is the corresponding author.
|
---
abstract: 'In inverse reinforcement learning (IRL), given a Markov decision process (MDP) and a set of demonstrations for completing a task, the objective is to learn a reward function to explain the demonstrations. However, it is challenging to apply IRL in tasks where a proper *memory structure* is the key to complete the task. To address this challenge, we develop an iterative algorithm that alternates between a *task inference module* that infers the high-level memory structure of the task and a *reward learning module* that learns a reward function with the inferred memory structure. In each iteration, the task inference module produces a series of queries to be answered by the demonstrator. Each query asks whether a sequence of high-level events leads to the completion of the task. The demonstrator then provides a demonstration executing the sequence to answer each query. After the queries are answered, the task inference module returns a *hypothesis deterministic finite automaton* (DFA) encoding the high-level memory structure to be used by the reward learning. The reward learning module incorporates the DFA states into the MDP states and creates a *product automaton*. Then the reward learning module proceeds to learn a Markovian reward function for this product automaton by performing *deep maximum entropy* IRL. At the end of each iteration, the algorithm computes an optimal policy with respect to the learned reward. This iterative process continues until the computed policy leads to *satisfactory* performance in completing the task. The experiments show that the proposed algorithm outperforms three IRL baselines in task performance.'
author:
- 'Farzan Memarian[^1], Zhe Xu, Bo Wu, Min Wen, Ufuk Topcu'
bibliography:
- 'arxiv.bib'
title: 'Active Task-Inference-Guided Deep Inverse Reinforcement Learning'
---
Introduction
============
In inverse reinforcement learning (IRL), [@ng2000algorithms; @abbeel2004apprenticeship; @ziebart2008maximum], given a reward free MDP and a set of demonstrations, the goal is to infer a reward function that can best explain the demonstrations.
Most existing IRL algorithms learn a Markovian reward function (i.e., a memoryless reward function that is independent from the history of visited states). However, it is challenging to apply IRL to tasks with complicated *memory structures*, as the whole history of previously visited states may need to be considered to determine the reward at the current state.
Another challenge for IRL is the dependence of the effectiveness of the learned reward function on the quality of the demonstrations provided beforehand. To address this challenge, active IRL methods [@lopes2009active; @odom] are proposed to actively query the demonstrator for samples at specific states or regions in the feature space. However, for tasks with complicated memory structures, queries on specific states or regions may not suffice as the high-level *event sequences* are more critical and should be queried instead for recovering the reward function.
In this paper, we develop an iterative algorithm that alternates between a *task inference module* that infers the high-level memory structure of the task and a *reward learning module* that learns a reward function with the inferred memory structure. In each iteration, the task inference module produces a series of queries to be answered by the demonstrator. Each query is a sequence of high-level events, and the demonstrator must execute the sequence in the environment and answer the query based on the *binary* outcome of whether the task is completed. The reward learning module incorporates the DFA states into the MDP states and creates a *product automaton*. Then the reward learning module proceeds to learn a Markovian reward function for this product automaton by performing *deep maximum entropy IRL*. At the end of each iteration, the algorithm computes an optimal policy with respect to the learned reward function. This iterative process continues until the computed policy leads to a *satisfactory* performance in completing the task.
In the reward learning module, we adapt the deep maximum entropy IRL algorithm [@wulfmeier2015maximum] to the proposed framework, and use a convolutional neural network (CNN) to represent the reward function. The reward function takes as input the states of the MDP and the DFA. Essentially, by using a CNN, we avoid the need for manual feature engineering. Moreover, CNNs are able to learn the most salient features corresponding to successful task completion. Hence the learned reward function has the potential for strong generalization performance.
To test the validity of the proposed algorithm, we create a *task-oriented navigation* environment. The experiments show that our algorithm can successfully infer the underlying memory structure of the task and use it to guide IRL. The training and test performances of the proposed algorithm outperform three baselines, i.e., a memoryless IRL method, a memory-based behavioral cloning method, and a memory augmented IRL method.
Here we list the contributions of the paper: **1)** We offer an algorithm for actively inferring the memory structure in the form of a DFA. **2)** We propose a method for inducing high-level task information encoded in the DFA in the IRL loop. The DFA tracks the progress in the execution of the task. **3)** We use deep reward learning in the proposed IRL framework, which improves generalizability of the proposed algorithm.
Background
==========
Generally, the problem of inverse reinforcement learning (IRL) can be described as follows: Given a reward-free MDP $\mathcal{M}$, and a set of demonstration trajectories $D$, infer a reward function $R$ that can optimally interpret the demonstrations in some pre-specified way. Different works on IRL can be distinguished by the following three aspects: First, the reward parameterization; second, the way to generate a policy with a given reward function; third, the interpretation of the demonstrations using the output policy. In this section, we briefly describe the maximum entropy inverse reinforcement learning (MaxEnt IRL) algorithm [@ziebart2008maximum]. We show the limitations of this algorithm for learning behaviors for complicated task structures, which motivate the proposed algorithm.
We adopt the following settings on environment model and demonstrations that are commonly used in IRL. The environment is modeled as a reward-free MDP $\mathcal{M} = \langle S, A,T, \rho, \eta \rangle$ where $S$ is a state space; $A$ is an action space; $T: S \times A \rightarrow \mathcal{D}(S)$ (where $\mathcal{D}(S)$ is the set of all probability distributions over $S$) is the transition function; $\rho \in \mathcal{D}(S)$ is an initial distribution over $S$; and $\eta: S\rightarrow E$ is a labeling function with $E$ as a finite set of events. Let $D = \{ \tau_1, \ldots, \tau_N\}$ be a set of demonstration trajectories, where $\tau_i = \{(s_{i,0}, a_{i,0}), \ldots, (s_{i, n_i}, a_{i, n_i})\}$ for all $i = 0, \ldots, n_i$. We define the task to be learned by a mapping $\mathcal{T}:E^*\rightarrow \{0,1\}$, where $E^*$ is the Kleene star of $E$, and $\mathcal{T}(\omega)=1$ denotes that a sequence of events $\omega\in E^*$ can complete the task.
The reward function is assumed to have a memory structure encoded by a deterministic finite automaton (DFA). A DFA $\mathcal{A}$ is a tuple $\langle Q_{\mathcal{A}}, \Sigma, \delta, q_0, F \rangle$ where $Q_{\mathcal{A}}$ is a set of states; $\Sigma$ is a set of input symbols (also called the alphabet); $\delta: Q_{\mathcal{A}} \times \Sigma \rightarrow Q_{\mathcal{A}}$ is a deterministic transition function; $q_0 \in Q_{\mathcal{A}}$ is the initial state; $F \subseteq Q_{\mathcal{A}}$ is a set of final states (also called *accepting* states). Given a finite sequence of input symbols $w = \sigma_0, \sigma_1, \ldots, \sigma_{k-1}$ in $\Sigma^k$ for some $k \in \mathbb{N}^+$, the DFA $\mathcal{A}$ generates a unique sequence of $k+1$ states $\tau_{\mathcal{A}} = q_0, q_1, \ldots, q_k$ in $Q_{\mathcal{A}}^{k+1}$ such that for each $t = 1, \ldots, k$, $q_t = \delta(q_{t-1}, \sigma_{t-1})$. We denote the last state $q_k$ by taking the sequence $w$ of inputs from $q_0$ as $\underline{\delta}(q_0, w)$. $w \in \Sigma^*$ is *accepted* by $\mathcal{A}$ if and only if $\underline{\delta}(q_0, w) \in F$. Let $\mathcal{L}(\mathcal{A}) \subseteq \Sigma^*$ be all the finite sequences of input symbols that are accepted by $\mathcal{A}$, which is also referred to as the *accepted language* of $\mathcal{A}$.
Maximum Entropy IRL
-------------------
In the original MaxEnt IRL algorithm [@ziebart2008maximum], the reward function is parameterized as a linear combination of a given set of feature functions. In other words, given a set of features $\{f_1, \ldots, f_K\}$ where $f_k: S \times A \rightarrow \mathbb{R}$ for $k = 1, \ldots, K$, the reward function $R_\theta$ is parameterized by $\theta = [\theta_1, \ldots, \theta_K]^\intercal$ and $R_\theta(s,a) = \sum_{k=1}^K \theta_k f_k(s,a)$. The basic assumption is that the expected total reward over the distribution of trajectories is the same as the empirical average reward over demonstration trajectories. With the principle of maximum entropy, it can be derived that the probability of generating any (finite-length) trajectory $\tau = s_0, a_0, \ldots, s_{|\tau|}, a_{|\tau|}$ is proportional to the exponent of the total reward of $\tau$: $$Pr(\tau | R_\theta) \propto \exp \frac{1}{|\tau|} \sum_{i = 0}^{|\tau|} R_\theta(s_i, a_i).
\label{eqn:maxentirl_finite}$$
While linear parameterization is commonly used in IRL literature [@abbeel2004apprenticeship; @ratliff2006maximum; @neu2007apprenticeship; @klein2012inverse], it suffers from several drawbacks. On the one hand, it requires human knowledge to provide properly designed reward features, which can be labor-intensive; on the other hand, if the given features fail to encode all the essential requirements to generate the demonstrations, there is no way to recover this flaw by learning from demonstrations. One way to deal with this problem is to use nonlinear reward models such as Gaussian process [@levine2011nonlinear], decision trees [@levine2010feature] or neural network to automatically construct reward features from expert demonstrations.
Active Task-Inference-Guided Deep Inverse Reinforcement\
Learning {#sec:TODIRL}
========================================================
In this section, we introduce the active task-inference-guided deep IRL (ATIG-DIRL), which iteratively infers the memory structure of tasks and incorporates the memory structure into reward learning.
Overview
--------
ATIG-DIRL, as illustrated in Algorithm \[alg:overall\], consists of a task inference module and a reward learning module. The task inference module utilizes L\* learning [@angluin1987learning], an active automaton inference algorithm, as the template to iteratively infer a DFA from queries and counterexamples. The inferreded DFA encodes the high-level temporal structure to help IRL recover reward functions. Following L\* learning, the task inference module generates two kinds of queries with Boolean answers, namely the *membership* query and the *conjecture* query. A membership query asks whether a sequence of high-level events can lead to task completion. We defer the details of answering membership queries to Sec. \[subsection:Constructing Hypothesis DFAs\]. After a number of membership queries, the inference engine outputs a hypothesis DFA and a set of corresponding demonstrations for answering the membership queries (line \[state:task-inferrence-module\] in Alg. \[alg:overall\]). The task inference module then asks a conjecture query about whether the hypothesis DFA can help recover a satisfactory reward function (line \[state:rewardlearningmodule\] in Alg. \[alg:irl\]). The conjecture query is to be answered by the reward learning module and the details are in Sec. \[sec:reward-learning-module\]. If the answer to the conjecture query is $True$, both the automaton inference process and the IRL are finished. Otherwise, the answer is $False$ and there exists a counterexample in the form of a sequence of high-level events to illustrate the difference between the conjectured DFA and the memory structure of the task. Such a counterexample will trigger a new iteration with next round of membership queries.
A reward-free MDP $\mathcal{M} = \langle S, A, T, \rho , \eta \rangle$, stopping threshold $\kappa$ A hypothesis DFA $\mathcal{A} = \langle Q_{\mathcal{A}}, \Sigma, \delta, q_0, F \rangle$, a reward network $R_\theta: S \times Q_{\mathcal{A}} \rightarrow \mathbb{R}$ and a policy $\pi_\theta: S \times Q_{\mathcal{A}} \rightarrow \mathcal{D}(A)$
Initialize success ratio $(\beta)$ as 0, counterexample $(CE)$ as null, set of demonstrations $(D)$ as empty set $\mathcal{A}$, $D$, $\gets$ TaskInferenceModule($\mathcal{M}$, $CE$) \[state:task-inferrence-module\] $R_\theta, \pi_\theta$, $\beta$, $CE$ $\gets$ RewardLearningModule($\mathcal{A}$, $\mathcal{M}$, $D$, $\kappa$) \[state:rewardlearningmodule\]
A reward-free MDP $\mathcal{M}$, a hypothesis DFA $\mathcal{A}$, a set of demonstrations $D = \{\tau_1, \ldots, \tau_N \}$, threshold for producing counterexamples $\kappa$ Reward network $R_\theta: S \times Q_{\mathcal{A}} \rightarrow \mathbb{R}$, policy $\pi_\theta: S \times Q_{\mathcal{A}} \rightarrow \mathcal{D}(A)$, success ratio ($\beta$), counterexample ($CE$) number of iterations between updating the success ratio $N$, stopping threshold $\epsilon$ Initialize the reward network parameter $\theta_0$; Stop $\leftarrow$ False; $t \leftarrow 0$ Use the parameter vector $\theta_t$ at the current time step $t$ to compute $Q_{\theta_t}$ and the policy $\pi_{\theta_t}$ via Eq. \[eqn:soft\_Q\] and Eq. \[eqn:soft\_pi\] Compute $\frac{\partial Q_\theta}{\partial \theta} \mid_{\theta = \theta_t}$ and $\frac{\partial \pi_\theta}{\partial \theta} \mid_{\theta = \theta_t}$ via Eq. \[eqn:dQ\] and Eq. \[eqn:dpi\] Compute $\frac{\partial L_D(\theta_t)}{\partial \theta_t}$ via Eq. \[eqn:dlD\] Update $\theta$: $\theta_{t+1} \gets \theta_t + \alpha_t \frac{\partial L_D(\theta)}{\partial \theta} \mid_{\theta = \theta_t}$ Compute the current success ratio $\beta_t$ at time step $t$ using Monte Carlo evaluation \[state:success-ratio\] **if** $\beta_t - \beta_{t-N} \leq \epsilon$ **then** *Stop* $\leftarrow$ *True* \[state:stopping\] $t\gets t+1$ Produce a $CE$ using Monte Carlo simulation \[state:counterexample\] $\beta\gets\beta_t$
Task Inference Module {#subsection:Constructing Hypothesis DFAs}
---------------------
To generate a hypothesis DFA $\mathcal{A} = \langle Q_{\mathcal{A}}, \Sigma, \delta, q_0, F \rangle$ where $\Sigma = E$, ATIG-DIRL produces a number of membership queries. Each membership query asks whether following a event sequence $\omega\in E^*$ leads to task completion, i.e. $\mathcal{T}(\omega)=1$. The task $\mathcal{T}$ is unknown, but given a sequence $\omega\in E^*$, one can observe $\mathcal{T}(\omega)$ from the environment. To answer a membership query, we rely on a demonstrator to make a demonstration in the MDP environment and generate a state sequence $\lambda\in S^*$ where $\eta(\lambda)=\omega$. Then if the event sequence $\omega$ completes the task, i.e. $\mathcal{T}(\omega)=1$, the answer to this membership is $True$, otherwise the answer is $False$. After answering the membership queries, ATIG-DIRL will generate a hypothesis DFA $\mathcal{A}$ following procedures in [@angluin1987learning].
After obtaining a hypothesis DFA $\mathcal{A}$, ATIG-DIRL asks a conjecture query that whether $\mathcal{A}$ is sufficient to recover the reward function. To answer this query, we follow the procedure introduced in Sec. \[sec:reward-learning-module\], which is summarized in Alg. \[alg:irl\].
**Demonstration trajectories.** When the task inference module asks a query $\omega$, if $\mathcal{T}(\omega)=1$, it means the query encodes an event sequence that leads to task completion. The demonstrator will then produce several demonstrations following the same event sequence and adds them to the set of demonstrations.
Reward Learning Module {#sec:reward-learning-module}
----------------------
This module is concerned with learning a reward function using the hypothesis DFA. Although previous deep MaxEnt IRL methods [@wulfmeier2015maximum; @finn2016guided; @wulfmeier2016watch; @wulfmeier2017large] can construct reward features automatically from demonstrations, they suffer from a fundamental limitation that the learned reward function is Markovian. As a result, the learned policy has to be independent from the history of states, which does not suffice for tasks with complicated memory structures. To address this issue, we propose a new maximum entropy deep IRL algorithm (which is inspired by the work in [@wulfmeier2015maximum]), as described in Alg. \[alg:irl\].
The key idea is to use the hypothesis DFA $\mathcal{A}$ and the MDP to create a product automaton and then learn a reward function over the state space of this product automaton. We define the product automaton as follows:
**Product automaton.** Let $\mathcal{M} = \langle S, A,T, \rho, \eta \rangle$ be a reward-free MDP and $\mathcal{A} = \langle Q_{\mathcal{A}}, \Sigma, \delta, q_0, F \rangle$ be a DFA. The product automaton $\mathcal{M}_{\rm{p}}:=\mathcal{M}\otimes\mathcal{A}=(Z,Z_0,\delta_{\rm{p}},\eta_{\rm{p}},F_{\rm{p}} )$ is a tuple such that, $Z=S\times Q_{\mathcal{A}}$ is a finite set of states. $Z_0$ is the initial set of states where for each $z_0=(s,q)\in Z_0$, $s\in S_0$, $q=\delta(q_0, \eta(s))$. $\delta_{\rm{p}}$ is the transition function of the product automaton defined as $$\delta_{\rm{p}}((s,q),a,(s',q'))=
\begin{cases}
T(s,a,s') ~~\mbox{if}~q'=\delta(q, \eta(s'));\\
0 ~~~~~~~~~~~~~~~~\mbox{otherwise};
\end{cases}$$ $\eta_{\rm{p}}((s,q))=\{q\}$ is a labeling function; and $F_{\rm{p}}=S\times F$ is a finite set of accepting states.
We apply the proposed IRL algorithm on the product automaton. As a result, the reward depends on both the current state $s$ in $\mathcal{M}$ and the current state $q$ in $\mathcal{A}$, which together form the current state of the product automaton $z\in Z$. DFA states can be considered as different stages in task implementation. Since the DFA state is an input to the reward function, the learned reward and the corresponding induced policy would be a function of the stage of the task.
Unlike MaxEnt IRL where the reward function is modeled as a linear combination of pre-specified features, we model the reward function as a neural network. The set of reward parameters are represented by $\theta$, which is the weight vector of the reward network. The objective is to maximize the posterior probability of observing the demonstration trajectories and reward parameters $\theta$ given a reward structure: $$\displaystyle
L(\theta) := \log Pr(D, \theta | R_\theta))
= \underbrace{\log Pr(D | R_\theta)}_{L_D} + \underbrace{\log P(\theta)}_{L_\theta}
\label{eqn:L}$$ $L_D$ is the log likelihood of the demonstration trajectories in $D$ given the reward function $R_\theta$. $L_\theta$ can be interpreted as either the logarithm of the prior distribution $P(\cdot)$ at $\theta$ or as a differentiable regularization term on $\theta$. In this work we assume a uniform prior distribution over $\theta$, so what remains is the maximization of $L_D$.
Since we apply IRL on the product automaton, the demonstration trajectories are projected onto the state space of the product automaton, i.e. $\tau_i = \{(z_{i,0}, a_{i,0}), \ldots, (z_{i, n_i}, a_{i, n_i})\}$, where $z_{i,j} = (s_{i,j},q_{i,j})$. Let $\pi_\theta$ be the policy corresponding to $R_\theta$, then $L_D$ can be expressed as $$L_D = \sum_{\tau_i \in D} \sum_{j=0}^{n_i - 1} \log \pi_\theta(a_{i,j} | z_{i,j}) + C,
\label{eqn:LD}$$ where $C$ is a constant that is dependent on $D$ and the transition dynamics $\delta_p$ of the product automaton.
The computation of $\pi_\theta$ given $R_\theta$ is essentially a maximum entropy reinforcement learning problem [@zhou2018infinite]. It can be proved [@zhou2018infinite] that for any $R_\theta(z,a)$, there exists a unique function $Q_\theta(z,a)$ which is the fixed point of $$\begin{aligned}
Q_\theta(z,a) &= R_\theta(z,a) \\ &+ \gamma \sum_{z' \in Z} \delta_p (z' | z , a) \log \sum_{a'} \exp(Q_\theta(z', a')),
\end{aligned}
\label{eqn:soft_Q}$$ where $\gamma$ is the discount factor which is a hyper-parameter. Note that $Q_\theta(z,a)$ is an implicit function of $\theta$, as $R_\theta(z,a)$ is parametrized by $\theta$ and $Q_\theta(z,a)$ is derived from $R_\theta(z,a)$ by Eq. \[eqn:soft\_Q\]. The policy $\pi_\theta$ can be derived from $Q_\theta$ as shown in the equation below $$\begin{aligned}
\pi_\theta(z | a) = \frac{Q_\theta(z,a)}{\sum_{a'} \exp(Q_\theta(z, a'))}.
\end{aligned}
\label{eqn:soft_pi}$$ To find the optimal $\theta$, we perform gradient ascent using the gradient of $L_D$ with respect to $\theta$ expressed as $$\begin{aligned}
\frac{\partial L_D}{\partial \theta}
=& \frac{\partial}{\partial \theta} \sum_{i=1}^N \sum_{l=0}^{n_i} \log \pi_\theta(z_{i,l}, a_{i,l})
\\= &\sum_{i=1}^N \sum_{l=0}^{n_i} \Big( \frac{\partial Q_\theta(z_{i,l}, a_{i,l})}{\partial \theta} - \sum_{a'} w_\theta(z_{i,l}, a') \Big).
\end{aligned}
\label{eqn:dlD}$$ where $w_\theta(z,a) := \pi_\theta(a | z) \frac{\partial Q_\theta(z,a)}{\partial \theta}$ for any $z\in Z, a \in A$. To compute the right hand side of Eq.\[eqn:dlD\], we compute the gradient of $\pi_\theta$ and $Q_\theta$ as $$\begin{aligned}
&\frac{\partial Q_\theta(z,a)}{\partial \theta}
= \frac{\partial R_\theta(z,a)}{\partial \theta} \\ &+ \gamma \sum_{z'} T(z' | z, a) \sum_{a'} \pi_\theta(a' | z') \frac{\partial Q_\theta(z',a')}{\partial \theta},
\end{aligned}
\label{eqn:dQ}$$
$$\begin{aligned}
\frac{\partial \pi_\theta(a | z)}{\partial \theta} =& z_\theta(z,a) - \pi_\theta(a | z) \sum_{a'} w_\theta(z, a').
\end{aligned}
\label{eqn:dpi}$$
Since $\gamma \in (0,1)$, it can be shown that for any $\theta$, there exists a unique solution $\frac{\partial Q_\theta(z,a)}{\partial \theta}$ to Eq. \[eqn:dQ\]. Therefore, there is also a unique solution $\frac{\partial \pi_\theta(a | z)}{\partial \theta}$ to Eq. \[eqn:dpi\].
Once we have performed a gradient ascent step using Eq.\[eqn:dlD\] to update the weight vector $\theta$ of the neural network, we have automatically updated the reward network $R_\theta$.
**Monte Carlo evaluation.** Evaluating the task performance of a reward network $R_\theta$, amounts to computing the *success ratio* of the optimal policy $\pi_\theta$ for $R_\theta$. We use the Monte Carlo approach to empirically compute the success ratio of the policy $\pi_\theta$. Concretely, we use $\pi_\theta$ to produce a set of state sequences $\lambda$, convert the state sequences to event sequences, i.e, $\eta(\lambda)=\omega$, and then observe $\mathcal{T}(\omega)$. The ratio of sequences with $\mathcal{T}(\omega)=1$ to the total number of sequences yields the success ratio (line \[state:success-ratio\] in Alg. \[alg:irl\]).
**Stopping criterion.** After every $N$ iterations of gradient ascent ($N$ is hyper-parameter), the module evaluates the success rate of the computed optimal policy $\pi_\theta$ for $R_\theta$, and stops the iterations once the success rate stops changing significantly according to a pre-specified threshold $\epsilon$ (line \[state:stopping\] in Alg.\[alg:irl\]).
**Counterexamples.** After we finish the iterations of gradient ascent, if the success ratio of the optimal policy $\pi_\theta$ for the obtained reward network $R_\theta$ is smaller than a threshold $\kappa$, then the module produces a counterexample to be used by the task inference module in the next iteration of Alg. \[alg:overall\]. To produce the counterexample, we apply Monte Carlo simulations and find an $\omega$ such that $\mathcal{T}(\omega) \neq \xi(\omega, \mathcal{A})$ where $\xi(\omega, \mathcal{A})$ is $1$ when $\omega$ is accepted by the DFA $\mathcal{A}$ and is $0$ otherwise (line \[state:counterexample\] in Alg. \[alg:overall\]).
Related Work
============
Recently, there have been interesting works on IRL with task information. The first attempt to incorporate task evaluation into IRL was to augment the demonstrations with evaluation of their task performance. [@el2016score] and [@burchfiel2016distance] augmented each demonstration trajectory with a score rated by experts. The Boolean labels and the continuous scores can be used to train a classification or a regression model to evaluate policies. Their experiment results showed that such data augmentation helps reduce the number of demonstration. But since the task is not explicitly defined, the learned policy evaluation model may be neither reliable nor interpretable. [@pan2018human] assumed that the experts provide a set of subgoal states for each demonstration. However, the learning agent does not understand how or why the demonstrator picks this set of critical subgoal states, especially if the number of subgoals are inconsistent over different demonstrations. Though the robot may recognize some similar states using the learned reward features in a new environment, it cannot tell if all of previous subgoals are still necessary or if they should be executed in the same order. With our method, the agent can search for a sequence of subgoals in the extended state space $S \times Q_{\mathcal{A}}$ that implements the task, which may not be necessarily the same as shown in training environments.
Several work has been done on policy learning with assumptions about the task structure. [@niekum2012learning] and [@michini2015bayesian] use Bayesian inference to segment unstructured demonstration trajectories. [@shiarlis2018taco] assumed that the expert performs a given sequence of (symbolic) subtasks in each demonstration. They solve the problem of temporal alignment for the demonstrations and policy learning for each subtask simultaneously. Another related work is hierarchical IRL (HIRL) [@krishnan2016hirl], where they assume each demonstration trajectory corresponds to a set of subtasks. The subtasks are separated by critical “transition states", i.e., states where transitions happen consistently across all the demonstrations. Transitions are defined based on changes in local linearity with respect to a Kernel function. Once the transition states are recognized, for each trajectory, the state space is augmented with features that capture the visitation history of the transition states. HIRL suffers from three limitations compared to our proposed method. First, the assumption that subtasks correspond to states where local linearity changes w.r.t a kernel function, is limited to certain classes of problems and hence HIRL may not be able to confidently detect the transition states. Second, it assumes that the task can always be decomposed into a sequence of subtasks. If a task requirement is such that there are two possible sequences of subgoals that can be followed to reach the final winning state, HIRL is unable to learn a decomposition for such a task. Third, their method does not benefit from the use of deep neural networks and automatic feature learning.
Experiments {#sec:experiment}
===========
We create the *task-oriented navigation* environment, which is a simulated environment that can model various navigation tasks. Different objects are present in the environment, such as {building, grass, tree, rock, barrel and tile}. A region is defined as any $3\times 3$ square neighborhood in the environment. Each region belongs to one of the types defined in Table. \[tab:region-def\]. Region types are used to define navigation tasks as we will see later. Visiting a region means visiting the center block of that region. Fig. \[fig:airsim-grid\] depicts one instantiation of the environment used as the training environment for our experiments.
![Training environment for task-oriented navigation.[]{data-label="fig:airsim-grid"}](./pics/grid-sim-quad.png){width=".5\linewidth"}
**Agent.** The agent is an aerial vehicle that can navigate in the environment. At each time step, it is located on top of an environment block, and it can choose either of the following actions {“up”, “down”, “left”, “right”}, and move one block in that direction.
**The navigation task.** For the rest of this section, we consider the following navigation task as the test bed for the experiments: {visit $R_0$, $R_1$, $R_2$, $R_3$, $R_0$ in this order. Any other order leads to failure}. Fig. \[fig:underlying-dfa\] depicts the DFA encoding this task. The DFA states are defined in Table \[tab:dfa-states\]. The goal of ATIG-DIRL is to infer an equivalent DFA and use it for reward learning. We have used the libalf library [@bollig2010libalf] to infer the DFA.
Type-0 region ($R_0$) a region with more than $6$ buildings
----------------------- ---------------------------------------
Type-1 region ($R_1$) a region with more than $6$ trees
Type-2 region ($R_2$) a region with more than $6$ barrels
Type-3 region ($R_3$) a region with more than $6$ stones
Irrelevant region none of the above
: Definition of region types.[]{data-label="tab:region-def"}
=\[fill=none, draw=black, text=black\] =\[fill=none, draw=none, text=black\]
(q0) at (0,0) [$q_0$]{}; (q1) \[right of=q0\] at (1,1.8) [$q_1$]{}; (q2) \[right of=q0\] at (4,1.8) [$q_2$]{}; (q3) \[right of=q0\] at (6,0) [$q_3$]{}; (q4) \[right of=q0\] at (4,-1.8) [$q_4$]{}; (q5) \[right of=q0\] at (1,-1.8) [$q_5$]{}; (qf) \[right of=q0\] at (2.5,0) [$q_f$]{};
(q0) edge \[bend left\] node [$0$]{} (q1) (q0) edge \[\] node [$1,2,3$]{} (qf) (q1) edge \[\] node [$1$]{} (q2) (q1) edge \[\] node \[above, sloped\] [$0,2,3$]{} (qf)
(q2) edge \[bend left\] node [$2$]{} (q3) (q2) edge node \[below, sloped\] [$0,1,3$]{} (qf)
(q3) edge \[bend left\] node \[below, sloped\] [$3$]{} (q4) (q3) edge \[\] node [$0,1,2$]{} (qf)
(q4) edge \[\] node \[\] [$0$]{} (q5) (q4) edge \[\] node \[below, sloped\] [$1,2,3$]{} (qf);
**$q_0$** None of $R_0, R_1, R_2, R_3$ have been visited
----------- -----------------------------------------------------------------------------------------------
**$q_1$** Only $R_0$ has been visited
**$q_2$** $R_0, R_1$ have been visited in this order
**$q_3$** $R_0, R_1, R_2$ have been visited in this order
**$q_4$** $R_0, R_1, R_2, R_3$ have been visited in this order
**$q_5$** $R_0, R_1, R_2, R_3, R_0$ have been visited in this order
**$q_f$** Failure state. Any scenario other than above transits the DFA into $q_f$ (an absorbing state)
: States of the DFA (depicted at Fig. \[fig:underlying-dfa\]) encoding the memory structure of the underlying task. []{data-label="tab:dfa-states"}
**Reward network.** The input to the convolutional layers of the reward network is the $7\times7$ neighborhood of the agent in the environment. Each block in the neighborhood is represented by a one hot encoded vector, specifying the object that occupies that block. There are 6 objects, so the input to convolutional layers is a $7\times7\times6$ tensor. The network has two convolutional layers, the first layer has $6$ kernels and the second layer has $8$ kernels, all kernels are of size $2\times2$ with a stride of $1$. After the convolutions, there is a flattening layer. And this is where the DFA state and the action are provided as input, by appending them to the output of the flattening layer. The DFA state and the action are represented by a one-hot-encoded vectors respectively of length 10 and 4. Note that the DFA has 7 states, but we use zero padding to make the one-hot-encoded vector be of size 10. Then there are two fully connected layers of size $214$ and $50$, and the output layer has 1 neuron. Each hidden layer is followed by a ReLU layer. **Baselines.** We implemented three baseline methods to compare with ATIG-DIRL: *memoryless IRL*, *IRL augmented with information bits (IRL-IB)* and *memory-based behavioural cloning (BC)*. The memoryless IRL is a basic deep MaxEnt IRL method [@wulfmeier2015maximum] that learns a reward function as a function of the states of the MDP. The IRL-IB method is a deep MaxEnt IRL method that uses a fixed size memory to augment the state space of the MDP. In our experiment, since there are 4 region types, the memory is a vector of size 4, where each element corresponds to one of the region types. If a region type has been visited, the value of the corresponding vector element is $1$, and $0$ otherwise. The memory-based BC method operates on the same extended state space as ATIG-DIRL, and uses the behavioural cloning [@pomerleau1991efficient] method to learn a policy that mimics the demonstrations. The memory-based BC method in essence is a simplified version of the method introduced in [@shiarlis2018taco] in which the task label is provided for each individual state in the demonstrations, and the agent does not need to learn the alignment between the demonstrations and *task sketches*.
### ATIG-DIRL vs baselines
The primary criterion we use in evaluating the performance of a trained model is the task success ratio. All methods are trained using the same training environment (Fig. \[fig:airsim-grid\]).
**Training performance.** The ATIG-DIRL was able to infer a DFA equivalent to the underling DFA (Fig. \[fig:underlying-dfa\]) in three iterations of the main algorithm (Alg. \[alg:overall\]), and the reported results correspond to the last call to Alg. \[alg:irl\] where the correct DFA is used in the IRL loop. The ATIG-DIRL algorithm and the memory-based BC method perform well at training time (Fig. \[fig:L-phi-compare-baseline-training\] and Fig. \[fig:BC-training\]) with ATIG-DIRL agent still outperforming the BC agent. The memoryless IRL method and the IRL-IB method, however, perform poorly even on the training environment (Fig. \[fig:L-phi-compare-baseline-training\]). As expected, the memoryless IRL method fails at this experiment, because it can only learn a memoryless reward function. The IRL-IB method, however, fails because its memory is unable to capture the memory structure required for the navigation task. Concretely, its memory is unable to keep track of repeated occurrences of the same high-level event.
**Test performance.** To compare the generalization capability of the four methods, we have tested them on 10 different randomly generated environments. The ATIG-DIRL method outperforms all the baselines on test cases (Fig. \[fig:gen-vs-baseline\]). The reason the memoryless IRL and the IRL-IB methods fail at generalization, is that they are not equipped with an appropriate memory structure. However, we have equipped the memory-based BC method with the same memory structure as the ATIG-DIRL method, the reason for the difference in generalizability between these two methods is as follows: the ATIG-DIRL algorithm learns a reward function which generalizes significantly better to test environments. The behavioral cloning agent, on the other hand, learns to shallowly imitate the demonstrations by directly learning a policy.
[0.45]{} ![Performances of ATIG-DIRL versus the baselines.[]{data-label="comparison baselins"}](./pics/success_ratio_training_baslines.png "fig:"){width="\textwidth"}
[0.45]{} ![Performances of ATIG-DIRL versus the baselines.[]{data-label="comparison baselins"}](./pics/success_ratio_orange0052.png "fig:"){width="\textwidth"}
[0.45]{} ![Performances of ATIG-DIRL versus the baselines.[]{data-label="comparison baselins"}](./pics/success_ratio_gen_overlay_compare_baseline.png "fig:"){width="\textwidth"}
Conclusion and Future Work
==========================
We have proposed a new IRL algorithm, active task-inference-guided deep IRL (ATIG-DIRL), that learns the memory structure of the task in the form of a deterministic finite automaton (DFA), and uses the DFA to extend the state space of the original MDP so that we can infer a Markovian reward function using deep MaxEnt IRL. The proposed algorithm learns a reward function over the extended state space obtained by composing the state spaces of the MDP and the inferred DFA. By modeling the reward functions as a convolutional neural network, the algorithm can automatically extract local features that are essential for task implementation. We show with experiments that the learned reward function can be used to generate policies that achieve near-perfect task performance in new environments without expert demonstrations, while methods such as memory-based behavior cloning, IRL with information bits, and memoryless IRL have poor generalization performance. For future work, we would like to apply our algorithm to more complex environment with continuous state spaces.
[^1]: Farzan Memarian, Zhe Xu, Bo Wu and Ufuk Topcu are with the Oden Institute for Computational Engineering and Sciences, University of Texas, Austin, Austin, TX 78712, Min When is with Google LLC. e-mails: farzan.memarian@utexas.edu, zhexu@utexas.edu, bwu3@utexas.edu, minwen@google.com, utopcu@utexas.edu.
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The Josephson plasma resonance (JPR) observed in the microwave absorption [@tsui], and in optical reflection [@marel; @shibata; @kakeshita; @dulic] and transmission measurements [@pim; @tors], has proven to be a powerful method to study the properties of highly anisotropic layered superconductors, such as vortex phases [@tsui; @kosh]. In particular, the spatial dispersion of the JPR, $\omega_p({\bf k},k_z)$, parallel (${\bf k}$) and perpendicular ($k_z$) of the superconducting layers reflects the inductive coupling of junctions due to intralayer currents and the charge coupling due to the variations of the electrochemical potential on the layers, respectively. Determining the latter is essential for understanding the coupled dynamics of the stack of intrinsic Josephson junctions in cuprate superconductors, e.g. with respect to coherence in THz emission [@tach; @sendai]. From a fundamental point of view, it contains unique information about the electronic structure of the superconducting CuO$_2$-layers, namely their compressibility, which is hard to obtain otherwise. Beyond this, from the damping of the JPR the $c$-axis quasiparticle (QP) conductivity at the JPR frequency can be extracted.
The parameter $\alpha$ characterizing the $c$-axis dispersion of the JPR in crystals with identical junctions is difficult to observe in bulk microwave, transport [@sendai] or optical experiments, because mainly modes with small $k_z$ are excited, although for grazing incidence the JPR peak amplitude can depend on $\alpha$ [@bh]. The latter is irrelevant for the experiments in [@marel; @shibata; @kakeshita; @dulic], which are performed with incidence parallel to the layers, which will be studied here. In the following we will show, how $\alpha$ can be extracted unambiguously for layered superconductors with a superstructure in $c$-direction from two independent measurements of the loss function and the magnetic field dependence of the plasma frequencies.
Recently, these experiments have been performed on SmLa$_{1-x}$Sr$_{x}$CuO$_{4-\delta}$, where magnetic Sm$_2$O$_2$ and nonmagnetic La$_{2-x}$Sr$_x$O$_{2-\delta}$ layers alternate in the barriers between the CuO$_2$-layers . In Refs. the reflectivity $R$ and transmission of the electromagnetic wave propagating along the layers and with the electric field along the $c$-axis (cf. Fig. 1a) were used to extract the effective dielectric function $\epsilon_{\rm eff}(\omega)$ with the help of the Fresnel formulas, e.g. $R(\omega)=
[1-\sqrt{\epsilon_{\rm eff}(\omega)}]/[1+\sqrt{\epsilon_{\rm eff}(\omega)}]$. The authors reported two peaks with quite similar widths in the loss function, $L(\omega)=~$Im$~[-1/\epsilon_{\rm eff}(\omega)]$, and a quite large ratio of the peak intensities $L(\omega_1)/L(\omega_2)$ between 10 and 20, see Fig. 2. In a two-junction model with two different plasma frequencies, but identical quasiparticle conductivities and no charge coupling the peak amplitudes in the loss function are quite similar, see Fig. 2 at $\alpha=0$. This disagreement with the experimental data cannot be explained by a strong frequency dependence of the conductivities, as the widths of the two peaks are too similar. Instead, it was argued in [@dulic; @dirk] that the $c$-axis coupling plays a crucial role for the peak intensities.
However, in the derivation of $\epsilon_{\rm eff}$ in Ref. dissipation was not introduced in the Maxwell equations, but arbitrarily in the final expression for $\epsilon_{\rm eff}$ calculated without dissipation. In this Letter we correct these results accounting for different tunneling conductivities in the junctions in accordance with their different critical current densities. We compare the theoretical $L(\omega)$ with the experimental data in [@shibata; @gorshunov] and extract at the JPR peaks the parameter $\alpha \approx 0.4$, which is the free electron value. Further, we show that the microwave absorption in the spatially uniform AC electric field applied along the $c$-axis (see Fig. 1b) is also determined by $L(\omega)$.
Recently, Pimenov et al. [@pim] measured the dependence of the plasma frequencies on the magnetic field ${\bf B}\parallel c$. We show that the field dependence of the peak positions in $L(\omega)$ alone allows us to extract $\alpha \approx 0.4$, which is an independent confirmation of the fit of $L(\omega)$ for $B=0$.
We also show that the QP conductivities extracted at two different frequencies do not show the $\omega$-dependence as anticipated for gapless $d$-wave pairing. We argue that the properties of the Cooper pair tunneling via the magnetic Sm ions may overshadow the $\omega$-dependence of the QP conductivity.
We consider the crystal with two alternating Josephson junctions characterized by different critical current densities, $J_{l}$, and the $c$-axis tunneling conductivities, $\sigma_l$, due to different tunneling matrix elements. We assume that all other parameters of the junctions are identical, as their distinction would lead to negligible corrections in the following. Without the charging effect the $c$-axis bare plasma frequencies $\omega_{0,l}$ are related to $J_{l}$ as $ \omega_{0,l}^2=8\pi^2csJ_{l}/\epsilon_{0}\Phi_0$, where $\Phi_0$ is the flux quantum, $\epsilon_{0}$ is the high frequency $c$-axis dielectric constant and $s$ is the interlayer distance. We neglect nonequilibrium effects [@sendai; @art] in the distribution function of the quasiparticles assuming that the plasma frequencies are well below the charge imbalance and energy relaxation rates.
To find the reflectivity $R(\omega)$ in parallel incidence and the microwave absorption, we use the Maxwell equations inside the crystals accounting for supercurrents inside the 2D layers at $z=ms$ and interlayer Josephson and quasiparticle currents determined by the difference of the electrochemical potentials in neighboring layers: $$\begin{aligned}
&&c\frac{\partial B_y}{\partial z}=
i\epsilon_{a0}\omega\left[E_x-\frac{\omega_{a0}^2}
{\omega^2}\sum_{m=1}^{N}E_xs \delta (z-ms)\right], \label{first} \\
&&\frac{\partial E_x}{\partial z}-ik_xE_z=i \frac{\omega}{c} B_y,
\ \ E_{z,m,m+1}= \int_{ms}^{(m+1)s} E_z\frac{dz}{s}, \\
&&ck_xB_y=-\omega\epsilon_0\left[
E_z-\sum_{m=1}^NA_mf_{m,m+1}(z)\right], \label{e} \\
&& \frac{{\tilde \omega}_l^2 es}{\omega_{0,l}^2} A_m= V_{m,m+1}
=esE_{z,m,m+1}+\mu_{m+1}-\mu_{m}. \label{last}\end{aligned}$$ Here $\mu_m$ is the chemical potential in the layer $m$, $V_{m,m+1}$ is the difference of the electrochemical potentials, $\omega_{a0}=c/\lambda_{ab}\sqrt{\epsilon_{a0}}$ is the in-plane plasma frequency and $\epsilon_{a0}$ is the high frequency in-plane dielectric constant. The function $f$ is defined as $f_{m,m+1}(z)=1$ at $ms<z<(m+1)s$ and zero outside this interval. To obtain Eq. (\[e\]) for small amplitude oscillations we expressed the supercurrent density $J_{m,m+1}^{(s)}=J_l\sin\varphi_{m,m+1}\approx J_l\varphi_{m,m+1}$ via the phase difference $\varphi_{m,m+1}=2iV_{m,m+1}/\hbar\omega$. Further, $\tilde{\omega}_l^2=\omega^2(1-i4\pi\sigma_l \omega / \omega_{0,l}^2
\epsilon_{0})^{-1}$ takes into account the dissipation due to QP tunneling currents, $J_{m,m+1}^{(qp)}=\sigma_lV_{m,m+1}/es$, which are depend on the different conductivities $\sigma_l$ in the junctions $l=1,2$ and the difference $V_{m,m+1}$ of the [*electrochemical*]{} potentials. We express the difference of the chemical potentials $\mu_m$ via the difference of the 2D charge densities, ${\rho}_m$, as $\mu_m-\mu_{m+1}=(4\pi s\alpha/\epsilon_0)({\rho}_m-{\rho}_{m+1})$, where the parameter $\alpha=(\epsilon_0/4\pi e s)(\partial \mu/\partial {\rho})$ characterizes the interlayer coupling. In the model of 2D free electrons we find $\partial \mu/\partial {\rho}=\pi\hbar^2/(e m^*)$. For an effective mass $m^*\approx 1-2 m_e$, as expected from ARPES [@norman], $s=6.3$ ${\rm \AA}$ and $\epsilon_0=20$ we estimate the order of $\alpha$ as $\approx 0.2-0.4$.
The solution of the Eqs. (\[first\])-(\[last\]) is $$\begin{aligned}
&&B_y(z)=\frac{\epsilon_{0}\omega}{ck_x}[c_m\exp(igz)+d_m\exp(-igz)]
-a\frac{ck_x}{\omega} A_m, \nonumber \\
&&E_x (z)=\frac{\epsilon_{0}\omega}{(\epsilon_{a0}a)^{1/2}ck_x}
[c_m\exp(igz)-d_m\exp(-igz)], \nonumber \\
&&E_z(z)=-[c_m\exp(igz)+d_m\exp(-igz)] +aA_m,\end{aligned}$$ where now $0\leq z\leq s$ in each junction, $g^2=\omega^2 \epsilon_{a0}/(c^2 a)$ and $a^{-1}=1-c^2k_x^2/\epsilon_{0}
\omega^2$. The Maxwell boundary conditions for $B_y$ and $E_x$ across the layers lead to a set of equations for $c_m$, $d_m$ and $A_m$, which we solve using the small parameters $b=gs/2 \sim s/\lambda_c\ll 1$ and $\beta=s^2/2\lambda_{ab}^2 a \ll 1$. In order to find the reflectivity for parallel incidence, cf. Fig. 1a, we consider the case $sk_z\ll b \lesssim \beta$, which is fulfilled for incident angles $\theta\ll 1$, and obtain the dispersion relation $k_x(\omega)$ for waves propagating inside the crystal (for details see [@longpaper]): $$\begin{aligned}
&&\frac{c^2k_x^2}{\omega^2\epsilon_{0}}=
\frac{\epsilon_{\rm eff}(w)}{\epsilon_{0}}=
\frac{r(w-v_1)(w-v_2)+iS}{r w^2-(1+r)(2\alpha+1/2)w+iS_1},
\label{r}\\
&&v_{1,2}=(1+r)(1+2\alpha)(1\mp \sqrt{1-p})/2r,
\label{v} \\
&&p=\frac{4r(1+4\alpha)}{(1+r)^2(1+2\alpha)^2},
\label{p} \\
&&S_1=w^{3/2} r (2 \alpha + 1/2) ({\tilde \sigma}_1 +
{\tilde \sigma}_2) , \\
&&S=w^{1/2} [(2\alpha+1) r w (\tilde{\sigma}_1+{\tilde \sigma}_2)-
(1+4\alpha)(\tilde{\sigma}_1+\tilde{\sigma}_2r)], \nonumber\end{aligned}$$ where $v_l=\omega^2_l/\omega^2_{0,1}$, $\omega_l$ are the JPR frequencies, $w=\omega^2/\omega_{0,1}^2$, $\tilde{\sigma}_l=4\pi\sigma_l/\epsilon_0\omega_{0,1}$ and $r=
\omega_{0,1}^2/\omega_{0,2}^2<1$.
The reflectivity coefficient is given by the usual Fresnel expression $R=(1-ck_x/\omega)/(1+ck_x/\omega)$. Note that for nonzero $\beta$ we obtain in principle two propagating modes inside the crystal, but the second mode gives a negligible contribution near the JPR peaks at $\beta\ll 1$, see [@longpaper]. The loss function is $$\begin{aligned}
L(w)&=&~{\rm Im}~[-1/\epsilon_{\rm eff}(\omega)]=
(w^{3/2}/ 2 \epsilon_{0}) \times \label{ab} \\
&&\frac{{\tilde \sigma}_1(wr-4\alpha-1)^2+
{\tilde \sigma}_2r^2(w-4\alpha-1)^2}{[r(w-v_1)(w-v_2)]^2+S^2}
\nonumber \end{aligned}$$ and shows resonances at the two transverse plasma bands, while the peak in Im$[\epsilon_{{\rm eff}}(\omega)]$ is at $v_T=(1+1/r)(2\alpha+1/2)$.
In $s$-wave superconductors with nonmagnetic ions in the barrier between the layers both the QP conductivities $\sigma_l$ and the critical current densities $J_{l}$ vary with $l$ only due to their proportionality to the squared matrix element for the interlayer tunneling in agreement with the doping dependence $\sigma_c(x)\propto \omega_p^2(x)$ found for La$_{2-x}$Sr$_x$CuO$_4$ [@uch]. We take the same relation, $\sigma_1 (\omega_1)/\sigma_2 (\omega_2)=r$ for SmLa$_{1-x}$Sr$_{x}$CuO$_{4-\delta}$, ignoring the frequency dependence of $\sigma_l$ due to the $d$-wave pairing [@hirsch], which decreases $\sigma_1
(\omega_1)/\sigma_2 (\omega_2)$ below $r$. On the other hand, we also neglect that due to the magnetism of the Sm ions the critical current density $J_1 \propto \omega_{0,1}^2$ is suppressed, while $\sigma_l$ is not affected, which suggests $\sigma_1/\sigma_2 >r$ without $d$-wave pairing [@bula]. The fact that our assumption $\sigma_1 = r \sigma_2$ turns out to be consistent with the experimental data suggests that both effects compensate each other.
In Fig. \[ratiopic\] we present the dependence of the ratio of the peak amplitudes in the loss function $L(\omega)$, i.e. $L(\omega_1)/L(\omega_2)$, vs. the ratio $\omega_1^2/\omega_2^2$ for different values of $\alpha$. Hence, this figure allows us to obtain the parameter $\alpha$ from the positions $\omega_{1,2}$ of the JPR peaks and their ratio of amplitudes. Fig. \[deltaalpha\] allows us to find the ratio of the squared bare frequencies, $r=\omega_{0,1}^2/\omega_{0,2}^2$ for a ratio $\omega_1/\omega_2$ when the parameter $\alpha$ is obtained from Fig. \[ratiopic\]. The comparison of $L(\omega)$ in Fig. \[losspic\] with the experimental data from Ref. gives the best fit for ${\tilde \sigma}_1 = r {\tilde \sigma}_2=0.12$ and $\epsilon_{0}=18$, $\alpha=0.44$, which is of the same order as the theoretical estimate for free electrons, cf. Tab. I for a fit of [@shibata; @dulic; @pim; @gorshunov]. In the case $\alpha=0$ a high ratio ${\tilde \sigma}_2 / {\tilde \sigma}_1 \approx 15$ of the conductivities is necessary to reproduce the ratio of the amplitudes, which fails to describe the shape of the resonance at $\omega_2$ correctly.
The loss function $L(\omega)$ also determines the microwave absorption of a crystal in a capacitor, which induces a uniform ($sk_z\rightarrow 0$, $k_x=0$) AC electric field $2{\cal E}\cos(\omega t)$ above and below the crystal, see Fig. 1b. The crystal excitations are longitudinal in this case, i.e. $E_x=B_y=0$ in Eqs. (\[first\])-(\[last\]). These equations together with the Poisson equations near the top and bottom layers determine for large $N$ the microwave absorption as $${\cal P}(\omega)=N^{-1}\sum_{m=1}^{N} \sigma_l|V_{m,m+1}/es|^2
=\frac{\omega}{4\pi}L(\omega) {\cal E}^2 .$$
Next we consider the dependence of the JPR peaks in $L(\omega)$ on the $c$-axis magnetic field $B$, which allows one to estimate the parameter $\alpha$, without relying on the absolute amplitude of the spectra. For Josephson junctions $B$ suppresses the critical current densities and the bare plasma frequencies, $\omega_{0,l}(B)$, and also broadens the resonance peaks due to the formation of pancake vortices randomly misaligned along the $c$-axis [@kosh]. From Eq. (\[v\]) we see that the ratio of the resonance frequencies, $$\omega_1^2/\omega_2^2=(1-\sqrt{1-p})/(1+\sqrt{1-p}),
\label{omegarat}$$ depends only on the quantity $p$ given by Eq. (\[p\]), which varies with $B$ only via the function $r(B)=\omega^2_{0,1}(B)/
\omega^2_{0,2}(B)$. When the functional form of $\omega_{0,l} (B)$ is known theoretically and $p(B)$ is obtained from the measurement of $\omega_1/\omega_2$ via Eq. (\[omegarat\]), one can fit unknown parameters in $\omega_{0,l}
(B)$ and obtain $\alpha$. It is pointed out that extracting $\alpha$ from $L(\omega,B)$ by fitting all parameters without further theoretical input is difficult due to the dependence of the effective $c$-axis conductivity on $B$ [@kosh].
The dependence $\omega_{0,l}(B)$ in the decoupled vortex liquid [@kosh] is known for crystals with equivalent junctions and can be used for alternating junctions as well, provided that the vortices in different layers are decoupled. This is suggested by $\hbar \omega_{0,l} \ll k T_c$ in SmLa$_{0.8}$Sr$_{0.2}$CuO$_{4-\delta}$ (cf. Tab. I) indicating Josephson coupling of the layers. Thus, for the vortex liquid phase with decoupled pancakes in high magnetic fields $B\gg B_J=\Phi_0 \lambda_{ab}^2/ (\lambda_c s)^2
\sim 1$ T the dependence of the plasma frequencies on $B$ is [@kosh] $$\omega_{0,l}^2(B)\approx\omega_{0,l}^4(0)\frac{\epsilon_{0}\Phi_0^3}{32B\pi^3
c^2sT}.$$ This gives $r(B\gg B_J)=r^2(B=0)$. Hence, we find $$p(B\gg B_J)=\frac{4r^2(0)(1+4\alpha)}{[1+r^2(0)]^2(1+2\alpha)^2},
\label{ddd}$$ while for $p(0)$ we use Eq. (\[p\]) with $r=r(0)$.
Consequently, the parameters $\alpha$ and $r(0)$ can be found analytically without fitting from $p(0)$ and $p(B\gg B_J)$ obtained from JPR frequency measurements using the data of Ref. [@pim]. The dependence $\omega_1^2,\omega_2^2\propto 1/B$ at fields above $B_J \approx 1$ T shows that in this field range the decoupled pancake liquid is present. We use the reported field dependence of the peak frequency in Im$[\epsilon_{{\rm eff}}(\omega)]$ for $\omega_2(B)$, as the authors noted that they are quite close and the difference will finally turn out to be $\approx 10$ %. Then we obtain (for $T \ll T_c$) $\omega_1^2/\omega_2^2\approx 0.31$ at $B=0$ and 0.21 at $B>B_J$ or $p(0)\approx 0.73$ and $p(B\gg B_J)\approx 0.57$ respectively. From Eqs. (\[p\]) and (\[ddd\]) we obtain $r(0)\approx 0.55$, $\alpha\approx 0.4$ and the bare frequencies are $\omega_{0,1}/ c=6.6$ cm$^{-1}$ and $\omega_{0,2}/ c=8.9$ cm$^{-1}$, cf. Tab. I. This estimate gives the parameters $\alpha$ and $r(0)$ similar to those obtained above from the fit of $L(\omega)$ at $B=0$ and confirms independently the relevance of the $c$-axis coupling $\alpha$ and our assumption ${\tilde \sigma}_1 \approx r {\tilde \sigma}_2$.
$x$ $T_c$ $\omega_{0,1}/ c$ $r$ $\sigma_{1}$\[$(\Omega$m)$^{-1}$\] $\alpha$ Ref.
------ ------- ------------------- ------ ------------------------------------ ---------- ---------
0.15 23K 10.9[cm]{}$^{-1}$ 0.42 10 0.36 \[3\]
0.15 30K 7.2[cm]{}$^{-1}$ 0.38 4.3 0.44 \[13\]
0.2 17K 6.6[cm]{}$^{-1}$ 0.55 3.9 0.40 \[5,6\]
: Extracted parameters for SmLa$_{1-x}$Sr$_{x}$CuO$_{4-\delta}$
In conclusion, we calculated the effective dielectric function $\epsilon_{\rm eff}$, Eq. (\[r\]), for alternating junctions with charge coupling $\alpha$ and frequency independent QP conductivities $\sigma_l\propto \omega_{0,l}^2$. This allows to describe satisfactory the optical properties of the SmLa$_{0.8}$Sr$_{0.2}$CuO$_{4-\delta}$ superconductor near the plasma frequencies. The parameter $\alpha\approx 0.4$ was extracted independently from the magnetic field dependence of the positions of the JPR peaks $\omega_{1,2}(B)$ and from the fit of the loss function $L(\omega)$ at $B=0$. Its value is expected to be universal in the cuprates and corresponds to the free electron value of the electronic compressibility, which can differ from the renormalized one particle density of states both for Fermi and non Fermi liquids [@nonfermi]. We also show that the extracted conductivities $\sigma_l$ differ according to the different tunneling matrix elements, $\sigma_1 / \sigma_2 \approx \omega_{0,1}^2 / \omega_{0,2}^2 $. This relation is anticipated for tunneling between $s$-wave superconductors via nonmagnetic ions. Its fulfillment in SmLa$_{0.8}$Sr$_{0.2}$CuO$_{4-\delta}$ might indicate that the gapless $d$-wave pairing in the layers reduces $\sigma_1(\omega_1)/\sigma_2(\omega_2)$ in a similar way as the magnetism of the Sm ions decreases $\omega_{0,1}^2/\omega_{0,2}^2$. This could be studied further by measuring the gap in the QP spectrum in the $I$-$V$ curve as in Ref. .
The authors thank M. Graf and W. Zwerger for useful discussions. The work was supported by the U.S. DOE (in Lehman College through Grant No. DE-FG02-93ER45487) and the NRCC of the Swiss NSF.
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|
---
abstract: 'We introduce a Prony-like method to recover a continuous domain 2-D piecewise smooth image from few of its Fourier samples. Assuming the discontinuity set of the image is localized to the zero level-set of a trigonometric polynomial, we show the Fourier transform coefficients of partial derivatives of the signal satisfy an annihilation relation. We present necessary and sufficient conditions for unique recovery of piecewise constant images using the above annihilation relation. We pose the recovery of the Fourier coefficients of the signal from the measurements as a convex matrix completion algorithm, which relies on the lifting of the Fourier data to a structured low-rank matrix; this approach jointly estimates the signal and the annihilating filter. Finally, we demonstrate our algorithm on the recovery of MRI phantoms from few low-resolution Fourier samples.'
author:
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bibliography:
- 'IEEEabrv.bib'
- 'root.bib'
title: |
Recovery of Piecewise Smooth Images\
from Few Fourier Samples
---
Introduction
============
The recovery of continuous domain parametric representations from few measurements using harmonic retrieval/linear prediction has received considerable attention in signal processing since Prony’s seminal work[@stoica1997introduction; @cheng2003review]. Extensive research has been devoted to the recovery of finite linear combination of exponentials with unknown continuous frequencies as well as linear combination of Diracs and non-uniform 1-D splines with unknown locations/knots [@vetterli2002sampling; @maravic2005sampling; @dragotti2007sampling]. Recently, convex algorithms that minimize atomic norm were also introduced to recover such continuous signals [@bhaskar2011atomic; @candes2013super]; these methods are off-the-grid continuous generalizations of compressed sensing theory, which can avoid discretization errors and hence are potentially more powerful. However, the direct extension of the above Prony-like and convex off-the-grid methods to 2-D piecewise smooth images is not straightforward. Specifically, the partial derivatives of piecewise smooth images can be thought of as a linear combination of a continuum of Diracs supported on the curves separating the regions, which current methods are not designed to handle.
Recently, Pan *et al.* [@pan2013sampling] introduced a complex analytic signal model for continuous domain 2-D images, such that the complex derivatives of the image are supported on a curve. Under the assumption that the curve is the zero-set of a band-limited function, the authors show the Fourier transform of the complex derivative of such a signal is annihilated by convolution with the Fourier coefficients of the band-limited function. This property is used to extend the finite-rate-of-innovation model [@vetterli2002sampling] to this class of 2-D signals. This work has several limitations. One problem is that a complex analytic signal model is not realistic for natural images (e.g., the only real-valued analytic functions are constant functions). In addition, if we are only measuring a finite number of Fourier samples, one can choose analytic functions such that the all of these coefficients vanish; i.e. the recovery of the signal from few Fourier coefficients of an arbitrary signal in this class is ill-posed. In this work, we address the above limitations by proposing an alternative signal model based on the a more realistic class of piecewise smooth functions. We show that we can generalize the annihilation property in [@pan2013sampling] to this class of functions, such that it is possible to recover an exact continuous domain representation of the edge set from few Fourier samples. We also determine necessary and sufficient conditions on the number of Fouier samples required for perfect recovery of the edge set in the caseof piecewise constant images.
To recover the full signal, we propose a single-step convex algorithm which can be thought of as jointly estimating the edge set and image amplitudes. This is fundamentally different than the two stage approach proposed in [@pan2013sampling], which we also investigated for super-resolution MRI in [@isbi2015]. Motivated by recently proposed algorithms for calibration-free parallel MRI recovery [@sake; @loraks], our new approach is based on the observation that the ideal Fourier samples of the signal lift to a structured low-rank matrix, which allows us to pose the recovery as a low-rank structured matrix completion problem. We demonstrate our algorithm on the recovery of MR phantoms from low-resolution Fourier samples.
Signal Model
============
Piecewise smooth signals
------------------------
In this paper, we consider the general class of 2-D *piecewise smooth* functions: $$\label{maineqn}
f({\mathbf}r) = \sum_{i=1}^n g_i({\mathbf}r)~ \chi_{\Omega_i}({\mathbf}r),~~\forall {\mathbf}r = (x,y)\in[0,1]^2,$$ where $\chi_{\Omega}$ denotes the characteristic function of the set $\Omega$: $$\chi_\Omega(\mathbf r) =
\begin{cases}
1 & \text{if } {\mathbf}r = (x,y) \in \Omega\\
0 & \text{else}.
\end{cases}
\label{eq:char_func}$$ Here we assume each $\Omega_i\subset [0,1]^2$ is a simply connected region with piecewise smooth boundary $\partial \Omega_i$. The functions $g_i$ in (\[maineqn\]) are smooth functions that vanish under of a collection of constant coefficient differential operators $\mathbf D = \{D_1,...,D_N\}$ within the region $\Omega_i$: $$D_j~ g_i({\mathbf}r) = 0, \forall {\mathbf}r \in \Omega_i; ~j=1,..,N.$$ We now show that the above class of functions is fairly general and includes many well-understood image models by appropriately choosing the set of differential operators $\mathbf D$.
###
We set ${\mathbf}D$ to $${\mathbf}D = \nabla = \{\partial_x, \partial_y\}.$$ Note that ${\mathbf}D g = 0$ if and only if $g = c_i$ for $c_i\in\mathbb{C}$. Hence, (\[maineqn\]) reduces to the well-known piecewise constant image model: $$f({\mathbf}r) = \sum_{i=1}^n c_i\,\chi_{\Omega_i}({\mathbf}r),~~\forall {\mathbf}r = (x,y)\in[0,1]^2,$$ This case will be the primary focus of this work due to its simplicity and provable guarantees.
###
Choosing ${\mathbf}D = \partial_{\bar{z}} = \partial_x + j \partial_y$, then ${\mathbf}D g = 0$ if and only if $g$ is complex analytic. Hence this model is equivalent to the one proposed in [@pan2013sampling]. As described above, this signal model is not very realistic for natural images.
###
Both the above cases consider only first-order differenial operators. One choice of a second-order differential operator is the Laplacian ${\mathbf}D = \Delta = \partial^2_{xx} + \partial^2_{yy}$. Then ${\mathbf}D g = 0$ if and only if $g$ is harmonic.
###
If we consider all second order partial derivatives ${\mathbf}D = \{\partial^2_{xx}, \partial^2_{xy}, \partial^2_{yy}\}$, then ${\mathbf}D g = 0$ if and only if $g$ is linear, i.e. $g({\mathbf}r) = \langle {\mathbf}a,{\mathbf}r\rangle + b$, for ${\mathbf}a \in \mathbb{C}^2$, $b\in \mathbb{C}$, and so $f$ has the expression $$f({\mathbf}r) = \sum_{i=1}^n \left(\langle{\mathbf}a_i,{\mathbf}r\rangle + b_i\right)~ \chi_{\Omega_i}({\mathbf}r),~~\forall {\mathbf}r = (x,y)\in[0,1]^2.$$
###
Generalizing the above case we may consider all $n$th order partial derivatives ${\mathbf}D = \{\partial^\alpha\}_{|\alpha|=n}$ where $\alpha$ is a multi-index. then ${\mathbf}D g = 0$ if and only if $g$ is a polynomial of degree at most $(n-1)$.
We will show that under certain assumptions on the edge set $C=\cup_{i=1}^n\partial \Omega_i$, the Fourier transform of derivatives of a piecewise smooth signal specified by (\[maineqn\]) satisfies an annihilation property. This will enable us to recover an exact continuous domain representation the edge set $C$ of a piecewise smooth signal from finitely many of its Fourier samples by solving a linear system.
Trigonometric polynomials and curves
------------------------------------
Following [@pan2013sampling], we will assume the edge set $C$ to be the zero-set of a band-limited periodic trigonometric polynomial $$\mu({\mathbf}r) = \sum_{{\mathbf}k\in\Lambda} c[{\mathbf}k]\, e^{j2\pi\langle {\mathbf}k,{\mathbf}r \rangle},\quad \forall {\mathbf}r\in{[0,1]}^2,
\label{eq:trigpoly}$$ where $c[{\mathbf}k]\in\mathbb{C}$ and $\Lambda$ is any finite subset of $\mathbb{Z}^2$; we call any function $\mu$ described by a *trigonometric polynomial*, and the zero-set $C: \{\mu=0\}$ a *trigonometric curve*. We also define the *degree* of a trigonometric polynomial $\mu$ to be the dimensions of the smallest rectangle that contains the frequency support set $\Lambda$, denoted as $deg(\mu) = (K,L)$. For trigonometric polynomials $\mu$ and $\nu$, we say $\nu$ *divides* $\mu$ or $\nu~|~\mu$ if $\mu = \nu \cdot \gamma$ where $\gamma$ is another trigonometric polynomial.
Using elementary results from algebraic geometry, we may show there is a unique minimal degree trigonometric polynomial associated with any trigonometric curve $C$, which we call the *minimal polynomial* for $C$:
For every trigonometric curve $C$ there is a unique (up to scaling) trigonometric polynomial $\mu_0$ with $C:\{\mu_0 = 0\}$ such that for any other trigonometric polynomial $\mu$ with $C:\{\mu = 0\}$ we have $deg(\mu_0) \leq deg (\mu)$ and $\mu_0~|~\mu$.
The following property of minimal polynomials is also important for our uniqueness results:
\[prop:musquared\] Let $C$ be the zero set of a trigonometric polynomial with minimal polynomial $\mu_0$. Suppose $\nu$ is a trigonometric polynomial such that $\nu = 0$ and $\nabla \nu = 0$ on $C$, then $\mu_0^2~|~\nu$. In particular, $\nabla \mu_0 = 0$ for at most finitely many points on $C$.
Annihilation property
=====================
We now show that the Fourier transform of the partial derivatives of piecewise smooth signals satisfy an annihilation property.
###
First we consider the case of a single characteristic function $\chi_\Omega$. Note that since $\chi_\Omega$ is non-smooth at the boundary, its derivatives are only defined in a distributional sense. Letting $\varphi$ denote any test function we have: $$\langle \partial_x \chi_\Omega, \varphi\rangle = - \langle \chi_\Omega, \partial_x \varphi\rangle = -\int_\Omega \partial_x \varphi \, d{\mathbf}r = -\oint_{\partial\Omega} \varphi\, dy$$ where the last step follows by Green’s theorem. Likewise, $$\langle \partial_y \chi_\Omega, \varphi\rangle = \oint_{\partial\Omega} \varphi\, dx$$ Hence $\partial_x \chi_\Omega$ and $\partial_y \chi_\Omega$ can be interpreted as a continuous stream of weighted Diracs supported on $\partial\Omega$. In particular, if $\psi$ is any smooth function that vanishes on $\partial\Omega$ then $$\psi\cdot\partial_x \chi_\Omega = \psi\cdot\partial_y \chi_\Omega = 0
\label{eq:spacedom}$$ where equality holds in the distributional sense. Assuming $\psi = \mu$ is a trigonometric polynomial, taking Fourier transforms of yields the following annihilation relation:
Let $f = \chi_\Omega$ with boundary $\partial\Omega$ given by the trigonometric curve $C: \{\mu = 0\}$. Let $D$ be any first order differential operator. Then the Fourier transform of $Df$ is annihilated by convolution with the Fourier coefficients $c[{\mathbf}k],{\mathbf}k \in \Lambda$ of $\mu$, that is $$\sum_{{\mathbf}k \in \Lambda} c[{\mathbf}k]\, \widehat{Df}({\boldsymbol}\omega - 2\pi{\mathbf}k) = 0, ~~\text{for all } {\boldsymbol}\omega \in \mathbb{R}^2.
\label{eq:annihilation}$$
Due to the above property, we call $\mu$ an *annihilating polynomial* for $Df$.
It is straightforward to extend the above proposition to piecewise constant functions $f = \sum_{i=1}^n c_i \chi_{\Omega_i}$, provided $\mu = 0$ on the union of the boundaries $C = \cup_{i=1}^n \partial \Omega_i$. Likewise, if $f = g \cdot \partial_\Omega$ where $Dg = 0$, then by the product rule $$D f = D g \cdot \chi_\Omega + g \cdot D\chi_\Omega = g \cdot D\chi_\Omega$$ which has support on $\partial\Omega$ and so, $\mu\cdot D f = 0$, which implies holds for $f$, and similarly for the linear combination $f = \sum_{i=1}^n g_i \cdot \chi_{\Omega_i}$, where $Dg_i = 0$ for all $i=1,...,n$.
###
Now consider the case where $D$ is any second-order differential operator. Let $f = g\cdot \chi_\Omega$ where $D g = 0$. We now show that $\mu^2$ is an annihilating polynomial for $Df$, where $\mu$ is any trigonometric polynomial that annihilates the partial derivatives of $\chi_\Omega$.
Let $\partial^2 = \partial_2\partial_1$ where $\partial_i \in \{\partial_x, \partial_y\}, i=1,2$. By the product rule we have: $$\partial^2 f = \partial^2 g \cdot \chi_\Omega + \partial_1 g \cdot \partial_2 \chi_\Omega + \partial_2 g \cdot \partial_1 \chi_\Omega + g \cdot \partial^2 \chi_\Omega.$$ Since $\partial_1 \chi_\Omega$ and $\partial_2 \chi_\Omega$ are annihilated by $\mu$, we have $$\mu^2 \cdot \partial^2 f = \chi_\Omega \cdot \mu^2 \cdot \partial^2 g + g \cdot \mu^2 \cdot \partial^2 \chi_\Omega.$$ Again by the product rule $$\mu^2 \cdot \partial^2 \chi_\Omega = \partial_2 (\mu^2 \cdot \partial_1 \chi_\Omega) - 2\,\mu \cdot \partial_2 \mu \cdot \partial_1 \chi_\Omega = 0,$$ which implies $$\mu^2 \cdot \partial^2 f = \chi_\Omega \cdot \mu^2 \cdot \partial^2 g$$ and so by linearity $$\mu^2 \cdot D f = \chi_\Omega \cdot \mu^2 \cdot D g = 0.$$ The above shows that $\mu^2$ is always sufficient to annihilate $D f$, where $D$ is second-order. However, using Prop. \[prop:musquared\], we may show that when $g$ does not vanish on $\partial\Omega$, then $\mu^2$ is also necessary for annhilation of $Df$, in the sense that if $\nu$ is any other trig polynomial satisfying $\nu \cdot Df = 0$, then $\mu_0^2~|~\nu$, where $\mu_0$ is the minimal polynomial for $\partial\Omega$. If $deg(\mu_0) = (K,L)$, this implies any annihilating polynomial $\nu$ for $Df$ has $deg(\nu) \geq (2K-1,2L-1)$.
###
A similar argument shows that when $D$ is any $n$th order differential operator, and $f = g\cdot \chi_\Omega$ where $D g = 0$, then $$\mu^n \cdot D f = 0.$$ This yields the following annihilation relation for higher-order differential operators:
Let $D$ be any $n$th order differential operator. Let $f = g \cdot \chi_\Omega$ with $Dg = 0$ and $\partial\Omega \subset \{\mu = 0\}$ for some trigonometric polynomial $\mu$. Then the Fourier transform of $Df$ is annihilated by convolution with the Fourier coefficients $d[{\mathbf}k], {\mathbf}k\in \Gamma$ of $\mu^n$, that is $$\sum_{{\mathbf}k \in \Gamma} d[{\mathbf}k]\, \widehat{Df}({\boldsymbol}\omega - 2\pi{\mathbf}k) = 0, ~~\text{for all } {\boldsymbol}\omega \in \mathbb{R}^2.
\label{eq:annihilation2}$$
Likewise, by linearity, is valid for linear combinations $f = \sum_i g_i \cdot \chi_{\Omega_i}$, where $Dg_i = 0$ and $\mu = 0$ on $\cup_i \partial\Omega_i$.
Recovery from finite Fourier samples
====================================
We now investigate necessary and sufficient conditions for the recovery of the filter coefficients describing the edge set from finitely many Fourier samples of the original signal $f$. For these results we restrict our attention to piecewise constant signals.
[Necessary conditions]{}
------------------------
For a piecewise constant signal $f$, from the annihilation condition we may form the linear system of equations: $$\begin{cases}
\sum_{{\mathbf}k \in \Lambda} d[{\mathbf}k] \widehat{f_x}\left(2\pi[{\mathbf}l - {\mathbf}k]\right) = 0,\\
\sum_{{\mathbf}k \in \Lambda} d[{\mathbf}k] \widehat{f_y}\left(2\pi[{\mathbf}l - {\mathbf}k]\right) = 0,
\end{cases}
~~\forall ~{\mathbf}l \in \Gamma.
\label{eq:annsys}$$ where $\widehat{f_x}$ and $\widehat{f_y}$ may be computed from samples of $\widehat{f}$ by $\widehat{f_x}({\boldsymbol}\omega) = -j\omega_x\cdot \widehat{f}({\boldsymbol}\omega)$, and $\widehat{f_y}({\boldsymbol}\omega) = -j\omega_y\cdot \widehat{f}({\boldsymbol}\omega)$. Supposing the sampling grid $\Omega$ is a rectangular of dimensions $(K',L')$, and the minimal polynomial for $C$ has degree $(K,L)$, with coefficients $c[{\mathbf}k]$ supported in $\Lambda$, then we may form at most $M = 2\cdot(K'-K +1)\cdot(L'-L+1)$ valid equations from . Therefore to solve for the at most $K\cdot L$ unknowns $c[{\mathbf}k]$, ${\mathbf}k \in \Lambda$, we require at least $M = K\cdot L$ equations. This gives the following necessary condition for recovery of $C$:
\[prop:necessity\] Let $f$ be piecewise constant such that the edge set $C$ has minimal polynomial $\mu$ of degree $(K,L)$. A necessary condition to recover the edge set $C$ from , is to collect samples of $\widehat{f}$ on a $(K',L')$ rectangular grid such that $$2\cdot(K'-K +1)\cdot(L'-L+1) \geq K\cdot L.$$
To illustrate this bound, suppose the minimal polynomial has degree $(K,K)$, and we take Fourier samples from a square region. Then this requires at least $1.71K\times 1.71K$ Fourier samples to recover the edge set $C$. Our numerical experiments on simulated data (see Fig. \[fig:sim\]) indicate the above necessary condition might also be *sufficient* for unique recovery; that is, we hypothesize the minimal filter coefficients $c[{\mathbf}k]$ are the only non-trivial solution to the system of equations .
\
[Sufficient conditions]{}
-------------------------
We now focus on sufficient conditions for the recovery of the edge set. Here we will use $m\Lambda$ to denote a dilation of the set $\Lambda$ by a factor of $m$: if $\Lambda = \{{\mathbf}(k,l): |k|\leq K, |l|\leq L\}$, then $m\Lambda = \{{\mathbf}(k,l): |k|\leq m\, K, |l|\leq m\,L\}$.
\[thm:unique1\] Let $f = \chi_\Omega$ be the characteristic function of a simply connected region $\Omega$ with boundary $\partial\Omega$ having minimal polynomial $\mu$ with coefficients $c[{\mathbf}k],{\mathbf}k \in \Lambda$. Then the $c[{\mathbf}k]$ can be uniquely recovered (up to scaling) as the only non-trivial solution to the equations $$\begin{cases}
\sum_{{\mathbf}k \in \Lambda} c[{\mathbf}k] \widehat{f_x}\left(2\pi[{\mathbf}l - {\mathbf}k]\right) = 0,\\
\sum_{{\mathbf}k \in \Lambda} c[{\mathbf}k] \widehat{f_y}\left(2\pi[{\mathbf}l - {\mathbf}k]\right) = 0,
\end{cases}
~~\forall ~{\mathbf}l \in 2 \Lambda.
\label{eq:unique1}$$
The proof entails showing any other trigonometric polynomial $\eta(r)$ having coefficients $d[{\mathbf}k]$, ${\mathbf}k \in \Lambda$ satisfying must vanish on $\partial\Omega$, from which it then follows that $\eta$ is a scalar multiple of the minimal polynomial $\mu$ by degree considerations. We also are able to show similar result holds for piecewise constant signals, provided the characteristic functions do not intersect:
\[thm:unique2\] Let $f({\mathbf}r) = \sum_{i=1}^n a_i \chi_{\Omega_i}({\mathbf}r)$ be piecewise constant, where the boundaries $\partial\Omega_i$ are described by non-intersecting trigonometric curves $\{\mu_i = 0\}$, where $\mu_i$ is the minimal polynomial for $\partial\Omega_i$. Then, the coefficients $d[{\mathbf}k]$,$k\in\Lambda$, of $\mu=\mu_1\cdots\mu_n$, and equivalently the edge set $C=\cup_{i=1}^n \partial \Omega_i$, can be uniquely recovered (up to scaling) as the only non-trivial solution of $$\begin{cases}
\sum_{{\mathbf}k \in \Lambda} d[{\mathbf}k] \widehat{f_x}\left(2\pi[{\mathbf}l - {\mathbf}k]\right) = 0,\\
\sum_{{\mathbf}k \in \Lambda} d[{\mathbf}k] \widehat{f_y}\left(2\pi[{\mathbf}l - {\mathbf}k]\right) = 0,
\end{cases}
~~\forall ~{\mathbf}l \in 2\Lambda.
\label{eq:unique2}$$
Note that to form the equations in and requires access to Fourier samples $\widehat{f}[{\mathbf}k]$ for all ${\mathbf}k\in 3 \Lambda$, which is greater than necessary number of samples given in Theorem \[prop:necessity\]. We conjecture that the uniqueness results in Theorems \[thm:unique1\] and \[thm:unique2\] can in fact be sharpened to the necessary number of samples, and extended to piecewise constant signals where the boundaries of the regions intersect.
Recovery Algorithms
===================
Up to now we have only considered the problem of recovering the edge set of a piecewise constant signal $f = \sum_{i=1}^n a_i \chi_{\Omega_i}$ from finite Fourier samples. In analogy with Prony’s method, once the edge set is determined it is theoretically possible to recover the signal amplitudes $a_i$ by substuting $f$ back into and solving a full rank system. However, this is not feasible in practice since it requires factoring a high degree multivariate polynomial into its irreducible factors. Instead we pursue approaches that allow us to pose the recovery as the solution to a convex optimization problem.
Curve-aware recovery
--------------------
Supposing we have access to an annihilating polynomial $\mu$ of the signal (equivalently, the edge set $C:\{\mu=0\}$), one approach is to pose the recovery as the weighted total variation minimization problem: $$f = \arg\min_g \int |\mu({\mathbf}r)\,\nabla g({\mathbf}r)|\, d{\mathbf}r \text{ subject to } \widehat{g}[{\mathbf}k] = \widehat{f}[{\mathbf}k], \forall {\mathbf}k \in \Gamma
\label{eq:TVmin}$$ Here, since $\mu = 0$ on the edge set $C$, the gradient of the image not penalized along the curve, which allows for the recovery of an image with sharp edges along $C$.
A version of this approach was investigated in an earlier work for the super-resolution recovery of MR signals from few Fourier samples [@isbi2015]. However, we found this scheme to have certain drawbacks, namely that it requires discretization onto a spacial grid, the optimization of many parameters, and is very sensitive to the estimate $\mu$. Hence we consider an alternate approach which does not rely on an explicit estimate of the annihilating polynomial $\mu$, but instead jointly recovers the image and the annihilating polynomial in a single stage algorithm.
Low-rank recovery
-----------------
The annhilation equations specified by can be represented in the matrix form as $$\underbrace{\left[\begin{array}{c}{\mathbf}T_x[\widehat{f}]\\ {\mathbf}T_y[\widehat{f}] \end{array}\right]}_{{\mathbf}T[\widehat{f}]} {\mathbf}d = {\mathbf}0
\label{matrixform}$$ where ${\mathbf}d$ is a vectorized version of the Fourier coefficients $d[{\mathbf}k], {\mathbf}k \in \Lambda$, and ${\mathbf}T_x$ and ${\mathbf}T_y$ are block Toeplitz matrices corresponding to the 2-D convolution of $d[{\mathbf}k]$ with the discrete samples of $j\omega_x\,\widehat f({\boldsymbol}\omega)$ and $j\omega_y\,\widehat f({\boldsymbol}\omega)$, respectively, for ${\boldsymbol}\omega = 2\pi {\mathbf}l$, ${\mathbf}l \in \Omega$. Specifically, if the coefficient support set $\Lambda$ has dimensions $K \times L$, each row of ${\mathbf}T_x(\hat {f})$ is the vectorized version of an $K\times L$ patch of $j\omega_x\,\widehat f({\boldsymbol}\omega)$, and likewise for ${\mathbf}T_y(\hat {f})$. The number of rows is equal to the number of distinct patches, which correspond to the number of annihilation equations. Note that for a given piecewise constant signal, a priori we do not know the degree of the minimal polynomial describing the edge set, which is needed to specify the size of ${\mathbf}T$. However, if $d[{\mathbf}k], {\mathbf}k \in \Lambda'$ corresponding to the trigonometric polynomial $\mu$ is a solution to whose support set $\Lambda'$ is strictly smaller than the assumed support set $\Lambda$, then any multiple $\nu = \mu \cdot \gamma$ having coefficients $e[{\mathbf}k] = (d \ast g)[{\mathbf}k]$ supported within $\Lambda$, is also a solution. This implies that if we consider a larger filter size than required by the minimal polynomial, i.e. more columns in ${\mathbf}T$ than the number of coefficients in ${\mathbf}d$, the matrix ${\mathbf}T$ will be low-rank.
The preceding discussion suggests we may pose the recovery of the signal as a structured low-rank matrix completion problem, entirely in the Fourier domain: $$\widehat{f} = \arg\min_{\widehat{g}}~\text{rank}({\mathbf}T[\widehat{g}]) \text{ subject to } \widehat{g}[{\mathbf}k] = \widehat{f}[{\mathbf}k], \forall {\mathbf}k \in \Gamma
\label{eq:rankmin}$$ We note this approach is still “off-the-grid” in the sense that we may recover a discrete image at any desired resolution by extrapolating $\widehat{f}$ to this resolution in Fourier space, and applying an inverse DFT. To address the case of noisy measurements and model mismatch we propose solving the convex relaxation of : $$\widehat{f} = \arg\min_{\widehat{g}}~\|{\mathbf}T[\widehat{g}]\|_* + \lambda\|P_{\Gamma}(\widehat{g} - \widehat{f})\|_2^2
\label{eq:nucnormmin}$$ where $\|\cdot\|_*$ denotes the nuclear norm, i.e. the absolute sum of singular values, $\lambda$ is a tunable parameter, and $P_{\Gamma}$ is the projection onto the sampling set $\Gamma$. A standard approach to solving is by an iterative singular value soft-thresholding algorithm, which requires an SVD of the estimate $T[\widehat{g}]$ at each step. Due to the size of $T[\widehat{g}]$, such an algorithm is computationally prohibitive in this case. Instead we use the SVD-free algorithm proposed in [@signoretto2013svd], which involves introducing auxiliary variables ${\mathbf}U \in \mathbb{C}^{M\times r}$ and ${\mathbf}V \in \mathbb{C}^{N\times r}$ via the well-known relation $\|{\mathbf}X\|_* = \min_{{\mathbf}X = {\mathbf}U {\mathbf}V^H} \|{\mathbf}U\|^2_F + \|{\mathbf}V\|^2_F$, and enforcing the constraint ${\mathbf}T[\widehat{g}] = {\mathbf}U {\mathbf}V^H$, with the ADMM algorithm.
In Fig. \[fig:lowrank\_SL\] we demonstrate the ability of the proposed algorithm to recover a piecewise constant signal from few of its uniform low-resolution Fourier samples. We experiment on simulated data obtained from analytical MRI phantoms derived in [@guerquin2012realistic]. We extrapolate from $65\times49=3185$ analytical Fourier samples of the Shepp-Logan phantom to a $256\times256$ grid ($\approx$20-fold undersampling), and recover the signal by performing a inverse DFT. Note that the ringing artifacts observed in the recovery are to be expected due to fact we are recovering exact Fourier coefficients of the signal, and could be removed with mild post-processing.
Conclusion
==========
We propose an extension of the annihilating filter method to a wide class of 2-D piecewise smooth functions whose edges are supported on level set of a band-limited function. This enables us to recover an exact continous domain representation of the edge set from few low-frequency Fourier samples. In the case of piecewise constant signals, we derive conditions of the necessary and sufficient number of Fourier samples to ensure exact recovery of the edge set. Lastly, we prosed one-stage algorithm to recover piecewise constant signals by extrapolating the signal in Fourier domain. We demonstrate that we may accurately recover MRI phantoms from few of their low-resolution Fourier samples.
|
---
author:
- |
Dipak Chaudhari\
Computer Science and Engg.\
IIT Bombay\
[dipakc@cse.iitb.ac.in]{} Om P. Damani\
Computer Science and Engg.\
IIT Bombay\
[damani@cse.iitb.ac.in]{} Srivatsan Laxman\
Microsoft Research India\
Bangalore\
[slaxman@microsoft.com]{}
bibliography:
- 'references.bib'
title: 'Lexical Co-occurrence, Statistical Significance, and Word Association'
---
|
---
abstract: |
Along with the prosperous data science activities, the importance of provenance during data science project lifecycle is recognized and discussed in recent data science systems research. Increasingly modern data science platforms today have non-intrusive and extensible provenance ingestion mechanisms to collect rich provenance and context information, handle modifications to the same file using distinguishable versions, and use [*graph*]{} data models (e.g., property graphs) and query languages (e.g., Cypher) to represent and manipulate the stored provenance/context information. Due to the schema-later nature of the metadata, multiple versions of the same files, and unfamiliar artifacts introduced by team members, the “provenance graph” is verbose and evolving, and hard to understand; using standard graph query model, it is difficult to compose queries and utilize this valuable information.
In this paper, we propose two high-level graph query operators to address the verboseness and evolving nature of such provenance graphs. First, we introduce a [**graph segmentation**]{} operator, which queries the *retrospective provenance* between a set of source vertices and a set of destination vertices via flexible boundary criteria to help users get insight about the derivation relationships among those vertices. We show the semantics of such a query in terms of a context-free grammar, and develop efficient algorithms that run orders of magnitude faster than state-of-the-art. Second, we propose a [**graph summarization**]{} operator that combines similar segments together to query *prospective provenance* of the underlying project. The operator allows tuning the summary by ignoring vertex details and characterizing local structures, and ensures the provenance meaning using path constraints. We show the optimal summary problem is PSPACE-complete and develop effective approximation algorithms. The operators are implemented on top of a property graph backend. We evaluate our query methods extensively and show the effectiveness and efficiency of the proposed methods.
author:
- '[^1]'
bibliography:
- 'main.bib'
title: Understanding Data Science Lifecycle Provenance via Graph Segmentation and Summarization
---
\[fig:example\]
Related Work {#sec:related_work}
============
Provenance studies can be roughly categorized in two types: data provenance and workflow provenance. Data provenance is discussed in dataflow systems, such as RDBMS, Pig Latin, and Spark [@survey_chiew@ftdb09; @lipstick@pvldb11; @titian@pvldb15], while workflow provenance studies address complex interactions among high-level conceptual components in various computational tasks, such as scientific workflows, business processes, and cybersecurity [@workflow_survey@cse08; @bpql_milo@vldb06; @linuxprov_abates@atc15]. Unlike query facilities in scientific workflow provenance systems [@workflow_survey@cse08], their processes are predefined in *workflow skeletons*, and multiple executions generate different instance-level provenance *run graphs* and have clear boundaries. Taking advantages of the skeleton, there are lines of research for advanced ancestry query processing, such as defining user views over such skeleton to aid queries on verbose run graphs [@zoom_penn@icde08], querying reachability on the run graphs efficiently [@reachability_optimallabeling_upenn@sigmod10], storing run graphs generated by the skeletons compactly [@compression_bertram@edbt09], and using visualization as examples to ease query construction [@visualization@vis05].
Most relevant work is querying evolving script provenance [@ingestions_noworkflow@ipaw14; @noworkflow_evolution@ipaw16]. Because script executions form clear boundary, query facilities to visualize and difference execution run graphs are proposed. In our context, as there are no clear boundaries of run graphs, it is crucial to design query facilities allowing the user to express the logical run graph segments and specify the boundary conditions first. Our method can also be applied on script provenance by segmenting within and summarizing across evolving run graphs.
Recently, there is emerging interest in developing systems for managing different aspects in the modeling lifecycle, such as building modeling lifecycle platforms [@google_tfx@kdd17], accelerating iterative modeling process [@columbus_tods16p], managing developed models [@modeldb_HILDA16p; @modelhub@icde17], organizing lifecycle provenance and metadata [@ground@cidr17; @provdb@hilda17; @amazon_metadata@learningsys17], auto-selecting models [@model_selection_arun_record15p], hosting pipelines and discovering reference models [@openml@kddexp13; @discovery@deem17], and assisting collaboration [@kandogan2015labbookp]. Issues of querying evolving and verbose provenance effectively are not considered in that work.
Parsing CFL on graphs and using it as query primitives has been studied in early theory work [@valiant_subcublic@jcss75; @cflr_to_datalog@pods90], later used widely in programming analysis [@reps_pa_reachability@islp97] and other domains such as bioinformatics [@alon@amia02] which requires high expressiveness language to constrain paths. Recently it is discussed as a graph query language [@cflpq@icdt14] and SPARQL extension [@cflsparql@iswc16] in graph databases. In particular, CFLR is a general formulation of many program analysis tasks on graph representations of programs. Most of the CFL used in program analysis is a Dyck language for matching parentheses [@reps_pa_reachability@islp97]. On provenance graphs, our work is the first to use CFL to constrain the path patterns to the best of our knowledge. CFL allows us to capture path similarities and constrain lineages in evolving provenance graphs. We envision many interesting provenance queries would be expressed in CFLR and support human-in-the-loop introspection of the underlying workflow.
Answering CFLR on graphs in databases has been studied in [@cflr_to_datalog@pods90], and shown equivalent to evaluating Datalog chain programs. Reps [@reps_pa_reachability@islp97; @reps_cflr_baseline@tcs00] describe a cubic time algorithm to answer CFLR and is widely used in program analysis. Later it is improved in [@chaudhuri_subcublic@popl08] to subcubic time. Because general CFLR is generalization of CFL parsing, it is difficult to improve [@reps_pa_reachability@islp97]. On specific data models and tasks, faster algorithms for Dyck language reachability are discussed in the PL community [@treecfl@esop09; @fastcfl@pldi13]. Our work can be viewed as utilizing provenance graph properties and rewriting CFG to improve CFLR evaluation.
Most work on graph summarization [@summary_tutorial@pvldb17] focuses on finding smaller representations for a very large graph by methods such as compression [@rastogi_mdl@sigmod08], attribute-aggregation [@yytian_summary@sigmod08] and bisimulation [@bisimulation@icde02]; while there are a few works aiming at combining a set of query-returned trees [@xmlresults@sigmod08] or graphs [@yinghui_summary@pvldb13] to form a compact representation. Our work falls into the latter category. Unlike other summarization techniques, our operator is designed for provenance graphs which include multiple types of vertices rather than a single vertex type [@graph_cube@sigmod11]; it works on query results rather than entire graph structure [@yytian_summary@sigmod08; @rastogi_mdl@sigmod08; @agg_lucmoreau@gam15]; the summarization requirements are specific to provenance graphs rather than returned trees [@xmlresults@sigmod08] or keyword search results [@yinghui_summary@pvldb13]. We also consider property aggregations and provenance types to allow tuning provenance meanings, which is not studied before to the best of our knowledge.
Conclusion {#sec:conclusion}
==========
We described the key challenges in querying provenance graphs generated in evolving workflows without predefined skeletons and clear boundaries, such as the ones collected by lifecyle management systems in collaborative analytics projects. At query time, as the users only have partial knowledge about the ingested provenance, due to the schema-later nature of the properties, multiple versions of the same files, unfamiliar artifacts introduced by team members, and enormous provenance records collected continuously. Just using standard graph query model is highly ineffective in utilizing the valuable information. We presented two graph query operators to address the verboseness and evolving nature of such provenance graphs. First, the segmentation operator allows the users to only provide the vertices they are familiar with and then induces a subgraph representing the retrospective provenance of the vertices of interest. We formulated the semantics of such a query in a context free language, and developed efficient algorithms on top of a property graph backend. Second, the summarization operator combines the results of multiple segmentation queries and preserves provenance meanings to help users understand similar and abnormal behavior. Extensive experiments on synthetic provenance graphs with different project characteristics show the operators and evaluation techniques are effective and efficient. The operators are also applicable for querying provenance graphs generated in other scenarios where there are no workflow skeletons, e.g., cybersecurity and system diagnosis.
Notation Table {#apdx:notation}
--------------
We summarize the notations used for defining operators’ semantics in the paper in Table \[tb:notations\].
$${\small{
\begin{aligned}
% \text{\sc Qd} &\rightarrow\quad v_j & \forall v_j \in \text{\segDestination} \\
% \text{\sc Lg} &\rightarrow\quad \text{\wasGeneratedByInv}\ \text{\sc Qd} \\
% &\quad|\quad \text{\wasGeneratedByInv}\ \text{\sc Re} \\
% \text{\sc Rg} &\rightarrow\quad \text{\sc Lg}\ \text{\wasGeneratedBy} \\
% \text{\sc La} &\rightarrow\quad \text{\activity}\ \text{\sc Rg} \\
% \text{\sc Ra} &\rightarrow\quad \text{\sc La}\ \text{\activity} \\
% \text{\sc Lu} &\rightarrow\quad \text{\usedInv}\ \text{\sc Ra} \\
% \text{\sc Ru} &\rightarrow\quad \text{\sc Lu}\ \text{\used} \\
% \text{\sc Le} &\rightarrow\quad \text{\entity}\ \text{\sc Ru} \\
% \text{\sc Re} &\rightarrow\quad \text{\sc Le}\ \text{\entity} \\
r_0: \text{\sc Qd} &\rightarrow\quad v_j & \forall v_j \in &\text{\segDestination} & \\
r_1: \text{\sc Lg} &\rightarrow\quad \text{\wasGeneratedByInv}\ \text{\sc Qd} & r_3: \text{\sc La} &\rightarrow\quad \text{\activity}\ \text{\sc Rg} & r_6: \text{\sc Ru} &\rightarrow\quad \text{\sc Lu}\ \text{\used} \\
&\quad|\quad \text{\wasGeneratedByInv}\ \text{\sc Re} & r_4: \text{\sc Ra} &\rightarrow\quad \text{\sc La}\ \text{\activity} & r_7: \text{\sc Le} &\rightarrow\quad \text{\entity}\ \text{\sc Ru} \\
r_2: \text{\sc Rg} &\rightarrow\quad \text{\sc Lg}\ \text{\wasGeneratedBy} & r_5: \text{\sc Lu} &\rightarrow\quad \text{\usedInv}\ \text{\sc Ra} & r_8: \text{\sc Re} &\rightarrow\quad \text{\sc Le}\ \text{\entity} \\
\end{aligned}
}}$$
Segmentation Operation {#apdx:seg_algs}
----------------------
: [@chaudhuri_subcublic@popl08] (shown in Alg. \[alg:cflrb\]) is a subcubic algorithm to solve general CFLR problem. Given a CFG, works on its normal form [@intr_automata@3ed], where each production has at most two RHS symbols, i.e., $N \rightarrow AB$ or $N \rightarrow A$. We show the normal form of in Fig. \[fig:simprov\_normalized\] (domain of LHS of each production rule is shown in the caption). At a high level, the algorithm traverses the graph and uses grammar as a guide to find new production facts $N(i,j)$, where $N$ is a LHS nonterminal, $i,j$ are graph vertices, and the found fact $N(i,j)$ denotes that there is a path from $i$ to $j$ whose path label satisfies $N$. To elaborate, similar to BFS, it uses a worklist $W$ (queue) to track newly found fact $N(i,j)$ and a *fast set* data structure $H$ with time complexity $O(n/log(n))$ for set diff/union and $O(1)$ for insert to memorize found facts.
In the beginning, all facts $F(i,j)$ from all single RHS symbol rules $F \rightarrow A$ are enqueued. In case ($r_0$: [Qd]{} in Fig. \[fig:simprov\_normalized\]), each $v_j \in$ is added to $W$ as $\text{\sc Qd}(v_j, v_j)$. From $W$, it processes one fact $F(i,j)$ at a time until $W$ is empty. When processing a dequeued fact $F(i,j)$, if $F$ appears in any rule in the following cases: $$\begin{aligned}
& N(i,j)\rightarrow F(i,j); \\
& N(i,v)\rightarrow F(i,j)A(j,v); \\
& N(u,j)\rightarrow A(u,i)F(i,j)
\end{aligned}$$ the new LHS fact $N(i,v)$ is derived by set diff $\{v \in A(j,v)\}\setminus \{v \in N(i,v)\}$ or $N(u,j)$ by $\{u \in A(u,i)\}\setminus \{u \in N(u,j)\}$ in $H$. As in , only the later two cases are present, in Alg. \[alg:cflrb\] line 4 6 and line 7 9 show the details of the algorithm. Row and Col accesses outgoing and incoming neighbors w.r.t. to a LHS symbol and is implemented using the fast set data structure. Then the new facts of $N$ are added to $H$ to avoid repetition and $W$ to explore it later. Once $W$ is empty, the start symbol $L$ facts $L(i,j)$ in $H$ include all vertices pairs $(i,j)$ which have a path with label that satisfies $L$. If a grammar has $k$ rules, then the worst case time complexity is $O(k^3 n^3 /log(n)$ and $W$ ad $H$ takes $O(k n^2)$ space. If path is needed, a parent table would be used similar to BFS using standard techniques. In (Fig. \[fig:simprov\_normalized\]), the start symbol is [Re]{}, $\forall v_i \in \text{\segSource}$, $\text{\sc Re}(v_i, v_t)$ facts include all $v_t$, s.t. between them there is $\text{\pathlabel}({\text{\segsubpath{i}{t}}}) \in \text{\cfglanguage{\similarPathPatternRule}}$.
: The proof of Lemma 1 is the following: On normal form (Fig. \[fig:simprov\_normalized\]), for $i \in [1,8]$, derives $r_i$ LHS facts by a $r_{i-1}$ LHS fact dequeued from $W$ (Note it also derives $r_1$ from $r_8$). For $i \in \{1,2\}$, $r_i(u,v)$ uses edges in the graph during the derivation, e.g., from $r_8$ LHS $\text{\sc Re}$ to $r_1: \text{\sc Lg}(u,v) \rightarrow \text{\wasGeneratedByInv}(u, k)\ \text{\sc Re}(k, v)$. As $\text{\sc Re}(k,v)$ can only be in the worklist $W$ once, we can see that each 3-tuple $(u,k,v)$ is formed only once on the RHS and there are at most $|\text{\wasGeneratedBy}||\text{\entity}|$ of such 3-tuples. To make sure $\text{\sc Lg}(u,v)$ is not found before, $H$ is checked. If not using fast set but a $O(1)$ time procedure for each instance $(u,k,v)$, then it takes $O(|\text{\wasGeneratedBy}||\text{\entity}|)$ to produce the LHS; on the other hand, if using a *fast set* on $u's$ domain for each $u$, for each $\text{\sc Re}(k,v)$, $O({|\text{\activity}|}/{\log|\text{\activity}|})$ time is required, thus it takes $O({|\text{\activity}||\text{\entity}|^2}/{\log|\text{\activity}|})$ in total. Applying similar analysis on $r_5$ and $r_6$ using to derive new facts, we can see it takes $O({|\text{\entity}||\text{\activity}|^2}/{\log|\text{\entity}|})$ with fast set and $O(|\text{\used}||\text{\activity}|)$ without fast set. Finally $r_3, r_4$ and $r_7, r_8$ can be viewed as following a vertex self-loop edge and do not affect the complexity result.
Query Evaluation Discussion
---------------------------
Validness of provenance graph is an important constraint [@prov_constraints@w3c_tr13; @prov_segmentation@tapp16]. In our system, the operator does not introduce new vertices or edge. As long as the original provenance graph is valid, the induced subgraph is valid. However, at query time, the boundaries criteria could possibly let the operator result exclude important vertices. As an interactive system, we leave it to the user to adjust the vertex set of interest and boundary criteria in their queries.
: For other purposes where the two-step approaches are not ideal, the exclusion constraints $\text{\boundaryCriteria}_v$ and $\text{\boundaryCriteria}_e$, and expansion criteria $\text{\boundaryCriteria}_x$ can be evaluated together using , and with small modifications on the grammar. In the label function $\mathcal{F}_v$ of $\text{\boundaryCriteria}_v$ can be applied at $r_0, r_3, r_4, r_7, r_8$ on or , while $\mathcal{F}_e$ of $\text{\boundaryCriteria}_e$ can be applied at rest of the rules involving and . For and , $\mathcal{F}_v$ and $\mathcal{F}_e$ can be applied together at $r'_1, r'_2$.
: We mainly focus on developing ad-hoc query evaluation schemes. As of now, the granularity of provenance in our context is at the level of commands executions, the number of activities are constrained by project members’ work rate. In case when the graph becomes extremely large, indexing techniques and incremental algorithms are more practical. We leave them as future steps.
Summarization Operation
-----------------------
We consider alternative of formulation of the summary graph. One way to combine segment graphs is to use context-free graph grammars (CFGG) [@bpql_milo@vldb06] which are able to capture recursive substructures. However without a predefined workflow skeleton CFGG, and due to the workflow noise resulting from the nature of analytics workload, inferring a minimum CFGG from a set of subgraphs is not only an intractable problem, but also possibly leads to complex graph grammars that are more difficult to be understood by the users [@subdue_cfgg@ijait04]. Instead, we view it as a graph summarization task by grouping vertices and edges in the set of segments to a .
Though requiring all paths in must exist in some segment may look strict and affect the compactness of the result, operator allows using the property aggregation () and provenance types () to tune the compactness of . Due to the rigidness and the utility of provenance, allowing paths that do not exist in any segment in the summary would cause misinterpretation of the provenance, thus would not be suitable for our context. In situations where extra paths in the summary graph is not an issue, problems with objectives such as minimizing the number of introduced extra paths, and minimizing the description length are interesting ones to be explored further. We leave them as future steps.
System Design Decision
----------------------
Note the design decision of using a general purpose native graph backend (Neo4j) for high-performance provenance ingestion may not be ideal, as the volume of ingested provenance records would be very large in some applications, e.g., whole-system provenance recording at kernel level [@pass_harvard@atc06; @linuxprov_abates@atc15] would generate GBs of data in minutes. The support of flexible and high performance graph ingestion on modern graph databases and efficient query evaluation remain an open question [@dda_microsoft@cidr17]. We leave the issue to support similar operators for general graph for our future steps. The proposed techniques in the paper focus on enabling better utilization of the ingested provenance information via novel query facilities and are orthogonal to the storage layer.
[^1]: Work done at UMD, now at Google.
|
---
abstract: |
This paper deals with a new Bayesian approach to the two-sample problem. More specifically, let $x=(x_1,\ldots,x_{n_1})$ and $y=(y_1,\ldots,y_{n_2})$ be two independent samples coming from unknown distributions $F$ and $G$, respectively. The goal is to test the null hypothesis $\mathcal{H}_0:~F=G$ against all possible alternatives. First, a Dirichlet process prior for $F$ and $G$ is considered. Then the change of their Cramér-von Mises distance from a priori to a posteriori is compared through the relative belief ratio. Many theoretical properties of the procedure have been developed and several examples have been discussed, in which the proposed approach shows excellent performance.
<span style="font-variant:small-caps;">Keywords:</span> Dirichlet process, hypothesis testing, relative belief inferences, two-sample problem.
[ **MSC 2000**]{} 62F15, 62N03
author:
- 'Luai Al-Labadi[^1]'
title: 'The Two-Sample Problem Via Relative Belief Ratio'
---
Introduction
============
For two independent samples, the *two-sample problem* is concerned to determine whether the two samples are generated from the same population. Although it is considered an old problem in statistics, it always attracts the attention of researchers due to it applications in different fields. For instance, in medical studies, one may want to asses the efficiency of a new drug to two groups of patients.
The two-sample problem can be stated formally as follows. Given two independent samples $x=(x_1,\ldots,x_{n_1}) \overset {i.i.d.} \sim F$ and $y=(y_1,\ldots,y_{n_2}) \overset {i.i.d.} \sim G$, with $F$ and $G$ being unknown continuous cumulative distribution functions (cdf’s), the aim is to test the null hypothesis $\mathcal{H}_0:~F=G$ against all other alternatives.
The methodology developed in this paper is Bayesian and it is inspired from the recent work of Al-Labadi and Evans (2018) for model checking. At first, two Dirichlet processes $DP(a_1, H_1)$ and $DP(a_2,H_2)$ are considered as priors for $F$ and $G$, respectively. Then the concentration of the posterior distribution of the distance between the two processes is compared to the concentration of the prior distribution of the distance between the two processes. If the posterior is more concentrated about the model than the prior, then this is evidence in favor of $\mathcal{H}_0$ and if the posterior is less concentrated, then this is evidence against $\mathcal{H}_0$. This comparison is made through a particular measure of evidence known as the *relative belief ratio*, which will indicate whether there is evidence for or against $\mathcal{H}_0$. Moreover, a calibration of this evidence is provided concerning whether there is strong or weak evidence for or against the hypothesis. The proposed methodology is simple, general and does not require obtaining a closed form of the relative belief ratio. More details about relative belief ratio are highlighted in Section 2 of this paper.
Developing procedures for hypothesis testing has recently given a considerable attention in the literature of Bayesian nonparametric inference. A main stream of these procedures has focused on embedding the suggested model as a null hypothesis in a larger family of distributions. Then priors are placed on the null and the alternative and a Bayes factor is computed. For instance, Florens, Richard, and Rolin (1996) used a Dirichlet process for the prior on the alternative. Carota and Parmigiani (1996), Verdinelli and Wasserman (1998), Berger and Guglielmi (2001) and McVinish, Rousseau, and Mengersen (2009) considered a mixture of Dirichlet processes, a mixture of Gaussian processes, a mixture of Pólya trees and a mixture of triangular distributions, respectively, for the prior on the alternative. Another approach for model testing is based on placing a prior on the true distribution generating the data and measuring the distance between the posterior distribution and the proposed one. Swartz (1999) and Al-Labadi and Zarepour (2013, 2014a) considered the Dirichlet process prior and used the Kolmogorov distance to derive a goodness-of-fit test for continuous models. Viele (2000) used the Dirichlet process and the Kullback-Leibler distance to test discrete models. Hsieh (2011) used the Pólya tree prior and the Kullback-Leibler distance to test continuous distributions. The work described above focuses only on goodness of fit tests and model checking. With regard to the two-sample problem, the literature is very scarce and scattered. Some exceptions include the remarkable work of Holmes, Caron, Griffin, and Stephens (2015) who developed a way to compute the Bayes factor for testing the null hypothesis through the marginal likelihood of the data with Pólya tree priors centered either subjectively or using an empirical procedure. Under the null hypothesis, they modeled the two samples to come from a single random measure distributed as a Pólya tree, whereas under the alternative hypothesis the two samples come from two separate Pólya tree random measures. Ma and Wong (2011) allowed the two distributions to be generated jointly through optional coupling of a Pólya tree prior. Borgwardt and Ghahramani (2009) discussed two-sample tests based on Dirichlet process mixture models and derived a formula to compute the Bayes factor in this case. An extension of the Bayes factor approach based on Pólya tree priors to cover censored and multivariate data was proposed by Chen and Hanson (2014). Huang and Ghosh (2014) considered the two-sample hypothesis testing problems under Pólya tree priors and Lehmann alternatives. Shang and Reilly (2017) introduced a class of tests, which use the connection between the Dirichlet process prior and the Wilcoxon rank sum test. They also extend their idea using the Dirichlet process mixture prior and developed a Bayesian counterpart to the Wilcoxon rank sum statistic and the weighted log rank statistic for right and interval censored data. In a recent work, Al-Labadi and Zarepour (2017) proposed a method based on the Kolmogorov distance and samples from the Dirichlet process to assess the equality of two unknown distributions, where the distance between two posterior Dirichlet processes is compared with a reference distance. The parameters of the two Dirichlet processes are chosen so that any discrepancy between the posterior distance and the reference distance is only attributed to the difference between the two samples.
In Section 3, the Dirichlet process prior $DP(a,H)$ is briefly reviewed. In Section 4, the Cramér-von Mises distance between two Dirichlet processes is considered and several of its theoretical properties are developed. Section 5 addresses setting parameters of the two Dirichlet processes. In Section 6, a computational algorithm of the approach is developed. Section 7 presents several examples where the behaviour of the approach is inspected. Finally, some concluding remarks are made in Section 8. The proofs are placed in the Appendix.
Relative Belief Ratios
======================
In this section, for the reader’s convenience, some background of relative belief ratios is provided. For more details about this topic consult, for example, Evans (2015). Let $\{f_{\theta}:\theta\in\Theta\}$ be a collection of densities on a sample space $\mathcal{X}$ and $\pi$ be a prior on $\Theta.$ The posterior distribution of $\theta$ given that data $x$ is $\pi(\theta\,|\,x)=\pi(\theta)f_{\theta}(x)/\int
_{\Theta}\pi(\theta)f_{\theta}(x)\,d\theta$. For an arbitrary parameter of interest $\psi=\Psi(\theta),$ the prior and posterior densities of $\psi$ are denoted by $\pi_{\Psi}$ and $\pi_{\Psi}(\cdot\,|\,x),$ respectively. The relative belief ratio for a value $\psi$ is then defined by $RB_{\Psi}(\psi\,|\,x)=\lim_{\delta\rightarrow0}\Pi_{\Psi
}(N_{\delta}(\psi\,)|\,x)/\Pi_{\Psi}(N_{\delta}(\psi\,))$, where $N_{\delta
}(\psi\,)$ is a sequence of neighbourhoods of $\psi$ converging nicely (see, for example, Rudin (1974)) to $\psi$ as $\delta\rightarrow0.$ Quit generally $$RB_{\Psi}(\psi\,|\,x)=\pi_{\Psi}(\psi\,|\,x)/\pi_{\Psi}(\psi), \label{relbel}$$ the ratio of the posterior density to the prior density at $\psi.$ That is, $RB_{\Psi}(\psi\,|\,x)$ is measuring how beliefs have changed that $\psi$ is the true value from *a priori* to *a posteriori*. Note that, a relative belief ratio is similar to a Bayes factor, as both are measures of evidence, but the latter measures this via the change in an odds ratio. A discussion about the relationship between relative belief ratios and Bayes factors is detailed in (Baskurt and Evans, 2013). In particular, when a Bayes factor is defined via a limit in the continuous case, the limiting value is the corresponding relative belief ratio.
By a basic principle of evidence, $RB_{\Psi}(\psi\,|\,x)>1$ means that the data led to an increase in the probability that $\psi$ is correct, and so there is evidence in favour of $\psi,$ while $RB_{\Psi}(\psi\,|\,x)<1$ means that the data led to a decrease in the probability that $\psi$ is correct, and so there is evidence against $\psi,$. Clearly, when $RB_{\Psi}(\psi\,|\,x)=1$, then there is no evidence either way.
Thus, the value $RB_{\Psi}(\psi_{0}\,|\,x)$ measures the evidence for the hypothesis $\mathcal{H}_{0}=\{\theta:\Psi(\theta)=\psi_{0}\}.$ It is also important to calibrate whether this is strong or weak evidence for or against $\mathcal{H}_{0}$. As suggested in Evans (2015), a useful calibration of $RB_{\Psi}(\psi_{0}\,|\,x)$ is obtained by computing the tail probability $$\Pi_{\Psi}(RB_{\Psi}(\psi\,|\,x)\leq RB_{\Psi}(\psi_{0}\,|\,x)\,|\,x).
\label{strength}$$ One way to view (\[strength\]) is as the posterior probability that the true value of $\psi$ has a relative belief ratio no greater than that of the hypothesized value $\psi_{0}.$ When $RB_{\Psi}(\psi_{0}\,|\,x)<1,$ so there is evidence against $\psi_{0},$ then a small value for (\[strength\]) indicates a large posterior probability that the true value has a relative belief ratio greater than $RB_{\Psi}(\psi_{0}\,|\,x)$ and so there is strong evidence against $\psi_{0}.$ When $RB_{\Psi}(\psi_{0}\,|\,x)>1,$ so there is evidence in favour of $\psi_{0},$ then a large value for (\[strength\]) indicates a small posterior probability that the true value has a relative belief ratio greater than $RB_{\Psi}(\psi_{0}\,|\,x))$ and so there is strong evidence in favour of $\psi_{0},$ while a small value of (\[strength\]) only indicates weak evidence in favour of $\psi_{0}.$
The Dirichlet Process
=====================
In this section, a concise summary of the Dirichlet process is given. Because of its attractive features, the Dirichlet process, formally introduced in Ferguson (1973), is considered the most well-known and widely used prior in Bayesian nonparametric inference. Consider a space $\mathfrak{X}$ with a $\sigma-$algebra $\mathcal{A}$ of subsets of $\mathfrak{X}$. Let $H$ be a fixed probability measure on $(\mathfrak{X},\mathcal{A})$, called the *base measure*, and $a$ be a positive number, called the *concentration parameter*. Following Ferguson (1973), a random probability measure $P=\left\{ P(A)\right\} _{A\in\mathcal{A}}$ is called a Dirichlet process on $(\mathfrak{X},\mathcal{A})$ with parameters $a$ and $H$, denoted by $DP(a,H)$, if for any finite measurable partition $\{A_{1},\ldots,A_{k}\}$ of $\mathfrak{X}$ with $k \ge 2$, $\left( P(A_{1}),\ldots\,P(A_{k})\right)\sim \text{Dirichlet}(aH(A_{1}),\ldots,$ $aH(A_{k}))$. It is assumed that if $H(A_{j})=0$, then $P(A_{j})=0$ with a probability one. Note that, for any $A\in\mathcal{A},$ $P(A) \sim \text{Beta}(aH(A),(1-H(A))$ and so ${E}(P(A))=H(A)\ \ $ and ${Var}(P(A))=H(A)(1-H(A))/(1+a).$ Thus, $G$ can be viewed as the center of the process. On the other hand, $a$ controls concentration, as the larger value of $a$, the more likely that $P$ will be close to $G$. We refer the reader to Al-Labadi and Abdelrazeq (2017) for additional interesting asymptotic properties of the Dirichlet process and other nonparametric priors.
A distinctive feature of the Dirichlet process, among many other nonparametric priors, is its conjugacy property. Specifically, if $x=(x_{1},\ldots,x_{n})$ is a sample from $P\sim DP(a,H)$, then the posterior distribution of $P$ is $P\,|\,x=P_x\sim DP(a+n,H_{x})$ where $$H_{x}=a(a+n)^{-1}H+n(a+n)^{-1}F_{n}, \label{DP_posterior}$$ with $F_{n}=n^{-1}\sum_{i=1}^{n}\delta_{{x}_{i}}$ and $\delta_{x_{i}}$ is the Dirac measure at $x_{i}.$ Notice that, $H_{x}$ is a convex combination of the prior base distribution and the empirical distribution. Clearly, $H_{x}\to H$ as $a \to \infty$ while $H_{x}\to F_n$ as $a \to 0$.
Following Ferguson (1973), $P\sim{DP}(a,H)$ has the following series representation $$P=\sum_{i=1}^{\infty}J_{i} \delta_{Y_{i}}, \label{series-dp}$$ where $\Gamma_{i}=E_{1}+\cdots+E_{i}$, $E_{i} \overset{i.i.d.}\sim \text{exponential}(1)$, $Y_{i} \overset{i.i.d.}\sim H$ independent of $\Gamma_{i}$, $L(x)=a\int_{x}^{\infty}t^{-1}e^{-t}dt,x>0,$ $L^{-1}(y)=\inf
\{x>0:L(x)\geq y\}$ and $J_{i}=L^{-1}(\Gamma_{i})/\sum_{i=1}^{\infty
}{L^{-1}(\Gamma_{i})}$. It follows clearly from (\[series-dp\]) that a realization of the Dirichlet process is a discrete probability measure. This is true even when the base measure is absolutely continuous. One could resemble the discreteness of $P$ with the discreteness of $F_n$. Note that, since data is always measured to finite accuracy, the true distribution being sampled from is discrete. This makes the discreteness property of $P$ with no practical significant limitation. Indeed, by imposing the weak topology, the support for the Dirichlet process is quite large. Specifically, the support for the Dirichlet process is the set of all probability measures whose support is contained in the support of the base measure. This means if the support of the base measure is $\mathfrak{X}$, then the space of all probability measures is the support of the Dirichlet process. In particular, if we have a normal base measure, then the Dirichlet process can choose any probability measure.
Zarepour and Al-Labadi (2012) derived the following series approximation with monotonically decreasing weights for the Dirichlet process $$P_{N}=\sum_{i=1}^{N}J_{i}\delta_{Y_{i}}, \label{eq11}$$ where $Y_{i}$ and $\Gamma
_{i}$ are as defined in (\[series-dp\]), $G_{a/N}$ be the co-cdf of the g$\text{amma}(a/N,1)$ distribution and $J_{i}={G_{a/N}^{-1}(\Gamma_{i}/\Gamma_{N+1})/}\sum
_{j=1}^{N}{G_{a/N}^{-1}(\Gamma_{j}/\Gamma_{N+1})}$. They proved that, as $N \to
\infty$, $P_{N}$ converges almost surely to (\[series-dp\]). Note that ${G_{a/N}^{-1}(p)}$ is the $(1-p)$-th quantile of the g$\text{amma}(a/N,1)$ distribution. This provides the following algorithm. $\bigskip$
**Algorithm A: Approximately generating a value from** $DP(a,H)$****
1\. Fix a relatively large positive integer $N$.****
2\. For $i=1,\ldots,N$, generate $Y_{i}\overset{i.i.d.} \sim H$. ****
3\. Independent of $\left( Y_{i}\right) _{1\leq
i\leq N}$, for $i=1,\ldots,N+1,$ generate $E_{i}\overset{i.i.d.} \sim\,$exponential$(1)$ and put $\Gamma_{i}=E_{1}+\cdots+E_{i}.$****
4\. For $i=1,\ldots,N$, compute $G_{a/N}^{-1}\left( {\Gamma_{i}}/{\Gamma_{N+1}}\right) .$****
5\. Use $P_N$ in (\[eq11\]) to obtain an approximate value from $DP(a,H)$.$\bigskip$
For other simulation methods for the Dirichlet process, see, for instance, Bondesson (1982), Sethuraman (1994), Wolpert and Ickstadt (1998) and Al-Labadi and Zarepour (2014b).
Throughout the paper, the notation $P$ could refer to either a probability measure or its corresponding cdf where the context determines the appropriate interpretation. That is, $P((-\infty,t])=P(t)$ for all $t \in \mathbb{R}$.
Cramér-von Mises Distance
=========================
A well-known and widely used distance between two distributions is the Cramér-von Mises Distance. For cdf’s $F$ and $G$ this is defined as $$d_{CvM}(F,G)=\int
_{-\infty}^{\infty}\left( F(x)-G(x)\right) ^{2}G(dx).$$
The next lemma demonstrates that, as sample sizes get large, the Cramér-von Mises distance between posterior distributions of Dirichlet processes converges to the Cramér-von Mises distance between the true distributions generated the data.
\[BSP3\] Given two independent samples $x=(x_1,\ldots,x_{n_1}) \overset {i.i.d.} \sim F$ and $y=(y_1,\ldots,y_{n_2}) \overset {i.i.d.} \sim G$, with $F$ and $G$ being continuous cdf’s. Let $P\sim DP(a_1,H_1)$, $Q\sim DP(a_2,H_2)$, $P|x=P_x$ and $Q|y=Q_y$. Then, as $n_1,n_2\to \infty$, $d(P_{x},Q_{y}) \overset{a.s.}\to d(F,G)$.
The next corollary shows that the posterior distribution of $d_{CvM}(P_x,Q_y)$ becomes concentrated around 0 as sample sizes increase if and only if $\mathcal{H}_{0}$ holds. The proof follows straightforwardly from Lemma \[BSP3\].
\[cvm4\] Let $x=(x_1,\ldots,x_{n_1}) \overset {i.i.d.} \sim F$ and $y=(y_1,\ldots,y_{n_2}) \overset {i.i.d.} \sim G$, with $F$ and $G$ being continuous cdf’s. Let $P\sim DP(a_1,H_1)$ and $Q\sim DP(a_2,H_2)$. As $n_1,n_2\to \infty$, (i) if $\mathcal{H}_{0}$ is true, then $d_{CvM}\left( P_{x},Q_{y}\right)
\overset{a.s.}{\rightarrow}0$ and (ii) if $\mathcal{H}_{0}$ is false, then $\lim\inf d_{CvM}(P_{x},Q_{y})\overset{a.s.}{>}0.$
The following result allows the use of the approximation (\[eq11\]) when considering the prior and posterior distributions of the Cramér-von Mises distance.
\[cvm3\] Let $P\sim DP(a_1,H_1)$ and $Q\sim DP(a_2,H_2)$. Let $P_{N_1}$ and $Q_{N_2}$ be two approximations of $P$ and $Q$, respectively, as defined in (\[eq11\]). Then, as $N_1,N_2 \to \infty$, $d_{CvM}\left( P_{N_1},Q_{N_2}\right) \overset{a.s.}{\rightarrow}d_{CvM}\left(
P,Q\right).$
The next lemma demonstrates that the distribution of the distance between two Dirichlet processes is independent from the base measures. This result will play a key role in the proposed approach.
\[BSP4\] Let $P \sim DP(a_1,H_1)$ and $Q\sim DP(a_1,H_2)$, where $H_1$ and $H_2$ are continuous. If $H_1=H_2$, then the distribution of $d_{CvM}\left(P,Q\right)$ does not depend on $H_1$ and $H_2$.
The Approach
============
Let $x=(x_1,\ldots,x_{n_1}) \overset {i.i.d.} \sim F$ and $y=(y_1,\ldots,y_{n_2}) \overset {i.i.d.} \sim G$ be independent samples with $F$ and $G$ being unknown continuous cdf’s. The goal to test the null hypothesis $\mathcal{H}_0:~F=G$. To this end, we use the priors $P\sim DP(a_1,H_1)$ and $Q \sim DP(a_2,H_2)$ so, by (\[DP\_posterior\]), $P|x \sim DP(a_1+n_1,H_x)$ and $Q|y \sim DP(a_1+n_1,H_y)$. From Lemma \[BSP3\], $D_{x,y}=d_{CvM}(P_x, Q_y)$ almost surely approximate $D=d_{CvM}(F,G)$. Thus, it looks clear that if $\mathcal{H}_{0}$ is true, then the posterior distribution of the distance between $P$ and $Q$ should be more concentrated about $0$ than the prior distribution of the distance between $P$ and $Q.$ For example, in Figure 1-a (see Example 1), since $\mathcal{H}_{0}$ is true, the plot of the posterior density of $D_{x,y}$ is much more concentrated about 0 than the the plot of the prior density of $D$. So, the proposed test includes a comparison of the concentrations of the prior and posterior distributions of $d_{CvM}$ via a relative belief ratio based on $d_{CvM}$ with the interpretation as discussed in$\ $Section 2.
The success of the approach depends significantly on a suitable selection of the parameters of $DP(a_1,H_1)$ and $DP(a_2,H_2)$. As illustrated below, inappropriate values of the parameters can lead to a failure in computing $d_{CvM}$. We discuss first setting values of $H_1$ and $H_2$. By Lemma \[BSP4\], the distribution of $d_{CvM}\left(P,Q\right)$ is independent from the choice of the base measures when $H_1=H_2$, where both need to be continuous. Thus, we suggest to set $H_1=H_2=N(0,1)$, although other choices of continuous distributions are certainly possible. An additional and important reason supporting the choice of $H_1=H_2$ is to avoid prior-data conflict (Evans and Moshonov, 2006; Al-Labadi and Evans, 2017). Prior-data conflict means that there is a tiny overlap between the effective support regions of $DP(a_1,H_1)$ and $DP(a_2,H_2)$. In this context, the existence of prior-data conflict can yield to a failure in computing the distribution of $d_{CvM}\left( P,Q\right) $ about 0. To avoid prior-data conflict, it is necessary that $H_1$ and $H_2$ share the same effective support (note that, $P$ and $Q$ have the same support as $H_1$ and $H_2$, respectively), which can certainly be secured by setting $H_1=H_2$. The effect of prior-data conflict is demonstrated in Section 7, Table \[tab2\].
The selection of $a_1$ and $a_2$ is also important. It is possible to consider several values of $a_1$ and $a_2$. In general, the values of $a_1$ and $a_2$ depends in $n_1$ and $n_2$, respectively. As indicated in Al-Labadi and Zarepour (2017), $a_i$ should be chosen to have a value at most $0.5n_i, i=1,2$ as otherwise the prior may become too influential. Holmes et al. (2015) recommend using values between 1 and 10 and checking the sensitivity of the results to the chosen values. The following algorithm outlines a procedure for selecting the concentration parameters.
**Algorithm B: Selection of concentration parameters**
1\. Start by setting $a_1=a_2=1$ and compute the relative belief ratio and its strength. Algorithm C in the next section addresses such computations.****
2\. Consider more concentrated priors by setting larger values of $a_1$ and $a_2$.
3\. Compute the corresponding relative belief ratio. There are two scenarios:
1. If the value of the relative belief ratio in step 1 is less (greater) than 1 and the new value is less (greater) than 1, then there is an evidence against (in favour) $\mathcal{H}_{0}$.****
2. If the value of the relative belief ratio in step 1 is greater than 1 and the new value is greater (less) than 1, then this is an evidence against (in favour) $\mathcal{H}_{0}$.****
Algorithm B is further explored in Table \[tab1\] of Section 7. In most cases, setting $a_1=a_2=1$ is found to be adequate. Holmes et al. (2015) recommend using values between 1 and 10 and checking the sensitivity of the results to the chosen values.
Computations
============
Closed forms of the densities of $D=d_{CvM}(P,Q)$ and $D_{x,y}=d_{CvM}(P_x,Q_y)$ are typically not available. Thus, the relative belief ratios need to be approximated via simulation. The following gives a computational algorithm to test $\mathcal{H}_{0}$. This algorithm is a revised version of Algorithm B of Al Labadi and Evans (2018). **Algorithm C: Relative belief algorithm for the two-sample problem**
1\. Use Algorithm A to (approximately) generate a $P$ from $DP(a_1=1,N(0,1))$ and a $Q$ from $DP(a_2=1,N(0,1))$. ****
2\. Compute $d_{CvM}(P,Q)$.****
3\. Repeat steps (1)-(2) to obtain a sample of $r_{1}$ values from the prior of $D$.****
4\. Use Algorithm A to (approximately) generate a $P_x$ from $DP(1+n_1,H_{x})$ and $Q_x$ from $DP(1+n_2,H_{y})$.****
5\. Compute $d_{CvM}(P_x,Q_y)$.****
6\. Repeat steps (4)-(5) to obtain a sample of $r_{2}$ values of $D_{x,y}$.****
7\. Let $M$ be a positive number. Let $\hat{F}_{D}$ denote the empirical cdf of $D$ based on the prior sample in (3) and for $i=0,\ldots,M,$ let $\hat{d}_{i/M}$ be the estimate of $d_{i/M},$ the $(i/M)$-th prior quantile of $D.$ Here $\hat{d}_{0}=0$, and $\hat{d}_{1}$ is the largest value of $d$. Let $\hat{F}_{D}(\cdot\,|\,x,y)$ denote the empirical cdf of $D$ based on the posterior sample in 6. For $d\in\lbrack\hat{d}_{i/M},\hat
{d}_{(i+1)/M})$, estimate $RB_{D}(d\,|\,x,y)={\pi_D(d|x,y)}/{\pi_D(d)}$ by $$\widehat{RB}_{D}(d\,|\,x,y)=M\{\hat{F}_{D}(\hat{d}_{(i+1)/M}\,|\,x,y)-\hat{F}_{D}(\hat{d}_{i/M}\,|\,x,y)\}, \label{rbest}$$ the ratio of the estimates of the posterior and prior contents of $[\hat
{d}_{i/M},\hat{d}_{(i+1)/M}).$ It follows that, we estimate $RB_{D}(0\,|\,x,y)={\pi_D(0|x,y)}/{\pi_D(0)}$ by $\widehat{RB}_{D}(0\,|\,x,y)=$ $M\widehat{F}_{D}(\hat{d}_{p_{0}}\,|\,x,y)$ where $p_{0}=i_{0}/M$ and $i_{0}$ is chosen so that $i_{0}/M$ is not too small (typically $i_{0}/M\approx0.05)$.****
8\. Estimate the strength $DP_{D}(RB_{D}(d\,|\,x,y)\leq RB_{D}(0\,|\,x,y)\,|\,x,y)$ by the finite sum $$\sum_{\{i\geq i_{0}:\widehat{RB}_{D}(\hat{d}_{i/M}\,|\,x,y)\leq\widehat{RB}_{D}(0\,|\,x,y)\}}(\hat{F}_{D}(\hat{d}_{(i+1)/M}\,|\,x,y)-\hat{F}_{D}(\hat
{d}_{i/M}\,|\,x,y)). \label{strest}$$
For fixed $M,$ as $r_{1}\rightarrow\infty,r_{2}\rightarrow\infty,$ then $\hat{d}_{i/M}$ converges almost surely to $d_{i/M}$ and (\[rbest\]) and (\[strest\]) converge almost surely to $RB_{D}(d\,|\,x)$ and $DP_{D}(RB_{D}(d\,|\,x,y)\leq RB_{D}(0\,|\,x,y)\,|\,x,y)$, respectively.
9\. As detailed in Algorithm B, repeat steps (1)-(8) for larger values of $a_1$ and $a_2$.
The following proposition establishes the consistency of the approach to the two-sample problem as sample size increases. So the procedure performs correctly as sample size increases when $\mathcal{H}_{0}$ is true. The proof follows immediately from Evans (2015), Section 4.7.1.
\[cvm6\]Consider the discretization $\{[0,d_{i_{0}/M}),[d_{i_{0}/M},d_{(i_{0}+1)/M}),\ldots,$$[d_{(M-1)/M},\infty)\}$. As $n_1,n_2\rightarrow\infty,$ (i) if $\mathcal{H}_{0}$ is true, then $$\begin{aligned}
& RB_{D}([0,d_{i_{0}/M})\,|\,x,y)\overset{a.s.}{\rightarrow}1/DP_{D}([0,d_{i_{0}/M})),\\
& RB_{D}([d_{i/M},d_{(i+1)/M})\,|\,x,y)\overset{a.s.}{\rightarrow}0\text{
whenever }i\geq i_{0},\\
& DP_{D}(RB_{D}(d\,|\,x,y)\leq RB_{D}(0\,|\,x,y)\,|\,x,y)\overset{a.s.}{\rightarrow}1,\end{aligned}$$ and (ii) if $\mathcal{H}_{0}$ is false and $d_{CvM}(P, Q)\geq d_{i_{0}/M}$, then $RB_{D}([0,d_{i_{0}/M})\,|\,x,y)\overset
{a.s.}{\rightarrow}0$ and $DP_{D}(RB_{D}(d\,|\,x,y)\leq RB_{D}(0\,|\,x,y)\,|\,x,y)\overset{a.s.}{\rightarrow}0.$
Examples
========
In this section, the approach is illustrated through three examples. In Examples 1 and 2, the methodology is assessed using simulated samples from a variety of distributions and in Example 3 an application to a real data set is presented.
The following notation is used for the distributions in the tables, namely, $N(\mu,\sigma)$ is the normal distribution with mean $\mu$ and standard deviation $\sigma$, $t_r$ is the $t$ distribution with $r$ degrees of freedom, exp$(\lambda)$ is the exponential distribution with mean $\lambda$ and $U(a,b)$ is the uniform distribution over $[a,b]$. For all cases, we set $N_1=N_2=1000$ in Algorithm A and $r_1=r_2=2000$, $M=20$ in Algorithm B. The results are also compared with the frequentist Cramér-von Mises (CvM) test. To calculate p-values of the CvM test, the **** function “cramer.test" is used. We also compared our results with the Bayesian nonparametric tests of Holmes et al. (2015) and Al-Labadi and Zarepour (2017). Since the obtained results are similar in these tests, we reported only the results of the new approach.
[**Example 1.**]{} \[example1\] Consider samples generated from the distributions in Table \[tab1\], where each sample is of size 50 (Case 1- Case 9). These distributions are also considered in Holmes et al. (2015) and Al-Labadi and Zarepour (2017). To study the sensitivity of the approach to the choice of concentration parameters, various values of $a_1$ and $a_1$ are considered. The results are reported in Table \[tab1\]. Recall that, we want $RB>1$ and the strength close to 1 when $\mathcal{H}_{0}$ is true and $RB<1$ and the strength close to 0 when $\mathcal{H}_{0}$ is false. It follows that, the methodology performs perfectly in all cases. For example, in Case 1, since $RB=9.4$ and strength$=1$, there is no reason to doubt that the two sampling distributions are not identical. On the other hand, in Case 2, since $RB=0$ and strength$=0$, the two samples are drawn from two different distributions. We point out that the standard Cramér-von Mises test failed to recognize the difference in Case 6 (i.e., $x\sim N(0,1)$ and $y\sim t_{0.5}$). Notice that, in all cases, the appropriate conclusion is attained with $a_1=a_2=1$. The other values of $a_1$ and $a_2$ considered in Table \[tab1\] support the reached conclusions.
Figure 1 provides plots of the density of the prior distance and the posterior distance for some cases in Example 1. It follows, for instance, from Figure 1 that the posterior density of the distance is more concentrated about 0 than the prior density of the distance when the two distributions are equal but not to the same degree otherwise.
Samples $a_1=a_2$ $RB$(Strength) p-values
----------------------------------------------- ----------- ---------------- ---------- --
$x\sim N(0,1)$, $y\sim N(0,1)$ 1 9.40(1) 0.2977
10 8.54(1)
20 4.48(0.776)
$x\sim N(0,1)$, $y\sim N(1,1)$ 1 0(0) 0.0000
10 0(0)
20 0(0)
$x\sim N(0,1)$, $y\sim N(0,2)$ 1 0(0) 0.0030
10 0.08(0.004)
20 0(0)
$x\sim N(0,1)$, $y\sim 0.5N(-2,1)+0.5N(2,1)$ 1 0(0) 0.0000
10 0(0)
20 0(0)
$x\sim N(0,1)$, $y\sim t_{3}$ 1 9.40(1) 0.4316
10 8.60(1)
20 5.78(1)
$x\sim N(0,1)$, $y\sim t_{0.5}$ 1 0(0) 0.1169
10 0.28(0.023)
20 0.04(0.002)
$\log x\sim N(0,1)$, $\log y\sim N(1,1)$ 1 0.02(0.001) 0.0000
10 0.02(0.001)
20 0.02(0.001)
$x\sim \text{exp}(1)$, $ y\sim \text{exp}(2)$ 1 0.10(0) 0.0020
10 0.12(0.006)
20 0.06(0.003)
$x\sim \text{exp}(1)$, $ y\sim \text{exp}(1)$ 1 9.02(1) 0.6134
10 7.30(1)
20 4.86(1)
: Relative belief ratios and strengths for testing the equality of the two distributions generating the samples in Example 1. p-values of the (frequentist) Cramér-von Mises test are also reported.[]{data-label="tab1"}
It is also interesting to consider the effect of prior-data conflict on the methodology. As discussed in Section 5, prior-data conflict will occur whenever there is only a tiny overlap between $H_1$ and $H_2$. Table \[tab2\] gives the outcomes when $x\sim N(0,1)$ and $y\sim N(1,1)$ for a particular sample of sizes $n_1=n_2=50$ with various choices of $H_1$ and $H_2$. Obviously, only when $H_1=H_2$ we get the correct conclusion. This illustrates the importance of setting $H_1=H_2$ in the priors $DP(a_1,H_1)$ and $DP(a_1,H_1)$.
\[c\][cllcc]{}Distribution & $H_1$ &$H_2$ &$RB$ (Strength)&p-value\
& $N(0,1)$ & $N(0,1)$ & $0\,(0)$&0.0000\
& $N(-5,1)$ & $N(5,1)$ & $20\,(1)$&\
&$U(10,20)$ & $N(0,1)$ & $20\,(1)$&\
& $U(10,20)$ & $U(10,20)$ & $0\,(0)$&\
Figure 2 also provides plots of the density of the prior distance and the posterior distance for the cases in Table \[tab2\]. It follows that the correct conclusion is only obtained when $H_1=H_2$.
[**Example 2.**]{} \[example2\] In this example, we explore the performance of the proposed test as sample sizes increase. We consider samples from the distributions $x\sim N(0,1)$, $y\sim N(0,1)$ (Case 1) and $x\sim N(0,1)$, $y\sim N(1,1)$ (Case 2). The results are summarized in Table \[tab3\]. It follows that the null hypothesis is not rejected in Case 1 but rejected in Case 2 . Clearly, the proposed approach works well even with small sample sizes.
--------------- ----------------- --------- -- ----------------- ---------
$RB$ (Strength) p-value $RB$ (Strength) p-value
$n_1=n_2=5$ 1.80(0.586) 0.7083 0.36(0.02) 0.1628
$n_1=n_2=10$ 1.24(0.250) 0.8132 0.48(0.064) 0.1359
$n_1=n_2=15$ 3.48(0.538) 0.9261 0.08(0.004) 0.0069
$n_1=n_2=20$ 2.64(0.422) 0.7103 0.12(0.010) 0.0170
$n_1=n_2=30$ 5.60(1) 0.5864 0.08(0.006) 0.0020
$n_1=n_2=50$ 9.40(1) 0.2977 0(0) 0.0030
$n_1=n_2=100$ 13.08(1) 0.4236 0(0) 0.0000
$n_1=n_2=200$ 17.88(1) 0.2697 0(0) 0.0000
--------------- ----------------- --------- -- ----------------- ---------
: Relative belief ratios and strengths versus p-values.[]{data-label="tab3"}
[**Example 3.**]{} \[example2\] The proposed approach of the two-sample problems is illustrated on the chickwts data in ****, where weights in grams are recorded for six groups of newly hatched chicks fed different supplements. The goal of this experiment was to measure and compare the effectiveness of various feed supplements on the growth rate of chickens. The first hypotheses of interest is to test whether the distributions of weight of chicks fed by soybean and linseed supplements differ. In the second hypothesis, we examine whether the distributions of weight of chick for sunflower and linseed groups differ. The ordered chick weights for the three samples are:
soybean: 158 171 193 199 230 243 248 248 250 267 271 316 327 329
linseed: 141 148 169 181 203 213 229 244 257 260 271 309
sunflower: 226 295 297 318 320 322 334 339 340 341 392 423
The values recorded in Table \[tab4\] do not support the evidence that the distributions of the weight of chicks fed by soybean and linseed supplements differ. On the other hand, they underline that the sunflower and linseed groups differ.
\[c\][cccc]{}Samples&$a_1=a_2$ & $RB$ (Strength)& p-value\
$x$: soybean & $y$: linseed &1 & $0.48\,(0.014)$&0.3487\
&2 & $2.50\,(0.717)$&\
&3 & $3.12\,(0.844)$&\
&4 & $ 3.14\,(0.843)$&\
&5& $3.34\,(0.833)$&\
$x$: soybean & $y$: sunflower &1 & $0\,(0)$&0\
&2 & $0\,(0)$&\
&3 & $0\,(0)$&\
&4 & $0\,(0)$&\
&5 & $0\,(0)$&\
Concluding Remarks
==================
A Bayesian approach for the two-sample problem based on the use of the Dirichlet process and relative belief has been developed. Implementing the approach is fairly simple and does not require obtaining a closed form of the relative belief ratio. Through several examples, it has been shown that the approach performs extremely well. While Cramér-von Mises distance has been used in this paper, other distance measures such as Anderson-Darling distance and the Kullback-Leibler distance are possible. It is also possible to extend the approach to cover the case of censored data.
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Proofs
======
**Proof of Lemma \[BSP3\]** For any cdf’s $P_{x}$ and $Q_{y}$, we have $d_{CvM}\left( P_{x},Q_{y}\right)
=\int_{-\infty}^{\infty}\left( P_{x}(z)-Q_{y}(z)\right) ^{2}Q_{y}(dz)$. Since $(P_{x}(z)-Q_{y}(z))^{2}\leq1$, $P_{x}(z)\overset{a.s.}{\rightarrow}F(z)$ and $Q_{y}(z)\overset{a.s.}{\rightarrow}G(z)$ (James, 2008; Al-Labadi and Abdelrazeq, 2017), the dominated convergence theorem completes the proof.
**Proof of Lemma \[cvm3\]** The proof is similar to the proof of Lemma 1. We include the proof for the sake of completeness. For cdf’s $P_{N_1}$ and $Q_{N_2}$, we have $d_{CvM}\left(P_{N_1},Q_{N_2}\right)
=\int_{-\infty}^{\infty}\left( P_{N_1}(z)-Q_{N_2}(z)\right) ^{2}$ $Q_{N_2}(dz)$. Since $(P_{N_1}(z)-Q_{N_2}(z))^{2}\leq1$, $P_{N_1}(z)\overset{a.s.}{\rightarrow}P(z)$ and $Q_{N_2}(z)\overset{a.s.}{\rightarrow}Q(z)$, the result is followed by the dominating convergence theorem.
**Proof of Lemma \[BSP4\]** Since $H_1$ is nondecreasing, we have $$\begin{aligned}
\theta_i < t \ \ \text{if and only if} \ \ H_1(\theta_i) < H_1(t).
\end{aligned}$$ It follows from (\[series-dp\]) that $$\begin{aligned}
P(t)=P\left((-\infty,t]\right)&=&\sum_{i=1}^{\infty} J_i \delta_{\theta_i}\left((-\infty,t]\right)=\sum_{i=1}^{\infty} J_i \delta_{H_1(\theta_i)}\left((0,H_1(t)]\right).
\end{aligned}$$ Observe that, since $(\theta_i)_{i \ge 1}$ is a sequence of i.i.d. random variables with continuous distribution $H_1,$ for $i \ge 1,$ we have $U_i\overset{d}=H_1(\theta_i)$, where $\left(U_i\right)_{i \ge 1}$ is a sequence of i.i.d. random variables with a uniform distribution on $[0,1]$. Hence, $P(t)=P_{\lambda}(H_1(t)),$ where $P_{\lambda_1}\sim DP(a_1,\lambda)$ and $\lambda$ is the Lebesgue measure on $[0,1]$. Similarly, $Q(t)=Q_{\lambda}(H_2(t)),$ where $Q_{\lambda}\sim DP(a_2,\lambda)$. Thus, $$\begin{aligned}
d_{CvM}(P,Q)&=&\int
_{-\infty}^{\infty}\left(P(t)-Q(t)\right) ^{2}Q(dt)\\
&=&\int_{-\infty}^{\infty}\left(P_{\lambda}(H_1(t))-Q_{\lambda}(H_2(t))\right)^{2}Q_{\lambda}(H_2(dt))
\end{aligned}$$ If $H_1=H_2=H$, and since $H$ is continuous, we have $$\begin{aligned}
d_{CvM}(P,Q) &=&\int_{-\infty}^{\infty}\left(P_{\lambda}(H(t))-Q_{\lambda}(H(t))\right)^{2}Q_{\lambda}(H(dt))\\
&=&\int_{0}^{1}\left(P_{\lambda}(z)-Q_{\lambda}(z)\right)^{2}Q_{\lambda}(dz).
\end{aligned}$$ This shows that the distribution of $d_{CvM}(P,Q)$ does not depend on the base measures $H_1$ and $H_2$ whenever $H_1=H_2$.
[^1]: [*Address for correspondence:*]{} Luai Al-Labadi, Department of Mathematics, University of Sharjah, P. O. Box 27272, Sharjah, UAE. E-mail: lallabadi@sharjah.ac.ae.
|
---
abstract: 'A classical result (often credited to Y. Medvedev) states that every language recognized by a finite automaton is the homomorphic image of a local language, over a much larger so-called local alphabet, namely the alphabet of the edges of the transition graph. Local languages are characterized by the value $k=2$ of the sliding window width in the McNaughton and Papert’s infinite hierarchy of strictly locally testable languages ($k$-slt). We generalize Medvedev’s result in a new direction, studying the relationship between the width and the alphabetic ratio telling how much larger the local alphabet is. We prove that every regular language is the image of a $k$-slt language on an alphabet of doubled size, where the width logarithmically depends on the automaton size, and we exhibit regular languages for which any smaller alphabetic ratio is insufficient. More generally, we express the trade-off between alphabetic ratio and width as a mathematical relation derived from a careful encoding of the states. At last we mention some directions for theoretical development and application.'
author:
- Stefano Crespi Reghizzi
- Pierluigi San Pietro
bibliography:
- 'FormalLangBib.bib'
title: 'From Regular to Strictly Locally Testable Languages [^1]'
---
Conclusion {#SectConclusion}
==========
We have generalized Medvedev’s homomorphic characterization of regular languages: instead of using as generator a local language over a large alphabet, which depends on the complexity of the regular language, we can use a strictly locally testable language over a smaller alphabet that does not depend on complexity, but just on the source alphabet. We have proved that the smallest alphabet one can use in the generator is the double of the alphabet of the regular language; thus, for instance, four symbols suffice to homomorphically generate any regular binary language.
In the main proof we have offered a specific and fairly optimized construction of the strictly locally testable language, for which we have derived the relationship between the width, the alphabetic ratio, and the complexity of the regular language. In our opinion, the construction should be of its own interest, as a new technique for simulating a NFA by means of a larger, yet strictly locally testable, machine. Our encoding is asymptotically optimal with respect to language complexity, and remains very close to the theoretical optimum for finite values of complexity. But it is an open technical question whether a different construction would yield better values for the alphabetic ratio and the width parameter.
Applications and developments of our result are conceivable in areas where a language characterization *à la* Medvedev has been found valuable, as in the next ones.\
Picture languages. A main family of 2-dimensional languages, the tiling systems [@GiammRestivo1997], is defined by a 2-dimensional Medvedev characterization. Does our result extend to 2D languages?\
Context-free languages. Combining our result with the Chomsky-Schutzenberger theorem it should be possible to obtain non-erasing homomorphic characterizations using a small alphabet.\
Consensual languages [@CrespiSpietro2001]. This generalization of finite-state machines motivated by modelling tightly connected concurrent computations uses homomorphism between words as its core mechanism.\
Information transmission for reducing the receiver cost was already mentioned in the introduction.
[*Acknowledgments:* ]{} Thanks to Aldo De Luca for suggesting relevant references.
[^1]: Extended Abstract.
|
---
abstract: '[The]{} availability of big data recorded from massively multiplayer online role-playing game (MMORPG) allows us to gain a deeper understanding the potential connection between individuals’ network positions and their economic outputs. We use a statistical filtering method to construct dependence networks from weighted friendship networks of individuals. We investigate the 30 distinct motif positions in the 13 directed triadic motifs which represent microscopic dependence among individuals. Based on the structural similarity of motif positions, we further classify individuals into different groups. The node position diversity of individuals is found to be positively correlated with their economic outputs. We also find that the economic outputs of leaf nodes are significantly lower than that of the other nodes in the same motif. Our findings shed light on understanding the influence of network structure on economic activities and outputs in social system.'
author:
- 'Wen-Jie Xie'
- 'Yan-Hong Yang'
- 'Ming-Xia Li'
- 'Zhi-Qiang Jiang'
- 'Wei-Xing Zhou'
bibliography:
- 'E:/Papers/Auxiliary/Bibliography.bib'
title: Individual position diversity in dependence socioeconomic network increases economic output
---
Introduction {#S1:Introduction}
============
Considerable studies have offered us a deep understanding of the influence of network structures on the dynamics of complex systems, such as the spreading of diseases and information [@Wang-Gonzalez-Hidalgo-Barabasi-2009-Science] and emerging of collaborations [@Newman-2001-PNAS]. However, the connection between network position and economic output is less studied. It is reported that specific network structures may enhance economic outputs [@Bhattacharya-Dugar-2014-MS; @Xie-Li-Jiang-Zhou-2014-SR; @Eagle-Macy-Claxton-2010-Science; @Bettencourt-Samaniego-Youn-2014-SR; @Ortman-Cabaniss-Sturm-Bettencourt-2015-SciAdv]. Eagle et al argue that the diversity of individual relationships within a community strongly correlates with economic development of communities [@Eagle-Macy-Claxton-2010-Science]. Furthermore, Bettencourt et al and Ortman et al find that the diversity of relationships is positively correlated with the productivity of individuals and communities [@Bettencourt-Samaniego-Youn-2014-SR; @Ortman-Cabaniss-Sturm-Bettencourt-2015-SciAdv].
Network motifs are building blocks of complex networks [@Milo-Itzkovitz-Kashtan-Levitt-ShenOrr-Ayzenshtat-Sheffer-Alon-2004-Science; @Milo-ShenOrr-Itzkovitz-Kashtan-Chklovskii-Alon-2002-Science; @Milo-Kashtan-Itzkovitz-Newman-Alon-2004-XXX]. It is found that network motifs of social networks may reflect the driving forces for forming social structures [@Kovanen-Kaski-Kertesz-Saramaki-2013-PNAS; @Klimek-Thurner-2013-NJP]. Similar to the friendship networks of US students [@Ball-Newman-2013-NS], Xie et al. studied triadic motifs in dependence networks of virtual societies and found that low level individuals have preference of forming links to high level individuals [@Xie-Li-Jiang-Zhou-2014-SR]. Their findings in virtual world are consistent with empirical findings in real society that “collaboration is easier when both partners share the same social status, and the probability of partnership formation decreases significantly as the status gap between the partners increases” [@Bhattacharya-Dugar-2014-MS].
Understanding the structure and function of social networks are of great importance to investigate economic activities and [[outputs]{}]{} in social systems [@Kovanen-Kaski-Kertesz-Saramaki-2013-PNAS; @Palla-Barabasi-Vicsek-2007-Nature; @Onnela-Saramaki-Hyvonen-Szabo-Lazer-Kaski-Kertesz-Barabasi-2007-PNAS; @Kumpula-Onnela-Saramaki-Kaski-Kertesz-2007-PRL; @Eagle-Penland-Lazer-2009-PNAS; @Jo-Pan-Kaski-2011-PLoS1; @Jiang-Xie-Li-Podobnik-Zhou-Stanley-2013-PNAS]. In real world, social networks are not large and samples are biased [@Ball-Newman-2013-NS; @Currarini-Jackson-Pin-2009-Em; @Currarini-Jackson-Pin-2010-PNAS], which hinders the empirical investigation of relationship between network structures and economic activities in social system. Some other empirical studies have shown that individuals’ or firms’ network positions are closely related to success measured in terms of economic production [@Uzzi-1996-ASR; @Guimera-Uzzi-Spiro-Amaral-2005-Science; @Cantner-Joel-2011-IUPJKM; @Garas-Tomasello-Schweitzer-2014-ArXiv]. In the era of big data, information technology provides us alternative methods to collect data of social relationships and economic activities, for example, massively multiplayer online role-playing games (MMORPGs) [@Jiang-Zhou-Tan-2009-EPL; @Jiang-Ren-Gu-Tan-Zhou-2010-PA; @Thurner-Szell-Sinatra-2012-PLoS1; @Szell-Sinatra-Petri-Thurner-Latora-2012-SR; @Szell-Thurner-2012-ACS], which enable us to study complex social and economic behaviours of human in online social systems [@Bainbridge-2007-Science; @Papagiannidis-Bourlakis-Li-2008-TFSC; @Williams-2010-CT]. Bainbridge et al also emphasized the scientific potential of virtual world for future research [@Bainbridge-2007-Science]. Empirical studies show that social behaviours in MMORPGs are representative of human behaviours in many aspects in real society [@Chesney-Chuah-Hoffmann-2009-JEBO; @Szell-Lambiotte-Thurner-2010-PNAS; @Szell-Thurner-2010-SN; @Klimek-Thurner-2013-NJP; @Szell-Thurner-2013-SR; @Grabowski-Kosinski-2008-APPA; @Grabowski-Kruszewska-2007-IJMPC].
In this paper, we construct dependence networks based on weighted friendship networks of individuals and identify 30 distinct motif positions in 13 directed triadic motifs which represent local dependence among individuals. Using the $k$-means algorithm, we further classify individuals into $k$ clusters based on the motif position profiles. Our results indicate that the motif position of individuals do have great influence on their economic outputs.
Materials and Methods {#section:Materials and Methods}
=====================
Data description
----------------
We use a huge database recorded from $124$ servers to investigate the potential connection between network structure and economic output for individuals within the virtual world of a popular Massively Multiplayer Online Role-Playing Game (MMORPG) in China. There are two opposing camps or societies in a virtual world residing in a server, thus giving us $248$ virtual societies. There exist great differences about the numbers of avatars among different virtual societies. The populations of virtual worlds vary from thousands up to fifty thousands. The distribution of the number of avatars in each virtual world is drawn in Fig. 1G. In each society, three professions have different skills. The advantage of warrior’s skill is the power of attack, the advantage of mage’s skill is the power of defense, and the advantage of priest’s skill is the ability to cure illness. An avatar can be a warrior, a priest, or a mage. To improve their skills, the avatars cooperate with friends to accomplish tasks. The more friends, the higher efficiency. Two individuals $i$ and $m$ are allowed to establish social ties to satisfy their desire of making friends and enhance their utility of collaborations. The strength of social ties is measured by the intimacy $I_{i,m}$, which increases according to the collaborative activities of $i$ and $m$ if they belong to the same society; otherwise, $I_{i,m}$ remains zero if $i$ and $m$ belong to two different societies. Hence the friendship networks of the two camps are essentially separated. As a measure of closeness to each friendship, the values of intimacy are recorded every day. When two individuals in the same society form a team and collaborate to accomplish a task, their intimacy increases. The evolving intimacy allows us to track the evolution of the cooperation behavior in the socioeconomic networks. Each individual can maintain a friendship list, denote as $\mathcal{F}_i$ for individual $i$. The social tie is symmetric: if $i \in \mathcal{F}_m$, then $m \in \mathcal{F}_i$. In addition to the friendship network, the game data contains other socioeconomic networks, such as the face to face trading networks between initiators and receivers, the vendor trading networks between vendors and costumers, the mail networks between senders and receivers, the mentor networks between students and mentors, the kill networks between killers and victims. In this paper, our focus is the friendship network. More details of the database can be found in our earlier works about the triadic motifs in dependence networks [@Xie-Li-Jiang-Zhou-2014-SR] and skill complementarity in collaboration networks [@Xie-Li-Jiang-Tan-Podobnik-Zhou-Stanley-2016-SR].
Economic outputs of individuals
-------------------------------
We measure the economic [[output by]{}]{} converting the virtual money and items into a standardized currency for each individual. There are two virtual currencies, [*Xingbi*]{} and [*Jinbi*]{}. [The [*Xingbi*]{} and [*Jinbi*]{} can be exchanged in the built-in platform in each virtual world]{}. [[The]{}]{} virtual system has an approximately stable exchange rate between [*Xingbi*]{} and the Chinese currency [*Renminbi*]{}. [[*Jinbi*]{} is produced by the economic activities of the individuals when they form a team and collaborate to accomplish the tasks. The currency [*Jinbi*]{} and virtual items, such as weapons, clothes, and medicines, are awarded to individuals when monsters are killed and tasks are accomplished.]{}
We convert the produced items and [*Jinbi*]{} to [*Xingbi*]{} to obtain the real economic [[output]{}]{} for each individual on each day. On average, the [[normal life span]{}]{} of virtual societies is close to 5 months [@Xie-Li-Jiang-Tan-Podobnik-Zhou-Stanley-2016-SR]. Therefore, we calculate the [[output]{}]{} of each individual for a fixed period of 145 days for all virtual societies, denoted as $y_{i}=\ln\sum_{t=1}^{t=145}y_{i,t}$.
{width="15cm"}
Construction of dependence network
----------------------------------
For a given friendship network, we construct a dependence network by removing the insignificant edges based on statistical validation [@Serrano-Boguna-Vespignani-2009-PNAS]. [First, we define the relative intimacy of individual $m$ in reference to all friends of individual $i$ as $w_{i,m} = I_{i,m}/\sum_{m=1}^{N}I_{i,m}$.]{} Obviously, $w_{i,m} \neq w_{m,i}$. Following the statistical validation [@Serrano-Boguna-Vespignani-2009-PNAS], a directed tie $i \rightarrow m$ is significant at the level of $\alpha$ if $$\alpha_{i,m} = 1-(k_{i}-1)\int_{0}^{w_{i,m}}(1-x)^{k_{i}-2}{\rm{d}}x < \alpha,
\label{Eq:Edges:Alpha:ij}$$ where $k_{i}$ is the degree of individual $i$. [ If $\alpha_{m,i}< \alpha$, the directed tie $m \rightarrow i$ is significant. For each society, we set the significant level $\alpha$ and remove the insignificant links,]{} resulting a directed dependence network. If link $i \to m $ is significant, it means that individual $m$ is relatively important to individual $i$ in $i$’s friends. [ In other words, individual $i$ depends individual $m$. By the disparity filter [@Serrano-Boguna-Vespignani-2009-PNAS], the dependence networks are the backbones of the original friendship networks.]{}
[ Fig. \[Fig:XCB:DependNet\] (A, B, C) show the topological structure of the dependence network constructed from a virtual society with significant level $\alpha=0.01, 0.05, 0.1$ respectively. We analysis the out-degree distribution $P(k_{\rm{out}})$ and in-degree distribution $P(k_{\rm{in}})$ of dependence networks for different significant level $\alpha=0.01, 0.05, 0.1$ in Fig. \[Fig:XCB:DependNet\] (D, E). The degree distribution of the dependence network is obviously different from the original friendship network ($\alpha=1$). Using the smaller significance level $\alpha$, the disparity filter reduces more edges. For different significant level $\alpha$, the fraction of nodes, edges, weight kept in the dependence network increase with the significant level $\alpha$ as shown in Fig. \[Fig:XCB:DependNet\] (F). And the average degree of the 248 dependence networks is monotone increasing with the significant level $\alpha$ in Fig. \[Fig:XCB:DependNet\] (I). For different significant level $\alpha=0.01, 0.05, 0.1$, the distribution $P(N)$ of the 248 dependence networks’ sizes $N$ have nearly the same shape in Fig. \[Fig:XCB:DependNet\] (G). The numbers of the avatars range from thousand to fifty thousands. Comparing with the global threshold filter, the disparity filter considers the relevant edges and ensures that the edges with small intimacy are not neglected. So we can find that the average intimacy kept in the dependence network is not monotone decreasing with the significant lever $\alpha$ in the Fig. \[Fig:XCB:DependNet\] (H). ]{}
Quantifying position ratio profile
----------------------------------
Following the method [@Milo-ShenOrr-Itzkovitz-Kashtan-Chklovskii-Alon-2002-Science] to identify motifs, we are able to identify 13 different directed triadic motifs, as shown in Fig. \[Fig:XCB\_Motifs\_Position30\], in dependence networks. These motifs uncover [[the]{}]{} dependence structures among individuals at the microscopic level. For example, motif $\leftarrow$$\rightarrow$ stands for the situation that one individual depends on the other two individuals, motif $\rightarrow$$\rightarrow$ means that one individual depends on another individual which in turn depends on the third individual, [[whereas]{}]{} motif $\rightarrow$$\rightarrow$$\rightarrow$ represents the situation that individual $a$ depends on individual $b$, individual $b$ depends on individual $c$, and individual $c$ also depends on individual $a$. We can further locate 30 distinct motif positions [@Stouffer-SalesPardo-Sirer-Bascompte-2012-Science] within the 13 different directed triadic motifs, as shown in Fig. \[Fig:XCB\_Motifs\_Position30\]. We directly enumerate the relative frequency $p_{i,j}$ that individual $i$ appears in position $j$ across the 13 motifs, which gives the motif position ratio profile $p_i=(p_{i,1},p_{i,2},...,p_{i,30})$ for individual $i$. Note that we have $\sum_{j=1}^{30}p_{i,j}=1$.
![Plots of 30 unique positions $j\in \{1,2,...,30\}$ in 13 directed triadic motifs.[]{data-label="Fig:XCB_Motifs_Position30"}](Fig2.eps){width="10cm"}
Hence, we first define the $z$-sore of occurrence frequency for position $j$: $$Z_{i,j}=\frac{p_{i,j}-\langle p_{i,j}\rangle}{\sigma(p_{i,j}) },
\label{Eq:Positions_Zscore}$$ where $\langle p_{i,j}\rangle=\sum_{i=1}^{N}p_{i,j}/N$ and $\sigma(p_{i,j})$ are the mean and standard deviation of $p_{i,j}$ over all individuals in the dependence network of a given virtual society [@Milo-Itzkovitz-Kashtan-Levitt-ShenOrr-Ayzenshtat-Sheffer-Alon-2004-Science; @Milo-Kashtan-Itzkovitz-Newman-Alon-2004-XXX]. The structural similarity between individual $i$ and $m$ are thus defined as the correlation coefficients between $Z_i$ and $Z_m$, such that, $$s_{i,m}=\frac{E[(Z_{i,j}-\langle Z_{i}\rangle)(Z_{m,j}-\langle Z_{m}\rangle)]}{E[(Z_{i,j}-\langle Z_{i}\rangle)^2]^{1/2}E[(Z_{m,j}-\langle Z_{m}\rangle)^2]^{1/2}},
\label{Eq:Positions_sij}$$ where $E[x]$ is the mathematical expectation of $x$.
Classifying individuals based on their position ratio profile
-------------------------------------------------------------
We employ the $k$-means algorithm to classify $N$ individuals based on their position ratio [[profiles $\{Z_i\}$]{}]{} in the dependence social networks. The position ratio profile of individual $i$ is wrote as $Z_i=(Z_{i,1},Z_{i,2},...,Z_{i,30})$. For each virtual society, we can get a matrix of position ratio profile, denoted by $\cal{Z}$, in which rows correspond to individuals and columns correspond to positions in triadic motifs. By adopting [[the]{}]{} $k$-means algorithm, we partition the individuals into $k$ clusters in terms of minimizing the sum, over all clusters, of the within-cluster sums of point-to-cluster-centroid distances. The partition algorithm is implemented in an iterative way. [ To evaluate the optimal number of clusters, we calculate the clustering evaluation object containing Davies-Bouldin index values and find the minimum criterion value. Then we can get the optimal number of clusters $k$.]{} The correlation distance defined as one minus the sample correlation between points (treated as $Z_i$) is used to capture the closeness between individuals in $k$-means algorithm, such as, $$d_{i,m}=1-s_{i,m}.
\label{Eq:Distance_sij}$$ [[The]{}]{} $k$-means algorithm returns the cluster indices of each individual and the $k$ cluster centroid locations in a $k$-by-$30$ matrix. Each centroid is the component-wise mean of the points in that cluster. We denote the $k$-th cluster centroid locations as $P_{k} =(P_{k,1},P_{k,2},...,P_{k,30})$, where $P_{k,j}=\sum_{i\in {\cal{C}}_{k}}p_{i,j}/c_k$ and $j\in \{1,2,...,30\}$. $c_k$ is the number of individuals in cluster ${\cal{C}}_k$. We have $\sum_{j=1}^{30}P_{k,j}=1$, and $P_{k}$ is the position ratio profile of cluster $k$.
Results
=======
Quantifying position ratio profile
----------------------------------
Individuals inside the virtual societies are involved in all kinds of social and economic activities. The theory of [structural]{} holes already tells us that the agent occupies special position in social network will have great influence on his economic performance [@Burt-2009]. Here, we empirically investigate the underlying connections between network structure and economic [[output]{}]{} for individuals in 248 virtual societies. Our data are game logs from a popular MMORPG in China (see data description in section \[section:Materials and Methods\]). For each society, we construct a dependence network of individuals, which only keeps the multiscale backbone of original friendship network (see construction of dependence network in section \[section:Materials and Methods\]). Fig. \[Fig:XCB:DependNet\] illustrates the dependence network from a virtual society [ and some basic statistics about the topological structure.]{} One can see that there are lots of in-stars in dependence network, which is in accordance with the findings [@Xie-Li-Jiang-Zhou-2014-SR] that the in-degree distribution is much fatter than out-degree distribution. Such phenomena can be explained by that in virtual societies the individuals with high levels may play a relatively important role in friend lists of low level individuals, which is supported by the strong preference of connecting high level individuals for low level individuals [@Xie-Li-Jiang-Zhou-2014-SR].
For a given dependence network, we can estimate a position ratio profile for each individual (see quantifying position ratio profile in section \[section:Materials and Methods\]). Xie et al. revealed that the open motifs have higher occurrence frequency than close motifs [@Xie-Li-Jiang-Zhou-2014-SR], suggesting that the positions in the open motifs may have higher occurrence frequency than positions in close motifs. Fig. \[Fig:XCB\_Motifs\_M30Position\_Block\] (A) shows the position ratio profile for all individuals in dependence networks in Fig. \[Fig:XCB:DependNet\]. [ There are more zeros in the position ratio profiles of individuals with low degrees than those with high degrees.]{} More extremely, there are lots of individuals occupying only one specific position ($j$ = 3, 9, 10, 16, 17, and 18) of triadic motifs in dependence networks, which leads to $p_{i, j} = 1$. The individuals with lager value of $p_{i,3},p_{i,9},p_{i,16}$ in their position ratio profile may correspond to leaf nodes in dependence networks. Note that the occurrence frequency of motif $\rightarrow$$\rightarrow$$\rightarrow$ is 0, resulting in that the position 21 is impossible to be observed.
{width="12cm"}
Similarity of position ratio profiles between individuals
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The triadic motifs are regarded as building blocks of complex networks [@Milo-ShenOrr-Itzkovitz-Kashtan-Chklovskii-Alon-2002-Science], suggesting that the 30 positions in triadic motifs may contain important [[structural]{}]{} information [[of]{}]{} nodes in complex [[networks]{}]{}. Here we utilize the occurrence frequency of the 30 different positions in dependence networks to represent the network structure profile for each individual. Based on these profiles, we can assess the structural similarity between individuals, which allows us further to group individuals. In dependence networks, it is observed that some motifs, for example, $\rightarrow$$\rightarrow$, $\rightarrow$$\leftarrow$, $\rightarrow$$\rightleftharpoons$, appears more frequency than other motifs [@Xie-Li-Jiang-Zhou-2014-SR].
By ordering the individuals according to the rule that nodes with large similarity are close to each other, we plot the structural similarity in Fig. \[Fig:XCB\_Motifs\_M30Position\_Block\] (D). The color bar stands for the value of similarities $s_{i,m}$. One intriguing observation is that there is a block-diagonal structure, strongly indicating the function of grouping individuals for position ratio profiles. This inspires us to further classify the individuals into clusters by $k$-means algorithm (see classifying individuals based on their position ratio profile in section \[section:Materials and Methods\]). [ In Fig. \[Fig:XCB\_Motifs\_M30Position\_Block\](A) and Fig. \[Fig:XCB\_Motifs\_M30Position\_Block\](D), there are some classes which are made up of some special individuals, such as individuals with $p_{i,3}=1$ in motif $\rightarrow$$\rightarrow$, or individuals with $p_{i,9}=1$ or $p_{i,10}=1$ in motif $\rightarrow$$\leftarrow$, or individuals with $p_{i,16}=1$, $p_{i,17}=1$, or $p_{i,18}=1$ in motif $\rightarrow$$\rightleftharpoons$. ]{}
The $k$-means algorithm is an iterative partitioning algorithm, which maximizes the similarity of the within-cluster. [ To evaluate the optimal number of clusters, we create a clustering evaluation object containing Davies-Bouldin index values in Fig. \[Fig:XCB\_Motifs\_M30Position\_Block\] (E). From the minimum criterion value, we can get the optimal number of clusters $k=41$. Based on the definition of position ratio profiles, there is a correlation among the 30 positions. From the dendrogram of the 30 position in Fig. \[Fig:XCB\_Motifs\_M30Position\_Block\] (B), we can find that a avatar at position 22 has a higher chance to be also in position 23 or 24 in the same motif. But there is no obvious classification among the 30 position. The block-diagonal structure can not be explained by this relation among the 30 position. An alternative method to elucidate the position significant profiles is to calculate the difference of position ratio profiles between the dependence networks and the reference randomized dependence networks [@Milo-ShenOrr-Itzkovitz-Kashtan-Chklovskii-Alon-2002-Science]. But this method is too computationally intensive for our 248 dependence networks and some of the networks’ sizes are larger than fifty thousands. ]{}
Relationship between position ratio profiles and economic outputs
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In social network, the individuals in key positions may have great impacts of network dynamics if some controlling strategies are applied on these key nodes. This leads to the conjecture that the position ratio profiles may have potential influence on the economic outputs for individuals. We estimate the economic [[output]{}]{} $y_{i}$ for each individual in a given world (see economic outputs of individuals in section \[section:Materials and Methods\]). To make these economic outputs comparable between different virtual worlds, we standardize the economic [[output]{}]{} within each society. The standardization of economic performance of individual $i$ is defined as [[output]{}]{} per capita, $Y_{i}=[y_{i}-\langle y_{i}\rangle]/\sigma(y_{i})$, where $\langle y_{i}\rangle$ and $\sigma(y_{i})$ are the mean and standard deviation of $y_{i}$ over all individuals in a given society. Therefore, the mean of economic performance of the class ${\cal{C}}_k$ can be estimated via $Y_{k}=\sum_{i\in {\cal{C}}_k}Y_{i}/c_{k,}$, where $c_{k}$ is the number of individuals in class ${\cal{C}}_k$.
Motivated by social diversity [@Eagle-Macy-Claxton-2010-Science], we define the individual position diversity $d_{i}$ as a function of Shannon entropy to quantify how individual appears in the 30 positions: $$d_{i}=-\sum_{j=1}^{30}p_{i,j}\ln p_{i,j}.
\label{Eq:Positions_H}$$
{width="15cm"}
[ Fig. \[Fig:XCB\_Motifs\_M30Position\_Block\] shows the results from the analysis of position ratio profiles for individuals in Fig. \[Fig:XCB:DependNet\] with significant level $\alpha=0.05$. The position ratio profiles of 4331 individuals are presented in Fig. \[Fig:XCB\_Motifs\_M30Position\_Block\] (A). To improve the visualization, the color bar represents the log value of position ratio profiles $p_{i,j}$. The position diversity $d_{i}=0$ means that the individual only occupy one triadic motif position. As shown in Fig. \[Fig:XCB\_Motifs\_M30Position\_Block\] (A), there are individuals with $p_{i,3}=1$ in motif $\rightarrow$$\rightarrow$, $p_{i,9}=1$, $p_{i,10}=1$ in motif $\rightarrow$$\leftarrow$, $p_{i,16}=1$, $p_{i,17}=1$, and $p_{i,18}=1$ in motif $\rightarrow$$\rightleftharpoons$. Fig. \[Fig:XCB\_Motifs\_M30Position\_Block\] (C, F) show the economic [[output]{}]{} $y_i$ and the individual position diversity $d_i$ respectively. The individuals with position diversity $d_{i}=0$ account for a large part of avatars in the dependence network of a given society. Fig. \[Fig:XCB\_Motifs\_M30Position\_Block\] (L) shows the scatter plot of economic [[output]{}]{} $y_{i}$ and position diversity $d_{i}$. The correlation coefficient $\rho$ is 0.63 and $p$-value is less than $10^{-6}$, implying that the correlation between $y_{i}$ and $d_{i}$ is positive and highly significant. Our results indicate that the individuals who appear in more triadic motif positions have higher economic outputs. It is easy to find that more active players have more friends (more in-degree and out-degree) and appear in more triadic motif positions. Here we show the plots of the relation between the economic [[output]{}]{} $y_i$ and the individuals’ in-degree $k_{\rm{in}}$, out-degree $k_{\rm{out}}$ in Fig. \[Fig:XCB\_Motifs\_M30Position\_Block\] (G, H) respectively. This agrees with Fuchs’ result [@Fuchs-Thurner-2014-PLoS1] that the output is correlated to both in- and out-degree. To avoid the impact of the in- and out-degree, we investigate the relation between the economic [[output]{}]{} $y_i$ and the position diversity $d_i$ of individuals with fixed $k_{\rm{in}}, k_{\rm{out}}= 1, 3, 5$ in Fig. \[Fig:XCB\_Motifs\_M30Position\_Block\] (J, K, L). And we can get the same conclusion that the individual position diversity increases economic [[output]{}]{} of individuals with fixed $k_{\rm{in}}$ or $k_{\rm{out}}$. What is more, we also find that the economic [[output]{}]{} is susceptible to asymmetries between individuals’ in- and out-degree in our dependence socioeconomic network. The individuals with high in-degree have higher economic outputs than with high out-degree. Because it is difficult to get the records and measure the activity of individuals accurately, we assume that the individuals with the same in- and out-degree have almost the same activity. ]{}
The $k$-means algorithm also gives cluster centroid locations denoted as $P_{k} =(P_{k,1},P_{k,2},...,P_{k,30})$ for each class ${\cal{C}}_{k}$. Similar to Eq. \[Eq:Positions\_H\], we define the cluster position diversity as $D_{k}=-\sum_{j=1}^{30}P_{k,j}\ln P_{k,j}$. [ In Fig. \[Fig:XCB\_Motifs\_M30Position\_Block\_Ys\](A-F), the clusters are sorted by the cluster position diversity $D_{k}$ in ascending order. Considering the impact of different significant level $\alpha$ on the relation between the economic [[output]{}]{} and position diversity, we analysis the dependence network from all the virtual society with different significant level $\alpha=0.01, 0.05, 0.1$. In Fig. \[Fig:XCB\_Motifs\_M30Position\_Block\_Ys\], the three columns from left to right correspond to the dependence networks with significant level $\alpha=0.01, 0.05, 0.1$ respectively. Fig. \[Fig:XCB\_Motifs\_M30Position\_Block\_Ys\] (A, B, C) show the plots of the average economic [[output]{}]{} $Y$ of classes and Fig. \[Fig:XCB\_Motifs\_M30Position\_Block\_Ys\] (D, E, F) illustrate the plots of the value of $P_k$ for different significant level $\alpha=0.01, 0.05, 0.1$ respectively. The clusters are sorted by the cluster position diversity $D_{k}$ in ascending order, so one can see that the economic [[output]{}]{} $Y$ increases with the cluster position diversity $D_{k}$. We can find that lots of clusters are comprised of individuals with $P_{k,3},P_{k,9},P_{k,16}$ equalling to one. Although these individuals have the same individual position diversity $d_i=0$, their economic outputs could be different. For each dependence network with a given $\alpha$, we calculate the Davies-Bouldin index values to evaluate the optimal number of clusters and get the optimal number of clusters $k$. The distributions of the optimal number $k$ are shown in Fig. \[Fig:XCB\_Motifs\_M30Position\_Block\_Ys\] (G, H, I). The average optimal number $k$ is between 50 and 60. ]{}
[ To compare the relation coefficients between the economic [[output]{}]{} and the individuals’ in-degree $k_{\rm{in}}$, out-degree $k_{\rm{out}}$, position diversity $d_i$, we denote the coefficients as $\rho_{k_{\rm{in}}}$, $\rho_{k_{\rm{out}}}$ and $\rho_{d_{i}}$ respectively and draw the distribution of the three kinds of coefficients for different significant level $\alpha$. When the significant levels $\alpha$ is smaller, there are greater difference between the three kinds of coefficients. At the same time, the asymmetries between individuals’ in- and out-degree in our dependence socioeconomic network is obvious with the significant level $\alpha=0.01$.]{}
Economic [[output]{}]{} of individuals with the position diversity $d_{i}=0$
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Here, we conduct a comparison of the economic [[output]{}]{} of individuals with individual position diversity $d_{i}=0$ in the dependence networks from all the virtual society with significant level $\alpha=0.01, 0.05, 0.1$. The individual position diversity $d_{i}=0$ means that the individual appears only in one triadic motif position. [ We perform $t$-tests on pairs of economic [[output]{}]{} of the individuals at two position in the same motif.]{} Fig. \[Fig:XCB\_Motifs\_M30Position\_Diversity\_Ys\_H\] shows the box plots of economic [[output]{}]{} of individuals with position diversity $d_{i}=0$ for 30 unique positions $j\in \{1,2,...,30\}$ in 13 directed triadic motifs. [ In Fig. \[Fig:XCB\_Motifs\_M30Position\_Diversity\_Ys\_H\], the superscript \*, \*\*, \*\*\* stand that the relationship is acceptable at the significant level $0.05$, $0.001$ and $0.0001$ respectively. For all the dependence networks with significant level $\alpha=0.01, 0.05, 0.1$, the results of the $t$-tests are similar. ]{}
{width="12cm"}
[In Fig. \[Fig:XCB\_Motifs\_M30Position\_Diversity\_Ys\_H\],]{} we perform $t$-tests on pairs of individual economic [[output]{}]{} of the individuals at two position [ in the dependence networks.]{} It is worthy stressing that the mean economic [[output]{}]{} of individual at position $1$ is greater than that of individuals within position $2$ in motif $\leftarrow$$\rightarrow$. [This relationship can be expressed as $1>2$ in Fig. \[Fig:XCB\_Motifs\_M30Position\_Diversity\_Ys\_H\].]{} The mean economic [[output]{}]{} of individuals at position $4$ and $5$ are greater than that of individuals at position $3$ in motif $\rightarrow$$\rightarrow$. [These relationship can be expressed as $4>3$ and $5>3$ in Fig. \[Fig:XCB\_Motifs\_M30Position\_Diversity\_Ys\_H\].]{} The mean economic [[output]{}]{} of individuals at position $10$ is greater than that of individuals at position $9$ in motif $\rightarrow$$\leftarrow$. [This relationship can be expressed as $10>9$ in Fig. \[Fig:XCB\_Motifs\_M30Position\_Diversity\_Ys\_H\].]{} The mean economic [[output]{}]{} of individuals at position $17$ and $18$ are greater than that of individuals at position $16$ in motif $\rightarrow$$\rightleftharpoons$. [These relationship can be expressed as $17>16$ and $18>16$ in Fig. \[Fig:XCB\_Motifs\_M30Position\_Diversity\_Ys\_H\].]{}
[[Comparing the output]{}]{} for individuals at different positions [ in the dependence networks with significant level $\alpha=0.05$]{}, the 30 positions can be classified into two groups. The first group contains the positions: , , , , , , , , , , , , , . The second group contains positions: , , , , , , , , , , , , , , . All the positions are sorted by means of individual economic [[output]{}]{} in ascending order. The mean economic [[output]{}]{} of individuals in the first group is less than 0, while in the second group the mean [[output]{}]{} is greater than 0. Such differences could result from that most of positions of the first group are in open motifs while most of positions in second group belong to close motifs.
Discussion
==========
In this paper we analyzed the relationship between friendship structure and economic [[output]{}]{} for individuals in a massively multiplayer online role-playing game. We [[found]{}]{} that the individual position diversity of individuals is positively correlated with their economic output and social status. We have developed a new approach to study structural similarity of individuals in networks and classify individuals into different clusters based on their position ratio profiles. It is found that the economic [[output]{}]{} of leaf nodes [[is]{}]{} significantly lower than the nodes at the other triadic motif positions. The individual position diversity positively correlates with the economic [[output]{}]{}. Our results are in consistent with the results from the physical society [@Bhattacharya-Dugar-2014-MS; @Ball-Newman-2013-NS]. Furthermore, Eagle et al. found that the diversity of individual relationships within a community strongly correlates with the economic development of the community [@Eagle-Macy-Claxton-2010-Science]. In a recent analysis, Xie et al. found the degree of skill complementarity is positively correlated with their output [@Xie-Li-Jiang-Tan-Podobnik-Zhou-Stanley-2016-SR].
Much evidence shows that the behaviour of individuals in virtual society [[is]{}]{} representative in many aspects of human behaviour in physical society. This is rational because individuals’ decision process is determined by players that are real human being. The invisible hand operates not only in modern societies but also in ancient societies, not only in real societies but also in virtual societies. Yee et al. argued that online environments such as MMORPGs could potentially be unique research platforms for the social sciences and clinical therapy, but it is crucial to firstly establish that social behavior and norms in virtual environments are comparable to those in the physical world [@Yee-Bailenson-Urbanek-Chang-Merget-2007-CPB]. To investigate the relation between friendship (or socioeconomic) networks in virtual and real worlds, Grabowski and Kruszewska conducted a survey among the players of an online game and construct the off-line network [@Grabowski-Kruszewska-2007-IJMPC]. They showed that the structure of the friendship network in virtual world was very similar to the structure of different social networks in real world.
We believe that our interdisciplinary work represents significant scientific evidence for understanding the behaviour of social systems. It involves topics that range from social science, network science, and economics to human dynamics. It also enriches our understanding on the formation of socio-economic networks and proposes a new method to classify nodes of complex networks for understanding of people’s economic behaviour from the big data of massive players. Our work sheds new light on the scientific research utility of virtual worlds for studying human behaviour in complex socio-economic systems.
This work was supported by the National Natural Science Foundation of China \[11505063, 11375064 and 11605062\]; the Shanghai Chenguang Program \[15CG29\]; the Ph.D. Programs Foundation of Ministry of Education of China \[20120074120028\]; and the Fundamental Research Funds for the Central Universities.
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abstract: 'Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces—so-called “topological semantics”. The first is classical higher-order logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic.'
author:
- 'S. Awodey[^1]'
- 'C. Butz[^2]'
title: 'Topological Completeness for Higher-Order Logic'
---
Introduction {#introduction .unnumbered}
============
Higher-order logic (also known as “type theory”) is logic that includes quantification over functions or relations. Many basic mathematical objects and theories can only be defined using this logic; the natural numbers and topological spaces are familiar examples. A more precise specification of what we call classical higher-order logic is given in §1 below.
As is well-known, higher-order theories are generally incomplete with respect to (standard) models in [$\mathbf{Sets}$]{}; that is, ${\ensuremath{\mathbb{T}}}\models\sigma$ does not imply ${\ensuremath{\mathbb{T}}}\vdash\sigma$ for ${\ensuremath{\mathbb{T}}}$ a theory in higher-order logic and $\vdash$ the entailment relation of any reasonable deductive calculus. It is by now also well-known that higher-order logic can be modeled in suitable generalized categories of sets, namely (elementary) topoi, and that with regard to such topos-valued semantics, standard higher-order deduction is complete (see for details).
Our results in this paper are concerned with topos models of a very special and natural kind, namely sheaves over topological spaces. If $X$ is a space, a model in the category $\operatorname{Sh}(X)$ of all sheaves on $X$ shall be called a *topological model*. We will show that higher-order logic is complete with respect to such models; for the reader unfamiliar with sheaf theory, we wish to emphasize their elementary topological character. Under the equivalence $\operatorname{Sh}(X){\ensuremath{\simeq}}\mathbf{Etale}/X$ a sheaf on a space $X$ is essentially the same thing as an étale space over $X$: a space $E$ equipped with a local homeomorphism $p : E\to X$ (also called an étale bundle). The various fibers $p^{-1}x$ of $E$ (the stalks of the sheaf) for the points $x\in X$ may be regarded as sets varying continuously over $X$. A morphism of étale spaces is just a continuous map $f : E\to E'$ over $X$, i.e. with $p'f=p$ as in the commutative triangle
E & \^[f]{} &&& E’\
&&\
&& X. &&\
Products, exponentials (“function spaces”), etc. of étale spaces of course agree with those calculated as sheaves. A topological model of a single-sorted theory thus consists of an étale space $p\colon
E\to X$ over a base space $X$ together with suitable operations, which are simply continuous maps over $X$.
As the reader who *is* familiar with sheaf theory will have noted, our topological models are just what are usually called “sheaves of […]{}s”, at least in the case of equational, first-order theories. Thus a topological model of the theory of groups is a sheaf of groups, and so on.
Despite the ultimately simple character of topological models, we use the more general language and methods of sheaf theory and topoi to study them. Our first theorem, proved in §3 below, asserts the completeness of standard, classical higher-order deduction $\vdash^{c}$ with respect to such topological semantics.
Let ${\ensuremath{\mathbb{T}}}$ be a higher-order theory. There exists a classical topological model $M$ of ${\ensuremath{\mathbb{T}}}$ such that, for any higher-order sentence $\sigma$ in the language of ${\ensuremath{\mathbb{T}}}$, $${\ensuremath{\mathbb{T}}}\vdash^{c}\sigma\qquad\text{if and only if}\qquad
M\models^{c}\sigma.$$ Moreover, the model $M$ has the property that every continuous function between the interpretations of type symbols is logically definable.
What permits theorem A to be true is our notion of a *classical* model. In an arbitrary topological sheaf topos $\operatorname{Sh}(X)$ there are two natural candidates for the interpretation of the type $2$ of formulas (or “propositions”, or “truth values”) of a higher-order theory; to wit, the sheaf $\Omega$ of open subsets of $X$ and the coproduct $1+1$. In the language of étale spaces, $1+1$ is the double covering $X\times 2\to X$. As detailed in §2 below, a classical model uses the latter to interpret the type of formulas. Function and power types are then interpreted as exponentials of sheaves (sometimes called “internal homs” or “sheaf-valued homs”). This standard treatment of exponentials is what chiefly distinguishes topological models from so-called Henkin models (see the appendix below for the exact relation between the two). Thus in particular, for any type $Z$ the power type $2^{Z}$ is interpreted as the sheaf of *complemented* subsheaves of the interpretation of $Z$. By further requiring of a classical model that the types be interpreted by so-called decidable sheaves, we can model classical higher-order logic in non-boolean topoi like $\operatorname{Sh}(X)$, which is impossible when interpreting the type $2$ by the subobject classifier $\Omega$. Indeed under that interpretation the analogue of theorem A fails—even permitting arbitrary Grothendieck topoi in place of topological sheaf topoi—as can be seen using Gödel incompleteness.
The issue of how to interpret the type of formulas of course vanishes when one considers the fragment of higher-order logic that results from omitting that type. This fragment—which we call $\lambda$-logic and describe in §4 below—may be regarded as a marriage of elementary logic and the $\lambda$-calculus. In addition to the usual propositional and quantificational language of elementary logic, it includes equations between and quantification over functions, functions of functions, etc. But since there is no type of formulas, there is no quantification over “propositional functions”, i.e. over relations.
Many familiar mathematical constructions, theorems, and proofs can be formalized in $\lambda$-logic. A simple example is Cayley’s theorem that every group is isomorphic to a group of permutations of its elements. The axiom of choice, in the familiar form $$\forall x\in X\exists y\in Y.\varphi(x, y) {\ensuremath{\Rightarrow}}\exists f\in Y^{X}\forall x\in X.\varphi(x, fx),$$ is also a statement of $\lambda$-logic. An example of a (non-elementary) $\lambda$-theory is synthetic differential geometry, applications to which of the present work shall be discussed elsewhere.
Our theorem B states the completeness of $\lambda$-logic with respect to topological models. More generally than theorem A, theorem B holds for standard, intuitionistic deductive entailment $\vdash$.
Let ${\ensuremath{\mathbb{T}}}$ be a $\lambda$-theory. There exists a topological model $M$ of ${\ensuremath{\mathbb{T}}}$ such that, for any $\lambda$-sentence $\sigma$ in the language of ${\ensuremath{\mathbb{T}}}$, $${\ensuremath{\mathbb{T}}}\vdash\sigma\qquad\text{if and only if}\qquad M\models\sigma.$$ Moreover, the model $M$ has the property that every continuous function between the interpretations of type symbols is logically definable.
Theorem B rests more squarely on one of the main supports of theorem A, namely a recent covering theorem for topoi due to the second author and I. Moerdijk. This covering theorem is the real heart of our completeness theorems; we sketch its application to our situation as an appendix to this paper. So as not to obscure the conceptual simplicity of this application, our treatment of the standard details of higher-order syntax and topos semantics is held quite brief.
Before getting down to business, we make two remarks on the statements of the completeness theorems. First, each has the form “there exists a model $M$ such that ${\ensuremath{\mathbb{T}}}\vdash\sigma$ just if $M\models\sigma$”, rather than the more familiar (for set-valued semantics) “${\ensuremath{\mathbb{T}}}\vdash\sigma$ just if for all models $M$, $M\models\sigma$ ”. The stronger form given here is made possible by considering models in topoi other than [$\mathbf{Sets}$]{}. The situation is analogous to that of the familiar Heyting-valued completeness theorem for first-order intuitionistic logic , which follows directly from our theorem B and indeed is the inspiration thereof. Second, and more substantially, the additional “Moreover…” clause of each theorem states a further property of the respective logical system that may be termed “definitional completeness”. It ensures that any function which is “present in every model” is logically definable. As in the case of deductive completeness, this definitional completeness is established in a strong form simply by exhibiting a *single* model in which *every* function of suitable type is definable. In light of the topological nature of the models at issue here, logical definability then coincides with continuity in that “minimal” model. For further discussion of this property (in the context of the $\lambda$-calculus) we refer to [@Awod:lambda].
#### Acknowledgments. {#acknowledgments. .unnumbered}
We have both benefitted greatly from conversations with Ieke Moerdijk on the spatial covering theorem and its logical applications. The Stefan Banach Mathematical Research Center in Warsaw, and the organizers of the Rasiowa memorial conference held there in December 1996, are thanked for supporting our collaboration.
Theories in classical higher-order logic
========================================
The systems of classical higher-order logic that we consider are essentially the same as those presented in , which in turn are modern formulations of [@Church:40]. We summarize one particular formulation for the reader’s convenience and for the special purposes of §4.
*Type symbols* are built up inductively from a given list of *basic type symbols* $X_{1}, \ldots, X_{n}$ and the *type of formulas* $2$ by the *type-forming operations* $Y\times Z$ and $Z^{Y}$.
*Terms* are built up inductively from *variables* and a given list of *basic terms* $c_{1}, \ldots, c_{m}$. Each variable and basic term has a type. The terms and their types are as follows, writing $\tau:Z$ for “$\tau$ is a term of type $Z$”.
- If $\tau_{1}:Z_{1}$ and $\tau_{2}:Z_{2}$, then $\langle\tau_{1},\tau_{2}\rangle:Z_{1}\times Z_{2}$.
- If $\tau:Z_{1}\times Z_{2}$, then $\pi_{1}(\tau):Z_{1}$ and $\pi_{2}(\tau):Z_{2}$.
- If $\tau:Z$ and $y$ is a variable of type $Y$, then $\lambda y.\tau:Z^{Y}$.
- If $\alpha:Z^{Y}$ and $\tau:Y$, then $\alpha(\tau):Z$.
- If $\tau,\tau':Z$, then $\tau = \tau' : 2$.
- If $\varphi,\psi:2$ and $y$ is a variable of type $Y$, then the following are terms of type $2$: $$\top,\ \bot,\ \neg\varphi,\ \varphi\wedge\psi,\ \varphi\vee\psi,\
\varphi{\ensuremath{\Rightarrow}}\psi,\ \forall y.\varphi,\ \exists y.\varphi.$$
A *basic language* (or *signature*) consists of basic type symbols $X_{1}, \ldots, X_{n}$ and basic constant symbols $c_{1}, \ldots, c_{m}$. A *theory* consists of a basic language and a list of sentences (closed formulas) $\sigma_{1}, \ldots, \sigma_{k}$ therein, called *axioms*. Given a theory ${\ensuremath{\mathbb{T}}}$, the *language* ${\ensuremath{\mathcal{L}}}({\ensuremath{\mathbb{T}}})$ of ${\ensuremath{\mathbb{T}}}$ is the set of terms in the basic language of ${\ensuremath{\mathbb{T}}}$.
The *entailment relation* $\varphi\vdash\psi$ between formulas is specified in the usual way by a deductive calculus. To include the possibility of “empty” types, it is convenient to give a family of entailment relations $\varphi\vdash_{\mathbf{x}}\psi$ indexed by lists $\mathbf{x} = (x_{1},\ldots,x_{i})$ of distinct variables including all those occurring free in $\varphi$ and $\psi$. These relations are generated by the following conditions (“rules of inference”):
1. *Order*
1. $\varphi\vdash_{\mathbf{x}}\varphi$
2. $\varphi\vdash_{\mathbf{x}}\psi
{\ \text{and}\ }\psi\vdash_{\mathbf{x}}\vartheta
{\quad\text{implies}\quad}\varphi\vdash_{\mathbf{x}}\vartheta$
3. $\varphi\vdash_{\mathbf{x},y}\psi
{\quad\text{implies}\quad}\varphi[\tau/y]\vdash_{\mathbf{x}}\psi[\tau/y]$
2. *Equality*
1. $\top\vdash_{\mathbf{x}}\tau=\tau$
2. $\tau = \tau' \vdash_{\mathbf{x}} \varphi[\tau/y]{\ensuremath{\Rightarrow}}\varphi[\tau'/y]$
3. $\vartheta\vdash_{\mathbf{x}}\varphi{\ensuremath{\Rightarrow}}\psi
{\ \text{and}\ }\vartheta\vdash_{\mathbf{x}}\psi{\ensuremath{\Rightarrow}}\varphi
{\quad\text{implies}\quad}\vartheta\vdash_{\mathbf{x}}\varphi=\psi$
4. $\forall y. \alpha(y)=\beta(y)\vdash_{\mathbf{x}} \alpha=\beta$
3. *Products*
1. $\top\vdash_{\mathbf{x}} \langle\pi_{1}\tau,\pi_{2}\tau\rangle=\tau$
2. $\top\vdash_{\mathbf{x}} \pi_{i}\langle\tau_{1},\tau_{2}\rangle=\tau_{i},\quad i=1,2$
4. *Exponents*
1. $\top\vdash_{\mathbf{x}}(\lambda x.\tau)(x)=\tau$
2. $\top\vdash_{\mathbf{x}}\lambda x.\alpha(x)=\alpha$
5. *Elementary logic*
1. $\bot\vdash_{\mathbf{x}}\varphi$
2. $\varphi\vdash_{\mathbf{x}}\top$
3. $\varphi\vdash_{\mathbf{x}}\neg\psi
{\quad\text{iff}\quad}\varphi\wedge\psi\vdash_{\mathbf{x}}\bot$
4. $\vartheta\vdash_{\mathbf{x}}\varphi
{\ \text{and}\ }\vartheta\vdash_{\mathbf{x}}\psi
{\quad\text{iff}\quad}\vartheta\vdash_{\mathbf{x}}\varphi\wedge\psi$
5. $\vartheta\vee\varphi\vdash_{\mathbf{x}}\psi
{\quad\text{iff}\quad}\vartheta\vdash_{\mathbf{x}}\psi
{\ \text{and}\ }\varphi\vdash_{\mathbf{x}}\psi$
6. $\vartheta\wedge\varphi\vdash_{\mathbf{x}}\psi
{\quad\text{iff}\quad}\vartheta\vdash_{\mathbf{x}}\varphi{\ensuremath{\Rightarrow}}\psi$
7. $\vartheta\vdash_{\mathbf{x},y}\varphi
{\quad\text{iff}\quad}\vartheta\vdash_{\mathbf{x}}\forall y.\varphi$
8. $\exists y.\vartheta\vdash_{\mathbf{x}}\varphi
{\quad\text{iff}\quad}\vartheta\vdash_{\mathbf{x},y}\varphi$
In the foregoing, the $\tau$’s are arbitrary terms; $\varphi$, $\psi$, $\vartheta$ are formulas; and $\alpha$, $\beta$ are terms of the same exponential type. In writing e.g.$\varphi[\tau/y]\vdash_{\mathbf{x}}\psi[\tau/y]$ in 1(c) it is assumed that $\varphi[\tau/y]$ and $\psi[\tau/y]$ are formulas with no free variables apart from $x_{1},\ldots,x_{i}$; so the term $\tau$ must have the same type as the variable $y$ and no other free variables. As usual, the substitution notation $\varphi[\tau/y]$ is understood to include a convention to avoid binding free variables in $\tau$.
A sentence $\sigma$ is called *provable* if $\top\vdash\sigma$, also written $\vdash\sigma$. For a theory [$\mathbb{T}$]{}, the notions of *${\ensuremath{\mathbb{T}}}$-entailment* and *${\ensuremath{\mathbb{T}}}$-provability* are given by adding the rules $\vdash\sigma$ for each axiom $\sigma$ of [$\mathbb{T}$]{}.
The *classical entailment* relation $\vdash^{c}$ results from $\vdash$ by adding the rule $$\vdash^{c}\ \forall p.p\vee\neg p.$$
\[succinct\] It is sometimes convenient to give a more succinct statement of the logical calculus by defining some of the logical primitives in terms of others. We mention one particularly simple primitive basis which will be useful in the next section (cf. ). Exponential types $Z^{Y}$ occur only in the form $2^{Y}$ (“power types”, usually written $P(Y)$); $\lambda$-terms $\lambda x.\varphi$ and evaluations $\alpha(\tau)$ are then restricted accordingly, and more naturally written $\{x|\varphi\}$ and $\tau\in\alpha$. Projection operators $\pi_{i}(\tau)$ are eliminated in favor of additional rules of inference. The logical operations $\top,\
\bot,\ \neg,\ \wedge,\ \vee,\ {\Rightarrow},\ \forall,\ \exists$ are defined in terms of $=$ and $\langle -, - \rangle,\ \{x| - \},\ \in$. We shall use the fact that this primitive basis suffices in the following way: to interpret the language of a theory it suffices to interpret the basic language, the type of formulas, product and power types, and the term-forming operations ${\langle -, - \rangle},\ {\{x| - \}},\ {\in},\ {=}$.
In the opposite direction, one can enlarge the primitive logical basis by including basic relation and function symbols in addition to basic constant symbols, although these are not needed in the presence of higher relation types. Relation symbols will be useful in §4, however, where there is no type of formulas; and both relation and function symbols are used in elementary logic, where there are no higher types at all.
Semantics in topoi
==================
Let ${\ensuremath{\mathbb{T}}}$ be a theory in classical higher-order logic, as defined in the foregoing section. It is fairly obvious how to interpret ${\ensuremath{\mathbb{T}}}$ in an arbitrary boolean topos [$\mathcal{B}$]{}: An *interpretation* $M$ of ${\ensuremath{\mathbb{T}}}$ in [$\mathcal{B}$]{} assigns to each basic type symbol $X$ an object $X_{M}$ of [$\mathcal{B}$]{}, and to the type $2$ of formulas, the coproduct $1_{{\ensuremath{\mathcal{B}}}}+1_{{\ensuremath{\mathcal{B}}}}$ in [$\mathcal{B}$]{} (which is the subobject classifier), $$2_{M} = 1_{{\ensuremath{\mathcal{B}}}}+1_{{\ensuremath{\mathcal{B}}}}.$$ The interpretation $M$ is then extended to product and power types by setting $$\begin{aligned}
(Y\times Z)_{M}& = Y_{M}\times Z_{M}&&\text{(product in {\ensuremath{\mathcal{B}}}),}\\
(2^{Y})_{M}& = (2_{M})^{(Y_{M})}&&\text{(exponential in {\ensuremath{\mathcal{B}}}).}
\end{aligned}$$ On terms, $M$ assigns to each basic constant symbol $c$ of ${\ensuremath{\mathbb{T}}}$, having say type $Z$, a morphism $$c_{M} \colon 1_{{\ensuremath{\mathcal{B}}}}{\ensuremath{\rightarrow}}Z_{M}$$ of [$\mathcal{B}$]{}, and variables are interpreted as identity morphisms. The interpretation is then extended inductively to all terms in ${\ensuremath{\mathcal{L}}}({\ensuremath{\mathbb{T}}})$ in the evident way, using the internal logic of [$\mathcal{B}$]{} (cf. , also for the external meaning of the logical operations thus modeled). For example, $$(\tau = \tau')_{M} = \delta\circ\langle\tau,\tau'\rangle_{M},$$ where $\delta\colon Z_{M}\times Z_{M}{\ensuremath{\rightarrow}}1_{{\ensuremath{\mathcal{B}}}}+1_{{\ensuremath{\mathcal{B}}}}$ classifies the diagonal morphism $\Delta = \langle 1_{Z_{M}}, 1_{Z_{M}}\rangle \colon
Z_{M}{\rightarrowtail}Z_{M}\times Z_{M}$, when $Z$ is the type of the terms $\tau,
\tau'$.
In particular, $M$ assigns to each formula $\varphi(y_{1}, \ldots, y_{n})$ with free variables $y_{i}$ of types $Y_{i}$ a morphism $$\varphi(y_{1}, \ldots, y_{n})_{M} \colon (Y_{1})_{M}\times\ldots\times (Y_{n})_{M}\longrightarrow 1_{{\ensuremath{\mathcal{B}}}}+1_{{\ensuremath{\mathcal{B}}}}$$ of [$\mathcal{B}$]{}. A sentence $\sigma$ is said to be *true in $M$*, written $M\models\sigma$, if $$\sigma_{M}=\operatorname{true}\colon 1_{{\ensuremath{\mathcal{B}}}}{\ensuremath{\rightarrow}}1_{{\ensuremath{\mathcal{B}}}}+1_{{\ensuremath{\mathcal{B}}}},$$ where $\operatorname{true}\colon 1_{{\ensuremath{\mathcal{B}}}}{\ensuremath{\rightarrow}}1_{{\ensuremath{\mathcal{B}}}}+1_{{\ensuremath{\mathcal{B}}}}$ is the first coproduct inclusion, which is the universal subobject. Of course, an interpretation $M$ is a *model* of ${\ensuremath{\mathbb{T}}}$ if each axiom of ${\ensuremath{\mathbb{T}}}$ is true in $M$.
Representing the category of models {#classtop}
-----------------------------------
Given models $M$ and $N$ of a theory ${\ensuremath{\mathbb{T}}}$ in a boolean topos [$\mathcal{B}$]{}, there is an evident notion of an isomorphism $h\colon M{\ensuremath{\stackrel{\sim}{\longrightarrow}}}N$ of ${\ensuremath{\mathbb{T}}}$-models, namely a family of isos $h=(h_{X}\colon X_{M}{\ensuremath{\stackrel{\sim}{\longrightarrow}}}X_{N})$ (indexed by the basic types $X$ of ${\ensuremath{\mathbb{T}}}$) that preserve the interpretations of the constant symbols of ${\ensuremath{\mathbb{T}}}$, in the obvious sense. Together with the evident composites and identities, one thus has for any theory ${\ensuremath{\mathbb{T}}}$ and any boolean topos [$\mathcal{B}$]{} a *category of models* of ${\ensuremath{\mathbb{T}}}$ in [$\mathcal{B}$]{}, denoted $$\operatorname{\mathbf{Mod}}_{{\ensuremath{\mathbb{T}}}}({\ensuremath{\mathcal{B}}}).$$
Observe that $\operatorname{\mathbf{Mod}}_{{\ensuremath{\mathbb{T}}}}({\ensuremath{\mathcal{B}}})$ is always a groupoid, i.e.a category in which every morphism is iso. For example, if ${\ensuremath{\mathbb{T}}}$ is the theory of topological spaces and [$\mathcal{B}$]{} is the topos [$\mathbf{Sets}$]{}, then $\operatorname{\mathbf{Mod}}_{{\ensuremath{\mathbb{T}}}}({\ensuremath{\mathcal{B}}})$ is the category of all topological spaces and homeomorphisms. One can of course also consider other morphisms of models, but the groupoid of isomorphisms suffices for our purposes.
A logical morphism between boolean topoi plainly preserves models and their morphisms. Such a functor $f \colon {\ensuremath{\mathcal{B}}}{\ensuremath{\rightarrow}}{\ensuremath{\mathcal{B'}}}$ therefore induces a functor $$\operatorname{\mathbf{Mod}}_{{\ensuremath{\mathbb{T}}}}(f)\colon \operatorname{\mathbf{Mod}}_{{\ensuremath{\mathbb{T}}}}({\ensuremath{\mathcal{B}}}){\ensuremath{\rightarrow}}\operatorname{\mathbf{Mod}}_{{\ensuremath{\mathbb{T}}}}({\ensuremath{\mathcal{B'}}})$$ (a groupoid homomorphism) on the associated categories of models.
Now, every theory ${\ensuremath{\mathbb{T}}}$ in classical higher-order logic has a (higher-order) *classifying topos*, a boolean topos ${\ensuremath{\mathcal{B}}}_{{\ensuremath{\mathbb{T}}}}$ determined uniquely (up to equivalence) by the property: for any boolean topos [$\mathcal{B}$]{} there is an equivalence of categories, natural in [$\mathcal{B}$]{}, $$\operatorname{\mathbf{Log}}({\ensuremath{\mathcal{B}}}_{{\ensuremath{\mathbb{T}}}}, {\ensuremath{\mathcal{B}}})\ {\ensuremath{\simeq}}\ \operatorname{\mathbf{Mod}}_{{\ensuremath{\mathbb{T}}}}({\ensuremath{\mathcal{B}}}),
\label{univprop}$$ where $\operatorname{\mathbf{Log}}({\ensuremath{\mathcal{B}}}_{{\ensuremath{\mathbb{T}}}}, {\ensuremath{\mathcal{B}}})$ is the category of logical morphisms ${\ensuremath{\mathcal{B}}}_{{\ensuremath{\mathbb{T}}}}{\ensuremath{\rightarrow}}{\ensuremath{\mathcal{B}}}$ and natural isomorphisms between them (cf.[@Awod:thesis]). The classifying topos ${\ensuremath{\mathcal{B}}}_{{\ensuremath{\mathbb{T}}}}$ can be constructed “syntactically” from ${\ensuremath{\mathcal{L}}}({\ensuremath{\mathbb{T}}})$ in the style of ; in particular, it is a small category (indeed, it is countable). In virtue of its universal mapping property , ${\ensuremath{\mathcal{B}}}_{{\ensuremath{\mathbb{T}}}}$ is freely generated as a boolean topos by the “universal model” $U_{{\ensuremath{\mathbb{T}}}}\in
\operatorname{\mathbf{Mod}}_{{\ensuremath{\mathbb{T}}}}({\ensuremath{\mathcal{B}}}_{{\ensuremath{\mathbb{T}}}})$ associated to the identity logical morphism ${\ensuremath{\mathcal{B}}}_{{\ensuremath{\mathbb{T}}}}{\ensuremath{\rightarrow}}{\ensuremath{\mathcal{B}}}_{{\ensuremath{\mathbb{T}}}}$ under . By the syntactic construction of ${\ensuremath{\mathcal{B}}}_{{\ensuremath{\mathbb{T}}}}$ this universal model has the following properties, which we record for later use:
\[univmodel\]
1. For any sentence $\sigma\in{\ensuremath{\mathcal{L}}}({\ensuremath{\mathbb{T}}})$, $${\ensuremath{\mathbb{T}}}\vdash^{c}\sigma\quad\text{just if}\quad U_{{\ensuremath{\mathbb{T}}}}\models\sigma.$$
2. For any types $Y$ and $Z$ and any morphism $f \colon
Y_{U_{{\ensuremath{\mathbb{T}}}}}{\ensuremath{\rightarrow}}Z_{U_{{\ensuremath{\mathbb{T}}}}}$ in ${\ensuremath{\mathcal{B}}}_{{\ensuremath{\mathbb{T}}}}$, there is a formula $\varphi(y,z)\in {\ensuremath{\mathcal{L}}}({\ensuremath{\mathbb{T}}})$ such that $$\mathrm{graph}(f)=\{\langle y,z\rangle \mid \varphi(y,z)\}_{U_{{\ensuremath{\mathbb{T}}}}}$$ (as subobjects of $Y_{U_{{\ensuremath{\mathbb{T}}}}}\times Z_{U_{{\ensuremath{\mathbb{T}}}}}$).
Observe that (i) of proposition \[univmodel\] and the universal mapping property together entail the soundness and completeness of the deductive calculus of §1 with respect to topos semantics:${\ensuremath{\mathbb{T}}}\vdash^{c}\sigma$ if and only if for every ${\ensuremath{\mathbb{T}}}$-model $M$, $M\models\sigma$.
Classical semantics
-------------------
We now extend the foregoing topos semantics for classical higher-order logic to non-boolean topoi. Let ${\ensuremath{\mathbb{T}}}$ be a fixed theory and [$\mathcal{E}$]{} an arbitrary topos. We begin with a bit of notation: Let $\operatorname{true}\colon
1_{{\ensuremath{\mathcal{E}}}}\to\Omega_{{\ensuremath{\mathcal{E}}}}$ be the subobject classifier in [$\mathcal{E}$]{}, and let us write $${\lvert-\rvert} = (\operatorname{true},\operatorname{false}) \colon
1_{{\ensuremath{\mathcal{E}}}}+1_{{\ensuremath{\mathcal{E}}}}\longrightarrow\Omega_{{\ensuremath{\mathcal{E}}}}$$ for the canonical map from the coproduct which, observe, is a monomorphism. An arbitrary morphism $\varphi \colon E{\ensuremath{\rightarrow}}\Omega_{{\ensuremath{\mathcal{E}}}}$ of [$\mathcal{E}$]{} factors through ${\lvert-\rvert}$ just if the subobject $S_{\varphi}{\rightarrowtail}E$ it classifies is *complemented*, i.e. if there is a subobject $S{\rightarrowtail}E$ with $S_{\varphi}+S{\ensuremath{\cong}}E$ (canonically). When this is the case, let us write $\overline{\varphi} \colon E{\ensuremath{\rightarrow}}1_{{\ensuremath{\mathcal{E}}}}+1_{{\ensuremath{\mathcal{E}}}}$ for the unique morphism such that $$\varphi = {\lvert\overline{\varphi}\rvert},$$ as indicated in
E &\
&& \_[[$\mathcal{E}$]{}]{}.\
Recall that an object $E$ of [$\mathcal{E}$]{} is said to be *decidable* if its diagonal $\Delta \colon E{\rightarrowtail}E\times E$ is complemented, thus just if $\overline{\delta} \colon E\times E {\ensuremath{\rightarrow}}1_{{\ensuremath{\mathcal{E}}}}+1_{{\ensuremath{\mathcal{E}}}}$ exists.
Next, we define an *interpretation* of the basic language of ${\ensuremath{\mathbb{T}}}$ in [$\mathcal{E}$]{} exactly as in a boolean topos; in particular the type $2$ of formulas is interpreted as $1_{{\ensuremath{\mathcal{E}}}}+1_{{\ensuremath{\mathcal{E}}}}$, which is plainly decidable. An interpretation $M$ such that for each type symbol $Z$ the object $Z_{M}$ in [$\mathcal{E}$]{} is decidable shall be called a *classical interpretation* (or *c-interpretation*).
Finally, by remark \[succinct\] any c-interpretation $M$ can be extended to all of ${\ensuremath{\mathcal{L}}}({\ensuremath{\mathbb{T}}})$ exactly as in a boolean topos, by interpreting the term-forming operations $\langle -, - \rangle,\ \{x| - \},\ \in$ as before and taking $\overline{\delta}\colon Z_{M}\times Z_{M}{\ensuremath{\rightarrow}}1_{{\ensuremath{\mathcal{E}}}}+1_{{\ensuremath{\mathcal{E}}}}$ to interpret $=$ at each type $Z$. Thus just as before a c-interpretation $M$ assigns to each formula $\varphi(y_{1}, \ldots, y_{n})$ with free variables $y_{i}$ of types $Y_{i}$ a morphism $$\varphi(y_{1}, \ldots, y_{n})_{M} \colon (Y_{1})_{M}\times\ldots\times (Y_{n})_{M} \longrightarrow 1_{{\ensuremath{\mathcal{E}}}}+1_{{\ensuremath{\mathcal{E}}}}.$$
The relation $\models^{c}$ of *satisfaction* for c-interpretations is defined by: $$M\models^{c}\sigma{\quad\text{iff}\quad}{\lvert\sigma_{M}\rvert} = \operatorname{true}.$$ Thus a c-interpretation $M$ satisfies a sentence $\sigma$ just if the following triangle commutes
1\_[[$\mathcal{E}$]{}]{} &\
&& \_[[$\mathcal{E}$]{}]{}.\
A *c-model* of the theory ${\ensuremath{\mathbb{T}}}$ is of course a c-interpretation that satisfies the axioms of ${\ensuremath{\mathbb{T}}}$. A c-interpretation $M$ is therefore a c-model just if for each axiom $\sigma$ the interpretation $$\sigma_{M}\colon 1_{{\ensuremath{\mathcal{E}}}} \longrightarrow 1_{{\ensuremath{\mathcal{E}}}}+1_{{\ensuremath{\mathcal{E}}}}$$ is the first coproduct inclusion, just as in the boolean case. Indeed, if the topos [$\mathcal{E}$]{} is boolean, then every object is decidable, and a c-model in [$\mathcal{E}$]{} is the same thing as a model.
If $M$ is a c-model then for any sentence $\sigma$, $${\ensuremath{\mathbb{T}}}\vdash^{c}\sigma{\quad\text{implies}\quad}M\models^{c}\sigma.$$
Consider the classifying topos ${\ensuremath{\mathcal{B}}}_{{\ensuremath{\mathbb{T}}}}$ with universal model $U_{{\ensuremath{\mathbb{T}}}}$. There is an evident functor $m : {\ensuremath{\mathcal{B}}}_{{\ensuremath{\mathbb{T}}}}\to{\ensuremath{\mathcal{E}}}$ with $M=m(U_{{\ensuremath{\mathbb{T}}}})$ and $$\sigma_{M}=\sigma_{m(U_{{\ensuremath{\mathbb{T}}}})}=m(\sigma_{U_{{\ensuremath{\mathbb{T}}}}})$$ for each sentence $\sigma$. Although $m$ is not logical if [$\mathcal{E}$]{} is not boolean, it still takes $\operatorname{true}: 1_{{\ensuremath{\mathcal{B}}}_{{\ensuremath{\mathbb{T}}}}}{\ensuremath{\rightarrow}}1_{{\ensuremath{\mathcal{B}}}_{{\ensuremath{\mathbb{T}}}}}+1_{{\ensuremath{\mathcal{B}}}_{{\ensuremath{\mathbb{T}}}}}$ to $\overline{\operatorname{true}} :
1_{{\ensuremath{\mathcal{E}}}}{\ensuremath{\rightarrow}}1_{{\ensuremath{\mathcal{E}}}}+1_{{\ensuremath{\mathcal{E}}}}$. The claim thus follows from the soundness of standard topos semantics (in particular, from proposition \[univmodel\]).
If the interpretation $Z_{M}$ is decidable then for any type $Y$ the canonical inclusion $(Z_{M})^{(Y_{M})}{\rightarrowtail}\Omega^{(Y_{M}\times
Z_{M})}$ factors as indicated in the following diagram.
&& (1\_[[$\mathcal{E}$]{}]{}+1\_[[$\mathcal{E}$]{}]{})\^[(Y\_[M]{}Z\_[M]{})]{}\
& & >[ [-]{}\^[(Y\_[M]{}Z\_[M]{})]{}]{}\
(Z\_[M]{})\^[(Y\_[M]{})]{} && \^[(Y\_[M]{}Z\_[M]{})]{}\
Thus even when defined in terms of power types as mentioned in remark \[succinct\], the exponential types $Z^{Y}$ are still interpreted as exponentials by a c-interpretation.
Topological completeness
========================
In this section we consider small topoi equipped with the finite epi topology. The covering families for this Grothendieck topology on a topos [$\mathcal{E}$]{} are finite families of morphisms $(C_{i}{\ensuremath{\rightarrow}}E)_{i}$ such that the canonical map $\coprod_{i}C_{i}{\ensuremath{\rightarrow}}E$ is epic. Of the following two technical lemmas, we omit the straightforward proof of the first; its second part is folklore.
\[lem:ypreserves\]
1. The finite epi topology is subcanonical.
2. For each morphism $e : E'\to E$ in [$\mathcal{E}$]{}, the sheafified Yoneda embedding $y:{\ensuremath{\mathcal{E}}}\to\operatorname{Sh}({\ensuremath{\mathcal{E}}})$ preserves not only the pullback functor $e^{*} : {\ensuremath{\mathcal{E}}}/E\to {\ensuremath{\mathcal{E}}}/E'$, but also its left and right adjoints, $$\Sigma_{e}\dashv e^{*}\dashv\Pi_{e} : {\ensuremath{\mathcal{E}}}/E'\to {\ensuremath{\mathcal{E}}}/E.$$ (Indeed, this is true for any subcanonical topology on a small category and any locally cartesian closed structure present there.)
\[lem:1\] Let $F\colon {\ensuremath{\mathcal{B}}} {\ensuremath{\rightarrow}}{\ensuremath{\mathcal{E}}} $ be a left-exact functor from a boolean topos [$\mathcal{B}$]{} to any topos [$\mathcal{E}$]{}. If $F$ is continuous for the finite epi topology then it preserves finite coproducts and first-order logic. If $F$ also preserves exponentials, then it preserves c-models.
An object of a topos has an empty covering family for the finite epi topology just if it is initial; so the continuous functor $F$ preserves initial objects. The coproduct inclusions $B_{1},
B_{2}{\rightarrowtail}B_{1}+B_{2}$ are a covering family of monos with $B_{1}\wedge B_{2}=0{\rightarrowtail}B_{1}+B_{2}$. Since $F$ is also left-exact, it then preserves coproducts as well. Moreover, it then preserves boolean complements of subobjects, whence it preserves negation $\neg$ since [$\mathcal{B}$]{} is boolean. The logical operations $\Rightarrow$ and $\forall$ are then also preserved, since in a boolean topos these can be constructed from negation and operations that are preserved by left-exact, continuous functors generally. Finally, if $F$ also preserves exponentials then by the foregoing it preserves the interpretations of all types and the associated term-forming operations, in addition to first-order logic; whence it clearly preserves c-models.
Theorem A will now follow by applying the covering theorem of the appendix, which states that every Grothendieck topos with enough points can be covered by a topological space via a connected, locally connected geometric morphism. We remind the reader that a Grothendieck topos ${\ensuremath{\mathcal{G}}}$ is said to have *enough points* if the geometric morphisms $p : {\ensuremath{\mathbf{Sets}}}\to{\ensuremath{\mathcal{G}}}$ are jointly surjective (i.e. if the inverse images $p^{*} : {\ensuremath{\mathcal{G}}}\to{\ensuremath{\mathbf{Sets}}}$ of these are jointly faithful), and that a geometric morphism $f^{*}\dashv f_{*}
\colon {\ensuremath{\mathcal{E}}}{\ensuremath{\rightarrow}}{\ensuremath{\mathcal{F}}}$ of topoi is *connected* if the inverse image functor $f^{*}$ is full and faithful, and *locally connected* (: “molecular”) if $f^{*}$ commutes with $\Pi$-functors.
Let ${\ensuremath{\mathbb{T}}}$ be a higher-order theory. There exists a topological space $X_{{\ensuremath{\mathbb{T}}}}$ and a c-model $M$ of ${\ensuremath{\mathbb{T}}}$ in $\operatorname{Sh}(X_{{\ensuremath{\mathbb{T}}}})$ such that:
1. for any sentence $\sigma\in {\ensuremath{\mathcal{L}}}({\ensuremath{\mathbb{T}}})$, $${\ensuremath{\mathbb{T}}}\vdash^{c}\sigma\qquad\text{if and only if}\qquad M\models^{c}\sigma;$$
2. given types $Y, Z$, every continuous function $f \colon Y_{M}{\ensuremath{\rightarrow}}Z_{M}$ over $X_{{\ensuremath{\mathbb{T}}}}$ is definable: there is a formula $\varphi(y,z)\in
{\ensuremath{\mathcal{L}}}({\ensuremath{\mathbb{T}}})$ such that $$\mathrm{graph}(f)=\{\langle y,z\rangle | \varphi(y,z)\}_{M}$$ (as subsheaves of $Y_{M}\times Z_{M}$).
First, one has the universal model $U_{{\ensuremath{\mathbb{T}}}}$ in the classifying topos ${\ensuremath{\mathcal{B}}}_{{\ensuremath{\mathbb{T}}}}$, as in §2.1. The Grothendieck topos $\operatorname{Sh}({\ensuremath{\mathcal{B}}}_{{\ensuremath{\mathbb{T}}}})$ of sheaves on ${\ensuremath{\mathcal{B}}}_{{\ensuremath{\mathbb{T}}}}$ for the finite epi topology is coherent, and so has enough points (cf. ). The covering theorem of the appendix therefore guarantees the existence of a topological space $X_{{\ensuremath{\mathbb{T}}}}$ and a connected, locally-connected geometric morphism $$m \colon \operatorname{Sh}(X_{{\ensuremath{\mathbb{T}}}}){\ensuremath{\rightarrow}}\operatorname{Sh}({\ensuremath{\mathcal{B}}}_{{\ensuremath{\mathbb{T}}}}).$$ The inverse image $m^{*}\colon \operatorname{Sh}({\ensuremath{\mathcal{B}}}_{{\ensuremath{\mathbb{T}}}}){\ensuremath{\rightarrow}}\operatorname{Sh}(X_{{\ensuremath{\mathbb{T}}}})$ of $m$ satisfies all hypotheses of the foregoing lemma \[lem:1\], as does the sheafified Yoneda embedding $$y \colon {\ensuremath{\mathcal{B}}}_{{\ensuremath{\mathbb{T}}}}{\ensuremath{\rightarrow}}\operatorname{Sh}({\ensuremath{\mathcal{B}}}_{{\ensuremath{\mathbb{T}}}}).$$ In particular, these functors preserve exponentials since they preserve $\Pi$-functors (using lemma \[lem:ypreserves\]). The composite $m^{*}\circ y\colon {\ensuremath{\mathcal{B}}}_{{\ensuremath{\mathbb{T}}}}{\ensuremath{\rightarrow}}\operatorname{Sh}(X_{{\ensuremath{\mathbb{T}}}})$ therefore also satisfies the hypotheses of lemma \[lem:1\], whence one has the c-model $$M = m^{*}\circ y(U_{{\ensuremath{\mathbb{T}}}})$$ in $\operatorname{Sh}(X_{{\ensuremath{\mathbb{T}}}})$. Since each of its factors is full and faithful, so is the functor $m^{*}\circ y$; the assertions (i) and (ii) thus follow from proposition \[univmodel\].
(Infinitary generalizations) Theorem A clearly applies equally to “theories” ${\ensuremath{\mathbb{T}}}$ with infinitely many type and/or constant symbols and/or axioms, since in such cases the foregoing proof can begin with a small topos ${\ensuremath{\mathcal{B}}}_{{\ensuremath{\mathbb{T}}}}$ which is a suitable colimit of classifying topoi for (finite) theories. We also merely mention that for the case of infinitary logic, with set-indexed meets and joins of formulas, a theorem analogous to theorem A holds, with complete Heyting algebras in place of topological spaces.
$\lambda$-logic
===============
What we call $\lambda$-logic differs from classical higher-order logic in that it has no type $2$ of formulas. Type symbols are now built up inductively from basic type symbols by the operations $-\times ?$ and $-^{?}$. Terms are built up inductively from variables, basic constant symbols, and just the term-forming operations $\langle
-,?\rangle$, $\pi_{1}(-)$, $\pi_{2}(-)$, $\lambda y.(-)$, and $?(-)$. Formulas are then constructed from terms and basic relation symbols in the customary way, using the language of first-order logic with equality. Finally, a *$\lambda$-theory* consists of (finitely many) basic type, constant, and relation symbols, and closed formulas in these parameters.
As rules of inference for the (intuitionistic) entailment relation $\varphi\vdash_{\mathbf{x}}\psi$ on formulas one may take a standard deductive calculus for (intuitionistic) many-sorted, first-order logic with equality, augmented by the usual rules for the (typed) $\lambda$-calculus. Indeed, the rules given in §1 above are suitable, under the omission of 2(c).
The notion of a model of a $\lambda$-theory in a topos is essentially the same as that already given in §2. It is, however, now more natural to interpret basic relation symbols and other formulas by subobjects (rather than their classifying morphisms), as is usually done for first-order logic (cf. ). In particular, the equality sign $=$ is interpreted in the standard way as a diagonal morphism, and since classical logic is not being assumed, the notion of a c-model is not required.
Deduction is clearly sound with respect to such semantics. To show that it is also complete—even with regard to just topological models—one can proceed as in the classical higher-order case in §3:
1. Construct the syntactic category ${\ensuremath{\mathcal{S}}}_{{\ensuremath{\mathbb{T}}}}$ of provable equivalence classes of formulas, to be equipped with the finite epi topology (which is sub-canonical).
2. Apply the sheafified Yoneda embedding $y \colon
{\ensuremath{\mathcal{S}}}_{{\ensuremath{\mathbb{T}}}}{\ensuremath{\rightarrow}}\operatorname{Sh}({\ensuremath{\mathcal{S}}}_{{\ensuremath{\mathbb{T}}}})$ (which preserves $\lambda$-logic by lemma \[lem:ypreserves\]) to get a full and faithful model in a Grothendieck topos with enough points.
3. Apply the covering theorem of the appendix to get a connected, locally connected geometric covering map $\operatorname{Sh}(X_{{\ensuremath{\mathbb{T}}}}){\ensuremath{\rightarrow}}\operatorname{Sh}({\ensuremath{\mathcal{S}}}_{{\ensuremath{\mathbb{T}}}})$ from a topological sheaf topos $\operatorname{Sh}(X_{{\ensuremath{\mathbb{T}}}})$.
We leave it to the reader to fill in the details of this sketch to provide the proof of the following.
Let ${\ensuremath{\mathbb{T}}}$ be a $\lambda$-theory. There exists a topological space $X_{{\ensuremath{\mathbb{T}}}}$ and a model $M$ of ${\ensuremath{\mathbb{T}}}$ in $\operatorname{Sh}(X_{{\ensuremath{\mathbb{T}}}})$ such that:
1. for any $\lambda$-sentence $\sigma$ in the language of ${\ensuremath{\mathbb{T}}}$, $${\ensuremath{\mathbb{T}}}\vdash\sigma\qquad\text{if and only if}\qquad M\models\sigma;$$
2. given types $Y, Z$, every continuous function $f \colon Y_{M}{\ensuremath{\rightarrow}}Z_{M}$ over $X_{{\ensuremath{\mathbb{T}}}}$ is definable: there is a $\lambda$-formula $\varphi(y,z)$ in the language of ${\ensuremath{\mathbb{T}}}$ such that $$\mathrm{graph}(f)=\{\langle y,z\rangle | \varphi(y,z)\}_{M}$$ (as subsheaves of $Y_{M}\times Z_{M}$).
Appendix: The spatial cover {#appendix-the-spatial-cover .unnumbered}
===========================
In the proofs of theorems A and B, use was made of the following covering theorem for topoi, which is part of theorem 13.5 of [@Butz:thesis] (also see ; cf. for a related result).
For any Grothendieck topos $\cal{G}$ with enough points there is a topological space $X_{\cal{G}}$ and a connected, locally connected geometric morphism $$\phi\colon\operatorname{Sh}(X_{\cal{G}})\to {\cal G}.$$
Thus in particular the inverse image functor $\phi^*\colon{\cal
G}\to\operatorname{Sh}(X_{\cal G})$ is fully faithful and preserves exponentials and the internal first-order logic of ${\cal G}$.
The purpose of this appendix is to describe the space $X_{\cal{G}}$ and the covering map $ \phi \colon\operatorname{Sh}(X_{\cal G})\to {\cal G}$ in the case of principal interest here, namely when ${\cal G}=\operatorname{Sh}({\cal B}_{\ensuremath{\mathbb{T}}})$ for ${\cal
B}_{\ensuremath{\mathbb{T}}}$ the small classifying topos of a (classical) higher-order theory, equipped with the finite epi topology. Thus we consider the situation of theorem A; that of theorem B of course has a similar description. Before going into details, let us mention that in fact there are many different spaces which will do the job, depending on various parameters that one is free to choose. We exhibit here just one such choice, intended to be illuminating.
To begin, recall from [@Hen50] that classical higher-order logic is complete with respect to [*general models*]{}, nowadays called [*Henkin models*]{}. The basic feature of a Henkin model $M$ of a theory ${\ensuremath{\mathbb{T}}}$ is that a function type $Z^Y$ (or power type $2^Y$) is interpreted by a [*subset*]{} $(Z^Y)_M\subset (Z_M)^{(Y_{M})}$ of the set of all functions from $Y_M$ to $Z_M$ (resp. of the power set $\wp Y_{M}$), rather than by the set itself. Of course, certain closure conditions also have to be satisfied. We mention only by the way that such models can be shown to arise “naturally” as images of the universal model $U_{{\ensuremath{\mathbb{T}}}}$ under continuous, left exact functors $\mathcal{B}_{{\ensuremath{\mathbb{T}}}}\to\mathbf{Sets}$, and that the said completeness can be inferred from this fact. For the following, it will be convenient to define the underlying set or *universe* ${\lvertM\rvert}$ of a Henkin model $M$ to be the (disjoint) union of the sets $Z_{M}$ for all types $Z$, $${\lvertM\rvert} =\bigcup\{Z_M\mid\mbox{$Z$ a type}\}.$$
To define the space $X_{\ensuremath{\mathbb{T}}}$ for the topos $\operatorname{Sh}({\cal B}_{\ensuremath{\mathbb{T}}})$, fix a *sufficient* set $S_{\ensuremath{\mathbb{T}}}$ of countable Henkin models $M$ of ${\ensuremath{\mathbb{T}}}$, i.e.$S_{\ensuremath{\mathbb{T}}}$ satisfies: $$\text{$M\models\sigma$ for all $M\in S_{\ensuremath{\mathbb{T}}}$}
\qquad\text{implies}\qquad{\ensuremath{\mathbb{T}}}\vdash\sigma$$ for all ${\ensuremath{\mathbb{T}}}$-sentences $\sigma$. For example, we could take (a set of representatives of) all countable Henkin models of ${\ensuremath{\mathbb{T}}}$ as the set $S_{\ensuremath{\mathbb{T}}}$. We then define a *labeling* of a Henkin model $M$ in $S_{\ensuremath{\mathbb{T}}}$ to be a partial function $$\mathbb{N}\supset \mathrm{dom}(\alpha) \stackrel{\alpha}{\longrightarrow}{\lvertM\rvert}$$ such that for each $a\in{\lvertM\rvert}$ the fiber $\alpha^{-1}(a)$ is infinite.
The points of the space $X_{\ensuremath{\mathbb{T}}}$ are labeled Henkin models in $S_{\ensuremath{\mathbb{T}}}$, i.e.pairs $$(M,\alpha)$$ where $M\in S_{\ensuremath{\mathbb{T}}}$ and $\alpha$ is a labeling of $M$. The topology is generated by basic open sets of the form $$U_{\varphi(\bar{z}),\bar{n}}= \{(M,\alpha)\mid
\begin{array}[t]{l}
\mbox{$\alpha(n_i)$ is defined and of type $Z_i$,}\\
\mbox{and $M\models\varphi(\alpha(n_1),\ldots,\alpha(n_m))$}
\quad\}
\end{array}$$ for $\varphi(\bar{z})=\varphi(z_1,\ldots,z_m)$ a ${\ensuremath{\mathbb{T}}}$-formula and $\bar{n}=(n_1,\ldots,n_m)$ a tuple of natural numbers.
To describe the covering map $ \phi \colon\operatorname{Sh}(X_{{\ensuremath{\mathbb{T}}}})\to \operatorname{Sh}({\cal
B}_{\ensuremath{\mathbb{T}}})$ we sketch the construction of the c-model $\Phi$ in $\operatorname{Sh}(X_{{\ensuremath{\mathbb{T}}}})$ induced by $\phi^*$. Here we use the equivalence, mentioned in the introduction, $\operatorname{Sh}(X_{\ensuremath{\mathbb{T}}})\simeq{\rm Etale}/X_{\ensuremath{\mathbb{T}}}$ of sheaves on $X_{\ensuremath{\mathbb{T}}}$ and étale bundles over $X_{\ensuremath{\mathbb{T}}}$. For each type $Z$ we have the set $$Z_\Phi=\sum_{(M,\alpha)\in X_{\ensuremath{\mathbb{T}}}}Z_M,$$ with the evident projection $$\pi_Z\colon Z_\Phi\to X_{\ensuremath{\mathbb{T}}}.$$ We generate a topology on $Z_\Phi$ by declaring to be open:
- the sets $\pi^{-1}_Z(U)$ for $U\subset X_{\ensuremath{\mathbb{T}}}$ open (thus making $\pi_Z$ continuous),
- the sets $V_n=\{(M,\alpha,a)\mid\mbox{$a\in Z_M$,
$\alpha(n)$ is defined, and
$\alpha(n)=a$}\}$.
It is easily checked that $\pi_Z$ then becomes a local homeomorphism (an étale map). The assignment $Z\mapsto Z_{\Phi}$ extends in the obvious way to a left exact, continuous functor ${\cal
B}_{\ensuremath{\mathbb{T}}}\to\operatorname{Sh}(X_{{\ensuremath{\mathbb{T}}}})$ that preserves exponentials, inducing the covering map $ \phi \colon\operatorname{Sh}(X_{{\ensuremath{\mathbb{T}}}})\to \operatorname{Sh}({\cal B}_{\ensuremath{\mathbb{T}}})$. Finally, the stalk $x^*\Phi$ of the c-model $\Phi$ at a point $x=(M,\alpha)$ of $X_{\ensuremath{\mathbb{T}}}$ is just the Henkin model $M$ itself, which gives the relationship between our results and [@Hen50].
[10]{}
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[^1]: Philosophy Department, Carnegie Mellon University, Pittsburgh PA 15213-3890, USA.
[^2]: BRICS, Basic Research in Computer Science, Centre of the Danish National Research Foundation, Computer Science Department, Aarhus University, Ny Munkegade, Bldg. 540, 8000 Aahrus C., Denmark.
|
[SLAC-PUB-16171\
December 2014\
]{}
[**Using pipe with corrugated walls for a sub-terahertz FEL**]{}
Gennady Stupakov\
SLAC National Accelerator Laboratory,\
2575 Sand Hill Road, Menlo Park, CA 64025
[*Submitted for publication to Physical Review Special Topics - Accelerators and Beams*]{}
Introduction
============
For applications in fields as diverse as chemical and biological imaging, material science, telecommunication, semiconductor and superconductor research, there is great interest in having a source of intense pulses of terahertz radiation. Laser-based sources of such radiation [@Auston:84; @You:93] are capable of generating several-cycle pulses with frequency over the range 10–70 THz and energy of 20 $\mu$J [@Sell:08]. In a beam-based sources, utilizing short, relativistic electron bunches [@Nakazato:89; @Carr:02] an electron bunch impinges on a thin metallic foil and generates coherent transition radiation (CTR). An implementation of this method at the Linac Coherent Light Source (LCLS) has obtained single-cycle pulses of radiation that is broad-band, centered on 10 THz, and contains $>0.1$ mJ of energy [@Daranciang:11]. Another beam-based method generates THz radiation by passing a bunch through a metallic pipe with a dielectric layer. As reported in [@Cook:09], this method was used to generate narrow-band pulses with frequency 0.4 THz and energy 10 $\mu$J.
It has been noted in the past, in the study of wall-roughness impedance [@bane99n; @bane00st], that a metallic pipe with corrugated walls supports propagation of a high-frequency mode that is in resonance with a relativistic beam. This mode can be excited by a beam whose length is a fraction of the wavelength. Similar to the dielectric-layer method, metallic pipe with corrugated walls can serve as a source of terahertz radiation [@terahertz12].
In this paper we study another option of excitation of the resonant mode in a metallic pipe with corrugated walls—via the mechanism of the free electron laser instability. This mechanism works if the bunch length is much longer than the wavelength of the radiation. While our focus will be on a metallic pipe with corrugated walls, most our results are also applicable to a dielectric-layer round geometries. The connection between the electrodynamic properties of the two types of structures can be found in Ref. [@stupakov12b].
Our analysis is carried out for relativistic electron beams with the Lorentz factor $\gamma \gg 1$. However, in some places we will keep small terms on the order of $1/\gamma^2$ to make our results valid for relatively moderate values of $\gamma \sim 5-10$. In particular, we will take into account that the particles’ velocity $v$ differs from the speed of light $c$ (in contrast to the approximation $v=c$ typically made in [@bane99n; @bane00st; @terahertz12]) . We will see that the FEL mechanism becomes much less efficient in the limit $\gamma \to \infty$, so the moderate values of $\gamma$ are of particular interest.
This paper is organized as follows. In Section \[sec:2\] we discuss the resonant frequency, the group velocity and the loss factor of the resonant mode whose phase velocity is equal to the velocity of the particle. Their derivations are given in Appendices \[app:1\] and \[app:2\]. In section \[sec:3\] we find the gain length and an estimate for the saturated power of an FEL in which a relativistic beam excites the resonant mode. In section \[sec:4\] we consider a practical numerical example of such an FEL. In section \[sec:5\] we discuss some of the effects that are not included in our analysis.
Wake in a round pipe with corrugated walls {#sec:2}
==========================================
We consider a round metallic pipe with inner radius $a$.
![Dimensions of a round corrugated pipe. An electron beam propagates along the axis of the pipe. The beam position $s$ in the pipe is measured along the axis with $s=0$ corresponding to the entrance to the pipe.[]{data-label="fig:1"}](corrug_pipe_1.pdf "fig:"){height="40.00000%"} ![Dimensions of a round corrugated pipe. An electron beam propagates along the axis of the pipe. The beam position $s$ in the pipe is measured along the axis with $s=0$ corresponding to the entrance to the pipe.[]{data-label="fig:1"}](corrug_pipe_2.pdf "fig:"){height="40.00000%"}
Small rectangular corrugations have depth $h$, period $p$ and gap $g$, as shown in Fig. \[fig:1\]. In the case when $h,p\ll a$ and $h\gtrsim p$, the fundamental resonant mode with the phase velocity equal to the speed of light, $v_{ph}=c$, has the frequency $\omega_0 =ck_0$ and the group velocity $v_{g0}$, where [@bane99n; @bane00st] $$\begin{aligned}
\label{eq:1}
k_0
=
\left(
\frac{2p}{agh}
\right)^{1/2}
,\qquad
1
-
\frac{v_{g0}}{c}
=
\frac{4gh}{ap}
.
\end{aligned}$$ Such a mode will be excited by an ultra-relativistic particle moving along the axes of the pipe with velocity $v=c$. Note that from the assumption $h,p\ll a$ follows the high-frequency nature of the resonant mode, $k_0\gg 1/a$.
As explained in the Introduction, in our analysis we would like to take into account the fact that the phase velocity of the resonant mode is smaller than the speed of light, $v_{ph} = v<c$. Calculation of the frequency and the group velocity of the resonant mode for this case is carried out in Appendix \[app:1\]. As follows from this calculation, the deviation of the resonant frequency and the group velocity from Eqs. is controlled by the parameter $$\begin{aligned}
\label{eq:2}
u = \frac{ak_0}{\gamma}
\end{aligned}$$ with $k_0$ defined by . The plot of the frequency $\omega_r$ of the resonant mode versus parameter $u$ is shown in Fig. \[fig:2\].
![Plot of the normalized frequency $\omega_r$ of the resonant wave as a function of the parameter $a\omega_0/c\gamma$.[]{data-label="fig:2"}](freq_vs_gamma.pdf){width="60.00000%"}
We see that decreasing the beam energy $\gamma$ increases the frequency $\omega_r$ of the mode. Note that because $k_0a\gg 1$ the deviation from the ultra-relativistic results can become important even for large values of gamma, $\gamma\sim k_0a$. The group velocity of the resonant mode for $u\sim 1$ also deviates from the limit $\gamma\to\infty$ given by . Calculations of the group velocity are given in Appendix \[app:1\] and the plot of $\Delta\beta_g = 1-v_g/c$ versus $u$ is shown in Fig. \[fig:3\].
![Plot of the ratio $\Delta\beta_g/\Delta\beta_{g0}$ (with $\Delta\beta_{g0} = 1-v_{g0}/c$ defined in ) versus the parameter $a\omega_0/c\gamma$. []{data-label="fig:3"}](group_velocity){width="60.00000%"}
A relativistic point charge entering the pipe at the longitudinal coordinate $s=0$ and moving along the pipe axis excites the resonant mode and generates a longitudinal wakefield. The standard description of this process in accelerator physics is based on the notion of the (longitudinal) wake $w(z)$ that depends on the distance between the source and the test charges measured in the direction of motion [@chao93]. In case of the resonant mode, this wake is localized behind the driving charge and is equal to $w(z) = 2\varkappa \cos(\omega_r z/c)$ where $\varkappa$ is the loss factor per unit length (see, e.g., [@stupakov12b; @stupakov_bane_dechirper12]). For our purposes, it is important to modify this wake taking into account that at any given distance $s$ from the entrance to the pipe, the wake extends behind the particle over a finite length; this makes the wake a function of two variables, $w(s,z)$. The distance at which the wake extends behind the charge can be obtained from a simple consideration: the wake propagates with the group velocity $v_g$ and when the charge travels distance $s$ with speed $v$ the wake emitted at $s=0$ lags behind the charge at the distance $\Delta z = s(1-v_g/v)$ (we assume $v_g<v$). Mathematically, this is expressed by the following equation: $$\begin{aligned}
\label{eq:3}
w(s,z)
=
\left\{
\begin{array}
{rl}
2\varkappa \cos(\omega_r z/c),& \mathrm{ for }\,\, -s(1-v_g/v) < z < 0\\
\varkappa,& \mathrm{ for }\,\, z = 0\\
0,& \mathrm{otherwize}
\end{array}
\right.
.
\end{aligned}$$ The sign of the wake is such that a positive wake corresponds to the energy loss, and a negative wake means the energy gain. Note that the wake is only non-zero for negative $z$, that is behind the source charge.
The loss factor $\varkappa_0$ in the limit $\gamma\to\infty$ is given by [@stupakov_bane_dechirper12] $$\begin{aligned}
\label{eq:4}
\varkappa_0
=
\frac{2}{a^2}
.
\end{aligned}$$ With account of finite, but large, value of $\gamma$ the loss factor is derived in Appendix \[app:2\]. It is plotted in Fig. \[fig:4\] again as a function of parameter $u$.
![Plot of the normalized loss factor $\varkappa/\varkappa_0$ factor versus parameter $u = ak_0/\gamma$.[]{data-label="fig:4"}](loss_factor.pdf){width="60.00000%"}
We see that the interaction of the mode with the beam decreases when $\gamma$ becomes small. This happens because the spot size of the relativistically compressed Coulomb field of the point charge field on the wall of the pipe has the size on the order of $a/\gamma$, and when $u\sim 1$, is comparable with the inverse wave number of the wake $c/\omega_0$. For $u\gtrsim 1$ the frequency content of the Coulomb field at wavenumbers $\sim\omega_0/c$ gets depleted, and the excitation of the resonant mode is suppressed.
1D FEL equations {#sec:3}
================
We now consider an electron beam of energy $\gamma mc^2$ with the transverse size much smaller than the pipe radius $a$ and with the uniform longitudinal current distribution propagating along a pipe with corrugated walls. Such a beam will be driving a resonant mode in the pipe, and if the pipe is long enough, it will become modulated and micro-bunched through the interaction with the mode. The mechanism of this interaction is exactly the same as in the free electron laser instability. In this section we describe an approach to calculate this instability, following the method developed in Ref. [@PAC03stupakov03kr]. The actual derivation is presented in Appendix \[app:3\].
The crucial step in the derivation is a modification of the standard Vlasov equation that describes evolution of the distribution function of the beam. This modification takes into account retardation effects associated with emission of the wake field. The distribution function of the beam $f(\eta,z,s)$ is a function of the relative energy deviation, $\eta = \Delta \gamma/\gamma_0$, with $\gamma_0$ corresponding to the averaged beam energy, longitudinal position inside the bunch $z$, and the distance $s$ from the entrance to the pipe. The evolution of $f$ is described by the Vlasov equation $$\begin{aligned}
\label{eq:5}
&
\frac{\partial f}{\partial s}
-
\alpha\eta
\frac{\partial f}{\partial z}
-
\frac{r_0}{\gamma}
\frac{\partial f}{\partial \eta}
\int_{-\infty}^\infty dz'
\int_{-\infty}^\infty d\eta'
w(s,z-z')
f\left(
\eta',z',s-v\frac{z'-z}{v-v_g}
\right)
=
0
,
\end{aligned}$$ where $\alpha = -\gamma^{-2}$ is the slip factor per unit length and $r_0 = e^2/mc^2$ is the classical electron radius. The distribution function $f$ is normalized so that $\int fd\eta$ gives the number of particles per unit length. The third argument of $f$ in the integrand of takes into account the retardation: the wake that is generated by a beam slice at coordinate $z'$ slips behind the slice with the velocity $v-v_g$ relative to the beam, and if it reaches the point $z$ when the beam arrives at location $s$, it should have been emitted at position $s-v(z'-z)/(v-v_g)$ [@PAC03stupakov03kr].
To establish a closer analogy with the standard FEL theory, it is convenient to introduce a new variable $k_w$ (an analog of the FEL undulator wave number) defined by the equation $$\begin{aligned}
\label{eq:6}
\frac{k_0}{k_w}
=
\frac{v}{v-v_g}
\approx
\frac{1}{\Delta \beta_g - \Delta \beta_{ph}}
,
\end{aligned}$$ where $\Delta \beta_g = 1 - v_g/c$ and $\Delta \beta_{ph} = 1 - v_{ph}/c$. In the ultra-relativistic limit $\gamma\to\infty$ using we find $$\begin{aligned}
\label{eq:7}
k_w
=
k_{w0}
\equiv
4
\left(
\frac{2gh}{a^3p}
\right)^{1/2}
.
\end{aligned}$$
Eq. is linearized assuming a small perturbation of the beam equilibrium $f_0(\eta)$, $f=f_0(\eta) + f_1(\eta, z, s)$, with $|f_1| \ll f_0$. In this analysis we assume a coasting beam with the equilibrium distribution function $f_0(\eta) = n_0 F(\eta)$, where $n_0$ is the number of particles per unit length of the beam. We seek the perturbation in the form $f_1\propto e^{ikz+q k_w s}$, where $k$ is the wavenumber and $q$ is the dimensionless propagation constant whose real part is responsible for the exponential growth (or decay, if ${{\rm Re\,}}q<0$) of the perturbation with $s$. The main result of the linear instability analysis is the dispersion relation that defines the propagation constant $q$ as a function of the frequency detuning $\nu = (ck - \omega_r)/\omega_r$. This dispersion relation is derived in Appendix \[app:3\] (it follows closely the derivation of Ref. [@PAC03stupakov03kr]), and is given by , $$\begin{aligned}
\label{eq:8}
\frac{1}{2}
\frac{(2\rho)^3}{q - i\nu}
\int_{-\infty}^\infty
d\eta
\frac{F'(\eta)}{q - i\alpha\eta({\omega_r}/{ck_w})}
=
1
\,,
\end{aligned}$$ where the parameter $\rho$ (an analog of the Pierce parameter [@bonifacio84pn]) is $$\begin{aligned}
\label{eq:9}
(2\rho)^3
=
\frac{2n_0\kappa c r_0}{k_w\gamma\omega_r}
.
\end{aligned}$$ Except for a slight notational difference, Eqs. and coincide with the standard equations of the 1D FEL theory [@huang07k].
For a cold beam, $F(\eta) = \delta(\eta)$ (here $\delta$ stands for the delta-function), and from we obtain $$\begin{aligned}
\label{eq:10}
q^2(q - i\nu)
=
-
\frac{i
\alpha\omega_r}{2ck_w}
(2\rho)^3
.
\end{aligned}$$ If follows from this equation that the fastest growth of the instability is achieved at zero detuning. Assuming $\nu=0$ we rewrite using the definition and $\alpha = -1/\gamma^2$, $$\begin{aligned}
\label{eq:11}
q^3
=
i
\frac{ n_0\kappa r_0}{k_w^2\gamma^3}
.
\end{aligned}$$ Among the three roots of this equation, there is one, which we denote $q_1$, with a positive real part. Introducing the power gain length $\ell = (2{{\rm Re\,}}q_1 k_w)^{-1}$, and using $n_0r_0 = I/I_A$, where $I$ is the beam current and $I_A = 17.5$ kA is the Alfven current, we obtain $$\begin{aligned}
\label{eq:12}
\ell
=
\frac{1}{\sqrt{3}}
\gamma
\left(
{\kappa k_w}
\frac{I}{I_A}
\right)^{-1/3}
.
\end{aligned}$$
In addition to the gain length, an important characteristic of the described FEL is the radiation power at saturation. Here we can use the result of the standard FEL theory, that the saturation occurs at the distance equal about 10-20 gain length, and the saturation power $P_{\mathrm{sat}}$ is $$\begin{aligned}
\label{eq:13}
P_{\mathrm{sat}}
\approx
\rho\gamma mc^2
\frac{I}{e}
.
\end{aligned}$$ In the next section we will consider a practical example of an FEL based on a pipe with corrugated walls and evaluate $\ell$ and $P_{\mathrm{sat}}$ for that example.
Numerical example {#sec:4}
=================
To give an illustrative example of a practical device we consider in this section a pipe with corrugated walls with the parameters close to those accepted in Ref. [@terahertz12]. Noting from Eq. that the gain length is proportional to the beam energy, and having in mind a compact device, we choose a relatively small beam energy of 5 MeV. The beam current is 100 A. The pipe and corrugation dimensions with the beam parameters are summarized in Table \[tab:1\].
-------------------- ----
Pipe radius, mm 2
Depth $h$, $\mu$m 50
Period $p$, $\mu$m 40
Gap $g$, $\mu$m 10
Bunch charge, nC 1
Energy, MeV 5
Bunch length, ps 10
-------------------- ----
: Corrugation and beam parameters
\[tab:1\]
Note that parameter $u$ defined by is $u=1.3$, and hence the deviation from the ultra-relativistic limit (corresponding to $u\ll 1$) is expected to be noticeable.
From Eq. we find that the frequency $\omega_r/2\pi$ of the resonant mode is $0.34$ THz. Using the results of the Appendices \[app:1\] and \[app:2\] we find the group velocity of the resonant mode, $\Delta \beta_g = 0.053$, and the loss factor $\kappa = 0.6(2/a^2)= 2.7$ kV/(pC m), and calculate the Pierce parameter $\rho=0.013$. This gives the gain length $\ell \approx 7$ cm, and the saturation power $P_{\mathrm{sat}} \approx 6.7$ MW.
It is interesting to point out that for a given pipe radius and corrugations, there is an optimal value of the beam energy that minimized the gain length. This follows from Eq. which shows that $\ell$ increases with $\gamma$ due to an explicit dependence $\ell\propto \gamma$, but $\ell$ also increases when $\gamma$ becomes too small due to the decrease of $\kappa$ shown in Fig. \[fig:4\]. As numerical minimization shows, the minimal value or $\ell$ is achieved for $u=1.9$ and is given by $$\begin{aligned}
\label{eq:15}
\ell
=
0.74
\frac{a^2k_0}{2\sqrt{3}}
\left(
\frac{I_A}{I}
\right)^{1/3}
\left(
\frac{ap}{2hg}
\right)^{1/6}
.
\end{aligned}$$ For the parameter considered above this gives the optimal value of the beam energy: $\gamma = 6.6$ with the corresponding gain length $\ell = 5.5$ cm.
Discussion {#sec:5}
==========
There are several issues of practical importance that were omitted in our analysis in preceding sections. Here will briefly discuss some of them leaving a more detailed study for a separate publication.
First, we used an approximation of a coasting beam, without taking into account the finite length of the bunch. This approximation assumes that the bunch length is much longer than the cooperation length of the instability $l_\mathrm{c}$ that is defined as the distance at which the point charge wake extends within the bunch when the particle travels one gain length $\ell$. Using Eq. we evaluate the coherence length as $l_\mathrm{c}\sim \ell(1-v_g/v)$. For the parameters considered in Section \[sec:4\] we find $l_\mathrm{c}\approx 3.3$ mm, or 11 ps. This is comparable with the bunch length of 10 ps, and hence the numerical estimates of the previous section should only be considered as crude estimates of the expected parameters of the FEL. A more accurate prediction for the selected set of parameters require computer simulations.
Second, we neglected the resistive wall losses that would cause the resonant mode to decay when it propagates in the pipe. The effect of the wall losses on the FEL instability can be estimated if we compare the gain length with the decay distance $l_\mathrm{d}$ of the resonant mode. An analytical formula for $l_\mathrm{d}$ is given in Ref. [@terahertz12]; using the formula we estimate that for our parameters $l_\mathrm{d}=66$ cm, which is much larger than the gain length calculated in the previous section. Hence, we conclude that the resistive wall effect is small.
Finally, we mention a deleterious effect of the transverse wake, that might cause the beam break-up instability. It is known that in a round pipe with corrugated walls, in addition to the resonant longitudinal wake, there is also a resonant dipole mode that creates a transverse wakefield. In the limit $\gamma\to\infty$, in a round pipe, the transverse mode has the same frequency as the longitudinal one. To mitigate the effect of the breakup instability, one has to apply a strong external transverse focusing on the beam and minimize the initial beam offset at the entrance to the pipe. It may also be advantageous to change the cross sections of the pipe from round to rectangular or elliptic, that will likely detune the transverse mode frequency from the longitudinal one. A more detailed study of the transverse instability is necessary.
Acknowledgments
===============
The author thanks M. Zolotorev and K. Bane and I. Kotelnikov for useful discussions.
This work was supported by Department of Energy contract DE-AC03-76SF00515.
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Resonant mode for moderate values of $\gamma$ {#app:1}
=============================================
In this Appendix we analyze properties of the resonant mode in a round pipe with corrugated walls assuming $\gamma\gg 1$ but keeping small terms on the order of $\gamma^{-2}$. The resonant mode in this case is defined as a mode that has the phase velocity $v_{ph} = c\sqrt{1-\gamma^{-2}}$. Our analysis is performed for the steady state wakefield; the modification due to the finite interaction length is done straightforwardly using Eq. .
It is shown in Ref. [@stupakov12b] that small wall corrugations can be treated as a thin material layer with some effective values of the dielectric permeability $\epsilon$ and magnetic permittivity $\mu$. Calculations of $\epsilon$ and $\mu$ for given values of the corrugation parameters are carried out in [@stupakov12b] where it is shown that $\mu = g/p$ and the effective dielectric permeability $\epsilon$ is typically small and can be neglected in comparison with $\mu$. The electrodynamical properties of the layer are expressed through the surface impedance $\zeta$ that relates the longitudinal component of the electric field with the azimuthal magnetic field on the wall, $$\begin{aligned}
\label{eq:16}
E_z|_{r=a}
=
-
\zeta
H_\theta|_{r=a}
,
\end{aligned}$$ where [@stupakov12b] $$\begin{aligned}
\label{eq:17}
\zeta(\omega,k_z)
=
ih
\frac{\omega}{c}
\left(
\frac{k_z^2c^2}{\omega^2}\epsilon^{-1}
-
\mu
\right)
.
\end{aligned}$$
To find the resonant mode we write an axisymmetric TM-like solution of Maxwell’s equations in the pipe with the time and $z$ dependences $\propto e^{-i\omega t +ik_z z}$ in the following form $$\begin{aligned}
\label{eq:18}
E_z
=
E_0
I_0(k_r r)
,\qquad
H_\theta
=
-
E_0
\frac{i\omega}{ck_r}
I_1(k_r r)
,
\end{aligned}$$ where $E_0$ is the field amplitude and $$\begin{aligned}
\label{eq:19}
k_r
=
\sqrt{k_z^2-\frac{\omega^2}{c^2}}
=
\frac{\omega}{c}
(\beta_{ph}^{-2}-1)^{1/2}
.
\end{aligned}$$ Here $\beta_{ph} = v_{ph}/c$ with $v_{ph} = \omega/k_z$ the phase velocity of the wave, $I_0$ and $I_1$ are the modified Bessel functions of the first kind, and we assume $k_z>\omega/c$ so that $\beta_{ph} < c$, and $k_r$ is real.
We now substitute into the boundary condition , to obtain $$\begin{aligned}
\label{eq:20}
k_r a
\frac{
I_0(k_r a)
}
{
I_1(k_r a)
}
=
\zeta
\frac{ia\omega }{c}
.
\end{aligned}$$ Taking into account that the phase velocity is close to the speed of light, $1-\beta_{ph} \ll 1$, we will use for $\zeta$ a simplified equation in which $k_z^2c^2/\omega^2$ us replaced by unity, $$\begin{aligned}
\label{eq:21}
\zeta\approx {ih\omega}{c^{-1}}(\epsilon^{-1}-\mu)
.
\end{aligned}$$ From we find $$\begin{aligned}
\label{eq:22}
x
\frac{
I_0(x)
}
{
I_1(x)
}
=
\frac{ha\omega^2 }{c^2}
(\mu-\epsilon^{-1})
,
\end{aligned}$$ with $$\begin{aligned}
\label{eq:23}
x
=
k_r a
=
a\sqrt{k_z^2-\frac{\omega^2}{c^2}}
.
\end{aligned}$$
Consider first an ultra-relativistic limit $\beta_{ph} \to 1$. In this limit, $x\to 0$ and $ \lim_{x\to 0} x { I_0(x) }/ { I_1(x) } = 2 $. Substituting this into we recover the standard result for the synchronous mode $$\begin{aligned}
\label{eq:24}
\frac{\omega_r}{c}
=
\frac{\omega_0}{c}
\equiv
\left[
\frac{2}{ha(\mu-\epsilon^{-1})}
\right]^{1/2}
.
\end{aligned}$$ Eq. is obtained from this expression by substituting $\mu = g/p$ and neglecting $\epsilon$ (see details in [@stupakov12b]).
We now assume $x\sim 1$ and write it as $$\begin{aligned}
\label{eq:25}
x
=
a
\frac{\omega}{c}
(\beta_{ph}^{-2}-1)^{1/2}
\approx
\frac{a\omega_r}{c \gamma}
,
\end{aligned}$$ where we used the resonant mode condition $\beta_{ph} = \sqrt{1-\gamma^{-2}} \approx 1-\frac{1}{2}\gamma^{-2}$. Using the notations $u={a\omega_0}/{c \gamma}$ and $y_r=\omega_r/\omega_0$ we rewrite , $$\begin{aligned}
\label{eq:26}
u
\frac{I_0(uy_r)}{I_1(uy_r)}
=
2y_r
.
\end{aligned}$$ This equation was solved numerically and the dependence $y_r(u)$ is plotted in Fig. \[fig:2\].
When $x\sim 1$ the group velocity of the resonant wave also deviates from the value given by the second equation in . To find the group velocity we first differentiate with respect to $\omega$: $$\begin{aligned}
\label{eq:27}
\frac{dx}{d\omega}
\frac{d}{dx}
x
\frac{
I_0(x)
}
{
I_1(x)
}
=
\frac{2ha\omega }{c^2}
(\mu-\epsilon^{-1})
,
\end{aligned}$$ and then use to find $dx/d\omega$, $$\begin{aligned}
\label{eq:28}
\frac{dx}{d\omega}
=
\frac{a^2\omega}{xc^2}
\left(
\frac{1}{\beta_{ph}\beta_g}
-
1
\right)
,
\end{aligned}$$ where $\beta_g = v_g/c = c^{-1}d\omega/dk_z$. Combining and yields $$\begin{aligned}
\label{eq:29}
\left(
\frac{1}{\beta_{ph}\beta_g}
-
1
\right)
=
\frac{2h }{a}
(\mu-\epsilon^{-1})
\left(
\frac{1}{x}
\frac{d}{dx}
x
\frac{I_0(x)}
{I_1(x)}
\right)^{-1}
.
\end{aligned}$$ We now use $\beta_{ph}\approx 1-\frac{1}{2}\gamma^{-2}$ and $\beta_g = 1-\Delta\beta_g$ and recalling that $\gamma \gg 1$ and $\Delta\beta_g \ll 1$ obtain $$\begin{aligned}
\label{eq:30}
\Delta\beta_g
=
\frac{2h }{a}
(\mu-\epsilon^{-1})
\left(
\frac{1}{x}
\frac{d}{dx}
x
\frac{I_0(x)}
{I_1(x)}
\right)^{-1}
\bigg|_{x=uy_r}
-
\frac{1}{2\gamma^2}
.
\end{aligned}$$ In the limit $\gamma\to \infty$ we have $x\to 0$ and one can find from $$\begin{aligned}
\label{eq:31}
\lim_{\gamma\to\infty}\Delta\beta_g
=
\Delta\beta_{g0}
\equiv
\frac{4h}{a}
(\mu-\epsilon^{-1}).
\end{aligned}$$ Again, neglecting $\epsilon$ and substituting $\mu=g/p$ one recovers the group velocity in Eq. . In the general case, we normalize $\Delta\beta_g$ by $\Delta\beta_{g0}$, $$\begin{aligned}
\label{eq:32}
\frac{\Delta\beta_g}{\Delta\beta_{g0}}
=
\frac{1}{2}
\left(
\frac{1}{x}
\frac{d}{dx}
x
\frac{I_0(x)}
{I_1(x)}
\right)^{-1}
\bigg|_{x=uy_r}
-
\frac{1}{16}
u^2
,
\end{aligned}$$ and using express it as a function of the parameter $u$. The plot of the ratio ${\Delta\beta_g}/{\Delta\beta_{g0}}$ as a function of the parameter $a\omega_0/c\gamma$ is shown in Fig. \[fig:3\].
Calculation of the loss factor {#app:2}
==============================
In this Appendix we calculate the excitation of the resonant mode by a relativistic charge moving in a pipe with corrugated walls assuming $\gamma\gg 1$ but keeping small terms on the order of $\gamma^{-2}$, and using the boundary condition .
Electric and magnetic fields of a point charge moving along the $z$ axis can be described with the electric potential $\varphi$ and the $z$-component $A_z$ of the vector potential ${\textnormal{\boldmath$A$}}$. In the Lorentz gauge, ${\partial}{\varphi}/{{\partial}ct}+{\partial}{A_z}/{{\partial}z} = 0$, they satisfy the wave equations: $$\begin{aligned}
\label{eq:33}
\nabla^2\varphi - \frac{1}{c^2}{\frac{\partial ^2\varphi}{\partial t^2}}
&=
-4\pi q\, \delta(z-vt)\,\delta(\pi r^2)
,\nonumber \\
\nabla^2A_z - \frac{1}{c^2}{\frac{\partial ^2A_z}{\partial t^2}}
&=
-4\pi q\, (v/c)\,\delta(z-vt)\,\delta(\pi r^2)
,\end{aligned}$$ where $r$ is the distance from the axis. We make the Fourier transformation in $z$ and time $$\begin{aligned}
\label{eq:34}
\hat \varphi(r,k_z,\omega)
&=
\int_{-\infty}^\infty
dt\,
dz\,
e^{-ik_zz+i\omega t}
\varphi(r,z,t)
,
\nonumber\\
\hat A_z(r,k_z,\omega)
&=
\int_{-\infty}^\infty
dt\,
dz\,
e^{-ik_zz+i\omega t}
A_z(r,z,t)
.
\end{aligned}$$ This transforms equations into $$\begin{aligned}
\label{eq:35}
\frac{1}{r}\frac{d}{dr}r\frac{d}{dr}
\hat \varphi
+\left(\frac{\omega^2}{c^2}-k_z^2\right)
\hat \varphi
&=
-8\pi^2 q\,\delta(\omega-k_zv)\,\delta(\pi r^2)
,\nonumber\\
\frac{1}{r}\frac{d}{dr}r\frac{d}{dr}
\hat A_z
+\left(\frac{\omega^2}{c^2}-k_z^2\right)
\hat A_z
&=
-8\pi^2 q\,(v/c)\,\delta(\omega-k_zv)\,\delta(\pi r^2)
.
\end{aligned}$$ A partial solution of these equations corresponding to the field in free space is $\hat\varphi = 4\pi q\delta(\omega-k_zv)K_0(|k_z|r/\gamma)$ and $\hat A_z = (v/c)\hat\varphi$, with $K_0$ the modified Bessel function of the second kind. To this partial solution we now add a general solution of the homogeneous equations bounded at $r\to 0$: $$\label{eq:36}
\hat\varphi
=
4\pi q\delta(\omega-k_zv)
\left[K_0(|k_z|r/\gamma)
+
\alpha I_0(|k_z| r/\gamma)
\right]
,
\qquad
\hat A_z = (v/c)\hat\varphi
,$$ where $\alpha$ will be found from the boundary condition.
The electric and magnetic fields involved into the boundary condition are $$\begin{aligned}
\label{eq:37}
\hat E_z
&=
-ik_z \hat\varphi
+
\frac{i\omega}{c}
\hat A_z
=
-\frac{ik_z}{\gamma^2}
\hat\varphi
\nonumber\\
&=
-\frac{4\pi q ik_z}{\gamma^2}
\delta(\omega-k_zv)
\left[K_0(|k_z|r/\gamma)
+
\alpha I_0(|k_z| r/\gamma)
\right]
,
\end{aligned}$$ and $$\begin{aligned}
\label{eq:38}
\hat H_\theta
&=
-
\frac{{\partial}\hat A_z}{{\partial}r}
=
-
\frac{4\pi |k_z|qv}{c\gamma}
\delta(\omega-k_zv)
\left[K_0'(|k_z|r/\gamma)
+
\alpha I_0'(|k_z| r/\gamma)
\right]
.
\end{aligned}$$ Substituting these equations into and using the expressions for the derivatives $$\begin{aligned}
\label{eq:39}
K'_0(x)
=
-
K_1(x)
,\qquad
I'_0(x)
=
I_1(x)
,
\end{aligned}$$ we obtain $$\begin{aligned}
\label{eq:40}
\alpha
&=
\frac{
\zeta(\omega)
\beta
K_1(|k_z|a/\gamma)
\mathrm{sign}(k_z)
-
i\gamma^{-1}
K_0(|k_z|a/\gamma)
}
{
i\gamma^{-1}
I_0(|k_z|a/\gamma)
+
\zeta(\omega)
\beta
I_1(|k_z|a/\gamma)
\mathrm{sign}(k_z)
}
.
\end{aligned}$$ In what follows we again will use the approximation for $\zeta$.
We now substitute $\alpha$ into , select only the second term proportional to $\alpha$ (the first term is singular on the axis and describes the vacuum electric field of the moving charge), set $r=0$ and $z=vt$ and make the inverse Fourier transformation. This gives the longitudinal electric field acting on the particle, $E_{z0}=E_{z}(z=vt,r=0,t)$. Using the notations $u={a\omega_0}/{c\gamma}$ and $y=\omega/\omega_0$, and replacing $v\approx c$, we obtain $$\begin{aligned}
\label{eq:41}
E_{z0}
&=
-
\frac{iq}{\pi c^2\gamma^{2}}\,
\int_{-\infty}^\infty
\omega
\alpha
\, d\omega
=
-
\frac{iq}{\pi a^2}
u^2
\int_{-\infty}^\infty
y
\frac{
2|y|K_1(u|y|)
+
u
K_0(u|y|)
}
{
-
u
I_0(u|y|)
+
2|y| I_1(u|y|)
}
dy
.
\end{aligned}$$
The integrand in has poles on the real axis $y$ when its denominator vanishes. As one can see, the poles are located at $y=\pm y_r$ with $y_r$, determined by Eq. , that is by the condition that the phase velocity of the mode is equal to the velocity of the particle. These poles should be bypassed in the complex plane $y$ in accordance with rule that is established in the theory of the Cherenkov radiation [@landau_lifshitz_ecm]. The rule can be easily understood if one introduces small losses into the boundary condition by adding an infinitesimally small positive real part $\epsilon>0$ to $\zeta$, $\zeta \to \zeta + \epsilon$. With account of $\epsilon$ the poles are shifted into the lower half plane of the complex variable $y$ and the integration path takes the shape shown in Fig. \[fig:6\].
![Complex plane of variable $y$. Shown by the solid black line is the integration path in .[]{data-label="fig:6"}](complex_plane_y.pdf){width="60.00000%"}
The integral reduces to the sum of the half-residues from the poles (with the negative sign), and is given by the following expression: $$\begin{aligned}
\label{eq:42}
E_{z0}
&=
-
\frac{2q}{ a^2}
K
\left(
\frac{ak_0}{\gamma}
\right)
,
\end{aligned}$$ where the factor $K$ is $$\begin{aligned}
\label{eq:43}
K(u)
=
u^2
y
\frac{
2
y
K_1(uy)
+
u
K_0(uy)
}
{
d[
-
u
I_0(uy)
+
2y
I_1(uy)
]
/dy
}
\bigg|_{y=y_r}
.
\end{aligned}$$ The loss factor is related to $E_{z0}$ through the equation $\varkappa=-E_{z0}/q$. It is easy to see that in the limit $u\to 0$ the factor $K\to 1$ and we reproduce the result for the loss factor in the limit $\gamma\to\infty$. The function $K(u)$ is plotted in Fig. \[fig:4\].
Derivation of the dispersion relation for the FEL instability of resonant mode in corrugated pipe {#app:3}
=================================================================================================
Starting with the Vlasov equation it is convenient to introduce new variables: $\bar{s} = k_w s$, where $k_w$ is defined by and $\theta = \omega_r z/c$, and consider $f$ as a function of $\bar{s}$ and $\theta$. We linearize Eq. (\[eq:5\]) assuming $f=f_0(\eta) + f_1(\eta, \theta, \bar{s})$ with $|f_1| \ll f_0$. Using notation $f_0(\eta) = n_0 F(\eta)$, where $n_0$ is the number of particles per unit length of the beam, we find $$\begin{aligned}
\label{eq:44}
&
\frac{\partial f_1}{\partial \bar s}
-
\alpha\eta
\frac{\omega_r}{ck_w}
\frac{\partial f_1}{\partial \theta}
-
(2\rho)^3
F'(\eta)
\int_{\theta}^{\theta+\bar s} d\theta'
\int_{-\infty}^\infty d\eta'
\nonumber\\
&\times
\tilde w(\theta'-\theta)
f_1
(\eta',\theta',\bar s-\theta'+\theta)
=
0
,
\end{aligned}$$ where $\rho$ is the Pierce parameter [@bonifacio84pn] given by $$\begin{aligned}
\label{eq:45}
(2\rho)^3
=
\frac{2n_0\kappa c r_0}{k_w\gamma\omega_r}
,
\end{aligned}$$ and $\tilde w$ is the dimensionless wake expressed as a function of the dimensionless argument $\theta$, $$\begin{aligned}
\label{eq:46}
\tilde w(\theta)
=
\cos(\theta)
.
\end{aligned}$$
We then introduce a new variable ${\bar s}' = \bar{s}+\theta - \theta'$, and rewrite Eq. (\[eq:44\]) in the following form $$\begin{aligned}
\label{eq:47}
&
\frac{\partial f_1}{\partial \bar s}
-
\alpha\eta
\frac{\omega_r}{ck_w}
\frac{\partial f_1}{\partial \theta}
-
(2\rho)^3
F'(\eta)
\int_{0}^{\bar s} d{\bar s}'
\int_{-\infty}^\infty d\eta'
\nonumber\\
&\times
\tilde w(\bar{s} - \bar{s}')
f_1
(\eta',\theta+\bar{s}-{\bar s}',{\bar s}')
=
0
\,.
\end{aligned}$$ We assume a sinusoidal modulation of the distribution function with the wavenumber $k$, $f_1 \propto e^{i k z} = e^{i (1+ \nu) \theta}$, where $\nu = (ck - \omega_r)/\omega_r$ with $|\nu|\ll 1$. We then define functions $\Phi_\nu$ and $K_\nu$ such that $$\begin{aligned}
f_1(\eta, \theta, {\bar s})
&=&
e^{i(1+\nu)\theta}
\Phi_\nu(\eta,{\bar s})
\,,
\nonumber\\
K_\nu({\bar s})
&=&
e^{-i(1+\nu){\bar s}}\tilde w (\bar s)
\,.
\end{aligned}$$ Eq. (\[eq:47\]) takes the form $$\begin{aligned}
\label{eq:48}
\frac{\partial \Phi_\nu}{\partial \bar{s}}
&
-
\alpha\eta
\frac{\omega_r}{ck_w}
i (1+\nu)
\Phi_\nu
=
(2\rho)^3
F'(\eta)
\int_0^{\bar s}
d \bar{s}'
K_\nu(\bar{s}' - \bar{s})
\nonumber\\
&\times
\int_{-\infty}^\infty
d\eta'
\Phi_\nu(\eta',\bar{s}' )
=
0
\,.
\end{aligned}$$ Laplace transforming Eq. (\[eq:48\]) we find $$\begin{aligned}
\label{eq:49}
-&\Phi_\nu(\eta,0)
-
\alpha\eta
\frac{\omega_r}{ck_w}
i (1+\nu)
\tilde{\Phi}_\nu(\eta,q)
=
(2\rho)^3
F'(\eta)
\tilde{K}_\nu(q)
\int_{-\infty}^\infty
d\eta'
\tilde{\Phi}_\nu(\eta',q )
\,,
\end{aligned}$$ where $$\begin{aligned}
\label{eq:50}
\tilde{\Phi}_\nu(\eta,q)
&=
\int_0^\infty
d{\bar s}
e^{-q {\bar s}}
\Phi_\nu(\eta,\bar{s})
\,,
\nonumber\\
\tilde{K}_\nu(q)
&=
\int_0^\infty
d{\bar s}
e^{-q {\bar s}}
K_\nu(-\bar{s})
=
\frac{1}{2}
\left(
\frac{1}{q - i\nu}
+
\frac{1}{q - i\nu - 2i}
\right)
\,.
\end{aligned}$$ Dividing Eq. (\[eq:49\]) by $q - i\alpha\eta (\omega_r/ck_w)(1+\nu)$ and integrating over $\eta$ yields $$\begin{aligned}
\label{eq:51}
\int_{-\infty}^\infty d\eta
\tilde{\Phi}_\nu(\eta,q )
=
\frac{
\int_{-\infty}^\infty
d\eta
\frac{\Phi_\nu(\eta,0)}
{q - i\alpha\eta(\omega_r/ck_w)(1+\nu)}
}
{
1-(2\rho)^3
\tilde{K}_\nu(q)
\int_{-\infty}^\infty
d\eta
\frac{F'(\eta)}{q - i\alpha\eta(\omega_r/ck_w)(1+\nu)}
}
\,.
\end{aligned}$$ The dispersion relation that defines the propagating constant $q$ of the mode is given by zeros of the denominator on the right hand side of this equation: $$\begin{aligned}
\label{eq:52}
(2\rho)^3
\tilde{K}_\nu(q)
\int_{-\infty}^\infty d\eta
\frac{F'(\eta)}{q - i\alpha\eta (\omega_r/ck_w)(1+\nu)}
=
1\,.
\end{aligned}$$ Rapid growth will be seen to correspond to $|\nu| \lesssim 2\rho$ and $q \sim 2\rho$. The second term in expression for $\tilde{K}_\nu$ in (\[eq:50\]) is not resonant and can be neglected, which gives $$\begin{aligned}
\label{eq:53}
\frac{1}{2}
(2\rho)^3
\frac{1}{q - i\nu}
\int_{-\infty}^\infty
d\eta
\frac{F'(\eta)}{q - i\alpha\eta({\omega_r}/{ck_w})}
=
1
\,,
\end{aligned}$$ where we neglected $\nu$ relative to unity in the denominator of the integrand of Eq. (\[eq:52\]).
|
---
abstract: |
We establish a reflection principle for three lattice walkers and use this principle to reduce the enumeration of the configurations of three vicious walkers to that of configurations of two vicious walkers. In the combinatorial treatment of two vicious walkers, we make connections to two-chain watermelons and to the classical ballot problem. Precisely, the reflection principle leads to a bijection between three walks $(L_1,
L_2, L_3)$ such that $L_2$ intersects both $L_1$ and $L_3$ and three walks $(L_1, L_2, L_3)$ such that $L_1$ intersects $L_3$. Hence we find a combinatorial interpretation of the formula for the generating function for the number of configurations of three vicious walkers, originally derived by Bousquet-Mélou by using the kernel method, and independently by Gessel by using tableaux and symmetric functions.
---
[**A Reflection Principle for Three Vicious Walkers** ]{}
William Y. C. Chen$^{1}$, Donna Q. J. Dou$^{2}$, Terence Y. J. Zhang$^{3}$
Center for Combinatorics, LPMC-TJKLC\
Nankai University\
Tianjin 300071, P.R. China
0.2 cm $^1$chen@nankai.edu.cn, $^2$qjdou@cfc.nankai.edu.cn, $^3$terry@mail.nankai.edu.cn
[**Keywords:**]{} vicious walkers, watermelon, Catalan numbers, Ballot numbers, reflection principle.
[**AMS Classification Numbers:**]{} 82B23; 05A15
Introduction
============
The vicious walker model was introduced by Fisher [@Fish] in 1984 and has drawn much attention. A walker is said to be vicious if he does not like to meet any other walker at any point. Formally speaking, a configuration of $r$ vicious walkers, called $r$ vicious walks, of length $n$, is an $r$-tuple of pairwise nonintersecting lattice walks of length $n$, consisting of up steps $U$ (i.e., $(1,1)$) and down steps $D$ (i.e., $(1,-1)$), starting from $(0,2i_1), (0,2i_2), \ldots, (0,2i_r)$ and ending at $(n,e_1),(n,e_2),\ldots, (n,e_r)$ where $i_r>\cdots> i_2>i_1=0$ and $e_r>\cdots >e_2>e_1$. Precisely, two lattice paths are said to be nonintersecting if they do not share any common points. In particular, a watermelon of length $n$ is a configuration consisting of $r$ chains, or paths, of length $n$ which start at the points $(0,0), (0,2), \ldots, (0,2r-2)$ and end at the points $(n,k),(n,k+2),\ldots, (n,k+2r-2)$ for some $k$. In other words, a watermelon is a vicious walker configuration starting at adjacent points and ending at adjacent points. Note that two lattice points are said to be adjacent if they are on the same vertical line and their $y$-coordinates differ by $2$. It is known that configurations of vicious walkers can be represented by tableaux. So the theory of symmetric functions can be employed to study vicious walkers, see [@Ges-Vie; @Gutt1; @Gutt4; @Hika; @Gutt2; @Gutt3].
The main objective of this paper is to present a combinatorial approach to the enumeration of configurations of three vicious walkers. Let us fix the starting points $(0,0)$, $(0, 2i)$ and $(0,
2i+2j)$. Let $V(i,j,n)$ be the set of three vicious walks $(L_1,
L_2, L_3)$ of length $n$, where $L_1$ is the path of the first walker starting from $(0,0)$, $L_2$ is the path of the second walker starting from $(0, 2i)$, and $L_3$ is the path of the third walker starting from $(0, 2i+2j)$. Define the generating function $V_{i,j}(t)$ to be $$\label{dvijt}
V_{i,j}(t) =\sum_{n=0}^\infty \, |V(i,j,n)| t^n,$$ where $|\cdot|$ denotes the cardinality of a set.
The enumeration of configurations of three vicious walkers has been solved independently by Bousquet-Mélou [@Melou] by using the kernel method, and by Gessel [@Gess] by using tableaux and symmetric functions. They obtained a formula for $V_{i,j}(t)$ in terms of the generating function of the Catalan numbers.
Let $C(t)$ be the generating function of the Catalan numbers $C_n={1\over n+1} {2n\choose n}$, that is, $$C(t)=\sum_{n=0}^\infty C_n t^n.$$ Recall that $C(t)$ satisfies the recurrence relation $$\label{cT-eq}
C(t) = 1 +t C^2(t).$$ Let $$\label{dtc}
D(t)=tC^2(t)=C(t)-1=\sum_{n=0}
C_{n+1}t^{n+1}.$$
The following elegant formula is due to Bousquet-Mélou [@Melou] and Gessel [@Gess].
\[mainthm2\] $$\label{maineqn2}
V_{i,j}(t)={1\over 1-8t} (1-D^i(2t))(1-D^j(2t)).$$
In view of the relation (\[dtc\]) and the identity $$\label{d2t}
\Big(\frac{1+D(t)}{1-D(t)}\Big)^2= { 1\over 1-4t},$$ Gessel derived the following form of the formula for $V_{i,j}(t)$.
\[Gessel [@Gess]\] \[mainthm\] For any $i,j\geq 1$, we have $$\label{maineqn}V_{i,j}(t)
=C^2(2t)\big(1+D(2t)+\cdots+D^{i-1}(2t)\big)
\big(1+D(2t)+\cdots+D^{j-1}(2t)\big).$$
Both Bousquet-Mélou [@Melou] and Gessel [@Gess] proposed the problem of finding a combinatorial interpretation of the formula for $V_{i,j}(t)$. The question of Bousquet-Mélou is concerned with the formula (\[maineqn2\]), while the question of Gessel is concerned with the formula in the form of (\[maineqn\]). In this paper, we will present a combinatorial interpretation of (\[maineqn2\]). As will be seen, the algebraic manipulations to transform the formula (\[maineqn2\]) to (\[maineqn\]) can be explained combinatorially. So we have obtained combinatorial interpretations of both formulas (\[maineqn2\]) and (\[maineqn\]).
We also take a different approach to the enumeration of configurations of two vicious walkers. By reformulating the problem in terms of pairs of intersecting walks, we give a decomposition of a pair of converging walks, that is, two walks that do not intersect until they reach the same ending point, into two-chain watermelons, or $2$-watermelons. Then we can use Labelle’s formula for the number of $2$-watermelons of length $n$ to derive the formula for the number of two vicious walks of length $n$. In the last section, we make a connection between pairs of converging walks and the classical ballot numbers, by applying the Labelle merging algorithm, in the form presented by Chen, Pang, Qu and Stanley [@CPQS],
The Reflection Principle
========================
In this section, we will establish a reflection principle so that we can reduce the enumeration of three vicious walkers to that of two vicious walkers. This reduction leads to a combinatorial interpretation of the formula for $V_{i,j}(t)$, as defined by (\[dvijt\]).
Let us recall some basic definitions. Two walks $L_1$ and $L_2$ are said to be intersecting, denoted $L_1\cap L_2\neq \emptyset$, if $L_1$ and $L_2$ share a common point. Let $U(i,j,n)$ be the set of all $3$-walks $(L_1,L_2,L_3)$ of length $n$, where $L_1$, $L_2$ and $L_3$ start from $(0,0)$, $(0,2i)$ and $(0,2i+2j)$ respectively. Let $$U_{i,j}(t) =\sum_{n=0}^\infty \, |U(i,j,n)| t^n.$$ It is obvious that $$\label{uij}
U_{i,j}(t)={ 1 \over 1-8t}.$$ We use $W_{12}(n)$, or $W_{12}$ for short, to denote the set of $3$-walks $(L_1,L_2,L_3)$ in $U(i,j,n)$ such that $L_1$ and $L_2$ are nonintersecting. Similarly, we use $W_{23}(n)$, or $W_{23}$ for short, to denote the set of $3$-walks $(L_1, L_2, L_3)$ in $U(i,j,n)$ such that $L_2$ and $L_3$ are nonintersecting. Clearly, the set $V(i,j,n)$ of three vicious walks of length $n$ can be expressed as $W_{12}\cap W_{23}$. By the principle of inclusion and exclusion, we see that $$\label{v-ijn}
|V(i,j,n)|=|W_{12} \cap W_{23}|=|W_{12}|+|W_{23}|-|W_{12}\cup
W_{23}|.$$ In order to compute $|W_{12}\cup W_{23}|$, we let $M_{12, 23}(n)$, or $M_{12, 23}$ for short, denote the set of $3$-walks $(L_1,L_2,L_3)$ in $U(i,j,n)$ such that $L_2$ intersects both $L_1$ and $L_3$. Clearly, we have $$\label{w1223}
|W_{12} \cup W_{23}| = |U(i,j,n)| - |M_{12, 23}|.$$
We are now in a position to establish a reflection principle to deal with the enumeration of $M_{12, 23}(n)$. Let $M_{13}(n)$, or $M_{13}$ for short, denote the set of $3$-walks $(L_1,L_2,L_3)$ in $U(i,j,n)$ such that $L_1$ intersects $L_3$. Then we have the following correspondence.
\[bij\] For $n\geq 1$, there exists a bijection between $M_{12, 23}(n)$ and $M_{13}(n)$.
We construct a map $\Phi$ from $M_{12, 23}(n)$ to $M_{13}(n)$ as follows. Let $(L_1,L_2,L_3)$ be a $3$-walk in $M_{12, 23}(n)$. We consider the following two cases. If $L_1\cap L_3\neq \emptyset$, then it is clear that $(L_1,L_2,L_3)\in M_{13}(n)$. In this case, we define $\Phi((L_1,L_2,L_3))=(L_1,L_2,L_3)$.
We may now assume that $L_1\cap L_3=\emptyset$. We first consider the case that $L_2$ meets $L_1$ before it meets $L_3$. Suppose that $P$ is the first intersection point of $L_2$ and $L_1$. We now conduct the usual reflection operation on $L_1$ and $L_2$, and denote the resulting paths by $L_1'$ and $L_2'$. Namely, $L_1'$ consists of the first segment of $L_1$ up to the point $P$ followed by the last segment of $L_2$ starting from the point $P$, and $L_2'$ consists of the first segment of $L_2$ up to the point $P$ followed by the last segment of $L_1$ starting from the point $P$. Figure \[fig-bijection\] is an illustration of the reflection.
Let $L_3'=L_3$ and $\Phi((L_1,L_2,L_3))=(L_1',L_2',L_3')$. It is clear that $L_1^{\prime}$ must meet $L_3^{\prime}$. Thus we have $(L_1^{\prime},L_2^{\prime},L_3^{\prime})\in M_{13}(n)$.
(280,80)
(-12,0)[$L_1$]{}(-12,25)[$L_2$]{}(-12,50)[$L_3$]{} (0,0)(20,-10)(40,20)(40,20)(60,40)(80,15)(80,15)(90,5)(120,20) (0,25)(30,4)(60,40)(60,40)(90,60)(120,30) (0,50)(40,90)(90,40)(90,40)(105,30)(120,50) (15,-6)[1]{}(90,8)[1]{} (15,13)[2]{}(60,35)[2]{}(92,45)[2]{}(118,30)[2]{} (15,57)[3]{}(92,35)[3]{}(118,48)[3]{}
(136,28)[$\stackrel{\Phi}\longrightarrow$]{} (168,0)[$L_1^{\prime}$]{}(168,25)[$L_2^{\prime}$]{}(168,50)[$L_3^{\prime}$]{} (180,0)(200,-10)(220,20)(220,20)(240,40)(260,15)(260,15)(270,5)(300,20) (180,25)(210,4)(240,40)(240,40)(270,60)(300,30) (180,50)(220,90)(270,40)(270,40)(295,30)(300,50) (225,25)(45,25) (220,26)[$P$]{}(41,26)[$P$]{} (195,-6)[1]{}(270,8)[2]{} (195,13)[2]{}(240,35)[1]{}(272,45)[1]{}(298,30)[1]{} (195,57)[3]{}(272,35)[3]{}(298,48)[3]{}
It is not difficult to see that the above procedure is reversible. We are still left with the case when $L_2$ intersects $L_3$ before meeting $L_1$. This case is analogous to the case that we have considered. Thus we have reached the conclusion that $\Phi$ is a bijection.
------------------------------------------------------------------------
Combining , and Theorem \[bij\], we obtain the following relation $$\label{v-ijn2}
|V(i,j,n)|=|W_{12}|+|W_{23}|+|M_{13}| -|U(i,j,n)|.$$
Let $W_{13}$ be the set of three walks $(L_1, L_2, L_3)$ in $U(i,j,n)$ such that $L_1$ never meets $L_3$, and define the generating functions for $|W_{12}|$, $|W_{23}|$ and $|W_{13}|$ by $W_{12}(t)$, $W_{23}(t)$ and $W_{13}(t)$ respectively. From (\[v-ijn2\]) it follows that $$\label{reduction 1}
|V(i,j,n)| = |W_{12}| + |W_{23}| -|W_{13}|.$$
\[redution-1\] $$\label{reduction-1}
V_{i,j}(t) = W_{12}(t) + W_{23}(t) -W_{13}(t).$$
The above formula can be viewed as a reduction of the three vicious walkers problem to that of two vicious walkers. Let $N(i,n)$ be the set of two vicious walks $(L_1,L_2)$ of length $n$ starting at $(0,0)$ and $(0,2i)$ respectively, and denote the corresponding generating function by $$N_i(t)=\sum_{n=0}^{\infty}|N(i,n)|t^n.$$
Bousquet-Mélou [@Melou] and Gessel [@Gess] obtained the following formula $$\label{tv2}
N_i(t)={1\over 1-4t}(1-D^i(t)).$$ As pointed out by Gessel [@Gess], the above formula for $N_{i}(2t)$ can be deduced from the formula (\[maineqn\]) for $V_{i,j}(t)$ by taking the limit $j \rightarrow¡¡\infty$, and by using the identity (\[d2t\]).
Using the above formula for $N_i(t)$, one can derive the following formulas for the generating functions $W_{12}(t)$, $W_{23}(t)$ and $W_{13}(t)$: $$\label{w123}
W_{12}(t) = {1-D^i(2t)\over 1-8t}, \; W_{23}(t)={1-D^j(2t)\over
1-8t}, \; W_{13}(t)={1-D^{i+j}(2t)\over 1-8t}.$$ Clearly, formula in Theorem \[mainthm2\] follows from the above formulas and the relation .
We note that Gessel [@Gess] obtained the following identity $$\label{vn}
V_{i,j}(t)=N_i(2t)+N_j(2t)-N_{i+j}(2t),$$ in accordance with the combinatorial statement derived from the reflection principle.
As to the question of finding a combinatorial interpretation of the generating function formula , the reflection principle (Theorem \[bij\]) along with the combinatorial interpretations of the formulas for $W_{12}(t)$, $W_{23}(t)$ and $W_{13}(t)$ can be considered as an answer because the principle of inclusion and exclusion for two sets can be easily justified combinatorially. In the next section, we will present a combinatorial treatment of the formula for two vicious walkers. Moreover, we note that one can give a combinatorial reasoning of the transformation from the formula to the formula .
It is to deduce from by utilizing the identity (\[d2t\]), which can be explained combinatorially in two steps. The first step is to show that $$\label{4n}
4^n= \sum_{k=0}^{2n} {2k \choose k}{2n-2k \choose n-k},$$ which is equivalent to the identity $$\sum_{n=0}^\infty {2n \choose n}t^n = {1 \over \sqrt{1-4t}}.$$ There are several combinatorial proofs of (\[4n\]), see, for example, Kleitman [@Klei] and Marta [@SvMa]. The second step is to show that $$\label{dt} {1+D(t)\over 1-D(t)}
=\sum_{n=0}^\infty {2n \choose n}t^n.$$ Note that ${1+D(t) \over 1-D(t)}$ can be written as ${C(t) \over 1-tC^2(t)}$. A combinatorial interpretation of the identity $${C(t)\over 1-tC^2(t)}=\sum_{n=0}^\infty {2n \choose n}t^n$$ is given by Chen, Li and Shapiro [@CLSh] in terms of doubly rooted plane trees and the butterfly decomposition.
Converging Walks and $2$-Watermelons
====================================
In this section, we present a different approach to the two vicious walkers problem by counting pairs of converging walks. A pair of walks is said to be converging if they never meet until they reach a common ending point. We will show that pairs of converging walks can be enumerated by applying Labelle’s formula for two-chain watermelons, or $2$-watermelons [@Labe]. Precisely, we will give a decomposition of a pair of converging walks into $2$-watermelons.
Recall that $M_{13}(n)$ is defined in the previous section. Let $M_{12}(n)$, or $M_{12}$ for short, be the set of 3-walks $(L_1,
L_2, L_3)$ in $U(i,j,n)$ such that $L_1$ intersects $L_2$. Similarly, we can define $M_{23}(n)$, or $M_{23}$ for short. Clearly, we have $$|M_{12}|= |U(i,j,n)|-|W_{12}|, \quad
|M_{23}|=|U(i,j,n)|-|W_{23}|.$$ From (\[v-ijn2\]) it follows that $$|V(i,j,n)|=|U(i,j,n)|
+|M_{13}|-|M_{12}|-|M_{23}|.$$
Let $M_{12}(t)$, $M_{23}(t)$ and $M_{13}(t)$ denote the generating functions for $|M_{12}(n)|$, $|M_{23}(n)|$ and $|M_{13}(n)|$, respectively.
We have \[intersect to vicious\] $$\label{vm}
V_{i,j}(t) = U_{i,j}(t)+M_{13}(t)- M_{12}(t) -M_{23}(t).$$
We will show that $M_{12}(t)$, $M_{13}(t)$ and $M_{23}(t)$ can be computed by using Labelle’s formula for $2$-watermelons.
\[watermelon\] The number of $2$-watermelons with each walk having $n$ steps is $C_{n+1}$.
By Labelle’s formula, one sees that the generating function of the number of $2$-watermelons equals $C^2(t)$. Note that $2$-watermelons of length $n$ correspond to pairs of converging walks of length $n+1$ with adjacent starting points. In general, let $T(i,n)$ be the set of pairs of converging walks $(L_1,L_2)$ of length $n$, where $L_1$ starts from $(0,0)$ and $L_2$ starts from $(0, 2i)$. Define $$T_i(t)=\sum_{n\geq
0}|T(i,n)|t^n.$$
\[b-prop\] For any $i\geq 1$, $T_i(t)=D^i(t)$.
Let $L_1=A_0A_1\ldots A_n$ and $L_2=B_0 B_1\ldots B_n$, where a walk is represented by a sequence of points. For $0\leq k\leq i$, let $j_k$ be the minimum index such that the difference of the $y$-coordinates of $(A_{j_k},B_{j_k})$ equals to $2i-2k$. It is clear that $j_0=0$ and $j_i=n$. We now decompose $(L_1,L_2)$ into $i$ 2-walks: $(L_1^{(1)},L_2^{(1)}),\ldots,(L_1^{(i)},L_2^{(i)})$, where $L_1^{(k)}=A_{j_{k-1}}A_{j_{k-1}+1} \ldots A_{j_k}$ and $L_2^{(k)}=B_{j_{k-1}} B_{j_{k-1}+1} \ldots B_{j_{k}}$. Figure \[decomposition\] is an illustration of the decomposition.
(90,130) (-35,15)[(1,0)[139]{}]{} (-20,0)[(0,1)[139]{}]{}
(-20,0)(15,0)[9]{} (-20,15)(15,0)[9]{} (-20,30)(15,0)[9]{} (-20,60)(15,0)[9]{} (-20,45)(15,0)[9]{} (-20,60)(15,0)[9]{} (-20,75)(15,0)[9]{} (-20,90)(15,0)[9]{} (-20,105)(15,0)[9]{} (-20,120)(15,0)[9]{} (-20,135)(15,0)[9]{} (-20,15)[(1,-1)[15]{}]{} (-5,0)[(1,1)[45]{}]{} (40,45)[(1,-1)[15]{}]{}(55,30)[(1,1)[15]{}]{} (70,45)[(1,-1)[15]{}]{}(85,30)[(1,1)[15]{}]{} (-20,105)[(1,1)[15]{}]{}(-5,120)[(1,-1)[60]{}]{} (55,60)[(1,1)[15]{}]{}(70,75)[(1,-1)[30]{}]{}
(-40,10)[$L_1$]{}(-39,106)[$L_2$]{} (-29,102)[$\small{B_{j_0}}$]{}(-28,7)[$\small{A_{j_0}}$]{} (23,93)[$\small{B_{j_1}}$]{}(23,22)[$\small{A_{j_1}}$]{} (38,77)[$\small{B_{j_2}}$]{}(35,35)[$\small{A_{j_2}}$]{} (100,40)[$\small{A_{j_3}(B_{j_3}})$]{}
(25,30)(25,60)(25,90)(40,45)(40,60)(40,75)
Observe that by the choice of $j_k$, the rightmost pair of steps in $(L_1^{(k)},L_2^{(k)})$ must be $(U,D)$. Moreover, if we delete this pair of steps, the resulting upper walk can be lowered $2i-2k$ units without intersecting the lower walk to form a $2$-watermelon. See Figure \[lower\] for an example.
(240,110) (-48,12)[(1,0)[48]{}]{} (-24,0)[(0,1)[96]{}]{}
(-24,0)(12,0)[4]{} (-24,12)(12,0)[4]{} (-24,24)(12,0)[4]{} (-24,36)(12,0)[4]{} (-24,48)(12,0)[4]{} (-24,60)(12,0)[4]{} (-24,72)(12,0)[4]{} (-24,84)(12,0)[4]{} (-24,96)(12,0)[4]{}
(-24,12)[(1,-1)[12]{}]{} (-12,0)[(1,1)[12]{}]{} (0,12)(6,18)(12,24) (-24,84)[(1,1)[12]{}]{}(-12,96)[(1,-1)[12]{}]{} (0,84)(6,78)(12,72) (-42,8)[$L_1^{(1)}$]{} (-42,78)[$L_2^{(1)}$]{}
(19,45)[$\rightarrow$]{}
(24,12)[(1,0)[36]{}]{} (36,0)[(0,1)[96]{}]{}
(36,0)(12,0)[3]{} (36,12)(12,0)[3]{} (36,24)(12,0)[3]{} (36,36)(12,0)[3]{} (36,48)(12,0)[3]{} (36,60)(12,0)[3]{} (36,72)(12,0)[3]{} (36,84)(12,0)[3]{} (36,96)(12,0)[3]{}
(36,12)[(1,-1)[12]{}]{} (48,0)[(1,1)[12]{}]{} (36,36)[(1,1)[12]{}]{}(48,48)[(1,-1)[12]{}]{}
(96,0)(12,0)[2]{} (96,12)(12,0)[2]{} (96,24)(12,0)[2]{} (96,36)(12,0)[2]{} (96,48)(12,0)[2]{} (96,60)(12,0)[2]{} (84,12)[(1,0)[24]{}]{}(96,0)[(0,1)[60]{}]{} (96,12)(102,18)(108,24) (96,60)(102,54)(108,48) (80,9)[$L_1^{(2)}$]{} (80,57)[$L_2^{(2)}$]{} (115,33)[$\rightarrow$]{} (132,33)[$\emptyset$]{}
(156,12)[(1,0)[60]{}]{}(168,0)[(0,1)[48]{}]{} (168,12)(12,0)[5]{} (168,24)(12,0)[5]{} (168,36)(12,0)[5]{} (168,48)(12,0)[5]{}
(168,24)[(1,-1)[12]{}]{}(180,12)[(1,1)[12]{}]{} (192,24)[(1,-1)[12]{}]{} (204,12)(210,18)(216,24)
(168,48)[(1,-1)[12]{}]{}(180,36)[(1,1)[12]{}]{} (192,48)[(1,-1)[12]{}]{} (204,36)(210,30)(216,24) (152,20)[$L_1^{(3)}$]{} (154,46)[$L_2^{(3)}$]{} (224,27)[$\rightarrow$]{}
(234,12)[(1,0)[48]{}]{}(246,0)[(0,1)[48]{}]{} (246,12)(12,0)[4]{} (246,24)(12,0)[4]{} (246,36)(12,0)[4]{} (246,48)(12,0)[4]{}
(246,24)[(1,-1)[12]{}]{} (258,12)[(1,1)[12]{}]{} (270,24)[(1,-1)[12]{}]{} (246,48)[(1,-1)[12]{}]{} (258,36)[(1,1)[12]{}]{} (270,48)[(1,-1)[12]{}]{}
By Proposition \[watermelon\], The generating function for the number of $2$-walks $(L_1^{(k)},L_2^{(k)})$ equals $D(t)=t\cdot
C^2(t)$. This completes the proof.
------------------------------------------------------------------------
Let $M(i,n)$ be the set of intersecting 2-walks $(L_1,L_2)$ of length $n$, where $L_1$ and $L_2$ start from $(0,0),(0,2i)$ respectively. Define $$M_i(t)=\sum_{n\geq 0}|M(i,n)|t^n.$$
Observe that every pair of intersecting paths $(L_1, L_2)$ can be decomposed into a pair of converging paths and a pair of arbitrary paths starting from the same point. Thus we have the following formula.
\[intersect\] For any $i\geq 1$, $$M_i(t) ={D^i(t)\over
{1-4t}}.$$
It is obvious that $$\label{twosum}M_i(t)+N_i(t)={1\over 1-4t}.$$ So the formula (\[tv2\]) for $N_i(t)$ can be deduced from the above formula. It is easy to see that $M_{12}(t)$, $M_{23}(t)$ and $M_{13}(t)$ can be computed by using the above formula for $M_i(t)$. So we get $$\label{gpq}
M_{12}(t)={D^i(2t)\over {1-8t}}, \quad M_{23}(t)={D^j(2t)\over
{1-8t}},\quad M_{13}(t)={D^{i+j}(2t)\over {1-8t}},$$ in agreement with (\[w123\]). Substituting (\[gpq\]) into (\[vm\]), we obtain Theorem \[mainthm2\].
Connection to the Ballot Numbers
================================
In this section, we put the Labelle merging algorithm in a more general setting, and show that the direct correspondence formulated by Chen, Pang, Qu and Stanley [@CPQS] leads to a connection between pairs of converging walks and the classical ballot numbers.
Let us recall the direct correspondence given in [@CPQS]. We will represent a walk as a sequence of steps rather than points. Let $(L_1, L_2)$ be a 2-watermelon of length $n$, and let $L_1=p_1p_2\cdots p_n$ and $L_2= q_1q_2\cdots q_n$, where $p_i,
q_i=U$ or $D$. Set $U'=D$ and $D'=U$. Using the direct correspondence in [@CPQS], the watermelon $(L_1, L_2)$ can be represented by a Dyck path of length $2n+2$: $$U q_1p_1' q_2 p_2'\cdots q_n p_n' D.$$ It is not difficult to see that the above correspondence is a bijection. Figure \[water2D\] gives an illustration.
(330,100) (0,0)(15,0)[8]{} (0,15)(15,0)[8]{} (0,30)(15,0)[8]{} (0,45)(15,0)[8]{} (0,60)(15,0)[8]{} (0,75)(15,0)[8]{} (0,90)(15,0)[8]{} (0,0)[(1,1)[30]{}]{}(30,30)[(1,-1)[15]{}]{} (45,15)[(1,1)[15]{}]{} (60,30)[(1,-1)[30]{}]{} (90,0)[(1,1)[15]{}]{}
(-13,-4)[$L_1$]{}(-13,26)[$L_2$]{}
(0,30)[(1,1)[30]{}]{}(30,60)[(1,-1)[15]{}]{} (45,45)[(1,1)[30]{}]{} (75,75)[(1,-1)[30]{}]{}
(134,40)[$\longleftrightarrow$]{} (180,60)(10,0)[17]{} (180,50)(10,0)[17]{} (180,40)(10,0)[17]{} (180,30)(10,0)[17]{}
(180,30)[(1,1)[20]{}]{} (200,50)[(1,-1)[10]{}]{} (210,40)[(1,1)[10]{}]{} (220,50)[(1,-1)[20]{}]{} (240,30)[(1,1)[20]{}]{} (260,50)[(1,-1)[10]{}]{} (270,40)[(1,1)[20]{}]{} (290,60)[(1,-1)[10]{}]{} (300,50)[(1,1)[10]{}]{} (310,60)[(1,-1)[30]{}]{}
Using the same idea, we may encode a pair of converging walks $(L_1,L_2)$ in $T(i,n)$ by a partial Dyck path $P$ in the sense that the starting point of $P$ is not necessarily the point $(0,0)$. We should note that the common definition of a partial Dyck path is a lattice path starting from the origin $(0,0)$ with up and down steps not going below the $x$-axis. Define $P(i,n)$ to be the set of all partial Dyck paths of length $2n$ which start from $(0,2i)$ and never return to the $x$-axis except for the final destination. The following proposition establishes the connection between converging walks and partial Dyck paths.
For $n\geq 1$, there exists a bijection between $T(i,n)$ and $P(i,n)$.
Given a pair of converging walks $(L_1,L_2)$ in $T(i,n)$, let $L_1=p_1p_2\cdots p_n$ and $L_2= q_1q_2\cdots q_n$, where $p_i,
q_i=U$ or $D$. Then $(L_1, L_2)$ can be represented by a partial Dyck path $P$ of length $2n$ starting from $(0,2i)$: $$P=q_1p_1' q_2 p_2'\cdots q_n p_n'.$$ Clearly, $P$ returns to the $x$-axis at the ending point and never touches the $x$-axis before the ending point, that is, $P\in
P(i,n)$. It is easy to verify that the above correspondence is a bijection. Figure \[merging\] is an illustration.
------------------------------------------------------------------------
(340,130) (-12,14)[(1,0)[108]{}]{} (0,2)[(0,1)[108]{}]{}
(0,12)(12,0)[9]{} (0,24)(12,0)[9]{} (0,36)(12,0)[9]{} (0,48)(12,0)[9]{} (0,60)(12,0)[9]{} (0,72)(12,0)[9]{} (0,84)(12,0)[9]{} (0,96)(12,0)[9]{} (0,108)(12,0)[9]{} (0,120)(12,0)[9]{}
(0,24)[(1,-1)[12]{}]{}(12,12)[(1,1)[36]{}]{} (48,48)[(1,-1)[12]{}]{}(60,36)[(1,1)[12]{}]{} (72,48)[(1,-1)[12]{}]{}(84,36)[(1,1)[12]{}]{}
(0,96)[(1,1)[12]{}]{}(12,108)[(1,-1)[48]{}]{} (60,60)[(1,1)[12]{}]{}(72,72)[(1,-1)[24]{}]{} (-12,20)[$L_1$]{}(-12,92)[$L_2$]{}
(132,12)[(1,0)[204]{}]{} (144,0)[(0,1)[120]{}]{} (133,81)[$2i$]{}
(144,12)(12,0)[17]{} (144,24)(12,0)[17]{} (144,36)(12,0)[17]{} (144,48)(12,0)[17]{} (144,60)(12,0)[17]{} (144,72)(12,0)[17]{} (144,84)(12,0)[17]{} (144,96)(12,0)[17]{} (144,108)(12,0)[17]{} (144,120)(12,0)[17]{}
(144,84)[(1,1)[24]{}]{}(168,108)[(1,-1)[84]{}]{} (252,24)[(1,1)[24]{}]{}(276,48)[(1,-1)[24]{}]{} (300,24)[(1,1)[12]{}]{}(312,36)[(1,-1)[24]{}]{}
(114,62)[$\leftrightarrow$]{} (144,77)[$P$]{}
It is well known that the number of partial Dyck paths in $P(i,n)$ is given by the classical ballot number. Here we give a decomposition of a partial Dyck path into Dyck paths in accordance with the generating function of $|T(i,n)|$ as given in Proposition \[b-prop\].
Given a partial Dyck path $P$ in $P(i,n)$, we can decompose $P$ into $i$ nonempty Dyck paths $P_1,\ldots,P_{i}$ via the following procedure. Let $P=A_0A_1\cdots A_{2n}$, where $P$ is represented by the sequence of points rather than steps. Let $j_0=0$, and for $1\leq k\leq i$, let $j_k$ be the minimum index such that the $y$-coordinate of $A_{j_k}$ is two less than that of $A_{j_{k-1}}$. Then we can decompose $P$ into $i$ segments $Q_1, Q_2, \ldots, Q_i$, where $Q_k$ is the segment of $P$ starting at $A_{j_{k-1}}$ and ending at $A_{j_k}$. Observe that by the choice of $j_k$, the rightmost two steps of $Q_k$ must be $DD$. Let $P_k$ denote the Dyck path obtained from $Q_k$ by deleting the last down step and adding an up step before the first step of $Q_k$. Evidently, $P_k$ is a nonempty Dyck path. This completes the proof.
------------------------------------------------------------------------
To conclude this paper, we note that $|T(i,n)|$ can be computed by using the Lagrange inversion formula, or by using the formula for the number of Dyck paths of length $2n+2i$ with $2i$ returns to the $x$-axis, see Deutsch [@Deut]. The explicit formula is as follows: $$|T(i,n)|={i\over n}{2n\choose n-i}.$$ We also note that $|T(i,n)|$ can be expressed as the classical ballot number $b(n+i-1,n-i)$, where $$b(n,i)=
{n+i\choose i}-{n+i\choose i-1} ={n+1-i\over n+1+i}{n+i+1\choose
i},$$ see, for example, Riordan [@Rior].
We would like to thank Ira Gessel for helpful discussions. This work was supported by the 973 Project, the PCSIRT Project of the Ministry of Education, the Ministry of Science and Technology, and the National Science Foundation of China.
[9]{}
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---
abstract: 'Current fluctuations in boundary-driven diffusive systems are, in many cases, studied using hydrodynamic theories. Their predictions are then expected to be valid for currents which scale inversely with the system size. To study this question in detail, we introduce a class of large-$N$ models of one-dimensional boundary-driven diffusive systems, whose current large deviation functions are exactly derivable for any finite number of sites. Surprisingly, we find that for some systems the predictions of the hydrodynamic theory may hold well beyond their naive regime of validity. Specifically, we show that, while a symmetric partial exclusion process exhibits non-hydrodynamic behaviors sufficiently far beyond the naive hydrodynamic regime, a symmetric inclusion process is well described by the hydrodynamic theory for arbitrarily large currents. We conjecture, and verify for zero-range processes, that the hydrodynamic theory captures the statistics of arbitrarily large currents for all models where the mobility coefficient as a function of density is unbounded from above. In addition, for the large-$N$ models, we prove the additivity principle under the assumption that the large deviation function has no discontinuous transitions.'
author:
- Yongjoo Baek
- Yariv Kafri
- Vivien Lecomte
bibliography:
- 'current\_LDF\_large-N.bib'
title: 'Extreme current fluctuations of boundary-driven systems in the large-$N$ limit'
---
=1
Introduction {#sec:intro}
============
One of the most fundamental ways to characterize the steady state of a system is through the statistical properties of currents. These have been studied both in and out of equilibrium and in both classical [@Derrida2007] and quantum systems [@pilgram_stochastic_2003; @Esposito2009; @genway_trajectory_2014]. Recently, much progress has been achieved in understanding the statistics of time-averaged currents, which are encoded in a corresponding large deviation functions (LDF), of boundary-driven diffusive systems in one dimension [@Derrida2004; @Bodineau2004; @Bertini2005a; @Bertini2006; @Harris2005; @Imparato2009; @Lecomte2010; @Shpielberg2015] as well as in other geometries [@bodineau_vortices_2008; @Akkermans2013]. Whereas exact microscopic solutions are often available for bulk-driven systems [@derrida_exact_1998; @prolhac_cumulants_2009; @lazarescu_exact_2011; @mallick_exact_2011; @lazarescu_matrix_2013; @lazarescu_physicists_2015; @ayyer_full_2015], the results for boundary-driven systems largely rest on the application of a hydrodynamic approach termed the macroscopic fluctuation theory (MFT) [@spohn_large_1991; @Bertini2002; @jordan_fluctuation_2004; @Bertini2015], with the notable exception of [@lazarescu_matrix_2013].
Being a hydrodynamic theory, the MFT is naively expected to yield the correct statistics of currents only when the current fluctuations are small enough for the hydrodynamic description to be valid. For example, consider a single-species diffusive system on the line $0 \leq x \leq \ell$, where $\ell$ denotes the length of the system. After coarse-graining and a diffusive rescaling ($x \to x/\ell$ and $t \to t/\ell^2$ [^1]), the hydrodynamic equation takes the form $$\label{eq:hydro}
\partial_t \rho(x) = -\partial_x J(x),$$ with $\rho(x)$ the coarse-grained density and $J(x)$ the coarse-grained current. Since we are interested in the $\ell \to \infty$ limit, this equation is not well defined for $J(x)$ which before the rescaling is not of the order of $1/\ell$. Thus, the statistics of currents obtained by the MFT are reliable only for current fluctuations of the order of $1/\ell$. The same conclusion can be reached by another argument more directly based on the MFT, which is discussed in Appendix \[app:hydro\_limit\].
In this paper we study the validity of the hydrodynamic approach in regions where it is expected to fail. Quite surprisingly, we find that there are classes of models where the hydrodynamic approach captures the statistics of currents much beyond its naive regime of validity. We give a simple explanation for this phenomena and based on it argue that this behavior is expected to be generic when the mobility diverges with the density of particles.
To obtain these results, we study current LDFs of boundary-driven systems whose lattice structure is preserved, keeping a finite number of sites $L$. Since the exact current LDFs of microscopic lattice models are difficult to obtain (with the exception of the zero-range-process [@Harris2005]), we consider a little-studied class of coarse-grained models, which we term [*large-$N$ models*]{}. A large-$N$ model consists of a one-dimensional chain of boxes, each of which holds a macroscopically large number of particles (controlled by $N$) and which relaxes instantaneously to local equilibrium. As such, it retains the lattice structure even after coarse-graining and can be thought of as an analog of the “boxed models” studied in [@Bunin2013; @Kafri2015]. In a manner similar to models of population dynamics [@Elgart2004; @Meerson2011] and lattice spin models in the large-spin limit [@Tailleur2007; @Tailleur2008], we rescale dynamical variables and hopping rates of the model by powers of $N$. This allows us to apply the standard saddle-point techniques in the $N \to \infty$ limit.
Thanks to simplifications arising from the assumption of a macroscopic number of particles at each box (site), the current LDFs of our large-$N$ models are exactly derivable even for a finite system with any number of sites $L$. By comparing the tail behaviors of the current LDFs in the large-$L$ limit with the predictions of the MFT approach, we can observe how and when non-hydrodynamic behaviors start to emerge. Interestingly, our formulation also shows that the same microscopic dynamics may produce [*different macroscopic models*]{} depending on how the microscopic variables are scaled with $N$.
We note that there were previous studies on models with multiple particles per site, such as partial exclusion processes [@schutz_non-abelian_1994], inclusion processes [@giardina_duality_2007], or both [@giardina_correlation_2010; @carinci_duality_2013]. These studies obtained exact expressions for particle density correlations on a finite lattice with $L$ sites. The corresponding density large deviations were studied in [@Tailleur2007; @Tailleur2008], but only after a gradient expansion in the $L \to \infty$ limit that washes away the lattice structure. To our knowledge, large deviation properties of these models at finite $L$ have not been properly explored [^2].
This paper is organized as follows. In Sec. \[sec:models\], we introduce two classes of large-$N$ models, which are the symmetric partial exclusion process (SPEP) and the symmetric inclusion process (SIP). It is shown that the latter becomes equivalent to the well-studied Kipnis–Marchioro–Presutti (KMP) model [@Kipnis1982; @Bertini2005b] after an appropriate rescaling by $N$. In Sec. \[sec:spep\_current\], we study current large deviations of the SPEP, which exhibits non-hydrodynamic behaviors for current fluctuations sufficiently far beyond the naive hydrodynamic regime expected by the argument given above. In addition, we also discuss the validity of the additivity principle. In Sec. \[sec:sip\_current\], we analyze current large deviations of the SIP for different large-$N$ limits, which in all cases exhibit hydrodynamic behaviors for arbitrarily large current fluctuations. Based on these results, in Sec. \[sec:criterion\] we propose a criterion for the persistence of hydrodynamic current fluctuations in the non-hydrodynamic regime, and confirm its validity for the symmetric zero-range process. Finally, we summarize our results and conclude in Sec. \[sec:conclusions\].
Large-$N$ models {#sec:models}
================
We now turn to introduce the large-$N$ versions of the SPEP and the SIP. Starting with the SPEP the microscopic model is defined and used to obtain a path-integral representation for the current cumulant generating function (CGF) along with the prescription for calculating it in the large-$N$ limit. The hydrodynamic limit of the model is then presented for completeness. The section closes by giving the corresponding results for the class of SIP models.
Microscopic dynamics {#ssec:models_micro}
--------------------
The models are defined on a one-dimensional chain of $L$ boxes which are in contact with two particle reservoirs denoted by $a$ and $b$ (see Fig. \[fig:models\] for an illustration). Each box is assumed to be in local equilibrium so that the state of box $k$ is completely specified by the number of particles $n_k$, for $k = 1,\,2,\,\ldots,\,L$. A particle hops from a box to an adjacent one with a rate (in arbitrary units) given by $$\begin{aligned}
\label{eq:micro_bulk_rates}
\text{SPEP:} &\qquad (n_k,\,n_l) \xrightarrow{n_k(N - n_l)} (n_k - 1,\, n_l + 1) \qquad \text{for $l = k \pm 1$}, \nonumber\\
\text{SIP:} &\qquad (n_k,\,n_l) \xrightarrow{n_k(N + n_l)} (n_k - 1,\, n_l + 1) \qquad \text{for $l = k \pm 1$},\end{aligned}$$ which reflects exclusion (‘attractive’) interactions between particles in the SPEP (SIP). It is clear that for the SPEP the range of $n_k$ is bounded from above and below ($0 \le n_k \le N$), while for the SIP $n_k$ is only bounded from below ($n_k \ge 0$). The hopping rates at the boundaries are defined similarly as: $$\begin{aligned}
{3}\label{eq:micro_boundary_rates}
\text{SPEP:} &\qquad n_1 &\xrightarrow{\alpha (N - n_1)} n_1 + 1, &\qquad n_1 &\xrightarrow{\gamma n_1} n_1 - 1, \nonumber\\
&\qquad n_L &\xrightarrow{\delta (N - n_L)} n_1 + 1, &\qquad n_L &\xrightarrow{\beta n_L} n_L - 1, \nonumber\\
\text{SIP:} &\qquad n_1 &\xrightarrow{\alpha (N + n_1)} n_1 + 1, &\qquad n_1 &\xrightarrow{\gamma n_1} n_1 - 1, \nonumber\\
&\qquad n_L &\xrightarrow{\delta (N + n_L)} n_1 + 1, &\qquad n_L &\xrightarrow{\beta n_L} n_L - 1.\end{aligned}$$ If the system is coupled only to reservoir $a$ (reservoir $b$), the average number of particles in each box relaxes to $\bar n_a$ ($\bar n_b$) as determined by $\alpha$ and $\gamma$ ($\beta$ and $\delta$). In what follows, we fix the contact rates to the reservoirs through $N/(\gamma + \alpha) = 1$, $N/(\beta + \delta) = 1$ for the SPEP, and $N/(\gamma - \alpha) = 1$, $N/(\beta - \delta) = 1$ for the SIP. The parameters $\bar{n}_a$ and $\bar{n}_b$ thus fully describe the coupling with the reservoirs: $$\begin{aligned}
\label{eq:micro_boundary_densities}
\text{SPEP:} &\qquad \alpha = \bar{n}_a, \quad \beta = N - \bar{n}_b, \quad \gamma = N - \bar{n}_a, \quad \delta = \bar{n}_b \;, \nonumber\\
\text{SIP:} &\qquad \alpha = \bar{n}_a, \quad \beta = N + \bar{n}_b, \quad \gamma = N + \bar{n}_a, \quad \delta = \bar{n}_b \;.\end{aligned}$$ This choice provides simpler expressions in the results presented below, without affecting the large-$L$ hydrodynamic behavior.
With these definitions it is natural to introduce density variables according to $$\label{eq:density_rescaling}
\rho_k \equiv \frac{n_k}{N}, \quad \bar\rho_a \equiv \frac{\bar n_a}{N}, \quad \bar\rho_b \equiv \frac{\bar n_b}{N}\;,$$ and rescale time as $t \to Nt$. Then the evolution of the average density profile, taken over some initial distribution and denoted by angular brackets, satisfies $$\label{eq:average_dyn}
\frac{\partial \langle \rho_k \rangle}{\partial t} = \langle \rho_{k-1}\rangle - 2 \langle \rho_k \rangle + \langle \rho_{k+1} \rangle$$ for any $k = 1,\,2,\,\ldots,\,L$ with $\rho_0 \equiv \bar\rho_a$ and $\rho_{L+1} \equiv \bar\rho_b$. We note that the discrete diffusion equation is also known to hold exactly for the standard Symmetric Simple Exclusion Process (SSEP), which corresponds to the SPEP with $N = 1$.
Under this rescaling, for the SPEP, $N$ is naturally interpreted as the capacity of each box. On the other hand, for the SIP the number of particles is not bounded from above. Therefore, $N$ does not admit a natural interpretation without specifying how both ${\bar n_a}$ and ${\bar n_b}$ scale with $N$. In fact, one can choose an alternate scaling and define densities for the SIP as $$\label{eq:density_rescaling_alpha}
\rho_k \equiv \frac{n_k}{N^{1+\alpha}}, \quad \bar\rho_a \equiv \frac{\bar n_a}{N^{1+\alpha}}, \quad \bar\rho_b \equiv \frac{\bar n_b}{N^{1+\alpha}}$$ with $t$ rescaled by $N$ as above and $\alpha>0$ (the rationale behind this constraint will become clear below). It is straightforward to check that is then unchanged. Interestingly, these two scaling choices for the SIP, as we show below, lead to different [*macroscopic*]{} theories. In what follows, when we also study the SIP rescaled by and refer to it as SIP(1+$\alpha$), in contrast to the SIP(1) whose scaling is defined in .
SPEP – current CGF and hydrodynamic limit {#ssec:models_spep}
-----------------------------------------
Our interest is in calculating the current CGF which encodes the statistics of the time-averaged density current $J$. We can obtain $J$, for example, by measuring the flux of particles from box $L$ to reservoir $b$ during an interval $t \in [0,\,T]$. The CGF is then defined through $$\label{eq:cgf_def}
e^{NT\psi_{N,L}(\lambda,\bar\rho_a,\bar\rho_b)} = {\left\langle}e^{N\lambda T J}{\right\rangle}\quad \text{for $T \gg 1$,}$$ where the average, denoted by angular brackets, is taken with fixed $\bar\rho_a$ and $\bar\rho_b$, and $\lambda$ is conjugate to the current $J$. Using standard methods (see Appendix \[app:path\]), we can write a path-integral representation of the CGF $$\label{eq:macro_path_integ}
e^{NT\psi_{N,L}(\lambda)} = \int {\mathcal{D}}{\boldsymbol{\rho}}{\mathcal{D}}{\hat{\boldsymbol{\rho}}}\,
\exp\left\{-N\int_{0}^{T} \mathrm{d}t \,\left[ {\hat{\boldsymbol{\rho}}}\cdot\dot{{\boldsymbol{\rho}}}
- H_L(\lambda;{\boldsymbol{\rho}},{\hat{\boldsymbol{\rho}}})\right] + o(N)\right\}$$ with ${\boldsymbol{\rho}}\equiv (\rho_1, \rho_2, \ldots, \rho_L)$ the density vector and ${\hat{\boldsymbol{\rho}}}\equiv (\hat\rho_1, \hat\rho_2, \ldots, \hat\rho_L)$ the auxiliary ‘momentum’ vector. For the SPEP, the Hamiltonian $H_L$ is given by $$\begin{aligned}
\label{eq:spep_H}
H^\mathrm{SPEP}_L(\lambda;{\boldsymbol{\rho}},{\hat{\boldsymbol{\rho}}})
&= \sum_{k=1}^{L-1} \left[ \rho_k (1-\rho_{k+1}) \left(e^{\hat{\rho}_{k+1} - \hat{\rho}_k}-1\right)
+ \rho_{k+1} (1-\rho_k) \left(e^{\hat{\rho}_k - \hat{\rho}_{k+1}}-1\right)\right] \nonumber\\
&\quad +\rho_1 (1-\bar{\rho}_a) \left(e^{- \hat{\rho}_1}-1\right) + \bar{\rho}_a (1-\rho_1) \left(e^{\hat{\rho}_1}-1\right) \nonumber\\
&\quad +\rho_L (1-\bar{\rho}_b) \left(e^{- \hat{\rho}_L + \lambda}-1\right) + \bar{\rho}_b (1-\rho_L) \left(e^{\hat{\rho}_L - \lambda}-1\right).\end{aligned}$$ When $N$ is very large (in the sense of $N \gg T \gg 1$), the large-$N$ CGF $\psi_L$ can be obtained using saddle-point asymptotics $$\label{eq:cgf_saddle}
\psi_L(\lambda) \equiv \lim_{N\to\infty} \psi_{N,L}(\lambda) = \lim_{T \to \infty} \frac{1}{T} \inf_{{\boldsymbol{\rho}},\,{\hat{\boldsymbol{\rho}}}}\int_{0}^{T} \mathrm{d}t \,
\left[ {\hat{\boldsymbol{\rho}}}\cdot\dot{{\boldsymbol{\rho}}} - H_L(\lambda;{\boldsymbol{\rho}},{\hat{\boldsymbol{\rho}}}) \right]$$ with the infimum taken over trajectories of ${\boldsymbol{\rho}}$ and ${\hat{\boldsymbol{\rho}}}$. As advertised above, this approximation requires only $N$ to be a large parameter, so its predictions hold for any value of $L$. The minimization principle is similar to that of the MFT approach [@Bertini2015] for the SSEP, with $N$, instead of $L$, playing the role of the large parameter governing the saddle-point. This allows us to keep track of the lattice structure at any finite $L$.
Assuming that the minimizing trajectory is time-independent, the saddle-point equations are given by $$\label{eq:time_indep_sol}
\frac{\partial {\boldsymbol{\rho}}}{\partial t} = \frac{\partial H_L}{\partial {\hat{\boldsymbol{\rho}}}} = 0, \quad \frac{\partial {\hat{\boldsymbol{\rho}}}}{\partial t} = -\frac{\partial H_L}{\partial {\boldsymbol{\rho}}} = 0.$$ The solutions of these equations, which we denote by ${\boldsymbol{\rho}}^*$ and ${\hat{\boldsymbol{\rho}}}^*$, are typically called the [*optimal profiles*]{} which support the current fluctuation $J$. Then the current CGF is obtained from as $$\label{eq:cgf_H}
\psi_L(\lambda) = H_L(\lambda;{\boldsymbol{\rho}}^*,{\hat{\boldsymbol{\rho}}}^*).$$ The additivity principle, proposed in [@Bodineau2004] (also independently studied in [@jordan_fluctuation_2004]), implies that the above assumption is applicable for any value of $\lambda$. Although counterexamples were found in periodic bulk-driven systems [@Bertini2005a; @bodineau_distribution_2005; @Bertini2006; @Hurtado2011; @Espigares2013], the principle was analytically shown to be true for any open boundary-driven diffusive system with a constant diffusion coefficient and a quadratic mobility coefficient [@Imparato2009] — without ruling out possible discontinuous transitions, which in turn were numerically discarded in [@Hurtado2009] for a specific model related to the SIP. As shown below, both the SPEP and the SIP correspond to this class of systems in the hydrodynamic limit. Thus we expect that the same principle is also applicable to our large-$N$ models, and discuss arguments supporting its validity in Sec. \[ssec:spep\_fin\_N\] and Appendix \[app:finiteNquantumfluctuationsSPEP\].
Finally, we show that under appropriate assumptions our large-$N$ models are well described by hydrodynamic theories. To see this, we first apply a diffusive scaling in terms of $L$, which involves writing the position of box $k$ as $x \equiv k/(L+1)$ (with the lattice spacing set to one) and rescaling time by $t \to t/(L+1)^2$. We also assume that differences between adjacent boxes, namely $\rho_{k+1} - \rho_k$ and ${\hat\rho}_{k+1} - {\hat\rho}_k$, scale as $1/(L+1)$. Then, in the $L \to \infty$ limit, the gradients $\partial_x \rho$ and $\partial_x {\hat\rho}$ are well defined, and can be approximated as $$\label{eq:hydro_path_integ}
e^{N (L+1)^2 T\psi(\lambda)} = \int {\mathcal{D}}\rho {\mathcal{D}}\hat\rho \,
\exp\left\{-N(L+1) \int_{0}^{T} \mathrm{d}t \left[ \left( \int_0^1 \mathrm{d}x \, {\hat\rho}\dot{\rho} \right) - H [\rho,{\hat\rho}] \right]\right\}.$$ Here the Hamiltonian $H [\rho,{\hat\rho}]$, which is no longer dependent on $\lambda$, is now a functional of continuous profiles $\rho(x)$ and ${\hat\rho}(x)$. The functional typically has the form of $$\label{eq:H_mft}
H[\rho,{\hat\rho}] = \int_0^1 \mathrm{d}x \,
\left[ -D(\rho)(\partial_x \rho)(\partial_x {\hat\rho}) + \frac{\sigma(\rho) (\partial_x {\hat\rho})^2}{2} \right]$$ with $D(\rho)$ the diffusion coefficient and $\sigma(\rho)$ the mobility coefficient. For the SPEP, these coefficients are given by $$D(\rho) = 1, \quad \sigma(\rho) = 2\rho(1-\rho),$$ respectively. We note that this $\sigma(\rho)$ is bounded from above, with the maximum value given by $\sigma(1/2) = 1/2$. Meanwhile, the rescaling of time speeds up the microscopic dynamics, so the leftmost ($k = 1$) and rightmost ($k = L$) boxes equilibrate with the coupled reservoirs (see *e.g.* Appendix B.2 of Ref. [@Tailleur2008]). Hence, the spatial boundary conditions are given by $$\rho(0) = \bar{\rho}_a, \quad \rho(1) = \bar{\rho}_b, \quad {\hat\rho}(0) = 0, \quad {\hat\rho}(1) = \lambda,$$ whose dependence on $\lambda$ keeps $\psi$ a function of $\lambda$.
In what follows we list the corresponding sets of results for the SIP(1) and the SIP(1+$\alpha$).
SIP(1) – current CGF and hydrodynamic limit {#ssec:models_sip}
-------------------------------------------
It is straightforward to repeat the above derivations for the SIP. We find, using the notation of , $$\begin{aligned}
\label{eq:sip1_H}
H^{\mathrm{SIP}(1)}_L (\lambda;{\boldsymbol{\rho}},{\hat{\boldsymbol{\rho}}}) &\equiv \sum_{k=1}^{L-1} \left[ \rho_k (1 + \rho_{k+1})
\left(e^{{\hat\rho}_{k+1} - {\hat\rho}_k}-1\right) + \rho_{k+1} (1 + \rho_k) \left(e^{{\hat\rho}_k - {\hat\rho}_{k+1}}-1\right)\right] \nonumber\\
&\quad+ \left[ \rho_1 (1 + \bar{\rho}_a) \left(e^{- {\hat\rho}_1}-1\right) + \bar{\rho}_a (1 + \rho_1) \left(e^{{\hat\rho}_1}-1\right)\right] \nonumber\\
&\quad+ \left[ \rho_L (1 + \bar{\rho}_b) \left(e^{- {\hat\rho}_L+\lambda}-1\right) + \bar{\rho}_b (1 + \rho_L) \left(e^{{\hat\rho}_L-\lambda}-1\right)\right].\end{aligned}$$ In addition, the corresponding hydrodynamic Hamiltonian in the large-$L$ limit is given by with $$\label{eq:coeff_sip1}
D(\rho) = 1, \quad \sigma(\rho) = 2\rho(1+\rho).$$ We note that $\sigma(\rho)$ in this case is not bounded from above.
SIP(1+$\alpha$) – current CGF and hydrodynamic limit {#ssec:models_sip1a}
----------------------------------------------------
For the SIP(1+$\alpha$) we similarly find, using the notation of , $$\begin{aligned}
\label{eq:sip1a_H}
H^{\mathrm{SIP}(1+\alpha)}_L(\lambda;{\boldsymbol{\rho}},{\hat{\boldsymbol{\rho}}})
&\equiv \sum_{k=1}^{L-1} \left[-(\rho_{k+1} - \rho_k)({\hat\rho}_{k+1} - {\hat\rho}_k) + \rho_k \rho_{k+1}({\hat\rho}_{k+1} - {\hat\rho}_k)^2\right] \nonumber\\
&\quad + (\bar\rho_a - \rho_1){\hat\rho}_1 + \bar\rho_a \rho_1 {\hat\rho}_1^2 + (\bar\rho_b - \rho_L)({\hat\rho}_L-\lambda) + \rho_L \bar\rho_b ({\hat\rho}_L-\lambda)^2.\end{aligned}$$ The hydrodynamic description of this model in the large-$L$ limit is given by with $$\label{eq:coeff_sip1a}
D(\rho) = 1, \quad \sigma(\rho) = 2\rho^2,$$ where $\sigma(\rho)$ is again not bounded from above. These transport coefficients are also shared by the Kipnis–Marchioro–Presutti (KMP) model of heat conduction [@Kipnis1982; @Bertini2005b]. It is notable that the same microscopic model produces different macroscopic behaviors depending on the reservoir properties.
Current large deviations in the SPEP {#sec:spep_current}
====================================
In what follows we first show that the scaled CGF of the time-averaged current in the SPEP in the large-$N$ limit is given by $$\label{eq:spep_cgf}
\psi^\mathrm{SPEP}_L(\lambda) = \begin{cases}
(L+1) \sinh^2 \left(\frac{1}{L+1} \mathrm{arcsinh} \sqrt{\omega^\mathrm{SPEP}}\right) &\text{if $\omega^\mathrm{SPEP} \ge 0$}, \\
-(L+1) \sin^2 \left(\frac{1}{L+1} \mathrm{arcsin} \sqrt{-\omega^\mathrm{SPEP}}\right) &\text{if $\omega^\mathrm{SPEP} < 0$}.
\end{cases}$$ where $$\label{eq:spep_omega}
\omega^\mathrm{SPEP} \equiv (1 - e^{-\lambda})\left[e^\lambda \bar\rho_a - \bar\rho_b - (e^\lambda - 1)\bar\rho_a\bar\rho_b\right].$$ Note that although the result depends explicitly on the sign of $\omega^\mathrm{SPEP}$, it is straightforward to verify that it is an analytic function of $\lambda$. After deriving this result, we compare to the predictions of the hydrodynamic theory. As we show, for large enough currents the two theories, as one might expect using the simple argument of the introduction, do not agree. Finally, we discuss finite-$N$ effects and their implications on the additivity principle.
Derivation of the scaled CGF {#ssec:spep_cgf}
----------------------------
\[ssec:ldfSPEPderivation\]
As stated above, assuming additivity, the problem of calculating the CGF in the large-$N$ limit is reduced to solving . To do this it is useful to use the canonical transformation [@DLS2002; @Tailleur2007; @Tailleur2008] $$\label{eq:spep_canonical}
\rho_k = F_k \left[1+(1 - F_k) \hat{F}_k\right], \qquad \hat{\rho}_k = \ln \left( 1 + \frac{\hat{F}_k}{1-F_k \hat{F}_k} \right) \;,$$ which can also be written as $$F_k = \frac{\rho_k}{e^{\hat\rho_k}(1-\rho_k) + \rho_k}\;, \qquad \hat{F}_k = (e^{\hat\rho_k}-1)(1-\rho_k) + (1-e^{-\hat\rho_k})\rho_k.$$ Then the Hamiltonian in the new set of coordinates, $K^\mathrm{SPEP}_L(\lambda,\bar\rho_a,\bar\rho_b;\mathbf{F},\hat{\mathbf{F}})$, is given by $$\begin{aligned}
\label{eq:spep_K}
K^\mathrm{SPEP}_L(\lambda,\bar\rho_a,\bar\rho_b;\mathbf{F},\hat{\mathbf{F}})
&= \sum_{k=1}^{L-1} \left[(\hat{F}_{k+1} - \hat{F}_k)(F_k - F_{k+1}) - \hat{F}_k\hat{F}_{k+1}(F_k-F_{k+1})^2\right]
+ \hat{F}_1 (\bar{\rho}_a - F_1) \nonumber\\
&\quad + e^{-\lambda}\left[\hat F_L - (e^\lambda - 1)(1-F_L\hat F_L) \right] \left[\bar{\rho}_b - F_L - F_L(1 - \bar{\rho}_b)(e^\lambda - 1)\right].\end{aligned}$$ where $\mathbf{F}=\left(F_1,F_2,\ldots, F_L\right)$ and $\hat{\mathbf{F}}=\left(\hat{F}_1,\hat{F}_2,\ldots, \hat{F}_L\right)$.
Note that the canonical transformation also adds temporal boundary conditions to the action which can be ignored in the $T \to \infty$ limit. The scaled CGF is then given by: $$\begin{aligned}
\label{eq:spep_cgf_K}
\psi_L(\lambda,\bar\rho_a,\bar\rho_b) = K^\mathrm{SPEP}_L(\lambda,\bar\rho_a,\bar\rho_b;\mathbf{F}^*,\hat{\mathbf{F}}^*),\end{aligned}$$ where $(\mathbf{F}^*,\hat{\mathbf{F}}^*)$ are solutions of $$\label{eq:spep_K_time_indep_sol}
\frac{\partial \mathbf{F}}{\partial t} = \frac{\partial K^\mathrm{SPEP}_L}{\partial \hat{\mathbf{F}}} = 0,
\qquad \frac{\partial \hat{\mathbf{F}}}{\partial t} = -\frac{\partial K^\mathrm{SPEP}_L}{\partial \mathbf{F}} = 0.$$
In what follows, we solve these equations using the methods used in [@Imparato2009]. To avoid cumbersome expressions we drop the $^*$ notation from the optimal profiles $(\mathbf{F}^*,\hat{\mathbf{F}}^*)$ and use $(\mathbf{F},\hat{\mathbf{F}})$. First, we choose the Ansatz $$\begin{aligned}
\label{eq:spep_dF_dFh_saddle}
\hat F_k = -A \sinh (kB), \qquad F_k = \bar\rho_a + \frac{1}{2A}\tanh \frac{kB}{2},\end{aligned}$$ where $A$ and $B$ are undetermined constants. It is easy to check that this Ansatz satisfies for $1 \le k \le L-1$. Then the constants $A$ and $B$ are determined by the remaining saddle-point equations $$\frac{\partial K^\mathrm{SPEP}_L}{\partial \hat F_L} = \frac{\partial K^\mathrm{SPEP}_L}{\partial F_L} = 0.$$ These equations imply $$\frac{\partial K^\mathrm{SPEP}_L}{\partial F_L} = -4A^2\cosh \frac{LB}{2} \frac{\partial K^\mathrm{SPEP}_L}{\partial \hat F_L},$$ from which we obtain $$\label{eq:spep_A_saddle}
A^2 = \mathcal{A}^\mathrm{SPEP} \equiv \frac{(e^\lambda - 1) [e^\lambda(\bar\rho_b - 1) - \bar\rho_b]}
{4 [1 + (e^\lambda - 1)\bar\rho_a] [e^\lambda\bar\rho_a(\bar\rho_b-1)-\bar\rho_a\bar\rho_b+\bar\rho_b]} \;.$$ Then one can show that $-\frac{1}{A}\frac{\partial K^\mathrm{SPEP}_L}{\partial \hat F_L} = 0$ has the form of $$\label{eq:spep_B_epsilon}
\sinh(LB - \varepsilon) + \sinh B = 2 \sinh \frac{(L+1)B - \varepsilon}{2} \cosh \frac{(L-1)B - \varepsilon}{2} = 0,$$ where $\varepsilon$ satisfies $$\label{eq:spep_omega2}
\sinh^2 \frac{\varepsilon}{2}
= \omega^\mathrm{SPEP} \equiv (1 - e^{-\lambda})\left[e^\lambda \bar\rho_a - \bar\rho_b - (e^\lambda - 1)\bar\rho_a\bar\rho_b\right].$$ Given $\varepsilon$, is solved by $$B = \frac{\varepsilon}{L+1}.$$
Thus we have found $A$ and $B$ up to the undetermined signs of $A$ and $\varepsilon$. These signs can be fixed by noting that the optimal density profile ${\boldsymbol{\rho}}^*$ must always be nonnegative and that the CGF must vanish at $\lambda = 0$. Without loss of generality, for $\bar\rho_a \ge \bar\rho_b$ the optimal profiles are given by $$\begin{aligned}
\label{eq:spep_profiles}
\hat F_k^\mathrm{SPEP} &=
\begin{cases}
-\sqrt{\mathcal{A}^\mathrm{SPEP}} \sinh \left( \frac{2k}{L+1} \, \mathrm{arcsinh} \sqrt{\omega^\mathrm{SPEP}}\right)
&\text{ if $\lambda < -\ln \left[\frac{\bar\rho_a(1-\bar\rho_b)}{\bar\rho_b(1-\bar\rho_a)}\right]$,} \\
-\sqrt{-\mathcal{A}^\mathrm{SPEP}} \sin \left( \frac{2k}{L+1} \, \mathrm{arcsin} \sqrt{-\omega^\mathrm{SPEP}}\right)
&\text{ if $-\ln \left[\frac{\bar\rho_a(1-\bar\rho_b)}{\bar\rho_b(1-\bar\rho_a)}\right] \le \lambda < 0$,} \\
\sqrt{\mathcal{A}^\mathrm{SPEP}} \sinh \left( \frac{2k}{L+1} \, \mathrm{arcsinh} \sqrt{\omega^\mathrm{SPEP}}\right)
&\text{ if $\lambda \ge 0$.}
\end{cases} \nonumber \\
F_k^\mathrm{SPEP} &=
\begin{cases}
\bar\rho_a + \frac{1}{2\sqrt{\mathcal{A}^\mathrm{SPEP}}} \tanh \left( \frac{k}{L+1} \, \mathrm{arcsinh} \sqrt{\omega^\mathrm{SPEP}}\right)
&\text{ if $\lambda < -\ln \left[\frac{\bar\rho_a(1-\bar\rho_b)}{\bar\rho_b(1-\bar\rho_a)}\right]$,} \\
\bar\rho_a - \frac{1}{2\sqrt{-\mathcal{A}^\mathrm{SPEP}}} \tan \left( \frac{k}{L+1} \, \mathrm{arcsin} \sqrt{-\omega^\mathrm{SPEP}}\right)
&\text{ if $-\ln \left[\frac{\bar\rho_a(1-\bar\rho_b)}{\bar\rho_b(1-\bar\rho_a)}\right] \le \lambda < 0$,} \\
\bar\rho_a - \frac{1}{2\sqrt{\mathcal{A}^\mathrm{SPEP}}} \tanh \left( \frac{k}{L+1} \, \mathrm{arcsinh} \sqrt{\omega^\mathrm{SPEP}}\right)
&\text{ if $\lambda \ge 0$.} \\
\end{cases} \nonumber \\\end{aligned}$$ We note that $\mathcal{A}^\mathrm{SPEP}$ and $\omega^\mathrm{SPEP}$ are negative for the intermediate range $-\ln \left[\frac{\bar\rho_a(1-\bar\rho_b)}{\bar\rho_b(1-\bar\rho_a)}\right] < \lambda < 0$ and nonnegative otherwise. The results for $\bar\rho_a < \bar\rho_b$ are easily obtained by a sign change $\lambda \to -\lambda$ and an exchange of $\bar\rho_a$ and $\bar\rho_b$. Using these results with , after some algebra one obtains .
Comparison with hydrodynamic results {#ssec:spep_hydro}
------------------------------------
We now compare the results of the large-$N$ limit with the predictions of the hydrodynamic theory. The latter has been derived in [@Bodineau2004; @Imparato2009] (for the SSEP which shares the same hydrodynamic theory) and can also be obtained by holding $\lambda$ fixed in and taking the large $L$ limit. The expression is given by $$\label{eq:spep_cgf_hydro}
\psi^\mathrm{SPEP}(\lambda) =
\begin{cases}
\frac{1}{L+1} \mathrm{arcsinh}^2 \sqrt{\omega^\mathrm{SPEP}} &\text{if $\omega^\mathrm{SPEP} \ge 0$}, \\
-\frac{1}{L+1} \mathrm{arcsin}^2 \sqrt{-\omega^\mathrm{SPEP}} &\text{if $\omega^\mathrm{SPEP} < 0$} \;,
\end{cases}$$ and the convergence to it is illustrated in Fig. \[fig:spep\_cgf\_lambda\_psi\]. In fact, one can show analytically that $$\psi^\mathrm{SPEP}_L(\lambda) - \psi^\mathrm{SPEP}(\lambda)
= \frac{\mathrm{arcsinh}^4\sqrt{\omega^\mathrm{SPEP}}}{3(L+1)^3} + O\left((L+1)^{-4}\right),$$ The sign of the leading correction term indicates that the lattice structure increases the magnitude of the current fluctuations.
To check the validity of the hydrodynamic predictions we next increase $\lambda$ as $\lambda \sim L^\zeta$. This gives $$\lim_{L \to \infty} \frac{\psi^\mathrm{SPEP}_L(\lambda)}{\psi^\mathrm{SPEP}(\lambda)} =
\begin{cases}
1 &\text{ if $\zeta < 1$,} \\
\frac{4}{\Lambda^2}\sinh^2 \frac{\Lambda}{2} &\text{ if $\zeta = 1$ with $\lambda=\Lambda L$} \\
\infty &\text{ if $\zeta > 1$.}
\end{cases}$$ This indicates that, as one would naively expect, the hydrodynamic description fails for sufficiently large currents. The threshold separating the hydrodynamic regime from the non-hydrodynamic regime is given by $\lambda \sim L$ (see Fig. \[fig:spep\_hydro\_breakdown\]).
As we later show, there are other models where the predictions of the hydrodynamic theory hold well beyond the naive expectation. To this end it is useful to see in detail how the predictions of the hydrodynamic limit fail for the SPEP. To do this, we note that Hamilton’s equation takes the form of $$\dot{\rho_k} = \frac{\partial H^\mathrm{SPEP}_L}{\partial {\hat\rho}_k} = J_{k-1,k} - J_{k,k+1},$$ where $J_{k,k+1}$ is the current from box $k$ to box $k+1$. The time-averaged current $J$ can be expressed in terms of the optimal profiles (again we drop the $^*$ notation) as $$\begin{aligned}
\label{eq:spep_J_profiles}
J &= \frac{1}{L+1}\sum_{k=0}^L\left[(\rho_k - \rho_{k+1})+\rho_k(1-\rho_{k+1})(e^{{\hat\rho}_{k+1}-{\hat\rho}_k}-1)
- \rho_{k+1}(1-\rho_k)(e^{{\hat\rho}_k-{\hat\rho}_{k+1}}-1)\right] \nonumber\\
&= \underbrace{\frac{\bar\rho_a-\bar\rho_b}{L+1}}_{= \langle J \rangle} + \underbrace{\frac{1}{L+1}
\sum_{k=0}^L\left[\rho_k(1-\rho_{k+1})(e^{{\hat\rho}_{k+1}-{\hat\rho}_k}-1) - \rho_{k+1}(1-\rho_k)(e^{{\hat\rho}_k-{\hat\rho}_{k+1}}-1)\right]}_{= \delta J}.\end{aligned}$$ Since the mean value $\langle J \rangle$ always scales as $1/L$, large values of $J$ are always dominated by the fluctuation $\delta J$ (see [@Meerson2014] for a similar observation). Next, note that, as shown in Fig. \[fig:spep\_profile\], a large $\delta J$ is supported by a plateau of the density profile close to $\rho = 1/2$ and a slope of the momentum profile which grows with $\lambda$ (and hence with $J$). In addition, as indicated by the data collapses in Fig. \[fig:spep\_rhoh\_collapse\], the momentum profile has the scaling form $${\hat\rho}_k(\lambda,L) \simeq \lambda g(k/L).$$ This implies that $${\hat\rho}_{k+1}(\lambda,L) - {\hat\rho}_k(\lambda,L) \simeq \frac{\lambda}{L} g'(k/L) \simeq L^{\zeta - 1} \partial_x g.$$ If $\zeta < 1$, the momentum gradient decreases with $L$. Then we can approximate $\delta J$ as $$\begin{aligned}
\label{eq:spep_J_hydro}
\delta J \simeq L^{\zeta - 1} \int_0^1 {\mathrm{d}}x\, 2 \rho(1-\rho) \partial_x g,\end{aligned}$$ whose integral form suggests that the current is blind to the lattice structure for any $\zeta < 1$. In other words, the current does not feel any difference between the case $\zeta = 0$ (which can be considered as proper hydrodynamic regime) and the case $0 < \zeta < 1$. Thus its fluctuations show hydrodynamic behaviors in both cases. On the other hand, if $\zeta \ge 1$, the momentum gradient increases with $L$. Then the approximate becomes invalid, and the current becomes sensitive to the lattice structure. Thus $\zeta = 1$, which corresponds to $J = O(L^0)$ by , is the threshold separating the hydrodynamic regime from the non-hydrodynamic one. We note that this threshold is larger than what one would naively expect from the simple argument given in Sec. \[sec:intro\], [*i.e.,*]{} $J = O(L^{-1})$.
Finite-$N$ corrections and the validity of the additivity principle {#ssec:spep_fin_N}
-------------------------------------------------------------------
In what follows, we analyze the leading finite-$N$ correction to the scaled CGF $\psi^\mathrm{SPEP}_L$. This provides a useful tool for numerical corroboration of our analytical results, and confirms the stability of the time-independent saddle-point profiles. The latter thus supports the validity of the additivity principle for the SPEP.
As explained in Appendix \[app:finiteNquantumfluctuationsSPEP\], one can integrate spatio-temporal fluctuations around the saddle-point optimal solutions. This is done by using a mapping (generalizing that of Ref. [@Lecomte2010]) between the CGF of the system with reservoirs at generic densities $\bar\rho_a$, $\bar\rho_b$ and the CGF for reservoirs at densities $\frac 12$. The resulting expression is finite and analytic, which proves that the additivity hypothesis is correct with respect to continuous phase transitions towards time-dependent profiles (which, if they had existed, would have implied an instability of ${\boldsymbol{\rho}}^*,{\hat{\boldsymbol{\rho}}}^*$, reflected in a singularity of the correction). The saddle-point contribution $\psi_L(\lambda)$ to the CGF is complemented by a $1/N$ correction: $$\label{eq:spep_cgf_fin_N_series}
\psi^\mathrm{SPEP}_{N,L}(\lambda)=\psi^\mathrm{SPEP}_L(\lambda) +N^{-1} \psi_L^{1,\mathrm{SPEP}}(\lambda) + o(N^{-1})$$ with, denoting $L'=L+1$, $$\begin{aligned}
\psi_L^{1,\mathrm{SPEP}}(\lambda)
&=
\sum_{p=1}^{L'-1}
\bigg\{
\boldsymbol c_\lambda
-
\cos\frac{p\pi}{2L'}
\sqrt{\frac 12 \boldsymbol c_\lambda\big(\boldsymbol c_\lambda+\cos\frac{p\pi}{L'}\big)}
-
\sin\frac{p\pi}{2L'}
\sqrt{\frac 12 \boldsymbol c_\lambda\big(\boldsymbol c_\lambda-\cos\frac{p\pi}{L'}\big)}
\bigg\}
\label{eq:respsiLhalfdensities_sym}\end{aligned}$$ where we defined $\boldsymbol c_\lambda= \cosh \frac{2\operatorname{arcsinh}\sqrt{\omega^{\text{SPEP}}}}{L'}$.
We numerically confirm our theoretical predictions by implementing a finite-$N$ propagator of the SPEP conditioned on a given value of $\lambda$. The eigenvalue with the largest real part corresponds to the scaled CGF $\psi^\mathrm{SPEP}_{N,L}$. As shown in Fig. \[fig:spep\_fin\_N\], our theory correctly predicts the leading-order behaviors of $\psi^\mathrm{SPEP}_{N,L} - \psi^\mathrm{SPEP}_L$.
We now detail how the large-$L$ limit (at fixed $\lambda$) of matches the MFT result obtained for the SSEP [@appert-rolland_universal_2008]. The $L\to\infty$ limit behavior of is not immediately extractable; following a procedure described in Appendix \[app:finiteNquantumfluctuationsSPEP\], one obtains $$\begin{aligned}
\psi_L^{1,\mathrm{SPEP}}(\lambda)
&=
\frac 1{8L^2} \mathcal F\big(\!-\!\mu(\lambda)\big) \ + \
O(L^{-3}).
\label{eq:largeLpsi1L}\end{aligned}$$ Here, with $\mu(\lambda)=\operatorname{arcsinh}^2\sqrt{\omega^{\text{SPEP}}}$, we recognize the universal scaling function $$\begin{aligned}
\label{eq:univ_scaling}
\mathcal F(u)
&= 4 \sum_{p=1}^{\infty}
\Big\{
(p\pi)^2+u-p\pi\sqrt{(p\pi)^2-2u }
\Big\}\end{aligned}$$ as the one also arising in MFT [@appert-rolland_universal_2008] and Bethe-Ansatz [@appert-rolland_universal_2008; @prolhac_cumulants_2009] studies of current fluctuations. The large-$L$ limit (at fixed $\lambda$) thus yields the same correction as in the MFT approach [@Imparato2009] for the SSEP. The universal scaling function $\mathcal F(u)$ is singular at a positive value $u_\text{c}=\pi^2/2$ of its argument, but this value is never reached for any real-valued $\lambda$ in . This confirms, as in the MFT context, that the additivity principle holds at large $L$.
Current large deviations of SIP {#sec:sip_current}
===============================
In this section, we derive the scaled CGF of the time-averaged current of the SIP in the large-$N$ limit. It is given by $$\label{eq:sip_cgf}
\psi^\mathrm{SIP}_L (\lambda,\bar\rho_a,\bar\rho_b) = \begin{cases}
(L+1) \sin^2 \left(\frac{1}{L+1} \mathrm{arcsin} \sqrt{\omega^\mathrm{SIP}}\right) &\text{if $\omega^\mathrm{SIP} \ge 0$}, \\
-(L+1) \sinh^2 \left(\frac{1}{L+1} \mathrm{arcsinh} \sqrt{-\omega^\mathrm{SIP}}\right) &\text{if $\omega^\mathrm{SIP} < 0$}
\end{cases}$$ with the differences between $\mathrm{SIP}(1)$ and $\mathrm{SIP}(1+\alpha)$ encoded in $$\label{eq:sip_omega}
\omega^\mathrm{SIP} = \begin{cases}
\omega^{\mathrm{SIP}(1)} \equiv (1 - e^{-\lambda})\left[e^\lambda \bar\rho_a - \bar\rho_b + (e^\lambda - 1)\bar\rho_a\bar\rho_b\right] &\text{ for $\mathrm{SIP}(1)$.} \\
\omega^{\mathrm{SIP}(1+\alpha)} \equiv \lambda\rho_a - \lambda\rho_b + \lambda^2 \rho_a \rho_b &\text{ for $\mathrm{SIP}(1+\alpha)$.}
\end{cases}$$ Again, one can easily verify that $\psi^\mathrm{SIP}_L$ does not have any singularity at $\omega^\mathrm{SIP} = 0$. A comparison of this result with the hydrodynamic theory shows that for both $\mathrm{SIP}(1)$ and $\mathrm{SIP}(1+\alpha)$ arbitrarily large current fluctuations are still correctly captured by the hydrodynamic theory, in contrast to the SPEP. We close the section with a discussion of finite-$N$ effects.
Derivation of the scaled CGF {#ssec:sip_cgf}
----------------------------
### $\mathrm{SIP}(1)$
Similarly to the SPEP, we first transform $H^{\mathrm{SIP}(1)}_L$ into a more convenient form. This is done by the canonical transformation $$\label{eq:sip1_canonical}
\rho_k = F_k \left[1+(1 + F_k) \hat{F}_k\right], \qquad {\hat\rho}_k = \ln \left( 1 + \frac{\hat{F}_k}{1+F_k \hat{F}_k} \right),$$ which can also be written as $$F_k = \frac{\rho_k}{e^{{\hat\rho}_k}(1+\rho_k)-\rho_k}, \qquad \hat{F}_k = (e^{{\hat\rho}_k}-1)(1+\rho_k) - (1-e^{-{\hat\rho}_k})\rho_k.$$ After this transformation, the Hamiltonian of the new variables is given by $$\begin{aligned}
\label{eq:sip1_K}
K^{\mathrm{SIP}(1)}_L(\lambda,\bar\rho_a,\bar\rho_b;\mathbf{F},\hat{\mathbf{F}})
&= \sum_{k=1}^{L-1} \left[(\hat{F}_{k+1} - \hat{F}_k)(F_k - F_{k+1}) + \hat{F}_k\hat{F}_{k+1}(F_k-F_{k+1})^2\right]
+ \hat{F}_1 (\bar{\rho}_a - F_1) \nonumber\\
&\quad + e^{-\lambda}\Big[\hat F_L - (e^\lambda - 1)(1+F_L\hat F_L) \Big] \Big[\bar{\rho}_b - F_L - F_L(1 + \bar{\rho}_b)(e^\lambda - 1)\Big].\end{aligned}$$ Comparing this expression with , we find the formal correspondence $$\label{eq:sip1_K_spep_K}
K^{\mathrm{SIP}(1)}_L (\lambda,\bar\rho_a,\bar\rho_b;\mathbf{F},\hat{\mathbf{F}})
= -K^\mathrm{SPEP}_L (\lambda,-\bar\rho_a,-\bar\rho_b;-\mathbf{F},\hat{\mathbf{F}}).$$ By examining the time-independent saddle-point equations derived from these Hamiltonians, we find a mapping between the optimal profiles of the SPEP and the $\mathrm{SIP}(1)$: $$\begin{aligned}
\label{eq:sip1_spep_mapping}
(\hat F^*_k)^{\mathrm{SIP}(1)} (\lambda,\bar\rho_a,\bar\rho_b) &= (\hat F^*_k)^\mathrm{SPEP} (\lambda,-\bar\rho_a,-\bar\rho_b), \nonumber\\
(F^*_k)^{\mathrm{SIP}(1)} (\lambda,\bar\rho_a,\bar\rho_b) &= -(F^*_k)^\mathrm{SPEP} (\lambda,-\bar\rho_a,-\bar\rho_b).\end{aligned}$$
This mapping can be used to obtain the optimal profiles and the scaled CGF of the $\mathrm{SIP}(1)$ from those of the SPEP. It should be noted that, when $\bar\rho_a$ and $\bar\rho_b$ are negative, we should reconsider the proper signs of $A$ and $\varepsilon$ in the optimal profiles of the SPEP. In this case, the optimal density profile ${\boldsymbol{\rho}}^*$ must always be nonpositive, so that it becomes nonnegative after the mapping to the $\mathrm{SIP}(1)$. Taking this into account, for $\bar\rho_a \ge \bar\rho_b$, the optimal profiles are given by (dropping $^*$) $$\begin{aligned}
\label{eq:sip1_profiles}
\hat F_k^{\mathrm{SIP}(1)} &=
\begin{cases}
-\sqrt{\mathcal{A}^{\mathrm{SIP}(1)}} \sin \left( \frac{2k}{L+1} \, \mathrm{arcsin} \sqrt{\omega^{\mathrm{SIP}(1)}}\right)
&\text{ if $-\ln \left(1+\frac{1}{\rho_b}\right) < \lambda < -\ln \left[\frac{\bar\rho_a(1+\bar\rho_b)}{\bar\rho_b(1+\bar\rho_a)}\right]$,} \\
-\sqrt{\mathcal{-A}^{\mathrm{SIP}(1)}} \sinh \left( \frac{2k}{L+1} \, \mathrm{arcsinh} \sqrt{-\omega^{\mathrm{SIP}(1)}}\right)
&\text{ if $-\ln \left[\frac{\bar\rho_a(1+\bar\rho_b)}{\bar\rho_b(1+\bar\rho_a)}\right] \le \lambda < 0$,} \\
\sqrt{\mathcal{A}^{\mathrm{SIP}(1)}} \sin \left( \frac{2k}{L+1} \, \mathrm{arcsin} \sqrt{\omega^{\mathrm{SIP}(1)}}\right)
&\text{ if $0 \le \lambda < \ln \left(1 + \frac{1}{\rho_a}\right)$,}
\end{cases} \nonumber \\
F_k^{\mathrm{SIP}(1)} &=
\begin{cases}
\bar\rho_a + \frac{1}{2\sqrt{\mathcal{A}^{\mathrm{SIP}(1)}}} \tan \left( \frac{k}{L+1} \, \mathrm{arcsin} \sqrt{\omega^{\mathrm{SIP}(1)}}\right)
&\text{ if $-\ln \left(1+\frac{1}{\rho_b}\right) < \lambda < -\ln \left[\frac{\bar\rho_a(1+\bar\rho_b)}{\bar\rho_b(1+\bar\rho_a)}\right]$,} \\
\bar\rho_a - \frac{1}{2\sqrt{-\mathcal{A}^{\mathrm{SIP}(1)}}} \tanh \left( \frac{k}{L+1} \, \mathrm{arcsinh} \sqrt{-\omega^{\mathrm{SIP}(1)}}\right)
&\text{ if $-\ln \left[\frac{\bar\rho_a(1+\bar\rho_b)}{\bar\rho_b(1+\bar\rho_a)}\right] \le \lambda < 0$,} \\
\bar\rho_a - \frac{1}{2\sqrt{\mathcal{A}^{\mathrm{SIP}(1)}}} \tan \left( \frac{k}{L+1} \, \mathrm{arcsin} \sqrt{\omega^{\mathrm{SIP}(1)}}\right)
&\text{ if $0 \le \lambda < \ln \left(1 + \frac{1}{\rho_a}\right)$,}
\end{cases}\end{aligned}$$ where we defined $$\begin{aligned}
\mathcal{A}^{\mathrm{SIP}(1)} (\lambda,\bar\rho_a,\bar\rho_b)
&\equiv -\mathcal{A}^\mathrm{SPEP} (\lambda,-\bar\rho_a,-\bar\rho_b) \nonumber\\
&= \frac{(e^\lambda - 1) [e^\lambda(\bar\rho_b + 1) - \bar\rho_b]}
{4 [1 - (e^\lambda - 1)\bar\rho_a] [e^\lambda\bar\rho_a(\bar\rho_b+1)-\bar\rho_a\bar\rho_b-\bar\rho_b]}, \label{eq:sip1_A_saddle} \\
\omega^{\mathrm{SIP}(1)} (\lambda,\bar\rho_a,\bar\rho_b)
&\equiv -\omega^\mathrm{SPEP} (\lambda,-\bar\rho_a,-\bar\rho_b). \label{eq:sip_omega_mapping-spep}\end{aligned}$$ Note that this definition of $\omega^{\mathrm{SIP}(1)}$ yields . The expressions for $\bar\rho_a < \bar\rho_b$ are obtained by $\lambda \to -\lambda$ and an exchange of $\bar\rho_a$ and $\bar\rho_b$.
Finally, due to and , the scaled CGFs of the SPEP and the $\mathrm{SIP}(1)$ are related by $$\psi^{\mathrm{SIP}(1)}_L (\lambda,\bar\rho_a,\bar\rho_b) = -\psi^\mathrm{SPEP}_L (\lambda,-\bar\rho_a,-\bar\rho_b)\;,$$ from which it is straightforward to derive .
Remarkably, the $\mathrm{SIP}(1)$ has a finite range of $\lambda$, whereas the SPEP has an unbounded range of $\lambda$. This is related to the fact that the domain of $\mathrm{arcsin}$ is limited to $[-1,1]$, while that of $\mathrm{arcsinh}$ is unlimited. As will be discussed later, the limited range of $\lambda$ is closely related to the persistence of hydrodynamic behaviors for extreme current fluctuations. Meanwhile, it should be noted that the limited range of $\lambda$ does not imply a limited range of the current being considered. The time-averaged current $J$ conditioned on $\lambda$, obtained from $\partial \psi^{\mathrm{SIP}(1)}_L/\partial \lambda$, still ranges from $-\infty$ to $\infty$ for both SPEP and $\mathrm{SIP}(1)$. In fact, using standard Legendre transform arguments it is easy to check that the limited range of definition of the CGF $\psi^{\mathrm{SIP}(1)}_L(\lambda)$ corresponds to exponential tails of the current distribution function.
### $\mathrm{SIP}(1+\alpha)$
We now turn to the case of $\mathrm{SIP}(1+\alpha)$. Again, the Hamiltonian, given by , can be simplified by a canonical transformation $$\label{eq:sip1a_canonical}
\rho_k = F_k (1+F_k\hat{F}_k), \qquad {\hat\rho}_k = \frac{\hat{F}_k}{1+F_k \hat{F}_k} \\$$ or $$F_k = \frac{\rho_k}{1+\rho_k{\hat\rho}_k}, \quad \hat{F}_k = {\hat\rho}_k(1+\rho_k{\hat\rho}_k),$$ which was also used in [@Tailleur2008] in the context of the equivalent KMP model. This transforms the Hamiltonian into $$\begin{aligned}
\label{eq:sip1a_K}
K^{\mathrm{SIP}(1+\alpha)}_L(\lambda,\bar\rho_a,\bar\rho_b;\mathbf{F},\hat{\mathbf{F}})
&= \sum_{k=1}^{L-1} \left[(\hat{F}_{k+1} - \hat{F}_k)(F_k - F_{k+1})
+ \hat{F}_k\hat{F}_{k+1}(F_k-F_{k+1})^2\right] + \hat{F}_1 (\bar{\rho}_a - F_1) \nonumber\\
&\quad + \Big[\hat F_L - \lambda(1+F_L\hat F_L) \Big] \Big[\bar{\rho}_b - F_L - \bar{\rho}_b\lambda F_L\Big].\end{aligned}$$ A comparison between this expression and shows $$\label{eq:sip1a_K_sip1_K}
K^{\mathrm{SIP}(1+\alpha)}_L (\lambda,\bar\rho_a,\bar\rho_b;\mathbf{F},\hat{\mathbf{F}})
= \lim_{N \to \infty} K^{\mathrm{SIP}(1)}_L
(N^{-\alpha}\lambda,N^\alpha\bar\rho_a,N^\alpha\bar\rho_b;N^\alpha\mathbf{F},N^{-\alpha}\hat{\mathbf{F}}).$$ The time-independent saddle-point equations of these Hamiltonians show that the optimal profiles of the $\mathrm{SIP}(1)$ and the $\mathrm{SIP}(1+\alpha)$ are related by (again dropping $^*$) $$\begin{aligned}
\label{eq:sip1a_sip1_mapping}
(\hat F_k)^{\mathrm{SIP}(1+\alpha)} (\lambda,\bar\rho_a,\bar\rho_b)
&= \lim_{N \to \infty} N^\alpha (\hat F_k)^{\mathrm{SIP}(1)} (N^{-\alpha}\lambda,N^\alpha\bar\rho_a,N^\alpha\bar\rho_b), \nonumber\\
(F_k)^{\mathrm{SIP}(1+\alpha)} (\lambda,\bar\rho_a,\bar\rho_b)
&= \lim_{N \to \infty} N^{-\alpha} (F_k)^{\mathrm{SIP}(1)} (N^{-\alpha}\lambda,N^\alpha\bar\rho_a,N^\alpha\bar\rho_b).\end{aligned}$$ Therefore, for $\bar\rho_a \ge \bar\rho_b$ the optimal profiles are obtained as $$\begin{aligned}
\label{eq:sip1a_profiles}
\hat F_k^{\mathrm{SIP}(1+\alpha)} &=
\begin{cases}
-\sqrt{\mathcal{A}^{\mathrm{SIP}(1+\alpha)}} \sin \left( \frac{2k}{L+1} \, \mathrm{arcsin} \sqrt{\omega^{\mathrm{SIP}(1+\alpha)}}\right)
&\text{ if $-\frac{1}{\bar\rho_b} < \lambda < \frac{1}{\bar\rho_a} - \frac{1}{\bar\rho_b}$,} \\
-\sqrt{\mathcal{-A}^{\mathrm{SIP}(1+\alpha)}} \sinh \left( \frac{2k}{L+1} \, \mathrm{arcsinh} \sqrt{-\omega^{\mathrm{SIP}(1+\alpha)}}\right)
&\text{ if $\frac{1}{\bar\rho_a} - \frac{1}{\bar\rho_b} \le \lambda < 0$,} \\
\sqrt{\mathcal{A}^{\mathrm{SIP}(1+\alpha)}} \sin \left( \frac{2k}{L+1} \, \mathrm{arcsin} \sqrt{\omega^{\mathrm{SIP}(1+\alpha)}}\right)
&\text{ if $0 \le \lambda < \frac{1}{\bar\rho_a}$,}
\end{cases} \nonumber \\
F_k^{\mathrm{SIP}(1+\alpha)} &=
\begin{cases}
\bar\rho_a + \frac{1}{2\sqrt{\mathcal{A}^{\mathrm{SIP}(1+\alpha)}}} \tan \left( \frac{k}{L+1} \, \mathrm{arcsin} \sqrt{\omega^{\mathrm{SIP}(1+\alpha)}}\right)
&\text{ if $-\frac{1}{\bar\rho_b} < \lambda < \frac{1}{\bar\rho_a} - \frac{1}{\bar\rho_b}$,} \\
\bar\rho_a - \frac{1}{2\sqrt{-\mathcal{A}^{\mathrm{SIP}(1+\alpha)}}} \tanh \left( \frac{k}{L+1} \, \mathrm{arcsinh} \sqrt{-\omega^{\mathrm{SIP}(1+\alpha)}}\right)
&\text{ if $\frac{1}{\bar\rho_a} - \frac{1}{\bar\rho_b} \le \lambda < 0$,} \\
\bar\rho_a - \frac{1}{2\sqrt{\mathcal{A}^{\mathrm{SIP}(1+\alpha)}}} \tan \left( \frac{k}{L+1} \, \mathrm{arcsin} \sqrt{\omega^{\mathrm{SIP}(1+\alpha)}}\right)
&\text{ if $0 \le \lambda < \frac{1}{\bar\rho_a}$,}
\end{cases}\end{aligned}$$ where we defined $$\begin{aligned}
\mathcal{A}^{\mathrm{SIP}(1+\alpha)} (\lambda,\bar\rho_a,\bar\rho_b)
&\equiv \lim_{N \to \infty} N^{2\alpha}\mathcal{A}^{\mathrm{SIP}(1)}
(N^{-\alpha}\lambda,N^\alpha\bar\rho_a,N^\alpha\bar\rho_b) \nonumber\\
&= \frac{\lambda(1+\lambda\rho_b)}{4 (1-\lambda\rho_a) (\rho_a - \rho_b + \lambda \rho_a \rho_b)}, \label{eq:sip1a_A_saddle} \\
\omega^{\mathrm{SIP}(1+\alpha)}
&\equiv \lim_{N \to \infty} \omega^{\mathrm{SIP}(1)} (N^{-\alpha}\lambda,N^\alpha\bar\rho_a,N^\alpha\bar\rho_b).\end{aligned}$$ Note that this definition of $\omega^{\mathrm{SIP}(1+\alpha)}$ leads to .
Using and , the scaled CGFs of the $\mathrm{SIP}(1)$ and the $\mathrm{SIP}(1+\alpha)$ are related by $$\psi^{\mathrm{SIP}(1+\alpha)}_L (\lambda,\bar\rho_a,\bar\rho_b)
= \lim_{N \to \infty} \psi^{\mathrm{SIP}(1)}_L (N^{-\alpha}\lambda,N^\alpha\bar\rho_a,N^\alpha\bar\rho_b),$$ which gives . As in the case of the $\mathrm{SIP}(1)$, $\mathrm{SIP}(1+\alpha)$ also has a finite range of $\lambda$, although the range of the current $J$ is unbounded.
Comparison with hydrodynamic results {#ssec:sip_hydro}
------------------------------------
For both $\mathrm{SIP}(1)$ and $\mathrm{SIP}(1+\alpha)$, in the $L \to \infty$ limit, the hydrodynamic expression of the scaled CGF can be written in a similar form $$\label{eq:sip_cgf_hydro}
\psi^\mathrm{SIP}(\lambda) = \begin{cases}
\frac{1}{L+1} \mathrm{arcsin}^2 \sqrt{\omega^\mathrm{SIP}} &\text{if $\omega^\mathrm{SIP} \ge 0$}, \\
-\frac{1}{L+1} \mathrm{arcsinh}^2 \sqrt{-\omega^\mathrm{SIP}} &\text{if $\omega^\mathrm{SIP} < 0$},
\end{cases}$$ where we used the superscript $\mathrm{SIP}$ to refer to both large-$N$ models. This expression is in agreement with the corresponding expression for the KMP model found in [@Imparato2009]. When $\lambda$ is fixed, the hydrodynamic limit is reached by $$\psi^\mathrm{SIP}_L(\lambda) - \psi^\mathrm{SIP}(\lambda) = -\frac{\mathrm{arcsin}^4\sqrt{\omega^\mathrm{SIP}}}{3L^3} + O(L^{-4}),$$ as illustrated in Fig. \[fig:sip\_cgf\]. In contrast to the SPEP, the lattice structure decreases the magnitude of fluctuations. Since the range of $\lambda$ is bounded and the two CGFs converge to each other throughout this range, we cannot find any scaling of $\lambda$ with $L$ that induces non-hydrodynamic current fluctuations.
\
In order to understand why hydrodynamic behaviors are still observed for arbitrarily large current fluctuations, we examine the optimal profiles of the $\mathrm{SIP}(1)$ as was done for the SPEP. Using the same argument applied to the SPEP, the time-averaged current $J$ of $\mathrm{SIP}(1)$ can be related to the optimal profiles by $$\begin{aligned}
\label{eq:sip1_J_profiles}
J &= \frac{1}{L+1}\sum_{k=0}^L\left[(\rho_k - \rho_{k+1})+\rho_k(1+\rho_{k+1})(e^{{\hat\rho}_{k+1}-{\hat\rho}_k}-1)
- \rho_{k+1}(1+\rho_k)(e^{{\hat\rho}_k-{\hat\rho}_{k+1}}-1)\right] \nonumber\\
&= \underbrace{\frac{\bar\rho_a-\bar\rho_b}{L+1}}_{= \langle J \rangle} + \underbrace{\frac{1}{L+1}
\sum_{k=0}^L\left[\rho_k(1+\rho_{k+1})(e^{{\hat\rho}_{k+1}-{\hat\rho}_k}-1) - \rho_{k+1}(1+\rho_k)(e^{{\hat\rho}_k-{\hat\rho}_{k+1}}-1)\right]}_{= \delta J},\end{aligned}$$ where $\delta J$ becomes dominant as $\lambda$ approaches its upper and lower bounds $$\lambda_\mathrm{max} = \ln \left(1 + \frac{1}{\rho_a}\right), \quad \lambda_\mathrm{min} = -\ln \left(1+\frac{1}{\rho_b}\right).$$ For convenience, let us denote by $\Delta \lambda$ both $|\lambda - \lambda_\mathrm{min}|$ and $|\lambda - \lambda_\mathrm{max}|$. As shown in Fig. \[fig:sip1\_profile\], a large $\delta J$ is supported by a growing density crest and a flattening momentum profile as $\Delta \lambda \to 0$. As the data collapses in Fig. \[fig:sip1\_profile\_collapse\] indicate, as $L \to \infty$, the optimal profiles have scaling forms $$\begin{aligned}
\label{eq:sip1_profile_scaling}
\rho_k(\Delta\lambda,L) &= \Delta\lambda^{-1/2} f(k/L), \nonumber\\
{\hat\rho}_k(\Delta\lambda,L) &= \Delta\lambda^{1/2} g(k/L).\end{aligned}$$ These imply that we can approximate $\delta J$ as $$\begin{aligned}
\label{eq:sip1_J_hydro}
\delta J &\simeq \frac{1}{L+1}\sum_{k=0}^L[\rho_k(1+\rho_{k+1}) + \rho_{k+1}(1+\rho_k)]({\hat\rho}_{k+1}-{\hat\rho}_k) \nonumber\\
&\simeq \frac{1}{\Delta\lambda^{1/2}L} \int_0^1 {\mathrm{d}}x\, 2 \rho(1+\rho) \partial_x g,\end{aligned}$$ which has an integral form for any small $\Delta \lambda$ corresponding to large $J$. Thus, $J$ exhibits hydrodynamic behaviors for arbitrarily large $J$. An almost identical argument also applies to the $\mathrm{SIP}(1+\alpha)$, whose optimal profiles have similar shapes and satisfy the scaling relation .
Finite-$N$ effects {#ssec:sip_fin_N}
------------------
### $\mathrm{SIP}(1)$
From and , we observe that the Hamiltonians of the SPEP and the $\mathrm{SIP}(1)$ are related by $$H^{\mathrm{SIP}(1)}_L (\lambda,\bar\rho_a,\bar\rho_b;{\boldsymbol{\rho}},{\hat{\boldsymbol{\rho}}}) = -H^\mathrm{SPEP}_L(\lambda,-\bar\rho_a,-\bar\rho_b;-{\boldsymbol{\rho}},{\hat{\boldsymbol{\rho}}}).$$ This suggests that the leading finite-$N$ correction to the scaled CGF $\psi^{\mathrm{SIP}(1)}_L$ can be obtained by a Gaussian approximation very similarly to the one applied to the SPEP in Sec. \[ssec:spep\_fin\_N\]. Consequently, the leading finite-$N$ correction is described by analogs of and , namely $$\psi^{\mathrm{SIP}(1)}_{N,L}(\lambda) = \psi^{\mathrm{SIP}(1)}_L(\lambda) + N^{-1} \psi^{1,\mathrm{SIP}(1)}_L(\lambda) + o(N^{-1})$$ with $$\begin{aligned}
\psi^{1,\mathrm{SIP}(1)}_L(\lambda,\bar\rho_a,\bar\rho_b) &= -\psi^{1,\mathrm{SPEP}}_L(\lambda,-\bar\rho_a,-\bar\rho_b) \nonumber\\
&=
-\sum_{p=1}^{L'-1}
\bigg\{
\boldsymbol c_\lambda
-
\cos\frac{p\pi}{2L'}
\sqrt{\frac 12 \boldsymbol c_\lambda\big(\boldsymbol c_\lambda+\cos\frac{p\pi}{L'}\big)}
-
\sin\frac{p\pi}{2L'}
\sqrt{\frac 12 \boldsymbol c_\lambda\big(\boldsymbol c_\lambda-\cos\frac{p\pi}{L'}\big)}
\bigg\},
\label{eq:respsiLhalfdensities_sym_SIP}\end{aligned}$$ where $\boldsymbol c_\lambda = \cos \frac{2\operatorname{arcsin}\sqrt{\omega^{\text{SIP}(1)}}}{L+1}$. The correction is again an analytic function of $\lambda$ within its domain, which proves the validity of the additivity principle with respect to continuous transitions, without ruling out possible discontinuous ones.
In the large-$L$ limit, using and the first equality of , we can also write $$\begin{aligned}
\psi_L^{1,\mathrm{SIP}(1)}(\lambda)
&=
-\frac 1{8L^2} \mathcal F\big(\nu(\lambda)\big) \ + \
O(L^{-3}),
\label{eq:sip_largeLpsi1L}\end{aligned}$$ with $\mathcal F(u)$ the universal scaling function defined in and $\nu(\lambda)=\operatorname{arcsin}^2\sqrt{\omega^{\text{SIP}(1)}}$. While $\mathcal F(u)$ is singular at $u_\text{c}=\pi^2/2$, $\nu(\lambda)$ cannot be greater than $\pi^2/4$ for any real-valued $\lambda$ in . This also confirms the validity of the additivity principle with respect to continuous transitions at large $L$.
### $\mathrm{SIP}(1+\alpha)$
Unlike the previous models, the leading finite-$N$ correction to $\psi^{\mathrm{SIP}(1+\alpha)}_L$ comes from a different origin. For this model, if we keep the leading finite-$N$ correction, the path integral in can be rewritten as $$e^{NT\psi^{\mathrm{SIP}(1+\alpha)}_{N,L}}
= \int {\mathcal{D}}{\boldsymbol{\rho}}{\mathcal{D}}{\hat{\boldsymbol{\rho}}}\, \exp\left\{-N\int_{0}^{T} \mathrm{d}t \,
\left[ {\hat{\boldsymbol{\rho}}}\cdot\dot{{\boldsymbol{\rho}}} - H^{\mathrm{SIP}(1+\alpha)}_L - N^{-\alpha}V^{\mathrm{SIP}(1+\alpha)}_L \right]\right\},$$ where $$\label{eq:sip1a_V_rho}
V^{\mathrm{SIP}(1+\alpha)}_L(\lambda,\bar\rho_a,\bar\rho_b;{\boldsymbol{\rho}},{\hat{\boldsymbol{\rho}}})
= \sum_{k=1}^{L-1} \frac{\rho_k + \rho_{k+1}}{2}\,({\hat\rho}_{k+1} - {\hat\rho}_k)^2
+ \frac{\bar\rho_a + \rho_1}{2}\,{\hat\rho}_1^2 + \frac{\bar\rho_b + \rho_L}{2}\,({\hat\rho}_L - \lambda)^2.$$ Applying a saddle-point approximation as before, we obtain $$\psi^{\mathrm{SIP}(1+\alpha)}_{N,L}(\lambda) = \psi^{\mathrm{SIP}(1+\alpha)}_L(\lambda) + N^{-\alpha} \psi^{1,\mathrm{SIP}(1+\alpha)}_L(\lambda) + o(N^{-\alpha})$$ with $$\label{eq:sip1a_saddle_rho}
\psi^{1,\mathrm{SIP}(1+\alpha)}_L(\lambda) = V^{\mathrm{SIP}(1+\alpha)}_L(\lambda;{\boldsymbol{\rho}}^*,{\hat{\boldsymbol{\rho}}}^*),$$ where ${\boldsymbol{\rho}}^*$ and ${\hat{\boldsymbol{\rho}}}^*$ are the optimal profiles determined in Sec. \[ssec:sip\_cgf\].
### Numerical results
We numerically confirm our theoretical predictions by constructing a matrix representation of the SIP conditioned on $\lambda$. Since it is impossible to implement the unbounded configuration space of this model, we introduce an artificial upper bound $M$ on the number of particles in each site. The matrix representation is such that any transition that violates this upper bound is forbidden, while the other transitions occur with the same rates as the original dynamics. We expect that if $M$ is sufficiently large, the effects of $M$ become irrelevant. The results for $\mathrm{SIP}(1)$ and $\mathrm{SIP}(1.5)$ shown in Fig. \[fig:sip\_finN\] are both in agreement with our predictions.
Criterion for persistent hydrodynamic behaviors {#sec:criterion}
===============================================
We have shown that current fluctuations of the SPEP have a non-hydrodynamic regime, while those of the SIP always behave according to the predictions of the hydrodynamic equations. As noted in Sec. \[sec:models\], one important difference between the SPEP and the SIP lies in whether the mobility coefficient $\sigma(\rho)$ is bounded from above. This suggests a connection between the presence of an upper bound on $\sigma(\rho)$ and hydrodynamic behaviors of current fluctuations. In order to investigate this connection, we examine how the optimal profiles depend on the time-averaged current $J$ within the naive hydrodynamic regime given by $J = O(L^{-1})$. An extrapolation of this dependence beyond the regime ([*i.e.*]{}, $J$ larger than $O(L^{-1})$) reveals whether non-hydrodynamic behaviors appear for sufficiently large $J$.
In the hydrodynamic limit, from Hamilton’s equations we have $$\label{eq:hamilton_hydro}
\frac{\partial \rho}{\partial t} = \frac{\delta H}{\delta \rho} = \partial_x \left[ D(\rho)\partial_x \rho - \sigma(\rho)\partial_x {\hat\rho}\right]$$ with $H$ given by . This gives a relation between $J$ and the optimal profiles through $$\begin{aligned}
\label{eq:J_profiles_hydro}
J = \frac{1}{L+1} \int_0^1 \mathrm{d}x\, \left[-D(\rho^*)\partial_x \rho^* + \sigma(\rho^*)\partial_x {\hat\rho}^*\right]
= \frac{G(\bar\rho_a) - G(\bar\rho_b)}{L+1} + \frac{1}{L+1} \int_0^1 \mathrm{d}x\, \left[\sigma(\rho^*)\partial_x {\hat\rho}^*\right],\end{aligned}$$ where $D(\rho) = G'(\rho)$. Then, as long as $G(\bar\rho_a)$ and $G(\bar\rho_b)$ are finite, $J$ beyond the naive hydrodynamic regime satisfies $$\begin{aligned}
\label{eq:J_profiles_hydro_approx}
J \simeq \frac{1}{L+1} \int_0^1 \mathrm{d}x\, \left[\sigma(\rho^*)\partial_x {\hat\rho}^*\right].\end{aligned}$$ In other words, in this regime $J$ is sensitive only to $\sigma(\rho^*)$ and $\partial_x {\hat\rho}^*$.
Note that when $\sigma(\rho)$ is bounded from above, an arbitrarily large $J$ can only be supported by an arbitrarily large $\partial_x {\hat\rho}^*$. This means that $\partial_x {\hat\rho}^*$ can no longer be expressed as a proper gradient for a sufficiently large $J$, in which case $J$ exhibits non-hydrodynamic behaviors, as was the case for the SPEP. Hence, the absence of an upper bound on $\sigma(\rho)$ is clearly a necessary condition for the persistence of hydrodynamic behaviors.
When $\sigma(\rho)$ is not bounded from above, a large $J$ can be supported by a large $\sigma(\rho^*)$ while $\partial_x {\hat\rho}^*$ remains well defined, so that $J$ is still blind to the lattice structure. Based on this possible scenario, we conjecture that the absence of an upper bound on $\sigma(\rho)$ is also a sufficient condition for the persistence of hydrodynamic behaviors. Although there is no rigorous proof yet, we can confirm this conjecture for the one-dimensional symmetric zero-range process, which provides a simple example of boundary-driven systems with non-constant $D(\rho)$ and unbounded $\sigma(\rho)$. An interested reader is referred to Appendix \[app:zrp\] for more discussions on this model.
Conclusions {#sec:conclusions}
===========
In this paper we introduced a class of large-$N$ models for one-dimensional boundary-driven diffusive systems. Using $N$ as a large parameter, we were able to obtain exact expressions for current large deviations on a finite lattice, without relying on a hydrodynamic approach. This allowed us to look at regimes where the hydrodynamic theory is naively expected to break down. Surprisingly, we found that there are classes of models, which we conjecture to be those with an unbounded $\sigma(\rho)$ as a function of $\rho$, where the predictions of the hydrodynamic theory always hold. It will be interesting to see if similar considerations also hold for models with a bulk bias and/or for large deviations of other additive observables, such as the activity.
In addition, we examined the finite-$N$ corrections and used them to argue that the additivity principle, assumed throughout the paper, is likely to hold for the models considered.
[*Acknowledgments:*]{} We are grateful for discussions with B. Derrida, M. R. Evans, B. Meerson, T. Sadhu, and H. Spohn. YB and YK were supported by an Israeli-Science-Foundation grant. YB is supported in part at the Technion by a fellowship from the Lady Davis Foundation. VL wishes to thank the hospitality of the Physics Department of Technion, Haifa, where part of the research was performed, and acknowledges support from LAABS Inphyniti CNRS project.
Current fluctuations in the hydrodynamic limit {#app:hydro_limit}
==============================================
The limited range of current fluctuations in the hydrodynamic limit, that we discuss in the Introduction, can also be seen from an argument more directly based on the MFT. The scaled CGF $\psi(\lambda)$ for current fluctuations is defined by $$e^{T_\mathrm{micro}\psi(\lambda)} = {\left\langle}e^{\lambda T_\mathrm{micro} J} {\right\rangle},$$ where $J$ is the mean current across a certain cross-section of the system averaged over a microscopic time interval $t \in [0, T_\mathrm{micro}]$. In the hydrodynamic limit, this expression can be written in a path integral form $${\left\langle}e^{\lambda T_\mathrm{micro} J} {\right\rangle}= \int {\mathcal{D}}\rho {\mathcal{D}}{\hat\rho}\, e^{-\ell \int_0^{T} {\mathrm{d}}t\, \mathcal{L}[\lambda;\rho,{\hat\rho}]},$$ where $T = T_\mathrm{micro}/\ell^2$ is the length of the time interval on a macroscopic scale. For $\ell \gg 1$, we can apply a saddle-point approximation to obtain $$\psi(\lambda) = -\frac{1}{T\ell} \inf_{\rho,{\hat\rho}} \int_0^{T} {\mathrm{d}}t\, \mathcal{L}[\lambda;\rho,{\hat\rho}].$$ The minimum action always has the form $$\inf_{\rho,{\hat\rho}} \int_0^{T} {\mathrm{d}}t\, \mathcal{L}[\lambda;\rho,{\hat\rho}] = -T f(\lambda),$$ from which we obtain $$\psi(\lambda) = \ell^{-1} f(\lambda).$$ Note that the Lagrange multiplier $\lambda$ and the conjugate current $J$ are related by $$J = \psi'(\lambda) = \ell^{-1} f'(\lambda).$$ Thus we recover the conclusion that $O(\ell^{-1})$ current fluctuations belong to the hydrodynamic regime.
Path Integral representation of the CGF {#app:path}
=======================================
These statistics of the current are encoded in the scaled cumulant generating function (CGF) $\psi_L$, which is defined by $$\label{eq:cgf_def_app}
e^{NT\psi_L(\lambda,\bar\rho_a,\bar\rho_b)} = {\left\langle}e^{N\lambda T J}{\right\rangle}= {\left\langle}e^{N^{-\alpha}\lambda \sum_s I_{L,L+1}(t_s)}{\right\rangle}.$$ Note that in the last step we divided the time interval $[0,\,T]$ into $M$ infinitesimal subintervals of length $\Delta t$, so that $t_s = s\Delta t$ for $s = 1,\,2,\,\ldots,\,M$, and $T = M\Delta t$. We also introduced the notation $$I_{k,k+1}(t_s) = \begin{cases}
+1 &\text{if a particle hops from $k$ to $k+1$ at $t \in [t_s,t_{s+1}]$}\\
-1 &\text{if a particle hops from $k+1$ to $k$ at $t \in [t_s,t_{s+1}]$}\\
0 &\text{otherwise}
\end{cases}$$ for $0 \le s \le M-1$.
From , the scaled CGF can be expressed in a path integral form $$\begin{aligned}
\label{eq:cgf_path_integ}
&e^{NT\psi_L(\lambda)} = {\left\langle}\int \prod_{s,\,k} \left[{\mathrm{d}}\rho_k(t_s) \,
\delta\left(\rho_k(t_{s+1}) - \rho_k(t_{s}) - \frac{I_{k-1,k}(t_s) - I_{k,k+1}(t_s)}{N^{1+\alpha}}\right)\right]
e^{N^{-\alpha} \lambda I_{L,L+1}(t_s)}{\right\rangle}\nonumber \\
&~ = \int \prod_{s} {\left\langle}\prod_{k = 1}^L\left[{\mathrm{d}}\rho_k(t_s) {\mathrm{d}}\hat \rho_k(t_s) \,
e^{-N\hat \rho_k(t_s)\left(\rho_k(t_{s+1}) - \rho_k(t_{s}) - \frac{I_{k-1,k}(t_s) - I_{k,k+1}(t_s)}{N^{1+\alpha}}\right)}\right]
e^{N^{-\alpha}\lambda I_{L,L+1}(t_s)}{\right\rangle}_{\mathbf{I}(t_s)},\end{aligned}$$ which has the standard Martin–Siggia–Rose (MSR) form [@Martin1973; @*Janssen1976; @*DeDominicis1976; @*DeDominicis1978] with auxiliary field variables ${\hat\rho}_1, {\hat\rho}_2, \ldots, {\hat\rho}_L$.
$\mathbf{I}(t_s) = \left(I_{0,1}(t_s), \ldots, I_{k,k+1}(t_s), \ldots, I_{L,L+1}(t_s)\right)$ represents a hopping event within the time interval $[t_s, t_{s+1}]$. The probability distribution of $\mathbf{I}(t_s)$ is given by $$\begin{aligned}
\mathbf{I}(t_s) = \begin{cases}
\left(0, \ldots, 0, I_{k,k+1} = 1, 0, \ldots, 0 \right) &\text{with prob. $n_k (N \mp n_{k+1}) \frac{\Delta t}{N}$}, \\
\left( 0, \ldots, 0, I_{k,k+1} = -1,0, \ldots,0 \right) &\text{with prob. $n_{k+1} (N \mp n_k) \frac{\Delta t}{N}$}, \\
\left(0,0,\ldots,0\right) &\text{with prob. $1 - \sum_k \left[N(n_k + n_{k+1}) \mp 2n_k n_{k+1}\right] \frac{\Delta t}{N}$},
\end{cases}\end{aligned}$$ where $k$ is any integer between $0$ and $L+1$, and the upper (lower) rates correspond to the SPEP (SIP). Choosing an appropriate value of the scaling exponent $\alpha$, we can evaluate the average $\langle \cdot \rangle_{\mathbf{I}(t_s)}$ (see [@Lefevre2007] for a general description of this procedure) and single out the leading-order component to obtain $$\label{eq:macro_path_integ_app}
e^{NT\psi_L(\lambda)} = \int {\mathcal{D}}{\boldsymbol{\rho}}{\mathcal{D}}{\hat{\boldsymbol{\rho}}}\,
\exp\left\{-N\int_{0}^{T} \mathrm{d}t \,\left[ {\hat{\boldsymbol{\rho}}}\cdot\dot{{\boldsymbol{\rho}}}
- H_L(\lambda;{\boldsymbol{\rho}},{\hat{\boldsymbol{\rho}}}) \right]\right\},$$ where the function $H_L$ contains all information about the dynamics. Note that the same result can also be derived by a different path-integral construction using $\mathrm{SU}(2)$ (for SPEP) or $\mathrm{SU}(1,1)$ (for SIP) coherent states, extending the one proposed in [@Tailleur2008] to the case of current large deviations.
Finite-$N$ corrections to the CGF for the SPEP arising from space-time fluctuations {#app:finiteNquantumfluctuationsSPEP}
===================================================================================
In this Appendix, we derive the leading finite-$N$ corrections to the CGF for the SPEP that we obtained in Sec. \[ssec:ldfSPEPderivation\] by a large-$N$ saddle-point approach. This analysis generalizes the MFT results [@Imparato2009] to the case of the lattice SPEP with finite $L$ and also allows us to discard the existence of a continuous phase transition in the CGF as $\lambda$ is varied, thus (partially) supporting the validity of the additivity principle for the SPEP. To avoid cumbersome expressions, in the following we drop the superscript SPEP.
Mapping to reservoirs at half densities
---------------------------------------
Determining the finite-$N$ corrections in principle amounts to integrating the quadratic fluctuations around the saddle-point solutions shown in . This is however rendered difficult by the nontrivial dependence of those solutions on the spatial index $k$ of the lattice. To bypass this issue, we generalize the approach presented in [@Lecomte2010]: we map the CGF (taken at $\lambda$) of a system in contact with reservoirs at densities $\bar\rho_a$ and $\bar\rho_b$ to the CGF of a system in contact with reservoirs at same densities $\frac 12$, but taken at a different value of $\lambda$: $$\psi_{N,L}(\lambda,\bar\rho_a,\bar\rho_b) =
\psi_{N,L}\big(\lambda=2\operatorname{arcsinh} \sqrt{\omega},\bar\rho_a=\tfrac12,\bar\rho_b=\tfrac12\big)
\label{eq:mmappingpsiLfiniteN}$$ This result arises from the $\mathrm{SU}(2)$ symmetry of the generating operator, whose eigenvalue of maximal real part yields the CGF: (*i*) the bulk part of this operator is left invariant by a $\mathrm{SU}(2)$ rotation as in [@Lecomte2010], but with spin $\frac N2$ instead of spin $\frac 12$ ; (*ii*) the terms describing the contact with reservoirs are affected by the rotation and yield for a well-chosen rotation.
The main advantage of this transformation is that at half densities $\bar\rho_a=\bar\rho_b=\tfrac12$, the saddle-point solutions shown in take a simple form: one has $$\hat F_k^*=\sinh\frac{\lambda k}{L+1},
\qquad
F_k^*=\frac 12-\frac 12\tanh \frac{\lambda k}{2(L+1)}.$$ In terms of the original variables ${\boldsymbol{\rho}}^*$ and ${\hat{\boldsymbol{\rho}}}^*$, the canonical transformation of gives $$\rho_k^*=\frac 12,
\qquad
\hat\rho_k^*=\lambda\frac{k}{L+1},
\label{eq:rhohatrhoksaddle}$$ which shows that the optimal density profile is flat, while the optimal momentum profile is linear. We note that the same behavior was also observed in the hydrodynamic limit [@Bodineau2004].
Small space-time fluctuations around saddle-point:
--------------------------------------------------
We thus first focus on the half-density case. One looks for a space-time perturbation around the saddle-point solutions of the form $$\rho_k(t)=\rho_k^* + N^{-\frac 12}\phi_k(t),
\qquad
\hat\rho_k(t)=\hat\rho_k^* + N^{-\frac 12} \hat\phi_k(t),
\label{eq:deffluctphi}$$ with $\phi_k(t)$ and $\hat\phi_k(t)$ of order $N^0$. The prefactor $N^{-\frac 12}$ is chosen so that when substituting into the action for ${\boldsymbol{\rho}}$, ${\hat{\boldsymbol{\rho}}}$ the temporal contribution to the action is $-\int_0^Tdt\,\hat{\boldsymbol \phi}\cdot\partial_t \boldsymbol \phi$, whose absence of prefactor facilitates further analysis. Expanding in powers of $N$, the total Hamiltonian decomposes as $$H_L(\lambda;{\boldsymbol{\rho}},{\hat{\boldsymbol{\rho}}})
=
\underbrace{
H_\lambda^\text{saddle}
}_{\textnormal{order $N^{0}$}}
+
\underbrace{
H_\lambda^\text{fluct}(\boldsymbol \phi, \hat {\boldsymbol \phi})
}_{\textnormal{order $N^{-1}$}}
\: + \ {\text{higher order terms}},$$ with $$H_\lambda^\text{saddle}
=H_L(\lambda;\boldsymbol \rho^*, \hat {\boldsymbol \rho}^*)
=(L+1)\sinh^2\frac{\lambda}{2(L+1)}$$ yielding the dominant contribution $\psi_L(\lambda)$ to the full CGF $\psi_{N,L}$. Meanwhile, $H_\lambda^\text{fluct}(\boldsymbol \phi, \hat {\boldsymbol \phi})$ has a quadratic form $$H_\lambda^\text{fluct}(\boldsymbol \phi, \hat {\boldsymbol \phi})
=
N^{-1}
\big(\begin{smallmatrix}
\boldsymbol \phi\\ \hat {\boldsymbol \phi}
\end{smallmatrix}\big)^T
\mathcal A \:
\big(\begin{smallmatrix}
\boldsymbol \phi\\ \hat {\boldsymbol \phi}
\end{smallmatrix}\big),
\label{eq:defmatAHfluct}$$ where $\mathcal A$ is a symmetric $2L\times 2L$ matrix defined by block structure $$\mathcal A =
\begin{pmatrix}
\mathcal A_{11} &\mathcal A_{12} \\ \mathcal A_{21}&\mathcal A_{22}
\end{pmatrix}
\label{eq:decompositionA2x2blocks}$$ with the $\mathcal A_{ij}$’s symmetric $L\times L$ matrices $$\begin{aligned}
\mathcal A_{11}
& = 2 \big(\tfrac12 \Delta + \mathbf 1\big) (1-\cosh\frac{\lambda}{L+1})
\\
\mathcal A_{12}=\mathcal A_{21}
&=\tfrac 12 \Delta\cosh\frac{\lambda}{L+1}
\\
\mathcal A_{22}=-\tfrac 12 \mathcal A_{12}
&=-\tfrac 14 \Delta\cosh\frac{\lambda}{L+1}.\end{aligned}$$ Here the $L\times L$ matrix $\Delta$ is the discrete Laplacian (with open boundaries)
$$\Delta
=
\begin{pmatrix}
-2&1 &0& &\ldots&0 \\
1&-2&1&0&\ldots&0 \\
0&\ddots&\ddots&\ddots&\ddots&0 \\
0&\ddots&\ddots&\ddots&\ddots&0 \\
0&\ldots&0&1&-2&1 \\
0&\ldots&&0&1&-2 \\
\end{pmatrix}.$$
The eigenvalues of $\Delta$ are $$\Delta_p = - 4 \cos^2\frac{p\pi}{2(L+1)}\;,
\qquad
1\leq p\leq L,$$ with corresponding orthonormal eigenvectors $$\mathbf V_p= \sqrt{\tfrac{2}{L+1}}\Big(\sin\frac{(L+1-p) k \pi}{L+1}\Big)_{1\leq k\leq L}.
\label{eq:defVpeigenvectorsDelta}$$
Corrections due to “quantum fluctuations”: mapping to independent bosons
------------------------------------------------------------------------
At the quadratic order, one has $$e^{NT\psi_{N,L}(\lambda)} \simeq
e^{N T H_\lambda^\text{saddle}}
\int\mathcal D\boldsymbol\phi \mathcal D \hat{\boldsymbol\phi}
\
e^{
-\int_0^T dt
\big[
\hat{\boldsymbol\phi}\cdot\partial_t\boldsymbol\phi
-N H_\lambda^\text{fluct}(\boldsymbol \phi, \hat {\boldsymbol \phi})
\big]\;.
}
\label{eq:PIQF}$$ To compute the path integral and evaluate the so-called “quantum fluctuations”, one can regard as a coherent-state path integral of a bosonic harmonic oscillator, whose ground state becomes dominant in the large $T$ limit. This leads to $$\psi_{N,L}(\lambda) = \psi_L(\lambda) - N^{-1} \min \operatorname{Sp} \mathbf H_\lambda + o(N^{-1}).
\label{eq:relationpsiLHoplambda}$$ The operator $\mathbf H_\lambda$ is such that its coherent-state path integral is given by ; thus, it can be written in the form $$-\mathbf H_\lambda =
\boldsymbol a^T \mathcal A_{11} \boldsymbol a+2{\boldsymbol a^\dag}^T \mathcal A_{12} \boldsymbol a+{\boldsymbol a^\dag}^T \mathcal A_{22} {\boldsymbol a^\dag},
\label{eq:defHopAij}$$ where the operators $a_1,\ldots,a_L$ are bosonic annihilation operators and $a^\dagger_1,\ldots,a^\dagger_L$ are their creation counterparts, with $[a_i,a_j^\dagger]=\delta_{ij}$. We remark that choosing a scaling other than $N^{-\frac 12}$ for the fluctuations in would leave the result (\[eq:relationpsiLHoplambda\]-\[eq:defHopAij\]) *unchanged* (the only important aspect being that the exponent is negative, allowing for a perturbation expansion).
The eigenvectors define an orthonormal matrix $O$ which renders the modes independent. Using $$O \Delta O^T = \operatorname{Diag}(\Delta_1,\ldots,\Delta_L)\equiv \widetilde\Delta,$$ one obtains $$\begin{aligned}
-\mathbf H_\lambda
&=
\boldsymbol a^TO^TO \mathcal A_{11}O^TO \boldsymbol a+\ldots
\\
&=
\boldsymbol b^T \widetilde {\mathcal A}_{11} \boldsymbol b+2{\boldsymbol b^\dag}^T \widetilde {\mathcal A}_{12} \boldsymbol b+{\boldsymbol b^\dag}^T \widetilde {\mathcal A}_{22} {\boldsymbol b^\dag},\end{aligned}$$ where $\boldsymbol b = O \boldsymbol a$ are new bosonic annihilation operators, and $\widetilde {\mathcal A}$ consists of four blocks as in , with each block $\widetilde {\mathcal A}_{ij}$ being a symmetric $L\times L$ matrix given by $$\begin{aligned}
\widetilde {\mathcal A}_{11}
& = 2 \big(\tfrac12 \widetilde\Delta + \mathbf 1\big) (1-\cosh\frac{\lambda}{L+1}),
\\
\widetilde {\mathcal A}_{12}=\widetilde {\mathcal A}_{21}
&=\tfrac 12 \widetilde\Delta\cosh\frac{\lambda}{L+1},
\\
\widetilde {\mathcal A}_{22}=-\tfrac 12 \widetilde {\mathcal A}_{12}
&=-\tfrac 14 \widetilde\Delta\cosh\frac{\lambda}{L+1}.\end{aligned}$$ Because each $\widetilde {\mathcal A}_{ij}$ is diagonal, the operator $\mathbf H_\lambda$ can be written as a sum of independent single-boson operators $$\begin{aligned}
\mathbf H_\lambda &= \sum_{p=1}^L \mathbf H_\lambda^{(p)},
\\
\mathbf H_\lambda^{(p)}
&=
\underbrace{
- 2 \big(\tfrac12 \Delta_p + 1\big) (1-\cosh\frac{\lambda}{L+1})}
_
{\equiv X_p}
b_p^2
\underbrace{
-
\Delta_p\cosh\frac{\lambda}{L+1}}
_
{\equiv 2 Z_p}
b^\dag_pb_p
+
\underbrace{
\tfrac 14 \Delta_p\cosh\frac{\lambda}{L+1}
}
_
{\equiv Y_p}
\big(b^\dag_p\big)^2.\end{aligned}$$ Besides (*e.g.*, through a generalized Bogoliubov transform), one finds that the ground state of every $\mathbf H_\lambda^{(p)}=X_pb_p^2+2Z_pb^\dag_pb_p+Y_p\big(b^\dag_p\big)^2$ (seen as a harmonic oscillator) is given by: $$\min \operatorname{Sp} \mathbf H_\lambda^{(p)}
=
\sqrt{Z_p^2-X_pY_p}-Z_p.$$
We finally obtain the result for $\bar\rho_a=\bar\rho_b=\frac 12$. Denoting $\boldsymbol c_\lambda=\cosh \frac{\lambda}{L+1}$, the correction $\psi_L^1(\lambda)$ in due to space-time fluctuations reads $$\begin{aligned}
\psi_L^1(\lambda)
&=
-\min \operatorname{Sp} \mathbf H_\lambda
\\
&=
-\sum_{p=1}^L \min \operatorname{Sp} \mathbf H_\lambda^{(p)}
\\
&=
-\sum_{p=1}^L
\bigg\{
\cos\frac{p\pi}{2(L+1)}
\sqrt{2\boldsymbol c_\lambda\big(\boldsymbol c_\lambda+\cos\frac{p\pi}{L+1}\big)}
- 2 \boldsymbol c_\lambda \cos^2\frac{p\pi}{2(L+1)}
\bigg\}
\label{eq:respsiLhalfdensities_app}\end{aligned}$$ For generic reservoir densities $\bar \rho_a$ and $\bar \rho_b$, one can use the mapping to find that the correction term still takes the form of , but now with $$\boldsymbol c_\lambda= \cosh
\frac{2\operatorname{arcsinh}\sqrt{\omega}}{L+1}.
$$
Averaging the $p$-th and $(L+1-p)$-th terms, the sum can be symmetrized as $$\begin{aligned}
\psi_L^1(\lambda)
&=
\sum_{p=1}^{L'-1}
\underbrace{
\bigg\{
\boldsymbol c_\lambda
-
\cos\frac{p\pi}{2L'}
\sqrt{\frac 12 \boldsymbol c_\lambda\big(\boldsymbol c_\lambda+\cos\frac{p\pi}{L'}\big)}
-
\sin\frac{p\pi}{2L'}
\sqrt{\frac 12 \boldsymbol c_\lambda\big(\boldsymbol c_\lambda-\cos\frac{p\pi}{L'}\big)}
\bigg\}
}_{\equiv \Sigma_{L'}(p)},
\label{eq:respsiLhalfdensities_sym_final}\end{aligned}$$ where we used a notation $L'=L+1$. This is our final result for the leading finite-$N$ correction to the CGF $\psi_L(\lambda)$. One checks that this expression is an analytic function of $\lambda$ at all system size $L$, indicating that the optimal profiles ${\boldsymbol{\rho}}^*,{\hat{\boldsymbol{\rho}}}^*$ are stable with respect to any small perturbations in space and time. This is consistent with the hypothesis of additivity that we assumed to derive ${\boldsymbol{\rho}}^*,{\hat{\boldsymbol{\rho}}}^*$, but there is still the possibility of discontinuous transitions (which, if they exist, can also be ruled out provided that they have a spinodal). Taking the large $L$ limit is not straightforward because (*i*) the summand $\Sigma_{L'}(p)$ in exhibits different scaling with $L$ depending on the value of $p$, and (*ii*) the range of $p$ itself depends on $L$. In particular, the sum cannot be approximated by a Riemann integral because the summand $\Sigma_{L'}(p)$, seen as function of a continuous variable $p \in (0,L')$, is not an analytic function. In fact, remains a discrete sum even in the large-$L$ limit, as we now explain.
We first note that for $L'$ even, $\Sigma_{L'}(L'/2)=0$. For any $L'$, thanks to the symmetry $\Sigma_{L'}(p)=\Sigma_{L'}(L'-p)$, one can thus restrict the sum as follows: $$\begin{aligned}
\psi_L^1(\lambda)
&=
2\sum_{p=1}^{\lfloor L'/2 \rfloor}
\Sigma_{L'}(p)\end{aligned}$$ At fixed $\lambda$, the leading-order term in $L'\to\infty$ gives (provided $1\leq p\leq\lfloor L'/2 \rfloor$) $$\Sigma_{L'}(p) =
\frac{1}{L'^2}
\frac 14
\Big\{
(p\pi)^2-\mu(\lambda)-p\pi\sqrt{(p\pi)^2+2\mu(\lambda) }
\Big\}
\ + \
O(L^{-3}),$$ with $\mu(\lambda)=\operatorname{arcsinh}^2\sqrt{\omega}$. Then, using the Euler-Maclaurin summation formula to control the rest (*i.e.*, the terms with $p>L'/2$), one finds $$\begin{aligned}
\psi_L^1(\lambda)
&= \frac{1}{2L^2}
\sum_{p=1}^{\infty}
\Big\{
(p\pi)^2-\mu(\lambda)-p\pi\sqrt{(p\pi)^2+2\mu(\lambda) }
\Big\}
\ + \
O(L^{-3})
\label{eq:respsi1L_largeL}
\\
&=
\frac 1{8L^2} \mathcal F\big(\!-\!\mu(\lambda)\big) \ + \
O(L^{-3}),\end{aligned}$$ where we recognize the universal scaling function $$\begin{aligned}
\mathcal F(u)
&= 4 \sum_{p=1}^{\infty}
\Big\{
(p\pi)^2+u-p\pi\sqrt{(p\pi)^2-2u }
\Big\}\end{aligned}$$ appearing in MFT and Bethe-Ansatz studies of current fluctuations [@appert-rolland_universal_2008; @prolhac_cumulants_2009]. The large-$L$ limit (at fixed $\lambda$) thus yields the same correction as does the MFT approach [@Imparato2009] for the SSEP. The finite-$L$ result however allows one to study large deviations regimes with $\lambda$ increasing as a function of $L$, which are not described by .
Another illustration is obtained by a direct expansion of the full result in powers of $\lambda$ at finite $L'$. A direct summation on $p$ then yields (focusing without loss of generality on the case $\bar\rho_a=\bar\rho_b=\tfrac 12$) $$\begin{aligned}
\psi^1_L(\lambda,\bar\rho_a=\bar\rho_b=\tfrac 12)
=
\
&\left[
\frac{1}{3 {L'} ^4}-\frac{1}{2 {L'} ^3}+\frac{1}{6 {L'} ^2}
\right]
\frac{\lambda^4}{16}
\nonumber
\\
+
&\left[
\frac{1}{5 {L'} ^6}-\frac{1}{2 {L'} ^5}+\frac{1}{3 {L'} ^4}-\frac{1}{30 {L'} ^2}
\right]
\frac{\lambda^6}{96}
\nonumber
\\
+&\left[
\frac{25}{168 {L'} ^8}+\frac{9}{80 {L'} ^7}+\frac{3}{20 {L'} ^6}-\frac{11}{80 {L'}
^4}+\frac{1}{42 {L'} ^2}
\right]
\frac{\lambda^8}{1152}
+ O(\lambda^{10}).
\label{eq:expansionlambdafiniteLpsi1}\end{aligned}$$ The dominant terms, of order $1/{L'}^2$, correspond as expected to the expansion in powers of $\lambda$ of the large-$L$ result . The other terms are the one provided at finite $L$ by the full expression . We note that if one scales $\lambda$ with $L$ as $\lambda\sim L^\zeta$ ($\zeta>0$), the expansion remains well defined only for $\zeta<1$. For $\zeta\geq 1$ there is thus a change of regime, as also occurs for the saddle-point contribution $\psi_L(\lambda)$ to the full CGF $\psi_{N,L}(\lambda)$ (see the corresponding discussion of the hydrodynamic behavior in Sec. \[ssec:spep\_hydro\]).
Symmetric zero-range process {#app:zrp}
============================
As a simple example supporting our conjecture on the relation between unbounded $\sigma(\rho)$ and hydrodynamic behaviors of current large deviations, we examine the symmetric zero-range process (ZRP) on an open one-dimensional system. In this model, a particle hops between neighboring sites at a rate $u(n_k)$ that depends only on the number of particles $n_k$ at the site of departure. More precisely, the bulk dynamics are given by $$\label{eq:zrp_bulk_rates}
(n_k,\,n_l) \xrightarrow{u(n_k)} (n_k - 1,\, n_l + 1) \qquad \text{for $l = k \pm 1$ with $k = 2,\ldots,L-1$},$$ while the boundary dynamics are given by $$\begin{aligned}
{2}\label{eq:zrp_boundary_rates}
n_1 &\xrightarrow{\alpha} n_1 + 1, &\qquad n_1 &\xrightarrow{\gamma u(n_1)} n_1 - 1, \nonumber\\
n_L &\xrightarrow{\delta} n_1 + 1, & n_L &\xrightarrow{\beta u(n_L)} n_L - 1.\end{aligned}$$ It can be shown [@spohn_large_1991; @Kipnis1999; @Bertini2002] that the hydrodynamic behaviors of the model are characterized by boundary conditions $$\label{eq:zrp_zab}
\bar z_a \equiv z(\bar\rho_a) = \frac{\alpha}{\gamma}, \quad \bar z_b \equiv = z(\bar\rho_b) = \frac{\delta}{\beta},$$ and transport coefficients $$\label{eq:zrp_coeff}
D(\rho) = z'(\rho), \quad \sigma(\rho) = 2z(\rho),$$ where the fugacity $z(\rho)$ is an increasing function of the particle density $\rho$ (see [@Evans2005], for example). We assume that $z(\rho)$ is not bounded from above; otherwise, a condensation transition occurs for sufficiently large current fluctuations [@Harris2005; @Harris2006; @Hirschberg2015], in which case we can no longer discuss the steady-state statistics of the currents. Given this assumption, $\sigma(\rho)$ is not bounded from above, so our conjecture predicts that the symmetric ZRP shows hydrodynamic behaviors for arbitrarily large current fluctuations. We check this prediction by comparing microscopic and hydrodynamic scaled CGFs for the time-averaged current, which are defined through $$e^{T\psi_L^\mathrm{ZRP}(\lambda)} \sim \langle e^{\lambda TJ} \rangle,
\quad e^{(L+1)^2 T\psi^\mathrm{ZRP}(\lambda)} \sim \langle e^{\lambda (L+1)^2 TJ} \rangle,$$ respectively. Note that $\langle \cdot \rangle$ denotes the average over all possible evolutions of the system during a time interval $t \in [0,\,T]$ in the former and $t \in [0,\,(L+1)^2T]$ in the latter. Since the exact microscopic expression was derived in [@Harris2005] as $$\label{eq:zrp_cgf_micro}
\psi^\mathrm{ZRP}_L(\lambda) = \frac{(1 - e^{-\lambda})(e^{\lambda}\alpha\beta-\gamma\delta)}{\gamma+\beta+\beta\gamma(L-1)},$$ here we present a derivation of the corresponding hydrodynamic expression only.
From and , in the hydrodynamic limit the effective Hamiltonian of the symmetric ZRP is $$\label{eq:zrp_h}
H^\mathrm{ZRP}[\rho,\hat\rho] = \int_0^1 \mathrm{d}x\, \left[ -z'(\rho)(\partial_x \rho)(\partial_x \hat\rho) + z(\rho) (\partial_x \hat\rho)^2 \right]$$ with boundary conditions given by $$\label{eq:zrp_bcs}
z(\rho(0)) = \bar z_a, \quad z(\rho(1)) = \bar z_b, \quad \hat\rho(0) = 0, \quad \hat\rho(1) = \lambda.$$ Assuming an additivity principle, the optimal profiles satisfy $$\begin{aligned}
\label{eq:zrp_ap}
\partial_t \rho = \frac{\delta H}{\delta \hat\rho} = \partial_x \left[ \partial_x z(\rho) - 2z(\rho)\partial_x \hat\rho \right] = 0, \nonumber\\
\partial_t \hat\rho = -\frac{\delta H}{\delta \rho} = -z'(\rho) \left[ \partial_x^2 \hat\rho + (\partial_x \hat\rho)^2 \right] = 0,\end{aligned}$$ which are solved by $$\begin{aligned}
\label{eq:zrp_profiles}
z(\rho^*(x)) &= -\left[(e^\lambda - 1)x + 1\right] \, \left[(\bar z_a - \bar z_b e^{-\lambda})x - 1\right], \nonumber\\
\hat\rho^*(x) &= \ln \left[ 1 + (e^\lambda - 1)x \right].\end{aligned}$$ Thus the hydrodynamic scaled CGF is obtained as $$\label{eq:zrp_cgf_hydro}
\psi^\mathrm{ZRP}(\lambda) = \frac{H[\rho^*,\hat\rho^*]}{L+1} = \frac{(e^\lambda - 1)\bar z_a - (1-e^{-\lambda})\bar z_b}{L+1}.$$ From , , and , we obtain $$\lim_{L \to \infty} \frac{\psi^\mathrm{ZRP}_L(\lambda)}{\psi^\mathrm{ZRP}(\lambda)}
= \lim_{L \to \infty} \frac{(e^\lambda - 1)\frac{\alpha}{\gamma} - (1-e^{-\lambda})\frac{\delta}{\beta}}{(e^\lambda - 1)\bar z_a - (1-e^{-\lambda})\bar z_b}
= 1,$$ which is true for any scaling of $\lambda$ with $L$. This confirms our prediction that the symmetric ZRP shows hydrodynamic behaviors for arbitrarily large current fluctuations.
We check whether the rationale behind our conjecture is also at work here. Following the procedure used for obtaining and , we obtain $$\begin{aligned}
J &= \frac{1}{L+1} \int_0^1 \mathrm{d}x\, \left[ -z'(\rho^*)\partial_x \rho^* + 2 z(\rho^*) \partial_x \hat\rho^* \right]
= \frac{\bar z_a - \bar z_b}{L+1} + \frac{2}{L+1} \int_0^1 \mathrm{d}x\, \left[z(\rho^*) \partial_x \hat\rho^* \right] \nonumber\\
&\simeq \frac{2}{L+1} \int_0^1 \mathrm{d}x\, \left[z(\rho^*) \partial_x \hat\rho^* \right].\end{aligned}$$ Thus $J$ beyond the naive hydrodynamic regime is dominated by $z(\rho^*)$ and $\partial_x {\hat\rho}^*$. Due to , these quantities satisfy $$\begin{aligned}
z(\rho^*) &\simeq \begin{cases}
e^\lambda x(1 - \bar z_a x) &\text{ if $\lambda > 0$ and $\lambda \gg 1$,} \\
e^{-\lambda} \bar z_b x (1-x) &\text{ if $\lambda < 0$ and $|\lambda| \gg 1$,}
\end{cases} \nonumber\\
\partial_x \hat\rho^* &= \frac{e^\lambda - 1}{1 + (e^\lambda - 1)x} \simeq \begin{cases}
\frac{1}{x} &\text{ if $\lambda > 0$ and $\lambda \gg 1$,} \\
\frac{1}{x-1} &\text{ if $\lambda < 0$ and $|\lambda| \gg 1$} \\
\end{cases}\end{aligned}$$ for $0 < x < 1$. Hence, a large $J$ is supported by a large $z(\rho^*)$, while $\partial_x \hat\rho^* = O(N^0)$ throughout the bulk region. We note that $\partial_x \hat\rho^*$ becomes arbitrarily large close to the boundaries, attaining the order of $L$ (corresponding to the threshold for non-hydrodynamic behaviors found in the SPEP) for $x \sim 1/L$ (for $\lambda > 0$) or $1- x \sim 1/L$ (for $\lambda < 0$). But one can easily see that $J$ has negligible contributions from these boundary regions compared to the bulk in the $L \to \infty$ limit. Therefore, the symmetric ZRP confirms our proposed scenario of how $J$ stays hydrodynamic for unbounded $\sigma(\rho)$.
[^1]: These notations indicate that the rescaled variables are defined as $\tilde{x} \equiv x/\ell$ and $\tilde{t} \equiv t/\ell^2$, and then renamed as $x$ and $t$, respectively. Other notations for rescaling schemes should be interpreted similarly.
[^2]: We note that there was a previous attempt to calculate the current LDF of a discrete system by applying a saddle-point approximation directly to the microscopic model [@Imparato2009]. This approximation, however, is not well controlled.
|
---
abstract: 'We study decay of correlations, the asymptotic distribution of hitting times and fluctuations of the return times for a robust class of multidimensional non-uniformly hyperbolic transformations. Oliveira and Viana [@OV07] proved that there is a unique equilibrium state $\mu$ for a large class of non-uniformly expanding transformations and Hölder continuous potentials with small variation. For an open class of potentials with small variation, we prove quasi-compactness of the Ruelle-Perron-Frobenius operator in a space $V_\theta$ of functions with essential bounded variation that strictly contain Hölder continuous observables. We deduce that the equilibrium states have exponential decay of correlations. Furthermore, we prove exponential asymptotic distribution of hitting times and log-normal fluctuations of the return times around the average $h_\mu(f)$.'
address: 'Paulo Varandas, Instituto de Matemática, Universidade Federal do Rio de Janeiro C. P. 68.530, 21.945-970, Rio de Janeiro, RJ-Brazil'
author:
- Paulo Varandas
title: Correlation decay and recurrence asymptotics for some robust nonuniformly hyperbolic maps
---
Introduction
============
Given a measure preserving discrete dynamical system it follows by earlier work of Poincaré that the orbit of almost every point will return to any arbitrary small neighborhood of it. Return times are strongly related to the complexity of the dynamical system and, in many cases, the entropy coincides with the exponential growth rate of the return times to decreasing sequences of nested sets. A related concept is the one of hitting times. Typical orbits will eventually visit any positive measure set with some frequency which coincides, in average, with the measure of the set. In particular return times reflect both hitting times statistics and the rate at which the system is mixing. Some questions that naturally arise are:
$1$. What is the distribution of the hitting times of the system when one considers sets of arbitrary small measure?
$2$. How do return times oscillate around their average?
In general, an answer to these questions involves a deep knowledge of the system’s chaotic features (expressed in terms of mixing properties) combined with information on the measure of sets in small scales. For the later it is usually enough the invariant measure to satisfy some local equilibrium property.
The study of hitting and first return times and their connection with hyperbolicity, speed of mixing and dimension theory achieved many recent developments and became an important ingredient to characterize the statistical properties of dynamical systems. Some of the first attempts to understand possible phenomena in a hyperbolic setting include works by Pitskel [@Pit91], for transitive Markov chains, by Hirata [@Hi93] for Axiom A diffeomorphisms and by Collet [@Col96] in the context of expanding maps of the interval with a spectral gap, where they proved that the distribution of hitting and return times is asymptotically exponential. The extension of these results beyond the scope of uniform hyperbolicity is as a challenge and gained special attention in the few past years following the recent interest and developments on the thermodynamical formalism for nonuniformly hyperbolic transformations. Nevertheless, and despite the effort of many authors, a general picture is still far from complete. Some recent contributions among many others include works by Collet, Galves [@CG93] on maps of the interval with indifferent fixed points; Galves, Schmitt [@GaSc97] for systems satisfying a $\varphi$-mixing condition; Haydn [@Hay99a; @Hay99b] for Julia sets of rational maps; Hirata, Saussol, Vaienti [@HSV99] on Poissonian laws for multiple return time statistics for some non hyperbolic maps of the interval; Paccaut [@Pac00] on weighted piecewise expanding maps of the interval; Abadi [@Aba01] for $\alpha$-mixing stationary processes; Collet [@Col01] on Poisson laws for non-uniformly hyperbolic maps that admit a Young tower; Bruin, Saussol, Troubetzkoy, Vaienti [@BSTV03] on the return time statistics via inducing; Saussol, Troubetzkoy, Vaienti [@STV03] on the relation between recurrence, dimension and Lyapunov exponents; and Bruin, Todd [@BT07b] on interval maps with positive Lyapunov exponent, just to mention some of the most recent advances. In particular, there are several evidences of an intricate relation between the asymptotic behavior of the distribution of hitting times and the memory loss of the system expressed in terms of good mixing properties. Finally, return time statistics are also useful to study the fluctuations of the return times in the Ornstein-Weiss formula for the metric entropy (see [@OW93]). In fact, it follows from the work of Saussol [@Sau01] that there is a strong connection between fluctuations of the return times and the fluctuations in the Shannon-MacMillan-Breiman’s theorem, which are easier to study if the invariant measure has a Gibbs property.
Here we deal with a robust class of multidimensional non-uniformly hyperbolic transformations introduced by Oliveira and Viana in [@OV07], that contain maps obtained as deformations by isotopy from expanding transformations as the ones considered in [@ABV00 Appendix]. Despite the existence of a (possibly non-generating) Markov partition many difficulties arise from the multidimensional character of the system and the absence of bounded distortion. Oliveira and Viana developed a thermodynamical formalism to show that there is a unique equilibrium state for every Hölder potential with small variation and, moreover, that is satisfies a weak Gibbs property. Our starting point to study the asymptotics of hitting and return times as statistical properties of the equilibrium states is to estimate the decay of correlations. That is, the velocity at which $$C_n(\Phi,\Psi)
=\Big| \int \Psi (\Phi \circ T^n) \, d\mu - \int \Phi \,d\mu \int \Psi \,d\mu\Big|$$ tends to zero as $n\to \infty$ for any observable $\Psi$ and $\Psi$ in some reasonable space $V_\theta$ of functions with essential bounded variation, that contain Hölder continuous observables and characteristic functions at cylinders of the partitions generated dynamically by the dynamics. For a related of potentials, we show that the Ruelle-Perron-Frobenius operator has a spectral gap in $V_\theta$ and deduce exponential decay of correlations and the central limit theorem. This mixing property is enough to obtain exponential return time statistics. Finally, a weak Gibbs property for the equilibrium states allow us to relate return time statistics with fluctuations of the measure of elements of dynamical partitions and return times around the metric entropy given by Shannon-MacMillan-Breiman and Ornstein-Weiss formulas and to obtain log-normal fluctuations of the return times. Let us point out that exponential return time statistics and log-normal fluctuations of return times are robust in this nonuniformly hyperbolic setting. Our approach to obtain exponential return time statistics, although similar in flavor with [@Pac00] and [@GaSc97], faces distinct difficulties that arise from the non hyperbolicity of the system. While the cornerstone in [@Pac00] was the non-Markov property of partitions defining piecewise expanding maps of the interval, our main difficulties lie in the lack of bounded distortion and that the diameter of cylinders in the Markov partition may not decrease to zero. For the sake of completeness, let us mention that Arbieto and Matheus [@AM] proved exponential decay of correlations the equilibrium states constructed in [@OV07] but considered different classes of potentials and observables.
A very interesting question is to obtain exponential return time statistics to balls instead of cylinders. Although this is possible to obtain for several cases in the one-dimensional setting, the study of return time statistics to balls presents itself as a major difficulty in this higher dimension setting. If such a result could be obtained it is most likely that our results can be extended to the more general context of [@VV1], where [@OV07] is generalized and no Markov partition is assumed to exist.
This article is organized as follows. In Section \[sec.statement\] we state our main results. In Section \[sec.preliminaries\] we introduce some notations and tools that will be used in the remaining of the paper. We prove the quasi-compactness of the Ruelle-Perron-Frobenius operator in $V_\theta$ in Section \[sec.spectral.gap\]. Finally, in Sections \[sec.exponentialdistribution\] and \[sec.fluctuations\] we deal with the asymptotic distribution of hitting times and fluctuations of the return times.
Statement of the Results {#sec.statement}
========================
Setting
-------
Let $M$ denote a compact Riemannian manifold. We say that a set $E
\subset M$ has *finite inner diameter* if there exists $L>0$ such that any two points in $E$ may be joined by a curve of length less than $L$ contained in $E$. Throughout, $f: M \to M$ will denote a $C^1$ local diffeomorphism satisfying conditions (H1) and (H2) below:
- There are $p \ge 1$, $q \ge 0$, and a family ${\mathcal{Q}}=\{Q_1\,, \dots, Q_q\,, Q_{q+1}, \dots, Q_{q+p}\}$ of pairwise disjoint open sets whose closures have finite inner diameter and cover the whole $M$, such that
- every $f|\bar Q_i$ is a homeomorphism onto its image
- if $f(Q_i) \cap Q_j \neq \emptyset$ then $f(Q_i) \supset
Q_j$ and, hence, $f(\bar{Q}_i) \supset \bar{Q}_j$
- there is $N\ge 1$ such that $f^N(Q_i)=M$ for every $i$.
- There are positive constants $\sigma>1$ and $L>0$ such that
- $\|Df(x)^{-1}\|\leq \sigma^{-1}$ for every $x\in {Q}_{q+1} \cup \dots \cup {Q}_{q+p}$
- $\|Df(x)^{-1}\| \leq L$ for every $x\in {Q}_1 \cup \cdots \cup {Q}_q$,
where $L$ is assumed to be close to be $1$ in order to satisfy the relations .
These conditions, which roughly mean that the transformation is expanding in some (topologically) large region of $M$ but may admit contracting behavior in the complement, are satisfied a large class of local diffeomorphisms obtained by a local bifurcation of an expanding transformation.
We will denote by $\phi: M \to \mathbb R$ an ${\alpha}$-Hölder continuous potential with small oscillation, in the sense that it satisfies
- $\sup\phi-\inf\phi < \log \deg(f) - \log q$.
and condition (H3b) stated at the beginning of Subsection \[subsec.spectral.gap\]. These conditions are clearly satisfied by an open class of potentials containing the constant ones. Note that (H1), (H2) and (H3a) are the assumptions in [@OV07].
Equilibrium states and conformal measures
-----------------------------------------
Given a continuous transformation $f:M\to M$ and a continuous potential $\phi:M \to \mathbb R$, an invariant probability measure $\mu$ is an *equilibrium state* for $f$ with respect to $\phi$ if it attains the supremum $${P_{{\operatorname{top}}}}(f,\phi)
= \sup\Big\{h_\eta(f)+\int \phi\,d\eta : \eta \; \text{is $f$-invariant}\Big\}$$ given by the variational principle for the pressure (see e.g. [@Wa82]). The *Ruelle-Perron-Frobenius operator* ${\mathcal{L}}_\phi$ is the linear operator that acts in the space $C(M)$ of continuous functions by $${\mathcal{L}}_\phi g (x)= \sum_{f(y)=x} e^{\phi(y)} g(y).$$ The action of the dual operator ${\mathcal{L}}_\phi^*$ on the space $\mathcal
M (M)$ of probability measures is given by $\int g \,d{\mathcal{L}}^*_\phi
\nu= \int {\mathcal{L}}_\phi g\, d\nu$ for every $g \in C(M)$. We say that a measure $\nu$ is *conformal* if there exists a strictly positive function $J_\nu f$ (Jacobian of $\nu$ with respect to $f$) such that $
\nu(f(A))= \int J_\nu f \, d\nu
$ for every measurable set $A$ such that $f\mid A$ is injective. It is nor difficult to see that any eigenmeasure $\nu$ for ${\mathcal{L}}_\phi^*$ associated to a positive eigenvalue $\lambda$ is a conformal measure for $f$ and that $J_\nu f=\lambda e^{-\phi}$.
A sequence of positive integers $(n_k)_{k\geq1}$ is *non-lacunary* if it is increasing and $n_{k+1}/n_k\to 1$ when $k$ tends to infinity. Consider the partition ${\mathcal{Q}}^{(n)}=\bigvee_{j=0}^{n-1} f^{-j}{\mathcal{Q}}$ and let $Q_n(x)$ be the element of ${{\mathcal{Q}}^{(n)}}$ that contains $x$.
A probability measure $\nu$ is a *non-lacunary Gibbs measure* if there is $K>0$ so that, for $\nu$-almost every $x \in M$ there exists some non-lacunary sequence $(n_k)_{k\geq 1}$, depending on $x$, such that $$\label{eq. Gibbs at hyperbolic times}
K^{-1} \leq \frac{\nu(Q_{n_k}(x))}
{\exp(-P\,n_k + S_{n_k}\phi(y))}\leq K$$ for every $y\in Q_{n_k}(x)$ and every $k\ge 1$.
Finally, we recall the notion of *hyperbolic time* introduced in [@ABV00]. We say that $n$ is a *$c$-hyperbolic time* for $x \in M$ if $$\label{eq. c-hyperbolic times}
\prod_{j=n-k}^{n-1} \| Df(f^j(x))^{-1}\| < e^{-ck}
\quad \text{for every} \; 1\leq k\leq n.$$ Since the constant $c$ will be fixed below, according to , we will refer to these simply as hyperbolic times. We say that $Q_n \in {{\mathcal{Q}}^{(n)}}$ is an *hyperbolic cylinder* if $n$ is a hyperbolic time for [**every**]{} point in $Q_n$. We denote by ${{\mathcal{Q}}^{(n)}}_h$ the set of hyperbolic cylinders of order $n$ and by $H$ the set of points that belong to the closure of infinitely many hyperbolic cylinders. We say that a probability measure $\nu$ is *expanding* if it satisfies $\nu(H)=1$. The next theorem summarizes the results by Oliveira, Viana [@OV07] in this non-uniformly hyperbolic setting:
\[t.OV2\] [@OV07] Assume that $f$ is a $C^1$ local diffeomorphism such that (H1) and (H2) hold and $\phi:M \to \mathbb R$ is an Hölder continuous potential that satisfies (H3a). Then there exists an expanding conformal measure $\nu$ such that ${\mathcal{L}}^*_\phi \nu={\lambda}\nu$, where ${\lambda}$ denotes the spectral radius of the operator ${\mathcal{L}}_\phi$ in the space $C(M)$. Moreover, there is a unique equilibrium state $\mu$ for $f$ with respect to $\phi$, it is absolutely continuous with respect to $\nu$ and it is a non-lacunary Gibbs measure.
Throughout, $\mu$ and $\nu$ will always denote the probability measures given above.
Statement of the main results
-----------------------------
First we introduce some necessary concepts. We consider a one parameter functional space $V_\theta$, introduced in [@Paccaut], using the reference partition ${\mathcal{Q}}$ and the conformal measure $\nu$. Given $\theta>0$ and $g \in L^\infty(\nu)$ define the $\theta$-variation of $g$ (with respect to ${\mathcal{Q}}$ and $\nu$) by $${\text{var}_{\theta}}(g)
= \sum_{n \geq 1} \;\theta^n \sum_{Q_n \in {\mathcal{Q}}^{(n)}}
e^{S_n \phi(Q_n)} {\overline}{\operatornamewithlimits{osc}}(g, {\mathcal{Q}}_n),$$ where $S_n\phi(Q_n)=\sup\{\sum_{j=0}^{n-1} \phi(f^j(x)): x \in
Q_n\}$ and ${\overline}{\operatornamewithlimits{osc}}(g, {\mathcal{Q}}_n)$ is the $\nu$-essential variation of $g$ in $Q_n$ defined in Subsection \[subsec.essential\]. Let $V_\theta$ be the space of functions with essential $\theta$-bounded variation: $$V_\theta = \{ g \in L^\infty(\nu) : \|g\|_\theta<\infty\},$$ where $\|\cdot\|_\theta=\|\cdot\|_\infty+{\text{var}_{\theta}}(\cdot)$. Paccaut [@Paccaut] proved that $V_\theta$ is a Banach space. We will say that $\mu$ satisfies *exponential decay of correlations* if there exist $C>0$ and $\xi \in (0,1)$ such that $$C_n(\Phi,\Psi)
:=
\Big|
\int \Phi (\Psi \circ f^n) d\mu - \int \Phi d\mu \int \Psi d\mu
\Big|
\leq C \xi^n \|\Phi\|_\theta \|\Psi\|_{L^1(\nu)}.$$ for every $\Psi \in L^1(\nu)$ and every $\Phi \in V_\theta$. A linear operator $L$ in a Banach space $B$ is *quasi-compact* if there exists an $L$-invariant decomposition $B=B_0\oplus B_1$ of the Banach space such that $B_0$ is finite dimensional and the spectrum of $L|_{B_0}$ is a finite number of eigenvalues of absolute value $r(L)$, and $r(L|_{B_1})<r(L)$. If $\dim(B_0)=1$ then we say that $L$ has a *spectral gap*. Our first main result is the following:
\[thm.spectralgap\] There exists a positive $\theta$ such that ${\mathcal{L}}_\phi$ is quasi-compact and has a spectral gap in the space $V_\theta$. Moreover, $\mu$ has exponential decay of correlations in $V_\theta$ and the density $d\mu/d\nu$ belongs to $V_\theta$.
Fix $\theta$ as above. The previous result implies that the correlation functions are summable. Consequently, the *asymptotic variance* ${\sigma}^2(\Phi)$ defined by $${\sigma}^2(\Phi)=\|\Phi\|_{L^1(\mu)}+2\sum_{j =1}^\infty \int \Phi \, (\Phi \circ f^j)\,d\mu$$ is well defined for every $\Phi\in V_\theta$. Standard computations involving the spectral gap property in the previous theorem (see e.g [@You98]) are enough to obtain the Central Limit Theorem:
\[c.CLT\] Assume that $\Phi \in V_\theta$ and that the asymptotic variance ${\sigma}^2(\Phi)$ is nonzero. Then the distribution of the random variables $$\frac{1}{{\sigma}(\Phi) \sqrt{n}}
\sum_{j=0}^{n-1} \Big( \Phi\circ f^j-\int \Phi \,d\mu \Big)$$ converges to the normal distribution ${\mathcal{N}}(0,1)$ as $n$ tends to infinity. Moreover, ${\sigma}(\Phi) = 0$ if and only if there exists a measurable function $\tilde \Phi$ such that $\Phi=\tilde \Phi \circ
f -\tilde \Phi$.
Our next aim is to study the asymptotics of hitting times. Given a set $Q$ consider the *hitting time* $\tau_{Q}$ defined as $$\tau_{Q}(x)=\inf\{k \geq 1 : f^k(x) \in Q\}.$$ We study the deviation of the hitting times from its average given by Kac’s formula: $\int \tau_Q \,d\mu = \mu(Q)^{-1}$. Indeed, we use that the potential $\phi$ belongs to $V_\theta$ (see Lemma \[l.Vtheta.contains.Holder\]) and the good mixing properties to show exponential return time statistics for the hitting time of most cylinders.
\[thm.first.entrance.distribution\] There are positive constants $K$ and $\beta$, and for every ${\varepsilon}>0$ there exists $N_{\varepsilon}\geq 1$ such that the following holds: for any $n \geq N_{\varepsilon}$ there exists a subset ${{\mathcal{Q}}^{(n)}}_{\varepsilon}$ of $n$-cylinders satisfying
1. $\mu(\cup\{Q_n: Q_n \in {{\mathcal{Q}}^{(n)}}_{\varepsilon}\}) \geq 1- {\varepsilon}$;
2. For every $Q \in {{\mathcal{Q}}^{(n)}}_{\varepsilon}$ $$\sup_{t\geq 0}
\Big| \mu\Big( \tau_{Q} > \frac{t}{\mu(Q)} \Big) - e^{-t} \Big|
\leq K e^{-\beta n}.$$
This theorem asserts that the distribution of the hitting times is asymptotically exponential and that the convergence is in a strong sense for the majority of the cylinders. This is very useful to study the fluctuations of the return times around the average in Ornstein-Weiss’s theorem. Since $\mu$ is ergodic, if $n \geq 1$ and $$R_n(x)= \inf\{k\geq 1: f^k(x) \in {{\mathcal{Q}}^{(n)}}(x)\}$$ denotes the *$n$th return time map* then Ornstein-Weiss’s theorem asserts that $$h_\mu(f,{\mathcal{Q}})=\lim_{n\to \infty} \frac 1n \log R_n(x),
\quad \text{for $\mu$-a.e. $x$.}$$ We shall see later on that the diameter of $Q_n(x)$ tends to zero as $n\to\infty$ at $\mu$-almost every $x$, which shows that ${\mathcal{Q}}$ is a generating partition for $\mu$. We study the fluctuation of the random variables $\log R_n$ around the average $n h_\mu(f)$.
\[thm.recurrence.fluctuations\] Assume that ${\sigma}^2(\phi)$ is positive. Then the following convergence in distribution holds: $$\frac{\log R_n -n h_\mu(f)}{ {\sigma}(\phi) \sqrt n} \xrightarrow[n \to
+\infty]{\mathcal D} {\mathcal{N}}(0,1),$$ where ${\mathcal{N}}(0,1)$ denotes the standard zero mean Gaussian.
Preliminaries {#sec.preliminaries}
=============
In this section we recall some necessary concepts that will be used later on. The reader may choose to omit this section in a first reading and to come back here when necessary.
Hyperbolic times
----------------
Here we collect some results from Section 3 and Section 4 in [@OV07] (see also [@VV1]), whose proofs we shall omit. Fix ${\varepsilon}_0>0$ such that $\sup \phi-\inf \phi < \log\deg(f)-\log q
-{\varepsilon}_0$ and set $P=\log {\lambda}$.
\[p.decreasing.measure\] There exists ${\gamma}_1>0$ such that the measure $\nu(Q_n) \leq
e^{-{\gamma}_1 n}$ for every cylinder $Q_n \in {{\mathcal{Q}}^{(n)}}$. There exists ${\gamma}\in (0,1)$ and $c_{\gamma}\leq\log q+{\varepsilon}_0$ such that the cardinality of cylinders in $$B(n,{\gamma})
=\big\{Q_n \in {\mathcal{Q}}^{(n)} \;|\; \#\{0\leq j \leq n-1 : f^j(Q_n) \subset
{\mathcal{Q}}_1 \cup \dots \cup {\mathcal{Q}}_q \}>{\gamma}n \big\}$$ is bounded from above by $\exp(c_{\gamma}n )$ for every large $n$. Moreover, the measure $\nu(B({\gamma},n))$ decreases exponentially fast as $n\to\infty$.
We are now in a position to state the precise condition on the constant $L>0$ in (H1) that is chosen in a [**different way**]{} from [@OV07]. Pick $c>0$ such that $$\label{eq.relation.expansion}
{\sigma}^{-(1-{\gamma})} < e^{-2c}<1
\quad\text{and}\quad
\log q + c{\alpha}+{\varepsilon}_0 < \log \deg(f),$$ where ${\alpha}>0$ denotes the Hölder exponent of the potential $\phi$. Assume that $L$ is sufficiently close to one such that $(\log L)^2
\leq 2 c^2$, $$\label{eq.relation.L}
{\sigma}^{-(1-{\gamma})} L^{\gamma}< e^{-2c}<1
\quad\text{and}\quad
\sup\phi-\inf\phi<\log\deg(f)-\log q -m\log L.$$
\[l.positive.frequency\][@OV07 Lemma 4.4] There exists $\tau\in(0,1)$ such that, for any $n\ge 1$ and any $x\notin B({\gamma},n)$, there exists $l>\tau n$ and integers $1\le n_1 <
\cdots < n_l$ such that $x$ belongs to the closure of an hyperbolic cylinder $Q_{n_i}\in{\mathcal{Q}}^{n_i}_h$ for every $i=1, \ldots, l$. Furthermore, $\tau \geq 2c/A$ where $A=\log L$.
Since $0<\tau<1$, our choice of $c$ in guarantees that $\log q + c\tau {\alpha}+{\varepsilon}_0 < \log \deg(f)$. The following lemma asserts backward distances contraction and a Gibbs property at hyperbolic times.
\[l.Gibbs.at.hyp.times\] Given $Q_n \in {{\mathcal{Q}}^{(n)}}_h$, $1 \leq j$ and $x,y$ in the closure of $Q_n$, $$d_{f^{n-j}(\bar{Q}^n)}(f^{n-j}(x),f^{n-j}(y))
\leq e^{-c j} {\operatorname{diam}}({\mathcal{Q}}).$$ Moreover, there exists $K>0$ (independent of $n$) such that every $y
\in {\overline}Q_n$ satisfies $$K^{-1} \leq \frac{\nu(Q_n)}{\exp( -P n + S_n \phi(y))} \leq K.$$
As an immediate consequence we obtain that the diameter of most cylinders decrease exponentially fast. More precisely,
\[c.diameter.vs.positive.frequency\] The diameter of every cylinder $Q_n \notin B(n)$ satisfies $${\operatorname{diam}}(Q_n) \leq e^{- c \tau n} {\operatorname{diam}}({\mathcal{Q}}).$$
Given $Q_n \not\in B(n)$, there exists a positive integer $k \geq
\tau n$ such that $k$ is a simultaneous hyperbolic time for every point in $Q_n$. By the mean value theorem $${\operatorname{diam}}(Q_n)
\leq e^{-c k} {\operatorname{diam}}(f^k(Q_n))
\leq e^{- c \tau n} {\operatorname{diam}}({\mathcal{Q}}),$$ which proves the corollary.
\[rem.cardinal.Qn\] Observe that $\# {\mathcal{Q}}^{(n)} \leq \#{\mathcal{Q}}\, \deg(f)^{n-1}$ for every positive integer $n$, as an easy consequence of the Markov property and that every point has $\deg(f)$ preimages. Indeed, given $n \in
\mathbb N$, the Markov assumption on ${\mathcal{Q}}$ implies that ${{\mathcal{Q}}^{(n)}}=f^{-n+1}({\mathcal{Q}})$. On the other hand, given $Q \in {\mathcal{Q}}$, $f^{-n}(Q)$ is the union of $\deg(f)^n$ cylinders. This shows that $\# {\mathcal{Q}}^{(n)} \leq \#{\mathcal{Q}}\, \deg(f)^{n}$ for every $n\ge 1$.
We say that a measure $\eta$ is *exact* if every element in the tail sigma-algebra ${\mathcal{B}}_\infty = \cap_{j\geq 0} f^{-j}{\mathcal{B}}$ is $\eta$-trivial in the sense that it has measure zero or one.
\[l.exactness\] $\mu$ is exact.
This is a direct consequence of [@VV1 Lemma 6.16].
Weak Gibbs property
-------------------
Since $\mu$ is absolutely continuous with respect to $\nu$ and the density $d\mu/d\nu$ is bounded away from zero and infinity (see [@OV07]), then $\mu$ satisfies the non-lacunary Gibbs property. Here we establish a criterium that relates the decay of the first hyperbolic time map with a weak Gibbs property similar to the one introduced by Yuri [@Yu99]. Let $n_1$ denote the first simultaneous hyperbolic time map.
\[l.weak.Gibbs\] There are almost everywhere defined function $(K_n)_{n \ge 1}$ such that $$K_n^{-1}(x)
\leq \frac{\nu(Q_n(x))}{\exp( -P n + S_n \phi(y))} \leq
K_n(x)$$ for $\nu$-almost every $x$ and every $y\in {\overline}Q_n(x)$, and $$\mu\Big( x \in M : K_n(x) > a(n) \Big)
\leq n\; \mu\Big( x \in M : n_1(x)> \frac{\log a(n)}{P+\sup|\phi|} \Big).$$ In particular, $\lim_{n} \frac1n \log K_n=0$ almost everywhere.
The proof of this lemma goes along the same ideas in [@OV07 Lemma 3.12]. Given $n \in \mathbb N$ and $x\in M$ set $K_n(x)=K\exp[(P+\sup|\phi|)(n_{i+1}(x)-n_i(x))]$, where $i$ is a positive integer such that $n_i(x) \leq n< n_{i+1}(x)$ and $n_i$ denotes the $i$th simultaneous hyperbolic time map. It is not hard to show that $K_n$ verifies the Gibbs relation above. Moreover, $$\begin{aligned}
\mu\Big(x \in M : K_n(x)> a(n)\Big) &
\leq \mu \Big( \bigcup_{i} \Big\{x: n_1(f^{n_i(x)}(x))> \frac{\log a(n)}{P+\sup|\phi|} \Big\}
\Big)\vspace{.2cm}\\
& \leq n \; \mu\Big( x\in M: n_1(x) > \frac{\log a(n)}{P+\sup|\phi|} \Big),\end{aligned}$$ where we made use of the invariance of $\mu$ and that $n_1(f^{n_i(x)}(x))\ge n_{i+1}(x)-n_i(x)$. The last claim in the lemma is a direct consequence of the decay estimates.
\[c.decay.Kn\] There exists $a\in \mathbb N$ such that $\mu$-almost every $x$ satisfies $K_n(x)< n^a$ for all but finitely many values of $n$.
We use the inclusion $\{n_1> \log a(n)\} \subset B({\gamma},\log a(n))$. If $a\in \mathbb N$ is large enough it follows from the previous result that $$\mu \Big( x \in M : K_n(x) > n^a \Big)
\leq n \exp\Big (- \frac{c_{\gamma}a}{P+\sup|\phi|} \log n\Big)
\leq n^{-2},$$ which is summable. Our claim follows directly from Borel-Cantelli’s lemma.
This result gives a sufficient condition to obtain sub-exponential growth of the sequence $(K_n(x))_{n \geq 1}$ in the weak Gibbs property that will be of particular interest for the proof of the log-normal fluctuations of the return times in Section \[sec.fluctuations\].
Essential oscillation and variation {#subsec.essential}
-----------------------------------
In this section we present some basic lemmas, needed for the proof of a Lasota-Yorke inequality in Subsection \[subsec.spectral.gap\]. Given $g \in L^{\infty}(\nu)$ and a set $E$ we define the *essential oscillation* ${\overline}{\operatornamewithlimits{osc}}(g,E)$ of $g$ on the set $E$ (with respect to $\nu$) as $${\overline}{\operatornamewithlimits{osc}}(g,E)=\nu-ess\sup \{ |g(x)-g(y)|: x,y \in E\}.$$ Analogously, ${\overline}\sup(g,E)$ and ${\overline}\inf(g,E)$ will denote, respectively, the essential supremum and essential infimum of $g$ in the set $E$. The following is an immediate consequence of the triangular inequality.
\[l.oscillation.inequality\] For every $g, h \in L^\infty(\nu)$ and any set $E$ it holds that $${\overline}{\operatornamewithlimits{osc}}(gh,E)
\leq {\overline}{\operatornamewithlimits{osc}}(g,E) {\overline}\sup(h, E)+{\overline}\sup(g, E) {\overline}{\operatornamewithlimits{osc}}(h, E).$$
In the next lemma we give an estimate on the oscillation of the $\alpha$-Hölder continuous potential $\phi$ in cylinders with positive frequency of hyperbolic times.
\[l.oscillation.hyp.times\] There exists $C_\phi>0$ such that $${\operatornamewithlimits{osc}}(e^\phi, Q_n)
\leq C_\phi e^{-c \tau {\alpha}n} {\operatorname{diam}}({\mathcal{Q}})^{\alpha}, \;
\text{for every}\; Q_n \not\in B(n).$$
Observe that $e^\phi$ is an $\alpha$-Hölder continuous function for some positive constant $C_\phi$. Therefore, it follows from Corollary \[c.diameter.vs.positive.frequency\] that $$| e^{\phi(x)}- e^{\phi(y)} |
\leq C_\phi {\operatorname{diam}}(Q_n)^{\alpha}\leq C_\phi e^{-c \tau {\alpha}n} {\operatorname{diam}}({\mathcal{Q}})^{\alpha}$$ for every $Q_n \not\in B({\gamma},n)$ and every $x, y \in Q_n$. This proves the lemma.
The following lemma plays a key role in the proof of the Lasota-Yorke inequality.
\[l.sup.osc.L1\] Given a positive $\nu$-measure set $E$ and $g\in L^\infty(\nu)$, $${\overline}\sup(g,E) \leq {\overline}{\operatornamewithlimits{osc}}(g,E) + \frac{1}{\nu(E)} \int_E |g| \,d\nu.$$
Observe that $|g(x)| \leq |g(x)-g(y)| +|g(y)| \leq
{\overline}{\operatornamewithlimits{osc}}(g,E)+|g(y)|$ for almost every $x,y \in E$. In particular, integrating both sides of the previous inequality with respect to $y$ it follows that $
|g(x)| \leq {\overline}{\operatornamewithlimits{osc}}(g, E) + \frac{1}{\nu(E)} \int_E |g| \,d\nu
$ for $\nu$-almost every $x\in E$. The lemma is now a direct consequence of the previous relation.
Denote by $f^k_{Q_k}$ the restriction of $f^k$ to the cylinder $Q_k$ and observe that it is a bijection onto its image. When no confusion is possible we will denote by $Q_{n+k}$ the cylinder $f^{-k}_{Q_k}(Q_n)$.
\[l.sup.concatenation\] Given any positive integers $k$, $n$ and cylinders $Q_k \in {{\mathcal{Q}}^{(k)}}$ and $Q_n \in {{\mathcal{Q}}^{(n)}}$ it holds that $$e^{S_{n+k}\phi(f^{-k}_{Q_n}(Q_n))}
\leq e^{S_n\phi(Q_n)} \, e^{S_k\phi(f^{-k}_{Q_k}(Q_n))}
\leq e^{(\sup\phi-\inf\phi) k} e^{S_{n+k}\phi(f^{-k}_{Q_k}(Q_n))}.$$
Fix $Q_k \in {{\mathcal{Q}}^{(k)}}$ and $Q_n\in{{\mathcal{Q}}^{(n)}}$. The first inequality is obvious. On the other hand, the Markov property implies that $f^n(Q_{n+k})= Q_k$. If $x \in Q_n$, $y \in Q_k$ are such that attain the maximum values in $e^{S_n \phi(Q_n)}$ and $e^{S_k
\phi(Q_k)}$, respectively, then $$e^{S_{n+k} \phi(Q_{n+k})} \geq e^{S_{n+k} \phi(f^{-k}_{Q_k}(x))}
= e^{S_k \phi( f^{-k}_{Q_k}(x))} \, e^{S_n \phi(x)}.$$ It follows immediately that $$e^{S_k \phi(Q_k)} \, e^{S_n \phi(Q_n)}
\leq e^{S_k \phi(y)-S_k \phi( f^{-k}_{Q_k}(x))} \, e^{S_{n+k} \phi(Q_{n+k})}
\leq e^{(\sup\phi-\inf\phi) k} e^{S_{n+k}\phi(Q_{n+k})},$$ which proves the lemma.
Since the diameter of cylinders $Q_n\notin B(n)$ decrease exponentially fast with $n$, the oscillation of an Hölder observable over such cylinders is also decreasing.
\[l.oscillation.concatenation\] Given $k \geq 1$ there exists $C_0>0$ (depending on $k$) such that, if $n$ is large enough then $${\operatornamewithlimits{osc}}(e^{S_k \phi}, f^{-k}_{Q_k}(Q_n))
\leq C_0 \sup(e^{S_k \phi}, f^{-k}_{Q_k}(Q_n)) \;e^{-c \tau {\alpha}n}.$$ for every $Q_k \in {\mathcal{Q}}^{(k)}$ and $Q_n \notin B(n)$.
Let $k \geq 1$ and $Q_k \in {\mathcal{Q}}^{(k)}$ be fixed. Since $\phi$ is ${\alpha}$-Hölder continuous there is $C>0$ so that $$|S_k \phi (x)-S_k \phi (y)|
\leq \sum_{j=0}^{k-1} |\phi (f^j(x))-\phi(f^j(y))|
\leq \sum_{j=0}^{k-1} C \, d(f^j(x),f^j(y))^\alpha$$ for any $x, y \in f^{-k}_{Q_k}(Q_n)$. The term in the right hand side is clearly bounded by $$C k \max_{0 \leq j \leq k-1} {\operatorname{diam}}(f^j(Q_{n+k}))^{\alpha}.$$ Recall that $\|Df(x)^{-1}\| \leq L$ for every $x\in M$ by (H1). In consequence, if $n$ is large enough then $
\exp( S_k \phi (x)-S_k \phi (y))
\leq \exp\Big[ C k \max\{1, L^k\}^{{\alpha}} {\operatorname{diam}}(Q_n)^{\alpha}\Big],
$ which is arbitrarily close to zero by Corollary \[c.diameter.vs.positive.frequency\]. Using that $e^t
\leq 2t$ for every $0<t<1$ we deduce that $
|\exp( S_k \phi (x))-\exp( S_k \phi (y))|
\leq e^{S_k\phi (Q_{n+k})} \, C_0 e^{-c\tau {\alpha}n},
$ where $$\label{eq.C0}
C_0(k)=2C k \max\{1, L^k\}^{{\alpha}}$$ is independent of $n$. Since $x$ and $y$ where chosen arbitrary this completes the proof of the lemma.
Spectral gap for the Ruelle-Perron-Frobenius operator {#sec.spectral.gap}
=====================================================
In this section we prove that the Ruelle-Perron-Frobenius operator has a spectral gap in the space $V_\theta$ of functions of essential bounded variation for special choices of the parameter $\theta$. As a consequence we show that the density $d\mu/d\nu$ belongs to $V_\theta$, and that the equilibrium state $\mu$ has exponential decay of correlations and satisfies a central limit theorem.
Continuity of the Ruelle-Perron-Frobenius operator
--------------------------------------------------
For notational simplicity we denote the Ruelle-Perron-Frobenius operator ${\mathcal{L}}_\phi$ simply by ${\mathcal{L}}$. First we show that the operator ${\mathcal{L}}$ is continuous in the Banach space $V_\theta$, provided that the parameter $\theta$ is small. More precisely,
\[l.continuous.operator\] If $\theta \,e^{\log \deg(f)+ \sup \phi} e^{-c\tau{\alpha}}<1$ then ${\mathcal{L}}$ is a continuous operator in $V_\theta$: there is a positive constant $C$ such that $
\|{\mathcal{L}}g\|_{\theta} \leq C \|g\|_{\theta},
$ for every $g \in V_\theta$.
Let $\theta>0$ be such that $\theta \,e^{\log \deg(f)+ \sup \phi}
e^{-c\tau{\alpha}} <1$. Given $g \in V_\theta$ we can write $${\mathcal{L}}g (x)
=\sum_{Q \in {\mathcal{Q}}} e^{\phi \circ f_Q^{-1}(x)} \, g\circ f_Q^{-1}(x)
1_{f(Q)}(x).$$ It is clear that $\|{\mathcal{L}}g\|_\infty$ is bounded from above by $\# {\mathcal{Q}}\,e^{\sup \phi} \|g\|_\infty$. Thus, to prove the lemma we are reduced to show that there exists a constant $C>0$ such that $$\label{eq.continuous.operator}
{\text{var}_{\theta}}({\mathcal{L}}g)
:=\sum_{n \geq 0} \theta^n \sum_{Q_n \in {{\mathcal{Q}}^{(n)}}} e^{S_n\phi(Q_n)} {\overline}{\operatornamewithlimits{osc}}({\mathcal{L}}g,Q_n)
\leq C \|g\|_\theta, \; \forall g \in V_\theta.$$ To bound the term involving the oscillation of ${\mathcal{L}}g$ notice that, for every $Q_n \in {{\mathcal{Q}}^{(n)}}$ $$\begin{gathered}
{\overline}{\operatornamewithlimits{osc}}({\mathcal{L}}g, Q_n)
\leq \sum_{Q \in {\mathcal{Q}}} {\overline}{\operatornamewithlimits{osc}}(e^{\phi \circ f_Q^{-1}} \, g\circ f_Q^{-1} , Q_n) \\
\leq \sum_{Q \in {\mathcal{Q}}}
\big[ {\operatornamewithlimits{osc}}(e^{\phi} , f_Q^{-1}(Q_n)) \;{\overline}\sup (g) + \sup (e^\phi) \;{\overline}{\operatornamewithlimits{osc}}(g, f_Q^{-1}(Q_n)) \big].\end{gathered}$$ Now we deal with the sum over elements $Q_n \in{{\mathcal{Q}}^{(n)}}$ in by dividing it in two parts, according to wether $Q_n$ belongs or not to $B(n)$. Since ${\overline}{\operatornamewithlimits{osc}}(h) \leq 2 {\overline}\sup (|h|)$ for every $h\in L^\infty(\nu)$ and $\# B({\gamma},n) \leq e^{(\log q + {\varepsilon}_0)n}$ for every large $n$, $$\begin{aligned}
\sum_{\substack{Q_n \in B(n)}} e^{S_n\phi(Q_n)} {\overline}{\operatornamewithlimits{osc}}({\mathcal{L}}g,Q_n)
& \leq \# B({\gamma},n) \, e^{\sup(\phi) n} \times 2 \# {\mathcal{Q}}\|e^\phi g \|_\infty \\
& \leq C_1 e^{(\log q + \sup \phi + {\varepsilon}_0) n} \|g\|_\infty,\end{aligned}$$ for some constant $C_1$ depending only on $\phi$ and ${\mathcal{Q}}$. On the other hand $$\begin{gathered}
\sum_{\substack{Q_n \not\in B(n)}} e^{S_n\phi(Q_n)} {\overline}{\operatornamewithlimits{osc}}({\mathcal{L}}g,Q_n) \\
\leq \sum_{\substack{Q_n \not\in B(n)}} \sum_{Q \in {\mathcal{Q}}}
e^{S_n\phi(Q_n)} \big[ {\operatornamewithlimits{osc}}(e^{\phi} , f_Q^{-1}(Q_n)) \;{\overline}\sup (g) + \sup (e^\phi) \;{\overline}{\operatornamewithlimits{osc}}(g, f_Q^{-1}(Q_n)) \big].\end{gathered}$$ Lemma \[l.oscillation.hyp.times\] implies that the right hand side above is bounded by $$\begin{aligned}
C_2 \sum_{\substack{Q_{n+1} \in {\mathcal{Q}}^{(n+1)} \\ f(Q_{n+1}) \not\in
B(n)}} e^{S_{n+1}\phi (Q_{n+1})} C_\phi e^{-c \tau {\alpha}n}
({\operatorname{diam}}{\mathcal{Q}})^{\alpha}\|g\|_\infty\\
+ C_2\sum_{\substack{Q_{n+1} \in {\mathcal{Q}}^{(n+1)} \\ f(Q_{n+1}) \not\in
B(n)}} e^{S_{n+1}\phi (Q_{n+1})} {\overline}{\operatornamewithlimits{osc}}(g, Q_{n+1})\end{aligned}$$ for some positive constant $C_2$ (depending only on $\phi$). We deduce that there is $C_3>0$ (depending on $\phi$, $\tau$, ${\alpha}$, ${\mathcal{Q}}$) such that ${\text{var}_{\theta}}({\mathcal{L}}g)$ is bounded from above by the sum of two terms: $$\tag{a}
\sum_{n=1}^\infty \theta^n \Big[
C_1 e^{(\log q+\sup\phi+{\varepsilon}_0) n}
+ C_3 e^{-c \tau {\alpha}n} \sum_{Q_{n+1} \in {\mathcal{Q}}^{(n+1)}} e^{S_{n+1}\phi (Q_{n+1})} \Big] \|g\|_\infty$$ and $$\tag{b}
\frac1\theta \sum_{n=0}^\infty \theta^{n+1}
C_2\sum_{Q_{n+1} \in {\mathcal{Q}}^{(n+1)}} e^{S_{n+1}\phi (Q_{n+1})}{\overline}{\operatornamewithlimits{osc}}(g,Q_{n+1}).$$ In particular it follows that ${\text{var}_{\theta}}({\mathcal{L}}g)$ is bounded by $$ C_2 \frac1{\theta}{\text{var}_{\theta}}(g)
+\sum_{n=0}^\infty \Big[ C_1\theta^n e^{(\log q + \sup\phi+{\varepsilon}_0)n}
+ C_3 \#{\mathcal{Q}}\theta^n e^{(\log \deg(f) + \sup\phi)n} e^{-c\tau {\alpha}n}
\Big] \|g\|_{\infty}.$$ Our choice of the parameter $\theta$ together with the relation $\log q + c\tau{\alpha}+{\varepsilon}_0 < \log \deg(f)$ guarantees the summability of the previous series and proves that ${\mathcal{L}}$ is a continuous operator in $V_\theta$.
Let us stress out that the proof of the previous lemma yields the existence of a constant $C'>0$ such that $
{\text{var}_{\theta}}({\mathcal{L}}g) \leq C_2\frac1{\theta}{\text{var}_{\theta}}(g) +C' \|g\|_\infty,
$ for every small $\theta$ . However, the term $\frac1{\theta}{\text{var}_{\theta}}(g)$ increases as $\theta$ gets smaller. In particular, the smaller $\theta$ is the higher the oscillations that may occur in elements of $V_\theta$.
Spectral gap and decay of correlations {#subsec.spectral.gap}
--------------------------------------
Here we prove a Lasota-Yorke inequality for the Ruelle-Perron-Frobenius operator. This will finally imply on exponential decay of correlations and central limit theorem for the equilibrium state. Throughout, let $\theta$ be fixed and such that $$\begin{cases}\tag{$\star$}
\theta \, e^{\log \deg(f)+ \inf\phi}>L^{\alpha}>1 \\
\theta \, e^{\log \deg(f)+ 2\inf\phi-\sup\phi}>L^{\alpha}>1 \\
\theta \, e^{\log \deg(f)+\sup\phi} e^{-c \tau {\alpha}}<1
\end{cases}$$ Some comments on $(\star)$ are in order. Notice that the third condition is the one required in Lemma \[l.continuous.operator\]. On the other hand, our choice of the parameter $\theta$ impose certain restrictions on the potential $\phi$. We are now in a position to state our second small variation condition on $\phi$: $$\label{eq.extra.conditions.potential}\tag{H3b}
\sup\phi-\inf\phi< \frac12 (c\tau-\log L){\alpha}$$
This condition is clearly satisfied by an open class of nearly constant potentials. Indeed, by construction, $c\tau \ge 2c^2/\log
L>\log L$. Note also that legitimates the choice in $(\star)$.
Let $r_\theta({\mathcal{L}}_\phi)$ and $r({\mathcal{L}}_\phi)$ denote, respectively, the spectral radius of ${\mathcal{L}}_\phi$ in the Banach spaces $V_\theta$ and $C(M)$. Since $\|{\mathcal{L}}^n 1\|_\theta \geq \|{\mathcal{L}}^n 1\|_\infty$ for every $n \in \mathbb N$ then clearly $$r_\theta({\mathcal{L}}_\phi)
= \lim_{n\to\infty} (\|{\mathcal{L}}_\phi^n\|_\theta)^\frac1n
\geq \lim_{n\to\infty} (\|{\mathcal{L}}_\phi^n 1\|_\infty)^\frac1n
\geq \deg(f) e^{\inf\phi},$$ which proves that $\deg(f)e^{\inf\phi}$ is simultaneously a lower bound for both spectral radius $r_\theta({\mathcal{L}}_\phi)$ and $r({\mathcal{L}}_\phi)$. For the time being, let ${\lambda}_0$ denote [**any**]{} positive number larger than $\deg(f)e^{\inf\phi}$. Observe that the previous choice of $\theta$ guarantees that $\theta {\lambda}_0>L^{\alpha}>1$ and $\theta {\lambda}_0 e^{-(\sup\phi-\inf\phi)}>L^{\alpha}>1$.
\[p.Lasota.Yorke\] There are $B_1$ and $\xi \in (0,1)$ such that, for every large $k
\geq 1$ there is $B_2(k)>0$ satisfying $${\text{var}_{\theta}}({\lambda}_0^{-k}{\mathcal{L}}^k g)
\leq B_1 \xi^k {\text{var}_{\theta}}(g)+B_2(k) \|g\|_{L^1(\nu)}.$$
Let $k \geq 1$ be fixed and let $Q_k$ denote the elements of the partition ${{\mathcal{Q}}^{(k)}}$. Observe that $${\mathcal{L}}^k g = \sum_{Q_k} e^{(S_k \phi) \circ f^{-k}_{Q_k}}\;
g \circ f^{-k}_{Q_k} \; 1_{f^k(Q_k)}.$$ Given $g \in V_\theta$, using ${\overline}\sup(g,Q_k)$ to majorate ${\overline}\sup(g,Q_{n+k})$ and Lemma \[l.sup.osc.L1\], it is not hard to see that ${\text{var}_{\theta}}({\lambda}_0^{-k}{\mathcal{L}}^k g)$ is bounded from above by the sum of the following three terms: $$\label{ub1}
({\lambda}_0\theta)^{-k} \sum_{n=0}^\infty \theta^{n+k} \sum_{Q_n} \sum_{Q_k}
e^{S_n\phi(Q_n)}{\operatornamewithlimits{osc}}(e^{S_k\phi},Q_{n+k})\,{\overline}{\operatornamewithlimits{osc}}(g,Q_k),$$ $$\label{ub2}
({\lambda}_0\theta)^{-k} \sum_{n=0}^\infty \theta^{n+k} \sum_{Q_n} \sum_{Q_k}
e^{S_n\phi(Q_n)}{\operatornamewithlimits{osc}}(e^{S_k\phi},Q_{n+k})\frac{1}{\nu(Q_k)}\int_{Q_k} g \,d\nu,$$ and $$\label{ub3}
({\lambda}_0\theta)^{-k} \sum_{n=0}^\infty \theta^{n+k} \sum_{Q_n} \sum_{Q_k}
e^{S_n\phi(Q_n)} e^{S_k\phi(Q_{n+k})} \,{\overline}{\operatornamewithlimits{osc}}(g,Q_{n+k}).$$ We deal with these three terms separately. First we treat by rewriting it as $$({\lambda}_0\theta)^{-k} \sum_{n=0}^\infty
\Big[ \theta^n \sum_{Q_n} e^{S_n\phi(Q_n)} \Big]
\Big[ \theta^k \sum_{Q_k} {\operatornamewithlimits{osc}}(e^{S_k\phi},Q_{n+k}){\overline}{\operatornamewithlimits{osc}}(g,Q_k) \Big]$$ and dividing the sum over elements in ${{\mathcal{Q}}^{(n)}}$ according to wether they belong or not to $B(n)$. Using ${\overline}{\operatornamewithlimits{osc}}(g,E) \leq 2
{\overline}\sup(g,E)$ it follows that is bounded by the sum of the two following terms: $$({\lambda}_0\theta)^{-k} \sum_{n=0}^\infty
\Big[ \theta^n \sum_{Q_n \in B(n)} e^{S_n\phi(Q_n)} \Big]
\Big[ \theta^k \sum_{Q_k} 2\sup(e^{S_k\phi},Q_k) \, {\overline}{\operatornamewithlimits{osc}}(g,Q_k) \Big]$$ and $$({\lambda}_0\theta)^{-k} \sum_{n=0}^\infty
\Big[ \theta^n \sum_{Q_n \not\in B(n)} e^{S_n\phi(Q_n)} \Big]
\Big[ \theta^k \sum_{Q_k} C_0(k)\sup(e^{S_k\phi},Q_k) e^{-c\tau{\alpha}n}\, {\overline}{\operatornamewithlimits{osc}}(g,Q_k)\Big],$$ where $C_0(k)$ is given by in Lemma \[l.oscillation.concatenation\]. Our choice on $\theta$ yields that the two previous terms are bounded from above by $C_0(k)({\lambda}_0\theta)^{-k}{\text{var}_{\theta}}(g)$ up to finite multiplicative constants. The constants involved are $2 \sum_{n=0}^\infty (\theta
e^{\log q +\sup\phi+{\varepsilon}_0})^n$ and $\sum_{n=0}^\infty (\theta
e^{\log \deg(f) +\sup\phi} e^{-c\tau{\alpha}})^n$, respectively.
On the one hand, is clearly bounded by $\|g\|_{L^1(\nu)}$ up to a multiplicative term obtained as the sum over all $n \geq 0$ of $${\lambda}_0^{-k} \max \nu(Q_k)^{-1} \, \theta^n \sum_{Q_n} \sum_{Q_k}
e^{S_n\phi(Q_n)}{\operatornamewithlimits{osc}}(e^{S_k\phi},Q_{n+k}).$$ Since the measure $\nu$ gives positive weight to any cylinder in ${{\mathcal{Q}}^{(k)}}$, this shows that there exists a positive constant $K_0(k)$ such that is bounded from above by $\|g\|_{L^1}$ up to the multiplicative term $$K_0(k) \sum_{n=0}^\infty \theta^n \sum_{Q_n} \sum_{Q_k}
e^{S_n\phi(Q_n)}{\operatornamewithlimits{osc}}(e^{S_k\phi},Q_{n+k}).$$ The part of the sum involving elements $Q_n \in{{\mathcal{Q}}^{(n)}}$ that belong to $B(n)$ is finite because those elements grow exponentially slow compared with the allowed size of the oscillations. Indeed, $$\sum_{n=0}^\infty \theta^n \sum_{Q_n \in B(n)} \sum_{Q_k}
e^{S_n\phi(Q_n)}{\operatornamewithlimits{osc}}(e^{S_k\phi},Q_{n+k})
\leq
2\#{{\mathcal{Q}}^{(k)}}e^{k \sup\phi} \sum_{n=0}^\infty \big(\theta e^{\log
q+\sup\phi+{\varepsilon}_0}\big)^n$$ is finite. In turn, the sum over elements $Q_n$ that do not belong $B(n)$ is also finite by Lemma \[l.oscillation.concatenation\]: $$\begin{aligned}
\sum_{n=0}^\infty \theta^n & \sum_{Q_n \not\in B(n)} \sum_{Q_k}
e^{S_n\phi(Q_n)}{\operatornamewithlimits{osc}}(e^{S_k\phi},Q_{n+k})
\\
& \leq \sum_{n=0}^\infty \theta^n \sum_{Q_n \not\in B(n)} \sum_{Q_k}
e^{S_n\phi(Q_n)} C_0 e^{S_k \phi(Q_{n+k})}\,e^{-c \tau {\alpha}n} \\
& \leq C_0 \#{{\mathcal{Q}}^{(k)}}e^{k \sup \phi} \sum_{n=0}^\infty \big[\theta
e^{\log \deg(f)+\sup\phi} \,e^{-c \tau {\alpha}}\big]^n <\infty.\\\end{aligned}$$ This shows that is bounded from above by $\|g\|_{L^1}$ up to a multiplicative constant $B_2(k)$. On the other hand Lemma \[l.sup.concatenation\] ensures that is bounded by $$\begin{aligned}
({\lambda}_0\theta e^{-(\sup\phi-\inf\phi)})^{-k} &
\sum_{n=0}^\infty \theta^{n+k}\sum_{Q_{n+k} \in {\mathcal{Q}}^{(n+k)}}
e^{S_{n+k}\phi(Q_{n+k})} \,{\overline}{\operatornamewithlimits{osc}}(g,Q_{n+k})\\
& \leq ({\lambda}_0\theta e^{-(\sup\phi-\inf\phi)})^{-k} {\text{var}_{\theta}}(g).\end{aligned}$$ It follows that $
{\text{var}_{\theta}}({\lambda}_0^{-k}{\mathcal{L}}^k g) \leq B_1 \xi^k {\text{var}_{\theta}}(g)+ B_2(k) \|g\|_{L^1},
$ for a constant $\xi$ is given by $
\xi =\max\{({\lambda}_0\theta)^{-1},({\lambda}_0\theta e^{-(\sup\phi-\inf\phi)})^{-1}\} \frac1k \log C_0(k).
$ Our first and second conditions on $\theta$ imply that $\xi$ is strictly smaller than one. This completes the proof of the proposition.
As a direct consequence we obtain the following:
\[c.Lasota.Yorke\] There are positive constanst $D_1, D_2$ and $\xi_1 \in (0,1)$ such that $${\text{var}_{\theta}}({\lambda}_0^{-n}{\mathcal{L}}^n g)
\leq D_1 \xi_1^n {\text{var}_{\theta}}(g)+D_2 \|g\|_{L^1(\nu)}.$$ for every $n \geq 1$.
Let $B_1$ and $\xi$ be given as in the previous proposition. Pick $k\geq 1$ such that $\xi_1:=\sqrt[k]{B_1 \xi^k}<1$ and, for any given $n \ge 1$, write $n=jk+r$ where $j$ is a positive integer and $0\leq r \leq k-1$. If one applies Proposition \[p.Lasota.Yorke\] recursively and note that ${\lambda}_0^{-1}{\mathcal{L}}$ does not increase the $L^(\nu)$-norm, because $\nu$ is conformal, it follows that $${\text{var}_{\theta}}({\lambda}_0^{-n}{\mathcal{L}}^n g)
\leq \xi_1^{kj} {\text{var}_{\theta}}({\lambda}_0^{-r}{\mathcal{L}}^r g) + B_2(k) \,\Big(\sum_{\ell \ge 0} \xi_1^\ell\Big) \,\|g\|_1.$$ Moreover, Proposition \[p.Lasota.Yorke\] also guarantees that $${\text{var}_{\theta}}({\lambda}_0^{-r}{\mathcal{L}}^n r)
\leq B_1 {\text{var}_{\theta}}(g) + \max_{1\le\ell\le k} B_2(\ell) \;\|g\|_1.$$ The corollary is then immediate taking $D_2=\max_{1\le\ell\le k}
B_2(\ell) +B_2(k) \,\Big(\sum_{\ell \ge 0} \xi_1^\ell\Big)$ and $D_1=B_1 \xi_1^{-k}$.
We proceed to prove that the Ruelle-Perron-Frobenius ${\mathcal{L}}_\phi$ is quasi-compact in the functional space $V_\theta$ for any parameter $\theta$ as above. Since ${\lambda}:=r({\mathcal{L}}_\phi)\geq \deg(f) e^{\inf\phi}$ the previous results hold with ${\lambda}_0={\lambda}$. First we show that the iterates of ${\lambda}^{-1}{\mathcal{L}}_\phi$ are well approximated by operators of finite rank. Let $A_n$ be the linear operator defined in $V_\theta$ by $$A_n(g)= {\lambda}^{-n}{\mathcal{L}}^n \Big(\mathbb E_\nu(g\mid {\mathcal{Q}}^{(n)})\Big)$$ for every $g \in V_\theta$, where $\mathbb E_\nu( \cdot \mid
{\mathcal{Q}}^{(n)})$ stands for the conditional expectation with respect to the partition ${\mathcal{Q}}^{(n)}$. Since the partitions ${{\mathcal{Q}}^{(n)}}$ have finitely many elements then it is not hard to see that each $A_n$ is a linear operator of finite rank, hence compact. In addition,
\[l.compact.approximation\] There is $C>0$ and $\xi_1 \in (0,1)$ such that $$\|{\lambda}^{-n} {\mathcal{L}}^n-A_n\|_\theta \leq C \xi_1^n.$$
First we bound the $L^\infty(\nu)$ part in $\|\cdot\|_\theta$. Given $g \in V_\theta$, $$\begin{aligned}
\|{\lambda}^{-n} {\mathcal{L}}^n g - A_n g &\|_\infty
= {\lambda}^{-n} \| {\mathcal{L}}^n \big( g -\mathbb E (g \mid {{\mathcal{Q}}^{(n)}})\big)\|_\infty \\
& = {\lambda}^{-n} \big\| \sum_{Q_n} e^{(S_n\phi)\circ f_{Q_n}^{-n}} \,
\big[ g\circ f_{Q_n}^{-n}-\mathbb E_\nu(g \mid {{\mathcal{Q}}^{(n)}})\circ f_{Q_n}^{-n}
\big] 1_{f^n(Q_n)} \,\big\|_\infty.\end{aligned}$$ Moreover, for any $Q_n \in {{\mathcal{Q}}^{(n)}}$ the difference between $g$ and $\mathbb E_\nu(g \mid {{\mathcal{Q}}^{(n)}})$ computed over the preimages of $f_{Q_n}^{-n}$ in the term above satisfies $$\big| g\circ f_{Q_n}^{-n}(x)-\mathbb E_\nu(g \mid \!{{\mathcal{Q}}^{(n)}})\circ f_{Q_n}^{-n}(x)\big|
\!\!\leq \frac{1}{\nu(Q_n)} \int_{Q_n} | g(f_{Q_n}^{-n}(x))-g(z) | \, d\nu(z)
\!\!\leq {\overline}{\operatornamewithlimits{osc}}(g,Q_n).$$ In particular, we deduce that the $L^\infty$ term involved in the computation of the norm $\|\cdot\|_\theta$ decreases exponentially fast: $$\|{\lambda}^{-n} {\mathcal{L}}^n g-A_n g\|_\infty
\leq{\lambda}^{-n} \sum_{Q_n} e^{S_n\phi(Q_n)} \, {\overline}{\operatornamewithlimits{osc}}(g,Q_n)
\leq (\theta{\lambda})^{-n} {\text{var}_{\theta}}(g).$$ The variation term in $\|\cdot\|_\theta$ can be bounded analogously, using Lemma \[l.oscillation.inequality\] as in the proof of the Lasota-Yorke inequality. Indeed, $$\begin{aligned}
{\text{var}_{\theta}}({\lambda}^{-n} {\mathcal{L}}^n g-A_n g) &
\leq {\lambda}^{-n} \sum_{k=0}^\infty \theta^k \sum_{Q_k} \sum_{Q_n}
e^{S_k\phi(Q_k)} {\overline}{\operatornamewithlimits{osc}}(e^{S_n\phi},Q_{n+k}) {\overline}\sup({\overline}g_n, Q_{n+k})
\\ & \\
& + {\lambda}^{-n} \sum_{k=0}^\infty \theta^k \sum_{Q_k} \sum_{Q_n}
e^{S_k\phi(Q_k)} e^{S_n\phi(Q_{n+k})} {\overline}{\operatornamewithlimits{osc}}({\overline}g_n,Q_{n+k}),\end{aligned}$$ where ${\overline}g_n = g -\mathbb E(g\,|\,{{\mathcal{Q}}^{(n)}})$ and $Q_{n+k}=f^{-n}_{Q_n}(Q_k)$. Since $\mathbb E_\nu(g \,|\,{{\mathcal{Q}}^{(n)}})$ is constant over the elements in ${{\mathcal{Q}}^{(n)}}$, clearly ${\overline}{\operatornamewithlimits{osc}}({\overline}g_n,Q_{n+k})={\overline}{\operatornamewithlimits{osc}}( g,Q_{n+k})$. In consequence the second term in the right hand side of the sum above coincides with , which in turn is bounded by $({\lambda}\theta
e^{-(\sup\phi-\inf\phi)})^{-n}{\text{var}_{\theta}}(g)$. On the other hand, the first term above can be bounded as in by $C({\lambda}\theta
L^{-{\alpha}})^{-n} {\text{var}_{\theta}}(g)$, for some positive constant $C$ that does not depend on $n$, because $${\overline}\sup({\overline}g_n, Q_{n+k})
\leq {\overline}\sup({\overline}g_n, Q_{n})
\leq {\overline}{\operatornamewithlimits{osc}}({\overline}g_n, Q_{n}) + \frac{1}{\nu(Q_n)} \int_{Q_n} {\overline}g_n \,d\nu$$ and $\int_{Q_n} {\overline}g_n \, d\nu=0$. In consequence, there exists $C>0$ and $0<\xi<1$ such that $${\text{var}_{\theta}}({\lambda}^{-n} {\mathcal{L}}^ng-A_n g)
\leq C \xi^n \|g\|_\theta.$$ Since this $\theta$-variation term also decreases exponentially fast as $n$ tends to infinity, this completes the proof of the lemma.
An interesting consequence of the previous lemma is that the spectral radius of the Ruelle-Perron-Frobenius operator ${\mathcal{L}}_\phi$ in the Banach spaces $C(M)$ and $V_\theta$ do coincide.
$r_\theta({\mathcal{L}}_\phi)=r({\mathcal{L}}_\phi)$.
The spectral radius $r({\mathcal{L}})$ of the linear operator ${\mathcal{L}}_\phi$ in the space $C(M)$ of continuous functions is clearly greater or equal than $\deg(f) e^{\inf\phi}$. Thus, the Lasota-Yorke inequality in Corollary \[c.Lasota.Yorke\] with ${\lambda}=r({\mathcal{L}})$ guarantees that there exists a uniform constant $C>0$ such that ${\text{var}_{\theta}}({\lambda}^{-n}
{\mathcal{L}}^{n} g) \leq C \|g\|_\theta$ for every $n \in \mathbb N$. Using $\|\cdot\|_1 \leq \|\cdot\|_\infty$, this proves that there exists a uniform constant $C>0$ such that $\|{\lambda}^{-n} {\mathcal{L}}^{n} g\|_\theta \leq
C \|g\|_\theta$ for every $g\in V_\theta$ and $n \in \mathbb N$. In consequence, the spectral radius $r_\theta({\lambda}^{-1}{\mathcal{L}}_\phi)$ of ${\mathcal{L}}_\phi$ in $V_\theta$ verifies $$r_\theta({\lambda}^{-1} {\mathcal{L}}_\phi) \leq 1.$$ On the other hand, using once more the conformality of the measure $\nu$, we get $$\|{\lambda}^{-n}{\mathcal{L}}^n 1\|_\theta
\geq \|{\lambda}^{-n}{\mathcal{L}}^n 1\|_\infty
\geq \|{\lambda}^{-n}{\mathcal{L}}^n 1\|_{L^1(\nu)}
=1$$ for every integer $n\geq 1$, which proves that $r_\theta({\lambda}^{-1}{\mathcal{L}}) \geq 1$. The two estimates above imply $
r_\theta({\lambda}^{-1}{\mathcal{L}}_\phi)
={\lambda}^{-1}r_\theta({\mathcal{L}}_\phi)
=1,
$ which shows that $r_\theta({\mathcal{L}}_\phi)={\lambda}=r({\mathcal{L}}_\phi)$ and completes the proof of the lemma.
We are now in a position to prove the quasi-compactness of the operator ${\lambda}^{-1}{\mathcal{L}}$ and, moreover, that it has a spectral gap.
$r_\theta({\lambda}^{-1}{\mathcal{L}}_\phi)=1$ and the spectrum ${\sigma}({\lambda}^{-1}{\mathcal{L}}_\phi)$ of the operator ${\lambda}^{-1}{\mathcal{L}}_\phi$ in $V_\theta$ satisfies $${\sigma}({\lambda}^{-1}{\mathcal{L}}_\phi)
\subseteq \big\{ z \in \mathbb C : |z| \leq 1\big\}.$$ Moreover, $1$ is a simple eigenvalue for ${\lambda}^{-1}{\mathcal{L}}_\phi$, there are no other eigenvalues of modulus one and the essential spectral radius $r_{\text{ess}}({\lambda}^{-1} {\mathcal{L}}_\phi)$ is strictly smaller than one. Furthermore, the density $d\mu/d\nu$ belongs to $V_\theta$.
Using Naussbaum’s formula for the essential spectral radius (see e.g. [@DSI] Page 709), which asserts that $$r_{\text{ess}}({\lambda}^{-1} {\mathcal{L}}_\phi)
= \lim_{n \to \infty} (\inf \{\|{\lambda}^{-n}{\mathcal{L}}^n-L\|_\theta : L \,\text{is compact operator}\,\})^\frac1n,$$ and Lemma \[l.compact.approximation\] it follows that $r_{\text{ess}}({\lambda}^{-1} {\mathcal{L}}_\phi) \leq \xi_1$ is strictly smaller than one. Hence, there is only a finite number of eigenvalues with finite-dimensional eigenspaces in $\{ z \in \mathbb {\sigma}({\lambda}^{-1}
{\mathcal{L}}) : |z| > r_{\text{ess}}\}$. Since $r_\theta({\lambda}^{-1}
{\mathcal{L}}_\phi)=1$ there must exist some eigenvalue on the unit circle, and we can write $${\lambda}^{-1}{\mathcal{L}}_\phi =\Pi_1
+ \sum_{\substack{z\in {\sigma}({\lambda}^{-1} {\mathcal{L}}_\phi) \\ |z| =1}} z \,\Pi_{z}
+ L_0$$ where $\Pi_z$ denotes the projection on the subspace associated to the eigenvalue $z \in \mathbb C$ and $r(L_0)<1$. Using that $\sum_{j=0}^{n-1} z^j$ is uniformly bounded in norm for every $n$ it follows that $$\Big\| \frac1n \sum_{j=0}^{n-1} {\lambda}^{-j}{\mathcal{L}}_\phi^j - \Pi_1
\Big\|_\theta \xrightarrow[n \to \infty]{} 0.$$ On the other hand, using $\|\cdot\|_\theta \geq \|\cdot\|_\infty
\geq \|\cdot\|_{L^1(\nu)}$ one gets $$\big\| \frac1n \sum_{j=0}^{n-1} {\lambda}^{-j}{\mathcal{L}}_\phi^j 1\big\|_\theta
\geq \big\| \frac1n \sum_{j=0}^{n-1} {\lambda}^{-j}{\mathcal{L}}_\phi^j 1 \big\|_{L^1(\nu)}
=1, \quad\text{for every $n \in \mathbb N$.}$$ This shows that $\Pi_1$ is nonzero and that $h=\Pi_1(1)$ is an eigenfunction for ${\lambda}^{-1}{\mathcal{L}}_\phi$ associated to the eigenvalue $1$. Up to a normalization in $L^1(\nu)$ it is not difficult to see that $\hat \mu=h\nu$ is an $f$-invariant probability measure: for every $g \in C(M)$ $$\int g\circ f \,d\hat \mu
= \int {\lambda}^{-1} {\mathcal{L}}_\phi( g\circ f h) \,d\nu
= \int {\lambda}^{-1} {\mathcal{L}}_\phi(h) g \,d\nu
= \int g \,d\hat \mu.$$ Since $\hat\mu$ is absolutely continuous with respect to $\nu$ then it is an equilibrium state. By uniqueness of the equilibrium state, $\hat\mu$ must coincide with $\mu$ and, in particular, $d\mu/d\nu=h$ belongs to $V_\theta$. In fact the same argument yields that $1$ is a simple eigenvalue, thus, the only eigenvalue of modulus one.
It remains only to rule out the existence of other eigenvalues of modulus one distinct from $1$. Let $z \in \mathbb C$ and $h' \in
V_\theta$ be such that ${\lambda}^{-1}{\mathcal{L}}_\phi h'=z h'$ and $|z|=1$. Since $h$ is bounded from below by some constant $C_1>0$ the function $\psi$ defined by $h'=h \psi$ belongs to $L^2(\mu)$ because $
\|\psi\|_{L^2(\mu)}
\leq \|h'\|_\infty^2 / {C_1^2}
<\infty.
$ The Koopman operator $U:L^2(\mu) \to L^2(\mu)$ acts on each $g \in
L^2(\mu)$ by $U(g)=g \circ f$. By construction $\psi$ is an eigenfunction for the dual operator $U^*$. Indeed, $U^*(\psi)=z
\psi$ because $$ (U\^\*) g d = (gf) d = h’(gf) d = \^[-1]{} \_(h’(gf)) d = (z h’) g d. $$ In consequence, $\psi=z^n U^n(\psi)=z^n \,(\psi \circ f^n)$ is measurable with respect to the sigma-algebra $f^{-n}({\mathcal{B}})$ for every $n \geq 1$. Since $\mu$ is exact (recall Lemma \[l.exactness\]) then $\psi$ must be constant, which proves that $h'$ belongs to the subspace generated by $h$ and consequently $z=1$. This shows that $1$ is the only eigenvalue of modulus one and completes the proof of the proposition.
\[c.correlation.decay\] There are $C>0$, $\xi \in (0,1)$ such that every $\Phi \in L^1(\nu)$ and $\Psi \in V_\theta$ satisfy $ C_n(\Phi,\Psi) \leq C \xi^n
\|\Psi\|_\theta \|\Phi\|_{L^1(\nu)}$. Moreover, $$\Big|\mu(Q_k \cap f^{-n}(Q_l)) - \mu(Q_k) \mu(Q_l)\Big|
\leq C \xi^n \mu(Q_l)$$ for every $n\ge 1$ and every pair of cylinders $Q_l \in {\mathcal{Q}}^{(l)}$ and $Q_k \in {{\mathcal{Q}}^{(k)}}$.
Given $\Phi \in L^1(\nu)$ and $\Psi \in V_\theta$, $$\int \Phi (\Psi \circ f^n) d\mu
= \int \Phi h (\Psi \circ f^n) d\nu
= \int ({\lambda}^{-n}{\mathcal{L}}^n)(\Phi h) \, \Psi \,d\nu$$ for every $n \geq 1$. Hence, $$\begin{gathered}
\Big| \int \Phi (\Psi \circ f^n) d\mu - \int \Phi d\mu \int \Psi d\mu \Big|
= \Big| \int \Big[ ({\lambda}^{-n}{\mathcal{L}}^n)(\Phi h)- h \int (\Phi h) \,d\nu \Big]
\, \Psi \,d\nu \Big| \\
\leq
\Big\| ({\lambda}^{-n}{\mathcal{L}}^n)(\Phi h)- h \int (\Phi h) \,d\nu
\Big\|_\theta \|\Psi\|_{L^1(\nu)}.\end{gathered}$$ Since $h \int (\Phi h) \,d\nu$ is the projection of $\Phi h$ in the one-dimensional eigenspace associated to the eigenvalue $1$ it is a consequence of the spectral gap that the previous term is bounded by $C \xi^n \|\Psi\|_\theta \|\Phi\|_{L^1(\nu)}$. On the other hand, the second claim is an immediate consequence of the first one provided that we show that the characteristic function $1_{Q}$ of a cylinder $Q \in {\mathcal{Q}}^{(k)}$ belongs to $V_\theta$. Since ${\overline}{\operatornamewithlimits{osc}}(\cdot) \leq 2 {\overline}\sup(\cdot)$, it is clear that $${\text{var}_{\theta}}(1_{Q})
= \sum_{n \geq 0} \theta^n \sum_{Q_n} e^{S_n\phi(Q_n)}
{\overline}{\operatornamewithlimits{osc}}(1_{Q}, Q_n) \leq 2 \sum_{n \geq 0} (\theta
e^{\sup\phi})^n$$ is finite. In consequence $\|1_{Q}\|_\theta= 1+{\text{var}_{\theta}}(1_{Q})$ is also finite, which proves our claim and finishes the proof of the corollary.
Finally, to complete the proof of Corollary \[c.CLT\] it is enough to prove the following:
\[l.Vtheta.contains.Holder\] $V_\theta$ contains the space of $\alpha$-Hölder observables.
Let $g$ be an $\alpha$-Hölder continuous observable. Since $\|g\|_\infty$ is finite it remains to estimate ${\text{var}_{\theta}}(g)$. Indeed, dividing the sum of the elements in ${{\mathcal{Q}}^{(n)}}$ according to wether they belong to $B(n)$ or not, it is not hard to check that there is $C_g>0$ such that $${\text{var}_{\theta}}(g)
\leq 2 \|g\|_\infty \sum_{n \geq 0} \theta^n \# B(n) e^{\sup\phi n}
+ C_g \sum_{n \geq 0} \theta^n e^{(\log\deg(f)+\sup\phi-c\tau{\alpha})n}.$$ This proves the lemma.
Exponential distribution of hitting times {#sec.exponentialdistribution}
=========================================
In this section we combine ideas from [@GaSc97] and [@Pac00] with the weak Gibbs property and the strong mixing properties of the equilibrium state $\mu$ to study the hitting times asymptotics in Theorem \[thm.first.entrance.distribution\]. For notational simplicity, given $Q\in{{\mathcal{Q}}^{(n)}}$ we set $$g_Q(t)=\mu\Big(\tau_Q > \frac{t}{\mu(Q)} \Big).$$ The strategy to prove Theorem \[thm.first.entrance.distribution\] is to consider a large subset ${{\mathcal{Q}}^{(n)}}_{\varepsilon}$ of $n$-cylinders such that $g_Q(\sqrt{\mu(Q)})$ behaves essentially as $\exp(-\mu(Q))$ for every $Q \in {{\mathcal{Q}}^{(n)}}_{\varepsilon}$ and to explore the strong mixing property of $\mu$ to obtain many instants of independence. We will need some preliminary results.
[@GaSc97 Lemma 2]\[l.Pac1\] For every measurable set $E$ and all positive $t$, $$\mu\Big(\tau_E \leq \frac{t}{\mu(E)}\Big) \leq t + \mu(E).$$
If $\gamma_1>0$ is given by Proposition \[p.decreasing.measure\] then we have the following:
\[l.special.cylinders\] There is $K>0$ such that for any ${\varepsilon}>0$ there is a subset ${{\mathcal{Q}}^{(n)}}_{\varepsilon}$ of $n$-cylinders of measure at least $1-{\varepsilon}$ and satisfying $$\label{eQ}
e^{-\sqrt{\mu(Q)} (1 + K e^{-{\gamma}_1 n/2})}
\leq
g_Q(\sqrt{\mu(Q)})
\leq
e^{-\sqrt{\mu(Q)} (1 - K e^{-{\gamma}_1 n/2})}$$ for every $Q \in {{\mathcal{Q}}^{(n)}}_{\varepsilon}$ and every large $n$.
First observe that if $n$ is large enough and $Q\in {{\mathcal{Q}}^{(n)}}$ arbitrary $$\begin{aligned}
-\log g_Q(\sqrt{\mu(Q)})
& = - \log \Big[1- \mu\Big(\tau_Q\leq \frac{\sqrt{\mu(Q)}}{\mu(Q)}\Big)\Big] \\
& \leq \mu\Big(\tau_Q\leq\frac{\sqrt{\mu(Q)}}{\mu(Q)}\Big) +
\Big[\mu\Big(\tau_Q\leq
\frac{\sqrt{\mu(Q)}}{\mu(Q)}\Big)\Big]^2.\end{aligned}$$ Then Proposition \[p.decreasing.measure\] and Lemma \[l.Pac1\] imply that the later sum is bounded from above by $\sqrt{\mu(Q)}(1+K
e^{-{\gamma}_1 n/2})$ for some positive constant $K$, which proves the lower bound in .
In the other direction, let ${{\mathcal{Q}}^{(n)}}_{\varepsilon}$ be the family of $n$-cylinders that have no self returns in the time interval $[1,\zeta n]$ for some $\zeta>0$: the $n$-cylinder $Q$ belongs to ${{\mathcal{Q}}^{(n)}}_{\varepsilon}$ if $f^j(Q)$ does not intersect $Q$ for every $1 \leq j
\leq \zeta n$. Any element in ${{\mathcal{Q}}^{(n)}}\setminus {{\mathcal{Q}}^{(n)}}_{\varepsilon}$ has short recurrence and is completely characterized by $\zeta n$ cylinders of the partition ${\mathcal{Q}}$. Consequently, $\# [{{\mathcal{Q}}^{(n)}}\setminus {{\mathcal{Q}}^{(n)}}_{\varepsilon}]\leq (\#{\mathcal{Q}})^{\zeta n}$ for every $n$. In particular, if $\zeta=\zeta({\varepsilon})$ is chosen small enough then $$\mu\Big( \cup\{Q \in {{\mathcal{Q}}^{(n)}}: Q \not\in {{\mathcal{Q}}^{(n)}}_{\varepsilon}\} \Big)
\leq (\# {\mathcal{Q}})^{\zeta n} e^{-{\gamma}_1 n}<{\varepsilon}$$ for every large $n$. We claim that every $Q\in {{\mathcal{Q}}^{(n)}}_{\varepsilon}$ verifies the upper bound in . On the one hand $$-\log g_Q(\sqrt{\mu(Q)})
\geq 1 - g_Q(\sqrt{\mu(Q)})
= \mu\Big(\tau_Q\leq \frac{\sqrt{\mu(Q)}}{\mu(Q)}\Big).$$ Following [@GaSc97 Lemma 3] and [@Pac00 Lemma 5.3], one gets $$\mu\Big(\tau_Q \leq \frac{t}{\mu(Q)}\Big)
\geq \frac{t^2}{t^2+\mu(Q)(1+t)+t(1+Ke^{-{\gamma}_1 n})}$$ for every $Q \in {{\mathcal{Q}}^{(n)}}_{\varepsilon}$. Together with the previous inequality this shows that $$-\log g_Q(\sqrt{\mu(Q)})
\geq \frac{\mu(Q)}{\mu(Q)[2+\sqrt{\mu(Q)}]+\sqrt{\mu(Q)}[1+Ke^{-{\gamma}_1 n}]}$$ which is larger than $\sqrt{\mu(Q)}(1-K e^{-{\gamma}_1 n/2})$. This completes the proof of the lemma.
The strong mixing property of $\mu$ guarantees some independence of the system.
\[l.independence\] There exists $C>$0 such that if $n \geq 1$ is sufficiently large then $$\sup_{s \geq \sqrt{\mu(Q)}}
\Big| g_Q(\sqrt{\mu(Q)}+s)- g_Q(\sqrt{\mu(Q)}) g_Q(s) \Big|
\leq C \mu(Q)^{3/4}$$ for every cylinder $Q \in {{\mathcal{Q}}^{(n)}}$.
This proof is similar to the one of [@Pac00 Lemma 5.4], which explores the strong mixing properties of the system. We include a brief sketch of the proof for completeness reasons.
Let $Q$ be any fixed cylinder of ${{\mathcal{Q}}^{(n)}}$. Given positive $t,s$ and a small $\Delta$ (to be chosen later on), by invariance of $\mu$ it follows that $|g_Q(t+s)-g_Q(t)\,g_Q(s)|$ is bounded from above by the sum of the following three terms: $$\label{eq.1}
\Bigg|g_Q(t+s)- \mu\Big( \tau_Q \notin [0,\frac{t}{\mu(Q)}] \cup [\frac{t}{\mu(Q)}+\Delta , \frac{t+s}{\mu(Q)}] \Big)\Bigg|,$$ $$\label{eq.2}
\Bigg|\mu\Big( \tau_Q \notin [0,\frac{t}{\mu(Q)}] \cup [\frac{t}{\mu(Q)}+\Delta , \frac{t+s}{\mu(Q)}] \Big)- g_Q(t) \; \mu\Big(\tau_Q \notin [\Delta , \frac{s}{\mu(Q)}] \Big)\Bigg|,$$ and $$\label{eq.3}
g_Q(t) \, \Bigg| \mu\Big(\tau_Q \notin [\Delta , \frac{s}{\mu(Q)}]
\Big) - \mu\Big(\tau_Q \notin [0 , \frac{s}{\mu(Q)}] \Big)\Bigg|.$$ Since is the measure of the set of points that do enter $Q$ in the time interval $[\frac{t}{\mu(Q)},
\frac{t}{\mu(Q)}+\Delta]$ then it is bounded by $\Delta \mu(Q)$. Similarly, is also bounded by $\Delta \mu(Q)$. For the remaining term, computations analogous to the ones in [@Pac00 page 356] guarantee that $$\eqref{eq.2}
= |\int g_1 \, (g_2\circ f^{{\Delta}+1}) \,d\mu -\int g_1 \,d\mu \; \int g_2 \,d\mu|
\leq C \xi^{\Delta+1} \|g_1\|_\theta \|g_2\|_1,$$ where $$g_1=1_{Q^c}\,\frac1h ({\lambda}^{-1}{\mathcal{L}}_{Q^c})^{[\frac{t}{\mu(Q)}]}(h)
\quad \text{and}\quad
g_2=\prod_{j=0}^{[\frac{s}{\mu(Q)}]-{\Delta}-1} 1_{Q^c} \circ f^j,$$ and ${\mathcal{L}}_Q(g)$ stands for ${\mathcal{L}}(g 1_Q)$. Clearly $\frac1h\in
V_\theta$, because $h \in V_\theta$ and $h$ is bounded away from zero. Observe that $\|g_2\|_1\le 1$ and that $\|{\lambda}^{-1}{\mathcal{L}}h\|_\infty =\|h\|_\infty$ is finite. Hence, using that ${\text{var}_{\theta}}(f_1
f_2) \le {\text{var}_{\theta}}(f_1) \|f_2\|_\infty+{\text{var}_{\theta}}(f_2) \|f_1\|_\infty$ then $$\begin{aligned}
\eqref{eq.2} & \leq C \xi^{{\Delta}+1} \Big\| 1_{Q^c}\,\frac1h ({\lambda}^{-1}{\mathcal{L}}_{Q^c})^{[\frac{t}{\mu(Q)}]}(h
)\Big\|_\theta\\
& \leq C \xi^{{\Delta}+1} \Bigg[
{\text{var}_{\theta}}(\frac1h) \Big\|1_{Q^c}\,({\lambda}^{-1}{\mathcal{L}}_{Q^c})^{[\frac{t}{\mu(Q)}]}(h
)\Big\|_\infty \\
& \qquad\qquad + \Big\|\frac1h \Big\|_\infty {\text{var}_{\theta}}\Big(1_{Q^c}\,
({\lambda}^{-1}{\mathcal{L}}_{Q^c})^{[\frac{t}{\mu(Q)}]}(h
)\Big) \Bigg] \\
& \leq C' \xi^{{\Delta}+1} \Big[1 + {\text{var}_{\theta}}\Big(({\lambda}^{-1}{\mathcal{L}}_{Q^c})^{[\frac{t}{\mu(Q)}]}(h
)\Big)\Big]\end{aligned}$$ for some positive constant $C'$. To estimate the term in the right hand side above we use (7) in page 358 of [@Pac00]: for every $N\ge 1$ $${\mathcal{L}}^N_{Q^c}(h)= h - \sum_{r=0}^{N-1} {\mathcal{L}}^r {\mathcal{L}}_{Q}(h)
+ \sum_{0\le i+j\le N-2} {\mathcal{L}}^i {\mathcal{L}}_{B_{i,j}}(h),$$ where $B_{i,j}= Q \cap f^{-1}(Q^c) \cap \dots \cap f^{-N+i+j+2}(Q^c)
\cap f^{-N+i+j+1}(Q)$ is a cylinder of order $n+N-i-j-1$ contained in $Q$. So, using the Lasota-Yorke inequality it is not hard to obtain $${\text{var}_{\theta}}({\lambda}^{-N}{\mathcal{L}}^N_{Q^c}(h)) \le C'' ( N+ N^2) \,{\text{var}_{\theta}}(h)$$ for some positive constant $C''$, and shows that $$|g_Q(t+s)-g_Q(t)\,g_Q(s)|
\le 2 \Delta \mu(Q) + C \xi^{\Delta+1} (1+2C''[\frac{t}{\mu(Q)}]^2)
\le C \mu(Q)^{\frac34}$$ for some $C>0$ provided that $t=\sqrt{\mu(Q)}$, $s\ge \sqrt{\mu(Q)}$ and $\Delta=\mu(Q)^{-\frac14}$. This completes the proof of the lemma.
We finish this section with the following:
Let $t \geq 0$, $n \geq 1$ and ${\varepsilon}>0$ be fixed, and let ${{\mathcal{Q}}^{(n)}}_{\varepsilon}$ be as in the previous lemma. Take $Q \in {{\mathcal{Q}}^{(n)}}_{\varepsilon}$ and set $k=[\frac{t}{\sqrt{\mu(Q)}}]$. The strategy is to divide the estimate on the distribution of the entrance time $g_Q(t)$ in blocks where some independence holds. Write $t=k \sqrt{\mu(Q)}+r$, with $0\leq r < \sqrt{\mu(Q)}$, and note that $$\begin{gathered}
\tag{$\dagger$}
\Big| g_Q(t)- e^{-t}\Big|
\leq \Big| g_Q(t)- g_Q(k \sqrt{\mu(Q)}) \Big|
+ \Big| g_Q(k \sqrt{\mu(Q)}) - g_Q(\sqrt{\mu(Q)} )^k \Big| \\
+ \Big| g_Q(\sqrt{\mu(Q)})^k - e^{-k \sqrt{\mu(Q)}}\Big|
+ \Big| e^{-k \sqrt{\mu(Q)}}- e^{-t} \Big|.\end{gathered}$$ The last term in the right hand side above is clearly bounded from above by $2 \sqrt{\mu(Q)}$, while we can use the invariance of $\mu$ to get $$\Big| g_Q(t)- g_Q(k \sqrt{\mu(Q)}) \Big|
= \mu\Big( \frac{k \sqrt{\mu(Q)}}{\mu(Q)}<\tau_Q \leq
\frac{t}{\mu(Q)} \Big)
\leq \mu\Big( 0 <\tau_Q \leq \frac{\sqrt{\mu(Q)}}{\mu(Q)} \Big),$$ which is bounded by $\sqrt{\mu(Q)}+\mu(Q)$ according to Lemma \[l.Pac1\]. Since $Q$ belongs to ${{\mathcal{Q}}^{(n)}}_{\varepsilon}$ then holds and the third term in $(\dagger)$ decays exponentially fast with $n$ because it is bounded from above by $$\begin{gathered}
2k \Big[ -\log g_Q(\sqrt{\mu(Q)}) - \sqrt{\mu(Q)}\Big] \;
\Big[ g_Q(\sqrt{\mu(Q)})^k + e^{-k \sqrt{\mu(Q)}} \Big] \\
\leq 2k \sqrt{\mu(Q)} K e^{-{\gamma}_1 n/2} \; \Big[ 2 e^{-k \sqrt{\mu(Q)} (1-Ke^{-{\gamma}_1 n/2})} \Big].
$$ Finally, we deal with the second term in the right hand side of $(\dagger)$. Indeed, it is not difficult (see e.g. [@GaSc97 Lemma 6]) to use induction in Lemma \[l.independence\] and obtain $$\label{eq.Galves.Schmitt}
\Big| g_Q(k \sqrt{\mu(Q)})- g_Q(\sqrt{\mu(Q)} )^k \Big|
\leq C \frac{\mu(Q)^{3/4}}{1-g_Q(\sqrt{\mu(Q)})}.$$ Using the last inequalities in the proof of Lemma \[l.special.cylinders\], every $Q\in{{\mathcal{Q}}^{(n)}}_{\varepsilon}$ satisfies $1-g_Q(\sqrt{\mu(Q)}) \geq \sqrt{\mu(Q)} (1-K e^{-{\gamma}_1 n})$, which guarantees that the second term in $(\dagger)$ satisfies $$\Big| g_Q(k \sqrt{\mu(Q)})- g_Q(\sqrt{\mu(Q)} )^k \Big|
\leq \frac{\mu(Q)^{3/4}}{\sqrt{\mu(Q)} (1-K e^{-{\gamma}_1 n})}
\leq \mu(Q)^{1/4}$$ and decreases exponentially fast with $n$. This completes the proof of the theorem.
Fluctuations of the return times {#sec.fluctuations}
================================
This section is devoted to the proof of Theorem \[thm.recurrence.fluctuations\]. We explore the exponential asymptotic distribution of hitting times, the weak Gibbs property of $\mu$ and the Central Limit Theorem to obtain the log-normal distribution of the return times. The following relation between hitting times and return times, similar to Lemma 4.1 in [@Pac00], is a consequence of the good mixing properties for $\mu$.
\[l.hitting.vs.recurrence\] Let $(t_n)$ be a sequence such that $\lim_{n\to\infty} t_n/n \to
+\infty$. Then $$\lim_{n\to\infty}
\Big| \mu (R_n>t_n) - \sum_{Q \in {{\mathcal{Q}}^{(n)}}} \mu(Q) \;\mu\Big(\tau_Q > t_n \Big) \Big|
=0.$$
Since $$\mu(R_n > t)
= \sum_{Q \in {{\mathcal{Q}}^{(n)}}} \mu(Q \cap \{\tau_Q > t\})$$ we will estimate the differences $\mu(Q \cap \{\tau_Q>t\})-\mu(Q)
\mu(\tau_Q>t)$ for elements $Q\in{{\mathcal{Q}}^{(n)}}$. In fact, given $k<n<r<t$ and $Q\in {{\mathcal{Q}}^{(n)}}$ write $$\begin{aligned}
(\ast) = \mu(Q \cap & \{\tau_Q>t\})-\mu(Q) \mu(\tau_Q>t) \nonumber \\
&\leq \mu(Q \cap \{\tau_Q>t\})-\mu(Q \cap \tau_Q\notin[r,t])\label{eq.indep1}\\
&+ \mu(Q \cap \tau_Q\notin[r,t])- \mu(Q) \mu(\tau_Q \notin[r,t]) \label{eq.indep2}\\
&+ \mu(Q) \Big[\mu(\tau_Q\notin[r,t])-\mu(\tau_Q>t)\Big] \label{eq.indep3}\end{aligned}$$ It is clear that coincides with $\mu(Q)
\mu(\tau_Q<r)$, which is bounded from above by $r \mu(Q)^2$, because $\mu(\tau_Q<r)\leq \mu(\cup_{j\leq r} f^{-j} Q)$. Moreover, by exponential decay of correlations $$|\eqref{eq.indep2}|
\leq |\mu( Q \cap f^{-r}(\cap_{j=0}^{t-r} f^{-j}(Q^c)) -\mu(Q) \mu(f^{-r}(\cap_{j=0}^{t-r}f^{-j}(Q^c)))|
\leq K \xi^r.$$ On the other hand, $|\eqref{eq.indep1}|$ is bounded from above by $\sum_{j=0}^r \mu(Q \cap f^{-j}(Q))$. To deal with this last term we will consider differently wether $Q$ belongs to the set $E_{<k}$ of $n$-cylinders such that there exists $0\leq i\leq k$ so that $f^{-i}(Q) \cap Q \neq \emptyset$, or not. In fact, summing up over elements in ${{\mathcal{Q}}^{(n)}}$ and using the exponential decay of correlations it follows that $$\begin{aligned}
\sum_{Q\in{{\mathcal{Q}}^{(n)}}} |\eqref{eq.indep1}|
\leq \sum_{Q\in E_{<k}} & \sum_{j=0}^r \mu(Q \cap f^{-j}(Q))
+ \sum_{Q\in E_{<k}^c} \sum_{j=0}^r \mu(Q \cap f^{-j}(Q))\\
&\leq r \mu (E_{<k}) +\sum_{Q\in E_{<k}^c} \sum_{j=k}^r [K' \xi^j +\mu(Q)] \;\mu(Q)\\
&\leq r \; \mu (E_{<k}) + r [K' \xi^k +e^{-{\gamma}_1 n}] \\\end{aligned}$$ where $K'$ is a constant that involves the constant $K$ from decay of correlations and an upper bound for $\|1_Q\|_\theta$. Using that $\mu (E_{<k}) \leq \#{{\mathcal{Q}}^{(k)}}e^{-{\gamma}_1 n}$ and the previous estimates we obtain $$\sum_{Q\in{{\mathcal{Q}}^{(n)}}} |(\ast)|
\leq r e^{-{\gamma}_1 n}
+ K \# {{\mathcal{Q}}^{(n)}}\xi^r
+ r \; \#{{\mathcal{Q}}^{(k)}}e^{-{\gamma}_1 n} + r [K' \xi^k +e^{-{\gamma}_1 n}].$$ The previous inequality holds for $r(n)=\min\{t_n, n^2\}$ and $k(n)=\Big[\frac{{\gamma}_1}{2\log\deg(f)} n \Big]$. In fact, we get $$\begin{aligned}
\Big| \mu & (R_n>t_n) - \sum_{Q \in {{\mathcal{Q}}^{(n)}}} \mu(Q) \;\mu\Big(\tau_Q > t_n \Big) \Big| \\
& \leq n^2 e^{-{\gamma}_1 n}
+ K \# {\mathcal{Q}}\deg(f)^n \xi^{r(n)}
+ n^2 \; \#{\mathcal{Q}}e^{-{\gamma}_1 n/2} + n^2 [K' \xi^{k(n)} +e^{-{\gamma}_1 n}]. \\\end{aligned}$$ The expression in the right hand side above tends to zero as $n\to\infty$. Indeed, note that the second term in the right hand side above tends to zero because $r(n)/n \to \infty$, by construction. This completes the proof of the lemma.
Let $\phi$ be an Hölder continuous potential as above such that ${\sigma}={\sigma}(\phi)>0$, and fix ${\varepsilon}>0$ arbitrary small. Given $t\in
\mathbb R$ and $n \ge 1$ $$\begin{aligned}
\mu\Big(\frac{\log R_n-n h_\mu(f)}{{\sigma}\sqrt n}>t\Big)
& =\mu(R_n > e^{n h_\mu(f)} e^{{\sigma}t \sqrt n}) \\
& = \sum_{Q \in {{\mathcal{Q}}^{(n)}}} \mu(Q \cap \{\tau_Q > e^{n h_\mu(f)} e^{{\sigma}t \sqrt n}\}).\end{aligned}$$ Let ${{\mathcal{Q}}^{(n)}}_{\varepsilon}$ be the family of cylinders given by Theorem \[thm.first.entrance.distribution\]. Since $\cup\{Q \in
{{\mathcal{Q}}^{(n)}}_{\varepsilon}\}$ has $\mu$-measure at least $1-{\varepsilon}$ and the first entrance time $\tau_Q$ of every cylinder $Q\in{{\mathcal{Q}}^{(n)}}_{\varepsilon}$ has exponential distribution up to a small error, then $$\mu\Big(\frac{\log R_n-n h_\mu(f)}{{\sigma}\sqrt n}>t\Big)
= \! \sum_{Q \in {{\mathcal{Q}}^{(n)}}_{\varepsilon}} \mu(Q) \;\mu\Big(\tau_Q > e^{n h_\mu(f)} e^{{\sigma}t \sqrt n}\Big)
+ {\mathcal{O}}(e^{-\beta n})+ {\mathcal{O}}({\varepsilon}).$$ This is a consequence of the previous Lemma \[l.hitting.vs.recurrence\] with $t_n=e^{nh_\mu(f)+{\sigma}t\sqrt n}$. Since ${\varepsilon}>0$ was chosen arbitrary, to show the log-normal distribution of the return times we are left to prove the convergence $$\label{eq.final.fluctuation}
\lim_{n \to \infty} \Bigg[ \sum_{Q \in {{\mathcal{Q}}^{(n)}}_{\varepsilon}} \mu(Q) \; e^{-\mu(Q)e^{n h_\mu(f)} e^{{\sigma}t \sqrt n}}\Bigg]
= \frac{1}{\sqrt{2\pi}}\int_{t}^\infty e^{-\frac{x^2}{2}} \,dx + {\mathcal{O}}({\varepsilon}).$$ By Lemma \[l.weak.Gibbs\] and Corollary \[c.decay.Kn\], there is $a\in \mathbb N$ and for $\mu$-almost every $x$ there is a sequence $(K_n)_n$ (depending on $x$) satisfying $K_n(x)\leq n^a$ for all but finitely many values of $n$ and such that $$K_n(x)^{-1}
\leq
\frac{\mu(Q_n(x))}{e^{-Pn + S_n \phi(x)}}
\leq
K_n(x)$$ for every $n \geq 1$. Using also $\mu(\cup \{Q : Q \notin
{{\mathcal{Q}}^{(n)}}_{\varepsilon}\})<{\varepsilon}$ and $h_\mu(f)+ \int \phi \,d\mu={P_{{\operatorname{top}}}}(f,\phi)=P$, for any $\rho>0$ $$\begin{aligned}
\sum_{Q \in {\mathcal{Q}}^{(n)}_{\varepsilon}} & \mu(Q) \; e^{-\mu(Q)e^{n h_\mu(f)}
e^{{\sigma}t \sqrt n}}
= \int_{\cup \{Q : Q \in {{\mathcal{Q}}^{(n)}}_{\varepsilon}\}} e^{-\mu(Q(x)) e^{n h_\mu(f)} e^{{\sigma}t \sqrt n}}
d\mu(x) \\
& \geq e^{-e^{-\rho {\sigma}\sqrt n}}
\Big[\mu\Big( x \in M \mid e^{-\mu(Q_n(x))e^{n h_\mu(f)} e^{{\sigma}t \sqrt n}} > e^{-e^{-\rho {\sigma}\sqrt n}}\Big) -{\varepsilon}\Big]\\
& \geq e^{-e^{-\rho {\sigma}\sqrt n}}
\Big[\mu\Big( x \in M \mid \frac{-S_n\phi(x)+ n \int \phi d\mu}{{\sigma}\sqrt n} > t + \rho + \frac{1}{\sqrt n}\log
K_n(x) \Big)-{\varepsilon}\Big].\end{aligned}$$ Since $\phi$ belongs to $V_\theta$ (recall Lemma \[l.Vtheta.contains.Holder\]) and it satisfies the Central Limit Theorem (see Corollary \[c.CLT\]), taking the limit as $n\to\infty$ and $\rho\to 0$ we obtain that $$\liminf_{n \to \infty} \Bigg[ \sum_{Q \in {{\mathcal{Q}}^{(n)}}_{\varepsilon}} \mu(Q) \; e^{-\mu(Q)e^{n h_\mu(f)} e^{{\sigma}t \sqrt n}}\Bigg]
\geq \frac{1}{\sqrt{2\pi}}\int_{t}^\infty e^{-\frac{x^2}{2}} \,dx -{\varepsilon}.$$ The upper estimate in is obtained analogously. Indeed, for any $\rho>0$ $$\begin{aligned}
\sum_{Q \in {\mathcal{Q}}^{(n)}_{\varepsilon}} & \mu(Q) \; e^{-\mu(Q)e^{n h_\mu(f)}
e^{{\sigma}t \sqrt n}}
= \int_{\cup \{Q : Q \in {{\mathcal{Q}}^{(n)}}_{\varepsilon}\}} e^{-\mu(Q(x)) e^{n h_\mu(f)} e^{{\sigma}t \sqrt n}} d\mu(x) \\
& \leq \mu\Big( x \in \cup \{Q : Q \in {{\mathcal{Q}}^{(n)}}_{\varepsilon}\} \mid e^{-\mu(Q_n(x))e^{n h_\mu(f)} e^{{\sigma}t \sqrt n}} > e^{-e^{-\rho {\sigma}\sqrt n}}\Big)\\
& \leq e^{e^{-\rho {\sigma}\sqrt n}}
\Big[\mu\Big( x \in M \mid e^{-\mu(Q_n(x))e^{n h_\mu(f)} e^{{\sigma}t \sqrt n}} > e^{-e^{-\rho {\sigma}\sqrt n}}\Big) +{\varepsilon}\Big]\\
& \leq e^{e^{-\rho {\sigma}\sqrt n}}
\Big[\mu\Big( x \in M \mid \frac{-S_n\phi(x)+ n \int \phi d\mu}{{\sigma}\sqrt n} > t + \rho - \frac{1}{\sqrt n}\log
K_n(x) \Big)+{\varepsilon}\Big].\end{aligned}$$ taking the limit as $n\to\infty$ and $\rho\to 0$ one gets $$\limsup_{n \to \infty} \Bigg[ \sum_{Q \in {{\mathcal{Q}}^{(n)}}_{\varepsilon}} \mu(Q) \; e^{-\mu(Q)e^{n h_\mu(f)} e^{{\sigma}t \sqrt n}}\Bigg]
\leq \frac{1}{\sqrt{2\pi}}\int_{t}^\infty e^{-\frac{x^2}{2}} \,dx +{\varepsilon},$$ which proves the upper bound in . The proof of the theorem is now complete.
**Acknowledgements:** Most of this work was done at IMPA, during the author’s PhD under the guidance of Marcelo Viana. The author greatly acknowledges IMPA excellent conditions and scientific environment. The author is grateful to V. Araújo for reading a preliminary version of this paper and making important suggestions, and to A. Galves for providing a copy of [@GaSc97]. This work was partially supported by Fundação para a Ciência e Tecnologia (FCT-Portugal) by the grant SFRH/BD/11424/2002) and by Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro - FAPERJ.
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---
abstract: 'We calculate the Raman spectrum of the two-dimensional (2D) spin-1/2 Heisenberg antiferromagnet by exact diagonalization and quantum Monte Carlo techniques on clusters of up to 144 sites and, on a 16-site cluster, by considering the phonon-magnon interaction which leads to random fluctuations of the exchange integral. Results are in good agreement with experiments on various high-$T_c$ precursors, such as La$_2$CuO$_4$ and YBa$_2$Cu$_3$O$_{6.2}$. In particular, our calculations reproduce the broad lineshape of the two-magnon peak, the asymmetry about its maximum, the existence of spectral weight at high energies, and the observation of nominally forbidden $A_{1g}$ scattering.'
address: |
$^1$Department of Physics, The University of Michigan, Ann Arbor, MI 48109-1120\
$^2$Department of Physics, Supercomputer Computations Research Institute and National High Magnetic Field Lab, Florida State University, Tallahassee, FL 32306
author:
- 'Franco Nori,$^1$ R. Merlin,$^1$ Stephan Haas,$^2$ Anders W. Sandvik,$^2$ and Elbio Dagotto$^2$'
title: |
Magnetic Raman Scattering in Two-Dimensional\
Spin-1/2 Heisenberg Antiferromagnets: Spectral Shape Anomaly and Magnetostrictive Effects
---
[*Introduction.—*]{} Raman scattering is a powerful technique to study electronic excitations in strongly correlated systems. Recently, much attention has been given to the anomalous magnetic scattering with a very broad and asymmetric lineshape observed in the Raman spectra of the parent insulating compounds of high-$T_c$ superconductors, such as La$_2$CuO$_4$, and YBa$_2$Cu$_3$O$_{6.2}$ at around 3230cm$^{-1}$ and 3080cm$^{-1}$, respectively [@experiment]. The selection rules associated with this peak are also anomalous. While the spin-pair excitations scatter predominantly in the allowed $B_{1g }$ channel, there is also a significant contribution in the nominally forbidden $A_{1g}$ configuration, as well as much weaker $B_{2g}$ and $A_{2g}$ scattering[@experiment].
Previous theoretical studies on the spin-1/2 Heisenberg model for 2D square lattices have computed the Raman spectra and its moments for a nearest-neighbor interaction[@elbio; @theory; @liu; @canali] and only the moments when spin interactions along the plaquette diagonal were also included.[@nnn] These show good agreement with experiments in regard to the position of the two-magnon peak, but they fail to account for the spectral [*shape*]{}, and its enhanced [*width*]{}. Several schemes have been considered to resolve this problem. Initially, from the analysis of the moments it was proposed that strong quantum fluctuations were responsible for the broadening (see, e.g., Ref. [@elbio; @nnn]). However, recent studies of spin-pair excitations in a [*spin*]{}–1 insulator, NiPS$_3$, show a width comparable to that of the spin–1/2 cuprates [@spin1]. This questions the view that the observed anomaly is due to large quantum fluctuations intrinsic to spin–1/2 systems. We remark that the measured widths are 3-4 times larger[@weber] than those predicted by Canali and Girvin [@canali] within spin-wave theory using the Dyson-Maleev transformation, even when processes involving up to four magnons are taken into account. The work by Canali and Girvin [@canali] and other groups[@4-spin; @marville] present convincing evidence that the observed anomalous features of the magnetic scattering [*cannot*]{} be satisfactorily explained by only considering quantum fluctuations.
In order to explain the observed anomalously broad and asymmetric lineshapes, it seems then necessary to invoke an additional process. Here, we consider the interaction between magnon pairs and phonons [@magnon-phonon]. This mechanism is motivated in part by recent experimental observations of a strong broadening of the $B_{1g}$ and an enhancement of the $A_{1g}$ scattering with increasing temperature[@knoll]. In our approach we consider the phonons as static lattice distortions which induce changes, $\delta J_{ij} $, in the exchange integral $J$ of the undistorted lattice. We calculate the Raman spectra for a [*nearest-neighbor*]{} Heisenberg model using a [*nearest-neighbor*]{} Raman operator in the quenched-phonon approximation which, like the Born-Oppenheimer approach, focuses on the fast (high-energy) magnon modes and freezes the slow (low-energy) phonons. This approximation is valid for the cuprates because there is a clear separation of energies between the magnetic and vibrational modes. For instance, in YBa$_2$Cu$_3$O$_6$ the characteristic Debye frequency is about 340cm$^{-1}$ while the two-magnon excitation is $\approx$ 3080cm$^{-1}$.
[*Raman Lineshape without phonon-magnon coupling.—*]{} The isotropic Heisenberg Hamiltonian is given by $
\ H_0 = J \sum_{<ij>} {\bf S}_i \cdot {\bf S}_j \ ,
$ where the notation is standard, and only nearest neighbor interaction is assumed. For the cuprates, the exchange integral is $J \simeq 1450K \simeq 0.12$eV. In our study, we obtained the ground state $| \phi_0 \rangle$ of $H_0$ on finite 2D square clusters with $N$ spins and periodic boundary conditions using a Lanczos ($N=16, 26$), and Quantum Monte Carlo (QMC) ($N=144$) algorithms. We studied zero and finite temperature spectra associated with the [*nearest*]{}-neighbor scattering operator \[1-4\] $$R = \sum_{<ij>}
({\bf E}_{inc} \cdot {\bf \widehat{\sigma}}_{ij} )
({\bf E}_{sc} \cdot {\bf \widehat{\sigma}}_{ij} )
{\bf S}_i \cdot {\bf S}_j ,$$ where ${\bf E}_{inc,sc}$ corresponds to the electric field of the incident and scattered photons, and ${\bf \widehat{\sigma}}_{ij}$ is the unit vector connecting sites $i$ and $j$. In the cuprates, and for nearest-neighbors only, the irreducible representations of $R$ are $B_{1g}$, $A_{1g}$, and $E$. We concentrate mainly on the dominant $B_{1g}$ scattering, e.g., ${\bf E}_{inc} \propto \widehat{x} + \widehat{y}$ and ${\bf E}_{sc} \propto \widehat{x} - \widehat{y}$.
The spectrum of the scattering operator can be written as $$I(\omega) = \sum_n | \langle \phi_n | R | \phi_0 \rangle |^2
\delta (\omega - (E_n - E_0)) ,$$ where $\phi_n$ denotes the eigenvectors of the Heisenberg model with energy $E_n$. When doing exact diagonalizations on small clusters, the dynamical spectrum $I(\omega)$ is extracted from a continued fraction expansion of the quantity $$I(\omega) = - \frac{1}{\pi} Im \langle \phi_0 |
R \frac{1}{\omega + E_0 + i\epsilon - H_0} R | \phi_0 \rangle ,$$ where $\epsilon$ is a small real number introduced in the calculation to shift the poles of Eq. 3 into the complex plane. In the QMC simulations, the imaginary-time correlator $ \langle R(\tau) R(0) \rangle $ is calculated and $I(\omega)$ is obtained by numerically continuing this function to real frequencies using a maximum entropy procedure[@gubernatis].
Our calculated $B_{1g}$ spectra are shown in Fig. 1(a). They were obtained from exact diagonalization ($N=16$) and QMC ($N=144$) studies of the Heisenberg Hamiltonian on square lattices. The two-magnon excitation observed experimentally lies around $3J$, which is in good agreement with the location of the main peak obtained from exact diagonalization in Fig. 1. The position of this peak can be understood in terms of the Ising model, which corresponds to the limit of the anisotropic Heisenberg Hamiltonian when no quantum fluctuations are present. In its ground state, the Ising spins align antiferromagnetically for $J > 0$. Within this model and for a 2D square lattice, the incoming light creates a local spin-pair flip at an energy $3J$ higher than the ground state energy. This argument remains approximately valid even in the presence of quantum fluctuations[@elbio; @theory; @liu; @canali]. Our results indicate that the two-magnon excitation is at $2.9757J$, $3.0370J$, and $~3.2J$ for the 16-, 26-, and 144-site square lattices, respectively. Finite-size effects are small because of the local nature of the Raman operator. For the 144-site lattice, the QMC calculation was carried out at a temperature $T=J/4$. The slight shift of the peak position, compared to the $T=0$ results for the smaller clusters, is consistent with the finite-$T$ exact diagonalization results of Ref.[@theory]. Statistical errors, absent in the exact diagonalization results but unavoidable in any stochastic simulation, enhance the width of the 144-site spectrum. These results confirm that neither finite-size effects nor finite temperature can account for the discrepancies with the experimental spectra.
[*Lineshape Anomaly.—*]{} The Raman spectra obtained from the pure Heisenberg model (see Fig. 1) shows good agreement with experiments in regard to the two-magnon peak position, but the calculated width is too small. We will consider here the coupling between the magnon pair and phonons [@magnon-phonon; @knoll] to account for the observed wide and asymmetric lineshape. Our mechanism relates to that proposed by Halley [@halley] to account for two-magnon infrared absorption in, e.g., MnF$_2$. Quantum and thermal fluctuations distort the lattice. The exchange coupling, which depends on the instantaneous positions of the ions, can be expanded in terms of the their displacements from equilibrium ${\bf u}$. Keeping only the dominant linear terms: $ J_{ij}({\bf r}) = J_{ij} =
J + \delta J_{ij} = J + {\bf u} \cdot \nabla J_{ij} ({\bf R}) $. Here, $\delta J_{ij} $ represents the instantaneous value of ${\bf u} \cdot \nabla J_{ij} ({\bf R})$, where ${\bf R}$ denotes the equilibrium position of the ion carrying the spin (located at ${\bf r=R+u}$). In the quenched-disorder approximation, the effective Hamiltonian is $$H_1 = \sum_{<ij>} (J + \delta J_{ij}) {\bf S}_i \cdot {\bf S}_j ,$$ where $| \delta J_{ij} | < J$ is a random variable corresponding to taking a snapshot of the lattice. This new Hamiltonian is no longer translational invariant. In our study, the random couplings $\delta J_{ij}$ were drawn from a Gaussian distribution $P(\delta J_{ij}) =
\exp{(- (\delta J_{ij})^2 / 2 \sigma )} / \sqrt{2 \pi \sigma } $. $I(\omega )$ was obtained as the quenched average over $m \simeq 1000$ realizations of the randomly distorted lattice. The quenched average of an operator $\hat{O}$ is defined by $ \langle \langle \hat{O} \rangle \rangle =
\frac{1}{m} \sum_{j=1}^m \langle \phi_0 (j)
|\hat{O} |\phi_0 (j) \rangle ,
$ where $\phi_0 (j)$ is the ground state of the $j$th realization of the disordered system.
In Fig. 1(b) we show the $B_{1g}$ Raman spectrum from Eq. (1) for a 16-sites square lattice with $\sigma \sim 0.4J$, which we found to agree best with experimental spectra [@experiment]. Our calculations do not consider the effect of frozen phonons on the scattering operator $R$. Notice that the coefficients pertaining to $R$ are generally unrelated to the matrix elements of the system’s Hamiltonian (e.g., $\partial J/ \partial Q$ in $H$ bears on $e^2/r$, while the corresponding terms $\propto Q S_i S_j$ in $R$ bear on the dipole moment). In particular, and unlike the case without phonons, the fully symmetric $A_{1g}$ component of the scattering operator does not commute with $H$.
We find that the three main features observed in the $B_{1g}$ configuration[@experiment], namely, the broad lineshape of the two-magnon peak, the asymmetry about its maximum, and the existence of spectral weight up to $\omega \sim 7J$ are well reproduced. Beyond the two-magnon peak, there is a continuum of phonon-multi-magnon excitations. The small feature around $\omega \simeq 5.5J$ (for $0 \leq \sigma \leq 0.3J$) is compatible with a four-magnon excitation.
[*Magnetostriction.—*]{} Since the effects of the phonon-magnon interaction (i.e., magnetostriction) have not been extensively studied by theoretical work in the cuprates, a few comments are in order. The coupling between the spin and strain degrees of freedom modifies both elastic and magnetic properties. In fact, there are extensive studies on the (sometimes very strong) influence of elasticity on magnetism[@mattis-shultz; @striction-1; @recent-prls]. Mattis and Shultz[@mattis-shultz] considered the influence of [*uniform compression*]{} (i.e., all bonds [*equally distorted*]{}) in their classic study of magnetothermomechanics. Their results were criticized[@striction-1] for ignoring the effects of phonons (i.e., [*local fluctuations*]{} in the bond lengths, which are taken into account in the present work). Recently, [*giant*]{} magnetostrictive effects have been reported in several high-$T_c$ superconductors[@striction-htcsc]. Also, important magnetostrictive effects have been reported in heavy-fermion[@striction-hf] and low-$T_c$[@striction-ltc] superconductors.
[*Superexchange-Phonon Coupling.—*]{} The width of the Gaussian distribution, $\sigma$, represents changes in $J$ due to large $incoherent$ atomic displacements. Thus, one can write $\sigma \sim |\langle \delta \ln J / \delta Q \rangle \langle Q \rangle |$ where $\langle Q \rangle$ is an average [*zero point motion*]{} (at $T=0$) and $ \langle \delta J / \delta Q \rangle $ is a weighted average of $\nabla J_{ij}$ with respect to the displacement of [*all*]{} the ions participating in the exchange. Parenthetically, it is trivial to treat the case $T \neq 0$ by increasing $\delta J$. Let $r$ be the Cu-Cu distance, $\upsilon$ the sound velocity, and $M$ an effective reduced mass for the ions. A simple calculation gives $ \langle Q \rangle / r \sim (M \upsilon r / \hbar)^{-1/2} \sim 0.05 $ which is consistent with X-ray measurements of the mean displacement of oxigen atoms normal to the layers[@d1; @d2]. While $\nabla J_{ij}$ is not known for most phonons, values for longitudinal acoustic modes can be gained from the $r$-dependence of $J$ in the form $J(r) \sim r^{-\alpha}$ or $\partial \ln J / \partial \ln r = - \alpha $[@Jpressure]. For conventional transition metal oxides and halides, $10 \leq \alpha \leq 14$[@Jpressure], in reasonable agreement with the theoretical estimate $\alpha=14$[@harrison]. For the cuprates, high-pressure Raman measurements[@aronson] and material trends[@lance] give, respectively, $\alpha \approx 5-7$ and $\alpha \approx 2-6$. These values translate into $\sigma \approx (0.1-0.35)J$. We emphasize that the relevant [*incoherent*]{} $\delta J$’s (or $\delta Q$’s) of our case are much larger than those in pressure studies involving [*coherent*]{} motion of ions (see, e.g., the discussion in p. 466 of [@halley]). Thus, we must use larger $\sigma$ ($\sigma \sim 0.4J$).
Finally, we would like to stress that not every kind of disorder gives rise to the observed broadening of the spectrum. For instance, disorder by point defects or twinning planes will not produce such an effect. Also, it is observed in experiments that the Raman linewidth broadens with increasing temperature[@knoll]. This is a strong indication of a phonon mechanism for the broadening.
[*$A_{1g}$ and $B_{2g}$ Symmetries.—*]{} For the $A_{1g}$ symmetry, the undistorted Raman operator commutes with the Heisenberg Hamiltonian, and [*no*]{} scattering can take place. However, the addition of disorder changes the commutator and can produce an $A_{1g}$ signal. Instead, the silent $B_{2g}$ channel remains forbidden within our [*nearest*]{}-neighbor Raman operator. Fig. 1(c) shows the comparison between our numerically obtained $A_{1g}$ spectra (for $\sigma \sim 0.4J$) and the experimental results [@experiment; @nnn]. The agreement between theory and experiments is reasonably good. We stress that the $A_{1g}$ scattering follows naturally from our model unlike approaches relying on additional hypotheses, like, for instance, diagonal-nearest-neighbor couplings[@nnn], 4-spin terms[@4-spin], new fermionic quasiparticles[@hsu], or spinons. For a detailed discussion of these and other proposed explanations of the lineshape anomaly, see [@canali; @marville].
[*Extensions.—*]{} The mid-infrared optical absorption in undoped lamellar copper oxides show broad features which are believed to originate from exciton-magnon absorption processes[@graybeal]. Instead, these results could be interpreted as due to the interaction between phonons and magnons[@infrared]. Also, excellent agreement with conductivity experiments has been recently achieved by the inclusion of phonon-induced strong-disorder[@fehrenbacher].
[*Summary.—*]{} We find that light scattering spectra by spin excitations is caused by intrinsic spin-spin interactions and by interactions with phonons. We provide strong evidence that the two-magnon Raman peak is strongly modified by coupling to low-energy phonons which randomly distort the lattice. Our calculations are in good agreement with experiments and provide a simple explanation of four puzzling features of the data: the broad lineshape of the two magnon peak, the asymmetry about its maximum, the existence of a spectral weight at high energies, and the observation of nominally forbidden $A_{1g}$ scattering. We thank J. Riera for his help. S.H. acknowledges support by the Supercomputer Computations Research Institute (SCRI). The work of E.D. and R.M. is partially supported by the donors of The Petroleum Research Fund, and the work of E.D. and A.S. by the ONR grant N00014-93-1-0495.
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abstract: 'Quasi-shuffle algebras have been a useful tool in studying multiple zeta values and related quantities, including multiple polylogarithms, finite multiple harmonic sums, and $q$-multiple zeta values. Here we show that two ideas previously considered only for multiple zeta values, the interpolated product of S. Yamamoto and the symmetric sum theorem, can be generalized to any quasi-shuffle algebra.'
author:
- |
Michael E. Hoffman\
U. S. Naval Academy\
Annapolis, MD 21402 USA\
`meh@usna.edu`
date: |
May 30, 2018\
Keywords: quasi-shuffle product, Hopf algebra, interpolated multiple zeta value\
2010 AMS Classification: 16T30, 11M32
title: 'Quasi-shuffle algebras and applications'
---
¶[P]{}
Introduction
============
Multiple zeta values and related quantities, although studied by Euler in the simplest cases, only began to receive systematic attention in the early 1990s. Suddenly they seemed to be everywhere: in high-energy physics, in knot theory, and in theoretical computer science. Many early papers on these quantities emphasized proofs of specific identities, and used methods of analysis. But from the beginning the importance of algebraic structure proved its importance.
In [@H97] the author recognized multiple zeta values as homomorphic images of quasi-symmetric functions, allowing the use of familiar results on symmetric functions in proving relations of multiple zeta values. This was generalized in [@H00], which introduced the quasi-shuffle product. (In the same year Li Guo and W. Keigher [@GK] independently introduced an essentially equivalent construction, but it took several years for the relation between the two to be generally recognized.)
Meanwhile, the circle of ideas around multiple zeta values and multiple polylogarithms continued to expand, and came to include examples that went beyond the framework of [@H00], particularly $q$-multiple zeta values. K. Ihara, J. Kajikawa, Y. Ohno and J. Okuda [@IKOO] generalized the definition of the quasi-shuffle product to include such cases, but neglected much of the algebraic machinery developed in [@H00], particularly the Hopf algebra structure and the linear maps induced by formal power series. In 2012 Ihara and the author set out to develop a generalization of the definition used in [@H00] while retaining the algebraic structures developed in that paper, and indeed extending them. This led to [@HI], which which presented such a generalization and applied it to an array of examples. We review this construction §2.
The methods introduced in [@HI] proved especially effective in treating the interpolated multiple zeta values (or $r$-MZVs) introduced by S. Yamamoto [@Y], which interpolate between ordinary multiple zeta values ($r=0$) and multiple zeta-star values $r=1$). Yamamoto showed that $r$-MZVs multiply according to an interpolated product; in §3 we define interpolated products on any quasi-shuffle algebra. A quasi-shuffle algebra with the interpolated product has a Hopf algebra structure, generalizing the results of [@HI].
The algebraic machinery of [@HI], which allows transparent proofs of many results in [@IKZ] and [@IKOO], is briefly introduced in §4 and applied to multiple zeta values in §5. We also give a new result for multiple zeta-half values (i.e., $r$-MZVs with $r=\frac12$). The same quasi-shuffle algebra that has the multiple zeta values as homomorphic images also has as images various “exotic” multiple zeta values, such as the multiple $t$-values [@H16], the Bessel-function zeta values introduced by T. V. Wakhare and C. Vignat [@WV1], and the Airy multiple zeta values, all discussed in §6.
In §7 we consider a different quasi-shuffle algebra, which has as its image the alternating or “colored” multiple zeta values. Finally, in §8 we show how the symmetric sum theorems given in [@H92] for multiple zeta values can be generalized to any quasi-shuffle algebra.
The basic construction
======================
We begin by reviewing the construction given in [@HI]. Let $A$ be a countable set $A$ of letters, $k$ a field. We assume there is a commutative, associative product $\op$ on $kA$.
Now let $\kA$ be the noncommutative polynomial algebra over $A$. So $\kA$ is the vector space over $k$ generated by “words” (monomials) $a_1a_2\cdots a_n$, with $a_i\in A$: for a word $w=a_1\cdots a_n$ we write $\ell(w)=n$ (and we set $\ell(1)=0$). Define a $k$-bilinear product $*$ on $\kA$ by making $1\in\kA$ the identity element for each product, and requiring that $*$ satisfy the relation $$\label{recur}
(aw)*(bv)=a(w*bv)+b(aw*v)+(a\op b)(w*v)$$ for all $a,b\in A$ and all monomials $w,v$ in $\kA$. Then $(\kA,*)$ is a commutative algebra. If the product $\op$ is identically zero, then $*$ coincides with the usual shuffle product $\sh$ on $\kA$. We will need the following lemma in the next section.
\[bare\] For letters $a,b$ and words $v,w$ such that $v\ne 1\ne w$, $$\label{one}
a\op(v*b)+ba\op v=(a\op v)*b+a\op bv,$$ $$\label{io}
(a\op v)*(b\op w)=a\op (v*(b\op w))+b\op((a\op v)*w)-a\op b\op(v*w),$$ and $$\begin{gathered}
\label{st}
a(v*(b\op w))+a\op(v*bw)+b((a\op v)*w)+b\op (av*w)=\\
av*(b\op w)+(a\op v)*bw+2 (a\op b)(v*w) .\end{gathered}$$
Writing $v=cv'$ for a letter $c$, Eq. (\[one\]) is $$a\op(cv'*b)+ba\op cv'=a\op cv'*b+a\op bcv',$$ or $$a\op c(v'*b)+a\op bv+a\op b\op v+ba\op v=a\op c(v'*b)
+ba\op v+a\op b\op v+a\op bv,$$ which is evidently true. Setting also $w=dw'$, the left- and right-hand sides of Eq. (\[io\]) are $$a\op c(v'*(b\op w))+b\op d((a\op v)*w')+a\op b\op c\op d(v'*w')$$ and $$\begin{gathered}
a\op c(v'*(b\op w))+a\op b\op d(v*w')+a\op c\op b\op d(v'*w')\\
+b\op a\op c(v'*w)+b\op d((a\op v)*w')+b\op a\op c\op d(v'*w')\\
-a\op b\op c(v'*w)-a\op b\op d(v*w')-a\op b\op c\op d(v'*w'),\end{gathered}$$ respectively, and these agree after cancellation. Using the same notation, we can rewrite the left-hand side of Eq. (\[st\]) as $$\begin{gathered}
a(v*(b\op w))+b((a\op v)*w)+a\op(c(v'*bw)+b(v*w)+c\op b(v'*w))\\
+b\op(a(v*w)+d(av*w')+a\op d(v*w'))\\
=a(v*(b\op w))+b((a\op v)*w)+a\op c(v'*bw)+a\op b(v*w)+a\op c\op b(v'*w)\\
+b\op a(v*w)+b\op d(av*w')+b\op a\op d(v*w')\end{gathered}$$ and the right-hand side of Eq. (\[st\]) as $$\begin{gathered}
a(v*(b\op w))+b\op d(av*w')+a\op b\op d(v*w')\\
+a\op c(v'*bw)+b((a\op v)*w)+a\op c\op b(v'*w)+2a\op b(v*w),\end{gathered}$$ and these evidently agree.
If $\De$ denotes the usual deconcatenation on $\kA$, i.e., $$\begin{gathered}
\De(a_1a_2\cdots a_n)=1\otimes a_1a_2\cdots a_n+a_1\otimes a_2\cdots a_n
+\dots+a_1\cdots a_{n-1}\otimes a_n\\
+a_1a_2\cdots a_n\otimes 1,\end{gathered}$$ then $(\kA,*,\De)$ is a Hopf algebra [@HI Thm. 4.2]. It is easy to see that it is a bialgebra, and (using the filtration of $\kA$ by word length) it is filtered connected; this makes existence of the antipode automatic.
For a composition $I=(i_1,\dots,i_m)$ of $n$ and a word $w=a_1\cdots a_n$ of $\kA$, define $$I[w]=(a_1\op\dots\op a_{i_1})(a_{i_1+1}\op\dots\op
a_{i_1+i_2})\cdots(a_{i_1+\dots+i_{m-1}+1}\op\dots\op a_n) .$$ Let $$f=c_1t+c_2t^2+c_3t^3+\cdots\in tk[[t]]$$ be a formal power series. We can define a $k$-linear map $\Psi_f:\kA\to\kA$ by $$\label{phif}
\Psi_f(w)=\sum_{I=(i_1,\dots,i_m)\in\CC(\ell(w))}c_{i_1}\cdots c_{i_m}I[w] ,$$ where $\CC(n)$ is the set of compositions of $n$. Then we have the following result.
\[comp\] \[[[@HI Thm. 3.1]]{}\] For $f,g\in k[[t]]$ as specified above, $\Psi_f\Psi_g=\Psi_{f\circ g}$.
Here are some examples. First, it is immediate from equation (\[phif\]) that $\Psi_t$ is the identity homomorphism of $\kA$. Also, $T=\Psi_{-t}$ sends a word $w$ to $(-1)^{\ell(w)}w$; evidently $T$ is an involution. We note that $\Si=\Psi_{\frac{t}{1-t}}$ and $\Si^{-1}=\Psi_{\frac{t}{1+t}}$ are given by $$\Si(w)=\sum_{I\in\CC(\ell(w))}I[w] \quad\text{and}\quad
\Si^{-1}(w)=\sum_{I\in\CC(\ell(w))}(-1)^{\ell(w)-\ell(I)}I[w] ,$$ where $\ell(I)$ is the number of parts of the composition $I$. Evidently $\Si(aw)=a\Si(w)+a\op\Si(w)$ for letters $a$ and words $w$, and (as in [@IKOO]) this property can be used to define $\Si$. While $\Si$ and $T$ are not inverses, it is easy to see from Theorem \[comp\] that $T\Si T=\Si^{-1}$, from which it follows that $\Si T$ and $T\Si$ are involutions.
From [@H00] we have the (inverse) functions $\exp=\Psi_{e^t-1}$ and $\log=\Psi_{\log(1+t)}$. As shown in [@H00 Theorem 2.5], $\exp$ is an algebra isomorphism from $(\kA,\sh)$ to $(\kA,*)$. We have the following identity.
$\Si=\exp T\log T$ .
This follows from Theorem \[comp\], since $\exp T=\Psi_{e^{-t}-1}$, $\log T=\Psi_{\log(1-t)}$, and $\log(1-t)$ composed with $e^{-t}-1$ gives $$\frac1{1-t}-1=\frac{1-(1-t)}{1-t}=\frac{t}{1-t} .$$
The interpolated product
========================
For any $r\in k$, define $\Si^r=\Psi_{\frac{t}{1-rt}}$; it then follows immediately from Theorem \[comp\] that $\Si^r\Si^s=
\Si^{r+s}$, and it is easily seen that $$\Si^r(aw)=a\Si^rw+ra\op\Si^rw$$ for any letter $a$ and word $w$. We now define the interpolated product $\prd$ by $$u\prd v=\Si^{-r}(\Si^ru*\Si^rv)$$ for any words $u,v$. Henceforth we shall treat both concatenation and $\op$ has having higher binding than $*$ and $\prd$, so the second identity of Lemma \[bare\] reads $$a\op v*b\op w=a\op (v*b\op w)+b\op(a\op v*w)-a\op b\op(v*w) .$$
\[decor\] Lemma \[bare\] remains true when $*$ is replaced by $\prd$.
For each identity, first replace $v$ and $w$ by $\Si^rv$ and $\Si^rw$ respectively and then apply $\Si^{-r}$ to both sides. After appropriate simplification and (in the case of identity (\[st\])) cancellation, the conclusion follows.
We now show that the product $\prd$ can be defined inductively by a rule similar to Eqn. (\[recur\]) for the quasi-shuffle product $*$. This rule was first given by Yamamoto [@Y] in the case of multiple zeta values.
The product $\prd$ can be specified by setting $1\prd w=w\prd 1=w$ for any word $w$, $a\prd b=ab+ba+(1-2t)a\op b$ for any letters $a,b$, and $$\begin{gathered}
av\prd bw = a(v\prd bw)+b(av\prd w)+(1-2r)a\op b(v\prd w)\\
+(r^2-r)a\op b\op (v\prd w)\end{gathered}$$ for any letters $a,b$ and words $v,w$ such that $vw\ne 1$.
Evidently $1\prd w=w\prd 1$ for any word $w$, and for letters $a,b$ we have $$\begin{gathered}
a\prd b=\Si^{-r}(a*b)=\Si^{-r}(ab+ba+a\op b)=ab-ra\op b+ba-rb\op a+a\op b\\
=ab+ba+(1-2r)a\op b.\end{gathered}$$ Now let $a,b$ be letters, $v\ne 1$ a word. Then $$\begin{gathered}
av\prd b=\Si^{-r}(\Si^r(av)*b)=\Si^{-r}(a\Si^rv*b+ra\op\Si^rv*b)=\\
\Si^{-r}(a(\Si^rv*b)+ba\Si^rv+a\op b\Si^rv+ra\op\Si^rv*b)=\\
a(v\prd b)-ra\op(v\prd b)+b\Si^{-r}a\Si^rv-rb\op\Si^{-r}a\Si^rv
+a\op bv-ra\op b\op v+ra\op v\prd b\\
=a(v\prd b)+bav+(1-r)a\op bv+(r^2-r)a\op b\op v
-r(a\op(v\prd b)+ba\op v-a\op v\prd b)\\
=a(v\prd b)+bav+(1-2r)a\op bv+(r^2-r)a\op b\op v,\end{gathered}$$ where we used Lemma 2 in the last step. Finally, let $a,b$ be letters, $v,w$ words with $v\ne 1\ne w$. Then $$\begin{gathered}
av\prd bw=\Si^{-r}(\Si^rav*\Si^rbw)=
\Si^{-r}((a\Si^rv+ra\op\Si^rv)*(b\Si^rw+rb\op\Si^rw))\\
=\Si^{-r}(a\Si^rv*b\Si^rw+ra\Si^rv*b\op\Si^rw+ra\op\Si^rv*b\Si^rw
+r^2a\op\Si^rv*b\op\Si^rw)\\
=\Si^{-r}(a(\Si^rv*b\Si^rw)+b(a\Si^rv*\Si^rw)+(a\op b)(\Si^rv*\Si^rw)
+ra\Si^rv*b\op\Si^rw\\
+ra\op\Si^rv*b\Si^rw+r^2a\op\Si^rv*b\op\Si^rw\\
=a(v\prd\Si^{-r}b\Si^rw)-ra\op(v\prd\Si^{-r}b\Si^rw)
+b(\Si^{-r}a\Si^rv\prd w)-rb\op(\Si^{-r}a\Si^rv\prd w)\\
+(a\op b)(v\prd w)-ra\op b\op(v\prd w)+r\Si^{-r}a\Si^rv\prd b\op w
+ra\op v\prd\Si^{-r}b\Si^rw+r^2a\op v\prd b\op w\\
=a(v\prd bw)-ra(v\prd b\op w)-ra\op(v\prd bw)+r^2a\op(v\prd b\op w)
+b(av\prd w)-rb(a\op v\prd w)\\
-rb\op(av\prd w)+r^2b\op(a\op v\prd w)+(a\op b)(v\prd w)
-ra\op b\op(v\prd w)+rav\prd b\op w\\
-r^2a\op v\prd b\op w+ra\op v\prd bw-r^2a\op v\prd b\op w+r^2a\op v\prd b\op w\\
=a(v\prd bw)+b(av\prd w)+(a\op b)(v\prd w)
-ra\op b\op(v\prd w)-ra(v\prd b\op w)\\
-ra\op(v\prd bw)-rb(a\op v\prd w)-rb\op(av\prd w)+rav\prd b\op w
+ra\op v\prd bw\\
+r^2(a\op(v\prd b\op w)+b\op(a\op v\prd w)-a\op v\prd b\op w)\\
=a(v\prd bw)+b(av\prd w)+(1-2r)(a\op b)(v\prd w)+(r^2-r)a\op b\op(v\prd w),\end{gathered}$$ where we used Lemma \[decor\] in the last step.
If $r=1$, we write $\star$ instead of $\,\overset{\scriptstyle{1}}*$. The product $\star$ has inductive rule $$av\star bw=a(v\star bw)+b(av\star w)-a\op b(v\star w),$$ which is of the same form as Eq. (\[recur\]). As noted in [@HI], $T:(\kA,\star)\to(\kA,*)$ and $T:(\kA,*)\to(\kA,\star)$ are isomorphisms. These are special cases of the following result.
$T:(\kA,\prd)\to(\kA,\overset{1-r}*)$ is an isomorphism.
First note that $\Si^s:(\kA,\prd)\to (\kA,\overset{r-s}*)$ is an isomorphism, and that $\Si^rT=T\Si^{-r}$ for all $r\in k$. Then $$\begin{gathered}
T(u\prd v)=T\Si^{-r}(\Si^ru*\Si^rv)=\Si^rT(\Si^ru*\Si^rv)=
\Si^r(T\Si^r\star T\Si^rv)=\\
\Si^r(\Si^{-r}Tu\star\Si^{-r}Tv)=Tu\overset{1-r}* Tv \end{gathered}$$ for $u,v\in\kA$, and the result follows.
In what follows, $R$ is the linear function on $\kA$ that reverses words, i.e., $R(a_1a_2\cdots a_n)=a_n a_{n-1}\cdots a_1$. We note that $R$ commutes with $\Psi_f$ for all $f\in tk[[t]]$ [@HI Prop. 4.3]. The following result generalizes [@HI Thm. 4.2].
$(\kA,\prd,\De)$ is a filtered connected Hopf algebra with antipode $\Si^{1-2r}TR$. Also, $\Si^r:(\kA,\prd,\De)\to(\kA,*,\De)$ is a Hopf algebra isomorphism.
To see that $(\kA,\prd,\De)$ is a Hopf algebra, the main thing to check is that $\De(w_1\prd w_2)=\De(w_1)\prd\De(w_2)$ for any two words $w_1$ and $w_2$. We do this inductively on the word length. We can assume $w_1\ne 1\ne w_2$, so let $w_1=au$ and $w_2=bv$ for letters $a,b$. Using Sweedler’s notation $$\De(u)=\sum u_{(1)}\otimes u_{(2)},\quad \De(v)=\sum v_{(1)}\otimes v_{(2)},$$ we have $$\De(au)=\sum au_{(1)}\otimes u_{(2)}+1\otimes au$$ and $$\De(bv)=\sum bv_{(1)}\otimes v_{(2)}+1\otimes bv$$ so that $$\begin{gathered}
\De(w_1)\prd \De(w_2)=\sum(au_{(1)}\prd bv_{(1)})\otimes (u_{(2)}\prd v_{(2)})
+\sum au_{(1)}\otimes(u_{(2)}\prd bv)\\
+\sum bv_{(1)}\otimes (au\prd v_{(2)})+1\otimes (au\prd bv)\\
=\sum a(u_{(1)}\prd bv_{(1)})\otimes(u_{(2)}\prd v_{(2)})
+\sum b(au_{(1)}\prd v_{(1)})\otimes(u_{(2)}\prd v_{(2)})+\\
(1-2r)\sum a\op b(u_{(1)}\prd v_{(1)})\otimes(u_{(2)}\prd v_{(2)})
+(r^2-r)\sum a\op b\op (u_{(1)}\prd v_{(1)})\otimes(u_{(2)}\prd v_{(2)})\\
+\sum au_{(1)}\otimes(u_{(2)}\prd w_2)+\sum bv_{(1)}\otimes(w_1\prd v_{(2)})
+1\otimes a(u\prd w_2)+1\otimes b(w_1\prd v)\\
+(1-2r)1\otimes a\op b(u\prd v)+(r^2-r)1\otimes a\op b\op (u\prd v) .\end{gathered}$$ Using the induction hypothesis, this is $$\begin{gathered}
(a\otimes 1)(\De(u\prd w_2))+1\otimes a(u\prd w_2)
+(b\otimes 1)(\De(w_1\prd v))+1\otimes b (w_1\prd v)\\
+(1-2r)(a\op b\otimes 1)\De(u\prd v)+(1-2r)(1\otimes a\op b)\De(u\prd v)\\
+(r^2-r)(a\op b\otimes 1)\op\De(u\prd v)
+(r^2-r)(1\otimes a\op b)\op\De(u\prd v)\end{gathered}$$ which can be recognized as $$\De(w_1\prd w_2)=\De(a(u\prd w_2)+b(w_1\prd v)+(1-2r)a\op b(u\prd v)
+(r^2-r)a\op b\op (u\prd v)) .$$
Now $\Si^r:(\kA,\prd)\to (\kA,*)$ is an algebra homomorphism by definition, and is also a coalgebra map for $\De$ [@HI Thm. 4.1]. Hence $\Si^r$ is a Hopf algebra isomorphism. Also, if $$w=\sum_{w} w_{(1)}\otimes w_{(2)}$$ for a nonempty word, then $$\Si^rw=\sum_{w} \Si^rw_{(1)}\otimes \Si^rw_{(2)}$$ and we have $$\sum_w S_*\Si^r w_{(1)}*\Si^r w_{(2)}=0$$ for $S_*=\Si TR$ the antipode of the Hopf algebra $(\kA,*,\De)$ [@HI Thm. 4.2]. Apply $\Si^{-r}$ to get $$\sum_w \Si^{1-r}TR\Si^r w_{(1)}\prd w_{(2)}=0;$$ but this shows that $\Si^{1-r}TR\Si^r=\Si^{1-2r}TR$ is the antipode of $(\kA,\prd,\De)$.
Of course if $r=0$ the Hopf algebra $(\kA,\prd,\De)$ is just $(\kA,*,\De)$; the antipode is $\Si TR$. If $r=1$ we get $(\kA,\star,\De)$, and the antipode is $\Si^{-1}TR=T\Si R$. For $r=\frac12$ the inductive rule for the product is $$av\phd bw=a(v\phd bw)+b(av\phd w)-\frac14 a\op b\op (v\phd w)$$ and the antipode is simply $TR$.
Algebraic formulas
==================
In [@IKZ] and [@IKOO] there are algebraic formulas involving $\exp$ and $\log$. These can be proved systematically from the following result of [@HI], where for $w\in\kA$ and $f=c_1t+c_2t^2+\cdots\in tk[[t]]$, $f_\bullet(\la w)$ denotes $$\la c_1w+\la^2 c_2w\bullet w+\la^3c_3w\bullet w\bullet w+\cdots \in \kA[[\la]]$$ for $\bullet=*,\sh,\star,\op$.
\[gsf\] \[[[@HI Thm. 5.1]]{}\] For any $f\in tk[[t]]$ and $w\in\kA$, $$\Psi_f\left(\frac1{1-\la w}\right)=\frac1{1-f_{\op}(\la w)} .$$
We write $\exp_\bullet(\la w)$ for $1+f_\bullet(\la w)$, $f=e^t-1$, and $\log_\bullet(\la w)$ for $f_\bullet(\la w)$, $f=\log(1+t)$. By applying Theorem \[gsf\] with $f=\log(1-t)$, we get $$\label{expg}
\exp_*(\log_{\op}(1+\la z))=\frac1{1-\la z} ,$$ and by applying it with $f=e^t-1$ we get $$\exp_*(\la z)=\exp\left(\frac1{1-\la z}\right)=\frac1{2-\exp_{\op}(\la z)} .$$ Another consequence of Theorem \[gsf\] is the following.
\[gsint\] \[[[@HI Cor. 5.5]]{}\] For any $z\in\kA$ and $r\in k$, $$\Si^r\left(\frac1{1-\la z}\right)*\frac1{1-r\la z}=\frac1{1-(1-r)z} .$$
Multiple zeta values
====================
For positive integers $i_1,\dots,i_k$ with $i_1>1$, the corresponding multiple zeta value is defined by $$\zt(i_1,\dots,i_k)=\sum_{n_1>n_2>\dots>n_k\ge 1}
\frac1{n_1^{i_1}n_2^{i_2}\cdots n_k^{i_k}} .$$ Let $A=\{z_1,z_2,\dots\}$, with the operation $z_i\op z_j=z_{i+j}$. The following result can be extracted from [@H97].
The Hopf algebra $(\Q\<A\>,*,\De)$ is isomorphic to the algebra $\QS$ of quasi-symmetric functions over $\Q$.
Let $\Q\<A\>^0$ be the subspace of $\Q\<A\>$ generated by 1 and all words that do not begin with $z_1$. Then $(\Q\<A\>^0,*)$ is a subalgebra of $(\Q\<A\>,*)$. We write $\QS^0$ for the corresponding subalgebra of $\QS$. The following fact was proved in [@H97].
The linear function $\zt:\QS^0\to\R$ defined by $\zt(z_{i_1}\cdots z_{i_k})=
\zt(i_1,\dots,i_k)$ is a homomorphism from $\QS^0$ to the reals with their usual multiplication.
If we take $z=z_k$ in Eq. (\[expg\]) above, we get $$\sum_{n\ge 0}\la^n z_k^n=\exp_*(\log_{\op}(1+\la z_k))=
\exp_*\left(\sum_{j\ge 1}\frac{(-1)^{j-1}\la^jz_{kj}}{j}\right),$$ or, after applying $\zt$, $$\sum_{n\ge 0}\la^n\zt(\{k\}_n)=\exp\left(
\sum_{j\ge 1}\frac{(-1)^{j-1}\la^j\zt(kj)}{j}\right) ,$$ where $\{k\}_n$ means $k$ repeated $n$ times. If $k=2$ the right-hand side is $$\exp\left(\sum_{j\ge 1}\frac{B_{2j}(2\pi)^{2j}}{(2j)(2j)!}\la^j\right)=
\frac{\sinh(\pi\la)}{\pi\la} ,$$ from which follows $$\label{repin1}
\zt(\{2\}_n)=\frac{\pi^{2n}}{(2n+1)!} ,$$ and a similar argument gives $$\label{repin2}
\zt(\{4\}_n)=\frac{2^{2n+1}\pi^{4n}}{(4n+2)!} .$$
Two remarkable results about multiple zeta values are (1) the “sum theorem,” i.e., the sum of all multiple zeta values of a fixed depth and weight $n$ is just $\zt(n)$, as in $$\zt(4,1,1)+\zt(3,2,1)+\zt(3,1,2)+\zt(2,3,1)+\zt(2,2,2)+\zt(2,1,3)=\zt(6),$$ and, (2) the “duality theorem,” i.e., there is an involution $\tau:\QS^0\to\QS^0$ so that $\zt(\tau(u))=\zt(u)$, as in $\zt(3,1,2)=\zt(2,3,1)$. To describe $\tau$ in terms of our algebraic setup, introduce two noncommuting variables $x$ and $y$, and set $z_i=x^{i-1}y$. Then $\QS^0$ is just the subspace of $\Q\<x,y\>$ generated by 1 and words that begin with $x$ and end with $y$: the function $\tau$ is the anti-isomorphism exchanging $x$ and $y$ (so, e.g., $\tau(z_3z_1z_2)=\tau(x^2y^2xy)=xyx^2y^2=
z_2z_3z_1$).
If we let $\zt^r=\zt\circ\Si^r$, then $\zt^r(w)$ is exactly the interpolated multiple zeta value as defined by Yamamoto [@Y]. Thus $\zt^0(w)=\zt(w)$ and $\zt^1(w)=\zt^{\star}(w)$ is the multiple zeta-star value defined by $$\zt^{\star}(i_1,\dots,i_k)=\sum_{n_1\ge n_2\ge\dots\ge n_k\ge 1}\frac1{n_1^{i_1}n_2^{i_2}
\cdots n_k^{i_k}} .$$ Yamamoto showed that the interpolated multiple zeta values satisfy the following version of the sum theorem, which is proved another way in [@HI].
If $n\ge 2$, then $$\sum_{\substack{i_1+\dots+i_l=n\\ i_1>1}}\zt^r(i_1,\dots,i_k)=
\zt(n)\sum_{k=0}^{l-1}r^n\binom{n-l-1+k}{k} .$$
Formulas for repeated values $\zt^r(\{m\}_n)$ can be obtained from those for $\zt(\{m\}_n)$: from Corollary \[gsint\] it follows that if $$Z(\la)=\sum_{n=0}^\infty \zt(\{m\}_n)\la^n,$$ then $$\sum_{n=0}^\infty \zt^r(\{m\}_n)\la^n=\frac{Z((1-r)\la)}{Z(-r\la)} .$$ Hence, e.g., $$\sum_{n=0}^\infty \zt^r(\{2\}_n)\la^n=\sqrt{\frac{r}{1-r}}
\frac{\sinh(\pi\sqrt{(1-r)\la})}{\sin(\pi\sqrt{r\la})} .$$
For interpolated multiple zeta values $\zt^r$ with $r=\frac12$ there is a “totally odd sum theorem.” This follows from two known results: the cyclic sum theorem and the two-one theorem. Define the cyclic sum operation on $\QS^0$ by $$\begin{gathered}
C(x^{i_1-1}yx^{i_2-1}y\cdots x^{i_k-1}y)=x^{i_1}yx^{i_2-1}y\cdots x^{i_k-1}y\\
+x^{i_2}yx^{i_3-1}y\cdots x^{i_k-1}yx^{i_1-1}y
+\dots+x^{i_k}yx^{i_1-1}y\cdots x^{i_{k-1}-1}y .\end{gathered}$$ Then the cyclic sum theorem for multiple zeta-star values [@OW] asserts that $$\zts(\tau C(w))=(n-1)\zt(n)$$ for any word $w\in\QS^0$ of degree $n-1$. The two-one formula [@Y; @Z] gives $$\zts((xy)^{j_1}y(xy)^{j_2}y\cdots (xy)^{j_l}y)
=2^l\zth(x^{2j_1}yx^{2j_2}y\cdots x^{2j_l}y)$$ for any sequence $(j_1,\dots,j_l)$ of nonnegative integers with $j_1>0$.
\[tost\] Let $n>2$, $l<n$ be positive integers of the same parity. Then $$\sum_{\substack{a_1+\dots+a_l=n\\ \text{$a_i$ odd},\ a_1>1}}\zth(a_1,\dots,a_l)
=\frac{n-1}{n-l}\binom{\frac{n+l}{2}-2}{l-1}\frac{\zt(n)}{2^{l-1}}
=\frac{n-1}{\frac{n+l}2-1}\binom{\frac{n+l}2-1}{l-1}\frac{\zt(n)}{2^l} .$$
By the two-one formula $$\sum_{\substack{a_1+\dots+a_l=n\\ \text{$a_i$ odd},\ a_1>1}}\zth(a_1,\dots,a_l)=
2^{-l}\sum_{\substack{j_1+\dots+j_l=\frac{n-l}2\\ j_i\ge 0,\ j_1\ge 1}}
\zts((xy)^{j_1}y(xy)^{j_2}y\cdots (xy)^{j_l}y) .$$ The latter sum has $$\label{bin}
\binom{\frac{n+l}2-2}{l-1}$$ terms. To see this, note that written in the sequence notation each term corresponds to a string $$\label{string}
\underbrace{2,\dots,2}_{j_1},1,\underbrace{2,\dots,2}_{j_2},1,\dots,
\underbrace{2,\dots,2}_{j_l},1$$ with $j_i\ge 0$, $j_1\ge 1$, and $\sum_{i=1}^lj_i=\frac{n-l}2$. Now the string (\[string\]) always starts with 2 and ends with 1, so we can think about the middle part: it has length $\frac{n+l}2-2$, and consists of $\frac{n-l}2-1$ twos and $l-1$ ones. To specify such a string, we need only give the $l-1$ positions where the ones go; so such strings are counted by the binomial coefficient (\[bin\]).
Now each word $u$ of the form $$\label{word1}
(xy)^{j_1}y(xy)^{j_2}y\cdots (xy)^{j_l}y$$ with $\sum_{i=1}^l j_i=\frac{n-l}2$ and $j_1>0$ has $$\tau(u)=x^{i_1-1}yx^{i_2-1}y\cdots x^{i_k-1}y$$ with $i_1>2$, $i_2,\dots,i_k>1$, $i_1+\dots+i_k=n$, and $k=\frac{n-l}2$. These are exactly the words that appear in $C(w)$ for $w$ of the form $x^{a_1-1}yx^{a_2-1}y\cdots x^{a_k-1}y$ with $a_1,\dots a_2>1$, $a_1+\dots+a_k=n-1$, and $k=\frac{n-1}2$. For any such $w$ the expansion of $\tau C(w)$ will have $\frac{n-l}2$ terms, so each term $\zts(u)$ contributes $$\frac2{n-l}(n-1)\zt(n),$$ and the result follows. (It may happen that $\frac2{n-l}\binom{\frac{n+l}2-2}{l-1}$ is not an integer, but the preceding sentence is still true since in that case there are duplications in one or more of the images under $\tau C$.)
From the definition of the zeta-half values we get the following corollary of Theorem \[tost\].
\[red\] The sum $$\sum_{\substack{a_1+\dots+a_l=n\\ \text{$a_i$ odd},\ a_1>1}}\zt(a_1,\dots,a_l)$$ is a rational linear combination of multiple zeta values of weight $n$ and depth less than $l$.
In the depth three case we can say more.
If $n$ is odd, the sum $$\sum_{\substack{a_1+a_2+a_3=n\\ \text{$a_i$ odd},\ a_1>1}}\zt(a_1,a_2,a_3)$$ is a polynomial in the ordinary zeta values with rational coefficients.
By Corollary \[red\], the sum can be written as a rational linear combination of single and double zeta values of weight $n$. But double zeta values of odd weight are known to be rational polynomials in the ordinary zeta values, and the conclusion follows.
“Exotic” multiple zeta values
=============================
In this section we give some examples of “exotic” homomorphic images of subalgebras of $\QS$. Our first example involves the multiple $t$-values as defined in [@H16]. For positive integers $i_1,\dots,i_k$ with $i_1>1$, let $$t(i_1,\dots,i_k)=\sum_{\substack{n_1>n_2>\dots>n_k\ge 1\\ \text{$n_j$ odd}}}
\frac1{n_1^{i_1}n_2^{i_2}\cdots n_k^{i_k}} .$$ Then $t:\QS^0\to\R$ defined by $t(z_{i_1}\cdots z_{i_k})=t(i_1,\dots,i_k)$ defines a homomorphism. The multiple $t$-values have obvious parallels with multiple zeta values; for example, it is evident that $t(n)=(1-2^{-n})\zt(n)$ for $n\ge 2$. Also, paralleling the identities (\[repin1\]) and (\[repin2\]) of the last section we have from [@H16] $$\label{repan}
t(\{2\}_n)=\frac{\pi^{2n}}{2^{2n}(2n)!},\quad
t(\{4\}_n)=\frac{\pi^{4n}}{2^{2n}(4n)!} .$$
Following Wakhare and Vignat [@WV1], we can take any function $G$ with real zeros $\{a_1,a_2,\dots\}$ such that $\lim_{n\to\infty} |a_n|=\infty$, and define a homomorphism $\zt_G:S\to\R$ by sending $z_{i_1}\cdots z_{i_l}$ to $$\zt_G(i_1,\dots,i_l)=
\sum_{n_1>n_2>\dots>n_k\ge 1}\frac1{a_{n_1}^{i_1}a_{n_2}^{i_2}\cdots a_{n_l}^{i_l}}$$ for some subalgebra $S$ of $\QS$ that depends on the growth rate of $|a_n|$ with $n$. Wakhare and Vignat consider the case where $a_n$ is the $n$th positive zero of the Bessel function $J_{\nu}$ of the first kind of order $\nu$. They obtain the remarkable formulas $$\begin{aligned}
\label{bes2}
\zt_{J_{\nu}}(\{2\}_n)&=\frac1{2^{2n}n!(\nu+1)(\nu+2)\cdots (\nu+n)},\\
\label{bes4}
\zt_{J_{\nu}}(\{4\}_n)&=\frac1{2^{4n}n!(\nu+1)\cdots (\nu+2n)(\nu+1)\cdots
(\nu+n)}.\end{aligned}$$ We note that since $$J_{\frac12}(z)=\sqrt{\frac2{\pi z}}\sin z\quad\text{and}\quad
J_{-\frac12}(z)=\sqrt{\frac2{\pi z}}\cos z$$ we have $$\pi^{|w|}\zt_{J_{\frac12}}(w)=\zt(w)\quad\text{and}\quad
\left(\frac{\pi}2\right)^{|w|}\zt_{J_{-\frac12}}(w)=t(w),$$ and thus Eqs. (\[bes2\]) and (\[bes4\]) imply Eqs. (\[repin1\]), (\[repin2\]), and (\[repan\]) above.
We can also choose $0>a_1>a_2>\cdots $ to be the zeros of the Airy function $\Ai(z)$. Now $\Ai(z)$ has the infinite product expansion [@VS p. 18] $$\label{aprod}
\Ai(z)=\Ai(0)e^{-\kappa z}\prod_{n=1}^\infty
\left(1-\frac{z}{a_n}\right)e^{\frac{z}{a_n}} ,$$ where $$\ka=\left|\frac{\Ai'(0)}{\Ai(0)}\right|=\frac{3^{\frac56}\Ga(\tfrac23)^2}
{2\pi} \approx 0.729011 .$$ Starting with Eq. (\[aprod\]), take logarithms and differentiate to get $$\frac{d}{dz}\log\Ai(z)=-\ka+\sum_{n=1}^\infty
\left[\frac1{a_n}+\frac1{z-a_n}\right] .$$ Then evidently $$\label{sern}
\frac{d^k}{dz^k}\log\Ai(z)=\sum_{n=1}^\infty\frac{(-1)^{k-1}(k-1)!}{(z-a_n)^k}$$ for $k\ge 2$. Since $\Ai''(z)=z\Ai(z)$, we have $$\label{d2}
\frac{d^2}{dz^2}\log\Ai(z)=z-\frac{\Ai'(z)^2}{\Ai(z)^2} .$$ Combining Eq. (\[sern\]) for $k=2$ and Eq. (\[d2\]), we have $$\label{d2a}
\sum_{n=1}^\infty\frac{-1}{(z-a_n)^2}=z-\frac{\Ai'(z)^2}{\Ai(z)^2} ,$$ which at $z=0$ gives $$\label{two}
\zt_{\Ai}(2)=\sum_{n=1}^\infty \frac1{a_n^2}=\ka^2 .$$ Repeated differentiation of $f(z)=\Ai'(z)/\Ai(z)$ gives the following result, originally due to Crandall [@C].
For all $n\ge 2$, $\zt_{\Ai}(n)$ is a rational polynomial in $\ka$ of degree $n$, with leading coefficient 1.
Also, from Eq. (\[aprod\]) it follows that $$\Ai(z)\Ai(-z)=\Ai(0)^2\prod_{k=1}^\infty\left(1-\frac{z^2}{a_k^2}\right)$$ and thus that $$\begin{gathered}
\sum_{n=0}^\infty \zt_{\Ai}(\{2\}_n)(-1)^nz^{2n}=\frac{\Ai(z)\Ai(-z)}{\Ai(0)^2}=\\
1-\ka^2z^2+\frac{\ka}{6}z^4-\frac1{60}z^6+\frac{\ka^2}{336}z^8-\frac{\ka}{6480}
z^{10}+\cdots .\end{gathered}$$ By comparison with the series [@R] $$\Ai(z)\Ai(-z)=\frac2{\sqrt{\pi}}\sum_{n\ge 0}\frac{(-1)^nz^{2n}}{12^{\frac{2n+5}{6}}
n!\Ga(\frac{2n+5}6)}$$ it can be seen that $\zt_{\Ai}(\{2\}_n)$ is rational if $n\equiv 0$ mod 3, a rational multiple of $\ka^2$ if $n\equiv 1$ mod 3, and a rational multiple of $\ka$ if $n\equiv 2$ mod 3. Further formulas for $\zt_{\Ai}(\{2\}_n)$ and also for $\zt_{\Ai}(\{4\}_n)$ were given by Wakhare and Vignat [@WV2].
Alternating multiple zeta values
================================
Let $r$ be a positive integer, $A=\{z_{m,j} |\ m\in\Z^+, i\in\{0,1,\dots,r-1\}\}$, with $z_{m,j}\op z_{n,k}=
z_{m+n,j+k}$, where addition in the second subscript is understood mod $r$. Then $(\Q\<A\>,*)$ is the “Euler algebra” $\E_r$ as defined in [@H00]. If we let $\E_r^0$ be the subalgebra generated by 1 and all words that do not begin with $z_{1,0}$, then there is a homomorphism $\ZZ_r:\E_r^0\to\C$ sending $z_{m_1,j_1}\cdots z_{m_k,j_k}$ to $$\sum_{n_1>\dots>n_k\ge 1}\frac{\ep^{n_1j_1}\cdots\ep^{n_kj_k}}{n_1^{m_1}\cdots n_k^{m_k}},$$ where $\ep=e^{\frac{2\pi i}{r}}$. Of course $\E_1$ is just $\QS$, with $\ZZ_1=\zt$. In the case $r=2$ the image of $\ZZ_r$ is real-valued, and $\ZZ_r$ sends a monomial to what is usually called an alternating or “colored” multiple zeta value. In this case we can adapt the sequence notation of multiple zeta values and write, e.g., $\zt(\bar1,2,\bar3)$ for $\ZZ_2(z_{1,1}z_{2,0}z_{3,1})$. Evidently $\zt(\bar1)=-\log2$ and $\zt(\bar k)=(2^{-k+1}-1)\zt(k)$ for $k\ge 2$. Generating functions for $\zt(\{\bar k\}_n)$ are discussed already in [@BBB]. A notable case is $$\label{b1}
\sum_{n=0}^\infty \zt(\{\bar 1\}_n)\la^n=\frac{\sqrt{\pi}}{\Ga(\frac{1-\la}2)
\Ga(1+\frac{\la}2)} .$$ The theory of interpolated products carries over to this case; for example $$\zt^r(\bar1,2,\bar3)=\zt(\bar1,3,\bar3)+r\zt(\bar3,\bar3)+r\zt(\bar1,\bar5)
+r^2\zt(6) .$$ We can generalize formulas like (\[b1\]) to interpolated alternating multiple zeta values: $$\sum_{n=0}^\infty \zt^r(\{\bar 1\}_n)\la^n=\frac{\Ga(\frac{1+r\la}2)
\Ga(1-\frac{r\la}2)}{\Ga(\frac{1-(1-r)\la}2)\Ga(1+\frac{(1-r)\la}2)} .$$ Some results for alternating multiple zeta values can be stated in terms of interpolated values, such as the following one of C. Glanois [@G].
If $s_1,\dots, s_r$ is a sequence of elements of $\{1,\bar2,3,\bar4,5,\dots\}$ with $s_1\ne 1$, then the interpolated alternating multiple zeta value $\zt^{\frac12}(s_1,\dots,s_r)$ is a rational linear combination of multiple zeta values.
Symmetric sum theorems
======================
The prototypical symmetric sum theorem was proved in [@H92].
\[proto\] \[[[@H92 Thm. 2.2]]{}\] If $k_1,\dots,k_n\ge 2$, then $$\sum_{\si\in S_n}\zt(k_{\si(1)},\dots,k_{\si(n)})=
\sum_{B=\{B_1,\dots,B_l\}\in\Pi_n}c(B)\prod_{m=1}^l\zt\left(\sum_{j\in B_m}k_j\right)$$ where $S_n$ is the symmetric group on $n$ letters, $\Pi_n$ is the set of partitions of the set $\{1,\dots,n\}$, and $$c(B)=(-1)^{k-l}(\card B_1-1)!(\card B_2-1)!\cdots (\card B_l-1)!$$ for $B=\{B_1,\dots,B_l\}\in\Pi_n$.
In fact, as noted in [@H15], this identity can be proved in $\QS$ by Möbius inversion and then (if all the $k_i\ge 2$) transferred to the reals via the homomorphism $\zt:\QS^0\to\R$. But in fact it can be generalized in two ways: first, it is true for [*any*]{} quasi-shuffle algebra $(\Q\<A\>,*)$, and second, we can extend it to the interpolated product. The result is as follows.
\[syform\] If $u_1,\dots,u_n\in A$, then in $(\Q\<A\>,\prd)$ $$\label{form}
\sum_{\si\in S_k} u_{\si(1)}u_{\si(2)}\cdots u_{\si(k)}=
\sum_{B=\{B_1,\dots,B_l\}\in\Pi_k}c_r(B)u_{B_1}\prd u_{B_2}\prd\cdots\prd u_{B_l} ,$$ where $u_{B_i}=\op_{j\in B_i}u_j$, $p_a(r)=(1-r)^a-(-r)^a$, and $$c_r(B)=(-1)^{k-l}\prod_{m=1}^l(\card B_m-1)!p_{\card B_m}(r)$$ for $B=\{B_1,\dots,B_l\}\in\Pi_k$.
We write $S(a,b)=ab+ba$, $S(a,b,c)=abc+acb+bac+bca+cab+cba$, and so on, so Eq. (\[form\]) is $$S(u_1,\dots,u_n)=\sum_{\substack{\text{partitions $\Pi=(P_1,\dots, P_l)$}\\
\text{of $\{1,\dots,n\}$}}}
c_r(\Pi)u_{P_1}\prd u_{P_2}\prd\cdots\prd u_{P_l} .$$ We proceed by induction on $n$. Take the $\prd$-product of both sides of Eq. (\[form\]) with $u_{n+1}$ to get $$\begin{gathered}
S(u_1,\dots,u_{n+1})+(1-2r)[S(u_1\op u_{n+1},u_2,\dots,u_n)
+S(u_1,u_2\op u_{n+1},\dots,u_n)\\
+\dots+S(u_1,\dots,u_{n-1},u_n\op u_{n+1})]
+2(r^2-r)[S(u_1\op u_2\op u_{n+1},u_3,\dots,u_n)\\
+S(u_1\op u_3\op u_{n+1},u_2,u_4,\dots,u_n)+\dots+
S(u_{n-1}\op u_n\op u_{n+1},u_1,\dots,u_{n-2})]\\
=\sum_{\substack{\text{partitions $\Pi=(P_1,\dots, P_l)$}\\
\text{of $\{1,\dots,n\}$}}}
c_r(\Pi)u_{P_1}\prd u_{P_2}\prd\cdots\prd u_{P_l}\prd u_{n+1}\end{gathered}$$ or $$\begin{gathered}
\label{mess}
S(u_1,\dots,u_{n+1})=-(1-2r)[S(u_1\op u_{n+1},u_2,\dots,u_n)
+S(u_1,u_2\op u_{n+1},\dots,u_n)\\
+\dots+S(u_1,\dots,u_{n-1},u_n\op u_{n+1})]
-2(r^2-r)[S(u_1\op u_2\op u_{n+1},u_3,\dots,u_n)\\
+S(u_1\op u_3\op u_{n+1},u_2,
u_4,\dots,u_n)+\dots+S(u_{n-1}\op u_n\op u_{n+1},u_1,\dots,u_{n-2})]\\
+\sum_{\substack{\text{partitions $\Pi=(P_1,\dots, P_l)$}\\
\text{of $\{1,\dots,n+1\}$ having}\\ \text{$\{n+1\}$ as a part}}}
c_r(\Pi)u_{P_1}\prd u_{P_2}\prd\cdots\prd u_{P_l} .\end{gathered}$$ We must show that the right-hand side of this equation coincides with $$\label{exp}
\sum_{\substack{\text{partitions $\Pi=(P_1,\dots, P_l)$}\\
\text{of $\{1,\dots,n+1\}$}}}
c_r(\Pi)u_{P_1}\prd u_{P_2}\prd\cdots\prd u_{P_l} ,$$ which we shall do by considering whether the cardinality of the part of $\Pi$ to which $n+1$ belongs is 1, 2, or $\ge 3$.
Note that there are three groups on terms on the right-hand side of Eq. (\[mess\]). If $\{n+1\}$ is a part of $\Pi$, the corresponding term in (\[exp\]) is contributed by the third group of terms on the right-hand side of (\[mess\]).
Suppose now that $n+1$ belongs to a part of cardinality 2 in $\Pi=(P_1,\dots,P_l)$, say $P_1$. The term corresponding to $\Pi$ in (\[exp\]) only arises (via the induction hypothesis) from the first group of terms on the right-hand side of (\[mess\]), and the coefficient of $u_{P_1}\cdots u_{P_l}$ is $$\begin{gathered}
-(1-2r)(-1)^{n-l}(\card P_2-1)!\cdots (\card P_l-1)!
p_{\card P_2}(t)\cdots p_{\card P_l}(t)\\
=(-1)^{n+1-l}(\card P_1-1)!\cdots (\card P_l-1)!
p_{\card P_1}(t)\cdots p_{\card P_l}(t) .\end{gathered}$$
Finally, suppose $n+1$ belongs to a part $P_1$ of $\Pi$ with cardinality $k\ge 3$. The term $u_{P_1}\cdots u_{P_l}$ arises from the first group of terms in $k-1$ ways, contributing coefficient $$-(k-1)(1-2r)(-1)^{n-l}p_{k-1}(r)(k-2)!C,$$ where $$C=(\card P_2-1)!\cdots (\card P_l-1)!p_{\card P_2}(t)\cdots p_{\card P_l}(t) .$$ The same term arises from the second group of terms in $\binom{k-1}{2}$ ways, contributing coefficient $$-\binom{k-1}{2}2(r^2-r)(-1)^{n-1-l}p_{k-2}(r)(k-3)!C,$$ and it suffices to show $$(1-2r)p_{k-1}(r)-(r^2-r)p_{k-2}(r)=p_k(r) ,$$ which is immediate.
Note that $p_a(0)=1$ and $p_a(1)=(-1)^{a-1}$, so $c_0(\Pi)=c(\Pi)$ and $c_1(\Pi)=|c(\Pi)|$, making Theorem \[syform\] reduce to $$\sum_{\si\in S_n}u_{\si(1)}u_{\si(2)}\cdots u_{\si(n)}=
\sum_{\substack{\text{partitions $\Pi=(P_1,\dots,P_l)$}\\ \text{of $\{1,\dots,n\}$}}}
c(\Pi)u_{P_1}* u_{P_2}*\cdots* u_{P_l}$$ in the case $r=0$; if $r=1$ we get $$\sum_{\si\in S_n}u_{\si(1)}u_{\si(2)}\cdots u_{\si(n)}=
\sum_{\substack{\text{partitions $\Pi=(P_1,\dots,P_l)$}\\ \text{of $\{1,\dots,n\}$}}}
|c(\Pi)|u_{P_1}\star u_{P_2}\star\cdots\star u_{P_l} .$$ Also, $$p_a\left(\frac12\right)=\begin{cases} 0,&\text{if $a$ even,}\\
2^{1-a},&\text{if $a$ odd,}\end{cases}$$ so that only partitions with all parts of odd cardinality appear when $r=\frac12$. In fact $$c_{\frac12}(\Pi)=\begin{cases} \left(\frac12\right)^{n-l}\prod_{i=1}^l
(\card P_i-1)!,&\text{if $\card P_1\cdots \card P_l$ is odd;}\\
0,&\text{otherwise.}\end{cases}$$
If in Theorem \[syform\] we take $A=\{z_1,z_2,\dots\}$ with $z_i\op z_j=z_{i+j}$ and $u_i=z_{k_i}$, $1\le i\le n$ (with $k_i\ne 1$ for all $i$), we get $$\label{syzt}
\sum_{\si\in S_n}\zt^r(k_{\si(1)},\dots,k_{\si(n)})=
\sum_{\substack{\text{partitions $\Pi=(P_1,\dots,P_l)$}\\ \text{of $\{1,\dots,n\}$}}}
c_r(\Pi)\prod_{j=1}^l\zt\left(\sum_{h\in P_j}k_h\right) ,$$ generalizing Theorem \[proto\]; in fact $r=0$ gives Theorem \[proto\] and $r=1$ gives the corresponding result for star-zeta values [@H92 Thm. 2.1]. Identity (\[syzt\]) holds with $t$ (or $\zt_{J_{\nu}}$ or $\zt_{\Ai}$) in place of $\zt$.
From Theorem \[syform\] we can obtain a result in terms of integer partitions.
\[repmzv\] If $u\in A$, then in $(\Q\<A\>,\prd)$ $$u^n=\sum_{\la\vdash n}\frac{\ep_{\la}}{z_{\la}}\prod_{j=1}^{\ell(\la)}
p_{\la_j}(r)u^{\op\la_1}\prd\cdots\prd u^{\op\la_l}$$ where $u^{\op n}$ means $\underbrace{u\op\cdots\op u}_n$ and (as in [@M]) $\ep_{\la}=(-1)^{n-\ell(\la)}$ and $z_{\la}=m_1(\la)!1^{m_1(\la)}m_2(\la)!2^{m_2(\la)}\cdots$, for $m_i(\la)$ the multiplicity of $i$ in $\la$.
Set $u_1=\dots=u_n=u$ in Theorem \[syform\] to get $$n!u^n=
\sum_{\substack{\text{partitions}\\ \Pi=(P_1,\dots,P_l)\\ \text{of $\{1,\dots,n\}$}}}(-1)^{n-l}(\la_1-1)!
\cdots (\la_l-1)!p_{\la_1}(r)\cdots p_{\la_l}(r)u_{i\la_1}\prd\cdots\prd u_{i\la_l},$$ where we write $\la_i=\card P_i$. Now the number of set partitions $(P_1,\dots,P_l)$ of $\{1,\dots,n\}$ corresponding to the integer partition $\la=(\la_1,\dots,\la_l)$ of $n$ is $$\frac1{m_1(\la)!m_2(\la)!\cdots}\binom{n}{\la_1}\binom{n-\la_1}{\la_2}\cdots
=\frac1{m_1(\la)!m_2(\la)!\cdots}\frac{n!}{\la_1!\la_2!\cdots \la_l!} .$$ Thus $u^n$ is $$\begin{gathered}
\sum_{\substack{\text{partitions}\\ \la=(\la_1,\dots,\la_l)\\ \text{of $n$}}}
\frac{(-1)^{n-l}(\la_1-1)!\cdots(\la_l-1)!}{m_1(\la)!m_2(\la)!\cdots
\la_1!\cdots\la_l!}
p_{\la_1}(r)\cdots p_{\la_l}(r) u^{\op\la_1}\prd\cdots\prd u^{\op\la_l}\\
=\sum_{\substack{\text{partitions}\\ \la=(\la_1,\dots,\la_l)\\ \text{of $n$}}}
\frac{\ep_{\la}}{z_{\la}}p_{\la_1}(r)\cdots p_{\la_l}(r)
u^{\op\la_1}\prd\cdots\prd u^{\op\la_l} .\end{gathered}$$
Applying $\zt^r$ to the corollary with $u=z_i$, $i\ge 2$, we obtain $$\zt(z_i^n)=\sum_{\la\vdash n}\frac{\ep_{\la}}{z_{\la}}\prod_{j=1}^{\ell(\la)}
p_{\la_j}(r)\zt(i\la_j) .$$ In the cases $r=0,1,\frac12$, this identity is respectively $$\begin{aligned}
\label{elem}
\zt(\{i\}_n)&=\sum_{\la\vdash n}\frac{\ep_{\la}}{z_{\la}}\prod_{j=1}^{\ell(\la)}
\zt(i\la_j)\\
\label{compl}
\zts(\{i\}_n)&=\sum_{\la\vdash n}\frac1{z_{\la}}\prod_{j=1}^{\ell(\la)}\zt(i\la_j)\\
\label{half}
\zth(\{i\}_n)&=\sum_{\substack{\la\vdash n\\ \text{all parts of $\la$ odd}}}
\frac1{2^{n-\ell(\la)}z_{\la}}\prod_{j=1}^{\ell(\la)}\zt(i\la_j) .\end{aligned}$$ Eqs. (\[compl\]) and (\[elem\]) are homomorphic images of the two parts of [@M Eq. ($2.14^{\prime}$)]. Eq. (\[half\]) is obtained a different way in [@HI] (see Eq. (41)).
We note that Eq. (\[syzt\]) applies to alternating multiple zeta values as well, provided we define addition on the set $\I=\{\dots,\bar2,\bar1,1,2,\dots\}$ of indices to agree with usual addition on $\{1,2,\dots,\}$ and extend it to $\I$ via $$\begin{aligned}
a+\bar b=\bar a+b&=\overline{a+b}\\
\bar a+\bar b&=a+b\end{aligned}$$ for positive integers $a,b$. Thus, e.g., $$\begin{gathered}
\zt^r(\bar1,2,\bar3)+\zt^r(\bar1,\bar3,2)+\zt^r(2,\bar1,\bar3)
+\zt^r(2,\bar3,\bar1)+\zt^r(\bar3,\bar1,2)+\zt^r(\bar3,2,\bar1)=\\
\zt(\bar1)\zt(2)\zt(\bar3)-(1-2r)(\zt(\bar3)^2+\zt(\bar1)\zt(\bar5))
+2(1-3r+3r^2)\zt(6).\end{gathered}$$ Eqs. (\[elem\]-\[half\]) also hold, provided we interpret $i\la_j$ in those formulas as the sum of $\la_j$ copies of $i\in\I$.
[9]{} D. J. Broadhurst, J. M. Borwein, and D. M. Bradley, Evaluation of $k$-fold Euler-Zagier sums: a compendium of results for arbitrary $k$, *Electron. J. Combin.* [**4(2)**]{} (1997), art. 5 (21 pp). R. E. Crandall, On the quantum zeta function, *J. Phys. A: Math. Gen.* [**29**]{} (1996), 6795-6816. P. Flajolet and G. Louchard, Analytic variations on the Airy distribution, *Algorithmica* [**31**]{} (2001), 361-377 C. Glanois, Periods of the motivic fundamental groupoid of $\P^1\setminus\{0,
\mu_N,\infty\}$, Thèse de doctorat, Université Pierre et Marie Curie, 2016. Li Guo and W. Keigher, Free Baxter algebras and shuffle products, *Adv. in Math* [**150**]{} (2000), 117-149. M. E. Hoffman, Multiple harmonic series, *Pacific J. Math.* [**152**]{} (1992), 275-290. M. E. Hoffman, The algebra of multiple harmonic series, *J. Algebra* [**194**]{} (1997), 477-495. M. E. Hoffman, Quasi-shuffle products, *J. Algebraic Combin.* [**11**]{} (2000), 49-68. M. E. Hoffman, Quasi-symmetric functions and mod p multiple harmonic sums, *Kyushu J. Math.* [**69**]{} (2015), 345-366. M. E. Hoffman, Odd variants of multiple zeta values, preprint [arXiv:1612.05232]{}. M. E. Hoffman and K. Ihara, Quasi-shuffle products revisited, *J. Algebra* [**481**]{} (2017), 293-326. K. Ihara, J. Kajikawa, Y. Ohno, and J. Okuda, Multiple zeta values and multiple star-zeta values, *J. Algebra* [**332**]{} (2011), 187-208. K. Ihara, M. Kaneko, and D. Zagier, Derivation and double shuffle relations for multiple zeta values, *Compos. Math.* [**142**]{} (2006), 307-338. I. G. Macdonald, *Symmetric Functions and Hall Polynomials*, 2nd ed., Oxford University Press, New York, 1995. Y. Ohno and N. Wakabayashi, Cyclic sum of multiple zeta values, *Acta Arith.* [**123**]{} (2006), 289-295. W. H. Reid, Integral representations for products of Airy functions, Z. angew. Math. Phys. [**46**]{} (1995), 159-170. O. Vallée and M. Soares, *Airy Functions and Applications to Physics*, World Scientific, Singapore, 2004. T. V. Wakhare and C. Vignat, Structural identities for generalized multiple zeta values, preprint [arXiv:1702.05534]{}. T. V. Wakhare and C. Vignat, Multiple zeta values for classical special functions, preprint. S. Yamamoto, Interpolation of multiple zeta values, *J. Algebra* [**385**]{} (2013), 102-114. J. Zhao, Identity families of multiple harmonic sums and multiple zeta-star values, *J. Math. Soc. Japan* [**68**]{} (2016), 1669-1694.
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author:
- 'Falong Shen, Gang Zeng'
bibliography:
- 'mybib.bib'
title: Fast Semantic Image Segmentation with High Order Context and Guided Filtering
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16SubNumber[1096]{}
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[H]{} ¶[P]{}
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[H]{} ¶[P]{}
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Biinvariant functions on the group of transformations leaving a measure quasiinvariant\
Neretin Yu.A.[^1]
[Let $\Gms$ be the group of transformations of a Lebesgue space leaving the measure quasiinvariant, let $\Ams$ be its subgroup consisting of transformations preserving the measure. We describe canonical forms of double cosets of $\Gms$ by the subgroup $\Ams$ and show that all continuous $\Ams$-biinvariant functions on $\Gms$ are functionals on of the distribution of a Radon–Nikodym derivative. ]{}
Statements
==========
[**The group $\Gms$.\[ss:1\]**]{} By $\R^\times$ we denote the multiplicative group of positive reals. By $t$ we denote the coordinate on $\R^\times$.
Let $M$ be a Lebesgue space (see [@Roh1]) with a continuous probabilistic measure $\mu$ (recall that any such space is equivalent to the segment $[0,1]$). Denote by $\Ams=\Ams(M)$ the group of all transformations (defined up to a.s.) preserving the measure $\mu$. By $\Gms=\Gms(M)$ we denote the group of transformations (defined up to a.s.) leaving the measure $\mu$ quasiinvariant.
The group $\Ams$ was widely discussed in connection with ergodic theory, the group $\Gms$, which is a topic of the present note, only occasionally was mentioned in the literature. However, it is an interesting object from the point of view of representations of infinite-dimensional groups (“large groups” in the terminology of A.M.Vershik), see [@Ner-poi], [@Ner-gauss].
[**The topology on $\Gms$.**]{} A separable topology on $\Gms$ was defined in [@Kech] 17.46, [@Ner-bist], [@Pes],§4.5 by different ways. One of the purposes of the present note is two show that these ways are equivalent.
The first way is following. Let $A$, $B\subset M$ be measurable subsets. For $g\in\Gms$ we define the distribution $$\kappa[g;A,B]$$ of the Radon–Nikodym derivative $g'$ on the set $A\cap g^{-1}(B)$. We say that a sequence $g_j\in\Gms$ converges to $g$, if for any measurable sets $A$, $B$ we have the following weak convergences of measures on $\R^\times$ $$\kappa[g_j;A,B] \to \kappa[g;A,B],\qquad t\kappa[g_j;A,B]\to t\kappa[g;A,B].
\label{eq:convergence}$$
[Remark 1.]{} Point out evident identities: $$\int_{\R^\times} \kappa[g;A,M](t)=\mu(A),\qquad
\int_{\R^\times} t\, \kappa[g;A,M](t)=\mu(gA)
\label{eq:id}
.$$
[Remark 2.]{} Consider a measurable finite partition $$\frh:M=M^1\cup M^2\cup \dots$$ of the space $M$. This gives us a matrix $S_{\alpha\beta}[g;\frh]:=\kappa[g;M^\alpha,M^\beta]$, composed of measures on $\R^\times$. If a partition $\frk$ is a refinement of $\frh$, we write $\frh \preccurlyeq\frk$. Consider a sequence of partitions $\frh_1 \preccurlyeq \frh_2 \preccurlyeq\dots$, generating the $\sigma$-algebra of the space[^2] $M$. A convergence $g_j\to g$ is equivalent to an element-wise convergence in the sense (\[eq:convergence\]) of all matrices $S[g_j;\frh_n]\to S[g;\frh_n]$.
\[pr:1\] The group $\Gms$ is a Polish group with respect to this topology, i.e., $\Gms$ is a separable topological group complete with respect to the two-side uniform structure and homeomorphic to a complete metric space[^3].
Let $1\le p\le \infty$, $s\in\R$. The group $\Gms$ acts in the space $L^p(M)$ by isometric transformations according the formula $$T_{1/p+is}f(x)=f(g(x))g'(x)^{1/p+is}
.$$ On the space $\cB(V)$ of operators of a Banach space $V$ we define in the usual way (see, e.g., [@RS], VI.1) the strong and weak topologies. Also, on the set $\cG\cL(V)$ of invertible operators we introduce a [*bi-strong*]{} topology, $A_j$ converges to $A$, if $A_j\to A$ and $A^{-1}_j\to A^{-1}$ strongly. The embedding $T_{1/p+is}:\Gms \to \cB(L^p)$ induces a certain topology on $\Gms$ from any operator topology on $\cB(V)$ or $\cG\cL(V)$.
\[pr:2\] [a)]{} Let $1<p<\infty$, $s\in\R$. A topology on $\Gms$ induced from any of three topologies [(]{}strong, weak, bi-strong[)]{} coincides with the topology defined above.
[b)]{} Let $p=1$, $s\in\R$. A topology on $\Gms$ induced from strong or bi-strong topology coincides with the topology defined above.
[c)]{} Let $1\le p<\infty$, $s\in\R$. Then the image of $\Gms$ в $\cG\cL(L^p(M))$ is closed in the bi-strong topology.
Point out that the coincidence of topologies is not surprising. It is known that two different Polish topologies on a group can not determine the same Borel structure, see [@Kech], 12.24. There are also theorems about automatic continuity of homomorphisms, see [@Kech], 9.10,
[**Double cosets $\Ams\setminus \Gms/\Ams$. Canonical forms.\[ss:predst\]**]{} We reformulate the problem of description of double cosets $\Ams\setminus \Gms/\Ams$ in the following way. Let $(P,\pi)$, $(R,\rho)$ be Lebesgue spaces with continuous probabilistic measures. Denote by $\Gms(P,R)$ the space of all bijections $g:P\to R$ (defined up to a.s.), such that images and preimages of sets of zero measure have zero measure. We wish to describe such bijections up to the equivalence $$g\sim u\cdot g\cdot v,\qquad \text{where $v\in\Ams(P)$, $u\in \Ams(R)$}
\label{eq:sim}$$ (clearly, such classes are in-to-one correspondence with double cosets $\Ams\setminus\Gms/\Ams$).
\[l:derivative\] Two elements $g_1$, $g_2\in\Gms(P,R)$ are contained in one class if and only if the Radon–Nikodym derivatives $g_1'$, $g'_2:P\to \R$ are equivalent with respect to the action of the group $\Ams(P)$, i.e., $g_2'(m)=g_1'(hm)$, where $h$ is an element of $\Ams(P)$.
An evident invariant of this action is the distribution $\nu$ of the Radon–Nikodym derivative $g'$ of the map $g$, $$\int_{\R^\times} d\nu(t)=1 ,\qquad \int_{\R^\times} t\,d\nu(t)=1.
\label{eq:nu}$$ This invariant is not exhaust, the problem is reduced to the Rokhlin theorem [@Roh2] on metric classification of functions, see discussion below, § \[ss:roh\]. The final answer is following.
Consider a countable number of copies $\R_1^\times$, $\R_2^\times$, …of half-line $\R^\times$. Consider one more more copy $\R_\infty^\times$. Consider the disjoint union $$\cL:=\R_1^\times\coprod \R_2^\times\coprod \R_3^\times\coprod \dots \coprod \bigl(\R_\infty^\times \times [0,1]\bigr).$$ Let $\nu_1$, $\nu_2$,…и $\nu_\infty$ be a family of measure on $\R^\times$ satisfying the following conditions
1\. $\nu_1$, $\nu_2$, …are continuous (but $\nu_\infty$ admits atoms).
2\. $\nu_1\ge\nu_2\ge\dots$
3\. The measure $\nu:=\nu_1+\nu_2+\dots+\nu_\infty$ satisfies (\[eq:nu\]).
Equip each $\R^\times_j$ with the measure $\nu_j$, equip $\R^\times_\infty\times[0,1]$ with the measure $\nu_\infty\times dx$, where $dx$ is the Lebesgue measure on the segment. Denote the resulting measure space by $\cL[\nu_1,\nu_2,\dots;\nu_\infty]$.
Consider the same measure on $\cL$ multiplied by $t$, we denote the resulting measure space by $\cL_*[\nu_1,\nu_2,\dots;\nu_\infty]$.
Consider the identity map $${\rm id}:\cL[\nu_1,\nu_2,\dots;\nu_\infty]\to \cL_*[\nu_1,\nu_2,\dots;\nu_\infty]
\label{eq:identity}$$ Evidently, the distribution of the Radon–Nikodym derivative of the map ${\rm id}$ coincides with $\nu$.
\[pr:3\] Any equivalence class [(\[eq:sim\])]{} contains a unique representative of the type [(\[eq:identity\])]{}.
Denote the double coset containing this representative by $S[\nu_1,\nu_2,\dots;\nu_\infty]$.
[**On closures of double cosets.**]{}
\[l:4\] Let a measure $\nu$ on $\R^\times$ satisfies [(\[eq:nu\])]{}, let $\nu=\nu^c+\nu^d$ be its decomposition into continuous and discrete parts. Then the closure of the double coset $S[\nu^c,0,0,\dots;\nu^d]$ contains all double cosets $S[\nu_1,\nu_2,\nu_3,\dots;\nu^c_\infty+\nu_d]$ with $\nu_1+\nu_2+\dots+\nu_\infty^c=\nu^c$.
[**Hausdorff quotient.**]{} Consider the space $\cM$ of all measures $\nu$ on $\R^\times$ satisfying (\[eq:nu\]). Say that $\nu^j\in\cM$ converges to $\nu$ if $\nu^j\to \nu$ and $t\nu^j\to t\nu$ weakly.
Consider a map $\Phi:\Gms\to \cM$ that for any $g$ assigns the distribution of its Radon–Nikodym derivative (i.e., $\Phi(g)=\kappa[g;M,M]$). In virtue of Theorem \[l:4\], preimages of points $\nu\in\cM$ are closures of double cosets $S[\nu^c,0,0,\dots;\nu^d]$.
\[th:quotient\] Let $f$ be a continuous map of $\Gms$ to a metric space $T$, moreover, $f$ let be constant on double cosets. Then $f$ has the form $f=q\circ \Phi$, where $q:\cM\to T$ is a continuous map.
[**A continuous section $\cM\to\Gms$.**]{} We say that a function $h:[0,1]\to[0,1]$ is contained in the class $\cG$, if
$\bullet$ $h$ is downward convex;
$\bullet$ $h(0)=0$, $h(1)=1$, and $h(x)>0$ for $x>0$.
Any such function is an element of the group $\Gms\bigl([0,1]\bigr)$.
\[pr:section\] Let $\nu\in \cM$. Then there is a unique function $\psi:[0,1]\to[0,1]$ of the class $\cG$ such that the distribution of the derivative $\psi'$ is $\nu$. Moreover, the map $\nu\mapsto\psi$ is a continuous map $\cM\to\Gms$.
[**A more general statement.**]{} Consider a finite or countable measurable partition of our measure space $M=\coprod_j M_j.$ Denote by $K$ the direct product $
K=\Ams(M_1)\times \Ams(M_2)\times \dots
$. Consider the double cosets $K\setminus \Gms/K$. Assign to each $g\in \Gms$ the matrix $\kappa_{ij}=\kappa[g;M_i,M_j]$ composed of measures on $\R^\times$. Denote by $\cS$ the set of matrices that can be obtained in this way, i.e., $$\sum_{j} \int_{\R^\times} d\kappa_{ij}(t)=\mu(M_i),
\qquad
\sum_{i} \int_{\R^\times} t\kappa \, d\mu_{ij}(t)=\mu(M_j).$$ Equip $\cS$ with element-wise convergence (\[eq:convergence\]). Denote by $\Psi$ the natural map $\Gms\to \cS$.
\[th:last\] Let $f$ be a continuous map from $\Gms$ to a metric space $T$. Then there exists a continuous map $q:\cS\to T$, such that $f=q\circ \Psi$.
Point out that this statement was actually used in [@Ner-bist], [@Ner-poi].
[**The structure of the note.**]{} The statements about topology on $\Gms$ are proved in §2, about double cosets in §3. Theorem \[th:quotient\] follows from Theorem \[l:4\]. However, as the referee pointed out, the first statement is simpler than the second (and it is more important). Therefore in the beginning of §3 we present a separate proof of Theorem \[th:quotient\].
The topology on the group $\Gms$
================================
Below we prove Propositions \[pr:1\] and \[pr:2\]. The main auxiliary statement is Lemma \[l:2.4\]. The remaining lemmas are proved in a straightforward way.
Notation:
$\bullet$ $\delta_a$ is an probabilistic atomic measure $\R^\times$ supported by a point $a$.
$\bullet$ $\{\cdot,\cdot\}_{pq}$ is the natural pairing of $L^p$ and $L^q$, where $1/p+1/q=1$;
$\bullet$ $\chi_A$ is the indicator function of a set $A\subset M$, i.e., $\chi_A(x)=1$ for $x\in A$ and $\chi_A(x)=0$ for $x\notin A$.
[**Preliminary remarks on the spaces $L^p$.**]{}
1\) Recall (see [@FJ], §3.3) that for $p\ne 2$ the group of isometries $\Isom\bigl(L^p(M)\bigr)$ of the space $L^p(M)$ consists of operators of the form $$R(g, \sigma)f(x)=\sigma(x) f(g(x))g'(x)^{1/p}
,
\label{eq:isometries}$$ where $g\in\Gms$, and $\sigma:M\to\C$ is a function whose absolute value equals 1.
2\) For $1<p<\infty$ the space $L^p$ is uniformly convex (see [@Koth], §26.7), therefore the restrictions of the strong and weak topologies to the unit sphere coincide. Therefore on the group of isometries $\Isom\bigl(L^p(M)\bigr)$ the weak and strong operator topologies coincide.
3\) Recall that for separable Banach spaces (in particular, for $L^p$ with $p\ne \infty$) the group of all isometries equipped with bi-strong topology is a Polish group, see [@Kech], 9.B9.
[**Preliminary remarks on the group $\Gms$.**]{}
1\) [*The invariance of the topology.*]{} Equip $\Gms$ with topology from Subsection 1.2. The product in $\Gms$ is separately continuous (this is a special case of Theorem 5.9 from [@Ner-boundary]). In particular, this implies that the topology on $\Gms$ is invariant with respect to left and right shifts.
The map $g\mapsto g^{-1}$ is continuous. Indeed, $$\kappa[g^{-1};B,A](t)=t^{-1}\kappa[g;A,B](t^{-1})
,$$ and this map transpose the convergences (\[eq:convergence\]).
2\) [*Separability of $\Gms$*]{}. For a measure $\kappa[g;A,B]$ consider the [*characteristic function*]{} $$\chi(z)=\int_{\R^\times} t^z d \kappa[g;A,B](t)
,
\label{eq:char}$$ continuous in the strip $0\le {\mathop {\mathrm {Re}}\nolimits}z\le 1$ and holomorphic in the open strip. The convergence of measures $\kappa$ is equivalent to point-wise convergence of characteristic functions uniform in each rectangle $$0\le {\mathop {\mathrm {Re}}\nolimits}z\le 1,\qquad -N\le {\mathop {\mathrm {Im}}\nolimits}z\le N,$$ [@Ner-boundary], Propositions 4.4-4.5. This convergence is separable. Next, by Remark 2 of §\[ss:1\], it suffices to verify the convergence of measures $\kappa[g_j;A,B]\to \kappa[g;A,B]$ for an appropriate countable set of pairs measurable subsets $(A,B)$.
3\) [*The action on Boolean algebra of sets.*]{}
\[eq:bigtriangleup\] Let $g_j\to g$ in $\Gms$. Then for any measurable set $A\subset M$ we have $$\mu(g_j A\bigtriangleup gA)\to 0
\label{eq:triangle}
.$$
[Proof.]{} By the invariance of the topology it suffices to consider $g=1$. Then $$\mu(g_j A\cap A)=\int\limits_{\R^\times} d\kappa[g_j^{-1}; A; A](t)
\to
\int\limits_{\R^\times} d\kappa[1; A; A](t)=
\int\limits_{\R^\times} \mu(A)\delta_0(t)=\mu(A)
;$$ $$\mu(g_j A)=\int\limits_{\R^\times} t\,d\kappa[g_j;A;M]\to
\int\limits_{\R^\times} t\,d\kappa[1;A;M]=\int\limits_{\R^\times} \mu(A) t\,\delta_0(t)=\mu(A)
.$$ Comparing two rows we get the desired statement. $\square$
[Remark.]{} The opposite is false. Let $M=[0,1]$, $$g_j(x)=x+\frac 1{2\pi n}\sin(2\pi n x).$$ Then for any $A\subset[0,1]$ we have $\mu(g_j(A)\bigtriangleup A)\to \mu(A)$. But there is no convergence $g_j\to 1$ in $\Gms$; $T_1(g_j)$ converges weakly to 1 in $L^1$, but there is no strong convergence. $\boxtimes$
4)[*The continuity of representations $T_{1/p+is}$.*]{}
For $p<\infty$ the homomorphisms $T_{1/p+is}:\Gms\to \Isom(L^p)$ are continuous with respect to the weak topology $\Isom(L^p)$.
[Proof.]{} Let $g_j\to g$ in $\Gms$. Consider ’matrix elements’ $$\{ T_{1/p+is}(g_j)\chi_A,\chi_B\}_{pq}=
\int\limits_{A\cap g_j^{-1}B} g_j'(x)^{1/p+is}d\mu(x)=
\int\limits_{\R^\times}t^{1/p+is}\,d\kappa[g_j;A,B](t)$$ Weak convergence of measures (\[eq:convergence\]) implies the convergence of characteristic functions (\[eq:char\]), our expression tends to $$\int\limits_{\R^\times}t^{1/p+is}\,d\kappa[g;A,B](t)=
\int\limits_{A\cap g^{-1}B} g'(x)^{1/p+is}d\mu(x)=\{ T_{1/p+is}(g)\chi_A,\chi_B\}_{pq}
,$$ as required. $\square$
Thus, for $1<p<\infty$ the maps $T_{1/p+is}:\Gms\to\Isom(L^p)$ are continuous with respect to the strong (=weak) topology. Keeping in mind the continuity of the map $g\mapsto g^{-1}$, we get that the maps $T_{1/p+is}$ are continuous with respect to the bi-strong topology.
The case $L^1$ must be considered separately.
Let $g_j\to g$. Then $T_{1+is} (g_j)\in \Isom(L^1)$ strongly converges to $T_{1+is} (g)$.
[Proof.]{} Without loss of generality, we can set $g=1$. It suffices to verify the convergence $\|T_{1+is} (g_j)\chi_A-\chi_A\|\to0$ for any measurable $A$. This equals $$\begin{gathered}
\int_M \bigl|\chi_A(g_j x) g'(x)^{1+is}-\chi_A(x)\bigr|\,d\mu(x)=\\=
\int_{A\cap g_j^{-1}A}\bigl| g'(x)^{1+is}-1\bigr|\,d\mu(x)+
\int_{A\setminus g_j^{-1}A}d\mu(x)+ \int_{ g_j^{-1}A\setminus A} g'(x)\,d\mu(x)
=\\=
\int_{\R^\times}|t^{1+is}-1|\,d\kappa[g_j;A,A](t)+
\mu\bigl(A\setminus g_j^{-1}A\bigr)+\mu(A\setminus g_j A)
\label{eq:L1}
.\end{gathered}$$ The second and the third summands tend to 0 by Lemma \[eq:bigtriangleup\], measures $\kappa[\dots]$ and $t\kappa[\dots]$ converge weakly to $\mu(A)\delta_0$, therefore the integral tends to 0. $\square$
[**The coincidence of topologies and the continuity of the multiplication.**]{}
\[l:2.4\] Let $1<p<\infty$. Let $T_{1/p+is}(g_j)$ weakly converge to $1$ in $\Isom(L_p)$. Then $g_j$ converges to $1$ in $\Gms$.
[Proof.]{} [*Step 1.*]{} Now it will be proved that $g'_j$ converges to 1 in $L^1(M)$. For this purpose, we notice that the following sequence of matrix elements must converge to 1: $$\{T_{1/p+is}(g_j)\, 1, 1\}_{pq}=\int\limits_M g'_j(x)^{1/p+is}d\mu(x)
=\int\limits_{\R^\times} t^{1/p+is} d\kappa[g_j;M,M](t).
\label{eq:me}$$ Estimate the integrand: $${\mathop {\mathrm {Re}}\nolimits}t^{1/p+is}\le t^{1/p}\le \frac 1 q+ \frac t p.$$ The second inequality means that the graph of upward convex function is lower than the tangent line at $t=1$. From another hand: $$\begin{gathered}
\int_{\R^\times}\Bigl( \frac 1 q+ \frac t p\Bigr)\, d\kappa[g_j;M,M](t)
=\\
=\frac 1 q\int_{\R^\times}d\kappa[g_j;M,M](t)+
\frac 1 p\int_{\R^\times}t\,d\kappa[g_j;M,M](t)=\frac 1 q+ \frac 1 p=1.\end{gathered}$$ Look to a deviation of integral (\[eq:me\]) from 1. The same reasoning with tangent line allows to estimate the difference $
\frac 1 q+ \frac t p-t^{1/p}
$. For any $\epsilon>0$ there is $\sigma>0$ such that $$\frac 1 q+ \frac t p-t^{1/p}>\begin{cases}
\sigma\qquad \text{for $t<1-\epsilon$};\\
\sigma t\qquad\text{for $t>1+\epsilon$}.
\end{cases}$$ Therefore $$1-{\mathop {\mathrm {Re}}\nolimits}\{T_{1/p+is}(g_j)\, 1, 1\}_{pq}>\sigma\int\limits_0^{1-\epsilon}d\kappa[g_j;M,M](t)+
\sigma
\int\limits_{1+\epsilon}^\infty t
\, d\kappa[g_j;M,M](t)
.$$ This must tend to 0, therefore $\kappa[g_j;M,M]$ and $t\cdot \kappa[g_j;M,M]$ tend to $\delta_0$ weakly. This implies the convergence $g'_j\to 1$ in the sense of $L^1$.
The remaining part of the proof is more-or-less automatic.
[*Step 2.*]{} Let $z$ be contained in the strip $0\le {\mathop {\mathrm {Re}}\nolimits}z\le 1$. Let us show that $(g'_j)^{z}$ tends to 1 in the sense of $L^1$. Let $\|g'-1\|_{L^1(M)}<\epsilon$. Then there is an uniform with respect to $g$ estimate $\|(g')^z-1\|_{L^1(M)}<\psi_z(\epsilon)$, where $\psi_z(\epsilon)$ tends to 0 as $\epsilon$ tends to 0. For this aim it is sufficient to notice that $$|a^z-1|< \begin{cases}
|z|\,(a-1) \qquad &\text{for $a>1$};\\
|z|\, 2^{-{\mathop {\mathrm {Re}}\nolimits}z+1} |a-1| \qquad &\text{for $1/2\le a\le 1$};\\
2 \qquad \text{for $0<a<1/2$},
\end{cases}$$ moreover, $g'<1/2$ can be only on the set of measure $\le 2\epsilon$.
In particular, for any subset $C\subset M$ we have $$\Bigl|\int_C g'(x)^z dx-\mu(C)\Bigr|\le\psi_z(\epsilon)
\label{eq:psi()}
.$$
[Step 3.]{} Now we use convergence of matrix elements: $$\{ T_{1/p+is}(g_j)\chi_A,\chi_B\}_{pq}
=\int\limits_{A\cap g_j^{-1}B} g_j'(x)^{1/p+is}d\mu(x)
\to
\{ \chi_A,\chi_B\}_{pq}= \mu(A\cap B).$$ By (\[eq:psi()\]), we have convergence $$\int_{A\cap g_j^{-1}B} g_j'(x)^{1/p+is}d\mu(x)-\mu(A\cap g_j^{-1}B)\to 0
.$$ Comparing two last convergences we get $\mu(A\cap g_j^{-1}B)
\to \mu(A\cap B)$.
[*Step 4.*]{} By the convergence $(g_j')^z$ in $L^1(M)$, we have $$\int t^z \,d\kappa[g_j;A,B](t)=\int_{A\cap g_j^{-1}B}g'(x)^z\,dx \to
\mu(A\cap B)$$ for each $z$; the point-wise convergence of characteristic functions implies weak converges (\[eq:convergence\]) of measures (see, [@Ner-boundary]), in our case, to $\mu(A\cap B)\delta_0$. $\square$
Thus the topology on $\Gms$ is induced from the strong operator topology of the spaces $L^p$. In separable Banach spaces the multiplication is continuous in the strong topology on bounded sets. Therefore, the multiplication in $\Gms$ is continuous.
Let operators $T_{1+is}(g_j)$ converge to $1$ in the strong operator topology of spaces $L^1$. Then $g_j\to 1$ in $\Gms$.
[Proof.]{} In (\[eq:L1\]) the first row must tend to zero. Therefore all summands of the last row tend 0, in particular the first one. This implies weak convergences of measures $\kappa[g_j;A,A]$ and $t\kappa[g_j;A,A]$ to $\mu(A)\delta_1$. Comparing this with (\[eq:id\]), we get convergences $\kappa[g_j;A,M\setminus A]$ and $t\kappa[g_j;A,M\setminus A]$ to 0. Now it is easy to derive the convergence of $g_j\to g$ in $\Gms$. $\square$
[**The completeness of $\Gms$.**]{} The group of isometries of a separable Banach space is a Polish group with respect to the bi-strong topology ([@Kech], 9.3.9). Let $p\ne 1$ $2$, $\infty$ и $s=0$. Then the isometries $T_{1/p}(g)$ are precisely isometries (\[eq:isometries\]) that send the cone of non-negative functions to itself. Obviously, the set of operators sending this cone to itself is weakly closed. Therefore, $\Gms$ is a closed subgroup in the group of all isometries and therefore it is complete.
$$\epsfbox{split.3}$$
[**Bi-strong closeness of the image.\[ss:closure\]**]{} The group $\Gms$ is closed in the group $\Isom(L^p)$, since it is complete with respect of the induced topology.
It is noteworthy that the group $\Isom(L^p)$ is not strongly closed in the space of bounded operators in $L^p$. The images of the groups $\Ams$ and $\Gms$ also are not closed.
[Example.]{} Let $p\ne\infty$. Consider an operator in $L^p$ of the form $$Rf(x)=
\begin{cases}
f(2x), \qquad\text{ for $0\le x\le 1/2$};
\\
f(2x-1), \qquad\text{for $1/2< x\le 1$};
\end{cases}$$ For any function $f$ we have $\|Rf\|=\|f\|$. However, this operator is not invertible. For the sequence $g_n\in \Ams$ from Fig. \[fig:1\] we have the strong convergence $T_{1/p}(g_n)$ to $R$. $\boxtimes$.
Weak closures for some subgroups $\Gms$ are discussed in [@Ner-poi], [@Ner-gauss].
Double cosets
=============
[**Proof of Theorem \[th:quotient\].**]{} Denote by $G^0\subset\Gms$ the group of transformations whose Radon–Nikodym derivative has only finite number of values. Obviously,
$\bullet$ The subgroup $G^0$ is dense in $\Gms$.
$\bullet$ Double cosets $\Ams\setminus G^0/\Ams$ are completely determined by the distribution of the Radon–Nikodym derivative.
Consider a measure $\kappa\in\cM$. Consider a sequence of discrete measures $\kappa_N\in \cM$ convergent to $\kappa$ and having the following property: Fix $N$ and cut the semi-axis $t>0$ into pieces of length $2^{-N}$. For any $j\in \N$ we require the following coincidence of measures of semi-intervals $$\int\limits_{
\frac {j-1}{2^N}<t\le \frac{j}{2^N}}\!\!
d\kappa(t)=
\!\!
\int\limits_{
\frac {j-1}{2^N}<t\le \frac{j}{2^N}}\!\!
d\kappa_N(t),\qquad
\int\limits_{
\frac {j-1}{2^N}<t\le \frac{j}{2^N}}
\!\!
t\cdot
d\kappa(t)=
\!\!
\int\limits_{
\frac {j-1}{2^N}<t\le \frac{j}{2^N}}
\!\!
t\cdot
d\kappa_N(t)$$
Consider $g\in \Gms$ whose distribution of the Radon–Nikodym derivative equals $\kappa$. Consider a sequence $g_N\in G^0$ convergent to $g$ such that a distribution of the Radon–Nikodym derivative of $g_N$ is $\kappa_N$. For this, we fix $N$ and for each $j$ consider the subset $A_j\subset M$, where the Radon–Nikodym derivative satisfies $$\frac {j-1}{2^N}<g'(x)\le \frac{j}{2^N}.$$ Set $B_j=g(A)$. Consider an arbitrary map $g_N\in G^0$ such that $g_N$ send $A_j$ to $ B_j$ and the distribution of the Radon–Nikodym derivative of $g_N$ coincides with the restriction of the measure $\kappa_N$ of the semi-interval $\bigl(\frac {j-1}{2^N}<t\le \frac{j}{2^N}]$. It easy to see that the sequence $g_N$ converges to $g$.
Now, let $f$ be a continuous function on $\Gms$ constant on double cosets. Пусть $g$ and $h\in\Gms$ have same distribution of Radon–Nikodym derivatives. Then $g_N$ and $h_N$ are contained in the same double coset, wherefore $f(g_N)=f(h_N)$. By continuity of $f$ we get $f(g)=f(h)$.
To avoid a proof of the continuity the map $q$ (see the statement of the theorem), we refer to Proposition \[pr:section\] (which is proved below independently of the previous considerations).
[**Proof of Proposition \[pr:3\].\[ss:roh\]**]{} Let $M\simeq[0,1]$ be a Lebesgue space. Invariants of measurable functions $f:M\to \R$ with respect to the action of $\Ams(M)$ were described by Rokhlin in [@Roh2]. To any function $f$ he assigns its distribution function $F(y)$, i.e., the measure of the set $M_y\subset M$ determined by the inequality $f(x)< y$. Also he assigns to $f$ a sequence of functions $F_1$, $F_2$, …, where $F_n(y)$ is the supremum of measures of all sets $A\subset M_y$, on which $f$ takes each value $\le n$ times. These data satisfy the following conditions:
$\bullet$ the function $F$ satisfies the usual properties of distribution functions: $F$ is a left-continuous non-decreasing function, $\lim_{y\to-\infty} f(y)=0$, $\lim_{y\to+\infty} f(y)=1$;
$\bullet$ $F_n$ are non-decreasing functions;
$\bullet$ $0\le F_1(y)\le F_2(y)\le\dots\le F(y)$;
$\bullet$ $F_k(y)-2 F_{k+1}(y)+ F_{k+2}(y)\ge 0$ for all $k$.
According [@Roh2], a function $f$ determined up to the action of the group $\Ams$ is uniquely defined by the invariants $F_1$, $F_2$,…, $F$. Moreover, for any collection of functions $F_1$, $F_2$,…, $F$ with above listed properties there exists $f$, whose invariants coincide with $F_1$, $F_2$,…, $F$.
Now we will describe canonical forms of functions $f$ under the action of the group $\Ams$. Consider a collection of continuous measures $\nu_1\le\nu_2\le\dots$ on $\R$ and the measure $\nu_\infty$ on $\R$ such that $\nu_1(\R)+\nu_2(\R)+\dots+\nu_\infty(\R)=1$. Denote by $t$ the coordinate on $\R$. Consider the disjoint union of the spaces with measures $$\cL=\Bigl((\R,\nu_1)\coprod(\R,\nu_2)\coprod\dots\Bigr)\coprod (\R\times[0,1],\nu_\infty \times ds)
,
\label{eq:for-roh}$$ where $ds$ is the Lebesgue measure on the segment $[0,1]$. Consider the function $f$ on $\cL$ that equals to $t$ on each copy of $\R$ and equals to $t$ on $\R\times[0,1]$.
The invariants of this function are $$F_n(y)=\sum_{j\le n}\nu_j(-\infty,y),\qquad F(y)=\sum_{1\le j<\infty} \nu_j(-\infty,y)+
\nu_\infty(-\infty,y)$$ It can be readily seen that measures $\nu_1$, $\nu_2$,…, $\nu_\infty$ admit a reconstruction from the invariants $F_1$, $F_2$,…, $F$. Moreover any admissible collection of invariants corresponds to a certain collection of measures $\nu_1$, $\nu_2$, …, $\nu_\infty$.
Now consider an element $g\in \Gms(P,R)$. Reduce the derivative $g':P\to\R^\times$ to the canonical form by a multiplication $g\mapsto gh$, where $h\in\Ams$. Since $g'(x)>0$, all the measures $\nu_j$, $\nu$ are supported by the half-line $t>0$. The integral of $g'$ is 1, therefore $$\sum_j\int t\,d\nu_j(t)+\int t\,d\nu_\infty(t)=1.
\label{eq:for-prob}$$ Now we assume $P=\cL$, see (\[eq:for-roh\]). Let $\cL_*$ be obtained from $\cL$ by a multiplication of the measure by $t$. In virtue of (\[eq:for-prob\]), this measure must be probabilistic. The map $g:\cL\to R$ can be regarded as a map $g_*:\cL_*\to R$. Since $g'=t$, for any measurable set $B\subset\cL$ the measure of $B$ in $\cL_*$ coincides with the measure $g(B)$. Therefore $g_*:\cL_*\to R$ preserves measure.
Thus $g$ is reduced to the canonical form.
[**Splitting of measures.**]{} We start a proof of Theorem \[l:4\]. Modify the notation for $\cL[\nu_1,\nu_2,\dots;\nu_\infty]$, $\cL_*[\nu_1,\nu_2,\dots;\nu_\infty]$ and $S[\nu_1,\nu_2,\dots;\nu_\infty]$ from (\[ss:predst\]). Now it is convenient to reject the condition $\nu_1\ge\nu_2\ge\dots$. Also, we weaken condition (\[eq:nu\]) and set $$\int_{\R^\times} d\nu(t)<\infty ,\qquad \int_{\R^\times} t\,d\nu(t)<\infty.
\label{eq:nu-1}$$
Let $\nu$ be a continuous measure on $\R^\times$ satisfying (\[eq:nu-1\]). Consider the space $\cL[\nu, 0,0,\dots;0]$. Represent $\nu$ as a sum $\nu=\nu_1+\nu_2$.
\[l:split\] The closure of the class $S[\nu, 0,0,\dots;0]$ contains $S[\nu_1, \nu_2,0,\dots;0]$.
$$\epsfbox{split.1}$$
[Proof.]{} Denote $$\cL:=\cL[\nu_1,0,\dots;0],\qquad \cL':=\cL[\nu_1, \nu_2,0,\dots;0]
.$$ The same measure spaces with the measure multiplied by $t$ we denote as $\cL_*$, $\cL_*'$ Now we will construct two sequences of measure preserving bijections $$\phi_n: \cL\to \cL',
\qquad
\psi_n: \cL_*\to \cL'_*
.$$ Cut $(\R^\times,\nu)$ by $2^n$ intervals $C_0$,…,$C_{2^n-1}$ by points $$a_k=\frac{k 2^{-n}}{1-k 2^{-n}} ,\qquad k=1,2,\dots,2^n-1
.$$ Denote this partition[^4] by $\frh_n$.
The exists a sub-interval $B_k\subset C_k$ such that $\nu(B_k)=\nu_1(C_k)$, $(t\cdot\nu)(B_k)=(t\cdot\nu_1)(C_k)$.
[Proof.]{} We have $C_k=[a_k,a_{k+1}]$. Consider segments $[a_k,u]$, $[v,a_{k+1}]\subset [a_k,a_{k+1}]$ such that $\nu[a_k,u]=\nu[v,a_{k+1}]=\nu_1[C_k]$. For any $z\in [a_k,v]$ there exists $z^\circ\le a_{k+1}$ such that $\nu[z,z^\circ]=\nu_1[C_k]$. It is easy to see that $$(t\cdot \nu)[a_k,u]\le\nu_1[C_k], \qquad (t\nu)[v,a_{k+1}]\ge (t\cdot\nu_1)[C_k].$$ Form continuity reasoning there exists $[z,z^\circ]$ satisfying the desired property. $\square$
For each $k$ consider arbitrary measure preserving maps $$(B_k,\nu)\to (C_k,\nu_1),\qquad (C_k\setminus B_k,\nu_2)\to (C_k,\nu_2).$$ This produces a map $\phi_n$ (see. Fig.\[fig:2\]). To obtain $\psi_n$ we take arbitrary measure preserving maps $$(B_k,t\cdot\nu)\to (C_k,t\cdot\nu_1),\qquad
(C_k\setminus B_k,t\cdot \nu_2)\to (C_k,t\cdot\nu_2).$$
Consider a map $$\theta_n:\psi_n\circ {\rm id}\circ \phi_n^{-1}:\cL'\to \cL_*'
.$$ The space $\cL'$ consists of two copies $\R^\times_1$, $\R^\times_2$ of the half-line $\R^\times$, each copy is cutted into segments $C_k$ . The map $\theta_n$ send each copy of a segment $C_k\subset\R^\times_1$, $C_k\subset\R^\times_2$ to itself, moreover the Radon–Nikodym derivative of $\theta_n$ takes values $C_k$ in limits $[a_{k},a_{k+1}]$.
It is easy to see that the sequence $\theta_n$ converges to the map ${\rm id}:\cL'\to \cL'_* $. $\square$
[**The spreading of measures.**]{} Denote $$\cL:=\cL[\nu,0,\dots,0], \qquad \cL''=\cL[0,0,\dots; \nu].$$ Let $\cL_*$, $\cL_*''$ be the same measure spaces with the measure multiplied by $t$. We construct a sequence of measure preserving bijections $$\xi_n: \cL\to\cL'',
\qquad
\zeta_n: \cL_*\to\cL_*''
.$$ For this aim, consider the same partitions $\frh_n$ of the space $(\R^\times,\nu)$. Consider arbitrary measure preserving maps[^5] $$(C_k,\nu)\to (C_k\times [0,1], \nu\times dx),\qquad
(C_k,t\nu)\to (C_k\times [0,1],(t\nu)\times dx)$$ This gives us the maps $\xi_n$ и $\zeta_n$. Consider the map $$\upsilon_n=\zeta_n\circ{\rm id}\circ \xi_n^{-1}: \cL''
\to \cL''_*.$$ The map $\upsilon_n$ sends each $C_k\times[0,1]$ to itself, its Radon–Nikodym derivative on $C_k\times[0,1]$ varies in the limits $[a_{k-1},a_k]$. Passing to a limit as $n\to\infty$, we get the identity map $\cL''\to \cL''_*$.
[**Proof of Theorem \[l:4\]**]{}. Пусть $\nu\in\cM$. Without loss of generality, we can assume that $\nu$ is continuous. Expand $\nu=\nu_1+\nu_2+\dots+\nu_\infty$. Set $$\cL^k=\cL\bigl[\nu_1,\dots,\nu_k,\sum_{j=k+1}^\infty\nu_j,0,0,\dots;\nu_\infty\bigr],
\qquad \cL^\infty:=\cL[\nu_1,\nu_2,\dots;\nu_\infty].$$ Let $\cL^k_*$, $\cL_*^\infty$ be the same measure spaces with measures multiplied by $t$. Let $\mathrm{id^k}:\cL^k\to \cL^k_*$, $\mathrm{id^\infty}:\cL^\infty\to \cL^\infty_*$ denote the identical maps.
Iterating arguments of the two previous subsections, we obtain that the closure of $S[\nu,0,\dots,0]$ contains elements $\mathrm{id}^k$ for any finite $k$. Consider a map $\alpha_k:\cL^k\to\cL^\infty$ constructed in the following way. It is identical on $\R_1^\times$, …, $\R_k^\times$ send the semi-line $\R^\times_{k+1}$ to $\coprod_{j\ge k+1} \R_j$ preserving the measure. In the same way we construct a map $\beta_k:\cL_*^k\to\cL_*^\infty$. It is easy to see that the sequence $$\chi_k:= \beta_k\circ \mathrm{id}_k\circ \alpha_k^{-1}:\cL^\infty\to \cL^\infty_*
.$$ converges to $\mathrm{id}^\infty$.
[**Construction of the function $\psi$.**]{} Here we obtain the continuous section $\cM\to\Gms$. Consider the distribution function $z=F(y)$ of the measure $\nu$ and the inverse function $y=G(z)$. If $y_0$ is a discontinuity point of $F$, we set $G(z)=y_0$ on the segment $[F(y_0-0), F(y_0)+y_0)$. If $F$ takes some value $z_0$ on a segment of nonzero length, then $G(z_0)$ is not defined. Further, we set $\psi(x)=\int_0^x G(z)\,dz$.
[**Proof of Proposition \[pr:section\].**]{} Let $\nu_j$ converges to $\nu$ in $\cM$, $y=\psi_j(x)$, $y=\psi(x)$ be the corresponding maps $[0,1]\to[0,1]$. We must prove that $\psi_j$ converges to $\psi$ in $\Gms$.
1\) Let $\nu\in\cM$. Consider the map $\R^\times\to[0,1]\times[0,1]$ given by the formula $$H:s\mapsto\bigl(\nu[(0,s)],(t\cdot\nu)[(0,s)]\bigr).$$ It easy to see that we get the graph of the functions $\psi$, from which we remove all straight segments. The convergence $\nu_j\to\nu$ means the point-wise convergence of the maps $H_j(s)\to H(s)$. From this it is easy to derive that $\psi_j$ converges to $\psi$ point-wise (See Fig. \[fig:3\]). In virtue of monotonicity and continuity of our functions, the point-wise convergence implies the uniform convergence.
$$\epsfbox{split.2}$$
2\) Let us show that derivatives $\psi_j'$ converge $\psi'$ a.s. Take a point $a$, where all derivatives $\psi_j'(a)$, $\psi'(a)$ are defined. Let $\ell_j$, $\ell$ - be tangent lines to graphs of $\psi_j$, $\psi$ at $a$. Suppose that $\psi_j'(a)$ does not converge to $\psi'(a)$. Choose a subsequence $\psi_{n_k}'(a)$ convergent to $\alpha\ne\psi'(a)$. Consider the limit line $\ell_{n_k}$, i.e., $$\ell^\circ:\qquad y= \alpha(x-a)+\psi(a)$$ It is easy to see (for more details, see [@Ale], Addendum, §6) that the graph $y=\psi(x)$ is located upper this line. I.e., $\ell^\circ$ is the second supporting line at $a$ (the first one was the tangent line), this contradicts to the existence of $\psi'(a)$.
3\) Now we prove a weak convergence of operators $T_{1/2}(\psi_j)$ в $L^2[0,1]$. Let $f$, $h$ be continuous functions. We must check that the following expressions approach zero $$\begin{aligned}
\Bigl|\int_0^1 f(\psi_j(x))\psi_j'(x)^{1/2}h(x)\,dx -
\int_0^1 f(\psi(x))\psi'(x)^{1/2}h(x)\,dx\Bigr|\le
\nonumber
\\
\le
\int_0^1 \Bigl|f(\psi_j(x))-f(\psi(x))\Bigr|\,\psi_j'(x)^{1/2}h(x)\,dx
\label{eq:al1}
+\\
+\int_0^1 \Bigl|f(\psi(x))(\psi_j'(x)^{1/2}-\psi'(x)^{1/2})\,h(x)\Bigr|\,dx
\label{eq:al2}\end{aligned}$$
In (\[eq:al1\]) the convergence $f(\psi_j(x))\to f(\psi(x))$ is uniform and $$\int_0^1 \psi_j^{1/2}(x)\le \int_0^1 (\psi_j^{1/2})^2(x) = 1.$$ By the Fatou Lemma, (\[eq:al1\]) tends to zero. Further notice that for functions $\psi\in\cG$ we have a priory estimation $$\psi'(x)\le \frac{1-\psi(x)}{1-x}\le \frac 1{1-x}
.$$ Hence the convergence in the integral (\[eq:al2\]) is dominated on each segment $[0,1-\epsilon]$. This implies that integrals $\int_0^{1-\epsilon}(\dots)$ approach zero. Further, denote $C=\bigl(\max |f(x)|\cdot \max |g(x)|\bigr)$, $$\begin{gathered}
\int_{1-\epsilon}^1 (\dots)\le C
\int_{1-\epsilon}^1 (\psi_j'(x)^{1/2}+ \psi'(x)^{1/2})\,dx
\le \epsilon C
\int_{1-\epsilon}^1( \psi_j'(x)+ \psi'(x))\,dx
=\\=
\epsilon C\bigl[(1-\psi_j(1-\epsilon)\bigr)+ \bigl(1-\psi(1-\epsilon)\bigr)\bigr]\end{gathered}$$ and this value is small for small $\epsilon$. $\square$
[**Proof of Theorem \[th:last\].**]{} Cut $M$ into pieces $A_{ij}:=M_i\cap g^{-1} M_j$, and also into pieces $B_{ij}=g M_{ij}=g(M_i)\cap M_j$. We get a collection of maps $A_{ij}\to B_{ij}$. Now the question is reduced to a canonical form of each map.
[cc]{}
Rohlin, V. A. [*On the fundamental ideas of measure theory.*]{} Mat. Sbornik N.S. 25(67), (1949). 107–150. Amer. Math. Soc. Translation 1952, (1952). no. 71, 55 pp.
Neretin, Yu. A. [*Spreading maps [(]{}polymorphisms[)]{}, symmetries of Poisson processes, and matching summation.*]{} J. Math. Sci. (N. Y.) 126 (2005), no. 2, 1077–1094
Neretin, Yu. [*Symmetries of Gaussian measures and operator colligations.*]{} J. Funct. Anal. 263 (2012), no. 3, 782–802.
Kechris, A. S. [*Classical descriptive set theory.*]{} Springer, New York, 1995.
Neretin, Yu.A. [*Categories of bistochastic measures and representations of some infinite- dimensional groups.*]{} Sbornik Math. 75, No.1, 197-219 (1993);
Pestov, V. [*Dynamics of infinite-dimensional groups. The Ramsey–Dvoretzky–Milman phenomenon.*]{} American Mathematical Society, Providence, RI, 2006.
Reed, M., Simon, B. [*Methods of modern mathematical physics. I. Functional analysis.*]{} Academic Press, New York-London, 1972.
Raikov, D. A. [*On the completion of topological groups.*]{} (Russian) Izvestia Akad. Nauk SSSR 10, (1946). 513-528.
Bourbaki, N. [*Éléments de mathématique. Première partie. (Fascicule III.) Livre III; Topologie générale. Chap. [3]{}: Groupes topologiques. Chap. [4]{}: Nombres réels.*]{} (French) Troisième édition revue et augmentée Actualités Sci. Indust., No. 1143. Hermann, Paris 1960.
Rokhlin, V. A.[*Metric classification of measurable functions*]{}. (Russian) Uspehi Mat. Nauk (N.S.) 12 (1957), no. 2(74), 169-174.
Fleming, R. J.; Jamison, J. E. [*Isometries on Banach spaces: function spaces.*]{} Chapman & Hall/CRC, Boca Raton, FL, 2003.
Köthe, G. [*Topological vector spaces. I.*]{} Springer, New York 1969.
Neretin Yu.A., [*On the boundary of the group of transformations leaving a measure quasi-invariant*]{} Sbornik: Mathematics(2013),204(8):1161
Aleksandrov, A. D. [*Intrinsic Geometry of Convex Surfaces.*]{} OGIZ, Moscow-Leningrad, 1948. English transl.: [*Alexandrov selected works. Part II.*]{} Edited by S. S. Kutateladze. Chapman & Hall/CRC, Boca Raton, FL, 2006.
Math.Dept., University of Vienna,
Institute for Theoretical and Experimental Physics
MechMath Dept., Moscow State University
neretin(frog)mccme.ru
URL: http://www.mat.univie.ac.at/$\sim$neretin/
[^1]: Supported by the grant FWF, P25142.
[^2]: As $\frh_n$ we can take a partition of the segment $M=[0,1]$ into $2^n$ pieces of type $[k2^{-n}, (k+1)2^{-n})$
[^3]: A metric is compatible with the topology of the group, but not with its algebraic structure; in particular a metric is not assumed to be invariant. A completeness of a group in the sense of two-side uniform structure (in Raikov’s sense [@Rai]) is defined (for metrizable groups) in the following way. Let double sequences $g_ig_j^{-1}$ and $g_i^{-1}g_j$ converge to 1 as $i$, $j\to\infty$. Then $g_i$ has a limit in the group.This definition is not equivalent to the definition of Bourbaki [@Bou], III.3.3, who requires a completeness with respect to both one-side uniform structures. The group $\Gms$ is not complete in the sense of Bourbaki.
[^4]: The only necessary for us property of partition is the following: a diameter of a partition on any finite interval $(0,M]$ tends to 0 as $n\to\infty$.
[^5]: Recall that any two Lebesgue spaces with continuous probabilistic measures are equivalent, see e.g., [@Roh1].
|
---
abstract: 'A preliminary version (`v0.2.1`) of the DANTON Monte-Carlo package is presented. DANTON allows the exclusive sampling of (decaying) $\tau$ generated by $\nu_\tau$ interactions with the Earth. The particles interactions with matter are simulated in detail, including transverse scattering. Detailed topography data of the Earth can be used as well. Yet, high Monte-Carlo efficiency is achieved by using a Backward Monte-Carlo technique. Some validation results are provided.'
author:
- |
Valentin Niess\
Université Clermont Auvergne, CNRS/IN2P3, LPC\
Clermont-Ferrand, France\
`niess@in2p3.fr`\
Olivier Martineau-Huynh\
LPNHE, Université Pierre et Marie Curie, Université Paris Diderot, CNRS/IN2P3\
Paris, France\
`martineau@lpnhe.in2p3.fr`\
bibliography:
- 'main.bib'
title: 'DANTON: a Monte-Carlo sampler of $\tau$ from $\nu_\tau$ interacting with the Earth'
---
Introduction
============
The present document is a brief technical note describing the DANTON (DecAyiNg Taus frOm Neutrinos) Monte-Carlo sampler. DANTON is used for the simulation of the Giant RAdio Neutrino Detector (GRAND, see e.g. @GRAND:ICRC2017). It is a generator of ultra relativistic $\tau$ originating from the interactions of $\nu_\tau$ of cosmic origin with the Earth. Motivations for this work, as well as a comprehensive description of the topic, can be found in @Schoorlemmer:2018. Comparisons to the NuTauSim[@NuTauSim:GitHub] Monte-Carlo code are also provided hereafter.
The main specificity of DANTON are the following:
(i) DANTON can be operated in both forward (classical) or Backward Monte-Carlo (BMC) mode. The latter is done following the approach described in @Niess:2018. The BMC mode allows DANTON to exclusively sample a given final $\tau$ state.
(ii) DANTON implements a detailed simulation of Physical processes, i.e. energy losses are stochastic. The transport of $\nu_\tau$ is done with ENT[@ENT:GitHub]. The cross-sections for $\nu$ Deep Inelastic Scattering (DIS) are computed by ENT from tabulated Parton Distribution Functions (PDF). The transport of $\tau$ is done with PUMAS[@PUMAS:pages]. The $\tau$ energy loss is read from material tables. Decays of $\tau$ are simulated with ALOUETTE[@ALOUETTE:GitHub], a C wrapper to TAUOLA[@TAUOLA] providing BMC decays.
(iii) DANTON can integrate a detailed topography from global models, e.g. ASTER[@ASTER] or SRTMGL1[@SRTMGL1]. This is done with the TURTLE library[@TURTLE:GitHub]. It also includes a US standard atmospheric model.
The DANTON package has been implemented in C99. Its source code is available from GitHub[@DANTON:GitHub]. Its functionalities are exposed to the end user as a dedicated library, `libdanton`, following an Object Oriented (OO) approach. A documentation of the library Application Programming Interface (API) is available online[@DANTON:API]. The API allows to efficiently integrate DANTON in higher level simulation schemes. An executable is provided as well: `danton`, encapsulating `libdanton`. It allows to configure and run DANTON simulations according to steering files in JSON format.
The DANTON library
==================
Initialisation and finalisation
-------------------------------
Prior to any usage, `libdanton` must be initialised. This is done by calling the [ [`danton_initialise`](https://niess.github.io/danton-docs/#HEAD/group/danton/danton_initialise)]{} function. The initialisation allows to provide data sets for the Physics. The PDF can be selected. It must be in Les Houches Accords Grid 1 format (`lhagrid1`). The $\tau$ energy loss tables for PUMAS can be selected as well. Their must conform to the PDG[@PhysRevD.98.030001] format. If a `NULL` argument is provided, then a default data set is loaded. This is CT14\_NLO[@CT14] for PDFs or self generated tables for PUMAS, with Photonuclear interactions modelled according to @Dutta:2001. For multithreaded usage, one must also provide a couple of [ [`danton_lock_cb`](https://niess.github.io/danton-docs/#HEAD/group/callback/danton_lock_cb)]{} callbacks in order to *lock* and *unlock* access to critical data sections. These callbacks might for example manage a mutex using pthread.
Finalising the library is done with the [ [`danton_finalise`](https://niess.github.io/danton-docs/#HEAD/group/danton/danton_finalise)]{} function. Any memory allocated by a `libdanton` function is automatically released at finalisation. Whenever one wants to explicitly release a `libdanton` object, one must call the generic [ [`danton_destroy`](https://niess.github.io/danton-docs/#HEAD/group/danton/danton_destroy)]{} function instead of the standard library function: `free`. This will correctly notify `libdanton` that the object has been released. Note that some objects require a specific destructor to be called, e.g. the `danton_context`.
Error handling
--------------
Whenever an error occurs when calling a `libdanton` function, it will return `EXIT_FAILURE` or a `NULL` pointer. A short description of the error is pilled up in an internal stack. It can be retrieved using the [ [`danton_error_pop`](https://niess.github.io/danton-docs/#HEAD/group/error/danton_error_pop)]{} function. The numbers of errors in the stack is provided by the [ [`danton_error_count`](https://niess.github.io/danton-docs/#HEAD/group/error/danton_error_count)]{} function. It is also possible to push an error to the stack, using the [ [`danton_error_push`](https://niess.github.io/danton-docs/#HEAD/group/error/danton_error_push)]{} function.
Setting the geometry
--------------------
Before running any simulation, the geometry must be defined. The Earth model is specified with the [ [`danton_earth_model`](https://niess.github.io/danton-docs/#HEAD/group/earth/danton_earth_model)]{} function. It requires to specify a geodesic: `PREM` for a spherical Earth, or `WGS84` for an elliptical one. A topography can be specified by providing a path to global elevation data. Note that elevation data must be given w.r.t. the sea level. Then, altitude values below zero -but above the topography- are under the sea, if this option is activated. Note that the topography path can encode a flat Earth model instead of a path to data, if it starts as `flat://`. Then, the following *float* value in the path string is taken as the constant ground level. The constituent material of the topography and its density can be customised as well. This might require to modify the PUMAS Material Description File (MDF), though.
Specifying a simulation context
-------------------------------
Each DANTON simulation requires to be executed within a specific [ [`struct danton_context`](https://niess.github.io/danton-docs/#HEAD/type)]{}. This allows to safely and efficiently run multiple simulations in parallel, provided that *lock* and *unlock* callbacks have been supplied. A new simulation context is created (destroyed) with the [ [`danton_context_create`](https://niess.github.io/danton-docs/#HEAD/group/context/danton_context_create)]{} ([ [`danton_context_destroy`](https://niess.github.io/danton-docs/#HEAD/group/context/danton_context_destroy)]{}) function. Then, the simulation can be directly configured by modifying the structure field values.
The final $\tau$ state to sample is specified by providing a [ [`struct danton_sampler`](https://niess.github.io/danton-docs/#HEAD/type)]{} to the simulation context. This sampler is created with the [ [`danton_sampler_create`](https://niess.github.io/danton-docs/#HEAD/group/sampler/danton_sampler_create)]{} function. The desired final state properties are then specified by modifying the exposed sampler data. Note that the final state must not necessarily be a $\tau$. A neutrino flux can be sampled instead, or all $\tau$, $\nu$ particles inclusively. Once done, one must call the [ [`danton_sampler_update`](https://niess.github.io/danton-docs/#HEAD/group/sampler/danton_sampler_update)]{} function in order to validate the changes.
In order to run a Monte-Carlo simulation one must also provide one or more primary $\nu$ flux model. This is done by attaching [ [`structdanton_primary`](https://niess.github.io/danton-docs/#HEAD/type)]{} objects to the context, one per $\nu$ type. The API specifies a generic primary object. The user might provide its own implementation, as long as it conforms to the API, i.e. inherits from the base object. Two primary types are provided by default. A discrete (constant) primary and a power-law spectrum. They are created with the [ [`danton_discrete_create`](https://niess.github.io/danton-docs/#HEAD/group/discrete/danton_discrete_create)]{} and [ [`danton_powerlaw_create`](https://niess.github.io/danton-docs/#HEAD/group/powerlaw/danton_powerlaw_create)]{} functions.
Running a simulation
--------------------
A Monte-Carlo simulation is run with the [ [`danton_run`](https://niess.github.io/danton-docs/#HEAD/group/danton/danton_run)]{} function. This function takes a simulation context as argument. The number of generated events must also be specified. In addition, one can request a specific number of events to be sampled. Note that the number of sampled events is generally lower than the number of generated events, even in a *backward* simulation. For example the backward sampled primary $\nu$ might have a too high energy such that the event is rejected. In addition to running a forward or backward Monte-Carlo, one can also operate a grammage scan of the geometry. This is specified on the simulation context.
Simulated events conforming to the sampler specifications can be recorded. For this purpose, a [ [`struct danton_recorder`](https://niess.github.io/danton-docs/#HEAD/type)]{} must have been attached to the context. The recorder is a generic API object as well. The user might implement his own one. A [ [`danton_text`](https://niess.github.io/danton-docs/#HEAD/group/text)]{} recorder is provided by `libdanton`. It allows to dump the sampled events to a text file. This recorder is created with the [ [`danton_text_create`](https://niess.github.io/danton-docs/#HEAD/group/text/danton_text_create)]{} function. If an empty path is provided for the output file, then the events are dumped to the standard output (`stdout`).
The DANTON executable
=====================
A DANTON simulation can be run directly with the `danton` executable. This executable encapsulates the `libdanton` API. Multithreading is supported with `pthread`. The executable requires a steering file as input, formated in JSON. Examples of steering files are provided with the sources, in the `share/cards` folder. A steering file must contain a single JSON object. The items of this object specify global configuration parameters or categories of sub-parameters, as nested JSON objects. A list of all parameters is given in table\[tab:cards\]. In the following we provide a brief summary of the available parameters. Note that most of these directly reflect the `libdanton` API.
Currently (`v0.2.1`), the `danton` executable doesn’t allow to use another Physics than the default one specified in `libdanton`. Nevertheless, the default Physics of `libdanton` can be changed at compilation time by editing the folowing variables in the Makefile: `DANTON_DEFAULT_PDF`, `DANTON_DEFAULT_MDF` and `DANTON_DEFAULT_DEDX`.
[\*3l]{} Key & Value & Description\
`decay` & `boolean` & If `true` the sampled taus are decayed.\
`events` & `integer` & The number of Monte-Carlo events to run.\
`longitudinal` & `boolean` & If `true` the transverse transport is disabled.\
`mode` & `string` & The run mode, one of `backward`, `forward` or `grammage`.\
`output-file` & `string, null` & The output file name or `null` for `stdout`.\
`requested` & `integer` & The requested number of valid Monte-Carlo events.\
\
` geodesic` & `string` & The geodesic model: `PREM` (spherical) or `WGS84`.\
` sea` & `boolean` & If `true` the PREM Earth is covered with sea.\
` topography` & `[string, integer]` & The topography data location and the in-memory stack size.\
\
` altitude` & `float, float[2]` & The altitude (range) of the sampled particles.\
` azimuth` & `float, float[2]` & The azimuth angle (range) of the sampled particles.\
` elevation` & `float, float[2]` & The elevation angle (range) of the sampled particles.\
` energy` & `float, float[2]` & The energy (range) of the sampled particles.\
` latitude` & `float` & The geodetic latitude of the sampled particles.\
` longitude` & `float` & The geodetic longitude of the sampled particles.\
` weight` & `{$particle:float}` & The elevation angle (range) of the sampled particles.\
\
` $particle` & `[$model, {...}]` & The primary spectrum model for the corresponding particle.\
\
` energy` & `float` & The total energy of the primary.\
` weight` & `float` & The weight of the primary, i.e. its integrated flux.\
\
` energy` & `float[2]` & The energy range of the primary spectrum.\
` exponent` & `float` & The exponent of the power law.\
` weight` & `float` & The weight of the primary, i.e. its integrated flux.\
\
` append` & `boolean` & If `true`, append to the output file.\
` path` & `string` & Path to the output file.\
` verbosity` & `integer` & Verbosity level for recording Monte-Carlo steps.\
Global simulation parameters
----------------------------
Global simulation parameters are located at the top level of the steering file. They allow to specify the operation mode: `backward`, `forward` Monte-Carlo or `grammage` scan. Scattering is enabled by setting the `longitudinal` flag to `false`. The default behaviour of the simulation is to print the sampled final states, on the fly, to the standard output (`stdout`). The output stream can be redirected to a file using the `output-file` global option.
Earth model
-----------
The Earth model parameters allow to configure the geometry of the simulation. They are located under the `earth-model` root item. The `geodesic` item allows to switch between a spherical Earth or an elliptic one (WGS84). The legacy PREM has an external layer of 3km of sea water. If the sea is disabled this layer is replaced with Standard Rock. Note also that the `topography` item can encode a flat Earth model, as for the library API. If global elevation data are provided instead, one must use the WGS84 ellipsoid, i.e. a non spherical Earth.
Particle sampler
----------------
The particle sampler parameters allow to specify the final state(s) to sample. They are located under the `particle-sampler` root item. Most of the sampler parameters can be provided as a single float for a constant value or as a list of two floats (`[min_value, max_value]`) for a range. The `weight` JSON object specifies the particles to sample as a nested JSON object. The valid particle names are `nu_tau`, `nu_tau~`, `nu_mu`, `nu_mu~`, `nu_e`, `nu_e~`, `tau` and `tau~`.
Primary $\nu$ flux
------------------
The primary $\nu$ flux is provided as a JSON object, under the `primary-flux` root item. For each item, the key must name a primary particle while the value specifies the corresponding flux model. Valid names are the same than previously, for the particle sampler. A flux model is given as a list of two items. The first one is the type of the model, `discrete` or `power-law`. The second one is a JSON object containing the model parameters. Those are described in table\[tab:cards\].
Recording Monte-Carlo steps
---------------------------
The Monte-Carlo steps can be recorded, e.g. for visualisation purpose. This is enabled by providing a `stepping` item at the root level of the steering file. An output file name must be specified with the `path` item. The verbosity level of recorded Monte-Carlo steps can be configured with the `verbosity` flag. Setting it to zero results in all steps being recorded. A higher value specifies a down-sampling period.
Selected results
================
Validation of the Backward Monte-Carlo
--------------------------------------
The BMC functionalities used by DANTON have been implemented as dedicated packages, ALOUETTE[@ALOUETTE:GitHub], ENT[@ENT:GitHub] and PUMAS[@PUMAS:pages]. These packages have been tested independently. In the present section, for purpose of illustration, we only present a few comparisons of the end-to-end *forward* and *backward* chains, obtained with DANTON. Specific cross-checks of PUMAS can be found in [@Niess:2018].
Figures \[fig:flux\_upward\], \[fig:flux\_downward\] and \[fig:flux\_1km\] show comparisons of the $\tau$ flux sampled above a spherical Earth, with a core density described by the Preliminary Reference Earth Model (PREM, see Table 1 of [@PREM]). It can be seen that the *forward* and *backward* results agree within statistical errors. These results have been obtained with version `v0.1.1` of DANTON and equivalent CPU time for both modes. The superior efficiency of the *backward* method is clearly visible here. Especially, the *backward* computed flux is much more accurate at both ends of the enrgy range considered. Fig. \[fig:flux\_1km\] is particularly interesting. In this case the $\tau$ flux is sampled at 1km above the ground. Two components are visible. Below $10^{17}\,\mathrm{eV}$ the $\tau$ flux is dominated by grazing $\nu_\tau$ interacting in the atmosphere. One recovers the results of fig. \[fig:flux\_downward\] in this case. At higher energies, the $\tau$ flux mostly results from $\nu_\tau$ interacting in the Earth crust, as in fig.\[fig:flux\_upward\]. This flux is however suppressed at lower energies, due to $\tau$ decaying in flight before reaching the altitude of 1km. Note that in *forward* mode, one hardly sees the first component, at lower energies, whereas in *backward* mode the whole spectrum is clearly resolved.
Comparison to NuTauSim
----------------------
NuTauSim and DANTON use similar ingredients but following different implementations. NuTauSim was written in C++. It relies on parametrisations and tabulations, e.g. for $\nu_\tau$ cross-sections, $\tau$ energy losses or $\tau$ decays. DANTON on the contrary glues together detailed (B)MC engines: ALOUETTE[@ALOUETTE:GitHub], ENT[@ENT:GitHub] and PUMAS[@PUMAS:pages]. Below is a commented list of the main differences between the two codes:
(i) The $\tau$ energy loss is deterministic in NuTauSim. This is inaccurate at extreme energies, above $10^{20}\,\mathrm{eV}$. At these energies, fluctuations in the $\tau$ radiative energy losses become important, extending its range.
(ii) NuTauSim allows to vary the cross-sections of physical processes within predefined models, using command line flags. DANTON, in principle allows any model to be pluged in, by changing the data set. However, this is less straightforward to do than in NuTauSim, since currently only one data set is packaged with DANTON.
(iii) NuTauSim simulates a flat spherical Earth, as a modified version of the PREM[@PREM]. There is no atmosphere neither. DANTON allows for a detailed simulation of the (elliptical) Earth surface according to world wide Digital Elevation Models (DEM), e.g. ASTER[@ASTER] or SRTMGL1[@SRTMGL1].
(iv) NuTauSim supports only a single material: Standard Rock. DANTON has different materials, i.e. cross-sections, for air, rocks and water. Note that the $\nu_\tau$ cross-sections and $\tau$ energy loss in rocks and water can differ by 10%.
(v) NuTauSim is a purely longitudinal MC. The scattering angle of the $\tau$ w.r.t. the primary $\nu_\tau$ direction is not simulated.
(vi) NuTauSim is 1kLOC, versus 15kLOC for DANTON, including its dependencies. DANTON is slower and harder (more error prone) to develop.
Detailed comparisons have been performed between the two codes. For these comparisons, NuTauSim was configured with *standard* $\nu_\tau$ cross-sections (`CCmode=0`, i.e. the middle parametrisation of @Connolly:2011) and $\tau$ energy loss (`ALLM(0)`). NuTauSim was hacked as well in order to use the standard PREM, as initially defined by [@PREM] and used in DANTON. Indeed, NuTauSim implements a modified version of the PREM with a 7km larger Earth radius. Note also that version `v0.1.1` of DANTON was used for the comparison, i.e. not the latest one. In particular, the CT14\_NNLO PDF have been used, instead of the CT14\_NLO PDF in the current `v0.2.1` version.
As an illustration, the comparison of the the $\tau$ flux emerging from the PREM spherical Earth, with 1deg of elevation, is shown on fig.\[fig:flux\_nts\]. Significant differences can be observed above $10^{20}$eV. These differences are due to fluctuations in the $\tau$ energy loss, as discussed previously. This was cross-checked by configuring PUMAS to use the Continuous Slowing Down Approximation (CSDA) instead of the detailed simulation for $\tau$ energy losses. Then, DANTON and NuTauSim yield very similar results, with a 10% systematic offset. This offset was checked to be consistent with the differences in the $\nu_\tau$ cross-sections and $\tau$ energy loss models used by the two codes. Other angles of emergence have been investigated as well. Overall, when using the same geometry, DANTON and NuTauSim are in very good agreement, within 10%. However, when using the default NuTauSim geometry, the emerging $\tau$ fluxes can differ by a factor of 10. The higher the $\tau$ energy, the higher the discrepancy.
The right part of fig.\[fig:flux\_nts\] also shows the CPU time needed in order to reach a 1% Monte-Carlo accuracy on the emerging flux. Note that in BMC, the Monte-Carlo events are necessarily weighted. Consequently, it is not relevant to compare the CPU time per Monte-Carlo event, since events are not equally important. Therefore, we instead compare the CPU time needed in order to reach a given accuracy, 1% in this case. It can be seen that in *forward* Monte-Carlo mode DANTON is significantly slower than NuTauSim, up to a factor of 10. This is not surprising considering the differences in complexity and level of details of the two codes. For example, DANTON implements a detailed stepping through the geometry, while NuTauSim maps the density distribution at initialisation. Indeed, in NuTauSim all Monte-Carlo events have the same initial direction and they don’t scatter. Therefore, they all go through the same density distribution.
In BMC mode, DANTON is more effcient than NuTauSim at lower energies, and equally efficient at extreme energies. Hence, the overburden resulting from doing a detailed simulation is counterbalanced by the gain in efficiency resulting from the backward sampling.
Conclusion
==========
In this brief technical note we presented a preliminary version (`v0.2.1`) of the DANTON Monte-Carlo package. The core functionalities of DANTON have been implemented as a dedicated library: `libdanton`. This allows a direct integration of DANTON in a higher level simulation. The DANTON package also provides an executable: `danton`, encapsulating `libdanton`. With this executable one can run DANTON simulations from the command line, according to steering files in JSON format.
A dedicated *backward* Monte-Carlo procedure allows DANTON to achieve high efficiency while performing a detailed simulation of the Physics and of the Earth topography. The validity of the procedure was cross-checked by comparison to classical Monte-Carlo results. When using identical geometries, good agreement was found with the NuTauSim simulation code, within theoretical systematics on the $\nu_\tau$ and $\tau$ cross-section models.
|
---
abstract: 'This paper explores distributionally robust zero-shot model-based learning and control using Wasserstein ambiguity sets. Conventional model-based reinforcement learning algorithms struggle to guarantee feasibility throughout the online learning process. We address this open challenge with the following approach. Using a stochastic model-predictive control (MPC) strategy, we augment safety constraints with affine random variables corresponding to the instantaneous empirical distributions of modeling error. We obtain these distributions by evaluating model residuals in real time throughout the online learning process. By optimizing over the worst case modeling error distribution defined within a Wasserstein ambiguity set centered about our empirical distributions, we can approach the nominal constraint boundary in a provably safe way. We validate the performance of our approach using a case study of lithium-ion battery fast charging, a relevant and safety-critical energy systems control application. Our results demonstrate marked improvements in safety compared to a basic learning model-predictive controller, with constraints satisfied at every instance during online learning and control.'
author:
- 'Aaron Kandel$^{1}$, and Scott J. Moura$^{1}$[^1] [^2]'
bibliography:
- './main.bib'
title: |
**Safe Zero-Shot Model-Based Learning and Control:\
A Wasserstein Distributionally Robust Approach**
---
=4
Introduction
============
This paper presents a novel application of Wasserstein ambiguity sets to robustify zero-shot learning and control. Stochastic optimal control is a longstanding topic in the controls literature, dating back decades to the original linear-quadratic Gaussian problem [@Karl00]. This field seeks to address optimal control under uncertainty. The rise in popularity of model-predictive control (MPC) has created a new application for robust and stochastic optimal control principles. For instance, foundational work by Kothare et al. addresses uncertainty in MPC optimization with linear matrix inequalities by allowing the state transition matrices to vary in time within a convex polytope [@Kothare00]. Within the past few years, stochastic optimal control has become connected to ongoing research in the burgeoning field of learning and control. Here, researchers seek guarantees on safety and performance when learning and controlling a dynamical system simultaneously. Work by Dean et al., for instance, explores safety and persistence of excitation for a learned constrained linear-quadratic regulator [@Dean00]. Within the space of MPC research, Rosolia and Borrelli derive recursive feasibility and performance guarantees for a learned iterative MPC controller [@Rosolia00]. Koller et al. also address the safety of a learned MPC controller when imperfect model knowledge and safe control exists [@Koller00]. The reality of this field, however, is that the *“learning”* moniker has only recently seen prolific use as a descriptor for research. Historically, the field of *adaptive* control, specifically adaptive MPC presents a host of relevant approaches from which we can glean meaningful insights to guide future research. A recent review presented in [@Hewing00] presents a strong, comprehensive description of the field of learning MPC within this context. Recent work has also explored recursive feasibility for adaptive MPC controllers based on recursive least-squares [@bujarbaruah2018adaptive] and set-membership parameter identification [@Tanaskovic00], although similar papers frequently possess limitations including a dependence on linear dynamical models.
In recent practice, distributionally robust optimization (DRO) has penetrated into learning and control research with the upside to potentially address shortcomings of existing work. DRO is a field of inquiry which seeks to guarantee robust solutions to optimization programs when the distributions of random variables are estimated from data. This uncertainty can involve the objective or the constraints of the optimization program. Uncertainty in both cases can pose significant challenges if unaccounted for, leading to suboptimal and potentially unsafe performance [@Nilim00]. Given that past work on adaptive MPC has considered potential accommodation of chance constraints [@bujarbaruah2018adaptive], incorporating a true distributionally robust approach possesses the potential to improve our capabilities of guaranteeing safety during learning. Within the context of control, these methods have been recently explored to address challenges of safety and performance imposed by uncertainty. For instance, Van Parys et al. address distributional uncertainty of a random exogenous disturbance process with a moment-based framework [@VanParys00]. Paulson et al. also apply polynomial chaos expansions to characterize distributional parametric uncertainty in a nonlinear model-predictive control application [@Paulson00].
Among the toolbox provided by DRO, Wasserstein ambiguity sets are a foremost tool. The Wasserstein metric (or “earth mover’s distance”) is a symmetric distance measure in the space of probability distributions. Wasserstein ambiguity sets account for distributional uncertainty in a random variable, frequently one approximated in a data-driven application. They accomplish this feat with out-of-sample performance guarantees by replacing the data-driven distribution of the random variable with the worst-case realization within a Wasserstein ball centered about the empirical distribution [@Esfahani00; @Gao00]. Expressions exist which map the quality of the empirical distribution with Wasserstein ball radii such that desired robustness characteristics are achieved without significant sacrifices to the performance of the solution [@Zhao00]. Within the control context, however, the Wasserstein distance metric has only recently began emerging as a valuable and widespread tool. Work by Yang et al. for instance explores the application of Wasserstein ambiguity sets for distributionally robust control subject to disturbance processes [@Yang00]. Overall, while Wasserstein ambiguity sets are seeing increased application in controls research, their true capabilities have yet to be fully exploited.
Beyond the scope of MPC, the bulk of distributionally robust learning and control research lies in the study of Markov decision processes (MDPs), where the dynamics are dictated by black-box transition probabilities. In this context, the objective is to learn a policy which maps the dynamical state directly to a desired control input. For distributionally robust processes, this policy is typically required to optimize the system with respect to worst-case returns, or worst-case realizations of the underlying state transition model [@NIPS2019_8942; @asadi2018lipschitz]. This is most commonly accomplished through dynamic programming methods including policy and value iteration [@Amin00]. The dependence on dynamic programming or tree search methods typically limits the scalability of such approaches [@NIPS2019_8942]. In general, this literature intersects more with pure episodic RL research than with conventional stochastic optimal control, meaning value and policy based methods are more common than MPC. Application of Wasserstein ambiguity sets to pure transition probability functions is therefore not control-oriented in the sense of consistently yielding a convex optimization program conducive for MPC. This is especially true when the transition model is developed via machine learning techniques including Gaussian processes or Bayesian neural networks [@Akbar00]. Some recent work has sought to address this shortcoming, although scalability still presents an open challenge for such approaches [@Yang03]. Application of Wasserstein ambiguity sets to robust optimal control still constitutes an open question in the literature, especially for zero-shot methods which do not possess inherent episodic design. The question of safety and feasibility also remains relatively unexplored.
This paper seeks to address key shortcomings in these areas of literature. Among those previously discussed, foremost is the lack of robust “zero-shot” methods for learning and control. Zero-shot methods describe modeling a process from highly limited data. In the context of learning and control, we use this keyword to characterize learning and controlling a system from scratch, rather than an episodic process more commonly seen in reinforcement learning approaches. We present a novel and simple-to-implement zero-shot model-based learning and control scheme based on MPC which provides strong probabilistic out-of-sample guarantees on safety. By developing Wasserstein ambiguity sets relating to empirical distributions of modeling error, we can conduct MPC with an imperfect snapshot model while maintaining confidence on our ability to satisfy nominal constraints. The Wasserstein ambiguity sets allow us to optimize with respect to constraint boundaries that are shifted into the safe region. As our empirical distributions improve, the offset variables tighten towards the nominal boundary in a provably safe way. We validate our approach by learning to safely fast charge a lithium-ion battery using a nonlinear equivalent circuit model. Battery fast charging presents a strong challenge for learning-based control methods, given that the optimal policy is a boundary solution which rides constraints until the terminal conditions are met. We learn the dynamical model online using recursive least-squares, and then conduct MPC using our DRO control framework. Our results demonstrate our control algorithm’s capability of providing safe MPC for a system whose parameters we learn from scratch.
Distributionally Robust Optimization
====================================
Stochastic Optimization with Chance Constraints
-----------------------------------------------
A chance constrained program includes probabilistic constraint statements, with random variables $\bf{R}$ with support $\Xi$. Consider $x(t) \in \mathbb{R}^{n}$ is the system state at timestep $t$, $u(t) \in \mathbb{R}^{p}$ is the control input, $\textbf{R} \in \mathbb{R}^{m}$ is the random variable in question, and $g(x(t), u(t), \bf{R}): \mathbb{R}^n \times \mathbb{R}^p \times \mathbb{R}^m \rightarrow \mathbb{R}^m$ is the vector of inequality constraints. The chance constraint is: $$\label{eqn:cc1}
\hat{\mathbb{P}} \big{[}g(x(t), u(t), \bf{R}) \leq 0\big{]} \geq 1 - \eta$$ where $\eta$ is our risk metric, or the probability of violating the constraint. The chance constraints discussed above depend on known distributions corresponding to each random variable. For many applications, we approximate these distributions using data to create an empirical CDF. In many data-driven applications, the true probability distribution $\mathbb{P}^*$ for the random variable $\bf{R}$ is unknown. Thus, our empirical distribution $\hat{\mathbb{P}}$ provides an approximation of $\mathbb{P}^*$ from data. Borel’s law of large numbers indicates that as the number of samples $\ell \rightarrow \infty$, $\hat{\mathbb{P}}\rightarrow \mathbb{P}^*$. This discrepancy characterizes distributional uncertainty in the random variable. This can affect our solution if $\hat{\mathbb{P}}$ is inaccurate [@Nilim00]. The literature presents several means by which we can accommodate this uncertainty. In the following subsection, we discuss the application of the Wasserstein distance within this context.
Wasserstein Ambiguity Sets
--------------------------
Depending on how accurate our empirical approximation is, we can say that it is a certain distance from characterizing the true underlying distribution. In probability and statistics, there are several methods to describe distance in the space of probability distributions. These include the various formulations of $\phi$-divergence, and the Wasserstein metric:
Given two marginal probability distributions $\mathbb{P}_1$ and $\mathbb{P}_2$ lying within the set of feasible probability distributions $\mathcal{P}(\Xi)$, the Wasserstein distance between them is defined by $$\mathcal{W}(\mathbb{P}_1, \mathbb{P}_2) = \underset{\Pi}\inf \bigg{\{} \int_{\Xi^2} ||\textbf{R}_1 - \textbf{R}_2 ||_a \Pi (d\textbf{R}_1, d\textbf{R}_2) \bigg{\}}$$ where $\Pi$ is a joint distribution of the random variables $\bf{R}_1$ and $\bf{R}_2$, and $a$ denotes any norm in $\mathbb{R}^n$.
The Wasserstein metric is also colloquially referred to as the “earth-movers distance.” This moniker is sourced from the representation of the Wasserstein distance as the minimum cost of transporting or redistributing mass from one distribution to another via non-uniform perturbation [@Yang00]. The Wasserstein distance allows us to replace the random variable with a “worst-case” realization sourced from a family of distributions within a certain Wasserstein distance of our empirical distribution. This family of distributions forms the Wasserstein ambiguity set. For instance, let us define the ambiguity set as $\mathbb{B}_\epsilon$, a ball of probability distributions with radius $\epsilon$ centered around our empirical CDF $\hat{\mathbb{P}}$: $$\label{eqn:wass1}
\mathbb{B}_\epsilon := \big{\{} \mathbb{P} \in \mathcal{P}(\Xi) \; | \; \mathcal{W}(\mathbb{P}, \hat{\mathbb{P}}) \leq \epsilon \big{\}}$$ where $\epsilon$ is the Wasserstein ball radius. Now, we can formulate the robust counterpart of the chance constraint in (\[eqn:cc1\]): $$\label{eqn:wass3}
\underset{\mathbb{P} \in \mathbb{B}_\epsilon}\inf \; \mathbb{P} \big{[} g(x(t), u(t), \bf{R}) \leq 0 \big{]} \geq 1 - \eta$$ The constraint shown in (\[eqn:wass3\]), while properly representing the exact process of applying Wasserstein ambiguity sets to chance-constrained programs, presents as an infinite dimensional nonconvex constraint. Ongoing research in statistics and robust optimization literature has pursued tractable reformulations of this constraint to facilitate computation.
Several expressions exist for the Wasserstein ball radius which, for a given confidence level $\beta$, is probabilistically guaranteed to contain the true distribution. We adopt the following formulation of $\epsilon$ from [@Zhao00] where $D$ is the diameter of the support of $\bf{R}$ composed of $\ell$ samples: $$\label{eqn:wass2}
\epsilon(\ell) = D \sqrt{\frac{2}{\ell} \log \bigg{(} \frac{1}{1-\beta} \bigg{)} }$$
What is important to note, beyond the computational challenges, is that this formulation of the chance constraint created by the Wasserstein radius defined in (\[eqn:wass2\]) affords the out-of-sample safety guarantee. This is principally due to the fact that the Wasserstein distance between two probability distributions bears no assumptions on the shape or support of each distribution. We demonstrate this feature by comparing a Wasserstein-based approach to one using $\phi$-divergence. If we were to utilize a $\phi$-divergence to reformulate (\[eqn:cc1\]): $$\mathbb{B}_\phi = \{\mathbb{P} \in \mathcal{P}(\Xi)\: | \: \phi(\mathbb{P}, \hat{\mathbb{P}}) \leq d \}$$ where $d$ is a distance-like hyperparameter, then existing equivalent reformulations simply perturb the risk level [@Jiang00]. However, perturbing the risk level provides much more limited out-of-sample guarantees because it limits the realization of the random variable to lie within a support that we have already observed. This finding is partially defined by the fact that the $\phi$-divergence between two probability distributions with different support is infinite. As a result, we adopt the Wasserstein distance metric for the remainder of this paper. We also adopt an equivalent reformulation of (\[eqn:wass3\]) detailed in [@Duan00]. This reformulation requires that the function $g(x(t), u(t), \bf{R})$ is linear in $\bf{R}$, and entails a scalar convex optimization program to derive. Importantly, the result is a conservative *convexity-preserving* approximation of (\[eqn:wass3\]). For an $m$-dimensional constraint function, the exact form of the ambiguity set is $\mathcal{V} = \operatorname*{conv}(\{r^{(1)}, ..., r^{(2^m)}\})$, where the vector $r$ is sourced from the optimization component of the overall procedure. The set of constraints we find to replace the infinite dimensional DRO chance constraint are: $$\begin{aligned}
&g(x(t),u(t)) + r^{(j)} \leq 0, &\forall \ j=1,...,2^m \label{eqn:ineq-reform}\end{aligned}$$ For complete and elegant discussion of this reformulation, we *highly* recommend the reader reference work in [@Duan00], specifically pages 5-7 of their paper.
Distributionally Robust Model-Based Learning and Control
========================================================
Model Predictive Control Formulation
------------------------------------
Next, we apply Wasserstein ambiguity sets to robustify a learning model predictive controller, based on the following mathematical optimization program formulation: $$\begin{aligned}
\min \quad & \sum_{k=t}^{t+N} J_k({x}(k),{u}(k)) \label{eqn:ftocp1} \\
\text{s. to:} \quad
& {x}(k+1) = f(x(k), u(k), \theta(t)) \\
& g({x}(k),{u}(k), \theta(t)) \leq 0 \label{eqn:ftocp3} \\
& {x}_0 = {x}(t) \label{eqn:ftocp5}\end{aligned}$$ where $k$ is the control horizon time index of length $N$; ${x}(k) \in {{\mathbb{R}}}^n$ is the vector of state variables at time $k$; ${u}(k) \in {{\mathbb{R}}}^p$ is the vector of inputs at time $k$; $\theta(t) \in \mathbb{R}^h$ is the estimate of the model parameters at time $t$; $J_k({x}(k),{u}(k)) : {{\mathbb{R}}}^n \times {{\mathbb{R}}}^p \rightarrow {{\mathbb{R}}}$ is the instantaneous cost at time $k$ as a function of the states and inputs; $f({x}(k),{u}(k),\theta(k)) : {{\mathbb{R}}}^n \times {{\mathbb{R}}}^p \times {{\mathbb{R}}}^h \rightarrow {{\mathbb{R}}}^n$ represents the linear or nonlinear system dynamics; and $g({x}(k),{u}(k),\theta(k)) : {{\mathbb{R}}}^n \times {{\mathbb{R}}}^p \times {{\mathbb{R}}}^h \rightarrow {{\mathbb{R}}}^m$ represents linear or nonlinear inequality constraints on the states and inputs.
Model Identification
--------------------
We assume the true model parameters $\theta^*$ are unknown. Several methods can be selected to learn $\theta$ online. In this paper, we focus on dynamical systems which are linear in the parameters, although this assumption is entirely unnecessary to obtain our algorithm’s safety guarantees. We make this distinction to allow recursive least-squares (RLS) adaptive filtering for online parameter identification. We assume full state measurements. Then, we apply discrete parameter updates governed by the following relations: $$\label{eqn:rls1}
F^{-1}(t+1) = F^{-1}(t) + \phi^T(t)\phi(t)$$ $$\label{eqn:rls2}
\hat{\theta}(t+1) = \hat{\theta}(t) + F(t+1)\phi(t)[y(t+1) - \hat{\theta}^T(t)\phi(t)]$$ We apply the Woodbury matrix identity to reformulate (\[eqn:rls1\]): $$\label{eqn:rls3}
F(t+1) = F(t) - \frac{F(t)\phi(t)\phi^T(t) F(t)}{1+\phi^T(t) F(t)\phi(t)}$$ which we can plug directly into our parameter update (\[eqn:rls2\]). At the start of online learning, we initialize the parameter vector to zero, i.e. $\hat{\theta}(0) = 0$, and the matrix $F_0 = 10^{10}I_h$, where $I_h$ is the identity matrix. Finally, we assume the regressors are subject to i.i.d. Gaussian measurement noise. This is a “zero-shot” approach insofar as we begin with no knowledge of the system parameters and must learn them rapidly online from limited data samples obtained through applying control to the system.
Safety and Robustness using Wasserstein Ambiguity Sets
-------------------------------------------------------
Now that we have outlined the distributionally robust chance constrained approach using the Wasserstein ambiguity set, we can describe how it fits within our robust control framework. Consider the nonlinear dynamical system $$x(t+1) = f(x(t), u(t), \theta(t))$$ with potentially nonlinear output equation $$y(t) = h(x(t), u(t), \theta(t))$$ First, we assume full state measurements where we learn the parameters $\theta(t)$ online subject to random additive measurement noise processes $d_x(t) \in D_x$ and $d_y(t) \in D_y$. These noise processes affect the regressors and output in our RLS learning algorithm, composed of measured states and outputs: $$\begin{aligned}
y_m(t) &= y(t) + d_y(t) \\
x_m(t) &= x(t) + d_x(t)\end{aligned}$$ Now, consider the constraint function: $$g(x(t), u(t), \theta(t)) \leq 0$$ with the following 1-step residual $\textbf{R}$ $$\textbf{R}_0 = g(x(t), u(t), \theta^*) - g(x(t), u(t), \theta(t))$$where $\theta^*$ is the true parameterization of the underlying system, and $\theta(t)$ is our current estimate of the model parameterization. At each time step, we must compare all historical data to model predictions given our latest learned parameterization $\theta(t)$. This step is crucial, as simply updating our empirical CDF with a single new residual at each timestep would fail to characterize the desired density function. For an RLS adaptive filter, if we assume persistence of excitation and zero-mean i.i.d. measurement noise, then as $t\rightarrow \infty$, $\theta(t) \rightarrow \theta^*$ which eliminates modeling error entirely with no model mismatch [@Sun00]. Under these conditions, the empirical CDF $\hat{\mathbb{P}}$ would characterize the measurement noise process as $t\rightarrow \infty$. Throughout online learning we expect these parameterizations to differ especially in the presence of measurement noise and subtle model mismatch. By considering these residuals in formulating the constraints, we can ostensibly guarantee safety in the face of these uncertainties: $$\label{eqn:ftocp3a}
g(x(t), u(t), \theta(t)) + \textbf{R}_0\leq 0$$ To accommodate distributional uncertainty in our estimate of $\hat{\mathbb{P}}$, we transform the constraint (\[eqn:ftocp3a\]) for each of $1\rightarrow N$ step residuals into a joint distributionally robust chance constraint via Wasserstein ambiguity set as follows: $$\begin{aligned}
\underset{\mathbb{P} \in \mathbb{B}_\epsilon} \inf \mathbb{P} &\left[
\begin{array}{r}
g(x(t), u(t), \theta(t)) + \bf{R}_0 \leq 0 \\
g(x(t+1), u(t+1), \theta(t)) + \bf{R}_1 \leq 0 \\
\vdots \\
g(x(t+N), u(t+N), \theta(t)) + \bf{R}_N \leq 0
\end{array}
\right] \\
& \geq 1 - \eta
\end{aligned}$$ Conventionally, inverting a joint chance constraint constitutes a significant open challenge in the literature. However, the reformulation we adopt from [@Duan00] presents a simple method to accommodate the constraint without inverting the CDF. Algorithm 1 provides an overview of the real-time implementation of our approach. As previously stated, the process for computing $r$ entails a simple scalar convex optimization program, which scales easily for high-dimensional problems.
At each time step, we compute model residuals with our most recent estimate $\theta(t)$ using our entire cumulative experience, compile a unique empirical distribution $\hat{\mathbb{P}}$ corresponding to each individual chance constraint, and compute the value of $r$ in to reformulate the distributionally robust chance constraints. We can begin the overall process with a small control horizon $N$, and gradually increase $N$ as we accumulate more and more data from experience. The residuals we compute are for horizon lengths of $1$ to $N$-steps, meaning the elements of $\bf{R}$ correspond to each of $i = 1,...,N$ step residuals. Then, we assemble a joint chance constraint where the elements of the column vector of the random variable are the $1\rightarrow N$ step residuals. We formulate the optimization program in this manner because, as we simulate farther along $N$ using our snapshot model parameterized by $\theta(t)$, the error of the state can potentially compound and affect the overall residual distribution. Finally, when we conduct MPC, we replace the nominal constraints with their distributionally robust counterparts: $$\begin{aligned}
\underset{\vec{u} \in \mathcal{U}}\min \quad & \sum_{k=t}^{t+N} J_k({x}(k),{u}(k)) \label{eqn:drftocp1}\\
\text{s. to:} \quad
& {x}(k+1) = f(x(k), u(k), \theta(t)) \\
& \begin{bmatrix}
g({x}(k),{u}(k), \theta(t)) \\
g({x}(k+1),{u}(k+1), \theta(t)) \\
\vdots \\
g({x}(k+N),{u}(k+N), \theta(t))
\end{bmatrix} + r^{(j)} \leq 0 \\ & {x}_0 = {x}(t) \label{eqn:drftocp5}\end{aligned}$$ Algorithm 1 describes our MPC architecture:
State space $\mathcal{X}$, Action space $\mathcal{U}$\
$u(t) =$ known safe input, $N=1$
Update the dynamical system model $\theta(t-1)\rightarrow\theta(t)$
Receding horizon increment rule (i.e. $N=min\{N_{targ}, round(\frac{t}{N_{targ}})+1\}$)
Obtain Wasserstein ambiguity set offset $r$:
$u(t) \leftarrow$ Solve MPC optimization program -
$x(t+1) = f(x(t), u(t), \theta^*)$
$y(t) = h(x(t), u(t), \theta^*)$
We must accommodate the following assumptions for our implementation to work effectively:
The problem (\[eqn:drftocp1\])-(\[eqn:drftocp5\]) with horizon $N$ and convex robust constraint offset $r$ admits a feasible solution.
Based on our initial estimate of the system parameters $\theta_0$, we assume we know a safe control input which we can apply at the first timestep.
Beyond these assumptions, the qualities of the Wasserstein ambiguity set make an intuitive fit for learned control. First, there exist simple expressions to define the Wasserstein ball radius $\epsilon(\ell)$ as a function of the amount of data we have collected. For example, in , note that as $\ell \rightarrow \infty$, $\epsilon(\ell) \rightarrow 0$, meaning as we collect more data samples to learn the dynamics, then the less conservative the distributionally robust chance constraint will be. The equivalent reformulation we adopt from [@Duan00] provides a convex approximation of the constraint. So, for convex MPC programs, our distributionally robust framework does not destroy the tractability of obtaining a fast online solution via polynomial-time optimization algorithms.
Persistence of Excitation (PoE)
-------------------------------
Perhaps the most evident challenge relates to the exploration-exploitation trade-off inherent to online learning. Past work considers nonstationary MDPs where exploration is not allowed along the temporal axis [@NIPS2019_8942]. In our case, obeying the MPC scheme ostensibly cannot guarantee PoE. To address this perceived shortcoming, we propose injecting additive noise into the control signal. This approach is similar to methods used in actor-critic based reinforcement learning algorithms [@lillicrap2015continuous], where noise is generated via an Ornstein-Uhlenbeck process. To satisfy the frequency condition for PoE with RLS, we simply apply a random Gaussian noise to the control input. We verify this noise process maintains feasibility by checking the satisfaction of the distributionally robust chance constraints with the added control input noise fixed for each snapshot model $\theta(t)$, giving the additional constraints: $$u^n = \vec{u} + \mathcal{N}$$ $${x}^n(t+1) = f(x^n(t), u^n(t), \theta(t))$$ We satisfy the same set of inequality constraints subject to the original and perturbed control input signals.
Case Study in Safe Online Lithium-Ion Battery Fast Charging
===========================================================
Equivalent Circuit Model of a Lithium-Ion Battery
-------------------------------------------------
Lithium-ion batteries can be modeled with varying degrees of complexity. The most complex dynamics models are based on electrochemistry. For example, the Doyle-Fuller-Newman (DFN) electrochemical battery model is a high-fidelity first-principles derived physics based model of the dynamics within a lithium-ion battery. Varying model-order reduction can be applied, yielding versions including the single particle model and the equivalent circuit model (ECM). For simplicity, this paper’s case study utilizes an ECM. The fast charging problem presents an ideal challenging landscape within which to evaluate our algorithm. Namely, the optimal trajectory is a boundary solution that rides the safe boundary of the voltage constraint. The relevant state variables in this model are the state of charge $SOC$ and capacitor voltages $V_{RC}$ in each of two RC pairs. The relevant constraint is on the terminal voltage $V$. This constraint prevents the battery from overheating or aging rapidly during charging and discharging. The state evolution laws are given by:
$$\begin{aligned}
SOC(t+1) &= SOC(t) + \frac{1}{Q}I(t)\cdot \Delta t \label{eqn:1a} \\
V_{\text{RC}_1}(t+1) &= V_{\text{RC}_1}(t) - \frac{\Delta t}{R_1 C_1}V_{\text{RC}_1}(t) + \frac{\Delta t}{C_1}I(t) \\
V_{\text{RC}_2}(t+1) &= V_{\text{RC}_2}(t) - \frac{\Delta t}{R_2 C_2}V_{\text{RC}_2}(t) + \frac{\Delta t}{C_2}I(t) \\
V(t)=V_{\text{ocv}}(&SOC(t)) + V_{\text{RC}_1}(t) + V_{\text{RC}_2}(t) + I(t) R_0 \label{eqn:2a}\end{aligned}$$
where $I(t)$ is the current input (which is the control variable for this problem), and $V_{OCV}$ is the open-circuit voltage function, which is conventionally obtained through experiments. The full experimental OCV curve is used to represent the true plant in the loop, and is obtained from a lithium-iron phosphate (LFP) battery cell [@Perez05]. In this paper, we conduct linear and nonlinear MPC case studies using (1) a cubic polynomial, and (2) a linear polynomial to approximate the experimental curve. These OCV relations take the form: $$\hat{V}_{\text{ocv}}(SOC) = a_0 + a_1SOC + a_2SOC^2 + a_3SOC^3$$ $$\hat{V}_{\text{ocv}}(SOC) = a_{0;\ell} + a_{1;\ell}SOC$$ This entire model is linear in the parameters, so we can learn the model online using RLS. With linear OCV, the model can be represented in linear state-space form, yielding a convex MPC program with the given formulation in (\[eqn::mpc\]-\[eqn::mpc2\]). We employ both linear and nonlinear MPC case studies for several reasons. We utilize a nonlinear MPC case study to demonstrate the versatility of our algorithmic framework in addressing safety for a more complex learned dynamical system. We use the linear MPC case study to illustrate our algorithm’s capability of preserving convexity with only marginal additional computational requirements. In both cases, our algorithm maintains feasibility subject to model mismatch sourced from the experimental OCV.
Parameter Description Value Units
------------ --------------------- ------- ------------------- --
$Q$ Charge Capacity 8280 $[\frac{1}{A.h}]$
$R_0$ Resistance 0.01 $[\Omega]$
$R_1$ Resistance 0.01 $[\Omega]$
$R_2$ Resistance 0.02 $[\Omega]$
$C_1$ Capacitance 2500 $[F]$
$C_2$ Capacitance 70000 $[F]$
$\Delta t$ Timestep 1 \[s\]
$N_{targ}$ Max Control Horizon 8 \[-\]
$\eta$ Risk Metric 0.01 \[-\]
$\beta$ Ambiguity Metric 0.99 \[-\]
$D$ Support Diameter 1 \[-\]
: Relevant Parameters[]{data-label="sample-table"}
-0.1in
Model-Predictive Control Formulation
------------------------------------
We utilize the following formulation of fast charging: $$\label{eqn::mpc}
\min_{I(k) \in \mathcal{U}} \sum_{k=t}^{t+N} (SOC(k) - SOC_{target})^2$$ subject to: $$\begin{aligned}
(\ref{eqn:1a})-(\ref{eqn:2a}), &\quad
SOC(0) = SOC_0 \\
V(k) \leq 3.6 V, &\quad
0 A \leq I(k) \leq 60 A\label{eqn::mpc2}\end{aligned}$$ We solve this problem using the YALMIP toolbox for MATLAB [@Lofberg00]. Specifically, we adopt the IPOPT solver to address the impact of the nonlinear OCV function in our model. The timestep $\Delta t=1$ second, $\eta=0.01$, $D=1$, $\beta = 0.99$, and $N_{targ}=8$ steps. In both the linear and nonlinear MPC cases, we use the full nonlinear plant in the loop with an experimental OCV curve.
Our baseline is a learning MPC controller with no DRO framework. We adopt the same problem formulation as if we were going to add the constant $r^{(i)}$ to the constraints, but we omit the DRO constant in the end to evaluate the impact it has on the robustness of the final control law.
Results
-------
Figure 1 illustrates learned safe fast charging results with the nonlinear model. We start from an initial SOC of 0.1, and we set a target SOC of 0.5. We apply i.i.d. zero-mean Gaussian measurement noise to our observations of the output voltage and regressors. To control for effects of measurement noise, we apply the same noise realizations to the regressors of both models. We assume imperfect knowledge that initial input currents $I_{init} \leq 30$A are safe. We add zero-mean Gaussian noise with $\sigma = 2.5$ Volts to the control input for PoE.
\[fig:res1\]
The MPC approach experiences several spikes in voltage early in the learning process, exhibiting clear violation of the safety constraint. Furthermore, it periodically violates the constraint throughout the charging. In constrast, our distributionally robust charging algorithm satisfies constraints at every instant in time. This is significant, especially since we are learning the model from scratch. The constraint offset tightens insofar as the mean square error (MSE) trends downwards with more data samples. The MSE attributable to the Gaussian noise is consistently accommodated by $r$. For further validation, we generate 10 independent runs of zero-shot learning and control. The percentage of unsafe time steps is $\frac{4}{3000}=0.133\%$, well within the risk tolerance $\eta=1\%$. Even with the added computation, our algorithm runs in real time with a nonlinear model.
We also compute 10 independent runs of our DRO learning MPC algorithm with a linear model. Figure 2 shows final charging results from one run. Since our voltage model is limited, the robust offset actually grows over time as our residuals from the past become larger. This is because the OCV model adapts over time, and old data becomes less representative. However, much like the nonlinear MPC case study, our DRO algorithm is able to maintain feasibility throughout the zero-shot run. Across all 10 runs, we observe zero timesteps with unsafe charging behavior when we add the DRO component to the MPC scheme. Table 2 shows a comparison of computation times for 10 runs of DRO-MPC using the linear model. Since the constraint reformulation requires a scalar convex optimization even for joint chance constraints, the additional computation will not scale.
In both linear and nonlinear cases, we are learning 17 and 19 model parameters, respectively. For brevity, we omit results of parametric evolution and convergence. Most of the parameters tend to converge close to their nominal values after 50-100 seconds.
Run DRO-MPC MPC
----- --------- --------
1 40.586 30.194
2 39.225 23.026
3 55.443 23.267
4 53.390 25.250
5 41.633 23.906
6 42.097 26.003
7 46.143 25.934
8 43.038 26.889
9 51.075 26.728
10 39.469 26.488
: Computational Comparison (Seconds) for DRO-MPC and MPC with a linear OCV model.[]{data-label="sample-table"}
-0.1in
\[fig:res1\]
Conclusion
==========
This paper presents a distributionally robust model-based control algorithm for zero-shot learning. It addresses the problem of safety during online learning and control, with zero knowledge of the true model parameters. We adopt a stochastic MPC formulation where we augment constraints with random variables corresponding to empirical distributions of modeling residuals. We apply Wasserstein ambiguity sets to optimize over the worst-case modeling error. This approach provides an out-of-sample safety guarantee which we validate through numerical experiments. For application to convex MPC problems, this added algorithmic framework preserves convexity with minimal additional computation.
Our results provide the basis for several meaningful insights. It is clear that the supporting research for Wasserstein ambiguity sets provide an ideal base for its application to online learning and control. Our numerical experiments indicate our approach is highly effective at providing probabilistic safety guarantees throughout online learning.
[^1]: This work is supported by a National Science Foundation Graduate Research Fellowship.
[^2]: $^1$Aaron Kandel and Scott Moura are with the are with the Energy, Controls, and Applications Lab (eCAL) at the University of California, Berkeley, Berkeley, CA, 94704 USA [{aaronkandel, smoura}@berkeley.edu]{}.
|
---
abstract: 'Sequence-to-sequence models have shown success in end-to-end speech recognition. However these models have only used shallow acoustic encoder networks. In our work, we successively train very deep convolutional networks to add more expressive power and better generalization for end-to-end ASR models. We apply network-in-network principles, batch normalization, residual connections and convolutional LSTMs to build very deep recurrent and convolutional structures. Our models exploit the spectral structure in the feature space and add computational depth without overfitting issues. We experiment with the WSJ ASR task and achieve 10.5% word error rate without any dictionary or language using a 15 layer deep network.'
address: |
$^1$Massachusetts Institute of Technology $^2$Carnegie Mellon University $^3$Google Brain\
[`yzhang87@mit.edu`, `williamchan@cmu.edu`, `ndjaitly@google.com`]{}
bibliography:
- 'cites.bib'
title: 'Very Deep Convolutional Networks for End-to-End Speech Recognition'
---
=1
Automatic Speech Recognition, End-to-End Speech Recognition, Very Deep Convolutional Neural Networks
References
==========
|
---
abstract: 'Dense image matching is a fundamental low-level problem in Computer Vision, which has received tremendous attention from both discrete and continuous optimization communities. The goal of this paper is to combine the advantages of discrete and continuous optimization in a coherent framework. We devise a model based on energy minimization, to be optimized by both discrete and continuous algorithms in a consistent way. In the discrete setting, we propose a novel optimization algorithm that can be massively parallelized. In the continuous setting we tackle the problem of non-convex regularizers by a formulation based on differences of convex functions. The resulting hybrid discrete-continuous algorithm can be efficiently accelerated by modern GPUs and we demonstrate its real-time performance for the applications of dense stereo matching and optical flow.'
author:
- |
Alexander Shekhovtsov, Christian Reinbacher, Gottfried Graber and Thomas Pock\
Institute for Computer Graphics and Vision, Graz University of Technology\
[{shekhovtsov,reinbacher,graber,pock}@icg.tugraz.at]{}
bibliography:
- 'bib/strings.bib'
- 'bib/discrete.bib'
- 'bib/books.bib'
- 'bib/continuous.bib'
- 'bib/suppl.bib'
title: 'Solving Dense Image Matching in Real-Time using Discrete-Continuous Optimization'
---
Details of Dual MM {#sec:discrete-details}
==================
|
---
bibliography:
- 'alvarobib.bib'
---
addtoreset[equation]{}[section]{}
\
We compute the tree-level potential between two parallel $p$-branes due to the exchange of scalars, gravitons and $(p+1)$-forms. In the case of BPS membranes in 4d $\mathcal{N}=1$ supergravity, this provides an interesting reinterpretation of the classical Cremmer et al. formula for the F-term scalar potential in terms of scalar, graviton and 3-form exchange. In this way, we present a correspondence between the scalar potential at every point in scalar field space and a system of two interacting BPS membranes. This could potentially lead to interesting implications for the swampland program by providing a concrete way to relate conjectures about the form of scalar potentials with conjectures regarding the spectrum of charged objects.
Introduction
============
It is well known that the 4 dimensional cosmological constant can be interpreted in terms of field strengths of 3-forms. Even though they do not propagate additional degrees of freedom, they can acquire non-vanishing vevs and give rise to a cosmological constant contribution. For this reason, 3-forms have been used in trying to solve the cosmological constant problem as in [@pioneros1; @pioneros2; @pioneros3; @pioneros4; @pioneros5]. More specifically, considering the membranes to which a 3-form naturally couples provides a mechanism for the cosmological constant to change when a membrane is crossed, as considered originally in [@BT; @BT2] and also in [@BP] within the context of String Theory. In fact, this relation works and has been studied not only for constant contributions but for more general scalar potentials including axions in [@morerecent2; @morerecent3; @morerecent4; @morerecent5; @KS; @KLS]. In String Theory, this has also been explored [@Dudas:2014pva; @Escobar:2015ckf; @imuv; @Carta:2016ynn; @Garcia_Valdecasas:2016voz; @Valenzuela:2016yny; @Blumenhagen:2017cxt] and in the context of type II compactifications with fluxes, it was shown in [@Bielleman:2015ina; @Herraez:2018vae] that the complete F-term flux potential can be expressed, after integrating out the 4-forms, as $$V=\dfrac{1}{2}Z^{AB}Q_AQ_B,$$ where $Z^{AB}$ includes the field dependence and can be obtained from the kinetic terms of the 3-forms. The $Q_A$ give the coupling of the corresponding 4-form. The fact that this $\mathcal{N}=1$ potential can be expressed completely in terms of 3-forms, which naturally couple to membranes, suggests that a direct relation between these objects can be drawn. In particular, the potential being a bilinear in the charges reminds of a membrane-membrane interaction, and this is precisely the relation that we study in this note. We particularize for the case two interacting BPS membranes in 4d and point out a correspondence between their different interactions and the different terms in the Cremmer et al. $\mathcal{N}=1$ F-term scalar potential (see eq. ). Moreover, no obstruction to the application of the same logic to codimension 1 branes in higher dimensions is expected.
This correspondence is interesting by itself, but it can also be useful in the context of the Swampland [@swampland] (see also [@vafafederico; @review] for interesting reviews), since it could provide a precise setup in which two types of conjectures may be related. On the one hand, hypothesis about the generic properties of the scalar potentials that can arise in QG have been the subject of a considerable study recently, including the idea that metastable de Sitter space cannot exist [@dS1; @dS3] or has to be sufficiently short-lived [@TCC], or the suggestion that stable non-susy AdS [@Ooguri:2016pdq], as well as scale separated AdS, belong to the Swampland [@lpv; @Gautason:2015tig]. On the other hand, a lot of progress has been made in clarifying conjectures about properties of the spectrum of QG theories, like the Weak Gravity Conjecture [@WGC] or the Swampland Distance Conjecture [@distance]. Besides, several connections between apparently different Swampland Conjectures have been gradually uncovered (see [@review] and references therein), realizing the idea of a web of interconnected conjectures, instead of a set of unrelated statements. In this context, the precise correspondence between the scalar potential and the interactions of two membranes could provide new ways to relate the restrictions on the scalar potentials with the properties of the membranes[^1].
In section \[section2\], we review the classical field theory calculation of the interaction between $p$-branes due to scalar, graviton and $(p+1)$-forms exchange, and use it to interpret a particular version of the WGC. Section \[section3\] presents the main result of this note, namely the correspondence between the different pieces of the Cremmer et al. scalar potential and the different interactions between a pair of flat BPS membranes. We leave the summary and outlook for the final section.
**Note added:** When finalizing this note we were informed that a related paper is about to appear [@LMMV], also pointing out the interpretation of the $\mathcal{N}=1$ F-term potential as a no-force condition for the membranes and studying implications for the swampland conjectures.
Interactions between D$p$-branes in D dimensions {#section2}
================================================
We begin by reviewing the field theory calculation of the tree-level potential between two parallel, infinite $p$-branes in $D$-dimensional Minkowski space [@Polchinski:1998rq; @Polchinski:1998rr][^2]. The goal of this section is to review this calculation in detail to fix the conventions and explicitly keep track of the units, which is crucial for comparison with Swampland Conjectures. For concreteness, let us consider a $D$-dimensional generalization of the low energy effective action of type II supergravity and include source terms corresponding to $Dp$-branes with charge $Q_p$ and tension $\tilde{T}_p$ (in string units) [@Polchinski:1998rq; @Polchinski:1998rr; @BOOK].
$$\label{Stotal}
S=S_{\bulk}+S_{\DBI}+S_{\CS},$$
where the different pieces take the form: $$\begin{aligned}
\label{Sbulk}
S_{\bulk} = & \displaystyle \dfrac{1}{2 \tilde{\kappa}_{D}^{2}} \, \int d^{D} x \sqrt{-\tilde{g}} \, e^{-2 \tilde{\phi}} \, \left(\tilde{R}+4(\partial \tilde{\phi})^{2}\right)-\frac{1}{4 \tilde{\kappa}_{D}^{2}} \int G_{p+2} \wedge \tilde{\star} G_{p+2} , \\
\label{SDBI}
S_{\DBI} = & -\tilde{T}_p \int_{WV} d^{p+1}\xi \, e^{-\tilde{\phi}} \, \left[-\det \left( \tilde{g}_{ab}+\mathcal{F}_{ab} \right) \right]^{1/2}, \\
\label{SCS}
S_{\CS} = & Q \int_{WV} C_{p+1} .\end{aligned}$$
This action is expressed in the string frame and the quantities are measured in string units[^3] Separating the dilaton into a background and a dynamical part, that is, $\tilde{\phi}=\bar{\phi}+\phi$ we can define the quantities $\kappa^2_D=\tilde{\kappa}^2_D e^{2\bar{\phi}}$ and $T_p=\tilde{T}_p e^{-\bar{\phi}}$, which are the gravity coupling constant and the effective tension of the membrane, respectively. Then, upon performing a Weyl rescaling of the metric $\tilde{g}_{\mu\nu}=e^{\frac{4}{D-2}\phi}g_{\mu \nu}$ we obtain the following expressions for $S_{bulk}$ and $S_{DBI}$ in the Einstein frame ($S_{CS}$ does not change since it is a topological term, independent of the bulk metric). $$\begin{aligned}
\label{SbulkE}
\displaystyle S_{\bulk} = & \dfrac{1}{2 \kappa_{D}^{2}} \, \int d^{D} x \sqrt{-g} \, \left(R+\dfrac{4}{D-2}(\partial \phi)^{2}\right)-\frac{e^{2\bar{\phi}}}{4 \kappa_{D}^{2}} \int G_{p+2} \wedge \star G_{p+2} , \\
\label{SDBIE}
S_{\DBI} = & -T_p \displaystyle \int_{WV} d^{p+1}\xi \, \exp{\left[\left(\frac{2p-D+4}{D-2} \right) \phi \right]} \, \sqrt{-\det \left( g_{ab} \right)}.\end{aligned}$$ We use $\kappa_D$ in this section, since it is more suitable for perturbative calculations but keep in mind that the Planck mass is given by $\kappa_D^{-2}=M_p^{D-2}$. From here, we can compute the tree-level scalar, graviton and $(p+1)$-form exchange between two membranes. The propagators associated to the graviton, scalar and $(p+1)$-form can be obtained from $S_{\bulk}$ and the interaction vertices between the brane and the fields fields from $S_{\DBI}$ (for the scalar and graviton) and $S_{\CS}$ (for the $(p+1)$-form).
The scalar plus graviton interaction
------------------------------------
Let us begin with the interaction due to the scalar and graviton exchange, which corresponds to the diagrams in fig. \[fig:NSexchange\]
![Diagramatic representation of the tree-level scalar and graviton exchange between two $p$-branes[]{data-label="fig:NSexchange"}](Images/membranescalarexchange.pdf "fig:"){height="58pt"} \[fig:scalarexchange\] $+$ ![Diagramatic representation of the tree-level scalar and graviton exchange between two $p$-branes[]{data-label="fig:NSexchange"}](Images/membranegravitonexchange.pdf "fig:"){height="58pt"} \[fig:gravitonexchange\]
Expanding the metric as a background plus a perturbation as $g_{\mu \nu}=\bar{g}_{\mu \nu}+ \kappa_D h_{\mu \nu}$ we compute $$\sqrt{-\det(g_{ab})}=\sqrt{-\det(\bar{g}_{ab})}\left(1+\dfrac{\kappa_D}{2}\bar{g}^{ab}h_{ab}+... \right),
\label{detexpansion}$$ where we have defined $|\bar{g}_{ab}|=-\det(\bar{g}_{ab})$. Using this expression and working in De Donder gauge, we can expand the Ricci scalar to obtain the graviton propagator in momentum space [@Polchinski:1998rq; @Polchinski:1998rr]: $$\langle h_{\mu \nu} h_{\alpha \beta} \rangle =-\dfrac{2 i}{k^2}\left( \bar{g}_{\mu \alpha} \bar{g}_{\nu \beta}+\bar{g}_{\mu \beta} \bar{g}_{\nu \alpha}-\dfrac{2}{D-2}\bar{g}_{\mu \nu} \bar{g}_{\alpha \beta} \right) .$$ Additionally, the scalar propagator takes the form $$\langle \phi \phi \rangle = -\dfrac{i \kappa_D^2}{k^2}\dfrac{(D-2)}{4}.$$
We calculate the relevant vertices by expanding $S_{DBI}$ to obtain $$\begin{split}
S_{\DBI} = & -T_p \int_{WV} d^{p+1} \xi\sqrt{|\bar{g}_{ab}|} -T_p \int_{WV} d^{p+1} \xi\sqrt{|\bar{g}_{ab}|} \left( \dfrac{2p-D+4}{D-2}\right) \phi + \\ & - T_p \int_{WV} d^{p+1} \xi\sqrt{|\bar{g}_{ab}|} \, \dfrac{\kappa_D}{2}\bar{g}^{ab}h_{ab}+
\end{split}$$ where the first term gives the usual contribution from the embedding of the worldvolume in the D-dimensional spacetime, the second gives the interaction between one scalar and the source and the third gives the interaction between the graviton and the energy-momentum tensor of the brane. The dots indicate interactions of the source with more than one field, which are not relevant for our computation. For a flat membrane in Minkowski space we can choose a set of coordinates such that $x^0=\xi^0, x^1=\xi^1 \, ... \ x^p=\xi^p $ and $x^{p+1}=...=x^{D-1}= \mathrm{const}$, impliying $g_{ab}=g_{\mu \nu} \delta_a^\mu \delta_b^\nu$.
The worldvolume metric then takes the form of the corresponding $(p+1)\times (p+1)$ block of the background D-dimensional metric. Using this we obtain the following Feynman rules for the relevant vertices $$\includegraphics[align=c, height=48pt]{Images/scalarvertex.pdf}
\label{fig:scalarvertex} \ = \ i T_p \sqrt{|\bar{g}_{ab}|} \left( \dfrac{2p-D+4}{D-2} \right),$$ $$\includegraphics[align=c, height=48pt]{Images/gravitonvertex.pdf}
\label{fig:gravitonvertex} \ = \ i T_p \sqrt{|\bar{g}_{ab}|} \dfrac{\kappa_D}{2}\bar{g}^{ab} \delta_a^\mu \delta_b^\nu. \qquad$$
The amplitude for the scalar and graviton interaction given in fig. \[fig:NSexchange\] then yields $$\label{As+ggen}
\mathcal{A}_{s+g}=\dfrac{T_p^2 |\bar{g}_{ab}| \kappa_D^2}{k^2} \left\{ \dfrac{(2p-D+4)^2}{4(D-2)}+\dfrac{(D-p-3)(p+1)}{D-2} \right\}= \dfrac{T_p^2 |\bar{g}_{ab}| \kappa_D^2}{k^2} \left\{ \dfrac{D-2}{4}\right\},$$ where $k$ indicates the momentum in the directions perpendicular to the membranes. The first term corresponds to the scalar interaction and the second to the graviton exchange. The final result is independent of $p$, as can be seen in the last step, but we will keep both terms in order to keep track of the scalar and the graviton pieces separately. Since we will be mainly interested in codimension 1 objects (i.e. $p=D-2$), let us particularize for that case, in which the amplitude takes the form: $$\mathcal{A}_{s+g}=\dfrac{T_p^2 |\bar{g}_{ab}| \kappa_D^2}{k^2}\left\{ \dfrac{D^2}{4(D-2)}-\dfrac{D-1}{D-2} \right\}.
\label{As+g}$$
In this case, the scalar contribution is always positive (i.e. attractive) but the contribution from the graviton exchange becomes negative, yielding a repulsive force, which only occurs in this particular case of codimension 1 objects. The potential between the two membranes can be calculated by taking the Fourier transform of this amplitude, and for the codimension 1 objects, it grows linearly with the distance, implying a distance independent long-range force.
The q-form interaction
----------------------
\[fig:qformexchange\]
{height="48pt"}
The $q$-form interaction between two $(q-1)$-branes corresponds to the diagram in fig. \[fig:qformexchange\]. To compute it we need the propagator of the $q$-form and its coupling to the membrane. We begin with the propagator and, in order to obtain the tensor structure, we first consider a canonically normalized $q$-form kinetic term and after we include the overall (background-field dependent) prefactors appearing in or . We consider then the kinetic part of the action of a q-form $A_q=\dfrac{1}{q!}A_{\mu_1 ...\mu_q }dx^{\mu_1}\wedge ... \wedge dx^{\mu_q}$, which takes the form $$S_{\mathrm{q,\, kin}}=-\dfrac{1}{2 (q+1)!}\int d^D x \sqrt{-g} F_{\mu_1...\mu_{q+1}}F^{\mu_1...\mu_{q+1}},$$ with $F_{\mu_1...\mu_{q+1}}=(q+1) \partial_{[\mu_1}A_{\mu_2 ...\mu_{q+1}] } $ [^4]. Integrating by parts and massaging the Lagrangian we obtain $$\label{qkinnogf}
\begin{split}
\mathcal{L}_{\mathrm{q,\, kin}} = & \dfrac{1}{2 q!}A_{[\mu_1...\mu_p]}\left\{ - \delta_{\nu_1}^{[\mu_1}...\delta_{\nu_q}^{\mu_q]} \partial_\rho \partial^\rho + q\, \delta_{\rho}^{\mu_1}\delta_{[\nu_1}^{\sigma}...\delta_{\nu_q]}^{\mu_q} \partial_\sigma \partial^\rho \right\}A^{[\nu_1...\nu_p]}.
\end{split}$$ In order to obtain the propagator, we need to invert the second variation of this part of the action with respect to the $q$-form field. To do so we choose a generalization of the Lorentz gauge [@Luscher] for $q$-forms, namely $\partial_\mu A^{[\mu \nu_2...\nu_q]}=0$, and implement it by means of the following gauge fixing term in the Lagrangian $\mathcal{L}_{gf}\sim \partial_\rho A^{[\rho \nu_2...\nu_q]}\partial^\sigma A_{[\sigma \nu_2...\nu_q]}$. By adjusting the constant in front we can cancel the second term in the last line of eq. , so that we are left with the task of inverting the operator $$K^{\mu_1...\mu_n}_{\nu_1...\nu_n}=\dfrac{1}{q!}\left(- \delta_{[\nu_1}^{[\mu_1}...\delta_{\nu_q] }^{\mu_q]} \partial_\rho \partial^\rho \right),$$ which yields the propagator (in momentum space) $$\langle A^{\nu_1...\nu_n} A_{\mu_1...\mu_n} \rangle = i \, \dfrac{q!}{k^2} \delta_{[\nu_1}^{[\mu_1}...\delta_{\nu_q] }^{\mu_q]} .$$ Note that this is normalized in such a way that the propagator from any independent component to itself (or any antisymmetric permutation thereof) coincides with the propagator of a scalar degree of freedom, as expected for a field with a canonical kinetic term. In order to calculate the propagator of the $C_{p+1}$ from eq. we just need to rescale the calculated propagator to account for the overall factors that prevent the kinetic term from being canonically normalized, obtaining $$\label{qformprop}
\langle C^{\nu_1...\nu_q} C_{\mu_1...\mu_q} \rangle =2i \, \dfrac{ \kappa_D^2\, e^{-2\bar{\phi}}}{k^2} q! \delta_{[\nu_1}^{[\mu_1}...\delta_{\nu_q] }^{\mu_q]}$$
To obtain the $q$-form brane vertex, we use eq. , which in components reads $$S_{CS}=\dfrac{Q}{q!}\int_{WV} d^q \xi\, C_{\mu_1...\mu_q} \left( \dfrac{\partial x^{\mu_1}}{\partial \xi^{a_1}}...\dfrac{\partial x^{\mu_q}}{\partial \xi^{a_q}} \right) \epsilon^{a_1...a_q}.$$ For a flat brane we obtain the following Feynman rule for the vertex $$\includegraphics[align=c, height=48pt]{Images/qformvertex.pdf}
\label{fig:qformvertex} \ = \ i \dfrac{Q}{q!}\epsilon^{a_1...a_q} \delta_{a_1}^{[\mu_1}...\delta_{a_q }^{\mu_q]}.$$ The $q$-form exchange of fig. \[fig:qformexchange\] yields the following amplitude $$\mathcal{A}_C= -\dfrac{Q^2}{(q!)^2}\dfrac{2\, \kappa_D^2 \, e^{-2\bar{\phi}}}{k^2}\, q!\, \epsilon^{a_1...a_q} \bar{g}_{a_1 b_1}...\bar{g}_{a_q b_q}\epsilon^{b_1...b_q}= -\dfrac{2 Q^2 |\bar{g}_{ab}| \kappa_D^2 e^{-2\bar{\phi}}}{k^2},
\label{AC}$$ where we have used $\epsilon^{a_1...a_q} \bar{g}_{a_1 b_1}...\bar{g}_{a_q b_q}\epsilon^{b_1...b_q}= q!\, |\bar{g}_{ab}|$ in the last step. Notice that this depends on the rank of the form and the spacetime dimension only implicitly via $g_{ab}$ and $k^2$, but not explicitly as in the scalar and graviton exchange.
Then, the three contributions cancel, and no net force is felt by the branes if $$\label{equality}
\left\{ \dfrac{D^2}{4(D-2)}-\dfrac{D-1}{D-2} \right\}T_p^2 =2 Q^2 e^{-2\bar{\phi}}.$$ This is what happens for single $Dp$-branes in 10 dimensions, which are BPS objects, upon substitution of the corresponding tensions and charges (i.e. $Q=\tilde{T}_p=T_p e^{\bar{\phi}}$). This cancellation is precisely expected for BPS objects, since they feel no net force. For the case of codimension 1 branes, the attractive contribution from the scalars compensates the repulsive force from graviton and $q$-form exchange.
Relation with the WGC for $q$-forms in the presence of dilaton-like couplings
-----------------------------------------------------------------------------
In this section, we make a small detour from the main goal of this note and explore the consistency of this calculation with the WGC. The aforementioned case of single D$p$-branes in 10 dimensions is a particular realization of the general claim that the extremal case in which the WGC inequality is saturated occurs for BPS objects [@Ooguri:2016pdq]. More generally, we can recover the results for the extremal form of the WGC for $q$-forms in the presence of dilaton-like couplings proposed in [@WGC16] by requiring that eqs. and cancel each other out.[^5]. This form of the WGC can be expressed, using our notation, as the following inequality $$\kappa_D^2 \left[\frac{\alpha^{2}}{2}+\frac{q(D-q-2)}{D-2}\right] T^{2} \, \leq \, e^2 \, Q^{2} \,
\label{WGCqforms}$$ This expression is obtained from the extremality bound of black branes in theories with gravity, a $q$-form and a dilaton-like scalar. The RHS comes from the $q$-form interaction, and in our case ,the gauge coupling can be read from eq. and equals $e^2=2 \kappa^2_D e^{-2\bar{\phi}}$. This matches exactly the RHS of . The second term of the LHS comes from the gravitational interaction, which depends on the spacetime dimensions and the rank of the form, and it can be checked that it matches the second term in the LHS of upon substitution of $q=p-1$. Finally, the first term on the LHS corresponds to the scalar interaction, whose coupling constant can be extracted from the kinetic terms of the $q$-form. In particular, for a theory with a $q$-form and a *conventionally normalized* scalar field $\phi$, a dilaton-like coupling is characterized by a term of the following form in the Lagrangian [@WGC16] $\mathcal{L}_{\mathrm{q,\, kin}} \sim e^{-\vec{\alpha}\cdot \vec{\phi}} F^{2}$, where $\alpha$ is the dilaton coupling constant. In our model, we need to identify the $\alpha$ from the kinetic terms for the $q$-forms, which are given in eq. and after the Weyl rescaling to go to the Einstein frame become $$\label{kinpform}
S_{\mathrm{q\, kin}}=-\dfrac{e^{2\bar{\phi}}}{4 \kappa_D^2} \dfrac{1}{(p+2)!}\int \sqrt{-g} \, d^D x \, e^{ \left(\frac{4p-2D+8}{D-2} \right) \phi } \, G_{\mu_1 ... \mu_{p+2}} G^{\mu_1 ... \mu_{p+2}}.$$ From the exponential, we can read the coupling to the dilaton $\phi$, and after conventionally normalizing its kinetic term via $\phi_c=\sqrt{\frac{8}{D-2}}\phi$ we obtain a kinetic term of the form described before with $$\label{alpha}
\dfrac{\alpha^2}{2}=\dfrac{(2p-D+4)^2}{4D-8}.$$ This exactly resembles the contribution from the scalar exchange in eq. . As advertised, this extremal form of the WGC, which was originally formulated from extremality arguments for BH’s, can be obtained from the requirement that the long-range interaction between charged objects cancels exactly (see also [@Palti; @Heidenreich:2019zkl] for more on this approach or [@Gonzalo:2020kke] for an alternative approach in terms of pair production).
Let us remark that the actionconsidered at the beginning of this section is really meaningful in $D=10$, since in this case it is a piece of the 10d type II effective action coupled to D$p$-branes. The extension to general $D$ serves for illustrative purposes, since it allows to capture some aspects of the typical $D$-dimensional effective actions, like the Lorentz index structure of the propagators. However, eq. should not be taken as a general expression, but rather as a check for the case in which a $D$- dimensional action can be recast into the form given in eqs. -. In particular, if we focus on particles in 4d (i.e. $p=0$ and $D=4$), this action could capture the coupling of particles coming from D$p$-branes wraping internal p-cycles to the real fields in the complex structure sector of type IIA compactifications, which is known to vanish [@irene; @Font:2019cxq]. In order to capture other couplings like the ones that would come from the Kähler sector in type IIA compactifications, or axions, one should take a more general 4d effective action, but this is not the main goal of this section so we will not elaborate more on this. Let us just mention that the kind of couplings that can be encoded in the $\alpha$ parameter, and therefore the ones that are included in , are also limited. In fact, they are restricted to the cases where the brane-scalar interaction, which is given by the derivative of the tension of the brane with respect to the field, is proportional to the tension itself.
The 4d $\mathcal{N}=1$ scalar potential in terms of membranes {#section3}
=============================================================
The main goal of this note is to point out a correspondence between the different terms of the 4d $\mathcal{N}=1$ F-term scalar potential and the interaction between two BPS membranes due to particle exchange.This scalar potential takes the following form in terms of the Kähler potential, $K$ and the superpotential, $W$ [@sugra] $$\label{Cremmeretal}
V=e^K\left( K^{I \bar{J}} D_I W D_{\bar{J}}\bar{W}- 3 |W|^2\right),$$ where $D_I W= \partial_I W + W \partial _I K$ and the $I, J$ indices run over all the complex scalar fields of the theory.
The scalar potential and the 3-form interaction
-----------------------------------------------
In order to make the correspondence precise, the first important ingredient is the fact that in type II flux compactifications, this scalar potential can also be expressed as the following bilinear [@Bielleman:2015ina; @Herraez:2018vae] $$V=\dfrac{1}{2} Z^{AB} Q_A Q_B,$$ where $Q_A$ is a vector containing the fluxes, and $Z^{AB}$ is the inverse of the matrix that appears in the kinetic term of 3-forms, which is of the form $\mathcal{S}_{\mathrm{3, \, kin}}= \frac{1}{2 \kappa_4^2} \int Z_{AB} F^A \wedge \star F^B $. This matrix encodes the dependence on the scalar fields (both the axions and the saxions), and the charge vectors encode all the information about the fluxes. This form of the potential has been argued to be valid fore more general $\mathcal{N}=1$ setups in [@Farakos:2017jme; @Farakos:2017ocw; @Bandos:2018gjp; @Bandos:2019wgy; @Lanza:2019xxg] and this is expected to be quite general at least at the level of any 4d EFT, since a cosmological constant term can be always rewritten in terms of 4-forms, so allowing for field dependent kinetic terms seems to be enough to encode this kind of scalar potentials[^6]. This bilinear expression is suggestive, since it resembles the form of an electric interaction between two charged objects, with charges $Q_A$, mediated by some mediator with propagator $\sim Z^{AB}$. In fact, this is the case when we consider the 3-form interaction between two flat membranes, whose couplings with the different 3-forms on the spectrum are given by a generalization of eq. . The propagator of the 3-forms takes the form $$\langle A^{A, \, \mu \nu \rho}\, A^B_{\alpha \beta \gamma} \rangle =i \, 3! \, \dfrac{ \kappa_4^2 }{k^2} \,Z^{AB} \, \delta_{[\alpha}^{\mu} \delta_{\beta}^{\nu} \delta_{\gamma] }^{\rho}$$ and vertex between the membrane and the vertex is the straighforward generalization of eq. , namely $$\includegraphics[align=c, height=48pt]{Images/3formvertex.pdf}
\label{fig:3formvertex} \ = \ i \dfrac{Q_A}{3!}\epsilon^{a b c} \delta_{a}^{[\mu}\delta_{b}^{\nu}\delta_{c}^{\rho]}.$$
\[fig:3formexchange\]
{height="48pt"}
Then the diagram in fig. \[fig:3formexchange\] gives an amplitude $$\mathcal{A}_{3-\mathrm{form}}=-\dfrac{ \kappa_4^2 |\bar{g}_{ab}| }{k^2} \, Z^{AB} Q_A Q_B,$$ which is proportional to the potential, as advertised.
The scalar and graviton interaction
-----------------------------------
We turn now to the computation of the diagrams corresponding to the exchange of gravitons and scalars. For flat BPS membranes, which preserve 1/2 of the supersymmetries of the Minkowski background along their worldvolume, the tensions takes the form [@Bandos:2018gjp; @Bandos:2019wgy; @Font:2019cxq; @Lanza:2019xxg] $$\label{TensionBPS}
T=2e^K |W|$$ where, $W$ corresponds to the superpotential sourced by the membrane [^7] and can be directly related to its charges $Q_A$ and the so called period vector. In type II we can interpret these membranes as coming from bound states of $Dp$ or $NS5$-branes wrapping internal cycles and their charges are related to the fluxes at the other side of the membrane.
The graviton exchange between two flat equal membranes of tension $T_p$ is universal and for codimension 1 objects is given by the second term in eq. . Upon substitution of eq. and $D=4$ we find $$\mathcal{A}_g=- \dfrac{2 \kappa_4^2 |g_{ab}|}{k^2}\, 3e^K |W|\, .$$
In order to calculate the contribution from the complex scalars, we recall that in a 4 dimensional $\mathcal{N}=1$ theory they have the following kinetic terms $$S_{\Phi,\, \mathrm{kin }}=\dfrac{1}{\kappa^2_4}\int \sqrt{-g}\, d^4 x \, K_{I\bar{J}}\partial_\mu \Phi^I \partial^\mu \bar{\Phi}^{\bar{J}},$$ and we call $\preal{(\Phi^I)}=X^I$ and $\pim{ (\Phi^I)}=Y^I$ with propagators (in momentum space) $$\langle X^I X^J \rangle= -\dfrac{i\kappa_4^2}{2 k^2} K^{I\bar{J}}, \qquad
\langle Y^I Y^J \rangle= -\dfrac{i\kappa_4^2}{2 k^2} K^{I\bar{J}}.$$
Additionally, the coupling between the scalars and the membrane can be obtained from the action [@Bandos:2018gjp; @Bandos:2019wgy; @Font:2019cxq; @Lanza:2019xxg] $$S_{mem}= -\int_{WV} \, d^3\xi \, \sqrt{|g_{ab}|} \, T\left( \Phi^I, \bar{\Phi}^{\bar{J}} \right) + S_{CS},$$ where the tension is given by eq. . The scalar-membrane vertex is obtained from the first derivative of the tension with respect to each of the (real) fields evaluated in the background. In order to relate this with eq. , we need to express everything in terms of the complex fields. To do so, we use that $K=K(\Phi^I + \bar{\Phi}^{\bar{J}})$ is a real function and $W=W(\Phi)$ is a holomorphic function to reexpress the derivatives of the tension with respect of $X^I$ and $Y^I$ in terms of derivatives with respect to $Z^I$ and $\bar{Z}^I$. From now on we restrict to a single complex scalar field $Z=X+iY$ for simplicity, but the generalization to more fields is straightforward.
![Diagramatic representation of the tree-level scalar exchange corresponding to the real and imaginary parts of a complex scalar[]{data-label="fig:ReImscalars"}](Images/membraneXexchange.pdf "fig:"){height="40pt"} \[fig:scalarexchangeX\] $+$ ![Diagramatic representation of the tree-level scalar exchange corresponding to the real and imaginary parts of a complex scalar[]{data-label="fig:ReImscalars"}](Images/membraneYexchange.pdf "fig:"){height="40pt"} \[fig:scalarexchangeY\]
The vertices we are interested in take the form
$$\begin{split}
\includegraphics[align=c, height=48pt]{Images/membraneX.pdf}
\label{fig:membraneX} =\, 2\sqrt{|g_{ab}|}\dfrac{ \partial T}{\partial X}\, = & \, i \sqrt{|g_{ab}|}\, e^{K/2}\left[ \dfrac{1}{2} \dfrac{\partial K}{\partial X} + \dfrac{1}{2 |W|} \left( \dfrac{ \partial W}{\partial X} \bar{W}+ \dfrac{ \partial \bar{W}}{\partial X} W \right) \right]= \\
= &\, 2i \sqrt{|g_{ab}|} \, e^{K/2}\left[ (\partial_Z K) |W|+ \dfrac{1}{ |W|} \preal{\left[(\partial_Z W) \bar{W} \right]} \right]
\end{split}$$
$$\begin{split}
\includegraphics[align=c, height=48pt]{Images/membraneY.pdf}
\label{fig:membraneY} =\, 2i\sqrt{|g_{ab}|} \dfrac{ \partial T}{\partial Y}\, = & \, i\sqrt{|g_{ab}|}\, e^{K/2}\left[ \dfrac{1}{2 |W|} \left( \dfrac{ \partial W}{\partial Y} \bar{W}+ \dfrac{ \partial \bar{W}}{\partial Y} W \right) \right]= \\
= &\, -2 i \sqrt{|g_{ab}|}\, e^{K/2} \dfrac{1}{ |W|} \pim {\left[(\partial_Z W) \bar{W} \right]}.
\end{split}$$
Having calculated the vertices and the propagators we can calculate the amplitude associated to scalar exchange, in fig. \[fig:ReImscalars\] which yields $$\mathcal{A}_{X+Y}= \dfrac{2 \kappa_4^2 |g_{ab}|}{ k^2} K^{Z\bar{Z}}\left[ \left( \dfrac{\partial T}{\partial X}\right)^2+ \left( \dfrac{\partial T}{\partial Y}\right)^2 \right] \, = \, \dfrac{2 \kappa_4^2 |g_{ab}|}{k^2} e^K (K^{Z\bar{Z}} D_Z W D_{\bar{Z}}\bar{W}).$$ and the interaction between the membranes then cancels if $$\label{Cremmeretal2}
\dfrac{1}{2}Z^{AB}Q_AQ_B\, =\, e^KK^{I \bar{J}} D_I W D_{\bar{J}}\bar{W}- 3 e^K |W|^2.$$ There is then a one to one correspondence between the three kinds of interactions of the membranes and eq. . In the membrane picture,the 3-form interactions correspond to the potential in and they equal the scalar and graviton interactions, which correspond to the two terms in the RHS of eq. , respectively. The upshot is that for every point in the scalar field space, we have a picture in which there is a potential, and another one with two BPS membranes in a Minkowski background which feel no net self-interactions. Then, whereas in the first picture the potential gets a contribution from the Kähler covariant derivatives of the superpotential and another one from the $-3|W|^2$ term; in the membrane picture these match the scalar and graviton interactions, respectively. In this language, for example, a supersymmetric vacuum of the potential would correspond in to a pair of membranes with no-scalar interaction. Notice, moreover, that in terms of the scalar potential description, this correspondence is valid even off-shell, since it is defined for every point in scalar field space, not only for the vacua of the potential.
Besides, in the membrane picture, the background is always Minkowski. This is the case because between the two membranes, the cosmological constant constant contribution (sourced by the membranes themselves) is encoded in the 3-form interaction, whose energy density between the membranes is canceled by the scalar and graviton interaction, resulting in a vanishing energy density between the membranes.
Summary and outlook {#s:summary}
===================
To sum up, we have studied the tree-level interaction between $p$-branes due to exchange of scalars, gravitons and $(p+1)$-forms. We have shown that, for the particular case of BPS membranes in a 4d Minkowski background, there is a correspondence between each interaction and each term in the $\mathcal{N}=1$ scalar potential. The fact that the scalar potential can be written in that form may be translated in the membrane picture to the requirement that the net force between the two membranes vanishes, that is, that the 3-form interaction cancels the scalar plus graviton contributions. From the point of view of the potential, this correspondence is valid off-shell (not only in the minima). This means that for every point in the scalar field space, characterized by a value of the scalar potential, there exists a corresponding membrane configuration with the same values for the scalar fields in which the self-interaction vanishes and whose 3-form interaction, or equivalently the scalar plus graviton interaction, equals the value of the potential in the initial picture. Let us remark that we have only worked out in detail the 4d case, but we expect similar arguments to apply for a relation between codimension 1 BPS objects and scalar potentials in more dimensions.
This correspondence, although interesting per se, might be useful for the swampland program (see [@PaltiTalkSP19; @LMMV]). In that context, lots of recent results suggest a very intricate web of swampland conjectures, in which apparently disconnected conjectures happen to be related or even imply each other in many different ways. In this respect, some swampland conjectures, like the WGC or the SDC make statements about the properties of the spectrum of consistent theories of QG, whereas others like the dS Conjecture or the ADC refer to the kinds of potentials or vacua that are allowed in QG. We believe that the correspondence explained in this note could help to uncover some connections between these two apparently different types of statements, since it relates configurations with a scalar potential with configurations of charged extended objects.
**Acknowledgments**
The author would like to thank L. Ibáñez, E. Palti and I. Valenzuela for useful discussions. This work is supported by the Spanish Research Agency (Agencia Estatal de Investigacion) through the grant IFT Centro de Excelencia Severo Ochoa SEV-2016-0597, and by the grants FPA2015-65480-P and PGC2018-095976-B-C21 from MCIU/AEI/FEDER, UE. The author is supported by the Spanish FPU Grant No. FPU15/05012.
[^1]: Some connections between these two kinds of statements have already been pointed out in [@PaltiTalkSP19], and also along the lines of potentials arising from integrating out towers of states in [@dS3; @review]
[^2]: In stringy terms, this corresponds to closed string exchange, which in the field theory limit reduces to the exchange of scalars and gravitons from the NSNS sector and $(p+1)$-forms from the RR sector [@Polchinski:1995mt]
[^3]: We denote all quantities in the string frame by tildes and use latin letters to refer to worldvolume indices whereas greek letters denote spacetime indices.
[^4]: Square brackets indicate antisymmetrization with a normalization factor of $1/q!$ in front, so that for an antisymmetric $q$-form we have $A_{\mu_1 ...\mu_q }=A_{[\mu_1 ...\mu_q ]}$
[^5]: This is just a manifestation of the special case in which arguments about long-range forces and extremality of black hole solutions coincide (for a more detailed discussion on the (in)equivalence of these two approaches see [@Palti; @Heidenreich:2019zkl]).
[^6]: There is an important caveat for this, namely the fact that the matrix $Z_{AB}$ needs to be invertible in order to be able to write the potential in terms of 4-forms.
[^7]: Notice that in our case we are considering a membrane separating a region where the superpotential vanishes identically from another one where the superpotential is generated by the membrane, so that the change in the superpotential is the superpotential itself.
|
---
abstract: 'It has been recognized that a heavily overparameterized artificial neural network exhibits surprisingly good generalization performance in various machine-learning tasks. Recent theoretical studies have made attempts to unveil the mystery of the overparameterization. In most of those previous works, the overparameterization is achieved by increasing the width of the network, while the effect of increasing the depth has been less well understood. In this work, we investigate the effect of increasing the depth within an overparameterized regime. To gain an insight into the advantage of depth, we introduce local and global labels as abstract but simple classification rules. It turns out that the locality of the relevant feature for a given classification rule plays an important role; our experimental results suggest that deeper is better for local labels, whereas shallower is better for global labels. We also compare the results of finite networks with those of the neural tangent kernel (NTK), which is equivalent to an infinitely wide network with a proper initialization and an infinitesimal learning rate. It is shown that the NTK does not correctly capture the depth dependence of the generalization performance, which indicates the importance of the feature learning, rather than the lazy learning.'
author:
- Takashi Mori$^1$
- 'Masahito Ueda$^{1,2,3}$'
bibliography:
- 'apsrevcontrol.bib'
- 'deep\_learning.bib'
- 'physics-quantum\_information.bib'
date: |
$^1$*RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan*\
$^2$*Department of Physics, Graduate School of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan*\
$^3$*Institute for Physics of Intelligence, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan*
title: 'Is deeper better? It depends on locality of relevant features'
---
Introduction {#sec:Intro}
============
Deep learning has achieved an unparalleled success in various tasks of artificial intelligence such as image classification [@Krizhevsky2012; @LeCun2015] and speech recognition [@Hinton2012]. Remarkably, in modern machine learning applications, impressive generalization performance has been observed in an *overparameterized* regime, in which the number of parameters in the network is much larger than that of training data samples. Contrary to what we learn in the classical learning theory, an overparameterized network fits random labels and yet generalizes very well without serious overfitting [@Zhang2017]. We do not have general theory that explains why deep learning works so well.
Recently, the learning dynamics and the generalization power of heavily overparameterized wide neural networks have extensively been studied. It has been reported that training of an overparameterized network easily achieves zero training error without getting stuck in local minima of the loss landscape [@Zhang2017; @Baity-Jesi2018]. Mathematically rigorous results have also been obtained [@Allen-Zhu2019_ICML; @Du2019]. From a different point of view, theory of the neural tangent kernel (NTK) has been developed as a new tool to investigate an overparameterized network with an infinite width [@Jacot2018; @Arora2019], which simply explains the reason why a sufficiently wide neural network can achieve a global minimum of the training loss.
As for generalization, “double-descent” phenomenon has attracted much attention [@Spigler2019; @Belkin2019]. The standard bias-variance tradeoff picture predicts a U-shaped curve of the test error [@Geman1992], but instead we find the double-descent curve, which tells us that the increased model capacity beyond the interpolation threshold results in improved performance. This finding triggered detailed studies on the behavior of the bias and variance in an overparameterized regime [@Neal2019; @dAscoli2020]. The double-descent phenomenon is not explained by traditional complexity measures such as the Vapnik-Chervonenkis dimension and the Rademacher complexity [@Mohri_text], and hence one seeks for new complexity measures of deep neural networks that can prove better generalization bounds [@Dziugaite2017; @Neyshabur2017; @Neyshabur2019; @Arora2018; @Nagarajan2017; @Perez2019].
These theoretical efforts mainly focus on the effect of increasing the network width, while benefits of the network depth remain unclear. It is known that expressivity of a deep neural network grows exponentially with the depth rather than the width [@Poole2016]. See also [@Bianchini2014; @Montufar2014]. However, it is far from clear whether exponential expressivity really leads to better generalization [@Ba2014; @Becker2020]. It is also nontrivial whether typical problems encountered in practice require such high expressivity. Although some works [@Eldan2016; @Safran2017] have shown that there exist simple and natural functions that are efficiently approximated by a network with two hidden layers but not by a network with one hidden layer, a recent work [@Malach2019] has demonstrated that a deep network can only learn functions that are well approximated by a shallow network by using a gradient-based optimization algorithm, which indicates that benefits of depth are not due to high expressivity of deep networks. Some other recent works have reported no clear advantage of the depth in an overparameterized regime [@Geiger2019a; @Geiger2019b].
To gain an insight into the advantage of the depth, in the present paper, we report our experimental study on the depth and width dependences of generalization in abstract but simple, well-controlled classification tasks with fully connected neural networks. We find that whether a deep network outperforms a shallow one depends on the property of relevant features for a given classification rule.
In this work, we introduce *local labels* and *global labels*, both of which give simple mappings between inputs and output class labels. By “local”, we mean that the label is determined only by a few components of the input vector. On the other hand, a global label is given by a sum of local terms and determined by all components of the input. Our experiments show strong depth-dependences of the generalization error for those simple input-output mappings. In particular, we find that *deeper is better for local labels, while shallower is better for global labels*. The implication of this result is that the depth is not always advantageous, but the locality of relevant features would give us a clue for understanding the advantage the depth brings about.
We also compare the generalization performance of a trained network of a finite width with that of the kernel method with the NTK. The latter corresponds to the infinite-width limit of a fully connected network with a proper initialization and an infinitesimal learning rate [@Jacot2018], which is referred to as the NTK limit. It is found that even if the width increases, in many cases the generalization error with an optimal learning rate does not converge to the NTK limit. In such a case, a finite-width network shows much better generalization compared with the kernel learning with the NTK. In the NTK limit, the network parameters stay close to their initial values during training, which is called the *lazy learning* [@Chizat2019], and hence the result mentioned above indicates the importance of the *feature learning*, in which network parameters change to learn relevant features.
Setting {#sec:setting}
=======
We consider a classification task with a training dataset $\mathcal{D}=\{(x_\mu,y_\mu):\mu=1,2,\dots,N\}$, where $x_\mu\in\mathbb{R}^d$ is an input data and $y_\mu\in\{1,2,\dots,K\}$ is its label. In this work, we consider the binary classification, $K=2$, unless otherwise stated.
Dataset
-------
Each input $x=(x^{(1)},x^{(2)},\dots,x^{(d)})^\mathrm{T}$ is a $d$-dimensional vector taken from i.i.d. Gaussian random variables of zero mean and unit variance, where $a^\mathrm{T}$ is the transpose of a vector $a$. For each input $x$, we assign a label $y$ according to one of the following rules.
### $k$-local label {#k-local-label .unnumbered}
We randomly fix integers $\{i_1,i_2,\dots,i_k\}$ with $1\leq i_1<i_2<\dots<i_k\leq d$. In the “$k$-local” label, the relevant feature is given by the product of the $k$ components of an input $x$, that is, the label $y$ is determined by $$y=\left\{
\begin{aligned}
&1& &\text{if }x^{(i_1)}x^{(i_2)}\dots x^{(i_k)}\geq 0; \\
&2& &\text{otherwise}.
\end{aligned}
\right.$$ This label is said to be local in the sense that $y$ is completely determined by just the $k$ components of an input $x$.[^1] For fully connected networks considered in this paper, without loss of generality, we can choose $i_1=1$, $i_2=2$,…$i_k=k$ because of the permutation symmetry with respect to indices of input vectors.
### $k$-global label {#k-global-label .unnumbered}
We again fix $1\leq i_1<i_2<\dots<i_k\leq d$. Let us define $$M=\sum_{j=1}^dx^{(j+i_1)}x^{(j+i_2)}\dots x^{(j+i_k)},$$ where the convention $x^{(d+i)}=x^{(i)}$ is used. The $k$-global label $y$ for $x$ is defined by $$y=\left\{
\begin{aligned}
&1& &\text{if }M\geq 0; \\
&2& &\text{otherwise}.
\end{aligned}
\right.$$ The relevant feature $M$ for this label is given by a uniform sum of the product of $k$ components of the input vector. Every component of $x$ contributes to this “$k$-global” label, in contrast to the $k$-local label with $k<d$.
Network architecture
--------------------
In the present work, we consider fully connected feedforward neural networks with $L$ hidden layers of width $H$. We call $L$ and $H$ the depth and the width of the network, respectively. The output of the network $f(x)$ for an input vector $x\in\mathbb{R}^d$ is determined as follows: $$\left\{
\begin{aligned}
&f(x)=z^{(L+1)}=w^{(L+1)}z^{(L)}+b^{(L+1)};\\
&z^{(l)}=\varphi\left(w^{(l)}z^{(l-1)}+b^{(l)}\right) \text{ for }l=1,2,\dots,L; \\
&z^{(0)}=x,
\end{aligned}
\right.$$ where $\varphi(x)=\max\{x,0\}$ is the component-wise ReLu activation function, $z^{(l)}$ is the output of the $l$th layer, and $$w^{(l)}\in\left\{
\begin{aligned}
&\mathbb{R}^{K\times H} \text{ for }l=L+1; \\
&\mathbb{R}^{H\times H} \text{ for }l=2,3,\dots,L; \\
&\mathbb{R}^{H\times d} \text{ for }l=1,
\end{aligned}
\right.
\qquad
b^{(l)}\in\left\{
\begin{aligned}
&\mathbb{R}^K \text{ for }l=L+1; \\
&\mathbb{R}^H \text{ for }l=1,2,\dots,L
\end{aligned}
\right.$$ are the weights and the biases, respectively. Let us denote by $w$ the set of all the weights and biases in the network. We focus on an overparameterized regime, i.e., the number of network parameters (the number of components of $w$) exceeds $N$, the number of training data points.
Supervised learning
-------------------
The network parameters $w$ are adjusted to correctly classify the training data. It is done by minimizing the softmax cross-entropy loss $L(w)$ given by $$L(w)=\frac{1}{N}\sum_{\mu=1}^N\ell\left(f(x_\mu),y_\mu\right), \quad
\ell\left(f(x),y\right)=-\ln\frac{e^{f_y(x)}}{\sum_{i=1}^Ke^{f_i(x)}}=-f_y(x)+\ln\sum_{i=1}^Ke^{f_i(x)},$$ where the $i$th component of $f(x)$ is denoted by $f_i(x)$. The main results of our paper do not change for other standard loss functions such as the mean-squared error.
The training of the network is done by the stochastic gradient descent (SGD) with learning rate $\eta$ and the mini-batch size $B$. That is, for each mini-batch $\mathcal{B}\subset\mathcal{D}$ with $|\mathcal{B}|=B$, the network parameter $w_t$ at time $t$ is updated as $$w_{t+1}=w_t-\eta\nabla_wL_B(w), \qquad L_B(w)=\frac{1}{B}\sum_{\mu\in\mathcal{B}}\ell(f(x_\mu),y_\mu).$$ Throughout the paper, we fix $B=50$. Meanwhile, we optimize $\eta>0$ before training (explain the detail later). Biases are initialized to be zero, and weights are initialized using the Glorot initialization [@Glorot2010].[^2]
The trained network classifies an input $x_\mu$ to the class $\hat{y}_\mu$ given by $\hat{y}_\mu=\operatorname*{\mathrm{arg\,max}}_{i\in\{1,2,\dots,K\}}f_i(x_\mu)$. Let us then define the training error as $$\mathcal{E}_\mathrm{train}=\frac{1}{N}\sum_{\mu=1}^N\left(1-\delta_{y_\mu,\hat{y}_\mu}\right),$$ that is the miss-classification rate for the training data $\mathcal{D}$. We train our network until $\mathcal{E}_\mathrm{train}=0$ is achieved, i.e., all the training data samples are correctly classified, which is possible in an overparameterized regime.
For a training dataset $\mathcal{D}$, we first perform the 10-fold cross validation to optimize the learning rate $\eta$ under the Bayesian optimization method [@Snoek2012], and then perform the training via the SGD by using the full training dataset. In the optimization of $\eta$, we try to minimize the miss-classification ratio for the validation data.
The generalization performance of a trained network is measured by computing the test error. We prepare the test data $\mathcal{D}_\mathrm{test}=\{(x_\mu',y_\mu'):\mu=1,2,\dots,N_\mathrm{test}\}$ independently from the training data $\mathcal{D}$. The test error $\mathcal{E}_\mathrm{test}$ is defined as the miss-classification ratio for $\mathcal{D}_\mathrm{test}$, i,.e., $$\mathcal{E}_\mathrm{test}=\frac{1}{N_\mathrm{test}}\sum_{\mu=1}^{N_\mathrm{test}}\left(1-\delta_{y_\mu',\hat{y}_\mu'}\right),$$ where $\hat{y}_\mu'=\operatorname*{\mathrm{arg\,max}}_if_i(x_\mu')$ is the prediction of our trained network. In our experiment discussed in Sec. \[sec:exp\], we fix $N_\mathrm{test}=10^5$.
Neural Tangent Kernel {#sec:NTK}
---------------------
Suppose a network of depth $L$ and width $H$ with the output $f(x,w)\in\mathbb{R}^K$. When the network is sufficiently wide and the learning rate is sufficiently small, the network parameters $w$ stay close to their randomly initialized values $w_0$ during training, and hence $f(x,w)$ is approximated by a linear function of $w-w_0$: $f(x,w)=f(x,w_0)+\nabla_wf(x,w)|_{w=w_0}\cdot(w-w_0)$. As a result, the minimization of the mean-squared error $L_\mathrm{MSE}=(1/N)\sum_{\mu=1}^N[f(x_\mu,w)-\vec{y}_\mu]^2$, where $\vec{y}\in\mathbb{R}^K$ is the one-hot representation of the label $y$, is equivalent to the kernel regression with the NTK $\Theta_{ij}^{(L)}(x,x')$ ($i,j=1,2,\dots, K$) that is defined as $$\Theta^{(L)}_{ij}(x,x')=\lim_{H\to\infty}\mathbb{E}_w\left[(\nabla_wf_i(x,w))^\mathrm{T}(\nabla_wf_j(x,w))\right],$$ where $\mathbb{E}_w$ denotes the average over random initializations of $w$ [@Jacot2018].
Let us consider a network whose biases $\{b^{(l)}\}$ and weights $\{w^{(l)}\}$ are randomly initialized as $b_i^{(l)}=\beta B_i^{(l)}$ with $B_i^{(l)}\sim\mathcal{N}(0,1)$ and $w_{ij}^{(l)}=\sqrt{2/n_{l-1}}W_{ij}^{(l)}$ with $W_{ij}^{(l)}\sim\mathcal{N}(0,1)$ for every $l$ respectively, where $n_l$ is the number of neurons in the $l$th layer, i.e., $n_0=d$, $n_1=n_2=\dots=n_L=H$. The parameter $\beta$ controls the impact of bias terms, and we set $\beta=0.1$ in our numerical experiment following Jacot et al. [@Jacot2018]. By using the ReLu activation function, we can give an explicit expression of the NTK that is suited for numerical calculations. Such formulas are given in Supplimentary Material.
It is shown that the NTK takes the form $\Theta_{ij}^{(L)}(x,x')=\delta_{i,j}\Theta^{(L)}(x,x')$, and the minimization of the mean-squared error with an infinitesimal weight decay yields the output function $$f^\mathrm{NTK}(x)=\sum_{\mu,\nu=1}^N\Theta^{(L)}(x,x_\mu)\left(K^{-1}\right)_{\mu\nu}\vec{y}_\nu,$$ where $K^{-1}$ is the inverse matrix of the Gram matrix $K_{\mu\nu}=\Theta^{(L)}(x_\mu,x_\nu)$. An input data $x$ is classified to $\hat{y}=\operatorname*{\mathrm{arg\,max}}_{i\in\{1,2,\dots,K\}}f_i^\mathrm{NTK}(x)$.
Experimental results {#sec:exp}
====================
We now present our experimental results. For each data point, the training dataset $\mathcal{D}$ is fixed and we optimize the learning rate $\eta$ via the 10-fold cross validation with the Bayesian optimization method (we used the package provided in [@Nogueira_github]). We used the optimized $\eta$ to train our network. At every 50 epochs we compute the training error $\mathcal{E}_\mathrm{train}$, and we stop the training if $\mathcal{E}_\mathrm{train}=0$. For the fixed dataset $\mathcal{D}$ and the optimized learning rate $\eta$, the training is performed 10 times and calculate the average and the standard deviation of test errors $\mathcal{E}_\mathrm{test}$.
1-local and 1-global labels
---------------------------
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\(a) 1-local \(b) 1-global
![Test error against the number of training data samples $N$ for several network architectures specified by the depth and width for (a) the 1-local label and (b) the 1-global label. Test errors calculated by the NTK of depth 1 and 7 are also plotted. Error bars are smaller than the symbols.[]{data-label="fig:1local_global"}](1local.eps "fig:"){width="0.49\linewidth"} ![Test error against the number of training data samples $N$ for several network architectures specified by the depth and width for (a) the 1-local label and (b) the 1-global label. Test errors calculated by the NTK of depth 1 and 7 are also plotted. Error bars are smaller than the symbols.[]{data-label="fig:1local_global"}](1global.eps "fig:"){width="0.49\linewidth"}
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In the 1-local and 1-global labels, the relevant feature is a linear function of the input vector. Therefore, in principle, even a linear network can correctly classify the data. Figure \[fig:1local\_global\] shows the generalization errors in nonlinear networks of the varying depth and width as well as those in the linear perceptron (the network of zero depth). The input dimension is set to be $d=1000$. We also plotted test errors calculated by the NTK, but we postpone the discussion about the NTK until Sec. \[sec:result\_NTK\].
Figure \[fig:1local\_global\] shows that in both 1-local and 1-global labels, the test error decreases with the network width, and a shallower network ($L=1$) shows better generalization compared with a deeper one ($L=7$). The linear perceptron shows the best generalization performance, which is natural because it is the simplest network that is capable of learning the relevant feature associated with the 1-local or 1-global label. Remarkably, test errors of nonlinear networks ($L=1$ and $L=7$) are not too large compared with those of the linear perceptron, although nonlinear networks are much more complex than the linear perceptron.
For a given network architecture, we do not see any important difference between the results for 1-local and 1-global labels, which would be explained by the fact that these labels are transformed to each other via the Fourier transformation of input vectors.
Opposite depth dependences for $k$-local and $k$-global labels with $k\geq 2$
-----------------------------------------------------------------------------
For $k\geq 2$, it turns out that experimental results show opposite depth dependences for $k$-local and $k$-global labels. Let us first consider $k$-local labels with $k\geq 2$. Figure \[fig:error\] (a) and (b) show test errors for varying $N$ in various networks for the 2-local and the 3-local labels, respectively. The input dimension $d$ is set to be $d=500$ in the 2-local label and $d=100$ in the 3-local label. We see that the test error strongly depends on the network depth. A deeper network ($L=7$) generalizes better than a shallower one ($L=1$). It should be noted that for $d=500$, the network of $L=1$ and $H=2000$ contains about $10^6$ trainable parameters, the number of which is much larger than that of trainable parameters ($\simeq 10^5$) in the network of $L=7$ and $H=100$. In spite of this fact, the latter outperforms the former in the 2-local label as well as in the 3-local label with large $N$, which implies that increasing the number of trainable parameters do not necessarily implies better generalization. In $k$-local labels with $k\geq 2$, the network depth is more strongly correlated to generalization compared with the network width.
From Fig. \[fig:error\] (b), it is obvious that the network of $L=7$ and $H=100$ fails to learn the 3-local label for small $N$. We also see that error bars of the test error are large in the network of $L=7$ and $H=100$. The error bar represents the variance due to initialization and training. By increasing the network width $H$, both variances and test errors decrease. This result is consistent with the recent observation in the lazy regime that increasing the network width results in better generalization because it reduces the variance due to initialization [@dAscoli2020].
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\(a) 2-local \(b) 3-local
![Test error against the number of training data samples $N$ for several network architectures specified by the depth and the width for (a) the 2-local label, (b) the 3-local label, (c) 2-global label, and (d) 3-global label. Error bars indicate the standard deviation of the test error for 10 iterations of the network initialization and the training. Test errors calculated by the NTK of the depth of 1 and 7 are also plotted.[]{data-label="fig:error"}](2local.eps "fig:"){width="0.49\linewidth"} ![Test error against the number of training data samples $N$ for several network architectures specified by the depth and the width for (a) the 2-local label, (b) the 3-local label, (c) 2-global label, and (d) 3-global label. Error bars indicate the standard deviation of the test error for 10 iterations of the network initialization and the training. Test errors calculated by the NTK of the depth of 1 and 7 are also plotted.[]{data-label="fig:error"}](3local.eps "fig:"){width="0.49\linewidth"}
\(c) 2-global \(d) 3-global
![Test error against the number of training data samples $N$ for several network architectures specified by the depth and the width for (a) the 2-local label, (b) the 3-local label, (c) 2-global label, and (d) 3-global label. Error bars indicate the standard deviation of the test error for 10 iterations of the network initialization and the training. Test errors calculated by the NTK of the depth of 1 and 7 are also plotted.[]{data-label="fig:error"}](2global.eps "fig:"){width="0.49\linewidth"} ![Test error against the number of training data samples $N$ for several network architectures specified by the depth and the width for (a) the 2-local label, (b) the 3-local label, (c) 2-global label, and (d) 3-global label. Error bars indicate the standard deviation of the test error for 10 iterations of the network initialization and the training. Test errors calculated by the NTK of the depth of 1 and 7 are also plotted.[]{data-label="fig:error"}](3global.eps "fig:"){width="0.49\linewidth"}
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Depth dependence of the test error for $N=10^4$ training samples with 2-local and 2-global labels. The dimension of input vectors is set to be $d=500$ in the 2-local label and $d=100$ in the 2-global label. The network width is fixed to be 500. An error bar indicates the standard deviation over 10 iterations of the training using the same dataset.[]{data-label="fig:depth"}](depth.eps){width="0.5\linewidth"}
Next, we consider $k$-global labels with $k=2$ and 3. The input dimension $d$ is set as $d=100$ for the 2-global label and $d=40$ for the 3-global label. We plot test errors against $N$ in Fig. \[fig:error\] for (c) the 2-global label and (d) the 3-global label. Again we find strong depth dependences, but now shallow networks ($L=1$) outperform deep ones ($L=7$), which is contrary to the results for $k$-local labels. For $L=7$, we also find strong width dependences; the test error of a wider network more quickly decreases with $N$. In particular, in the 3-global label, an improvement of the generalization with $N$ is subtle for $L=7$ and $H=100$. By increasing the width, the decrease of the test error with $N$ becomes much faster \[see the result for $L=7$ and $H=500$ in Fig. \[fig:error\] (d)\].
To see more details of the effect of depth, we also plot the depth dependence of the test error for fixed training data samples. We prepare $N=10000$ training data samples for the 2-local and 2-global labels, respectively. The input dimension is $d=500$ for the 2-local label and $d=100$ for the 2-global label. By using the prepared training data samples, networks of the depth $L$ and the width $H=500$ are trained up to $L=10$. The test errors of trained networks are shown in Fig. \[fig:depth\]. In the 2-local label, the test error decreases with $L$, whereas the test error increases with $L$ in the 2-global label. Thus, Fig. \[fig:depth\] clearly shows the opposite depth dependences for local and global labels.
Comparison between finite networks and NTKs {#sec:result_NTK}
-------------------------------------------
In Figs. \[fig:1local\_global\] and \[fig:error\], test errors calculated by using the NTK are also plotted. In the case of $k=1$ (Fig. \[fig:1local\_global\]), the generalization performance of the NTK is comparable with that of finite networks. For the 2-global label \[Fig. \[fig:error\] (c)\], the test error obtained by the NTK is comparable or lower than that of finite networks.
The crucial difference is seen in the case of $k$-local label with $k=2$ and 3 and the 3-global label. In Fig. \[fig:error\] (a) and (b), we see that the NTK almost completely fails to classify the data, although finite networks succeed in doing so. In the case of the 3-global label, the NTK of depth $L=7$ correctly classifies the data, while the NTK of depth $L=1$ fails \[see Fig. \[fig:error\] (d)\]. In those cases, the test error calculated by a finite network does not seem to converge to that obtained by the NTK as the network width increases.
The NTK has been proposed as a theoretical tool to investigate the infinite-width limit, but it should be kept in mind that the learning rate has to be sufficiently small to achieve the NTK limit [@Jacot2018; @Arora2019]. The discrepancy between a wide network and the NTK in Fig. \[fig:error\] stems from the strong learning-rate dependence of the generalization performance. In our experiment, the learning rate has been optimized by performing the 10-fold cross validation. If the optimized learning rate is not small enough for each width, the trained network may not be described by the NTK even in the infinite-width limit.
In Fig. \[fig:lr\_dep\] (a), we plot the learning-rate dependence of the test error for the 2-local label and the 2-global label in the network of the depth $L=1$ and the width $H=2000$. We observe a sharp learning-rate dependence in the case of the 2-local label in contrast to the case of the 2-global label. In Fig. \[fig:lr\_dep\] (b), we compare the learning-rate dependences of the test error for $L=1$ and $L=7$ in the case of the 3-global label (in both cases $H=2000$). We see that the learning-rate dependence for $L=1$ is much stronger than that for $L=7$, which is consistent with the fact that the NTK fails only for $L=1$. It should be noted that Fig. \[fig:lr\_dep\] (b) shows that the deep network ($L=7$) outperforms the shallow one ($L=1$) in the regime of small learning rates, while the shallow one performs better than the deep one at their optimal learning rates.
Figure \[fig:lr\_dep\] also shows that the test error for a sufficiently small learning rate approaches the one obtained by the corresponding NTK. Therefore, the regime of small learning rates is identified as a lazy learning regime, while larger learning rates correspond to a feature learning regime. Sharp learning-rate dependences found here provide theoretical and practical importance of the feature learning.
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(a) (b)
![Learning-rate dependence of the test error. (a) Numerical results for the 2-local and 2-global labels in the network with the depth of 1 and the width of 2000. (b) Numerical results for the 3-global label in the networks with the depth of 1 and 7 (the network width is set at 2000 for both cases). The dotted lines show the test error calculated by the NTK. When the learning rate is sufficiently small, the test error in a finite network approaches that of the corresponding NTK. Each data is plotted up to the maximum learning rate beyond which the zero training error is not achieved within 2500 epochs (in some cases training fails due to divergence of network parameters during the training). Error bars indicate the standard deviation over 10 iterations of the training. []{data-label="fig:lr_dep"}](lr_dep_2.eps "fig:"){width="0.49\linewidth"} ![Learning-rate dependence of the test error. (a) Numerical results for the 2-local and 2-global labels in the network with the depth of 1 and the width of 2000. (b) Numerical results for the 3-global label in the networks with the depth of 1 and 7 (the network width is set at 2000 for both cases). The dotted lines show the test error calculated by the NTK. When the learning rate is sufficiently small, the test error in a finite network approaches that of the corresponding NTK. Each data is plotted up to the maximum learning rate beyond which the zero training error is not achieved within 2500 epochs (in some cases training fails due to divergence of network parameters during the training). Error bars indicate the standard deviation over 10 iterations of the training. []{data-label="fig:lr_dep"}](lr_dep_3global.eps "fig:"){width="0.49\linewidth"}
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Conclusion
==========
In this work, we have studied the effect of increasing the depth in classification tasks. Instead of using real data, we have employed abstract setting with random inputs and simple classification rules because such a simple setup helps us understand under what situations deeper networks perform better or worse. We find that the locality of the relevant feature for a given classification rule plays a key role.
We note that the advantage of the depth in local labels is not due to high expressivity of deep networks. If a network can accurately classify the data with the $k$-local label and the input dimension $d=k$, it can *in principle* classify the data with an arbitrarily large input dimension $d\geq k$. This is because the $k$-local label depends only on the $k$ components among $d$ components. By using this fact, it is confirmed that a small network with one hidden layer of the width of about 10-100 can express the 2-local label and the 3-local label almost perfectly.[^3] In other words, learning the $k$-local label for small $k$ does not require high expressive power. Nevertheless, a deeper network outperforms a shallower one.
It is also an interesting observation that shallower networks do better than deeper ones for the $k$-global label. This result shows that the depth is not always beneficial. In future studies, we hope to investigate which properties of the data other than locality studied here result in (dis)advantage of the depth.
Broader Impact {#broader-impact .unnumbered}
==============
It is an important practical problem to find neural architecture designing principles depending on specific machine-learning tasks. Although our paper is motivated by a theoretical question, i.e., why deep networks perform so well compared with shallow networks, this work will impact the practical problem mentioned above. Our work indicates that a deeper architecture is better for local features, whereas a shallower architecture is better for global features. It is expected that local features are important in typical image classification tasks, and our work suggests that a deep architecture should be used for such a task, which is consistent with our experience. Furthermore, if we could find a transformation of local features to global ones, then even shallow networks should be able to classify the data with a great accuracy after the transformation, which is of practical merit. It would be also important to identify problems where labels are effectively global (e.g., where Fourier analysis works well, finance, weather forecasting). The present research can lead to better solutions in these fields. In short, our theoretical work suggests a guiding principle for future studies on architecture designing principles.
Explicit expression of the NTK
==============================
We consider a network whose biases $\{b^{(l)}\}$ and weights $\{w^{(l)}\}$ are randomly initialized as $b_i^{(l)}=\beta B_i^{(l)}$ with $B_i^{(l)}\sim\mathcal{N}(0,1)$ and $w_{ij}^{(l)}=\sqrt{2/n_{l-1}}W_{ij}^{(l)}$ with $W_{ij}^{(l)}\sim\mathcal{N}(0,1)$ for every $l$, where $n_l$ is the number of neurons in the $l$th layer, i.e., $n_0=d$, $n_1=n_2=\dots=n_L=H$. In the infinite-width limit $H\to\infty$, the pre-activation $f^{(l)}=w^{(l)}z^{(l-1)}+b^{(l)}$ at every hidden layer tends to an i.i.d. Gaussian process with covariance $\Sigma^{(l-1)}:\mathbb{R}^d\times\mathbb{R}^d\to\mathbb{R}$ that is defined recursively as $$\left\{
\begin{split}
&\Sigma^{(0)}(x,x')=\frac{x^\mathrm{T}x'}{d}+\beta^2; \\
&\Lambda^{(l)}(x,x')=\begin{pmatrix}
\Sigma^{(l-1)}(x,x) & \Sigma^{(l-1)}(x,x') \\ \Sigma^{(l-1)}(x',x) & \Sigma^{(l-1)}(x',x')
\end{pmatrix}; \\
&\Sigma^{(l)}(x,x')=2\mathbb{E}_{(u,v)\sim\mathcal{N}(0,\Lambda^{(l)})}\left[\varphi(u)\varphi(v)\right]+\beta^2
\end{split}
\right.$$ for $l=1,2,\dots, L$. We also define $$\dot{\Sigma}^{(l)}(x,x')=2\mathbb{E}_{(u,v)\sim\mathcal{N}(0,\Lambda^{(l)})}\left[\dot{\varphi}(u)\dot{\varphi}(v)\right],$$ where $\dot{\varphi}$ is the derivative of $\varphi$. The NTK is then expressed as $\Theta_{ij}^{(L)}(x,x')=\delta_{i,j}\Theta^{(L)}(x,x')$, where $$\Theta^{(L)}(x,x')=\sum_{l=1}^{L+1}\left(\Sigma^{(l-1)}(x,x')\prod_{l'=l}^{L+1}\dot{\Sigma}^{(l')}(x,x')\right).
\label{eq:NTK}$$ The derivation of this formula is given by Arora et al. [@Arora2019].
Using the ReLu activation function $\varphi(u)=\max\{u,0\}$, we can further calculate $\Sigma^{(l)}(x,x')$ and $\dot{\Sigma}^{(l)}(x,x')$, obtaining $$\Sigma^{(l)}(x,x')=\frac{\sqrt{\det\Lambda^{(l)}}}{\pi}+\frac{\Sigma^{(l-1)}(x,x')}{\pi}\left[\frac{\pi}{2}+\arctan\left(\frac{\Sigma^{(l-1)}(x,x')}{\sqrt{\det\Lambda^{(l)}}}\right)\right]+\beta^2
\label{eq:Sigma}$$ and $$\dot{\Sigma}^{(l)}(x,x')=\frac{1}{2}\left[1+\frac{2}{\pi}\arctan\left(\frac{\Sigma^{(l-1)}(x,x')}{\sqrt{\det\Lambda^{(l)}}}\right)\right].
\label{eq:dot_Sigma}$$ For $x=x'$, we obtain $\Sigma^{(l)}(x,x)=\Sigma^{(0)}(x,x)+l\beta^2=\|x\|^2/d+(l+1)\beta^2$. By solving eqs. (\[eq:Sigma\]) and (\[eq:dot\_Sigma\]) iteratively, the NTK in eq. (\[eq:NTK\]) is obtained.[^4]
[^1]: The locality here does not necessarily imply that $k$ points $i_1,i_2,\dots,i_k$ are spatially close to each other. Such a use of the terminology “$k$-local” has been found in the field of quantum computation [@Kempe2006].
[^2]: We also tried the He initialization [@He2015] and confirmed that results are similar to the ones obtained by the Glorot initialization, in particular when input vectors are normalized as $\|x\|=1$.
[^3]: This fact does not mean at all that such a small network can *actually* learn the local label for large $d$ by a gradient-based algorithm.
[^4]: When $\beta=0$ (no bias), the equations are further simplified; $\Sigma^{(l)}=\frac{\|x\|\|x'\|}{d}\cos\theta^{(l)}$ and $\dot{\Sigma}^{(l)}=1-\frac{\theta^{(l-1)}}{\pi}$, where $\theta^{(0)}\in[0,\pi]$ is the angle between $x$ and $x'$, and $\theta^{(l)}$ is iteratively determined by $$\cos\theta^{(l)}=\frac{1}{\pi}\left[\sin\theta^{(l-1)}+(\pi-\theta^{(l-1)})\cos\theta^{(l-1)}\right].$$
|
---
abstract: 'We study theoretically the effect of the giant resonance in Xe on the phase difference between the consecutive high order resonantly-enhanced harmonics and calculate the duration of the attosecond pulses produced by these harmonics. For certain conditions resonantly-induced dephasing compensates the phase difference which is intrinsic for the off-resonance harmonics. We find these conditions analytically and compare them with the numerical results. This harmonic synchronization allows attosecond pulse shortening in conjunction with the resonance-induced intensity increase by more than an order of magnitude; the latter enhancement relaxes the requirements for the UV filtering needed for the attosecond pulse production. Using a two-color driving field allows further increase of the intensity. In particular, a caustic-like feature in the harmonic spectrum leads to the generation efficiency growth up to two orders of magnitude, however accompanied by an elongation of the XUV pulse.'
author:
- 'V.V. Strelkov[^1]'
title: 'Attosecond pulse production using resonantly-enhanced high-order harmonics'
---
[ Attosecond pulse production using high order harmonics generated by intense laser field [@Paul; @Tzallas] is essentially based on the phase-locking of the harmonics. This phase-locking is well understood [@Antoine; @Salieres2001] for the case when there are no resonances affecting the process. However, recently much attention has been paid to the role of resonances in high harmonic generation (HHG) in gases [@Gilbertson; @Xe-Kr; @caustic2; @Rothhardt] and plasma plumes [@Ganeev0; @Haessler-atto; @Rosenthal] (for a review of earlier studies see also [@Ganeev1; @Ganeev2]). It was shown that when the high harmonic frequency is close to the transition to an excited quasi-stable state of the generating particle the harmonic can be much more intense than the off-resonant ones. For the HHG in plasma plumes such enhancement can be as high as an order of magnitude of even more.]{} The XUV generation efficiency enhancement due to the giant resonance in Xe was predicted in [@Frolov] and observed in [@Xe-Kr; @caustic2]. Namely, the XUV near 100 eV in the spectral region of about 20 eV is more intense than the lower-frequency XUV, and the enhancement near the center of the resonance is approximately an order of magnitude.
The broadband resonant enhancement potentially allows generating attosecond pulses using resonant harmonics. This approach is interesting not only because of the higher generation efficiency of the resonant HH, but also because it essentially reduces the requirements for harmonic filtering (the resonant region is naturally standing out). However, the phase-locking of resonant HH differs from the one of the non-resonant HH [@Haessler-atto], so the attosecond pulse production in the former case [is not straightforward]{}. In this Rapid Communication we investigate this aspect of resonant HHG both numerically and analytically. We study the effect of the resonance on the phase difference between the neighbor harmonics and calculate the duration of the attosecond pulses produced by resonant harmonics.
The time-dependent three-dimensional Schrödinger equation (TDSE) is solved numerically for a single-active electron atom in an external laser field. The method of the numerical TDSE solution is described in [@JPB_num].The model atomic potential is (atomic units are used throughout): V(r)= - a\_0 ( -)+a\_1 ( -)
The first term is the binding potential of the atomic core, and the second one is the barrier providing a bound quasi-static state with positive energy. The potential is similar to the one used earlier in the resonant HHG calculations [@Strelkov_4step; @Tudorovskaya; @HHG_Fano]. Moreover, a double-barrier effective potential was found in [@Kapoor] describing autoionization in time-dependent density-functional theory.
In the potentials that we used in [@Strelkov_4step; @HHG_Fano] the first term is a soft-Coulomb potential, whereas in the potential (\[model\_potential\]) it is Gaussian. Such potential does not have the Coulomb ’tails’ and thus does not support Rydberg states; however, it provides more freedom to simulate the properties of the desired atom. Choosing the parameters $a_0$, $a_1$, $b_0$, $b_1$, and $r_0$ of the model potential we reproduce the ionization energy of Xe atom, [frequency and width of the giant resonance so that the frequency and width of the resonantly enhanced region in the calculated HH spectrum is close to those observed ]{} in Ref. [@Xe-Kr]. The parameters of the potential used in our calculations are $a_0=7.1$, $b_0=1.0$, $a_1=4.5$, $b_1=0.5$ and $r_0=1.23$ a.u.. Throughout this paper we simulate only the shortest electronic trajectory [(if not specially stated otherwise)]{}, suppressing the others with properly defined absorbing region in the numerical box, as it was done in [@harmonic_ellipticity_PRA].
![(color online) Emission time calculated via numerical TDSE solution (symbols) and the estimates of this time as a classical return time of the electron in the simple-man model (solid lines). The driving laser intensities and wavelengths are shown in the graph; the two-color field is given by eq. (\[two-color\]) with $\alpha=1$ and $\phi=\pi/2$. The inset shows harmonic intensities near the resonance.[]{data-label="fig1"}](Fig1_new_with_inset.pdf){width="0.8\columnwidth"}
The laser field intensity is switched on smoothly during 4 optical cycles, then it is constant during 4 cycles, and then decreases during 4 cycles; the shape of the laser pulse is described in [@Strelkov_2006]. [We are using either a single-color driving field or a two-color one. The two-color field is given by ]{} E(t)=E\_0 f(t) \[(-i \_l t)+ (-i 2 \_l t +i )\]+ c.c.
We calculate the harmonic spectrum and find the spectral phase differences between the consecutive harmonics. Then we calculate the emission times $t_e (q \omega_l) =(\varphi_{q+1}- \varphi_{q-1})/(2 \omega_{l})$, where $q$ is an even number, $\varphi_{q \pm 1}$ are the harmonic spectral phases; $t_e$ characterizes the time instant when an attosecond pulse formed by a group of harmonics with the central frequency $q \omega_l$ is emitted [@Mairesse_2003]. The results are shown in Fig \[fig1\] for different laser intensities and wavelengths together with the classical electronic return times $t_r (q \omega_l)$ calculated within the simple-man approach [@simple-man1; @simple-man2]. Times $t_e$ and $t_r$ in general are close to each other [@Antoine]; the agreement is very good under the conditions of our calculations which are well within the tunneling regime of ionization. However, the pronounced deviation from the classical prediction can be seen in the cut-off region and in the resonant region. The phase-locking of the harmonics near the cut-off was studied recently in details in [@Khokhlova_2016]; in this paper we will study the harmonic phase-locking in the resonant region.
![(color online) Close-up of the harmonic emission times near the resonance. The fields’ parameters are the same as in Fig. \[fig1\].[]{data-label="fig2"}](Fig2b_new.pdf){width="0.7\columnwidth"}
The inset in Fig. \[fig1\] shows the harmonic spectrum near the resonance. The group of harmonics around approx. 96 eV are enhanced, and the harmonic intensity in the center of the resonance is about an order of magnitude higher than far from it. The width of the resonantly-enhanced harmonic group is 17 eV (FWHM).
Fig. \[fig2\] shows the emission times for these harmonics. To calculate these emission times we used harmonic phases averaged over 10 laser intensities in the vicinity (within $\pm 2 \%$) of the intensity presented in the figure; this is done to reduce the numerical noise. We can see that the emission time is affected by the resonance: the resonant harmonics are emitted [*later*]{} than they would be emitted in the absence of the resonance. This result is in agreement with the published experimental results for HHG in Sn$^+$ [@Haessler-atto], as well as analytical and numerical studies [@Tudorovskaya; @HHG_Fano]. The found delay time for the harmonics near the center of the resonant line (68 as) is close to the lifetime of the quasi-stable state (77 as). This result can be well understood within the four-step model of the resonant HHG [@Strelkov_4step]: the resonant XUV emission is delayed with respect to the non-resonant one, and the delay time is the time which the system stays in the quasi-stable state after rescattering.
The resonant-induced delay of the XUV emission smoothly decreases with the increase of the detuning from the resonance. So, in the spectral region above the resonance this ’resonant attochirp’ can compensate the usual ’free-motion attochirp’ (the one caused by the free electronic motion before rescattering). Let us estimate both these chirps and their influence on the attosecond pulse duration.
To do this we consider the chirped Gaussian attopulse:
[l]{} F(t)=(-i t) \_[-]{}\^[+]{} ( -2 (2) ( )\^2 )\
( i ’\^2 ) {-i ’ (t-t\_r())} d ’
where $\Omega$ is the central frequency of the pulse and $t_r(\Omega)$ is the emission time of the pulse. Let us assume that the chirp of the pulse is only due to the variation of the emission frequency described by the simple-man model (below we denote this chirp as the ’free-motion-induced attochirp’). Thus the derivative of the spectral phase ($\varphi \equiv \frac{K}{2} \omega'^2+t_r(\Omega) \omega' $) over the frequency $\omega' $ is the classical electronic return time $t_r(\omega')$. So $K=\partial t_r / \partial \omega'$. From Fig. \[fig1\] we can see that $t_r$ is an almost linear function of $\omega'$ except for the lowest and the highest part of the plateau. This linear approximation can be found from the solution of the Newton’s equation for the electron in the simple-man model. We find that approximately K=1/(2 \_l U\_p) where $U_p$ is the ponderomotive energy. The duration of the chirped pulse (\[pulse\]) depends on its spectral width $ \Delta \omega$. The shortest duration is achieved when $
\Delta \omega = \sqrt{4 \ln(2)/K}
$. Substituting (\[K\]) in the latter equation we find that = 2 The duration (FWHM of intensity) of this pulse is = 2 Estimates (\[Dw1\]) and (\[duration\]) agree very well with the numerical TDSE calculations for the off-resonant harmonics. Similar estimate of the shortest attosecond pulse duration was found in [@Platonenko_1997]. Note that equation (\[duration\]) shows that $\tau \propto \sqrt{\omega_l /I}$ where $I$ is the laser intensity. [ Since the maximum laser intensity is practically limited by the target ionization, this equation shows that the minimum attopulse duration decreases with the laser frequency decrease.]{}
The ’resonantly-induced attochirp’ can be estimated taking into account the delay in the resonant XUV emission. As we discussed above, this delay is the lifetime of the resonance (denoted below as $\Delta t $) for the XUV in the center of the resonance, and it vanishes within the width of the resonance $\Gamma$. So the resonantly-induced attochirp is $K_{res}= - \Delta t / \Gamma$. Having in mind that $\Delta t=1/\Gamma$ we find that $
K_{res}=-\Gamma^{-2}
$. From this equation and equation (\[K\]) we find that $K=-K_{res}$ for =
![(color online). The attosecond pulses calculated via equation (\[attosecond\_pulses\]) using resonantly-enhanced harmonics below the center of the resonance (red lines with triangles), above the center of the resonance (blue lines with diamonds), all resonantly-enhanced harmonics (dashed black line) and the group of off-resonant harmonics chosen to minimize the attopulse duration (violet line with circles), the latter attopulse is multiplied by 10; see text for more details. The inset shows the harmonic spectrum; the spectral regions used to calculate the attosecond pulses are highlighted with the corresponding colors. The laser wavelength is 1800 nm and the intensity is $2 \times 10^{14}$ W/cm$^2$.[]{data-label="fig3"}](Fig3a_new.pdf){width="0.8\columnwidth"}
![(color online). The attosecond pulses formed by all harmonics higher than 60 eV (thin black line) and higher than 190 eV (thick navy line); the latter is multiplied by 10. The instantaneous strength of the laser field is shown with dashed line. The laser parameters are the same as in Fig. \[fig3\][]{data-label="fig4"}](Fig3b_new.pdf){width="0.7\columnwidth"}
The conditions of our calculations [for the single-color driving field]{} were chosen so that the latter equation is approximately satisfied: we can see that in Fig. \[fig2\] the free-motion-induced attochirp is compensated with the resonantly-induced one, so the group of harmonics above the central frequency of the resonance have approximately the same phases. [In contrast to this, the parameters of the two-color field used in our calculations lead to smaller free-motion induced attochirp, so the resonantly-induced one dominates.]{}
In Figs. \[fig3\] and \[fig4\] we show the attosecond pulses calculated using XUV from different spectral regions. Namely, using the complex amplitudes of the microscopic response $d(\omega)$ calculated via numerical TDSE solution we find the XUV intensity: I(t)= | \_[=-]{}\^ M() d() (-i t) d | \^2 where the used spectrum mask $M(\omega)$ is either a Gaussian $M_{G}(\omega)=\exp\{-2 \ln(2)((\omega-\Omega)/\Delta \omega)^2\}$ or a step-like function: $M_{step}(\omega)=\theta(\omega-\omega_{low})\theta(\omega_{high}-\omega)$. In Fig. \[fig3\] we present the attopulses formed by resonant harmonics below the resonance (calculated using $M_{step}$ with $\omega_{low}=60$eV and $\omega_{high}=96$eV), above it ($\omega_{low}=96$eV and $\omega_{high}=130$eV), and all the resonant harmonics ($\omega_{low}=60$eV and $\omega_{high}=130$eV). We can see that the attosecond pulse formed by the harmonics above the resonance is much shorter than the one formed by those below the resonance. This is because the above-resonant harmonics are in phase, whereas those below the resonance have significant phase differences (see Fig. \[fig2\]), as it was discussed above. In the same figure we show the attopulse formed by the off-resonance harmonics calculated using $M_{G}$ with the central frequency $\Omega=155$ eV and the width $\Delta \omega$=15.2 eV. The latter is found numerically to minimize the pulse duration; the found width and duration are very close to the predictions of eq. (\[Dw1\]) and (\[duration\]), respectively. We can see that this pulse is slightly longer and much weaker than the one formed by the above-resonance harmonics. Moreover, if all the resonant harmonics are used to produce the attopulse, its duration does not increase dramatically, see dashed line in the Fig. \[fig3\].
[Metal foils or multilayer mirrors are usually applied as spectral filters [@Paul; @Tzallas; @filter; @Goulielmakis] to obtain attosecond pulses; such filter, in particular, can transmit well all UV higher than certain frequency. To simulate the attosecond pulses obtained with such filter we use in equation (\[attosecond\_pulses\]) the step-like mask with $\omega_{high}$ which is much higher than the cut-off frequency. The results obtained using $\omega_{low}$ well-below and well-above the resonance are shown in Fig. \[fig4\].]{} Again, in the latter case this number is chosen to minimize the duration of the attosecond pulse. In spite of this optimization, we can see that this attopulse using off-resonant harmonics is longer and more than an order of magnitude weaker than the resonant one. Note that the parameters of the attosecond pulse formed by the resonant harmonics are not very sensitive to $\omega_{low}$ as long as it is well-below resonance; this is natural because the off-resonant harmonics are much weaker than the resonant ones. This means that practically there is much freedom in choosing such filter as long as it transmits the resonant harmonics. Moreover, if the absorption edge of the filter is far from the resonance, the filter dispersion (which is usually pronounced only in the vicinity of the absorption edge) would not affect the attosecond pulse duration. So the duration of 165 as found in our calculations is close to the one which can be experimentally obtained using harmonics enhanced by the giant resonance in Xe.
![(color online). Harmonic intensities generated in single-color (blue circles) and two-color (green diamonds) fields, see text for more details, and the classical return time for the two-color case. The inset shows the driving field strength (dotted line) and the resonant XUV pulse intensity (solid line) as functions of time.[]{data-label="fig5"}](Fig4.pdf){width="0.8\columnwidth"}
[Making similar calculations for the HHG by the two-color field with the parameters $\alpha$ and $\phi$ considered above we find that the attopulse formed by above-resonant XUV is longer, and the one formed by below-resonant XUV is shorter than in the single-color field. This is the result of the attochirp behavior shown in Fig. \[fig2\]: the absolute value of the total attochirp above the resonance is higher in the two-color case, below the resonance the relation is reversed. Note that this leads to smaller dephasing between the harmonics near the very center of the resonance. Since these harmonics are the most intense ones, this results in even shorter attosecond pulse than in the single-color field. Namely, the attosecond pulses formed by all XUV with frequency higher than $\omega_{low}=60$eV can be as short as 105 fs in the two-color case.]{}
[As it was shown both theoretically [@JPB_num] and experimentally [@Colosimo; @Shiner] the harmonic yield rapidly decreases with the decrease of the driving wavelength. So the perspective of the generation efficiency increase using two-color field [@Eichmann; @Kim; @Emelina] is especially important for the middle-infrared drivers considered here. To achieve maximum resonant harmonic intensity in the two-color field we chose the parameters $\alpha$ and $\phi$ of the latter so that the resonant frequency coincides with the caustic-like feature [@caustic1; @caustic2] in the dependence of the returning electron energy on the return time. Due to this feature almost all detached electrons return back with the energy close to the one of the quasi-stable state. This leads to a further increase of the resonant harmonic generation efficiency. Fig. \[fig5\] shows the results calculated for the fundamental intensity $2.2 \times 10^{14}$ W/cm$^2$, $\alpha=1/3$, $\phi=1.0$ (here we take into account all the electronic trajectories in the TDSE). We compare the HHG efficiency in the two-color field with that in the single-color field having the intensity equal to the sum of the intensities of the fundamental and the second harmonic in the two-color case. Fig. \[fig5\] shows that the gain from using the two-color field with the proper parameters can be about two orders of magnitude. Together with the resonance-induced enhancement this provides the level of conversion which can be interesting for using such harmonics as an efficient source of coherent XUV in the range of 100 eV. However, the inset in Fig. \[fig5\] shows that the generation efficiency increase using such caustic-like feature leads to a loss of the attosecond nature of the emitted XUV: the calculated XUV pulse is of approximately 2 fs duration. This value is close to the time interval when the classical electrons return with energy close to the resonant one.]{}
[Thus in this paper we find conditions for which the free-motion-induced attochirp can be compensated by the resonantly-induced attochirp, leading to phase synchronization of a group of resonant harmonics. It is shown that attopulses with duration of 165 as can be obtained using resonantly-enhanced harmonics generated in Xe. This duration is smaller than the minimal duration of the attosecond pulse formed by the off-resonant harmonics; it can be further reduced down to almost hundred attoseconds using the two-color driver. Resonant HHG enhancement leads to an increase of the attopulse intensity by more than an order of magnitude and relaxes the requirements for the XUV filtering: only harmonics much lower than the resonance should be suppressed by the filter. Using two-color field with specific parameters providing ’caustic-induced’ enhancement of the resonant harmonics provides further (almost two-orders of magnitude) increase of the XUV intensity at the cost of increasing the pulse duration to above 1 fs. ]{}
[Note that the giant resonances are observed also in other atoms, ions and molecules. The detailed investigation of their applicability for attosecond pulse production can be a natural development of the present study. However, our findings, in particular, the estimate of the laser field parameters required for the phase synchronization of the resonant harmonics given by eq. (\[result\]) should be applicable for other resonances as well.]{}
This study was funded by RSF (grant N 16-12-10279).
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[^1]: strelkov.v@gmail.com
|
---
abstract: |
The folk result in Kyle-Back models states that the value function of the insider remains unchanged when her admissible strategies are restricted to absolutely continuous ones. In this paper we show that, for a large class of pricing rules used in current literature, the value function of the insider can be finite when her strategies are restricted to be absolutely continuous and infinite when this restriction is not imposed. This implies that the folk result doesn’t hold for those pricing rules and that they are not consistent with equilibrium. We derive the necessary conditions for a pricing rule to be consistent with equilibrium and prove that, when a pricing rule satisfies these necessary conditions, the insider’s optimal strategy is absolutely continuous, thus obtaining the classical result in a more general setting.
This, furthermore, allows us to justify the standard assumption of absolute continuity of insider’s strategies since one can construct a pricing rule satisfying the necessary conditions derived in the paper that yield the same price process as the pricing rules employed in the modern literature when insider’s strategies are absolutely continuous.
address:
- 'Department of Statistics, London School of Economics and Political Science, 10 Houghton st, London, WC2A 2AE, UK'
- 'Department of Mathematics, London School of Economics and Political Science, 10 Houghton st, London, WC2A 2AE, UK'
author:
- Umut Çetin
- Albina Danilova
bibliography:
- 'ref.bib'
title: 'On pricing rules and optimal strategies in general Kyle-Back models'
---
Introduction
============
The canonical model of markets with asymmetric information is due to Kyle [@Kyle], where he studies a market for a single risky asset whose price is determined in equilibrium. Kyle’s model is set up in discrete time and it has been extended to a continuous time framework by Back [@Back] and is commonly referred to as a Kyle-Back model in subsequent literature. In this type of models there are typically three types of agents participating in the market: non-strategic noise traders, a strategic risk-neutral informed trader (insider) with private information regarding the future value of the asset, and a number of risk-neutral market makers competing for the total demand. The goal of market makers is to set the [*pricing rule*]{} so that the resulting price is rational, which in particular entails finite expected profit for the insider trading at these prices. On the other hand, the objective of the insider is to maximise her expected final wealth given the pricing rule set by the market makers. Thus, this type of modelling can be viewed as a game with asymmetric information between the market makers and the insider and the goal is to find an equilibrium of this game.
Apart from extending the Kyle’s model to continuous time the most important contribution of Back [@Back] was to establish that, when the market maker sets the price to be a harmonic function of total order, the insider’s value function is finite and the optimal control solving the insider’s optimisation problem is absolutely continuous. This implies that the set of admissible controls of the insider can be reduced to absolutely continuous ones. This restriction significantly simplifies the problem of finding an equilibrium since it allows one to employ a PDE approach to the insider’s optimal control problem that yields a system of PDEs that the value function of the insider and the pricing rule of the market maker have to satisfy in equilibrium.
Back’s result was the original justification for restricting the set of admissible controls of the insider to absolutely continuous ones and this restriction is now standard in the asymmetric information literature (see, e.g., [@Back-Baruch], [@Cho], [@CDF], [@CCD], [@C18], [@CFNO], [@CNF], and [@MSZ]). In this paper we show that if we extend the class of pricing rules beyond harmonic functions of total order to include ones used in the recent literature, e.g. in the papers cited above, then the value function of the insider is infinite and her optimal control is not absolutely continuous. In particular, this is true for the pricing rules in the aforementioned papers. Since the value function of the insider is infinite, those pricing rules can not be equilibrium pricing rules.
However, since the infinite profit is due to penalty imposed on discontinuous strategies or strategies with additional martingale part being insufficient to offset the profit made due to private information, one can modify this penalty to ensure optimality of absolutely continuous strategies, while warranting the same price process when insider’s strategy is absolutely continuous. This is precisely what we do in this paper by establishing a class of pricing rules that yield the same price process as the models cited before when the trading strategy of the insider is absolutely continuous but produce a finite value for the insider when her strategies are allowed to have jumps or martingale parts. We show that for this class of pricing rules the set of admissible controls of the insider can be reduced to absolutely continuous ones.
To the best of our knowledge, this paper is the first one to identify this class of pricing rules consistent with an equilibrium. Moreover, it is also the first one since [@Back] that justifies the restriction of insider’s controls to absolutely continuous ones in a general setting. In this it closes the gap between the assumption of absolutely continuous controls and its justification in the modern literature that employs more general pricing rules.
The paper is structured as follows. In Section 2 we describe the model and introduce the set of pricing rules that generalise the pricing rules employed in the current literature. In Section 3 we analyse the optimisation problem of the insider and establish a subset of these pricing rules that yield a finite value to this problem. In particular, Theorem \[t:jmprice\] derives the necessary conditions on the pricing rule that ensure that the insider cannot achieve infinite profits by employing discontinuous strategies and/or strategies with a martingale part. Theorem \[t:gzero\] establishes a PDE condition on the pricing rule that is necessary for the existence of equilibrium. In Section \[s:abs\] we demonstrate that, under the conditions for the pricing rule derived in Section 3, the restriction of admissible controls to absolutely continuous ones produces the same value function. Moreover, as a by-product we obtain the familiar sufficient conditions on the pricing rule and the trading strategy in order for the equilibrium to exist.
Model setup
===========
As in [@Back] we will assume that the trading will take place over the time interval $[0,1]$. Let $(\Omega , \cG , (\cG_t)_{t \in [0,1]} , \bbQ)$ be a filtered probability space satisfying the usual conditions, The time-1 value of the traded asset is given by $f(Z_1)$, which will become public knowledge at $t=1$ to all market participants, where $Z$ is a continuous and adapted process, and $f$ is a measurable increasing function.
Three types of agents trade in the market. They differ in their information sets and objectives as follows.
- *Noise/liquidity traders* trade for liquidity reasons, and their total demand at time $t$ is given by a standard $(\cG_t)$-Brownian motion $
B$ independent of $Z$.
- *Market makers* only observe the total demand $$Y=\theta+B,$$ where $\theta$ is the demand process of the informed trader. The admissibility condition imposed later on $\theta$ will entail in particular that $Y$ is a semimartingale.
They set the price of the risky asset via a [*Bertrand competition*]{} and clear the market. Similar to [@Back-Baruch] we assume that the market makers set the price as a function of weighted total order process at time $t$, i.e. we consider pricing functionals $
S\left( Y_{[0,t]},t\right) $ of the following form $$\label{mm:e:rule_mm}
S\left( Y_{[0,t]},t\right) = H\left(t, X_t\right), \qquad \forall t\in [0,1)$$ where $X$ is adapted to the filtration generated by $B$ and $Z$ and is the unique strong solution of a certain SDE whose coefficients and drivers are constructed by the market makers as made precise in Definition \[mm:d:prule\]. Moreover, a pricing rule has to be admissible in the sense of Definition \[mm:d:prule\], which will entail $S$ being a semimartingale.
- *The informed investor* observes the price process $
S_{t}=H\left(t, X_t\right)$ and her private signal $Z$. Since she is risk-neutral, her objective is to maximize the expected final wealth, i.e. &&\_E\^[0,z]{}, \[ins\_obj\]\
W\_[1]{}\^[ ]{}&=& (f(Z\_[1]{})-S\_[1-]{})\_[1-]{}+\_[0]{}\^[1-]{}\_[s-]{}dS\_[s]{}. \[mm:eq:insW\] In above $\mathcal{A}$ is the set of admissible trading strategies for the given pricing rule[^1] observable by the insider, which will be defined in Definition \[mm:d:iadm\]. Moreover, $E^{0,z}$ is the expectation with respect to $P^{0,z}$, which is the regular conditional distribution of $(X_s, Z_s; s\leq 1)$ given $X_0=0$ and $Z_0=z$, which exists due to Theorem 44.3 in [@Bauer].
Thus, the insider maximises the expected value of her final wealth $W_{1}^{\theta }$, where the first term on the right hand side of equation ( \[ins\_obj\]) is the contribution to the final wealth due to a potential differential between the market price and the fundamental value at the time of information release, and the second term is the contribution to the final wealth coming from the trading activity.
Given the above market structure, we can now precisely define the filtrations of the market makers and of the informed trader. As we shall require them to satisfy the usual conditions, we first define the probability measures that will be used in the completion of their filtrations.
First define $\cF:=\sigma(B_t,Z_t; t \leq 1)$ and let $Q^{0,z}$ be the regular conditional distribution of $(B,Z)$ given $B_0=0$ and $Z_0=z$. Observe that any $P^{0,z}$-null set is also $Q^{0,z}$-null in view of the assumption on $X$. Due to the measurability of regular conditional distributions one can define the probability measure $\bbP$ on $(\Om, \cF)$ by \[mm:d:bbP\] (E)=\_ Q\^[0,z]{}(E) (Z\_0dz), for any $E \in \cF$.
While $Q^{0,z}$ is how the informed trader assign likelihood to the events generated by $B$ and $Z$, $\bbP$ is the probability distribution of the market makers who do not observe $Z_0$ exactly. Thus, the market makers’ filtration, denoted by $\cF^M$, will be the right-continuous augmentation with the $\bbP$-null sets of the filtration generated by $Y$. In particular $\cF^M$ satisfies the usual conditions.
On the other hand, since the informed trader knows the value of $Z_0$ perfectly, it is plausible to assume that her filtration is augmented with the $Q^{0,z}$-null sets. However, this will make the modelling cumbersome since the filtration will have an extra dependence on the value of $Z_0$ purely for technical reasons. Another natural choice is to consider the null sets that belong to every $Q^{0,z}$, i.e. the sets that are elements of the following \[mm:e:nullI\] \^I:={E: Q\^[0,z]{}(E)=0, z}. These null sets will correspond to the [*a priori*]{} beliefs that the informed trader has about the model before she is given the private information about $Z_0$ and, thus, can be used as a good benchmark for comparison. Therefore we assume that the informed trader’s filtration, denoted by $\cF^I$, is the right continuous augmentation of the filtration generated by $S$ and $Z$ with the sets of $\cN^I$. Note that the resulting filtration is [*not*]{} complete.
We are finally in a position to give a rigorous definition of the rational expectations equilibrium of this market, i.e. a pair consisting of an *admissible* pricing rule and an *admissible* trading strategy such that: *a)* given the pricing rule the trading strategy is optimal, *b)* given the trading strategy, the pricing rule is [*rational*]{} in the following sense: \[mm:d:mm\_obj\] H(t,X\_t)=S\_t=, where $\bbE$ corresponds to the expectation operator under $\bbP$. To formalize this definition of equilibrium, we first define the sets of admissible pricing rules and trading strategies.
\[mm:d:prule\] An [*admissible pricing rule*]{} is any pair $(H,w,c,j)$ fulfilling the following conditions:
1. $w : [0,1]\times \bbR \to (0,\infty)$ is a function in $C^{1,2}([0,1] \times \bbR)$;
2. Given a Brownian motion, $\beta$, on some filtered probability space, there exists a unique strong solution to $$d\tilde{X}_t=w(t,\tilde{X}_t)d\beta_t, \qquad \tilde{X}_0=0.$$
3. $H \in C^{1,2}([0,1) \times \bbR)$.
4. $x \mapsto H (t,x)$ is strictly increasing for every $t\in [0,1)$;
5. $c:[0,1]\times \bbR \to \bbR$ is locally Lipschitz;
6. $j:[0,1]\times \bbR \times\bbR \to \bbR$ is continuous.
Suppose that $(H,w,c,j)$ is an admissible pricing rule and the market makers face the total demand, $Y=B+\theta$, where $\theta$ is an admissible trading strategy in the sense of Definition \[mm:d:iadm\]. Then the price set by the market makers is $S_t=H(t,X_t)$, where $X$ is the unique strong solution[^2] of $$\label{mm:eq:signal_mm1}
dX_t = w (t,X_{t-}) dY^c_t+dC_t+ J_t ,\quad X_0 =0,$$ over the time interval $[0,1]$ on $(\Om, \cF, (\cF^M_t), \bbP)$. In above dC\_t&=&(+c(t,X\_[t-]{}))w(t,X\_[t-]{})(d\[Y,Y\]\^c\_t-dt),\
J\_t&=&K\_w\^[-1]{}(t,j(t,X\_[t-]{},Y\_t)+K\_w(t,X\_[t-]{})+Y\_t)-X\_[t-]{},\
K\_w(t,x)&=&\_0\^xdy + \_0\^t w\_x(s,0)ds. \[mm:e:Kwdef\]
Note that in view of Definition \[mm:d:prule\] the set of admissible $\theta$ for which there exists a strong solution to (\[mm:eq:signal\_mm1\]) is not empty.
Moreover, if $\theta$ is absolutely continuous, $dX_t=w(t,X_t)dY_t$ and the price set by the market makers agrees with the one set in the standard literature. That is, the choice of $c$ and $j$ does not affect the market price if the insider’s trading strategy is restricted to be absolutely continuous. This implies that (\[mm:eq:signal\_mm1\]) defines a set of pricing rules for general strategies of the insider that are consistent with the ones used in the literature under the assumption that the insider is only allowed to follow absolutely continuous strategies. Thus, if one can identify the functions $c$ and $j$ for which the optimal strategy of the insider is absolutely continuous, one recovers the equilibria obtained in the previous studies with the modification of the pricing rule given by those $c$ and $j$.
\[mm:r:Kwubdd\] Observe that the existence of a unique strong solution in Definition \[mm:d:prule\] implies $\min\{\bbP(K_w(t,\tilde{X}_t)>y), \bbP(K_w(t,\tilde{X}_t)<-y)\}>0$. In particular $K_w(t,\cdot):\bbR \to\bbR$ is onto for every $t \in [0,1]$.
Indeed, $\bbP(K_w(t,\tilde{X}_t)>y)\geq \bbP(K_w(t,\tilde{X}_t)>y, \sup_{s\leq t}|\tilde{X}_s|\leq n)$ for some large enough $n$. On the other hand, application of Ito’s formula yields $$K(t,\tilde{X}_t)= \beta_t -\int_0^t G(s,\tilde{X}_s)ds,$$ where $G(t,x)=\int_0^x g(t,y)dy$ and $g(t,x):=\frac{w_t(t,x)}{w^2(t,x)}+\half w_{xx}(t,x)$ are continuous.
Thus, the law of $K(t\wedge \tau_n,\tilde{X}_{t\wedge \tau_n})_{t\in [0,1]}$ is equivalent to that of $(\beta_{t \wedge \nu_n})_{t \in [0,1]}$, where $\tau_n=\inf\{t\geq 0:|\tilde{X}_t|\geq n\}$ and $\nu_n=\{t\geq 0: |K^{-1}_w(t,\beta_t)|\geq n\}$ by Girsanov’s theorem. Therefore, $\bbP(K_w(t,\tilde{X}_t)>y, \sup_{s\leq t}|\tilde{X}_s|\leq n)>0$ yielding the claim. Similarly, $\bbP(K_w(t,\tilde{X}_t)<-y, \sup_{s\leq t}|\tilde{X}_s|\leq n)>0$. Consequently, $\min\{\bbP(\tilde{X}_t>x), \bbP(\tilde{X}_t<-x)\}>0$ for all $t \in [0,1]$ by choosing $y=K(t,x)$.
\[mm:r:insfilt\] The strict monotonicity of $H$ in the space variable implies $H$ is invertible prior to time $1$, thus, the filtration of the insider is generated by $X$ and $Z$. Note that jumps of $Y$ can be inferred from the jumps of $X$ via (\[mm:eq:signal\_mm1\]) and the form of $J$. Moreover, since $K_w \in C^{1,2}$ under the hypothesis on $w$, an application of Ito’s formula yields $$dK_w(t,X_t)=dY^c_t -\half w_x(t,X_{t-})dt +K_w(t,X_{t})-K_w(t,X_{t-})+\frac{\partial}{\partial t}K_w(t,X_{t-})dt.$$ Thus, one can also obtain the dynamics of $Y^c$ by observing $X$. Hence, the natural filtrations of $X$ and $Y$ coincide. This in turn implies that $(\cF^{S,Z}_t)=(\cF^{B,Z}_t)$, i.e. the insider has full information about the market.
In view of the above one can take $\cF^I_t=\cF^{B,Z}_t$ for all $t\in [0,1]$.
\[mm:d:iadm\] An $\cF^{B,Z}$-adapted $\theta$ is said to be an admissible trading strategy for a given pricing rule $(H,w,c,j)$ if
1. $\theta$ is a semimartingale[^3] on $(\Om, \cF, (\cF^{B,Z}_t), Q^{0,z})$ for each $z \in \bbR$;
2. There exists a unique strong solution, $X$, to (\[mm:eq:signal\_mm1\]) over the time interval $[0,1]$ on $(\Om, \cF, (\cF^{B,Z}_t), \bbP)$, where $Y
=B+\theta$;
3. and no doubling strategies are allowed, i.e. for all $z \in \bbR$ $$E^{0,z}\left[ \int_{0}^{1}H^{2}\left(t,X_t\right)dt\right] <\infty.
\label{mm:e:theta_cond_2}$$ The set of admissible trading strategies for the given pricing rule $(H,w,c,j)$ is denoted with $\mathcal{A}(H,w,c,j)$. For the notational brevity, we will also denote by $\mathcal{A}(H,w):= \mathcal{A}(H,w,0,0)$.
Observe that the jumps of $\theta$ are summable since $\theta$ is a semimartingale in a Brownian filtration and, thus, the local martingale in its decomposition is continuous. This in particular implies that that $Y$ is a semimartingale with summable jumps and the price process is well-defined.
Now we can formally define the market equilibrium as follows.
\[eqd\] A couple $((H^{\ast}, w^*, c^*,j^*), \theta^{\ast})$ is said to form an equilibrium if $(H^{\ast},w^*, c^*,j^*)$ is an admissible pricing rule, $\theta^{\ast} \in \cA(H^{\ast},w^*, c^*,j^*))$, and the following conditions are satisfied:
1. [*Market efficiency condition:*]{} given $\theta^{\ast}$, $(H^{\ast},w^*, c^*,j^*)$ is a rational pricing rule, i.e. it satisfies (\[mm:d:mm\_obj\]).
2. [*Insider optimality condition:*]{} given $(H^{\ast},w^*, c^*,j^*)$, $\theta^{\ast}$ solves the insider optimization problem for all $z$: $$E^{0,z}[W^{\theta^{\ast}}_1] = \sup_{\theta \in \cA(H^{\ast},w^*, c^*,j^*)} E^{0,z} [W^{\theta}_1]<\infty.$$
The above setup uses the standard definition of equilibirum as in Back [@Back]. The difference lies in the generalisation of the set of admissible pricing rules that in particular includes the ones used in the current literature (see, e.g., [@Back-Baruch], [@Cho], [@CDF], [@CCD], [@C18], [@CFNO], [@CNF], and [@MSZ]). Moreover, the signal of the insider is not assumed to be Markovian contrary to the common assumption.
On equilibrium pricing rule
===========================
In this section we will show that (\[mm:eq:signal\_mm1\]) is a necessary condition for the pricing rule to be compatible with the equilibrium, since any other choice of market maker’s weighting of the signal will result in the infinite profit for the insider.
In what follows we will assume that $H$ and $w$ satisfy w\_t(t,x)+w\_[xx]{}(t,x)&=&w\^2(t,x)g(t,x), \[mm:eq:pdewg\]\
H\_t(t,x)+w\^2(t,x)H\_[xx]{}(t,x)&=&0, \[mm:e:pdeh\] where $g$ is a continuous function. These PDEs (with $g\equiv 0$) can be obtained via a formal derivation of HJB equations associated with the insider optimisation problem as in [@CCD] and [@CDBook].
We will demonstrate that $w$ must satisfy (\[mm:eq:pdewg\]) with $g\equiv 0$ for the equilibrium to exist[^4].
The next theorem computes the expected final wealth of the insider in our general setup. We will use this representation to solve the optimisation problem for the insider. In particular this representation will provide an upper bound on the value function when $g$ vanishes and the trading strategies are continuous.
\[mm:t:iWealth\] Let $(H,w,c,j)$ be an admissible pricing rule such that $H$ satisfies (\[mm:e:pdeh\]) and $w$ solves (\[mm:eq:pdewg\]), where $g$ is a continuous function on $[0,1]\times \bbR$. Assume $\theta\in \mathcal{A}(H,w,c,j)$. Then E\^[0,z]{}&=&E\^[0,z]{}, where $$\label{mm:e:generalG_a}
\Psi^a(t,x):=\int_{\xi(t,a)} ^x \frac{H(t,u)-a}{w(t,u)}du+\frac{1}{2}\int_t^1H_x(s,\xi(s,a))w(s,\xi(s,a))ds$$ and $\xi(t,a)$ is the unique solution of $H(t,\xi(t,a))=a$.
Moreover, \[mm:eq:uljumps\]
\^[a]{}(t,X\_t)-(H(t,X\_t)-a)\_t(H(t,X\_[t]{})-a)j(t,X\_[t-]{},Y\_t)\
\^[a]{}(t,X\_t)-(H(t,X\_t)-a)\_t(H(t,X\_[t-]{})-a)j(t,X\_[t-]{},Y\_t)-H(t,X\_t)\_t.
Using Ito’s formula for general semimartingales (see, e.g. Theorem II.32 in [@Pro]) we obtain $$dH(t, X_t)=H_x(t,X_{t-})w(t,X_{t-})dY^c_t +dFV_t,$$ where $FV$ is of finite variation. Therefore, \[, S\]\^c\_t =\_0\^t H\_x(s,X\_[s-]{})w(s,X\_[s-]{}){d\[B,\]\_s+d\[,\]\^c\_s}. \[mm:eq:THQV\] Moreover, integrating (\[mm:eq:insW\]) by parts (see Corollary 2 of Theorem II.22 in [@Pro]) we get \[mm:eq:Wibp\] W\^\_1=f(Z\_1)\_[1-]{}-\_0\^[1-]{}H(t,X\_[t-]{}))d\_t -\[,H(, X)\]\_[1-]{} since the jumps of $\theta$ are summable. Moreover, direct calculations lead to $$\label{mm:eq:pdepsi}
\Psi^a_t+\frac{1}{2}w(t,x)^2\Psi^a_{xx}=-\int_{\xi(t,a)} ^x (H(t,u)-a)g(t,u)du.$$ Ito’s formula in conjunction with above yields \^a(1-,X\_[1-]{})&=&\^a(0,0)+ \_0\^[1-]{}H(t,X\_[t-]{})(dB\_t +d\_t) -a (B\_1 +\_[1-]{})\
&&+ \_0\^[1-]{}w(t,X\_[t-]{})H\_x(t,X\_[t-]{})(d\[Y,Y\]\^c\_t-dt)\
&&+ \_[0<t<1]{}{\^[a]{}(t,X\_t)-\^a(t,X\_[t-]{})-(H(t,X\_[t-]{})-a)\_t}\
&&+\_0\^[1-]{}(H(t,X\_[t-]{})-a) c(t,X\_[t-]{})(d\[Y,Y\]\^c\_t-dt) -\_0\^[1-]{}\_[(t,a)]{} \^[X\_[t-]{}]{} (H(t,u)-a)g(t,u)dudt Combining the above and (\[mm:eq:Wibp\]) and noting that the stochastic integral with respect to $B$ is a true martingale we deduce E\^[0,z]{}&=&E\^[0,z]{}. Note that since $w$ is positive and $H$ is increasing, we have \^[a]{}(t,X\_t)-\^[a]{}(t,X\_[t-]{})-(H(t,X\_t)-a)\_t&=&\_[X\_[t-]{}]{}\^[X\_t]{}du -(H(t,X\_t)-a)\_t\
&&(H(t,X\_t)-a)\_[X\_[t-]{}]{}\^[X\_t]{}du -(H(t,X\_t)-a)\_t\
&=&(H(t,X\_t)-a)j(t,X\_[t-]{},Y\_t). Similarly, $$\Psi^{a}(t,X_t)-\Psi^{a}(t,X_{t-})-(H(t,X_t)-a)\Delta\theta_t\geq (H(t,X_{t-})-a)j(t,X_{t-},\Delta Y_t)-\Delta H(t,X_t)\Delta \theta_t.$$
\[r:absfinval\] The representation of the expected profit given by the above theorem shows that the absolutely continuous strategies deliver expected wealth bounded by $E^{0,z}[\Psi^{f(Z_1)}(0,0)]$ when $g\equiv 0$. Similarly, if the optimisation problem $$\inf_X E^{0,z}\left[\int_0^{1-}\int_{\xi(t,f(Z_1))} ^{X_t} (H(t,u)-f(Z_1))|g(t,u)|dudt\right]$$ has a finite value, the value function of the insider is also bounded when she is restricted to use absolutely continuous strategies. In what follows we will impose conditions sufficient for this to hold.
In view of the above remark we make the following assumption.
\[mm:a:g\] For every $z \in \bbR$ there exists an $x \in \bbR$ such that $$E^{0,z} \left[\int_0^{1-}\int_{\xi(t,f(Z_1))} ^{x} (H(t,u)-f(Z_1))|g(t,u)|dudt\right]<\infty.$$
Under this assumption we show below that the insider achieves infinite profits unless the pricing rule penalises the jumps and martingale parts correctly. The assumptions on the random variable $f(Z_1)$ are quite general and are satisfied in the available literature. In particular they are satisfied in a large class of diffusion models. The condition that $c=j=0$, however, is not satisfied in the available literature unless $w$ is constant.
\[t:jmprice\] Assume the Assumption \[mm:a:g\] holds and $(H,w,c,j)$ is an admissible pricing rule such that $H$ satisfies (\[mm:e:pdeh\]) and $w$ solves (\[mm:eq:pdewg\]), where $g$ is a continuous function on $[0,1]\times \bbR$. Assume further that the random variable $f(Z_1)$ is such that :
- \[mm:eq:sqint\] E\^[0,z]{}+E\^[0,z]{}<, z ,
- $\lim_{z \rar -\infty}E^{0,z}[-f(Z_1)]=\lim_{z \rar \infty}E^{0,z}[f(Z_1)]=\infty$,
- and $\limsup_{z \rar -\infty}E^{0,z}[f(Z_1)\chf_{[f(Z_1)>k]}]<\infty$, $\liminf_{z \rar \infty}E^{0,z}[f(Z_1)\chf_{[f(Z_1)<k]}]>-\infty$.
Then there exists a set $E$ such that $\bbQ(E)>0$ and for any $z\in E$ we have $$\sup_{\theta \in \cA(H,w,c,j)}E^{0,z}[W_1^{\theta}]=\infty$$ unless $c$ and $j$ are identically $0$.
Suppose $c(t,x)\neq 0$ for some $t<1$ and $x$. Since $c$ is continuous, there exist $\nu_1<\nu_2<1$ and $x_1<x_2$ such that $|c(t,x)|>\eps$ for some $\eps>0$ on $[\nu_1,\nu_2]\times[x_1,x_2]$. Moreover, by the continuity of $K_w^{-1}$ there exists $t_1<t_2<1$ and $y_1<y_2$ such that $K_w^{-1}(t,y) \in [x_1,x_2]$ for all $(t,y)\in [t_1,t_2]\times [y_1,y_2]$.
We shall construct a continuous trading strategy to achieve arbitrarily large profits for some realisation of $Z_1$. This construction will be done in three stages. The first stage will utilise Lemma \[mm:l:appr\] to bring $X$ inside $[K_w^{-1}(t_1,y_1),[K_w^{-1}(t_1,y_2)]$ at time $t_1$. The second stage will keep $K_w(t,X_t)$ inside the interval $[y_1,y_2]$ with arbitrarily large quadratic variation. The final stage will keep $X$ bounded up to time $1$.
Observe that for any continuous semimartingale $\theta$ and $G(t,x):=\int_0^{x}g(t,y)dy$ $$dK_w(t,X_t)=dY_t+ c(t,X_t)(d[Y,Y]_t-dt)-G(t,X_t)dt.$$
[**Stage 1:**]{} To obtain a bounded $X^{\eps}$ satisfying $K_w(t_1,X^{\eps}_{t_1})\in (y_1,y_2)$ apply Lemma \[mm:l:appr\] to $$x(t)=\frac{(K_w^{-1}(t_1,y_1)+K_w^{-1}(t_1,y_2))t}{2 t_1} \mbox{ and } \eps=\frac{(K_w^{-1}(t_1,y_2)-K_w^{-1}(t_1,y_1))}{4}.$$ Set $X=X^{\eps}$ on $[0,t_1]$.
[**Stage 2:**]{} Fix a $y \in (y_1,y_2)$. Consider interval $[t_1, t_2]$ and the solutions of $$dR_t=(b+1)dB_t +(b+1)^2 \left(\frac{1}{R_t-y_1}\chf_{[R_t \leq y]}-\frac{1}{y_2-R_t}\chf_{[R_t>y]}\right)dt+ (b^2+2b)c(t, K_w^{-1}(t,R_t))dt.$$ and observe that pathwise uniqueness holds until the exit time from $(y_1,y_2)$ since $c$ is locally Lipschitz and $K_w^{-1}$ is continuously differentiable. Thus, if we can show the existence of a weak solution that never exits $(y_1,y_2)$, we will arrive at a strong solution that stays in $(y_1,y_2)$. Indeed, since $c(t,K_w^{-1}(t,x))$ is bounded for all $(t,x) \in (t_1,t_2)\times (y_1,y_2)$, by means of Girsanov transformation, weak solutions of above are the same as those of \[mm:eq:neverexit\] dU\_t=qd\_t + q\^2(\_[\[U\_t y\]]{}-\_[\[U\_t>y\]]{})dt, which are unique in law and never exit $(y_1,y_2)$ by Proposition 3.1 in [@rtr]. Define $X_t:=K_w^{-1}(t,R_t)$ and observe that $$d\theta_t=b dB_t + (b+1)^2 \left(\frac{1}{R_t-y_1}\chf_{[R_t \leq y]}-\frac{1}{y_2-R_t}\chf_{[R_t>y]}\right)dt + G(t,K_w^{-1}(t,R_t))dt.$$
[**Stage 3:**]{} Finally consider the interval $(t_2,1]$. Apply Lemma \[mm:l:appr\] to $x(t)=X_{t_2}$ and $\eps$ as before to get $|X^{\eps}|$ and set $X=X^{\eps}$.
Observe that $X$ constructed above is bounded by a determinstic constant, which in turn implies the boundedness of $H(t,X_t)$. Thus, $\theta \in \cA(H,w,c,j)$. Recall from Theorem \[mm:t:iWealth\] that
E\^[0,z]{}&&E\^[0,z]{} since $\Psi^{f(Z_1)}(0,0)\geq 0$ and $d[\theta,\theta]_t=0$ for $t \in [0,1]\backslash [t_1,t_2]$. Moreover, as $dK_w(1,u)=\frac{1}{w(1,u)}$, $$\Psi^{f(Z_1)}(1-,X_{1-})\leq (H(1,X_{1-})-f(Z_1))(K_w(1,X_{1-})-K_w(1,\xi(1,f(Z_1))))\nn$$ since $H$ and $K_w$ are increasing functions. Since $X$ is bounded and (\[mm:eq:sqint\]) holds, we deduce \[mm:eq:psibound\] E\^[0,z]{}\[\^[f(Z\_1)]{}(1-,X\_[1-]{})\]<\_1(z)<. Also observe that \_0\^[1-]{}\_[(t,f(Z\_1))]{} \^[X\_[t-]{}]{} (H(t,u)-f(Z\_1))|g(t,u)|dudt&=&\_0\^[1-]{}\_[(t,f(Z\_1))]{} \^[x(z)]{} (H(t,u)-f(Z\_1))|g(t,u)|dudt\
&&+\_0\^[1-]{}\_[x(z)]{} \^[X\_[t-]{}]{} (H(t,u)-f(Z\_1))|g(t,u)|dudt. Thus, for some constant $\ell_2(z)$ independent of $b$ due to Assumption \[mm:a:g\] and $X$ taking values in a bounded interval, \[mm:e:gintegrable\] E\^[0,z]{}\_2(z) <.
Therefore, E\^[0,z]{}&& (z) +E\^[0,z]{}\
&& (z) + b\^2(m\_1+m E\^[0,z]{}\[f(Z\_1)\]+(M-m)E\^[0,z]{}\[f(Z\_1)\_[\[f(Z\_1)<0\]]{}\])\
&&+ b(m\_3+2m E\^[0,z]{}\[f(Z\_1)\]+2(M-m)E\^[0,z]{}\[f(Z\_1)\_[\[f(Z\_1)<0\]]{}\])\
&=& (z) + b\^2(m\_1+M E\^[0,z]{}\[f(Z\_1)\]+(m-M)E\^[0,z]{}\[f(Z\_1)\_[\[f(Z\_1)0\]]{}\])\
&&+ b(m\_3+2M E\^[0,z]{}\[f(Z\_1)\]+2(m-M)E\^[0,z]{}\[f(Z\_1)\_[\[f(Z\_1)0\]]{}\]), where the constants $m$ and $M$ such that $mM>0$ and $M\geq -\int_{t_1}^{t_2}c(t,X_t)dt\geq m$ exist due to the continuity of $c$, boundedness of $X$ and that $c(t,X_t)$ is bounded away from $0$ on $[t_1,t_2]$ by construction.
Observe that the coefficient of $b^2$ in above can be made positive for large enough $z$ (resp. small enough $z$) if $m>0$ (resp. if $M<0$) due to our assumption on the random variable $f(Z_1)$. This implies that insider’s wealth can be made arbitrarily large for such $z$ by making $b$ arbitrarily large. This yields the claim that $c$ must be $0$ for insider’s profit to be finite.
Next, suppose $c\equiv0$, but $j(t,x,\kappa)\neq 0$ for some $t<1$ and $x,\kappa$. Without loss of generality, assume $j(t,x,\kappa)> 0$ (the proof in the case $j(t,x,\kappa)<0$ is similar). Since $j$ is continuous, there exist $t_1<t_2<1$, $x_1<x_2$, and $\kappa_1<\kappa_2$ such that $j(t,x,\kappa)>\delta$ for some $\delta>0$ on $[t_1,t_2]\times[x_1,x_2]\times[\kappa_1,\kappa_2]$.
We will construct a strategy that achieves an arbitrarily large profit for some realisations of $Z_1$. This will be again done in three stages: first, we will bring $X$ inside the interval $[x_1,x_2]$ at time $t_1$ via Lemma \[mm:l:appr\]. On the interval $[t_1,t_2]$ we will construct a process with an arbitrary number of jumps each of which will give positive contribution to the final utility. Finally, we will keep $X$ in the interval $[x_1,x_2]$ after time $t_2$.
Fix $0<\eps< \frac{x_2-x_1}{2}$
- On the interval $[0,t_1)$ let $x(t)=\frac{x_1+x_2}{2t_1}t$ and apply Lemma \[mm:l:appr\] with $\eps$ as above to obtain a bounded process $X$ such that $X_{t_1 -}\in (x_1,x_2)$. The associated $\theta^{\eps}$ will be used as insider’s strategy on $[0,t_1)$.
- Next, we iteratively construct the process of jumps on $[t_1,t_2]$.
To this end, consider $s_i=t_1+i\frac{t_2-t_1}{n}$, for $i=0,n$ and set $\theta _{t_1}=\theta _{s_0}=\theta _{s_0-}+\kappa_1$. Observe that $j(s_0,X_{s_0-}, \kappa_1)>\delta$, and $X_{s_0}= K_w^{-1}(s_0,j(s_0,X_{s_0-}, \kappa_1)+K_w(s_0,X_{s_0-})+\kappa_1)$.
Suppose we already constructed the process $\theta$ (and $X$) on $[0,s_i]$ and $i<n$, then on the interval $(s_i, s_{i+1})$ consider $x(t)=X_{s_i}+\frac{\frac{x_1+x_2}{2}-X_{s_i}}{s_{i+1}-s_i}(t-s_i)$ and apply Lemma \[mm:l:appr\] with $\eps$ as above. Similar to Stage 1 the associated $\theta^{\eps}$ will be used as the trading strategy and $X$ satisfies $X_{s_{i+1}-}\in[x_1,x_2]$. Finally, for $i< n-1$, set $\theta _{s_{i+1}}=\theta _{s_{i+1}-}+\kappa_1$. We again obtain $j(s_{i+1},X_{s_{i+1}-}, \kappa_1)>\delta$, and $X_{s_{i+1}}= K_w^{-1}(s_{i+1}, j(s_{i+1},X_{s_{i+1}-}, \kappa_1)+K_w(s_{i+1},X_{s_{i+1}-})+\kappa_1)$.
If $i=n-1$, define $\theta _{s_{i+1}}=\theta _{s_{i+1}-}$ and $X_{s_{i+1}}=X _{s_{i+1}-}$.
Thus, we constructed a process $\theta$ with $n$ jumps such that $[\theta,\theta]^c\equiv 0$.
- On the interval $[t_2,1]$ use Lemma \[mm:l:appr\] to construct $X$ to stay in the interval $[x_1,x_2]$ by using $x(t)=\frac{x_1+x_2}{2}$ with the same $\eps$.
Thus, we constructed a process $\theta$ and $X$ with $n$ jumps such that $[\theta,\theta]^c\equiv 0$ and $X$ is taking values in $(m, M)$ for some $m<M$. Since $X$ is bounded, $H(t,X_t)$ is also bounded and therefore $\theta \in \cA(H,w,c,j)$. Moreover, due to Theorem \[mm:t:iWealth\], (\[mm:eq:uljumps\]) and the fact that $\Psi^{f(Z_1)}(0,0)\geq 0$ we have
E\^[0,z]{}& & E\^[0,z]{}. Using the computations that led to (\[mm:eq:psibound\])and (\[mm:e:gintegrable\]) we obtain $$E^{0,z}\left[\Psi^{f(Z_1)}(1-,X_{1-})+\int_0^{1-}\int_{\xi(t,f(Z_1))} ^{X_{t-}} (H(t,u)-f(Z_1))|g(t,u)|dudt\right]\leq \ell_1(z)<\infty, \quad \forall z.$$ Note that the constant $\ell_1$ is independent of $n$.
Furthermore, since the jumps of $\theta$ are of size $\kappa_1$, the jumps of $X$ are uniformly bounded and jumps occur only at $s_i$, we have $\Delta H(t,X_t)\Delta \theta_t\leq \ell_2<\infty.$
Combining the above estimates with the expression for wealth, we get $$E^{0,z}\left[W_1^{\theta}\right]\geq E^{0,z}\left[\sum_{i=0}^{n-1}(H(s_i,X_{s_i-})-f(Z_1))j(t,X_{s_i-},\kappa_1) \right]-\ell_1(z)-\ell_2.$$ Let $j_m=\min_{(t,x)\in[t_1,t_2]\times[x_1,x_2]}j(t,x,\kappa_1)>\delta$, $\infty>j_M=\max_{(t,x)\in[t_1,t_2]\times[x_1,x_2]}j(t,x,\kappa_1)$, $h_m=\min_{(t,x)\in[t_1,t_2]\times[x_1,x_2]}H(t,x)>-\infty$, $h_M=\max_{(t,x)\in[t_1,t_2]\times[x_1,x_2]}H(t,x)<\infty$. We have $$\begin{aligned}
E^{0,z}\left[(H(s_i,X_{s_i-})-f(Z_1))j(t,X_{s_i-},\kappa_1) \right]&\geq&E^{0,z}\left[(h_m-f(Z_1))j(t,X_{s_i-},\kappa_1) \right]\\
&=&E^{0,z}\left[(h_m-f(Z_1))j(t,X_{s_i-},\kappa_1)\chf_{[h_m<f(Z_1)]} \right]\\
&&+E^{0,z}\left[(h_m-f(Z_1))j(t,X_{s_i-},\kappa_1)\chf_{[h_m\geq f(Z_1)]} \right]\\
& \geq &(j_M-j_m) \left(\min\{0,h_m\}-E^{0,z}\left[f(Z_1)\chf_{[h_m<f(Z_1)]}\right]\right)\\
&& +j_m \left(h_m -E^{0,z}\left[f(Z_1)\right]\right).
\end{aligned}$$ Since $\lim_{z \rar -\infty}E^{0,z}[f(Z_1)]=-\infty$ and $\limsup_{z \rar -\infty}E^{0,z}[f(Z_1)\chf_{[f(Z_1)>k]}]<\infty$, there exists a constant $z_1$ such that for any $z<z_1$ we will have $$E^{0,z}\left[(H(s_i,X_{s_i-})-f(Z_1))j(t,X_{s_i-},\kappa_1) \right]>1$$ for all $i$. Thus we will have $E^{0,z}\left[W_1^{\theta}\right]\geq n-\ell_1(z)-\ell_2,$ and letting $n\rar\infty$ will complete the proof.
The following theorem allows a glimpse into the optimal strategy of the insider. It shows that if the pricing rule satisfies $c=j=0$, it is not optimal for the insider to use jumps.
\[t:thetacts\] Suppose that there exists an equilibrium $((H^*,w^*),\theta^*)$, where $(H^*,w^*)$ satisfy (\[mm:e:pdeh\]) and (\[mm:eq:pdewg\]). Assume $Z_0=Z_1$, $\lim_{z \rar \infty}f(z)=-\lim_{z \rar -\infty}f(z) =\infty$, and $\frac{g}{H_y}$ as a function on $[0,1]\times \bar{\bbR}$ with values in $\bar{\bbR}$ is continuous. Then $P^{0,z}(\om:\theta^*_t(\om) \in C([0,1) ))=1$.
The proof of this theorem is postponed to the Appendix. It relies on the following lemma that will also be useful in the proof of Theorem \[t:gzero\]. This lemma shows that only a class of weighting functions that satisfy a further condition on $g$ can be supported in an equilibrium. Thus it allows us to restrict the set of admissible pricing rules further.
\[mm:l:gbbelow\] Suppose $Z_0=Z_1$, $\lim_{z \rar \infty}f(z)=-\lim_{z \rar -\infty}f(z) =\infty$, and there exists an equilibrium $((H,w),\theta)$, where $H$ solves (\[mm:e:pdeh\]) and $w$ satisfies (\[mm:eq:pdewg\]). Assume further that $\frac{g}{H_y}$ as a function on $[0,1]\times \bar{\bbR}$ with values in $\bar{\bbR}$ is continuous. Then $\frac{g}{H_y}$ is bounded from below on $[0,1]\times \bbR$.
Let us first show that the existence of equilibrium implies $H(t,\infty)=-H(t,-\infty)=\infty$ for all $t \in [0,1]$. Indeed, if there exists $\hat{t}$ such that $H(\hat{t},x)\geq h$ for all $x \in \bbR$, then for all $s \leq \hat{t}$ we have $H(s,x)\geq h$ for all $x \in \bbR$. This follows from the fact that $Y$ is a Brownian motion in the market maker’s filtration in equilibrium and, thus, the random variable $X_t$ has full support due to Remark \[mm:r:Kwubdd\]. That $Y$ is a Brownian motion is a consequence of rationality, $H_x>0$ and $H$ satisfies (\[mm:e:pdeh\]).
However, uniform boundedness of the price from below on $[0,\hat{t}]$ gives an unbounded profit for the insider contradicting the definition of equilibrium. Indeed, since $f$ is unbounded from below there exists a $z \in \bbR$ such that $f(z)< h$. Consider the trading strategy $$d\theta_t=-n \chf_{[0,\hat{t}]}(t)dt$$ and note that integrating by parts the associated final wealth we obtain $$W_1=-n\int_0^{\hat{t}}\left(f(z)-H(t,X_t)\right)dt\geq n (h-f(z))\hat{t}\rar\infty \mbox{ as } n \rar \infty.$$
Therefore, we can assume that $H(t,\infty)=-H(t,-\infty)=\infty$ for all $t \in [0,1]$.
Next denote $\frac{g}{H_y}(t,H^{-1}(t,x))$ by $\tilde{g}(t,x)$ and observe that $$\int_{\xi(t,a)}^{\infty}(H(t,u)-a) g(t,u)du=\int_{a}^{\infty}(u-a) \tilde{g}(t,u)du.$$
Suppose that for all $t \in [0,1]$ we have $\lim_{u \rar \infty} \frac{g}{H_y}(t,H^{-1}(t,u))=\lim_{u \rar \infty} \tilde{g}(t,u)\geq 0$ as well as $\lim_{u \rar -\infty} \tilde{g}(t,u)\geq 0$, and consider $A_n:=\{(t,u)\in [0,1]\times \bbR:\tilde{g}(t,u)\leq -n\}$. Clearly, $A_n$ is closed for each $n\geq 1$. Moreover, it is also bounded. Indeed, suppose there exists a sequence $(t_m,u_m)_{m\geq 1}\subset A_n$ such that $\lim_{m\rar \infty}t_m=t\leq 1$ and $\lim_{m \rar \infty}u_m=\infty$ or $-\infty$. Then, $\lim_{m\rar \infty}\tilde{g}(t_m,u_m)\geq 0$ due to the joint continuity of $\tilde{g}$, which is a contradiction. Thus $A_n$s are compact and their intersection would be nonempty by the nested set property if $\tilde{g}$ were not bounded from below. However, if $(\hat{t},\hat{u})\in [0,1]\times \bbR$ belongs to the intersection, $\tilde{g}(\hat{t},\hat{u})=-\infty$. Thus, the intersection must be empty and, therefore, $\tilde{g}$ is bounded from below.
Next suppose that there exists a $\hat{t}$ such that either $\lim_{u \rar \infty} \tilde{g}(\hat{t},u)< 0$ or $\lim_{u \rar -\infty} \tilde{g}(\hat{t},u)< 0$. Without loss of generality assume the former and observe that this leads to $ \tilde{g}(t,x)< -c$ for all $(t,x)$ in $[t_1,t_2]\times [x_1,\infty)$ for some $c>0$ and $t_1, t_2$ in $[0,1]$ and $x_1 \in \bbR$ due to the joint continuity of $\tilde{g}$. Note that $x_1$ can be assumed to satisfy $H(t,x_1)\geq f(z)$ for all $t \in [t_1,t_2]$ due to the continuity and monotonicity of $H$.
Let $x^n:[0,1] \to \bbR$ be the piecewise linear function defined by $x^n(0)=0, x^n(t_1)=x_1 +\half, x^n(\frac{3t_1+t_2}{4})=x_1 + n+\half, x^n(t_2)=x_1+ \half, x^n(1)=f(z)$ and $x^n(t)=x_1+ n+ \half$ for all $t \in [\frac{3t_1 +t_2}{4},\frac{t_1+3t_2}{4}]$. Consider $\eps =\frac{1}{4}$ and an application of Lemma \[mm:l:appr\] yields an existence of an admissible strategy $\theta^n$ that is continuous and of finite variation and satisfies $\sup_{t\in [0,1]}|X^n_t-x^n(t)|<\frac{1}{4}$, where $X^n$ is as in the Lemma. The wealth associated with this strategy is given by $$W^n_1:=\Psi^{f(z)}(0,0)-E^{0,z}\left[\int_0^1\int_{f(z)}^{H(t,X^n_t)}(u-f(z))\tilde{g}(t,u)dudt\right]$$ by Theorem \[mm:t:iWealth\]. Since $X^n$ is uniformly bounded on $[0,t_1]$ and$[t_2,1]$, we only need to consider the above integral on $[t_1,t_2]$. Moreover, continuity of $\tilde{g}$ implies W\^n\_1 &&- E\^[0,z]{}- E\^[0,z]{}\
&& + E\^[0,z]{}, where the second and third inequality are due to $\tilde{g}$ being less than or equal to $-c$ on $[t_1,t_2]\times [x_1,\infty)$. Since $H(t, X^n_t)\geq f(z)$ on $[\frac{3t_1 +t_2}{4},\frac{t_1+3t_2}{4}]$, it follows from the monotone convergence theorem that $W^n_1\rar \infty$ as $n \rar \infty$, which contradicts the definition of equilibrium.
\[mm:r:noH\] Note that in view of the above lemma we can take $H$ to be the identity function. Indeed, if $(H,w)$ is a pricing rule satisfying the conditions of above lemma, then $(\tilde{H}, \tilde{w})$, where $\tilde{H}(t,x)=x$ and $\tilde{w}(t,x)=H_x(t,H^{-1}(t,x))w(t,H^{-1}(t,x))$ is also a pricing rule satisfying the conditions of the lemma with $\tilde{g}(t,x)=\frac{g(t,H^{-1}(t,x))}{H_x(t,H^{-1}(t,x))}$. Moreover, $$\int_{\xi(t,a)}^{x}(H(t,u)-a) g(t,u)du=\int_{a}^{H(t,x)}(u-a) \tilde{g}(t,u)du,$$ and $H(t,\infty)=-H(t,-\infty)=\infty$ for all $t \in [0,1]$.
Moreover, consider $S_t:=H(t,X_t)$, where $X$ is the unique solution of (\[mm:eq:signal\_mm1\]) with $c=j=0$. Then $$dS_t= \tilde{w}(t,S_{t-})dY^c_t+ \frac{\tilde{w}(t,S_{t-})\tilde{w}_x(t,S_{t-})}{2}\left(d[Y,Y]^c_t-dt\right)+(H(t,X_{t})-H(t,X_{t-}),$$ and, therefore, $S$ satisfies (\[mm:eq:signal\_mm1\]) with $w=\tilde{w}$. Indeed, $$S_{t-}+H(t,X_{t})-H(t,X_{t-})=K_{\tilde{w}}^{-1}(t,K_{\tilde{w}}(t,S_{t-})+\Delta Y_t).$$ The above is equivalent to $$\Delta Y_t= K_{\tilde{w}}(t, H(t,X_t))-K_{\tilde{w}}(t,S_{t-})=\int_{S_{t-}}^{S_t}\frac{1}{\tilde{w}(t,y)}dy=\int_{X_{t-}}^{X_t}\frac{1}{w(t,y)}dy,$$ which holds in view of dynamics of $X$. Therefore, without loss of generality we can assume $H$ is identity.
The following theorem provides the justification that the weighting function must satisfy the PDE (\[mm:eq:pdewg\]) with $g=0$.
\[t:gzero\] Suppose that there exists an equilibrium $((H^*,w^*),\theta^*)$, where $(H^*,w^*)$ satisfies (\[mm:e:pdeh\]) and (\[mm:eq:pdewg\]). Assume $Z_0=Z_1$, $\lim_{z \rar \infty}f(z)=-\lim_{z \rar -\infty}f(z) =\infty$, and $\frac{g}{H_y}$ as a function on $[0,1]\times \bar{\bbR}$ with values in $\bar{\bbR}$ is continuous.
Assume further that there exists a set $E$ such that $\bbQ(E)=1$ and for all $z\in E$ there exists a continuous function $s^{f(z)}$ of finite variation such that \[mm:eq:max\_g\] s\^[f(z)]{}(t):=\_x -\_[(t,f(z))]{}\^x (H\^\*(t,y)-f(z)) g(t,y)dy=\_x \_[(t,f(z))]{}\^x (H\^\*(t,y)-f(z)) g(t,y)dy for all $t\in[0,\nu]$. Then $g\equiv 0$ for all $t\in[0,\nu)$.
Note that $\theta^*$ is continuous in view of Theorem \[t:thetacts\]. Observe that $H^*$ can be taken equal to identity in view of Remark \[mm:r:noH\]. We will also restrict our attention to $z\in E$.
Fix $z\in E$. We show that $X^*$ must be such that $X^*_t=\arg \min_x \int_{f(z)}^x (y-f(z)) g(t,y)dy$ for all $t\in(0,\nu)$. Suppose not, i.e. assume that there exists $t\in (0,\nu)$ and $\delta>0$ such that $$P^{0,z}\left[\int_{f(z)}^{X^*_t} (y-f(z)) g(t,y)dy -\int_{f(z)}^{s^{f(z)}(t)} (y-f(z)) g(t,y)dy>\delta\right]>0.$$ since both $X^*$ and $s^{f(z)}$ are continuous on $[0,\nu]$ and $\int_{f(z)}^{X^*_u} (y-f(z)) g(u,y)dy -\int_{f(z)}^{s^{f(z)}(u)} (y-f(z)) g(u,y)dy\geq 0$ for all $u\in[0,\nu]$ we will have $$P^{0,z}\left[\int_0^{\nu}\int_{f(z)}^{X^*_t} (y-f(z)) g(t,y)dydt -\int_0^{\nu}\int_{f(z)}^{s^{f(z)}(t)} (y-f(z)) g(t,y)dydt>0\right]>0.$$ This implies \[mm:eq:estsast\] E\^[0,z]{}=>0
For any $\eps>0$ consider $s^{\eps}$ such that $s^{\eps}(0)=0$, $s^{\eps}=s^{f(z)}$ on $[\eps,\nu)$, and $s^{\eps}$ is continuous and monotone on $[0,\eps]$. Due to Lemma \[mm:l:appr\] there exists $X^{\eps}$ such that $$\sup_{t\in [0,\nu]}|s^{\eps}(t)-X^{\eps}_t|<\eps.$$ We will use the corresponding $\theta^{\eps}$ as the insider’s strategy on $[0,\nu]$.
On $[\nu, \nu+\eps]$ set $d\theta^{\eps}=-dB_t+\frac{f(z)-X^{\eps}_{\nu}}{\eps w(t,X^{\eps}_t)}dt$ and note that $X^{\eps}$ remains bounded on $[\nu,\nu+\eps]$ and $X^{\eps}_{\nu+\eps}=f(z)$.
Now consider the interval $[\nu+\eps,\nu+2 \eps]$ and introduce $$dR_t^{\eps}=dB_t+ \frac{R^*_t-R^{\eps}_t}{\nu+2\eps- t}dt,$$ with $R^{\eps}_{\nu+\eps}=K_w(\nu+\eps,f(z))$. It is easy to see that the solution to the above SDE on $[\nu+\eps,\nu+2\eps)$ is given by $$R_t^{\eps}=(\nu+2\eps-t)\left[R^{\eps}_{\nu+\eps}+\int_{\nu+\eps}^t\frac{1}{\nu+2\eps-s}dB_s +\int_{\nu+\eps}^t\frac{R^*_s}{(\nu+2\eps-s)^2}ds\right],$$ where $R^*_t=K_w(t,X^*_t)$. Observe that the first two terms in the square brackets multiplies by $(\nu+2\eps-t)$ converge two $0$ as $t \rar \nu+2\eps$ in view of Exercise IX.2.12 in [@RY]. Moreover, on $[R^*_{\nu+2\eps}\neq 0]$ an application of L’Hospital’s rule shows that the third term multiplied by $(\nu+2\eps-t)$ converges to $R^*_{\nu +2\eps}$. Similarly, on $[R^*_{\nu+2\eps}=0]$, $(\nu+2\eps-t)\int_{\nu+\eps}^t\frac{|R^*_s|}{(\nu+2\eps-s)^2}ds\rar 0$. Therefore, $R^{\eps}_{\nu+2 \eps}=R^*_{\nu+2\eps},$ a.s.. Note that if we define $\tau_R:=\inf\{t\geq \nu+\eps: R^*_t=R^{\eps}_t\}$, then $R^{\eps}$ is a semimartingale on $[\nu+\eps,\tau_R]$. Indeed, $$\int_{\nu+\eps}^{\tau_R}\frac{|R^*_t-R^{\eps}_t|}{(\nu+2\eps-t)}dt=\left|\int_{\nu+\eps}^{\tau_R}\frac{R^*_t-R^{\eps}_t}{(\nu+2\eps-t)}dt\right|=\left|B_{\tau_R}-R^{\eps}_{\tau_R}-B_{\nu+\eps}+R^{\eps}_{\nu+\eps}\right|<\infty.$$
Next we define $\tilde{X_t}= K_w^{-1}(t,R^{\eps}_t)$ for $t \in [\nu+\eps,\nu+2\eps]$ and set $$X^{\eps}_t=\left\{\ba{ll}
f(z)+(\tilde{X_t}-f(z))^+, & \mbox{ if } X^*_{\nu+\eps}\geq f(z); \\
f(z)-(\tilde{X_t}-f(z))^-, & \mbox{ if } X^*_{\nu+\eps}< f(z),
\ea \right . \quad \forall t \in [\nu+\eps, \tau],$$ where $\tau=\inf\{t\geq \nu+\eps: X^*_t= f(z)\}\wedge\tau_R$. First note that $\tau \leq \nu+2\eps$ and for all $t\in [\nu+\eps,\tau]$ we have either $X^*_t\geq X^{\eps}_t\geq f(z)$ or $f(z)\geq X^{\eps}_t\geq X^*_t$ depending on $X^*_{\nu+\eps}$. Moreover, $X^{\eps}_{\tau}=X^*_{\tau}$ so we can set $X^{\eps}_t=X^*_t$ for $t \in [\tau, 1]$.
$X^{\eps}$ on $[\nu+\eps,\tau]$ satisfies $$dX^{\eps}_t=w(t,X^{\eps}_t)(dB_t +d\theta^{\eps}_t),$$ where in case $X^*_{\nu+\eps}\geq f(z)$ $$d\theta^{\eps}_t=\chf_{[X^{\eps}_t >f(z)]}\left( G(t,X_t^{\eps})+\frac{R^*_t-R^{\eps}_t}{\nu+2\eps-t}\right)dt +\frac{1}{2}d\tilde{L}_t -\chf_{[X^{\eps}_t=f(z)]}dB_t,$$ and $\tilde{L}$ is the local time of $\tilde{X}$ at $f(z)$ in view of Theorem 68 in Chap. IV of [@Pro]. Similarly, if $X^*_{\nu+\eps}< f(z)$, $$d\theta^{\eps}_t=\chf_{[X^{\eps}_t <f(z)]}\left( G(t,X_t^{\eps})+\frac{R^*_t-R^{\eps}_t}{\nu+2\eps-t}\right)dt -\frac{1}{2}d\tilde{L}_t -\chf_{[X^{\eps}_t=f(z)]}dB_t.$$
Clearly, $\theta^{\eps}$ is admissible since $X^{\eps}$ is bounded on $[0,\nu+\eps]$ and $|X^{\eps}_t -f(z)|\leq |X^*_t-f(z)|$ for all $t \in [\nu+\eps,1]$ and $X^*$ is admissible.
We shall show that the above strategy will outperform $\theta^*$ for small enough $\eps$. E\^[0,z]{}\[W\^\_1-W\^\*\_1\]&& E\^[0,z]{}+\
&&-\_1- E\^[0,z]{}+, where the first inequality is due to (\[mm:eq:estsast\]) and the boundedness of $X^{\eps}$ on $[\nu,\nu+\eps]$ in conjunction with the following estimate: -\_\^[1]{}w(t,X\_t\^)d\[\^,\^\]\_t+\_[0]{}\^[1]{}w(t,X\_t\^[\*]{})d\[\^[\*]{},\^[\*]{}\]\_t&=& -\_\^w(t,X\_t\^)d\[\^,\^\]\_t+\_[0]{}\^w(t,X\_t\^[\*]{})d\[\^[\*]{},\^[\*]{}\]\_t\
&& -\_\^[+]{}w(t,X\_t\^)dt+\_[+]{}\^\_[\[X\^\_t=f(z)\]]{}w(t,f(z))dt.
The second inequality is due to the fact that the first, second and the fourth terms are integrals of continuous functions on compact domains whose measures are proportional to $\eps$, and the sixth is positive since by construction either $X^*\geq X^{\eps}\geq f(z)$ or $X^*\leq X^{\eps}\leq f(z)$ on $[\nu+\eps, \nu+2\eps]$ and $g$ is bounded from below by some constant $-C$ in view of Lemma \[mm:l:gbbelow\]. Finally, the above lower estimate converges to $\delta$ as $\eps \rar 0$ due to the square integrability of $X^{\eps}$ and $X^{\eps}$.
Thus, $\theta^*$ should be such that $X^*_t= \arg \min_x \int_{f(z)}^x (y-f(z)) g(t,y)dy$ for all $t\in(0,\nu)$ for $t\in[0,\nu]$.
Define $$E_t(s):=\{a\in\bbR: s=\arg \min
_x \int_{a}^x (y-a) g(t,y)dy\},$$ a set of realisations of insider’s signal that allow the insider to set $X^*_t=s$.
In what follows we will show that : i) $E_t(s)$ is a connected set for all $t\in[0,\nu]$, and ii) for all $E_t(s)$ such that $s\in E_t(s)$ and $\{s\}\neq E_t(s)$ we have $g(t,y)=0$ for all $y\in E_t(s)$.
This will imply that $g\equiv 0$ for all $t\in[0,\nu)$. Indeed, suppose there exists $(t,a)\in [0,\nu)\times \bbR$ such that $g(t,a)\neq 0$. Since $g$ is continuous, there exists a set $[a_1,a_2]\subseteq \bbR$ such that $g(t,a)\neq 0$ for all $a\in [a_1,a_2]$. Due to ii) we must have $X^*_t=a$ on the set $\{f(Z_1)=a\}$ for all $a\in [a_1,a_2]$. Indeed, suppose on the set $\{f(Z_1)=a\}$ we have $X^*_t=s\neq a$. Since $s=X^*_t= \arg \min_x \int_{f(z)}^x (y-f(z)) g(t,y)dy$, ii) yields that $s\notin E_t(s)$ (since $a\in E_t(s)$), but then we have a contradiction to the rationality of the pricing rule as $s=X_t^*=\bbE[f(Z_1)|\cF_t^M]=\bbE[f(Z_1)\chf_{[f(Z_1)\in E_t(s)]}|\cF_t^M]\in E_t(s)$, where the last inclusion is due to i). Thus, we have $X^*_t=a$ on the set $\{f(Z_1)=a\}$ for all $a\in [a_1,a_2]$, or, equivalently, $X^*_t=f(Z_1)$ for all $Z_1\in f^{-1}([a_1,a_2])$, but this means that the rational pricing rule must satisfy $X^*_s= X^*_t$ for any $s\geq t$ which is not consistent with the definition of the pricing rule since $w>0$.
- [**$\mathbf{E_t(s)}$ is connected:**]{} Suppose $a_1,a_2\in E_t(s)$ and $\lambda\in[0,1]$. We need to show that $a=\lambda a_1+(1-\lambda)a_2\in E_t(s)$. Indeed, for any $r\in \bbR$ we have: $$\begin{aligned}
\int_{a}^r (y-a) g(t,y)dy-\int_{a}^s (y-a) g(t,y)dy&=& \int_{s}^r (y-a) g(t,y)dy\\
&=&\lambda\left(\int_{a_1}^r (y-a_1) g(t,y)dy-\int_{a_1}^s (y-a_1) g(t,y)dy\right)\\
&&+(1-\lambda)\left(\int_{a_2}^r (y-a_2) g(t,y)dy-\int_{a_2}^s (y-a_2) g(t,y)dy\right)\geq 0,
\end{aligned}$$ where the last inequality is due to the fact that for $i=1,2$ $$\int_{a_i}^s (y-a_i) g(t,y)dy=\min_{r\in\bbR}\int_{a_i}^r (y-a_i) g(t,y)dy.$$ This implies that $\int_{a}^r (y-a) g(t,y)dy\geq \int_{a}^s (y-a) g(t,y)dy$ for all $r\in \bbR$ and therefore $a\in E_t(s)$.
- [**[$\mathbf{ s\in E_t(s)}$ and $\mathbf{ \{s\}\neq E_t(s)}$ $\mathbf{ \Rightarrow}$ $\mathbf{ g(t,y)=0}$ for all $\mathbf{ y\in E_t(s)}$.]{}**]{} Suppose there exist $(t,s)\in[0,\nu]\times \bbR$ such that: $s\in E_t(s)$, $E_t(s)\neq \{s\}$, and $g(t,\tilde{y})\neq 0$ for some $\tilde{y}\in E_t(s)$.
Since $s\in E_t(s)$ for any $r\in \bbR$ we will have $$\int_{s}^r (y-s) g(t,y)dy\geq \int_{s}^s (y-s) g(t,y)dy=0.$$ Let $G(t,y)=\int_s^yg(t,u)du$. We have $$\begin{aligned}
0\leq \int_{s}^r (y-s) g(t,y)dy&=& \int_{s}^r (y-s) dG(t,y)=(r-s) G(t,r)-\int_{s}^r G(t,y)dy \nn\\
&=&(r-s)(G(t,r)-G(t,\psi)) \label{mm:eq:GonR}
\end{aligned}$$ for any $r\in \bbR$ and some $\psi\in [r\wedge s, r\vee s]$.
Note that $\tilde{y}\neq s$ since for any $a\in E_t(s)$ we have $s=\arg \min_x \int_{a}^x (y-a) g(t,y)dy$ and therefore the first order conditions imply $g(t,s)=0$ as we can choose $a\neq s$.
Suppose $\tilde{y}>s$. Let $G(t,y^*)=\min_{y\in[s,\tilde{y}]} G(t,y)$. Due to (\[mm:eq:GonR\]) there exists $\psi\in [s, y^*]$ such that \[mm:eq:G\_min\] (y\^\*-s) G(t,y\^\*)-\_[s]{}\^[y\^\*]{} G(t,y)dy=(y\^\*-s)(G(t,y\^\*)-G(t,))0 and therefore $G(t,y^*)\geq G(t,\psi)$. Thus, $G(t,y^*)= G(t,\psi)$ and (\[mm:eq:G\_min\]) implies $$\int_{s}^{y^*} G(t,y)dy=(y^*-s)G(t,\psi)=(y^*-s)G(t,y^*)=\int_{s}^{y^*} \min_{y\in[s,y^*]}G(t,y)dy.$$ This yields $G(t,y)=const$ for $y\in[s,y^*]$ and in particular $G(t,s)=G(t,y^*)=\min_{y\in[s,\tilde{y}]} G(t,y)$. Since $G(t,y)=const$ for $y\in[s,y^*]$ and $g$ is continuous, we have $g(t,y)=0$ for $y\in[s,y^*]$. Thus $\tilde {y}\neq y^*$ which implies $G(t,s)=\min_{y\in[s,\tilde{y}]} G(t,y)<G(t,\tilde{y})$.
Since $\tilde{y}\in E_t(s)$ we have $$\begin{aligned}
(s-\tilde{y})(G(t,s)-G(t,\phi))&=&(s-\tilde{y}) dG(t,s)-\int_{\tilde{y}}^s G(t,y)dy=\int_{\tilde{y}}^s (y-\tilde{y}) dG(t,y)\\
& =&\int_{\tilde{y}}^s (y-\tilde{y}) g(t,y)dy\leq \int_{\tilde{y}}^{\tilde{y}} (y-\tilde{y}) g(t,y)dy=0
\end{aligned}$$ for some $\phi\in [ s, \tilde{y}]$. Thus $G(t,\phi)=G(t,s)=\min_{y\in[0,\tilde{y}]}G(t,y)$ and therefore $$\int_s^{\tilde{y}} \min_{y\in[0,\tilde{y}]}G(t,y)dy= (\tilde{y}-s)G(t,\phi)=\int_s^{\tilde{y}} G(t,y)dy.$$ Since $G$ is continuous it implies that $G(t,y)=G(t,s)$ for all $y\in[0,\tilde{y}]$ which contradicts the above result that $G(t,s)<G(t,\tilde{y})$. Thus, we can not have $\tilde{y}\geq s$. Suppose $\tilde{y}<s$. Let $G(t,y^*)=\max_{y\in[\tilde{y},s]} G(t,y)$. Due to (\[mm:eq:GonR\]) there exists $\psi\in [ y^*,s]$ such that \[mm:eq:G\_max\] (y\^\*-s) G(t,y\^\*)-\_[s]{}\^[y\^\*]{} G(t,y)dy=(y\^\*-s)(G(t,y\^\*)-G(t,))0 and therefore $G(t,y^*)\leq G(t,\psi)$. Thus, $G(t,y^*)= G(t,\psi)$ and (\[mm:eq:G\_max\]) implies $$\int_{s}^{y^*} G(t,y)dy=(y^*-s)G(t,\psi)=(y^*-s)G(t,y^*)=\int_{s}^{y^*} \max_{y\in[y^*,s]}G(t,y)dy.$$ This yields $G(t,y)=const$ for $y\in[y^*,s]$ and in particular $G(t,s)=G(t,y^*)=\max_{y\in[\tilde{y},s]} G(t,y)$. Since $G(t,y)=const$ for $y\in[y^*,s]$ and $g$ is continuous, we have $g(t,y)=0$ for $y\in[y^*,s]$. Thus $\tilde {y}\neq y^*$ which implies $G(t,s)=\max_{y\in[\tilde{y},s]} G(t,y)>G(t,\tilde{y})$.
Since $\tilde{y}\in E_t(s)$ we have $$\begin{aligned}
(s-\tilde{y})(G(t,s)-G(t,\phi))&=&(s-\tilde{y}) dG(t,s)-\int_{\tilde{y}}^s G(t,y)dy=\int_{\tilde{y}}^s (y-\tilde{y}) dG(t,y)\\
& =&\int_{\tilde{y}}^s (y-\tilde{y}) g(t,y)dy\leq \int_{\tilde{y}}^{\tilde{y}} (y-\tilde{y}) g(t,y)dy=0
\end{aligned}$$ for some $\phi\in [ s, \tilde{y}]$. Thus $G(t,\phi)=G(t,s)=\max_{y\in[\tilde{y},s]}G(t,y)$ and therefore $$\int_s^{\tilde{y}} \min_{y\in[0,\tilde{y}]}G(t,y)dy= (\tilde{y}-s)G(t,\phi)=\int_s^{\tilde{y}} G(t,y)dy.$$ Since $G$ is continuous it implies that $G(t,y)=G(t,s)$ for all $y\in[0,\tilde{y}]$ which contradicts the above result that $G(t,s)<G(t,\tilde{y})$. Thus, there doesn’t exist $\tilde{y}\in E_t(s)$ such that $g(t,\tilde{y})\neq 0$.
Absolute continuity of the optimal strategy of the insider {#s:abs}
==========================================================
In this section we will show that the equations (\[mm:eq:pdewg\])-(\[mm:e:pdeh\]) with $g=0$ imply the insider must use continuous strategies of finite variation under a mild further condition on the pricing rule. To understand this assumption in (\[e:MQVbd\]) suppose $K_w(1,H^{-1}(1,f(Z_1)))$ is $P^{0,z}$-integrable, define \[e:defM\] M\_t:=E\^[0,z]{}and observe that $M$ is independent of $B$. We will further assume that \[e:MQV\] d\[M,M\]\_t=\^2\_t dt for some measurable process $\tilde{\sigma}$.
This process $M$ will be used by the insider to drive the market price to its fundamental value. Under the optimality conditions of the theorem below $K_w(1,H^{-1}(1,f(Z_1)))=Y_1$. Thus, $M$ corresponds to the insider’s expectation of the final total demand using her own private information only. Not using public information ensures $M$ is independent of $B$.
The condition (\[e:MQVbd\]) is in fact an assumption on the quadratic variation of the signal and is satisfied in the Markovian framework employed in the earlier Kyle-Back models (see, among others, [@Danilova], [@BP], [@CCD],[@Wu], and [@CCD2]).
\[mm:t:AC\] Suppose that \[mm:e:fsqint\] E\^[0,z]{}(f\^2(Z\_1))<, z , i.e. $f(Z_1)$ is square integrable for any initial condition of $Z$. Let $(H,w)$ be an admissible pricing rule satisfying (\[mm:eq:pdewg\]) and (\[mm:e:pdeh\]) with $g=0$. Then $\theta \in \cA(H,w)$ is an optimal strategy if
1. $\theta$ is continuous and of finite variation,
2. and $H(1-,X_{1-})=f(Z_1), \, P^{0,z}$-a.s.,
where $$X_t= \int_0^t w(s,X_s)\{dB_s +d\theta_s\}.$$
Moreover, if we further assume that \[mm:e:Hextra\] E\^[0,z]{}<, z , and $M$ defined by (\[e:defM\]) satisfies (\[e:MQV\]) with $\tilde{\sigma}$ such that \[e:MQVbd\] \_[t 1]{} \^2\_t (1-t)\^[-1]{}=0 for some $\alpha \in (1,2)$, then for any $\theta \in \cA(H,w)$, there exists a sequence of admissible [*absolutely continuous*]{} strategies, $(\theta^n)_{n \geq 1}$, such that $$E^{0,z}\left[W_1^{\theta}\right]\leq \lim_{ n \rar \infty} E^{0,z}\left[W_1^{\theta^n}\right].$$
In view of (\[mm:eq:uljumps\]) and since $w$ is positive and $H$ is increasing, we have $$E^{0,z}\left[W_1^{\theta}\right]\leq E^{0,z}\left[\Psi^{f(Z_1)}(0,0)-\Psi^{f(Z_1)}(1-,X_{1-})\right]$$ Note the inequality above becomes equality if and only if $\Delta \theta_t=0$ due to the strict monotonicity of $H$. Moreover, $\Psi^{f(Z_1)}(1-,X_{1-})\geq 0$ with an equality if and only if $H(1-, X_{1-})=f(Z_1)$. Therefore, $E^{0,z}\left[W_1^{\theta}\right]\leq E^{0,z}\left[\Psi^{f(Z_1)}(0,0)\right]$ for all admissible $\theta$s and equality is reached if and only if the following two conditions are met:
- $\theta$ is continuous and of finite variation.
- $H(1-,X_{1-})=f(Z_1), \, P^{0,z}$-a.s..
Hence, the proof will be complete if one can find a sequence of absolutely continuous admissible strategies, $(\theta^n)_{n \geq 1}$ such that $\lim_{n \rar \infty}E^{0,z}\left[W_1^{\theta^n}\right]=E^{0,z}\left[\Psi^{f(Z_1)}(0,0)\right]$.
Consider the bridge process, $Y$, that starts at $0$ and ends up at $M_1$ at $t=1$: $$Y_t:=B_t +\int_0^t \frac{M_s-Y_s}{1-s}ds=(1-t)\left(\int_{0}^t\frac{1}{1-s}dB_s +\int_0^t \frac{M_s}{(1-s)^2}ds\right).$$ It is easy to check that the above converges a.s. to $M_1$ using the continuity of $M$ and L’Hospital rule since $(1-t)\int_{0}^t\frac{1}{1-s}dB_s$ is the Brownian bridge from $0$ to $0$ as in Exercise IX.2.12 in [@RY].
To establish the semimartingale property of $Y$ first observe that $$Y_t-M_t = (1-t)\int_0^t \frac{1}{1-s}\{dB_s-dM_s\}.$$ Thus, by Theorem V.1.6 in [@RY], there exists a Brownian motion $\tilde{\beta}$ such that $$\int_0^1\frac{|M_t-Y_t|}{1-t}dt=\int_0^1 |\tilde{\beta}_{\tau_t}|dt,$$ where $$\tau_t=\int_0^t\frac{1+\tilde{\sigma}^2_s}{(1-s)^2}ds.$$ Observe that $\tau_{1}=\infty$, $P^{0,z}$-a.s.. Thus by the law of iterated logarithm for Brownian motion (see Corollary II.1.12 in [@RY]), we have $$\frac{|\tilde{\beta}_{\tau_t}|}{\sqrt{ \tau_t \log \log \tau_t}}<C \; \forall t \in [0,1], \, P^{0,z}\mbox{-a.s.}$$ for some finite random variable $C$. Therefore, \[e:Ysemibd\] \_0\^1dt<C \_0\^1 dtC\_0\^1\_t\^dt, for all $\eps>0$. Thus, $Y$ is a semimartingale.
Note that (\[e:MQVbd\]) implies that for any $n>1$ there exists $\delta>0$ such that for any $s \in [1-\delta,1]$, $\tilde{\sigma}^2_s (1-s)^{\alpha-1}<\frac{1}{n}$. Therefore, for $t \geq 1-\delta$, $$(1-t)^{\alpha}\int_0^t \frac{1+\tilde{\sigma}^2_s}{(1-s)^2}ds\leq (1-t)^{\alpha-1}-(1-t)^{\alpha} + (1-t)^{\alpha}\left(\int_0^{1-\delta} \frac{1+\tilde{\sigma}^2_s}{(1-s)^2}ds+ \frac{1}{n}\left((1-t)^{-\alpha}-\delta^{-\alpha}\right)\right),$$ which in turn yields $\lim_{t \rar 1}\tau_t (1-t)^{\alpha}=0$. Hence, $$\int_0^1\tau_t^{\frac{1+\eps}{2}}dt\leq \tilde{C}\int_0^1 (1-t)^{-\frac{\alpha(1+\eps)}{2}}dt<\infty$$ for any $\eps <\frac{2}{\alpha} -1$. This proves the semimartingale property of $Y$ in view of (\[e:Ysemibd\]).
Next define the stopping times $$\tau^{n}:=\inf\{t:|Y_t|\geq n\}$$ with the convention that $\inf \emptyset =1$, and introduce the sequence of trading strategies, $\theta^{n}$ given by $$d\theta^{n}_t=\chf_{[\tau^{n}\geq t]}\frac{M_t-Y_t}{1-t}dt + \chf_{[\tau^{n,m}< t]}d\tilde{\theta}^n_t,$$ where $\tilde{\theta}^n$ is the continuous and of finite variation process given by Lemma \[mm:l:appr\] to keep $Y^n_t \in (-1-n,1+n)$ for $t\geq\tau^n$ via choosing $x(t)=Y_{\tau^n}\frac{1-t}{1-\tau^n}$ and $\eps=w=1$. This will also ensure that $Y^n_1 \in (-1,1)$ on $[\tau^n<1]$. Thus, the total demand process $Y^n$ corresponding to $\theta^{n}$ satisfies
1. $\sup_{t\in [0,1]}|Y^{n}_t|\leq n+1$, a.s.;
2. $Y^{n}_1\chf_{[\tau^{n}=1]}=Y_1\chf_{[\tau^{n}=1]}=K_w(1,H^{-1}(1,f(Z_1)))\chf_{[\tau^{n}=1]}$, a.s..
3. $Y^{n}_1\chf_{[\tau^{n}<1]} \in (-1,1)$.
In view of Remark \[mm:r:Kwubdd\] and the continuity of $H(1, K_w^{-1}(1,\cdot))$ we deduce that $H(t, K_w^{-1}(t,Y^{n}_t))$ is bounded uniformly in $t$ yielding $\theta^{n}$ admissible for each $n$.
Recall that since $\theta^{n}$ is absolutely continuous, we have $$E^{0,z}[W^{\theta^{n}}]=E^{0,z}\left[\Psi^{f(Z_1)}(0,0)-\Psi^{f(Z_1)}(1,K_w^{-1}(1,Y^{n}_{1}))\right].$$ On the other hand, \^[f(Z\_1)]{}(1,K\_w\^[-1]{}(1,Y\^[n]{}\_[1]{})) && (H(1,K\_w\^[-1]{}(1,Y\^[n]{}\_1))-f(Z\_1))(Y\^[n]{}\_1-K\_w(1,H\^[-1]{}(1,f(Z\_1))))\
&=&\_[\[\^[n]{}<1\]]{}(H(1,K\_w\^[-1]{}(1,Y\^[n]{}\_1))-f(Z\_1))(Y\^[n]{}\_1-K\_w(1,H\^[-1]{}(1,f(Z\_1)))). Since $f(Z_1)K_w(1,H^{-1}(1,f(Z_1)))$ is integrable and $Y^{n}_1$ is uniformly bounded, applying the dominated convergence theorem yields $$\lim_{n \rar \infty}E^{0,z}[W^{\theta^{n}}]=E^{0,z}\left[\Psi^{f(Z_1)}(0,0)\right],$$ i.e. the expected wealth corresponding to our sequence of admissible strategies converges to the upper limit of the value function.
Auxiliary results
=================
\[mm:l:appr\] Consider bounded stopping times $S \leq T$ and let $x:[S,T] \mapsto \bbR$ be continuous, adapted, and of finite variation. Then for any $\eps>0$ there exists an adapted process $\theta^{\eps}$ that is continuous and of finite variation on $[S,T]$ such that there exists a strong solution to $$dX^{\eps}_t=w(t,X_t)\{dB_t +d\theta^{\eps}_t\}$$ with $X^{\eps}_S=x(S)$ for a given $w:[0,1]\times \bbR\to (0,\infty) \in C^{1,2}$ . Moreover, $X^{\eps}$ satisfies $$\sup_{r \in [S,T]}|X^{\eps}_r-x(r)|<\eps.$$
Define $y(t):=K_{w}(t,x(t))$ and observe that $y$ is continuous and of finite variation. Moreover, introduce the stochastic process $U^{\delta}$ with $U^{\delta}_S=0$ and $$dU^{\delta}_t= dB_t + \left(\frac{1}{U^{\delta}_t+\delta}\chf_{[U^{\delta}_t\leq 0]}-\frac{1}{\delta-U^{\delta}_t}\chf_{[U_t^{\delta}>0]}\right)dt,$$ which stays in $(-\delta,\delta)$ in view of Proposition 3.1 in [@rtr].
Next define $R^{\delta}:=U^{\delta}+y$ on $[S,T]$ and set $X^{\delta}_t=K^{-1}_{w}(t,R^{\delta}_t)$. Thus, $$dX^{\delta}_t= w(t,X^{\delta}_t) \left\{dB_t +d\theta^{\delta}_t\right\},$$ where $$d\theta^{\delta}_t=\left(G(t,K_w^{-1}(t,R_t^{\delta}))+\frac{1}{U^{\delta}_t+\delta}\chf_{[U^{\delta}_t\leq 0]}-\frac{1}{\delta-U^{\delta}_t}\chf_{[U_t^{\delta}>0]}\right)dt+dy_t,$$ where $G(t,x):=\int_0^{x}g(t,y)dy$. Therefore, \[mm:eq:seps-xeps\] \_[t ]{}|y(t)-K\_w(t,X\^\_t)|<. Choosing $\delta$ small enough we thus obtain $$\sup_{t \in [S,T]}|x(t)-X^{\delta}_t|=\sup_{t \in [S,T]}|K_w^{-1}(t,y(t))-X^{\delta}_t)|<\eps.$$ due to the uniform continuity of $K^{-1}_w$ on compacts.
Let $g:[0,1]\times \bar{\bbR}\to \bar{\bbR}$ be continuous such that $g:[0,1]\times \bbR\to \bbR$ is also continuous. Suppose that $\lim_{u \rar \infty}g(t,u) \geq 0$ and $\lim_{u \rar -\infty}g(t,u) \geq 0$ for every $t \in [0,1]$. Then $g$ is bounded from below.
Consider sets $(E_n)_{n \geq 1}$ $$E_n:=\{(t,u)\in [0,1]\times \bbR:g(t,u)\leq -n \}.$$ Clearly, $A_n$s are closed. They are also bounded. Indeed, if there exists a sequence $(t_m,u_m)\in A_n$ such that $u_m \rar \infty$ or $u_m \rar -\infty$. Then, $\lim_{m\rar \infty}g(t_m,u_m) \leq -n$, contradicting the hypotheses on joint continuity and the limits at $\pm \infty$. Thus, $A_n$s are compact. Therefore, if all $A_n$s are non-empty, then $\cap_{n \geq 1} A_n \neq \emptyset$ by the nested set property (see Corollary to Theorem 2.36 in [@RudinMA]). By construction $g(t,u)=-\infty$ for any $(t,u)$ in this intersection, which contradicts our continuity assumption on $g$, Therefore, $A_n$ must be empty for all $n>N$ for some $N$.
First observe that $H^*$ can be taken equal to identity in view of Remark \[mm:r:noH\]. Let $\nu:=\inf\{t\geq 0: \Delta X_t>0\}$ and suppose $P^{0,z}(\nu<1)>0$. We will construct a strategy $\theta^{\eps}$ that agrees with $\theta^*$ on $[0,\nu)$. Let $\eps>0$ and choose $\eps(\nu)=\eps \wedge \frac{1-\nu}{3}$.
On $[\nu, \nu+\eps(\nu)]$ set $X_{\nu}^{\eps}=X^*_{\nu-}, \, d\theta^{\eps}=-dB_t+\frac{f(z)-X^{\eps}_{\nu}}{\eps(\nu) w(t,X^{\eps}_t)}dt$ and note that $|X^{\eps}|\leq |X^*_{\nu-}|+|f(z)|$ on $[\nu,\nu+\eps(\nu)]$ as well as $X^{\eps}_{\nu+\eps(\nu)}=f(z)$.
Now consider the interval $[\nu+\eps(\nu),\nu+2 \eps(\nu)]$ and introduce $$dR_t^{\eps}=dB_t+ \frac{R^*_t-R^{\eps}_t}{\nu+2\eps(\nu)- t}dt,$$ with $R^{\eps}_{\nu+\eps(\nu)}=K_w(\nu+\eps(\nu),f(z))$. It is easy to see that the solution to the above SDE on $[\nu+\eps(\nu),\nu+2\eps(\nu))$ is given by $$R_t^{\eps}=(\nu+2\eps(\nu)-t)\left[R^{\eps}_{\nu+\eps(\nu)}+\int_{\nu+\eps(\nu)}^t\frac{1}{\nu+2\eps(\nu)-s}dB_s +\int_{\nu+\eps(\nu)}^t\frac{R^*_s}{(\nu+2\eps(\nu)-s)^2}ds\right],$$ by $(\nu+2\eps(\nu)-t)$ converge two $0$ as $t \rar \nu+2\eps(\nu)$ in view of Exercise IX.2.12 in [@RY]. Moreover, on $[R^*_{\nu+2\eps(\nu)}\neq 0]$ an application of L’Hospital’s rule shows that the third term multiplied by $(\nu+2\eps(\nu)-t)$ converges to $R^*_{(\nu +2\eps(\nu))-}$. Similarly, on $[R^*_{(\nu+2\eps(\nu))-}=0]$, $(\nu+2\eps(\nu)-t)\int_{\nu+\eps(\nu)}^t\frac{|R^*_s|}{(\nu+2\eps(\nu)-s)^2}ds\rar 0$. Therefore, $R^{\eps}_{\nu+2 \eps(\nu)}=R^*_{(\nu+2\eps(\nu))-},$ a.s.. Note that if we define $$\tau_R:=\inf\{t\geq \nu+\eps(\nu): \operatorname{sgn}(R^*_t-R^{\eps}_t)\neq \operatorname{sgn}(R^*_{\nu+\eps(\nu)}-R^{\eps}_{\nu+\eps(\nu)})\}\wedge \inf\{t\geq \nu+\eps(\nu): R^*_t=R^{\eps}_t\},$$ where $\operatorname{sgn}(x)=\chf_{x>0}-\chf_{x\leq 0}$, then $R^{\eps}$ is a semimartingale on $[\nu+\eps(\nu),\tau_R]$. Indeed, $$\int_{\nu+\eps(\nu)}^{\tau_R}\frac{|R^*_t-R^{\eps}_t|}{(\nu+2\eps(\nu)-t)}dt=\left|\int_{\nu+\eps(\nu)}^{\tau_R}\frac{R^*_t-R^{\eps}_t}{(\nu+2\eps(\nu)-t)}dt\right|=\left|B_{\tau_R}-R^{\eps}_{\tau_R}-B_{\nu+\eps(\nu)}+R^{\eps}_{\nu+\eps(\nu)}\right|<\infty.$$
Next we define $\tilde{X_t}= K_w^{-1}(t,R^{\eps}_t)$ for $t \in [\nu+\eps(\nu),\nu+2\eps(\nu))$ and set $$X^{\eps}_t=\left\{\ba{ll}
f(z)+(\tilde{X_t}-f(z))^+, & \mbox{ if } X^*_{\nu+\eps(\nu)}\geq f(z); \\
f(z)-(\tilde{X_t}-f(z))^-, & \mbox{ if } X^*_{\nu+\eps(\nu)}< f(z),
\ea \right . \quad \forall t \in [\nu+\eps(\nu), \tau),$$ where $$\tau=\inf\{t\geq \nu+\eps(\nu): X^*_t= f(z)\}\wedge \inf\{t\geq \nu+\eps(\nu): \operatorname{sgn}(X^*_t- f(z))\neq \operatorname{sgn}(X^*_{\nu+\eps(\nu)}- f(z))\}\wedge\tau_R.$$
$X^{\eps}$ on $[\nu+\eps(\nu),\tau)$ satisfies $$dX^{\eps}_t=w(t,X^{\eps}_t)(dB_t +d\theta^{\eps}_t),$$ where in case $X^*_{\nu+\eps(\nu)}\geq f(z)$ $$d\theta^{\eps}_t=\chf_{[X^{\eps}_t >f(z)]}\left( G(t,X_t^{\eps})+\frac{R^*_t-R^{\eps}_t}{\nu+2\eps(\nu)-t}\right)dt +\frac{1}{2}d\tilde{L}_t -\chf_{[X^{\eps}_t=f(z)]}dB_t,$$ and $\tilde{L}$ is the local time of $\tilde{X}$ at $f(z)$ in view of Theorem 68 in Chap. IV of [@Pro]. Similarly, if $X^*_{\nu+\eps(\nu)}< f(z)$, $$d\theta^{\eps}_t=\chf_{[X^{\eps}_t <f(z)]}\left( G(t,X_t^{\eps})+\frac{R^*_t-R^{\eps}_t}{\nu+2\eps(\nu)-t}\right)dt -\frac{1}{2}d\tilde{L}_t -\chf_{[X^{\eps}_t=f(z)]}dB_t.$$ Next pick an $n\geq 1$ and consider $\hat{\theta}$, which is given by $$\hat{\theta}_t=\chf_{[t<\nu]}\theta^*_t+ \chf_{[t\geq \nu]}\left(\theta^*_t\chf_{[X^*_{\nu-}>n]}+\theta^{\eps}_t\chf_{[X^*_{\nu-}\leq n]}\right).$$ This strategy is clearly admissible and will outperform $\theta^*$ for small enough $\eps$ and large enough $n$ by following the reasoning and calculations that led to the analogous conclusion in Theorem \[t:gzero\].
[^1]: Note that this implies the insider’s optimal trading strategy takes into account the *feedback effect*, i.e. that prices react to her trading strategy.
[^2]: Following Kurtz [@Kurtz07] $X$ is a strong solution of (\[mm:eq:signal\_mm1\]) if there exists a measurable mapping, $\varphi$, such that $X:=\varphi(Y)$ satisfies (\[mm:eq:signal\_mm1\]). Note that $Y$ may jump only due to the discontinuities in $\theta$.
[^3]: Note that due to the incompleteness of the stochastic basis we follow the notion of semimartingale from Jacod and Shiryaev [@JS] that only requires the right-continuity of filtrations.
[^4]: If the equilibrium is inconspicuous as in most of the literature, the stated PDE for $H$ will follow from the standard filtering theory.
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abstract: 'We consider a Szilard engine in one dimension, consisting of a single particle of mass $m$, moving between a piston of mass $M$, and a heat reservoir at temperature $T$. In addition to an external force, the piston experiences repeated elastic collisions with the particle. We find that the motion of a heavy piston ($M \gg m$), can be described effectively by a Langevin equation. Various numerical evidences suggest that the frictional coefficient in the Langevin equation is given by $\gamma = (1/X)\sqrt{8 \pi m k_BT}$, where $X$ is the position of the piston measured from the thermal wall. Starting from the exact master equation for the full system and using a perturbation expansion in $\epsilon= \sqrt{m/M}$, we integrate out the degrees of freedom of the particle to obtain the effective Fokker-Planck equation for the piston albeit with a different frictional coefficient. Our microscopic study shows that the piston is never in equilibrium during the expansion step, contrary to the assumption made in the usual Szilard engine analysis — nevertheless the conclusions of Szilard remain valid.'
author:
- Deepak Bhat
- Abhishek Dhar
- Anupam Kundu
- Sanjib Sabhapandit
bibliography:
- 'reference.bib'
title: 'Non-equilibrium dynamics of the piston in the Szilard engine'
---
*Introduction*.– The Szilard engine, a simple realization of the Maxwell demon, is a paradigmatic model designed to address the conceptual foundations of the second law of thermodynamics [@Szilard]. In apparent violation of the second law, this model envisages a thought experiment, where the system working in a cycle extracts work from a single heat reservoir by using the information about the initial state of the system. Recently there has been renewed interest in this problem due to important developments in the areas of stochastic thermodynamics of small systems [@Bustamante] and fluctuation theorems [@Jarzynski2; @Jarzynski; @Evans; @Seifert; @Kurchan]. Moreover, recent developments of technology has made it possible to realize this thought experiment in the laboratory [@Toyabe; @Koski; @Berut].
In spite of the importance of the Szilard problem, surprisingly there have been very few microscopic studies [@Hatano; @Hondou] of its dynamics in the original set-up. In Szilard’s analysis, the piston is assumed to be ideal, having infinite mass and its motion is then described by a quasi-static deterministic process. However a realistic piston has a large but finite mass. As a result, its motion is strongly affected by fluctuations that we expect to be important in small systems. Hence it is crucial to understand the stochastic dynamics of the piston. This is the main aim of this Letter.
The basic model consists of a single hard point particle of mass $m$ confined to move in one dimension, between a piston and a heat reservoir at temperature $T$ (see Fig. \[f1\]). The piston itself is taken to be another hard point particle of mass $M>>m$. Let $x$ and $v$ ($X$ and $V$) respectively be the position and velocity of the particle (piston). On collisions with the thermal wall at $x=0$, the particle emerges with a velocity $v>0$, chosen independently at each time from the Rayleigh distribution $f(v)=\beta m v e^{-\beta m v^2/2}$, where $\beta=(k_BT)^{-1}$. The collision between the particle and the piston is taken to be elastic. In between collisions with the wall and the piston, the particle moves ballistically. The piston, apart from collisions with the particle, also experiences an external force $-\mathcal{U}'(X)$. This dynamics takes the system to the Gibb’s equilibrium state $P_\text{eq}=Z^{-1}\exp{[ -\beta (mv^2/2+M V^2/2 + \mathcal{U}(X))]} \theta(x) \theta(X-x)$, where $Z$ is the partition function. However the dynamics of the relaxation process is non-trivial, as can be seen even when the piston is held fixed [@Bhat].
Our set-up is similar to that of the well-known adiabatic piston problem, where one usually considers the deterministic motion of a heavy piston in presence of a gas of thermodynamically large number of small particles with [@Neishtadt; @Gruber; @Mansour; @Cerino; @Hondou; @Proesmans; @Baule; @Hoppenau; @Chernov; @Chernov2; @Lebowitz] or without reservoirs [@Sinai; @Wright]. Here we look at the stochastic dynamics of the piston in the presence of a single particle gas, as required in the Szilard set-up, where fluctuations play an important role.
Our main finding is that, in the limit of heavy piston mass ($M>>m$), the effective stochastic dynamics is given by the Langevin equation $$\begin{aligned}
M\frac{dV}{dt} = -\mathcal{U}'(X)+\frac{k_BT}{X}-\gamma(X) V +\sqrt{2\gamma(X) k_BT}~\eta(t),~ ~~~\label{eom}\end{aligned}$$ where $\eta(t)$ is Gaussian white noise with $\langle \eta(t)\rangle=0$ and $\langle\eta(t) \eta(t')\rangle=\delta(t-t')$, and the space-dependent dissipation is given by $$\begin{aligned}
\gamma(X)= \frac{1}{X}\sqrt{8\pi mk_BT}=:\gamma_{\rm sp}.\label{gamma}\end{aligned}$$ The second term on the rhs of Eq. is the pressure term while the frictional and noise terms satisfy the fluctuation-dissipation relation. All of these three terms arise from the repeated collisions of the particle with the piston. The stationary distribution corresponding to the above Langevin equation is given by $\Psi_\text{eq}(X,V)=\mathcal{Z}^{-1}\exp{[-\beta (MV^2/2+\mathcal{U}_\text{eff}(X))]}$, where $\mathcal{U}_\text{eff}= \mathcal{U}- k_B T \ln X$ is the effective potential and $\mathcal{Z}$ is the partition function. This is consistent with the equilibrium distribution $P_\text{eq}(x,v,X,V)$ of the full system. If we replace our single particle gas by a equilibrium gas at finite density $\nu$, then the friction coefficient is given by [@kampen1961power] $\nu \sqrt{8 m k_B T/\pi}$. Naively extending this result to our one-particle gas, by setting $\nu=1/X$, would suggest $\gamma(X)=(1/X) \sqrt{8 m k_B T/\pi} =: \gamma_{\rm gas}$, which is different from Eq. . Our numerical results, however, strongly indicates that Eq. with $\gamma(X)$ as in Eq. describes the piston dynamics more accurately.
[*Numerical results*]{}.– We compute various physical quantities related to the piston, from the exact microscopic dynamics (EMD) of particle-piston system and compare them with the corresponding results obtained from the effective Langevin equation (LE) . In the EMD simulation we start with the piston at a fixed position $X_0$ and velocity $V=0$. On the other hand, the initial position $x$ and velocity $v$ of the small particle are chosen from the equilibrium distribution $p_\text{eq}(x,v)=(1/X_0)~e^{-m v^2/(2k_B T)}/\sqrt{2 \pi k_B T/m}$ with $0<x<X_0$. Starting from this initial condition we follow the collisional dynamics. Note that there is a stochastic component to the dynamics due to the collisions between particle and heat bath. Finally we compute observables related to the piston, by averaging over the initial configurations (of particle) as well as the trajectories. In our LE simulations we start from the same fixed initial conditions $(X=X_0,V=0)$ for the piston and then average over noise realizations. In all our simulations we have set $k_B T=1$ and $m=1$.
In Fig. \[f2\](a) and (b), we plot the average position $\langle X \rangle$ and average kinetic energy $\langle \mathcal{E} \rangle$ of the piston as a function of time, obtained from the EMD and LE simulations, for the case of a constant force $-F$ ($\mathcal{U}=FX,~F>0$), directed towards the bath. We see excellent agreement between the EMD and the LE with $\gamma(X)$ from Eq. , whereas the LE with $\gamma_{\rm gas}$ shows significant deviations. We see damped oscillations and an eventual approach to the expected equilibrium values $\langle X \rangle_\text{eq}= 2k_BT/F$. Note that the mean position does not correspond to the minimum of the effective potential, $X_\text{min}=k_BT/F$, expected from a naive pressure balance. This is basically because of equilibration in a asymmetric potential and can also be understood in terms of a two-particle system in a constant-pressure ensemble [@Hondou]. From Fig.\[f2\](c) we can see that the time period of oscillations scales as $\sqrt{M}$ and this is again due to the motion in the effective potential $\mathcal{U}_{\rm eff}$. Finally the inset in Fig. \[f2\](c) shows that the equilibration time scale $\sim M$, as can be inferred from the Langevin equation Eq. .
The prediction $\gamma(X)= \gamma_{\rm gas}$ is made for a heavy particle interacting with a many-particle gas in equilibrium. A question naturally arises whether $\gamma(X)= \gamma_{\rm gas}$ is better than Eq. in describing correlations when the heavy particle is in equilibrium with the single particle gas. This leads us to investigate the velocity auto-correlation $\langle V(0)V(t) \rangle_{\rm eq}$ for the piston, as shown in Fig. \[f2\](d). We plot this correlation quantity as a function of time, as obtained by simulating the EMD and from the LE with both $\gamma_{\rm sp}$ and $\gamma_{\rm gas}$. Once again we observe that $\gamma_{\rm sp}$ works much better than $\gamma_{\rm gas}$. As in the non-stationary case \[Fig. \[f2\](a) and (b)\], here too the oscillation period scales as $\sqrt{M}$ while the relaxation time scales as $M$.
\
\
So far we have looked at averages of time-instantaneous quantities and found that the Langevin description (with $\gamma_{\rm sp}$) agrees very well with the EMD. It is natural to ask if this agreement will continue to hold for time-integrated quantities, e.g physical observables such as heat and work. These are quantities that are of direct relevance in the original context of the Szilard engine. We consider a protocol where the piston is released from an initial position $X_i$ and velocity $V_i$, and allowed to evolve till it reaches a specified final position $X_f$ for the first time. Note that the time $\tau$ required for this process is a random variable. For a given realization of the trajectory $\{X(t),V(t): 0 \le t \le \tau \}$, the work done by the piston and the change in the kinetic energy are respectively $$\begin{aligned}
\mathcal{W}= \int^{X_f}_{X_i}F dX ~~{\text{and}}~~
\Delta \mathcal{E}=\frac{1}{2} M V_f^2-\frac{1}{2} M V_i^2~, \label{energy} \end{aligned}$$ where $V_f$ is the final velocity (which is random) of the piston. From the first law of thermodynamics, the amount of heat absorbed in the process is given by $$\begin{aligned}
\mathcal{Q}= \Delta \mathcal{E} + \mathcal{W}~. \end{aligned}$$ We numerically compute these quantities from trajectories, with $X_i=L/2, V_i=0$ and $X_f=L$, generated by both the EMD and the LE . In Fig. \[trajaverages\] we plot the averages of these quantities as a function of the constant applied force $F$ and find good agreement between the EMD and the LE. This further confirms the validity of the Langevin description in .
We now discuss some interesting aspects related to Fig. \[trajaverages\]. In (a) where we plot $\langle \mathcal{Q} \rangle$ as a function of $F$, we see that, at zero force and in the $M \to \infty$ limit, the amount of heat absorbed $\mathcal{Q} = \langle \Delta \mathcal{E} \rangle=k_BT \ln(2)$. This can be understood from the LE , since we see that $\mathcal{Q}=\int_{L/2}^L dX k_B T/X +O(M^{-1/2})$, where the sub-leading correction term comes from the dissipative and noise terms. As we apply force on the piston, an amount of work $\mathcal{W}=FL/2$ is done by the system. Note that it is independent of trajectories and also of $M$. On the other hand, we see in Fig. (\[trajaverages\]b) that the change in average kinetic energy of the piston $\langle \Delta \mathcal{E} \rangle$ decreases with increasing $F$, as expected. In the $M \to \infty$ limit, $\Delta \mathcal{E}= \mathcal{Q}-\mathcal{W}=k_B T \ln (2) - FL/2$, as seen in Fig. (\[trajaverages\]b), for the largest mass case. It then follows that at a critical value of the force $F_c=2k_B T \ln 2/L$, the change in energy vanishes and all the heat absorbed gets converted to work, as in the ideal Szilard engine. Note that in the original single molecule Szilard engine, the work and heat computations are carried out under the assumption that the system is always in equilibrium and described by the equation of state $\langle X \rangle = k_BT/F$. However, as pointed out recently by Hondou [@Hondou], in the single particle case the equation of state is in fact $\langle X \rangle = 2k_BT/F$ and this seems to contradict the basic premise of the Szilard calculation. This equation of state in fact follows on noting that $\langle 1/X \rangle = 2/\langle X \rangle$ with the average taken over $P_{\text eq}$. One of the finding of our microscopic study is that in the large piston mass limit, there is a critical force $F_c$, when one can convert $k_BT \ln 2$ amount of heat completely into work and this happens [*while the system is never in equilibrium*]{} — hence Szilard’s conclusions remain valid.
Figure (\[trajaverages\]c) shows that the time required to reach $X_f=L$ grows rapidly beyond $F_c$, and this growth is faster for larger $M$. We note that, for $M\to \infty$, the (scaled) time required for the piston to reach $X_f=L$, given by $\mathcal{T}/\sqrt{M}=\sqrt{1/2}\int^{L}_{L/2} [\mathcal{U}_{eff}(L/2)-\mathcal{U}_{eff}(X)]^{-1/2}dX$, diverges at $F=F_c$. For large but finite $M$, the piston finally reaches $X_f=L$ after a large time because of the noise and dissipation terms. It is easy to see that the piston gains energy $\mathcal{O}(M^{-1/2})$ from the noise term during the time period $\mathcal{O}(M^{1/2})$ required to reach $L$ \[as seen in Fig. (\[trajaverages\]c)\]. Interestingly, we see that $\Delta \mathcal{E}$ for finite masses has a minimum at $F=F_c$ and beyond $F_c$, the $\Delta \mathcal{E}$ increases.
Qualitatively similar thermodynamic features can also be seen for other choices of the external force, for instance harmonic force like $F=-\kappa(X-\frac{L}{2})$ or $F= -\frac{\mu}{X}$. Following the aforementioned procedure for work extraction process of Szilárd engine, we can show that, by suitably choosing the parameters ($\kappa=\kappa_c=8 k_BT \log(2)/L^2$ or $\mu=\mu_c=k_BT$) we can get $\langle \Delta \mathcal{E} \rangle=0$, $\langle \Delta \mathcal{Q} \rangle=k_BT\ln(2)$ and work extracted is $k_BT \ln(2)$ in the limit $\epsilon=\sqrt{m/M} \rightarrow 0$.
So far we have presented various numerical evidence which suggests that is the correct effective Langevin description of the piston motion. We now present a possible theoretical derivation based on van-Kampen’s $\Omega-$expansion method [@VANKAMPEN2007219], where $\epsilon=\sqrt{m/M}$ is the expansion parameter. We start with the master equation for the joint probability distribution function $P(x,X,v,V,t)$ as follows [@Hondou]: $$\begin{aligned}
\frac{\partial P}{\partial t} = \mathcal{L}P~~~~;~~~~\mathcal{L}=\mathcal{L}_d+ \mathcal{L}_r+\mathcal{L}_c~. \label{me}\end{aligned}$$ Here $$\begin{aligned}
\mathcal{L}_dP=-v\frac{\partial P}{\partial x} -V\frac{\partial P}{\partial X}
+\frac{F}{M}\frac{\partial P}{\partial V} \end{aligned}$$ corresponds to the deterministic evolution of the particle and piston, in between collisions. The collision of the small particle with the thermal reservoir is represented by the term $$\begin{aligned}
\begin{split}
\mathcal{L}_rP &=\int dv' \left[\mathcal{R}_r(v|v')P(x,X,v',V) \right. \\
& \left.
- \mathcal{R}_r(v'|v)P(x,X,v,V) \right]~,
\end{split} \label{collwall} \end{aligned}$$ with $\mathcal{R}_r(v|v')=\delta(x) (-v') ~\theta(-v')f(v)$. The first term in corresponds to gain in probability from events in which a particle with a negative velocity hits the reservoir and emerges with a velocity $v$ with probability $f(v)$. The second term corresponds to loss of probability when a particle with velocity $v$ hits the reservoir. Finally the elastic collisions between the particle and piston are represented by the term $$\begin{aligned}
\begin{split}
\mathcal{L}_cP &= \int dv' dV' \left[\mathcal{R}_c(v,V |v',V')P(x,X,v',V') \right. \\ &\left. - \mathcal{R}_c(v',V'|v,V)P(x,X,v,V) \right],~
\end{split} \label{collpiston}\end{aligned}$$ with $ \mathcal{R}_c(v,V|v',V')= \delta(X-x)\theta(v'-V') (v'-V') \delta\left[v'-\frac{(\epsilon^2-1)v+2V}{1+\epsilon^2}\right] \delta\left[V'-\frac{2\epsilon^2 v+(1-\epsilon^2)V}{1+\epsilon^2} \right] $. The $\delta-$function constraints on the velocities arise from the momentum and energy conservation during the elastic collisions. Our aim now is to integrate out the particle-degrees of freedom, to get an effective Fokker-Planck equation for the piston.
We first rescale the velocity of the piston and the particle as $U=V\sqrt{\beta M}$, $u=v\sqrt{\beta m}$ and time as $\tau=t/\sqrt{\beta m}$. We write the joint distribution of the rescaled variables $Q(x,X,u,U,\tau)$ as $$\begin{aligned}
Q(x,X,u,U,\tau)=\Phi(x,u,\tau|X,U)\Psi(X,U,\tau), \label{Q}\end{aligned}$$ where $\Psi(X,U,\tau)$ is the marginal distribution of the position and velocity of the piston and $\Phi(x,u,\tau|X,U)$ is the distribution of the position and velocity of the small particle conditioned on given piston configuration. Inserting this form of $Q(x,X,u,U,\tau)$ in , and performing some simplifications we obtain a master equation $$\begin{aligned}
\frac{\partial Q}{\partial \tau}=&&\mathcal{L}_\epsilon Q \label{Q-me}\end{aligned}$$ where the expansion parameter $\epsilon$ is explicit [@SM]. To proceed further we expand the operator $\mathcal{L}$ and the distributions $\Phi$, $\Psi$ in powers of $\epsilon$ as follows : $$\begin{aligned}
\mathcal{L}&=\mathcal{L}_0+\epsilon \mathcal{L}_1 + \epsilon^2 \mathcal{L}_2 +\mathcal{O}(\epsilon^3), \label{FP-O}\\
\Phi&=\Phi_0+\epsilon \Phi_1 + \epsilon^2 \Phi_2 + \mathcal{O}(\epsilon^3), \label{phi}\\
\Psi&=\Psi_0+\epsilon \Psi_1 + \epsilon^2 \Psi_2 +\mathcal{O}(\epsilon^3). \label{psi}\end{aligned}$$ Inserting these in the master equation , we look at the resulting equation at different orders of $\epsilon$. At each order we integrate the particle position $x$ and velocity $u$ to get following time evolution equations for $\Psi_0,~\Psi_1$ and $\Psi_2$. $$\begin{aligned}
%\begin{split}
&\frac{\partial \Psi_0}{\partial \tau}=0 \nonumber \\
&\frac{\partial \Psi_1}{\partial\tau} + U\frac{\partial \Psi_0}{\partial X} - \beta F\frac{\partial \Psi_0}{\partial
U} =-\frac{1}{X} \frac{\partial \Psi_0}{\partial U} \\
&\frac{\partial \Psi_2}{\partial\tau} + U\frac{\partial \Psi_1}{\partial X} + \left[-\beta F
+\frac{1}{X}\right]\frac{\partial \Psi_1}{\partial U}= \gamma(X)\frac{\partial }{\partial U}\left[U\Psi_0 +
\frac{\partial \Psi_0}{\partial U} \right] \nonumber
%\end{split}
\label{psinotdot}\end{aligned}$$ which provide the equation for $\Psi = \Psi_0 + \epsilon \Psi_1 + \epsilon^2 \Psi_2 + \mathcal{O}(\epsilon^3)$. In the original variables, we find $$\begin{aligned}
%\begin{split}
\frac{\partial \Psi}{\partial t} &=& - V\frac{\partial \Psi}{\partial X} - \frac{1}{M} \left(-F+\frac{k_BT}{X}\right)\frac{\partial
\Psi}{\partial V} \nonumber \\
&+& \frac{\gamma(X,t)}{M} \frac{\partial \left(V\Psi \right) }{\partial V} +\frac{\gamma(X,t) k_BT}{M^2} \frac{\partial^2 \Psi}{\partial V^2}
%\end{split}
\label{psi-FP}\end{aligned}$$ where the expression of $\gamma(X,t)$ is given in terms of an inverse Laplace transform in . The small and large time asymptotic forms of $\gamma(X,t)$ are $$\begin{aligned}
\gamma(X,t) ~\sim
\begin{cases}
& \frac{1}{X}\sqrt{\frac{8 mk_BT}{\pi}} ~~~;~~~t\rightarrow 0~~\nonumber\\
& \nonumber \\
& \frac{\sqrt{mk_BT}}{X}\log(t) ~~~;~~~t \rightarrow \infty.
\end{cases}
\label{gamma-p}\end{aligned}$$ Note that the Fokker-Planck Eq. corresponds precisely to the Langevin equation . However, our previously presented numerical evidences suggest that the friction coefficient is given by Eq. which is different from the prediction in Eq. .
At short times the predicted $\gamma(X)$ is same as $\gamma_{\text gas}$ while at large times it diverges logarithmically. This divergence indicates the breakdown of the perturbation theory, which however provides the correct form of equation of motion. The possible reasons for this breakdown are the following: (i) The perturbation theory here implicitly assumes a separation of time scales. As pointed out in a recent paper, for the case of a fixed piston the small particle shows a slow power-law relaxation to equilibrium [@Bhat]. This suggests that there may be no time scale separation between the particle and the piston. (ii) In addition, we have not taken the multiple collisions into account. These two means of breakdown arise because in the perturbation expansions in Eqs. -, we have implicitly assumed that the velocity $v$ of the small particle is always of order $\sqrt{k_BT}$. This is not always true because it is possible that the small particle can emerge from the bath with very small value ($\sim \epsilon$) and then the above perturbation expansion fails. Hence the derivation of exact expression for $\gamma$ is non-trivial. More details about the derivation of Eq.\[psi\] are given in the supplementary material.
[*Conclusion:*]{} In this Letter we looked at the non-equilibrium dynamics of the Szilard engine. We find that in the limit of large mass of the piston, it’s effective stochastic dynamics is given by a Langevin equation (Eq. \[eom\]) with a space dependent friction coefficient $\gamma(X)$ (Eq. \[gamma\]). To arrive at this equation we integrated out the particle degrees of freedom in the full master equation following the $\Omega$-expansion method. While this perturbation method correctly provides the form of the Langevin equation, it does not give the right form for the friction term, and we argue that this arises due to rare events in the small particle’s dynamics. However our extensive numerical studies suggests that the form of the friction coefficient $\gamma(X)$ given in Eq. is in fact accurate. To verify this form further, we have also considered the situation where the piston interacts with one thermal particle on each side. In this case also we find excellent numerical agreement (see [@SM]). Finally we used the Langevin equation description to study the thermodynamics of work extraction process in the Szilard engine. We found that, in the limit $M\rightarrow \infty$, and at a crucial value of the force $F_c$, the work extracted from the engine becomes equal to the heat absorbed, $k_BT \ln (2)$ — even though the piston remains out of equilibrium during the entire process. The analytic derivation of the result $\gamma(X)=\gamma_{sp}=(1/X)\sqrt{8\pi mk_BT}$ for the dissipation constant remains an interesting open problem.
Acknowledgements
================
D.B. acknowledges Arghya Dutta, Udo Seifert, Onuttom Narayan, Tridibh Sadhu and Satya Majumdar for discussions. A.D., A.K. and S.S. acknowledge support of the Indo-French Centre for the promotion of advanced research (IFCPAR) under Project No. 5604-2. A. K. acknowledges support from DST grant under project No. ECR/2017/000634.
|
---
abstract: 'In this article we prove that a semialgebraic map $\pi:M\to N$ is a branched covering if and only if its associated spectral map is a branched covering. In addition, such spectral map has a neat behavior with respect to the branching locus, the ramification set and the ramification index. A crucial result to prove this is the characterization of the prime ideals whose fiber under the previous spectral map is a singleton.'
address: 'Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 MADRID (SPAIN)'
author:
- 'E. Baro'
- 'Jose F. Fernando'
- 'J.M. Gamboa'
date: '23/01/2020'
title: |
Spectral maps associated to\
semialgebraic branched coverings
---
[^1]
Introduction {#s1}
============
The primary goal of semialgebraic geometry is to study the set of solutions of a finite system of polynomial inequalities in a finite number of variables with coefficients in the field ${{\mathbb R}}$ of real numbers or, more generally, in an arbitrary real closed field. Frequently, one wants to do this without using polynomial data, as it happens in classical algebraic geometry, where one often avoids working explicitly with the systems of polynomials equalities and non-equalities involved. After the pioneer work of Delfs-Knebusch [@dk2], where they introduced locally semialgebraic spaces and locally semialgebraic maps between them, real algebraic geometers realized the need of constructing their abstract counterpart.
A subset $M\subset{{\mathbb R}}^m$ is a *basic semialgebraic* if it can be written as $$M:=\{x\in{{\mathbb R}}^m:\ f(x)=0,g_1(x)>0,\ldots,g_{\ell}(x)>0\}$$ for some polynomials $f,g_1,\ldots,g_{\ell}\in{{\mathbb R}}[{{\tt x}}_1,\ldots,{{\tt x}}_m]$. The finite unions of basic semialgebraic sets are called [*semialgebraic sets*]{}. A continuous map $f:M\to N$ between semialgebraic sets $M\subset{{\mathbb R}}^m$ and $N\subset{{\mathbb R}}^n$ is [*semialgebraic*]{} if its graph is a semialgebraic subset of ${{\mathbb R}}^{m+n}$. In general, semialgebraic map refers to a (non necessarily continuous) map whose graph is semialgebraic. However, as most of semialgebraic functions appearing in this work are continuous, we omit for the sake of readability the continuity condition when referring to them. By Tarski-Seidenberg’s Theorem semialgebraic sets can be characterized as the first order definable sets in the pure field structure of $\mathbb{R}$ (see [@dries]).
The sum and product of functions defined pointwise endow the set ${\mathcal S}(M)$ of semialgebraic functions on $M$ with a natural structure of commutative ${{\mathbb R}}$-algebra with unit. The subset ${\mathcal S}^*(M)$ of bounded semialgebraic functions on $M$ is an ${{\mathbb R}}$-subalgebra of ${\mathcal S}(M)$. We use the notation ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ to refer indistinctly to both rings and we will denote ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M):={\operatorname{Spec}}({\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M))$ the Zariski spectra of ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ endowed with the Zariski topology. Recall that $M$ is a dense subset of ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)$. We denote ${\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)$ the maximal spectrum of ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$, that is, the set of closed points of ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)$. As ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ is a Gelfand ring (that is, each prime ideal of ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ is contained in a unique maximal ideal of ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$), there exists a natural retraction ${\tt r}_N:{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)\to{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)$, which is continuous. As it is well-known, ${\operatorname{\beta}}(M)$ and ${\operatorname{\beta^*}}(M)$ are homeomorphic (see for instance [@fg1 Thm.3.5]).
Each semialgebraic map $\pi:M\to N$ has associated a homomorphism of ${{\mathbb R}}$-algebras $\varphi_{\pi}^{{\text{\tiny$\displaystyle\diamond$}}}:{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(N)\to{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M),\ g\mapsto g\circ\pi$. Thus, one has morphisms $$\begin{aligned}
&{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi):{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)\to{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N),\ {{\mathfrak p}}\mapsto(\varphi_{\pi}^{{\text{\tiny$\displaystyle\diamond$}}})^{-1}({{\mathfrak p}}),\\
&{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(\pi):={\tt r}_N\circ{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)|_{{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)}:{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)\to{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)\to{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(N),\end{aligned}$$ which are continuous and ‘extend’ $\pi:M\to N$.
Morphisms between algebraic varieties over algebraically closed fields induce homomorphisms between their coordinate rings and these induce morphisms between their Zariski spectra. This is the classical approach to study morphisms between ‘geometric varieties’ via the ‘abstract morphisms’ between affine schemes. In the real setting it was not clear neither which are the right rings of functions to deal with nor which should be the ‘real affine schemes’. However, since the pioneer works [@br] by Brumfiel and [@cc] by Carral and Coste, it was realized that (continuous) semialgebraic functions provide a fruitful setting.
In addition, rings of semialgebraic functions present a key property: their Zariski and real spectra are canonically homeomorphic. In this way the theory of the real spectrum introduced by Coste and Roy [@bcr §7] provides powerful tools to understand the interplay between the geometric and the abstract settings. Last but not least it is worthwhile mention [@s1; @s2; @s3; @s4; @s5; @s6], where Schwartz developed much more that the abstraction of the geometric locally semialgebraic spaces studied in [@dk2] by Delfs and Knebusch.
The papers and books cited right now have a foundational nature. On the other hand, much more recently, the articles [@bfg; @fe1; @fe2; @fe3; @fg1; @fg2; @fg3; @fg4; @fg5; @fg6] are devoted to understand more in detail the relationship between a semialgebraic map $\pi:M\to N$ and its spectral counterpart ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi):{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)\to{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)$. This article focuses on this question when $\pi:M\to N$ is a semialgebraic branched covering.
Branched coverings constitute a common and useful tool in many subjects in Mathematics that appears often in Knot Theory, Orbifolds (quotients of manifolds under the discontinuous action of a group), (complex) Algebraic Geometry, (complex) Analytic Geometry, Riemann surfaces, etc. Given two topological spaces $X$ and $Y$, a continuous map $\pi:X\to Y$ is a *finite quasi-covering *if it is a separated, open, closed, surjective map whose fibers are finite (§\[quasi\]). Inspired by the theory of analytic coverings, we propose the following notion of *branched covering *(§\[branched\]) adapted to the definition of a good ramification index function. Roughly speaking, a finite quasi-covering $\pi:X\to Y$ is a *branched covering *if $\pi$ locally behaves as a covering with a constant number of sheets (after removing certain subset with dense complementary called the *ramification set *${\mathcal R}_{\pi}$). The ramification set ${\mathcal R}_{\pi}$ is the image under $\pi$ of the *branching locus *${\mathcal B}_{\pi}$, which is the set of points of $X$ at which $\pi$ is not a local homeomorphism. If the fibers at the points of $Y\setminus{\mathcal R}_{\pi}$ have constant cardinality $d$, we say that $\pi$ is a *$d$-branched covering*. There is a well-defined notion of *ramification index *at a point $x\in M$. Intuitively, it is the number of sheets that $\pi$ has close to $x$. A preliminary example that shows the subtleties of the definition of branched covering is Example \[notbranched\].**************
One first goal is to analyze the notions above in the semialgebraic setting. In the semialgebraic context maps with similar properties have been already studied. For instance, Schwartz characterized openness of semialgebraic maps $\pi:M\to N$ with finite fibers [@s1 Thm.13]: *a semialgebraic map $\pi$ with finite fibers is open if and only if the homomorphism $\varphi_{\pi}:{\mathcal S}(N)\to{\mathcal S}(M)$ is flat*.**
The main result of this article is the following. It analyzes the behavior of the spectral map associated to a semialgebraic branched covering.
\[bc\] Let $M\subset{{\mathbb R}}^m$ and $N\subset{{\mathbb R}}^n$ be semialgebraic sets and let $\pi:M\to N$ be a semialgebraic map. The following assertions are equivalent:
- $\pi$ is a branched covering.
- ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi):{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)\to{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)$ is a branched covering.
- ${\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(\pi):{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)\to{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(N)$ is a branched covering.
In addition, if such is the case, then:
- ${\mathcal B}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}({\mathcal B}_\pi)$ and ${\mathcal B}_{{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(\pi)}={\operatorname{Cl}}_{{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)}({\mathcal B}_\pi)$.
- ${\mathcal R}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)}({\mathcal R}_\pi)$ and ${\mathcal R}_{{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(\pi)}={\operatorname{Cl}}_{{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(N)}({\mathcal R}_\pi)$.
The openness, closedness and surjectivity of ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)$ follow from [@fg3]. As it is well-known, the topological space ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)$ is in general not Hausdorff. Thus, it is not clear why the spectral map ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)$ of a semialgebraic branched covering $\pi:M\to N$ should be a separable map. We will prove this in Proposition \[sep2\] through an analysis of the effect over ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ of symmetric polynomials via the semialgebraic branched covering $\pi:M\to N$ (as it is done in [@gr Thm. 12, Ch. III] with analytic coverings). Once this is proved (and consequently ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)$ is a finite quasi-covering), it remains to be shown that the spectral map ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)$ is in fact a branched covering. To ease the presentation of this fact we include in Section \[s2\] several (maybe known) technical results of topological nature that make the proof of Theorem \[bc\] more readable.
Another important tool is the study of the set of points of $X$ at which ‘there is a complete collapse of the fibers of a finite quasi-covering $\pi:X\to Y$’. More precisely, the *collapsing set *${\mathcal{C}}_\pi$ of a finite quasi-covering $\pi:X\to Y$ is the set of points $x\in X$ such that the fiber $\pi^{-1}(\pi(x))$ is a singleton. Given a semialgebraic $d$-branched covering $\pi:M\to N$, our purpose is to characterize the collapsing set ${\mathcal{C}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}$ of the spectral map ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi):{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)\to{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)$. To that aim we introduce (§\[mu\]) a map $\mu^{{\text{\tiny$\displaystyle\diamond$}}}:{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)\to{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(N)$, where $\mu^{{\text{\tiny$\displaystyle\diamond$}}}(f)(y)$ is ‘intuitively’ defined as the weighted arithmetic mean with respect to the ramification index of the values of $f\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ on the points of the finite fiber $\pi^{-1}(y)$. The homomorphism $\varphi^{{\text{\tiny$\displaystyle\diamond$}}}_{\pi}$ endows ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ with a natural structure of ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(N)$-module and the map $\mu^{{\text{\tiny$\displaystyle\diamond$}}}$ is a homomorphism of ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(N)$-modules.**
\[colapse\] Let $\pi:M\to N$ be a semialgebraic $d$-branched covering. Then
- ${\mathcal{C}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}$ is the set of prime ideals of ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ that contain $\ker(\mu^{{\text{\tiny$\displaystyle\diamond$}}})$.
- ${\mathcal{C}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}({\mathcal{C}}_\pi)$.
- ${\mathcal{C}}_{{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(\pi)}$ is the set of maximal ideals of ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ that contain $\ker(\mu^{{\text{\tiny$\displaystyle\diamond$}}})$.
- ${\mathcal{C}}_{{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(\pi)}={\operatorname{Cl}}_{{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)}({\mathcal{C}}_\pi)$.
Semialgebraic branched coverings were implicitly introduced by Brumfiel in [@br]. Given a semialgebraic set $M\subset{{\mathbb R}}^m$ and a closed equivalence relation $E\subset M\times M$ such that the projection $\pi:E\to M$ is proper, it was proved in [@br Thm. 1.4] the existence of a semialgebraic set $N\subset{{\mathbb R}}^m$, a surjective semialgebraic map $\pi:M\to N$ and a homeomorphism $g:M/E\to N$ such that $\pi=g\circ\rho$, where $\rho:M\to M/E$ is the natural projection. Scheiderer studied more general quotients in [@sch]. The map $\pi$ is usually a semialgebraic branched covering. We present an enlightening related example of this situation at the end of this paper (the Bezoutian covering). Another source of examples are algebraic morphisms between complex algebraic curves [@bcg1; @bcg2] (after taking their real intrinsic structures).
Branched coverings {#s2}
==================
We begin this section with some general topological facts. For each subset $A$ of a topological space $X$ we denote ${\operatorname{Cl}}_X(A)$ and ${\operatorname{Int}}_X(A)$ the closure and the interior of $A$ in $X$. In addition, $\#(A)$ denotes the cardinality of $A$. The following results are straightforward, but very useful for our discussion below.
\[trivial\] Let $\pi:X\to Y$ be a surjective map and let $Z\subset Y$. Denote $T:=\pi^{-1}(Z)$. Then for each set $A\subset X$, we have $\pi(A\cap T)=\pi(A)\cap\pi(T)$. In addition,
- If $\pi$ is open, $\pi|_T:T\to Z$ is open.
- If $\pi$ is closed, $\pi|_T:T\to Z$ is closed.
\[opcl\] Let $\pi:X\to Y$ be a continuous map and let $A\subset X$ and $B\subset Y$. Then
- If $\pi$ is open, ${\operatorname{Cl}}_X(\pi^{-1}(B))=\pi^{-1}({\operatorname{Cl}}_Y(B))$.
- If $\pi$ is closed, $\pi({\operatorname{Cl}}_X(A))={\operatorname{Cl}}_Y(\pi(A))$.
Finite quasi-coverings {#quasi}
----------------------
A continuous map $\pi:X\to Y$ is *separated *if each pair of points in the same fiber admit disjoint neighborhoods in $X$. A *finite quasi-covering *is a separated, open and closed surjective map $\pi:X\to Y$ whose fibers are finite.****
\[fqc\] Let $\pi:X\to Y$ be a finite quasi-covering and let $Z\subset Y$. Denote $T:=\pi^{-1}(Z)$. Then $\pi|_{T}:T\to Z$ is a finite quasi-covering by Lemma \[trivial\].
We define next some special neighborhoods related to the points of the spaces that appear in a finite quasi-covering.
\[disting\] Let $\pi:X\to Y$ be a finite quasi-covering and let $y\in Y$ be such that its fiber $\pi^{-1}(y):=\{x_1,\dots,x_r\}$ has $r$ distinct points. Let $W^{x_1},\dots,W^{x_r}\subset X$ be pairwise disjoint open neighborhoods of $x_1,\ldots,x_r$. Then there exists an open neighborhood $V_0\subset Y$ of $y$ such that for each open neighborhood $V\subset V_0$ of $y$ there exist pairwise disjoint open neighborhoods $U^{x_1},\dots,U^{x_r}\subset X$ of the points $x_1,\dots,x_r$ satisfying $U^{x_i}\subset W^{x_i}$, $$\label{cubierta}
\pi^{-1}(V)=\bigsqcup_{j=1}^rU^{x_j}\quad\text{and}\quad\pi(U^{x_j})=V\quad\text{for}\quad 1\leq j\leq r.$$ In addition, $\#(\pi^{-1}(z))\geq r$ for each $z\in V$. *We say that $V$ is a *distinguished neighborhood of $y$ *(with respect to $\pi$) and $U^{x_1},\ldots, U^{x_r}$ is a family of *characteristic neighborhoods *with respect to $V$. For any $x\in X$ we say that $U$ is a *characteristic neighborhood *of $x$ if $U$ is a member of a family of characteristic neighborhoods with respect to a distinguished open neighborhood of $\pi(x)$.*******
The existence of pairwise disjoint open neighborhoods $W^{x_1},\dots,W^{x_r}\subset X$ of the points $x_1,\ldots,x_r$ is guaranteed because $\pi$ is a separated map. As $\pi$ is an open and closed map, $$V_0:=\Big(Y\setminus\pi\Big(X\setminus\bigcup_{j=1}^rW^{x_j}\Big)\Big)\cap\bigcap_{j=1}^r\pi(W^{x_j})\subset Y$$ is an open neighborhood of $y$ and $\pi^{-1}(V_0)\subset\bigcup_{j=1}^rW^{x_j}$. Let $V\subset V_0$ be an open neighborhood of $y$. Define $U^{x_j}:=W^{x_j}\cap\pi^{-1}(V)\subset W^{x_j}\subset X$, which is an open neighborhood of $x_j$. Then $U^{x_i}\cap U^{x_j}=\varnothing$ if $i\neq j$ and the reader can check that equalities follow. Once this is checked the last part of the statement is clear.
\[intersection\] Let $\pi:X\to Y$ be a finite quasi-covering and let $y\in Y$. Let $\pi^{-1}(y):=\{x_1,\ldots,x_r\}$ and let $W_1,\ldots,W_k$ be open neighborhoods of $x_1,\ldots,x_k$ for some $k\leq r$. Suppose that $V$ is a distinguished neighborhood of $y$ and let $U_1,\ldots,U_r$ be characteristic neighborhoods of $x_1,\dots,x_r$ with respect to $V$. Then there exist a distinguished open neighborhood $\widetilde{V}\subset V$ of $y$ and characteristic neighborhoods $\widetilde{U}_i$ with respect to $\widetilde{V}$ (for $i=1,\ldots,r$) satisfying:
- $\widetilde{U}_i\subset U_i\cap W_i$ for $i=1,\ldots,k$
- $\widetilde{U}_i\subset U_i$ for $i=k+1,\ldots,r$
- $\widetilde{U}_i=\pi^{-1}(\widetilde{V})\cap U_i\cap W_i$ for $i=1,\ldots,k$
- $\widetilde{U}_i=\pi^{-1}(\widetilde{V})\cap U_i$ for $i=k+1,\ldots,r$.
Indeed, it is enough to apply Lemma \[disting\] to the family $U_1\cap W_1,\ldots, U_k\cap W_k,U_{k+1},\ldots, U_r$. Note that for each $z\in \widetilde{V}$, we have $\#(\pi^{-1}(z)\cap \widetilde{U}_i)=\#(\pi^{-1}(z)\cap U_i)$.
\[ccs0\] Let $\pi:X\to Y$ be a finite quasi-covering and let $y\in Y$. Write $\pi^{-1}(y)=\{x_1,\ldots,x_r\}$. Let $V$ be a distinguished neighborhood of $y$ and let $U^{x_1},\ldots,U^{x_r}$ be a family of characteristic neighborhoods with respect to $V$. Assume that $V$ is connected. Then $U^{x_1},\ldots,U^{x_r}$ are the connected components of $\pi^{-1}(V)$.
Each $U^{x_j}$ is open and closed in $\pi^{-1}(V)$. Suppose that $U^{x_1}$ is not connected. Then there exist two disjoint open and closed subsets $C_1,C_2$ of $U^{x_1}$ such that $U^{x_1}=C_1\cup C_2$. Assume $x_1\in C_1$. As $C_j$ is open and closed in $U^{x_1}$ and the restriction map $\pi|_{\pi^{-1}(V)}:\pi^{-1}(V)\to V$ is open and closed, $\pi(C_j)$ is an open and closed subset of $V$. As $V$ is connected, $\pi(C_j)=V$ for $j=1,2$. As $x_1\notin C_2$ and $\pi(C_2)=V$, we deduce $C_2\cap\{x_2,\ldots,x_r\}$ is not empty, which is a contradiction because $C_2\subset U^{x_1}$. Consequently, $U^{x_j}$ is connected for $j=1,\ldots,r$. As $\pi^{-1}(V)=\bigsqcup_{j=1}^rU^{x_j}$, we conclude $U^{x_1},\ldots,U^{x_r}$ are the connected components of $\pi^{-1}(V)$, as required.
If $\pi:X\to Y$ is a finite quasi-covering, the *branching locus *of $\pi$ is the closed set $\mathcal{B}_\pi$ of all points belonging to $X$ at which $\pi$ is not a local homeomorphism. The *ramification set *of $\pi$ is the closed set ${\mathcal R}_{\pi}:=\pi({\mathcal B}_{\pi})\subset Y$. The *regular locus *of $\pi$ is the open set $$X_{{\operatorname{reg}}}:=X\setminus\pi^{-1}({\mathcal R}_{\pi})\subset X.$$ We say that $\pi:X\to Y$ is a *$d$-unbranched covering *(for some integer $d\geq1$) if $X_{{\operatorname{reg}}}=X$ and the cardinality of each fiber is equal to $d$. In case we do not want to specify the integer $d$, we will say that $\pi$ is an *unbranched covering*. It is important to keep in mind that the fibers of an unbranched covering have constant cardinality (see Examples \[notbranched\]).**********
\[genbranch\] Let $\pi:X\to Y$ be a finite quasi-covering. Then $y\in Y\setminus{\mathcal R}_{\pi}$ if and only if there exists an open neighborhood $W\subset Y$ of $y$ such that the cardinality of the fiber $\pi^{-1}(z)$ for each $z\in W$ is constant.
Let $y\in Y$ and denote $\pi^{-1}(y):=\{x_1,\ldots,x_r\}$. Let $V$ be a distinguished neighborhood of $y$ with respect to $\pi$ and let $U^{x_1},\ldots,U^{x_r}$ be a family of characteristic neighborhoods. By Remark \[fqc\] the restriction map $\pi|_{\pi^{-1}(V)}:\pi^{-1}(V)=\bigsqcup_{i=1}^rU^{x_i}\to V$ is an open and closed map with finite fibers. Each $U^{x_i}$ is an open and closed subset of $\pi^{-1}(V)$. Thus, the restriction $\pi|_{U^{x_i}}:U^{x_i}\to V$ is an open and closed surjective map for each $i=1,\ldots,r$. Hence, $\pi|_{U^{x_i}}:U^{x_i}\to V$ is a homeomorphism if and only if it is injective.
If $y\in Y\setminus{\mathcal R}_{\pi}$ there exists an open neighborhood $W_i\subset U_i$ of $x_i$ such that $\pi|_{W_i}:W_i\to \pi(W_i)$ is a homeomorphism for each $i=1,\ldots,r$. Thus, by Remark \[intersection\] there exists an open distinguished neighborhood $\widetilde{V}\subset V$ of $y$ and a family $\widetilde{U}^{x_1},\ldots,\widetilde{U}^{x_r}$ of characteristic neighborhoods which satisfy $\widetilde{U}_i\subset U_i$ for $i=1,\ldots,r$. Thus, each $\pi|_{\widetilde{U}_i}$ is a homeomorphism, so the cardinality of $\pi^{-1}(z)$ is constant for $z\in \widetilde{V}$ as required.
Conversely, if there exists a neighborhood $W\subset Y$ of $y$ such that the cardinality of each fiber $\pi^{-1}(z)$ for $z\in W$ is constant, we replace $V$ by $W\cap V$. Now, each restriction $\pi|_{U^{x_i}}$ is injective. Thus, each restriction $\pi|_{U^{x_i}}:U^{x_i}\to V$ is a homeomorphism, so $y\in Y\setminus{\mathcal R}_{\pi}$, as required.
\[max\] Let $\pi:X\to Y$ be a finite quasi-covering and suppose that $d:=\sup\{\#(\pi^{-1}(y)):\ y\in Y\}<+\infty$. Let $y\in Y$ be such that $\#(\pi^{-1}(y))=d$. Then $y\in Y\setminus{\mathcal R}_{\pi}$.
Denote $\pi^{-1}(y):=\{x_1,\ldots,x_d\}$. Let $V$ be a distinguished neighborhood of $y$ with respect to $\pi$. By Lemma \[disting\] $d=\#(\pi^{-1}(y))\leq\#(\pi^{-1}(z))\leq d$ for each $z\in V$, so the cardinality of the fiber $\pi^{-1}(z)$ for each $z\in V$ is constant. By Lemma \[genbranch\] $y\in Y\setminus{\mathcal R}_{\pi}$, as required.
We finish this part with a topological property of certain finite quasi-coverings that will be used in the proof of Theorem \[bc\].
\[nowhere\] Let $\pi:X\to Y$ be a finite quasi-covering such that $X_{{\operatorname{reg}}}$ is dense in $X$. If $Z$ is a closed nowhere dense subset of $X$ then $\pi(Z)$ is a closed nowhere dense subset of $Y$.
As $\mathcal{R}_\pi$ is a nowhere closed subset of $Y$, if $\text{int}_Y(\pi(Z)\setminus \mathcal{R}_\pi)=\varnothing$, then $\text{int}_Y(\pi(Z))=\varnothing$. Thus, we can assume $X_{{\operatorname{reg}}}=X$.
Suppose there exists a non-empty open subset $V$ of $Y$ contained in $\pi(Z)$. As $X_{{\operatorname{reg}}}=X$, we can assume that there exist open subsets $U_1,\ldots,U_r$ of $X$ such that $\pi^{-1}(V)=\bigsqcup_{i=1}^r U_i $ and each restriction $\pi|_{U_i}:U_i\to V$ is a homeomorphism (Remark \[intersection\]). Thus, each $\widetilde{Z}_i:=\pi(U_i\cap Z)\subset V$ is a closed nowhere dense subset of $V$, whereas $V=\bigcup_{i=1}^{r} \widetilde{Z}_i$ is a non-empty open subset of $Y$, which is a contradiction.
Branched coverings {#branched}
------------------
Let $\pi:X\to Y$ be a finite quasi-covering, let $y\in Y$ and pick $x\in\pi^{-1}(y)$. Let $V$ be an open neighborhood of $y$ and let $U$ be an open and closed subset of $\pi^{-1}(V)$. By Lemma \[trivial\] the restriction map $\pi|_U:U\to \pi(U)$ is also a finite quasi-covering and $\mathcal{B}_{\pi|_U}=\mathcal{B}_{\pi}\cap U$, so that $\mathcal{R}_{\pi|_U}\subset \mathcal{R}_{\pi}$ and $X_{\text{reg}}\cap U\subset U_{\text{reg}}$.
With the notations introduced above:
\(i) A characteristic neighborhood $U$ of $x$ with respect to a distinguished neighborhood $V$ of $y$ such that the restriction $\pi|_{U_{{\operatorname{reg}}}}:U_{{\operatorname{reg}}}\to V\setminus{\mathcal R}_{\pi|_U}$ is an unbranched covering is called an *exceptional neighborhood of $x$ *(with respect to $V$). If each member of a family of characteristic neighborhoods with respect to $V$ is exceptional, then $V$ is a *special neighborhood of $y$*. In that case, we say that such a family is a *family of exceptional neighborhoods with respect to $V$*.******
The number $b_\pi(x)$ of sheets of $\pi|_{U_{{\operatorname{reg}}}}$ (the common cardinality of its fibers) is called the *ramification index *of $x$ relative to $U$. We will show in Lemma \[indexwell\] that $b_\pi(x)$ does not depend on $U$.**
\(ii) We say that $\pi$ is a *branched covering *if $X_{{\operatorname{reg}}}$ is a dense subset of $X$ and each $y\in Y$ admits a special neighborhood.**
\(iii) Let $d$ be a positive integer and let $\pi:X\to Y$ be a branched covering. For each $y\in Y$ there exists an open neighborhood $V$ of $y$ such that the cardinality of the fibers of the points in $(Y\setminus{\mathcal R}_\pi)\cap V$ is constant (Lemma \[genbranch\]). We say that $\pi$ is a *$d$-branched covering *if this constant equals $d$ for each point $y\in Y\setminus{\mathcal R}_\pi$.**
\[intersection:branched\] Let $\pi:X\to Y$ be a finite quasi-covering, let $y\in Y$ and denote $\pi^{-1}(y):=\{x_1,\ldots,x_r\}$. If each $x_i$ has an exceptional neighborhood $U_i$ for $i=1,\ldots,r$ and $U_1,\ldots,U_r$ are pairwise disjoint, then $y$ has a special neighborhood as small as required (note that we are not assuming that $U_1,\ldots,U_r$ is a family of characteristic neighborhoods with respect to its image, but that each $U_i$ belongs to a possibly different family of characteristic neighborhoods).
Indeed, by Remark \[intersection\] let $\widetilde{U}_1,\ldots,\widetilde{U}_r$ be a family of characteristic neighborhoods with respect to a distinguished neighborhood $\widetilde{V}$ of $y$ as small as required such that each $\widetilde{U}_i\subset U_i$. Recall that $\widetilde{U}_i=U_i\cap\pi^{-1}(\widetilde{V})$. We claim: $\mathcal{R}_{\pi|_{\widetilde{U}_i}}=\mathcal{R}_{\pi|_{U_i}}\cap\widetilde{V}$.
As $\mathcal{B}_{\pi|_{\widetilde{U}_i}}=\mathcal{B}_{\pi|_{U_i}}\cap\widetilde{U}_i=\mathcal{B}_{\pi|_{U_i}}\cap\pi^{-1}(\widetilde{V})$, we have by Lemma \[trivial\] $$\mathcal{R}_{\pi|_{\widetilde{U}_i}}=\pi(\mathcal{B}_{\pi|_{\widetilde{U}_i}})=\pi(\mathcal{B}_{\pi|_{U_i}}\cap\pi^{-1}(\widetilde{V}))=\pi(\mathcal{B}_{\pi|_{U_i}})\cap\widetilde{V}=\mathcal{R}_{\pi|_{U_i}}\cap\widetilde{V}.$$
We conclude $$\begin{gathered}
\widetilde{U}_{i,{\operatorname{reg}}}=\widetilde{U}_i\setminus\pi^{-1}(\mathcal{R}_{\pi|_{\widetilde{U}_i}})=\widetilde{U}_i\setminus\pi^{-1}(\mathcal{R}_{\pi|_{U_i}}\cap \widetilde{V})\\=(U_i\setminus\pi^{-1}(\mathcal{R}_{\pi|_{U_i}}))\cap\pi^{-1}(\widetilde{V})=U_{i,{\operatorname{reg}}}\cap\pi^{-1}(\widetilde{V})=U_{i,{\operatorname{reg}}}\cap\widetilde{U}_i\end{gathered}$$ for each $i=1,\ldots,r$. It follows that $$\pi|_{\widetilde{U}_{i,{\operatorname{reg}}}}:\widetilde{U}_{i,{\operatorname{reg}}}=U_{i,{\operatorname{reg}}}\cap\pi^{-1}(\widetilde{V})\to(\pi(U_i)\setminus\mathcal{R}_{\pi|_{U_i}})\cap\widetilde{V}$$ is an unbranched covering for each $i=1,\ldots,r$, as required.
\[indexwell\] Let $\pi:X\to Y$ be a branched covering and let $x\in X$. Then the ramification index $b_\pi(x)$ of $x$ values the same with respect to all exceptional neighborhoods of $x$.
We may assume that $\pi$ is a $d$-branched covering. Let $y\in Y$ and $\pi^{-1}(y):=\{x_1,\ldots,x_r\}$. Let $V$ be a special open neighborhood of $y$ and let $U_1,\ldots,U_r$ be a family of characteristic neighborhoods of $x_1,\ldots,x_r$ with respect to $V$. Let $V'$ be another special open neighborhood of $y$ and let $U'_1,\ldots,U'_r$ be a family of characteristic neighborhoods of $x_1,\ldots,x_r$ with respect to $V'$. We denote by $b^{U_i}_{\pi}(x_i)$ the ramification index of $x_i$ with respect to $U_i$, that is, $b^{U_i}_{\pi}(x_i)=\#(\pi^{-1}(z)\cap U_i)$ for each $z\in V\setminus \mathcal{R}_\pi$. Similarly, we consider the ramification index $b^{U'_i}_{\pi}(x_i)$ of $x_i$ with respect to $U'_i$. After reordering the indexes, it is enough to show: $b^{U_1}_{\pi}(x_1)=b^{U'_1}_{\pi}(x_1)$ (that is, the case $i=1$).
By Remark \[intersection\] there exists a distinguished open neighborhood $\widetilde{V}\subset V$ of $y$ and characteristic neighborhoods $\widetilde{U}_1,\ldots\widetilde{U}_r$ with respect to $\widetilde{V}$ such that $\widetilde{U}_1=\pi^{-1}(\widetilde{V})\cap U_1\cap U'_1$ and $\widetilde{U}_i=\pi^{-1}(\widetilde{V})\cap U_i$ for $i=2,\ldots,r$.
In particular, for each $z\in \widetilde{V}\setminus \mathcal{R}_{\pi}$ we have $b^{U_i}_{\pi}(x_i)=\# (\pi^{-1}(z)\cap U_i)=\# (\pi^{-1}(z)\cap \widetilde{U_i})$ for $i=2,\ldots,r$ and $$\begin{aligned}
b^{U'_1}_{\pi}(x_1)=\ &\# (\pi^{-1}(z)\cap U'_1)\geq\# (\pi^{-1}(z)\cap U_1\cap U'_1)=\# (\pi^{-1}(z)\cap\widetilde{U}_1)=\\
=\ &d-\sum^r_{i=2}\# (\pi^{-1}(z)\cap\widetilde{U}_i)=d-\sum^r_{i=2}b^{U_i}_{\pi}(x_i)=b^{U_1}_{\pi}(x_1). \end{aligned}$$ We have proved $b^{U_1}_{\pi}(x_1)\leq b^{U'_1}_{\pi}(x_1)$. The same argument shows $b^{U'_1}_{\pi}(x_1)\leq b^{U_1}_{\pi}(x_1)$, so $b^{U_1}_{\pi}(x_1)= b^{U'_1}_{\pi}(x_1)$, as required.
\[cuenta\] (i) Let $\pi:X\to Y$ be a branched covering. Let $y\in Y$ and let $V$ be a special neighborhood of $y$. Then the restriction map $\pi|_{\pi^{-1}(V)}:\pi^{-1}(V)\to V$ is a $d_y$-branched covering where $d_y:=\sum_{x\in\pi^{-1}(y)}b_\pi(x)$. Thus, $d_y=\#(\pi^{-1}(w))$ for each $w\in V\setminus\mathcal{R}_{\pi|_{\pi^{-1}(V)}}$ and $d_z=d_y$ for each $z\in V$.
\(ii) Let $\pi:X\to Y$ be a branched covering. If $Y$ is connected then $\pi$ is a $d$-branched covering for some $d\geq1$.
By Remark \[cuenta\](i) the set $Y_d:=\{y\in Y:d_y=d\}$ is open for each $d\in \mathbb{N}$. Consequently, each set $Y_d$ is also closed. As $Y$ is connected, we deduce that $Y=Y_d$ for some $d\in \mathbb{N}$.
\(iii) Let $\pi:X\to Y$ be a finite quasi-covering. Let $y\in Y$ and $x\in\pi^{-1}(y)$. Let $U\subset X$ be a characteristic neighborhood of $x$ with respect to the distinguished neighborhood $V\subset Y$ of $y$. Let $G$ be an open dense subset of $U$ such that the cardinality of the fibers of the restriction $\pi|_{G}:G\to \pi(G)$ is constant and equal to $d\in {{\mathbb N}}$. Then $\pi|_{U_{{\operatorname{reg}}}}:U_{{\operatorname{reg}}}\to V\setminus{\mathcal R}_{\pi|_U}$ is a $d$-unbranched covering and $U$ is an *exceptional neighborhood of $x$ *with respect to $V$.**
Indeed, it is clear that the branching locus of the finite quasi-covering $\pi|_{U_{{\operatorname{reg}}}}:U_{{\operatorname{reg}}}\to V\setminus{\mathcal R}_{\pi|_U}$ is empty. We claim: *$\#(\pi^{-1}(z)\cap U)=d$ for each $z\in\pi(G)$*.**
Let $z\in\pi(G)$. By hypothesis $\#(\pi^{-1}(z)\cap G)=d$. Assume $m:=\#(\pi^{-1}(z)\cap U)>d$ and let $V_0\subset\pi(G)$ be a distinguished neighborhood of $z$. Let $U_{01},\ldots,U_{0m}$ be a family of characteristic neighborhoods with respect to $V_0$. As $G$ is dense in $U$, the intersection $G\cap U_{0i}$ is dense in $U_{0i}$, so by Lemmas \[trivial\] and \[opcl\] $\bigcap_{i=1}^m\pi(G\cap U_{0i})$ is a dense open subset of $V_0$. If $z'\in\bigcap_{i=1}^m\pi(G\cap U_{0i})$, then $d=\#(\pi^{-1}(z')\cap G)\geq m>d$, which is a contradiction.
Let us prove next: *$\#(\pi^{-1}(y)\cap U)=d$ for each $y\in V\setminus{\mathcal R}_{\pi|_U}$*. Once this is shown the statement follows.**
Let $y\in V\setminus{\mathcal R}_{\pi|_U}$. By Lemma \[genbranch\] there exists an open neighborhood $W\subset V$ such that the cardinality of the fiber $\pi^{-1}(z)\cap U$ for each $z\in W$ is a constant $c$. By Remark \[intersection\] we can assume that $\pi^{-1}(W)\cap U_{{\operatorname{reg}}}=\bigsqcup^c_{i=1} U_i$ where each $\pi|_{U_i}$ is a homeomorphism onto $W$. As $G$ is an open dense subset of $U$, we deduce that $G\cap U_i$ is an open dense subset of $U_i$ for each $i=1,\ldots,c$. Thus, $\bigcap^c_{i=1} \pi(G\cap U_i)$ is a dense open subset of $W$. If $z\in \bigcap^c_{i=1}\pi(G\cap U_i)$, then the fiber $\pi^{-1}(z)\cap U$ has $d$ elements, so $c=d$. Thus, $\pi|_{U_{{\operatorname{reg}}}}:U_{{\operatorname{reg}}}\to V\setminus{\mathcal R}_{\pi|_U}$ is a $d$-unbranched covering.
\(iv) Let $\pi:X\to Y$ be a branched covering. Let $y\in Y$ and write $\pi^{-1}(y):=\{x_1,\ldots,x_r\}$. Let $W^{x_i}\subset X$ be an open neighborhood of $x_i$ for each $i=1,\ldots,r$. Then there exists an open special neighborhood $V\subset Y$ of $y$ and a family of exceptional neighborhoods $U^{x_1},\ldots,U^{x_r}$ with respect to $V$ such that $U^{x_i}\subset W^{x_i}$ for $i=1,\ldots,r$.
Let $V_0\subset Y$ be a special neighborhood of $y$ and let $U_0^{x_i}$ be the corresponding exceptional neighborhood of $x_i$. By Remark \[intersection\] there exists a distinguished neighborhood $V\subset V_0$ of $y$ and a family of characteristic neighborhoods $U^{x_i}\subset U_0^{x_i}\cap W^{x_i}$ of $x_i$ with respect to $V$ such that $\pi^{-1}(V)\cap U_0^{x_i}\cap W^{x_i}=U^{x_i}$ for each $i=1,\ldots,r$. Let us check: *$V$ is a special neighborhood of $y$ and (to that end) that each $U^{x_i}$ is an exceptional neighborhood of $x_i$*.**
As $U^{x_i}$ is open and closed in $\pi^{-1}(V)$, we have $\mathcal{B}_{\pi|_{U^{x_i}}}=\mathcal{B}_{\pi|_{U_0^{x_i}}}\cap U^{x_i}$. In addition, as $\pi(U^{x_i})=V$, we deduce by Lemma \[trivial\] $\mathcal{R}_{\pi|_{U^{x_i}}}=\mathcal{R}_{\pi|_{U_0^{x_i}}}\cap V$. Thus, it is enough to show: *$\#(\pi^{-1}(z)\cap U^{x_i}_i)=\#(\pi^{-1}(z)\cap U^{x_i}_0)$ for each $z\in V$*, which follows from the last line of Remark \[intersection\].**
\(v) Let $\pi:X\to Y$ be a branched covering. Let $U$ be an exceptional neighborhood of a point $x\in U$. Then $b_{\pi}(x):=\max\{\#(\pi^{-1}(y)\cap U):\ y\in \pi(U)\}$. In addition, for each $x'\in U$ we have $b_{\pi}(x')\leq b_{\pi}(x)$. Thus, for each $e\in \mathbb{N}$ the set $\{b_{\pi}\leq e\}$ is open in $M$, so $b_{\pi}$ is upper semi-continuous.
By definition $b_{\pi}(x)=\#(\pi^{-1}(y))$ for each $y\in\pi(U)\setminus\mathcal{R}_{\pi|_U}$. Pick $y'\in V:=\pi(U)$ and denote $\pi^{-1}(y')\cap U=\{x'_1,\ldots,x'_\ell\}$. By Remark \[cuenta\](iv) there exist a special neighborhood $V'\subset V$ of $y'$ and exceptional neighborhoods $U'_1,\ldots,U'_\ell\subset U$ of $x'_1,\ldots,x'_\ell$. Each intersection $U_{{\operatorname{reg}}}\cap U'_i$ is a dense open subset of $U'_{i}$, so $\bigcap^\ell_{i=1} \pi(U_{{\operatorname{reg}}}\cap U'_i)$ is a dense open subset of $V'$. Consequently, $\ell\leq b_{\pi}(x)$.
Next, given $x'\in U$ there exists by Remark \[cuenta\](iv) an exceptional neighborhood $U'$ of $x'$ contained in $U$. It follows that $$b_{\pi}(x')=\max\{\#(\pi^{-1}(y)\cap U'):\ y\in \pi(U')\}\leq \max\{\#(\pi^{-1}(y)\cap U):\ y\in \pi(U)\}=b_{\pi}(x).$$
\(vi) Let $\pi:X\to Y$ be a branched covering and let $x\in X$. Then $b_{\pi}(x)=1$ if and only if $x\notin \mathcal{B}_{\pi}$.
If $x\notin \mathcal{B}_{\pi}$, then by Remark \[intersection\] and Remark \[cuenta\](iv) there exists an exceptional neighborhood $U$ of $x$ such that $\pi|_U$ is a homeomorphism, so $b_{\pi}(x)=1$. Conversely, if $b_{\pi}(x)=1$, then by Remark \[cuenta\](v) for any exceptional neighborhood $U$ of $x$ we have that $1=\max\{ \#(\pi^{-1}(y)\cap U):\ y\in \pi(U)\}$, so $\pi|_U$ is a homeomorphism.
### Behavior of branched coverings under restriction
We analyze next how branched coverings behave under restriction.
\[restr\] Let $\pi:X\to Y$ be a branched covering. Let $W\subset Y$ be an open set. Let $W\subset Z\subset{\operatorname{Cl}}(W)$ and denote $T:=\pi^{-1}(Z)$. Then $\pi|_{T}:T\to Z$ is a branched covering, ${\mathcal B}_{\pi|_T}={\mathcal B}_\pi\cap T$, $T_{{\operatorname{reg}}}=X_{{\operatorname{reg}}}\cap T$ and ${\mathcal R}_{\pi|_T}={\mathcal R}_\pi\cap Z$.
First, we show: *If $U$ is an open subset of $X$ and $A$ is a dense subset of $U$, then $A\cap T$ is dense in $U\cap T$*.**
By Lemma \[opcl\] the inverse image $\pi^{-1}(W)$ is dense in $T$. As $A$ is dense in $U$ and $\pi^{-1}(W)$ is open, $A\cap\pi^{-1}(W)$ is dense in $U\cap\pi^{-1}(W)$, so $A\cap T$ is dense in $U\cap T$.
By Remark \[fqc\] $\pi|_{T}$ is a finite quasi-covering. Observe that ${\mathcal B}_{\pi|_T}\subset{\mathcal B}_\pi\cap T$, so ${\mathcal R}_{\pi|_T}\subset{\mathcal R}_\pi\cap Z$ and $X_{{\operatorname{reg}}}\cap T\subset T_{{\operatorname{reg}}}$. As $X_{{\operatorname{reg}}}$ is dense in $X$, it follows $T_{{\operatorname{reg}}}$ is dense in $T$.
Pick a point $y\in Z$ and write $\pi^{-1}(y):=\{x_1,\ldots,x_r\}\subset T$. Let $V$ be a special neighborhood of $y$ and $U_1,\ldots,U_r$ be a family of exceptional neighborhoods of $x_1,\ldots,x_r$ with respect to $V$. Thus, each restriction $\pi|_{U_{i,{\operatorname{reg}}}}:U_{i,{\operatorname{reg}}}\to V\setminus{\mathcal R}_{\pi|_{U_i}}$ is a $(b_\pi(x_i))$-unbranched covering. Regarding $\pi|_T:T\to Z$, the open set $V\cap Z$ is a distinguished neighborhood of $y$ and $U_1\cap T,\ldots,U_r\cap T$ is a family of characteristic neighborhoods with respect to $V\cap Z$. In addition, $$\pi|_{(U_i\cap T)_{{\operatorname{reg}}}}:(U_i\cap T)_{{\operatorname{reg}}}\to(V\cap Z)\setminus{\mathcal R}_{\pi|_{U_i\cap T}}$$ is also a $(b_\pi(x_i))$-unbranched covering by Remark \[cuenta\](iii) because $U_{i,{\operatorname{reg}}}\cap T$ is a dense open subset of $U_i\cap T$ and the cardinality of the fibers of $\pi|_{U_{i,{\operatorname{reg}}}\cap T}$ equals $b_\pi(x_i)$. We conclude that $U_1\cap T,\ldots,U_r\cap T$ is a family of exceptional neighborhoods with respect to the special neighborhood $V\cap T$. In particular, $\pi|_{\pi^{-1}(V)\cap T}:\pi^{-1}(V)\cap T\to V\cap Z$ is a $d$-branched covering where $d:=\sum_{i=1}^rb_\pi(x_i)$.
By Remark \[cuenta\](v) $x_i\in{\mathcal B}_{\pi|_T}$ if and only if $b_\pi(x_i)>1$. Therefore ${\mathcal B}_{\pi|_T}={\mathcal B}_\pi\cap T$. In particular, ${\mathcal R}_{\pi|_T}={\mathcal R}_\pi\cap Z$ by Lemma \[trivial\]. Finally, $$T_{{\operatorname{reg}}}=T\setminus \pi^{-1}(\mathcal{R}_{\pi|_T})=T\setminus (\pi^{-1}(\mathcal{R}_{\pi})\cap\pi^{-1}(Z))=T\setminus \pi^{-1}(\mathcal{R}_{\pi})=T\cap X_{{\operatorname{reg}}},$$ as required.
The following is a straightforward consequence.
\[rcc\] Let $\pi:X\to Y$ be a map and assume that $Y$ has finitely many connected components $Y_1,\ldots,Y_r$. Denote $X_i:=\pi^{-1}(Y_i)$ for each $i=1,\ldots,r$. Then $\pi:X\to Y$ is a branched covering if and only for each $i=1,\ldots, k$ there exists an integer $d_i\geq 1$ such that ${\pi|_{X_i}:X_i\to Y_i}$ is a $d_i$-branched covering.
Each connected component of $Y$ is open and closed, so the result follows from Remark \[cuenta\](ii) and Lemma \[restr\].
\[opcl2\] Let $\pi:X\to Y$ be a branched covering. Let $T\subset X$ be an open and closed set and denote $Z:=\pi(T)$. Then $\pi|_T:T\to Z$ is a branched covering, ${\mathcal B}_{\pi|_T}={\mathcal B}_\pi\cap T$, ${\mathcal R}_{\pi|_T}\subset{\mathcal R}_\pi\cap Z$ and $X_{{\operatorname{reg}}}\cap T\subset T_{{\operatorname{reg}}}$.
First, observe that $Z$ is open and closed in $Y$. As $T$ is an open and closed subset of $\pi^{-1}(Z)$, the restriction map $\pi|_T:T\to Z$ is a finite quasi-covering with ${\mathcal B}_{\pi|_T}={\mathcal B}_\pi\cap T$, ${\mathcal R}_{\pi|_T}\subset{\mathcal R}_\pi\cap Z$ and $T\cap X_{{\operatorname{reg}}}\subset T_{{\operatorname{reg}}}$. As $X_{{\operatorname{reg}}}$ is dense in $X$ and $T$ is open, $X_{{\operatorname{reg}}}\cap T$ is dense in $T$, so $T_{{\operatorname{reg}}}$ is dense in $T$.
Pick $y\in Z$ and write $\pi^{-1}(y):=\{x_1,\ldots,x_r\}$. We may assume $\pi^{-1}(y)\cap T=\{x_1,\ldots,x_s\}$ for some $s\leq r$. Let $V\subset Y$ be a special neighborhood of $y$ and $U_1,\ldots,U_r$ be a family of exceptional neighborhoods of $x_1,\ldots,x_r$ with respect to $V$. As each set $U_1\cap T,\ldots, U_s\cap T$ is open, we can assume by Remark \[cuenta\](iv) that $U_1,\ldots,U_s\subset T$, so $V\subset Z$. As $\pi|_{U_{i,{\operatorname{reg}}}}:U_{i,{\operatorname{reg}}}\to V\setminus{\mathcal R}_{\pi|_{U_i}}$ is an unbranched covering for $i=1,\ldots,s$, we conclude that $V$ is a special neighborhood with respect to $\pi|_T:T\to Z$ and $U_1,\ldots,U_s$ are exceptional neighborhoods. Thus, $\pi|_T$ is a branched covering, as required.
### Some special branched coverings
We propose next some mild sufficient conditions under which we can guarantee that a finite quasi-covering is a branched covering.
\[sc\] Let $\pi:X\to Y$ be a finite quasi-covering such that $d:=\sup\{\#(\pi^{-1}(y)):\ y\in Y\}<+\infty$ and for each $y\in Y$ there exists a distinguished neighborhood $V$ of $y$ such that $V\cap(Y\setminus{\mathcal R}_\pi)$ is connected. Then $\pi$ is a branched covering.
We show first: *$X_{{\operatorname{reg}}}=X\setminus\pi^{-1}({\mathcal R}_{\pi})$ is dense in $X$*. It is enough to check: *$Y\setminus{\mathcal R}_{\pi}$ is dense in $Y$*.****
Suppose that ${\mathcal R}_{\pi}$ contains a non-empty open subset $V\subset Y$. Define $V_k:=\{y\in V: \#(\pi^{-1}(y))=k\}$ and note that $V=V_1\sqcup\cdots\sqcup V_d$. In particular, for some $1\leq k\leq d$ the interior ${\operatorname{Int}}(V_k)\cap V$ of $V_k$ in $V$ is not empty. By Lemma \[genbranch\] we get ${\operatorname{Int}}(V_k)\cap V\subset Y\setminus{\mathcal R}_\pi$, which is a contradiction.
Fix $y\in Y$. By hypothesis there exists a distinguished neighborhood $V$ of $y$ such that ${V\cap(Y\setminus{\mathcal R}_\pi)}$ is connected. As $Y\setminus{\mathcal R}_\pi$ is dense in $Y$, the intersection $V\cap(Y\setminus{\mathcal R}_\pi)$ is dense in $V$, so $V$ is connected. Write $\pi^{-1}(y):=\{x_1,\ldots,x_r\}$.
Let $U_1,\ldots,U_r$ be a family of characteristic neighborhoods of $x_1,\ldots,x_r$ with respect to $V$. Then $\pi|_{U_i}:U_i\to V$ is a finite quasi-covering and ${\mathcal R}_{\pi|_{U_i}}\subset{\mathcal R}_{\pi}\cap V$ for $i=1,\ldots,r$. Thus, each difference $V\setminus {\mathcal R}_{\pi|_{U_i}}$ is connected. By Lemma \[genbranch\] the cardinality of the fibers of the restriction map $\pi|_{U_{i,{\operatorname{reg}}}}:U_{i,{\operatorname{reg}}}\to V\setminus {\mathcal R}_{\pi|_{U_i}}$ is locally constant. As $V\setminus {\mathcal R}_{\pi|_{U_i}}$ is connected, $\pi|_{U_{i,{\operatorname{reg}}}}$ is a $k_i$-unbranched covering for some $k_i\in{{\mathbb N}}$, so $U_i$ is an exceptional neighborhood of $x_i$, as required.
The previous situation is quite common. It arises for instance when one considers the underlying real structure of a complex irreducible analytic germ and analyzes local parameterization theorem [@gr Ch.2.B & Ch.3.B] (as a consequence of Noether’s normalization lemma [@am Ch.5.Ex.16]). Analogously, it appears when consider the underlying real structure of Noether’s normalization lemma of a complex irreducible algebraic set [@am Ch.5.Ex.16].
Collapsing set of a finite quasi-covering {#s3}
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The main purpose of this section is to analyze the set of points at which there exists a complete collapse of the fibers of a semialgebraic branched covering. This notion will be crucial for the purposes of the paper.
Let $\pi:X\to Y$ be a finite quasi-covering. We define the *collapsing set of $\pi$ *as ${\mathcal{C}}_\pi:=\{x\in X:\pi^{-1}(\pi(x))=\{x\}\}$.**
\(i) The collapsing set of a finite quasi-covering $\pi:X\to Y$ is a closed subset of $X$.
By Lemma \[disting\] the cardinality of the fibers of the points close to a given point $y\in Y$ is greater than or equal to the cardinality of $\pi^{-1}(y)$. Thus, the set $S$ of points whose fiber contains at least two points is an open subset of $Y$. Thus, ${\mathcal C}_\pi=\pi^{-1}(X\setminus S)$ is a closed subset of $X$.
\(ii) If $\pi:X\to Y$ is a $d$-branched covering, its collapsing set is ${\mathcal{C}}_{\pi}=\{x\in X:\ b_\pi(x)=d\}$.
We will need also the following result.
\[colapseinB\] Let $\pi:X\to Y$ be a finite quasi-covering. Let $C\subset X$ and $D\subset Y$ be such that $\pi|_C:C\to D$ is a $d$-branched covering with $d>1$. Then $\mathcal{C}_{\pi|_C}\subset \mathcal{B}_\pi$.
Let $x\in \mathcal{C}_{\pi|_C}$ and suppose that $x\notin \mathcal{B}_\pi$. Then there exists an open neighborhood $W$ of $x$ such that $\pi|_W:W\to \pi(W)$ is a homeomorphism. As $\pi|_C:C\to D$ is an open continuous map, $\pi|_{W\cap C}:W\cap C\to\pi(W\cap C)$ is an open continuous bijective map, so it is a homeomorphism and $\pi(W\cap C)$ is an open subset of $D$. Thus, by Remark \[cuenta\](vi) we deduce $b_{\pi|_C}(x)=1$. As $x\in \mathcal{C}_{\pi|_C}$, we conclude $b_{\pi|_C}(x)=d$, so $d=1$, which is a contradiction.
Semialgebraic branched coverings
--------------------------------
As one can expect a *semialgebraic finite quasi-covering *is a finite quasi-covering that is in addition a semialgebraic map. As semialgebraic sets are Hausdorff spaces, we deduce the following from Lemma \[disting\].**
\[semiquasi\] Let $\pi:M\to N$ be an open, closed, surjective semialgebraic map with finite fibers between semialgebraic sets $M$ and $N$. Then $\pi$ is a finite quasi-covering and each point $y\in N$ admits a basis of distinguished semialgebraic neighborhoods with respect to $\pi$.
Concerning the branching locus and ramification set we have the following.
\[ss\] Let $\pi:M\to N$ be a semialgebraic finite quasi-covering.
- The cardinality of the fibers of $\pi$ is bounded by a common constant.
- $\mathcal{B}_\pi,M_{{\operatorname{reg}}},\mathcal{C}_{\pi}\subset M$ and ${\mathcal R}_{\pi}\subset N$ are semialgebraic sets.
\(i) This is a direct consequence of the cell decomposition of semialgebraic sets [@dries Corollary 3.7].
\(ii) A point $x\in\mathcal{B}_\pi$ if and only if the restriction $\pi|_V:V\to\pi(V)$ is not a homeomorphism for each open semialgebraic neighborhood $V\subset M$ of $x$. As $\pi$ is open, continuous and surjective, we deduce $x\in\mathcal{B}_\pi$ if and only if the restriction $\pi|_V$ is not injective on each open semialgebraic neighborhood $V\subset M$ of $x$. Thus, a point $x\in M$ belongs to ${\mathcal B}_{\pi}$ if and only if for each ${\varepsilon}>0$ there exist points $u_1,u_2\in M$ satisfying $\|x-u_i\|<{\varepsilon}$ such that $u_1\neq u_2$ and $\pi(u_1)=\pi(u_2)$. Consequently, ${\mathcal B}_{\pi}$ is a semialgebraic subset of $M$. Therefore, ${\mathcal R}_{\pi}=\pi({\mathcal B}_{\pi})$ is a semialgebraic subset of $N$ and $M_{{\operatorname{reg}}}:=M\setminus\pi^{-1}({\mathcal R}_{\pi})$ is a semialgebraic subset of $M$. The set $\mathcal{C}_{\pi}=\{x\in M:\ \pi^{-1}(x)=\{x\}\}$ is straightforwardly semialgebraic, as required.
A *semialgebraic branched covering *is a map $\pi:M\to N$ that is simultaneously a branched covering and a semialgebraic map. Lemma \[indexwell\], Corollary \[rcc\] and Lemma \[sc\] apply readily in the semialgebraic case. In order to show some subtleties hidden in the definition of branched coverings we provide next an example of a semialgebraic finite quasi-covering that is not a semialgebraic branched covering.**
\[notbranched\] (i) Consider the semialgebraic subsets of ${{\mathbb R}}^2$ defined by $$\begin{split}
M_1:&=([-2,0]\times\{3/2\})\cup\Big\{(x,y)\in{{\mathbb R}}^2:\ 0\leq x\leq 2,\ y=\frac{3\pm x}{2}\Big\},\\
M_2:&=([0,2]\times\{3\})\cup\Big\{(x,y)\in{{\mathbb R}}^2:\ -2\leq x\leq 0,\ y=\frac{6\pm x}{2}\Big\}
\end{split}$$ and $N:=[-2,2]\times\{0\}$. The projection $\pi:M:=M_1\cup M_2\to N,\ (x,y)\mapsto x$ is a semialgebraic finite quasi-covering, but it is not a semialgebraic branched covering.
The branching locus of $\pi$ is $\mathcal{B}_\pi:=\{p_1:=(0,3/2),p_2:=(0,3)\}$, so the ramification set is ${\mathcal R}_{\pi}:=\pi({\mathcal B}_{\pi})=\{q:=(0,0)\}$. We have $\#(\pi^{-1}(q))=2$ and $\#(\pi^{-1}(y))=3$ for each point $y\in N\setminus\{q\}$.
The regular locus of $\pi$ is the dense subset $M_{{\operatorname{reg}}}=M\setminus\{{p_1,p_2}\}$ of $M$. Suppose that $\pi$ is a semialgebraic branched covering. Then there exists a distinguished open semialgebraic neighborhood $V$ of $q$ in $N$ and open semialgebraic neighborhoods $U_i$ of $p_i$ such that for $i=1,2$ the restriction $\pi|_{M_{{\operatorname{reg}}}\cap U_i}:M_{{\operatorname{reg}}}\cap U_i\to (N\setminus{\mathcal R}_{\pi})\cap V$ is a semialgebraic unbranched covering. This is false because
- the cardinality of the fibers of $\pi|_{M_{{\operatorname{reg}}}\cap U_1}$ of the points belonging to $(N\setminus{\mathcal R}_{\pi})\cap V\cap\{x<0\}$ equals $1$,
- the cardinality of the fibers of $\pi|_{M_{{\operatorname{reg}}}\cap U_1}$ of the points belonging to $(N\setminus{\mathcal R}_{\pi})\cap V\cap\{x>0\}$ equals $2$.
\(ii) A similar pathology can be achieved in the (more restrictive) real algebraic case if one considers $$X:=\{x=(y-2)^2\}\cup\{x=-(y+2)^2\}\cup\{y=2\}\cup\{y=-2\},\quad Y:=\{y=0\}$$ and $\pi:X\to Y,\ (x,y)\mapsto x$. The previous map is a finite quasi-covering, the general fiber has $4$ points, but it is not a branched covering.
(2,5.3) node[$M_2$]{}; (2,1.7) node[$M_1$]{}; (2,0.3) node[$N$]{};
; ; ; ; ; ; ; coordinates[(0,1.5)(0,3)(0,0)]{};
Let $\pi:M\to N$ be a semialgebraic branched covering. The ramification index function $b_\pi:M\to{{\mathbb N}}\subset{{\mathbb R}}$ has semialgebraic graph.
Let $d:=\max\{\#(\pi^{-1}(y)):\ y\in N\}<+\infty$. Observe that $b_\pi(M)\subset\{1,\ldots,d\}$. For each $k=1,\ldots,d$ define ${\mathcal B}_{\pi,k}=\{x\in M:\ b_\pi(x)=k\}$, ${\mathcal B}_{\pi,k}^*=\{x\in M:\ b_\pi(x)\geq k\}$ and ${\mathcal B}_{\pi,d+1}^*=\varnothing$. As ${\mathcal B}_{\pi,k}={\mathcal B}_{\pi,k}^*\setminus{\mathcal B}_{\pi,k+1}^*$ for $k=1,\ldots,d$, to prove that $b_\pi$ is a semialgebraic map, it is enough to check: *${\mathcal B}_{\pi,k}^*$ is a semialgebraic set for each $k=1,\ldots,d$*.**
It holds that $$\begin{gathered}
{\mathcal B}_{\pi,k}^*=\{x\in M:\ \forall{\varepsilon}>0,\ \exists u_1,\ldots,u_k\in M,\ \|x-u_i\|<{\varepsilon}\\
(\forall i=1,\ldots,k),\ u_i\neq u_j,\ \pi(u_1)=\cdots=\pi(u_k)\}.\end{gathered}$$ As ${\mathcal B}_{\pi,k}^*$ is described by a first order formula, we deduce that it is a semialgebraic set for $k=1,\ldots,d+1$, as required.
Even though we will not use it in the sequel, in the following result we analyze further the semialgebraic nature of the ramification index. Recall that a finite semialgebraic partition $\{M_\ell\}_{\ell=1}^r$ of a semialgebraic set $M\subset{{\mathbb R}}^n$ is *compatible *with a semialgebraic subset $S\subset M$ if $S$ is the union of those $M_\ell$ that meet $S$.**
\[partition\] Let $\pi:M\to N$ be a semialgebraic branched covering and denote $d:=\max\{\#(\pi^{-1}(y)):\ y\in N\}<+\infty$. Consider the semialgebraic sets $B_m:=\{b_\pi=m\}$ for $m=1,\ldots,d$. Then there exists a finite semialgebraic partition $\{M_{k\ell}\}_{k,\ell}$ of $M$ and a finite semialgebraic partition $\{N_k\}$ of $N$ such that:
- $\{M_{k\ell}\}_{k,\ell}$ is compatible with $\{B_m\}_{m=1}^d$.
- $\{N_k\}$ is compatible with ${\mathcal R}_\pi$.
- $\pi^{-1}(N_k)=\bigsqcup_\ell M_{k\ell}$ and $\pi|_{M_{k\ell}}:M_{k\ell}\to N_k$ is a semialgebraic homeomorphism.
- If $x\in M$, $N_k\subset N\setminus{\mathcal R}_\pi$ and $\pi(x)\in{\operatorname{Cl}}(N_k)$, then $b_\pi(x)=\#(\{\ell:\ x\in{\operatorname{Cl}}(M_{k\ell})\})$.
By Hardt’s trivialization theorem [@bcr 9.3.2] and Lemma \[ss\] there exist:
- a semialgebraic partition $\{P_1,\dots,P_r\}$ of $N$,
- positive integers $s_j\geq1$ for $j=1,\ldots,r$ and
- semialgebraic homeomorphisms $\theta_j:P_j\times\{1,\ldots,s_j\}\to \pi^{-1}(P_j)$
such that for $1\leq j\leq r$ we have the following commutative diagram $$\xymatrix{
P_j\times\{1,\ldots,s_j\}\ar@{->}[r]^(0.625){\quad\theta_j\quad}\ar@{->}[rd]^{\pi_j\quad}&\pi^{-1}(P_j)\ar@{->}[d]^{\pi\quad}\\
&P_j}$$ where $\pi_j:P_j\times\{1,\ldots,s_j\}\to P_j$ is the projection onto $P_j$. Denote $T_{ji}:=\theta_j(P_j\times\{i\})$ for $i=1,\ldots,s_j$. Fix $j=1,\ldots,r$ and consider the semialgebraic sets $$P_{j,m_1,\ldots,m_{s_j}}:=\pi(T_{j1}\cap B_{m_1})\cap\cdots\cap\pi(T_{js_j}\cap B_{m_{s_j}})$$ where $m_1,\ldots,m_{s_j}\in\{1,\ldots,d\}$. We claim: *the semialgebraic sets $P_{j,m_1,\ldots,m_{s_j}}$ are pairwise disjoint*.**
Suppose that $y\in P_{j,m_1,\ldots,m_{s_j}}\cap P_{j,m_1',\ldots,m_{s_j}'}$. As $\pi|_{T_{ji}}:T_{ji}\to P_j$ is a semialgebraic homeomorphism, there exists a unique $x_i\in T_{ji}$ such that $\pi(x_i)=y$. Observe that $x_i\in B_{m_i}\cap B_{m_i'}$. As the semialgebraic sets $B_m$ are pairwise disjoint, we deduce $m_i=m_i'$ for $i=1,\ldots,s_j$, as claimed.
Denote $\{N_k\}_k$ the collection of the connected components of the semialgebraic sets $P_{j,m_1,\ldots,m_{s_j}}$ that are non-empty and let $M_{k\ell}$ be the connected components of $\pi^{-1}(N_k)$. Observe that $\pi|_{M_{k\ell}}:M_{k\ell}\to N_k$ is a semialgebraic homeomorphism. In addition, *each $M_{k\ell}$ is contained in some $B_m$ for each pair $(k,\ell)$*, so $\{M_{k\ell}\}_{k,\ell}$ is compatible with $\{B_m\}_{m=1}^d$.**
Indeed, fix a pair $(k,\ell)$ and let $P_{j,m_1,\ldots,m_{s_j}}\subset P_j$ be such that $N_k$ is one of its connected components. As $M_{k\ell}$ is a connected component of $\pi^{-1}(N_k)$, $\pi^{-1}(P_j)=\bigsqcup_{i=1}^{s_j}T_{ji}$ and each $T_{ji}$ is open and closed in $\pi^{-1}(P_j)$, there exists $i=1,\ldots,s_i$ such that $M_{k\ell}\subset T_{ji}$. As $P_{j,m_1,\ldots,m_{s_j}}=\pi(T_{j1}\cap B_{m_1})\cap\cdots\cap\pi(T_{js_j}\cap B_{m_{s_j}})$ and $\pi|_{T_{ji}}:T_{ji}\to P_j$ is a homeomorphism, we conclude $M_{k\ell}\subset T_{ji}\cap B_{m_i}\subset B_{m_i}$, as claimed.
As ${\mathcal R}_\pi=\pi({\mathcal B}_\pi)=\bigcup_{m=2}^d\pi(B_m)$ (use Remark \[cuenta\](v)), we deduce that $\{N_k\}_k$ is compatible with ${\mathcal R}_\pi$. We claim: *We can also assume that for every open neighborhood $V_x$ of a point $x\in {\operatorname{Cl}}_N(N_k)$ there exists an open neighborhood $x\in W^x\subset V_x$ such that $W^x\cap N_k$ is connected.***
Indeed, we may assume that $N$ is bounded. By the triangulation theorem [@bcr Thm.9.2.1] there exists a triangulation $\phi:|K|\to {\operatorname{Cl}}_{{{\mathbb R}}^n}(N)$ compatible with the semialgebraic partition $\{N_k\}_k$ of $N$. We identify $N$ with $|K|$ and consider the new semialgebraic partition of $N$ given by $\{\sigma^0\}_{\sigma\subset N}$. Observe that for each $x\in |K|$ and each open ball $W^x$ centered in $x$, the set $W_x\cap\sigma^0$ is convex, so in particular connected.
Finally, we show (iv). Pick a point $x\in M$ and let $m$ be such that $x\in B_m$. Let $N_k\subset N\setminus{\mathcal R}_\pi$ be such that $\pi(x)\in{\operatorname{Cl}}(N_k)$. Let $V\subset N$ be a special neighborhood of $y=\pi(x)$ and let $U\subset M$ be an exceptional neighborhood of $x$ with respect to $V$. Thus, $\pi|_{U_{{\operatorname{reg}}}}:U_{{\operatorname{reg}}}\to V\setminus{\mathcal R}_{\pi|_U}$ is an unbranched covering of $m$ sheets. Shrinking $V$ we may assume that each non-empty intersection $V\cap N_k\neq\varnothing$ is connected (use the previous claim). For each $M_{k\ell}$ the restriction $\pi|_{M_{k\ell}}$ is a homeomorphism, so $\pi^{-1}(V)\cap M_{k\ell}$ is connected too. As $U$ is open and closed in $\pi^{-1}(V)$, it follows that $U\cap M_{k\ell}$ is an open and closed subset of $\pi^{-1}(V)\cap M_{k\ell}$. Thus, either $U\cap M_{k\ell}=\pi^{-1}(V)\cap M_{k\ell}$ or $U\cap M_{k\ell}=\varnothing$. Hence, $x\in{\operatorname{Cl}}(M_{k\ell})$ if and only if $U\cap M_{k\ell}=\pi^{-1}(V)\cap M_{k\ell}$. As $N_k\subset N\setminus \mathcal{R}_\pi$, the cardinality of the family of sets $M_{k\ell}$ such that $U\cap M_{k\ell}=\pi^{-1}(V)\cap M_{k\ell}$ equals $m$, as required.
Branched coverings and spectral maps
====================================
In this section we analyze the properties of the spectral maps of semialgebraic finite quasi-coverings and branched coverings. These results will be applied in Sections \[mu\] and \[s5\]. One of the main results of this section is Proposition \[sep2\], where we prove that if $\pi:M\to N$ is a semialgebraic branched covering, the spectral map ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)$ is separable and in fact it is a finite quasi-covering.
Preliminaries on rings of semialgebraic functions {#prelim}
-------------------------------------------------
Let $M\subset{{\mathbb R}}^m$ be a semialgebraic set and let $f\in{\mathcal S}(M)$. We denote $${{Z}}(f):=\{x\in M:\ f(x)=0\}\quad\text{and}\quad{{D}}(f):=M\setminus{{Z}}(f).$$ If $N$ is a closed semialgebraic subset of $M$, the semialgebraic function $g:={\operatorname{dist}}(\cdot,N)\in{\mathcal S}(M)$ satisfies $Z(g)=N$. In fact, substituting $g$ by $\frac{g}{1+g^2}$ we may assume in addition that $g$ is bounded. The restriction homomorphism ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)\to{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(N), f\mapsto f|_N$ is by [@dk1 Thm. 3] surjective. Denote $$\begin{split}
{{\EuScript Z}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(f):&=\{{{\mathfrak p}}\in{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M):\ f\in{{\mathfrak p}}\},\\
{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(f):&={\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)\setminus{{\EuScript Z}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(f)=\{{{\mathfrak p}}\in{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M):\ f\notin{{\mathfrak p}}\}.
\end{split}$$ The semialgebraic set $M$ is identified with a dense subspace of ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)$ via the embedding $${\tt j}_M:M\hookrightarrow{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M),\ x\mapsto{{\mathfrak m}}^{{\text{\tiny$\displaystyle\diamond$}}}_x,$$ where ${{\mathfrak m}}^{{\text{\tiny$\displaystyle\diamond$}}}_x:=\{f\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M):\ f(x)=0\}$ is the maximal ideal of ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ associated to $x$. In particular, ${{Z}}(f)=M\cap{{\EuScript Z}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(f)$ and ${{D}}(f)=M\cap{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(f)$.
We denote ${\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)\subset{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)$ the set of maximal ideals of ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$. As ${{\mathfrak m}}_x\in{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)$ for each point $x\in M$, we have $M\subset{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)$. Denote $\partial M:={\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)\setminus M$ and recall that ${\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)$ is a Hausdorff space. In addition, ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ is a Gelfand ring, that is, each prime ideal ${{\mathfrak p}}\in{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)$ is contained in a unique prime ideal ${{\mathfrak m}}\in{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)$. This fact provides a natural retraction ${\tt r}_M:{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)\to{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)$, which is continuous [@mo Thm.1.2].
If $\pi:M\to N$ be a semialgebraic map, the induced maps $$\begin{aligned}
&{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi):{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)\to{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N),\ {{\mathfrak p}}\mapsto\varphi_{\pi}^{-1}({{\mathfrak p}}),\\
&{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(\pi):={\tt r}_N\circ{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)|_{{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)}:{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)\to{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)\to{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(N)\end{aligned}$$ are continuous, ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)|_{M}=\pi$ and ${\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(\pi)|_{M}=\pi$. We recall here the following result from [@fg3 Thm.1.6] and [@fg6 Rem.4.2].
\[propern\] Let $\pi:M\to N$ be a semialgebraic map.
- If $\pi$ is open, closed and surjective, then ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)$ is open, closed and surjective.
- Suppose that $\pi$ is surjective. Then it is proper if and only if ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)(\partial M)=\partial N$.
Observe that ${\mathcal W}_M:=\{f\in{\mathcal S}^*(M):\ Z(f)=\varnothing\}$ is a multiplicatively closed subset of ${\mathcal S}^*(M)$ and ${\mathcal S}(M)={\mathcal W}_M^{-1}{\mathcal S}^*(M)$. This is so because each function $f\in{\mathcal S}(M)$ can be written as $f=g/h$, where $$g:=\frac{f}{1+f^2}\in{\mathcal S}^*(M)\quad\text{and}\quad h:=\frac{1}{1+f^2}\in{\mathcal W}_M.$$ As ${\mathcal S}(M)={\mathcal W}_M^{-1}{\mathcal S}^*(M)$, there exists a bijection (which is in fact a homeomorphism [@fg2 Lem.3.2]) $$\begin{aligned}
\phi&:{\operatorname{Spec}}(M)\to{\mathfrak S}(M),\ {{\mathfrak q}}\mapsto{{\mathfrak q}}\cap{\mathcal S}^*(M),\\
\phi^{-1}&:{\mathfrak S}(M)\to{\operatorname{Spec}}(M),\ {{\mathfrak q}}'\mapsto{\mathcal W}_M^{-1}{{\mathfrak q}}'\end{aligned}$$ (which preserves inclusions between prime ideals) where ${\mathfrak S}(M)$ is the set of prime ideals of ${\mathcal S}^*(M)$ that do not meet ${\mathcal W}_M$.
\[r:maximal\] Recall that the map $\beta(M)\mapsto\beta^*(M)$ that maps each maximal ideal ${{\mathfrak m}}$ of ${\mathcal S}(M)$ to the unique maximal ideal ${{\mathfrak m}}^*$ of $\mathcal{S}^*(M)$ that contains ${{\mathfrak m}}\cap\mathcal{S}^*(M)$ is a homeomorphism [@fe1 2.5.2]. In addition, by [@fe1 Prop.5.1] the following property holds: *Let ${{\mathfrak p}}\in{\operatorname{Spec}}^*(M)$ be a prime ideal. If ${{\mathfrak m}}^*\in\beta^*(M)$ is the unique maximal ideal of ${\mathcal S}^*(M)$ that contains ${{\mathfrak p}}$ and ${{\mathfrak m}}\in\beta(M)$ is the unique maximal ideal of ${\mathcal S}(M)$ such that ${{\mathfrak m}}\cap\mathcal{S}^*(M)\subset{{\mathfrak m}}^*$, then either ${{\mathfrak p}}\subset{{\mathfrak m}}\cap\mathcal{S}^*(M)$ or ${{\mathfrak m}}\cap\mathcal{S}^*(M)\subset{{\mathfrak p}}$*.**
Given a semialgebraic map $\pi:M\to N$ the induced homomorphism $\varphi_{\pi}:{\mathcal S}(N)\to{\mathcal S}(M)$ maps ${\mathcal S}^*(N)$ to ${\mathcal S}^*(M)$ and we denote $\varphi^*_{\pi}:{\mathcal S}^*(N)\to{\mathcal S}^*(M)$ the restricted homomorphism.
\[mintomin\] Let $\pi:M\to N$ be an open, closed and surjective semialgebraic map. Then
- The induced homomorphism $\varphi^{{\text{\tiny$\displaystyle\diamond$}}}_{\pi}:{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(N)\to{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ satisfies the going-down property.
- The spectral map ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi):{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)\to{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)$ maps minimal prime ideals of ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ to minimal prime ideals of ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(N)$.
\(i) The bounded case is afforded in [@fg3 3.6], so let us analyze what happens with the homomorphism $\varphi_{\pi}:{\mathcal S}(N)\to{\mathcal S}(M)$. Let ${{\mathfrak p}}_1\subset{{\mathfrak p}}_2$ be prime ideals of ${\mathcal S}(N)$ and let ${{\mathfrak q}}_2$ be a prime ideal of ${\mathcal S}(M)$ such that ${{\mathfrak p}}_2=\varphi_{\pi}^{-1}({{\mathfrak q}}_2)$. As ${\mathcal S}(N)={\mathcal W}_N^{-1}{\mathcal S}^*(N)$, we write ${{\mathfrak p}}_i={\mathcal W}_N^{-1}{{\mathfrak p}}_i'$, where ${{\mathfrak p}}_i':={{\mathfrak p}}_i\cap{\mathcal S}^*(N)$ is a prime ideal of ${\mathcal S}^*(N)$ that does not intersect ${\mathcal W}_N$. Analogously, we write ${{\mathfrak q}}_2={\mathcal W}_M^{-1}{{\mathfrak q}}_2'$, where ${{\mathfrak q}}_2':={{\mathfrak q}}_i\cap{\mathcal S}^*(M)$ is a prime ideal of ${\mathcal S}^*(M)$ that does not intersect ${\mathcal W}_M$. By [@fg3 3.6] the homomorphism $\varphi^*_{\pi}:{\mathcal S}^*(N)\to{\mathcal S}^*(M)$ satisfies the going-down property, so there exists a prime ideal ${{\mathfrak q}}_1'$ of ${\mathcal S}^*(M)$ such that ${{\mathfrak q}}_1'\subset{{\mathfrak q}}_2'$ and ${{\mathfrak p}}_1'=(\varphi^*_{\pi})^{-1}({{\mathfrak q}}_1')$. In addition, ${{\mathfrak q}}_1'\cap{\mathcal W}_M\subset{{\mathfrak q}}_2'\cap{\mathcal W}_M=\varnothing$. Thus, ${{\mathfrak q}}_1:={\mathcal W}_M^{-1}{{\mathfrak q}}_1'$ is a prime ideal of ${\mathcal S}(M)$ lying over ${{\mathfrak p}}_1$ via $\varphi_{\pi}$.
\(ii) Let ${{\mathfrak p}}$ be a minimal prime ideal of ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ and suppose that ${{\mathfrak q}}:={\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)({{\mathfrak p}})$ is not a minimal prime ideal of ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(N)$. Then there exists a prime ideal ${{\mathfrak q}}_1\subsetneq{{\mathfrak q}}$. By part (i) there exists a prime ideal ${{\mathfrak p}}_1\subsetneq{{\mathfrak p}}$ such that ${{\mathfrak q}}_1:={\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)({{\mathfrak p}}_1)$, which is a contradiction because ${{\mathfrak p}}$ is a minimal prime ideal of ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$, as required.
Addition of radical and prime ideals
------------------------------------
We need some results concerning the addition of radical and prime ideals of rings of semialgebraic functions.
\[suma\] Let ${{\mathfrak a}}$ be a radical ideal of ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ and let ${{\mathfrak p}}$ be a prime ideal of ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$. Let $f,g\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$. It holds:
- If $|f|\leq |g|$ and $g\in{{\mathfrak a}}$, then $f\in{{\mathfrak a}}$.
- The sum of two radical ideals of ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ is a radical ideal.
- The sum ${{\mathfrak a}}+{{\mathfrak p}}$ is either ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ or a prime ideal of ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$.
\(i) The semialgebraic function defined as $$h:M\to{{\mathbb R}},\ x\mapsto
\begin{cases}
0&\text{if $x\in Z(g)$,}\\[4pt]
f^2(x)/g(x)&\text{if $x\in D(g)$}\\
\end{cases}$$ satisfies $f^2=hg\in{{\mathfrak a}}$ and it is bounded in case $f$ is bounded. As ${{\mathfrak a}}$ is a radical ideal, $f\in{{\mathfrak a}}$.
\(ii) This follows from [@s2] because ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ is a real closed ring (we refer the reader to [@s2 §III.1] and [@s4 Thm. 5.12]).
\(iii) By (ii) ${{\mathfrak b}}:={{\mathfrak a}}+{{\mathfrak p}}$ is a radical ideal. Let ${{\EuScript P}}:=\{{{\mathfrak q}}\in{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M):\ {{\mathfrak b}}\subset{{\mathfrak q}}\}$. Then ${{\mathfrak b}}=\bigcap_{{{\mathfrak q}}\in{{\EuScript P}}}{{\mathfrak q}}$. The set ${{\EuScript P}}$ is by [@fg2 (3.1.4)] and [@fg5 Thm.1.1] a finite chain with respect to the inclusion. Thus, either ${{\mathfrak b}}={\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ or ${{\mathfrak b}}$ is a prime ideal, as required.
Recall that an ideal ${{\mathfrak a}}$ of the ring ${\mathcal S}(M)$ is a *$z$-ideal *if for each pair of functions $f,g\in{\mathcal S}(M)$ such that $f\in{{\mathfrak a}}$ and $Z(f)\subset Z(g)$, it holds $g\in{{\mathfrak a}}$. In particular, $z$-ideals are radical ideals.**
\[min\] Let $M\subset{{\mathbb R}}^n$ be a semialgebraic set.
- Let ${{\mathfrak a}}_1,{{\mathfrak a}}_2$ be two $z$-ideals of ${\mathcal S}(M)$. Then the sum ${{\mathfrak a}}_1+{{\mathfrak a}}_2$ is either ${\mathcal S}(M)$ or a $z$-ideal of ${\mathcal S}(M)$.
- Let ${{\mathfrak p}}_1,\dots,{{\mathfrak p}}_k$ be minimal prime ideals of ${\mathcal S}(M)$. Then the sum ${{\mathfrak p}}:={{\mathfrak p}}_1+\cdots+{{\mathfrak p}}_k$ is ${\mathcal S}(M)$ or a prime $z$-ideal of ${\mathcal S}(M)$.
\(i) Suppose that ${{\mathfrak a}}_1+{{\mathfrak a}}_2\neq{\mathcal S}(M)$ and let $f,g\in{\mathcal S}(M)$ be such that $f\in{{\mathfrak a}}_1+{{\mathfrak a}}_2$ and $Z(f)\subset Z(g)$. Then there exist functions $f_i\in{{\mathfrak a}}_i$ such that $f=f_1+f_2$. As $Z(f_1)\cap Z(f_2)\subset Z(f)\subset Z(g)$, the function $$h:N:=Z(f_1)\cup Z(f_2)\to{{\mathbb R}},\ x\mapsto
\begin{cases}
0&\text{if $x\in Z(f_1)$},\\
g(x)&\text{if $x\in Z(f_2)$}\\
\end{cases}$$ is a well-defined semialgebraic function. By [@dk1] there exists $H\in{\mathcal S}(M)$ such that $H|_N=h$. We have $Z(f_1)\subset Z(H)$ and $Z(f_2)\subset Z(g-H)$. As each ${{\mathfrak a}}_i$ is a $z$-ideal, $H\in{{\mathfrak a}}_1$ and $g-H\in{{\mathfrak a}}_2$. Thus, $g=H+(g-H)\in{{\mathfrak a}}_1+{{\mathfrak a}}_2$.
\(ii) We prove the statement by induction on $k$. By [@fe1 Cor.4.7] all minimal prime ideals of ${\mathcal S}(M)$ are $z$-ideals. Suppose $k\geq2$ and let ${{\mathfrak q}}:={{\mathfrak p}}_1+\cdots+{{\mathfrak p}}_{k-1}$. By induction hypothesis either ${{\mathfrak q}}={\mathcal S}(M)$ or ${{\mathfrak q}}$ is a prime $z$-ideal. By part (i) the sum ${{\mathfrak p}}={{\mathfrak q}}+{{\mathfrak p}}_k$ is either ${\mathcal S}(M)$ or a $z$-ideal. In the last case ${{\mathfrak p}}$ is by Lemma \[suma\](iii) a prime ideal, as required.
Symmetric polynomials and semialgebraic $d$-branched coverings
--------------------------------------------------------------
As it is done in [@gr Thm. 12, Ch. III] with analytic coverings, we analyze the effect over ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ of symmetric polynomials via a semialgebraic $d$-branched covering $\pi:M\to N$.
\[sym\] Let $\pi:M\to N$ be a semialgebraic $d$-branched covering and let $\sigma\in{{\mathbb R}}[{{\tt x}}_1,\ldots,{{\tt x}}_d]$ be a symmetric polynomial. Let $f\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ and define $$\sigma(f):N\to{{\mathbb R}},\ y\mapsto\sigma(f(x_1),\overset{b_\pi(x_1)}{\ldots},f(x_1),\ldots,f(x_r),\overset{b_\pi(x_r)}{\ldots},f(x_r))$$ if $\pi^{-1}(y)=\{x_1,\ldots,x_r\}$ (recall that $b_\pi(x_1)+\cdots+b_\pi(x_r)=d$). Then $\sigma(f)\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(N)$.
As $\sigma$ is a symmetric polynomial, $\sigma$ is a well-defined function. We prove first: *$\sigma(f)$ is continuous on $N$*.**
Pick a point $y\in Y$ and write $\pi^{-1}(y):=\{x_1,\ldots,x_r\}$. Fix ${\varepsilon}>0$. As $\sigma:{{\mathbb R}}^d\to{{\mathbb R}}$ is continuous at the point $$p:=(p_1,\ldots,p_d):=(f(x_1),\overset{b_\pi(x_1)}{\ldots},f(x_1),\ldots,f(x_r),\overset{b_\pi(x_r)}{\ldots},f(x_r)),$$ there exists $\delta>0$ such that if $q:=(q_1,\ldots,q_d)\in{{\mathbb R}}^d$ and $|p_i-q_i|<\delta$ for $i=1,\ldots,d$, then $|\sigma(p)-\sigma(q)|<{\varepsilon}$. As $f$ is continuous at $x_1,\ldots,x_r$, there exist open neighborhoods $A^{x_i}$ of $x_i$ such that $|f(z_i)-f(x_i)|<\delta$ for each $z_i\in A^{x_i}$ for $i=1,\ldots,r$. Let $V\subset N$ be a special neighborhood of $y$ and $U^{x_1},\ldots,U^{x_r}$ a family of exceptional neighborhoods with $U^{x_i}\subset A^{x_i}$ for $i=1,\ldots,r$ (use Remark \[cuenta\](iv)). Pick a point $y'\in V$ and write $\pi^{-1}(y'):=\{z_{11},\ldots,z_{1s_1},\ldots,z_{r1},\ldots,z_{rs_r}\}$ where $z_{ij}\in U^{x_i}$ for $j=1,\ldots,s_i$ and $i=1,\ldots,r$. Using the fact that $\pi|_{M_{{\operatorname{reg}}}}:M_{{\operatorname{reg}}}\to N\setminus{\mathcal R}_\pi$ is an unbranched covering, the reader can check that $\sum_{j=1}^{r_i}b_\pi(z_{ij})=b_\pi(x_i)$ for $i=1,\ldots,r$ (Remark \[cuenta\](v) can be useful). As $x_i,z_{ij}\in U^{x_i}\subset A^{x_i}$, we have $|f(x_i)-f(z_{ij})|<\delta$ for $j=1,\ldots,s_i$ and $i=1,\ldots,r$. As $\sum_{j=1}^{r_i}b_\pi(z_{ij})=b_\pi(x_i)$ for $i=1,\ldots,r$, we conclude $$\begin{split}
|\sigma(f)(y)-\sigma(f)(y')|&=|\sigma(f(x_1),\overset{b_\pi(x_1)}{\ldots},f(x_1),\ldots,f(x_r),\overset{b_\pi(x_r)}{\ldots},f(x_r))\\
&-\sigma(f(z_{11}),\overset{b_\pi(z_{11})}{\ldots},f(z_{11}),\ldots,f(z_{1s_1}),\overset{b_\pi(z_{1s_1})}{\ldots},f(z_{1s_1}),\\&\ldots,f(z_{r1}),\overset{b_\pi(z_{r1})}{\ldots},f(z_{r1}),\ldots,f(z_{rs_r}),\overset{b_\pi(z_{rs_r})}{\ldots},f(z_{rs_r}))|<{\varepsilon}.
\end{split}$$ Thus, $\sigma(f)$ is continuous.
In addition, if $f$ is bounded, then it is straightforward to check that $\sigma(f)$ is also bounded because finite sums of finite products of bounded values is a bounded value.
We prove next: *$\sigma(f)$ has semialgebraic graph*.**
The restriction $\pi|_{M_{{\operatorname{reg}}}}:M_{{\operatorname{reg}}}\to N\setminus{\mathcal R}_\pi$ is a semialgebraic map. For each $y\in N\setminus{\mathcal R}_\pi$ there exist exactly $d$ different points $x_1,\ldots,x_d\in M$ such that $\pi(x_i)=y$ and $\sigma(f)(y)=\sigma(f(x_1),\ldots,f(x_d))$ (as the polynomial $\sigma$ is symmetric the order of the evaluation is not relevant). Thus, the graph of $\sigma(f)|_{N\setminus{\mathcal R}_\pi}$ is a first order definable set, so $\sigma(f)|_{N\setminus{\mathcal R}_\pi}$ is a semialgebraic map. As $\sigma(f)$ is a continuous map, the set $N\setminus{\mathcal R}_\pi$ is dense in $N$ and $\sigma(f)|_{N\setminus{\mathcal R}_\pi}=\sigma(f|_{M\setminus\pi^{-1}({\mathcal R}_\pi}))$ is a semialgebraic function on $N\setminus{\mathcal R}_\pi$, we conclude that the graph $\Gamma(\sigma(f))$ of $\sigma(f)$ is the semialgebraic set ${\operatorname{Cl}}_{M\times{{\mathbb R}}}(\Gamma(\sigma(f)|_{N\setminus{\mathcal R}_\pi}))$. Thus, $\sigma(f)$ is a semialgebraic function, as required.
Separate spectral maps
----------------------
We prove next a separation result for certain pair of points in ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)$, which will allow us to prove in Proposition \[sep2\] that the spectral map associated to a semialgebraic branched covering is separated.
\[sep\] Let ${{\mathfrak p}}_1,{{\mathfrak p}}_2\in{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)$ be such that ${{\mathfrak p}}_1\not\subset{{\mathfrak p}}_2$ and ${{\mathfrak p}}_2\not\subset{{\mathfrak p}}_1$. Then there exist $f_1,f_2\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ such that ${{\mathfrak p}}_i\in{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(f_i)$ and $f_1f_2=0$. In particular, ${{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(f_1)\cap{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(f_2)=\varnothing$.
Let $g_1\in{{\mathfrak p}}_2\setminus{{\mathfrak p}}_1$ and let $g_2\in{{\mathfrak p}}_1\setminus{{\mathfrak p}}_2$. Observe that $(g_i-|g_i|)(g_i+|g_i|)=0\in{{\mathfrak p}}_i$ and changing $g_i$ by $-g_i$ if necessary, we may assume $g_i-|g_i|\in{{\mathfrak p}}_i$ for $i=1,2$. This means that $g_i+{{\mathfrak p}}_i$ defines a positive element of the real closed field $\kappa({{\mathfrak p}}_i):={\rm qf}({\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)/{{\mathfrak p}}_i)$ for $i=1,2$. Let $h:=g_1-g_2$ and observe that $h\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)\setminus({{\mathfrak p}}_1\cup{{\mathfrak p}}_2)$. In addition, $h+{{\mathfrak p}}_1$ is strictly positive in $\kappa({{\mathfrak p}}_1)$ and $h+{{\mathfrak p}}_2$ is strictly negative in $\kappa({{\mathfrak p}}_2)$. Define $f_1:=h+|h|\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ and $f_2:=h-|h|\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ and note that $f_1f_2=0\in {{\mathfrak p}}_i$ for $i=1,2$. As $h+{{\mathfrak p}}_1$ is strictly positive in $\kappa({{\mathfrak p}}_1)$, we deduce $f_2\in{{\mathfrak p}}_1$. As $h+{{\mathfrak p}}_2$ is strictly negative in $\kappa({{\mathfrak p}}_2)$, we deduce $f_1\in{{\mathfrak p}}_2$. If $f_1,f_2\in{{\mathfrak p}}_i$, then $h\in{{\mathfrak p}}_i$, which is a contradiction. We conclude $f_i\not\in{{\mathfrak p}}_i$, so ${{\mathfrak p}}_i\in{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(f_i)$ and $f_1f_2=0$, as required.
The previous type of neighborhoods in ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)$ are used below to show the local connectedness of the Zariski spectra of rings of semialgebraic functions.
\[neigh\] Let $M\subset{{\mathbb R}}^m$ be a semialgebraic set and let $f\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$. Denote $D:=D(f)\subset M\subset {\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)$. Then ${{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(f)\subset{\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(D)$.
Denote $C:={\operatorname{Cl}}_M(D)\subset {\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)$ and let ${{\mathfrak p}}\in{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(f)$. We claim: *the kernel of the restriction homomorphism $\psi:{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)\to{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(C),\ g\mapsto g|_C$ is contained in ${{\mathfrak p}}$*.**
Let $h\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ be such that $h|_C=0$. As $hf=0$ and ${{\mathfrak p}}\in{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(f)$, we deduce $h\in{{\mathfrak p}}$.
Now, by [@fg2 Lem.4.3] ${{\mathfrak p}}\in{\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(C)$ and we conclude ${{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(f)\subset{\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(C)$, as required.
\[ccD\] Let $M\subset{{\mathbb R}}^m$ be a semialgebraic set and let $g\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$. Let $E_1,\ldots,E_s$ be the connected components of $D(g)$. Then the connected components of ${{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(g)$ are ${\mathcal V}_i:={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(E_i)\cap{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(g)$ for $i=1,\ldots,s$. In addition, ${\mathcal V}_i={{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(g)\setminus\bigcup_{j\neq i}{\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(E_j)$ is an open subset of ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)$.
It is enough to show: *each ${\mathcal V}_i$ is connected, ${{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(g)=\bigcup_{i=1}^s{\mathcal V}_i$ and ${\mathcal V}_j\cap{\mathcal V}_k=\varnothing$ if $j\neq k$*.**
As $E_i$ is connected, ${\mathcal V}_i={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(E_i)\cap{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(g)$ is also connected. By Lemma \[neigh\] $$\begin{gathered}
{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(g)={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}({{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(g))\cap{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(g)\\
={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}\Big(\bigcup_{i=1}^sE_i\Big)\cap{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(g)
=\bigcup_{i=1}^s{\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(E_i)\cap{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(g)=\bigcup_{i=1}^s{\mathcal V}_i.\end{gathered}$$
If $j\neq k$, we have by [@fg2 Lem.4.5] $$\begin{split}
{\mathcal V}_j\cap{\mathcal V}_k&={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(E_j)\cap{\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(E_k)\cap{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(g)\\
&={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}({\operatorname{Cl}}(E_j)\cap{\operatorname{Cl}}(E_k))\cap{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(g)\\
&={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}({\operatorname{Cl}}(E_j)\cap{\operatorname{Cl}}(E_k)\cap D(g))\cap{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(g)=\varnothing,
\end{split}$$ as required.
\[lczs\] Let $M\subset{{\mathbb R}}^m$ be a semialgebraic set. Then ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)$ is locally connected.
Let ${{\mathfrak p}}\in{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)$ and let ${\mathcal W}\subset{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)$ be an open neighborhood of ${{\mathfrak p}}$. Let $g\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ be such that ${{\mathfrak p}}\in{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(g)\subset{\mathcal W}$. By Lemma \[ccD\] ${{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(g)$ has finitely many connected components ${\mathcal V}_i$, which are open subsets of ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)$. We may assume ${{\mathfrak p}}\in{\mathcal V}_1$, which is a connected open subset of ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)$ contained in ${\mathcal W}$, as required.
The following result shows (among other things) that spectral maps associated to semialgebraic $d$-branched coverings are separated.
\[sep2\] Let $\pi:M\to N$ be a semialgebraic $d$-branched covering. Then
- $\varphi^{{\text{\tiny$\displaystyle\diamond$}}}_\pi:{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(N)\to{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ is an integral homomorphism.
- ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)$ is a finite quasi-covering.
- If ${{\mathfrak m}}^{{\text{\tiny$\displaystyle\diamond$}}}\in{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)$, we have ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)({{\mathfrak m}}^{{\text{\tiny$\displaystyle\diamond$}}})\in{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(N)$. In addition, ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)|_{{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)}={\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(\pi)$.
- If ${{\mathfrak n}}^{{\text{\tiny$\displaystyle\diamond$}}}\in{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(N)$, it holds ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({{\mathfrak n}}^{{\text{\tiny$\displaystyle\diamond$}}})$ is a finite subset of ${\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)$.
- If $y\in N$ and $\pi^{-1}(y):=\{x_1,\ldots,x_r\}$, we have ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({{\mathfrak n}}^{{\text{\tiny$\displaystyle\diamond$}}}_y):=\{{{\mathfrak m}}^{{\text{\tiny$\displaystyle\diamond$}}}_{x_1},\ldots,{{\mathfrak m}}^{{\text{\tiny$\displaystyle\diamond$}}}_{x_r}\}$.
\(i) Let $f\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$. Let $\sigma_k\in{{\mathbb Z}}[{\tt x}_1,\dots,{\tt x}_d]$ be the $k$th elementary symmetric form (for $1\leq k\leq d$) and consider the functions $$\sigma_k(f):N\to{{\mathbb R}},\ y\mapsto \sigma_k(f(x_{1}),\dots,f(x_{d})),$$ where $\pi^{-1}(y):=\{x_1,\dots,x_d\}$. By Lemma \[sym\] each $\sigma_k(f)\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(N)$. As $f$ is a root of the polynomial $${\tt p}({\tt t}):={\tt t}^d+\sum_{k=1}^d(-1)^k\sigma_k(f){\tt t}^{d-k}\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(N)[{\tt t}],$$ we conclude $f$ is integral over ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(N)$ via $\varphi^{{\text{\tiny$\displaystyle\diamond$}}}_{\pi}$. This means that $\varphi^{{\text{\tiny$\displaystyle\diamond$}}}_{\pi}$ is an integral homomorphism.
\(ii) By Lemma \[propern\](ii) the spectral map ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)$ is open, closed and surjective. In addition, ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)$ has finite fibers by [@s1 Prop.11]. We prove next that ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)$ is separated.
Given ${{\mathfrak p}}_1,{{\mathfrak p}}_2\in{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)$ with ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)({{\mathfrak p}}_1)={\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)({{\mathfrak p}}_2)$ we have by (i) and [@am Cor.5.9] ${{\mathfrak p}}_1\not\subset{{\mathfrak p}}_2$ and ${{\mathfrak p}}_2\not\subset{{\mathfrak p}}_1$. Thus, by Lemma \[sep\] ${{\mathfrak p}}_1$ and ${{\mathfrak p}}_2$ have disjoint open neighborhoods.
\(iii) As ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)$ is closed, it maps closed points to closed points and the statement follows readily.
\(iv) By Lemma \[opcl\](i) we have $${\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}({\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({{\mathfrak n}}^{{\text{\tiny$\displaystyle\diamond$}}}))={\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}({{\mathfrak n}}^{{\text{\tiny$\displaystyle\diamond$}}}))={\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({{\mathfrak n}}^{{\text{\tiny$\displaystyle\diamond$}}}).$$ Write ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({{\mathfrak n}}^{{\text{\tiny$\displaystyle\diamond$}}}):=\{{{\mathfrak p}}_1,\ldots,{{\mathfrak p}}_r\}$. We have $$\{{{\mathfrak p}}_1,\ldots,{{\mathfrak p}}_r\}={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(\{{{\mathfrak p}}_1,\ldots,{{\mathfrak p}}_r\})=\bigcup_{i=1}^r{\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}({{\mathfrak p}}_i).$$ Let ${{\mathfrak m}}^{{\text{\tiny$\displaystyle\diamond$}}}_i$ be the unique maximal ideal of ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ that contains ${{\mathfrak p}}_i$. Then ${{\mathfrak m}}_i^{{\text{\tiny$\displaystyle\diamond$}}}\in{\operatorname{Cl}}_{{\operatorname{Spec}}(M)}({{\mathfrak p}}_i)\subset\{{{\mathfrak p}}_1,\ldots,{{\mathfrak p}}_r\}$ for $i=1,\ldots,r$. As ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)$ is separated, ${{\mathfrak p}}_i\not\subset{\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}({{\mathfrak p}}_j)$ if $i\neq j$. We conclude ${{\mathfrak p}}_i={{\mathfrak m}}_i^{{\text{\tiny$\displaystyle\diamond$}}}$ for $i=1,\ldots,r$, so ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({{\mathfrak n}}^{{\text{\tiny$\displaystyle\diamond$}}})=\{{{\mathfrak m}}_1^{{\text{\tiny$\displaystyle\diamond$}}},\ldots,{{\mathfrak m}}_r^{{\text{\tiny$\displaystyle\diamond$}}}\}\subset{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)$.
\(v) As $\pi:M\to N$ is a closed map with finite fibers, it is proper so by Lemma \[propern\](ii) ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)(\partial M)\cap N=\varnothing$. By (iv) ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({{\mathfrak n}}_y^{{\text{\tiny$\displaystyle\diamond$}}})\subset{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)$ is a finite set. Thus, ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({{\mathfrak n}}_y^{{\text{\tiny$\displaystyle\diamond$}}})=\{{{\mathfrak m}}_{x_1}^{{\text{\tiny$\displaystyle\diamond$}}},\ldots,{{\mathfrak m}}_{x_r}^{{\text{\tiny$\displaystyle\diamond$}}}\}$, as required.
Next result points out the good properties of the minimal elements of the collapsing set in the ${\mathcal S}$-case.
\[minz\] Let $\pi:M\to N$ be a semialgebraic $d$-branched covering. Then each minimal element ${{\mathfrak P}}\in{\mathcal{C}}_{{\operatorname{Spec}}(\pi)}$ is a $z$-ideal of ${\mathcal S}(M)$.
If ${{\mathfrak P}}$ is a minimal prime ideal of ${\mathcal S}(M)$, the statement follows from [@fe1 Cor.4.7]. Thus, we may assume ${{\mathfrak P}}$ is not a minimal prime ideal.
Let ${{\mathfrak Q}}_1$ be a minimal prime ideal of ${\mathcal S}(M)$ contained in ${{\mathfrak P}}$. By Lemma \[mintomin\](ii) its image ${{\mathfrak q}}:={\operatorname{Spec}}(\pi)({{\mathfrak Q}}_1)\subset{\operatorname{Spec}}(\pi)({{\mathfrak P}}):={{\mathfrak p}}$ is a minimal prime ideal of ${\mathcal S}(N)$ and ${\operatorname{Spec}}(\pi)^{-1}({{\mathfrak p}})=\{{{\mathfrak P}}\}$ because ${{\mathfrak P}}\in{\mathcal{C}}_{{\operatorname{Spec}}(\pi)}$. Write ${\operatorname{Spec}}(\pi)^{-1}({{\mathfrak q}}):=\{{{\mathfrak Q}}_1,\dots,{{\mathfrak Q}}_\ell\}$ for some $\ell\leq d$. As ${\operatorname{Spec}}(\pi)$ is separated, the fiber ${\operatorname{Spec}}(\pi)^{-1}({{\mathfrak q}})$ has the trivial topology. We claim: *${{\mathfrak Q}}_j$ is a minimal prime ideal of ${\mathcal S}(M)$ for $1\leq j\leq\ell$*.**
Assume ${{\mathfrak Q}}_j$ is not a minimal prime ideal of ${\mathcal S}(M)$ for some $j=2,\ldots,\ell$ and let ${{\mathfrak P}}'$ be a prime ideal of ${\mathcal S}(M)$ strictly contained in ${{\mathfrak Q}}_j$. Then ${\operatorname{Spec}}(\pi)({{\mathfrak P}}')\subset{\operatorname{Spec}}(\pi)({{\mathfrak Q}}_j)={{\mathfrak q}}$. As the latter is a minimal prime ideal of ${\mathcal S}(N)$, we have ${\operatorname{Spec}}(\pi)({{\mathfrak P}}')={{\mathfrak q}}$, that is, ${{\mathfrak P}}'\in{\operatorname{Spec}}(\pi)^{-1}({{\mathfrak q}})$. This is a contradiction because the fiber ${\operatorname{Spec}}(\pi)^{-1}({{\mathfrak q}})$ does not contain a pair of prime ideals such that ${{\mathfrak P}}'\subsetneq{{\mathfrak Q}}_j$ (recall that ${\operatorname{Spec}}(\pi)$ is by Proposition \[sep2\](ii) a finite quasi-covering).
We prove next: *${{\mathfrak Q}}_j\subset{{\mathfrak P}}$ for $1\leq j\leq\ell$*.**
As ${{\mathfrak q}}={\operatorname{Spec}}(\pi)({{\mathfrak Q}}_j)={\operatorname{Spec}}(\pi)({{\mathfrak Q}}_1)\subset{\operatorname{Spec}}(\pi)({{\mathfrak P}}):={{\mathfrak p}}$ and ${\operatorname{Spec}}(\pi)$ is a closed continuous map, we have $${{\mathfrak p}}\in{\operatorname{Cl}}_{{\operatorname{Spec}}(N)}(\{{\operatorname{Spec}}(\pi)({{\mathfrak Q}}_j)\})={\operatorname{Spec}}(\pi)({\operatorname{Cl}}_{{\operatorname{Spec}}(M)}(\{{{\mathfrak Q}}_j\})).$$ As ${\operatorname{Spec}}(\pi)^{-1}({{\mathfrak p}})=\{{{\mathfrak P}}\}$, this implies that ${{\mathfrak P}}\in{\operatorname{Cl}}_{{\operatorname{Spec}}(M)}(\{{{\mathfrak Q}}_j\})$, so ${{\mathfrak Q}}_j\subset{{\mathfrak P}}$.
By Lemma \[min\](ii) the sum ${{\mathfrak Q}}_1+\cdots+{{\mathfrak Q}}_\ell$ is a prime $z$-ideal contained in ${{\mathfrak P}}$. To prove that ${{\mathfrak P}}$ is a prime $z$-ideal, it is enough to check: ${{\mathfrak P}}={{\mathfrak Q}}_1+\cdots+{{\mathfrak Q}}_\ell$. As ${{\mathfrak P}}$ is a minimal element of ${\mathcal{C}}_{{\operatorname{Spec}}(\pi)}$, it suffices to show: ${{\mathfrak Q}}_1+\cdots+{{\mathfrak Q}}_\ell\in{\mathcal{C}}_{{\operatorname{Spec}}(\pi)}$. Denote ${{\mathfrak q}}':={\operatorname{Spec}}(\pi)({{\mathfrak Q}}_1+\cdots+{{\mathfrak Q}}_\ell)$ and let us prove: *${{\mathfrak Q}}_1+\cdots+{{\mathfrak Q}}_\ell$ is the unique point in the fiber ${\operatorname{Spec}}(\pi)^{-1}({{\mathfrak q}}')$*.**
Pick a point ${{\mathfrak q}}'_1\in{\operatorname{Spec}}(\pi)^{-1}({{\mathfrak q}}')$. As $${{\mathfrak q}}={\operatorname{Spec}}(\pi)({{\mathfrak Q}}_1)\subset{\operatorname{Spec}}(\pi)({{\mathfrak Q}}_1+\cdots+{{\mathfrak Q}}_\ell)={{\mathfrak q}}'={\operatorname{Spec}}(\pi)({{\mathfrak q}}'_1),$$ there exists by Lemma \[mintomin\](i) a point in the fiber of ${{\mathfrak q}}$ contained in ${{\mathfrak q}}'_1$. Thus, ${{\mathfrak Q}}_k\subset{{\mathfrak q}}'_1$ for some index $1\leq k\leq\ell$. Consequently, ${{\mathfrak q}}'_1$ and ${{\mathfrak Q}}_1+\cdots+{{\mathfrak Q}}_\ell$ are prime ideals of ${\mathcal S}(M)$ containing ${{\mathfrak Q}}_k$. As the prime ideals of ${\mathcal S}(M)$ containing ${{\mathfrak Q}}_k$ constitute a chain [@fg2 3.1.4], either ${{\mathfrak q}}'_1\subset{{\mathfrak Q}}_1+\cdots+{{\mathfrak Q}}_\ell$ or ${{\mathfrak Q}}_1+\cdots+{{\mathfrak Q}}_\ell\subset{{\mathfrak q}}'_1$. As the fiber ${\operatorname{Spec}}(\pi)^{-1}({{\mathfrak q}}')$ has the trivial topology, ${{\mathfrak q}}'_1={{\mathfrak Q}}_1+\cdots+{{\mathfrak Q}}_\ell$, so ${{\mathfrak Q}}_1+\cdots+{{\mathfrak Q}}_\ell\in{\mathcal{C}}_{{\operatorname{Spec}}(\pi)}$, as required.
Proof of Theorem \[colapse\] {#mu}
============================
To get a better understanding of the finite quasi-covering ${\operatorname{Spec}}(\pi)$ induced by a semialgebraic branched covering $\pi:M\to N$ we will prove Theorem \[colapse\], which provides a precise description of the subset ${\mathcal{C}}_{{\operatorname{Spec}}(\pi)}$. Its proof does not involve Theorem \[bc\]. We need the following notion.
Let $\pi:M\to N$ be a semialgebraic $d$-branched covering and let $b_{\pi}:M\to{{\mathbb Z}}$ be the branching index of $\pi$. We define the map $$\mu^{{\text{\tiny$\displaystyle\diamond$}}}:{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)\to{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(N),\ f\mapsto\mu^{{\text{\tiny$\displaystyle\diamond$}}}(f):=\tfrac{1}{d}\sigma_1(f)=\tfrac{1}{d}\sum_{x\in\pi^{-1}(y)}b_{\pi}(x)f(x),$$ where $\sigma_1({{\tt x}}_1,\ldots,{{\tt x}}_d):={{\tt x}}_1+\cdots+{{\tt x}}_d$ is the first elementary symmetric form in $d$ variables.
\[module\] (i) If $b_{\pi}(x)=1$ for each point in the fiber of a point $y\in N$, then $\pi^{-1}(y)$ consists of $d$ points, so $\mu^{{\text{\tiny$\displaystyle\diamond$}}}(f)(y)$ is the arithmetic mean of the values of $f$ on the points of $\pi^{-1}(y)$. In general, $\mu^{{\text{\tiny$\displaystyle\diamond$}}}(f)(y)$ is a weighted arithmetic mean of the values of $f$ on $\pi^{-1}(y)$.
\(ii) The homomorphism $\varphi_{\pi}$ endows ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ with a natural structure of ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(N)$-module and the map $\mu^{{\text{\tiny$\displaystyle\diamond$}}}:{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)\to{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(N)$ is a homomorphism of ${\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(N)$-modules.
For each $g\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(N)$ and each $y\in N$ we have $$\begin{split}
\big(\mu^{{\text{\tiny$\displaystyle\diamond$}}}(g\circ\pi)\big)(y)&=\tfrac{1}{d}\sum_{x\in\pi^{-1}(y)}b_{\pi}(x)(g\circ\pi)(x)\\
&=\tfrac{1}{d}\sum_{x\in\pi^{-1}(y)}b_{\pi}(x)g(y)=g(y)\Big(\tfrac{1}{d}\sum_{x\in\pi^{-1}(y)}b_{\pi}(x)\Big)=g(y),
\end{split}$$ so $\mu^{{\text{\tiny$\displaystyle\diamond$}}}(g\circ\pi)=g$.
We will also need the following result.
\[homeo:fe\_FG\] Let $N\subset M\subset {{\mathbb R}}^n$ be semialgebraic sets and let ${\tt j}:N\hookrightarrow M$ be the inclusion. Then there exists $h\in \mathcal{S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ with $Z(h)\subset{\operatorname{Cl}}_M(N)$ nowhere dense in ${\operatorname{Cl}}_M(N)$ such that $${\operatorname{Spec}}^{{\text{\tiny$\displaystyle\diamond$}}}(N)\setminus {\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}({\tt j})^{-1}({{\EuScript Z}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(h))={\operatorname{Spec}}^{{\text{\tiny$\displaystyle\diamond$}}}(N)\setminus {{\EuScript Z}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(h|_N)$$ is homeomorphic to ${\operatorname{Cl}}_{{\operatorname{Spec}}^{{\text{\tiny$\displaystyle\diamond$}}}(M)}(N)\setminus {{\EuScript Z}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(h)$ via ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}({\tt j})$. In addition, if $N$ is locally compact, we may assume $Z(h)={\operatorname{Cl}}_M(N)\setminus N$. If $N$ is closed in $M$, then ${\operatorname{Spec}}^{{\text{\tiny$\displaystyle\diamond$}}}(N)$ is homeomorphic to ${\operatorname{Cl}}_{{\operatorname{Spec}}^{{\text{\tiny$\displaystyle\diamond$}}}(M)}(N)$ via ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}({\tt j})$.
Let $H\in \mathcal{S}^*({{\mathbb R}}^n)$ be such that $Z(H)={\operatorname{Cl}}_{{{\mathbb R}}^n}({\operatorname{Cl}}_{{{\mathbb R}}^n}(N)\setminus N)$ and define $h:=H|_{M}\in {\mathcal S}^*(M)$. The difference ${\operatorname{Cl}}_{{{\mathbb R}}^n}(N)\setminus Z(H)=N\setminus Z(H)$ is a dense subset of $N$ (see [@fe3 §2.2]). In particular, ${\operatorname{Cl}}_M(N)\setminus Z(h)=N\setminus Z(H)$ is also a dense subset of ${\operatorname{Cl}}_M(N)$.
Let ${\tt j}_1:N\hookrightarrow{\operatorname{Cl}}_M(N)$ be the inclusion. We claim: *$${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}({\tt j}_1)|:{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)\setminus{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}({\tt j}_1)^{-1}({{\EuScript Z}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(h|_{{\operatorname{Cl}}_{M}(N)}))\to {\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}({\operatorname{Cl}}_M(N))\setminus{{\EuScript Z}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(h|_{{\operatorname{Cl}}_{M}(N)})$$ is a homeomorphism*.**
Indeed, in the $\mathcal{S}$-case the claim is a straightforward consequence of [@fe3 Lem.1.1]. In the $\mathcal{S}^*$-case, we have by [@fe3 Thm.1.2] that the restriction map $${\operatorname{Spec^*}}({\tt j}_1)|':{\operatorname{Spec^*}}(N)\setminus{\operatorname{Spec^*}}({\tt j}_1)^{-1}({{\EuScript Z}})\to{\operatorname{Spec^*}}({\operatorname{Cl}}_M(N))\setminus{{\EuScript Z}},$$ where ${{\EuScript Z}}:={\operatorname{Cl}}_{{\operatorname{Spec^*}}({\operatorname{Cl}}_M(N))}({\operatorname{Cl}}_M(N)\setminus N)$, is a homeomorphism. As ${{\EuScript Z}}\subset {{\EuScript Z}}^*(h|_{{\operatorname{Cl}}_M(N)})$, also the restriction of ${\operatorname{Spec^*}}({\tt j}_1)$ to ${\operatorname{Spec^*}}(N)\setminus {\operatorname{Spec^*}}({\tt j}_1)^{-1}({{\EuScript Z}})$ is a homeomorphism. Note that ${\operatorname{Spec^*}}({\tt j}_1)^{-1}({{\EuScript Z}}^*(h|_{{\operatorname{Cl}}_{M}(N)}))={{\EuScript Z}}^*(h|_N)$.
Next, let ${\tt j}_2:{\operatorname{Cl}}_M(N)\hookrightarrow M$ be the inclusion. As ${\operatorname{Cl}}_M(N)$ is closed in $M$, ${\operatorname{Spec^*}}({\operatorname{Cl}}_M(N))$ is by [@fg2 Cor.4.6] homeomorphic to ${\operatorname{Cl}}_{{\operatorname{Spec^*}}(M)}({\operatorname{Cl}}_M(N))={\operatorname{Cl}}_{{\operatorname{Spec^*}}(M)}(N)$ via ${\operatorname{Spec^*}}({\tt j}_2)$. Thus, $${\operatorname{Spec^*}}({\tt j}_2)|:{\operatorname{Spec^*}}({\operatorname{Cl}}_M(N))\setminus{{\EuScript Z}}^*(h|_{{\operatorname{Cl}}_M(N)})\to{\operatorname{Cl}}_{{\operatorname{Spec^*}}(M)}(N)\setminus{{\EuScript Z}}^*(h)$$ is a homeomorphism. Thus, the composition ${\operatorname{Spec^*}}({\tt j})|={\operatorname{Spec^*}}({\tt j}_2)|\circ {\operatorname{Spec^*}}({\tt j}_1)|$ is a homeomorphism too.
If $N$ is locally compact, ${\operatorname{Cl}}_{\mathbb{R}^n}(N)\setminus N$ is a closed subset of ${{\mathbb R}}^n$. Thus, $Z(h)=Z(H)\cap M=({\operatorname{Cl}}_{\mathbb{R}^n}(N)\setminus N)\cap M={\operatorname{Cl}}_M(N)\setminus N$. Finally, if $N$ is closed in $M$, then ${\operatorname{Cl}}_M(N)=N$ and ${\operatorname{Spec}}^{{\text{\tiny$\displaystyle\diamond$}}}(N)$ is by [@fg2 Cor.4.6] homeomorphic to ${\operatorname{Cl}}_{{\operatorname{Spec}}^{{\text{\tiny$\displaystyle\diamond$}}}(M)}(N)$ via ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}({\tt j})$, as required.
We are ready to prove Theorem \[colapse\].
Consider the commutative diagram $$\xymatrix{
{\mathcal S}^*(N)\ar[r]^{\varphi^*_\pi}\ar@{^{(}->}[d]&{\mathcal S}^*(M)\ar@{^{(}->}[d]\\
{\mathcal S}(N)\ar[r]^{\varphi_\pi}&{\mathcal S}(M)
}$$ As $\pi$ is surjective, $f\in{\mathcal S}(N)$ is bounded if and only if $\varphi_\pi(f)=f\circ\pi\in{\mathcal S}(M)$ is bounded. If ${{\mathfrak q}}\in{\operatorname{Spec}}(M)$, then $${\operatorname{Spec^*}}(\pi)({{\mathfrak q}}\cap{\mathcal S}^*(M))=(\varphi_\pi^*)^{-1}({{\mathfrak q}}\cap{\mathcal S}^*(M))=\varphi_\pi^{-1}({{\mathfrak q}})\cap{\mathcal S}^*(N)={\operatorname{Spec}}(\pi)({{\mathfrak q}})\cap{\mathcal S}^*(N).$$
We begin with claims \[mec\] and \[mec2\] below that will allow us to make a reduction from the ${\mathcal S}^*$-case to the ${\mathcal S}$-case. Along the proof we will make use of Remark \[r:maximal\] without mention.
Pick ${{\mathfrak p}}\in{\mathcal{C}}_{{\operatorname{Spec^*}}(\pi)}$. Let ${{\mathfrak m}}^*\in\beta^*(M)$ be the unique maximal ideal of ${\mathcal S}^*(M)$ that contains ${{\mathfrak p}}$ and let ${{\mathfrak m}}\in \beta(M)$ be the unique maximal ideal of ${\mathcal S}(M)$ such that ${{\mathfrak m}}\cap{\mathcal S}^*(M)\subset{{\mathfrak m}}^*$. Fix ${{\mathfrak p}}_0\in{\mathcal{C}}_{{\operatorname{Spec^*}}(\pi)}$ a minimal element of ${\mathcal{C}}_{{\operatorname{Spec^*}}(\pi)}$ contained in ${{\mathfrak p}}$.
#### {#mec}
We claim: ${{\mathfrak p}}_0\subset{{\mathfrak m}}\cap{\mathcal S}^*(M)$.
Suppose ${{\mathfrak m}}\cap{\mathcal S}^*(M)\subsetneq{{\mathfrak p}}_0$, so ${{\mathfrak m}}\cap{\mathcal S}^*(M)\not\in{\mathcal{C}}_{{\operatorname{Spec^*}}(\pi)}$. Thus, there exists a prime ideal ${{\mathfrak q}}'\in{\operatorname{Spec^*}}(M)$ such that ${{\mathfrak q}}'\neq{{\mathfrak m}}\cap{\mathcal S}^*(M)$ and ${\operatorname{Spec^*}}(\pi)({{\mathfrak q}}')={\operatorname{Spec^*}}(\pi)({{\mathfrak m}}\cap{\mathcal S}^*(M))$. As ${\operatorname{Spec^*}}(\pi)$ is a separated map, ${{\mathfrak q}}'\not\subset{{\mathfrak m}}\cap{\mathcal S}^*(M)$ and ${{\mathfrak m}}\cap{\mathcal S}^*(M)\not\subset{{\mathfrak q}}'$.
Let ${{\mathfrak m}}_1^*\in\beta^*(M)$ be the unique maximal ideal of ${\mathcal S}^*(M)$ that contains ${{\mathfrak q}}'$ and let ${{\mathfrak m}}_1\in\beta(M)$ be the unique maximal ideal of ${\mathcal S}(M)$ such that ${{\mathfrak m}}_1\cap{\mathcal S}^*(M)\subset{{\mathfrak m}}_1^*$. Let us show: ${{\mathfrak q}}'={{\mathfrak m}}_1\cap \mathcal{S}^*(M)$.
If ${{\mathfrak m}}_1\cap{\mathcal S}^*(M)\subset{{\mathfrak q}}'$, then $$\begin{gathered}
{\operatorname{Spec}}(\pi)({{\mathfrak m}}_1)\cap{\mathcal S}^*(N)={\operatorname{Spec^*}}(\pi)({{\mathfrak m}}_1\cap{\mathcal S}^*(M))\\
\subset{\operatorname{Spec^*}}(\pi)({{\mathfrak q}}')={\operatorname{Spec^*}}(\pi)({{\mathfrak m}}\cap{\mathcal S}^*(M))={\operatorname{Spec}}(\pi)({{\mathfrak m}})\cap{\mathcal S}^*(N).\end{gathered}$$ As ${\operatorname{Spec}}(\pi)$ is a closed map, ${{\mathfrak n}}_1:={\operatorname{Spec}}(\pi)({{\mathfrak m}}_1)$ and ${{\mathfrak n}}:={\operatorname{Spec}}(\pi)({{\mathfrak m}})$ are maximal ideals of ${\mathcal S}(N)$. Let ${{\mathfrak n}}^*\in\beta^*(M)$ be the unique maximal ideal of ${\mathcal S}^*(N)$ that contains ${{\mathfrak n}}\cap{\mathcal S}^*(N)$. Thus, ${{\mathfrak n}}_1\cap{\mathcal S}^*(N)\subset{{\mathfrak n}}\cap{\mathcal S}^*(N)\subset{{\mathfrak n}}^*$. We conclude ${{\mathfrak n}}_1^*={{\mathfrak n}}^*$, so ${{\mathfrak n}}_1={{\mathfrak n}}$ and ${\operatorname{Spec^*}}(\pi)({{\mathfrak m}}_1\cap{\mathcal S}^*(M))={\operatorname{Spec^*}}(\pi)({{\mathfrak q}}')$. As ${\operatorname{Spec^*}}(\pi)$ is separated, ${{\mathfrak m}}_1\cap{\mathcal S}^*(M)={{\mathfrak q}}'$.
Otherwise, ${{\mathfrak q}}'\subset{{\mathfrak m}}_1\cap{\mathcal S}^*(M)$. Thus, there exists a prime ideal ${{\mathfrak q}}:={\mathcal W}_M^{-1}{{\mathfrak q}}'\subset{{\mathfrak m}}_1$ of ${\mathcal S}(M)$ such that ${{\mathfrak q}}'={{\mathfrak q}}\cap{\mathcal S}^*(M)$ and $$\begin{aligned}
&{\operatorname{Spec}}(\pi)({{\mathfrak q}})\cap{\mathcal S}^*(N)={\operatorname{Spec^*}}(\pi)({{\mathfrak q}}')={\operatorname{Spec^*}}(\pi)({{\mathfrak m}}\cap{\mathcal S}^*(M))={\operatorname{Spec}}(\pi)({{\mathfrak m}})\cap{\mathcal S}^*(N),\\
&{\operatorname{Spec}}(\pi)({{\mathfrak q}})\cap{\mathcal S}^*(N)={\operatorname{Spec^*}}(\pi)({{\mathfrak q}}')\subset{\operatorname{Spec^*}}(\pi)({{\mathfrak m}}_1\cap{\mathcal S}^*(M))={\operatorname{Spec}}(\pi)({{\mathfrak m}}_1)\cap{\mathcal S}^*(N).\end{aligned}$$ Consequently, $${\operatorname{Spec}}(\pi)({{\mathfrak q}})\cap{\mathcal S}^*(N)={\operatorname{Spec}}(\pi)({{\mathfrak m}})\cap{\mathcal S}^*(N)\subset{\operatorname{Spec}}(\pi)({{\mathfrak m}}_1)\cap{\mathcal S}^*(N).$$ As in the previous case, ${\operatorname{Spec}}(\pi)({{\mathfrak m}})={\operatorname{Spec}}(\pi)({{\mathfrak m}}_1)$ and we deduce that ${{\mathfrak q}}'={{\mathfrak m}}_1\cap{\mathcal S}^*(M)$.
Next, as ${{\mathfrak q}}'\not\subset{{\mathfrak m}}\cap{\mathcal S}^*(M)$ and ${{\mathfrak m}}\cap{\mathcal S}^*(M)\not\subset{{\mathfrak q}}'$, we deduce ${{\mathfrak m}}_1\neq{{\mathfrak m}}$, so ${{\mathfrak m}}_1^*\neq{{\mathfrak m}}^*$. In addition, $${\operatorname{Spec^*}}(\pi)({{\mathfrak m}}_1\cap{\mathcal S}^*(M))={\operatorname{Spec^*}}(\pi)({{\mathfrak m}}\cap{\mathcal S}^*(M))\subset{\operatorname{Spec^*}}(\pi)({{\mathfrak m}}_1^*)\cap{\operatorname{Spec^*}}(\pi)({{\mathfrak m}}^*).$$ As ${\operatorname{Spec^*}}(\pi)$ is a closed map and ${\mathcal S}^*(N)$ is Gelfand, we conclude ${\operatorname{Spec^*}}(\pi)({{\mathfrak m}}_1^*)={\operatorname{Spec^*}}(\pi)({{\mathfrak m}}^*)$, so ${{\mathfrak m}}^*\not\in{\mathcal{C}}_{{\operatorname{Spec^*}}(\pi)}$. By Lemma \[disting\] and since ${{\mathfrak m}}^*\in{\operatorname{Cl}}_{{\operatorname{Spec^*}}(M)}({{\mathfrak p}}_0)$, $$\#({\operatorname{Spec^*}}(\pi)^{-1}({\operatorname{Spec^*}}(\pi)({{\mathfrak p}}_0)))\geq\#({\operatorname{Spec^*}}(\pi)^{-1}({\operatorname{Spec^*}}(\pi)({{\mathfrak m}}^*)))\geq2,$$ which is a contradiction because ${{\mathfrak p}}_0\in{\mathcal{C}}_{{\operatorname{Spec^*}}(\pi)}$.
#### {#mec2}
As ${{\mathfrak p}}_0\subset{{\mathfrak m}}\cap{\mathcal S}^*(M)$, there exists a unique prime ideal ${{\mathfrak q}}_0:={\mathcal W}_M^{-1}{{\mathfrak p}}_0\subset{{\mathfrak m}}$ such that ${{\mathfrak p}}_0={{\mathfrak q}}_0\cap{\mathcal S}^*(M)$. We claim: *${{\mathfrak q}}_0\in{\mathcal{C}}_{{\operatorname{Spec}}(\pi)}$*.**
Pick ${{\mathfrak q}}_1\in{\operatorname{Spec}}(M)$ such that ${\operatorname{Spec}}(\pi)({{\mathfrak q}}_0)={\operatorname{Spec}}(\pi)({{\mathfrak q}}_1)$. Thus, $$\begin{gathered}
{\operatorname{Spec^*}}(\pi)({{\mathfrak q}}_0\cap{\mathcal S}^*(M))={\operatorname{Spec}}(\pi)({{\mathfrak q}}_0)\cap{\mathcal S}^*(N)\\
={\operatorname{Spec}}(\pi)({{\mathfrak q}}_1)\cap{\mathcal S}^*(N)={\operatorname{Spec^*}}(\pi)({{\mathfrak q}}_1\cap{\mathcal S}^*(M)).\end{gathered}$$ As ${{\mathfrak p}}_0\in{\mathcal{C}}_{{\operatorname{Spec^*}}(\pi)}$, it follows ${{\mathfrak q}}_1\cap{\mathcal S}^*(M)={{\mathfrak q}}_0\cap{\mathcal S}^*(M)={{\mathfrak p}}_0$, so ${{\mathfrak q}}_0={{\mathfrak q}}_1$.
Once we have showed claims \[mec\] and \[mec2\], we are ready to prove the different assertions in the statement:
\(i) Define ${\mathcal T}^{{\text{\tiny$\displaystyle\diamond$}}}:=\{{{\mathfrak p}}\in{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M):\ \ker(\mu^{{\text{\tiny$\displaystyle\diamond$}}})\subset{{\mathfrak p}}\}$, which is a closed subset of ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)$.
We prove first: ${\mathcal{C}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}\subset{\mathcal T}^{{\text{\tiny$\displaystyle\diamond$}}}$. Pick ${{\mathfrak p}}\in{\mathcal{C}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}$ and let ${{\mathfrak p}}_0\in{\mathcal{C}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}$ be a minimal element of ${\mathcal{C}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}$ contained in ${{\mathfrak p}}$. By \[mec\] and \[mec2\] it is enough to consider the ${\mathcal S}$-case. As ${{\mathfrak p}}\in{\operatorname{Cl}}_{{\operatorname{Spec}}(M)}(\{{{\mathfrak p}}_0\})$ and $\mathcal{T}$ is closed, it is enough to check: ${{\mathfrak p}}_0\in{\mathcal T}$. To that end, pick $f\in\ker(\mu)$ and let us show: $f\in{{\mathfrak p}}_0$.
Consider the non-negative functions $h_1:=|f|-f$ and $h_2:=|f|+f$. As $h_1h_2=0\in{{\mathfrak p}}_0$, we may assume $h_1\in{{\mathfrak p}}_0$. As $\pi$ is open, closed and surjective, $$g_1:N\to{{\mathbb R}},\ y\mapsto\sup\{h_1(x):\ x\in\pi^{-1}(y)\}$$ is by [@fg3 Const.3.1] a semialgebraic function. By [@fg3 Eq.$(\ast)$ in Proof Thm.1.5] it holds $${{\operatorname{Spec}}(\pi)}({{\EuScript D}}(h_1))={{\EuScript D}}(g_1).$$ As ${{\mathfrak p}}_0\notin{{\EuScript D}}(h_1)$ and $\{{{\mathfrak p}}_0\}={{\operatorname{Spec}}(\pi)}^{-1}({{\operatorname{Spec}}(\pi)}({{\mathfrak p}}_0))$, we deduce $${{\operatorname{Spec}}(\pi)}({{\mathfrak p}}_0)\notin{{\operatorname{Spec}}(\pi)}({{\EuScript D}}(h_1))={{\EuScript D}}(g_1),$$ so $g_1\circ\pi\in{{\mathfrak p}}_0$. By Corollary \[minz\] ${{\mathfrak p}}_0$ is a $z$-ideal, so to prove that $f\in{{\mathfrak p}}_0$ it is enough to show: $Z(g_1\circ\pi)\subset Z(f)$.
Suppose there exists a point $x\in Z(g_1\circ\pi)$ such that $f(x)\neq0$. As $g_1(\pi(x))=0$, the function $h_1$ vanishes identically on the fiber $\pi^{-1}(\pi(x))$ (recall that $h_1\geq0$ on $M$). Thus, $f(z)\geq0$ for each $z\in\pi^{-1}(\pi(x))$ and $f(x)>0$. Hence, $$\mu(f)(\pi(x))=\tfrac{1}{d}\sum_{z\in \pi^{-1}(\pi(x))}b_{\pi}(z)f(z)>0,$$ which is a contradiction because $f\in\ker(\mu)$.
Next, we prove the converse inclusion: ${\mathcal T}^{{\text{\tiny$\displaystyle\diamond$}}}\subset{\mathcal{C}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}$. Suppose there exists ${{\mathfrak p}}\in{\mathcal T}^{{\text{\tiny$\displaystyle\diamond$}}}\setminus{\mathcal{C}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}$. Then there exists ${{\mathfrak p}}_1\in{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)\setminus\{{{\mathfrak p}}\}$ such that ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)({{\mathfrak p}})={\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)({{\mathfrak p}}_1)$. As ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)$ is a separated map, we achieve a contradiction if we show: ${{\mathfrak p}}\subset{{\mathfrak p}}_1$.
Pick $f\in{{\mathfrak p}}$. If we prove $f^2\in{{\mathfrak p}}_1$, then $f\in{{\mathfrak p}}_1$, so we assume $f$ is non-negative. By Remark \[module\](ii) we have $\mu^{{\text{\tiny$\displaystyle\diamond$}}}(\mu^{{\text{\tiny$\displaystyle\diamond$}}}(f)\circ\pi)=\mu^{{\text{\tiny$\displaystyle\diamond$}}}(f)$. Thus, $f-(\mu^{{\text{\tiny$\displaystyle\diamond$}}}(f)\circ\pi)\in\ker(\mu^{{\text{\tiny$\displaystyle\diamond$}}})\subset{{\mathfrak p}}$, so $$\mu^{{\text{\tiny$\displaystyle\diamond$}}}(f)\circ\pi=f-(f-(\mu^{{\text{\tiny$\displaystyle\diamond$}}}(f)\circ\pi))\in{{\mathfrak p}}.$$ Hence, $\mu^{{\text{\tiny$\displaystyle\diamond$}}}(f)\in{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)({{\mathfrak p}})={\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)({{\mathfrak p}}_1)$. Consequently, $\mu^{{\text{\tiny$\displaystyle\diamond$}}}(f)\circ\pi\in{{\mathfrak p}}_1$. In addition, for each $x\in M$ we have $0\leq f(x)\leq d\cdot(\mu^{{\text{\tiny$\displaystyle\diamond$}}}(f)\circ\pi)(x)$ and $d\cdot(\mu^{{\text{\tiny$\displaystyle\diamond$}}}(f)\circ\pi)\in{{\mathfrak p}}_1$. By Lemma \[suma\](i) we conclude $f\in{{\mathfrak p}}_1$.
Just for the record, let us show:
#### {#mec3}
*The prime ideal ${{\mathfrak q}}_0\in{\mathcal{C}}_{{\operatorname{Spec}}(\pi)}$ introduced in *\[mec2\] *is minimal in ${\mathcal{C}}_{{\operatorname{Spec}}(\pi)}$*.****
Suppose there exists ${{\mathfrak q}}_1\subset{{\mathfrak q}}_0$ such that ${{\mathfrak q}}_1\in{\mathcal{C}}_{{\operatorname{Spec}}(\pi)}$. By (i) we have $\ker(\mu)\subset{{\mathfrak q}}_1$. Therefore $\ker(\mu^*)=\ker(\mu)\cap \mathcal{S}^*(M)\subset {{\mathfrak q}}_1\cap\mathcal{S}^*(M)$, so ${{\mathfrak q}}_1\cap\mathcal{S}^*(M)\in {\mathcal{C}}_{{\operatorname{Spec^*}}(\pi)}$. As ${{\mathfrak q}}_1\cap\mathcal{S}^*(M)\subset{{\mathfrak q}}_0\cap\mathcal{S}^*(M)$ and ${{\mathfrak q}}_0\cap\mathcal{S}^*(M)\in {\mathcal{C}}_{{\operatorname{Spec}}(\pi)}$ is minimal, it follows ${{\mathfrak q}}_1\cap\mathcal{S}^*(M)= {{\mathfrak q}}_0\cap\mathcal{S}^*(M)$, so ${{\mathfrak q}}_1={{\mathfrak q}}_0$.
\(ii) Recall that for each $x_1\in M$ we have ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)({{\mathfrak m}}_{x_1}))=\{{{\mathfrak m}}_{x_1},\ldots,{{\mathfrak m}}_{x_r}\}$ where $\pi^{-1}(\pi(x_1))=\{x_1,\ldots,x_r\}$. Thus, ${\mathcal{C}}_\pi={\mathcal{C}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}\cap M$, so ${\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}({\mathcal{C}}_\pi)\subset{\mathcal{C}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}$. Let us prove next the converse inequality. Pick ${{\mathfrak p}}\in{\mathcal{C}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}$ and let ${{\mathfrak p}}_0\in{\mathcal{C}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}$ be a minimal element of ${\mathcal{C}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}$ contained in ${{\mathfrak p}}$. If we prove that ${{\mathfrak p}}_0\in{\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}({\mathcal{C}}_\pi)$, then ${{\mathfrak p}}\in{\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(\{{{\mathfrak p}}_0\})\subset{\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}({\mathcal{C}}_\pi)$. By \[mec\] and \[mec2\] it is enough to consider the ${\mathcal S}$-case. By Corollary \[minz\] the prime ideal ${{\mathfrak p}}_0$ is a $z$-ideal and by [@fg2 Lem.4.1] ${{\mathfrak P}}_0:={\operatorname{Spec}}(\pi)({{\mathfrak p}}_0)$ is also a $z$-ideal.
Let $f\in{{\mathfrak P}}_0$ be such that ${\tt d}:=\dim(Z(f))=\min\{\dim(Z(g)):\ g\in{{\mathfrak P}}_0\}$ and denote $Z:=Z(f)$. As ${{\mathfrak P}}_0$ is a $z$-ideal, ${{\mathfrak P}}_0\in{\operatorname{Cl}}_{{\operatorname{Spec}}(N)}(Z)$ (use [@fg2 Lem.4.3]). Denote $T:=\pi^{-1}(Z)$ and consider the restriction map $\pi|_T:T\to Z$. By Hardt’s trivialization theorem [@bcr 9.3.2] there exist:
- a semialgebraic partition $\{A_1,\dots,A_r\}$ of $Z$,
- semialgebraic sets $P_1,\dots,P_r\subset{{\mathbb R}}^p$ and
- semialgebraic homeomorphisms $\theta_{\ell}:A_{\ell}\times P_{\ell}\to \pi^{-1}(A_{\ell})$
such that for $1\leq\ell\leq r$ we have the following commutative diagram $$\xymatrix{
A_{\ell}\times P_{\ell}\ar@{->}[r]^{\quad\theta_{\ell}\quad}\ar@{->}[rd]^{\pi_{\ell}\quad}&\pi^{-1}(A_{\ell})\ar@{->}[d]^{\pi|_{\pi^{-1}(A_\ell)}\quad}\\
&A_{\ell}}$$ where $\pi_{\ell}:A_{\ell}\times P_{\ell}\to A_{\ell}$ is the projection onto $A_{\ell}$. Taking a semialgebraic triangulation of $Z$ compatible with $A_1,\ldots,A_r$ we may assume that each $A_i$ is locally compact. As $\pi$ has finite fibers, each $P_\ell$ is a finite set.
As ${\operatorname{Cl}}_{{\operatorname{Spec}}(N)}(Z)=\bigcup_{\ell=1}^r{\operatorname{Cl}}_{{\operatorname{Spec}}(N)}(A_\ell)$, we may assume ${{\mathfrak P}}_0\in{\operatorname{Cl}}_{{\operatorname{Spec}}(N)}(A_1)$. By [@fg2 Lem.4.3] it follows that for each $g\in\mathcal{S}(N)$ such that $Z(g)={\operatorname{Cl}}_N(A_1)$ we have $g\in{{\mathfrak P}}_0$, so $\dim(A_1)=\dim({\operatorname{Cl}}_N(A_1))\geq {\tt d}=\dim(Z)\geq\dim(A_1)$, that is, $\dim(A_1)={\tt d}$. As $A_1$ is locally compact, the semialgebraic set $C:={\operatorname{Cl}}_N(A_1)\setminus A_1$ is closed in $N$.
By Lemma \[homeo:fe\_FG\] there exists $h\in {\mathcal S}(N)$ with $Z(h)=C$ such that the inclusion ${\tt j}:A_1\to N$ induces a homeomorphism $${\operatorname{Spec}}({\tt j})|:{\operatorname{Spec}}(A_1)\setminus {{\EuScript Z}}(h|_{A_1})\to {\operatorname{Cl}}_{{\operatorname{Spec}}(N)}(A_1)\setminus {{\EuScript Z}}(h).$$ As ${{\mathfrak P}}_0$ is a $z$-ideal and $\dim(Z(h))=\dim(C)<{\tt d}$, we have ${{\mathfrak P}}_0\not\in{{\EuScript Z}}(h)$. In particular, there exists $${{\mathfrak P}}_0'\in{\operatorname{Spec}}(A_1)\setminus {{\EuScript Z}}(h|_{A_1})$$ such that ${\operatorname{Spec}}({\tt j})({{\mathfrak P}}_0')={{\mathfrak P}}_0$.
Next, consider the ${\tt d}$-dimensional subset $\pi^{-1}(A_1)$ of $M$ (recall that $\pi$ has finite fibers). As $\pi:M\to N$ is closed and has finite fibers, it is a proper map, so $\pi^{-1}(A_1)$ is locally compact. Thus, $C':={\operatorname{Cl}}_M(\pi^{-1}(A_1))\setminus \pi^{-1}(A_1)$ is a closed subset of $M$. By Lemma \[opcl\] we have $C'=\pi^{-1}(C)$ and ${\operatorname{Cl}}_{{\operatorname{Spec}}(M)}(C')={\operatorname{Spec}}(\pi)^{-1}({\operatorname{Cl}}_{{\operatorname{Spec}}(N)}(C))$. As ${{\mathfrak P}}_0\not\in{{\EuScript Z}}(h)$, we deduce ${{\mathfrak P}}_0\not\in{\operatorname{Cl}}_{{\operatorname{Spec}}(N)}(C)$ and ${{\mathfrak p}}_0\notin {\operatorname{Cl}}_{{\operatorname{Spec}}(M)}(C')$.
As before, there exists $h'\in {\mathcal S}(M)$ with $Z(h')=C'$ such that the inclusion ${\tt i}:\pi^{-1}(A_1)\to M$ induces a homeomorphism $${\operatorname{Spec}}({\tt i})|:{\operatorname{Spec}}(\pi^{-1}(A_1))\setminus {{\EuScript Z}}(h'|_{\pi^{-1}(A_1)})\to {\operatorname{Cl}}_{{\operatorname{Spec}}(N)}(\pi^{-1}(A_1))\setminus {{\EuScript Z}}(h').$$ As ${{\mathfrak p}}_0$ is $z$-ideal and ${{\mathfrak p}}_0\notin {\operatorname{Cl}}_{{\operatorname{Spec}}(M)}(C')$, it follows from [@fg2 Lem.4.3] that ${{\mathfrak p}}_0\notin {{\EuScript Z}}(h')$. Thus, there exists $${{\mathfrak p}}_0'\in{\operatorname{Spec}}(\pi^{-1}(A_1))\setminus {{\EuScript Z}}(h'|_{\pi^{-1}(A_1)})$$ such that ${\operatorname{Spec}}({\tt i})({{\mathfrak p}}_0')={{\mathfrak p}}_0$.
Suppose that ${{\mathfrak p}}_0\not\in{\operatorname{Cl}}_{{\operatorname{Spec}}(M)}({\mathcal{C}}_\pi)$. Observe that $A_1\cap\pi({\mathcal{C}}_\pi)\neq\varnothing$ if and only if $\#(P_1)=1$. If such is the case, then $A_1\subset\pi({\mathcal{C}}_\pi)$. Thus, $$\begin{gathered}
{{\mathfrak P}}_0={\operatorname{Spec}}(\pi)({{\mathfrak p}}_0)\in{\operatorname{Cl}}_{{\operatorname{Spec}}(N)}(A_1)\subset{\operatorname{Cl}}_{{\operatorname{Spec}}(N)}(\pi({\mathcal{C}}_\pi))\\
={\operatorname{Cl}}_{{\operatorname{Spec}}(N)}({\operatorname{Spec}}(\pi)({\mathcal{C}}_\pi))={\operatorname{Spec}}(\pi)({\operatorname{Cl}}_{{\operatorname{Spec}}(M)}({\mathcal{C}}_\pi)).\end{gathered}$$ As $\{{{\mathfrak p}}_0\}={\operatorname{Spec}}(\pi)^{-1}({{\mathfrak P}}_0)$, we deduce ${{\mathfrak p}}_0\in{\operatorname{Cl}}_{{\operatorname{Spec}}(M)}({\mathcal{C}}_\pi)$, against our assumption. Consequently, $\#(P_1)\geq2$ and $A_1\cap\pi({\mathcal{C}}_\pi)=\varnothing$.
Consider the commutative diagram $$\xymatrix{
{\operatorname{Spec}}(A_1)\times P_1\ar@{->}[rr]^{\quad{\operatorname{Spec}}(\theta_1)\quad}\ar@{->}[rrd]_{{\operatorname{Spec}}(\pi_1)\quad}&&{\operatorname{Spec}}(\pi^{-1}(A_1))\ar@{->}[d]^{{\operatorname{Spec}}(\pi|_{\pi^{-1}(A_1)})\quad} \ar@{->}[rr]^{\quad{\operatorname{Spec}}({\tt i})\quad}&&{\operatorname{Spec}}(M)\ar@{->}[d]^{{\operatorname{Spec}}(\pi)}\\
&&{\operatorname{Spec}}(A_1) \ar@{->}[rr]^{\quad{\operatorname{Spec}}({\tt j})\quad}&& {\operatorname{Spec}}(N)
}$$ where ${\operatorname{Spec}}(\theta_1)$ is a homeomorphism. Thus, there exists ${{\mathfrak p}}'_1\in{\operatorname{Spec}}(\pi^{-1}(A_1))$ such that ${{\mathfrak p}}'_1\neq {{\mathfrak p}}'_0$ and ${\operatorname{Spec}}(\pi|_{\pi^{-1}(A_1)})({{\mathfrak p}}_0')={\operatorname{Spec}}(\pi|_{\pi^{-1}(A_1)})({{\mathfrak p}}_1')={{\mathfrak P}}_0'$. Observe that ${{\mathfrak p}}'_1\notin {{\EuScript Z}}(h'|_{\pi^{-1}(A_1)})$. Define $${{\mathfrak p}}_1:={\operatorname{Spec}}({\tt i})({{\mathfrak p}}_1')\in{\operatorname{Cl}}_{{\operatorname{Spec}}(M)}(\pi^{-1}(A_1))\setminus {{\EuScript Z}}(h').$$ As ${{\mathfrak p}}_1'\neq{{\mathfrak p}}_0'$ and ${\operatorname{Spec}}({\tt i})|$ is bijective, we deduce ${{\mathfrak p}}_1\neq{{\mathfrak p}}_0$. As ${\operatorname{Spec}}(\pi)({{\mathfrak p}}_1)={{\mathfrak P}}_0={\operatorname{Spec}}(\pi)({{\mathfrak p}}_0)$ we have ${{\mathfrak p}}_0\not\in{\mathcal{C}}_{{\operatorname{Spec}}(\pi)}$, which is a contradiction. Consequently, ${{\mathfrak p}}_0\in{\operatorname{Cl}}_{{\operatorname{Spec}}(M)}({\mathcal{C}}_\pi)$.
\(iii) and (iv) follow from the equality ${\mathcal{C}}_{{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(\pi)}={\mathcal{C}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}\cap{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)$ (that follows from Proposition \[sep2\](iii) and (iv)).
Proof of Theorem \[bc\] {#s5}
=======================
In this section we prove Theorem \[bc\]. Before that we introduce the following statement that will help us to reduce the $\mathcal{S}$-case to the $\mathcal{S}^*$-case.
\[RedStoS\*\] Let $\pi:M\to N$ be a semialgebraic branched covering. Denote $$\mathfrak{S}(M):=\{{{\mathfrak p}}\in{\operatorname{Spec^*}}(M):\ {{\mathfrak p}}\cap {\mathcal W}_M=\varnothing\}.$$ By [@fg2 Lem.3.2] the map $$\rho_M:{\operatorname{Spec}}(M)\to\mathfrak{S}(M),\ {{\mathfrak q}}\to {{\mathfrak q}}\cap \mathcal{S}^*(M)$$ is a homeomorphism whose inverse map is $\rho_M^{-1}:\mathfrak{S}(M)\to{\operatorname{Spec}}(M),\ {{\mathfrak p}}\to {{\mathfrak p}}\mathcal{S}(M)$. In particular, we have the commutative diagram $$\xymatrix{
{\operatorname{Spec}}(M)\ar[r]^{\rho_M}\ar_{{\operatorname{Spec}}(\pi)}[d]&\mathfrak{S}(M)\ar[d]^{{\operatorname{Spec^*}}(\pi)}\\
{\operatorname{Spec}}(N)\ar[r]^{\rho_N}& \mathfrak{S}(N)
}$$
\(i) If $\mathcal{B}_{{\operatorname{Spec^*}}(\pi)}={\operatorname{Cl}}_{{\operatorname{Spec^*}}(M)}(\mathcal{B}_\pi)$, then $\mathcal{B}_{{\operatorname{Spec}}(\pi)}\subset {\operatorname{Cl}}_{{\operatorname{Spec}}(M)}(\mathcal{B}_\pi)$.
Indeed, let $\mathcal{U}$ be an open neighborhood of a prime ideal ${{\mathfrak p}}\in \mathcal{B}_{{\operatorname{Spec}}(\pi)}$. Let us show: $\mathcal{U}\cap \mathcal{B}_\pi\neq \varnothing$.
Note that $\rho_M({{\mathfrak p}})\in \mathcal{B}_{{\operatorname{Spec^*}}(\pi)}={\operatorname{Cl}}_{{\operatorname{Spec^*}}(M)}(\mathcal{B}_\pi)$ and $\rho_M(\mathcal{U})=\mathcal{V}\cap \mathfrak{S}(M)$ for some open neighborhood $\mathcal{V}$ of $\rho_M({{\mathfrak p}})$ in ${\operatorname{Spec^*}}(M)$. Thus, there exists $x\in \mathcal{B}_\pi$ such that ${{\mathfrak m}}^*_x\in \mathcal{V}$. As ${{\mathfrak m}}^*_x\in\mathfrak{S}(M)$, we deduce ${{\mathfrak m}}_x\in \mathcal{U}$, as required.
\(ii) We will prove in Theorem \[bc\] that ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi):{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)\to{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)$ is a branched covering. Assume this for a while and let us prove: *For each ${{\mathfrak p}}\in{\operatorname{Spec}}(M)$ it holds $b_{{\operatorname{Spec}}(\pi)}({{\mathfrak p}})\leq b_{{\operatorname{Spec^*}}(\pi)}(\rho_M({{\mathfrak p}}))$*.**
Indeed, let $\mathcal{V}\subset{\operatorname{Spec^*}}(M)$ be an exceptional neighborhood of $\rho_M({{\mathfrak p}})$ with respect to the map ${\operatorname{Spec^*}}(\pi)$. As $\mathcal{U}':=\rho_M^{-1}(\mathfrak{S}(M)\cap \mathcal{V})$ is an open neighborhood of ${{\mathfrak p}}\in {\operatorname{Spec}}(M)$, there exists by Remark \[cuenta\](iv) an exceptional neighborhood $\mathcal{U}\subset\mathcal{U}'$ of ${{\mathfrak p}}$ with respect to ${\operatorname{Spec}}(\pi)$. By Remark \[cuenta\](v) we have $$\begin{gathered}
b_{{\operatorname{Spec}}(\pi)}({{\mathfrak p}})=\max\{\#(\pi^{-1}({{\mathfrak q}})\cap\mathcal{U}):\ {{\mathfrak q}}\in{\operatorname{Spec}}(\pi)(\mathcal{U})\}\\
\leq\max\{\#(\pi^{-1}({{\mathfrak q}}')\cap\mathcal{V}):\ {{\mathfrak q}}'\in {\operatorname{Spec^*}}(\pi)(\mathcal{V})\}=b_{{\operatorname{Spec^*}}(\pi)}(\rho_M({{\mathfrak p}})), \end{gathered}$$ as required.
The implication (ii) $\Longrightarrow$ (iii) follows from Proposition \[sep2\](iii) and (iv) and the density of ${\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)$ in ${\operatorname{Spec}}(M)$, whereas the implication (iii) $\Longrightarrow$ (i) follows from Proposition \[sep2\](v) and the density of $M$ in ${\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)$. The equalities ${\mathcal B}_{{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(\pi)}={\operatorname{Cl}}_{{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(M)}({\mathcal B}_\pi)$ and ${\mathcal R}_{{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(\pi)}={\operatorname{Cl}}_{{\operatorname{\beta^{{\text{\tiny$\displaystyle\diamond$}}}}}(N)}({\mathcal R}_\pi)$ follow from the equalities ${\mathcal B}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}({\mathcal B}_\pi)$ and ${\mathcal R}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)}({\mathcal R}_\pi)$ (once they are proved) together with Proposition \[sep2\](iv).
\(i) $\Longrightarrow$ (ii). Let $N_1,\ldots,N_r$ be the connected components of $N$ and denote $M_i:=\pi^{-1}(N_i)$. By Lemma \[rcc\] there exist integers $d_i\geq1$ such that $\pi|_{M_i}:M_i\to N_i$ is a $d_i$-branched covering. In addition, by [@fg2 Cor.4.7] ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N_1),\ldots,{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N_r)$ are the connected components of ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)$. By [@fg2 Cor.4.6] and Corollary \[rcc\] it is enough to prove Theorem \[bc\] for the $d_i$-branched coverings $\pi|_{M_i}:M_i\to N_i$. Thus, we may assume from the beginning that $N$ is connected.
By Remark \[cuenta\](ii) $\pi:M\to N$ is a $d$-branched covering for some integer $d\geq1$. By Proposition \[sep2\](ii) ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi):{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)\to{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)$ is a finite quasi-covering.
#### {#section}
Let us show now: *the fibers of ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)$ have no more than $d$ points*.**
Otherwise, there exists ${{\mathfrak q}}\in{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)$ such that $\#({\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({{\mathfrak q}}))>d$. As ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)$ is separated, there exists by Lemma \[disting\] an open neighborhood $\mathcal{V}$ of ${{\mathfrak q}}$ in ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)$ such that $\#({\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({{\mathfrak p}}))>d$ for each ${{\mathfrak p}}\in\mathcal{V}$. As $N$ is dense in ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)$, there exists ${{\mathfrak n}}_y\in\mathcal{V}\cap N$. Write $\pi^{-1}(y):=\{x_1,\ldots,x_r\}$. By Proposition \[sep2\](v) ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({{\mathfrak n}}_y)=\{{{\mathfrak m}}_{x_1},\ldots,{{\mathfrak m}}_{x_r}\}$, so $d<r$, which is a contradiction because $\pi$ is a $d$-branched covering.
#### {#section-1}
We claim: ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)\setminus{\mathcal R}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}=\{{{\mathfrak q}}\in{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N):\ \#({\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({{\mathfrak q}}))=d\}$.
The inclusion right to left follows from Lemma \[disting\] and Corollary \[max\]. To prove the converse inclusion let ${{\mathfrak q}}\in{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)\setminus{\mathcal R}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}$. Then by Lemma \[genbranch\] there exists an open neighborhood $\mathcal{W}$ of ${{\mathfrak q}}$ such that the cardinality of the fibers of the points in $\mathcal{W}$ is a constant $c$. As $N\setminus \mathcal{R}_\pi$ is dense in ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)$ the intersection $\pi^{-1}(\mathcal{W})\cap (N\setminus \mathcal{R}_\pi)$ is non-empty, so $c=d$, as claimed.
#### {#pa:regular}
We check next: *${{\mathfrak m}}_x\in{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}_{{\operatorname{reg}}}$ for $x\in M_{{\operatorname{reg}}}$*.**
By the previous remark it is enough to check that $\#(\pi^{-1}(\pi(x)))=d$, which is true by Proposition \[sep2\](v) because $x\in M_{{\operatorname{reg}}}$.
#### {#unbr}
Consequently, the restriction $${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)|_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)_{{\operatorname{reg}}}}:{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)_{{\operatorname{reg}}}\to{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)\setminus{\mathcal R}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}$$ is a $d$-unbranched covering. As $M_{{\operatorname{reg}}}$ is dense in $M$, it follows from \[pa:regular\] $${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)_{{\operatorname{reg}}}={\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)\setminus{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({\mathcal R}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)})$$ is dense in ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)$.
#### {#neigh1}
Let ${{\mathfrak q}}\in{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)$ and write ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({{\mathfrak q}}):=\{{{\mathfrak p}}_1,\ldots,{{\mathfrak p}}_r\}$. We claim: *there exist $g\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(N)$ and $f_1,\ldots,f_r\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ such that ${{\mathfrak q}}\in{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(g)$, ${{\mathfrak p}}_i\in{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(f_i)$, $f_if_j=0$ if $i\neq j$, $${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(g))=\bigsqcup_{i=1}^r{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(f_i)\quad\text{and}\quad{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)({{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(f_i))={{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(g)$$ for $i=1,\ldots,r$. ***
For each pair $1\leq i<j\leq r$ there exist by Lemma \[sep\] semialgebraic functions $f_{ij},f_{ji}\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ such that ${{\mathfrak p}}_i\in {{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(f_{ij})$, ${{\mathfrak p}}_j\in {{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(f_{ji})$ and $f_{ij}f_{ji}=0$. For each $i=1,\ldots,r$ define $h_i:=\prod_{k,\,k\neq i}f_{ik}\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$. It holds ${{\mathfrak p}}_i\in{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(h_i)$ and $h_ih_j=0$ if $i\neq j$. Observe that ${{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(h_i)\cap{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(h_j)=\varnothing$ if $i\neq j$. Define $$\mathcal{W}:=\Big({\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)\setminus{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)\Big({\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)\setminus\bigcup_{i=1}^r{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(h_i)\Big)\Big)\cap \bigcap_{i=1}^r{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)\big({{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(h_i)\big),$$ which is an open neighborhood of ${{\mathfrak q}}$ in ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)$ such that $${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}(\mathcal{W})\subset\bigcup_{i=1}^r{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(h_i).$$ Let $g\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(N)$ be such that ${{\mathfrak q}}\in{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(g)\subset \mathcal{W}$. As ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(g))={{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(g\circ\pi)$ it follows $${{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(h_i)\cap{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(g))={{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(h_i(g\circ\pi)).$$ If we define $f_i:=h_i(g\circ\pi)$ for $i=1,\ldots,r$, then the reader can check now straightforwardly that the claim follows.
#### {#pa:branched}
We claim: *${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi):{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)\to{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)$ is a $d$-branched covering map*.**
By Remark \[intersection\] it is enough to show that each ${{\mathfrak q}}\in \mathcal{R}_\pi$ has a special neighborhood. Write ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({{\mathfrak q}}):=\{{{\mathfrak p}}_1,\ldots,{{\mathfrak p}}_r\}$ where $r<d$. Let $g\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(N)$ and $f_1,\ldots,f_r\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ be as in \[neigh1\] for ${{\mathfrak q}}$ and ${{\mathfrak p}}_1,\ldots,{{\mathfrak p}}_r$.
Let $E_1,\ldots,E_s$ be the connected components of $D(g)$. By Lemma \[ccD\] the connected components ${\mathcal V}_i:={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)}(E_i)\cap{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(g)$ of ${{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(g)$ for $i=1,\ldots,s$ are open subsets of ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)$.
We may assume ${{\mathfrak q}}\in{\mathcal V}:={\mathcal V}_1$. If ${\mathcal U}:={\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({\mathcal V})$ and ${\mathcal U}_i:={\mathcal U}\cap{{\EuScript D}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(f_i)$, we have ${\mathcal U}=\bigsqcup_{i=1}^r{\mathcal U}_i$ and ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)({\mathcal U}_i)={\mathcal V}$. Observe that ${{\mathfrak p}}_i\in{\mathcal U}_i$ for $i=1,\ldots,r$.
By \[unbr\] and Lemma \[restr\] the restriction $${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)|_{{\mathcal U}\cap{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)_{{\operatorname{reg}}}}:{\mathcal U}\cap{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)_{{\operatorname{reg}}}\to{\mathcal V}\setminus{\mathcal R}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}$$ is a $d$-unbranched covering. By Lemma \[opcl2\] the restriction $${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)|_{{\mathcal U}_1\cap{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)_{{\operatorname{reg}}}}:{\mathcal U}_1\cap{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)_{{\operatorname{reg}}}\to{\mathcal V}\setminus{\mathcal R}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}$$ is a branched covering with empty ramification set. If we prove that there exists an open dense subset ${\mathcal G}$ of $\mathcal{U}_1$ such that the cardinality of the fibers of $${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)|_{{\mathcal G}}:\mathcal{G}\to {\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)({\mathcal G})$$ is a constant $e\in {{\mathbb N}}$, we deduce by Remark \[cuenta\](iii) that ${\mathcal U}_1$ is an exceptional neighborhood of ${{\mathfrak p}}_1$ with respect to ${\mathcal V}$. As this can be done with each ${\mathcal U}_i$, we deduce that ${\mathcal V}$ is a special neighborhood of ${{\mathfrak q}}$ and ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)$ is a $d$-branched covering.
To finish the proof of Theorem \[bc\] it only remains to prove the equalities $\mathcal{B}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}({\mathcal B}_\pi)$ and ${\mathcal R}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)}({\mathcal R}_\pi).$
#### {#section-2}
We claim: *there exists an open dense subset ${\mathcal G}$ of ${\mathcal U}_1$ such that the cardinality of the fibers of the restriction map ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)|_{\mathcal G}$ is a constant $e\in {{\mathbb N}}$.***
Denote $E:=E_1$, $D:=\pi^{-1}(E)$ and $D_i:=D(f_i)\cap D$ for $i=1,\ldots,r$. By \[neigh1\] and Proposition \[sep2\](v) we have $D=\bigsqcup_{i=1}^rD_i$ and $\pi(D_i)=E$ for $i=1,\ldots,r$. By Lemma \[restr\] the restriction $\pi|_D:D\to E$ is a $d$-branched covering. By Remark \[cuenta\](ii) and Lemma \[opcl2\] $\pi|_{D_1}:D_1\to E$ is an $e$-branched covering for some integer $e\geq1$. By \[unbr\] applied to $\pi|_{D_1}$, $$\label{d1}
{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi|_{D_1}):{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(D_1)_{{\operatorname{reg}}}\to{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(E)\setminus{\mathcal R}_{\pi|_{D_1}}$$ is an $e$-unbranched covering. As $E$ is dense in ${\mathcal V}$ and ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)|_{\mathcal U}:{\mathcal U}\to{\mathcal V}$ is by Lemma \[opcl\] open, closed and surjective, we deduce that $D$ is dense in ${\mathcal U}$. As $D\cap{\mathcal U}_1=D_1$, we deduce that $D_1$ is dense in ${\mathcal U}_1
$.
Let ${\tt i}: D_1\to M$ and ${\tt j}: E \to N$ be the inclusions. Consider the commutative diagrams $${\small\xymatrix{
D_1\ar[d]_{\pi|_{D_1}}\ar@{^{(}->}[r]^(0.5){\tt i}& M\ar@<0ex>[d]^\pi&&{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(D_1)\ar[d]_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi|_{D_1})}\ar[rr]^(0.5){{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}({\tt i})}&&{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)\ar@<0ex>[d]^{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}\\
E\ar@{^{(}->}[r]^(0.5){\tt j}& N&&{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(E)\ar[rr]^(0.5){{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}({\tt j})}&&{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)
}}$$ By Lemma \[homeo:fe\_FG\] there exist $a\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(M)$ and $b\in{\mathcal S}^{{\text{\tiny$\displaystyle\diamond$}}}(N)$ such that $$\begin{aligned}
&{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}({\tt i})|:{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(D_1)\setminus{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}({\tt i})^{-1}({{\EuScript Z}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(a))\to{\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(D_1)\setminus{{\EuScript Z}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(a),\\
&{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}({\tt j})|:{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(E)\setminus{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}({\tt j})^{-1}({{\EuScript Z}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(b))\to{\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)}(E)\setminus{{\EuScript Z}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(b),\end{aligned}$$ are homeomorphisms,
- ${\operatorname{Cl}}_M(D_1)\setminus Z(a)$ is dense in ${\operatorname{Cl}}_M(D_1)$ and
- ${\operatorname{Cl}}_N(E)\setminus Z(b)$ is dense in ${\operatorname{Cl}}_N(E)$.
In particular, ${\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(D_1)\setminus{{\EuScript Z}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(a)$ is dense in ${\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(D_1)$ and ${\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)}(E)\setminus{{\EuScript Z}^{{{\text{\tiny$\displaystyle\diamond$}}}}}(b)$ is dense in ${\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)}(E)$.
As $D_1$ is dense in ${\mathcal U}_1$ and $E$ is dense in $\mathcal{V}$, we deduce $$\begin{aligned}
{\mathcal U}_1&\subset{\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}({\mathcal U}_1)={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(D_1),\\
{\mathcal V}&\subset{\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}({\mathcal V})={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(E).\end{aligned}$$ Define ${{\EuScript Z}}_1:=\mathcal{U}_1\cap {{\EuScript Z}}^{{\text{\tiny$\displaystyle\diamond$}}}(a)$ and ${{\EuScript Z}}_2:=\mathcal{V}\cap {{\EuScript Z}}^{{\text{\tiny$\displaystyle\diamond$}}}(b)$, which are closed nowhere dense subsets of $\mathcal{U}_1$ and $\mathcal{V}$. As ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)|_{\mathcal{U}_1}:{\mathcal{U}_1}\to {\mathcal{V}}$ is a finite quasi-covering and ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)_{{\operatorname{reg}}}\cap {\mathcal{U}_1}\subset \mathcal{U}_{1,{\operatorname{reg}}}$ is dense in $\mathcal{U}_1$, we have by Lemma \[nowhere\] that ${\operatorname{Spec}}(\pi)({{\EuScript Z}}_1)$ is a closed nowhere dense subset of $\mathcal{V}$. Thus, $${\mathcal G}:=\big(\mathcal{U}_1\cap {\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)_{{\operatorname{reg}}}\big) \setminus \big({\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)({{\EuScript Z}}_1)\cap {{\EuScript Z}}_2)\big)$$ is an open dense subset of $\mathcal{U}_1$. As the spectral map is an $e$-unbranched covering, we deduce (via the homeomorphisms ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}({\tt i})|$ and ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}({\tt j})|$) that the cardinality of the fibers of the restriction $${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)|_{{\mathcal G}}:{\mathcal G}\to {\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)({\mathcal G})\subset \mathcal{V}\setminus{\mathcal R}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}$$ is also $e$, as claimed.
#### {#closureofB}
We prove next: $\mathcal{B}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}({\mathcal B}_\pi)$.
The inclusion $\mathcal{B}_{\pi}\subset \mathcal{B}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}$ is clear. As $\mathcal{B}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}$ is a closed subset of ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)$, it holds ${\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}({\mathcal B}_\pi)\subset \mathcal{B}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}$. To prove the converse, pick ${{\mathfrak p}}_1\in \mathcal{B}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}$ and let us show: ${{\mathfrak p}}_1\in{\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}({\mathcal B}_\pi)$. By Remark \[RedStoS\*\](i) it is enough to prove it in the $\mathcal{S}^*$-case.
Denote ${{\mathfrak q}}:={\operatorname{Spec^*}}({{\mathfrak p}}_1)$ and ${\operatorname{Spec^*}}(\pi)^{-1}({{\mathfrak q}})=\{{{\mathfrak p}}_1,\ldots,{{\mathfrak p}}_r\}$. As ${\operatorname{Spec^*}}(\pi):{\operatorname{Spec^*}}(M)\to{\operatorname{Spec^*}}(N)$ is a $d$-branched covering, there exist a special neighborhood $\mathcal{V}$ of ${{\mathfrak q}}$ and corresponding exceptional neighborhoods $\mathcal{U}_1,\ldots,\mathcal{U}_r$ of ${{\mathfrak p}}_1,\ldots,{{\mathfrak p}}_r$. Let $h_i \in{\mathcal S}^*(M)$ be such that ${{\mathfrak p}}_i\in{{\EuScript D}}^*(h_i)\subset \mathcal{U}_i$ for $i=1,\ldots,r$. Arguing as in the proof of \[neigh1\], we obtain functions $g\in{\mathcal S}^*(N)$ and $f_1,\ldots,f_r \in{\mathcal S}^*(M)$ such that ${{\EuScript D}}^*(g)$ is a special neighborhood of ${{\mathfrak q}}$ and ${{\mathfrak p}}_i\in {{\EuScript D}}^*(f_i)$ for $i=1,\ldots,r$ are exceptional neighborhoods with respect to ${{\EuScript D}}^*(g)$ (see Remark \[cuenta\](iv)). In particular, ${\operatorname{Spec^*}}(\pi)|_{{{\EuScript D}}^*(f_1)}:{{\EuScript D}}^*(f_1)\to{{\EuScript D}}^*(g)$ is an $e$-branched covering whose collapsing set contains ${{\mathfrak p}}_1$. Note that $b_{{\operatorname{Spec^*}}(\pi)}({{\mathfrak p}}_1)=e$ and as ${{\mathfrak p}}_1\in{\mathcal B}_{{\operatorname{Spec^*}}(\pi)}$, we deduce $e>1$. We also point out: ${\operatorname{Cl}}_{{\operatorname{Spec^*}}(M)}({{\EuScript D}}^*(f_1))\cap {{\EuScript D}}^*(f_j)=\varnothing$ for each $j\neq 1$ and by Lemma \[neigh\] ${\operatorname{Cl}}_{{\operatorname{Spec^*}}(M)}({{\EuScript D}}^*(f_1))={\operatorname{Cl}}_{{\operatorname{Spec^*}}(M)}(D(f_1))$.
Next, consider the open subset $D(g)$ of $N$ and note that $\pi^{-1}(D(g))=\bigsqcup_{i=1}^r D(f_i)$. Thus, by Lemmas \[restr\] and \[opcl2\] we have that $\pi|_{D(f_1)}: D(f_1)\to D(g)$ is a branched covering. By Proposition \[sep2\] it is an $e$-branched covering.
Let ${\tt i}:D(f_1)\hookrightarrow M$ and ${\tt j}:D(g)\hookrightarrow N$ be the inclusions. Denote $Z:={\operatorname{Cl}}_M(D(f_1))\setminus D(f_1)$ and note that as ${\operatorname{Cl}}_M(D(f_1))\cap D(f_j)=\varnothing$ for $j\neq 1$, we have by Lemma \[opcl\] that $\pi(Z)={\operatorname{Cl}}_N(D(g))\setminus D(g)$. By Lemma \[homeo:fe\_FG\] $${\operatorname{Spec^*}}({\tt i})|:{\operatorname{Spec^*}}(D(f_1))\setminus{\operatorname{Spec^*}}({\tt i})^{-1}({{\EuScript Z}})\to{\operatorname{Cl}}_{{\operatorname{Spec^*}}(M)}(D(f_1))\setminus{{\EuScript Z}}$$ is a homeomorphism, where ${{\EuScript Z}}:={\operatorname{Cl}}_{{\operatorname{Spec^*}}(M)}(Z)$. Similarly, $${\operatorname{Spec^*}}({\tt j})|:{\operatorname{Spec^*}}(D(g))\setminus{\operatorname{Spec^*}}({\tt j})^{-1}({{\EuScript Z}}')\to{\operatorname{Cl}}_{{\operatorname{Spec^*}}(N)}(D(g))\setminus{{\EuScript Z}}'$$ is a homeomorphism, where ${{\EuScript Z}}':={\operatorname{Cl}}_{{\operatorname{Spec^*}}(N)}(\pi(Z))={\operatorname{Spec^*}}(\pi)({{\EuScript Z}})$.
We claim: *${{\mathfrak p}}_1\in {\operatorname{Cl}}_{{\operatorname{Spec^*}}(M)}(D(f_1))\setminus {{\EuScript Z}}$ and in particular ${{\mathfrak q}}={\operatorname{Spec^*}}({{\mathfrak p}}_1)\in {\operatorname{Cl}}_{{\operatorname{Spec^*}}(N)}(D(g))\setminus {{\EuScript Z}}'$.***
Suppose ${{\mathfrak p}}_1\in {{\EuScript Z}}$. As ${{\mathfrak p}}_1\in {{\EuScript D}}^*(f_1)$, we deduce ${{\EuScript D}}^*(f_1)\cap Z\neq \varnothing$, so there exists $x\in {\operatorname{Cl}}_M(D(f_1))\setminus D(f_1)$ such that ${{\mathfrak m}}^*_x\in {{\EuScript D}}^*(f_1)$, which is a contradiction because ${{\EuScript D}}^*(f_1)\cap M=D(f_1)$.
We have the following commutative diagrams $$\xymatrix{
D(f_1)\ar@{^{(}->}[r]^{{\tt i}}\ar[d]_{\pi|_{D(f_1)}}&M\ar[d]^\pi& {\operatorname{Spec^*}}(D(f_1))\setminus {\operatorname{Spec^*}}({\tt i})^{-1}({{\EuScript Z}})\ar[rr]^(0.55){{\operatorname{Spec^*}}({\tt i})|}\ar[d]_{{\operatorname{Spec^*}}(\pi|_{D(f_1)})}&& {\operatorname{Cl}}_{{\operatorname{Spec^*}}(M)}(D(f_1))\setminus {{\EuScript Z}}\ar@<-5ex>[d]_{{\operatorname{Spec^*}}(\pi)|}\\
D(g)\ar@{^{(}->}[r]^{{\tt j}}&N&{\operatorname{Spec^*}}(D(g))\setminus {\operatorname{Spec^*}}({\tt j})^{-1}({{\EuScript Z}}')\ar[rr]^(0.55){{\operatorname{Spec^*}}({\tt j})|}&&{\operatorname{Cl}}_{{\operatorname{Spec^*}}(N)}(D(g))\setminus {{\EuScript Z}}'
}$$ where the maps ${\operatorname{Spec^*}}({\tt i})|$ and ${\operatorname{Spec^*}}({\tt j})|$ are (as proved above) homeomorphisms. As $${\operatorname{Spec^*}}(\pi)^{-1}({{\mathfrak q}})\cap {\operatorname{Cl}}_{{\operatorname{Spec^*}}(M)}(D(f_1))=\{{{\mathfrak p}}_1\}$$ we deduce from Theorem \[colapse\] that ${\operatorname{Spec^*}}({\tt i})^{-1}({{\mathfrak p}}_1)\in \mathcal{C}_{{\operatorname{Spec^*}}(\pi|_{D(f_1)})}={\operatorname{Cl}}_{{\operatorname{Spec^*}}(D(f_1))}(\mathcal{C}_{\pi|_{D(f_1)}})$. By Lemma \[colapseinB\] we conclude ${{\mathfrak p}}_1\in {\operatorname{Cl}}_{{\operatorname{Spec^*}}(M)}(\mathcal{C}_{\pi|_{D(f_1)}})\subset {\operatorname{Cl}}_{{\operatorname{Spec^*}}(M)}(\mathcal{B}_{\pi})$.
#### {#section-3}
By Lemma \[opcl\] $$\begin{gathered}
{\mathcal R}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}={\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)({\mathcal B}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)})={\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)({\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}({\mathcal B}_\pi))\\
={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)}({\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)({\mathcal B}_\pi))={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)}(\pi({\mathcal B}_\pi))={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)}({\mathcal R}_\pi).\end{gathered}$$ Just for the record, by Lemma \[opcl\] $$\begin{gathered}
{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({\mathcal R}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)})={\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)}({\mathcal R}_\pi))\\
={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)}({\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({\mathcal R}_\pi))={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(\pi^{-1}({\mathcal R}_\pi)).\end{gathered}$$ This means that ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)_{{\operatorname{reg}}}={\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)\setminus{\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(\pi^{-1}({\mathcal R}_\pi))$, as required.
\[ri\] Let $\pi:M\to N$ be a semialgebraic $d$-branched covering and let ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi):{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)\to{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(N)$ be the associated spectral map, which is by Theorem \[bc\] a $d$-branched covering.
\(i) Fix an integer $e\geq2$ and let us check: $$\{b_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}\geq e\}={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(\{b_\pi\geq e\}).$$ The latter shows the neat behavior of $b_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}$ with respect to $b_\pi$, because $$\begin{gathered}
\{b_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}=e\}=\{b_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}\geq e\}\setminus\{b_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}\geq e+1\}\\
={\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(\{b_\pi\geq e\})\setminus{\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(\{b_\pi\geq e+1\}).\end{gathered}$$
Let ${{\mathfrak p}}\in{\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(\{b_\pi\geq e\})$ and let ${\mathcal U}$ be an exceptional neighborhood of ${{\mathfrak p}}$. Then ${\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)|_{\mathcal U}:{\mathcal U}\to{\mathcal V}:={\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)({\mathcal U})$ is an $(b_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}({{\mathfrak p}}))$-branched covering. In particular, there exists $x\in \{b_\pi\geq e\}$ such that ${{\mathfrak m}}^{{\text{\tiny$\displaystyle\diamond$}}}_x\in \mathcal{U}$. As $\mathcal{U}\cap M$ is an open neighborhood of $x\in M$, there exists by Remark \[cuenta\](iv) an exceptional neighborhood $U$ of $x$ such that $U\subset \mathcal{U}\cap M$. By Proposition \[sep2\] and Remark \[cuenta\](v) we deduce $$\begin{gathered}
e\leq b_\pi(x)=\max\{\#(\pi^{-1}(y)\cap U):y\in \pi(U)\}\\
\leq\max\{\#({\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)^{-1}({{\mathfrak q}})\cap \mathcal{U}):{{\mathfrak q}}\in {\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)(\mathcal{U})\}=b_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}({{\mathfrak p}}).\end{gathered}$$
Let us show the converse inclusion. In the $\mathcal{S}^*$-case, we showed in \[closureofB\] inside the proof of Theorem \[bc\] that $$\{b_{{\operatorname{Spec^*}}(\pi)}= e\}\subset {\operatorname{Cl}}_{{\operatorname{Spec^*}}(M)}(\{b_\pi= e\}),$$ so $\{b_{{\operatorname{Spec^*}}(\pi)}\geq e\}\subset{\operatorname{Cl}}_{{\operatorname{Spec^*}}(M)}(\{b_\pi\geq e\})$. In the $\mathcal{S}$-case we obtain by Remark \[RedStoS\*\](ii) $$\{{{\mathfrak p}}:\ b_{{\operatorname{Spec}}(\pi)}({{\mathfrak p}})\geq e\}\subset\{{{\mathfrak p}}:\ b_{{\operatorname{Spec^*}}(\pi)}(\rho_M({{\mathfrak p}}))\geq e\}\subset\{{{\mathfrak p}}:\ \rho_M({{\mathfrak p}})\in{\operatorname{Cl}}_{{\operatorname{Spec^*}}(M)}(\{b_\pi\geq e\})\},$$ so $\{b_{{\operatorname{Spec}}(\pi)}({{\mathfrak p}})\geq e\}\subset{\operatorname{Cl}}_{{\operatorname{Spec}}(M)}(\{b_\pi\geq e\})$, as required.
\(ii) For each $x\in M$ we have $b_{\pi}(x)=b_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}({{\mathfrak m}}^{{\text{\tiny$\displaystyle\diamond$}}}_x)$.
Indeed, if we denote $e:=b_{\pi}(x)$, then ${{\mathfrak m}}^{{\text{\tiny$\displaystyle\diamond$}}}_x\in {\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(\{b_\pi\geq e\})$. As $\{b_\pi\geq e+1\}$ is by Remark \[cuenta\](v) a closed subset of $M$, we have ${\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(\{b_\pi\geq e+1\})\cap M=\{b_\pi\geq e+1\}$, so ${{\mathfrak m}}_x\in{\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(\{b_\pi\geq e\})\setminus{\operatorname{Cl}}_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(M)}(\{b_\pi\geq e+1\})=\{b_{{\operatorname{Spec^{{{\text{\tiny$\displaystyle\diamond$}}}}}}(\pi)}=e\}$, as required.
Bezoutian covering {#A}
==================
Let ${\mathcal S}_n$ denote the symmetric group in $n$ symbols. For each $\gamma\in{\mathcal S}_n$ consider the semialgebraic homeomorphism $${\widehat\gamma}:{{\mathbb R}}^n\to{{\mathbb R}}^n,\ x:=(x_1,\dots,x_n)\mapsto(x_{\gamma(1)}\dots,x_{\gamma(n)}),$$ and define the following equivalence relation $E$ in ${{\mathbb R}}^n$: $$E:=\bigcup_{\gamma\in{\mathcal S}_n}\{(x,z)\in{{\mathbb R}}^n\times{{\mathbb R}}^n:\ z={\widehat\gamma}(x)\},$$ which is a closed semialgebraic subset of ${{\mathbb R}}^n\times{{\mathbb R}}^n$. In addition, $\pi_1:E\to{{\mathbb R}}^n,\ (x,z)\mapsto x$ is a proper map because $\pi_1^{-1}([-r,r]^n)\subset[-r,r]^n\times[-r,r]^n$ for each real number $r>0$.
According to [@br Thm. 1.4] there exist a semialgebraic set $N$, a surjective semialgebraic map $f:{{\mathbb R}}^n\to N$ and a homeomorphism $g:{{\mathbb R}}^n/E\to N$ such that $f=g\circ\pi$, where $\pi:{{\mathbb R}}^n\to{{\mathbb R}}^n/E$ is the natural projection. We claim: *the semialgebraic set $N$ and the maps $f$ and $g$ admit a very precise description*.**
Let $\sigma_k\in{{\mathbb Z}}[{\tt x}_1,\dots,{\tt x}_n]$ be the elementary symmetric forms for $1\leq k\leq n$ and consider the polynomial map $$\sigma:{{\mathbb R}}^n\to{{\mathbb R}}^n,\ x\mapsto(\sigma_1(x),\dots,\sigma_n(x)).$$ Then $N:=\sigma({{\mathbb R}}^n)$ is a semialgebraic set, the semialgebraic map $(f:=)\sigma:{{\mathbb R}}^n\to N$ is surjective and *$(g:=)\ol{\sigma}:{{\mathbb R}}^n/E\to N,\ [x]\mapsto\sigma(x)$ is a well-defined bijection*.**
If $[x]=[z]$, there exists $\gamma\in{\mathcal S}_n$ such that $z={\widehat\gamma}(x)$. Hence $\sigma(x)=\sigma(z)$ because each component $\sigma_k$ of $\sigma$ is a symmetric polynomial. To prove that $\ol{\sigma}$ is injective pick $x,z\in{{\mathbb R}}^n$ such that $\sigma(x)=\sigma(z)$. Then $$\prod_{i=1}^n({{\tt t}}-x_i)={{\tt t}}^n+\sum_{k=1}^{n}(-1)^k\sigma_k(x){{\tt t}}^{n-k}={{\tt t}}^n+\sum_{k=1}^{n}(-1)^k\sigma_k(z){{\tt t}}^{n-k}=\prod_{i=1}^n({{\tt t}}-z_i).$$ Thus, there exists $\gamma\in{\mathcal S}_n$ such that $z={\widehat\gamma}(x)$, so $[x]=[z]$.
Consequently, $\sigma^{-1}(\sigma(z))=\{\gamma(z):\ \gamma\in{\mathcal S}_n\}$ for each $z\in{{\mathbb R}}^n$. We have the following commutative diagram: $$\xymatrix{
{{\mathbb R}}^n\ar[r]^(0.4)\pi\ar[rd]_\sigma&{{\mathbb R}}^n/E\ar[d]^{\ol{\sigma}}\\
&N
}$$ Note that $\ol{\sigma}$ is continuous and let us see: *$\ol{\sigma}$ is a homeomorphism*.**
For each $u:=(u_1,\ldots,u_n)\in{{\mathbb R}}^n$ consider the polynomial $$f_u({\tt t}):={\tt t}^n+\sum_{k=1}^{n}(-1)^ku_k{\tt t}^{k}.$$ Denote $\zeta_1(u),\dots,\zeta_n(u)$ the real parts of the (complex) roots of the polynomial $f_u$. Each value $\zeta_i(u)$ is repeated according to the multiplicity of the corresponding root. We index such values in such a way that $\zeta_1(u)\leq\cdots\leq\zeta_n(u)$. By [@gj §13.3] the functions $\zeta_1,\ldots,\zeta_n:{{\mathbb R}}^n\to{{\mathbb R}}$ are continuous. As $N$ is exactly the set of points $a\in{{\mathbb R}}^n$ such that $f_a$ has $n$ real roots, the map $$\label{sectt}
s:N\to{{\mathbb R}}^n,\ a\mapsto(\zeta_1(a),\dots,\zeta_n(a))$$ is a continuous section of $\sigma$. In particular, $\ol{\sigma}^{-1}=\pi\circ s$ is continuous, so $\ol{\sigma}$ is a homeomorphism.
We prove next: *$\zeta_1,\dots,\zeta_n$ have semialgebraic graph*, so $s:N\to{{\mathbb R}}^n$ is a semialgebraic map.**
Let ${\tt u}:=({\tt u}_1,\dots,{\tt u}_n)$ and ${\tt z}$ be variables, ${\tt i}:=\sqrt{-1}$. Consider the non-zero polynomial $${\tt P}({\tt u},{\tt z}):={\tt z}^n+\sum_{j=1}^n{\tt u}_j{\tt z}^{n-j}\in{{\mathbb Z}}[{\tt u},{\tt z}].$$
If we write ${\tt z}:={\tt x}+{\tt i}{\tt y}$, we have $${\tt P}({\tt u},{\tt z})=({\tt x}+{\tt i}{\tt y})^n+\sum_{j=1}^n{\tt u}_j({\tt x}+{\tt i}{\tt y})^{n-j}={\tt P}_1({\tt u},{\tt x},{\tt y})+{\tt i}{\tt P}_2({\tt u},{\tt x},{\tt y})$$ for certain non-zero polynomials ${\tt P}_1,{\tt P}_2\in{{\mathbb Z}}[{\tt u},{\tt x},{\tt y}]$. Let $\zeta_j(u)+{\tt i}\eta_j(u)\in{{\mathbb C}}$ be the roots of $f_u$ for $u\in{{\mathbb R}}^n$ (where $1\leq j\leq n$). Then $${\tt P}_1(u,\zeta_j(u),\eta_j(u))+{\tt i}{\tt P}_2(u,\zeta_j(u),\eta_j(u))={\tt P}(u,\zeta_j(u)+{\tt i}\eta_j(u))=f_u(\zeta_j(u)+{\tt i}\eta_j(u))=0.$$ Consequently, $${\tt P}_1(u,\zeta_j(u),\eta_j(u))=0,\ {\tt P}_2(u,\zeta_j(u),\eta_j(u))=0.$$ Let ${\tt R}({\tt u},{\tt x})\in{{\mathbb Z}}[{\tt u},{\tt x}]$ be the resultant, with respect to ${\tt y}$, of the polynomials ${\tt P}_1({\tt u},{\tt x},{\tt y})$ and ${\tt P}_2({\tt u},{\tt x},{\tt y})$. For each $u\in{{\mathbb R}}^n$ the real number $\eta_j(u)$ is a common root of ${\tt P}_1(u,\zeta_j(u),{\tt y})$ and ${\tt P}_2(u,\zeta_j(u),{\tt y})$, so ${\tt R}(u,\zeta_j(u))=0$ for $u\in{{\mathbb R}}^n$ and $1\leq j\leq n$. Thus, $\zeta_j$ has semialgebraic graph, as claimed.
For each $p\in{{\mathbb R}}^n$ the cardinality of the fiber $\sigma^{-1}(\sigma(p))$ is less than or equal to ${\operatorname{ord}}({\mathcal S}_n)=n!$. The equality is achieved if the coordinates of $x$ are pairwise distinct. Let us check: *$\sigma:{{\mathbb R}}^n\to N$ is a semialgebraic finite quasi-covering*. It is enough to show: *it is an open and closed map*, or equivalently, *$\pi:{{\mathbb R}}^n\to{{\mathbb R}}^n/E$ is an open and closed map*.******
Let $A$ be an open (resp. closed) subset of ${{\mathbb R}}^n$. Then the union $$\pi^{-1}(\pi(A))=\bigcup_{\gamma\in{\mathcal S}_n}{\widehat\gamma}^{-1}(A)$$ is an open (resp. closed) subset of ${{\mathbb R}}^n$, so $\pi(A)$ is open (resp. closed) in ${{\mathbb R}}^n/E$.
As $\sigma^{-1}(\sigma(z))=\{\gamma(z):\ \gamma\in{\mathcal S}_n\}$ for each $z\in{{\mathbb R}}^n$, the collapsing set of $\sigma$ is $${\mathcal{C}}_{\sigma}=\{(z,\ldots,z)\in{{\mathbb R}}^n:\ z\in{{\mathbb R}}\},$$ whereas the branching set of $\sigma$ is $${\mathcal B}_{\sigma}=\bigcup_{1\leq i<j\leq n}\{(x_1,\dots,x_n)\in{{\mathbb R}}^n:\ x_i=x_j\},$$ which is a finite union of hyperplanes of ${{\mathbb R}}^n$ (and it is nowhere dense in ${{\mathbb R}}^n$).
The inclusion right to left is clear. Suppose conversely that the coordinates of $x:=(x_1,\ldots,x_n)\in \mathbb{R}^n$ are pairwise distinct. Let $I_i\subset{{\mathbb R}}$ be an open interval that contains $x_i$ and satisfies $I_i\cap I_j=\varnothing$ if $i\neq j$. The restriction of $\sigma$ to $\prod_{i=1}^nI_i$ is a homeomorphism onto its image. Thus, $x\not\in{\mathcal B}_{\sigma}$.
Observe that $\sigma^{-1}(\sigma({\mathcal B}_{\sigma}))={\mathcal B}_{\sigma}$, so $\mathbb{R}^n_{\text{reg}}=\mathbb{R}^n\setminus{\mathcal B}_{\sigma}$.
The restriction map $\sigma|_{\mathbb{R}^n_{\text{reg}}}:\mathbb{R}^n_{\text{reg}}\to N\setminus{\mathcal R}_{\sigma}$ is an $(n!)$-unbranched covering. Let us show next: *$\sigma$ is an $(n!)$-branched covering*.**
For each $\gamma\in {\mathcal S}_n$ consider the semialgebraic section $s_\gamma:=\widehat{\gamma}\circ s:N\to {{\mathbb R}}^n$ of $\sigma$, where the semialgebraic map $s$ was defined in . Pick $a\in N$ and write $$f_a({\tt t}):={\tt t}^n+\sum_{k=1}^{n}(-1)^ka_k{\tt t}^{k}=({\tt t}-b_1)^{k_1}\cdots ({\tt t}-b_\ell)^{k_{\ell}}$$ (where $k_1+\cdots+k_\ell=n$). The cardinality of $\pi^{-1}(a)$ is $d:=\frac{n!}{k_1!\cdots k_\ell!}$ and $\sigma^{-1}(a)=\{s_\gamma(a):\ \gamma\in{\mathcal S}_n\}$. Write $\pi^{-1}(a):=\{p_1,\ldots,p_d\}$ and let $V$ be a connected open semialgebraic neighborhood of $a$ in $N$ such that there exist pairwise disjoint connected open semialgebraic neighborhoods $U_1,\ldots,U_d$ of $p_1,\ldots,p_d$ satisfying $\sigma(U_i)=V$ and $\sigma^{-1}(V)=\bigsqcup_{i=1}^dU_i$ (use Lemma \[disting\]). Define ${\mathcal S}_{n,i}:=\{\gamma\in{\mathcal S}_n:\ s_\gamma(a)=p_i\}$. Thus, ${\mathcal S}_{n,i}\cap{\mathcal S}_{n,j}=\varnothing$ for $i\neq j$ and ${\mathcal S}_n=\bigsqcup_{i=1}^d{\mathcal S}_{n,i}$. In addition, if $i\neq j$, there exists $\gamma_{ij}\in {\mathcal S}_n$ such that $\widehat{\gamma}_{ij}(p_i)=p_j$. The map ${\mathcal S}_{n,i}\to{\mathcal S}_{n,j},\ \gamma\mapsto\gamma_{ij}\circ\gamma$ is a bijection. We deduce that the cardinality of each ${\mathcal S}_{n,i}$ equals $r:=k_1!\cdots k_\ell!$. In addition, $U_i=s_\gamma(V)$ for each $\gamma\in {\mathcal S}_{n,i}$ and each $i=1,\ldots,d$. The reader can check that ${\mathcal B}_{\sigma|_{U_i}}={\mathcal B}_\sigma\cap U_i$, ${\mathcal R}_{\sigma|_{U_i}}={\mathcal R}_\sigma\cap V$ and $U_{i,{\operatorname{reg}}}=U_i\cap \mathbb{R}^n_{\text{reg}}$. The restriction map $\sigma|_{U_{i,{\operatorname{reg}}}}:U_{i,{\operatorname{reg}}}=U_i\cap \mathbb{R}^n_{\text{reg}}\to V\setminus{\mathcal R}_{\sigma|_{U_i}}=V\setminus{\mathcal R}_\sigma$ is an unbranched semialgebraic covering of $r$ sheets (the ramification index at each point $p_i$ is equal to $r$). Consequently, $V$ is a special neighborhood of $a$ and $U_1,\ldots,U_d$ are the corresponding exceptional neighborhoods for $p_1,\ldots,p_d$.
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[^1]: Authors supported by Spanish GRAYAS MTM2014-55565-P, Spanish STRANO MTM2017-82105-P and Grupos UCM 910444
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abstract: 'Matveev and Piergallini independently showed that, with a small number of known exceptions, any triangulation of a three-manifold can be transformed into any other triangulation of the same three-manifold with the same number of vertices, via a sequence of 2-3 and 3-2 moves. We can interpret this as showing that the Pachner graph of such triangulations is connected. In this paper, we extend this result to show that (again with a small number of known exceptions), the subgraph of the Pachner graph consisting of triangulations without degree one edges is also connected, for single-vertex triangulations of closed manifolds, and ideal triangulations of manifolds with non-spherical boundary components.'
author:
- Henry Segerman
bibliography:
- '../henrybib.bib'
title: 'Connectivity of triangulations without degree one edges under 2-3 and 3-2 moves'
---
Introduction
============
Matveev [@matveev_2-3_connectivity_paper; @matveev] and (independently) Piergallini [@piergallini] show that the set of triangulations of a three-manifold is connected under 2-3 and 3-2 moves, as long as we ignore the small number of triangulations of manifolds that consist of only a single tetrahedron. For these, no 2-3 or 3-2 move can be applied. Said another way, we consider the *Pachner graph* $\mathfrak{T}(M)$, whose vertices are the triangulations of a given manifold $M$, where two vertices are connected by an edge if the corresponding triangulations are related by a 2-3 move. Matveev’s and Piergallini’s result then says that $\mathfrak{T}(M)$ is connected, again if we ignore the one-tetrahedron triangulations.
A natural question to ask is whether this remains true when we impose conditions on the triangulations. In other words we consider the connectivity of the subgraph of $\mathfrak{T}(M)$ corresponding to a given property. For example, we could consider the subgraph of geometric triangulations. Hoffman noted that the subgraph of geometric triangulations of the figure 8 knot complement is not connected; in fact there are isolated geometric triangulations (this observation is mentioned in [@dadd_duan]). However, geometricity is a very strong property. There are many weaker properties, corresponding to larger subgraphs, which may be connected. Of particular interest are the properties of *1-efficiency* and of *having essential edges*. The *3D index* is a recent quantum object mapping ideal triangulations to Laurent series, introduced by Dimofte, Gukov, and Gaiotto [@dimofte_gaiotto_gukov_1; @dimofte_gaiotto_gukov_2]. Garoufalidis [@garoufalidis_3D_index] showed that the 3D index is invariant under 2-3 and 3-2 moves, provided that it is defined on both sides of the move. In [@ghrs_index_structures], Garoufalidis, Hodgson, Rubinstein and the author show that the 3D index is defined only on 1-efficient triangulations. Therefore, if we knew that the subgraph of $\mathfrak{T}(M)$ corresponding to 1-efficient triangulations were connected, we would have an invariant of the manifold. Unfortunately this is not known. Also in [@ghrs_index_structures], we define an extremely small subgraph of the 1-efficient triangulations that we can prove is connected, and since the subgraph is determined purely by the topology of the manifold, we have an invariant. However, this is a somewhat unsatisfying workaround. A very similar story holds for the *1-loop invariant*, as defined by Dimofte and Garoufalidis [@dimofte_garoufalidis]. In this case, the triangulations are required to have solutions to Thurston’s gluing equations corresponding to the complete hyperbolic structure, which is implied by the triangulation having essential edges. Again it is not known if the triangulations with essential edges form a connected subgraph of $\mathfrak{T}(M)$.
In this paper, we answer the connectivity question in the affirmative for the subgraph of triangulations with the property of *having no degree one edges*.
Let $M$ be an oriented three-manifold other than the lens space L(4,1). If $M$ is closed, let $\mathfrak{T}(M)$ denote the set of single vertex triangulations of $M$. If $M$ has boundary, let $\mathfrak{T}(M)$ denote the set of ideal triangulations of $M$. Exclude from $\mathfrak{T}(M)$ any triangulations consisting of a single tetrahedron. Then the subgraph of $\mathfrak{T}(M)$ consisting of triangulations without degree one edges is connected under 2-3 and 3-2 moves. \[main theorem\]
The restrictions on the manifold being orientable, and the triangulation being either ideal or having only a single vertex are likely not very serious. Most probably, a few other special cases would need to be ruled out. For brevity however, in this paper we restrict to these cases. The restriction ruling out L(4,1) is a special case, similar to the cases of single-tetrahedron triangulations. Most of the triangulations of the manifold L(4,1) that do not have degree one edges are in one large connected component, but there is also an infinite family of isolated triangulations without degree one edges, for which any 2-3 move would introduce a degree one edge. These triangulations have no degree three edges, and so there are no 3-2 moves to consider. We discuss this family in Remark \[L41\_triangs\].
The property of having no degree one edges is very weak, but is a prerequisite for a triangulation to have essential edges, or to be 1-efficient, or for most other properties of triangulations that have been investigated. We might hope that it will be possible to build up to connectivity of these stronger properties from weaker properties such as having no degree one edges.
Definitions and preliminaries {#defns_and_prelims}
=============================
A *model triangulation* is a set of identical oriented $3$–simplices together with a set of orientation-reversing face pairings. A face cannot be glued to itself, and each face must be paired with another face. We refer to the quotient space after making the identifications given by the face pairings as a *triangulation*.
The simplices prior to identification, and their vertices, edges, and so on, are called *model cells*. The map of a model cell to its image in the triangulation need not be a homeomorphism – this occurs whenever a cell is glued to itself in some fashion.
An *ideal triangulation* is the result of removing the vertices from a triangulation, producing a manifold with boundary. For a closed manifold $M$, we define $\mathfrak{T}(M)$ to be the set of single-vertex triangulations. For a manifold $M$ with boundary, we define $\mathfrak{T}(M)$ to be the set of ideal triangulations of $M$, and we assume that $M$ has no spherical boundary components.
A triangulation as defined above (i.e. with material vertices, rather than an ideal triangulation) is topologically a manifold if and only if the *link* of each vertex (i.e. the boundary of a small closed regular neighbourhood of the vertex) is a sphere.
The *degree* of an edge $e$, written $\deg(e)$, is the number of model edges that are identified to form $e$.
In this paper, we will translate freely back and forth between a triangulation and its dual *special spine*.
A *spine* of a manifold $M$ with non-empty boundary is a compact subpolyhedron $P$, such that $M$ collapses to $P$. For a closed manifold $M$, a spine is a spine of the complement of an open ball in $M$. If we are given a triangulation of a manifold, we obtain the dual (special) spine by inserting into each tetrahedron a *butterfly*. See Figure \[butterfly\]. A spine is *simple* if every point on a spine has a neighbourhood homeomorphic to one of the pictures shown in Figure \[simple\_spine\_points\].
A connected component of the set of non-singular points is a *2-stratum*. A connected component of the set of triple points is a *1-stratum*. A simple spine is *special* if each 2-stratum is an open disk and each 1-stratum is an open interval.
For further details on special spines and the moves on them described in this paper, see Matveev’s book, *Algorithmic topology and classification of 3-manifolds* [@matveev].
Let ${\mathcal{T}}$ be a triangulation with at least two tetrahedra. A *2-3 move* can be performed on any pair of distinct tetrahedra of ${\mathcal{T}}$ that share a triangular face ${\bigtriangleup}$. We remove ${\bigtriangleup}$ and the two tetrahedra, and replace them with three tetrahedra arranged around a new edge, which has endpoints the two vertices not on ${\bigtriangleup}$. See Figure \[2-3\]. A *3-2 move* is the reverse of a 2-3 move, and can be performed on any triangulation with a degree three edge $e$, where the three tetrahedra incident to $e$ are distinct (i.e. the three model edges that are identified to form $e$ are edges of distinct model tetrahedra).
$e$ at 121 145 0-2 at 274 147 ![The 0-2 move, shown acting on a cross-section through the book of tetrahedra around the edge $e$. Edges in the plane of the cross-section are drawn with a thick line, while intersections with triangular faces are drawn with a thin line. The intersections with the two triangular faces the move is performed on are drawn with a dashed line. One half book is on the left, the other on the right.[]{data-label="0-2_cross-section"}](Figures/0-2_move_cross-section2 "fig:"){width="50.00000%"}
The focus of this paper is on connectivity of triangulations under the 2-3 and 3-2 moves. In order to organise the argument we will also consider a number of other moves, including the 0-2 and 2-0 moves, and (later) the V-move.
\[0-2\_move\] Let ${\mathcal{T}}$ be a triangulation. A *0-2 move* can be performed on any pair of distinct triangular faces of ${\mathcal{T}}$ that share an edge $e$. Around the edge $e$, the tetrahedra of ${\mathcal{T}}$ are arranged in a cyclic sequence, which we call a *book of tetrahedra*. (Note that tetrahedra may appear more than once in the book.) The two triangles and $e$ separate the book into two *half–books*. We unglue the tetrahedra that are identified across the two triangles, duplicating the triangles and also duplicating $e$. We glue into the resulting hole a pair of tetrahedra glued to each other in such a way that there is a degree two edge between them. See Figure \[0-2\]. In Figure \[0-2\_cross-section\] we see the effect of the 0-2 move on the book of tetrahedra around $e$. The figure shows a cross-section through the book, perpendicular to $e$. A *2-0 move* is the reverse of a 0-2 move, and can be performed on any triangulation with a degree two edge, where the two tetrahedra incident to that edge are distinct, and the two edges opposite the degree two edge are not identified.
The 0-2 move is also called the [*lune*]{} move in the dual language of special spines.
\[connectivity\] Let ${\mathcal{T}}$ and ${\mathcal{T}}'$ be two triangulations of a manifold, both of which have at least two tetrahedra, and which have the same number of vertices as each other. Then ${\mathcal{T}}$ and ${\mathcal{T}}'$ are connected by a sequence of 2-3 and 3-2 moves.
Our goal is to modify a path of triangulations given by the above theorem to avoid degree one edges.
Anatomy of a degree one edge
============================
By definition, a degree one edge is formed from a single model edge $e$. The model tetrahedron that has $e$ as an edge has the two model faces incident to $e$ paired with each other. See Figure \[make\_deg\_1\_edge\]. Note that only one triangle of a triangulation containing a degree one edge is incident to that edge.
$e$ at 251 39 $e'$ at 273.5 65 ![A degree one edge is formed by gluing two faces of a tetrahedron to each other by “closing the book” around the edge. Edges of the tetrahedron are drawn with a thick line, while “horizon” lines of a triangular face that curves away from us are drawn with a thin line.[]{data-label="make_deg_1_edge"}](Figures/make_degree_1_edge "fig:"){width="80.00000%"}
In order to avoid introducing degree one edges when performing 2-3 and 3-2 moves, it will be useful to know under what conditions a degree one edge can arise. A 2-3 move alters the degrees of *at most* nine edges of the triangulation: the nine edges of the two tetrahedra shown in the upper left of Figure \[2-3\]. When a 2-3 move is performed, the degrees of the three “equatorial” edges go down by one, and the degrees of the other six edges go up by one. We say *“at most* nine edges” here, because there may be identifications among the edges, due to gluings not shown in Figure \[2-3\]. Thus, in general, a 2-3 move can reduce the degree of an edge of the triangulation by up to three (if the three equatorial edges are identified with each other), while a 3-2 move can reduce the degree of an edge of the triangulation by up to six. However, any such large jumps down in degree cannot result in a degree one edge, as we will see in Corollary \[must\_go\_from\_2\_to\_1\], which is derived from the following lemma.
\[ways\_to\_create\_deg\_one\] There are two ways in which a degree one edge can be created, either via a 3-2 move or a 2-3 move. The two possibilities are shown in Figure \[deg2\_to\_deg1\_edge\]. In both cases, all of the tetrahedra shown are distinct (i.e. each tetrahedron shown corresponds to a distinct model tetrahedron). The triangulations before and after these moves may or may not have identifications amongst the boundary faces of the collections of tetrahedra shown.
We consider the reverse move: starting with a tetrahedron $T$ incident to a degree one edge $e$, and considering which 2-3 or 3-2 moves can be applied to remove the degree one edge.
First, consider the possible 3-2 moves that involve $T$ (and therefore can affect $e$). In Figure \[make\_deg\_1\_edge\], $e$ is labelled, as is its opposite edge $e'$. A 3-2 move can only be applied to a degree three edge, all of whose incident tetrahedra are distinct (i.e. three distinct model tetrahedra have model edges identified to form the degree three edge). Therefore we cannot apply a 3-2 move to $e$, since its degree is incorrect, or to the other two edges incident to our tetrahedron, since the model tetrahedron has two of its model edges identified at each of these two edges. So the only possible edge that a 3-2 move can be applied to is $e'$. This move is shown in reverse in Figure \[deg2\_to\_deg1\_2-3\]. A 3-2 move results in two distinct tetrahedra, so we must start with two distinct tetrahedra when producing a degree one edge as in Figure \[deg2\_to\_deg1\_2-3\].
Second, consider the possible 2-3 moves that involve $T$ (and therefore can affect $e$). Such a move must be performed on a triangle of the triangulation, whose two model triangles are on distinct model tetrahedra. Therefore we cannot use the one triangle incident to $e$, and must use one of the other two faces of $T$. Up to symmetry these give the same configuration. The reverse move is shown in Figure \[deg2\_to\_deg1\_3-2\]. The result of the 2-3 move consists of three distinct tetrahedra, so we must start with three distinct tetrahedra when producing a degree one edge as in Figure \[deg2\_to\_deg1\_3-2\].
\[must\_go\_from\_2\_to\_1\] A degree one edge comes from reducing the degree of a degree two edge by one. No single 2-3 or 3-2 move can convert any edge of degree higher than two into a degree one edge.
Lemma \[ways\_to\_create\_deg\_one\] lists the two ways to produce a degree one edge, and in both cases the edge in question has degree two beforehand.
It is possible that a 3-2 move can make two degree one edges at once. A 2-3 move can also make more than one degree one edge, but only in a triangulation with multiple material vertices, or both ideal and material vertices.
Proof of the main theorem
=========================
In this section, we prove Theorem \[main theorem\]. We first describe the general strategy of the proof, with details to come in the subsequent pages.
We are given two triangulations ${\mathcal{T}}$ and ${\mathcal{T}}'$ of a manifold $M$, neither of which has a degree one edge, and we must find a path of 2-3 and 3-2 moves which connects from one to the other, and which does not pass through a triangulation with a degree one edge. Theorem \[connectivity\] gives us the existence of a path $\gamma = ({\mathcal{T}}= {\mathcal{T}}_1, \ldots, {\mathcal{T}}_N = {\mathcal{T}}')$ connected by 2-3 and 3-2 moves, although some of the intermediate triangulations ${\mathcal{T}}_i, 1<i<N$ may have degree one edges. Our plan is to modify the path to detour around any such triangulations.
First, observe that the tetrahedron $T$ incident to a degree one edge $e$ is unchanged for the entire lifetime of the degree one edge. Any 2-3 or 3-2 move that alters $T$ is one of the moves shown in Figure \[deg2\_to\_deg1\_edge\], as we saw in the proof of Lemma \[ways\_to\_create\_deg\_one\]. This means in particular that the triangle ${\bigtriangleup}$ incident to $e$ is also unchanged for the lifetime of the degree one edge.
Suppose at first that there are no degree one edges in the path $\gamma$ from ${\mathcal{T}}_1$ to ${\mathcal{T}}_i$. Suppose that a single degree one edge $e$ is introduced in the triangulation ${\mathcal{T}}_{i+1}$, persists until ${\mathcal{T}}_{j-1}$, and then is removed in ${\mathcal{T}}_j$. Let the single triangle incident to $e$ be called ${\bigtriangleup}$. We take a detour from ${\mathcal{T}}_i$, performing a sequence of moves (which do not go through any triangulations with degree one edges) that results in a triangulation $\overline{{\mathcal{T}}_i}$. This triangulation is identical to ${\mathcal{T}}_i$, other than that the triangle ${\bigtriangleup}$ has been unglued, and a triangulation of a three-ball with boundary consisting of two triangles is glued onto the two revealed faces of the triangulation. Said another way, we unglue the triangle ${\bigtriangleup}$, and insert a *triangular pillow*. Our triangular pillow must not have any degree one edges inside of it. The act of gluing it in where ${\bigtriangleup}$ was increases the degree of the three edges incident to ${\bigtriangleup}$.
We now continue the path $\gamma$, with $\overline{{\mathcal{T}}_i}$ in place of ${\mathcal{T}}_i$. The next move no longer produces a degree one edge, since the degree of $e$ has been increased by the insertion of the triangular pillow. As we continue this parallel path to the original $\gamma$, no moves alter the triangular pillow, by our previous observation about the fact that ${\bigtriangleup}$ is unchanged for the lifetime of the degree one edge.
We continue, up until we get to the triangulation $\overline{{\mathcal{T}}_j}$, which is the same as ${\mathcal{T}}_j$ with ${\bigtriangleup}$ unglued and the triangular pillow inserted into the resulting hole. After this move, we remove the triangular pillow by another sequence of moves, that converts $\overline{{\mathcal{T}}_j}$ back into ${\mathcal{T}}_j$ (this is precisely the reverse process to that of inserting the triangular pillow). Having completed our detour, we continue with the path $\gamma$, until we reach ${\mathcal{T}}_N = {\mathcal{T}}'$. The resulting path then has no triangulations with degree one edges.
If we can perform the above detour, then we can similarly deal with paths with multiple degree one edges, even multiple produced on the same move. All we need to do is apply the move of inserting a triangular pillow for each degree one edge in turn, leaving it there for the lifetime of the degree one edge, then remove it immediately afterwards.
Given this strategy, all we need to do is describe our triangular pillow, and show how it can be inserted into the triangle incident to an edge $e$ just before the degree of $e$ is to be reduced to one, all without introducing any degree one edges in the process.
The triangular pillow {#triangular_pillow}
---------------------
First, we describe the triangulation we use for our triangular pillow. Figure \[make\_bird\_beak\] shows a *bird beak*, which is two tetrahedra arranged around a degree two edge. On the right we draw the bird beak in a suggestive manner – rotating the upper half down so that it looks more like a real-life bird’s beak. In Figure \[combine\_two\_beaks\], we glue together two bird beaks, interleaving the mandibles of each beak. The resulting triangulation of the three-ball has two triangular faces on its boundary, and has no degree one edges in its interior. Ungluing a triangle ${\bigtriangleup}$ of a triangulation and inserting this triangular pillow adds three, three, and eight to the degrees of the edges incident to ${\bigtriangleup}$.
![A bird beak, is two tetrahedra arranged around a degree two edge.[]{data-label="make_bird_beak"}](Figures/make_bird_beak){width="50.00000%"}
$3$ at 200 36 $3$ at 238 36 $8$ at 219 9 ![Our triangulated triangular pillow is formed from two bird beaks, interleaved with each other. On the right, the contributions to the degrees of the three edges incident to the triangular pillow are shown. The four edges internal to the triangular pillow have degrees two, two, three and three.[]{data-label="combine_two_beaks"}](Figures/combine_two_beaks "fig:"){width="65.00000%"}
The V-move
----------
We must insert our triangular pillow using 2-3 and 3-2 moves. As a first step, we introduce a bird beak. We use Matveev’s *V-move* [@matveev Figure 1.13]. The effect of the V-move is shown in Figure \[triangulation\_V-move\]: it wraps a bird beak around two faces of a tetrahedron. Note that there is a symmetry in the resulting three tetrahedra: we can also see this as wrapping a bird beak around the opposite two faces of the same tetrahedron. There are three different possible V-moves to apply in a tetrahedron, because there are three pairs of opposite edges.
![The V-move wraps a bird beak around two faces of a tetrahedron. []{data-label="triangulation_V-move"}](Figures/triangulation_V-move){width="50.00000%"}
The V-move can be implemented using 2-3 and 3-2 moves, assuming that the triangulation has more than one tetrahedron. The process is easier to understand in the dual setting of special spines. Figure \[triangulation\_and\_spine\_V-move\] shows the V-move again, with both the triangulation and the special spine shown. Vertices, edges and faces of the dual spine correspond to tetrahedra, faces and edges of the triangulation.
Let ${\bigtriangleup}$ be a face of a triangulation which has no degree one edges. Suppose that ${\bigtriangleup}$ is incident to two distinct tetrahedra $T_1, T_2$, and suppose that the edges of ${\bigtriangleup}$ (which may not be distinct) all have degree at least three. Then we can perform any of the three V-moves in either $T_1$ or $T_2$ by sequences of 2-3 and 3-2 moves, none of which introduce a degree one edge at any point. \[V-move\_lemma\]
Figure \[V-move\], adapted from [@matveev Figure 1.15] shows the process of implementing the V-move with 2-3 and 3-2 moves, using the dual spine to the triangulation. By inspection, we can check that only the three faces marked with a $*$ have their degree reduced below their starting value during the process, and only by one, during the first 2-3 move. (In the dual picture, these three faces correspond to the three edges of the triangle ${\bigtriangleup}$.) Of the new faces added, the smallest degree we see is two. Moreover, by Corollary \[must\_go\_from\_2\_to\_1\], even if there were identifications between the faces marked $*$, the only way that such a face could become degree one is if it were degree two at the start. By assumption, this is not the case.
$*$ at 65 205 $*$ at 56 187 $*$ at 63 166 2-3 at 345 201 2-3 at 530 190 2-3 at 530 64 3-2 at 345 50 ![The V-move is a composition of 2-3 and 3-2 moves.[]{data-label="V-move"}](Figures/V-move "fig:"){width="90.00000%"}
Rotating the mandibles of a bird beak
-------------------------------------
With the V-move, we introduce a bird beak, wrapped around two faces and across their common edge $e$ of a tetrahedron. We can think of the introduction of the bird beak as a 0-2 move (as in Definition \[0-2\_move\]), and so it splits the book of tetrahedra around $e$ into two half-books, one of which contains only the one tetrahedron we wrapped the bird beak around.
We will also need to be able to rotate the mandibles of the bird beak around in the split book of tetrahedra, effectively moving tetrahedra from one half-book to the other. This will let us close the mandibles on each other and make our triangular pillow. The move shown in Figure \[move\_beak\] achieves this goal, allowing us to move one mandible of the beak from one side of a tetrahedron to the other, assuming that this third tetrahedron is distinct from the two tetrahedra of the bird beak. The move is shown in the dual picture in Figure \[move\_beak\_spine\], which is adapted from [@matveev Fig. 1.18].
The move shown in Figure \[move\_beak\] does not allow us to rotate the two mandibles of a single bird beak *past each other*. Here the configuration to consider involves a model triangle ${\bigtriangleup}$, two of whose edges are identified to form a single edge $e$ of the triangulation. If we perform a 0-2 move, inserting a bird beak and splitting apart the book of tetrahedra around $e$, it can happen that the two mandibles of the beak are glued to each other across ${\bigtriangleup}$, and we may want to swap their order. The move shown in [@matveev Fig. 1.19] allows us to rotate mandibles past each other in such a configuration, but we will not require such a move in this paper. \[move\_beak\_past\_self\]
Note that we do not want to rotate one mandible to close onto the other mandible of a single bird beak, as this would result in a degree one edge where the beak is folded over onto itself.
Assume that we have a triangulation with a bird beak, and that collapsing the bird beak via a 2-0 move would not result in a triangulation with a degree one edge. Then we can perform the move of rotating the bird beak past a tetrahedron that is not one of the two tetrahedra of the bird beak, by performing a 2-3 followed by a 3-2 move, neither of which introduce a degree one edge, unless this move results in a bird beak folded over onto itself. \[rotate\_bird\_beak\]
See Figure \[move\_beak\_spine\]. The faces marked $*$ have degree at least two, since otherwise we have a bird beak folded over onto itself. All other faces in the sequence above have degree at least two, either by directly counting vertices in the diagram, or because the corresponding face after performing the 2-0 move has degree at least two, by assumption.
2-3 at 341 458 3-2 at 723 458
2-0 at 356 265 2-0 at 710 265
$*$ at 41 432 $*$ at 908 341
![The dual picture of rotating a bird beak past a tetrahedron.[]{data-label="move_beak_spine"}](Figures/rotate_bird_beak_spine2){width="90.00000%"}
If we start with a triangulation with no degree one edges and then perform a V-move, this lemma allows us to rotate the mandibles of the resulting bird beak to any position we wish without introducing any degree one edges, as long as we move the mandibles of the beak through a part of the book of tetrahedra for which all tetrahedra are distinct (to avoid the issues discussed in Remark \[move\_beak\_past\_self\]), and we do not close the beak onto itself.
Inserting a triangular pillow
-----------------------------
We have our triangulation ${\mathcal{T}}_i$, which has a triangle ${\bigtriangleup}$, with incident edge $e$ which is degree two, and which in ${\mathcal{T}}_{i+1}$ becomes degree one. Let the other two edges of ${\bigtriangleup}$ be called $e'$ and $e''$, and suppose that the two tetrahedra incident to $e$ are also glued along the edges $\bar{e}'$ and $\bar{e}''$. See Figure \[near\_tri\_and\_degree\_2\].
${\bigtriangleup}$ at 41 74 $e$ \[t\] at 45 58 $e'$ \[bl\] at 95 87 $\bar{e}'$ \[tl\] at 95 28 $f$ \[l\] at 69 34 $e''$ \[br\] at 35 87 $\bar{e}''$ \[tr\] at 35 28 $v$ \[l\] at 127 58 ![The vicinity of the triangle ${\bigtriangleup}$, with edge $e$ of degree two.[]{data-label="near_tri_and_degree_2"}](Figures/near_tri_and_degree_2 "fig:"){width="30.00000%"}
Our plan is to use two V-moves, on tetrahedra incident to each of $e'$ and $e''$, to introduce bird beaks that split the books of tetrahedra around $e'$ and $e''$. We then use Lemma \[rotate\_bird\_beak\] to rotate the mandibles of the two beaks around until they form our triangular pillow. In order to apply Lemma \[V-move\_lemma\] and so apply the V-move, splitting the book of tetrahedra around $e'$ say, we need to find a triangle incident to $e'$, all of whose edges are degree at least three. In particular, we require that $e'$ has degree at least three – we first check that this is the case.
Suppose not, and assume that $e'$ has degree two. Then considering Figure \[near\_tri\_and\_degree\_2\], we see that the triangulation would have a vertex $v$ with spherical link, with only the two tetrahedra incident to $e$ incident to it. But there must then be another vertex of the triangulation, since not all vertices of the two tetrahedra are at $v$. In the closed case this contradicts the assumption in Theorem \[main theorem\] that the triangulation has only a single vertex. In the case that $M$ has boundary, the spherical link of $v$ gives another contradiction, since we assume that $M$ has no spherical boundary components.
So we know that $e'$ has degree at least three, as do $e'', \bar{e}',$ and $\bar{e}''$ by the same argument. Next, we need to find a triangle incident to $e'$, all of whose edges have degree at least three. Consider Figure \[near\_tri\_and\_degree\_2\] again. We know that the edge $\bar{e}'$ has degree at least three. If the edge $f$ also has degree at least three then we are done: we have found a triangle incident to $e'$ with three edges of degree at least three. If not, then $f$ must have degree two (since there are no degree one edges), and $f$ is incident to another tetrahedron, $T$, say, which is incident to $e', \bar{e}', e'', \bar{e}''$ and one other edge, say $f_1$, which is opposite $f$ in $T$. The tetrahedron $T$ has two triangles that are incident to $e'$, one of which we have already checked, and the other, which has edges $e', e'',$ and $f_1$. We saw above that $\bar{e}''$ has degree at least three, and so if $f_1$ also has degree three, then we are again done. If not, then again $f_1$ must have degree two, and we continue with another tetrahedron incident to $e', \bar{e}', e'', \bar{e}''$ and one other edge, say $f_2$, which is opposite $f_1$ in the new tetrahedron.
We continue in this fashion, building a stack of tetrahedra, until we either find a triangle incident to $e'$ and two other edges of degree at least three, or the stack wraps around to join onto the back side of the two tetrahedra we started with, and with the final degree two edge being the edge opposite $e$ in the back tetrahedron shown in Figure \[near\_tri\_and\_degree\_2\]. Note that this is the only way in which the stack of tetrahedra can glue to itself.
Since the manifold is orientable, there are four cases to consider, depending on the angle by which the front of the stack of tetrahedra we have built is glued onto the back.
- If the stack is glued with no rotation (and so there must be an even number of tetrahedra in the stack), then the four vertices of the tetrahedra are distinct after gluing with spherical vertex links, which again contradicts the hypotheses given in Theorem \[main theorem\].
- If the stack is glued with a half-turn rotation (and so again there must be an even number of tetrahedra in the stack), then there are two distinct vertices after gluing, each with spherical vertex links, and again we contradict the hypotheses given in Theorem \[main theorem\].
- If the stack is glued with either a quarter turn or a three quarters turn, then we have a single-vertex triangulation of the manifold L(4,1), which is the exception listed in Theorem \[main theorem\].
The above argument constructs the triangulations of L(4,1) mentioned in the introduction. Each is formed from a stack of an odd number of tetrahedra, with degree two edges between each pair of tetrahedra in the stack. The two triangles at the top of the stack are glued to the two triangles at the bottom of the stack, with a quarter (or equivalently, three-quarter) turn. No edge has degree three, so no 3-2 move is possible. Moreover, every possible 2-3 move introduces a degree one edge. \[L41\_triangs\]
So, we find a triangle incident to $e'$ for which all edges have degree at least three. Then using Lemma \[V-move\_lemma\], we perform a V-move that results in a bird beak that splits apart the book of tetrahedra around $e'$. Figure \[first\_V-move\] shows the result in the case that the edge $f$ has degree at least three – we get a bird beak wrapped around one of the two tetrahedra incident to $e$. Otherwise, we obtain a bird beak wrapped around a tetrahedron somewhere further up the stack. By using Lemma \[rotate\_bird\_beak\], we can rotate the mandibles of such a beak one by one down the stack (noting that all tetrahedra of the stack are distinct), until it is in the position shown in Figure \[first\_V-move\]. By moving the lower mandible before the upper one, we can ensure that we do not close the bird beak onto itself when rotating its mandibles, which would produce a degree one edge.
$e$ \[t\] at 45 58 $\bar{e}'$ \[tl\] at 95 28 $f$ \[l\] at 69 34 $e''$ \[br\] at 35 87 $\bar{e}''$ \[tr\] at 35 28
Next, we need to add the other bird beak needed to make our triangular pillow. As before, we know that edges $e''$ and $\bar{e}''$ have degree at least three, and after our previous moves, $f$ now has degree at least three as well. So we can apply Lemma \[V-move\_lemma\] here, splitting apart the book of tetrahedra around $e''$ and obtaining Figure \[second\_V-move\]. Finally, using Lemma \[rotate\_bird\_beak\], we close the two bird beaks around each others mandibles, producing our triangular pillow. See Figure \[close\_bird\_beaks\].
$e$ \[t\] at 45 58 $\bar{e}'$ \[tl\] at 95 28 $f$ \[l\] at 69 34 $\bar{e}''$ \[tr\] at 35 28
To summarise, in this section we described a procedure to introduce a triangular pillow without introducing a degree one edge at any stage. This then completes the proof of Theorem \[main theorem\].
|
---
abstract: 'We construct start-products on the co-adjoint orbit of the Lie group $\Aff({\bf C})$ of affine transformations of the complex straight line and apply them to obtain the irreducible unitary representations of this group. These results show effectiveness of the Fedosov quantization even for groups which are neither nilpotent nor exponential. Together with the result for the group $\Aff({\bf R})$ \[see DH\], we have thus a description of quantum $\overline{MD}$ co-adjoint orbits.'
address: 'Institute of Mathematics, National Center for Natural Sciences and Technology, P. O. Box 631, Bo Ho, 10.000, Hanoi, Vietnam'
author:
- Do Ngoc Diep and Nguyen Viet Hai
date: 'Version of August 10, 1999'
title: 'Quantum Co-Adjoint Orbits of the Group of Affine Transformations of the Complex Straight Line'
---
[^1]
Introduction
============
The notion of $\star$-products was a few years ago introduced and played a fundamental role in the basic problem of quantization, see e.g. references \[AC1,AC2,F,G…\], as a new approach to quantization on arbitrary symplectic manifolds. In \[DH\] we have constructed star-products on upper half-plane, obtained operator $\hat{\ell}_z, z \in \aff({\bf R}) = \Lie\Aff({\bf R})$ and we have proved that the representation $\exp(\hat{ \ell}_z) = \exp(\alpha\frac{\partial}{\partial s} + i{\beta}e^s)$ of group $\Aff_0({\bf R})$ coincides with representation $T_{\Omega_\pm}$ obtained from the orbit method or Mackey small subgroup method. One of the advantage of this group, with which the computation is rather accessible is the fact that its connected component $\Aff_0(\mathbf R)$ is exponential. We could use therefore the canonical coordinates for Kirillov form on the orbits.
It is natural to consider the same problem for the group $\Aff({\bf C})$. We can expect that the calculations and final expressions could be similar to the corresponding in real line case, but this group $\Aff({\bf C})$ is no more exponential, i.e exponential map $$\exp : \aff({\bf C}) \to \widetilde{\Aff}({\bf C})$$ is no longer a global diffeomorphism and the general theory of D. Arnal and J. Cortet [@arnalcortet1], [@arnalcortet2] and others could not be directly applicable. We overcame these difficulties by a way rather different which could indicate new ideas for more general non-exponential groups: To overcome the main difficulty in applying the deformation quantization to this group, we replace the global diffeomorphism in Arnal-Cortet’s setting by a local diffeomorphism. With this replacement, we need to pay attention on complexity of the the symplectic Kirillov form in new coordinates. We then computed the inverse image of the Kirillov form on appropriate local charts. The question raised here is how to choose a good local chart in order to have as possible simple calculation. The calculations we proposed are realized by using complex analysis on very simple complex domain.
Our main result is the fact that by an exact computation we had found out an explicit star-product formula (Proposition 3.5) on the local chart. This means that the functional algebras on co-adjoint orbits admit a suitable deformation, or in other words, we obtained quantum co-adjoint orbits of this group as exact models of new quantum objects, say “quantum punctured complex planes" $(\mathbf C^2 \setminus L)_q$. Then, by using the Fedosov deformation quantization, it is not hard to obtain the full list of irreducible unitary representations (Theorem 4.2) of the group $\Aff({\bf C})$, although the computation in this case, like by using the Mackey small subgroup method or modern orbit method, is rather delicate. The infinitesimal generators of those exact model of infinite dimensional irreducible unitary representations, nevertheless, are given by rather simple formulae.
We introduce some preliminary result in §2. The operators $\hat {\ell}_A$ which define the representation of the Lie algebra $\aff({\bf C})$ are found in §3. In particular, we obtain the unitary representations of Lie group $\widetilde{\Aff}$([**C**]{}) in Theorem 4.3 §4.
Preliminary Results
===================
Recall that the Lie algebra ${\mathfrak g} = \aff({\bf C})$ of affine transformations of the complex straight line is described as follows, see \[D\].
It is well-known that the group $\Aff({\bf C})$ is a four (real) dimensional Lie group which is isomorphism to the group of matrices: $$\Aff({\bf C}) \cong \left\{\left(\begin{array}{cc} a & b \\ 0 & 1 \end{array}\right) \vert a,b \in {\bf C}, a \ne 0 \right\}$$
The most easy method is to consider $X$,$Y$ as complex generators, $X=X_1+iX_2$ and $Y=Y_1+iY_2$. Then from the relation $[X,Y]=Y$, we get$ [X_1,Y_1]-[X_2,Y_2]+i([X_1Y_2]+[X_2,Y_1]) = Y_1+iY_2$. This mean that the Lie algebra $\aff({\bf C})$ is a real 4-dimensional Lie algebra, having 4 generators with the only nonzero Lie brackets: $[X_1,Y_1] - [X_2,Y_2]=Y_1$; $[X_2,Y_1] + [X_1,Y_2] = Y_2$ and we can choose another basic noted again by the same letters to have more clear Lie brackets of this Lie algebra: $$[X_1,Y_1] = Y_1; [X_1,Y_2] = Y_2; [X_2,Y_1] = Y_2; [X_2,Y_2] = -Y_1$$
[The exponential map $$\exp: {\bf C} \longrightarrow {\bf C}^{*} := {\bf C} \backslash \{0\}$$ giving by $z \mapsto e^z$ is just the covering map and therefore the universal covering of ${\bf} C^*$ is $\widetilde {\bf C}^* \cong {\bf C}$. As a consequence one deduces that $$\widetilde {\Aff}({\bf C}) \cong {\bf C} \ltimes{\bf C} \cong \{(z,w) \vert z,w \in {\bf C} \}$$ with the following multiplication law: $$(z,w)(z^{'},w^{'}) := (z+z',w+e^{z}w')$$ ]{}
The co-adjoint orbit of $\widetilde\Aff({\bf C})$ in ${\mathfrak g}^*$ passing through $F \in {\mathfrak g}^*$ is denoted by $$\Omega_{F} := K(\widetilde {\Aff}({\bf C})) F = \{K(g)F \vert g \in \widetilde \Aff({\bf C})\}$$ Then, (see \[D\]):
1. Each point $(\alpha,0,0,\delta)$ is 0-dimensional co-adjoint orbit $\Omega_{(\alpha,0,0,\delta)}$
2. The open set $\beta^{2}+\gamma^{2} \ne $ 0 is the single 4-dimensional co-adjoint orbit $\Omega_{F} = \Omega_{\beta^{2}+\gamma^{2} \ne 0} $. We shall also use $\Omega_{F}$ in form $\Omega_{F} \cong {\bf C} \times {{\bf C}}^*$.
[ Let us denote: $$\mathbf H_{k} = \{w=q_{1}+iq_{2} \in {\bf C} \vert -\infty< q_1<+\infty ; 2k\pi < q_2< 2k\pi+2\pi\}; k=0,\pm1,\dots$$ $$L=\{{\rho}e^{i\varphi} \in {\bf C} \vert 0< \rho < +\infty; \varphi = 0\} \mbox{ and } {\bf C} _{k } = {\bf C} \backslash L$$ is a univalent sheet of the Riemann surface of the complex variable multi-valued analytic function $\Ln(w)$, ($k=0,\pm 1,\dots$) Then there is a natural diffeomorphism $w \in mathbf H_{k} \longmapsto e^{w} \in {\bf C}_k$ with each $k=0,\pm1,\dots.$ Now consider the map: $${\bf C} \times {\bf C} \longrightarrow \Omega_F = {\bf C} \times {\bf C}^*$$ $$(z,w) \longmapsto (z,e^w),$$ with a fixed $k \in \mathbf Z$. We have a local diffeomorphism $$\varphi_k: {\bf C} \times {\bf H}_k \longrightarrow {\bf C} \times {\bf C}_k$$ $$(z,w) \longmapsto (z,e^w)$$ This diffeomorphism $\varphi_k$ will be needed in the all sequel. ]{}
On ${\bf C}\times {\bf H}_k$ we have the natural symplectic form $$\omega = \frac{1 }{2}[dz \wedge dw+d\overline {z} \wedge d\overline {w}],$$ induced from $\mathbf C^2$. Put $z=p_1+ip_2,w=q_1+iq_2$ and $(x^1,x^2,x^3,x^4)=(p_1,q_1,p_2,q_2) \in {\bf R}^4$, then $$\omega = dp_1 \wedge dq_1-dp_2 \wedge dq_2.$$ The corresponding symplectic matrix of $\omega$ is $$\wedge = \left(\begin{array}{cccc} 0 & -1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & -1 & 0 \end{array}\right)
\mbox{ and }
\wedge^{-1} = \left(\begin{array}{cccc} 0 & 1 & 0& 0 \\
-1 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 \\
0 & 0 & 1 & 0 \end{array}\right)$$
We have therefore the Poisson brackets of functions as follows. With each $f,g \in {\bf C}^{\infty}(\Omega)$ $$\{f,g\} = \wedge^{ij}\frac{\partial f }{\partial x^i}\frac{\partial g}{\partial x^j} =
\wedge^{12}\frac{\partial f }{\partial p_1}\frac{\partial g}{\partial q_1}+
\wedge^{21}\frac{\partial f }{\partial q_1}\frac{\partial g}{\partial p_1} +
\wedge^{34}\frac{\partial f}{\partial p_2}\frac{\partial g}{\partial q_2} +
\wedge^{43}\frac{\partial f}{\partial q_2}\frac{\partial g}{\partial p_2} =$$ $$\ \ \ \ \ \ \ =\frac{\partial f }{\partial p_1}\frac{\partial g}{\partial q_1} -
\frac{\partial f}{\partial q_1}\frac{\partial g}{\partial p_1} -
\frac{\partial f}{\partial p_2}\frac{\partial g }{\partial q_2} +
\frac{\partial f}{\partial q_2}\frac{\partial g }{\partial p_2} =$$ $$\ \ =2\Bigl[\frac{\partial f}{\partial z}\frac{\partial g}{\partial w} -
\frac{\partial f}{\partial w}\frac{\partial g}{\partial z} +
\frac{\partial f}{\partial \overline z}\frac{\partial g}{\partial \overline{w}} -
\frac{\partial f}{\partial \overline w}\frac{\partial g}{\partial \overline z}\Bigr]$$
Fixing the local diffeomorphism $\varphi_k (k \in {\bf Z})$, we have:
1. For any element $A \in \aff(\mathbf C)$, the corresponding Hamiltonian function $\widetilde{A}$ in local coordinates $(z,w)$ of the orbit $\Omega_F$ is of the form $$\widetilde A\circ\varphi_k(z,w) = \frac{1}{2} [\alpha z +\beta e^w + \overline{\alpha} \overline{z} + \overline{\beta}e^{\overline {w}}]$$
2. In local coordinates $(z,w)$ of the orbit $\Omega_F$, the symplectic Kirillov form $\omega_F$ is just the standard form (1).
[*Proof*]{}. 1$^o$ Each element $ F \in \Omega \subset (\aff({\bf C}))^*$ is of the form $$F = zX^{*} +e^{w}Y^{*} = \left(\begin{matrix}z & 0 \cr e^{w} & 0 \end{matrix}\right)$$ in local Darboux coordinates $(z,w)$. From this implies $$\widetilde{A}(F) = \langle F,A\rangle = \Re \tr (F.A) =$$ $$=\Re\tr \left(\begin{matrix}\alpha z & \beta z \cr \alpha e ^{w} & \beta e^ {w}\end{matrix}\right) = \frac{1}{ 2} [\alpha z +\beta e^w + \overline{\alpha} \overline{z} + \overline{\beta}e^{\overline {w}}]$$
2$^o$ Using the definition of the Poisson brackets$\{, \}$, associated to a symplectic form $\omega$, we have $$\{\widetilde {A},f \} = \alpha\frac{\partial f }{\partial w} - \beta e^{w} \frac{\partial f} {\partial z} -\overline{\beta} e^ {\overline{w}} \frac{\partial f }{\partial {\overline{z}}}+ \overline{\alpha}\frac{\partial f}{\partial \overline w}$$ Let us from now on denote by $\xi_A$ the Hamiltonian vector field (symplectic gradient) corresponding to the Hamiltonian function $\tilde{A}$, $A\in \aff(\mathbf C)$.
Now we consider two vector fields: $$\xi_A = \alpha_1\frac{\partial }{\partial w} - \beta_1 e^{w} \frac{\partial }{\partial z} -\overline{\beta_1} e^ {\overline{w}} \frac{\partial }{\partial {\overline{z}}}+ \overline{\alpha_1}\frac{\partial }{\partial{\overline{w}}} ; \xi_B = \alpha_2\frac{\partial }{\partial w} - \beta_2 e^{w} \frac{\partial }{\partial z} -\overline{\beta_2} e^ {\overline{w}}\frac{\partial }{\partial {\overline{z}}}+ \overline{\alpha_2}\frac{\partial }{\partial{\overline{w}} },$$ where $A = \left(\begin{matrix}\alpha_1& \beta_1 \cr 0 & 0 \end{matrix}\right); B = \left(\begin{matrix}\alpha_2& \beta_2 \cr 0 & 0 \end{matrix}\right) \in \aff ({\bf C})$. It is easy to check that $$\xi_A \otimes \xi_B = \beta_1 \beta_2 e ^{2w} \frac{\partial }{\partial z} \otimes \frac{\partial }{\partial z}+
\alpha_1 \alpha_2 \frac{\partial }{\partial w} \otimes \frac{\partial }{\partial w}+
\overline {\beta_1} \overline{\beta_2} e ^{2\overline w}\frac{\partial }{\partial{\overline { z}}} \otimes \frac{\partial }{\partial{\overline {z}}} + \overline {\alpha_1} \overline{\alpha_2} \frac{\partial }{\partial{\overline { w}}} \otimes \frac{\partial }{\partial{\overline {w}}} +$$ $$+ ( \alpha_1 \beta_2 - \alpha_2\beta_1 )e ^{w}\frac{\partial }{\partial z} \otimes \frac{\partial }{\partial w}+ ( \overline{\alpha_1} \overline{ \beta_2} -\overline{\alpha_2}\overline{\beta_1} )e ^{\overline{w}}\frac{\partial }{\partial{\overline z}} \otimes \frac{\partial }{\partial{\overline w}}+( \alpha_1 \overline{ \beta_2} -\alpha_2\overline{\beta_1} )e ^{\overline{w}}\frac{\partial }{\partial{\overline z}} \otimes \frac{\partial }{\partial w}+$$ $$+( \overline{\alpha_1} \beta_2 -\overline{\alpha_2}\beta_1 )e ^{w}\frac{\partial }{\partial z} \otimes \frac{\partial }{\partial{\overline w}}+(\beta_1\overline{\beta_2} - \overline{\beta_1}\beta_2 )e^ {w + \overline{w}} \frac{\partial }{\partial z} \otimes \frac{\partial }{\partial{\overline z}}+
(\alpha_1\overline{\alpha_2} - \overline{\alpha_1}\alpha_2 ) \frac{\partial }{\partial w} \otimes \frac{\partial }{\partial{\overline w}}.$$ Thus, $$\langle\omega,\xi_A \otimes \xi_B\rangle = \frac{1}{2} \Bigl[( \alpha_1 \beta_2 - \alpha_2\beta_1 )e ^{w} +
( \overline{\alpha_1} \overline{ \beta_2} -\overline{\alpha_2}\overline{\beta_1} )e ^{\overline{w}} \Bigr] = \Re \tr(F.[A,B]) = \langle F,[A,B]\rangle.$$ The proposition is proved. $\square$
Computation of Operators $\hat{\ell}_A^{(k)}$.
==============================================
\[Proposition 3.1\] With $A,B \in \aff({\bf C})$, the Moyal $\star$-product satisfies the relation: $$i \widetilde{A} \star i \widetilde{B} - i \widetilde{B} \star i \widetilde{A} = i[\widetilde{A,B} ]$$
[*Proof*]{}. Consider two arbitrary elements $A=\alpha_1X+\beta_1Y$; $B=\alpha_2X+\beta_2Y$. Then the corresponding Hamiltonian functions are: $$\widetilde{A} = \frac{1}{2} [\alpha_1 z +\beta_1 e^{w} + \overline{\alpha_1}\overline z + \overline{\beta_1} e^ {\overline w}];\widetilde{B} =\frac{1}{2}[\alpha_2 z +\beta_2 e^{w} + \overline{\alpha_2}\overline z + \overline{\beta_2} e^ {\overline w}]$$ It is easy, then, to see that:
$$\begin{array}{rcl}P^0(\widetilde{A},\widetilde{B}) &=& \widetilde{A}. \widetilde{B}\\
P^{1}(\widetilde{A},\widetilde{B}) &=& \{ \widetilde{A}. \widetilde{B} \} = 2\Bigl[\frac{\partial\widetilde{A}}{\partial z}\frac{\partial \widetilde{B}}{\partial w} -
\frac{\partial\widetilde {A}}{\partial w}\frac{\partial\widetilde {B}}{\partial z} +
\frac{\partial\widetilde {A}}{\partial \overline z}\frac{\partial \widetilde {B}}{\partial \overline w} -
\frac{\partial\widetilde {A}}{\partial \overline w}\frac{\partial \widetilde{ B}}{\partial \overline z}\Bigr] \\
&=& \frac{1}{2} \Bigl[( \alpha_1 \beta_2 - \alpha_2\beta_1 )e ^{w} +
( \overline{\alpha_1} \overline{ \beta_2} -\overline{\alpha_2}\overline{\beta_1} )e ^{\overline{w}} \Bigr]\\
\mbox{ and }& & \\
P^{r}(\widetilde{A},\widetilde{B}) &=& 0, \ \forall r \ge 2.\end{array}$$
Thus, $$i\widetilde{A} \star i \widetilde{B} - i \widetilde{B} \star i \widetilde{A} =\frac{1 }{2i}\Bigl[ P^1(i\widetilde{A},i\widetilde{B}) - P^1(i\widetilde{B},i\widetilde{A})\Bigr] =$$ $$= \frac{i }{ 2} \Bigl[( \alpha_1 \beta_2 - \alpha_2\beta_1 )e ^{w} + ( \overline{\alpha_1} \overline{ \beta_2}- \overline{\alpha_2}\overline{\beta_1} )e ^{\overline{w}} \Bigr]$$ on one hand.
On the other hand, because of $[A,B] = (\alpha_1\beta_2-\alpha_2\beta_1)Y$ we have $$i[\widetilde{A,B}] = i\langle F,[A,B]\rangle =\frac{i }{2} \Bigl[( \alpha_1 \beta_2 - \alpha_2\beta_1 )e ^{w} + ( \overline{\alpha_1} \overline{ \beta_2} - \overline{\alpha_2}\overline{\beta_1} )e ^{\overline w} \Bigr]$$ The Proposition is hence proved. $\square$
For each $A \in \hbox{aff}({\bf C}$), the corresponding Hamiltonian function is $$\widetilde{A} = \frac{1}{2} [\alpha z + \beta e^{w} + \overline{\alpha}\overline z + \overline{\beta} e^ {\overline w}]$$ and we can consider the operator ${\ell}_A^{(k)}$ acting on dense subspace $L^2({\bf R}^2\times ({\bf R}^2)^*,\frac{dp_1dq_1dp_2dq_2}{(2\pi)^2} )^{\infty}$ of smooth functions by left $\star$-multiplication by $i \widetilde{A}$, i.e: ${\ell}_A^{(k)} (f) = i \widetilde{A} \star f$. Because of the relation in Proposition 3.1, we have
\[Consequence 3.2\] $${\ell}_{[A,B]}^{(k)} = {\ell}_A^{(k)} \star {\ell}_B^{(k)} - {\ell}_B^{(k)} \star {\ell}_A^{(k)} := {\Bigl[ {\ell}_A^{(k)}, {\ell}_B^{(k)}\Bigr]}^{\star}$$
From this it is easy to see that, the correspondence $A \in \aff({\bf C}) \longmapsto {\ell}_A^{(k)} = $i$\widetilde {A} \star$. is a representation of the Lie algebra $\aff({\bf C}$) on the space N$\bigl[[\frac{i}{2}]\bigr] $ of formal power series, see \[G\] for more detail.
Now, let us denote ${\mathcal F}_z$(f) the partial Fourier transform of the function f from the variable $z=p_1+ip_2$ to the variable $\xi=\xi_1+i\xi_2$, i.e: $${\mathcal F}_z(f )(\xi,w) = \frac{1 }{2\pi} \iint_{R^2} e^{-iRe(\xi \overline{z})} f(z,w)dp_1dp_2$$
Let us denote by $$\mathcal F_z^{-1}(f )(\xi,w) = \frac{1}{2\pi} \iint_{R^2} e^{iRe(\xi \overline{z})} f( \xi,w)d \xi_1d\xi_2$$ the inverse Fourier transform.
\[Lemma 3.2\] Putting $g = g(z,w) = {\mathcal F}_z^{-1} (f )(z,w)$ we have:
1. $$\partial_z g = \frac{i}{2}\overline\xi g \ ; \partial_z^{r} g = {(\frac{i}{2}\overline\xi)}^r g, r=2,3,\dots$$
2. $$\partial_{\overline z} g = \frac{i}{2}\xi g \ ; \partial_{\overline z}^{r} g = {(\frac{i}{2}\xi)}^r g, r=2,3,\dots$$
3. $${\mathcal F}_z(zg) = 2i\partial_{\overline\xi}{\mathcal F}_z(g) = 2i\partial_{\overline \xi}f \ ; {\mathcal F}_z(\overline{z}g) = 2i\partial_{\xi}{\mathcal F}_z(g) = 2i\partial_{\xi}f$$
4. $$\partial_w g = \partial_w ({\mathcal F}_z^{-1}(f)) = {{\mathcal F}_z} ^{-1}(\partial_{w}f);\ \partial_{\overline w}g = \partial_{\overline w} ({\mathcal F}_z^{-1}(f) = {{\mathcal F}_z}^{-1}(\partial_{\overline w}f)$$
[*Proof*]{}. First we remark that $$\partial_z = \frac{1 }{2}(\partial_{p_1} - i\partial_{p_2});\quad \partial_{\overline z} = \frac{1 }{2}(\partial_{p_1} + i\partial_{p_2})$$ we obtain 1$^o$,2$^o$.
3$^o$ $${\mathcal F}_z(zg) = \frac{1}{2\pi} \iint e^{-i(p_1\xi_1 + p_2\xi_2)}p_1g(z,w)dp_1dp_2 +$$ $$+ i\frac{1}{2\pi} \iint e^{-i(p_1\xi_1 + p_2\xi_2)}p_2g(z,w)dp_1dp_2$$ $$= i \partial_{\xi_1}{\mathcal F}_z(g) + i^2 \partial_{\xi_2} {\mathcal F}_z(g) = (i \partial_{\xi_1} - \partial_{\xi_2}) {\mathcal F}_z(g) = 2i \partial_{\overline \xi} {\mathcal F}_z(g) = 2i \partial_{\overline \xi}f.$$ $${\mathcal F}_z (\overline {z}g) = \frac{1}{2\pi} \iint e^{-i(p_1 \xi_1 + p_2 \xi_2)} p_1 g(z,w)dp_1 dp_2 -$$ $$-i\frac{1}{2\pi} \iint e^{-i(p_1 \xi_1 + p_2 \xi_2)}p_2 g(z,w)dp_1 dp_2 =$$ $$=2i \partial_{\xi} {\mathcal F}_z (g) = 2i \partial_{\xi}f.$$
4$^o$ The proof is straightforward.
The Lemma \[Lemma 3.2\] is therefore proved. $\square$
We also need another Lemma which will be used in the sequel.
\[Lemma 3.3\] With $g = {\mathcal F}_z^{-1}$$(f)($$z,w)$, we have:
1. $${\mathcal F}_z(P^0(\widetilde{A},g)) = i(\alpha \partial_{\overline \xi} + \overline {\alpha} \partial_{\xi})f + \frac{1}{ 2} \beta e^w f + \frac{1}{2} \overline {\beta} e^{\overline w} f.$$
2. $${\mathcal F}_z(P^1(\widetilde{A},g)) = \overline {\alpha} \partial_{\overline w}f + \alpha \partial_{w}f - \overline {\beta} e^{\overline w} (\frac{i}{2} \xi)f - \beta e^w (\frac{i}{2}\overline {\xi})f.$$
3. $${\mathcal F}_z(P^r(\widetilde{A},g)) = {(-1)}^r.2^{r-1}[\overline \beta{ e^{\overline w}} (\frac{i}{2}\xi)^r + \beta e^w (\frac{i}{2}\overline \xi)^r]f \ \ \ \ \ \ \forall r \ge 2.$$
[*Proof*]{}. Applying Lemma \[Lemma 3.2\] we obtain 1$^o $ $$P^0(\widetilde {A},g) = \widetilde {A}.g = \frac{1}{2} [\alpha zg + \beta e^w g + \overline \alpha \overline z g + \overline \beta e^{\overline w} g].$$ Thus, $${\mathcal F}_z(P^0(\widetilde{A},g)) = \frac{1}{2} [\alpha {\mathcal F}_z(zg) + \beta e^w {\mathcal F}_z(g) + \overline \alpha {\mathcal F}_z(\overline z g) + \overline \beta e^{\overline w} {\mathcal F}_z(g)] =$$ $$\frac{1}{2} [2i\alpha \partial_{\overline \xi} {\mathcal F}_z(g) + 2i\overline \alpha \partial_{\xi} {\mathcal F}_z(g) + \beta e^w {\mathcal F}_z(g) + \overline \beta e^{\overline w} \mathcal F_z(g)] =$$ $$=i(\alpha \partial_{\overline \xi} + \overline {\alpha} \partial_{\xi})f +
\frac{1 }{ 2} \beta e^w f + \frac{1}{2} \overline {\beta} e^{\overline w} f.$$
2$^o$ $$(P^1(\widetilde{A},g)) = \wedge^{12} \partial_{p_1} \widetilde {A} \partial_{q_1}g + \wedge^{21} \partial_{q_1} \widetilde {A} \partial_{p_1}g + \wedge^{34} \partial_{p_2} \widetilde {A} \partial_{q_2}g + \wedge^{43} \partial_{q_2} \widetilde {A} \partial_{p_2}g$$ $$= \overline \alpha \partial_{\overline w}g + \alpha \partial_w g - \overline \beta e^{\overline w} \partial_{\overline z}g - \beta e^w \partial_zg.$$ This implies that: $${\mathcal F}_z(P^1(\widetilde{A},g)) = \overline {\alpha} \partial_{\overline w} {\mathcal F}_z(g) + \alpha \partial_{w} {\mathcal F}_z(g) - \overline {\beta} e^{\overline w} \partial_{\overline z} {\mathcal F}_z(g) - \beta e^w \partial_{z} \mathcal F_z(g) =$$ $$= \overline {\alpha} \partial_{\overline w}f + \alpha \partial_{w}f - \overline {\beta} e^{\overline w} (\frac{i}{2} \xi)f - \beta e^w (\frac{i}{2}\overline {\xi})f.$$
3$^o$ $$(P^2(\widetilde{A},g)) = \wedge^{21} \wedge^{21} \partial_{q_1q_1} \widetilde {A} \partial_{p_1p_1}g + \wedge^{21} \wedge^{43} \partial_{q_1q_2} \widetilde {A} \partial_{p_1p_2}g + \wedge^{43} \wedge^{21} \partial_{q_2q_1} \widetilde {A} \partial_{p_2p_1}g +$$ $$+ \wedge^{43} \wedge^{43} \partial_{q_2q_2} \widetilde {A} \partial_ {p_2p_2}g = \frac{1}{2}\bigl[(\overline \beta e^{\overline w} + \beta e^w - \beta e^w + \overline \beta e^{\overline w} + \overline \beta e^{\overline w} - \beta e^w + \beta e^w + \overline \beta e^{\overline w}) \partial_{\overline z}^2 g +$$ $$+ (\overline \beta e^{\overline w} + \beta e^w + \beta e^w - \overline \beta e^{\overline w} - \overline \beta e^{\overline w} + \beta e^w + \beta e^w + \overline \beta e^{\overline w}) \partial_z^2 g \bigr] =$$ $$= 2\overline \beta e^{\overline w} \partial_{\overline z}^2 g + 2\beta e^w \partial_z^2 g.$$ This implies also that: $${\mathcal F}_z(P^2(\widetilde{A},g)) = 2\overline \beta e^{\overline w} {\mathcal F}_z(\partial_{\overline z}^2 g) + 2\beta e^w {\mathcal F}_z( \partial_z^2 g) = 2\overline \beta e^{\overline w} (\frac{i}{2}\xi)^2f + 2 \beta e^w (\frac{i}{2}\overline \xi)^2f.$$ By analogy, $$P^3(\widetilde {A},g) = (-1)^3[4\overline \beta e^{\overline w} \partial_{\overline z}^3g + 4\beta e^w \partial_z^3g].$$ $${\mathcal F}_z(P^3(\widetilde{A},g)) = (-1)^3.2^2[\overline \beta e^{\overline w} (\frac{i}{2}\xi)^3f + \beta e^w (\frac{i}{2}\overline \xi)^3f]$$ and with $r \ge 4$ $$P^r(\widetilde {A},g) = {(-1)}^r.2^{r-1}[\overline \beta{ e^{\overline w}} \partial_{\overline z}^r g + \beta e^w \partial_z^rg].$$ $${\mathcal F}_z(P^r(\widetilde{A},g)) = {(-1)}^r.2^{r-1}[\overline \beta{ e^{\overline w}} (\frac{i}{2}\xi)^r + \beta e^w (\frac{i}{2}\overline \xi)^r]f.$$
The Lemma \[Lemma 3.3\] is therefore proved. $\square$
\[Proposition 3.4\] For each $A = \left(\begin{matrix}\alpha & \beta \cr 0 & 0 \cr\end{matrix}\right) \in \aff({\bf C}) $ and for each compactly supported $C^{\infty}$-function $f \in C_0^{\infty}({\bf C} \times {\bf H}_k)$, we have: $${\ell}_A^{(k)}{f} := {\mathcal F}_z \circ \ell_A^{(k)} \circ {\mathcal F}_z^{-1}(f) = [\alpha (\frac{1}{2} \partial_w - \partial_{\overline \xi})f + \overline \alpha (\frac{1 }{2}\partial_{\overline w} - \partial_\xi)f +$$ $$+\frac{i}{2}(\beta e^{w-\frac{1}{2}\overline \xi} + \overline \beta e^{\overline w - \frac{1}{2} \xi})f]$$
[*Proof*]{}. Applying Lemma \[Lemma 3.3\], we have: $${\ell}_A^{(k)}(f):= {\mathcal F}_Z(i \widetilde{A} \star {\mathcal F}_z^{-1} (f))= i{\sum_{r \ge 0}\frac{1}{r!}(\frac{1}{2i})^r \mathcal F_z \Bigl( P ^r( \widetilde{A}, {\mathcal F}_z^{-1} (f))\Bigr)} =$$ $$=i\Bigl\{[i( \alpha \partial_{\overline \xi}+ \overline{\alpha}\partial _{\xi}) f + \frac{1}{2} \beta e^{w} f +\frac{1}{2} \overline{\beta} e^{\overline{w}} f ]+\frac{1}{1!} (\frac{1}{2i})[ \overline{\alpha}\partial _{\overline{ w}} f + \alpha\partial _{w}f - \overline{\beta} e^{\overline{w}} (\frac{i}{2}\xi)f -$$ $$- \beta e^{w} (\frac{i}{2} \overline{\xi})f] + \frac{1}{2!} (\frac{-1}{2i})^22[ \overline{\beta} e^{\overline{w}} (\frac{i}{2}\xi)^2 f + \beta e^{w} ({\frac{i}{2} \overline{\xi})}^2 f] + \dots +$$ $$+\frac{1}{r!} (\frac{-1}{2i})^r 2^ {r-1}[ \overline{\beta} e^{\overline{w}} ({\frac{i}{2}\xi)}^r f + \beta e^{w} ({\frac{i}{2} \overline{\xi})}^{r} f]+ \dots \Bigr\}$$ $$=-( \alpha \partial_{\overline \xi} - \overline{\alpha}\partial _{\xi}) f+\frac{1}{2}( \overline{ \alpha} \partial_{\overline w}+ \alpha\partial_{w}) f+ i \Bigl\{ \Bigl[ \frac{1}{2} \beta e^{w} +\frac{1}{2} \overline{\beta} e^{\overline{w}} -\frac{1}{2}\overline{\beta} e^{\overline{w}} (\frac{1}{2}\xi)- \frac{1}{2}\beta e^{w} (\frac{1}{2} \overline{\xi})\Bigr]f+$$ $$+ \frac{1}{2}. \frac{1}{2!}\Bigl[ \overline{\beta} e^{\overline{w}} ({\frac{-1}{2}\overline\xi)}^2 + \beta e^{w} ({\frac{-1}{2} \xi)}^2 \Bigr]f+ \dots +
\frac{1}{2} \frac{1}{k!}\Bigl[\overline{\beta} e^{\overline{w}} ({\frac{-1}{2}\xi)}^{r} + \beta e^{w} ({\frac{-1}{2} \overline{\xi})}^{r} \Bigr]f+\dots\Bigr\}$$ $$= \Bigl[\alpha( \frac{1}{2}\partial_{w} - \partial _{\overline{\xi}}) +\overline \alpha( \frac{1}{2}\partial_{\overline w } - \partial_\xi) + \frac{i}{2} \beta e^{w}e^{-\frac{1}{2}\overline{\xi}} + \frac{i}{2} \overline{\beta} e^{\overline {w}}e^{-\frac{1}{2}\xi}\Bigr]f$$ $$= \Bigl[\alpha( \frac{1}{2}\partial_w - \partial _{\overline \xi}) +\overline \alpha( \frac{1}{2}\partial_{\overline w} - \partial_\xi) + \frac{i}{2}({\beta e^{w-\frac{1}{2}\overline \xi}} + \overline{\beta} e^{\overline w-\frac{1}{2}\xi})\Bigr]f$$ The Proposition is therefore proved. $\square$
\[Remark 3.5\][Setting new variables u = $w - \frac{1}{ 2}\overline{\xi}$;$v = w + \frac{1 }{2}{\overline{\xi}}$ we have $$\hat {\ell}_A^{(k)}(f) = \alpha\frac{ \partial f }{\partial u}+ \overline{\alpha}\frac{\partial f }{\partial{\overline{u}}}+ \frac{i }{2}(\beta e^{u}+\overline{\beta}e^{\overline{u}})f \vert_{(u,v)}$$ i.e $\hat {\ell}_A^{(k)} = \alpha\frac{ \partial }{\partial u}+ \overline{\alpha}\frac{ \partial }{\partial{\overline{u}}}+ \frac{i }{2}(\beta e^{u}+\overline{\beta}e^{\overline{u}})$,which provides a ( local) representation of the Lie algebra aff([**C**]{}). ]{}
The Irreducible Representations of $\widetilde{\Aff}({\bf C})$
===============================================================
Since $\hat {\ell}_A^{(k)}$ is a representation of the Lie algebra $\widetilde{\hbox {Aff}} ({\bf C})$, we have: $$\exp(\hat {\ell}_A^{(k)}) = \exp\bigl(\alpha\frac{ \partial }{\partial {u}}+ \overline{\alpha}\frac{ \partial }{\partial{\overline{u}}}+ \frac{i }{2}(\beta e^{u}+\overline{\beta}e^{\overline{u}})\bigr)$$ is just the corresponding representation of the corresponding connected and simply connected Lie group $\widetilde\Aff ({\bf C})$.
Let us first recall the well-known list of all the irreducible unitary representations of the group of affine transformation of the complex straight line, see \[D\] for more details.
\[Theorem 4.1\] Up to unitary equivalence, every irreducible unitary representation of $\widetilde{\hbox {Aff}} ({\bf C})$ is unitarily equivalent to one the following one-to-another nonequivalent irreducible unitary representations:
1. The unitary characters of the group, i.e the one dimensional unitary representation $U_{\lambda},\lambda \in {\bf C}$, acting in ${\bf C}$ following the formula $U_{\lambda}(z,w) = e^{{i\Re(z\overline{\lambda})}}, \forall (z,w) \in \widetilde{\Aff} ({\bf C}), \lambda \in {\bf C}.$
2. The infinite dimensional irreducible representations $T_{\theta},\theta \in {\mathbf S}^1$, acting on the Hilbert space $L^{2}(\mathbf R\times \mathbf S ^1)$ following the formula: $$\Bigr[T_{\theta}(z,w)f\Bigl](x) = \exp \Bigr(i(\Re(wx)+2\pi\theta[\frac{\Im(x+z) }{2\pi}])\Bigl)f(x\oplus z),$$ Where $(z,w) \in\widetilde{\Aff}({\bf C})$ ; $x \in {\bf R}\times {\mathbf S} ^1= {\bf C} \backslash \{0\}; f \in L^{2}({\bf R}\times {\mathbf S} ^1);$ $$x\oplus z = Re(x+z) +2 \pi i \{\frac{\Im(x+z) }{2\pi}\}$$
In this section we will prove the following important Theorem which is very interesting for us both in theory and practice.
\[Theorem 4.2\] The representation $\exp(\hat {\ell}_A^{(k)})$ of the group $\widetilde{\Aff}({\bf C})$ is the irreducible unitary representation $T_\theta$ of $\widetilde{\Aff}({\bf C})$ associated, following the orbit method construction, to the orbit $\Omega$, i.e: $$\exp(\hat {\ell}_A^{(k)})f(x) = [T_\theta (\exp A)f](x),$$ where $f \in L^{2}({\bf R}\times {\mathbf S} ^1) ; A = \begin{pmatrix}\alpha & \beta \cr 0 & 0 \cr\end{pmatrix} \in \aff({\bf C}) ; \theta \in {\mathbf S}^1 ; k = 0, \pm1,\dots$
[*Proof*]{}. Putting $x = e^u \in {\bf C} \backslash \{0\} = {\bf R} \times $${\mathbf S}^1 $ and recall that $$\begin{pmatrix}a & b \cr 0 & 1\cr\end{pmatrix} = \exp(A) = \exp \begin{pmatrix}\alpha & \beta \cr 0 & 0 \end{pmatrix},$$
we can rewrite (7) as following:
$$[T_\theta (\exp A)f](e^u) = \exp \Bigl( i(\Re(\frac{e^\alpha -1}{\alpha}\beta e^u)+2\pi\theta[\frac{\Im e^{u+\alpha}}{2\pi}])\Bigr) f(e^{u \oplus \alpha}),$$ where $$u \oplus \alpha = \Re(u+\alpha)+ 2\pi i\{\frac{\Im(u+\alpha) }{2\pi}\} = u+\alpha - 2\pi i[\frac{\Im(u+\alpha) }{2\pi}].$$
Therefore, for the one-parameter subgroup $\exp tA$, $t \in {\bf R}$, we have the action formula:
$$\bigl[T_\theta (\exp tA)f\bigr](e^u) = \exp \Bigl(i(\Re{\frac{e^{t\alpha}-1}{\alpha}\beta e^u}+2\pi\theta[\frac{\Im e^{u+t\alpha} }{2\pi}])\Bigr)f(e^{u \oplus t\alpha})$$
By a direct computation: $$\frac{\partial }{\partial t} \bigl([T_\theta (\exp tA)f](e^u)\bigr) =$$ $$=\frac{\partial }{\partial t} \Bigl(\exp\Bigl(\frac{i}{2}(\frac{e^{t\alpha} -1}{\alpha}\beta e^u + \frac{e^{t\overline \alpha} -1}{ \overline \alpha}\overline \beta e^{\overline u})+ 2\pi\theta i[\frac{\Im{ e^{u+t\alpha}}}{2\pi}]\Bigr)\Bigr) + f(e^{u+t\alpha - 2\pi i [\frac{\Im(u+t\alpha)}{2\pi}]})$$ $$+\exp \bigl(\frac{i }{2}(\frac{e^{t\alpha} -1}{\alpha}\beta e^u + \frac{e^{t\overline \alpha} -1}{\overline \alpha}\overline \beta e^{\overline u})+ 2\pi\theta i[\frac{\Im e^{u+t\alpha}}{2\pi}]\bigr)\frac{\partial }{\partial t}f(e^{u+t\alpha - 2\pi i [\frac{\Im (u+t\alpha)}{2\pi }]}) =$$ $$=\frac{i }{2}(\beta e^{u+t\alpha}+\overline \beta e^{\overline u + t\overline \alpha)} \bigl[T_\theta (\exp tA)f\bigr](e^u) +$$ $$+ \exp \Bigl(i(\Re(\frac{e^{t\alpha} -1}{\alpha}\beta e^u)+2\pi\theta i[\frac{\Im e^{u+t\alpha}}{2\pi}]\Bigr)\alpha e^{u \oplus t\alpha} \frac{\partial f }{\partial u}$$ on one hand.
On the other hand, we have: $$\hat {\ell}_A^{(k)}([T_\theta (\exp tA)f](e^u)=$$ $$=\alpha\frac{\partial }{\partial u}\bigl([T_\theta (\exp tA)f](e^u)\bigr) + \overline \alpha\frac{\partial }{\partial {\overline u}}\bigl([T_\theta (\exp tA)f](e^u)\bigr) +$$ $$+\frac{i }{2}(\beta e^{u}+\overline \beta e^{\overline u})\bigl[T_\theta (\exp tA)f](e^u)\bigr] =$$ $$=\alpha\frac{i }{2}(\frac{e^{t\alpha}-1}{\alpha}\beta e^u) \exp \bigl( i(\Re(\frac {e^{t\alpha} -1}{\alpha}\beta e^u)+2\pi\theta[\frac{\Im e^{u+t\alpha}}{2\pi}]\bigr)\Bigr)f(e^{u \oplus t\alpha})+$$ $$+ \alpha \exp \bigl(i(\Re(\frac{e^{t\alpha} -1}{\alpha}\beta e^u)+2\pi\theta[\frac{\Im e^{u+t\alpha} }{2\pi}]\bigr)\Bigr)e^{u \oplus t\alpha}\frac{\partial f }{\partial u} +$$ $$+ \overline \alpha \frac{i }{2}(\frac{e^{t\overline \alpha} - 1}{\overline \alpha}\overline \beta e^{\overline u})\exp \bigl(i(\Re(\frac{e^{t\alpha} -1}{\alpha}\beta e^u)+2\pi\theta[\frac{\Im e^{u+t\alpha} }{2\pi}]\bigr)\Bigr) f(e^{u \oplus t\alpha}) +$$ $$+ \frac{i }{2}(\overline \beta e^{\overline u}+ \beta e^u)[T_\theta(\exp tA)f](e^u) =$$ $$= \frac{i }{2}(\beta e^{u + t\alpha} + \overline \beta e^{\overline u + t\overline \alpha})[T_\theta(\exp tA)f](e^u)+$$ $$+\exp \bigl(i(\Re(\frac{e^{t\alpha} -1}{\alpha}\beta e^u)+2\pi\theta[\frac{\Im e^{u+t\alpha} }{2\pi}]\bigr)\Bigr)\alpha e^{u \oplus \alpha t}\frac{\partial f }{\partial u}$$ From (8) and (9) implies that : $$\frac{\partial }{\partial t}[T_\theta(\exp tA)f](x) = \hat {\ell}_A^{(k)}\bigl([T_\theta(\exp tA)f](x)\bigr) \ \ \ \ \ \forall x \in {\bf R} \times {\mathbf S}^1.$$ Remark $$T_\theta(\exp tA)f](e^u) \vert_{t=0} = \exp(2 \pi i[\frac{\Im e^u }{{2\pi}}]\theta)f(e^{u - 2\pi i[\frac{\Im u }{2\pi}]})
= f(e^u).$$ This means that: $\exp(\hat {\ell}_A^{(k)})f(x)$ and $[T_\theta(\exp tA)f](x)$ together are the solution of the Cauchy problem $$\left\{\begin{array}{rcl}\frac{\partial }{\partial t}u(t,x) &=& \hat {\ell}_A^{(k)}u(t,x);\\
u(0,x) &=& id.\end{array}\right.$$ The operator $\hat {\ell}_A^{(k)}$ is behaved good enough, so that the Cauchy problem has an unique solution. From this uniqueness we deduce that $\exp(\hat {\ell}_A^{(k)})f(x) \equiv [T_\theta(\exp tA)f](x) \ \forall x \in {\bf R} \times {\bf S}^1.$ The Theorem is hence proved. $\square$
\[Remark 4.3\]
We say that a real Lie algebra ${\mathfrak g}$ is in the class $\overline{MD}$ if every K-orbit is of dimension, equal 0 or dim ${\mathfrak g}$. Further more, one proved that (\[D, Theorem 4.4\]) Up to isomorphism, every Lie algebra of class $\overline {MD}$ is one of the following:
1. Commutative Lie algebras.
2. Lie algebra $\aff({\bf R})$ of affine transformations of the real straight line
3. Lie algebra $\aff({\bf C})$ of affine transformations of the complex straight line.
Thus, by calculation for the group of affine transformations of the real straight line $\Aff({\bf R})$ in \[DH\] and here for the group affine transformations of the complex straight line $\Aff({\bf C})$ we obtained a description of the quantum $\overline {MD}$ co-adjoint orbits.
[label]{}
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B.Fedosov, [*Deformation Quantization and Index Theory*]{}, Akademie der Wissenschaften Verlag, 1993.
S. Gutt, [*Deformation Quantization*]{}, ICTP Workshop on Representation Theory of Lie groups, SMR 686/14/1993.
I.M.Gel’fand and M.A.Naimark, [*Unitary representations of the groups of affine transformation of the straight line*]{}, Dokl. AN SSSR, [**55**]{}(1947) No7,571-574.
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[^1]: This work was supported in part by the National Foundation for Fundamental Science Research of Vietnam and the Alexander von Humboldt Foundation, Germany
|
---
abstract: '[We consider the a posteriori error analysis of fully discrete approximations of parabolic problems based on conforming $hp$-finite element methods in space and an arbitrary order discontinuous Galerkin method in time. Using an equilibrated flux reconstruction, we present a posteriori error estimates yielding guaranteed upper bounds on the $L^2(H^1)$-norm of the error, without unknown constants and without restrictions on the spatial and temporal meshes. It is known from the literature that the analysis of the efficiency of the estimators represents a significant challenge for $L^2(H^1)$-norm estimates. Here we show that the estimator is bounded by the $L^2(H^1)$-norm of the error plus the temporal jumps under the one-sided parabolic condition $h^2 \lesssim \tau$. This result improves on earlier works that required stronger two-sided hypotheses such as $h \simeq \tau$ or $h^2\simeq \tau$; instead our result now encompasses the practically relevant case for computations and allows for locally refined spatial meshes. The constants in our bounds are robust with respect to the mesh and time-step sizes, the spatial polynomial degrees, and also with respect to refinement and coarsening between time-steps, thereby removing any transition condition.]{}'
author:
- |
[Alexandre Ern[^1], Iain Smears[^2] and Martin Vohralík[^3]]{}\
Université Paris-Est, CERMICS (ENPC), 77455 Marne-la-Vallée cedex 2, France\
& INRIA Paris, 2 Rue Simone Iff, 75589 Paris, France.
- Alexandre Ern
- Iain Smears
- Martin Vohralík
title:
- 'Equilibrated flux a posteriori error estimates in $L^2(H^1)$-norms for high-order discretizations of parabolic problems'
- 'Equilibrated flux a posteriori error estimates in $L^2(H^1)$-norms for high-order discretizations of parabolic problems[^4]'
---
Parabolic partial differential equations, a posteriori error estimates, guaranteed upper bound, polynomial-degree robustness, high-order methods
Introduction {#sec:intro}
============
We consider the heat equation $$\label{eq:parabolic}
\begin{aligned}
{\partial}_t u - \Delta u = f & & & \text{in }{\Omega}\times(0,T),\\
u = 0 & & &\text{on }{\partial{\Omega}}\times (0,T),\\
u(0) = u_0 & & &\text{in }{\Omega},
\end{aligned}$$ where ${\Omega}\subset {\mathbb{R}}^d $, $1\leq d \leq 3$, is a bounded, connected, polytopal open set with Lipschitz boundary, and $T>0$ is the final time. We assume that $f \in L^2(0,T;L^2({\Omega}))$, and that $u_0\in L^2({\Omega})$. We are interested here in the a posteriori error analysis in the $L^2(H^1)$-norm of fully discrete numerical methods for . In particular, we consider an arbitrary-order discontinuous Galerkin finite element method (DGFEM) in time, coupled with a conforming $hp$-FEM in space. We recall that a posteriori error estimates should ideally provide *guaranteed upper bounds* on the error, without unknown constants. Otherwise, if the estimators constitute an upper bound on the error up to an unknown constant, then we say instead that the estimators are *reliable*. Furthermore, the estimators should be *locally efficient*, meaning that the local estimators should lie below the error measured in a local space-time neighbourhood, up to a generic constant. Finally, the estimators should ideally be *robust*, with all constants in the bounds being independent of all discretization parameters. Furthermore, on a practical side it is highly desirable that the estimators be locally computable. We refer the reader to [@Verfurth2013[@Verfurth2013]]{} for an introduction to these concepts. Our motivation for considering the heat equation as a model problem is that the posteriori error estimates developed in this context serve as a starting point for extensions to diverse applications, for example nonlinear problems (see [@AmreinWihler2016 [@DiPietroVohralikSoleiman2015; @DolejsiErnVohralik2013; @Kreuzer2013][@AmreinWihler2016; @DiPietroVohralikSoleiman2015; @DolejsiErnVohralik2013; @Kreuzer2013]]{}), as well as playing a central role in adaptive algorithms (see [@ChenFeng2004 [@GaspozKreuzerSiebertZiegler2016; @KreuzerMollerSchmidtSiebert2012][@ChenFeng2004; @GaspozKreuzerSiebertZiegler2016; @KreuzerMollerSchmidtSiebert2012]]{}). For nonconforming discretization methods in space, we refer to [@ErnVohralik2010 [@GeorgoulisLakkisVirtanen2011; @NicaiseSoualem2005][@ErnVohralik2010; @GeorgoulisLakkisVirtanen2011; @NicaiseSoualem2005]]{}.
The literature shows that the structure of parabolic problems leads to several outstanding challenges facing the central goals in a posteriori error estimation. In particular, several difficulties arise in the analysis of the efficiency and robustness of the estimators. To explain some of the challenges, first recall that the a posteriori error analysis of parabolic problems admits a range of norms in which to measure the error: for instance, these include the $L^2(H^1)$-norm (see [@Picasso1998 [@Verfurth1998][@Picasso1998; @Verfurth1998]]{}), $L^2(L^2)$-norm (see [@Verfurth1998a[@Verfurth1998a]]{}), $L^\infty(L^2)$-norms and $L^\infty(L^\infty)$-norms (see [@ErikssonJohnson1995[@ErikssonJohnson1995]]{}), $L^\infty(L^2)\cap L^2(H^1)$-norms (see [@LakkisMakridakis2006 [@MakridakisNochetto2003; @SchotzauWihler2010][@LakkisMakridakis2006; @MakridakisNochetto2003; @SchotzauWihler2010]]{}), and also the $L^2(H^1)\cap H^1(H^{-1})$-norms (see [@BergamBernardiMghazli2005 [@ErnVohralik2010; @GaspozKreuzerSiebertZiegler2016; @NicaiseSoualem2005; @Repin2002; @Verfurth2003][@BergamBernardiMghazli2005; @ErnVohralik2010; @GaspozKreuzerSiebertZiegler2016; @NicaiseSoualem2005; @Repin2002; @Verfurth2003]]{}). To our knowledge, efficiency results have so far only been attained in the case of the $L^2(H^1)$ norm under restrictions linking mesh and time-step sizes, whereas in the $L^2(H^1)\cap H^1(H^{-1})$ norm, such restrictions have been removed. It is important to observe that these two functional settings admit an inf-sup theory for the continuous problem that establishes an equivalence between appropriate norms of the error and of the residual. Although no analysis of efficiency is yet available in the setting of other norms, the optimal order of convergence of the estimators has nonetheless been observed in [@LakkisMakridakis2006 [@MakridakisNochetto2003][@LakkisMakridakis2006; @MakridakisNochetto2003]]{} for instance.
A posteriori error estimators in the $L^2(H^1)$-norm for a class of nonlinear parabolic problem have been studied in [@Verfurth1998[@Verfurth1998]]{}. In particular, the analysis in [@Verfurth1998[@Verfurth1998]]{} found that the ratio between the constants in the upper and lower bounds for the error by the estimators depends on $1 + \tau h^{-2} + \tau^{-1} h^2 + {\lvert\log h\rvert}$, see [@Verfurth1998 [Prop. 4.1][@Verfurth1998 Prop. 4.1]]{}, where $h$ denotes the spatial mesh size and $\tau$ denotes the time-step size, and thus the efficiency of the estimators is subject to the assumption that $\tau \simeq h^2$. [@Picasso1998[@Picasso1998]]{} studied implicit Euler discretizations of the heat equation: under the assumption that $\tau \simeq h$, he showed that the spatial estimator can be bounded from above by the $L^2(H^1)$-norm of the error plus the temporal jump estimator; in particular, the temporal jump estimator, denoted there by $\varepsilon_K^n$ defined in [@Picasso1998 [eq. (2.11)][@Picasso1998 eq. (2.11)]]{}, appears on the right-hand side of the lower bound [@Picasso1998 [eq. (2.24)][@Picasso1998 eq. (2.24)]]{}. In both [@Picasso1998 [@Verfurth1998][@Picasso1998; @Verfurth1998]]{}, the two-sided restrictions between the time-step and mesh sizes have the disadvantage of necessarily requiring that the meshes must be quasi-uniform, and thus theoretically prohibiting adaptive refinement.
Starting with [@Verfurth2003[@Verfurth2003]]{}, one approach to removing these two-sided restrictions has been to consider a different functional framework for the a posteriori error analysis, namely by estimating the $L^2(H^1)\cap H^1(H^{-1})$-norm of the error. Part of the justification of this approach is to be found in the observation in [@Verfurth2003 [p. 198, Par. (5)][@Verfurth2003 p. 198, Par. (5)]]{}, showing that the estimators of [@Picasso1998 [@Verfurth1998][@Picasso1998; @Verfurth1998]]{} are upper bounds to not only the $L^2(H^1)$-norm of the error, but also the $L^2(H^1) \cap H^1(H^{-1})$-norm of the error, up to data oscillation. It was then shown in [@Verfurth2003[@Verfurth2003]]{} that these estimators are efficient, locally-in-time yet only globally-in-space, with respect to the $L^2(H^1)\cap H^1(H^{-1})$-norm of the error, without requiring conditions between mesh and time-step sizes; see also [@BergamBernardiMghazli2005[@BergamBernardiMghazli2005]]{}. Given that the estimators used in both frameworks are the same up to data oscillation, it is of course natural that more general efficiency results are obtainable in when including the $H^1(H^{-1})$ part of the norm, since it allows for the appearance of additional terms on the right-hand side in the efficiency bounds.
Recently, [@ErnSmearsVohralik2016[@ErnSmearsVohralik2016]]{} developed we developed in [@ErnSmearsVohralik2016[@ErnSmearsVohralik2016]]{} a posteriori error estimators, based on equilibrated fluxes, for arbitrary order discretizations of parabolic problems within the $L^2(H^1)\cap H^1(H^{-1})$-norm setting, that are guaranteed, locally efficient, and robust. In particular, the analysis does not require any coupling between mesh and time-step sizes, and overcomes the problem of obtaining local-in-space and local-in-time efficiency by considering a natural extension of the $L^2(H^1)\cap H^1(H^{-1})$-norm to the time-nonconforming approximation space. The estimators are robust not only with respect to the mesh and time-step sizes, but also with respect to the polynomial degrees in space and time, and also with respect to mesh coarsening and refinement, thereby removing the so-called transition conditions previously encountered [@Verfurth2003[@Verfurth2003]]{}. These results are built upon the analysis for elliptic problems in [@BraessPillweinSchoberl2009 [@ErnVohralik2010; @ErnVohralik2015; @ErnVohralik2016a][@BraessPillweinSchoberl2009; @ErnVohralik2010; @ErnVohralik2015; @ErnVohralik2016a]]{}.
In this work, we present a posteriori error estimates for the $L^2(H^1)$-norm of the error, which are based on the same locally computable equilibrated flux as in [@ErnSmearsVohralik2016[@ErnSmearsVohralik2016]]{}, thereby showing that the same methodology can be used in the $L^2(H^1)$-norm estimates as for the $L^2(H^1)\cap H^1(H^{-1})$-norm. Our main contributions, presented in Theorem \[thm:X\_norm\_guaranteed\_efficiency\] in section \[sec:results\] below, include guaranteed upper bounds for the $L^2(H^1)$-norm of the error, and local-in-space-and-time lower bounds for the spatial estimator under the one-sided condition $h^2 \lesssim \tau$. We therefore remove the need for the two-sided conditions encountered previously, and we note that the assumptions in [@Picasso1998 [@Verfurth1998][@Picasso1998; @Verfurth1998]]{} were stronger than our assumption. We emphasize that the regime where $h^2 \lesssim \tau$ is the one of practical interest in computations, since implicit methods offer the possibility for large time-steps. Our lower bound is similar to [@Picasso1998[@Picasso1998]]{} in at least one respect, namely that the right-hand side of our lower bound includes the temporal jump estimator, since it does not appear possible to show in general that this estimator is locally bounded from above by the $L^2(H^1)$-norm of the error. Furthermore, we show that the constant of the lower bound is robust with respect to the spatial polynomial degree, and is also robust with respect to refinement and coarsening of the meshes, thereby allowing us to remove the so-called transition conditions. We also show that our results imply local-in-space and local-in-time efficiency when considered in the framework of the augmented norms that were proposed in [@AkrivisMakridakisNochetto2009 [@MakridakisNochetto2006; @SchotzauWihler2010][@AkrivisMakridakisNochetto2009; @MakridakisNochetto2006; @SchotzauWihler2010]]{}.
Our analysis rests upon the following key ingredients. First, in section \[sec:infsup\], we present the inf-sup identity which relates the $L^2(H^1)$-norm of the error to an appropriate dual norm of the residual on test functions in a subspace of $L^2(H^1)\cap H^1(H^{-1})$. After setting the notation for the class of finite element methods in section \[sec:fem\], we recall the construction of the equilibrated flux from our earlier work [@ErnSmearsVohralik2016[@ErnSmearsVohralik2016]]{} in section \[sec:flux\_equilibration\]. We state the main results in section \[sec:results\]. Section \[sec:upper\_bound\] uses the inf-sup framework to prove the guaranteed upper bounds and the proof of the lower bounds is the subject of section \[sec:efficiency\]. It is based on the combination of two key ideas. The first is to take advantage of the semi-discreteness in time of the test functions appearing in the fundamental efficiency result of [@ErnSmearsVohralik2016 [Lemma 8.2][@ErnSmearsVohralik2016 Lemma 8.2]]{} in order to gain control over a negative norm on the time derivatives of the test functions; see Lemma \[lem:X\_norm\_main\_estimate\] below. The second idea is to appeal to a specific pointwise-in-time identity for the discontinuous Galerkin time-stepping method, see Lemma \[lem:time\_dg\_exact\] below. Thus, we employ the definition of the numerical scheme for proving the lower bounds, which is somewhat unusual for a posteriori error analysis. The combination of these two ideas then yields the lower bounds stated in section \[sec:results\] under the relaxed hypothesis that $h^2 \lesssim \tau$ only.
Throughout this paper, the notation $a\lesssim b$ means that $a\leq C b$, with a generic constant $C$ that depends possibly on the shape-regularity of the spatial meshes and the space dimension ${d}$, but is otherwise independent of the mesh-size, time-step size, as well as the spatial and temporal polynomial degrees, or on refinement and coarsening between time-steps.
Inf-sup theory {#sec:infsup}
==============
Recall that ${\Omega}\subset {\mathbb{R}}^d $, $1\leq d \leq 3$ is a bounded, connected, polyhedral open set with Lipschitz boundary. For an arbitrary open subset $\omega\subset {\Omega}$, we use $(\cdot,\cdot)_{{\omega}}$ to denote the $L^2$-inner product for scalar- or vector-valued functions on $\omega$, with associated norm ${\lVert\cdot\rVert}_{{\omega}}$. In the special case where ${\omega}= {\Omega}$, we drop the subscript notation, i.e. ${\lVert\cdot\rVert}\coloneqq{\lVert\cdot\rVert}_{{\Omega}}$.
The starting point of the analysis is the weak formulation of problem where the time derivative has been cast onto a test function, using integration by parts in time. In particular, the solution space $X$ and test space ${Y_T}$ are defined by $$\begin{aligned}
X & \coloneqq {L^2(0,T;H^1_0({\Omega}))},
\\ {Y_T}&\coloneqq \{ {\varphi}\in {L^2(0,T;H^1_0({\Omega}))}\cap {H^1(0,T;H^{-1}({\Omega}))}, {\varphi}(T) =0 \}.
\end{aligned}$$ The spaces $X$ and ${Y_T}$ are equipped with the norms $$\label{eq:XYnorms}
\begin{aligned}
{\lVertv\rVert}_X^2 & \coloneqq \int_0^T {\lVert\nabla v\rVert}^2 \,{\mathrm{d}}t & & \forall\, v \in X,\\
{\lVert{\varphi}\rVert}_{{Y_T}}^2 & \coloneqq \int_0^T {\lVert{\partial}_t {\varphi}\rVert}_{H^{-1}({\Omega})}^2 + {\lVert\nabla {\varphi}\rVert}^2 \,{\mathrm{d}}t + {\lVert{\varphi}(0)\rVert}^2 & & \forall\, {\varphi}\in {Y_T}.
\end{aligned}$$ Let the bilinear form ${{\mathcal{B}}_X}\colon X\times {Y_T}{\rightarrow}{\mathbb{R}}$ be defined by $$\begin{aligned}
{{\mathcal{B}}_X}(v,{\varphi}) \coloneqq \int_0^T - {\langle {\partial}_t {\varphi},v \rangle} + (\nabla v,\nabla {\varphi}) \,{\mathrm{d}}t & && \forall\,v\in X,\; {\varphi}\in {Y_T},
\end{aligned}$$ where ${\langle \cdot,\cdot \rangle}$ denotes the duality pairing between $H^{-1}({\Omega})$ and $H^1_0({\Omega})$. Then, problem admits the following weak formulation: find $u \in X$ such that $$\label{eq:X_formulation}
\begin{aligned}
{{\mathcal{B}}_X}(u,{\varphi}) = \int_0^T (f,{\varphi})\,{\mathrm{d}}t + (u_0,{\varphi}(0)) & && \forall\, {\varphi}\in {Y_T}.
\end{aligned}$$ The well-posedness of is well-known and can be shown by Galerkin’s method, see for instance the textbook [@Wloka1987[@Wloka1987]]{}. Note that in this weak formulation, the initial condition $u(0)=u_0$ is expressed as a natural condition, appearing in , rather than as an essential condition imposed by the choice of solution space.
Problem admits an alternative weak formulation where the test space is $X$ and the trial space is ${L^2(0,T;H^1_0({\Omega}))}\cap {H^1(0,T;H^{-1}({\Omega}))}$ The two formulations possess the same solution, although they lead to different quantitative relations between the norm of the error and of the residual.
The following result states an inf–sup stability result for the bilinear form ${{\mathcal{B}}_X}$. This inf–sup stability result has the interesting and important property of taking the form of an identity, which is advantageous for the sharpness of a posteriori error analysis, and shows that the choice of norms for the spaces $X$ and ${Y_T}$ in above are optimal.
\[thm:inf\_sup\_parabolic\] For every $v \in X$, we have $$\begin{aligned}
{\lVertv\rVert}_{X} & = \sup_{{\varphi}\in Y_T\setminus\{0\}} \frac{ {{\mathcal{B}}_X}(v,{\varphi}) }{{\lVert{\varphi}\rVert}_{Y_T}}. \label{eq:infsup_X}
\end{aligned}$$
The arguments in the proof of [@ErnSmearsVohralik2016 [Theorem 2.1][@ErnSmearsVohralik2016 Theorem 2.1]]{} can be used to show the following inf-sup identity: for any ${\varphi}\in {Y_T}$, we have $$\label{eq:adjoint_inf_sup}
\begin{aligned}
{\lVert{\varphi}\rVert}_{{Y_T}} = \sup_{v\in X\setminus\{0\}} \frac{{{\mathcal{B}}_X}(v,{\varphi})}{{\lVertv\rVert}_X} .
\end{aligned}$$ So, immediately implies the lower bound ${\lVertv\rVert}_X \geq \sup_{{\varphi}\in {Y_T}\setminus\{0\}} {{\mathcal{B}}_X}(v,{\varphi}) / {\lVert{\varphi}\rVert}_{{Y_T}}$ for any fixed $v\in X$. To obtain the converse bound, let ${\varphi}_* \in {Y_T}$ denote the solution of ${{\mathcal{B}}_X}(w,{\varphi}_*) = \int_0^T (\nabla w, \nabla v) \,{\mathrm{d}}t $ for all $w \in X$. This problem can simply be seen as a backward-in-time parabolic problem with final time condition ${\varphi}_*(T)=0$. Hence, we have ${\lVertv\rVert}_X^2 = {{\mathcal{B}}_X}(v,{\varphi}_*)$ and implies that ${\lVert{\varphi}_*\rVert}_{{Y_T}} = {\lVertv\rVert}_X$. This immediately shows that ${\lVertv\rVert}_X \leq \sup_{{\varphi}\in {Y_T}\setminus\{0\}} {{\mathcal{B}}_X}(v,{\varphi}) / {\lVert{\varphi}\rVert}_{{Y_T}}$, and completes the proof of .
In order to estimate the error between the solution $u$ of and its approximation, we define the residual functional ${\mathcal{R}}_X \colon X{\rightarrow}[{Y_T}]^\prime$ by $$\label{eq:R_Y_def}
{\langle {\mathcal{R}}_X(v),{\varphi}\rangle}_{[{Y_T}]^\prime\times {Y_T}} \coloneqq {\mathcal{B}}_X(u-v,{\varphi})= \int_0^T (f,{\varphi}) + {\langle {\partial}_t {\varphi},v \rangle} - (\nabla v,\nabla {\varphi}) \,{\mathrm{d}}t + (u_0,{\varphi}(0)),$$ where $v\in X$ and ${\varphi}\in {Y_T}$, and where the equality follows simply from . The dual norm of the residual ${\lVert{\mathcal{R}}_X(v)\rVert}_{[{Y_T}]^\prime}$ is naturally defined by $${\lVert{\mathcal{R}}_X(v)\rVert}_{[{Y_T}]^\prime}\coloneqq \sup_{{\varphi}\in {Y_T}\setminus\{0\}} \frac{{\langle {\mathcal{R}}_X(v),{\varphi}\rangle}}{{\lVert{\varphi}\rVert}_{{Y_T}}}.$$ Theorem \[thm:inf\_sup\_parabolic\] implies the following *equivalence between the error and dual norm of the residual*: $$\label{eq:X_error_residual_equivalence}
\begin{aligned}
{\lVertu-v\rVert}_{X} = {\lVert{\mathcal{R}}_X(v)\rVert}_{[{Y_T}]^\prime} & & &\forall\,v\in X.
\end{aligned}$$
Finite element approximation {#sec:fem}
============================
The time interval $(0,T)$ is partitioned into sub-intervals $I_n\coloneqq (t_{n-1},t_n) $, with $1\leq n \leq N$, where it is assumed that $[0,T]=\bigcup_{n=1}^N\overline{I_n}$, and that $\{t_n\}_{n=0}^N$ is strictly increasing with $t_0=0$ and $t_N = T$. For each interval $I_n $, we let $\tau_n \coloneqq t_n-t_{n-1}$ denote the local time-step size. No special assumptions are made about the relative sizes of the time-steps to each other. A temporal polynomial degree $q_n\geq 0$ is associated to each time-step $I_n$, and we gather all the polynomial degrees in the vector $\bm q = (q_n)_{n=1}^N$. For a general vector space $V$, we shall write ${\mathcal{Q}}_{q_n}\left(I_n;V\right)$ to denote the space of $V$-valued univariate polynomials of degree at most $q_n$ over the time-step interval $I_n$.
Meshes
------
For each $0\leq n \leq N$, let ${\mathcal{T}}^n$ denote a matching simplicial mesh of the domain ${\Omega}$, where we assume shape-regularity of the meshes uniformly with respect to $n$. We consider here only matching simplicial meshes for simplicity, although we indicate that mixed simplicial–parallelepipedal meshes, possibly containing hanging nodes, can also be treated: see [@DolejsiErnVohralik2016[@DolejsiErnVohralik2016]]{} for instance. The mesh ${\mathcal{T}}^0$ will be used to approximate the initial datum $u_0$. For each element $K\in{\mathcal{T}}^n$, let $h_K\coloneqq \operatorname{diam}K$ denote the diameter of $K$. We associate a local spatial polynomial degree $p_K\geq 1$ to each $K\in {\mathcal{T}}^n$, and we gather all spatial polynomial degrees in the vector $\bm p_n= (p_K)_{K\in{\mathcal{T}}^n}$. In order to keep our notation sufficiently simple, the dependence of the local spatial polynomial degrees $p_K$ on the time-step is kept implicit, although we bear in mind that the polynomial degrees may change between time-steps.
Approximation spaces
--------------------
Given a general matching simplicial mesh ${\mathcal{T}}$ and given a vector of polynomial degrees $\bm p=(p_K)_{K\in{\mathcal{T}}}$, $p_K\geq 1$ for all $K\in{\mathcal{T}}$, we define the $H^1_0({\Omega})$-conforming $hp$-finite element space ${V_h}({\mathcal{T}},\bm p)$ by $$\label{eq:conforming_space_def}
{V_h}({\mathcal{T}},\bm p)\coloneqq \left\{v_h \in H^1_0({\Omega}),\; {\left. v_h\right|_{K}} \in {\mathcal{P}}_{p_K}(K)\quad\forall\,K\in{\mathcal{T}}\right\},$$ where ${\mathcal{P}}_{p_K}(K)$ denotes the space of polynomials of total degree at most $p_K$ on $K$. To shorten the notation, let ${V^n}\coloneqq V_h({\mathcal{T}}^n,\bm{p}_n)$ for each $0\leq n \leq N$. Let $\Pi_{h} u_0 \in V^0$ denote an approximation to the initial datum $u_0$, a typical choice being the $L^2$-orthogonal projection of $u_0$ onto $V^0$. Given the collection of time intervals $\{I_n\}_{n=1}^N$, the vector $\bm q$ of temporal polynomial degrees, and the $hp$-finite element spaces $\{{V^n}\}_{n=0}^N$, the finite element space ${V_{h\tau}}$ is defined by $$\label{eq:space_time_fem}
{V_{h\tau}}\coloneqq \left\{ v_{{{h\tau}}}|_{(0,T)}\in X,\; {\left. v_{{{h\tau}}}\right|_{I_n}} \in {\mathcal{Q}}_{q_n}(I_n;{V^n}) \quad\forall\, n=1,\dots,N,\; v_{{{h\tau}}}(0)\in V^0 \right\}.$$ Functions in ${V_{h\tau}}$ are generally discontinuous with respect to the time-variable at the temporal partition points. We take them to be left-continuous: for all $1\leq n \leq N$, we define $v_{{{h\tau}}}(t_n)$ as the trace at $t_n$ of the restriction ${\left. v_{{{h\tau}}}\right|_{I_n}}$. Moreover, functions in ${V_{h\tau}}$ also have a well-defined value at $t_0=0$. For all $0\leq n < N$, we denote the right-limit of $v_{{{h\tau}}}\in{V_{h\tau}}$ at $t_n$ by $v_{{{h\tau}}}(t_n^+)$. Then, the temporal jump operators ${\llparenthesis}\cdot {\rrparenthesis}_n$ are defined by $$\label{eq:jump_operators}
{\llparenthesis}v_{{{h\tau}}} {\rrparenthesis}_n \coloneqq v_{{{h\tau}}}(t_n)-v_{{{h\tau}}}(t_n^+), \quad 0\leq n \leq N-1.$$
Refinement and coarsening
-------------------------
Similarly to other works, e.g., [@Verfurth2003 [p. 196][@Verfurth2003 p. 196]]{}, we assume that we have at our disposal a common refinement mesh ${\widetilde{{\mathcal{T}}^{n}}}$ of ${\mathcal{T}}^{n-1}$ and ${\mathcal{T}}^n$ for each $1\leq n \leq N$, as well as associated polynomial degrees $\widetilde{\bm{p}}_n=(p_{{\widetilde{K}}})_{{\widetilde{K}}\in{\widetilde{{\mathcal{T}}^{n}}}}$, such that ${V^{n-1}}+ {V^n}\subset {\widetilde{{V^n}}}\coloneqq {V_h}({\widetilde{{\mathcal{T}}^{n}}},\widetilde{\bm{p}}_n) $. For a function $v_{{{h\tau}}}\in {V_{h\tau}}$, we observe that ${\llparenthesis}v_{{{h\tau}}} {\rrparenthesis}_{n-1} \in {\widetilde{{V^n}}}$ for each $1\leq n \leq N$ since $v_{{{h\tau}}}(t_{n-1})\in {V^{n-1}}$, $v_{{{h\tau}}}(t_{n-1}^+)\in {V^n}$, and ${V^{n-1}}+ {V^n}\subset {\widetilde{{V^n}}}$. It is assumed that the shape-regularity of ${\widetilde{{\mathcal{T}}^{n}}}$ is equivalent up to constants to those of ${\mathcal{T}}^{n-1}$ and ${\mathcal{T}}^n$, and that every element ${\widetilde{K}}\in{\widetilde{{\mathcal{T}}^{n}}}$ is wholly contained in a single element $K^\prime\in{\mathcal{T}}^{n-1}$ and a single element $K^{\prime\prime}\in{\mathcal{T}}^n$. We emphasize that we do not require any assumptions on the relative coarsening or refinement between successive spaces ${V^{n-1}}$ and ${V^n}$. In particular, we do not need the transition condition assumption from [@Verfurth2003 [p. 196, 201][@Verfurth2003 p. 196, 201]]{}, which requires a uniform bound on the ratio of element sizes between ${\widetilde{{\mathcal{T}}^{n}}}$ and ${\mathcal{T}}^n$.
Numerical method
----------------
The numerical scheme consists of finding $u_{{{h\tau}}} \in {V_{h\tau}}$ such that $u_{{{h\tau}}}(0)=\Pi_h u_0$, and such that $$\label{eq:num_scheme}
\int_{I_n} ({\partial}_t u_{{{h\tau}}}, v_{{{h\tau}}} ) + (\nabla u_{{{h\tau}}}, \nabla v_{{{h\tau}}}) \,{\mathrm{d}}t - \left({\llparenthesis}u_{{{h\tau}}} {\rrparenthesis}_{n-1}, v_{{{h\tau}}}(t_{n-1}^+) \right) = \int_{I_n} (f,v_{{{h\tau}}}) \,{\mathrm{d}}t$$ for all test functions $v_{{h\tau}}\in {\mathcal{Q}}_{q_n}(I_n;{V^n})$, for each time-step interval $I_n$, $n=1,\dots, N$. Here the time derivative ${\partial}_t u_{{{h\tau}}}$ is understood as the piecewise time-derivative on each time-step interval $I_n$. The numerical solution $u_{{{h\tau}}}\in {V_{h\tau}}$ can thus be obtained by solving the fully discrete problem on each successive time-step. At each time-step, this requires solving a linear system that is symmetric only in the case $q_n=0$; this can be performed efficiently in practice for arbitrary orders following [@Smears2016[@Smears2016]]{}. Note further that the initial condition $u_{{{h\tau}}}(0)=\Pi_h u_0$ does not guarantee that the right-limit $u_{{{h\tau}}}(0^+)$ should equal $\Pi_h u_0$.
Reconstruction operator {#sec:reconstruction}
-----------------------
For each time-step interval $I_n$ and each nonnegative integer $q$, let $L_q^n$ denote the polynomial on $I_n$ obtained by mapping the standard $q$-th Legendre polynomial under an affine transformation of $(-1,1)$ to $I_n$. It follows that $L_q^n(t_n) =1$ for all $q\geq 0$, and $L_q^n(t_{n-1})=(-1)^q$, and that the mapped Legendre polynomials $\{L_q^n\}_{q\geq 0}$ are $L^2$-orthogonal on $I_n$, and satisfy $\int_{I_n}{\lvertL_q^n\rvert}^2\,{\mathrm{d}}t = \frac{\tau_n}{2q+1}$ for all $q\geq 0$. Following [@MakridakisNochetto2006[@MakridakisNochetto2006]]{} (see also [@Smears2016 [Remark 2.3][@Smears2016 Remark 2.3]]{}), we introduce the reconstruction operator ${\mathcal{I}}$ defined on ${V_{h\tau}}$ by $$\label{eq:def_radau_reconstruction}
\begin{aligned}
{\left. ({\mathcal{I}}v_{{{h\tau}}})\right|_{I_n}} \coloneqq {\left. v_{{{h\tau}}} \right|_{I_n}} + \frac{(-1)^{q_n} }{2} \left( L_{q_n}^n - L_{q_n+1}^n \right) {\llparenthesis}v_{{{h\tau}}} {\rrparenthesis}_{n-1} & & & \forall\,v_{{{h\tau}}}\in{V_{h\tau}}.
\end{aligned}$$ It is clear that ${\mathcal{I}}$ is a linear operator on ${V_{h\tau}}$. Furthermore, the definition ensures that ${\left. {\mathcal{I}}v_{{{h\tau}}} \right|_{I_n}}(t_n) = v_{{{h\tau}}}(t_n)$, and that $ {\left. {\mathcal{I}}v_{{{h\tau}}} \right|_{I_n}}(t_{n-1}^+) = v_{{{h\tau}}}(t_{n-1})$ for all $1\leq n\leq N$. This implies that ${\mathcal{I}}v_{{{h\tau}}}$ is continuous with respect to the temporal variable at the interval partition points $\{t_n\}_{n=0}^{N-1}$ and hence ${\mathcal{I}}v_{{{h\tau}}} \in H^1(0,T;H^1_0({\Omega}))$. Furthermore, ${\left. {\mathcal{I}}v_{{{h\tau}}} \right|_{I_n}} \in {\mathcal{Q}}_{q_n+1}\big( I_n;{\widetilde{{V^n}}}\big)$ for any $v_{{{h\tau}}}\in{V_{h\tau}}$, where we recall that ${V^{n-1}}+ {V^n}\subset {\widetilde{{V^n}}}$. It is well-known from [@ErnSchieweck2016 [@MakridakisNochetto2006; @Smears2016][@ErnSchieweck2016; @MakridakisNochetto2006; @Smears2016]]{} that we may rewrite the numerical scheme as $$\label{eq:num_scheme_equiv}
\int_{I_n} ({\partial}_t {{\mathcal{I}}u_{{{h\tau}}}},v_{{{h\tau}}}) + (\nabla u_{{{h\tau}}}, \nabla v_{{{h\tau}}})
\,{\mathrm{d}}t = \int_{I_n} (f,v_{{{h\tau}}}) \,{\mathrm{d}}t \quad \forall\,v_{{h\tau}}\in {\mathcal{Q}}_{q_n}(I_n;{V^n}).$$ Note also that ${{\mathcal{I}}u_{{{h\tau}}}}(0)=\Pi_h u_0$.
Construction of the equilibrated flux {#sec:flux_equilibration}
=====================================
The a posteriori error estimates presented in this paper are based on a discrete and locally computable $\bm{H}(\operatorname{div})$-conforming flux ${\bm{\sigma}_{{{h\tau}}}}$ that satisfies the key equilibration property $$\label{eq:sigma_th_equilibration}
\begin{aligned}
{\partial}_t {{\mathcal{I}}u_{{{h\tau}}}}+ \nabla{\cdot} {\bm{\sigma}_{{{h\tau}}}}= f_{{{h\tau}}} & & &\text{in }{\Omega}\times(0,T),
\end{aligned}$$ where ${{\mathcal{I}}u_{{{h\tau}}}}$ is defined in section \[sec:reconstruction\], and $f_{{{h\tau}}}\approx f$ is an approximation of the data that is defined in below. We call ${\bm{\sigma}_{{{h\tau}}}}$ an equilibrated flux. The construction of ${\bm{\sigma}_{{{h\tau}}}}$ given here is exactly the same as in [@ErnSmearsVohralik2016[@ErnSmearsVohralik2016]]{}. This has the practical benefit that a single construction of the equilibrated flux can be used for both a posteriori error estimates in the $L^2(H^1)\cap H^1(H^{-1})$-norm and also in the $L^2(H^1)$-norm.
Local mixed finite element spaces
---------------------------------
For each $1\leq n \leq N$, let ${\mathcal{V}^n}$ denote the set of vertices of the mesh ${\mathcal{T}}^n$, where we distinguish the set of interior vertices ${\mathcal{V}^n_{\mathrm{int}}}$ and the set of boundary vertices ${\mathcal{V}^n_{\mathrm{ext}}}$. For each ${{\bm{a}}}\in {\mathcal{V}^n}$, let ${\psi_{{{\bm{a}}}}}$ denote the hat function associated with ${{\bm{a}}}$, and let ${{{\omega}_{{{\bm{a}}}}}}$ denote the interior of the support of ${\psi_{{{\bm{a}}}}}$, with associated diameter $h_{{{{\omega}_{{{\bm{a}}}}}}}$. Furthermore, let ${\widetilde{{\mathcal{T}}^{{{\bm{a}}}}}}$ denote the restriction of the mesh ${\widetilde{{\mathcal{T}}^{n}}}$ to ${{{\omega}_{{{\bm{a}}}}}}$. Recalling that the common refinement spaces ${\widetilde{{V^n}}}$ were obtained with a vector of polynomial degrees $\widetilde{\bm{p}}_n = (p_{{\widetilde{K}}})_{{\widetilde{K}}\in {\widetilde{{\mathcal{T}}^{n}}}}$, we associate to each ${{\bm{a}}}\in {\mathcal{V}^n}$ the fixed polynomial degree $$\label{eq:patch_polynomial_degree}
p_{{{\bm{a}}}} \coloneqq \max_{{\widetilde{K}}\in {\widetilde{{\mathcal{T}}^{{{\bm{a}}}}}}} (p_{{\widetilde{K}}}+1).$$ For a polynomial degree $p\geq 0$, let the piecewise polynomial (discontinuous) spaces ${\mathcal{P}}_{p}({\widetilde{{\mathcal{T}}^{{{\bm{a}}}}}})$ and ${\bm{RTN}}_p({\widetilde{{\mathcal{T}}^{{{\bm{a}}}}}})$ be defined by $$\begin{aligned}
{\mathcal{P}}_{p}({\widetilde{{\mathcal{T}}^{{{\bm{a}}}}}}) &\coloneqq \{ q_h \in L^2({{{\omega}_{{{\bm{a}}}}}}),\quad q_h|_{{\widetilde{K}}} \in {\mathcal{P}}_{p}({\widetilde{K}})\quad\forall\,{\widetilde{K}}\in{\widetilde{{\mathcal{T}}^{{{\bm{a}}}}}}\},
\\ {\bm{RTN}}_p({\widetilde{{\mathcal{T}}^{{{\bm{a}}}}}}) &\coloneqq \{ \bm{v}_h \in \bm{L}^2({{{\omega}_{{{\bm{a}}}}}}),\quad \bm{v}_h|_{{\widetilde{K}}} \in {\bm{RTN}}_{p}({\widetilde{K}})\quad\forall{\widetilde{K}}\in{\widetilde{{\mathcal{T}}^{{{\bm{a}}}}}}\},\end{aligned}$$ where ${\bm{RTN}}_{p}({\widetilde{K}}) \coloneqq \bm{{\mathcal{P}}}_{p}({\widetilde{K}}) + {\mathcal{P}}_{p}({\widetilde{K}})\bm{x}$ denotes the Raviart–Thomas–Nédélec space of order $p$ on the simplex ${\widetilde{K}}$. It is important to notice that whereas the patch ${{{\omega}_{{{\bm{a}}}}}}$ is subordinate to the elements of the mesh ${\mathcal{T}}^n$ around the vertex ${{\bm{a}}}\in{\mathcal{V}^n}$, the spaces ${\mathcal{P}}_{p}({\widetilde{{\mathcal{T}}^{{{\bm{a}}}}}})$ and ${\bm{RTN}}_p({\widetilde{{\mathcal{T}}^{{{\bm{a}}}}}})$ are subordinate to the submesh elements in ${\widetilde{{\mathcal{T}}^{{{\bm{a}}}}}}$; of course, in the absence of coarsening, this distinction vanishes. We now introduce the local spatial mixed finite element space ${\bm{V}_h^{{\bm{a}}}}$, defined by $$\begin{aligned}
{\bm{V}_h^{{\bm{a}}}}& \coloneqq
\begin{cases}
\left\{\bm{v}_h \in {\bm{H}(\operatorname{div},{{{\omega}_{{{\bm{a}}}}}})}\cap {\bm{RTN}_{p_{{{\bm{a}}}}}({\widetilde{{\mathcal{T}}^{{{\bm{a}}}}}})},\; \bm{v}_h\cdot \bm{n} =0\text{ on }{\partial}{{{\omega}_{{{\bm{a}}}}}}\right\} & \text{if }{{\bm{a}}}\in{\mathcal{V}^n_{\mathrm{int}}},\\
\left\{\bm{v}_h \in {\bm{H}(\operatorname{div},{{{\omega}_{{{\bm{a}}}}}})}\cap {\bm{RTN}_{p_{{{\bm{a}}}}}({\widetilde{{\mathcal{T}}^{{{\bm{a}}}}}})},\; \bm{v}_h\cdot \bm{n} =0\text{ on }{\partial}{{{\omega}_{{{\bm{a}}}}}}\setminus{\partial{\Omega}}\right\}& \text{if }{{\bm{a}}}\in{\mathcal{V}^n_{\mathrm{ext}}}.
\end{cases}\end{aligned}$$ We then define the space-time mixed finite element space $$\label{eq:spacetime_mixed_space_def}
\begin{aligned}
{\bm{V}_{h\tau}^{{{\bm{a}}},n}}\coloneqq {\mathcal{Q}}_{q_n}(I_n;{\bm{V}_h^{{\bm{a}}}}),
\end{aligned}$$ where we recall that ${\mathcal{Q}}_{q_n}\left(I_n;{\bm{V}_h^{{\bm{a}}}}\right)$ denotes the space of ${\bm{V}_h^{{\bm{a}}}}$-valued univariate polynomials of degree at most $q_n$ over the time-step interval $I_n$.
Data approximation {#sec:data_approximation}
------------------
Our a posteriori error estimates given in section \[sec:results\] involve certain approximations of the source term $f$ appearing in . It is helpful to define these approximations here. For each $1\leq n \leq N$ and for each ${{\bm{a}}}\in{\mathcal{V}^n}$, let ${\Pi_{{{h\tau}}}^{{{\bm{a}}},n}}$ be the $L^2_{{\psi_{{{\bm{a}}}}}}$-orthogonal projection from $L^2(I_n;L^2_{{\psi_{{{\bm{a}}}}}}({{{\omega}_{{{\bm{a}}}}}}))$ onto ${\mathcal{Q}}_{q_n}(I_n;{{\mathcal{P}}_{p_{{{\bm{a}}}}-1}({\widetilde{{\mathcal{T}}^{{{\bm{a}}}}}})})$, where $L^2_{{\psi_{{{\bm{a}}}}}}({{{\omega}_{{{\bm{a}}}}}})$ is the space of measurable functions $v$ on ${{{\omega}_{{{\bm{a}}}}}}$ such that $\int_{{{{\omega}_{{{\bm{a}}}}}}} {\psi_{{{\bm{a}}}}}{\lvertv\rvert}^2\,{\mathrm{d}}x<\infty$. In other words, the projection operator ${\Pi_{{{h\tau}}}^{{{\bm{a}}},n}}$ is defined by $\int_{I_n} ({\psi_{{{\bm{a}}}}}{\Pi_{{{h\tau}}}^{{{\bm{a}}},n}}v, q_{{{h\tau}}})_{{{{\omega}_{{{\bm{a}}}}}}}\,{\mathrm{d}}t = \int_{I_n} ({\psi_{{{\bm{a}}}}}v , q_{{{h\tau}}})_{{{{\omega}_{{{\bm{a}}}}}}}\,{\mathrm{d}}t$ for all $\, q_{{{h\tau}}} \in {\mathcal{Q}}_{q_n}(I_n;{{\mathcal{P}}_{p_{{{\bm{a}}}}-1}({\widetilde{{\mathcal{T}}^{{{\bm{a}}}}}})})$. We adopt the convention that ${\Pi_{{{h\tau}}}^{{{\bm{a}}},n}}v$ is extended by zero from ${{{\omega}_{{{\bm{a}}}}}}\times I_n$ to $\Omega\times(0,T)$ for all $v\in L^2(I_n;L^2_{{\psi_{{{\bm{a}}}}}}({{{\omega}_{{{\bm{a}}}}}}))$. Then, we define $f_{{{h\tau}}}$ by $$\label{eq:f_discrete_approx}
f_{{{h\tau}}} \coloneqq \sum_{n=1}^N\sum_{{{\bm{a}}}\in{\mathcal{V}^n}}{\psi_{{{\bm{a}}}}}\,{\Pi_{{{h\tau}}}^{{{\bm{a}}},n}}f.$$ See [@ErnSmearsVohralik2016[@ErnSmearsVohralik2016]]{} for further remarks concerning the approximation properties of $f_{{{h\tau}}}$. In particular, it is shown there that $f_{{{h\tau}}}$ is a data approximation that is at least of same order as the one used in the numerical scheme .
Flux reconstruction {#sec:flux_reconstruction_def}
-------------------
For each $1\leq n \leq N$ and each ${{\bm{a}}}\in{\mathcal{V}^n}$, let the scalar function ${g_{{{h\tau}}}^{{{\bm{a}}},n}}\in {\mathcal{Q}}_{q_n}(I_n;{{\mathcal{P}}_{p_{{{\bm{a}}}}}({\widetilde{{\mathcal{T}}^{{{\bm{a}}}}}})}) $ and vector field ${\bm{\tau}_{{{h\tau}}}^{{{\bm{a}}},n}}\in {\mathcal{Q}}_{q_n}(I_n;{\bm{RTN}_{p_{{{\bm{a}}}}}({\widetilde{{\mathcal{T}}^{{{\bm{a}}}}}})}) $ be defined by
\[eq:tau\_g\_def\] $$\begin{aligned}
{\bm{\tau}_{{{h\tau}}}^{{{\bm{a}}},n}}&\coloneqq {\psi_{{{\bm{a}}}}}\nabla u_{{{h\tau}}}|_{{{{\omega}_{{{\bm{a}}}}}}\times I_n},\label{eq:tau_def} \\
{g_{{{h\tau}}}^{{{\bm{a}}},n}}&\coloneqq {\psi_{{{\bm{a}}}}}\,\left({\Pi_{{{h\tau}}}^{{{\bm{a}}},n}}f - {\partial}_t {{\mathcal{I}}u_{{{h\tau}}}}\right)|_{{{{\omega}_{{{\bm{a}}}}}}\times I_n} - \nabla {\psi_{{{\bm{a}}}}}\cdot \nabla u_{{{h\tau}}}|_{{{{\omega}_{{{\bm{a}}}}}}\times I_n}.\label{eq:g_def}\end{aligned}$$
For interior vertices, the numerical scheme implies that $$\label{eq:gthan_mean_value_zero}
\begin{aligned}
({g_{{{h\tau}}}^{{{\bm{a}}},n}}(t),1)_{{{{\omega}_{{{\bm{a}}}}}}} = 0 & & & \forall\,t\in I_n.
\end{aligned}$$
\[def:flux\_construction\_1\]Let $u_{{h\tau}}\in {V_{h\tau}}$ be the numerical solution of . For each time-step interval $I_n$ and for each vertex ${{\bm{a}}}\in{\mathcal{V}}$, let the space ${\bm{V}_{h\tau}^{{{\bm{a}}},n}}$ be defined by . Let ${g_{{{h\tau}}}^{{{\bm{a}}},n}}$ and ${\bm{\tau}_{{{h\tau}}}^{{{\bm{a}}},n}}$ be defined by . Let ${\bm{\sigma}_{{{h\tau}}}^{{{\bm{a}}},n}}\in {\bm{V}_{h\tau}^{{{\bm{a}}},n}}$ be defined by $$\label{eq:stha_minimization_def}
{\bm{\sigma}_{{{h\tau}}}^{{{\bm{a}}},n}}\coloneqq \operatorname*{argmin}_{\substack{ \bm{v}_h \in {\bm{V}_{h\tau}^{{{\bm{a}}},n}}\\ \nabla{\cdot} \bm{v}_h = {g_{{{h\tau}}}^{{{\bm{a}}},n}}}}\int_{I_n} {\lVert\bm{v}_h + {\bm{\tau}_{{{h\tau}}}^{{{\bm{a}}},n}}\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2\,{\mathrm{d}}t.$$ Then, after extending ${\bm{\sigma}_{{{h\tau}}}^{{{\bm{a}}},n}}$ by zero from ${{{\omega}_{{{\bm{a}}}}}}\times I_n$ to ${\Omega}\times (0,T)$ for each ${{\bm{a}}}\in {\mathcal{V}}$ and for each $1\leq n\leq N$, we define $$\label{eq:flux_reconstruction_1}
{\bm{\sigma}_{{{h\tau}}}}\coloneqq \sum_{n=1}^N \sum_{{{\bm{a}}}\in {\mathcal{V}^n}} {\bm{\sigma}_{{{h\tau}}}^{{{\bm{a}}},n}}.$$
Note that ${\bm{\sigma}_{{{h\tau}}}^{{{\bm{a}}},n}}\in {\bm{V}_{h\tau}^{{{\bm{a}}},n}}$ is well-defined for all ${{\bm{a}}}\in{\mathcal{V}^n}$: in particular, for interior vertices ${{\bm{a}}}\in{\mathcal{V}^n_{\mathrm{int}}}$, we use to guarantee the compatibility of the datum ${g_{{{h\tau}}}^{{{\bm{a}}},n}}$ with the constraint $\nabla{\cdot}{\bm{\sigma}_{{{h\tau}}}^{{{\bm{a}}},n}}= {g_{{{h\tau}}}^{{{\bm{a}}},n}}$. The following key result is quoted from [@ErnSmearsVohralik2016[@ErnSmearsVohralik2016]]{}.
\[thm:sigma\_th\_equilibration\] Let the flux reconstruction ${\bm{\sigma}_{{{h\tau}}}}$ be given by Definition [\[def:flux\_construction\_1\]]{}, and let $f_{{{h\tau}}}$ be defined in . Then ${\bm{\sigma}_{{{h\tau}}}}\in L^2(0,T;{\bm{H}(\operatorname{div})})$ and the equilibration identity holds.
Moreover, for the purpose of implementation, it is known that on each patch of the mesh and at each time-step, the solution of the minimization problem decouples into $q_n+1$ independent spatial mixed finite element linear systems, which helps to reduce the cost of computing the flux ${\bm{\sigma}_{{{h\tau}}}}$.
Main results {#sec:results}
============
We introduce the following a posteriori error estimators and data oscillation terms:
\[eq:estimators\] $$\begin{aligned}
[{\eta_{\mathrm{F},K}^{n}}]^2 &\coloneqq \int_{I_n}{\lVert{\bm{\sigma}_{{{h\tau}}}}+ \nabla u_{{{h\tau}}} \rVert}_{K}^2 \,{\mathrm{d}}t , \label{eq:etaEq_def} \\
[{\eta_{\mathrm{J},K}^n}]^2 & \coloneqq \int_{I_n} {\lVert\nabla(u_{{{h\tau}}} - {{\mathcal{I}}u_{{{h\tau}}}})\rVert}_{K}^2 \, {\mathrm{d}}t, \label{eq:etaJ_def}\\
[{\eta_{\mathrm{osc},{{h\tau}}}^n}]^2 & \coloneqq \frac{1+\sqrt{2}}{2} \int_{I_n} \sum_{{\widetilde{K}}\in{\widetilde{{\mathcal{T}}^{n}}}} \left[\frac{\tau_n}{\pi}+\frac{h_{{\widetilde{K}}}^2}{\pi^2} \right] {\lVertf-f_{{{h\tau}}} \rVert}_{{\widetilde{K}}}^2 \,{\mathrm{d}}t, \label{eq:etaOscTh_def} \\
{\eta_{\mathrm{osc},\mathrm{init}}}& \coloneqq {\lVertu_0-\Pi_h u_0\rVert},\end{aligned}$$
where, $K\in {\mathcal{T}}^n$, $1\leq n \leq N$, the equilibrated flux ${\bm{\sigma}_{{{h\tau}}}}$ is prescribed in Definition \[def:flux\_construction\_1\], and where the data approximation $f_{{{h\tau}}}$ is defined in section \[sec:data\_approximation\]. The total estimator for the error is defined by $$\label{eq:etaX_def}
[\eta_X]^2 \coloneqq \sum_{n=1}^N \left\{\left[\sum_{K\in{\mathcal{T}}^n} \left\{ [{\eta_{\mathrm{F},K}^{n}}]^2+[{\eta_{\mathrm{J},K}^n}]^2\right\}\right]^{\frac{1}{2}} + {\eta_{\mathrm{osc},{{h\tau}}}^n}\right\}^2 + [{\eta_{\mathrm{osc},\mathrm{init}}}]^2.$$ The flux estimator ${\eta_{\mathrm{F},K}^{n}}$ and the temporal jump estimator ${\eta_{\mathrm{J},K}^n}$ are the two main estimators. In particular, the flux estimator ${\eta_{\mathrm{F},K}^{n}}$ measures the lack of $\bm{H}(\operatorname{div})$-conformity of $\nabla u_{{{h\tau}}}$, and the temporal jump estimator ${\eta_{\mathrm{J},K}^n}$ measures the lack of temporal conformity of $u_{{{h\tau}}}$. Indeed, ${\eta_{\mathrm{J},K}^n}$ is related to the jump ${\llparenthesis}u_{{{h\tau}}} {\rrparenthesis}_{n-1}$, since it was shown in [@SchotzauWihler2010 [@ErnSmearsVohralik2016][@SchotzauWihler2010; @ErnSmearsVohralik2016]]{} that ${\eta_{\mathrm{J},K}^n}$ can be equivalently rewritten as $$\label{eq:jump_equivalent_form}
{\eta_{\mathrm{J},K}^n}= \sqrt{\tfrac{\tau_n ({q_n}+1)}{(2q_n+1)(2q_n+3)}}\, {\lVert\nabla {\llparenthesis}u_{{{h\tau}}} {\rrparenthesis}_{n-1}\rVert}_K.$$ Given that ${\eta_{\mathrm{F},K}^{n}}$ and ${\eta_{\mathrm{J},K}^n}$ respectively measure the lack of spatial and temporal conformity of the approximate solution, it is common in the literature to call ${\eta_{\mathrm{F},K}^{n}}$ the spatial estimator and ${\eta_{\mathrm{J},K}^n}$ the temporal estimator. However, such terminology must not be interpreted as stating that these estimators bound the errors due respectively to the spatial and temporal discretization.
\[thm:X\_norm\_guaranteed\_efficiency\] Let $u \in X$ be the weak solution of , and let $u_{{{h\tau}}}\in {V_{h\tau}}$ denote the solution of the numerical scheme . Let $\eta_X$ be defined by . Then, we have the following $X$-norm a posteriori error estimate: $$\label{eq:Xnorm_guaranteed_upper}
{\lVertu-u_{{{h\tau}}}\rVert}_X \leq \eta_X.$$ If $K\in {\mathcal{T}}^n$, $1\leq n \leq N$, is an element such that $h_{{{{\omega}_{{{\bm{a}}}}}}}^2 \leq \gamma_{{{\bm{a}}}} \, \tau_n$ for each ${{\bm{a}}}\in{\mathcal{V}}_K$, with ${\mathcal{V}}_K$ the set of vertices of the element $K$, with some constant $\gamma_{{{\bm{a}}}}>0$, where $h_{{{{\omega}_{{{\bm{a}}}}}}}$ denotes the diameter of the patch ${{{\omega}_{{{\bm{a}}}}}}$, then we have the local lower bound for the flux estimator ${\eta_{\mathrm{F},K}^{n}}$ $$\label{eq:Xnorm_local_efficiency}
[{\eta_{\mathrm{F},K}^{n}}]^2 \leq C^2_{\gamma_{{{\bm{a}}}},q_n} \sum_{{{\bm{a}}}\in{\mathcal{V}}_{K}} \left\{ \int_{I_n}{\lVert\nabla(u-u_{{{h\tau}}})\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2 + {\lVert\nabla(u_{{{h\tau}}}-{{\mathcal{I}}u_{{{h\tau}}}})\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2 {\mathrm{d}}t + [{\eta_{\mathrm{osc}}^{{{\bm{a}}},n}}]^2 \right\},$$ where the local data ocillation ${\eta_{\mathrm{osc}}^{{{\bm{a}}},n}}$ is defined by $$\label{eq:tetaOsca_def}
[{\eta_{\mathrm{osc}}^{{{\bm{a}}},n}}]^2\coloneqq \int_{I_n} {\lVert f - {\Pi_{{{h\tau}}}^{{{\bm{a}}},n}}f\rVert}_{H^{-1}({{{\omega}_{{{\bm{a}}}}}})}^2 {\mathrm{d}}t.$$ Furthermore, under the hypothesis that there exists $\gamma>0$ such that $h_{{{{\omega}_{{{\bm{a}}}}}}}^2 \leq \gamma \, \tau_n$ for every ${{\bm{a}}}\in{\mathcal{V}^n}$ and every $1\leq n \leq N$, then we have the global lower bound $$\label{eq:Xnorm_global_efficiency}
\sum_{n=1}^N \sum_{K\in{\mathcal{T}}^n} [{\eta_{\mathrm{F},K}^{n}}]^2 \leq C_{\gamma,q_n}^2\left\{ {\lVertu-u_{{{h\tau}}}\rVert}_X^2 + {\lVertu_{{{h\tau}}}-{{\mathcal{I}}u_{{{h\tau}}}}\rVert}_X^2 + \sum_{n=1}^N \sum_{{{\bm{a}}}\in{\mathcal{V}^n}} [{\eta_{\mathrm{osc}}^{{{\bm{a}}},n}}]^2 \right\}.$$ The constants $C_{\gamma_{{{\bm{a}}}},q_n}$ in and $C_{\gamma,q_n}$ in satisfy $C_{\gamma,q_n} \lesssim (q_n+1)^{\frac{1}{2}} + \gamma (q_n+1)^{\frac{5}{2}}$, and may depend on the shape regularity of ${\mathcal{T}}^n$ and ${\widetilde{{\mathcal{T}}^{n}}}$ and on the dimension ${d}$, but otherwise do not depend on the mesh-size, time-step size, spatial polynomial degrees, or on refinement and coarsening between time-steps.
The proof of Theorem \[thm:X\_norm\_guaranteed\_efficiency\] is given in several stages throughout the following sections. In the first stage, we give the proof of the upper bound immediately after the helpful data oscillation estimate of Lemma \[lem:Xnorm\_data\_oscillation\] below in section \[sec:upper\_bound\]. In the second stage, we show the lower bounds and in section \[sec:efficiency\].
\[rem:jump\_estimator\] In the local lower bound , we have $\int_{I_n} {\lVert\nabla(u_{{{h\tau}}}-{{\mathcal{I}}u_{{{h\tau}}}})\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2{\mathrm{d}}t= \sum_{K\subset {{{\omega}_{{{\bm{a}}}}}}} [{\eta_{\mathrm{J},K}^n}]^2$, see also , where the sum is over all elements $K$ of ${\mathcal{T}}^n$ contained in ${{{\omega}_{{{\bm{a}}}}}}$. Similarly, in the global lower bound , the term ${\lVertu_{{{h\tau}}}-{{\mathcal{I}}u_{{{h\tau}}}}\rVert}_X^2 = \sum_{n=1}^N\sum_{K\in {\mathcal{T}}^n} [{\eta_{\mathrm{J},K}^n}]^2$ appears. Thus our result here is comparable to those in [@Picasso1998[@Picasso1998]]{} where the jump estimator also appears on the right-hand side of the local lower bounds. The reason for the appearance of this term can be essentially traced back to the lack of Galerkin orthogonality for the temporal reconstruction ${{\mathcal{I}}u_{{{h\tau}}}}$, see . Furthermore, in [@ErnSmearsVohralik2016[@ErnSmearsVohralik2016]]{} it was shown that the (time-local but space-global) jump estimators are bounded from above by the (time-local space-global) $L^2(H^1)\cap H^1(H^{-1})$-norm of the error, up to possible data oscillation.
As pointed out by the remark in [@Verfurth2003 [p. 198, Par. (5)][@Verfurth2003 p. 198, Par. (5)]]{} concerning the equivalence of residual-based estimators for both $L^2(H^1)$ and $L^2(H^1)\cap H^1(H^{-1})$ norms, it is important to observe that in the absence of data oscillation, the estimator $\eta_X$ defined above in is equivalent up to constants to the $L^2(H^1)\cap H^1(H^{-1})$-norm estimator defined in [@ErnSmearsVohralik2016 [Eq. (5.10)][@ErnSmearsVohralik2016 Eq. (5.10)]]{}. However, an important difference between these estimators concerns the data oscillation. Indeed, it is known since [@Verfurth2003[@Verfurth2003]]{} that $L^2(H^1)\cap H^1(H^{-1})$ estimators generally contain a data oscillation term that can be of same temporal order as the error. By comparison, the data oscillation term features an additional half-order with respect to the time-step size. Therefore we expect that the $X$-norm estimator given above may be of special use in situations with significant data oscillation in time.
Theorem \[thm:X\_norm\_guaranteed\_efficiency\] is our main result on a posteriori error estimation of ${\lVertu-u_{{{h\tau}}}\rVert}_X$. Several authors have also considered various augmented norms and error measures, see e.g. [@AkrivisMakridakisNochetto2009 [@MakridakisNochetto2006; @SchotzauWihler2010][@AkrivisMakridakisNochetto2009; @MakridakisNochetto2006; @SchotzauWihler2010]]{}. For instance, we can define the error measure $$\label{eq:error_measures}
{{\mathcal{E}}_X}\coloneqq \max\left\{{\lVertu-u_{{{h\tau}}}\rVert}_X, {\lVertu-{{\mathcal{I}}u_{{{h\tau}}}}\rVert}_X \right\}.$$ The choice in $\eqref{eq:error_measures}$ is only one of many possibilities; for instance we could equally well consider ${\lVertu-u_{{{h\tau}}}\rVert}_X + {\lVertu_{{{h\tau}}}-{{\mathcal{I}}u_{{{h\tau}}}}\rVert}_X$. The interest of this approach is that the bounds , and immediately yield a global upper bound and local-in-time and local-in-space efficiency with respect to this error measure, see Corollary \[cor:error\_measures\] below. However, it is important to note that it does not appear possible to show in general an equivalence between ${{\mathcal{E}}_X}$ and ${\lVertu-u_{{{h\tau}}}\rVert}_X$, see Remark \[rem:jump\_estimator\].
\[cor:error\_measures\] Let ${{\mathcal{E}}_X}$ be defined by . Then, we have the guaranteed upper bound $${{\mathcal{E}}_X}\leq 2\, \eta_X,$$ If $K\in {\mathcal{T}}^n$, $1\leq n \leq N$, is an element such that $h_{{{{\omega}_{{{\bm{a}}}}}}}^2 \leq \gamma_{{{\bm{a}}}} \, \tau_n$ for each ${{\bm{a}}}\in{\mathcal{V}}_K$ with some constant $\gamma_{{{\bm{a}}}}>0$, where $h_{{{{\omega}_{{{\bm{a}}}}}}}$ denotes the diameter of the patch ${{{\omega}_{{{\bm{a}}}}}}$, then we have the local efficiency bound $$[{\eta_{\mathrm{F},K}^{n}}]^2 + [{\eta_{\mathrm{J},K}^n}]^2 \leq C_{\gamma_{{{\bm{a}}}},q_n}^2 \sum_{{{\bm{a}}}\in{\mathcal{V}}_K} \left\{ [{{\mathcal{E}}_X^{{{\bm{a}}},n}}]^2 + [{\eta_{\mathrm{osc}}^{{{\bm{a}}},n}}]^2 \right\}.$$ where the local error measures ${{\mathcal{E}}_X^{{{\bm{a}}},n}}$, ${{\bm{a}}}\in{\mathcal{V}^n}$, are defined by $$\label{eq:augmented_local_efficiency}
[{{\mathcal{E}}_X^{{{\bm{a}}},n}}]^2 \coloneqq \max\left\{\int_{I_n}{\lVert\nabla(u-u_{{{h\tau}}})\rVert}^2_{{{{\omega}_{{{\bm{a}}}}}}}\,{\mathrm{d}}t, \int_{I_n}{\lVert\nabla(u-{{\mathcal{I}}u_{{{h\tau}}}})\rVert}^2_{{{{\omega}_{{{\bm{a}}}}}}}\,{\mathrm{d}}t \right\}.$$ Furthermore, under the hypothesis that there exists $\gamma>0$ such that $h_{{{{\omega}_{{{\bm{a}}}}}}}^2 \leq \gamma \, \tau_n$ for every ${{\bm{a}}}\in{\mathcal{V}^n}$ and every $1\leq n \leq N$, then we have the global efficiency bound $$\label{eq:augmented_global_efficiency}
\sum_{n=1}^N\sum_{K\in{\mathcal{T}}^n} [{\eta_{\mathrm{F},K}^{n}}]^2 + [{\eta_{\mathrm{J},K}^n}]^2\leq C_{\gamma,q_n}^2 \left\{[{{\mathcal{E}}_X}]^2 + \sum_{n=1}^N \sum_{{{\bm{a}}}\in{\mathcal{V}^n}} [{\eta_{\mathrm{osc}}^{{{\bm{a}}},n}}]^2 \right\}.$$
Proof of the guaranteed upper bound {#sec:upper_bound}
====================================
We will make use of the following preparatory lemmas.
Let $I_n$ be a given time interval, and let ${\varphi}\in L^2(I_n;H^1_0({\Omega}))\cap H^1(I_n;H^{-1}({\Omega}))$ be an arbitrary function. Let ${\varphi}^n\in H^1_0({\Omega})$, the time-mean value of ${\varphi}$ over $I_n$, be defined by ${\varphi}^n \coloneqq \tfrac{1}{\tau_n}\int_{I_n} {\varphi}\,{\mathrm{d}}t$. Then
\[eq:bochner\_bounds\] $$\begin{aligned}
\int_{I_n} {\lVert\nabla {\varphi}^n\rVert}^2\,{\mathrm{d}}t & \leq \int_{I_n}{\lVert\nabla {\varphi}\rVert}^2\,{\mathrm{d}}t, \label{eq:approx_2}
\\ \int_{I_n} {\lVert{\varphi}-{\varphi}^n\rVert}^2 \,{\mathrm{d}}t &\leq \frac{\tau_n}{\pi}\left(\int_{I_n} {\lVert{\partial}_t {\varphi}\rVert}^2_{H^{-1}({\Omega})}\,{\mathrm{d}}t\right)^{\frac{1}{2}}\left(\int_{I_n}{\lVert\nabla {\varphi}\rVert}^2\,{\mathrm{d}}t \right)^{\frac{1}{2}}. \label{eq:approx_1}\end{aligned}$$
The bound is simply the stability of the $L^2$-projection with respect to time; thus it remains only to show . It is well-known that there exists a maximal sequence $\{\psi_k\}_{k=1}^\infty$ that is orthonormal in the $L^2({\Omega})$-inner product and orthogonal in the $H^1_0({\Omega})$ inner product: i.e. $(\psi_k,\psi_j)=\delta_{kj}$ and $(\nabla\psi_k,\nabla\psi_j) = \lambda_{k} \delta_{kj}$, with $\{\lambda_{k}\}_{k=1}^\infty \subset {\mathbb{R}}_{>0}$. Then, we have ${\varphi}= \sum_{k=1}^{\infty} \alpha_k \psi_k$ and ${\varphi}^n = \sum_{k=1}^\infty \overline{\alpha_k} \psi_k$, with real-valued $\alpha_k \in H^1(I_n)$ and $\overline{\alpha_k} = \frac{1}{\tau_n}\int_{I_n} \alpha_k {\mathrm{d}}t$. Thus we may use the Poincaré inequality for real-valued functions to obtain $$\begin{split}
\int_{I_n}{\lVert{\varphi}-{\varphi}^n\rVert}^2{\mathrm{d}}t &= \sum_{k=1}^\infty {\lVert\alpha_k-\overline{\alpha_k}\rVert}_{L^2(I_n)}^2 \leq \frac{\tau_n}{\pi} \sum_{k=1}^\infty {\lvert\alpha_k\rvert}_{H^1(I_n)}{\lVert\alpha_k\rVert}_{L^2(I_n)}
\\ & \leq \frac{\tau_n}{\pi}\left(\sum_{k=1}^\infty \frac{1}{\lambda_k}{\lvert\alpha_k\rvert}_{H^1(I_n)}^2 \right)^{\frac{1}{2}}\left(\sum_{k=1}^\infty \lambda_k {\lVert\alpha_k\rVert}_{L^2(I_n)}^2 \right)^{\frac{1}{2}}.
\end{split}$$ We then deduce from the identities $\int_{I_n}{\lVert{\partial}_t {\varphi}\rVert}_{H^{-1}({\Omega})}^2{\mathrm{d}}t = \sum_{k=1}^\infty \frac{1}{\lambda_k}{\lvert\alpha_k\rvert}_{H^1(I_n)}^2$ and $\int_{I_n}{\lVert\nabla {\varphi}\rVert}^2{\mathrm{d}}t = \sum_{k=1}^\infty \lambda_k {\lVert\alpha_k\rVert}_{L^2(I_n)}^2$.
\[lem:Xnorm\_data\_oscillation\] Let $f\in L^2(0,T;L^2({\Omega}))$, let $f_{{{h\tau}}}$ be defined by , and let ${\varphi}\in {L^2(0,T;H^1_0({\Omega}))}\cap {H^1(0,T;H^{-1}({\Omega}))}$ be an arbitrary function. Then, for each $1\leq n \leq N$, $$\label{eq:X_data_oscillation_estimate}
\begin{aligned}
{\left\lvert\int_{I_n} (f-f_{{{h\tau}}},{\varphi})\,{\mathrm{d}}t \right\rvert} \leq {\eta_{\mathrm{osc},{{h\tau}}}^n}\left(
\int_{I_n} {\lVert{\partial}_t {\varphi}\rVert}_{H^{-1}({\Omega})}^2+{\lVert\nabla {\varphi}\rVert}^2 \,{\mathrm{d}}t \right)^{\frac{1}{2}} .
\end{aligned}$$
For a given function ${\varphi}\in {L^2(0,T;H^1_0({\Omega}))}\cap {H^1(0,T;H^{-1}({\Omega}))}$, we define ${\varphi}^n$ the time-mean value of ${\varphi}$ over $I_n$ as ${\varphi}^n \coloneqq \tfrac{1}{\tau_n}\int_{I_n} {\varphi}\,{\mathrm{d}}t\in H^1_0({\Omega})$, and we define ${\varphi}^n_{{\widetilde{K}}}$ the space-mean value of ${\varphi}^n$ over ${\widetilde{K}}$ as ${\varphi}^n_{{\widetilde{K}}}|_{{\widetilde{K}}} \coloneqq \tfrac{1}{{\lvert{\widetilde{K}}\rvert}}\int_{{\widetilde{K}}}{\varphi}^n {\mathrm{d}}x$, where $1\leq n\leq N$ and ${\widetilde{K}}\in{\widetilde{{\mathcal{T}}^{n}}}$. Now, we note that the definition of $f_{{{h\tau}}}$ in implies that $f-f_{{{h\tau}}}$ has zero mean value over each space-time element ${\widetilde{K}}\times I_n$. Therefore, we obtain $$\int_{I_n}(f-f_{{{h\tau}}},{\varphi})\,{\mathrm{d}}t = \int_{I_n}(f-f_{{{h\tau}}},{\varphi}-{\varphi}^n) + \sum_{{\widetilde{K}}\in{\widetilde{{\mathcal{T}}^{n}}}} (f-f_{{{h\tau}}},{\varphi}^n-{\varphi}^n_{{\widetilde{K}}})_{{\widetilde{K}}}\,{\mathrm{d}}t\eqqcolon A+B.$$ Then, we apply the bounds and to obtain $$\begin{aligned}
{\lvertA\rvert} &\leq \left(\int_{I_n} \sum_{{\widetilde{K}}\in{\widetilde{{\mathcal{T}}^{n}}}} \frac{\tau_{n}}{\pi} {\lVertf-f_{{{h\tau}}}\rVert}_{{\widetilde{K}}}^2\,{\mathrm{d}}t\right)^{\frac{1}{2}} \left(\int_{I_n}{\lVert{\partial}_t {\varphi}\rVert}^2_{H^{-1}({\Omega})}\,{\mathrm{d}}t\right)^{\frac{1}{4}}\left(\int_{I_n}{\lVert\nabla {\varphi}\rVert}^2\,{\mathrm{d}}t \right)^{\frac{1}{4}},
\\
{\lvertB\rvert} &\leq \left(\int_{I_n} \sum_{{\widetilde{K}}\in{\widetilde{{\mathcal{T}}^{n}}}} \frac{h^2_{{\widetilde{K}}}}{\pi^2} {\lVertf-f_{{{h\tau}}}\rVert}_{{\widetilde{K}}}^2\,{\mathrm{d}}t\right)^{\frac{1}{2}} \left(\int_{I_n}{\lVert\nabla {\varphi}\rVert}^2\,{\mathrm{d}}t\right)^{\frac{1}{2}}.\end{aligned}$$ Then, the Cauchy–Schwarz inequality and the Young inequality $ ab + b^2 \leq \frac{1+\sqrt{2}}{2}(a^2 + b^2)$ for all $a$, $b\in {\mathbb{R}}$, imply that the bound holds.
#### Proof of the upper bound .
Recall from on the equivalence of norms and residuals that ${\lVertu-u_{{{h\tau}}}\rVert}_X={\lVert{\mathcal{R}}_X(u_{{{h\tau}}})\rVert}_{[Y_T]'}$, so we turn our attention to bounding ${\langle {\mathcal{R}}_X(u_{{{h\tau}}}),{\varphi}\rangle}$ for an arbitrary test function ${\varphi}\in Y_T$. By adding and subtracting $\int_{0}^T ({\partial}_t {{\mathcal{I}}u_{{{h\tau}}}}+ \nabla\cdot {\bm{\sigma}_{{{h\tau}}}},{\varphi})\,{\mathrm{d}}t$ and recalling the flux equilibration identity , we get$$\begin{gathered}
\label{eq:Xnorm_upper_1}
{\langle {\mathcal{R}}_X(u_{{{h\tau}}}),{\varphi}\rangle} = \int_{0}^T(f-f_{{{h\tau}}},{\varphi})+{\langle {\partial}_t {\varphi},u_{{{h\tau}}}-{{\mathcal{I}}u_{{{h\tau}}}}\rangle}-({\bm{\sigma}_{{{h\tau}}}}+\nabla u_{{{h\tau}}},\nabla {\varphi})\,{\mathrm{d}}t \\+ (u_0-\Pi_h u_0,{\varphi}(0)),\end{gathered}$$ where we have used integration by parts with respect to time for the time derivative ${\partial}_t {{\mathcal{I}}u_{{{h\tau}}}}$, noting that ${{\mathcal{I}}u_{{{h\tau}}}}(0)=\Pi_h u_0$ and that ${\varphi}(T)=0$, and also where we have used integration by parts over ${\Omega}$ for the flux ${\bm{\sigma}_{{{h\tau}}}}\in L^2(0,T;H(\operatorname{div},{\Omega}))$. Employing the shorthand notation ${\lVert{\varphi}\rVert}_{Y(I_n)}^2\coloneqq \int_{I_n}{\lVert{\partial}_t{\varphi}\rVert}_{H^{-1}({\Omega})}^2 +{\lVert\nabla{\varphi}\rVert}^2\,{\mathrm{d}}t$, we then use Lemma \[lem:Xnorm\_data\_oscillation\] and the Cauchy–Schwarz inequality to bound $$\label{eq:Xnorm_upper_2}
\begin{split}
& \int_{0}^T(f-f_{{{h\tau}}},{\varphi})+{\langle {\partial}_t {\varphi},u_{{{h\tau}}}-{{\mathcal{I}}u_{{{h\tau}}}}\rangle}-({\bm{\sigma}_{{{h\tau}}}}+\nabla u_{{{h\tau}}},\nabla {\varphi})\,{\mathrm{d}}t
\\ & \leq \sum_{n=1}^N \left\{\left[\int_{I_n} {\lVert{\bm{\sigma}_{{{h\tau}}}}+ \nabla u_{{{h\tau}}}\rVert}^2+ {\lVert\nabla(u_{{{h\tau}}}-{{\mathcal{I}}u_{{{h\tau}}}})\rVert}^2\,{\mathrm{d}}t \right]^{\frac{1}{2}} + {\eta_{\mathrm{osc},{{h\tau}}}^n}\right\}{\lVert{\varphi}\rVert}_{Y(I_n)}
\\ & = \sum_{n=1}^N \left\{\left[\sum_{K\in{\mathcal{T}}^n} \left\{ [{\eta_{\mathrm{F},K}^{n}}]^2+[{\eta_{\mathrm{J},K}^n}]^2\right\}\right]^{\frac{1}{2}} + {\eta_{\mathrm{osc},{{h\tau}}}^n}\right\} {\lVert{\varphi}\rVert}_{Y(I_n)}.
\end{split}$$ We then combine and with the Cauchy–Schwarz inequality to find that ${\langle {\mathcal{R}}_X(u_{{{h\tau}}}),{\varphi}\rangle} \leq \eta_X {\lVert{\varphi}\rVert}_{Y_T}$; since ${\varphi}\in Y_T$ was arbitrary, we obtain ${\lVertu-u_{{{h\tau}}}\rVert}_X\leq \eta_X$ as a result of , thereby completing the proof of .
Proof of the bounds and {#sec:efficiency}
=========================
We start by observing that ${\bm{\sigma}_{{{h\tau}}}}|_{K\times I_n} = \sum_{{{\bm{a}}}\in{\mathcal{V}}_K} {\bm{\sigma}_{{{h\tau}}}^{{{\bm{a}}},n}}|_{K\times I_n}$, and thus $$\int_{I_n} [{\eta_{\mathrm{F},K}^{n}}]^2 \,{\mathrm{d}}t = \int_{I_n} {\lVert {\textstyle\sum}_{{{\bm{a}}}\in {\mathcal{V}}_K} ({\bm{\sigma}_{{{h\tau}}}^{{{\bm{a}}},n}}+ {\psi_{{{\bm{a}}}}}\nabla u_{{{h\tau}}})\rVert}_{K}^2\,{\mathrm{d}}t \leq {\lvert{\mathcal{V}}_K\rvert} \sum_{{{\bm{a}}}\in{\mathcal{V}}_K} \int_{I_n}{\lVert{\bm{\sigma}_{{{h\tau}}}^{{{\bm{a}}},n}}+ {\psi_{{{\bm{a}}}}}\nabla u_{{{h\tau}}}\rVert}_K^2 \,{\mathrm{d}}t, \label{eq:X_norm_local_efficiency_1}$$ where we recall that ${\mathcal{V}}_K$ stands for the vertices of the element $K$ and ${\lvert{\mathcal{V}}_K\rvert}$ stands for its cardinality. We shall now bound the right-hand side of . For each $1\leq n\leq N$ and each ${{\bm{a}}}\in {\mathcal{V}^n}$, we introduce the patch residual functional ${{\mathcal{R}}_{{{h\tau}}}^{{{\bm{a}}},n}}\colon L^2(I_n,H^1_0({{{\omega}_{{{\bm{a}}}}}})){\rightarrow}{\mathbb{R}}$ defined by $$\label{eq:patch_residual}
\begin{aligned}
{\langle {{\mathcal{R}}_{{{h\tau}}}^{{{\bm{a}}},n}},{\varphi}\rangle} =\int_{I_n} \big({\Pi_{{{h\tau}}}^{{{\bm{a}}},n}}f - {\partial}_t {{\mathcal{I}}u_{{{h\tau}}}},{\varphi}\big)_{{{{\omega}_{{{\bm{a}}}}}}} -\big(\nabla u_{{{h\tau}}},\nabla {\varphi}\big)_{{{\omega}_{{{\bm{a}}}}}}{\mathrm{d}}t & & & \forall\,{\varphi}\in L^2(I_n;H^1_0({{{\omega}_{{{\bm{a}}}}}})).
\end{aligned}$$
The essential result that forms the starting point for our analysis is the following abstract efficiency result first shown in [@ErnSmearsVohralik2016 [Lemma 8.2][@ErnSmearsVohralik2016 Lemma 8.2]]{}, which is an application of a more general underlying key result in [@ErnSmearsVohralik2016c [Theorem 1.2][@ErnSmearsVohralik2016c Theorem 1.2]]{} concerning the existence of polynomial-degree robust liftings of piecewise polynomial data into discrete subspaces of ${\bm{H}(\operatorname{div})}$, which itself is based on the fundamental results of [@CostabelMcintosh2010 [@BraessPillweinSchoberl2009][@CostabelMcintosh2010; @BraessPillweinSchoberl2009]]{}.
\[lem:flux\_reconstruction\_stability\] Let ${\bm{\sigma}_{{{h\tau}}}^{{{\bm{a}}},n}}$ denote the patch-wise flux reconstructions of Definition [\[def:flux\_construction\_1\]]{}, and let ${{\mathcal{R}}_{{{h\tau}}}^{{{\bm{a}}},n}}$ denote the local patch residual defined by . Then, we have $$\label{eq:flux_reconstruction_stability}
\left(\int_{I_n}{\lVert{\bm{\sigma}_{{{h\tau}}}^{{{\bm{a}}},n}}+ {\psi_{{{\bm{a}}}}}\nabla u_{{{h\tau}}}\rVert}_{{{\omega}_{{{\bm{a}}}}}}^2\,{\mathrm{d}}t\right)^{\frac{1}{2}} \lesssim \sup_{{\varphi}\in {\mathcal{Q}}_{q_n}(I_n;H^1_0({{{\omega}_{{{\bm{a}}}}}}))\setminus\{0\}} \frac{{\langle {{\mathcal{R}}_{{{h\tau}}}^{{{\bm{a}}},n}},{\varphi}\rangle}}{\left(\int_{I_n}{\lVert\nabla {\varphi}\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2\,{\mathrm{d}}t\right)^{\frac{1}{2}}},$$ where ${\mathcal{Q}}_{q_n}(I_n;H^1_0({{{\omega}_{{{\bm{a}}}}}}))$ denotes the space of $H^1_0({{{\omega}_{{{\bm{a}}}}}})$-valued univariate polynomials of degree at most $q_n$ on $I_n$. In particular, the constant in does not depend on the mesh-size, time-step size, spatial and temporal polynomial degrees, or on refinement and coarsening between time-steps.
As explained above in the introduction, our analysis of the efficiency of the equilibrated flux estimator ${\eta_{\mathrm{F},K}^{n}}$ relies on two original ideas. We now detail the first one, which is based on the key observation that the set of test functions appearing in are polynomials with respect to the time variable. Hence, in order to obtain estimates on the efficiency of the estimators with respect to the $X$-norm of the error, we shall show that the set of test functions appearing in can be restricted to functions vanishing at the end-points of the time interval and thereby lying in the test space ${Y_T}$ through a bubble-in-time argument, provided that $h_{{{{\omega}_{{{\bm{a}}}}}}}^2 \lesssim \tau_n$.
We start by defining the space ${H^1_{\dagger}({{{\omega}_{{{\bm{a}}}}}})}$ through $$\label{eq:Hstar}
{H^1_{\dagger}({{{\omega}_{{{\bm{a}}}}}})}\coloneqq \begin{cases}
\{ v \in H^1({{{\omega}_{{{\bm{a}}}}}}), \quad (v,{\psi_{{{\bm{a}}}}})_{{{{\omega}_{{{\bm{a}}}}}}}=0 \} & \text{if }{{\bm{a}}}\in{\mathcal{V}^n_{\mathrm{int}}}, \\
\{ v \in H^1({{{\omega}_{{{\bm{a}}}}}}), \quad {\left. v\right|_{{\partial}{{{\omega}_{{{\bm{a}}}}}}\cap {\partial{\Omega}}}} =0 \} & \text{if }{{\bm{a}}}\in{\mathcal{V}^n_{\mathrm{ext}}}.
\end{cases}$$ Recall that the dual norm ${\lVert\cdot\rVert}_{{[{H^1_{\dagger}({{{\omega}_{{{\bm{a}}}}}})}]^\prime}}$ of ${H^1_{\dagger}({{{\omega}_{{{\bm{a}}}}}})}$ is defined by ${\lVert\Phi\rVert}_{{[{H^1_{\dagger}({{{\omega}_{{{\bm{a}}}}}})}]^\prime}}=\sup{\langle \Phi,v \rangle}$, where the supremum is taken among all test functions $v\in {H^1_{\dagger}({{{\omega}_{{{\bm{a}}}}}})}$ such that ${\lVert\nabla v\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}=1$. The motivation for working with the space ${H^1_{\dagger}({{{\omega}_{{{\bm{a}}}}}})}$ is that the ${\psi_{{{\bm{a}}}}}$-weighted mean value of the function $u-{{\mathcal{I}}u_{{{h\tau}}}}$ possesses special properties derived from the numerical scheme; in particular, see Lemma \[lem:time\_dg\_exact\] and the discussion surrounding below.
\[lem:X\_norm\_main\_estimate\] Let ${{\bm{a}}}\in{\mathcal{T}}^n$, $1\leq n \leq N$, and suppose that there exists a constant $\gamma$ such that the patch diameter $h_{{{{\omega}_{{{\bm{a}}}}}}}$ and $\tau_n$ satisfy $ h_{{{{\omega}_{{{\bm{a}}}}}}}^2/\tau_n \leq \gamma_{{{\bm{a}}}} $. Then, $$\label{eq:X_norm_efficiency_main_estimate}
\left(\int_{I_n}{\lVert{\bm{\sigma}_{{{h\tau}}}^{{{\bm{a}}},n}}+ {\psi_{{{\bm{a}}}}}\nabla u_{{{h\tau}}}\rVert}_{{{\omega}_{{{\bm{a}}}}}}^2\,{\mathrm{d}}t\right)^{\frac{1}{2}} \leq C_{\gamma_{{{\bm{a}}}},q_n} \sup_{\substack{{\varphi}\in {\mathcal{Q}}_{q_n+2}(I_n;H^1_0({{{\omega}_{{{\bm{a}}}}}})) \\ \cap H^1_0(I_n;H^1_0({{{\omega}_{{{\bm{a}}}}}}))}} \frac{{\langle {{\mathcal{R}}_{{{h\tau}}}^{{{\bm{a}}},n}},{\varphi}\rangle}}{\left( \int_{I_n}{\lVert{\partial}_t {\varphi}\rVert}_{{[{H^1_{\dagger}({{{\omega}_{{{\bm{a}}}}}})}]^\prime}}^2+{\lVert\nabla {\varphi}\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2\,{\mathrm{d}}t\right)^{\frac{1}{2}}},$$ $$\begin{gathered}
\label{eq:X_norm_efficiency_main_estimate}
\left(\int_{I_n}{\lVert{\bm{\sigma}_{{{h\tau}}}^{{{\bm{a}}},n}}+ {\psi_{{{\bm{a}}}}}\nabla u_{{{h\tau}}}\rVert}_{{{\omega}_{{{\bm{a}}}}}}^2\,{\mathrm{d}}t\right)^{\frac{1}{2}} \\ \leq C_{\gamma_{{{\bm{a}}}},q_n} \sup_{\substack{{\varphi}\in {\mathcal{Q}}_{q_n+2}(I_n;H^1_0({{{\omega}_{{{\bm{a}}}}}})) \\ \cap H^1_0(I_n;H^1_0({{{\omega}_{{{\bm{a}}}}}}))}} \frac{{\langle {{\mathcal{R}}_{{{h\tau}}}^{{{\bm{a}}},n}},{\varphi}\rangle}}{\left( \int_{I_n}{\lVert{\partial}_t {\varphi}\rVert}_{{[{H^1_{\dagger}({{{\omega}_{{{\bm{a}}}}}})}]^\prime}}^2+{\lVert\nabla {\varphi}\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2\,{\mathrm{d}}t\right)^{\frac{1}{2}}},\end{gathered}$$ where $H^1_0(I_n;H^1_0({{{\omega}_{{{\bm{a}}}}}}))$ denotes the space of functions in $H^1(I_n;H^1_0({{{\omega}_{{{\bm{a}}}}}}))$ that vanish at both endpoints $t_{n-1}$ and $t_n$ of the time interval $I_n$. In particular, the constant $C_{\gamma_{{{\bm{a}}}},q_n}$ in satisfies $C_{\gamma_{{{\bm{a}}}},q_n} \lesssim (q_n+1)^{\frac{1}{2}} + \gamma_{{{\bm{a}}}} (q_n+1)^{\frac{5}{2}}$, and may depend on the shape regularity of ${\mathcal{T}}^n$ and ${\widetilde{{\mathcal{T}}^{n}}}$ and on the space dimension ${d}$, but otherwise does not depend on the mesh-size, time-step size, spatial polynomial degrees, or on refinement and coarsening between time-steps.
The starting point for the proof is Lemma \[lem:flux\_reconstruction\_stability\]. Keeping in mind the right-hand side of , for each ${\varphi}\in {\mathcal{Q}}_{q_n}(I_n;H^1_0({{{\omega}_{{{\bm{a}}}}}}))$, we shall construct a new function ${\varphi}_*\in {\mathcal{Q}}_{q_n+2}(I_n;H^1_0({{{\omega}_{{{\bm{a}}}}}}))$ defined by $${\varphi}_* \coloneqq {\varphi}- {\varphi}(t_{n-1}^+) \frac{(-1)^{q_n+1}}{2}(L^n_{q_n+1}-L^n_{q_n+2}) - {\varphi}(t_n) \frac{1}{2}(L^n_{q_n+1}+L^n_{q_n+2}).$$ It follows from the fact that $L^n_q(t_{n-1})=(-1)^q$ and that $L^n_q(t_n)=1$ for all $q\geq 0$ that ${\varphi}_*(t_{n-1}^+)={\varphi}_*(t_n)=0$ and hence ${\varphi}_* \in H^1_0(I_n;H^1_0({{{\omega}_{{{\bm{a}}}}}}))$. Recalling that the functions ${\Pi_{{{h\tau}}}^{{{\bm{a}}},n}}f$, $ {\partial}_t {{\mathcal{I}}u_{{{h\tau}}}}$, and $\nabla u_{{{h\tau}}}$ appearing in are polynomials of degree at most $q_n$ in time, it also follows from the orthogonality of the Legendre polynomials that $${\langle {{\mathcal{R}}_{{{h\tau}}}^{{{\bm{a}}},n}},{\varphi}_* \rangle} = {\langle {{\mathcal{R}}_{{{h\tau}}}^{{{\bm{a}}},n}},{\varphi}\rangle}.$$ It is then seen that we shall obtain as a result of provided that we can bound $\int_{I_n}{\lVert{\partial}_t {\varphi}_*\rVert}_{{[{H^1_{\dagger}({{{\omega}_{{{\bm{a}}}}}})}]^\prime}}^2+{\lVert\nabla {\varphi}_*\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2\,{\mathrm{d}}t$ in terms of $\int_{I_n} {\lVert\nabla {\varphi}\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2 \,{\mathrm{d}}t$. First, the triangle inequality and the properties of the Legendre polynomials imply that $$\label{eq:X_norm_efficiency_1}
\int_{I_n}{\lVert\nabla{\varphi}_*\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2\,{\mathrm{d}}t \lesssim \int_{I_n} {\lVert\nabla{\varphi}\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2 \,{\mathrm{d}}t + \tfrac{\tau_n}{q_n+1}\left({\lVert\nabla{\varphi}(t_{n-1})\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2+{\lVert\nabla{\varphi}(t_n)\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2\right),$$ where the constant is independent of all other quantities. Now, the key point is that we have the inverse inequality $$\label{eq:discrete_inverse_inequalities}
\max_{t\in I_n}{\lVert\nabla{\varphi}(t)\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2 \lesssim \tfrac{(q_n+1)^2}{\tau_n} \int_{I_n}{\lVert\nabla{\varphi}\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2\,{\mathrm{d}}t,$$ where the constant is independent of all other quantities since ${\varphi}\in {\mathcal{Q}}_{q_n}(I_n;H^1_0({{{\omega}_{{{\bm{a}}}}}}))$ is discrete with respect to time. Note in particular that the inverse inequality is valid even though ${\mathcal{Q}}_{q_n}(I_n;H^1_0({{{\omega}_{{{\bm{a}}}}}}))$ is itself an infinite dimensional space, see Remark \[rem:inverse\_inequality\] below. Therefore, we find from and that $$\int_{I_n}{\lVert\nabla{\varphi}_*\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2\,{\mathrm{d}}t \lesssim (q_n+1) \int_{I_n}{\lVert\nabla{\varphi}\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2\,{\mathrm{d}}t.$$
To bound $\int_{I_n} {\lVert{\partial}_t {\varphi}_*\rVert}^2_{{[{H^1_{\dagger}({{{\omega}_{{{\bm{a}}}}}})}]^\prime}}{\mathrm{d}}t$, we recall that ${\varphi}_*(t) \in H^1_0({{{\omega}_{{{\bm{a}}}}}})$ for all $t\in I_n$, and therefore satisfies the Poincaré inequality ${\lVert{\varphi}_*(t)\rVert}_{{{{\omega}_{{{\bm{a}}}}}}} \lesssim h_{{{{\omega}_{{{\bm{a}}}}}}} {\lVert\nabla {\varphi}_*(t)\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}$ for all $t\in I_n$. Furthermore, we also have a similar Poincaré inequality for all test functions $v\in {H^1_{\dagger}({{{\omega}_{{{\bm{a}}}}}})}$. Therefore, we find that ${\lVert{\varphi}_*(t)\rVert}_{{[{H^1_{\dagger}({{{\omega}_{{{\bm{a}}}}}})}]^\prime}} \lesssim h_{{{{\omega}_{{{\bm{a}}}}}}}^2 {\lVert\nabla {\varphi}_*(t)\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}$, for all $t\in I_n$. Thus, we obtain, using an inverse inequality in time, $$\begin{gathered}
\int_{I_n}{\lVert{\partial}_t{\varphi}_*\rVert}_{{[{H^1_{\dagger}({{{\omega}_{{{\bm{a}}}}}})}]^\prime}}^2\,{\mathrm{d}}t \lesssim \tfrac{(q_n+1)^4}{\tau_n^2}\int_{I_n}{\lVert{\varphi}_*\rVert}_{{[{H^1_{\dagger}({{{\omega}_{{{\bm{a}}}}}})}]^\prime}}^2 \,{\mathrm{d}}t
\\ \lesssim \tfrac{(q_n+1)^4 h_{{{{\omega}_{{{\bm{a}}}}}}}^4 }{\tau_n^2} \int_{I_n}{\lVert\nabla {\varphi}_{*}\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2\,{\mathrm{d}}t
\lesssim \gamma_{{{\bm{a}}}}^2 (q_n+1)^5 \int_{I_n}{\lVert\nabla {\varphi}\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2\,{\mathrm{d}}t,
$$ where we have used the hypothesis that $ h_{{{{\omega}_{{{\bm{a}}}}}}}^2/ \tau_n\leq \gamma_{{{\bm{a}}}}$ in the last inequality. Hence, we have shown that $$\label{eq:X_norm_efficiency_2}
\int_{I_n} {\lVert{\partial}_t {\varphi}_*\rVert}_{{[{H^1_{\dagger}({{{\omega}_{{{\bm{a}}}}}})}]^\prime}}^2 + {\lVert\nabla {\varphi}_*\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2 \,{\mathrm{d}}t \leq C^2_{\gamma_{{{\bm{a}}}},q_n} \int_{I_n}{\lVert\nabla {\varphi}\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2\,{\mathrm{d}}t,$$ where the constant $C_{\gamma_{{{\bm{a}}}},q_n} \lesssim (q_n+1)^{\frac{1}{2}} + \gamma_{{{\bm{a}}}} (q_n+1)^{\frac{5}{2}}$. The bound then follows from and the identity ${\langle {{\mathcal{R}}_{{{h\tau}}}^{{{\bm{a}}},n}},{\varphi}\rangle}={\langle {{\mathcal{R}}_{{{h\tau}}}^{{{\bm{a}}},n}},{\varphi}_* \rangle}$ given above.
\[rem:inverse\_inequality\] The proof of the inverse inequalities appearing above in can be found simply by expanding the function ${\varphi}$ in any orthogonal basis $\{ \psi_k \}_{k=1}^{\infty}$ of $H^1_0({{{\omega}_{{{\bm{a}}}}}})$ as ${\varphi}(t) = \sum_{k=1}^\infty c_k(t) \psi_k$, where the coefficient functions $c_k$ are real-valued polynomials of degree at most $q_n$, for all $k\geq 1$, and then by applying coefficient-wise known inverse inequalities for real-valued functions.
Lemma \[lem:X\_norm\_main\_estimate\] constitutes the first step towards the local lower bound . In particular, we see that the test functions in are bounded in $H^1(I_n;{[{H^1_{\dagger}({{{\omega}_{{{\bm{a}}}}}})}]^\prime})$ norm. In order to exploit this property, we use a second key idea for our analysis, which is to employ the following special property of the time-discretization scheme. Together, these two ingredients allows us to obtain the lower bounds assuming only that $h^2 \lesssim \tau$, rather than the stronger requirements used in [@Picasso1998 [@Verfurth1998][@Picasso1998; @Verfurth1998]]{}.
\[lem:time\_dg\_exact\] For each $1\leq n\leq N$ and each interior vertex ${{\bm{a}}}\in{\mathcal{V}^n_{\mathrm{int}}}$, the functions ${{\mathcal{I}}u_{{{h\tau}}}}$ and $u_{{{h\tau}}}$ satisfy $$\label{eq:scheme_pointwise_identity}
\begin{aligned}
{\langle {\partial}_t{{\mathcal{I}}u_{{{h\tau}}}},{\psi_{{{\bm{a}}}}}\rangle} + (\nabla u_{{{h\tau}}},\nabla {\psi_{{{\bm{a}}}}})= ({\Pi_{{{h\tau}}}^{{{\bm{a}}},n}}f,{\psi_{{{\bm{a}}}}}) & & &\text{pointwise in } I_n,
\end{aligned}$$ where ${\Pi_{{{h\tau}}}^{{{\bm{a}}},n}}f$ was defined in section \[sec:data\_approximation\].
Since ${{\bm{a}}}\in{\mathcal{V}^n_{\mathrm{int}}}$, it follows that $\phi {\psi_{{{\bm{a}}}}}\in {\mathcal{Q}}_{q_n}(I_n;{V^n})$ for any polynomial $\phi$ in time of degree at most $q_n$ over $I_n$. Therefore, the numerical scheme implies that, for any real-valued polynomial $\phi$ in time of degree at most $q_n$, $$\int_{I_n}\phi \left[(f,{\psi_{{{\bm{a}}}}})-({\partial}_t {{\mathcal{I}}u_{{{h\tau}}}},{\psi_{{{\bm{a}}}}})-(\nabla u_{{{h\tau}}},\nabla {\psi_{{{\bm{a}}}}})\right]{\mathrm{d}}t =0.$$ Furthermore, the definition of ${\Pi_{{{h\tau}}}^{{{\bm{a}}},n}}$ implies that $\int_{I_n} \phi (f,{\psi_{{{\bm{a}}}}}) {\mathrm{d}}t = \int_{I_n} \phi ({\Pi_{{{h\tau}}}^{{{\bm{a}}},n}}f, {\psi_{{{\bm{a}}}}}) {\mathrm{d}}t$ for any real-valued polynomial $\phi$ in time of degree at most $q_n$. Since the function $t\mapsto ({\partial}_t {{\mathcal{I}}u_{{{h\tau}}}}(t),{\psi_{{{\bm{a}}}}})+(\nabla u_{{{h\tau}}}(t),\nabla {\psi_{{{\bm{a}}}}})- ({\Pi_{{{h\tau}}}^{{{\bm{a}}},n}}f(t),{\psi_{{{\bm{a}}}}})$ is a real-valued polynomial of degree at most $q_n$ over $I_n$, it follows that it vanishes everywhere in $I_n$. We therefore obtain .
We now give the proof of the bounds and under the hypothesis stated in Theorem \[thm:X\_norm\_guaranteed\_efficiency\].
#### Proof of the bounds and {#proof-of-the-bounds-and}
The proof consists in bounding the right-hand side of so as to show that, for each ${{\bm{a}}}\in{\mathcal{V}^n}$, we have the bound $$\begin{gathered}
\label{eq:X_norm_local_efficiency_2}
\int_{I_n} {\lVert{\bm{\sigma}_{{{h\tau}}}^{{{\bm{a}}},n}}+ {\psi_{{{\bm{a}}}}}\nabla u_{{{h\tau}}}\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2\,{\mathrm{d}}t
\leq C_{\gamma_{{{\bm{a}}}},q_n}^2 \left\{ \int_{I_n} {\lVert\nabla(u-{{\mathcal{I}}u_{{{h\tau}}}})\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2 + {\lVert\nabla(u-u_{{{h\tau}}})\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2\,{\mathrm{d}}t \right. \\
\left. + \int_{I_n}{\lVert f - {\Pi_{{{h\tau}}}^{{{\bm{a}}},n}}f\rVert}_{H^{-1}({{{\omega}_{{{\bm{a}}}}}})}^2 \,{\mathrm{d}}t \right\},\end{gathered}$$ where $C_{\gamma_{{{\bm{a}}}},q_n} \lesssim (q_n+1)^{\frac{1}{2}} + \gamma (q_n+1)^{\frac{5}{2}}$. Then, once is known, it is then straightforward to show and from .
To show , we will treat first the more difficult case where ${{\bm{a}}}\in {\mathcal{V}^n_{\mathrm{int}}}$ is an interior node. It will be convenient to denote ${\overline{\psi}_{{{\bm{a}}}}}\coloneqq {\psi_{{{\bm{a}}}}}/ {\lVert{\psi_{{{\bm{a}}}}}\rVert}_{L^1({{{\omega}_{{{\bm{a}}}}}})}$ the renormalized hat function associated with ${{\bm{a}}}$. Let ${\varphi}\in {\mathcal{Q}}_{q_n+2}(I_n;H^1_0({{{\omega}_{{{\bm{a}}}}}}))\cap H^1_0(I_n;H^1_0({{{\omega}_{{{\bm{a}}}}}}))$ be a fixed but arbitrary test function, such that $\int_{I_n}{\lVert{\partial}_t {\varphi}\rVert}_{{[{H^1_{\dagger}({{{\omega}_{{{\bm{a}}}}}})}]^\prime}}^2+{\lVert\nabla {\varphi}\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2\,{\mathrm{d}}t=1$. It follows that the zero-extension of ${\varphi}$ to ${\Omega}\times (0,T)$ belongs to ${Y_T}$, and therefore, we may use the weak formulation in the definition of ${{\mathcal{R}}_{{{h\tau}}}^{{{\bm{a}}},n}}$ from to find that $$\label{eq:patch_residual_1}
{\langle {{\mathcal{R}}_{{{h\tau}}}^{{{\bm{a}}},n}},{\varphi}\rangle} = \int_{I_n}-( u-{{\mathcal{I}}u_{{{h\tau}}}}, {\partial}_t {\varphi})_{{{{\omega}_{{{\bm{a}}}}}}} + (\nabla(u-u_{{{h\tau}}}),\nabla {\varphi})_{{{{\omega}_{{{\bm{a}}}}}}} + ({\Pi_{{{h\tau}}}^{{{\bm{a}}},n}}f - f,{\varphi})_{{{{\omega}_{{{\bm{a}}}}}}}\,{\mathrm{d}}t.$$ Note that, in general, $u-{{\mathcal{I}}u_{{{h\tau}}}}$ fails to belong to ${H^1_{\dagger}({{{\omega}_{{{\bm{a}}}}}})}$ when ${{\bm{a}}}\in{\mathcal{V}^n_{\mathrm{int}}}$ is an interior node because we can not generally guarantee that $(u-{{\mathcal{I}}u_{{{h\tau}}}},{\psi_{{{\bm{a}}}}})_{{{{\omega}_{{{\bm{a}}}}}}}=0$ a.e. in time; thus, ${\lvert(u-{{\mathcal{I}}u_{{{h\tau}}}},{\partial}_t {\varphi})_{{{{\omega}_{{{\bm{a}}}}}}}\rvert}\not\leq {\lVert\nabla (u-{{\mathcal{I}}u_{{{h\tau}}}})\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}{\lVert{\partial}_t {\varphi}\rVert}_{{[{H^1_{\dagger}({{{\omega}_{{{\bm{a}}}}}})}]^\prime}}$ in general. To overcome this obstacle, we introduce the auxiliary function $$\label{eq:auxiliary_error_function}
e_{{{\bm{a}}}} \coloneqq u-{{\mathcal{I}}u_{{{h\tau}}}}- (u-{{\mathcal{I}}u_{{{h\tau}}}},{\overline{\psi}_{{{\bm{a}}}}})_{{{{\omega}_{{{\bm{a}}}}}}},$$ that is, we subtract the ${\overline{\psi}_{{{\bm{a}}}}}$-weighted average of $u-{{\mathcal{I}}u_{{{h\tau}}}}$ from $u-{{\mathcal{I}}u_{{{h\tau}}}}$. It follows from the definition that $e_{{{\bm{a}}}}(t)\in {H^1_{\dagger}({{{\omega}_{{{\bm{a}}}}}})}$ and that ${\lVert\nabla e_{{{\bm{a}}}}(t)\rVert}_{{{{\omega}_{{{\bm{a}}}}}}} = {\lVert\nabla(u-{{\mathcal{I}}u_{{{h\tau}}}})(t)\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}$ for almost all $t\in I_n$. We now show how to reformulate the patch residual ${\langle {{\mathcal{R}}_{{{h\tau}}}^{{{\bm{a}}},n}},{\varphi}\rangle}$ in terms of the auxiliary function $e_{{{\bm{a}}}}$. First, we may choose the test function $ {\overline{\psi}_{{{\bm{a}}}}}({\varphi},1)_{{{{\omega}_{{{\bm{a}}}}}}} \in {Y_T}$ in , and use Fubini’s theorem and linearity of integration to find that $$\label{eq:mean_value_term_1}
\begin{split}
\int_{I_n} - ( (u,{\overline{\psi}_{{{\bm{a}}}}})_{{{{\omega}_{{{\bm{a}}}}}}} , {\partial}_t {\varphi})_{{{{\omega}_{{{\bm{a}}}}}}} {\mathrm{d}}t &= \int_{I_n} - {\langle u,{\partial}_t({\overline{\psi}_{{{\bm{a}}}}}({\varphi},1)_{{{{\omega}_{{{\bm{a}}}}}}}) \rangle} {\mathrm{d}}t
\\ & = \int_{I_n} (f, {\overline{\psi}_{{{\bm{a}}}}})_{{{{\omega}_{{{\bm{a}}}}}}}({\varphi},1)_{{{{\omega}_{{{\bm{a}}}}}}}- (\nabla u,\nabla {\overline{\psi}_{{{\bm{a}}}}})_{{{{\omega}_{{{\bm{a}}}}}}}({\varphi},1)_{{{{\omega}_{{{\bm{a}}}}}}} {\mathrm{d}}t.
\end{split}$$ Next, we multiply by $({\varphi},1)_{{{{\omega}_{{{\bm{a}}}}}}}$ and integrate by parts over $I_n$ and obtain $$\label{eq:mean_value_term_2}
\int_{I_n} - ( ({{\mathcal{I}}u_{{{h\tau}}}},{\overline{\psi}_{{{\bm{a}}}}})_{{{{\omega}_{{{\bm{a}}}}}}}, {\partial}_t {\varphi})_{{{{\omega}_{{{\bm{a}}}}}}} {\mathrm{d}}t = \int_{I_n} ({\Pi_{{{h\tau}}}^{{{\bm{a}}},n}}f,{\overline{\psi}_{{{\bm{a}}}}})_{{{{\omega}_{{{\bm{a}}}}}}} ({\varphi},1)_{{{{\omega}_{{{\bm{a}}}}}}} - (\nabla u_{{{h\tau}}},\nabla{\overline{\psi}_{{{\bm{a}}}}})_{{{{\omega}_{{{\bm{a}}}}}}} ({\varphi},1)_{{{{\omega}_{{{\bm{a}}}}}}} {\mathrm{d}}t.$$ The combination of with and shows that ${\langle {{\mathcal{R}}_{{{h\tau}}}^{{{\bm{a}}},n}},{\varphi}\rangle}=\sum_{i=1}^5 R_i$, where the quantities $R_i$, $1\leq i \leq 5$, are defined by $$\begin{gathered}
R_1 \coloneqq \int_{I_n} - ( e_{{{\bm{a}}}} , {\partial}_t {\varphi})_{{{{\omega}_{{{\bm{a}}}}}}}\,{\mathrm{d}}t ,
\\ R_2 \coloneqq \int_{I_n} (\nabla(u-u_{{{h\tau}}}),\nabla{\varphi})_{{{{\omega}_{{{\bm{a}}}}}}}\,{\mathrm{d}}t,
\qquad R_3 \coloneqq - \int_{I_n} (\nabla(u-u_{{{h\tau}}}),\nabla {\overline{\psi}_{{{\bm{a}}}}})_{{{{\omega}_{{{\bm{a}}}}}}}({\varphi},1)_{{{{\omega}_{{{\bm{a}}}}}}}\,{\mathrm{d}}t,
\\ R_4 \coloneqq \int_{I_n} (f- {\Pi_{{{h\tau}}}^{{{\bm{a}}},n}}f,{\overline{\psi}_{{{\bm{a}}}}})_{{{{\omega}_{{{\bm{a}}}}}}}({\varphi},1)_{{{{\omega}_{{{\bm{a}}}}}}}\,{\mathrm{d}}t,
\qquad R_5 \coloneqq - \int_{I_n} ( f - {\Pi_{{{h\tau}}}^{{{\bm{a}}},n}}f,{\varphi})_{{{{\omega}_{{{\bm{a}}}}}}}\,{\mathrm{d}}t .\end{gathered}$$ Using the fact that $\int_{I_n} {\lVert{\partial}_t {\varphi}\rVert}^2_{{[{H^1_{\dagger}({{{\omega}_{{{\bm{a}}}}}})}]^\prime}}\,{\mathrm{d}}t\leq 1$, where we recall that ${H^1_{\dagger}({{{\omega}_{{{\bm{a}}}}}})}$ is defined in , and that ${\lVert\nabla e_{{{\bm{a}}}}\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}={\lVert\nabla(u-{{\mathcal{I}}u_{{{h\tau}}}})\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}$, we find that ${\lvertR_1\rvert}^2\leq \int_{I_n}{\lVert\nabla(u-{{\mathcal{I}}u_{{{h\tau}}}})\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2\,{\mathrm{d}}t$. Next, we find that ${\lvertR_2\rvert}^2 \leq \int_{I_n}{\lVert\nabla(u-u_{{{h\tau}}})\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2\,{\mathrm{d}}t$. To bound $R_3$ and $R_4$, we apply the Cauchy–Schwarz inequality and use the Poincaré inequality on $H^1_0({{{\omega}_{{{\bm{a}}}}}})$ to obtain $${\lvertR_3\rvert}^2 + {\lvertR_4\rvert}^2 \lesssim \int_{I_n} \frac{h_{{{{\omega}_{{{\bm{a}}}}}}}^2 {\lvert{{{\omega}_{{{\bm{a}}}}}}\rvert} {\lVert\nabla {\psi_{{{\bm{a}}}}}\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2}{{\lVert{\psi_{{{\bm{a}}}}}\rVert}_{L^1({{{\omega}_{{{\bm{a}}}}}})}^2} \left[ {\lVert\nabla(u-u_{{{h\tau}}})\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2+{\lVertf-{\Pi_{{{h\tau}}}^{{{\bm{a}}},n}}f\rVert}_{H^{-1}({{{\omega}_{{{\bm{a}}}}}})}^2\right] \,{\mathrm{d}}t,$$ where ${\lvert{{{\omega}_{{{\bm{a}}}}}}\rvert}$ denotes the measure of ${{{\omega}_{{{\bm{a}}}}}}$. Since there is a constant depending only on the shape-regularity of the elements of the patch ${{{\omega}_{{{\bm{a}}}}}}$ such that $h_{{{{\omega}_{{{\bm{a}}}}}}} {\lvert{{{\omega}_{{{\bm{a}}}}}}\rvert}^{1/2}{\lVert\nabla {\psi_{{{\bm{a}}}}}\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}\lesssim {\lVert{\psi_{{{\bm{a}}}}}\rVert}_{L^1({{{\omega}_{{{\bm{a}}}}}})}$, we find that ${\lvertR_3\rvert}^2 +{\lvertR_4\rvert}^2 \lesssim \int_{I_n} {\lVert\nabla(u-u_{{{h\tau}}})\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}^2+{\lVertf-{\Pi_{{{h\tau}}}^{{{\bm{a}}},n}}f\rVert}_{H^{-1}({{{\omega}_{{{\bm{a}}}}}})}^2{\mathrm{d}}t$. Finally, it is straightforward to show that ${\lvertR_5\rvert}^2 \leq \int_{I_n} {\lVert f -{\Pi_{{{h\tau}}}^{{{\bm{a}}},n}}f\rVert}_{H^{-1}({{{\omega}_{{{\bm{a}}}}}})}^2\,{\mathrm{d}}t $. Therefore, the above bounds on the quantities $R_i$ imply for the case where ${{\bm{a}}}\in{\mathcal{V}^n_{\mathrm{int}}}$ is an interior vertex.
The analogous result for the case where ${{\bm{a}}}\in{\mathcal{V}^n_{\mathrm{ext}}}$ is a boundary vertex poses fewer difficulties than the case of interior vertices, owing to the fact that $u-{{\mathcal{I}}u_{{{h\tau}}}}\in {H^1_{\dagger}({{{\omega}_{{{\bm{a}}}}}})}$ for a.e. $t\in I_n$, since $u$ and ${{\mathcal{I}}u_{{{h\tau}}}}$ are both in $X$ and therefore have vanishing trace on ${\partial}{{{\omega}_{{{\bm{a}}}}}}\cap {\partial{\Omega}}$.
Using the triangle inequality ${\lVert\nabla(u-{{\mathcal{I}}u_{{{h\tau}}}})\rVert}_{{{{\omega}_{{{\bm{a}}}}}}} \leq {\lVert\nabla(u-u_{{{h\tau}}})\rVert}_{{{{\omega}_{{{\bm{a}}}}}}} + {\lVert\nabla(u_{{{h\tau}}}-{{\mathcal{I}}u_{{{h\tau}}}})\rVert}_{{{{\omega}_{{{\bm{a}}}}}}}$, it is then straightforward to obtain and from and .
Funding {#funding .unnumbered}
=======
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No 647134 GATIPOR).
[^1]: Email:alexandre.ern@enpc.fr
[^2]: Corresponding author. Email:iain.smears@inria.fr
[^3]: Email:martin.vorhalik@inria.fr
[^4]: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No 647134 GATIPOR).
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abstract: 'The limited capacity of distribution grids for hosting renewable generation is one of the main challenges towards the energy transition. Local energy markets, enabling direct exchange of energy between prosumers, help to integrate the growing number of residential photovoltaic panels by scheduling flexible demand for balancing renewable energy locally. Nevertheless, existing scheduling mechanisms do not take into account the phases to which households are connected, increasing network unbalance and favoring bigger voltage rises/drops and higher losses. In this paper, we reduce network unbalance by leveraging market transactions information to dynamically allocate houses to phases using solid state switches. We propose cost effective mechanisms for the selection of households to switch and for their optimal allocation to phases. Using load flow analysis we show that only 6% of houses in our case studies need to be equipped with dynamic switches to counteract the negative impact of local energy markets while maintaining all the benefits. Combining local energy markets and dynamic phase switching we improve both overall load balancing and network unbalance, effectively augmenting DER hosting capacity of distribution grids.'
author:
- José Horta
- Daniel Kofman
- David Menga
- Mathieu Caujolle
bibliography:
- 'references.bib'
title: Augmenting DER hosting capacity of distribution grids through local energy markets and dynamic phase switching
---
<ccs2012> <concept> <concept\_id>10010405.10010481.10010482.10010486</concept\_id> <concept\_desc>Applied computing Command and control</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010583.10010662.10010668.10010672</concept\_id> <concept\_desc>Hardware Smart grid</concept\_desc> <concept\_significance>300</concept\_significance> </concept> <concept> <concept\_id>10010147.10010178.10010199.10010202</concept\_id> <concept\_desc>Computing methodologies Multi-agent planning</concept\_desc> <concept\_significance>100</concept\_significance> </concept> </ccs2012>
|
---
abstract: 'Quasiparticle spectra of potentially half-metallic Co$_2$MnSi and Co$_2$FeSi Heusler compounds have been calculated within the one-shot $GW$ approximation in an all-electron framework without adjustable parameters. For Co$_2$FeSi the many-body corrections are crucial: a pseudogap opens and good agreement of the magnetic moment with experiment is obtained. Otherwise, however, the changes with respect to the density-functional-theory starting point are moderate. For both cases we find that photoemission and x-ray absorption spectra are well described by the calculations. By comparison with the $GW$ density of states, we conclude that the Kohn-Sham eigenvalue spectrum provides a reasonable approximation for the quasiparticle spectrum of the Heusler compounds considered in this work.'
author:
- Markus Meinert
- Christoph Friedrich
- Günter Reiss
- Stefan Blügel
title: '$GW$ study of the half-metallic Heusler compounds Co$_2$MnSi and Co$_2$FeSi'
---
Introduction
============
Heusler compounds [@Heusler] attract ever-growing experimental and theoretical attention, largely because a vast number of such compounds have been predicted to be half-metallic ferromagnets (HMF), i.e., the compounds behave like a metal for one spin channel and like a semiconductor for the other.[@Groot83; @Kuebler83; @Ishida95; @Galanakis02] Peculiar electronic transport properties are expected from such materials, e.g., huge magnetoresistive effects in giant and tunnel magnetoresistive (GMR, TMR) devices.
Heusler compounds are ternary intermetallic compounds with the general chemical formula $X_2YZ$, where $X$ and $Y$ are transition-metal atoms and $Z$ is a main group element. They form the cubic L2$_1$ structure (space group Fm$\bar{3}$m) with a four atom basis. The half-metals among the Heusler compounds follow the Slater-Pauling rule, which connects the magnetic moment per formula unit $m$ and the number of valence electrons $N_\mathrm{V}$ via [@Galanakis02] $$m = N_\mathrm{V} - 24.$$
Most theoretical studies of these materials have been based on density functional theory [@HK; @KS] (DFT) in the Kohn-Sham formalism so far,[@Graf11] which gives access to ground-state properties, such as the total energy, atomic forces, magnetic moments etc. It relies on a mapping of the real system onto a fictitious system of noninteracting electrons moving in an effective potential. The half-metallic nature found experimentally for some Heusler compounds is predicted correctly by DFT, together with a quantitative explanation of the Slater-Pauling behavior. However, it is questionable whether the Kohn-Sham eigenvalue spectrum can be taken as the excitation spectrum of the real system. Strictly speaking, there is no theoretical justification for such an interpretation. In fact, while the band structure often resembles the experimentally determined dispersions remarkably well, there are important quantitative discrepancies. For example, the fundamental band gaps of semiconductors and insulators are usually underestimated by a factor of 2 or more. This raises the question if the half-metal band gap is also subject to this underestimation. Studies on Co$_2$MnSi indicate that this is not so: the experimental gap is not larger than about 1eV as inferred from tunnel spectroscopy and x-ray absorption experiments.[@Kubota09; @Sakuraba10; @Kallmayer09] This value is very close to the calculated Kohn-Sham gap. The bandwidth of metals and the exchange splitting of ferromagnets are two other important spectral quantities which are often unsatisfactorily described by Kohn-Sham DFT.[@Yamasaki03; @Schilfgaarde06]
There are several approaches that allow to go beyond Kohn-Sham DFT in this respect. For example, DFT$+U$ and DFT+DMFT (dynamical mean field theory in a correlated subspace) employ an effective, partially screened interaction parameter, the Hubbard $U$ parameter, that acts between electrons in the subspace of localized states while the rest is treated on the level of DFT.[@Minar11] The $U$ parameter itself is taken in its static limit. Dynamical screening effects of the itinerant electrons are thus neglected. Furthermore, the Hubbard $U$ parameter is usually taken as an empirical parameter that is fitted to experiment, and the artificial separation into localized and itinerant electrons requires a double-counting correction, which is not uniquely defined. LDA+DMFT calculations on half-metals suggest the presence of nonquasiparticle states inside the half-metal gap, which may destroy the half-metallic character of a material.[@Chioncel03; @Chioncel08; @Chioncel09]
Another method that allows to obtain physical electron addition and removal energies is the $GW$ approximation for the electronic self-energy within many-body perturbation theory.[@Hedin65; @Aryasetiawan98] In contrast to DFT, the $GW$ method is designed for spectral properties, such as the band structure. Typically, it opens the gap of semiconductors and insulators and gives good agreement with experiments. We apply this method to Co$_2$MnSi and Co$_2$FeSi, two prototypical and potentially half-metallic Heusler compounds, to study the effect of many-body corrections on their band structures. In particular, we will answer the question if the $GW$ approximation increases the half-metal band gap as in the case of semiconductors and insulators or not. As already mentioned above, an increase may worsen the good agreement to experiment achieved by Kohn-Sham DFT.
Co$_2$MnSi and Co$_2$FeSi are particularly interesting because of their large magnetic moments and Curie temperatures. They are known to form the L2$_1$ structure with a low degree of chemical disorder;[@Balke06] this allows accurate comparison between experiment and theory. The half-metallic character and integer magnetic moment of Co$_2$MnSi is already predicted by DFT.[@Ishida95] For Co$_2$FeSi, DFT calculations predict a significantly reduced magnetic moment with respect to experiment and the Slater-Pauling value.[@Balke06] DFT$+U$ and DFT+DMFT find a magnetic moment in accordance with the Slater-Pauling rule and experiment with $U$ parameters of 1.8 and 3eV, respectively.[@Balke06; @Chadov09] However, DFT$+U$ deteriorates the spectral properties of Co$_2$FeSi compared to conventional DFT calculations.[@Meinert12] It is the aim of this work to investigate to which extent many-body corrections within the $GW$ method modify or confirm the predictions made by DFT calculations and, in particular, whether the $GW$ approximation is able to rectify the magnetic moment of Co$_2$FeSi without deteriorating the spectral properties.
Method
======
In this work we present one-shot $GW$ calculations, which yield the quasiparticle energies $E_{n \mathbf{k}}^\sigma$ as corrections on the Kohn-Sham energies $\epsilon_{n \mathbf{k}}^\sigma$, $$\label{qpe}
E_{n \mathbf{k}}^\sigma = \epsilon_{n \mathbf{k}}^\sigma + \left\langle \phi_{n \mathbf{k}}^\sigma \left| \Sigma^\sigma_\mathrm{xc}(E_{n \mathbf{k}}^\sigma) - v_\mathrm{xc}^\sigma \right| \phi_{n \mathbf{k}}^\sigma \right\rangle,$$ where $\phi_{n \mathbf{k}}^\sigma$ are the Kohn-Sham wavefunctions and $n$, $\mathbf{k}$ and $\sigma$ are the band index, Bloch vector, and electron spin, respectively. The quasiparticle correction contains the exchange-correlation potential $v_\mathrm{xc}^\sigma$, for which we employ the PBE functional,[@PBE] and the $GW$ self-energy operator, which is given in formal notation by $\Sigma_\mathrm{xc}^\sigma = iG^\sigma W$,[@Hedin65] where $G^\sigma$ and $W$ are the Kohn-Sham Green function and screened Coulomb potential, respectively. The latter is approximated by the random-phase approximation $W=v(1-vP)^{-1}$ with the polarization function $P = -i \sum_\sigma G^\sigma G^\sigma$ and the bare Coulomb interaction $v$. Notably, $W$ does not depend on spin: quasiparticles of both spin directions interact via the same screened potential.
We use the <span style="font-variant:small-caps;">fleur</span> [@fleur] and <span style="font-variant:small-caps;">spex</span> [@Friedrich10] programs for the DFT and $GW$ calculations, respectively. These codes are based on the highly precise all-electron full-potential linearized augmented-plane-wave (FLAPW) method. Transition-metal 3*s*, 3*p* and Si 2*s*, 2*p* semicore states are treated with local orbitals, although their effect on the spectra is small. The muffin-tin radii are set to 2.25 and 2.31bohr for the transition-metal atoms and Si, respectively. We employ plane-wave and angular momentum cutoff parameters of $k_\mathrm{max} = 4.0\,\mathrm{bohr}^{-1}$ and $l_\mathrm{max} = 8$. The DFT calculations are performed on 256 $\mathbf{k}$ points in the irreducible wedge to obtain a reliable starting point.
The $GW$ calculations are performed with a $10 \times 10 \times 10$ $\mathbf{k}$-point mesh that contains 47 points in the irreducible wedge with cutoff parameters for the mixed product basis $L_\mathrm{max} = 4$ and $G_\mathrm{max}' = 3.5\,\mathrm{bohr}^{-1}$, and an additional cutoff $\sqrt{4\pi/v_\mathrm{min}} = 4.5\,\mathrm{bohr}^{-1}$ for the correlation part of the self-energy; see Ref. for details. We find that 50 empty bands are sufficient to converge the quasiparticle spectra to better than 0.05eV. This is also the estimated accuracy of the $\mathbf{k}$-point sampling. The self-energy is evaluated with a contour integration in the complex frequency plane, and Eq. \[qpe\], which is nonlinear in energy, is solved on an energy mesh with spline interpolation between the points.
The densities of states (DOS) curves are obtained with tetrahedron integration and convoluted with a Gaussian of 0.1eV full-width at half-maximum. Binding energies are always taken relative to the corresponding Fermi energy, which is determined by the condition that the DOS integrates to the total number of electrons from $-\infty$ to the Fermi energy. All calculations are based on the experimental lattice constant of 5.64Å for both compounds.[@Balke06]
Results
=======
In Fig. \[Fig1\] we present the PBE and $G W$ DOS of Co$_2$MnSi and Co$_2$FeSi. In both cases, the main effect of the quasiparticle corrections are downshifts of the Si *s* states (between $-9$ and $-12$eV) by 0.9eV and the hybrid *p-d* states (between $-4$ and $-8$eV) by 0.8eV-0.5eV—see, e.g., Ref. for partial DOS plots. Additionally, the exchange splitting of these states is reduced.
![\[Fig1\] Kohn-Sham and $GW$ DOS of Co$_2$MnSi and Co$_2$FeSi.](Fig1){width="8.6cm"}
m$^\mathrm{PBE}$ m$^{GW}$ m$^\mathrm{exp}$ $E_{\Gamma \rightarrow \Gamma}^\mathrm{PBE}$ $E_{\Gamma \rightarrow \Gamma}^{GW}$ $E_{\Gamma \rightarrow \mathrm{X}}^\mathrm{PBE}$ $E_{\Gamma \rightarrow \mathrm{X}}^{GW}$ $E_{\downarrow\uparrow}^\mathrm{PBE}$ $E_{\downarrow\uparrow}^{GW}$ $E_{\downarrow\uparrow}^\mathrm{exp}$
------------ ------------------ ---------- ------------------ ---------------------------------------------- -------------------------------------- -------------------------------------------------- ------------------------------------------ --------------------------------------- ------------------------------- ---------------------------------------
Co$_2$MnSi 5.00 5.00 4.97 0.86 0.99 0.82 0.95 0.37 0.17 0.25, 0.35
Co$_2$FeSi 5.52 5.89 5.97 0.94 0.92 – – – – –$^,$
The binding energies of the occupied *d* states of Co$_2$MnSi remain largely unchanged. While the absolute values of the 3*d* quasiparticle energies do change due to the exactly cancelled self-interaction error, the Fermi energy changes likewise so that the difference remains more or less the same. Close to the Fermi energy we find a small increase of the exchange splitting by 0.2eV in Co$_2$MnSi, which places the $GW$ Fermi energy closer to the minority valence band minimum. In addition, the minority gap (given by the $\Gamma \rightarrow \mathrm{X}$ transition) is slightly enhanced from 0.82eV to 0.95eV, and the unoccupied minority *d* states are rigidly pushed up in energy. The only small quasiparticle correction of the minority gap is noteworthy in view of the fact that semiconductor and insulator gaps usually increase considerably (and rightly so) when treated within the $GW$ approximation. Thus, the apprehension that $GW$ might worsen the agreement with experiment is proved wrong with this result. This aspect will be analyzed in more detail in the next section.
A similar effect is encountered for Co$_2$FeSi, but the increase of the exchange splitting and the shift of the unoccupied *d* states are larger than in Co$_2$MnSi. This places the Fermi energy in the middle of a minority pseudogap, which accommodates a light band of Fe $t_{2g}$ character.
![\[Fig2\] Quasiparticle shifts as function of the Kohn-Sham energy.](Fig2){width="8.6cm"}
The quasiparticle shifts are displayed in Fig. \[Fig2\]. We see that the *d* states are pushed up in energy; the occupied states move closer to $E_\mathrm{F}$ and the unoccupied states away from it. Also the increase of the exchange splitting around the Fermi energy becomes visible. For Co$_2$FeSi, the states close to the Kohn-Sham Fermi energy are pushed up in energy by as much as 0.85eV. These are mostly of Fe *d* character with 25 - 50% admixture of Co *d* character.
Table \[Tab1\] compares the magnetic moments, the minority $\Gamma \rightarrow \Gamma$ and $\Gamma \rightarrow \mathrm{X}$ transition energies, and the minority spin flip gaps from the Kohn-Sham and quasiparticle calculations and from experiments. The magnetic moment of Co$_2$MnSi is the same in PBE and $GW$ and matches the experimental value very well.[@Balke06] With the Fermi energy located in the pseudogap, the magnetic moment of Co$_2$FeSi is increased from 5.52$\mu_\mathrm{B}$/f.u. to 5.89$\mu_\mathrm{B}$/f.u., improving the agreement with the experimental value of 5.97$\mu_\mathrm{B}$/f.u. considerably.[@Balke06] Hence, the one-shot $GW$ approach manages to correct the magnetic moment. We note that the orbital magnetic moment [@Chadov09; @Meinert12] is not taken into account in our calculations.
The minority spin flip gap, i.e., the energy required to promote an electron from the minority valence band maximum to a majority state at the Fermi energy, is nonzero for Co$_2$MnSi but zero for Co$_2$FeSi due to the minority pseudogap. From tunnel spectroscopy of magnetic tunnel junctions one deduces a spin flip gap for Co$_2$MnSi between 0.25 and 0.35eV.[@Kubota09; @Sakuraba10] Both theoretical values are in fair agreement with these experimental numbers. Co$_2$FeSi does not have a spin flip gap in the experiment,[@Kubota09; @Oogane09] which agrees with both calculations. The minority $\Gamma \rightarrow \Gamma$ transition energy is increased for Co$_2$MnSi by 0.13eV, whereas it essentially remains the same in the case of Co$_2$FeSi. This is very different from the DFT$+U$ ($U=1.8$eV) result, where the $\Gamma \rightarrow \Gamma$ gap of Co$_2$FeSi increases to 1.8eV.[@Balke06]
![\[Fig3\] Comparison of experimental high-energy x-ray photoemission spectra and total DOS of Co$_2$MnSi and Co$_2$FeSi. Experimental data taken from Ref. .](Fig3){width="8.6cm"}
We compare our calculated quasiparticle spectra with experimental high energy x-ray photoemission spectra (HXPS) taken at 7.935keV.[@Fecher07] The full valence band spectra are given in Fig. \[Fig3\], with the features discussed in the following marked by arrows. We compare only peak positions, as a detailed analysis of the peak heights would require the calculation of the transition matrix elements, which is beyond the scope of this paper. For both materials, the main features of the spectra are reproduced by the calculations. The valence band minima of Co$_2$MnSi and Co$_2$FeSi at $-12.4$eV and $-12.8$eV, respectively, are accurately reproduced by the $GW$ calculations. Also the maxima of the emission from the Si *s* states are about correct. The emission maxima of the *p-d* hybrid states are in good agreement with the PBE calculation, whereas $GW$ places them too low in energy compared to experiment, while their onset is described better. It is difficult to assign the individual structures between $-5$eV and $E_\mathrm{F}$ in the experimental spectra to the various peaks in the quasiparticle spectra. However, the overall agreement seems to be reasonable in both cases. The plasmon frequency calculated within the random-phase approximation amounts to 4.7eV and 6.0eV for Co$_2$MnSi and Co$_2$FeSi, respectively, in agreement with previous calculations.[@Picozzi06; @Kumar09] These energies are well within the valence band region, indicating that the measured x-ray photoemission spectra might be affected by plasmon satellites.
![\[Fig4\] Comparison of experimental high-energy x-ray photoemission spectra and total DOS of Co$_2$MnSi and Co$_2$FeSi close to the Fermi energy. The horizontal dashed line denotes the additional background added to the theoretical spectra. Experimental data taken from Ref. .](Fig4){width="8.6cm"}
Additional high-resolution HXPS spectra taken close to the Fermi energy are shown in Fig. \[Fig4\]. Both spectra are well described by the $GW$ calculation. For Co$_2$MnSi, the main feature at $-1.25$eV, arising from a Co-Mn majority *d* state, is placed 0.1eV too high in $GW$. The shoulder at $-0.7$eV arises from a pure Co minority *d* state and is reproduced by $GW$. The structure at $-0.3$eV in the experimental Co$_2$MnSi spectrum might be related to the minority valence band maximum, which appears at about the same energy in PBE and $GW$, see the spin-flip gap values in Table \[Tab1\]. Strangely, the $GW$ DOS does not show a structure at this energy in contrast to the PBE DOS. A comparison with Fig. \[Fig1\] reveals that while the minority DOS drops at $-0.3$eV, the majority DOS happens to increase at exactly the same energy so as to compensate the decrease from the minority states. However, we note that even a small difference in the transition matrix elements of spin-up and spin-down states, which have been neglected in the present work, are expected to produce a structure in the $GW$ spectrum at the correct energy.
The photoemission spectrum of Co$_2$FeSi close to the Fermi energy in Fig. \[Fig4\] is well described by the $GW$ calculation and improves on the PBE result. The features are less pronounced than for Co$_2$MnSi; however, the shoulder at $-1.3$eV and the shape of the spectrum below $-0.6$eV are reproduced.
![\[Fig5\] Top row: experimental spin-averaged x-ray absorption spectra at the Co, Fe, and Mn L$_3$ absorption edges. Bottom row: site-resolved $GW$ *d* electron DOS. The absorption maxima are aligned with the theoretical DOS maxima. Experimental data taken from Refs. and .](Fig5){width="8.6cm"}
Now we turn to the unoccupied states. We focus on the transition-metal *d* states, which can be mapped out element-specifically by soft x-ray absorption spectroscopy at the L$_3$ edges (using the $2p\rightarrow3d$ transitions). In Fig. \[Fig5\] we compare the experimental L$_3$ absorption spectra of Co, Mn, and Fe in Co$_2$MnSi and Co$_2$FeSi with the corresponding $GW$ *d* electron DOS. The absorption maxima are aligned with the DOS maxima. The shapes of the spectra agree with the computed DOS; also, the alignments with the Fermi energy seem reasonable, and the hybridizations are visible in spite of the large lifetime broadening of the spectra. For a detailed comparison of the energy levels one would have to take into account the interaction of the core hole with the photoelectron, i.e., an exciton. This effect is of the order of $0.3\dots0.5$eV, and it affects the final states in dependence on their symmetry and localization.[@Meinert11; @Kallmayer09] A consistent treatment of the optical absorption process would require solving the Bethe-Salpeter equation.[@Laskowski10] Kallmayer *et al.* have taken the exciton binding energy as 0.5eV and assumed the exciton to effect a rigid shift of the unoccupied *d* states towards the Fermi level. With these assumptions, they find that the maximum DOS of Co should be at 0.9eV and 0.6eV above $E_\mathrm{F}$ for Co$_2$MnSi and Co$_2$FeSi, respectively.[@Kallmayer09] These values agree with our calculated $GW$ values within 0.1eV, while the Kohn-Sham spectrum shows a larger discrepancy, see Fig. \[Fig1\].
The unoccupied minority $d$ states of Co$_2$FeSi are mostly shifted rigidly upwards in the $GW$ calculation. It was recently shown, that x-ray magnetic linear dichroism spectra of Co$_2$FeSi can be described by a DFT calculation with the PBE functional plus a rigid shift of the $d$ states.[@Meinert12] We conclude that the spectrum of unoccupied states is described correctly within the $GW$ approximation.
Role of the screening
=====================
In the $GW$ approximation, the screened Coulomb interaction $W(\mathbf{r},\mathbf{r}';\epsilon)$ is the key ingredient. Intuitively, one may expect that the similarity of the PBE and $GW$ results arises from the metallic screening of the majority spin channel. To test this conjecture, we have computed the $GW$ gap of Co$_2$MnSi without metallic screening. We also analyze the importance of local-field effects and briefly discuss results from a one-shot PBE0 hybrid functional scheme.[@Ernzerhof99; @Perdew96]
Neglecting screening altogether, i.e., replacing $W$ by the bare Coulomb interaction $v$, we obtain the (non-selfconsistent) Hartree-Fock gap of 9.65 eV, a gross overestimation. Now we allow for screening effects but suppress the metallic screening. We achieve this by replacing polarization contributions from the majority spin channel, where metallic screening takes place, by the polarization arising from the minority spin electrons, i.e, we use $P = 2 P_\downarrow$. This enforces a long-range $W$ also in the static limit, since the electrons cannot flow freely in the gapped minority channel, which would enable them to screen test charges completely. Employing this artificial semiconductor-like polarization, which exhibits a finite dielectric constant of $\varepsilon_\infty=14$, we obtain only a slightly larger minority energy gap of 0.97eV. On the other hand, setting $P = 2 P_\uparrow$ reduces the gap to 0.86eV. Clearly, the majority electrons generate a more effective screening, but the differences in the gap values are relatively small. Long-range metallic screening does not seem to contribute significantly to the total screening, and screening taking place at short distances seems to be more effective.
To investigate this further, we exclude local-field effects. Local-field effects arise from density fluctuations of a different wave length than their generating fields. These couplings are related to the offdiagonal elements of the polarization matrix $P$ represented in a plane-wave basis. (We employ, instead, a basis of eigenvectors of the Coulomb matrix represented in the mixed product basis, which are, however, reasonably close to plane waves.) Setting these offdiagonal elements to zero implies that the screened interaction $W(\mathbf{r},\mathbf{r}';\epsilon)$ only depends on the difference $\left|\mathbf{r} - \mathbf{r}'\right|$ rather than on the absolute positions $\mathbf{r}$ and $\mathbf{r}'$. This is equivalent to saying that the charge density within the unit cell and its screening are homogeneous.[@Hybertsen86] The resulting energy gap of 1.65eV is nearly twice as large as the Kohn-Sham value. Also, the low-lying *s* and *p-d* states are affected significantly: they shift by about 0.5eV upwards in energy with respect to the PBE result, at odds with experiment. Furthermore, the exchange splitting of the occupied *d* states increases and the minority spin-flip gap vanishes. Thus, the charge inhomogeneity plays a crucial role for the screening properties. We note, that Damewood and Fong found similarly small changes of the half-metallic gaps of zincblende CrAs, MnAs, and MnC in the $GW$ approximation with respect to PBE calculations, and a similar behaviour of the gap size with respect to the local-field effects.[@Damewood11]
In recent years, potentials derived from hybrid functionals, e.g., PBE0,[@Ernzerhof99] have often been used as an approximation to the electronic self-energy. Being nonlocal they fulfill an important condition of the self-energy. Hybrid functionals have been shown to overcome the typical underestimation of band gaps within Kohn-Sham DFT. However, dynamical effects are not taken into account, and screening is only considered in an average way by the parameter that mixes the nonlocal and local parts. In the PBE0 functional this mixing parameter is universally taken to be 0.25.[@Perdew96] Since the $GW$ approximation contains the bare exchange exactly, we can easily calculate a one-shot (non-selfconsistent) PBE0 energy spectrum. We find that while PBE0 gives similar results for the binding energies of the low-lying *s* and *p-d* states as the $GW$ approximation, it completely fails in determining the minority gap, for which it yields 3.03eV. Also, the exchange splitting is strongly overestimated in both cases. Thus, only a dynamical self-energy can simultaneously describe states close to the Fermi energy and far away equally well.
Conclusions
===========
We have presented one-shot $GW$ calculations of the (potentially) half-metallic Heusler compounds Co$_2$MnSi and Co$_2$FeSi. The $GW$ quasiparticle spectra are qualitatively similar to the Kohn-Sham eigenvalue spectra, but show important quantitative differences. In particular, the $GW$ approximation predicts an electronic structure with a minority pseudogap in the case of Co$_2$FeSi, which corrects the magnetic moment per unit cell to nearly an integral number, consistent with available experimental data.
The quasiparticle spectra are in good agreement with photoemission and x-ray absorption data for both compounds. The electronic screening is effective at short distances and charge inhomogeneities play an important role for the screening. Furthermore, it has been shown that the PBE0 hybrid potential cannot be used as an approximate self-energy: it even yields worse results than the local PBE potential.
So far, most theoretical studies of Heusler compounds have been based on the Kohn-Sham band structure. In this work, we have demonstrated that it can, in fact, represent a reasonable approximation to the many-body quasiparticle spectrum, which confirms previous successful calculations of spectral properties of Heusler compounds within DFT.
Financial support by the Deutsche Forschungsgemeinschaft (DFG) is acknowledged. We are grateful for helpful discussions with Ersoy Şaşioğlu and Jan Schmalhorst.
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abstract: 'Observations from the Hinode/XRT telescope and STEREO/SECCHI/EUVI are utilized to study polar coronal jets and plumes. The study focuses on the temporal evolution of both structures and their relationship. The data sample, spanning April 7-8 2007, shows that over $90\%$ of the 28 observed jet events are associated with polar plumes. EUV images (STEREO/SECCHI) show plume haze rising from the location of approximately $70\%$ of the polar X-ray (Hinode/XRT) and EUV jets, with the plume haze appearing minutes to hours after the jet was observed. The remaining jets occurred in areas where plume material previously existed causing a brightness enhancement of the latter after the jet event. Short-lived, jet-like events and small transient bright points are seen (one at a time) at different locations within the base of pre-existing long-lived plumes. X-ray images also show instances (at least two events) of collimated-thin jets rapidly evolving into significantly wider plume-like structures that are followed by the delayed appearance of plume haze in the EUV. These observations provide evidence that X-ray jets are precursors of polar plumes, and in some cases cause brightenings of plumes. Possible mechanisms to explain the observed jet and plume relationship are discussed.'
author:
- 'N.-E. Raouafi, G. J. D. Petrie, A. A. Norton, C. J. Henney, and S. K. Solanki'
title: Evidence for polar jets as precursors of polar plume formation
---
Introduction
============
Recent space missions, such as Hinode [@Kosugi07] and STEREO [@Kaiser08], and ground-based facilities such as SOLIS [@Keller03] provide a set of data unprecedented in quality and cadence. The complementary observations from the different instruments provide the necessary spatial, temporal and temperature coverage to observe the dynamics of jets and polar plumes, helping to form a more complete picture of these structures.
X-ray jets occur almost everywhere in the solar corona [see @Shibata92], in particular in the polar holes. They are characterized by their transient nature and often appear as collimated high-temperature emissive beam guided by open magnetic flux [length of $10^5-10^6$ km and collimated widths of $\sim10^4$ km; see @Cirtain07]. [@Cirtain07] reported that the plasma outflow speeds within X-ray jets range from $\sim100$ to $\sim1000$ km s$^{-1}$ and that Alfvén waves are responsible for the high outflow velocities.
In contrast, polar coronal plumes are observed to be hazy in nature without sharp edges, as seen in extreme ultra-violet (EUV) images from SOHO/EIT [@Boudin95] and STEREO/SECCHI/EUVI [@Howard08]. Plumes are also observed to be significantly wider than X-ray jets [$\sim20-40$ Mm; see @Wilhelm06] and reach several solar radii in height [see @DeForest97]. Plumes are brighter, cooler and the plasma outflows are smaller than in inter-plumes [see @DeForest97; @Wilhelm98; @Raouafi07].
Recent studies of jets and polar plumes [X-ray and EUV; see @Wang98; @MInsertis08] treat these coronal structures independently and the relationship between them is not investigated. The present research is motivated by the fact that polar X-ray and EUV jets and plumes usually share common properties. Both are episodic in nature and occur at magnetic field concentrations that coincide with the chromospheric network where both structures form through flux emergence [see @Canfield96; @Wang98]. Studying the relationship between jets and plumes is important to understand their formation processes, evolution and the eventual contributions to the solar wind and heating of the plasma in the polar coronal holes. The present work is motivated by the observations of polar jets evolving into plumes such as the one shown in Fig. \[EUV\_fig\_jet\_plume\_070407\_2200\]. The aim of the paper is to investigate the relationship between these prominent coronal structures.
Observations and Data Analysis
==============================
The XRT telescope [@Golub07] on Hinode provides high resolution images ($\approx1-2\arcsec$ depending on the location within the field of view) of the solar corona at temperatures ranging from 1-20 MK. Observations of the southern coronal hole from XRT were utilized to study the evolution of polar X-ray jets and their relation with plumes. The data cover several time intervals on April 7-8, 2007 (07 April: 03:30 - 06:59 UT and 18:29 - 23:59 UT; 08 April: 11:49 - 17:59 UT and 21:30 - 22:59 UT) with a cadence of less than a minute. The data were corrected for instrumental effects utilizing XRT-calibration procedures.
A total of 28 X-ray jets were identified, with at least two recurring events within an hour. Most of the events are characterized by sharp collimated beams. The observed jets have different properties with regards to brightness, spatial extension, lifetime, and evolution. The bright point at the base of each jet is enhanced in brightness with every eruption and then fades after the jet is no longer observed.
In addition, 171 [Å]{} images from the STEREO/SECCHI satellite “A” were utilized to study EUV features in relation to the identified X-ray events. Particular attention was given to the presence of plume material during or after the eruption of jets. The choice of 171 [Å]{} was dictated by the adequate temperature corresponding to polar plume emissions.
Results
=======
Fig. \[solis\_070407\_jet\_plume\] displays the LOS-chromospheric magnetogram (Ca 8542 [Å]{}) of the south pole on April 7, 2007 recorded by the SOLIS/VSM instrument [@Henney08]. Spatial location of the X-ray jets on April 7$^{\rm{th}}$ and 8$^{\rm{th}}$ are marked by ‘+’ and ‘$\times$’ signs, respectively. No SOLIS/VSM chromospheric magnetograms were available for April 8, 2007. The solar rotation effect on the events’ spatial locations has been corrected using the model by @Howard90. It is clear that most jet events, in particular those of April 7$^{\rm{th}}$, are rooted in or near magnetic flux concentrations. At the base of bright jets are relatively large flux elements of one polarity surrounded by more diffuse flux of the opposite polarity (see Figs. \[solis\_070407\_jet\_plume\] and \[fig\_letter\_080407\_0332UT\]). Weaker and short-lived jets are based in areas of more diffuse magnetic flux.
Top panels of Figs. \[fig\_letter\_080407\_0332UT\]-\[fig\_letter\_080408\_2240UT\] show a sample of nine X-ray jets recorded by Hinode/XRT on April 7$^{\rm{th}}$ and 8$^{\rm{th}}$ 2007, respectively. The different events are indexed xj$_i$ ($i=1-9$) according to the time of their appearance. Although the brevity of the polar observation sequences did not allow us to determine the real lifetime of several events, jet lifetimes are estimated to range from minutes to a few tens of minutes with a number of events recurring within an hour, such as the event xj$_1$.
The middle and bottom panels of Figs. \[fig\_letter\_080407\_0332UT\]-\[fig\_letter\_080408\_2240UT\] display EUV images of the southern polar region corresponding to the X-ray observations. The data cover time intervals spreading over several hours after the disappearance of the X-ray events. A number of X-ray jet events are also present in EUV images (i.e., xj$_1$ and xj$_2$ in Fig. \[fig\_letter\_080407\_0332UT\]a and corresponding EUV structure in Fig. \[fig\_letter\_080407\_0332UT\]d; similarly xj$_7$ in Fig. \[fig\_letter\_080408\_2240UT\]a and Fig. \[fig\_letter\_080408\_2240UT\]g-h). Some of these events look brighter and sharper in EUV than in X-ray (see xj$_2$ in Fig. \[fig\_letter\_080407\_0332UT\]a and EUV counterpart in Fig. \[fig\_letter\_080407\_0332UT\]d), perhaps for plasma temperature reasons. This highlights that X-ray and EUV jet events are contiguous when plasma conditions allow emission in both temperature ranges.
[ccc]{} xj$_1$ (Fig.\[fig\_letter\_080407\_0332UT\]a) & Fig.\[fig\_letter\_080407\_0332UT\]d & Fig.\[fig\_letter\_080407\_0332UT\]e-i\
xj$_2$ (Fig.\[fig\_letter\_080407\_0332UT\]a) & Fig.\[fig\_letter\_080407\_0332UT\]d & Fig.\[fig\_letter\_080407\_0332UT\]e-f\
xj$_5$ (Fig.\[fig\_letter\_080407\_0332UT\]c) & – & Fig.\[fig\_letter\_080407\_0332UT\]g\
xj$_6$ (Fig.\[fig\_letter\_080408\_2240UT\]a-c) & Fig.\[fig\_letter\_080408\_2240UT\]d & Fig.\[fig\_letter\_080408\_2240UT\]e-i\
xj$_7$ (Fig.\[fig\_letter\_080408\_2240UT\]a) & – & Fig.\[fig\_letter\_080408\_2240UT\]g-i\
xj$_9$ (Fig.\[fig\_letter\_080408\_2240UT\]c) & – & Fig.\[fig\_letter\_080408\_2240UT\]g-i
The EUV data show that a significant number of polar jet eruptions are followed by rising polar plume haze with a time delay ranging from minutes to hours. Table \[corr\_jet\_plume\] summarizes the correlation and corresponding figures between the different X-ray and EUV events. A good example of plume haze appearing after a jet is given by the event xj$_6$, where collimated plasma emission is observed both in X-ray and in EUV images (see Fig. \[fig\_letter\_080408\_2240UT\]). The xj$_6$ event first appeared in X-ray images earlier than 21:31 UT (no X-ray data available to determine the exact start time). This event dimmed around 21:47 UT and reappeared again around 21:58 UT. The collimated EUV emission lasted longer than the X-ray one and evolved gradually into a wider and hazy structure that lasted for several hours, showing a polar plume with time-varying emission. Events xj$_3$, xj$_4$ and xj$_8$ were adjacent to off-limb plume emission locations. Cases of polar jets erupting within the base of ongoing plumes resulted in emission enhancement of the latter (compare P$_{07}$ in Fig. \[fig\_letter\_080407\_0332UT\]f & h and Fig. \[fig\_letter\_080407\_0332UT\]i; and P$_{08}$ in Fig. \[fig\_letter\_080408\_2240UT\]d-i).
Discussion
==========
X-ray and EUV observations indicate that more than 90% of the jets observed in the southern polar hole on April 7-8, 2007 are associated with plume haze. 70% of these jets are followed by polar plumes with a time delay ranging from minutes to tens of minutes. Emission of pre-existing plumes is enhanced after every jet eruption within their base. A number of prominent plumes (e.g., P$_{07}$ and P$_{08}$) show evidence for short lived, jet-like events in the EUV that occur within the plume base (see the sharp structures Fig. \[fig\_letter\_080407\_0332UT\](f) & (h) and the several bright points in panel (i)). Jet-like events ensure the continuous rise of haze and may contribute to the change in plume brightness [see @DeForest97].
The event xj$_7$ in Fig. \[fig\_letter\_080408\_2240UT\] is an interesting case. It was observed in X-rays from 21:58 - 22:16 UT on April 8, 2007. Fig. \[fig\_letter\_080408\_2240UT\](d-e) shows an EUV collimated structure similar to the one observed in X-rays more than three hours earlier. This may be caused by the plasma being heated to several MK and then becoming visible in X-rays, then gradually cooling down until it appears in the EUV range. More data needs to be analyzed to confirm the plausibility of this hypothesis.
The event xj$_9$, illustrated by Fig. \[fig\_letter\_080408\_2240UT\](c), is also peculiar and lasted less than 30 minutes. A narrow, collimated beam of plasma rose from the left edge of the large bright point with a shape typical of X-ray jets. It evolved rapidly and after 4-5 minutes the base width of the emission began to widen to cover the whole bright point. The width of the emitting structure exceeded 20 Mm, which is the typical width of polar plumes [see @Wilhelm06]. EUV images showed a faint haze several hours after the X-ray event (see Fig. \[fig\_letter\_080408\_2240UT\]d-i). GONG magnetograms show that the flux at the base of xj$_9$ weakened during the event’s lifetime. We believe that the initial jet event evolved into a plume due to significant emerging magnetic flux causing a catastrophic magnetic reconnection on a relatively short time scale but over a large spatial area. This may allow dissipation of the magnetic energy budget of the structure over a short period of time with an associated ejection of a significant amount of material over a relative large spatial scale, unlike other jet-plume events that develop over intervals of several hours. This type of event is recorded twice in the data set utilized here.
It is likely that jets play a key role in the formation process of polar plumes. Both coronal structures share numerous common characteristics, i.e., a magnetic field of mixed polarities at the base, leading to magnetic reconnection. We believe that the magnetic flux emergence causes the jet, opening of previously closed flux results in plume. Jet eruption seems to be the result of gradually emerging magnetic flux from the solar interior that suddenly reconnects on a small scale with the ambient photospheric field, leading to a collimated beam of plasma rising in the corona [e.g., @Yokoyama95]. EUV images show that coronal plume haze is observed following the jet events. They also provide evidence for several small bright points and short-lived, jet-like events within the base of the plume. These may be the results of magnetic reconnection at smaller spatio-temporal scales that modulate and sporadically brighten pre-existing polar plumes. This is most often seen in long-lived polar plumes, since several phases of reconnection can develop in a single long-lived structure. However, fast opening of magnetic flux can allow a plume to develop almost immediately such as in the case of the xj$_9$ event.
The transition from fast, impulsive, magnetically-driven dynamics of reconnection to the thermal expansion of newly liberated gas along opened magnetic field could explain the time delay observed between the jet and plume events. On the one hand, the jet eruption is the result of fast and explosive dissipation of magnetic energy on a short time scale. On the other hand, the plume might be a result of a pressure gradient within the open flux, which would lift the plume material in the corona. This hypothesis is supported by the fact that plasma outflow velocities in plumes are measured to be rather low up to $\sim1~R_\sun$ above the solar surface. The continuous emergence of magnetic flux at a slow rate and relatively large scale might ultimately create a sizable bundle of newly opened flux, allowing in turn a significant plume of escaping plasma to develop.
It is beyond us to simulate the development of a jet into a plume in an MHD model. However, some basic physics of such a development can be anticipated.
If a bipolar field emerges into a unipolar, open field region, then the two fields are not, in general, exactly parallel across the boundary between them. Then, according to Parker’s (1994) theory, a magnetic tangential discontinuity forms and current dissipation and field reconnection become inevitable at this boundary. Any two non-parallel fields can be resolved into parallel and anti-parallel components. The anti-parallel components will mutually annihilate at the discontinuity. The dissipated magnetic energy is partially converted to kinetic and thermal energy, which would cause a jet of energized plasma to escape along the open field next to the dissipating current sheet. Whenever some quantity of open flux is locally annihilated along the current sheet an equal quantity of closed flux must become open for magnetic flux continuity (${\bf\nabla}\cdot{\bf B}=0$). This open flux can allow a plume of thermally expanding plasma, formerly trapped by its closed field, to escape.
A jet model with a single magnetic neutral point such as Yokoyama & Shibata’s (1996) anenome jet model (see their Figure 1) could also result in a plume. Energy gained from emerging flux is converted to kinetic and thermal energy at the X-type neutral point during reconnection producing a jet of energized plasma. When the field has reconnected, there is a bundle of newly-opened magnetic flux through which hitherto trapped coronal plasma can escape as a plume.
The present results would benefit from future, more extensive analysis of larger data samples recorded by different instruments in a simultaneous fashion over large time intervals.
The authors would like to thank the anonymous referee and J. W. Harvey for helpful comments on the manuscript. The National Solar Observatory (NSO) is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. SOLIS data used here are produced cooperatively by NSF/NSO and NASA/LWS. NER’s work is supported by NASA grant NNH05AA12I. NER is a member of the coronal polar plume study team sponsored by the International Space Science Institute (ISSI), Bern, Switzerland.
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{width="50.00000%"}
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---
abstract: 'Optimum experimental design theory has recently been extended for parameter estimation in copula models. However, the choice of the correct dependence structure still requires wider analyses. In this work the issue of copula selection is treated by using discrimination design techniques. The new proposed approach consists in the use of $D_s$-optimality following an extension of corresponding equivalence theory. We also present some examples and highlight the strength of such a criterion as a way to discriminate between various classes of dependences.'
address:
- 'Institute of Science and Technology, 3400 Klosterneuburg, Austria'
- |
Department of Applied Statistics, Johannes Kepler University of Linz,\
4040 Linz, Austria
author:
- 'E. PERRONE'
- 'A. RAPPOLD'
- 'W.G. MÜLLER'
title: Optimal discrimination design for copula models
---
Copula selection ,Design discrimination ,Stochastic dependence.
Introduction
============
One of the most important tasks in copula modeling is to decide which specific copula to employ. For that purpose a rather general approach is to use omnibus goodness-of-fit tests that require minimum assumptions, for recent reviews see, e.g., [@Berg_2009], [@Genest_et_al_2009], or [@Fermanian_2013]. Other more specific avenues consist in applying graphical tools ([@michiels+d_13]) or information based criteria ([@gronneberg+h_14]). In fully parametric models, as considered in this paper, the latter can be formulated in terms of functions of the Fisher information matrices, which will allow us to generate optimal designs for copula model discrimination.
Design optimization is generally largely employed in many applied fields as a convenient tool to improve drawing informative experiments. Recently, in [@Perrone_16] the theory of $D$-optimality has been extended to a wider class of models for the usage of copulas. Although the employment of such functions allows for a substantial flexibility in modeling, it also leads to the natural question of their (proper) choice. As stated, developments of powerful goodness-of-fit tests and strategies to avoid the wrong choice of the dependence constitute a considerable part of the literature on copulas. The issue of model choice or discrimination is in principle also a well known part of (optimum) experimental design theory and several criteria (e.g., $D_s$-optimality, $T$-optimality, $KL$-optimality) have been proposed (see [@dette+t_09; @lopez-fidalgo_2007; @studden_80], and [@deldossi_16] for a special application to copula models).
In this work we first extend the general theory of $D_A$-optimality to copula models. Then, we present the usage of the $D_s$-criterion to discriminate between various classes of dependences and possible scenarios. Finally, we show through some examples possible real applications.
Theoretical framework
=====================
In this section we provide the extension for the $D_A$-criterion of a Kiefer-Wolfowitz type equivalence theorem, assuming the dependence described by a copula model. We then illustrate the basic idea of the new approach through a motivating example already analyzed in [@Perrone_16].
$D$-, $D_A$-, and $D_s$-optimality
----------------------------------
Let us consider a vector $\mathbf{x}^T = (x_1, \ldots, x_r) \in \mathcal{X}$ of control variables, where $\mathcal{X} \subset \mathbb{R}^r$ is a compact set. The results of the observations and of the expectations in a regression experiment are the vectors $$\mathbf{y}(\mathbf{x}) = (y_1(\mathbf{x}), y_2(\mathbf{x})),$$ $$\mathbf{E}[\mathbf{Y}(\mathbf{x})] = \mathbf{E}[(Y_1,Y_2)] = \boldsymbol{\eta}(\mathbf{x},\boldsymbol{\beta}) = (\eta_1(\mathbf{x},\boldsymbol{\beta}),\eta_2(\mathbf{x},\boldsymbol{\beta})),$$ where $\boldsymbol{\beta}=(\beta_1, \ldots,\beta_k)$ is a certain unknown parameter vector to be estimated and $\eta_i \; (i = 1,2)$ are known functions.
Let us call $F_{Y_i}(y_i(\mathbf{x}, \boldsymbol{\beta}))$ the margins of each $Y_i$ for all $i\in\{1,2\}$ and $f_{\mathbf{Y}}(\mathbf{y}(\mathbf{x}, \boldsymbol{\beta}), \boldsymbol{\alpha})$ the joint probability density function of the random vector $\mathbf{Y}$, where $\boldsymbol{\alpha}=({\alpha}_1,\ldots, {\alpha}_l)$ is the unknown copula parameter vector.
According to Sklar’s theorem ([@nelsen_06]), let us assume that the dependence between $Y_1$ and $Y_2$ is modeled by a copula function $$C_{\boldsymbol{\alpha}}(F_{Y_1}(y_1(\mathbf{x}, \boldsymbol{\beta})), F_{Y_2}(y_2(\mathbf{x}, \boldsymbol{\beta}))).$$
The Fisher Information Matrix $m(\mathbf{x}, \boldsymbol{\beta}, \boldsymbol{\alpha})$ for a single observation is a $(k +l) \times (k +l)$ matrix whose elements are $$\label{Eq:FIM}
\mathbf{E} \left( - \dfrac{\partial^2}{\partial \gamma_i \partial \gamma_j} \log \Big[ \dfrac{\partial^2}{\partial y_1 \partial y_2} C_{\alpha}(F_{Y_1}(y_1(\mathbf{x}, \boldsymbol{\beta})), F_{Y_2}(y_2(\mathbf{x}, \boldsymbol{\beta})))\Big] \right)$$ where $\boldsymbol{\gamma}=\{{\gamma}_1,\ldots,{\gamma}_{k+l}\}=\{{\beta}_1,\ldots,{\beta}_k,{\alpha}_1,, \ldots, {\alpha}_l\}$.
The aim of design theory is to quantify the amount of information on both sets of parameters $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$, respectively, from the regression experiment embodied in the Fisher Information Matrix.
For a concrete experiment with $N$ independent observations at $n \le N$ support points $\mathbf{x_1},\ldots,\mathbf{x_n}$, the corresponding information matrix $M(\xi, \boldsymbol{\gamma})$ then is $$M(\xi, \boldsymbol{\gamma}) = N^{-1} \sum\limits_{i=1}^n w_i \; m(\mathbf{x_i},\boldsymbol{\gamma}),$$ where $w_i$ and $\xi$ are such that: $$\sum\limits_{i=1}^n w_i = 1, \quad
\xi = \left \{
\begin{array}{cccc}
\mathbf{x_1} & \ldots & \mathbf{x_n} \\
w_1 & \ldots & w_n
\end{array}
\right \}.$$ The approximate design theory is concerned with finding $\xi^*(\boldsymbol{\gamma})$ such that it maximizes some scalar function $\phi(M(\xi,\boldsymbol{\gamma}))$, i.e., the so-called design criterion.
In [@Perrone_16], we have developed the theory for the well known criterion of $D$-optimality, i.e., the criterion $\phi (M(\xi,\boldsymbol{\gamma})) = \log \det M(\xi,\boldsymbol{\gamma}) $, if $M(\xi,\boldsymbol{\gamma})$ is non-singular. In this work, we consider the case when the primary interest is in certain meaningful parameter contrasts. Such contrasts are element of the vector $A^T\boldsymbol{\gamma}$, where $A^T$ is an $s \times (k+l)$ matrix of rank $s < (k+l)$. If $M(\xi, \boldsymbol{\gamma})$ is non-singular, then the variance matrix of the least-square estimator of $A^T\boldsymbol{\gamma}$ is proportional to $A^T \{ M(\xi, \boldsymbol{\gamma}) \}^{-1} A$ and then a natural criterion, generalization of the $D$-optimality for this context, would be of maximizing $\log \det[A^T \{ M(\xi, \boldsymbol{\gamma}) \}^{-1} A]^{-1}$. This criterion is called *$D_A$-optimality* ([@silvey_80]).
The following Theorem shows a generalization for the $D_A$-optimality of the Kiefer-Wolfowitz type equivalence theorem already proved in [@Perrone_16] for $D$-optimality. We have omitted the proof as it is, albeit a little more elaborate, fully analogous.
\[Th:1\] For a localized parameter vector $(\tilde{\boldsymbol{\gamma}})$, the following properties are equivalent:
1. $\xi^*$ is $D_A$-optimal;
2. for every $\mathbf{x} \in \mathcal{X}$, the next inequality holds: $$\textnormal{ tr }[ M(\xi^*, \tilde{\boldsymbol{\gamma}})^{-1} A (A^T M(\xi^*, \tilde{\boldsymbol{\gamma}})^{-1} A)^{-1} A^T M(\xi^*, \tilde{\boldsymbol{\gamma}})^{-1} m(\mathbf{x}, \tilde{\boldsymbol{\gamma}})]\leq s;$$
3. over all $\xi \in \Xi$, the design $\xi^*$ minimizes the function $$\max\limits_{x \in \mathcal{X}}\textnormal{ tr }[M(\xi^*, \tilde{\boldsymbol{\gamma}})^{-1} A (A^T M(\xi^*, \tilde{\boldsymbol{\gamma}})^{-1} A)^{-1} A^T M(\xi^*, \tilde{\boldsymbol{\gamma}})^{-1} m(\mathbf{x}, \tilde{\boldsymbol{\gamma}})].$$
Although we here extend the theory to the general case of $D_A$-optimality, in the following our interest is in the first $s < (k+l)$ parameters, only. In such a case, $M(\xi, \boldsymbol{\gamma})$ can be written as: $$M(\xi, \boldsymbol{\gamma}) = \left (
\begin{array}{cc}
M_{11} & M_{12} \\
M_{12}^T & M_{22}
\end{array}
\right ),$$ where $M_{11}$ is the $(s \times s)$ minor related to the estimated parameters. Therefore, the simplified criterion is to maximize the function $\phi_s (M(\xi, \boldsymbol{\gamma})) = \log \det (M_{11} - M_{12}M_{22}^{-1}M_{12}^T)$, which is called *$D_s$-optimality*. We now have
\[Co:2\] $D_s$-optimality follows as a particular case of Theorem \[Th:1\] by the choice $A^T = (I_s \; 0)$.
Given the characterization of Corollary \[Co:2\], two designs $\xi$ and $\xi^*$ can be compared by means of a ratio called *$D_s$-Efficiency* defined as follows: $$\left(\dfrac{\det[M_{11}(\xi, \tilde{\boldsymbol{\gamma}}) - M_{12}(\xi, \tilde{\boldsymbol{\gamma}})M_{22}^{-1}(\xi, \tilde{\boldsymbol{\gamma}})M_{12}^T(\xi, \tilde{\boldsymbol{\gamma}})]}{\det[
M_{11}(\xi^*, \tilde{\boldsymbol{\gamma}}) - M_{12}(\xi^*, \tilde{\boldsymbol{\gamma}})M_{22}^{-1}(\xi^*, \tilde{\boldsymbol{\gamma}})M_{12}^T(\xi^*,\tilde{\boldsymbol{\gamma}})]}\right)^{1/s}.$$ In the next section we will describe the usage of $D_s$-optimality in the sense of discrimination through a simple example originally reported in [@fedorov_71].
A motivating example
--------------------
Let us assume that for each design point $x \in [0,1]$, we observe an independent pair of random variables $Y_1$ and $Y_2$, such that $$E[Y_1(x)] = \beta_1 + \beta_2 x + \beta_3 x^2 ,$$ $$E[Y_2(x)] = \beta_4 x + \beta_5 x^3 + \beta_6 x^4.$$ The model is then linear in the parameter vector $\boldsymbol{\beta}$ and has dependence described by the product copula with Gaussian margins.
$x$ $w$
-------- --------
0.0000 0.1502
0.3414 0.0854
0.7901 0.3419
1.0000 0.4226
![Design points (first column), weights (second column), sensitivity function (continuous line) and weights (bars) of the $D_s$-optimal design for $\beta_1,\dots,\beta_6$.[]{data-label="fig:Fed"}](fed4Ds.pdf)
This example has already been generalized in [@Perrone_16] where various dependences through copula functions have been introduced and the corresponding $D$-optimal designs have been computed. In order to illustrate the usage of $D_s$-optimality in this context, let us assume the dependence between $Y_1$ and $Y_2$ described by a Clayton copula with $\alpha_1 = 18$, corresponding to a Kendall’s $\tau$ (see equation (\[eq:tau\])) value of $0.9$.
Even though the low losses in $D$-efficiency reported in [@Perrone_16] suggest that the impact of the assumed dependence is completely negligible, one might aim at verifying whether the information related to the dependence structure is only carried by the estimation of $\alpha_1$. Essentially, one might focus on the six marginal parameters entirely disregarding the estimation of the dependence parameter $\alpha_1$. This can be done in practice by applying the $D_s$-optimality to the parameter vector $\boldsymbol{\beta}$.
Figure \[fig:Fed\] shows the $D_s$-optimal design corresponding to this case. Comparing the $D$-optimal design of the product copula, assuming no dependence, with the $D_s$-optimal design for only the vector $\boldsymbol{\beta}$, the loss in $D_s$-efficiency is of $8\%$. This shows that the dependence structure itself can substantially affect the design even if the dependence parameter $\alpha_1$ is ignored in the estimation.
In more complex models, a similar approach can be used to identify informative designs to specific properties of interest. In the following, we highlight the usefulness of flexible copula models through the application of the $D_s$-criterion to a subclass of meaningful model parameters. We construct in this way designs which better reflect the strength and the structure of a specific dependence and can then be used to discriminate between classes of copulas.
Bivariate binary case
=====================
Copulas: combinations and measure of association
------------------------------------------------
As already mentioned above, the problem of specifying a probability model for dependent random variables can be simplified by expressing the corresponding 2-dimensional joint distribution $F_{Y_1Y_2}$ in terms of its two margins $F_{Y_1}$ and $F_{Y_2}$, and an associated [2-copula]{} (or dependence function) $C$, implicitly defined through the functional identity stated by Sklar’s Theorem [@sklar_59]. Copula theory allows the practitioner to gain in flexibility as for example any finite *convex linear combination* of 2- copulas $C_i$’s is itself a 2-copula. In particular, for $k \in \mathbb{N}$, let $C$ be given by $$C(u,v)= \sum_{i=1}^{k} \lambda_i C_i (u,v),$$ where $\lambda_i \geq 0$ for all indexes, and $\sum_{i=1}^{k} \lambda_i = 1$. Then, $C$ is a 2-copula.
Another useful property of copulas is that considering $Y_1$ and $Y_2$ two continuous random variables whose copula is $C_{\alpha_1}$, the measure of association Kendall’s $\tau$ is related to the expectation of the random variable $W = C_{\alpha_1}(U,V)$, which is $$\label{eq:tau}
\tau = 4 \int\limits_{I}\int\limits_{I} C_{\alpha_1}(u,v) d C_{\alpha_1}(u,v) - 1.$$ The relation in Equation (\[eq:tau\]) results in a correspondence between the copula parameter $\alpha_1$ and a fixed $\tau$ value. Such a relationship can be used in the construction of extremely flexible models, as shown in the next example.
The example
-----------
We analyze an example with potential applications in clinical trials already examined in [@denman+al_11] and [@Perrone_16]. We consider a bivariate binary response $(Y_{i1}, Y_{i2})$, $i=1, \ldots, n$ with four possible outcomes $\{ (0,0),(0,1),(1,0),(1,1)\}$ where $1$ usually represents a success and $0$ a failure (of, e.g., a drug treatment where $Y_1$ and $Y_2$ might be efficacy and toxicity). For a single observation denote the joint probabilities of $Y_1$ and $Y_2$ by $p_{y_1,y_2} = \mathbb{P} (Y_1 = y_1, Y_2 = y_2)$ for $(y_1,y_2 \in\{0,1\})$. Now, define $$\begin{array}{cccccc}
p_{11} & = & C_{\boldsymbol{\alpha}} (\pi_1, \pi_2), & p_{10} & = & \pi_1 - p_{11}, \\
p_{01} & = & \pi_2 - p_{11}, & p_{00} & = & 1 - \pi_1 - \pi_2 + p_{11}.
\end{array}$$
A particular case of the introduced model has already been analyzed in [@heise+m_1996]. In that work, the authors assume the marginal probabilities of success given by the models $$\log \left( \dfrac{\pi_i}{1 - \pi_i} \right) = \beta_{i1} + \beta_{i2} x, \qquad i=1,2$$ with $x \in [0,10]$ and ‘localized’ parameters $\tilde{\boldsymbol{\beta_1}}=(-1, 1)$ and $\tilde{\boldsymbol{\beta_2}}=(-2, 0.5)$. Let us now allow the strength of the dependence itself be dependent upon the regressor $x$. As in our context only positive associations make sense we consider in the following the corresponding Kendall’s $\tau$ modeled by a logistic: $$\tau(x, \alpha_1) = \dfrac{e^{\alpha_1 x - c}}{1 + e^ {\alpha_1 x - c}},$$ where $c$ is a constant chosen such that $\tau$ takes values in $[\epsilon,1]$ for $\alpha_1 \in [0,1]$. For our computations we choose $\epsilon=0.05$ and we select three values for $\alpha_1$ such that the $\tau$ ranges are $I_1 = [0.05,0.3], \; I_2= [0.05,0.9]$, and $I_3= [0.05,0.95]$.
Then, using the relationship from equation (\[eq:tau\]) that associates the Kendall’s $\tau$ with the copula parameter, we model $p_{11}$ by pair convex combinations of Joe, Frank, Clayton, and Gumbel copulas by linking the two copulas $C_1$ and $C_2$ at the same $\tau$ values through the functions $h_1$ and $h_2$: $$C(\pi_1,\pi_2; \alpha_1, \alpha_2) = \alpha_2 C_1(\pi_1,\pi_2; \; h_1(x,\alpha_1)) + (1-\alpha_2) C_2(\pi_1,\pi_2; \, h_2(x,\alpha_1)).$$ Notice that the construction is more general and any convex combination of standard copulas from the R package ’copula’ can be considered through the package ’docopulae’ ([@Docopulae]).
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![Sensitivity functions (continuous lines) and weights (bars) for $D$-optimal (left column) and $D_s$-optimal (right column) designs for Clayton-Gumbel (first line) and Frank-Gumbel (second line) with $\tau \in I_2=[0.05,0.9]$ and $\alpha_2=0.5$.[]{data-label="fig:Biv_case"}](bin1_clay_gumb_5197225_50.pdf "fig:") ![Sensitivity functions (continuous lines) and weights (bars) for $D$-optimal (left column) and $D_s$-optimal (right column) designs for Clayton-Gumbel (first line) and Frank-Gumbel (second line) with $\tau \in I_2=[0.05,0.9]$ and $\alpha_2=0.5$.[]{data-label="fig:Biv_case"}](bin1_clay_gumb_5197225_50Ds.pdf "fig:")
![Sensitivity functions (continuous lines) and weights (bars) for $D$-optimal (left column) and $D_s$-optimal (right column) designs for Clayton-Gumbel (first line) and Frank-Gumbel (second line) with $\tau \in I_2=[0.05,0.9]$ and $\alpha_2=0.5$.[]{data-label="fig:Biv_case"}](bin1_fran_gumb_5197225_50.pdf "fig:") ![Sensitivity functions (continuous lines) and weights (bars) for $D$-optimal (left column) and $D_s$-optimal (right column) designs for Clayton-Gumbel (first line) and Frank-Gumbel (second line) with $\tau \in I_2=[0.05,0.9]$ and $\alpha_2=0.5$.[]{data-label="fig:Biv_case"}](bin1_fran_gumb_5197225_50Ds.pdf "fig:")
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-------------------- ---------------- ---------------- ---------------- ---------------- ---------------- ----------------
$\tilde{\alpha}_2$ $\tau \in I_1$ $\tau \in I_2$ $\tau \in I_3$ $\tau \in I_1$ $\tau \in I_2$ $\tau \in I_3$
0.1 34.94 38.80 41.37 49.85 49.45 45.10
0.5 42.36 38.20 41.83 43.65 39.27 39.03
0.9 55.11 47.23 44.15 37.87 34.65 37.78
$\tilde{\alpha}_2$ $\tau \in I_1$ $\tau \in I_2$ $\tau \in I_3$ $\tau \in I_1$ $\tau \in I_2$ $\tau \in I_3$
0.1 35.92 36.35 39.01 47.13 48.29 46.17
0.5 45.37 43.17 45.53 37.65 34.41 34.37
0.9 49.92 48.72 45.36 38.51 34.19 36.26
-------------------- ---------------- ---------------- ---------------- ---------------- ---------------- ----------------
: \[tab1\] Losses in $D_s$-efficiency in percent for $I_1 = [0.05,0.3], \; I_2= [0.05,0.9]$, and $I_3= [0.05,0.95]$.
In this model, the impact of the dependence structure and the association level is reflected by two different parameters, as the $\alpha_1$ parameter is only related to the measure of association Kendall’s $\tau$, while the $\alpha_2$ parameter is strictly related to the structure of the dependence. Therefore, applying the $D_s$-criterion on $\alpha_2$, we find a design for discriminating, in this specific model, between the two copulas considered.
We compare the design obtained for different $\tau$ intervals and localized values for $\alpha_2$ with the $D$-optimal design obtained for the same localized values (Figure \[fig:Biv\_case\]). Analyzing the rather high losses in $D_s$-efficiency reported in Table \[tab1\], it shows that the $D$-criterion alone is not sufficient when we require information about the structure of the model.
In this scenario, an interesting question is whether the obtained $D_s$-optimal designs are robust with respect to the initial model assumptions. To analyze this aspect, we computed the $D_s$-efficiencies for cross-comparisons of $D_s$-optimal designs. In Table \[tab2\], the results for $\tau \in I_2$ and $\tilde{\alpha}_2 = 0.5$ are reported (see Figure \[fig:Biv\_case\], also). Looking at the table, one can notice that the losses correspondent to the assumed combination Clayton-Gumbel are in general lower, not exceeding $16\%$. This means that such a combination provides good results in order to discriminate between all the considered dependences. Further studies in this direction would lead to the development of new design techniques to construct robust and stable designs for discrimination between various classes of dependences.
-- ----------- ----------- ----------- -----------
${0.00}$ ${28.44}$ ${7.43}$ ${19.07}$
${16.09}$ ${0.00}$ ${30.17}$ ${19.51}$
${4.25}$ ${34.27}$ ${0.00}$ ${13.51}$
${15.13}$ ${13.97}$ ${9.52}$ ${0.00}$
-- ----------- ----------- ----------- -----------
: Losses in $D_s$-efficiency in percent for $\tau \in I_2$ and $\tilde{\alpha}_2=0.5$ by comparing the true copula model with the assumed one.[]{data-label="tab2"}
Bivariate Weibull function
==========================
In this section we extend an example originally reported in [@Kim_15]. After providing a brief overview of the theoretical framework, we construct original asymmetric copula models and we apply $D_s$-optimality to discriminate between symmetric and asymmetric scenarios.
Copulas and exchangeability
---------------------------
A copula $C$ is said to be *exchangeable* (or *symmetric*) if it does not change under any permutation of its arguments. In particular, if $(U,V)$ is a random pair distributed according to an exchangeable copula $C$, then $$\mathbb{P}(V\le v\mid U\le u)=\mathbb{P}(U\le v\mid V\le u),$$ for all $(u,v)\in[0,1]^2$. Consequently, the conditional distributions of $(V\mid U\le u)$ and $(U\mid V\le u)$ are equal. This indicates that a causality relationship between $U$ and $V$ leads to non-exchangeability.
Possible ways of quantifying non-exchangeability in copula models have been provided in the literature ([@klement+m_06; @Nelsen_07]). Although some classes of bivariate copulas can directly deal with non-exchangeability ([@Capera_00; @Charpentier_14; @Klement_05; @DeBaets_07]), many other copulas largely used in modeling belong to the class of exchangeable ones. An example is given by the well-known family of Archimedean copulas (see [@DurSem15; @Genest_86; @nelsen_06]), which are not suitable to model many situations that might arise in real scenarios (see, for instance, [@Genest_13]).
To overcome these restrictions, a possibility is to apply transformations which commute exchangeable copulas into non-exchangeable ones ([@Durante_2007; @Frees_98; @Khoudraji_95]). In the next example, we apply the Khoudraji’s asymmetrization described in [@Khoudraji_95]. In particular, we modify a given exchangeable copula $C_{\alpha_1}$, with parameter $\alpha_1$, into the copula $C=C_{\alpha_1,\alpha_2,\alpha_3}$ defined, for every $(u,v)\in [0,1]^2$, by $$\label{Eq:AC}
C(u,v) = u^{\alpha_2} v^{\alpha_3} C_{\alpha_1}(u^{1-\alpha_2}, v^{1-\alpha_3}),$$ where $\alpha_2,\;\alpha_3 \in[0,1]$. For $\alpha_2\neq \alpha_3$, $C$ is non-exchangeable.
First investigations on the changes in the geometry of the $D$-optimal design for such transformations have been carried out in [@Durante_16], where a theoretical overview of exchangeability in the copula theory is also given. In the following we instead present the usage of the $D_s$-optimality to discriminate between symmetric and asymmetric models.
The Weibull case
----------------
We now analyze an example originally reported in [@Kim_15]. We assume two dependent binary outcomes, $U$ and $V$, for two system components, respectively. Considering $0$ indicating no failure and $1$ indicating failure, the outcome probabilities given a stress $x$ can be written as: $$p_{uv}(x, \boldsymbol{\gamma}) = \mathbb{P}(U=u,V=v\mid x, \boldsymbol{\gamma}),$$ with $u,v \in \{ 0,1\}$ and where $\boldsymbol{\gamma}$ denotes a vector of all the model parameters.
Let $Y$ and $Z$ denote the amount of damage on component 1 and component 2, respectively, and let $f(y,z\mid x,\boldsymbol{\gamma})$ be the bivariate Weibull regression model. Suppose that failures are defined by dichotomizing damage measurements $Y$ and $Z$: $$\begin{array}{c}
U = \left\{ \begin{array}{cc} 0 & \text{ (no failure for component 1), if } Y < \zeta_1, \\ 1 & \text{ (failure for component 1), otherwise}\end{array}\right. \\
\\
V = \left\{ \begin{array}{cc} 0 & \text{ (no failure for component 2), if } Z < \zeta_2, \\ 1 & \text{ (failure for component 2), otherwise}\end{array}\right.
\end{array}$$ where $\zeta_1$ and $\zeta_2$ are predetermined cut-off values. Then, the probabilities of success and failure are: $$\begin{array}{c}
p_{00} = \int_0^{\zeta_1} \int_0^{\zeta_2} f(y,z\mid x,\boldsymbol{\gamma}) \,d y\,d z, \quad p_{01} = \int_0^{\zeta_1} \int_{\zeta_2}^{\infty} f(y,z\mid x,\boldsymbol{\gamma}) \,d y\,d z, \\
\\
p_{10} = \int_{\zeta_1}^{\infty} \int_0^{\zeta_2} f(y,z\mid x,\boldsymbol{\gamma}) \,d y\,d z, \quad
p_{11} = \int_{\zeta_1}^{\infty} \int_{\zeta_2}^{\infty} f(y,z\mid x,\boldsymbol{\gamma}) \,d y\,d z.
\end{array}$$ Now, considering $f(y,z\mid x,\boldsymbol{\gamma})$ defined as follows: [$$f(y,z) = \left\{ \begin{array}{lc}
\beta_1(\beta_3 + \beta_5) \kappa^2 (yz)^{\kappa - 1} \text{exp}\{ -(\beta_3 + \beta_5) z^{\kappa} - (\beta_1 + \beta_2 - \beta_5) y^{\kappa}\} & \text{for } 0 < y < z < \infty; \\
\beta_2(\beta_3 + \beta_4) \kappa^2 (yz)^{\kappa - 1} \text{exp}\{ -(\beta_3 + \beta_4) y^{\kappa} - (\beta_1 + \beta_2 - \beta_4) z^{\kappa}\} & \text{for } 0 < z < y < \infty; \\
\beta_3 \kappa (y)^{\kappa - 1} \text{exp}\{ -(\beta_1 + \beta_2 + \beta_3) & \text{for } 0 < y = z < \infty.
\end{array}\right.$$ ]{} The marginal survival functions of the bivariate Weibull density are weighted univariate Weibull survival functions: [$$\mathbb{P}(Y \geq y) = \dfrac{\beta_2} {\beta_1 + \beta_2 - \beta_4} \text{exp} \{ - (\beta_3 + \beta_4) y^{\kappa}\} + \left(1- \dfrac{\beta_2}{\beta_1 + \beta_2 - \beta_4}\right) \text{exp} \{ - (\beta_1 + \beta_2 + \beta_3) y^{\kappa} \}$$ $$\mathbb{P}(Z \geq z) = \dfrac{\beta_1} {\beta_1 + \beta_2 - \beta_5} \text{exp} \{ - (\beta_3 + \beta_5) z^{\kappa}\} + \left(1- \dfrac{\beta_1} {\beta_1 + \beta_2 - \beta_5}\right) \text{exp} \{ - (\beta_1 + \beta_2 + \beta_3) z^{\kappa} \}$$ ]{} In [@Kim_15], the authors set $\zeta_1=0.8$ and $\zeta_2 = 0.7$. Moreover, they consider the following predictor functions: $$\left\{ \begin{array}{rll}
-\log (\beta_3 + \beta_5) & = & \theta_0 + \theta_1 x , \\
-\log (\beta_3 + \beta_4) & = & \theta_0 + \theta_2 x , \\
-\log (\beta_1 + \beta_2 + \beta_3) & = & \theta_0 + \theta_3 x.
\end{array}\right.$$ with $x\in[0,1]$.
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![Sensitivity functions (continuous lines) and design weights (bars) of the $D$-optimal design for the Weibull case as reported in [@Kim_15] (left), and for asymmetric Clayton with $(\tilde{\alpha}_1,\tilde{\alpha}_2,\tilde{\alpha}_3)= (1.5,0.4,0)$ (right); $,p_{00}$; $,p_{11}$; $,p_{0.}$; $,p_{.0}$ []{data-label="fig:Wei_case"}](wei0.pdf "fig:") ![Sensitivity functions (continuous lines) and design weights (bars) of the $D$-optimal design for the Weibull case as reported in [@Kim_15] (left), and for asymmetric Clayton with $(\tilde{\alpha}_1,\tilde{\alpha}_2,\tilde{\alpha}_3)= (1.5,0.4,0)$ (right); $,p_{00}$; $,p_{11}$; $,p_{0.}$; $,p_{.0}$ []{data-label="fig:Wei_case"}](wei3_clay_1_5_4_0.pdf "fig:")
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![Sensitivity functions (continuous lines) and design weights (bars) of $D$-optimal designs (first row) and $D_s$-optimal designs (second row) for the Weibull case for asymmetric Clayton with $(\tilde{\alpha}_1,\tilde{\alpha}_2,\tilde{\alpha}_3)= (2,0.4,0.2)$ (left column), and for $(\tilde{\alpha}_1,\tilde{\alpha}_2,\tilde{\alpha}_3)= (3.6,0.6,0)$ (right column); $,p_{00}$; $,p_{11}$; $,p_{0.}$; $,p_{.0}$[]{data-label="fig:DsWei_case"}](wei3_clay_2_4_2.pdf "fig:") ![Sensitivity functions (continuous lines) and design weights (bars) of $D$-optimal designs (first row) and $D_s$-optimal designs (second row) for the Weibull case for asymmetric Clayton with $(\tilde{\alpha}_1,\tilde{\alpha}_2,\tilde{\alpha}_3)= (2,0.4,0.2)$ (left column), and for $(\tilde{\alpha}_1,\tilde{\alpha}_2,\tilde{\alpha}_3)= (3.6,0.6,0)$ (right column); $,p_{00}$; $,p_{11}$; $,p_{0.}$; $,p_{.0}$[]{data-label="fig:DsWei_case"}](wei3_clay_3_6_6_0.pdf "fig:")
![Sensitivity functions (continuous lines) and design weights (bars) of $D$-optimal designs (first row) and $D_s$-optimal designs (second row) for the Weibull case for asymmetric Clayton with $(\tilde{\alpha}_1,\tilde{\alpha}_2,\tilde{\alpha}_3)= (2,0.4,0.2)$ (left column), and for $(\tilde{\alpha}_1,\tilde{\alpha}_2,\tilde{\alpha}_3)= (3.6,0.6,0)$ (right column); $,p_{00}$; $,p_{11}$; $,p_{0.}$; $,p_{.0}$[]{data-label="fig:DsWei_case"}](wei3_clay_2_4_2_Ds.pdf "fig:") ![Sensitivity functions (continuous lines) and design weights (bars) of $D$-optimal designs (first row) and $D_s$-optimal designs (second row) for the Weibull case for asymmetric Clayton with $(\tilde{\alpha}_1,\tilde{\alpha}_2,\tilde{\alpha}_3)= (2,0.4,0.2)$ (left column), and for $(\tilde{\alpha}_1,\tilde{\alpha}_2,\tilde{\alpha}_3)= (3.6,0.6,0)$ (right column); $,p_{00}$; $,p_{11}$; $,p_{0.}$; $,p_{.0}$[]{data-label="fig:DsWei_case"}](wei3_clay_3_6_6_0_Ds.pdf "fig:")
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In [@Kim_15], the asymmetry in the causality has been reflected by different cut points, e.g., unequal values for $\zeta_1$ and $\zeta_2$, and different initial failure rates $\beta_1$ and $\beta_2$ as well as different coefficients $\theta_1$ and $\theta_2$ of the predictor.
In our example, we additionally allow asymmetry of the phenomenon to appear in the dependence structure. In particular, such an asymmetry is introduced through the transformation presented in equation (\[Eq:AC\]), adding new parameters in the process. Going into details, we introduce two parameters $\nu_1$, and $\nu_2$ such that the following is satisfied: $$\left\{ \begin{array}{l}
\theta_1 = \theta_2 + \nu_1,\\ \beta_1 = \beta_2 + \nu_2. \end{array}\right.$$ The vector $(\nu_1, \nu_2)$ then quantifies the dissimilarity of the margins. For our study, we assume the joint dependence to be described by the asymmetric Clayton copula with three parameters $\alpha_1, \; \alpha_2$ and $\alpha_3$, constructed according to equation (\[Eq:AC\]). In this context, we apply $D_s$-optimality to the parameters $\boldsymbol{\mu} = (\nu_1, \nu_2, \alpha_2, \alpha_3)$ which denote the total asymmetry of the phenomenon, both from the marginals and the joint dependence. In such a way, we find designs which are more informative to the asymmetry and are then suitable to discriminate between exchangeable models and non-exchangeable ones. The used parameter setting corresponds to two Kendall’s tau values: 0.5 and 0.25, respectively. The initial values of the parameters ${\alpha}_1, {\alpha}_2,$ and ${\alpha}_3$ are the same as used in [@Durante_16], while the other parameter values are $\tilde{\theta}_0=-2,\; \tilde{\theta}_2=5,\; \tilde{\theta}_3=2,\; \tilde{\nu}_1=-1,\; \tilde{\nu}_2=0.1,\; \tilde{\beta}_2=0.2,$ and $\tilde{\kappa}=2$.
The $D$-optimal designs obtained spread weight to four design points, slightly differing in their distribution. Figure \[fig:Wei\_case\] shows a representative design for our model side by side with the $D$-optimal design for the Weibull case as reported in [@Kim_15]. The maximal and minimal values of the loss in $D$-efficiency by comparing the design reported in [@Kim_15] and the $D$-optimal designs for our models are reported in Table \[tab4\]. A full table with the losses of such comparison for each set of initial values of ${\alpha}_1, \; {\alpha}_2,$ and $ {\alpha}_3$ is available in the supplementary material. The results suggest that in every case it would be advantageous to choose one of our models as generally more informative and robust.
--------------- ------ ------- ------- -------
Assumed Model min max min max
Weibull 0.00 0.00 17.78 71.65
Our Models 9.43 10.18 0.00 3.37
--------------- ------ ------- ------- -------
: \[tab4\] Losses in $D$-efficiency in percent for crossed comparison between the optimal design find for the Weibull model as reported in [@Kim_15] and all our models.
We are now interested in verifying whether the $D$-optimal design is informative enough to discriminate between asymmetry and symmetry. To this aim, we compare $D_s$-optimal designs for $\boldsymbol{\mu}$ to the corresponding $D$-optimal designs (Figure \[fig:DsWei\_case\]). In this case, the loss in $D_s$-efficiency never exceeds $5\%$. In contrast to the binary case, such a result indicates that the $D$-optimal design is already quite adequate for discriminating between symmetric and asymmetric models.
Discussion
==========
In this paper we embed the issue of the choice of the copula in the framework of discrimination design. We present a new methodology based on the $D_s$-optimality to construct design that discriminate between various dependences. Through some examples we highlight the strength of the proposed technique due to the usage of the copula properties. In particular, the proposed approach allows to check the robustness of the $D$-optimal design in the sense of discrimination and to construct more informative designs able to distinguish between classes of dependences.
All the shown results are obtained by the usage of the R package ’docopulae’ ([@Docopulae]). Although we here compare just few possible dependences, the general construction is much wider. The R package ’docopulae’ allows the interested reader to run designs assuming a broad variety of dependence structures. It then provides a strong computational tool to the usage of copula models in real applications.
The innovative approach we present in this paper is promising as it breaks new ground in the field of experimental design. In the future, we aim at generalizing other discrimination criteria such as $T-$optimality and $KL-$optimality to flexible copula models ([@dette+t_09; @ucinski_05; @lopez-fidalgo_2007]). Furthermore, powerful compound criteria might be developed for such models (see, for instance, [@atkinson_08; @dette_93; @tommasi_09]). In addition, the construction of multistage design procedures that allow for discrimination and estimation might be of great interest in special applications such as clinical trial studies ([@dragalin_08; @mueller_96]).
Acknowledgments {#acknowledgments .unnumbered}
===============
This work has been supported by the project ANR-2011-IS01-001-01 “DESIRE" and Austrian Science Fund (FWF) I 833-N18.
We thank F. Durante, L. Pronzato, J. Rendas and E.P. Klement for fruitful discussions.
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|
---
abstract: 'We construct energy-optimized resonating valence bond wave functions as a means to sketch out the zero-temperature phase diagram of the square-lattice quantum Heisenberg model with competing nearest- ($J_1)$ and next-nearest- ($J_2$) interactions. Our emphasis is not on achieving an accurate representation of the magnetically disordered intermediate phase ( on a relative coupling $g = J_2/J_1 \sim 1/2$ and whose exact nature is still controversial) but on exploring whether and how the Marshall sign structure breaks down in the vicinity of the phase boundaries. Numerical evaluation of two- and four-spin correlation functions is carried out stochastically using a worm algorithm that has been modified to operate in either of two modes: one in which the sublattice is fixed beforehand and another in which the worm manipulates the current so as to sample various sign conventions. Our results suggest that the disordered phase evolves continuously out of the $(\pi,\pi)$ Néel phase and largely inherits its Marshall sign structure; on the other hand, the transition from the magnetically ordered $(\pi,0)$ phase is strongly first order and involves an abrupt change in the sign structure and spatial symmetry as the result of a level crossing.'
author:
- Xiaoming Zhang
- 'K. S. D. Beach'
date: 'March 7, 2013'
title: |
Resonating valence bond trial wave functions with both\
static and dynamically determined Marshall sign structure
---
\[SEC:introduction\] Introduction
===================================
Simple spin models have contributed significantly to our understanding of quantum magnetism. They consist of mutually interacting spin-$S$ objects arranged in a lattice and are meant to describe the behavior of localized electrons in a crystalline environment. Such models are generally viewed as effective, low-energy descriptions, descended from their electronic parent models by a process of integrating out the gapped charge degrees of freedom.[@MacDonald88]
A tremendous variety of spin interactions can arise. In particular, a “$t/U$”-style power series from the strong correlation limit generates (or at least motivates) an increasingly complicated zoo of multi-spin interaction terms.[@Fujimoto05; @Lauchli05; @Sandvik07; @Beach07a; @Majumdar12] Nonetheless, we know that even the leading order term in the expansion, corresponding to Heisenberg models with just two-spin interactions, can display highly nontrivial physics if the exchange interactions are sufficiently frustrating.[@Diep05; @Lacroix11] In that case, the ground state may be a magnetically disordered, spin-rotation-invariant state—either liquid[@Anderson73] or solid[@Affleck88; @Read89]—having no classical .
Otherwise, conventional magnetic order (at some ordering vector $\mathbf{Q}$) is a generic feature of the ground state for Heisenberg models in spatial dimension greater than one.[@Misguich02; @Hastings04] The absence of frustration is connected to three inter-related properties: (i) the existence of a bipartite such that all antiferromagnetic interactions connect sites in opposite sublattices, (ii) strict adherence to a Marshall sign rule,[@Marshall55] and (iii) the possibility of transforming mechanistically to a basis in which all amplitudes of the wave function are real and nonnegative. The last of these is why nonfrustrated models can be easily simulated using quantum Monte Carlo approaches. [@Beard96; @Syliuasen02; @Evertz03]
For the $S=1/2$ case, all three properties are conceptually unified in the language of *valence bonds*.[@Rumer32; @Pauling33; @Hulthen38; @Fazekas74; @Beach06] The collinear, $\mathbf{Q}$-ordered ground state of a nonfrustrated Heisenberg model can be described in a bipartite valence bond basis[@Beach06; @Beach08] in which the AB sublattice coincides with the alternating pattern laid out by $\mathbf{Q}$ and only spins in opposite sublattices are bound into singlet pairs. In terms of such a basis $\mathcal{V}_{\text{AB}} = \{ \lvert v \rangle \}$, the ground state has an expansion $\lvert \psi \rangle = \sum_v \psi(v) \lvert v \rangle$ in which each amplitude $\psi(v)$ is real and nonnegative. The exact amplitudes can be obtained numerically by projection.[@Liang90; @Santoro99; @Sandvik05; @Sandvik10; @Banerjee10]
It is also possible to find extremely good approximate values of the form $\psi(v) \approx \prod_{[i,j]\in v} h(\mathbf{r}_{ij})$, where $h(\mathbf{r}) > 0$ is a function of the vector connecting bond endpoints. This resonating valence bond (RVB) ansatz, due to Liang, Doucot, and Anderson, [@Liang88] strictly enforces the geometric tiling constraint on the singlet bonds but ignores additional bond-bond correlations.[@Lin12] For a magnetically ordered state, one can show that factorizability into individual bond amplitudes is the correct assumption.[@Beach07b; @Hasselmann06] Moreover, for nonfrustrated systems, the amplitudes exhibit power-law decay, and hence the wave function contains bonds on all length scales.
As a specific and illustrative example, we consider the square-lattice $J_1$–$J_2$ model for spin half. It has two nonfrustrated limits. The model with anitferromagnetic nearest- interactions only ($J_1 = 1$, $J_2 = 0$) exhibits a Néel ordered ground state whose staggered moment is roughly 60% of its fully polarized, classical value. The state is almost perfectly captured by an RVB wave function whose bond amplitudes are computed as $h(\mathbf{r}) = \sum_{\mathbf{q}}e^{i\mathbf{q}\cdot\mathbf{r}}\bigl[1-(1-\gamma_{\mathbf{q}}^2)^{1/2}\bigr]/\gamma_{\mathbf{q}}$. Here, $\gamma_{\mathbf{q}} = (\cos q_x + \cos q_y)/2$, and the wave-vector sum is taken over a Brillouin zone reduced with respect to $\mathbf{Q} = (\pi,\pi)$. The opposite limit, with *next*-nearest- interactions dominating ($J_1 = 0^+$, $J_2 = 1$), is equivalent to two interpenetrating nearest- Heisenberg antiferromagnets rotated $45^{\circ}$. The spin directions in the two otherwise disjoint subsystems lock to each other[@Chandra90] provided that $J_1$ is not strictly zero. In this case, the ground state is equally well described by the RVB wave function, but with the substitution of $\gamma_{\mathbf{q}} = \cos q_x \cos q_y$ and a Brillouin zone defined modulo $\mathbf{Q} = (\pi,0)$ or $\mathbf{Q} = (0,\pi)$.
What we present in this paper is an attempt to interpolate between these two limits—through the entire range of relative couplings that are highly frustrated—using the RVB state as a variational wave function. Our approach is inspired by Ref. , but there are several important differences. The first is simply the scale of the calculation: we have simulated a large number of lattice sizes up to $L=32$ on a dense grid of relative coupling values ($g=J_2/J_1$ ranging from 0 to 1 in steps of $\delta g = 0.01$). Second, we do not require that $h(\mathbf{r})$ respect the full $C_4$ symmetry of the square lattice. Rather, we impose only the $x$- and $y$-axis reflection symmetry, giving the amplitudes an opportunity either to acquire (over the course of the energy optimization) the full symmetry or to settle into a state that looks different under $90^{\circ}$ rotation. Third, we explore the space of AB sublattice s by which the bipartite valence bond basis is constructed.
As in Ref. , we make use of an unbiased, stochastic optimization scheme. Changes to the $h(\mathbf{r})$ values are made in the downhill direction of the local energy gradient. Step sizes are randomized, and their magnitude decreases on a power-law schedule. We do not attempt to guide the optimization, other than to ensure that none of the bond amplitudes goes negative; nor do we impose any constraints on the variational parameters based on any prior knowledge (gleaned, e.g., from mean-field theory[@Beach07b] or from a master-equation analysis[@Beach09]).
We discover the following. At this level of approximation, the $J_1$–$J_2$ model does indeed support a magnetically disordered intermediate phase. But its width is much smaller than expected: the phase boundaries are found to be at $g_{\text{c}1} \doteq 0.54(1)$ and $g_{\text{c}2} \doteq 0.5891(3)$. The transitions are unambiguously second- and first-order, respectively, with the ground state achieving the full $C_4$ symmetry for all $g < g_{\text{c}2}$. As the system is tuned up from $g=0$, increasing frustration eventually extinguishes the $(\pi,\pi)$ ordered moment at $g_{\text{c}1}$ in a continuous fashion.
The disappearance of magnetic order is preceded by a failure of the Marshall sign rule at $g_{\text{M}1} \doteq 0.398(4)$, in agreement with the scenario first outlined by Richter and co-workers.[@Richter94] Still, even though the rule is not strictly obeyed beyond $g_{\text{M}1}$, the Marshall structure inherited from the $g=0$ model remains largely intact throughout the intermediate phase. This is true in the sense that continuing to define the bipartite bond basis from a *checkerboard* sublattice decomposition produces only a microscopic number of negative $h(\mathbf{r})$ values—only $h(\pm 1,\pm 2)$ and $h(\pm 2,\pm 1)$ initially. Moreover, when we allow the AB pattern to arise on its own within the simulation (described in detail in Secs. \[SECT:dynamic\] and \[SECT:results\]), the checkerboard pattern is the one selected whenever $g < g_{\text{c}2}$.
On the other hand, the RVB state at large $g$ explicitly breaks the $90^\circ$ rotation symmetry and has a Marshall sign structure based on a *stripe* sublattice decomposition. As the coupling is tuned down from the $g=\infty$ limit, the $(\pi,0)$ ordered moment is not strongly affected, and it persists with only weak variation (never dropping below 47% of its fully polarized value) down to $g_{\text{c}2}$, where the spatially symmetric, checkerboard-based RVB wave function takes over as the lowest energy state. This state in the region $g_{\text{c}1} < g < g_{\text{c}2}$ is, as far as we can tell, featureless. It exhibits no long-range spin or dimer order, and it breaks no symmetries. It is not, however, a “short-range RVB state” in the usual sense, since it is not made up of predominantly short bonds. Its amplitude function $h(\mathbf{r})$ is highly anisotropic (as anticipated elsewhere[@Beach09]) and remains long ranged along the principal spatial axes. Spin correlations appear to be critical and to display circular symmetry at long distances, despite the anisotropy of the bond weights. Dimer correlations decay either exponentially or with a high power law. This is in stark contrast to the usual short-bond-only RVB state, often referred to as the nearest- RVB (NNRVB), which has spin correlations that decay exponentially [@Liang88] and dimer correlations that decay algebraically.[@Albuquerque10; @Tang11] Moreover, the presence of long bonds implies an absence of the topological order[@Albuquerque10; @Tang11] that is characteristic of a purely short-range RVB state in two dimensions.
Model and method
==================
Frustrated Hamiltonian
------------------------
The spin-half, square-lattice Heisenberg model with frustrating interactions has a Hamiltonian $$H = J_1\sum_{\langle i,j \rangle}\mathbf{S}_i\cdot \mathbf{S}_j+J_2\sum_{\langle \langle i,j \rangle \rangle}\mathbf{S}_i\cdot \mathbf{S}_j,$$ where $J_1>0$ and $J_2>0$ are the antiferromagnetic exchange couplings. The summations range over pairs of adjacent sites $\langle i,j \rangle$ and over farther pairs $\langle \langle i,j \rangle \rangle$ that sit diagonally across a plaquette. The ratio $g=J_2/J_1$ is the key tuning parameter at zero temperature. In the classical version of this model ($S\to \infty$), two magnetic phases meet at exactly $g=0.5$, separated by a first-order transition.[@Moreo90; @Chubukov91; @Ferrer93; @Ceccatto93]
In the $S=1/2$ problem, the two magnetically ordered ground states obtain for values $g \lesssim0.4$ and $g\gtrsim 0.6$,[@Dagotto89; @Schulz96; @Oitma96; @Bishop98; @Singh99; @Sushkov01] and a magnetically disordered phase intervenes. (There is, however, a good deal of disagreement over the exact positions of the critical points; cf. Refs. and , which put the lower critical point as low as 0.35 and as high as 0.45.) The physics of the phase in the intermediate region is not known with complete certainty, but it is commonly believed to be short ranged and not to exhibit any kind of conventional magnetic order. One possibility is a crystalline arrangement of valence bonds, a state with broken translational symmetry in which singlet formation s an enlargement of the unit cell beyond that of the underlying square lattice.[@Gelfand89; @Gelfand90; @Singh90; @Zhitomirsky96; @Leung96; @Kotov99; @Kotov00; @Capriotti00; @Takao03; @Mambrini06; @Murg09; @Reuther10; @Reuther11; @Yu12] A featureless spin liquid that does not break any symmetries is another possibility.[@Chandra88; @Figueirido90; @Oguchi90; @Locher90; @Schulz92; @Zhong93; @Zhang03; @Capriotti01; @Capriotti03; @Captiotti04a; @Capriotti04b]
The case for a spin liquid ground state has been advanced by recent tensor product[@Wang11] and density matrix renormalization group (DMRG)[@Jiang11] calculations and by a variational approach based on the entangled-plaquette ansatz.[@Mezzacapo12] With regard to the DMRG result, Sandvik has suggested that the use of a cylindrical geometry complicates the detection of crystalline order.[@Sandvik12] His numerical experiments seem to indicate that the mixture of open and closed boundary conditions significantly raises the crossover length scale $\xi$ beyond which bond order takes hold (i.e., where the finite size scaling of the dimer-dimer correlations is truly in the asymptotic regime). Such questions are difficult to resolve. Unlike in three-dimensional systems, where crystalline bond order, if it is present, is almost always strong,[@Beach07a; @Block12] in two dimensions it is quite delicate and can easily be disguised by a $U(1)$ effective symmetry for system sizes $L \lesssim \xi$. (See Sects. III and IV of Ref. and references therein.) Here, we attempt to make the best of this unsatisfactory state of affairs. We simply take the point of view that, for the lattice sizes (up to $L=32$) we can simulate, the liquid and the weakly ordered bond crystal are indistinguishable.
RVB trial wave function
-----------------------
In quantum Heisenberg models, competing interactions that frustrate the order have the potential to stabilize exotic quantum phases, but they also render the problem computationally intractable on large lattices. Frustrating interactions of even infinitesimal strength cause a sign problem[@Loh90] that makes quantum Monte Carlo calculations unfeasible. Moreover, the size of the Hilbert space grows exponentially with system size and is thus beyond the capability of exact diagonalization calculations if we want to get near the thermodynamic limit. (The record for spin half has recently jumped from 42 sites[@Richter04; @Richter10; @Nakano11; @Lauchli11] to 48 sites,[@Lauchli12] a terribly impressive technical feat that nonetheless limits us to two-dimensional length scales $\sim\!\sqrt{48}$ that are quite small.) An approximate method based on good trial wave functions is therefore one of the few remaining possibilities for large systems.
We consider a lattice of $2N$ spins and a factorizable RVB wave function of the form $$\lvert \psi \rangle = \sum_v \prod_{[i,j] \in v} h(\mathbf{r}_{ij}) \lvert v \rangle,$$ where the sum is over all partitions of the lattice into $N$ directed pairs $v = ( [i_1,j_1], [i_2,j_2], \ldots, [i_N,j_N] )$. To every such *dimer covering* $v$, there is a corresponding singlet product state; e.g., $$\lvert v \rangle = \frac{1}{2^{N/2}}\bigotimes_{[i,j] \in v} \Bigl( \lvert \uparrow_{i} \downarrow_{j} \rangle
- \lvert \downarrow_{i} \uparrow_{j} \rangle \Bigr)$$ in the $S=1/2$ case. The set $\mathcal{V} = \{ \lvert v \rangle \}$ of all possible singlet product states is both overcomplete and nonorthogonal and constitutes the so-called valence bond basis.
We can now break up the lattice into two sublattices—groups of sites A and B, equal in number—and restrict ourselves to a reduced basis in which valence bonds connect only sites in opposite sublattices (i.e., $v \in \mathcal{V}_{\text{AB}} \simeq S_N$, rather than $v \in \mathcal{V} \simeq S_{2N}/Z_2^N$). We adopt the convention that each bond $[i,j]$ is arranged with site $i$ in sublattice A and site $j$ in sublattice B. This has the advantage of rendering the overlap strictly positive: $\langle v | v' \rangle = 2^{N_l(C)- N}$, where $N_l(C)$ is the number of loops in the double dimer covering $C = (v,v')$. (In this “bosonic” convention, the singlets are AB directed bonds. In the complementary “fermionic” convention, the bonds are directionless and all signs are moved into the overlaps.[@Anderson87; @Capriotti01; @Yunoki04; @Cano10; @Li12])
To start, we consider two families of trial state, each built using a bipartite bond basis consistent with one of two static choices of sublattice , viz., the checkerboard and stripe patterns shown in Fig. \[FIG:patterns\]. Later in the paper, we go on to describe a procedure in which the trial state is built using an unrestricted bond basis and the sublattice (and hence the Marshall sign convention) is determined dynamically.
The RVB wave function is quite expressive. Its degrees of freedom are the full set of $h(\mathbf{r})$ values with the bond vector $\mathbf{r}$ spanning all lengths and orientations that can be achieved on an $L \times L$ cluster with periodic boundary conditions and that are unique up to whatever symmetries are enforced. (Still, the total number of parameters grows only linearly with the number of spins, which is radically slower than the number of states in the total spin singlet sector.) Previous calculations of this kind[@Lou07; @Beach09] considered only the checkerboard AB pattern and imposed on $h(\mathbf{r}) = h(x,y)$ the full symmetry of the lattice, such that $h(x,y) = h(|x|,|y|) = h(|y|,|x|)$. In this calculation, we impose a less restrictive condition, $h(x,y) = h(|x|,|y|)$, that respects reflection symmetry across the lines $x=0$ and $y=0$ but not across the lines $y=\pm x$. For the checkerboard pattern, the number of free parameters is $(L/2-\eta)(L/2-1)$, where $\eta = (L/2 \mod 2)$ distinguishes between $L/2$ even and odd. For the stripe pattern, the count is only slightly higher: $(L/2+1)(L/2-1) = L^2/4-1$.
To recapitulate, our work involves a basis choice. We do not construct the trial wave functions from the largest possible set of valence bond states in which the spins are joined in all possible ways. Instead, we obtain a more restricted basis by dividing the system into two groups of sites (A and B) and keeping only states in which bonds connect A sites and B sites (bipartite bonds). No approximation is involved in this basis choice since the restricted basis is so massively overcomplete that even this subset still spans the relevant part of the Hilbert space.
But in assigning A and B labels to the sites, we are making a choice about the form of the trial wave function. By working with the checkerboard and stripe AB patterns, we are in essence adapting the trial wave function to $g = 0$ and $\infty$, respectively, and taking advantage of the Marshall sign rules that exist in those two limits. We are not biasing the wave function, however, at least not in the sense that we are building in magnetic order. The wave functions constructed from either AB pattern are fully capable of representing nonmagnetic states.
Sampling algorithm
------------------
Every measurement $\langle \hat{O} \rangle = \langle \psi | \hat{O} | \psi \rangle / \langle \psi | \psi \rangle$ is equivalent to $\langle\!\langle O \rangle\!\rangle$, an ensemble average of the appropriate estimator $O:~\!C \to O(C)$ in the gas of fluctuating loops described by $$\label{EQ:Z0}
Z = \frac{1}{q^N} \sum_{C} q^{N_l(C)} \prod_{[i,j] \in C} h_{ij}.$$ As before, $C = (v,v')$ is a loop configuration arising from the superposition of two dimer coverings, and $N_l(C)$ counts the number of loops. The value $q=2$ is the loop fugacity appropriate for $S=1/2$. When Marshall’s theorem holds, the bond amplitudes satisfy $h_{ij} \ge 0$ and thus every term in Eq. is nonnegative. This model is amenable to Monte Carlo simulation. We now outline a simple and efficient algorithm for performing the stochastic sampling.
As a formal trick (in the spirit of Ref. ), we enlarge the phase space from $\Phi_0$ to $\Phi_0 \times \Phi_1 \times \cdots \times \Phi_N$, where $\Phi_n$ is the set of configurations in which $2n$ free endpoints have been introduced by breaking $n$ valence bonds. (The system has been converted to one of both closed loops and open strings.) We take the partition function to be $$Z = \frac{1}{q^N} \sum_{C} q^{N_l(C)}\sigma^{N_s(C)}
\prod_{[i,j] \in C} h_{ij}.$$ The configurations $C$ are now assembled from all possible partial coverings $v = ( [i_1,j_1], [i_2,j_2], \ldots, [i_n,j_n] )$ of variable length $0 \le n \le N$, and $\sigma$ is introduced as a fugacity for the open strings \[numbering $N_s(C) = N-n$\]. The loop-only sector corresponds to the original partition function, $Z_0 = \langle 1 \rangle_{\Phi_0}$. (In each string sector there is a Green’s function defined by the string endpoints: $G_{ij} = \langle \delta_{i,\alpha_1} \delta_{j,\beta_1} \rangle_{\Phi_1}$, $G_{ij;kl} = \langle \delta_{i,\alpha_1} \delta_{j,\beta_1} \delta_{k,\alpha_2}
\delta_{l,\beta_2} \rangle_{\Phi_2}$, etc. Here, $\alpha_n$ and $\beta_n$ denote the positions of the head and tail of the $n^{\text{th}}$ string. It is worth emphasizing that these $2n$-point Green’s functions do not coincide with expectation values of the physical spin operators. In general, we must take all measurements in the $\Phi_0$ configuration space using the loop estimators derived in Ref. .)
We will consider a process that involves breaking a single valence bond ($\Phi_0 \to \Phi_1$) to produce an open string whose two endpoints (the “head” and “tail”) serve as walkers subject to Monte Carlo updates. The walkers move via a series of two-step motions that involve drawing a new bond and erasing an old one. When the walkers meet, the loop is closed ($\Phi_1 \to \Phi_0$). Figure \[FIG:updates\] shows an example circuit. The fives successive steps shown in panels (b)–(f) produce an overall change in the relative weight $$\label{EQ:step5}
\frac{\sigma}{h_{5,4}} \times
\frac{h_{1,4}}{qh_{1,2}} \times
\frac{h_{5,8}}{qh_{7,8}} \times
\frac{qh_{3,2}}{h_{3,6}} \times
\frac{h_{7,6}}{\sigma}.$$
Since we have chosen the bond amplitudes $h_{ij}$ to be nonnegative, we can define a local amplitude $H_i = \sum_j h_{ij}$ and a total overall amplitude $\mathsf{H} = \sum_{i} H_i = \sum_{ij} h_{ij}$. These definitions will be useful in the derivations that follow.
To begin, let us consider processes that take the system from the space of loops to the space of loops and one string. We move from a configuration $C \sim [i,j]$ to a configuration $C' \sim (i)(j)$ by breaking a bond $[i,j]$ and thus leaving string endpoints $(i)$ and $(j)$. The transition probabilities for breaking and repairing the bond obey the detailed balance equation $$W^{\text{break}}_{[i,j]} P(i) \pi_C = W^{\text{repair}}_{(i)(j)}P(j | i) \pi_{C'}.$$ Here $P(i)$ is the probability of choosing a site $i$ whose bond we want to break, and $P(j|i)$ is the probability of choosing $j$ given a walker (string endpoint) at site $i$. $\pi_C$ and $\pi_{C'}$ represent the likelihood of the system being found in configurations $C$ and $C'$. Their ratio is given by $$\frac{\pi_{C'}}{\pi_C} = \frac{\sigma}{h_{ij}}.$$ If we choose which bond to break according to the distribution of local bond weight $P(i) = H_i/\mathsf{H}$ and choose walker movements according to the distribution $P(j | i) = h_{ij} / H_i$, then $$\delta = \frac{W^{\text{break}}_{[i,j]} }{W^{\text{repair}}_{(i)(j)}} = \frac{P(j | i) \pi_{C'}}{P(i) \pi_C}
= \frac{\sigma}{\mathsf{H}}.$$ We are free to choose $\sigma = \mathsf{H}$, in which case the transition probabilities $W^{\text{break}}_{[i,j]}$ and $W^{\text{repair}}_{(i)(j)}$ are equal and unit-valued.
For motion of the walkers within $\Phi_1$, we need to know the transition rates between configurations $C\sim (i)[j,k]$ and $C' \sim [i,j](k)$. This represents a process in which a walker at $i$ draws a new bond to some site $j$ and then erases the preexisting bond connecting $j$ to $k$, thus leaving the walker at site $k$. The detailed balance equation is $$W^{\text{walk}}_{i\to k} P(j | i) \pi_C = W^{\text{walk}}_{k \to i} P(j | k) \pi_{C'}.$$ The ratio $$\frac{\pi_{C'}}{\pi_C} = q^{\delta N_l} \frac{h_{ij}}{h_{jk}}$$ depends on $\delta N_l = N_l(C')-N_l(C) = \pm 1$ (or 0 if the moves do not respect a fixed lattice bipartition; see discussion in Sect. \[SECT:dynamic\]). As before, we attempt moves according to the distribution $P(j | i) = h_{ij} / H_i$. Then, $$\delta = \frac{ W^{\text{walk}}_{i\to k} }{ W^{\text{walk}}_{k\to i} } = \frac{P(j|k)}{ P(j|i) } \frac{\pi_{C'}}{\pi_C}
= \frac{H_i}{H_k} q^{\pm 1},$$ which can be solved in the usual way as $W^{\text{walk}}_{i\to k} = \delta/(1+\delta)$ or $W^{\text{walk}}_{i\to k} = \min(1,\delta)$.
Note that the transition rate does not depend on the ratio of bond amplitudes, as it would if we had, for example, selected a site uniformly with $P(j | i) = 1/N$. The ratio $h_{ij}/h_{jk}$ may fluctuate wildly over many orders of magnitude, so subsuming it into the sampling maximizes the efficiency of the algorithm.
In the case of a translationally invariant system, the amplitude for pairing spins at $i$ and $j$ must be a function of the vector $\mathbf{r}_{ij}$ connecting the two sites; i.e., $h_{ij} = h(\mathbf{r}_{ij})$. Hence, $H = \mathsf{H}/N = H_i = \sum_{\mathbf{r}} h(\mathbf{r})$ for all $i$, which implies that $P(i) = H_i/\mathsf{H} \to 1/N$ is uniform and $P(j | i) = h_{ij}/H_{i} \to h(\mathbf{r}_{ij})/H$. The algorithm can be summarized as follows:
1. Pick any valence bond $[i,j]$ (by choosing $i$ uniformly from the set of A sublattice sites and then selecting its partner site in $v$ or $v'$) and break it. The resulting string has endpoints at $\mathbf{R} = \mathbf{r}_i$ and $\mathbf{R}' = \mathbf{r}_j$.
2. To move the head, choose a new bond vector $\mathbf{r}$ from the distribution $h(\mathbf{r})/H$. So long as $\mathbf{R}+\mathbf{r} \neq \mathbf{R}'$, attempt to draw a new bond from $\mathbf{R}$ to $\mathbf{R}+\mathbf{r} = \mathbf{r}_k$ (for some $k$). The bond $[k,l]$ that already exists at that site is then erased and the walker is moved to $\mathbf{r}_l$. The move is accepted with probability 1/2 if its effect is to join another loop to the string and with probability 1 otherwise.
3. Otherwise, if $\mathbf{R} + \mathbf{r} = \mathbf{R}'$, close the open string by drawing a new valence bond from $\mathbf{R}$ to $\mathbf{R}'$.
The worm algorithm described here is ergodic and guaranteed to have a high acceptance rate. This is in contrast to the original bond-swap scheme proposed in Ref. , wherein two A-site or B-site bond endpoints sitting diagonally across a plaquette are swapped using Metropolis sampling. This antiquated algorithm runs into difficulty when the function $h(\mathbf{r})$ is short ranged. In particular, short bonds that are adjacent but not sharing a common plaquette generate long bonds under rearrangement, so whenever the amplitudes for long bonds become small, the acceptance rate can become correspondingly small. Worse, there are typically many trapping configurations from which the simulation cannot emerge. The worm algorithm does not suffer from these problems, because it can traverse any local barriers by stepping outside the space of closed loops. (We make no claims of novelty in this regard. Other approaches to overcome the sampling difficulty have been presented elsewhere.[@Albuquerque10; @Tang11; @Sandvik06])
\[SECT:dynamic\]Fluctuating sublattice assignment
-------------------------------------------------
The discussion in the previous section was specific to the case in which (i) the AB pattern is regular and (ii) the $\mathbf{r}$ vectors that have nonzero $h(\mathbf{r})$ only connect sites in opposite sublattices. If those conditions hold, there are only two possible consequences to the motion of the open string: a loop is joined to the string ($\delta N_l = -1$) or a loop is split off from it ($\delta N_l = +1$). In both cases, represented in Fig. \[FIG:dynamic\] by panels (a)$\to$(b) and (d)$\to$(e), the AB pattern itself is left undisturbed.
More generally, as the open string propagates it lays down a chain of singlet bonds whose alternating site labels may be at odds with the traversed sites’ current AB assignments. A simple workaround is to flip the sublattice labels as required to correct the mismatch. The relevant processes are now those in which a moving open string absorbs a closed loop ($\delta N_l = -1$) or reorganizes itself without impinging on any additional sites ($\delta N_l = 0$). The first case is depicted in Fig. \[FIG:dynamic\] by panels (a)$\to$(c) and the second by (d)$\to$(f) or (d)$\to$(g). A crucial consideration is that, since the singlets are directional, flipping sublattice labels along a loop segment has the effect of reversing a chain of singlet bonds. If an odd number of singlets is effected, the overall sign of the wave function will change. This is true for all $\delta N_l = 0$ worm steps.
The sublattice mismatch can either be a temporary condition—lasting only until the worm updates succeed in laying down a global AB pattern that is an invariant of the worm motion—or it may be that the motion described by a given $h(\mathbf{r})$ is incompatible with any static AB site . For example, consider the one-parameter family of short-range states on the square lattice described by $h(\pm 1,0) = h(0,\pm 1) = \cos\theta$ and $h(\pm 1,\pm 1) = \sin\theta$ (with $0 \le \theta \le \pi/4$). Regardless of the initial sublattice —it can be any random assignment having an equal number of A and B labels—the simulation will dynamically establish the checkerboard pattern provided that $\theta = 0$. We keep track of the AB pattern by measuring a function $\Lambda(\mathbf{Q})=\sum_{\mathbf{r},\mathbf{r'}} e^{i\mathbf{Q}\cdot (\mathbf{r}-\mathbf{r'})} \langle\!\langle \lambda(\mathbf{r}) \lambda(\mathbf{r'}) \rangle\!\rangle$, where $\lambda(\mathbf{r})$ takes the value $-1$ or $1$ depending on the current sublattice assignment at site $\mathbf{r}$. If $\theta = 0$, $\Lambda(\mathbf{Q})$ starts off broad but systematically flows toward the distribution consisting of a single delta function peak at $\mathbf{Q} = (\pi,\pi)$; once that is achieved, the pattern ceases to evolve. Similar is exhibited at $\theta = \pi/4$, where the system settles into a static pattern with either $\mathbf{Q}=(\pi,0)$ or $\mathbf{Q}=(0,\pi)$. Only in those two extreme cases is the sublattice pattern eventually static and the simulation sign-problem free.
\[SECT:results\]Results
=======================
n $Z$ $-L^2\,C_1$ $L^2\,C_2$ $L^4\,M^2(\pi,\pi)$ $L^4\,M^2(\pi,0)$ $L^2\,\lvert D \rvert$ $L^4\, 4D^2 / 3$
---- ------------ ------------- ------------- --------------------- ------------------- ------------------------ ------------------
0 1559232 22241280 9902080 113983488 17383424 4376064 102133760
1 13008384 194568192 104726528 1117618176 139902976 28540928 645455872
2 66018816 997232640 585695232 6104383488 709410816 127591424 2844606464
3 223842816 3395051520 2137292800 21861335040 2381496320 389861376 8395677696
4 568694016 8564477952 5689352192 57653526528 6069354496 932687872 19758309376
5 1108661760 16547069952 11594661888 116342292480 11792498688 1697314816 35459866624
6 1767412224 25797685248 18932629504 189239033856 18888998912 2580870144 53692563456
7 2302253568 32679444480 25148850176 250229981184 24519589888 3165620224 65523884032
8 2528419968 34418749440 27661209600 275349995520 27030159360 3329164288 68794482688
9 2302253568 29878050816 25148850176 250229981184 24519589888 2857185280 58976903168
10 1767412224 21512073216 18932629504 189239033856 18888998912 2089987072 43192369152
11 1108661760 12538503168 11594661888 116342292480 11792498688 1223688192 25387999232
12 568694016 5848903680 5689352192 57653526528 6069354496 594391040 12360392704
13 223842816 2070282240 2137292800 21861335040 2381496320 218601472 4580990976
14 66018816 528863232 585695232 6104383488 709410816 60980224 1313734656
15 13008384 84836352 104726528 1117618176 139902976 11331584 254992384
16 1559232 6254592 9902080 113983488 17383424 1074688 31887360
As a test of the worm implementation, we compare its output to *analytical* results obtained for the $4\times 4$ lattice. We exploit the fact that the bipartite valence bond basis $\mathcal{V}_{\text{AB}}$ for $2N$ spins is isomorphic to the set of permutations on $N$ elements.[@Beach06] Hence, the basis states have a natural lexical ordering via the Lehmer code[@Lehmer60; @Knuth73] and can easily be enumerated. For $4\times 4 = 16$ sites, the total number of the states is only $8! = 40\,320$, which means that expectation values of the trial wave function can be evaluated exactly at very little computational cost. Moreover, we can carry out the calculation symbolically. Each observable takes the form of a rational function of order \[16/16\]: $$\label{EQ:rational_polynomial}
\langle \hat{O} \rangle = \frac{O(x)}{Z(x)} = \frac{3}{4} \frac{\sum_{k=0}^{16} o_k x^k}{\sum_{l=0}^{16} z_l x^l}.$$ The argument of the polynomials appearing in the numerator and denominator is the real-valued ratio $x = h(2,1)/h(1,0)$, and the coefficients $o_k$ and $z_k$ are all integers. Specific values for various observables are listed in Table \[TAB:coefficients\].
For this test we have on the nearest- and next-nearest- spin correlation functions, $C_1=\frac{1}{L^2}\sum_{\langle i,j\rangle}\langle\mathbf{S}_i\cdot \mathbf{S}_{j}\rangle $ and $C_2=\frac{1}{L^2}\sum_{\langle \langle i,j\rangle \rangle} \langle \mathbf{S}_i\cdot \mathbf{S}_{j} \rangle$; the $\mathbf{Q} = (\pi,\pi)$ staggered and $\mathbf{Q} = (\pi,0)$ stripe magnetization, $M^{2}({\mathbf{Q}})=\frac{1}{L^4}\sum_{\mathbf{r,r'}}(-1)e^{i\mathbf{Q}\cdot(\mathbf{r}-\mathbf{r}')}\langle \mathbf{S}_{\mathbf{r}}\cdot \mathbf{S}_{\mathbf{r}'}\rangle$; and the order parameter for a columnar dimer crystal, $D^2=\frac{1}{L^4}\sum_{\mathbf{r,r'}}(-1)^{\mathbf{e}_x\cdot(\mathbf{r}+\mathbf{r}')}\langle (\mathbf{S}_{\mathbf{r}}\cdot
\mathbf{S}_{\mathbf{r}+\mathbf{e}_x})(\mathbf{S}_{\mathbf{r}'}\cdot \mathbf{S}_{\mathbf{r}'+\mathbf{e}_x})\rangle$. We have verified that the worm algorithm, conventional bond swap Monte Carlo, and exact evaluation give consistent results for all these quantities.
The comparison of the energetics is shown in Fig. \[FIG:factoradic\]. Note that in Figs. \[FIG:factoradic\](a) and \[FIG:factoradic\](b), the stochastic evaluation of $C_1$ and $C_2$ continues to work in some range of $x < 0$ but breaks down as $x$ becomes strongly negative. For the symbolic result, the determination of the best energy is carried out by considering the two-parameter function $\mathcal{E}(x,g)/J_1L^2 = C_1(x) + gC_2(x)$, which is known exactly by way of Eq. . For every value of the relative coupling strength $g$, the optimal value of $x$ \[Fig. \[FIG:factoradic\](c)\] is the one that produces the lowest energy \[Fig. \[FIG:factoradic\](d)\] according to $$\label{EQ:energy_min_4x4}
E(g) = \mathcal{E}(x_{\text{opt}},g) = \underset{x}{\text{min}}\, \mathcal{E}(x,g).$$ In practice, Eq. represents a root-finding problem in $x$ for $\partial \mathcal{E}(x,g)/\partial x = 0$; this is solved via Newton-Raphson. We find that the optimized value $x_{\text{opt}}$ is positive for weak frustration. It decreases monotonically from its nonfrustrated value, $x_{\text{opt}} = 0.2780138519$, and drops below zero when the coupling strength exceeds $g = 0.40756$. This marks the point at which the Marshall sign rule first fails. For reference (it may be of use in benchmarking RVB calculations accomplished by other methods, e.g., Ref. ), we report that the specific values $x_{\text{opt}} = 0.006787458952$, $-0.03777121711$, $-0.07881072679$, and $-0.1128184711$ obtain at coupling strengths $g = 0.40$, $0.45$, $0.50$, and $0.55$.
Having established confidence in our numerical implementation, we proceed with unbiased optimization calculations using a static sublattice assignment on lattices up to size $L=32$. Convergence is limited by statistical uncertainty in the (energy to bond count) correlation function that determines the local energy gradient,[@Lou07] and it is difficult to optimize reliably for larger system sizes. (See Appendix \[SEC:optimization\] for more details.) We first consider the checkerboard AB pattern. At $g=0$, the bond amplitudes are given an initial value $$h(x,y) = \bigl[\min(x,L-x)^2 + \min(y,L-y)^2\bigr]^{-3/2}$$ for $\lvert x \rvert + \lvert y \rvert$ odd and zero otherwise. The new set of amplitudes obtained from this first run serves as the input for the next optimization process. That is to say, we daisy chain the calculations, at each step using the converged result at $g$ to seed the simulation at $g + \delta g$. An analogous procedure is carried out for the stripe AB pattern, starting from $g=\infty$ and stepping the relative coupling down.
One finds that the two sets of simulations do not join smoothly but instead meet with strongly opposite slopes $dE/dg$. A careful extrapolation to the thermodynamic limit, presented in Fig. \[FIG:crossings\], puts the location of the energy level crossing at $g_{\text{c}2} \doteq 0.5891(3)$. As Fig. \[FIG:phases\] makes clear, this point represents the rightmost edge of an intermediate phase that is magnetically disordered. The leftmost edge sits at $g_{\text{c}1} \doteq 0.54(1)$, where the $\mathbf{Q}=(\pi,\pi)$ antiferromagnetism vanishes in a continuous fashion. As a rough gauge of the quality of the RVB trial wave function, we note that for $g=0.5$ the energy density extrapolates to $E_{\text{RVB}} = -0.49023(2)$ in the thermodynamic limit. This result is bracketed by the energies of the best projected entangled pair states (PEPS) with bond dimension $D=3$ \[$E_{\text{PEPS}} = -0.48612(2)$; see Ref. \] and $D=9$ \[$E_{\text{PEPS}} = -0.4943(7)$; see Ref. \].
An important detail is that the optimizations are carried out with the bond amplitudes constrained to have $x$- and $y$-axis reflection symmetry but not necessarily $90^{\circ}$ rotation symmetry. In the case of the checkerboard simulation, the amplitudes nonetheless realize the full lattice symmetry under optimization up to large values of the relative coupling. For small lattice sizes $L=4, 6, 8$, the symmetry breaks down beyond values $g \approx 0.51, 0.55, 0.57$. For all larger sizes, that point is pushed well to the right of $g_{\text{c}2}$. This means that, in the thermodynamic limit, $h(\mathbf{r})$ shares a common symmetry across both the staggered magnetic phase and the disordered intermediate phase. But it experiences a sudden break at the onset of stripe magnetic order, dropping from $C_4$ to $C_2$.
In the vicinity of $g=0$, the optimized bond amplitudes are positive definite and an almost perfect function of bond length. As the frustration increases, the amplitudes begin to deviate from circular symmetry, developing strong lobes of weight along the x and y axes. Bonds not aligned along those preferred directions become increasingly short ranged, and the eight knight’s move bonds, those symmetry equivalent to $h(2,1)$, eventually trend through zero to negative values. The extrapolation shown in Fig. \[FIG:marshall\] pinpoints the breakdown of the Marshall sign rule at $g_{\text{M}1} \doteq 0.398(4)$. What this suggests is that there is strict adherence to a checkerboard Marshall sign rule only below $g_{\text{M}1}$; in the range $g_{\text{M}1} < g < g_{\text{c}2}$, the sign rule is violated, even though the overall sign structure is still partially consistent with the checkerboard pattern. \[There is no indication that the amplitudes of any other bond type are on track to change sign. Attempts to extrapolate the amplitudes next most likely to turn negative, viz. $h(4,1)$ and $h(6,1)$, put their vanishing points deep in the intermediate phase or beyond it.\] We find that the on the large coupling side is not comparable. There, the coupling at which bond amplitudes first go negative scales as $g_{\text{M}2} \sim L^4 $ and hence does not converge in the thermodynamic limit. We interpret this to mean that the static stripe pattern is only ever a weak description of the Marshall sign structure. See Fig. \[FIG:amplitudes\].
We have attempted to confirm this picture by running simulations in which the Marshall sign structure is determined dynamically. More specifically, we want to verify that the strongly first-order transition at $g_{\text{c}2}$ is not merely an artifact of two static, incompatible sublattice conventions colliding. And we would like to see if any pattern other than checkerboard or stripe could emerge on its own. If permitted, might the system’s sublattice structure smoothly interpolate over some range of $g$, with the peak in $\Lambda(\mathbf{Q})$ migrating from $(\pi,\pi)$ to $(\pi,0)$? Or perhaps with the peak in $\Lambda(\mathbf{Q})$ broadening into incoherence? We follow the procedure outlined in Sect. \[SECT:dynamic\], whereby the sublattice is no longer fixed and the worm motion itself is allowed to reconfigure the current AB pattern. Our approach is to simulate for various $g$ values—with *no* daisy chaining—in each case starting from a random AB pattern and a random loop configuration. The bond amplitudes are initialized with $h(\mathbf{r})$ forming a broad peak around $\mathbf{r} = \mathbf{0}$ and having no zero entries. We perform a crude simulation in which the signs associated with the worm updates are thrown away. (See Appendix \[SEC:sign\_problem\].) Otherwise, the optimization of $h(\mathbf{r})$ proceeds as before. What we find is a result that exactly tracks the state of lower energy produced by assuming one of the two static AB patterns. The simulation flows to the checkerboard for all $g < g_{\text{c}2}$ and to the stripe for all $g >g_{\text{c}2}$; the peak in $\Lambda(\mathbf{Q})$ jumps discontinuously. Obviously we should not read too much into a result that follows from an uncontrolled approximation (sampling by ignoring the signs), but it does give us a sense that the stability of the checkerboard pattern through the intermediate phase and the abrupt change in Marshall sign structure at $g_{\text{c}2}$ might be genuine features of the model.
The optimized state in the intermediate phase is definitely not a bond crystal. For a given lattice, the dimer correlations are somewhat enhanced in the strongly frustrated region, but with increasing lattice size they show clear convergence to zero. Still, spatially resolved dimer correlations do give us important information. One can see in Fig. \[FIG:dimer\_pattern\] that the optimized state shows the same pattern of dimer correlation and anticorrelation as the NNRVB, but it decays much faster as a function of dimer separation. The comparison is made more explicit in Fig. \[FIG:dimer\_corr\], which shows correlations along a line and a stack of dimers. The functions measured are $$\label{EQ:dimer_correlation_functions}
\begin{split}
C_\text{line}(d) &= \langle \hat{B}(0,0)\hat{B}(d,0) \rangle - \langle \hat{B}(0,0)\rangle\langle \hat{B}(d,0) \rangle, \\
C_\text{stack}(d) &= \langle \hat{B}(0,0)\hat{B}(0,d) \rangle - \langle \hat{B}(0,0)\rangle\langle \hat{B}(0,d) \rangle,
\end{split}$$ which we have expressed in terms of the $x$-directed bond operator $\hat{B}(x,y) = \mathbf{S}(x,y)\cdot\mathbf{S}(x+1,y)$.
\[SEC:conclusions\] Conclusions
===============================
We have used an optimized valence bond trial wave function to study the square-lattice $J_1$–$J_2$ Heisenberg model, with an eye to both mapping out the zero-temperature phase diagram and determining how the Marshall sign structure breaks down near the phase boundaries. In the first instance, we fix the AB sublattice to coincide with the order that exists at small and large coupling. For each lattice size, the intermediate phase is approached in two independent simulations (or, rather, chains of history-dependent simulations) by evolving the states progressively out of the two ordered phases, minimizing their energy at each step. These simulations are fully non-sign-problematic, since the AB pattern is fixed and the bond amplitudes are restricted to be positive.
Finite-size scaling of the dimer order parameter suggests that there is no long-range dimer order at any value of $g$. This is as expected, since the trial state explicitly ignores bond-bond correlations beyond those generated by the hardcore tiling constraint. Measurements of the staggered magnetization show clear evidence of a continuous phase transition in which the staggered magnetization vanishes at $g_{\text{c}1} \doteq 0.54(1)$. On the other edge of the intermediate phase, an energy level crossing at $g_{\text{c}2} \doteq 0.5891(3)$ results in the sudden disappearance of the otherwise robust stripe magnetization. This is accompanied by the restoration of the system’s rotational symmetry. (Since the trial state is least able to describe the intermediate phase—again, because of its lack of explicit bond-bond correlations—we should probably view $g_{\text{c}1}$ and $g_{\text{c}2}$ as upper and lower bounds, respectively, on the true positions of the phase boundaries.) We have also performed calculations (approximate and uncontrolled, but suggestive) in which no sublattice is put in by hand and the AB pattern is allowed to emerge dynamically. We find that, regardless of the initial sublattice assignment, the simulation reliably settles into the checkerboard pattern for all $g < g_{\text{c}2}$ and the stripe for all $g > g_{\text{c}2}$. Taken together, our results point to the checkerboard AB pattern being the best choice throughout the intermediate phase. Hence, within the context of our particular trial wave function scheme, we surmise that the state beyond $g_{\text{c}1}$ is a “bosonic” spin liquid with the lowest-lying magnetic excitations at $(\pi,\pi)$.
Figure \[FIG:amplitudes\] gives a quick summary of our results. We observe that at high frustration the bond amplitudes take on a highly anisotropic form. This is quite different from the long-bond to short-bond picture that is usually invoked. Recall that Liang, Ducot, and Anderson studied long-range RVB states on the square lattice with amplitudes $h \sim r^{-p}$ that decay as a power law in the bond length $r$. [@Liang88] In that framework, the state becomes magnetically disordered when $p$ exceeds a critical value of 3.3, [@Havilio99; @Havilio00; @Beach07b] and the entire family of states in the range $p > 3.3$ is continuously connected to $p=\infty$, which is the (short-bond-only) NNRVB. The intermediate phase state obtained in our simulations is of a quite different character: (i) the state is magnetically disordered not because its bond amplitudes are uniformly short ranged but because they have become short ranged over some sufficiently large angular interval of bond orientation; (ii) its spin and dimer correlations are distinct from those of the NNRVB; and (iii) the presence of many system-spanning bonds implies that the usual topological invariant for short-ranged RVB states, defined by the parity of bond cuts along a reference line,[@Diep05; @Albuquerque10] is almost certainly not a good quantum number.
This work was supported by a Discovery grant from NSERC of Canada.
\[SEC:optimization\]Numerical optimization of the RVB bond amplitudes
=====================================================================
The RVB ansatz assumes that the quantum amplitude $\psi(v)$ associated with each valence bond state $| v \rangle$ is of the factorizable form $$\label{EQ:rvb_ansatz}
\psi(v) \approx \prod_{[i,j] \in v} h(\mathbf{r}_{ij}) \equiv \prod_{\tilde{\mathbf{r}}} h(\tilde{\mathbf{r}})^{n(\tilde{\mathbf{r}};v)}.$$ The first product ranges over all pairs of spins forming a singlet bond. The second ranges over the minimal set of vectors $\tilde{\mathbf{r}}$ that are inequivalent under whatever lattice symmetries have been enforced. The whole-number exponent $n(\tilde{\mathbf{r}};v)$ represents how many times a bond amplitude $h(\mathbf{r})$, with $\mathbf{r}$ symmetry-equivalent to $\tilde{\mathbf{r}}$, appears in the product for a given $v$. \[Hence, $\sum_{\tilde{\mathbf{r}}} n(\tilde{\mathbf{r}}) = L^2/2 = N$, the number of bonds appearing in $| v \rangle$.\]
Accordingly, the energy expectation value is $$\label{EQ:energy_expectation_value}
E = \frac{\langle \psi | \hat{H} | \psi \rangle}{\langle \psi | \psi \rangle}
= \frac{\sum_{C} H(C) w(C) }{\sum_{C} w(C)} \equiv \langle\!\langle H \rangle\!\rangle,$$ where $H(C) = \langle v | \hat{H} | v' \rangle / \langle v | v' \rangle$ is the loop estimator of the Hamiltonian. The notation $\langle\!\langle \cdot \rangle\!\rangle$ denotes averaging with respect to the Monte Carlo weight $$\label{EQ:Monte_Carlo_weight}
w(C) = \langle v | v' \rangle\psi(v)\psi(v') = \pm q^{N_l(C)}\prod_{\tilde{\mathbf{r}}} h(\tilde{\mathbf{r}})^{n(\tilde{\mathbf{r}};C)}.$$ Here, each configuration $C = (v,v')$ is a superposition of two dimer coverings, and the sum $n(\tilde{\mathbf{r}};C) \equiv n(\tilde{\mathbf{r}};v)+n(\tilde{\mathbf{r}};v')$ is the combined count of $\tilde{\mathbf{r}}$-type bonds in states $| v \rangle$ and $| v' \rangle$. The $\pm$ on the right-hand-side of Eq. acknowledges that the configuration weight may be negative if the sublattice pattern is not fixed. By way of the identity $$\frac{\partial w(C)}{\partial h(\tilde{\mathbf{r}})} = \frac{n(\tilde{\mathbf{r}};C)w(C)}{h(\tilde{\mathbf{r}})},$$ we find that the downhill direction in the energy landscape by $\{ h(\tilde{\mathbf{r}}) \}$ is related to the energy to bond count correlation function $$\label{EQ:energy_bond_corr_func}
G_k(\tilde{\mathbf{r}}) \equiv -\frac{\partial E}{\partial \log h(\tilde{\mathbf{r}})} = \langle\!\langle H \rangle\!\rangle_k \langle\!\langle n(\tilde{\mathbf{r}}) \rangle\!\rangle_k
- \langle\!\langle H n(\tilde{\mathbf{r}}) \rangle\!\rangle_k.$$ In anticipation of Eq. , we have used $\langle\!\langle \cdot \rangle\!\rangle_k$ to denote averaging with respect to the $k^{\text{th}}$ Monte Carlo bin.
Our optimization procedure is carried out as follows. For a given logarithmic amplitude $\lambda^{(1)} = \log h(\tilde{\mathbf{r}})$, we generate a sequence of (not always energy-reducing) steps $$\label{EQ:opt_scheme}
\lambda^{(k+1)}:= \frac{\lambda^{(k)}R\,\delta \lambda}{k^{1/3}} \operatorname{sgn}G_k.$$ $R$ is a random number chosen from the uniform distribution on the interval $[0,1]$, and $k = 1, 2, \ldots, 1000$ counts the steps taken through the landscape. The $1/3$ power ensures that the step size envelope decreases by a factor 10 over the course of 1000 steps. The optimization is run repeatedly with restarts for step sizes beginning at $\delta \lambda = 0.1$ and reduced by successive powers of two until convergence is achieved.
The most serious difficulty is that the correlation function estimates $G_k(\tilde{\mathbf{r}})$ become increasingly noisy for large system sizes, to the point where the determination of $\operatorname{sgn}G_k(\tilde{\mathbf{r}})$ is no longer reliable. The problem is most acute for the longest bonds, which appear least frequently and thus have the worst statistics. (The bond amplitudes, which represent the probability of a given type of bond appearing during the Monte Carlo sampling, fall off rapidly as a function of bond length.)
In small amounts, this noise does not interfere with the energy optimization. It simply overlays a randomizing motion, somewhat akin to the effect of nonzero temperature in simulated annealing. Nonetheless, good convergence requires that the noise fall below a certain threshold (set by the depth and curvature of the well in which the energy minimum sits.) In practice, mitigating the noise means taking the Monte Carlo bin size large enough so that the longest bonds in the system (with length $\lvert \tilde{\mathbf{r}} \rvert \sim L$) appear often enough in the sampling. This consideration sets the limit on the systems sizes we can optimize.
\[SEC:sign\_problem\]Sign-problematic simulations
=================================================
The energy computed by ignoring signs \[i.e., by sampling with respect to the *magnitude* of Eq. \] is $$E^\star = \frac{\sum_{C} H(C) \lvert w(C)\rvert }{\sum_{C} \lvert w(C)\rvert} \equiv \llbracket H \rrbracket.$$ Making the substitution $w = \lvert w\rvert \operatorname{sgn}w$, we can rewrite Eq. as the ratio of averages $$E = \frac{\sum_{C} H(C) \lvert w(C)\rvert \operatorname{sgn}w(C)}{\sum_{C} \lvert w(C)\rvert \operatorname{sgn}w(C)} \equiv
\frac{\llbracket H \operatorname{sgn}w \rrbracket}{\llbracket \operatorname{sgn}w \rrbracket};$$ hence, the energy discrepancy $\Delta E = E^\star - E$ takes the form of a correlation function $$\Delta E = E^\star - E =
\frac{\llbracket H \rrbracket \cdot \llbracket \operatorname{sgn}w \rrbracket- \llbracket H \operatorname{sgn}w \rrbracket }{ \llbracket \operatorname{sgn}w \rrbracket }.$$
If the $\operatorname{sgn}w$ term fluctuates within the simulation so that $\llbracket \operatorname{sgn}w \rrbracket \approx 0$, evaluation of $\Delta E$ is impossible due to large statistical uncertainties. Despite this, the actual value of $\Delta E$ may itself be small if there is only a weak correlation between the sign and the energy estimator. Moreover, $\Delta E$ is identically zero if the $h(\mathbf{r})$ values evolve to produce a static sublattice . So, at the very least, we can view as a rigorous result the fact that no new static pattern emerged over the course of our simulations.
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---
abstract: 'The neutrino oscillation experiment KamLAND has provided us with the first evidence for $\bar \nu_e$ disappearance, coming from nuclear reactors. We have combined their data with all solar neutrino data, assuming two flavor neutrino mixing, and obtained allowed parameter regions which are compatible with the so-called large mixing angle MSW solution to the solar neutrino problem. The allowed regions in the plane of mixing angle and mass squared difference are now split into two islands at 99% C.L. We have speculated how these two islands can be distinguished in the near future. We have shown that a 50% reduction of the error on SNO neutral-current measurement can be important in establishing in each of these islands the true values of these parameters lie. We also have simulated KamLAND positron energy spectrum after 1 year of data taking, assuming the current best fitted values of the oscillation parameters, combined it the with current solar neutrino data and showed how these two split islands can be modified.'
author:
- 'H. Nunokawa$^1$'
- 'W. J. C. Teves$^2$'
- 'R. Zukanovich Funchal$^2$'
title: |
-2.0cm [IFUSP-DFN/02-078\
-0.4cm IFT-P.001/2003]{}\
Determining the oscillation parameters by Solar neutrinos and KamLAND
---
Introduction {#sec:intro}
============
-0.3cm Many solar and atmospheric neutrino experiments have collected data in the last decades, giving evidence that neutrinos produced in the Sun and in the Earth’s atmosphere suffer flavor conversion. While the atmospheric neutrino results [@atmnuobs] may be understood by $\nu_\mu \to \nu_\tau$ conversion driven by a neutrino mass squared difference within the experimental reach of the accelerator based neutrino oscillation experiment K2K [@k2k], the mass squared difference needed to explain the solar neutrino data was, until quite recently, before the Kamioka Liquid scintillator AntiNeutrino Detector (KamLAND) [@kamland] has started its operation, too small to be inspected by a terrestrial neutrino oscillation experiment.
A number of different fits, assuming standard neutrino oscillations induced by mass and mixing [@fits] as well as other exotic flavor conversion mechanisms [@exotics], have been performed using the combined solar neutrino data from Homestake [@homestake], GALLEX/GNO [@gallex; @gno], SAGE [@sage], Super-Kamiokande-I [@superk] and SNO [@sno]. These analyses selected some allowed areas in the free parameter region of each investigated mechanism, but did not allow one to establish beyond reasonable doubt which is the mechanism and what are the values of the parameters that are responsible for solar $\nu_e$ flavor conversion. After the first result of the KamLAND (or KL hereafter) experiment [@kamland] this picture has changed drastically.
In the first part of this paper, we present the allowed region for the oscillation parameters in two generations for the entire set of solar neutrino data, for KamLAND data alone and for KamLAND result combined with all solar neutrino data, showing that this last result finally establishes the so called large mixing angle (LMA) Mikheyev-Smirnov-Wolfenstein (MSW) [@msw] solution as the final answer to the long standing solar neutrino problem [@bahcall], definitely discarding all the other mass induced or more exotic solutions. (For the first discussions on the complete “MSW triangle” which includes the LMA region, see Ref. [@mswtriangle].) In the second part, we speculate on the possibility of further constraining the oscillation parameters in the near future. For instance, we point out the importance of SNO neutral-current (NC) data in further constraining the LMA MSW solution. In particular, we discuss the consequence of a significant reduction (50 %) of the SNO neutral-current data uncertainty. Finally, we simulate the expected inverse $\beta$-decay $e^+$ energy spectrum after 1 year of KamLAND data taking, based on the best fitted values of the oscillation parameters. We combine this with the current solar neutrino data in order to show how the allowed parameter region can be modified.
-2.7cm
Determination of Oscillation Parameters {#sec:analysis}
=======================================
-0.3cm KamLAND has observed about 40% suppression of $\bar
\nu_e$ flux with respect to the theoretically expected one [@kamland], which is compatible with neutrino oscillations in vacuum in two generations. In this case the relevant oscillation parameters, which must be determined by the fit to experimental data, are a mass squared difference ($\Delta m^2$) and a mixing angle ($\theta$). We first obtained the allowed region in the ($\tan^2\theta$, $\Delta m^2$) plane compatible with all solar neutrino experimental data, then with KamLAND data alone, and finally we combine these two sets of data. -1.7cm
Solar Neutrino Experiments {#subsec:solar}
--------------------------
-0.3cm We have determined the parameter region allowed by the solar neutrino rates measured by Homestake [@homestake], GALLEX/GNO [@gallex; @gno], SAGE [@sage] and SNO (elastic scattering, charged-current and neutral-current reactions) [@sno] (6 data points) as well as by the Super-Kamiokande-I zenith spectrum data [@superk] (44 data points), assuming neutrino oscillations in two generations.
We have computed the $\nu_e \to \nu_e$ survival probability, properly taking into account the neutrino production distributions in the Sun according to the Standard Solar Model [@ssm], the zenith-angle exposure of each experiment, as well as the Earth matter effect as in Ref. [@exotics], except that here we solved the neutrino evolution equation entirely numerically. We then have estimated the allowed parameter region by minimizing the $\chi^2_\odot$ function which is defined as $$\label{chi}
\chi^2_\odot = \sum_{i,j=1,...,50}
\left[R_i^{\text{th}}-R_i^{\text{obs}} \right] \,
\left[\sigma_\odot^2 \right]^{-1}_{ij} \,
\left[R_j^{\text{th}}-R_j^{\text{obs}} \right]\,,$$ where $R_i^{\text{th}}$ and $R_i^{\text{obs}}$ denote the theoretically expected and observed event rates, respectively, which run through all 50 data points mentioned above, and $\sigma_\odot$ is the $50\times 50$ correlated error matrix, defined in a similar way as in Ref. [@exotics]. In this work we have treated the $^8$B neutrino flux as a free parameter.
In Figure \[fig1\] we show the region, in the $(\tan^2 \theta,\Delta m^2)$ plane, allowed by the Super-Kamiokande-I zenith spectrum data as well as by the rates of all other solar neutrino experiments at 90%, 95%, 99% and 99.73% C.L. In our fit we obtained a $\chi^2_{\odot}(\text{min})=37.7$ for 47 d.o.f (83 % C.L.), corresponding to the global best fit values $\Delta m^2= 7.5 \times 10^{-5}$ eV$^2$ and $\tan^2 \theta=0.42$.
KamLAND {#subsec:kam}
-------
-0.3cm KamLAND [@kamland] is a reactor neutrino oscillation experiment searching for $\bar \nu_e$ oscillation from over 16 power reactors in Japan and South Korea, mostly located at distances that vary from 80 to 344 km from the Kamioka mine, allowing KamLAND to probe the LMA MSW neutrino oscillation solution to the solar neutrino problem.
The KamLAND detector consists of about 1 kton of liquid scintillator surrounded by photomultiplier tubes that register the arrival of $\bar \nu_e$ through the inverse $\beta$-decay reaction $\bar \nu_e + p \to e^+ + n$, by measuring $e^+$ and the 2.2 MeV $\gamma$-ray from neutron capture of a proton in delayed coincidence. The $e^+$ annihilate in the detector, producing the total visible energy $E$ which is related to the incoming $\bar \nu_e$ energy, $E_\nu$, as $E=E_\nu-(m_n-m_p)+m_e$, where $m_n$, $m_p$ and $m_e$ are respectively, the neutron, proton and electron mass.
After 145.1 days of data taking, which corresponds to 162 ton yr exposure, KamLAND has measured 54 inverse $\beta$-decay events, where 87 were expected without neutrino conversion. These events are distributed in 13 bins of 0.425 MeV above the analysis threshold of 2.6 MeV (applied to contain the background under about 1 event).
We have theoretically computed the expected number of events in the $i$-th bin, $N_i^{\text{theo}}$, as $$N_i^{\text{theo}} = \int dE_\nu \, \sigma(E_\nu) \sum_k \phi_k(E_\nu)
P_{\nu_e \to \nu_e} \int_i dE \,R(E,E^\prime),$$ where $R(E,E^\prime)$ is the energy resolution function, $E$ the observed and $E^\prime$ the true $e^+$ energy, with the energy resolution $7.5\%/\sqrt{E(\text{MeV})}$. Here $\sigma(E_\nu)$ is the neutrino interaction cross-section and $\phi_k$ is the neutrino flux from the $k$-th power reactor, we have included all reactors with baseline smaller than 350 km in the sum. $P_{\nu_e \to \nu_e} \equiv P_{\bar \nu_e \to \bar \nu_e}$ (if CPT is conserved, which we will assume here) is the familiar neutrino survival probability in vacuum (the matter effect is negligible here), which is equal to one in case of no oscillation, and explicitly depends on $\Delta m^2$ and $\tan^2 \theta$.
We were able to compute the region, in the $(\tan^2 \theta,\Delta m^2)$ plane, allowed by the KamLAND spectrum data, by minimizing with respect to these free parameters, the $\chi^2_{\text{KL}}$ function defined as $\chi^2_{\text{KL}} = \chi^2_{\text{G}} + \chi^2_{\text{P}}$ with $$\chi^2_{\text{G}}= \displaystyle \sum_{i}
\frac{( N_i^{\text{theo}} - N_i^{\text{obs}})^2}{\sigma_i^2},$$ and $$\chi^2_{\text{P}}= \sum_{j} 2(N_j^{\text{theo}} - N_j^{\text{obs}}) +
2 \, N_j^{\text{obs}} \ln \displaystyle \frac{N_j^{\text{obs}}}
{N_j^{\text{theo}}},$$ where $\sigma_i = \sqrt{ N_i^{\text{obs}}+(0.0642\, N_i^{\text{obs}})^2}$ is the statistical plus systematic uncertainty in the number of events in the $i$-th bin and the sum in $i (j)$ is done over the bins having 4 or more (less than 4) events. We have also computed the allowed regions using purely Gaussian or Poissonian $\chi^2$ functions and found that the hybrid $\chi^2$ definition above could reproduce better KamLAND’s allowed regions [@kamland]. Therefore, we have prefered to use it in our paper (see also Ref. [@valle]).
Using this $\chi^2_{\text{KL}}$ we have computed the allowed region at 90%, 95%, 99% and 99.73% C.L. shown in Fig. \[fig2\], which are quite consistent with the ones obtained by the KamLAND group in Fig. 6 of Ref. [@kamland]. In our fit we obtained a $\chi^2_{\text{KL}}(\text{min})=5.4$ for 11 d.o.f (91 % C.L.), corresponding to the best fit values $\Delta m^2= 7.0 \times 10^{-5}$ eV$^2$ and $\tan^2 \theta=0.79$.
-0.5cm
Combined Results {#subsec:comb}
----------------
-0.3cm Combining the results of all solar experiments with KamLAND data we have obtained the allowed region showed in Fig. \[fig3\]. The minimum value of $\chi^2_{\text{tot}}=\chi^2_\odot+\chi^2_{\text{KL}}$ for the combined fit is $\chi^2_{\text{tot}}(\text{min})=43.6$ for 60 d.o.f (94.5 % C.L.), corresponding to the best fit values $\Delta m^2= 7.1 \times 10^{-5}$ eV$^2$ and $\tan^2 \theta=0.42$. We observe that there are two separated regions which are allowed at 99 % C.L.: a lower one in $\Delta m^2$ (region 1) where the global best fit point is located, and an upper one (region 2) where the local best fit values are $\Delta m^2= 1.5 \times 10^{-4}$ eV$^2$ and $\tan^2 \theta=0.41$, corresponding to $\chi^2_{\text{loc}}(\text{min})=49.2$. We observe that depending on the definition of $\chi^2_{\text{KL}}$ (gaussian, poisson or hybrid) used, a third tiny region above $\Delta m^2= 2 \times 10^{-4}$ eV$^2$ appears at 99.73% C.L. However, apart from this small change, the combined allowed region is not essentially affected by the $\chi^2_{\text{KL}}$ used.
In Fig. \[fig4\] we show the theoretically predicted energy spectra at KamLAND for no oscillation, the best fit values of the oscillation parameters for KamLAND data alone and for KamLAND combined with solar data in regions 1 and 2. We note that the fourth energy bin, which is for the moment below the analysis cut, can be quite important in determining the values of the oscillation parameters in the future.
Future Perspectives {#sec:fut}
===================
-0.3cm In this section we consider the effect of possible experimental improvements which can help in determining the oscillation parameters with more accuracy in the future. We first consider a reduction of the error in the SNO neutral-current measurement then an increase of event statistics in KamLAND.
Effect of reducing SNO neutral-current error {#subsec:sno}
--------------------------------------------
-0.3cm In order to constrain the solar neutrino oscillation parameters even more, in particular, to decide in which of the 99% C.L. islands $\Delta m^2$ really lie, we have investigated the effect of increasing the SNO neutral-current data precision to twice its current value. We have re-calculated the region, in the $(\tan^2 \theta,\Delta m^2)$ plane, allowed by all current solar neutrino data, artificially decreasing the SNO NC measurement error but keeping the current central value, as well as the other solar neutrino data, unchanged. The result can be seen in Fig. \[fig5\]. The best fit point and the value of $\chi^2_\odot$(min) remain practically unchanged with respect to the result obtained in Sec. \[subsec:solar\], but the allowed region shrinks significantly. This is because the $^8B$ neutrino flux normalization, which can be directly inferred from SNO NC measurements, gets more constrained. Combining this with KamLAND data we obtain the allowed region shown in Fig. \[fig6\]. We observe that this allowed region is substantially smaller compared to the one shown in Fig. \[fig3\]. Moreover, region 2 only remains at 99% C.L.
Effect of increasing KamLAND statistics {#subsec:kamfut}
---------------------------------------
-0.3cm We simulate the expected KamLAND spectrum after one year of data taking for three distinct assumptions. We have generated KamLAND future data compatible with the best fitted values of $\Delta m^2$ and $\tan^2 \theta$ obtained for : (a) KamLAND data alone, (b) KamLAND and current solar neutrino data in region 1 and (c) KamLAND and current solar neutrino data in region 2. We have also included an extra bin, corresponding to the fourth bin in Fig. \[fig4\]. We have re-calculated the region allowed by the combined fit with the current solar neutrino data in each case.
The results of our calculations can be seen in Figs. \[fig7\]-\[fig9\]. If the future KamLAND result is close to the current one (see Fig. \[fig7\]), values of $\tan^2 \theta$ larger than the ones allowed now will be possible and region 2 will be excluded at 99% C.L. For this case we have obtained $\chi^2_{\text{tot}}$(min) $=42.1$. On the other hand, if the future KamLAND data are more compatible with the current best fit point of solar neutrino data (see Fig. \[fig8\]), the global allowed region will diminish substantially with respect to Fig. \[fig3\] and region 2 will only remain at 99% C.L. For this case we have obtained $\chi^2_{\text{tot}}$(min) $=39.1$. Finally, if after one year KamLAND data is more compatible with region 2 (see Fig. \[fig9\]) then one should observe an increase towards larger values of $\Delta m^2$ in the combined allowed region with respect to the one shown in Fig. \[fig3\]. In this case region 1 and 2 will have similar statistical significance, corresponding to $\chi^2_{\text{tot}}$(min) $ \sim 44.0$.
Discussions and Conclusion {#sec:conclusions}
==========================
-0.3cm
We have performed a combined analysis of the complete set of solar neutrino data with the recent KamLAND result in a two neutrino flavor oscillation scheme. We have obtained, in agreement with other groups [@outros], two distinct islands, denominated regions 1 and 2, in the $(\tan^2 \theta, \Delta m^2)$ plane, which are the most probable regions where the true values of these parameters lie. Region 1, where the global best fit point was found, is around $\Delta m^2=7.1 \times 10^{-5}$ eV$^2$, while region 2 is around $\Delta m^2=1.5 \times 10^{-4}$ eV$^2$.
We have considered two possible future improvements in the determination of the neutrino oscillation parameters. First, we have investigated the effect of a 50 % decrease in the error of the SNO NC measurement. We have shown that this would substantially reduce the allowed parameter region when combined with KamLAND data. In particular, region 2 would not be allowed at 95% C.L. anymore.
Second, we have studied what can happen in the near future, when KamLAND collects 1 year of data. We have simulated the expected KamLAND spectrum including an extra lower bin, corresponding to the fourth bin in Fig. \[fig3\]. Three different cases were studied in combination with the present solar neutrino data. In the first case, we have assumed that the future KamLAND spectrum will be compatible with oscillation parameter values at the best fit point for the present KamLAND data alone. This is the most restrictive case for region 2. In the second case, we have considered that future data will be more compatible with the present best fit point for the solar neutrino experiments. In this case, the combined allowed region will be much smaller than the present one and region 2 will be only allowed at 99% C.L. Finally, in the third case, we have assumed that the future KamLAND data will be compatible with the local best fit point in region 2. In this case, the combined allowed region will suffer an increase towards larger values of $\Delta m^2$ and region 1 and 2 will both have similar statistical significance.
This work was supported by Funda[ç]{}[ã]{}o de Amparo [à]{} Pesquisa do Estado de S[ã]{}o Paulo (FAPESP) and Conselho Nacional de Ci[ê]{}ncia e Tecnologia (CNPq).
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-0.2cm ![Regions in $(\tan^2 \theta,\Delta m^2)$ plane allowed by KamLAND data alone. The best fit point is marked by a cross.[]{data-label="fig2"}](sol.eps "fig:") -.8cm
-0.2cm -.6cm ![Regions in $(\tan^2 \theta,\Delta m^2)$ plane allowed by KamLAND data alone. The best fit point is marked by a cross.[]{data-label="fig2"}](kamland.eps "fig:") -0.7cm
-0.5cm
2.0cm -1.5cm ![Region allowed by all the solar neutrino experiments combined with KamLAND (KL) data. The region below (above) $\Delta m^2=10^{-4}$ eV$^2$ is referred to as region 1 (2). The best fit points in each region are also marked by cross (global best) and plus (local best).[]{data-label="fig3"}](comb.eps "fig:") -.6cm
-0.5cm
-0.5cm
-2.2cm -1.0cm ![Expected positron energy spectra at KamLAND (KL) for no oscillation, the best fit values of the oscillation parameters for KamLAND data alone and KamLAND data combined with the solar neutrino data in regions 1 and 2 of Fig. \[fig3\]. The KamLAND data [@kamland] is also shown as solid circles with error bars. The energy threshold at 2.6 MeV is marked by a vertical line.[]{data-label="fig4"}](kamhist.eps "fig:") -6.cm
0.6cm
-0.2cm -.5cm ![Same as Fig. \[fig3\] but decreasing the SNO neutral-current data error to half of its current value.[]{data-label="fig6"}](sol_nc.eps "fig:") -.8cm
-0.2cm -0.2cm ![Same as Fig. \[fig3\] but decreasing the SNO neutral-current data error to half of its current value.[]{data-label="fig6"}](comb_nc.eps "fig:") -0.8cm
-0.8cm
-0.2cm -1.cm ![Same as Fig. \[fig7\] but for the KL + Solar neutrino global best fit $\Delta m^2=7.1\times 10^{-5}$ eV$^2$ and $\tan^2\theta=0.42$ in region 1.[]{data-label="fig8"}](sol_kam_sim1.eps "fig:") -0.6cm
-0.2cm -0.4cm ![Same as Fig. \[fig7\] but for the KL + Solar neutrino global best fit $\Delta m^2=7.1\times 10^{-5}$ eV$^2$ and $\tan^2\theta=0.42$ in region 1.[]{data-label="fig8"}](sol_kam_sim2.eps "fig:") -0.6cm
-0.3cm
-0.2cm 0.cm ![Same as Fig. \[fig7\] but for the KL + Solar neutrino local best fit $\Delta m^2=1.5\times 10^{-4}$ eV$^2$ and $\tan^2\theta=0.41$ in region 2.[]{data-label="fig9"}](sol_kam_sim3.eps "fig:") -0.6cm
-0.3cm
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abstract: 'In large scale distributed storage systems (DSS) deployed in cloud computing, correlated failures resulting in simultaneous failure (or, unavailability) of blocks of nodes are common. In such scenarios, the stored data or a content of a failed node can only be reconstructed from the available live nodes belonging to available blocks. To analyze the resilience of the system against such block failures, this work introduces the framework of Block Failure Resilient (BFR) codes, wherein the data (e.g., file in DSS) can be decoded by reading out from a same number of codeword symbols (nodes) from each available blocks of the underlying codeword. Further, repairable BFR codes are introduced, wherein any codeword symbol in a failed block can be repaired by contacting to remaining blocks in the system. Motivated from regenerating codes, file size bounds for repairable BFR codes are derived, trade-off between per node storage and repair bandwidth is analyzed, and BFR-MSR and BFR-MBR points are derived. Explicit codes achieving these two operating points for a wide set of parameters are constructed by utilizing combinatorial designs, wherein the codewords of the underlying outer codes are distributed to BFR codeword symbols according to projective planes.'
author:
- '\'
title: Repairable Block Failure Resilient Codes
---
Introduction
============
Increasing demand for storing and analyzing *big-data* as well as several applications of cloud computing systems require efficient cloud computing infrastructures. One inevitable nature of the storage systems is node failures. In order to provide resilience against failures, redundancy is introduced in the storage. Classical redundancy schemes range from *replication* to *erasure coding*. Erasure coding allows for better performance in terms of reliability and redundancy compared to replication, however repair bandwidth in reconstructing a failed node is higher. Regenerating codes are proposed to overcome this problem in the seminal work of Dimakis et al. [@Dimakis:Network10]. In such a model of distributed storage systems (DSS), the file ${{\cal M}}$ is encoded to $n$ nodes such that any $k\leq n$ nodes (each with $\alpha$ symbols) allow for reconstructing the file and any $d\geq k$ nodes (with $\beta\leq \alpha$ symbols from each) reconstruct a failed node with a repair bandwidth $\gamma=d\beta$. The trade-off between per node storage ($\alpha$) and repair bandwidth ($\gamma$) is characterized and two ends of the trade-off are named as minimum storage regenerating (MSR) and minimum bandwidth regenerating (MBR) points [@Dimakis:Network10]. Several explicit codes have been proposed to achieve these points recently [@Tamo:Zigzag13; @Rashmi:Optimal11; @Dimakis:Survey11]. Another metric for an efficient repair is repair degree $d$, and regenerating codes necessarily have $d\geq k$. Codes with locality and locally repairable codes with regeneration properties [@Gopalan:Locality12; @Papailiopoulos:Locally12; @Oggier:Self11; @Rawat:Optimal14; @Kamath:Codes12; @Kamath:Explicit13] allow for a small repair degree, wherein failed nodes are reconstructed via local connections. Instances of such codes are recently considered in DSS [@Sathiamoorthy:XORing13; @Huang:Erasure12].
In large-scale distributed storage systems (such as GFS [@Ghemawat:Google03]), *correlated failures* are unavoidable. As analyzed in [@Ford:Availability10], these simultaneous failures of multiple nodes affect the performance of computing systems severely. The analysis in [@Ford:Availability10] further shows that these correlated failures arise due to *failure domains*. For example, nodes connected to the same power source or nodes belonging to the same rack exhibit these failure bursts. The unavailability periods are transient, and largest failure bursts almost always have significant rack-correlation. In order to overcome from failures having such patterns, a different approach is needed.
In this paper, we develop a framework to analyze resilience against block failures in DSS with node repair efficiencies. We consider a DSS with a single failure domain, where nodes belonging to the same failure group constitute a block of the codeword. We introduce block failure resilient (BFR) codes, which allow for data collection from any $b_c= b-\rho$ blocks, where $b$ is the number of blocks, and $\rho$ is the resilience parameter of the code. Considering a load-balancing among blocks, a same number of nodes are contacted within these $b_c$ blocks. (A total of $k=k_cb_c$ nodes and downloading $\alpha$ - i.e., all - symbols from each.) This constitutes data collection property of BFR codes. ($\rho=0$ case can be considered as a special case of batch codes introduced in [@Ishai:Batch04].) Then, we introduce repairability in BFR codes, where any node of a failed block can be reconstructed from any $d_r$ of any remaining $b_r\leq b-1$ blocks. (A total of $d=d_rb_r$ nodes and downloading $\beta$ symbols from each.) As introduced in [@Dimakis:Network10], we utilize graph expansion of DSS employing these repairable codes, and derive file size bounds and characterize BFR-MBR and BFR-MSR points. (We note that the blocks in our model can be used to model racks in DSS. Such a model is related to the work [@Gaston:realistic13] which differentiates between within-rack communication and cross-rack communication. Our focus here would correspond to the case where within rack communication is much higher than the cross-rack communication, as no nodes from the failed rack can be contacted to regenerate a node.) We construct explicit codes achieving these points for a wide set of parameters. For a system with $b=2$ blocks case, we show that achieving both MSR and MBR properties simultaneously is asymptotically possible. (This is somewhat similar to the property of Twin codes [@Rashmi:Enabling11], but here the data collection property is different.) Then, for a system with $b\geq 3$ blocks case, we consider utilizing multiple codewords, which are placed into DSS via a combinatorial design based codeword placement algorithm. We show this technique establishes optimal codes for a wide set of parameter ranges.
The paper is organized as follows. Section II introduces model and preliminaries. Section III is devoted to the analysis of file size bounds. Code constructions are provided in Section IV. Section V includes extensions and concluding remarks.
Background and Preliminaries {#sec:Background}
============================
Block failure resilient codes and repairability
-----------------------------------------------
Consider a code ${{\cal C}}$ which maps ${{\cal M}}$ symbols (over ${\mbox{\bb F}}_q$) in ${{\bf f}}$ (file) to length $n$ codewords (nodes) ${{\bf c}}=(c_1,\cdots,c_n)$ with $c_i\in {\mbox{\bb F}}_q^\alpha$ for $i=1,\cdots,n$. These codewords are distributed into $b$ blocks each with block capacity $c=\frac{b}{n}$ nodes per block. We have the following definition.
An $(n,b,{{\cal M}},k,\rho,\alpha)$ block failure resilient (BFR) code encodes ${{\cal M}}$ elements in ${\mbox{\bb F}}_q$ (${{\bf f}}$) to $n$ codeword symbols (each in ${\mbox{\bb F}}_q^\alpha$) that are grouped into $b$ blocks such that ${{\bf f}}$ can be decoded by accessing to any $\frac{k}{b-\rho}$ nodes of from each of the $b-\rho$ blocks.
We remark that, in the above, $\rho$ represents the resilience parameter of the BFR code, i.e., the code can tolerate $\rho$ block erasures. Due to this data collection (file decoding) property of the code, we denote the number of blocks accessed as $b_c=b-\rho$ and number of nodes accessed per block as $k_c=\frac{k}{b_c}$. Noting that $k_c\leq c$ should be satisfied, we differentiate between *partial* block access, $k_c<c$, and *full* block access $k_c=c$. Throughout the paper, we assume $n|b$. i.e., $c$ is integer, and $(b-\rho)|k$, i.e., $k_c$ is integer.
Remarkably, any MDS array code [@McWilliams:Theory77] can be utilized as BFR codes for the full access case. In fact, such an approach will be optimal in terms of minimum distance, and therefore for resilience $\rho$. However, for $k_c<c$, MDS array codes may not result in an optimal code. Constructing optimal BFR codes in terms of the trade-off between resilience $\rho$ and code rate $\frac{{{\cal M}}}{n\alpha}$ will be studied elsewhere. In this work, we focus on repairable BFR codes, as defined in the following.
An $(n,b,{{\cal M}},k,\rho,\alpha,d,\sigma,\beta)$ block failure resilient regenerating code (BFR-RC) is an $(n,b,{{\cal M}},k,\rho,\alpha)$ BFR code (data collection property) with the following repair property: Any node of a failed block can be reconstructed by accessing to any $d_r=\frac{d}{b-\sigma}$ nodes of any $b_r=b-\sigma$ blocks and downloading $\beta$ symbols from each of these $d=b_rd_r$ nodes.
We assume $(b-\rho)|d$, i.e., $d_r$ is integer. (Note that $d_r$ should necessarily satisfy $\frac{d}{b-\sigma}=d_r\leq c=\frac{n}{b}$ in our model.) We consider the trade-off between the *repair bandwidth* $\gamma=d\beta$ and *per node storage* $\alpha$ similar to the seminal work [@Dimakis:Network10]. In particular, we define $\alpha_{\textrm{BFR-MSR}}=\frac{{{\cal M}}}{k}$ as the minimum per node storage and $\gamma_{\textrm{BFR-MBR}}=\alpha$ as the minimum repair bandwidth for an $(n,b,{{\cal M}},k,\rho,\alpha,d,\sigma,\beta)$ BFR-RC. When deriving this trade-off, we focus on systems having $d_r=\frac{d}{b-\sigma}\geq k_c=\frac{k}{b-\rho}$, i.e., data collection process contacts to less number of nodes per block as compared to symbol regeneration. (We note that, similar to regenerating codes, without loss of generality, one should only consider systems that satisfy $d\geq k$, i.e., $d_r(b-\sigma)\geq k_c (b-\rho)$. Therefore, our $d_r\geq k_c$ assumption can be made without loss of generality for systems having $\rho\leq \sigma$.)
Information flow graph
----------------------
The operation of a DSS employing such codes can be modeled by a multicasting scenario over an information flow graph [@Dimakis:Network10], which has three types of nodes: 1) Source node ($S$): Contains original file ${{\bf f}}$. 2) Storage nodes, each represented as $x_i$ with two sub-nodes($(x^{\rm in}_i,x^{\rm out}_i)$), where $x^{\rm in}$ is the sub-node having the connections from the live nodes, and $x^{\rm out}$ is the storage sub-node, which stores the data and is contacted for node repair or data collection (edges between each $x^{\rm in}_i$ and $x^{\rm out}_i)$ has $\alpha$-link capacity). 3) Data collector ($\rm{DC}$) which contacts $x^{\rm{out}}$ sub-nodes of $k$ live nodes (with edges each having $\infty$-link capacity). (As described above, for BFR codes these $k$ nodes can be any $\frac{k}{b-\rho}$ nodes from each of the $b-\rho$ blocks.) Then, for a given graph ${{\cal G}}$ and DCs $\rm{DC}_i$, the file size can be bounded using the max flow-min cut theorem for multicasting utilized in network coding [@Ho:random06; @Dimakis:Network10].
\[lma:MFMCforMulticast\] $${{\cal M}}\leq \min_{{{\cal G}}} \min_{\rm{DC}_i} \rm{max flow}(S \to \rm{DC}_i,{{\cal G}}),$$ where $\rm{flow}(S \to \rm{DC}_i,{{\cal G}})$ represents the flow from the source node $S$ to $\rm{DC}_i$ over the graph ${{\cal G}}$.
Therefore, $\mathcal{M}$ symbol long file can be delivered to a DC, only if the min cut is at least $\mathcal{M}$. In the next section, similar to Dimakis et al., [@Dimakis:Network10], we consider $k$ successive node failures and evaluate the min-cut over possible graphs, and obtain a file size bound for a DSS operating with BFR-RC.
Block designs and projective planes
-----------------------------------
We first provide the definition of balanced incomplete block designs (BIBDs) [@Stinson:Combinatorial04].
A $(v,\kappa,\lambda)$-BIBD has $v$ points distributed into blocks of size $\kappa$ such that any pair of points are contained in $\lambda$ blocks.
\[thm:BIBDcorollary\] For a $(v,\kappa,\lambda)$-BIBD,
- Every point occurs in $r=\frac{\lambda (v-1)}{\kappa-1}$ blocks.
- The design has exactly $b=\frac{vr}{\kappa}=\frac{\lambda(v^2-v)}{\kappa^2-\kappa}$ blocks.
In the achievable schemes of this work, we utilize a special class of block designs that are called projective planes.
A $(v=p^2+p+1,\kappa=p+1,\lambda=1)$-BIBD with $p\geq 2$ is called a projective plane of order $p$.
Projective planes have the property that every pair of blocks intersect at a unique point (as $\lambda=1$). In addition, due to Corollary \[thm:BIBDcorollary\], in projective planes, every point occurs in $r=p+1$ blocks, and there are $b=v=p^2+p+1$ blocks.
File Size Bound for Repairable BFR Codes {#sec:FileSize}
========================================
Information flow graph analysis, similar to that of considered in [@Dimakis:Network10], can be performed to obtain file size bounds for repairable BFR codes. In this paper, we focus on the case $\sigma=1$, i.e., regeneration of a node in a failed block is performed by contacting to all remaining live blocks. In the following, we first analyze $\rho=0$ case, i.e., data collector connects all the blocks to reconstruct the data.
$\rho=0$, $b=2$ case
--------------------
![Repair process for $b=2$ (two blocks) case.[]{data-label="fig:Two-block"}](two-block){width="0.6\columnwidth"}
Consider $b=2$-block case as in Fig. \[fig:Two-block\] and assume $2|k$. From Lemma \[lma:MFMCforMulticast\], the file size ${{\cal M}}$ can be upper bounded with the repair procedure shown in Fig. \[fig:Two-block\], which displays one of the “minimum-cut” scenarios, wherein any two consecutive node failures belong to different blocks. Assuming $k$ is even and $d\geq \frac{k}{2}$, $${{\cal M}}\leq \sum_{i=0}^{\frac{k}{2}-1}\min(\alpha,(d-i)\beta) + \sum_{i=1}^{\frac{k}{2}}\min(\alpha,(d-i)\beta).
\label{eq:min-cut_two}$$ Achieving this upper bound with equality would yield maximum possible file size. One particular instance is shown in Fig. \[fig:Two-block\], and we note that the order of failed nodes does not matter as the sum of the cut would be the same with different order of failures as long as we consider connection from data collector to $\frac{k}{2}$ repaired nodes from each block.
For MSR point, $\alpha=\alpha_{\textrm{BFR-MSR}}=\frac{{{\cal M}}}{k}$. In the bound , we then have $\alpha_{\textrm{BFR-MSR}} \leq (d-\frac{k}{2})\beta_{\textrm{BFR-MSR}}$. Achieving equality would give the minimum repair bandwidth for the MSR case. Hence, BFR-MSR point is given by $$(\alpha_{\textrm{BFR-MSR}},\gamma_{\textrm{BFR-MSR}}) = (\frac{{{\cal M}}}{k},\frac{2{{\cal M}}d}{2kd-k^2}).
\label{eq:min-cut_two-MSR-values}$$
BFR-MBR codes, on the other hand, have the property that $d\beta=\alpha$ with minimum possible $d\beta$ while achieving the equality in . Inserting $d\beta=\alpha$ in , we obtain that $$(\alpha_{\textrm{BFR-MBR}},\gamma_{\textrm{BFR-MBR}}) = (\frac{4{{\cal M}}d}{4dk-k^2},\frac{4{{\cal M}}d}{4dk-k^2}).
\label{eq:min-cut_two-MBR-values}$$ Same analysis can be done for odd values of $k$ as well, $$(\alpha_{\textrm{BFR-MSR}},\gamma_{\textrm{BFR-MSR}}) =
\begin{cases}
(\frac{M}{k},\frac{2Md}{2kd-k^2-k}), \textrm{ if $k$ is odd} \\
(\frac{M}{k},\frac{2Md}{2kd-k^2}), \textrm{ o.w.}
\end{cases}
\label{eq:MSR_cases}$$ $$(\alpha_{\textrm{BFR-MBR}},\gamma_{\textrm{BFR-MBR}}) =
\begin{cases}
(\frac{4Md}{4dk-k^2+1},\frac{4Md}{4dk-k^2+1}), \textrm{ if $k$ is odd} \\
(\frac{4Md}{4dk-k^2},\frac{4Md}{4dk-k^2}), \textrm{ o.w.}
\end{cases}
\label{eq:MBR_cases}$$ Here, we compare $\gamma_{\textrm{BFR-MSR}}$ and $\gamma_{\textrm{MBR}}$. We have $\gamma^{\textrm{k-odd}}_{\textrm{BFR-MSR}} \geq \gamma^{\textrm{k-even}}_{\textrm{BFR-MSR}} \geq \gamma_{\textrm{MBR}}=\frac{2{{\cal M}}d}{k(2d-k+1)}$, and, if we have $2d-k \gg 1$, then $\gamma^{\textrm{k-odd}}_{\textrm{BFR-MSR}} \approx \gamma^{\textrm{k-even}}_{\textrm{BFR-MSR}} \approx \gamma_{\textrm{MBR}}$. This implies that BFR-MSR codes with $b=2$ achieves repair bandwidth of MBR and per-node storage of MSR codes simultaneously for systems with $d \gg 1$. We provide the generalization of these bounds to $b \geq 2$ case in the following.
$\rho=0$, $b\geq2$ case
-----------------------
The same steps described above can be used to derive the file size bound for $b$-blocks.
The optimal file size is given by $$\begin{aligned}
\begin{split}
{{\cal M}}= & \sum_{i=0}^{\frac{k}{b}-1}\min(\alpha,(d-(b-1)i)\beta) \\
& + \sum_{i=0}^{\frac{k}{b}-1}\min(\alpha,(d-1-(b-1)i)\beta) + \ldots \\
& + \sum_{i=0}^{\frac{k}{b}-1}\min(\alpha,(d-(b-1)-(b-1)i)\beta).
\end{split}\end{aligned}$$
BFR-MSR and BFR-MBR points are as follows, $$(\alpha_{\textrm{BFR-MSR}},\gamma_{\textrm{BFR-MSR}}) = \left(\frac{{{\cal M}}}{k},\frac{{{\cal M}}d}{kd-\frac{k^2(b-1)}{b}}\right)
\label{eq:min-cut_b-MSR-values}$$
$$(\alpha_{\textrm{BFR-MBR}},\gamma_{\textrm{BFR-MBR}}) = \left(\frac{{{\cal M}}d}{kd-\frac{k^2(b-1)}{2b}},\frac{{{\cal M}}d}{kd-\frac{k^2(b-1)}{2b}}\right)
\label{eq:min-cut_b-MBR-values}$$
We observe that $\gamma_{\textrm{BFR-MSR}} \leq \gamma_{\textrm{MSR}}=\frac{{{\cal M}}d}{k(d-k+1)}$ for $b \leq k$, which is the case here as $b \mid k$. Also, we have $\frac{\gamma_{\textrm{BFR-MSR}}}{\gamma_{\textrm{MBR}}} = \frac{d-\frac{k-1}{2}}{d-k\frac{b-1}{b}} \geq 1$ when $b \geq \frac{2k}{k+1}$ which is always true. Hence, $\gamma_{\textrm{BFR-MSR}}$ is between $\gamma_{\textrm{MSR}}$ and $\gamma_{\textrm{MBR}}$.
$\rho>0$ case
-------------
If we restrict data collector to connect $b_c<b$ blocks (i.e., $\rho>0$), but keep the repair process same as before, the above analysis follows and corresponding MSR and MBR points are given by replacing $b$ in and with $b_{c}=b-\rho$ - for systems satisfying $d_r\geq k_c$. (This follows as the repair from these $\rho$ blocks will not contribute to the cut between the source $S$ and DC.)
BFR-MSR and BFR-MBR Code Constructions {#sec:CodeConst}
======================================
Transpose code for b=2 case
---------------------------
One instance of BFR codes is given in the Fig. \[fig:Transpose\]. We set $\alpha=d=\frac{n}{2}$, and store the transpose of the first block’s symbols in the second block. The repair of a failed node $i$ in the first block can be performed by connecting all the nodes in the second block and downloading only $1$ symbol from each node. That is, $d\beta=\alpha$. Further, we set ${{\cal M}}=kd-(\frac{k}{2})^2$, and use an $[N=\alpha^2,K={{\cal M}}]$ MDS code to encode file ${{\bf f}}$ into symbols denoted with $x_{i,j}$, $i,j=1,...,\alpha$. BFR data collection property allows for reconstructing the file, as connecting any $\frac{k}{2}$ nodes from each block assures at least $K$ distinct symbols. This code is a BFR-MBR code for $\beta=1$ (scalar code), as the optimal file size in , i.e., ${{\cal M}}=kd-(\frac{k}{2})^2$, is achieved with $d\beta=\alpha$. A similar code to this construction is Twin codes introduced in [@Rashmi:Enabling11], where the nodes are split into two types and a failed node of a a given type is regenerated by connecting to nodes only in the other type. However, Twin codes, as opposed to our model, do not have balanced node connection for data collection. In particular, DC connects to only (a subset of $k$ nodes from) a single type. On the other hand, BFR codes, for $b=2$ case, connects to $\frac{k}{2}$ nodes from each block.
![Transpose code is a two-block BFR-MBR code.[]{data-label="fig:Transpose"}](2-block_trans){width="0.60\columnwidth"}
Block design based regenerating code symbol placement
-----------------------------------------------------
Consider that the file ${{\cal F}}$ of size ${{\cal M}}$ contains 3 sub-files ${{\cal F}}_{1}$, ${{\cal F}}_{2}$ and ${{\cal F}}_{3}$ each of size $\tilde{{{\cal M}}}$. We encode these sub-files with $[\tilde{n}=10,\tilde{k}=4,\tilde{d}=5,\tilde{\alpha},\tilde{\beta}]$ regenerating code $\tilde{{{\cal C}}}$, represent the resulting symbols with ${{\cal P}}_{1}=p_{1,1:\tilde{n}}$ for ${{\cal F}}_1$, ${{\cal P}}_{2}=p_{2,1:\tilde{n}}$ for ${{\cal F}}_2$, and ${{\cal P}}_{3}=p{_{3,1:\tilde{n}}}$ for ${{\cal F}}_3$. These symbols are grouped in a specific way placed into nodes within blocks as represented in Fig. \[fig:3-block BFR-RC\], where each node contains two symbols each coming from two of the different sets ${{\cal P}}_{1},{{\cal P}}_{2},{{\cal P}}_{3}$. We set the sub-code $\tilde{{{\cal C}}}$ parameters as $[{{\cal M}}=3\tilde{{{\cal M}}}, k=\frac{3}{2}\tilde{k}, d=2\tilde{d}, \alpha=2\tilde{\alpha}, \beta=\tilde{\beta}]$.
Assume Block 1 is unavailable and its first node, which contains codeword $c_{1}$, has to be reconstructed. Due to underlying regenerating code, contacting $5$ nodes of Block 2 and accessing to $p_{1,6:10}$ repairs $p_{1,1}$. Similarly, $p_{2,1}$ can be reconstructed from Block 3. Any node failures can be handled similarly, by connecting to remaining 2 blocks and repairing each symbol of lost node by connecting $\tilde{d}$ nodes in a block. As we have $k=6$, DC, connecting to 2 nodes from each block, obtains $12$ symbols which has 4 different symbols from each of ${{\cal P}}_{1}$, ${{\cal P}}_{2}$ and ${{\cal P}}_{3}$. As the embedded regenerating code has $\tilde{k}=4$, all $3$ sub-files can be recovered.
We generalize the BFR-RC construction above utilizing projective planes. First, the file ${{\bf f}}$ of size ${{\cal M}}$ is partitioned into $v$ parts, ${{\cal M}}_{1}$, ${{\cal M}}_{2}$,...,${{\cal M}}_{v}$. Each part, of size $\tilde{{{\cal M}}}$, then encoded using $[\tilde{n},\tilde{k},\tilde{d},\tilde{\alpha},\tilde{\beta}]$ regenerating code $\tilde{{{\cal C}}}$. We represent the resulting symbols with ${{\cal P}}_{i}=p_{i,1:\tilde{n}}$ for $i=1,\cdots, v$. We then consider index of each part as a point in a $(v=p^2+p+1,\kappa=p+1,\lambda=1)$ projective plane. (Indices of symbol sets ${{\cal P}}_{{\cal J}}$ and points ${{\cal J}}$ of projective plane are used interchangeably in the following.) We perform the placement of each point in the system using this projective plane mapping. (The setup in Fig. \[fig:3-block BFR-RC\] can be considered as a toy model. Although the combinatorial design with blocks given by $\{p_1,p_2\},\{p_1,p_3\},\{p_2,p_3\}$ has projective plane properties, it is not considered as an instance of a projective plane.) In this placement, total of $\tilde{n}$ nodes from each partition ${{\cal P}}_{i}$ are distributed to $r$ blocks evenly, each block contains $\frac{\tilde{n}}{r}$ nodes where each node stores $\alpha=\kappa\tilde{\alpha}$ symbols. Note that blocks of projective plane give the indices of parts ${{\cal P}}_i$ stored in the nodes of the corresponding block in DSS. That is, all nodes in a block stores symbols from unique subset of ${{\cal P}}=\{{{\cal P}}_1,\cdots,{{\cal P}}_v\}$ of size $\kappa$. Overall, the system can store a file of size ${{\cal M}}=v\tilde{{{\cal M}}}$ with $b=v$ blocks. We set the sub-code $\tilde{{{\cal C}}}$ parameters as $$M=v\tilde{M}, k=\frac{b}{r}\tilde{k}, d=\kappa\tilde{d}, \alpha=\kappa\tilde{\alpha}, \beta=\tilde{\beta}
\label{eq:assignment-b}$$ where we choose parameters to satisfy $r-1 \mid \tilde{d}$, $r \mid \tilde{n}$ and $r \mid \tilde{k}$.
*Node Repair:* Consider that one of the nodes in a block is to be repaired. Note that the failed node contains $\kappa$ symbols, each coming from a distinct subfile’s regenerating codeword. Using projective planes’ property that any $2$ blocks has only $1$ point in common, any remaining block can help for in the regeneration of $1$ symbol of the failed node. Furthermore, as any point has a repetition degree of $r$, one can connect to $r-1$ blocks, $\frac{\tilde{d}}{r-1}$ nodes per block, to repair one symbol of a failed node. Combining these two, node regeneration is performed by connecting $(r-1)\kappa$ blocks. Substituting $\kappa=p+1$ and $r=p+1$, connecting to $p^{2}+p=b-1$ blocks allows for reconstructing any node of a failed block.
*Data Collection:* DC, connects $\frac{\tilde{k}}{r}$ nodes per block from all $b_{c}=b$ blocks, i.e., a total of $k=\frac{b}{r}\tilde{k}$ nodes each having encoded symbols of $\kappa$ subfiles. These total of $v\tilde{k}$ symbols include $\tilde{k}$ symbols from each subfile, from which all subfiles, hence the file ${{\bf f}}$, can be decoded.
![Three-block BFR-RC via projective plane symbol placement.[]{data-label="fig:3-block BFR-RC"}](3-block){width="0.6\columnwidth"}
### BFR-MSR
To construct a BFR-MSR code, we set each subcode $\tilde{{{\cal C}}}$ as an MSR code, which has $$\tilde{\alpha}=\frac{\tilde{{{\cal M}}}}{\tilde{k}}, \tilde{d}\tilde{\beta}=\frac{\tilde{{{\cal M}}}\tilde{d}}{\tilde{k}(\tilde{d}-\tilde{k}+1)}.$$
This, together with , results in the following parameters of our BFR-MSR construction $$\alpha=\tilde{\alpha}\kappa=\frac{{{\cal M}}}{k}, d\beta=\kappa \tilde{d}\tilde{\beta}=\frac{{{\cal M}}d}{k(d-\frac{k(p+1)^{2}}{p^2+p+1}+p+1)}.
\label{eq:bfr-msr}$$
We remark that if we utilize ZigZag codes[@Tamo:Zigzag13] as the sub-code $\tilde{{{\cal C}}}$ above, we have $[\tilde{n},\tilde{k},\tilde{d}=\tilde{n}-1,\tilde{\alpha}=\tilde{r}^{\tilde{k}-1},\tilde{\beta}=\tilde{r}^{\tilde{k}-2}, \tilde{r}=\tilde{n}-\tilde{k}]$, and having $\tilde{d}=\tilde{n}-1$ requires connecting to $1$ node per block for repairs in our block model. On the other hand, product matrix MSR codes [@Rashmi:Optimal11] can be used as the sub-code $\tilde{{{\cal C}}}$ for any $\tilde{d} \geq 2\tilde{k}-2$, for which we do not necessarily have $\frac{\tilde{d}}{r-1}=1$. We observe from and that MSR point is achieved for $\tilde{k}=p+1$, meaning $k=b$.
### BFR-MBR
To construct a BFR-MBR code, we set each subcode $\tilde{{{\cal C}}}$ as a product matrix MBR code [@Rashmi:Optimal11], which has $$\tilde{\alpha}=\tilde{d}\tilde{\beta}= \frac{2\tilde{{{\cal M}}}\tilde{d}}{\tilde{k}(2\tilde{d}-\tilde{k}+1)}.$$
This, together with , results in the following parameters of our BFR-MSR construction $$\alpha=d\beta=\frac{2{{\cal M}}d}{k(2d-\frac{k(p+1)^2}{p^2+p+1}+p+1)}.
\label{eq:bfr-mbr}$$
From and , MBR point is achieved for $\tilde{k}=p+1$.
Extensions and concluding remarks {#sec:Discussion}
=================================
$\rho>0$ case
-------------
In the above, we considered the cases where DC connects all $b$ blocks in file reconstruction. In order to support $b_{c}<b$, we consider employing Gabidulin codes [@Gabidulin:Theory85] as an outer code similar to the constructions provided in [@Rawat:Optimal14; @Kamath:Explicit13]. We briefly discuss our approach here. Detailed results will be provided elsewhere. $[N,K,D=N-K+1]_{q^{m}}$ Gabidulin code $C^{Gab}$, $m\geq N$, has a codeword $( f(g_{1}),f(g_{2}),...,f(g_{N})) \in {\mbox{\bb F}}_{q^{m}}^N$, where $f(x)$ is a linearized polynomial over ${\mbox{\bb F}}_{q^{m}}$ of $q$-degree $K-1$ with $K$ message symbols as its coefficients and $g_{1},g_{2},...,g_{N} \in {\mbox{\bb F}}_{q^{m}}$ are linearly independent over $F_{q}$ [@Gabidulin:Theory85].
Given evaluations of $f(\cdot)$ at any $K$ linearly independent (over ${\mbox{\bb F}}_{q}$) points in ${\mbox{\bb F}}_{q^{m}}$, one can reconstruct the message vector.
Here, before partitioning the message into $v$ parts, we encode the file with a Gabidulin code first, then partition the resulting codeword into $v$ parts and follow remaining steps as before. With this approach, decoding the message at DC follows by obtaining at least $K$ independent evaluations from $k$ nodes, $k_c=\frac{k}{b_{c}}$ nodes per block from a total of $b_{c}=b-\rho$ blocks. As considered in [@Rawat:Optimal14; @Kamath:Explicit13], the number of such evaluations can be derived from the rank accumulation profile of the inherent MSR/MBR codes $\tilde{{{\cal C}}}$ as in the following $$\tilde{a_{j}} =
\begin{cases}
\tilde{\alpha},& \text{if $\tilde{{{\cal C}}}$ is MSR and } 1\leq j\leq \tilde{k} \\
\tilde{\alpha}-(j-1)\tilde{\beta}, & \text{if $\tilde{{{\cal C}}}$ is MBR and } 1\leq j\leq \tilde{k} \\
0, & \text{if $\tilde{{{\cal C}}}$ is MSR/MBR and } \tilde{k}+1\leq j\leq \tilde{n} \\
\end{cases}$$ Note that because of projective plane property, connecting $b-1$ blocks would result in getting $k_cr$ evaluations for $v-\kappa$ points and $k_c (r-1)$ evaluations for $\kappa$ points. Hence DC can decode the message by using an outer Gabidulin code if $$\sum_{t=1}^{v-\kappa}\sum_{j=1}^{k_c r}\tilde{a}_{j} + \sum_{t=1}^{\kappa}\sum_{j=1}^{k_c (r-1)}\tilde{a}_{j}\geq K.$$ Similarly, for $b_{c}=b-2$, decoding at DC is possible if $$\sum_{t=1}^{v-(2\kappa-1)}\sum_{j=1}^{k_c r}\tilde{a}_{j} + \sum_{t=1}^{2\kappa-2}\sum_{j=1}^{k_c (r-1)}\tilde{a}_{j}+\sum_{j=1}^{k_c (r-2)}\tilde{a}_{j}\geq K.$$ With such an approach, for $b_{c}\leq b-3$ there are multiple collection possibilities for DC. For example, by connecting $b-3$ blocks DC can observe either a) $k_cr$ evaluations for $v-(3\kappa-2)$, $k_c(r-1)$ evaluations for $3(\kappa-1)$ points and $k_c(r-3)$ evaluations for 1 point, or b) $k_cr$ evaluations for $v-(3\kappa-3)$, $k_c(r-1)$ evaluations for $3(\kappa-2)$ points and $k_c(r-2)$ evaluations for 3 points. Therefore, we need to ensure that minimum rank accumulations of all cases is at least $K$.
Concluding remarks
------------------
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|
---
abstract: 'We present a new interatomic potential for solids and liquids called Spectral Neighbor Analysis Potential (SNAP). The SNAP potential has a very general form and uses machine-learning techniques to reproduce the energies, forces, and stress tensors of a large set of small configurations of atoms, which are obtained using high-accuracy quantum electronic structure (QM) calculations. The local environment of each atom is characterized by a set of bispectrum components of the local neighbor density projected on to a basis of hyperspherical harmonics in four dimensions. The bispectrum components are the same bond-orientational order parameters employed by the GAP potential [@Bartok2010]. The SNAP potential, unlike GAP, assumes a linear relationship between atom energy and bispectrum components. The linear SNAP coefficients are determined using weighted least-squares linear regression against the full QM training set. This allows the SNAP potential to be fit in a robust, automated manner to large QM data sets using many bispectrum coefficients. The calculation of the bispectrum components and the SNAP potential are implemented in the LAMMPS parallel molecular dynamics code. We demonstrate that a previously unnoticed symmetry property can be exploited to reduce the computational cost of the force calculations by more than one order of magnitude. We present results for a SNAP potential for tantalum, showing that it accurately reproduces a range of commonly calculated properties of both the crystalline solid and the liquid phases. In addition, unlike simpler existing potentials, SNAP correctly predicts the energy barrier for screw dislocation migration in BCC tantalum.'
address:
- 'Multiscale Science Department, Sandia National Laboratories, P.O. Box 5800, MS 1322, Albuquerque, NM 87185'
- 'Optimization and Uncertainty Quantification Department, Sandia National Laboratories, P.O. Box 5800, MS 1318, Albuquerque, NM 87185'
- 'Scalable Algorithms Department, Sandia National Laboratories, P.O. Box 5800, MS 1322, Albuquerque, NM 87185'
- 'Computational Materials and Data Science Department, Sandia National Laboratories, P.O. Box 5800, MS 1411, Albuquerque, NM 87185'
- 'Materials Science and Engineering Department, Drexel University, Philadelphia, PA 19104'
author:
- 'A.P. Thompson'
- 'L.P. Swiler'
- 'C.R. Trott'
- 'S.M. Foiles'
- 'G.J. Tucker'
bibliography:
- 'snap\_jcp.bib'
nocite: '[@*]'
title: 'A Spectral Analysis Method for Automated Generation of Quantum-Accurate Interatomic Potentials'
---
interatomic potential ,machine learning ,spectral neighbor analysis potential ,SNAP ,Gaussian approximation potentials ,molecular dynamics
Introduction {#sec:intro}
============
Classical molecular dynamics simulation (MD) is a powerful approach for describing the mechanical, chemical, and thermodynamic behavior of solid and fluid materials in a rigorous manner [@Griebel2007]. The material is modeled as a large collection of point masses (atoms) whose motion is tracked by integrating the classical equations of motion to obtain the positions and velocities of the atoms at a large number of timesteps. The forces on the atoms are specified by an interatomic potential that defines the potential energy of the system as a function of the atom positions. Typical interatomic potentials are computationally inexpensive and capture the basic physics of electron-mediated atomic interactions of important classes of materials, such as molecular liquids and crystalline metals. Efficient MD codes running on commodity workstations are commonly used to simulate systems with $N = 10^5 - 10^6$ atoms, the scale at which many interesting physical and chemical phenomena emerge. Quantum molecular dynamics (QMD) is a much more computationally intensive method for solving a similar physics problem [@Schultz2005]. Instead of assuming a fixed interatomic potential, the forces on atoms are obtained by explicitly solving the quantum electronic structure of the valence electrons at each timestep. Because MD potentials are short-ranged, the computational complexity of MD generally scales as $O(N)$, whereas QMD calculations require global self-consistent convergence of the electronic structure, whose computational cost is $O(N^\alpha_e)$, where $2 < \alpha < 3$ and $N_e$ is the number of electrons. For the same reasons, MD is amenable to spatial decomposition on parallel computers, while QMD calculations allow only limited parallelism.
As a result, while high accuracy QMD simulations have supplanted MD in the range $N =10-100$ atoms, QMD is still intractable for $N > 1000$, even using the largest supercomputers. Conversely, typical MD potentials often exhibit behavior that is inconsistent with QMD simulations. This has led to great interest in the development of MD potentials that match the QMD results for small systems, but can still be scaled to the interesting regime $N = 10^5 - 10^6$ atoms [@Bartok2010; @Artrith2012; @Li2003]. These quantum-accurate potentials require many more floating point operations per atom compared to conventional potentials, but they are still short-ranged. So the computational cost remains $O(N)$, but with a larger algorithm pre-factor.
In this paper, we present a new quantum-accurate potential called SNAP. It is designed to model the migration of screw dislocations in tantalum metal under shear loading, the fundamental process underlying plastic deformation in body-centered cubic metals. In the following section we explain the mathematical structure of the potential and the way in which we fit the potential parameters to a database of quantum electronic structure calculations. We follow that with a brief description of the implementation of the SNAP potential in the LAMMPS code. We demonstrate that a previously unnoticed symmetry property can be exploited to reduce the computational cost of the force calculations by more than one order of magnitude. We then present results for the SNAP potential that we have developed for tantalum. We find that this new potential accurately reproduces a range of properties of solid and liquid tantalum. Unlike simpler potentials, it correctly matches quantum MD results for the screw dislocation core structure and minimum energy pathway for displacement of this structure, properties that were not included in the training database.
Mathematical Formulation {#sec:formulation}
========================
Bispectrum coefficients {#subsec:bispectrum}
-----------------------
The quantum mechanical principle of near-sightedness tells us that the electron density at a point is only weakly affected by atoms that are not near. This provides support for the common assumption that the energy of a configuration of atoms is dominated by contributions from clusters of atoms that are near to each other. It is reasonable then to seek out descriptors of local structure and build energy models based on these descriptors. Typically, this is done by identifying geometrical structures, such as pair distances and bond angles, or chemical structures, such as bonds. Interatomic potentials based on these approaches often produce useful qualitative models for different types of materials, but it can be difficult or impossible to adjust these potentials to accurately reproduce known properties of specific materials. Recently, Bart[ó]{}k et al. have studied several infinite classes of descriptor that are related to the density of neighbors in the spherically symmetric space centered on one atom [@Bartok2010; @BartokThesis2010; @Bartok2013]. They demonstrated that by adding descriptors of successively higher order, it was possible to systematically reduce the mismatch between the potential and the target data. One of these descriptors, the bispectrum of the neighbor density mapped on to the 3-sphere, forms the basis for their Gaussian Approximation Potential (GAP) [@Bartok2010]. We also use the bispectrum as the basis for our SNAP potential. We derive this bispectrum below, closely following the notation of Ref.[@Bartok2013].
The density of neighbor atoms around a central atom $i$ at location $\textbf{r}$ can be considered as a sum of $\delta$-functions located in a three-dimensional space:
$$\rho_i ({\bf r}) = \delta({\bf r}) + \sum_{r_{ii'} < R_{cut}}{f_c(r_{ii'}) w_{i'} \delta({\bf r}-{\bf r}_{ii'})}$$
where ${\bf r}_{ii'}$ is the vector joining the position of the central atom $i$ to neighbor atom $i'$. The $w_{i'}$ coefficients are dimensionless weights that are chosen to distinguish atoms of different types, while the central atom is arbitrarily assigned a unit weight. The sum is over all atoms $i'$ within some cutoff distance $R_{cut}$. The switching function $f_c(r)$ ensures that the contribution of each neighbor atom goes smoothly to zero at $R_{cut}$. The angular part of this density function can be expanded in the familiar basis of spherical harmonic functions $Y^l_m(\theta,\phi)$, defined for $l = 0,1,2,\ldots$ and $m = -l,-l+1,\ldots,l-1,l$ [@Varshalovich1987]. The radial component is often expanded in a separate set of radial basis functions that multiply the spherical harmonics. Bart[ó]{}k et al. made a different choice, mapping the radial distance $r$ on to a third polar angle $\theta_0$ defined by,
$$\theta_0 = \theta_0^{max}\frac{r}{R_{cut}}$$
The additional angle $\theta_0$ allows the set of points $(\theta, \phi, r)$ in the 3D ball of possible neighbor positions to be mapped on to the set of points $(\theta, \phi, \theta_0)$ that are a subset of the 3-sphere. Points south of the latitude $\theta_0^{max}$ are excluded. It is advantageous to use most of the 3-sphere, while still excluding the region near the south pole where the configurational space becomes highly compressed.
The natural basis for functions on the 3-sphere is formed by the 4D hyperspherical harmonics $U^j_{m,m'}(\theta_0,\theta,\phi)$, defined for $j=0,\frac{1}{2},1,\ldots$ and $m,m' = -j,-j+1,\ldots,j-1,j$ [@Varshalovich1987]. These functions also happen to be the elements of the unitary transformation matrices for spherical harmonics under rotation by angle $2\theta_0$ about the axis defined by $(\theta, \phi)$. When the rotation is parameterized in terms of the three Euler angles, these functions are better known as $D^j_{m,m'}(\alpha, \beta, \gamma)$, the Wigner $D$-functions, which form the representations of the $SO(3)$ rotational group [@Meremianin2006; @Varshalovich1987]. Dropping the atom index $i$, the neighbor density function can be expanded in the $U^j_{m,m'}$ functions
$$\rho({\bf r}) = \sum_{j=0,\frac{1}{2},\ldots}^{\infty}\sum_{m=-j}^{j}\sum_{m'=-j}^{j} u^j_{m,m'} U^j_{m,m'}(\theta_0,\theta,\phi)$$
where the expansion coefficients are given by the inner product of the neighbor density with the basis function. Because the neighbor density is a weighted sum of $\delta$-functions, each expansion coefficient can be written as a sum over discrete values of the corresponding basis function,
$$u^j_{m,m'} = U^j_{m,m'}(0,0,0) + \sum_{r_{ii'} < R_{cut}}{f_c(r_{ii'}) w_{i'} U^j_{m,m'}(\theta_0,\theta,\phi)}$$
The expansion coefficients $u^j_{m,m'}$ are complex-valued and they are not directly useful as descriptors, because they are not invariant under rotation of the polar coordinate frame. However, the following scalar triple products of expansion coefficients can be shown to be real-valued and invariant under rotation [@Bartok2013].
$$B_{j_1,j_2,j} = \\
\sum_{m_1,m'_1=-j_1}^{j_1}\sum_{m_2,m'_2=-j_2}^{j_2}\sum_{m,m'=-j}^{j} (u^j_{m,m'})^*
{H\!\!{\tiny\begin{array}{l}j m m' \\ j_1 \!m_1 \!m'_1 \\ j_2 m_2 m'_2 \end{array}}}
u^{j_1}_{m_1,m'_1} u^{j_2}_{m_2,m'_2}$$
The constants ${H\!\!{\tiny\begin{array}{l}j m m' \\ j_1 \!m_1 \!m'_1 \\ j_2 m_2 m'_2 \end{array}}}$ are coupling coefficients, analogous to the Clebsch-Gordan coefficients for rotations on the 2-sphere. These invariants are the components of the bispectrum. They characterize the strength of density correlations at three points on the 3-sphere. The lowest-order components describe the coarsest features of the density function, while higher-order components reflect finer detail. An analogous bispectrum can be defined on the 2-sphere in terms of the spherical harmonics. In this case, the components of the bispectrum are a superset of the second and third order bond-orientational order parameters developed by Steinhardt et al. [@Steinhardt1983]. These in turn are specific instances of the order parameters introduced in Landau’s theory of phase transitions [@Landau1980].
The coupling coefficients are non-zero only for non-negative integer and half-integer values of $j_1, j_2,$ and $j$ satisfying the conditions $\| j_1-j_2 \| \leq j \leq j_1 + j_2$ and $j_1+j_2 - j$ not half-integer [@Meremianin2006]. In addition, $B_{j_1,j_2,j}$ is symmetric in $j_1$ and $j_2$. Hence the number of distinct non-zero bispectrum coefficients with indices $j_1, j_2, j$ not exceeding a positive integer $J$ is $(J+1)^3$. Furthermore, it is proven in the appendix that bispectrum components with reordered indices are related by the following identity:
$$\label{eqn:symm}
\frac{B_{j_1,j_2,j}}{2j+1} = \frac{B_{j,j_2,j_1}}{2 j_1+1} = \frac{B_{j_1,j,j_2}}{2 j_2+1}.$$
We can exploit this equivalence by further restricting $j_2 \leq j_1 \leq j$, in which case the number of distinct bispectrum coefficients drops to $(J+1) (J + 2) (J + \frac{3}{2}) / 3$, a three-fold reduction in the limit of large $J$.
SNAP Potential Energy Function {#subsec:snap}
------------------------------
Given the bispectrum components as descriptors of the neighborhood of each atom, it remains to express the potential energy of a configuration of $N$ atoms in terms of these descriptors. We write the energy of the system containing $N$ atoms with positions $\textbf{r}^N$ as the sum of a reference energy $E_{ref}$ and a local energy $E_{local}$
$$E(\textbf{r}^N) = E_{ref}(\textbf{r}^N) + E_{local}(\textbf{r}^N).$$
The reference energy includes known physical phenomena, such as long-range electrostatic interactions, for which well-established energy models exist. $E_{local}$ must capture all the additional effects that are not accounted for by the reference energy. Following Bart[ó]{}k et al. [@Bartok2010; @Bartok2013] we assume that the local energy can be decomposed into separate contributions for each atom,
$$E_{local}(\textbf{r}^N) = \sum_{i=1}^{N} E_i (\textbf{q}_i)$$
where $E_i$ is the local energy of atom $i$, which depends on the set of descriptors $\textbf{q}_i$, in our case the set of $K$ bispectrum components $\textbf{B}^i = \{ B_1^i, \ldots, B_K^i \} $. The original GAP formulation of Bart[ó]{}k et al. [@Bartok2010] expressed the local energy in terms of a Gaussian process kernel. For the materials that we have examined so far, we have found that energies and forces obtained from quantum electronic structure calculations can be accurately reproduced by linear contributions from the lowest-order bispectrum components, with linear coefficients that depend only on the chemical identity of the central atom:
$$\label{eqn:snapatomenergy}
E^i_{SNAP}(\textbf{B}^i) = \beta^{\alpha_i}_0 + \sum_{k=1}^K \beta_k^{\alpha_i} B_k^i = \beta^{\alpha_i}_0 + \boldsymbol\beta^{\alpha_i}\cdot {\bf B}^i$$
where $\alpha_i$ is the chemical identity of atom $i$ and $\beta_k^{\alpha}$ are the linear coefficients for atoms of type $\alpha$. Hence the problem of generating the interatomic potential has been reduced to that of choosing the best values for the linear SNAP coefficients. Since our goal is to reproduce the accuracy of quantum electronic structure calculations for materials under a range of conditions, it makes sense to select SNAP coefficients that accurately reproduce quantum calculations for small configurations of atoms representative of these conditions.
In quantum methods such as density functional theory [@Schultz2005] the most readily computed properties are total energy, atom forces, and stress tensor. The linear form of the SNAP energy allows us to write all of these quantities explicitly as linear functions of the SNAP coefficients. We restrict ourselves here to the case of atoms of a single type, but the results are easily extended to the general case of multiple atom types. In the case of total energy, the SNAP contribution can be written in terms of the bispectrum components of the atoms
$$\label{eqn:snapenergy}
E_{SNAP}(\textbf{r}^N) = N \beta_0 + \boldsymbol\beta\cdot\sum_{i=1}^{N} {\bf B}^i$$
where $\boldsymbol\beta$ is the $K$-vector of SNAP coefficients and $\beta_0$ is the constant energy contribution for each atom. $ {\bf B}^i$ is the $K$-vector of bispectrum components for atom $i$. The contribution of the SNAP energy to the force on atom $j$ can be written in terms of the derivatives of the bispectrum components w.r.t. ${\bf r}_j$, the position of atom $j$
$$\label{eqn:snapforce}
{\bf F}^j_{SNAP} = -\nabla_j E_{SNAP} = - \boldsymbol\beta\cdot \sum_{i=1}^{N} \frac{\partial {\bf B}^i}{\partial {\bf r}_j},$$
where ${\bf F}^j_{SNAP}$ is the force on atom $j$ due to the SNAP energy. Finally, we can write the contribution of the SNAP energy to the stress tensor
$$\label{eqn:snapstress}
{\bf W}_{SNAP} = - \sum_{j=1}^{N} {\bf r}_j \otimes \nabla_j E_{SNAP} = - \boldsymbol\beta\cdot \sum_{j=1}^{N} {\bf r}_j \otimes \sum_{i=1}^{N} \frac{\partial {\bf B}^i}{\partial {\bf r}_j}$$
where ${\bf W}_{SNAP}$ is the contribution of the SNAP energy to the stress tensor and $\otimes$ is the Cartesian outer product operator.
All three of these expressions consist of the vector $\boldsymbol\beta$ of SNAP coefficients multiplying a vector of quantities that are calculated from the bispectrum components of atoms in a configuration. This linear structure greatly simplifies the task of finding the best choice for $\boldsymbol\beta$. We can define a system of linear equations whose solution corresponds to an optimal choice for $\boldsymbol\beta$, in that it minimizes the sum of square differences between the above expressions and the corresponding quantum results defined for a large number of different atomic configurations. This is described in more detail in the following section.
Automated Potential Generation Methodology
==========================================
The previous section outlined the SNAP formulation. In practice, one needs to determine the values of the SNAP coefficients, $\boldsymbol\beta$. This section presents how we solve for the $K$-vector $\boldsymbol\beta$ of SNAP coefficients using a least-squares formulation.
Formulation of the Linear Least Squares Problem {#subsec:linear}
-----------------------------------------------
The fitting problem is overdetermined, in the sense that the number of data points that we are fitting to far exceeds the number of SNAP coefficients. The cost of evaluating the bispectrum components $B_{j_1,j_2,j}$ increases strongly with the order of the indices $j$, $j_1$, and $j_2$. For this reason, $K$ is limited to the range $10 - 100$. In contrast, with the availability of high-performance computers and highly optimized electronic structure codes, it is not difficult to generate data for hundreds or thousands of configurations of atoms. Note that we refer to a *configuration* as a set of atoms located at particular positions in a quantum mechanical calculation. In most cases, the atoms define an infinite repeating structure with specified periodic lattice vectors. For a particular configuration $s$, containing $N_s$ atoms, the electronic structure calculation yields $3 N_s + 7$ data values: the total energy, the $3 N_s$ force components, and the 6 independent components of the stress tensor. The same quantities are calculated for the reference potential. In addition, the bispectrum components and derivatives for each atom in the configuration are calculated. We can use all of this data to construct the following set of linear equations.
$$\begin{bmatrix}
\vdots & \vdots \\
N_s & \sum_{i=1}^{N_s}{\bf B}^i \\
\vdots & \vdots \\
0 & -\sum_{i=1}^{N_s}\frac{\partial {\bf B}^i}{\partial r_j^\alpha}
\\
\vdots & \vdots \\
0 & -\sum_{j=1}^{N_s}r_j^{\alpha}\sum_{i=1}^{N_s}\frac{\partial {\bf B}^i}{\partial r_j^\beta} &
\\
\vdots & \vdots \\
\end{bmatrix} \cdot
\begin{bmatrix}
\beta_0 \\
\boldsymbol\beta\\
\end{bmatrix}
=
\begin{bmatrix}
\vdots \\
E_s^{qm}-E_s^{ref}\\
\vdots \\
F^{qm}_{j,\alpha}-F^{ref}_{j,\alpha} \\
\vdots\\
W_{\alpha\beta,s}^{qm}-W_{\alpha\beta,s}^{ref} \\
\vdots\\
\end{bmatrix}
\label{eqn:abmatrix}$$
This matrix formulation is of the type ${\bf A}\cdot\boldsymbol\beta={\bf y}$ which can be solved for the coefficients $\boldsymbol\beta$. The optimal solution $\hat{\boldsymbol\beta}$ for this set of equations is [@Strang1980]: $$\hat{\boldsymbol\beta} = \underset{\boldsymbol\beta}{\operatorname{argmin}} \| ({\bf A}\cdot\boldsymbol\beta-{\bf y}) \|^2 ={\bf A}^{-1}\cdot{\bf y}$$ In practice, we do not explicitly take the inverse of the ${\bf A}$ matrix, but instead use a QR factorization to solve for $\boldsymbol\beta$. We have found the linear solve to obtain the optimal SNAP coefficients to be very fast and not poorly conditioned.
We have also added the capability to perform weighted least squares to weight certain rows more than others. That is, we add a vector of weights ${\bf w}$ to the minimization formulation:
$$\hat{\boldsymbol\beta} = \underset{\boldsymbol\beta}{\operatorname{argmin}} \| {\bf w} \circ ({\bf A} \cdot \boldsymbol\beta-{\bf y}) \|^2$$
where $\circ$ is used to denote element by element multiplication by the weight vector. Thus, each row in the ${\bf A}$ matrix and the ${\bf y}$ vector are multiplied by a weight specified for that row. In this way, we are able to specify weights per configuration type (e.g. BCC crystals, liquids, etc.) and per quantity of interest (e.g. energy, force, stress tensor). We have found that the ability to weight different rows in the ${\bf A}$ matrix is critical to ensure the regression works well. One reason is that the total energy, forces, and stress components can vary considerably in relative magnitude, depending on what units they are expressed in. However, the more important reason is that it is desirable to control the relative influence of different configurations, depending on the material properties that are of greatest importance. For example, if we want the SNAP potential to more accurately reproduce BCC elastic constants, we can increase the weight on the stress components of strained BCC configurations. We also found it helpful to convert all extensive quantities to intensive quantities, in order to counteract overweighting of configurations with large $N_s$. Total energy rows were scaled by the number of atoms and stress tensor components rows were scaled by the cell volume.
![Flowchart of the optimization loop around the generation of the SNAP potential in LAMMPS[]{data-label="fig:flowDakota"}](SNAPDakota)
SNAP software implementation {#subsec:sw_implement}
----------------------------
We describe the Python framework to generate the SNAP fit within the LAMMPS software tool. We start with quantum mechanical (QM) training data, generated from ab initio calculations, which can be obtained from a wide variety of quantum electronic structure software packages. The training data is first converted to a set of files, one per configuration, using standardized JavaScript Object Notation (JSON) format. The JSON files contain all relevant information such as atom coordinates, atom types, periodic cell dimensions, energies, forces, and stress tensors. Note that the performance of the SNAP potential will depend on the comprehensiveness of the configurations in the training set. In the example of tantalum below, we are interested in material plasticity and stability as well as elastic constants and the lattice parameter. For this reason, the training data included configurations for all the important crystal structures, generalized stacking faults, free surfaces, liquid structures, and randomly deformed primitive cells of the ground-state crystal.
Once the training data is generated, pre-processing and post-processing scripts must be run to generate the SNAP potential in the LAMMPS software. The pre-processing script converts the training data from the JSON format into the native LAMMPS input format. LAMMPS is then used to generate the bispectrum components for the training data configurations, as well as calculating the reference potential. The LAMMPS output is used to generate the energy, force and stress tensor rows of the $A$ matrix defined in Eq. \[eqn:abmatrix\]. The LAMMPS output for the reference potential is combined with the training data to generate the righthand side vector, which is the difference between the training data and the reference potential. A standard linear algebra library is used to perform QR factorization, yielding the least-squares optimal SNAP linear coefficients, ${\hat{\boldsymbol\beta}}$. The steps involved in the generation of the SNAP potential are shown in the yellow box on the right side of Fig. \[fig:flowDakota\].
Optimization of hyperparameters governing the SNAP potential
------------------------------------------------------------
Each SNAP potential developed in this manner depends on the particular values chosen for a set of “hyperparameters.” These include things such as the weights for each training configuration, the list of bispectrum components to be calculated, and the cutoff distance defining the neighborhood of an atom. It is not obvious how to choose values of these hyperparameters which will lead to an “optimal” SNAP potential in the sense of minimizing the SNAP prediction error with respect to energy errors, force errors, or other quantities.
To determine the optimal hyperparameters governing the SNAP potential, we have used an optimization framework and placed it around the SNAP calculation to generate many instances of SNAP potentials. The optimization framework we use is the DAKOTA software [@Dakota2011], which is a toolkit of optimization and uncertainty quantification methods designed to interface to scientific computing codes. The process of generating a SNAP potential within an optimization loop to optimize the governing parameters is shown in Fig. \[fig:flowDakota\]. DAKOTA varies input parameters such as the weights per configuration group and the cutoff distance. These values are then specified in the SNAP generation and the SNAP potential is calculated. Once the SNAP coefficients are generated, they are used to predict the energy and forces of the QM training data. The errors in the SNAP prediction (defined as the difference between SNAP configuration energies vs. QM configuration energies, SNAP forces vs. QM forces, etc.) are then aggregated into an objective function which is returned to DAKOTA and used as the quantity which DAKOTA tries to optimize.
Using this optimization framework, we can identify the SNAP potential that has the “best” result, according to minimizing an objective function. We have examined various objective functions. Currently the objective function we use involves a weighted sum of the errors with respect to energies, forces, and stress tensors, as well as errors with respect to the elastic constants. The objective function is not trivial to define: the resulting SNAP potential can be sensitive to which error measures are weighted more in the objective function.
Implementation {#sec:implementation}
==============
A detailed account of the SNAP implementation and its optimization for specific computing platforms is given elsewhere [@Trott2014]. The force on each atom due to the SNAP potential is formally given by Eq. \[eqn:snapforce\]. In order to perform this calculation efficiently, we use a neighbor list, as is standard practice in the LAMMPS code [@LAMMPS; @Plimpton1995]. This list identifies all the neighbors of a given atom $i$. In order to avoid negative and half-integer indices, we have switched notation from $u^j_{m,m'}$ to $u^\eta_{\mu,\mu'}$, where $\eta=2 j$, $\mu = m+j$, and $\mu' = m'+j$. Analogous transformations are used for ${H\!\!{\tiny\begin{array}{l}j m m' \\ j_1 \!m_1 \!m'_1 \\ j_2 m_2 m'_2 \end{array}}}$ and $B_{j_1,j_2,j}$. This also allows us to reclaim the symbol $j$ for indexing the neighbor atoms of atom $i$. Finally, boldface symbols with omitted indices such as ${\bf u}_i$ are used to indicate a finite multidimensional arrays of the corresponding indexed variables.
Fig. \[fig:calcsnap\] gives the resulting force computation algorithm, where Calc\_U() calculates all expansion coefficients $u^\eta_{\mu,\mu'}$ for an atom $i$ while Calc\_Z() calculates the partial sums $Z_{\eta_1,\eta_2,\eta}^{\mu,\mu'} $, which are defined as
$$\label{eqn:Z}
Z_{j_1,j_2,j}^{m,m'} = \\
\sum_{m_1,m'_1=-j_1}^{j_1}\sum_{m_2,m'_2=-j_2}^{j_2}
H_{j_1,m_1,m'_1,j_2,m_2,m'_2}^{j,m,m'} u^{j_1}_{m_1,m'_1} u^{j_2}_{m_2,m'_2} .$$
In the loop over neighbors, first the derivatives of ${\bf u}_i$ with respect to the distance vector between atoms $i$ and $j$ are computed in Calc\_dUdR() and then the derivatives of $B^i$ are computed in Calc\_dBdR(), which is the most computationally expensive part of the algorithm. For the parameter sets used in this study, Calc\_dBdR() is responsible for approximately 90% of all floating point and memory operations. Thus, we concentrate our description on this function. In Fig. \[fig:calcdbidrjold\] we show the original algorithm based on Eq. \[eqn:dbisold\]. In Fig. \[fig:calcdbidrjnew\] we show the improved algorithm based on Eq. \[eqn:dbisnew\] that takes advantage of the symmetry relation Eq. \[eqn:symm\]. The elimination of the two innermost nested loops reduces the scaling of the computational complexity of the bispectrum component from $O(J^4)$ to $O(J^2)$ where $J$ is the upper limit on $j_1$, $j_2$, and $j$.
Scaling Results {#subsec:scaling}
---------------
Both of these algorithms have been implemented in the LAMMPS parallel molecular dynamics package [@LAMMPS; @Plimpton1995]. In Fig. \[fig:timing\] we compare the performance of the old and new algorithms. Timings are based on a 10,000 step MD simulations of BCC tantalum crystal using the SNAP potential described in Section \[sec:tantalum\]. The calculations were performed on Sandia’s Chama high-performance cluster with a dual socket Intel Sandy Bridge processor with 16 cores on each node. Three different system sizes were used, containing 512, 4096, and 32768 atoms. Each system size was run on 8, 16, 32, 64, 128, and 256 nodes. We plot the time required to calculate one MD time step versus the number of atoms per node. In this form, results for the three different system sizes are almost indistinguishable, indicating that both single node performance and strong scaling efficiency are determined primarily by the number of atoms per node. When the number of atoms per node is large, the parallel scaling is close to ideal, and the improved algorithm is consistently about 16x faster than the old algorithm. As the number of atoms per node decreases, this margin also decreases, due to the parallel efficiency of the new algorithm decreasing more rapidly. This is seen more clearly in Fig. \[fig:efficiency\] which shows the parallel efficiency of each calculation relative to the timing for the largest system on 8 nodes. Both the old and new algorithms show a decrease in parallel efficiency as the number of atoms per node decreases, but this happens at an earlier point for the new algorithm, because there is a lot less computation per atom, while the amount of communication is unchanged.
![CPU time per MD time step versus atoms per node for benchmark simulations of BCC tantalum using the old and improved implementations of the SNAP potential. Results are shown for systems containing 512, 4096, and 32768 atoms. The number of nodes ranged from 8 to 256 nodes.[]{data-label="fig:timing"}](old_new_time.pdf)
![Parallel efficiency versus atoms per node for benchmark simulations of BCC tantalum using the old and improved implementations of the SNAP potential. Results are shown for systems containing 512, 4096, and 32768 atoms. The number of nodes ranged from 8 to 256 nodes.[]{data-label="fig:efficiency"}](old_new_efficiency.pdf)
SNAP Potential for Tantalum {#sec:tantalum}
===========================
Training Data {#subsec:trainingdata}
-------------
The training set data as well as the validation data for these potentials were computed using density functional theory (DFT) electronic structure calculations as implemented in the Vienna Ab initio Simulation Package (VASP) [@Kresse1996]. The pseudopotentials employed are of the projector augmented wave (PAW) form [@Kresse1999] with the exchange correlation (XC) energy evaluated using the Perdew, Burke and Ernzerhof (PBE) formulation [@Perdew1996] of the generalized gradient approximation (GGA). The pseudopotential employed treats 11 electrons $(5p5d6s)$ as valent. Compared to the use of pseudopotentials with 5 valence states, this improved the prediction of the elastic constants and is expected to be important in geometries with highly compressed bond lengths. A plane-wave cut-off energy of 500 eV was used to insure the convergence of the computed stresses and forces in addition to the energies. The k-space integrations were performed using the Monkhorst-Pack (MK) integration scheme [@Monkhorst1976] with mesh sizes that depended on the particular geometry and chosen to obtain convergence of the energy, forces and stress tensor. The lattice predicted for Ta is 0.332 nm compared to the experimental value of 0.331 nm. The computed elastic constants, $C_{11}$, $C_{12}$ and $C_{44}$ are 267.5, 159.7, and 71.1 GPa. This compares well with the experimental results of 266.3, 158.2, and 87.4 GPa by Featherston and Neighbours [@Featherston1963] and 266, 161.1, and 82.5 GPa by Katahara et al. [@Katahara1976].
A variety of different types of atomic configurations were used to construct the full set of training data, and these are summarized in Table \[tab:trainingdata\]. Configurations of different types were chosen to adequately sample the important regions of the potential energy surface. Configurations of type “Displaced” were constructed by randomly displacing atoms from their equilibrium lattice sites in supercells of the A15, BCC, and FCC crystal structures. The maximum displacement in any direction was limited to 10% of the nearest neighbor distance. Configurations of type“Elastic” were constructed by applying random strains to primitive cells of the BCC crystal. The maximum linear strains were limited to 0.5%. The configurations of type “GSF" consist of both relaxed and unrelaxed generalized stacking faults on the (110) and (112) crystallographic planes in the $\langle111\rangle$ direction. The configurations of type “Liquid" were taken from a high-temperature quantum molecular dynamics simulations of molten tantalum. The configurations of type “Surface" consisted of relaxed and unrelaxed (100), (110), (111), and (112) BCC surfaces. For each type of configuration we specified a weight for the energy, force, and stress. We set the force and energy weights of the“Elastic" configurations to zero and we set the stress weights of all other configurations to zero.
Type $N_{conf}$ $N_{atoms}$ Energy Force Stress
--------------- ------------ ------------- -------- ------- --------
Displaced A15 9 64 100 1 -
Displaced BCC 9 54 100 1 -
Displaced FCC 9 48 100 1 -
Elastic BCC 100 2 - - 0.0001
GSF 110 22 24 100 1 -
GSF 112 22 30 100 1 -
Liquid 3 100 100 1 -
Surface 7 30 100 1 -
: DFT data used to fit the SNAP potential for Tantalum[]{data-label="tab:trainingdata"}
In addition to the training data and way in which different quantities were weighted, the quality of the SNAP potential also was somewhat dependent on the choices made for the reference potential and the SNAP hyperparameters. Because the training data did not sample highly compressed configurations, it was important that the reference potential provide a good physical description of Pauli repulsion that dominates the interaction at close separation. We chose the Ziegler-Biersack-Littmark (ZBL) empirical potential that has been found to correctly correlate the high-energy scattering of ions with their nuclear charge $Z_{zbl}$ [@Ziegler1985]. Because the ZBL potential decays rapidly with radial separation, we used a switching function to make the energy and force go smoothly to zero at a distance $R_{zbl,o}$, while leaving the potential unchanged for distances less than $R_{zbl,i}$. The values for these three parameters are given in Table \[tab:snapparams\]. For the neighbor density switching function, we use the same functional form as Bart[ó]{}k et al. [@Bartok2010].
$$\begin{aligned}
\label{eqn:f_c}
f_c(r) & = & \frac{1}{2}(\cos(\pi r/R_{cut}) + 1), r \leq R_{cut} \\
& = & 0, r > R_{cut}\end{aligned}$$
The DAKOTA package was used to optimize the value of $R_{cut}$ so as to minimize the error in the energies, forces, and elastic constants relative to the training data. The resultant value of $R_{cut} = 4.67637$ is physically reasonable, as it includes the 14 nearest neighbors in the BCC crystal, and the first coordination shell in the melt. The effect of using fewer or more bispectrum components was examined experimentally by varying $J$. We found that the fitting errors decreased monotonically with increasing $J$, but the marginal improvement also decreased. We found that truncating at $J = 3$ provided a good trade off between accuracy and computational efficiency. The full set of ZBL and SNAP parameters values are given in Table \[tab:snapparams\], while the values of the SNAP linear coefficients corresponding to each bispectrum component are listed in Table \[tab:snapcoefficients\]. [@LAMMPS]
------------------ --------------
$J$ 3
$R_{cut}$ 4.67637 Å
$\theta_0^{max}$ $0.99363\pi$
$R_{zbl,i}$ 4.0 Å
$R_{zbl,o}$ 4.8 Å
$Z_{zbl}$ 73.0
------------------ --------------
: SNAP and ZBL potential parameters used to model tantalum. [@LAMMPS][]{data-label="tab:snapparams"}
$k$ $2 j_1$ $2 j_2$ $2 j$ $\beta_k$
----- --------- --------- ------- -----------
0 -2.92477
1 0 0 0 -0.01137
2 1 0 1 -0.00775
3 1 1 2 -0.04907
4 2 0 2 -0.15047
5 2 1 3 0.09157
6 2 2 2 0.05590
7 2 2 4 0.05785
8 3 0 3 -0.11615
9 3 1 4 -0.17122
10 3 2 3 -0.10583
11 3 2 5 0.03941
12 3 3 4 -0.11284
13 3 3 6 0.03939
14 4 0 4 -0.07331
15 4 1 5 -0.06582
16 4 2 4 -0.09341
17 4 2 6 -0.10587
18 4 3 5 -0.15497
19 4 4 4 0.04820
20 4 4 6 0.00205
21 5 0 5 0.00060
22 5 1 6 -0.04898
23 5 2 5 -0.05084
24 5 3 6 -0.03371
25 5 4 5 -0.01441
26 5 5 6 -0.01501
27 6 0 6 -0.00599
28 6 2 6 -0.06373
29 6 4 6 0.03965
30 6 6 6 0.01072
: SNAP linear coefficients for tantalum. [@LAMMPS][]{data-label="tab:snapcoefficients"}
Validation Results {#subsec:validationresults}
------------------
One of the crucial requirements for interatomic potential models is that they predict the correct minimum energy crystal structure and that the energetics of competing crystal structures be qualitatively correct. Fig. \[fig:evsv\] plots the energy per atom computed with the SNAP potential as a function of volume for the BCC, FCC, A15, and HCP phases. The energy of diamond structure Ta was also computed. As expected, it was found to be substantially ($\sim$2.8 eV/atom) higher than the BCC phase and is not included on the plot. In addition, energies computed from density functional theory are included as crosses. It is seen that the relative energy of these phases is correctly predicted. The BCC phase is the most stable throughout the volume range considered, with the A15 phase somewhat higher. The FCC phase is next with a minimum energy about 0.2 eV/atom above that of BCC. Note that these energy differences are very consistent with the DFT calculations. The SNAP predicted HCP energy is also shown. Note that the SNAP potential is able to differentiate HCP and FCC crystal structures which are structurally very similar. The SNAP potential predicts that the HCP structure is higher in energy than the FCC structure. This is a prediction in that no HCP data was used in the potential construction. The relative energies of HCP and FCC are in agreement with our DFT calculations, which show that lowest energy HCP structure (not shown) lies 0.04 eV/atom above the minimum energy FCC structure. Further, the SNAP potential predicts the HCP $c/a$ ratio to be 1.72, which is considerably greater than ideal value $c/a \approx 1.63$. The DFT calculations for HCP (not shown) predict an even larger value of $c/a$ = 1.77.
Melting point and Liquid structure
----------------------------------
The melting point predicted by the SNAP potential has been determined. An atomistic slab was created and brought to temperature above the melting point using a Langevin thermostat until the surface of the slab was melted. The molecular dynamics simulation was then continued in an NVE ensemble. This resulted in a system containing two solid-liquid interfaces. The temperature of the MD system now fluctuated around the equilibrium melting temperature. It is important that the solid phase be at the correct melting point density. This was ensured through a simple iterative procedure. An estimate of the melting point was obtained for an assumed lattice constant, the lattice constant of the solid at that temperature was determined from an NPT simulation of the solid, and the melting point was determined with the interfacial area determined by this lattice constant. This process was iterated until the assumed and predicted melting points agreed. This procedure predicted a melting point of 2790 K. This value is in reasonable agreement with the experimental melting point of 3293 K. It should be noted that the melting point is typically a difficult quantity for interatomic potentials to predict accurately. Further, the comparison with experiment is not a direct test of the agreement of the SNAP potential with DFT calculations. The experimental value reflects contributions of the free energy of electronic excitations. Further, the DFT prediction for the melting point is not known.
While configurations that correspond to molten Ta were used in the training set, it is important to determine if the potentials actually reproduce the correct distribution of spatial density correlations in the liquid state. The liquid is an important test of potential models since it samples configurations that are far from those of the equilibrium solid crystals. In particular, the liquid structure depends strongly on the repulsive interactions that occur when two atoms approach each other. We calculated the pair correlation function of the liquid, $g(r)$, both using DFT and from the SNAP potential. These simulations were performed for the same temperature, 3250 K, and atomic density of 49.02 atom/nm$^3$. The DFT simulation treated 100 atoms for a period of 2 ps while the SNAP simulations considered a cell containing 1024 atoms and averaged over 500 ps. Fig. \[fig:liquidstructure\] compares $g(r)$ obtained in the two simulations. The agreement is excellent except perhaps in the region of the first minimum. Note that there is substantially more statistical uncertainty in the DFT result due to the short simulation time and the DFT structure can not be determined beyond about 0.6 nm, due to the smaller simulation cell. These results indicate that the SNAP potential provides a good representation of the molten structure.
Planar and Point Defects
------------------------
A key metric for the applicability of an interatomic potential model is its ability to describe common defects in a material. Table \[tab:defectenergies\] presents a comparison of the energy associated with surfaces, unstable stacking faults, vacancies and self-interstitial atoms. The results obtained from the SNAP potential are compared to our DFT calculations and also against two other interatomic potential models, the embedded atom method (EAM) model developed by Zhou et al. [@Zhou2004] and the angular dependent potential (ADP) due to Mishin and Lozovoi [@Mishin2006].
The training data includes the surfaces and unstable stacking faults presented in Table \[tab:defectenergies\]. The unstable stacking fault energy is the maximum of the energy associated with translating half of the crystal in order to create a stacking fault on the specified plane. This energy is expected to be related to the structure and behavior of dislocations. The overall magnitude of the surface energies is in good agreement with the DFT data. The SNAP potential correctly predicts that the (110) surface is the lowest energy surface plane though it does not correctly predict the relative energies of the other higher energy surface planes. The surface energies predicted by the SNAP potential are generally somewhat better than those predicted by the EAM and ADP potentials. All of the potentials predict unstable stacking fault energies less than the DFT results though all of the potentials predict that the unstable stacking fault energy on the (112) plane is higher than on the (110) plane.
The point defect energies were not included in the training data for the SNAP potential. The predicted vacancy formation energy is 0.15 eV lower than the density function value. The EAM and ADP potentials are in better agreement with the DFT value. However, both of those potentials were also explicitly fit to the vacancy formation energy. The formation energy for self-interstitials are presented for four different configurations. Self-interstitial energies and preferred geometries are challenging quantities for interatomic potential models since they typically involve bond lengths which are substantially shorter than equilibrium values. The overall values for the SNAP potential are in reasonable accord with the DFT, but they incorrectly indicate that the $\langle$110$\rangle$ dumbbell is more favorable than the Crowdion configuration predicted by the DFT calculations.
DFT SNAP EAM ADP
----------------------------------------------------------- ------- ------- ------- -------
Lattice Constant (Å) 3.320 3.316 3.303 3.305
$B$ (Mbar) 1.954 1.908 1.928 1.971
$C' = \frac{1}{2}(C_{11} - C_{12})$ (Mbar) 50.7 59.6 53.3 51.0
$C_{44}$ (Mbar) 75.3 73.4 81.4 84.6
Vacancy Formation Energy (eV) 2.89 2.74 2.97 2.92
\(100) Surface Energy (J/m$^2$) 2.40 2.68 2.34 2.24
\(110) Surface Energy (J/m$^2$) 2.25 2.34 1.98 2.13
\(111) Surface Energy (J/m$^2$) 2.58 2.66 2.56 2.57
\(112) Surface Energy (J/m$^2$) 2.49 2.60 2.36 2.46
\(110) Relaxed Unstable SFE (J/m$^2$) 0.72 1.14 0.75 0.58
\(112) Relaxed Unstable SFE (J/m$^2$) 0.84 1.25 0.87 0.74
Self-Interstitial $-$ Octahedral Site (eV) 6.01 7.10 5.06 7.61
Self-Interstitial $-$ Crowdion (eV) 4.73 5.74 5.09 7.02
Self-Interstitial $-$ $\langle 100 \rangle$ dumbbell (eV) 6.12 6.89 5.24 7.59
Self-Interstitial $-$ $\langle$110$\rangle$ dumbbell (eV) 5.63 5.43 4.93 6.99
: Calculated lattice constant, elastic constants, and formation energies for various crystal surfaces and defects using the SNAP, EAM [@Zhou2004] , and ADP [@Mishin2006] potentials, compared to DFT calculations.[]{data-label="tab:defectenergies"}
Dislocations
------------
![Comparison of screw dislocation migration energy barrier calculated using DFT and the SNAP, EAM [@Zhou2004], and ADP [@Mishin2006] potentials.[]{data-label="fig:dislocationbarrier"}](dislocationbarrier.png){width="95.00000%"}
Fig. \[fig:dislocationbarrier\] shows the screw dislocation migration energy barrier calculated using DFT and the SNAP, EAM [@Zhou2004] , and ADP [@Mishin2006] potentials. One of the dominant deformation mechanisms for metallic materials is the motion of dislocations. For the case of body-centered cubic materials such as tantalum, the screw dislocations are known to play a crucial role. The structure and motion of screw dislocations in BCC metals has been examined for many years and is discussed in detail by Gröger et al. [@Groger2008] and by references therein. A crucial feature of screw dislocations in BCC metals is the Peierls barrier which is the energy barrier to move the dislocation to its next stable configuration. Unlike face-centered-cubic metals where the Peierls barrier is generally negligible, the barrier in the case of BCC metals is substantial and plays a significant role in mechanical deformation. As has been shown recently by Weinberger et al. [@Weinberger2013], many empirical potentials for BCC metals predict qualitatively incorrect Peierls barriers. DFT calculations show that the Peierls has a simple shape with a single hump while many empirical potentials predict a transition path with two maxima and a metastable intermediate state. The Peierls barrier computed via the SNAP potential is shown in Fig. \[fig:dislocationbarrier\] along with the DFT barrier and the barriers predicted by the EAM and ADP potentials. In all cases the barriers were computed based on a dislocation dipole configuration as described by Weinberger et al. [@Weinberger2013]. While the ADP and EAM potentials both predict incorrect barriers with an intermediate metastable state, the SNAP potential predicts a barrier with a single maximum in agreement with the DFT results. Further, the magnitude of the barrier is in excellent agreement with the DFT prediction.
Summary {#sec:summary}
=======
In this paper we have introduced a new class of interatomic potentials based on the 4D bispectrum components first proposed by Bart[ó]{}k et al. [@Bartok2010]. Our SNAP potentials differ from Bart[ó]{}k’s GAP potentials primarily in the use of an explicit linear dependence of the energy on the bispectrum components. We have developed a powerful machine-learning methodology for fitting SNAP potentials to large data sets of high-accuracy quantum electronic structure calculations. We have demonstrated the effectiveness of this approach for tantalum. The new SNAP potential accurately represents a wide range of energetic properties of solid tantalum phases, as well as both the structure of molten tantalum and its melting point. Most importantly, we have found that the SNAP potential for tantalum correctly predicts the size and shape of the Peierls barrier for screw dislocation motion in BCC tantalum. This is a critical property for describing plasticity in tantalum under shear loading and is not correctly described by other published potentials for tantalum [@Zhou2004; @Mishin2006].
One possible drawback of the SNAP methodology is computational cost when used in large-scale atomistic simulations. Even with the algorithmic improvements described in this paper, the cost of SNAP is one to two orders of magnitude more expensive than its competitors. However, it is important to note the high computational intensity also makes the method very amenable to massively parallel algorithms. For a fixed size problem, the SNAP potential has been shown to scale to a much greater extent than simpler potentials [@Trott2014]. We anticipate that atomistic simulations of materials behavior using high-accuracy computationally-intensive potentials such as SNAP will become more common as access to petascale computing resources increases.
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors acknowledge helpful discussions with Stan Moore and Jonathan Moussa on the symmetry properties of bispectrum components. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.
Proof of additional symmetry relation on bispectrum components {#sec:bispectrumsymmetry}
==============================================================
The bispectrum components for an arbitrary function defined on the 3-sphere are given by a sum over triple products of the expansion coefficients
$$\label{eqn:bispectrum}
B_{j_1,j_2,j} = \\
\sum_{m,m'}^{j} (u^j_{m,m'})^*\sum_{m_1,m'_1}^{j_1}\sum_{m_2,m'_2}^{j_2}
{H\!\!{\tiny\begin{array}{l}j m m' \\ j_1 \!m_1 \!m'_1 \\ j_2 m_2 m'_2 \end{array}}}
u^{j_1}_{m_1,m'_1} u^{j_2}_{m_2,m'_2}$$
where $u^{j}_{m,m'}$ are expansion coefficients, given by the inner product with the corresponding basis function $U^{j}_{m,m'}$. The constants ${H\!\!{\tiny\begin{array}{l}j m m' \\ j_1 \!m_1 \!m'_1 \\ j_2 m_2 m'_2 \end{array}}}$ are the Clebsch-Gordan coupling coefficients that can be written as products of the more well-known Clebsch-Gordan coefficients for functions on the 2-sphere
$${H\!\!{\tiny\begin{array}{l}j m m' \\ j_1 \!m_1 \!m'_1 \\ j_2 m_2 m'_2 \end{array}}} = C_{j_1,m_1,j_2,m_2}^{j,m} C_{j_1,m'_1,j_2,m'_2}^{j,m'}$$
It follows from several elementary symmetry properties of the Clebsch-Gordan coefficients [@Varshalovich1987] that coupling coefficients for which the indices $j_2$ and $j$ are interchanged satisfy the following identity
$$\frac{{H\!\!{\tiny\begin{array}{l}j m m' \\ j_1 \!m_1 \!m'_1 \\ j_2 m_2 m'_2 \end{array}}}}
{(2j+1)(-1)^{j+m} (-1)^{j+m'} } =
\frac{{H\!\!{\tiny\begin{array}{l}j_2, -m_2, -m'_2 \\ j_1, \!m_1, \!m'_1 \\ j, -m, -m' \end{array}}}}
{(2j_2+1)(-1)^{j_2-m_2} (-1)^{j_2-m'_2}}$$
Similarly, the 4D hyperspherical harmonics satisfy the following symmetry sign-inversion relation [@Varshalovich1987]
$$U^{j}_{m,m'} = (-1)^{m'-m} (U^{j}_{-m,-m'})^*$$
It follows from linearity of the inner product that the same relation holds for the expansion coefficients $u^{j}_{m,m'}$. Substituting both of these symmetry relations in Eq. \[eqn:bispectrum\] gives,
$$B_{j_1,j_2,j} = (-1)^\epsilon \frac{2 j + 1}{2 j_2 + 1}\\
\sum_{m,m'}^{j} u^{j}_{-m,-m'} \sum_{m_1,m'_1}^{j_1} \sum_{m_2,m'_2}^{j_2}
{H\!\!{\tiny\begin{array}{l}j_2, -m_2, -m'_2 \\ j_1, \!m_1, \!m'_1 \\ j, -m, -m' \end{array}}}
u^{j_1}_{m_1,m'_1} (u^{j_2}_{-m_2,-m'_2})^*$$
where the parity exponent $\epsilon$ is given by the following expression in integer and half-integer indices
$$\begin{aligned}
\epsilon & = & (m'-m) + (m'_2-m_2) + (j+m) + (j+m') + (j_2-m_2) + ( j_2-m'_2) \nonumber \\
& = & 2(j+m') + 2(j_2- m_2)\end{aligned}$$
Both of the quantities in parentheses are integers for all values of $j$, $m'$, $j_2$, and $m_2$ and so the factor $(-1)^\epsilon$ is unity. Finally, reversing the order of the summations over $m_2$, $m'_2$, $m$, and $m'$ yields
$$B_{j_1,j_2,j} = \frac{2 j + 1}{2 j_2 + 1}\\
\sum_{m_2,m'_2}^{j_2} (u^{j_2}_{m_2,m'_2})^* \sum_{m_1,m'_1}^{j_1}\sum_{m,m'}^{j}
{H\!\!{\tiny\begin{array}{l}j_2 m_2 m'_2 \\ j_1 \!m_1 \!m'_1 \\ j m m' \end{array}}}
u^{j_1}_{m_1,m'_1} u^{j}_{m,m'}$$
Comparison with the original Eq. \[eqn:bispectrum\] we see that the nested sum on the right-hand side is $B_{j_1,j,j_2}$. A similar result can be obtained by interchanging $j_1$ and $j$, and so we find that $B_{j_1,j_2,j}$, $B_{j,j_2,j_1}$, and $B_{j,j,j_2}$ are all related by
$$\label{eqn:symm2}
\frac{B_{j_1,j_2,j}}{2j+1} = \frac{B_{j,j_2,j_1}}{2 j_1+1} = \frac{B_{j_1,j,j_2}}{2 j_2+1}.$$
In addition to eliminating redundant bispectrum components, this relation greatly simplifies the calculation of forces. In the original calculation, the spatial derivative of the bispectrum components was written as
$$\begin{aligned}
\label{eqn:dbisold}
\nabla B_{j_1,j_2,j} & = &
\sum_{m,m'}^{j} (\nabla u^j_{m,m'})^*\sum_{m_1,m'_1}^{j_1}
{H\!\!{\tiny\begin{array}{l}j m m' \\ j_1 \!m_1 \!m'_1 \\ j_2 m_2 m'_2 \end{array}}}
u^{j_1}_{m_1,m'_1} u^{j_2}_{m_2,m'_2} \nonumber \\
& & + \sum_{m,m'}^{j} (u^j_{m,m'})^*\sum_{m_1,m'_1}^{j_1}
{H\!\!{\tiny\begin{array}{l}j m m' \\ j_1 \!m_1 \!m'_1 \\ j_2 m_2 m'_2 \end{array}}}
\nabla u^{j_1}_{m_1,m'_1} u^{j_2}_{m_2,m'_2} \nonumber \\
& & + \sum_{m,m'}^{j} (u^j_{m,m'})^*\sum_{m_1,m'_1}^{j_1}
{H\!\!{\tiny\begin{array}{l}j m m' \\ j_1 \!m_1 \!m'_1 \\ j_2 m_2 m'_2 \end{array}}}
u^{j_1}_{m_1,m'_1} \nabla u^{j_2}_{m_2,m'_2} \end{aligned}$$
where the symbol $\nabla$ denotes the derivative of what follows with respect to the position of some neighbor atom. We have dropped the double summation over $m_2$ and $m'_2$, as the coupling coefficients are non-zero only for $m_2 = m-m_1$, likewise for $m'_2$. In the first term, the inner double sum over $m_1$ and $m'_1$ contains no derivatives, and so can be pre-calculated. Hence, for the highest order bispectrum component ($j_1=j_2=j=J$), the computational complexity of this term is $O(J^2)$. However, in the second and third term, the inner double sums contain derivatives and so must be calculated separately for each neighbor atom and for each entry in the outer double sums over $m$ and $m'$. As a result, the computation complexity of the second and third terms is $O(J^4)$. Using the symmetry relation given by Eq. \[eqn:symm\], we can rewrite this as
$$\begin{aligned}
\label{eqn:dbisnew}
\nabla B_{j_1,j_2,j} & = & \hspace{0.24in}
\sum_{m,m'}^{j} (\nabla u^j_{m,m'})^*\sum_{m_1,m'_1}^{j_1}
{H\!\!{\tiny\begin{array}{l}j m m' \\ j_1 \!m_1 \!m'_1 \\ j_2 m_2 m'_2 \end{array}}}
u^{j_1}_{m_1,m'_1} u^{j_2}_{m_2,m'_2} \\
& & \hspace{-0.5in} + \frac{2 j + 1}{2 j_1 + 1} \sum_{m_1,m'_1}^{j_1} (\nabla u^{j_1}_{m_1,m'_1})^*\sum_{m,m'}^{j}
{H\!\!{\tiny\begin{array}{l}j_1 m_1 m'_1 \\ j \!m \!m' \\ j_2 m_2 m'_2 \end{array}}}
u^{j}_{m,m'} u^{j_2}_{m_2,m'_2} \nonumber \\
& & \hspace{-0.5in} + \frac{2 j + 1}{2 j_2 + 1} \sum_{m_2,m'_2}^{j_2} (\nabla u^{j_2}_{m_2,m'_2})^*\sum_{m_1,m'_1}^{j_1}
{H\!\!{\tiny\begin{array}{l}j_2 m_2 m'_2 \\ j_1 \!m_1 \!m'_1 \\ j m m' \end{array}}}
u^{j_1}_{m_1,m'_1} u^{j}_{m,m'} \nonumber\end{aligned}$$
Written in this way, all of the inner double sums are free of derivatives and can be pre-calculated. This has the effect of reducing overall the computational complexity from $O(J^4)$ to $O(J^2)$. For typical cases this results in more than one order of magnitude reduction in computational cost.
|
---
abstract: 'An X-ray spectrograph consisting of radially ruled off-plane reflection gratings and silicon pore optics was tested at the Max Planck Institute for extraterrestrial Physics PANTER X-ray test facility. The silicon pore optic (SPO) stack used is a test module for the Arcus small explorer mission, which will also feature aligned off-plane reflection gratings. This test is the first time two off-plane gratings were actively aligned to each other and with a SPO to produce an overlapped spectrum. The gratings were aligned using an active alignment module which allows for the independent manipulation of subsequent gratings to a reference grating in three degrees of freedom using picomotor actuators which are controllable external to the test chamber. We report the line spread functions of the spectrograph and the actively aligned gratings, and plans for future development.'
author:
- 'Hannah Marlowe, Randall L. McEntaffer, Ryan Allured, Casey DeRoo, Drew M. Miles, Benjamin D. Donovan, James H. Tutt, Vadim Burwitz, Benedikt Menz, Gisela D. Hartner, Randall K. Smith, Ramses Günther, Alex Yanson, Giuseppe Vacanti, Marcelo Ackermann'
bibliography:
- 'refs.bib'
title: 'Performance Testing of a Novel Off-plane Reflection Grating and Silicon Pore Optic Spectrograph at PANTER'
---
[: Hannah Marlowe, University of Iowa, Physics and Astronomy, 210 Van Allen Hall, Iowa City, Iowa, USA, 52242; Tel: +1 319-355-1835; Fax: +1 319-335-1753; E-mail: ]{}
Introduction {#sect:intro}
============
Arcus[@Smith14] is a proposed X-ray spectrograph to be installed on the International Space Station. This spectrograph consists of silicon pore optics (SPO[@Beijersbergen04]) and blazed, radially ruled off-plane reflection gratings optimized for the feature-rich soft X-ray regime. The mission utilizes the SPO being developed for ESA’s Athena mission by cosine Research and reflection gratings being developed in collaboration between the University of Iowa and MIT/Lincoln Labs, and the mission . The Arcus mission will answer key science questions related to structure formation in the Universe, supermassive black hole feedback, and stellar life cycles. To meet its science objectives, Arcus will have a resoltuion of $\lambda/\Delta \lambda$ > 2500 and effective area > 600 cm$^{2}$ in the critical science bandpass around the O VII and O VIII lines (22.6 – 25 Å ). The mission will have a minimum resolution and effective area of $\lambda/\Delta \lambda$ > 1300 and > 130 cm$^{2}$ over the entire bandpass (8 – 52 Å ) with $\lambda/\Delta \lambda$ reaching $\sim$3000 at the longest wavelengths.
Performance testing of aligned off-plane reflection gratings with a SPO module was carried out at the PANTER[@Burwitz13] test facility of the Max Planck Institute for extraterrestrial Physics (MPE) in October 2014. During the tests, a radially ruled, off-plane reflection grating was aligned to the SPO test module. A second grating was then actively aligned to the first reference grating. This test was the first time that off-plane diffraction gratings were aligned with an SPO, and that two off-plane gratings were aligned to one another *in situ*. This paper describes the experimental setup and results of the test campaign, specifically the comparison between the line spread functions of the SPO module and the first order Mg-K line from the aligned gratings. The components of the spectrograph are detailed in §\[sect:telescope\], an overview of the alignment procedure is given in §\[sect:exp\], CCD image reduction steps and measured line widths are presented in §\[sect:reduction\]–\[sect:results\], and discussion of the results in §\[sect:disc\].
Spectrograph Assembly {#sect:telescope}
=====================
The spectrograph assembly tested at the PANTER facility consists of a SPO telescope, off-plane reflection gratings, and a CCD detector at the grating focal plane. A schematic of the SPO, grating, and focal plane positions within the test chamber is shown in Figure \[fig:sch\]. An image of the components installed in the PANTER vacuum chamber during initial grating alignment with a laser is presented in Figure \[fig:spo\] where the SPO stack is closest to the camera and the grating module is visible in the background.
![ \[fig:sch\] Bottom view of the test chamber SPO and grating integration.](fig1.png){width=".8\textwidth"}
![ \[fig:spo\] The SPO as viewed from the source direction during optical alignment. The active alignment grating module is visible in the background. ](fig2.jpg){width=".7\textwidth"}
Silicon Pore Optics
-------------------
SPO have been developed for the past 10 years by a consortium led by cosine Research, and have become the main technology for the X-ray mirror of the Athena mission [@Willingale13]. SPO make use of industry standard super polished silicon wafers. These wafers are first diced into mirrors plates of the desired rectangular shape, then each plate is wedged using a thin deposition of material on each side so that a focusing optics is formed when multiple plates are stacked onto a conical mandrel. Before being stacked, the plates are ribbed, leaving a thin membrane used to reflect the X-rays and a number of ribs that are used to bond to the next plate. This results in pores in the SPO stack, through which the X-rays can reflect and travel to the focal plane detector. If necessary to meet science requirements, plates can be coated to increase their reflectivity. An SPO stack is very stiff, light-weight, and the stacking process is such that the figure of the mandrel is reproduced with high fidelity so that by combining two stacks in a mirror module, a high resolution imaging system can be built.
For this campaign, a single SPO stack was built. The stack consists of 13 plates, with radii between 450 and 439 mm, width of 66 mm, and length of 22 mm and its geometry approximates that of a parabolic reflector, and t. With a focal length of 8 m, the wedge on each plate was tuned to deliver the required 10 arcsecond ($''$) change in incidence angle between consecutive plates, resulting in a confocal system. For on-axis measurements, the incidence angle is of the order of 1.5$^{\circ}$.
The SPO stack is shown in Figure \[fig:spo\_close\] prior to installation in the PANTER vacuum chamber. Due to time and budget constraints, the SPO module for this test was shaped using a simple Aluminum mandrel rather than one made of high quality polished fused silica. Therefore, it is important to note that the properties of this SPO module, while qualitatively similar to those in the planned Arcus design, are not characteristic of the state of the art in SPO manufacturing.
![ \[fig:spo\_close\] The SPO module prior to installation in the PANTER chamber. ](fig3.jpg){width=".7\textwidth"}
Gratings
--------
A diagram of the off-plane grating geometry is shown in Figure \[fig:grating\_geom\]. In the off-plane mount, light that is incident onto the gratings at a grazing angle and roughly parallel to the groove direction is diffracted into an arc. The diffraction equation for the off-plane mount is: \[equ:grating\] $$\sin \alpha + \sin \beta = \dfrac{n \lambda}{d \sin \gamma}$$ where $\gamma$ is the polar angle of the incident X-rays defined from the groove axis at the point of impact, $d$ is the line spacing of the grooves, $\alpha$ represents the azimuthal angle along a cone with half-angle $\gamma$, and $\beta$ is the azimuthal angle of the diffracted light. The grooves are radially ruled such that the spacing between adjacent grooves decreases towards the focus to match the convergence of the telescope.
![ \[fig:grating\_geom\] Geometry of the off-plane grating mount[@McEntaffer13]](fig4.png){width=".4\textwidth"}
For this test, two gratings were actively aligned together to demonstrate a technique for aligning nested diffraction gratings in an Active Alignment Module (AAM). The AAM consists of slots for the grating wafers and an exterior skeleton into which 5 picomotors are mounted in order to align sequential gratings to an initially installed fixed reference grating. The AAM is shown in Figure \[fig:aam\], where two of the five picomotors (used to control grating yaw in this test) are visible on the top of the module. For this test only 1 additional grating was installed and aligned to the reference grating, though the procedure could be repeated to add more gratings as desired. The general installation and alignment procedure is as follows: a reference grating is bonded into the first wafer slot of the AAM. The active wafer is then installed into the adjacent wafer slot with springs between the active wafer and reference wafer surfaces and between the second wafer and the base of the AAM. Finally, the picomotor cage is installed and the 5 picomotors actuate to apply force against the active wafer which is balanced by the surface and base springs. One can now actuate the active grating and finely align it *in situ* using incident X-rays. The step size of each picomotor is approximately 30 nm, which translates to angular step sizes of $\sim$0.1$''$ in roll, pitch, and yaw. A detailed overview of the active alignment is given by Allured et al. [@AlluredIP].
![ \[fig:aam\] The AAM as viewed from the X-ray source direction. The top two picomotors control the active grating yaw while the three motors on the face of the grating (image right) actuate roll and pitch. ](fig5.png){width=".7\textwidth"}
Detectors
---------
Three detectors were in use at the focal plane in the PANTER chamber during these test, TRoPIC, the ROSAT Position Sensitive Proportional Counter (PSPC[@Briel86]), and PIXI. TRoPIC is a single photon counting detector with 75 $\upmu$m pixels operated in frame-store mode. TRoPIC is a prototype of the eROSITA detector and is identical apart from its smaller pixel format of 256$\times$256 compared to 384$\times$384 [@Meidinger09]. The PANTER facility also has a spare of the ROSAT PSPC detector which we utilized for macro imaging of orders and for rough alignment due to its large active area (80 mm diameter) though relatively course spatial resolution of $\sim$250 $\upmu$m[@Pfeffermann87; @Pfeffermann03]. PIXI is a Peltier and water cooled Princeton Instruments PI-MTE-1300B integrating in-vacuum CCD with 20 $\upmu$m pixels in a 1340$\times$1300 format [@Burwitz13]. All three detectors are shown in Figure \[fig:det\]. PSPC and TRoPIC are mounted onto the same translation stages and were able to image the 0 and $\pm1^{\rm st}$ orders. PIXI was mounted on a separate vertical translation stage sharing the other movements with TRoPIC and PSPC and was able to reach negative orders. In this paper we focus on measurements taken with the PIXI detector.
![ \[fig:det\] The X-ray detectors at the PANTER facility. From left to right: PIXI, TRoPIC, and PSPC. ](fig6.jpg){width=".7\textwidth"}
PANTER Testing {#sect:exp}
==============
To characterize the spectral resolving power of the grating spectrograph assembly, the gratings and SPO were installed into the detector chamber of the PANTER test facility. The PANTER beamline consists of a multi-target electron impact X-ray source at the head of a 1 m diameter 120 m long vacuum chamber. The beamline ends in a 3.5 m by 12 m test chamber which easily accommodates the SPO, gratings, and detectors.
The gratings were mounted into the PANTER chamber within the AAM with six degrees of freedom relative to the SPO and CCD. The SPO light is incident on the gratings at an angle of 1.5$^{\circ}$. This incidence angle was set using the separation between the SPO focus and the zeroth order reflection of an optical laser mounted at the head of the beamline and has an uncertainty of approximately 1 cm over the 8 m throw ($\sim$4$'$). The chamber was then evacuated, and the PSPC was used find the various diffraction orders and to initially zero the reference grating yaw. Rough alignment was accomplished with PSPC followed by fine alignment with PIXI and TRoPIC. The active grating was aligned to the reference grating by manipulating the AAM picomotors on the active grating (as discussed in §\[sect:telescope\]) which allowed for independent roll, pitch, and yaw adjustment. Once both gratings were aligned, we bonded the active grating in place and remeasured the alignment of the gratings using the PIXI detector.
Reduction {#sect:reduction}
=========
This section outlines the reduction steps taken to extract LSF measurements from the CCD images taken at PANTER. For the scope of this paper, we focus on images taken with PIXI after the active grating was aligned and bonded into place.
The preprocessing of the images is as follows:
1. The dark frame is subtracted from the integrated image. The dark frame being a 1340$\times$1300 array where each pixel is the median value of $N$ dark exposures taken consecutively.
2. The background distribution is determined by fitting a normal distribution to the pixels in the non-illuminated edges of the image.
3. The dark-corrected images are thresholded by setting to zero any pixels which fall below 3 times the background $\sigma$ value.
The result of preprocessing is an image whose pixels represent integrated ADC counts.
To fit the LSF of each spectral line, the image is first cropped in $x$ and $y$ around the spectral line. A rotation is then applied to the image via a rotation matrix to account for misalignment with respect to the CCD. The rotation angle is found by performing a least squares fit of the form $y=mx+b$, where $y$ is in the dispersion direction, is applied to the image weighted by each pixel value. This fit is applied for an array of rotation angles, yielding a relationship between the fitted slope and the rotation angle. The best rotation is found as the intercept of a line fit to the rotation angles versus the slope at each angle. This angle is the amount by which the image should be rotated to minimize the slope of the spectral line thereby aligning it with respect to the image x and y axis.
Figure \[fig:CCDims\] shows the cropped and rotated CCD images of the SPO focus and of the first order Mg-K (1.25 keV) line. A ‘V’-like structure is apparent in the SPO focus (Figure \[fig:CCDims\] (Left)). Similar structure to the SPO focus also appears, as expected, in the grating focus (Figure \[fig:CCDims\] (Right)). This structure causes the spectral lines to be poorly fit by simple Gaussian models. Therefore, we describe the line width by the half-energy width (HEW) in the dispersion direction. The CCD image is first flattened in the non-dispersion direction and the HEW of the line is calculated from its cumulative distribution function (CDF) where the HEW contains the central 50% of the line’s integrated ADC counts. The CDF is fit with a spline interpolation, allowing the HEW boundaries fall between pixel values.
![ \[fig:CCDims\] (Left) PIXI image of the SPO focus; (right) PIXI image of the 1$^{\rm st}$ order Mg-K line. ](fig7.png){width=".5\textwidth"}
Measured SPO and Telescope Line Spread Function {#sect:results}
===============================================
Figure \[fig:spo\_lsf\] shows the width of the SPO focus in the grating dispersion direction (dashed line) overplotted with the first order Mg-K line (solid line). The Mg-K line is presented due to its natural line width ($\Delta E/E =\mathrm{0.36~eV/1254~eV}$) which is much narrower in dispersion than the width of the SPO focus. The HEW of the SPO focus is found to be 2.29$''$ and the HEW of the Mg-K line is found to be 1.78$''$.
![ \[fig:spo\_lsf\] Mg-K 1$^{\rm st}$ order PIXI line profile (solid curve) overplotted with the SPO focus line profile (dashed curve). The HEW are found to be 2.29$''$ and 1.78$''$ for the SPO and Mg-K lines, respectively.](fig8.png){width=".7\textwidth"}
Discussion {#sect:disc}
==========
We have demonstrated a spectrograph composed of silicon pore optics (SPO) and actively aligned off-plane reflection gratings at the PANTER test facility. The half energy width (HEW) of the SPO module is measured to be 2.29$''$ while the HEW of the actively aligned gratings was measured to be 1.78$''$. We therefore find a narrower line width for the first order Mg-K line than for the SPO focus. A narrower line width for the grating focus could be attributed to the grating sub-aperturing the light from the SPO. While the SPO focus is an integration over all of the SPO plates, the unmasked regions of the gratings only intercept light from approximately two SPO plates. Thus, misalignment between SPO plates will have a larger impact on the width of the total SPO focus than the grating reflection.
Future development will improve the performance of the telescope. For example, the SPO plates were shaped using a simple Aluminum mandrel. Further iterations will improve on the optic quality by the use of high quality polished fused silica mandrels. We can conclude that the current resolution of the telescope is not limited by the grating resolution or grating alignment, and that higher resolutions can be achieved with further iterations of the mirror assembly.
[**Hannah Marlowe**]{} is a graduate research assistant at the University of Iowa. She received her bachelors degree in astrophysics from Agnes Scott College in 2011. Her current research interests include off-plane X-ray diffraction, X-ray polarimetry, and X-ray spectroscopy.
Biographies and photographs of the other authors are not available.
|
---
abstract: 'We perform a refined count of BPS states in the compactification of M-theory on $K3 \times T^2$, keeping track of the information provided by both the $SU(2)_L$ and $SU(2)_R$ angular momenta in the $SO(4)$ little group. Mathematically, this four variable counting function may be expressed via the motivic Donaldson-Thomas counts of $K3 \times T^2$, simultaneously refining Katz, Klemm, and Pandharipande’s motivic stable pairs counts on $K3$ and Oberdieck-Pandharipande’s Gromov-Witten counts on $K3 \times T^2$. This provides the first full answer for motivic curve counts of a compact Calabi-Yau threefold. Along the way, we develop a Hodge-elliptic genus for Calabi-Yau manifolds – a new counting function for BPS states that interpolates between the Hodge polynomial and the elliptic genus of a Calabi-Yau.'
---
Shamit Kachru$^1$ and Arnav Tripathy$^2$
$^1$Stanford Institute for Theoretical Physics
Department of Physics, Stanford University, Palo Alto, CA 94305, USA
$^2$Department of Mathematics, Harvard University
Cambridge, MA 02138, USA
Introduction and summary
========================
This paper is motivated by natural enumerative questions in physics and mathematics, related to degeneracies of rotating black holes on the one hand, and enumerative geometry of Calabi-Yau manifolds and allied questions in geometry and number theory on the other. Therefore, we will provide a brief introduction to both the physical and mathematical problems of interest.
Entropy counts
--------------
Understanding exact counts of BPS states in string vacua has been an ongoing project of significant interest for the past twenty years. As one primary application, computing protected indices and increasing the string coupling until the states collapse into black holes is the predominant approach for obtaining microscopic accounts of black hole entropy. The first work in this direction was that of Strominger and Vafa [@StromingerVafa] counting BPS states in type IIB string theory on $K3 \times S^1$ (or, equivalently, M-theory on $K3 \times T^2$). Since then, much effort has been devoted to refining these counts, often in the related four-dimensional string vacuum given by compactifying type IIA or IIB string theory on $K3 \times T^2$ (for reviews, see [@Senreview; @Atishreview]); the counts of BPS states in the 4d and 5d theories are related via the 4d-5d lift, as explained in [@SSY]. In such 4d $\mc{N} = 4$ theories, we have two classes of BPS states – those that preserve half of the supersymmetry, and those preserving only a quarter. The $1/2$-BPS states may be assumed to carry only electric charge in a suitable U-duality frame, whereas the $1/4$-BPS states are dyons, carrying both electric and magnetic charges. As such, an unflavored count of $1/2$-BPS states should depend on only one quantum number, namely the single charge, while the count for $1/4$-BPS states takes slightly more care. The electric and magnetic charges $Q_e$ and $Q_m$ in the charge lattice transform under the action of the U-duality group, and there are precisely three U-duality invariant combinations, namely $Q_e \cdot Q_e, Q_e \cdot Q_m,$ and $Q_m \cdot Q_m$. So, an accounting of the $1/4$-BPS states (without flavoring by any other information) should be a generating function in three variables. A precise form for this function was first proposed by Dijkgraaf-Verlinde-Verlinde (DVV) in [@DVV]. We will review the formulae for the 1/2-BPS and 1/4-BPS counts in the next section.
While the existing story of 1/4-BPS counts therefore involves a partition function tracking three gradings, from at least two perspectives it seems that a four-variable function would be more natural. The first perspective is that of entropy counts. The little group of a massive particle is $SO(4) \sim SU(2)_L \times SU(2)_R$. In [@BMPV], black hole solutions with two charges ($Q_F$ and $Q_H$) and two angular momenta (in two two-planes in ${\mathbb R}^4$) are discussed. In the limit that $K3$ is small compared to the circle, the degeneracy of supersymmetric ground states can be computed in an effective string theory (wrapping the $S^1$). This theory has $(4,4)$ supersymmetry, and the angular momenta map to the left and right fermion number charges (i.e., quantum numbers under the $U(1)$ R-symmetries embedded in the left/right $SU(2)$ worldsheet R-symmetries guaranteed to exist by $(4,4)$ superconformal invariance) via $$J_1 = F_L + F_R$$ $$J_2 = F_L - F_R$$ Counting supersymmetric solutions in space-time should be related to studying world-volume BPS states where the right-movers are kept in a Ramond ground state. The supersymmetric black holes constructed in supergravity have $|J_1| = |J_2| = J$, but it was already mentioned in [@BMPV] that from the effective string perspective, one can keep track of the $F_R$ quantum numbers of Ramond ground states for the right movers. So one should expect a more refined counting to allow independent tracking of the $F_R$ quantum number, and hence two angular momenta. It is this counting that we detail in this paper.
It is important to stress that, as noted in [@BMPV], because the right-moving Ramond ground states have $F_R$ bounded by the central charge of the effective string, this additional quantum number can be viewed as small hair on top of the macroscopic supergravity quantum numbers. In fact, given a count of BPS states at some point in moduli space, only some of these will become microstates of a single-center black hole as one increases the string coupling. Others may be related to hair which dresses the horizon, to multi-center solutions, or even to solutions of distinct horizon topology. For basic calculations extracting black hole microstate counts from BPS state counts by subtracting counts of the hair, see [@Sameer; @Senhair; @Murthyhair]; for a discussion of the fact that one has vanishing angular momentum for 4d black hole microstates, see [@zero] and references therein. In light of these developments, it is important to stress that our conjectural BPS counting function is a count of the full spectrum of BPS states, without considerations of hair versus black hole microstates.
There is also a second development in the literature that suggests the existence of a more general counting function. The black holes above generically have both $Q_F$ and $Q_H$, but you may count the subsector of spinning black holes only charged under one; for example, only having $Q_F$. This subsector corresponds in four dimensions to only counting the states with $Q_m^2 = 0$ (but generically carrying both $Q_e^2$ and $Q_e \cdot Q_m$ charges). The generating function for degeneracies of such special spinning 5d black holes (with $|J_1| = |J_2|$) was computed by Katz, Klemm, and Vafa (KKV) in [@KKV] (in the 4d picture, the angular momentum becomes $Q_e \cdot Q_m$). A refinement to account for the possibility of turning on $|J_1| \neq| J_2|$ was described recently by Katz, Klemm and Pandharipande (KKP) [@KKP]. It gives rise to a three-variable generating function, but clearly describes black holes not carrying $Q_m^2$ (or its 5d lift). From this perspective as well, it seems that there is a natural role for a more general function allowing $Q_m^2 \neq 0$, but keeping the general angular momenta allowed by the KKP generating function. We will construct the four-variable generating function $\Phi$ which correctly reduces to both the DVV counting function when one turns off the second angular momentum quantum number, and the KKP counting function when one turns off $Q_m^2$.
Mathematical questions
----------------------
These counts have also been of interest mathematically in the curve-counting literature. Depending on the particular U-duality frame one chooses, it can be more natural to interpret them in terms of Gromov-Witten invariants arising from the topological string, Gopakumar-Vafa invariants, or Donaldson-Thomas invariants. The connections between these invariants are described in detail in [@GV; @INOV; @Kapustin] in the physics literature, and in [@MNOP] in the mathematics literature. All of the BPS state counts above have interpretations in one or more of these frames and therefore lend themselves to mathematical interpretation. The original $1/2$-BPS state count may be interpreted as the genus $0$ Gromov-Witten theory of $K3$, as formulated in Yau-Zaslow [@Yau-Zaslow] and proven in [@KMPS]. The partially flavored KKV count of spinning black holes corresponds to turning on a chemical potential tracking the source genus in the topological string; interpreted as the full Gromov-Witten theory of $K3$, this statement is discussed and proven in work of Pandharipande-Thomas [@PT]. More recently, the $1/4$-BPS state counts of $K3 \times T^2$ were interpreted as a precise mathematical conjecture in terms of Gromov-Witten invariants in [@OP] and discussed further in terms of Donaldson-Thomas invariants in [@Bryan]. Refining further corresponds to considering motivic invariants in the sense of [@KS]; indeed, the connection between flavoring with respect to a little group spin and considering the motivic lift of Donaldson-Thomas invariants is explained via wall-crossing in Dimofte-Gukov [@Gukov]. It follows that the geometric interpretation of the KKP counting function is as the full motivic generating function for $K3$. This leaves open the question of the full motivic curve counts for $K3 \times T^2$ (though some comments appear in the final section of [@OP]). The four-variable function $\Phi$ we compute here constitutes a physically motivated proposal for these counts.
A perhaps surprising feature of these counting functions (sometimes explained through S-duality, which itself is poorly understood) is that they are related to automorphic functions. It is not difficult to see that the unflavored $1/2$-BPS count is modular for the usual modular group $SL(2, \mb{Z})$; it is more interesting that the KKV and KKP counts are Jacobi and multivariable Jacobi forms, respectively. That the $1/4$-BPS state count – which yields the Igusa cusp form – is automorphic for the group $Sp(4, \mb{Z})$ or $O(2, 3, \mb{Z})$ is still more surprising, and many attempts have been made to explain this curious automorphy, notably including [@Gaiotto] and [@Dabholkar]. We offer no new interpretations of the automorphy of the BPS state counts in this paper, merely noting that the $1/4$-BPS state counts very often seem to be a lift of the $1/2$-BPS state counts in a precise mathematical sense; this holds not just for $K3\times T^2$ compactification, but for a wide variety of 4d $N=4$ supersymmetric models obtained as so-called CHL strings (which involve quotients of $K3 \times T^2$ by a Nikulin involution on $K3$ and a shift on the torus) [@Sen]. A suitable version of our function $\Phi$ appears to be automorphic for $O(2, 4, \mb{Z})$, which again requires physical explanation; we will not make further comment on this curious automorphy here.
There is also a connection to wall-crossing phenomena. The refined BPS counts jump as we move in the moduli space of our theory, undergoing wall-crossing transformations (as discussed for example in [@KS] or [@Gukov]). In fact, the wall-crossing for the dyon counting function of DVV was carefully investigated in [@Cheng] to establish the validity of the formula not just in one chamber of moduli space but throughout all of moduli space. It likely will be interesting to carry out the same analysis for our refined count $\Phi$ to match the wall-crossing for the refined BPS index on the one hand with the mathematical jumping of the coefficients in our (conjecturally) automorphic function $\Phi$ as we cross walls on its domain.
A final connection of mathematical interest is to moonshine. Some numerical coincidences suggested to the authors of [@KKP] that their invariants may exhibit an analogue of Mathieu moonshine [@EOT]. This story was developed in [@Kachru], where the KKP invariants were re-expressed as traces in a moonshine module for Conway’s largest sporadic group. Our further refined function $\Phi$ should also naturally be expected to play a role in moonshine; we leave this connection for future work.
The plan of the paper is as follows. In the next section, we review the known formulae for BPS state counts before going on to state an analogous formula for our refined state-counting function $\Phi$. In section $3$, we define the Hodge-elliptic genus. Section $4$ proposes a formula for the refined counting function $\Phi$ in terms of the Hodge-elliptic genus, by considering the D1-D5 system in type IIB string theory and applying an argument following the strategy of Dijkgraaf, Moore, Verlinde, and Verlinde (DMVV) [@DMVV]. Finally, section $5$ has a discussion of the interpretation of these results in the type IIA D0-D2-D6 frame; mathematicians will likely be most interested in this section.
Statements of BPS state and black hole counts
=============================================
We recall in this section many of the counts previously computed. First, if we denote by $c_n$ the number of $1/2$-BPS states with charge squaring to $2n - 2$, we may easily compute the generating function $$f(q) = \sum c_n q^{n-1},$$ for example by working in the dual frame given by heterotic string theory compactified on $T^6$. Here, the $1/2$-BPS states are ground states on the right and hence simply given by excitations of the $24$ left-moving bosonic oscillators [@Harvey]. One may hence easily compute $$f(q) = \frac{1}{\eta(\tau)^{24}} = \frac{1}{\Delta(\tau)},$$ where $q = e^{2 \pi i \tau}$ and the $\eta$ function is given by $$\eta(\tau) = q^{1/24} \prod_{n=1}^{\infty} (1 - q^n).$$
Starting to flavor by the angular momenta, we next have the KKV count, where we write the generating function in terms of $c^r_n$, the number of $1/2$-BPS multiplets with $SU(2)_L$ representation labeled by spin $r$ and charge whose square is $2n - 2$; now, we write the generating function as $$\sum c^r_n q^{n-1} y^{[r]} = \frac{1}{\phi_{KKV}(\tau, z)},$$ where we use the notation $$j^{[\ell]} = j^{-2 \ell} + j^{-2(\ell - 1)} + \cdots + j^{2(\ell - 1)} + j^{2 \ell}.$$ We now have $\phi_{KKV}$ a Jacobi form in terms of $\tau, z$ under the relations $q = e^{2 \pi i \tau}, y = e^{2 \pi i z}$. In fact is the named Jacobi form $$\phi_{10,1}(\tau, z) = (y^{1/2}-y^{-1/2})^2 ~q \prod_{n=1}^{\infty} (1 - q^n)^{20} (1 - q^n y)^2 (1 - q^n y^{-1})^2,$$ where the subscripts refer to the weight and index of the Jacobi form.
Finally, the fully flavored count for states with $Q_H=0$, if we denote $c^{r_L, r_R}_n$ the number of representations now with spin $r_L$ under $SU(2)_L$ and $r_R$ under $SU(2)_R$, is given by $$\sum c^{r_L, r_R}_n q^{n-1} y^{[r_L]} u^{[r_R]} = \frac{1}{\phi_{KKP}(\tau, z, \nu)},$$ where we have now also introduced $u = e^{2 \pi i \nu}$ and $\phi_{KKP}$ is the multivariable Jacobi form given by $$\begin{aligned}
\nonumber
\phi_{KKP}(\tau, z, \nu) &=& q~ (y^{1/2}u^{1/2}-y^{-1/2}u^{-1/2})(y^{-1/2}u^{1/2}-y^{1/2}u^{-1/2}) \\
\nonumber
&&\prod_{n=1}^{\infty} (1 - q^n)^{20} (1 - q^n uy) (1 - q^n uy^{-1})
(1 - q^n u^{-1}y) (1 - q^n u^{-1}y^{-1})\end{aligned}$$
Moving to the $1/4$-BPS state counts, if we denote by $c_{n, \ell, m}$ the number of multiplets with charges satisfying $$\frac{1}{2} Q_e \cdot Q_e = n, Q_e \cdot Q_m = \ell, \frac{1}{2} Q_m \cdot Q_m = m,$$ we have the generating function $$\sum c_{n, \ell, m} p^{n} y^{\ell} q^{m} = \frac{1}{\Phi_{10}(\sigma, z, \tau)}.$$ We have now also introduced $p = e^{2 \pi i \sigma}$ and in fact $\Phi_{10}(\sigma, z, \tau)$ is the (unique) Siegel cusp form of degree $2$, weight $10$, and full level. We have a similar product-type formula for this generating function, given by the following multiplicative lift: $$\Phi_{10}(\sigma, z, \tau) = pqy \prod_{(n,m,\ell) > 0} (1 - p^n y^{\ell} q^m)^{c(nm, \ell)}~.$$ Here, $(n,m,\ell) > 0$ means $n, m \ge 0$ and $\ell$ runs over all integers, with the caveat that if $n = m = 0$, then $\ell < 0$. The $c(m, \ell)$ are the coefficients of the elliptic genus $$Z_{EG}(K3)(\tau, z) = \sum c(m, \ell) q^m y^{\ell},$$ with the elliptic genus being independently defined as $$Z_{EG}(\tau, z) = \tr_{RR} \Big( (-1)^{F_L + F_R} y^{F_L} q^{L_0 - {c\over 24}} \bar q^{\bar L_0 - {\bar c \over 24}} \Big).$$ $F_L, F_R$ are the left and right fermion numbers and $L_0, \bar L_0$ are the left and right moving Hamiltonians, with the trace being taken over the entire Ramond-Ramond sector of the theory. Note however that due to the $(-1)^{F_R}$ insertion, the trace localizes to the right Ramond ground states, as all other states pair up and annihilate in this index. This observation is crucial to the extension we define next. Note furthermore that just as the $1/4$-BPS state count was a multiplicative lift of the elliptic genus in the sense that the coefficients of the elliptic genus appear as exponents in a product formula for the generating function, the $1/2$-BPS state counts are also multiplicative lifts of simpler expressions, namely the (symmetrized) Euler characteristic, $\chi_y$ genus, and Hodge polynomial of $K3$ respectively. Explicitly, these are given by $$\begin{aligned}
\chi(K3) &=& 24 \\ \chi_y(K3) &=& 2y^{-1} + 20 + 2y \\ \chi_{Hodge}(K3) &=& u^{-1}y^{-1} + u^{-1}y + 20 + uy^{-1} + uy. \end{aligned}$$ We will return to this observation in section $4$ when we revisit the DMVV argument; for now, we simply comment that we will express the refined count of $1/4$-BPS states in terms of a multiplicative lift of an interpolation between the Hodge polynomial and the elliptic genus, which we term the Hodge-elliptic genus.
The Hodge-elliptic genus
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The definition is simple: $$Z_{HEG}(\tau, z, \nu) = \tr \Big((-1)^{F_L + F_R} y^{F_L} u^{F_R} q^{L_0 - {c \over 24}} \Big),$$ where now the trace is crucially taken over only those states whose right-moving part is a Ramond ground state. This definition makes it unclear that we have an index, i.e. that this quantity is invariant under deformation, and indeed, this quantity should jump in a prescribed semicontinuous fashion as we vary in moduli space for a general Calabi-Yau. In fact, it is manifest from the definition that wherever the (left) chiral algebra enhances as one moves in CFT moduli space, the formula will jump – at those points, extra states exist with no right-moving excitation. An index related to ours – roughly, $\left({\partial^2 \over \partial \nu^2} Z_{HEG}\right) \vert_{\nu = 0}$ – was studied previously in [@MMS].
We can also propose a purely mathematical definition. Recall that the elliptic genus [@Warner; @EGrefs; @Yamada] may be reformulated as a holomorphic Euler characteristic as follows: $$\begin{gathered}
Z_{EG}(\tau, z. \nu) = i^{r - D} q^{(r - D) / 12} y^{-r / 2} \chi \Big( X, \bigotimes_{n=1}^{\infty} \Lambda_{-yq^{n-1}} \mc{E} \otimes \bigotimes_{n=1}^{\infty} \Lambda_{-y^{-1}q^n} \mc{E}^{\vee} \\ \otimes \bigotimes_{n=1}^{\infty} \Sym_{q^n} TX \otimes \bigotimes_{n=1}^{\infty} \Sym_{q^n} T^*X \Big). \end{gathered}$$ Here we give the formula relevant for a general (0,2) supersymmetric CFT, whose specifying data includes a holomorphic bundle $\mc{E}$ on $X$, where we use the notation $$\Lambda_t \mc{E} = \bigoplus_{s = 0}^{\infty} t^s \Lambda^s \mc{E}$$ and $$\Sym_t \mc{E} = \bigoplus_{s = 0}^{\infty} t^s \Sym^s \mc{E}$$ for formal power series of bundles. Here $r$ denotes the (complex) dimension of $X$ and $D$ the rank of the bundle $\mc{E}$, while $\chi(X, -)$ denotes the holomorphic Euler characteristic for (formal power series of) holomorphic bundles $$\chi(X, \mc{E}) = \sum_{j = 0}^r (-1)^j \dim H^j(X, \mc{E}).$$
Following the same reasoning, we may upgrade the above formula to one for the Hodge-elliptic genus as follows: $$\begin{gathered}
Z_{HEG}(\tau, z, \nu) = i^{r - D} q^{(r - D) / 12} y^{-r / 2} \sum_{j=0}^r (-u)^j \dim H^j \Big( X, \bigotimes_{n=1}^{\infty} \Lambda_{-yq^{n-1}} \mc{E} \otimes \bigotimes_{n=1}^{\infty} \Lambda_{-y^{-1}q^n} \mc{E}^{\vee} \\ \otimes \bigotimes_{n=1}^{\infty} \Sym_{q^n} TX \otimes \bigotimes_{n=1}^{\infty} \Sym_{q^n} T^*X \Big). \end{gathered}$$ Indeed, mathematicians may take this as a definition of the Hodge-elliptic genus in some level of generality.
As an example, it is easy to compute the Hodge-elliptic genus of an abelian variety starting from this expression: adopting the convention as usual that if the holomorphic bundle $\mc{E}$ is not specified, we take it to be the tangent bundle $TX$, the triviality of the tangent bundle of an abelian variety implies we are merely taking power series in the trivial bundle. In particular, the ranks of the cohomology of the trivial bundle do not depend on the complex moduli of our abelian variety by Hodge theory as $$\dim H^j(X, \mc{O}) = h^{0, j}(X)$$ does not vary in complex moduli. As such, we may assume our abelian variety is a product of elliptic curves; just as with the elliptic genus, the Hodge-elliptic genus is multiplicative, allowing us to reduce to the case of an elliptic curve. Finally, we have for an elliptic curve $E$ that $$\begin{aligned}
Z_{HEG}(E)(\tau, z, \nu) &=& y^{-1/2} \Big( \sum_{j=0}^1 (-u)^j \dim H^j(E, \mc{O}_E) \Big) \times \\ && \prod_{n=1}^{\infty} (1 - yq^{n-1}) \prod_{n=1}^{\infty}(1 - y^{-1}q^n) \Big(\prod_{n=1}^{\infty} (1 + q^n + q^{2n} + \cdots)\Big)^{2} \\ &=& y^{-1/2}u^{-1/2} (1 - u) \prod_{n=1}^{\infty} \frac{(1 - yq^{n-1})(1 - y^{-1}q^n)}{(1 - q^n)^2} \\ &=& 4 \frac{\theta_1(\tau, z)}{\theta_1^*(\tau, 0)} u_-,\end{aligned}$$ where we will use the notation $$u_{\pm} = \frac{u^{-1/2} \pm u^{1/2}}{2}$$ and the Jacobi theta function $$\theta_1(\tau, z) = - q^{1/8} y^{-1/2} \prod_{n=1}^{\infty} (1 - yq^{n-1}) (1 - q^n)(1 - y^{-1}q^n),$$ where we denote $$\theta_1^*(\tau, 0) = -2q^{1/8} \prod_{n=1}^{\infty} (1 - q^n)^3.$$ We hence easily have by the above reasoning that $$Z_{HEG}(T^{2r}) = \Big( 4 \frac{\theta_1(\tau, z)}{\theta_1^*(\tau, 0)} u_- \Big)^r,$$ where we just denote the dependence on the underlying manifold $T^{2r}$ as we have already argued the complex structure of the $r$-dimensional abelian variety is irrelevant.
Returning to the case of $K3$, we may also argue from the expression in terms of cohomology ranks that the Hodge-elliptic genus is invariant under deformation of complex structure. Recall from Bochner’s formula that global sections of powers of the tangent bundle of a Kahler-Einstein manifold are necessarily parallel tensor fields, i.e. invariant under parallel transport and hence determined by the invariance of the holonomy representation on the fibre at the identity; see for example [@Kobayashi] for an overview. As such, the rank of the global sections in the formula above are completely determined by linear algebra and hence obviously deformation-invariant; similarly, by Serre duality, the same argument applies to the ranks of the second cohomology groups and as the holomorphic Euler characteristics are necessarily invariant in flat families, the same must be true for the first cohomology ranks, showing that the Hodge-elliptic genus is in fact deformation-invariant for any Kahler-Einstein surface as expressed above.
It is important to note that the purely mathematical definition above follows from the trace definition in a CFT only for CFTs which describe large radius sigma models with target $X$. Away from large radius, as noted previously, the chiral algebra of the sigma model can be enhanced (on loci of proper codimension) by additional chiral currents. (These will always be of higher spin for compact Calabi-Yau targets, as such spaces admit no continuous isometries). This will lead to jumps in the trace definition of the genus; the mathematical definition only captures the strict large-volume limit. The jumps cancel in the limit that one takes $u \to 1$ because the chiral currents come paired with superpartners that then cancel in the index; this recovers the constancy of the elliptic genus. Away from points with enhanced chiral algebra – i.e., at generic points in the moduli space of superconformal theories – the Hodge-elliptic genus will be constant on the moduli space.
We close by giving the result of a sample calculation. We choose a particularly simple point in moduli space – a Kummer orbifold point where $K3$ is considered as a resolution of $T^4$ by the inversion $\mb{Z}/2$-action (and where the $T^4$ is a product of two square $T^2$s with unit volume). Just as the elliptic genus of $K3$ may be expressed in terms of theta functions as [@EOTY] $$Z_{EG}(K3)(\tau, z) = 8 \Big( \Big( \frac{\theta_2(\tau, z)}{\theta_2(\tau, 0)} \Big)^2 + \Big( \frac{\theta_3(\tau, z)}{\theta_3(\tau, 0)} \Big)^2 + \Big( \frac{\theta_4(\tau, z)}{\theta_4(\tau, 0)} \Big)^2 \Big),$$ the Hodge-elliptic genus is now given as a sum over all four sectors of $\mb{Z}/2$-periodicity conditions we may put on the torus (morphisms from its fundamental group to $\mb{Z}/2$), as now the right-moving fermion zero-mode no longer causes the first contribution to cancel. In fact, we calculate $$Z_{HEG}(K3)(\tau, z, \nu) = 8 \Big( \Big( \frac{\theta_1(\tau, z)}{\theta_1^*(\tau, 0)} u_- \Big)^2 + \Big( \frac{\theta_2(\tau, z)}{\theta_2(\tau, 0)} u_+ \Big)^2 + \Big( \frac{\theta_3(\tau, z)}{\theta_3(\tau, 0)} \Big)^2 + \Big( \frac{\theta_4(\tau, z)}{\theta_4(\tau, 0)} \Big)^2 \Big).$$ This is the value of the Hodge-elliptic genus for a particularly symmetric orbifold K3. A conjecture for the form of the genus for ${\it generic}$ values of the K3 moduli, where there is no enhanced worldsheet chiral algebra, has been put forward by Katrin Wendland [@Wendland]. We return to the importance of $Z_{HEG}(K3)$ in the next section.
The D1-D5 system and symmetric powers
=====================================
We now explain how to obtain the refined count of spinning BPS states from a lift of the Hodge-elliptic genus. First, we revisit the DMVV argument calculating the free energy of a gas of D1 branes dissolved inside a D5 wrapping $K3 \times S^1$ in a IIB compactification, where the D1s wrap the auxiliary $S^1$ and are localized to points in the $K3$. As usual, ignoring the trivial dynamics of the six-dimensional $U(1)$ gauge-theory from the worldvolume of the D5, the dynamics of this system with $N$ D1s is given by a nonlinear sigma-model mapping to the orbifold symmetric power $(K3)^n$ modulo the symmetric group $S_n$-action by permutation of the factors. We will denote this orbifold by $\Sym^n K3$ but mathematicians should be aware we mean the orbifold conformal field theory as opposed to the sigma-model of maps to this singular space itself; equivalently, by deforming in the Kahler moduli space to resolve singularities, we may consider the sigma-model to the crepant resolution $\Hilb^n K3$. As such, mathematicians may mentally substitute $\Hilb^n K3$ below for all mentions of $\Sym^n K3$. In any case, DMVV are able to argue that the unrefined BPS state count in five dimensions is given by $$\sum_{n=0}^{\infty} p^n Z_{EG}(\Sym^n K3),$$ which they then compute via elegant arguments entailing how the string states in the orbifold CFTs arrange themselves into long strings in the original sigma-model, allowing for immediate multiplicative lift formulas that apply not just to the elliptic genus but also to many other indices of manifolds. Indeed, by the same arguments, we have all of the following identities: $$\begin{aligned}
\sum_{n=0}^{\infty} p^n \chi(\Sym^n K3) &=& \prod_{k=1}^{\infty} \frac{1}{(1 - p^k)^{24}} = \frac{p}{\Delta(\sigma)} \\ \sum_{n=0}^{\infty} p^n \chi_{-y}(\Sym^n K3) &=& \prod_{k=1}^{\infty} \frac{1}{(1 - yp^k)^2(1 - p^k)^{20}(1 - y^{-1}p^k)^2} = \frac{p(-y + 2 - y^{-1})}{\phi_{10, 1}(\sigma, z)} \\ \sum_{n=0}^{\infty} p^n Z_{EG}(\Sym^n K3) &=& \prod_{r > 0, s \ge 0, t} \frac{1}{(1 - q^s y^t p^r)^{c(rs, t)}} = \frac{p \phi_{10, 1}(\tau, z)}{\Phi_{10}(\sigma, z, \tau)}, \end{aligned}$$ where we define the coefficients $c(n, \ell)$ via the expansion $$Z_{EG}(K3) = \sum_{n, \ell} c(n, \ell) q^n y^{\ell}.$$
The last formula in particular makes clear the nature of this multiplicative lift: we use the coefficients of the index on the original manifold $K3$ as exponents in the product representation for the generating function of the index over all symmetric powers; indeed, the above two formulas are special cases but also correspond to the simple observations $\chi(K3) = 24$ and $\chi_{-y}(K3) = 2y^{-1} + 20 + 2y$, making clear where the exponents come from in the first two formulae. The reason that the automorphic representations on the right-hand side of the above product formulas require some correction factors in the numerator is due to the ranges of the parameters in the product of $r > 0$ and $s \ge 0$; those correction factors disappear if we were to instead have products over $r, s \ge 0$, as is the case if we count the four-dimensional BPS states rather than the five-dimensional BPS states. Indeed, those states with $r = 0$ are exactly those introduced (or removed) under the 4d-5d lift as the kinematic factors for the center-of-mass of the $D5$ brane in the five-dimensional frame.
The multiplicative lift property is made clear from DMVV’s analysis of the Hilbert space of the symmetric power CFTs, from which all these formulas follow by simply taking traces. In fact, the arguments work equally well if we only consider the subspace of the Hilbert space given by Ramond ground states on the right, so that we may also take the traces that define the symmetrized Hodge polynomial or Hodge-elliptic genus of $K3$ and derive the following formulas: $$\begin{aligned}
\sum_{n=0}^{\infty} p^n \chi_{Hodge}(\Sym^n K3) &=& \prod_{k=1}^{\infty} \frac{1}{(1 - uyp^k)(1 - u^{-1}yp^k)(1 - p^k)^{20}(1 - uy^{-1}p^k)(1 - u^{-1}y^{-1}p^k)} \\ &=& \frac{p(u-y-y^{-1}+u^{-1})}{\phi_{KKP}(\sigma, z, \nu)} \\ \sum_{n=0}^{\infty} p^n Z_{HEG}(\Sym^n K3) &=& \prod_{r > 0, s \ge 0, t, v} \frac{1}{(1 - q^s y^t p^r u^v)^{c(rs, t, v)}} = \frac{p \phi_{KKP}(\tau, z, \nu)}{\Phi(\sigma, z, \tau, \nu)},\end{aligned}$$ where now we have the coefficients of the Hodge-elliptic genus $$\begin{aligned}
Z_{HEG}(K3)(\tau, z, \nu) &=& 8 \Big( \Big( \frac{\theta_1(\tau, z)}{\theta_1^*(\tau, 0)} u_- \Big)^2 + \Big( \frac{\theta_2(\tau, z)}{\theta_2(\tau, 0)} u_+ \Big)^2 + \Big( \frac{\theta_3(\tau, z)}{\theta_3(\tau, 0)} \Big)^2 + \Big( \frac{\theta_4(\tau, z)}{\theta_4(\tau, 0)} \Big)^2 \Big) \\ &=& \sum_{n, \ell, m} c(n, \ell, m) q^n y^{\ell} u^m.\end{aligned}$$ Here, we have implicitly defined our four-variable counting function $$\Phi(\sigma, z, \tau, \nu) = pqy \prod (1 - q^s y^t p^r u^v)^{c(rs, t, v)}$$ as given by a multiplicative lift in terms of the Fourier coefficients $c(n,\ell,m)$ of $Z_{HEG}(K3)$. As in the DMVV argument that gave rise to the definition of $\Phi_{10}$, the index set in the product representation for $\Phi$ requires some care. The correct index set over which we take the product is over all $r, s \ge 0$ and all $t, v$ except that if $r = s = 0$, we restrict to $t < 0$. We see that $\Phi$ is simply related to $\Phi_{10}$ when $u=1$ and $\phi_{KKP}$ when $q \to 0$.
To relate these symmetric power sigma-models back to the D-brane BPS state counts, we have only to note that the extra $U(1)$ charge we wish to turn on in the BPS state count is precisely the same as that which we turn on in the Hodge-elliptic genus. Then the five-dimensional refined BPS state count is precisely given by the formula above for the multiplicative lift of the Hodge-elliptic genus. For completeness, we also note the four-dimensional count of BPS states, as differing by the $r = 0$ modes: denoting $c_{n, \ell, m, r}$ the number of four-dimensional $1/4$-BPS multiplets with charges labelled as before and spin $r$, we have $$\sum c_{n, \ell, m, r} p^n y^{\ell} q^m u^{[r]} = \frac{1}{\Phi(\sigma, z, \tau, \nu)}.$$
Technically speaking, the above product representation for $\Phi$ only converges in one chamber of moduli space and must be analytically continued elsewhere as a meromorphic function. The BPS state counts as extracted from a contour integral procedure will hence jump as the contours cross over poles and a wall-crossing analysis will ensue. Close to the point of definition, however, the above formula holds on the nose and gives the precise refined BPS counting function.
Here, we see explicitly that in the 5d state count, the new variable parametrizes hair which is small compared to the charges visible in supergravity. The factor of $\phi_{KKP}$ in the numerator causes the powers of $u$ to be strictly bounded in terms of the powers of $p$: in no term that occurs in the expansion of the five-dimensional count does the exponent of $u$ exceed that of $p$, irrespective of the exponents of $q$ or $y$. This fact is clear from the original definition of the Hodge-elliptic genus but also is necessary for this count of spinning black holes to be physically meaningful. Indeed, in order to refine this count by extra hair such as this new $U(1)$ charge, standard supergravity literature informs us this hair must be microscopic, i.e. not visible at the semiclassical level; the BPS black holes in supergravity have $|J_1| = |J_2|$. The 4d counting formula, where the automorphic correction factor in the numerator of the 5d formula vanishes, doesn’t limit the powers of $u$ in the same way; this should likely be interpreted as counting of additional multicenter or small black hole states with large angular momentum in the 4d picture.
The IIA frame
=============
Adding a circle and T-dualizing the D1-D5 system in type IIB, or starting from M-theory and compactifying along the M-theory circle, allows us to consider this system as a D0-D2-D6 brane configuration in IIA string theory. The corresponding state counts have well-established mathematical interpretations as (weighted) Euler characteristics of the relevant moduli spaces. Typically we could also interpret these invariants in terms of the topological string, but unfortunately the refinement we conduct in this paper does not yet have a mathematical interpretation in the topological string: motivic Gromov-Witten invariants are yet to be defined. As such, we phrase the mathematically formulated versions of our conjectures in terms of motivic Donaldson-Thomas invariants to retain at least some degree of mathematical precision. We defer full precision and some checks to later work.
We first review the ordinary Donaldson-Thomas invariants of $K3 \times T^2$ as discussed in [@Bryan; @Oberdieck]. Here, for $X$ the Calabi-Yau threefold given as a product of some $K3$ surface $S$ with an elliptic curve $E$, we first consider the Hilbert scheme of subschemes of $X$ with fixed numerical invariants as follows: $$\Hilb^{\beta, n}(X) = \{Z \subset X \Big| [Z] = \beta, \chi(\mc{O}_Z) = n\},$$ where $\beta \in H_2(X ; \mb{Z})$ is a curve class in $S$, $Z$ is a subscheme of $X$, and $[Z]$ represents its underlying class in homology. Note that $X$ has a natural $E$-action by translation; this action is inherited by $\Hilb^{\beta, n}(X)$ and so taking a (weighted) Euler characteristic directly would vanish, corresponding to the extra supersymmetry physically. Eliminating the superfluous fermionic zero-modes corresponds mathematically to instead taking the weighted Euler characteristic of the quotient of this Hilbert scheme by the $E$-action, and we have the reduced Donaldson-Thomas invariants $$\mathrm{DT}_{\beta, n}(X) = \int_{\Hilb^{\beta, n}(X)/E} \nu d\chi,$$ where $\nu$ is the usual Behrend function and we integrate with respect to Euler characteristic. We now conjecture that we may define $\mathrm{DT}_{h, d, n}(X)$ as $\mathrm{DT}_{\beta + dE, n}(X)$ where $\beta$ is a (primitive) class satisfying $\beta^2 = 2h - 2$; Bryan [@Bryan] argues for this invariance using deformation-invariance, sketching a proof using the machinery of shifted symplectic structures. Assembling these invariants into a generating function as $$\mathrm{DT}(X) = \sum_{h, d \ge 0, n} \mathrm{DT}_{h, d, n}(X) q^{h-1} p^{d - 1} (-y)^n,$$ we in fact have the mathematical conjecture $$\mathrm{DT}(X) = -\frac{1}{\Phi_{10}(\sigma, z, \tau)}.$$
In these terms, it is clear how to extend to a motivic version, modulo some natural conjectures. First, we note following [@Pantev; @Bussi] that the Hilbert scheme used to define Donaldson-Thomas invariants has a motivic incarnation $$DT^{mot}_{\beta, n}(X) \in K^{\hat{\mu}}(\mathrm{Var})[\mb{L}^{-1}],$$ the Grothendieck ring of varieties with action by roots of unity as extended by inverting the Tate class. Unfortunately, a difficulty arises here when choosing orientation data, roughly corresponding to the choice of spin structure on the relevant moduli spaces of sheaves. In previous work along these lines such as in [@KKP], all relevant moduli spaces are conjecturally simply-connected and so there is a single choice for orientation datum. Here, we seem to have four choices, and it remains to explore their compatibility as we vary in moduli space. We make no further reference to pinning down this detail here, merely conjecturing that there exists some consistent choice such that the following conjectures hold.
We now further conjecture a canonical quotient by the $E$-action by translation, i.e. we conjecture a natural reduced motivic Donaldson-Thomas invariant related to the above via $$DT^{mot}_{\beta, n}(X) = \mathrm{DT}^{mot}_{\beta, n}(X) \cdot [E]$$ as an equation of classes in $K^{\hat{\mu}}(\mathrm{Var})[\mb{L}^{-1}]$. Taking the motivic measure given by the Euler characteristic should now return the reduced Donaldson-Thomas invariants; if we instead take the (symmetrized) Poincaré polynomial, which we denote by $$P_{\beta, n}(X) = P(\mathrm{DT}^{mot}_{\beta, n}(X)) \in \mb{Q}[u, u^{-1}],$$ we now further conjecture that this Laurent polynomial only depends on $h, d,$ and $n$, where as before we take a curve class $\beta + d E$ for $\beta$ a curve class in $K3$ satisfying $\beta^2 = 2h - 2$. Assuming these conjectures, we may assemble the generating function $$\mathrm{DT}^{mot}(X) = \sum_{h, d \ge 0, n} P_{h, d, n}(X) q^{h-1} p^{d - 1} (-y)^n,$$ which we conjecture should be given by $$\mathrm{DT}^{mot}(X) = -\frac{1}{\Phi(\sigma, z, \tau, \nu)}$$ for $\Phi$ the multiplicative lift of $Z_{HEG}(K3)$ as above.
In §3, we saw that the natural field theory definition of the Hodge-elliptic genus has ‘jumps’ at points in moduli space with enhanced chiral algebra (while admitting a ‘generic’ answer that is constant away from such points). This implies a similar jumping phenomenon for the motivic Donaldson-Thomas invariants, which has been seen explicitly in computations on non-commutative Calabi-Yau threefolds [@Jump]. In particular, varying in the complex or Kähler moduli space should correspond to varying the algebraic structure or the stability structure on the derived category, which may *a priori* lead to different motivic Donaldson-Thomas counts. Our conjecture implies that the jumping behavior of these refined curve counts should match precisely to where the conformal field theory enjoys extra currents as we vary in the K3 $\sigma$-model moduli space. Following [@Wendland], the computation $Z_{HEG}$ for a generic K3 $\sigma$-model is now known following certain natural assumptions, and so lifting this function should give the motivic Donaldson-Thomas invariants for generic complex and Kähler parameters. The four-variable function $\Phi$ we give earlier is for a generic K3 surface of Kummer type, and so we expect the corresponding lift to give the motivic Donaldson-Thomas invariants (with the correct choice of orientation datum) for precisely these surfaces.
**[Acknowledgements]{}**
We would like to thank G. Oberdieck, N. Paquette, C. Vafa, R. Vakil, R. Volpato, and K. Wendland for helpful conversations, and K. Wendland of informing us about her conjecture for the generic value of $Z_{HEG}(K3)$. We are especially grateful to S. Murthy and A. Sen for discussions of BPS state counts and black hole hair, and to D. Maulik and G. Oberdieck for helpful comments on the role of orientation data in defining motivic Donaldson-Thomas invariants. S.K. was supported in part by the National Science Foundation under grant PHY-1316699.
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|
---
author:
- Melissa Rinaldin
- 'Ruben W. Verweij'
- Indrani Chakraborty
- 'Daniela J. Kraft'
bibliography:
- 'references.bib'
title: '[**Colloidal supported lipid bilayers for self-assembly**]{}'
---
Introduction {#introduction .unnumbered}
============
Colloidal supported lipid bilayers (CSLBs) are used in a diverse range of research areas and applications [@Troutier2007], from drug delivery [@Carmona-Ribeiro2012; @Li2014; @Savarala2010], bio-sensing [@Castellana2006; @Chemburu2010], membrane manipulation [@Brouwer2015] and cell biology [@Sackmann2007a; @Madwar2015; @Mashaghi2013] to fundamental studies on lipid phase separation [@Rinaldin2018; @Fonda2018] and self-assembly [@VanDerMeulen2013; @VanDerMeulen2014; @VanderMeulen2015; @Chakraborty2016a]. The presence of a lipid bilayer around nano- or micrometer-sized solid particles or droplets provides biomimetic properties and a platform for further functionalization. One intriguing recent example used DNA-based linkers to functionalize the lipid bilayer thereby enabling self-assembly of the underlying colloidal particles or droplets into flexible structures. [@VanDerMeulen2013; @Feng2013a; @VanDerMeulen2014; @VanderMeulen2015; @Chakraborty2016a; @McMullen2018]. Within such a structure, the colloidal elements can move over each other’s surface while remaining strongly and specifically bonded. This new type of bonding enables fundamental studies on structures with internal degrees of flexibility, such as the self-assembly of novel crystal phases and their phase transitions [@Kohlstedt2013; @Ortiz2014; @Smallenburg; @Hu2018]. Furthermore, these complex colloids have great potential for smart drug delivery techniques [@Carmona-Ribeiro2012], photonic band-gap materials [@Joannopoulos1997; @Lin2005] and wet computing [@Phillips2014a].
CSLBs are particularly suitable and versatile building blocks for the assembly of floppy structures, because they combine the best qualities of free standing bilayers (vesicles) and colloids. Vesicles, upon applications of linkers [@Hadorn2010], can connect into flexible structures, but are unstable to small disturbances, heterogenous in size and easily deformable. Colloidal particles are available in diverse materials and with a variety of stable shapes, and can be assembled after functionalization with surface-bound DNA linkers.[@Wang2012; @Wang2015a]. However, the obtained structures are often rigid due to the immobility of the linking groups on the particles’ surface and are non-equilibrium structures due to a “hit-and-stick” aggregation process [@Schade2013]. Emulsions coated with lipid monolayers and DNA-linkers that are mobile on the droplet interface posses both interaction specificity and bond flexibility [@Feng2013a; @Zhang2017; @McMullen2018]. Therefore, they assemble into flexible structures in a controlled fashion, but their shape is limited to spheres and they deform upon binding. Conversely, CSLBs consist of colloidal particles which provide a stable support for the lipid bilayer that is tunable in shape, size and material. The range of shapes for colloidal particles comprises, among others, spheres, cubes, rods, and (a)symmetric dumbbell particles, and their sizes range from hundreds of nanometers to several micrometers. They can be produced reliably with a narrow size distribution and are commercially available. Additionally, CLSBs feature a lipid bilayer on the surface of the colloids which creates a liquid film for molecules, such as DNA linkers, to freely move in. This allows for binding particles specifically, and yet non-rigidly, making the assembly of floppy structures possible [@VanDerMeulen2013; @VanDerMeulen2014; @VanderMeulen2015; @Chakraborty2016a; @Zhang2017].
To obtain flexible instead of rigid structures, it is vital that the linker molecules which are inserted into the lipid bilayer are free to move over the surface of the CSLBs. Their lateral mobility relies on the fluidity and homogeneity of the bilayer, which in turn depend on the linker concentration [@Chakraborty2016a] and lipid composition. The lipids need to be in the fluid state under experimental conditions, and this may be impeded by bilayer-surface interactions. Similarly, the success of experiments studying the phase separation of lipid bilayers on anisotropic colloidal supports relies on the fluidity and homogeneity of the bilayer [@Rinaldin2018; @Fonda2018]. Finally, controlling the self-assembly pathway through complementary DNA linkers implies that all other non-specific interactions need to be suppressed. In other words, CSLBs need to have sufficient colloidal stability. To the best of our knowledge, these requirements of membrane homogeneity and fluidity plus colloidal stability have not been studied simultaneously. However, they are of key importance for using CSLBs in self-assembly and model membrane studies, while possibly having wider implications for all other applications.
Here, we carefully characterize every stage in the preparation of CSLBs specifically related to these three properties. First, we study the effect of the material properties of the particles and the use of polymers on the membrane fluidity and homogeneity. Then, we investigate the influence of lipopolymers and inert double-stranded DNA on the colloidal stability of the CSLBs. Subsequently, we include DNA-based linkers connected to hydrophobic anchors and characterize their diffusion in the bilayer. Finally, we show that when using the optimal experimental parameters determined by this study, CSLBs self-assemble into flexibly linked structures that are freely-jointed.
Experimental Section {#experimental-section .unnumbered}
====================
Reagents {#reagents .unnumbered}
--------
### Chemicals {#chemicals .unnumbered}
(POPC), , (TopFluor-Cholesterol), (DOPC), (DOPE-PEG(2000)), DOPE-PEG(3000) and DOPE-PEG(5000) were purchased from Avanti Polar Lipids. (HEPES, $\geq$) and (, , $\geq$) were purchased from Carl Roth. (, extra pure), (, ), (, extra pure), (TEMED, ), (APS, ), (, ) and (, , extra pure) were purchased from Acros Organics. III, (, 28-30 ), (TPM, ), Pluronic F-127, (, $\geq$), ethanol ($\geq$), sodium dodecyl sulfate (SDS, $\geq$), polyvinylpyrrolidone (PVP, average Mw ), itaconic acid ($\geq$), () and acetic acid () were purchased from Sigma-Aldrich. Magnesium chloride (, for analysis) was purchased from Merck. All solutions were prepared with Milli-Q water (Milli-Q Gradient A10).
### Buffers {#buffers .unnumbered}
HEPES buffer type 1 was made with , , , and HEPES. HEPES buffer type 2 consisted of HEPES, , and . HEPES buffer type 3 consisted of HEPES, and . The buffers were prepared by mixing all reagents in the appropriate amounts in fresh Milli-Q water. After mixing, the pH was adjusted to 7.4 using .
### Particles {#particles .unnumbered}
Commercial silica spheres () were synthesized by Microparticles GmbH, using a Stöber method where tetraethoxysilane (TEOS) reacts with water and bases in an ethanolic solution (sol-gel process). Commercial polystyrene particles () were obtained from Sigma Aldrich. Hematite cubic particles () were made following the protocol of Sugimoto et al. [@Sugimoto1992] and coated according to Rossi et al. [@Rossi2011]. Polystyrene-3-(Trimethoxysilyl)propyl methacrylate (Polystyrene-TPM) particles () with varying asperity were synthesized and coated with silica following the protocol of Meester et al. [@Meester2016]. TPM particles () were made following the protocol of Van der Wel et al. [@VanDerWel2017TPM]. TPM particles functionalized with carboxyl groups (), or amino groups () were prepared by synthesizing according to [@VanDerWel2017TPM] and then functionalizing according to [@DohertyTBP]. Briefly, amine or carboxylic acid groups were incorporated onto the TPM surface by addition of either 3-aminopropyltriethoxysilane or itaconic acid, respectively, during the emulsification stage. Polystyrene particles with carboxyl groups () were synthesized according to Appel et al. [@Appel2013]. Polystyrene-TPM particles of spherical, symmetric and asymmetric dumbbell shape were made and coated with silica following the protocols reported in Rinaldin et al. [@Rinaldin2018].
### DNA oligonucleotides {#dna-oligonucleotides .unnumbered}
All DNA strands were synthesized as single stranded DNA, purified using reverse phase high-performance liquid chromatography (HPLC-RP) and checked using matrix-assisted laser desorption/ionization time-of-flight mass spectrometry (Maldi-TOF MS) by Kaneka Eurogentec S.A. We used double-stranded inert DNA for steric stabilization and double-stranded DNA linkers with a sticky end for binding. Both types of DNA have a hydrophobic anchor (double Stearyl/double cholesterol for linker and double cholesterol for inert DNA) connected to a short carbon chain which is then connected to the oligonucleotide. The linkers are additionally functionalized with a fluorescent dye. All strands, including all functionalizations, are listed in Table S1 of the Supporting Information. These DNA strands were hybridized by mixing the single strands in a 1:1 molar ratio in HEPES buffer type 3. The solution was then heated in an oven to for , after which the oven was turned off and the solution was allowed to cool down slowly overnight in the closed oven. Strand 1 and 2 are hybridized to form “ Inert DNA”, 3 and 4 are hybridized to form “ Inert DNA”, 4 and 5 form “ Linker A”, 4 and 6 make “ Linker A$'$”, 7 and 8 form “ Linker A” and, finally, 7 and 9 are hybridized to form “ Linker A$'$”. The linkers “A” have a single-stranded sticky end (indicated by cursive text in Table S1 of the Supporting Information) that is complementary to the single-stranded end of linkers “A$'$”.
Preparation of CSLBs {#preparation-of-cslbs .unnumbered}
--------------------
Typically, CSLBs were made by spontaneous spreading and fusion of small unilamellar vesicles (SUVs) on the particle surface. An SUV dispersion prepared via either extrusion or sonication was mixed with the particles, allowing a bilayer to spread on the surface for at least one hour. Subsequently the CSLBs were washed to remove excess SUVs. We observed no substantial differences in the obtained CSLBs between the two methods presented here.
### CSLB preparation: method 1 {#cslb-preparation-method-1 .unnumbered}
of a mixture of DOPE-Rhodamine () and varying amounts of POPC and PEGylated lipids was dried by two hours of vacuum desiccation and then re-suspended to a dispersion with HEPES buffer type 1. The solution was vortexed for to produce multilamellar vesicles. Then, the vesicle dispersion was extruded 21 times with a mini extruder (Avanti Polar Lipids) equipped with two gas-tight syringes (Hamilton), two drain discs and one nucleopore track-etch membrane (Whatman). The pore size of the membrane was 0.05 or 0.1 for experiments with DOPE-PEG(2000) and DOPE-PEG(3000-5000), respectively. The as-prepared of SUVs were added to of of particles dispersed in HEPES buffer 1. The particles were gently rotated for . The resulting dispersion was centrifuged at for and the supernatant replaced with HEPES buffer type 1 to remove any SUVs present in the dispersion.
This method was used for all experiments regarding the influence of particle material, surface roughness and the effect of polymer insertion on the spreading and mobility of the lipid bilayer.
### CSLB preparation: method 2 {#cslb-preparation-method-2 .unnumbered}
Typically, a lipid mixture of DOPC, DOPE-PEG(2000) and DOPE-Rhodamine or TopFluor-Cholesterol in chloroform was dried overnight in a glass vial covered with aluminum foil under vacuum desiccation. We investigated different PEGylated lipid lengths and molar ratios. After drying, HEPES buffer type 2 or 3 was added to reach a concentration of . The dispersion was vortexed for , after which it became turbid. It was then transferred to a plastic test tube and ultrasonicated using a tip sonicator (Branson Sonifier SFX150) set to of its maximum amplitude for a total time of using a pulsed sequence ( on/ off, total on time ) on ice to prevent heating. The SUV dispersion was then centrifuged for at to sediment larger vesicles and titania particles originating from the tip [@Cremer1999]. SUVs were taken from the top to isolate the smallest vesicles.
of SUVs in HEPES buffer 2 or 3 were mixed with of particles in Milli-Q water, leading to a surface ratio of SUVs:particles of 8:1. The dispersion was gently rotated for . The particles were centrifuged at for and the supernatant was replaced with HEPES buffer type 2 or 3 to remove any remaining free SUVs from the dispersion. Alternatively, the particles were allowed to sediment by gravity for instead of centrifuging and the supernatant was replaced.
This method was used for all experiments regarding the colloidal stability of CSLBs, the mobility of inserted DNA and the mobility of self-assembled CSLB clusters.
Coating CSLBs with DNA for self-assembly {#coating-cslbs-with-dna-for-self-assembly .unnumbered}
----------------------------------------
After the particles were coated with a lipid bilayer using method 2, various amounts of inert and/or linker DNA were added and the dispersion was gently rotated for . To remove any remaining free DNA strands in solution, the particles were washed by centrifugation for at , or alternatively, by sedimentation by gravity for , and the supernatant was replaced three times by HEPES buffer type 2 or three.
We characterize the amount of dsDNA that we add as a surface density $\sigma_{\mathrm{DNA}}$, which we define as $$\begin{aligned}
\sigma_{\mathrm{DNA}}
&= \frac{N_{\mathrm{DNA}}}{A_{\mathrm{CSLB}}}, \label{eq:sigma_dna} \end{aligned}$$ where $N_{\mathrm{DNA}}$ is the total number of dsDNA strands and $A_{\mathrm{CSLB}}$ is the total surface area of the CSLBs. The total number of dsDNA strands and particles were estimated from the reported stock concentrations. In this calculation, we assume that all the added dsDNA strands are distributed homogeneously over all particles and that no dsDNA remains in solution. We typically used $\sigma_{\mathrm{DNA}}=$ dsDNA linkers to obtain flexible structures.
Particle clusters were formed by mixing two particle types coated with complementary DNA linkers in a 1:1 number ratio in a round metal sample holder on a polyacrylamide coated cover slip (see [@Wel2016] for details). The polyacrylamide coating keeps the particles from sticking to the glass surface, allowing them to cluster via diffusion limited aggregation.
Sample characterization {#sample-characterization .unnumbered}
-----------------------
The samples were imaged with an inverted confocal microscope (Nikon Ti-E) equipped with a Nikon A1R confocal scanhead with galvano and resonant scanning mirrors. A oil immersion objective ($\mathrm{NA} = 1.49$) was used. A laser was employed to excite the Lissamine Rhodamine dye, a laser was used to excite the TopFluor-Cholesterol dye. The excitation light passed through a quarter wave plate to prevent polarization of the dyes. 500-550 and 565-625 filters were used to separate the emitted light from the TopFluor and the Rhodamine dyes, respectively.
The charge of the particles in MilliQ water was determined via zeta potential measurements using a Malvern Zetasizer Nano ZS.
Fluorescence recovery after photobleaching (FRAP) {#fluorescence-recovery-after-photobleaching-frap .unnumbered}
-------------------------------------------------
We used fluorescence recovery after photobleaching (FRAP) to check the mobility of the lipids in a CSLB. A circular area of the fluorescent sample was bleached, the recovery signal was collected and normalized as I\_[corr]{}(t) = , \[eq:i\_norm\] where $I_{\rm corr}(t)$ is the measured intensity $I(t)$ normalized with respect to the intensity just before bleaching $I(t=0)$ and corrected for bleaching through measurement of the intensity of a non-bleached reference area, $I_{\rm ref}(t)$. Additionally, we subtracted the background signal from $I$ and $I_{\rm ref}$. We found that the signal can be fitted using the following expression: I\_[corr]{}(t) = A(1-e\^[-]{}) \[recovery\] where $A$ is the extent of the recovery, $t-t_{0}$ is the time elapsed since the beginning of the recovery process and $\tau$ the recovery time.
While there is a simple relation linking $\tau$ to $D$ for circular bleaching areas on planar surfaces [@Axelrod1976], we are not aware of a similar expression for a spherical surface that is partly bleached from the side, as is the case in our experiments. Therefore, we quantify the lateral mobility in terms of the recovery time $\tau$ only.
All FRAP experiments on silica particles were performed using particles, unless stated otherwise. To measure the lateral mobility of DNA linkers using FRAP, no fluorescently labeled lipids were used and instead, we used a high linker DNA concentration ($\sigma_{\mathrm{DNA}}=$ ) that provided a sufficiently bright fluorescent signal.
Particle stability analysis {#sec:stab_tracking .unnumbered}
---------------------------
To estimate the colloidal stability of particles, we rotated the particles () in a test tube for at least , thereby allowing them to aggregate. We then immobilized some of the clusters on a glass substrate, allowing us to take a “snapshot” of the cluster distribution at that time. Particles were located in bright-field microscopy images of these sedimented, semi-dilute (volume fraction $\phi\approx0.001$) samples. The cluster sizes were determined by using the `bandpass`, `locate` and `cluster` functions from TrackPy [@trackpy]. Erroneously tracked positions were corrected manually. The separation distance below which particles are considered to be part of a cluster was chosen to be $1.1D$, where $D$ is the particle diameter. This can lead to a small overestimation of the number of clusters when particles are close together but have not irreversibly aggregated. We defined the fraction of single particles $f_{\mathrm{single}}$ as the number of detected clusters with a size of 1 (i.e. single particles) divided by the total number of individual particles. The error on this fraction was estimated as the standard deviation of the average cluster size divided by the square root of the total number of particles. For each measurement, we analyzed between individual particles.
Trimer flexibility analysis {#trimer-flexibility-analysis .unnumbered}
---------------------------
We have analyzed three linearly-connected CSLBs that were functionalized with inert dsDNA and linker dsDNA. To quantify the mobility of the self-assembled trimers, we tracked the position of the three individual particles in bright-field movies as a function of time and calculated the opening angle $\theta$ between them. For tracking and calculating $\theta$, we used a custom algorithm that is depicted in Figure S2 of the Supporting Information.
First, the user selects the particles of interest from the first frame (see Figure S2 A). This increases the computational efficiency of tracking because it reduces the number of tracked features and allows for cropping of all frames. We then iterate over all frames to identify the positions of the selected particles. Each current frame is inverted, so that all particles have a ring of high intensity around them (see Figure S2 B). The frame is converted to polar coordinates with the current provisional particle position at the origin (see Figure S2 C), where the provisional position is the one that the user selected for the first frame and the previous tracked position for all subsequent frames. For each row (each polar angle), the position of maximum intensity is found (see Figure S2 D) and these coordinates are then converted back to the original Cartesian coordinate system of the frame (see Figure S2 E). A circle is fitted to these coordinates using a least squares method (see Figure S2 F). After all three particles are found in this frame, the opening angle between them is determined using simple trigonometry (see Figure S2 G). From the opening angles of all the frames, we calculated the mean squared displacement of the angle (MSAD or “joint flexibility” $J$) [@Chakraborty2016a].
We analyzed the free energy of trimers as function of opening angle using two methods: 1) by converting the histogram to the free energy using Boltzmann weighing and 2) using a maximum likelihood estimation method of angular displacements [@Wel2016; @Wel2017]. We confirmed that both methods agreed and show only the result of the maximum likelihood method, because it allows us to estimate the error in our measurement. We now describe these methods in detail.
### Trimer free energy: Boltzmann weighing {#trimer-free-energy-boltzmann-weighing .unnumbered}
We obtained a histogram of opening angles between with a bin width of . We then mirrored and averaged the data around and converted this to a probability density function. From the probability density function we determined the free energy using Boltzmann weighing, $$\begin{aligned}
\frac{F}{k_B T} &= -\ln{P} + \frac{F_0}{k_B
T} \label{eq:boltzmann}, \end{aligned}$$ where $F$ is the free energy, $k_B$ is the Boltzmann constant, $T$ is the temperature, $P$ is the probability density and $F_0$ is a constant offset to the free energy, which we chose at a reference point () so that the free energy is equal to zero there.
### Trimer free energy: maximum likelihood estimation {#trimer-free-energy-maximum-likelihood-estimation .unnumbered}
While the Boltzmann weighing method is very straightforward, it gives no information about the experimental error. To estimate the error, we used an analysis that is based on a maximum likelihood method in which particle displacements are modelled [@Wel2016; @Wel2017], which we adapted for our experimental system. We used (angular) displacements because for Brownian particles they are uncorrelated in time, in contrast to positions (or values of the opening angle). This means that using this method, we can obtain reliable results even for a limited number of particles [@Sarfati2017].
To summarize, we followed the method outlined in chapter 3.4.2 of [@Wel2017]: we find the maximum likelihood estimate of the local force field $F(\theta)$ by using a model for the transition probability $P$: $$\begin{aligned}
P(\theta_1,t+\tau|\theta_0,t)=&\left(4\pi D \tau\right)^{-\frac{1}{2}}\\\nonumber &\exp\left( -\frac{(\Delta\theta-\beta D F(\theta) \tau)^2}{4\pi D \tau} \right)
\label{eq:transition_prob}\end{aligned}$$ where $\theta_0$ is the opening angle at a time $t$ and $\theta_1$ is the angle at a later time $t+\tau$, $\tau$ is the time between measurements, $D$ is the diffusion coefficient determined from the mean squared displacement, $\Delta\theta=\theta_1-\theta_0$ and $\beta$ is the Boltzmann constant times the temperature. A Baysian method was used to find the maximum likelihood estimate by using emcee [@emcee] and the error was determined as the standard deviation of the chain of Markov Chain Monte Carlo (MCMC) samples. We determined the free energy up to an arbitrary choice of a reference energy by numerical integration of this force. This free energy was then mirrored around and averaged to determine the free energy between and . We observed a boundary effect inherent to the analysis for angles smaller than $+\sqrt{2J\tau}$ (where $J$ is the joint flexibility) leading to a slight overestimation of the free energy for those angles.
![**Overview of the experimental system** **Step 1)** Micrometer sized colloidal particles are coated with a lipid bilayer by adding small unilamellar vesicles (SUVs) that rupture and spread on the particle surface. We varied the composition of the lipids, as well as the material and shape of the particles. **Step 2)** DNA linkers with hydrophobic anchors can be added to make particles that are functionalized with DNA with complementary sticky ends. When the lipid bilayer is fluid, the linkers can diffuse over the particle surface and therefore also the linked particles can slide over each other. \[fig:overview\]](Figures_draft/figure0.pdf){width="0.8\linewidth"}
Results and Discussion {#results-and-discussion .unnumbered}
======================
We will now characterize every step in the formation of CSLBs. In the first section of the results we study the homogeneity and mobility of the lipid bilayer on colloidal particles made from different materials, and therefore various surface functionalities and degrees of roughness. Furthermore, we investigate the effect of PEGylated lipids on the homogeneity and mobility of the lipid bilayer and their use as steric stabilisers to prevent unspecific aggregation.
Having found conditions that yield colloidal particles with a homogeneous and mobile bilayer, we subsequently introduce double-stranded DNA connected to a hydrophobic anchor into the bilayer, as shown in .2. We employ DNA constructs both with and without single-stranded sticky ends, to investigate their use in DNA-mediated binding and their effect on colloidal stability, respectively. Finally, we demonstrate that CSLBs can be used for self-assembly by employing DNA linkers with complementary single-stranded sequences. We use FRAP to measure the lateral mobility of DNA linkers on the particle surface inside and outside the bond area. In this way, we show that they are mobile if the bilayer is fluid and that, in this case, the particles can freely roll over each other’s surfaces when bonded.
Lipid bilayer coating of colloidal particles {#lipid-bilayer-coating-of-colloidal-particles .unnumbered}
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To use CSLBs in self-assembly studies or as model membrane systems, it is critical that a homogeneous and fluid bilayer forms on the colloidal particles. This implies successful assembly of both leaflets of the bilayer and lateral mobility of the lipids, and hence proteins, linkers, and larger lipid domains, in the membrane. The formation of lipid bilayers on solid supports can be achieved by deposition of SUVs under physiological conditions, as shown in .1. A combination of electrostatic and Van der Waals forces lead to spreading and fusion of the liposomes on the surface of the supports [@Richter2006; @Sackmann1996; @Raedler1995; @GoZen]. Between the surface of the support and the bilayer a thin layer of water remains, allowing the lipids to laterally diffuse in the absence of other motion-restricting forces. Previous studies on planar SLBs reported that there are many factors which can prevent homogeneity and mobility of the bilayer [@Machan2010]. These factors are related to the surface that is coated (its surface charge, chemical composition and roughness), the aqueous medium (pH and ionic strength), the SUVs (composition, charge, size, transition temperature) and the temperature at which the lipid coating happens [@Richter2006; @Jing2014]. Here we will study how some of these factors, that are inherent to the use of solid particles, influence the formation of supported lipid bilayers on colloidal substrates.
### Influence of the chemical properties of the particle surface {#influence-of-the-chemical-properties-of-the-particle-surface .unnumbered}
[0.95]{}[X S\[retain-explicit-plus\] l l]{} **Material** & **Zeta potential \[mV\]** & **Homogeneous** & **Mobile**\
Silica spheres (Stöber method, Microparticles GmbH) & -566 & yes& yes\
Hematite cubic particles [@Sugimoto1992]& +395 & no & no\
Silica-coated hematite cubic particles [@Rossi2011] & -326 & yes& yes\
Polystyrene spheres (Sigma Aldrich) & -382 & no & no\
Polystyrene spheres with carboxyl groups [@Appel2013] & -431 & no & no\
Silica-coated Polystyrene-TPM anisotropic particles [@Rinaldin2018] & -331 & yes & yes\
TPM spheres [@VanDerWel2017TPM] & -421 & yes & no\
TPM spheres with carboxyl groups [@DohertyTBP] & -461 & no & no\
TPM spheres with amino groups [@DohertyTBP] & -124 & no & no\
The available variety of colloids with anisotropic shapes makes them attractive for self-assembly and model membrane studies. Current synthetic procedures tailored to obtain different shapes, however, typically rely on the use of specific materials and therefore yield colloids with different surface properties. We have selected a range of particles of different shapes and commonly used materials to test for membrane homogeneity and mobility after coating with SUVs. In particular, we tested silica spheres prepared by a sol-gel method, commercially available polystyrene spheres, polystyrene spheres with carboxyl groups made using a surfactant-free dispersion polymerization method [@Appel2013], 3-(trimethoxysilyl)propyl methacrylate (TPM) spheres [@VanDerWel2017TPM], TPM spheres functionalized with carboxyl and amino groups [@DohertyTBP], silica-coated polystyrene-TPM spheres and symmetric and asymmetric dumbbells [@Rinaldin2018]; as well as hematite cubes [@Sugimoto1992] and silica-coated hematite particles [@Rossi2011]. Silica-coated polystyrene-TPM dumbbells and hematite cubes were obtained by depositing a silica layer following the Stöber method [@Castillo2014].
After coating, we visually inspect the lipid-coated particles using confocal microscopy and consider bilayers to be homogenous if more than 50% of the particles do not show defects in the bilayer. We characterize the bilayer fluidity by measuring the mobility of the fluorescently-labeled lipids on the colloid surface using FRAP. After bleaching, we observe the recovery of the fluorescence intensity due to the diffusion of the dyed lipids in and out of the bleached area. We consider the lipids and thus the bilayer to be mobile if the intensity signal recovers homogeneously in the bleached area, otherwise we consider them to be (partially) pinned to the surface.
Our first observation was that only particles that possess a silica surface exhibited homogeneous and mobile bilayers (). We did not succeed in coating colloids made from polystyrene or hematite with a homogeneous bilayer. However, once such substrates were first coated with a silica shell, the bilayer was found to be both homogeneous and mobile. Unexpectedly, particles made from an organosilica compound (TPM) whose surfaces are similar to silica [@VanDerWel2017TPM] only showed homogeneous, but not mobile bilayers. Since silica, TPM and polystyrene colloids were all negatively charged, we conclude that the chemical composition of the substrate and not only the surface charge plays a fundamental role in the homogeneity and fluidity of the bilayer. These results agree with previous experiments on planar SLBs, in which silica-based surfaces were found to be one of the most reliable supports for homogeneous and mobile lipid bilayers [@Richter2006].
Since colloidal particles are often functionalized with different groups on the surface, we furthermore have characterized the bilayer on particles equipped with surface groups commonly used in colloidal science (). While TPM particles with an unmodified silica-like surface showed homogeneous bilayers, we found that functionalization with negatively charged carboxyl or positively charged amino group prevented spreading and fusion of the lipid vesicles. Likewise, functionalization of polystyrene spheres with carboxyl groups did not enhance the homogeneity of the lipid bilayer. A previous study on planar SLBs reported that the spreading of SUVs depends on the combination of the molecular ions in the buffer and the type and density of surface charge [@Cha2006]. While amino-functionalized surfaces are hence expected to disrupt the spreading of SUVS in the presence of the negatively charged HEPES molecules of the buffer, the observation of inhomogenous bilayers on carboxyl-functionalized surfaces can likely be allocated to an insufficiently dense surface coverage. We conclude that similar to planar SLBS, the homogeneity and fluidity of the bilayer of CSLBs is dependant on a complex interplay of the chemical and physical properties of the lipids and the particle’s surface.
### Influence of particle curvature differences {#influence-of-particle-curvature-differences .unnumbered}
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Another factor that may influence the successful formation of a homogeneous and mobile bilayer is the variation in curvature of the colloidal substrate, which may hinder spreading and fusion of SUVs. Curvature differences can originate from the overall anisotropic shape of the particles or from surface roughness. As discussed before, we found that particles with a comparably slowly varying curvature, such as hematite cubes or symmetric and asymmetric dumbbells (see ), had a fluid and homogeneous bilayer after coating, if they featured a silica surface clean of any polymer residues from synthesis. Particles with rough surfaces however have a much higher and frequent variation in curvature. To investigate the effect of large curvature differences, we prepared two batches of polystyrene particles which only differed in their surface roughness and coated them with a silica layer following a Stöber method [@Meester2016]. In we show that particles with some roughness (A) can be homogeneously coated with a bilayer (B) while particles with very rough surfaces (C) show an inhomogeneous bilayer (D). FRAP experiments confirmed that the bilayer on the “smooth” surface is not only homogeneous, but also mobile, while the inhomogeneous bilayer on the rough particle is immobile as indicated by the non-recovering intensity signal. We conclude that the roughness of the surface plays an important role in both bilayer homogeneity and mobility.
### Influence of free and grafted polymers {#influence-of-free-and-grafted-polymers .unnumbered}
![**Effect of PVP on CSLB formation.** FRAP experiment on a group of three cubes with **A)** and without **B)** PVP. Only the sample without PVP shows recovery of the signal in the bleached area. Scale bars are . **Influence of F127 in linker inclusion.** **C)** Control image of the fluorescence of the undyed CSLBs. **D)** Fluorescence intensity of CSLBs coated with linker dsDNA. **E)** The same sample as in D), but imaged after dispersion in F-127 solution. The fluorescence on the particles was found to be significantly less homogeneous than in D). **F)** The same sample as in D), but imaged after dispersion in F-127 solution. The fluorescence intensity is comparable to the uncoated control in C) so we conclude that all dsDNA has been removed from the bilayer by F-127. Scale bar is . \[fig:PVP\] ](Figures_draft/figure2.pdf)
Polymers or surfactants are often employed to stabilize colloidal particles in solution [@DeGennes1987; @Upadhyayula2012a; @VanDerWel2017], but may influence the formation and mobility of the bilayer in CSLBs. Here, we test how the presence of, for example, leftover polymers from particle synthesis, affects the bilayer. We compare a sample of silica-coated hematite cubes with and without PVP, a polymer commonly used in colloidal syntheses and conservation. To remove the PVP from the surface after synthesis, we calcinated the colloids at for 2h. shows that cubes with PVP posses an inhomogeneous bilayer and the ones without it feature a bilayer that homogeneously covers the surface ( B). As expected for Stöber silica surfaces, the bilayer on the colloids for which the PVP was removed is also mobile, as indicated by the recovery of the fluorescence intensity.
Moreover, the presence of polymers may not only affect the bilayer’s properties, but also the incorporation of functional groups such as DNA linkers into it. We tested this by preparing CSLBs with fluorescently labeled DNA linkers connected to double cholesterol anchors and transferring an aliquot of this dispersion to a HEPES solution containing 5% w/w of Pluronic F-127, a polymer that is commonly used for the stabilization of colloidal particles. While the fluorescent signal of the CSLBs with and without F-127 were initially equal, already 15 minutes after mixing we observed less dsDNA fluorescence on the CSLBs with F-127 compared to particles without it. After , the fluorescence intensity of the CSLBs with F-127 was comparable to that of control particles not coated with linker dsDNA ( C-F). We therefore conclude that F-127 removed the cholesterol-anchored linker DNA from the bilayer, in line with recent experiments on emulsion droplets coated with mobile DNA linkers [@VanDerWel2018].
### Influence of PEGylated lipids on bilayer homogeneity and mobility {#influence-of-pegylated-lipids-on-bilayer-homogeneity-and-mobility .unnumbered}
{width="\linewidth"}
The presence of polymers in SLBs is not always detrimental, but may even improve bilayer mobility. Previous studies on planar SLBs showed that membranes can be supported by polymers covalently bound to lipids (lipopolymer tethers) [@naumann2002polymer; @Tanaka2005; @Tanaka2006; @Wagner2000; @Deverall2008]. Since lipopolymer tethers increase the thickness of the water layer between the solid support and the bilayer [@naumann2002polymer; @Tanaka2005; @Tanaka2006] they are thought to reduce the friction between the substrate and the bilayer, allowing for higher diffusivity of lipid molecules and linkers.
Inspired by this, we study how a specific lipopolymer tether affects the spreading and the fluidity of the bilayer in CSLBs. We used the lipopolymer DOPE-PEG, a phospholipid with a covalently bound PEG molecule. We employed PEGylated lipids with three different molecular weights: 2000, 3000 and 5000 in varying concentrations. It is important to note that PEGylated lipids were introduced in the system during the SUV formation by mixing them with the other lipids. This means that once the bilayer is formed, they are present in both leaflets.
We report in A-B the effect of varying concentration and molecular weight of the lipopolymers on the spreading and the mobility of the bilayer. In the absence of PEGylated lipids, the bilayer on the CSLBs was observed to be fluid. At increased concentrations of DOPE-PEG, the bilayer became inhomogeneous, which indicates insufficient spreading and fusion of the SUVs. This effect appeared at lower molar fractions for lipopolymers with higher molecular weights of the DOPE-PEG. For completeness, we note that a small fraction of particles in samples that are labeled as inhomogeneously coated do exhibit a homogeneous, but nevertheless immobile, bilayer. We believe that the reason for the observed inhomogeneity is two-fold. On the one hand, higher concentrations of lipopolymers lead to an increased steric stabilization, that prevents fusion of the SUVs and hinders the van der Waals interactions between the SUVs and the substrate that aid spreading. On the other hand, PEGylated lipids in the brush state increase the bending rigidity of the SUV membrane, thereby preventing rupture and spreading on the surface [@lipowsky1995bending].
For fluid membranes, we quantified the mobility of the lipids by calculating the recovery time from FRAP experiments, which is the time it takes a bleached area to recover its fluorescence intensity. We find that the diffusion of the lipids is faster for PEGylated lipids with a lower molecular weight and increases with decreasing amount of the lipopolymers, see C. This latter result agrees with a study performed on planar supported lipid bilayers [@naumann2002polymer]. In the presence of lipopolymers, we find the shortest recovery time (3.2 $\pm$ 0.02 s), e.g. highest diffusivity, for of DOPE-PEG(2000). The decrease of the diffusion coefficient with the amount of lipopolymer indicates that the PEGylated lipids are pinned to the surface and in this way hinder the mobility of the other lipids.
We emphasize that the mobility of supported lipid bilayers in presence of polymers is dependent on many factors and one may not extend our results to other types of polymers, lipid bilayers or physiological environments [@Deverall2008]. The complex interplay between polymers and the chemical properties of the colloidal surface can lead to surprising results. For example, and in contrast to what we reported above, we found that an homogenous bilayer on cubic silica particles could only be obtained by using both PEGylated lipids (DOPE-PEG(5000)) and a negatively charged lipid (GM1). Interestingly, at high concentrations of PEGylated lipids the bilayer is very homogeneous but not mobile (D). This is in contrast to silica spheres coated with the same concentrations of lipopolymers and only zwitterionic lipids, which do not possess a homogeneous bilayer, see A. We indicated this state in which the bilayer is homogeneous, but not fluid, with blue squares in D. A possible origin of this unusual behavior could be the different porosity, surface chemistry and charge of the silica cubes [@Castillo2014] compared to the silica spheres ().
Stabilizing CSLBs against aspecific aggregation {#stabilizing-cslbs-against-aspecific-aggregation .unnumbered}
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To build specific colloidal structures from the bottom up, careful control over the interactions between the particles is required. On the one hand, specific attractive interactions may be employed to control which particles interact. This specific binding can be achieved by using dsDNA linkers with complementary sticky ends [@VanDerMeulen2013; @Chakraborty2016a; @Geerts2010]. On the other hand, the particles need to be prevented from binding to each other aspecifically: that is, not via dsDNA linker interactions but via other attractive forces that act indiscriminately between all particles, such as Van der Waals forces. In other words, it is crucial to be able to control the colloidal stability of the CSLBs [@Valignat2005].
In our experiments, stabilization by repulsive electrostatic interactions is not a feasible route because surface charges are screened by the counterions in the HEPES buffer that is needed to allow the complementary DNA sticky ends to bind [@Geerts2010]. The ionic strength of the buffer must be higher than so that clusters are formed via DNA-mediated attractions [@Biancaniello2007]. At these salt concentrations, even the bare silica particles are no longer stabilized by their negatively charged surface groups. Indeed, we found that both the bare silica particles and the silica particles coated with a lipid bilayer aggregated in all buffers, as was previously observed [@Nordlund2009]. The fraction of single particles determined from light microscopy images was $f_{\mathrm{single}}=$ for uncoated silica particles after one hour of mixing in the buffer, while they were previously stable in deionised water. We therefore explored different ways to sterically stabilize the particles using higher concentrations of PEGylated lipids, SDS and inert dsDNA strands.
### Stabilization using SDS {#stabilization-using-sds .unnumbered}
SDS is a surfactant with amphiphilic properties, consisting of a polar headgroup and a hydrocarbon tail, that has been shown to stabilize emulsion droplets coated with lipid monolayers [@Zhang2017]. Inspired by these findings, we added SDS to the CSLBs after bilayer coating to increase their stability. However, in contrast to lipid coated emulsion droplets we found no significant increase in stability when we varied the SDS concentration between . In fact, the highest concentration of led to a decrease in particle stability from $f_{\mathrm{single}}=0.67$ without SDS to $f_{\mathrm{single}}=0.45$ at . This is likely caused by the disruptive effect that SDS can have on lipid bilayers. In a study on large unilamellar vesicles (LUVs) made from POPC, it was found that SDS can completely solubilize the vesicles above concentrations of [@Tan2002a]. While this concentration is higher than the concentrations that we used here, we already observed some damage to the bilayer. The resulting inhomogeneous coating may allow aspecific “sticking” on patches that are not covered with a lipid bilayer and a subsequent decrease in overall particle stability.
### Stabilization using PEG {#stabilization-using-peg .unnumbered}
![**Steric stabilization of CSLBs.** **A)** Higher concentrations of DOPE-PEG(2000) lead to a higher fraction of single particles in the absence of linker DNA. Stability is shown after one washing cycle of at and overnight rotation. We hypothesize that above , the packing density of PEG on the surface of the membrane is high enough for the PEG to be in the brush state, making it more effective as a steric stabilizer. **B)** Increasing the number of dsDNA strands on the particle surface increases the particle’s stability for two different lengths of inert dsDNA ( and ). **C)** Centrifugation and redispersion with a solution containing dsDNA affects the fraction of single particles. After centrifuging particles that were initially stable ($f_{\mathrm{single}}=0.95$, $\sigma_{\mathrm{DNA}}=$ dsDNA) $3\times$ at for , we observed aspecific aggregation ($f_{\mathrm{single}}=0.51$) in the absence of dsDNA strands in solution while increasing the concentration of dsDNA () in the washing solution could preserve stability. \[fig:stability\]](Figures_draft/figure4.pdf){width="0.95\linewidth"}
In contrast to SDS, PEGylated lipids can provide colloidal stability through steric repulsions between the PEG moieties while also being easily integrated into the bilayer through their lipid tails [@DeGennes1987; @Upadhyayula2012a; @VanDerWel2017]. To test their use for colloidal stabilization, we coated the particles with SUVs that contain a small fraction of the following PEGylated lipids: DOPE-PEG(2000), DOPE-PEG(3000) and DOPE-PEG(5000). Since we include these lipopolymers during SUV preparation, they are part of both the inner and outer leaflet, as depicted in .1. At low concentrations, we observed no significant change in the stability of the particles upon an increase in the concentration of PEGylated lipids. For example, for DOPE-PEG(2000) stability remained constant below as shown in A). For concentrations between and for DOPE-PEG(2000), and for DOPE-PEG(3000) and and for DOPE-PEG(5000), the average fraction of unclustered particles lay between $f_{\mathrm{single}}=0.5$ and $f_{\mathrm{single}}=0.77$, with no clear trend observed for different polymer lengths or concentrations. For all measurements, we verified that the spreading of the SUVs was successful. We believe that the stability of the particles did not improve significantly because at these concentrations, the grafted PEG was in the mushroom state instead of the brush state, and therefore not sufficient to provide steric stability [@TanjaDrobek2005; @Meng2004]. Therefore, we increased the concentration of DOPE-PEG(2000) above . Indeed, as shown in , the colloidal stability increases above this concentration, likely due to a transition from the mushroom to the brush state, as was shown for similar lipopolymers (DSPE-PEG(2000)) around [@Garbuzenko2005].
While the colloidal stability can be improved by increasing the concentration of PEGylated lipids, the bilayer is not fluid at the concentrations required for colloidal stability as we showed earlier (see B). Therefore, embedded DNA linkers will also not be mobile in the bilayer and it is not possible to form reconfigurable structures from CSLBs stabilized by PEGylated lipids only.
### Stabilization using inert dsDNA {#sec:stability_dna .unnumbered}
Since PEGylated lipids cannot be used to provide steric stability because they reduce the fluidity of the membrane, we explored an alternative route to stabilize the CSLBs. Inspired by numerical findings that inert double stranded DNA (dsDNA) can also act as a steric stabilizer via excluded volume interactions between DNA strands on different particles [@Angioletti-Uberti2014], we employed dsDNA strands with a double cholesterol anchor at one end to functionalize the CSLBs with this DNA via hydrophobic insertion of the cholesterol into the bilayer [@VanDerMeulen2014], see .2.
We varied the surface concentration $\sigma_{\mathrm{DNA}}$ (see ) of two inert dsDNA strands with different lengths and measured its effect on the particle stability, which is shown in B). The stability was determined after the particles were coated with dsDNA and rotated for one hour. For both the dsDNA and the dsDNA, we found that increasing the number of grafted dsDNA strands per particle led to an increase in particle stability from $f_{\mathrm{single}}=0.68$ without dsDNA to $f_{\mathrm{single}}=$ above $\sigma_{\mathrm{DNA}}=$ . We found that the dsDNA is slightly more efficient as a steric stabilizer than the dsDNA, as can be seen in B). This is expected, because for the longer DNA, excluded volume interactions between particles start to become important already at larger particle separations than for the shorter DNA. Additionally, excluded volume interactions between DNA strands on the same particle force the DNA to extend outwards already at lower concentrations for the longer DNA strands as compared to the shorter DNA strands. Therefore, the repulsion between the particles also has a longer range, leading to better colloidal stability. However, at concentrations above $\sigma_{\mathrm{DNA}}=$ , the difference between the and dsDNA is less pronounced, because the particles are so densely coated that adding longer or more DNA strands will not stabilize the particles any further.
To use these particles in self-assembly studies, specific interactions need to be present as well, which we here induce by adding dsDNA *linkers*, that is, dsDNA strands with a sticky end and double cholesterol anchors. After the particles are functionalised, any excess linker DNA left in the solvent needs to be removed to reduce unwanted aggregation or saturation of the complementarily functionalized particles. To remove excess linker DNA, we washed and redispersed the particles in buffer solution three times and measured the particle stability afterwards. Unexpectedly, it decreased from $f_{\mathrm{single}}=0.95$ before washing to $f_{\mathrm{single}}=0.51$ after washing. To detect whether the partial removal of the stabilizing inert dsDNA during washing had caused this aggregation, we washed the particles in a HEPES buffer that contained various concentrations of inert dsDNA. As shown in C), increasing the concentration of dsDNA in the washing solution led to an increase in particle stability after washing, therefore confirming our hypothesis. Including of inert DNA in the washing solution allowed us to preserve the particle stability ($f_{\mathrm{single}}=0.91$).
However, washing the particles with such high concentrations of dsDNA proved detrimental to the bilayer and led to the formation of membrane tubes between long (see Figure S4 in the Supporting Information). These membrane tubes are highly curved surfaces that are only formed under specific conditions, for example a difference in spontaneous curvature between the inner and the outer membrane leaflets [@Lipowsky2013]. In our case, since dsDNA is only added after the formation of the bilayer it is only present in the outer leaflet and hence induces a difference in spontaneous curvature which causes formation of tubes. If the DNA would be present on both leaflets, no tube formation is expected. We tested this by mixing the dsDNA with the SUVs before coating the particles but found no fluorescence signal on the particles’ surface, implying the absence of a bilayer coating. We believe that the dsDNA sterically stabilizes the SUVs and prevents spreading and fusion of SUVs on the particle surface.
To summarize, we found that inert double-stranded DNA can impart colloidal stability to the CSLBs. Asymmetric distribution of DNA causes the formation of membrane tubes at high coating concentrations (above approximately ), but is unavoidable if colloidal stability needs to be preserved during repeated washing cycles.
Linker functionalization for self-assembly {#linker-functionalization-for-self-assembly .unnumbered}
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To be able to employ CSLBs in self-assembly experiments, we induce specific attractive interactions by using two sets of dsDNA linkers with complementary single-stranded sticky ends that can form bonds via hybridization. We use two hydrophobic anchors (either cholesterol or stearyl) per dsDNA complex to insert the dsDNA linker into the outer leaflet (see .2), while at the same time adding inert dsDNA strands for steric stabilization. We use double hydrophobic anchors because dsDNA with a single hydrophobic anchor is less strongly confined to the bilayer [@VanDerMeulen2014]. The dsDNA is attached to the double hydrophobic anchor with a tetraethylene glycol (TEG) or hexaethylene glycol (HEG) spacer to allow it to swivel with respect to the anchor. We label the dsDNA linkers with a fluorescent dye to image them using confocal microscopy. Full details on the DNA strands we used can be found in Table S1 of the Supporting Information.
Previous experiments [@Chakraborty2016a] have shown that several interesting structures, such as flexible colloidal polymers and molecules, can be obtained via self-assembly of CSLBs. In order to form these reconfigurable structures, not only the lipids in the bilayer should be mobile but also the grafted linker DNA should be mobile on the surface of the membrane. We can quantify the mobility of dsDNA on the surface of the bilayer by measuring the FRAP of fluorescently labeled DNA. Note that for these experiments we did not employ fluorescent lipids. For a successful recovery after bleaching of the DNA linkers in the binding patch between two particles, two requirements need to be fulfilled: the DNA linkers outside of the binding patch have to be mobile and the bleached linkers inside the binding patch have to be able to unbind, to allow unbleached linkers to diffuse into the binding patch. We calculate the melting temperature of the sticky end using an approximate formula [@Nakano1999] and find a melting temperature $T_m =
$ , meaning that at the probability for the sticky end to form a duplex is $P($$)=0.68$ based on melting temperature considerations only. Therefore, the sticky ends continuously bind and unbind in our experiments, making fluorescence recovery possible, while the sheer number of linkers in the patch area keeps the particles always bound.
We confirmed the mobility of linker DNA on the particle surface using FRAP experiments, shown in . Note that the whole cluster is immobilized on the (non-passivated) glass coverslip to enable FRAP on these micrometer-sized particles. Therefore these clusters do not show translational diffusion or cluster flexibility. In A), a representative FRAP experiment on a single sphere is shown. We measured an average recovery time of from independent measurements on 16 different particles. Outside the bond area ( B), we measured an average recovery time of from independent measurements on 15 different clusters. Within the error, the diffusion of DNA outside of the bond area is the same as on free particles as expected. Inside the bond area ( C), we measured recovery times of with an average recovery time of from independent measurements on 12 different clusters. This is longer than the recovery times we measured outside the bond area and on single particles, which indicates the diffusion is slower in the bond area. However, the recovery time inside the bond area varied greatly between different clusters indicating that diffusion into and out of the bond area can be different. A likely cause is the spread in the DNA concentration between individual particles in a batch. Higher DNA concentrations imply a higher concentration of bonded linkers in the bond area. This will sterically hinder the diffusion of unbonded linkers inside the patch area and will thus lead to longer recovery times. Furthermore, we hypothesize that the linker concentration in the patch area slowly increases as a function of time after binding, so that the recovery time depends on the time that has elapsed after the formation of the cluster, which we did not control. In summary, we have shown here that the dsDNA linkers are mobile on each part of the (un)bound particle, which is a prerequisite for creating flexibly linked clusters.
Mobility of self-assembled structures {#mobility-of-self-assembled-structures .unnumbered}
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![**Mobility of self-assembled trimers** **A)** Mean squared angular displacement (MSAD) of the opening angle $\theta$ for a mobile trimer. The MSAD is linear and we find a joint flexibility $J=$ ($J=$ when averaging over all experiments). **B, C)** Free energy as function of opening angle $\theta$ for particles with **B)** smooth bilayers and **C)** bilayers that have membrane tubes. The grey shaded area marks one standard deviation confidence interval. We analyzed 53 clusters with smooth bilayers and 18 clusters with membrane tubes. In both cases, we find no preference for specific opening angles within the experimental error, meaning the particles are freely-jointed. Note that the slight repulsion at small angles is caused by boundary effects inherent to our analysis method.\[fig:trimer\_theta\]](Figures_draft/Figure7.pdf)
The mobility of individual dsDNA linkers on the surface does not necessarily imply that bonded clusters of DNA functionalized CSLBs are also reconfigurable. For example, for emulsion droplets functionalized with DNA linkers, the large linker patch that is formed between particles can slow down the motion when the supporting fluid is inhomogeneous [@VanDerWel2018] and colloidal joints lose their flexibility with an increasing concentration of dsDNA linkers in the bond area [@Chakraborty2016a].
To measure the flexibility of larger structures, we assembled CSLBs with complementary dsDNA linkers and imaged chains of three CSLBs, so called trimers, over time. We extracted the position of the individual particles and the opening angle $\theta$ (see inset A) for all frames and calculated the mean squared angular displacement (MSAD) to characterize the flexibility [@Chakraborty2016a]. To investigate the influence of the membrane homogeneity on the structural flexibility, we compared trimers assembled from CSLBs with homogeneous, fluid bilayer to trimers with bilayers that had spontaneously formed membrane tubes. In the following, we will only show the free energy landscape for $\theta$ from due to the symmetry of a trimer.
For trimers made from CSLBs with smooth lipid bilayers, we found that the particles ($D=$ ) move with respect to each other over the full angular range. We analyzed the opening angle $\theta$ for different clusters by tracking the individual particles and calculating $\theta$ for all frames (see inset A). The average value of the flexibility of the trimers is $J=$ (or ) and agrees well with previous experiments [@Chakraborty2016a]. The spread in the flexibility that we observe is likely caused by the spread in dsDNA linker density on the particles. We then determined the free energy using the maximum likelihood estimation of angular displacements method (see Experimental Section), as shown in B). We found no preference in the opening angle that is significant with respect to the thermal energy $k_B T$, indicating that the surface is smooth enough to allow the particles to move over one another without restrictions. We observed a boundary effect inherent to the analysis for angles smaller than $+\sqrt{2J\tau}\approx$ (where $J$ is the joint flexibility) leading to a slight overestimation of the free energy for those angles. In previous experiments, we observed a small preference (0.9 $K_B T$ free energy difference) in opening angle around [@Chakraborty2016a]. We do not find the same preference in angle for the experiments presented here. When re-analyzing the data from [@Chakraborty2016a] using our maximum likelihood method, we noted that the observed free energy difference is within the experimental error (see Figure S3 in the Supporting Information) and therefore we conclude that there is no preference in opening angle. This is also in line with recent experiments on flexibly linked emulsion droplets, which are freely-jointed like the CSLBs we present here [@McMullen2018].
To test the robustness of this structural flexibility in the presence of membrane inhomogeneities, we prepared particles with a large number of membrane tubes by washing with inert dsDNA. As shown in C), this did not alter the flexibility of the clusters: they still expore the full angular range and do not show a significant preferred opening angle, similar to the particles coated with a smooth membrane. Surprisingly, this means that particles stabilized by dsDNA can in principle be used for self-assembly studies despite the fact that the high concentration of dsDNA causes membrane tubes to form, because the tubes do not significantly alter the relative motion of clustered particles.
Summary and Conclusions {#summary-and-conclusions .unnumbered}
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We investigated various factors in the preparation protocol of colloidal supported lipid bilayers (CSLBs) in view of their emerging use in self-assembly and model membrane studies. Specifically, we focused on realizing a homogeneous and fluid bilayer, while achieving colloidal stability and functionalization with DNA linkers at the same time.
Similar to what has been reported for flat supported lipid bilayers, we found that the quality of the lipid bilayer on colloidal particles critically depends on the material of the particle’s surface. The bilayer was not fluid on particles made from polystyrene (with or without carboxyl groups), hematite and TPM particles (with or without carboxyl or amino groups). Colloids featuring a silica surface, on the other hand, were able to host a fluid and homogenous bilayer, at least in the absence of any polymer residues from the synthesis. We furthermore observed that the variation in the substrate curvature does not affect the bilayer formation if it is sufficiently gentle, while excessive surface roughness can hinder the spreading and fusion of SUVs.
Use of PEGylated lipids in the bilayer increased the colloidal stability, but affected the bilayer homogeneity and mobility negatively. Addition of the amphiphilic surfactant SDS led to a disintegration of the bilayer. A better way to provide colloidal stability is by steric stabilisation by excluded volume effects through the insertion of double-stranded inert DNA. Increasing the concentration of dsDNA leads to an increase in colloidal stability.
Finally, we demonstrated that these CSLBs can be functionalized with surface-mobile DNA linkers and assembled them into flexible structures of freely-jointed particles. We found that local bilayer inhomogeneities in the form of membrane tubes do not affect the free energy landscape of the connected particles.
CSLBs with fluid, homogeneous membranes and surface-mobile binding groups have great promise in a wide range of applications and fundamental studies. The fact that the bonded particles can flexibly move with respect to each other opens the door to overcoming equilibration issues previously encountered in hit-and-stick processes and assembling structures with internal deformation modes. This enables studying the impact of structural flexibility on the phase behaviour, such as the formation of crystals with new lattices or properties [@Kohlstedt2013; @Ortiz2014; @Smallenburg; @Hu2018], and the experimental realization of information elements for wet computing [@Phillips2014a]. CSLBs with increased membrane fluidity also mimic biological membranes more closely which may be advantageous for model membrane and cell biology studies [@Sackmann2007a; @Madwar2015; @Mashaghi2013; @Rinaldin2018; @Fonda2018] smart drug delivery [@Carmona-Ribeiro2012; @Li2014; @Savarala2010] and bio-sensing applications [@Castellana2006; @Chemburu2010].
#### Supporting Information {#supporting-information .unnumbered}
The Supporting Information consists of 4 figures and 1 table: large fields of view of microscopy images; a table listing all employed DNA strands; a schematic representation of the particle tracking algorithm; a graph showing data from a previous paper re-analyzed using a different method; and a microscopy picture of three CSLBs that feature membrane tubes.
#### Acknowledgments {#acknowledgments .unnumbered}
We thank Rachel Doherty and Vera Meester for help with the colloidal syntheses and electron microscopy imaging, and Piermarco Fonda and Luca Giomi for useful discussions. This work was supported by the Netherlands Organization for Scientific Research (NWO/OCW), as part of the Frontiers of Nanoscience program and VENI grant 680-47-431. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement no. 758383).
#### Author Contributions {#author-contributions .unnumbered}
MR and RWV contributed equally to the work. MR and RWV performed the experiments and the data analysis. IC performed exploratory experiments. MR, RWV and DJK conceived the experiments and wrote the paper.
{width="3.25in"}
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address: |
^a^ Pix4D SA, EPFL Innovation Park, Building F, 1015 Lausanne, Switzerland\
^b^ University of Minnesota\
(carlos.becker, elena.rosinskaya, emmanuel.dangelo, christoph.strecha)@pix4d.com\
haeni001@umn.edu\
author:
- 'C. Becker ^a^, N. H[ä]{}ni ^b^[^1] , E. Rosinskaya ^a^, E. d’Angelo ^a^, C. Strecha ^a^'
bibliography:
- 'bibliography.bib'
title: Classification of Aerial Photogrammetric 3D Point Clouds
---
=1
[0.9]{}
[^1]: This work was done while the author was working at Pix4D SA
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---
abstract: 'We investigate the dripline mirror nuclei, $^{11}$Li and $^{11}$O, located on the neutron and proton dripline, respectively. We calculate the lowest four states, $3/2^-$, $1/2^+$, $3/2^+$ and $5/2^+$, built on double occupancy in the nuclear $s_{1/2}$ and $p_{1/2}$ valence single-particle states. We use the hyperspherical adiabatic expansion method to solve the three-body problem for a frozen nuclear core surrounded by two identical nucleons. The four analogue states in $^{11}$O are obtained with precisely the same interactions as used for the four states in $^{11}$Li, except for addition of the Coulomb interaction from the charge of the substituted valence protons. Surprisingly the four energies deviate from each other only by less than a few hundred keV. Any of them could then turn out to be the ground state, due to the uncertainty related to the angular momentum and parity dependence of the three-body potential. Still, our calculations marginally favor the $1/2^+$ state. The structures of these four states in $^{11}$O deviate substantially from the analogue states in the mirror, $^{11}$Li.'
author:
- 'E. Garrido$^{1}$, A.S. Jensen$^{2}$'
title: 'Few-body structures in the mirror nuclei, $^{11}$O and $^{11}$Li'
---
Introduction
============
Nuclear structure varies tremendously from the many-body leptodermous features of heavy nuclei to the individual properties of light nuclei [@boh75; @zel17; @sie87; @mye69; @tho04; @fre07]. The unexpected observed increased jump in radius from lighter Li-isotopes to $^{11}$Li [@tan85; @tan85b] triggered the research on halo structures [@han87] in a number of subfields of physics [@jen04]. Few-body structure was especially efficient to describe the gross features of halos, simply because the degrees of freedom essentially decouple into two groups, where only a few nucleons determine the low-energy properties [@jen04].
The overall properties of $^{11}$Li are established as a three-body system with constituents of two neutrons and $^{9}$Li [@joh90]. In this connection the spin-spin splitting of the $s_{1/2}$ and $p_{1/2}$ single-neutron states coupled to the $3/2^-$ ground state of $^{9}$Li is crucial for the halo properties [@kat99; @gar02a]. These halo structures are consistent with reaction information [@gar96; @gar97; @cas17] even after the binding energy has been measured with better accuracy [@smi08]. The space spanned by these single-particle states provides the $3/2^-$ ground-state as well as the dipole-excited states of $1/2^+$, $3/2^+$ and $5/2^+$ [@gar02].
The present investigation is triggered by the recent experiments [@web19] on the mirror nucleus, $^{11}$O, which was preceded by related experiments on $^{10}$N [@hoo17; @lep02], and quickly followed up by theoretical papers on these and a few neighboring nuclei [@wan19; @mor19; @for19]. Several previous publications on $^{11}$O and $^{10}$N are available [@aoy97; @til04; @cha12]. The comparison of isobaric analogue structures is a classical nuclear discipline, which has provided strong support for the generalization of the isospin concept from nucleons to nuclei [@boh75; @zel17; @sie87]. Since halo structures only occur near threshold for $s$ and $p$ valence-nucleons, as in $^{11}$Li, the structure may be strongly influenced even for small energy changes. Thus, mirror nuclei on the driplines are most likely to exhibit larger differences than stable nuclear mirrors [@jen04].
The mirror pair, $^{11}$Li and $^{11}$O, are located on the neutron dripline and slightly outside the proton dripline, respectively. Still, both are accessible by experiments, which for $^{11}$O largely is possible due to the Coulomb barrier. The major effect is from the Coulomb interaction of the additional protons, which has both direct and indirect influence. The same structure in both nuclei produces an energy difference from the additional charge. However, the structure itself is modified by this extra Coulomb interaction and in turn resulting in a modified energy.
The total effect of the additional Coulomb interaction is quantified in the Thomas-Ehrman shift [@ehr51; @tho52], which is defined as the energy difference (apart from the neutron-proton mass difference) between analogue states in mirror nuclei. This energy shift may depend on the state, and probably therefore is especially sensitive towards variation of halo structure between analogue states in $^{11}$Li and $^{11}$O [@aue00; @gri02; @gar04]. The possible structure variation between these analogue states may lead to sizable state-dependent Thomas-Ehrman shifts. This may even change the sequence of ground and excited states built on these valence configurations.
The recent experimental activity towards $^{11}$O and $^{10}$N is an opportunity to compare properties in mirrors each located around different nucleon driplines. This has earlier proved to be informative. Previous theoretical investigations already provided a number of details on these nuclei. However, they are a little random as essentially all are incomplete in descriptions of the low-lying states supported by the valence nucleon $s_{1/2}$ and $p_{1/2}$ single-particle states, since only the $3/2^-$-state is considered. The only exception is Ref.[@wan19], where the positive parity state $5/2^+$ is also investigated.
In general, the connection between these mirror nuclei is not particularly well explored. Furthermore, the previous results are for some reason quantitatively deviating, either due to different methods, interactions, or perhaps accuracy of some kind. We therefore decided to investigate these low-lying nuclear states by use of our well established few-body method, which due to the phenomenological input also is both simple and accurate. Thus, in the present paper we report on detailed studies of low-energy properties of $^{11}$O in comparison to similar investigations of $^{11}$Li. Our purpose is two-fold, that is first to discuss few-body properties of the specific $^{11}$O-nucleus, and second to look for general conclusions by studying this mirror of the prototype of a halo nucleus. Larger differences can be expected for such dripline structures in comparison and in contrast to stable mirror nuclei.
The paper, in section II, first briefly presents the applied hyperspherical adiabatic expansion method [@nie01], the degrees-of-freedom, and the choice of interaction form. Section III describes the choice of parameters and the derived properties of the subsystems, $^{10}$Li and $^{10}$N. Section IV, V and VI are devoted to the computed three-body properties of $^{11}$O specifically in comparison to $^{11}$Li. In section VII we present a summary and the conclusion.
Sketch of the method {#method}
====================
The three-body calculations will be performed using the well-established hyperspherical adiabatic expansion method described in detail in [@nie01]. In this method the three-body wave function, with total angular momentum $J$ and projection $M$, is written as: $$\Psi^{JM}=\frac{1}{\rho^{5/2}} \sum_n f^J_n(\rho) \Phi^{JM}_n(\rho,\Omega),
\label{exp0}$$ where $\rho$ is the hyperradius, $\Omega$ collects the five hyperangles as defined for instance in [@nie01], and $f^J_n(\rho)$ are the radial expansion functions. The basis set $\{\Phi^{JM}_n(\rho,\Omega)\}$ used in the expansion above is formed by the eigenfunctions of the angular part of the Schrödinger (or Faddeev) equations, $$\left[ \hat{\Lambda}^2 + \frac{2m\rho^2}{\hbar^2}(V_{12}+V_{13}+V_{23}) \right] \Phi_n^{JM}=\lambda_n(\rho) \Phi_n^{JM}(\rho,\Omega),
\label{angf}$$ where $\hat{\Lambda}$ is the grand-angular momentum operator [@nie01], $V_{ij}$ is the interaction between particles $i$ and $j$, $m$ is the normalization mass used to define the Jacobi coordinates [@nie01], and $\lambda_n(\rho)$ is the eigenvalue associated to the angular eigenfunction $\Phi_n^{JM}(\rho,\Omega)$.
In practice, Eq.(\[angf\]) is solved after the expansion $$\Phi^{JM}_n(\rho,\Omega)=\sum_q C_q^{(n)}(\rho)
\left[ {\cal Y}_{\ell_x\ell_y}^{K L}(\Omega) \otimes \chi_{s_x s_y}^S \right]^{JM},
\label{exp1}$$ where $q$ collects all the quantum numbers $\{K,\ell_x,\ell_y,L,s_x,S\}$, where $\ell_x$ and $\ell_y$ are the relative orbital angular momenta between two of the particles, and between the third particle and the center-of-mass of the first two, respectively. The total orbital angular momentum $L$ results from the coupling of $\ell_x$ and $\ell_y$. The quantum number $K$ is the so-called hypermomentum, which is defined as $K=2\nu +\ell_x +\ell_y$, with $\nu=0,1,2,\cdots$. The dependence on these quantum numbers, $\ell_x$, $\ell_y$, $L$, and $K$, is contained in the usual hyperspherical harmonics, ${\cal Y}_{\ell_x\ell_y}^{K L}(\Omega)$, whose definition can also be found in [@nie01], and which satisfy $\hat{\Lambda}^2 {\cal Y}_{\ell_x\ell_y}^{K L}=
K(K+4){\cal Y}_{\ell_x\ell_y}^{K L}$. In the same way, $s_x$ is the total spin of two of the particles, which couples to the spin of the third particle, $s_y$, to give the total spin $S$. The total spin function is represented in Eq.(\[exp1\]) by $\chi_{s_x s_y}^S$. Finally $L$ and $S$ couple to the total three-body angular momentum $J$ with projection $M$.
Obviously the definition of the $\bm{x}$ and $\bm{y}$ coordinates (the Jacobi coordinates) is not unique, since for three-body systems three different sets of Jacobi coordinates can be formed [@nie01]. When solving the Schrödinger equation a choice has to be made, which means that only one of the internal two-body subsystems is treated in its natural coordinate. In this work, however, we solve instead the Faddeev equations, which have the nice property of treating all the three possible sets of Jacobi coordinates on the same footing [@nie01].
The radial functions, $f^J_n(\rho)$, in Eq.(\[exp0\]) are obtained after solving the set of coupled equations $$\begin{aligned}
\lefteqn{
\hspace*{-1cm}
\left[
-\frac{\partial^2}{\partial \rho^2}+\frac{\lambda_n(\rho)+\frac{15}{4}}{\rho^2} -\frac{2mE}{\hbar^2}
\right] f_n^J( \rho)=
} \nonumber \\ & &
\sum_{n'} \left(2P_{nn'}(\rho)\frac{\partial}{\partial\rho} +Q_{nn'}(\rho) \right) f_{n'}^J(\rho),
\label{radf}\end{aligned}$$ where $E$ is the three-body energy, and $\lambda_n(\rho)$ is obtained from the angular equation (\[angf\]). The explicit form and properties of the coupling functions $P_{nn'}(\rho)$ and $Q_{nn'}(\rho)$ can be found in [@nie01].
The set of equations (\[radf\]) has to be solved imposing to the radial wave functions the appropriate asymptotic behaviour. This is particularly simple for bound states, due to the asymptotic exponential fall-off of the radial wave functions. In order to exploit the simplicity of this asymptotic behaviour, we compute resonances (understood as poles of the $S$-matrix) by means of the complex scaling method [@ho83; @moi98]. In this method the three-body energy is allowed to be complex, and the radial coordinates are rotated into the complex plane by an arbitrary angle $\theta$ ($\rho
\rightarrow \rho e^{i\theta}$). Under this transformation, and provided that $\theta$ is sufficiently large, the resonance wave function behaves asymptotically as a bound state, i.e., it decays exponentially at large distances, and its complex energy, $E=E_R-i\Gamma_R/2$, gives the resonance energy, $E_R$, and the resonance width, $\Gamma_R$.
Being more specific, after the complex scaling transformation, the Eqs.(\[radf\]) are solved by imposing a box boundary condition. The continuum spectrum is then discretized, and the corresponding discrete energies appear in the complex energy plane rotated by an angle equal to $2\theta$ [@ho83; @moi98]. The resonances show up as discrete points, independent of the complex scaling angle, and out of the cut corresponding to continuum states.
Note that an accurate enough solution of the three-body problem requires convergence at two different levels. First, one needs convergence in the expansion of the angular eigenfunctions in Eq.(\[exp1\]), which is necessary in order to obtain sufficient accuracy in the $\lambda_n$-eigenvalues in the radial equations (\[radf\]). A correct convergence requires inclusion of the relevant $\{\ell_x,\ell_y,L,s_x,S\}$-components, and, for each of them, a sufficiently large value, $K_{max}$, of the hypermomentum $K$ is also needed. Second, one has to reach convergence as well in the expansion in Eq.(\[exp0\]), which implies a sufficiently large number of adiabatic terms.
Typically, the convergence in the expansion (\[exp0\]) is rather fast, and for bound states and resonances (after the complex scaling transformation) four or five terms are usually enough. However, the expansion (\[exp1\]) is more demanding, especially when dealing with particles with non-zero spin, since the number of components can increase significantly in accordance with a given total three-body angular momentum $J$. Also, for extended systems, for which the $\lambda_n$-functions have to be accurately computed at large distances, the required maximum value of the hypermomentum, $K_{max}$, can be rather large.
Given a three-body system, the key quantities determining its properties are the two-body potentials entering in Eq.(\[angf\]). In this work we shall assume that the nucleon-nucleon interaction is the GPT potential described in [@gog70].
For the core-nucleon potential we choose an interaction, adjusted independently for the different partial waves, each term of the form: $$V_{Nc}^{(\ell)}(r)=V_c^{(\ell)}(r)+V_{ss}^{(\ell)}(r) \bm{s}_c\cdot (\bm{\ell}+\bm{s}_N)+V_{so}^{(\ell)} \bm{\ell}\cdot \bm{s}_N,
\label{eq1}$$ where $\bm{\ell}$ is the relative orbital angular momentum between the core and the nucleon, whose intrinsic spins are denoted by $\bm{s}_c$ and $\bm{s}_N$, respectively. As shown in [@gar03], this spin-operator structure, which is consistent with the mean-field description of the nucleons in the core, is crucial for a correct implementation of the Pauli principle.
Obviously, when the interaction involves two charged particles, the Coulomb potential should be added to the interactions described above. In this work we shall describe the core as a uniformly charged sphere with radius equal to the charge radius, which for $^9$C will be taken equal to 2.5 fm. We assume all nucleons are point-like particles.
The core-nucleon system {#pots}
=======================
For the case of $^{11}$Li ($^9$Li+$n$+$n$) and its mirror partner, $^{11}$O ($^9$C+$p$+$p$), it is clear that the essential ingredient is the nuclear part of the core-nucleon interaction. Due to the charge symmetry of the strong interaction, these potentials will be the same for both, $^{10}$Li ($^9$Li+$n$) and $^{10}$N ($^9$C+$p$), since also the $^{9}$Li and $^{9}$C cores are mirror nuclei. Table \[tab1a\] contains the parameters used in this work for the potential form given in Eq.(\[eq1\]) with the $s$- and $p$-state parameters from Ref.[@mor19]. The radial shapes are for convenience chosen to be Gaussians with the same range in all terms. The actual shape is unimportant as long as it is of short range with a range consistent with the core-size.
$\ell$ $S_{c}^{(\ell)}$ $S_{ss}^{(\ell)}$ $S_{so}^{(\ell)}$
-------- ------------------ ------------------- -------------------
0 $-5.4$ $-4.5$ –
1 260.75 1.0 300
2 260 $-9.0$ $-300$
: The strength parameters, $S_i^{(\ell)}$, in MeV for the Gaussian core-nucleon potentials, $V_i^{(\ell)}=S_i^{(\ell)}
e^{-r^2/b^2}$, defined in Eq.(\[eq1\]), with the $s$ and $p$ partial waves as in Ref.[@mor19] (also denoted P1I in Ref.[@cas17]). We choose the same numerical value, $b = 2.55$ fm, for the range parameter, $b$, in all terms and partial waves. []{data-label="tab1a"}
The two all-decisive properties of the nucleon-core system are the positions of the two-body resonances, and the exclusion of Pauli forbidden states occupied by the core-nucleons. The first property is achieved by the numerical values specified in Table \[tab1a\]. The second property is fulfilled by use of the shallow $s$-wave potential without a bound state, and a large and inverse (positive) sign of the $p$-wave spin-orbit strength, which places the $p_{3/2}$-shell at an unreachable high energy. In this way, by construction, the valence-nucleon can not occupy the Pauli forbidden $s_{1/2}$- and $p_{3/2}$-shells, which already are occupied by the six neutrons or the six protons in the $^9$Li or $^9$C-core.
![(a): Core-nucleon potentials for $s_{1/2}$ states, $1^-$ (red) and $2^-$ (black), in $^{10}$Li (solid lines) and $^{10}$N (dashed lines). (b): The same as in panel (a), but for the $p_{1/2}$ states, $1^+$ (red) and $2^+$ (black).[]{data-label="figpot"}](pots.eps){width="\linewidth"}
The resulting nucleon-core potentials are shown in Fig. \[figpot\] for the $s_{1/2}$-states, $1^-$ (red) and $2^-$ (black) in panel (a), and for the $p_{1/2}$-states, $1^+$ (red) and $2^+$ (black) in panel (b). The solid and dashed lines refer, respectively, to the $^{10}$Li and $^{10}$N-cases. The difference between them arises from the Coulomb repulsion entering in the $^ 9$C-proton interaction for $^{10}$N. The $s$-waves in the left panel of Fig. \[figpot\] reveal our choices for $^{10}$Li of an attractive $2^-$-potential placing a virtual nucleon-core state very close to zero energy, while in contrast the $1^-$-potential is very small and slightly repulsive. The same potentials for $^{10}$N are pushed up by the Coulomb repulsion, where the $2^-$-potential still has an attractive short-range part, whereas the $1^-$-potential is clearly overall repulsive. The $p$-wave potentials in the right panel of Fig. \[figpot\] all have an attractive short-range part leading to more or less known $p$-wave resonances in both $^{10}$Li and $^{10}$N.
As shown in [@gar02a; @gar97], the main properties of $^{11}$Li, as well as the behavior of the momentum distributions, are essentially determined by the energy of the centroid of the spin-splitted $s$- and $p$- doublets. Therefore, the subsequent three-body results would remain basically unchanged with the opposite order of the $1^-$ and $2^-$-virtual states and of the $1^+$ and $2^+$-resonances.
$^{10}$Li-properties
--------------------
$\delta(E_R)=\frac{\pi}{2}$
------- ---------- ------------ ---------- ----------------------------- --------------- --------------- -------
$E_R$ $\Gamma_R$ $E_R$ $\Gamma_R$ $E_R$ $\Gamma_R$ $E_R$
1$^-$ – – – – – – –
2$^-$ $-0.020$ – $-0.028$ – – – –
1$^+$ 0.32 0.19 0.42 0.19 $0.42\pm0.05$ $0.15\pm0.07$ 0.37
2$^+$ 0.58 0.49 0.71 0.40 $0.80\pm0.08$ $0.30\pm0.10$ 0.78
4$^-$ 3.95 2.45 4.13 3.12 $4.47\pm0.10$ $0.7\pm0.2$ 4.88
: For $^{10}$Li, the second column shows the energies of the 1$^-$ and 2$^-$ virtual states, and energies and widths of the $1^+$, $2^+$ and $4^-$ resonances in $^{10}$Li obtained with the two-body potentials described in the text. The third column shows the energies and widths obtained in Ref.[@kat99]. In the fourth column the available experimental data are given [@boh93]. The last column gives the resonance energies computed as $\delta(E_R)=\pi/2$. All the energies, $E_R$, and widths, $\Gamma_R$, are given in MeV.[]{data-label="tab10Li"}
The $^{10}$Li-properties are determined by the potentials given by the solid curves in Figs. \[figpot\]a and \[figpot\]b. The computed spectrum is shown in the second column of Table \[tab10Li\]. The ground state is a virtual $2^-$-state resulting from the coupling of an $s_{1/2}$ valence-neutron with the $3/2^-$ ground-state of the core, whose energy is about $-20$ keV. The corresponding potential is given by the solid black curve in Fig. \[figpot\]a. Due to the repulsive character of the potential shown by the solid red curve in the same figure, the $1^-$ $s$-wave partner appears at high energy in the continuum. The $p$-wave resonant-states, 1$^+$ and $2^+$, produced by the $p$-wave potential barriers (solid curves in Fig. \[figpot\]b), are found at 0.32 MeV and 0.58 MeV, respectively, with corresponding widths of 0.19 MeV and 0.49 MeV.
![Complex energies of the $p_{1/2}$-resonances in $^{10}$Li, panel (a), and the $s_{1/2}$- and $p_{1/2}$-resonances in $^{10}$N, panel (b), after a two-body complex scaling calculation using the potential described in Sect. \[pots\].[]{data-label="fig1"}](10N.eps){width="\linewidth"}
The virtual state is obtained by finding the energy providing the correct divergent asymptotic behaviour produced by the poles of the $S$-matrix located on the negative imaginary axis in the complex momentum plane. The resonances are also obtained as poles of the $S$-matrix by means of the complex scaling method [@ho83; @moi98], which simplifies the numerical calculation by giving rise to an exponential fall-off of the complex rotated resonance wave functions. As mentioned in Sect. \[method\], the complex rotated two-body problem is solved after discretization of the continuum by means of a box boundary condition.
The corresponding discrete energies appear in the complex energy plane rotated by an angle equal to twice the angle used for the complex scaling coordinate transformation. The resonances appear as discrete points, independent of the complex scaling angle, out of the cut (lines) corresponding to continuum states. This is shown for $^{10}$Li in Fig. \[fig1\]a. As we can see, a complex scaling angle of $\theta=0.3$ rads is enough to capture the $1^+$ and $2^+$-resonances. As shown in Table \[tab10Li\], together with the $1^+$ and $2^+$-states, the core-neutron potential described in Sect. \[pots\] gives rise to a $4^-$-resonance (with the valence neutron in the $d_{5/2}$-state) at 3.95 MeV with a width of 2.45 MeV. For the sake of clarity in the figure, this resonance is not shown in Fig. \[fig1\]a.
The computed spectrum can be compared to the one obtained in Ref.[@kat99] (third column of Table \[tab10Li\]), where a microscopic coupled-channel calculation is performed. The similar virtual $s$-states in the two calculations are dictated by a demand to reproduce measured two-neutron halo properties of $^{11}$Li in subsequent calculations. In the fourth column of the table we give the available experimental data. Note that the ones of the $4^-$-state are actually in Ref.[@boh93] assigned preliminary to angular momentum and parity, $2^-$. However, as suggested in [@kat99], the calculations might indicate that they could actually correspond to the $4^-$-resonance.
Although in both this work and Ref.[@kat99], the energies are computed as poles of the $S$-matrix, the agreement with the experimental $1^+$ and $2^+$-energies [@boh93] seems to be worse in our calculation. The $p$-states deviate somewhat by more than about 100 keV in centroid energy of the two spin-split $p$-states (0.5 MeV in this work and 0.6 MeV in [@kat99]) . However, in [@kat99] the calculation is performed by fitting the energy of the $1^+$$S$-matrix pole to the experimental energy of the $1^+$-resonance, whereas in our case the experimental energies are better reproduced by the energies for which the corresponding phase shifts are equal to $\pi/2$. As seen in the last column in Table \[tab10Li\], when computed in this way, our potential gives rise to 1$^+$ and $2^+$-energies equal to 0.37 MeV and 0.78 MeV, respectively, as well as to a $4^-$-energy of 4.88 MeV.
The differences between resonance energies obtained through the different mathematical definition reflect an intrinsic uncertainty, which only can be resolved by comparing calculations of directly measured observables like specific scattering cross sections. In this connection, it is important that the $^9$Li-neutron interaction used in the present work also leads to reproduction of the experimental excitation energy spectrum of $^{10}$Li after the breakup reaction, $d(^9\mbox{Li},p)^{10}$Li, initiated by a $^{9}$Li laboratory energy of $11.1$ MeV/A, see Ref.[@mor19].
$^{10}$N-properties
-------------------
$\delta(E_R)=\frac{\pi}{2}$
------- ------- ------------ ------- ------------ ----------------------------- --------------------- --------------------- --------------------- -------
$E_R$ $\Gamma_R$ $E_R$ $\Gamma_R$ $E_R$ $\Gamma_R$ $E_R$ $\Gamma_R$ $E_R$
1$^-$ – – – – – – $1.9^{+0.2}_{-0.2}$ $2.5^{+2.0}_{-1.5}$ –
2$^-$ 1.74 3.94 1.51 3.47 – – $2.2^{+0.2}_{-0.2}$ $3.1^{+0.9}_{-0.7}$ –
1$^+$ 2.62 1.68 2.84 1.89 $2.6^{+0.4}_{-0.4}$ $2.3^{+1.6}_{-1.6}$ – – 3.51
2$^+$ 2.89 2.21 3.36 2.82 – – – – 4.45
: For $^{10}$N, energies and widths, in MeV, of the 1$^-$, 2$^-$, $1^+$, and $2^+$ resonances obtained in our calculation (second column), the theoretical values given in Ref.[@aoy97] (third column), and the experimental values given in Refs.[@lep02; @hoo17] (fourth and fifth columns). The last column gives the resonance energies computed as $\delta(E_R)=\pi/2$. (\*) Although in [@lep02] the observed resonance was assigned to be an $s$-wave resonance, as indicated in Ref.[@til04], it is very likely the energy and width corresponding to the $1^+$ state. []{data-label="tab1"}
The mirror nucleus, $^{10}$N, is now assumed to have exactly the same potentials as $^{10}$Li, except for the Coulomb interaction arising from the substituted valence-proton. We assume point-like protons and a spherical and homogeneously charged $^9$C-core. With these interactions we now compute the spectrum of $^{10}$N as described by a $^9$C-core and a proton. The immediate consequence of the Coulomb repulsion is that all the core-nucleon potentials are pushed up in energy, reducing the depth of the potentials, and increasing the potential barriers. This is precisely as seen by comparing the dashed and solid curves in Fig. \[figpot\].
For the $s_{1/2}$-states the Coulomb barrier implies that $1^-$ and $2^-$-states in principle might appear as resonances. However, the overall repulsive behaviour of the $1^-$-potential (dashed red curve in Fig. \[figpot\]) does not exhibit any barrier, and only $2^-$ resonant-states are then possible. As for $^{10}$Li, resonances in $^{10}$N are obtained after a complex scaling calculation. The results are shown in Fig. \[fig1\]b, where the resonances are the isolated points out of the cut (line) associated with the continuum states. The ground state, the $s$-wave $2^-$-state, is clearly broader than the 1$^+$ and $2^+$ $p$-states, and therefore requires a larger angle in the complex scaling transformation in order to be captured in the calculation. In particular, the calculation shown in the figure has been made using a complex scaling angle equal to 0.5 rads for the $2^-$-state, and 0.3 rads for the $1^+$ and $2^+$-states.
The resonance energies and widths obtained for $^{10}$N are collected in the second column of Table \[tab1\], where the results are compared to the values given in Refs.[@aoy97; @lep02; @hoo17]. Our results are very consistent with the theoretical values given in [@aoy97], where the complex scaling method also is used. The slightly different energies are due to a core-nucleon strong interaction producing also slightly different energies for the $^{10}$Li-states, see Table \[tab10Li\]. The experimental value in Ref.[@lep02] was initially assigned to a 1$^-$-state, but in [@til04] it is suggested that this resonance is very likely the mirror of the probable $1^+$-state at 0.24 MeV in $^{10}$Li.
Finally, in Ref.[@hoo17] (fifth column in Table \[tab1\]) two resonances have been measured with energies around 2.0 MeV, which are assigned by the authors to states with angular momentum and parity, $1^-$ and $2^-$. In this reference the authors mention as well an excited $1^-$ or $2^-$ resonant-state with an energy of $2.8\pm 0.2$ MeV. As seen in Fig. \[fig1\]b, we have not found any trace of such an excited state with negative parity. It is in any case striking, that the three energies reported in [@hoo17] agree reasonably well with the three energies obtained in this work for the $2^-$, $1^+$, and $2^ +$-states.
Due to its large resonance width, the $2^-$-phase shift never reaches the value of $\pi/2$, and therefore the energy of this resonance can not be extracted in this way. In contrast, this is possible for the $1^+$ and $2^+$-states, and this happens for energies equal to 3.51 MeV and 4.45 MeV, respectively, that is clearly larger than the energies obtained as poles of the $S$-matrix (last column in Table \[tab1\]). Again these different definitions reflect in themselves an inherent uncertainty in the resonance parameters.
The core-nucleon-nucleon system
===============================
After discussing the two-body properties of $^{10}$Li and $^{10}$N, we now investigate the effects they determine for the structure and properties of the three-body mirror nuclei, $^{11}$Li and $^{11}$O. We shall do this in three different steps, first in this section we present the energy spectra, and in the two following sections we discuss the properties of the different states, respectively the internal structure of the wave functions, and the spatial distribution of the three constituents.
Energy spectrum of $^{11}$Li
----------------------------
![Complex resonance energies for the computed $1/2^+$ (open brown circles), $3/2^+$ (solid green squares), and $5/2^+$ (open blue squares) states in $^{11}$Li, panel (a), and $^{11}$O, panel (b), where the $3/2^-$resonance (solid red circles) is also shown. The calculations have been performed with a complex scaling angle $\theta=0.3$ rads.[]{data-label="fig2"}](11O.eps){width="\linewidth"}
The potentials used to describe $^{11}$Li are the same as used in Ref.[@cas17], where the properties of the computed $3/2^-$ ground-state wave function in $^{11}$Li are described. An effective three-body force is used to fit the experimental two-neutron separation energy of $369.15(65)$ keV [@smi08], which leads to a charge root-mean-square radius of 3.42 fm, also in agreement with the experimental value reported in [@smi08]. In this work an attractive Gaussian three-body force with a range of 5 fm and a strength of $-0.6$ MeV has been used. In addition, as shown in Ref.[@cas17], the computed $^{11}$Li ground-state wave function permits reproduction of the experimental energy-integrated angular differential cross section for the $^{11}$Li$(p, d)^{10}$Li reaction at 5.7 MeV/A.
In Ref.[@gar02] the electric dipole excitations in $^{11}$Li, i.e. the $1/2^+$, $3/2^+$, and $5/2^+$ resonant-states, were investigated by means of the complex scaling method [@ho83; @moi98]. It was found that the energies of these three resonances are pretty close to each other, with specific values depending slightly on the properties of the core-neutron interaction. In any case, they lie in the energy interval between 0.3 MeV and 0.7 MeV above the three-body threshold.
$^{11}$Li (Comp.) $\Delta_c^{(1)}$ $^{11}$O (Estim.)
----------------- ------------------- ------------------ -------------------
$\frac{3}{2}^-$ $-0.37$ 4.93 4.56
$\frac{1}{2}^+$ $0.39-i0.15$ $2.69-i0.77$ $3.08-i0.92$
$\frac{3}{2}^+$ $0.35-i0.09$ $3.41-i0.86$ $3.76-i0.95$
$\frac{5}{2}^+$ $0.47-i0.22$ $3.89-i1.20$ $4.36-i1.42$
: Computed complex energies, $E_R-i\Gamma_R/2$, of the $\frac{3}{2}^-$, $\frac{1}{2}^+$, $\frac{3}{2}^+$, and $\frac{5}{2}^+$ states in $^{11}$Li, the Coulomb shift for each of them as defined in Eq.(\[eq2\]), and the estimated complex energies of the corresponding states in $^{11}$O. All the values are given in MeV.[]{data-label="tab0"}
When the specific interactions used in this work are employed, the complex scaling method reveals the existence of $1/2^+$, $3/2^+$, and $5/2^+$-resonances. The results are shown in Fig. \[fig2\]a, where specific dots appear clearly separated from the straight line corresponding to the background continuum states. The resonances of interest are indicated by the arrows in the figure. The precise computed values for the resonance energies and widths, $(E_R,
\Gamma_R)$, are $(0.39, 0.30)$ MeV, $(0.35, 0.18)$ MeV, and $(0.47,
0.44)$ MeV for the $1/2^+$, $3/2^+$, and $5/2^+$-states, respectively. This is also given in the second column in Table \[tab0\] as a complex number for each state.
Energy spectrum of $^{11}$O
---------------------------
As expected, due to the Coulomb repulsion, the ground state in $^{11}$O is not bound. Therefore, in this case all the states, $J^\pi=3/2^ -$, $1/2^ +$, $3/2^ +$, and $5/2^+$, will be computed by means of the complex scaling method. We have used a complex scaling angle of $\theta=0.30$ rads, and the result of the calculation is shown in Fig. \[fig2\]b. The straight line, rotated by an angle equal to $2\theta$, contains the discretized continuum states, and the points out of this line correspond to the different resonances. The lowest $3/2^-$, $1/2^+$, $3/2^+$, and $5/2^+$-states are indicated by the corresponding arrows.
As seen in the figure, in all the cases a second resonance is found in the vicinity of $E_R=5$ MeV. In the 5/2$^+$-case even a third resonance around 5.5 MeV is seen. In order to make the plot clean, the cuts associated to the two-body resonances, i.e. two-body resonance plus the third particle in the continuum [@moi98], are not shown in the figure.
----------------- ------- ------------ ------- ------------ ---------------- ------------ ------- ------------
$E_R$ $\Gamma_R$ $E_R$ $\Gamma_R$ $E_R$ $\Gamma_R$ $E_R$ $\Gamma_R$
$\frac{3}{2}^-$ 4.74 2.75 4.16 1.30 $3.21\pm 0.84$ – 4.75 2.51
4.97 5.07 4.85 1.33 – – – –
$\frac{1}{2}^+$ 3.77 2.74 – – – –
5.02 4.87 – – – –
$\frac{3}{2}^+$ 3.79 2.84 – – – –
4.93 4.65 – – – –
$\frac{5}{2}^+$ 4.16 3.38 4.65 1.06 – – – –
4.89 5.36 6.28 1.96 – – – –
----------------- ------- ------------ ------- ------------ ---------------- ------------ ------- ------------
: For $^{11}$O, the second column gives the computed energies, $E_R$, and widths, $\Gamma_R$, of the $3/2^-$, $1/2^+$, $3/2^+$ and $5/2^+$states. The last three columns show the results given in Refs.[@web19; @wan19], Ref.[@cha12], and Ref.[@for19], respectively. Both, energies and widths, are given in MeV.[]{data-label="tab3"}
The precise values of the resonant energies, $E_R$, and widths, $\Gamma_R$, are given in the second column of Table \[tab3\] for the two lowest resonances for each of the computed $J^ \pi$-states. As seen in the table, the $1/2^+$, $3/2^+$, and $5/2^+$-energies are similar to each other, especially the 1/2$^+$ and $3/2^+$-states, which are almost degenerate. Therefore, very likely one of these states should actually be the ground state, since the $3/2^-$-energy is at least 0.6 MeV higher.
At this point, we emphasize that the computed resonance energies and widths have been obtained without inclusion of any three-body force in the radial Eqs.(\[radf\]). When a short-range effective three-body potential is included, we have observed that the effect is clearly bigger for the 3/2$^-$-state than for the positive parity resonances. This is an indication that for the $3/2^-$-case the core and the two valence-protons are clearly closer to each other than for the $1/2^+$, $3/2^+$, and $5/2^+$-states.
In particular, if we use a Gaussian three-body force with a range of 5 fm and an attractive strength of $-4$ MeV, the energies and widths of the positive parity resonances remain essentially unchanged, whereas for the lowest $3/2^-$-state we get an energy of 3.72 MeV (with a width of 1.18 MeV), similar to the one of the other resonances. The conclusion is then, that from the pure three-body calculation it is difficult, or even not possible, to determine which $J^\pi$-state is actually the ground state in $^{11}$O. In any case, it seems clear that we cannot exclude the possibility that another state than the $3/2^-$-state becomes the ground state in $^{11}$O.
In Table \[tab3\] we compare our results with previous works. In all of them, the ground state is assigned to an angular momentum and parity, $3/2^-$. Only in Refs.[@web19; @wan19] the positive parity state $5/2^+$ is also considered. In these references, [@web19; @wan19], the computed resonances are clearly narrower than the ones obtained in the present work, and the fact that the two energies given for the lowest 3/2$^-$ and $5/2^+$-states are similar to ours, but with the quantum numbers exchanged, is probably just an accidental coincidence. The reason for this difference is difficult to determine. On the one hand, in our work the core is assumed to be spherical in contrast to Refs.[@web19; @wan19]. Thus, we have ignored the possibly significant role played by the deformation. On the other hand, in Refs.[@web19; @wan19] the calculations are performed taking a maximum value for the hypermomentum, $K_{max}$, equal to $20$. As we shall discuss later, this value might be too small to guarantee that convergence has been reached in the calculations.
Our three-body approach treats the constituents as inert particles with central two-body interactions, and it is therefore insensitive to deformation. In any case, as discussed in [@mis97], for sufficiently weakly bound systems, or in other words, provided the valence nucleons are located at relatively large distance from the core, “the quadrupole deformation of the resulting halo is completely determined by the intrinsic structure of a weakly bound orbital, irrespective of the shape of the core”. Furthermore, our phenomenological choice of interaction parameters necessarily accounts for at least part of the effects of the core-deformation.
A further comparison is found in another publication [@cha12], where a clearly lower energy, 3.21 MeV, although with a large error bar, is given for the $3/2^-$-state. However, this energy has not been computed, but obtained as an extrapolation using the isobaric multiplet mass-equation, whose coefficients are determined after the shell-model computed energies for $^{11}$Li, $^{11}$Be, and $^{11}$B. In addition to these energy properties, it was argued in Ref.[@for19], that the experimental breakup data given in [@web19], can be as well reproduced by only the ground state of $^{11}$O, whose energy and width are given in Table \[tab3\] to be 4.75 MeV and 2.51 MeV, respectively, very similar to our lowest $3/2^-$ energy and width.
Coulomb shift
-------------
Due to the charge symmetry of the strong interaction, the $^{11}$O-states computed in the previous subsection have been obtained simply by adding the Coulomb potential to the nuclear interactions used to describe $^{11}$Li. The repulsive character of the Coulomb interaction, between the two valence-protons and between each of the valence-protons and the core, is obviously the reason for the increase of the energies.
An estimate of how much these energies should be modified by the Coulomb repulsion can be obtained as the first-order perturbative value of the Coulomb shift: $$\Delta_c^{(1)}=\langle \Psi(\mbox{$^{11}$Li}) | V_{coul}| \Psi(\mbox{$^{11}$Li}) \rangle,
\label{eq2}$$ where $ \Psi(\mbox{$^{11}$Li})$ is the three-body $^{11}$Li wave function corresponding to a given state, but where the valence-neutrons are replaced by protons, and the charge of the core is assumed to be the one of the mirror nucleus, $^9$C. Therefore, $\Psi(\mbox{$^{11}$Li})$ in Eq.(\[eq2\]) represents an artificial $^{11}$O wave function, which is assumed to have the same structure as the corresponding $^{11}$Li-state. The $V_{coul}$-potential is the resulting Coulomb interaction between the three pairs of charged particles, where the protons are point-like and the $^9$C-core is spherical and uniformly charged corresponding to the root-mean-square radius of about $2.5$ fm. The first order Coulomb shift, $\Delta_c^{(1)}$, is then the diagonal contribution to the Coulomb shift.
As already mentioned, the $1/2^+$, $3/2^+$, and $5/2^+$-states in $^{11}$Li have been obtained by means of the complex scaling method. The wave functions are then complex rotated, and the corresponding value of $\Delta_c^{(1)}$ has to be obtained after complex rotation of the Coulomb potential. In this way $\Delta_c^{(1)}$ will be a complex quantity, whose imaginary part can be interpreted as the uncertainty in the energy shift [@moi98]. In other words, the imaginary part of $\Delta_c^{(1)}$ in the $1/2^+$, $3/2^+$, and $5/2^+$-cases permits us to estimate as well the change in the width of the resonance.
In Table \[tab0\] we give the computed energies of the $^{11}$Li-states (second column) together with the computed values of $\Delta_c^{(1)}$ for the four states considered (third column). The results depend slightly on how the Coulomb potential is constructed, but the overall size and relations are very well determined. As we can see, the value of $\Delta_c^{(1)}$ is substantially larger for the $3/2^-$-state than for the other three states, which again is an indication of the smaller size of the $3/2^-$-state, since the smaller the system the larger the Coulomb repulsion, and therefore the larger the value of $\Delta_c^{(1)}$.
When this shift is added to the $^{11}$Li-energies, we obtain the estimate for the energies of the $^{11}$O-states given in the last column of the table. In the case of the $3/2^-$-state, since the $^{11}$Li wave function is real, the shift $\Delta_c^{(1)}$ is also real, and an estimate of the width in the $3/2^-$-state in $^{11}$O is not possible in this way. As we can see, the estimated energies given in the last column of Table \[tab0\] are quite reasonable, pretty close to the computed energies given in Table \[tab3\] for the lowest $3/2^-$, $1/2^+$, $3/2^+$, and $5/2^+$ states. The only exception is perhaps the $1/2^+$-state, where a difference of about 0.7 MeV is found.
These similarities show that the variations in the energy shift due to the structure differences, as expected, are relatively small. The difference between the energy shift, $\Delta_c^{(1)}$, for a given state and the experimental energy shift, $\Delta_c$, between the experimental energies is a measure of the structure effect of the Thomas-Ehrman shift, that is $\Delta_{TE}=\Delta_c - \Delta_c^{(1)}$. Recent calculations of $\Delta_{TE}$ concerning different light mirror nuclei are available in the literature. For example, in Ref.[@aue00] the shift between the mirror system, $^{11}$Be and $^{11}$N, was investigated. In [@gri02] the same was done for $^{12}$Be-$^{12}$O and $^{16}$C-$^{16}$Ne, and in Ref.[@gar04] this shift was computed for the case of $^{17}$N and $^{17}$Ne. In all the cases the value of $\Delta_{TE}$ was obtained to be of no more than a few hundreds of keV. These sizes are consistent with the energy difference between the lowest $J^\pi$-energies obtained in our calculation (Table \[tab3\]), and the estimated energies given in the last column of Table \[tab0\].
Three-body wave functions
=========================
----------------- ---------- ---------- ----- ------- ------- ----------- ----------- ----------
$J^\pi$ $\ell_x$ $\ell_y$ $L$ $s_x$ $S$ $K_{max}$ $^{11}$Li $^{11}$O
$\frac{3}{2}^-$ 0 0 0 1 $3/2$ 120 22% 5%
0 0 0 2 $3/2$ 120 35% 7%
1 1 0 1 $3/2$ 60 6% 11%
1 1 0 2 $3/2$ 80 10% 19%
1 1 1 1 $1/2$ 60 4% 9%
1 1 1 1 $3/2$ 40 5% 11%
1 1 1 2 $3/2$ 40 3% 7%
1 1 1 2 $5/2$ 80 12% 28%
$\frac{1}{2}^+$ 0 1 1 1 $1/2$ 121 $<1$% 5%
0 1 1 1 $3/2$ 201 30% 37%
1 0 1 1 $1/2$ 121 6% 6%
1 0 1 1 $3/2$ 121 7% 6%
1 0 1 2 $3/2$ 201 42% 41%
1 2 1 1 $3/2$ 61 7% 5%
2 1 1 1 $3/2$ 41 1% $<1$%
$\frac{3}{2}^+$ 0 1 1 1 $1/2$ 101 1% $<1$%
0 1 1 1 $3/2$ 101 1% 1%
0 1 1 2 $3/2$ 161 9% 7%
0 1 1 2 $5/2$ 201 34% 32%
1 0 1 1 $1/2$ 101 1% 1%
1 0 1 1 $3/2$ 161 8% 7%
1 0 1 2 $3/2$ 161 5% 4%
1 0 1 2 $5/2$ 201 36% 43%
1 2 1 2 $5/2$ 41 2% 4%
$\frac{5}{2}^+$ 0 1 1 2 $3/2$ 201 24% 23%
0 1 1 2 $5/2$ 201 25% 20%
1 0 1 1 $3/2$ 201 25% 26%
1 0 1 2 $3/2$ 81 3% 2%
1 0 1 2 $5/2$ 201 23% 25%
1 2 1 2 $3/2$ 31 $<1$% 2%
1 2 1 2 $5/2$ 31 $<1$% 2%
----------------- ---------- ---------- ----- ------- ------- ----------- ----------- ----------
: Dominant components (larger than 1% probability) in the lowest $3/2^-$, $1/2^+$, $3/2^+$, and $5/2^+$ wave functions in $^{11}$Li and $^{11}$O in the Jacobi set with the $\bm{x}$-coordinate defined between the core and one of the valence nucleons. The quantum numbers are as defined below Eq.(\[exp1\]). Note that the core has negative parity.[]{data-label="tabcom0"}
The calculation of the $^{11}$Li and $^{11}$O three-body states has been made including all the components satisfying $\ell_x,\ell_y \leq
7$, where $\ell_x$ and $\ell_y$ are the relative angular momenta between two of the particles, and between their center-of-mass and the third particle, respectively. The maximum value of the hypermomentum, $K_{max}$, has to be sufficiently large to reach convergence, but for all partial waves it has been taken to be at least $20$. In Table \[tabcom0\] we give the partial wave decomposition and the components with probability larger than 1% for each of the lowest $J^\pi$-states. Note here that the use of the complex scaling method permits us to normalize the resonance wave functions as described in Ref.[@moi98]. The quantum numbers in the table are as described below Eq.(\[exp1\]), with the $\bm{x}$-Jacobi coordinate defined between the core and one of the valence nucleons.
As seen in the table, for these components the $K_{max}$-value used is pretty large, very likely overdoing the work of getting a well-converged three-body solution, especially for $^{11}$Li. A careful analysis of each individual component could certainly reduce the $K_{max}$-value. As a test, we have performed the same calculations using $K_{max}=20$ for all the components. These less accurate calculations result in an increase of the three-body energies by up to 0.4 MeV for the $^{11}$Li-states and by up to 1 MeV for the $^{11}$O-resonances. The only exception is the lowest $3/2^-$-state (bound in the case of $^{11}$Li), for which the increase in energy is of about 50 keV for $^{11}$Li, and of about 200 keV for $^{11}$O. This is once more reflecting the smaller size of the lowest $3/2^-$-state compared to the positive parity states, since the closer the particles are to each other, the smaller is the basis necessary to reach convergence.
----------------- ---------- ------- ------- ---------- ------- ----------- ----------
$J^\pi$ $\ell_x$ $j_N$ $j_x$ $\ell_y$ $j_y$ $^{11}$Li $^{11}$O
$\frac{3}{2}^-$ 0 1/2 1 0 $1/2$ 22% 5%
0 1/2 2 0 $1/2$ 35% 7%
1 1/2 1 1 $1/2$ 15% 32%
1 1/2 2 1 $1/2$ 25% 53%
$\frac{1}{2}^+$ 0 1/2 1 1 $1/2$ 34% 41%
1 1/2 1 0 $1/2$ 52% 51%
1 1/2 2 2 $3/2$ 3% 2%
1 3/2 0 0 $1/2$ 1% 1%
1 3/2 1 0 $1/2$ 2% 1%
1 3/2 1 2 $3/2$ 1% 1%
1 3/2 2 2 $3/2$ 3% 2%
$\frac{3}{2}^+$ 0 1/2 1 1 $1/2$ 2% 2%
0 1/2 2 1 $1/2$ 45% 39%
1 1/2 1 0 $1/2$ 51% 52%
1 1/2 3 2 $3/2$ $<1$% 3%
$\frac{5}{2}^+$ 0 1/2 2 1 $1/2$ 49% 44%
1 1/2 1 2 $3/2$ $<1$% 1%
1 1/2 2 0 $1/2$ 49% 52%
1 1/2 2 2 $3/2$ $<1$% 1%
----------------- ---------- ------- ------- ---------- ------- ----------- ----------
: The same as Table \[tabcom0\] but in the coupling scheme where the core-neutron relative orbital angular momentum $\ell_x$ couples to the spin of the neutron to provide the angular momentum $j_N$, which couples to the spin of the core to the total core-neutron angular momentum $j_x$. The relative orbital angular momentum between the core-neutron center-of-mass and the second neutron, $\ell_y$, couples to the spin of the second neutron to give the angular momentum $j_y$. Both, $j_x$ and $j_y$ couple to the total three-body angular momentum $J$.[]{data-label="tabcom2"}
The same decomposition is shown in Table \[tabcom2\], but in a coupling scheme more consistent with the mean-field quantum numbers, where the core-nucleon relative orbital angular momentum, $\ell_x$, couples to the spin of the nucleon to provide the angular momentum $j_N$, which in turn couples to the spin of the core to give the total core-nucleon angular momentum, $j_x$. The relative orbital angular momentum between the core-nucleon center-of-mass and the second nucleon, $\ell_y$, couples to the spin of the second nucleon to give the angular momentum, $j_y$. Both $j_x$ and $j_y$ couple to the total three-body angular momentum, $J$.
As we can see, the structure of the $3/2^-$-state changes substantially due to the Coulomb repulsion. In the case of $^{11}$Li the $3/2^-$ (bound) ground state contains about 40% of core-neutron $p$-wave contribution. More precisely, 15% of the wave function corresponds to $^{10}$Li in the $1^+$-state, and 25% to $^{10}$Li in the $2^+$-state (Table \[tabcom2\]). With respect to the $s$-wave contribution, even if the $1^-$-state in $^{10}$Li is lying high in the continuum, 22% corresponds to $^{10}$Li populating that state.
These characteristics are however very different when analyzing the $3/2^-$-state in $^{11}$O. The Coulomb repulsion, which pushes up the $s$-wave core-proton 2$^-$-state by more than 1.5 MeV, turns out to be crucial producing a drastic reduction of the $s$-wave contribution. In fact, as seen in the upper part of Tables \[tabcom0\] and \[tabcom2\], the $p$-wave components give 85% of the wave function, whereas the $s$-wave contribution reduces now from almost 60% in $^{11}$Li to only about 12% in $^{11}$O.
This result is in contrast to Ref.[@wan19], where the $29\%$ $s$-wave contribution in the $3/2^-$-state in $^{11}$O is even higher than the 25% given for $^{11}$Li. This low $s$-wave content in the $^{11}$Li ground-state wave function seems to disagree with previous results in [@gar96; @gar97; @cas17], where it is shown that the agreement with experimental momentum distributions and angular differential cross sections requires a $p$-wave content of about $35\%-40\%$ in the $^{11}$Li ground-state, or, equivalently, $60\%-65\%$ $s$-wave contribution.
For both nuclei, $^{11}$Li and $^{11}$O, the contribution of $d$-waves in the present work is far from substantial, in total of about 3% in both cases, and none of the $d$-wave components provides more than 1% of the norm. This result seems to contradict the measured increase of about 8.8% [@neu08] of the quadrupole moment in $^{11}$Li relative to that of $^{9}$Li, which in shell-model calculations in Ref.[@suz03] is explained as due to a significant $d$-wave contribution of similar size as the one corresponding to $p$-waves.
The small probability of $d$-waves may be related to the lack of deformation of the frozen core as seen by the argument. If the structure of a given deformation is expanded on another, say body-fixed, deformation, there must be partial wave components corresponding to this deformation. However, our three-body model provides the full wave function corresponding to that obtained with deformation after, not before, projection of angular momentum and parity. Thus, our model can only say something about the inserted frozen core-structure and the calculated valence-structure.
However, as discussed in [@neu08], the measured quadrupole moment could instead be related to an about $10\%$ increase of the charge-radius in $^{11}$Li compared to the one of $^{9}$Li. This increase can be interpreted as due to the neutron halo. The two neutrons in the zero angular momentum ground state produce a distortion of the $^{9}$Li-core, which effectively corresponds to an increase of the core-radius. The initially slightly deformed $^{9}$Li-core is otherwise in principle maintained in the subsequent three-body calculations without significant effect as argued in [@mis97]. Using such an increased radius, the neutron-core interaction still must be adjusted to the previously described specific desired properties. These all-decisive phenomenologically obtained interactions guarantee the same resulting three-body structure as obtained with the bare $^{9}$Li-radius. This is consistent with [@neu08], where it is stated that there is a striking analogy between the quadrupole moment and the root-mean-square charge-radius without any additional change of wave function structure.
In any case, a detailed analysis of the quadrupole moment of $^{11}$Li requires to take into account the different sources contributing to the measured value, since the two valence neutrons are mostly on the same side of $^{9}$Li. First the contribution from the original $^{9}$Li quadrupole moment, second the one from the rotation of the $^9$Li core around the three-body center of mass, and third and fourth the one from increased radius and induced deformation of $^{9}$Li from the two valence neutrons. From Ref.[@boh75] we know that neutral nucleons polarize the charged core by an amount of the same order as if they were charged. Thus, our model is consistent with all available $^{11}$Li data, but for now we leave the complicated quantitative quadrupole moment calculation for future investigations.
Concerning the $1/2^+$, $3/2^+$, and $5/2^+$-resonances, they are all almost completely given by $sp$-interference terms. Only minor contributions from $pd$-interferences are seen, with the largest, as given in the tables, reaching up to 7% in the $1/2^+$-case. The presence of low-lying $p$-resonances in $^{10}$Li and $^{10}$N makes the $pd$-interferences more likely than the $dp$-ones, whose weight is always smaller than 1%. In general, we can see that the structure of the three positive parity resonances does not change significantly by moving from $^{11}$Li to $^{11}$O. The weight of the different components remains essentially the same in both cases.
An important difference, seen in Table \[tabcom2\], between the structure of the different $J^+$-resonances, is that only the $1/2^+$-state has substantial contributions from $s$-waves ($\ell_x=0$) in the nucleon-core $1^-$-state ($j_x=1$) in either $^{10}$Li or $^{10}$N. In contrast, for both the $3/2^+$ and $5/2^+$-resonances only the 2$^-$ $s$-state ($\ell_x=0$, $j_x=2$) in $^ {10}$Li or $^{10}$N is substantially populated. In the next section, we shall discuss this difference as responsible for the very different spatial structure of these states. Keep in mind that a similar weight of the different partial-wave components does not necessarily imply a similar spatial distribution of the constituents, which is in fact determined by the radial wave functions $f_n^J(\rho)$ in Eq.(\[exp0\]) and the expansion coefficients $C_q^{(n)}(\rho)$ in Eq.(\[exp1\]). A different $\rho$-dependence can still provide a similar weight of the partial waves after integration of the square of the wave function.
Three-body spatial structure
============================
Let us examine now the spatial structure of the $^{11}$Li and $^{11}$O-states. A clean indication of how the core and valence nucleons are distributed in space is reflected by the different two-body root-mean-square (rms) radii, which in turn permit us to obtain a clear picture of the most probable inter-particle distances. We therefore first discuss the radii or second radial moments, and afterwards the origin in the structures of the wave functions.
Radii
-----
Since the complex scaling method has been used in the calculations, the corresponding resonance three-body wave functions are complex rotated. As a consequence, for the resonant states, the rms radii have to be obtained as the expectation value of the square of complex rotated radial distance ($r\rightarrow
re^{i\theta}$). Therefore, the rms radii are in this case complex quantities, and as described in [@moi98], the imaginary part is a measure of the uncertainty of the computed value.
[|c|cc|cc|]{}\
$J^\pi$ & $\langle r_{nn}^2\rangle^{1/2}$ & $\langle r_{c,nn}^2\rangle^{1/2}$ & $\langle r_{cn}^2\rangle^{1/2}$ & $\langle r_{n,cn}^2\rangle^{1/2}$\
$\frac{3}{2}^-$ & 6.4 & 5.0 & 5.9 & 5.7\
$\frac{1}{2}^+$ &$22.3+i6.3$ & $11.7+i3.0$ & $16.5+i4.4$ & $17.1+i4.9$\
$\frac{3}{2}^+$ & $13.9+i4.8$ & $7.4+i3.4$ & $10.4+i4.1$ & $10.7+i4.1$\
$\frac{5}{2}^+$ & $9.4+i4.3$ & $3.1+i0.9$ & $6.8+i2.0$ & $7.0+i2.2$\
\
[|c|cc|cc|]{}\
$J^\pi$ & $\langle r_{pp}^2\rangle^{1/2}$ & $\langle r_{c,pp}^2\rangle^{1/2}$ & $\langle r_{cp}^2\rangle^{1/2}$ & $\langle r_{p,cp}^2\rangle^{1/2}$\
$\frac{3}{2}^-$ & $5.1+i 3.9$ & $2.8+i2.1$ & $3.9+i3.0$ & $3.9+i3.0$\
$\frac{1}{2}^+$ & $12.8+i5.0$ & $6.6+i3.0$ & $9.3+i4.0$ & $9.4+i4.0$\
$\frac{3}{2}^+$ & $12.2+i5.4$ & $6.5+i2.9$ & $9.0+i4.1$ & $9.0+i4.1$\
$\frac{5}{2}^+$ & $10.7+i6.2$ & $5.2+i3.1$ & $7.5+i4.5$ & $7.5+i4.6$\
In Table \[tabr2\] we give the rms radii, $\langle
r_{ij}^2\rangle^{1/2}$ and $\langle r_{k,ij}^2\rangle^{1/2}$, for the different $^{11}$Li (upper part) and $^{11}$O (lower part) states. From the right part of the table we notice that for all the states in both $^{11}$Li and $^{11}$O, the distances, $\langle
r_{cN}^2\rangle^{1/2}$ and $\langle r_{N,cN}^2\rangle^{1/2}$, are similar to each other, where $N$ can be either neutrons ($n$) or protons ($p$). Since the core is about nine times heavier than the nucleon, the value of $\langle r_{N,cN}^2\rangle^{1/2}$ is not far from the distance between the core and the second nucleon, which implies that the two valence-nucleons are roughly at the same distance from the core in all the $J^\pi$-states.
Looking now into the left part of Table \[tabr2\], we can see that for the bound $3/2^-$-state in $^{11}$Li the neutron-neutron distance is similar to the core-neutron distance, which indicates an equilateral triangular structure with a particle-particle distance of about 6 fm. However, for the resonant states in $^{11}$Li and $^{11}$O the situation is slightly different, since the nucleon-nucleon distance is roughly 1.4 times larger than the core-nucleon distance. This structure corresponds to an isosceles triangle, where the unequal side (the nucleon-nucleon distance) is about 40% bigger than the two equal sides given by the core-nucleon distance.
It is also interesting to note that the $3/2^-$-state for both nuclei is clearly smaller than the positive-parity resonances. This was already anticipated by the larger effect of the three-body force, the larger value of $\Delta_c^{(1)}$ (Table \[tab0\]), and the smaller $K_{max}$-values required to get convergence for the $3/2^-$-states. This is related to the facts, that in $^{11}$Li the $3/2^-$-state is bound, and in $^{11}$O the $3/2^-$ wave function has a clearly dominant contribution from two valence-protons in a relative $p$-wave with respect to the core (85% according to Table \[tabcom2\]). In this structure the potential barrier prevents the protons from moving too far away from the core (see the dashed curves in Fig. \[figpot\]b).
On the other hand, as mentioned when discussing Table \[tabcom2\], the $1/2^+$, $3/2^+$, and $5/2^+$-resonances are almost entirely $sp$-structures, which implies that one of the halo nucleons is always populating a core-nucleon $s$-state. As seen in Fig. \[figpot\]a, the $s$-wave potentials do not feel any confining barrier, except the $2^-$-potential in $^{10}$N (dashed black curve), for which the potential barrier is almost a factor of 2 lower than for the $p$-potentials for the same system. As a consequence, the positive-parity resonances are, as seen in Table \[tabr2\], significantly bigger than for the 3/2$^-$-states.
Also, as already mentioned and shown in Table \[tabcom2\], the contribution of the nucleon-core $1^-$-state ($\ell_x=0$, $j_x=1$) to the $J^+$-resonances is only substantial for the $1/2^+$-state, whereas for the $3/2^+$ and $5/2^+$-resonances basically all the $\ell_x=0$ contribution arises through the 2$^-$ state ($j_x=2$). In the case of $^{11}$Li, since $^{10}$Li shows a very low-lying $2^-$ virtual state, the $^{11}$Li-resonances with a large $2^-$-component ($3/2^+$ and $5/2^+$) will show a tendency to keep the neutron close to the core, leading therefore to a system smaller than the $1/2^+$-state, where the $1^-$-components dominates. This is actually seen in the upper part of Table \[tabr2\], where the rms radii for the $1/2^+$-state are significantly larger than those of the $3/2^+$ and $5/2^+$-resonances.
In the case of $^{11}$O, the $2^-$-state in $^{10}$N is not that low anymore, and it is actually rather broad (see Table \[tab1\]), being then close to disappear into the continuum. The confining effect of the $s$-wave $2^-$-resonance disappears, and the $1/2^+$, $3/2^+$, and $5/2^+$-resonances in $^{11}$O have a similar size, see Table \[tabr2\].
Structure
---------
![Real part of the structure function $F(r_{cn},r_{n,cn})$, as defined in Eq.(\[funF\]), in fm$^{-2}$, for the four computed states in $^{11}$Li. A complex scaling angle $\theta=0.30$ rads has been used.[]{data-label="fig0"}](3D.eps){width="\columnwidth"}
![The same as Fig. \[fig0\] for the $^{11}$O states.[]{data-label="fig4"}](3D-O.eps){width="\columnwidth"}
The origin of the average distance results discussed above can be visualized by means of the structure function $$\begin{aligned}
\lefteqn{ \hspace*{-7mm}
F(r_{cN},r_{N,cN})=} \nonumber \\ & &
r_{cN}^2 r_{N,cN}^2 \int \left(\Psi(\bm{r}_{cN},\bm{r}_{N,cN})\right)^2 d\Omega_{cN} d\Omega_{N,cN},
\label{funF}\end{aligned}$$ where $N$ represents either the neutron for $^{11}$Li or the proton for $^{11}$O, $\Psi$ is the complex rotated three-body wave function of a given $J^\pi$-state, and $\Omega_{cN}$ and $\Omega_{N,cN}$ are the angles defining the directions of $\bm{r}_{cN}$ and $\bm{r}_{N,cN}$, respectively. Note that following the normalization criteria described in Ref.[@moi98], the definition of the structure function above is made in terms of the square of the wave function, and not in terms of the square of the modulus of the wave function. In principle, the function $F$ depends on the complex scaling angle, but since $\Psi$ is normalized to 1 it is obvious that $F$ satisfies that $$\int F(r_{cN},r_{N,cN}) dr_{cN} dr_{N,cN}=1,
\label{funF1}$$ which implies that the integral of the imaginary part is equal to zero.
The real part of the structure function $F$ is shown in Figs. \[fig0\] and \[fig4\] for all the computed states in $^{11}$Li and $^{11}$O, respectively. For the resonances the complex scaling angle $\theta=0.3$ rads has been used.
For $^{11}$Li, the $3/2^-$ ground-state wave function is rather confined, with a high peak centered around average distances determined by $r_{cn}\approx r_{n,cn}\approx 6$ fm, as expected from the rms radii shown in Table \[tabr2\] for this state. When looking at the 1/2$^+$, $3/2^+$, and $5/2^+$-resonances, we can see that the wave function is progressively developing a tail along the $r_{cn}$- and $r_{n,cn}$-axis, which in turn can be related to the lack of barrier in the $s$-wave potential. The main difference in the structure function for these three states, is the presence of a peak at relatively small core-neutron distances for the $3/2^+$ and $5/2^+$-states. As explained above this is attributed to the important contribution of the $2^-$-states, which present a very low-lying virtual state. In the $1/2^+$-state this virtual state does not contribute, the peak then disappears, and the wave function shows mainly two wide peaks each located along the two axes.
For $^{11}$O, we see in Fig. \[fig4\] that the $3/2^-$-resonance shows a structure function with a peak apparently similar to the one of the $3/2^+$ and $5/2^+$-states, although in this case the peak is essentially only of $p$-wave character. For the $1/2^+$, $3/2^+$, and $5/2^+$-resonances, the $s$-wave contributes significantly, but due the Coulomb repulsion, which pushes up the $2^-$-resonance in $^{10}$N, the $s$-wave potential is not able to keep the $s$-wave proton sufficiently close to the core, and the peak observed for the $3/2^+$ and $5/2^+$-states in $^{11}$Li disappears.
Summary and conclusions
=======================
We have calculated three-body properties of the four lowest excited bound states or resonances for the two light mirror nuclei, $^{11}$O and $^{11}$Li. The phenomenological interactions are chosen to reproduce all known properties of $^{11}$Li combined with consistent information about the subsystem, $^{10}$Li. The only difference in interactions is that the Coulomb potentials are added in $^{11}$O from the charges of the two protons and the $^{9}$C-core. We use the established hyperspherical adiabatic expansion method combined with complex rotation to separate the resonances from the background continuum structure.
The nuclei, $^{11}$O and $^{11}$Li, are special by being non-identical mirrors on the neutron and proton driplines, that is at opposite sides of the beta-stability curve. The effect of the Coulomb interactions is rather small for most nuclei, except for a substantial translation of the absolute energies. However, these smaller effects sometimes carry signals about features of interest in a better understanding of many-body nuclear structure. In general the importance lies in change of structure, which requires theoretical models beyond the mean-field. A prominent example is the Thomas-Ehrman shift, but in general nuclei at the driplines are expected to maximize such structure variation.
In this report, we predict the properties of $^{11}$O and $^{10}$N from knowledge of $^{11}$Li and its two-body subsystem, $^{10}$Li. Other investigations are available, but to our knowledge none consider systematically all four lowest-lying excited states/resonances, and their relation to the properties of the nucleon-core subsystems. Furthermore, we use phenomenological interactions, which should enhance the reliability of our predictions. We emphasize that the interactions in the present work are able to reproduce all known features of $^{10}$Li and $^{11}$Li.
We first investigate the two-body nucleon-core subsystems, proton-$^{9}$C and neutron-$^{9}$Li. In realistic calculations the spin-spin splitting is essential, that is coupling of the $3/2^-$-core and the $1/2^{\pm}$-proton angular momenta and parities. The sequence of the resulting states of $1^-$ and $2^-$ is not experimentally determined. Fortunately, the only two crucial properties are, first the degeneracy weighted centroid energy, and second that one of these states is unbound with a marginally negative virtual energy.
The two-body potentials turn out to have attractive pockets at short range for $2^-$, while overall repulsive for the $1^-$-state. Both receive additional repulsion from the Coulomb potentials in the $^{10}$N-nucleus. By construction, the $2^-$-potential for $^{10}$Li has a marginally unbound virtual state. Both the $2^+$ and $1^+$-potentials have attractive short-range pockets for both $^{10}$Li and $^{10}$N. The resonance energies of course follow the pattern of the potentials with less than $0.6$ MeV for all $^{10}$Li states and about $2$ MeV higher energies for $^{10}$N.
The computed three-body energy of $^{11}$Li is fine-tuned to reproduce precisely the measured ground state value, while the three positive parity excited states of both positions and widths are experimentally unknown, but predicted to be very similar. For $^{11}$O, we find in contrast to $^{11}$Li that the $3/2^-$-state is about $1$ MeV higher than the three positive-parity states. The two lowest resonances, 1/2$^+$ and $3/2^+$, are very similar, and it is therefore as likely that one of these is the ground state. This would be a qualitative difference between important properties of these mirror nuclei. This predicted energy sequence in $^{11}$O is consistent with a perturbation estimate of the Coulomb shift. However, it is important to keep in mind that the uncertainty introduced by the unknown three-body interaction, which is seen to play a more relevant role in the 3/2$^-$ state than in the positive-parity states, could modify the ordering in the energy spectrum.
The structure of the wave functions is in principle revealed by the partial wave decomposition. It is striking that the positive parity resonances all are of very similar partial wave content in $^{11}$Li and $^{11}$O. In contrast, the $3/2^-$-state in $^{11}$O is almost entirely made of $p$-waves of both nucleon-core two-body states, whereas $p$-waves in $^{11}$Li only contribute about $40\%$ and $s$-waves correspondingly by $57\%$.
The spatial distributions of the nucleons surrounding the core also differ substantially in the two nuclei. For the $3/2^-$-state, the two nucleons are symmetrically distributed in one peak in both cases, but about $30\%$ closer to the core and more smeared out in the $^{11}$O-resonance than in the $^{11}$Li bound-state. The positive parity resonances in $^{11}$O all three exhibit two peaks in their density distributions corresponding to one proton close and one further away from the core. In $^{11}$Li, these two peaks coincide for the $3/2^+$ and $5/2^+$-resonances, whereas they remain for $1/2^+$, but with much larger tails. This fact shows that, even if the partial wave content is similar (as shown in Table \[tabcom2\] for the 3/2$^+$ and 5/2$^+$ resonances), the spatial distribution of the constituents can be different.
In conclusion, all these detailed predictions are beyond present laboratory tests, but still displaying essential properties, which in turn should inspire to new experimental investigations. The substantial differences between the two mirror nuclei are all due to the additional Coulomb interaction. This is a new experience in nuclear physics, where the Coulomb interaction generally is believed to influence nuclear structure only marginally. In summary, we have learned that mirror nuclei not necessarily have very similar structure. Furthermore, we have seen that dripline nuclei still can deliver new information about nuclear structure.
We want to thank H.O.U. Fynbo and K. Riisager for drawing attention to these systems and subsequent continuous discussions. We also thank J. Casal for constructive comments and discussions. This work has been partially supported by the Spanish Ministry of Science, Innovation and University MCIU/AEI/FEDER,UE (Spain) under Contract No. FIS2018-093636-B-I00.
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|
---
author:
- |
P.-H. Heenen,$^{*1}$ P. Bonche,$^{*2}$, S. Ćwiok$^{*3}$, W. Nazarewicz$^{*4-*6}$ and A. Valor$^{*1}$\
\
1 Physique Nucléaire Théorique, University of Brussels, Belgium\
2 SPhT, CEA, Saclay, France.\
3 Institute of Physics, Warsaw University of Technology, Poland\
4 Department of Physics, University of Tennessee, Knoxville, USA\
5 Physics Division, Oak Ridge National Laboratory, USA\
6 Institute of Theoretical Physics, University of Warsaw, Poland
title: 'Skyrme mean-field studies of nuclei far from the stability line'
---
-2.5pc -2.5pc -3pc .8pc
Introduction {#introduction .unnumbered}
============
Mean-field calculations on a 3-dimensional mesh were introduced in nuclear physics in the late 1970s in the studies of nuclear collisions within the time-dependent Hartree-Fock method$^{1)}$. In these early applications, schematic nuclear interactions were used, without spin-orbit and Coulomb terms, and applications were limited to collisions between light nuclei. The same technique has later been applied in a systematic study of the spectrocopic properties of Zr and neighboring isotopes$^{2)}$. New developments have then permitted the use of effective interactions of the Skyrme type, including Coulomb and spin-orbit terms, and pairing correlations were considered the BCS approximation with a seniority force. There are several advantages of the coordinate-space technique which make it very attractive. It allows one to describe very general intrinsic deformation with a good numerical accuracy. It also provides a very flexible computation scheme, suitable for generalizations (e.g., introduction of correlations beyond the mean-field theory). The main developments of the method performed after the completion of the Zr study were the introduction of a cranking constraint to describe rotating nuclei$^{3)}$, the introduction of the configuration mixing using the generator coordinate method$^{4)}$, the particle-number projected HF+BCS calculations$^{5)}$, and improvements in the treatment of the pairing correlations by the use of the HFB method$^{6)}$ with density-dependent zero-range interactions$^{7)}$. Let us also mention a theoretical study of the properties of coordinate-space calculations$^{8)}$ which introduced powerful ways to interpolate wave functions discretized on a 3D mesh.
In this note, we shall first present a very representative application of the method to odd-N superheavy elements$^{9)}$ (SHE). We shall then describe new developments$^{10)}$ related to the restoration of the rotational and particle number symmetries; these improvements are expected to be extremely useful in future studies of medium-mass nuclei far from stability.
Structure of Odd-$N$ Superheavy Elements {#structure-of-odd-n-superheavy-elements .unnumbered}
========================================
There is no consensus among theorists with regard to the center of the shell stability in the region of spherical SHE. For the neutrons, most calculations predict a magic gap at $N$=184. However, because of different treatments of the large Coulomb potential and spin-orbit interaction, various models yield different predictions for the position of the magic proton gap (cf. discussion in Ref. $^{11)}$). In this context, the synthesis of the $N$=175 isotones reported by the Dubna/Livermore (DL) collaboration ($Z$=114) $^{12)}$ and the Berkeley/Oregon (BO) team ($Z$=118) $^{13)}$ is particularly important. Due to the large neutron number of the compound nucleus, the observed $\alpha$ chains do not terminate at some known systems. Consequently, until the charge-mass identification is made, one can use theoretical arguments to support/disprove the experimental assignment.
We have performed a large-scale self-consistent study of properties of even-even, $A$-odd, and odd-odd heaviest nuclei. In our work, we have used the Hartree-Fock-Bogoliubov (HFB) method with a Skyrme interaction in the particle-hole (ph) channel and a delta force in the pairing channel. The details of the calculations closely follow Ref. $^{11)}$. The HFB equations have been solved in the coordinate space according to the method of Ref. $^{7)}$. In the ph channel, the Skyrme effective interaction SLy4 has been used. This parameterization$^{14)}$ has, in particular, been adjusted to reproduce long isotopic sequences; hence, one can expect it to have good isospin properties. The pairing strengths of the interaction have then been adjusted to reproduce the average proton and neutron pairing gaps in even-even nuclei around $^{254}$Fm. The properties of one-quasiparticle states were calculated by means of the self-consistent blocking.
Nucleus Orbital Energy (MeV) $\beta_2$
------------- ------------------------- -------------- -----------
$^{289}$114 \[707\]$\frac{15}{2}^-$ 0 0.12
(9.64) \[611\]$\frac{1}{2}^+$ 0.52 0.11
\[604\]$\frac{7}{2}^+$ 0.79 0.13
$^{285}$112 \[611\]$\frac{1}{2}^+$ 0 0.14
(8.88) \[611\]$\frac{3}{2}^+$ 0.60 0.14
\[707\]$\frac{15}{2}^-$ 0.62 0.13
\[606\]$\frac{11}{2}^+$ 0.65 0.15
\[604\]$\frac{9}{2}^+$ 0.72 0.15
$^{281}$110 \[604\]$\frac{9}{2}^+$ 0 0.19
(9.32) \[606\]$\frac{11}{2}^+$ 0.07 0.19
\[611\]$\frac{1}{2}^+$ 0.12 0.18
\[611\]$\frac{3}{2}^+$ 0.59 0.17
\[613\]$\frac{5}{2}^+$ 0.65 0.17
$^{277}$108 \[611\]$\frac{1}{2}^+$ 0 0.21
\[604\]$\frac{9}{2}^+$ 0.04 0.20
\[613\]$\frac{5}{2}^+$ 0.31 0.21
\[716\]$\frac{13}{2}^-$ 0.36 0.21
: Predicted structure of the lowest one-quasiparticle excitations in the $^{289}$114 $\alpha$-decay chain. Each excitation is characterized by the Nilsson quantum numbers of the odd neutron, excitation energy, and quadrupole deformation. The number in parentheses indicates the $Q_\alpha$ value for the ground-state to ground-state transition.
Nucleus Orbital Energy (MeV) $\beta_2$
------------- ------------------------- -------------- -----------
$^{293}$118 \[707\]$\frac{15}{2}^-$ 0 0.11
(11.59) \[611\]$\frac{1}{2}^+$ 0.16 0.10
\[602\]$\frac{5}{2}^+$ 0.84 0.12
\[604\]$\frac{7}{2}^+$ 0.98 0.10
$^{289}$116 \[611\]$\frac{1}{2}^+$ 0 0.13
(10.18) \[606\]$\frac{11}{2}^+$ 0.62 0.14
\[611\]$\frac{3}{2}^+$ 0.66 0.13
\[604\]$\frac{9}{2}^+$ 0.69 0.14
\[707\]$\frac{15}{2}^-$ 0.72 0.13
$^{285}$114 \[606\]$\frac{11}{2}^+$ 0 0.16
(10.60) \[611\]$\frac{1}{2}^+$ 0.04 0.16
\[611\]$\frac{3}{2}^+$ 0.15 0.16
\[604\]$\frac{9}{2}^+$ 0.16 0.16
$^{281}$112 \[611\]$\frac{1}{2}^+$ 0 0.19
(10.85) \[604\]$\frac{9}{2}^+$ 0.07 0.19
\[611\]$\frac{3}{2}^+$ 0.37 0.19
\[613\]$\frac{5}{2}^+$ 0.41 0.19
: Same as in Table 1, except for the $^{293}$118 $\alpha$-decay chain.
The calculated one-quasiparticle structures and the g.s.-to-g.s. values of $Q_\alpha$ in the $^{289}114$ and $^{293}118$ $\alpha$-decay chains are shown in Tables 1 and 2, respectively.
According to our calculations, the g.s-to-g.s. $\alpha$ decays of $^{289}114$ and $^{293}118$ are structurally forbidden since the g.s. properties of parent and daughter nuclei differ dramatically. The allowed transitions to the excited \[707\]$\frac{15}{2}^-$ level in $^{285}112$ and $^{289}116$ correspond to $Q_\alpha$=9.0 MeV and $Q_\alpha$=10.9 MeV, respectively, and are considerably lower than the experimental values (DL: 9.9 MeV; BO: 12.6 MeV) Consequently, the most probable candidates for the first $\alpha$ transitions are the 10.2 MeV ($^{289}114$) and 11.8 MeV ($^{293}118$) lines associated with the allowed \[611\]$\frac{1}{2}^+$$\rightarrow$\[611\]$\frac{1}{2}^+$ decays. By the same token, the \[611\]$\frac{1}{2}^+$ g.s. of $N$=173 isotones is expected to decay to the excited \[611\]$\frac{1}{2}^+$ level in the $N$=171 daughters. For $^{285}112$, the corresponding $Q_\alpha$ energy, 8.76 MeV, is very close to the experimental energy of the second $\alpha$ transition, 8.84 MeV, reported by the DL group. For the $^{289}116$ decay we obtain $Q_\alpha$=10.14 MeV; i.e., we underestimate the energy of the second $\alpha$ particle (11.8MeV) from the BO experiment. This is the worst deviation from experiment obtained in our calculations.
The \[611\]$\frac{1}{2}^+$ level is also expected to be the ground state of the $N$=169 isotones, and our prediction for the allowed \[611\]$\frac{1}{2}^+$$\rightarrow$\[611\]$\frac{1}{2}^+$ decays is 9.4 MeV for $^{281}110$ (DL: 9.0MeV) and 10.6 MeV for $^{285}114$ (BO: 11.5 MeV).
For the $\alpha$ transitions in the $^{281}112$$\rightarrow$$\cdots$$\rightarrow$$^{265}104$ chain, allowing the positive-parity Nilsson orbitals with very similar quantum numbers, we obtain the following values of $Q_\alpha$: 10.7 MeV (BO: 10.8 MeV), 11.1 MeV (BO: 10.3 MeV), 9.9 MeV (BO: 9.9 MeV), 8.0-8.5 MeV (BO: 8.9 MeV). An alternative route is possible that involves $\alpha$ transitions between $j_{15/2}$ orbitals. Here, for the last two $Q_\alpha$ values we obtain 9.8 MeV and 8.7 MeV. In both cases we obtain good agreement with BO data.
The calculated equilibrium deformations of one-quasiparticle states in $N$=175 isotones are rather small ($\beta_2$$\approx$0.11), and they increase along the $\alpha$-decay chain. This trend reflects the influence of the $N$=184 magic neutron gap for the heaviest systems and the deformed $N$=162 gap for the lightest nuclei in the chain. As far as $Q_\alpha$ values are concerned, our calculations are consistent with the recent experimental findings.
Correlations beyond mean field in Mg isotopes {#correlations-beyond-mean-field-in-mg-isotopes .unnumbered}
=============================================
The cranking method is widely used in nuclear spectroscopy to describe high-spin states. Applications based on effective nuclear interactions have been particularly successful in the description of superdeformed rotational bands in several regions of the mass table. In the cranking method, a rotational band is generated by the rotation of a deformed intrinsic state. Since cranking states are not eigenstates of the angular momentum, this causes some problems in determining, e.g., transition rates in nuclei which are not very well deformed.
Another limitation of the cranking model appears in nuclei which are soft with respect to the variation of a collective variable. In this case, one expects the interference of the zero-point vibrational mode with the rotational motion which leads to variations in the nuclear structure along the yrast line.
In ref $^{5)}$, we have presented a method to restore the particle number symmetry within the HF+BCS theory that allows us to perform a configuration mixing of projected wave functions in the direction of selected collective variables. The method presented in this section generalizes it by the inclusion of a restoration of the rotational symmetry. It enables us to describe details of collective spectra and transition rates.
The starting point of the method are wave functions $ |\Phi{_\alpha}\rangle$ generated by mean-field calculations with a constraint on a collective coordinate ${\alpha}$. Wave functions with good angular momentum and particle numbers, $$\label{first}
|\Phi,JM\alpha\rangle = \frac{1}{N_{x}}\sum_{K}g_K
{\hat{P}^{J}_{MK}\hat{P}^{Z}\hat{P}^{N} |\Phi_{\alpha}\rangle},$$ are obtained by means of the projection operators $\hat{P}$. In Eq. (\[first\]), $N_x$ is a normalization factor, depending on $x=J,M,N,Z,\alpha$.
Using the projected state $|\Phi,JM\alpha\rangle$ as a generating function, configuration mixing along the collective variable $\alpha$ is performed for each angular momentum: $$|\Psi,JM\rangle =\sum_{\alpha}f_{\alpha}^{JM}|
\Phi,JM\alpha\rangle.$$ The weight functions $f_{\alpha}^{JM}$ are found by requiring that the expectation value of the energy, $$\label{E10a}
E^{JM}={\langle{\Psi, JM}\vert\hat H\vert{\Psi, JM}\rangle
\over\langle{\Psi, JM}\vert{\Psi, JM}\rangle},$$ is stationary with respect to an arbitrary variation $\delta f_{\alpha}^{JM}$. This prescription leads to the discretized Hill-Wheeler equation $$\label{E11}
\sum_{\alpha}({\mathcal H}^{JM}_{\alpha,\alpha'}
-E^{JM}_k{\mathcal I}^{JM}_{\alpha,\alpha'})
f^{JM,k}_{\alpha'}=0,$$ in which the Hamiltonian kernel ${\mathcal H}^{JM}$ and the overlap kernel ${\mathcal I}^{JM}$ are defined as $$\label{E12}
{\mathcal H}^{JM}_{\alpha,\alpha'}=
\langle{\Phi JM\alpha}\vert\hat H\vert{\Phi JM\alpha'}\rangle\quad,
{\mathcal I}^{JM}_{\alpha,\alpha'}=
\langle{\Phi JM\alpha}\vert{\Phi JM\alpha'}\rangle.$$ The kernels (\[E12\]) are obtained by integrating the matrix elements between rotated wave functions over three Euler angles and two gauge angles. Besides these kernels, one can determine transition probabilities between different eigenstates of the Hill-Wheeler equation. This requires the calculation of matrix elements of electromagnetic operators.
Currently, in order to save computing time and to test the method, certain symmetry restrictions have been imposed on the mean-field wave functions. Namely, they have been assumed to be axially symmetric and time-reversal invariant. In this way, the integration over the Euler angles is limited to a single angle. Pairing correlations are treated in the BCS approximation.
In the following examples, a Skyrme force is used in the particle-hole channel and a density-dependent zero-range interaction in the pairing channel. As in ref$^{5)}$, in the calculation of non-diagonal matrix elements, the density dependence of the Skyrme interaction is generalized to a dependence on the mixed density.
[*Application to $^{24}$Mg*]{}
The results shown in this section have been obtained using the HF+BCS wave functions generated with an axial quadrupole constraint. The Lipkin-Nogami prescription has been used to improve the treatment of pairing correlations. The variation of the energy as a function of prolate and oblate deformations is plotted in Fig. 1 for the Sly4 Skyrme interaction and a surface pairing interaction having strength $G$=1000MeV fm$^3$ for both neutrons and protons. In addition to the prolate absolute minimum, the mean-field curve presents a shoulder at an oblate deformation around 50fm$^2$. The triple projection creates an oblate minimum at the position of the shoulder for $J$ up to 6$^+$. For greater values of $J$, the weights of the intrinsic wave functions for deformations below –200fm$^2$ are very small. Consequently, the projected energy curves do not exhibit any oblate mimima.
For each value of the angular momentum, we have performed a configuration-mixing calculation including quadrupole moments between –350fm$^2$ and 450fm$^2$. This corresponds to intrinsic configurations excited by about 30MeV with respect to the prolate minimum. The spectrum that is generated in this way (represented by bars) is plotted at the quadrupole moment corresponding to the largest component of the collective wave function. The value of this quadrupole moment is very close to the minimum of the projected energy curve. Moreover, the energy of this minimum is only slightly modified by the configuration mixing. The largest gain, $\sim$800keV, is obtained for the 0$^+$ state, but it is reduced at higher spins. Several excited states are found at low energy for each spin value. Except for the second 0$^+$ and 10$^+$, the wave functions of yrare states are peaked around the oblate secondary minimum.
In the same figure, theoretical transition probabilities along the yrast line are compared with the data. In the GCM calculation, the minima of projected energy curves have been used. The $B(E2;2_1^+\rightarrow 0_1^+)$ rate calculated between the minima of the two curves is very close to the experimental value. The configuration mixing causes a spreading of the collective wave function on the quadrupole moment and decreases the value of the $B(E2)$ rate. Since for spins greater than zero wave functions do not have low quadrupole-moment components, the configuration mixing does not affect significantly the transition probabilities. Here, the agreement between calculations and experiment is excellent. One has to note, however, that the inclusion of triaxiality is expected to modify these predictions to some extent.
[*Application to $^{32}$Mg*]{}
Altough $^{32}$Mg corresponds to the $N$=20 shell closure, there is some experimental evidence that it is deformed in its ground state. The excitation energy of its first 2$^+$ state is only around 900keV, i.e., significantly lower than in lighter Mg isotopes. Also the $B(E2;2_1^+\rightarrow 0_1^+)$ value is unusually large.
Figure 2 shows the calculated energy curves for $^{32}$Mg. The ground state calculated in the HF+BCS approximation is spherical (see discussion in ref.$^{15}$). However, unlike in the $^{24}$Mg case, the restoration of broken symmetries modifies the topology of the energy curve in a more dramatic way. Namely, the resulting 0$^+$ ground state corresponds to the projection of a deformed intrinsic state with a quadrupole moment of 100fm$^2$. The minima for the higher-$J$ values can be associated with even larger deformations. The configuration mixing leads to small energy gains. The collective 0$^+$ wave function is spread over a large range of quadrupole moments between –200fm$^2$ and 200fm$^2$, while for other $J$-values the weights of the HF+BCS wave functions have well-pronounced maxima around 200fm$^2$.
The calculated spectrum and transition probabilities are compared to the experimental data in Fig. 2. It is seen that our calculations do not reproduce the pronounced collectivity of $^{32}$Mg. Compared to the data, the energy of the 2$^+$ is too high and the $B(E2)$ value is too low. It should be noted, however, that the delicate balance between the spherical minimum and the deformed intruder configuration in this nucleus is sensitive to the effective interaction used$^{15)}$.
This research was supported in part by the U.S. Department of Energy under Contract Nos. DE-FG02-96ER40963 (University of Tennessee), DE-FG05-87ER40361 (Joint Institute for Heavy Ion Research), DE-AC05-96OR22464 with Lockheed Martin Energy Research Corp. (Oak Ridge National Laboratory), the Polish Committee for Scientific Research (KBN) under Contract No. 2 P03B 040 14, the NATO grant CRG 970196, the Rector Grant of Warsaw University of Technology, and by the PAI-P3-043 of the Belgian Office for Scientific Policy.
References {#references .unnumbered}
==========
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abstract: 'The fragmentation of excited hypernuclear system formed in heavy ion collisions has been described by the canonical thermodynamical model extended to three component systems. The multiplicity distribution of the fragments has been analyzed in detail and it has been observed that the hyperons have the tendency to get attached to the heavier fragments. Another important observation is the phase coexistence of the hyperons, a phenomenon which is linked to liquid gas phase transition in strange matter.'
author:
- 'S. Mallik and G. Chaudhuri'
title: Liquid gas phase transition in hypernuclei
---
The physics of hypernuclei is an important area of study in the regime of high energy heavy ion collisions. It has been observed that baryons and mesons(strange hadrons) are produced abundantly in high energy heavy ion reactions. Hypernuclei are formed when the strange hyperons or baryons are captured by the nuclei. The $\Lambda$-nucleon interactions are well studied and the potential depth of $\Lambda$ hyperons is such that bound $\Lambda$ hypernuclear states exist. The only bound $\Sigma$ hypernucleus known so far is which is bound by isospin forces [@Millener]. Several $\Xi$ hypernuclear states are reported in the literature and hence its interaction with nucleon seems to be attractive. On the other hand the hyperon-hyperon interaction is not really well known; a few double $\Lambda$ hypernuclear states have been reported. The interaction between other pairs of hyperons as $\Lambda$ $\Xi$ or $\Xi$ $\Xi$ is not known experimentally [@Bielich1]. The formation of multi-strange nuclei is especially important in order to study the properties of strange matter. Deep understanding of strange matter is extremely important for the formulation of models of strong interaction [@Papaz]. Another important application of this study is the core of neutron stars [@Bielich] where the hyperons are expected to be produced in abundance at high density nuclear matter. The stability of hypernuclei beyond the neutron and proton driplines(normal nuclear chart) is also a fascinating subject which is important in recent day activities [@Botvina1; @Botvina2; @Botvina3]. The knowledge of structure of normal nuclei [@Hashimoto] as well as the extension of the nuclear chart into the strangeness sector [@Schaffner; @Greiner; @Samanta] gets valuable input from the results of hypernuclei study. Another important area in the study of intermediate energy heavy ion collisions is the phenomenon of phase coexistence or liquid gas phase transition [@Siemens; @Pochodzalla; @Dasgupta_Phase_transition]. The appearance of ’liquid-like’ as well as ’gas-like’ fragments simultaneously over a temperature interval is linked to first order phase transition. Whether this phase-coexistence will still persist in the presence of hyper-fragments(strange fragments) is the object of investigation in this work.\
The canonical thermodynamical model has already been extended to three component systems [@Dasgupta_hyperon1] i.e, inclusion of hyperons(usually $\Lambda$ in addition to the neutrons and the protons). Due to fragmentation of the PLF, normal (non-strange) components as well as hypernuclei will be formed. In previous works [@Dasgupta_hyperon1; @Dasgupta_hyperon2], the total number of strange particles was confined to 2. In this work we include the possibility of existence of multiple (more than 2) strange particles.\
In a recent paper, [@Dasgupta_hyperon2] a hybrid model based on participant-spectator picture combined with the Canonical Thermodynamical Model(CTM) model has been used to determine the production cross section of a hypernucleus in high energy heavy-ion collisions,. For heavy ion collisions in 3-10 GeV range, the following scenario (backed by experiments) is applicable. For a general impact parameter, there is a region of violent collision called the participating region. In addition there is a mildly excited projectile like fragment(PLF) and also a mildly excited target like fragment(TLF). Physics of both PLF and TLF are similar for symmetric collisions; here we concentrate our analysis on PLF. Because of excitation energy (usually characterized by a temperature, T) PLF will break up into many fragments [@Mallik2; @Mallik3; @Mallik9] and the velocities of the fragments are centered around the velocity of the projectile. In fixed target experiments they are emitted in a forward cone and are more easily recognizable. In the participating region, apart from original neutrons and protons, particles (pions, $\Lambda$’s, etc.) are produced. The produced $\Lambda$’s have an extended rapidity range. Those produced in the rapidity range close to that of the projectile and having total momenta in the PLF frame up to the Fermi momenta can be trapped in the PLF and form hypernuclei [@Wakai; @Saito; @Gaitanos]. At higher energies multiple hyperons can get attached to the PLF. In this work we consider a maximum number of eight hyperons being attached to the PLF. The fragmentation of the PLF into different composites(strange and non-strange) is calculated using the three component CTM [@Dasgupta_hyperon1; @Dasgupta_hyperon2].\
The main motivation of this work is to analyze the composition of the fragments produced from fragmentation of PLF which initially has multiple hyperons attached to it. The important feature which emerges from the results is that hyperons have greater affinity of getting attached to the higher mass fragments. The most striking feature of the distribution of the hyperons is the phase coexistence, a feature which has already been observed in the case of normal(non-strange) fragments [@Siemens; @Pochodzalla; @Dasgupta_Phase_transition]. The typical ’U’ shaped distribution observed in the fragmentation of non-strange(normal) nuclei is also exhibited by the fragmentation of the strange nuclei irrespective of the amount of strangeness content. One can infer that phase transition which is a characteristic feature of fragmentation of normal nuclei also persists in the case of hyperfragments.\
[**[*Theoretical formalism:-*]{}**]{} The Canonical Thermodynamical Model (CTM) for two kinds of particles (neutron and proton) is well-known and has had long usage [@Das1]. This has been extended to three kinds of particles (neutron, proton and $\Lambda$) few years back [@Dasgupta_hyperon1; @Dasgupta_hyperon2]. In this section, the 3-component Canonical Thermodynamical Model is discussed briefly. Assuming that a system with $A_0$ baryons, $Z_0$ protons and $H_0$ hyperons at temperature $T$, has expanded to a higher than normal volume, the partitioning into different composites can be calculated according to the rules of equilibrium statistical mechanics. The canonical partition function is given by $$Q_{A_0,Z_0,H_0}=\sum\prod\frac{(\omega_{a,z,h})^{n_{a,z,h}}}{n_{a,z,h}!}$$ Here the product is over all fragments of one break up channel and sum is over all possible channels of break-up (the number of such channel is enormous); $\omega_{a,z,h}$ is the partition function of one composite with $a$ baryons $z$ protons and $h$ hyperons whereas $n_{a,z,h}$ is the number of this composite in the given channel. The one-body partition function $\omega_{a,z,h}$ is a product of two parts: one arising from the translational motion and another is the intrinsic partition function of the composite: $$\omega_{a,z,h}=\frac{V}{h^3}(2\pi T)^{3/2}\{(a-h)m_n+hm_h\}^{3/2}\times z_{a,z,h}(int)$$ Here $m_n$ and $m_h$ are masses of nucleon (we use 938 MeV) and hyperon (we use 1116 MeV for $\Lambda$ hyperon) respectively. $V$ is the volume available for translational motion; $V$ will be less than $V_f$, the volume to which the system has expanded at break up. We use $V = V_f - V_0$ , where $V_0$ is the normal nuclear volume. Since hyperfragments are generally studied from PLF, hence we have considered $V_f = 3V_0$. The average number of composites with $a$ baryons, $z$ protons and $h$ hyperons can be written as $$<n_{a,z,h}>=\frac{\omega_{a,z,h}Q_{A_0-a,Z_0-z,H_0-h}}{Q_{A_0,Z_0,H_0}}$$ Each allowed break up channel in Eq. 1 must satisfy, total baryon, proton and hyperon conservation i.e. $$\begin{aligned}
\sum an_{a,z,h} &=& A_0 \nonumber \\
\sum zn_{a,z,h} &=& Z_0 \nonumber \\
\sum hn_{a,z,h} &=& H_0\end{aligned}$$ Substituting Eq.(3) in these three constraint conditions, three different recursion relations [@Chase] can be obtained. Any one recursion relation can be used for calculating $Q_{A_0,Z_0,H_0}$. For example $$Q_{A_0,Z_0,H_0}=\frac{1}{A_0}\sum_{a,z,h}a\omega_{a,z,h}Q_{A_0-a,Z_0-z,H_0-h}$$ Therefore calculation of any partition function using this recursion relation will require very short computational time and then substituting those in Eq. 3 one can calculate the average multiplicity $\langle n_{a,z,h}\rangle$ easily.\
To construct $z_{int}(a,z,h)$, experimental binding energies are used for low mass nuclei and hypernuclei, and for higher masses a liquid drop formula is used. The neutron, proton and $\Lambda$ particles are taken as the fundamental blocks therefore $z_{int}(1,0,0)$=$z_{int}(1,1,0)$=$z_{int}(1,0,1)$=1. For deuteron, triton, $^3$He and $^4$He we use $z_{a,z,0}(int)=(2s_{a,z,0}+1)\exp(-\beta e_{a,z,0}(gr))$ where $\beta=1/T, E_{I,J}(gr)$ is the ground state energy (taken from experimental data) and $(2s_{I,J}+1)$ is the experimental spin degeneracy of the ground state. For $1<a\le8$, the ground state binding energies and excited state energies are taken from experimental data [@Dasgupta_hyperon2]. For heavier nuclei and hypernuclei, liquid-drop formula is used for calculating ground state energy [@Botvina3]. This is given by $$\begin{aligned}
e_{a,z,h}(gr)=-16a+\sigma(T)a^{2/3}+0.72kz^2/(a^{1/3})\nonumber\\
+25(a-h-2z)^2/(a-h)
-10.68h+21.27h/(a^{1/3})\end{aligned}$$
where $\sigma(T)$ is the surface tension which is given by $\sigma(T)=\sigma_{0}\{(T_{c}^2-T^2)/(T_{c}^2+T^2)\}^{5/4}$ with $\sigma_{0}=18.0$ MeV and $T_{c}=18.0$ MeV and k is the correction factor in Coulomb energy which incorporates the effect of its long-range behavior by Wigner-Seitz approximation as in Ref. [@Bondorf1]. We include all nuclei within drip lines in constructing the partition function. Another useful parametrization in liquid drop formula for hypernuclei was proposed by Samanta et. al. [@Samanta]. A comparative study of these two formula in the case of hyperfragmentation was described in Ref. [@Botvina3] and finally the one used here was chosen because it produces results closer to the experimental data.\
The study of the liquid-drop model formula (which has been used in our model), reveals that by adding hyperons the stability of the fragments increase for mass numbers $a>8$. Hence this implies that the hyperon-nucleon interaction is attractive for this mass range. For $a\le8$, this liquid-drop formula is not suitable and so we have used experimental binding energies for these lower mass nuclei or hypernuclei. It is known that, $^4$H or $^5$He are not stable, but when one $\Lambda$ is added the corresponding nuclei ($^4_{\Lambda}$H or $^5_{\Lambda}$He) become stable. Hence this establishes the attractive nature of the hyperon-nucleon interaction. However from this work, it is difficult to comment about hyperon-hyperon interaction as here it is not possible to isolate it from the other two interactions (nucleon-nucleon and nucleon-hyperon).\
In addition to the liquid-drop formula we have also included the contribution to $z_{int}(a,z,h)$ due from the excited states. This gives a multiplicative factor $exp (r(T)Ta/\epsilon_0)$ where we have introduced a correction term $r(T)=\frac{12}{12+T}$ to the expression used in Ref. [@Bondorf1]. This slows down the increase of $z_{int}(a,z,h)$ due to excited states as $T$ increases.\
[**[*Results and Discussions:-*]{}**]{}We have computed the average number of normal and hyperfragments of different mass, charge and strangeness by the canonical thermodynamical model. The fragmenting hypernuclei is assumed to have mass number $A_0 = 128$, charge $Z_0= 50$ and total strangeness $H_0=8$. We have calculated the strangeness distribution of the hyperfragments in a very much similar way one calculates the charge or mass distribution of the fragments. Fig. 1 shows this distribution of hyperfragments ($\langle n_h\rangle=\sum_{a,z}\langle n_{a,z,h}\rangle$) at different temperatures (excitation energies). At lowest temperature $3$ MeV, the distribution resembles ’U’ shape. This nature is very much similar to what one obtains in the case of mass distribution of normal fragments at low temperature. This ’U’ shape of mass distribution for normal fragments and the lowering down of the height of the maxima on the higher mass side as temperature is increased is usually linked to first-order phase transition or phase coexsistence[@Das1; @Dasgupta_Phase_transition; @Chaudhuri1; @Chaudhuri2]. Similar feature also emerges in the case of strange fragments or hyperfragments. With similar reasoning as in the case of normal (non-strange) fragments, we can associate this phenomenon in hyperfragments (see Fig. 1) with phase coexistence or liquid-gas phase transition. There is existence of hyperfragments with small strangeness content
{width="5.2cm"}
{width="\columnwidth"}
{width="6.5cm"}
as well as large strangeness content at the same time. This $\langle n_h\rangle$ vs $h$ plot is similar to $\langle n_a\rangle$ vs $a$ plot [@Das1] at different temperatures. As we increase the temperature, the so called ’U’ shape gradually flattens and finally at higher temperature, it changes to monotonically decreasing pattern as is seen from the figure. This can be inferred as disappearing of one phase as the temperature is increased. Though the difference in strangeness content between the two phases is not very much in the present case, but still we can refer to this pattern as phase coexistence in the hyperfragments. This calculation is confined to a maximum number of $8$ hyperons but we believe that if it is extended to larger number of hyperons, the pattern will remain same and will confirm our inference from this figure.\
In order to further analyze the distribution of strangeness content in different fragments of varying mass, we have calculated their mass distribution with different $h$ values separately. Fig. 2 displays this multiplicity distribution at two different temperatures $3$ and $6$ MeV. First let us concentrate on the lower temperature , that is $3$ MeV. For $h=0$ that is for the normal fragments with no strangeness , the nature of the curve is monotonically decreasing which shows that cross-section of formation of heavier fragments with no strangeness content is extremely less. This can be interpreted by the fact that the hyperons tend to get attached to the heavier fragments at lower temperature and hence most of the heavier fragments are strange. This is confirmed by the other plots in the same figure which shows the mass distribution for fragments with different strangeness content, i.e, $h=2,4,6$ or $8$. More is the mass number of a fragment, greater is the probability of more hyperons getting attached to it. On the contrary, the strangeness content of lower mass fragments is comparatively less. The multiplicity of fragments with $h = 0$ or $h = 2$ is much more for lower values of $a$. For small strangeness content, the multiplicity decreases as one increases $a$. The right side of this figure shows the similar plot for a higher temperature $T=6$ MeV. As the temperature increases, fragments with higher mass decrease for obvious reasons. Lighter mass fragments are predominant at higher temperatures and they contain little or no strangeness.\
{width="6.5cm"}
The “rise and fall” nature of intermediate mass fragment (IMF) multiplicities is also an important signature of liquid gas phase transition for normal nuclei [@Dasgupta_Phase_transition; @Peaslee; @Ogilvie; @Tsang]. In this article, our aim is to investigate how IMF and HMF (heavier mass fragment) multiplicities change with temperature for hypernuclei with different strangeness content. Fig. 3 shows the variation of the average number of intermediate mass fragments \[$\langle n_{IMF}(h)\rangle=\sum_{z=3}^{20}\langle n_{a,z,h}\rangle$\] with temperature for different $h$ content. For $h = 0$, $\langle N_{IMF}\rangle$ increases with $T$. This implies that the multiplicity of ordinary intermediate mass fragments increase monotonically with temperature. For $h=1$ or for higher values of $h$, the multiplicity first increases, reaches a peak at a certain temperature and then decreases. Though the trend is similar for different $h$ values, the exact nature of the variation is different. The multiplicity at higher temperatures is more for fragments with lesser strangeness content, i.e, with lower values of $h$. Naturally, more strange is the fragment, less is the multiplicity of IMF which once again establishes the tendency of hyperons to get preferentially attached to the heavier mass fragments.\
Fig. 4 shows the variation of multiplicity of heavier mass fragments ($\langle n_{HMF}(h)\rangle=\sum_{z\rangle {20}}\langle n_{a,z,h}\rangle$)with temperature for different $h$ values. Since the heavier mass fragments are predominantly strange, hence they have maximum multiplicity for $h$ =8. This can be easily understood refereing to fig. 2(left panel).\
{width="5.2cm"}
Fig 5. shows the variation of $\langle a_h\rangle$($=\sum_{a,z}a\langle n_{a,z,h}\rangle/\sum_{a,z}\langle n_{a,z,h}\rangle$) with $h$ for two different temperatures. At the lower temperature $3$ MeV, the steep increase of $\langle a_h\rangle$ with $h$ signifies once again the tendency attachment of more number of hyperons to the heavier fragments. At lower excitation energy(temperature), formation of heavier fragments is dominant and that is being reflected in the plot. Average mass of ordinary fragments(with no strangeness) is much less as compared to the strange ones. But this feature drastically changes at higher temperature where the variation of $\langle a_h\rangle$ vs h is much flatter. This is mainly due to the fact that heavier mass fragments are dominant at lower temperature and their formation is far less probable as one increases temperature. The average value of $a_h$ for $h$ =8 is about $5$ times more in case of $3$ MeV than $7$ MeV.\
{width="5.2cm"}
Fig 6. shows the variation of $\langle n_a\rangle$ with mass number $a$(mass distribution) for fragmentation of nuclei with $H$=0 and $H$ = 8 at $T$ = 3 MeV. The important feature is that both the curves are very similar in nature. If we concentrate on the fragmentation of the ordinary nuclei with no strangeness, the mass distribution displays an ’U’ shaped variation which is expected at lower temperature (3 MeV). This shape gradually disappears as the temperature is increased. This feature indicates liquid gas phase transition or phase co-existence i.e. existence of ’liquid-like’(heavier) and gas-like’ (lighter) fragments. This phenomenon has been well studied for non strange fragments in both statistical [@Dasgupta_Phase_transition; @Das1; @Chaudhuri1; @Chaudhuri2] and dynamical [@Mallik10] models as well as in experimental observations [@Pochodzalla; @Borderie] and hence we will not elaborate here. Our main motivation is to investigate the fragmentation of a nucleus with considerable amount of strangeness $H$ = 8. It is quite amazing that the nature of mass distribution is similar and the two curves are pretty close to each other. This establishes the fact that the first order phase transition (co-existence) still persists in the presence of hyperfragments. This feature is independent of the strangeness content of the fragments.\
[**[*Summary and Conclusion:-*]{}**]{} The fragmentation of a nucleus with multiple hyperons attached to it has been studied with the motivation to analyze the fragmentation pattern. The results clearly point to the affinity of the hyperons getting preferentially attached to the higher mass (heavier) fragments. Another important feature which emerges from the mass distribution is coexistence of liquid-like and gas-like hyper-fragments in a certain temperature interval. This phase coexistence is indicative of first order phase transition occurring in the fragmentation of nuclei with multiple hyperons. Above the transition temperature, the heavier fragments disappear giving rise to lower mass fragments with less hyperons being attached to them. This establishes the occurrence of phase transition in hyper-fragments, a phenomenon which has already been observed in case of ordinary non strange fragments.\
[**[*Acknowledgement:-*]{}**]{} The authors are gratefully acknowlege important discussions with Dr. D. N. Basu of VECC.
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abstract: 'We extend the Eliashberg-Thurston theorem on approximations of taut oriented $C^2$-foliations of 3-manifolds by both positive and negative contact structures to a large class of taut oriented $C^0$-foliations. These $C^0$-foliations can therefore be approximated by weakly symplectically fillable, universally tight, contact structures. This allows applications of $C^2$-foliation theory to contact topology and Floer theory to be generalized and extended to constructions of $C^0$-foliations.'
address:
- 'Department of Mathematics, University of Georgia, Athens, GA 30602'
- 'Department of Mathematics, Washington University, St. Louis, MO 63130'
author:
- 'William H. Kazez'
- Rachel Roberts
title: 'Approximating $C^0$-foliations'
---
[^1]
Introduction
============
In [@ET], Eliashberg and Thurston introduce the notion of [*confoliation*]{} and prove that when $k\ge 2$, taut, transversely oriented $C^k$-foliations can be $C^k$-approximated by a pair of $C^k$ contact structures, one positive and one negative. It follows that any contact structure sufficiently close to the plane field of such a foliation is weakly symplectically fillable and universally tight. For the most part, they restrict attention to confoliations which are at least $C^1$. For their main theorem, they restrict attention to confoliations which are at least $C^2$. Their assumption that 2-plane fields and 1-forms are $C^1$ is necessary for it to be possible to take derivatives. Their assumption that 2-plane fields and 1-forms are $C^2$ is necessary for Sacksteder’s Theorem ([@S], see also Theorem 1.2.5 of [@ET]) to apply. A weakening of this $C^2$ assumption in the neighborhood of compact leaves is used by Kronheimer, Mrowka, Ozsváth, and Szabó in [@KMOS] to show that the methods of [@ET] apply to those foliations constructed by Gabai in [@G1; @G2; @G3] which are $C^{\infty}$ except along torus leaves. In this paper we show that many of the techniques of Eliashberg and Thurston extend to transversely oriented, taut, $C^0$-foliations satisfying a natural transitivity condition.
Eliashberg and Thurston’s proof that sufficiently smooth transversely oriented taut foliations of 3-manifolds can be perturbed to weakly symplectically fillable contact structures gives a direct connection between foliation theory and symplectic topology via contact topology. This connection has been most spectacularly exploited by Kronheimer and Mrowka [@KrMr] in their proof of the Property P Conjecture.
Since the contact structures produced by Eliashberg-Thurston are weakly symplectically fillable and universally tight, their theorem is important in contact topology. To apply their theorem, of course one must start with a taut oriented foliation. The issue that we seek to address is that most new constructions of foliations produce foliations that are not smooth enough to apply the Eliashberg-Thurston theorem.
Constructions of foliations which fail even to be $C^1$ can be found in [@DL; @DR; @G1; @g1; @g2; @g3; @G2; @G3; @Ga; @KRo; @KR2; @Li; @Li2; @LR; @R; @R1; @R2]. In each of these papers, foliations are constructed using branched surfaces. Foliations constructed using branched surfaces will have smooth leaves but will often vary only continuously in a transverse direction because of the role and nonsmooth nature of leaf blow-up, Operation 2.1.1 of [@Ga]. (To obtain the smooth foliations found in [@G1; @g1; @g2; @g3; @G2; @G3], Gabai avoids appealing to Operation 2.1.1 and instead takes advantage of the fact that the branched surfaces involved in his construction are finite depth.)
The main results of this paper are, roughly, that $C^0$-foliations satisfying a simple transitivity condition can be approximated by weakly symplectically fillable contact structures. Such foliations include most of those found in [@DL; @DR; @g1; @g2; @g3; @G3; @Ga; @KRo; @KR2; @LR; @R; @R1; @R2] and many of those found in [@G1; @G2; @Li; @Li2]. In proving this, we give new methods for both initiating and propagating contact structures when beginning with a foliation. Before giving a more precise indication of our results, we give a brief description of some of the main ideas in Eliashberg and Thurston’s proof.
Eliashberg and Thurston interpolate between the notions of foliation and contact structure by introducing confoliations. This structure restricts to a contact structure on an open set, but is equal to the original foliation elsewhere. Their first step is to create a contact zone in the manifold, that is, to approximate the foliation so that it is a contact structure on a non-empty set. One place they accomplish this is in a neighborhood of a curve with attracting holonomy. We obtain a related result, Theorem \[attracting\], for the larger class of curves for which the holonomy has a contracting interval. We also introduce the notion of $L$-bracketed foliation in Definition \[transtranv\] and show how a contact zone can be naturally introduced about a regular neighborhood of the corresponding link $L$. This has applications to spines of open book decompositions, and more generally to manifolds obtained by surgery or filling.
Their next step is to propagate the contact structure throughout the manifold using smooth foliation charts. See Lemma \[ETlemma\], and Corollary \[approximate\]. We use a similar strategy, but we rely on the local existence of smooth approximations to a given $C^0$-foliation.
To prove that the contact structure they produce is weakly symplectically fillable, they use the crucial structure of a smooth volume preserving flow $\Phi$ transverse to the taut foliation. Such a smooth flow is the starting point for our work, for they exist even for $C^0$-foliations.
There are several notions that describe the relationship between a foliation $\mathcal F$ and “nearby” contact structures $\xi$. If there is a continuous family of contact structures $\xi_t$ such that $\xi_t \to \mathcal F$, we say $\mathcal F$ can be [*perturbed*]{} or [*deformed*]{} to a contact structure. A weaker notion is the existence of a sequence, $\xi_n \to \mathcal F$, in which case $\mathcal F$ can be [*approximated*]{} by contact structures. Both notions of convergence can be refined by defining convergence using a $C^k$ norm on tangent planes for values of $k$ ranging from 0 to $\infty$. We use the flow $\Phi$ transverse to $\mathcal F$ to define a weaker, topological notion of approximation.
\[Phiapprox\] Given an oriented foliation $\mathcal F$ and a positively transverse flow $\Phi$, we say an oriented 2-plane field $\chi$, typically a contact structure or a confoliation, is a [*$\Phi$-approximation*]{} of $\mathcal F$ if $\chi$ is positively transverse to $\Phi$. We say that two oriented 2-plane fields are [*$\Phi$-close*]{} if both are positively transverse to $\Phi$.
One application of our work is to complete the proof of a theorem of [@HKM2]. Honda, Kazez, and Matić show that sufficiently large fractional Dehn twisting for an open book decomposition with connected binding implies that the canonically associated contact structure is weakly symplectically fillable. Their proof requires the existence of contact structures approximating $C^0$-foliations constructed by Roberts in [@R2], and thus needs a stronger version of the Eliashberg-Thurston theorem.
Another application of our work is to prove that 3-manifolds containing taut, oriented $C^0$-foliations satisfying our transitivity condition cannot be L-spaces, thus extending the result of Theorem 1.4 in [@OS]. (See also Theorem 2.1 in [@KMOS] and Theorem 41.4.1 in [@KM].)
Our result also strengthens a result of Baldwin and Etnyre [@BE]. They give a set of examples showing that when an open book decomposition has multiple binding components, no fixed lower bounds on fractional Dehn twisting can guarantee weak symplectic fillabilty. This can now be viewed as a non-existence theorem for taut, oriented $C^0$-foliations satisfying our transitivity condition.
We thank Larry Conlon, John Etnyre, and Ko Honda for many helpful conversations. We would also like to thank the referee for several helpful suggestions and corrections.
An overview
===========
The transition from taut foliations to tight contact structures involves two auxiliary structures, volume preserving flows and symplectic topology. We summarize the results we need from each field as follows.
(Theorem II.20, [@Sullivan]; see also Theorem A1, [@Hass]) Suppose $\mathcal F$ is a taut codimension-1 $C^0$-foliation of a smooth closed Riemannian 3-manifold $M$. Then there is a volume-preserving smooth flow $\Phi$ transverse to $\mathcal F$.
For clarity, we break the theorem found in [@ET] into two statements:
\[weaklysymplectic\] (Corollaries 3.2.2, 3.2.4 and 3.2.8, [@ET]; see also Theorem 41.3.2, [@KM]) Let $M$ be a smooth closed Riemannian 3-manifold with a volume preserving flow $\Phi$. Suppose there exist a smooth positive contact structure $\xi_+$ and a smooth negative contact structure $\xi_-$, both of which are transverse to $\Phi$. Then each of $\xi_{\pm}$ is weakly symplectically fillable and universally tight. Moreover, if $\xi$ is any smooth (positive or negative) contact structure transverse to $\Phi$, then $\xi$ is weakly symplectically fillable and universally tight.
[**Remark:**]{} The statement of Theorem \[weaklysymplectic\] is meant to emphasize two things. First, given a smooth positive (respectively, negative) contact structure transverse to a volume preserving flow, it is sufficient to produce a negative (respectively, positive) contact structure also transverse to the flow to conclude both are weakly symplectically fillable and universally tight. Next, once such $\xi_+$ and $\xi_-$ are shown to exist, any contact structure $\xi$ transverse to $\Phi$ is necessarily weakly symplectically fillable and universally tight.
\[weaklysymplectic2\] (Corollary 3.2.5, [@ET]; see also Theorem 41.3.2, [@KM]) Let $M$ be a smooth closed oriented 3-manifold which contains a taut, oriented $C^2$-foliation $\mathcal F$. There exist a smooth positive contact structure $\xi_+$ and a smooth negative contact structure $\xi_-$, both $C^0$-close to $\mathcal F$.
After giving background definitions and facts about foliations in § \[foliation basics\], we describe in §\[transitive flow boxes\] how flow boxes can be organized with an eye towards spreading an initial contact structure throughout the ambient manifold. Contact structures are propagated from one flow box to the next via a collection of local extension theorems described in §\[basicsection\]. This leads to an inductive construction of the desired contact structure in §\[theconstructionsection\]. Throughout, it is helpful to keep in mind the following
[**Guiding Principle:**]{} Constructions must be kept transverse to the flow. Moreover, when constructing a positive contact structure, the slope of the characteristic foliation of a partially constructed confoliation must be greater than or equal the slope of the intersection of the given foliation of $\mathcal F$ and vertical boundary of our flow boxes, with equality allowed only where $\mathcal F$ is smooth. When constructing a negative contact structure, the slope inequality is reversed.
To explain this principle more formally, suppose that a closed oriented 3-manifold $M$ is expressed as a union of smooth submanifolds $V$ and $W$, possibly with corners, with $\partial V=\partial W$. Suppose, moreover, that $W$ admits a codimension-1 foliation $\mathcal F_W$. These submanifolds will be chosen so that their common boundary decomposes into [*horizontal*]{} and [*vertical*]{} portions, that is, portions tangent and transverse, respectively, to $\mathcal F_W$. If a confoliation $\xi_V$ has been constructed on $V$ so that it is tangent to the horizontal portion of $\partial V$, transverse to the vertical portion, and contact on certain prescribed portions of $V$, then we call $V$ a [*contact zone*]{}.
The Guiding Principle is a statement that the confoliation $\xi_V$ on $V$ must [*dominate*]{} the foliation $\mathcal F_W$ along the vertical boundary (see Definition \[compatible\]) both for an initial choice of $V$, and also for subsequent choices as $V$ is expanded to all of $M$, and $W$ is shrunk correspondingly.
To expand a contact zone $V$ to the entire manifold $M$, we use the following structure.
A foliation $\mathcal F_W$ is [*$V$-transitive*]{} if every point in $W$ can be connected by a path in a leaf of $\mathcal F$ to a point of $V$.
We will see in Theorem \[main1\] that the following structure is very useful.
\[tridecomposition\] A closed 3-manifold $M$ admits a [*positive (respectively, negative) $(\xi_V,\mathcal F_W,\Phi)$ decomposition*]{} if $M$ can be decomposed as a union $$M=V\cup W,$$ where the horizontal portion of $\partial W$ is tangent to $\mathcal F_W$ and the vertical portion of $\partial V$ is tangent to $\Phi$, and
1. $\mathcal F_W$ is a $V$-transitive oriented foliation of $W$,
2. $\xi_V$ is a smooth contact structure defined on $V$ which positively (respectively, negatively) dominates $\mathcal F_W$,
3. for some choice of Riemannian metric, $M$ admits a volume preserving flow $\Phi$ transverse to both $\xi_V$ and $\mathcal F_W$.
Note that the existence of a $(\xi_V,\mathcal F_W,\Phi)$ decomposition does not require the existence of a codimension-1 foliation defined on all of $M$.
[**Theorem \[main1\].**]{} [*If $M$ admits a positive $(\xi_V,\mathcal F_W,\Phi)$ decomposition, then $M$ admits a smooth positive contact structure which agrees with $\xi_V$ on $V$ and is $\Phi$-close to $\mathcal F_W$ on $W$. The analogous result holds if $M$ admits a negative $(\xi_{V'},\mathcal F_{W'},\Phi)$ decomposition. If $M$ admits both a positive $(\xi_V,\mathcal F_W,\Phi)$ decomposition and a negative $(\xi_{V'},\mathcal F_{W'},\Phi)$ decomposition, then these contact structures are weakly symplectically fillable.*]{}
If a closed oriented 3-manifold admits both a positive $(\xi_V,\mathcal F_W,\Phi)$ and a negative $(\xi_{V'},\mathcal F_{W'},\Phi)$ decomposition, does it contain a taut oriented foliation transverse to $\Phi$?
Given a splitting $M=V\cup W$ and a flow $\Phi$, an oriented, codimension-1 foliation $\mathcal F$ is [*compatible*]{} with $(V, W, \Phi)$ if $\mathcal F$ is transverse to $\Phi$, and the common boundary $\partial V=\partial W$ decomposes into subsurfaces which are either horizontal or vertical with respect to $\mathcal F$.
An oriented codimension-1 foliation $\mathcal F$ of a 3-manifold $M$ is [*bracketed*]{} if, for some volume preserving flow $\Phi$,
1. $\mathcal F$ is compatible with some $(V,W,\Phi)$ decomposition of $M$ for which there exist $\mathcal F_W$ and $\xi_V$ such that $(\xi_V,\mathcal F_W,\Phi)$ is a positive decomposition, and
2. $\mathcal F$ is compatible with some $(V',W',\Phi)$ decomposition of $M$ for which there exist $\mathcal F_{W'}$ and $\xi_{V'}$ such that $(\xi_{V'},\mathcal F_{W'},\Phi)$ is a negative decomposition.
When we wish to specify the flow $\Phi$, $\mathcal F$ is called [*$\Phi$-bracketed*]{}.
Let $\mathcal F$ be an oriented codimension-1 foliation of a 3-manifold $M$ which is $\Phi$-bracketed. Then there exist a smooth positive contact structure $\xi_+$ and a smooth negative contact structure $\xi_-$, both $\Phi$-close to $\mathcal F$.
Sometimes $\mathcal F_W$ and $\mathcal F_{W'}$ are obtained by restricting $\mathcal F$ to $W$ and $W'$ respectively, and sometimes they are not. Very roughly speaking, when $W=W'$, we think of the restriction of $\mathcal F$ to $W$ as being [*bracketed by*]{} $\mathcal F_{W'}$ and $\mathcal F_W$ as a generalization of the situation in which the slope of $\mathcal F$ along boundary components of $W$ lies between the corresponding boundary slopes of $\mathcal F_{W'}$ and $\mathcal F_W$.
As noted in Corollary \[smoothisbracketed\], all taut, oriented $C^2$-foliations apart from the product foliation $S^1\times S^2$ are bracketed. In Example \[s1timess2\], we show that the product foliation of $S^1\times S^2$ is not bracketed. In this paper, we show that many taut, oriented $C^0$-foliations are bracketed. For each bracketed foliation considered in this paper, it is possible to choose $V=V'$ and $W=W'$.
\[bracketconj\] Let $\mathcal F$ be a taut oriented $C^0$-foliation of a closed oriented 3-manifold $M\ne S^1\times S^2$. Then $\mathcal F$ is bracketed.
We have the following two closely related questions.
Suppose $\mathcal F$ is a taut, oriented foliation with no torus leaf. Is $\mathcal F$ $C^0$-close to a taut, oriented smooth foliation?
Suppose $\mathcal F$ is a taut, oriented foliation with no torus leaf. Is $\mathcal F$ $\Phi$-close to a taut, oriented smooth foliation for some volume preserving flow $\Phi$?
Establishing the initial contact zone is of fundamental importance. In the context of $C^0$-foliation theory we introduce, in §\[attracting holonomy\], the notion of [*holonomy with a contracting interval*]{} and define what we mean by [*attracting neighborhood*]{}. This is significantly weaker than the more familiar notion of linear attracting holonomy, yet it suffices to build an initial contact zone. The precise definition appears as Definition \[contracting\]. As a corollary to Theorem \[main1\], we obtain:
[**Theorem \[attracting\].**]{} [*Let $\mathcal F$ be a taut $C^0$-foliation transverse to a flow $\Phi$. If $V$ is a disjoint union of attracting neighborhoods, and $\mathcal F$ is $V$-transitive, then $\mathcal F$ is bracketed and hence can be $\Phi$-approximated by a pair of weakly symplectically fillable and universally tight, contact structures, one positive and one negative.* ]{}
When working with $C^0$-foliations, it can be difficult, or even impossible, to establish the existence of sufficient attracting holonomy. Therefore, we introduce a different way of creating an initial contact zone. Roughly speaking, instead of looking for loops tangent to the foliation and satisfying a nice property, we look for loops *transverse* to the foliation and satisfying a nice property. We make this precise in Definition \[transtranv\], where we define [*$L$-bracketed foliation*]{}. As a corollary to Theorem \[main1\], we obtain:
[**Theorem \[transitivemain\].**]{} [*Suppose $\mathcal F$ is a taut oriented codimension-1 foliation in $M$, and that $\mathcal F$ is $L$-bracketed for some link $L$. Then $\mathcal F$ is bracketed and hence can be $\Phi$-approximated by a pair of smooth contact structures $\xi_{\pm}$, one positive and one negative. These contact structures are necessarily weakly symplectically fillable and universally tight.* ]{}
In § \[OBresults\], we consider the important special case that $\mathcal F$ is transverse to a flow $\Phi$ that has been obtained by removing a fibred link $L$ and doing a Dehn filling of a volume preserving suspension flow. In this case, $L$ forms the binding of an open book decomposition $(S,h)$ of $M$ and the contact structure $\xi_{(S,h)}$ compatible with $(S,h)$ is $\Phi$-close to $\mathcal F$.
In [@HKM2], Honda, Kazez and Matić introduced the use of foliations $\Phi$-close to $\xi_{(S,h)}$ as a way of establishing universal tightness of $\xi_{(S,h)}$. In particular, they appealed to $C^0$-foliations constructed in [@R1; @R2] to claim that $\xi_{(S,h)}$ is universally tight whenever the binding of $(S,h)$ is connected and the fractional Dehn twist coefficient at least one. Although the foliations constructed in [@R1; @R2] are not smooth, and therefore the proof in [@HKM2] contained a gap, they are $L$-bracketed, and hence Theorem \[transitivemain\] reveals that the conclusions of [@HKM2] are correct.
In §\[Open book\] we also include some background material relating language arising in the theory of open books with language arising in the theory of foliations. In particular, we give a translation between coordinates used in each subject together with a summary of our results related to open book decompositions.
To make the paper more self-contained there is an appendix containing an overview of the relationship between volume preserving flows and closed dominating 2-forms, and giving some standard definitions from symplectic topology. Most of this material is present either implicitly or explicitly in [@ET].
We close this section with an application of Theorem \[main1\] to the study of L-spaces.
(Definition 1.1, [@OS2]) A closed three-manifold $Y$ is called an [*L-space*]{} if $H_1(Y;\mathbb Q)=0$ and $\widehat{HF}(Y)$ is a free abelian group of rank $|H_1(Y;\mathbb Z)|$.
(Theorem 1.4, [@OS]) An L-space has no symplectic semi-filling with disconnected boundary; and all its symplectic fillings have $b_2^+(W)=0$. In particular, $Y$ admits no taut smooth foliation.
In other words, Ozsváth and Szabó show that if $Y$ is an L-space then there is no symplectic manifold $(X,\omega)$ with weakly convex boundary such that $|\partial X|>1$ and $Y$ is one of the boundary components. So an L-space cannot contain a pair of $\Phi$-close contact structures, $\xi_+$ positive and $\xi_-$ negative, where $\Phi$ is a volume preserving flow. (For details, see the Appendix.) Theorem \[main1\] thus implies the following.
An L-space $Y$ admits no bracketed foliation.
In particular, the foliations constructed in [@KRo; @LR; @R1; @R2] never exist in an L-space.
Foliation basics {#foliation basics}
================
\[folndefn\] Let $M$ be a smooth closed 3-manifold, and let $k$ be a non-negative integer or infinity. A *[$C^k$ codimension-1 foliation]{} $\mathcal F$ of (or in) $M$ is a union of disjoint connected surfaces $L_i$, called the *[leaves]{} of $\mathcal F$, such that:**
1. $\cup_i L_i = M$, and
2. there exists a $C^k$ atlas $\mathcal A$ on $M$ which contains all $C^{\infty}$ charts and with respect to which $\mathcal F$ satisfies the following local product structure:
- for every $p\in M$, there exists a coordinate chart $(U,(x,y,z))$ in $\mathcal A$ about $p$ such that $U\approx \mathbb R^3$ and the restriction of $\mathcal F$ to $U$ is the union of planes given by $z = $ constant.
When $k=0$, the tangent plane field $T\mathcal F$ is required to be $C^0$.
The extra hypothesis in the $k=0$ case, that $T\mathcal F$ is $C^0$, implies that there is a continuous, and hence there is a smooth, 1-dimensional foliation transverse to $\mathcal F$. It follows (see Proposition \[transcont\]) that the leaves of $\mathcal F$ are therefore smoothly immersed in $M$. Such foliations are called $C^{0+}$ in [@CC]. It follows from Proposition \[transcont\] that for $k\ge 1$, the condition that $\mathcal F$ is $C^k$ is equivalent to the condition that $T\mathcal F$ is $C^k$.
A frequently used technique for constructing foliations is to start with a branched surface embedded in $M$ that has product complementary regions. Since the embedding may be smoothed, a foliation resulting from thickening the branched surface and extending across the complementary regions can be constructed to be $C^0$.
Definition \[folndefn\] extends in an obvious way to define a codimension-1 foliation on a compact oriented smooth 3-manifold with non-empty boundary, where we insist that for each torus boundary component $T$, either $T$ is a leaf of $\mathcal F$, or $\mathcal F$ is everywhere transverse to $T$, and that any non-torus boundary component is a leaf of $\mathcal F$.
Recall that a smooth structure with corners on a topological 3-manifold $M$ with nonempty boundary is a maximal collection of smoothly compatible charts with corners whose domains cover $M$, where a chart with corners is an open set diffeomorphic to one of $\mathbb R^3$, $\{(x,y,z)\}| z\ge 0\}$, $\{(x,y,z)\}| y,z\ge 0\}$, or $\{(x,y,z)\}| x,y, z\ge 0\}$. Notice that the boundary of a manifold with corners naturally admits a stratification as a disjoint union of 0-, 1-, and 2-dimensional manifolds. The 0- and 1-manifolds of this stratification are referred to as the corners of $M$.
Definition \[folndefn\] extends further in an obvious way to define a codimension-1 foliation on a compact smooth 3-manifold $M$ with corners, where we insist that $\partial M$ can be written as a union of two compact piecewise linear surfaces $\partial_v M$ and $\partial_h M$, where the intersection $\partial_v M\cap\partial_h M$ is a union of corners of $M$, the components of $\partial_h M$ are contained in leaves of $\mathcal F$, and $\partial_v M$ is everywhere transverse to $\mathcal F$.
A [*flow*]{} is an oriented 1-dimensional foliation of $M$; namely, a decomposition $\Phi$ of a smooth compact 3-manifold $M$ into a disjoint union of connected 1-manifolds, called the [*flow curves*]{} of $\Phi$, such that there exists a $C^k$ atlas $\mathcal A$ on $M$ which contains all $C^{\infty}$ charts and with respect to which $\Phi$ satisfies the following local product structure:
- for every $p\in M$, there exists a coordinate chart $(U,(x,y,z))$ in $\mathcal A$ about $p$ such that $U\approx \mathbb R^3$, and the restriction of $\Phi$ to $U$ is the union of lines given by $(x,y) = $ constant.
When $M$ has boundary a disjoint union of tori, we insist that for each torus boundary component $T$, either $\Phi$ is everywhere tangent to $T$ or $\Phi$ is everywhere transverse to $T$. When $M$ is smooth with corners, we insist that $\partial M$ can be written as a union of two compact piecewise linear surfaces $\partial_v M$ and $\partial_h M$, where the intersection $\partial_v M\cap\partial_h M$ is a union of corners of $M$, $\partial_h M$ is everywhere transverse to $\Phi$, and $\partial_v M$ is everywhere tangent to $\Phi$.
Flows and oriented codimension-1 foliations coexist in interesting ways. A good overview can be found in [@CC]. In particular, given an oriented $C^k$-foliation $\mathcal F$, for any $k$, of an oriented 3-manifold $M$, possibly with non-empty boundary and possibly with corners, there is a $C^{\infty}$ flow everywhere transverse to $\mathcal F$. From this we have the following:
\[Proposition 5.1.4 of [@CC]\]\[transcont\] Let $M$ be a smooth compact oriented 3-manifold, possibly with corners. Let $R$ denote any one of $\mathbb R^2$, the closed upper half-plane in $\mathbb R^2$, or the closed upper right quadrant of $\mathbb R^2$. Given an oriented codimension-1 $C^k$-foliation $\mathcal F$ and smooth flow $\Phi$ transverse to $\mathcal F$, there exists a smooth biregular cover for $(M,\mathcal F,\Phi)$; namely, for every $p\in M$ there is a smooth coordinate chart $(U,(x,y,z))$, where $U\approx R\times \mathbb R$, and
1. the restriction of $\Phi$ to $U$ is the union of lines given by $(x,y) = $ constant, and
2. the restriction of $\mathcal F$ to $U$ is a $C^k$ family of $C^{\infty}$ graphs over $R$.
The second condition emphasizes the fact that $C^0$-foliations are leafwise smooth and transversely $C^0$.
When the oriented foliation $\mathcal F$ is taut, and $M$ is Riemannian, the smooth transverse flow can be chosen to be volume preserving.
(Theorem II.20, [@Sullivan]; see also Theorem A1, [@Hass])\[volpreserve\] Let $\mathcal F$ be a codimension-1, taut $C^0$-foliation of a closed smooth Riemannian 3-manifold $M$. Then there is a volume-preserving smooth flow everywhere transverse to $\mathcal F$. Equivalently, there is a smooth closed 2-form dominating $\mathcal F$.
Given a 3-manifold $M$ containing a taut $C^0$-foliation, one can ask whether there is a closely related $C^{\infty}$-foliation. Interpreting ‘closely related’ to mean any of $C^0$-$\epsilon$-close for some fixed $\epsilon>0$, $\Phi$-close, or topologically conjugate results in questions for which the answers are very little understood. There are certainly 3-manifolds which contain Reebless $C^0$-foliations but not Reebless $C^{\infty}$-foliations (Theorem D, [@BNR]). The existence of a taut sutured manifold hierarchy guarantees the existence of two types of foliation, one $C^0$ and finite depth and the other $C^{\infty}$ ([@G1; @G2; @G3]). Fixing a Riemannian metric and some $\epsilon>0$, these two types of foliation are not necessarily $C^0$-$\epsilon$-close. However, since they are carried by a common transversely oriented branched surface, they are $\Phi$-close.
We will take advantage of the fact that it is always possible [*locally*]{} to $C^0$-approximate $(\mathcal F,T\mathcal F)$ by $(\tilde{\mathcal F}, T\tilde{\mathcal F})$, for some locally defined smooth foliation $\tilde{\mathcal F}$.
\[smoothapprox\] Let $D$ be a smooth disk with corners and let $\mathcal F$ be a $C^0$-foliation of $D^2\times [0,1]$ which is positively transverse to the smooth 1-dimensional foliation by intervals $\{ (x,y)\}\times [0,1],\,\, (x,y)\in D^2$. Given any $\epsilon>0$, there is a smooth foliation $\tilde{\mathcal F}$ which is positively transverse to the smooth 1-dimensional foliation by intervals $\{ (x,y)\}\times [0,1],\,\, (x,y)\in D^2$, and satisfies $(\tilde{\mathcal F},T\tilde{\mathcal F})$ is $C^0$ $\epsilon$-close to $(\mathcal F, T\mathcal F)$. Moreover, if $\mathcal F$ is smooth on some compact $\mathcal F$-saturated subset, then we may choose $\tilde{\mathcal F}$ to equal $\mathcal F$ on this subset.
By identifying $D$ with a subset of the plane, a point $p$ in a leaf of $\mathcal F$ determines both a point in $\mathbb R^3$, and by choosing a unit vector perpendicular to the tangent plane of the leaf, a point $\bf{u}_p$ in $T\mathbb R^3$. The standard metric on $T\mathbb R^3 = \mathbb R^6$ is used to measure the distance between two leaves of $\mathcal F$ as follows.
By Proposition \[transcont\], we may assume that the leaves of $\mathcal F$ are given by the graphs of $z=f_{\theta}(x,y)$, for some continuous family of smooth functions $f_{\theta} : D\to [0,1]$, $0\le\theta\le 1$. For any two such leaves, $L_1$ and $L_2$ say, given by $z=f_{\theta_1}(x,y)$ and $z=f_{\theta_2}(x,y)$ respectively, define the distance between them to be the maximum distance, computed in $T\mathbb R^3$, between $(x,y,f_{\theta_1}(x,y),{\bf u}_{(x,y,f_{\theta_1}(x,y))})$ and $(x,y,f_{\theta_2}(x,y),{\bf u}_{(x,y,f_{\theta_2}(x,y))})$ for $(x,y) \in D$. Since $D$ is compact, uniform continuity guarantees that $d$ is continuous and hence a metric on the leaf space of $\mathcal F$.
For any $\theta \in [0,1]$, let $U_{\theta}$ denote the subset of $D\times [0,1]$ which is the union of all graphs $z=f_{\theta}(x,y)$ which are of d-distance strictly less than $\epsilon/2$ from the leaf $z = f_{\theta}(x,y)$. Since $U_{\theta}$ is the pullback of an $\epsilon/2$ $d$-neighborhood in the leaf space, $U_{\theta}$ is open in $D\times[0,1]$.
Pick a finite cover of $D\times[0,1]$ by $U_{\theta_0},U_{\theta_1},\cdots,U_{\theta_r}$ for some $r\ge 0$ and $0=\theta_0<\theta_1<\cdots <\theta_r = 1$. Now let $\tilde{\mathcal F}$ be the foliation of $D\times[0,1]$ which includes the leaves given by the graphs of $f_{\theta_i}$ and, for each $i, 0\le i\le r-1$, the leaves given by the graphs of a damped straight line homotopy between $f_{\theta_i}$ and $f_{\theta_{i+1}}$. Thus if $g$ is a smooth homeomorphism of $[0,1]$ with derivatives at $0$ and $1$ vanishing to infinite order, the leaves of $\tilde{\mathcal F}$ are $z=(1-g(t))f_{\theta_i}(x,y) + g(t)f_{\theta_{i+1}}(x,y)$, $0\le t\le 1$, on the subset of $D\times [0,1]$ bounded by the graphs of $f_{\theta_i}$ and $f_{\theta_{i+1}}$. By construction, $\tilde{\mathcal F}$ is smooth.
Moreover, $(\tilde{\mathcal F},T\tilde{\mathcal F})$ and $(\mathcal F,T\mathcal F)$ are $\epsilon$-close. To see this, recall that a normal vector to a graph $z=f(x,y)$ is given by ${\bf n}_f = \langle -f_x, -f_y, 1\rangle$, and a straight-line homotopy between $f_{\theta_1}$ and $f_{\theta_2}$ induces a straight-line homotopy between ${\bf n}_{f_{\theta_1}}$ and ${\bf n}_{f_{\theta_2}}$. Normalizing this straight-line homotopy of normal vectors gives a geodesic on the unit sphere joining ${\bf u}_{f_{\theta_1}}$ and ${\bf u}_{f_{\theta_2}}$. Since the leaves given by $z = f_{\theta_i}(x,y)$ and $z = f_{\theta_{i+1}}(x,y)$ are of $d$-distance at most $\epsilon/2$, it follows immediately from the triangle inequality that the leaves given by $z=f_{(1-g(t))\theta_i + g(t)\theta_{i+1}}(x,y)$ and $z=(1-g(t))f_{\theta_i}(x,y) + g(t)f_{\theta_{i+1}}(x,y)$ are of $d$-distance strictly less than $\epsilon$. So$(\tilde{\mathcal F},T\tilde{\mathcal F})$ and $(\mathcal F,T\mathcal F)$ are $\epsilon$-close.
If $\mathcal F$ is smooth on some compact $\mathcal F$-saturated subset $A$ of $D\times [0,1]$, each component of $A$ is bounded by graphs of the form $f_{\theta}$. By compactness of $A$, $\partial A$ contains only finitely many such $f_{\theta}$. For each $z=f_{\theta}$ in $\partial A$, include $\theta$ in the list $\theta_0,\theta_1,\cdots,\theta_r$ and modify $\mathcal F$ only on the complement of $A$.
Next we recall Operation 2.1.1 of [@Ga]. Let $L_1,\dots, L_m$ be distinct leaves of a $C^0$-foliation $\mathcal F$. Modify $\mathcal F$ by thickening each of the leaves $L_j$. Thus, each $L_j$ is blown up to an $I$-bundle $L_j\times [-1,1]$. Let $\mathcal F'$ denote the resulting foliation. We highlight the following observation.
\[blowup\] The leaves $L_1, \dots, L_m$ may be thickened so that the foliation $\mathcal F'$ is $C^0$ and the restriction of $\mathcal F'$ to $$L_j\times (-1,1)\subset M$$ is a smooth foliation for each $j$.
Transitive flow box decompositions {#transitive flow boxes}
==================================
\[flowboxdefn\]
A [*flow box*]{}, $F$, for a $C^0$-foliation $\mathcal F$ and smooth transverse flow $\Phi$, is a smooth chart with corners that is of the form $D\times I$, where $D$ is a polygon (a disk with at least three corners), $\Phi$ intersects $F$ in the arcs $\{(x,y)\}\times I$, and $\mathcal F$ intersects $F$ in disks which are everywhere transverse to $\Phi$ and hence can be thought of as graphs over $D$. In particular, $D\times \partial I$ lies in leaves of $\mathcal F$, each component of $\mathcal F\cap F$ is a smoothly embedded disk, and these disks vary continuously in the $I$ direction.
The [*vertical boundary*]{} of $F$, denoted $\partial_v F$, is $\partial D \times I$. The [*horizontal boundary*]{} of $F$ is $D \times \partial I$ and is denoted $\partial_h F$. An arc in $M$ is [*vertical*]{} if it is a subarc of a flow line and [*horizontal*]{} if it contained in a leaf of $\mathcal F$.
It is often useful to view the disk $D$ as a 2-cell with $\partial D$ the cell complex obtained by letting the vertices correspond exactly to the corners of $D$. Similarly, it is useful to view the flow box $F$ as a 3-cell possessing the product cell complex structure of $D\times I$. Then the horizontal boundary $\partial_h F$ is a union of two (horizontal) 2-cells and the vertical boundary $\partial_v F$ is a union of $c$ (vertical) 2-cells, where $c$ is the number of corners of $D$.
A subset $R$ of $F$ is called a [*vertical rectangle*]{} if it has the form $\alpha\times [a,b]$, where $0\le a<b\le 1$ and $\alpha$ is either a 1-cell of $\partial D$ or else a properly embedded arc in $D$ connecting distinct vertices of $D$. A subset $e$ of $F$ is called an [*edge*]{} if it is a compact interval contained in a 1-cell of $F$.
Given a vector $\vec w$ tangent to $\partial_v F$, we choose a [*slope*]{} convention such that the leaves of $\mathcal F \cap \partial_v F$ have slope $0$, the flow lines have slope $\infty$, and the sign of the slope of $\vec w$ is computed as viewed from outside of $F$.
Given a codimension-1 leafwise smooth foliation $\mathcal F$ and transverse smooth flow $\Phi$, let $V$ be a compact codimension-0 sub-manifold of $M$, with $\partial V = \partial_v V \cup\partial_h V$, where $\partial_v V$ is a union of flow arcs or circles, and $\partial_hV$ is a union of subsurfaces of leaves of $\mathcal F$. In the case that $\partial V = \partial_vV$, $\mathcal F$ and $\Phi$ need only be defined on the complement of $V$. A [*flow box decomposition*]{} of $M$ [*rel*]{} $V$ is a decomposition of $M\setminus \text{int} V$ as a finite union $M = V\cup (\cup_iF _i)$ where
1. Each $F _i$ is a standard flow box for $\mathcal F$.
2. If $i \neq j$, the interiors of $F _i$ and $F _j$ are disjoint.
3. If $F _i$ and $F _j$ are different flow boxes, then their intersection is connected and either empty, a 0-cell, an edge, a vertical rectangle, or a subdisk of $\partial_h F_i \cap \partial_h F_j$.
\[transitive\] We call a flow box decomposition $M= V\cup F_1 \cup \dots \cup F_n$ [*transitive*]{} if $V_0=V$, $V_i = V_{i-1} \cup F_i$, and for $i=1,\dots, n$,
1. each 2-cell of $\partial_v F_i$ has interior disjoint from $\partial_h F_j$ for all $j<i$,
2. $V_{i-1}\cap F_i$ is a union of horizontal subsurfaces and vertical 2-cells of $F_i$, together possibly with some 0- and 1-cells, and
3. $V_{i-1}\cap F_i$ contains a vertical 2-cell of $F_i$.
\[transitiveflowbox\] If $M$ is $V$-transitive, then there is a transitive flow box decomposition of $M$ rel $V$.
Since $M$ is $V$-taut, for each point $x\in M\backslash V$, there exists $\gamma$ an embedded arc in the leaf containing $x$ that connects $x$ to $V$. By taking a regular neighborhood of $\gamma$ in its leaf and flowing along it, create a flow box $F$ with the property that it has one vertical 2-cell contained in $\partial_v V$. The point $x$ may or may not be in the interior of $F$. Using compactness of $M$, pick a finite collection of flow boxes, of the sort just described, $F_1, F_2,\dots, F_r$ that cover $M\backslash V$. Assume no proper subset of the $F_i, 1\le i\le r,$ cover.
Next, let $L_1, L_2, \dots, L_m$ be the collection of leaves of $\mathcal F$ that contain the horizontal boundaries of all $F_i$.
We proceed by induction to show that $V\cup F_1\cup\dots\cup F_i$ admits a flow box decomposition with respect to $V$ for every $i, 1\le i\le r$. Certainly, $V\cup F_1$ does. So suppose that $V\cup F_1\cup\dots\cup F_{i-1}$ admits a flow box decomposition with respect to $V$. After renaming and reindexing as necessary, we assume that this flow box decomposition is given by $V\cup F_1\cup\dots\cup F_{i-1}$. We show that $V_i = V\cup F_1\cup\dots\cup F_{i-1}\cup F_i$ also admits a flow box decomposition with respect to $V$.
Begin by slightly increasing the size of $F_i$ in $M\setminus \text{int} V$, as necessary, so that $F_i$ is still a flow box and, for all $j<i$, $\partial_v F_j$ and $\partial_v F_i$ are transverse away from $V$, where they may overlap tangentially. Notice that this ensures that $ V\cup F_1\cup\dots\cup F_{i-1}\cup F_i$ is a codimension-0 submanifold with corners and piecewise vertical and horizontal boundary. Also, cut $F_i$ open along those (horizontal disk) components of $(\cup_i L_i)\cap F_i$ which have non-empty intersection with $\partial_h F_j$ for some $j<i$. Denote the resulting flow boxes by $F_i^1,\dots,F_i^s$; so $F_i = F_i^1\cup\dots\cup F_i^s$.
Consider $F_j\cap F_i^1$ for some $j<i$. Since $\partial_v F_j$ and $\partial_v F_i^1$ are transverse away from $V$, each component of $F_j\cap F_i^1$ is a flow box. Consider any such component, $X$ say, from the point of view of the flow box $F_i^1=D_i\times [c,d]$. Notice that $X=D\times [c,d]$, where $D$ is a subdisk (with corners) of $D_i$, and $X\cap \partial_v F_i^1$ is a non-empty union of vertical 2-cells.
Now, for all $j<i$, remove $F_j\cap F_i^1$ from $F_i^1$. Taking the closure of the result we get a union $G_1\cup\dots\cup G_b$ of flow boxes, where each $G_k\cap V_{i-1}$ is a union of horizontal subsurfaces and vertical 2-cells of $G_k$, together possibly with some 0- and 1-cells, and contains a vertical 2-cell of $G_k$. Notice that $G_k\cap G_l=\emptyset$ if $k\ne l$ and, by subdividing each $G_k$ along finitely many vertical rectangles as necessary, we may assume $G_k\cap F_j$ is connected for all $j<i$. The resulting union $$V\cup F_1\cup\dots\cup F_{i-1}\cup G_1\cup\dots\cup G_b$$ is then a transitive flow box decomposition of $V\cup F_1\cup\dots\cup F_{i-1}\cup F_i^1$ with respect to $V$.
Repeat this process for each $a, 2\le a\le s$, to obtain a transitive flow box decomposition of $V\cup F_1\cup\dots\cup F_i$ with respect to $V$.
Basic extension results {#basicsection}
=======================
In this section, we collect together an assortment of confoliation extension results important for the inductive construction to be described in Section \[theconstructionsection\]. For the most part, it will be possible to restrict attention to flow boxes diffeomorphic to one of the flow boxes $F$, $G$ or $H$, where $F$, $G$ and $H$ are defined as follows.
Let $F$ denote the flow box given by $$F=\{|x| \le 1, 0 \le y \le 1, |z| \le 1\}.$$ Let $G$ denote the flow box given by $\Delta \times [0,1]$, where $\Delta$ is the region in the $xy$-plane bounded by the triangle with vertices $$(-3/2,1), (3/2,1) \mbox{ and } (0,-1/2).$$ Let $\Delta^{(0)}$ denote the $0$-skeleton $$\Delta^{(0)} = \{(-3/2,1), (3/2,1), (0,-1/2)\}.$$ Let $H=F\cap G$, a flow box with hexagonal horizontal cross-section. Notice that $H$ is diffeomorphic to the complement in $G$ of an open neighborhood of the $1$-skeleton of $\partial_v G$.
We begin with the following elementary, and very useful, observations of Eliashberg-Thurston, [@ET].
(Proposition 1.1.5, [@ET])\[ETlemma\] Let $\eta$ be a $C^k$-confoliation with $k \ge 1$ and domain $F$ given by a 1-form $dz - a(x,y,z) dx$. Then $$\dfrac {\partial a}{\partial y}(x,y,z) \ge 0$$ at all points of $F$, and $$\dfrac {\partial a}{\partial y}(x,y,z) > 0$$ where $\eta$ is contact.
Eliashberg and Thurston use this lemma to approximate the confoliation by a contact structure with the following corollary.
(Lemma 2.8.2, [@ET])\[approximate\] Let $\eta$ be a $C^k$-confoliation, with $k \ge 1$ and domain $F$, given by a 1-form $dz - a(x,y,z) dx$. If $\eta$ is contact in a neighborhood of $y=1$ in $F$, then $\eta$ can be approximated by a confoliation $\hat\eta$ which coincides with $\eta$ together with all of its derivatives along the boundary $\partial F$ and which is contact inside $F$.
It is enough to approximate $a(x,y,z)$ along each interval $\{x\} \times [0,1] \times \{z\}$ by a function $\hat{a}(x,y,z)$ that is strictly monotonic for $(x,z)\in(-1,1) \times (-1,1)$ but is damped to agree smoothly with $a(x,y,z)$ on $\partial F$.
\[dominate\] If $\alpha$ and $\beta$ are families of curves transverse to $\partial/\partial z$ and contained in a vertical 2-cell $R$ of the vertical boundary of a flow box, we say $\alpha$ [*strictly dominates*]{} $\beta$ along $A\subset R$ if at every $p\in A$, the slope of the tangent to $\alpha$ at $p$ is greater than the slope of the tangent to $\beta$ at $p$. It must be specified if the comparison of slopes is made from inside or outside of the flow box. If $\alpha$ strictly dominates $A$, and $\alpha$ and $\beta$ are the characteristic foliations of 2-plane fields $\xi_1$ and $\xi_2$ respectively, we also say that $\xi_1$ strictly dominates $\xi_2$ along $R$. If $\xi_2 = T\mathcal F$ for some codimension-1 foliation $\mathcal F$, we also say that $\xi_1$ strictly dominates $\mathcal F$ along $R$.
The statement of Lemma \[ETlemma\] raises the question of whether flow box coordinates can always be chosen so that the contact form can be written as $dz-a(x,y,z)dx$. The next lemma points out that this is the case and gives a simple condition for a contact structure to dominate in such coordinates.
Let $U$ be a regular neighborhood in $F$ of the union of $x=\pm 1$ and $z=\pm 1$. Let $\eta$ be a $C^k$-confoliation with $k \ge 1$ defined in a neighborhood $V$ of $y=1$ in $F$ which is everywhere transverse to the vertical segments $(x,y) =$ constant, horizontal in $\overline{U}$, and contact on $V\setminus \overline{U}$. Then, after smoothly reparametrizing $F$ as necessary, we may assume that $\eta$ is given by a 1-form $$dz - a(x,y,z) dx$$ with
1. $a(x,y,z)=0$ on $V\cap \overline{U}$, and
2. $\dfrac {\partial a}{\partial y}(x,y,z) > 0$ on $V\setminus \overline{U}$.
Moreover, the characteristic foliation of $\eta$ along the complement of $\overline{U}$ in $y=1$ strictly dominates the horizontal foliation, when viewed from inside $F$, if and only if $a(x,y,z)>0$ in $V'\setminus \overline{U}$, for some neighbourhood $V'\subset V$ of $y=0$.
Since $dz - a(x,y,z) dx$ vanishes on $\partial/\partial y$, it is enough to choose coordinates $x, y$ for leaves so that curves with constant $x$ coordinate are Legendrian. At points where $\eta$ is transverse to the horizontal foliation, there is a unique Legendrian direction. At all other points, any direction is Legendrian. The coordinate $y$ can be constructed by choosing a section of the Legendrian directions.
![Each figure shows a $z=0$ slice capturing the flow box setup of one of Corollaries \[extend3\]–\[extend8\]. Plus signs are positioned on the side from which the contact structure dominates the horizontal foliation. Dashes, for instance along $U$, show where the confoliation slope is 0. The smooth foliation acts as a transport mechanism for contact structures in the direction shown by the arrows.[]{data-label="extend"}](extend){width="5in"}
\[extend3\] Let $U$ be a regular neighborhood in $F$ of the union of $x=\pm 1$ and $z=\pm 1$. Let $\eta$ be a $C^k$-confoliation with $k \ge 1$ defined in a neighborhood $V$ of $y=1$ in $F$ by a 1-form $$dz - a(x,y,z) dx$$ with
1. $a(x,y,z)=0$ on $V\cap \overline{U}$, and
2. $a(x,y,z)>0$ and $\dfrac {\partial a}{\partial y}(x,y,z) > 0$ on $V\setminus \overline{U}$.
Then $\eta$ can be extended to a $C^k$-confoliation $\hat\eta$ on $F$ that agrees with the horizontal foliation of $F$ in $\overline{U}$, is contact on the complement of $\overline{U}$, and strictly dominates, when viewed from outside $F$, the horizontal foliation on the complement in $y=0$ of $\overline{U}$.
It is enough to extend $a(x,y,z)$ along each interval $\{x_0\} \times [0,1] \times \{z_0\}$ to a $C^k$ function $\hat{a}(x,y,z)$ such that
1. $\hat{a}(x,y,z)=0$ in $\overline{U}$, and
2. $\hat{a}(x,y,z)>0$ and $\dfrac {\partial \hat{a}}{\partial y}(x,y,z) > 0$ outside $\overline{U}$.
\[extend4\] Let $U$ be a regular neighborhood in $F$ of the union of $x=\pm 1$ and $z=\pm 1$. Let $\eta$ be a $C^k$-confoliation with $k \ge 1$ defined in a neighborhood $V$ of $y=1$ in $F$ by a 1-form $$dz - a(x,y,z) dx$$ with
1. $a(x,y,z)=0$ on $V\cap \overline{U}$, and
2. $a(x,y,z)>0$ and $\dfrac {\partial a}{\partial y}(x,y,z) > 0$ on $V\setminus \overline{U}$.
Then $\eta$ can be extended to a $C^k$-confoliation $\hat\eta$ on $F$ that agrees with the horizontal foliation of $F$ in $\overline{U}$, is contact on the complement of $\overline{U}$ in the interior of $F$, and smoothly agrees with the horizontal foliation at $y=0$.
Proceed as in the proof of Corollary \[extend3\] except insist that $$\hat{a}(x,0,z)\equiv 0.$$
\[extend5\] Let $U$ be a regular neighborhood in $F$ of the union of $x=\pm 1$ and $z=\pm 1$. Let $\eta$ be a $C^k$-confoliation with $k \ge 1$ defined in a neighborhood $V$ of the union of $y=0$ and $y=1$ in $F$. Suppose that when viewed from inside $F$, $\eta$ dominates the horizontal foliation along the vertical faces given by $y=0$ and $y=1$, with strict domination in the complement of $\overline{U}$. Then $\eta$ can be extended to a $C^k$-confoliation $\hat\eta$ on $F$ that agrees with the horizontal foliation of $F$ in $\overline{U}$, and is contact on the complement of $\overline{U}$.
Decompose $F$ as a union of two flow boxes diffeomorphic to $F$ by cutting open along the plane $y=1/2$. Apply Corollary \[extend4\] to each of the resulting flow boxes.
The point of “smoothly agrees with” in the next corollary is that flow boxes are brick-like objects that, when sensibly glued together, should define a smooth confoliation. Thus we require smooth convergence of the confoliation to horizontal at $y=0$.
\[extend6\] Let $U$ be a regular neighborhood in $F$ of the union of $x=\pm 1$ and $z=\pm 1$. Let $\eta$ be a $C^k$-confoliation with $k \ge 1$ defined in a neighborhood $V$ of $y=1$ in $F$ by a 1-form $$dz - a(x,y,z) dx$$ with
1. $a(x,y,z)=0$ on $V\cap \overline{U}$, and
2. $a(x,y,z)>0$ and $\dfrac {\partial a}{\partial y}(x,y,z) > 0$ on $V\setminus \overline{U}$.
Then $\eta$ can be extended to a $C^k$-confoliation $\hat\eta$ on $H$ that agrees with the horizontal foliation of $H$ in $\overline{U}$, is contact on the complement of $\overline{U}$ in the interior of $H$, smoothly agrees with the horizontal foliation at $y=0$, and dominates the horizontal foliation in the complement of $\overline{U}$ along the lines $$y = x -1/2 \mbox{ and } y = -1/2-x.$$
This follows from Corollary \[extend4\].
\[extend7\] Let $U$ be a regular neighborhood in $G$ of the union of $z=\pm 1$ and $\Delta^{(0)}\times [-1,1]$. Let $S$ denote the complement in $\partial_v G$ of the 2-cell given by $y=1$. Let $\eta$ be a $C^k$-confoliation with $k \ge 1$ defined in a neighborhood $V$ of $S$ in $G$ such that
1. $\eta$ is contact on $V\setminus \overline{U}$,
2. $\eta$ agrees with the horizontal foliation on $V\cap\overline{U}$, and
3. $\eta$ strictly dominates the horizontal foliation along $S\setminus \overline{U}$, when viewed from inside $G$.
Then $\eta$ can be extended to a $C^k$-confoliation $\hat\eta$ on $G$ that agrees with the horizontal foliation of $G$ in $\overline{U}$, is contact on the complement of $\overline{U}$ in the interior of $G$, and, when viewed from outside $G$, dominates the horizontal foliation in the complement of $\overline{U}$ along the line $y=1$.
Let $\alpha$ be a smooth arc properly embedded in $\Delta$ which connects the vertices $(3/2,1)$ and $(0,-1/2)$ and is not tangent to a side of the triangle $\Delta$ at its endpoints. Let $R=\alpha\times [-1,1]$, a vertical rectangle in $G$. Decompose $G$ along $R$ into two flow boxes as $G=G'\cup F'$, where $G'$ is diffeomorphic to $G$ and $F'$ is diffeomorphic to $F$. First apply Corollary \[extend6\] to $G'$ and then apply Corollary \[extend3\] to $F'$.
\[extend8\] Let $U$ be a regular neighborhood in $G$ of the union of $z=\pm 1$ and $\Delta^{(0)}\times [-1,1]$. Let $\eta$ be a $C^k$-confoliation with $k \ge 1$ defined in a neighborhood $V$ of $\partial_v G$ in $G$ such that
1. $\eta$ is contact on $V\setminus \overline{U}$,
2. $\eta$ agrees with the horizontal foliation on $V\cap\overline{U}$, and
3. $\eta$ strictly dominates the horizontal foliation along $\partial_v G \setminus \overline{U}$, when viewed from inside $G$.
Then $\eta$ can be extended to a $C^k$-confoliation $\hat\eta$ on $G$ that agrees with the horizontal foliation of $G$ in $\overline{U}$ and is contact on the complement of $\overline{U}$ in the interior of $G$.
This time we cut $G$ open along two vertical rectangles and consider instead the resulting union of flow boxes. This time we cut $G$ open along two disjoint vertical rectangles to obtain $$G=G'\cup F'\cup F^{''},$$ where $G'$ is diffeomorphic to $G$ and each of $F'$ and $F^{''}$ is diffeomorphic to $F$. Then apply Corollary \[extend6\] to $G'$. Finally, apply Corollary \[extend3\] to each of $F'$ and $F^{''}$.
We now consider a case where the initial confoliation is defined on the entire vertical boundary of a solid cylinder. Let $\mathcal L$ be a smooth 1-dimensional foliation on the cylinder $S^1 \times I$ such that the boundary components of $S^1\times I$ are leaves of $\mathcal L$. Let $h:I\to I$ be the holonomy map of $\mathcal L$.
\[cylinderextend\] If $h'(z)<0$ for $z\in(0,1)$, there is a confoliation on $D^2 \times I$ that is contact on $D^2 \times (0,1)$, tangent on $D^2 \times \partial I$, has characteristic foliation $\mathcal L$, and is everywhere transverse to $\partial/\partial z$.
This will follow from Lemmas \[extension1\]–\[extension3\].
\[extension1\] If $h'(z)<0$ for $z\in(0,1)$, then there exists a smooth 1-dimensional foliation $\mathcal K$ of $S^1 \times I$ with the same holonomy map $h:I\to I$ such that for every $(\theta, z)\in S^1 \times (0,1)$, the slope, $s(\theta,z)$, of $\mathcal K$ at $(\theta,z)$ is negative.
As a first approximation, let $\mathcal K_1$ be the foliation of $[0,2\pi] \times I$ given by connecting each point $(0,z)$ to $(2\pi,h(z))$ by a straight line. Next create $S^1 \times I$ by identifying $(0,z)$ and $(2\pi,z)$ for $z\in I$. Let $\mathcal K_2$ denote the image of $\mathcal K_1$ in $S^1\times I$. This has the desired properties, except that $\mathcal K_2$ is not smooth along $\{0\} \times I$. Carefully rounding these corners (see for example, Lemma 4.7 of [@Milnor]), yields the desired smooth foliation.
\[extension2\] There is a diffeomorphism, $F$, of $D^2 \times I$ that is the identity map on $D^2 \times \partial I$ and takes $\mathcal L$ to $\mathcal K$.
Let $\theta = 0$ be a base point for $S^1$ so that the holonomy map for each foliation is $h:\{0\} \times I \to \{0\} \times I$. Let $f$ be the diffeomorphism of $S^1 \times I$ such that $f$ restricts to the identity map on $\{0\} \times I$, preserves the $S^1$ coordinate, and maps $\mathcal L$ to $\mathcal K$.
Define $F:D^2 \times I\to D^2 \times I$ by $F(r,\theta,z)=(r, t(r)f(\theta,z) + (1-t(r))(\theta,z))$ where $t$ is a diffeomorphism of the interval smoothly damped at the endpoints.
\[extension3\] There is a confoliation $\xi$ on $D^2 \times I$ that is contact on $D^2 \times (0,1)$, tangent on $D^2 \times \partial I$, has characteristic foliation $\mathcal K$, and is everywhere transverse to $\partial/\partial z$.
Using cylindrical coordinates, define $\alpha = dz - r^2s(\theta,z)d\theta$. Then $d\alpha=-2rs(\theta,z)drd\theta-r^2s_z(\theta,z)dzd\theta$, from which it follows that $$\alpha \wedge d\alpha=-2s(\theta,z)rdrd\theta dz.$$ Setting $\xi=\text{ker}(\alpha)$ has the desired properties.
The inductive construction of $\xi$. {#theconstructionsection}
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In this section, we show how a $C^0$-foliation can be used to propagate a contact structure across $M$. Before describing this procedure, we highlight the role of smoothness in the approach used by Eliashberg and Thurston in performing this propagation.
First, recall Lemma \[ETlemma\]. Roughly speaking, a clever choice of foliation coordinates permits a confoliation along a Legendrian curve to be described by a monotone function. By taking advantage of a beginning contact zone, such a function can be approximated by a strictly monotone function, thereby creating a larger contact zone. This argument can be repeated on overlapping regions covering the manifold. The issue is whether strict monotonicity attained on a given region can be preserved under subsequent approximations. This is precisely where smoothness of the foliation becomes important, guaranteeing that derivatives are globally defined and continuous, and thus allowing one to preserve strict monotonicity under subsequent approximations.
We circumvent the issue of monotonicity with carefully chosen, minimally overlapping, flow boxes and a more discrete propagation technique, which we now describe.
Recall Definition \[dominate\] and the slope convention chosen in Definition \[flowboxdefn\]. Let $F$ be a flow box and let $R$ be a rectangle in $\partial_v F$. Let $\chi_{\xi_1}$ and $ \chi_{\xi_2}$ denote the characteristic foliations induced on $R$ by two 2-plane fields $\xi_1$ and $\xi_2$ defined in a neighborhood of $R$ and positively transverse to $\Phi$. We write $\chi_{\xi_1}<_p\chi_{\xi_2}$ if the unit vector tangent to $\chi_{\xi_1}$ at $p$ has slope less than the unit vector tangent to $\chi_{\xi_2}$ at $p$. Similarly, we write $\chi_{\xi_1}=_p\chi_{\xi_2}$ if the unit vector tangent to $\chi_{\xi_1}$ at $p$ has slope equal to the unit vector tangent to $\chi_{\xi_2}$ at $p$. We write $\chi_{\xi_1}<\chi_{\xi_2}$ if $\chi_{\xi_1}<_p\chi_{\xi_2}$ for all $p\in \text{int}(R)$. Similarly, we write $\chi_{\xi_1}\le\chi_{\xi_2}$ if $\chi_{\xi_1}\le_p\chi_{\xi_2}$ for all $p\in \text{int}(R)$.
Let $M$ be a closed oriented 3-manifold with smooth flow $\Phi$. Suppose that $M$ decomposes as a union $$M=V\cup W,$$ where $V$ and $W$ are smooth 3-manifolds, possibly with corners, and $\partial V=\partial W$. We say that this decomposition is [*compatible with the flow $\Phi$*]{} if $\partial V$ (and hence $\partial W$) decomposes as a union of compact subsurfaces $\partial_v V\cup \partial_h V$, where $\partial_v V$ is a union of flow segments of $\Phi$ and, $\partial_h V$ is transverse to $\Phi$. In the presence of a foliation transverse to the flow, the notation $\partial_h V$ will be used for the portion of $\partial V$ tangent to the foliation.
\[compatible1\] Let $M$ be a closed oriented 3-manifold with smooth flow $\Phi$. Suppose that $M$ can be expressed as a union $$M=V\cup W,$$ where $V$ and $W$ are smooth 3-manifolds, possibly with corners, such that $\partial V=\partial W$. Suppose also that this decomposition is compatible with $\Phi$, that $V$ admits a smooth contact structure $\xi_V$, and that $W$ admits a $C^0$-foliation $\mathcal F_W$. We say that $(V,\xi_V)$ is [*$\Phi$-compatible*]{} with $(W,\mathcal F_W)$, and that $M$ admits a positive $(\xi_V,\mathcal F_W,\Phi)$ decomposition, if the following are satisfied:
1. $\xi_V$ and $\mathcal F_W$ are (positively) transverse to $\Phi$ on their domains of definition,
2. $\xi_V$ is tangent to $\partial_h V$, and
3. $\chi_{\xi_V}<\chi_{T\mathcal F_W}$ on the interior of $\partial_v V$, when viewed from outside $V$.
The main result of the section is Theorem \[main1\]. The starting point is a transitive flow box decomposition $M= V\cup F_1 \cup \dots \cup F_n$ for $\mathcal F$. Set $V_0 = V$ and $V_i = V\cup F_1\cup...\cup F_i$ for each $i, 1\le i\le n$. For each $i, 0\le i\le n$, set $W_i = M\setminus \text{int}(V_i)$. Thus, for $0\le i \le n$, $\mathcal F$ is compatible with $(V_i,W_i,\Phi)$. When $i=0$, $\partial_h V_i = \emptyset$, and when $i=n$, $V_n=M$ and hence $\partial V_n=\emptyset$.
\[smoooth\] After possibly blowing up finitely many leaves of $\mathcal F$, we may assume that
1. $\mathcal F$ is smooth in a neighborhood of the horizontal faces $\partial_h F_i$, and
2. $\mathcal F$ is smooth in a neighborhood of the vertical edges of each $F_i$.
We will define $\xi$ inductively and in a piecewise fashion over each $F_i$. To guarantee that the resulting confoliation is everywhere smooth, we add a [*smoothly foliated collar*]{} about the horizontal boundary and the vertical 1-simplices of each $F_i$ as follows. Let $L_1, L_2, \dots, L_m$ be the collection of leaves of $\mathcal F$ that contain the horizontal boundaries of all $F_i$. Modify the original foliation $\mathcal F$ by thickening each of the leaves $L_j$. Thus each $L_j$ is replaced with a product of leaves $L_j \times [-1,1]$. The thickening should be performed so that for each $i$, if a component of $\partial_h F_i$ was originally contained in $L_j$, then it is now contained in $L_j \times \{0\}$. As noted in Lemma \[blowup\], we may assume the interior of each thickening, $L_j\times (-1,1)$, is smoothly immersed. Note that $M= V\cup F_1 \cup \dots \cup F_n$ remains a transitive flow box decomposition of $M$ with respect to this new foliation. Moreover, $\mathcal F \cap F_i$ is smooth in a neighourhood of $\partial_h F_i$ for each $i$.
The vertical edges of the $F_i$ are smooth transverse arcs, thus $\mathcal F$ can be smoothed on $D^2 \times I$ product neighborhoods of these edges.
Now we fix a preferred regular neighborhood of the union of the horizontal 2-cells and the vertical 1-cells of the flow box decomposition. We do this as follows. For each $i, 1\le i \le n$, choose a regular neighborhood of $ \partial_h F_i$ which is contained in the thickening $L_i\times (-1,1)$, and choose a regular neighborhood of the vertical 1-cells of $F_i$on which $\mathcal F$ is smooth. Choose these neighborhoods so that the union $U$ of these neighborhoods is a regular neighborhood of the union of all horizontal 2-cells and all vertical 1-cells of the flow box decomposition. We refer to both this preferred neighborhood $U$ and the restriction of $\mathcal F$ to $U$ as the [*smoothly foliated collar*]{}.
Next, we modify Definition \[compatible1\] slightly to account for the smoothly foliated collar.
\[compatible\] Let $0\le i<n$ and let $\xi_i$ be a smooth confoliation on $V_i$ such that $\xi_i$ is tangent to $\partial_h V_i$ and everywhere transverse to $\partial_v V_i$. (Note that by smoothness, we may consider $\xi_i$ to be defined on an open neighborhood of $ V_i$.) We say the smooth confoliation $\xi_i$ [*dominates*]{} $\mathcal F$ if $\chi_{\xi_i} \ge \chi_{\mathcal F}$ on $\partial_v V_i$ when viewed from outside $V_i$, with equality permitted only on the closure, $\overline{U}$, of the smoothly foliated collar. Use [*strictly dominates*]{} if the inequality is strict.
Let $\xi_i$ be a smooth confoliation defined on $V_i$. We say that $(V_i,\xi_i)$ is [*smoothly $\Phi$-compatible*]{} with $(W_i,\mathcal F)$ if the following are satisfied:
1. $\xi_i$ and $\mathcal F_W$ are (positively) transverse to $\Phi$ on their domains of definition,
2. $\xi_i$ is tangent to $\partial_h V_i$
3. $\xi_i = T\mathcal F$ on $\overline{U}\cap V_i$,
4. $\xi_i$ is a contact structure on $V_i\setminus U$, and
5. $\xi_i$ dominates $\mathcal F$ (on $\partial_v V_i$).
\[inductthis\]Let $M = V \cup F_1 \cup \dots \cup F_n$ be a transitive flow box decomposition of $M$, and let $\xi_i$, $i\ge 0$, be a smooth confoliation defined on $V_i$ that is smoothly $\Phi$-compatible with $(W_i, \mathcal F)$. Then there is a smooth confoliation $\xi_{i+1}$ defined on $V_{i+1}$ that is smoothly $\Phi$-compatible with $(W_{i+1}, \mathcal F)$ and restricts to $\xi_i$ on $V_i$.
Since any polygon admits a triangulation, any transitive flow box decomposition can be chosen to consist only of flow boxes diffeomorphic to the flow box $G=\Delta\times [0,1]$ defined in Section \[basicsection\]. In particular, we may assume that $F_{i+1}$ is diffeomorphic to $G$. Such a diffeomorphism preserves the foliation and flow directions, that is, slopes $0$ and $\infty$; thus we will make slope comparisons and approximations without reference to the change of coordinates.
By hypothesis, $\xi$ strictly dominates $\mathcal F$ on $X=\partial_v V_i \setminus U$. By compactness, there exists $\epsilon>0$ such that $$\text{slope}(\xi_i)-\text{slope}(\mathcal F)> 3\epsilon$$ on $X\cap \partial_v F_{i+1}$.
By Proposition \[smoothapprox\], we may approximate the restriction of $\mathcal F$ to $F_{i+1}$ by a smooth foliation $\tilde{\mathcal F}$ such that $(\tilde{\mathcal F}, T\tilde{\mathcal F})$ is $C^0$-close to $(\mathcal F,T\mathcal F)$. Choose this approximation so that $\xi_i$ dominates $\tilde{\mathcal F}$ on $\partial_v(F_{i+1}) \cap V_i$, and so that $$|\text{slope}(\tilde{\mathcal F})-\text{slope}(\mathcal F)|<\epsilon$$ on all of $\partial_v F_{i+1}$. It follows that $$\text{slope}(\xi_i)-\text{slope}(\tilde{\mathcal F}) > 2\epsilon$$ on $X\cap \partial_v F_{i+1}$. Choose smooth coordinates $(x,y,z)$ on $F_{i+1}$ so that the leaves of $\tilde{\mathcal F}$ are horizontal, given by $z=$ constant. (Note that although this change of coordinates might change slope values, it doesn’t affect the relative values of slopes.)
Next consider the number of 2-cells contained in $\partial_v(F_{i+1}) \cap V_i$. From the definition of a transitive flow box decomposition there is at least one. Depending on whether there are exactly one, two, or three such 2-cells, apply the corresponding Corollary \[extend6\], \[extend7\], or \[extend8\] to smoothly extend $\xi_i$ across $F_{i+1}$ and call the resultant confoliation $\xi_{i+1}$. Smoothness of the glued confoliation is assured by the construction of the confoliations near the boundaries of their flow boxes.
In constructing the extensions of Corollaries \[extend5\]–\[extend8\] the starting point is a 1-form on $\partial_v F_{i+1}$ given by $dz-a(x,y,z)dx$ with $a(x,y,z)>0$ on $ X \cap \partial_v F_{i+1}$. This is then extended across $F_{i+1}$ while keeping $a(x,y,z)>0$ and also $\frac{\partial a}{\partial y}(x,y,z)>0$. Given any $\delta>0$, these constructions can be performed while keeping the change in $a(x,y,z)$ along Legendrian curves less than $\delta$. In other words, the extension $\xi_{i+1}$ can be chosen so that the change in $\text{slope}(\xi_{i+1})$ along Legendrian curves is less than $\epsilon$. Thus $$\text{slope}(\xi_{i+1})-\text{slope}(\tilde{\mathcal F}) > \epsilon$$ on $\partial_v V_{i+1} -(N_h\cup N_v)$, and consequently, on this set we also have $$\text{slope}(\xi_{i+1})-\text{slope}({\mathcal F}) > 0.$$
\[inductthiscor\] Let $M = V \cup F_1 \cup \dots \cup F_n$ be a transitive flow box decomposition of $M$, and let $\xi_i$, $i\ge 0$, be a smooth confoliation defined on $V_i$ that is smoothly $\Phi$-compatible with $(W_i, \mathcal F)$ and lies within $\epsilon$ of $\mathcal F$ on the intersection of the domain of $\mathcal F$ with $V_i$. Then there is a smooth confoliation $\xi_{i+1}$ defined on $V_{i+1}$ that is smoothly $\Phi$-compatible with $(W_{i+1}, \mathcal F)$, restricts to $\xi_i$ on $V_i$, and lies within $3\epsilon$ of $T\mathcal F$ on the intersection of the domain of $\mathcal F$ with $V_{i+1}$.
It follows from Proposition \[smoothapprox\] that a smooth foliation $\tilde{\mathcal F}$ may be chosen on $F_{i+1}$ so that $T\tilde{\mathcal F}$ is within $\epsilon$ of $T\mathcal F$. Restricting attention to $\partial_v V\cap F_{i+1}$, $T\tilde{\mathcal F}$ lies within $2\epsilon$ of $\xi_i$. From the proof of Proposition \[inductthis\], $\xi_{i+1}$ is constructed to be as close or closer to $T\tilde{\mathcal F}$ on $F_{i+1}$ than $\xi_i$ is on $\partial_v V\cap F_{i+1}$. Thus $\xi_{i+1}$ is within $2\epsilon$ of $T\tilde{\mathcal F}$, and hence within $3\epsilon$ of $T\mathcal F$.
\[almost\] Let $M = V \cup F_1 \cup \dots \cup F_n$ be a transitive flow box decomposition of $M$, and let $\xi_0$ be a smooth contact structure defined on $V$ that is compatible with $(W, \mathcal F)$. Then there is a smooth contact structure defined on $M$ that agrees with $\xi_0$ in $V$ and is $\Phi$-close to $\mathcal F$ on $W$.
Inductively applying Proposition \[inductthis\] produces a transitive smooth confoliation that is $\Phi$-close to $\mathcal F$ on $W$ and agrees with $\xi_0$ in $V$. By Proposition 2.8.1 of [@ET] (see also [@Et2]), this transitive smooth confoliation can be smoothly deformed into a smooth contact structure. Thus there is a smooth contact structure defined on $M$ that agrees with $\xi_0$ in $V$ and is $\Phi$-close to $\mathcal F$ on $W$.
Let $M = V \cup F_1 \cup \dots \cup F_n$ be a transitive flow box decomposition of $M$, and let $\xi_0$ be a smooth confoliation defined on $V$ that is $\Phi$-compatible with $(W, \mathcal F)$. Then there is a smooth contact structure defined on $M$ that agrees with $\xi_0$ in $V$ and is $\Phi$-close to $\mathcal F$ on $W$. If $\xi_0$ is within $\epsilon$ of $T\mathcal F$ along $\partial V=\partial W$, then $\xi$ can be chosen to lie within $f(n)\epsilon$ of $T\mathcal F$ in $W$, for some positive function $f$.
This follows immediately from Corollary \[inductthiscor\].
\[phitoC0\] Let $M$ be a closed oriented 3-manifold. Suppose that $M$ can be expressed as a union $$M=V\cup W,$$ such that $\partial V=\partial W$, $V$ admits a smooth contact structure $\xi_0$, $W$ admits a $C^0$-foliation $\mathcal F$, and $(V,\xi_0)$ and $(W,\mathcal F)$ are $\Phi$-compatible. Then $\xi_0$ can be modified in an arbitrarily small collar neighborhood of $\partial V$ so that the restriction of $\xi_0$ to $\partial_v V$ lies arbitrarily $C^0$-close to $T\mathcal F$.
Fix any $\epsilon>0$. Let $X=(\partial_v V) - (N_h\cup N_v).$ Let $X\times (-\delta,0]$ be a collar neighborhood of $X$ in $V$, with $X=X\times \{0\}$. Pick any smooth line field $l$ on $X$ which is tangent to $\mathcal F$ along the smoothly foliated collar, is dominated by the projection of $\xi_0|_{X\times \{ -\delta\}}$ to $X$, dominates the projection of $\mathcal F|_{X\times \{0\}}$ to $X$, and lies with $\epsilon$ of $\mathcal F|_{X\times \{0\}}$. Replace the restriction of $\xi_0$ to $X\times [-\delta,0]$ by a straight line homotopy between $\xi_0$ restricted to $X\times \{-\delta\}$ and $l$, damped to fit smoothly with the original $\xi_0$ as defined on the complement of the collar.
\[main1\] If $M$ admits a positive $(\xi_V,\mathcal F_W,\Phi)$ decomposition, then $M$ admits a smooth positive contact structure which agrees with $\xi_V$ on $V$ and is $\Phi$-close to $\mathcal F_W$ on $W$. Moreover, $M$ also admits a smooth contact structure which agrees with $\xi_0$ on the complement in $V$ of a collar neighborhood of $\partial V$ and is arbitrarily $C^0$-close to $\mathcal F$ on $W$. The analogous result holds if $M$ admits a negative $(\xi_{V'},\mathcal F_{W'},\Phi)$ decomposition. If $M$ admits both a positive $(\xi_V,\mathcal F_W,\Phi)$ decomposition and a negative $(\xi_{V'},\mathcal F_{W'},\Phi)$ decomposition, then these contact structures are weakly symplectically fillable.
This follows immediately from Proposition \[transitiveflowbox\], Corollary \[almost\], and Lemma \[phitoC0\].
Attracting holonomy {#attracting holonomy}
===================
This section gives a generalization of the Eliashberg-Thurston result on perturbing foliations in a neighborhood of a curve in a leaf with sometimes attracting holonomy (Proposition 2.5.1, [@ET]) to the larger class of $C^0$-foliations with holonomy containing a [*contracting interval*]{}. Holonomy with a contracting interval will be defined below in terms of the existence of a particular smooth submanifold with corners. Holonomy with a contracting interval is a weaker condition than holonomy with an attracting leaf.
Let $P$ be the prism in $\mathbb R^3$ given by $|x|\le 1$, $|y|\le 1$ and $|z|\le y/2 +3/2$. The slanted top and bottom of $P$ is denoted by $\partial_h P$, and the rest of $\partial P$ is denoted by $\partial_v P$. Let $V$ be the solid torus given by identifying each pair of points $(x,-1,z)$ and $(x,1,z)$ where both are in $P$. Then $\partial_h V$ is defined to be the image of $\partial_h P$, and $\partial_v V$ is defined to be the portion of $\partial V$ that is in the image of $\partial_v P$.
\[contracting\]
The holonomy of a $C^0$-foliation $\mathcal F$ transverse to a flow $\Phi$ in a 3-manifold $M$ has a [*contracting interval*]{} if there is a subset of $M$ diffeomorphic to $V$, called an [*attracting neighborhood*]{}, such that
1. $\partial_h V$ is mapped into leaves of $\mathcal F$ and
2. vertical intervals in $V$ are mapped to flow lines of $\Phi$.
\[attracting\] Let $\mathcal F$ be a taut oriented $C^0$-foliation transverse to a flow $\Phi$. If $V$ is a disjoint union of attracting neighborhoods, and $\mathcal F$ is $V$-transitive, then $\mathcal F$ can be $\Phi$-approximated both by a smooth positive contact structure and by a smooth negative contact structure. These contact structures are therefore weakly symplectically fillable and universally tight.
In light of Theorem \[main1\], it is enough to construct a smooth confoliation $\xi$ on $V$ such that $(V, \xi)$ is compatible with $(W,\mathcal F)$, in the sense of Definition \[compatible\]. It is sufficient to consider the case that $\xi$ is positive. The construction of the desired $\xi$ is the same on each component of $V$, so we will treat $V$ as if it is connected. Let $W$ denote the closure of the complement of $V$ in $M$.
![Foliated neighborhoods of horizontal faces and vertical edges are shown only in $Q$ though they exist in $P$ and $V=P/\sim$ as well.[]{data-label="QPV"}](QPV){width="5in"}
Consider the transformation of the prism $P$ to the cube $$Q=[-1,1] \times [-1,1] \times [-1,1]$$ that fixes $(x,y)$ and linearly scales the $z$-coordinate. View $V$ as the quotient $$V=Q/{\sim}$$ of $Q$ obtained by identifying $$(x,-1,z)\sim (x,1,z/2).$$ Notice that $\partial_h Q$ is mapped into leaves of $\mathcal F$, and vertical intervals in $Q$ are mapped to flow lines of $\Phi$. Moreover, we may assume that the original parametrization of $P$ was chosen so that $\mathcal F$ meets the $y=\pm 1$ sides of $\partial_v Q$ in horizontal lines. Since $\mathcal F$ is $C^0$, the leaves of the foliation meet each of the $x=\pm 1$ sides in a continuous family of smooth graphs.
To facilitate smooth gluings, thicken the leaf or leaves of $\mathcal F$ which meet $\partial_h Q$. Fix $0< \epsilon< 1/4$. Choose the thickening of the leaves of $\mathcal F$ intersecting $\partial_h V$ to replace the leaves $z=\pm 1$ in $Q$ with a disjoint union, $J$, of $I$-bundles in $Q$, with $J$ containing $[-1,1]\times[-1,1]\times [-1,-1+\epsilon]$ and $[-1,1]\times[-1,1]\times [1-\epsilon,1]$ as components. Assume also that $\mathcal F$ meets an $\epsilon$-neighborhood, $N$, of the quotient of the vertical edges of $Q$ in a smooth horizontal foliation. Let $U_Q = \text{int} (J\cup N)$; $U_Q$ is an $\epsilon$-neighbourhood of the union of $\partial_h Q$ with the 1-cells of $\partial_v Q$. We assume also that the pull-back of $\mathcal F$ to $\overline{U_Q}$ is horizontal. We will abuse notation and let $U_Q$ also refer to the projection of $U_Q$ to $M$.
Since $Q$ is a flow box it is amenable to the constructions of Section \[basicsection\]. We will $\Phi$-approximate $\mathcal F$ in $Q$, and thus in $V$, by a $C^{\infty}$-confoliation $\xi_0$ which smoothly respects both the identification $\sim$ and the gluing of $(V,\xi_0)$ and $(W,\mathcal F)$ along $\partial_h V=\partial_h W$.
The confoliation $\xi_0$ will be chosen to agree with $\mathcal F$ on $U_Q$ and to be a contact structure on $Q\setminus \overline{U_Q}$ which, when viewed from outside $Q$, is strictly dominated by $\mathcal F$ on the $y=-1$ side of $\partial_v Q$ and strictly dominates $\mathcal F$ on the remaining three sides of $\partial_v Q$. See Figure \[thecube\]. Since the interior of the $y=-1$ side lies in the interior of $V$, the resulting $(V,\xi_0)$ will be compatible with $(W,\mathcal F)$ in the sense of Definition \[compatible\].
![Some of the choices of flow lines of the vector fields $X_A,...$ and their relationship to $\mathcal F$ are shown. Not enough detail is drawn to show that the holonomy given by flowing from left to right, that is counterclockwise about $\partial_v Q$, is decreasing.[]{data-label="thecube"}](cube){width="5in"}
As a first step in constructing $\xi_0$, we define $\xi_0$ along $\partial_v Q$. The vertical boundary $\partial_v Q$ consists of four vertical faces. Let $A$ denote the face $y=-1$, let $B$ denote the face $x=1$, let $C$ denote the face $y=1$, and let $D$ denote the face $x=-1$.
We construct $\xi_0$ by first specifying smooth unit vector fields $X_A, X_B, X_C,$ and $X_D$ along the faces $A, B, C, $ and $D$ respectively, and then declaring $\xi_0$ along $\partial_v Q$ to be the 2-plane field which is normal to $\partial_v Q$ and contains the corresponding tangent vector $X_A, X_B, X_C,$ or $ X_D$. We will choose $X_A, X_B, X_C,$ and $X_D$ to be horizontal on $\overline{U_Q}$ and hence in a neighborhood of the vertical 1-simplices of $\partial_v Q$; in particular, the 2-plane field $\xi_0$ will therefore be well-defined on the vertical edges.
Begin by defining the vector field $X_B$. Choose $X_B=X_B(1,y,z)$ to be a smooth unit vector field which satisfies the following
1. $X_B$ dominates $\mathcal F$,
2. $X_B$ has positive slope when both $y$ and $z$ lie in $(-1+\epsilon, 1-\epsilon)$, and
3. $X_B =\partial/\partial y$ when $y$ or $z$ lies in $ [-1,-1+\epsilon]\cup [1-\epsilon,1]$.
Let $\Psi_B$ denote the flow generated by $X_B$. Abusing notation a bit, we denote by $\Psi_B(y,z)$ the intersection of of the flow line of $\Psi_B$ that starts at $(1,-1,z)$ with $\{(1,y)\}\times[-1,1]$. (We use this notation when referring to all flows in this section.) Let $f_B: [-1,1]\to [-1,1]$ denote the diffeomorphism given by $f_B(z) = \Psi_B(1,z)$.
Note that since $X_B$ has positive slope whenever both $t$ and $z$ lie in $(-1+\epsilon, 1-\epsilon)$, $f_B(z)>z$ whenever $z$ lies in $(-1+\epsilon, 1-\epsilon)$. Indeed, by choosing the slope of $X_B$ to be great enough, we may guarantee that $f_B(-1/2)\ge 1/2$. Rechoose $X_B$ as necessary so that $X_B$ satisfies (1)–(3) and also:\
(4) $f_B(-1/2)\ge 1/2$.\
Now choose $X_D$ along the side $x=-1$ by setting $X_D(-1,y,z) = X_B(1,-y,z)$.
Notice that if $f :[-1,1]\to [-1,1]$ is any orientation preserving diffeomorphism, then there is a smooth flow $\Psi=\Psi(x,z)$ on $[-1,1]\times [-1,1]$ such that $\Psi(1,z) = f(z)$; simply set $$\Psi(x,z)=\frac{1-t(x)}{2} z + \frac{1+t(x)}{2} f(z)$$ where $t(x)$ is a smooth function of $x$ that is $-1$ for $x\in[-1,-1+\epsilon]$, $1$ for $x\in [1-\epsilon, 1]$, and has positive derivative for all other $x$. Thus, by specifying a diffeomorphism $f:[-1,1]\to [-1,1]$, we specify a family of smooth flows $\Psi(x,z)$ and corresponding smooth unit tangent vector fields $X$.
We take advantage of this to define the vector field $X_A$ along the side $y = -1$. Let $f_A: [-1,1]\to [-1,1]$ be the diffeomorphism given by $$f_A^{-1}(z)=u\circ f_B \circ f_B\circ f_B(z),$$ where $u: [-1,1]\to [-1,1]$ is a diffeomorphism which is the identity on $z\in [-1,-1+\epsilon]\cup [1-\epsilon,1]$ and strictly increasing elsewhere. Let $\Psi_A$ be a smooth flow on $[-1,1]\times [-1,1]$ which satisfies the following:
1. $\Psi_A$ has negative slope whenever $z\in (-1+\epsilon, 1-\epsilon)$,
2. $\Psi_A$ has unit tangent vector field given by $\partial/\partial x$ when $z\in [-1,-1+\epsilon]\cup [1-\epsilon,1]$
3. $ \Psi_A(1,z) = f_A(z)$.
Let $X_A$ be the smooth unit tangent vector field to $\Psi_A$. Recall that $\mathcal F$ is horizontal, and hence is dominated by $X_A$, along $A$.
Similarly, we use a diffeomorphism $f_C:[-1,1]\to [-1,1]$ to define a smooth vector field $X_C$ along the side $y=1$. Let $f_C:[-1,1]\to [-1,1]$ be a diffeomorphism which satisfies:
1. $f_C(z)=1/2 f_A(2z)$ when $|z| \le 1/2$,
2. $f_C(z) = z$ when $z\in [-1,-1+\epsilon]\cup [1-\epsilon,1]$, and
3. $f_C(z)<f_B(z)$ when $ 1/2 < |z| < 1-\epsilon$.
Since $f_A(z)=z$ whenever $|z|\ge 1-\epsilon$, $f_C(z)=z$ for $(1-\epsilon)/2 \le |z|\le 1/2$. Therefore, $f_C[-1/2,1/2]=[-1/2,1/2]$. Since $f_B(-1/2)>1/2$, it follows that $f_C(z)\le f_B(z)$ for all $z$, with equality only when $|z|\ge 1-\epsilon$. So $f_B\circ f_C\circ f_B(z)<f_B\circ f_B\circ f_B (z)$, and hence $$f_A\circ f_B\circ f_C\circ f_B(z)< z,$$ for all $z \in (-1+\epsilon, 1-\epsilon).$
So the diffeomorphism $f_A\circ f_B\circ f_C\circ f_B : [-1,1]\to [-1,1]$ is strictly decreasing on $(-1+\epsilon,1-\epsilon)$ and hence Proposition \[cylinderextend\] applies.
\[smoothisbracketed\] Let $\mathcal F$ be a taut, oriented, $C^2$-foliation of a closed oriented 3-manifold. Suppose $\mathcal F$ is not the product foliation $S^1\times S^2$. Then $\mathcal F$ is bracketed.
Let $\Phi$ be a volume preserving flow transverse to $\mathcal F$. As noted in [@ET], either $\mathcal F$ has a lot of nontrivial linear holonomy or it is $C^0$ close, and hence $\Phi$-close, to a foliation $\mathcal F'$ which has a lot of nontrivial linear holonomy. In other words, there exists a disjoint union $V$ of attracting neighborhoods such that one of $\mathcal F$ or $\mathcal F'$ is $V$-transitive. In either case it follows that $\mathcal F$ is bracketed.
$L$-bracketed foliations {#L bracketed}
========================
In this section, we introduce a new method for $\Phi$-approximating a foliation $\mathcal F$ by a pair of transitive confoliations, one positive and one negative. This method applies whenever there exists a link transverse to $\mathcal F$ which satisfies the condition given in Definition \[transtranv\]. We also remark on some consequences yielding 3-manifolds containing weakly symplectically fillable contact structures.
Before stating Definition \[transtranv\], we make some preliminary observations.
\[flowextend\] Suppose $\mathcal F$ is a taut oriented codimension-1 foliation in $M$. Let $L$ be any link transverse to $\mathcal F$. Then there is a choice of metric on $M$ and volume preserving flow $\Phi$ everywhere transverse to $\mathcal F$ such that $L$ is contained as a union of closed orbits of $\Phi$. Moreover, given a choice of regular neighborhood $N(L) = \cup_i D_i\times S^1$ of $L$, the metric on $M$ and $\Phi$ can be chosen so that $\Phi$ is a trivial product $\{p\}\times S^1$, $p\in D_i$, on this regular neighborhood; in particular, $\Phi$ restricts to a flow on $\partial N(L)$.
This follows immediately from the proof of Theorem A1 found in [@Hass].
So any link transverse to $\mathcal F$ can be extended to a volume preserving flow $\Phi$ transverse to $\mathcal F$. Alternatively, we may begin with a volume preserving flow $\Phi$ and let $L$ be a collection of closed orbits of $\Phi$. Without loss of generality, we will restrict attention to the case that this flow $\Phi$ restricts to a flow on $\partial N(L)$ for some choice of regular neighborhood $N(L)$ of $L$. In either case, $\Phi$ determines a preferred, possibly non-compact, curve on each component of $\partial N(L)$:
\[torusfoln\] Let $T$ be a framed torus and let $\Phi$ be a flow on $T$. Then either $\Phi$ contains a simple closed curve of some rational slope $m_{\Phi}$ or $\Phi$ is topologically conjugate either to a foliation by lines of some irrational slope $m_{\Phi}$ or to a Denjoy blowup of a foliation by lines of some irrational slope $m_{\Phi}$. In each case, the slope $m_{\Phi}$ is uniquely determined (by $\Phi$).
As long as no leaf of $\Phi$ has slope $1/0$, the framing determines a unique realization of $\Phi$ as a suspension of some homeomorphism $f$ of $S^1$ and the Poincaré rotation number of $f$ determines the slope $m_{\Phi}$. See, for example, 4.3.1, 5.1.1 and 5.1.3 of [@HH].
![Slope convention on a component of $\partial N(L)$ in which slopes are designated as viewed from inside $N(L)$. []{data-label="slopeconvention"}](slopeconvention){width="3in"}
Denote this preferred isotopy class of curves, represented by either a simple closed leaf or an immersed $\mathbb R$, by $m_{\Phi}^T$. We are interested in the case that this torus $T$ is a component of $\partial N(L)$. In this case, there is also the isotopy class of the meridian, $\nu^T$ say, and $\nu^T\ne m_{\Phi}^T$. We shall call an isotopy class of a nontrivial curve $C$ in $T$ [*positive*]{} if it has positive slope with respect to $\langle \nu, m_{\Phi}\rangle$ when viewed from inside $N$. Similarly, we shall call an isotopy class of curves [*negative*]{} if it has negative slope with respect to $\langle \nu, m_{\Phi}\rangle$ when viewed from inside $N$. This convention is illustrated in Figure \[slopeconvention\].
Note that if $\mathcal F$ is an oriented codimension-1 foliation which intersects a torus $T$ transversely, then $\mathcal F\cap T$ is a flow on $T$, and $m_{\mathcal F}^T$ denotes the preferred isotopy class of this flow.
Define a triple $(M,\mathcal F, \Phi)$ to be [*coherent*]{} if the foliation $\mathcal F$ is taut and oriented, the flow $\Phi$ is volume preserving and positively transverse to $\mathcal F$, and the boundary of M, if nonempty, is a union of flow lines.
Let $L$ be a link in $M$. A foliation $\mathcal F$ is [*$L$-taut*]{} if $\mathcal F\pitchfork L$, and $\mathcal F$ is $L$-transitive, that is, each leaf of $\mathcal F$ has nonempty intersection with $L$. Similarly, if $\partial M\ne \emptyset$, then $\mathcal F$ is [*$\partial M$-taut*]{} if $\mathcal F$ intersects $\partial M$ transversely, with no Reeb annuli, and $\mathcal F$ is $\partial M$-transitive, that is, each leaf of $\mathcal F$ has nonempty intersection with $\partial M$.
Recall that a foliation $\mathcal F_0$ is said to [*realize slope*]{} $m$ along a framed torus boundary component $T$ if $\partial \mathcal F_0\cap T$ consists of parallel curves, not necessarily compact, of slope $x$. When $\mathcal F_0$ is oriented, these curves $\partial \mathcal F_0\cap T$ are necessarily consistently oriented. Notice that the condition that a foliation $\mathcal F_0$ be $\partial M_0$-taut is weaker than the condition that $\partial F_0$ realizes slope $m_{\mathcal F_0}^T$ for each component $T$ of $\partial M_0$; in other words, nontrivial holonomy is possible for $\mathcal F_0\cap \partial M_0$.
\[transtranv\] Suppose $\mathcal F$ is a taut oriented codimension-1 foliation in $M$. Let $L$ be a link in $M$ which is transverse to $\mathcal F$ and let $M_0$ equal $M\setminus \text{int}N(L)$. Let $\mathcal F_0$ denote the restriction of $\mathcal F$ to $M_0$. The foliation $\mathcal F$ is [*$L$-bracketed*]{} if, for some choice of metric on $M_0$, there is a volume preserving flow $\Phi_0$ on $M_0$ such that $(M_0,\mathcal F_0,\Phi_0)$ is coherent and the following property is satisfied:
$M_0$ contains a pair of foliations $\mathcal F_{\pm}$ such that
1. $(M_0,\mathcal F_+,\Phi_0)$ and $(M_0,\mathcal F_-,\Phi_0)$ are coherent,
2. $\mathcal F_{\pm}$ are $\partial M_0$-taut, and
3. for each component $T$ of $\partial M_0$, $m_{\mathcal F_-}^T$ is negative and $m_{\mathcal F_+}^T$ is positive with respect to $\langle \nu^T, \Phi|_T\rangle$, where $\nu^T$ is the meridian slope of component $T$ (and hence is the slope of $\mathcal F_0\cap T$).
To make the flow explicit, we also say that $M$ contains an [*$(\mathcal F,\Phi)$-transitive*]{} link $L$, where $\Phi$ is the flow $\Phi_0$ blown down to a (volume preserving) flow on $M$.
The notion of $L$-bracketed is a special case of a bracketed foliation $\mathcal F$. The decomposition is given by setting $V=V'=N(L)$, $W=W'=M_0$. There is a canonical choice of positive or negative contact structure on $N(L)$ which is given by perturbing the meridional disks, and then the requirement is to find foliations $\mathcal F_W=\mathcal F_+$ and $\mathcal F_{W'}=\mathcal F_-$ on $M_0$.
\[transitivemain\] Suppose $\mathcal F$ is a taut oriented codimension-1 foliation in $M$ and that $\mathcal F$ is $L$-bracketed for some link $L$. Then $\mathcal F$ can be $\Phi$-approximated by a pair of smooth contact structures $\xi_{\pm}$, one positive and one negative. These contact structures are necessarily weakly symplectically fillable and universally tight.
Set $V= N(L)$, and let $W$ denote the closure of the complement of $V$. Define a contact structure $\xi_0$ on $N(L)$ so that each component of $L$ is a transverse knot and each component of $N(L)$ is a standard positive contact neighborhood of its core. Choosing the rate of rotation of the contact planes along each meridional disk to be small guarantees that the characteristic foliation of $\xi_0$ along $\partial N(L)$ is close to the meridian. It follows that we may choose $\xi_0$ so that it is strictly dominated by $\mathcal F_+$. Apply Theorem \[main1\] to obtain $\xi_+$. Similarly, each component of $N(L)$ can be modeled using the standard negative radial model, and Theorem \[main1\] can be applied to obtain $\xi_-$.
Since $\xi_{\pm}$ are both positively transverse to the volume preserving flow $\Phi$, they are weakly symplectically fillable.
Since transitive links are somewhat mysterious, it is natural to ask:
Given a foliation $\mathcal F$, does there exist a link $L$ for which $\mathcal F$ is $L$-bracketed?
It is not clear how the answer to this might change if the link is required to be connected.
Given a foliation $\mathcal F$, does there exist a knot $K$ for which $\mathcal F$ is $K$-bracketed?
\[s1timess2\] The product foliation $\mathcal F$ on $S^1 \times S^2$ is an example of a foliation which is not $L$-bracketed for any link $L$. The existence of such a link would imply, by Theorem \[transitivemain\], that $\mathcal F$ can be approximated by a tight contact structure $\xi$. This would imply that the underlying 2-plane bundles of $\mathcal F$ and $\xi$ are equivalent. The Euler class of the foliation, $e(\mathcal F)$, evaluated on a spherical leaf $S^2$ of $\mathcal F$ equals 2. On the other hand, this $S^2$ is homotopic to a convex surface $S'$, [@Gi1], in a tight contact structure. It follows that $S'$ has a connected dividing set, [@Gi2], and that $e(\xi)$ vanishes on it.
A taut foliation $\mathcal F$ is certainly $L$-taut for some link $L$. In fact, it is $L$-taut for some knot $L$. Moreover, as noted above, for some choice of metric there is a volume preserving flow $\Phi$ transverse to $\mathcal F$ and containing $L$ as an orbit or union of closed orbits. And often, although not necessarily, foliations on $M_0$ that are $\Phi_0$-close to $\mathcal F$ will also be $L$-taut. So a key question is the existence of a pair of $\Phi_0$-close foliations $\mathcal F_{\pm}$ in $M_0$ such that $m_{\mathcal F_-}^T< \text{slope}\, \partial \mathcal F_0|_T< m_{\mathcal F_+}^T$ for each component $T$.
Consider the case that $M_0$ is any compact orientable manifold with boundary a nonempty union of $b$ tori. Suppose that $\Phi_0$ is a volume preserving flow which is tangent to $\partial M_0$ and that $B$ is a transversely oriented branched surface transverse to $\Phi_0$.
If $B$ fully carries a set of foliations which are $\partial M_0$-taut and realize a nonempty open set $J$ of boundary slopes (if $b=1$) or multi-slopes (if $b\ge 2$), then Dehn-filling $M_0$ along any rational slope or multi-slope in $J$ results in a foliation which is $L$-bracketed, where $L$ is the link which is the core of the Dehn filling. Examples of such foliations can be found in the papers [@DL; @DR; @g1; @g2; @g3; @G3; @KRo; @KR2; @Li2; @LR; @R; @R1; @R2].
One can ask whether the foliations constructed by Dehn filling more than one torus can be $K$-bracketed for some knot $K$.
In [@G1; @G2; @G3] Gabai constructs foliations in closed manifolds $M$ with $H_2(M)\ne 0$. These foliations are fully carried by finite depth branched surfaces. One can ask whether such foliations are $L$-bracketed for some link $L$.
Finally, we note that the proof of Theorem \[transitivemain\] doesn’t actually require the existence of the foliation $\mathcal F_0$. More precisely, we have the following.
\[tranvlink\] Suppose $(M,\xi)$ is a contact 3-manifold. Let $L$ be a transverse link in $(M,\xi)$. Let $M_0$ equal $M\setminus \text{int}N(L)$. The contact structure $\xi$ is [*$L$-bracketed*]{} if, for some choice of metric on $M_0$, there is a volume preserving flow $\Phi_0$ on $M_0$, tangent to $\partial M_0$, such that the following property is satisfied:
$M_0$ contains a pair of foliations $\mathcal F_{\pm}$ such that
1. $(M_0,\mathcal F_+,\Phi_0)$ and $(M_0,\mathcal F_-,\Phi_0)$ are coherent,
2. $\mathcal F_{\pm}$ are $\partial M_0$-taut, and
3. for each component $T$ of $\partial M_0$, $m_{\mathcal F_-}^T$ is negative and $m_{\mathcal F_+}^T$ is positive with respect to $\langle \nu^T, \Phi|_T\rangle$, where $\nu^T$ is the meridian slope of component $T$.
\[transitivemaincontact\] Suppose $(M,\xi)$ is a contact 3-manifold and that $\xi$ is $L$-bracketed for some transverse link $L$. Then $\xi$ can be $\Phi$-approximated by a pair of smooth contact structures $\xi_{\pm}$, one positive and one negative. These contact structures are necessarily weakly symplectically fillable and universally tight.
Open book decompositions {#Open book}
========================
An interesting class of $L$-bracketed foliations is obtained by considering the special case that $L$ is a fibered link in $M$ and $\mathcal F$ is transverse to a flow $\Phi$ obtained by surgery from a volume preserving suspension flow of the corresponding fibre bundle complement of $L$. In this case $L$ forms the binding of an open book decomposition $(S,h)$ of $M$ and the contact structure $\xi_{(S,h)}$ compatible with $(S,h)$ is $\Phi$-close to $\mathcal F$.
For completeness, we begin with some standard definitions. Since we are relating ideas from the world of codimension-1 foliations and the world of contact structures, we will also provide some translations between the terminologies of these two worlds. The main results of this section appear in Subsection \[OBresults\].
Open book decompositions {#open-book-decompositions}
------------------------
Let $S$ be a compact surface with nonempty boundary. A pair $(S,h)$, where $h$ is a homeomorphism that restricts to the identity map on $\partial S$, determines a closed 3-manifold $M=S\times [0,1]/\approx$ where the equivalence relation $\approx$ identifies $(x,1)\approx(h(x),0)$ for all $x\in S$ and $(x,s)\approx(x,t)$ for all $x\in \partial S$ and $s,t\in [0,1]$. The singular fibration with pages $S\times\{t\}$ is called the [*open book determined by the data $(S,h)$,*]{} and we write $M=(S,h)$.
Surface bundles over $S^1$
--------------------------
Corresponding to an open book decomposition of $M$ is a description of $M$ as a Dehn surgery along the binding $L=\cup l_i$ by meridional multislope $(\nu_1,...\nu_b)$. Conversely, corresponding to such a Dehn filling description of $M$, we have a corresponding open book description of $M$. Since many existing constructions of foliations are described from the Dehn surgery perspective, it is useful to consider this correspondence more carefully.
Let $M_0$ denote the compact complement of $L$; so $M_0= M\setminus \text{int} N(L)$, where $N(L)$ is a regular neighbourhood of $L$, and $M_0$ is homeomorphic to $S\times [0,1]/h$.
Notice that if $S$ is a disk, then necessarily $h$ is isotopic rel boundary to the identity map. If $S$ is an annulus, then $h$ is isotopic rel boundary to some power of the Dehn twist about the core of $S$. Otherwise, $S$ is hyperbolic. We therefore lose little by restricting attention to the case that $S$ is hyperbolic and will now do so.
Recall Thurston’s classification of surface automorphisms.
\[Thurston\] [@Th; @CB; @FLP] Let $S$ be an oriented hyperbolic surface with geodesic boundary, and let $h\in \text{Homeo}(S,\partial S)$. Then $h$ is freely isotopic to either
\(1) a pseudo-Anosov homeomorphism $\theta$,
\(2) a periodic homeomorphism $\theta$, in which case there is a hyperbolic metric for which $S$ has geodesic boundary and such that $\theta$ is an isometry of $S$, or
\(3) a reducible homeomorphism $h'$ that fixes, setwise, a maximal collection of disjoint simple closed geodesic curves $\{C_j\}$ in $S$.
Recall that a pseudo-Anosov homeomorphism has finitely many prong singularities and is smooth and hyperbolic elsewhere [@FLP]. To avoid overlap in the cases, we refer to a map as reducible only if it is not periodic. Since we will be considering homeomorphisms $h$ in the context of open books $(S,h)$, we will be considering only homeomorphisms $h$ which fix $\partial S$ pointwise. Therefore, given a reducible map, splitting $S$ along $\cup_jC_j$ gives a collection of surfaces $S_1,\dots, S_n \subset S$ with geodesic boundary that are fixed by $h'$. Maximality of $\{C_j\}$ implies that applying Thurston’s classification theorem to each $h'|_{S_i}\in \text{Homeo}(S_i, \partial S_i)$ produces either a pseudo-Anosov or periodic representative. So we may assume that $h'$ is either periodic or pseudo-Anosov away from some small neighborhood of the $C_i$.
\[Thurstonrep\] Let $S$ be hyperbolic and $h\in \text{Homeo}(S,\partial S)$. If conclusion (1) or conclusion (2) of Theorem \[Thurston\] is satisfied, call $\theta$ *the Thurston representative* of $h$. If instead conclusion (3) holds, let $\theta: (S,\partial S)\to (S,\partial S)$ denote the piecewise continuous function uniquely determined by the following constraints:
- $\theta$ restricted to each component of the complement of the union $\cup C_i$ is freely isotopic to the restriction of $h'$ to this component and is either periodic or pseudo-Anosov.
- $\theta$ restricted to each simple closed geodesic $C_i$ is freely isotopic to the restriction of $h'$ to $C_i$ and is a periodic isometry, and
Again, refer to $\theta$ as *the Thurston representative* of $h$.
Now consider again the open book decomposition $M=(S,h)$ and let $\theta$ denote the Thurston representative of $h$. When $\theta$ is periodic or pseudo-Anosov, the link complement $M_0 = M\setminus \text{int} N(L)$ is also homeomorphic to the mapping torus $S\times [0,1]/(x,1)\sim (\theta(x),0)$ of $\theta$, and in the discussions which follow, we will typically view $M_0$ as the mapping torus of $\theta$. When $\theta$ is reducible and so only piecewise continuous, we will typically view $M_0$ as the union along essential tori of the mapping tori of the extension to $\cup S_i$ of the restriction of $\theta$ to the complement of the union $\cup C_i$.
Let $\Theta_0$ be the flow obtained by integrating the vector field $\partial/\partial t$, where points of $M_0$ are given by $[(x,t)], x\in S, t\in [0,1]$. We will refer to this flow as either the [*suspension flow of $\theta$*]{} or the [*Thurston flow (associated to $h$)*]{}.
Since $\theta$ is area preserving with respect to some metric on the fibre, $\Theta_0$ is volume preserving with respect to some choice of metric on $M_0$.
Notice that the suspension flow $\Theta_0$ is pseudo-Anosov (respectively, periodic) when $\theta$ is pseudo-Anosov (respectively, periodic). In particular, when $\theta$ is periodic, all orbits of $\Theta_0$ are closed. When $\theta$ is pseudo-Anosov, there are an even number of alternately attracting and repelling closed orbits of $\Theta_0|{\partial N(l_i)}$ for each $i$. When $\theta$ is periodic or pseudo-Anosov, this flow is continuous and the orbits are smoothly embedded. When $\theta$ is reducible, this flow is not continuous, but orbits of the flow are smoothly embedded.
Since many closed manifolds, together with corresponding open book decompositions, can be realized by Dehn filling $M_0$, it is useful to have canonical framings on the boundary components of $M_0$ which are defined independently from $M$. As in [@R2], we will use the Thurston flow $\Theta_0$ to define these canonical coordinate systems on $\partial M_0$.
The Thurston flow framing on surface bundles over $S^1$
-------------------------------------------------------
\[coordinatesgalore\] Let $\partial_iM_0$ denote the $i$-th boundary component of $M_0$. Choose an oriented identification $\partial_i M_0 \sim \mathbb R^2/\mathbb Z^2$ by choosing oriented curves $\lambda_i$ and $\mu_i$, so that $\lambda_i$ has slope $0$ and $\mu_i$ has slope $\infty$, as follows. Let $\lambda_i = \partial (S\times \{0\})$, with orientation induced by the orientation on $S$. Let $\gamma_i$ be a closed orbit of the flow $\Theta_0$ restricted to $\partial_iM_0$. Choose $\mu_i$ to be an oriented simple closed curve which has algebraic intersection number $< \lambda_i,\mu_i> = 1$ and which minimizes the geometric intersection number $|\gamma_i\cap\mu_i|$. This choice is unique except in the case that the geometric intersection number $|\gamma_i\cap\lambda_i| = 2$. In this case we choose $\mu_i$ so that $\gamma_i$ has slope $+2$. Call the resulting framing the *Thurston flow framing* on $\partial_iM_0$. Slopes expressed in terms of the flow framing will be said to be given in [*Thurston flow coordinates*]{}. This was originally called the [*natural framing*]{} or [*natural coordinates*]{} by Roberts in [@R2]. In this section, we are beginning with a fixed fibering and hence the associated Thurston flow coordinates are well defined. In general, different choices of fibering can lead to nonisotopic closed orbits $\gamma_i$, and hence to different Thurston flow coordinates. Notice also that in these coordinates, the slope of $\gamma_i$ always satisfies $$1/ (\mbox{slope } \gamma_i) \in (-1/2,1/2].$$
Now let’s consider the relationship between the flow framing and the fractional Dehn twist coefficient defined by Honda, Kazez, and Matić in [@HKMRV1]. First recall the definition of fractional Dehn twist coefficient.
[@HKMRV1] Fixing a component $C_i$ of $\partial S$ and restricting the flow $\Theta_0$ to the component of $\partial N(L)$ corresponding to $C_i$, $\Theta_0$ necessarily has periodic orbits. Let $\gamma_i$ be one such, and write $$\gamma_i = p_i\lambda_i+q_i\nu_i$$ where $\lambda_i = C_i$, $\nu_i$ is the meridian (oriented so that $<\lambda_i,\nu_i>=1$), and $p_i$ and $q_i$ are relatively prime integers with $q_i>0$. The [*fractional Dehn twist coefficient*]{} of $h$ with respect to the component $C_i$ of $\partial S$ is given by $$c_i(h)=p_i/q_i\, .$$ In particular, when $\gamma_i=\nu_i, (p_i,q_i)=(0,1)$ and $c_i(h)=0.$
Recall that $M$ is obtained from $M_0$ by $(\nu_1,...,\nu_b)$ Dehn filling along the boundary components of $M_0$. Beginning with the open book decomposition of $M$ and the associated fractional Dehn twist coefficients $c_i=p_i/q_i, 1\le i\le b,$ (as above, expressed in $(\lambda_i,\nu_i)$ coordinates), it is sometimes useful to express the slopes of $\nu_i$ and $\gamma_i$ in terms of the Thurston flow coordinates, $(\lambda_i,\mu_i)$. We now describe how to do this. In flow coordinates, $\lambda_i$ has slope $0$. Since $|\lambda_i\cap\nu_i| = 1$, it follows that, in flow coordinates, $\nu_i$ has slope $1/k_i$ for some integer $k_i$. As noted in [@HKM2], the integer $k_i$ is uniquely determined by the fractional Dehn twist coefficient $c_i(h)$ for each $i, 1\le i \le b$. This relationship can be very simply stated:
\[translation1\][**(Coordinate translation I)**]{} Let $c_i = c_i(h)$ and let $n_i$ be the integer determined by the condition $$c_i\in (n_i-1/2,n_i+1/2].$$ In other words, $$n_i=\lceil c_i -1/2 \rceil,$$ the integer nearest to $c_i$, with ties in the case $c_i\in \mathbb Z+1/2$ broken by rounding down.
Then $k_i = -n_i$ and so $\nu_i$ has slope $-1/n_i$. Moreover, $\gamma_i$ has slope $1/(c_i-n_i)$ and so
1. if $c_i = n_i$, then $\gamma_i=\mu_i$ has slope $1/0$;
2. if $c_i > n_i$, then $\gamma_i$ has positive slope; and
3. if $c_i<n_i$, then $\gamma_i$ has negative slope.
The meridian $\nu_i$ has slope $1/k_i$ and so $\nu_i = \mu_i + k_i\lambda_i$. So $$\gamma_i = p_i\lambda_i+ q_i\nu_i = p_i\lambda_i + q_i(\mu_i + k_i\lambda_i) = q_i\mu_i +(p_i+k_iq_i)\lambda_i$$ has slope ${q_i}/(p_i+k_iq_i) = {1}/(c_i+k_i)$ and $|\gamma_i\cap\mu_i| = |p_i+k_iq_i|$. By definition of flow coordinates, $k_i$ is chosen to minimize $|\gamma_i\cap\mu_i| = |p_i+k_iq_i|$, and hence to minimize $|c_i + k_i|$. There is a unique such minimizing $k_i$ unless $c_i\in \mathbb Z + 1/2$. In this case, $k_i$ is chosen so that $\gamma_i$ has slope $2/1$; namely, so that $c_i = -k_i + 1/2$. So $k_i$ is the unique integer satisfying $c_i\in (-k_i-1/2,-k_i + 1/2]$.
Conversely, given the fibered 3-manifold $M_0$ and meridional Dehn filling slopes $\nu_i, 1\le i\le b,$ in terms of the Thurston flow coordinates $(\lambda_i,\mu_i)$, it is often useful to express the slopes $\nu_i$ and $\gamma_i$ in terms of the associated open book coordinates $(\lambda_i,\nu_i)$. We have the following.
\[translation2\][**(Coordinate translation (II)**]{} Suppose $M_0$ is fibered, with the boundary components $\partial_i M_0$ given the Thurston flow framing $(\lambda_i,\mu_i)$ for each $i$. As above, let $\nu_i$ be a meridional slope and let $\gamma_i$ be a closed orbit of the Thurston flow on $\partial_i M_0$. In terms of the Thurston framing, $\nu_i=-1/n_i$ and $\gamma_i=r_i/s_i$, for some integers $n_i,r_i$ and $s_i$. Again as above, let $M$ be the manifold obtained by $(\nu_1,...,\nu_b)$ filling $M_0$ and let $(S,h)$ be the open book decomposition of $M$ determined by the fibering of $M_0$. Then, in terms of the open book framing $(\lambda_i,\nu_i)$ on $\partial_i M_0 = \partial N(l_i)$, $$\nu_i=1/0, \,\,\, \mu_i = 1/n_i \,\,\, \mbox{ and } \gamma_i = r_i/(n_i r_i + s_i).$$ In particular, the fractional Dehn twist coefficient along or $S \cap \partial_iM_0$ is given by $$c_i(h) = n_i+s_i/r_i,$$where, as noted in Definition \[coordinatesgalore\], $s_i/r_i\in (-1/2,1/2]$.
To eliminate the ugliness of subscripts, focus on a particular boundary component, $\partial_i M_0$, and drop all reference to $i$.
Two right handed framings are related by a unique transformation in $SL_2(\mathbb Z)$. Notice that $$A=
\left[ {\begin{array}{cc}
1 & n \\
0 & 1 \\
\end{array} } \right]$$ is the element in $SL_2(\mathbb Z)$ which maps the pair $({1 \choose 0},{-n \choose 1})$ to the pair $({1 \choose 0},{0 \choose 1})$.
Hence, with the correspondence ${a\choose b} \mapsto b/a$, $A$ describes the translation from $(\lambda,\mu)$ coordinates to $(\lambda,\nu)$ coordinates. The slope computations follow immediately.
Dehn filling the Thurston flow
------------------------------
Dehn surgery on Anosov flows is defined by Goodman in [@Go]. (See also [@Fried].) This definition generalizes naturally to the setting of pseudo-Anosov flows and permits us to consider the effect of Dehn filling of Thurston flows. We define Dehn filling of a Thurston flow as follows. Let $Y$ be any closed 3-manifold obtained by Dehn filling $M_0$. For each $i, 1\le i\le b$, let $X_i$ denote the solid torus in $Y$ bounded by $\partial_i M_0$ and let $\kappa_i$ denote the core of $X_i$. As long as the surgery coefficient along $\partial_i M_0$ is not $\gamma_i$, it is possible to blow down $X_i$ to its core and obtain a flow $\Theta$ defined on $Y$. Notice that the cores $\kappa_i$ are closed orbits of $\Theta$. Also, either $\Theta_0$ is periodic in a neighborhood of $\partial_i M_0$ and therefore so is $\Theta$ in a neighborhood of $\kappa_i$, or else $\Theta_0$ is pseudo-Anosov (with possibly a single prong pair along $\kappa_i$) in a neighborhood of $\partial_i M_0$ and therefore so is $\Theta$ in a neighborhood of $\kappa_i$. We shall refer to this flow $\Theta$ as the [*surgered Thurston flow*]{}. Notice that since $\Theta_0$ is volume preserving for some metric, so is $\Theta$.
Suppose $M$ has an open book decomposition $(S,h)$ with binding $L$ and corresponding Dehn filling description $M=M_0(\nu_1,\cdots,\nu_b)$. Let $\Theta_0$ denote the Thurston flow on the complement of $L$. Then $\Theta_0$ extends to a flow $\Theta$ on $M$ if and only if all fractional Dehn twist coefficients $c_i$ are nonzero.
Notice that $$c_i(h)=0 \iff p_i = 0, q_i = 1 \iff \gamma_i = \nu_i \implies \nu_i = \mu_i = \gamma_i.$$ In particular, if at every component $C_i$ of $\partial S$ the fractional Dehn twist coefficient $c_i(h)\ne 0$, then $\nu_i\ne\gamma_i$ and so it is possible to blow down the flow $\Theta_0$ to the surgered Thurston flow $\Theta$ on $M$. Otherwise it is not.
Notice that the binding $L=\cup l_i$ inherits orientations both from the open book structure and from the flow $\Theta$.
![ Curves on a boundary component of $M_0$.[]{data-label="postrans"}](postrans){width="4.2in"}
\[bindingorientation\] When $c_i>0$, these orientations agree on $l_i$. When $c_i<0$, these orientations do not agree on $l_i$.
The orientation of $\Theta$ restricted to the binding is determined by the sign of the slope of $\gamma_i$ as expressed in $(\lambda_i,\nu_i)$ coordinates. This is illustrated in Figure \[postrans\].
Contact structures supported by an open book
--------------------------------------------
In [@Gi], Giroux defined the notion of contact structure supported by an open book: a positive (respectively, negative) contact structure $\xi$ on $M$ is [*supported by, or compatible with, an open book decomposition $(S,h)$ of $M$*]{} if $\xi$ can be isotoped through contact structures so that there is a contact 1-form $\alpha$ for $\xi$ such that
1. $d\alpha$ is a positive area form on each page $S_t$ of the open book and
2. $\alpha>0$ (respectively, $\alpha<0$) on the binding $l$.
([@Gi]) Two contact structures supported by the same open book are contact isotopic.
We may therefore abuse language and refer to [*the*]{} contact structure compatible with the open book decomposition $(S,h)$.
Let $(\xi_+)_{(S,h)}$ denote the positive contact structure compatible with the open book decomposition $(S,h)$. Let $(\xi_-)_{(S,h)}$ denote the negative contact structure compatible with the open book decomposition $(S,h)$.
\[posnegc\] Suppose all fractional Dehn twist coefficients are nonzero and let $\Theta$ denote the surgered Thurston flow.
1. $(\xi_+)_{(S,h)}$ is positively transverse to $\Theta$ if and only if all fractional Dehn twist coefficients are positive.
2. $(\xi_-)_{(S,h)}$ is positively transverse to $\Theta$ if and only if all fractional Dehn twist coefficients are negative.
This follows immediately from Lemma \[bindingorientation\].
Any contact structure $(M,\xi)$ is supported by infinitely many open book decompositions. Honda, Kazez, and Matić proved in [@HKM2] that if there is a compatible open book decomposition with nonpositive fractional Dehn twist coefficient $c_i$ for some $i$, then necessarily $\xi$ is overtwisted. In fact, they show the following.
\[Theorem 1.1 of [@HKMRV1]\] A contact structure $(M,\xi)$ is tight if and only if all of its compatible open book decompositions $(S,h)$ have fractional Dehn twist coefficients $c_i \ge 0$ for $1\le i\le |\partial S|$.
Let $S'$ be the smallest invariant subsurface of $S$ for the Thurston representative of $h$; thus $S'=S$ if and only if $h$ is not reducible. If $c_i=0$ for some common boundary component of $S$ and $S'$, there are two possibilities for $h|S'$. The first is that it is periodic, and it follows that it is equal to the identity map. The second possibility is that $h|S'$ is pseudo-Anosov, but this immediately implies $\xi$ is overtwisted [@HKMRV1]. Since our primary focus is studying tightness for pseudo-Anosov maps it is enough to consider open book decompositions for which all fractional Dehn twist coefficients $c_i$ are positive (or, in the case that the contact structure is negative, to a consideration of open book decompositions for which all fractional Dehn twist coefficients $c_i$ are negative).
Foliations compatible with an open book decomposition {#OBresults}
-----------------------------------------------------
Let $\mathcal F$ be an oriented foliation of $M$. Let $(S,h)$ be an open book decomposition of $M$, with binding $L$. Let $\Phi$ be the surgered Thurston flow associated to $h$. If $\mathcal F$ is everywhere transverse to $\Phi$, then we say that $\mathcal F$ is [*compatible with the open book*]{} $(S,h)$.
\[oppfoln\] Let $(S,h)$ be an open book decomposition of $M$, with binding $L$, and surgered Thurston flow $\Phi$. Set $M_0=M\setminus \text{int} N(L)$ and $\Phi_0$ to be the Thurston flow associated to $h$.
1. Suppose all fractional Dehn twist coefficients are positive. If there exists a foliation $\mathcal F_-$ in $M_0$ which is $L$-taut, is transverse to $\Phi_0$ and satisfies $m_{\mathcal F_-}^T < 0$ with respect to $\langle \nu^T,\Phi|_T \rangle$ for all components $T$ of $\partial M_0$, then $(\xi_+)_{(S,h)}$ is weakly symplectically fillable.
2. Suppose all fractional Dehn twist coefficients are negative. If there exists a foliation $\mathcal F_+$ in $M_0$ which is $L$-taut, is transverse to $\Phi_0$ and satisfies $m_{\mathcal F_+}^T > 0$ with respect to $\langle \nu^T,\Phi|_T \rangle$ for all components $T$ of $\partial M_0$, then $(\xi_-)_{(S,h)}$ is weakly symplectically fillable.
Consider case (1). Since $(\xi_+)_{(S,h)}$ is transverse to $\Phi$, it suffices to show that there is a negative contact structure $\xi_-$ which is transverse to $\Phi$. This follows immediately from the assumptions by Theorem \[main1\].
Case (2) follows similarly.
Set $\mathcal G_0$ to be the fibration $S\times [0,1]/h$ in the complement of $L$. Note that $\mathcal G_0\cap T$ has either positive or negative slope with respect to $\langle \nu^T,\Phi|_T \rangle$, on each component $T$ of $\partial N(L)$. For completeness, we note the following:
Let $T_i$ be a component of of $\partial N(L)$. Then $\mathcal G_0\cap T_i$ has positive (respectively, negative) slope with respect to $\langle \nu^{T_i},\Phi|_{T_i}\rangle$ if the corresponding fractional Dehn twist $c_i$ is positive (respectively, negative).
Consider the relative slope values of $\gamma, \lambda$, and $\nu$ on any boundary component $T_i$ of $\partial M_0$. This is captured in Figure \[postrans\]. Notice that $\lambda$ represents the slope of $\partial \mathcal G_0$ and $\gamma$ represents the slope of $\Phi$. Recalling the slope convention, illustrated in Figure \[slopeconvention\], we see that $\mathcal G_0\cap T_i$ has positive (respectively, negative) slope with respect to $\langle \nu^{T_i},\Phi|_{T_i}\rangle$ if the corresponding fractional Dehn twist $c_i$ is positive (respectively, negative).
So $\mathcal G_0$ extends to a positive confoliation transverse to $\Phi$ when $c_i>0$ for all $i$ and to a negative confoliation transverse to $\Phi$ when $c_i<0$ for all $i$. In other words, and unsurprisingly, $\mathcal G_0$ as a foliation playing the role of $\mathcal F_{+}$ (respectively, $\mathcal F_-$) gives a second way of establishing the existence of $\xi_+$ (respectively $\xi_-$).
Finally, we use Proposition \[translation1\] to rephrase Proposition \[oppfoln\] in terms of Thurston flow coordinates. For simplicity of exposition, we restrict attention to the case that all fractional Dehn twist coefficients are positive. There is a symmetric statement in the case that all fractional Dehn twist coefficients are negative.
\[oppfolnTh\] Let $(S,h)$ be an open book decomposition of $M$, with binding $L$. Suppose all fractional Dehn twist coefficients $c_i, 1\le i\le b,$ are positive. For each $i$, let $n_i$ be the integer nearest to $c_i$, with ties in the case $c_i\in \mathbb Z+1/2$ broken by rounding down. Set $M_0=M\setminus \text{int} N(L)$. Let $\Phi_0$ denote the Thurston flow associated to $h$ and let $\Phi$ denote the surgered Thurston flow on $M$.
Suppose there exists a foliation $\mathcal F_-$ in $M_0$ which is $L$-taut, is transverse to $\Phi$, and such that, for each component $T_i$ of $\partial M_0$, $c_i$, $n_i$ and $$m_i = m^{T_i}_{\mathcal F_-}$$ satisfy one of the following:
1. $c_i=n_i$ and $m_i\in (-\infty,-\frac{1}{n_i})$,
2. $c_i>n_i$ and $m_i\in (\frac{1}{c_i-n_i},\infty] \cup [-\infty,-\frac{1}{n_i})$, or
3. $c_i<n_i$ and $m_i\in (\frac{1}{c_i-n_i},-\frac{1}{n_i})$.
Thus $m_{\mathcal F_-}^T<0$ with respect to $\langle \nu^T,\Phi|_T \rangle$ for all components $T$ of $\partial M_0$, and consequently, $(\xi_+)_{(S,h)}$ is weakly symplectically fillable.
The boundary slope of $\mathcal F_-$ on the $i^{th}$ boundary component lies between $\Phi$ and $\nu_i$ as shown in Figure \[slopeconvention\]. By Proposition \[translation1\] the slope of $\Phi$ is $\frac{1}{c_i-n_i}$ while the slope of $\nu_i$ is $-\frac{1}{n_i}$. The form of the intervals given, depends on a case by case analysis of whether or not they contain slope $\infty$. Weak symplectic fillability of $(\xi_+)_{(S,h)}$ follows from Proposition \[oppfoln\].
One can translate the results of [@HKM2; @R2] into the current context as follows.
\[existence\] [@R2] When the binding $L$ is connected, $c>0$, and the monodromy $h$ has pseudo-Anosov representative, there are $\partial M_0$-taut foliations in $M_0$ transverse to $\Phi_0$ which realize all slopes in an interval $J$ as follows:
1. $c=n$ and $J=(-\infty,\infty)$, or
2. $c>n$ and $J=(-\infty,1)$, or
3. $c<n$ and $J=(-1,\infty)$.
The next corollary follows by intersecting the intervals where foliations exist in Theorem \[existence\] with the intervals where they are required in Proposition \[oppfolnTh\].
There exists $\mathcal F_-$ as described in Proposition \[oppfolnTh\], if one of the following is true
1. $c=n$ and $(-\infty,-\frac{1}{n})\cap (-\infty,\infty)\ne\emptyset$, or
2. $c>n$ and $((\frac{1}{c-n},\infty] \cup [-\infty,-\frac{1}{n}))\cap (-\infty,1)\ne\emptyset$, or
3. $c<n$ and $ (-\frac{1}{n-c},-\frac{1}{n})\cap (-1,\infty)\ne\emptyset$.
The case when the fractional Dehn twist coefficient is greater than or equal to $1$ is of particular interest since
If $c \ge 1$ there exists $\mathcal F_-$ as described in Proposition \[oppfolnTh\].
If $c>0$ then $n\ge 0$, with $n=0$ only when $c\in (0,1/2]$. Thus $c \ge 1$ implies $n\ge 1$, and it follows that the intersections in Cases (1) and (2) are nonempty. In Case (3), $1 \le c < n$, and the intersection is again nonempty.
This corollary is exactly what is needed to complete the proof of a theorem of Honda, Kazez, and Matić in [@HKM2] that was one of the original motivations for this work.
If $(S,h)$ is an open book decomposition such that $S$ has connected boundary, $h$ is isotopic to a pseudo-Anosov homeomorphism, and the fractional Dehn twist coefficient of $h$ is greater than or equal to $1$, then the contact structure canonically associated to the open book decomposition, $\xi(S,h)$, is weakly symplectically fillable.
The proof strategy of [@HKM2] used a single foliation $\mathcal F$ defined on all of $(S,h)$ as constructed by Roberts, [@R1; @R2] with boundary slope related to open book data. Next they wanted to apply the Eliashberg-Thurston theorem to produce a weakly symplectically fillable contact structure $\xi$. Finally they argued that the two contact structures $\xi(S,h)$ and $\xi$ were necessarily equivalent.
There is a somewhat surprising aspect that arises in addressing the issue of lack of smoothness of $\mathcal F$. It is that we do not use the same foliation. To complete their proof with our strategy, we require the existence of two foliations $\mathcal F^+$ and $\mathcal F^-$, both of which exist on the complement of the binding of $(S,h)$, by the work of Roberts, and have boundary slopes on either side of the boundary slope of $\mathcal F$. One foliation is used to the produce a positive contact structure, the other a negative contact structure, and both are necessary to conclude that the approximating contact structure is weakly symplectically fillable.
The notion of $\Phi$-approximating contact structure that we produce is sufficient to conclude that $\xi(S,h)$ and $\xi$ are equivalent using the argument of [@HKM2].
When $M=(S,h)$ has binding which is not connected, our results can be applied to the Kalelkar-Roberts constructions of $\partial M_0$-taut foliations in $M_0$ transverse to $\Phi_0$.
\[Theorem 1.1,[@KRo]\] There are $\partial M_0$-taut oriented foliations in $M_0$ transverse to $\Phi_0$ and realizing a neighborhood of rational boundary multislopes about the boundary multislope of the fibration.
Suppose $M$ has open book decomposition $(S,h)$ and fractional Dehn twist coefficients $c_i, 1\le i\le b$. There are constants $A_i= A_i(M_0), 1\le i\le b$, dependent on $M_0$ such that if $c_i>A_i$, then $\xi_{(S,h)}$ is weakly symplectically fillable.
Work of Baldwin and Etnyre [@BE] implies that any such constants $A_i$ must depend on $M_0$, at least in the case that the page $S$ has genus one.
There exist open books whose fractional Dehn twist coefficients are arbitrarily large, but whose compatible contact structures are not $C^0$ close to smooth orientable taut foliations.
There exist open books whose fractional Dehn twist coefficients are arbitrarily large, but whose compatible contact structures are not $\Phi$-close to taut oriented bracketed $C^0$-foliations, for any volume preserving flow $\Phi$.
Some symplectic topology {#symplectic}
========================
This section contains an overview of the relationship between foliations, volume preserving flows, symplectic topology, and contact topology that is summarized in Theorem \[weaklysymplectic\].
Let $\mathcal F$ be a transversely oriented, taut $C^0$-foliation in $M$. Fix a metric on $M$, and let $\Phi$ be a volume preserving flow transverse to $\mathcal F$. The starting point for the interconnections we will describe is a carefully chosen 2-form.
(See Section 3.2 of [@ET].) Let $\xi$ be a co-oriented $C^k$ 2-plane field on a smooth 3-manifold $M$ with $k\ge 0$. A smooth closed 2-form $\omega$ on $M$ is said to [*dominate*]{} $\xi$ if $\omega|\xi$ does not vanish (i.e., if $p\in M$ and $X_p,Y_p$ is a basis for $\xi_p$, then $\omega_p(X_p,Y_p)\ne 0$). A smooth closed 2-form $\omega$ on $M$ is said to [*positively dominate*]{} $\xi$ if $\omega|\xi$ is positive (i.e., for all $p\in M$, if $X_p,Y_p$ is a positively oriented basis for $\xi_p$, then $\omega_p(X_p,Y_p)> 0$).
To produce such a dominating closed 2-form, let $\Omega$ be the volume form on $M$ preserved by the smooth flow $\Phi$, and let $X$ be the vector field which generates $\Phi$. Define $$\omega = X \lrcorner\Omega.$$ Recall that $\Phi$ is volume preserving if and only if $\mathcal L_X\Omega = 0$, where $\mathcal L_X$ denotes the Lie derivative with respect to $X$. (See for example, Proposition 18.16 of [@Lee].) By Cartan’s Formula (see, for example, Proposition 18.13 of [@Lee]), $$\mathcal L_X\Omega = X\lrcorner (d\Omega) + d(X\lrcorner \Omega) = d(X\lrcorner \Omega) = d\omega.$$ It follows that $\omega$ is closed. From its definition, $\omega$ is killed by the flow direction $X$, thus a co-oriented 2-plane field $\xi$ is positively dominated by $\omega$ if and only if it is everywhere positively transverse to $\Phi$.
A closed 2-form dominating $T\mathcal F$ can be produced directly from a taut foliation [@Sullivan; @Hass] thereby eliminating the need to choose a metric, a volume form, and a flow preserving the volume form. We have chosen to emphasize the volume preserving flow since it clarifies the local nature of our foliations in flow boxes.
Specialize now to the case of Theorem \[weaklysymplectic\] in which $\xi$ is a contact structure positively transverse to $\Phi$. Choose a 1-form $\alpha$ such that ker$\,\alpha=\xi$ and $\alpha \wedge \omega>0$. Define a 2-form $\tilde{\omega}$ on $M \times [-1,1]$ using the projection map $p$ and the formula
$$\label{sympform}
\tilde{\omega} = p^{\star}(\omega) + \epsilon d(t\alpha).$$
Direct computation shows that if $\epsilon$ is positive and small enough, $(M\times[-1,1],\tilde{\omega})$ is a symplectic manifold with boundary, that is, $\tilde{\omega}$ is a non-degenerate 2-form. The important role of the positive and negative contact structures, $\xi_+$ and $\xi_-$, of Theorem \[weaklysymplectic\] will be described after the next definition.
A boundary component $Y$ of a symplectic manifold $(W,\tilde{\omega})$ is called *[weakly convex]{} if $Y$ admits a positive contact structure dominated by $\tilde{\omega}|_Y$.*
This is precisely the structure that the boundary components of $(M\times[-1,1],\tilde{\omega})$ have. The restriction of $\tilde{\omega}$ positively dominates $\xi_+$ on $M\times\{1\}$. Because of boundary orientations, $\xi_-$ defines a positive contact structure on $M\times\{-1\}$ that is positively dominated by the restriction of $\tilde{\omega}$. Moreover, $\tilde{\omega}$ restricts to $\omega$ on $M\times\{0\}$, thus $M \times[-1,0]$ also has weakly convex boundary.
Since both boundary components of either $(M\times[-1,1], \tilde{\omega})$ or the restriction of $\tilde{\omega}$ to $M\times [-1,0]$ are weakly convex, they give examples of weak symplectic fillings.
A [*weak symplectic filling*]{} of a contact manifold $(M,\xi)$ is a symplectic manifold $(W,\tilde{\omega})$ with $\partial W = M$ (as oriented manifolds) such that $\tilde{\omega}|_{\xi}> 0$. A contact manifold $(M,\xi)$ which admits a weak symplectic filling is called *[weakly symplectically fillable]{}.*
A [*strong symplectic filling*]{} of a contact manifold $(M,\xi)$ is a symplectic manifold $(W,\tilde{\omega})$ with $\partial W = M$ (as oriented manifolds) where $\xi = \text{ker} \, \alpha$ for a 1-form $\alpha$ satisfying $d\alpha = \tilde{\omega}|_M$. A contact manifold $(M,\xi)$ which admits a strong symplectic filling is called *[strongly symplectically fillable]{}.*
In general, strong symplectic fillability is a stronger condition than weak symplectic fillability [@Eli2]. However, by Lemma 1.1 of [@OO], when $M$ is a rational homology sphere, $(M,\xi)$ is weakly symplectically fillable if and only if it is strongly symplectically fillable.
A contact manifold is said to be weakly (or strongly) [*semi-fillable*]{} if it is one boundary component of a weak (or strong) symplectic filling.
A weakly (or strongly) semi-fillable contact manifold is weakly (or strongly) fillable [@Eli; @Et].
The following fundamental theorem gives an example of the importance of weak symplectic fillability in contact topology.
\[[@Gr],[@Eli1],[@ET]\] \[EGr\] Weakly symplectically fillable contact structures are tight. Weakly symplectically fillable contact structures that are $\Phi$-close to taut foliations are universally tight.
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[^1]: This work was partially supported by a grant from the Simons Foundation (\#244855 to William Kazez)
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---
abstract: 'The detection and characterization of accretion processes in the disks surrounding young stars may be directly relevant to studies of planet formation. Especially the study of systems with very low accretion rates ($<<$ 10$^{-10}$ M$_\odot$ yr$^{-1}$) is important, since at those rates radial mixing becomes inefficient and disk material will have to be dissipated into larger bodies at its present location. In these proceedings, we compare the different methods of tracing accretion onto Herbig Ae/Be stars and conclude that high-resolution infrared spectroscopy is currently the only reliable method that offers the required sensitivity to shed light on this problem.'
author:
- 'Mario E. van den Ancker'
title: 'Tracing Accretion Onto Herbig Ae/Be Stars using Near-Infrared Spectroscopy'
---
Accretion onto YSOs: Why Should We Care?
========================================
The mass accretion rate is thought to be a key parameter in the evolution of young stellar objects (YSOs). As a function of time, the accretion rate traces the build-up of material onto the young star and severely affects the evolution of the disk itself. Local disk structure is affected by the rate of mass flow, which in turn is determined by the rate of gravitational energy release. Several studies (e.g. Bertout et al. 1988; Hartigan et al. 1995; Hartmann et al, 1998) have shown that accretion rates around typical T Tauri stars stars are low: of the order 10$^{-8}$–10$^{-10}$ M$_\odot$ yr$^{-1}$. Compared to typical disk-masses of $<$ 0.1 M$_\odot$, and disk lifetimes of $<$ 10 Myr, these low accretion rates imply that at the evolutionary stage of a typical T Tauri star, disk accretion is no longer important for building up the mass of the central star. Furthermore, at these accretion rates, the heating of the disk is dominated by reprocessing of light from the central star: the disk is passive.
However, there are several reasons why it is especially important to study accretion in YSOs with the lowest accretion rates: (1) if the accretion rate is observed to be lower than $\sim$ 10$^{-7}$ M$_\odot$ yr$^{-1}$ (the disk mass (typically $<$ 0.1 M$_\odot$)/the disk dissipation timescale ($<$ 10 Myr)), the disk cannot disappear due to accretion onto the central star(s). The conclusion that a significant mass fraction of the disk may be coagulated into larger bodies, such as comets or full-fledged planets, seems justified. (2) recent studies of infrared emission from dust in the disks surrounding Herbig Ae/Be stars (e.g. Bouwman et al. 2003) have shown that crystalline silicates are found at temperatures of a few hundred K; temperatures that are much lower than their sublimation temperature of $\sim$ 1500 K. Therefore they cannot have formed at their present location. It is currently believed that this provides strong evidence for the importance of radial mixing in Herbig stars. However, radial mixing can only occur with the required efficiency in the presence of mass accretion rates larger than a few times 10$^{-9}$ M$_\odot$ yr$^{-1}$. If accretion rates are proven to be lower than this limit, radial mixing becomes too inefficient to affect disk structure as a whole: not only must the mass in these disks be dissipated into larger bodies, but they also have to form from the radial distribution of material as it is observed
In these [*proceedings*]{} we will discuss the different methods of tracing accretion onto Young Stellar Objects and conclude that high-resolution infrared spectroscopy may be the most sensitive method in our current arsenal of determining accretion rates.
Methods to Trace Accretion
==========================
![Observed Spectral Energy Distribution of the Herbig Ae/Be star V590 Mon (dots) fitted to a Kurucz stellar photosphere model (solid line) appropriate for its spectral type. The top panel shows the fit adopting a foreground extinction with larger than interstellar dust grains ($R_V$ = $A_V/E(B-V)$ = 4.2), resulting in a good fit to the UV-optical SED. The bottom panel shows the appearance of an virtual UV-excess above photospheric levels when attempting to fit this SED with a normal interstellar dust composition ($R_V$ = 3.1).[]{data-label="eps1"}](hires2003_F1.ps){width=".78\textwidth"}
Currently employed methods to trace accretion onto YSO can roughly be distributed into two categories: those that study the continuum emission from disk (IR) and accretion shock (UV), and those that attempt to directly study infalling gas through emission lines in the optical or infrared, or its associated free-free emission at radio wavelengths. For accretion rates at which the disk becomes passive (i.e. the majority of disk energy comes from reprocessed starlight rather than the viscous dissipation of accretion energy), the derivation of accretion rates from infrared continuum emission becomes dependent on the details of the disk structure, and hence exceedingly difficult to determine. The commonly used determination of accretion luminosities from UV excesses presumes that one has a good knowledge of stellar photospheric parameters, and of the properties of circumstellar extinction, for which one needs to correct. As the latter is often anomalous, e.g. due to differences in chemical composition or dust particle sizes, a degeneracy occurs between UV excesses commonly attributed to the accretion shock, and extinction properties (Fig. 1). Additionally, for very low mass accretion rates, typical uncertainties in stellar classification and the associated intrinsic stellar colours may prohibit the reliable determination of smaller UV excesses in commonly used photometric systems.
![Examples of detected lines in the IRTF spectra showing the lines of Br$\alpha$ (4.05 $\mu$m), Pf$\gamma$ (3.74 $\mu$m), Br$\gamma$ (2.67 $\mu$m) and the CO band-head around 2.3–2.4 $\mu$m.[]{data-label="eps2"}](hires2003_F2.ps){height=".82\textwidth"}
![Correlation between accretion luminosity as derived from UV excesses versus Br$\gamma$ luminosity for T Tauri stars (black dots; from Muzerolle et al. 1998), and Herbig Ae/Be stars (grey dots; this study).[]{data-label="eps3"}](hires2003_F3.ps){height=".55\textwidth"}
Methods that rely on the emission of infalling gas that gets heated directly by viscous dissipation of energy appear to be more reliable tracers of accretion. The most commonly used of these may be radio continuum emission due to free-free transitions in H-ions (e.g. Panagia 1991; Nisini et al. 1995). However, current radio telescope sensitivities limit this method to accretion rates larger then $\sim$ 10$^{-8}$ M$_\odot$ yr$^{-1}$. Therefore we conclude that the study of emission lines, and in particular infrared hydrogen recombination lines, may be the only reliable method currently available to trace the low accretion rates directly relevant to planet formation.
Near-IR Spectroscopy of Herbig Ae/Be stars
==========================================
We obtained new 1.9–4.1 $\mu$m spectra of 26 Herbig Ae/Be stars – young massive (2–10 $M_\odot$) stars surrounded by disks – using SpeX, a medium-resolution ($R$ = 1000–2000) cross-dispersed spectrograph mounted on NASA’s [*Infrared Telescope Facility*]{} (Rayner et al. 2003). Commonly detected lines in these data include IR hydrogen recombination lines such as Br$\alpha$, Pf$\gamma$ and Br$\gamma$, as well as the CO band-heads around 2.3–2.4 $\mu$m (Fig. 2). All detected emission lines appear unresolved at our moderate (a few hundred km s$^{-1}$) spectral resolution.
![Plot of derived accretion rates from Br$\gamma$ line fluxes versus the continuum excess in the $K$-band (2.2 $\mu$m). We also plot an empirical division between disks which are dominated by viscous dissipation of energy due to accretion (active disks), and systems in which the dust is mainly heated by re-processing of light from the central star (passive disks).[]{data-label="eps4"}](hires2003_F4.ps){height="93.00000%"}
The strength of these lines is expected to be a good tracer of the emission measure of infalling hydrogen gas, and hence be directly correlated to accretion. For example Muzerolle et al. (1998) and Nisini et al. (these proceedings) found a strong correlation between Br$\gamma$ line strength and accretion luminosity in samples of low-mass exposed and embedded YSOs, respectively. Using ultraviolet excesses derived from archive [*International Ultraviolet Explorer*]{} data, we find that the higher-mass Herbig Ae/Be stars with strong Br$\gamma$ line emission exhibit the same tight correlation between UV excess and hydrogen recombination line strength found for their lower-mass counterparts (Fig. 3). A comparison between radio continuum data and infrared recombination line fluxes (not shown here) shown a similar tight correlation.
The accretion luminosities derived from the relation seen in Fig. 3 can easily be transferred to accretion rates using some simple assumptions about the accretion radius ($\dot M = L_{\rm acc} R_{\rm acc}/M_{\star}$). Note that, whereas we were unable to conclusively detect UV excesses in sources with $\dot M$ $<$ 10$^{-7}$ M$_\odot$ yr$^{-1}$, we detected Br$\gamma$ line fluxes as small as a few times 10$^{-16}$ W m$^{-2}$, corresponding to accretion rates as low as 10$^{-9}$ M$_\odot$ yr$^{-1}$.
As an interesting side-note to this, we note that Herbig stars with high accretion rates invariably have mid-infrared spectra which show the well-known 10 $\mu$m silicate feature in absorption (Acke & van den Ancker 2004). We interpret this difference in 10 $\mu$m silicate appearance as a reflection of the different temperature structure of active disks – heated by viscous dissipation of accretion energy in the mid-plane – versus that of passive disks – heated by absorption of light from the central star hitting the disk surface. The Br$\gamma$ probe of accretion rates suggests that in Herbig Ae/Be stars the transition between passive and active disks occurs at accretion rates of $\sim$ 2 $\times$ 10$^{-7}$ M$_\odot$ yr$^{-1}$ (Fig. 4).
The Need for Higher Spectral Resolution
=======================================
In the preceeding sections, we have shown that, in analogy to what is found for their lower-mass counterparts, infrared hydrogen recombination lines appear to be sensitive probes of mass accretion in Herbig Ae/Be stars. Since we did not resolve the detected lines, the only information available to us were line strengths. It is conceivable that other processes occurring in these Herbig stars (e.g. outflows, compact H[ii]{} regions) can also contribute to the total hydrogen recombination line flux of the system. The tight correlation between UV excesses and Br$\gamma$ luminosity illustrated in Fig. 3 demonstrates that this pollution by other processes is apparently not important for systems with high accretion rates. However, at present we are unable to fully assess whether this will also be the case when studying the systems with lower accretion rates. Therefore our derived accretion rates below 10$^{-7}$ M$_\odot$ yr$^{-1}$ should at this moment be regarded upper limits to the true accretion rate.
New observations with higher resolution are needed to remedy this unsatisfactory situation. Those observations should be able to distinguish the characteristic P Cygni profiles of infalling matter, and to separate those broad lines from narrow lines that may be produced by a compact H[ii]{} region, and hence clarify whether we can truly attribute all the flux in the infrared hydrogen recombination lines to accretion processes.
At ESO, two interesting new instruments will soon become available with which we may seek the answer to these questions: CRIRES, an $R$ $>$ 100,000 spectrograph at the VLT through which these questions may be addressed through spatially unresolved high-spectral resolution observations, and AMBER at the VLTI ($R$ = 10,000), whose unique combination of high spatial and spectral resolution may allow us to probe the accretion regions of young stars in unprecedented detail. Both instruments will have the ability to push back our detection limits for accretion rates to well below 10$^{-10}$ M$_\odot$ yr$^{-1}$, the realm directly relevant for planet formation theories.
[8.]{}
Acke B., van den Ancker M.E., 2004, A&A, submitted
Bertout C., Basri G., Bouvier J., 1988, ApJ 330, 350
Bouwman J., de Koter A., Dominik C., Waters L.B.F.M., 2003, A&A 401, 577
Hartigan P., Kenyon S.J., Hartmann L., Strom S.E., Edwards S., Welty A., Stauffer J., 1991, ApJ 382, 617
Hartmann L., Calvet N., Gullbring E., D’Alessio P., 1998, ApJ 495, 385
Muzerolle J., Hartmann L., Calvet N., 1998, AJ 116, 2965
B., [Milillo]{} A., [Saraceno]{} P., [Vitali]{} F., 1995, A&A 302, 169
N., 1991, [*“NATO ASI: Physics of Star Formation and Early Stellar Evolution”*]{}, Dordrecht, Kluwer
Rayner J.T., Toomey D.W., Onaka P.M., Denault A.J., Stahlberger W.E., Vacca W.D., Cushing M.C., Wang S., 2003, PASP 155, 362
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---
author:
- '[^1]'
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title: |
Integrating LHCb workflows on HPC resources:\
status and strategies
---
Introduction {#sec:intro}
============
is constantly looking for ways to opportunistically expand its distributed computing resources, beyond those pledged by the sites of the Worldwide LHC Computing Grid (WLCG). One way of doing so is by integrating High Performance Computing (HPC) supercomputers in the grid, managed via [@DIRAC-SW] and its extension, [@LHCbDIRAC-SW].
’s interest in using HPC sites is mainly for running (MC) simulation jobs. MC simulation jobs are, in fact, by far, the largest consumers of the share of WLCG compute resources (will be more than 90% in Run3). When it comes to distributed computing, the strategy is to use any new compute resources to run more MC simulation. The fraction of non-simulations jobs and CPU, in , is small enough that we can rely, for them, on the currently existing pledged resources.
This paper describes the work that was performed in to be able to run MC simulation jobs on the Marconi-A2 HPC facility at CINECA in Bologna, Italy. An allocation on this supercomputer became available to in mid-2019 within the context of a joint application of the Italian LHC community for a PRACE grant on this resource, as described more in detail in another presentation at this conference [@Boccali2019]. This work leveraged on the close collaboration between CINECA and CNAF, the Italian Tier-1 site for WLCG, which is managed by the Istituto Nazionale di Fisica Nucleare (INFN) and is also located in Bologna.
Two different challenges had to be addressed to integrate the LHCb simulation workflow on the Marconi-A2 system, powered by many-core Intel Knights Landing (KNL) processors, with limited RAM memory per hardware thread: first, the MC software application, Gauss [@Clemencic_2011], had to be re-engineered to use multi-processing (MP) or multi-threading (MT) to have a lower memory footprint per thread; second, the framework had to modified to be able to submit MP or MT jobs on batch queues, as this was the first time these types of jobs were used in distributed computing (even if this is expected to be the norm in the future, also for other types of workflows such as event reconstruction).
This paper is organized as follows. Section \[sec:hpcs\] describes the generic challenges for integrating computing workflows on HPC resources. Section \[sec:marconi\] introduces the Marconi-A2 HPC at CINECA. Section \[sec:gaussmp\] describes the work that was done to commission multi-processing applications in the Gauss simulation software. In section \[sec:dirac\] the project is briefly introduced, with a focus on the capabilities of the Workload Management System. Section \[sec:distributed\] gives more details about the distributed computing challenges and solutions for on Marconi-A2. Finally, conclusions are given in Section \[sec:conclusions\].
Challenges of High Performance Computer systems {#sec:hpcs}
===============================================
Using HPC facilities, in but more generally for any HEP experiment, poses two rather distinct types of challenges:
- Software architecture challenges: the compute power of HPC supercomputers may come from a range of different processor architectures, including multi-core and many-core x86 CPUs, non-x86 CPUs (ARM, Power9) and accelerators (GPUs, FPGAs). For an LHC experiment, being able to efficiently exploit these resources may require significant changes to its software applications, which are generally designed for the setup of a traditional WLCG worker node, based on an x86 CPU with at least 2 GB RAM available per hardware thread. In addition, HPCs provide extremely fast inter-node connectivity, often used for parallel processing using MPI, while most HEP software applications generally use the individual nodes independently of one another, as if HPCs were just very large clusters.
- Distributed computing challenges: HPC sites usually have strict site policies for security reasons and may therefore be characterized by limited or absent external connectivity, ad-hoc operating systems, limited storage on local disks, restrictive policies for user authentication/authorization and for batch queues. This differs from the configuration of traditional sites in WLCG, which provide full access to remote services like the CernVM File System (CernVM-FS, often abbreviated CVMFS[@bib:CVMFS]) for software installation, uniform operating systems and the capability to use user-level virtualization.
More generally, unlike WLCG sites which provide a relatively uniform computing environment, HPC centers may differ significantly from one another. Some HPCs are easier to exploit than others, e.g. already uses Piz Daint at CSCS [@Sciacca_2017], which looks like a traditional Grid site providing a cluster of nodes powered by x86 CPUs. The collaboration of the experiments with the local system administrators and performance experts is in any case essential to address the specific issues of each HPC center, and has proved to be mutually beneficial.
All in all, HPCs are not the most natural fit for HEP computing today. Because of the large amounts of resources dedicated to scientific computing that are currently deployed at HPC centers now, and of their predicted further increase in the future, it is however essential that LHC experiments continue to work on adapting their software and computing infrastructures to be able to efficiently exploit these resources in the near future.
Marconi-A2: a KNL based partition of the CINECA supercomputer {#sec:marconi}
=============================================================
Marconi [@marconi] is a supercomputer at CINECA, available for the Italian and European research community. Currently ranked number 19 in the top500.org list [@top500], Marconi provides its compute capacity through several independent partitions. The Marconi-A2 partition, which has been used for the work described in this paper, consists of nodes equipped with one Xeon Phi 7250 (KNL) at 1.4 GHz, with 96 GB of on board RAM. This is an x86 many-core CPU with 68 physical cores, supporting 4-way hyperthreading. Keeping into account that approximately 10 GB of memory are reserved for the O/S, this means that just over 300 MB of RAM are available per hardware thread if all 272 threads are used. This is much lower than the 2 GB (or more) per thread available at WLCG sites, motivating the effort to implement MP and MT approaches in the Gauss software, as described in the next Section \[sec:gaussmp\].
In January 2020, most of the A2 partition (which included 3600 nodes in 2019) was switched off, to upgrade Marconi’s compute capacity by replacing the KNLs in Marconi-A2 by the GPUs in a new Marconi100 partition. Some of the KNLs in A2 are however still available at the time of writing in March 2020, and will be used for our work until the end of the granted allocation. While the efficient exploitation of KNLs already required some software development effort, as described in the next section, it should be noted that the work we describe could not have been performed on the GPU-based partition, as neither event generation nor detector simulation are yet possible in the software on GPUs.
The default computing environment on Marconi is also quite different from that normally found at WLCG sites. Thanks to the excellent collaboration between the Italian experiment contacts and the site managers and sysadmins at CINECA and CNAF, many essential ad-hoc changes were deployed for the LHC experiments [@Boccali2019]. In particular: CVMFS mounts and Squids were provided; external outgoing networking was partially opened at CINECA, with routing active to the IP ranges of CERN, FermiLab and CNAF; the Singularity [@singularity] container management tool was deployed; a HTCondor-CE (the HTCondor Computing Element) was allowed on a CINECA edge node, for submitting jobs to the internal SLURM batch system connected to Marconi-A2. The only change that was needed to allow MP/MT MC simulation workflows on Marconi-A2 was therefore the implementation of MP/MT job submission in , as described in Section \[sec:distributed\].
Multi-process MC simulation on Marconi-A2: GaussMP {#sec:gaussmp}
==================================================
The software for running simulations, Gauss [@Clemencic_2011], is used for both event generation [@Belyaev_2011] and detector simulation, the latter using internally the Geant4 [@AGOSTINELLI2003250] simulation toolkit. Currently, MC simulation jobs in execute both steps sequentially on the same worker node, therefore they need basically no input data (only a configuration file). This implies that, unlike workflows such as event reconstruction or stripping, the management of input data files is not an issue for MC simulation. In addition, both event generation and detector simulation software applications are compute-intensive, rather than I/O-intensive. Optimising their performance is mainly a problem of an efficient use of CPU and memory.
Until recently, has only used a single-process (SP), single-threaded (ST), version of the Gauss software for all of its MC simulation productions. Only x86 architectures are currently supported, although ports to ARM have been worked on. The typical memory footprint of these applications, around 1.4 GB [@corti2019], has so far not been an issue on traditional WLCG nodes, where 2 GB per thread are available. On many-core CPUs like the KNL used in Marconi-A2, however, these workflows are very inefficient, because the limited memory available per thread (300 MB if 4 threads are used on each of the 68 physical cores of a KNL, or 600 MB if only 2 threads are used) effectively limits the maximum number of SP/ST Gauss instances that can be executed simultaneously, i.e. the number of KNL threads that can be filled. To reduce the memory footprint per thread and be able to fill a larger number of KNL threads, multi-processing (MP) or multi-threading (MT) approaches are needed.
In the long term, the solution will be to base its MC simulations on Gaussino [@muller2019], a MT implementation of Gauss. In spite of its very fast recent progress, however, Gaussino was still not ready for meaningful productions or tests when the Marconi-A2 allocation started. As a temporary solution, the software work targeting the Marconi A2 timescales focused on the test and commissioning of GaussMP, a MP-based version of Gauss. This leveraged on GaudiMP [@gaudimp2014; @nrphd2014], a MP version of the event processing framework Gaudi, which already existed but had never been used in production.
The software work on GaussMP had two main aspects: extensive functional testing and bug-fixing, and performance testing and optimization. Functional testing and bug-fixing essentially targeted, and achieved, a validation of results requiring identical results in the SP and MP versions of Gauss, when simulating the same set of events, starting from the same random number seeds. In other words, rather than requiring a physics validation of results by comparing physics distributions within MC statistical errors, the MP software was validated by requiring that some event-by-event properties (numbers of particles and vertices etc.) should be the same in MP and SP applications. This implied careful checks in both the event generation and detector simulation steps of the application.
Performance testing and optimization essentially consisted in running several identical copies of an application on a given worker node, in SP mode or using different MP configurations, to understand which configuration maximises the total throughput of the entire node, i.e. the number of events processed per unit wall-clock time. Memory usage was also monitored to provide an interpretation of throughput results. The tests were performed both on a reference node at CERN, using a traditional hardware setup based on two multi-core Haswell CPUs with 2 GB per hardware thread (2x8 physical cores with 2-way hyperthreading and 64 GB RAM), and on a KNL node from Marconi-A2 (68 physical cores with 4-way hyper-threading and 96 GB RAM).
The results of these tests are shown in Fig. \[fig:haswell\] for the reference node at CERN and in Fig. \[fig:knl\] for the KNL on Marconi-A2. The memory footprint of the physics process used for this specific test (event generation and detector simulation of $B^+\rightarrow J/\psi\,K^+$ production, including spillover from minimum bias production in adjacent collisions) is around 900 MB per process/thread in SP mode. On the reference node, with 64 GB RAM, 32 instances of an SP application can be used to fill all 32 threads, and this is the configuration providing throughput (6.0 events per minute): several MP configurations provide similar, but slightly lower, integrated throughput (for instance, 5.8 events per minute for 8 instances of GaussMP with 4 processes each), because of the overhead involved in the extra processes used by GaussMP. On the KNL node, however, at most 85 SP instances can be launched, because some processes are killed by the out-of-memory monitor if more instances are launched. Using GaussMP results in a lower memory footprint per hardware thread, allowing a larger number of KNL threads to be filled: in particular, the maximum throughput on the KNL is achieved when 8 GaussMP application instances are executed in parallel, each using 17 processes. This corresponds to using 2 hardware thread per KNL core (i.e. 136 in total), not 4: many failures are observed when trying to use 4 threads per KNL core (i.e. 272 in total).
The highest GaussMP throughput achieved on the KNL (3.6 events per minute) is only moderately higher ($\sim$15%) than that achieved using SP Gauss (3.2 events per minute), because the forking strategy used does not optimize the use of copy-on-write to minimize the memory footprint. Currently, new worker processes are forked after job initialization but before the first event [@gaudimp2014; @nrphd2014], where the magnetic field map is read from disk [@bmklhcb2019]; forking workers after processing the first event would make it possible to share a larger amount of memory across workers and reduce the overall memory footprint, as recently demonstrated by ATLAS in their new AthenaMP forking strategy [@elms]. Looking forward, however, software efforts in the simulation area will focus on Gaussino, the long-term MT solution, rather than on GaussMP, the temporary MP solution.
It is also interesting to note that, in absolute terms, the throughput per thread achieved in the configuration maximising the total node throughput is a factor 7 lower on the KNL (0.026 events per minute per core, for 3.6 events per minute on 136 threads) than on the reference Haswell node (0.188 events per minute per thread, for 6.0 events per minute on 32 threads). This can only be partly explained in terms of the lower clock speed of the KNL cores, and is probably also due to the memory access patterns of the Gauss application on the two architectures, but no specific studies have been performed to understand this better.
The project {#sec:dirac}
===========
[@DIRAC-SW] is a software framework that enables communities to interact with distributed computing resources. It builds a layer between users and resources, hiding diversities across computing, storage, catalog, and queuing resources. has been adopted by several HEP and non-HEP experiment communities [@DIRAC-HEP-CHEP2016], with different goals, intents, resources and workflows: it is experiment agnostic, extensible, and flexible [@GPUDIRAC]. uses for managing all its distributed computing activities. is an open source project, which was started around 2002 as an project. Following interest of adoption from other communities its code was made available under open licence in 2009. Now, it is hosted on GitHub [@dirac-github] and is released under the GPLv3 license.
The Workload Management System (WMS) is in charge of exploiting distributed computing resources. In other words, it manages jobs, and pilot jobs [@DIRAC-PILOTS_2016] (from here on simply called “pilots”). The emergence of new distributed computing resources (private and commercial clouds, High Performance Computing clusters, volunteer computing, etc) changed the traditional landscape of computing for offline processing. It is therefore crucial to provide a very versatile and flexible system for handling distributed computing (production and user data analysis). If we restrict for a moment our vision to LHC experiments, and we analyze the amount of CPU cycles they used in the last year, we can notice that all of them have consumed more CPU-hours than those official reserved (pledged) to them by WLCG. Each community found ways to exploit “opportunistic”, i.e. non-pledged, compute resources (CPUs, or even GPUs). Such resources may be private to the experiment (e.g. the “online” computing farm - often simply called “High Level Trigger” farm) or public; resources may sometimes be donated free of charge, like in the case of volunteer computing, or not, like public commercial cloud providers. Integrating non-grid resources is common to all communities that have been using WLCG in the past, and still do. Communities that use want to exploit all possible CPU or GPU cycles. Software products like aim to make this easy, and the pilot is the *federator* of each and every computing resource. With , transparent access to the computing resources is realized by implementing the pilot model.
Distributed computing challenges on Marconi-A2: fat nodes {#sec:distributed}
=========================================================
The integration of new compute resources into the distributed computing framework, based on DIRAC, is generally an easy task when: first, worker nodes (WNs) have outbound network connectivity; second, the CVMFS endpoints are mounted on the WNs; and, third, the WN O/S is an SLC6 (Scientific Linux CERN 6) or CC7 (CERN CentOS 7) compatible flavor, or Singularity containers are available. As discussed in Sec. \[sec:hpcs\], none of these conditions would normally be satisfied on Marconi, but all three were eventually met on Marconi-A2 specifically for the LHC experiments, thanks to the good collaboration between the experiment contacts and the site managers.
![Allocating jobs within a fat node: the DIRAC pilot, when using the Pool inner Computing Element, realizes a de-facto partitioning of the node, and parallel jobs matching. The figure above shows a theoretical allocation of jobs to logical processors, with each box representing a logical processor. Each color represents a different application. []{data-label="fig:fat"}](fat_node.pdf){width="5cm"}
As a consequence, on the distributed computing side, the main challenge had to address was that each job slot provided by the HTCondorCE represents a whole KNL node, with 68 physical cores and up to 4 hardware threads per core, i.e. a total of [nproc]{}=272 logical processors (assuming that 4-way hyper-threading is enabled). Rather than implementing a quick ad-hoc solution for Marconi-A2, this was addressed in by developing a generic mechanism for managing “fat nodes”, as shown schematically in Figs. \[fig:fat\] and \[fig:match\]. In terminology, this problem, of how to subdivide a fat node and allocate its resources to several jobs, is simply called “matching”. It is worth noting that this was never done in before, as only jobs running SP/ST software workloads and using a single logical processor were used.
For a proper job allocation, needs to “partition” the node for optimal memory and throughput (and maybe only use a subset of the logical processors). For this to happen, we have developed the *Pool* “inner” Computing Element, with which it is possible to execute parallel jobs matching. In the following, by “processor” we mean a “logical processor” (whose number is [nproc]{} in total); by “single-processor” jobs we mean single-threaded, single-process software application workloads, requiring a single logical processor, while by “multi-processor” job we mean a software application workload that uses more than one logical processor, whether the application is implemented using a multi-process approach (like GaussMP) or a multi-threaded approach (like Gaussino), or a combination of both.
From a user’s perspective, it is possible to describe the jobs precisely enough to satisfy all use cases below:
- certain jobs may be able to run only in single-processor mode
- certain jobs may be able to run only in multi-processor mode (i.e., they need at least 2 logical processors)
- certain multi-processor jobs may need a fixed amount of logical processors (in our specific case on Marconi-A2, we chose to submit only GaussMP jobs, at most 8 simultaneously, using 17 logical processors per job, to maximize the whole node throughput)
- certain jobs may be able to run both in single-processor or multi-processor mode, depending on what is possible on the WN/Queue/CE
- for certain jobs we may want to specify a maximum number of processors to use
At the same time, from a resource provider’s perspective, it is possible to describe CEs and Queues precisely enough to satisfy all use cases below:
- may give their users the possibility to run on their resources:
- only single processor jobs
- both single and multi processor jobs
- may ask their users to distinguish clearly between single and multi processor jobs
- may need to know the exact number of processors a job is requesting
- may ask for only “wholeNode” jobs
Summary and outlook {#sec:conclusions}
===================
In summary, both of the challenges involved in the integration of LHCb MC simulation workflows on the Marconi-A2 HPC have been addressed: a multi-process version of the Gauss software framework with reduced memory footprint per thread has been commissioned, and the functionality of managing fat nodes has been added to the LHCdDIRAC distributed computing framework and has been successfully tested in a dedicated certification environment. At the time of writing in March 2020, however, the new functionality has not yet been deployed in production, because of the timescales involved in the LHCb software release process. This is the reason why no results of production use of Marconi A2 by LHCb for MC simulation using GaussMP are shown in this paper. As soon as the new is released within the next few weeks, however, the remaining LHCb allocation on the Marconi-A2 KNLs will be used to launch the first production jobs of MC simulation, using software workflows based on GaussMP, as well as using Gaussino if available on time.
More generally, this effort at integrating a new HPC resource into the LHCb software and computing was extremely valuable. On the software application side, it was useful to highlight some of the challenges ahead in the use of non-traditional compute architectures (which may well be GPUs in the not-so distant future). On the distributed computing side, it was useful to pave the way to the more routine use of multi-threaded software applications on the grid, which will soon become the norm. Last but not least, the collaboration with the other LHC experiments and with the local site managers and sysadmins at CINECA and CNAF was an essential ingredient of this effort, and a pleasant and fruitful experience for which we thank them, and that we look forward to repeating in the future.
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[^1]:
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---
abstract: 'Foams are surfaces with branch lines at which three sheets merge. They have been used in the categorification of $sl(3)$ quantum knot invariants and also in physics. The $2D$-TQFT of surfaces, on the other hand, is classified by means of commutative Frobenius algebras, where saddle points correspond to multiplication and comultiplication. In this paper, we explore algebraic operations that branch lines derive under TQFT. In particular, we investigate Lie bracket and bialgebra structures. Relations to the original Frobenius algebra structures are discussed both algebraically and diagrammatically.'
author:
- |
J. Scott Carter\
University of South Alabama\
- |
Masahico Saito\
University of South Florida
title: Algebraic Structures Derived from Foams
---
Introduction
============
Frobenius algebras have been used extensively in the study of categorification of the Jones polynomial [@Kh06], via $2$-dimensional Topological Quantum Field Theory ($2D$-TQFT, [@Kock]). For categorifications of other knot invariants, $2$-dimensional complexes called foams have been used instead [@Kh03; @MV06]. Although $2D$-TQFT has been characterized [@Kock] in terms of commutative Frobenius algebras, foams have not been algebraically characterized in terms of TQFT. Relations to Lie algebras, for example, have been suggested [@Kh03; @MV06] through their boundaries which are called webs and that are trivalent graphs. Foams have branch curves along which three sheets meet. Similar complexes appear as spines of $3$-manifolds, and have been used for quantum invariants [@CFS; @ChFS; @KSS; @TV].
Herein we study the types of algebraic operations that appear along the branch curves of foams in relation to $2D$-TQFT. Recall that a $2D$-TQFT is a functor from the category of $2$-dimensional cobordisms to a category of $R$-modules (for some suitable ring $R$) that assigns an $R$-module to each connected component (circle) on the boundary of a surface, and an $R$-module homomorphism to a surface. In the case of a foam, we examine the associated algebraic operations that might be associated to branching circles in relation to the Frobenius algebra structure that occurs on the unbranched surfaces. Specifically, we identify and study Lie algebra and bialgebra structures in relation to branch curves, and study their relations to the Frobenius algebra structure.
After reviewing necessary materials in Section \[prelimsec\], a Lie algebra structure for the branch curves is studied in Section \[Liesec\], and comultiplications of bialgebras are examined in Section \[bialgsec\]. The foam skein theory based on the bialgebra case is also defined in Section \[bialgsec\].
Preliminary {#prelimsec}
===========
Algebraic structures we investigate include Frobenius algebras, Lie algebras and bialgebras. We restrict to the following situations.
A [*Lie*]{} algebra is a module $A$ over a unital commutative ring $R$ with a binary operation $[\ , \ ]: A \times A \rightarrow A$ that is bilinear, skew symmetric ($[x, y]=-[y,x]$ for $x, y \in A$) and satisfies the Jacobi identity ($[x,[y,z]]+[z,[x,y]]+[y,[z,x]]=0$ for $x,y,z\in A$).
A [*Frobenius algebra*]{} is an algebra over $R$ (that comes with associative linear multiplication $\mu: A \otimes A \rightarrow A$ and unit $\eta: R \rightarrow A$) with a non-degenerate form $\epsilon: A \rightarrow R$ that is associative ($\epsilon(x\otimes yz)=\epsilon(xy\otimes z)$ for $x,y,z \in A$). There is an induced co-associative comultiplication $\Delta: A \rightarrow A \otimes A$. See [@CCEKS] for diagrams for Frobenius algebras which we will use in this paper. A [*bialgebra*]{} is an algebra $A$ over $R$ with a comultiplication $\Delta: A \rightarrow A \otimes A$ that is an algebra homomorphism ($\Delta(xy)=\Delta(x)\Delta(y)$) and a counit $\epsilon: A \rightarrow R$ such that $(\epsilon \otimes {\rm id})\Delta= {\rm id}
= ({\rm id}
\otimes \epsilon )\Delta$. The following are typical examples.
\[trancpolyex\] [Let $A=A_N$ be the Frobenius algebra of truncated polynomial $A_N=R [X] / (X^N)$ for a commutative unital ring $R$, with counit (Frobenius form) $\epsilon$ determined by $\epsilon (x^{N-1})=1$ and $\epsilon (x^i)=0$ for $i \ne N-1$. The comultiplication $\Delta $ is determined by $\Delta(1) = \sum_{i=0}^{N-1} X^i \otimes X^{N-1-i}$. Diagrammatically, this is represented by a “neck cutting” relation [@BN], which we call [*a $\Delta(1)$-relation*]{} to distinguish the specific relation given in [@BN] for $N=2$. See the right of Fig. \[CNrel\] for a diagrammatic representation of the $\Delta(1)$-relation in this case. ]{}
In general, the $\Delta(1)$-relation is also described as follows (see [@Kh03; @Kock]). For a commutative Frobenius algebra $A$ over a unital ring $R$ of finite rank and with a non-degenerate Frobenius form $\epsilon$, there is a basis $\{ x_i\}$ and a dual basis $\{ y_i\}$, $i=1, \ldots, n$, such that $\epsilon (x_i y_i)=\delta_{i,j}$, the Kronecker delta, and $x=\sum_i y_i \epsilon (x_i x)$. This situation is depicted in Fig. \[tube\], where the identity map $x \mapsto x$ in the LHS corresponds to the annular cobordism in the left of the figure, and the sum involving the Frobenius form $\epsilon$ is depicted in the right of the figure.
![The $\Delta(1)$-relation[]{data-label="tube"}](tube){width="1.5in"}
\[MVex\] [The Frobenius algebra structure on $A={\mathbb{Z}}[a,b,c] [X] / (X^3-aX^2-bX-c)$ is presented in [@MV06] as follows. The multiplication and the unit are defined by those for polynomials, the Frobenius form (counit) $\epsilon$ is defined by $\epsilon(1)=\epsilon(X)=0$, $\epsilon(X^2)=-1$. The comultiplication is accordingly computed as $$\begin{aligned}
\Delta (1) &=& -( 1 \otimes X^2 + X \otimes X + X^2 \otimes 1 )
+ a ( 1 \otimes X + X \otimes 1) + b (1 \otimes 1), \\
\Delta (X) &=& - (X \otimes X^2 + X^2 \otimes X) + a (X \otimes X) - c (1 \otimes 1), \\
\Delta (X^2) &=& - (X^2 \otimes X^2) - b( X\otimes X) -c (1 \otimes X+ X\otimes 1) .\end{aligned}$$ ]{}
![Operation on a branch circle[]{data-label="branch"}](branch){width="2in"}
We fix a $2D$-TQFT such that a connected circle corresponds to $A$. For TQFTs we refer to [@Kock]. We follow definitions of foams in [@Kh03; @MV06], except that facets of foams are decorated by basis elements of $A$, in a general way as in [@Kaiser]. A foam without boundary is called closed.
We briefly summarize their definitions. ${\bf Foam}_A$ is the category of formal linear combination over $R$ of cobordisms of compact $2$-dimensional complexes in $3$-space with the following data. (1) Boundaries are planar graphs with trivalent rigid vertices. (2) For an interior point of a cobordism, the neighborhood of each point is homeomorphic to either Euclidean $2$-space (a facet) or a [*branch curve*]{} where three facets of half planes meet. (3) Each facet is oriented, and the induced orientation on the branch curve is consistent among three facets that share the curve. (4) A cyclic order of facets are specified using the orientation of $3$-space as depicted in Fig. \[branch\]. (5) Each facet has a basis element of $A$ assigned. (6) An annular cobordism as depicted in the left of Fig. \[tube\] is equivalent to the linear combination as depicted on the right. (7) Values $\theta(\alpha, \beta, \gamma) \in A$ of the theta foam, as depicted in Fig. \[theta\] are specified.
In [@Kh03; @MV06], it was shown that the values in $A$ of closed foams are well-defined for values of the theta foams, as long as the [*cyclic symmetry condition*]{} $\theta(\alpha, \beta, \gamma)=\theta( \beta, \gamma, \alpha)=\theta(\gamma,\alpha, \beta)$ is satisfied.
![The theta foam[]{data-label="theta"}](theta){width=".8in"}
By $2D$-TQFT for a chosen $A$, the two circles on the left Fig. \[branch\] are mapped to the factors of $A \otimes A$. For the cyclic order along the oriented branch circle as depicted, make a correspondence between the facet labeled $1$, $2$, $3$, respectively, to the first, second, and the target factor of $A \otimes A \rightarrow A$. Thus the cobordism near a branch circle as depicted in the figure induces a linear map $A \otimes A \rightarrow A$ under the chosen TQFT and the values of theta foams. Denote this map by ${{\bf{m}} }: A \otimes A \rightarrow A$. The goal of this paper is to investigate this map.
In terms of maps among tensor products of $A$s, we use planar graphs regularly used in knot theory, as well as Frobenius algebras as in [@CCEKS]. In particular, the Frobenius form (the counit) is depicted by a maximum, unit by a minimum, (co)multiplications by trivalent vertices. In this convention, diagrams are read from bottom to top, corresponding to the domain and range of maps. The map ${{\bf{m}} }$ corresponding to theta foams has a specified cyclic order, as indicated on the right of Fig. \[branch\]. The map ${{\bf{m}} }$ is defined with this specific order, and the map with the opposite order, depicted by a diagram with the opposite arrow, represent the map ${{\bf{m}} }\circ \tau$, where $\tau: A \otimes A \rightarrow A \otimes A$ is the map induced from the transposition $\tau(x \otimes y) = y \otimes x$.
Lie algebras {#Liesec}
============
In this section we show that there are infinitely many TQFTs under which Lie algebra structures are induced from the branch circle operation. Since our goal is to exhibit a Lie bracket, in this section we use the notation $[\ , \ ] : A \times A \rightarrow A$, instead of ${{\bf{m}} }: A \otimes A \rightarrow A$.
For any commutative unital ring $R$ and a positive odd integer $N > 1$, there exist a Frobenius algebra $A$ over $R$ and values of the theta foams in ${\bf Foam}_A$ such that the branch circle operation ${{\bf{m}} }$ induces a non-trivial Lie algebra structure on $A$.
[*Proof.*]{} Let $A=R[X]/(X^N)$ for an odd integer $N>1$. For simplicity we denote $\theta (X^a, X^b, X^c)$ by $\theta (a,b,c)$ in this proof.
Let $N>3$. Define $\theta( a,b,c)=1$ if $a=0$, $b+c=N$ and $1<b<c$, as well as all cyclic permutations of such $(a,b,c)$. Define $\theta( a,b,c) =-1$ if $a=0$, $b+c=N$ and $1<c<b$, as well as all cyclic permutations of such $(a,b,c)$. Finally define $\theta( a,b,c)= 0$ for all the other cases. For $N=3$, replace the conditions $1<b<c$ and $1<c<b$, respectively, by $b<c$ and $c<b$. We show that these theta foam values induce Lie brackets as desired.
![Evaluating bracket[]{data-label="CNrel"}](CNrel){width="3in"}
The operation $[X^j, X^k]$ is evaluated, using the $\Delta(1)$-relation, by $$[X^j, X^k]= \sum_{i=0}^{N-1} \theta (X^i, X^j, X^k) X^{N-1-i}.$$ This calculation is depicted in Fig. \[CNrel\]. Since $\theta (i,j,k) =0$ unless $i+j+k=N$, we have $[X^j, X^k]= \theta (i,j,k) X^{N-1-i}$ where $i=N-(j+k)$ and $N-1-i=j+k-1$, so that $[X^j, X^k]= \theta ({N-(j+k)}, j, k) X^{j+k-1}$. Note that if $j+k>N$, then the RHS is understood to be zero from the definition of $\theta$. From the definition of $\theta$ by cyclic ordering, the skew symmetry of $[\ , \ ]$ is clear. We show the Jacobi identity $$[ X^j, [ X^k, X^\ell ] ] + [ X^\ell, [ X^j, X^k]] + [ X^k, [ X^\ell, X^j]] =0$$ case by case. First we compute $$\begin{aligned}
[ X^j, [ X^k, X^{\ell } ]]
&=& \theta ({N -(k+\ell )}, k, \ell) \ \theta( N+1-(j+k+\ell), j, k +\ell -1) , \\ {}
[ X^\ell, [ X^j, X^k]] &=& \theta ({N -(j+k )},
j, k)\ \theta( N+1-(j+k+\ell), \ell, j+k-1), \\ {}
[ X^k, [ X^\ell, X^j]] &=& \theta ({N -(\ell+j )},
\ell , j)\ \theta( N+1-(j+k+\ell), k, \ell +j -1),
\end{aligned}$$ hence it is sufficient to prove that the sum of the right-hand sides is zero.
[**Case 1**]{}: $j+k+\ell > N+1$.
In this case, the second factors of the RHS are zero, so that all terms are zero.
[**Case 2**]{}: $j+k+\ell \leq N+1$ and $k + \ell > N$.
This case implies that $j=0$ and $k+\ell=N+1$. Since $N+1$ is even, $k$ and $\ell$ have the same parity. The first factor $\theta ({N -(k+\ell)}, k, \ell) $ is $0$ since $N -(k+\ell)=-1$. (When the arguments of $\theta$ are out of range, then $\theta=0$. ) Suppose $k=\ell=(N+1)/2$. Then the second and the third terms are $$\begin{aligned}
\lefteqn{
\theta ({N -(j+k)},
j, k)\ \theta( N+1-(j+k+\ell), \ell, j+k-1) } \\
&=& \theta ((N-1)/2, 0, (N+1)/2 )\ \theta (0, (N+1)/2, (N-1)/2 ) = (-1) (-1) = 1, \\
\lefteqn{
\theta ({N -(\ell+j)},
\ell , j)\ \theta( N+1-(j+k+\ell), k, \ell +j -1) } \\
&=& \theta ((N-1)/2, (N+1)/2, 0 )\ \theta (0, (N+1)/2, (N-1)/2 ) = (1) (-1)=-1,
\end{aligned}$$ as desired. Hence assume $k < \ell$ without loss of generality. For the second and third terms, we have $$\begin{aligned}
\lefteqn{
\theta ({N -(j+k)}
j, k) \ \theta( N+1-(j+k+\ell), \ell, j+k-1) } \\
&=& \theta ( \ell -1 , 0, k) \ \theta (0, \ell, k-1) = (1) (-1) = -1, \\
\lefteqn{
\theta ({N -(\ell+j)},
\ell , j)\ \theta( N+1-(j+k+\ell), k, \ell +j -1) }\\
&=& \theta ( k -1 , \ell, 0)\ \theta (0, k, \ell-1) = (1) (1) = 1,
\end{aligned}$$ as desired.
[**Case 3**]{}: $j+k+\ell \leq N+1$ and $k + \ell \leq N$.
First we check the case where two of $j, k, \ell$ are the same. Suppose $j=k$. Then the second term is zero, as $\theta(N-(j+k), j, k)=0$. Furthermore, for the first and third terms, we have $\theta ({N -(k+\ell)}, k, \ell) = - \theta ({N -(k+\ell)}, \ell , j)$ and $$\theta( N+1-(j+k+\ell), j, k +\ell -1)=
\theta( N+1-(j+k+\ell), k, \ell +j -1)$$ as desired. The other cases $(k=\ell, j=\ell)$ are checked similarly. Hence we may assume $j < k < \ell$.
Since $\theta$ vanishes unless one of the entries is $0$, the first factors of the RHS are zero if $N> k+ \ell$ and $0 < j<k<\ell$. hence we may assume that $k+\ell = N$ or $j=0$.
We continue to examine specific subcases. Suppose that $j=0$ and $k + \ell < N$. The RHS becomes: $$\begin{aligned}
\lefteqn{
\theta(N-(k+\ell), k, \ell)\theta(N+1-(k+\ell),0,k+\ell-1)} \\
&+& \theta(N-k, 0, k)\theta(N+1-(k+\ell),\ell,k-1) \\
& +& \theta(N-\ell, \ell,0)\theta(N+1-(k+\ell),k,\ell-1) .
\end{aligned}$$
If $j=0$ and $k+\ell = N$, we have $$\theta(0, k, \ell)\theta(1,0,N-1) +
\theta(\ell, 0, k)\theta(1,\ell,k-1) +
\theta(k, \ell,0)\theta(1,k,\ell-1) .$$
If $1<k$, then the sum is $0$ since $\theta(1,0,N-1)=0$ and the arguments of the second and the third factors are all non-zero. If $k=1$, then $\theta(0,1,N-1)=\theta(\ell,0,1)=\theta(1, \ell,0)$, so the sum is $0$.
Now suppose that $k+\ell<N$. The first term is $0$ and the second factors in the sum have arguments that are all non-zero unless $k=1$. If $k=1$, we have $$\theta(N-1,0,1)\theta(N-\ell,\ell, 0)+ \theta (N-\ell, \ell, 0)\theta(N-\ell,1,\ell-1)=0 .$$
Finally, suppose that $j\ne0$, so that $k+\ell=N$. The RHS becomes: $$\theta(0,N-\ell, \ell)\theta(1-j,j, N-1) + \theta(0,j,N-\ell)\theta(1-j,\ell, N-1) + \theta(N-(\ell +j),\ell, j)\theta(1-j,N-\ell,\ell+j-1).$$ If $1<j$, then the first argument of all the second factors is negative, so the sum is $0$. If $j=1$, then each term has a factor that is $0$. $\Box$
Since the original motivation came from the foams in [@Kh03; @MV06], we examine the Frobenius algebra in [@MV06] closely. In this case, the multiplication that is induced by branch circles also satisfies the Jacobi identity.
\[fromVaz\] Let $A={\mathbb{Z}}[a,b,c] [X] / (X^3-aX^2-bX-c)$ with Frobenius structure defined as in Example \[MVex\] from [@MV06]. The branch curve operation $[ \ , \ ] $ is skew-symmetric and satisfies the Jacobi identity: $$[ U , [ V, W ] ]
+
[ V , [ W, U] ]
+
[ W, [ U, V ] ]$$ for any $U,V,W \in A$.
[*Proof.*]{} This is confirmed by calculations. From the axioms of $A$ and the theta foam values that are given in [@MV06]: $$\theta(1,X,X^2)=\theta(X^2,1,X)=\theta(X,X^2,1)=1=-\theta(1,X^2,X)
=-\theta(X,1,X^2)=
-
\theta(X^2,X,1)$$ while $\theta=0$ for any other arguments, we compute using the $\Delta(1)$ relation for Example \[MVex\]:
$$\begin{aligned}
[1, X ] &=& -1 , \\ {}
[1, X^2 ] &=& X- a , \\ {}
[X, X^2] &=& -X^2 + aX + b . \end{aligned}$$
Then one computes $$\begin{aligned}
[ 1, [X, X^2]] &=& -X , \\ {}
[X, [X^2, 1]] &=& a , \\ {}
[X^2, [1, X]] &=& X-a ,\end{aligned}$$ as desired. In general, we consider cyclic permutations of $X^j,$ $X^k$, and $X^\ell$ in the expression $[X^j ,[X^k, X^\ell]]$. Since the bracket is skew-symmetric, then we need only consider the cases in which $j$, $k$, and $\ell$ are distinct. The remaining case follows by skew-symmetry. $\Box$
![Upside-down operation[]{data-label="branchup"}](branchup){width="2in"}
We define the operation ${{\bf{ \Delta}} }: A \rightarrow A \otimes A $ that is associated to the left of Fig. \[branchup\], a diagram that is up-side down of Fig. \[branch\], in which one circle branches into two from bottom to top. A cyclic order is specified in the figure. If we specify the ordered tensor factors assigned to each sheet by $A_i$, $i=1,2,3$, then the operation is defined as ${{\bf{ \Delta}} }: A_1 \rightarrow A_3 \otimes A_2 $. A planar diagram representing this operation is depicted in the right of the figure. Imitating Sweedler notation $\Delta(u)=\sum u_{(1)} \otimes u_{(2)} $ for comultiplication, we denote ${{\bf{ \Delta}} }(u)=\sum u_{ ((1))} \otimes u_{((2))}.$ The next lemma relates this operation to the unit map, and diagrammatic formulations are given in Fig. \[leftright\].
Let $A={\mathbb{Z}}[a,b,c] [X] / (X^3-aX^2-bX-c)$, with $\Delta(1)$-condition defined as in Example \[MVex\]. The map ${{\bf{ \Delta}} }$ is computed as follows. $$\begin{aligned}
{{\bf{ \Delta}} }(1) &=& 1 \otimes X - X \otimes 1 , \\
{{\bf{ \Delta}} }(X) &=& a( 1 \otimes X - X \otimes 1 ) - (1 \otimes X^2 - X^2 \otimes 1 ), \\
{{\bf{ \Delta}} }(X^2) &=& (a^2 + b) ( 1 \otimes X - X \otimes 1 ) - a (1 \otimes X^2 - X^2 \otimes 1 )
+ (X \otimes X^2 - X^2 \otimes X ) . \end{aligned}$$
![${{\bf{ \Delta}} }$ can be defined from left or right[]{data-label="leftright"}](leftright){width="1.7in"}
Direct calculations show
${{\bf{ \Delta}} }(V ) = \sum [ V, 1_{(1)} ] \otimes 1_{(2)}= \sum 1_{(1)} \otimes [ 1_{(2)}, V ]$.
The diagram for this relation is depicted in Fig. \[leftright\]. Other relations that follow are depicted in Fig. \[otherrels\].
![Other symmetric relations[]{data-label="otherrels"}](otherrels){width="4in"}
The following relations hold for maps in Frobenius algebras and maps associated to branch circles. Here we used the notation ${{\bf{m}} }$ instead of $[\ , \ ]$ to formulate in tensor products. The equalities are diagrammatically represented in Fig. \[webskeins\].
![Web skein relations[]{data-label="webskeins"}](webskeins){width="5in"}
\[skeinprop\] For $A={\mathbb{Z}}[a,b,c]/(X^3-aX^2-bX-c)$ with $\theta$ values as above, the map ${{\bf{ \Delta}} }: A \rightarrow A \otimes A$ satisfies the following identities: $$\begin{aligned}
({{\bf{m}} }\otimes {|})({|}\otimes {{\bf{ \Delta}} }) &=&
\Delta(1) ( \epsilon \mu )- \tau , \\
( \ ({{\bf{m}} }\otimes {|})({|}\otimes {{\bf{ \Delta}} }) \ ) ^2 &=& {|}+ \Delta(1) ( \epsilon \mu ), \\
{{\bf{m}} }{{\bf{ \Delta}} }&=& 2 \ {|}. \end{aligned}$$
[*Proof.*]{} The first and the third equalities are verified by calculations on basis elements. For all $X^i$ and $X^j$, it is computed as $[X^i, X^j_{((1))}] \otimes X^j_{((2))} = X^j \otimes X^i + \epsilon (X^{i+j} ) \Delta (1)$. The second relation is diagrammatically computed as in Fig. \[webskeinproof\]. Note that the [*handle element*]{} $\epsilon\mu\Delta(1)$ is $3$. $\Box$
![Proof of the skein relation[]{data-label="webskeinproof"}](webskeinproof){width="3.5in"}
The skein relations stated in Proposition \[skeinprop\], as planar diagrams (instead of surface skein relation), coincide with those described in [@Kh03] as a description of Kuperberg’s invariant [@Kup91], with the choice of $q=1$.
Thus, the operation at branch curve of the foam used to categorify the quantum $sl(3)$ invariant satisfies the skein relations at the classical limit of the invariant.
![A surface skein relation in [@Kh03; @MV06] []{data-label="sqr"}](sqr){width="4in"}
The second relation in Proposition \[skeinprop\] is related to the local surface skein relation in [@Kh03; @MV06] as follows. Their local relation is depicted in Fig. \[sqr\]. Notice the negative signs, as well as resemblance to our relation. After performing their relations locally, move the holes of each term along the $S^1$ factor to the other side. Then one obtains a tube connecting two sheets. Then perform the [*bamboo cutting relation*]{}, that is computed by applying $\Delta(1)$-relation three times. In this case, one computes that it is the negative of the original bamboo segment. These negative signs cancel, and we obtain our equation. Thus, our relation follows from theirs, or algebraically as we have shown.
We also point out that the second and the third relation in Proposition \[skeinprop\] have interpretations in ${\bf Foam}_A$. One simply takes the product of these diagrams with $S^1$ to obtain foams, and the equalities hold in ${\bf Foam}_A$. The first equality, however, is not realized in ${\bf Foam}_A$, as the intersection of surfaces are not allowed in ${\bf Foam}_A$.
![Cutting a bamboo segment[]{data-label="bambocut"}](bamboocut){width="4in"}
Bialgebras {#bialgsec}
==========
In this section, we investigate functors whose image of branch curves induce bialgebra structure for group algebras. Let $G$ be a group. Let $A=R[G]$ be the group ring with a commutative unital ring $R$. It is well known that $A$ has a commutative Hopf algebra structure defined as follows (see, for example, [@Kock]). Define ${{\bf{ \Delta}} }: A \rightarrow A \otimes A$ by linearly extending ${{\bf{ \Delta}} }(x)=x \otimes x$. (This is different from the comultiplication as a Frobenius algebra $\Delta(x)= \sum_{x=yz} y \otimes z$.) The unit map is defined as the same as the Frobenius unit map $\eta (1)=1_G$, where $1_G$ is the identity element of $G$. (The counit map as a Frobenius algebra is defined by $\epsilon (1_G)=1$ and $\epsilon(x)=0$ for $x \neq 1_G$.) The following shows that there is a strong requirement for group algebras to give bialgebra structures through branch curves.
![Comultiplication by theta foams[]{data-label="bialg"}](bialg){width="2.2in"}
\[bialgprop\] Let $G$ be an abelian group. For any unital ring $R$, the branch circle operation ${{\bf{m}} }$ induces a bialgebra structure on $A$ if and only if every non-identity element of $G$ has order $2$.
[*Proof.*]{} The $\Delta(1)$-relation is written as $\Delta(1)=\sum_{y \in G} y \otimes y^{-1}$, and the reducing ${{\bf{ \Delta}} }$ into the theta foam is depicted in Fig. \[bialg\]. For ${{\bf{ \Delta}} }(x)=x \otimes x$ to hold in the figure, we have $y=z=x$, and the value of the theta foam being $\theta(x, y^{-1}, z^{-1})=1$ for $y=z=x$ and $0$ otherwise.
For $\theta$ to satisfy the cyclic symmetry, this condition is satisfied if and only if $x^{-1}=x$ ($x$ having order $2$) for any $x\in G$, and in this case, the theta foam values are determined by $\theta (x,x,x)=1$ for any $x \in G$ and $0$ otherwise. $\Box$
![The compatibility condition of a bialgebra[]{data-label="compatisfce"}](compatisfce){width="2.2in"}
[The condition of a bialgebra that the comultiplication is an algebra homomorphism (also called a compatibility condition) ${{\bf{ \Delta}} }(a b)={{\bf{ \Delta}} }(a) {{\bf{ \Delta}} }(b)$ for $a, b \in A$, is represented by surfaces in Fig. \[compatisfce\]. ]{}
![Surface skein relations[]{data-label="skein"}](skein){width="3.3in"}
In [@AF], skein modules for $3$-manifolds based on embedded surfaces modulo the surface skein relations described in [@BN] were defined and studied. Surface skein modules were generalized in [@Kaiser] using general commutative Frobenius algebras. Such notions are directly generalized to foams, with various skein relations at hand. Skein modules for $sl(3)$ foams are analogously defined using the local skein relations given in [@Kh03; @MV06], for example.
Here we propose local skein relations based on the foams in Proposition \[bialgprop\] with the bialgebra on branch curves for the group ring ${\mathbb{Z}}[x]/(x^2-1)$. Considering that the move characteristic to bialgebras is the compatibility condition as depicted in Fig. \[compatisfce\], we take a local change that happens at the saddle point of this move, as depicted in the top of Fig. \[skein\] (labeled as saddle) as a local surface skein relation. Other relations in Fig. \[skein\] are those coming from Frobenius algebra structure and the theta foam values as before.
Thus the skein module $ {\bf F}(M) $ in this case can be defined to be the isotopy classes of foams in a given $3$-manifold $M$ modulo the local surface skein relations in Fig. \[skein\]. Although computations of this skein module in general is out of the scope of this paper, it seems interesting to look into relations between foams and the topology of $3$-manifolds.
Acknowledgments {#acknowledgments .unnumbered}
----------------
JSC was supported in part by NSF Grant DMS \#0603926. MS was supported in part by NSF Grants DMS \#0603876 and \#0900671.
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abstract: 'We report on a new class of magnetoresistance oscillations observed in a high-mobility two-dimensional electron gas (2DEG) in GaAs-Al$_x$Ga$_{1-x}$As heterostructures. Appearing in a weak magnetic field ($B<$ 0.3 T) and only in a narrow temperature range (2 K $<T<$ 9 K), these oscillations are periodic in $1/B$ with a frequency proportional to the electron Fermi wave vector, $k_F$. We interpret the effect as a magnetophonon resonance of the 2DEG with leaky interface-acoustic phonon modes carrying a wave vector $q=2k_F$. Calculations show a few branches of such modes on the GaAs-Al$_x$Ga$_{1-x}$As interface, and their velocities are in quantitative agreement with the data.'
author:
- 'M. A. Zudov'
- 'I. V. Ponomarev'
- 'A. L. Efros'
- 'R. R. Du'
- 'J. A. Simmons'
- 'J. L. Reno'
- '(Received 2 May 2000)'
title: 'New Class of Magnetoresistance Oscillations: Interaction of a Two-Dimensional Electron Gas with Leaky Interface Phonons'
---
[^1]
There are several classes of transverse magnetoresistance (MR) oscillations known to exist in a two-dimensional homogeneous electron gas (2DEG). The most common of these are the Shubnikov-de Haas oscillations (SdH), which arise from a magnetic field $B$-induced modulation of the density of states at the Fermi level $E_F$. They become more pronounced with decreasing temperature $T$. The magnetophonon resonance (MPR) [@mpr; @tsui] is a source of another class of oscillations resulting from the absorption of bulk longitudinal optical phonons. These resonances appear under the condition $\omega_{LO}=l\omega_c$, where $\omega_{LO}$ and $\omega_c=eB/mc$ are the optical phonon and cyclotron frequencies respectively, $l$ is an integer, and $m$ is the effective mass of the carriers. These oscillations are only seen at relatively high $T\sim 100-180$K [@tsui]. Both SdH and MPR are periodic in $1/B$, but the SdH frequency (reciprocal period) scales with electron density as $n_e$, whereas MPR is $n_e$-independent.
In this Letter, we report on a new class of MR oscillations [@zudov] observed in a high-mobility 2DEG in GaAs-Al$_x$Ga$_{1-x}$As heterostructures. Unrelated to either of the above origins, these novel oscillations are still periodic in $1/B$, but they appear [*only*]{} in a narrow temperature range (2 K $<T<$ 9 K), and their frequency scales with $\sqrt {n_e}$. We interpret the data in terms of a magnetophonon resonance mediated by thermally excited [*leaky interface-acoustic phonon*]{} (LIP) modes. In principle, the surface modes might provide a good explanation as well, but in our case 2DEG is located so far from the surface ($\sim 0.5$ $\mu$m) that no such interaction is possible.
The leaky interface modes have been studied for a few decades in connection with the Earth’s crust [@maradudin]. The term “leaky” shows that the waves propagate at a small angle with the interface, so that the energy radiates away from the boundary. For some specific parameters these waves may not be leaky [@stoneley], but for the interface under study all of them are leaky. Despite the fact that LIP is commonly present in layered material systems [@zinin], it has so far not been considered on the GaAs-Al$_x$Ga$_{1-x}$As interface. Due to radiation of energy, the frequency and velocity of leaky waves are complex: $u=\omega/q=u_R-iu_I$ with $u_I \ll u_R$.
The novel oscillations can be explained by a simple momentum selection rule which is derived later in the paper. It states that at high Landau levels (LLs) the electrons interact predominantly with the interface phonons carrying a wave vector $q=2k_F$, where $k_F$ is the Fermi wave vector of the 2DEG at zero $B$ field. The condition for resonant absorption or emission of an interface phonon is then given by $$2k_F u_R= l \omega _c,\,\,\,\, \,\,\,\,l=1,2,3,...\,.
\label{con}$$
We claim that Eq. (\[con\]) determines the values of $B$ for the [*maxima*]{} in these new MR oscillations. It shows that the oscillations are periodic in $1/B$ with a frequency $f=2k_Fumc/e$. Evidently, the bulk phonons can not account for the resonance, since their frequency depends on $q_z$, while the selection rule includes lateral momentum only.
Our primary samples are lithographically defined Hall bars cleaved from modulation-doped GaAs-Al$_{0.3}$Ga$_{0.7}$As heterostructures of high-mobility $\mu \approx 3\times 10^6$ cm$^2$/Vs. The wafers are grown by molecular-beam epitaxy on the (001) GaAs substrate. At low $T$, the density of the 2DEG, $n_e$ (in units of $10^{11}$ cm$^{-2}$ throughout the text), can be tuned by a combination of illumination from light-emitting diode and the NiCr front gate potential. The experiments were performed in a variable-temperature $^4$He cryostat equipped with a superconducting magnet, employing a standard low-frequency lock-in technique for resistance measurement.
In Fig. \[fig1\] we show the normalized low-field magnetoresistivity $\rho_{xx}(B)/\rho_{xx}(0)$ measured at $T = 4$ K, for the electron density $n_e=$ 2.05, 2.27, and 2.55, respectively. In addition to the damped SdH commonly seen in a 2DEG at this $T$, the traces reveal new oscillations that appear only at $B<0.3$ T. The amplitude of the oscillations is about 2-3 % in these traces.
Three aspects of the observation should be highlighted. First, the oscillations are roughly periodic in inverse magnetic field, $1/B$. The arrows next to the traces indicate the $\rho_{xx} (B_l)$ [*maxima*]{} (indexed as $l$ = 1, 2, 3, 4) in this oscillatory structure. In the inset we plot the order of the oscillations, $l$ (and $-d^2\rho_{xx}/dB^2$), vs. $1/B$ for $n_e=$ 2.55 and observe a linear dependence. Such periodic oscillations have been seen for all $n_e$ (from $\sim 1.5$ to 3) studied. Second, with increasing $n_e$ the features shift orderly towards higher $B$. Finally, the oscillatory structure is accompanied by a negative MR background, apparently in the same $B$ range where the oscillations take place.
We have now measured over a dozen specimen (from five wafers), of both the Hall bar (width from 10 $\mu$m to 500 $\mu$m) and the square (5 mm x 5 mm) geometries, and consistently observed similar oscillatory structures. On the other hand, the significance of the ubiquitous MR background remains unclear. Either negative or positive MR has been observed, and its strength (and even the sign) is largely specimen and cooling-cycle dependent. In the following we shall focus on the analysis of the oscillatory structure, in particular, its dependence on $n_e$ and $T$.
To further quantify our results, we have performed fast Fourier transform (FFT) on the resistance data. As an example, Fig. \[fig2\] shows the FFT power spectra obtained from the three traces in Fig. \[fig1\] [@fft]. Surprisingly, such analysis has uncovered two frequencies, marked by $A$ and $B$. Peak $A$ corresponds to the main period, conforming to the simple fit in Fig. \[fig1\]. Peak $B$ is somewhat weaker, and occurs at $f_B\approx 1.5 f_A$. The shift of the doublet with increasing $n_e$ is marked by three arrows for the main peak. The FFT data have revealed a striking linear relation between the frequencies of oscillations and the electron Fermi wave vector. We plot (see the inset) $f^2$ of the FFT peaks against the electron density, $n_e$, which has been varied from 1.47 to 2.95 in the same specimen. Since $k_F =\sqrt{2\pi n_e}$, the observed linearity indicates that $f \propto k_F$.
Such a linear dependence distinguishes the new oscillations from SdH, as $f_{SdH} \propto k_F^2$, and is exactly what one expects from the phonon resonance scenario proposed here. As such, the oscillatory structure must be viewed as resulting from the resonance of the 2DEG with two branches of the interface modes. Using Eq. \[con\] and a single known material parameter, the GaAs band electron mass $m \approx 0.068 m_e$, we fit the data (solid line in the inset) and deduce a velocity for the slow (fast) mode $u_A \approx$ 2.9 km/s ($u_B \approx 4.4$ km/s). Within the experimental error of 10% the data from several specimen collapse on the same lines, indicating that the new oscillations are generic in high-mobility 2DEG in GaAs-Al$_x$Ga$_{1-x}$As heterostructures.
The $T$-dependence of the oscillations is consistent with a [*thermally excited*]{} phonon-scattering model. Fig. \[fig3\] shows the $\rho_{xx}(B)$ at selected temperatures (1.9 K $<T<$ 9.1 K ) where the evolution of the oscillations is clearly seen. Notice first (see inset) that $\rho_{xx}(0)$ grows linearly with $T$, indicating that acoustic-phonon scattering dominates the electron mobility in this temperature range [@mendez; @stormer]. Considering the interface phonon modes of interest here, we use the value of the slow mode $u_A=2.9$ km/s to estimate a characteristic temperature, $T_c$, from $k_BT_c =\hbar u_A(2k_F)$. The value of $T_c\approx$ 5 K can qualitatively account for the temperature dependence of the main features of the oscillations. While the SdH gradually diminishes as $T$ increases, the oscillations are best developed at $T \approx 3-7$ K and are strongly damped at both higher and lower $T$. At $T\ll T_c$ the number of interface phonons carrying $q=2k_F$ becomes small and therefore the amplitudes diminish. At high $T$ the smearing of the LLs prevails and the oscillations disappear as well.
We now turn to the details of the theoretical explanation of the novel oscillations. We have performed calculations [@efros] of LIP modes for the GaAs-Al$_{0.3}$Ga$_{0.7}$As interface on the basal (001) plane. In the anisotropic case the speed of LIPs depends on angle between $q$ and the \[100\] direction. Using the elastic moduli of the bulk lattices [@bulk] we found a series of modes with weak anisotropy and a small imaginary part of the velocity ($u_I/u_R<0.03$). We have studied the modes within the interval of velocities 2.4$-$6.0 km/s. Two close groups of modes have been found, one within the interval of 3$-$3.5 km/s and the other within 4.2$-$4.5 km/s. These modes may be responsible for the two periods of oscillations which have been observed. The frequencies of the other modes found are too high to be detected in our experiment. Note that different modes may interact with electrons with different strengths.
To calculate the transverse conductivity due to the scattering of the 2DEG by the LIPs, we employ a 2D analog of the formula, first derived by Titeica [@titeica]: $$\begin{aligned}
\label{Titfor}
&&\sigma_{xx}=\frac{4\pi e^2}{A m^2 k_BT \omega _c^2}
\sum_{n,n'}\sum_{k_y,k_y'}\sum_{q_x,q_y}|I_{nn'}(q\lambda)|^2 q_y^2
|C(q)|^2\nonumber\\ &&\times
N_{l}f_{n}(1-f_{n'})\delta_{k_y-k_y'+q_y}\delta(\omega _c(n'-n)-qu).\end{aligned}$$ Here $A$ is the area, $N_l=\left(\exp(\hbar\omega /k_BT)-1\right)^{-1}$, $f_{n}=\left(\exp\left((E_n-\mu)/k_BT\right)+1\right)^{-1}$, $\lambda=\sqrt{\hbar c/eB}$ is the magnetic length, and $|C(q)|^2\equiv v(q)/A$ is the square modulus of the 2DEG-LIP interaction, which has a power law dependence on $q$. This formula can be interpreted in the following way. A 2D electron in a magnetic field has a wave function which is a product of a plane wave in the $y$ direction and an oscillatory wave function, centered at the position $x_0=-c\hbar k_y/eB$: $\Psi=\exp(ik_y y)\phi_n(x-x_0)$, where $n$ is the LL index. In the absence of scattering the electric current may flow only in the $y$-direction, providing the Hall effect. A transverse conductivity appears because an electron transfers wave vector $q_y=k_{y}'-k_y$ to a scatterer. This is equivalent to a jump in the $x$-direction at a distance $\Delta x_0=c\hbar q_y/eB$. In Eq. (\[Titfor\]) this physics is applied to electron-interface phonon scattering. The mechanism of the 2DEG-LIP interaction, which may be either deformation potential or piezoelectric interaction, is not particularly important for our purpose. The overall scattering is of the same order as the bulk phonon scattering, since the energy densities of both excitations are of the same order in the vicinity of the interface.
If the interface phonon has no attenuation, the square of matrix element $I_{nn'}$ is given by [@sdh] $$\begin{aligned}
\label{matel}
|I_{n,n+l}(b)|^2 &=& \left|\int_{-\infty}^{+\infty}e^{iq_x x}\phi_n(x-x_0) \phi_{n+l}(x-x_0')\,dx\right|^2\nonumber\\
&=&\frac {n!}{(n+l)!} \left(\frac{b^2}{2}\right)^{l} e^{-\frac{b^2}{2}}
\left[L_n^l\left(\frac{b^2}{2}\right)\right]^2,\end{aligned}$$ where $b=q\lambda$ and $L_n^l(x)$ is the generalized Laguerre polynomial. Substituting summation over wave vectors by integration in Eq. (\[Titfor\]) one obtains $$\begin{aligned}
\label{interm}
&&\sigma_{xx}=\frac{e^2}{2\pi\hbar}\frac{1}{m k_BT \omega _c}
\sum_{n,l}N_{l}f_{n}(1-f_{n+l})\nonumber\\ && \times
\int_0^{\infty}dq\, q^3 v(q)\left| I_{n\,n+l}(q\lambda) \right|^2
\delta\left(\omega _c l-qu\right).\end{aligned}$$ Taking into account the imaginary part of the LIP frequency, $\omega=q(u_R-iu_I)$, we can substitute for the $\delta$-function in Eq. (\[interm\]) a Gaussian distribution with appropriate dispersion $\sigma=qu_I$. Since the dispersion is small we can set $q=\omega _c l/u_R$ everywhere except for the strongly oscillating function $|I_{nl}(q\lambda)|^2$. Then after averaging we obtain for the transverse conductivity $$\label{fineq}
\sigma_{xx}=\frac{e^2/2\pi\hbar}{mu\omega _c k_BT}
\sum_{l,n}v\left(\frac {\omega _c l}{u}\right) F_{nl} N_l
f_n(1-f_{n+l}),$$ where the function $F_{nl}$ can be expressed as a series of Hermite polynomials of imaginary argument: $$\begin{aligned}
F_{nl} & = & \frac{(\omega_c l/u)^3}{\sqrt{1+\alpha^{-1}}} \exp\left(-\frac{\alpha}{1+\alpha}\frac{\hbar\omega _c l^2}{2mu^2}\right) \nonumber \\
& &\times \sum_{k,j}^n \frac{n!(-1)^l}{(n+l)!k!j!}
\left(\begin{matrix} {n+l} \\ {n-k} \end{matrix} \right)
\left(\begin{matrix} {n+l} \\ {n-j} \end{matrix} \right)
% \left(\begin{array}{c} {n+l} \\ {n-j} \end{array} \right)
% \left( \begin{array} {c} {n+l} \\ {n-k} \end{array} \right)
\nonumber \\
& &\times \frac{H_{2(k+j+l)}\left(il\sqrt{\frac{\hbar\omega_c}{2mu^2}}\frac{\alpha} {\sqrt{1+\alpha}}\right)}{[2\sqrt{1+\alpha}]^{2(k+l+j)}},
%\binom {n+l}{n-k} \binom {n+l}{n-j}
%\frac{H_{2(k+j+l)}\left(il\sqrt{\frac{\hbar\omega_c}{2mu^2}}\frac{\alpha} {\sqrt{1+\alpha}}\right)}{[2\sqrt{1+\alpha}]^{2(k+l+j)}},\end{aligned}$$ with $\alpha=(u/\sigma \lambda)^2$. Hereafter, we assume that $u$ is the real part of the LIP velocity. In Fig. \[fig4\] we plot $F_{nl}$ for $n=17$ and $l=1$ as a function of $B$ for LIP with $\sigma =\omega_c u_I/u_R$ (solid line) and in the limit $\sigma =0$ (dashed line).
As we can see, once attenuation is introduced, only one strong peak remains that corresponds to Eq. (\[con\]) at $l=1$. This means that only phonons with wave vector $q=2\sqrt{2n}/\lambda$ effectively interact with electrons under the condition $n\gg 1$. In fact, due to the Fermi distribution in Eq. (\[fineq\]) only the values of $n\approx E_F/\hbar\omega_c$ are important. Then, indeed, $2\sqrt{2n}/\lambda= 2\sqrt{2\pi n_e}=2k_F$, and we arrive at Eq. (\[con\]) for $l=1$. The same conclusion holds for any $l\ll n$.
This is an important result of our work. It can be interpreted from the following semi-classical consideration (see the inset in Fig. \[fig4\]). Let us consider $n\sim n' \gg 1$. Since the square of the matrix element in Eq. (\[matel\]) depends on $q$ only, we can put $q_x=0$. Then the integrand in (\[matel\]) is an overlap of two oscillatory wave functions shifted with respect to each other. In the vicinity of the turning point the wave function always has a maximum since the momentum is small and the particle spends most of its time there. There are three possibilities. Cases 1 and 3 in the inset show situations when turning points are apart from each other, and case 2 occurs when the turning points coincide in space. Obviously, in case 2 the overlap integral has a maximum. This occurs when $m\omega_c^2(\Delta x_0)^2/8=n\hbar\omega_c$, which is equivalent to the above condition $q\lambda\approx 2\sqrt{2n}$. Note that the other maxima in Fig. \[fig4\] can be smeared very easily because their widths are proportional to $n^{-1/2}$, while the first maximum near the turning point can be approximated by an Airy function and its width is independent of $n$. Thus, the maxima in $F_{nl}(B)$ for different $l$ give rise to oscillations in $\rho_{xx}(B)$.
As a whole, the results provide good agreement with our experimental data. In particular, the slow mode $u_A$ can be identified with the lower bunch of modes calculated here. Within the experimental uncertainty we are unable to find any anisotropy for the velocity, therefore, the data must be viewed as an average over all directions. Likewise, the velocities of fast modes coincide with $u_B=$ 4.4 km/s, but this should be taken with caution. Since our experiments have so far been centered on a temperature range around 5 K, a positive identification of the fast mode awaits for a more detailed $T$-dependence study at higher temperatures.
In conclusion, we have discovered a new class of magneto-oscillations in a high-mobility 2DEG and interpreted it as a magnetophonon resonance with leaky interface-acoustic phonons. Owning to their 2D characteristics, the leaky interface modes play a unique role in the scattering of 2D electrons in GaAs-Al$_x$Ga$_{1-x}$As heterostructures and quantum wells. This role has never been studied before.
The experimental work (M.A.Z. and R.R.D.) is supported by NSF grant DMR-9705521. R.R.D. also acknowledges an Alfred P. Sloan Research Fellowship and thanks M. E. Raikh for helpful conversations. The theoretical work (I.V.P. and A.L.E.) is supported by a seed grant of the University of Utah. A.L.E. is grateful to R. L. Willett for insightful discussions. The work at Sandia is supported by the US DOE under contract DE-AC04-94AL85000.
V. L. Gurevich and Yu. A. Firsov, Sov. Phys. JETP [**13**]{}, 137 (1961). D. C. Tsui [*et al.*]{}, Phys. Rev. Lett. [**44**]{}, 341 (1980). M. A. Zudov, Ph. D. thesis, University of Utah, August 1999, unpublished; M. A. Zudov [*et al.*]{}, APS Bulletin [**44**]{}, 1572 (1999); I. V. Ponomarev [*et al.*]{}, APS Bulletin [**45**]{}, 361 (2000). see, [*e.g.*]{}, A. A. Maradudin, in [*Surface Phonons*]{}, ed. by W. Kress and F. W. de Wette (Springer-Verlag, Berlin, 1991) and references therein. R. Stoneley, Proc. R. Soc. (London) [**A 106**]{}, 416 (1924). For example, leaky interface-acoustic modes on the N$_3$Si$_4$-GaAs (001) interface have been observed in Brillouin spectroscopy, P. Zinin, [*et al.*]{}, Phys. Rev. B [**60**]{}, 2844 (1999). The FFT was performed on $-d^2\rho_{xx}/dB^2$ to enhance the signal/noise ratio. The broader SdH width reflects the fact that the data were from a limited $B$ range 0 $<B<$ 0.4 T. E. E. Mendez, P. J. Price, and M. Heiblum, Appl. Phys. Lett. [**45**]{}, 294 (1984). H. L. Stormer, [*et al.*]{}, Phys. Rev. B [**41**]{}, 1278 (1990). I. V. Ponomarev and A. L. Efros, Phys. Rev. B [**63**]{} 165305 (2001). S. H. Simon, Phys. Rev. B [**54**]{}, 13878 (1996). S. Titeica, Annal d. Phys. [**22**]{}, 128 (1935). T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. [**54**]{}, 437 (1982).
[^1]: Present address: Stanford Picosecond Free Electron Laser Center, Stanford University, Stanford, CA 94305
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abstract: 'Let $K$ be a rationally null-homologous knot in a three-manifold $Y$. We construct a version of knot Floer homology in this context, including a description of the Floer homology of a three-manifold obtained as Morse surgery on the knot $K$. As an application, we express the Heegaard Floer homology of rational surgeries on $Y$ along a null-homologous knot $K$ in terms of the filtered homotopy type of the knot invariant for $K$. This has applications to Dehn surgery problems for knots in $S^3$. In a different direction, we use the techniques developed here to calculate the Heegaard Floer homology of an arbitrary Seifert fibered three-manifold.'
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abstract: 'We study $U(1)$ symmetries dual to Betti multiplets in the $AdS_4/CFT_3$ correspondence for M2 branes at Calabi-Yau four-fold singularities. Analysis of the boundary conditions for vector fields in $AdS_4$ allows for a choice where wrapped M5 brane states carrying non-zero charge under such symmetries can be considered. We begin by focusing on isolated toric singularities without vanishing six-cycles, and study in detail the cone over $Q^{111}$. The boundary conditions considered are dual to a CFT where the gauge group is $U(1)^2 \times SU(N)^4$. We find agreement between the spectrum of gauge-invariant baryonic-type operators in this theory and wrapped M5 brane states. Moreover, the physics of vacua in which these symmetries are spontaneously broken precisely matches a dual gravity analysis involving resolutions of the singularity, where we are able to match condensates of the baryonic operators, Goldstone bosons and global strings. We also argue more generally that theories where the resolutions have six-cycles are expected to receive non-perturbative corrections from M5 brane instantons. We give a general formula relating the instanton action to normalizable harmonic two-forms, and compute it explicitly for the $Q^{222}$ example. The holographic interpretation of such instantons is currently unclear.'
---
.5cm
**Baryonic symmetries and M5 branes in\
\[3.4mm\]**
the AdS$_4$/CFT$_3$ correspondence
\
1.3cm
[Nessi Benishti$^{1}$, Diego Rodríguez-Gómez$^{2}$, and James Sparks$^{3}$]{}\
1: [*Rudolf Peierls Centre for Theoretical Physics,\
University of Oxford,\
1 Keble Road, Oxford OX1 3NP, U.K.*]{}\
0.5cm
2: [*Queen Mary, University of London,\
Mile End Road, London E1 4NS, U.K.*]{}\
0.5cm
3: [*Mathematical Institute, University of Oxford,\
24-29 St Giles’, Oxford OX1 3LB, U.K.*]{}\
.8cm
------------------------------------------------------------------------
width 5cm
Introduction {#sec:intro}
============
Over the last two years there have been major advances towards understanding the $AdS_4/CFT_3$ duality. Elaborating on [@Gustavsson:2007vu; @Bagger:2007vi], Aharony, Bergman, Jafferis and Maldacena [@Aharony:2008ug] proposed a theory conjectured to be dual to M2 branes probing a $\mathbb{C}^4/\mathbb{Z}_k$ singularity, where ${\mathbb{Z}}_k$ acts with weights $(1,1,-1,-1)$ on the coordinates of ${\mathbb{C}}^4$. This low energy theory on the worldvolume of $N$ coincident M2 branes is a $U(N)_k\times U(N)_{-k}$ quiver Chern-Simons (CS) theory, with a marginal quartic superpotential whose coefficient is related by the high degree of SUSY to the CS coupling. Indeed, for generic CS coupling $k$ the theory enjoys $\mathcal{N}=6$ SUSY, and as such possesses an $SO(6)_R$ symmetry which is manifest in the potential [@Benna:2008zy]. The theory is then automatically conformal at the quantum level. For $k=1,\, 2$ the SUSY is enhanced to $\mathcal{N}=8$. In field theory it has been argued [@Aharony:2008ug] that this enhancement is due to quantum effects where ’t Hooft monopole operators play a key rôle. Indeed, the ABJM theory has just the right structure [@Benna:2009xd] for these monopole operators to have appropriate quantum numbers that then allow for such a symmetry enhancement.
Motivated by this progress in understanding the maximally SUSY case, it is natural to consider M2 branes moving in less symmetric spaces, leading to versions of the duality with reduced SUSY. Inspired by ABJM [@Aharony:2008ug], the theories considered are $\prod_{a=1}^G U(N)_{k_a}$ quiver CS (QCS) theories with bifundamental matter. The rôle of the CS levels is far from trivial, and it has been argued in [@Gaiotto:2009mv; @Gaiotto:2009yz] that the sum $\sum_{a=1}^G k_a$ corresponds to the Type IIA SUGRA Romans mass parameter. In this paper we will focus entirely on the case in which the CS levels sum to zero; the Romans mass then vanishes and the system admits an M-theory lift.
Since the kinetic terms for the gauge fields are given by the CS action, the only classically dimensionful parameters are the superpotential couplings. Strong gauge dynamics is then conjectured to drive the theory to a superconformal IR fixed point. In [@Jafferis:2008qz; @Martelli:2008si; @Hanany:2008cd], a general analysis of the moduli spaces of such superconformal gauge theories was presented. In particular, it is crucial that the CS levels sum to zero if there is to be a so-called geometric branch of the moduli space which is a Calabi-Yau four-fold cone, where the branes are interpreted as moving. In parallel to the ABJM case, multiplying the vector of CS levels by an integer orbifolds this moduli space in a certain way. More precisely, the theory with $k=\mathrm{gcd}\{k_a\}$ has a (abelian) moduli space which is a ${\mathbb{Z}}_k$ quotient of the moduli space of the theory with CS levels $\{k_a/k\}$. Generically this group will not act freely away from the tip of the cone. In this case one might expect additional gauge symmetries at such fixed points. This has recently motivated [@Benini:2009qs; @Jafferis:2009th] the consideration of dual field theories which involve fundamental, as well as bifundamental, matter. It is however fair to say that, at present, there is no comprehensive understanding of these constructions.
On general grounds, the presence of global symmetries is of great help in classifying the spectrum of a gauge theory. One particularly important such global symmetry is the R-symmetry. In three dimensions a theory preserving $\mathcal{N}$ supersymmetries admits the action of an $SO\left(\mathcal{N}\right)$ R-symmetry. Thus the existence of a non-trivial R-symmetry, which can then provide important constraints on the dynamics, requires that we focus on $\mathcal{N}\ge2$, implying there is at least a $U(1)_R$. In particular it then follows that, assuming the theory flows to an IR superconformal fixed point, the scaling dimensions of chiral primary operators coincide with their R-charges. We note that, generically, the $\mathcal{N}=2$ theories considered have classically irrelevant superpotentials. Strong gauge dynamics is required to give large anomalous dimensions, thus making it possible to reach a non-trivial IR fixed point. However, in three dimensions there are few independent field theory checks on the existence of such a fixed point. For example, there is no useful analogous version of $a$-maximization [@Intriligator:2003jj], which for four-dimensional $\mathcal{N}=1$ theories allows one to determine the R-charge in the superconformal algebra at the IR fixed point. This places the conjectured dualities on a much weaker footing than their four-dimensional cousins in Type IIB string theory.
The $\mathcal{N}=2$ QCS theories that we consider are expected to be dual to M2 branes moving in a Calabi-Yau four-fold cone over a seven-dimensional Sasaki-Einstein base $Y$, thus giving rise to an $AdS_4\times Y$ near horizon dual geometry. Such Sasaki-Einstein manifolds $Y$ will typically have non-trivial topology, implying the existence of Kaluza-Klein (KK) modes obtained by reduction of SUGRA fields along the corresponding homology cycles. Of particular interest are five-cycles, on which one can reduce the M-theory six-form potential to obtain $b_2(Y)=\dim H_2(Y,{\mathbb{R}})$ vector fields in $AdS_4$. These vector fields are part of short multiplets of the KK reduction on $Y$, known as *Betti multiplets* [@D'Auria:1984vv; @D'Auria:1984vy] (for a discussion relevant to the cases we will consider, see also [@Fabbri:1999hw; @Merlatti:2000ed]). In analogy with the Type IIB case, where these symmetries are well-known to correspond to global baryonic symmetries [@Klebanov:1999tb], we will sometimes employ the same terminology here and refer to these as baryonic $U(1)$s.
In this paper we set out to study the above symmetries in the $AdS_4/CFT_3$ correspondence. In the rather better-understood $AdS_5/CFT_4$ correspondence in Type IIB string theory, from the field theory point of view these baryonic $U(1)$ symmetries appear as non-anomalous combinations of the diagonal $U(1)$ factors inside the $U(N)$ gauge groups.[^1] The key point is that, in four dimensions, abelian gauge fields are IR free and thus become global symmetries in the IR. However, this is no longer true in three dimensions, thus raising the question of the fate of these abelian symmetries. From the gravity perspective, in the dual $AdS_4$ the vector fields admit two admissible fall-offs at the boundary of $AdS_4$ [@Witten:2003ya; @Marolf:2006nd]. This is in contrast to the $AdS_5$ case where only one of them, that which leads to the interpretation as dual to a global current, is allowed. That the two behaviours are permitted implies that the corresponding boundary symmetries remain either gauged or ungauged, respectively, defining in each case a different boundary CFT. This issue is closely related to the gauge groups being either $U(N)$ or $SU(N)$ in the case at hand. From the point of view of the QCS theory with $U(N)$ gauge groups, at lowest CS level $k=1$ there is no real distinction between $U(N)$ and $U(1)\times SU(N)$ gauge groups [@Aharony:2008ug; @Imamura:2008nn; @Lambert:2010ji] . Therefore the discussion in [@Witten:2003ya] can be applied to the abelian part of the symmetry. In this way it is possible to connect the $SU(N)$ and the $U(N)$ theories in a rather precise manner, while keeping track of the corresponding action on the gravity side, which amounts to selecting one particular fall-off for the vector fields in $AdS_4$. This provides motivation to look at the $SU(N)$ version of the theory as dual to a particular choice of boundary conditions in the dual gravity picture.
In the first part of this paper we focus on the simplest class of examples, namely isolated toric Calabi-Yau four-fold singularities with no vanishing six-cycles (no exceptional divisors in a crepant resolution). These are discussed in more detail in [§\[sec:classification\]]{}. In particular we study in detail the example of $\mathcal{C}(Q^{111})$. A dual $U(N)^4$ QCS field theory was proposed for this singularity in [@Franco:2008um], and further studied in [@Franco:2009sp] where the non-abelian chiral ring of the theory (at large $k$) was shown to precisely match the coordinate ring of the variety. Motivated by the analysis of the behaviour of gauge fields in $AdS_4$, we will choose boundary conditions where the $b_2(Q^{111})=2$ Betti multiplets are dual to global symmetries. This amounts to focusing on a certain version of the theory with gauge group $U(1)^2\times SU(N)^4$. On the other hand, gauge fields in $AdS_4$ can have [*a priori*]{} both electric sources, corresponding to wrapped M5 branes, and magnetic sources, corresponding to wrapped M2 branes. It turns out that the boundary conditions necessary to define the $AdS/CFT$ correspondence allow for just one of the two types at a time [@Witten:2003ya]. In particular, the chosen $U(1)^2\times SU(N)^4$ quantization allows only for electric sources; that is, wrapped supersymmetric M5 branes. In turn, these correspond to baryonic operators [@Imamura:2008ji] in the field theory that are charged under the global symmetries. We will analyse this correspondence in detail, finding the expected agreement.
On the other hand, magnetic sources correspond to M2 branes [@Imamura:2008ji]. While in the $AdS$ geometry these wrap non-supersymmetric cycles, we can also consider resolutions of the corresponding cone where there are supersymmetric wrapped M2 branes. Along the lines of [@Klebanov:2007us; @Klebanov:2007cx], we will identify the relevant operator, responsible for the resolution, which is acquiring a VEV. Very much as in reference [@Klebanov:2007cx], it is possible to find an interpretation of these solutions as spontaneous symmetry breaking (SSB) through the explicit appearance of a Goldstone boson in the SUGRA dual.
A natural next step is to enlarge the class of singularities under consideration by allowing dual geometries with exceptional six-cycles. One such example is a $\mathbb{Z}_2$ orbifold of $\mathcal{C}(Q^{111})$ known as $\mathcal{C}(Q^{222})$. A dual field theory candidate has been proposed in [@Franco:2009sp; @Amariti:2009rb; @Davey:2009sr]. Further tests of this theory were performed in [@Franco:2009sp], where it was shown that its chiral ring matches the gravity computation at large $k$. The interpretation of such six-cycles is somewhat obscure holographically. Indeed, such six-cycles, when resolved, can support M5 brane instantons leading to non-perturbative corrections [@Witten:1996bn]. In the second part of this paper we set up the study of such corrections by finding a general expression for the Euclidean action of such branes in terms of normalizable harmonic two-forms, and compute this explicitly for $Q^{222}$. We leave a full understanding of such non-perturbative effects from the gauge theory point of view for future work.
The organization of this paper is as follows. In [§\[sec:2\]]{} we review the Freund-Rubin-type solutions which are eleven-dimensional $AdS_4\times Y$ backgrounds. We then turn to KK reduction of the SUGRA six-form potential on five-cycles in $Y$, leading to the Betti multiplets of interest. General analysis of gauge fields in $AdS_4$ shows that two possible fall-offs are admissible. We then review the construction in [@Witten:2003ya] relating these different boundary conditions for a single abelian gauge field in $AdS_4$ to the action of $SL(2,\mathbb{Z})$. In [§\[sec:3\]]{} we turn in more detail to the field theory description. We start by reviewing general aspects of $U(N)$ QCS theories that have appeared in the literature, before turning in [§\[sec:Q111\]]{} to the example of interest. We then propose a set of boundary conditions dual to the $U(1)^2\times SU(N)^4$ theory. We identify the ungauged $U(1)$s via the electric M5 branes wrapping holomorphic divisors in the geometry. In [§\[sec:4\]]{} we turn to the spontaneous breaking of these baryonic symmetries. We compute on the gravity side the baryonic condensate and identify the Goldstone boson of the SSB. In [§\[sec:5\]]{} we initiate the study of exceptional six-cycles. We compute the warped volume of a Euclidean brane in the resolved $\mathcal{C}(Q^{222})$ geometry. By extending our results on warped volumes to arbitrary geometries, both for the baryonic condensate and the Euclidean brane, we find general formulae for such warped volumes. We end with some concluding comments in [§\[sec:6\]]{}. Finally, a number of relevant calculations and formulae are collected in the appendices.
**Note added:** as this paper was being finalized the preprint [@Klebanov:2010tj] appeared, which has partial overlap with our results.
$AdS_4$ backgrounds and abelian symmetries {#sec:2}
==========================================
We begin by reviewing general properties of Freund-Rubin $AdS_4$ backgrounds, and also introduce the $Q^{111}$ and $Q^{222}=Q^{111}/{\mathbb{Z}}_2$ examples of main interest. KK reduction of the M-theory potentials on topologically non-trivial cycles leads to gauge symmetries in $AdS_4$. We review their dynamics in the $AdS/CFT$ context and the sources allowed, depending on the chosen quantization. Of central relevance for our purposes will be wrapped supersymmetric M5 branes.
Freund-Rubin solutions {#sec:FR}
----------------------
The $AdS_4$ backgrounds of interest are of Freund-Rubin type, with eleven-dimensional metric and four-form given by \[AdSbackground\] s\^2\_[11]{} &= &R\^2(s\^2(AdS\_4) + s\^2(Y)) ,\
G &=& R\^3 (AdS\_4) . Here the $AdS_4$ metric is normalized so that $R_{\mu\nu} = -3g_{\mu\nu}$. The Einstein equations imply that $Y$ is an Einstein manifold of positive Ricci curvature, with metric normalized so that $R_{ij} = 6g_{ij}$. The flux quantization condition \_Y \_[11]{} G = N , then leads to the relation R = 2\_p()\^[1/6]{} , where $\ell_p$ denotes the eleven-dimensional Planck length.
As is well-known, such solutions arise as the near-horizon limit of $N$ M2 branes placed at the tip $r=0$ of the Ricci-flat cone \[cone\] s\^2([C]{}(Y)) = r\^2 + r\^2 s\^2(Y) . More precisely, the eleven-dimensional solution is \[background\] s\^2\_[11]{} &=& h\^[-2/3]{} s\^2(\^[1,2]{}) + h\^[1/3]{} s\^2(X) ,\
G &=& \^3 x h\^[-1]{} , where in the case at hand we take the eight-manifold $X={\cal C}(Y)$ with conical metric (\[cone\]). Placing $N$ Minkowski space-filling M2 branes at $r=0$ leads, after including their gravitational back-reaction, to the warp factor \[warping\] h = 1 + . In the near-horizon limit, near to $r=0$, the background (\[background\]) approaches the $AdS_4$ background (\[AdSbackground\]). In fact the warp factor $h=R^6/r^6$ is precisely the $AdS_4$ background in a Poincaré slicing. More precisely, writing \[z\_coordinate\] z= , s\^2(AdS\_4) = z\^[-2]{}(z\^2 + s\^2(\^[1,2]{})) , leads to the metric (\[AdSbackground\]).
Supersymmetries $\mathcal{N}$ $Y$ ${\cal C}(Y)$
------------------------------- --------------------- --------------------------------------
1 weak $G_2$ holonomy $Spin(7)$ holonomy
2 Sasaki-Einstein $SU(4)$ holonomy (Ricci-flat Kähler)
3 3-Sasakian $Sp(2)$ holonomy (hyperKähler)
: Relation between the number of supersymmetries $\mathcal{N}$ in $AdS_4$ and the special Einstein geometry of $Y$ and its cone ${\cal C}(Y)$.[]{data-label="table"}
We shall be interested in solutions of this form preserving supersymmetry in $AdS_4$. The well-known result [@Acharya:1998db] is summarized in Table \[table\]. As mentioned in the introduction, in general $\mathcal{N}$ supersymmetries leads to the R-symmetry group $SO(\mathcal{N})$, and thus supersymmetry provides a strong constraint on the spectrum only for $\mathcal{N}\geq 2$. We hence restrict attention to the $\mathcal{N}=2$ Sasaki-Einstein case, which includes the $\mathcal{N}=3$ geometry as a special case. It is then equivalent to say that the cone metric on ${\cal C}(Y)$ is Kähler as well as as Ricci-flat, [*i.e.*]{} Calabi-Yau. Geometries with $\mathcal{N}\geq 4$ supersymmetries are necessarily quotients of $S^7$.
Only a decade ago the only known examples of such Sasaki-Einstein seven-manifolds were homogeneous spaces. Since then there has been dramatic progress. 3-Sasakian manifolds, with $\mathcal{N}=3$, may be constructed via an analogue of the hyperKähler quotient, leading to rich infinite classes of examples [@BG]. For $\mathcal{N}=2$ supersymmetry one could take $Y$ to be one of the explicit $Y^{p,k}$ manifolds constructed in [@Gauntlett:2004hh], and further studied in [@Res; @Martelli:2008rt], or any of their subsequent generalizations. These $\mathcal{N}=2$ examples are all toric, meaning that the isometry group contains $U(1)^4$ as a subgroup. In fact, toric Sasaki-Einstein manifolds are now completely classified thanks to the general existence and uniqueness result in [@FOW]. At the other extreme, there are also non-explicit metrics in which $U(1)_R$ is the only isometry [@BG].
However, for our purposes it will be sufficient to focus on two specific homogeneous examples, namely $Q^{111}$ and $Q^{222}=Q^{111}/{\mathbb{Z}}_2$, with ${\mathbb{Z}}_2\subset U(1)_R$ being along the R-symmetry of $Q^{111}$. These will turn out to be simple enough so that everything can be computed explicitly, and yet at the same time we shall argue that many of the features seen in these cases hold also for the more general geometries mentioned above. In both cases the isometry group is $SU(2)^3\times U(1)_R$, and in local coordinates the explicit metrics are \[Qiiimetric\] s\^2 = (+\_[i=1]{}\^3 \_i\_i)\^2 + \_[i=1]{}\^3 (\_i\^2 + \^2\_i\_i\^2) . Here $(\theta_i,\phi_i)$ are standard coordinates on three copies of $S^2=\mathbb{CP}^1$, $i=1,2,3$, and $\psi$ has period $4\pi$ for $Q^{111}$ and period $2\pi$ for $Q^{222}$. The two Killing spinors are charged under $\partial_\psi$, which is dual to the $U(1)_R$ symmetry. The metric (\[Qiiimetric\]) shows very explicitly the regular structure of a $U(1)$ bundle over the standard Kähler-Einstein metric on $\mathbb{CP}^1\times\mathbb{CP}^1\times\mathbb{CP}^1$, where $\psi$ is the fibre coordinate and the Chern numbers are $(1,1,1)$ and $(2,2,2)$ respectively. These are hence natural generalizations[^2] to seven dimensions of the $T^{11}$ and $T^{22}$ manifolds.
$C$-field modes {#sec:Cfield}
---------------
One might wonder whether it is possible to turn on an internal $G$-flux $G_Y$ on $Y$, in addition to the $G$-field in (\[AdSbackground\]), and still preserve supersymmetry, [*i.e.*]{} G = R\^3 (AdS\_4) + G\_Y . In fact necessarily $G_Y=0$. This follows from the results of [@Becker:1996gj]: for any warped Calabi-Yau four-fold background with metric of the form (\[background\]), one can turn on a $G$-field $G_X$ on $X$ without changing the Calabi-Yau metric on $X$ only if $G_X$ is self-dual. But for a cone, with $G_X=G_Y$ a pull-back from the base $Y$, this obviously implies that $G_X=0$.
However, more precisely the $G$-field in M-theory determines a class[^3] in $H^4(Y,{\mathbb{Z}})$. The differential form part of $G$ captures only the image of this in $H^4(Y,{\mathbb{R}})$, and so $G_Y=0$ still allows for a topologically non-trivial $G$-field classified by the torsion part $H^4_{\mathrm{tor}}(Y,{\mathbb{Z}})$. This is also captured, up to gauge equivalence, by the holonomy of the corresponding flat $C$-field through dual torsion three-cycles in $Y$. There are hence $|H^4_{\mathrm{tor}}(Y,{\mathbb{Z}})|$ physically distinct $AdS_4$ Freund-Rubin backgrounds associated to the same geometry, which should thus correspond to physically inequivalent dual SCFTs. In a small number of examples with proposed Chern-Simons quiver duals, including the original ABJ(M) theory, different choices of this torsion $G$-flux have been argued to be dual to changing the *ranks* in the quiver [@Martelli:2009ga; @Aharony:2008gk]. However, the related Seiberg-like dualities are currently very poorly understood in examples without Hanany-Witten-type brane duals. In particular, for example, one can compute $H^4(Q^{111},{\mathbb{Z}})\cong{\mathbb{Z}}_2$, implying there are two distinct M-theory backgrounds with the same $Q^{111}$ geometry but different $C$-fields. This is an important aspect of the $AdS_4/CFT_3$ duality that we shall not discuss any further in this paper.
More straightforwardly, if one has $b_3(Y)=\dim H_3(Y,{\mathbb{R}})$ three-cycles in $Y$ then one can also turn on a closed three-form $C$ with non-zero periods through these cycles. Including large gauge transformations, this gives a space $U(1)^{b_3(Y)}$ of such flat $C$-fields. Since these are continuously connected to each other they would be dual to marginal deformations in the dual field theory. Indeed, the *harmonic* three-forms on a Sasaki-Einstein seven-manifold are in fact paired by an almost complex structure [@Boyer:1998sf] and thus $b_3(Y)$ is always even, allowing these to pair naturally into complex parameters as required by $\mathcal{N}=2$ supersymmetry. However, for the class of toric singularities studied in this paper, including $Q^{111}$ and $Q^{222}$, it is straightforward[^4] to show that $b_3(Y)=0$ and there are hence no such marginal deformations associated to the $C$-field.
Finally, since $H_6(Y,{\mathbb{R}})=0$ for any positively curved Einstein seven-manifold, there are never periods of the dual potential $C_6$ through six-cycles in $Y$.
Baryonic symmetries and wrapped branes {#sec:baryons}
--------------------------------------
Of central interest in this paper will be symmetries associated to the topology of $Y$, and the corresponding charged BPS states associated to wrapped M branes. By analogy with the corresponding situation in $AdS_5\times Y_5$ in Type IIB string theory, we shall refer to these symmetries as baryonic symmetries; the name will turn out to be justified.
Denote by $b_2(Y)=\dim H_2(Y,{\mathbb{R}})$ the second Betti number of $Y$. By Poincare duality we have $\dim H_5(Y,{\mathbb{R}})=\dim H_2(Y,{\mathbb{R}})=b_2(Y)$. Let $\alpha_1,\ldots,\alpha_{b_2(Y)}$ be a set of dual harmonic five-forms with integer periods. Then for the $AdS_4 \times Y$ Freund-Rubin background we may write the KK ansatz C\_6 = \_[I=1]{}\^[b\_2(Y)]{} \_I\_I , \[3-form-to-global\] where $T_5={2\pi}/{(2\pi \ell_p)^6}$ is the M5 brane tension. This gives rise to $b_2(Y)$ massless $U(1)$ gauge fields $\mathcal{A}_I$ in $AdS_4$. For a supersymmetric theory these gauge fields of course sit in certain multiplets, known as *Betti multiplets*. See, for example, [@D'Auria:1984vv; @D'Auria:1984vy; @Fabbri:1999hw; @Merlatti:2000ed].
### Vector fields in $AdS_4$, boundary conditions and dual CFTs
The $AdS/CFT$ duality requires specifying the boundary conditions for the fluctuating fields in $AdS$. In particular, vector fields in $AdS_4$ admit different sets of boundary conditions [@Witten:2003ya; @Marolf:2006nd] leading to different boundary CFT´s. In order to see this, let us consider a vector field in $AdS_{d+1}$. Using the straightforward generalization to $AdS_{d+1}$ of the coordinates in (\[z\_coordinate\]), in the gauge $A_z=0$ the bulk equations of motion set $$\label{gauge_field_In_AdS4}
A_{\mu}=a_{\mu}+j_{\mu}\, z^{d-2} \ ,$$ where $a_{\mu},\, j_{\mu}$ satisfy the free Maxwell equation in Lorentz gauge in the Minkowski space. It is not hard to see that in $d<4$ both behaviours have finite action, and thus can be used to define a consistent $AdS/CFT$ duality.
Let us now concentrate on the case of interest $d=3$, where both quantizations are allowed. In order to have a well-defined variational problem for the gauge field in $AdS_4$ we should be careful with the boundary terms when varying the action. In general, we have $$\delta S = \int \Big\{ \frac{\partial \sqrt{\det g}\,\mathcal{L}}{\partial A_M}-\partial_N\frac{\partial \sqrt{\det g}\,\mathcal{L}}{\partial\partial_NA_M}\Big\}\, \delta A_M +\partial_N\Big\{ \frac{\partial\sqrt{\det g}\, \mathcal{L}}{\partial\partial_NA_M}\, \delta A_M\Big\} \ .$$ The bulk term gives the equations of motion whose solution behaves as (\[gauge\_field\_In\_AdS4\]). In turn, the boundary term can be seen to reduce to $$\delta S_B=- \frac{1}{2}\,\int_{\mathrm{Boundary}} \, j_{\mu}\delta a^{\mu} \, {\mathrm{d}}^3 x~.$$ Therefore, in order to have a well-posed variational problem, we need to demand $\delta a_{\mu}=0$; that is, we need to impose boundary conditions where $a_{\mu}$ is fixed in the boundary.
On the other hand, since in $d=3$ both behaviours for the gauge field have finite action, we can consider adding suitable boundary terms such that the action becomes [@Marolf:2006nd] $$\label{dynamical_a}
S=\frac{1}{4}\int \sqrt{\det g}\, F_{AB}\, F^{AB} +\frac{1}{2}\,\int_{\mathrm{Boundary}}\, \sqrt{\det g}\, A^{\mu} \, F_{r\mu}|_{\mathrm{Boundary}} \, {\mathrm{d}}^3x .$$ The boundary term is now $$\delta S_B=\frac{1}{2}\,\int_{\mathrm{Boundary}} \, a_{\mu}\delta j^{\mu} \, {\mathrm{d}}^3 x~,$$ so that we need to impose the boundary condition $\delta j_{\mu}=0$; that is, fix the boundary value of $j_{\mu}$.
Defining $\vec{B}=\frac{1}{2}\epsilon^{\mu\nu\rho}\, F_{\nu\rho}$ and $\vec{E}=F_{\mu r}$, we have $$B^{\mu}=\epsilon^{\mu\nu\rho}\partial_{\nu}a_{\rho}+\epsilon^{\mu\nu\rho}\partial_{\nu}j_{\rho}\, z\,, \qquad E^{\mu}=j^{\mu}\, z^2 \ .$$ The two sets of boundary conditions then correspond to either setting $E_{\mu}=0$ while leaving $a_{\mu}$ unrestricted, or setting $B_{\mu}=0$ while leaving $j_{\mu}$ unrestricted.
At this point we note that $a_{\mu},\, j_{\mu}$ are naturally identified, respectively, with a dynamical gauge field and a global current in the boundary. In accordance with this identification, eq. (\[gauge\_field\_In\_AdS4\]) and the usual $AdS/CFT$ prescription shows each field to have the correct scaling dimension for this interpretation: for a gauge field $\Delta(a_{\mu})=1$, while for a global current $\Delta(j_{\mu})=2$. Therefore, the quantization $E_{\mu}=0$ is dual to a boundary CFT where the $U(1)$ gauge field is dynamical; while the quantization $B_{\mu}=0$ is dual to a boundary CFT where the $U(1)$ is ungauged and is instead a global symmetry. Furthermore, as discussed in [@Klebanov:1999tb] for the scalar counterpart, once the improved action is taken into account the two quantizations are Legendre transformations of one another [@Klebanov:2010tj], as can be seen by *e.g.* computing the free energy in each case.
One can consider electric-magnetic duality in the bulk theory, which exchanges $E_{\mu}\leftrightarrow B_{\mu}$ thus exchanging the two boundary conditions for the $AdS_4$ gauge field quantization. This action translates in the boundary theory into the so-called $\mathcal{S}$ *operation* [@Witten:2003ya]. This is an operation on three-dimensional CFTs with a global $U(1)$ symmetry, taking one such CFT to another. In addition, it is possible to construct a $\mathcal{T}$ *operation*, which amounts, from the bulk perspective, to a shift of the bulk $\theta$-angle by $2\pi$. Following [@Witten:2003ya], we can be more precise in defining these actions in the boundary CFT. Starting with a three-dimensional CFT with a global $U(1)$ current $J^{\mu}$, one can couple this global current to a background gauge field $A$ resulting in the action $S[A]$. The $\mathcal{S}$ operation then promotes $A$ to a dynamical gauge field and adds a BF coupling of $A$ to a new background field $B$, while the $\mathcal{T}$ operation instead adds a CS term for the background gauge field $A$: $$\mathcal{S}:\, S[A]\,\rightarrow\, S[A]+\frac{1}{2\pi} \int B\wedge {\mathrm{d}}A~,\qquad \mathcal{T}:\,S[A]\,\rightarrow\, S[A]+\frac{1}{4\pi}\int A\wedge {\mathrm{d}}A \ .$$ As shown in [@Witten:2003ya], these two operations generate the group $SL(2,\mathbb{Z})$.[^5] In turn, as discussed above, the $\mathcal{S}$ and $\mathcal{T}$ operations have the bulk interpretation of exchanging $E_{\mu}\leftrightarrow B_{\mu}$ and shifting the bulk $\theta$-angle by $2\pi$, respectively. It is important to stress that these actions on the bulk theory change the boundary conditions. Because of this, the dual CFTs living on the boundary are different.
### Boundary conditions and sources for gauge fields: M5 branes in toric manifolds {#s:bc}
We are interested in gauge symmetries in $AdS_4$ associated to the topology of $Y$; that is, arising from KK reductions as in (\[3-form-to-global\]). All Kaluza-Klein modes, and hence their dual operators, carry zero charge under these $b_2(Y)$ $U(1)$ symmetries. However, there are operators associated to wrapped M branes that do carry charge under this group. In particular, an M5 brane wrapped on a five-manifold $\Sigma_5\subset Y$, such that the cone ${\cal C}(\Sigma_5)$ is a complex divisor in the Kähler cone ${\cal C}(Y)$, is supersymmetric and leads to a BPS particle propagating in $AdS_4$. Since the M5 brane is a source for $G$, this particle is electrically charged under the $b_2(Y)$ massless $U(1)$ gauge fields $\mathcal{A}_I$. One might also consider M2 branes wrapped on two-cycles in $Y$. However, such wrapped M2 branes are supersymmetric only if the cone ${\cal C}(\Sigma_2)$ over the two-submanifold $\Sigma_2\subset Y$ is calibrated in the Calabi-Yau cone, and there are no such calibrating three-forms. Thus these particles, although topologically stable, are not BPS. They are magnetically charged under the $U(1)^{b_2(Y)}$ gauge fields in $AdS_4$ [@Imamura:2008ji].
As discussed above, the $AdS/CFT$ duality instructs us to choose, for each $U(1)$ gauge field, a set of boundary conditions where either $E_{\mu}$ or $B_{\mu}$ vanishes. Clearly, only the latter possibility allows for the existence of the SUSY electric M5 branes, otherwise forbidden by the boundary conditions. In turn, this quantization leaves, in the boundary theory, the $U(1)$ symmetry as a global symmetry. Therefore, in this case we should expect to find operators in the field theory that are charged under the global baryonic symmetries and dual to the M5 brane states. We turn to this point in the next section. We note that, with this choice of boundary condition, the rôle of the Betti multiplets is very similar to their $AdS_5$ counterparts, giving rise to global baryonic symmetries in the boundary theory, and hence motivating the use of the same name in the case at hand.
For toric manifolds there is a canonical set of such wrapped M5 brane states, where ${\cal C}(\Sigma_5)$ are taken to be the toric divisors. Each such state leads to a corresponding dual chiral primary operator that is charged under the $U(1)^{b_2(Y)}$ global symmetries and will also have definite charge under the $U(1)^4$ flavour group dual to the isometries of $Y$. We refer the reader to the standard literature for a thorough introduction to toric geometry. However, the basic idea is simple to state. The cone ${\cal C}(Y)$ fibres over a polyhedral cone in ${\mathbb{R}}^4$ with generic fibre $U(1)^4$. This polyhedral cone is by definition a convex set of the form $\bigcap\{\mathbf{x}\cdot \mathbf{v}_\alpha\geq 0\}\subset{\mathbb{R}}^4$, where $\mathbf{v}_\alpha\in{\mathbb{Z}}^4$ are integer vectors. This set of vectors is precisely the set of charge vectors specifying the $U(1)$ subgroups of $U(1)^4$ that have complex codimension one fixed point sets. These fixed point sets are, by definition, the toric divisors referred to above. The Calabi-Yau condition implies that, with a suitable choice of basis, we can write $\mathbf{v}_\alpha=(1,\mathbf{w}_\alpha)$, with $\mathbf{w}_\alpha\in{\mathbb{Z}}^3$. If we plot these latter points in ${\mathbb{R}}^3$ and take their convex hull, we obtain the *toric diagram*.
For the $Q^{111}$ example the toric divisors are given by taking $\Sigma_5=\{\theta_i=0\}$ or $\Sigma_5=\{\theta_i=\pi\}$, for any $i=1,2,3$, which are 6 five-manifolds in $Y$. The toric diagram for $Q^{111}$ is shown in Figure \[fig:toricdiagramQ111\], where one sees clearly these 6 toric divisors as the 6 external vertices. Notice that for $Q^{111}$ the full isometry group may be used to rotate $\{\theta_i=0\}$ into $\{\theta_i=\pi\}$, specifically using the $i$th copy of $SU(2)$ in the $SU(2)^3\times U(1)_R$ isometry group. In fact these two five-manifolds are two points in an $S^2$ family of such five-manifolds related via the isometry group. Similar comments apply also to $Q^{222}$.
Baryonic symmetries in QCS theories {#sec:3}
===================================
In the previous section we discussed the rôle of vector fields in $AdS_4$. In particular, we have shown that there is a choice of boundary conditions where the Betti multiplets corresponding to (\[3-form-to-global\]) are dual to global currents in the boundary theory. From the bulk perspective, this translates into the possibility of having electric M5 brane states in the theory, in a consistent manner. On general grounds, we expect these states to be dual to certain operators in the boundary theory charged under the global $U(1)^{b_2(Y)}$. In this section we turn to a more precise field theoretic description of this. We begin with a brief review of the $U(N)$ theories considered in the literature, before turning to our $\mathcal{C}(Q^{111})$ example and considering the rôle of the abelian symmetries in this case.
$U(N)$ QCS theories
-------------------
Let us start by considering the $\prod_{a=1}^G U(N_a)$ theories. The Lagrangian, in $\mathcal{N}=2$ superspace notation, for a theory containing an arbitrary number of bifundamentals $X_{ab}$ in the representation $(\square_a,\, \bar{\square}_b)$ under the $(a,\, b)$-th gauge groups and a choice of superpotential $W$, reads \[action\] = &&\^4\
&& + \^2 W(X\_[ab]{}) + . Here $k_a\in{\mathbb{Z}}$ are the CS levels for the vector multiplet $V_a$. For future convenience we define $k={\rm gcd}\{k_a\}$.
The classical vacuum moduli space (VMS) is determined in general by the following equations [@Martelli:2008si; @Hanany:2008cd] $$\begin{aligned}
\label{VMSeqns}
{\nonumber}\partial_{X_{ab}} W &=& 0~,\\
{\nonumber}\mu_a := -\sum\limits_{b=1}^G {X_{ba}}^{\dagger} {X_{ba}} +
\sum\limits_{c=1}^G {X_{ac}} {X_{ac}}^{\dagger} &=&
\frac{k_a\sigma_a}{2\pi}~, \\
\label{DF} \sigma_a X_{ab} - X_{ab} \sigma_b &=& 0~,\end{aligned}$$ where $\sigma_a$ is the scalar component of $V_a$. Following [@Martelli:2008si], upon diagonalization of the fields using $SU(N)$ rotations, one can focus on the branch where $\sigma_a=\sigma$, $\forall a$, so that the last equation is immediately satisfied.[^6] Under the assumption that $\sum_{a=1}^G k_a=0$, the equations for the moment maps $\mu_a$ boil down to a system of $G-2$ independent equations for the bifundamental fields, analogous to D-term equations. Since for toric superpotentials the set of F-flat configurations, determining the so-called master space, is of dimension $G+2$, upon imposing the $G-2$ D-terms and dividing by the associated gauge symmetries we have a ${\rm dim}_\mathbb{C}\mathcal{M}=4$ moduli space $\mathcal{M}$ where the M2 branes move.
However, due to the peculiarities of the CS kinetic terms, extra care has to be taken with the diagonal part of the gauge symmetry. At a generic point of the moduli space the gauge group is broken to $N$ copies of $U(1)^{G}$. The diagonal gauge field $\mathcal{B}_G=\sum_{a=1}^G\mathcal{A}_a$ is completely decoupled from the matter fields, and only appears coupled to $\mathcal{B}_{G-1}=k^{-1}\,\sum_{a=1}^Gk_a\,\mathcal{A}_a$ through $$\label{SBG}
S(\mathcal{B}_G)=\frac{k}{4\pi\, G}\int (\mathcal{B}_{G-1})_{\mu}\,\epsilon^{\mu\nu\rho}\, (\mathcal{G}_G)_{\nu\rho}~.$$ Since $\mathcal{B}_G$ appears only through its field strength, it can be dualized into a scalar $\tau$. Following the standard procedure, it is easy to see that integrating out $\mathcal{G}_G={\mathrm{d}}\mathcal{B}_G$ sets $$\label{identification}
\mathcal{B}_{G-1}=\frac{G}{k}\, {\mathrm{d}}\tau~,$$ such that the relevant part of the action becomes a total derivative $$S(\mathcal{B}_G)=\int {\mathrm{d}}\Big(\frac{\tau}{2\pi}\,\mathcal{G}_G\Big)~.$$ Around a charge $n\in{\mathbb{Z}}$ monopole in the diagonal $U(1)$ gauge field $\mathcal{B}_G$ we then have $\int \mathcal{G}_G=2\pi\,G\, n$, so that $\tau$ must have period $2\pi/G$ in order for the above phase to be unobservable [@Martelli:2008si]. Gauge transformations of $\mathcal{B}_{G-1}$ then allow one to gauge-fix $\tau$ to a particular value via (\[identification\]), but this still leaves a residual discrete set of ${\mathbb{Z}}_k$ gauge symmetries that leave this gauge choice invariant. The space of solutions to (\[DF\]) is then quotiented by gauge transformations where the parameters $\theta_a$ satisfy $\sum_{a=1}^G k_a\, \theta_a =0$, together with the residual discrete ${\mathbb{Z}}_k$ gauge transformations generated by $\theta_a = 2\pi/k$ for all $a$. Altogether this leads to a $U(1)^{G-2}\times {\mathbb{Z}}_k$ quotient. We refer to [@Martelli:2008si] for further discussion, and to [@Franco:2009sp] for a discussion in the context of the $Q^{111}$ theory in particular.
An alternative point of view has recently appeared in the literature [@Benini:2009qs; @Jafferis:2009th], in which the existence of two special monopole operators $T, \tilde{T}$ is noted. These monopole operators, which have charges $\pm (k_1,\cdots, k_G)$ respectively under each gauge group, are conjectured to satisfy a relation in the chiral ring of the form $T\, \tilde{T}=1~$. In this approach the moduli space is defined as the chiral ring of the abelian theory enhanced by the operators $T, \tilde{T}$, together with the constraint.
The $\mathcal{C}(Q^{111})$ theory {#sec:Q111}
---------------------------------
### The theory and its moduli space
A field theory candidate dual to M2 branes probing $\mathcal{C}(Q^{111})/{\mathbb{Z}}_k$ was proposed in [@Franco:2008um] and further studied in [@Franco:2009sp]. The proposal in those references is a $U(N)^4$ Chern-Simons gauge theory with CS levels $(k,\, k,\, -k,\, -k)$, with matter content summarized by the quiver in Figure \[fig:quiverdiagramQ111\].
![The quiver diagram for a conjectured dual of $\mathcal{C}(Q^{111})$.[]{data-label="fig:quiverdiagramQ111"}](./quiver_Q111)
In addition, there is a superpotential given by $$\label{WQ111}
W={\rm Tr}\, \Big(\, C_2\, B_1\, A_1\, B_2\, C_1\, A_2\,-\,C_2\,B_1\, A_2\, B_2\, C_1\, A_1\, \Big)\ .$$ As expected for a field theory dual to $N$ point-like branes moving in $\mathcal{C}(Q^{111})/{\mathbb{Z}}_k$, the moduli space contains a branch which is the symmetric product of $N$ copies of this conical singularity. To see this, let us begin with the abelian theory in which all the gauge groups are $U(1)$. As shown in [@Franco:2009sp], after integrating out the auxiliary $\sigma$ scalar the geometric branch of the moduli space with $k=1$ is described by $G-2=2$ D-term equations. Recalling the special rôle played by $\mathcal{B}_{G-1}=\mathcal{B}_3,\, \mathcal{B}_G=\mathcal{B}_4$, it is useful to introduce the following basis for the $U(1)$ gauge fields: \[redefinedGF\] &&\_[I]{}=(\_[1]{}-\_[2]{}+\_[3]{}-\_[4]{}) , \_[II]{}=(\_[1]{}-\_[2]{}-\_[3]{}+\_[4]{}) ,\
&&\_3=\_[1]{}+\_[2]{}-\_[3]{}-\_[4]{} ,\_4=\_[1]{}+\_[2]{}+\_[3]{}+\_[4]{} . Then the two D-terms to impose are those for $\mathcal{A}_I,\,\mathcal{A}_{II}$. In turn, the charge matrix is \[gaugedU(1)charges\]
[l | c c c c c c]{} & A\_1 & A\_2 & B\_1 & B\_2 & C\_1 & C\_2\
U(1)\_[I]{} & 1 & 1 & 0 & 0 & -1 & -1\
U(1)\_[II]{} & -1 & -1 & 1 & 1 & 0 & 0\
U(1)\_[\_3]{} & 0 & 0 & -2 & 2 & -2 & 2\
U(1)\_[\_4]{} & 0 & 0 & 0 & 0 & 0 & 0
. Notice the appeareance of the $SU(2)^3$ global symmetry, under which the pairs $A_i$, $B_i$, $C_i$ transform as doublets under each of the respective factors.
Since for the abelian theory the superpotential is identically zero, one can determine the abelian moduli space by constructing the gauge-invariants with respect to the gauge transformations for $\mathcal{A}_I,\, \mathcal{A}_{II}$. Borrowing the results from [@Franco:2009sp], for CS level $k=1$ these are $$\begin{array}{lclclcl}
w_1 = A_1\,B_2\,C_1~, & \ & w_2 = A_2\,B_1\,C_2~, & \ & w_3 = A_1\,B_1\,C_2~, & \ & w_4 = A_2\,B_2\,C_1~, \\
w_5 = A_1\,B_1\,C_1~, & \ & w_6 = A_2\,B_1\,C_1~, & \ & w_7 = A_1\,B_2\,C_2~, & \ & w_8 = A_2\,B_2\,C_2 \, .
\end{array}
\label{ws}$$ One can then check explicitly that these satisfy the 9 relations defining $\mathcal{C}(Q^{111})$ as an affine variety: $$\begin{array}{cccccccc}
w_1\,w_2 - w_3\,w_4 & = & w_1\,w_2 - w_5\,w_8 & = & w_1\,w_2 - w_6\,w_7 & = & 0~, & \\
w_1\,w_3 - w_5\,w_7 & = & w_1\,w_6 - w_4\,w_5 & = & w_1\,w_8 - w_4\,w_7 & = & 0~, & \\
w_2\,w_4 - w_6\,w_8 & = & w_2\,w_5 - w_3\,w_6 & = & w_2\,w_7 - w_3\,w_8 & = & 0~.
\label{eq_Q11}
\end{array}$$ This is an affine toric variety, with toric diagram given by Figure \[fig:toricdiagramQ111\]. Indeed, we also notice that for the abelian theory the description of the moduli space as a $U(1)^2$ Kähler quotient of ${\mathbb{C}}^6$ with coordinates $\{A_i,B_i,C_i\}$ is precisely the minimal gauged linear sigma model (GLSM) description. Thus the 6 toric divisors in Figure \[fig:toricdiagramQ111\], discussed in [§\[sec:baryons\]]{}, are defined by $\{A_i=0\}$, $\{B_i=0\}$, $\{C_i=0\}$, $i=1,2$.
For CS level $k>1$ one obtains an $\mathcal{N}=2$ supersymmetric ${\mathbb{Z}}_k\subset U(1)_{\mathcal{B}_{G-1}}$ orbifold of $\mathcal{C}(Q^{111})$. Notice that $\{w_i\mid i=1,\ldots,4\}$ are invariant under this action, while $\{w_5,w_6\}$ and $\{w_7,w_8\}$ are rotated with equal and opposite phase. On the other hand, for the non-abelian theory with $N>1$ it was shown in [@Franco:2009sp] that for large $k$, where the use of still poorly-understood monopole operators is evaded, upon using the F-terms of the full non-abelian superpotential (\[WQ111\]) the chiral ring matches that expected for the corresponding orbifold. In this case the chiral primaries at the non-abelian level are just the usual gauge-invariants given by $$\label{nonabelianchirals}
{\rm Tr}\, \Big( \prod_{a=1}^r\, X^{\pm}_{i_a}\Big)\ ,\quad \mbox{where}\quad X^+_i=A_i\, C_2\, B_1\ , \quad X^-_i=A_i\, B_2\, C_1~.$$ An important subtlety in this theory is that $U(1)_{\mathcal{B}_{G-1}}$ does not act freely on $Q^{111}$: it fixes two disjoint copies of $S^3$ inside $Q^{111}$, as explained in [@Benini:2009qs]. Indeed, using (\[gaugedU(1)charges\]) one sees that the corresponding two cones ${\mathbb{C}}^2=\mathcal{C}(S^3)$ are parametrized respectively by $\{w_1,w_4\}$ and $\{w_2,w_3\}$, with in each case all other $w_i=0$. Thus for $k>1$ the horizon $Y=Q^{111}/{\mathbb{Z}}_k$ has orbifold singularities in codimension four. This means that the SUGRA approximation cannot be trusted for $k>1$. In fact these are $A_{k-1}$ singularities which can support “fractional” M2 branes wrapping the collapsed cycles, and one expects an $SU(k)$ gauge theory to be supported on these $S^3$s. A different perspective can be obtained by interpreting $U(1)_{\mathcal{B}_{G-1}}$ as the M-theory circle and reducing to Type IIA. This results in $k$ D6 branes wrapping these two $S^3$ submanifolds. From now on we will therefore assume that $k=1$.
### Gauged versus global abelian subgroups and $SL(2,\mathbb{Z})$
At $k=1$ the orbifold identification due to the CS terms is trivial. Indeed, in this case there is no real distinction between $U(N)$ and $SU(N)\times U(1)$ gauge groups, as discussed in [@Aharony:2008ug; @Imamura:2008nn; @Lambert:2010ji] for the ABJM theory and orbifolds of it. We shall argue that ungauging some of the $U(1)$s is dual to a particular choice of boundary conditions on the gravity side. That is, we apply the general discussion in [§\[sec:baryons\]]{} to the $b_2(Q^{111})=2$ $U(1)$ gauge fields, and argue that the associated $U(1)$ symmetries are those in $SU(N)\times U(1)$, for appropriate gauge group factors. This raises the important problem of how to identify the relevant two $U(1)$ symmetries dual to the Betti multiplets in the QCS theory proposed above. The key is to recall that the boundary conditions which amount to ungauging these $U(1)$s in turn allow for the existence of supersymmetric M5 branes on the gravity side. As discussed in [§\[sec:baryons\]]{}, from an algebro-geometric point of view the corresponding divisors are easy to identify. In turn we notice that, for the abelian theory, the fields $\{A_i, B_i, C_i\}$ are also the minimal GLSM coordinates. Setting each to zero therefore gives one of the 6 toric divisors that may be wrapped by an M5 brane. The charges of the resulting M5 brane states under $U(1)^{b_2(Y)}$ are then the *same* as the charges of these fields under the $U(1)_I\times U(1)_{II}$ we quotient by in forming the abelian moduli space – this was shown for the D3 brane case in [@Franco:2005sm], and the same argument applies here also. This strongly suggests that the gauge symmetries $U(1)_I$, $U(1)_{II}$ should in fact be dual to the Betti multiplets discussed in [§\[sec:baryons\]]{}.
Once we have identified the relevant abelian symmetries, we can consider acting with the $\mathcal{S}$ and $\mathcal{T}$ operations. We schematically write the action of the $U(N)^4$ $Q^{111}$ theory (which we will denote as $S_U$), separating the $U(1)$ sector from the rest, as $$\label{S_gauged}
S_U\sim \int \mathcal{B}_3\wedge {\mathrm{d}}\mathcal{B}_4+ \mathcal{A}_I\wedge {\mathrm{d}}\mathcal{A}_{II}+\int \mathcal{L}_R \ ,$$ where $\int \mathcal{L}_R$ stands for the remaining terms. We can then consider a theory without the gauge fields $\mathcal{A}_I$, $\mathcal{A}_{II}$, constructed schematically as $S_{SU}=\int \mathcal{B}_3\wedge {\mathrm{d}}\mathcal{B}_4+\int \mathcal{L}_R$. By construction, this theory has exactly 2 global symmetries satisfying all the properties expected as dual to Betti multiplets. Following [@Witten:2003ya], we can introduce a background gauge field for one of them, which we can call $\mathcal{A}_I$. Then, as reviewed in [§\[sec:baryons\]]{}, the $\mathcal{S}$-operation amounts to regarding this field as dynamical, while at the same time introducing a coupling to another background field $\mathcal{C}_I$ as $$S_{SU}\rightarrow S_{SU}[\mathcal{A}_I]+\int \mathcal{C}_I\wedge {\mathrm{d}}\mathcal{A}_I \ .$$ We can introduce yet another background gauge field $\mathcal{A}_{II}$ for the second global symmetry and perform yet another $\mathcal{S}$-operation. However, this time we will choose to regard $\mathcal{C}_I-\mathcal{A}_{II}$ as the background gauge field on which to act with the $\mathcal{S}$-generator. This results in $$S_{SU}[\mathcal{A}_I]+\int \mathcal{C}_I\wedge {\mathrm{d}}\mathcal{A}_I \rightarrow S_{SU}[\mathcal{A}_I,\, \mathcal{A}_{II}]+\int \mathcal{C}_I\wedge {\mathrm{d}}\mathcal{A}_I + \int \mathcal{C}_{II}\wedge {\mathrm{d}}(\mathcal{C}_I-\mathcal{A}_{II}) \ .$$ Integrating by parts yields $$S_{SU}[\mathcal{A}_I,\, \mathcal{A}_{II}]+\int \mathcal{C}_I\wedge {\mathrm{d}}(\mathcal{C}_{II}+\mathcal{A}_I) - \int \mathcal{C}_{II}\wedge {\mathrm{d}}\mathcal{A}_{II} \ .$$ Since $\mathcal{C}_I$ only appears linearly, its functional integral gives rise to a delta functional setting $\mathcal{C}_{II}=-\mathcal{A}_I$, which leads to an action of the precise form (\[S\_gauged\]). We have therefore been able to establish a connection between a theory where the gauge group is $U(1)^2\times SU(N)^4$, and whose action is $S_{SU}$, with the original $U(N)^4$ theory, whose action is given by $S_U$, via repeated action with the $\mathcal{S}$-operation.
More generally, the whole of $SL(2,{\mathbb{Z}})$ will act on the boundary conditions for the bulk gauge fields, leading in general to an infinite orbit of CFTs for each $U(1)$ gauge symmetry in $AdS_4$. This is a rich structure that deserves considerable further investigation. In this paper, however, we will content ourselves to study the particular choice of boundary conditions described by the $S_{SU}$ theory. Since the dual to the $\mathcal{S}$ operation is the exchange of the $E_{\mu}\leftrightarrow B_{\mu}$ boundary conditions, we expect the gravity dual to the $S_{SU}$ theory to still be $AdS_4\times Q^{111}$, but with an appropriate choice of boundary conditions. In turn, these boundary conditions allow for the existence of the electrically charged M5 branes which we used to identify the symmetries. These M5 branes would not be allowed in the quantization $E_{\mu}=0$, which in turn would be dual to a CFT where the corresponding $U(1)$ factors would remain gauged. In agreement, the dual operators which we will propose below would not be gauge-invariant in that case.
Let us now consider the effect of the $U(1)^2\times SU(N)$ gauge group on the construction of the moduli space. The diagonalization of the $\sigma_a$ auxiliary fields in the equations defining the moduli space (\[DF\]) relies on the non-abelian part of the gauge symmetry, and therefore it applies even if we consider ungauging some of the diagonal $U(1)$ factors. More crucially, in order to obtain the correct four-fold moduli space we needed the $S(\mathcal{B}_4)$ piece (\[SBG\]) of the CS action so that, upon dualizing the $\mathcal{B}_4$ field, the dual scalar $\tau$ is gauge-fixed via gauge transformations of $\mathcal{B}_{3}$. Thus provided we leave $\mathcal{B}_{4}$ and $\mathcal{B}_{3}$ gauged, with the same CS action, all of this discussion is unaffected if we ungauge the remaining $U(1)_I$, $U(1)_{II}$. Correspondingly, we will still have the 8 gauge-invariants (\[ws\]), which will give rise to the same 9 equations defining $\mathcal{C}(Q^{111})$ as a non-complete intersection as “mesonic” moduli space. The remarks on the non-abelian chiral ring elements spanned by (\[nonabelianchirals\]) are also unchanged. However, with only a $U(1)_{\mathcal{B}_3}\times U(1)_{\mathcal{B}_4}\times SU(N)^4$ gauge symmetry we also have additional chiral primary operators, charged under the now global $U(1)_I$, $U(1)_{II}$. Indeed, we have the following “baryonic” type operators: \_[A\_[I\_1...I\_N]{}]{}&=& \^[i\_1i\_N]{} \_[j\_1j\_N]{} (A\_[I\_1]{})\^[j\_1]{}\_[i\_1]{}(A\_[I\_N]{})\^[j\_N]{}\_[i\_N]{} ,\
\_[B\_i]{}&=& \^[i\_1i\_N]{} \_[j\_1j\_N]{} (B\_i)\^[j\_1]{}\_[i\_1]{}(B\_i)\^[j\_N]{}\_[i\_N]{}\^[(-1)\^[i-1]{}N]{} ,\
\_[C\_i]{}&=& \^[i\_1i\_N]{} \_[j\_1j\_N]{} (C\_i)\^[j\_1]{}\_[i\_1]{}(C\_i)\^[j\_N]{}\_[i\_N]{}\^[(-1)\^[i-1]{}N]{} . \[b-operators\] In particular, for the 6 fields in the quiver there is a canonical set of 6 baryonic operators given by determinants of these fields, dressed by appropriate powers of the disorder operators ${\mathrm{e}}^{{\mathrm{i}}\tau}$ to obtain gauge-invariants under $\mathcal{B}_3$. These operators are in 1-1 correspondence with the toric divisors in the geometry. This is precisely the desired mapping between baryonic operators in the field theory and M5 branes wrapping such toric submanifolds, with one M5 brane state for each divisor. Indeed, the charges of these operators under the two baryonic $U(1)$s are $$\begin{array}{c | c c c}
& \mathscr{B}_{A_{I_1..I_N}} & \mathscr{B}_{B_i} & \mathscr{B}_{C_i} \\ \hline
U(1)_{I} & N & 0 & -N \\
U(1)_{II} & -N & N & 0 \\
\end{array}~.$$ These are precisely the charges of M5 branes, wrapped on the five-manifolds corresponding to the divisors $\{A_i=0\}$, $\{B_i=0\}$, $\{C_i=0\}$, under the two $U(1)^{b_2(Y)}$ symmetries in $AdS_4$. Indeed, recall that the two two-cycles in $Q^{111}$ may be taken to be the anti-diagonal $S^2$s in two factors of $\mathbb{CP}^1\times \mathbb{CP}^1\times \mathbb{CP}^1$, at $\psi=0$. Let us choose these to be the anti-diagonal in the first and third factor, and second and first factor, respectively. The charge of an M5 brane wrapped on a five-cycle $\Sigma_5\subset Y$ under each $U(1)$ is then the intersection number of $\Sigma_5$ with each corresponding two-cycle. Thus with this basis choice, the charges of the operator associated to an M5 brane wrapped on the base of one of the 6 toric divisors $\{A_i=0\}$, $\{B_i=0\}$, $\{C_i=0\}$ are precisely those listed in the above table.
Being chiral primary, the conformal dimensions of these operators are given by $N\, \Delta[X]=N\, R[X]$, $R[X]$ being the R-charge of the field $X$. The conformal dimension of an M5 brane wrapping a supersymmetric five-cycle $\Sigma_5\subset Y$ is given by the general formula [@Gubser:1998fp] $$\Delta[\Sigma_5] = \frac{N\pi {\mathrm{vol}}(\Sigma_5)}{6{\mathrm{vol}}(Y)}~.$$ These volumes are easily computed for the $Q^{111}$ metric (\[Qiiimetric\]): ${\mathrm{vol}}(Q^{111})=\frac{\pi^4}{8}$, ${\mathrm{vol}}(\Sigma_5)=\frac{\pi^3}{4}$, where $\Sigma_5$ is any of the 6 toric five-cycles. From this one obtains $\Delta[\Sigma_5]=\frac{N}{3}$ in each case, giving conformal dimensions $\Delta[X]=\frac{1}{3}$ for each field. With this R-charge assignment we see that the superpotential (\[WQ111\]) has R-charge 2, precisely as it must at a superconformal fixed point. Indeed, the converse argument was applied in [@Hanany:2008fj] to obtain this R-charge assignment. We thus regard this as further evidence in support of our claim in this section, as well as further support for these theories as candidate SCFT duals to $AdS_4\times Q^{111}$.
QCS theories dual to isolated toric Calabi-Yau four-fold singularities {#sec:gen}
----------------------------------------------------------------------
We would like to apply the preceding discussion to more general $\mathcal{N}=2$ CS-matter theories dual to M2 branes probing Calabi-Yau four-fold cones. With the exception of ${\mathbb{C}}^4$, the apex of the cone always corresponds to a *singular point* $p$ in the toric variety. The simplest class of such singularities occurs when this singular point $p$ is isolated; that is, when no other singular loci intersect it. In this case the base of the cone $Y$ is a smooth Sasakian seven-manifold. The condition for the singular point $p$ to be isolated was given in [@Lerman2] and interpreted in terms of the toric diagram in [@Benishti:2009ky].
An additional ingredient is the possible presence of vanishing six-cycles at the tip of the cone. In terms of the toric data, these six-cycles are signalled by internal lattice points in the toric diagram. These codimension 2 cycles, in very much the same spirit as their four-cycle Type IIB counterparts (see *e.g.* [@Benvenuti:2005qb]), represent a further degree of complexity. We postpone the analysis of geometries with exceptional six-cycles to [§\[sec:5\]]{}.
We study such isolated Calabi-Yau singularities without vanishing six-cycles in more detail in [§\[sec:classification\]]{}, in particular classifying the singularities with 4 or 5 external vertices in the toric diagram. In the cases where a Lagrangian description of the M2 brane theory exists, it turns out that for all these cases one can construct an appropriate toric superpotential, so that there is a toric gauge theory which realizes at the abelian level the minimal GLSM. This toric gauge theory has $b_2(Y)+2$ gauge group factors, and can be promoted to have $U(N)$ gauge groups. Such quiver Chern-Simons theories have been considered in the past in [@Aharony:2008ug; @Franco:2008um; @Benishti:2009ky].
We would like to generalize our proposal to this simplest class of isolated singularities with no vanishing six-cycles. Indeed, we expect that a similar sequence of $\mathcal{T}$ and $\mathcal{S}$ operations amounts to ungauging of precisely $b_2(Y)$ $U(1)$ factors. In very much the same spirit as in the $\mathcal{C}(Q^{111})$ example, this should correspond to a particular choice of boundary conditions in the dual $AdS_4$. Furthermore, we conjecture the gauge group to be $U(1)^2\times SU(N)^{b_2(Y)+2}$, the two $U(1)$ factors being those corresponding to the $\mathcal{B}_{G}$ and $\mathcal{B}_{G-1}$ gauge fields. This way we are naturally left with $b_2(Y)$ global $U(1)$ symmetries which exactly correspond to the $b_2(Y)$ expected $U(1)$ baryonic symmetries. Furthermore, the M5 branes would be naturally identified with the corresponding baryonic operators, constructed in a similar manner as in the $\mathcal{C}(Q^{111})$ example.
As a final remark, for the $\mathcal{Q}^{111}$ case one can see that the total moduli space can be described as two extra complex directions that are fibred over the mesonic moduli space. The resulting variety is often called the *master space*. More generally one can see that the moduli spaces of the class of theories that we described above can be described as $b_2(Y)$ baryonic directions fibred over the mesonic moduli space. This is implied by the fact that the number of gauge nodes in these theories is $b_2(Y)+2$.
Gravity duals of baryonic symmetry breaking {#sec:4}
===========================================
In [§\[sec:baryons\]]{} we explained that for each $U(1)$ gauge field in $AdS_4$, arising from reduction of $C_6$ along five-cycles in $Y$, there are two different $AdS$ quantizations: one of these gives rise to a conserved $U(1)$ current in the dual CFT, while the other instead gives rise to a *dynamical* $U(1)$ gauge field. As discussed in [§\[sec:baryons\]]{}, in this paper we content ourselves with studying the quantization which is more closely analogous to the case in Type IIB string theory, in which the $b_2(Y)$ baryonic $U(1)$ gauge fields (\[3-form-to-global\]) in $AdS_4$ are dual to conserved currents in the dual SCFT. As we have argued, in this theory M5 branes wrapped on supersymmetric cycles in $Y$ should appear as chiral primary baryonic-type operators in the dual SCFT. Indeed, at least for toric theories with appropriate smooth supergravity horizons $Y$ we expect the dual SCFT to be described by a QCS theory with gauge group $U(1)^2\times SU(N)^{b_2(Y)+2}$. The M5 brane states are then the usual gauge-invariant determinant-like operators in these theories, as we discussed in detail for the $Q^{111}$ theory in the previous section.
We may then study the gravity duals to vacua in which the $b_2(Y)$ global $U(1)$s are (spontaneously) broken. On general grounds, these should correspond to supergravity solutions constructed from resolutions of the corresponding cone over $Y$. The baryonic operators are charged under the global baryonic symmetries, and vacua in which these operators obtain a VEV lead to spontaneous symmetry breaking. By giving this VEV we pick a point in the moduli space of the theory, which at the same time introduces a scale and thus an RG flow, whose endpoint will be a different SCFT. The supergravity dual of this RG flow was first discussed in the Type IIB context by Klebanov-Witten [@Klebanov:1999tb], and to some extent in the M-theory context in [@Benishti:2009ky]. In this section we begin by discussing in detail these gravity solutions for the case of $Q^{111}$. The physics in fact very closely resembles the physics in the Type IIB context. We then describe how to generalize this discussion for general Calabi-Yau four-fold singularities. In particular, we will obtain a general formula for M5 brane condensates, or indeed more generally still a formula for the on-shell action of a wrapped brane in a warped Calabi-Yau background. Essentially this formula appeared in [@Baumann:2006th], where it was checked in some explicit examples. Here we provide a general proof of this formula.
We emphasize that the interpretation of the gravity backgrounds considered in this section is only for the special choice of dual CFT in which the $b_2(Y)$ baryonic $U(1)$ gauge fields in $AdS_4$ are dual to global symmetries in the dual SCFT. More generally, different choices of boundary conditions will imply that some, or all, of the M5 brane states considered here are absent. It is then clearly very interesting to ask what is the dual field theory interpretation of these gravity backgrounds in such situations. Again, we leave this for future work.
Resolutions of $\mathcal{C}(Q^{111})$ {#sec:resQ111}
-------------------------------------
In this section we consider the warped resolved gravity backgrounds for $Q^{111}$. We begin by discussing this in the context of the GLSM, and then proceed to construct corresponding explicit supergravity solutions.
### Algebraic analysis {#sec:GKZQ111}
The toric singularity $\mathcal{C}(Q^{111})$ may be described by a GLSM with 6 fields, $a_i$, $b_i$, $c_i$, $i=1,2$, and gauge group $U(1)^2$. This is also the same as the abelian QCS theory presented in [§\[sec:3\]]{}, but without the CS terms. The charge matrix is
[c|cccccc]{} & a\_1 & a\_2 & b\_1 & b\_2 & c\_1 & c\_2\
U(1)\_1 & -1 & -1 & 1 & 1 & 0 & 0\
U(1)\_2 & -1 & -1 & 0 & 0 & 1 & 1
. The singular cone $\mathcal{C}(Q^{111})$ is the moduli space of this GLSM where the FI parameters $\zeta_1=\zeta_2=0$ are both zero. However, more generally we may allow $\zeta_1, \zeta_2\in {\mathbb{R}}$, leading to different (partial) resolutions of the singularity. In fact since there are no internal points in the toric diagram in Figure \[fig:toricdiagramQ111\], this GLSM in fact describes *all* possible (partial) resolutions of the singular cone.
It is straightforward to analyse the various cases. Suppose first that $\zeta_1,\zeta_2>0$ are both positive. We may write the two D-terms of the GLSM as \[Dpos\] |b\_1|\^2 + |b\_2|\^2 &=& \_1 + |a\_1|\^2 + |a\_2|\^2>0 ,\
|c\_1|\^2 + |c\_2|\^2 &=& \_2 + |a\_1|\^2 + |a\_2|\^2>0 . In particular, for $a_1=a_2=0$ we obtain $\mathbb{CP}^1\times\mathbb{CP}^1$ where the Kähler class of each factor is proportional to $\zeta_1$ and $\zeta_2$, respectively. Here $b_i$ and $c_i$ may be thought of as homogeneous coordinates on the $\mathbb{CP}^1$s. Altogether, this describes the total space of the bundle $\mathcal{O}(-1,-1)\oplus \mathcal{O}(-1,-1)\rightarrow\mathbb{CP}^1\times\mathbb{CP}^1$, with $a_1, a_2$ the two fibre coordinates on the ${\mathbb{C}}^2$ fibres.
Suppose instead that $\zeta_1<0$. We may then rewrite the D-terms as |a\_1|\^2 + |a\_2|\^2 &=& -\_1 + |b\_1|\^2 + |b\_2|\^2>0 ,\
|c\_1|\^2 + |c\_2|\^2 &=& \_2-\_1 + |b\_1|\^2 + |b\_2|\^2 . Provided also $\zeta_2-\zeta_1>0$, we hence obtain precisely the same geometry as when $\zeta_1,\zeta_2>0$, but with the $\mathbb{CP}^1\times\mathbb{CP}^1$ zero section now parametrized by $a_i$ and $c_i$ and with Kähler classes proportional to $-\zeta_1$ and $\zeta_2-\zeta_1$, respectively. There is a similar situation with $\zeta_2<0$ and $\zeta_1-\zeta_2>0$.
![The GKZ fan for $Q^{111}$ is ${\mathbb{R}}^2$, divided into three cones.[]{data-label="fig:GKZ1"}](./GZK1crop)
Hence in total there are three different resolutions of $\mathcal{C}(Q^{111})$, corresponding to choosing which of the three $\mathbb{CP}^1$s in $Q^{111}$ collapses at the zero section in $\mathcal{O}(-1,-1)\oplus \mathcal{O}(-1,-1)\rightarrow\mathbb{CP}^1\times\mathbb{CP}^1$. We label these three $\mathbb{CP}^1$s as $\mathbb{CP}^1_a$, $a=1,2,3$, which in $Q^{111}$ are parametrized by $c_i$, $b_i$, $a_i$, respectively. This is shown in Figure \[fig:GKZ1\], which is known more generally as the *GKZ fan*. Notice there is a $\Sigma_3$ permutation symmetry of the three $\mathbb{CP}^1$s in $Q^{111}$ and the three different resolutions are permuted by this symmetry.
The boundary edges between the regions correspond to collapsing another of the $\mathbb{CP}^1$s, leading only to a partial resolution of the singularity. Thus, for example, take $\zeta_1>0$ but $\zeta_2=0$. The D-terms are now |b\_1|\^2 + |b\_2|\^2 &=& \_1 + |a\_1|\^2 + |a\_2|\^2>0 ,\
|c\_1|\^2 + |c\_2|\^2 &=& |a\_1|\^2 + |a\_2|\^2 . The second line describes the conifold singularity, which is then fibred over a $\mathbb{CP}^1$, parametrized by the $b_i$, of Kähler class $\zeta_1$.
### Supergravity analysis {#sec:sugraQ111}
For each of the resolutions of $\mathcal{C}(Q^{111})$ described above there is a corresponding Ricci-flat Kähler metric that is asymptotic to the cone metric over $Q^{111}$. More precisely, there is a unique such metric for each choice of Kähler class, or equivalently FI parameter $\zeta_1,\zeta_2\in {\mathbb{R}}$. As we shall discuss later in this section, this is guaranteed by a general theorem that has only just been proven in the mathematics literature. However, for $Q^{111}$ these metrics may in fact be written down explicitly. Denoting the (partially) resolved Calabi-Yau generically by $X$, the Calabi-Yau metrics are given by $$\begin{aligned}
\label{resolvedQ111}
{\mathrm{d}}s^2(X) &=&\kappa(r)^{-1}{\mathrm{d}}r^2+\kappa(r)\frac{r^2}{16}\Big({\mathrm{d}}\psi+\sum_{i=1}^3 \cos\theta_i {\mathrm{d}}\phi_i\Big)^2+\frac{(2a+r^2)}{8}\Big({\mathrm{d}}\theta_2^2+\sin^2\theta_2 {\mathrm{d}}\phi_2^2\Big)\nonumber \\ &&+\frac{(2b+r^2)}{8}\Big({\mathrm{d}}\theta_3^2+\sin^2\theta_3{\mathrm{d}}\phi_3^2\Big)+\frac{r^2}{8}\Big({\mathrm{d}}\theta_1^2+\sin^2\theta_1{\mathrm{d}}\phi_1^2\Big)~,\end{aligned}$$ where $$\label{kappa}
\kappa(r)=\frac{(2A_-+r^2)(2A_++r^2)}{(2a+r^2)(2b+r^2)}~,$$ $a$ and $b$ are arbitrary constants, and we have also defined $$A_\pm=\frac{1}{3}\Big(2a+2b\pm\sqrt{4a^2-10ab+4b^2}\Big)~.$$ One easily sees that at large $r$ the metric approaches the cone over the $Q^{111}$ metric (\[Qiiimetric\]). This way of writing the resolved metric breaks the $\Sigma_3$ symmetry, since it singles out the $\mathbb{CP}^1$ parametrized by $(\theta_1,\phi_1)$ as that collapsing at $r=0$. Here we have an exceptional $\mathbb{CP}^1\times\mathbb{CP}^1$, parametrized by $(\theta_2,\phi_2)$, $(\theta_3,\phi_3)$, with Kähler classes proportional to $a>0$ and $b>0$, respectively. Thus setting $\{a=0, b>0\}$, or $\{b=0,a>0\}$, leads to a partial resolution with a residual $\mathbb{CP}^1$ family of conifold singularities at $r=0$. We shall examine this in more detail below. For further details of this solution we refer the reader to [§\[sec:Q111-geometry\]]{}.
We are interested in studying supergravity backgrounds corresponding to M2 branes localized on one of these resolutions of $\mathcal{C}(Q^{111})$. We thus consider the following ansatz for the background sourced by such M2 branes $$\begin{aligned}
\label{metricagain}
{\mathrm{d}}s^2_{11}&=&h^{-2/3}\, {\mathrm{d}}s^2\left({\mathbb{R}}^{1,2}\right) +h^{1/3}{\mathrm{d}}s^2(X)~, \nonumber\\ G&=&{\mathrm{d}}^3 x \wedge {\mathrm{d}}h^{-1}~,\end{aligned}$$ where ${\mathrm{d}}s^2(X)$ is the Calabi-Yau metric (\[resolvedQ111\]). If we place $N$ spacetime-filling M2 branes at a point $y\in X$, we must then also solve the equation \_x h\[y\] = \^8(x-y) , for the warp factor $h=h[y]$. Here $\Delta$ is the scalar Laplacian on $X$. Having the explicit form of the metric we can compute this Laplacian and solve for the warp factor to obtain the full supergravity solution. This is studied in detail in [§\[sec:D\]]{}.
In the remainder of this subsection let us analyse the simplified case in which we partially resolve the cone, setting $a=0$ and $b>0$. This corresponds to one of the boundary lines in the GKZ fan in Figure \[fig:GKZ1\], with the point on the boundary ${\mathbb{R}}_{> 0}$ labelled by the metric parameter $b>0$. Here one can solve explicitly for the warp factor in the case where we put the $N$ M2 branes at the north pole of the exceptional $\mathbb{CP}^1$ parametrized by $(\theta_3,\phi_3)$; this is the point with coordinates $y=\{r=0, \theta_3=0\}$. Notice the choice of north pole is here without loss of generality, due to the $SU(2)$ isometry acting on the third copy of $\mathbb{CP}^1$. We denote the corresponding warp factor in this case as simply $h[y=\{r=0,\theta_3=0\}]\equiv h$. As shown in [§\[sec:Q111-warp\]]{}, $h=h(r,\theta_3)$ is then given explicitly in terms of hypergeometric functions by $$\begin{aligned}
h(r,\theta_3) &=&\sum_{l=0}^\infty\,H_{l}(r)\, P_{l}(\cos\theta_3)~,\nonumber \\
H_{l}(r)&=& \mathcal{C}_{l}\, \Big(\frac{8b}{3r^2}\Big)^{3(1+\beta)/2}\, _2F_1\left(-\tfrac{1}{2}+\tfrac{3}{2}\beta,\tfrac{3}{2}+\tfrac{3}{2}\beta,1+3\beta,-\tfrac{8b}{3r^2}\right)~,
\label{Q111-warp-factor}\end{aligned}$$ where $P_{l}$ denotes the $l$th Legendre polynomial, $$\beta=\beta(l)=\sqrt{1+\frac{8}{9}l(l+1)}~,$$ and the normalization factor $\mathcal{C}_{l}$ is given by \_[l]{}&=&()\^3 (2l+1) R\^6 ,\
\[anothereqn\] R\^6&=&=\^2 N \_p\^6 . In the field theory this solution corresponds to breaking one combination of the two global $U(1)$ baryonic symmetries, rather than both of them. This will become clear in the next subsection. The resolution of the cone can be interpreted in terms of giving an expectation value to a certain operator $\mathcal{U}$ in the field theory. This operator is contained in the same multiplet as the current that generates the broken baryonic symmetry, and couples to the corresponding $U(1)$ gauge field in $AdS_4$. Since a conserved current has no anomalous dimension, the dimension of $\mathcal{U}$ is uncorrected in going from the classical description to supergravity [@Klebanov:1999tb]. According to the general $AdS/CFT$ prescription [@Klebanov:1999tb], the VEV of the operator $\mathcal{U}$ is dual to the subleading correction to the warp factor. For large $r$ we can write $$h(r,\theta_3)\sim\sum_{l=0}^\infty\, \mathcal{C}_{l}\, \Big(\frac{8b}{3r^2}\Big)^{3(1+\beta)/2}\, P_{l}(\cos\theta_3)~.$$ Expanding the sum we then have $$h(r,\theta_3)\sim \frac{R^6}{r^6}\left(1+\frac{18b\, \cos\theta_3}{5r^2}+\cdots\right)~.$$ In terms of the $AdS_4$ coordinate $z= r^{-2}$ we have that the leading correction is of order $z$, which indicates that the dual operator $\mathcal{U}$ is dimension 1. This is precisely the expected result, since this operator sits in the same supermultiplet as the broken baryonic current, and thus has a protected dimension of 1. Furthermore, its VEV is proportional to $b$, the metric resolution parameter, which reflects the fact that in the conical ($AdS$) limit in which $b=0$ this baryonic current is not broken, and as such $\langle \mathcal{U}\rangle=0$.
The moduli space of the field theory in the new IR is equivalent to the geometry close to the branes. Recall that we placed the $N$ M2 branes at the north pole $\{\theta_3=0\}$ of the exceptional $(\theta_3,\phi_3)$ sphere at $r=0$. Defining $\tilde{\psi}=\psi+\phi_3$ and introducing the new radial variables $\tilde{r}=\frac{b}{2}\,\theta_3$, $\rho=\frac{\sqrt{3}}{2}\, r$, the geometry close to the branes becomes to leading order $$\begin{aligned}
{\mathrm{d}}s^2&=&{\mathrm{d}}\rho^2+\frac{\rho^2}{9}\Big({\mathrm{d}}\tilde{\psi}+\sum_{i=1}^2 \cos\theta_i{\mathrm{d}}\phi_i\Big)^2+\frac{\rho^2}{6}\Big({\mathrm{d}}\theta_2^2+\sin^2\theta_2{\mathrm{d}}\phi_2^2\Big)\nonumber \\ &&+\frac{\rho^2}{6}\Big({\mathrm{d}}\theta_1^2+\sin^2\theta_1{\mathrm{d}}\phi_1^2\Big)+\Big({\mathrm{d}}\tilde{r}^2+\tilde{r}^2{\mathrm{d}}\phi_3^2\Big)~,
\label{c_x_conifold}\end{aligned}$$ which is precisely the Ricci-flat Kähler metric of $\mathcal{C}(T^{11})\times \mathbb{C}$, in accordance with the discussion in the previous subsection.
Higgsing the $Q^{111}$ field theory {#sec:HiggsQ111}
-----------------------------------
We have argued that the warped resolved supergravity solutions described in the previous section are dual to spontaneous symmetry breaking in the SCFT in which the M5 brane states appear as baryonic-type operators. Let us study this in more detail in the field theory described in [§\[sec:3\]]{}. In this SCFT the symmetries $U(1)_I$ and $U(1)_{II}$ in (\[gaugedU(1)charges\]) are global, rather than gauge, symmetries, with the corresponding conserved currents coupling to the baryonic $U(1)$ gauge fields in $AdS_4$. By inspection of this charge matrix we conclude that it is possible to give a VEV to the $A_i$, $B_i$ and $C_i$ fields. These VEVs then break the corresponding baryonic $U(1)$ symmetries. In particular, by giving a VEV to any pair of fields ($A$s, $B$s or $C$s) we break only one particular $U(1)$ baryonic symmetry, leaving another combination unbroken. In this section we will examine the resulting Higgsings of the gauge theory obtained by giving VEVs to different pairs of fields, and compare with the gravity results of the previous section.
As explained in [§\[sec:2\]]{}, at each of the two poles for each copy of $\mathbb{CP}^1$ in the Kähler-Einstein base of $Q^{111}$, there is a supersymmetric five-cycle that may be wrapped by an M5 brane. Altogether these are 6 M5 brane states, corresponding to the toric divisors of $\mathcal{C}(Q^{111})$. Each pair are acted on by one of the $SU(2)$ factors in the isometry group $SU(2)^3\times U(1)_R$, rotating one into the other. Quantizing the BPS particles in $AdS_4$ one obtains dual baryonic-type operators given by (\[b-operators\]). In particular, consider the M5 branes that sit at a point on the copy of $\mathbb{CP}^1=S^2$ with coordinates $(\theta_3,\phi_3)$. In the next section we will compute the VEV of these M5 brane operators in the partially resolved gravity background described by (\[Q111-warp-factor\]), showing that the baryonic operator dual to the M5 brane at $\theta_3=0$ vanishes, while that at the opposite pole $\theta_3=\pi$ is non-zero and proportional to the resolution parameter $b$ (see equation (\[vev\])). Considering the $A$ fields in the field theory this corresponds to the fact that, after breaking the baryonic symmetry by giving diagonal VEVs to these fields, it is possilbe to use the $SU(2)$ flavour symmetry to find one combination of $A$ fields with zero VEV, and an orthogonal combination with non-vanishing VEV. Let us assume for example that $\langle A_1\rangle=0$ and $\langle A_2\rangle=b\, I_{N\times N}$. Thus only one baryonic operator in (\[b-operators\]) has non-vanishing VEV, namely $\langle\mathscr{B}_{A_2}\rangle = b^{N\, \Delta_{A_2}}$.[^7] This situation was analyzed in [@Franco:2009sp], where it was shown that the effective field theory in the IR has CS quiver
with superpotential $$\label{con1}
W\,=\, {\rm Tr}\,\Phi\, \Big(\,C_2\, B_1\, B_2\, C_1\,-\,B_2\, C_1\, C_2\, B_1\,\Big)~.$$ As shown in [@Franco:2009sp], the moduli space of this theory is indeed $\mathcal{C}(T^{11})\times{\mathbb{C}}$. This is of course expected from the gravity analysis of equation (\[c\_x\_conifold\]). Any other VEV for the $A$ fields corresponds to placing the M2 branes on $SU(2)$-equivalent points on the blown-up $\mathbb{CP}^1$, and therefore results in the same near horizon geometry.
The manifest symmetry exhibited by the Lagrangian of the $Q^{111}$ field theory is $SU(2) \times U(1)_R$, which is only a subgroup of the full $SU(2)^3 \times U(1)_R$ symmetry which is expected from the isometry of the $Q^{111}$ manifold. Therefore, in contrast to the situation with the $A$ fields, we see that different VEVs for the pairs of $B$ or $C$ fields result in different theories in the IR. Giving a non-vanishing VEV to $C_1$ and $C_2$, for example, results in the CS quiver
with superpotential $$\label{zeroCS}
W\,=\, {\rm Tr}\,\Big(\,A_2\, B_1\, B_2\, A_1\,-\,B_2\, A_2\, A_1\, B_1\,\Big)~,$$ while when the VEV of only one field is non-vanishing we instead obtain the CS quiver
with superpotential $$\label{con2}
W\,=\, {\rm Tr}\,C_2\, \Big(\,A_2\, B_1\, B_2\, A_1\,-\, A_2\, A_1\, B_1\,B_2\,\Big)~.$$ Of course, both cases correspond geometrically to blowing up the same $\mathbb{CP}^1$, as can be seen from the explicit construction of the field theory moduli space. However, recall that the position of the M2 branes depends on the VEVs of $C_1$ and $C_2$. While in the gravity picture all locations on the exceptional $\mathbb{CP}^1$ are $SU(2)$-equivalent, in the field theory since only part of the global symmetery is manifest we obtain different theories for different VEVs. The supergravity analysis hence suggests that the theories obtained in the IR above are dual. Indeed, one can check that the moduli spaces of these theories are the same, the details appearing in [@Davey:2009qx], for example.
The QCS theory for $\mathcal{C}(T^{11})\times{\mathbb{C}}$ in (\[zeroCS\]) has zero CS levels, and there is therefore no tunable coupling parameter in this theory. This may be understood as follows. After blowing up the cone to the partial resolution and placing the stack of M2 branes at a residual singular point on the exceptional $\mathbb{CP}^1$, the tangent cone geometry at this point is $\mathcal{C}(T^{11})\times{\mathbb{C}}$. In the field theory the $S^1$ that rotates the copy of ${\mathbb{C}}$ corresponds to the M-theory circle. Shrinking the size of this circle in the M-theory supergravity solution corresponds to a Type IIA limit describing a black D2 brane solution with no smooth near horizon. We should therefore not expect to find a dual field theory with a weak coupling limit. On the other hand, in the remaining two field theories (\[con1\]), (\[con2\]) for $\mathcal{C}(T^{11})\times{\mathbb{C}}$ described in this section the M-theory circle involves a $U(1)$ that acts also on $\mathcal{C}(T^{11})$.[^8]
As a final comment in this section, notice that in general we will have an infinite set of CFTs dual to the $Q^{111}$ geometry, with different boundary conditions on the baryonic gauge fields (\[3-form-to-global\]) in $AdS_4$. From our earlier discussion, these will have the same QCS theories in the $SU(N)$ sector, but different $U(1)$ sectors. In particular, in general different combinations of the diagonal $U(1)$s in $U(N)$ may be gauged, and with different $U(1)$ CS levels from the levels $k_a$ of the $SU(N)$ factors. These $U(1)$ sectors will in general behave differently under Higgsing. In particular, a global $U(1)$ can be sponteneously broken, while a gauged $U(1)$ can be Higgsed. It will be interesting to investigate this general structure in both the field theory and dual supergravity solutions, although we leave this for future work, focusing instead on the $U(1)^2\times SU(N)^4$ theory.
Baryonic condensates and M5 branes in the $Q^{111}$ solution {#sec:M5Q111}
------------------------------------------------------------
In the previous section we discussed the RG flow triggered by giving a VEV to one of the fields, and hence baryonic operators, in the $Q^{111}$ theory with gauge group $U(1)^2\times SU(N)^4$. In this section we calculate the VEV of this baryonic operator in the corresponding gravity solution described by (\[Q111-warp-factor\]). In order to do this we follow the analogous calculation in the Type IIB context, discussed in [@Martelli:2008cm; @Klebanov:2007us]. In this prescription, to determine the VEV of a baryonic operator dual to an M5 brane wrapped on a five-submanifold $\Sigma_5\subset Y$ in the (partially) resolved supergravity background, we compute the Euclidean action of an M5 brane which wraps a minimal six-dimensional manifold in $X$ which at large $r$ asymptotes to the cone over $\Sigma_5$. In this section we present an explicit example, although later we will present a general formula for such VEVs.
Again, we focus on the partially resolved background for $Q^{111}$ described by (\[Q111-warp-factor\]). We are interested in computing the VEV of the operator that was carrying charge under the baryonic symmetry before it was broken. As described in [§\[sec:2\]]{}, this symmetry originates on the supergravity side to a mode (\[3-form-to-global\]) of the six-form potential $C_6$ along a five-cycle in the Sasaki-Einstein manifold $Q^{111}$. Consider a Euclidean M5 brane that is wrapped on the six-manifold at a fixed point in the $(\theta_3,\phi_3)$ copy of $\mathbb{CP}^1$, and is coordinatized by $\{r,\psi,\theta_1,\theta_2,\phi_1,\phi_2\}$ in the partial resolution of $\mathcal{C}(Q^{111})$. This six-manifold is a divisor in the partially resolved background, and hence this wrapped M5 brane worldvolume is a calibrated submanifold. The M5 brane carries charge under the $U(1)$ gauge field in $AdS_4$ that descends from the corresponding harmonic five-form in $Q^{111}$. We calculate the Euclidean action of this wrapped M5 brane up to a radial cut-off $r=r_c$, identifying ${\mathrm{e}}^{-S(r_c)}$ with the classical field dual to the baryonic operator, as in [@Klebanov:2007us]. Explicitly, the action is given by S(r\_c)=T\_[5]{}\_[rr\_c]{} h \^6x , \[m5-action\] where $T_{5}=2\pi/(2\pi\ell_p)^6$ is the M5 brane tension, the warp factor $h$ is given by (\[Q111-warp-factor\]), and $\det g_6$ is the determinant of the metric induced on the M5 brane worldvolume. This induced metric is $$\begin{aligned}
{\mathrm{d}}s_6^2&=&\kappa^{-1}{\mathrm{d}}r^2+\kappa\frac{r^2}{16}\Big({\mathrm{d}}\psi+\sum_{i=1}^2 \cos\theta_i{\mathrm{d}}\phi_i\Big)^2+\frac{r^2}{8}\Big({\mathrm{d}}\theta_2^2+\sin^2\theta_2{\mathrm{d}}\phi_2^2\Big)\nonumber \\ &&+\frac{r^2}{8}\Big({\mathrm{d}}\theta_1^2+\sin^2\theta_1{\mathrm{d}}\phi_1^2\Big)~,\end{aligned}$$ where recall that $\kappa(r)$ is given by (\[kappa\]). A straightforward calculation hence gives = . \[detg6\] Subtituting these results into (\[m5-action\]) then gives S(r\_c)&=&\_1 \_2 \_1 \_2 \_1 \_2 \_0\^[r\_c]{} r r\^5 \_[l=0]{}\^H\_[l]{}(r) P\_[l]{}(\_3)\
&=& . \[mem-action\] Let us evaluate the integrals separately. The first diverges in the absence of the cut-off $r_c$, since \_0\^[r\_c]{} r r\^5 H\_0 (r) . The second integral is finite and can be calculated straightforwardly: \_0\^ r r\^5 \_[l=1]{}\^H\_[l]{}(r) P\_[l]{}(\_3)&=&\_[l=1]{}\^P\_[l]{}(\_3)\
&=& -(+) . Recall here that $R$ is given by (\[anothereqn\]). Subtituting these results into then gives \^[-S(r\_c)]{}=\^[[7N]{}/[18]{}]{} ()\^()\^[N]{} . \[vev\] The interpertation of this result is along the same lines as the discussion in the case of the conifold in [@Klebanov:2007us]. We will therefore keep our discussion brief and refer the reader to [@Klebanov:2007us] for further details. Since the $AdS_4$ radial coordinate is related to $r$ as $z=r^{-2}$, we see that the operator dimension is $\Delta = \frac{N}{3}$, as anticipated by our field theory discussion. Indeed, this provides a non-trivial check of the R-charge assignment required for the theory to have an IR superconformal fixed point, supporting the conjecture that the theory is indeed dual to M2 branes moving in $\mathcal{C}(Q^{111})$.
For the remaining M5 branes, wrapped on Euclidean six-submanifolds at a point in either the $(\theta_1,\phi_1)$ or $(\theta_2,\phi_2)$ copies of $\mathbb{CP}^1$, we note that ${\mathrm{e}}^{-S(r_c)}=0$. This is simply because the M5 brane worldvolumes intersect the M2 brane stack on the exceptional $\mathbb{CP}^1$ parametrized by $(\theta_3,\phi_3)$ in these cases, and hence the worldvolume action (\[m5-action\]) is logarithmically divergent near to the M2 branes – for further discussion of this in the D3 brane case, see [@Martelli:2008cm]. Thus the dual gravity analysis of the partially resolved supergravity solution is in perfect agreement with the proposed $Q^{111}$ field theory, at least with the boundary conditions we study in this section.
Wrapped branes and the phase of the condensate {#sec:wrapped}
----------------------------------------------
In the resolved, or partially resolved, geometry one can consider various different kinds of stable wrapped branes in the $r\sim 0$ region. These shed further light on the physical interpretation of the supergravity solutions.
In the fully resolved case with $a>0$ and $b>0$ one could consider an M5 brane wrapping the exceptional $\mathbb{CP}^1\times\mathbb{CP}^1$ at $r=0$ and filling the spacetime directions $x_0$, $x_1$. This is a domain wall in the Minkowski three-space in (\[metricagain\]). Its tension, given by the energy of the probe brane, is $$E_{\mathrm{M5}}=\mathcal{T}_{\mathrm{wall}}=\frac{abT_5\pi^2}{16}~.$$ Notice that the warp factor cancels out in this computation, and the brane remains of finite tension even if the stack of M2 branes is placed at $r=0$.
Of more interest for us is to consider an M2 brane wrapping an exceptional $\mathbb{CP}^1$, either in the resolved or partially resolved background. In the former case notice there are homologically two choices of such $\mathbb{CP}^1$ inside $\mathbb{CP}^1\times\mathbb{CP}^1$ at $r=0$. Again, in either case this is a calibrated cycle. This leads to a point particle in the Minkowski three-space, moving along the time direction $x_0$, whose mass is $$E_{\mathrm{M2}}=m_{\mathrm{point}}=\frac{bT_2\pi}{4}~.$$ Again, its energy remains finite even when the M2 brane stack is at $r=0$. This wrapped M2 brane is the analogue of the global string that was discussed in [@Klebanov:2007cx] for the conifold theory in Type IIB. In our case the worldline of this point particle is linked by a circle, as it lives in a three dimensional spacetime. As we shall explain below, there are certain light fields/particles that have monodromies around this circle.
Consider three-form fluctuations of the form $$\label{fluctuateC3}
\delta C_3=A\wedge \beta~,$$ where $\beta$ is a two-form on the Ricci-flat Kähler manifold $X$, and $\star_8$ denotes the Hodge dual on $X$. Demanding that $A$ is a massless gauge field in the Minkowski three-space leads to the equations $$\label{betaeqn}
{\mathrm{d}}\beta=0\ ,\qquad {\mathrm{d}}(\, h\star_8\beta)=0~.$$ In particular $\beta$ is closed and hence defines a cohomology class in $H^2(X,{\mathbb{R}})$; we shall be interested in the case where this class is Poincaré dual to the two-cycle wrapped by the particle-like M2 brane. In three dimensions the gauge field $A$ is dual to a periodic scalar, which can be identified as the Goldstone boson of the spontaneous symmetry breaking. Indeed, the M2 particle is a magnetic source for this pseudoscalar. The pseudoscalar modes will therefore have unit monodromy around a circle linking the M2 particle worldline. As in [@Klebanov:2007cx], this Goldstone boson is expected to appear as a phase, through the Wess-Zumino action of the Euclidean M5 brane, in the baryonic condensate. We shall see that this is indeed the case.
It thus remains to construct appropriate two-forms $\beta$ satisfying (\[betaeqn\]) in the warped resolved backgrounds for $Q^{111}$. This will occupy us for the remainder of this subsection. It will turn out to be simpler to use the metric in form given in (\[Q111Cvetic\]), which we reproduce here for convenience: $$\begin{aligned}
{\mathrm{d}}s^2&=&{U}^{-1}{\mathrm{d}}\varrho^2+{U}\varrho\Big({\mathrm{d}}\psi+\sum_{i=1}^3 \cos\theta_i{\mathrm{d}}\phi_i\Big)^2+(l_2^2+\varrho)\,\Big({\mathrm{d}}\theta_2^2+\sin^2\theta_2{\mathrm{d}}\phi_2^2\Big)\nonumber \\ &&+(l_3^2+\varrho)\,\Big({\mathrm{d}}\theta_3^2+\sin^2\theta_3{\mathrm{d}}\phi_3^2\Big)+\varrho\, \Big({\mathrm{d}}\theta_1^2+\sin^2\theta_1{\mathrm{d}}\phi_1^2\Big)~,\end{aligned}$$ where $$U(\varrho)= \frac{3\, \varrho^3+4\, \varrho^2\, (l_2^2+l_3^2)+6\, l_2^2\, l_3^2\varrho}{6\,(l_2^2+ \varrho)\, (l_3^2+ \varrho)}~.$$ The constants $l_2$, $l_3$ are related to the constants $a,b$ in (\[resolvedQ111\]) via $a=4l_2^2$, $b=4l_3^2$.
### Harmonic forms: unwarped case
As a warm-up, we begin with the unwarped case in which the warp factor $h\equiv 1$. It is convenient to introduce the following vielbein e\_[\_i]{}=\_i ,e\_[\_i]{}=\_i ,g\_5=+ \_[i=1]{}\^3 \_i . A natural ansatz for two two-forms $\beta_i$, $i=2,3$, is then $$\beta_i=e_{\theta_i}\wedge e_{\phi_i}+{\mathrm{d}}(f_i\, g_5)~, \qquad i=2, 3~,$$ where $f_i$ is a function of the radial coordinate $\varrho$. Without loss of generality we will focus on the case $i=2$ and, in order not to clutter notation, drop the subscript in $f_i$. After some algebra one finds the following equation for $f$ $$\begin{aligned}
\label{eqnforf}
&&\frac{\varrho\, (\varrho+l_3^2)}{(\varrho+l_2^2)}(1-f)- \frac{(\varrho+l_2^2)\, (\varrho+l_3^2)}{\varrho}f- \frac{(\varrho+l_2^2)\, \varrho}{(\varrho+l_3^2)}f {\nonumber}\\
&&+\partial_{\varrho}\Big(\varrho\,(\varrho+l_2^2)\, (\varrho+l_3^2)\, \partial_{\varrho}f\Big)=0~.\end{aligned}$$ Since the corresponding $\delta C_3$ fluctuation in (\[fluctuateC3\]) couples to the M2 brane wrapped over the $\mathbb{CP}^1$ at $\varrho = 0$, $\beta$ should approach here the volume form of the finite sized $\mathbb{CP}^1$. For this we require the boundary condition that $f(\varrho = 0)$ vanishes. On the other hand, as we will see later, in the large $\varrho$ region $\beta$ should asymptote to a harmonic two-form $\omega_2$ on the singular cone.
It turns out that equation (\[eqnforf\]) can be solved exactly. Using the above boundary conditions to fix integration constants, we have $$f=\frac{\varrho^2+\frac{3}{2}\, l_3^2\, \rho}{3\, (\varrho+l_2^2)\, (\varrho+l_3^2)}~.$$ Given this $f$, at small $\varrho$ one can check that $\beta$ asymptotes to $e_{\theta_2}\wedge e_{\phi_2}$, which is the volume form of the submanifold that the M2 brane is wrapped on, while at large $\varrho$ instead $\beta$ asymptotes to a harmonic two-form $\omega_2$ on the singular cone which is simply the pull-back of a harmonic two-form on $Q^{111}$.
### Harmonic forms: warped case
Let us now turn to the warped case. Since with the M2 brane stack at $\varrho=0$ the warp factor is then a function of $(\varrho, \theta_2, \theta_3)$, it is natural to consider the following ansatz for the two-form $\beta$: $$\beta=e_{\theta_2}\wedge e_{\phi_2}+{\mathrm{d}}\left(\, f_0\, g_5+\sum_{i=2}^3 f_i\, e_{\phi_i}\, \right)~,$$ where now $f_{\mu}=f_{\mu}(\varrho,\,\theta_2,\,\theta_3)$, and $\mu=0,2$ or $3$.
The second equation in (\[betaeqn\]) implies that the $f_{\mu}$ must satisfy $$\begin{aligned}
&&\partial_{\rho}\Big(\frac{h\,\sqrt{\det g}\,U(\rho)}{\ell_j^2+\rho}\partial_{\rho} f_j \Big)+\sum_{i=2}^3 \frac{1}{\sin{\theta_i}} \partial_{\theta_i}\Big( \frac{h\,\sqrt{\det g}\,\sin{\theta_i}}{(\ell_j^2+\varrho)(\ell_i^2+\varrho)} \partial_{\theta_i}f_j \Big)-\\ {\nonumber}&&-\partial_{\theta_j}\Big( \frac{h\,\sqrt{\det g}\,f_0}{(\ell_j^2+\varrho)^2}\Big)+\partial_{j}\Big( \frac{h\,\sqrt{\det g}\cot{\theta_j}}{(\ell_j^2+\varrho)^2}\Big)f_j +\partial_{\theta_j}\Big( \frac{h\,\sqrt{\det g}}{(\ell_j^2+\varrho)^2}\Big)\delta_{j,2}=0~,\end{aligned}$$ for $\mu=j$, and $$\begin{aligned}
&&\partial_{\varrho}\Big(h\, \sqrt{\det g}\, \partial_{\varrho}f_0\Big)+\sum_{i=1}^3\frac{1}{\sin\theta_i}\partial_{\theta_i}\Big(\frac{U^{-1}(\varrho)\, \sqrt{\det g}\, h}{(\varrho+l_i^2)}\, \sin\theta_i\,\partial_{\theta_i}\, f_0\Big) \\ \nonumber
&& +\sum_{i=2}^3\frac{\sqrt{ \det g}\, h}{(\varrho+l_i^2)^2\, \sin\theta_i}\partial_{\theta_i}\Big(\sin\theta_i\, f_i\Big)-\sum_{i=1}^3 \frac{\sqrt{\det g}\, h}{(\varrho+l_i^2)^2}f_0+\frac{h\, \sqrt{\det g}}{(\varrho+l_2^2)^2}=0~,\end{aligned}$$ for $\mu=0$. Note that $l_1\equiv 0$ and $\sqrt{\det g}=\varrho\, (\varrho+l_2^2)\, (\varrho+l_3^2)$.
We have three equations for three functions $f_{\mu}$, so we expect this system to contain the desired solution for $\beta$. Furthermore, if we consider the unwarped case where $h\equiv 1$ we can consistently set $f_i=0$ and assume $f_0=f_0(\varrho)$. In this case the second equation reduces to $$\begin{aligned}
&&\partial_{\varrho}\Big(\sqrt{\det g}\, \partial_{\varrho}f_0\Big) -\sum_{i=2}^3 \frac{\sqrt{\det g}}{(\varrho+l_i^2)^2}f_0+\frac{\sqrt{\det g}}{(\varrho+l_2^2)^2}=0~.\end{aligned}$$ This is precisely the equation obtained above. Furthermore, these equations can be seen to arise from minimizing the action &&I=\_X \_8h= \_0\^ \_2 \_0\^ \_3 \_0\^ h . With the boundary conditions above, one can check that $I$ is finite. We shall give a more general argument for existence of this two-form later in this section.
Going back to the original physical motivation, the warped harmonic form $\beta$ allows one to construct a three-form fluctuation $\delta C=A\wedge \beta$ that satisfies the linearized SUGRA equations. Consider the Hodge dual $$\delta G_{7}=\star_3 {\mathrm{d}}A\wedge h\star_8\beta~.$$ As noted above, the three-dimensional gauge field $A$ is dual to a periodic scalar via $\star_3{\mathrm{d}}A={\mathrm{d}}p$. Thus we can write $$\delta G_{7}={\mathrm{d}}p\,\wedge h\star_8\beta={\mathrm{d}}\left(p\, h\star_8\beta\right)~,$$ thus obtaining a local form for the six-form fluctuation. At very large $\varrho$, $\beta$ can be approximated to leading order by $$\beta\sim \frac{2}{3}\, e_{\theta_2}\wedge e_{\phi_2} - \frac{1}{3}\, e_{\theta_1}\wedge e_{\phi_1}- \frac{1}{3}\, e_{\theta_3}\wedge e_{\phi_3}~,$$ so that $$\delta C_{6}\supset p \,\,\frac{{\mathrm{d}}\varrho}{\varrho^2}\wedge {\mathrm{d}}\psi \wedge e_{\theta_1}\wedge e_{\phi_1}\wedge e_{\theta_2}\wedge e_{\phi_2}~.$$ This form of the local potential couples to the baryonic M5 brane through the Wess-Zumino term, thus reinforcing the identification of the scalar $p$ as the phase of the baryonic condensate and Goldstone boson. Furthermore, at large $\varrho$ we can also write $$\delta G_{7}\supset {\mathrm{d}}(\, \varrho^{-1}\, {\mathrm{d}}p\wedge {\mathrm{d}}\psi\wedge e_{\theta_1}\wedge e_{\phi_1}\wedge e_{\theta_2}\wedge e_{\phi_2})~.$$ Then, following [@Klebanov:2007cx], we note the appearance of the volume form of the submanifold wrapped by the baryonic M5 brane, with the appropriate decay for a conserved current in three dimensions of dimension 2. This motivates the identification $$\langle J^B_{\mu}\rangle \sim \partial_{\mu}p~.$$
General warped resolved Calabi-Yau backgrounds {#sec:generalwarped}
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Much of our discussion of the resolved $Q^{111}$ backgrounds can be extended to general warped resolutions of Calabi-Yau cones. In the remainder of this section we describe what is known about such generalizations. In particular in the next subsection we present a novel method for computing M5 brane condensates in such backgrounds, or more generally the worldvolume actions of branes in warped Calabi-Yau geometries.
### Gravity backgrounds
As for the $Q^{111}$ case we are interested in M-theory backgrounds of the form \[background2\] s\^2\_[11]{} &=& h\^[-2/3]{} s\^2(\^[1,2]{}) + h\^[1/3]{} s\^2(X) ,\
G &=& \^3 x h\^[-1]{} , where $X$ is a Ricci-flat Kähler eight-manifold that is asymptotic to a cone metric over some Sasaki-Einstein seven-manifold $Y$. Placing $N$ spacetime-filling M2 branes at a point $y\in X$ leads to the warp factor equation \[green\] \_x h\[y\] = \^8(x-y) . Here $\Delta h = {\mathrm{d}}^* {\mathrm{d}}h = - \nabla^i\nabla_i h$ is the scalar Laplacian of $X$ acting on $h$. Thus $h[y](x)$ is simply the Green’s function on $X$. More generally, one could pick different points $y_i\in X$, with $N_i$ M2 branes at $y_i$, such that $\sum_i N_i=N$. Then $h[\{y_i,N_i\}](x)$ will be a sum of Green’s functions, weighted by $N_i$. We shall regard this as an obvious generalization. There are thus two steps involved in constructing such a solution: choose a Calabi-Yau metric on $X$, and then solve for the Green’s function. If the latter is chosen so that it vanishes at infinity then the SUGRA solution (\[background2\]) will be asymptotically $AdS_4 \times Y$ with $N$ units of $G_7$ flux through $Y$.
If $Y$ is a Sasaki-Einstein manifold then $\mathcal{C}(Y)$ defines an isolated singularity at the tip of the cone $r=0$. We may then take a resolution $\pi:X\rightarrow \mathcal{C}(Y)$, which defines our manifold $X$ as a complex manifold. The map $\pi$ is a biholomorphism of complex manifolds on the complement of the singular point $\{r=0\}$, so that in $X$ the singular point is effectively replaced by a higher-dimensional locus, which is called the exceptional set. We require that the holomorphic $(4,0)$-form on $\mathcal{C}(Y)$ extends to a smooth holomorphic $(4,0)$-form on $X$. Such resolutions are said to be “crepant”, and they are not always guaranteed to exist, even for toric singularities. In the latter case one can typically only partially resolve so that $X$ has at worst orbifold singularities. Having chosen such an $X$ we must then find a Calabi-Yau metric on $X$ that approaches the given cone metric asymptotically. Fortunately, mathematicians have very recently proved that you can *always* find such a metric. Essentially, this is a non-compact version of Yau’s theorem with a “Dirichlet” boundary condition, where we have a fixed Sasaki-Einstein metric at infinity on $\partial X=Y$ and ask to fill it in with a Ricci-flat Kähler metric. There are a number of papers that have developed this subject in recent years [@crep1; @crep2; @crep3; @crep4; @crep5; @crep6], but the most recent [@goto; @vanC] prove the strongest possible result: that in each Kähler class in $H^2(X,{\mathbb{R}})\cong H^{1,1}(X,{\mathbb{R}})$ (see [@crep6]) there is a unique Calabi-Yau metric that is asymptotic to a fixed given Sasaki-Einstein metric on $Y=\partial X$. Note this result *assumes* the existence of the Sasaki-Einstein metric – it does not prove it.
The crepant (partial) resolutions of toric singularities are well understood, being described by toric geometry and hence fans of polyhedral cones. The extended Kähler cone for such resolutions is known as the GKZ fan, or secondary fan. The fan is a collection of polyhedral cones living in ${\mathbb{R}}^{b_2(X)}$, glued together along their boundaries, such that each cone corresponds to a particular choice of topology for $X$. Implicit here is the fact that $b_2(X)$ is independent of which topology for $X$ we choose. A point inside the polyhedral cone corresponding to a given $X$ is a Kähler class on $X$. The boundaries between cones correspond to partial resolutions, where there are further residual singularities, and there is a topology change as one crosses a boundary from one cone into another. The GKZ fan for $Y=Q^{111}$ was described in Figure \[fig:GKZ1\]. If we combine this description with the above existence results, we see that the GKZ fan is in fact classifying the space of resolved asymptotically conical Calabi-Yau metrics.
Having chosen a particular resolution and Kähler class, hence metric, we must then find the warp factor $h$ satisfying (\[green\]). This amounts to finding the Green’s function on $X$, and this always exists and is unique using very general results in Riemannian geometry. A general discussion in the Type IIB context may be found in [@Martelli:2007mk], and the comments here apply equally to the M-theory setting. In the warped metric (\[background2\]) the point $y\in X$ is effectively sent to infinity, and the geometry has two asymptotically $AdS_4$ regions: one near $r=\infty$ that is asymptotically $AdS_4\times Y$, and one near to the point $y$, which if $y$ is a smooth point is asymptotically $AdS_4\times S^7$. For further discussion, see [@Martelli:2007mk; @Martelli:2008cm; @Benishti:2009ky].
If one places the $N$ M2 branes at the same position $y\in X$, the moduli space is naturally a copy of $X$. The $b_2(X)$ Kähler moduli are naturally complexified by noting that $H^6(X,Y,{\mathbb{R}})\cong H_2(X,{\mathbb{R}})\cong {\mathbb{R}}^{b_2(X)}$ by Poincaré duality, and that this group classifies the periods of $C_6$ through six-cycles in $X$, which may either be closed or have a boundary five-cycle on $Y=\partial X$. More precisely, taking into account large gauge transformations leads to the torus $H^2(X,Y,{\mathbb{R}})/H^2(X,Y,{\mathbb{Z}})\cong U(1)^{b_2(X)}$. Altogether this moduli space of SUGRA solutions should be matched to the full moduli space of the dual SCFT. At least for toric $X$ one can prove quite generally via an exact sequence that $b_2(X)=b_2(Y)+b_6(X)$, where $b_6(X)$ is also the number of irreducible exceptional divisors in the resolution. In toric language, this is the number of internal lattice points in the toric diagram. We shall discuss such examples in [§\[sec:5\]]{}: the presence of calibrated six-cycles is expected to lead to M5 brane instanton corrections in these backgrounds.
### Harmonic two-forms
Recall we are also interested in fluctuations of the form $$\delta C_3=A\wedge \beta~,$$ where $A$ leads to a massless gauge field in the Minkowski three-space if $$\label{betaagain}
{\mathrm{d}}\beta=0\ ,\qquad {\mathrm{d}}(\, h\star_8\beta)=0~.$$ For trivial warp factor $h\equiv 1$ this just says that $\beta$ is harmonic. It is a general result that if we also impose that $\beta$ is $L^2$ normalizable, or equivalently that $A$ has finite kinetic energy in three dimensions, then such forms are guaranteed to exist and are in 1-1 correspondence with $H^2(X,Y,{\mathbb{R}})\cong H_6(X,{\mathbb{R}})$ [@hausel]. Thus there are always $b_6(X)$ $L^2$ normalizable harmonic two-forms $\beta$ in the unwarped case.
However, this case isn’t the physical case for applications to $AdS/CFT$. Instead we should look for solutions to (\[betaagain\]) where $h$ is the Green’s function on $X$. Again, fortunately there are mathematical results that we may appeal to to guarantee existence of such forms. These are again described in the Type IIB context in [@Martelli:2008cm]. In the warped case $\beta$ is harmonic with respect to the metric $h^{1/2} {\mathrm{d}}s^2(X)$. This manifold has an asymptotically conical end with boundary $S^7$ (or more generally $Y_{IR}$ if the M2 brane stack is placed at a singular point with horizon $Y_{IR}$), and an isolated conical singularity with horizon metric $\tfrac{1}{4}{\mathrm{d}}s^2(Y)$. The number of $L^2$ harmonic two-forms on such a space is in fact known, and is $b_2(X)$. To see this requires combining a number of mathematical results that are described in [@Martelli:2008cm]. In particular, since $b_2(X)=b_2(Y)+b_6(X)$ there is a corresponding harmonic form, and hence Goldstone mode, for each of the $b_2(Y)$ baryonic $U(1)$ symmetries. Indeed, these $b_2(Y)$ harmonic forms can be seen to asymptote to the harmonic two-forms on $Y$ at $r=\infty$. Thus the analysis at the end of [§\[sec:wrapped\]]{} carries over in much more general backgrounds. We shall analyse the asymptotics of the $b_6(X)$ unwarped $L^2$ harmonic forms in more detail in [§\[sec:5\]]{}, where they will be given a very different interpretation.
Baryonic condensates: M5 branes in general warped geometries {#sec:M5general}
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As discussed in [§\[sec:baryons\]]{}, M5 branes wrapped on five-manifolds $\Sigma_5\subset Y$ lead, with appropriate choice of quantization of the gauge fields in $AdS_4$, to scalar operators in the dual SCFT that are charged under the $U(1)^{b_2(Y)}$ baryonic symmetry group. We have already described how to compute the VEV of such an operator in the (partially) resolved $Q^{111}$ background. More generally one should compute the action of a Euclidean M5 brane which is wrapped on a minimal six-submanifold $D\subset X$ with boundary $\partial D=\Sigma_5$. Similar computations, in some specific examples, have been performed in [@Baumann:2006th]. In this section we explain how this Euclidean action may be computed exactly, in general, in the case where $D$ is a divisor. This is essentially a technical computation that may be skipped if the reader is not interested in the details: the final formula is (\[thefinalcountdown\]).
Let suppose that we are given a warped background (\[background2\]), where $X$ has an asymptotically conical Calabi-Yau metric and the warp factor satisfying (\[green\]) is given, with a specific choice of point $y\in X$ where the stack of $N$ M2 branes are located. We would like to compute (-T\_5\_D h\[y\] \^6x) . Here $T_5=2\pi/(2\pi \ell_p)^6$ is the M5 brane tension, and the integrand is the worldvolume action (in the absence of a self-dual two-form). Thus $g_{D}$ denotes the pull-back of the unwarped metric to the worldvolume $D$. We also assume that $D$ is a divisor, in order to preserve supersymmetry. Then $D$ is also minimal and the integral is \_D h\[y\] \^6 x = \_D h\[y\] , where $\omega$ denotes the Kähler form of the unwarped metric, pulled back to $D$.
Before beginning our computation, we note that on a Kähler manifold the scalar Laplacian $\Delta h = {\mathrm{d}}^* {\mathrm{d}}h$ can be written as \[KahlerLaplace\] h = -\^c h = -2|h . Here ${\mathrm{d}}^c \equiv I\circ {\mathrm{d}}= {\mathrm{i}}(\bar{\partial}-\partial)$, where $I$ is the complex structure tensor. The contraction $\omega\lrcorner {\mathrm{d}}{\mathrm{d}}^c h$ is then in local complex coordinates \^c h = 4\^[i|[j]{}]{} = -h , where =\_[i|[j]{}]{}z\^i|[z]{}\^[|[j]{}]{} .
For simplicity we shall study the case in which $D$ is described globally by the equation $D=\{f=0\}$, where $f$ is a global holomorphic function on $X$. This means that the homology class of $D$ is trivial, and hence in fact the wrapped M5 brane carries zero charge under $U(1)^{b_2(Y)}$. The cases studied in [@Baumann:2006th] are of this form. More generally, since $D$ is a complex divisor it defines an associated holomorphic line bundle $\mathcal{L}_D$ over $X$. Then we may take $D$ to be the zero set of a holomorphic section of $\mathcal{L}_D$, with a simple zero along $D$. To extend the computation below to this case would require combining the argument we give here with the arguments in [§\[sec:5\]]{}.
Thus, suppose that $f$ is a holomorphic function with a simple zero along $D$, and introduce the two-form \_D \^c |f| = - | |f|\^2 . Let us examine its properties. First, note that |f|\^2 = f + |[f]{} . Since $f$ is holomorphic, and thus $\bar{\partial}f =0$ by definition, this shows that away from the locus $f=0$, which is the divisor $D$, in fact $\eta_D=0$. On the other hand, locally along $D$ we can write $f = zg$ where $z$ is a local coordinate normal to $D$, and $g=g(z,w_1,w_2,w_3)$, where $w_1,w_2,w_3$ are local complex coordinates along $D$ and $g$ has no zero in this local chart. We may then write $z=r {\mathrm{e}}^{{\mathrm{i}}\theta}$, and note that \^c r = \^2(0) z|[z]{} . This is just the elementary statement that $(1/2\pi)\log r$ is the unit Green’s function in dimension two (the local transverse space to $D$). Thus we have shown that $\eta_D$ is zero away from $D$, and is a unit delta function supported along $D$.
Using these properties of $\eta_D$ we may hence write VT\_5\_D h\[y\] = \_X \^c |f| . Note in particular that $T_5 h[y]/2\pi$ is $N$ times the unit Green’s function (with unit delta function source), [*i.e.*]{} \_x () = \^8(x-y) . We then integrate by parts \[eq1\] V = \_[X=Y]{} \^c |f| - \_X () \^c |f| . We will deal with the boundary terms later, focusing for now on the integrals over $X$. First note that = - I() . holds for any one-form $\gamma$. Using this we may write V &=& -\_X () \^c |f| +\
&=& \_X \^c() |f| + . Here we have used that the metric is of course Hermitian, so that $g_X(\gamma,\delta)=g_X(I\gamma,I\delta)$. We then again integrate by parts \[eq2\] V = \_X \^c ()|f| + , where explicitly now = \_[X=Y]{} { \^c |f| - \^c () |f|} . In fact this boundary integral is divergent – a key physical point in interpreting it holographically. For now let us deal with the integral over $X$ in (\[eq2\]). Using (\[KahlerLaplace\]) we may write -\^c () = () = \^8(x-y) , and thus V = -\_X N \^8(x-y) |f| \^8 x = - N |f(y)|+ .
Let us turn now to the boundary terms. In order to render this finite, we cut off the integral at some large $r=r_c$, and write the boundary integral as \[bterms\] = \_[Y\_[r\_c]{}]{} . We require that $D$ is asymptotically conical, so that at large $r_c$ it approaches a cone over a compact five-manifold $\Sigma_5\subset Y$. In the cone geometry, a conical divisor with trivial homology class is specified as the zero set of a homogeneous function under $r\partial_r$. Thus we take |f| = Ar\^(1+…) , where $A$ is homogeneous degree zero ([*i.e.*]{} a function on $Y$) and the $\ldots$ are terms that go to zero as $r\rightarrow\infty$. Thus $f$ has asymptotic homogeneous degree $\lambda>0$. Now, the volume form on $Y$ is (Y)= , where $\eta={\mathrm{d}}^c \log r$ and $\omega_{\mathrm{cone}} = \frac{1}{2}{\mathrm{d}}(r^2 \eta)$ is the Kähler form on the cone over $Y$. Asymptotically, = \_ + O(r\^4) . The $O(r^4)$ follows since the leading correction to the cone metric is a harmonic two-form on $Y$, which is down by a factor of $r^{-2}$ relative to the cone metric. We also have = (1 + …) . (\[warping\]). Thus the first term in (\[bterms\]) is *convergent* and gives \_[r\_c]{} \_[Y\_[r\_c]{}]{} \^c |f| = (Y) = . Note here that the function $A$ does not contribute to the integral as it is independent of $r$, and thus $J(r\partial_r)\lrcorner {\mathrm{d}}^c \log A=0$. On the other hand, the second term in (\[bterms\]) is divergent, the leading divergent piece being $N\lambda \log r_c~.$ Provided the $\ldots$ terms in $|f|$ and $h$ fall off as $o(r^{-\epsilon})$, for some $\epsilon>0$, then in fact this is the only divergence (since any positive power of $r$ grows faster than $\log r$). There is also a finite part, namely $N\int_Y \log A/{\mathrm{vol}}(Y)$. However, the important point is that this depends only on asymptotic data.
Let us interpret this divergence. Suppose that the Sasaki-Einstein manifold $Y$ is quasi-regular, meaning that it is a $U(1)$ bundle over a Kähler-Einstein orbifold $Z$.[^9] Then asymptotically $f$ is, in its dependence on $Z$, a holomorphic section of $L^k$ for some integer $k\in{\mathbb{Z}}_{>0}$, where $L=K_Z^{-1/I}$ with $I=I(Z)\in{\mathbb{Z}}_{>0}$ being the orbifold Fano index of $Z$. Here $K_Z$ denotes the orbifold canonical bundle of $Z$, and $I$ is by definition the largest integer so that the root $L$ is defined. It follows that = , where $4=\dim_{\mathbb{C}}X$. The five-manifold $\Sigma_5\subset Y$ is then the total space of a $U(1)$ bundle over an orbifold surface $S\subset Z$, with the Poincaré dual to $S$ being represented by $c_1(L^k)=kc_1(Z)/I$. The Kähler-Einstein condition on $Z$ gives \[KE\] 8\[\_Z\] = 2c\_1(Z)H\^[1,1]{}(Z,) , where $\omega_Z=(1/2){\mathrm{d}}\eta$ denotes the Kähler form of $Z$.
Now, the conformal dimension of the operator dual to an M5 brane wrapped on $\Sigma_5$ is given by the general formula (\[\_5\]) = . In the quasi-regular case at hand, the length of the $U(1)$ circle cancels in the numerator and denominator and we can write this as (\[\_5\]) = = = . Here in the last step we used the Kähler-Einstein condition (\[KE\]). Thus we have the general result that the divergent part of the integral is Nr\_c = 2(\[\_5\]) r\_c = -(\[\_5\]) z , where we have changed to the usual $AdS_4$ coordinate $r_c^2 = 1/z$. Thus we see that the divergent part of the integral is precisely such that we can interpret its coefficient as the VEV of the operator $\mathcal{O}[\Sigma_5]$ in this background. This coefficient is, from the above computations, \[thefinalcountdown\] (-V\_) = |f(y)|\^N(--\_Y (|f|/r\^)) . This is an exact result for the VEV, or regularized exponential of the M5 brane action, in terms of the defining function $f$ of the divisor $D$. The integral is understood in the limit $Y=Y_{r_c}$ as $r=r_c\rightarrow \infty$, which is convergent as the integrand is independent of $r$ in this limit. Notice, in particular, that if one multiplies $f$ by a constant, this constant drops out of the formula, as it should.
Six-cycles and non-perturbative superpotentials {#sec:5}
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So far our discussion has mainly focused on the $Q^{111}$ theory and its warped resolved supergravity solutions. However, this solution is somewhat special in that the resolved Calabi-Yau manifolds are “small resolutions” – that is, there are no exceptional divisors, or six-cycles, in the resolved solution. An isolated toric Calabi-Yau four-fold singularity will typically have exceptional divisors when it is resolved.[^10] The irreducible components of these divisors are in 1-1 correspondence with the internal lattice points in the toric diagram, and a simple homology calculation shows that these generate the group $H_6(X,{\mathbb{R}})$ of six-cycles. This immediately raises the question of what is the $AdS/CFT$ interpretation of such six-cycles, as we mentioned briefly at the end of section \[sec:generalwarped\]. Again, in order to make our discussion concrete we will begin by focusing on a simple example, namely the cone over $Q^{222}$. This is a $\mathbb{Z}_2$ orbifold of $Q^{111}$ in which the free ${\mathbb{Z}}_2$ quotient is along the R-symmetry $U(1)$ fibre. Further details about the geometry of this manifold are contained in appendix \[sec:Q222-geometry\].
The $\mathcal{C}(Q^{222})$ theory
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A candidate dual field theory to $\mathcal{C}(Q^{222})$ was proposed in [@Franco:2009sp]. In fact there are (at least) two possible toric phases for this theory. The quivers of the two phases are
with superpotentials &&W\_I=[Tr]{} \_[ij]{} \_[mn]{} X\_[12]{}\^i X\_[23]{}\^m X\_[34]{}\^j X\_[41]{}\^n ,\
&&W\_[II]{} = [Tr]{}( \_[ij]{} \_[mn]{} X\_[32]{}\^i X\_[24]{}\^m X\_[43]{}\^[jn]{}-\_[ij]{} \_[mn]{} X\_[31]{}\^m X\_[14]{}\^i X\_[43]{}\^[jn]{}) . Following the same prescription for $\mathcal{C}(Q^{111})$, we will consider quantizations of the gauge fields in $AdS_4$ such that the gauge groups are $U(1)_G\times U(1)_{G-1}\times SU(N)^4$. As shown in [@Franco:2009sp], the moduli spaces of both phases give the desired cone over $Q^{222}$. Furthermore, with $SU(N)$ gauge groups we have two global $U(1)$ baryonic symmetries, which precisely match the $b_2(Q^{222})=2$ gauge fields we find in $AdS_4$ from the gravitational point of view. Thus, as far as the moduli space and baryonic symmetries are concerned, the $\mathcal{C}(Q^{222})$ case closely resembles its $\mathcal{C}(Q^{111})$ cousin.
However, inspection of the $\mathcal{C}(Q^{222})$ toric diagram in Figure \[fig:toricdiagramQ222\]
![The toric diagram for $\mathcal{C}(Q^{222})$.[]{data-label="fig:toricdiagramQ222"}](./toric_diagram_Q222)
shows an interior lattice point, signaling the possibility of blowing up a six-cycle. We shall discuss the geometry of such resolutions in more detail later in this section. However, this immediately raises the question of the field theory interpretation of this six-cycle. As we discuss in the next subsection, such six-cycles have been shown to be responsible for non-perturbative superpotentials in Calabi-Yau compactifications via wrapped Euclidean M5 branes. We are interested in such contributions to a non-perturbative superpotential in warped Calabi-Yau backgrounds. The warping here is induced by the back-reaction of point-like M2 brane sources on the Calabi-Yau.[^11]
In previous sections we have argued that, for toric singularities with no vanishing six-cycles, we can choose a set of boundary conditions which amounts to ungauging all but the $\mathcal{B}_{G-1},\, \mathcal{B}_G$ $U(1)$ symmetries. This gives rise to a field theory moduli space which is roughly a $({\mathbb{C}}^*)^{b_2(Y)}$ fibration over the mesonic space. Here the fibres are the global $U(1)$s we do not quotient by, and their corresponding D-terms that we do not impose. On the SUGRA side, the mesonic part of the moduli space arises from the $N$ M2 branes that are free to move in the geometry. The $b_2(Y)$ ${\mathbb{C}}^*$s are instead naturally the $b_2(Y)=b_2(X)$ Kähler classes plus the corresponding periodic scalars coming from modes of $C_3$ on associated two-cycles. More generally we have $b_2(X)=b_2(Y)+b_6(X)$, where $X$ is a Calabi-Yau resolution of $Y$. We can at this point consider the same operation for geometries with six-cycles, so that $b_6(X)\neq 0$. However, in this case the field theory moduli space seems to be too small. Let us consider the $\mathcal{C}(Q^{222})$ example. In field theory we still have a $({\mathbb{C}}^*)^2$ fibration over Sym$^N$ $\mathcal{C}(Q^{222})$ as moduli space, since there are again only 4 nodes in the quiver. On the other hand, the gravity side has 3 Kähler classes and corresponding periodic scalars coming from modes of $C_3$ on the two-cycles (see [§\[s:GKZ\]]{}). This immediately implies that the classical field theory moduli space cannot possibly match all the gravity solutions. However, as argued in the next subsections, geometries with exceptional six-cycles may have non-perturbative corrections which could potentially fix this mismatch.
Non-perturbative superpotentials from six-cycles
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The presence of exceptional six-cycles in warped resolved Calabi-Yau backgrounds raises the very interesting possibility that Euclidean M5 branes (EM5) may wrap such cycles. Indeed, the toric geometries under consideration do not have three-cycles, either in the boundary Sasaki-Einstein manifold $Y$, or in the Calabi-Yau resolution $X$ of $\mathcal{C}(Y)$. Thus there are no cycles on which to wrap Euclidean M2 branes, leaving only EM5 branes as possible instantonic branes in these backgrounds. More precisely, such EM5 branes may be wrapped on the irreducible components of the exceptional divisor in the Calabi-Yau resolution. Such cycles, being complex submanifolds, are automatically supersymmetric. Moreover, as already mentioned, the irreducible components are in 1-1 correspondence with the generating homology classes in $H_{6}(X,{\mathbb{R}})$.
A very similar situation was considered in [@Witten:1996bn], where compactifications of M-theory to three dimensions on a Calabi-Yau four-fold were discussed. In that reference the rôle of Euclidean instantonic M5 branes, and their possible contribution to non-perturbative superpotentials, was studied in detail. In order to generate such contributions, the number of zero modes, which includes the Goldstinos of the SUSYs broken by the brane, must be appropriate to saturate the superspace measure. In particular, such instantons must wrap cycles without infinitesimal holomorphic deformations, since the superpartners of the deformation moduli would provide additional fermionic zero modes. In [@Witten:1996bn] it was shown that the appropriate zero mode counting in the case of an M5 brane wrapping a divisor $D$ in this set-up requires the necessary condition that \[Todd\] (D,\_D)\_[i=0]{}\^3 (-1)\^i H\^i(D,\_D)=1 . \[arithmetic-genus\] This is the *arithmetic genus* of $D$. Assuming this necessary condition is satisfied, the structure of the non-perturbative superpotential generated by an EM5 brane also requires one to understand its dependence on the Kähler moduli. As explained in [@Witten:1996bn], the dependence on these Kähler moduli is known exactly, being encoded entirely in the semi-classical term ${\mathrm{e}}^{-V + {\mathrm{i}}\phi}$. Here $V$ denotes the volume of the six-cycle, while $\phi$ is the expected linear multiplet superpartner to $V$ and is given by the period of $C_6$ through $D$. This latter structure is determined from holomorphy of the superpotential.
It is well-known that for *any* smooth compact toric manifold the arithmetic genus in (\[Todd\]) is indeed equal to 1. This suggests that EM5 branes would typically generate non-perturbative contributions to the superpotential in such cases. However, at this point we should recall that the situation in [@Witten:1996bn] is slightly different from the one at hand. Firstly, our Calabi-Yau four-fold is non-compact, so that gravity is decoupled from the point of view of reduction to Minkowski three-space. Secondly, our set-up contains also point-like M2 branes, and moreover in the warped solutions the back-reaction of these M2 branes is also included. This leads to the asymptotically $AdS_4$ backgrounds discussed in section \[sec:FR\]. It should however be possible to extend the analysis of [@Witten:1996bn] to such warped cases.
Let us first briefly discuss a similar situation in the more controlled Type IIB scenario. In that case one can consider a Calabi-Yau three-fold singularity with colour and fractional D branes wrapping the collapsed cycles at the singularity, leading to a four-dimensional $\mathcal{N}=1$ SUSY field theory at the singularity. In addition one can consider instantonic Euclidean Ep branes. The various types of strings stretching between these branes can give rise to non-perturbative contributions to the superpotential of the field theory at the Calabi-Yau singularity. Exactly as for the M-theory case, in order for this non-perturbative superpotential to be generated at all the right number of zero modes must be present. An important remark here is that the Ep-Ep sector sees the full $\mathcal{N}=2$ Calabi-Yau three-fold background, thus generically leading to too many zero modes to saturate the $\mathcal{N}=1$ superspace measure. Therefore in order for a non-perturbative superpotential to be generated, some method of eliminating these extra zero modes is required. On the other hand, the situation in M-theory is very different since the colour M2 branes do not break any further the SUSYs of the Calabi-Yau four-fold background. From this point of view, we then expect EM5 instantons to generically contribute to non-perturbative corrections to the superpotential in three dimensions.
Nevertheless, the structure and interpretation of such corrections is far from clear. We can think of the gravitational background as a warped Calabi-Yau four-fold compactification, albeit one which is asymptotically an $AdS_4$ background. As such, we expect to be able to promote all moduli, both Kähler and those related to the positions of the M2 branes in the colour stack, to spacetime fields in the Minkowski three-space. These will be dynamical fields provided the fluctuations are normalizable in the warped metric; at least for the Kähler moduli this is expected to be the case, as discussed in the Type IIB context in [@Martelli:2008cm]. However, finding an ansatz for such a reduction is far from trivial, and at the time of writing there is no complete proposal for such an ansatz which would allow one to compute the precise form of the non-pertubative superpotential for these modes. The most recent paper on this subject is [@Frey:2008xw], where the authors consider only the universal Kähler modulus in a warped compactification. On the other hand, following the more controlled Type IIB case, one might expect that computing the warped volume of the Euclidean brane is the dual “closed membrane channel” of the picture above, described in terms of M2 branes in the blown up Calabi-Yau four-fold in the presence of the EM5 branes. In the IIB case it has been explicitly checked [@Baumann:2006th] in some simple situations how the computation of open string diagrams in the relevant sector [@Berg:2004ek; @Berg:2005ja] can be reproduced though the computation of warped volumes, which are then interpreted as a resummation of such open string diagrams. Of course, it should be stressed that in the M-theory scenario at hand this can be taken only as a heuristic picture. In any case, one would expect that a general *holographic* interpretation of the superpotential should also be available, since the warped background is asymptotically $AdS_4$. One natural suggestion is that this might come from considering the boundary behaviour of the six-form fluctuation sourced by the EM5 branes. We leave a more complete investigation of these issues for further work. Instead in this paper we focus on computing the warped volumes of the EM5 branes as a function of the moduli. Understanding precisely how this is related to non-perturbative corrections in these warped resolved geometries will require further work.
EM5 instantons in the resolved $Q^{222}$ background {#s:GKZ}
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There is a unique resolution of $\mathcal{C}(Q^{222})$ where one blows up the six-cycle corresponding to the internal lattice point in Figure \[fig:toricdiagramQ222\]. This gives a Calabi-Yau four-fold $X$ which is the total space of the canonical bundle $\mathcal{O}(-2,-2,-2)\rightarrow\mathbb{CP}^1\times\mathbb{CP}^1\times\mathbb{CP}^1$. There is a Kähler class for each factor in the zero section exceptional divisor $(\mathbb{CP}^1)^3$, leading to a GKZ fan which is $({\mathbb{R}}_{\geq 0})^3$, as shown in Figure \[fig:GKZ2\].
![The GKZ fan for $Q^{222}$ is $({\mathbb{R}}_{\geq 0})^3$. The axes $\zeta_a$, $a=1,2,3$, may be identified with the Kähler classes of each factor in $\mathbb{CP}^1\times\mathbb{CP}^1\times\mathbb{CP}^1$, or equivalently the FI parameters in the GLSM.[]{data-label="fig:GKZ2"}](./GKZ2crop)
By the general theorem mentioned earlier we know that there will be a unique Ricci-flat Kähler metric, which is asymptotic to the cone metric over $Q^{222}$, for each choice of Kähler class. Again, in this case one can write these metrics explicitly: $$\begin{aligned}
{\mathrm{d}}s^2&=&\kappa(r)^{-1}{\mathrm{d}}r^2+\kappa(r)\frac{r^2}{16}\Big({\mathrm{d}}\psi+\sum_{i=1}^3 \cos\theta_i {\mathrm{d}}\phi_i\Big)^2+\frac{(2a+r^2)}{8}\Big({\mathrm{d}}\theta_2^2+\sin^2\theta_2 {\mathrm{d}}\phi_2^2\Big)\nonumber \\ &&+\frac{(2b+r^2)}{8}\Big({\mathrm{d}}\theta_3^2+\sin^2\theta_3 {\mathrm{d}}\phi_3^2\Big)+\frac{r^2}{8}\Big({\mathrm{d}}\theta_1^2+\sin^2\theta_1 {\mathrm{d}}\phi_1^2\Big)~,\end{aligned}$$ where now $$\kappa(r)=\frac{r^8+\frac{8}{3}\, (a+b)\, r^6+8\, a\,b \,r^4-16\, k}{r^4\, (2\,a+r^2)\,(2\,b+r^2)} \ .$$ Here $a$, $b$ and $k$ are arbitrary constants, and correspond to the choice of Kähler classes $\zeta_a$, $a=1,2,3$. In particular, setting $a=b=0$ implies that all three $\mathbb{CP}^1$s have the same volume, and the metric simplifies considerably. In this case it is convenient to define $r_\star^8=16k$, so that the exceptional divisor is at the radial position $r=r_\star$. For further details about these metrics we refer the reader to appendix \[sec:D\].
Again, it is also possible to solve explicitly for the warp factor for these metrics. In the simplified case with $a=b=0$, one can place the stack of $N$ M2 branes at an arbitrary radial position $r=r_0\geq r_\star$ and solve for the Green’s function. Again, we refer the reader to appendix \[sec:D\] for details of this warp factor. From now on we focus exclusively on the case $a=b=0$, where $\zeta_1=\zeta_2=\zeta_3$ is parametrized by the radius $r_\star>0$.
### Warped volumes
We are interested in computing the warped volume of the compact exceptional divisor $(\mathbb{CP}^1)^3$ at $r=r_{\star}$, with the stack of $N$ M2 branes at the position $y=(r_0,\xi_0)$. Here $\xi_0$ denotes the point in the copy of $Q^{222}$ at radius $r=r_0$. Thus we define $$S=T_5\, \int_D\, \sqrt{\det g_D}\, h\, {\mathrm{d}}^6 x$$ where $D$ is the exceptional divisor and $h$ is the (pull back of the) warp factor. Here the latter is given by the expression (\[general-warp\]) in terms of the mode expansion discussed in appendix \[sec:D\]. The determinant of the metric pulled back to the divisor is simply $$\sqrt{\det g_D}=\frac{r_{\star}^6}{8^3}\, \sin\theta_1\, \sin\theta_2\, \sin\theta_3~.$$ After subtituting these results into the worldvolume action one obtains $$S=\frac{{T_5}\,r_{\star}^6}{8^3}\, \sum_I\, Y_I(\xi_0)^*\,\psi_I(r_{\star})\,\int_D\, \sin\theta_1\, \sin\theta_2\, \sin\theta_3 \, Y_I(\xi)\, {\mathrm{d}}^6 x~.$$ Explicitly, the integral reads $$\begin{aligned}
\int_D\, \sin\theta_1\, \sin\theta_2\, \sin\theta_3 \, Y_I(\xi)\, {\mathrm{d}}^6 x=\mathcal{C}_{I}\,\prod_{i=1}^3 \int_{\theta_i=0}^\pi\int_{\phi_i=0}^{2\pi} \sin\theta_i\, {\mathrm{e}}^{{\mathrm{i}}\, m_i\,\phi_i}J_{0,\, l_i,\, m_i}(\theta_i)\, {\mathrm{d}}\theta_i\, {\mathrm{d}}\phi_i~.\end{aligned}$$ Here $\mathcal{C}_I$ is a normalization constant that ensures the mode $Y_I$ has unit norm. The $\phi_i$ integrals vanish unless $m_i=0$. Then $J_{0,\,l_i,\, 0}(\theta_i)$ reduces to a Legendre polynomial ${\rm P}_{l_i}(\cos\theta_i)$, so that $$\int_{\theta_i=0}^\pi \sin\theta_i\, {\rm P}_{l_i}(\cos\theta_i)\, {\mathrm{d}}\theta_i=2\, \delta_{l_i,\,0}~.$$ Therefore $$\int_D\, \sin\theta_1\, \sin\theta_2\, \sin\theta_3 \, Y_I(\xi)\, {\mathrm{d}}^6 x=2^6\, \pi^3\, \mathcal{C}_{0}\,\prod_{i=1}^3 \delta_{l_i,\,0}\,\delta_{m_i,\,0}~.$$ Subtituting this back into the expression for $S$ one finds $$S=\frac{2^6\, \pi^3\,{T}_5\,r_\star^6}{8^3}\, |\mathcal{C}_0|^2 \,\psi_{0,0,0}(r_{\star})~.$$ From , (\[warp-Q222-II\]) and (\[warp-Q222-III\]) we have $$\psi_{0,0,0}(r_{\star})=\frac{1}{r_0^2}\, _2F_1\Big(\frac{3}{4},\, 1,\, \frac{7}{4},\Big(\frac{r_{\star}^2}{r_0^2}\Big)^4\Big)\, , \quad\, |\mathcal{C}_0|^2=\frac{2^9\, \pi^2\, N\, \ell_p^6}{3}~.$$ Finally, substituting the explicit M5 brane tension results in the warped volume $$S=\frac{2\,N}{3}\,\frac{r_{\star}^6}{r_0^6}\, _2F_1\Big(\frac{3}{4},\, 1,\, \frac{7}{4},\Big(\frac{r_{\star}^2}{r_0^2}\Big)^4\Big)
\label{Q222-action}~.$$ Note that for $r_0 \rightarrow r_{\star}$ we get $S\sim -\frac{N}{2}\,\log(r_0-r_{\star})=-\log\,\rho^N$, where for the last equality we used (\[gamma-equ\]) and (\[gamma-equ-II\]).
### The $L^2$ harmonic two-form
The key observation in this subsection is that the warped volume (\[Q222-action\]) of the exceptional divisor is closely related to the $L^2$ normalizable two-form $\beta$ which is Poincaré dual to the six-cycle. The claim is that the precise relation between the two is (for $N=1$) =| S . \[beta-S\] Here the derivatives are regarded as acting on the coordinates of the point $y=(r_0,\xi_0)$, which recall is the location of the stack of M2 branes. We shall prove this claim in full generality in [§\[s:general-V\]]{}. Here we first prove it in the current explicit example, where it is convenient to use the coordinate system in section \[coordinates\]. From we see that $r_0^8=r_{\star}^8+\rho^4$, so in the new coordinate system we can write (setting $N=1$) $$S=\frac{2}{3}\,\frac{r_{\star}^6}{(\rho^4+r_{\star}^8)^{\frac{3}{4}}}\, _2F_1\Big(\frac{3}{4},\, 1,\, \frac{7}{4},\frac{r_{\star}^8}{\rho^4+r_{\star}^8}\Big)~.$$ After some algebra it can be shown that $$(\bar{\partial}-\partial)S={\mathrm{i}}\,\frac{r_{\star}^6}{(\rho^4+r_{\star}^8)^{{3}/{4}}}\,g_5~,$$ where g\_5=+\_[i=1]{}\^3\_i \_i , and therefore (\[beta-S\]) reads $$\beta=\partial \bar{\partial}\,\left(\frac{1}{\pi \, {\mathrm{i}}}\,S\right)=\frac{1}{2}{\mathrm{d}}(\bar{\partial}-\partial)\left(\frac{1}{\pi \, {\mathrm{i}}}\,S\right)=\frac{1}{2 \pi}{\mathrm{d}}\left(\frac{r_{\star}^6}{(\rho^4+r_{\star}^8)^{{3}/{4}}}\,g_5\right)~.$$ Going back to the original coordinates we then have $$\beta=\frac{1}{2 \pi}\,{\mathrm{d}}\left(\frac{r_{\star}^6}{r_0^6}\,g_5\right)~.$$ This is easily checked to be a harmonic two-form with respect to the unwarped metric, and also $L^2$ with respect to this metric. To see that $\beta$ is indeed Poincare dual to the six-cycle, we choose the following closed form on $X=\mathcal{O}(-2,-2,-2)\rightarrow (\mathbb{CP}^1)^3$ $$\mu = \frac{\sin\,\theta_1\,\sin\,\theta_2\,\sin\,\theta_3}{2^6 \, \pi^3} {\mathrm{d}}\theta_1 \wedge {\mathrm{d}}\phi_1 \wedge {\mathrm{d}}\theta_2 \wedge {\mathrm{d}}\phi_2 \wedge {\mathrm{d}}\theta_3 \wedge {\mathrm{d}}\phi_3~.$$ It is easy to see that $$\int_D\,\mu = 1 \, , \quad \, \int_X\,\beta\,\wedge\,\mu = 1~.$$ Hence $\beta$ is $L^2$ normalizable and Poincaré dual to the divisor, as claimed.
### Critical points
Formally, the superpotential that is induced by the instanton action that we have calculated in the previous subsections is given by \[supQ222\] W=\^[-S]{}= , where $S$ is given in (\[Q222-action\]). This is something of a formal statement, since in reality what we have computed is the on-shell Euclidean action of the wrapped M5 brane as a function of the moduli of the SUGRA background. Here essentially $r_\star$ is a Kähler modulus, while $r_0$ is a modulus associated to the position of the stack of M2 branes. On the other hand, the superpotential should be a function of the corresponding spacetime fields in Minkowski space, obtained by promoting these moduli to dynamical fields. In the unwarped case there is essentially no distinction between the two, but in the warped case it is not known how to do this at present, and the situation is much less clear. Note also that we have only computed the real part of $S$, and hence absolute value of $W$. Thus the best we can do is to examine the critical points of ${\mathrm{e}}^{-S}$ interpreted directly as a superpotential on the SUGRA moduli space. It is straightforward to compute \_[r\_0]{}S= . For $r_{\star}>0$ there are no critical points of $S$. In order for ${\mathrm{d}}W=0$ we must then necessarily have $r_0=r_{\star}$, which gives $S=+\infty$ and $W=0$ on this locus. In this case the branes move only on the six-cycle. Clearly, this is always a solution since the “superpotential” (\[supQ222\]) is identically zero if the branes are moving on the divisor $D$. For $r_{\star}=0$ there are formally no contributions of instantons to the superpotential. However, notice this is a singular limit of the SUGRA solution in which the six-cycle is blown down. In the absence of instantons of the course M2 branes are free to propagate on the cone $\mathcal{C}(Q^{222})$ in which the six-cycle is blown down.
In [§\[sec:EH\]]{} we have performed the same computation of the warped volume in the much simpler toy example of the Eguchi-Hanson manifold. Interestingly, the same qualitative results on the behaviour of the warped volume (basically the non-perturbative superpotential) are obtained.
EM5 brane instantons: general discussion {#s:general-V}
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In this section we describe how the above calculations for $Q^{222}$ generalize to more general Calabi-Yaus.
### Geometric set-up
Throughout this section we assume we are given an asymptotically conical Kähler manifold $X$ of complex dimension $n$, with metric $g=g_X$ and Kähler two-form $\omega$. This means that the manifold $X$ is non-compact, and that the metric $g$ asymptotically approaches the cone metric ${\mathrm{d}}r^2 + r^2 g_Y$, where $Y=\partial X$ is the compact base of the cone. The metric $g_Y$ is then Sasakian, by definition. In fact this is slightly too general for the situation we are interested in. More precisely, we want $X$ to be a *resolution* of the singularity at $r=0$ of the cone $\mathcal{C}(Y)\cong ({\mathbb{R}}_{\geq 0}\times Y)\cup \{r=0\}$. This means that there is a proper birational map $\pi:X\rightarrow \mathcal{C}(Y)$ which is a biholomorphism on the restriction $X\setminus E\rightarrow \{r>0\}\subset \mathcal{C}(Y)$. Less formally, the resolution $X$ replaces the singular point $\{r=0\}$ of the cone by the *exceptional set* $E=\pi^{-1}(r=0)$. For physical applications we require the metric $g$ on $X$ to be Ricci-flat and hence Calabi-Yau. Then by definition $X$ above is a *crepant* resolution of the cone singularity. In fact this won’t really affect the computations that follow.
Of particular interest for us in this section are the exceptional divisors in $X$. These are the irreducible (prime) divisors of $E=\pi^{-1}(r=0)$. Call these irreducible components $E_i$. Since $X$ is contractable onto $E$ we have H\_[2n-2]{}(X,)H\_[2n-2]{}(E,)\_i H\_[2n-2]{}(E\_i,)\^[b\_[2n-2]{}(X)]{} , so that $i=1,\ldots,b_{2n-2}(X)=\dim H_{2n-2}(X,{\mathbb{R}})$. Thus the exceptional divisors $E_i$ generate the homology of $X$ in codimension two. Notice that if $b_{2n-2}(X)=0$ then the resolution has no exceptional divisors and the resolution is said to be *small*. For example, the resolved conifold is a small resolution of the cone over $T^{1,1}$, since the exceptional set is $E\cong \mathbb{CP}^1$; similarly, the resolutions of $Q^{111}$ considered in [§\[sec:4\]]{} are small.
### $L^2$ harmonic two-forms
Another key result for us is that in the above situation \^2\_[L\^2]{}(X,g)H\^2(X,Y,)H\_[2n-2]{}(X,) , for $n>2$. Here $\mathcal{H}^2_{L^2}(X,g)$ denotes the $L^2$ normalizable harmonic two-forms on $X$ (which thus depends on the metric $g$). That is, the codimension two cycles in $X$ are 1-1 with the prime exceptional divisors, and these are also 1-1 with the $L^2$ normalizable harmonic two-forms on $X$, as long as $n>2$. This result about $L^2$ harmonic forms holds in general for complete asymptotically conical manifolds, and was proven in [@hausel]. In dimension $n=2$, instead $\mathcal{H}^2_{L^2}(X) \cong \mathrm{Im}\left[H^2(X,Y,{\mathbb{R}})\rightarrow
H^2(X,{\mathbb{R}})\right]$. For example, for the Eguchi-Hanson manifold the map $H^2(X,Y,{\mathbb{Z}})\rightarrow H^2(X,{\mathbb{Z}})$ is multiplication by 2, so that there is a unique $L^2$ harmonic two-form, up to scale.
We shall need some more information about these harmonic forms. First, we note that the harmonic two-forms are of Hodge type $(1,1)$. This is because they are Poincaré dual to divisors in $X$, so if there was a $(0,2)$ part of the harmonic form it would be cohomologically trivial and hence[^12] identically zero by the result of [@hausel].
Pick a particular $D=E_i$ and normalize the associated $L^2$ harmonic two-form so that it is Poincaré dual to $D$. We may then think of the harmonic form as the curvature of a Hermitian line bundle $\mathcal{L}=\mathcal{L}_D$ – the divisor bundle for $D$. If $s$ is a *local* nowhere zero holomorphic section of $\mathcal{L}$ over an open set $U\subset X$, and $H$ is the Hermitian metric on $\mathcal{L}$, then we may write the harmonic form as \[betaform\] \_U = | H(s,s) . This is a standard result. Notice that $\beta$ here does not depend on the choice of local holomorphic section $s$. We will be interested in the special case where we take $U$ as large as possible, which is $U = X\setminus D$. By construction, the line bundle $\mathcal{L}_D$ is trivial over $U$, with trivializing nowhere zero holomorphic section $s$. We pick an $s$, and write $H=H(s,s)$ for the corresponding real function on $X\setminus D$.
We next note that $\log H$ is itself a harmonic function on $X\setminus D$. To see this, suppose generally we have a harmonic $(1,1)$-form $\beta$. Because $(X,g)$ is complete, this means $\beta$ is both closed and co-closed. The co-closed condition involves computing the Hodge dual of $\beta$, which is = -\^[n-2]{} + ()\^[n-1]{} . Here we have simply used that ${\mathrm{vol}}= \omega^n/n!$ and that $\beta$ is type $(1,1)$. Thus $\beta^{ij}\omega_{im}\omega_{jn}=\beta_{mn}$. By definition, $\omega\lrcorner\beta=(1/2!)\omega_{ij}\beta^{ij}$. Thus if $\beta$ is closed, then $\star\beta$ is co-closed if and only if $\omega\lrcorner\beta$ is closed, [*i.e.*]{} constant.
Thus we learn that for a harmonic $(1,1)$-form $\beta$, $\omega\lrcorner\beta$ is in fact constant. Now, for an $L^2$ form on an asymptotically conical manifold this constant is in fact necessarily zero. To see why, we must look at the asymptotics of $\beta$. This was studied in appendix A of [@Martelli:2008cm]. Here we have a two-form, so $p=2$ in Table 4 of that reference, and we are interested in $L^2_{\infty}$, so that the form is normalizable at infinity. For $n>2$, $p<n$ so the normalizable modes are of type II and type III$^-$. However, the type II modes are constructed from harmonic one-forms on the base $Y$, and there are not any of these as $(Y,g_Y)$ is a positive curvature Einstein manifold so $b_1(Y)=0$ by Myers’ theorem. Thus asymptotically, the two-form $\beta$ is to leading order of the form \~(r\^[2-n-]{}\_) , where $\beta_\mu$ is a massive co-closed one-form along $Y$ \_Y \_= , and = . Consider now $\omega\lrcorner\beta$. Since asymptotically $\omega\sim (1/2){\mathrm{d}}(r^2\eta)$, where $\eta={\mathrm{i}}(\bar\partial-\partial)\log r$ is the contact one-form of the Sasakian manifold $(Y,g_Y)$, we have \~r\^[-n-]{} . Since $\nu>0$ it follows that $\omega\lrcorner\beta\rightarrow 0$ at infinity. Since we have also shown that $\omega\lrcorner\beta$ is constant, it follows that this constant is zero. If we write $\beta$ as in (\[betaform\]) on $X\setminus D$, then $\omega\lrcorner\beta=0$ is equivalent to saying that $\log H$ is harmonic. This just follows from the form of the scalar Laplacian on a Kähler manifold: $\Delta f = -2\omega\lrcorner{\mathrm{i}}\partial\bar{\partial} f$.
As an aside comment that will be important below, the first non-zero eigenvalue $\mu$ is bounded below by $4(n-1)$. In fact, this is saturated precisely by *Killing* one-forms – see, for example, [@DNP]. Thus $\mu\geq 4(n-1)$ and correspondingly $\nu \geq \sqrt{(n-2)^2+4(n-1)} = n$.
We thus conclude that $\log H$ is a harmonic function. It will be crucial in what follows that $\log H$ is not in fact defined everywhere on $X$. By construction, it is defined only on $X\setminus D$. Along $D$ in fact $\log H$ is singular. This is simply because $D$ is the zero set of $s$, which has a simple zero along $D$ by assumption. Thus if $z=\rho\, {\mathrm{e}}^{{\mathrm{i}}\theta}$ is a local complex coordinate normal to $D$, with $D$ at $\rho=0$, then $\log H$ blows up near to $D$ like $\log \rho^2 = 2 \log \rho$.
### The instanton action
An instantonic brane wrapped on an exceptional divisor $D=E_i$ is calibrated and supersymmetric – for example, a D3 brane for $n=3$ in Type IIB string theory or an M5 brane for $n=4$ in M-theory. These are in 1-1 correspondence with the homology classes $H_{2n-2}(X,{\mathbb{R}})$, and moreover there is a unique $L^2$ harmonic two-form associated to each irreducible exceptional divisor, which is Poincaré dual to the divisor, as discussed above.
Let $G[y](x)$ denote the Green’s function on $X$, with a fixed (Ricci-flat) Kähler metric, normalized so that \_x G\[y\](x) = 2 \^[2n]{}(x-y) . Consider the on-shell action of an instantonic brane, given by the following Green’s function weighted volume of $D$: V = \_D G\[y\] \^[2n-2]{}x = \_D G\[y\] . This is the relevant formula both for D3 branes and M5 branes, where the warp factor is $h= NG/T$, with $N$ the number of spacetime filling branes and $T$ the tension of the wrapped instanton brane. The warped volume $V$ depends on the source point $y\in X$, so $V=V(y)$. Of course, it also depends on the choice of Kähler metric. If we consider the Calabi-Yau case, we know that there is a unique metric in each Kähler class, so we may think of $V$ also as a function of the Kähler class: $V=V(y;[\omega])$. Then we claim that V(y) = -H(y) . Of course, $\log H$ depends implicitly on the Kähler metric since the associated harmonic form does also. Notice this result actually provides a *formula* for the $L^2$ harmonic two-forms on an asymptotically conical Kähler manifold, in terms of the Green’s function on $X$. This is a new mathematical result, as far as we are aware.[^13]
The strategy for proving this claim involves three steps: (i) show that $V(y)$ is a harmonic function on $X\setminus D$, (ii) show that $V$ diverges as $-\log \rho$ along $D$, (iii) show that the two-form ${\mathrm{i}}\partial\bar{\partial} V$ is $L^2$. These steps show that the latter two-form is an $L^2$ harmonic form that is Poincaré dual to $D$. We may then appeal to the uniqueness of such a form.
Step (i). This is straightforward. We want to compute $\Delta_y$ acting on $V$. Using that the Green’s function $G[y](x)$ is symmetric in its arguments, so $G[y](x)=G[x](y)$, then provided $y\notin D$ \_y V= \_[D]{}(\_y G\[y\](x)) \^[2n-2]{}x = 0 . The last step follows since $y\notin D$. This shows that $V(y)$, interpreted as a function on $X\setminus D$, is indeed harmonic.
Step (ii). Near to the source point $y$ we have \[Gnearsource\] G\[y\](x) = (1+[o]{}(1)) , where $\rho$ denotes geodesic distance from $y$, where we regard the latter as fixed. Here ${\mathrm{vol}}(S^{2n-1})$ is the volume of a unit $(2n-1)$-sphere, which appears in this computation as a small sphere around the point $y$. The divergence is a local question, so this is really a question in Euclidean space. Let us take the local model ${\mathbb{C}}^{n-1}\times {\mathbb{C}}$, with complex coordinate $z$ on ${\mathbb{C}}$. We suppose that the divisor $D$ is locally $z=0$ in this patch, and that the point $y$ is $(0,z)$ in this coordinate system. The integral over $D$ is finite outside a ball $B_\delta$ of fixed radius $\delta>0$ in ${\mathbb{C}}^{n-1}\times\{0\}$, so we would like to analyse the divergence in the integral \_[B\_]{} R\^[2n-3]{}R (S\^[2n-3]{}) , as $|z|\rightarrow 0$. This is easily seen to be (-|z|\^2) . We next need the ratio of volumes of spheres: (S\^[2n-1]{})= , so that the divergence is (-|z\^2|) = -|z|\^2 = - |z| . This shows that $V$ diverges as $-\log |z|=-\log \rho$ near to $D$, as claimed.
Step (iii). We consider the two-form ${\mathrm{d}}{\mathrm{d}}^c V(y)$ as the point $y$ tends to infinity. Since we have already shown this two-form is harmonic, it will have an asymptotic expansion as in Appendix A of [@Martelli:2008cm]. There are three types of modes, I, II and III. As already mentioned, there are in fact no modes of type II since $Y$ has no harmonic one-forms. The modes of type I are pull-backs of harmonic two-forms on $Y$, which are not $L^2$. The modes of type III$^{\pm}$ are of the form (r\^[2-n]{}\_) , with $\nu=\sqrt{(n-2)^2+\mu^2}$ as above. The III$^{-}$ modes are $L^2$, while III$^{+}$ are not. Thus we must show that ${\mathrm{d}}{\mathrm{d}}^c V$ has leading term of type III$^{-}$, so that it is normalizable at infinity and hence normalizable.
Since $V$ is globally defined on $X\setminus D$, we immediately see that there can be no mode of type I since near infinity ${\mathrm{d}}{\mathrm{d}}^c V$ is exact. Thus we are reduced to analysing the asymptotic $r$-dependence of the one-form ${\mathrm{d}}^c V$.
If we regard the point $x$ as fixed, then as $y$ tends to infinity we have G\[y\](x) = (1+o(1)). Then = \^c \_D G\[y\] , gives to leading order \[asym\] \~-(r\^[2-2n]{}) . We thus conclude that 2-n= 2-2n . From the comment above, this means that we indeed have a normalizable mode III$^{-}$, and moveover that $\nu=n$ and hence $\mu=4(n-1)$ saturates the lower bound on the eigenvalue. The Killing one-form $\eta$ is of course dual to the R-symmetry.
This completes our proof.
As a final check on the last asymptotic formula, we can compare with the Eguchi-Hanson result (\[EHasym\]). These indeed agree, noting that $\mathrm{vol}(D)=\pi c^2$ and ${\mathrm{vol}}(Y)=\pi^2$ in this case.
### The superpotential {#s:W}
To conclude our results thus far, we have proven in general that \^[-V]{}(y,\[\]) = , where $(1/2\pi {\mathrm{i}}) \partial\bar{\partial} \log H$ is the unique $L^2$ harmonic two-form that is Poincaré dual to the divisor $D$ wrapped by the instanton. The on-shell action of this instanton is $V$. Notice that $\sqrt{H}$ has a simple zero along $D$, as expected on general grounds. In fact locally $H=H(s,s)={\mathrm{e}}^{2g}|s|^2$, where $g$ is function and $s$ is a holomorphic section of the divisor bundle $\mathcal{L}_D$. Again, this was expected from arguments in [@Ganor:1996pe], where the phase that pairs with $V$, coming from the Wess-Zumino term in the action, was studied. Thus the result presented here is rather complimentary to the discussion in reference [@Ganor:1996pe]. Restoring the factor of $N$, our computation hence shows that, formally at least, we have the superpotential W = \^[-NV]{}(y;\[\]) = = \^[Ng]{} |s|\^N . This is interpreted as a function of both the Kähler class, and also the position of the stack of M2 branes $y\in X$, and generalizes the result (\[supQ222\]) we derived explicitly for $Q^{222}$. A critical point of this $W$ requires either a critical point of $V$, or else $V=\infty$. Since $V=-\frac{1}{2}\log H$, the first case requires a critical point of the harmonic function $\log H$. By the maximum principle, notice that such a critical point cannot be either a local maximum or a local minimum of $\log H$.
Conclusions {#sec:6}
===========
In this paper we set out to study abelian symmetries in the context of the $AdS_4/CFT_3$ correspondence. In particular, we considered gauge fields in $AdS_4$ arising from KK reduction of the SUGRA potentials over the $b_2(Y)$ topologically non-trivial cycles (sometimes called Betti multiplets in the literature). In contrast to its better-understood $AdS_5/CFT_4$ relative, the case at hand displays many more subtleties. The key difference resides in the fact that gauge fields in $AdS_4$ admit, in a consistent manner, quantizations with either of two possible fall-offs at the boundary, implying that the gauge field can be dual to either a global symmetry or to a dynamical gauge field in the boundary CFT. From the bulk perspective, electric-magnetic duality in the four-dimensional electromagnetic theory in $AdS_4$ amounts to exchanging these two boundary fall-offs. In addition, from the bulk perspective one can shift the $\theta$-angle by $2\pi$. Following [@Witten:2003ya], these two actions translate into particular operations on the boundary theory, the $\mathcal{T}$ and $\mathcal{S}$ operations reviewed in [§\[sec:baryons\]]{}, which then generate the group $SL(2,\mathbb{Z})$. As stressed in the main text, these actions exchange different boundary conditions for the gauge field in $AdS_4$. Correspondingly, the dual boundary CFTs are different. Indeed, the whole of $SL(2,\mathbb{Z})$ acts on the boundary conditions for the bulk gauge fields, leading in general to an infinite orbit of CFTs for each $U(1)$ gauge symmetry in $AdS_4$. Understanding the structure of such orbits is a very interesting problem which we postpone for further work.
In this paper we have contented ourselves with studying the particular case of M2 branes moving in $\mathcal{C}(Q^{111})$.[^14] In [@Franco:2008um; @Franco:2009sp] a $U(N)^4$ dual theory was proposed and further studied. In [§\[sec:Q111\]]{} we proposed a choice of quantization for the abelian vector fields in the Betti multiplets such that precisely two $U(1)$s are ungauged, leading to the gauge group $U(1)^2\times SU(N)^4$. This leaves precisely two global symmetries that may be identified with the two gauge fields coming from KK reduction of the SUGRA six-form potential over five-cycles in $Q^{111}$. A key point in that identification is that the corresponding boundary conditions in the bulk $AdS_4$ allow for electric wrapped M5 brane states. These M5 branes can be easily identified in terms of the toric geometry of the variety. Since the field theory realizes the minimal GLSM, it is then straightforward to identify the relevant $U(1)$ symmetries in the QCS theory. In turn, this allows one to construct dual baryonic operators to such M5 branes. It is then natural to consider the spontaneous breaking of such symmetries, where the operator dual to such an M5 brane acquires a VEV. We analyzed in detail such SSB in [§\[sec:4\]]{}. In particular, we have been able to compute the VEV of the baryonic condensate, with precise agreement with field theory expectations. We stress that this is a non-trivial check of the dual theory, as this suggests that it does admit an IR superconformal fixed point with the correct properties (specifically, R-charges) to be dual to M2 branes moving in $\mathcal{C}(Q^{111})$. Along the lines of [@Klebanov:2007us; @Klebanov:2007cx], we have also been able to identify the Goldstone boson of this SSB. However, a comprehensive understanding of these resolutions in the context of the actions on the boundary conditions is still lacking. We postpone this for further work. An interesting by-product of our computation is the finding of general expressions for warped volumes in Calabi-Yau backgrounds, which are potentially of interest for other, similar computations.
It is natural to extend our analysis and consider backgrounds with exceptional six-cycles, as we briefly considered in [§\[sec:5\]]{}. Upon resolution, Euclidean M5 branes can be wrapped on these exceptional divisors. As opposed to the Type IIB counterpart case for four-cycles, the M2 branes sourcing the background do not break SUSY any futher than that preserved by the Euclidean brane in the resolution of the cone. Thus, in very much the same spirit as in [@Witten:1996bn], it is natural to expect that these Euclidean branes contribute as non-perturbative effects to the superpotential, even in the warped case. Nevertheless, a comprehensive understanding of these issues is lacking. In this paper we have however taken some first steps towards understanding this by computing the warped volume of such branes. In particular we studied in detail the example $\mathcal{C}(Q^{222})$, which is a certain $\mathbb{Z}_2$ orbifold of $\mathcal{C}(Q^{111})$, as well as the Eguchi-Hanson manifold. In extending our findings to more general geometries we have found expressions which might be of relevance in other contexts.
It is fair to say that the $AdS_4/CFT_3$ correspondence still hides many mysteries. In this paper we have scratched the surface of a few of them. We hope to be able to report on further progress in the near future.
Acknowledgments {#acknowledgments .unnumbered}
---------------
N. B. would like to thank the ISEF for their support; this work was completed with the support of a University of Oxford Clarendon Fund Scholarship. D. R-G. acknowledges financial support from the European Commission through Marie Curie OIF grant contract No. MOIF-CT-2006-38381, Spanish Ministry of Science through the research grant No. FPA2009-07122 and Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042). J. F. S. is supported by a Royal Society University Research Fellowship.
Isolated toric Calabi-Yau four-fold singularities with no vanishing six-cycles {#sec:classification}
==============================================================================
In [@Benishti:2009ky] it was shown that the toric diagram of an isolated Calabi-Yau four-fold singularity should satisfy the following conditions:
1. All the faces of the polytope should be triangular.
2. No lattice point should appear on faces or edges of the polytope.
Recall that, as explained in [§\[s:bc\]]{}, the $v_{\alpha}$ vectors define an affine toric Calabi-Yau four-fold. This definition is unique up to a unimodular transformation $\mathcal{R}$, where $\mathcal{R} \in GL(4,{\mathbb{Z}})$ and $\det \mathcal{R}=\pm 1$. The vectors $v_{\alpha}$ may be written as $v_{\alpha} = (1,w_{\alpha})$ for an appropriate choice of basis, where $w_{\alpha} \in {\mathbb{Z}}^3$ are the vertices of the three-dimensional toric diagram. We will be interested in singularities with no vanishing six-cycles. We will therefore demand, in addition, that no lattice points appear inside the polytope. These toric diagrams are known as *lattice-free* polytopes. Such lattice-free polytopes in three dimensions are characterized by the fact that they have *width* one (see for example [@kantor-1997] and references therein). This is sometimes referred to as Howe’s theorem, and is translated into the fact that the vertices of any lattice-free polytope are sitting in adjacent planes, [*i.e*]{} two lattice planes with no lattice points inbetween. These planes can be chosen to be $\{z=0\}$ and $\{z=1\}$.[^15] We want to start by showing that any three-dimensional lattice-free polytope with more than four vertices describes a cone over a seven-dimensional simply-connected Sasakian manifold.
From [@Lerman2] we learn that the first and second homotopy groups of a toric Sasakian manifold $Y$ can be read straightforwardly from the toric diagram. The results for Calabi-Yau four-folds are \_1(Y)\^4/L , \_2(Y)\^[d-4]{} , \[homotopy\] where $L=\mathrm{span}_{\mathbb{Z}}\{v_\alpha\}$ is the span over ${\mathbb{Z}}$ of the space of $d$ external vertices of the toric diagram. According to the Hurewicz Theorem $H_2(Y)\cong\pi_2(Y)$ whenever $H_1(Y)\cong\pi_1(Y)/[\pi_1(Y),\pi_1(Y)]$ is trivial. Therefore we see from that for simply-connected Sasakian manifolds $b_2(Y)=d-4$, which is also the number of gauge groups in the minimal GLSM describing this geometry. This immediately suggests that the number of gauge nodes of the corresponding field theory, in the case that the latter is identified with the minimal GLSM, is $b_2(Y)+2$. The two additional gauge nodes correspond to the $U(1)$s which are not quotiented by in forming the moduli space. Note from that $Y$ is simply-connected if and only if the external vertices span ${\mathbb{Z}}^4$.
For polytopes with more than four vertices, three of the vertices must be co-planar. Thus the matrix that describes four of the vertices can be written as follows A= (
[cccc]{} 1 & 1 & 1 & 1\
0 & x\_1 & x\_2 & x\_3\
0 & y\_1 & y\_2 & y\_3\
0 & 0 & 0 & 1
) , where each column corresponds to a vertex (the $x$, $y$ and $z$ coordinates correspond to the 2nd, 3rd and 4th rows, respectively). It is easy to see that by an $\mathcal{R}$ transformation this can be brought into the form B= (
[cccc]{} 1 & 1 & 1 & 1\
0 & 1 & 0 & 0\
0 & 0 & 1 & 0\
0 & 0 & 0 & 1
) . To see that the matrices $A$ and $B$ are related by an $\mathcal{R}$ transformation first note that $|\det (A^{-1}\,B)|=1/|(x_1\,y_2-x_2\,y_1)|$. The denominator of the latter is just the area of the parallelogram made up of two identical triangles defined by the first three columns in $A$. The area of this triangle is $1/2$, as this is the condition for a lattice-free triangle in two dimensions.
The four vertices described by $B$ span ${\mathbb{Z}}^4$. Therefore any three-dimensional lattice-free polytope with more than four vertices corresponds to a simply-connected Sasakian seven-manifold. This not always true for diagrams with four vertices, since in this case each plane can contain two points.
To start our analysis we note that lattice-free polytopes with 4 vertices correspond to a type of orbifold singularity that have been discussed intensively in the literature (see [*e.g.*]{} Section 3.1 in [@Morrison:1998cs]). This is a supersymmetric ${\mathbb{C}}^4/{\mathbb{Z}}_k$ orbifold corresponding to an isolated singularity that cannot be resolved, with the orbifold weights in this case being $(1,-1,q,-q)$ with $\gcd \{k,q\}=1$. As already noted in [@Jafferis:2009th], if $q>1$, for any choice of $U(1)$ isometry to reduce on, one can show that $Y$ reduces in Type IIA to a space with orbifold singularities. Therefore, it seems that these $AdS_4 \times Y$ solutions are dual to field theories with no Lagrangian description. Thus we are left with the ABJM orbifolds, obtained by taking $q=1$, for which there are of course already field theory candidates.
We continue with the classification of diagrams with five vertices, where recall that we have shown that the first four vertices are described by $B$. Since the fifth vertex should be in the $\{z=1\}$ plane (to prevent a face with 4 vertices) it can be written as $(1,x,y,1)$ with $x,y>0$.[^16] The only way to break the lattice-free condition would be if there were points between $(1,x,y,1)$ and $(1,0,0,1)$. Thus we have to require $\gcd\{x,y\}=1$. This concludes the classification of polytopes with five vertices. There are no additional $\mathcal{R}$ transformations that connect between diagrams in this set; as we show later, the corresponding GLSM charge matrix that describes this toric diagram is unique for any choice of $x$ and $y$.
Toric diagrams with 6 vertices are potentially more complicated, although we note that these include the $Q^{111}$ example studied in detail in this paper. 6 vertices is also the maximal number since, otherwise, it is not possible to arrange the vertices in two adjacent planes with the constraint that all faces are triangular.
We continue now with a discussion of the geometries that correspond to the polytopes obtained above. Recall that, given a toric diagram, one can recover the corresponding Calabi-Yau four-fold via Delzant’s construction. In physics terms, this would be called a GLSM description of the four-fold. Let us discuss the toric diagrams with five vertices described above. The GLSM charge matrix can be computed by taking the null-space of the $G$-matrix, obtaining \[Qt-matrix\] Q\_t=(x+y , -x , -y , -1 , 1 ) . Since the GLSM charge matrix contains one gauge group, we find that the corresponding quiver should have 3 nodes. However, it is not possible to find a QCS field theory for every value of $x$ and $y$. First, note that there are no zero entries in $Q_t$, therefore there should be no adjoint fields in the quiver. The most general way to construct a quiver with 3 nodes and 5 fields with no adjoints, such that there is an equal number of in-going and out-going arrows at each node, is given in Figure \[General\_quiver\].
![The unique quiver that contains 3 nodes, 5 fields and no adjoints.[]{data-label="General_quiver"}](./General-3-node.eps)
This quiver was also discussed in [@Hanany:2008gx]. Since we are interested in field theories which reproduce the minimal GLSM, we must have a toric superpotential which vanishes in the abelian case. The natural candidate is $$W = \mathrm{Tr}\, \epsilon^{ij} C_iDC_jAB~,
\label{WW}$$ where $\epsilon^{ij}$ is the usual alternating symbol.
In this theory the only contribution to the GLSM matrix comes from the D-term, which in this case reduces to Q\_=(-k\_1 - k\_2, -k\_1 - k\_2, k\_1 + k\_2, k\_2, k\_1 ) , Note that two of the entries are equal while in general there are no equal entries in $Q_t$. Therefore the only hope to reproduce $Q_t$ (up to an overall minus sign) is to choose $k_1+k_2=\pm\,1$. Substituting this back into $Q_{\mathrm{quiver}}$ we obtain Q\_=(1, 1, 1, 1-k\_1, k\_1 ) . Obviously we can reproduce $Q_t$ only for $x=1$ or $y=1$. For other values, the geometries are not captured by the quiver that we have written. Indeed, it seems that the $AdS_4 \times Y$ spaces, where $Y$ is the base of the corresponding isolated Calabi-Yau singularity, reduce in Type IIA to singular spaces, for any choice of $U(1)$ isometry on $Y$. Thus the corresponding M2 brane theories apparently do not admit a Lagrangian description, according to [@Jafferis:2009th].
Geometry of $\mathcal{C}(Q^{111})$ and its resolutions {#sec:Q111-geometry}
======================================================
The cone over $Q^{111}$ is a non-complete intersection defined by 8 $w_i\in \mathbb{C}$ such that $$\begin{aligned}
\nonumber
&& w_1w_2-w_3w_4=w_1w_2-w_5w_8=w_1w_2-w_6w_7=0\ ,\\ \nonumber
&& w_1w_3-w_5w_7=w_1w_6-w_4w_5=w_1w_8-w_4w_7=0\ ,\\ \nonumber
&& w_2w_4-w_6w_8=w_2w_5-w_3w_6=w_2w_7-w_3w_8=0\ .\nonumber\end{aligned}$$
One can check that these equations can be solved in general by taking $$\begin{tabular}{l l}
$w_1=\rho {\mathrm{e}}^{\frac{{\mathrm{i}}}{2}(\psi+\phi_1+\phi_2+\phi_3)}\cos\frac{\theta_1}{2}\cos\frac{\theta_2}{2}\cos\frac{\theta_3}{2}\ ,$ & $w_2=\rho {\mathrm{e}}^{\frac{{\mathrm{i}}}{2}(\psi-\phi_1-\phi_2-\phi_3)}\sin\frac{\theta_1}{2}\sin\frac{\theta_2}{2}\sin\frac{\theta_3}{2}\ ,$\\
$w_3=\rho {\mathrm{e}}^{\frac{{\mathrm{i}}}{2}(\psi+\phi_1-\phi_2-\phi_3)}\cos\frac{\theta_1}{2}\sin\frac{\theta_2}{2}\sin\frac{\theta_3}{2}\ ,$ & $w_4=\rho {\mathrm{e}}^{\frac{{\mathrm{i}}}{2}(\psi-\phi_1+\phi_2+\phi_3)}\sin\frac{\theta_1}{2}\cos\frac{\theta_2}{2}\cos\frac{\theta_3}{2}\ ,$\\
$w_5=\rho {\mathrm{e}}^{\frac{{\mathrm{i}}}{2}(\psi+\phi_1+\phi_2-\phi_3)}\cos\frac{\theta_1}{2}\cos\frac{\theta_2}{2}\sin\frac{\theta_3}{2}\ ,$ &$w_6=\rho {\mathrm{e}}^{\frac{{\mathrm{i}}}{2}(\psi-\phi_1+\phi_2-\phi_3)}\sin\frac{\theta_1}{2}\cos\frac{\theta_2}{2}\sin\frac{\theta_3}{2}\ ,$\\
$w_7=\rho {\mathrm{e}}^{\frac{{\mathrm{i}}}{2}(\psi+\phi_1-\phi_2+\phi_3)}\cos\frac{\theta_1}{2}\sin\frac{\theta_2}{2}\cos\frac{\theta_3}{2}\ ,$ & $w_8=\rho {\mathrm{e}}^{\frac{{\mathrm{i}}}{2}(\psi-\phi_1-\phi_2+\phi_3)}\sin\frac{\theta_1}{2}\sin\frac{\theta_2}{2}\cos\frac{\theta_3}{2}\ .$
\end{tabular}
\label{coordinates}$$ Since the cone over $Q^{111}$ is both Ricci-flat and Kähler the metric can be written as $g_{m\bar{n}} = \partial_m\,\partial_{\bar{n}}K$, where $K$ is the Kähler potential. In the singular case, the most general such Kähler potential which one could write, compatible with the $SU(2)^3\times U(1)_R$ symmetry, is $$K=F(\rho^2)\ .$$ We may resolve $\mathcal{C}(Q^{111})$ by blowing up a copy of $\mathbb{CP}^1\times\mathbb{CP}^1$, as explained in the main text. We take the corresponding Kähler potential to be $$K=F(\rho^2)+a\log(1+|\lambda_1|^2)+b\log(1+|\lambda_2|^2)\ .$$ Here $a, b$ are the resolution parameters of the two $\mathbb{CP}^1$s, coordinatized respectively by $$\lambda_1=\frac{w_2}{w_6}={\mathrm{e}}^{-{\mathrm{i}}\phi_2}\tan\frac{\theta_2}{2}\ ,\quad \lambda_1=\frac{w_5}{w_1}={\mathrm{e}}^{-{\mathrm{i}}\phi_3}\tan\frac{\theta_3}{2}\ .$$ The Ricci tensor for a Kähler manifold is related to the determinant of the metric as $R_{\bar{a}b}=-\bar{\partial}_{\bar{a}}\partial_b\log{\rm det}\, g$. For the case at hand, the determinant of the metric reads $${\rm det}\, g= (F'+\rho^2F'')(a+\rho^2 F')(b+\rho^2F')F' \ ,$$ where $'\equiv\frac{{\mathrm{d}}}{{\mathrm{d}}\rho^2}$. It is useful to define $\gamma=\rho^2\, F'$. Then Ricci flatness implies $$\gamma'\gamma(a+\gamma)(b+\gamma)=\frac{\rho^2}{32}\ .$$ Integrating this expression and setting the integration constant to zero we obtain $$\gamma^4+\frac{4}{3}(a+b)\gamma^3+2ab\gamma^2=\frac{\rho^4}{16} \ .$$ In terms of $\gamma$ the metric then becomes $$\begin{aligned}
{\mathrm{d}}s^2&=&\gamma'\, {\mathrm{d}}\rho^2+\frac{\rho^2 \gamma'}{4}\Big({\mathrm{d}}\psi+\sum_{i=1}^3 \cos\theta_i {\mathrm{d}}\phi_i\Big)^2+\frac{(a+\gamma)}{4}\Big({\mathrm{d}}\theta_2^2+\sin^2\theta_2 {\mathrm{d}}\phi_2^2\Big)\nonumber \\ &&+\frac{(b+\gamma)}{4}\Big({\mathrm{d}}\theta_3^2+\sin^2\theta_3 {\mathrm{d}}\phi_3^2\Big)+\frac{\gamma}{4}\Big({\mathrm{d}}\theta_1^2+\sin^2\theta_1 {\mathrm{d}}\phi_1^2\Big)\ .\end{aligned}$$ It is now convenient to introduce a new radial coordinate $r$ as $r^2=2\gamma$. In terms of this $r$ the metric becomes $$\begin{aligned}
\label{Q111}
{\mathrm{d}}s^2&=&\kappa^{-1}{\mathrm{d}}r^2+\kappa\frac{r^2}{16}\Big({\mathrm{d}}\psi+\sum_{i=1}^3 \cos\theta_i {\mathrm{d}}\phi_i\Big)^2+\frac{(2a+r^2)}{8}\Big({\mathrm{d}}\theta_2^2+\sin^2\theta_2 {\mathrm{d}}\phi_2^2\Big)\nonumber \\ &&+\frac{(2b+r^2)}{8}\Big({\mathrm{d}}\theta_3^2+\sin^2\theta_3 {\mathrm{d}}\phi_3^2\Big)+\frac{r^2}{8}\Big({\mathrm{d}}\theta_1^2+\sin^2\theta_1 {\mathrm{d}}\phi_1^2\Big)\ ,
\label{Q111-resolved-metric}\end{aligned}$$ with $$\kappa=\frac{2\rho^2\gamma'}{\gamma}\ .$$ After some algebra, this is $$\label{kappaQ111}
\kappa=\frac{(2A_-\,+r^2)(2A_+\,+r^2)}{(2a+r^2)(2b+r^2)}\ ,$$ where $$A_{\pm}=\frac{1}{3}\Big(2a+2b \pm \sqrt{4a^2-10ab+4b^2}\Big)\ .$$ This metric has appeared in the literature in a slightly different form [@Cvetic:2000db; @Cvetic:2001ma].[^17] In the main text we make use of yet another form which can be obtained by redefining $r=\sqrt{8\,\varrho}$, $a=4\,l_2^2$ and $b=4\,l_3^2$. After these redefinitions the metric becomes $$\begin{aligned}
\label{Q111Cvetic}
{\mathrm{d}}s^2&=&{U}^{-1}{\mathrm{d}}\varrho^2+{U}\varrho\Big({\mathrm{d}}\psi+\sum_{i=1}^3 \cos\theta_i {\mathrm{d}}\phi_i\Big)^2+(l_2^2+\varrho)\,\Big({\mathrm{d}}\theta_2^2+\sin^2\theta_2 {\mathrm{d}}\phi_2^2\Big)\nonumber \\ &&+(l_3^2+\varrho)\,\Big({\mathrm{d}}\theta_3^2+\sin^2\theta_3 {\mathrm{d}}\phi_3^2\Big)+\varrho\, \Big({\mathrm{d}}\theta_1^2+\sin^2\theta_1 {\mathrm{d}}\phi_1^2\Big)\ ;\end{aligned}$$ where
$${U}=\frac{3\, \varrho^3+4\, \varrho^2\, (l_2^2+l_3^2)+6\, l_2^2\, l_3^2\varrho}{6\,(l_2^2+ \varrho)\, (l_3^2+ \varrho)} \ .$$
Geometry of $\mathcal{C}(Q^{222})$ and its resolutions {#sec:Q222-geometry}
======================================================
The cone $\mathcal{C}(Q^{222})$ is a $\mathbb{Z}_2$ orbifold of $\mathcal{C}(Q^{111})$, where ${\mathbb{Z}}_2\subset U(1)_R$ acts on the fibre coordinate $\psi$. Thus we can construct the variety by starting with the $w_i$ holomorphic coordinates and taking the desired orbifold. In particular, it follows that the metric is just that of $\mathcal{C}(Q^{111})$ with $\psi\in[0,2\pi]$.
We are interested in resolving this conical singularity. The computation of the metric is formally similar to that for $\mathcal{C}(Q^{111})$. Thus, after defining $\gamma=\rho^2\, F'$, we have the equation $$\gamma'\, \gamma\, (a+\gamma)\, (b+\gamma)=\frac{\rho^2}{32} \ .$$ This equation is integrated into $$\gamma^4+\frac{4}{3}(a+b)\, \gamma^3+2\,a\,b\,\gamma^2-k=\frac{\rho^4}{16} \ ,
\label{gamma-equ}$$ where we have left a non-zero integration constant $k$. If $k=0$ we reduce to the $\mathcal{C}(Q^{111})$ case (locally). Following the same steps as for $\mathcal{C}(Q^{111})$, we introduce a new radial variable r\^2=2 , \[gamma-equ-II\] such that the metric reduces to $$\begin{aligned}
\label{Q222}
{\mathrm{d}}s^2&=&\kappa^{-1}{\mathrm{d}}r^2+\kappa\frac{r^2}{16}\Big({\mathrm{d}}\psi+\sum_{i=1}^3 \cos\theta_i {\mathrm{d}}\phi_i\Big)^2+\frac{(2a+r^2)}{8}\Big({\mathrm{d}}\theta_2^2+\sin^2\theta_2 {\mathrm{d}}\phi_2^2\Big)\nonumber \\ &&+\frac{(2b+r^2)}{8}\Big({\mathrm{d}}\theta_3^2+\sin^2\theta_3 {\mathrm{d}}\phi_3^2\Big)+\frac{r^2}{8}\Big({\mathrm{d}}\theta_1^2+\sin^2\theta_1 {\mathrm{d}}\phi_1^2\Big)\ ,\end{aligned}$$ where now $$\label{kappaQ222}
\kappa=\frac{r^8+\frac{8}{3}\, (a+b)\, r^6+8\, a\,b \,r^4-16\, k}{r^4\, (2\,a+r^2)\,(2\,b+r^2)} \ .$$
Let us consider the case in which $a=b=0$ and $k\ne 0$. Then $$\kappa=1-\frac{16\,k}{r^8} \ .$$ Defining $r_{\star}^8=16\, k$, we have that close to $r_{\star}$, the metric approaches $${\mathrm{d}}s^2=\frac{r_{\star}}{8\, (r-r_{\star})}\, {\mathrm{d}}r^2+\frac{r_{\star}}{2}\, (r-r_{\star})\, \Big({\mathrm{d}}\psi+\sum_{i=1}^3 \cos\theta_i\, {\mathrm{d}}\phi_i\Big)^2+\frac{r_{\star}^2}{8}\, \sum_{i=1}^3 {\mathrm{d}}\theta_i^2+\sin^2\theta_i\,{\mathrm{d}}\phi_i^2 \ .$$ Introducing a new radial variable $u=\sqrt{\frac{r_{\star}}{2}\, (r-r_{\star})}$ we have $${\mathrm{d}}s^2={\mathrm{d}}u^2+u^2\, \Big({\mathrm{d}}\psi+\sum_{i=1}^3 \cos\theta_i\, {\mathrm{d}}\phi_i\Big)^2+\frac{r_{\star}^2}{8}\, \sum_{i=1}^3 {\mathrm{d}}\theta_i^2+\sin^2\theta_i\,{\mathrm{d}}\phi_i^2 \ .$$ Here in order to avoid a conical singularity the period of $\psi$ must be $2\pi$. Thus the metric (\[Q222\]) is the resolution of $\mathcal{C}(Q^{222})$. Indeed, by redefinition of variables we can recover the metric for the resolved $\mathcal{C}(Q^{222})$ that appeared in [@Cvetic:2000db; @Cvetic:2001ma].[^18]
The warp factor for resolutions of $\mathcal{C}(Q^{111})$ and $\mathcal{C}(Q^{222})$ {#sec:D}
====================================================================================
We are interested in studying supergravity backgrounds corresponding to M2 branes localized on the space $X$, which will be the resolution of either $\mathcal{C}(Q^{111})$ or $\mathcal{C}(Q^{111})$ described in [§\[sec:Q111-geometry\]]{} and [§\[sec:Q222-geometry\]]{} respectively. After placing $N$ spacetime-filling M2 branes at a point $y$ in the resolved space $X$ we must solve the Green’s equation \_x h\[y\] = \^8(x-y) , for the warp factor $h=h[y]$. Here $\Delta$ is the scalar Laplacian on $X$ and $g_X$ is the metric on the resolved cone. Using the explicit form of the Laplacian we can write $$\frac{1}{\sqrt{\det g}}\,
\partial_i\Big(\sqrt{\det g}\, g^{ij}\partial_j h\Big)=-\frac{(2\pi \ell_p)^6N}{\sqrt{\det g}} \delta^8(x-y) \ .$$ Since we can choose coordinates such that the metrics we are considering are formally identical $$\begin{aligned}
{\mathrm{d}}s^2&=&\kappa^{-1}{\mathrm{d}}r^2+\kappa\frac{r^2}{16}\Big({\mathrm{d}}\psi+\sum_{i=1}^3 \cos\theta_i {\mathrm{d}}\phi_i\Big)^2+\frac{(2a+r^2)}{8}\Big({\mathrm{d}}\theta_2^2+\sin^2\theta_2 {\mathrm{d}}\phi_2^2\Big)\nonumber \\ &&+\frac{(2b+r^2)}{8}\Big({\mathrm{d}}\theta_3^2+\sin^2\theta_3 {\mathrm{d}}\phi_3^2\Big)+\frac{r^2}{8}\Big({\mathrm{d}}\theta_1^2+\sin^2\theta_1 {\mathrm{d}}\phi_1^2\Big)\ ,\end{aligned}$$ we have that the Laplacian in both cases can be written as $$\frac{1}{\sqrt{\det g}}\,\partial_i\Big(\sqrt{\det g}\, g^{ij}\partial_j h\Big)=\frac{\partial_r\Big(r^3(2a+r^2)(2b+r^2)\kappa\,\partial_rh\Big)}{r^3(2a+r^2)(2b+r^2)}+\mathbf{A}h \ ,$$ where the angular Laplacian $\mathbf{A}$ is $$\label{AngLap}
\mathbf{A}h=\frac{8}{r^2}\Delta_1h+\frac{8}{2a+r^2}\Delta_2 h+\frac{8}{2b+r^2}\Delta_3h+\frac{16}{r^2\kappa}\partial^2_{\psi}h \ ,$$ with $$\Delta_i=\frac{1}{\sin\theta_i}\partial_{\theta_i}(\sin\theta_i\partial_{\theta_i})+\Big(\frac{1}{\sin\theta_i}\partial_{\phi_i}-\cot\theta_i\partial_{\psi}\Big)^2 \ .$$ As we show in (\[delta\]) in the next subsection, we can expand the delta function in terms of eigenfunctions of the Laplacian such that $$\frac{1}{\sqrt{\det g}} \delta^8(x-y)=\frac{1}{r^3(r^2+2a)(r^2+2b)}\delta(r-r_0)\sum_I\, Y_I(\xi_0)^*Y_I(\xi) \ ,$$ where we denote collectively the angular coordinates as $\xi$, and define $x=(r,\xi)$ and $y=(r_0,\xi_0)$. Then, the equation for the warp factor reads $$\frac{1}{f}\partial_r\Big(f\,\partial_rh\Big)+\kappa^{-1}\,\mathbf{A}h=- \frac{(2\pi \ell_p)^6N}{f}\delta(r-r_0)\sum_I\, Y_I(\xi_0)^*Y_I(\xi) \ ,$$ where we have defined for simplicity $$\label{f-def}
f=r^3(2a+r^2)(2b+r^2)\kappa \ .$$ We can now expand $h$ in eigenfunctions of the angular Laplacian. Since $\xi_0$ is just a point (not a variable), we can write $$h=\sum_I \psi_I(r)\, Y_I(\xi_0)^*\, Y_I(\xi) \ .
\label{general-warp}$$ Then the radial equation we have to solve reduces to $$\frac{1}{f}\partial_r\Big(f\,\partial_r\psi_I\Big)-\kappa^{-1}\,E_I\,\psi_I=- \frac{(2\pi \ell_p)^6N}{f}\delta(r-r_0) \ ,
\label{radial-to-solve}$$ where $E_I$ is the angular eigenvalue of $Y_I$, to which we now turn.
Angular eigenfunctions in $Q^{111}$
-----------------------------------
We want to consider (\[AngLap\]) with fixed $r$ and construct eigenfunctions of such an operator. For this we first concentrate on each of the $\Delta_i$ operators. For each $\theta$, these look like $$\Delta_{i} =\frac{1}{\sin\theta }\partial_{\theta}(\sin\theta\partial_{\theta})+\Big(\frac{1}{\sin\theta}\partial_{\phi}-\cot\theta\partial_{\psi}\Big)^2 \ .$$ Note that these angular Laplacians are the same as those for the conifold. As such, many technical details and results can be borrowed from [@Klebanov:2007us].
We consider the following function $$Y=J(\theta)\, {\mathrm{e}}^{{\mathrm{i}}\, m\, \phi}\, {\mathrm{e}}^{{\mathrm{i}}\, {R\psi}/{2}} \ .$$ It is obvious that $$\Delta_{i} Y=\Big\{\frac{1}{\sin\theta}\partial_{\theta}(\sin\theta\partial_{\theta} J)-\Big(\frac{m}{\sin\theta}-\cot\theta\frac{R}{2}\Big)^2 J\Big\}J^{-1} Y \ .$$ Therefore it is interesting to consider the following eigenfunctions $$\frac{1}{\sin\theta}\partial_{\theta}(\sin\theta\partial_{\theta} J)-\Big(\frac{m}{\sin\theta}-\cot\theta\frac{R}{2}\Big)^2 J=-E J \ .$$ This equation has two solutions, given in terms of hypergeometric functions $$\label{sol_A}
J^A_{l,m,R}=\sin^m\theta\, \cot^{\frac{R}{2}}\frac{\theta}{2}\, \,_2F_1\Big(-l+m,1+l+m;1+m-\frac{R}{2};\sin^2\frac{\theta}{2}\Big) ~,$$ and $$\label{sol_B}
J^B_{l,m,R}=\sin^{\frac{R}{2}}\theta\, \cot^{m}\frac{\theta}{2}\, \,_2F_1\Big(-l+\frac{R}{2},1+l+\frac{R}{2};1-m+\frac{R}{2};\sin^2\frac{\theta}{2}\Big) \ ,$$ where we have introduced a labelling for the quantum numbers distinguishing the eigenfunctions. If $m\ge \frac{R}{2}$ solution (\[sol\_A\]) is non-singular while if $m\le \frac{R}{2}$ it is (\[sol\_B\]) that is the non-singular solution. Both have eigenvalue under each $\Delta_i$ operator given by $$E=l(l+1)-\frac{R^2}{4} \ .$$ Given these results, we can consider the functions $$\label{Y-function}
Y_{I}=\mathcal{C}_I\,J_{l_1,m_1,R}(\theta_1)J_{l_2,m_2,R}(\theta_2)J_{l_3,m_3,R}(\theta_3)\, {\mathrm{e}}^{{\mathrm{i}}\, (m_1\phi_1+m_2\phi_2+m_3\phi_3)}\, {\mathrm{e}}^{{\mathrm{i}}\, {R\psi}/{2}} \ ,$$ where the multi-index $I$ stands for $\{(l_1,m_1),(l_2,m_2),(l_3,m_3),R\}$ and where $\mathcal{C}_I$ is just a normalization factor such that the norm of $Y_I$ is one. It is now clear that $$\mathbf{A}Y_I=-E_I Y_I \ ,$$ with $$E_I=\frac{8\,l_1(l_1+1)}{r^2}+\frac{8\,l_2(l_2+1)}{r^2+2a}+\frac{8\,l_3(l_3+1)}{r^2+2b}+2R^2\Big(\frac{2}{r^2\kappa}-\frac{1}{r^2}-\frac{1}{r^2+2a}-\frac{1}{r^2+2b}\Big) \ .$$ We now note that the $Y_I$ are also eigenfunctions of the singular cone. Indeed, we can consider the Laplacian on the unit $Q^{111}$, namely $\mathbf{\tilde{A}}|_{a=0=b;r=1}$. Then $$\mathbf{\tilde{A}}Y_I=-\tilde{E}_I Y_I \ ,$$ with $$\tilde{E}_I=8\,l_1(l_1+1)+8\,l_2(l_2+1)+8\,l_3(l_3+1)-2R^2 \ .$$ Therefore, the $Y_I$ are also normalized eigenfunctions for the $\mathbf{\tilde{A}}$ operator. Being eigenfunctions of a Hermitian operator, these satisfy $$\int {\mathrm{d}}^7\xi\, \sqrt{\det \tilde{g}}\, Y_I(\xi)^*Y_J(\xi)=\delta_{I-J} \ ,$$ and therefore $$\sum_I\, Y_I(\xi_1)^*Y_I(\xi_2)=\frac{1}{\sqrt{\det \tilde{g}}}\, \delta^7(\xi_1-\xi_2) \ ,$$ where we use $\xi$ to generically parametrize the angular coordinates and $\tilde{g}$ stands for the angular part of the metric. One can check very easily that $$\sqrt{\det g}=r^3(r^2+2a)(r^2+2b)\, \sqrt{\det \tilde{g}} \ .$$ Therefore, if we denote $x=(r,\xi)$ and $y=(r_0,\xi_0)$ we have $$\frac{1}{\sqrt{\det g}}\, \delta^8(x-y)=\frac{1}{r^3(r^2+2a)(r^2+2b)}\delta(r-r_0)\frac{1}{\sqrt{\det \tilde{g}}}\, \delta^7(\xi-\xi_0) \ .$$ Using the completeness relation above we may hence write $$\label{delta}
\frac{1}{\sqrt{\det g}}\, \delta^8(x-y)=\frac{1}{r^3(r^2+2a)(r^2+2b)}\delta(r-r_0)\sum_I\, Y_I(\xi_0)^*Y_I(\xi) \ .$$
Angular eigenfunctions in $Q^{222}$
-----------------------------------
Since $\mathcal{C}(Q^{222})$ is a $\mathbb{Z}_2$ orbifold of $\mathcal{C}(Q^{111})$ along $\psi$ it is clear that the local computation of the previous subsection will not be changed. Thus, we just have to take care of global issues. Recall that the wavefunctions in $\mathcal{C}(Q^{111})$ are $$Y_{I}=\mathcal{C}_I\,J_{l_1,m_1,R}(\theta_1)J_{l_2,m_2,R}(\theta_2)J_{l_3,m_3,R}(\theta_3)\, {\mathrm{e}}^{{\mathrm{i}}\, (m_1\phi_1+m_2\phi_2+m_3\phi_3)}\, {\mathrm{e}}^{{\mathrm{i}}\, {R\psi}/{2}} \ .$$ Since now $\psi\in [0,2\pi]$, it is clear that the well-behaved $Y_I$ will be those for which $R$ is even; that is, $R=2\, \tilde{R}$. Therefore, dropping the tilde, the angular wavefunctions in $\mathcal{C}(Q^{222})$ are $$Y_{I}=\mathcal{C}_I\,J_{l_1,m_1,R}(\theta_1)J_{l_2,m_2,R}(\theta_2)J_{l_3,m_3,R}(\theta_3)\, {\mathrm{e}}^{{\mathrm{i}}\, (m_1\phi_1+m_2\phi_2+m_3\phi_3)}\, {\mathrm{e}}^{{\mathrm{i}}\, R\psi} \ ,$$ such that $$\mathbf{A}Y_I=-E_I Y_I \ ,$$ with $$E_I=\frac{8\,l_1(l_1+1)}{r^2}+\frac{8\,l_2(l_2+1)}{r^2+2a}+\frac{8\,l_3(l_3+1)}{r^2+2b}+8R^2\Big(\frac{2}{r^2\kappa}-\frac{1}{r^2}-\frac{1}{r^2+2a}-\frac{1}{r^2+2b}\Big) \ .$$
The warp factor for $Q^{111}$ {#sec:Q111-warp}
-----------------------------
We now want to use the results derived so far to compute explicitly the warp factor for the resolution of the $\mathcal{C}(Q^{111})$ space. We will consider the stack of branes to be sitting on the exceptional locus, where both the $U(1)$ fibre and the $(\theta_1,\phi_1)$ sphere shrink to zero size. This means that $h=h(r,\theta_2,\theta_3)$, which in turn implies that $R$ and $l_1$ in (\[Y-function\]) vanish. Then, under these assumptions, the multi-index $I$ takes the values $I=\{(l_2,m_2),(l_3,m_3)\}$. Indeed, we will assume the branes are located at the north pole of each of the two two-spheres. As such, we should consider also $m_2=m_3=0$. Therefore, for such cases the angular eigenfunctions $J^A$ and $J^B$ coincide and reduce, for each sphere, to Legendre polynomials $$J_{l,0,0}=\,_2F_1(-l,1+l;1;\sin^2\tfrac{\theta}{2})=P_l(\cos\theta) \ ,$$ such that for the case at hand where only $l_2,l_3\ne 0$ $$Y_{l_2,l_3}=\mathcal{C}_{l_2,l_3}\, P_{l_2}(\cos\theta_2)\, P_{l_3}(\cos\theta_3) \ ,$$ and $$E_I=\frac{8\,l_2(l_2+1)}{r^2+2a}+\frac{8\,l_3(l_3+1)}{r^2+2b} \ .$$ Thus, from (\[radial-to-solve\]) we see that the equation to solve reads $$\frac{1}{f}\partial_r\Big(f\,\partial_r\psi_I\Big)-\Big(\frac{8\,l_2(l_2+1)}{r^2+2a}+\frac{8\,l_3(l_3+1)}{r^2+2b}\Big)\kappa^{-1}\,\psi_I=-\frac{(2\pi \ell_p)^6N}{f}\, \delta(r) \ .$$ We are interested in the simplified case in which, say, only $b\ne 0$. Under such assumption, also the $(\theta_2,\phi_2)$ sphere shrinks to zero, so that we can also consider $l_2=0$. Then the corresponding angular wavefunctions are $$Y_{l_3}=\mathcal{C}_{l_3}\, P_{l_3}(\cos\theta_3) \ .$$ Also from (\[kappaQ111\]) and (\[f-def\]) we see that $$\kappa=\frac{r^2+\frac{8b}{3}}{r^2+2b}\ ,\qquad f=r^5\, (r^2+\frac{8b}{3}) \ .$$ Then the equation to solve reduces to $$\partial_r\Big[r^5\, (r^2+\frac{8b}{3})\, \partial_r\psi_I\Big]-8\,r^5\, l_3(l_3+1)\,\psi_I+(2\pi \ell_p)^6N\delta(r)=0 \ .$$ The two solutions are $$\begin{aligned}
\psi_I^{(1)}& \sim &\Big(\frac{8b}{3r^2}\Big)^{\frac{3}{2}(1-\beta)}\, _2F_1(-\frac{1}{2}-\frac{3}{2}\beta,\frac{3}{2}-\frac{3}{2}\beta,1-3\beta,-\frac{8b}{3r^2})\ , \\ \nonumber
\psi_I^{(2)}& \sim &\Big(\frac{8b}{3r^2}\Big)^{\frac{3}{2}(1+\beta)}\, _2F_1(-\frac{1}{2}+\frac{3}{2}\beta,\frac{3}{2}+\frac{3}{2}\beta,1+3\beta,-\frac{8b}{3r^2}) \ ,\end{aligned}$$ with $$\beta=\sqrt{1+\frac{8}{9}l_3(l_3+1)} \ .$$ Since $\beta\ge 1$, for large $r$ only the $\psi_I^{(2)}$ solutions decay at infinity, and these are therefore the solutions of interest.
We can now state the result for the warp factor, which turns out to be $$h=\sum_{l_3}\, \mathcal{C}_{l_3}\, \Big(\frac{8b}{3r^2}\Big)^{\frac{3}{2}(1+\beta)}\, _2F_1\Big(-\frac{1}{2}+\frac{3}{2}\beta,\frac{3}{2}+\frac{3}{2}\beta,1+3\beta,-\frac{8b}{3r^2}\Big)\, P_{l_3}(\cos\theta_3) \ ,$$ where we collect all normalization factors in $\mathcal{C}_{l_3}$.
The warp factor for $Q^{222}$ {#s:warp_factor-Q222}
-----------------------------
We will compute the warp factor for $N$ M2 branes in arbitrary location. As was shown above, the equation to solve is (\[radial-to-solve\]) where now we should use (\[kappaQ222\]) for $\kappa$. We will be interested in the simpler case in which $a=b=0$. Moreover, as we explain in the main text, the interesting contribution is that coming from $R=l_i=0$. Under this simplification, the equation to solve now reads $$\partial_r\Big(r^7\, (1-\frac{r_{\star}^8}{r^8})\, \partial_r\psi_I\Big)=-(2\pi \ell_p)^6N\, \delta(r-r_0) \ .$$ Solving for $r>r_0$ we obtain $$\psi_>=\frac{1}{r^6}\, _2F_1\Big(\frac{6}{8},\, 1,\, \frac{7}{4},\Big(\frac{r_{\star}^2}{r^2}\Big)^4\Big) \ ,$$ and for $r<r_0$ instead $$\psi_<=\frac{1}{r_0^6}\, _2F_1\Big(\frac{3}{4},\, 1,\, \frac{7}{4},\Big(\frac{r_{\star}^2}{r_0^2}\Big)^4\Big) \, _2F_1\Big(0,\, \frac{6}{8},\, \frac{3}{4},\Big(\frac{r^2}{r_{\star}^2}\Big)^4\Big) \ .
\label{warp-Q222-I}$$ Indeed, the leading term for large $r$ corresponds to $l_i=0$, and in this limit $$h\sim \frac{ |\mathcal{C}_0|^2}{r^6} \equiv \frac{R^6}{r^6} \ .
\label{warp-Q222-II}$$ Recall that for $\mathcal{C}(Q^{222})$ $$R^6= \frac{2^9\, \pi^2\, N\, l_p^6}{3} \ .
\label{warp-Q222-III}$$
The Eguchi-Hanson manifold as a toy model {#sec:EH}
=========================================
In this appendix we illustrate, with the aid of a simple toy model, some of the computations derived in [§\[s:general-V\]]{}.
Harmonic forms
--------------
In this subsection we derive the normalizable harmonic two-form in the Eguchi-Hanson manifold. This two-form will be important later on when we show how the warped volume of the exceptional divisor can be inferred from it. The Eguchi-Hanson metric is $${\mathrm{d}}s^2=\frac{{\mathrm{d}}r^2}{1-\frac{c^4}{r^4}}+\frac{r^2}{4}\, \Big(1-\frac{c^4}{r^4}\Big)\, \Big({\mathrm{d}}\psi+\cos\theta \, {\mathrm{d}}\phi\Big)^2+\frac{r^2}{4}\,\Big({\mathrm{d}}\theta^2+\sin^2\theta\, {\mathrm{d}}\phi^2\Big) \ .$$ To avoid a conical singularity at $r=c$ when $c>0$ we have to take $\psi\in[0,2\pi]$. It is natural to define the one-form vielbein $$g_5={\mathrm{d}}\psi+\cos\theta\, {\mathrm{d}}\phi\ , \qquad e_{\theta}={\mathrm{d}}\theta\ , \qquad e_{\phi}=\sin\theta\, {\mathrm{d}}\phi \ .$$ Then a natural ansatz for a closed and co-closed two-form is $$\tilde\beta=e_{\theta}\wedge e_{\phi}+{\mathrm{d}}(f\, g_5)\ , \qquad f=f(r) \ .$$ It is immediate that this form is closed. The Hodge dual is $$\star\tilde\beta=\frac{2}{r}\, (1-f)\, {\mathrm{d}}r\wedge g_5+\frac{r}{2}\, \frac{{\mathrm{d}}f}{{\mathrm{d}}r}\, e_{\theta}\wedge e_{\phi} \ .$$ Thus co-closedness implies $$\frac{{\mathrm{d}}}{{\mathrm{d}}r}\Big(r\, \frac{{\mathrm{d}}f}{{\mathrm{d}}r}\Big)+\frac{4}{r}(1-f)=0 \ .$$ Choosing the solution decaying at infinity, this can be integrated into $$f=1+\frac{A}{r^2} \ ,$$ $A$ being an integration constant which we will fix to $A=-1$. Thus the self-dual harmonic two-form is $$\tilde\beta=\frac{1}{r^2}\, \sin\theta\, {\mathrm{d}}\theta\wedge {\mathrm{d}}\phi+\frac{2}{r^3}\, {\mathrm{d}}r\wedge g_5 \ .$$ We may define a normalized two-form $\hat{\beta}$ as $\hat{\beta}=\frac{ c^2}{2\sqrt{2}\,\pi}\, \tilde\beta$. Changing now to the following coordinate system $$z_1=(r^4-c^4)^{1/4}\, {\mathrm{e}}^{\frac{{\mathrm{i}}}{2}\, (\psi+\phi)}\, \sin\frac{\theta}{2}\qquad z_2=(r^4-c^4)^{1/4}\, {\mathrm{e}}^{\frac{{\mathrm{i}}}{2}\, (\psi-\phi)}\, \cos\frac{\theta}{2} \ ,$$ the normalized two-form now reads $$\hat{\beta}= \partial_{z_i}\partial_{\bar{z}_j}\, S_{\beta}\,{\mathrm{d}}z_i\wedge {\mathrm{d}}\bar{z}_j\ ,\qquad S_{\beta}={\mathrm{i}}\frac{1}{\pi\, \sqrt{2}}\, {\rm arctanh}\frac{c^2}{r^2} \ .$$ It is then easy to see that $$\int_{X_{EH}} \hat{\beta}\wedge \star \hat{\beta}=1 \ .$$
Laplace operator
----------------
Prior to the computation of the warped volume, in this subsection we derive the warp factor for the Eguchi-Hanson manifold. Thus, we are interested in solving the Green’s equation $$\Box_{EH}\, h=\frac{1}{\sqrt{\det g}}\, \partial_i\Big(\sqrt{\det g}\, g^{ij}\, \partial_j\, h\Big)=\mathcal{N}\, \frac{\delta^4(x-p)}{\sqrt{\det g}} \ ,
\label{laplace}$$ where $p$ is some arbitrary point in the Eguchi-Hanson manifold and $\mathcal{N}$ is a normalization factor. After some algebra one can see that $$\Box_{EH}\, h=\frac{1}{r^3}\, \partial_r\Big(r^3\, f\, \partial_r\, h\Big)+\frac{4}{r^2}\,\Delta\, h+\frac{4}{r^2\, f}\, \partial_{\psi}^2\, h \ ,$$ where we have introduced the operator $$\Delta=\frac{1}{\sin\theta}\partial_{\theta}\Big(\sin\theta\,\partial_{\theta}\Big)+\Big(\frac{\partial_{\phi}}{\sin\theta}-\cot\theta\, \partial_{\psi}\Big)^2 \ ,$$ and defined for simplicity $$f=1-\frac{c^4}{r^4} \ .$$ Coming back to the Green’s equation, denoting collectively the angular coordinates by $\xi$ and $x=(r,\xi)$ and $p=(r_p,\xi_p)$, we can write (\[laplace\]) as $$\Box_{EH}\, h=\frac{8\,\mathcal{N}}{r^3}\, \delta(r-r_p)\, \frac{\delta(\psi-\psi_p)\, \delta(\theta-\theta_p)\, \delta(\phi-\phi_p)}{\sin\theta} \ .
\label{laplace2}$$
It is now useful to think of the Eguchi-Hanson manifold as a resolution of a $\mathbb{Z}_2$ orbifold of $\mathbb{C}^2$ which has been resolved with the parameter $c$. From this point of view, the metric on the singular cone would be simply $${\mathrm{d}}s^2={\mathrm{d}}r^2+r^2\Big\{ \frac{1}{4}\Big({\mathrm{d}}\psi+\cos\theta \, {\mathrm{d}}\phi\Big)^2+\frac{1}{4}\,\Big({\mathrm{d}}\theta^2+\sin^2\theta\, {\mathrm{d}}\phi^2\Big)\Big\} \ ,$$ where $\psi\in [0,2\pi]$. The Laplacian in this cone is $$\Box_{C}\, h=\frac{1}{r^3}\, \partial_r\Big(r^3\, \partial_r\, h\Big)+\frac{1}{r^2}\, \mathbf{A}\, h \ ,$$ where we have introduced the angular Laplacian $\mathbf{A}$ $$\mathbf{A}=4\Delta +4\, \partial_{\psi}^2 \ .$$ As usual, with the normalized eigenfunctions of $\mathcal{A}$ $$\mathcal{A}\, Y_{R,\,l,\,m}=-E_{R,\, l,\,m}\, Y_{R,\,l,\,m} \ ,$$ we can construct a representation of the delta function on the base of the cone. This delta function is precisely the part appearing in the original Green’s problem we are interested in. Thus $$\sum_{R,\, l,\, m}\, Y_{R,\,l,\,m}(\xi_p)^*\, Y_{R,\,l,\,m}(\xi)=\frac{\delta(\psi-\psi_p)\, \delta(\theta-\theta_p)\, \delta(\phi-\phi_p)}{\sin\theta} \ .
\label{deltaEH}$$
In order to find the explicit form for the $Y_{R,\,l,\,m}(\xi)$ eigenfunctions we start by writing $$Y_{R,\,l,\,m}(\xi)={\mathrm{e}}^{{\mathrm{i}}\,R\,\psi}\, {\mathrm{e}}^{{\mathrm{i}}\, m\, \phi}\, J_{l,\,m}(\theta) \ ,$$ where $R,\, m\in \mathbb{Z}$. Then, the $J_{l,\,m}$ functions satisfy $$\frac{1}{\sin\theta}\partial_{\theta}\Big(\sin\theta\, \partial_{\theta} J_{l,\,m}\Big)-\Big(\frac{m}{\sin\theta}-R\, \cot\theta\Big)^2\, J_{l,\,m}=-\frac{E_{R,\,l,\,m}}{4}\, J_{l,\,m} \ .$$ One can then verify that the solutions to this equation are $$J_{l,\,m}^A(\theta)=\sin^m\theta\, \cot^R\frac{\theta}{2}\, _2F_1\Big(-l+m,\, 1+l+m,\, 1+m-R,\, \sin^2\frac{\theta}{2}\Big)~,$$ and $$J_{l,\,m}^B(\theta)=\sin^R\theta\, \cot^m\frac{\theta}{2}\, _2F_1\Big(-l+R,\, 1+l+R,\, 1-m+R,\, \sin^2\frac{\theta}{2}\Big) \ ,$$ and that for both $$E_{R,\,l,\,m}=4\,\Big(l(l+1)-R^2\Big) \ .$$ If $m\ge R$ the solution $J^A_{l,\, m}$ is singular, while if $m\le R$ it is the solution $J^B_{l,\,m}$ that becomes singular. Of course, for $R=m$ both solutions coincide. Because of this, depending on $R$ we should use one or the other.
Finally, the normalized eigenfunctions which we are after are $$Y_{R,\,l,\,m}(\xi)=\mathcal{A}_{R,\,l,\,m}\, {\mathrm{e}}^{{\mathrm{i}}\,R\,\psi}\, {\mathrm{e}}^{{\mathrm{i}}\, m\, \phi}\, J_{l,\,m}(\theta) \ ,$$ where $\mathcal{A}_{R,\,l,\,m}$ encodes the normalization. It is now clear that we should expand $h$ as $$h=\sum_{R,\, l,\,m}\, \psi_{R,\, l}(r)\, Y_{R,\,l,\,m}(\xi_p)^*\, Y_{R,\,l,\,m}(\xi) \ .
\label{warp_factor}$$ After substituting this and (\[deltaEH\]) into (\[laplace2\]), a straightforward computation shows that $ \psi_{R,\, l}$ satisfies $$\frac{1}{r^3}\,\partial_r\Big(r^3\, f\, \partial_r\, \psi_{R,\, l}\Big)-\Big\{\frac{4\,\Big(l(l+1)-R^2\Big)}{r^2}+\frac{4\,R^2}{r^2\, f}\Big\} \psi_{R,\, l}=\frac{8\, \mathcal{N}}{r^3}\, \delta(r-r_p) \ .$$ The two solutions of this equation are $$\psi^{(1)} \sim \Big(1-\frac{c^4}{r^4}\Big)^{\frac{|R|}{2}}\, \Big(\frac{r}{c}\Big)^{2\,l}\, _2F_1\Big(\frac{-l+|R|}{2},\frac{1-l+|R|}{2},\frac{1}{2}-l,\frac{c^4}{r^4}\Big)~,$$ and $$\label{warp1}
\psi^{(2)} \sim \Big(1-\frac{c^4}{r^4}\Big)^{\frac{|R|}{2}}\, \Big(\frac{c}{r}\Big)^{2+2\,l}\, _2F_1\Big(\frac{1+l+|R|}{2},\frac{2+l+|R|}{2},\frac{3}{2}+l,\frac{c^4}{r^4}\Big) \ .$$ First we want to check which is the regular solution for $r>r_p$. For that we check the large $r$ limit $$\psi^{(1)}\rightarrow \Big(\frac{r}{c}\Big)^{2\,l}\ ,\qquad \psi^{(2)}\rightarrow \Big(\frac{c}{r}\Big)^{2+2\,l} \ .$$ Thus we conclude that the solution to use for $r> r_p$ is $\psi^{(2)}$.
In order to find the $r<r_p$ solution, it is better to re-write the solutions of the above equation as $$\tilde{\psi}^{(1)} \sim \Big(\frac{r^4}{c^4}-1\Big)^{\frac{|R|}{2}}\, _2F_1\Big(\frac{-l+|R|}{2},\, \frac{1+l+|R|}{2},\, \frac{1}{2},\, \frac{r^4}{c^4}\Big)~,$$ and $$\tilde{\psi}^{(2)} \sim \frac{r^2}{c^2}\, \Big(\frac{r^4}{c^4}-1\Big)^{\frac{|R|}{2}}\, _2F_1\Big(\frac{1-l+|R|}{2},\, \frac{2+l+|R|}{2},\, \frac{3}{2},\, \frac{r^4}{c^4}\Big) \ .$$ The regular solution appears as a linear combination of these two solutions. To see this, let us define $$\begin{aligned}
\label{combained}
\psi&=&\Big(\frac{r^4}{c^4}-1\Big)^{\frac{|R|}{2}}\, \Big[A_1\, _2F_1\Big(\frac{-l+|R|}{2},\, \frac{1+l+|R|}{2},\, \frac{1}{2},\, \frac{r^4}{c^4}\Big)+\\ \nonumber && A_2\, \frac{r^2}{c^2}\, _2F_1\Big(\frac{1-l+|R|}{2},\, \frac{2+l+|R|}{2},\, \frac{3}{2},\, \frac{r^4}{c^4}\Big)\Big] \ .\end{aligned}$$ In order to ensure finiteness as $r$ tends to $c$, we have to set $$A_2=-A_1\,\frac{2\,\Gamma\Big(\frac{1-l+|R|}{2}\Big)\, \Gamma\Big(\frac{2+l+|R|}{2}\Big)}{\Gamma\Big(\frac{-l+|R|}{2}\Big)\, \Gamma\Big(\frac{1+l+|R|}{2}\Big)} \ ,$$ for modes where $|R| \ \geqslant l$ and the parity of $R$ and $l$ is the same. The other modes should be set to zero. Subtituting this result into (\[combained\]) and suppressing the overall factor $A_1$, we obtain $$\begin{aligned}
\label{warp2}
\psi& \sim &\Big(\frac{r^4}{c^4}-1\Big)^{\frac{|R|}{2}}\, \Big[_2F_1\Big(\frac{-l+|R|}{2},\, \frac{1+l+|R|}{2},\, \frac{1}{2},\, \frac{r^4}{c^4}\Big)\\ \nonumber && -\frac{2\,\Gamma\Big(\frac{1-l+|R|}{2}\Big)\, \Gamma\Big(\frac{2+l+|R|}{2}\Big)}{\Gamma\Big(\frac{-l+|R|}{2}\Big)\, \Gamma\Big(\frac{1+l+|R|}{2}\Big)}\, \frac{r^2}{c^2}\, _2F_1\Big(\frac{1-l+|R|}{2},\, \frac{2+l+|R|}{2},\, \frac{3}{2},\, \frac{r^4}{c^4}\Big)\Big] \ ,\end{aligned}$$ which is the well-behaved solution for $r<r_p$.
In the following subsections it will become clear that the interesting mode for us is the one with $R=l=0$. With this choice (\[warp1\]) and (\[warp2\]) become $$\begin{aligned}
\psi_> &=& A\,\Big( \frac{c}{r}\Big)^2\,_2F_1\Big(\frac{1}{2},\, 1,\, \frac{3}{2},\, \frac{c^4}{r^4}\Big)~,\end{aligned}$$ and $$\begin{aligned}
\label{EH-sol}
\psi_< &=& A\,\Big( \frac{c}{r_p}\Big)^2\,_2F_1\Big(\frac{1}{2},\, 1,\, \frac{3}{2},\, \frac{c^4}{r_p^4}\Big) \,_2F_1\Big(0,\, \frac{1}{2},\, \frac{1}{2},\, \frac{r^4}{c^4}\Big) \ ,\end{aligned}$$ where $A$ is a normalization constant. To normalize these solutions we consider the $r \rightarrow \infty$ limit in (\[warp\_factor\]). We see that in this limit $h \simeq \, A\, |\mathcal{A}_{0,\,0,\,0}|^2 \,c^2/r^2$. Thus, if we integrate (\[laplace\]) we get $$A\, |\mathcal{A}_{0,\,0,\,0}|^2\,c^2=\frac{\mathcal{N}}{2\,{\mathrm{vol}}(\Omega_{EH})} \ ,$$ where $\Omega_{EH}\cong S^3/{\mathbb{Z}}_2$ is the the base of the Eguchi-Hanson manifold at infinity. Explicitly this volume is $${\mathrm{vol}}(\Omega_{EH})=\int_{\Omega_{EH}} \sqrt{\det g}=\frac{1}{8}\int_0^{2\,\pi}{\mathrm{d}}\,\psi\,\int_0^{2\,\pi}\,{\mathrm{d}}\,\phi\,\int_0^{\pi}\,\sin\,\theta\,{\mathrm{d}}\,\theta=\pi^2 \ .$$ Thus, in the convention in which $\mathcal{N}=2\,\pi$ we have $$\label{EH-sol-norm}
A=\frac{1}{\pi\,c^2\, |\mathcal{A}_{0,\,0,\,0}|^2} \ .$$
Warped volumes {#s:EH-V}
--------------
We are now interested in computing the warped volume of the blown-up divisor in the Eguchi-Hanson manifold. This reads $$S=\int_D\, \sqrt{\det g_D}\, h \, {\mathrm{d}}^2x~,$$ where $D$ is the blown-up $S^2=\mathbb{CP}^1$ at $r=c$, and $g_D$ and $h$ are the pull-backs of the metric and warp factor to $D$, respectively. After substituting (\[warp\_factor\]) and $$\sqrt{\det g_D}=\frac{c^2}{4} \sin\theta \ ,$$ into this expression one obtains $$S=\frac{c^2}{4}\,\sum_{l,\,m}\, \psi_l(c)\,\int_{D}\, Y_{0,\,l,\,m}(\xi_p)^*\, Y_{0,\,l,\,m}(\xi)\, {\mathrm{d}}^2\xi \ .$$ We now proceed by evaluating the integral $$\int_{D}\, Y_{0,\,l,\,m}(\xi_p)^*\, Y_{0,\,l,\,m}(\xi)\, {\mathrm{d}}^2\xi=|\mathcal{A}_{0,\,l,\,m}|^2\, {\mathrm{e}}^{-{\mathrm{i}}\,m\,\phi_p}\,J_{l,\,m}(\theta_p)\,\int\, {\mathrm{d}}\theta\, {\mathrm{d}}\phi\, \sin\theta\, {\mathrm{e}}^{{\mathrm{i}}\,m\,\phi}\, J_{l,\,m}(\theta) \ .$$ The $\phi$ integral forces that only the $m=0$ term contributes. Thus $$\int_{D}\, Y_{0,\,l,\,m}(\xi_p)^*\, Y_{0,\,l,\,m}(\xi)\, {\mathrm{d}}^2 \xi=2\pi\,\delta_{m,0}\,|\mathcal{A}_{0,\,l,\,0}|^2\,J_{l,\,0}(\theta_p)\,\int_0^\pi\, {\mathrm{d}}\theta\, \sin\theta\, J_{l,\,0}(\theta) \ .$$ Furthermore, the $J_{l,0}(\theta)$ are just Legendre polynomials in $\cos\theta$. One can then see that all integrals are zero except for the $l=0$ mode. Thus $$\int_{D}\,Y_{0,\,l,\,m}(\xi_p)^*\, Y_{0,\,l,\,m}(\xi)\, {\mathrm{d}}^2 \xi=4\pi\,\delta_{m,0}\, \delta_{l,0}\,|\mathcal{A}_{0,\,0,\,0}|^2 \ .$$ So finally $$S=\pi\,c^2\,|\mathcal{A}_{0,\,0,\,0}|^2\, \psi_<(c) \ .$$ $\psi_<(c)$ can be read from (\[EH-sol\]) together with the normalization at (\[EH-sol-norm\]). Neglecting the label $p$ in $r_p$ we obtain our final extremely simple result: $$S={\rm arctanh} \frac{c^2}{r^2} \ .$$
Harmonic forms from the warped volume
-------------------------------------
In this subsection we show how to rederive the harmonic two-form, that was derived in the first subsection, using the warped volume just computed. As shown in the first subsection, the two-form = |S\_is harmonic, where S\_= . Recall here that $r\geq c$, with the exceptional divisor $D=\{r=c\}$ being a copy of $\mathbb{CP}^1$. Moreover, $\hat{\beta}$ is normalized so that \_[X\_[EH]{}]{} = 1 , where $X_{EH}=\mathcal{O}(-2)\rightarrow\mathbb{CP}^1$ is the Eguchi-Hanson manifold. The first claim is that the correctly normalized Poincaré dual to $D=\mathbb{CP}^1$ is = . Indeed, then \_[X\_[EH]{}]{} = -\_[X\_[EH]{}]{} = 2 . We require the 2 here since this is the Euler number of the normal bundle to the exceptional $\mathbb{CP}^1$. Thus, in our notation in the main text, H = (2)S\_= -2 = . The radial coordinate near to $D=\mathbb{CP}^1$ is $\rho=\sqrt{r-c}$, so that $D$ is at $\rho=0$. Thus we see that, near to $D$, $\log H$ blows up as $\log \rho^2$, precisely as claimed in the main text.
Also notice that $\beta$ may be written \[EHasym\] = -(r\^[-2]{}) . Here $\eta=\tfrac{1}{2}\left({\mathrm{d}}\psi-\cos\theta{\mathrm{d}}\phi\right)$. Thus the mode $\beta_\mu=\eta$ in this case, which in particular is a Killing one-form. Thus $\mu=4$ and hence $2-n-\nu=-\nu=-\sqrt{\mu}=-2$, which is the power of $r$ above. This confirms the claims about the asymptotic expansion in this case.
[99]{}
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[^1]: For a complete discussion of the Type IIB case we refer to *e.g.* [@Franco:2005sm; @Martelli:2007mk; @Martelli:2008cm].
[^2]: The other natural such generalization is the homogeneous space $V_{5,2}=SO(5)/SO(3)$, which has been studied in detail in [@Martelli:2009ga].
[^3]: This is true since the membrane global anomaly described in [@Witten:1996md] is always zero on a seven-manifold $Y$ that is spin.
[^4]: There are, however, examples: the Calabi-Yau four-fold hypersurfaces $\sum_{i=1}^5 z_i^d=0$, where $d=3, 4$, are known to have Calabi-Yau cone metrics, and these have $b_3(Y)=10$, $60$, respectively [@Boyer:1998sf].
[^5]: Even though we are explicitly discussing the effect of $SL(2,\mathbb{Z})$ on the vector fields, since these are part of a whole Betti multiplet we expect a similar action on the other fields of the multiplet. We leave this investigation for future work.
[^6]: We stress that there might be, and indeed even in the $Q^{111}$ example there are, other branches of the moduli space where the condition $\sigma_a=\sigma$ for all $a$ is not met, and yet still the bosonic potential is minimized.
[^7]: As anticipated in [§\[sec:3\]]{}, at the IR superconformal fixed point the dimensions of the chiral fields are expected to be different from the free field fixed point. That is why generically the VEV of the baryonic operator is $\langle\mathscr{B}_{A_2}\rangle = b^{N\, \Delta_{A_2}}$.
[^8]: The M-theory circle can be deduced by computing the ${\mathbb{Z}}_k$ orbifold action on the moduli space in each case.
[^9]: In fact the irregular case can be approximated arbitrarily closely by the quasi-regular case.
[^10]: As already mentioned, generically one can at best partially resolve such singularities so that the remaining singularities are all of orbifold type.
[^11]: It might be possible to generalize this to the case with SUSY $G$-flux, in which the flux also sources the warp factor.
[^12]: Notice this result definitely fails for spaces that are not asymptotically conical. A good example is the Taub-NUT space, which has no two-cycles but does have an $L^2$ harmonic two-form.
[^13]: We thank Tamas Hausel for discussions on this.
[^14]: Although as described in [§\[sec:gen\]]{}, and further elaborated in [§\[sec:classification\]]{}, we expect similar results to hold for other toric isolated four-fold singularities with no exceptional six-cycles.
[^15]: If there are more vertices in one plane we choose it to be the $\{z=0\}$ plane without loss of generality.
[^16]: As can be easily seen from (\[Qt-matrix\]), toric diagrams with triangular faces obtained by picking other values of $x$ and $y$ are related by $\mathcal{R}$ transformations.
[^17]: This form can be recovered by defining $r=\sqrt{2}\rho$, $a=l_2^2$ and $b=l_3^2$.
[^18]: This form can be recovered by starting with (\[Q222\]) and redefining $r=\sqrt{8\, \rho+r_{\star}}\ , l_1^2=\frac{2\,a+r_{\star}}{8}\ , l_2^2=\frac{2\,b+r_{\star}}{8}\ , l_3^2=\frac{r_{\star}}{8} \ .$
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abstract: 'We present a “modern” approach to the Erdős–Ko–Rado theorem for $Q$-polynomial distance-regular graphs and apply it to the twisted Grassmann graphs discovered in 2005 by van Dam and Koolen.'
address: 'Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI 53706, U.S.A.'
author:
- Hajime Tanaka
title: 'The Erdős–Ko–Rado theorem for twisted Grassmann graphs'
---
Introduction
============
The 1961 theorem of Erdős, Ko and Rado [@EKR1961QJMO] asserts that the largest possible families $Y$ of $d$-subsets of a $v$-set such that $|x\cap y|\geqslant t$ for all $x,y\in Y$ where $v>(t+1)(d-t+1)$ are the families of all $d$-subsets containing some fixed $t$-subset. In fact, the exact bound $v>(t+1)(d-t+1)$ was obtained later by Wilson [@Wilson1984C] as an application of Delsarte’s linear programming method [@Delsarte1973PRRS]. It is natural to think of this theorem as a result about (vertex) subsets of the Johnson graphs $J(v,d)$, and analogous theorems are known for several other families of distance-regular graphs, e.g., Hamming graphs $H(d,q)$ $(q\geqslant t+2)$ [@Moon1982JCTA], Grassmann graphs $J_q(v,d)$ $(v\geqslant 2d)$ [@Hsieh1975DM; @FW1986JCTA; @Fu1999DM; @Tanaka2006JCTA], bilinear forms graphs $\mathrm{Bil}_q(d,e)$ $(d\leqslant e)$ [@Huang1987DM; @Fu1999DM; @Tanaka2006JCTA].
In this note, we first distill common algebraic techniques found in some of the proofs of these “Erdős–Ko–Rado theorems” into a unified approach for general $Q$-polynomial distance-regular graphs $\Gamma$.[^1] Our approach is also “modern” in the sense that it is based on and motivated by the theory of two parameters, *width* $w$ and *dual width* $w^*$, of a subset $Y$ of $\Gamma$ introduced in 2003 by Brouwer et al. [@BGKM2003JCTA]. In this setting, the “$t$-intersecting” condition amounts to requiring $w\leqslant d-t$ where $d$ is the diameter of $\Gamma$, and we shall view the Erdős–Ko–Rado theorem as characterizing those subsets $Y$ with $w=d-t$ and $w^*=t$ by their sizes among all $t$-intersecting families. There are two steps involved: (1) construction of a specific feasible solution to the dual of a linear programming problem; (2) classification of the *descendents* [@Tanaka2010pre] of $\Gamma$, i.e., those subsets having the property $w+w^*=d$. We demonstrate this approach by deriving the Erdős–Ko–Rado theorem for the *twisted Grassmann graphs* $\tilde{J}_q(2d+1,d)$ discovered in 2005 by van Dam and Koolen [@DK2005IM].
A “modern” approach to the Erdős–Ko–Rado theorem for $Q$-polynomial distance-regular graphs {#sec: modern approach}
===========================================================================================
Let $\Gamma=(X,R)$ be a finite connected simple graph with diameter $d$ and path-length distance $\partial$, and $\mathbb{R}^{X\times X}$ the set of real matrices with rows and columns indexed by $X$. For each $i$ $(0\leqslant i\leqslant d)$, let $A_i\in\mathbb{R}^{X\times X}$ be the adjacency matrix of the distance-$i$ graph $\Gamma_i$ of $\Gamma$, so $A_0=I$ and $\sum_{i=0}^dA_i=J$, the all ones matrix. We say $\Gamma$ is *distance-regular* if $\bm{A}:=\mathrm{span}\{A_0,A_1,\dots,A_d\}$ is closed under ordinary matrix multiplication; or equivalently, $\bm{A}$ is a (commutative) algebra. (The reader is referred to [@BI1984B; @BCN1989B; @Godsil1993B] for background material on distance-regular graphs.) Throughout this note, suppose $\Gamma$ is distance-regular. We call $\bm{A}$ the *Bose–Mesner algebra* of $\Gamma$. It is semisimple (as it is closed under transposition) and therefore has a basis $\{E_i\}_{i=0}^d$ consisting of the primitive idempotents; we always set $E_0=|X|^{-1}J$. Note that $\bm{A}$ is also closed under entrywise multiplication, denoted $\circ$. We shall assume $\Gamma$ is $Q$-*polynomial* with respect to the ordering $\{E_i\}_{i=0}^d$, i.e., $E_1\circ E_i$ is a linear combination of $E_{i-1},E_i,E_{i+1}$ with nonzero coefficients for $E_{i-1},E_{i+1}$ $(0\leqslant i\leqslant d)$, where $E_{-1}=E_{d+1}=0$. Let $Q=(Q_{ij})_{0\leqslant i,j\leqslant d}$ be the *second eigenmatrix* of $\Gamma$: $$E_j=\frac{1}{|X|}\sum_{i=0}^dQ_{ij}A_i \quad (0\leqslant j\leqslant d).$$
Let $Y$ be a nonempty subset of $X$ and $\chi\in\mathbb{R}^X$ its (column) characteristic vector. Brouwer et al. [@BGKM2003JCTA] defined the *width* $w$ and *dual width* $w^*$ of $Y$ as follows: $$w=\max\{i:\chi^{\mathsf{T}}A_i\chi\ne 0\}, \quad w^*=\max\{i:\chi^{\mathsf{T}}E_i\chi\ne 0\}.$$ They showed (among other results) that $$\label{fundamental inequality}
w+w^*\geqslant d.$$ We call $Y$ a *descendent* [@Tanaka2010pre] of $\Gamma$ if $w+w^*=d$. It should be remarked that every descendent is a so-called completely regular code (cf. [@KLM2010P]), and that the induced subgraph is a $Q$-polynomial distance-regular graph provided it is connected; see [@BGKM2003JCTA Theorems 1–3]. See also [@Tanaka2010pre] for more information on descendents.
Now fix an integer $t$ $(0<t<d)$ and suppose $w\leqslant d-t$; in other words, $Y$ is “$t$-intersecting”. We recall the inner distribution $\bm{e}=(e_0,e_1,\dots,e_d)$ of $Y$: $$e_i=\frac{1}{|Y|}\chi^{\mathsf{T}}A_i\chi, \quad (\bm{e}Q)_i=\frac{|X|}{|Y|}\chi^{\mathsf{T}}E_i\chi \quad (0\leqslant i\leqslant d).$$ It follows that $|Y|=(\bm{e}Q)_0$ and $$\begin{gathered}
e_0=1, \quad e_1\geqslant 0,\dots,e_{d-t}\geqslant 0,\quad e_{d-t+1}=\dots=e_d=0, \\
(\bm{e}Q)_1\geqslant 0,\dots,(\bm{e}Q)_d\geqslant 0.\end{gathered}$$ (Observe that the $E_i$ are positive semidefinite.) Following [@Delsarte1973PRRS], we view these as a linear programming maximization problem. A vector $\bm{f}=(f_0,f_1,\dots,f_d)$ satisfying , below gives a feasible solution to its dual problem: $$\begin{gathered}
f_0=1, \quad f_1=\dots=f_t=0, \quad f_{t+1}>0,\dots,f_d>0, \label{constraint 1} \\
(\bm{f}Q^{\mathsf{T}})_1=\dots=(\bm{f}Q^{\mathsf{T}})_{d-t}=0. \label{constraint 2}\end{gathered}$$ Indeed, we have $$|Y|=(\bm{e}Q)_0\leqslant\bm{e}Q\bm{f}^{\mathsf{T}}=(\bm{f}Q^{\mathsf{T}})_0$$ with equality if and only if $(\bm{e}Q)_{t+1}=\dots=(\bm{e}Q)_d=0$, i.e., $w^*\leqslant t$. By virtue of , it follows that
\[EKR for general Gamma\] Let $Y$ be a nonempty subset of $X$ with $w\leqslant d-t$. Suppose there is a vector $\bm{f}=(f_0,f_1,\dots,f_d)$ satisfying , . Then $|Y|\leqslant(\bm{f}Q^{\mathsf{T}})_0$, and equality holds if and only if $Y$ is a descendent of $\Gamma$ with $w=d-t$ and $w^*=t$.
The vector $\bm{f}$ above is of independent interest from the point of view of *Leonard systems*[^2] [@Terwilliger2006N] and will be discussed in detail in a future paper. Here we mention that $\bm{f}$ can be found for the following graphs:
$(\bm{f}Q^{\mathsf{T}})_0$
------------------------------------------------------------ ---------------------------------------
$J(v,d)$ ($v>(t+1)(d-t+1)$) $\binom{v-t}{d-t}$
$H(d,q)$ ($t=d-1$; or $q\geqslant d$; or $q=d-1$, $t<d-2$) $q^{d-t}$
$J_q(v,d)$ ($v\geqslant 2d$) ${\genfrac{[}{]}{0pt}{}{v-t}{d-t}}_q$
$\mathrm{Bil}_q(d,e)$ ($d\leqslant e$) $q^{(d-t)e}$
For $\Gamma=J(v,d)$ or $J_q(v,d)$ (with $v,d$ as in the table), Wilson and Frankl [@Wilson1984C; @FW1986JCTA] constructed a matrix $B\in\bm{A}$ such that (i) $B_{xy}=0$ if $\partial(x,y)\leqslant d-t$; (ii) $B+I-{\genfrac{[}{]}{0pt}{}{v-t}{d-t}}^{-1}J$ is positive semidefinite and its $i^{\text{th}}$ eigenvalue $\lambda_i$ is positive precisely when $t+1\leqslant i\leqslant d$, where we interpret ${\genfrac{[}{]}{0pt}{}{m}{n}}$ as $\binom{m}{n}$ for $J(v,d)$ and ${\genfrac{[}{]}{0pt}{}{m}{n}}_q$ for $J_q(v,d)$. We define $\bm{f}$ by $f_0=1$, $f_1=\dots=f_t=0$, and $f_i={\genfrac{[}{]}{0pt}{}{v-t}{d-t}}{\genfrac{[}{]}{0pt}{}{v}{d}}^{-1}\lambda_i$ for $t+1\leqslant i\leqslant d$. For $\Gamma=\mathrm{Bil}_q(d,e)$ $(d\leqslant e)$, Delsarte [@Delsarte1978JCTA] constructed a *Singleton system*, i.e., a subset whose inner distribution $\bm{e}'=(e_0',e_1',\dots,e_d')$ satisfies $e_1'=\dots=e_t'=0$ and $(\bm{e}'Q)_1=\dots=(\bm{e}'Q)_{d-t}=0$. It follows that $e_{t+1}',\dots,e_d'$ are positive; see [@Tanaka2006JCTA §4]. We define $\bm{f}=\bm{e}'\cdot\mathrm{diag}(k_0,k_1,\dots,k_d)^{-1}$ where $k_i$ is the valency of $\Gamma_i$ $(0\leqslant i\leqslant d)$. For $\Gamma=H(d,q)$, a subset having the above properties is known as an MDS code [@MS1977B Chapter 11]. MDS codes may not exist for some $d,q,t$, but still $\bm{e}'$ makes sense and is uniquely determined. If $t=d-1$ or $q\geqslant d$, or if $q=d-1$ and $t<d-2$, then it follows that $e_{t+1}',\dots,e_d'$ are positive; see e.g., [@EGS2011IEEE Appendix]. We again define $\bm{f}=\bm{e}'\cdot\mathrm{diag}(k_0,k_1,\dots,k_d)^{-1}$.
Concerning the conclusion of Lemma \[EKR for general Gamma\], the classification of descendents has been done for the 15 known infinite families of $Q$-polynomial distance-regular graphs with so-called classical parameters and with unbounded diameter, including the above 4 families; see [@BGKM2003JCTA; @Tanaka2006JCTA; @Tanaka2010pre]. Moon [@Moon1982JCTA] showed that the upper bound $q^{d-t}$ for $H(d,q)$ and the characterization of its descendents as optimal intersecting families are valid under the (in general) weaker assumption $q\geqslant t+2$. Dual polar graphs discussed in [@Tanaka2006JCTA] do not always possess $\bm{f}$ even for the case $t=1$ [@Stanton1980SIAMb]; see [@PSV2011JCTA], however, for a description of optimal $1$-intersecting families.
The Erdős–Ko–Rado theorem for twisted Grassmann graphs
======================================================
Let $q$ be a prime power and fix a hyperplane $H$ of $\mathbb{F}_q^{2d+1}$. Let $X_1$ be the set of $(d+1)$-dimensional subspaces of $\mathbb{F}_q^{2d+1}$ not contained in $H$, and $X_2$ the set of $(d-1)$-dimensional subspaces of $H$. The *twisted Grassmann graph* $\Gamma=\tilde{J}_q(2d+1,d)$ [@DK2005IM] has vertex set $X=X_1\cup X_2$, and two vertices $x,y\in X$ are adjacent if $\dim x+\dim y-2\dim x\cap y=2$. It has the same parameters (i.e., the structure constants of $\bm{A}$) as $J_q(2d+1,d)$. The twisted Grassmann graphs provide the first known family of non-vertex-transitive distance-regular graphs with unbounded diameter. See [@FKT2006IIG; @BFK2009EJC; @MT2009pre] for more information.
The Erdős–Ko–Rado theorem for $\tilde{J}_q(2d+1,d)$ can now be rapidly obtained. Note that $J_q(2d+1,d)$ and $\tilde{J}_q(2d+1,d)$ share the same $Q$. Hence we may use the vector $\bm{f}$ for $J_q(2d+1,d)$ constructed in §\[sec: modern approach\], and Lemma \[EKR for general Gamma\] applies. The descendents of $\tilde{J}_q(2d+1,d)$ have recently been classified by the author [@Tanaka2010pre Theorem 8.20]. To summarize:
Let $Y$ be a nonempty subset of $\tilde{J}_q(2d+1,d)$ with width $w\leqslant d-t$, where $0<t<d$. Then $|Y|\leqslant{\genfrac{[}{]}{0pt}{}{2d+1-t}{d-t}}_q$, and equality holds if and only if $Y=\{x\in X_2:u\subseteq x\}$ for some subspace $u$ of $H$ with $\dim u=t-1$.
Acknowledgements {#acknowledgements .unnumbered}
================
The author would like to thank the Department of Mathematics at the University of Wisconsin–Madison for its hospitality throughout the period in which this work was done. Support from the JSPS Excellent Young Researchers Overseas Visit Program is also gratefully acknowledged.
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[^1]: $Q$-polynomial distance-regular graphs are thought of as finite/combinatorial analogues of symmetric spaces of rank one; see [@BI1984B pp. 311–312].
[^2]: Leonard systems provide a linear algebraic framework characterizing the terminating branch of the Askey scheme [@KLS2010B] of (basic) hypergeometric orthogonal polynomials.
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abstract: |
As declarative query processing techniques expand in scope — to the Web, data streams, network routers, and cloud platforms — there is an increasing need for *adaptive* query processing techniques that can re-plan in the presence of failures or unanticipated performance changes. A status update on the data distributions or the compute nodes may have significant repercussions on the choice of which query plan should be running. Ideally, new system architectures would be able to make *cost-based decisions* about reallocating work, migrating data, etc., and react quickly as real-time status information becomes available. Existing cost-based query optimizers are not incremental in nature, and must be run “from scratch” upon each status or cost update. Hence, they generally result in adaptive schemes that can only react slowly to updates.
An open question has been whether it is possible to build a cost-based *re-optimization* architecture for adaptive query processing in a streaming or repeated query execution environment, e.g., by *incrementally* updating optimizer state given new cost information. We show that this can be achieved beneficially, especially for stream processing workloads. Our techniques build upon the recently proposed approach of formulating query plan enumeration as a set of *recursive datalog queries*; we develop a variety of novel optimization approaches to ensure effective pruning in both static and incremental cases. We implement our solution within an existing research query processing system, and show that it effectively supports cost-based initial optimization as well as frequent adaptivity.
author:
- |
Mengmeng Liu\
\
\
\
Zachary G. Ives\
\
\
\
Boon Thau Loo\
\
\
\
bibliography:
- 'zives-short.bib'
- 'boon.bib'
- 'mengmeng.bib'
title: 'Enabling Incremental Query Re-Optimization'
---
Introduction
============
\[sec:intro\]
The problem of supporting rapid *adaptation* to runtime conditions during query processing — *adaptive query processing* [@aqp-survey] — is of increasing importance in today’s data processing environments. Consider declarative cloud data processing systems [@boom-analytics; @asterix; @DBLP:journals/pvldb/MelnikGLRSTV10] and data stream processing [@aurora; @stanford-stream; @telegraphcq] platforms, where data properties and the status of cluster compute nodes may be constantly changing. Here it is very difficult to effectively choose a good plan for query execution: data statistics may be unavailable or highly variable; cost parameters may change due to resource contention or machine failures; and in fact a *combination* of query plans might perform better than any single plan. Similarly, in conventional DBMSs there may be a need to perform *self-tuning* so the performance of a query or set of queries can be improved [@db2-mqreopt].
To this point, query optimization techniques in adaptive query processing systems fall into three general classes: (1) operator-specific techniques that can adapt the order of evaluation for filtering operators [@stanford-aqp]; (2) eddies [@DBLP:conf/sigmod/AvnurH00; @stairs] and related flow heuristics, which are highly adaptive but also continuously devote resources to exploring *all* plans and require fully pipelined execution; (3) approaches that use a cost-based query re-optimizer to re-estimate plan costs and determine whether the system should change plans [@DBLP:conf/sigmod/KabraD98; @db2-mqreopt; @tukwila-04; @cape-04]. Of these, the last is the most flexible, e.g., in that it supports complex query operators like aggregation, as well as expensive adaptations like data repartitioning across a cluster. Perhaps most importantly, a cost-based engine allows the system to spend the majority of its resources on query execution once the various cost parameters have been properly calibrated. Put another way, it can be applied to highly complex plans and has the potential to provide significant benefit if a cost estimation error was made, but it should incur little overhead if a good plan was chosen. Unfortunately, to this point cost-based techniques have not been able to live up to their potential, because the cost-based re-optimization step has been too expensive to perform frequently.
Our goal in this paper is to explore whether *incremental* techniques for re-optimization can be developed, where an optimizer would only re-explore query plans whose costs were affected by an updated cardinality or cost value; and whether such incremental techniques could be used to facilitate more efficient adaptivity.
Our long-term goal is to develop adaptive techniques for complex OLAP-style queries (which contain operators not amenable to the use of eddies) being executed across a data-partitioned cluster, as in [@boom-analytics; @asterix] . However, in this paper we focus on developing incremental re-optimization techniques that we evaluate within a single-node (local) query engine, in two main contexts. (1) We address the problem of adaptive query processing in *data stream management systems* where data may be bursty, and its distributions may vary over time — meaning that different query plans may be preferred over different segments. Here it is vital to optimize frequently based on recent data distribution and cost information, ideally as rapidly as possible. (2) We address query re-optimization in traditional OLAP settings when the same query (or highly similar queries) gets executed frequently, as in a prepared statement. Here we may wish to re-optimize the plan after each iteration, given increasingly accurate information about costs, and we would like this optimization to have minimal overhead.
The main contribution of this paper is to show for the first time how an *incremental* re-optimizer can be developed, and how it can be useful in adaptive query processing scenarios matching the application domains cited above. Our incremental re-optimizer implements the basic capabilities of a modern database query optimizer, and could easily be extended to support other more advanced features; our main goal is to show that an incremental optimizer following our model can be competitive with a standard optimizer implementation for *initial* optimization, and significantly faster for *repeated* optimization. Moreover, in contrast to randomized or heuristics-based optimization methods, we **still guarantee the discovery of the best plan** according to the cost model. Since our work is oriented towards adaptive query processing, we evaluate the system in a variety of settings in conjunction with a basic pipelined query engine for stream and stored data.
We implement the approach using a novel approach, which is based on the observation that query optimization is essentially a recursive process involving the derivation and subsequent pruning of state (namely, alternative plans and their costs). If one is to build an *incremental* re-optimizer, this requires preservation of state (i.e., the optimizer memoization table) across optimization runs — but moreover, it must be possible to determine what plans have been *pruned* from this state, and to re-derive such alternatives and test whether they are now viable.
One way to achieve such “re-pruning” capabilities is to carefully define a semantics for how state needs to be tracked and recomputed in an optimizer. However, we observe that this task of “re-pruning” in response to updated information looks remarkably similar to the database problem of *view maintenance* through aggregation [@springerlink:10.1007/BFb0014149] and recursion as studied in the database literature [@gms93-dred]. In fact, recent work [@evita-raced] has shown that query optimization can itself be captured in recursive datalog. Thus, rather than inventing a custom semantics for incrementally maintaining state within a query optimizer, we instead adopt the approach of developing an incremental re-optimizer expressed *declaratively*.
More precisely, we express the optimizer as a recursive datalog program consisting of a set of rules, and leverage the existing database query processor to actually execute the declarative program. In essence, this is optimizing a query optimizer using a query processor. Our implementation approaches the performance of conventional procedural optimizers for reasonably-sized queries. Our implementation recovers the initial overhead during subsequent re-optimizations by leveraging *incremental view maintenance* [@gms93-dred; @recursive-views] techniques. It only recomputes portions of the search space and cost estimates that might be affected by the cost updates. Frequently, this is only a small portion of the overall search space, and hence we often see order-of-magnitude performance benefits.
Our approach achieves pruning levels that rival or best bottom-up (as in System-R [@systemR]) and top-down (as in Volcano [@DBLP:journals/debu/Graefe95a; @volcano]) plan enumerations with branch-and-bound pruning. We develop a variety of novel *incremental* and *recursive* optimization techniques to capture the kinds of pruning used in a conventional optimizer, and more importantly, to generalize them to the incremental case. Our techniques are of broader interest to incremental evaluation of recursive queries as well. Empirically, we see updates on only a small portion of the overall search space, and hence we often see order-of-magnitude performance benefits of incremental re-optimization. We also show that our re-optimizer fits nicely into a complete adaptive query processing system, and measure both the performance and quality, the latter demonstrated well in the yielded query plans, of our incremental re-optimization techniques on the Linear Road stream benchmark. We make the following contributions:
The first query optimizer that prunes yet supports incremental re-optimization.
A rule-based, declarative approach to query (re)optimization. Our implementation decouples plan enumerations and cost estimations, relaxing traditional restrictions on search order and pruning.
Novel strategies to prune the state of an executing recursive query, such as a declarative optimizer: *aggregate selection* with *tuple source suppression*; *reference counting*; and *recursive bounding*.
A formulation of query re-optimization as an *incremental view maintenance* problem, for which we develop novel algorithms.
An implementation over a query engine developed for recursive stream processing [@recursive-views], with a comprehensive evaluation of performance against alternative approaches, over a diverse workload.
Demonstration that incremental re-optimization can be incorporated to good benefit in existing cost-based adaptive query processing techniques [@tukwila-04; @cape-04].
Declarative Query Optimization
==============================
\[sec:declarative\]
Achieving Pruning
=================
\[sec:sip\]
Incremental Re-Optimization
===========================
\[sec:incremental\]
Evaluation
==========
\[sec:eval\]
Related Work
============
\[sec:related\]
Our work takes a first step towards supporting continuous adaptivity in a distributed (e.g., cloud) setting where correlations and runtime costs may be unpredictable at each node. Fine-grained adaptivity has previously only been addressed in the query processing literature via heuristics, such as flow rates [@DBLP:conf/sigmod/AvnurH00; @viglas-rate], that continuously “explore” alternative plans rather than using long-term cost estimates. Exploration adds overhead even when a good plan has been found; moreover, for joins and other stateful operators, the flow heuristics has been shown to result in state that later incurs significant costs [@stairs]. Other strategies based on filter reordering [@stanford-aqp] are provably optimal, but only work for selection-like predicates. Full-blown cost-based re-optimization can avoid these future costs but was previously only possible on a coarse-grained (high multiple seconds) interval [@tukwila-04; @cape-04]. Our use of declarative techniques to specify the optimizer was inspired in part by the Evita Raced [@evita-raced] system. However, their work aims to construct an entire optimizer using reprogrammable datalog rules, whereas our goal is to effectively perform incremental maintenance of the output query plan. We seek to fully match the pruning techniques of conventional optimizers following the System R [@systemR] and Volcano [@volcano] models. Our results show for the first time that a declarative optimizer *can* be competitive with a procedural one, even for one-time “static” optimizations, and produce large benefits for future optimizations.
Conclusions and Future Work
===========================
\[sec:conclusion\]
To build large-scale, pipelined query processors that are reactive to conditions across a cluster, we must develop new adaptive query processing techniques. This paper represents the first step towards that goal: namely, a fully cost-based architecture for incrementally re-optimizing queries. We have made the following contributions:
A rule-based, declarative approach to query (re)optimization in adaptive query processing systems.
Novel optimization techniques to prune the optimizer state: *aggregate selection*, *reference counting*, and *recursive bounding*.
A formulation of query re-optimization as an incremental view maintenance problem, for which we develop novel incremental algorithms to deal with insertions, deletions and updates over runtime cost parameters.
An implementation over the ASPEN query engine [@recursive-views], with a comprehensive evaluation of performance against alternative approaches, over a diverse workload, showing order-of-magnitude speedups for incremental re-optimization.
We believe this basic architecture leaves a great deal of room for future exploration. We plan to study how our declarative execution model parallelizes across multi-core hardware and clusters, and how it can be extended to consider the cost of *changing* a plan given existing query execution state.
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abstract: 'In a diffusion process on a network, how many nodes are expected to be influenced by a set of initial spreaders? This natural problem, often referred to as *influence estimation*, boils down to computing the marginal probability that a given node is active at a given time when the process starts from specified initial condition. Among many other applications, this task is crucial for a well-studied problem of *influence maximization*: finding optimal spreaders in a social network that maximize the influence spread by a certain time horizon. Indeed, influence estimation needs to be called multiple times for comparing candidate seed sets. Unfortunately, in many models of interest an exact computation of marginals is \#P-hard. In practice, influence is often estimated using Monte-Carlo sampling methods that require a large number of runs for obtaining a high-fidelity prediction, especially at large times. It is thus desirable to develop analytic techniques as an alternative to sampling methods. Here, we suggest an algorithm for estimating the influence function in popular *independent cascade* model based on a scalable *dynamic message-passing* approach. This method has a computational complexity of a single Monte-Carlo simulation and provides an upper bound on the expected spread on a general graph, yielding exact answer for treelike networks. We also provide dynamic message-passing equations for a stochastic version of the *linear threshold* model. The resulting saving of a potentially large sampling factor in the running time compared to simulation-based techniques hence makes it possible to address large-scale problem instances.'
bibliography:
- 'biblio.bib'
title: Scalable Influence Estimation Without Sampling
---
Influence Estimation, Dynamic Message-Passing, Independent Cascade Model, Linear Threshold Model, Message Passing Algorithm
Introduction
============
Accurate prediction of an outcome of information diffusion from a given set of initial spreaders is a challenging problem known as *influence estimation*. This is one of the first natural questions that one would like to answer when studying a particular spreading process. Examples include estimation of the size of epidemic outbreak for determining the necessary quarantine and vaccination measures [@hethcote2000mathematics]; forecasting the impact of cascading failures in critical infrastructures for determining the set of control actions preventing the outage [@dobson2007complex]; or prediction of the outcome of a marketing or political campaign for an optimal use of limited budget and resources [@Domingos:2001:MNV:502512.502525]. This last application motivates a popular *influence maximization* problem, pivotal for efficient marketing, opinion setting and other spreading processes within social networks, where the task is broadly defined as identifying a given number of individual constituents who will maximize the spread of information, or influence, within a certain time window [@Domingos:2001:MNV:502512.502525]. This problem was first mathematically formulated for the Independent Cascade (IC) [@goldenberg2001talk] and Linear Threshold (LT) [@granovetter1978threshold] models by Kempe et al. in [@kempe2003maximizing]. A remarkable result for the NP-hard seed selection problem states that a simple greedy algorithm guarantees a $(1-1/e)$ approximation to the optimal solution provided the oracle value for the influence function. The challenge is that the exact computation of the influence function has been proved to be \#P-hard [@chen:2010:SIM:1835804.1835934] itself. Henceforth, in practice one typically resorts to approximation methods for carrying out the influence estimation task. Existing approaches to influence estimation can be broadly classified into two categories: sampling-based techniques and analytical methods, with or without rigorous guarantees. Without aiming at providing an extensive survey of existing methods, in the Related Work section below we point out most relevant references to the present work. In this paper, we build on *dynamic message-passing* (DMP) approach and develop two low complexity algorithms for an accurate estimation of the influence function in IC model in an arbitrary time frame: <span style="font-variant:small-caps;">DMPest</span> and <span style="font-variant:small-caps;">DMPinf</span>, for estimating the influence function at finite and infinite time horizons, respectively. <span style="font-variant:small-caps;">DMPest</span> and <span style="font-variant:small-caps;">DMPinf</span> correctly deal with the exclusion of the influence of the node to be updated, and estimate the spread without any restrictions on the values of transmission probabilities. Importantly, both algorithms scale only linearly with the number of edges in the graph for arbitrary initial conditions, including the probabilistic seeding assignment. We prove that DMP-estimated influence is exact on tree graphs and on graphs with the size of the loops that is larger than the time horizon of interest, and still provides an upper bound on exact influence on general loopy graphs. As the main focus of this work, we provide numerical evidence for the accuracy and scalability of the DMP-based approach, and conclude with a few remarks on perspectives of using DMP method for reducing the number of simulations in sampling-based approaches, as well as on possible extensions.
Related work
============
Sampling methods
----------------
Monte-Carlo sampling represents a natural class of methods used for estimating the influence function [@kempe2003maximizing; @Chen:2009:EIM:1557019.1557047; @du2013scalable; @cohen2014sketch; @lucier2015influence; @nguyen2017outward]. The crucial bottleneck within this approach consists in the time complexity of Monte-Carlo simulations that grows with the number of graph edges $\vert E \vert$, campaign deadline $T$ and the number of runs $R$ that are required to achieve a desired accuracy of influence estimate. The work [@du2013scalable] estimates the number of samples required to achieve a certain accuracy in continuous-time independent cascade model, which however depends on the actual unknown influence function. The sketching approach of [@cohen2014sketch] construct oracles guarantee constant-factor approximation approximations to the influence functions through samples of influence graphs, however the number of required samples can be quadratic in the size of the graph. Recent works [@lucier2015influence] and [@nguyen2017outward] provide advanced algorithms for reducing sampling requirements, and explore the possibility of distributed computations that is essential for scaling up the problem size. However, even with these improved algorithms, the required effective sampling factor $R$ scales at least linearly with the size of the network instance. Hence, it is natural to search for methods that might potentially save on these factors necessary for sampling methods.
Analytical methods
------------------
Given the potentially high cost of sampling methods, it is desirable to develop approximate influence estimation methods that do not rely on sampling, but still provide guarantees on the quality of the approximation. <span style="font-variant:small-caps;">SteadyStateSpread</span> algorithm for approximating influence based on a direct application of the model probability rule has been suggested in [@aggarwal2011flow]. However, this algorithm does not correctly account for the activation of the node to be updated, and hence is not exact even on tree graphs. This fact has been noticed in subsequent works [@zhang2013probabilistic] where inclusion-exclusion terms in the update step have been considered, and in [@yang2012approximation] where an approximation based on the linearization of the IC model update rule has been exploited. The paper [@zhou2015upper] provides an upper bound on the influence function through an approximate relation of spread at subsequent times via matrix of transmission probabilities. This bound constitutes a basis for the <span style="font-variant:small-caps;">UBLF</span> algorithm for influence maximization [@zhou2013ublf]. However, this approach suffers from similar shortcomings as equations in [@aggarwal2011flow], and overestimates the spread even on tree graphs, while asymptotic approximation is obtained only for small transmission probabilities and large number of nodes in the graph. Moreover, convergence analysis of the series requires transmission probabilities to be strictly smaller than one, and the resulting algorithm requires inversion of a system-large matrix which is potentially time-consuming. Lastly, several other spectral methods have been developed for upper-bounding the influence at infinite time based on finding the spectral radius of the adjacency matrix [@draief2006thresholds], Hazard matrix [@lemonnier2014tight], or its refinement accounting for the sensitive edges [@lee2016spectral].
Several analytical approaches have been developed for particular cases or approximations to the IC model. For instance, [@kimura2006tractable] treats the variant of the IC model where propagation only occurs through shortest paths on the network. [@liu2014influence] provides an upper bound on the spread, but for the linear model where influence flowing into a node is a linear combination of influences flowing from its neighbors, instead of a product leading to non-linear expressions in the original IC model. Both approaches work as approximations for IC model only for small enough propagation probabilities. Exact equations for estimating influence of a single seed on a tree graph have been derived in [@jung2012irie]; however, the general case of an arbitrary set of active nodes in the approach of [@jung2012irie] requires the use of an oracle influence estimation algorithm or sampling as a subroutine.
The main challenge in the analytical approach is due to the fact influence estimation represents an NP-hard inference problem of computation of marginal probabilities in loopy graphical models. However, there exists a class of graphical model algorithms that are specifically tailored to tackle this problem. These techniques are related to loopy belief propagation method, and are commonly referred to as *message-passing algorithms* [@Mezard:2009:IPC:1592967; @Wainwright:2008:GME:1498840.1498841]. Recently, some of these methods have been generalized for selected dynamical models, leading to heuristic algorithms for approximating activation probabilities in several discrete and continuous-time epidemic, threshold and rumor spreading models [@PhysRevE.82.016101; @altarelli2013large; @PhysRevX.4.021024; @shrestha2014message; @PhysRevE.91.012811]. Message passing-type equations that are most intimately related to the infinite-time case in the present work have appeared in recent work focusing on the computation of the cascade size distribution for homogeneous [@burkholz2019efficient] and heterogeneous [@burkholz2019cascade] activation probabilities, and in [@abbe2017nonbacktracking] where similar expressions have been used for establishing upper bounds on the spread at infinite time using the non-backtracking walks approach, previously used for community detection problems [@krzakala2013spectral]. Finally, as noted in [@lokhov2016reconstructing], the finite-time equations for the Independent Cascade model presented in this work are related to the dynamic equations for the susceptible-infected-recovered model [@volz2008sir; @PhysRevE.82.016101; @miller2011note; @PhysRevX.4.021024; @PhysRevE.91.012811] used for modeling some epidemic spreading processes, in the limit of deterministic recovery rate; as we will see below, a particular dynamical rule for the Independent Cascade model allows one to obtain simplified equations that contain only one type of dynamical variables, or messages, which make them practical for analysis and implementation.
Additional Related Work
-----------------------
A large body of work has been focused on improving the scalability of sampling methods, but acting as a subroutine for drastically reducing the number of unnecessary computations inside the greedy approach to the influence maximization problem; see [@Leskovec:2007:COD:1281192.1281239; @goyal2011celf++; @Borgs:2014:MSI:2634074.2634144; @Tang:2014:IMN:2588555.2593670; @Tang:2015:IMN:2723372.2723734] for a non-exhaustive list. Most of these methods are related to influence estimation in the sense that they attempt to carefully prune the nodes that are not important from the influence maximization perspective, see [@doi:10.2200/S00527ED1V01Y201308DTM037] and [@v011a004] for recent reviews. However, they do not necessarily provide guarantees for the influence estimation problem considered here. Notice that in the present paper, we focus on the task of accurate influence estimation as a general problem that has applications for problems beyond influence maximization, for instance identifying the origin of the influence spread from measurements at sparsely located sensors [@shah2011rumors; @lokhov2014inferring], or estimation of model parameters from partially observed samples [@lokhov2016reconstructing; @lokhov2015efficient]. In addition, there exists a large number of heuristic methods that aim at estimating the influence by completely neglecting the dynamic model, e.g. those based on variants of random selection, weighted degree distributions and node centralities. Although these techniques may scale well, we do not review them here as their performance is not guaranteed and significantly varies in quality depending on the considered model and setting. Finally, let us also mention recent papers that address the problem of direct estimation of influence functions from samples [@NIPS2015_5989; @du2014influence]. This line of research is in some sense orthogonal to the present contribution where we assume a well-defined model and develop an analytical method for scalable estimation of the influence function.
Model and Problem Definition {#sec:Problem}
============================
Model
-----
We study the popular model used in the studies of influence estimation and maximization, the discrete-time Independent Cascade model [@kempe2003maximizing]. An instance of the IC model is defined on a graph $G=(V,E)$ with $V$ and $E \subseteq V \times V$ denoting the set of nodes and pairwise edges, respectively. At any time, a node $i$ can be in either inactive or active state. A node $i$ activated at a time $t$ has a single chance to activate its neighbor $j$ at the subsequent time $t+1$ with probability $b_{ij}$ associated with the edge $(i,j) \in E$. Each realization of this diffusion process can be interpreted in terms of a live-edge graph [@kempe2003maximizing] defined on the set of nodes $V$ and described by a set of binary random variables $\mathbf{d} = \{ d_{ij} \}_{(i,j) \in E}$ associated with edges in the graph. Each edge $(i,j) \in E$ is declared *live* randomly with probability $b_{ij}$ (in which case we set $d_{ij} = 1$), and *blocked* otherwise (and thus characterized by $d_{ij} = 0$). Given the *seeds*, i.e. a set $S$ of active nodes at initial time $t=0$, the set of eventually influenced nodes is given by the set of nodes that are connected to nodes in $S$ via the live edges. We will say that node $i$ is *reachable* from node $j$ in time $t$ on graph $G$ if there exists a path connecting $i$ and $j$ such that $d_{ij} = 1$ along all edges of this path and its length is smaller or equal to $t$. Additionally, for the reasons that will become clear later, in the definition of reachable nodes we exclude paths that traverse the same edge multiple times.
Another model that has been considered in the context of influence estimation and maximization is the Linear Threshold model [@granovetter1978threshold], where to each node $i$ is associated a certain activation threshold $\theta_i \in [0,1]$, and $i$ activates if the following condition is satisfied: $\sum_{j \in \partial i} b_{ji} x_{j} \geq \theta_i$, where $\partial i$ denotes the set of neighbors of $i$, entries of $\mathbf{b}$ satisfy $\sum_{j \in \partial i} b_{ji} \leq 1$, and $x_{j} = 1$ if $j$ is active and $x_{j} = 0$ otherwise. We do not focus on this model in the present work because the original version of this model assumes a deterministic dynamics [@granovetter1978threshold; @kempe2003maximizing], where influence can be easily estimated in linear time. It is however possible to generalize this model by introducing a stochastic activation rule and non-deterministic initial conditions. For completeness, in Appendix \[app:LTM\] we discuss the dynamic message-passing equations for the LT model, variants of which have previously appeared in the literature for different settings related to this model, see [@ohta2010universal; @altarelli2013large; @shrestha2014message; @PhysRevE.91.012811].
Influence function
------------------
The object of our main interest is the so-called *influence function* $\sigma(S)$, that represents the expected number of ultimately influenced nodes averaged over the stochasticity of transmission probabilities $\mathbf{b} = \{ b_{ij} \}_{ij \in E}$, or, equivalently, over the realization of $\mathbf{d}$. As discussed above, this object is crucial in the classical *influence maximization* problem that attempts to find the set of seeds of size $k$ leading to the maximum value of the influence. Although the influence function is essentially defined in the large time limit when the diffusion process stops, in many real-world scenarios with a very large number of nodes it might be desirable to predict the expected outcome at finite time horizon $T$, which may for instance represent the deadline of a marketing campaign [@du2013scalable]. Let $\sigma_{t}(S)$ be the expected number of active nodes at time $t$; with this definition, the original influence function $\sigma(S)$ can be equivalently denoted as $\sigma_{\infty}(S)$.
Classical formulations of the influence estimation and maximization problems assume a deterministic selection of the initial influence set of nodes. At the same time, it is easy to think of real-world applications with limited allocation budget, where an access to a subset of nodes for initial seeding is limited or influencing any desirable node is too costly; there is always a possibility that the initial seed might “change its mind”. Instead, one may imagine a scenario of massive targeted advertisement campaign that attempts to reach specific nodes by spending more or less resources on implementing the initial seeding. Formally, assume that each node $i$ at initial time is activated independently with probability $p_i(0)$, so that the sum of these probabilities over the entire graph is equal to the total available budget: $\sum_{i \in V} p_i(0) = k$. In this setting, the influence function should be generalized to take into account an arbitrary initial condition $\mathbf{p}_{0} = (p_{1}(0), \ldots, p_{\vert V \vert}(0))$ of the type described above. It may therefore be advantageous to develop influence estimation method that can cope with this extension to probabilistic initial condition. Obviously, the case of classical seeds represents a particular case where probabilities $p_i(0)$ for respective nodes are set to one or zero.
Marginal activation probabilities
---------------------------------
Let us denote $p_i(t)$ the probability that node $i$ is active at time $t$, and $t_i$ the time at which node $i$ gets activated (with a convention $t_i = T+1$ if node $i$ does not get activated before the time horizon $T$). Then, given the initial condition $\mathbf{p}_{0}$, the influence function at time $t$ can be expressed as a sum over marginal probabilities that nodes get activated before time $t$: $$\sigma_{t}(\mathbf{p_{0}}) = \mathbb{E} \left[\sum_{i \in V}\mathds{1}[t_{i} \leq t]\right] = \sum_{i \in V} p_i(t),
\label{eq:exact_influence}$$ where the expectation is taken over the realizations of $t_{i}$. Therefore, for producing an accurate estimation of the influence function $\sigma_{t}(\mathbf{p}_{0})$, it is crucial to estimate marginal probabilities $p_i(t)$. In the next section, we introduce our *dynamic message-passing* approach to the computation of $p_i(t)$ in the IC model.
Dynamic Message-Passing Method {#sec:DMP}
==============================
In this section, we introduce dynamic message-passing equations that will serve as a foundation for our influence estimation algorithm. We will first start with a derivation on tree graphs, and will then state approximate equations that can be used in the case of loopy graphs for estimating marginal activation probabilities $p_i(t)$.
Derivation of DMP equations on trees
------------------------------------
As a starting point, we notice that for each node $i \in V$ the marginal probability $p_i(t)$ can be exactly expressed as $$p_{i}(t) = 1 - \left[ 1 - p_{i}(0) \right] q_{i}(t).
\label{eq:exact_marginal_p}$$ The equation is true for any graph, and straightforwardly conveys the following meaning: the probability that node $i$ is active at time $t$ is given by one minus the probability that $i$ has not been activated by time $t$. This last event is given by the probability $(1 - p_{i}(0))$ that node $i$ was not active initially times $q_{i}(t)$ that denotes the probability that node $i$ was not activated by its neighbors before time $t$.
Unfortunately, it is hard to compute the quantity $q_{i}(t)$ on a general graph. However, it is possible to compute $q_{i}(t)$ exactly on tree graphs, whereby $$\begin{aligned}
q_{i}(t) = \prod_{j\in \partial i}q_{j \rightarrow i}(t),
\label{eq:tree_q_computation}\end{aligned}$$ where $\partial i$ denotes the set of neighbors of $i$ on the graph $G$, and $q_{j \rightarrow i}(t)$ represents the probability that node $i$ did not get activated by its neighbor $j$ through the edge $(ji) \in E$ by time $t$. Note that by definition $q_{j \rightarrow i}(t)$ is conditioned on the fact that node $i$ is not active at time $t$.
Figure \[fig:q\_factorization\], Left provides an explanation why the expression is exact for tree graphs. Indeed, if node $i$ is not active, the influence of different branches centered in node $i$ are independent as on a tree by definition they do not have overlapping edges. Therefore, the expression for the marginal can be rewritten as follows: $$\begin{aligned}
p_{i}(t) = 1 - \left[ 1 - p_{i}(0) \right] \prod_{j\in \partial i}q_{j \rightarrow i}(t).
\label{eq:tree_marginal_p}\end{aligned}$$
![[**Left:**]{} Illustration of computation of $q_i(t)$. On a tree (dashed paths not present), contribution of different branches are independent, and hence expressions and are exact. On a loopy graph (dashed paths may exist), the quantities $q_{j \rightarrow i}(t)$ for $j \in \partial i$ are not independent, and factorization becomes only an approximation. [**Right:**]{} Illustration of computation of $q^{(i)}_{j}(t)$. Node $i$ has been removed from the graph, and does not influence $j$. On a tree (dashed paths not present), contribution of different remaining branches are independent, and hence expressions and are exact. On a loopy graph (dashed paths may exist), the quantities $q_{l \rightarrow j}(t)$ for $l \in \partial j \backslash i$ are not independent, and factorization becomes only an approximation. Both approximations and are good in treelike networks, where dashed paths are long, or in the case where transmission probabilities along the dashed path are small.[]{data-label="fig:q_factorization"}](q_marginal "fig:"){width="0.45\columnwidth"} ![[**Left:**]{} Illustration of computation of $q_i(t)$. On a tree (dashed paths not present), contribution of different branches are independent, and hence expressions and are exact. On a loopy graph (dashed paths may exist), the quantities $q_{j \rightarrow i}(t)$ for $j \in \partial i$ are not independent, and factorization becomes only an approximation. [**Right:**]{} Illustration of computation of $q^{(i)}_{j}(t)$. Node $i$ has been removed from the graph, and does not influence $j$. On a tree (dashed paths not present), contribution of different remaining branches are independent, and hence expressions and are exact. On a loopy graph (dashed paths may exist), the quantities $q_{l \rightarrow j}(t)$ for $l \in \partial j \backslash i$ are not independent, and factorization becomes only an approximation. Both approximations and are good in treelike networks, where dashed paths are long, or in the case where transmission probabilities along the dashed path are small.[]{data-label="fig:q_factorization"}](q_message "fig:"){width="0.39\columnwidth"}
Let us now introduce a different quantity, $p_{j \rightarrow i}(t)$, that denotes the probability that $j$ is activated conditioned on the fact that $i$ is not. As previously, $p_{j \rightarrow i}(t)$ can be thought of as an equivalent of $p_{j}(t)$ defined on a graph where $i$ has been deleted together with all its adjacent edges. Therefore, similarly to , $p_{j \rightarrow i}(t)$ can be defined as $$p_{j \rightarrow i}(t) = 1 - \left[ 1 - p_{j}(0) \right] q^{(i)}_{j}(t),
\label{eq:exact_p}$$ where $q^{(i)}_{j}(t)$ is defined in the cavity graph from which node $i$ has been removed. The definition is valid for any graph, but, again, it is difficult to compute the quantity $q^{(i)}_{j}(t)$ on general graph. However, equivalently to , on tree graphs we have $$\begin{aligned}
q^{(i)}_{j}(t) = \prod_{l\in \partial j \backslash i}q_{l \rightarrow j}(t).
\label{eq:tree_q_j_computation}\end{aligned}$$ Equation mimics expression , but for $q^{(i)}_{j}(t)$ instead of $q_{i}(t)$. The difference is that the product in the right hand side of runs over $\partial j \backslash i$ that denotes the set of neighbors of $j$ except $i$; this is due to the definitions of $p_{j \rightarrow i}(t)$ and $q^{(i)}_{j}(t)$ that assume that $i$ is not active at time $t$ (or, alternatively, that $i$ has been removed from the graph), see Figure \[fig:q\_factorization\], Right for an illustration of this concept.
Now, similarly to , the evolution of $p_{j \rightarrow i}(t)$ for times $t \geq 0$ on tree graphs simply reads: $$p_{j \rightarrow i}(t)=1 - \left[ 1 - p_{j}(0) \right] \prod_{l\in \partial j \backslash i}q_{l \rightarrow j}(t).
\label{eq:tree_p_through_q}$$
At this point, let us notice that $q_{j \rightarrow i}(t+1)$ can be expressed as $p_{j \rightarrow i}(t)$ by recalling the definition of these two quantities: $$q_{j \rightarrow i}(t+1) = 1 - b_{ji}p_{j \rightarrow i}(t).
\label{eq:q_through_p}$$ Indeed, the probability that $i$ did not get activated by its neighbor $j$ through the edge $(ji) \in E$ by time $t+1$ is given by one minus that both of the following events occurred: node $j$ got activated by time $t$ (this happens with probability $p_{j \rightarrow i}(t)$), and the edge $(ji)$ was live, i.e. $d_{ji} = 1$ (this happens with probability $b_{ji}$).
Now substituting relation in , we finally obtain the following equations for $t>0$: $$\begin{aligned}
&p_{j \rightarrow i}(t)=1 - \left[ 1 - p_{j}(0) \right] \prod_{l\in \partial j \backslash i}\left[ 1 - b_{lj} p_{l \rightarrow j}(t-1) \right], \label{eq:tree_p}
\\
&p_{i}(t) = 1 - \left[ 1 - p_{i}(0) \right] \prod_{j\in \partial i}\left[ 1 - b_{ji} p_{j \rightarrow i}(t-1) \right].
% \label{eq:tree_marginal_p}\end{aligned}$$ Once the conditional probabilities $p_{j \rightarrow i}(t)$ are obtained by running the system of equations on graph edges , the marginal probabilities $p_i(t)$ are estimated through equation . Combined with the initialization $$\begin{aligned}
p_{i \rightarrow j}(0) = p_{i}(0) \quad \forall (i,j) \in E,
\label{eq:tree_initialization}\end{aligned}$$ equations - constitute our *dynamic message-passing equations* for computing the marginals in IC model on tree graphs. The reason for this name is that can be interpreted as passing “dynamic messages” (conditional probabilities $p_{j \rightarrow i}(t)$) along the edges of the graph.
DMP equations on arbitrary graphs
---------------------------------
Definitions , and are valid on arbitrary graphs. However, equations and are no longer exact on graphs with loops: as explained in Figure \[fig:q\_factorization\], the factorization over branches rooted at node $i$ is in general no longer valid because quantities $q_{j \rightarrow i}(t)$ are not independent anymore due to potential presence of loops. However, we can still use these expressions as an approximation. A priori, this approximation is of a good quality as long as either of the two conditions are met: (i) the graph is locally treelike, i.e. the length of a typical loop is large; (ii) transmission probabilities are small. In both of these cases, the mutual influence between different $q_{j \rightarrow i}(t)$ becomes negligeble and vanishes while propagating through the loop. In the field of message-passing algorithms, these conditions are usually formally referred to as correlation decay properties, and can be proven for certain models [@Wainwright:2008:GME:1498840.1498841]. Condition (i) is typically met for random graphs for large enough $N$; for instance, the smallest loop in a sparse Erdős-Rényi random graph with coordination number $c$ scales as $\log_c N$ [@Mezard:2009:IPC:1592967].
Taking the considerations above into account, we essentially use equations and as a definition of the algorithm defined on a general graph with loops, replacing exact marginals $p_{i}(t)$ and messages $p_{j \rightarrow i}(t)$ by their estimates that we denote by $\widehat{p}_{i}(t)$ and $\widehat{p}_{j \rightarrow i}(t)$[^1]. For instance, it means that we employ the following approximation for $p_{i}(t)$: $$\begin{aligned}
p_{i}(t) \approx \widehat{p}_{i}(t) = 1 - \left[ 1 - p_{i}(0) \right] \prod_{j\in \partial i}\widehat{q}_{j \rightarrow i}(t),
\label{eq:approx_marginal_p}\end{aligned}$$ where $\widehat{q}_{j \rightarrow i}(t) = 1 - b_{ji}\widehat{p}_{j \rightarrow i}(t)$ represents an estimate of $q_{j \rightarrow i}(t)$.
In the end of the day, DMP equations for IC model on arbitrary graphs read for all $i \in V$ and $(i,j) \in E$: $$\begin{aligned}
&\widehat{p}_{j \rightarrow i}(t)=1 - \left[ 1 - p_{j}(0) \right] \prod_{l\in \partial j \backslash i}\left[ 1 - b_{lj} \widehat{p}_{l \rightarrow j}(t-1) \right], \label{eq:p}
\\
&\widehat{p}_{i}(t) = 1 - \left[ 1 - p_{i}(0) \right] \prod_{j\in \partial i}\left[ 1 - b_{ji} \widehat{p}_{j \rightarrow i}(t-1) \right].
\label{eq:marginal_p}\end{aligned}$$ Estimates $\widehat{p}_{j \rightarrow i}(t)$ are obtained by running the system of equations on graph edges starting with the initialization $$\begin{aligned}
\widehat{p}_{i \rightarrow j}(0) = p_{i}(0) \quad \forall (i,j) \in E.
\label{eq:initialization}\end{aligned}$$ Marginal probabilities $p_i(t)$ are then estimated through equation .
At this point, let us make a connection to existing analytical results for the IC model in the literature. The <span style="font-variant:small-caps;">SteadyStateSpread</span> algorithm of [@aggarwal2011flow] is based on update equations that are similar to , except that $\widehat{p}_{j \rightarrow i}(t-1)$ in the right hand side of the equation is replaced by the estimate of the full marginal probability $\widehat{p}_{j}(t-1)$. Although the resulting expression is similar to the update rule in IC model for a single sampling run, it is easy to see that it leads to incorrect dependencies for activation probabilities even on a tree graph. In particular, it creates an “echo chamber” effect, where node $j$ can influence node $i$ while being influenced by $i$ at the previous step, a situation clearly prohibited by the model. The algorithms of [@kimura2006tractable] and [@zhou2015upper] can suffer from similar effects, while DMP equations correctly excludes this kind of effect via dynamic messages. Interestingly, this effect has been accounted for in an earlier work [@jung2012irie], where equations for estimating influence of a single seed on a tree graph have been derived; however, the authors did not derive the respective equations for the case of an arbitrary set of active nodes, and instead wrote approximate equations that require the use of a different influence estimation algorithm or sampling as a subroutine. In the case of linearized scheme similar to the one in [@liu2014influence], one should be careful and start from DMP equations - that provide a better approximation to the underlying marginal probabilities. It is worth noticing that most analytical methods reported better results in case of small transition probabilities. We anticipate that this may be possible due to the correlation decay of the type described above, and represents a natural behavior of approximation algorithms. Finally, as discussed in the Introduction, expressions similar to the large-time limit (discussed below) of the DMP equations introduced above have appeared as subroutines in [@abbe2017nonbacktracking; @burkholz2019cascade], while finite-time DMP equations are equivalent to those of the SIR model [@PhysRevE.82.016101; @PhysRevE.91.012811] in the limit of deterministic recovery rate, but involve a singe dynamic message due to the peculiarity of dynamic rules of the IC model.
Influence Estimation with DMP {#sec:Estimation}
=============================
In this section, we use DMP equations derived in the previous section to estimate the influence function under IC model.
Influence estimation at finite time
-----------------------------------
DMP equations represent the central part of the <span style="font-variant:small-caps;">DMPest</span> algorithm that we use to provide an estimation $\widehat{\sigma}_{T}(\mathbf{p}_{0})$ of the influence function value , see Algorithm \[alg:DMPest\]. It is easy to see that for influence estimation at finite time horizon $T$, the computational complexity of <span style="font-variant:small-caps;">DMPest</span> is $O(\vert E \vert T)$, i.e. proportional to the complexity of a single Monte-Carlo simulation of the IC dynamics.
Graph $G = (V,E)$, time horizon $T$, transition probabilities $\mathbf{b} = \{ b_{ij} \}_{(ij) \in E}$, initial conditions $\mathbf{p}_{0} = (p_{1}(0), \ldots, p_{\vert V \vert}(0))$ Initialize $\widehat{p}_{i \rightarrow j}(0) = p_{i}(0)$ as in Compute $\widehat{p}_{j \rightarrow i}(t)$ through Compute $\widehat{p}_{i}(T)$ using $\widehat{\sigma}_{T}(\mathbf{p}_{0}) \leftarrow \sum_{i \in V} \widehat{p}_i(T)$ $\widehat{\sigma}_{T}(\mathbf{p}_{0})$
\[alg:DMPest\]
Let us now outline some properties of the DMP equations -. The following theorem suggests that influence estimate obtained through <span style="font-variant:small-caps;">DMPest</span> is exact when $G$ is a tree.
The influence estimate $\widehat{\sigma}_{T}(\mathbf{p}_{0})$ output by <span style="font-variant:small-caps;">DMPest</span> is exact if the underlying graph $G$ is a tree. \[th:tree\]
Directly follows from the derivation of DMP equations in tree networks in the previous section.
Alternatively, the statement of the theorem follows from the fact that DMP equations - can be derived using the general belief propagation approach on time trajectories [@PhysRevE.91.012811] for models with progressive dynamics, while belief propagation equations provide exact marginal probabilities on tree graphs. Perhaps more interestingly, it is possible to show that <span style="font-variant:small-caps;">DMPest</span> algorithm provides an upper bound on the influence value on general loopy graphs, as stated in the following theorem.
For general graphs, the estimate $\widehat{\sigma}_{T}(\mathbf{p}_{0})$ output by <span style="font-variant:small-caps;">DMPest</span> represents an upper bound on the influence function value $\sigma_{T}(\mathbf{p}_{0})$. \[th:bound\]
The proof of the Theorem \[th:bound\] is given in the Appendix \[app:proof\_theorem2\]. Theorem \[th:bound\] states that <span style="font-variant:small-caps;">DMPest</span> can be used to quantify the efficacy of investment in a marketing campaign, providing a guarantee that the spread will not rise above the bound given by <span style="font-variant:small-caps;">DMPest</span>. On the other hand, a careful examination of the proof of Theorem \[th:bound\] conveys that in some common cases (such as locally treelike networks or sparse random graphs) <span style="font-variant:small-caps;">DMPest</span> can provide exact predictions for the influence even in the presence of loops.
Let $L$ denote the length of the shortest loop in the graph $G$. <span style="font-variant:small-caps;">DMPest</span> provides an exact estimate of the influence if $L \geq 2T + 1$. \[corollary\]
In the spirit of the Corollary \[corollary\], it is possible to use spanning trees to construct lower bounds on the influence function. Indeed, in practice, application of <span style="font-variant:small-caps;">DMPest</span> to any spanning tree of the original graph $G$ will provide such a lower bound as it will neglect possible correlations coming from the loops; a similar approach has been previously used in [@abbe2017nonbacktracking] using directed acyclic graphs.
Influence estimation at infinite time
-------------------------------------
DMP equations - have low algorithmic complexity compared to sampling methods, essentially saving a potentially very large multiplicative factor that is needed for gaining accuracy from simulations. Still, the algorithmic complexity of <span style="font-variant:small-caps;">DMPest</span> is linearly proportional to $T$. If one is interested in estimating the influence function to a certain precision $\epsilon$ at infinite time, there is no need to use <span style="font-variant:small-caps;">DMPest</span> with a large artificial bound on $T$: in fact, one can save this factor in computational complexity, as stated in the following Observation.
Influence at infinite time in the IC model can be estimated with an algorithmic complexity $O(\vert E \vert)$ only through the large time limit of DMP equations. \[obs:DMPinf\]
Indeed, taking the $t \to \infty$ limit in equations -, we immediately get the following fixed point equations for conditional probabilities: $$\begin{aligned}
\widehat{p}_{j \rightarrow i}(\infty)=1 - \left[ 1 - p_{j}(0) \right] \prod_{l\in \partial j \backslash i}\left[ 1 - b_{lj} \widehat{p}_{l \rightarrow j}(\infty) \right], \label{eq:p_infty}\end{aligned}$$ These equations can be solved self-consistently, for example by iteration until a certain tolerance $\epsilon$ on the difference between subsequent updates is met. The marginal probabilities that are used to estimate the influence can be computed as follows: $$\begin{aligned}
\widehat{p}_{i}(\infty) = 1 - \left[ 1 - p_{i}(0) \right] \prod_{j\in \partial i}\left[ 1 - b_{ji} \widehat{p}_{j \rightarrow i}(\infty) \right].
\label{eq:marginal_p_infty}\end{aligned}$$ The pseudocode for the <span style="font-variant:small-caps;">DMPinf</span> algorithm based on equations that allows one to estimate influence at infinite time is given in Algorithm \[alg:DMPinf\].
Graph $G = (V,E)$, transition probabilities $\mathbf{b} = \{ b_{ij} \}_{(ij) \in E}$, initial conditions $\mathbf{p}_{0} = (p_{1}(0), \ldots, p_{\vert V \vert}(0))$, tol $\epsilon$ Initialize $\widehat{p}_{i \rightarrow j}(\infty) = p_i(0)$ $\widehat{p}^{\text{new}}_{j \rightarrow i}(\infty) \leftarrow \{ \widehat{p}^{\text{old}}_{l \rightarrow j}(\infty) \}_{l\in \partial j \backslash i}$ through iteration Compute $\widehat{p}_{i}(\infty)$ using $\widehat{\sigma}_{\infty}(\mathbf{p}_{0}) \leftarrow \sum_{i \in V} \widehat{p}_i(\infty)$ $\widehat{\sigma}_{\infty}(\mathbf{p}_{0})$
\[alg:DMPinf\]
Numerical Results {#sec:Numerics}
=================
In this section, our goal is to test the performance of <span style="font-variant:small-caps;">DMPest</span> and <span style="font-variant:small-caps;">DMPinf</span> in practice for real-world networks. Thanks to Theorem \[th:tree\], we know that these algorithms are exact on tree graphs, so we do not present results on tree networks here. As explained in Section \[sec:DMP\], existing heuristic approaches [@aggarwal2011flow; @kimura2006tractable; @zhou2015upper; @liu2014influence; @jung2012irie] do not provide exact answer even in the case of tree graphs. We will focus on testing <span style="font-variant:small-caps;">DMPest</span> and <span style="font-variant:small-caps;">DMPinf</span> on loopy graph instances; Monte Carlo simulations with a large sampling factor will be used as a benchmark tool to produce an estimate of the ground truth marginal probabilities. Notice that this is a more detailed information computed by our algorithms compared to a single sum representing the resulting influence .
We first illustrate the accuracy of <span style="font-variant:small-caps;">DMPinf</span> on a small real-world social network [@opsahl2009clustering]. We chose random transmission probabilities $\mathbf{b}$ distributed independently and uniformly at random for each edge in the interval $[0,1]$ in order to test the impact of heterogeneous parameters (that can be both arbitrarily small and large) on the accuracy of the message-passing procedure. The estimated marginal probabilities are plotted in Figure \[fig:dmp-accuracy-Irvine\], Left against the “ground truth” obtained via $10^{6}$ Monte Carlo simulations for $k=20$ seeds chosen at random, and the topology of the network is sketched in Figure \[fig:dmp-accuracy-Irvine\], Right. It is remarkable that despite the presence of short loops, DMP shows an excellent agreement with the ground truth, saving a huge sampling factor for generating prediction compared to sampling approach. In a synthetic experiment presented in Figure \[fig:dmp-scaling\], we show that <span style="font-variant:small-caps;">DMPest</span> is indeed scalable to very large network instances with sizes beyond hundred of millions of nodes. Moreover, in agreement with the derivations in the Section \[sec:Estimation\], we observe a linear-time scaling of <span style="font-variant:small-caps;">DMPest</span> with the number of edges in the network; in Figure \[fig:dmp-scaling\], this is shown for the family of random regular graphs, where $\vert E \vert \propto \vert V \vert$.
![[**Left:**]{} Scatter plot representing marginals predicted by <span style="font-variant:small-caps;">DMPinf</span> versus “ground truth” obtained by averaging $10^{6}$ Monte Carlo simulations for $T=\infty$ and $k=20$ randomly-selected seeds on a Facebook-like social network with $1899$ nodes and $20,296$ edges that represents an online community for students at University of California, Irvine [@opsahl2009clustering]. The per-node error $\Delta p_{i}(T) = \frac{1}{N} \sum_{j=1}^{N} \vert \widehat{p}_{j}(T) - p^{MC}_{j}(T) \vert$ is equal to $0.0044$ in this example. [**Right:**]{} The topology of the social network used in simulations. In this representation, high-degree nodes are placed on the periphery. This network contains a large number of loops of short length.[]{data-label="fig:dmp-accuracy-Irvine"}](Irvine2 "fig:"){width="0.48\columnwidth"}![[**Left:**]{} Scatter plot representing marginals predicted by <span style="font-variant:small-caps;">DMPinf</span> versus “ground truth” obtained by averaging $10^{6}$ Monte Carlo simulations for $T=\infty$ and $k=20$ randomly-selected seeds on a Facebook-like social network with $1899$ nodes and $20,296$ edges that represents an online community for students at University of California, Irvine [@opsahl2009clustering]. The per-node error $\Delta p_{i}(T) = \frac{1}{N} \sum_{j=1}^{N} \vert \widehat{p}_{j}(T) - p^{MC}_{j}(T) \vert$ is equal to $0.0044$ in this example. [**Right:**]{} The topology of the social network used in simulations. In this representation, high-degree nodes are placed on the periphery. This network contains a large number of loops of short length.[]{data-label="fig:dmp-accuracy-Irvine"}](SocialNetwork "fig:"){width="0.37\columnwidth"}
![Linear computational time scaling of <span style="font-variant:small-caps;">DMPest</span> as a function of number of nodes obtained for $T=10$ on random regular graphs of degree $3$. Influence estimation on a graph of size $10^8$ takes only $20$ sec with <span style="font-variant:small-caps;">DMPest</span>, while MC simulations this task is essentially intractable.[]{data-label="fig:dmp-scaling"}](running_time){width="0.66\columnwidth"}
------------------------------------------------ --------- --------- ------------------- ---------- -----------
Network $N$ $M$ $\Delta p_{i}(T)$ DMP $10^4$ MC
name nodes edges error runtime runtime
UC Irvine social [@opsahl2009clustering] 1899 20,296 0.003002 0.08 sec 10.2 sec
GR collaborations [@leskovec2007graph] 5242 14,484 0.009101 0.04 sec 18.8 sec
Internet autonomous [@nr] 22,963 48,436 0.001593 1.15 sec 1.6 min
Gnutella P2P [@Ripeanu:2002:MGN:613352.613670] 62,586 147,892 0.000363 0.7 sec 5.9 min
Web-sk graph [@boldi2004ubicrawler] 121,422 334,419 0.002523 1.7 sec 20.1 min
Amazon co-purchasing [@yang2015defining] 262,111 899,792 0.000469 6.1 sec 68.3 min
------------------------------------------------ --------- --------- ------------------- ---------- -----------
In Table 1, we provide an extensive benchmarking of the accuracy of <span style="font-variant:small-caps;">DMPest</span> on a number of real-world social and web networks. For these tests, we had to limit the size of networks considered by hundred of thousands, in order to be able to run at least $10^4$ Monte-Carlo simulations for estimating the marginal probabilities. We see that <span style="font-variant:small-caps;">DMPest</span> yields impressively accurate results even on these graphs with loops. Notice that the sampling-based approach becomes prohibitively expensive already for graphs with tens of thousands of nodes, which makes it hard to use them in applications where influence estimation subroutine needs to be called many times. At the same time, <span style="font-variant:small-caps;">DMPest</span> requires only a single run of the DMP equations, and thus results in extremely small running times for these real instances.
Conclusions {#sec:Conclusions}
===========
In this paper, we presented an analytical approach to influence estimation based on dynamic message-passing approach. This method provides an estimation of the influence function with an algorithmic complexity $O(\vert E \vert T)$ for finite-time horizon problems and $O(\vert E \vert)$ for influence estimation at infinite time, which makes it possible to apply the developed algorithms to large-scale problems. Importantly, developed algorithms provide an exact estimation of the influence function on tree graphs, and an upper bound on the influence value on general loopy graphs. DMP-based algorithms should be especially accurate on sparse locally treelike graphs due to diverging size of loops. These practical aspects of the DMP algorithm were at the focus of this work: demonstration of an excellent performance of the algorithm on a variety of real-world network instances, both in terms of the quality of predictions and of the computational complexity.
Due to the upper bounds provided by <span style="font-variant:small-caps;">DMPest</span> and <span style="font-variant:small-caps;">DMPinf</span>, these algorithms can be straightforwardly used in applications where estimation needs to be called many times, e.g. for pruning nodes that have a week influence spreading potential. Corollary \[corollary\] suggests an even more interesting use of <span style="font-variant:small-caps;">DMPest</span> in conjunction with MC sampling methods. Indeed, we know that at finite times <span style="font-variant:small-caps;">DMPest</span> will provide exact influence estimation for nodes that have a treelike neighborhood, and will only make a mistakes in the regions with loops. Therefore, it is possible to use <span style="font-variant:small-caps;">DMPest</span> to save a potentially significant number of samples for providing very accurate influence estimation in sparse regions, and use MC sampling in dense regions of the graph. Moreover, lower bounds that can be obtained in practice by running <span style="font-variant:small-caps;">DMPest</span> and <span style="font-variant:small-caps;">DMPinf</span> on spanning trees of the graph, can further reduce the use of MC simulations if the DMP-estimated marginals appear to be tight. Empirical exploration of these directions is left for future work.
An explicit algorithmic form of the influence functions paves a way towards developing new heuristic DMP-based optimization algorithms for influence maximization problems, similar to how message-passing equations have been used for other models in previous work [@altarelli2013large; @LokhovE8138]. In particular, it would be interesting to explore the settings of non-deterministic seeding as DMP equations are valid for arbitrary factorized probabilistic initial condition, which generalizes the case of fixed seeds. Future work should also focus on the development of robust DMP-based framework for dealing with uncertainty in parameters, which should be particularly useful for applications to the robust version of the influence maximization problem [@He:2016:RIM:2939672.2939760; @Chen:2016:RIM:2939672.2939745].
Dynamic message-passing equations for the stochastic linear threshold model {#app:LTM}
===========================================================================
As discussed in Section \[sec:Problem\], another popular model considered in the context of influence estimation and maximization is the Linear Threshold model. The deterministic version of LT model described in does not present any difficulty from the influence estimation perspective, as running a single simulation is sufficient for evaluating the influence function. However, an algorithm that allows for an analytical estimation of the influence even in the deterministic case can still be relevant for inference, optimization, or learning applications; message-passing equations for the this version of the LT model appeared in [@altarelli2013large], and have essentially the same structure as dynamical equations previously derived for an equivalent (in a certain limit) model from statistical physics, zero-temperature random field Ising model ($T=0$ RFIM), see [@ohta2010universal] for more details. Here, we consider a stochastic version of this model, where the activation of node $i$ happens with probability $\eta_{i}$ when the threshold condition $\sum_{j \in \partial i} b_{ji} x_{j} \geq \theta_i$ is satisfied, and for potentially stochastic initial condition, similarly to the setting discussed above for the IC model. Dynamic message-passing equations for this generalized model have been studied in [@shrestha2014message] for continuous time and in [@PhysRevE.91.012811] for discrete time (in the form of the equivalent $T=0$ RFIM). Here, for completeness we state the DMP equations for the discrete-time LT model as defined above. The notations are equivalent to the ones used in the DMP equations for the IC model. $$\begin{aligned}
\nonumber
p_{i}(t+1) & = (1 - \eta_{i}) p_{i}(t)
\\
& + \eta_{i} \sum_{\{x_k\}_{k \in \partial i}} \hspace{-0.2cm} \mathbbm{1} \left[ \sum_{k \in \partial i} b_{ki}x_k \geq \theta_i \right] \prod_{k \in \partial i: x_k=1} \hspace{-0.1cm} p_{k \to i}(t) \hspace{-0.1cm} \prod_{k \in \partial i: x_k=0} \hspace{-0.1cm} \left[ 1 - p_{k \to i}(t) \right];
\label{eq:LTM_marginal}
\\
\nonumber
p_{i \to j}(t+1) & = (1 - \eta_{i}) p_{i \to j}(t)
\\
& + \eta_{i} \sum_{\{x_k\}_{k \in \partial i \backslash j}} \hspace{-0.2cm} \mathbbm{1} \left[ \sum_{k \in \partial i \backslash j} b_{ki}x_k \geq \theta_i \right] \prod_{k \in \partial i \backslash j: x_k=1} \hspace{-0.1cm} p_{k \to i}(t) \hspace{-0.1cm} \prod_{k \in \partial i \backslash j: x_k=0} \hspace{-0.1cm} \left[ 1 - p_{k \to i}(t) \right],
\label{eq:LTM_message}\end{aligned}$$ supplemented with the initial condition $p_{i \to j}(0) = p_{i}(0)$ for all $i$ and $j$, where $p_{i}(0)$ is a (in general stochastic) initial condition for the node $i$.
Similarly to the case of the DMP equations for the IC model, equations - are exact on tree graphs, see [@PhysRevE.91.012811] for details. Interestingly, it is shown in [@shrestha2014message] that no upper bound through mechanism discussed in Theorem \[th:bound\] exists for the stochastic LT model, although empirical evaluation of the predictions of the DMP equations still shows good agreement when compared to the marginal probabilities of the model. The work [@khim2016computing] derives bounds for the LT model based on the spectral bound approaches, similar to the ones for the IC model discussed in the Introduction.
Proof of Theorem 2 {#app:proof_theorem2}
==================
Before proceeding with the proof of the Theorem \[th:bound\], let us give an equivalent representation for the probability in terms of a live-edge graph: $$q_{i}(t) = \left\langle \prod_{l \in \mathcal{N}^i_t[\mathbf{d}]} \left[1 - p_{l}(0) \right] \right\rangle
\label{eq:q_counting}$$ Here, $\mathcal{N}^i_t[\mathbf{d}]$ denotes the set of nodes from which node $i$ is reachable in time $t$ given a particular realization of $\mathbf{d}$, and $\left\langle \cdot \right\rangle$ is an average with respect to the realizations of $\mathbf{d}$. Equation has the following meaning: the probability that node $i$ did not get activated by time $t$ is given by the average over realizations in which all nodes that are reachable from $i$ were not active at initial time.
Similarly, in the live-edge representation, the probability $q_{j \rightarrow i}(t)$ can be expressed as follows: $$q_{j \rightarrow i}(t) = \left\langle \prod_{l \in \mathcal{N}^{i \leftarrow j}_t[\mathbf{d}]} \left[1 - p_{l}(0) \right] \right\rangle,
\label{eq:q_message_counting}$$ where $\mathcal{N}^{i \leftarrow j}_t[\mathbf{d}]$ is the set of nodes from which node $i$ is reachable in time $t$ given a particular realization of $\mathbf{d}$ with an additional constraint that the corresponding path ends on the edge $(ji) \in E$. Recall that by our definition we exclude paths that cross $i$: first arriving from another neighbor $l \in \partial i$ and $l \neq j$, reaching $j$ and then coming back to $i$, which is consistent with $q_{j \rightarrow i}(t)$ being conditioned on $i$ being non-active. Alternatively, one can think of $\mathcal{N}^{i \leftarrow j}_t[\mathbf{d}]$ as a reachable set defined on a *cavity* graph where all edges outgoing from $i$ have been deleted.
The proof of Theorem \[th:bound\] also makes use of the following two Lemmas that we state below.
Let $f_1(x_1, \ldots, x_n)$, $\ldots$, $f_m(x_1, \ldots, x_n)$ be comonotonic functions, i.e. simultaneously non-increasing or non-decreasing in each of their arguments. Then $$\left\langle \prod_{i=1}^{m} f_i(x_1, \ldots, x_n) \right\rangle \geq \prod_{i=1}^{m} \left\langle f_i(x_1, \ldots, x_n) \right\rangle,$$ where $\left\langle \cdot \right\rangle$ denotes an average over the distribution of random variables $x_1, \ldots, x_n$. \[lemma:chebyshev\]
On general graphs with loops, estimates obtained through satisfy $\widehat{p}_{j \rightarrow i}(t) \geq p_{j \rightarrow i}(t)$ for all $(i,j) \in E$. \[lemma:underestimation\]
The proofs of Lemmas \[lemma:chebyshev\] and \[lemma:underestimation\] are given in the Appendix \[app:proofs\_lemmas\]. The proof technique is similar to [@PhysRevE.82.016101].
The proof starts with the exact expression , and unfolds with the analysis of each of the approximation steps in , and , making use of the live-edge graph representation. Let us first notice that using definitions of the sets $\mathcal{N}^i_t[\mathbf{d}]$ and $\mathcal{N}^{i \leftarrow j}_t[\mathbf{d}]$, we have for all realizations of random variables $\mathbf{d}$: $$\vert \mathcal{N}^i_t[\mathbf{d}] \vert \leq \sum_{j \in \partial i} \vert \mathcal{N}^{i \leftarrow j}_t[\mathbf{d}] \vert.
\label{eq:overcounting_sets}$$ An illustration for this observation is given in Figure \[fig:reachability\]. Assume that $t=3$, and that the realization of $\mathbf{d}$ is the one shown in Figure \[fig:reachability\]: all edges in the vicinity of $i$ are live except the edge $(i,j_3)$. By definition, the reachable sets read for this case: $\mathcal{N}^i_{t=3}[\mathbf{d}] = \{j_1, j_2, l_1, l_2\}$; $\mathcal{N}^{i \leftarrow j_1}_{t=3}[\mathbf{d}] = \{j_1, l_1, l_2\}$; $\mathcal{N}^{i \leftarrow j_2}_{t=3}[\mathbf{d}] = \{j_2, l_1, l_2\}$; and $\mathcal{N}^{i \leftarrow j_3}_{t=3}[\mathbf{d}] = \emptyset$. Notice that $l_1$ and $l_2$ appear in both sets reachable through edges $(i,j_1)$ and $(i,j_2)$, and therefore . On this example, we see that as long as the realization of live edges forms a loop of size smaller than $2t+1$ in the vicinity of $i$, equation will be verified.
![Example used for illustrating the statements of Theorem \[th:bound\]. Given the estimation for node $i$ at $t=3$ and for a given realization of live edges represented by $\mathbf{d}$, both red nodes will be counted twice in the right hand side of , but only once in the left hand side of this expression. []{data-label="fig:reachability"}](Reachability){width="0.37\columnwidth"}
As a consequence of , the following relation $$\left\langle \prod_{l \in \mathcal{N}^i_t[\mathbf{d}]} \left[1 - p_{l}(0) \right] \right\rangle \geq \left\langle \prod_{j \in \partial i} \prod_{l \in \mathcal{N}^{i \leftarrow j}_t[\mathbf{d}]} \left[1 - p_{l}(0) \right] \right\rangle
\label{eq:overcounting}$$ is naturally satisfied because for some $l$ (e.g. $l_1$ and $l_2$ in Figure \[fig:reachability\]) the terms $\left[1 - p_{l}(0) \right] \leq 1$ are counted several times.
The proof of the Theorem follows from the following chain of inequalities: $$\begin{aligned}
p_{i}(T) & \stackrel{(a)}= 1 - \left[ 1 - p_{i}(0) \right] \left\langle \prod_{l \in \mathcal{N}^i_T[\mathbf{d}]} \left[1 - p_{l}(0) \right] \right\rangle
\label{eq:theorem_(a)}
\\
& \stackrel{(b)}\leq 1 - \left[ 1 - p_{i}(0) \right] \left\langle \prod_{j \in \partial i} \prod_{l \in \mathcal{N}^{i \leftarrow j}_T[\mathbf{d}]} \left[1 - p_{l}(0) \right] \right\rangle \\
& \stackrel{(c)}\leq 1 - \left[ 1 - p_{i}(0) \right] \prod_{j \in \partial i} \left\langle \prod_{l \in \mathcal{N}^{i \leftarrow j}_T[\mathbf{d}]} \left[1 - p_{l}(0) \right] \right\rangle \\
& \stackrel{(d)}= 1 - \left[ 1 - p_{i}(0) \right] \prod_{j \in \partial i} q_{j \rightarrow i}(T) \\
& \stackrel{(e)}= 1 - \left[ 1 - p_{i}(0) \right] \prod_{j \in \partial i} \left[ 1 - b_{ji} p_{j \rightarrow i}(T-1) \right]
\label{eq:theorem(e)}
\\
& \stackrel{(f)}\leq 1 - \left[ 1 - p_{i}(0) \right] \prod_{j \in \partial i} \left[ 1 - b_{ji} \widehat{p}_{j \rightarrow i}(T-1) \right] \\
& \stackrel{(g)}= \widehat{p}_{i}(T),
\label{eq:theorem_(g)}\end{aligned}$$ where $(a)$ is simply the definition ; $(b)$ follows from ; $(c)$ follows from the application of Lemma \[lemma:chebyshev\]; $(d)$ is due to the definition of $q_{j \rightarrow i}(t)$ ; $(e)$ expresses the application of the relation ; $(f)$ follows from Lemma \[lemma:underestimation\]; and $(g)$ is the definition of the marginal probability . The statement of the theorem immediately follows from the application of - in : $$\sigma_{T}(\mathbf{p_{0}}) = \sum_{i \in V} p_i(T) \leq \sum_{i \in V} \widehat{p}_i(T) = \widehat{\sigma}_{T}(\mathbf{p_{0}}).$$
Proofs of technical Lemmas {#app:proofs_lemmas}
==========================
The proof for general $n$ and $m$ straightforwardly follows from a subsequent application of the well-known result for the expectation of the product of two comonotonic functions $f(x)$ and $g(x)$ [@Chebyshev]: $$\left\langle f(x) g(x) \right\rangle \geq \left\langle f(x) \right\rangle \left\langle g(x) \right\rangle
\label{eq:chebyshev_two_functions}$$ The simplest way to prove consists in observing that due to the comonotonocity $$\left[ f(x) - f(y) \right] \left[ g(x) - g(y) \right] \geq 0$$ for any $x$ and $y$, and applying expectation to the last expression.
Starting from definition for $p_{j \rightarrow i}(t)$, let us rewrite the probability $q^{(i)}_{j}(t)$ in the live-edge representation: $$q^{(i)}_{j}(t) = \left\langle \prod_{l \in \mathcal{N}^{j,(i)}_t[\mathbf{d}]} \left[1 - p_{l}(0) \right] \right\rangle,$$ where $\mathcal{N}^{j,(i)}_t[\mathbf{d}]$ is the generalization of the reachable set for $j$ in the cavity graph where node $i$ has been removed. The first steps of Lemma’s proof closely follow - while working with the reachable sets in the graph where $i$ is in cavity. Repeating the same arguments as in the in the proof of Theorem \[th:bound\], we prove the following relation: $$p_{j \rightarrow i}(t) \leq 1 - \left[ 1 - p_{j}(0) \right] \prod_{l\in \partial j \backslash i}\left[ 1 - b_{lj} p_{l \rightarrow j}(t-1) \right].
\label{eq:lemma_step}$$ The rest of the proof follows by induction. Due to the initializations , we have $p_{j \rightarrow i}(t) = \widehat{p}_{j \rightarrow i}(t)$ for all $(i,j) \in E$. Let us assume that $p_{j \rightarrow i}(t-1) \leq \widehat{p}_{j \rightarrow i}(t-1)$. Then, substituting $\widehat{p}_{l \rightarrow j}(t-1)$ instead of $p_{l \rightarrow j}(t-1)$ in the right hand side of , we obtain $$p_{j \rightarrow i}(t) \leq \left[ 1 - p_{j}(0) \right] \prod_{l\in \partial j \backslash i}\left[ 1 - b_{lj} \widehat{p}_{l \rightarrow j}(t-1) \right].
\label{eq:lemma_step1}$$ But from the DMP update equation , the right hand side of is equal to $\widehat{p}_{j \rightarrow i}(t)$, from which Lemma’s statement follows: $p_{j \rightarrow i}(t) \leq \widehat{p}_{j \rightarrow i}(t)$.
[^1]: This is similar to how loopy belief propagation algorithm relates to belief propagation algorithm derived on a tree graph [@Wainwright:2008:GME:1498840.1498841].
|
---
abstract: 'We present the first plasma simulations obtained with the code [[dHybridR]{}]{}, a hybrid particle-in-cell code with fluid electrons and both thermal and energetic ions that retain relativistic dynamics. [[dHybridR]{}]{} is constructed to study astrophysical and space-physics problems where a few energetic non-thermal particles (i.e., cosmic rays, CRs) affect the overall dynamics of a non-relativistic plasma, such as CR-driven instabilities, collisionless shocks, magnetic reconnection, turbulence, etc. In this method paper we provide some applications to linear (resonant/non-resonant CR streaming instability) and strongly non-linear (parallel shocks) problems that show the capabilities of the code. In particular, we provide the first self-consistent hybrid runs that show the acceleration of relativistic ions at non-relativistic shocks; CRs develop a power-law in momentum, which translates to a broken power law in energy that exhibits a steepening around the ion rest mass, as predicted by the theory of diffusive shock acceleration. We present examples of 2D [[dHybridR]{}]{} runs relevant for fast shocks in radio supernovae, whose evolution can be followed in real time, and 3D runs of low-Mach-number heliospheric shocks, which can be compared with in-situ spacecraft observations.'
author:
- 'Colby C. Haggerty'
- Damiano Caprioli
bibliography:
- 'Total.bib'
title: '*dHybridR*: a Hybrid–Particle-in-Cell Code Including Relativistic Ion Dynamics'
---
Introduction {#sec:intro}
============
Understanding the generation and dynamical effects of non-thermal, high-energy particles (Cosmic Rays, CRs) in astrophysical plasmas has been an important question since their discovery in the early 20$^{\rm th}$ century [see, e.g., @baade+34; @Fermi49; @chen+75; @krymskii77; @axford+77; @bell78a; @bell78b; @blandford+78 for some representative seminal papers on the acceleration of Galactic CRs]. CRs are ubiquitous throughout the universe and in the Galactic interstellar medium are in equipartition with the thermal plasma and the magnetic fields, despite being very few in number, about $10^{-9}$ times less abundant than thermal protons [e.g., @yoasthull+14 and references therein].
Self-consistent modeling of the non-linear interplay between CRs, thermal plasma, and magnetic fields is a challenging problem and requires kinetic numerical approaches; moreover, such a non-linear physics inherently spans multiple length and time scales. For instance, the gyroradius of a GeV particle is about $10^{12}$cm in the $\mu$G magnetic field typical of heliospheric and interstellar media, significantly larger than electron/ion skin depths, which are of the order of $10^5-10^7$cm for typical densities of about 1 cm$^{-3}$. Accelerators can be several orders of magnitude larger: $\sim 10^{9}$cm for the Earth bow shock, a fraction to a few astronomical units for interplanetary shocks, tens of pc for Galactic supernova remnants and even a few Mpc for radio relics in galaxy clusters.
Fully kinetic plasma models (like Particle-In-Cell, hereafter PIC, or Vlasov codes) can accurately model all of the relevant physics in collisionless systems by evolving the 6-dimensional phase space distribution function of both ions and electrons [e.g., @bl91; @bell+06; @valentini+07; @lapenta12; @palmroth+18]. However, these fully-kinetic simulations require grid sizes and time steps that resolve both the electron and ion dynamics, and because a electron is a factor of 1836 lighter than a proton, the characteristic scales of the electron dynamics are significantly smaller than the ions’. Having to resolve the electron scales limits the ability of such approaches to model the long-term evolution of the ions and, especially, of the CRs.
The hybrid model, which treats ions as kinetic macro-particles that satisfy the Vlasov equation with phase space trajectories evolved by the Lorentz force equation and electrons as a fluid that keeps the system charge neutral, can bridge thermal and non-thermal regimes at the expense of the detailed kinetic electron physics. Hybrid models [see @winske+96; @Lipatov02 for reviews] have been used to study many different plasma problems including shocks [e.g., @winske85; @quest88; @burgess89; @giacalone+92; @giacalone04; @gargate+12; @burgess+13; @burgess+16; @caprioli15p], turbulence [e.g., @karimabadi+14; @matthaeus+15; @pecora+18; @arzamasskiy+19] and magnetic reconnection [e.g., @mandt+94; @shay+01; @le+09].
An implicit assumption of the hybrid model, however, is that the speed of light is taken to be infinitely large, in order to neglect Maxwell’s correction in the Ampère law (see Section \[sec:hybrid\] for more details), which forces the ion dynamics to be non-relativistic. This restriction is significant for modeling the physics of CRs and may raise concerns when simulations are compared with observations.
Alternative approaches have used a kinetic description of CRs, while treating the thermal population as a magneto-hydrodynamical (MHD) fluid [e.g., @zachary+86; @lb00; @bai+15; @vanmarle+18; @mignone+18; @dubois+19]. While these MHD-PIC simulations can capture some CR physics, the gap between thermal and energetic particles requires the injection of CRs in the system to be externally prescribed, rather than modeled from first principles.
In this work we present the first –to our knowledge– hybrid code that includes relativistic ion dynamics, [[dHybridR]{}]{}, which is built upon the massively-parallel Newtonian code *dHybrid* [@gargate+07]. In Section \[sec:hybrid\] we outline the basics of the code and we argue that the set of systems with both thermal and CR populations can be modeled this way without violating any of the hybrid approximations. In Section \[sec:instability\] we compare [[dHybridR]{}]{} simulations of CR streaming instabilities with linear theory predictions and show that the physics of CRs and thermal plasma interaction are being correctly modeled. In Section \[sec:shocks\], we investigate the acceleration of CRs in parallel shocks and the lack thereof in oblique shocks. Finally, in Section \[sec:3D\], we show a 3D simulation of an oblique, low Mach number shock with parameters comparable to the Earth’s bow shock, which exhibits features consistent with very recent [*in-situ*]{} observations [@johlander+16b; @johlander+18].
Hybrid and [[dHybridR]{}]{} {#sec:hybrid}
===========================
The hybrid model for simulating collisionless plasma physics is fundamentally a Monte-Carlo approach to solving the Vlasov–Maxwell system of equations: $$\begin{gathered}
\label{eq:vlas}
\frac{\partial f_s}{\partial t} + {{\bf v}}\cdot {{\bf \nabla}}+ \frac{q_s}{m_s}({{\bf E}} + \frac{\bf v}{c} \times {{\bf B}})\cdot {{\bf \nabla}}_v f = 0\\
\frac{\partial {{\bf B}}}{\partial t} = -c {{\bf \nabla}}\times {{\bf E}}\\
\frac{\partial {{\bf E}}}{\partial t} = c {{\bf \nabla}}\times {{\bf B}} - 4\pi {{\bf J}}\\\label{eq:amp}
{{\bf \nabla}}\cdot {{\bf B}} = 0\\
{{\bf \nabla}}\cdot {{\bf E}} = \sum_s q_sn_s\label{eq:gauss}\end{gathered}$$ where ${{\bf E}}$ and ${{\bf B}}$ are the electric and magnetic fields, $f_s({{\bf x}}, {{\bf v}}, t)$ is the phase-space distribution function for a given species $s$ of particles with charge $q_s$ and mass $m_s$, $n_s \equiv \int f_s d^3v$ is the number density of species $s$ and ${{\bf J}} \equiv \sum_s q_s n_s{{\bf V}}_s$ is the total current, where ${{\bf V}}_s \equiv \int vf_sd^3v/n_s$ is the bulk velocity of each species. In this work only electron–proton plasmas will be considered, but ions with arbitrary mass and charge can be easily accounted for [e.g., @caprioli+17].
The motivation of the hybrid model is to simulate kinetic ion dynamics (i.e., Equation \[eq:vlas\]) on larger length and time scales at the expense of kinetically modeling electron dynamics. In practice this is done by assuming that the electron mass is negligibly small compared to the ion mass. In this way, electrons are treated as a massless charge-neutralizing fluid that enforces quasi-neutrality in system. This corresponds to $n_i = n_e$ and hence to ${{\bf \nabla}}\cdot {{\bf E}} = 0$ (Equation \[eq:gauss\]) and ${{\bf J}} = en_i({{\bf V_i}} - {{\bf V_e}})$.
The evolution of the ions in time are described by Equation \[eq:vlas\]. In practice this is done by approximating the ion distribution function with a large number of *macro-particles* whose motion in phase space is determined by the Lorentz force. For a given set of electromagnetic fields, the macro-particle position and velocity can be advanced in time. The updated positions and velocities can be interpolated onto a grid, returns a fluid density and bulk flow; note that because the electrons are taken to be massless, they do not contribute to the bulk flow.
The electric field, ${\bf E}$, is determined by multiplying the Vlasov equation for the electrons (Equation \[eq:vlas\]) by $m_e {{\bf v}}$ and integrating it over all of velocity space, which yields: $$m_e n_e \left(\frac{\partial}{\partial t} - {{\bf V}}_e \cdot {{\bf \nabla}}\right){{\bf V}}_e = -{{\bf \nabla}}P_e -e n_e{{\bf E}} - \frac{e n_e {{\bf V}}_e}{c}\times {{\bf B}}.$$ Here we introduced the (isotropic) electron pressure $P_e$, which encompasses higher order moments of the electron distribution function. Reapplying the assumption that $n_i = n_e$ in the limit $m_e\ll m_i$, we derive an effective Ohm’s law for the electric field: $$\begin{aligned}
{\bf E}& = &-\frac{{\bf V}_e}{c} \times {\bf B} - \frac{1}{en}{\bm{\nabla}}P_e\\
& =& -\frac{{\bf V}_i}{c} \times {\bf B} + \frac{{\bf J}}{enc} \times {\bf B} - \frac{1}{en}{\bm{\nabla}}P_e\label{eq:ohms}\end{aligned}$$
The next assumption required for the hybrid model is to neglect the displacement current in Ampère’s law (i.e., the time derivative of the electric field) such that ${\bm \nabla} \times {\bf B} = \frac{4\pi}{c}{\bf J}$; this usually referred to as the radiation-free limit, or the *Darwin approximation*. This assumption is often equated with taking the speed of light to be much larger than any other velocity in the system; however, we show below how it may hold even when a small number of relativistic particles are present. Ultimately omitting this term from the hybrid model neglects the role of light waves. Finally, the electron pressure is prescribed by an equation of state, often taken as isotropic and polytropic. This electric field can then be used in Faraday’s law to update the magnetic field and thus yielding a closed set of equations describing the evolution of the systems.
While the behavior of the ions is fully detailed in the hybrid model, the electrons physical description and evolution is more [*ad hoc*]{}. This is evident in choosing the most physically appropriate value of $\gamma_{\rm eff}$ for a polytropic electron equation of state, $P_e \propto n^{\gamma_{\rm eff}}$. It could be argued that the electrons should be adiabatic and so $\gamma_{\rm eff} = 5/3$. However, if the adiabatic description is used in shocks with a large Mach number, electrons cannot increases their entropy at the shock and the downstream electron pressure may end up being orders of magnitude smaller than the ion pressure. If one asserts that the electron and ion downstream pressure should be in equipartition, the Rankine-Hugoniot jump condition may be used to calculate what $\gamma_{\rm eff}$ should be [see the appendix of @caprioli+18 for more details]. More complicated, anisotropic, prescriptions may be needed when dealing with magnetic reconnection [e.g., @le+09]. In this work we will use the equipartition equation of state for shock simulations and the adiabatic one for CR streaming simulations.
Along with the disparate length and time scales, plasma systems can also span multiple scales in velocity space, ranging from thermal particles that make up the bulk of the plasma to CRs with kinetic energies orders of magnitude larger than their rest mass. An implicit assumption of the Darwin model is that the bulk velocities of the system are small relative to the speed of light, and because of this hybrid codes have traditionally not included relativistic effects for the macro-particle ions. However, since this approximation is based on *bulk motions* being small relative to the speed of light, even plasma systems with a non-relativistic background and a small number of relativistic particles (or CRs) can be modeled in this limit. This can be seen from a scaling argument of Ampère’s law with Maxwell’s correction: $${\bm \nabla} \times {\bf B} = \frac{4\pi}{c}{\bf J} + \frac{1}{c}\frac{\partial {\bf E}}{\partial t}
\ \to\ \frac{B}{\lambda} \colon \frac{4\pi J}{c} \colon \frac{E}{c\tau}$$ where derivatives have been replaced by $\lambda$ and $\tau$, which correspond to characteristic length and time scales of the systems we are interested in studying, and the colon ($\colon$) denotes an order of magnitude comparison. We can see from the scaling of Faraday’s law that $E/B \sim V/c$, where $V = \lambda/\tau$ is the characteristic velocity of the system. Using this and that $J \sim enV$ we can simplify our scaling equation to $$1 \colon \frac{\lambda}{d_i}\frac{V}{v_A} \colon \left ( \frac{V}{c} \right )^2$$ where $d_i = c/\omega_{pi} = \sqrt{c^2m_i/4\pi n e^2}$ is the ion inertial length (skin depth) and $v_A = B/\sqrt{4\pi m_i n}$ is the Alfvén speed. Neglecting the displacement current is no longer appropriate when the third term becomes comparable to the other two and so we find that this approximation is good as long as $$\label{eq:speed_cond}
\left(\frac{V}{c}\right)^2 \ll 1,\ {\rm and} \ \frac{Vv_A}{c^2} \ll 1$$ where we used 1 for $\lambda/d_i$, which is the strictest value that can be used for hybrid simulations. The systems that we aim to study are composed of a background ion thermal population with number density $n_{i}$, characteristic velocity $V_{\rm bkg} \ll c$ and a high-energy CR population with $n_{\rm cr} \ll n_{i}$ and $v_{\rm cr} \sim c$. The composite background + CR populations bulk flow speed can be estimated as $$\label{eq:char_flow}
\frac{V}{c} = \frac{n_{i}V_{\rm bkg} + n_{\rm cr}c}{c(n_{i} + n_{\rm cr})}
\approx \frac{V_{\rm bkg}}{c} + \frac{n_{\rm cr}}{n_{i}}$$ From Equations \[eq:speed\_cond\] and \[eq:char\_flow\], we find three conditions the systems must meet for this approximation to be valid:
- $V_{\rm bkg}\ll c$, i.e., bulk flows cannot be relativistic;
- $n_{\rm cr}\ll n_{n_i}$, i.e., the CR number density must be negligible relative to the gas number density;
- $v_A\ll c$, i.e., magnetic field energy density must be much smaller than the rest mass energy density.
The last condition is derived by taking the bulk flow velocity to be Alfvénic; note that $B/\sqrt{4\pi m_i n}$ can even exceed $c$, in which case the dispersion relation for an Alfvén wave needs to be modified by including the displacement current term, thus violating one of the previously outlined assumptions for hybrid [@krall-trivel]. These conditions are satisfied for many systems in space and astrophysical plasmas where CR acceleration, transport and scattering are important.
To study these types of problems we have developed [[dHybridR]{}]{}, a hybrid simulation code that retains the fully relativistic ion dynamics. [[dHybridR]{}]{} is a generalization of the *dHybrid* code [@gargate+07], where the relativistic Lorentz force is used for the ion macro-particle evolution, i.e., $$m_i\frac{d\gamma {\bf v}}{dt} = q{\bf E} + \frac{q{\bf v}}{c}\times {\bf B}$$ where $\gamma$ is the Lorentz factor of a given macro-particle and given by $\gamma = 1/\sqrt{1 - (v/c)^2}$. This is implemented in the code using the well documented relativistic Boris algorithm [see @bl91 for details].
The equations that govern both the electromagnetic fields and the particle dynamics are normalized to arbitrary magnetic field, $B_0$, and number density, $n_0$. Lengths are scaled to the ion inertial length based on this density, $L_0 \equiv d_{i0} = c/\omega_{pi0}$, and time to the inverse ion gyro-frequency based on this magnetic field, $t_0 \equiv \Omega_{ci0}^{-1} = \frac{cm_i}{eB_0}$. Velocities are normalized to the ratio of the length and time normalizations and so a velocity of unity corresponds to the Alfvén speed in the reference magnetic field and density, $v_0 \equiv L_0/t_0 = B_0/\sqrt{4\pi m_i n_0}$. Electric fields are normalized to $B_0v_0/c$ and temperatures and energies to $m_i v_0^2$. Throughout this work simulations are initialized such that the unshocked/background plasma have a magnetic field, density and ion/electron temperature of unity and so the simulation units are effectively normalized to the background/upstream plasma parameters, i.e., $v_0 = v_A = v_{th}$ and $d_{i0} = d_i = c/\omega_{pi} = r_{g,th}$, the gyroradius of the thermal ions. By normalizing the discretized equations in this way, the speed of light only appears as the ratio $c/v_0$ and then only occurs in the Lorentz factor, $\gamma(v) = 1/\sqrt{1 - (v/v_A)^2(v_A/c)^2}$, in the Lorentz force equation. The magnetic field is evolved using a two-step Lax-Wendroff scheme that is second-order accurate in space and time [@bl91; @Hockney]. Further details about the non-relativistic implementation of [[dHybridR]{}]{} are described in [@gargate+07].
The remainder of this paper is dedicated to the demonstration and validation of [[dHybridR]{}]{} simulating CR generation and transport for selected plasma systems, in which a small number of highly-energetic ions affects the dynamics. We will examine the non-resonant streaming instability (commonly referred to as the Bell instability), the resonant streaming instability, and different regimes of collisionless shocks.
In particular, we will study the transition from non-relativistic to relativistic CR energies in fast non-relativistic shocks; since the required timestep is inversely proportional to $c/v_A$, we initially focus on shock environments where $v_A$ is rather large, such as radio supernovae, where $V_{\rm sh}\sim 0.1c$, $B_0 \sim 0.1{\rm G}$ and $n_0 \sim 10^3 {\rm cm}^{-1}$ at the peak of the synchrotron emission [e.g., @cf06]. These parameters correspond to Alfvénic mach numbers $M_A\equiv V_{\rm sh}/v_A\sim 10$ and $c/v_A \sim 100$. Then, we show simulations of lower-Mach number shocks which are more applicable to heliospheric systems, such as planetary bow shocks and interplanetary shocks triggered by coronal mass ejections, where plasma speeds vary between several hundreds to thousands of km s$^{-1}$, corresponding to Mach numbers ranging from 1 to 10 and $c/v_A \gtrsim 10^4$ [@sheeley+85; @cane+03]. Despite the limited spatial extent of such heliospheric systems, trans-relativistic and even relativistic particles can be produced in such environments, too [e.g., @reames99; @tylka+05; @wilson+16; @reames13; @desai+16a].
There are numerous astrophysical systems where $c/v_A$ and $c/v_{\rm sh}$ are considerably larger than the simulations presented in this work; however, as long as there is a clear separation of scales between the thermal/Alfvénic speed, the speed of the shock, and the speed of light, the underlying physics can be studied fruitfully. This idea implies that the physical results from these simulations, and [[dHybridR]{}]{} in general, are potentially applicable to many different astrophysical systems.
Resonant and Non-Resonant Streaming Instability {#sec:instability}
===============================================
To verify that [[dHybridR]{}]{} correctly simulates the physics relevant to systems with CRs, we present two simulations of the CR-driven streaming instability. This occurs when a population of low density energetic CRs drift relative to a thermal population, driving the amplification of magnetic fluctuations perpendicular to the mean field. The characteristics of the instability are controlled by the CR current density: in the weak current limit, CRs trigger the growth of modes that are gyro-resonant with themselves [resonant streaming instability; e.g., @kulsrud+69; @skilling75a; @bell78a; @zweibel03]. In the strong current limit, instead, the return current in background electrons that is needed to enforce charge neutrality drives modes with wavelengths shorter than the CR gyro-radius [non-resonant or Bell instability; e.g., @bell04; @weidl+19a].
The kinetic theory of these instabilities and the transition between the two has been detailed, e.g., by [@amato+09] for a CR distribution $\propto p^{-4}$ in momentum. In this case the boundary between the two regimes is defined by the parameter [see @amato+09]: $$\label{eq:ineq}
\Bar{\sigma} \equiv \frac{4\pi}{c}\frac{r_L}{B}J_{\rm cr} = \frac{n_{\rm cr}}{n_i}\frac{p_{\rm min}v_d}{m_i v_A^2},$$ where $r_L$ is the gyro-radius of the particles with the minimum momentum in the CR distribution, $p_{\rm min}$, and $J_{\rm cr} = e n_{\rm cr} v_d$ is the CR current, defined by the CR number density $n_{\rm cr}$ and their drift velocity $v_d$. For $\Bar{\sigma} \gg 1$ the non-resonant mode grows faster than the resonant one, while for $\Bar{\sigma} \ll 1$ they grow at the same rate [@bell04; @amato+09]. In the resonant regime, because CRs have a velocity spread much larger than the drift velocity, both right- and left-handed magnetic fluctuations are driven, while in the non-resonant case only electron-driven right-handed modes are amplified.
We set up [[dHybridR]{}]{} simulations of the CR streaming instabilities with different $n_{\rm CR}\propto \Bar{\sigma}$ and test both the strong and the weak current regimes. This allows us to probe the non-trivial coupling between CRs, magnetic fields, and thermal background plasma both in a MHD-like (non-resonant) and a purely kinetic (resonant) scenario.
We consider two simulations in periodic domains of size $[L_x,L_y]=[10^4, 5] d_i$ with a uniform magnetic field ${\bf B}=B_0{\bf x}$ and a stationary background population of protons with thermal speed equal to $v_A$. Superimposed on the background population is a lower-density CR population with a power-law distribution in momentum space $f(p) \propto p^{-4}$ extending from $p_{\rm min}/m_i c = 1$ to $p_{\rm max}/m_i c = 10^4$, which is isotropic in a frame moving with a drift velocity $v_d = 10 v_A$. The box transverse size makes the simulations effectively 2D for the thermal background, i.e., it is larger than the gyroradius of thermal ions, but actually 1D in terms of the CR length scales. In both simulations, the speed of light is set to be $c = 100 v_A$ and there are two grid cells per $d_i$; with 225 and 100 macro-particles per cell used for the background and CR populations, respectively. The CR number density relative to the background population is adjusted to trigger either the non-resonant ($n_{\rm cr}/n_i = 10^{-2}$) or the resonant ($n_{\rm cr}/n_i = 10^{-4}$) instability [@bell04; @amato+09]. The time step is chosen to be $dt = 2.5\times 10^{-3} \Omega_{ci}^{-1}$ based on the initial magnetic field such that CRs with $\gamma \gg 1$ and $v \approx c$ do not move more than 1 grid space during each time step.
Each simulation is initialized with a mean magnetic field and no electric fields. However, because of numerical noise inherent to the finite sampling of the ion distribution, there are initially density and bulk flow fluctuations. These fluctuations generate electric fields through Ohm’s law (Equation \[eq:ohms\]), which produces perpendicular magnetic perturbations that act as seeds for the unstable modes. The amplitude of this noise is controlled by the number of macro-particles per cell and for the simulations presented in this work the noise floor is on the order of $\left < B_\perp^2\right >_{\rm noise} \sim 10^{-4} B_0^2$. Changing the number of particles per cell alters the initial noise and changes the time that it takes to achieve saturation, but does not affect either the wavelength or the growth rate of the fastest growing modes.
For the non-resonant (or Bell) regime, the fastest growing mode is right handed (hereafter $k_{\rm max}^+$) and its corresponding growth rate, $\gamma^+_{\rm max}$, reads [@bell04]: $$\frac{\gamma^+_{\rm max}}{\Omega_{ci}} = k^+_{\rm max}d_i =
\frac{1}{2}\frac{n_{\rm cr}}{n_i}\frac{v_d}{v_A},
\label{eq:bell}.$$ Instead, in the resonant regime the fastest growing modes have no preferential helicity and their wavenumbers and growth rate read: $$\label{eq:res}
k^{\pm}_{\rm max}d_i = \frac{m_iv_A}{p_0};\qquad
\frac{\gamma^\pm_{\rm max}}{\Omega_{ci}} \approx
\frac{\pi}{8}\frac{n_{\rm cr}}{n_i}\frac{v_d}{v_A},$$ where the $\pm$ superscripts refer to the right and left handed modes, respectively; Equation \[eq:res\] is calculated by Taylor-expanding equation 28 in [@amato+09] in terms of the small parameter $n_{\rm cr}v_d p_0/(n_im_iv_A^2)$ and keeping only the linear term.
{width="48.00000%"}
, as a function of wave number $k$ and time for a 1D simulation of the non-resonant streaming instability. Top panel: spectrum as a function of $k d_i$, where each color corresponds to a different time in the simulation; the vertical black dashed line corresponds to the $k_{\rm max}$ predicted by the linear theory (Equation \[eq:bell\]). Middle and bottom panels: Magnetic power in both right-handed ($|F_+|^2 = |F_x + iF_z|^2$) and left-handed ($|F_-|^2 =|F_x - iF_z|^2$) modes as a function of time; the dashed lines show the growth rates predicted by the linear theory [equation 28 of @amato+09].\[fig:bell\]
![As in Figure \[fig:bell\] for a 1D simulation of the resonant streaming instability. The theoretical expectations are from Equation \[eq:res\].[]{data-label="fig:ressonant"}](fig_resonant3.png){width="48.00000%"}
To compare these predictions with the simulations, we introduce $F_{i} = {\rm FFT}[B_{i}]$, for $i=y,z$, where ${\rm FFT}$ is the discreet fast Fourier transform calculated along the $x$ direction. The magnetic power spectrum $|F_y|^2+|F_z|^2$ is plotted in the first panels of Figure \[fig:bell\] and Figure \[fig:ressonant\] for the non-resonant and resonant cases, respectively. In both figures the color corresponds to different times in the simulation and the black dashed line shows $k_{\rm max}$ predicted by Eq.\[eq:bell\] and \[eq:res\]. There is good agreement between theory and simulation for the location of the fastest growing modes.
The second and third panels of Figure \[fig:bell\] and Figure \[fig:ressonant\] show the value of the magnetic power in right ($|F_+|^2 = |F_y + iF_z|^2$) and left ($|F_-|^2 = |F_y - iF_z|^2$) handed modes as a function of time for the value of $k_{\rm max}$ denoted by the black dashed line in the first panel. The magnetic energy is expected to increase exponentially in time as $|F_\pm|^2 \propto e^{2\gamma^\pm_{\rm max} t}$ and the black dashed line corresponds to the $2\gamma_{\rm max}$ given by Equation \[eq:bell\] and \[eq:res\]; there is a general agreement between theory and simulations in both the non-resonant and resonant cases. Note that the black dashed line in the bottom panel of Figure \[fig:bell\] is calculated using Equation 28 in [@amato+09].
It is worth noting the differences in the time and length scales of the two instabilities simulated. The resonant instability stems out from a gyro-resonant interaction with the CR population [e.g., @kulsrud+69] and amplifies magnetic fluctuations on scales comparable to the CR gyroradius.
Note that, since $\Bar{\sigma}\ll 1$ for the resonant instability, the growth rates are small compared to the cyclotron frequency of the background population (Equation \[eq:res\]); yet, [[dHybridR]{}]{} can accurately capture this phenomenon over more than $10^4$ cyclotron times (about $4\times 10^6$ time steps).
Recent works have tackled the study of the non-resonant instability with PIC and hybrid simulations [e.g., @ohira+09; @riquelme+09; @gargate+10] and of the resonant instability with PIC and PIC-MHD simulations [e.g. @Bai+19; @holcomb+19; @weidl+19b]; these studies have generally found results consistent with theory for the fasting growing mode and corresponding growth rate for the linear phase. Nevertheless, the saturation of the CR streaming instability is a complex and non-linear physical phenomenon that is not yet completely understood. A detailed examination of properties of the two CR streaming instabilities using [[dHybridR]{}]{}, as well as a more thorough comparison with previous works, is in preparation [see @Zacharegkas+19; @haggerty+19p for preliminary results]. The agreement between simulations and the linear theory verifies that [[dHybridR]{}]{} can accurately model the physical coupling of the thermal background plasma and a drifting CR population for quasi-linear problems, both in the strong and week current regimes.
Non-Relativistic Shocks {#sec:shocks}
=======================
Setup and Simulation Parameters
-------------------------------
Shock simulations were performed with [[dHybridR]{}]{} following the set up described in [@gargate+12]. The simulations are performed in 2.5D (2D in real space, and 3D in momentum space) on a regular Cartesian grid, with periodic boundary conditions in the $y$ direction (transverse to the shock), open on the right boundary ($+x$ direction or normal and upstream of the shock), and a conducting reflecting wall on the left boundary ($-x$ direction and downstream of the shock). The derivative along x of $E_x,\ B_y,\ {\rm and}\ B_z$ through the left boundary is zero, while $E_y = E_z = 0$ and $B_x = B_x(t=0)$ in the wall. The shock is formed by initializing the plasma with a bulk flow in the $-x$ direction; the plasma closest to the left wall is reflected and begins streaming in the $+x$ direction. This configuration is unstable and within $\sim10 \Omega_{ci}^{-1}$ a shock forms and travels upstream. Across the shock fluid quantities satisfy the Rankine–Hugoniot (RH) jump conditions. For the simulations in this study, the initial/upstream magnetic field and density are set to unity and the initial magnetic field points in the first quadrant of the $x,y$ plane, the shock angle is measured relative to the positive x direction (normal to the shock) $\vartheta_{Bn}$ (e.g., for a parallel shock ${\bf B} = B_0 {\bf \hat{x}}$ and $\vartheta_{Bn} = 0$). The initial ion thermal velocity is equal to the upstream Alfvén speed and the electron temperature is equal to the ion temperature ($T_0 = T_i = T_e$). Following previous hybrid shock simulations [e.g., @gargate+12; @caprioli+14a], a polytropic index for the electron equation of state is selected so that the downstream electron thermal energy will be half of the upstream kinetic energy in the shock frame [also see @caprioli+18 for more details].
Run $M$ $c/V_A$ $L_x/d_i$ $L_\perp/d_i$ $\Delta x/d_i$ $\Delta t \Omega_{ci}$ $\vartheta_{Bn}^{\circ}$
----- ----- --------- ---------------- --------------- ---------------- ------------------------ -------------------------- --
A 20 200 $8\times 10^5$ 200 0.5 .0025 0
B 15 50 $10^5$ 150 0.5 .005 0
C 30 10000 $10^4$ 2700 0.5 .0025 70
3D 5 100 1000 100 0.5 .02 70
: Parameters for the shock simulations presented in this work. From left to right: Alfvénic mach number (i.e., $v_{\rm sh}/V_A$), speed of light, longitudinal ($L_x$) and transverse ($L_\perp$) box sizes, spacial grid resolution, time step and angle of the initial magnetic field relative to the upstream plasma bulk flow. Note, the time step in simulations C and 3D are set by the speed of fastest particles in the simulation, not the speed of light.[]{data-label="tab:shocks"}
Shocks are parametrized by their Alfvénic and sonic Mach numbers, $M_A = v_{\rm sh}/v_A$ and $M_s = v_{\rm sh}/v_s = v_{\rm sh}/\sqrt{2\gamma k_B T_0/m_i}$, where $v_{\rm sh}$ is the upstream velocity in the lab/simulation frame (i.e., in the frame where the downstream medium is at rest). The choice of temperature in these simulations links the two Mach numbers, $M_A = \sqrt{10/3}M_s$ and in this work we will reference the Mach number as simply $M \equiv M_A \simeq M_s$. We use $2$ grid cells per $d_{i}$ and 4 particles per cell. The time step is chosen such that the fastest ion will travel at most one grid cell in one time step. For the parallel shock simulations this corresponds to $\Delta x/ \Delta t < p_{\rm max}/\gamma_{\rm max} \lesssim c$ and $\Delta x/ \Delta t < 3v_{sh}$ for the perpendicular case.
Simulations were run for thousands of cyclotron times to model the CRs transition from non-relativistic to relativistic energies. The largest and longest run of these simulations is shown in Figure \[fig:overview\] at the end of the simulation, which shows various plasma/fluid quantities around the shock. For this run we used $M = 20$, $c = 200 V_{A0}$ and $[L_x,L_y] = [8\times 10^5, 200] d_{i0}$. The speed of light limiting the fastest speed in our simulation allowed us to run unprecedentedly-long hybrid simulations of non-relativistic shocks, up to $\sim 6000 \Omega_{ci}^{-1}$ before the highest-energy CRs began to escape from the box.
Momentum and Energy
-------------------
Consistent with results from previous non-relativistic hybrid simulations of parallel shocks [e.g., @giacalone+97; @burgess+12; @caprioli+14a; @caprioli+14b; @caprioli+14c], we find that thermal ions can be spontaneously energized into an extended power-law distribution. Figure \[fig:p\] shows the post-shock distribution function as a function of both the ion velocity normalized to $c$ (first panel) and the ion momentum normalized $m_ic$ (the second panel). The majority of ions are thermally heated by the shock, forming the Gaussian peak around $p/m_i \sim v \sim 0.1c$; the black dashed lines correspond to a Gaussian with temperature reduced by $20\%$ with respect to the one predicted by the RH conditions for a mono-atomic ideal gas. The deviation form the prediction is consistent with the amount of energy (about $10-20\%$ of the shock ram energy) that is channeled in the non-thermal power-law distribution that develops beyond $v\sim 0.2c$, whose extent increases with time (color code). The velocity spectra cuts off at $v \leq c$ as expected, however the momentum continues to extend with the same slope beyond $p\gtrsim m_i c$. The distributions shown in Figure \[fig:p\] are multiplied by $v^{-4}$ and $p^{-4}$, respectively. The very reason why the spectrum looks a bit steeper than $p^{-4}$ has profound physical reasons, which will be discussed in a forthcoming paper [see @caprioli+19p for a preliminary discussions].
While the momentum spectra shows a nearly constant power law slope, the energy spectrum should have different slopes in the non-relativistic and relativistic regimes. The energy distribution is linked to the momentum distribution through the conservation of the phase-space volume: $f(E) = 4\pi p^2 f(p) dp/dE$. In the non-relativistic regime, $E \propto p^2$ and so for a momentum power-law index of $q$, the energy distribution should go as $f(E) \propto E^{(1-q)/2}$, i.e., $E^{-1.5}$ for $q = 4$. In the relativistic regime, $E \propto p$ and thus the kinetic energy distribution should scale as $f(E) \propto E^{2-q}$, i.e., the canonical $E^{-2}$ for $q = 4$.
The energy spectrum for our benchmark simulation is shown in Figure \[fig:E\], where the spectrum is multiplied by $E^{1.5}$ in the top panel and $E^2$ in the bottom panel in an attempt to emphasize the agreement with the expected slopes in both the non-relativistic and the relativistic regimes. In this run some particles became relativistic, with $\gamma\gtrsim 5$, but running such a large simulation long enough for the power-law tail to extend beyond $\sim 10 m_ic^2$ is computationally impractical. Thus, in order to see the transition in the energy power-law slope more clearly, we performed a simulation with a smaller speed of light relative to the shock velocity and Alfvén speed (Run B in Table \[tab:shocks\]); the reduced separation of scales allows us to investigate the trans-relativistic regime more easily. Figure \[fig:Ecutoff\] shows the momentum and energy distribution for Run B at $t=2000 \Omega_{ci}^{-1}$ (top and bottom panels, respectively). In the first panel the distribution is fitted with to a power-law $\propto p^{-4}$ multiplied by an exponential cut off at $p_{\rm max} = 9 m_i c$. The bottom panel, shows the energy distribution, multiplied by $E^2$; the black and red dashed lines correspond to the relativistic and classical power-law predictions based on the fit curve from the top panel. In essence, the black line shows the shape of the distribution if $E = m_ic^2(\sqrt{p^2/c^2 + 1} -1)$ and the red line for $E = p^2/2m_i$. In the non-relativistic regime, both the black and red predictions agree well with the measured spectrum; as the distribution extends into the relativistic regime ($E \gtrsim 2 m_i c$), though, there is a clear steepening to the slope of $-2$, followed by the exponential cutoff, which agrees well with the black line prediction. The classical prediction (red line), however, continues to increase for nearly an order of magnitude in energy beyond the actual energy cut-off.
This analysis shows, for the first time in hybrid simulations, how non-relativistic shocks can accelerate particles to ultra-relativistic energies (with Lorentz factors up to $\gamma\gtrsim 20$ in our case), also confirming that DSA produces power-laws *in momentum space* across the non-relativistic and relativistic regimes. These results are consistent with those obtained for both electrons and ions in 1D full-PIC simulations of non-relativistic shocks [@park+15] and for electrons in full-PIC simulations of trans-relativistic shocks [@crumley+19].
One important astrophysical application that stems put from these preliminary runs is relevant for young SNe. In fact, if we consider the typical values for the magnetic field inferred in type Ib & Ic supernova with Wolf-Rayet star progenitors [@cf06], the inverse cyclotron time $\Omega_{ci}^{-1}$ would be on the order of milliseconds. Both the A & B Runs have $M_A$ and $c/v_A$ typical of these systems and so physically these simulations are modeling a few seconds of fast radio SNe. Notably, in these simulations, thermal protons are accelerated to multi-GeV energies in a matter of seconds, which has implications for the generation of $\gamma-$rays and neutrinos, as discussed below.
Rate of Maximum Energy Increase {#sec:maxE}
-------------------------------
An important question regarding DSA is what is the maximum energy, ${E_{\rm max}}(t)$, of the particles produced by a shock with a given speed and magnetic field in a finite amount of time [e.g., @drury83; @lagage+83a; @blasi+07]. When the magnetic field perturbations responsible for particle diffusion is self-generated by the CRs, ${E_{\rm max}}$ is determined by the current in CRs streaming in the upstream medium, ${J_{\rm cr}}$; such a current can be estimated as the number density ${n_{\rm cr}}$ of particles close to the instantaneous ${E_{\rm max}}$, times their velocity, ${v_{\rm cr}}$. For a momentum spectrum $f(p)\propto p^{-4}$, in the non-relativistic regime, one has ${n_{\rm cr}}\propto p_{\rm max}^3f(p)\propto {E_{\rm max}}^{-1/2}$ and ${v_{\rm cr}}\propto {E_{\rm max}}^{1/2}$, so that ${J_{\rm cr}}= e{n_{\rm cr}}{v_{\rm cr}}\simeq $ is constant in time. Conversely, in the relativistic regime ${n_{\rm cr}}\propto {E_{\rm max}}^{-1}$ and ${v_{\rm cr}}\simeq c$, so that ${J_{\rm cr}}\propto {E_{\rm max}}^{-1}$; therefore, the current decreases when the maximum CR energy increases and one may expect a slower amplification of the magnetic field, and in turn a slower rate of increase of ${E_{\rm max}}$. Note that this effect may be partially compensated by the fact that the CR precursor becomes larger when ${E_{\rm max}}$ increases, so that the time available for growing the field (of the order of one advection time on a CR diffusion length) also increases. The net effect in general depends on whether most of the field growth is provided by escaping or diffusing particles, and on the details of the instability saturation [@caprioli+14b].
A change in the rate of increase of $E_{\rm max}$ when ions become relativistic was first investigated by [@bai+15] using a MHD-PIC approach. Note that such a framework requires to specify a priori the fraction of particles that effectively become CRs but —when acceleration becomes efficient— this fraction has to decrease with time to avoid an energy runaway. Since a quantitative theory of how this occurs is still lacking, MHD-PIC methods cannot investigate the long-term evolution of the shock self-consistently.
To quantify the change in the maximum energy we define ${E_{\rm max}}$ as the exponential cutoff of the CR distribution, taken in the form $f(E) \sim E^{-q} e^{-E/{E_{\rm max}}}$. Following [@bai+15], we calculate ${E_{\rm max}}$ by integrating over the energy distribution function: $${E_{\rm max}}\sim \frac{\int E^{4} f(E) dE}{\int E^{3} f(E) dE}.\label{eq:Emax}$$ Since $f(E)$ has an energy slope between $1.5$ and $2$, the integral differs from ${E_{\rm max}}$ by a constant of order unity. Figure \[fig:Emax\] shows the maximum energy as a function of time for Run A, where ${E_{\rm max}}(t)\propto t$ can be fitted with a broken linear function with a change of slope in the trans-relativistic regime. The rate of increase of ${E_{\rm max}}$ is about $3.9\times 10^{-4} m_ic^2\Omega_{ci}$ and $2.4\times 10^{-4} m_ic^2\Omega_{ci}$ below and above the rest mass energy, respectively. This decrease by nearly a factor of two is quantitatively consistent with the reduction found in [@bai+15], further supporting the idea that the decrease is due to a reduction in CR current in the relativistic regime.
The connection between the self-generated diffusion and the growth of the maximum CR energy can be made more explicit by measuring the average diffusion coefficient upstream of the shock. For DSA, the return time upstream is typically the bottleneck of the acceleration rate. Such a diffusion coefficient $D(E)$ is estimated using the approach outlined in [@caprioli+14c], namely: $$D(E) \simeq \frac{v_{\rm sh}}{f_{\rm sh}(E)}\int_{\rm shock}^{x_0} f(x, E) dx \label{eq:diff_coef}$$ where $x_0$ is a position far enough upstream that the CR population is negligible and $f_{\rm sh}(E)$ is the CR distribution function just downstream of the shock.
Figure \[fig:diff\] shows the time evolution of the diffusion coefficient normalized to the Bohm diffusion coefficient ($D_B\equiv vr_L/2$) for Run A. As discussed in [@caprioli+14c], for $M=20$ the diffusion coefficient is about an order of magnitude larger than Bohm, which is consistent with having self-generated magnetic fluctuations —at scales resonant with the CRs— that are approximately an order of magnitude smaller than the initial upstream magnetic field. As the simulation evolves in time and the maximum energy transitions into the relativistic regime and we can see a change in the diffusion coefficient. We find the value of the diffusion coefficient is consistently larger at relativistic energies, $D(m_ic^2) \sim 10 - 20 D_B$, than at non-relativistic energies, $D(m_ic^2/5) \sim 5 - 10 D_B$. The rate of maximum CR energy increase for non-relativistic shocks can be written as [@caprioli+14c]: $$\frac{{E_{\rm max}}(t)}{\frac12 m_i v_{\rm sh}^2} \approx \frac13 \frac{D_B({E_{\rm max}})}{D({E_{\rm max}})}\Omega_{ci} t.$$ The increase in the diffusion coefficient as CRs transition to relativistic energies is consistent with the reduction in the rate of change of ${E_{\rm max}}$, as seen in Figure \[fig:Emax\]. The increase of the diffusion coefficient by an approximate factor of 2 agrees with the reduction of the slope by a comparable factor.
In previous hybrid simulations, the self-generated diffusion coefficient normalized to the Bohm coefficient has been linked to the Mach number by $D/D_B \propto 1/\sqrt{M}$ [@caprioli+14b; @caprioli+14c]. Using this scaling along with the measured rate of increase of the maximum energy from our simulation, we can calculate a prediction for the maximum energy as a function of time: $$\frac{{E_{\rm max}}}{\rm GeV} \approx 20 \left( \beta_{\rm sh}^5\frac{n}{{\rm cm}^{-3}} \frac{B}{{\rm Gauss}} \right)^{1/2} \frac{t}{{\rm s}}$$ where $\beta_{sh} \equiv v_{\rm sh}/c$. Again, for the typical values of fast radio supernovae, with $\beta_{\rm sh} \gtrsim 0.01$, CRs with GeV energies will be reached within seconds, and TeV CRs will be produced in about an hour. If the circumstellar medium is dense enough, multi-TeV neutrinos[^1] in the range of sensitivity of Ice Cube could be produced in a matter of days after the SN explosion.
Acceleration Efficiency
-----------------------
We consider the evolution of the fraction of shock energy that is transferred to CRs as a function of time. Following [@caprioli+14a] and [@caprioli+15], we distinguish the CRs as the ions that achieved energies $E\gtrsim 10 E_{sh}$ and define the acceleration efficiency ${\varepsilon_{\rm cr}}$ as the fraction of the energy density in these particles normalized by the total energy density. $$\varepsilon_{cr} = \frac{\int_{10E_{sh}}^\infty Ef(E)dE}{\int_0^{\infty} Ef(E)dE}$$
Figure \[fig:acc\] shows that an acceleration efficiency on the order of $10\%$ is reached within the first hundred inverse cyclotron times, and then remains nearly constant throughout the entire simulation, consistent with what was seen in the non-relativistic case [@caprioli+14a]. The vertical black dashed line denotes when ${E_{\rm max}}\sim m_ic^2$, the color corresponds to ${E_{\rm max}}/ m_ic^2$ as shown in Figure \[fig:Emax\].
From this it is clear that ${\varepsilon_{\rm cr}}$ is unaffected as the CR population transitions from non-relativistic to relativistic energies, and that the canonical value of $\sim 10\%$ quoted by [@caprioli+14a] should be considered the asymptotic one. In this respect, it is worth stressing that in the non-relativistic regime the efficiency ${\varepsilon_{\rm cr}}\propto E^2 f(E)\propto E^{1/2}$ is typically dominated by the highest-energy CRs, while in the relativistic regime there is about the same energy density per decade. Since ${\varepsilon_{\rm cr}}$ saturates well before CRs become trans-relativistic, it is necessary for the shock to “be aware” of the efficient CR acceleration; such a CR feedback will be discussed in greater detail in forthcoming works, but here we mention that the pressure in the CR precursor affects the dynamics shock front, which reacts by injecting fewer particles into DSA.
Until this moment we have not discussed oblique or perpendicular shocks. This is because it has previously been found in classical hybrid simulations that shocks with $\vartheta_{Bn}\gtrsim 50^{\circ}$, thermal ions are not energized enough to initiate the DSA process [@caprioli+14a; @caprioli+15]. Note that, if the injection issue is overcome, for instance when pre-energized CR seeds are present [@caprioli+18], or the presence of external plasma turbulence, acceleration at oblique shocks proceeds unhindered, even more rapidly than at quasi-parallel shocks [e.g., @jokipii87; @giacalone05]
Recently, PIC-MHD simulations of very oblique shocks ($\vartheta_{Bn}\gtrsim 70^{\circ}$) have suggested that thermal particle injection and DSA will eventually occur for simulations run long enough [@vanmarle+18]. We have tested this claim with the full-hybrid [[dHybridR]{}]{} code and did not recover such a result. Figure \[fig:no\_perp\_inj\] shows a simulation perform with the same initial parameters as the quasi-perpendicular, $M=30$ simulation discussed in [@vanmarle+18]. The simulation is $[L_x;L_\perp]=[10^4; 2.7\times 10^3]d_{i0}$ in size with two cells per skin depth in each direction and was run for a comparable amount of time ($600 \Omega_{ci}^{-1}$). Using 4 particles per cell, the [[dHybridR]{}]{} simulation has approximately $4/3 \times M\times 16\times 2700\simeq 1.72\times 10^6$ macro-particles impinging on the shock per unit cyclotron time, where the factor of $r/(r-1)\simeq 4/3$ comes from the conversion of the upstream flow speed from the simulation to the shock frame. For the canonical 1% injection efficiency [@caprioli+15], in our simulation $\sim
1.7\times 10^4$ CR particles are produced per unit time, which returns a statistics comparable with the $\sim 10^4$ rate used by [@vanmarle+18].
The top panel shows the energy density distribution as a function of $x$, in which energy is normalized to the shock energy. Downstream of the shock ($x<0$), ions are heated up to supra-thermal energies ($E \lesssim 10 E_{sh}$), but there is no DSA tail, and no energetic particles upstream ($x>0$). The bottom panel of Figure \[fig:no\_perp\_inj\] shows a 2D plot of the magnitude of the magnetic field, which reveals the canonical downstream compression, with additional some small-scale deviations (which we discuss below); the upstream magnetic field, instead, is unperturbed. These results stress how a self-consistent model for ion injection can only be provided by full-hybrid simulations.
3D Simulations {#sec:3D}
==============
Finally, we present a quasi-perpendicular 3D shock simulation with a smaller Mach number ($M = 5$), identified as Run 3D in Table \[tab:shocks\]. The conditions in this simulation are quite similar to typical heliospheric shocks, such as the Earth’s bow shock, which is formed by the supersonic/super-Alfvénic solar wind, traveling at speed $\gtrsim 100$km s$^{-1}$ and impinging on the Earth’s magnetosphere [@sheeley+85; @cane+03]. For typical solar wind conditions, the ion temperature is of the order of $10$eV and thermal and magnetic pressure are comparable to each other, which corresponds to $M\approx 5-10$ [e.g., @schwartz+88; @wilsoniii+18p]; also interplanetary shocks triggered by coronal mass ejections typically span the same range of Mach numbers [e.g., @wilson+19]. [[dHybridR]{}]{} is well suited to study low-Mach-number heliospheric shocks because in this systems ions can be accelerated to trans-relativistic energies, and because the relevant sizes and scales can be modeled to scale at a reasonable computational cost. In Fig \[fig:3D\] we present an orthographic projection of $B_z$, where $z$ is the direction normal to the upstream flow and the mean upstream magnetic field; therefore, $B_z$ is the self-generated component of the magnetic field. Upstream of the shock there are no indications of magnetic field amplification, in agreement with the 2D simulation. However, downstream some magnetic structures can be observed: there is a clear rippling of the magnetic field along the shock interface, which is produced by shock reformation, consistent with what has been previously found in observations [@johlander+16b; @johlander+18] and simulations [@lb03; @caprioli+15; @burgess+16].
{width="\textwidth"}
The black line in Figure \[fig:3D\] represents the trajectory of a synthetic probe through the simulation box, mimicking in-situ spacecraft observations, and Figure \[fig:3D\_cut\] shows the magnetic field measured by such a probe. The trajectory is diagonal through the shock interface, with only a small component normal to the upstream magnetic field ($0.681{\bf \hat{x}} + 0.727{\bf \hat{y}} - 0.091{\bf \hat{z}}$ intersecting a point in the middle of the $y-z$ plane at $x = 312.5 d_i$). From this cut, the periodic structure of the ripples can be clearly seen; considering that the direction of propagation is primarily in the $y$ direction, the wave number can be estimated to be on the order of $k \Omega_{ci}/v_{\rm sh} \sim k r_g \sim 1$, where $r_g$ is the gyroradius of the downstream population.
This is a great example of how [[dHybridR]{}]{} simulations can be directly compared with [*in-situ*]{} measured heliospheric plasma phenomena.
Conclusion
==========
In this work we presented the first results from [[dHybridR]{}]{}, a hybrid plasma simulation code that includes relativistic ion dynamics. We detail how relativistic ion motion is included in the code and how for specific systems of interest, the assumptions required for hybrid simulations are not violated. This novel simulation software can be used to help understand, from first principles, numerous different open problems involving space and astrophysical plasmas. The code is well suited to study many astrophysical systems where a high energy, low density CR population interacts with a non-relativistic thermal background population.
To verify that [[dHybridR]{}]{} can correctly model physical systems of interest, we simulated CR-driven non-resonant and resonant streaming instabilities. In both test cases, the location in $k$ space and the value of the maximum growth rate found in simulations agreed remarkably well with the linear prediction. Then, we moved to use [[dHybridR]{}]{} to model strongly non-linear problems such as DSA at non-relativistic collisionless shocks, similar to those found in the heliosphere, in SN remnants, and in galaxy clusters. In particular, we presented simulations with parameters relevant to fast SN shocks (radio SNe, Figure \[fig:overview\]) as well as heliospheric shocks such as the Earth’s bow shock (Figure \[fig:3D\]).
We performed unprecedentedly-long simulations of parallel shocks in which ions achieve Lorentz factors as large as $\gamma \gtrsim 20$, attesting for the first time in full hybrid simulations that DSA produces a power-law tail in momentum across the trans-relativistic regime, which implies an energy distribution that follows a broken power law that steepens by 0.5 in slope. When CRs become relativistic, the increase of the maximum particle energy is still linear in time, but with a rate reduced by a factor of $\sim 2$; such a reduction is a consequence of the saturation of the velocity of escaping particles to $c$.
The acceleration efficiency (i.e., the fraction of the shock energy channelled into non-thermal particles with energy $E\gtrsim 10 E_{sh}$) was found to reach about $10\%$ within tens of cyclotron times and remain nearly constant as the high energy population transitions into the relativistic regime. These results are directly applicable to fast radio SNe, where we predict GeV/TeV CRs to be produced within seconds/days. With the current sensitivity of $\gamma$-ray and neutrino telescopes, such a delay could be measured for a Galactic SN.
Finally, we presented a 3D simulation produced with [[dHybridR]{}]{} with conditions comparable to the Earth’s bow shock with a quasi-perpendicular configuration. We showed that [[dHybridR]{}]{} reproduces both qualitatively and quantitatively the shock rippling that has been found with [*in-situ*]{} satellite observations [@johlander+18].
In summary, this work presents, to the authors’ knowledge, the first hybrid simulations to include relativistic ion dynamics, which is a critical tool for studying the inherently multi-scale nature of CR/thermal ion interplay in space and astrophysical plasmas.
We would like to thank Luis Gargaté for providing the original version of [dHybrid]{}. This research was partially supported by NASA (grant NNX17AG30G, 80NSSC18K1218, and 80NSSC18K1726) and NSF (grants AST-1714658 and AST-1909778). Simulations were performed on computational resources provided by the University of Chicago Research Computing Center, the NASA High-End Computing Program through the NASA Advanced Supercomputing Division at Ames Research Center, and XSEDE TACC (TG-AST180008).
[^1]: Hadronic neutrinos and $\gamma$-rays of energy $E$ are produced by parent protons of energy $\sim 10E$.
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